LMFDB - Embedded newform 4010.2.a.g.1.2

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			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.41421 q^{6} +2.82843 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.41421 q^{6} +2.82843 q^{7} -1.00000 q^{8} -1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} +1.41421 q^{12} -3.41421 q^{13} -2.82843 q^{14} +1.41421 q^{15} +1.00000 q^{16} -1.41421 q^{17} +1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +4.00000 q^{21} +2.00000 q^{22} +0.585786 q^{23} -1.41421 q^{24} +1.00000 q^{25} +3.41421 q^{26} -5.65685 q^{27} +2.82843 q^{28} -8.48528 q^{29} -1.41421 q^{30} -4.00000 q^{31} -1.00000 q^{32} -2.82843 q^{33} +1.41421 q^{34} +2.82843 q^{35} -1.00000 q^{36} +1.75736 q^{37} +2.00000 q^{38} -4.82843 q^{39} -1.00000 q^{40} +5.65685 q^{41} -4.00000 q^{42} -11.3137 q^{43} -2.00000 q^{44} -1.00000 q^{45} -0.585786 q^{46} -6.82843 q^{47} +1.41421 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} -3.41421 q^{52} +6.24264 q^{53} +5.65685 q^{54} -2.00000 q^{55} -2.82843 q^{56} -2.82843 q^{57} +8.48528 q^{58} +11.6569 q^{59} +1.41421 q^{60} +8.00000 q^{61} +4.00000 q^{62} -2.82843 q^{63} +1.00000 q^{64} -3.41421 q^{65} +2.82843 q^{66} -11.0711 q^{67} -1.41421 q^{68} +0.828427 q^{69} -2.82843 q^{70} -5.17157 q^{71} +1.00000 q^{72} -10.4853 q^{73} -1.75736 q^{74} +1.41421 q^{75} -2.00000 q^{76} -5.65685 q^{77} +4.82843 q^{78} -6.34315 q^{79} +1.00000 q^{80} -5.00000 q^{81} -5.65685 q^{82} +4.48528 q^{83} +4.00000 q^{84} -1.41421 q^{85} +11.3137 q^{86} -12.0000 q^{87} +2.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} -9.65685 q^{91} +0.585786 q^{92} -5.65685 q^{93} +6.82843 q^{94} -2.00000 q^{95} -1.41421 q^{96} -13.4142 q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	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	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	−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
$13$	−3.41421	−0.946932	−0.473466	−	0.880812i	$-0.656997\pi$
$13$	−3.41421	−0.946932	−0.473466	+	0.880812i	$0.656997\pi$
$14$	−2.82843	−0.755929
$15$	1.41421	0.365148
$16$	1.00000	0.250000
$17$	−1.41421	−0.342997	−0.171499	−	0.985184i	$-0.554861\pi$
$17$	−1.41421	−0.342997	−0.171499	+	0.985184i	$0.554861\pi$
$18$	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
$23$	0.585786	0.122145	0.0610725	−	0.998133i	$-0.480548\pi$
$23$	0.585786	0.122145	0.0610725	+	0.998133i	$0.480548\pi$
$24$	−1.41421	−0.288675
$25$	1.00000	0.200000
$26$	3.41421	0.669582
$27$	−5.65685	−1.08866
$28$	2.82843	0.534522
$29$	−8.48528	−1.57568	−0.787839	−	0.615882i	$-0.788800\pi$
$29$	−8.48528	−1.57568	−0.787839	+	0.615882i	$0.788800\pi$
$30$	−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
$37$	1.75736	0.288908	0.144454	−	0.989512i	$-0.453857\pi$
$37$	1.75736	0.288908	0.144454	+	0.989512i	$0.453857\pi$
$38$	2.00000	0.324443
$39$	−4.82843	−0.773167
$40$	−1.00000	−0.158114
$41$	5.65685	0.883452	0.441726	−	0.897150i	$-0.354366\pi$
$41$	5.65685	0.883452	0.441726	+	0.897150i	$0.354366\pi$
$42$	−4.00000	−0.617213
$43$	−11.3137	−1.72532	−0.862662	−	0.505781i	$-0.831205\pi$
$43$	−11.3137	−1.72532	−0.862662	+	0.505781i	$0.831205\pi$
$44$	−2.00000	−0.301511
$45$	−1.00000	−0.149071
$46$	−0.585786	−0.0863695
$47$	−6.82843	−0.996028	−0.498014	−	0.867169i	$-0.665937\pi$
$47$	−6.82843	−0.996028	−0.498014	+	0.867169i	$0.665937\pi$
$48$	1.41421	0.204124
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	−3.41421	−0.473466
$53$	6.24264	0.857493	0.428746	−	0.903425i	$-0.358955\pi$
$53$	6.24264	0.857493	0.428746	+	0.903425i	$0.358955\pi$
$54$	5.65685	0.769800
$55$	−2.00000	−0.269680
$56$	−2.82843	−0.377964
$57$	−2.82843	−0.374634
$58$	8.48528	1.11417
$59$	11.6569	1.51759	0.758797	−	0.651328i	$-0.225788\pi$
$59$	11.6569	1.51759	0.758797	+	0.651328i	$0.225788\pi$
$60$	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
$65$	−3.41421	−0.423481
$66$	2.82843	0.348155
$67$	−11.0711	−1.35255	−0.676273	−	0.736651i	$-0.736406\pi$
$67$	−11.0711	−1.35255	−0.676273	+	0.736651i	$0.736406\pi$
$68$	−1.41421	−0.171499
$69$	0.828427	0.0997309
$70$	−2.82843	−0.338062
$71$	−5.17157	−0.613753	−0.306876	−	0.951749i	$-0.599284\pi$
$71$	−5.17157	−0.613753	−0.306876	+	0.951749i	$0.599284\pi$
$72$	1.00000	0.117851
$73$	−10.4853	−1.22721	−0.613605	−	0.789613i	$-0.710281\pi$
$73$	−10.4853	−1.22721	−0.613605	+	0.789613i	$0.710281\pi$
$74$	−1.75736	−0.204289
$75$	1.41421	0.163299
$76$	−2.00000	−0.229416
$77$	−5.65685	−0.644658
$78$	4.82843	0.546712
$79$	−6.34315	−0.713660	−0.356830	−	0.934169i	$-0.616142\pi$
$79$	−6.34315	−0.713660	−0.356830	+	0.934169i	$0.616142\pi$
$80$	1.00000	0.111803
$81$	−5.00000	−0.555556
$82$	−5.65685	−0.624695
$83$	4.48528	0.492324	0.246162	−	0.969229i	$-0.420831\pi$
$83$	4.48528	0.492324	0.246162	+	0.969229i	$0.420831\pi$
$84$	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
$91$	−9.65685	−1.01231
$92$	0.585786	0.0610725
$93$	−5.65685	−0.586588
$94$	6.82843	0.704298
$95$	−2.00000	−0.205196
$96$	−1.41421	−0.144338
$97$	−13.4142	−1.36201	−0.681004	−	0.732280i	$-0.738456\pi$
$97$	−13.4142	−1.36201	−0.681004	+	0.732280i	$0.738456\pi$
$98$	−1.00000	−0.101015
$99$	2.00000	0.201008
$100$	1.00000	0.100000
$101$	−4.48528	−0.446302	−0.223151	−	0.974784i	$-0.571634\pi$
$101$	−4.48528	−0.446302	−0.223151	+	0.974784i	$0.571634\pi$
$102$	2.00000	0.198030
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	3.41421	0.334791
$105$	4.00000	0.390360
$106$	−6.24264	−0.606339
$107$	15.0711	1.45698	0.728488	−	0.685059i	$-0.240224\pi$
$107$	15.0711	1.45698	0.728488	+	0.685059i	$0.240224\pi$
$108$	−5.65685	−0.544331
$109$	−9.17157	−0.878477	−0.439239	−	0.898370i	$-0.644752\pi$
$109$	−9.17157	−0.878477	−0.439239	+	0.898370i	$0.644752\pi$
$110$	2.00000	0.190693
$111$	2.48528	0.235892
$112$	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
$115$	0.585786	0.0546249
$116$	−8.48528	−0.787839
$117$	3.41421	0.315644
$118$	−11.6569	−1.07310
$119$	−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
$127$	3.41421	0.302962	0.151481	−	0.988460i	$-0.451596\pi$
$127$	3.41421	0.302962	0.151481	+	0.988460i	$0.451596\pi$
$128$	−1.00000	−0.0883883
$129$	−16.0000	−1.40872
$130$	3.41421	0.299446
$131$	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
$134$	11.0711	0.956395
$135$	−5.65685	−0.486864
$136$	1.41421	0.121268
$137$	17.4142	1.48780	0.743898	−	0.668293i	$-0.232975\pi$
$137$	17.4142	1.48780	0.743898	+	0.668293i	$0.232975\pi$
$138$	−0.828427	−0.0705204
$139$	16.8284	1.42737	0.713684	−	0.700468i	$-0.247025\pi$
$139$	16.8284	1.42737	0.713684	+	0.700468i	$0.247025\pi$
$140$	2.82843	0.239046
$141$	−9.65685	−0.813254
$142$	5.17157	0.433989
$143$	6.82843	0.571022
$144$	−1.00000	−0.0833333
$145$	−8.48528	−0.704664
$146$	10.4853	0.867768
$147$	1.41421	0.116642
$148$	1.75736	0.144454
$149$	7.65685	0.627274	0.313637	−	0.949543i	$-0.398453\pi$
$149$	7.65685	0.627274	0.313637	+	0.949543i	$0.398453\pi$
$150$	−1.41421	−0.115470
$151$	6.48528	0.527765	0.263882	−	0.964555i	$-0.414997\pi$
$151$	6.48528	0.527765	0.263882	+	0.964555i	$0.414997\pi$
$152$	2.00000	0.162221
$153$	1.41421	0.114332
$154$	5.65685	0.455842
$155$	−4.00000	−0.321288
$156$	−4.82843	−0.386584
$157$	4.58579	0.365986	0.182993	−	0.983114i	$-0.441421\pi$
$157$	4.58579	0.365986	0.182993	+	0.983114i	$0.441421\pi$
$158$	6.34315	0.504634
$159$	8.82843	0.700140
$160$	−1.00000	−0.0790569
$161$	1.65685	0.130578
$162$	5.00000	0.392837
$163$	15.5563	1.21847	0.609234	−	0.792991i	$-0.291477\pi$
$163$	15.5563	1.21847	0.609234	+	0.792991i	$0.291477\pi$
$164$	5.65685	0.441726
$165$	−2.82843	−0.220193
$166$	−4.48528	−0.348125
$167$	2.24264	0.173541	0.0867704	−	0.996228i	$-0.472345\pi$
$167$	2.24264	0.173541	0.0867704	+	0.996228i	$0.472345\pi$
$168$	−4.00000	−0.308607
$169$	−1.34315	−0.103319
$170$	1.41421	0.108465
$171$	2.00000	0.152944
$172$	−11.3137	−0.862662
$173$	−18.4853	−1.40541	−0.702705	−	0.711481i	$-0.748025\pi$
$173$	−18.4853	−1.40541	−0.702705	+	0.711481i	$0.748025\pi$
$174$	12.0000	0.909718
$175$	2.82843	0.213809
$176$	−2.00000	−0.150756
$177$	16.4853	1.23911
$178$	10.0000	0.749532
$179$	−15.6569	−1.17025	−0.585124	−	0.810944i	$-0.698954\pi$
$179$	−15.6569	−1.17025	−0.585124	+	0.810944i	$0.698954\pi$
$180$	−1.00000	−0.0745356
$181$	−18.9706	−1.41007	−0.705035	−	0.709172i	$-0.749069\pi$
$181$	−18.9706	−1.41007	−0.705035	+	0.709172i	$0.749069\pi$
$182$	9.65685	0.715814
$183$	11.3137	0.836333
$184$	−0.585786	−0.0431847
$185$	1.75736	0.129204
$186$	5.65685	0.414781
$187$	2.82843	0.206835
$188$	−6.82843	−0.498014
$189$	−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
$193$	2.10051	0.151198	0.0755988	−	0.997138i	$-0.475913\pi$
$193$	2.10051	0.151198	0.0755988	+	0.997138i	$0.475913\pi$
$194$	13.4142	0.963084
$195$	−4.82843	−0.345771
$196$	1.00000	0.0714286
$197$	18.9706	1.35160	0.675798	−	0.737087i	$-0.263799\pi$
$197$	18.9706	1.35160	0.675798	+	0.737087i	$0.263799\pi$
$198$	−2.00000	−0.142134
$199$	18.8284	1.33471	0.667356	−	0.744739i	$-0.267426\pi$
$199$	18.8284	1.33471	0.667356	+	0.744739i	$0.267426\pi$
$200$	−1.00000	−0.0707107
$201$	−15.6569	−1.10435
$202$	4.48528	0.315583
$203$	−24.0000	−1.68447
$204$	−2.00000	−0.140028
$205$	5.65685	0.395092
$206$	0	0
$207$	−0.585786	−0.0407150
$208$	−3.41421	−0.236733
$209$	4.00000	0.276686
$210$	−4.00000	−0.276026
$211$	−0.343146	−0.0236231	−0.0118116	−	0.999930i	$-0.503760\pi$
$211$	−0.343146	−0.0236231	−0.0118116	+	0.999930i	$0.503760\pi$
$212$	6.24264	0.428746
$213$	−7.31371	−0.501127
$214$	−15.0711	−1.03024
$215$	−11.3137	−0.771589
$216$	5.65685	0.384900
$217$	−11.3137	−0.768025
$218$	9.17157	0.621177
$219$	−14.8284	−1.00201
$220$	−2.00000	−0.134840
$221$	4.82843	0.324795
$222$	−2.48528	−0.166801
$223$	−6.34315	−0.424768	−0.212384	−	0.977186i	$-0.568123\pi$
$223$	−6.34315	−0.424768	−0.212384	+	0.977186i	$0.568123\pi$
$224$	−2.82843	−0.188982
$225$	−1.00000	−0.0666667
$226$	6.00000	0.399114
$227$	−9.41421	−0.624843	−0.312422	−	0.949944i	$-0.601140\pi$
$227$	−9.41421	−0.624843	−0.312422	+	0.949944i	$0.601140\pi$
$228$	−2.82843	−0.187317
$229$	9.31371	0.615467	0.307734	−	0.951473i	$-0.400429\pi$
$229$	9.31371	0.615467	0.307734	+	0.951473i	$0.400429\pi$
$230$	−0.585786	−0.0386256
$231$	−8.00000	−0.526361
$232$	8.48528	0.557086
$233$	16.2426	1.06409	0.532045	−	0.846716i	$-0.321424\pi$
$233$	16.2426	1.06409	0.532045	+	0.846716i	$0.321424\pi$
$234$	−3.41421	−0.223194
$235$	−6.82843	−0.445437
$236$	11.6569	0.758797
$237$	−8.97056	−0.582701
$238$	4.00000	0.259281
$239$	2.48528	0.160759	0.0803797	−	0.996764i	$-0.474387\pi$
$239$	2.48528	0.160759	0.0803797	+	0.996764i	$0.474387\pi$
$240$	1.41421	0.0912871
$241$	4.34315	0.279767	0.139883	−	0.990168i	$-0.455327\pi$
$241$	4.34315	0.279767	0.139883	+	0.990168i	$0.455327\pi$
$242$	7.00000	0.449977
$243$	9.89949	0.635053
$244$	8.00000	0.512148
$245$	1.00000	0.0638877
$246$	−8.00000	−0.510061
$247$	6.82843	0.434482
$248$	4.00000	0.254000
$249$	6.34315	0.401981
$250$	−1.00000	−0.0632456
$251$	−1.31371	−0.0829205	−0.0414603	−	0.999140i	$-0.513201\pi$
$251$	−1.31371	−0.0829205	−0.0414603	+	0.999140i	$0.513201\pi$
$252$	−2.82843	−0.178174
$253$	−1.17157	−0.0736562
$254$	−3.41421	−0.214227
$255$	−2.00000	−0.125245
$256$	1.00000	0.0625000
$257$	28.8284	1.79827	0.899134	−	0.437674i	$-0.144197\pi$
$257$	28.8284	1.79827	0.899134	+	0.437674i	$0.144197\pi$
$258$	16.0000	0.996116
$259$	4.97056	0.308856
$260$	−3.41421	−0.211741
$261$	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
$265$	6.24264	0.383482
$266$	5.65685	0.346844
$267$	−14.1421	−0.865485
$268$	−11.0711	−0.676273
$269$	−23.7990	−1.45105	−0.725525	−	0.688196i	$-0.758403\pi$
$269$	−23.7990	−1.45105	−0.725525	+	0.688196i	$0.758403\pi$
$270$	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
$273$	−13.6569	−0.826550
$274$	−17.4142	−1.05203
$275$	−2.00000	−0.120605
$276$	0.828427	0.0498655
$277$	−21.7574	−1.30727	−0.653637	−	0.756809i	$-0.726757\pi$
$277$	−21.7574	−1.30727	−0.653637	+	0.756809i	$0.726757\pi$
$278$	−16.8284	−1.00930
$279$	4.00000	0.239474
$280$	−2.82843	−0.169031
$281$	25.7990	1.53904	0.769519	−	0.638623i	$-0.220496\pi$
$281$	25.7990	1.53904	0.769519	+	0.638623i	$0.220496\pi$
$282$	9.65685	0.575057
$283$	−0.928932	−0.0552193	−0.0276096	−	0.999619i	$-0.508790\pi$
$283$	−0.928932	−0.0552193	−0.0276096	+	0.999619i	$0.508790\pi$
$284$	−5.17157	−0.306876
$285$	−2.82843	−0.167542
$286$	−6.82843	−0.403773
$287$	16.0000	0.944450
$288$	1.00000	0.0589256
$289$	−15.0000	−0.882353
$290$	8.48528	0.498273
$291$	−18.9706	−1.11207
$292$	−10.4853	−0.613605
$293$	−9.75736	−0.570031	−0.285016	−	0.958523i	$-0.591999\pi$
$293$	−9.75736	−0.570031	−0.285016	+	0.958523i	$0.591999\pi$
$294$	−1.41421	−0.0824786
$295$	11.6569	0.678688
$296$	−1.75736	−0.102144
$297$	11.3137	0.656488
$298$	−7.65685	−0.443550
$299$	−2.00000	−0.115663
$300$	1.41421	0.0816497
$301$	−32.0000	−1.84445
$302$	−6.48528	−0.373186
$303$	−6.34315	−0.364404
$304$	−2.00000	−0.114708
$305$	8.00000	0.458079
$306$	−1.41421	−0.0808452
$307$	−14.3431	−0.818607	−0.409303	−	0.912398i	$-0.634228\pi$
$307$	−14.3431	−0.818607	−0.409303	+	0.912398i	$0.634228\pi$
$308$	−5.65685	−0.322329
$309$	0	0
$310$	4.00000	0.227185
$311$	3.31371	0.187903	0.0939516	−	0.995577i	$-0.470050\pi$
$311$	3.31371	0.187903	0.0939516	+	0.995577i	$0.470050\pi$
$312$	4.82843	0.273356
$313$	10.4853	0.592663	0.296332	−	0.955085i	$-0.404237\pi$
$313$	10.4853	0.592663	0.296332	+	0.955085i	$0.404237\pi$
$314$	−4.58579	−0.258791
$315$	−2.82843	−0.159364
$316$	−6.34315	−0.356830
$317$	−11.8995	−0.668342	−0.334171	−	0.942512i	$-0.608456\pi$
$317$	−11.8995	−0.668342	−0.334171	+	0.942512i	$0.608456\pi$
$318$	−8.82843	−0.495074
$319$	16.9706	0.950169
$320$	1.00000	0.0559017
$321$	21.3137	1.18962
$322$	−1.65685	−0.0923329
$323$	2.82843	0.157378
$324$	−5.00000	−0.277778
$325$	−3.41421	−0.189386
$326$	−15.5563	−0.861586
$327$	−12.9706	−0.717274
$328$	−5.65685	−0.312348
$329$	−19.3137	−1.06480
$330$	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	$0.517505\pi$
$332$	4.48528	0.246162
$333$	−1.75736	−0.0963027
$334$	−2.24264	−0.122712
$335$	−11.0711	−0.604877
$336$	4.00000	0.218218
$337$	21.7990	1.18747	0.593733	−	0.804662i	$-0.297653\pi$
$337$	21.7990	1.18747	0.593733	+	0.804662i	$0.297653\pi$
$338$	1.34315	0.0730575
$339$	−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
$345$	0.828427	0.0446010
$346$	18.4853	0.993775
$347$	−1.41421	−0.0759190	−0.0379595	−	0.999279i	$-0.512086\pi$
$347$	−1.41421	−0.0759190	−0.0379595	+	0.999279i	$0.512086\pi$
$348$	−12.0000	−0.643268
$349$	−18.3431	−0.981886	−0.490943	−	0.871192i	$-0.663348\pi$
$349$	−18.3431	−0.981886	−0.490943	+	0.871192i	$0.663348\pi$
$350$	−2.82843	−0.151186
$351$	19.3137	1.03089
$352$	2.00000	0.106600
$353$	−13.2132	−0.703268	−0.351634	−	0.936138i	$-0.614374\pi$
$353$	−13.2132	−0.703268	−0.351634	+	0.936138i	$0.614374\pi$
$354$	−16.4853	−0.876183
$355$	−5.17157	−0.274479
$356$	−10.0000	−0.529999
$357$	−5.65685	−0.299392
$358$	15.6569	0.827490
$359$	−12.9706	−0.684560	−0.342280	−	0.939598i	$-0.611199\pi$
$359$	−12.9706	−0.684560	−0.342280	+	0.939598i	$0.611199\pi$
$360$	1.00000	0.0527046
$361$	−15.0000	−0.789474
$362$	18.9706	0.997071
$363$	−9.89949	−0.519589
$364$	−9.65685	−0.506157
$365$	−10.4853	−0.548825
$366$	−11.3137	−0.591377
$367$	−10.2426	−0.534661	−0.267331	−	0.963605i	$-0.586142\pi$
$367$	−10.2426	−0.534661	−0.267331	+	0.963605i	$0.586142\pi$
$368$	0.585786	0.0305362
$369$	−5.65685	−0.294484
$370$	−1.75736	−0.0913608
$371$	17.6569	0.916698
$372$	−5.65685	−0.293294
$373$	−8.14214	−0.421584	−0.210792	−	0.977531i	$-0.567604\pi$
$373$	−8.14214	−0.421584	−0.210792	+	0.977531i	$0.567604\pi$
$374$	−2.82843	−0.146254
$375$	1.41421	0.0730297
$376$	6.82843	0.352149
$377$	28.9706	1.49206
$378$	16.0000	0.822951
$379$	−15.6569	−0.804239	−0.402119	−	0.915587i	$-0.631726\pi$
$379$	−15.6569	−0.804239	−0.402119	+	0.915587i	$0.631726\pi$
$380$	−2.00000	−0.102598
$381$	4.82843	0.247368
$382$	4.00000	0.204658
$383$	−9.17157	−0.468645	−0.234323	−	0.972159i	$-0.575287\pi$
$383$	−9.17157	−0.468645	−0.234323	+	0.972159i	$0.575287\pi$
$384$	−1.41421	−0.0721688
$385$	−5.65685	−0.288300
$386$	−2.10051	−0.106913
$387$	11.3137	0.575108
$388$	−13.4142	−0.681004
$389$	14.1421	0.717035	0.358517	−	0.933523i	$-0.383282\pi$
$389$	14.1421	0.717035	0.358517	+	0.933523i	$0.383282\pi$
$390$	4.82843	0.244497
$391$	−0.828427	−0.0418954
$392$	−1.00000	−0.0505076
$393$	14.1421	0.713376
$394$	−18.9706	−0.955723
$395$	−6.34315	−0.319158
$396$	2.00000	0.100504
$397$	22.2843	1.11842	0.559208	−	0.829028i	$-0.311105\pi$
$397$	22.2843	1.11842	0.559208	+	0.829028i	$0.311105\pi$
$398$	−18.8284	−0.943784
$399$	−8.00000	−0.400501
$400$	1.00000	0.0500000
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	15.6569	0.780893
$403$	13.6569	0.680296
$404$	−4.48528	−0.223151
$405$	−5.00000	−0.248452
$406$	24.0000	1.19110
$407$	−3.51472	−0.174218
$408$	2.00000	0.0990148
$409$	2.34315	0.115861	0.0579306	−	0.998321i	$-0.481550\pi$
$409$	2.34315	0.115861	0.0579306	+	0.998321i	$0.481550\pi$
$410$	−5.65685	−0.279372
$411$	24.6274	1.21478
$412$	0	0
$413$	32.9706	1.62238
$414$	0.585786	0.0287898
$415$	4.48528	0.220174
$416$	3.41421	0.167396
$417$	23.7990	1.16544
$418$	−4.00000	−0.195646
$419$	−30.9706	−1.51301	−0.756505	−	0.653987i	$-0.773095\pi$
$419$	−30.9706	−1.51301	−0.756505	+	0.653987i	$0.773095\pi$
$420$	4.00000	0.195180
$421$	26.8284	1.30754	0.653769	−	0.756694i	$-0.273187\pi$
$421$	26.8284	1.30754	0.653769	+	0.756694i	$0.273187\pi$
$422$	0.343146	0.0167041
$423$	6.82843	0.332009
$424$	−6.24264	−0.303169
$425$	−1.41421	−0.0685994
$426$	7.31371	0.354350
$427$	22.6274	1.09502
$428$	15.0711	0.728488
$429$	9.65685	0.466237
$430$	11.3137	0.545595
$431$	5.65685	0.272481	0.136241	−	0.990676i	$-0.456498\pi$
$431$	5.65685	0.272481	0.136241	+	0.990676i	$0.456498\pi$
$432$	−5.65685	−0.272166
$433$	−41.3137	−1.98541	−0.992705	−	0.120568i	$-0.961528\pi$
$433$	−41.3137	−1.98541	−0.992705	+	0.120568i	$0.961528\pi$
$434$	11.3137	0.543075
$435$	−12.0000	−0.575356
$436$	−9.17157	−0.439239
$437$	−1.17157	−0.0560439
$438$	14.8284	0.708530
$439$	0.970563	0.0463224	0.0231612	−	0.999732i	$-0.492627\pi$
$439$	0.970563	0.0463224	0.0231612	+	0.999732i	$0.492627\pi$
$440$	2.00000	0.0953463
$441$	−1.00000	−0.0476190
$442$	−4.82843	−0.229665
$443$	−9.89949	−0.470339	−0.235170	−	0.971954i	$-0.575565\pi$
$443$	−9.89949	−0.470339	−0.235170	+	0.971954i	$0.575565\pi$
$444$	2.48528	0.117946
$445$	−10.0000	−0.474045
$446$	6.34315	0.300357
$447$	10.8284	0.512167
$448$	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
$453$	9.17157	0.430918
$454$	9.41421	0.441831
$455$	−9.65685	−0.452720
$456$	2.82843	0.132453
$457$	21.3137	0.997013	0.498507	−	0.866886i	$-0.333882\pi$
$457$	21.3137	0.997013	0.498507	+	0.866886i	$0.333882\pi$
$458$	−9.31371	−0.435201
$459$	8.00000	0.373408
$460$	0.585786	0.0273124
$461$	17.1716	0.799760	0.399880	−	0.916568i	$-0.369052\pi$
$461$	17.1716	0.799760	0.399880	+	0.916568i	$0.369052\pi$
$462$	8.00000	0.372194
$463$	2.92893	0.136119	0.0680595	−	0.997681i	$-0.478319\pi$
$463$	2.92893	0.136119	0.0680595	+	0.997681i	$0.478319\pi$
$464$	−8.48528	−0.393919
$465$	−5.65685	−0.262330
$466$	−16.2426	−0.752426
$467$	17.8995	0.828290	0.414145	−	0.910211i	$-0.364081\pi$
$467$	17.8995	0.828290	0.414145	+	0.910211i	$0.364081\pi$
$468$	3.41421	0.157822
$469$	−31.3137	−1.44593
$470$	6.82843	0.314972
$471$	6.48528	0.298826
$472$	−11.6569	−0.536550
$473$	22.6274	1.04041
$474$	8.97056	0.412032
$475$	−2.00000	−0.0917663
$476$	−4.00000	−0.183340
$477$	−6.24264	−0.285831
$478$	−2.48528	−0.113674
$479$	−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
$482$	−4.34315	−0.197825
$483$	2.34315	0.106617
$484$	−7.00000	−0.318182
$485$	−13.4142	−0.609108
$486$	−9.89949	−0.449050
$487$	−1.85786	−0.0841879	−0.0420939	−	0.999114i	$-0.513403\pi$
$487$	−1.85786	−0.0841879	−0.0420939	+	0.999114i	$0.513403\pi$
$488$	−8.00000	−0.362143
$489$	22.0000	0.994874
$490$	−1.00000	−0.0451754
$491$	3.31371	0.149546	0.0747728	−	0.997201i	$-0.476177\pi$
$491$	3.31371	0.149546	0.0747728	+	0.997201i	$0.476177\pi$
$492$	8.00000	0.360668
$493$	12.0000	0.540453
$494$	−6.82843	−0.307225
$495$	2.00000	0.0898933
$496$	−4.00000	−0.179605
$497$	−14.6274	−0.656129
$498$	−6.34315	−0.284243
$499$	−17.6569	−0.790429	−0.395215	−	0.918589i	$-0.629330\pi$
$499$	−17.6569	−0.790429	−0.395215	+	0.918589i	$0.629330\pi$
$500$	1.00000	0.0447214
$501$	3.17157	0.141695
$502$	1.31371	0.0586337
$503$	3.51472	0.156714	0.0783568	−	0.996925i	$-0.475033\pi$
$503$	3.51472	0.156714	0.0783568	+	0.996925i	$0.475033\pi$
$504$	2.82843	0.125988
$505$	−4.48528	−0.199592
$506$	1.17157	0.0520828
$507$	−1.89949	−0.0843595
$508$	3.41421	0.151481
$509$	−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
$511$	−29.6569	−1.31194
$512$	−1.00000	−0.0441942
$513$	11.3137	0.499512
$514$	−28.8284	−1.27157
$515$	0	0
$516$	−16.0000	−0.704361
$517$	13.6569	0.600628
$518$	−4.97056	−0.218394
$519$	−26.1421	−1.14751
$520$	3.41421	0.149723
$521$	0.343146	0.0150335	0.00751674	−	0.999972i	$-0.497607\pi$
$521$	0.343146	0.0150335	0.00751674	+	0.999972i	$0.497607\pi$
$522$	−8.48528	−0.371391
$523$	−22.3848	−0.978818	−0.489409	−	0.872054i	$-0.662787\pi$
$523$	−22.3848	−0.978818	−0.489409	+	0.872054i	$0.662787\pi$
$524$	10.0000	0.436852
$525$	4.00000	0.174574
$526$	−8.00000	−0.348817
$527$	5.65685	0.246416
$528$	−2.82843	−0.123091
$529$	−22.6569	−0.985081
$530$	−6.24264	−0.271163
$531$	−11.6569	−0.505864
$532$	−5.65685	−0.245256
$533$	−19.3137	−0.836570
$534$	14.1421	0.611990
$535$	15.0711	0.651579
$536$	11.0711	0.478197
$537$	−22.1421	−0.955504
$538$	23.7990	1.02605
$539$	−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
$543$	−26.8284	−1.15132
$544$	1.41421	0.0606339
$545$	−9.17157	−0.392867
$546$	13.6569	0.584459
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	17.4142	0.743898
$549$	−8.00000	−0.341432
$550$	2.00000	0.0852803
$551$	16.9706	0.722970
$552$	−0.828427	−0.0352602
$553$	−17.9411	−0.762934
$554$	21.7574	0.924382
$555$	2.48528	0.105494
$556$	16.8284	0.713684
$557$	−21.1127	−0.894574	−0.447287	−	0.894391i	$-0.647610\pi$
$557$	−21.1127	−0.894574	−0.447287	+	0.894391i	$0.647610\pi$
$558$	−4.00000	−0.169334
$559$	38.6274	1.63377
$560$	2.82843	0.119523
$561$	4.00000	0.168880
$562$	−25.7990	−1.08826
$563$	−16.9706	−0.715224	−0.357612	−	0.933870i	$-0.616409\pi$
$563$	−16.9706	−0.715224	−0.357612	+	0.933870i	$0.616409\pi$
$564$	−9.65685	−0.406627
$565$	−6.00000	−0.252422
$566$	0.928932	0.0390459
$567$	−14.1421	−0.593914
$568$	5.17157	0.216994
$569$	−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
$571$	−22.9706	−0.961288	−0.480644	−	0.876916i	$-0.659597\pi$
$571$	−22.9706	−0.961288	−0.480644	+	0.876916i	$0.659597\pi$
$572$	6.82843	0.285511
$573$	−5.65685	−0.236318
$574$	−16.0000	−0.667827
$575$	0.585786	0.0244290
$576$	−1.00000	−0.0416667
$577$	24.3431	1.01342	0.506709	−	0.862117i	$-0.330862\pi$
$577$	24.3431	1.01342	0.506709	+	0.862117i	$0.330862\pi$
$578$	15.0000	0.623918
$579$	2.97056	0.123452
$580$	−8.48528	−0.352332
$581$	12.6863	0.526316
$582$	18.9706	0.786355
$583$	−12.4853	−0.517088
$584$	10.4853	0.433884
$585$	3.41421	0.141160
$586$	9.75736	0.403073
$587$	37.4558	1.54597	0.772984	−	0.634425i	$-0.218763\pi$
$587$	37.4558	1.54597	0.772984	+	0.634425i	$0.218763\pi$
$588$	1.41421	0.0583212
$589$	8.00000	0.329634
$590$	−11.6569	−0.479905
$591$	26.8284	1.10357
$592$	1.75736	0.0722270
$593$	37.0122	1.51991	0.759954	−	0.649977i	$-0.225221\pi$
$593$	37.0122	1.51991	0.759954	+	0.649977i	$0.225221\pi$
$594$	−11.3137	−0.464207
$595$	−4.00000	−0.163984
$596$	7.65685	0.313637
$597$	26.6274	1.08979
$598$	2.00000	0.0817861
$599$	−15.1716	−0.619894	−0.309947	−	0.950754i	$-0.600311\pi$
$599$	−15.1716	−0.619894	−0.309947	+	0.950754i	$0.600311\pi$
$600$	−1.41421	−0.0577350
$601$	−28.6274	−1.16774	−0.583868	−	0.811848i	$-0.698462\pi$
$601$	−28.6274	−1.16774	−0.583868	+	0.811848i	$0.698462\pi$
$602$	32.0000	1.30422
$603$	11.0711	0.450849
$604$	6.48528	0.263882
$605$	−7.00000	−0.284590
$606$	6.34315	0.257673
$607$	5.45584	0.221446	0.110723	−	0.993851i	$-0.464683\pi$
$607$	5.45584	0.221446	0.110723	+	0.993851i	$0.464683\pi$
$608$	2.00000	0.0811107
$609$	−33.9411	−1.37536
$610$	−8.00000	−0.323911
$611$	23.3137	0.943172
$612$	1.41421	0.0571662
$613$	48.3848	1.95424	0.977121	−	0.212683i	$-0.0682200\pi$
$613$	48.3848	1.95424	0.977121	+	0.212683i	$0.0682200\pi$
$614$	14.3431	0.578842
$615$	8.00000	0.322591
$616$	5.65685	0.227921
$617$	35.5563	1.43144	0.715722	−	0.698385i	$-0.246098\pi$
$617$	35.5563	1.43144	0.715722	+	0.698385i	$0.246098\pi$
$618$	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
$621$	−3.31371	−0.132975
$622$	−3.31371	−0.132868
$623$	−28.2843	−1.13319
$624$	−4.82843	−0.193292
$625$	1.00000	0.0400000
$626$	−10.4853	−0.419076
$627$	5.65685	0.225913
$628$	4.58579	0.182993
$629$	−2.48528	−0.0990947
$630$	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	$0.630328\pi$
$632$	6.34315	0.252317
$633$	−0.485281	−0.0192882
$634$	11.8995	0.472589
$635$	3.41421	0.135489
$636$	8.82843	0.350070
$637$	−3.41421	−0.135276
$638$	−16.9706	−0.671871
$639$	5.17157	0.204584
$640$	−1.00000	−0.0395285
$641$	11.8579	0.468357	0.234179	−	0.972194i	$-0.424760\pi$
$641$	11.8579	0.468357	0.234179	+	0.972194i	$0.424760\pi$
$642$	−21.3137	−0.841185
$643$	−6.34315	−0.250149	−0.125075	−	0.992147i	$-0.539917\pi$
$643$	−6.34315	−0.250149	−0.125075	+	0.992147i	$0.539917\pi$
$644$	1.65685	0.0652892
$645$	−16.0000	−0.629999
$646$	−2.82843	−0.111283
$647$	28.3848	1.11592	0.557960	−	0.829868i	$-0.311584\pi$
$647$	28.3848	1.11592	0.557960	+	0.829868i	$0.311584\pi$
$648$	5.00000	0.196419
$649$	−23.3137	−0.915143
$650$	3.41421	0.133916
$651$	−16.0000	−0.627089
$652$	15.5563	0.609234
$653$	−20.6274	−0.807213	−0.403607	−	0.914933i	$-0.632244\pi$
$653$	−20.6274	−0.807213	−0.403607	+	0.914933i	$0.632244\pi$
$654$	12.9706	0.507189
$655$	10.0000	0.390732
$656$	5.65685	0.220863
$657$	10.4853	0.409070
$658$	19.3137	0.752927
$659$	−18.9706	−0.738988	−0.369494	−	0.929233i	$-0.620469\pi$
$659$	−18.9706	−0.738988	−0.369494	+	0.929233i	$0.620469\pi$
$660$	−2.82843	−0.110096
$661$	32.2843	1.25571	0.627856	−	0.778329i	$-0.283933\pi$
$661$	32.2843	1.25571	0.627856	+	0.778329i	$0.283933\pi$
$662$	2.00000	0.0777322
$663$	6.82843	0.265194
$664$	−4.48528	−0.174063
$665$	−5.65685	−0.219363
$666$	1.75736	0.0680963
$667$	−4.97056	−0.192461
$668$	2.24264	0.0867704
$669$	−8.97056	−0.346822
$670$	11.0711	0.427713
$671$	−16.0000	−0.617673
$672$	−4.00000	−0.154303
$673$	46.1838	1.78025	0.890127	−	0.455713i	$-0.150616\pi$
$673$	46.1838	1.78025	0.890127	+	0.455713i	$0.150616\pi$
$674$	−21.7990	−0.839666
$675$	−5.65685	−0.217732
$676$	−1.34315	−0.0516595
$677$	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
$679$	−37.9411	−1.45605
$680$	1.41421	0.0542326
$681$	−13.3137	−0.510182
$682$	−8.00000	−0.306336
$683$	−39.5563	−1.51358	−0.756791	−	0.653657i	$-0.773234\pi$
$683$	−39.5563	−1.51358	−0.756791	+	0.653657i	$0.773234\pi$
$684$	2.00000	0.0764719
$685$	17.4142	0.665363
$686$	16.9706	0.647939
$687$	13.1716	0.502527
$688$	−11.3137	−0.431331
$689$	−21.3137	−0.811988
$690$	−0.828427	−0.0315377
$691$	11.0294	0.419580	0.209790	−	0.977747i	$-0.432722\pi$
$691$	11.0294	0.419580	0.209790	+	0.977747i	$0.432722\pi$
$692$	−18.4853	−0.702705
$693$	5.65685	0.214886
$694$	1.41421	0.0536828
$695$	16.8284	0.638339
$696$	12.0000	0.454859
$697$	−8.00000	−0.303022
$698$	18.3431	0.694298
$699$	22.9706	0.868826
$700$	2.82843	0.106904
$701$	−51.5980	−1.94883	−0.974414	−	0.224759i	$-0.927841\pi$
$701$	−51.5980	−1.94883	−0.974414	+	0.224759i	$0.927841\pi$
$702$	−19.3137	−0.728949
$703$	−3.51472	−0.132560
$704$	−2.00000	−0.0753778
$705$	−9.65685	−0.363698
$706$	13.2132	0.497285
$707$	−12.6863	−0.477117
$708$	16.4853	0.619555
$709$	−18.9706	−0.712454	−0.356227	−	0.934399i	$-0.615937\pi$
$709$	−18.9706	−0.712454	−0.356227	+	0.934399i	$0.615937\pi$
$710$	5.17157	0.194086
$711$	6.34315	0.237887
$712$	10.0000	0.374766
$713$	−2.34315	−0.0877515
$714$	5.65685	0.211702
$715$	6.82843	0.255369
$716$	−15.6569	−0.585124
$717$	3.51472	0.131260
$718$	12.9706	0.484057
$719$	10.4853	0.391035	0.195518	−	0.980700i	$-0.437361\pi$
$719$	10.4853	0.391035	0.195518	+	0.980700i	$0.437361\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	15.0000	0.558242
$723$	6.14214	0.228428
$724$	−18.9706	−0.705035
$725$	−8.48528	−0.315135
$726$	9.89949	0.367405
$727$	−48.1838	−1.78704	−0.893518	−	0.449026i	$-0.851771\pi$
$727$	−48.1838	−1.78704	−0.893518	+	0.449026i	$0.851771\pi$
$728$	9.65685	0.357907
$729$	29.0000	1.07407
$730$	10.4853	0.388078
$731$	16.0000	0.591781
$732$	11.3137	0.418167
$733$	16.1421	0.596223	0.298112	−	0.954531i	$-0.403643\pi$
$733$	16.1421	0.596223	0.298112	+	0.954531i	$0.403643\pi$
$734$	10.2426	0.378063
$735$	1.41421	0.0521641
$736$	−0.585786	−0.0215924
$737$	22.1421	0.815616
$738$	5.65685	0.208232
$739$	−33.3137	−1.22546	−0.612732	−	0.790291i	$-0.709930\pi$
$739$	−33.3137	−1.22546	−0.612732	+	0.790291i	$0.709930\pi$
$740$	1.75736	0.0646018
$741$	9.65685	0.354753
$742$	−17.6569	−0.648204
$743$	45.3553	1.66393	0.831963	−	0.554831i	$-0.187217\pi$
$743$	45.3553	1.66393	0.831963	+	0.554831i	$0.187217\pi$
$744$	5.65685	0.207390
$745$	7.65685	0.280525
$746$	8.14214	0.298105
$747$	−4.48528	−0.164108
$748$	2.82843	0.103418
$749$	42.6274	1.55757
$750$	−1.41421	−0.0516398
$751$	−6.62742	−0.241838	−0.120919	−	0.992662i	$-0.538584\pi$
$751$	−6.62742	−0.241838	−0.120919	+	0.992662i	$0.538584\pi$
$752$	−6.82843	−0.249007
$753$	−1.85786	−0.0677043
$754$	−28.9706	−1.05505
$755$	6.48528	0.236024
$756$	−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	$0.630921\pi$
$758$	15.6569	0.568683
$759$	−1.65685	−0.0601400
$760$	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	$0.303053\pi$
$762$	−4.82843	−0.174915
$763$	−25.9411	−0.939132
$764$	−4.00000	−0.144715
$765$	1.41421	0.0511310
$766$	9.17157	0.331382
$767$	−39.7990	−1.43706
$768$	1.41421	0.0510310
$769$	−16.1421	−0.582100	−0.291050	−	0.956708i	$-0.594005\pi$
$769$	−16.1421	−0.582100	−0.291050	+	0.956708i	$0.594005\pi$
$770$	5.65685	0.203859
$771$	40.7696	1.46828
$772$	2.10051	0.0755988
$773$	1.02944	0.0370263	0.0185131	−	0.999829i	$-0.494107\pi$
$773$	1.02944	0.0370263	0.0185131	+	0.999829i	$0.494107\pi$
$774$	−11.3137	−0.406663
$775$	−4.00000	−0.143684
$776$	13.4142	0.481542
$777$	7.02944	0.252180
$778$	−14.1421	−0.507020
$779$	−11.3137	−0.405356
$780$	−4.82843	−0.172885
$781$	10.3431	0.370107
$782$	0.828427	0.0296245
$783$	48.0000	1.71538
$784$	1.00000	0.0357143
$785$	4.58579	0.163674
$786$	−14.1421	−0.504433
$787$	48.0416	1.71250	0.856250	−	0.516562i	$-0.172789\pi$
$787$	48.0416	1.71250	0.856250	+	0.516562i	$0.172789\pi$
$788$	18.9706	0.675798
$789$	11.3137	0.402779
$790$	6.34315	0.225679
$791$	−16.9706	−0.603404
$792$	−2.00000	−0.0710669
$793$	−27.3137	−0.969938
$794$	−22.2843	−0.790839
$795$	8.82843	0.313112
$796$	18.8284	0.667356
$797$	−34.2843	−1.21441	−0.607206	−	0.794545i	$-0.707710\pi$
$797$	−34.2843	−1.21441	−0.607206	+	0.794545i	$0.707710\pi$
$798$	8.00000	0.283197
$799$	9.65685	0.341635
$800$	−1.00000	−0.0353553
$801$	10.0000	0.353333
$802$	−1.00000	−0.0353112
$803$	20.9706	0.740035
$804$	−15.6569	−0.552175
$805$	1.65685	0.0583964
$806$	−13.6569	−0.481042
$807$	−33.6569	−1.18478
$808$	4.48528	0.157792
$809$	6.28427	0.220943	0.110472	−	0.993879i	$-0.464764\pi$
$809$	6.28427	0.220943	0.110472	+	0.993879i	$0.464764\pi$
$810$	5.00000	0.175682
$811$	−44.2843	−1.55503	−0.777516	−	0.628864i	$-0.783520\pi$
$811$	−44.2843	−1.55503	−0.777516	+	0.628864i	$0.783520\pi$
$812$	−24.0000	−0.842235
$813$	0	0
$814$	3.51472	0.123191
$815$	15.5563	0.544915
$816$	−2.00000	−0.0700140
$817$	22.6274	0.791633
$818$	−2.34315	−0.0819262
$819$	9.65685	0.337438
$820$	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	$0.613479\pi$
$822$	−24.6274	−0.858980
$823$	−14.9289	−0.520390	−0.260195	−	0.965556i	$-0.583787\pi$
$823$	−14.9289	−0.520390	−0.260195	+	0.965556i	$0.583787\pi$
$824$	0	0
$825$	−2.82843	−0.0984732
$826$	−32.9706	−1.14719
$827$	30.6274	1.06502	0.532510	−	0.846424i	$-0.321249\pi$
$827$	30.6274	1.06502	0.532510	+	0.846424i	$0.321249\pi$
$828$	−0.585786	−0.0203575
$829$	−1.85786	−0.0645263	−0.0322631	−	0.999479i	$-0.510271\pi$
$829$	−1.85786	−0.0645263	−0.0322631	+	0.999479i	$0.510271\pi$
$830$	−4.48528	−0.155686
$831$	−30.7696	−1.06738
$832$	−3.41421	−0.118367
$833$	−1.41421	−0.0489996
$834$	−23.7990	−0.824092
$835$	2.24264	0.0776098
$836$	4.00000	0.138343
$837$	22.6274	0.782118
$838$	30.9706	1.06986
$839$	9.17157	0.316638	0.158319	−	0.987388i	$-0.449393\pi$
$839$	9.17157	0.316638	0.158319	+	0.987388i	$0.449393\pi$
$840$	−4.00000	−0.138013
$841$	43.0000	1.48276
$842$	−26.8284	−0.924569
$843$	36.4853	1.25662
$844$	−0.343146	−0.0118116
$845$	−1.34315	−0.0462056
$846$	−6.82843	−0.234766
$847$	−19.7990	−0.680301
$848$	6.24264	0.214373
$849$	−1.31371	−0.0450864
$850$	1.41421	0.0485071
$851$	1.02944	0.0352887
$852$	−7.31371	−0.250564
$853$	−11.8579	−0.406006	−0.203003	−	0.979178i	$-0.565070\pi$
$853$	−11.8579	−0.406006	−0.203003	+	0.979178i	$0.565070\pi$
$854$	−22.6274	−0.774294
$855$	2.00000	0.0683986
$856$	−15.0711	−0.515118
$857$	−7.17157	−0.244976	−0.122488	−	0.992470i	$-0.539087\pi$
$857$	−7.17157	−0.244976	−0.122488	+	0.992470i	$0.539087\pi$
$858$	−9.65685	−0.329680
$859$	23.6569	0.807161	0.403581	−	0.914944i	$-0.367765\pi$
$859$	23.6569	0.807161	0.403581	+	0.914944i	$0.367765\pi$
$860$	−11.3137	−0.385794
$861$	22.6274	0.771140
$862$	−5.65685	−0.192673
$863$	5.27208	0.179464	0.0897318	−	0.995966i	$-0.471399\pi$
$863$	5.27208	0.179464	0.0897318	+	0.995966i	$0.471399\pi$
$864$	5.65685	0.192450
$865$	−18.4853	−0.628518
$866$	41.3137	1.40390
$867$	−21.2132	−0.720438
$868$	−11.3137	−0.384012
$869$	12.6863	0.430353
$870$	12.0000	0.406838
$871$	37.7990	1.28077
$872$	9.17157	0.310589
$873$	13.4142	0.454002
$874$	1.17157	0.0396290
$875$	2.82843	0.0956183
$876$	−14.8284	−0.501006
$877$	32.3848	1.09356	0.546778	−	0.837278i	$-0.315854\pi$
$877$	32.3848	1.09356	0.546778	+	0.837278i	$0.315854\pi$
$878$	−0.970563	−0.0327549
$879$	−13.7990	−0.465428
$880$	−2.00000	−0.0674200
$881$	5.11270	0.172251	0.0861256	−	0.996284i	$-0.472551\pi$
$881$	5.11270	0.172251	0.0861256	+	0.996284i	$0.472551\pi$
$882$	1.00000	0.0336718
$883$	−10.3431	−0.348075	−0.174037	−	0.984739i	$-0.555681\pi$
$883$	−10.3431	−0.348075	−0.174037	+	0.984739i	$0.555681\pi$
$884$	4.82843	0.162398
$885$	16.4853	0.554147
$886$	9.89949	0.332580
$887$	−6.04163	−0.202858	−0.101429	−	0.994843i	$-0.532341\pi$
$887$	−6.04163	−0.202858	−0.101429	+	0.994843i	$0.532341\pi$
$888$	−2.48528	−0.0834006
$889$	9.65685	0.323880
$890$	10.0000	0.335201
$891$	10.0000	0.335013
$892$	−6.34315	−0.212384
$893$	13.6569	0.457009
$894$	−10.8284	−0.362157
$895$	−15.6569	−0.523351
$896$	−2.82843	−0.0944911
$897$	−2.82843	−0.0944384
$898$	14.0000	0.467186
$899$	33.9411	1.13200
$900$	−1.00000	−0.0333333
$901$	−8.82843	−0.294118
$902$	11.3137	0.376705
$903$	−45.2548	−1.50599
$904$	6.00000	0.199557
$905$	−18.9706	−0.630603
$906$	−9.17157	−0.304705
$907$	−38.3848	−1.27455	−0.637273	−	0.770638i	$-0.719938\pi$
$907$	−38.3848	−1.27455	−0.637273	+	0.770638i	$0.719938\pi$
$908$	−9.41421	−0.312422
$909$	4.48528	0.148767
$910$	9.65685	0.320122
$911$	53.2548	1.76441	0.882206	−	0.470864i	$-0.156058\pi$
$911$	53.2548	1.76441	0.882206	+	0.470864i	$0.156058\pi$
$912$	−2.82843	−0.0936586
$913$	−8.97056	−0.296882
$914$	−21.3137	−0.704995
$915$	11.3137	0.374020
$916$	9.31371	0.307734
$917$	28.2843	0.934029
$918$	−8.00000	−0.264039
$919$	16.9706	0.559807	0.279904	−	0.960028i	$-0.409697\pi$
$919$	16.9706	0.559807	0.279904	+	0.960028i	$0.409697\pi$
$920$	−0.585786	−0.0193128
$921$	−20.2843	−0.668389
$922$	−17.1716	−0.565516
$923$	17.6569	0.581182
$924$	−8.00000	−0.263181
$925$	1.75736	0.0577816
$926$	−2.92893	−0.0962507
$927$	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
$932$	16.2426	0.532045
$933$	4.68629	0.153422
$934$	−17.8995	−0.585689
$935$	2.82843	0.0924995
$936$	−3.41421	−0.111597
$937$	49.8995	1.63015	0.815073	−	0.579359i	$-0.196697\pi$
$937$	49.8995	1.63015	0.815073	+	0.579359i	$0.196697\pi$
$938$	31.3137	1.02243
$939$	14.8284	0.483907
$940$	−6.82843	−0.222719
$941$	−36.7696	−1.19865	−0.599327	−	0.800505i	$-0.704565\pi$
$941$	−36.7696	−1.19865	−0.599327	+	0.800505i	$0.704565\pi$
$942$	−6.48528	−0.211302
$943$	3.31371	0.107909
$944$	11.6569	0.379398
$945$	−16.0000	−0.520480
$946$	−22.6274	−0.735681
$947$	5.45584	0.177291	0.0886456	−	0.996063i	$-0.471746\pi$
$947$	5.45584	0.177291	0.0886456	+	0.996063i	$0.471746\pi$
$948$	−8.97056	−0.291350
$949$	35.7990	1.16208
$950$	2.00000	0.0648886
$951$	−16.8284	−0.545699
$952$	4.00000	0.129641
$953$	54.2843	1.75844	0.879220	−	0.476416i	$-0.158064\pi$
$953$	54.2843	1.75844	0.879220	+	0.476416i	$0.158064\pi$
$954$	6.24264	0.202113
$955$	−4.00000	−0.129437
$956$	2.48528	0.0803797
$957$	24.0000	0.775810
$958$	16.0000	0.516937
$959$	49.2548	1.59052
$960$	1.41421	0.0456435
$961$	−15.0000	−0.483871
$962$	6.00000	0.193448
$963$	−15.0711	−0.485658
$964$	4.34315	0.139883
$965$	2.10051	0.0676177
$966$	−2.34315	−0.0753895
$967$	−59.4975	−1.91331	−0.956655	−	0.291224i	$-0.905938\pi$
$967$	−59.4975	−1.91331	−0.956655	+	0.291224i	$0.905938\pi$
$968$	7.00000	0.224989
$969$	4.00000	0.128499
$970$	13.4142	0.430704
$971$	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
$973$	47.5980	1.52592
$974$	1.85786	0.0595298
$975$	−4.82843	−0.154633
$976$	8.00000	0.256074
$977$	30.9706	0.990836	0.495418	−	0.868655i	$-0.335015\pi$
$977$	30.9706	0.990836	0.495418	+	0.868655i	$0.335015\pi$
$978$	−22.0000	−0.703482
$979$	20.0000	0.639203
$980$	1.00000	0.0319438
$981$	9.17157	0.292826
$982$	−3.31371	−0.105745
$983$	−20.9706	−0.668857	−0.334429	−	0.942421i	$-0.608543\pi$
$983$	−20.9706	−0.668857	−0.334429	+	0.942421i	$0.608543\pi$
$984$	−8.00000	−0.255031
$985$	18.9706	0.604452
$986$	−12.0000	−0.382158
$987$	−27.3137	−0.869405
$988$	6.82843	0.217241
$989$	−6.62742	−0.210740
$990$	−2.00000	−0.0635642
$991$	49.4558	1.57102	0.785508	−	0.618851i	$-0.212402\pi$
$991$	49.4558	1.57102	0.785508	+	0.618851i	$0.212402\pi$
$992$	4.00000	0.127000
$993$	−2.82843	−0.0897574
$994$	14.6274	0.463953
$995$	18.8284	0.596901
$996$	6.34315	0.200990
$997$	35.6569	1.12926	0.564632	−	0.825343i	$-0.309018\pi$
$997$	35.6569	1.12926	0.564632	+	0.825343i	$0.309018\pi$
$998$	17.6569	0.558918
$999$	−9.94113	−0.314523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.g.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.g.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.41421 q^{6} +2.82843 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.41421 q^{6} +2.82843 q^{7} -1.00000 q^{8} -1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} +1.41421 q^{12} -3.41421 q^{13} -2.82843 q^{14} +1.41421 q^{15} +1.00000 q^{16} -1.41421 q^{17} +1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +4.00000 q^{21} +2.00000 q^{22} +0.585786 q^{23} -1.41421 q^{24} +1.00000 q^{25} +3.41421 q^{26} -5.65685 q^{27} +2.82843 q^{28} -8.48528 q^{29} -1.41421 q^{30} -4.00000 q^{31} -1.00000 q^{32} -2.82843 q^{33} +1.41421 q^{34} +2.82843 q^{35} -1.00000 q^{36} +1.75736 q^{37} +2.00000 q^{38} -4.82843 q^{39} -1.00000 q^{40} +5.65685 q^{41} -4.00000 q^{42} -11.3137 q^{43} -2.00000 q^{44} -1.00000 q^{45} -0.585786 q^{46} -6.82843 q^{47} +1.41421 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} -3.41421 q^{52} +6.24264 q^{53} +5.65685 q^{54} -2.00000 q^{55} -2.82843 q^{56} -2.82843 q^{57} +8.48528 q^{58} +11.6569 q^{59} +1.41421 q^{60} +8.00000 q^{61} +4.00000 q^{62} -2.82843 q^{63} +1.00000 q^{64} -3.41421 q^{65} +2.82843 q^{66} -11.0711 q^{67} -1.41421 q^{68} +0.828427 q^{69} -2.82843 q^{70} -5.17157 q^{71} +1.00000 q^{72} -10.4853 q^{73} -1.75736 q^{74} +1.41421 q^{75} -2.00000 q^{76} -5.65685 q^{77} +4.82843 q^{78} -6.34315 q^{79} +1.00000 q^{80} -5.00000 q^{81} -5.65685 q^{82} +4.48528 q^{83} +4.00000 q^{84} -1.41421 q^{85} +11.3137 q^{86} -12.0000 q^{87} +2.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} -9.65685 q^{91} +0.585786 q^{92} -5.65685 q^{93} +6.82843 q^{94} -2.00000 q^{95} -1.41421 q^{96} -13.4142 q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) 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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