LMFDB - Embedded newform 425.2.a.a.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:	no (minimal twist has level 85)
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)

$f(q)$

$=$

\(q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{6} +2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{21} -2.00000 q^{22} -6.00000 q^{23} -6.00000 q^{24} +2.00000 q^{26} +4.00000 q^{27} -2.00000 q^{28} -6.00000 q^{29} -10.0000 q^{31} -5.00000 q^{32} -4.00000 q^{33} +1.00000 q^{34} -1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{39} +10.0000 q^{41} +4.00000 q^{42} -4.00000 q^{43} -2.00000 q^{44} +6.00000 q^{46} -12.0000 q^{47} +2.00000 q^{48} -3.00000 q^{49} +2.00000 q^{51} +2.00000 q^{52} +10.0000 q^{53} -4.00000 q^{54} +6.00000 q^{56} +6.00000 q^{58} +8.00000 q^{59} -14.0000 q^{61} +10.0000 q^{62} +2.00000 q^{63} +7.00000 q^{64} +4.00000 q^{66} -8.00000 q^{67} +1.00000 q^{68} +12.0000 q^{69} -2.00000 q^{71} +3.00000 q^{72} +14.0000 q^{73} +2.00000 q^{74} +4.00000 q^{77} -4.00000 q^{78} -14.0000 q^{79} -11.0000 q^{81} -10.0000 q^{82} -4.00000 q^{83} +4.00000 q^{84} +4.00000 q^{86} +12.0000 q^{87} +6.00000 q^{88} +6.00000 q^{89} -4.00000 q^{91} +6.00000 q^{92} +20.0000 q^{93} +12.0000 q^{94} +10.0000 q^{96} -2.00000 q^{97} +3.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	2.00000	0.577350
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	−2.00000	−0.426401
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−6.00000	−1.22474
$25$	0	0
$26$	2.00000	0.392232
$27$	4.00000	0.769800
$28$	−2.00000	−0.377964
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−5.00000	−0.883883
$33$	−4.00000	−0.696311
$34$	1.00000	0.171499
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	4.00000	0.617213
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	6.00000	0.884652
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	2.00000	0.288675
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	2.00000	0.277350
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	6.00000	0.801784
$57$	0	0
$58$	6.00000	0.787839
$59$	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
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	10.0000	1.27000
$63$	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	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	1.00000	0.121268
$69$	12.0000	1.44463
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	3.00000	0.353553
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	0	0
$77$	4.00000	0.455842
$78$	−4.00000	−0.452911
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	−10.0000	−1.10432
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	4.00000	0.431331
$87$	12.0000	1.28654
$88$	6.00000	0.639602
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	6.00000	0.625543
$93$	20.0000	2.07390
$94$	12.0000	1.23771
$95$	0	0
$96$	10.0000	1.02062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	3.00000	0.303046
$99$	2.00000	0.201008
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	−2.00000	−0.198030
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−10.0000	−0.971286
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	−4.00000	−0.384900
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	−2.00000	−0.188982
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	−2.00000	−0.184900
$118$	−8.00000	−0.736460
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−7.00000	−0.636364
$122$	14.0000	1.26750
$123$	−20.0000	−1.80334
$124$	10.0000	0.898027
$125$	0	0
$126$	−2.00000	−0.178174
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	3.00000	0.265165
$129$	8.00000	0.704361
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	8.00000	0.691095
$135$	0	0
$136$	−3.00000	−0.257248
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−12.0000	−1.02151
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	24.0000	2.02116
$142$	2.00000	0.167836
$143$	−4.00000	−0.334497
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	6.00000	0.494872
$148$	2.00000	0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	−4.00000	−0.322329
$155$	0	0
$156$	−4.00000	−0.320256
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	14.0000	1.11378
$159$	−20.0000	−1.58610
$160$	0	0
$161$	−12.0000	−0.945732
$162$	11.0000	0.864242
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	4.00000	0.310460
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	−12.0000	−0.925820
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	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	$0.426753\pi$
$174$	−12.0000	−0.909718
$175$	0	0
$176$	−2.00000	−0.150756
$177$	−16.0000	−1.20263
$178$	−6.00000	−0.449719
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	4.00000	0.296500
$183$	28.0000	2.06982
$184$	−18.0000	−1.32698
$185$	0	0
$186$	−20.0000	−1.46647
$187$	−2.00000	−0.146254
$188$	12.0000	0.875190
$189$	8.00000	0.581914
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	−14.0000	−1.01036
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	3.00000	0.214286
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−2.00000	−0.142134
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	6.00000	0.422159
$203$	−12.0000	−0.842235
$204$	−2.00000	−0.140028
$205$	0	0
$206$	12.0000	0.836080
$207$	−6.00000	−0.417029
$208$	2.00000	0.138675
$209$	0	0
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	−10.0000	−0.686803
$213$	4.00000	0.274075
$214$	2.00000	0.136717
$215$	0	0
$216$	12.0000	0.816497
$217$	−20.0000	−1.35769
$218$	−2.00000	−0.135457
$219$	−28.0000	−1.89206
$220$	0	0
$221$	2.00000	0.134535
$222$	−4.00000	−0.268462
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	−10.0000	−0.668153
$225$	0	0
$226$	−6.00000	−0.399114
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	−18.0000	−1.18176
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−8.00000	−0.520756
$237$	28.0000	1.81880
$238$	2.00000	0.129641
$239$	−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
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	7.00000	0.449977
$243$	10.0000	0.641500
$244$	14.0000	0.896258
$245$	0	0
$246$	20.0000	1.27515
$247$	0	0
$248$	−30.0000	−1.90500
$249$	8.00000	0.506979
$250$	0	0
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	−2.00000	−0.125988
$253$	−12.0000	−0.754434
$254$	8.00000	0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−8.00000	−0.498058
$259$	−4.00000	−0.248548
$260$	0	0
$261$	−6.00000	−0.371391
$262$	−6.00000	−0.370681
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	0	0
$267$	−12.0000	−0.734388
$268$	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
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	1.00000	0.0606339
$273$	8.00000	0.484182
$274$	−2.00000	−0.120824
$275$	0	0
$276$	−12.0000	−0.722315
$277$	30.0000	1.80253	0.901263	−	0.433273i	$-0.142641\pi$
$277$	30.0000	1.80253	0.901263	+	0.433273i	$0.142641\pi$
$278$	−14.0000	−0.839664
$279$	−10.0000	−0.598684
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−24.0000	−1.42918
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	4.00000	0.236525
$287$	20.0000	1.18056
$288$	−5.00000	−0.294628
$289$	1.00000	0.0588235
$290$	0	0
$291$	4.00000	0.234484
$292$	−14.0000	−0.819288
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	−6.00000	−0.348743
$297$	8.00000	0.464207
$298$	−6.00000	−0.347571
$299$	12.0000	0.693978
$300$	0	0
$301$	−8.00000	−0.461112
$302$	12.0000	0.690522
$303$	12.0000	0.689382
$304$	0	0
$305$	0	0
$306$	1.00000	0.0571662
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	−4.00000	−0.227921
$309$	24.0000	1.36531
$310$	0	0
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	12.0000	0.679366
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	14.0000	0.787562
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	20.0000	1.12154
$319$	−12.0000	−0.671871
$320$	0	0
$321$	4.00000	0.223258
$322$	12.0000	0.668734
$323$	0	0
$324$	11.0000	0.611111
$325$	0	0
$326$	−2.00000	−0.110770
$327$	−4.00000	−0.221201
$328$	30.0000	1.65647
$329$	−24.0000	−1.32316
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	4.00000	0.219529
$333$	−2.00000	−0.109599
$334$	2.00000	0.109435
$335$	0	0
$336$	4.00000	0.218218
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	9.00000	0.489535
$339$	−12.0000	−0.651751
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	−20.0000	−1.07990
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−6.00000	−0.322562
$347$	26.0000	1.39575	0.697877	−	0.716218i	$-0.254128\pi$
$347$	26.0000	1.39575	0.697877	+	0.716218i	$0.254128\pi$
$348$	−12.0000	−0.643268
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	−10.0000	−0.533002
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	16.0000	0.850390
$355$	0	0
$356$	−6.00000	−0.317999
$357$	4.00000	0.211702
$358$	24.0000	1.26844
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−26.0000	−1.36653
$363$	14.0000	0.734809
$364$	4.00000	0.209657
$365$	0	0
$366$	−28.0000	−1.46358
$367$	34.0000	1.77479	0.887393	−	0.461014i	$-0.152514\pi$
$367$	34.0000	1.77479	0.887393	+	0.461014i	$0.152514\pi$
$368$	6.00000	0.312772
$369$	10.0000	0.520579
$370$	0	0
$371$	20.0000	1.03835
$372$	−20.0000	−1.03695
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	−36.0000	−1.85656
$377$	12.0000	0.618031
$378$	−8.00000	−0.411476
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	−8.00000	−0.409316
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−6.00000	−0.306186
$385$	0	0
$386$	18.0000	0.916176
$387$	−4.00000	−0.203331
$388$	2.00000	0.101535
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	−9.00000	−0.454569
$393$	−12.0000	−0.605320
$394$	2.00000	0.100759
$395$	0	0
$396$	−2.00000	−0.100504
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−6.00000	−0.300753
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	−16.0000	−0.798007
$403$	20.0000	0.996271
$404$	6.00000	0.298511
$405$	0	0
$406$	12.0000	0.595550
$407$	−4.00000	−0.198273
$408$	6.00000	0.297044
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	12.0000	0.591198
$413$	16.0000	0.787309
$414$	6.00000	0.294884
$415$	0	0
$416$	10.0000	0.490290
$417$	−28.0000	−1.37117
$418$	0	0
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	−2.00000	−0.0973585
$423$	−12.0000	−0.583460
$424$	30.0000	1.45693
$425$	0	0
$426$	−4.00000	−0.193801
$427$	−28.0000	−1.35501
$428$	2.00000	0.0966736
$429$	8.00000	0.386244
$430$	0	0
$431$	−38.0000	−1.83040	−0.915198	−	0.403005i	$-0.867966\pi$
$431$	−38.0000	−1.83040	−0.915198	+	0.403005i	$0.867966\pi$
$432$	−4.00000	−0.192450
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	20.0000	0.960031
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	28.0000	1.33789
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−2.00000	−0.0951303
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	16.0000	0.757622
$447$	−12.0000	−0.567581
$448$	14.0000	0.661438
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	−6.00000	−0.282216
$453$	24.0000	1.12762
$454$	−2.00000	−0.0938647
$455$	0	0
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	−10.0000	−0.467269
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−26.0000	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$462$	8.00000	0.372194
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−14.0000	−0.648537
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	2.00000	0.0924500
$469$	−16.0000	−0.738811
$470$	0	0
$471$	12.0000	0.552931
$472$	24.0000	1.10469
$473$	−8.00000	−0.367840
$474$	−28.0000	−1.28608
$475$	0	0
$476$	2.00000	0.0916698
$477$	10.0000	0.457869
$478$	8.00000	0.365911
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	6.00000	0.273293
$483$	24.0000	1.09204
$484$	7.00000	0.318182
$485$	0	0
$486$	−10.0000	−0.453609
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	−42.0000	−1.90125
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	20.0000	0.901670
$493$	6.00000	0.270226
$494$	0	0
$495$	0	0
$496$	10.0000	0.449013
$497$	−4.00000	−0.179425
$498$	−8.00000	−0.358489
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	4.00000	0.178707
$502$	−28.0000	−1.24970
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	6.00000	0.267261
$505$	0	0
$506$	12.0000	0.533465
$507$	18.0000	0.799408
$508$	8.00000	0.354943
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	11.0000	0.486136
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	−8.00000	−0.352180
$517$	−24.0000	−1.05552
$518$	4.00000	0.175750
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	6.00000	0.262613
$523$	−32.0000	−1.39926	−0.699631	−	0.714504i	$-0.746652\pi$
$523$	−32.0000	−1.39926	−0.699631	+	0.714504i	$0.746652\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	−8.00000	−0.348817
$527$	10.0000	0.435607
$528$	4.00000	0.174078
$529$	13.0000	0.565217
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	−20.0000	−0.866296
$534$	12.0000	0.519291
$535$	0	0
$536$	−24.0000	−1.03664
$537$	48.0000	2.07135
$538$	−10.0000	−0.431131
$539$	−6.00000	−0.258438
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	16.0000	0.687259
$543$	−52.0000	−2.23153
$544$	5.00000	0.214373
$545$	0	0
$546$	−8.00000	−0.342368
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	−2.00000	−0.0854358
$549$	−14.0000	−0.597505
$550$	0	0
$551$	0	0
$552$	36.0000	1.53226
$553$	−28.0000	−1.19068
$554$	−30.0000	−1.27458
$555$	0	0
$556$	−14.0000	−0.593732
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	10.0000	0.423334
$559$	8.00000	0.338364
$560$	0	0
$561$	4.00000	0.168880
$562$	6.00000	0.253095
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	−24.0000	−1.01058
$565$	0	0
$566$	6.00000	0.252199
$567$	−22.0000	−0.923913
$568$	−6.00000	−0.251754
$569$	−38.0000	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$569$	−38.0000	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$570$	0	0
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	4.00000	0.167248
$573$	−16.0000	−0.668410
$574$	−20.0000	−0.834784
$575$	0	0
$576$	7.00000	0.291667
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	−1.00000	−0.0415945
$579$	36.0000	1.49611
$580$	0	0
$581$	−8.00000	−0.331896
$582$	−4.00000	−0.165805
$583$	20.0000	0.828315
$584$	42.0000	1.73797
$585$	0	0
$586$	30.0000	1.23929
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	−6.00000	−0.247436
$589$	0	0
$590$	0	0
$591$	4.00000	0.164538
$592$	2.00000	0.0821995
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	−8.00000	−0.328244
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−12.0000	−0.491127
$598$	−12.0000	−0.490716
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	8.00000	0.326056
$603$	−8.00000	−0.325785
$604$	12.0000	0.488273
$605$	0	0
$606$	−12.0000	−0.487467
$607$	−18.0000	−0.730597	−0.365299	−	0.930890i	$-0.619033\pi$
$607$	−18.0000	−0.730597	−0.365299	+	0.930890i	$0.619033\pi$
$608$	0	0
$609$	24.0000	0.972529
$610$	0	0
$611$	24.0000	0.970936
$612$	1.00000	0.0404226
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	12.0000	0.483494
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	−24.0000	−0.965422
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	−6.00000	−0.240578
$623$	12.0000	0.480770
$624$	−4.00000	−0.160128
$625$	0	0
$626$	2.00000	0.0799361
$627$	0	0
$628$	6.00000	0.239426
$629$	2.00000	0.0797452
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	−42.0000	−1.67067
$633$	−4.00000	−0.158986
$634$	10.0000	0.397151
$635$	0	0
$636$	20.0000	0.793052
$637$	6.00000	0.237729
$638$	12.0000	0.475085
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−4.00000	−0.157867
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	0	0
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	−33.0000	−1.29636
$649$	16.0000	0.628055
$650$	0	0
$651$	40.0000	1.56772
$652$	−2.00000	−0.0783260
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	4.00000	0.156412
$655$	0	0
$656$	−10.0000	−0.390434
$657$	14.0000	0.546192
$658$	24.0000	0.935617
$659$	−44.0000	−1.71400	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000	−1.71400	−0.856998	+	0.515319i	$0.827673\pi$
$660$	0	0
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	−16.0000	−0.621858
$663$	−4.00000	−0.155347
$664$	−12.0000	−0.465690
$665$	0	0
$666$	2.00000	0.0774984
$667$	36.0000	1.39393
$668$	2.00000	0.0773823
$669$	32.0000	1.23719
$670$	0	0
$671$	−28.0000	−1.08093
$672$	20.0000	0.771517
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	−6.00000	−0.231111
$675$	0	0
$676$	9.00000	0.346154
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	12.0000	0.460857
$679$	−4.00000	−0.153506
$680$	0	0
$681$	−4.00000	−0.153280
$682$	20.0000	0.765840
$683$	−18.0000	−0.688751	−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000	−0.688751	−0.344375	+	0.938832i	$0.611909\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	−20.0000	−0.763048
$688$	4.00000	0.152499
$689$	−20.0000	−0.761939
$690$	0	0
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	−6.00000	−0.228086
$693$	4.00000	0.151947
$694$	−26.0000	−0.986947
$695$	0	0
$696$	36.0000	1.36458
$697$	−10.0000	−0.378777
$698$	−22.0000	−0.832712
$699$	−28.0000	−1.05906
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	8.00000	0.301941
$703$	0	0
$704$	14.0000	0.527645
$705$	0	0
$706$	−14.0000	−0.526897
$707$	−12.0000	−0.451306
$708$	16.0000	0.601317
$709$	−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
$711$	−14.0000	−0.525041
$712$	18.0000	0.674579
$713$	60.0000	2.24702
$714$	−4.00000	−0.149696
$715$	0	0
$716$	24.0000	0.896922
$717$	16.0000	0.597531
$718$	−20.0000	−0.746393
$719$	14.0000	0.522112	0.261056	−	0.965324i	$-0.415929\pi$
$719$	14.0000	0.522112	0.261056	+	0.965324i	$0.415929\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	19.0000	0.707107
$723$	12.0000	0.446285
$724$	−26.0000	−0.966282
$725$	0	0
$726$	−14.0000	−0.519589
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	−12.0000	−0.444750
$729$	13.0000	0.481481
$730$	0	0
$731$	4.00000	0.147945
$732$	−28.0000	−1.03491
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−34.0000	−1.25496
$735$	0	0
$736$	30.0000	1.10581
$737$	−16.0000	−0.589368
$738$	−10.0000	−0.368105
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	0	0
$741$	0	0
$742$	−20.0000	−0.734223
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	60.0000	2.19971
$745$	0	0
$746$	10.0000	0.366126
$747$	−4.00000	−0.146352
$748$	2.00000	0.0731272
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	12.0000	0.437595
$753$	−56.0000	−2.04075
$754$	−12.0000	−0.437014
$755$	0	0
$756$	−8.00000	−0.290957
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	2.00000	0.0726433
$759$	24.0000	0.871145
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−16.0000	−0.579619
$763$	4.00000	0.144810
$764$	−8.00000	−0.289430
$765$	0	0
$766$	0	0
$767$	−16.0000	−0.577727
$768$	34.0000	1.22687
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	−4.00000	−0.144056
$772$	18.0000	0.647834
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−6.00000	−0.215387
$777$	8.00000	0.286998
$778$	−26.0000	−0.932145
$779$	0	0
$780$	0	0
$781$	−4.00000	−0.143131
$782$	−6.00000	−0.214560
$783$	−24.0000	−0.857690
$784$	3.00000	0.107143
$785$	0	0
$786$	12.0000	0.428026
$787$	6.00000	0.213877	0.106938	−	0.994266i	$-0.465895\pi$
$787$	6.00000	0.213877	0.106938	+	0.994266i	$0.465895\pi$
$788$	2.00000	0.0712470
$789$	−16.0000	−0.569615
$790$	0	0
$791$	12.0000	0.426671
$792$	6.00000	0.213201
$793$	28.0000	0.994309
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−6.00000	−0.212664
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	6.00000	0.212000
$802$	−18.0000	−0.635602
$803$	28.0000	0.988099
$804$	−16.0000	−0.564276
$805$	0	0
$806$	−20.0000	−0.704470
$807$	−20.0000	−0.704033
$808$	−18.0000	−0.633238
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	12.0000	0.421117
$813$	32.0000	1.12229
$814$	4.00000	0.140200
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	38.0000	1.32864
$819$	−4.00000	−0.139771
$820$	0	0
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	4.00000	0.139516
$823$	34.0000	1.18517	0.592583	−	0.805510i	$-0.298108\pi$
$823$	34.0000	1.18517	0.592583	+	0.805510i	$0.298108\pi$
$824$	−36.0000	−1.25412
$825$	0	0
$826$	−16.0000	−0.556711
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	6.00000	0.208514
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	−60.0000	−2.08138
$832$	−14.0000	−0.485363
$833$	3.00000	0.103944
$834$	28.0000	0.969561
$835$	0	0
$836$	0	0
$837$	−40.0000	−1.38260
$838$	−18.0000	−0.621800
$839$	26.0000	0.897620	0.448810	−	0.893627i	$-0.351848\pi$
$839$	26.0000	0.897620	0.448810	+	0.893627i	$0.351848\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−26.0000	−0.896019
$843$	12.0000	0.413302
$844$	−2.00000	−0.0688428
$845$	0	0
$846$	12.0000	0.412568
$847$	−14.0000	−0.481046
$848$	−10.0000	−0.343401
$849$	12.0000	0.411839
$850$	0	0
$851$	12.0000	0.411355
$852$	−4.00000	−0.137038
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	28.0000	0.958140
$855$	0	0
$856$	−6.00000	−0.205076
$857$	−34.0000	−1.16142	−0.580709	−	0.814111i	$-0.697225\pi$
$857$	−34.0000	−1.16142	−0.580709	+	0.814111i	$0.697225\pi$
$858$	−8.00000	−0.273115
$859$	−24.0000	−0.818869	−0.409435	−	0.912339i	$-0.634274\pi$
$859$	−24.0000	−0.818869	−0.409435	+	0.912339i	$0.634274\pi$
$860$	0	0
$861$	−40.0000	−1.36320
$862$	38.0000	1.29429
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	−20.0000	−0.680414
$865$	0	0
$866$	−34.0000	−1.15537
$867$	−2.00000	−0.0679236
$868$	20.0000	0.678844
$869$	−28.0000	−0.949835
$870$	0	0
$871$	16.0000	0.542139
$872$	6.00000	0.203186
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	0	0
$876$	28.0000	0.946032
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	−6.00000	−0.202490
$879$	60.0000	2.02375
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	3.00000	0.101015
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	24.0000	0.806296
$887$	22.0000	0.738688	0.369344	−	0.929293i	$-0.379582\pi$
$887$	22.0000	0.738688	0.369344	+	0.929293i	$0.379582\pi$
$888$	12.0000	0.402694
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−22.0000	−0.737028
$892$	16.0000	0.535720
$893$	0	0
$894$	12.0000	0.401340
$895$	0	0
$896$	6.00000	0.200446
$897$	−24.0000	−0.801337
$898$	−18.0000	−0.600668
$899$	60.0000	2.00111
$900$	0	0
$901$	−10.0000	−0.333148
$902$	−20.0000	−0.665927
$903$	16.0000	0.532447
$904$	18.0000	0.598671
$905$	0	0
$906$	−24.0000	−0.797347
$907$	−30.0000	−0.996134	−0.498067	−	0.867139i	$-0.665957\pi$
$907$	−30.0000	−0.996134	−0.498067	+	0.867139i	$0.665957\pi$
$908$	−2.00000	−0.0663723
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−38.0000	−1.25900	−0.629498	−	0.777002i	$-0.716739\pi$
$911$	−38.0000	−1.25900	−0.629498	+	0.777002i	$0.716739\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	6.00000	0.198462
$915$	0	0
$916$	−10.0000	−0.330409
$917$	12.0000	0.396275
$918$	4.00000	0.132020
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−32.0000	−1.05444
$922$	26.0000	0.856264
$923$	4.00000	0.131662
$924$	8.00000	0.263181
$925$	0	0
$926$	−36.0000	−1.18303
$927$	−12.0000	−0.394132
$928$	30.0000	0.984798
$929$	58.0000	1.90292	0.951459	−	0.307775i	$-0.0995844\pi$
$929$	58.0000	1.90292	0.951459	+	0.307775i	$0.0995844\pi$
$930$	0	0
$931$	0	0
$932$	−14.0000	−0.458585
$933$	−12.0000	−0.392862
$934$	28.0000	0.916188
$935$	0	0
$936$	−6.00000	−0.196116
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	16.0000	0.522419
$939$	4.00000	0.130535
$940$	0	0
$941$	34.0000	1.10837	0.554184	−	0.832394i	$-0.313030\pi$
$941$	34.0000	1.10837	0.554184	+	0.832394i	$0.313030\pi$
$942$	−12.0000	−0.390981
$943$	−60.0000	−1.95387
$944$	−8.00000	−0.260378
$945$	0	0
$946$	8.00000	0.260102
$947$	−26.0000	−0.844886	−0.422443	−	0.906389i	$-0.638827\pi$
$947$	−26.0000	−0.844886	−0.422443	+	0.906389i	$0.638827\pi$
$948$	−28.0000	−0.909398
$949$	−28.0000	−0.908918
$950$	0	0
$951$	20.0000	0.648544
$952$	−6.00000	−0.194461
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	−10.0000	−0.323762
$955$	0	0
$956$	8.00000	0.258738
$957$	24.0000	0.775810
$958$	18.0000	0.581554
$959$	4.00000	0.129167
$960$	0	0
$961$	69.0000	2.22581
$962$	−4.00000	−0.128965
$963$	−2.00000	−0.0644491
$964$	6.00000	0.193247
$965$	0	0
$966$	−24.0000	−0.772187
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	−21.0000	−0.674966
$969$	0	0
$970$	0	0
$971$	−8.00000	−0.256732	−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000	−0.256732	−0.128366	+	0.991727i	$0.540973\pi$
$972$	−10.0000	−0.320750
$973$	28.0000	0.897639
$974$	22.0000	0.704925
$975$	0	0
$976$	14.0000	0.448129
$977$	−34.0000	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$978$	4.00000	0.127906
$979$	12.0000	0.383522
$980$	0	0
$981$	2.00000	0.0638551
$982$	16.0000	0.510581
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	−60.0000	−1.91273
$985$	0	0
$986$	−6.00000	−0.191079
$987$	48.0000	1.52786
$988$	0	0
$989$	24.0000	0.763156
$990$	0	0
$991$	−58.0000	−1.84243	−0.921215	−	0.389053i	$-0.872802\pi$
$991$	−58.0000	−1.84243	−0.921215	+	0.389053i	$0.872802\pi$
$992$	50.0000	1.58750
$993$	−32.0000	−1.01549
$994$	4.00000	0.126872
$995$	0	0
$996$	−8.00000	−0.253490
$997$	62.0000	1.96356	0.981780	−	0.190022i	$-0.0608559\pi$
$997$	62.0000	1.96356	0.981780	+	0.190022i	$0.0608559\pi$
$998$	14.0000	0.443162
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.a.a.1.1		1
3.2	odd	2		3825.2.a.l.1.1		1
4.3	odd	2		6800.2.a.v.1.1		1
5.2	odd	4		425.2.b.c.324.1		2
5.3	odd	4		425.2.b.c.324.2		2
5.4	even	2		85.2.a.a.1.1	✓	1
15.14	odd	2		765.2.a.a.1.1		1
17.16	even	2		7225.2.a.d.1.1		1
20.19	odd	2		1360.2.a.b.1.1		1
35.34	odd	2		4165.2.a.l.1.1		1
40.19	odd	2		5440.2.a.x.1.1		1
40.29	even	2		5440.2.a.e.1.1		1
85.4	even	4		1445.2.d.a.866.1		2
85.64	even	4		1445.2.d.a.866.2		2
85.84	even	2		1445.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.a.1.1	✓	1	5.4	even	2
425.2.a.a.1.1		1	1.1	even	1	trivial
425.2.b.c.324.1		2	5.2	odd	4
425.2.b.c.324.2		2	5.3	odd	4
765.2.a.a.1.1		1	15.14	odd	2
1360.2.a.b.1.1		1	20.19	odd	2
1445.2.a.c.1.1		1	85.84	even	2
1445.2.d.a.866.1		2	85.4	even	4
1445.2.d.a.866.2		2	85.64	even	4
3825.2.a.l.1.1		1	3.2	odd	2
4165.2.a.l.1.1		1	35.34	odd	2
5440.2.a.e.1.1		1	40.29	even	2
5440.2.a.x.1.1		1	40.19	odd	2
6800.2.a.v.1.1		1	4.3	odd	2
7225.2.a.d.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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