LMFDB - Embedded newform 310.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [310,2,Mod(1,310)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(310, 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("310.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 310 = 2 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	310.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:	$2.47536246266$
Analytic rank:	$0$
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	$\chi$	$=$	310.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} +2.00000 q^{12} -2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} -4.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} -4.00000 q^{27} -4.00000 q^{29} -2.00000 q^{30} -1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} -4.00000 q^{38} -1.00000 q^{40} +6.00000 q^{41} +2.00000 q^{43} +2.00000 q^{44} -1.00000 q^{45} -4.00000 q^{46} +2.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} +8.00000 q^{53} -4.00000 q^{54} -2.00000 q^{55} -8.00000 q^{57} -4.00000 q^{58} +8.00000 q^{59} -2.00000 q^{60} -1.00000 q^{62} +1.00000 q^{64} +4.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} +1.00000 q^{72} +6.00000 q^{73} -8.00000 q^{74} +2.00000 q^{75} -4.00000 q^{76} -4.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} +6.00000 q^{83} -2.00000 q^{85} +2.00000 q^{86} -8.00000 q^{87} +2.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} -4.00000 q^{92} -2.00000 q^{93} +4.00000 q^{95} +2.00000 q^{96} -2.00000 q^{97} -7.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
$2$	1.00000	0.707107
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.00000	0.816497
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	2.00000	0.577350
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.00000	0.426401
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	2.00000	0.408248
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	−2.00000	−0.365148
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	4.00000	0.696311
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−1.00000	−0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	2.00000	0.301511
$45$	−1.00000	−0.149071
$46$	−4.00000	−0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	2.00000	0.288675
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	4.00000	0.560112
$52$	0	0
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	−4.00000	−0.544331
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−8.00000	−1.05963
$58$	−4.00000	−0.525226
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	−2.00000	−0.258199
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	2.00000	0.242536
$69$	−8.00000	−0.963087
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−8.00000	−0.929981
$75$	2.00000	0.230940
$76$	−4.00000	−0.458831
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000	−0.111803
$81$	−11.0000	−1.22222
$82$	6.00000	0.662589
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	2.00000	0.215666
$87$	−8.00000	−0.857690
$88$	2.00000	0.213201
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−4.00000	−0.417029
$93$	−2.00000	−0.207390
$94$	0	0
$95$	4.00000	0.410391
$96$	2.00000	0.204124
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−7.00000	−0.707107
$99$	2.00000	0.201008
$100$	1.00000	0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	4.00000	0.396059
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	8.00000	0.777029
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	−4.00000	−0.384900
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	−2.00000	−0.190693
$111$	−16.0000	−1.51865
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	−8.00000	−0.749269
$115$	4.00000	0.373002
$116$	−4.00000	−0.371391
$117$	0	0
$118$	8.00000	0.736460
$119$	0	0
$120$	−2.00000	−0.182574
$121$	−7.00000	−0.636364
$122$	0	0
$123$	12.0000	1.08200
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	0	0
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	4.00000	0.345547
$135$	4.00000	0.344265
$136$	2.00000	0.171499
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	−8.00000	−0.681005
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	6.00000	0.496564
$147$	−14.0000	−1.15470
$148$	−8.00000	−0.657596
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	2.00000	0.163299
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−4.00000	−0.324443
$153$	2.00000	0.161690
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	−4.00000	−0.318223
$159$	16.0000	1.26888
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	−11.0000	−0.864242
$163$	24.0000	1.87983	0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000	1.87983	0.939913	+	0.341415i	$0.110906\pi$
$164$	6.00000	0.468521
$165$	−4.00000	−0.311400
$166$	6.00000	0.465690
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−2.00000	−0.153393
$171$	−4.00000	−0.305888
$172$	2.00000	0.152499
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	−8.00000	−0.606478
$175$	0	0
$176$	2.00000	0.150756
$177$	16.0000	1.20263
$178$	−6.00000	−0.449719
$179$	14.0000	1.04641	0.523205	−	0.852207i	$-0.324736\pi$
$179$	14.0000	1.04641	0.523205	+	0.852207i	$0.324736\pi$
$180$	−1.00000	−0.0745356
$181$	−24.0000	−1.78391	−0.891953	−	0.452128i	$-0.850665\pi$
$181$	−24.0000	−1.78391	−0.891953	+	0.452128i	$0.850665\pi$
$182$	0	0
$183$	0	0
$184$	−4.00000	−0.294884
$185$	8.00000	0.588172
$186$	−2.00000	−0.146647
$187$	4.00000	0.292509
$188$	0	0
$189$	0	0
$190$	4.00000	0.290191
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	2.00000	0.144338
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−7.00000	−0.500000
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	2.00000	0.142134
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	1.00000	0.0707107
$201$	8.00000	0.564276
$202$	2.00000	0.140720
$203$	0	0
$204$	4.00000	0.280056
$205$	−6.00000	−0.419058
$206$	8.00000	0.557386
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	8.00000	0.549442
$213$	0	0
$214$	−8.00000	−0.546869
$215$	−2.00000	−0.136399
$216$	−4.00000	−0.272166
$217$	0	0
$218$	−18.0000	−1.21911
$219$	12.0000	0.810885
$220$	−2.00000	−0.134840
$221$	0	0
$222$	−16.0000	−1.07385
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	14.0000	0.931266
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	−8.00000	−0.529813
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	4.00000	0.263752
$231$	0	0
$232$	−4.00000	−0.262613
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	0	0
$236$	8.00000	0.520756
$237$	−8.00000	−0.519656
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	−2.00000	−0.129099
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	−7.00000	−0.449977
$243$	−10.0000	−0.641500
$244$	0	0
$245$	7.00000	0.447214
$246$	12.0000	0.765092
$247$	0	0
$248$	−1.00000	−0.0635001
$249$	12.0000	0.760469
$250$	−1.00000	−0.0632456
$251$	14.0000	0.883672	0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000	0.883672	0.441836	+	0.897096i	$0.354327\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	16.0000	1.00393
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	0	0
$261$	−4.00000	−0.247594
$262$	20.0000	1.23560
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	4.00000	0.246183
$265$	−8.00000	−0.491436
$266$	0	0
$267$	−12.0000	−0.734388
$268$	4.00000	0.244339
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	4.00000	0.243432
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	18.0000	1.08742
$275$	2.00000	0.120605
$276$	−8.00000	−0.481543
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	2.00000	0.119952
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	4.00000	0.234888
$291$	−4.00000	−0.234484
$292$	6.00000	0.351123
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	−14.0000	−0.816497
$295$	−8.00000	−0.465778
$296$	−8.00000	−0.464991
$297$	−8.00000	−0.464207
$298$	−14.0000	−0.810998
$299$	0	0
$300$	2.00000	0.115470
$301$	0	0
$302$	4.00000	0.230174
$303$	4.00000	0.229794
$304$	−4.00000	−0.229416
$305$	0	0
$306$	2.00000	0.114332
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	1.00000	0.0567962
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	−4.00000	−0.225018
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	16.0000	0.897235
$319$	−8.00000	−0.447914
$320$	−1.00000	−0.0559017
$321$	−16.0000	−0.893033
$322$	0	0
$323$	−8.00000	−0.445132
$324$	−11.0000	−0.611111
$325$	0	0
$326$	24.0000	1.32924
$327$	−36.0000	−1.99080
$328$	6.00000	0.331295
$329$	0	0
$330$	−4.00000	−0.220193
$331$	−30.0000	−1.64895	−0.824475	−	0.565899i	$-0.808529\pi$
$331$	−30.0000	−1.64895	−0.824475	+	0.565899i	$0.808529\pi$
$332$	6.00000	0.329293
$333$	−8.00000	−0.438397
$334$	12.0000	0.656611
$335$	−4.00000	−0.218543
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−13.0000	−0.707107
$339$	28.0000	1.52075
$340$	−2.00000	−0.108465
$341$	−2.00000	−0.108306
$342$	−4.00000	−0.216295
$343$	0	0
$344$	2.00000	0.107833
$345$	8.00000	0.430706
$346$	−2.00000	−0.107521
$347$	−22.0000	−1.18102	−0.590511	−	0.807030i	$-0.701074\pi$
$347$	−22.0000	−1.18102	−0.590511	+	0.807030i	$0.701074\pi$
$348$	−8.00000	−0.428845
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	16.0000	0.850390
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	14.0000	0.739923
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	−24.0000	−1.26141
$363$	−14.0000	−0.734809
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	−4.00000	−0.208514
$369$	6.00000	0.312348
$370$	8.00000	0.415900
$371$	0	0
$372$	−2.00000	−0.103695
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	4.00000	0.206835
$375$	−2.00000	−0.103280
$376$	0	0
$377$	0	0
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	4.00000	0.205196
$381$	32.0000	1.63941
$382$	−8.00000	−0.409316
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	2.00000	0.101797
$387$	2.00000	0.101666
$388$	−2.00000	−0.101535
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	−7.00000	−0.353553
$393$	40.0000	2.01773
$394$	8.00000	0.403034
$395$	4.00000	0.201262
$396$	2.00000	0.100504
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	8.00000	0.399004
$403$	0	0
$404$	2.00000	0.0995037
$405$	11.0000	0.546594
$406$	0	0
$407$	−16.0000	−0.793091
$408$	4.00000	0.198030
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	−6.00000	−0.296319
$411$	36.0000	1.77575
$412$	8.00000	0.394132
$413$	0	0
$414$	−4.00000	−0.196589
$415$	−6.00000	−0.294528
$416$	0	0
$417$	4.00000	0.195881
$418$	−8.00000	−0.391293
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−16.0000	−0.778868
$423$	0	0
$424$	8.00000	0.388514
$425$	2.00000	0.0970143
$426$	0	0
$427$	0	0
$428$	−8.00000	−0.386695
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−4.00000	−0.192450
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	8.00000	0.383571
$436$	−18.0000	−0.862044
$437$	16.0000	0.765384
$438$	12.0000	0.573382
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	−2.00000	−0.0953463
$441$	−7.00000	−0.333333
$442$	0	0
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	−16.0000	−0.759326
$445$	6.00000	0.284427
$446$	−8.00000	−0.378811
$447$	−28.0000	−1.32435
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	1.00000	0.0471405
$451$	12.0000	0.565058
$452$	14.0000	0.658505
$453$	8.00000	0.375873
$454$	8.00000	0.375459
$455$	0	0
$456$	−8.00000	−0.374634
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	4.00000	0.186908
$459$	−8.00000	−0.373408
$460$	4.00000	0.186501
$461$	−8.00000	−0.372597	−0.186299	−	0.982493i	$-0.559649\pi$
$461$	−8.00000	−0.372597	−0.186299	+	0.982493i	$0.559649\pi$
$462$	0	0
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	−4.00000	−0.185695
$465$	2.00000	0.0927478
$466$	−26.0000	−1.20443
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	12.0000	0.552931
$472$	8.00000	0.368230
$473$	4.00000	0.183920
$474$	−8.00000	−0.367452
$475$	−4.00000	−0.183533
$476$	0	0
$477$	8.00000	0.366295
$478$	12.0000	0.548867
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	−2.00000	−0.0912871
$481$	0	0
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	−7.00000	−0.318182
$485$	2.00000	0.0908153
$486$	−10.0000	−0.453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	0	0
$489$	48.0000	2.17064
$490$	7.00000	0.316228
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	12.0000	0.541002
$493$	−8.00000	−0.360302
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	−1.00000	−0.0449013
$497$	0	0
$498$	12.0000	0.537733
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	−1.00000	−0.0447214
$501$	24.0000	1.07224
$502$	14.0000	0.624851
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	−8.00000	−0.355643
$507$	−26.0000	−1.15470
$508$	16.0000	0.709885
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	−4.00000	−0.177123
$511$	0	0
$512$	1.00000	0.0441942
$513$	16.0000	0.706417
$514$	10.0000	0.441081
$515$	−8.00000	−0.352522
$516$	4.00000	0.176090
$517$	0	0
$518$	0	0
$519$	−4.00000	−0.175581
$520$	0	0
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	−4.00000	−0.175075
$523$	38.0000	1.66162	0.830812	−	0.556553i	$-0.187876\pi$
$523$	38.0000	1.66162	0.830812	+	0.556553i	$0.187876\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	−24.0000	−1.04645
$527$	−2.00000	−0.0871214
$528$	4.00000	0.174078
$529$	−7.00000	−0.304348
$530$	−8.00000	−0.347498
$531$	8.00000	0.347170
$532$	0	0
$533$	0	0
$534$	−12.0000	−0.519291
$535$	8.00000	0.345870
$536$	4.00000	0.172774
$537$	28.0000	1.20829
$538$	0	0
$539$	−14.0000	−0.603023
$540$	4.00000	0.172133
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	−20.0000	−0.859074
$543$	−48.0000	−2.05988
$544$	2.00000	0.0857493
$545$	18.0000	0.771035
$546$	0	0
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	2.00000	0.0852803
$551$	16.0000	0.681623
$552$	−8.00000	−0.340503
$553$	0	0
$554$	0	0
$555$	16.0000	0.679162
$556$	2.00000	0.0848189
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	−1.00000	−0.0423334
$559$	0	0
$560$	0	0
$561$	8.00000	0.337760
$562$	10.0000	0.421825
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	−14.0000	−0.588984
$566$	−24.0000	−1.00880
$567$	0	0
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	8.00000	0.335083
$571$	−26.0000	−1.08807	−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000	−1.08807	−0.544033	+	0.839064i	$0.683103\pi$
$572$	0	0
$573$	−16.0000	−0.668410
$574$	0	0
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	−13.0000	−0.540729
$579$	4.00000	0.166234
$580$	4.00000	0.166091
$581$	0	0
$582$	−4.00000	−0.165805
$583$	16.0000	0.662652
$584$	6.00000	0.248282
$585$	0	0
$586$	−18.0000	−0.743573
$587$	−2.00000	−0.0825488	−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000	−0.0825488	−0.0412744	+	0.999148i	$0.513142\pi$
$588$	−14.0000	−0.577350
$589$	4.00000	0.164817
$590$	−8.00000	−0.329355
$591$	16.0000	0.658152
$592$	−8.00000	−0.328798
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	−8.00000	−0.328244
$595$	0	0
$596$	−14.0000	−0.573462
$597$	32.0000	1.30967
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	2.00000	0.0816497
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	4.00000	0.162758
$605$	7.00000	0.284590
$606$	4.00000	0.162489
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	0	0
$611$	0	0
$612$	2.00000	0.0808452
$613$	−44.0000	−1.77714	−0.888572	−	0.458738i	$-0.848302\pi$
$613$	−44.0000	−1.77714	−0.888572	+	0.458738i	$0.848302\pi$
$614$	−16.0000	−0.645707
$615$	−12.0000	−0.483887
$616$	0	0
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	16.0000	0.643614
$619$	−34.0000	−1.36658	−0.683288	−	0.730149i	$-0.739451\pi$
$619$	−34.0000	−1.36658	−0.683288	+	0.730149i	$0.739451\pi$
$620$	1.00000	0.0401610
$621$	16.0000	0.642058
$622$	8.00000	0.320771
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−22.0000	−0.879297
$627$	−16.0000	−0.638978
$628$	6.00000	0.239426
$629$	−16.0000	−0.637962
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	−4.00000	−0.159111
$633$	−32.0000	−1.27189
$634$	2.00000	0.0794301
$635$	−16.0000	−0.634941
$636$	16.0000	0.634441
$637$	0	0
$638$	−8.00000	−0.316723
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	−16.0000	−0.631470
$643$	−2.00000	−0.0788723	−0.0394362	−	0.999222i	$-0.512556\pi$
$643$	−2.00000	−0.0788723	−0.0394362	+	0.999222i	$0.512556\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	−8.00000	−0.314756
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	−11.0000	−0.432121
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	24.0000	0.939913
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	−36.0000	−1.40771
$655$	−20.0000	−0.781465
$656$	6.00000	0.234261
$657$	6.00000	0.234082
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	−4.00000	−0.155700
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−30.0000	−1.16598
$663$	0	0
$664$	6.00000	0.232845
$665$	0	0
$666$	−8.00000	−0.309994
$667$	16.0000	0.619522
$668$	12.0000	0.464294
$669$	−16.0000	−0.618596
$670$	−4.00000	−0.154533
$671$	0	0
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	14.0000	0.539260
$675$	−4.00000	−0.153960
$676$	−13.0000	−0.500000
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	28.0000	1.07533
$679$	0	0
$680$	−2.00000	−0.0766965
$681$	16.0000	0.613121
$682$	−2.00000	−0.0765840
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	−4.00000	−0.152944
$685$	−18.0000	−0.687745
$686$	0	0
$687$	8.00000	0.305219
$688$	2.00000	0.0762493
$689$	0	0
$690$	8.00000	0.304555
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	−22.0000	−0.835109
$695$	−2.00000	−0.0758643
$696$	−8.00000	−0.303239
$697$	12.0000	0.454532
$698$	10.0000	0.378506
$699$	−52.0000	−1.96682
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	2.00000	0.0753778
$705$	0	0
$706$	30.0000	1.12906
$707$	0	0
$708$	16.0000	0.601317
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	−6.00000	−0.224860
$713$	4.00000	0.149801
$714$	0	0
$715$	0	0
$716$	14.0000	0.523205
$717$	24.0000	0.896296
$718$	−8.00000	−0.298557
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	−3.00000	−0.111648
$723$	−4.00000	−0.148762
$724$	−24.0000	−0.891953
$725$	−4.00000	−0.148556
$726$	−14.0000	−0.519589
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−6.00000	−0.222070
$731$	4.00000	0.147945
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	4.00000	0.147643
$735$	14.0000	0.516398
$736$	−4.00000	−0.147442
$737$	8.00000	0.294684
$738$	6.00000	0.220863
$739$	26.0000	0.956425	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956425	0.478213	+	0.878244i	$0.341285\pi$
$740$	8.00000	0.294086
$741$	0	0
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	−2.00000	−0.0733236
$745$	14.0000	0.512920
$746$	−26.0000	−0.951928
$747$	6.00000	0.219529
$748$	4.00000	0.146254
$749$	0	0
$750$	−2.00000	−0.0730297
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	0	0
$753$	28.0000	1.02038
$754$	0	0
$755$	−4.00000	−0.145575
$756$	0	0
$757$	12.0000	0.436147	0.218074	−	0.975932i	$-0.430023\pi$
$757$	12.0000	0.436147	0.218074	+	0.975932i	$0.430023\pi$
$758$	16.0000	0.581146
$759$	−16.0000	−0.580763
$760$	4.00000	0.145095
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	32.0000	1.15924
$763$	0	0
$764$	−8.00000	−0.289430
$765$	−2.00000	−0.0723102
$766$	16.0000	0.578103
$767$	0	0
$768$	2.00000	0.0721688
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	20.0000	0.720282
$772$	2.00000	0.0719816
$773$	−28.0000	−1.00709	−0.503545	−	0.863969i	$-0.667971\pi$
$773$	−28.0000	−1.00709	−0.503545	+	0.863969i	$0.667971\pi$
$774$	2.00000	0.0718885
$775$	−1.00000	−0.0359211
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	−36.0000	−1.29066
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	−8.00000	−0.286079
$783$	16.0000	0.571793
$784$	−7.00000	−0.250000
$785$	−6.00000	−0.214149
$786$	40.0000	1.42675
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	8.00000	0.284988
$789$	−48.0000	−1.70885
$790$	4.00000	0.142314
$791$	0	0
$792$	2.00000	0.0710669
$793$	0	0
$794$	−14.0000	−0.496841
$795$	−16.0000	−0.567462
$796$	16.0000	0.567105
$797$	24.0000	0.850124	0.425062	−	0.905164i	$-0.360252\pi$
$797$	24.0000	0.850124	0.425062	+	0.905164i	$0.360252\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	−6.00000	−0.212000
$802$	38.0000	1.34183
$803$	12.0000	0.423471
$804$	8.00000	0.282138
$805$	0	0
$806$	0	0
$807$	0	0
$808$	2.00000	0.0703598
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	11.0000	0.386501
$811$	24.0000	0.842754	0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000	0.842754	0.421377	+	0.906886i	$0.361547\pi$
$812$	0	0
$813$	−40.0000	−1.40286
$814$	−16.0000	−0.560800
$815$	−24.0000	−0.840683
$816$	4.00000	0.140028
$817$	−8.00000	−0.279885
$818$	6.00000	0.209785
$819$	0	0
$820$	−6.00000	−0.209529
$821$	20.0000	0.698005	0.349002	−	0.937122i	$-0.386521\pi$
$821$	20.0000	0.698005	0.349002	+	0.937122i	$0.386521\pi$
$822$	36.0000	1.25564
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	8.00000	0.278693
$825$	4.00000	0.139262
$826$	0	0
$827$	−14.0000	−0.486828	−0.243414	−	0.969923i	$-0.578267\pi$
$827$	−14.0000	−0.486828	−0.243414	+	0.969923i	$0.578267\pi$
$828$	−4.00000	−0.139010
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	−6.00000	−0.208263
$831$	0	0
$832$	0	0
$833$	−14.0000	−0.485071
$834$	4.00000	0.138509
$835$	−12.0000	−0.415277
$836$	−8.00000	−0.276686
$837$	4.00000	0.138260
$838$	−24.0000	−0.829066
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−22.0000	−0.758170
$843$	20.0000	0.688837
$844$	−16.0000	−0.550743
$845$	13.0000	0.447214
$846$	0	0
$847$	0	0
$848$	8.00000	0.274721
$849$	−48.0000	−1.64736
$850$	2.00000	0.0685994
$851$	32.0000	1.09695
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	−8.00000	−0.273434
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	42.0000	1.43302	0.716511	−	0.697576i	$-0.245738\pi$
$859$	42.0000	1.43302	0.716511	+	0.697576i	$0.245738\pi$
$860$	−2.00000	−0.0681994
$861$	0	0
$862$	−24.0000	−0.817443
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	−4.00000	−0.136083
$865$	2.00000	0.0680020
$866$	26.0000	0.883516
$867$	−26.0000	−0.883006
$868$	0	0
$869$	−8.00000	−0.271381
$870$	8.00000	0.271225
$871$	0	0
$872$	−18.0000	−0.609557
$873$	−2.00000	−0.0676897
$874$	16.0000	0.541208
$875$	0	0
$876$	12.0000	0.405442
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	40.0000	1.34993
$879$	−36.0000	−1.21425
$880$	−2.00000	−0.0674200
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	−7.00000	−0.235702
$883$	−26.0000	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$884$	0	0
$885$	−16.0000	−0.537834
$886$	−28.0000	−0.940678
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	−16.0000	−0.536925
$889$	0	0
$890$	6.00000	0.201120
$891$	−22.0000	−0.737028
$892$	−8.00000	−0.267860
$893$	0	0
$894$	−28.0000	−0.936460
$895$	−14.0000	−0.467968
$896$	0	0
$897$	0	0
$898$	6.00000	0.200223
$899$	4.00000	0.133407
$900$	1.00000	0.0333333
$901$	16.0000	0.533037
$902$	12.0000	0.399556
$903$	0	0
$904$	14.0000	0.465633
$905$	24.0000	0.797787
$906$	8.00000	0.265782
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	8.00000	0.265489
$909$	2.00000	0.0663358
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	−8.00000	−0.264906
$913$	12.0000	0.397142
$914$	2.00000	0.0661541
$915$	0	0
$916$	4.00000	0.132164
$917$	0	0
$918$	−8.00000	−0.264039
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	4.00000	0.131876
$921$	−32.0000	−1.05444
$922$	−8.00000	−0.263466
$923$	0	0
$924$	0	0
$925$	−8.00000	−0.263038
$926$	36.0000	1.18303
$927$	8.00000	0.262754
$928$	−4.00000	−0.131306
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	2.00000	0.0655826
$931$	28.0000	0.917663
$932$	−26.0000	−0.851658
$933$	16.0000	0.523816
$934$	8.00000	0.261768
$935$	−4.00000	−0.130814
$936$	0	0
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	0	0
$939$	−44.0000	−1.43589
$940$	0	0
$941$	−60.0000	−1.95594	−0.977972	−	0.208736i	$-0.933065\pi$
$941$	−60.0000	−1.95594	−0.977972	+	0.208736i	$0.933065\pi$
$942$	12.0000	0.390981
$943$	−24.0000	−0.781548
$944$	8.00000	0.260378
$945$	0	0
$946$	4.00000	0.130051
$947$	−14.0000	−0.454939	−0.227469	−	0.973785i	$-0.573045\pi$
$947$	−14.0000	−0.454939	−0.227469	+	0.973785i	$0.573045\pi$
$948$	−8.00000	−0.259828
$949$	0	0
$950$	−4.00000	−0.129777
$951$	4.00000	0.129709
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	8.00000	0.259010
$955$	8.00000	0.258874
$956$	12.0000	0.388108
$957$	−16.0000	−0.517207
$958$	24.0000	0.775405
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	1.00000	0.0322581
$962$	0	0
$963$	−8.00000	−0.257796
$964$	−2.00000	−0.0644157
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	−7.00000	−0.224989
$969$	−16.0000	−0.513994
$970$	2.00000	0.0642161
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	−10.0000	−0.320750
$973$	0	0
$974$	32.0000	1.02535
$975$	0	0
$976$	0	0
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	48.0000	1.53487
$979$	−12.0000	−0.383522
$980$	7.00000	0.223607
$981$	−18.0000	−0.574696
$982$	2.00000	0.0638226
$983$	8.00000	0.255160	0.127580	−	0.991828i	$-0.459279\pi$
$983$	8.00000	0.255160	0.127580	+	0.991828i	$0.459279\pi$
$984$	12.0000	0.382546
$985$	−8.00000	−0.254901
$986$	−8.00000	−0.254772
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	−2.00000	−0.0635642
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	−1.00000	−0.0317500
$993$	−60.0000	−1.90404
$994$	0	0
$995$	−16.0000	−0.507234
$996$	12.0000	0.380235
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	10.0000	0.316544
$999$	32.0000	1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.2.a.b.1.1	✓	1
3.2	odd	2		2790.2.a.h.1.1		1
4.3	odd	2		2480.2.a.c.1.1		1
5.2	odd	4		1550.2.b.e.249.2		2
5.3	odd	4		1550.2.b.e.249.1		2
5.4	even	2		1550.2.a.a.1.1		1
8.3	odd	2		9920.2.a.bg.1.1		1
8.5	even	2		9920.2.a.d.1.1		1
31.30	odd	2		9610.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.a.b.1.1	✓	1	1.1	even	1	trivial
1550.2.a.a.1.1		1	5.4	even	2
1550.2.b.e.249.1		2	5.3	odd	4
1550.2.b.e.249.2		2	5.2	odd	4
2480.2.a.c.1.1		1	4.3	odd	2
2790.2.a.h.1.1		1	3.2	odd	2
9610.2.a.a.1.1		1	31.30	odd	2
9920.2.a.d.1.1		1	8.5	even	2
9920.2.a.bg.1.1		1	8.3	odd	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:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	310.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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