LMFDB - Embedded newform 4410.2.a.bx.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(1,4410)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4410, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4410.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4410.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:	$35.2140272914$
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:	$\Z[a_1, \ldots, a_{11}]$
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	$\chi$	$=$	4410.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} -3.41421 q^{11} -1.17157 q^{13} +1.00000 q^{16} -3.41421 q^{17} -4.82843 q^{19} +1.00000 q^{20} -3.41421 q^{22} +3.65685 q^{23} +1.00000 q^{25} -1.17157 q^{26} -5.41421 q^{29} -7.41421 q^{31} +1.00000 q^{32} -3.41421 q^{34} -3.41421 q^{37} -4.82843 q^{38} +1.00000 q^{40} +3.65685 q^{41} -7.07107 q^{43} -3.41421 q^{44} +3.65685 q^{46} -2.58579 q^{47} +1.00000 q^{50} -1.17157 q^{52} -6.48528 q^{53} -3.41421 q^{55} -5.41421 q^{58} +0.828427 q^{59} +1.65685 q^{61} -7.41421 q^{62} +1.00000 q^{64} -1.17157 q^{65} +5.89949 q^{67} -3.41421 q^{68} +10.4853 q^{71} -12.8284 q^{73} -3.41421 q^{74} -4.82843 q^{76} +13.6569 q^{79} +1.00000 q^{80} +3.65685 q^{82} -0.485281 q^{83} -3.41421 q^{85} -7.07107 q^{86} -3.41421 q^{88} -14.4853 q^{89} +3.65685 q^{92} -2.58579 q^{94} -4.82843 q^{95} -14.4853 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} - 4 q^{11} - 8 q^{13} + 2 q^{16} - 4 q^{17} - 4 q^{19} + 2 q^{20} - 4 q^{22} - 4 q^{23} + 2 q^{25} - 8 q^{26} - 8 q^{29} - 12 q^{31} + 2 q^{32} - 4 q^{34} - 4 q^{37} - 4 q^{38} + 2 q^{40} - 4 q^{41} - 4 q^{44} - 4 q^{46} - 8 q^{47} + 2 q^{50} - 8 q^{52} + 4 q^{53} - 4 q^{55} - 8 q^{58} - 4 q^{59} - 8 q^{61} - 12 q^{62} + 2 q^{64} - 8 q^{65} - 8 q^{67} - 4 q^{68} + 4 q^{71} - 20 q^{73} - 4 q^{74} - 4 q^{76} + 16 q^{79} + 2 q^{80} - 4 q^{82} + 16 q^{83} - 4 q^{85} - 4 q^{88} - 12 q^{89} - 4 q^{92} - 8 q^{94} - 4 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 - 4 * q^11 - 8 * q^13 + 2 * q^16 - 4 * q^17 - 4 * q^19 + 2 * q^20 - 4 * q^22 - 4 * q^23 + 2 * q^25 - 8 * q^26 - 8 * q^29 - 12 * q^31 + 2 * q^32 - 4 * q^34 - 4 * q^37 - 4 * q^38 + 2 * q^40 - 4 * q^41 - 4 * q^44 - 4 * q^46 - 8 * q^47 + 2 * q^50 - 8 * q^52 + 4 * q^53 - 4 * q^55 - 8 * q^58 - 4 * q^59 - 8 * q^61 - 12 * q^62 + 2 * q^64 - 8 * q^65 - 8 * q^67 - 4 * q^68 + 4 * q^71 - 20 * q^73 - 4 * q^74 - 4 * q^76 + 16 * q^79 + 2 * q^80 - 4 * q^82 + 16 * q^83 - 4 * q^85 - 4 * q^88 - 12 * q^89 - 4 * q^92 - 8 * q^94 - 4 * q^95 - 12 * q^97

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−3.41421	−1.02942	−0.514712	−	0.857363i	$-0.672101\pi$
$11$	−3.41421	−1.02942	−0.514712	+	0.857363i	$0.672101\pi$
$12$	0	0
$13$	−1.17157	−0.324936	−0.162468	−	0.986714i	$-0.551945\pi$
$13$	−1.17157	−0.324936	−0.162468	+	0.986714i	$0.551945\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.41421	−0.828068	−0.414034	−	0.910261i	$-0.635881\pi$
$17$	−3.41421	−0.828068	−0.414034	+	0.910261i	$0.635881\pi$
$18$	0	0
$19$	−4.82843	−1.10772	−0.553859	−	0.832611i	$-0.686845\pi$
$19$	−4.82843	−1.10772	−0.553859	+	0.832611i	$0.686845\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−3.41421	−0.727913
$23$	3.65685	0.762507	0.381253	−	0.924471i	$-0.375493\pi$
$23$	3.65685	0.762507	0.381253	+	0.924471i	$0.375493\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.17157	−0.229764
$27$	0	0
$28$	0	0
$29$	−5.41421	−1.00539	−0.502697	−	0.864463i	$-0.667659\pi$
$29$	−5.41421	−1.00539	−0.502697	+	0.864463i	$0.667659\pi$
$30$	0	0
$31$	−7.41421	−1.33163	−0.665816	−	0.746116i	$-0.731916\pi$
$31$	−7.41421	−1.33163	−0.665816	+	0.746116i	$0.731916\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.41421	−0.585533
$35$	0	0
$36$	0	0
$37$	−3.41421	−0.561293	−0.280647	−	0.959811i	$-0.590549\pi$
$37$	−3.41421	−0.561293	−0.280647	+	0.959811i	$0.590549\pi$
$38$	−4.82843	−0.783274
$39$	0	0
$40$	1.00000	0.158114
$41$	3.65685	0.571105	0.285552	−	0.958363i	$-0.407823\pi$
$41$	3.65685	0.571105	0.285552	+	0.958363i	$0.407823\pi$
$42$	0	0
$43$	−7.07107	−1.07833	−0.539164	−	0.842201i	$-0.681260\pi$
$43$	−7.07107	−1.07833	−0.539164	+	0.842201i	$0.681260\pi$
$44$	−3.41421	−0.514712
$45$	0	0
$46$	3.65685	0.539174
$47$	−2.58579	−0.377176	−0.188588	−	0.982056i	$-0.560391\pi$
$47$	−2.58579	−0.377176	−0.188588	+	0.982056i	$0.560391\pi$
$48$	0	0
$49$	0	0
$50$	1.00000	0.141421
$51$	0	0
$52$	−1.17157	−0.162468
$53$	−6.48528	−0.890822	−0.445411	−	0.895326i	$-0.646942\pi$
$53$	−6.48528	−0.890822	−0.445411	+	0.895326i	$0.646942\pi$
$54$	0	0
$55$	−3.41421	−0.460372
$56$	0	0
$57$	0	0
$58$	−5.41421	−0.710921
$59$	0.828427	0.107852	0.0539260	−	0.998545i	$-0.482826\pi$
$59$	0.828427	0.107852	0.0539260	+	0.998545i	$0.482826\pi$
$60$	0	0
$61$	1.65685	0.212138	0.106069	−	0.994359i	$-0.466173\pi$
$61$	1.65685	0.212138	0.106069	+	0.994359i	$0.466173\pi$
$62$	−7.41421	−0.941606
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.17157	−0.145316
$66$	0	0
$67$	5.89949	0.720738	0.360369	−	0.932810i	$-0.382651\pi$
$67$	5.89949	0.720738	0.360369	+	0.932810i	$0.382651\pi$
$68$	−3.41421	−0.414034
$69$	0	0
$70$	0	0
$71$	10.4853	1.24437	0.622187	−	0.782869i	$-0.286244\pi$
$71$	10.4853	1.24437	0.622187	+	0.782869i	$0.286244\pi$
$72$	0	0
$73$	−12.8284	−1.50145	−0.750727	−	0.660613i	$-0.770297\pi$
$73$	−12.8284	−1.50145	−0.750727	+	0.660613i	$0.770297\pi$
$74$	−3.41421	−0.396894
$75$	0	0
$76$	−4.82843	−0.553859
$77$	0	0
$78$	0	0
$79$	13.6569	1.53652	0.768258	−	0.640140i	$-0.221124\pi$
$79$	13.6569	1.53652	0.768258	+	0.640140i	$0.221124\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	3.65685	0.403832
$83$	−0.485281	−0.0532666	−0.0266333	−	0.999645i	$-0.508479\pi$
$83$	−0.485281	−0.0532666	−0.0266333	+	0.999645i	$0.508479\pi$
$84$	0	0
$85$	−3.41421	−0.370323
$86$	−7.07107	−0.762493
$87$	0	0
$88$	−3.41421	−0.363956
$89$	−14.4853	−1.53544	−0.767718	−	0.640787i	$-0.778608\pi$
$89$	−14.4853	−1.53544	−0.767718	+	0.640787i	$0.778608\pi$
$90$	0	0
$91$	0	0
$92$	3.65685	0.381253
$93$	0	0
$94$	−2.58579	−0.266704
$95$	−4.82843	−0.495386
$96$	0	0
$97$	−14.4853	−1.47076	−0.735379	−	0.677656i	$-0.762996\pi$
$97$	−14.4853	−1.47076	−0.735379	+	0.677656i	$0.762996\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	0.100000
$101$	−9.65685	−0.960893	−0.480446	−	0.877024i	$-0.659525\pi$
$101$	−9.65685	−0.960893	−0.480446	+	0.877024i	$0.659525\pi$
$102$	0	0
$103$	−5.65685	−0.557386	−0.278693	−	0.960380i	$-0.589901\pi$
$103$	−5.65685	−0.557386	−0.278693	+	0.960380i	$0.589901\pi$
$104$	−1.17157	−0.114882
$105$	0	0
$106$	−6.48528	−0.629906
$107$	10.8284	1.04682	0.523412	−	0.852080i	$-0.324659\pi$
$107$	10.8284	1.04682	0.523412	+	0.852080i	$0.324659\pi$
$108$	0	0
$109$	14.4853	1.38744	0.693719	−	0.720246i	$-0.255971\pi$
$109$	14.4853	1.38744	0.693719	+	0.720246i	$0.255971\pi$
$110$	−3.41421	−0.325532
$111$	0	0
$112$	0	0
$113$	4.34315	0.408569	0.204284	−	0.978912i	$-0.434513\pi$
$113$	4.34315	0.408569	0.204284	+	0.978912i	$0.434513\pi$
$114$	0	0
$115$	3.65685	0.341003
$116$	−5.41421	−0.502697
$117$	0	0
$118$	0.828427	0.0762629
$119$	0	0
$120$	0	0
$121$	0.656854	0.0597140
$122$	1.65685	0.150005
$123$	0	0
$124$	−7.41421	−0.665816
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.1421	−1.25491	−0.627456	−	0.778652i	$-0.715904\pi$
$127$	−14.1421	−1.25491	−0.627456	+	0.778652i	$0.715904\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.17157	−0.102754
$131$	6.48528	0.566622	0.283311	−	0.959028i	$-0.408567\pi$
$131$	6.48528	0.566622	0.283311	+	0.959028i	$0.408567\pi$
$132$	0	0
$133$	0	0
$134$	5.89949	0.509639
$135$	0	0
$136$	−3.41421	−0.292766
$137$	−4.48528	−0.383203	−0.191602	−	0.981473i	$-0.561368\pi$
$137$	−4.48528	−0.383203	−0.191602	+	0.981473i	$0.561368\pi$
$138$	0	0
$139$	1.31371	0.111427	0.0557137	−	0.998447i	$-0.482257\pi$
$139$	1.31371	0.111427	0.0557137	+	0.998447i	$0.482257\pi$
$140$	0	0
$141$	0	0
$142$	10.4853	0.879905
$143$	4.00000	0.334497
$144$	0	0
$145$	−5.41421	−0.449626
$146$	−12.8284	−1.06169
$147$	0	0
$148$	−3.41421	−0.280647
$149$	12.7279	1.04271	0.521356	−	0.853339i	$-0.325426\pi$
$149$	12.7279	1.04271	0.521356	+	0.853339i	$0.325426\pi$
$150$	0	0
$151$	3.17157	0.258099	0.129049	−	0.991638i	$-0.458807\pi$
$151$	3.17157	0.258099	0.129049	+	0.991638i	$0.458807\pi$
$152$	−4.82843	−0.391637
$153$	0	0
$154$	0	0
$155$	−7.41421	−0.595524
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	13.6569	1.08648
$159$	0	0
$160$	1.00000	0.0790569
$161$	0	0
$162$	0	0
$163$	7.07107	0.553849	0.276924	−	0.960892i	$-0.410685\pi$
$163$	7.07107	0.553849	0.276924	+	0.960892i	$0.410685\pi$
$164$	3.65685	0.285552
$165$	0	0
$166$	−0.485281	−0.0376651
$167$	6.58579	0.509623	0.254812	−	0.966991i	$-0.417987\pi$
$167$	6.58579	0.509623	0.254812	+	0.966991i	$0.417987\pi$
$168$	0	0
$169$	−11.6274	−0.894417
$170$	−3.41421	−0.261858
$171$	0	0
$172$	−7.07107	−0.539164
$173$	20.1421	1.53138	0.765689	−	0.643211i	$-0.222398\pi$
$173$	20.1421	1.53138	0.765689	+	0.643211i	$0.222398\pi$
$174$	0	0
$175$	0	0
$176$	−3.41421	−0.257356
$177$	0	0
$178$	−14.4853	−1.08572
$179$	−10.7279	−0.801843	−0.400921	−	0.916113i	$-0.631310\pi$
$179$	−10.7279	−0.801843	−0.400921	+	0.916113i	$0.631310\pi$
$180$	0	0
$181$	−3.51472	−0.261247	−0.130623	−	0.991432i	$-0.541698\pi$
$181$	−3.51472	−0.261247	−0.130623	+	0.991432i	$0.541698\pi$
$182$	0	0
$183$	0	0
$184$	3.65685	0.269587
$185$	−3.41421	−0.251018
$186$	0	0
$187$	11.6569	0.852434
$188$	−2.58579	−0.188588
$189$	0	0
$190$	−4.82843	−0.350291
$191$	9.31371	0.673916	0.336958	−	0.941520i	$-0.390602\pi$
$191$	9.31371	0.673916	0.336958	+	0.941520i	$0.390602\pi$
$192$	0	0
$193$	20.1421	1.44986	0.724931	−	0.688821i	$-0.241871\pi$
$193$	20.1421	1.44986	0.724931	+	0.688821i	$0.241871\pi$
$194$	−14.4853	−1.03998
$195$	0	0
$196$	0	0
$197$	0.343146	0.0244481	0.0122241	−	0.999925i	$-0.496109\pi$
$197$	0.343146	0.0244481	0.0122241	+	0.999925i	$0.496109\pi$
$198$	0	0
$199$	−18.2426	−1.29319	−0.646593	−	0.762835i	$-0.723807\pi$
$199$	−18.2426	−1.29319	−0.646593	+	0.762835i	$0.723807\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−9.65685	−0.679454
$203$	0	0
$204$	0	0
$205$	3.65685	0.255406
$206$	−5.65685	−0.394132
$207$	0	0
$208$	−1.17157	−0.0812340
$209$	16.4853	1.14031
$210$	0	0
$211$	−23.7990	−1.63839	−0.819195	−	0.573515i	$-0.805579\pi$
$211$	−23.7990	−1.63839	−0.819195	+	0.573515i	$0.805579\pi$
$212$	−6.48528	−0.445411
$213$	0	0
$214$	10.8284	0.740216
$215$	−7.07107	−0.482243
$216$	0	0
$217$	0	0
$218$	14.4853	0.981067
$219$	0	0
$220$	−3.41421	−0.230186
$221$	4.00000	0.269069
$222$	0	0
$223$	10.8284	0.725125	0.362563	−	0.931959i	$-0.381902\pi$
$223$	10.8284	0.725125	0.362563	+	0.931959i	$0.381902\pi$
$224$	0	0
$225$	0	0
$226$	4.34315	0.288902
$227$	24.9706	1.65735	0.828677	−	0.559727i	$-0.189094\pi$
$227$	24.9706	1.65735	0.828677	+	0.559727i	$0.189094\pi$
$228$	0	0
$229$	5.17157	0.341747	0.170874	−	0.985293i	$-0.445341\pi$
$229$	5.17157	0.341747	0.170874	+	0.985293i	$0.445341\pi$
$230$	3.65685	0.241126
$231$	0	0
$232$	−5.41421	−0.355461
$233$	23.6569	1.54981	0.774906	−	0.632076i	$-0.217797\pi$
$233$	23.6569	1.54981	0.774906	+	0.632076i	$0.217797\pi$
$234$	0	0
$235$	−2.58579	−0.168678
$236$	0.828427	0.0539260
$237$	0	0
$238$	0	0
$239$	−19.6569	−1.27150	−0.635748	−	0.771897i	$-0.719308\pi$
$239$	−19.6569	−1.27150	−0.635748	+	0.771897i	$0.719308\pi$
$240$	0	0
$241$	−19.0711	−1.22848	−0.614238	−	0.789121i	$-0.710536\pi$
$241$	−19.0711	−1.22848	−0.614238	+	0.789121i	$0.710536\pi$
$242$	0.656854	0.0422242
$243$	0	0
$244$	1.65685	0.106069
$245$	0	0
$246$	0	0
$247$	5.65685	0.359937
$248$	−7.41421	−0.470803
$249$	0	0
$250$	1.00000	0.0632456
$251$	14.3431	0.905331	0.452666	−	0.891680i	$-0.350473\pi$
$251$	14.3431	0.905331	0.452666	+	0.891680i	$0.350473\pi$
$252$	0	0
$253$	−12.4853	−0.784943
$254$	−14.1421	−0.887357
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.89949	−0.243244	−0.121622	−	0.992577i	$-0.538810\pi$
$257$	−3.89949	−0.243244	−0.121622	+	0.992577i	$0.538810\pi$
$258$	0	0
$259$	0	0
$260$	−1.17157	−0.0726579
$261$	0	0
$262$	6.48528	0.400662
$263$	−10.0000	−0.616626	−0.308313	−	0.951285i	$-0.599764\pi$
$263$	−10.0000	−0.616626	−0.308313	+	0.951285i	$0.599764\pi$
$264$	0	0
$265$	−6.48528	−0.398388
$266$	0	0
$267$	0	0
$268$	5.89949	0.360369
$269$	14.9706	0.912771	0.456386	−	0.889782i	$-0.349144\pi$
$269$	14.9706	0.912771	0.456386	+	0.889782i	$0.349144\pi$
$270$	0	0
$271$	−1.75736	−0.106752	−0.0533760	−	0.998574i	$-0.516998\pi$
$271$	−1.75736	−0.106752	−0.0533760	+	0.998574i	$0.516998\pi$
$272$	−3.41421	−0.207017
$273$	0	0
$274$	−4.48528	−0.270966
$275$	−3.41421	−0.205885
$276$	0	0
$277$	−31.4142	−1.88750	−0.943749	−	0.330664i	$-0.892727\pi$
$277$	−31.4142	−1.88750	−0.943749	+	0.330664i	$0.892727\pi$
$278$	1.31371	0.0787910
$279$	0	0
$280$	0	0
$281$	−21.4558	−1.27995	−0.639974	−	0.768396i	$-0.721055\pi$
$281$	−21.4558	−1.27995	−0.639974	+	0.768396i	$0.721055\pi$
$282$	0	0
$283$	14.9706	0.889908	0.444954	−	0.895554i	$-0.353220\pi$
$283$	14.9706	0.889908	0.444954	+	0.895554i	$0.353220\pi$
$284$	10.4853	0.622187
$285$	0	0
$286$	4.00000	0.236525
$287$	0	0
$288$	0	0
$289$	−5.34315	−0.314303
$290$	−5.41421	−0.317934
$291$	0	0
$292$	−12.8284	−0.750727
$293$	−25.3137	−1.47884	−0.739421	−	0.673243i	$-0.764901\pi$
$293$	−25.3137	−1.47884	−0.739421	+	0.673243i	$0.764901\pi$
$294$	0	0
$295$	0.828427	0.0482329
$296$	−3.41421	−0.198447
$297$	0	0
$298$	12.7279	0.737309
$299$	−4.28427	−0.247766
$300$	0	0
$301$	0	0
$302$	3.17157	0.182504
$303$	0	0
$304$	−4.82843	−0.276929
$305$	1.65685	0.0948712
$306$	0	0
$307$	−26.6274	−1.51971	−0.759853	−	0.650094i	$-0.774729\pi$
$307$	−26.6274	−1.51971	−0.759853	+	0.650094i	$0.774729\pi$
$308$	0	0
$309$	0	0
$310$	−7.41421	−0.421099
$311$	−21.4558	−1.21665	−0.608325	−	0.793688i	$-0.708158\pi$
$311$	−21.4558	−1.21665	−0.608325	+	0.793688i	$0.708158\pi$
$312$	0	0
$313$	−7.65685	−0.432791	−0.216395	−	0.976306i	$-0.569430\pi$
$313$	−7.65685	−0.432791	−0.216395	+	0.976306i	$0.569430\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	13.6569	0.768258
$317$	14.9706	0.840831	0.420415	−	0.907332i	$-0.361884\pi$
$317$	14.9706	0.840831	0.420415	+	0.907332i	$0.361884\pi$
$318$	0	0
$319$	18.4853	1.03498
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	16.4853	0.917266
$324$	0	0
$325$	−1.17157	−0.0649872
$326$	7.07107	0.391630
$327$	0	0
$328$	3.65685	0.201916
$329$	0	0
$330$	0	0
$331$	15.7990	0.868391	0.434196	−	0.900819i	$-0.357033\pi$
$331$	15.7990	0.868391	0.434196	+	0.900819i	$0.357033\pi$
$332$	−0.485281	−0.0266333
$333$	0	0
$334$	6.58579	0.360358
$335$	5.89949	0.322324
$336$	0	0
$337$	−26.4853	−1.44275	−0.721373	−	0.692547i	$-0.756488\pi$
$337$	−26.4853	−1.44275	−0.721373	+	0.692547i	$0.756488\pi$
$338$	−11.6274	−0.632448
$339$	0	0
$340$	−3.41421	−0.185162
$341$	25.3137	1.37081
$342$	0	0
$343$	0	0
$344$	−7.07107	−0.381246
$345$	0	0
$346$	20.1421	1.08285
$347$	17.6569	0.947870	0.473935	−	0.880560i	$-0.342833\pi$
$347$	17.6569	0.947870	0.473935	+	0.880560i	$0.342833\pi$
$348$	0	0
$349$	20.4853	1.09655	0.548276	−	0.836297i	$-0.315284\pi$
$349$	20.4853	1.09655	0.548276	+	0.836297i	$0.315284\pi$
$350$	0	0
$351$	0	0
$352$	−3.41421	−0.181978
$353$	−11.8995	−0.633346	−0.316673	−	0.948535i	$-0.602566\pi$
$353$	−11.8995	−0.633346	−0.316673	+	0.948535i	$0.602566\pi$
$354$	0	0
$355$	10.4853	0.556501
$356$	−14.4853	−0.767718
$357$	0	0
$358$	−10.7279	−0.566988
$359$	−32.8284	−1.73262	−0.866309	−	0.499508i	$-0.833514\pi$
$359$	−32.8284	−1.73262	−0.866309	+	0.499508i	$0.833514\pi$
$360$	0	0
$361$	4.31371	0.227037
$362$	−3.51472	−0.184730
$363$	0	0
$364$	0	0
$365$	−12.8284	−0.671471
$366$	0	0
$367$	0.485281	0.0253315	0.0126657	−	0.999920i	$-0.495968\pi$
$367$	0.485281	0.0253315	0.0126657	+	0.999920i	$0.495968\pi$
$368$	3.65685	0.190627
$369$	0	0
$370$	−3.41421	−0.177497
$371$	0	0
$372$	0	0
$373$	21.7574	1.12655	0.563277	−	0.826268i	$-0.309541\pi$
$373$	21.7574	1.12655	0.563277	+	0.826268i	$0.309541\pi$
$374$	11.6569	0.602762
$375$	0	0
$376$	−2.58579	−0.133352
$377$	6.34315	0.326689
$378$	0	0
$379$	4.48528	0.230393	0.115197	−	0.993343i	$-0.463250\pi$
$379$	4.48528	0.230393	0.115197	+	0.993343i	$0.463250\pi$
$380$	−4.82843	−0.247693
$381$	0	0
$382$	9.31371	0.476531
$383$	29.6985	1.51752	0.758761	−	0.651369i	$-0.225805\pi$
$383$	29.6985	1.51752	0.758761	+	0.651369i	$0.225805\pi$
$384$	0	0
$385$	0	0
$386$	20.1421	1.02521
$387$	0	0
$388$	−14.4853	−0.735379
$389$	16.0416	0.813343	0.406671	−	0.913574i	$-0.366689\pi$
$389$	16.0416	0.813343	0.406671	+	0.913574i	$0.366689\pi$
$390$	0	0
$391$	−12.4853	−0.631408
$392$	0	0
$393$	0	0
$394$	0.343146	0.0172874
$395$	13.6569	0.687151
$396$	0	0
$397$	−8.34315	−0.418730	−0.209365	−	0.977838i	$-0.567140\pi$
$397$	−8.34315	−0.418730	−0.209365	+	0.977838i	$0.567140\pi$
$398$	−18.2426	−0.914421
$399$	0	0
$400$	1.00000	0.0500000
$401$	4.97056	0.248218	0.124109	−	0.992269i	$-0.460393\pi$
$401$	4.97056	0.248218	0.124109	+	0.992269i	$0.460393\pi$
$402$	0	0
$403$	8.68629	0.432695
$404$	−9.65685	−0.480446
$405$	0	0
$406$	0	0
$407$	11.6569	0.577809
$408$	0	0
$409$	−26.8701	−1.32864	−0.664319	−	0.747449i	$-0.731279\pi$
$409$	−26.8701	−1.32864	−0.664319	+	0.747449i	$0.731279\pi$
$410$	3.65685	0.180599
$411$	0	0
$412$	−5.65685	−0.278693
$413$	0	0
$414$	0	0
$415$	−0.485281	−0.0238215
$416$	−1.17157	−0.0574411
$417$	0	0
$418$	16.4853	0.806321
$419$	40.2843	1.96802	0.984008	−	0.178126i	$-0.0570034\pi$
$419$	40.2843	1.96802	0.984008	+	0.178126i	$0.0570034\pi$
$420$	0	0
$421$	8.82843	0.430271	0.215136	−	0.976584i	$-0.430981\pi$
$421$	8.82843	0.430271	0.215136	+	0.976584i	$0.430981\pi$
$422$	−23.7990	−1.15852
$423$	0	0
$424$	−6.48528	−0.314953
$425$	−3.41421	−0.165614
$426$	0	0
$427$	0	0
$428$	10.8284	0.523412
$429$	0	0
$430$	−7.07107	−0.340997
$431$	−16.1421	−0.777539	−0.388770	−	0.921335i	$-0.627100\pi$
$431$	−16.1421	−0.777539	−0.388770	+	0.921335i	$0.627100\pi$
$432$	0	0
$433$	−32.8284	−1.57763	−0.788817	−	0.614628i	$-0.789306\pi$
$433$	−32.8284	−1.57763	−0.788817	+	0.614628i	$0.789306\pi$
$434$	0	0
$435$	0	0
$436$	14.4853	0.693719
$437$	−17.6569	−0.844642
$438$	0	0
$439$	−14.2426	−0.679764	−0.339882	−	0.940468i	$-0.610387\pi$
$439$	−14.2426	−0.679764	−0.339882	+	0.940468i	$0.610387\pi$
$440$	−3.41421	−0.162766
$441$	0	0
$442$	4.00000	0.190261
$443$	−0.970563	−0.0461128	−0.0230564	−	0.999734i	$-0.507340\pi$
$443$	−0.970563	−0.0461128	−0.0230564	+	0.999734i	$0.507340\pi$
$444$	0	0
$445$	−14.4853	−0.686668
$446$	10.8284	0.512741
$447$	0	0
$448$	0	0
$449$	−38.8284	−1.83243	−0.916213	−	0.400691i	$-0.868770\pi$
$449$	−38.8284	−1.83243	−0.916213	+	0.400691i	$0.868770\pi$
$450$	0	0
$451$	−12.4853	−0.587909
$452$	4.34315	0.204284
$453$	0	0
$454$	24.9706	1.17193
$455$	0	0
$456$	0	0
$457$	0.142136	0.00664882	0.00332441	−	0.999994i	$-0.498942\pi$
$457$	0.142136	0.00664882	0.00332441	+	0.999994i	$0.498942\pi$
$458$	5.17157	0.241652
$459$	0	0
$460$	3.65685	0.170502
$461$	2.68629	0.125113	0.0625565	−	0.998041i	$-0.480075\pi$
$461$	2.68629	0.125113	0.0625565	+	0.998041i	$0.480075\pi$
$462$	0	0
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	−5.41421	−0.251349
$465$	0	0
$466$	23.6569	1.09588
$467$	38.8284	1.79677	0.898383	−	0.439214i	$-0.144743\pi$
$467$	38.8284	1.79677	0.898383	+	0.439214i	$0.144743\pi$
$468$	0	0
$469$	0	0
$470$	−2.58579	−0.119273
$471$	0	0
$472$	0.828427	0.0381314
$473$	24.1421	1.11006
$474$	0	0
$475$	−4.82843	−0.221543
$476$	0	0
$477$	0	0
$478$	−19.6569	−0.899084
$479$	30.8284	1.40859	0.704293	−	0.709909i	$-0.251264\pi$
$479$	30.8284	1.40859	0.704293	+	0.709909i	$0.251264\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	−19.0711	−0.868663
$483$	0	0
$484$	0.656854	0.0298570
$485$	−14.4853	−0.657743
$486$	0	0
$487$	−9.17157	−0.415604	−0.207802	−	0.978171i	$-0.566631\pi$
$487$	−9.17157	−0.415604	−0.207802	+	0.978171i	$0.566631\pi$
$488$	1.65685	0.0750023
$489$	0	0
$490$	0	0
$491$	18.9289	0.854251	0.427125	−	0.904192i	$-0.359526\pi$
$491$	18.9289	0.854251	0.427125	+	0.904192i	$0.359526\pi$
$492$	0	0
$493$	18.4853	0.832535
$494$	5.65685	0.254514
$495$	0	0
$496$	−7.41421	−0.332908
$497$	0	0
$498$	0	0
$499$	−31.1127	−1.39280	−0.696398	−	0.717656i	$-0.745215\pi$
$499$	−31.1127	−1.39280	−0.696398	+	0.717656i	$0.745215\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	14.3431	0.640166
$503$	25.4142	1.13316	0.566582	−	0.824005i	$-0.308265\pi$
$503$	25.4142	1.13316	0.566582	+	0.824005i	$0.308265\pi$
$504$	0	0
$505$	−9.65685	−0.429724
$506$	−12.4853	−0.555038
$507$	0	0
$508$	−14.1421	−0.627456
$509$	−35.5980	−1.57785	−0.788926	−	0.614488i	$-0.789363\pi$
$509$	−35.5980	−1.57785	−0.788926	+	0.614488i	$0.789363\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−3.89949	−0.171999
$515$	−5.65685	−0.249271
$516$	0	0
$517$	8.82843	0.388274
$518$	0	0
$519$	0	0
$520$	−1.17157	−0.0513769
$521$	20.8284	0.912510	0.456255	−	0.889849i	$-0.349190\pi$
$521$	20.8284	0.912510	0.456255	+	0.889849i	$0.349190\pi$
$522$	0	0
$523$	−24.6274	−1.07688	−0.538441	−	0.842663i	$-0.680986\pi$
$523$	−24.6274	−1.07688	−0.538441	+	0.842663i	$0.680986\pi$
$524$	6.48528	0.283311
$525$	0	0
$526$	−10.0000	−0.436021
$527$	25.3137	1.10268
$528$	0	0
$529$	−9.62742	−0.418583
$530$	−6.48528	−0.281703
$531$	0	0
$532$	0	0
$533$	−4.28427	−0.185572
$534$	0	0
$535$	10.8284	0.468154
$536$	5.89949	0.254819
$537$	0	0
$538$	14.9706	0.645427
$539$	0	0
$540$	0	0
$541$	−43.9411	−1.88918	−0.944588	−	0.328258i	$-0.893539\pi$
$541$	−43.9411	−1.88918	−0.944588	+	0.328258i	$0.893539\pi$
$542$	−1.75736	−0.0754850
$543$	0	0
$544$	−3.41421	−0.146383
$545$	14.4853	0.620481
$546$	0	0
$547$	−7.27208	−0.310932	−0.155466	−	0.987841i	$-0.549688\pi$
$547$	−7.27208	−0.310932	−0.155466	+	0.987841i	$0.549688\pi$
$548$	−4.48528	−0.191602
$549$	0	0
$550$	−3.41421	−0.145583
$551$	26.1421	1.11369
$552$	0	0
$553$	0	0
$554$	−31.4142	−1.33466
$555$	0	0
$556$	1.31371	0.0557137
$557$	−6.68629	−0.283307	−0.141654	−	0.989916i	$-0.545242\pi$
$557$	−6.68629	−0.283307	−0.141654	+	0.989916i	$0.545242\pi$
$558$	0	0
$559$	8.28427	0.350387
$560$	0	0
$561$	0	0
$562$	−21.4558	−0.905060
$563$	−15.7990	−0.665848	−0.332924	−	0.942954i	$-0.608035\pi$
$563$	−15.7990	−0.665848	−0.332924	+	0.942954i	$0.608035\pi$
$564$	0	0
$565$	4.34315	0.182718
$566$	14.9706	0.629260
$567$	0	0
$568$	10.4853	0.439953
$569$	−0.686292	−0.0287708	−0.0143854	−	0.999897i	$-0.504579\pi$
$569$	−0.686292	−0.0287708	−0.0143854	+	0.999897i	$0.504579\pi$
$570$	0	0
$571$	−4.68629	−0.196115	−0.0980576	−	0.995181i	$-0.531263\pi$
$571$	−4.68629	−0.196115	−0.0980576	+	0.995181i	$0.531263\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	0	0
$575$	3.65685	0.152501
$576$	0	0
$577$	36.6274	1.52482	0.762410	−	0.647095i	$-0.224016\pi$
$577$	36.6274	1.52482	0.762410	+	0.647095i	$0.224016\pi$
$578$	−5.34315	−0.222246
$579$	0	0
$580$	−5.41421	−0.224813
$581$	0	0
$582$	0	0
$583$	22.1421	0.917034
$584$	−12.8284	−0.530844
$585$	0	0
$586$	−25.3137	−1.04570
$587$	−10.1421	−0.418611	−0.209305	−	0.977850i	$-0.567120\pi$
$587$	−10.1421	−0.418611	−0.209305	+	0.977850i	$0.567120\pi$
$588$	0	0
$589$	35.7990	1.47507
$590$	0.828427	0.0341058
$591$	0	0
$592$	−3.41421	−0.140323
$593$	39.6985	1.63022	0.815111	−	0.579305i	$-0.196676\pi$
$593$	39.6985	1.63022	0.815111	+	0.579305i	$0.196676\pi$
$594$	0	0
$595$	0	0
$596$	12.7279	0.521356
$597$	0	0
$598$	−4.28427	−0.175197
$599$	−32.6274	−1.33312	−0.666560	−	0.745451i	$-0.732234\pi$
$599$	−32.6274	−1.33312	−0.666560	+	0.745451i	$0.732234\pi$
$600$	0	0
$601$	46.3848	1.89207	0.946037	−	0.324058i	$-0.105047\pi$
$601$	46.3848	1.89207	0.946037	+	0.324058i	$0.105047\pi$
$602$	0	0
$603$	0	0
$604$	3.17157	0.129049
$605$	0.656854	0.0267049
$606$	0	0
$607$	31.1127	1.26283	0.631413	−	0.775447i	$-0.282475\pi$
$607$	31.1127	1.26283	0.631413	+	0.775447i	$0.282475\pi$
$608$	−4.82843	−0.195819
$609$	0	0
$610$	1.65685	0.0670841
$611$	3.02944	0.122558
$612$	0	0
$613$	19.4142	0.784133	0.392066	−	0.919937i	$-0.371760\pi$
$613$	19.4142	0.784133	0.392066	+	0.919937i	$0.371760\pi$
$614$	−26.6274	−1.07460
$615$	0	0
$616$	0	0
$617$	−39.1127	−1.57462	−0.787309	−	0.616559i	$-0.788526\pi$
$617$	−39.1127	−1.57462	−0.787309	+	0.616559i	$0.788526\pi$
$618$	0	0
$619$	−0.828427	−0.0332973	−0.0166486	−	0.999861i	$-0.505300\pi$
$619$	−0.828427	−0.0332973	−0.0166486	+	0.999861i	$0.505300\pi$
$620$	−7.41421	−0.297762
$621$	0	0
$622$	−21.4558	−0.860301
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−7.65685	−0.306029
$627$	0	0
$628$	−10.0000	−0.399043
$629$	11.6569	0.464789
$630$	0	0
$631$	15.4558	0.615287	0.307644	−	0.951502i	$-0.400460\pi$
$631$	15.4558	0.615287	0.307644	+	0.951502i	$0.400460\pi$
$632$	13.6569	0.543240
$633$	0	0
$634$	14.9706	0.594557
$635$	−14.1421	−0.561214
$636$	0	0
$637$	0	0
$638$	18.4853	0.731839
$639$	0	0
$640$	1.00000	0.0395285
$641$	18.6274	0.735739	0.367869	−	0.929877i	$-0.380087\pi$
$641$	18.6274	0.735739	0.367869	+	0.929877i	$0.380087\pi$
$642$	0	0
$643$	−6.97056	−0.274892	−0.137446	−	0.990509i	$-0.543889\pi$
$643$	−6.97056	−0.274892	−0.137446	+	0.990509i	$0.543889\pi$
$644$	0	0
$645$	0	0
$646$	16.4853	0.648605
$647$	13.6152	0.535270	0.267635	−	0.963520i	$-0.413758\pi$
$647$	13.6152	0.535270	0.267635	+	0.963520i	$0.413758\pi$
$648$	0	0
$649$	−2.82843	−0.111025
$650$	−1.17157	−0.0459529
$651$	0	0
$652$	7.07107	0.276924
$653$	45.1127	1.76540	0.882698	−	0.469940i	$-0.155725\pi$
$653$	45.1127	1.76540	0.882698	+	0.469940i	$0.155725\pi$
$654$	0	0
$655$	6.48528	0.253401
$656$	3.65685	0.142776
$657$	0	0
$658$	0	0
$659$	−28.5858	−1.11354	−0.556772	−	0.830665i	$-0.687960\pi$
$659$	−28.5858	−1.11354	−0.556772	+	0.830665i	$0.687960\pi$
$660$	0	0
$661$	22.3431	0.869048	0.434524	−	0.900660i	$-0.356917\pi$
$661$	22.3431	0.869048	0.434524	+	0.900660i	$0.356917\pi$
$662$	15.7990	0.614045
$663$	0	0
$664$	−0.485281	−0.0188326
$665$	0	0
$666$	0	0
$667$	−19.7990	−0.766620
$668$	6.58579	0.254812
$669$	0	0
$670$	5.89949	0.227917
$671$	−5.65685	−0.218380
$672$	0	0
$673$	10.9706	0.422884	0.211442	−	0.977391i	$-0.432184\pi$
$673$	10.9706	0.422884	0.211442	+	0.977391i	$0.432184\pi$
$674$	−26.4853	−1.02017
$675$	0	0
$676$	−11.6274	−0.447208
$677$	15.4558	0.594016	0.297008	−	0.954875i	$-0.404011\pi$
$677$	15.4558	0.594016	0.297008	+	0.954875i	$0.404011\pi$
$678$	0	0
$679$	0	0
$680$	−3.41421	−0.130929
$681$	0	0
$682$	25.3137	0.969312
$683$	26.3431	1.00799	0.503996	−	0.863706i	$-0.331863\pi$
$683$	26.3431	1.00799	0.503996	+	0.863706i	$0.331863\pi$
$684$	0	0
$685$	−4.48528	−0.171374
$686$	0	0
$687$	0	0
$688$	−7.07107	−0.269582
$689$	7.59798	0.289460
$690$	0	0
$691$	1.02944	0.0391616	0.0195808	−	0.999808i	$-0.493767\pi$
$691$	1.02944	0.0391616	0.0195808	+	0.999808i	$0.493767\pi$
$692$	20.1421	0.765689
$693$	0	0
$694$	17.6569	0.670245
$695$	1.31371	0.0498318
$696$	0	0
$697$	−12.4853	−0.472914
$698$	20.4853	0.775379
$699$	0	0
$700$	0	0
$701$	7.75736	0.292991	0.146496	−	0.989211i	$-0.453201\pi$
$701$	7.75736	0.292991	0.146496	+	0.989211i	$0.453201\pi$
$702$	0	0
$703$	16.4853	0.621754
$704$	−3.41421	−0.128678
$705$	0	0
$706$	−11.8995	−0.447843
$707$	0	0
$708$	0	0
$709$	21.1127	0.792904	0.396452	−	0.918055i	$-0.370241\pi$
$709$	21.1127	0.792904	0.396452	+	0.918055i	$0.370241\pi$
$710$	10.4853	0.393506
$711$	0	0
$712$	−14.4853	−0.542859
$713$	−27.1127	−1.01538
$714$	0	0
$715$	4.00000	0.149592
$716$	−10.7279	−0.400921
$717$	0	0
$718$	−32.8284	−1.22515
$719$	−33.4558	−1.24769	−0.623846	−	0.781547i	$-0.714431\pi$
$719$	−33.4558	−1.24769	−0.623846	+	0.781547i	$0.714431\pi$
$720$	0	0
$721$	0	0
$722$	4.31371	0.160540
$723$	0	0
$724$	−3.51472	−0.130623
$725$	−5.41421	−0.201079
$726$	0	0
$727$	−13.4558	−0.499050	−0.249525	−	0.968368i	$-0.580274\pi$
$727$	−13.4558	−0.499050	−0.249525	+	0.968368i	$0.580274\pi$
$728$	0	0
$729$	0	0
$730$	−12.8284	−0.474801
$731$	24.1421	0.892929
$732$	0	0
$733$	6.97056	0.257464	0.128732	−	0.991679i	$-0.458909\pi$
$733$	6.97056	0.257464	0.128732	+	0.991679i	$0.458909\pi$
$734$	0.485281	0.0179121
$735$	0	0
$736$	3.65685	0.134793
$737$	−20.1421	−0.741945
$738$	0	0
$739$	−2.82843	−0.104045	−0.0520227	−	0.998646i	$-0.516567\pi$
$739$	−2.82843	−0.104045	−0.0520227	+	0.998646i	$0.516567\pi$
$740$	−3.41421	−0.125509
$741$	0	0
$742$	0	0
$743$	45.6569	1.67499	0.837494	−	0.546447i	$-0.184020\pi$
$743$	45.6569	1.67499	0.837494	+	0.546447i	$0.184020\pi$
$744$	0	0
$745$	12.7279	0.466315
$746$	21.7574	0.796594
$747$	0	0
$748$	11.6569	0.426217
$749$	0	0
$750$	0	0
$751$	−41.7990	−1.52527	−0.762633	−	0.646831i	$-0.776094\pi$
$751$	−41.7990	−1.52527	−0.762633	+	0.646831i	$0.776094\pi$
$752$	−2.58579	−0.0942939
$753$	0	0
$754$	6.34315	0.231004
$755$	3.17157	0.115425
$756$	0	0
$757$	21.3553	0.776173	0.388087	−	0.921623i	$-0.373136\pi$
$757$	21.3553	0.776173	0.388087	+	0.921623i	$0.373136\pi$
$758$	4.48528	0.162913
$759$	0	0
$760$	−4.82843	−0.175145
$761$	8.62742	0.312744	0.156372	−	0.987698i	$-0.450020\pi$
$761$	8.62742	0.312744	0.156372	+	0.987698i	$0.450020\pi$
$762$	0	0
$763$	0	0
$764$	9.31371	0.336958
$765$	0	0
$766$	29.6985	1.07305
$767$	−0.970563	−0.0350450
$768$	0	0
$769$	12.9289	0.466229	0.233115	−	0.972449i	$-0.425108\pi$
$769$	12.9289	0.466229	0.233115	+	0.972449i	$0.425108\pi$
$770$	0	0
$771$	0	0
$772$	20.1421	0.724931
$773$	−18.9706	−0.682324	−0.341162	−	0.940005i	$-0.610820\pi$
$773$	−18.9706	−0.682324	−0.341162	+	0.940005i	$0.610820\pi$
$774$	0	0
$775$	−7.41421	−0.266326
$776$	−14.4853	−0.519991
$777$	0	0
$778$	16.0416	0.575120
$779$	−17.6569	−0.632622
$780$	0	0
$781$	−35.7990	−1.28099
$782$	−12.4853	−0.446473
$783$	0	0
$784$	0	0
$785$	−10.0000	−0.356915
$786$	0	0
$787$	3.02944	0.107988	0.0539939	−	0.998541i	$-0.482805\pi$
$787$	3.02944	0.107988	0.0539939	+	0.998541i	$0.482805\pi$
$788$	0.343146	0.0122241
$789$	0	0
$790$	13.6569	0.485889
$791$	0	0
$792$	0	0
$793$	−1.94113	−0.0689314
$794$	−8.34315	−0.296087
$795$	0	0
$796$	−18.2426	−0.646593
$797$	−7.85786	−0.278340	−0.139170	−	0.990269i	$-0.544443\pi$
$797$	−7.85786	−0.278340	−0.139170	+	0.990269i	$0.544443\pi$
$798$	0	0
$799$	8.82843	0.312327
$800$	1.00000	0.0353553
$801$	0	0
$802$	4.97056	0.175517
$803$	43.7990	1.54563
$804$	0	0
$805$	0	0
$806$	8.68629	0.305962
$807$	0	0
$808$	−9.65685	−0.339727
$809$	−16.4853	−0.579592	−0.289796	−	0.957088i	$-0.593587\pi$
$809$	−16.4853	−0.579592	−0.289796	+	0.957088i	$0.593587\pi$
$810$	0	0
$811$	24.6274	0.864786	0.432393	−	0.901685i	$-0.357669\pi$
$811$	24.6274	0.864786	0.432393	+	0.901685i	$0.357669\pi$
$812$	0	0
$813$	0	0
$814$	11.6569	0.408573
$815$	7.07107	0.247689
$816$	0	0
$817$	34.1421	1.19448
$818$	−26.8701	−0.939490
$819$	0	0
$820$	3.65685	0.127703
$821$	28.9289	1.00963	0.504813	−	0.863229i	$-0.331561\pi$
$821$	28.9289	1.00963	0.504813	+	0.863229i	$0.331561\pi$
$822$	0	0
$823$	10.3431	0.360539	0.180270	−	0.983617i	$-0.442303\pi$
$823$	10.3431	0.360539	0.180270	+	0.983617i	$0.442303\pi$
$824$	−5.65685	−0.197066
$825$	0	0
$826$	0	0
$827$	23.7990	0.827572	0.413786	−	0.910374i	$-0.364206\pi$
$827$	23.7990	0.827572	0.413786	+	0.910374i	$0.364206\pi$
$828$	0	0
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	−0.485281	−0.0168444
$831$	0	0
$832$	−1.17157	−0.0406170
$833$	0	0
$834$	0	0
$835$	6.58579	0.227911
$836$	16.4853	0.570155
$837$	0	0
$838$	40.2843	1.39160
$839$	24.2843	0.838386	0.419193	−	0.907897i	$-0.362313\pi$
$839$	24.2843	0.838386	0.419193	+	0.907897i	$0.362313\pi$
$840$	0	0
$841$	0.313708	0.0108175
$842$	8.82843	0.304248
$843$	0	0
$844$	−23.7990	−0.819195
$845$	−11.6274	−0.399995
$846$	0	0
$847$	0	0
$848$	−6.48528	−0.222705
$849$	0	0
$850$	−3.41421	−0.117107
$851$	−12.4853	−0.427990
$852$	0	0
$853$	51.6569	1.76870	0.884349	−	0.466827i	$-0.154603\pi$
$853$	51.6569	1.76870	0.884349	+	0.466827i	$0.154603\pi$
$854$	0	0
$855$	0	0
$856$	10.8284	0.370108
$857$	−51.8995	−1.77285	−0.886426	−	0.462869i	$-0.846820\pi$
$857$	−51.8995	−1.77285	−0.886426	+	0.462869i	$0.846820\pi$
$858$	0	0
$859$	−30.9706	−1.05670	−0.528351	−	0.849026i	$-0.677189\pi$
$859$	−30.9706	−1.05670	−0.528351	+	0.849026i	$0.677189\pi$
$860$	−7.07107	−0.241121
$861$	0	0
$862$	−16.1421	−0.549803
$863$	28.9706	0.986169	0.493085	−	0.869981i	$-0.335869\pi$
$863$	28.9706	0.986169	0.493085	+	0.869981i	$0.335869\pi$
$864$	0	0
$865$	20.1421	0.684853
$866$	−32.8284	−1.11556
$867$	0	0
$868$	0	0
$869$	−46.6274	−1.58173
$870$	0	0
$871$	−6.91169	−0.234194
$872$	14.4853	0.490534
$873$	0	0
$874$	−17.6569	−0.597252
$875$	0	0
$876$	0	0
$877$	−41.5563	−1.40326	−0.701629	−	0.712542i	$-0.747544\pi$
$877$	−41.5563	−1.40326	−0.701629	+	0.712542i	$0.747544\pi$
$878$	−14.2426	−0.480666
$879$	0	0
$880$	−3.41421	−0.115093
$881$	−52.1421	−1.75671	−0.878357	−	0.478006i	$-0.841360\pi$
$881$	−52.1421	−1.75671	−0.878357	+	0.478006i	$0.841360\pi$
$882$	0	0
$883$	20.5269	0.690786	0.345393	−	0.938458i	$-0.387746\pi$
$883$	20.5269	0.690786	0.345393	+	0.938458i	$0.387746\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−0.970563	−0.0326067
$887$	−45.8995	−1.54115	−0.770577	−	0.637347i	$-0.780032\pi$
$887$	−45.8995	−1.54115	−0.770577	+	0.637347i	$0.780032\pi$
$888$	0	0
$889$	0	0
$890$	−14.4853	−0.485548
$891$	0	0
$892$	10.8284	0.362563
$893$	12.4853	0.417804
$894$	0	0
$895$	−10.7279	−0.358595
$896$	0	0
$897$	0	0
$898$	−38.8284	−1.29572
$899$	40.1421	1.33882
$900$	0	0
$901$	22.1421	0.737661
$902$	−12.4853	−0.415714
$903$	0	0
$904$	4.34315	0.144451
$905$	−3.51472	−0.116833
$906$	0	0
$907$	−56.3259	−1.87027	−0.935135	−	0.354290i	$-0.884722\pi$
$907$	−56.3259	−1.87027	−0.935135	+	0.354290i	$0.884722\pi$
$908$	24.9706	0.828677
$909$	0	0
$910$	0	0
$911$	25.5980	0.848099	0.424049	−	0.905639i	$-0.360608\pi$
$911$	25.5980	0.848099	0.424049	+	0.905639i	$0.360608\pi$
$912$	0	0
$913$	1.65685	0.0548339
$914$	0.142136	0.00470143
$915$	0	0
$916$	5.17157	0.170874
$917$	0	0
$918$	0	0
$919$	−19.4558	−0.641789	−0.320895	−	0.947115i	$-0.603984\pi$
$919$	−19.4558	−0.641789	−0.320895	+	0.947115i	$0.603984\pi$
$920$	3.65685	0.120563
$921$	0	0
$922$	2.68629	0.0884683
$923$	−12.2843	−0.404342
$924$	0	0
$925$	−3.41421	−0.112259
$926$	36.0000	1.18303
$927$	0	0
$928$	−5.41421	−0.177730
$929$	−52.9117	−1.73598	−0.867988	−	0.496585i	$-0.834587\pi$
$929$	−52.9117	−1.73598	−0.867988	+	0.496585i	$0.834587\pi$
$930$	0	0
$931$	0	0
$932$	23.6569	0.774906
$933$	0	0
$934$	38.8284	1.27050
$935$	11.6569	0.381220
$936$	0	0
$937$	18.9706	0.619741	0.309871	−	0.950779i	$-0.399714\pi$
$937$	18.9706	0.619741	0.309871	+	0.950779i	$0.399714\pi$
$938$	0	0
$939$	0	0
$940$	−2.58579	−0.0843391
$941$	−56.0000	−1.82555	−0.912774	−	0.408465i	$-0.866064\pi$
$941$	−56.0000	−1.82555	−0.912774	+	0.408465i	$0.866064\pi$
$942$	0	0
$943$	13.3726	0.435471
$944$	0.828427	0.0269630
$945$	0	0
$946$	24.1421	0.784929
$947$	−38.6274	−1.25522	−0.627611	−	0.778527i	$-0.715967\pi$
$947$	−38.6274	−1.25522	−0.627611	+	0.778527i	$0.715967\pi$
$948$	0	0
$949$	15.0294	0.487876
$950$	−4.82843	−0.156655
$951$	0	0
$952$	0	0
$953$	45.1716	1.46325	0.731625	−	0.681707i	$-0.238762\pi$
$953$	45.1716	1.46325	0.731625	+	0.681707i	$0.238762\pi$
$954$	0	0
$955$	9.31371	0.301385
$956$	−19.6569	−0.635748
$957$	0	0
$958$	30.8284	0.996021
$959$	0	0
$960$	0	0
$961$	23.9706	0.773244
$962$	4.00000	0.128965
$963$	0	0
$964$	−19.0711	−0.614238
$965$	20.1421	0.648398
$966$	0	0
$967$	40.9706	1.31752	0.658762	−	0.752351i	$-0.271080\pi$
$967$	40.9706	1.31752	0.658762	+	0.752351i	$0.271080\pi$
$968$	0.656854	0.0211121
$969$	0	0
$970$	−14.4853	−0.465094
$971$	−9.65685	−0.309903	−0.154952	−	0.987922i	$-0.549522\pi$
$971$	−9.65685	−0.309903	−0.154952	+	0.987922i	$0.549522\pi$
$972$	0	0
$973$	0	0
$974$	−9.17157	−0.293876
$975$	0	0
$976$	1.65685	0.0530346
$977$	−29.3137	−0.937829	−0.468914	−	0.883244i	$-0.655355\pi$
$977$	−29.3137	−0.937829	−0.468914	+	0.883244i	$0.655355\pi$
$978$	0	0
$979$	49.4558	1.58062
$980$	0	0
$981$	0	0
$982$	18.9289	0.604046
$983$	−14.3848	−0.458803	−0.229402	−	0.973332i	$-0.573677\pi$
$983$	−14.3848	−0.458803	−0.229402	+	0.973332i	$0.573677\pi$
$984$	0	0
$985$	0.343146	0.0109335
$986$	18.4853	0.588691
$987$	0	0
$988$	5.65685	0.179969
$989$	−25.8579	−0.822232
$990$	0	0
$991$	−12.1421	−0.385708	−0.192854	−	0.981227i	$-0.561774\pi$
$991$	−12.1421	−0.385708	−0.192854	+	0.981227i	$0.561774\pi$
$992$	−7.41421	−0.235402
$993$	0	0
$994$	0	0
$995$	−18.2426	−0.578331
$996$	0	0
$997$	8.34315	0.264230	0.132115	−	0.991234i	$-0.457823\pi$
$997$	8.34315	0.264230	0.132115	+	0.991234i	$0.457823\pi$
$998$	−31.1127	−0.984855
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.a.bx.1.1	yes	2
3.2	odd	2		4410.2.a.bp.1.2	✓	2
7.6	odd	2		4410.2.a.bu.1.1	yes	2
21.20	even	2		4410.2.a.bs.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4410.2.a.bp.1.2	✓	2	3.2	odd	2
4410.2.a.bs.1.2	yes	2	21.20	even	2
4410.2.a.bu.1.1	yes	2	7.6	odd	2
4410.2.a.bx.1.1	yes	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:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} -3.41421 q^{11} -1.17157 q^{13} +1.00000 q^{16} -3.41421 q^{17} -4.82843 q^{19} +1.00000 q^{20} -3.41421 q^{22} +3.65685 q^{23} +1.00000 q^{25} -1.17157 q^{26} -5.41421 q^{29} -7.41421 q^{31} +1.00000 q^{32} -3.41421 q^{34} -3.41421 q^{37} -4.82843 q^{38} +1.00000 q^{40} +3.65685 q^{41} -7.07107 q^{43} -3.41421 q^{44} +3.65685 q^{46} -2.58579 q^{47} +1.00000 q^{50} -1.17157 q^{52} -6.48528 q^{53} -3.41421 q^{55} -5.41421 q^{58} +0.828427 q^{59} +1.65685 q^{61} -7.41421 q^{62} +1.00000 q^{64} -1.17157 q^{65} +5.89949 q^{67} -3.41421 q^{68} +10.4853 q^{71} -12.8284 q^{73} -3.41421 q^{74} -4.82843 q^{76} +13.6569 q^{79} +1.00000 q^{80} +3.65685 q^{82} -0.485281 q^{83} -3.41421 q^{85} -7.07107 q^{86} -3.41421 q^{88} -14.4853 q^{89} +3.65685 q^{92} -2.58579 q^{94} -4.82843 q^{95} -14.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} - 4 q^{11} - 8 q^{13} + 2 q^{16} - 4 q^{17} - 4 q^{19} + 2 q^{20} - 4 q^{22} - 4 q^{23} + 2 q^{25} - 8 q^{26} - 8 q^{29} - 12 q^{31} + 2 q^{32} - 4 q^{34} - 4 q^{37} - 4 q^{38} + 2 q^{40} - 4 q^{41} - 4 q^{44} - 4 q^{46} - 8 q^{47} + 2 q^{50} - 8 q^{52} + 4 q^{53} - 4 q^{55} - 8 q^{58} - 4 q^{59} - 8 q^{61} - 12 q^{62} + 2 q^{64} - 8 q^{65} - 8 q^{67} - 4 q^{68} + 4 q^{71} - 20 q^{73} - 4 q^{74} - 4 q^{76} + 16 q^{79} + 2 q^{80} - 4 q^{82} + 16 q^{83} - 4 q^{85} - 4 q^{88} - 12 q^{89} - 4 q^{92} - 8 q^{94} - 4 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 - 4 * q^11 - 8 * q^13 + 2 * q^16 - 4 * q^17 - 4 * q^19 + 2 * q^20 - 4 * q^22 - 4 * q^23 + 2 * q^25 - 8 * q^26 - 8 * q^29 - 12 * q^31 + 2 * q^32 - 4 * q^34 - 4 * q^37 - 4 * q^38 + 2 * q^40 - 4 * q^41 - 4 * q^44 - 4 * q^46 - 8 * q^47 + 2 * q^50 - 8 * q^52 + 4 * q^53 - 4 * q^55 - 8 * q^58 - 4 * q^59 - 8 * q^61 - 12 * q^62 + 2 * q^64 - 8 * q^65 - 8 * q^67 - 4 * q^68 + 4 * q^71 - 20 * q^73 - 4 * q^74 - 4 * q^76 + 16 * q^79 + 2 * q^80 - 4 * q^82 + 16 * q^83 - 4 * q^85 - 4 * q^88 - 12 * q^89 - 4 * q^92 - 8 * q^94 - 4 * q^95 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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