LMFDB - Embedded newform 425.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,2,Mod(1,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, 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("425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.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:	$3.39364208590$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} -4.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} -1.00000 q^{16} +1.00000 q^{17} -2.00000 q^{18} -6.00000 q^{19} -1.00000 q^{21} -4.00000 q^{22} +3.00000 q^{24} -1.00000 q^{26} +5.00000 q^{27} -1.00000 q^{28} -7.00000 q^{31} +5.00000 q^{32} +4.00000 q^{33} +1.00000 q^{34} +2.00000 q^{36} -4.00000 q^{37} -6.00000 q^{38} +1.00000 q^{39} -2.00000 q^{41} -1.00000 q^{42} +4.00000 q^{43} +4.00000 q^{44} -6.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -1.00000 q^{51} +1.00000 q^{52} +11.0000 q^{53} +5.00000 q^{54} -3.00000 q^{56} +6.00000 q^{57} +8.00000 q^{59} +10.0000 q^{61} -7.00000 q^{62} -2.00000 q^{63} +7.00000 q^{64} +4.00000 q^{66} +8.00000 q^{67} -1.00000 q^{68} +7.00000 q^{71} +6.00000 q^{72} +4.00000 q^{73} -4.00000 q^{74} +6.00000 q^{76} -4.00000 q^{77} +1.00000 q^{78} -11.0000 q^{79} +1.00000 q^{81} -2.00000 q^{82} -8.00000 q^{83} +1.00000 q^{84} +4.00000 q^{86} +12.0000 q^{88} -6.00000 q^{89} -1.00000 q^{91} +7.00000 q^{93} -6.00000 q^{94} -5.00000 q^{96} -16.0000 q^{97} -6.00000 q^{98} +8.00000 q^{99} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−3.00000	−1.06066
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−2.00000	−0.471405
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	−1.00000	−0.196116
$27$	5.00000	0.962250
$28$	−1.00000	−0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	5.00000	0.883883
$33$	4.00000	0.696311
$34$	1.00000	0.171499
$35$	0	0
$36$	2.00000	0.333333
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−6.00000	−0.973329
$39$	1.00000	0.160128
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−1.00000	−0.154303
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−1.00000	−0.140028
$52$	1.00000	0.138675
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	−3.00000	−0.400892
$57$	6.00000	0.794719
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−7.00000	−0.889001
$63$	−2.00000	−0.251976
$64$	7.00000	0.875000
$65$	0	0
$66$	4.00000	0.492366
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	0	0
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	6.00000	0.707107
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	6.00000	0.688247
$77$	−4.00000	−0.455842
$78$	1.00000	0.113228
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	12.0000	1.27920
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	7.00000	0.725866
$94$	−6.00000	−0.618853
$95$	0	0
$96$	−5.00000	−0.510310
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	−6.00000	−0.606092
$99$	8.00000	0.804030
$100$	0	0
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	−1.00000	−0.0990148
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	11.0000	1.06841
$107$	−19.0000	−1.83680	−0.918400	−	0.395654i	$-0.870518\pi$
$107$	−19.0000	−1.83680	−0.918400	+	0.395654i	$0.870518\pi$
$108$	−5.00000	−0.481125
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	−1.00000	−0.0944911
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	6.00000	0.561951
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	8.00000	0.736460
$119$	1.00000	0.0916698
$120$	0	0
$121$	5.00000	0.454545
$122$	10.0000	0.905357
$123$	2.00000	0.180334
$124$	7.00000	0.628619
$125$	0	0
$126$	−2.00000	−0.178174
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−3.00000	−0.265165
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	−4.00000	−0.348155
$133$	−6.00000	−0.520266
$134$	8.00000	0.691095
$135$	0	0
$136$	−3.00000	−0.257248
$137$	−11.0000	−0.939793	−0.469897	−	0.882721i	$-0.655709\pi$
$137$	−11.0000	−0.939793	−0.469897	+	0.882721i	$0.655709\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	7.00000	0.587427
$143$	4.00000	0.334497
$144$	2.00000	0.166667
$145$	0	0
$146$	4.00000	0.331042
$147$	6.00000	0.494872
$148$	4.00000	0.328798
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	18.0000	1.45999
$153$	−2.00000	−0.161690
$154$	−4.00000	−0.322329
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	21.0000	1.67598	0.837991	−	0.545684i	$-0.183730\pi$
$157$	21.0000	1.67598	0.837991	+	0.545684i	$0.183730\pi$
$158$	−11.0000	−0.875113
$159$	−11.0000	−0.872357
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−8.00000	−0.620920
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	3.00000	0.231455
$169$	−12.0000	−0.923077
$170$	0	0
$171$	12.0000	0.917663
$172$	−4.00000	−0.304997
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	−8.00000	−0.601317
$178$	−6.00000	−0.449719
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	−1.00000	−0.0741249
$183$	−10.0000	−0.739221
$184$	0	0
$185$	0	0
$186$	7.00000	0.513265
$187$	−4.00000	−0.292509
$188$	6.00000	0.437595
$189$	5.00000	0.363696
$190$	0	0
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	−7.00000	−0.505181
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	6.00000	0.428571
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	8.00000	0.568535
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	15.0000	1.05540
$203$	0	0
$204$	1.00000	0.0700140
$205$	0	0
$206$	6.00000	0.418040
$207$	0	0
$208$	1.00000	0.0693375
$209$	24.0000	1.66011
$210$	0	0
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	−11.0000	−0.755483
$213$	−7.00000	−0.479632
$214$	−19.0000	−1.29881
$215$	0	0
$216$	−15.0000	−1.02062
$217$	−7.00000	−0.475191
$218$	−4.00000	−0.270914
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	4.00000	0.268462
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	5.00000	0.334077
$225$	0	0
$226$	−12.0000	−0.798228
$227$	25.0000	1.65931	0.829654	−	0.558278i	$-0.188538\pi$
$227$	25.0000	1.65931	0.829654	+	0.558278i	$0.188538\pi$
$228$	−6.00000	−0.397360
$229$	1.00000	0.0660819	0.0330409	−	0.999454i	$-0.489481\pi$
$229$	1.00000	0.0660819	0.0330409	+	0.999454i	$0.489481\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−8.00000	−0.520756
$237$	11.0000	0.714527
$238$	1.00000	0.0648204
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	5.00000	0.321412
$243$	−16.0000	−1.02640
$244$	−10.0000	−0.640184
$245$	0	0
$246$	2.00000	0.127515
$247$	6.00000	0.381771
$248$	21.0000	1.33350
$249$	8.00000	0.506979
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	19.0000	1.18519	0.592594	−	0.805502i	$-0.298104\pi$
$257$	19.0000	1.18519	0.592594	+	0.805502i	$0.298104\pi$
$258$	−4.00000	−0.249029
$259$	−4.00000	−0.248548
$260$	0	0
$261$	0	0
$262$	−15.0000	−0.926703
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	−6.00000	−0.367884
$267$	6.00000	0.367194
$268$	−8.00000	−0.488678
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	−1.00000	−0.0606339
$273$	1.00000	0.0605228
$274$	−11.0000	−0.664534
$275$	0	0
$276$	0	0
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	−13.0000	−0.779688
$279$	14.0000	0.838158
$280$	0	0
$281$	9.00000	0.536895	0.268447	−	0.963294i	$-0.413489\pi$
$281$	9.00000	0.536895	0.268447	+	0.963294i	$0.413489\pi$
$282$	6.00000	0.357295
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	−7.00000	−0.415374
$285$	0	0
$286$	4.00000	0.236525
$287$	−2.00000	−0.118056
$288$	−10.0000	−0.589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	16.0000	0.937937
$292$	−4.00000	−0.234082
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	6.00000	0.349927
$295$	0	0
$296$	12.0000	0.697486
$297$	−20.0000	−1.16052
$298$	21.0000	1.21650
$299$	0	0
$300$	0	0
$301$	4.00000	0.230556
$302$	−18.0000	−1.03578
$303$	−15.0000	−0.861727
$304$	6.00000	0.344124
$305$	0	0
$306$	−2.00000	−0.114332
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	4.00000	0.227921
$309$	−6.00000	−0.341328
$310$	0	0
$311$	−27.0000	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$312$	−3.00000	−0.169842
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	21.0000	1.18510
$315$	0	0
$316$	11.0000	0.618798
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	−11.0000	−0.616849
$319$	0	0
$320$	0	0
$321$	19.0000	1.06048
$322$	0	0
$323$	−6.00000	−0.333849
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−11.0000	−0.609234
$327$	4.00000	0.221201
$328$	6.00000	0.331295
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	8.00000	0.439057
$333$	8.00000	0.438397
$334$	8.00000	0.437741
$335$	0	0
$336$	1.00000	0.0545545
$337$	−24.0000	−1.30736	−0.653682	−	0.756770i	$-0.726776\pi$
$337$	−24.0000	−1.30736	−0.653682	+	0.756770i	$0.726776\pi$
$338$	−12.0000	−0.652714
$339$	12.0000	0.651751
$340$	0	0
$341$	28.0000	1.51629
$342$	12.0000	0.648886
$343$	−13.0000	−0.701934
$344$	−12.0000	−0.646997
$345$	0	0
$346$	6.00000	0.322562
$347$	7.00000	0.375780	0.187890	−	0.982190i	$-0.439835\pi$
$347$	7.00000	0.375780	0.187890	+	0.982190i	$0.439835\pi$
$348$	0	0
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	−20.0000	−1.06600
$353$	19.0000	1.01127	0.505634	−	0.862748i	$-0.331259\pi$
$353$	19.0000	1.01127	0.505634	+	0.862748i	$0.331259\pi$
$354$	−8.00000	−0.425195
$355$	0	0
$356$	6.00000	0.317999
$357$	−1.00000	−0.0529256
$358$	6.00000	0.317110
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−22.0000	−1.15629
$363$	−5.00000	−0.262432
$364$	1.00000	0.0524142
$365$	0	0
$366$	−10.0000	−0.522708
$367$	23.0000	1.20059	0.600295	−	0.799779i	$-0.295050\pi$
$367$	23.0000	1.20059	0.600295	+	0.799779i	$0.295050\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	0	0
$371$	11.0000	0.571092
$372$	−7.00000	−0.362933
$373$	19.0000	0.983783	0.491891	−	0.870657i	$-0.336306\pi$
$373$	19.0000	0.983783	0.491891	+	0.870657i	$0.336306\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	18.0000	0.928279
$377$	0	0
$378$	5.00000	0.257172
$379$	25.0000	1.28416	0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000	1.28416	0.642082	+	0.766636i	$0.278071\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	−10.0000	−0.511645
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	24.0000	1.22157
$387$	−8.00000	−0.406663
$388$	16.0000	0.812277
$389$	−34.0000	−1.72387	−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−34.0000	−1.72387	−0.861934	+	0.507020i	$0.830747\pi$
$390$	0	0
$391$	0	0
$392$	18.0000	0.909137
$393$	15.0000	0.756650
$394$	14.0000	0.705310
$395$	0	0
$396$	−8.00000	−0.402015
$397$	4.00000	0.200754	0.100377	−	0.994949i	$-0.467995\pi$
$397$	4.00000	0.200754	0.100377	+	0.994949i	$0.467995\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	−8.00000	−0.399004
$403$	7.00000	0.348695
$404$	−15.0000	−0.746278
$405$	0	0
$406$	0	0
$407$	16.0000	0.793091
$408$	3.00000	0.148522
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	0	0
$411$	11.0000	0.542590
$412$	−6.00000	−0.295599
$413$	8.00000	0.393654
$414$	0	0
$415$	0	0
$416$	−5.00000	−0.245145
$417$	13.0000	0.636613
$418$	24.0000	1.17388
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−25.0000	−1.21843	−0.609213	−	0.793007i	$-0.708514\pi$
$421$	−25.0000	−1.21843	−0.609213	+	0.793007i	$0.708514\pi$
$422$	−25.0000	−1.21698
$423$	12.0000	0.583460
$424$	−33.0000	−1.60262
$425$	0	0
$426$	−7.00000	−0.339151
$427$	10.0000	0.483934
$428$	19.0000	0.918400
$429$	−4.00000	−0.193122
$430$	0	0
$431$	31.0000	1.49322	0.746609	−	0.665263i	$-0.231681\pi$
$431$	31.0000	1.49322	0.746609	+	0.665263i	$0.231681\pi$
$432$	−5.00000	−0.240563
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	−7.00000	−0.336011
$435$	0	0
$436$	4.00000	0.191565
$437$	0	0
$438$	−4.00000	−0.191127
$439$	−33.0000	−1.57500	−0.787502	−	0.616312i	$-0.788626\pi$
$439$	−33.0000	−1.57500	−0.787502	+	0.616312i	$0.788626\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	−1.00000	−0.0475651
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	−21.0000	−0.993266
$448$	7.00000	0.330719
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	12.0000	0.564433
$453$	18.0000	0.845714
$454$	25.0000	1.17331
$455$	0	0
$456$	−18.0000	−0.842927
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	1.00000	0.0467269
$459$	5.00000	0.233380
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	4.00000	0.186097
$463$	12.0000	0.557687	0.278844	−	0.960337i	$-0.410049\pi$
$463$	12.0000	0.557687	0.278844	+	0.960337i	$0.410049\pi$
$464$	0	0
$465$	0	0
$466$	−8.00000	−0.370593
$467$	22.0000	1.01804	0.509019	−	0.860755i	$-0.330008\pi$
$467$	22.0000	1.01804	0.509019	+	0.860755i	$0.330008\pi$
$468$	−2.00000	−0.0924500
$469$	8.00000	0.369406
$470$	0	0
$471$	−21.0000	−0.967629
$472$	−24.0000	−1.10469
$473$	−16.0000	−0.735681
$474$	11.0000	0.505247
$475$	0	0
$476$	−1.00000	−0.0458349
$477$	−22.0000	−1.00731
$478$	−8.00000	−0.365911
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	18.0000	0.819878
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	−16.0000	−0.725775
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	−30.0000	−1.35804
$489$	11.0000	0.497437
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−2.00000	−0.0901670
$493$	0	0
$494$	6.00000	0.269953
$495$	0	0
$496$	7.00000	0.314309
$497$	7.00000	0.313993
$498$	8.00000	0.358489
$499$	13.0000	0.581960	0.290980	−	0.956729i	$-0.406019\pi$
$499$	13.0000	0.581960	0.290980	+	0.956729i	$0.406019\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	−8.00000	−0.357057
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	6.00000	0.267261
$505$	0	0
$506$	0	0
$507$	12.0000	0.532939
$508$	16.0000	0.709885
$509$	5.00000	0.221621	0.110811	−	0.993842i	$-0.464655\pi$
$509$	5.00000	0.221621	0.110811	+	0.993842i	$0.464655\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	−11.0000	−0.486136
$513$	−30.0000	−1.32453
$514$	19.0000	0.838054
$515$	0	0
$516$	4.00000	0.176090
$517$	24.0000	1.05552
$518$	−4.00000	−0.175750
$519$	−6.00000	−0.263371
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	15.0000	0.655278
$525$	0	0
$526$	16.0000	0.697633
$527$	−7.00000	−0.304925
$528$	−4.00000	−0.174078
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−16.0000	−0.694341
$532$	6.00000	0.260133
$533$	2.00000	0.0866296
$534$	6.00000	0.259645
$535$	0	0
$536$	−24.0000	−1.03664
$537$	−6.00000	−0.258919
$538$	10.0000	0.431131
$539$	24.0000	1.03375
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	2.00000	0.0859074
$543$	22.0000	0.944110
$544$	5.00000	0.214373
$545$	0	0
$546$	1.00000	0.0427960
$547$	5.00000	0.213785	0.106892	−	0.994271i	$-0.465910\pi$
$547$	5.00000	0.213785	0.106892	+	0.994271i	$0.465910\pi$
$548$	11.0000	0.469897
$549$	−20.0000	−0.853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−11.0000	−0.467768
$554$	−30.0000	−1.27458
$555$	0	0
$556$	13.0000	0.551323
$557$	−37.0000	−1.56774	−0.783870	−	0.620925i	$-0.786757\pi$
$557$	−37.0000	−1.56774	−0.783870	+	0.620925i	$0.786757\pi$
$558$	14.0000	0.592667
$559$	−4.00000	−0.169182
$560$	0	0
$561$	4.00000	0.168880
$562$	9.00000	0.379642
$563$	−26.0000	−1.09577	−0.547885	−	0.836554i	$-0.684567\pi$
$563$	−26.0000	−1.09577	−0.547885	+	0.836554i	$0.684567\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	−24.0000	−1.00880
$567$	1.00000	0.0419961
$568$	−21.0000	−0.881140
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	0	0
$571$	−3.00000	−0.125546	−0.0627730	−	0.998028i	$-0.519994\pi$
$571$	−3.00000	−0.125546	−0.0627730	+	0.998028i	$0.519994\pi$
$572$	−4.00000	−0.167248
$573$	10.0000	0.417756
$574$	−2.00000	−0.0834784
$575$	0	0
$576$	−14.0000	−0.583333
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	1.00000	0.0415945
$579$	−24.0000	−0.997406
$580$	0	0
$581$	−8.00000	−0.331896
$582$	16.0000	0.663221
$583$	−44.0000	−1.82229
$584$	−12.0000	−0.496564
$585$	0	0
$586$	6.00000	0.247858
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	−6.00000	−0.247436
$589$	42.0000	1.73058
$590$	0	0
$591$	−14.0000	−0.575883
$592$	4.00000	0.164399
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	−20.0000	−0.820610
$595$	0	0
$596$	−21.0000	−0.860194
$597$	0	0
$598$	0	0
$599$	−44.0000	−1.79779	−0.898896	−	0.438163i	$-0.855629\pi$
$599$	−44.0000	−1.79779	−0.898896	+	0.438163i	$0.855629\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	4.00000	0.163028
$603$	−16.0000	−0.651570
$604$	18.0000	0.732410
$605$	0	0
$606$	−15.0000	−0.609333
$607$	−27.0000	−1.09590	−0.547948	−	0.836512i	$-0.684591\pi$
$607$	−27.0000	−1.09590	−0.547948	+	0.836512i	$0.684591\pi$
$608$	−30.0000	−1.21666
$609$	0	0
$610$	0	0
$611$	6.00000	0.242734
$612$	2.00000	0.0808452
$613$	−47.0000	−1.89831	−0.949156	−	0.314806i	$-0.898061\pi$
$613$	−47.0000	−1.89831	−0.949156	+	0.314806i	$0.898061\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	12.0000	0.483494
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	−6.00000	−0.241355
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	−27.0000	−1.08260
$623$	−6.00000	−0.240385
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	−10.0000	−0.399680
$627$	−24.0000	−0.958468
$628$	−21.0000	−0.837991
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	33.0000	1.31267
$633$	25.0000	0.993661
$634$	−14.0000	−0.556011
$635$	0	0
$636$	11.0000	0.436178
$637$	6.00000	0.237729
$638$	0	0
$639$	−14.0000	−0.553831
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	19.0000	0.749870
$643$	1.00000	0.0394362	0.0197181	−	0.999806i	$-0.493723\pi$
$643$	1.00000	0.0394362	0.0197181	+	0.999806i	$0.493723\pi$
$644$	0	0
$645$	0	0
$646$	−6.00000	−0.236067
$647$	4.00000	0.157256	0.0786281	−	0.996904i	$-0.474946\pi$
$647$	4.00000	0.157256	0.0786281	+	0.996904i	$0.474946\pi$
$648$	−3.00000	−0.117851
$649$	−32.0000	−1.25611
$650$	0	0
$651$	7.00000	0.274352
$652$	11.0000	0.430793
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	4.00000	0.156412
$655$	0	0
$656$	2.00000	0.0780869
$657$	−8.00000	−0.312110
$658$	−6.00000	−0.233904
$659$	−2.00000	−0.0779089	−0.0389545	−	0.999241i	$-0.512403\pi$
$659$	−2.00000	−0.0779089	−0.0389545	+	0.999241i	$0.512403\pi$
$660$	0	0
$661$	−5.00000	−0.194477	−0.0972387	−	0.995261i	$-0.531001\pi$
$661$	−5.00000	−0.194477	−0.0972387	+	0.995261i	$0.531001\pi$
$662$	−14.0000	−0.544125
$663$	1.00000	0.0388368
$664$	24.0000	0.931381
$665$	0	0
$666$	8.00000	0.309994
$667$	0	0
$668$	−8.00000	−0.309529
$669$	2.00000	0.0773245
$670$	0	0
$671$	−40.0000	−1.54418
$672$	−5.00000	−0.192879
$673$	−38.0000	−1.46479	−0.732396	−	0.680879i	$-0.761598\pi$
$673$	−38.0000	−1.46479	−0.732396	+	0.680879i	$0.761598\pi$
$674$	−24.0000	−0.924445
$675$	0	0
$676$	12.0000	0.461538
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	12.0000	0.460857
$679$	−16.0000	−0.614024
$680$	0	0
$681$	−25.0000	−0.958002
$682$	28.0000	1.07218
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	−12.0000	−0.458831
$685$	0	0
$686$	−13.0000	−0.496342
$687$	−1.00000	−0.0381524
$688$	−4.00000	−0.152499
$689$	−11.0000	−0.419067
$690$	0	0
$691$	35.0000	1.33146	0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000	1.33146	0.665731	+	0.746191i	$0.268120\pi$
$692$	−6.00000	−0.228086
$693$	8.00000	0.303895
$694$	7.00000	0.265716
$695$	0	0
$696$	0	0
$697$	−2.00000	−0.0757554
$698$	19.0000	0.719161
$699$	8.00000	0.302588
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	−5.00000	−0.188713
$703$	24.0000	0.905177
$704$	−28.0000	−1.05529
$705$	0	0
$706$	19.0000	0.715074
$707$	15.0000	0.564133
$708$	8.00000	0.300658
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	22.0000	0.825064
$712$	18.0000	0.674579
$713$	0	0
$714$	−1.00000	−0.0374241
$715$	0	0
$716$	−6.00000	−0.224231
$717$	8.00000	0.298765
$718$	−10.0000	−0.373197
$719$	17.0000	0.633993	0.316997	−	0.948427i	$-0.397326\pi$
$719$	17.0000	0.633993	0.316997	+	0.948427i	$0.397326\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	17.0000	0.632674
$723$	−18.0000	−0.669427
$724$	22.0000	0.817624
$725$	0	0
$726$	−5.00000	−0.185567
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	3.00000	0.111187
$729$	13.0000	0.481481
$730$	0	0
$731$	4.00000	0.147945
$732$	10.0000	0.369611
$733$	−5.00000	−0.184679	−0.0923396	−	0.995728i	$-0.529435\pi$
$733$	−5.00000	−0.184679	−0.0923396	+	0.995728i	$0.529435\pi$
$734$	23.0000	0.848945
$735$	0	0
$736$	0	0
$737$	−32.0000	−1.17874
$738$	4.00000	0.147242
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	11.0000	0.403823
$743$	21.0000	0.770415	0.385208	−	0.922830i	$-0.374130\pi$
$743$	21.0000	0.770415	0.385208	+	0.922830i	$0.374130\pi$
$744$	−21.0000	−0.769897
$745$	0	0
$746$	19.0000	0.695639
$747$	16.0000	0.585409
$748$	4.00000	0.146254
$749$	−19.0000	−0.694245
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	6.00000	0.218797
$753$	8.00000	0.291536
$754$	0	0
$755$	0	0
$756$	−5.00000	−0.181848
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	25.0000	0.908041
$759$	0	0
$760$	0	0
$761$	−39.0000	−1.41375	−0.706874	−	0.707339i	$-0.749895\pi$
$761$	−39.0000	−1.41375	−0.706874	+	0.707339i	$0.749895\pi$
$762$	16.0000	0.579619
$763$	−4.00000	−0.144810
$764$	10.0000	0.361787
$765$	0	0
$766$	6.00000	0.216789
$767$	−8.00000	−0.288863
$768$	17.0000	0.613435
$769$	33.0000	1.19001	0.595005	−	0.803722i	$-0.297150\pi$
$769$	33.0000	1.19001	0.595005	+	0.803722i	$0.297150\pi$
$770$	0	0
$771$	−19.0000	−0.684268
$772$	−24.0000	−0.863779
$773$	45.0000	1.61854	0.809269	−	0.587439i	$-0.199864\pi$
$773$	45.0000	1.61854	0.809269	+	0.587439i	$0.199864\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	48.0000	1.72310
$777$	4.00000	0.143499
$778$	−34.0000	−1.21896
$779$	12.0000	0.429945
$780$	0	0
$781$	−28.0000	−1.00192
$782$	0	0
$783$	0	0
$784$	6.00000	0.214286
$785$	0	0
$786$	15.0000	0.535032
$787$	15.0000	0.534692	0.267346	−	0.963601i	$-0.413853\pi$
$787$	15.0000	0.534692	0.267346	+	0.963601i	$0.413853\pi$
$788$	−14.0000	−0.498729
$789$	−16.0000	−0.569615
$790$	0	0
$791$	−12.0000	−0.426671
$792$	−24.0000	−0.852803
$793$	−10.0000	−0.355110
$794$	4.00000	0.141955
$795$	0	0
$796$	0	0
$797$	−17.0000	−0.602171	−0.301085	−	0.953597i	$-0.597349\pi$
$797$	−17.0000	−0.602171	−0.301085	+	0.953597i	$0.597349\pi$
$798$	6.00000	0.212398
$799$	−6.00000	−0.212265
$800$	0	0
$801$	12.0000	0.423999
$802$	6.00000	0.211867
$803$	−16.0000	−0.564628
$804$	8.00000	0.282138
$805$	0	0
$806$	7.00000	0.246564
$807$	−10.0000	−0.352017
$808$	−45.0000	−1.58309
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	0	0
$811$	7.00000	0.245803	0.122902	−	0.992419i	$-0.460780\pi$
$811$	7.00000	0.245803	0.122902	+	0.992419i	$0.460780\pi$
$812$	0	0
$813$	−2.00000	−0.0701431
$814$	16.0000	0.560800
$815$	0	0
$816$	1.00000	0.0350070
$817$	−24.0000	−0.839654
$818$	19.0000	0.664319
$819$	2.00000	0.0698857
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	11.0000	0.383669
$823$	53.0000	1.84746	0.923732	−	0.383040i	$-0.125123\pi$
$823$	53.0000	1.84746	0.923732	+	0.383040i	$0.125123\pi$
$824$	−18.0000	−0.627060
$825$	0	0
$826$	8.00000	0.278356
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	0	0
$831$	30.0000	1.04069
$832$	−7.00000	−0.242681
$833$	−6.00000	−0.207888
$834$	13.0000	0.450153
$835$	0	0
$836$	−24.0000	−0.830057
$837$	−35.0000	−1.20978
$838$	12.0000	0.414533
$839$	23.0000	0.794048	0.397024	−	0.917808i	$-0.370043\pi$
$839$	23.0000	0.794048	0.397024	+	0.917808i	$0.370043\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−25.0000	−0.861557
$843$	−9.00000	−0.309976
$844$	25.0000	0.860535
$845$	0	0
$846$	12.0000	0.412568
$847$	5.00000	0.171802
$848$	−11.0000	−0.377742
$849$	24.0000	0.823678
$850$	0	0
$851$	0	0
$852$	7.00000	0.239816
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	57.0000	1.94822
$857$	−20.0000	−0.683187	−0.341593	−	0.939848i	$-0.610967\pi$
$857$	−20.0000	−0.683187	−0.341593	+	0.939848i	$0.610967\pi$
$858$	−4.00000	−0.136558
$859$	18.0000	0.614152	0.307076	−	0.951685i	$-0.400649\pi$
$859$	18.0000	0.614152	0.307076	+	0.951685i	$0.400649\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	31.0000	1.05586
$863$	56.0000	1.90626	0.953131	−	0.302558i	$-0.0978405\pi$
$863$	56.0000	1.90626	0.953131	+	0.302558i	$0.0978405\pi$
$864$	25.0000	0.850517
$865$	0	0
$866$	−34.0000	−1.15537
$867$	−1.00000	−0.0339618
$868$	7.00000	0.237595
$869$	44.0000	1.49260
$870$	0	0
$871$	−8.00000	−0.271070
$872$	12.0000	0.406371
$873$	32.0000	1.08304
$874$	0	0
$875$	0	0
$876$	4.00000	0.135147
$877$	16.0000	0.540282	0.270141	−	0.962821i	$-0.412930\pi$
$877$	16.0000	0.540282	0.270141	+	0.962821i	$0.412930\pi$
$878$	−33.0000	−1.11370
$879$	−6.00000	−0.202375
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	12.0000	0.404061
$883$	−2.00000	−0.0673054	−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000	−0.0673054	−0.0336527	+	0.999434i	$0.510714\pi$
$884$	1.00000	0.0336336
$885$	0	0
$886$	6.00000	0.201574
$887$	−43.0000	−1.44380	−0.721899	−	0.691998i	$-0.756731\pi$
$887$	−43.0000	−1.44380	−0.721899	+	0.691998i	$0.756731\pi$
$888$	−12.0000	−0.402694
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−4.00000	−0.134005
$892$	2.00000	0.0669650
$893$	36.0000	1.20469
$894$	−21.0000	−0.702345
$895$	0	0
$896$	−3.00000	−0.100223
$897$	0	0
$898$	36.0000	1.20134
$899$	0	0
$900$	0	0
$901$	11.0000	0.366463
$902$	8.00000	0.266371
$903$	−4.00000	−0.133112
$904$	36.0000	1.19734
$905$	0	0
$906$	18.0000	0.598010
$907$	−36.0000	−1.19536	−0.597680	−	0.801735i	$-0.703911\pi$
$907$	−36.0000	−1.19536	−0.597680	+	0.801735i	$0.703911\pi$
$908$	−25.0000	−0.829654
$909$	−30.0000	−0.995037
$910$	0	0
$911$	1.00000	0.0331315	0.0165657	−	0.999863i	$-0.494727\pi$
$911$	1.00000	0.0331315	0.0165657	+	0.999863i	$0.494727\pi$
$912$	−6.00000	−0.198680
$913$	32.0000	1.05905
$914$	3.00000	0.0992312
$915$	0	0
$916$	−1.00000	−0.0330409
$917$	−15.0000	−0.495344
$918$	5.00000	0.165025
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	−14.0000	−0.461065
$923$	−7.00000	−0.230408
$924$	−4.00000	−0.131590
$925$	0	0
$926$	12.0000	0.394344
$927$	−12.0000	−0.394132
$928$	0	0
$929$	−56.0000	−1.83730	−0.918650	−	0.395072i	$-0.870720\pi$
$929$	−56.0000	−1.83730	−0.918650	+	0.395072i	$0.870720\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	8.00000	0.262049
$933$	27.0000	0.883940
$934$	22.0000	0.719862
$935$	0	0
$936$	−6.00000	−0.196116
$937$	25.0000	0.816714	0.408357	−	0.912822i	$-0.366102\pi$
$937$	25.0000	0.816714	0.408357	+	0.912822i	$0.366102\pi$
$938$	8.00000	0.261209
$939$	10.0000	0.326338
$940$	0	0
$941$	34.0000	1.10837	0.554184	−	0.832394i	$-0.313030\pi$
$941$	34.0000	1.10837	0.554184	+	0.832394i	$0.313030\pi$
$942$	−21.0000	−0.684217
$943$	0	0
$944$	−8.00000	−0.260378
$945$	0	0
$946$	−16.0000	−0.520205
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	−11.0000	−0.357263
$949$	−4.00000	−0.129845
$950$	0	0
$951$	14.0000	0.453981
$952$	−3.00000	−0.0972306
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	−22.0000	−0.712276
$955$	0	0
$956$	8.00000	0.258738
$957$	0	0
$958$	−12.0000	−0.387702
$959$	−11.0000	−0.355209
$960$	0	0
$961$	18.0000	0.580645
$962$	4.00000	0.128965
$963$	38.0000	1.22453
$964$	−18.0000	−0.579741
$965$	0	0
$966$	0	0
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	−15.0000	−0.482118
$969$	6.00000	0.192748
$970$	0	0
$971$	−32.0000	−1.02693	−0.513464	−	0.858111i	$-0.671638\pi$
$971$	−32.0000	−1.02693	−0.513464	+	0.858111i	$0.671638\pi$
$972$	16.0000	0.513200
$973$	−13.0000	−0.416761
$974$	4.00000	0.128168
$975$	0	0
$976$	−10.0000	−0.320092
$977$	13.0000	0.415907	0.207953	−	0.978139i	$-0.433320\pi$
$977$	13.0000	0.415907	0.207953	+	0.978139i	$0.433320\pi$
$978$	11.0000	0.351741
$979$	24.0000	0.767043
$980$	0	0
$981$	8.00000	0.255420
$982$	20.0000	0.638226
$983$	−21.0000	−0.669796	−0.334898	−	0.942254i	$-0.608702\pi$
$983$	−21.0000	−0.669796	−0.334898	+	0.942254i	$0.608702\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	0	0
$987$	6.00000	0.190982
$988$	−6.00000	−0.190885
$989$	0	0
$990$	0	0
$991$	5.00000	0.158830	0.0794151	−	0.996842i	$-0.474695\pi$
$991$	5.00000	0.158830	0.0794151	+	0.996842i	$0.474695\pi$
$992$	−35.0000	−1.11125
$993$	14.0000	0.444277
$994$	7.00000	0.222027
$995$	0	0
$996$	−8.00000	−0.253490
$997$	4.00000	0.126681	0.0633406	−	0.997992i	$-0.479825\pi$
$997$	4.00000	0.126681	0.0633406	+	0.997992i	$0.479825\pi$
$998$	13.0000	0.411508
$999$	−20.0000	−0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.a.c.1.1	yes	1
3.2	odd	2		3825.2.a.f.1.1		1
4.3	odd	2		6800.2.a.q.1.1		1
5.2	odd	4		425.2.b.a.324.2		2
5.3	odd	4		425.2.b.a.324.1		2
5.4	even	2		425.2.a.b.1.1	✓	1
15.14	odd	2		3825.2.a.k.1.1		1
17.16	even	2		7225.2.a.h.1.1		1
20.19	odd	2		6800.2.a.i.1.1		1
85.84	even	2		7225.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.b.1.1	✓	1	5.4	even	2
425.2.a.c.1.1	yes	1	1.1	even	1	trivial
425.2.b.a.324.1		2	5.3	odd	4
425.2.b.a.324.2		2	5.2	odd	4
3825.2.a.f.1.1		1	3.2	odd	2
3825.2.a.k.1.1		1	15.14	odd	2
6800.2.a.i.1.1		1	20.19	odd	2
6800.2.a.q.1.1		1	4.3	odd	2
7225.2.a.b.1.1		1	85.84	even	2
7225.2.a.h.1.1		1	17.16	even	2

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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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