LMFDB - Embedded newform 4010.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4010,2,Mod(1,4010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4010, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4010.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4010 = 2 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4010.a (trivial)

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:	yes
Analytic conductor:	$32.0200112105$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4010.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.41421 q^{6} -2.82843 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.41421 q^{6} -2.82843 q^{7} -1.00000 q^{8} -1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} -1.41421 q^{12} -0.585786 q^{13} +2.82843 q^{14} -1.41421 q^{15} +1.00000 q^{16} +1.41421 q^{17} +1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +4.00000 q^{21} +2.00000 q^{22} +3.41421 q^{23} +1.41421 q^{24} +1.00000 q^{25} +0.585786 q^{26} +5.65685 q^{27} -2.82843 q^{28} +8.48528 q^{29} +1.41421 q^{30} -4.00000 q^{31} -1.00000 q^{32} +2.82843 q^{33} -1.41421 q^{34} -2.82843 q^{35} -1.00000 q^{36} +10.2426 q^{37} +2.00000 q^{38} +0.828427 q^{39} -1.00000 q^{40} -5.65685 q^{41} -4.00000 q^{42} +11.3137 q^{43} -2.00000 q^{44} -1.00000 q^{45} -3.41421 q^{46} -1.17157 q^{47} -1.41421 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} -0.585786 q^{52} -2.24264 q^{53} -5.65685 q^{54} -2.00000 q^{55} +2.82843 q^{56} +2.82843 q^{57} -8.48528 q^{58} +0.343146 q^{59} -1.41421 q^{60} +8.00000 q^{61} +4.00000 q^{62} +2.82843 q^{63} +1.00000 q^{64} -0.585786 q^{65} -2.82843 q^{66} +3.07107 q^{67} +1.41421 q^{68} -4.82843 q^{69} +2.82843 q^{70} -10.8284 q^{71} +1.00000 q^{72} +6.48528 q^{73} -10.2426 q^{74} -1.41421 q^{75} -2.00000 q^{76} +5.65685 q^{77} -0.828427 q^{78} -17.6569 q^{79} +1.00000 q^{80} -5.00000 q^{81} +5.65685 q^{82} -12.4853 q^{83} +4.00000 q^{84} +1.41421 q^{85} -11.3137 q^{86} -12.0000 q^{87} +2.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} +1.65685 q^{91} +3.41421 q^{92} +5.65685 q^{93} +1.17157 q^{94} -2.00000 q^{95} +1.41421 q^{96} -10.5858 q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^8 - 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{9} - 2 q^{10} - 4 q^{11} - 4 q^{13} + 2 q^{16} + 2 q^{18} - 4 q^{19} + 2 q^{20} + 8 q^{21} + 4 q^{22} + 4 q^{23} + 2 q^{25} + 4 q^{26} - 8 q^{31} - 2 q^{32} - 2 q^{36} + 12 q^{37} + 4 q^{38} - 4 q^{39} - 2 q^{40} - 8 q^{42} - 4 q^{44} - 2 q^{45} - 4 q^{46} - 8 q^{47} + 2 q^{49} - 2 q^{50} - 4 q^{51} - 4 q^{52} + 4 q^{53} - 4 q^{55} + 12 q^{59} + 16 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} - 4 q^{69} - 16 q^{71} + 2 q^{72} - 4 q^{73} - 12 q^{74} - 4 q^{76} + 4 q^{78} - 24 q^{79} + 2 q^{80} - 10 q^{81} - 8 q^{83} + 8 q^{84} - 24 q^{87} + 4 q^{88} - 20 q^{89} + 2 q^{90} - 8 q^{91} + 4 q^{92} + 8 q^{94} - 4 q^{95} - 24 q^{97} - 2 q^{98} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^8 - 2 * q^9 - 2 * q^10 - 4 * q^11 - 4 * q^13 + 2 * q^16 + 2 * q^18 - 4 * q^19 + 2 * q^20 + 8 * q^21 + 4 * q^22 + 4 * q^23 + 2 * q^25 + 4 * q^26 - 8 * q^31 - 2 * q^32 - 2 * q^36 + 12 * q^37 + 4 * q^38 - 4 * q^39 - 2 * q^40 - 8 * q^42 - 4 * q^44 - 2 * q^45 - 4 * q^46 - 8 * q^47 + 2 * q^49 - 2 * q^50 - 4 * q^51 - 4 * q^52 + 4 * q^53 - 4 * q^55 + 12 * q^59 + 16 * q^61 + 8 * q^62 + 2 * q^64 - 4 * q^65 - 8 * q^67 - 4 * q^69 - 16 * q^71 + 2 * q^72 - 4 * q^73 - 12 * q^74 - 4 * q^76 + 4 * q^78 - 24 * q^79 + 2 * q^80 - 10 * q^81 - 8 * q^83 + 8 * q^84 - 24 * q^87 + 4 * q^88 - 20 * q^89 + 2 * q^90 - 8 * q^91 + 4 * q^92 + 8 * q^94 - 4 * q^95 - 24 * q^97 - 2 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.41421	0.577350
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	−1.00000	−0.353553
$9$	−1.00000	−0.333333
$10$	−1.00000	−0.316228
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.41421	−0.408248
$13$	−0.585786	−0.162468	−0.0812340	−	0.996695i	$-0.525886\pi$
$13$	−0.585786	−0.162468	−0.0812340	+	0.996695i	$0.525886\pi$
$14$	2.82843	0.755929
$15$	−1.41421	−0.365148
$16$	1.00000	0.250000
$17$	1.41421	0.342997	0.171499	−	0.985184i	$-0.445139\pi$
$17$	1.41421	0.342997	0.171499	+	0.985184i	$0.445139\pi$
$18$	1.00000	0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	1.00000	0.223607
$21$	4.00000	0.872872
$22$	2.00000	0.426401
$23$	3.41421	0.711913	0.355956	−	0.934503i	$-0.384155\pi$
$23$	3.41421	0.711913	0.355956	+	0.934503i	$0.384155\pi$
$24$	1.41421	0.288675
$25$	1.00000	0.200000
$26$	0.585786	0.114882
$27$	5.65685	1.08866
$28$	−2.82843	−0.534522
$29$	8.48528	1.57568	0.787839	−	0.615882i	$-0.211200\pi$
$29$	8.48528	1.57568	0.787839	+	0.615882i	$0.211200\pi$
$30$	1.41421	0.258199
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	2.82843	0.492366
$34$	−1.41421	−0.242536
$35$	−2.82843	−0.478091
$36$	−1.00000	−0.166667
$37$	10.2426	1.68388	0.841940	−	0.539571i	$-0.181414\pi$
$37$	10.2426	1.68388	0.841940	+	0.539571i	$0.181414\pi$
$38$	2.00000	0.324443
$39$	0.828427	0.132655
$40$	−1.00000	−0.158114
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	−4.00000	−0.617213
$43$	11.3137	1.72532	0.862662	−	0.505781i	$-0.168795\pi$
$43$	11.3137	1.72532	0.862662	+	0.505781i	$0.168795\pi$
$44$	−2.00000	−0.301511
$45$	−1.00000	−0.149071
$46$	−3.41421	−0.503398
$47$	−1.17157	−0.170891	−0.0854457	−	0.996343i	$-0.527231\pi$
$47$	−1.17157	−0.170891	−0.0854457	+	0.996343i	$0.527231\pi$
$48$	−1.41421	−0.204124
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	−0.585786	−0.0812340
$53$	−2.24264	−0.308050	−0.154025	−	0.988067i	$-0.549224\pi$
$53$	−2.24264	−0.308050	−0.154025	+	0.988067i	$0.549224\pi$
$54$	−5.65685	−0.769800
$55$	−2.00000	−0.269680
$56$	2.82843	0.377964
$57$	2.82843	0.374634
$58$	−8.48528	−1.11417
$59$	0.343146	0.0446738	0.0223369	−	0.999751i	$-0.492889\pi$
$59$	0.343146	0.0446738	0.0223369	+	0.999751i	$0.492889\pi$
$60$	−1.41421	−0.182574
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	4.00000	0.508001
$63$	2.82843	0.356348
$64$	1.00000	0.125000
$65$	−0.585786	−0.0726579
$66$	−2.82843	−0.348155
$67$	3.07107	0.375191	0.187595	−	0.982246i	$-0.439931\pi$
$67$	3.07107	0.375191	0.187595	+	0.982246i	$0.439931\pi$
$68$	1.41421	0.171499
$69$	−4.82843	−0.581274
$70$	2.82843	0.338062
$71$	−10.8284	−1.28510	−0.642549	−	0.766245i	$-0.722123\pi$
$71$	−10.8284	−1.28510	−0.642549	+	0.766245i	$0.722123\pi$
$72$	1.00000	0.117851
$73$	6.48528	0.759045	0.379522	−	0.925183i	$-0.376088\pi$
$73$	6.48528	0.759045	0.379522	+	0.925183i	$0.376088\pi$
$74$	−10.2426	−1.19068
$75$	−1.41421	−0.163299
$76$	−2.00000	−0.229416
$77$	5.65685	0.644658
$78$	−0.828427	−0.0938009
$79$	−17.6569	−1.98655	−0.993276	−	0.115773i	$-0.963065\pi$
$79$	−17.6569	−1.98655	−0.993276	+	0.115773i	$0.963065\pi$
$80$	1.00000	0.111803
$81$	−5.00000	−0.555556
$82$	5.65685	0.624695
$83$	−12.4853	−1.37044	−0.685219	−	0.728337i	$-0.740293\pi$
$83$	−12.4853	−1.37044	−0.685219	+	0.728337i	$0.740293\pi$
$84$	4.00000	0.436436
$85$	1.41421	0.153393
$86$	−11.3137	−1.21999
$87$	−12.0000	−1.28654
$88$	2.00000	0.213201
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	1.00000	0.105409
$91$	1.65685	0.173686
$92$	3.41421	0.355956
$93$	5.65685	0.586588
$94$	1.17157	0.120839
$95$	−2.00000	−0.205196
$96$	1.41421	0.144338
$97$	−10.5858	−1.07482	−0.537412	−	0.843320i	$-0.680598\pi$
$97$	−10.5858	−1.07482	−0.537412	+	0.843320i	$0.680598\pi$
$98$	−1.00000	−0.101015
$99$	2.00000	0.201008
$100$	1.00000	0.100000
$101$	12.4853	1.24233	0.621166	−	0.783679i	$-0.286659\pi$
$101$	12.4853	1.24233	0.621166	+	0.783679i	$0.286659\pi$
$102$	2.00000	0.198030
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0.585786	0.0574411
$105$	4.00000	0.390360
$106$	2.24264	0.217825
$107$	0.928932	0.0898033	0.0449016	−	0.998991i	$-0.485703\pi$
$107$	0.928932	0.0898033	0.0449016	+	0.998991i	$0.485703\pi$
$108$	5.65685	0.544331
$109$	−14.8284	−1.42031	−0.710153	−	0.704048i	$-0.751374\pi$
$109$	−14.8284	−1.42031	−0.710153	+	0.704048i	$0.751374\pi$
$110$	2.00000	0.190693
$111$	−14.4853	−1.37488
$112$	−2.82843	−0.267261
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−2.82843	−0.264906
$115$	3.41421	0.318377
$116$	8.48528	0.787839
$117$	0.585786	0.0541560
$118$	−0.343146	−0.0315891
$119$	−4.00000	−0.366679
$120$	1.41421	0.129099
$121$	−7.00000	−0.636364
$122$	−8.00000	−0.724286
$123$	8.00000	0.721336
$124$	−4.00000	−0.359211
$125$	1.00000	0.0894427
$126$	−2.82843	−0.251976
$127$	0.585786	0.0519801	0.0259901	−	0.999662i	$-0.491726\pi$
$127$	0.585786	0.0519801	0.0259901	+	0.999662i	$0.491726\pi$
$128$	−1.00000	−0.0883883
$129$	−16.0000	−1.40872
$130$	0.585786	0.0513769
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	2.82843	0.246183
$133$	5.65685	0.490511
$134$	−3.07107	−0.265300
$135$	5.65685	0.486864
$136$	−1.41421	−0.121268
$137$	14.5858	1.24615	0.623074	−	0.782163i	$-0.285884\pi$
$137$	14.5858	1.24615	0.623074	+	0.782163i	$0.285884\pi$
$138$	4.82843	0.411023
$139$	11.1716	0.947560	0.473780	−	0.880643i	$-0.342889\pi$
$139$	11.1716	0.947560	0.473780	+	0.880643i	$0.342889\pi$
$140$	−2.82843	−0.239046
$141$	1.65685	0.139532
$142$	10.8284	0.908701
$143$	1.17157	0.0979718
$144$	−1.00000	−0.0833333
$145$	8.48528	0.704664
$146$	−6.48528	−0.536726
$147$	−1.41421	−0.116642
$148$	10.2426	0.841940
$149$	−3.65685	−0.299581	−0.149791	−	0.988718i	$-0.547860\pi$
$149$	−3.65685	−0.299581	−0.149791	+	0.988718i	$0.547860\pi$
$150$	1.41421	0.115470
$151$	−10.4853	−0.853280	−0.426640	−	0.904422i	$-0.640303\pi$
$151$	−10.4853	−0.853280	−0.426640	+	0.904422i	$0.640303\pi$
$152$	2.00000	0.162221
$153$	−1.41421	−0.114332
$154$	−5.65685	−0.455842
$155$	−4.00000	−0.321288
$156$	0.828427	0.0663273
$157$	7.41421	0.591719	0.295859	−	0.955232i	$-0.404394\pi$
$157$	7.41421	0.591719	0.295859	+	0.955232i	$0.404394\pi$
$158$	17.6569	1.40470
$159$	3.17157	0.251522
$160$	−1.00000	−0.0790569
$161$	−9.65685	−0.761067
$162$	5.00000	0.392837
$163$	−15.5563	−1.21847	−0.609234	−	0.792991i	$-0.708523\pi$
$163$	−15.5563	−1.21847	−0.609234	+	0.792991i	$0.708523\pi$
$164$	−5.65685	−0.441726
$165$	2.82843	0.220193
$166$	12.4853	0.969046
$167$	−6.24264	−0.483070	−0.241535	−	0.970392i	$-0.577651\pi$
$167$	−6.24264	−0.483070	−0.241535	+	0.970392i	$0.577651\pi$
$168$	−4.00000	−0.308607
$169$	−12.6569	−0.973604
$170$	−1.41421	−0.108465
$171$	2.00000	0.152944
$172$	11.3137	0.862662
$173$	−1.51472	−0.115162	−0.0575810	−	0.998341i	$-0.518339\pi$
$173$	−1.51472	−0.115162	−0.0575810	+	0.998341i	$0.518339\pi$
$174$	12.0000	0.909718
$175$	−2.82843	−0.213809
$176$	−2.00000	−0.150756
$177$	−0.485281	−0.0364760
$178$	10.0000	0.749532
$179$	−4.34315	−0.324622	−0.162311	−	0.986740i	$-0.551895\pi$
$179$	−4.34315	−0.324622	−0.162311	+	0.986740i	$0.551895\pi$
$180$	−1.00000	−0.0745356
$181$	14.9706	1.11275	0.556377	−	0.830930i	$-0.312191\pi$
$181$	14.9706	1.11275	0.556377	+	0.830930i	$0.312191\pi$
$182$	−1.65685	−0.122814
$183$	−11.3137	−0.836333
$184$	−3.41421	−0.251699
$185$	10.2426	0.753054
$186$	−5.65685	−0.414781
$187$	−2.82843	−0.206835
$188$	−1.17157	−0.0854457
$189$	−16.0000	−1.16383
$190$	2.00000	0.145095
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	−1.41421	−0.102062
$193$	21.8995	1.57636	0.788180	−	0.615445i	$-0.211024\pi$
$193$	21.8995	1.57636	0.788180	+	0.615445i	$0.211024\pi$
$194$	10.5858	0.760015
$195$	0.828427	0.0593249
$196$	1.00000	0.0714286
$197$	−14.9706	−1.06661	−0.533304	−	0.845924i	$-0.679050\pi$
$197$	−14.9706	−1.06661	−0.533304	+	0.845924i	$0.679050\pi$
$198$	−2.00000	−0.142134
$199$	13.1716	0.933708	0.466854	−	0.884334i	$-0.345387\pi$
$199$	13.1716	0.933708	0.466854	+	0.884334i	$0.345387\pi$
$200$	−1.00000	−0.0707107
$201$	−4.34315	−0.306342
$202$	−12.4853	−0.878461
$203$	−24.0000	−1.68447
$204$	−2.00000	−0.140028
$205$	−5.65685	−0.395092
$206$	0	0
$207$	−3.41421	−0.237304
$208$	−0.585786	−0.0406170
$209$	4.00000	0.276686
$210$	−4.00000	−0.276026
$211$	−11.6569	−0.802491	−0.401245	−	0.915971i	$-0.631423\pi$
$211$	−11.6569	−0.802491	−0.401245	+	0.915971i	$0.631423\pi$
$212$	−2.24264	−0.154025
$213$	15.3137	1.04928
$214$	−0.928932	−0.0635005
$215$	11.3137	0.771589
$216$	−5.65685	−0.384900
$217$	11.3137	0.768025
$218$	14.8284	1.00431
$219$	−9.17157	−0.619757
$220$	−2.00000	−0.134840
$221$	−0.828427	−0.0557260
$222$	14.4853	0.972188
$223$	−17.6569	−1.18239	−0.591195	−	0.806529i	$-0.701344\pi$
$223$	−17.6569	−1.18239	−0.591195	+	0.806529i	$0.701344\pi$
$224$	2.82843	0.188982
$225$	−1.00000	−0.0666667
$226$	6.00000	0.399114
$227$	−6.58579	−0.437114	−0.218557	−	0.975824i	$-0.570135\pi$
$227$	−6.58579	−0.437114	−0.218557	+	0.975824i	$0.570135\pi$
$228$	2.82843	0.187317
$229$	−13.3137	−0.879795	−0.439897	−	0.898048i	$-0.644985\pi$
$229$	−13.3137	−0.879795	−0.439897	+	0.898048i	$0.644985\pi$
$230$	−3.41421	−0.225127
$231$	−8.00000	−0.526361
$232$	−8.48528	−0.557086
$233$	7.75736	0.508202	0.254101	−	0.967178i	$-0.418221\pi$
$233$	7.75736	0.508202	0.254101	+	0.967178i	$0.418221\pi$
$234$	−0.585786	−0.0382941
$235$	−1.17157	−0.0764250
$236$	0.343146	0.0223369
$237$	24.9706	1.62201
$238$	4.00000	0.259281
$239$	−14.4853	−0.936975	−0.468487	−	0.883470i	$-0.655201\pi$
$239$	−14.4853	−0.936975	−0.468487	+	0.883470i	$0.655201\pi$
$240$	−1.41421	−0.0912871
$241$	15.6569	1.00855	0.504273	−	0.863544i	$-0.331760\pi$
$241$	15.6569	1.00855	0.504273	+	0.863544i	$0.331760\pi$
$242$	7.00000	0.449977
$243$	−9.89949	−0.635053
$244$	8.00000	0.512148
$245$	1.00000	0.0638877
$246$	−8.00000	−0.510061
$247$	1.17157	0.0745454
$248$	4.00000	0.254000
$249$	17.6569	1.11896
$250$	−1.00000	−0.0632456
$251$	21.3137	1.34531	0.672655	−	0.739957i	$-0.265154\pi$
$251$	21.3137	1.34531	0.672655	+	0.739957i	$0.265154\pi$
$252$	2.82843	0.178174
$253$	−6.82843	−0.429300
$254$	−0.585786	−0.0367555
$255$	−2.00000	−0.125245
$256$	1.00000	0.0625000
$257$	23.1716	1.44540	0.722702	−	0.691160i	$-0.242900\pi$
$257$	23.1716	1.44540	0.722702	+	0.691160i	$0.242900\pi$
$258$	16.0000	0.996116
$259$	−28.9706	−1.80014
$260$	−0.585786	−0.0363289
$261$	−8.48528	−0.525226
$262$	−10.0000	−0.617802
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−2.82843	−0.174078
$265$	−2.24264	−0.137764
$266$	−5.65685	−0.346844
$267$	14.1421	0.865485
$268$	3.07107	0.187595
$269$	15.7990	0.963281	0.481641	−	0.876369i	$-0.340041\pi$
$269$	15.7990	0.963281	0.481641	+	0.876369i	$0.340041\pi$
$270$	−5.65685	−0.344265
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	1.41421	0.0857493
$273$	−2.34315	−0.141814
$274$	−14.5858	−0.881160
$275$	−2.00000	−0.120605
$276$	−4.82843	−0.290637
$277$	−30.2426	−1.81710	−0.908552	−	0.417772i	$-0.862811\pi$
$277$	−30.2426	−1.81710	−0.908552	+	0.417772i	$0.862811\pi$
$278$	−11.1716	−0.670026
$279$	4.00000	0.239474
$280$	2.82843	0.169031
$281$	−13.7990	−0.823179	−0.411589	−	0.911369i	$-0.635026\pi$
$281$	−13.7990	−0.823179	−0.411589	+	0.911369i	$0.635026\pi$
$282$	−1.65685	−0.0986642
$283$	−15.0711	−0.895882	−0.447941	−	0.894063i	$-0.647843\pi$
$283$	−15.0711	−0.895882	−0.447941	+	0.894063i	$0.647843\pi$
$284$	−10.8284	−0.642549
$285$	2.82843	0.167542
$286$	−1.17157	−0.0692766
$287$	16.0000	0.944450
$288$	1.00000	0.0589256
$289$	−15.0000	−0.882353
$290$	−8.48528	−0.498273
$291$	14.9706	0.877590
$292$	6.48528	0.379522
$293$	−18.2426	−1.06575	−0.532873	−	0.846195i	$-0.678888\pi$
$293$	−18.2426	−1.06575	−0.532873	+	0.846195i	$0.678888\pi$
$294$	1.41421	0.0824786
$295$	0.343146	0.0199787
$296$	−10.2426	−0.595341
$297$	−11.3137	−0.656488
$298$	3.65685	0.211836
$299$	−2.00000	−0.115663
$300$	−1.41421	−0.0816497
$301$	−32.0000	−1.84445
$302$	10.4853	0.603360
$303$	−17.6569	−1.01436
$304$	−2.00000	−0.114708
$305$	8.00000	0.458079
$306$	1.41421	0.0808452
$307$	−25.6569	−1.46431	−0.732157	−	0.681136i	$-0.761486\pi$
$307$	−25.6569	−1.46431	−0.732157	+	0.681136i	$0.761486\pi$
$308$	5.65685	0.322329
$309$	0	0
$310$	4.00000	0.227185
$311$	−19.3137	−1.09518	−0.547590	−	0.836747i	$-0.684455\pi$
$311$	−19.3137	−1.09518	−0.547590	+	0.836747i	$0.684455\pi$
$312$	−0.828427	−0.0469005
$313$	−6.48528	−0.366570	−0.183285	−	0.983060i	$-0.558673\pi$
$313$	−6.48528	−0.366570	−0.183285	+	0.983060i	$0.558673\pi$
$314$	−7.41421	−0.418408
$315$	2.82843	0.159364
$316$	−17.6569	−0.993276
$317$	7.89949	0.443680	0.221840	−	0.975083i	$-0.428794\pi$
$317$	7.89949	0.443680	0.221840	+	0.975083i	$0.428794\pi$
$318$	−3.17157	−0.177853
$319$	−16.9706	−0.950169
$320$	1.00000	0.0559017
$321$	−1.31371	−0.0733241
$322$	9.65685	0.538155
$323$	−2.82843	−0.157378
$324$	−5.00000	−0.277778
$325$	−0.585786	−0.0324936
$326$	15.5563	0.861586
$327$	20.9706	1.15967
$328$	5.65685	0.312348
$329$	3.31371	0.182691
$330$	−2.82843	−0.155700
$331$	−2.00000	−0.109930	−0.0549650	−	0.998488i	$-0.517505\pi$
$331$	−2.00000	−0.109930	−0.0549650	+	0.998488i	$0.517505\pi$
$332$	−12.4853	−0.685219
$333$	−10.2426	−0.561293
$334$	6.24264	0.341582
$335$	3.07107	0.167790
$336$	4.00000	0.218218
$337$	−17.7990	−0.969573	−0.484786	−	0.874633i	$-0.661103\pi$
$337$	−17.7990	−0.969573	−0.484786	+	0.874633i	$0.661103\pi$
$338$	12.6569	0.688442
$339$	8.48528	0.460857
$340$	1.41421	0.0766965
$341$	8.00000	0.433224
$342$	−2.00000	−0.108148
$343$	16.9706	0.916324
$344$	−11.3137	−0.609994
$345$	−4.82843	−0.259954
$346$	1.51472	0.0814318
$347$	1.41421	0.0759190	0.0379595	−	0.999279i	$-0.487914\pi$
$347$	1.41421	0.0759190	0.0379595	+	0.999279i	$0.487914\pi$
$348$	−12.0000	−0.643268
$349$	−29.6569	−1.58750	−0.793748	−	0.608247i	$-0.791873\pi$
$349$	−29.6569	−1.58750	−0.793748	+	0.608247i	$0.791873\pi$
$350$	2.82843	0.151186
$351$	−3.31371	−0.176873
$352$	2.00000	0.106600
$353$	29.2132	1.55486	0.777431	−	0.628968i	$-0.216522\pi$
$353$	29.2132	1.55486	0.777431	+	0.628968i	$0.216522\pi$
$354$	0.485281	0.0257924
$355$	−10.8284	−0.574713
$356$	−10.0000	−0.529999
$357$	5.65685	0.299392
$358$	4.34315	0.229542
$359$	20.9706	1.10678	0.553392	−	0.832921i	$-0.313333\pi$
$359$	20.9706	1.10678	0.553392	+	0.832921i	$0.313333\pi$
$360$	1.00000	0.0527046
$361$	−15.0000	−0.789474
$362$	−14.9706	−0.786835
$363$	9.89949	0.519589
$364$	1.65685	0.0868428
$365$	6.48528	0.339455
$366$	11.3137	0.591377
$367$	−1.75736	−0.0917334	−0.0458667	−	0.998948i	$-0.514605\pi$
$367$	−1.75736	−0.0917334	−0.0458667	+	0.998948i	$0.514605\pi$
$368$	3.41421	0.177978
$369$	5.65685	0.294484
$370$	−10.2426	−0.532490
$371$	6.34315	0.329320
$372$	5.65685	0.293294
$373$	20.1421	1.04292	0.521460	−	0.853276i	$-0.325388\pi$
$373$	20.1421	1.04292	0.521460	+	0.853276i	$0.325388\pi$
$374$	2.82843	0.146254
$375$	−1.41421	−0.0730297
$376$	1.17157	0.0604193
$377$	−4.97056	−0.255997
$378$	16.0000	0.822951
$379$	−4.34315	−0.223092	−0.111546	−	0.993759i	$-0.535580\pi$
$379$	−4.34315	−0.223092	−0.111546	+	0.993759i	$0.535580\pi$
$380$	−2.00000	−0.102598
$381$	−0.828427	−0.0424416
$382$	4.00000	0.204658
$383$	−14.8284	−0.757697	−0.378849	−	0.925459i	$-0.623680\pi$
$383$	−14.8284	−0.757697	−0.378849	+	0.925459i	$0.623680\pi$
$384$	1.41421	0.0721688
$385$	5.65685	0.288300
$386$	−21.8995	−1.11465
$387$	−11.3137	−0.575108
$388$	−10.5858	−0.537412
$389$	−14.1421	−0.717035	−0.358517	−	0.933523i	$-0.616718\pi$
$389$	−14.1421	−0.717035	−0.358517	+	0.933523i	$0.616718\pi$
$390$	−0.828427	−0.0419490
$391$	4.82843	0.244184
$392$	−1.00000	−0.0505076
$393$	−14.1421	−0.713376
$394$	14.9706	0.754206
$395$	−17.6569	−0.888413
$396$	2.00000	0.100504
$397$	−34.2843	−1.72068	−0.860339	−	0.509722i	$-0.829748\pi$
$397$	−34.2843	−1.72068	−0.860339	+	0.509722i	$0.829748\pi$
$398$	−13.1716	−0.660231
$399$	−8.00000	−0.400501
$400$	1.00000	0.0500000
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	4.34315	0.216616
$403$	2.34315	0.116720
$404$	12.4853	0.621166
$405$	−5.00000	−0.248452
$406$	24.0000	1.19110
$407$	−20.4853	−1.01542
$408$	2.00000	0.0990148
$409$	13.6569	0.675288	0.337644	−	0.941274i	$-0.390370\pi$
$409$	13.6569	0.675288	0.337644	+	0.941274i	$0.390370\pi$
$410$	5.65685	0.279372
$411$	−20.6274	−1.01748
$412$	0	0
$413$	−0.970563	−0.0477583
$414$	3.41421	0.167799
$415$	−12.4853	−0.612878
$416$	0.585786	0.0287205
$417$	−15.7990	−0.773680
$418$	−4.00000	−0.195646
$419$	2.97056	0.145121	0.0725607	−	0.997364i	$-0.476883\pi$
$419$	2.97056	0.145121	0.0725607	+	0.997364i	$0.476883\pi$
$420$	4.00000	0.195180
$421$	21.1716	1.03184	0.515920	−	0.856637i	$-0.327450\pi$
$421$	21.1716	1.03184	0.515920	+	0.856637i	$0.327450\pi$
$422$	11.6569	0.567447
$423$	1.17157	0.0569638
$424$	2.24264	0.108912
$425$	1.41421	0.0685994
$426$	−15.3137	−0.741952
$427$	−22.6274	−1.09502
$428$	0.928932	0.0449016
$429$	−1.65685	−0.0799937
$430$	−11.3137	−0.545595
$431$	−5.65685	−0.272481	−0.136241	−	0.990676i	$-0.543502\pi$
$431$	−5.65685	−0.272481	−0.136241	+	0.990676i	$0.543502\pi$
$432$	5.65685	0.272166
$433$	−18.6863	−0.898006	−0.449003	−	0.893530i	$-0.648221\pi$
$433$	−18.6863	−0.898006	−0.449003	+	0.893530i	$0.648221\pi$
$434$	−11.3137	−0.543075
$435$	−12.0000	−0.575356
$436$	−14.8284	−0.710153
$437$	−6.82843	−0.326648
$438$	9.17157	0.438235
$439$	−32.9706	−1.57360	−0.786800	−	0.617209i	$-0.788263\pi$
$439$	−32.9706	−1.57360	−0.786800	+	0.617209i	$0.788263\pi$
$440$	2.00000	0.0953463
$441$	−1.00000	−0.0476190
$442$	0.828427	0.0394043
$443$	9.89949	0.470339	0.235170	−	0.971954i	$-0.424435\pi$
$443$	9.89949	0.470339	0.235170	+	0.971954i	$0.424435\pi$
$444$	−14.4853	−0.687441
$445$	−10.0000	−0.474045
$446$	17.6569	0.836076
$447$	5.17157	0.244607
$448$	−2.82843	−0.133631
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	1.00000	0.0471405
$451$	11.3137	0.532742
$452$	−6.00000	−0.282216
$453$	14.8284	0.696700
$454$	6.58579	0.309086
$455$	1.65685	0.0776745
$456$	−2.82843	−0.132453
$457$	−1.31371	−0.0614527	−0.0307263	−	0.999528i	$-0.509782\pi$
$457$	−1.31371	−0.0614527	−0.0307263	+	0.999528i	$0.509782\pi$
$458$	13.3137	0.622109
$459$	8.00000	0.373408
$460$	3.41421	0.159189
$461$	22.8284	1.06323	0.531613	−	0.846987i	$-0.321586\pi$
$461$	22.8284	1.06323	0.531613	+	0.846987i	$0.321586\pi$
$462$	8.00000	0.372194
$463$	17.0711	0.793360	0.396680	−	0.917957i	$-0.370162\pi$
$463$	17.0711	0.793360	0.396680	+	0.917957i	$0.370162\pi$
$464$	8.48528	0.393919
$465$	5.65685	0.262330
$466$	−7.75736	−0.359353
$467$	−1.89949	−0.0878981	−0.0439491	−	0.999034i	$-0.513994\pi$
$467$	−1.89949	−0.0878981	−0.0439491	+	0.999034i	$0.513994\pi$
$468$	0.585786	0.0270780
$469$	−8.68629	−0.401096
$470$	1.17157	0.0540406
$471$	−10.4853	−0.483136
$472$	−0.343146	−0.0157946
$473$	−22.6274	−1.04041
$474$	−24.9706	−1.14694
$475$	−2.00000	−0.0917663
$476$	−4.00000	−0.183340
$477$	2.24264	0.102683
$478$	14.4853	0.662541
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	1.41421	0.0645497
$481$	−6.00000	−0.273576
$482$	−15.6569	−0.713150
$483$	13.6569	0.621408
$484$	−7.00000	−0.318182
$485$	−10.5858	−0.480676
$486$	9.89949	0.449050
$487$	−30.1421	−1.36587	−0.682935	−	0.730479i	$-0.739297\pi$
$487$	−30.1421	−1.36587	−0.682935	+	0.730479i	$0.739297\pi$
$488$	−8.00000	−0.362143
$489$	22.0000	0.994874
$490$	−1.00000	−0.0451754
$491$	−19.3137	−0.871615	−0.435808	−	0.900040i	$-0.643537\pi$
$491$	−19.3137	−0.871615	−0.435808	+	0.900040i	$0.643537\pi$
$492$	8.00000	0.360668
$493$	12.0000	0.540453
$494$	−1.17157	−0.0527116
$495$	2.00000	0.0898933
$496$	−4.00000	−0.179605
$497$	30.6274	1.37383
$498$	−17.6569	−0.791223
$499$	−6.34315	−0.283958	−0.141979	−	0.989870i	$-0.545347\pi$
$499$	−6.34315	−0.283958	−0.141979	+	0.989870i	$0.545347\pi$
$500$	1.00000	0.0447214
$501$	8.82843	0.394425
$502$	−21.3137	−0.951277
$503$	20.4853	0.913394	0.456697	−	0.889622i	$-0.349032\pi$
$503$	20.4853	0.913394	0.456697	+	0.889622i	$0.349032\pi$
$504$	−2.82843	−0.125988
$505$	12.4853	0.555588
$506$	6.82843	0.303561
$507$	17.8995	0.794944
$508$	0.585786	0.0259901
$509$	−32.0000	−1.41838	−0.709188	−	0.705020i	$-0.750938\pi$
$509$	−32.0000	−1.41838	−0.709188	+	0.705020i	$0.750938\pi$
$510$	2.00000	0.0885615
$511$	−18.3431	−0.811453
$512$	−1.00000	−0.0441942
$513$	−11.3137	−0.499512
$514$	−23.1716	−1.02205
$515$	0	0
$516$	−16.0000	−0.704361
$517$	2.34315	0.103051
$518$	28.9706	1.27289
$519$	2.14214	0.0940293
$520$	0.585786	0.0256884
$521$	11.6569	0.510696	0.255348	−	0.966849i	$-0.417810\pi$
$521$	11.6569	0.510696	0.255348	+	0.966849i	$0.417810\pi$
$522$	8.48528	0.371391
$523$	14.3848	0.629002	0.314501	−	0.949257i	$-0.398163\pi$
$523$	14.3848	0.629002	0.314501	+	0.949257i	$0.398163\pi$
$524$	10.0000	0.436852
$525$	4.00000	0.174574
$526$	−8.00000	−0.348817
$527$	−5.65685	−0.246416
$528$	2.82843	0.123091
$529$	−11.3431	−0.493180
$530$	2.24264	0.0974141
$531$	−0.343146	−0.0148913
$532$	5.65685	0.245256
$533$	3.31371	0.143533
$534$	−14.1421	−0.611990
$535$	0.928932	0.0401612
$536$	−3.07107	−0.132650
$537$	6.14214	0.265053
$538$	−15.7990	−0.681143
$539$	−2.00000	−0.0861461
$540$	5.65685	0.243432
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	−21.1716	−0.908559
$544$	−1.41421	−0.0606339
$545$	−14.8284	−0.635180
$546$	2.34315	0.100277
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	14.5858	0.623074
$549$	−8.00000	−0.341432
$550$	2.00000	0.0852803
$551$	−16.9706	−0.722970
$552$	4.82843	0.205512
$553$	49.9411	2.12371
$554$	30.2426	1.28489
$555$	−14.4853	−0.614866
$556$	11.1716	0.473780
$557$	41.1127	1.74200	0.871000	−	0.491282i	$-0.163472\pi$
$557$	41.1127	1.74200	0.871000	+	0.491282i	$0.163472\pi$
$558$	−4.00000	−0.169334
$559$	−6.62742	−0.280310
$560$	−2.82843	−0.119523
$561$	4.00000	0.168880
$562$	13.7990	0.582075
$563$	16.9706	0.715224	0.357612	−	0.933870i	$-0.383591\pi$
$563$	16.9706	0.715224	0.357612	+	0.933870i	$0.383591\pi$
$564$	1.65685	0.0697661
$565$	−6.00000	−0.252422
$566$	15.0711	0.633484
$567$	14.1421	0.593914
$568$	10.8284	0.454351
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	−2.82843	−0.118470
$571$	10.9706	0.459104	0.229552	−	0.973296i	$-0.426274\pi$
$571$	10.9706	0.459104	0.229552	+	0.973296i	$0.426274\pi$
$572$	1.17157	0.0489859
$573$	5.65685	0.236318
$574$	−16.0000	−0.667827
$575$	3.41421	0.142383
$576$	−1.00000	−0.0416667
$577$	35.6569	1.48441	0.742207	−	0.670171i	$-0.233779\pi$
$577$	35.6569	1.48441	0.742207	+	0.670171i	$0.233779\pi$
$578$	15.0000	0.623918
$579$	−30.9706	−1.28709
$580$	8.48528	0.352332
$581$	35.3137	1.46506
$582$	−14.9706	−0.620550
$583$	4.48528	0.185761
$584$	−6.48528	−0.268363
$585$	0.585786	0.0242193
$586$	18.2426	0.753597
$587$	−13.4558	−0.555382	−0.277691	−	0.960670i	$-0.589569\pi$
$587$	−13.4558	−0.555382	−0.277691	+	0.960670i	$0.589569\pi$
$588$	−1.41421	−0.0583212
$589$	8.00000	0.329634
$590$	−0.343146	−0.0141271
$591$	21.1716	0.870882
$592$	10.2426	0.420970
$593$	−45.0122	−1.84843	−0.924215	−	0.381873i	$-0.875279\pi$
$593$	−45.0122	−1.84843	−0.924215	+	0.381873i	$0.875279\pi$
$594$	11.3137	0.464207
$595$	−4.00000	−0.163984
$596$	−3.65685	−0.149791
$597$	−18.6274	−0.762369
$598$	2.00000	0.0817861
$599$	−20.8284	−0.851026	−0.425513	−	0.904952i	$-0.639906\pi$
$599$	−20.8284	−0.851026	−0.425513	+	0.904952i	$0.639906\pi$
$600$	1.41421	0.0577350
$601$	16.6274	0.678246	0.339123	−	0.940742i	$-0.389870\pi$
$601$	16.6274	0.678246	0.339123	+	0.940742i	$0.389870\pi$
$602$	32.0000	1.30422
$603$	−3.07107	−0.125064
$604$	−10.4853	−0.426640
$605$	−7.00000	−0.284590
$606$	17.6569	0.717261
$607$	−45.4558	−1.84500	−0.922498	−	0.386002i	$-0.873856\pi$
$607$	−45.4558	−1.84500	−0.922498	+	0.386002i	$0.873856\pi$
$608$	2.00000	0.0811107
$609$	33.9411	1.37536
$610$	−8.00000	−0.323911
$611$	0.686292	0.0277644
$612$	−1.41421	−0.0571662
$613$	11.6152	0.469134	0.234567	−	0.972100i	$-0.424633\pi$
$613$	11.6152	0.469134	0.234567	+	0.972100i	$0.424633\pi$
$614$	25.6569	1.03543
$615$	8.00000	0.322591
$616$	−5.65685	−0.227921
$617$	4.44365	0.178895	0.0894473	−	0.995992i	$-0.471490\pi$
$617$	4.44365	0.178895	0.0894473	+	0.995992i	$0.471490\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	−4.00000	−0.160644
$621$	19.3137	0.775032
$622$	19.3137	0.774409
$623$	28.2843	1.13319
$624$	0.828427	0.0331636
$625$	1.00000	0.0400000
$626$	6.48528	0.259204
$627$	−5.65685	−0.225913
$628$	7.41421	0.295859
$629$	14.4853	0.577566
$630$	−2.82843	−0.112687
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	17.6569	0.702352
$633$	16.4853	0.655231
$634$	−7.89949	−0.313729
$635$	0.585786	0.0232462
$636$	3.17157	0.125761
$637$	−0.585786	−0.0232097
$638$	16.9706	0.671871
$639$	10.8284	0.428366
$640$	−1.00000	−0.0395285
$641$	40.1421	1.58552	0.792760	−	0.609535i	$-0.208644\pi$
$641$	40.1421	1.58552	0.792760	+	0.609535i	$0.208644\pi$
$642$	1.31371	0.0518479
$643$	−17.6569	−0.696318	−0.348159	−	0.937435i	$-0.613193\pi$
$643$	−17.6569	−0.696318	−0.348159	+	0.937435i	$0.613193\pi$
$644$	−9.65685	−0.380533
$645$	−16.0000	−0.629999
$646$	2.82843	0.111283
$647$	−8.38478	−0.329640	−0.164820	−	0.986324i	$-0.552704\pi$
$647$	−8.38478	−0.329640	−0.164820	+	0.986324i	$0.552704\pi$
$648$	5.00000	0.196419
$649$	−0.686292	−0.0269393
$650$	0.585786	0.0229764
$651$	−16.0000	−0.627089
$652$	−15.5563	−0.609234
$653$	24.6274	0.963745	0.481873	−	0.876241i	$-0.339957\pi$
$653$	24.6274	0.963745	0.481873	+	0.876241i	$0.339957\pi$
$654$	−20.9706	−0.820014
$655$	10.0000	0.390732
$656$	−5.65685	−0.220863
$657$	−6.48528	−0.253015
$658$	−3.31371	−0.129182
$659$	14.9706	0.583170	0.291585	−	0.956545i	$-0.405817\pi$
$659$	14.9706	0.583170	0.291585	+	0.956545i	$0.405817\pi$
$660$	2.82843	0.110096
$661$	−24.2843	−0.944549	−0.472274	−	0.881452i	$-0.656567\pi$
$661$	−24.2843	−0.944549	−0.472274	+	0.881452i	$0.656567\pi$
$662$	2.00000	0.0777322
$663$	1.17157	0.0455001
$664$	12.4853	0.484523
$665$	5.65685	0.219363
$666$	10.2426	0.396894
$667$	28.9706	1.12174
$668$	−6.24264	−0.241535
$669$	24.9706	0.965418
$670$	−3.07107	−0.118646
$671$	−16.0000	−0.617673
$672$	−4.00000	−0.154303
$673$	−30.1838	−1.16350	−0.581749	−	0.813368i	$-0.697632\pi$
$673$	−30.1838	−1.16350	−0.581749	+	0.813368i	$0.697632\pi$
$674$	17.7990	0.685591
$675$	5.65685	0.217732
$676$	−12.6569	−0.486802
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	−8.48528	−0.325875
$679$	29.9411	1.14903
$680$	−1.41421	−0.0542326
$681$	9.31371	0.356902
$682$	−8.00000	−0.306336
$683$	−8.44365	−0.323087	−0.161544	−	0.986866i	$-0.551647\pi$
$683$	−8.44365	−0.323087	−0.161544	+	0.986866i	$0.551647\pi$
$684$	2.00000	0.0764719
$685$	14.5858	0.557294
$686$	−16.9706	−0.647939
$687$	18.8284	0.718349
$688$	11.3137	0.431331
$689$	1.31371	0.0500483
$690$	4.82843	0.183815
$691$	44.9706	1.71076	0.855380	−	0.518000i	$-0.173323\pi$
$691$	44.9706	1.71076	0.855380	+	0.518000i	$0.173323\pi$
$692$	−1.51472	−0.0575810
$693$	−5.65685	−0.214886
$694$	−1.41421	−0.0536828
$695$	11.1716	0.423762
$696$	12.0000	0.454859
$697$	−8.00000	−0.303022
$698$	29.6569	1.12253
$699$	−10.9706	−0.414945
$700$	−2.82843	−0.106904
$701$	27.5980	1.04236	0.521181	−	0.853446i	$-0.325492\pi$
$701$	27.5980	1.04236	0.521181	+	0.853446i	$0.325492\pi$
$702$	3.31371	0.125068
$703$	−20.4853	−0.772617
$704$	−2.00000	−0.0753778
$705$	1.65685	0.0624007
$706$	−29.2132	−1.09945
$707$	−35.3137	−1.32811
$708$	−0.485281	−0.0182380
$709$	14.9706	0.562231	0.281116	−	0.959674i	$-0.409296\pi$
$709$	14.9706	0.562231	0.281116	+	0.959674i	$0.409296\pi$
$710$	10.8284	0.406384
$711$	17.6569	0.662184
$712$	10.0000	0.374766
$713$	−13.6569	−0.511453
$714$	−5.65685	−0.211702
$715$	1.17157	0.0438143
$716$	−4.34315	−0.162311
$717$	20.4853	0.765037
$718$	−20.9706	−0.782614
$719$	−6.48528	−0.241860	−0.120930	−	0.992661i	$-0.538588\pi$
$719$	−6.48528	−0.241860	−0.120930	+	0.992661i	$0.538588\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	15.0000	0.558242
$723$	−22.1421	−0.823475
$724$	14.9706	0.556377
$725$	8.48528	0.315135
$726$	−9.89949	−0.367405
$727$	28.1838	1.04528	0.522639	−	0.852554i	$-0.324948\pi$
$727$	28.1838	1.04528	0.522639	+	0.852554i	$0.324948\pi$
$728$	−1.65685	−0.0614071
$729$	29.0000	1.07407
$730$	−6.48528	−0.240031
$731$	16.0000	0.591781
$732$	−11.3137	−0.418167
$733$	−12.1421	−0.448480	−0.224240	−	0.974534i	$-0.571990\pi$
$733$	−12.1421	−0.448480	−0.224240	+	0.974534i	$0.571990\pi$
$734$	1.75736	0.0648653
$735$	−1.41421	−0.0521641
$736$	−3.41421	−0.125850
$737$	−6.14214	−0.226248
$738$	−5.65685	−0.208232
$739$	−10.6863	−0.393102	−0.196551	−	0.980494i	$-0.562974\pi$
$739$	−10.6863	−0.393102	−0.196551	+	0.980494i	$0.562974\pi$
$740$	10.2426	0.376527
$741$	−1.65685	−0.0608661
$742$	−6.34315	−0.232864
$743$	−25.3553	−0.930197	−0.465099	−	0.885259i	$-0.653981\pi$
$743$	−25.3553	−0.930197	−0.465099	+	0.885259i	$0.653981\pi$
$744$	−5.65685	−0.207390
$745$	−3.65685	−0.133977
$746$	−20.1421	−0.737456
$747$	12.4853	0.456813
$748$	−2.82843	−0.103418
$749$	−2.62742	−0.0960037
$750$	1.41421	0.0516398
$751$	38.6274	1.40953	0.704767	−	0.709439i	$-0.251051\pi$
$751$	38.6274	1.40953	0.704767	+	0.709439i	$0.251051\pi$
$752$	−1.17157	−0.0427229
$753$	−30.1421	−1.09844
$754$	4.97056	0.181017
$755$	−10.4853	−0.381598
$756$	−16.0000	−0.581914
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	4.34315	0.157750
$759$	9.65685	0.350522
$760$	2.00000	0.0725476
$761$	32.0000	1.16000	0.580000	−	0.814617i	$-0.303053\pi$
$761$	32.0000	1.16000	0.580000	+	0.814617i	$0.303053\pi$
$762$	0.828427	0.0300107
$763$	41.9411	1.51837
$764$	−4.00000	−0.144715
$765$	−1.41421	−0.0511310
$766$	14.8284	0.535773
$767$	−0.201010	−0.00725806
$768$	−1.41421	−0.0510310
$769$	12.1421	0.437857	0.218928	−	0.975741i	$-0.429744\pi$
$769$	12.1421	0.437857	0.218928	+	0.975741i	$0.429744\pi$
$770$	−5.65685	−0.203859
$771$	−32.7696	−1.18017
$772$	21.8995	0.788180
$773$	34.9706	1.25780	0.628902	−	0.777485i	$-0.283505\pi$
$773$	34.9706	1.25780	0.628902	+	0.777485i	$0.283505\pi$
$774$	11.3137	0.406663
$775$	−4.00000	−0.143684
$776$	10.5858	0.380008
$777$	40.9706	1.46981
$778$	14.1421	0.507020
$779$	11.3137	0.405356
$780$	0.828427	0.0296624
$781$	21.6569	0.774943
$782$	−4.82843	−0.172664
$783$	48.0000	1.71538
$784$	1.00000	0.0357143
$785$	7.41421	0.264625
$786$	14.1421	0.504433
$787$	−0.0416306	−0.00148397	−0.000741985	−	1.00000i	$-0.500236\pi$
$787$	−0.0416306	−0.00148397	−0.000741985		1.00000i	$0.500236\pi$
$788$	−14.9706	−0.533304
$789$	−11.3137	−0.402779
$790$	17.6569	0.628203
$791$	16.9706	0.603404
$792$	−2.00000	−0.0710669
$793$	−4.68629	−0.166415
$794$	34.2843	1.21670
$795$	3.17157	0.112484
$796$	13.1716	0.466854
$797$	22.2843	0.789349	0.394675	−	0.918821i	$-0.370857\pi$
$797$	22.2843	0.789349	0.394675	+	0.918821i	$0.370857\pi$
$798$	8.00000	0.283197
$799$	−1.65685	−0.0586153
$800$	−1.00000	−0.0353553
$801$	10.0000	0.353333
$802$	−1.00000	−0.0353112
$803$	−12.9706	−0.457721
$804$	−4.34315	−0.153171
$805$	−9.65685	−0.340359
$806$	−2.34315	−0.0825338
$807$	−22.3431	−0.786516
$808$	−12.4853	−0.439231
$809$	−50.2843	−1.76790	−0.883950	−	0.467581i	$-0.845126\pi$
$809$	−50.2843	−1.76790	−0.883950	+	0.467581i	$0.845126\pi$
$810$	5.00000	0.175682
$811$	12.2843	0.431359	0.215680	−	0.976464i	$-0.430803\pi$
$811$	12.2843	0.431359	0.215680	+	0.976464i	$0.430803\pi$
$812$	−24.0000	−0.842235
$813$	0	0
$814$	20.4853	0.718009
$815$	−15.5563	−0.544915
$816$	−2.00000	−0.0700140
$817$	−22.6274	−0.791633
$818$	−13.6569	−0.477501
$819$	−1.65685	−0.0578952
$820$	−5.65685	−0.197546
$821$	−20.0000	−0.698005	−0.349002	−	0.937122i	$-0.613479\pi$
$821$	−20.0000	−0.698005	−0.349002	+	0.937122i	$0.613479\pi$
$822$	20.6274	0.719464
$823$	−29.0711	−1.01335	−0.506677	−	0.862136i	$-0.669126\pi$
$823$	−29.0711	−1.01335	−0.506677	+	0.862136i	$0.669126\pi$
$824$	0	0
$825$	2.82843	0.0984732
$826$	0.970563	0.0337702
$827$	−14.6274	−0.508645	−0.254323	−	0.967119i	$-0.581852\pi$
$827$	−14.6274	−0.508645	−0.254323	+	0.967119i	$0.581852\pi$
$828$	−3.41421	−0.118652
$829$	−30.1421	−1.04688	−0.523440	−	0.852063i	$-0.675352\pi$
$829$	−30.1421	−1.04688	−0.523440	+	0.852063i	$0.675352\pi$
$830$	12.4853	0.433370
$831$	42.7696	1.48366
$832$	−0.585786	−0.0203085
$833$	1.41421	0.0489996
$834$	15.7990	0.547074
$835$	−6.24264	−0.216035
$836$	4.00000	0.138343
$837$	−22.6274	−0.782118
$838$	−2.97056	−0.102616
$839$	14.8284	0.511934	0.255967	−	0.966685i	$-0.417606\pi$
$839$	14.8284	0.511934	0.255967	+	0.966685i	$0.417606\pi$
$840$	−4.00000	−0.138013
$841$	43.0000	1.48276
$842$	−21.1716	−0.729621
$843$	19.5147	0.672123
$844$	−11.6569	−0.401245
$845$	−12.6569	−0.435409
$846$	−1.17157	−0.0402795
$847$	19.7990	0.680301
$848$	−2.24264	−0.0770126
$849$	21.3137	0.731485
$850$	−1.41421	−0.0485071
$851$	34.9706	1.19878
$852$	15.3137	0.524639
$853$	−40.1421	−1.37444	−0.687220	−	0.726449i	$-0.741169\pi$
$853$	−40.1421	−1.37444	−0.687220	+	0.726449i	$0.741169\pi$
$854$	22.6274	0.774294
$855$	2.00000	0.0683986
$856$	−0.928932	−0.0317502
$857$	−12.8284	−0.438211	−0.219105	−	0.975701i	$-0.570314\pi$
$857$	−12.8284	−0.438211	−0.219105	+	0.975701i	$0.570314\pi$
$858$	1.65685	0.0565641
$859$	12.3431	0.421143	0.210571	−	0.977578i	$-0.432468\pi$
$859$	12.3431	0.421143	0.210571	+	0.977578i	$0.432468\pi$
$860$	11.3137	0.385794
$861$	−22.6274	−0.771140
$862$	5.65685	0.192673
$863$	30.7279	1.04599	0.522995	−	0.852336i	$-0.324815\pi$
$863$	30.7279	1.04599	0.522995	+	0.852336i	$0.324815\pi$
$864$	−5.65685	−0.192450
$865$	−1.51472	−0.0515020
$866$	18.6863	0.634986
$867$	21.2132	0.720438
$868$	11.3137	0.384012
$869$	35.3137	1.19794
$870$	12.0000	0.406838
$871$	−1.79899	−0.0609564
$872$	14.8284	0.502154
$873$	10.5858	0.358275
$874$	6.82843	0.230975
$875$	−2.82843	−0.0956183
$876$	−9.17157	−0.309879
$877$	−4.38478	−0.148063	−0.0740317	−	0.997256i	$-0.523587\pi$
$877$	−4.38478	−0.148063	−0.0740317	+	0.997256i	$0.523587\pi$
$878$	32.9706	1.11270
$879$	25.7990	0.870178
$880$	−2.00000	−0.0674200
$881$	−57.1127	−1.92418	−0.962088	−	0.272740i	$-0.912070\pi$
$881$	−57.1127	−1.92418	−0.962088	+	0.272740i	$0.912070\pi$
$882$	1.00000	0.0336718
$883$	−21.6569	−0.728811	−0.364406	−	0.931240i	$-0.618728\pi$
$883$	−21.6569	−0.728811	−0.364406	+	0.931240i	$0.618728\pi$
$884$	−0.828427	−0.0278630
$885$	−0.485281	−0.0163126
$886$	−9.89949	−0.332580
$887$	42.0416	1.41162	0.705810	−	0.708401i	$-0.250583\pi$
$887$	42.0416	1.41162	0.705810	+	0.708401i	$0.250583\pi$
$888$	14.4853	0.486094
$889$	−1.65685	−0.0555691
$890$	10.0000	0.335201
$891$	10.0000	0.335013
$892$	−17.6569	−0.591195
$893$	2.34315	0.0784104
$894$	−5.17157	−0.172963
$895$	−4.34315	−0.145175
$896$	2.82843	0.0944911
$897$	2.82843	0.0944384
$898$	14.0000	0.467186
$899$	−33.9411	−1.13200
$900$	−1.00000	−0.0333333
$901$	−3.17157	−0.105660
$902$	−11.3137	−0.376705
$903$	45.2548	1.50599
$904$	6.00000	0.199557
$905$	14.9706	0.497638
$906$	−14.8284	−0.492641
$907$	−1.61522	−0.0536326	−0.0268163	−	0.999640i	$-0.508537\pi$
$907$	−1.61522	−0.0536326	−0.0268163	+	0.999640i	$0.508537\pi$
$908$	−6.58579	−0.218557
$909$	−12.4853	−0.414111
$910$	−1.65685	−0.0549242
$911$	−37.2548	−1.23431	−0.617154	−	0.786842i	$-0.711714\pi$
$911$	−37.2548	−1.23431	−0.617154	+	0.786842i	$0.711714\pi$
$912$	2.82843	0.0936586
$913$	24.9706	0.826405
$914$	1.31371	0.0434536
$915$	−11.3137	−0.374020
$916$	−13.3137	−0.439897
$917$	−28.2843	−0.934029
$918$	−8.00000	−0.264039
$919$	−16.9706	−0.559807	−0.279904	−	0.960028i	$-0.590303\pi$
$919$	−16.9706	−0.559807	−0.279904	+	0.960028i	$0.590303\pi$
$920$	−3.41421	−0.112563
$921$	36.2843	1.19561
$922$	−22.8284	−0.751814
$923$	6.34315	0.208787
$924$	−8.00000	−0.263181
$925$	10.2426	0.336776
$926$	−17.0711	−0.560990
$927$	0	0
$928$	−8.48528	−0.278543
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	−5.65685	−0.185496
$931$	−2.00000	−0.0655474
$932$	7.75736	0.254101
$933$	27.3137	0.894211
$934$	1.89949	0.0621534
$935$	−2.82843	−0.0924995
$936$	−0.585786	−0.0191470
$937$	30.1005	0.983341	0.491670	−	0.870781i	$-0.336387\pi$
$937$	30.1005	0.983341	0.491670	+	0.870781i	$0.336387\pi$
$938$	8.68629	0.283617
$939$	9.17157	0.299303
$940$	−1.17157	−0.0382125
$941$	36.7696	1.19865	0.599327	−	0.800505i	$-0.295435\pi$
$941$	36.7696	1.19865	0.599327	+	0.800505i	$0.295435\pi$
$942$	10.4853	0.341629
$943$	−19.3137	−0.628941
$944$	0.343146	0.0111684
$945$	−16.0000	−0.520480
$946$	22.6274	0.735681
$947$	−45.4558	−1.47712	−0.738558	−	0.674190i	$-0.764493\pi$
$947$	−45.4558	−1.47712	−0.738558	+	0.674190i	$0.764493\pi$
$948$	24.9706	0.811006
$949$	−3.79899	−0.123320
$950$	2.00000	0.0648886
$951$	−11.1716	−0.362263
$952$	4.00000	0.129641
$953$	−2.28427	−0.0739948	−0.0369974	−	0.999315i	$-0.511779\pi$
$953$	−2.28427	−0.0739948	−0.0369974	+	0.999315i	$0.511779\pi$
$954$	−2.24264	−0.0726082
$955$	−4.00000	−0.129437
$956$	−14.4853	−0.468487
$957$	24.0000	0.775810
$958$	16.0000	0.516937
$959$	−41.2548	−1.33219
$960$	−1.41421	−0.0456435
$961$	−15.0000	−0.483871
$962$	6.00000	0.193448
$963$	−0.928932	−0.0299344
$964$	15.6569	0.504273
$965$	21.8995	0.704970
$966$	−13.6569	−0.439402
$967$	39.4975	1.27015	0.635077	−	0.772449i	$-0.280969\pi$
$967$	39.4975	1.27015	0.635077	+	0.772449i	$0.280969\pi$
$968$	7.00000	0.224989
$969$	4.00000	0.128499
$970$	10.5858	0.339889
$971$	8.00000	0.256732	0.128366	−	0.991727i	$-0.459027\pi$
$971$	8.00000	0.256732	0.128366	+	0.991727i	$0.459027\pi$
$972$	−9.89949	−0.317526
$973$	−31.5980	−1.01298
$974$	30.1421	0.965816
$975$	0.828427	0.0265309
$976$	8.00000	0.256074
$977$	−2.97056	−0.0950367	−0.0475184	−	0.998870i	$-0.515131\pi$
$977$	−2.97056	−0.0950367	−0.0475184	+	0.998870i	$0.515131\pi$
$978$	−22.0000	−0.703482
$979$	20.0000	0.639203
$980$	1.00000	0.0319438
$981$	14.8284	0.473435
$982$	19.3137	0.616325
$983$	12.9706	0.413697	0.206848	−	0.978373i	$-0.433679\pi$
$983$	12.9706	0.413697	0.206848	+	0.978373i	$0.433679\pi$
$984$	−8.00000	−0.255031
$985$	−14.9706	−0.477002
$986$	−12.0000	−0.382158
$987$	−4.68629	−0.149166
$988$	1.17157	0.0372727
$989$	38.6274	1.22828
$990$	−2.00000	−0.0635642
$991$	−1.45584	−0.0462464	−0.0231232	−	0.999733i	$-0.507361\pi$
$991$	−1.45584	−0.0462464	−0.0231232	+	0.999733i	$0.507361\pi$
$992$	4.00000	0.127000
$993$	2.82843	0.0897574
$994$	−30.6274	−0.971443
$995$	13.1716	0.417567
$996$	17.6569	0.559479
$997$	24.3431	0.770955	0.385478	−	0.922717i	$-0.374037\pi$
$997$	24.3431	0.770955	0.385478	+	0.922717i	$0.374037\pi$
$998$	6.34315	0.200789
$999$	57.9411	1.83318

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.g.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.g.1.1	✓	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4010 = 2 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4010.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.41421 q^{6} -2.82843 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.41421 q^{6} -2.82843 q^{7} -1.00000 q^{8} -1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} -1.41421 q^{12} -0.585786 q^{13} +2.82843 q^{14} -1.41421 q^{15} +1.00000 q^{16} +1.41421 q^{17} +1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +4.00000 q^{21} +2.00000 q^{22} +3.41421 q^{23} +1.41421 q^{24} +1.00000 q^{25} +0.585786 q^{26} +5.65685 q^{27} -2.82843 q^{28} +8.48528 q^{29} +1.41421 q^{30} -4.00000 q^{31} -1.00000 q^{32} +2.82843 q^{33} -1.41421 q^{34} -2.82843 q^{35} -1.00000 q^{36} +10.2426 q^{37} +2.00000 q^{38} +0.828427 q^{39} -1.00000 q^{40} -5.65685 q^{41} -4.00000 q^{42} +11.3137 q^{43} -2.00000 q^{44} -1.00000 q^{45} -3.41421 q^{46} -1.17157 q^{47} -1.41421 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} -0.585786 q^{52} -2.24264 q^{53} -5.65685 q^{54} -2.00000 q^{55} +2.82843 q^{56} +2.82843 q^{57} -8.48528 q^{58} +0.343146 q^{59} -1.41421 q^{60} +8.00000 q^{61} +4.00000 q^{62} +2.82843 q^{63} +1.00000 q^{64} -0.585786 q^{65} -2.82843 q^{66} +3.07107 q^{67} +1.41421 q^{68} -4.82843 q^{69} +2.82843 q^{70} -10.8284 q^{71} +1.00000 q^{72} +6.48528 q^{73} -10.2426 q^{74} -1.41421 q^{75} -2.00000 q^{76} +5.65685 q^{77} -0.828427 q^{78} -17.6569 q^{79} +1.00000 q^{80} -5.00000 q^{81} +5.65685 q^{82} -12.4853 q^{83} +4.00000 q^{84} +1.41421 q^{85} -11.3137 q^{86} -12.0000 q^{87} +2.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} +1.65685 q^{91} +3.41421 q^{92} +5.65685 q^{93} +1.17157 q^{94} -2.00000 q^{95} +1.41421 q^{96} -10.5858 q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^8 - 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{9} - 2 q^{10} - 4 q^{11} - 4 q^{13} + 2 q^{16} + 2 q^{18} - 4 q^{19} + 2 q^{20} + 8 q^{21} + 4 q^{22} + 4 q^{23} + 2 q^{25} + 4 q^{26} - 8 q^{31} - 2 q^{32} - 2 q^{36} + 12 q^{37} + 4 q^{38} - 4 q^{39} - 2 q^{40} - 8 q^{42} - 4 q^{44} - 2 q^{45} - 4 q^{46} - 8 q^{47} + 2 q^{49} - 2 q^{50} - 4 q^{51} - 4 q^{52} + 4 q^{53} - 4 q^{55} + 12 q^{59} + 16 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} - 4 q^{69} - 16 q^{71} + 2 q^{72} - 4 q^{73} - 12 q^{74} - 4 q^{76} + 4 q^{78} - 24 q^{79} + 2 q^{80} - 10 q^{81} - 8 q^{83} + 8 q^{84} - 24 q^{87} + 4 q^{88} - 20 q^{89} + 2 q^{90} - 8 q^{91} + 4 q^{92} + 8 q^{94} - 4 q^{95} - 24 q^{97} - 2 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^8 - 2 * q^9 - 2 * q^10 - 4 * q^11 - 4 * q^13 + 2 * q^16 + 2 * q^18 - 4 * q^19 + 2 * q^20 + 8 * q^21 + 4 * q^22 + 4 * q^23 + 2 * q^25 + 4 * q^26 - 8 * q^31 - 2 * q^32 - 2 * q^36 + 12 * q^37 + 4 * q^38 - 4 * q^39 - 2 * q^40 - 8 * q^42 - 4 * q^44 - 2 * q^45 - 4 * q^46 - 8 * q^47 + 2 * q^49 - 2 * q^50 - 4 * q^51 - 4 * q^52 + 4 * q^53 - 4 * q^55 + 12 * q^59 + 16 * q^61 + 8 * q^62 + 2 * q^64 - 4 * q^65 - 8 * q^67 - 4 * q^69 - 16 * q^71 + 2 * q^72 - 4 * q^73 - 12 * q^74 - 4 * q^76 + 4 * q^78 - 24 * q^79 + 2 * q^80 - 10 * q^81 - 8 * q^83 + 8 * q^84 - 24 * q^87 + 4 * q^88 - 20 * q^89 + 2 * q^90 - 8 * q^91 + 4 * q^92 + 8 * q^94 - 4 * q^95 - 24 * q^97 - 2 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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