LMFDB - Embedded newform 315.2.bf.c.109.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,2,Mod(109,315)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("315.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	315.bf (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.51528766367$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 8x^{14} + 26x^{12} - 96x^{10} - 781x^{8} - 2400x^{6} + 16250x^{4} + 125000x^{2} + 390625 $ x^16 + 8x^14 + 26x^12 - 96x^10 - 781x^8 - 2400x^6 + 16250x^4 + 125000*x^2 + 390625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			109.5
Root			$-2.19280 - 0.437757i$ of defining polynomial
Character	$\chi$	$=$	315.109
Dual form			315.2.bf.c.289.5

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.515448 - 0.297594i) q^{2} +(-0.822876 + 1.42526i) q^{4} +(0.717291 - 2.11790i) q^{5} +(1.68480 - 2.03996i) q^{7} +2.16991i q^{8} +O(q^{10})$	\(q+(0.515448 - 0.297594i) q^{2} +(-0.822876 + 1.42526i) q^{4} +(0.717291 - 2.11790i) q^{5} +(1.68480 - 2.03996i) q^{7} +2.16991i q^{8} +(-0.260548 - 1.30513i) q^{10} +(2.68967 - 4.65864i) q^{11} +4.07992i q^{13} +(0.261348 - 1.55288i) q^{14} +(-1.00000 - 1.73205i) q^{16} +(4.27384 + 2.46750i) q^{17} +(-0.322876 - 0.559237i) q^{19} +(2.42832 + 2.76510i) q^{20} -3.20172i q^{22} +(5.30473 - 3.06269i) q^{23} +(-3.97099 - 3.03830i) q^{25} +(1.21416 + 2.10299i) q^{26} +(1.52110 + 4.07992i) q^{28} -2.42832 q^{29} +(-2.50000 + 4.33013i) q^{31} +(-4.78929 - 2.76510i) q^{32} +2.93725 q^{34} +(-3.11193 - 5.03149i) q^{35} +(0.760548 - 0.439102i) q^{37} +(-0.332851 - 0.192172i) q^{38} +(4.59564 + 1.55646i) q^{40} -11.2814 q^{41} +10.7945i q^{43} +(4.42652 + 7.66697i) q^{44} +(1.82288 - 3.15731i) q^{46} +(-1.87919 + 1.08495i) q^{47} +(-1.32288 - 6.87386i) q^{49} +(-2.95102 - 0.384343i) q^{50} +(-5.81496 - 3.35727i) q^{52} +(-6.85108 - 3.95547i) q^{53} +(-7.93725 - 9.03805i) q^{55} +(4.42652 + 3.65587i) q^{56} +(-1.25167 + 0.722653i) q^{58} +(-4.16518 + 7.21430i) q^{59} +(-0.177124 - 0.306788i) q^{61} +2.97594i q^{62} +0.708497 q^{64} +(8.64086 + 2.92649i) q^{65} +(2.01222 + 1.16176i) q^{67} +(-7.03367 + 4.06089i) q^{68} +(-3.10138 - 1.66738i) q^{70} +2.42832 q^{71} +(7.82718 + 4.51902i) q^{73} +(0.261348 - 0.452669i) q^{74} +1.06275 q^{76} +(-4.97188 - 13.3357i) q^{77} +(2.96863 + 5.14181i) q^{79} +(-4.38560 + 0.875514i) q^{80} +(-5.81496 + 3.35727i) q^{82} +11.6556i q^{83} +(8.29150 - 7.28164i) q^{85} +(3.21236 + 5.56398i) q^{86} +(10.1088 + 5.83633i) q^{88} +(-2.68967 - 4.65864i) q^{89} +(8.32288 + 6.87386i) q^{91} +10.0808i q^{92} +(-0.645751 + 1.11847i) q^{94} +(-1.41600 + 0.282682i) q^{95} -6.71453i q^{97} +(-2.72749 - 3.14944i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{4}+O(q^{10}) $ 16 * q + 8 * q^4	$ 16 q + 8 q^{4} + 8 q^{10} - 16 q^{16} + 16 q^{19} - 16 q^{25} - 40 q^{31} - 80 q^{34} + 8 q^{40} + 8 q^{46} - 24 q^{61} + 96 q^{64} - 56 q^{70} + 144 q^{76} - 16 q^{79} + 48 q^{85} + 112 q^{91} + 32 q^{94}+O(q^{100}) $ 16 * q + 8 * q^4 + 8 * q^10 - 16 * q^16 + 16 * q^19 - 16 * q^25 - 40 * q^31 - 80 * q^34 + 8 * q^40 + 8 * q^46 - 24 * q^61 + 96 * q^64 - 56 * q^70 + 144 * q^76 - 16 * q^79 + 48 * q^85 + 112 * q^91 + 32 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.515448	−	0.297594i	0.364477	−	0.210431i	−0.306566	−	0.951849i	$-0.599180\pi$
$2$	0.515448	−	0.297594i	0.364477	−	0.210431i	0.671043	+	0.741419i	$0.265847\pi$
$3$		0			0
$3$		0			0
$4$	−0.822876	+	1.42526i	−0.411438	+	0.712631i
$5$	0.717291	−	2.11790i	0.320782	−	0.947153i
$5$	0.717291	−	2.11790i	0.320782	−	0.947153i
$6$		0			0
$7$	1.68480	−	2.03996i	0.636796	−	0.771033i
$7$	1.68480	−	2.03996i	0.636796	−	0.771033i
$8$			2.16991i			0.767178i
$9$		0			0
$10$	−0.260548	−	1.30513i	−0.0823924	−	0.412718i
$11$	2.68967	−	4.65864i	0.810965	−	1.40463i	−0.101224	−	0.994864i	$-0.532276\pi$
$11$	2.68967	−	4.65864i	0.810965	−	1.40463i	0.912189	−	0.409770i	$-0.134391\pi$
$12$		0			0
$13$			4.07992i			1.13157i	0.824554	+	0.565783i	$0.191426\pi$
$13$			4.07992i			1.13157i	−0.824554	+	0.565783i	$0.808574\pi$
$14$	0.261348	−	1.55288i	0.0698483	−	0.415025i
$15$		0			0
$16$	−1.00000	−	1.73205i	−0.250000	−	0.433013i
$17$	4.27384	+	2.46750i	1.03656	+	0.598457i	0.918856	−	0.394592i	$-0.129114\pi$
$17$	4.27384	+	2.46750i	1.03656	+	0.598457i	0.117701	+	0.993049i	$0.462447\pi$
$18$		0			0
$19$	−0.322876	−	0.559237i	−0.0740728	−	0.128298i	0.826610	−	0.562775i	$-0.190266\pi$
$19$	−0.322876	−	0.559237i	−0.0740728	−	0.128298i	−0.900683	+	0.434478i	$0.856933\pi$
$20$	2.42832	+	2.76510i	0.542989	+	0.618294i
$21$		0			0
$22$		−	3.20172i		−	0.682608i
$23$	5.30473	−	3.06269i	1.10611	−	0.638615i	0.168293	−	0.985737i	$-0.446174\pi$
$23$	5.30473	−	3.06269i	1.10611	−	0.638615i	0.937820	+	0.347122i	$0.112841\pi$
$24$		0			0
$25$	−3.97099	−	3.03830i	−0.794197	−	0.607660i
$26$	1.21416	+	2.10299i	0.238116	+	0.412430i
$27$		0			0
$28$	1.52110	+	4.07992i	0.287460	+	0.771033i
$29$	−2.42832			−0.450928			−0.225464	−	0.974252i	$-0.572390\pi$
$29$	−2.42832			−0.450928			−0.225464	+	0.974252i	$0.572390\pi$
$30$		0			0
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	−0.998322	−	0.0579057i	$-0.981558\pi$
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	0.549309	+	0.835619i	$0.314891\pi$
$32$	−4.78929	−	2.76510i	−0.846634	−	0.488804i
$33$		0			0
$34$	2.93725			0.503735
$35$	−3.11193	−	5.03149i	−0.526013	−	0.850477i
$36$		0			0
$37$	0.760548	−	0.439102i	0.125033	−	0.0721880i	−0.436179	−	0.899860i	$-0.643669\pi$
$37$	0.760548	−	0.439102i	0.125033	−	0.0721880i	0.561212	+	0.827672i	$0.310335\pi$
$38$	−0.332851	−	0.192172i	−0.0539956	−	0.0311744i
$39$		0			0
$40$	4.59564	+	1.55646i	0.726635	+	0.246097i
$41$	−11.2814			−1.76185			−0.880927	−	0.473252i	$-0.843080\pi$
$41$	−11.2814			−1.76185			−0.880927	+	0.473252i	$0.843080\pi$
$42$		0			0
$43$			10.7945i			1.64614i	0.567941	+	0.823069i	$0.307740\pi$
$43$			10.7945i			1.64614i	−0.567941	+	0.823069i	$0.692260\pi$
$44$	4.42652	+	7.66697i	0.667324	+	1.15584i
$45$		0			0
$46$	1.82288	−	3.15731i	0.268768	−	0.465520i
$47$	−1.87919	+	1.08495i	−0.274109	+	0.158257i	−0.630753	−	0.775983i	$-0.717254\pi$
$47$	−1.87919	+	1.08495i	−0.274109	+	0.158257i	0.356645	+	0.934240i	$0.383921\pi$
$48$		0			0
$49$	−1.32288	−	6.87386i	−0.188982	−	0.981981i
$50$	−2.95102	−	0.384343i	−0.417337	−	0.0543544i
$51$		0			0
$52$	−5.81496	−	3.35727i	−0.806390	−	0.465569i
$53$	−6.85108	−	3.95547i	−0.941068	−	0.543326i	−0.0507729	−	0.998710i	$-0.516168\pi$
$53$	−6.85108	−	3.95547i	−0.941068	−	0.543326i	−0.890295	+	0.455384i	$0.849502\pi$
$54$		0			0
$55$	−7.93725	−	9.03805i	−1.07026	−	1.21869i
$56$	4.42652	+	3.65587i	0.591519	+	0.488536i
$57$		0			0
$58$	−1.25167	+	0.722653i	−0.164353	+	0.0948890i
$59$	−4.16518	+	7.21430i	−0.542260	+	0.939221i	0.456514	+	0.889716i	$0.349098\pi$
$59$	−4.16518	+	7.21430i	−0.542260	+	0.939221i	−0.998774	+	0.0495053i	$0.984236\pi$
$60$		0			0
$61$	−0.177124	−	0.306788i	−0.0226784	−	0.0392802i	0.854463	−	0.519512i	$-0.173886\pi$
$61$	−0.177124	−	0.306788i	−0.0226784	−	0.0392802i	−0.877142	+	0.480231i	$0.840553\pi$
$62$			2.97594i			0.377945i
$63$		0			0
$64$	0.708497			0.0885622
$65$	8.64086	+	2.92649i	1.07177	+	0.362987i
$66$		0			0
$67$	2.01222	+	1.16176i	0.245832	+	0.141931i	0.617854	−	0.786293i	$-0.288002\pi$
$67$	2.01222	+	1.16176i	0.245832	+	0.141931i	−0.372022	+	0.928224i	$0.621336\pi$
$68$	−7.03367	+	4.06089i	−0.852958	+	0.492456i
$69$		0			0
$70$	−3.10138	−	1.66738i	−0.370686	−	0.199290i
$71$	2.42832			0.288189			0.144094	−	0.989564i	$-0.453973\pi$
$71$	2.42832			0.288189			0.144094	+	0.989564i	$0.453973\pi$
$72$		0			0
$73$	7.82718	+	4.51902i	0.916102	+	0.528912i	0.882389	−	0.470520i	$-0.155934\pi$
$73$	7.82718	+	4.51902i	0.916102	+	0.528912i	0.0337124	+	0.999432i	$0.489267\pi$
$74$	0.261348	−	0.452669i	0.0303811	−	0.0526217i
$75$		0			0
$76$	1.06275			0.121905
$77$	−4.97188	−	13.3357i	−0.566599	−	1.51975i
$78$		0			0
$79$	2.96863	+	5.14181i	0.333997	+	0.578499i	0.983292	−	0.182038i	$-0.0582693\pi$
$79$	2.96863	+	5.14181i	0.333997	+	0.578499i	−0.649295	+	0.760537i	$0.724936\pi$
$80$	−4.38560	+	0.875514i	−0.490325	+	0.0978854i
$81$		0			0
$82$	−5.81496	+	3.35727i	−0.642155	+	0.370748i
$83$			11.6556i			1.27936i	0.768639	+	0.639682i	$0.220934\pi$
$83$			11.6556i			1.27936i	−0.768639	+	0.639682i	$0.779066\pi$
$84$		0			0
$85$	8.29150	−	7.28164i	0.899340	−	0.789804i
$86$	3.21236	+	5.56398i	0.346398	+	0.599979i
$87$		0			0
$88$	10.1088	+	5.83633i	1.07760	+	0.622155i
$89$	−2.68967	−	4.65864i	−0.285104	−	0.493815i	0.687530	−	0.726156i	$-0.258695\pi$
$89$	−2.68967	−	4.65864i	−0.285104	−	0.493815i	−0.972634	+	0.232341i	$0.925362\pi$
$90$		0			0
$91$	8.32288	+	6.87386i	0.872474	+	0.720577i
$92$			10.0808i			1.05100i
$93$		0			0
$94$	−0.645751	+	1.11847i	−0.0666042	+	0.115362i
$95$	−1.41600	+	0.282682i	−0.145279	+	0.0290026i
$96$		0			0
$97$		−	6.71453i		−	0.681758i	−0.940107	−	0.340879i	$-0.889275\pi$
$97$		−	6.71453i		−	0.681758i	0.940107	−	0.340879i	$-0.110725\pi$
$98$	−2.72749	−	3.14944i	−0.275518	−	0.318141i
$99$		0			0
$100$	7.59800	−	3.15956i	0.759800	−	0.315956i
$101$	−1.73686	+	3.00832i	−0.172824	+	0.299339i	−0.939406	−	0.342807i	$-0.888622\pi$
$101$	−1.73686	+	3.00832i	−0.172824	+	0.299339i	0.766582	+	0.642146i	$0.221956\pi$
$102$		0			0
$103$	−6.30608	+	3.64082i	−0.621357	+	0.358740i	−0.777397	−	0.629010i	$-0.783460\pi$
$103$	−6.30608	+	3.64082i	−0.621357	+	0.358740i	0.156040	+	0.987751i	$0.450127\pi$
$104$	−8.85305			−0.868113
$105$		0			0
$106$	−4.70850			−0.457330
$107$	4.45643	−	2.57292i	0.430820	−	0.248734i	−0.268876	−	0.963175i	$-0.586652\pi$
$107$	4.45643	−	2.57292i	0.430820	−	0.248734i	0.699696	+	0.714441i	$0.253319\pi$
$108$		0			0
$109$	−6.61438	+	11.4564i	−0.633543	+	1.09733i	0.353279	+	0.935518i	$0.385067\pi$
$109$	−6.61438	+	11.4564i	−0.633543	+	1.09733i	−0.986822	+	0.161810i	$0.948267\pi$
$110$	−6.78091	−	2.29656i	−0.646534	−	0.218969i
$111$		0			0
$112$	−5.21812	−	0.878205i	−0.493066	−	0.0829825i
$113$		−	15.8219i		−	1.48840i	−0.667958	−	0.744199i	$-0.732831\pi$
$113$		−	15.8219i		−	1.48840i	0.667958	−	0.744199i	$-0.267169\pi$
$114$		0			0
$115$	−2.68143	−	13.4317i	−0.250044	−	1.25251i
$116$	1.99820	−	3.46099i	0.185529	−	0.321345i
$117$		0			0
$118$			4.95813i			0.456432i
$119$	12.2342	−	4.56120i	1.12151	−	0.418125i
$120$		0			0
$121$	−8.96863	−	15.5341i	−0.815330	−	1.41219i
$122$	−0.182597	−	0.105422i	−0.0165315	−	0.00954448i
$123$		0			0
$124$	−4.11438	−	7.12631i	−0.369482	−	0.639962i
$125$	−9.28316	+	6.23080i	−0.830311	+	0.557300i
$126$		0			0
$127$			4.07992i			0.362034i	0.983480	+	0.181017i	$0.0579390\pi$
$127$			4.07992i			0.362034i	−0.983480	+	0.181017i	$0.942061\pi$
$128$	9.94376	−	5.74103i	0.878913	−	0.507441i
$129$		0			0
$130$	5.32482	−	1.06301i	0.467017	−	0.0932324i
$131$	−3.90383	−	6.76163i	−0.341079	−	0.590766i	0.643554	−	0.765400i	$-0.277459\pi$
$131$	−3.90383	−	6.76163i	−0.341079	−	0.590766i	−0.984633	+	0.174634i	$0.944126\pi$
$132$		0			0
$133$	−1.68480	−	0.283551i	−0.146091	−	0.0245870i
$134$	1.38293			0.119467
$135$		0			0
$136$	−5.35425	+	9.27383i	−0.459123	+	0.795224i
$137$	3.42554	+	1.97774i	0.292663	+	0.168969i	0.639142	−	0.769088i	$-0.279289\pi$
$137$	3.42554	+	1.97774i	0.292663	+	0.168969i	−0.346479	+	0.938058i	$0.612623\pi$
$138$		0			0
$139$	7.22876			0.613135			0.306568	−	0.951849i	$-0.400819\pi$
$139$	7.22876			0.613135			0.306568	+	0.951849i	$0.400819\pi$
$140$	9.73192	−	0.295034i	0.822498	−	0.0249349i
$141$		0			0
$142$	1.25167	−	0.722653i	0.105038	−	0.0606437i
$143$	19.0069	+	10.9736i	1.58944	+	0.917661i
$144$		0			0
$145$	−1.74181	+	5.14293i	−0.144650	+	0.427097i
$146$	5.37934			0.445197
$147$		0			0
$148$			1.44531i			0.118803i
$149$	−1.47551	−	2.55566i	−0.120878	−	0.209367i	0.799236	−	0.601017i	$-0.205238\pi$
$149$	−1.47551	−	2.55566i	−0.120878	−	0.209367i	−0.920114	+	0.391650i	$0.871904\pi$
$150$		0			0
$151$	4.46863	−	7.73989i	0.363652	−	0.629863i	−0.624907	−	0.780699i	$-0.714863\pi$
$151$	4.46863	−	7.73989i	0.363652	−	0.629863i	0.988559	+	0.150836i	$0.0481965\pi$
$152$	1.21349	−	0.700610i	0.0984272	−	0.0568270i
$153$		0			0
$154$	−6.53137	−	5.39426i	−0.526313	−	0.434682i
$155$	7.37754	+	8.40071i	0.592578	+	0.674761i
$156$		0			0
$157$	10.1088	+	5.83633i	0.806772	+	0.465790i	0.845834	−	0.533447i	$-0.179104\pi$
$157$	10.1088	+	5.83633i	0.806772	+	0.465790i	−0.0390619	+	0.999237i	$0.512437\pi$
$158$	3.06034	+	1.76689i	0.243468	+	0.140566i
$159$		0			0
$160$	−9.29150	+	8.15984i	−0.734558	+	0.645092i
$161$	2.68967	−	15.9815i	0.211976	−	1.25952i
$162$		0			0
$163$	−1.52110	+	0.878205i	−0.119141	+	0.0687863i	−0.558386	−	0.829581i	$-0.688579\pi$
$163$	−1.52110	+	0.878205i	−0.119141	+	0.0687863i	0.439245	+	0.898367i	$0.355246\pi$
$164$	9.28316	−	16.0789i	0.724893	−	1.25555i
$165$		0			0
$166$	3.46863	+	6.00784i	0.269218	+	0.466299i
$167$			12.6351i			0.977733i	0.872359	+	0.488867i	$0.162590\pi$
$167$			12.6351i			0.977733i	−0.872359	+	0.488867i	$0.837410\pi$
$168$		0			0
$169$	−3.64575			−0.280442
$170$	2.10687	−	6.22081i	0.161589	−	0.477114i
$171$		0			0
$172$	−15.3849	−	8.88249i	−1.17309	−	0.677284i
$173$	−8.06457	+	4.65608i	−0.613138	+	0.353995i	−0.774193	−	0.632950i	$-0.781844\pi$
$173$	−8.06457	+	4.65608i	−0.613138	+	0.353995i	0.161055	+	0.986945i	$0.448510\pi$
$174$		0			0
$175$	−12.8883	+	2.98172i	−0.974267	+	0.225397i
$176$	−10.7587			−0.810965
$177$		0			0
$178$	−2.77277	−	1.60086i	−0.207828	−	0.119989i
$179$	8.33035	−	14.4286i	0.622640	−	1.07844i	−0.366352	−	0.930476i	$-0.619394\pi$
$179$	8.33035	−	14.4286i	0.622640	−	1.07844i	0.988992	−	0.147968i	$-0.0472731\pi$
$180$		0			0
$181$	12.6458			0.939951			0.469976	−	0.882679i	$-0.344263\pi$
$181$	12.6458			0.939951			0.469976	+	0.882679i	$0.344263\pi$
$182$	6.33563	+	1.06628i	0.469628	+	0.0790380i
$183$		0			0
$184$	6.64575	+	11.5108i	0.489931	+	0.848586i
$185$	−0.384440	−	1.92573i	−0.0282646	−	0.141582i
$186$		0			0
$187$	22.9904	−	13.2735i	1.68123	−	0.970656i
$188$		−	3.57113i		−	0.260451i
$189$		0			0
$190$	−0.645751	+	0.567102i	−0.0468477	+	0.0411419i
$191$	5.37934	+	9.31728i	0.389235	+	0.674175i	0.992347	−	0.123482i	$-0.0394060\pi$
$191$	5.37934	+	9.31728i	0.389235	+	0.674175i	−0.603112	+	0.797657i	$0.706073\pi$
$192$		0			0
$193$	−10.8694	−	6.27543i	−0.782394	−	0.451715i	0.0548839	−	0.998493i	$-0.482521\pi$
$193$	−10.8694	−	6.27543i	−0.782394	−	0.451715i	−0.837278	+	0.546777i	$0.815854\pi$
$194$	−1.99820	−	3.46099i	−0.143463	−	0.248485i
$195$		0			0
$196$	10.8856	+	3.77089i	0.777544	+	0.269349i
$197$		−	5.91453i		−	0.421393i	−0.977552	−	0.210697i	$-0.932427\pi$
$197$		−	5.91453i		−	0.421393i	0.977552	−	0.210697i	$-0.0675732\pi$
$198$		0			0
$199$	8.82288	−	15.2817i	0.625437	−	1.08329i	−0.363019	−	0.931782i	$-0.618254\pi$
$199$	8.82288	−	15.2817i	0.625437	−	1.08329i	0.988456	−	0.151507i	$-0.0484126\pi$
$200$	6.59283	−	8.61667i	0.466183	−	0.609291i
$201$		0			0
$202$			2.06751i			0.145470i
$203$	−4.09124	+	4.95368i	−0.287149	+	0.347680i
$204$		0			0
$205$	−8.09203	+	23.8928i	−0.565172	+	1.66874i
$206$	−2.16697	+	3.75330i	−0.150980	+	0.261505i
$207$		0			0
$208$	7.06663	−	4.07992i	0.489983	−	0.282892i
$209$	−3.47371			−0.240282
$210$		0			0
$211$	−14.9373			−1.02832			−0.514161	−	0.857693i	$-0.671897\pi$
$211$	−14.9373			−1.02832			−0.514161	+	0.857693i	$0.671897\pi$
$212$	11.2752	−	6.50972i	0.774382	−	0.447090i
$213$		0			0
$214$	1.53137	−	2.65242i	0.104683	−	0.181315i
$215$	22.8616	+	7.74277i	1.55915	+	0.528052i
$216$		0			0
$217$	4.62128	+	12.3953i	0.313713	+	0.841449i
$218$			7.87360i			0.533267i
$219$		0			0
$220$	19.4130	−	3.87548i	1.30882	−	0.261285i
$221$	−10.0672	+	17.4369i	−0.677194	+	1.17293i
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	−13.7097	+	5.11131i	−0.916017	+	0.341514i
$225$		0			0
$226$	−4.70850	−	8.15536i	−0.313205	−	0.542486i
$227$	−3.06034	−	1.76689i	−0.203122	−	0.117273i	0.394989	−	0.918686i	$-0.370748\pi$
$227$	−3.06034	−	1.76689i	−0.203122	−	0.117273i	−0.598111	+	0.801413i	$0.704082\pi$
$228$		0			0
$229$	−1.14575	−	1.98450i	−0.0757134	−	0.131139i	0.825683	−	0.564135i	$-0.190790\pi$
$229$	−1.14575	−	1.98450i	−0.0757134	−	0.131139i	−0.901396	+	0.432995i	$0.857457\pi$
$230$	−5.37934	−	6.12538i	−0.354703	−	0.403895i
$231$		0			0
$232$		−	5.26923i		−	0.345942i
$233$	−6.66848	+	3.85005i	−0.436867	+	0.252225i	−0.702268	−	0.711913i	$-0.747829\pi$
$233$	−6.66848	+	3.85005i	−0.436867	+	0.252225i	0.265401	+	0.964138i	$0.414496\pi$
$234$		0			0
$235$	0.949892	+	4.75817i	0.0619641	+	0.310389i
$236$	−6.85484	−	11.8729i	−0.446212	−	0.772862i
$237$		0			0
$238$	4.94870	−	5.99188i	0.320776	−	0.388396i
$239$	8.85305			0.572656			0.286328	−	0.958132i	$-0.407565\pi$
$239$	8.85305			0.572656			0.286328	+	0.958132i	$0.407565\pi$
$240$		0			0
$241$	−14.1144	+	24.4468i	−0.909187	+	1.57476i	−0.0939910	+	0.995573i	$0.529962\pi$
$241$	−14.1144	+	24.4468i	−0.909187	+	1.57476i	−0.815196	+	0.579185i	$0.803371\pi$
$242$	−9.24572	−	5.33802i	−0.594337	−	0.343141i
$243$		0			0
$244$	0.583005			0.0373231
$245$	−15.5070	−	2.12885i	−0.990708	−	0.136007i
$246$		0			0
$247$	2.28164	−	1.31731i	0.145177	−	0.0838182i
$248$	−9.39597	−	5.42477i	−0.596645	−	0.344473i
$249$		0			0
$250$	−2.93074	+	5.97427i	−0.185356	+	0.377846i
$251$	16.1380			1.01862			0.509311	−	0.860583i	$-0.329900\pi$
$251$	16.1380			1.01862			0.509311	+	0.860583i	$0.329900\pi$
$252$		0			0
$253$		−	32.9505i		−	2.07158i
$254$	1.21416	+	2.10299i	0.0761832	+	0.131953i
$255$		0			0
$256$	2.70850	−	4.69126i	0.169281	−	0.293203i
$257$	1.36375	−	0.787360i	0.0850682	−	0.0491141i	−0.456862	−	0.889537i	$-0.651027\pi$
$257$	1.36375	−	0.787360i	0.0850682	−	0.0491141i	0.541931	+	0.840423i	$0.317694\pi$
$258$		0			0
$259$	0.385622	−	2.29129i	0.0239614	−	0.142374i
$260$	−11.2814	+	9.90735i	−0.699641	+	0.614428i
$261$		0			0
$262$	−4.02444	−	2.32351i	−0.248631	−	0.143547i
$263$	−12.4887	−	7.21033i	−0.770084	−	0.444608i	0.0628207	−	0.998025i	$-0.479990\pi$
$263$	−12.4887	−	7.21033i	−0.770084	−	0.444608i	−0.832905	+	0.553417i	$0.813324\pi$
$264$		0			0
$265$	−13.2915	+	11.6727i	−0.816491	+	0.717046i
$266$	−0.952811	+	0.355232i	−0.0584206	+	0.0217806i
$267$		0			0
$268$	−3.31161	+	1.91196i	−0.202289	+	0.116792i
$269$	−12.2342	+	21.1902i	−0.745931	+	1.29199i	0.203827	+	0.979007i	$0.434662\pi$
$269$	−12.2342	+	21.1902i	−0.745931	+	1.29199i	−0.949758	+	0.312984i	$0.898671\pi$
$270$		0			0
$271$	−11.1144	−	19.2507i	−0.675150	−	1.16939i	−0.976425	−	0.215857i	$-0.930745\pi$
$271$	−11.1144	−	19.2507i	−0.675150	−	1.16939i	0.301275	−	0.953537i	$-0.402588\pi$
$272$		−	9.87000i		−	0.598457i
$273$		0			0
$274$	2.35425			0.142225
$275$	−24.8350	+	10.3274i	−1.49761	+	0.622765i
$276$		0			0
$277$	14.8938	+	8.59894i	0.894882	+	0.516660i	0.875536	−	0.483152i	$-0.160508\pi$
$277$	14.8938	+	8.59894i	0.894882	+	0.516660i	0.0193459	+	0.999813i	$0.493842\pi$
$278$	3.72605	−	2.15123i	0.223473	−	0.129022i
$279$		0			0
$280$	10.9179	−	6.75261i	0.652467	−	0.403545i
$281$	3.99641			0.238406			0.119203	−	0.992870i	$-0.461966\pi$
$281$	3.99641			0.238406			0.119203	+	0.992870i	$0.461966\pi$
$282$		0			0
$283$	−20.4393	−	11.8007i	−1.21499	−	0.701476i	−0.251150	−	0.967948i	$-0.580809\pi$
$283$	−20.4393	−	11.8007i	−1.21499	−	0.701476i	−0.963843	+	0.266472i	$0.914142\pi$
$284$	−1.99820	+	3.46099i	−0.118572	+	0.205372i
$285$		0			0
$286$	13.0627			0.772416
$287$	−19.0069	+	23.0135i	−1.12194	+	1.35845i
$288$		0			0
$289$	3.67712	+	6.36897i	0.216301	+	0.374645i
$290$	0.632693	+	3.16927i	0.0371530	+	0.186106i
$291$		0			0
$292$	−12.8816	+	7.43719i	−0.753838	+	0.435228i
$293$		−	15.1894i		−	0.887371i	−0.896182	−	0.443686i	$-0.853671\pi$
$293$		−	15.1894i		−	0.887371i	0.896182	−	0.443686i	$-0.146329\pi$
$294$		0			0
$295$	12.2915	+	13.9962i	0.715639	+	0.814889i
$296$	0.952811	+	1.65032i	0.0553810	+	0.0959228i
$297$		0			0
$298$	−1.52110	−	0.878205i	−0.0881147	−	0.0508730i
$299$	12.4955	+	21.6429i	0.722635	+	1.25164i
$300$		0			0
$301$	22.0203	+	18.1865i	1.26923	+	1.04825i
$302$		−	5.31935i		−	0.306094i
$303$		0			0
$304$	−0.645751	+	1.11847i	−0.0370364	+	0.0641489i
$305$	−0.776796	+	0.155075i	−0.0444792	+	0.00887956i
$306$		0			0
$307$			20.3996i			1.16427i	0.813093	+	0.582133i	$0.197782\pi$
$307$			20.3996i			1.16427i	−0.813093	+	0.582133i	$0.802218\pi$
$308$	23.0981	+	3.88739i	1.31614	+	0.221505i
$309$		0			0
$310$	6.30274	+	2.13462i	0.357971	+	0.121238i
$311$	−1.73686	+	3.00832i	−0.0984881	+	0.170586i	−0.911059	−	0.412276i	$-0.864734\pi$
$311$	−1.73686	+	3.00832i	−0.0984881	+	0.170586i	0.812571	+	0.582862i	$0.198067\pi$
$312$		0			0
$313$	7.82718	−	4.51902i	0.442418	−	0.255430i	−0.262205	−	0.965012i	$-0.584450\pi$
$313$	7.82718	−	4.51902i	0.442418	−	0.255430i	0.704623	+	0.709582i	$0.251116\pi$
$314$	6.94743			0.392066
$315$		0			0
$316$	−9.77124			−0.549675
$317$	−15.2485	+	8.80372i	−0.856441	+	0.494466i	−0.862819	−	0.505513i	$-0.831303\pi$
$317$	−15.2485	+	8.80372i	−0.856441	+	0.494466i	0.00637790	+	0.999980i	$0.497970\pi$
$318$		0			0
$319$	−6.53137	+	11.3127i	−0.365687	+	0.633388i
$320$	0.508199	−	1.50053i	0.0284092	−	0.0838819i
$321$		0			0
$322$	−3.36961	−	9.03805i	−0.187781	−	0.503671i
$323$		−	3.18678i		−	0.177317i
$324$		0			0
$325$	12.3960	−	16.2013i	0.687608	−	0.898687i
$326$	−0.522697	+	0.905337i	−0.0289495	+	0.0501420i
$327$		0			0
$328$		−	24.4795i		−	1.35166i
$329$	−0.952811	+	5.66142i	−0.0525302	+	0.312124i
$330$		0			0
$331$	−8.79150	−	15.2273i	−0.483225	−	0.836970i	0.516590	−	0.856233i	$-0.327201\pi$
$331$	−8.79150	−	15.2273i	−0.483225	−	0.836970i	−0.999814	+	0.0192633i	$0.993868\pi$
$332$	−16.6122	−	9.59108i	−0.911715	−	0.526379i
$333$		0			0
$334$	3.76013	+	6.51274i	0.205745	+	0.356361i
$335$	3.90383	−	3.42836i	0.213289	−	0.187311i
$336$		0			0
$337$			12.2398i			0.666742i	0.942796	+	0.333371i	$0.108186\pi$
$337$			12.2398i			0.666742i	−0.942796	+	0.333371i	$0.891814\pi$
$338$	−1.87919	+	1.08495i	−0.102215	+	0.0590137i
$339$		0			0
$340$	3.55537	+	17.8094i	0.192817	+	0.965853i
$341$	13.4483	+	23.2932i	0.728268	+	1.26140i
$342$		0			0
$343$	−16.2512	−	8.88249i	−0.877482	−	0.479610i
$344$	−23.4230			−1.26288
$345$		0			0
$346$	−2.77124	+	4.79993i	−0.148983	+	0.258046i
$347$	16.7948	+	9.69651i	0.901594	+	0.520536i	0.877717	−	0.479179i	$-0.159066\pi$
$347$	16.7948	+	9.69651i	0.901594	+	0.520536i	0.0238772	+	0.999715i	$0.492399\pi$
$348$		0			0
$349$	−25.8745			−1.38503			−0.692515	−	0.721403i	$-0.743498\pi$
$349$	−25.8745			−1.38503			−0.692515	+	0.721403i	$0.743498\pi$
$350$	−5.75593	+	5.37241i	−0.307667	+	0.287168i
$351$		0			0
$352$	−25.7632	+	14.8744i	−1.37318	+	0.792807i
$353$	−1.36375	−	0.787360i	−0.0725849	−	0.0419069i	0.463268	−	0.886218i	$-0.346677\pi$
$353$	−1.36375	−	0.787360i	−0.0725849	−	0.0419069i	−0.535853	+	0.844311i	$0.680010\pi$
$354$		0			0
$355$	1.74181	−	5.14293i	0.0924458	−	0.272959i
$356$	8.85305			0.469211
$357$		0			0
$358$		−	9.91625i		−	0.524090i
$359$	−9.54451	−	16.5316i	−0.503740	−	0.872503i	−0.999991	−	0.00432406i	$-0.998624\pi$
$359$	−9.54451	−	16.5316i	−0.503740	−	0.872503i	0.496251	−	0.868179i	$-0.334710\pi$
$360$		0			0
$361$	9.29150	−	16.0934i	0.489026	−	0.847019i
$362$	6.51823	−	3.76330i	0.342590	−	0.197795i
$363$		0			0
$364$	−16.6458	+	6.20595i	−0.872474	+	0.325280i
$365$	15.1852	−	13.3357i	0.794829	−	0.698023i
$366$		0			0
$367$	16.1455	+	9.32160i	0.842787	+	0.486583i	0.858211	−	0.513298i	$-0.171576\pi$
$367$	16.1455	+	9.32160i	0.842787	+	0.486583i	−0.0154235	+	0.999881i	$0.504910\pi$
$368$	−10.6095	−	6.12538i	−0.553057	−	0.319307i
$369$		0			0
$370$	−0.771243	−	0.878205i	−0.0400950	−	0.0456557i
$371$	−19.6117	+	7.31173i	−1.01819	+	0.379606i
$372$		0			0
$373$	20.7088	−	11.9562i	1.07226	−	0.619069i	0.143461	−	0.989656i	$-0.454177\pi$
$373$	20.7088	−	11.9562i	1.07226	−	0.619069i	0.928798	+	0.370587i	$0.120843\pi$
$374$	7.90024	−	13.6836i	0.408512	−	0.707563i
$375$		0			0
$376$	−2.35425	−	4.07768i	−0.121411	−	0.210290i
$377$		−	9.90735i		−	0.510255i
$378$		0			0
$379$	2.29150			0.117707			0.0588533	−	0.998267i	$-0.481256\pi$
$379$	2.29150			0.117707			0.0588533	+	0.998267i	$0.481256\pi$
$380$	0.762298	−	2.25079i	0.0391051	−	0.115463i
$381$		0			0
$382$	5.54553	+	3.20172i	0.283734	+	0.163814i
$383$	−12.3061	+	7.10491i	−0.628811	+	0.363044i	−0.780291	−	0.625416i	$-0.784929\pi$
$383$	−12.3061	+	7.10491i	−0.628811	+	0.363044i	0.151481	+	0.988460i	$0.451596\pi$
$384$		0			0
$385$	−31.8100	+	0.964353i	−1.62119	+	0.0491480i
$386$	−7.47012			−0.380219
$387$		0			0
$388$	9.56997	+	5.52523i	0.485842	+	0.280501i
$389$	−15.4465	+	26.7542i	−0.783171	+	1.35649i	0.146915	+	0.989149i	$0.453066\pi$
$389$	−15.4465	+	26.7542i	−0.783171	+	1.35649i	−0.930086	+	0.367343i	$0.880268\pi$
$390$		0			0
$391$	30.2288			1.52873
$392$	14.9156	−	2.87052i	0.753354	−	0.144983i
$393$		0			0
$394$	−1.76013	−	3.04863i	−0.0886740	−	0.153588i
$395$	13.0192	−	2.59907i	0.655067	−	0.130774i
$396$		0			0
$397$	−27.5060	+	15.8806i	−1.38049	+	0.797023i	−0.992217	−	0.124523i	$-0.960260\pi$
$397$	−27.5060	+	15.8806i	−1.38049	+	0.797023i	−0.388268	+	0.921546i	$0.626927\pi$
$398$		−	10.5025i		−	0.526445i
$399$		0			0
$400$	−1.29150	+	9.91625i	−0.0645751	+	0.495813i
$401$	−19.6117	−	33.9685i	−0.979363	−	1.69631i	−0.664715	−	0.747097i	$-0.731447\pi$
$401$	−19.6117	−	33.9685i	−0.979363	−	1.69631i	−0.314648	−	0.949208i	$-0.601886\pi$
$402$		0			0
$403$	−17.6666	−	10.1998i	−0.880035	−	0.508088i
$404$	−2.85843	−	4.95095i	−0.142212	−	0.246319i
$405$		0			0
$406$	−0.634637	+	3.77089i	−0.0314965	+	0.187146i
$407$		−	4.72416i		−	0.234168i
$408$		0			0
$409$	12.7915	−	22.1555i	0.632499	−	1.09552i	−0.354540	−	0.935041i	$-0.615363\pi$
$409$	12.7915	−	22.1555i	0.632499	−	1.09552i	0.987039	−	0.160480i	$-0.0513041\pi$
$410$	2.93933	+	14.7236i	0.145163	+	0.727148i
$411$		0			0
$412$		−	11.9838i		−	0.590398i
$413$	7.69938	+	20.6515i	0.378862	+	1.01619i
$414$		0			0
$415$	24.6853	+	8.36044i	1.21175	+	0.410398i
$416$	11.2814	−	19.5399i	0.553115	−	0.958023i
$417$		0			0
$418$	−1.79052	+	1.03376i	−0.0875771	+	0.0505627i
$419$	17.7061			0.864999			0.432500	−	0.901634i	$-0.357632\pi$
$419$	17.7061			0.864999			0.432500	+	0.901634i	$0.357632\pi$
$420$		0			0
$421$	−9.22876			−0.449782			−0.224891	−	0.974384i	$-0.572203\pi$
$421$	−9.22876			−0.449782			−0.224891	+	0.974384i	$0.572203\pi$
$422$	−7.69938	+	4.44524i	−0.374800	+	0.216391i
$423$		0			0
$424$	8.58301	−	14.8662i	0.416828	−	0.721967i
$425$	−9.47434	−	22.7836i	−0.459573	−	1.10517i
$426$		0			0
$427$	−0.924256	−	0.155551i	−0.0447279	−	0.00752766i
$428$			8.46878i			0.409354i
$429$		0			0
$430$	14.0881	−	2.81247i	0.679391	−	0.135629i
$431$	10.3286	−	17.8896i	0.497509	−	0.861711i	−0.502487	−	0.864585i	$-0.667581\pi$
$431$	10.3286	−	17.8896i	0.497509	−	0.861711i	0.999996	+	0.00287387i	$0.000914782\pi$
$432$		0			0
$433$		−	33.8287i		−	1.62570i	−0.582472	−	0.812851i	$-0.697914\pi$
$433$		−	33.8287i		−	1.62570i	0.582472	−	0.812851i	$-0.302086\pi$
$434$	6.07080	+	5.01387i	0.291408	+	0.240674i
$435$		0			0
$436$	−10.8856	−	18.8544i	−0.521327	−	0.902964i
$437$	−3.42554	−	1.97774i	−0.163866	−	0.0946079i
$438$		0			0
$439$	15.1144	+	26.1789i	0.721370	+	1.24945i	0.960451	+	0.278450i	$0.0898206\pi$
$439$	15.1144	+	26.1789i	0.721370	+	1.24945i	−0.239081	+	0.971000i	$0.576846\pi$
$440$	19.6117	−	17.2231i	0.934952	−	0.821079i
$441$		0			0
$442$			11.9838i			0.570009i
$443$	−14.5505	+	8.40071i	−0.691313	+	0.399130i	−0.804104	−	0.594489i	$-0.797354\pi$
$443$	−14.5505	+	8.40071i	−0.691313	+	0.399130i	0.112791	+	0.993619i	$0.464021\pi$
$444$		0			0
$445$	−11.7958	+	2.35484i	−0.559175	+	0.111630i
$446$		0			0
$447$		0			0
$448$	1.19368	−	1.44531i	0.0563960	−	0.0682843i
$449$	22.5627			1.06480			0.532401	−	0.846492i	$-0.321290\pi$
$449$	22.5627			1.06480			0.532401	+	0.846492i	$0.321290\pi$
$450$		0			0
$451$	−30.3431	+	52.5559i	−1.42880	+	2.47476i
$452$	22.5503	+	13.0194i	1.06068	+	0.612383i
$453$		0			0
$454$	−2.10326			−0.0987111
$455$	20.5281	−	12.6964i	0.962371	−	0.595218i
$456$		0			0
$457$	27.7754	−	16.0361i	1.29928	−	0.750139i	0.318999	−	0.947755i	$-0.396653\pi$
$457$	27.7754	−	16.0361i	1.29928	−	0.750139i	0.980279	+	0.197617i	$0.0633201\pi$
$458$	−1.18115	−	0.681937i	−0.0551915	−	0.0318648i
$459$		0			0
$460$	21.3502	+	7.23090i	0.995459	+	0.337143i
$461$	−16.1380			−0.751622			−0.375811	−	0.926696i	$-0.622636\pi$
$461$	−16.1380			−0.751622			−0.375811	+	0.926696i	$0.622636\pi$
$462$		0			0
$463$			17.5090i			0.813712i	0.913492	+	0.406856i	$0.133375\pi$
$463$			17.5090i			0.813712i	−0.913492	+	0.406856i	$0.866625\pi$
$464$	2.42832	+	4.20597i	0.112732	+	0.195257i
$465$		0			0
$466$	−2.29150	+	3.96900i	−0.106152	+	0.183860i
$467$	35.5868	−	20.5460i	1.64676	−	0.950757i	0.668412	−	0.743791i	$-0.266974\pi$
$467$	35.5868	−	20.5460i	1.64676	−	0.950757i	0.978348	−	0.206966i	$-0.0663591\pi$
$468$		0			0
$469$	5.76013	−	2.14752i	0.265978	−	0.0991632i
$470$	1.90562	+	2.16991i	0.0878998	+	0.100090i
$471$		0			0
$472$	−15.6544	−	9.03805i	−0.720550	−	0.416010i
$473$	50.2875	+	29.0335i	2.31222	+	1.33496i
$474$		0			0
$475$	−0.416995	+	3.20172i	−0.0191330	+	0.146905i
$476$	−3.56630	+	21.1902i	−0.163461	+	0.971252i
$477$		0			0
$478$	4.56329	−	2.63461i	0.208720	−	0.120504i
$479$	6.33215	−	10.9676i	0.289323	−	0.501122i	−0.684325	−	0.729177i	$-0.739903\pi$
$479$	6.33215	−	10.9676i	0.289323	−	0.501122i	0.973648	+	0.228055i	$0.0732364\pi$
$480$		0			0
$481$	1.79150	+	3.10297i	0.0816855	+	0.141483i
$482$			16.8014i			0.765284i
$483$		0			0
$484$	29.5203			1.34183
$485$	−14.2207	−	4.81628i	−0.645729	−	0.218696i
$486$		0			0
$487$	20.7088	+	11.9562i	0.938404	+	0.541788i	0.889460	−	0.457014i	$-0.151081\pi$
$487$	20.7088	+	11.9562i	0.938404	+	0.541788i	0.0489441	+	0.998802i	$0.484414\pi$
$488$	0.665702	−	0.384343i	0.0301349	−	0.0173984i
$489$		0			0
$490$	−8.62660	+	3.51749i	−0.389710	+	0.158904i
$491$	36.2724			1.63695			0.818476	−	0.574541i	$-0.194819\pi$
$491$	36.2724			1.63695			0.818476	+	0.574541i	$0.194819\pi$
$492$		0			0
$493$	−10.3782	−	5.99188i	−0.467413	−	0.269861i
$494$	0.784045	−	1.35801i	0.0352759	−	0.0610996i
$495$		0			0
$496$	10.0000			0.449013
$497$	4.09124	−	4.95368i	0.183517	−	0.222203i
$498$		0			0
$499$	−2.50000	−	4.33013i	−0.111915	−	0.193843i	0.804627	−	0.593780i	$-0.202365\pi$
$499$	−2.50000	−	4.33013i	−0.111915	−	0.193843i	−0.916542	+	0.399937i	$0.869032\pi$
$500$	−1.24164	−	18.3581i	−0.0555277	−	0.821000i
$501$		0			0
$502$	8.31830	−	4.80257i	0.371264	−	0.214349i
$503$		−	15.8219i		−	0.705463i	−0.935725	−	0.352731i	$-0.885253\pi$
$503$		−	15.8219i		−	0.705463i	0.935725	−	0.352731i	$-0.114747\pi$
$504$		0			0
$505$	5.12549	+	5.83633i	0.228081	+	0.259713i
$506$	−9.80586	−	16.9842i	−0.435924	−	0.755042i
$507$		0			0
$508$	−5.81496	−	3.35727i	−0.257997	−	0.148955i
$509$	−10.7587	−	18.6346i	−0.476870	−	0.825963i	0.522779	−	0.852468i	$-0.324895\pi$
$509$	−10.7587	−	18.6346i	−0.476870	−	0.825963i	−0.999649	+	0.0265057i	$0.991562\pi$
$510$		0			0
$511$	22.4059	−	8.35347i	0.991178	−	0.369536i
$512$			19.7400i			0.872393i
$513$		0			0
$514$	0.468627	−	0.811686i	0.0206702	−	0.0358019i
$515$	3.18759	+	15.9672i	0.140462	+	0.703597i
$516$		0			0
$517$			11.6727i			0.513363i
$518$	−0.483106	−	1.29580i	−0.0212264	−	0.0569341i
$519$		0			0
$520$	−6.35021	+	18.7499i	−0.278475	+	0.822236i
$521$	8.33035	−	14.4286i	0.364959	−	0.632128i	−0.623810	−	0.781576i	$-0.714416\pi$
$521$	8.33035	−	14.4286i	0.364959	−	0.632128i	0.988770	+	0.149448i	$0.0477495\pi$
$522$		0			0
$523$	2.01222	−	1.16176i	0.0879882	−	0.0508000i	−0.455360	−	0.890307i	$-0.650490\pi$
$523$	2.01222	−	1.16176i	0.0879882	−	0.0508000i	0.543349	+	0.839507i	$0.317156\pi$
$524$	12.8495			0.561331
$525$		0			0
$526$	−8.58301			−0.374237
$527$	−21.3692	+	12.3375i	−0.930856	+	0.537430i
$528$		0			0
$529$	7.26013	−	12.5749i	0.315658	−	0.546735i
$530$	−3.37736	+	9.97212i	−0.146703	+	0.433161i
$531$		0			0
$532$	1.79052	−	2.16796i	0.0776288	−	0.0939930i
$533$		−	46.0271i		−	1.99365i
$534$		0			0
$535$	−2.25263	−	11.2838i	−0.0973897	−	0.487842i
$536$	−2.52090	+	4.36633i	−0.108886	+	0.188597i
$537$		0			0
$538$			14.5633i			0.627867i
$539$	−35.5810	−	12.3256i	−1.53258	−	0.530901i
$540$		0			0
$541$	−14.7915	−	25.6196i	−0.635936	−	1.10147i	−0.986316	−	0.164866i	$-0.947281\pi$
$541$	−14.7915	−	25.6196i	−0.635936	−	1.10147i	0.350380	−	0.936608i	$-0.386053\pi$
$542$	−11.4578	−	6.61514i	−0.492153	−	0.284145i
$543$		0			0
$544$	−13.6458	−	23.6351i	−0.585057	−	1.01335i
$545$	19.5191	+	22.2262i	0.836108	+	0.952065i
$546$		0			0
$547$		−	32.6394i		−	1.39556i	−0.716313	−	0.697779i	$-0.754172\pi$
$547$		−	32.6394i		−	1.39556i	0.716313	−	0.697779i	$-0.245828\pi$
$548$	−5.63758	+	3.25486i	−0.240826	+	0.139041i
$549$		0			0
$550$	−9.72777	+	12.7140i	−0.414794	+	0.542126i
$551$	0.784045	+	1.35801i	0.0334015	+	0.0578530i
$552$		0			0
$553$	15.4906	+	2.60706i	0.658729	+	0.110864i
$554$	10.2360			0.434885
$555$		0			0
$556$	−5.94837	+	10.3029i	−0.252267	+	0.436939i
$557$	1.57869	+	0.911455i	0.0668911	+	0.0386196i	0.533073	−	0.846070i	$-0.321037\pi$
$557$	1.57869	+	0.911455i	0.0668911	+	0.0386196i	−0.466181	+	0.884689i	$0.654371\pi$
$558$		0			0
$559$	−44.0405			−1.86272
$560$	−5.60286	+	10.4215i	−0.236764	+	0.440389i
$561$		0			0
$562$	2.05994	−	1.18931i	0.0868934	−	0.0501679i
$563$	7.51678	+	4.33981i	0.316794	+	0.182901i	0.649963	−	0.759966i	$-0.274784\pi$
$563$	7.51678	+	4.33981i	0.316794	+	0.182901i	−0.333168	+	0.942867i	$0.608118\pi$
$564$		0			0
$565$	−33.5091	−	11.3489i	−1.40974	−	0.477452i
$566$	−14.0472			−0.590449
$567$		0			0
$568$			5.26923i			0.221092i
$569$	−11.5427	−	19.9926i	−0.483896	−	0.838132i	0.515933	−	0.856629i	$-0.327445\pi$
$569$	−11.5427	−	19.9926i	−0.483896	−	0.838132i	−0.999829	+	0.0184967i	$0.994112\pi$
$570$		0			0
$571$	−2.26013	+	3.91466i	−0.0945835	+	0.163823i	−0.909435	−	0.415847i	$-0.863485\pi$
$571$	−2.26013	+	3.91466i	−0.0945835	+	0.163823i	0.814851	+	0.579670i	$0.196819\pi$
$572$	−31.2806	+	18.0599i	−1.30791	+	0.755121i
$573$		0			0
$574$	−2.94837	+	17.5186i	−0.123062	+	0.731213i
$575$	−30.3704	−	3.95547i	−1.26653	−	0.164955i
$576$		0			0
$577$	−19.1877	−	11.0780i	−0.798793	−	0.461183i	0.0442559	−	0.999020i	$-0.485908\pi$
$577$	−19.1877	−	11.0780i	−0.798793	−	0.461183i	−0.843049	+	0.537837i	$0.819242\pi$
$578$	3.79073	+	2.18858i	0.157674	+	0.0910329i
$579$		0			0
$580$	−5.89674	−	6.71453i	−0.244849	−	0.278806i
$581$	23.7769	+	19.6373i	0.986432	+	0.814694i
$582$		0			0
$583$	−36.8542	+	21.2778i	−1.52635	+	0.881237i
$584$	−9.80586	+	16.9842i	−0.405769	+	0.702813i
$585$		0			0
$586$	−4.52026	−	7.82932i	−0.186730	−	0.323426i
$587$		−	44.2789i		−	1.82758i	−0.406182	−	0.913792i	$-0.633140\pi$
$587$		−	44.2789i		−	1.82758i	0.406182	−	0.913792i	$-0.366860\pi$
$588$		0			0
$589$	3.22876			0.133039
$590$	10.5008	+	3.55642i	0.432311	+	0.146415i
$591$		0			0
$592$	−1.52110	−	0.878205i	−0.0625166	−	0.0360940i
$593$	21.9170	−	12.6538i	0.900022	−	0.519628i	0.0228149	−	0.999740i	$-0.492737\pi$
$593$	21.9170	−	12.6538i	0.900022	−	0.519628i	0.877207	+	0.480112i	$0.159404\pi$
$594$		0			0
$595$	−0.884698	−	29.1825i	−0.0362691	−	1.19636i
$596$	4.85664			0.198936
$597$		0			0
$598$	12.8816	+	7.43719i	0.526767	+	0.304129i
$599$	−12.2342	+	21.1902i	−0.499875	+	0.865809i	−1.00000		0.000144290i	$-0.999954\pi$
$599$	−12.2342	+	21.1902i	−0.499875	+	0.865809i	0.500125	+	0.865953i	$0.333287\pi$
$600$		0			0
$601$	23.1033			0.942402			0.471201	−	0.882026i	$-0.343821\pi$
$601$	23.1033			0.942402			0.471201	+	0.882026i	$0.343821\pi$
$602$	16.7625	+	2.82111i	0.683188	+	0.114980i
$603$		0			0
$604$	7.35425	+	12.7379i	0.299240	+	0.518299i
$605$	−39.3328	+	7.85215i	−1.59911	+	0.319235i
$606$		0			0
$607$	16.1455	−	9.32160i	0.655325	−	0.378352i	−0.135169	−	0.990823i	$-0.543158\pi$
$607$	16.1455	−	9.32160i	0.655325	−	0.378352i	0.790493	+	0.612471i	$0.209824\pi$
$608$			3.57113i			0.144828i
$609$		0			0
$610$	−0.354249	+	0.311103i	−0.0143431	+	0.0125962i
$611$	−4.42652	−	7.66697i	−0.179078	−	0.310172i
$612$		0			0
$613$	25.4938	+	14.7188i	1.02968	+	0.594488i	0.916894	−	0.399131i	$-0.130688\pi$
$613$	25.4938	+	14.7188i	1.02968	+	0.594488i	0.112789	+	0.993619i	$0.464022\pi$
$614$	6.07080	+	10.5149i	0.244997	+	0.424348i
$615$		0			0
$616$	28.9373	−	10.7885i	1.16592	−	0.434682i
$617$		−	13.8628i		−	0.558096i	−0.960277	−	0.279048i	$-0.909981\pi$
$617$		−	13.8628i		−	0.558096i	0.960277	−	0.279048i	$-0.0900189\pi$
$618$		0			0
$619$	6.50000	−	11.2583i	0.261257	−	0.452510i	−0.705319	−	0.708890i	$-0.749196\pi$
$619$	6.50000	−	11.2583i	0.261257	−	0.452510i	0.966576	+	0.256379i	$0.0825296\pi$
$620$	−18.0440	+	3.60219i	−0.724665	+	0.144668i
$621$		0			0
$622$			2.06751i			0.0828997i
$623$	−14.0350	−	2.36208i	−0.562301	−	0.0946347i
$624$		0			0
$625$	6.53747	+	24.1301i	0.261499	+	0.965204i
$626$	2.68967	−	4.65864i	0.107501	−	0.186197i
$627$		0			0
$628$	−16.6366	+	9.60515i	−0.663873	+	0.383287i
$629$	4.33394			0.172806
$630$		0			0
$631$	3.16601			0.126037			0.0630184	−	0.998012i	$-0.479927\pi$
$631$	3.16601			0.126037			0.0630184	+	0.998012i	$0.479927\pi$
$632$	−11.1573	+	6.44165i	−0.443812	+	0.256235i
$633$		0			0
$634$	−5.23987	+	9.07572i	−0.208102	+	0.360443i
$635$	8.64086	+	2.92649i	0.342902	+	0.116134i
$636$		0			0
$637$	28.0448	−	5.39723i	1.11118	−	0.213846i
$638$			7.77479i			0.307807i
$639$		0			0
$640$	−5.02635	−	25.1779i	−0.198684	−	0.995243i
$641$	−11.0200	+	19.0872i	−0.435265	+	0.753900i	−0.997317	−	0.0732011i	$-0.976679\pi$
$641$	−11.0200	+	19.0872i	−0.435265	+	0.753900i	0.562053	+	0.827102i	$0.310012\pi$
$642$		0			0
$643$			39.0979i			1.54187i	0.636913	+	0.770935i	$0.280211\pi$
$643$			39.0979i			1.54187i	−0.636913	+	0.770935i	$0.719789\pi$
$644$	20.5645	+	16.9842i	0.810356	+	0.669273i
$645$		0			0
$646$	−0.948368	−	1.64262i	−0.0373130	−	0.0646281i
$647$	21.1866	+	12.2321i	0.832931	+	0.480893i	0.854855	−	0.518867i	$-0.173646\pi$
$647$	21.1866	+	12.2321i	0.832931	+	0.480893i	−0.0219243	+	0.999760i	$0.506979\pi$
$648$		0			0
$649$	22.4059	+	38.8081i	0.879508	+	1.52335i
$650$	1.56809	−	12.0399i	0.0615056	−	0.472244i
$651$		0			0
$652$		−	2.89061i		−	0.113205i
$653$	−24.8271	+	14.3339i	−0.971558	+	0.560929i	−0.899711	−	0.436486i	$-0.856223\pi$
$653$	−24.8271	+	14.3339i	−0.971558	+	0.560929i	−0.0718473	+	0.997416i	$0.522889\pi$
$654$		0			0
$655$	−17.1206	+	3.41785i	−0.668958	+	0.133547i
$656$	11.2814	+	19.5399i	0.440463	+	0.762905i
$657$		0			0
$658$	1.19368	+	3.20172i	0.0465344	+	0.124816i
$659$	−17.7061			−0.689732			−0.344866	−	0.938652i	$-0.612076\pi$
$659$	−17.7061			−0.689732			−0.344866	+	0.938652i	$0.612076\pi$
$660$		0			0
$661$	−9.32288	+	16.1477i	−0.362618	+	0.628073i	−0.988391	−	0.151933i	$-0.951450\pi$
$661$	−9.32288	+	16.1477i	−0.362618	+	0.628073i	0.625773	+	0.780005i	$0.284784\pi$
$662$	−9.06312	−	5.23260i	−0.352248	−	0.203371i
$663$		0			0
$664$	−25.2915			−0.981501
$665$	−1.80903	+	3.36485i	−0.0701510	+	0.130483i
$666$		0			0
$667$	−12.8816	+	7.43719i	−0.498777	+	0.287969i
$668$	−18.0083	−	10.3971i	−0.696763	−	0.402277i
$669$		0			0
$670$	0.991961	−	2.92890i	0.0383228	−	0.113153i
$671$	−1.90562			−0.0735657
$672$		0			0
$673$			12.2398i			0.471808i	0.971776	+	0.235904i	$0.0758051\pi$
$673$			12.2398i			0.471808i	−0.971776	+	0.235904i	$0.924195\pi$
$674$	3.64248	+	6.30896i	0.140303	+	0.243012i
$675$		0			0
$676$	3.00000	−	5.19615i	0.115385	−	0.199852i
$677$	−26.2232	+	15.1399i	−1.00784	+	0.581875i	−0.910558	−	0.413382i	$-0.864348\pi$
$677$	−26.2232	+	15.1399i	−1.00784	+	0.581875i	−0.0972799	+	0.995257i	$0.531014\pi$
$678$		0			0
$679$	−13.6974	−	11.3127i	−0.525657	−	0.434140i
$680$	15.8005	+	17.9918i	0.605921	+	0.689954i
$681$		0			0
$682$	13.8638	+	8.00429i	0.530874	+	0.306500i
$683$	−29.7990	−	17.2044i	−1.14023	−	0.658309i	−0.193739	−	0.981053i	$-0.562061\pi$
$683$	−29.7990	−	17.2044i	−1.14023	−	0.658309i	−0.946486	+	0.322744i	$0.895395\pi$
$684$		0			0
$685$	6.64575	−	5.83633i	0.253921	−	0.222995i
$686$	−11.0200	+	0.257794i	−0.420746	+	0.00984264i
$687$		0			0
$688$	18.6965	−	10.7945i	0.712799	−	0.411535i
$689$	16.1380	−	27.9518i	0.614809	−	1.06488i
$690$		0			0
$691$	−7.14575	−	12.3768i	−0.271837	−	0.470836i	0.697495	−	0.716590i	$-0.254298\pi$
$691$	−7.14575	−	12.3768i	−0.271837	−	0.470836i	−0.969332	+	0.245754i	$0.920965\pi$
$692$		−	15.3255i		−	0.582588i
$693$		0			0
$694$	11.5425			0.438147
$695$	5.18512	−	15.3098i	0.196683	−	0.580733i
$696$		0			0
$697$	−48.2147	−	27.8368i	−1.82626	−	1.05439i
$698$	−13.3370	+	7.70010i	−0.504811	+	0.291453i
$699$		0			0
$700$	6.35577	−	20.8229i	0.240226	−	0.787030i
$701$	−25.8513			−0.976390			−0.488195	−	0.872735i	$-0.662344\pi$
$701$	−25.8513			−0.976390			−0.488195	+	0.872735i	$0.662344\pi$
$702$		0			0
$703$	−0.491125	−	0.283551i	−0.0185231	−	0.0106943i
$704$	1.90562	−	3.30064i	0.0718209	−	0.124397i
$705$		0			0
$706$	−0.937254			−0.0352740
$707$	3.21060	+	8.61155i	0.120747	+	0.323871i
$708$		0			0
$709$	−3.17712	−	5.50294i	−0.119319	−	0.206667i	0.800179	−	0.599762i	$-0.204738\pi$
$709$	−3.17712	−	5.50294i	−0.119319	−	0.206667i	−0.919498	+	0.393094i	$0.871405\pi$
$710$	−0.632693	−	3.16927i	−0.0237445	−	0.118940i
$711$		0			0
$712$	10.1088	−	5.83633i	0.378844	−	0.218726i
$713$			30.6269i			1.14699i
$714$		0			0
$715$	36.8745	−	32.3834i	1.37903	−	1.21107i
$716$	13.7097	+	23.7459i	0.512355	+	0.887425i
$717$		0			0
$718$	−9.83940	−	5.68078i	−0.367203	−	0.212005i
$719$	9.80586	+	16.9842i	0.365697	+	0.633406i	0.988888	−	0.148664i	$-0.0474974\pi$
$719$	9.80586	+	16.9842i	0.365697	+	0.633406i	−0.623191	+	0.782070i	$0.714164\pi$
$720$		0			0
$721$	−3.19738	+	18.9982i	−0.119077	+	0.707531i
$722$		−	11.0604i		−	0.411625i
$723$		0			0
$724$	−10.4059	+	18.0235i	−0.386732	+	0.669839i
$725$	9.64282	+	7.37796i	0.358126	+	0.274011i
$726$		0			0
$727$		−	40.5432i		−	1.50366i	−0.659355	−	0.751832i	$-0.729170\pi$
$727$		−	40.5432i		−	1.50366i	0.659355	−	0.751832i	$-0.270830\pi$
$728$	−14.9156	+	18.0599i	−0.552811	+	0.669343i
$729$		0			0
$730$	3.85855	−	11.3929i	0.142811	−	0.421670i
$731$	−26.6353	+	46.1337i	−0.985143	+	1.70632i
$732$		0			0
$733$	9.07885	−	5.24168i	0.335335	−	0.193606i	−0.322872	−	0.946443i	$-0.604648\pi$
$733$	9.07885	−	5.24168i	0.335335	−	0.193606i	0.658207	+	0.752837i	$0.271315\pi$
$734$	11.0962			0.409568
$735$		0			0
$736$	−33.8745			−1.24863
$737$	10.8244	−	6.24947i	0.398722	−	0.230202i
$738$		0			0
$739$	5.96863	−	10.3380i	0.219559	−	0.380288i	−0.735114	−	0.677944i	$-0.762871\pi$
$739$	5.96863	−	10.3380i	0.219559	−	0.380288i	0.954673	+	0.297655i	$0.0962047\pi$
$740$	3.06101	+	1.03671i	0.112525	+	0.0381101i
$741$		0			0
$742$	−7.93289	+	9.60515i	−0.291226	+	0.352616i
$743$			10.6760i			0.391666i	0.980637	+	0.195833i	$0.0627410\pi$
$743$			10.6760i			0.391666i	−0.980637	+	0.195833i	$0.937259\pi$
$744$		0			0
$745$	−6.47099	+	1.29183i	−0.237079	+	0.0473289i
$746$	7.11619	−	12.3256i	0.260542	−	0.451273i
$747$		0			0
$748$			43.6898i			1.59746i
$749$	2.25955	−	13.4258i	0.0825623	−	0.490569i
$750$		0			0
$751$	−13.7288	−	23.7789i	−0.500969	−	0.867705i	−0.999999	−	0.00111978i	$-0.999644\pi$
$751$	−13.7288	−	23.7789i	−0.500969	−	0.867705i	0.499030	−	0.866585i	$-0.333690\pi$
$752$	3.75839	+	2.16991i	0.137054	+	0.0791284i
$753$		0			0
$754$	−2.94837	−	5.10672i	−0.107373	−	0.185976i
$755$	−13.1870	−	15.0159i	−0.479924	−	0.546483i
$756$		0			0
$757$			40.7992i			1.48287i	0.671023	+	0.741436i	$0.265855\pi$
$757$			40.7992i			1.48287i	−0.671023	+	0.741436i	$0.734145\pi$
$758$	1.18115	−	0.681937i	0.0429013	−	0.0247691i
$759$		0			0
$760$	−0.613394	−	3.07260i	−0.0222501	−	0.111455i
$761$	−2.25955	−	3.91366i	−0.0819087	−	0.141870i	0.822161	−	0.569255i	$-0.192768\pi$
$761$	−2.25955	−	3.91366i	−0.0819087	−	0.141870i	−0.904070	+	0.427385i	$0.859435\pi$
$762$		0			0
$763$	12.2268	+	32.7949i	0.442638	+	1.18726i
$764$	−17.7061			−0.640584
$765$		0			0
$766$	−4.22876	+	7.32442i	−0.152791	+	0.264642i
$767$	−29.4338	−	16.9936i	−1.06279	−	0.613603i
$768$		0			0
$769$	24.6458			0.888749			0.444374	−	0.895841i	$-0.353426\pi$
$769$	24.6458			0.888749			0.444374	+	0.895841i	$0.353426\pi$
$770$	−16.1094	+	9.96353i	−0.580542	+	0.359061i
$771$		0			0
$772$	17.8883	−	10.3278i	0.643813	−	0.371706i
$773$	−16.5799	−	9.57241i	−0.596338	−	0.344296i	0.171262	−	0.985226i	$-0.445216\pi$
$773$	−16.5799	−	9.57241i	−0.596338	−	0.344296i	−0.767599	+	0.640930i	$0.778549\pi$
$774$		0			0
$775$	23.0837	−	9.59913i	0.829191	−	0.344811i
$776$	14.5699			0.523030
$777$		0			0
$778$			18.3872i			0.659213i
$779$	3.64248	+	6.30896i	0.130505	+	0.226042i
$780$		0			0
$781$	6.53137	−	11.3127i	0.233711	−	0.404799i
$782$	15.5813	−	8.99590i	0.557188	−	0.321693i
$783$		0			0
$784$	−10.5830	+	9.16515i	−0.377964	+	0.327327i
$785$	19.6117	−	17.2231i	0.699972	−	0.614719i
$786$		0			0
$787$	12.6122	+	7.28164i	0.449575	+	0.259562i	0.707651	−	0.706562i	$-0.249755\pi$
$787$	12.6122	+	7.28164i	0.449575	+	0.259562i	−0.258076	+	0.966125i	$0.583088\pi$
$788$	8.42976	+	4.86693i	0.300298	+	0.173377i
$789$		0			0
$790$	5.93725	−	5.21412i	0.211238	−	0.185510i
$791$	−32.2760	−	26.6568i	−1.14760	−	0.947805i
$792$		0			0
$793$	1.25167	−	0.722653i	0.0444482	−	0.0256622i
$794$	−9.45193	+	16.3712i	−0.335436	+	0.580993i
$795$		0			0
$796$	14.5203	+	25.1498i	0.514657	+	0.891412i
$797$			3.53378i			0.125173i	0.998040	+	0.0625865i	$0.0199349\pi$
$797$			3.53378i			0.125173i	−0.998040	+	0.0625865i	$0.980065\pi$
$798$		0			0
$799$	−10.7085			−0.378839
$800$	10.6170	+	25.5314i	0.375368	+	0.902673i
$801$		0			0
$802$	−20.2176	−	11.6727i	−0.713910	−	0.412176i
$803$	42.1050	−	24.3093i	1.48585	−	0.857858i
$804$		0			0
$805$	−31.9179	−	17.1598i	−1.12496	−	0.604804i
$806$	−12.1416			−0.427670
$807$		0			0
$808$	−6.52778	−	3.76882i	−0.229647	−	0.132587i
$809$	22.4702	−	38.9194i	0.790009	−	1.36833i	−0.135952	−	0.990715i	$-0.543409\pi$
$809$	22.4702	−	38.9194i	0.790009	−	1.36833i	0.925961	−	0.377620i	$-0.123257\pi$
$810$		0			0
$811$	40.2288			1.41262			0.706311	−	0.707901i	$-0.250358\pi$
$811$	40.2288			1.41262			0.706311	+	0.707901i	$0.250358\pi$
$812$	−3.69370	−	9.90735i	−0.129624	−	0.347680i
$813$		0			0
$814$	−1.40588	−	2.43506i	−0.0492761	−	0.0853487i
$815$	0.768880	+	3.85145i	0.0269327	+	0.134911i
$816$		0			0
$817$	6.03666	−	3.48527i	0.211196	−	0.121934i
$818$		−	15.2267i		−	0.532389i
$819$		0			0
$820$	−27.3948	−	31.1941i	−0.956667	−	1.08934i
$821$	8.59170	+	14.8813i	0.299852	+	0.519360i	0.976102	−	0.217313i	$-0.0697292\pi$
$821$	8.59170	+	14.8813i	0.299852	+	0.519360i	−0.676250	+	0.736673i	$0.736396\pi$
$822$		0			0
$823$	41.6870	+	24.0680i	1.45312	+	0.838957i	0.998657	−	0.0518109i	$-0.0164993\pi$
$823$	41.6870	+	24.0680i	1.45312	+	0.838957i	0.454459	+	0.890768i	$0.349833\pi$
$824$	−7.90024	−	13.6836i	−0.275218	−	0.476691i
$825$		0			0
$826$	10.1144	+	8.35347i	0.351924	+	0.290654i
$827$			31.0112i			1.07837i	0.842189	+	0.539183i	$0.181267\pi$
$827$			31.0112i			1.07837i	−0.842189	+	0.539183i	$0.818733\pi$
$828$		0			0
$829$	15.5516	−	26.9362i	0.540131	−	0.935533i	−0.458766	−	0.888557i	$-0.651708\pi$
$829$	15.5516	−	26.9362i	0.540131	−	0.935533i	0.998896	−	0.0469761i	$-0.0149585\pi$
$830$	15.2120	−	3.03683i	0.528016	−	0.105410i
$831$		0			0
$832$			2.89061i			0.100214i
$833$	11.3075	−	32.6420i	0.391782	−	1.13098i
$834$		0			0
$835$	26.7599	+	9.06305i	0.926063	+	0.313640i
$836$	2.85843	−	4.95095i	0.0988610	−	0.171232i
$837$		0			0
$838$	9.12657	−	5.26923i	0.315272	−	0.182022i
$839$	−42.6972			−1.47407			−0.737035	−	0.675855i	$-0.763775\pi$
$839$	−42.6972			−1.47407			−0.737035	+	0.675855i	$0.763775\pi$
$840$		0			0
$841$	−23.1033			−0.796664
$842$	−4.75694	+	2.74642i	−0.163935	+	0.0946480i
$843$		0			0
$844$	12.2915	−	21.2895i	0.423091	−	0.732815i
$845$	−2.61507	+	7.72133i	−0.0899610	+	0.265622i
$846$		0			0
$847$	−46.7994	−	7.87629i	−1.60804	−	0.270633i
$848$			15.8219i			0.543326i
$849$		0			0
$850$	−11.6638	−	8.92426i	−0.400065	−	0.306100i
$851$	2.68967	−	4.65864i	0.0922006	−	0.159696i
$852$		0			0
$853$		−	2.63461i		−	0.0902075i	−0.998982	−	0.0451037i	$-0.985638\pi$
$853$		−	2.63461i		−	0.0902075i	0.998982	−	0.0451037i	$-0.0143618\pi$
$854$	−0.522697	+	0.194874i	−0.0178863	+	0.00666846i
$855$		0			0
$856$	5.58301	+	9.67005i	0.190823	+	0.330515i
$857$	41.4393	+	23.9250i	1.41554	+	0.817262i	0.995903	−	0.0904296i	$-0.0288240\pi$
$857$	41.4393	+	23.9250i	1.41554	+	0.817262i	0.419637	+	0.907692i	$0.362157\pi$
$858$		0			0
$859$	8.53137	+	14.7768i	0.291087	+	0.504177i	0.974067	−	0.226260i	$-0.0726499\pi$
$859$	8.53137	+	14.7768i	0.291087	+	0.504177i	−0.682980	+	0.730437i	$0.739317\pi$
$860$	−29.8477	+	26.2124i	−1.01780	+	0.893835i
$861$		0			0
$862$		−	12.2949i		−	0.418765i
$863$	38.1317	−	22.0153i	1.29802	−	0.749411i	0.317956	−	0.948105i	$-0.397003\pi$
$863$	38.1317	−	22.0153i	1.29802	−	0.749411i	0.980061	+	0.198695i	$0.0636702\pi$
$864$		0			0
$865$	4.07646	+	20.4197i	0.138604	+	0.694291i
$866$	−10.0672	−	17.4369i	−0.342098	−	0.592531i
$867$		0			0
$868$	−21.4693	−	3.61327i	−0.728716	−	0.122642i
$869$	31.9385			1.08344
$870$		0			0
$871$	−4.73987	+	8.20970i	−0.160604	+	0.278175i
$872$	−24.8594	−	14.3526i	−0.841846	−	0.486040i
$873$		0			0
$874$	−2.35425			−0.0796336
$875$	−2.92972	+	29.4350i	−0.0990426	+	0.995083i
$876$		0			0
$877$	−27.2843	+	15.7526i	−0.921324	+	0.531927i	−0.884057	−	0.467379i	$-0.845199\pi$
$877$	−27.2843	+	15.7526i	−0.921324	+	0.531927i	−0.0372670	+	0.999305i	$0.511865\pi$
$878$	15.5813	+	8.99590i	0.525845	+	0.303597i
$879$		0			0
$880$	−7.71710	+	22.7858i	−0.260143	+	0.768108i
$881$	45.9857			1.54930			0.774649	−	0.632392i	$-0.217927\pi$
$881$	45.9857			1.54930			0.774649	+	0.632392i	$0.217927\pi$
$882$		0			0
$883$			21.8449i			0.735140i	0.929996	+	0.367570i	$0.119810\pi$
$883$			21.8449i			0.735140i	−0.929996	+	0.367570i	$0.880190\pi$
$884$	−16.5681	−	28.6968i	−0.557246	−	0.965179i
$885$		0			0
$886$	−5.00000	+	8.66025i	−0.167978	+	0.290947i
$887$	−0.0323423	+	0.0186729i	−0.00108595	+	0.000626973i	−0.500543	−	0.865712i	$-0.666866\pi$
$887$	−0.0323423	+	0.0186729i	−0.00108595	+	0.000626973i	0.499457	+	0.866339i	$0.333533\pi$
$888$		0			0
$889$	8.32288	+	6.87386i	0.279140	+	0.230542i
$890$	−5.37934	+	4.72416i	−0.180316	+	0.158354i
$891$		0			0
$892$		0			0
$893$	1.21349	+	0.700610i	0.0406080	+	0.0234450i
$894$		0			0
$895$	−24.5830	−	27.9923i	−0.821719	−	0.935681i
$896$	5.04180	−	29.9574i	0.168435	−	1.00081i
$897$		0			0
$898$	11.6299	−	6.71453i	0.388095	−	0.224067i
$899$	6.07080	−	10.5149i	0.202472	−	0.350693i
$900$		0			0
$901$	−19.5203	−	33.8101i	−0.650314	−	1.12638i
$902$			36.1197i			1.20266i
$903$		0			0
$904$	34.3320			1.14187
$905$	9.07069	−	26.7824i	0.301520	−	0.890278i
$906$		0			0
$907$	0.760548	+	0.439102i	0.0252536	+	0.0145802i	0.512574	−	0.858643i	$-0.328692\pi$
$907$	0.760548	+	0.439102i	0.0252536	+	0.0145802i	−0.487320	+	0.873223i	$0.662025\pi$
$908$	5.03657	−	2.90786i	0.167144	−	0.0965008i
$909$		0			0
$910$	6.80277	−	12.6534i	0.225509	−	0.419456i
$911$	−1.56809			−0.0519532			−0.0259766	−	0.999663i	$-0.508270\pi$
$911$	−1.56809			−0.0519532			−0.0259766	+	0.999663i	$0.508270\pi$
$912$		0			0
$913$	54.2991	+	31.3496i	1.79704	+	1.03752i
$914$	9.54451	−	16.5316i	0.315704	−	0.546816i
$915$		0			0
$916$	3.77124			0.124605
$917$	−20.3706	−	3.42836i	−0.672698	−	0.113214i
$918$		0			0
$919$	−4.43725	−	7.68555i	−0.146372	−	0.253523i	0.783512	−	0.621376i	$-0.213426\pi$
$919$	−4.43725	−	7.68555i	−0.146372	−	0.253523i	−0.929884	+	0.367853i	$0.880093\pi$
$920$	29.1456	−	5.81845i	0.960902	−	0.191828i
$921$		0			0
$922$	−8.31830	+	4.80257i	−0.273949	+	0.158164i
$923$			9.90735i			0.326104i
$924$		0			0
$925$	−4.35425	−	0.567102i	−0.143167	−	0.0186462i
$926$	5.21057	+	9.02497i	0.171230	+	0.296579i
$927$		0			0
$928$	11.6299	+	6.71453i	0.381771	+	0.220415i
$929$	−6.68608	−	11.5806i	−0.219363	−	0.379948i	0.735250	−	0.677796i	$-0.237065\pi$
$929$	−6.68608	−	11.5806i	−0.219363	−	0.379948i	−0.954613	+	0.297848i	$0.903731\pi$
$930$		0			0
$931$	−3.41699	+	2.95920i	−0.111987	+	0.0969840i
$932$		−	12.6724i		−	0.415100i
$933$		0			0
$934$	12.2288	−	21.1808i	0.400137	−	0.693058i
$935$	−11.6211	−	58.2123i	−0.380052	−	1.90375i
$936$		0			0
$937$		−	0.255999i		−	0.00836312i	−0.999991	−	0.00418156i	$-0.998669\pi$
$937$		−	0.255999i		−	0.00836312i	0.999991	−	0.00418156i	$-0.00133104\pi$
$938$	2.32996	−	2.82111i	0.0760758	−	0.0921126i
$939$		0			0
$940$	−7.56329	−	2.56154i	−0.246687	−	0.0835482i
$941$	15.9692	−	27.6595i	0.520582	−	0.901675i	−0.479131	−	0.877743i	$-0.659048\pi$
$941$	15.9692	−	27.6595i	0.520582	−	0.901675i	0.999714	−	0.0239319i	$-0.00761847\pi$
$942$		0			0
$943$	−59.8447	+	34.5513i	−1.94881	+	1.12515i
$944$	16.6607			0.542260
$945$		0			0
$946$	34.5608			1.12367
$947$	25.3102	−	14.6128i	0.822470	−	0.474853i	−0.0287974	−	0.999585i	$-0.509168\pi$
$947$	25.3102	−	14.6128i	0.822470	−	0.474853i	0.851268	+	0.524732i	$0.175834\pi$
$948$		0			0
$949$	−18.4373	+	31.9343i	−0.598499	+	1.03663i
$950$	0.737872	+	1.77441i	0.0239397	+	0.0575696i
$951$		0			0
$952$	9.89739	+	26.5470i	0.320776	+	0.860394i
$953$			57.6827i			1.86852i	0.356587	+	0.934262i	$0.383940\pi$
$953$			57.6827i			1.86852i	−0.356587	+	0.934262i	$0.616060\pi$
$954$		0			0
$955$	23.5916	−	4.70968i	0.763407	−	0.152402i
$956$	−7.28496	+	12.6179i	−0.235612	+	0.408093i
$957$		0			0
$958$		−	7.53764i		−	0.243530i
$959$	9.80586	−	3.65587i	0.316648	−	0.118054i
$960$		0			0
$961$	3.00000	+	5.19615i	0.0967742	+	0.167618i
$962$	1.84685	+	1.06628i	0.0595449	+	0.0343783i
$963$		0			0
$964$	−23.2288	−	40.2334i	−0.748148	−	1.29583i
$965$	−21.0872	+	18.5189i	−0.678822	+	0.596145i
$966$		0			0
$967$		−	24.2235i		−	0.778976i	−0.921032	−	0.389488i	$-0.872652\pi$
$967$		−	24.2235i		−	0.778976i	0.921032	−	0.389488i	$-0.127348\pi$
$968$	33.7076	−	19.4611i	1.08340	−	0.625503i
$969$		0			0
$970$	−8.76333	+	1.74946i	−0.281373	+	0.0561716i
$971$	−11.1888	−	19.3795i	−0.359065	−	0.621919i	0.628740	−	0.777616i	$-0.283571\pi$
$971$	−11.1888	−	19.3795i	−0.359065	−	0.621919i	−0.987805	+	0.155697i	$0.950238\pi$
$972$		0			0
$973$	12.1790	−	14.7464i	0.390442	−	0.472747i
$974$	14.2324			0.456035
$975$		0			0
$976$	−0.354249	+	0.613577i	−0.0113392	+	0.0196401i
$977$	50.1696	+	28.9654i	1.60507	+	0.926686i	0.990452	+	0.137859i	$0.0440221\pi$
$977$	50.1696	+	28.9654i	1.60507	+	0.926686i	0.614615	+	0.788827i	$0.289311\pi$
$978$		0			0
$979$	−28.9373			−0.924839
$980$	15.7945	−	20.3498i	0.504538	−	0.650051i
$981$		0			0
$982$	18.6965	−	10.7945i	0.596631	−	0.344465i
$983$	−5.85252	−	3.37896i	−0.186667	−	0.107772i	0.403755	−	0.914867i	$-0.367705\pi$
$983$	−5.85252	−	3.37896i	−0.186667	−	0.107772i	−0.590421	+	0.807095i	$0.701038\pi$
$984$		0			0
$985$	−12.5264	−	4.24244i	−0.399124	−	0.135175i
$986$	−7.13259			−0.227148
$987$		0			0
$988$			4.33592i			0.137944i
$989$	33.0601	+	57.2617i	1.05125	+	1.82082i
$990$		0			0
$991$	8.19738	−	14.1983i	0.260398	−	0.451023i	−0.705949	−	0.708262i	$-0.749479\pi$
$991$	8.19738	−	14.1983i	0.260398	−	0.451023i	0.966348	+	0.257239i	$0.0828127\pi$
$992$	23.9464	−	13.8255i	0.760300	−	0.438959i
$993$		0			0
$994$	0.634637	−	3.77089i	0.0201295	−	0.119605i
$995$	−26.0365	−	29.6474i	−0.825411	−	0.939885i
$996$		0			0
$997$	−27.5060	−	15.8806i	−0.871123	−	0.502943i	−0.00340163	−	0.999994i	$-0.501083\pi$
$997$	−27.5060	−	15.8806i	−0.871123	−	0.502943i	−0.867721	+	0.497051i	$0.834416\pi$
$998$	−2.57724	−	1.48797i	−0.0815811	−	0.0471009i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bf.c.109.5	yes	16
3.2	odd	2	inner	315.2.bf.c.109.4	yes	16
5.4	even	2	inner	315.2.bf.c.109.3	✓	16
7.2	even	3	inner	315.2.bf.c.289.3	yes	16
7.3	odd	6		2205.2.d.r.1324.5		8
7.4	even	3		2205.2.d.p.1324.6		8
15.14	odd	2	inner	315.2.bf.c.109.6	yes	16
21.2	odd	6	inner	315.2.bf.c.289.6	yes	16
21.11	odd	6		2205.2.d.p.1324.3		8
21.17	even	6		2205.2.d.r.1324.4		8
35.4	even	6		2205.2.d.p.1324.4		8
35.9	even	6	inner	315.2.bf.c.289.5	yes	16
35.24	odd	6		2205.2.d.r.1324.3		8
105.44	odd	6	inner	315.2.bf.c.289.4	yes	16
105.59	even	6		2205.2.d.r.1324.6		8
105.74	odd	6		2205.2.d.p.1324.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bf.c.109.3	✓	16	5.4	even	2	inner
315.2.bf.c.109.4	yes	16	3.2	odd	2	inner
315.2.bf.c.109.5	yes	16	1.1	even	1	trivial
315.2.bf.c.109.6	yes	16	15.14	odd	2	inner
315.2.bf.c.289.3	yes	16	7.2	even	3	inner
315.2.bf.c.289.4	yes	16	105.44	odd	6	inner
315.2.bf.c.289.5	yes	16	35.9	even	6	inner
315.2.bf.c.289.6	yes	16	21.2	odd	6	inner
2205.2.d.p.1324.3		8	21.11	odd	6
2205.2.d.p.1324.4		8	35.4	even	6
2205.2.d.p.1324.5		8	105.74	odd	6
2205.2.d.p.1324.6		8	7.4	even	3
2205.2.d.r.1324.3		8	35.24	odd	6
2205.2.d.r.1324.4		8	21.17	even	6
2205.2.d.r.1324.5		8	7.3	odd	6
2205.2.d.r.1324.6		8	105.59	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.bf (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.515448 - 0.297594i) q^{2} +(-0.822876 + 1.42526i) q^{4} +(0.717291 - 2.11790i) q^{5} +(1.68480 - 2.03996i) q^{7} +2.16991i q^{8} +O(q^{10})\)	\(q+(0.515448 - 0.297594i) q^{2} +(-0.822876 + 1.42526i) q^{4} +(0.717291 - 2.11790i) q^{5} +(1.68480 - 2.03996i) q^{7} +2.16991i q^{8} +(-0.260548 - 1.30513i) q^{10} +(2.68967 - 4.65864i) q^{11} +4.07992i q^{13} +(0.261348 - 1.55288i) q^{14} +(-1.00000 - 1.73205i) q^{16} +(4.27384 + 2.46750i) q^{17} +(-0.322876 - 0.559237i) q^{19} +(2.42832 + 2.76510i) q^{20} -3.20172i q^{22} +(5.30473 - 3.06269i) q^{23} +(-3.97099 - 3.03830i) q^{25} +(1.21416 + 2.10299i) q^{26} +(1.52110 + 4.07992i) q^{28} -2.42832 q^{29} +(-2.50000 + 4.33013i) q^{31} +(-4.78929 - 2.76510i) q^{32} +2.93725 q^{34} +(-3.11193 - 5.03149i) q^{35} +(0.760548 - 0.439102i) q^{37} +(-0.332851 - 0.192172i) q^{38} +(4.59564 + 1.55646i) q^{40} -11.2814 q^{41} +10.7945i q^{43} +(4.42652 + 7.66697i) q^{44} +(1.82288 - 3.15731i) q^{46} +(-1.87919 + 1.08495i) q^{47} +(-1.32288 - 6.87386i) q^{49} +(-2.95102 - 0.384343i) q^{50} +(-5.81496 - 3.35727i) q^{52} +(-6.85108 - 3.95547i) q^{53} +(-7.93725 - 9.03805i) q^{55} +(4.42652 + 3.65587i) q^{56} +(-1.25167 + 0.722653i) q^{58} +(-4.16518 + 7.21430i) q^{59} +(-0.177124 - 0.306788i) q^{61} +2.97594i q^{62} +0.708497 q^{64} +(8.64086 + 2.92649i) q^{65} +(2.01222 + 1.16176i) q^{67} +(-7.03367 + 4.06089i) q^{68} +(-3.10138 - 1.66738i) q^{70} +2.42832 q^{71} +(7.82718 + 4.51902i) q^{73} +(0.261348 - 0.452669i) q^{74} +1.06275 q^{76} +(-4.97188 - 13.3357i) q^{77} +(2.96863 + 5.14181i) q^{79} +(-4.38560 + 0.875514i) q^{80} +(-5.81496 + 3.35727i) q^{82} +11.6556i q^{83} +(8.29150 - 7.28164i) q^{85} +(3.21236 + 5.56398i) q^{86} +(10.1088 + 5.83633i) q^{88} +(-2.68967 - 4.65864i) q^{89} +(8.32288 + 6.87386i) q^{91} +10.0808i q^{92} +(-0.645751 + 1.11847i) q^{94} +(-1.41600 + 0.282682i) q^{95} -6.71453i q^{97} +(-2.72749 - 3.14944i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4}+O(q^{10}) \) 16 * q + 8 * q^4	\( 16 q + 8 q^{4} + 8 q^{10} - 16 q^{16} + 16 q^{19} - 16 q^{25} - 40 q^{31} - 80 q^{34} + 8 q^{40} + 8 q^{46} - 24 q^{61} + 96 q^{64} - 56 q^{70} + 144 q^{76} - 16 q^{79} + 48 q^{85} + 112 q^{91} + 32 q^{94}+O(q^{100}) \) 16 * q + 8 * q^4 + 8 * q^10 - 16 * q^16 + 16 * q^19 - 16 * q^25 - 40 * q^31 - 80 * q^34 + 8 * q^40 + 8 * q^46 - 24 * q^61 + 96 * q^64 - 56 * q^70 + 144 * q^76 - 16 * q^79 + 48 * q^85 + 112 * q^91 + 32 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more