LMFDB - Embedded newform 968.2.i.q.753.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,2,Mod(9,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 6]))

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

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

chi := DirichletCharacter("968.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	968.i (of order $5$, degree $4$, 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:	$7.72951891566$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{5})$
Coefficient field:	8.0.1305015625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 5x^{6} - 9x^{5} + 29x^{4} + 36x^{3} + 80x^{2} + 64x + 256 $ x^8 - x^7 + 5x^6 - 9x^5 + 29x^4 + 36x^3 + 80x^2 + 64x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			753.2
Root			$0.482546 + 1.48512i$ of defining polynomial
Character	$\chi$	$=$	968.753
Dual form			968.2.i.q.9.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.26332 + 0.917858i) q^{3} +(1.10058 - 3.38724i) q^{5} +(2.52665 - 1.83572i) q^{7} +(-0.173529 - 0.534068i) q^{9} +O(q^{10})$	\(q+(1.26332 + 0.917858i) q^{3} +(1.10058 - 3.38724i) q^{5} +(2.52665 - 1.83572i) q^{7} +(-0.173529 - 0.534068i) q^{9} +(1.58313 + 4.87236i) q^{13} +(4.49939 - 3.26900i) q^{15} +(-0.618034 + 1.90211i) q^{17} +(-3.23607 - 2.35114i) q^{19} +4.87689 q^{21} +2.43845 q^{23} +(-6.21702 - 4.51693i) q^{25} +(1.71861 - 5.28935i) q^{27} +(-4.14468 + 3.01129i) q^{29} +(-1.71861 - 5.28935i) q^{31} +(-3.43723 - 10.5787i) q^{35} +(6.11742 - 4.44457i) q^{37} +(-2.47214 + 7.60845i) q^{39} +(-0.908612 - 0.660145i) q^{41} +7.12311 q^{43} -2.00000 q^{45} +(-6.47214 - 4.70228i) q^{47} +(0.850968 - 2.61901i) q^{49} +(-2.52665 + 1.83572i) q^{51} +(3.78429 + 11.6468i) q^{53} +(-1.93019 - 5.94050i) q^{57} +(-6.31661 + 4.58929i) q^{59} +(-0.347059 + 1.06814i) q^{61} +(-1.41884 - 1.03085i) q^{63} +18.2462 q^{65} +9.56155 q^{67} +(3.08055 + 2.23815i) q^{69} +(-2.68371 + 8.25960i) q^{71} +(4.14468 - 3.01129i) q^{73} +(-3.70820 - 11.4127i) q^{75} +(3.43723 + 10.5787i) q^{79} +(5.66312 - 4.11450i) q^{81} +(-0.270975 + 0.833976i) q^{83} +(5.76271 + 4.18686i) q^{85} -8.00000 q^{87} +2.68466 q^{89} +(12.9443 + 9.40456i) q^{91} +(2.68371 - 8.25960i) q^{93} +(-11.5254 + 8.37371i) q^{95} +(4.80878 + 14.7999i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{3} - 3 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) $ 8 * q - q^3 - 3 * q^5 - 2 * q^7 - 3 * q^9	$ 8 q - q^{3} - 3 q^{5} - 2 q^{7} - 3 q^{9} - 2 q^{13} + 7 q^{15} + 4 q^{17} - 8 q^{19} + 72 q^{21} + 36 q^{23} - 3 q^{25} - 7 q^{27} - 2 q^{29} + 7 q^{31} + 14 q^{35} + 11 q^{37} + 16 q^{39} + 6 q^{41} + 24 q^{43} - 16 q^{45} - 16 q^{47} - 22 q^{49} + 2 q^{51} - 8 q^{53} - 4 q^{57} + 5 q^{59} - 6 q^{61} - 20 q^{63} + 80 q^{65} + 60 q^{67} - 13 q^{69} + 5 q^{71} + 2 q^{73} + 24 q^{75} - 14 q^{79} + 14 q^{81} + 10 q^{83} + 6 q^{85} - 64 q^{87} - 28 q^{89} + 32 q^{91} - 5 q^{93} - 12 q^{95} - 27 q^{97}+O(q^{100}) $ 8 * q - q^3 - 3 * q^5 - 2 * q^7 - 3 * q^9 - 2 * q^13 + 7 * q^15 + 4 * q^17 - 8 * q^19 + 72 * q^21 + 36 * q^23 - 3 * q^25 - 7 * q^27 - 2 * q^29 + 7 * q^31 + 14 * q^35 + 11 * q^37 + 16 * q^39 + 6 * q^41 + 24 * q^43 - 16 * q^45 - 16 * q^47 - 22 * q^49 + 2 * q^51 - 8 * q^53 - 4 * q^57 + 5 * q^59 - 6 * q^61 - 20 * q^63 + 80 * q^65 + 60 * q^67 - 13 * q^69 + 5 * q^71 + 2 * q^73 + 24 * q^75 - 14 * q^79 + 14 * q^81 + 10 * q^83 + 6 * q^85 - 64 * q^87 - 28 * q^89 + 32 * q^91 - 5 * q^93 - 12 * q^95 - 27 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/968\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$849$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{5}\right)$

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			0
$2$		0			0
$3$	1.26332	+	0.917858i	0.729380	+	0.529925i	0.889367	−	0.457194i	$-0.151145\pi$
$3$	1.26332	+	0.917858i	0.729380	+	0.529925i	−0.159987	+	0.987119i	$0.551145\pi$
$4$		0			0
$5$	1.10058	−	3.38724i	0.492194	−	1.51482i	−0.329089	−	0.944299i	$-0.606742\pi$
$5$	1.10058	−	3.38724i	0.492194	−	1.51482i	0.821284	−	0.570520i	$-0.193258\pi$
$6$		0			0
$7$	2.52665	−	1.83572i	0.954982	−	0.693835i	0.00300226	−	0.999995i	$-0.499044\pi$
$7$	2.52665	−	1.83572i	0.954982	−	0.693835i	0.951980	+	0.306160i	$0.0990443\pi$
$8$		0			0
$9$	−0.173529	−	0.534068i	−0.0578431	−	0.178023i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$	1.58313	+	4.87236i	0.439080	+	1.35135i	0.888847	+	0.458204i	$0.151507\pi$
$13$	1.58313	+	4.87236i	0.439080	+	1.35135i	−0.449767	+	0.893146i	$0.648493\pi$
$14$		0			0
$15$	4.49939	−	3.26900i	1.16174	−	0.844052i
$16$		0			0
$17$	−0.618034	+	1.90211i	−0.149895	+	0.461330i	−0.997608	−	0.0691254i	$-0.977979\pi$
$17$	−0.618034	+	1.90211i	−0.149895	+	0.461330i	0.847713	+	0.530456i	$0.177979\pi$
$18$		0			0
$19$	−3.23607	−	2.35114i	−0.742405	−	0.539389i	0.151058	−	0.988525i	$-0.451732\pi$
$19$	−3.23607	−	2.35114i	−0.742405	−	0.539389i	−0.893463	+	0.449136i	$0.851732\pi$
$20$		0			0
$21$	4.87689			1.06423
$22$		0			0
$23$	2.43845			0.508451			0.254226	−	0.967145i	$-0.418179\pi$
$23$	2.43845			0.508451			0.254226	+	0.967145i	$0.418179\pi$
$24$		0			0
$25$	−6.21702	−	4.51693i	−1.24340	−	0.903386i
$26$		0			0
$27$	1.71861	−	5.28935i	0.330747	−	1.01794i
$28$		0			0
$29$	−4.14468	+	3.01129i	−0.769648	+	0.559182i	−0.901854	−	0.432040i	$-0.857794\pi$
$29$	−4.14468	+	3.01129i	−0.769648	+	0.559182i	0.132207	+	0.991222i	$0.457794\pi$
$30$		0			0
$31$	−1.71861	−	5.28935i	−0.308672	−	0.949995i	−0.978281	−	0.207282i	$-0.933538\pi$
$31$	−1.71861	−	5.28935i	−0.308672	−	0.949995i	0.669609	−	0.742714i	$-0.266462\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.43723	−	10.5787i	−0.580998	−	1.78813i
$36$		0			0
$37$	6.11742	−	4.44457i	1.00570	−	0.730683i	0.0423960	−	0.999101i	$-0.486501\pi$
$37$	6.11742	−	4.44457i	1.00570	−	0.730683i	0.963302	+	0.268418i	$0.0865009\pi$
$38$		0			0
$39$	−2.47214	+	7.60845i	−0.395859	+	1.21833i
$40$		0			0
$41$	−0.908612	−	0.660145i	−0.141901	−	0.103097i	0.514570	−	0.857448i	$-0.327952\pi$
$41$	−0.908612	−	0.660145i	−0.141901	−	0.103097i	−0.656471	+	0.754351i	$0.727952\pi$
$42$		0			0
$43$	7.12311			1.08626			0.543132	−	0.839648i	$-0.317238\pi$
$43$	7.12311			1.08626			0.543132	+	0.839648i	$0.317238\pi$
$44$		0			0
$45$	−2.00000			−0.298142
$46$		0			0
$47$	−6.47214	−	4.70228i	−0.944058	−	0.685898i	0.00533600	−	0.999986i	$-0.498301\pi$
$47$	−6.47214	−	4.70228i	−0.944058	−	0.685898i	−0.949394	+	0.314087i	$0.898301\pi$
$48$		0			0
$49$	0.850968	−	2.61901i	0.121567	−	0.374144i
$50$		0			0
$51$	−2.52665	+	1.83572i	−0.353801	+	0.257052i
$52$		0			0
$53$	3.78429	+	11.6468i	0.519812	+	1.59982i	0.774353	+	0.632754i	$0.218076\pi$
$53$	3.78429	+	11.6468i	0.519812	+	1.59982i	−0.254541	+	0.967062i	$0.581924\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.93019	−	5.94050i	−0.255659	−	0.786838i
$58$		0			0
$59$	−6.31661	+	4.58929i	−0.822353	+	0.597474i	−0.917386	−	0.398000i	$-0.869705\pi$
$59$	−6.31661	+	4.58929i	−0.822353	+	0.597474i	0.0950325	+	0.995474i	$0.469705\pi$
$60$		0			0
$61$	−0.347059	+	1.06814i	−0.0444363	+	0.136761i	−0.970813	−	0.239837i	$-0.922906\pi$
$61$	−0.347059	+	1.06814i	−0.0444363	+	0.136761i	0.926377	+	0.376598i	$0.122906\pi$
$62$		0			0
$63$	−1.41884	−	1.03085i	−0.178758	−	0.129875i
$64$		0			0
$65$	18.2462			2.26316
$66$		0			0
$67$	9.56155			1.16813			0.584065	−	0.811707i	$-0.301461\pi$
$67$	9.56155			1.16813			0.584065	+	0.811707i	$0.301461\pi$
$68$		0			0
$69$	3.08055	+	2.23815i	0.370854	+	0.269441i
$70$		0			0
$71$	−2.68371	+	8.25960i	−0.318497	+	0.980234i	0.655793	+	0.754940i	$0.272334\pi$
$71$	−2.68371	+	8.25960i	−0.318497	+	0.980234i	−0.974291	+	0.225294i	$0.927666\pi$
$72$		0			0
$73$	4.14468	−	3.01129i	0.485098	−	0.352444i	−0.318198	−	0.948024i	$-0.603078\pi$
$73$	4.14468	−	3.01129i	0.485098	−	0.352444i	0.803296	+	0.595580i	$0.203078\pi$
$74$		0			0
$75$	−3.70820	−	11.4127i	−0.428187	−	1.31782i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	3.43723	+	10.5787i	0.386718	+	1.19020i	0.935226	+	0.354051i	$0.115196\pi$
$79$	3.43723	+	10.5787i	0.386718	+	1.19020i	−0.548508	+	0.836146i	$0.684804\pi$
$80$		0			0
$81$	5.66312	−	4.11450i	0.629235	−	0.457166i
$82$		0			0
$83$	−0.270975	+	0.833976i	−0.0297434	+	0.0915408i	−0.964826	−	0.262889i	$-0.915325\pi$
$83$	−0.270975	+	0.833976i	−0.0297434	+	0.0915408i	0.935083	+	0.354429i	$0.115325\pi$
$84$		0			0
$85$	5.76271	+	4.18686i	0.625054	+	0.454128i
$86$		0			0
$87$	−8.00000			−0.857690
$88$		0			0
$89$	2.68466			0.284573			0.142287	−	0.989825i	$-0.454555\pi$
$89$	2.68466			0.284573			0.142287	+	0.989825i	$0.454555\pi$
$90$		0			0
$91$	12.9443	+	9.40456i	1.35693	+	0.985866i
$92$		0			0
$93$	2.68371	−	8.25960i	0.278287	−	0.856481i
$94$		0			0
$95$	−11.5254	+	8.37371i	−1.18248	+	0.859125i
$96$		0			0
$97$	4.80878	+	14.7999i	0.488258	+	1.50270i	0.827206	+	0.561899i	$0.189929\pi$
$97$	4.80878	+	14.7999i	0.488258	+	1.50270i	−0.338948	+	0.940805i	$0.610071\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.618034	+	1.90211i	0.0614967	+	0.189267i	0.977085	−	0.212850i	$-0.0682745\pi$
$101$	0.618034	+	1.90211i	0.0614967	+	0.189267i	−0.915588	+	0.402117i	$0.868274\pi$
$102$		0			0
$103$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$103$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$104$		0			0
$105$	5.36741	−	16.5192i	0.523806	−	1.61211i
$106$		0			0
$107$	−10.8160	−	7.85829i	−1.04562	−	0.759689i	−0.0742474	−	0.997240i	$-0.523655\pi$
$107$	−10.8160	−	7.85829i	−1.04562	−	0.759689i	−0.971375	+	0.237551i	$0.923655\pi$
$108$		0			0
$109$	−12.2462			−1.17297			−0.586487	−	0.809959i	$-0.699490\pi$
$109$	−12.2462			−1.17297			−0.586487	+	0.809959i	$0.699490\pi$
$110$		0			0
$111$	11.8078			1.12074
$112$		0			0
$113$	0.354711	+	0.257713i	0.0333684	+	0.0242436i	0.604344	−	0.796723i	$-0.293435\pi$
$113$	0.354711	+	0.257713i	0.0333684	+	0.0242436i	−0.570976	+	0.820967i	$0.693435\pi$
$114$		0			0
$115$	2.68371	−	8.25960i	0.250257	−	0.770212i
$116$		0			0
$117$	2.32746	−	1.69100i	0.215173	−	0.156333i
$118$		0			0
$119$	1.93019	+	5.94050i	0.176940	+	0.544565i
$120$		0			0
$121$		0			0
$122$		0			0
$123$	−0.541951	−	1.66795i	−0.0488660	−	0.150394i
$124$		0			0
$125$	−7.73546	+	5.62014i	−0.691880	+	0.502681i
$126$		0			0
$127$	−1.93019	+	5.94050i	−0.171276	+	0.527134i	−0.999444	−	0.0333464i	$-0.989384\pi$
$127$	−1.93019	+	5.94050i	−0.171276	+	0.527134i	0.828168	+	0.560481i	$0.189384\pi$
$128$		0			0
$129$	8.99878	+	6.53800i	0.792299	+	0.575639i
$130$		0			0
$131$	−13.3693			−1.16808			−0.584041	−	0.811724i	$-0.698529\pi$
$131$	−13.3693			−1.16808			−0.584041	+	0.811724i	$0.698529\pi$
$132$		0			0
$133$	−12.4924			−1.08323
$134$		0			0
$135$	−16.0248	−	11.6427i	−1.37920	−	1.00205i
$136$		0			0
$137$	−2.60762	+	8.02544i	−0.222784	+	0.685660i	0.775725	+	0.631072i	$0.217385\pi$
$137$	−2.60762	+	8.02544i	−0.222784	+	0.685660i	−0.998509	+	0.0545881i	$0.982615\pi$
$138$		0			0
$139$	12.2348	−	8.88914i	1.03775	−	0.753967i	0.0679020	−	0.997692i	$-0.478369\pi$
$139$	12.2348	−	8.88914i	1.03775	−	0.753967i	0.969844	+	0.243725i	$0.0783695\pi$
$140$		0			0
$141$	−3.86037	−	11.8810i	−0.325102	−	1.00056i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	5.63839	+	17.3532i	0.468243	+	1.44110i
$146$		0			0
$147$	3.47892	−	2.52759i	0.286937	−	0.208472i
$148$		0			0
$149$	−1.31215	+	4.03839i	−0.107496	+	0.330838i	−0.990308	−	0.138888i	$-0.955647\pi$
$149$	−1.31215	+	4.03839i	−0.107496	+	0.330838i	0.882812	+	0.469726i	$0.155647\pi$
$150$		0			0
$151$	−7.57994	−	5.50715i	−0.616846	−	0.448165i	0.234972	−	0.972002i	$-0.424500\pi$
$151$	−7.57994	−	5.50715i	−0.616846	−	0.448165i	−0.851819	+	0.523837i	$0.824500\pi$
$152$		0			0
$153$	1.12311			0.0907977
$154$		0			0
$155$	−19.8078			−1.59100
$156$		0			0
$157$	3.59078	+	2.60885i	0.286575	+	0.208209i	0.721780	−	0.692122i	$-0.243324\pi$
$157$	3.59078	+	2.60885i	0.286575	+	0.208209i	−0.435205	+	0.900331i	$0.643324\pi$
$158$		0			0
$159$	−5.90936	+	18.1872i	−0.468643	+	1.44234i
$160$		0			0
$161$	6.16109	−	4.47630i	0.485562	−	0.352781i
$162$		0			0
$163$	−1.23607	−	3.80423i	−0.0968163	−	0.297970i	0.890906	−	0.454187i	$-0.150070\pi$
$163$	−1.23607	−	3.80423i	−0.0968163	−	0.297970i	−0.987723	+	0.156217i	$0.950070\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.47214	−	7.60845i	−0.191300	−	0.588760i	−1.00000		0.000538710i	$-0.999829\pi$
$167$	−2.47214	−	7.60845i	−0.191300	−	0.588760i	0.808700	−	0.588221i	$-0.200171\pi$
$168$		0			0
$169$	−10.7164	+	7.78593i	−0.824339	+	0.598918i
$170$		0			0
$171$	−0.694117	+	2.13627i	−0.0530805	+	0.163365i
$172$		0			0
$173$	9.90739	+	7.19814i	0.753245	+	0.547265i	0.896831	−	0.442373i	$-0.145863\pi$
$173$	9.90739	+	7.19814i	0.753245	+	0.547265i	−0.143586	+	0.989638i	$0.545863\pi$
$174$		0			0
$175$	−24.0000			−1.81423
$176$		0			0
$177$	−12.1922			−0.916425
$178$		0			0
$179$	5.20881	+	3.78442i	0.389325	+	0.282861i	0.765179	−	0.643818i	$-0.222651\pi$
$179$	5.20881	+	3.78442i	0.389325	+	0.282861i	−0.375854	+	0.926679i	$0.622651\pi$
$180$		0			0
$181$	−0.406463	+	1.25096i	−0.0302121	+	0.0929834i	−0.965026	−	0.262156i	$-0.915567\pi$
$181$	−0.406463	+	1.25096i	−0.0302121	+	0.0929834i	0.934813	+	0.355139i	$0.115567\pi$
$182$		0			0
$183$	−1.41884	+	1.03085i	−0.104884	+	0.0762027i
$184$		0			0
$185$	−8.32210	−	25.6128i	−0.611853	−	1.88309i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−5.36741	−	16.5192i	−0.390422	−	1.20160i
$190$		0			0
$191$	8.44488	−	6.13557i	0.611050	−	0.443954i	−0.238734	−	0.971085i	$-0.576732\pi$
$191$	8.44488	−	6.13557i	0.611050	−	0.443954i	0.849784	+	0.527131i	$0.176732\pi$
$192$		0			0
$193$	2.81919	−	8.67659i	0.202930	−	0.624555i	−0.796862	−	0.604161i	$-0.793508\pi$
$193$	2.81919	−	8.67659i	0.202930	−	0.624555i	0.999792	−	0.0203931i	$-0.00649179\pi$
$194$		0			0
$195$	23.0509	+	16.7474i	1.65071	+	1.19931i
$196$		0			0
$197$	14.4924			1.03254			0.516271	−	0.856425i	$-0.327320\pi$
$197$	14.4924			1.03254			0.516271	+	0.856425i	$0.327320\pi$
$198$		0			0
$199$	−12.4924			−0.885564			−0.442782	−	0.896629i	$-0.646009\pi$
$199$	−12.4924			−0.885564			−0.442782	+	0.896629i	$0.646009\pi$
$200$		0			0
$201$	12.0793	+	8.77615i	0.852010	+	0.619022i
$202$		0			0
$203$	−4.94427	+	15.2169i	−0.347020	+	1.06802i
$204$		0			0
$205$	−3.23607	+	2.35114i	−0.226017	+	0.164211i
$206$		0			0
$207$	−0.423142	−	1.30230i	−0.0294104	−	0.0905159i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−2.62430	−	8.07677i	−0.180664	−	0.556028i	0.819182	−	0.573533i	$-0.194428\pi$
$211$	−2.62430	−	8.07677i	−0.180664	−	0.556028i	−0.999847	+	0.0175052i	$0.994428\pi$
$212$		0			0
$213$	−10.9715	+	7.97128i	−0.751757	+	0.546183i
$214$		0			0
$215$	7.83955	−	24.1277i	0.534653	−	1.64549i
$216$		0			0
$217$	−14.0521	−	10.2094i	−0.953917	−	0.693061i
$218$		0			0
$219$	8.00000			0.540590
$220$		0			0
$221$	−10.2462			−0.689235
$222$		0			0
$223$	9.55268	+	6.94043i	0.639695	+	0.464766i	0.859745	−	0.510723i	$-0.170622\pi$
$223$	9.55268	+	6.94043i	0.639695	+	0.464766i	−0.220050	+	0.975488i	$0.570622\pi$
$224$		0			0
$225$	−1.33351	+	4.10413i	−0.0889009	+	0.273609i
$226$		0			0
$227$	18.7070	−	13.5914i	1.24163	−	0.902094i	0.243921	−	0.969795i	$-0.421566\pi$
$227$	18.7070	−	13.5914i	1.24163	−	0.902094i	0.997706	+	0.0677010i	$0.0215664\pi$
$228$		0			0
$229$	4.53781	+	13.9659i	0.299867	+	0.922895i	0.981543	+	0.191241i	$0.0612513\pi$
$229$	4.53781	+	13.9659i	0.299867	+	0.922895i	−0.681676	+	0.731654i	$0.738749\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.27724	+	7.00864i	0.149187	+	0.459151i	0.997526	−	0.0703033i	$-0.0223967\pi$
$233$	2.27724	+	7.00864i	0.149187	+	0.459151i	−0.848338	+	0.529454i	$0.822397\pi$
$234$		0			0
$235$	−23.0509	+	16.7474i	−1.50367	+	1.09248i
$236$		0			0
$237$	−5.36741	+	16.5192i	−0.348651	+	1.07304i
$238$		0			0
$239$	−3.94549	−	2.86657i	−0.255213	−	0.185423i	0.452821	−	0.891601i	$-0.350418\pi$
$239$	−3.94549	−	2.86657i	−0.255213	−	0.185423i	−0.708034	+	0.706179i	$0.750418\pi$
$240$		0			0
$241$	−29.1231			−1.87598			−0.937992	−	0.346657i	$-0.887317\pi$
$241$	−29.1231			−1.87598			−0.937992	+	0.346657i	$0.887317\pi$
$242$		0			0
$243$	−5.75379			−0.369106
$244$		0			0
$245$	−7.93465	−	5.76486i	−0.506926	−	0.368303i
$246$		0			0
$247$	6.33251	−	19.4895i	0.402928	−	1.24008i
$248$		0			0
$249$	−1.10780	+	0.804864i	−0.0702040	+	0.0510062i
$250$		0			0
$251$	0.482546	+	1.48512i	0.0304581	+	0.0937403i	0.965130	−	0.261771i	$-0.0843066\pi$
$251$	0.482546	+	1.48512i	0.0304581	+	0.0937403i	−0.934672	+	0.355512i	$0.884307\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	3.43723	+	10.5787i	0.215248	+	0.662464i
$256$		0			0
$257$	−9.50901	+	6.90870i	−0.593156	+	0.430953i	−0.843443	−	0.537219i	$-0.819475\pi$
$257$	−9.50901	+	6.90870i	−0.593156	+	0.430953i	0.250287	+	0.968172i	$0.419475\pi$
$258$		0			0
$259$	7.29760	−	22.4597i	0.453451	−	1.39558i
$260$		0			0
$261$	2.32746	+	1.69100i	0.144066	+	0.104670i
$262$		0			0
$263$	−19.1231			−1.17918			−0.589591	−	0.807702i	$-0.700711\pi$
$263$	−19.1231			−1.17918			−0.589591	+	0.807702i	$0.700711\pi$
$264$		0			0
$265$	43.6155			2.67928
$266$		0			0
$267$	3.39159	+	2.46413i	0.207562	+	0.150803i
$268$		0			0
$269$	−6.40859	+	19.7236i	−0.390739	+	1.20257i	0.541492	+	0.840706i	$0.317860\pi$
$269$	−6.40859	+	19.7236i	−0.390739	+	1.20257i	−0.932231	+	0.361864i	$0.882140\pi$
$270$		0			0
$271$	−23.0509	+	16.7474i	−1.40024	+	1.01733i	−0.405587	+	0.914056i	$0.632933\pi$
$271$	−23.0509	+	16.7474i	−1.40024	+	1.01733i	−0.994653	+	0.103277i	$0.967067\pi$
$272$		0			0
$273$	7.72074	+	23.7620i	0.467281	+	1.43814i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	5.56231	+	17.1190i	0.334207	+	1.02858i	0.967112	+	0.254353i	$0.0818624\pi$
$277$	5.56231	+	17.1190i	0.334207	+	1.02858i	−0.632905	+	0.774229i	$0.718138\pi$
$278$		0			0
$279$	−2.52665	+	1.83572i	−0.151266	+	0.109901i
$280$		0			0
$281$	−5.02036	+	15.4511i	−0.299489	+	0.921733i	0.682187	+	0.731178i	$0.261029\pi$
$281$	−5.02036	+	15.4511i	−0.299489	+	0.921733i	−0.981676	+	0.190556i	$0.938971\pi$
$282$		0			0
$283$	16.1803	+	11.7557i	0.961821	+	0.698804i	0.953573	−	0.301162i	$-0.0973744\pi$
$283$	16.1803	+	11.7557i	0.961821	+	0.698804i	0.00824833	+	0.999966i	$0.497374\pi$
$284$		0			0
$285$	−22.2462			−1.31775
$286$		0			0
$287$	−3.50758			−0.207046
$288$		0			0
$289$	10.5172	+	7.64121i	0.618660	+	0.449483i
$290$		0			0
$291$	−7.50917	+	23.1109i	−0.440195	+	1.35478i
$292$		0			0
$293$	−2.72583	+	1.98043i	−0.159245	+	0.115698i	−0.664554	−	0.747240i	$-0.731378\pi$
$293$	−2.72583	+	1.98043i	−0.159245	+	0.115698i	0.505309	+	0.862938i	$0.331378\pi$
$294$		0			0
$295$	8.59307	+	26.4468i	0.500308	+	1.53979i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.86037	+	11.8810i	0.223251	+	0.687096i
$300$		0			0
$301$	17.9976	−	13.0760i	1.03736	−	0.753688i
$302$		0			0
$303$	−0.965093	+	2.97025i	−0.0554431	+	0.170636i
$304$		0			0
$305$	3.23607	+	2.35114i	0.185297	+	0.134626i
$306$		0			0
$307$	32.4924			1.85444			0.927220	−	0.374516i	$-0.122191\pi$
$307$	32.4924			1.85444			0.927220	+	0.374516i	$0.122191\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7.89098	−	5.73313i	−0.447456	−	0.325096i	0.341134	−	0.940015i	$-0.389189\pi$
$311$	−7.89098	−	5.73313i	−0.447456	−	0.325096i	−0.788591	+	0.614918i	$0.789189\pi$
$312$		0			0
$313$	−3.03077	+	9.32774i	−0.171309	+	0.527235i	−0.999446	−	0.0332913i	$-0.989401\pi$
$313$	−3.03077	+	9.32774i	−0.171309	+	0.527235i	0.828137	+	0.560526i	$0.189401\pi$
$314$		0			0
$315$	−5.05329	+	3.67143i	−0.284721	+	0.206862i
$316$		0			0
$317$	−4.38564	−	13.4976i	−0.246322	−	0.758102i	−0.995416	−	0.0956380i	$-0.969511\pi$
$317$	−4.38564	−	13.4976i	−0.246322	−	0.758102i	0.749094	−	0.662464i	$-0.230489\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−6.45132	−	19.8551i	−0.360077	−	1.10820i
$322$		0			0
$323$	6.47214	−	4.70228i	0.360119	−	0.261642i
$324$		0			0
$325$	12.1658	−	37.4424i	0.674836	−	2.07693i
$326$		0			0
$327$	−15.4709	−	11.2403i	−0.855544	−	0.621589i
$328$		0			0
$329$	−24.9848			−1.37746
$330$		0			0
$331$	34.9309			1.91997			0.959987	−	0.280044i	$-0.0903491\pi$
$331$	34.9309			1.91997			0.959987	+	0.280044i	$0.0903491\pi$
$332$		0			0
$333$	−3.43526	−	2.49586i	−0.188251	−	0.136772i
$334$		0			0
$335$	10.5233	−	32.3873i	0.574947	−	1.76950i
$336$		0			0
$337$	−13.5418	+	9.83872i	−0.737671	+	0.535949i	−0.891981	−	0.452073i	$-0.850685\pi$
$337$	−13.5418	+	9.83872i	−0.737671	+	0.535949i	0.154310	+	0.988023i	$0.450685\pi$
$338$		0			0
$339$	0.211571	+	0.651149i	0.0114910	+	0.0353656i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	4.09799	+	12.6123i	0.221271	+	0.681001i
$344$		0			0
$345$	10.9715	−	7.97128i	0.590687	−	0.429159i
$346$		0			0
$347$	−7.02662	+	21.6257i	−0.377209	+	1.16093i	0.564767	+	0.825250i	$0.308966\pi$
$347$	−7.02662	+	21.6257i	−0.377209	+	1.16093i	−0.941976	+	0.335680i	$0.891034\pi$
$348$		0			0
$349$	−26.0877	−	18.9538i	−1.39644	−	1.01458i	−0.995123	−	0.0986387i	$-0.968551\pi$
$349$	−26.0877	−	18.9538i	−1.39644	−	1.01458i	−0.401321	−	0.915937i	$-0.631449\pi$
$350$		0			0
$351$	28.4924			1.52081
$352$		0			0
$353$	−24.0540			−1.28026			−0.640132	−	0.768265i	$-0.721120\pi$
$353$	−24.0540			−1.28026			−0.640132	+	0.768265i	$0.721120\pi$
$354$		0			0
$355$	25.0236	+	18.1807i	1.32811	+	0.964932i
$356$		0			0
$357$	−3.01409	+	9.27640i	−0.159522	+	0.490959i
$358$		0			0
$359$	3.63445	−	2.64058i	0.191819	−	0.139364i	−0.487731	−	0.872994i	$-0.662175\pi$
$359$	3.63445	−	2.64058i	0.191819	−	0.139364i	0.679550	+	0.733630i	$0.262175\pi$
$360$		0			0
$361$	−0.927051	−	2.85317i	−0.0487922	−	0.150167i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.63839	−	17.3532i	−0.295127	−	0.908307i
$366$		0			0
$367$	−18.5515	+	13.4784i	−0.968379	+	0.703568i	−0.955081	−	0.296343i	$-0.904233\pi$
$367$	−18.5515	+	13.4784i	−0.968379	+	0.703568i	−0.0132971	+	0.999912i	$0.504233\pi$
$368$		0			0
$369$	−0.194892	+	0.599815i	−0.0101457	+	0.0312251i
$370$		0			0
$371$	30.9418	+	22.4806i	1.60642	+	1.16713i
$372$		0			0
$373$	8.24621			0.426973			0.213486	−	0.976946i	$-0.431518\pi$
$373$	8.24621			0.426973			0.213486	+	0.976946i	$0.431518\pi$
$374$		0			0
$375$	−14.9309			−0.771027
$376$		0			0
$377$	−21.2336	−	15.4271i	−1.09359	−	0.794538i
$378$		0			0
$379$	0.0594042	−	0.182827i	0.00305139	−	0.00939120i	−0.949519	−	0.313709i	$-0.898428\pi$
$379$	0.0594042	−	0.182827i	0.00305139	−	0.00939120i	0.952571	+	0.304318i	$0.0984284\pi$
$380$		0			0
$381$	−7.89098	+	5.73313i	−0.404267	+	0.293717i
$382$		0			0
$383$	−0.634713	−	1.95345i	−0.0324323	−	0.0998165i	0.933530	−	0.358499i	$-0.116712\pi$
$383$	−0.634713	−	1.95345i	−0.0324323	−	0.0998165i	−0.965962	+	0.258683i	$0.916712\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.23607	−	3.80423i	−0.0628329	−	0.193380i
$388$		0			0
$389$	−2.88136	+	2.09343i	−0.146091	+	0.106141i	−0.658430	−	0.752642i	$-0.728779\pi$
$389$	−2.88136	+	2.09343i	−0.146091	+	0.106141i	0.512339	+	0.858783i	$0.328779\pi$
$390$		0			0
$391$	−1.50704	+	4.63820i	−0.0762145	+	0.234564i
$392$		0			0
$393$	−16.8898	−	12.2711i	−0.851976	−	0.618997i
$394$		0			0
$395$	39.6155			1.99327
$396$		0			0
$397$	10.4924			0.526600			0.263300	−	0.964714i	$-0.415189\pi$
$397$	10.4924			0.526600			0.263300	+	0.964714i	$0.415189\pi$
$398$		0			0
$399$	−15.7820	−	11.4663i	−0.790086	−	0.574031i
$400$		0			0
$401$	9.42268	−	29.0000i	0.470546	−	1.44819i	−0.381325	−	0.924441i	$-0.624532\pi$
$401$	9.42268	−	29.0000i	0.470546	−	1.44819i	0.851871	−	0.523751i	$-0.175468\pi$
$402$		0			0
$403$	23.0509	−	16.7474i	1.14824	−	0.834249i
$404$		0			0
$405$	−7.70406	−	23.7107i	−0.382818	−	1.17819i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−6.95054	−	21.3916i	−0.343682	−	1.05774i	−0.962286	−	0.272041i	$-0.912301\pi$
$409$	−6.95054	−	21.3916i	−0.343682	−	1.05774i	0.618603	−	0.785703i	$-0.287699\pi$
$410$		0			0
$411$	−10.6605	+	7.74529i	−0.525843	+	0.382047i
$412$		0			0
$413$	−7.53522	+	23.1910i	−0.370784	+	1.14116i
$414$		0			0
$415$	2.52665	+	1.83572i	0.124028	+	0.0901117i
$416$		0			0
$417$	23.6155			1.15646
$418$		0			0
$419$	−32.4924			−1.58736			−0.793679	−	0.608336i	$-0.791837\pi$
$419$	−32.4924			−1.58736			−0.793679	+	0.608336i	$0.791837\pi$
$420$		0			0
$421$	−2.01641	−	1.46501i	−0.0982739	−	0.0714002i	0.537563	−	0.843223i	$-0.319345\pi$
$421$	−2.01641	−	1.46501i	−0.0982739	−	0.0714002i	−0.635837	+	0.771823i	$0.719345\pi$
$422$		0			0
$423$	−1.38823	+	4.27255i	−0.0674983	+	0.207738i
$424$		0			0
$425$	12.4340	−	9.03386i	0.603139	−	0.438206i
$426$		0			0
$427$	1.08390	+	3.33590i	0.0524536	+	0.161436i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.38150	−	25.7956i	−0.403723	−	1.24253i	−0.921957	−	0.387292i	$-0.873411\pi$
$431$	−8.38150	−	25.7956i	−0.403723	−	1.24253i	0.518234	−	0.855239i	$-0.326589\pi$
$432$		0			0
$433$	18.3523	−	13.3337i	0.881954	−	0.640777i	−0.0518137	−	0.998657i	$-0.516500\pi$
$433$	18.3523	−	13.3337i	0.881954	−	0.640777i	0.933768	+	0.357880i	$0.116500\pi$
$434$		0			0
$435$	−8.80464	+	27.0979i	−0.422150	+	1.29925i
$436$		0			0
$437$	−7.89098	−	5.73313i	−0.377477	−	0.274253i
$438$		0			0
$439$	−4.49242			−0.214412			−0.107206	−	0.994237i	$-0.534190\pi$
$439$	−4.49242			−0.214412			−0.107206	+	0.994237i	$0.534190\pi$
$440$		0			0
$441$	−1.54640			−0.0736380
$442$		0			0
$443$	−9.15430	−	6.65099i	−0.434934	−	0.315998i	0.348685	−	0.937240i	$-0.386628\pi$
$443$	−9.15430	−	6.65099i	−0.434934	−	0.315998i	−0.783619	+	0.621242i	$0.786628\pi$
$444$		0			0
$445$	2.95468	−	9.09358i	0.140065	−	0.431077i
$446$		0			0
$447$	−5.36434	+	3.89742i	−0.253724	+	0.184342i
$448$		0			0
$449$	−11.2935	−	34.7577i	−0.532971	−	1.64032i	−0.747992	−	0.663707i	$-0.768982\pi$
$449$	−11.2935	−	34.7577i	−0.532971	−	1.64032i	0.215021	−	0.976609i	$-0.431018\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−4.52113	−	13.9146i	−0.212421	−	0.653765i
$454$		0			0
$455$	46.1017	−	33.4949i	2.16128	−	1.57026i
$456$		0			0
$457$	−7.37368	+	22.6939i	−0.344926	+	1.06157i	0.616697	+	0.787201i	$0.288470\pi$
$457$	−7.37368	+	22.6939i	−0.344926	+	1.06157i	−0.961623	+	0.274373i	$0.911530\pi$
$458$		0			0
$459$	8.99878	+	6.53800i	0.420027	+	0.305168i
$460$		0			0
$461$	−1.12311			−0.0523082			−0.0261541	−	0.999658i	$-0.508326\pi$
$461$	−1.12311			−0.0523082			−0.0261541	+	0.999658i	$0.508326\pi$
$462$		0			0
$463$	15.3153			0.711764			0.355882	−	0.934531i	$-0.384180\pi$
$463$	15.3153			0.711764			0.355882	+	0.934531i	$0.384180\pi$
$464$		0			0
$465$	−25.0236	−	18.1807i	−1.16044	−	0.843110i
$466$		0			0
$467$	8.74524	−	26.9151i	0.404681	−	1.24548i	−0.516480	−	0.856299i	$-0.672758\pi$
$467$	8.74524	−	26.9151i	0.404681	−	1.24548i	0.921161	−	0.389182i	$-0.127242\pi$
$468$		0			0
$469$	24.1587	−	17.5523i	1.11554	−	0.810489i
$470$		0			0
$471$	2.14176	+	6.59165i	0.0986869	+	0.303727i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	9.49876	+	29.2342i	0.435833	+	1.34136i
$476$		0			0
$477$	5.56352	−	4.04214i	0.254736	−	0.185077i
$478$		0			0
$479$	4.94427	−	15.2169i	0.225910	−	0.695278i	−0.772288	−	0.635272i	$-0.780888\pi$
$479$	4.94427	−	15.2169i	0.225910	−	0.695278i	0.998198	−	0.0600061i	$-0.0191120\pi$
$480$		0			0
$481$	31.3402	+	22.7700i	1.42899	+	1.03822i
$482$		0			0
$483$	11.8920			0.541107
$484$		0			0
$485$	55.4233			2.51664
$486$		0			0
$487$	−12.0793	−	8.77615i	−0.547367	−	0.397685i	0.279447	−	0.960161i	$-0.409849\pi$
$487$	−12.0793	−	8.77615i	−0.547367	−	0.397685i	−0.826814	+	0.562476i	$0.809849\pi$
$488$		0			0
$489$	1.93019	−	5.94050i	0.0872860	−	0.268639i
$490$		0			0
$491$	11.1270	−	8.08427i	0.502157	−	0.364838i	−0.307683	−	0.951489i	$-0.599554\pi$
$491$	11.1270	−	8.08427i	0.502157	−	0.364838i	0.809840	+	0.586651i	$0.199554\pi$
$492$		0			0
$493$	−3.16625	−	9.74473i	−0.142601	−	0.438880i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8.38150	+	25.7956i	0.375962	+	1.15709i
$498$		0			0
$499$	23.4492	−	17.0369i	1.04973	−	0.762675i	0.0775704	−	0.996987i	$-0.475284\pi$
$499$	23.4492	−	17.0369i	1.04973	−	0.762675i	0.972161	+	0.234312i	$0.0752838\pi$
$500$		0			0
$501$	3.86037	−	11.8810i	0.172469	−	0.530804i
$502$		0			0
$503$	−25.5775	−	18.5831i	−1.14044	−	0.828581i	−0.153263	−	0.988185i	$-0.548978\pi$
$503$	−25.5775	−	18.5831i	−1.14044	−	0.828581i	−0.987181	+	0.159604i	$0.948978\pi$
$504$		0			0
$505$	7.12311			0.316974
$506$		0			0
$507$	−20.6847			−0.918638
$508$		0			0
$509$	14.8052	+	10.7566i	0.656227	+	0.476777i	0.865387	−	0.501105i	$-0.167073\pi$
$509$	14.8052	+	10.7566i	0.656227	+	0.476777i	−0.209160	+	0.977882i	$0.567073\pi$
$510$		0			0
$511$	4.94427	−	15.2169i	0.218722	−	0.673156i
$512$		0			0
$513$	−17.9976	+	13.0760i	−0.794612	+	0.577319i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	5.90936	+	18.1872i	0.259392	+	0.798327i
$520$		0			0
$521$	−1.06413	+	0.773138i	−0.0466205	+	0.0338718i	−0.610851	−	0.791745i	$-0.709173\pi$
$521$	−1.06413	+	0.773138i	−0.0466205	+	0.0338718i	0.564231	+	0.825617i	$0.309173\pi$
$522$		0			0
$523$	3.70820	−	11.4127i	0.162148	−	0.499042i	−0.836666	−	0.547713i	$-0.815499\pi$
$523$	3.70820	−	11.4127i	0.162148	−	0.499042i	0.998815	+	0.0486712i	$0.0154987\pi$
$524$		0			0
$525$	−30.3197	−	22.0286i	−1.32326	−	0.961406i
$526$		0			0
$527$	11.1231			0.484530
$528$		0			0
$529$	−17.0540			−0.741477
$530$		0			0
$531$	3.54711	+	2.57713i	0.153932	+	0.111838i
$532$		0			0
$533$	1.77802	−	5.47218i	0.0770145	−	0.237026i
$534$		0			0
$535$	−38.5218	+	27.9877i	−1.66544	+	1.21001i
$536$		0			0
$537$	3.10685	+	9.56190i	0.134070	+	0.412626i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	7.37368	+	22.6939i	0.317019	+	0.975685i	0.974915	+	0.222576i	$0.0714467\pi$
$541$	7.37368	+	22.6939i	0.317019	+	0.975685i	−0.657896	+	0.753109i	$0.728553\pi$
$542$		0			0
$543$	−1.66170	+	1.20730i	−0.0713104	+	0.0518100i
$544$		0			0
$545$	−13.4779	+	41.4808i	−0.577331	+	1.77684i
$546$		0			0
$547$	−34.1779	−	24.8317i	−1.46134	−	1.06173i	−0.983011	−	0.183547i	$-0.941242\pi$
$547$	−34.1779	−	24.8317i	−1.46134	−	1.06173i	−0.478331	−	0.878180i	$-0.658758\pi$
$548$		0			0
$549$	0.630683			0.0269169
$550$		0			0
$551$	20.4924			0.873007
$552$		0			0
$553$	28.1041	+	20.4189i	1.19511	+	0.868298i
$554$		0			0
$555$	12.9954	−	39.9957i	0.551624	−	1.69772i
$556$		0			0
$557$	−3.03688	+	2.20642i	−0.128677	+	0.0934891i	−0.650262	−	0.759710i	$-0.725341\pi$
$557$	−3.03688	+	2.20642i	−0.128677	+	0.0934891i	0.521585	+	0.853199i	$0.325341\pi$
$558$		0			0
$559$	11.2768	+	34.7064i	0.476957	+	1.46792i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7.56857	−	23.2937i	−0.318977	−	0.981711i	−0.974086	−	0.226177i	$-0.927377\pi$
$563$	−7.56857	−	23.2937i	−0.318977	−	0.981711i	0.655109	−	0.755535i	$-0.272623\pi$
$564$		0			0
$565$	1.26332	−	0.917858i	0.0531484	−	0.0386146i
$566$		0			0
$567$	6.75565	−	20.7917i	0.283711	−	0.873171i
$568$		0			0
$569$	−21.7439	−	15.7978i	−0.911550	−	0.662280i	0.0298562	−	0.999554i	$-0.490495\pi$
$569$	−21.7439	−	15.7978i	−0.911550	−	0.662280i	−0.941406	+	0.337274i	$0.890495\pi$
$570$		0			0
$571$	−16.4924			−0.690186			−0.345093	−	0.938568i	$-0.612153\pi$
$571$	−16.4924			−0.690186			−0.345093	+	0.938568i	$0.612153\pi$
$572$		0			0
$573$	16.3002			0.680950
$574$		0			0
$575$	−15.1599	−	11.0143i	−0.632210	−	0.459328i
$576$		0			0
$577$	4.80878	−	14.7999i	0.200192	−	0.616129i	−0.799684	−	0.600421i	$-0.795000\pi$
$577$	4.80878	−	14.7999i	0.200192	−	0.616129i	0.999877	−	0.0157079i	$-0.00500018\pi$
$578$		0			0
$579$	11.5254	−	8.37371i	0.478980	−	0.348000i
$580$		0			0
$581$	0.846284	+	2.60460i	0.0351098	+	0.108057i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−3.16625	−	9.74473i	−0.130908	−	0.402895i
$586$		0			0
$587$	19.8148	−	14.3963i	0.817844	−	0.594198i	−0.0982503	−	0.995162i	$-0.531325\pi$
$587$	19.8148	−	14.3963i	0.817844	−	0.594198i	0.916094	+	0.400963i	$0.131325\pi$
$588$		0			0
$589$	−6.87446	+	21.1574i	−0.283257	+	0.871776i
$590$		0			0
$591$	18.3086	+	13.3020i	0.753115	+	0.547170i
$592$		0			0
$593$	−3.36932			−0.138361			−0.0691806	−	0.997604i	$-0.522038\pi$
$593$	−3.36932			−0.138361			−0.0691806	+	0.997604i	$0.522038\pi$
$594$		0			0
$595$	22.2462			0.912006
$596$		0			0
$597$	−15.7820	−	11.4663i	−0.645913	−	0.469283i
$598$		0			0
$599$	−4.94427	+	15.2169i	−0.202017	+	0.621746i	0.797805	+	0.602915i	$0.205994\pi$
$599$	−4.94427	+	15.2169i	−0.202017	+	0.621746i	−0.999823	+	0.0188306i	$0.994006\pi$
$600$		0			0
$601$	3.03688	−	2.20642i	0.123877	−	0.0900018i	−0.524122	−	0.851643i	$-0.675606\pi$
$601$	3.03688	−	2.20642i	0.123877	−	0.0900018i	0.647999	+	0.761642i	$0.275606\pi$
$602$		0			0
$603$	−1.65921	−	5.10652i	−0.0675683	−	0.207954i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	14.1721	+	43.6171i	0.575226	+	1.77036i	0.635408	+	0.772176i	$0.280832\pi$
$607$	14.1721	+	43.6171i	0.575226	+	1.77036i	−0.0601823	+	0.998187i	$0.519168\pi$
$608$		0			0
$609$	−20.2132	+	14.6857i	−0.819079	+	0.595096i
$610$		0			0
$611$	12.6650	−	38.9789i	0.512372	−	1.57692i
$612$		0			0
$613$	9.59635	+	6.97216i	0.387593	+	0.281603i	0.764468	−	0.644661i	$-0.223002\pi$
$613$	9.59635	+	6.97216i	0.387593	+	0.281603i	−0.376876	+	0.926264i	$0.623002\pi$
$614$		0			0
$615$	−6.24621			−0.251872
$616$		0			0
$617$	−2.49242			−0.100341			−0.0501706	−	0.998741i	$-0.515976\pi$
$617$	−2.49242			−0.100341			−0.0501706	+	0.998741i	$0.515976\pi$
$618$		0			0
$619$	−15.3154	−	11.1273i	−0.615578	−	0.447243i	0.235796	−	0.971802i	$-0.424230\pi$
$619$	−15.3154	−	11.1273i	−0.615578	−	0.447243i	−0.851374	+	0.524559i	$0.824230\pi$
$620$		0			0
$621$	4.19075	−	12.8978i	0.168169	−	0.517571i
$622$		0			0
$623$	6.78318	−	4.92827i	0.271762	−	0.197447i
$624$		0			0
$625$	−1.35019	−	4.15547i	−0.0540077	−	0.166219i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4.67330	+	14.3829i	0.186337	+	0.573485i
$630$		0			0
$631$	34.0224	−	24.7187i	1.35441	−	0.984036i	0.355631	−	0.934626i	$-0.384266\pi$
$631$	34.0224	−	24.7187i	1.35441	−	0.984036i	0.998779	−	0.0494097i	$-0.0157340\pi$
$632$		0			0
$633$	4.09799	−	12.6123i	0.162880	−	0.501294i
$634$		0			0
$635$	17.9976	+	13.0760i	0.714212	+	0.518905i
$636$		0			0
$637$	14.1080			0.558977
$638$		0			0
$639$	4.87689			0.192927
$640$		0			0
$641$	37.4576	+	27.2146i	1.47949	+	1.07491i	0.977721	+	0.209908i	$0.0673163\pi$
$641$	37.4576	+	27.2146i	1.47949	+	1.07491i	0.501767	+	0.865003i	$0.332684\pi$
$642$		0			0
$643$	−2.83587	+	8.72792i	−0.111836	+	0.344196i	−0.991274	−	0.131818i	$-0.957919\pi$
$643$	−2.83587	+	8.72792i	−0.111836	+	0.344196i	0.879438	+	0.476013i	$0.157919\pi$
$644$		0			0
$645$	32.0496	−	23.2854i	1.26195	−	0.916863i
$646$		0			0
$647$	4.19075	+	12.8978i	0.164755	+	0.507065i	0.999018	−	0.0443021i	$-0.0141064\pi$
$647$	4.19075	+	12.8978i	0.164755	+	0.507065i	−0.834263	+	0.551367i	$0.814106\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−8.38150	−	25.7956i	−0.328497	−	1.01101i
$652$		0			0
$653$	−28.4589	+	20.6766i	−1.11368	+	0.809137i	−0.983239	−	0.182319i	$-0.941640\pi$
$653$	−28.4589	+	20.6766i	−1.11368	+	0.809137i	−0.130442	+	0.991456i	$0.541640\pi$
$654$		0			0
$655$	−14.7140	+	45.2851i	−0.574924	+	1.76943i
$656$		0			0
$657$	−2.32746	−	1.69100i	−0.0908027	−	0.0659720i
$658$		0			0
$659$	11.6155			0.452477			0.226238	−	0.974072i	$-0.427357\pi$
$659$	11.6155			0.452477			0.226238	+	0.974072i	$0.427357\pi$
$660$		0			0
$661$	41.8078			1.62613			0.813067	−	0.582170i	$-0.197796\pi$
$661$	41.8078			1.62613			0.813067	+	0.582170i	$0.197796\pi$
$662$		0			0
$663$	−12.9443	−	9.40456i	−0.502714	−	0.365243i
$664$		0			0
$665$	−13.7489	+	42.3148i	−0.533160	+	1.64090i
$666$		0			0
$667$	−10.1066	+	7.34286i	−0.391328	+	0.284317i
$668$		0			0
$669$	5.69779	+	17.5360i	0.220289	+	0.677981i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−10.2690	−	31.6046i	−0.395839	−	1.21827i	−0.928306	−	0.371817i	$-0.878735\pi$
$673$	−10.2690	−	31.6046i	−0.395839	−	1.21827i	0.532467	−	0.846451i	$-0.321265\pi$
$674$		0			0
$675$	−34.5763	+	25.1211i	−1.33084	+	0.966913i
$676$		0			0
$677$	6.40859	−	19.7236i	0.246302	−	0.758040i	−0.749117	−	0.662437i	$-0.769522\pi$
$677$	6.40859	−	19.7236i	0.246302	−	0.758040i	0.995420	−	0.0956028i	$-0.0304779\pi$
$678$		0			0
$679$	39.3185	+	28.5666i	1.50891	+	1.09628i
$680$		0			0
$681$	36.1080			1.38366
$682$		0			0
$683$	6.73863			0.257847			0.128923	−	0.991655i	$-0.458848\pi$
$683$	6.73863			0.257847			0.128923	+	0.991655i	$0.458848\pi$
$684$		0			0
$685$	24.3142	+	17.6653i	0.928997	+	0.674956i
$686$		0			0
$687$	−7.08603	+	21.8086i	−0.270349	+	0.832048i
$688$		0			0
$689$	−50.7566	+	36.8768i	−1.93367	+	1.40490i
$690$		0			0
$691$	−3.07349	−	9.45923i	−0.116921	−	0.359846i	0.875422	−	0.483360i	$-0.160584\pi$
$691$	−3.07349	−	9.45923i	−0.116921	−	0.359846i	−0.992343	+	0.123514i	$0.960584\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−16.6442	−	51.2256i	−0.631350	−	1.94310i
$696$		0			0
$697$	1.81722	−	1.32029i	0.0688322	−	0.0500095i
$698$		0			0
$699$	−3.55604	+	10.9444i	−0.134502	+	0.413954i
$700$		0			0
$701$	40.8492	+	29.6787i	1.54285	+	1.12095i	0.948514	+	0.316735i	$0.102587\pi$
$701$	40.8492	+	29.6787i	1.54285	+	1.12095i	0.594340	+	0.804214i	$0.297413\pi$
$702$		0			0
$703$	−30.2462			−1.14076
$704$		0			0
$705$	−44.4924			−1.67568
$706$		0			0
$707$	5.05329	+	3.67143i	0.190049	+	0.138078i
$708$		0			0
$709$	0.677438	−	2.08494i	0.0254417	−	0.0783016i	−0.937529	−	0.347906i	$-0.886893\pi$
$709$	0.677438	−	2.08494i	0.0254417	−	0.0783016i	0.962971	+	0.269604i	$0.0868929\pi$
$710$		0			0
$711$	5.05329	−	3.67143i	0.189513	−	0.137689i
$712$		0			0
$713$	−4.19075	−	12.8978i	−0.156945	−	0.483027i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−2.35333	−	7.24280i	−0.0878866	−	0.270487i
$718$		0			0
$719$	−28.6580	+	20.8213i	−1.06876	+	0.776503i	−0.975690	−	0.219156i	$-0.929670\pi$
$719$	−28.6580	+	20.8213i	−1.06876	+	0.776503i	−0.0930749	+	0.995659i	$0.529670\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−36.7919	−	26.7309i	−1.36830	−	0.994132i
$724$		0			0
$725$	39.3693			1.46214
$726$		0			0
$727$	−23.3153			−0.864718			−0.432359	−	0.901702i	$-0.642319\pi$
$727$	−23.3153			−0.864718			−0.432359	+	0.901702i	$0.642319\pi$
$728$		0			0
$729$	−24.2582	−	17.6246i	−0.898454	−	0.652765i
$730$		0			0
$731$	−4.40232	+	13.5490i	−0.162826	+	0.501126i
$732$		0			0
$733$	0.908612	−	0.660145i	0.0335603	−	0.0243830i	−0.570879	−	0.821034i	$-0.693397\pi$
$733$	0.908612	−	0.660145i	0.0335603	−	0.0243830i	0.604439	+	0.796651i	$0.293397\pi$
$734$		0			0
$735$	−4.73270	−	14.5658i	−0.174568	−	0.537266i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.812926	+	2.50193i	0.0299040	+	0.0920349i	0.964895	−	0.262638i	$-0.0845924\pi$
$739$	0.812926	+	2.50193i	0.0299040	+	0.0920349i	−0.934991	+	0.354673i	$0.884592\pi$
$740$		0			0
$741$	25.8885	−	18.8091i	0.951039	−	0.690971i
$742$		0			0
$743$	−3.31842	+	10.2130i	−0.121741	+	0.374680i	−0.993293	−	0.115622i	$-0.963114\pi$
$743$	−3.31842	+	10.2130i	−0.121741	+	0.374680i	0.871552	+	0.490303i	$0.163114\pi$
$744$		0			0
$745$	12.2348	+	8.88914i	0.448250	+	0.325673i
$746$		0			0
$747$	0.492423			0.0180168
$748$		0			0
$749$	−41.7538			−1.52565
$750$		0			0
$751$	−4.49939	−	3.26900i	−0.164185	−	0.119287i	0.502659	−	0.864485i	$-0.332355\pi$
$751$	−4.49939	−	3.26900i	−0.164185	−	0.119287i	−0.666844	+	0.745197i	$0.732355\pi$
$752$		0			0
$753$	−0.753522	+	2.31910i	−0.0274599	+	0.0845127i
$754$		0			0
$755$	−26.9963	+	19.6140i	−0.982498	+	0.713826i
$756$		0			0
$757$	4.86819	+	14.9827i	0.176937	+	0.544557i	0.999717	−	0.0238039i	$-0.00757773\pi$
$757$	4.86819	+	14.9827i	0.176937	+	0.544557i	−0.822779	+	0.568361i	$0.807578\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.58313	−	4.87236i	−0.0573883	−	0.176623i	0.918253	−	0.395993i	$-0.129600\pi$
$761$	−1.58313	−	4.87236i	−0.0573883	−	0.176623i	−0.975642	+	0.219370i	$0.929600\pi$
$762$		0			0
$763$	−30.9418	+	22.4806i	−1.12017	+	0.813851i
$764$		0			0
$765$	1.23607	−	3.80423i	0.0446901	−	0.137542i
$766$		0			0
$767$	−32.3607	−	23.5114i	−1.16848	−	0.848948i
$768$		0			0
$769$	−25.6155			−0.923720			−0.461860	−	0.886953i	$-0.652818\pi$
$769$	−25.6155			−0.923720			−0.461860	+	0.886953i	$0.652818\pi$
$770$		0			0
$771$	−18.3542			−0.661009
$772$		0			0
$773$	−32.9582	−	23.9456i	−1.18543	−	0.861262i	−0.192652	−	0.981267i	$-0.561709\pi$
$773$	−32.9582	−	23.9456i	−1.18543	−	0.861262i	−0.992773	+	0.120005i	$0.961709\pi$
$774$		0			0
$775$	−13.2070	+	40.6469i	−0.474408	+	1.46008i
$776$		0			0
$777$	29.8340	−	21.6757i	1.07029	−	0.777611i
$778$		0			0
$779$	1.38823	+	4.27255i	0.0497387	+	0.153080i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	8.80464	+	27.0979i	0.314652	+	0.968400i
$784$		0			0
$785$	12.7887	−	9.29157i	0.456450	−	0.331630i
$786$		0			0
$787$	9.19443	−	28.2975i	0.327746	−	1.00870i	−0.642440	−	0.766336i	$-0.722078\pi$
$787$	9.19443	−	28.2975i	0.327746	−	1.00870i	0.970186	−	0.242362i	$-0.0779222\pi$
$788$		0			0
$789$	−24.1587	−	17.5523i	−0.860071	−	0.624878i

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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.i (of order \(5\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.26332 + 0.917858i) q^{3} +(1.10058 - 3.38724i) q^{5} +(2.52665 - 1.83572i) q^{7} +(-0.173529 - 0.534068i) q^{9} +O(q^{10})\)	\(q+(1.26332 + 0.917858i) q^{3} +(1.10058 - 3.38724i) q^{5} +(2.52665 - 1.83572i) q^{7} +(-0.173529 - 0.534068i) q^{9} +(1.58313 + 4.87236i) q^{13} +(4.49939 - 3.26900i) q^{15} +(-0.618034 + 1.90211i) q^{17} +(-3.23607 - 2.35114i) q^{19} +4.87689 q^{21} +2.43845 q^{23} +(-6.21702 - 4.51693i) q^{25} +(1.71861 - 5.28935i) q^{27} +(-4.14468 + 3.01129i) q^{29} +(-1.71861 - 5.28935i) q^{31} +(-3.43723 - 10.5787i) q^{35} +(6.11742 - 4.44457i) q^{37} +(-2.47214 + 7.60845i) q^{39} +(-0.908612 - 0.660145i) q^{41} +7.12311 q^{43} -2.00000 q^{45} +(-6.47214 - 4.70228i) q^{47} +(0.850968 - 2.61901i) q^{49} +(-2.52665 + 1.83572i) q^{51} +(3.78429 + 11.6468i) q^{53} +(-1.93019 - 5.94050i) q^{57} +(-6.31661 + 4.58929i) q^{59} +(-0.347059 + 1.06814i) q^{61} +(-1.41884 - 1.03085i) q^{63} +18.2462 q^{65} +9.56155 q^{67} +(3.08055 + 2.23815i) q^{69} +(-2.68371 + 8.25960i) q^{71} +(4.14468 - 3.01129i) q^{73} +(-3.70820 - 11.4127i) q^{75} +(3.43723 + 10.5787i) q^{79} +(5.66312 - 4.11450i) q^{81} +(-0.270975 + 0.833976i) q^{83} +(5.76271 + 4.18686i) q^{85} -8.00000 q^{87} +2.68466 q^{89} +(12.9443 + 9.40456i) q^{91} +(2.68371 - 8.25960i) q^{93} +(-11.5254 + 8.37371i) q^{95} +(4.80878 + 14.7999i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{3} - 3 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) 8 * q - q^3 - 3 * q^5 - 2 * q^7 - 3 * q^9	\( 8 q - q^{3} - 3 q^{5} - 2 q^{7} - 3 q^{9} - 2 q^{13} + 7 q^{15} + 4 q^{17} - 8 q^{19} + 72 q^{21} + 36 q^{23} - 3 q^{25} - 7 q^{27} - 2 q^{29} + 7 q^{31} + 14 q^{35} + 11 q^{37} + 16 q^{39} + 6 q^{41} + 24 q^{43} - 16 q^{45} - 16 q^{47} - 22 q^{49} + 2 q^{51} - 8 q^{53} - 4 q^{57} + 5 q^{59} - 6 q^{61} - 20 q^{63} + 80 q^{65} + 60 q^{67} - 13 q^{69} + 5 q^{71} + 2 q^{73} + 24 q^{75} - 14 q^{79} + 14 q^{81} + 10 q^{83} + 6 q^{85} - 64 q^{87} - 28 q^{89} + 32 q^{91} - 5 q^{93} - 12 q^{95} - 27 q^{97}+O(q^{100}) \) 8 * q - q^3 - 3 * q^5 - 2 * q^7 - 3 * q^9 - 2 * q^13 + 7 * q^15 + 4 * q^17 - 8 * q^19 + 72 * q^21 + 36 * q^23 - 3 * q^25 - 7 * q^27 - 2 * q^29 + 7 * q^31 + 14 * q^35 + 11 * q^37 + 16 * q^39 + 6 * q^41 + 24 * q^43 - 16 * q^45 - 16 * q^47 - 22 * q^49 + 2 * q^51 - 8 * q^53 - 4 * q^57 + 5 * q^59 - 6 * q^61 - 20 * q^63 + 80 * q^65 + 60 * q^67 - 13 * q^69 + 5 * q^71 + 2 * q^73 + 24 * q^75 - 14 * q^79 + 14 * q^81 + 10 * q^83 + 6 * q^85 - 64 * q^87 - 28 * q^89 + 32 * q^91 - 5 * q^93 - 12 * q^95 - 27 * q^97

Introduction

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