LMFDB - Embedded newform 675.2.k.a.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(199,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("675.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.k (of order $6$, degree $2$, not 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:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			199.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	675.199
Dual form			675.2.k.a.424.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 0.500000i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(2.59808 - 1.50000i) q^{7} +3.00000i q^{8} +O(q^{10})$	\(q+(0.866025 - 0.500000i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(2.59808 - 1.50000i) q^{7} +3.00000i q^{8} +(-1.00000 - 1.73205i) q^{11} +(1.73205 + 1.00000i) q^{13} +(1.50000 - 2.59808i) q^{14} +(0.500000 + 0.866025i) q^{16} +4.00000i q^{17} +8.00000 q^{19} +(-1.73205 - 1.00000i) q^{22} +(2.59808 + 1.50000i) q^{23} +2.00000 q^{26} +3.00000i q^{28} +(0.500000 + 0.866025i) q^{29} +(-4.33013 - 2.50000i) q^{32} +(2.00000 + 3.46410i) q^{34} +4.00000i q^{37} +(6.92820 - 4.00000i) q^{38} +(2.50000 - 4.33013i) q^{41} +(-6.92820 + 4.00000i) q^{43} +2.00000 q^{44} +3.00000 q^{46} +(6.06218 - 3.50000i) q^{47} +(1.00000 - 1.73205i) q^{49} +(-1.73205 + 1.00000i) q^{52} +2.00000i q^{53} +(4.50000 + 7.79423i) q^{56} +(0.866025 + 0.500000i) q^{58} +(7.00000 - 12.1244i) q^{59} +(-3.50000 - 6.06218i) q^{61} -7.00000 q^{64} +(-2.59808 - 1.50000i) q^{67} +(-3.46410 - 2.00000i) q^{68} -2.00000 q^{71} +4.00000i q^{73} +(2.00000 + 3.46410i) q^{74} +(-4.00000 + 6.92820i) q^{76} +(-5.19615 - 3.00000i) q^{77} +(-3.00000 - 5.19615i) q^{79} -5.00000i q^{82} +(-7.79423 + 4.50000i) q^{83} +(-4.00000 + 6.92820i) q^{86} +(5.19615 - 3.00000i) q^{88} -15.0000 q^{89} +6.00000 q^{91} +(-2.59808 + 1.50000i) q^{92} +(3.50000 - 6.06218i) q^{94} +(-1.73205 + 1.00000i) q^{97} -2.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4}+O(q^{10}) $ 4 * q - 2 * q^4	$ 4 q - 2 q^{4} - 4 q^{11} + 6 q^{14} + 2 q^{16} + 32 q^{19} + 8 q^{26} + 2 q^{29} + 8 q^{34} + 10 q^{41} + 8 q^{44} + 12 q^{46} + 4 q^{49} + 18 q^{56} + 28 q^{59} - 14 q^{61} - 28 q^{64} - 8 q^{71} + 8 q^{74} - 16 q^{76} - 12 q^{79} - 16 q^{86} - 60 q^{89} + 24 q^{91} + 14 q^{94}+O(q^{100}) $ 4 * q - 2 * q^4 - 4 * q^11 + 6 * q^14 + 2 * q^16 + 32 * q^19 + 8 * q^26 + 2 * q^29 + 8 * q^34 + 10 * q^41 + 8 * q^44 + 12 * q^46 + 4 * q^49 + 18 * q^56 + 28 * q^59 - 14 * q^61 - 28 * q^64 - 8 * q^71 + 8 * q^74 - 16 * q^76 - 12 * q^79 - 16 * q^86 - 60 * q^89 + 24 * q^91 + 14 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$

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 $

$2$	0.866025	−	0.500000i	0.612372	−	0.353553i	−0.161521	−	0.986869i	$-0.551640\pi$
$2$	0.866025	−	0.500000i	0.612372	−	0.353553i	0.773893	+	0.633316i	$0.218307\pi$
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.59808	−	1.50000i	0.981981	−	0.566947i	0.0791130	−	0.996866i	$-0.474791\pi$
$7$	2.59808	−	1.50000i	0.981981	−	0.566947i	0.902867	+	0.429919i	$0.141458\pi$
$8$			3.00000i			1.06066i
$9$		0			0
$10$		0			0
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	0.674967	−	0.737848i	$-0.264158\pi$
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	−0.976478	+	0.215615i	$0.930824\pi$
$12$		0			0
$13$	1.73205	+	1.00000i	0.480384	+	0.277350i	0.720577	−	0.693375i	$-0.243877\pi$
$13$	1.73205	+	1.00000i	0.480384	+	0.277350i	−0.240192	+	0.970725i	$0.577210\pi$
$14$	1.50000	−	2.59808i	0.400892	−	0.694365i
$15$		0			0
$16$	0.500000	+	0.866025i	0.125000	+	0.216506i
$17$			4.00000i			0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$			4.00000i			0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$		0			0
$19$	8.00000			1.83533			0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000			1.83533			0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$		0			0
$22$	−1.73205	−	1.00000i	−0.369274	−	0.213201i
$23$	2.59808	+	1.50000i	0.541736	+	0.312772i	0.745782	−	0.666190i	$-0.232076\pi$
$23$	2.59808	+	1.50000i	0.541736	+	0.312772i	−0.204046	+	0.978961i	$0.565409\pi$
$24$		0			0
$25$		0			0
$26$	2.00000			0.392232
$27$		0			0
$28$			3.00000i			0.566947i
$29$	0.500000	+	0.866025i	0.0928477	+	0.160817i	0.908708	−	0.417432i	$-0.137070\pi$
$29$	0.500000	+	0.866025i	0.0928477	+	0.160817i	−0.815861	+	0.578249i	$0.803736\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	−4.33013	−	2.50000i	−0.765466	−	0.441942i
$33$		0			0
$34$	2.00000	+	3.46410i	0.342997	+	0.594089i
$35$		0			0
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	6.92820	−	4.00000i	1.12390	−	0.648886i
$39$		0			0
$40$		0			0
$41$	2.50000	−	4.33013i	0.390434	−	0.676252i	−0.602072	−	0.798441i	$-0.705658\pi$
$41$	2.50000	−	4.33013i	0.390434	−	0.676252i	0.992507	+	0.122189i	$0.0389915\pi$
$42$		0			0
$43$	−6.92820	+	4.00000i	−1.05654	+	0.609994i	−0.924473	−	0.381246i	$-0.875495\pi$
$43$	−6.92820	+	4.00000i	−1.05654	+	0.609994i	−0.132068	+	0.991241i	$0.542162\pi$
$44$	2.00000			0.301511
$45$		0			0
$46$	3.00000			0.442326
$47$	6.06218	−	3.50000i	0.884260	−	0.510527i	0.0121990	−	0.999926i	$-0.496117\pi$
$47$	6.06218	−	3.50000i	0.884260	−	0.510527i	0.872060	+	0.489398i	$0.162783\pi$
$48$		0			0
$49$	1.00000	−	1.73205i	0.142857	−	0.247436i
$50$		0			0
$51$		0			0
$52$	−1.73205	+	1.00000i	−0.240192	+	0.138675i
$53$			2.00000i			0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$			2.00000i			0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$		0			0
$55$		0			0
$56$	4.50000	+	7.79423i	0.601338	+	1.04155i
$57$		0			0
$58$	0.866025	+	0.500000i	0.113715	+	0.0656532i
$59$	7.00000	−	12.1244i	0.911322	−	1.57846i	0.0991242	−	0.995075i	$-0.468396\pi$
$59$	7.00000	−	12.1244i	0.911322	−	1.57846i	0.812198	−	0.583382i	$-0.198271\pi$
$60$		0			0
$61$	−3.50000	−	6.06218i	−0.448129	−	0.776182i	0.550135	−	0.835076i	$-0.314576\pi$
$61$	−3.50000	−	6.06218i	−0.448129	−	0.776182i	−0.998264	+	0.0588933i	$0.981243\pi$
$62$		0			0
$63$		0			0
$64$	−7.00000			−0.875000
$65$		0			0
$66$		0			0
$67$	−2.59808	−	1.50000i	−0.317406	−	0.183254i	0.332830	−	0.942987i	$-0.391996\pi$
$67$	−2.59808	−	1.50000i	−0.317406	−	0.183254i	−0.650236	+	0.759733i	$0.725330\pi$
$68$	−3.46410	−	2.00000i	−0.420084	−	0.242536i
$69$		0			0
$70$		0			0
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	2.00000	+	3.46410i	0.232495	+	0.402694i
$75$		0			0
$76$	−4.00000	+	6.92820i	−0.458831	+	0.794719i
$77$	−5.19615	−	3.00000i	−0.592157	−	0.341882i
$78$		0			0
$79$	−3.00000	−	5.19615i	−0.337526	−	0.584613i	0.646440	−	0.762964i	$-0.276257\pi$
$79$	−3.00000	−	5.19615i	−0.337526	−	0.584613i	−0.983967	+	0.178352i	$0.942924\pi$
$80$		0			0
$81$		0			0
$82$		−	5.00000i		−	0.552158i
$83$	−7.79423	+	4.50000i	−0.855528	+	0.493939i	−0.862512	−	0.506036i	$-0.831110\pi$
$83$	−7.79423	+	4.50000i	−0.855528	+	0.493939i	0.00698436	+	0.999976i	$0.497777\pi$
$84$		0			0
$85$		0			0
$86$	−4.00000	+	6.92820i	−0.431331	+	0.747087i
$87$		0			0
$88$	5.19615	−	3.00000i	0.553912	−	0.319801i
$89$	−15.0000			−1.59000			−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000			−1.59000			−0.794998	+	0.606612i	$0.792528\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$	−2.59808	+	1.50000i	−0.270868	+	0.156386i
$93$		0			0
$94$	3.50000	−	6.06218i	0.360997	−	0.625266i
$95$		0			0
$96$		0			0
$97$	−1.73205	+	1.00000i	−0.175863	+	0.101535i	−0.585348	−	0.810782i	$-0.699042\pi$
$97$	−1.73205	+	1.00000i	−0.175863	+	0.101535i	0.409484	+	0.912317i	$0.365709\pi$
$98$		−	2.00000i		−	0.202031i
$99$		0			0
$100$		0			0
$101$	−9.00000	−	15.5885i	−0.895533	−	1.55111i	−0.833143	−	0.553058i	$-0.813461\pi$
$101$	−9.00000	−	15.5885i	−0.895533	−	1.55111i	−0.0623905	−	0.998052i	$-0.519872\pi$
$102$		0			0
$103$	−6.92820	−	4.00000i	−0.682656	−	0.394132i	0.118199	−	0.992990i	$-0.462288\pi$
$103$	−6.92820	−	4.00000i	−0.682656	−	0.394132i	−0.800855	+	0.598858i	$0.795621\pi$
$104$	−3.00000	+	5.19615i	−0.294174	+	0.509525i
$105$		0			0
$106$	1.00000	+	1.73205i	0.0971286	+	0.168232i
$107$			3.00000i			0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$			3.00000i			0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$		0			0
$109$	−5.00000			−0.478913			−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000			−0.478913			−0.239457	+	0.970907i	$0.576969\pi$
$110$		0			0
$111$		0			0
$112$	2.59808	+	1.50000i	0.245495	+	0.141737i
$113$	−6.92820	−	4.00000i	−0.651751	−	0.376288i	0.137376	−	0.990519i	$-0.456133\pi$
$113$	−6.92820	−	4.00000i	−0.651751	−	0.376288i	−0.789127	+	0.614231i	$0.789466\pi$
$114$		0			0
$115$		0			0
$116$	−1.00000			−0.0928477
$117$		0			0
$118$		−	14.0000i		−	1.28880i
$119$	6.00000	+	10.3923i	0.550019	+	0.952661i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$	−6.06218	−	3.50000i	−0.548844	−	0.316875i
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$			5.00000i			0.443678i	0.975083	+	0.221839i	$0.0712060\pi$
$127$			5.00000i			0.443678i	−0.975083	+	0.221839i	$0.928794\pi$
$128$	2.59808	−	1.50000i	0.229640	−	0.132583i
$129$		0			0
$130$		0			0
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	−0.966803	−	0.255524i	$-0.917752\pi$
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	0.704692	+	0.709514i	$0.251085\pi$
$132$		0			0
$133$	20.7846	−	12.0000i	1.80225	−	1.04053i
$134$	−3.00000			−0.259161
$135$		0			0
$136$	−12.0000			−1.02899
$137$	10.3923	−	6.00000i	0.887875	−	0.512615i	0.0146279	−	0.999893i	$-0.495344\pi$
$137$	10.3923	−	6.00000i	0.887875	−	0.512615i	0.873247	+	0.487278i	$0.162010\pi$
$138$		0			0
$139$	−8.00000	+	13.8564i	−0.678551	+	1.17529i	0.296866	+	0.954919i	$0.404058\pi$
$139$	−8.00000	+	13.8564i	−0.678551	+	1.17529i	−0.975417	+	0.220366i	$0.929275\pi$
$140$		0			0
$141$		0			0
$142$	−1.73205	+	1.00000i	−0.145350	+	0.0839181i
$143$		−	4.00000i		−	0.334497i
$144$		0			0
$145$		0			0
$146$	2.00000	+	3.46410i	0.165521	+	0.286691i
$147$		0			0
$148$	−3.46410	−	2.00000i	−0.284747	−	0.164399i
$149$	−8.50000	+	14.7224i	−0.696347	+	1.20611i	0.273377	+	0.961907i	$0.411859\pi$
$149$	−8.50000	+	14.7224i	−0.696347	+	1.20611i	−0.969724	+	0.244202i	$0.921474\pi$
$150$		0			0
$151$	1.00000	+	1.73205i	0.0813788	+	0.140952i	0.903842	−	0.427865i	$-0.140734\pi$
$151$	1.00000	+	1.73205i	0.0813788	+	0.140952i	−0.822464	+	0.568818i	$0.807401\pi$
$152$			24.0000i			1.94666i
$153$		0			0
$154$	−6.00000			−0.483494
$155$		0			0
$156$		0			0
$157$	12.1244	+	7.00000i	0.967629	+	0.558661i	0.898513	−	0.438948i	$-0.144649\pi$
$157$	12.1244	+	7.00000i	0.967629	+	0.558661i	0.0691164	+	0.997609i	$0.477982\pi$
$158$	−5.19615	−	3.00000i	−0.413384	−	0.238667i
$159$		0			0
$160$		0			0
$161$	9.00000			0.709299
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	2.50000	+	4.33013i	0.195217	+	0.338126i
$165$		0			0
$166$	−4.50000	+	7.79423i	−0.349268	+	0.604949i
$167$	7.79423	+	4.50000i	0.603136	+	0.348220i	0.770274	−	0.637713i	$-0.220119\pi$
$167$	7.79423	+	4.50000i	0.603136	+	0.348220i	−0.167139	+	0.985933i	$0.553453\pi$
$168$		0			0
$169$	−4.50000	−	7.79423i	−0.346154	−	0.599556i
$170$		0			0
$171$		0			0
$172$		−	8.00000i		−	0.609994i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$	1.00000	−	1.73205i	0.0753778	−	0.130558i
$177$		0			0
$178$	−12.9904	+	7.50000i	−0.973670	+	0.562149i
$179$	−2.00000			−0.149487			−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000			−0.149487			−0.0747435	+	0.997203i	$0.523814\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$	5.19615	−	3.00000i	0.385164	−	0.222375i
$183$		0			0
$184$	−4.50000	+	7.79423i	−0.331744	+	0.574598i
$185$		0			0
$186$		0			0
$187$	6.92820	−	4.00000i	0.506640	−	0.292509i
$188$			7.00000i			0.510527i
$189$		0			0
$190$		0			0
$191$	4.00000	+	6.92820i	0.289430	+	0.501307i	0.973674	−	0.227946i	$-0.0732010\pi$
$191$	4.00000	+	6.92820i	0.289430	+	0.501307i	−0.684244	+	0.729253i	$0.739868\pi$
$192$		0			0
$193$	8.66025	+	5.00000i	0.623379	+	0.359908i	0.778183	−	0.628037i	$-0.216141\pi$
$193$	8.66025	+	5.00000i	0.623379	+	0.359908i	−0.154805	+	0.987945i	$0.549475\pi$
$194$	−1.00000	+	1.73205i	−0.0717958	+	0.124354i
$195$		0			0
$196$	1.00000	+	1.73205i	0.0714286	+	0.123718i
$197$		−	12.0000i		−	0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$		−	12.0000i		−	0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$		0			0
$202$	−15.5885	−	9.00000i	−1.09680	−	0.633238i
$203$	2.59808	+	1.50000i	0.182349	+	0.105279i
$204$		0			0
$205$		0			0
$206$	−8.00000			−0.557386
$207$		0			0
$208$			2.00000i			0.138675i
$209$	−8.00000	−	13.8564i	−0.553372	−	0.958468i
$210$		0			0
$211$	11.0000	−	19.0526i	0.757271	−	1.31163i	−0.186966	−	0.982366i	$-0.559865\pi$
$211$	11.0000	−	19.0526i	0.757271	−	1.31163i	0.944237	−	0.329266i	$-0.106801\pi$
$212$	−1.73205	−	1.00000i	−0.118958	−	0.0686803i
$213$		0			0
$214$	1.50000	+	2.59808i	0.102538	+	0.177601i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−4.33013	+	2.50000i	−0.293273	+	0.169321i
$219$		0			0
$220$		0			0
$221$	−4.00000	+	6.92820i	−0.269069	+	0.466041i
$222$		0			0
$223$	−16.4545	+	9.50000i	−1.10187	+	0.636167i	−0.936713	−	0.350100i	$-0.886148\pi$
$223$	−16.4545	+	9.50000i	−1.10187	+	0.636167i	−0.165161	+	0.986267i	$0.552814\pi$
$224$	−15.0000			−1.00223
$225$		0			0
$226$	−8.00000			−0.532152
$227$	−3.46410	+	2.00000i	−0.229920	+	0.132745i	−0.610535	−	0.791989i	$-0.709046\pi$
$227$	−3.46410	+	2.00000i	−0.229920	+	0.132745i	0.380615	+	0.924734i	$0.375712\pi$
$228$		0			0
$229$	7.50000	−	12.9904i	0.495614	−	0.858429i	−0.504373	−	0.863486i	$-0.668276\pi$
$229$	7.50000	−	12.9904i	0.495614	−	0.858429i	0.999987	+	0.00505719i	$0.00160976\pi$
$230$		0			0
$231$		0			0
$232$	−2.59808	+	1.50000i	−0.170572	+	0.0984798i
$233$		−	24.0000i		−	1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$		−	24.0000i		−	1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$		0			0
$235$		0			0
$236$	7.00000	+	12.1244i	0.455661	+	0.789228i
$237$		0			0
$238$	10.3923	+	6.00000i	0.673633	+	0.388922i
$239$	4.00000	−	6.92820i	0.258738	−	0.448148i	−0.707166	−	0.707048i	$-0.750027\pi$
$239$	4.00000	−	6.92820i	0.258738	−	0.448148i	0.965904	+	0.258900i	$0.0833599\pi$
$240$		0			0
$241$	5.50000	+	9.52628i	0.354286	+	0.613642i	0.986996	−	0.160748i	$-0.0513906\pi$
$241$	5.50000	+	9.52628i	0.354286	+	0.613642i	−0.632709	+	0.774389i	$0.718057\pi$
$242$		−	7.00000i		−	0.449977i
$243$		0			0
$244$	7.00000			0.448129
$245$		0			0
$246$		0			0
$247$	13.8564	+	8.00000i	0.881662	+	0.509028i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		−	6.00000i		−	0.377217i
$254$	2.50000	+	4.33013i	0.156864	+	0.271696i
$255$		0			0
$256$	8.50000	−	14.7224i	0.531250	−	0.920152i
$257$	5.19615	+	3.00000i	0.324127	+	0.187135i	0.653231	−	0.757159i	$-0.273413\pi$
$257$	5.19615	+	3.00000i	0.324127	+	0.187135i	−0.329104	+	0.944294i	$0.606747\pi$
$258$		0			0
$259$	6.00000	+	10.3923i	0.372822	+	0.645746i
$260$		0			0
$261$		0			0
$262$			6.00000i			0.370681i
$263$	13.8564	−	8.00000i	0.854423	−	0.493301i	−0.00771799	−	0.999970i	$-0.502457\pi$
$263$	13.8564	−	8.00000i	0.854423	−	0.493301i	0.862141	+	0.506669i	$0.169123\pi$
$264$		0			0
$265$		0			0
$266$	12.0000	−	20.7846i	0.735767	−	1.27439i
$267$		0			0
$268$	2.59808	−	1.50000i	0.158703	−	0.0916271i
$269$	25.0000			1.52428			0.762138	−	0.647414i	$-0.224150\pi$
$269$	25.0000			1.52428			0.762138	+	0.647414i	$0.224150\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$	−3.46410	+	2.00000i	−0.210042	+	0.121268i
$273$		0			0
$274$	6.00000	−	10.3923i	0.362473	−	0.627822i
$275$		0			0
$276$		0			0
$277$	−10.3923	+	6.00000i	−0.624413	+	0.360505i	−0.778585	−	0.627539i	$-0.784062\pi$
$277$	−10.3923	+	6.00000i	−0.624413	+	0.360505i	0.154172	+	0.988044i	$0.450729\pi$
$278$			16.0000i			0.959616i
$279$		0			0
$280$		0			0
$281$	−7.50000	−	12.9904i	−0.447412	−	0.774941i	0.550804	−	0.834634i	$-0.314321\pi$
$281$	−7.50000	−	12.9904i	−0.447412	−	0.774941i	−0.998217	+	0.0596933i	$0.980988\pi$
$282$		0			0
$283$	−18.1865	−	10.5000i	−1.08108	−	0.624160i	−0.149890	−	0.988703i	$-0.547892\pi$
$283$	−18.1865	−	10.5000i	−1.08108	−	0.624160i	−0.931187	+	0.364542i	$0.881225\pi$
$284$	1.00000	−	1.73205i	0.0593391	−	0.102778i
$285$		0			0
$286$	−2.00000	−	3.46410i	−0.118262	−	0.204837i
$287$		−	15.0000i		−	0.885422i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$		0			0
$292$	−3.46410	−	2.00000i	−0.202721	−	0.117041i
$293$	10.3923	+	6.00000i	0.607125	+	0.350524i	0.771839	−	0.635818i	$-0.219337\pi$
$293$	10.3923	+	6.00000i	0.607125	+	0.350524i	−0.164714	+	0.986341i	$0.552670\pi$
$294$		0			0
$295$		0			0
$296$	−12.0000			−0.697486
$297$		0			0
$298$			17.0000i			0.984784i
$299$	3.00000	+	5.19615i	0.173494	+	0.300501i
$300$		0			0
$301$	−12.0000	+	20.7846i	−0.691669	+	1.19800i
$302$	1.73205	+	1.00000i	0.0996683	+	0.0575435i
$303$		0			0
$304$	4.00000	+	6.92820i	0.229416	+	0.397360i
$305$		0			0
$306$		0			0
$307$		−	7.00000i		−	0.399511i	−0.979846	−	0.199756i	$-0.935985\pi$
$307$		−	7.00000i		−	0.399511i	0.979846	−	0.199756i	$-0.0640148\pi$
$308$	5.19615	−	3.00000i	0.296078	−	0.170941i
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	12.1244	−	7.00000i	0.685309	−	0.395663i	−0.116543	−	0.993186i	$-0.537181\pi$
$313$	12.1244	−	7.00000i	0.685309	−	0.395663i	0.801852	+	0.597522i	$0.203848\pi$
$314$	14.0000			0.790066
$315$		0			0
$316$	6.00000			0.337526
$317$	−29.4449	+	17.0000i	−1.65379	+	0.954815i	−0.678294	+	0.734791i	$0.737280\pi$
$317$	−29.4449	+	17.0000i	−1.65379	+	0.954815i	−0.975494	+	0.220024i	$0.929386\pi$
$318$		0			0
$319$	1.00000	−	1.73205i	0.0559893	−	0.0969762i
$320$		0			0
$321$		0			0
$322$	7.79423	−	4.50000i	0.434355	−	0.250775i
$323$			32.0000i			1.78053i
$324$		0			0
$325$		0			0
$326$	−2.00000	−	3.46410i	−0.110770	−	0.191859i
$327$		0			0
$328$	12.9904	+	7.50000i	0.717274	+	0.414118i
$329$	10.5000	−	18.1865i	0.578884	−	1.00266i
$330$		0			0
$331$	3.00000	+	5.19615i	0.164895	+	0.285606i	0.936618	−	0.350352i	$-0.113938\pi$
$331$	3.00000	+	5.19615i	0.164895	+	0.285606i	−0.771723	+	0.635959i	$0.780605\pi$
$332$		−	9.00000i		−	0.493939i
$333$		0			0
$334$	9.00000			0.492458
$335$		0			0
$336$		0			0
$337$	−6.92820	−	4.00000i	−0.377403	−	0.217894i	0.299285	−	0.954164i	$-0.403252\pi$
$337$	−6.92820	−	4.00000i	−0.377403	−	0.217894i	−0.676688	+	0.736270i	$0.736585\pi$
$338$	−7.79423	−	4.50000i	−0.423950	−	0.244768i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			15.0000i			0.809924i
$344$	−12.0000	−	20.7846i	−0.646997	−	1.12063i
$345$		0			0
$346$		0			0
$347$	−3.46410	−	2.00000i	−0.185963	−	0.107366i	0.404128	−	0.914702i	$-0.367575\pi$
$347$	−3.46410	−	2.00000i	−0.185963	−	0.107366i	−0.590091	+	0.807337i	$0.700908\pi$
$348$		0			0
$349$	−2.50000	−	4.33013i	−0.133822	−	0.231786i	0.791325	−	0.611396i	$-0.209392\pi$
$349$	−2.50000	−	4.33013i	−0.133822	−	0.231786i	−0.925147	+	0.379610i	$0.876058\pi$
$350$		0			0
$351$		0			0
$352$			10.0000i			0.533002i
$353$	−20.7846	+	12.0000i	−1.10625	+	0.638696i	−0.937856	−	0.347024i	$-0.887192\pi$
$353$	−20.7846	+	12.0000i	−1.10625	+	0.638696i	−0.168397	+	0.985719i	$0.553859\pi$
$354$		0			0
$355$		0			0
$356$	7.50000	−	12.9904i	0.397499	−	0.688489i
$357$		0			0
$358$	−1.73205	+	1.00000i	−0.0915417	+	0.0528516i
$359$	24.0000			1.26667			0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000			1.26667			0.633336	+	0.773877i	$0.281685\pi$
$360$		0			0
$361$	45.0000			2.36842
$362$	−6.06218	+	3.50000i	−0.318621	+	0.183956i
$363$		0			0
$364$	−3.00000	+	5.19615i	−0.157243	+	0.272352i
$365$		0			0
$366$		0			0
$367$	−20.7846	+	12.0000i	−1.08495	+	0.626395i	−0.932227	−	0.361874i	$-0.882137\pi$
$367$	−20.7846	+	12.0000i	−1.08495	+	0.626395i	−0.152721	+	0.988269i	$0.548804\pi$
$368$			3.00000i			0.156386i
$369$		0			0
$370$		0			0
$371$	3.00000	+	5.19615i	0.155752	+	0.269771i
$372$		0			0
$373$	−8.66025	−	5.00000i	−0.448411	−	0.258890i	0.258748	−	0.965945i	$-0.416690\pi$
$373$	−8.66025	−	5.00000i	−0.448411	−	0.258890i	−0.707159	+	0.707055i	$0.750023\pi$
$374$	4.00000	−	6.92820i	0.206835	−	0.358249i
$375$		0			0
$376$	10.5000	+	18.1865i	0.541496	+	0.937899i
$377$			2.00000i			0.103005i
$378$		0			0
$379$	26.0000			1.33553			0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000			1.33553			0.667765	+	0.744372i	$0.267251\pi$
$380$		0			0
$381$		0			0
$382$	6.92820	+	4.00000i	0.354478	+	0.204658i
$383$	31.1769	+	18.0000i	1.59307	+	0.919757i	0.992777	+	0.119974i	$0.0382810\pi$
$383$	31.1769	+	18.0000i	1.59307	+	0.919757i	0.600289	+	0.799783i	$0.295052\pi$
$384$		0			0
$385$		0			0
$386$	10.0000			0.508987
$387$		0			0
$388$		−	2.00000i		−	0.101535i
$389$	−16.5000	−	28.5788i	−0.836583	−	1.44900i	−0.892735	−	0.450582i	$-0.851216\pi$
$389$	−16.5000	−	28.5788i	−0.836583	−	1.44900i	0.0561516	−	0.998422i	$-0.482117\pi$
$390$		0			0
$391$	−6.00000	+	10.3923i	−0.303433	+	0.525561i
$392$	5.19615	+	3.00000i	0.262445	+	0.151523i
$393$		0			0
$394$	−6.00000	−	10.3923i	−0.302276	−	0.523557i
$395$		0			0
$396$		0			0
$397$		−	34.0000i		−	1.70641i	−0.521575	−	0.853206i	$-0.674655\pi$
$397$		−	34.0000i		−	1.70641i	0.521575	−	0.853206i	$-0.325345\pi$
$398$	−3.46410	+	2.00000i	−0.173640	+	0.100251i
$399$		0			0
$400$		0			0
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	−0.998350	−	0.0574304i	$-0.981709\pi$
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	0.548911	+	0.835881i	$0.315043\pi$
$402$		0			0
$403$		0			0
$404$	18.0000			0.895533
$405$		0			0
$406$	3.00000			0.148888
$407$	6.92820	−	4.00000i	0.343418	−	0.198273i
$408$		0			0
$409$	7.00000	−	12.1244i	0.346128	−	0.599511i	−0.639430	−	0.768849i	$-0.720830\pi$
$409$	7.00000	−	12.1244i	0.346128	−	0.599511i	0.985558	+	0.169338i	$0.0541630\pi$
$410$		0			0
$411$		0			0
$412$	6.92820	−	4.00000i	0.341328	−	0.197066i
$413$		−	42.0000i		−	2.06668i
$414$		0			0
$415$		0			0
$416$	−5.00000	−	8.66025i	−0.245145	−	0.424604i
$417$		0			0
$418$	−13.8564	−	8.00000i	−0.677739	−	0.391293i
$419$	−13.0000	+	22.5167i	−0.635092	+	1.10001i	0.351404	+	0.936224i	$0.385704\pi$
$419$	−13.0000	+	22.5167i	−0.635092	+	1.10001i	−0.986496	+	0.163787i	$0.947629\pi$
$420$		0			0
$421$	−17.0000	−	29.4449i	−0.828529	−	1.43505i	−0.899192	−	0.437555i	$-0.855845\pi$
$421$	−17.0000	−	29.4449i	−0.828529	−	1.43505i	0.0706626	−	0.997500i	$-0.477489\pi$
$422$		−	22.0000i		−	1.07094i
$423$		0			0
$424$	−6.00000			−0.291386
$425$		0			0
$426$		0			0
$427$	−18.1865	−	10.5000i	−0.880108	−	0.508131i
$428$	−2.59808	−	1.50000i	−0.125583	−	0.0725052i
$429$		0			0
$430$		0			0
$431$	30.0000			1.44505			0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000			1.44505			0.722525	+	0.691345i	$0.242982\pi$
$432$		0			0
$433$			28.0000i			1.34559i	0.739827	+	0.672797i	$0.234907\pi$
$433$			28.0000i			1.34559i	−0.739827	+	0.672797i	$0.765093\pi$
$434$		0			0
$435$		0			0
$436$	2.50000	−	4.33013i	0.119728	−	0.207375i
$437$	20.7846	+	12.0000i	0.994263	+	0.574038i
$438$		0			0
$439$	14.0000	+	24.2487i	0.668184	+	1.15733i	0.978412	+	0.206666i	$0.0662612\pi$
$439$	14.0000	+	24.2487i	0.668184	+	1.15733i	−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$		0			0
$442$			8.00000i			0.380521i
$443$	−12.9904	+	7.50000i	−0.617192	+	0.356336i	−0.775775	−	0.631010i	$-0.782641\pi$
$443$	−12.9904	+	7.50000i	−0.617192	+	0.356336i	0.158583	+	0.987346i	$0.449307\pi$
$444$		0			0
$445$		0			0
$446$	−9.50000	+	16.4545i	−0.449838	+	0.779142i
$447$		0			0
$448$	−18.1865	+	10.5000i	−0.859233	+	0.496078i
$449$	−26.0000			−1.22702			−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000			−1.22702			−0.613508	+	0.789689i	$0.710242\pi$
$450$		0			0
$451$	−10.0000			−0.470882
$452$	6.92820	−	4.00000i	0.325875	−	0.188144i
$453$		0			0
$454$	−2.00000	+	3.46410i	−0.0938647	+	0.162578i
$455$		0			0
$456$		0			0
$457$	17.3205	−	10.0000i	0.810219	−	0.467780i	−0.0368128	−	0.999322i	$-0.511721\pi$
$457$	17.3205	−	10.0000i	0.810219	−	0.467780i	0.847032	+	0.531542i	$0.178387\pi$
$458$		−	15.0000i		−	0.700904i
$459$		0			0
$460$		0			0
$461$	4.50000	+	7.79423i	0.209586	+	0.363013i	0.951584	−	0.307388i	$-0.0994551\pi$
$461$	4.50000	+	7.79423i	0.209586	+	0.363013i	−0.741998	+	0.670402i	$0.766122\pi$
$462$		0			0
$463$	31.1769	+	18.0000i	1.44891	+	0.836531i	0.998417	−	0.0562469i	$-0.0179134\pi$
$463$	31.1769	+	18.0000i	1.44891	+	0.836531i	0.450497	+	0.892778i	$0.351247\pi$
$464$	−0.500000	+	0.866025i	−0.0232119	+	0.0402042i
$465$		0			0
$466$	−12.0000	−	20.7846i	−0.555889	−	0.962828i
$467$			20.0000i			0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$			20.0000i			0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$		0			0
$469$	−9.00000			−0.415581
$470$		0			0
$471$		0			0
$472$	36.3731	+	21.0000i	1.67421	+	0.966603i
$473$	13.8564	+	8.00000i	0.637118	+	0.367840i
$474$		0			0
$475$		0			0
$476$	−12.0000			−0.550019
$477$		0			0
$478$		−	8.00000i		−	0.365911i
$479$	9.00000	+	15.5885i	0.411220	+	0.712255i	0.995023	−	0.0996406i	$-0.0317693\pi$
$479$	9.00000	+	15.5885i	0.411220	+	0.712255i	−0.583803	+	0.811895i	$0.698436\pi$
$480$		0			0
$481$	−4.00000	+	6.92820i	−0.182384	+	0.315899i
$482$	9.52628	+	5.50000i	0.433910	+	0.250518i
$483$		0			0
$484$	3.50000	+	6.06218i	0.159091	+	0.275554i
$485$		0			0
$486$		0			0
$487$			16.0000i			0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$			16.0000i			0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	18.1865	−	10.5000i	0.823266	−	0.475313i
$489$		0			0
$490$		0			0
$491$	10.0000	−	17.3205i	0.451294	−	0.781664i	−0.547173	−	0.837020i	$-0.684296\pi$
$491$	10.0000	−	17.3205i	0.451294	−	0.781664i	0.998467	+	0.0553560i	$0.0176294\pi$
$492$		0			0
$493$	−3.46410	+	2.00000i	−0.156015	+	0.0900755i
$494$	16.0000			0.719874
$495$		0			0
$496$		0			0
$497$	−5.19615	+	3.00000i	−0.233079	+	0.134568i
$498$		0			0
$499$	−16.0000	+	27.7128i	−0.716258	+	1.24060i	0.246214	+	0.969216i	$0.420813\pi$
$499$	−16.0000	+	27.7128i	−0.716258	+	1.24060i	−0.962472	+	0.271380i	$0.912520\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			7.00000i			0.312115i	0.987748	+	0.156057i	$0.0498784\pi$
$503$			7.00000i			0.312115i	−0.987748	+	0.156057i	$0.950122\pi$
$504$		0			0
$505$		0			0
$506$	−3.00000	−	5.19615i	−0.133366	−	0.230997i
$507$		0			0
$508$	−4.33013	−	2.50000i	−0.192118	−	0.110920i
$509$	−21.5000	+	37.2391i	−0.952971	+	1.65059i	−0.214026	+	0.976828i	$0.568658\pi$
$509$	−21.5000	+	37.2391i	−0.952971	+	1.65059i	−0.738945	+	0.673766i	$0.764676\pi$
$510$		0			0
$511$	6.00000	+	10.3923i	0.265424	+	0.459728i
$512$		−	11.0000i		−	0.486136i
$513$		0			0
$514$	6.00000			0.264649
$515$		0			0
$516$		0			0
$517$	−12.1244	−	7.00000i	−0.533229	−	0.307860i
$518$	10.3923	+	6.00000i	0.456612	+	0.263625i
$519$		0			0
$520$		0			0
$521$	−11.0000			−0.481919			−0.240959	−	0.970535i	$-0.577462\pi$
$521$	−11.0000			−0.481919			−0.240959	+	0.970535i	$0.577462\pi$
$522$		0			0
$523$		−	29.0000i		−	1.26808i	−0.773300	−	0.634041i	$-0.781395\pi$
$523$		−	29.0000i		−	1.26808i	0.773300	−	0.634041i	$-0.218605\pi$
$524$	−3.00000	−	5.19615i	−0.131056	−	0.226995i
$525$		0			0
$526$	8.00000	−	13.8564i	0.348817	−	0.604168i
$527$		0			0
$528$		0			0
$529$	−7.00000	−	12.1244i	−0.304348	−	0.527146i
$530$		0			0
$531$		0			0
$532$			24.0000i			1.04053i
$533$	8.66025	−	5.00000i	0.375117	−	0.216574i
$534$		0			0
$535$		0			0
$536$	4.50000	−	7.79423i	0.194370	−	0.336659i
$537$		0			0
$538$	21.6506	−	12.5000i	0.933425	−	0.538913i
$539$	−4.00000			−0.172292
$540$		0			0
$541$	−39.0000			−1.67674			−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000			−1.67674			−0.838370	+	0.545101i	$0.816491\pi$
$542$	−6.92820	+	4.00000i	−0.297592	+	0.171815i
$543$		0			0
$544$	10.0000	−	17.3205i	0.428746	−	0.742611i
$545$		0			0
$546$		0			0
$547$	25.1147	−	14.5000i	1.07383	−	0.619975i	0.144604	−	0.989490i	$-0.453809\pi$
$547$	25.1147	−	14.5000i	1.07383	−	0.619975i	0.929225	+	0.369514i	$0.120476\pi$
$548$			12.0000i			0.512615i
$549$		0			0
$550$		0			0
$551$	4.00000	+	6.92820i	0.170406	+	0.295151i
$552$		0			0
$553$	−15.5885	−	9.00000i	−0.662889	−	0.382719i
$554$	−6.00000	+	10.3923i	−0.254916	+	0.441527i
$555$		0			0
$556$	−8.00000	−	13.8564i	−0.339276	−	0.587643i
$557$		−	30.0000i		−	1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$		−	30.0000i		−	1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$		0			0
$559$	−16.0000			−0.676728
$560$		0			0
$561$		0			0
$562$	−12.9904	−	7.50000i	−0.547966	−	0.316368i
$563$	−18.1865	−	10.5000i	−0.766471	−	0.442522i	0.0651433	−	0.997876i	$-0.479250\pi$
$563$	−18.1865	−	10.5000i	−0.766471	−	0.442522i	−0.831614	+	0.555354i	$0.812583\pi$
$564$		0			0
$565$		0			0
$566$	−21.0000			−0.882696
$567$		0			0
$568$		−	6.00000i		−	0.251754i
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	0.922032	−	0.387113i	$-0.126528\pi$
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	−0.796266	+	0.604947i	$0.793194\pi$
$570$		0			0
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	0.308443	+	0.951243i	$0.400192\pi$
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	−0.978022	+	0.208502i	$0.933141\pi$
$572$	3.46410	+	2.00000i	0.144841	+	0.0836242i
$573$		0			0
$574$	−7.50000	−	12.9904i	−0.313044	−	0.542208i
$575$		0			0
$576$		0			0
$577$			10.0000i			0.416305i	0.978096	+	0.208153i	$0.0667451\pi$
$577$			10.0000i			0.416305i	−0.978096	+	0.208153i	$0.933255\pi$
$578$	0.866025	−	0.500000i	0.0360219	−	0.0207973i
$579$		0			0
$580$		0			0
$581$	−13.5000	+	23.3827i	−0.560074	+	0.970077i
$582$		0			0
$583$	3.46410	−	2.00000i	0.143468	−	0.0828315i
$584$	−12.0000			−0.496564
$585$		0			0
$586$	12.0000			0.495715
$587$	−28.5788	+	16.5000i	−1.17957	+	0.681028i	−0.955916	−	0.293640i	$-0.905133\pi$
$587$	−28.5788	+	16.5000i	−1.17957	+	0.681028i	−0.223659	+	0.974668i	$0.571800\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−3.46410	+	2.00000i	−0.142374	+	0.0821995i
$593$		−	20.0000i		−	0.821302i	−0.911793	−	0.410651i	$-0.865302\pi$
$593$		−	20.0000i		−	0.821302i	0.911793	−	0.410651i	$-0.134698\pi$
$594$		0			0
$595$		0			0
$596$	−8.50000	−	14.7224i	−0.348174	−	0.603054i
$597$		0			0
$598$	5.19615	+	3.00000i	0.212486	+	0.122679i
$599$	5.00000	−	8.66025i	0.204294	−	0.353848i	−0.745613	−	0.666379i	$-0.767843\pi$
$599$	5.00000	−	8.66025i	0.204294	−	0.353848i	0.949908	+	0.312531i	$0.101177\pi$
$600$		0			0
$601$	−1.00000	−	1.73205i	−0.0407909	−	0.0706518i	0.844909	−	0.534910i	$-0.179654\pi$
$601$	−1.00000	−	1.73205i	−0.0407909	−	0.0706518i	−0.885700	+	0.464258i	$0.846321\pi$
$602$			24.0000i			0.978167i
$603$		0			0
$604$	−2.00000			−0.0813788
$605$		0			0
$606$		0			0
$607$	−35.5070	−	20.5000i	−1.44119	−	0.832069i	−0.443257	−	0.896394i	$-0.646177\pi$
$607$	−35.5070	−	20.5000i	−1.44119	−	0.832069i	−0.997929	+	0.0643251i	$0.979511\pi$
$608$	−34.6410	−	20.0000i	−1.40488	−	0.811107i
$609$		0			0
$610$		0			0
$611$	14.0000			0.566379
$612$		0			0
$613$		−	44.0000i		−	1.77714i	−0.458738	−	0.888572i	$-0.651698\pi$
$613$		−	44.0000i		−	1.77714i	0.458738	−	0.888572i	$-0.348302\pi$
$614$	−3.50000	−	6.06218i	−0.141249	−	0.244650i
$615$		0			0
$616$	9.00000	−	15.5885i	0.362620	−	0.628077i
$617$	31.1769	+	18.0000i	1.25514	+	0.724653i	0.972125	−	0.234464i	$-0.0753335\pi$
$617$	31.1769	+	18.0000i	1.25514	+	0.724653i	0.283011	+	0.959117i	$0.408667\pi$
$618$		0			0
$619$	−2.00000	−	3.46410i	−0.0803868	−	0.139234i	0.823029	−	0.567999i	$-0.192282\pi$
$619$	−2.00000	−	3.46410i	−0.0803868	−	0.139234i	−0.903416	+	0.428765i	$0.858949\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−38.9711	+	22.5000i	−1.56135	+	0.901443i
$624$		0			0
$625$		0			0
$626$	7.00000	−	12.1244i	0.279776	−	0.484587i
$627$		0			0
$628$	−12.1244	+	7.00000i	−0.483814	+	0.279330i
$629$	−16.0000			−0.637962
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$	15.5885	−	9.00000i	0.620076	−	0.358001i
$633$		0			0
$634$	−17.0000	+	29.4449i	−0.675156	+	1.16940i
$635$		0			0
$636$		0			0
$637$	3.46410	−	2.00000i	0.137253	−	0.0792429i
$638$		−	2.00000i		−	0.0791808i
$639$		0			0
$640$		0			0
$641$	16.5000	+	28.5788i	0.651711	+	1.12880i	0.982708	+	0.185164i	$0.0592817\pi$
$641$	16.5000	+	28.5788i	0.651711	+	1.12880i	−0.330997	+	0.943632i	$0.607385\pi$
$642$		0			0
$643$	7.79423	+	4.50000i	0.307374	+	0.177463i	0.645751	−	0.763548i	$-0.276544\pi$
$643$	7.79423	+	4.50000i	0.307374	+	0.177463i	−0.338377	+	0.941011i	$0.609878\pi$
$644$	−4.50000	+	7.79423i	−0.177325	+	0.307136i
$645$		0			0
$646$	16.0000	+	27.7128i	0.629512	+	1.09035i
$647$		−	17.0000i		−	0.668339i	−0.942513	−	0.334169i	$-0.891544\pi$
$647$		−	17.0000i		−	0.668339i	0.942513	−	0.334169i	$-0.108456\pi$
$648$		0			0
$649$	−28.0000			−1.09910
$650$		0			0
$651$		0			0
$652$	3.46410	+	2.00000i	0.135665	+	0.0783260i
$653$	−3.46410	−	2.00000i	−0.135561	−	0.0782660i	0.430686	−	0.902502i	$-0.358272\pi$
$653$	−3.46410	−	2.00000i	−0.135561	−	0.0782660i	−0.566247	+	0.824236i	$0.691605\pi$
$654$		0			0
$655$		0			0
$656$	5.00000			0.195217
$657$		0			0
$658$		−	21.0000i		−	0.818665i
$659$	4.00000	+	6.92820i	0.155818	+	0.269884i	0.933357	−	0.358951i	$-0.116865\pi$
$659$	4.00000	+	6.92820i	0.155818	+	0.269884i	−0.777539	+	0.628835i	$0.783532\pi$
$660$		0			0
$661$	7.00000	−	12.1244i	0.272268	−	0.471583i	−0.697174	−	0.716902i	$-0.745559\pi$
$661$	7.00000	−	12.1244i	0.272268	−	0.471583i	0.969442	+	0.245319i	$0.0788928\pi$
$662$	5.19615	+	3.00000i	0.201954	+	0.116598i
$663$		0			0
$664$	−13.5000	−	23.3827i	−0.523902	−	0.907424i
$665$		0			0
$666$		0			0
$667$			3.00000i			0.116160i
$668$	−7.79423	+	4.50000i	−0.301568	+	0.174110i
$669$		0			0
$670$		0			0
$671$	−7.00000	+	12.1244i	−0.270232	+	0.468056i
$672$		0			0
$673$	5.19615	−	3.00000i	0.200297	−	0.115642i	−0.396497	−	0.918036i	$-0.629774\pi$
$673$	5.19615	−	3.00000i	0.200297	−	0.115642i	0.596794	+	0.802395i	$0.296441\pi$
$674$	−8.00000			−0.308148
$675$		0			0
$676$	9.00000			0.346154
$677$	−36.3731	+	21.0000i	−1.39793	+	0.807096i	−0.994176	−	0.107772i	$-0.965628\pi$
$677$	−36.3731	+	21.0000i	−1.39793	+	0.807096i	−0.403755	+	0.914867i	$0.632295\pi$
$678$		0			0
$679$	−3.00000	+	5.19615i	−0.115129	+	0.199410i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	12.0000i		−	0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$		−	12.0000i		−	0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$		0			0
$685$		0			0
$686$	7.50000	+	12.9904i	0.286351	+	0.495975i
$687$		0			0
$688$	−6.92820	−	4.00000i	−0.264135	−	0.152499i
$689$	−2.00000	+	3.46410i	−0.0761939	+	0.131972i
$690$		0			0
$691$	7.00000	+	12.1244i	0.266293	+	0.461232i	0.967901	−	0.251330i	$-0.0808679\pi$
$691$	7.00000	+	12.1244i	0.266293	+	0.461232i	−0.701609	+	0.712562i	$0.747535\pi$
$692$		0			0
$693$		0			0
$694$	−4.00000			−0.151838
$695$		0			0
$696$		0			0
$697$	17.3205	+	10.0000i	0.656061	+	0.378777i
$698$	−4.33013	−	2.50000i	−0.163898	−	0.0946264i
$699$		0			0
$700$		0			0
$701$	−23.0000			−0.868698			−0.434349	−	0.900745i	$-0.643022\pi$
$701$	−23.0000			−0.868698			−0.434349	+	0.900745i	$0.643022\pi$
$702$		0			0
$703$			32.0000i			1.20690i
$704$	7.00000	+	12.1244i	0.263822	+	0.456954i
$705$		0			0
$706$	−12.0000	+	20.7846i	−0.451626	+	0.782239i
$707$	−46.7654	−	27.0000i	−1.75879	−	1.01544i
$708$		0			0
$709$	−20.5000	−	35.5070i	−0.769894	−	1.33349i	−0.937620	−	0.347661i	$-0.886976\pi$
$709$	−20.5000	−	35.5070i	−0.769894	−	1.33349i	0.167727	−	0.985834i	$-0.446357\pi$
$710$		0			0
$711$		0			0
$712$		−	45.0000i		−	1.68645i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.00000	−	1.73205i	0.0373718	−	0.0647298i
$717$		0			0
$718$	20.7846	−	12.0000i	0.775675	−	0.447836i
$719$	6.00000			0.223762			0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000			0.223762			0.111881	+	0.993722i	$0.464312\pi$
$720$		0			0
$721$	−24.0000			−0.893807
$722$	38.9711	−	22.5000i	1.45036	−	0.837363i
$723$		0			0
$724$	3.50000	−	6.06218i	0.130076	−	0.225299i
$725$		0			0
$726$		0			0
$727$	−19.9186	+	11.5000i	−0.738739	+	0.426511i	−0.821611	−	0.570049i	$-0.806924\pi$
$727$	−19.9186	+	11.5000i	−0.738739	+	0.426511i	0.0828714	+	0.996560i	$0.473591\pi$
$728$			18.0000i			0.667124i
$729$		0			0
$730$		0			0
$731$	−16.0000	−	27.7128i	−0.591781	−	1.02500i
$732$		0			0
$733$	29.4449	+	17.0000i	1.08757	+	0.627909i	0.932929	−	0.360061i	$-0.117244\pi$
$733$	29.4449	+	17.0000i	1.08757	+	0.627909i	0.154642	+	0.987971i	$0.450578\pi$
$734$	−12.0000	+	20.7846i	−0.442928	+	0.767174i
$735$		0			0
$736$	−7.50000	−	12.9904i	−0.276454	−	0.478832i
$737$			6.00000i			0.221013i
$738$		0			0
$739$	2.00000			0.0735712			0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000			0.0735712			0.0367856	+	0.999323i	$0.488288\pi$
$740$		0			0
$741$		0			0
$742$	5.19615	+	3.00000i	0.190757	+	0.110133i
$743$	−25.1147	−	14.5000i	−0.921370	−	0.531953i	−0.0372984	−	0.999304i	$-0.511875\pi$
$743$	−25.1147	−	14.5000i	−0.921370	−	0.531953i	−0.884072	+	0.467351i	$0.845209\pi$
$744$		0			0
$745$		0			0
$746$	−10.0000			−0.366126
$747$		0			0
$748$			8.00000i			0.292509i
$749$	4.50000	+	7.79423i	0.164426	+	0.284795i
$750$		0			0
$751$	−5.00000	+	8.66025i	−0.182453	+	0.316017i	−0.942715	−	0.333599i	$-0.891737\pi$
$751$	−5.00000	+	8.66025i	−0.182453	+	0.316017i	0.760263	+	0.649616i	$0.225070\pi$
$752$	6.06218	+	3.50000i	0.221065	+	0.127632i
$753$		0			0
$754$	1.00000	+	1.73205i	0.0364179	+	0.0630776i
$755$		0			0
$756$		0			0
$757$			26.0000i			0.944986i	0.881334	+	0.472493i	$0.156646\pi$
$757$			26.0000i			0.944986i	−0.881334	+	0.472493i	$0.843354\pi$
$758$	22.5167	−	13.0000i	0.817842	−	0.472181i
$759$		0			0
$760$		0			0
$761$	7.50000	−	12.9904i	0.271875	−	0.470901i	−0.697467	−	0.716617i	$-0.745690\pi$
$761$	7.50000	−	12.9904i	0.271875	−	0.470901i	0.969342	+	0.245716i	$0.0790230\pi$
$762$		0			0
$763$	−12.9904	+	7.50000i	−0.470283	+	0.271518i
$764$	−8.00000			−0.289430
$765$		0			0
$766$	36.0000			1.30073
$767$	24.2487	−	14.0000i	0.875570	−	0.505511i
$768$		0			0
$769$	2.50000	−	4.33013i	0.0901523	−	0.156148i	−0.817423	−	0.576038i	$-0.804598\pi$
$769$	2.50000	−	4.33013i	0.0901523	−	0.156148i	0.907575	+	0.419890i	$0.137931\pi$
$770$		0			0
$771$		0			0
$772$	−8.66025	+	5.00000i	−0.311689	+	0.179954i
$773$			24.0000i			0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$			24.0000i			0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$		0			0
$775$		0			0
$776$	−3.00000	−	5.19615i	−0.107694	−	0.186531i
$777$		0			0
$778$	−28.5788	−	16.5000i	−1.02460	−	0.591554i
$779$	20.0000	−	34.6410i	0.716574	−	1.24114i
$780$		0			0
$781$	2.00000	+	3.46410i	0.0715656	+	0.123955i
$782$			12.0000i			0.429119i
$783$		0			0
$784$	2.00000			0.0714286
$785$		0			0
$786$		0			0
$787$	24.2487	+	14.0000i	0.864373	+	0.499046i	0.865474	−	0.500953i	$-0.167017\pi$
$787$	24.2487	+	14.0000i	0.864373	+	0.499046i	−0.00110111	+	0.999999i	$0.500350\pi$
$788$	10.3923	+	6.00000i	0.370211	+	0.213741i
$789$		0			0

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.k (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(2.59808 - 1.50000i) q^{7} +3.00000i q^{8} +O(q^{10})\)	\(q+(0.866025 - 0.500000i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(2.59808 - 1.50000i) q^{7} +3.00000i q^{8} +(-1.00000 - 1.73205i) q^{11} +(1.73205 + 1.00000i) q^{13} +(1.50000 - 2.59808i) q^{14} +(0.500000 + 0.866025i) q^{16} +4.00000i q^{17} +8.00000 q^{19} +(-1.73205 - 1.00000i) q^{22} +(2.59808 + 1.50000i) q^{23} +2.00000 q^{26} +3.00000i q^{28} +(0.500000 + 0.866025i) q^{29} +(-4.33013 - 2.50000i) q^{32} +(2.00000 + 3.46410i) q^{34} +4.00000i q^{37} +(6.92820 - 4.00000i) q^{38} +(2.50000 - 4.33013i) q^{41} +(-6.92820 + 4.00000i) q^{43} +2.00000 q^{44} +3.00000 q^{46} +(6.06218 - 3.50000i) q^{47} +(1.00000 - 1.73205i) q^{49} +(-1.73205 + 1.00000i) q^{52} +2.00000i q^{53} +(4.50000 + 7.79423i) q^{56} +(0.866025 + 0.500000i) q^{58} +(7.00000 - 12.1244i) q^{59} +(-3.50000 - 6.06218i) q^{61} -7.00000 q^{64} +(-2.59808 - 1.50000i) q^{67} +(-3.46410 - 2.00000i) q^{68} -2.00000 q^{71} +4.00000i q^{73} +(2.00000 + 3.46410i) q^{74} +(-4.00000 + 6.92820i) q^{76} +(-5.19615 - 3.00000i) q^{77} +(-3.00000 - 5.19615i) q^{79} -5.00000i q^{82} +(-7.79423 + 4.50000i) q^{83} +(-4.00000 + 6.92820i) q^{86} +(5.19615 - 3.00000i) q^{88} -15.0000 q^{89} +6.00000 q^{91} +(-2.59808 + 1.50000i) q^{92} +(3.50000 - 6.06218i) q^{94} +(-1.73205 + 1.00000i) q^{97} -2.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4}+O(q^{10}) \) 4 * q - 2 * q^4	\( 4 q - 2 q^{4} - 4 q^{11} + 6 q^{14} + 2 q^{16} + 32 q^{19} + 8 q^{26} + 2 q^{29} + 8 q^{34} + 10 q^{41} + 8 q^{44} + 12 q^{46} + 4 q^{49} + 18 q^{56} + 28 q^{59} - 14 q^{61} - 28 q^{64} - 8 q^{71} + 8 q^{74} - 16 q^{76} - 12 q^{79} - 16 q^{86} - 60 q^{89} + 24 q^{91} + 14 q^{94}+O(q^{100}) \) 4 * q - 2 * q^4 - 4 * q^11 + 6 * q^14 + 2 * q^16 + 32 * q^19 + 8 * q^26 + 2 * q^29 + 8 * q^34 + 10 * q^41 + 8 * q^44 + 12 * q^46 + 4 * q^49 + 18 * q^56 + 28 * q^59 - 14 * q^61 - 28 * q^64 - 8 * q^71 + 8 * q^74 - 16 * q^76 - 12 * q^79 - 16 * q^86 - 60 * q^89 + 24 * q^91 + 14 * q^94

Introduction

L-functions

Modular forms

Varieties

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