LMFDB - Embedded newform 5070.2.b.q.1351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.b (of order $2$, degree $1$, 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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.q.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +2.82843i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +2.82843i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.65685i q^{11} -1.00000 q^{12} +2.82843 q^{14} -1.00000i q^{15} +1.00000 q^{16} +4.82843 q^{17} -1.00000i q^{18} +2.82843i q^{19} +1.00000i q^{20} +2.82843i q^{21} -5.65685 q^{22} -8.48528 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -2.82843i q^{28} -3.17157 q^{29} -1.00000 q^{30} -4.00000i q^{31} -1.00000i q^{32} -5.65685i q^{33} -4.82843i q^{34} +2.82843 q^{35} -1.00000 q^{36} -0.343146i q^{37} +2.82843 q^{38} +1.00000 q^{40} -3.65685i q^{41} +2.82843 q^{42} +1.65685 q^{43} +5.65685i q^{44} -1.00000i q^{45} +8.48528i q^{46} -8.00000i q^{47} +1.00000 q^{48} -1.00000 q^{49} +1.00000i q^{50} +4.82843 q^{51} -9.31371 q^{53} -1.00000i q^{54} -5.65685 q^{55} -2.82843 q^{56} +2.82843i q^{57} +3.17157i q^{58} -13.6569i q^{59} +1.00000i q^{60} +6.00000 q^{61} -4.00000 q^{62} +2.82843i q^{63} -1.00000 q^{64} -5.65685 q^{66} +5.65685i q^{67} -4.82843 q^{68} -8.48528 q^{69} -2.82843i q^{70} -5.65685i q^{71} +1.00000i q^{72} +2.48528i q^{73} -0.343146 q^{74} -1.00000 q^{75} -2.82843i q^{76} +16.0000 q^{77} +13.6569 q^{79} -1.00000i q^{80} +1.00000 q^{81} -3.65685 q^{82} -17.6569i q^{83} -2.82843i q^{84} -4.82843i q^{85} -1.65685i q^{86} -3.17157 q^{87} +5.65685 q^{88} -4.34315i q^{89} -1.00000 q^{90} +8.48528 q^{92} -4.00000i q^{93} -8.00000 q^{94} +2.82843 q^{95} -1.00000i q^{96} +8.82843i q^{97} +1.00000i q^{98} -5.65685i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{10} - 4 q^{12} + 4 q^{16} + 8 q^{17} - 4 q^{25} + 4 q^{27} - 24 q^{29} - 4 q^{30} - 4 q^{36} + 4 q^{40} - 16 q^{43} + 4 q^{48} - 4 q^{49} + 8 q^{51} + 8 q^{53} + 24 q^{61} - 16 q^{62} - 4 q^{64} - 8 q^{68} - 24 q^{74} - 4 q^{75} + 64 q^{77} + 32 q^{79} + 4 q^{81} + 8 q^{82} - 24 q^{87} - 4 q^{90} - 32 q^{94}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^10 - 4 * q^12 + 4 * q^16 + 8 * q^17 - 4 * q^25 + 4 * q^27 - 24 * q^29 - 4 * q^30 - 4 * q^36 + 4 * q^40 - 16 * q^43 + 4 * q^48 - 4 * q^49 + 8 * q^51 + 8 * q^53 + 24 * q^61 - 16 * q^62 - 4 * q^64 - 8 * q^68 - 24 * q^74 - 4 * q^75 + 64 * q^77 + 32 * q^79 + 4 * q^81 + 8 * q^82 - 24 * q^87 - 4 * q^90 - 32 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$.

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	− 1.00000i	− 0.408248i
$7$	2.82843i	1.06904i	0.845154	+	0.534522i	$0.179509\pi$
$7$	2.82843i	1.06904i	−0.845154	+	0.534522i	$0.820491\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	− 5.65685i	− 1.70561i	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	− 5.65685i	− 1.70561i	0.522233	−	0.852803i	$-0.325099\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	2.82843	0.755929
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	4.82843	1.17107	0.585533	−	0.810649i	$-0.300885\pi$
$17$	4.82843	1.17107	0.585533	+	0.810649i	$0.300885\pi$
$18$	− 1.00000i	− 0.235702i
$19$	2.82843i	0.648886i	0.945905	+	0.324443i	$0.105177\pi$
$19$	2.82843i	0.648886i	−0.945905	+	0.324443i	$0.894823\pi$
$20$	1.00000i	0.223607i
$21$	2.82843i	0.617213i
$22$	−5.65685	−1.20605
$23$	−8.48528	−1.76930	−0.884652	−	0.466252i	$-0.845604\pi$
$23$	−8.48528	−1.76930	−0.884652	+	0.466252i	$0.845604\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	− 2.82843i	− 0.534522i
$29$	−3.17157	−0.588946	−0.294473	−	0.955660i	$-0.595144\pi$
$29$	−3.17157	−0.588946	−0.294473	+	0.955660i	$0.595144\pi$
$30$	−1.00000	−0.182574
$31$	− 4.00000i	− 0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$	− 4.00000i	− 0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 5.65685i	− 0.984732i
$34$	− 4.82843i	− 0.828068i
$35$	2.82843	0.478091
$36$	−1.00000	−0.166667
$37$	− 0.343146i	− 0.0564128i	−0.999602	−	0.0282064i	$-0.991020\pi$
$37$	− 0.343146i	− 0.0564128i	0.999602	−	0.0282064i	$-0.00897957\pi$
$38$	2.82843	0.458831
$39$	0	0
$40$	1.00000	0.158114
$41$	− 3.65685i	− 0.571105i	−0.958363	−	0.285552i	$-0.907823\pi$
$41$	− 3.65685i	− 0.571105i	0.958363	−	0.285552i	$-0.0921770\pi$
$42$	2.82843	0.436436
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	5.65685i	0.852803i
$45$	− 1.00000i	− 0.149071i
$46$	8.48528i	1.25109i
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	1.00000	0.144338
$49$	−1.00000	−0.142857
$50$	1.00000i	0.141421i
$51$	4.82843	0.676115
$52$	0	0
$53$	−9.31371	−1.27934	−0.639668	−	0.768651i	$-0.720928\pi$
$53$	−9.31371	−1.27934	−0.639668	+	0.768651i	$0.720928\pi$
$54$	− 1.00000i	− 0.136083i
$55$	−5.65685	−0.762770
$56$	−2.82843	−0.377964
$57$	2.82843i	0.374634i
$58$	3.17157i	0.416448i
$59$	− 13.6569i	− 1.77797i	−0.457935	−	0.888985i	$-0.651411\pi$
$59$	− 13.6569i	− 1.77797i	0.457935	−	0.888985i	$-0.348589\pi$
$60$	1.00000i	0.129099i
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−4.00000	−0.508001
$63$	2.82843i	0.356348i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−5.65685	−0.696311
$67$	5.65685i	0.691095i	0.938401	+	0.345547i	$0.112307\pi$
$67$	5.65685i	0.691095i	−0.938401	+	0.345547i	$0.887693\pi$
$68$	−4.82843	−0.585533
$69$	−8.48528	−1.02151
$70$	− 2.82843i	− 0.338062i
$71$	− 5.65685i	− 0.671345i	−0.941979	−	0.335673i	$-0.891036\pi$
$71$	− 5.65685i	− 0.671345i	0.941979	−	0.335673i	$-0.108964\pi$
$72$	1.00000i	0.117851i
$73$	2.48528i	0.290880i	0.989367	+	0.145440i	$0.0464598\pi$
$73$	2.48528i	0.290880i	−0.989367	+	0.145440i	$0.953540\pi$
$74$	−0.343146	−0.0398899
$75$	−1.00000	−0.115470
$76$	− 2.82843i	− 0.324443i
$77$	16.0000	1.82337
$78$	0	0
$79$	13.6569	1.53652	0.768258	−	0.640140i	$-0.221124\pi$
$79$	13.6569	1.53652	0.768258	+	0.640140i	$0.221124\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	−3.65685	−0.403832
$83$	− 17.6569i	− 1.93809i	−0.246881	−	0.969046i	$-0.579406\pi$
$83$	− 17.6569i	− 1.93809i	0.246881	−	0.969046i	$-0.420594\pi$
$84$	− 2.82843i	− 0.308607i
$85$	− 4.82843i	− 0.523716i
$86$	− 1.65685i	− 0.178663i
$87$	−3.17157	−0.340028
$88$	5.65685	0.603023
$89$	− 4.34315i	− 0.460373i	−0.973147	−	0.230186i	$-0.926066\pi$
$89$	− 4.34315i	− 0.460373i	0.973147	−	0.230186i	$-0.0739335\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	8.48528	0.884652
$93$	− 4.00000i	− 0.414781i
$94$	−8.00000	−0.825137
$95$	2.82843	0.290191
$96$	− 1.00000i	− 0.102062i
$97$	8.82843i	0.896391i	0.893936	+	0.448195i	$0.147933\pi$
$97$	8.82843i	0.896391i	−0.893936	+	0.448195i	$0.852067\pi$
$98$	1.00000i	0.101015i
$99$	− 5.65685i	− 0.568535i
$100$	1.00000	0.100000
$101$	12.1421	1.20819	0.604094	−	0.796913i	$-0.293535\pi$
$101$	12.1421	1.20819	0.604094	+	0.796913i	$0.293535\pi$
$102$	− 4.82843i	− 0.478086i
$103$	−9.65685	−0.951518	−0.475759	−	0.879576i	$-0.657827\pi$
$103$	−9.65685	−0.951518	−0.475759	+	0.879576i	$0.657827\pi$
$104$	0	0
$105$	2.82843	0.276026
$106$	9.31371i	0.904627i
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	−1.00000	−0.0962250
$109$	− 3.17157i	− 0.303782i	−0.988397	−	0.151891i	$-0.951464\pi$
$109$	− 3.17157i	− 0.303782i	0.988397	−	0.151891i	$-0.0485362\pi$
$110$	5.65685i	0.539360i
$111$	− 0.343146i	− 0.0325700i
$112$	2.82843i	0.267261i
$113$	−10.4853	−0.986372	−0.493186	−	0.869924i	$-0.664168\pi$
$113$	−10.4853	−0.986372	−0.493186	+	0.869924i	$0.664168\pi$
$114$	2.82843	0.264906
$115$	8.48528i	0.791257i
$116$	3.17157	0.294473
$117$	0	0
$118$	−13.6569	−1.25722
$119$	13.6569i	1.25192i
$120$	1.00000	0.0912871
$121$	−21.0000	−1.90909
$122$	− 6.00000i	− 0.543214i
$123$	− 3.65685i	− 0.329727i
$124$	4.00000i	0.359211i
$125$	1.00000i	0.0894427i
$126$	2.82843	0.251976
$127$	1.65685	0.147022	0.0735110	−	0.997294i	$-0.476580\pi$
$127$	1.65685	0.147022	0.0735110	+	0.997294i	$0.476580\pi$
$128$	1.00000i	0.0883883i
$129$	1.65685	0.145878
$130$	0	0
$131$	22.1421	1.93457	0.967284	−	0.253697i	$-0.0816467\pi$
$131$	22.1421	1.93457	0.967284	+	0.253697i	$0.0816467\pi$
$132$	5.65685i	0.492366i
$133$	−8.00000	−0.693688
$134$	5.65685	0.488678
$135$	− 1.00000i	− 0.0860663i
$136$	4.82843i	0.414034i
$137$	5.31371i	0.453981i	0.973897	+	0.226990i	$0.0728886\pi$
$137$	5.31371i	0.453981i	−0.973897	+	0.226990i	$0.927111\pi$
$138$	8.48528i	0.722315i
$139$	17.6569	1.49763	0.748817	−	0.662776i	$-0.230622\pi$
$139$	17.6569	1.49763	0.748817	+	0.662776i	$0.230622\pi$
$140$	−2.82843	−0.239046
$141$	− 8.00000i	− 0.673722i
$142$	−5.65685	−0.474713
$143$	0	0
$144$	1.00000	0.0833333
$145$	3.17157i	0.263385i
$146$	2.48528	0.205683
$147$	−1.00000	−0.0824786
$148$	0.343146i	0.0282064i
$149$	7.65685i	0.627274i	0.949543	+	0.313637i	$0.101547\pi$
$149$	7.65685i	0.627274i	−0.949543	+	0.313637i	$0.898453\pi$
$150$	1.00000i	0.0816497i
$151$	12.0000i	0.976546i	0.872691	+	0.488273i	$0.162373\pi$
$151$	12.0000i	0.976546i	−0.872691	+	0.488273i	$0.837627\pi$
$152$	−2.82843	−0.229416
$153$	4.82843	0.390355
$154$	− 16.0000i	− 1.28932i
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−17.3137	−1.38178	−0.690892	−	0.722958i	$-0.742782\pi$
$157$	−17.3137	−1.38178	−0.690892	+	0.722958i	$0.742782\pi$
$158$	− 13.6569i	− 1.08648i
$159$	−9.31371	−0.738625
$160$	−1.00000	−0.0790569
$161$	− 24.0000i	− 1.89146i
$162$	− 1.00000i	− 0.0785674i
$163$	− 11.3137i	− 0.886158i	−0.896483	−	0.443079i	$-0.853886\pi$
$163$	− 11.3137i	− 0.886158i	0.896483	−	0.443079i	$-0.146114\pi$
$164$	3.65685i	0.285552i
$165$	−5.65685	−0.440386
$166$	−17.6569	−1.37044
$167$	− 24.9706i	− 1.93228i	−0.258018	−	0.966140i	$-0.583069\pi$
$167$	− 24.9706i	− 1.93228i	0.258018	−	0.966140i	$-0.416931\pi$
$168$	−2.82843	−0.218218
$169$	0	0
$170$	−4.82843	−0.370323
$171$	2.82843i	0.216295i
$172$	−1.65685	−0.126334
$173$	−13.3137	−1.01222	−0.506111	−	0.862468i	$-0.668917\pi$
$173$	−13.3137	−1.01222	−0.506111	+	0.862468i	$0.668917\pi$
$174$	3.17157i	0.240436i
$175$	− 2.82843i	− 0.213809i
$176$	− 5.65685i	− 0.426401i
$177$	− 13.6569i	− 1.02651i
$178$	−4.34315	−0.325533
$179$	−24.4853	−1.83012	−0.915058	−	0.403322i	$-0.867855\pi$
$179$	−24.4853	−1.83012	−0.915058	+	0.403322i	$0.867855\pi$
$180$	1.00000i	0.0745356i
$181$	−3.65685	−0.271812	−0.135906	−	0.990722i	$-0.543394\pi$
$181$	−3.65685	−0.271812	−0.135906	+	0.990722i	$0.543394\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	− 8.48528i	− 0.625543i
$185$	−0.343146	−0.0252286
$186$	−4.00000	−0.293294
$187$	− 27.3137i	− 1.99738i
$188$	8.00000i	0.583460i
$189$	2.82843i	0.205738i
$190$	− 2.82843i	− 0.205196i
$191$	−11.3137	−0.818631	−0.409316	−	0.912393i	$-0.634232\pi$
$191$	−11.3137	−0.818631	−0.409316	+	0.912393i	$0.634232\pi$
$192$	−1.00000	−0.0721688
$193$	− 14.4853i	− 1.04267i	−0.853351	−	0.521337i	$-0.825434\pi$
$193$	− 14.4853i	− 1.04267i	0.853351	−	0.521337i	$-0.174566\pi$
$194$	8.82843	0.633844
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 9.31371i	− 0.663574i	−0.943354	−	0.331787i	$-0.892348\pi$
$197$	− 9.31371i	− 0.663574i	0.943354	−	0.331787i	$-0.107652\pi$
$198$	−5.65685	−0.402015
$199$	−21.6569	−1.53521	−0.767607	−	0.640921i	$-0.778553\pi$
$199$	−21.6569	−1.53521	−0.767607	+	0.640921i	$0.778553\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	5.65685i	0.399004i
$202$	− 12.1421i	− 0.854318i
$203$	− 8.97056i	− 0.629610i
$204$	−4.82843	−0.338058
$205$	−3.65685	−0.255406
$206$	9.65685i	0.672825i
$207$	−8.48528	−0.589768
$208$	0	0
$209$	16.0000	1.10674
$210$	− 2.82843i	− 0.195180i
$211$	23.3137	1.60498	0.802491	−	0.596664i	$-0.203508\pi$
$211$	23.3137	1.60498	0.802491	+	0.596664i	$0.203508\pi$
$212$	9.31371	0.639668
$213$	− 5.65685i	− 0.387601i
$214$	4.00000i	0.273434i
$215$	− 1.65685i	− 0.112997i
$216$	1.00000i	0.0680414i
$217$	11.3137	0.768025
$218$	−3.17157	−0.214806
$219$	2.48528i	0.167940i
$220$	5.65685	0.381385
$221$	0	0
$222$	−0.343146	−0.0230304
$223$	− 5.17157i	− 0.346314i	−0.984894	−	0.173157i	$-0.944603\pi$
$223$	− 5.17157i	− 0.346314i	0.984894	−	0.173157i	$-0.0553968\pi$
$224$	2.82843	0.188982
$225$	−1.00000	−0.0666667
$226$	10.4853i	0.697471i
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	− 2.82843i	− 0.187317i
$229$	− 24.1421i	− 1.59536i	−0.603083	−	0.797679i	$-0.706061\pi$
$229$	− 24.1421i	− 1.59536i	0.603083	−	0.797679i	$-0.293939\pi$
$230$	8.48528	0.559503
$231$	16.0000	1.05272
$232$	− 3.17157i	− 0.208224i
$233$	−22.4853	−1.47306	−0.736530	−	0.676405i	$-0.763537\pi$
$233$	−22.4853	−1.47306	−0.736530	+	0.676405i	$0.763537\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	13.6569i	0.888985i
$237$	13.6569	0.887108
$238$	13.6569	0.885242
$239$	− 16.0000i	− 1.03495i	−0.855697	−	0.517477i	$-0.826871\pi$
$239$	− 16.0000i	− 1.03495i	0.855697	−	0.517477i	$-0.173129\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	− 17.3137i	− 1.11527i	−0.830085	−	0.557637i	$-0.811708\pi$
$241$	− 17.3137i	− 1.11527i	0.830085	−	0.557637i	$-0.188292\pi$
$242$	21.0000i	1.34993i
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	1.00000i	0.0638877i
$246$	−3.65685	−0.233153
$247$	0	0
$248$	4.00000	0.254000
$249$	− 17.6569i	− 1.11896i
$250$	1.00000	0.0632456
$251$	−5.17157	−0.326427	−0.163213	−	0.986591i	$-0.552186\pi$
$251$	−5.17157	−0.326427	−0.163213	+	0.986591i	$0.552186\pi$
$252$	− 2.82843i	− 0.178174i
$253$	48.0000i	3.01773i
$254$	− 1.65685i	− 0.103960i
$255$	− 4.82843i	− 0.302368i
$256$	1.00000	0.0625000
$257$	−0.828427	−0.0516759	−0.0258379	−	0.999666i	$-0.508225\pi$
$257$	−0.828427	−0.0516759	−0.0258379	+	0.999666i	$0.508225\pi$
$258$	− 1.65685i	− 0.103151i
$259$	0.970563	0.0603078
$260$	0	0
$261$	−3.17157	−0.196315
$262$	− 22.1421i	− 1.36795i
$263$	−0.485281	−0.0299237	−0.0149619	−	0.999888i	$-0.504763\pi$
$263$	−0.485281	−0.0299237	−0.0149619	+	0.999888i	$0.504763\pi$
$264$	5.65685	0.348155
$265$	9.31371i	0.572137i
$266$	8.00000i	0.490511i
$267$	− 4.34315i	− 0.265796i
$268$	− 5.65685i	− 0.345547i
$269$	2.48528	0.151530	0.0757651	−	0.997126i	$-0.475860\pi$
$269$	2.48528	0.151530	0.0757651	+	0.997126i	$0.475860\pi$
$270$	−1.00000	−0.0608581
$271$	− 15.3137i	− 0.930242i	−0.885247	−	0.465121i	$-0.846011\pi$
$271$	− 15.3137i	− 0.930242i	0.885247	−	0.465121i	$-0.153989\pi$
$272$	4.82843	0.292766
$273$	0	0
$274$	5.31371	0.321013
$275$	5.65685i	0.341121i
$276$	8.48528	0.510754
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	− 17.6569i	− 1.05899i
$279$	− 4.00000i	− 0.239474i
$280$	2.82843i	0.169031i
$281$	19.6569i	1.17263i	0.810083	+	0.586315i	$0.199422\pi$
$281$	19.6569i	1.17263i	−0.810083	+	0.586315i	$0.800578\pi$
$282$	−8.00000	−0.476393
$283$	6.34315	0.377061	0.188530	−	0.982067i	$-0.439628\pi$
$283$	6.34315	0.377061	0.188530	+	0.982067i	$0.439628\pi$
$284$	5.65685i	0.335673i
$285$	2.82843	0.167542
$286$	0	0
$287$	10.3431	0.610537
$288$	− 1.00000i	− 0.0589256i
$289$	6.31371	0.371395
$290$	3.17157	0.186241
$291$	8.82843i	0.517532i
$292$	− 2.48528i	− 0.145440i
$293$	28.6274i	1.67243i	0.548401	+	0.836216i	$0.315237\pi$
$293$	28.6274i	1.67243i	−0.548401	+	0.836216i	$0.684763\pi$
$294$	1.00000i	0.0583212i
$295$	−13.6569	−0.795133
$296$	0.343146	0.0199449
$297$	− 5.65685i	− 0.328244i
$298$	7.65685	0.443550
$299$	0	0
$300$	1.00000	0.0577350
$301$	4.68629i	0.270113i
$302$	12.0000	0.690522
$303$	12.1421	0.697547
$304$	2.82843i	0.162221i
$305$	− 6.00000i	− 0.343559i
$306$	− 4.82843i	− 0.276023i
$307$	− 10.3431i	− 0.590315i	−0.955449	−	0.295157i	$-0.904628\pi$
$307$	− 10.3431i	− 0.590315i	0.955449	−	0.295157i	$-0.0953720\pi$
$308$	−16.0000	−0.911685
$309$	−9.65685	−0.549359
$310$	4.00000i	0.227185i
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	2.97056	0.167906	0.0839531	−	0.996470i	$-0.473245\pi$
$313$	2.97056	0.167906	0.0839531	+	0.996470i	$0.473245\pi$
$314$	17.3137i	0.977069i
$315$	2.82843	0.159364
$316$	−13.6569	−0.768258
$317$	− 2.68629i	− 0.150877i	−0.997150	−	0.0754386i	$-0.975964\pi$
$317$	− 2.68629i	− 0.150877i	0.997150	−	0.0754386i	$-0.0240357\pi$
$318$	9.31371i	0.522287i
$319$	17.9411i	1.00451i
$320$	1.00000i	0.0559017i
$321$	−4.00000	−0.223258
$322$	−24.0000	−1.33747
$323$	13.6569i	0.759888i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−11.3137	−0.626608
$327$	− 3.17157i	− 0.175388i
$328$	3.65685	0.201916
$329$	22.6274	1.24749
$330$	5.65685i	0.311400i
$331$	− 8.48528i	− 0.466393i	−0.972430	−	0.233197i	$-0.925081\pi$
$331$	− 8.48528i	− 0.466393i	0.972430	−	0.233197i	$-0.0749186\pi$
$332$	17.6569i	0.969046i
$333$	− 0.343146i	− 0.0188043i
$334$	−24.9706	−1.36633
$335$	5.65685	0.309067
$336$	2.82843i	0.154303i
$337$	22.9706	1.25129	0.625643	−	0.780109i	$-0.284837\pi$
$337$	22.9706	1.25129	0.625643	+	0.780109i	$0.284837\pi$
$338$	0	0
$339$	−10.4853	−0.569482
$340$	4.82843i	0.261858i
$341$	−22.6274	−1.22534
$342$	2.82843	0.152944
$343$	16.9706i	0.916324i
$344$	1.65685i	0.0893316i
$345$	8.48528i	0.456832i
$346$	13.3137i	0.715749i
$347$	1.65685	0.0889446	0.0444723	−	0.999011i	$-0.485839\pi$
$347$	1.65685	0.0889446	0.0444723	+	0.999011i	$0.485839\pi$
$348$	3.17157	0.170014
$349$	− 16.1421i	− 0.864069i	−0.901857	−	0.432034i	$-0.857796\pi$
$349$	− 16.1421i	− 0.864069i	0.901857	−	0.432034i	$-0.142204\pi$
$350$	−2.82843	−0.151186
$351$	0	0
$352$	−5.65685	−0.301511
$353$	17.3137i	0.921516i	0.887526	+	0.460758i	$0.152422\pi$
$353$	17.3137i	0.921516i	−0.887526	+	0.460758i	$0.847578\pi$
$354$	−13.6569	−0.725854
$355$	−5.65685	−0.300235
$356$	4.34315i	0.230186i
$357$	13.6569i	0.722797i
$358$	24.4853i	1.29409i
$359$	28.2843i	1.49279i	0.665505	+	0.746393i	$0.268216\pi$
$359$	28.2843i	1.49279i	−0.665505	+	0.746393i	$0.731784\pi$
$360$	1.00000	0.0527046
$361$	11.0000	0.578947
$362$	3.65685i	0.192200i
$363$	−21.0000	−1.10221
$364$	0	0
$365$	2.48528	0.130086
$366$	− 6.00000i	− 0.313625i
$367$	14.3431	0.748706	0.374353	−	0.927286i	$-0.377865\pi$
$367$	14.3431	0.748706	0.374353	+	0.927286i	$0.377865\pi$
$368$	−8.48528	−0.442326
$369$	− 3.65685i	− 0.190368i
$370$	0.343146i	0.0178393i
$371$	− 26.3431i	− 1.36767i
$372$	4.00000i	0.207390i
$373$	−25.3137	−1.31069	−0.655347	−	0.755328i	$-0.727478\pi$
$373$	−25.3137	−1.31069	−0.655347	+	0.755328i	$0.727478\pi$
$374$	−27.3137	−1.41236
$375$	1.00000i	0.0516398i
$376$	8.00000	0.412568
$377$	0	0
$378$	2.82843	0.145479
$379$	24.4853i	1.25772i	0.777517	+	0.628862i	$0.216479\pi$
$379$	24.4853i	1.25772i	−0.777517	+	0.628862i	$0.783521\pi$
$380$	−2.82843	−0.145095
$381$	1.65685	0.0848832
$382$	11.3137i	0.578860i
$383$	18.3431i	0.937291i	0.883386	+	0.468645i	$0.155258\pi$
$383$	18.3431i	0.937291i	−0.883386	+	0.468645i	$0.844742\pi$
$384$	1.00000i	0.0510310i
$385$	− 16.0000i	− 0.815436i
$386$	−14.4853	−0.737281
$387$	1.65685	0.0842226
$388$	− 8.82843i	− 0.448195i
$389$	−10.4853	−0.531625	−0.265812	−	0.964025i	$-0.585640\pi$
$389$	−10.4853	−0.531625	−0.265812	+	0.964025i	$0.585640\pi$
$390$	0	0
$391$	−40.9706	−2.07197
$392$	− 1.00000i	− 0.0505076i
$393$	22.1421	1.11692
$394$	−9.31371	−0.469218
$395$	− 13.6569i	− 0.687151i
$396$	5.65685i	0.284268i
$397$	− 26.2843i	− 1.31917i	−0.751630	−	0.659585i	$-0.770732\pi$
$397$	− 26.2843i	− 1.31917i	0.751630	−	0.659585i	$-0.229268\pi$
$398$	21.6569i	1.08556i
$399$	−8.00000	−0.400501
$400$	−1.00000	−0.0500000
$401$	6.97056i	0.348093i	0.984737	+	0.174047i	$0.0556844\pi$
$401$	6.97056i	0.348093i	−0.984737	+	0.174047i	$0.944316\pi$
$402$	5.65685	0.282138
$403$	0	0
$404$	−12.1421	−0.604094
$405$	− 1.00000i	− 0.0496904i
$406$	−8.97056	−0.445202
$407$	−1.94113	−0.0962180
$408$	4.82843i	0.239043i
$409$	− 7.65685i	− 0.378607i	−0.981919	−	0.189304i	$-0.939377\pi$
$409$	− 7.65685i	− 0.378607i	0.981919	−	0.189304i	$-0.0606230\pi$
$410$	3.65685i	0.180599i
$411$	5.31371i	0.262106i
$412$	9.65685	0.475759
$413$	38.6274	1.90073
$414$	8.48528i	0.417029i
$415$	−17.6569	−0.866741
$416$	0	0
$417$	17.6569	0.864660
$418$	− 16.0000i	− 0.782586i
$419$	−5.17157	−0.252648	−0.126324	−	0.991989i	$-0.540318\pi$
$419$	−5.17157	−0.252648	−0.126324	+	0.991989i	$0.540318\pi$
$420$	−2.82843	−0.138013
$421$	− 4.14214i	− 0.201875i	−0.994893	−	0.100938i	$-0.967816\pi$
$421$	− 4.14214i	− 0.201875i	0.994893	−	0.100938i	$-0.0321843\pi$
$422$	− 23.3137i	− 1.13489i
$423$	− 8.00000i	− 0.388973i
$424$	− 9.31371i	− 0.452314i
$425$	−4.82843	−0.234213
$426$	−5.65685	−0.274075
$427$	16.9706i	0.821263i
$428$	4.00000	0.193347
$429$	0	0
$430$	−1.65685	−0.0799006
$431$	16.0000i	0.770693i	0.922772	+	0.385346i	$0.125918\pi$
$431$	16.0000i	0.770693i	−0.922772	+	0.385346i	$0.874082\pi$
$432$	1.00000	0.0481125
$433$	−10.9706	−0.527212	−0.263606	−	0.964630i	$-0.584912\pi$
$433$	−10.9706	−0.527212	−0.263606	+	0.964630i	$0.584912\pi$
$434$	− 11.3137i	− 0.543075i
$435$	3.17157i	0.152065i
$436$	3.17157i	0.151891i
$437$	− 24.0000i	− 1.14808i
$438$	2.48528	0.118751
$439$	22.6274	1.07995	0.539974	−	0.841682i	$-0.318434\pi$
$439$	22.6274	1.07995	0.539974	+	0.841682i	$0.318434\pi$
$440$	− 5.65685i	− 0.269680i
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	41.6569	1.97918	0.989588	−	0.143926i	$-0.0459728\pi$
$443$	41.6569	1.97918	0.989588	+	0.143926i	$0.0459728\pi$
$444$	0.343146i	0.0162850i
$445$	−4.34315	−0.205885
$446$	−5.17157	−0.244881
$447$	7.65685i	0.362157i
$448$	− 2.82843i	− 0.133631i
$449$	− 30.2843i	− 1.42920i	−0.699532	−	0.714602i	$-0.746608\pi$
$449$	− 30.2843i	− 1.42920i	0.699532	−	0.714602i	$-0.253392\pi$
$450$	1.00000i	0.0471405i
$451$	−20.6863	−0.974079
$452$	10.4853	0.493186
$453$	12.0000i	0.563809i
$454$	−4.00000	−0.187729
$455$	0	0
$456$	−2.82843	−0.132453
$457$	− 15.1716i	− 0.709696i	−0.934924	−	0.354848i	$-0.884533\pi$
$457$	− 15.1716i	− 0.709696i	0.934924	−	0.354848i	$-0.115467\pi$
$458$	−24.1421	−1.12809
$459$	4.82843	0.225372
$460$	− 8.48528i	− 0.395628i
$461$	− 14.0000i	− 0.652045i	−0.945362	−	0.326023i	$-0.894291\pi$
$461$	− 14.0000i	− 0.652045i	0.945362	−	0.326023i	$-0.105709\pi$
$462$	− 16.0000i	− 0.744387i
$463$	35.7990i	1.66372i	0.554985	+	0.831860i	$0.312724\pi$
$463$	35.7990i	1.66372i	−0.554985	+	0.831860i	$0.687276\pi$
$464$	−3.17157	−0.147237
$465$	−4.00000	−0.185496
$466$	22.4853i	1.04161i
$467$	15.3137	0.708634	0.354317	−	0.935125i	$-0.384713\pi$
$467$	15.3137	0.708634	0.354317	+	0.935125i	$0.384713\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	8.00000i	0.369012i
$471$	−17.3137	−0.797774
$472$	13.6569	0.628608
$473$	− 9.37258i	− 0.430952i
$474$	− 13.6569i	− 0.627280i
$475$	− 2.82843i	− 0.129777i
$476$	− 13.6569i	− 0.625961i
$477$	−9.31371	−0.426445
$478$	−16.0000	−0.731823
$479$	11.3137i	0.516937i	0.966020	+	0.258468i	$0.0832177\pi$
$479$	11.3137i	0.516937i	−0.966020	+	0.258468i	$0.916782\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−17.3137	−0.788618
$483$	− 24.0000i	− 1.09204i
$484$	21.0000	0.954545
$485$	8.82843	0.400878
$486$	− 1.00000i	− 0.0453609i
$487$	0.485281i	0.0219902i	0.999940	+	0.0109951i	$0.00349992\pi$
$487$	0.485281i	0.0219902i	−0.999940	+	0.0109951i	$0.996500\pi$
$488$	6.00000i	0.271607i
$489$	− 11.3137i	− 0.511624i
$490$	1.00000	0.0451754
$491$	9.85786	0.444879	0.222440	−	0.974946i	$-0.428598\pi$
$491$	9.85786	0.444879	0.222440	+	0.974946i	$0.428598\pi$
$492$	3.65685i	0.164864i
$493$	−15.3137	−0.689695
$494$	0	0
$495$	−5.65685	−0.254257
$496$	− 4.00000i	− 0.179605i
$497$	16.0000	0.717698
$498$	−17.6569	−0.791223
$499$	16.4853i	0.737983i	0.929433	+	0.368991i	$0.120297\pi$
$499$	16.4853i	0.737983i	−0.929433	+	0.368991i	$0.879703\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 24.9706i	− 1.11560i
$502$	5.17157i	0.230819i
$503$	−40.4853	−1.80515	−0.902575	−	0.430533i	$-0.858326\pi$
$503$	−40.4853	−1.80515	−0.902575	+	0.430533i	$0.858326\pi$
$504$	−2.82843	−0.125988
$505$	− 12.1421i	− 0.540318i
$506$	48.0000	2.13386
$507$	0	0
$508$	−1.65685	−0.0735110
$509$	14.6863i	0.650958i	0.945549	+	0.325479i	$0.105526\pi$
$509$	14.6863i	0.650958i	−0.945549	+	0.325479i	$0.894474\pi$
$510$	−4.82843	−0.213806
$511$	−7.02944	−0.310964
$512$	− 1.00000i	− 0.0441942i
$513$	2.82843i	0.124878i
$514$	0.828427i	0.0365404i
$515$	9.65685i	0.425532i
$516$	−1.65685	−0.0729389
$517$	−45.2548	−1.99031
$518$	− 0.970563i	− 0.0426441i
$519$	−13.3137	−0.584407
$520$	0	0
$521$	−6.97056	−0.305386	−0.152693	−	0.988274i	$-0.548795\pi$
$521$	−6.97056	−0.305386	−0.152693	+	0.988274i	$0.548795\pi$
$522$	3.17157i	0.138816i
$523$	34.6274	1.51415	0.757076	−	0.653327i	$-0.226627\pi$
$523$	34.6274	1.51415	0.757076	+	0.653327i	$0.226627\pi$
$524$	−22.1421	−0.967284
$525$	− 2.82843i	− 0.123443i
$526$	0.485281i	0.0211593i
$527$	− 19.3137i	− 0.841318i
$528$	− 5.65685i	− 0.246183i
$529$	49.0000	2.13043
$530$	9.31371	0.404562
$531$	− 13.6569i	− 0.592657i
$532$	8.00000	0.346844
$533$	0	0
$534$	−4.34315	−0.187946
$535$	4.00000i	0.172935i
$536$	−5.65685	−0.244339
$537$	−24.4853	−1.05662
$538$	− 2.48528i	− 0.107148i
$539$	5.65685i	0.243658i
$540$	1.00000i	0.0430331i
$541$	− 2.48528i	− 0.106851i	−0.998572	−	0.0534253i	$-0.982986\pi$
$541$	− 2.48528i	− 0.106851i	0.998572	−	0.0534253i	$-0.0170139\pi$
$542$	−15.3137	−0.657780
$543$	−3.65685	−0.156931
$544$	− 4.82843i	− 0.207017i
$545$	−3.17157	−0.135855
$546$	0	0
$547$	23.3137	0.996822	0.498411	−	0.866941i	$-0.333917\pi$
$547$	23.3137	0.996822	0.498411	+	0.866941i	$0.333917\pi$
$548$	− 5.31371i	− 0.226990i
$549$	6.00000	0.256074
$550$	5.65685	0.241209
$551$	− 8.97056i	− 0.382159i
$552$	− 8.48528i	− 0.361158i
$553$	38.6274i	1.64260i
$554$	26.0000i	1.10463i
$555$	−0.343146	−0.0145657
$556$	−17.6569	−0.748817
$557$	33.3137i	1.41155i	0.708437	+	0.705774i	$0.249400\pi$
$557$	33.3137i	1.41155i	−0.708437	+	0.705774i	$0.750600\pi$
$558$	−4.00000	−0.169334
$559$	0	0
$560$	2.82843	0.119523
$561$	− 27.3137i	− 1.15319i
$562$	19.6569	0.829174
$563$	41.6569	1.75563	0.877814	−	0.479002i	$-0.159002\pi$
$563$	41.6569	1.75563	0.877814	+	0.479002i	$0.159002\pi$
$564$	8.00000i	0.336861i
$565$	10.4853i	0.441119i
$566$	− 6.34315i	− 0.266622i
$567$	2.82843i	0.118783i
$568$	5.65685	0.237356
$569$	−20.3431	−0.852829	−0.426415	−	0.904528i	$-0.640224\pi$
$569$	−20.3431	−0.852829	−0.426415	+	0.904528i	$0.640224\pi$
$570$	− 2.82843i	− 0.118470i
$571$	12.9706	0.542801	0.271401	−	0.962466i	$-0.412513\pi$
$571$	12.9706	0.542801	0.271401	+	0.962466i	$0.412513\pi$
$572$	0	0
$573$	−11.3137	−0.472637
$574$	− 10.3431i	− 0.431715i
$575$	8.48528	0.353861
$576$	−1.00000	−0.0416667
$577$	− 27.4558i	− 1.14300i	−0.820601	−	0.571501i	$-0.806361\pi$
$577$	− 27.4558i	− 1.14300i	0.820601	−	0.571501i	$-0.193639\pi$
$578$	− 6.31371i	− 0.262616i
$579$	− 14.4853i	− 0.601988i
$580$	− 3.17157i	− 0.131692i
$581$	49.9411	2.07191
$582$	8.82843	0.365950
$583$	52.6863i	2.18204i
$584$	−2.48528	−0.102842
$585$	0	0
$586$	28.6274	1.18259
$587$	42.6274i	1.75942i	0.475509	+	0.879711i	$0.342264\pi$
$587$	42.6274i	1.75942i	−0.475509	+	0.879711i	$0.657736\pi$
$588$	1.00000	0.0412393
$589$	11.3137	0.466173
$590$	13.6569i	0.562244i
$591$	− 9.31371i	− 0.383115i
$592$	− 0.343146i	− 0.0141032i
$593$	− 11.6569i	− 0.478690i	−0.970935	−	0.239345i	$-0.923067\pi$
$593$	− 11.6569i	− 0.478690i	0.970935	−	0.239345i	$-0.0769326\pi$
$594$	−5.65685	−0.232104
$595$	13.6569	0.559876
$596$	− 7.65685i	− 0.313637i
$597$	−21.6569	−0.886356
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	6.68629	0.272740	0.136370	−	0.990658i	$-0.456456\pi$
$601$	6.68629	0.272740	0.136370	+	0.990658i	$0.456456\pi$
$602$	4.68629	0.190999
$603$	5.65685i	0.230365i
$604$	− 12.0000i	− 0.488273i
$605$	21.0000i	0.853771i
$606$	− 12.1421i	− 0.493241i
$607$	−4.97056	−0.201749	−0.100874	−	0.994899i	$-0.532164\pi$
$607$	−4.97056	−0.201749	−0.100874	+	0.994899i	$0.532164\pi$
$608$	2.82843	0.114708
$609$	− 8.97056i	− 0.363506i
$610$	−6.00000	−0.242933
$611$	0	0
$612$	−4.82843	−0.195178
$613$	34.2843i	1.38473i	0.721548	+	0.692364i	$0.243431\pi$
$613$	34.2843i	1.38473i	−0.721548	+	0.692364i	$0.756569\pi$
$614$	−10.3431	−0.417415
$615$	−3.65685	−0.147459
$616$	16.0000i	0.644658i
$617$	− 2.00000i	− 0.0805170i	−0.999189	−	0.0402585i	$-0.987182\pi$
$617$	− 2.00000i	− 0.0805170i	0.999189	−	0.0402585i	$-0.0128181\pi$
$618$	9.65685i	0.388456i
$619$	− 29.1716i	− 1.17250i	−0.810129	−	0.586252i	$-0.800603\pi$
$619$	− 29.1716i	− 1.17250i	0.810129	−	0.586252i	$-0.199397\pi$
$620$	4.00000	0.160644
$621$	−8.48528	−0.340503
$622$	− 24.0000i	− 0.962312i
$623$	12.2843	0.492159
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 2.97056i	− 0.118728i
$627$	16.0000	0.638978
$628$	17.3137	0.690892
$629$	− 1.65685i	− 0.0660631i
$630$	− 2.82843i	− 0.112687i
$631$	− 22.3431i	− 0.889467i	−0.895663	−	0.444733i	$-0.853298\pi$
$631$	− 22.3431i	− 0.889467i	0.895663	−	0.444733i	$-0.146702\pi$
$632$	13.6569i	0.543240i
$633$	23.3137	0.926637
$634$	−2.68629	−0.106686
$635$	− 1.65685i	− 0.0657503i
$636$	9.31371	0.369313
$637$	0	0
$638$	17.9411	0.710296
$639$	− 5.65685i	− 0.223782i
$640$	1.00000	0.0395285
$641$	−40.6274	−1.60469	−0.802343	−	0.596863i	$-0.796414\pi$
$641$	−40.6274	−1.60469	−0.802343	+	0.596863i	$0.796414\pi$
$642$	4.00000i	0.157867i
$643$	39.5980i	1.56159i	0.624786	+	0.780796i	$0.285186\pi$
$643$	39.5980i	1.56159i	−0.624786	+	0.780796i	$0.714814\pi$
$644$	24.0000i	0.945732i
$645$	− 1.65685i	− 0.0652386i
$646$	13.6569	0.537322
$647$	8.48528	0.333591	0.166795	−	0.985992i	$-0.446658\pi$
$647$	8.48528	0.333591	0.166795	+	0.985992i	$0.446658\pi$
$648$	1.00000i	0.0392837i
$649$	−77.2548	−3.03252
$650$	0	0
$651$	11.3137	0.443419
$652$	11.3137i	0.443079i
$653$	14.2843	0.558987	0.279493	−	0.960148i	$-0.409834\pi$
$653$	14.2843	0.558987	0.279493	+	0.960148i	$0.409834\pi$
$654$	−3.17157	−0.124018
$655$	− 22.1421i	− 0.865165i
$656$	− 3.65685i	− 0.142776i
$657$	2.48528i	0.0969601i
$658$	− 22.6274i	− 0.882109i
$659$	−24.4853	−0.953811	−0.476906	−	0.878955i	$-0.658242\pi$
$659$	−24.4853	−0.953811	−0.476906	+	0.878955i	$0.658242\pi$
$660$	5.65685	0.220193
$661$	20.1421i	0.783438i	0.920085	+	0.391719i	$0.128120\pi$
$661$	20.1421i	0.783438i	−0.920085	+	0.391719i	$0.871880\pi$
$662$	−8.48528	−0.329790
$663$	0	0
$664$	17.6569	0.685219
$665$	8.00000i	0.310227i
$666$	−0.343146	−0.0132966
$667$	26.9117	1.04202
$668$	24.9706i	0.966140i
$669$	− 5.17157i	− 0.199945i
$670$	− 5.65685i	− 0.218543i
$671$	− 33.9411i	− 1.31028i
$672$	2.82843	0.109109
$673$	12.6274	0.486751	0.243376	−	0.969932i	$-0.421745\pi$
$673$	12.6274	0.486751	0.243376	+	0.969932i	$0.421745\pi$
$674$	− 22.9706i	− 0.884793i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	23.6569	0.909207	0.454603	−	0.890694i	$-0.349781\pi$
$677$	23.6569	0.909207	0.454603	+	0.890694i	$0.349781\pi$
$678$	10.4853i	0.402685i
$679$	−24.9706	−0.958282
$680$	4.82843	0.185162
$681$	− 4.00000i	− 0.153280i
$682$	22.6274i	0.866449i
$683$	22.3431i	0.854937i	0.904030	+	0.427468i	$0.140594\pi$
$683$	22.3431i	0.854937i	−0.904030	+	0.427468i	$0.859406\pi$
$684$	− 2.82843i	− 0.108148i
$685$	5.31371	0.203026
$686$	16.9706	0.647939
$687$	− 24.1421i	− 0.921080i
$688$	1.65685	0.0631670
$689$	0	0
$690$	8.48528	0.323029
$691$	− 11.7990i	− 0.448855i	−0.974491	−	0.224427i	$-0.927949\pi$
$691$	− 11.7990i	− 0.448855i	0.974491	−	0.224427i	$-0.0720512\pi$
$692$	13.3137	0.506111
$693$	16.0000	0.607790
$694$	− 1.65685i	− 0.0628933i
$695$	− 17.6569i	− 0.669763i
$696$	− 3.17157i	− 0.120218i
$697$	− 17.6569i	− 0.668801i
$698$	−16.1421	−0.610989
$699$	−22.4853	−0.850471
$700$	2.82843i	0.106904i
$701$	28.1421	1.06291	0.531457	−	0.847085i	$-0.321645\pi$
$701$	28.1421	1.06291	0.531457	+	0.847085i	$0.321645\pi$
$702$	0	0
$703$	0.970563	0.0366055
$704$	5.65685i	0.213201i
$705$	−8.00000	−0.301297
$706$	17.3137	0.651610
$707$	34.3431i	1.29161i
$708$	13.6569i	0.513256i
$709$	− 12.8284i	− 0.481782i	−0.970552	−	0.240891i	$-0.922560\pi$
$709$	− 12.8284i	− 0.481782i	0.970552	−	0.240891i	$-0.0774396\pi$
$710$	5.65685i	0.212298i
$711$	13.6569	0.512172
$712$	4.34315	0.162766
$713$	33.9411i	1.27111i
$714$	13.6569	0.511095
$715$	0	0
$716$	24.4853	0.915058
$717$	− 16.0000i	− 0.597531i
$718$	28.2843	1.05556
$719$	18.3431	0.684084	0.342042	−	0.939685i	$-0.388882\pi$
$719$	18.3431	0.684084	0.342042	+	0.939685i	$0.388882\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	− 27.3137i	− 1.01722i
$722$	− 11.0000i	− 0.409378i
$723$	− 17.3137i	− 0.643904i
$724$	3.65685	0.135906
$725$	3.17157	0.117789
$726$	21.0000i	0.779383i
$727$	−21.9411	−0.813751	−0.406876	−	0.913484i	$-0.633382\pi$
$727$	−21.9411	−0.813751	−0.406876	+	0.913484i	$0.633382\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 2.48528i	− 0.0919844i
$731$	8.00000	0.295891
$732$	−6.00000	−0.221766
$733$	11.6569i	0.430556i	0.976553	+	0.215278i	$0.0690657\pi$
$733$	11.6569i	0.430556i	−0.976553	+	0.215278i	$0.930934\pi$
$734$	− 14.3431i	− 0.529415i
$735$	1.00000i	0.0368856i
$736$	8.48528i	0.312772i
$737$	32.0000	1.17874
$738$	−3.65685	−0.134611
$739$	14.1421i	0.520227i	0.965578	+	0.260113i	$0.0837600\pi$
$739$	14.1421i	0.520227i	−0.965578	+	0.260113i	$0.916240\pi$
$740$	0.343146	0.0126143
$741$	0	0
$742$	−26.3431	−0.967087
$743$	− 20.2843i	− 0.744158i	−0.928201	−	0.372079i	$-0.878645\pi$
$743$	− 20.2843i	− 0.744158i	0.928201	−	0.372079i	$-0.121355\pi$
$744$	4.00000	0.146647
$745$	7.65685	0.280525
$746$	25.3137i	0.926801i
$747$	− 17.6569i	− 0.646031i
$748$	27.3137i	0.998688i
$749$	− 11.3137i	− 0.413394i
$750$	1.00000	0.0365148
$751$	11.3137	0.412843	0.206422	−	0.978463i	$-0.433818\pi$
$751$	11.3137	0.412843	0.206422	+	0.978463i	$0.433818\pi$
$752$	− 8.00000i	− 0.291730i
$753$	−5.17157	−0.188463
$754$	0	0
$755$	12.0000	0.436725
$756$	− 2.82843i	− 0.102869i
$757$	−47.9411	−1.74245	−0.871225	−	0.490884i	$-0.836674\pi$
$757$	−47.9411	−1.74245	−0.871225	+	0.490884i	$0.836674\pi$
$758$	24.4853	0.889345
$759$	48.0000i	1.74229i
$760$	2.82843i	0.102598i
$761$	16.3431i	0.592439i	0.955120	+	0.296219i	$0.0957259\pi$
$761$	16.3431i	0.592439i	−0.955120	+	0.296219i	$0.904274\pi$
$762$	− 1.65685i	− 0.0600215i
$763$	8.97056	0.324756
$764$	11.3137	0.409316
$765$	− 4.82843i	− 0.174572i
$766$	18.3431	0.662765
$767$	0	0
$768$	1.00000	0.0360844
$769$	14.0000i	0.504853i	0.967616	+	0.252426i	$0.0812286\pi$
$769$	14.0000i	0.504853i	−0.967616	+	0.252426i	$0.918771\pi$
$770$	−16.0000	−0.576600
$771$	−0.828427	−0.0298351
$772$	14.4853i	0.521337i
$773$	30.6863i	1.10371i	0.833940	+	0.551855i	$0.186080\pi$
$773$	30.6863i	1.10371i	−0.833940	+	0.551855i	$0.813920\pi$
$774$	− 1.65685i	− 0.0595544i
$775$	4.00000i	0.143684i
$776$	−8.82843	−0.316922
$777$	0.970563	0.0348187
$778$	10.4853i	0.375916i
$779$	10.3431	0.370582
$780$	0	0
$781$	−32.0000	−1.14505
$782$	40.9706i	1.46510i
$783$	−3.17157	−0.113343
$784$	−1.00000	−0.0357143
$785$	17.3137i	0.617953i
$786$	− 22.1421i	− 0.789784i
$787$	− 24.0000i	− 0.855508i	−0.903895	−	0.427754i	$-0.859305\pi$
$787$	− 24.0000i	− 0.855508i	0.903895	−	0.427754i	$-0.140695\pi$
$788$	9.31371i	0.331787i
$789$	−0.485281	−0.0172765
$790$	−13.6569	−0.485889
$791$	− 29.6569i	− 1.05448i
$792$	5.65685	0.201008
$793$	0	0
$794$	−26.2843	−0.932794
$795$	9.31371i	0.330323i
$796$	21.6569	0.767607
$797$	28.6274	1.01404	0.507018	−	0.861936i	$-0.330748\pi$
$797$	28.6274	1.01404	0.507018	+	0.861936i	$0.330748\pi$
$798$	8.00000i	0.283197i
$799$	− 38.6274i	− 1.36654i
$800$	1.00000i	0.0353553i
$801$	− 4.34315i	− 0.153458i
$802$	6.97056	0.246139
$803$	14.0589	0.496127
$804$	− 5.65685i	− 0.199502i
$805$	−24.0000	−0.845889
$806$	0	0
$807$	2.48528	0.0874860
$808$	12.1421i	0.427159i
$809$	−9.31371	−0.327453	−0.163726	−	0.986506i	$-0.552351\pi$
$809$	−9.31371	−0.327453	−0.163726	+	0.986506i	$0.552351\pi$
$810$	−1.00000	−0.0351364
$811$	30.1421i	1.05843i	0.848487	+	0.529217i	$0.177514\pi$
$811$	30.1421i	1.05843i	−0.848487	+	0.529217i	$0.822486\pi$
$812$	8.97056i	0.314805i
$813$	− 15.3137i	− 0.537075i
$814$	1.94113i	0.0680364i
$815$	−11.3137	−0.396302
$816$	4.82843	0.169029
$817$	4.68629i	0.163953i
$818$	−7.65685	−0.267716
$819$	0	0
$820$	3.65685	0.127703
$821$	22.2843i	0.777726i	0.921296	+	0.388863i	$0.127132\pi$
$821$	22.2843i	0.777726i	−0.921296	+	0.388863i	$0.872868\pi$
$822$	5.31371	0.185337
$823$	19.0294	0.663324	0.331662	−	0.943398i	$-0.392391\pi$
$823$	19.0294	0.663324	0.331662	+	0.943398i	$0.392391\pi$
$824$	− 9.65685i	− 0.336412i
$825$	5.65685i	0.196946i
$826$	− 38.6274i	− 1.34402i
$827$	− 1.65685i	− 0.0576145i	−0.999585	−	0.0288072i	$-0.990829\pi$
$827$	− 1.65685i	− 0.0576145i	0.999585	−	0.0288072i	$-0.00917090\pi$
$828$	8.48528	0.294884
$829$	30.6863	1.06578	0.532889	−	0.846185i	$-0.321106\pi$
$829$	30.6863	1.06578	0.532889	+	0.846185i	$0.321106\pi$
$830$	17.6569i	0.612878i
$831$	−26.0000	−0.901930
$832$	0	0
$833$	−4.82843	−0.167295
$834$	− 17.6569i	− 0.611407i
$835$	−24.9706	−0.864142
$836$	−16.0000	−0.553372
$837$	− 4.00000i	− 0.138260i
$838$	5.17157i	0.178649i
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	2.82843i	0.0975900i
$841$	−18.9411	−0.653142
$842$	−4.14214	−0.142747
$843$	19.6569i	0.677018i
$844$	−23.3137	−0.802491
$845$	0	0
$846$	−8.00000	−0.275046
$847$	− 59.3970i	− 2.04090i
$848$	−9.31371	−0.319834
$849$	6.34315	0.217696
$850$	4.82843i	0.165614i
$851$	2.91169i	0.0998114i
$852$	5.65685i	0.193801i
$853$	− 18.2843i	− 0.626042i	−0.949746	−	0.313021i	$-0.898659\pi$
$853$	− 18.2843i	− 0.626042i	0.949746	−	0.313021i	$-0.101341\pi$
$854$	16.9706	0.580721
$855$	2.82843	0.0967302
$856$	− 4.00000i	− 0.136717i
$857$	15.1716	0.518251	0.259126	−	0.965844i	$-0.416566\pi$
$857$	15.1716	0.518251	0.259126	+	0.965844i	$0.416566\pi$
$858$	0	0
$859$	−29.9411	−1.02158	−0.510789	−	0.859706i	$-0.670647\pi$
$859$	−29.9411	−1.02158	−0.510789	+	0.859706i	$0.670647\pi$
$860$	1.65685i	0.0564983i
$861$	10.3431	0.352493
$862$	16.0000	0.544962
$863$	− 28.2843i	− 0.962808i	−0.876499	−	0.481404i	$-0.840127\pi$
$863$	− 28.2843i	− 0.962808i	0.876499	−	0.481404i	$-0.159873\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	13.3137i	0.452680i
$866$	10.9706i	0.372795i
$867$	6.31371	0.214425
$868$	−11.3137	−0.384012
$869$	− 77.2548i	− 2.62069i
$870$	3.17157	0.107526
$871$	0	0
$872$	3.17157	0.107403
$873$	8.82843i	0.298797i
$874$	−24.0000	−0.811812
$875$	−2.82843	−0.0956183
$876$	− 2.48528i	− 0.0839699i
$877$	− 39.2548i	− 1.32554i	−0.748822	−	0.662771i	$-0.769380\pi$
$877$	− 39.2548i	− 1.32554i	0.748822	−	0.662771i	$-0.230620\pi$
$878$	− 22.6274i	− 0.763638i
$879$	28.6274i	0.965579i
$880$	−5.65685	−0.190693
$881$	−46.2843	−1.55936	−0.779678	−	0.626180i	$-0.784617\pi$
$881$	−46.2843	−1.55936	−0.779678	+	0.626180i	$0.784617\pi$
$882$	1.00000i	0.0336718i
$883$	−8.68629	−0.292317	−0.146158	−	0.989261i	$-0.546691\pi$
$883$	−8.68629	−0.292317	−0.146158	+	0.989261i	$0.546691\pi$
$884$	0	0
$885$	−13.6569	−0.459070
$886$	− 41.6569i	− 1.39949i
$887$	−23.5147	−0.789547	−0.394773	−	0.918778i	$-0.629177\pi$
$887$	−23.5147	−0.789547	−0.394773	+	0.918778i	$0.629177\pi$
$888$	0.343146	0.0115152
$889$	4.68629i	0.157173i
$890$	4.34315i	0.145583i
$891$	− 5.65685i	− 0.189512i
$892$	5.17157i	0.173157i
$893$	22.6274	0.757198
$894$	7.65685	0.256084
$895$	24.4853i	0.818453i
$896$	−2.82843	−0.0944911
$897$	0	0
$898$	−30.2843	−1.01060
$899$	12.6863i	0.423112i
$900$	1.00000	0.0333333
$901$	−44.9706	−1.49819
$902$	20.6863i	0.688778i
$903$	4.68629i	0.155950i
$904$	− 10.4853i	− 0.348735i
$905$	3.65685i	0.121558i
$906$	12.0000	0.398673
$907$	48.2843	1.60325	0.801626	−	0.597825i	$-0.203968\pi$
$907$	48.2843	1.60325	0.801626	+	0.597825i	$0.203968\pi$
$908$	4.00000i	0.132745i
$909$	12.1421	0.402729
$910$	0	0
$911$	−8.97056	−0.297208	−0.148604	−	0.988897i	$-0.547478\pi$
$911$	−8.97056	−0.297208	−0.148604	+	0.988897i	$0.547478\pi$
$912$	2.82843i	0.0936586i
$913$	−99.8823	−3.30562
$914$	−15.1716	−0.501831
$915$	− 6.00000i	− 0.198354i
$916$	24.1421i	0.797679i
$917$	62.6274i	2.06814i
$918$	− 4.82843i	− 0.159362i
$919$	−25.9411	−0.855719	−0.427859	−	0.903845i	$-0.640732\pi$
$919$	−25.9411	−0.855719	−0.427859	+	0.903845i	$0.640732\pi$
$920$	−8.48528	−0.279751
$921$	− 10.3431i	− 0.340818i
$922$	−14.0000	−0.461065
$923$	0	0
$924$	−16.0000	−0.526361
$925$	0.343146i	0.0112826i
$926$	35.7990	1.17643
$927$	−9.65685	−0.317173
$928$	3.17157i	0.104112i
$929$	− 45.5980i	− 1.49602i	−0.663687	−	0.748011i	$-0.731009\pi$
$929$	− 45.5980i	− 1.49602i	0.663687	−	0.748011i	$-0.268991\pi$
$930$	4.00000i	0.131165i
$931$	− 2.82843i	− 0.0926980i
$932$	22.4853	0.736530
$933$	24.0000	0.785725
$934$	− 15.3137i	− 0.501080i
$935$	−27.3137	−0.893254
$936$	0	0
$937$	−28.6274	−0.935217	−0.467608	−	0.883936i	$-0.654884\pi$
$937$	−28.6274	−0.935217	−0.467608	+	0.883936i	$0.654884\pi$
$938$	16.0000i	0.522419i
$939$	2.97056	0.0969407
$940$	8.00000	0.260931
$941$	− 21.0294i	− 0.685540i	−0.939419	−	0.342770i	$-0.888635\pi$
$941$	− 21.0294i	− 0.685540i	0.939419	−	0.342770i	$-0.111365\pi$
$942$	17.3137i	0.564111i
$943$	31.0294i	1.01046i
$944$	− 13.6569i	− 0.444493i
$945$	2.82843	0.0920087
$946$	−9.37258	−0.304729
$947$	41.6569i	1.35367i	0.736137	+	0.676833i	$0.236648\pi$
$947$	41.6569i	1.35367i	−0.736137	+	0.676833i	$0.763352\pi$
$948$	−13.6569	−0.443554
$949$	0	0
$950$	−2.82843	−0.0917663
$951$	− 2.68629i	− 0.0871090i
$952$	−13.6569	−0.442621
$953$	56.1421	1.81862	0.909311	−	0.416117i	$-0.136609\pi$
$953$	56.1421	1.81862	0.909311	+	0.416117i	$0.136609\pi$
$954$	9.31371i	0.301542i
$955$	11.3137i	0.366103i
$956$	16.0000i	0.517477i
$957$	17.9411i	0.579954i
$958$	11.3137	0.365529
$959$	−15.0294	−0.485326
$960$	1.00000i	0.0322749i
$961$	15.0000	0.483871
$962$	0	0
$963$	−4.00000	−0.128898
$964$	17.3137i	0.557637i
$965$	−14.4853	−0.466298
$966$	−24.0000	−0.772187
$967$	24.4853i	0.787394i	0.919240	+	0.393697i	$0.128804\pi$
$967$	24.4853i	0.787394i	−0.919240	+	0.393697i	$0.871196\pi$
$968$	− 21.0000i	− 0.674966i
$969$	13.6569i	0.438721i
$970$	− 8.82843i	− 0.283464i
$971$	32.4853	1.04250	0.521251	−	0.853403i	$-0.325465\pi$
$971$	32.4853	1.04250	0.521251	+	0.853403i	$0.325465\pi$
$972$	−1.00000	−0.0320750
$973$	49.9411i	1.60104i
$974$	0.485281	0.0155494
$975$	0	0
$976$	6.00000	0.192055
$977$	19.6569i	0.628878i	0.949278	+	0.314439i	$0.101816\pi$
$977$	19.6569i	0.628878i	−0.949278	+	0.314439i	$0.898184\pi$
$978$	−11.3137	−0.361773
$979$	−24.5685	−0.785214
$980$	− 1.00000i	− 0.0319438i
$981$	− 3.17157i	− 0.101261i
$982$	− 9.85786i	− 0.314577i
$983$	− 13.6569i	− 0.435586i	−0.975995	−	0.217793i	$-0.930114\pi$
$983$	− 13.6569i	− 0.435586i	0.975995	−	0.217793i	$-0.0698858\pi$
$984$	3.65685	0.116576
$985$	−9.31371	−0.296759
$986$	15.3137i	0.487688i
$987$	22.6274	0.720239
$988$	0	0
$989$	−14.0589	−0.447046
$990$	5.65685i	0.179787i
$991$	58.9117	1.87139	0.935696	−	0.352808i	$-0.114773\pi$
$991$	58.9117	1.87139	0.935696	+	0.352808i	$0.114773\pi$
$992$	−4.00000	−0.127000
$993$	− 8.48528i	− 0.269272i
$994$	− 16.0000i	− 0.507489i
$995$	21.6569i	0.686568i
$996$	17.6569i	0.559479i
$997$	38.6863	1.22521	0.612604	−	0.790390i	$-0.290122\pi$
$997$	38.6863	1.22521	0.612604	+	0.790390i	$0.290122\pi$
$998$	16.4853	0.521832
$999$	− 0.343146i	− 0.0108567i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.q.1351.2		4
13.5	odd	4		5070.2.a.bc.1.1		2
13.8	odd	4		390.2.a.h.1.2	✓	2
13.12	even	2	inner	5070.2.b.q.1351.3		4
39.8	even	4		1170.2.a.o.1.2		2
52.47	even	4		3120.2.a.bc.1.1		2
65.8	even	4		1950.2.e.o.1249.1		4
65.34	odd	4		1950.2.a.bd.1.1		2
65.47	even	4		1950.2.e.o.1249.4		4
156.47	odd	4		9360.2.a.ch.1.1		2
195.8	odd	4		5850.2.e.bk.5149.3		4
195.47	odd	4		5850.2.e.bk.5149.2		4
195.164	even	4		5850.2.a.cl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.h.1.2	✓	2	13.8	odd	4
1170.2.a.o.1.2		2	39.8	even	4
1950.2.a.bd.1.1		2	65.34	odd	4
1950.2.e.o.1249.1		4	65.8	even	4
1950.2.e.o.1249.4		4	65.47	even	4
3120.2.a.bc.1.1		2	52.47	even	4
5070.2.a.bc.1.1		2	13.5	odd	4
5070.2.b.q.1351.2		4	1.1	even	1	trivial
5070.2.b.q.1351.3		4	13.12	even	2	inner
5850.2.a.cl.1.1		2	195.164	even	4
5850.2.e.bk.5149.2		4	195.47	odd	4
5850.2.e.bk.5149.3		4	195.8	odd	4
9360.2.a.ch.1.1		2	156.47	odd	4

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +2.82843i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +2.82843i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.65685i q^{11} -1.00000 q^{12} +2.82843 q^{14} -1.00000i q^{15} +1.00000 q^{16} +4.82843 q^{17} -1.00000i q^{18} +2.82843i q^{19} +1.00000i q^{20} +2.82843i q^{21} -5.65685 q^{22} -8.48528 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -2.82843i q^{28} -3.17157 q^{29} -1.00000 q^{30} -4.00000i q^{31} -1.00000i q^{32} -5.65685i q^{33} -4.82843i q^{34} +2.82843 q^{35} -1.00000 q^{36} -0.343146i q^{37} +2.82843 q^{38} +1.00000 q^{40} -3.65685i q^{41} +2.82843 q^{42} +1.65685 q^{43} +5.65685i q^{44} -1.00000i q^{45} +8.48528i q^{46} -8.00000i q^{47} +1.00000 q^{48} -1.00000 q^{49} +1.00000i q^{50} +4.82843 q^{51} -9.31371 q^{53} -1.00000i q^{54} -5.65685 q^{55} -2.82843 q^{56} +2.82843i q^{57} +3.17157i q^{58} -13.6569i q^{59} +1.00000i q^{60} +6.00000 q^{61} -4.00000 q^{62} +2.82843i q^{63} -1.00000 q^{64} -5.65685 q^{66} +5.65685i q^{67} -4.82843 q^{68} -8.48528 q^{69} -2.82843i q^{70} -5.65685i q^{71} +1.00000i q^{72} +2.48528i q^{73} -0.343146 q^{74} -1.00000 q^{75} -2.82843i q^{76} +16.0000 q^{77} +13.6569 q^{79} -1.00000i q^{80} +1.00000 q^{81} -3.65685 q^{82} -17.6569i q^{83} -2.82843i q^{84} -4.82843i q^{85} -1.65685i q^{86} -3.17157 q^{87} +5.65685 q^{88} -4.34315i q^{89} -1.00000 q^{90} +8.48528 q^{92} -4.00000i q^{93} -8.00000 q^{94} +2.82843 q^{95} -1.00000i q^{96} +8.82843i q^{97} +1.00000i q^{98} -5.65685i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{10} - 4 q^{12} + 4 q^{16} + 8 q^{17} - 4 q^{25} + 4 q^{27} - 24 q^{29} - 4 q^{30} - 4 q^{36} + 4 q^{40} - 16 q^{43} + 4 q^{48} - 4 q^{49} + 8 q^{51} + 8 q^{53} + 24 q^{61} - 16 q^{62} - 4 q^{64} - 8 q^{68} - 24 q^{74} - 4 q^{75} + 64 q^{77} + 32 q^{79} + 4 q^{81} + 8 q^{82} - 24 q^{87} - 4 q^{90} - 32 q^{94}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^10 - 4 * q^12 + 4 * q^16 + 8 * q^17 - 4 * q^25 + 4 * q^27 - 24 * q^29 - 4 * q^30 - 4 * q^36 + 4 * q^40 - 16 * q^43 + 4 * q^48 - 4 * q^49 + 8 * q^51 + 8 * q^53 + 24 * q^61 - 16 * q^62 - 4 * q^64 - 8 * q^68 - 24 * q^74 - 4 * q^75 + 64 * q^77 + 32 * q^79 + 4 * q^81 + 8 * q^82 - 24 * q^87 - 4 * q^90 - 32 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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