LMFDB - Embedded newform 370.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [370,2,Mod(1,370)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(370, 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("370.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 370 = 2 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	370.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:	$2.95446487479$
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	$\chi$	$=$	370.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} +2.00000 q^{12} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -3.00000 q^{22} +2.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} -4.00000 q^{27} +1.00000 q^{28} -3.00000 q^{29} -2.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} -3.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} +3.00000 q^{41} -2.00000 q^{42} -1.00000 q^{43} +3.00000 q^{44} +1.00000 q^{45} -2.00000 q^{46} +4.00000 q^{47} +2.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} +13.0000 q^{53} +4.00000 q^{54} +3.00000 q^{55} -1.00000 q^{56} -12.0000 q^{57} +3.00000 q^{58} +2.00000 q^{60} -15.0000 q^{61} -3.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -6.00000 q^{66} +3.00000 q^{68} +4.00000 q^{69} -1.00000 q^{70} -2.00000 q^{71} -1.00000 q^{72} +1.00000 q^{74} +2.00000 q^{75} -6.00000 q^{76} +3.00000 q^{77} -8.00000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -3.00000 q^{82} -4.00000 q^{83} +2.00000 q^{84} +3.00000 q^{85} +1.00000 q^{86} -6.00000 q^{87} -3.00000 q^{88} -18.0000 q^{89} -1.00000 q^{90} +2.00000 q^{92} +6.00000 q^{93} -4.00000 q^{94} -6.00000 q^{95} -2.00000 q^{96} -7.00000 q^{97} +6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	2.00000	0.577350
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−1.00000	−0.267261
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−1.00000	−0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	1.00000	0.223607
$21$	2.00000	0.436436
$22$	−3.00000	−0.639602
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	−2.00000	−0.408248
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	1.00000	0.188982
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	−2.00000	−0.365148
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	−1.00000	−0.176777
$33$	6.00000	1.04447
$34$	−3.00000	−0.514496
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	6.00000	0.973329
$39$	0	0
$40$	−1.00000	−0.158114
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	−2.00000	−0.308607
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	3.00000	0.452267
$45$	1.00000	0.149071
$46$	−2.00000	−0.294884
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	2.00000	0.288675
$49$	−6.00000	−0.857143
$50$	−1.00000	−0.141421
$51$	6.00000	0.840168
$52$	0	0
$53$	13.0000	1.78569	0.892844	−	0.450367i	$-0.148707\pi$
$53$	13.0000	1.78569	0.892844	+	0.450367i	$0.148707\pi$
$54$	4.00000	0.544331
$55$	3.00000	0.404520
$56$	−1.00000	−0.133631
$57$	−12.0000	−1.58944
$58$	3.00000	0.393919
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	2.00000	0.258199
$61$	−15.0000	−1.92055	−0.960277	−	0.279050i	$-0.909981\pi$
$61$	−15.0000	−1.92055	−0.960277	+	0.279050i	$0.909981\pi$
$62$	−3.00000	−0.381000
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	3.00000	0.363803
$69$	4.00000	0.481543
$70$	−1.00000	−0.119523
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−1.00000	−0.117851
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	1.00000	0.116248
$75$	2.00000	0.230940
$76$	−6.00000	−0.688247
$77$	3.00000	0.341882
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	−3.00000	−0.331295
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	2.00000	0.218218
$85$	3.00000	0.325396
$86$	1.00000	0.107833
$87$	−6.00000	−0.643268
$88$	−3.00000	−0.319801
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	2.00000	0.208514
$93$	6.00000	0.622171
$94$	−4.00000	−0.412568
$95$	−6.00000	−0.615587
$96$	−2.00000	−0.204124
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	6.00000	0.606092
$99$	3.00000	0.301511
$100$	1.00000	0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−6.00000	−0.594089
$103$	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
$105$	2.00000	0.195180
$106$	−13.0000	−1.26267
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	−4.00000	−0.384900
$109$	−3.00000	−0.287348	−0.143674	−	0.989625i	$-0.545892\pi$
$109$	−3.00000	−0.287348	−0.143674	+	0.989625i	$0.545892\pi$
$110$	−3.00000	−0.286039
$111$	−2.00000	−0.189832
$112$	1.00000	0.0944911
$113$	−7.00000	−0.658505	−0.329252	−	0.944242i	$-0.606797\pi$
$113$	−7.00000	−0.658505	−0.329252	+	0.944242i	$0.606797\pi$
$114$	12.0000	1.12390
$115$	2.00000	0.186501
$116$	−3.00000	−0.278543
$117$	0	0
$118$	0	0
$119$	3.00000	0.275010
$120$	−2.00000	−0.182574
$121$	−2.00000	−0.181818
$122$	15.0000	1.35804
$123$	6.00000	0.541002
$124$	3.00000	0.269408
$125$	1.00000	0.0894427
$126$	−1.00000	−0.0890871
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−1.00000	−0.0883883
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	6.00000	0.522233
$133$	−6.00000	−0.520266
$134$	0	0
$135$	−4.00000	−0.344265
$136$	−3.00000	−0.257248
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	−4.00000	−0.340503
$139$	3.00000	0.254457	0.127228	−	0.991873i	$-0.459392\pi$
$139$	3.00000	0.254457	0.127228	+	0.991873i	$0.459392\pi$
$140$	1.00000	0.0845154
$141$	8.00000	0.673722
$142$	2.00000	0.167836
$143$	0	0
$144$	1.00000	0.0833333
$145$	−3.00000	−0.249136
$146$	0	0
$147$	−12.0000	−0.989743
$148$	−1.00000	−0.0821995
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−2.00000	−0.163299
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	6.00000	0.486664
$153$	3.00000	0.242536
$154$	−3.00000	−0.241747
$155$	3.00000	0.240966
$156$	0	0
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	8.00000	0.636446
$159$	26.0000	2.06193
$160$	−1.00000	−0.0790569
$161$	2.00000	0.157622
$162$	11.0000	0.864242
$163$	−5.00000	−0.391630	−0.195815	−	0.980641i	$-0.562735\pi$
$163$	−5.00000	−0.391630	−0.195815	+	0.980641i	$0.562735\pi$
$164$	3.00000	0.234261
$165$	6.00000	0.467099
$166$	4.00000	0.310460
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−2.00000	−0.154303
$169$	−13.0000	−1.00000
$170$	−3.00000	−0.230089
$171$	−6.00000	−0.458831
$172$	−1.00000	−0.0762493
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	6.00000	0.454859
$175$	1.00000	0.0755929
$176$	3.00000	0.226134
$177$	0	0
$178$	18.0000	1.34916
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	1.00000	0.0745356
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−30.0000	−2.21766
$184$	−2.00000	−0.147442
$185$	−1.00000	−0.0735215
$186$	−6.00000	−0.439941
$187$	9.00000	0.658145
$188$	4.00000	0.291730
$189$	−4.00000	−0.290957
$190$	6.00000	0.435286
$191$	21.0000	1.51951	0.759753	−	0.650211i	$-0.225320\pi$
$191$	21.0000	1.51951	0.759753	+	0.650211i	$0.225320\pi$
$192$	2.00000	0.144338
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	7.00000	0.502571
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	−3.00000	−0.213201
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	10.0000	0.703598
$203$	−3.00000	−0.210559
$204$	6.00000	0.420084
$205$	3.00000	0.209529
$206$	−8.00000	−0.557386
$207$	2.00000	0.139010
$208$	0	0
$209$	−18.0000	−1.24509
$210$	−2.00000	−0.138013
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	13.0000	0.892844
$213$	−4.00000	−0.274075
$214$	−2.00000	−0.136717
$215$	−1.00000	−0.0681994
$216$	4.00000	0.272166
$217$	3.00000	0.203653
$218$	3.00000	0.203186
$219$	0	0
$220$	3.00000	0.202260
$221$	0	0
$222$	2.00000	0.134231
$223$	23.0000	1.54019	0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000	1.54019	0.770097	+	0.637927i	$0.220208\pi$
$224$	−1.00000	−0.0668153
$225$	1.00000	0.0666667
$226$	7.00000	0.465633
$227$	13.0000	0.862840	0.431420	−	0.902151i	$-0.358013\pi$
$227$	13.0000	0.862840	0.431420	+	0.902151i	$0.358013\pi$
$228$	−12.0000	−0.794719
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−2.00000	−0.131876
$231$	6.00000	0.394771
$232$	3.00000	0.196960
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	−16.0000	−1.03931
$238$	−3.00000	−0.194461
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	2.00000	0.129099
$241$	24.0000	1.54598	0.772988	−	0.634421i	$-0.218761\pi$
$241$	24.0000	1.54598	0.772988	+	0.634421i	$0.218761\pi$
$242$	2.00000	0.128565
$243$	−10.0000	−0.641500
$244$	−15.0000	−0.960277
$245$	−6.00000	−0.383326
$246$	−6.00000	−0.382546
$247$	0	0
$248$	−3.00000	−0.190500
$249$	−8.00000	−0.506979
$250$	−1.00000	−0.0632456
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	1.00000	0.0629941
$253$	6.00000	0.377217
$254$	4.00000	0.250982
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	2.00000	0.124515
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	−3.00000	−0.185695
$262$	12.0000	0.741362
$263$	−1.00000	−0.0616626	−0.0308313	−	0.999525i	$-0.509815\pi$
$263$	−1.00000	−0.0616626	−0.0308313	+	0.999525i	$0.509815\pi$
$264$	−6.00000	−0.369274
$265$	13.0000	0.798584
$266$	6.00000	0.367884
$267$	−36.0000	−2.20316
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	4.00000	0.243432
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	−8.00000	−0.483298
$275$	3.00000	0.180907
$276$	4.00000	0.240772
$277$	−20.0000	−1.20168	−0.600842	−	0.799368i	$-0.705168\pi$
$277$	−20.0000	−1.20168	−0.600842	+	0.799368i	$0.705168\pi$
$278$	−3.00000	−0.179928
$279$	3.00000	0.179605
$280$	−1.00000	−0.0597614
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	−8.00000	−0.476393
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	−2.00000	−0.118678
$285$	−12.0000	−0.710819
$286$	0	0
$287$	3.00000	0.177084
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	3.00000	0.176166
$291$	−14.0000	−0.820695
$292$	0	0
$293$	3.00000	0.175262	0.0876309	−	0.996153i	$-0.472070\pi$
$293$	3.00000	0.175262	0.0876309	+	0.996153i	$0.472070\pi$
$294$	12.0000	0.699854
$295$	0	0
$296$	1.00000	0.0581238
$297$	−12.0000	−0.696311
$298$	−18.0000	−1.04271
$299$	0	0
$300$	2.00000	0.115470
$301$	−1.00000	−0.0576390
$302$	−16.0000	−0.920697
$303$	−20.0000	−1.14897
$304$	−6.00000	−0.344124
$305$	−15.0000	−0.858898
$306$	−3.00000	−0.171499
$307$	34.0000	1.94048	0.970241	−	0.242140i	$-0.0778494\pi$
$307$	34.0000	1.94048	0.970241	+	0.242140i	$0.0778494\pi$
$308$	3.00000	0.170941
$309$	16.0000	0.910208
$310$	−3.00000	−0.170389
$311$	25.0000	1.41762	0.708810	−	0.705399i	$-0.249232\pi$
$311$	25.0000	1.41762	0.708810	+	0.705399i	$0.249232\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	3.00000	0.169300
$315$	1.00000	0.0563436
$316$	−8.00000	−0.450035
$317$	21.0000	1.17948	0.589739	−	0.807594i	$-0.299231\pi$
$317$	21.0000	1.17948	0.589739	+	0.807594i	$0.299231\pi$
$318$	−26.0000	−1.45801
$319$	−9.00000	−0.503903
$320$	1.00000	0.0559017
$321$	4.00000	0.223258
$322$	−2.00000	−0.111456
$323$	−18.0000	−1.00155
$324$	−11.0000	−0.611111
$325$	0	0
$326$	5.00000	0.276924
$327$	−6.00000	−0.331801
$328$	−3.00000	−0.165647
$329$	4.00000	0.220527
$330$	−6.00000	−0.330289
$331$	30.0000	1.64895	0.824475	−	0.565899i	$-0.191471\pi$
$331$	30.0000	1.64895	0.824475	+	0.565899i	$0.191471\pi$
$332$	−4.00000	−0.219529
$333$	−1.00000	−0.0547997
$334$	−12.0000	−0.656611
$335$	0	0
$336$	2.00000	0.109109
$337$	12.0000	0.653682	0.326841	−	0.945079i	$-0.394016\pi$
$337$	12.0000	0.653682	0.326841	+	0.945079i	$0.394016\pi$
$338$	13.0000	0.707107
$339$	−14.0000	−0.760376
$340$	3.00000	0.162698
$341$	9.00000	0.487377
$342$	6.00000	0.324443
$343$	−13.0000	−0.701934
$344$	1.00000	0.0539164
$345$	4.00000	0.215353
$346$	−9.00000	−0.483843
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	−6.00000	−0.321634
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−3.00000	−0.159901
$353$	19.0000	1.01127	0.505634	−	0.862748i	$-0.331259\pi$
$353$	19.0000	1.01127	0.505634	+	0.862748i	$0.331259\pi$
$354$	0	0
$355$	−2.00000	−0.106149
$356$	−18.0000	−0.953998
$357$	6.00000	0.317554
$358$	−24.0000	−1.26844
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	−1.00000	−0.0527046
$361$	17.0000	0.894737
$362$	−2.00000	−0.105118
$363$	−4.00000	−0.209946
$364$	0	0
$365$	0	0
$366$	30.0000	1.56813
$367$	13.0000	0.678594	0.339297	−	0.940679i	$-0.389811\pi$
$367$	13.0000	0.678594	0.339297	+	0.940679i	$0.389811\pi$
$368$	2.00000	0.104257
$369$	3.00000	0.156174
$370$	1.00000	0.0519875
$371$	13.0000	0.674926
$372$	6.00000	0.311086
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	−9.00000	−0.465379
$375$	2.00000	0.103280
$376$	−4.00000	−0.206284
$377$	0	0
$378$	4.00000	0.205738
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	−6.00000	−0.307794
$381$	−8.00000	−0.409852
$382$	−21.0000	−1.07445
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	−2.00000	−0.102062
$385$	3.00000	0.152894
$386$	10.0000	0.508987
$387$	−1.00000	−0.0508329
$388$	−7.00000	−0.355371
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	6.00000	0.303046
$393$	−24.0000	−1.21064
$394$	18.0000	0.906827
$395$	−8.00000	−0.402524
$396$	3.00000	0.150756
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	−8.00000	−0.401004
$399$	−12.0000	−0.600751
$400$	1.00000	0.0500000
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	0	0
$403$	0	0
$404$	−10.0000	−0.497519
$405$	−11.0000	−0.546594
$406$	3.00000	0.148888
$407$	−3.00000	−0.148704
$408$	−6.00000	−0.297044
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	−3.00000	−0.148159
$411$	16.0000	0.789222
$412$	8.00000	0.394132
$413$	0	0
$414$	−2.00000	−0.0982946
$415$	−4.00000	−0.196352
$416$	0	0
$417$	6.00000	0.293821
$418$	18.0000	0.880409
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	2.00000	0.0975900
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	3.00000	0.146038
$423$	4.00000	0.194487
$424$	−13.0000	−0.631336
$425$	3.00000	0.145521
$426$	4.00000	0.193801
$427$	−15.0000	−0.725901
$428$	2.00000	0.0966736
$429$	0	0
$430$	1.00000	0.0482243
$431$	−31.0000	−1.49322	−0.746609	−	0.665263i	$-0.768319\pi$
$431$	−31.0000	−1.49322	−0.746609	+	0.665263i	$0.768319\pi$
$432$	−4.00000	−0.192450
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	−3.00000	−0.144005
$435$	−6.00000	−0.287678
$436$	−3.00000	−0.143674
$437$	−12.0000	−0.574038
$438$	0	0
$439$	37.0000	1.76591	0.882957	−	0.469454i	$-0.155549\pi$
$439$	37.0000	1.76591	0.882957	+	0.469454i	$0.155549\pi$
$440$	−3.00000	−0.143019
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−34.0000	−1.61539	−0.807694	−	0.589601i	$-0.799285\pi$
$443$	−34.0000	−1.61539	−0.807694	+	0.589601i	$0.799285\pi$
$444$	−2.00000	−0.0949158
$445$	−18.0000	−0.853282
$446$	−23.0000	−1.08908
$447$	36.0000	1.70274
$448$	1.00000	0.0472456
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	−1.00000	−0.0471405
$451$	9.00000	0.423793
$452$	−7.00000	−0.329252
$453$	32.0000	1.50349
$454$	−13.0000	−0.610120
$455$	0	0
$456$	12.0000	0.561951
$457$	9.00000	0.421002	0.210501	−	0.977594i	$-0.432490\pi$
$457$	9.00000	0.421002	0.210501	+	0.977594i	$0.432490\pi$
$458$	6.00000	0.280362
$459$	−12.0000	−0.560112
$460$	2.00000	0.0932505
$461$	−1.00000	−0.0465746	−0.0232873	−	0.999729i	$-0.507413\pi$
$461$	−1.00000	−0.0465746	−0.0232873	+	0.999729i	$0.507413\pi$
$462$	−6.00000	−0.279145
$463$	12.0000	0.557687	0.278844	−	0.960337i	$-0.410049\pi$
$463$	12.0000	0.557687	0.278844	+	0.960337i	$0.410049\pi$
$464$	−3.00000	−0.139272
$465$	6.00000	0.278243
$466$	−4.00000	−0.185296
$467$	−37.0000	−1.71216	−0.856078	−	0.516847i	$-0.827106\pi$
$467$	−37.0000	−1.71216	−0.856078	+	0.516847i	$0.827106\pi$
$468$	0	0
$469$	0	0
$470$	−4.00000	−0.184506
$471$	−6.00000	−0.276465
$472$	0	0
$473$	−3.00000	−0.137940
$474$	16.0000	0.734904
$475$	−6.00000	−0.275299
$476$	3.00000	0.137505
$477$	13.0000	0.595229
$478$	−9.00000	−0.411650
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	−2.00000	−0.0912871
$481$	0	0
$482$	−24.0000	−1.09317
$483$	4.00000	0.182006
$484$	−2.00000	−0.0909091
$485$	−7.00000	−0.317854
$486$	10.0000	0.453609
$487$	34.0000	1.54069	0.770344	−	0.637629i	$-0.220085\pi$
$487$	34.0000	1.54069	0.770344	+	0.637629i	$0.220085\pi$
$488$	15.0000	0.679018
$489$	−10.0000	−0.452216
$490$	6.00000	0.271052
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	6.00000	0.270501
$493$	−9.00000	−0.405340
$494$	0	0
$495$	3.00000	0.134840
$496$	3.00000	0.134704
$497$	−2.00000	−0.0897123
$498$	8.00000	0.358489
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	1.00000	0.0447214
$501$	24.0000	1.07224
$502$	−6.00000	−0.267793
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	−1.00000	−0.0445435
$505$	−10.0000	−0.444994
$506$	−6.00000	−0.266733
$507$	−26.0000	−1.15470
$508$	−4.00000	−0.177471
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	−6.00000	−0.265684
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	24.0000	1.05963
$514$	22.0000	0.970378
$515$	8.00000	0.352522
$516$	−2.00000	−0.0880451
$517$	12.0000	0.527759
$518$	1.00000	0.0439375
$519$	18.0000	0.790112
$520$	0	0
$521$	−23.0000	−1.00765	−0.503824	−	0.863806i	$-0.668074\pi$
$521$	−23.0000	−1.00765	−0.503824	+	0.863806i	$0.668074\pi$
$522$	3.00000	0.131306
$523$	−44.0000	−1.92399	−0.961993	−	0.273075i	$-0.911959\pi$
$523$	−44.0000	−1.92399	−0.961993	+	0.273075i	$0.911959\pi$
$524$	−12.0000	−0.524222
$525$	2.00000	0.0872872
$526$	1.00000	0.0436021
$527$	9.00000	0.392046
$528$	6.00000	0.261116
$529$	−19.0000	−0.826087
$530$	−13.0000	−0.564684
$531$	0	0
$532$	−6.00000	−0.260133
$533$	0	0
$534$	36.0000	1.55787
$535$	2.00000	0.0864675
$536$	0	0
$537$	48.0000	2.07135
$538$	10.0000	0.431131
$539$	−18.0000	−0.775315
$540$	−4.00000	−0.172133
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	10.0000	0.429537
$543$	4.00000	0.171656
$544$	−3.00000	−0.128624
$545$	−3.00000	−0.128506
$546$	0	0
$547$	23.0000	0.983409	0.491704	−	0.870762i	$-0.336374\pi$
$547$	23.0000	0.983409	0.491704	+	0.870762i	$0.336374\pi$
$548$	8.00000	0.341743
$549$	−15.0000	−0.640184
$550$	−3.00000	−0.127920
$551$	18.0000	0.766826
$552$	−4.00000	−0.170251
$553$	−8.00000	−0.340195
$554$	20.0000	0.849719
$555$	−2.00000	−0.0848953
$556$	3.00000	0.127228
$557$	−40.0000	−1.69485	−0.847427	−	0.530912i	$-0.821850\pi$
$557$	−40.0000	−1.69485	−0.847427	+	0.530912i	$0.821850\pi$
$558$	−3.00000	−0.127000
$559$	0	0
$560$	1.00000	0.0422577
$561$	18.0000	0.759961
$562$	24.0000	1.01238
$563$	−11.0000	−0.463595	−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000	−0.463595	−0.231797	+	0.972764i	$0.574461\pi$
$564$	8.00000	0.336861
$565$	−7.00000	−0.294492
$566$	20.0000	0.840663
$567$	−11.0000	−0.461957
$568$	2.00000	0.0839181
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	12.0000	0.502625
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	0	0
$573$	42.0000	1.75458
$574$	−3.00000	−0.125218
$575$	2.00000	0.0834058
$576$	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	$0.285320\pi$
$578$	8.00000	0.332756
$579$	−20.0000	−0.831172
$580$	−3.00000	−0.124568
$581$	−4.00000	−0.165948
$582$	14.0000	0.580319
$583$	39.0000	1.61521
$584$	0	0
$585$	0	0
$586$	−3.00000	−0.123929
$587$	35.0000	1.44460	0.722302	−	0.691577i	$-0.243084\pi$
$587$	35.0000	1.44460	0.722302	+	0.691577i	$0.243084\pi$
$588$	−12.0000	−0.494872
$589$	−18.0000	−0.741677
$590$	0	0
$591$	−36.0000	−1.48084
$592$	−1.00000	−0.0410997
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	12.0000	0.492366
$595$	3.00000	0.122988
$596$	18.0000	0.737309
$597$	16.0000	0.654836
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−2.00000	−0.0816497
$601$	21.0000	0.856608	0.428304	−	0.903635i	$-0.359111\pi$
$601$	21.0000	0.856608	0.428304	+	0.903635i	$0.359111\pi$
$602$	1.00000	0.0407570
$603$	0	0
$604$	16.0000	0.651031
$605$	−2.00000	−0.0813116
$606$	20.0000	0.812444
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	6.00000	0.243332
$609$	−6.00000	−0.243132
$610$	15.0000	0.607332
$611$	0	0
$612$	3.00000	0.121268
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	−34.0000	−1.37213
$615$	6.00000	0.241943
$616$	−3.00000	−0.120873
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	−16.0000	−0.643614
$619$	−37.0000	−1.48716	−0.743578	−	0.668649i	$-0.766873\pi$
$619$	−37.0000	−1.48716	−0.743578	+	0.668649i	$0.766873\pi$
$620$	3.00000	0.120483
$621$	−8.00000	−0.321029
$622$	−25.0000	−1.00241
$623$	−18.0000	−0.721155
$624$	0	0
$625$	1.00000	0.0400000
$626$	−26.0000	−1.03917
$627$	−36.0000	−1.43770
$628$	−3.00000	−0.119713
$629$	−3.00000	−0.119618
$630$	−1.00000	−0.0398410
$631$	−41.0000	−1.63218	−0.816092	−	0.577922i	$-0.803864\pi$
$631$	−41.0000	−1.63218	−0.816092	+	0.577922i	$0.803864\pi$
$632$	8.00000	0.318223
$633$	−6.00000	−0.238479
$634$	−21.0000	−0.834017
$635$	−4.00000	−0.158735
$636$	26.0000	1.03097
$637$	0	0
$638$	9.00000	0.356313
$639$	−2.00000	−0.0791188
$640$	−1.00000	−0.0395285
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	−4.00000	−0.157867
$643$	11.0000	0.433798	0.216899	−	0.976194i	$-0.430406\pi$
$643$	11.0000	0.433798	0.216899	+	0.976194i	$0.430406\pi$
$644$	2.00000	0.0788110
$645$	−2.00000	−0.0787499
$646$	18.0000	0.708201
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	11.0000	0.432121
$649$	0	0
$650$	0	0
$651$	6.00000	0.235159
$652$	−5.00000	−0.195815
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	6.00000	0.234619
$655$	−12.0000	−0.468879
$656$	3.00000	0.117130
$657$	0	0
$658$	−4.00000	−0.155936
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	6.00000	0.233550
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	−30.0000	−1.16598
$663$	0	0
$664$	4.00000	0.155230
$665$	−6.00000	−0.232670
$666$	1.00000	0.0387492
$667$	−6.00000	−0.232321
$668$	12.0000	0.464294
$669$	46.0000	1.77846
$670$	0	0
$671$	−45.0000	−1.73721
$672$	−2.00000	−0.0771517
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	−12.0000	−0.462223
$675$	−4.00000	−0.153960
$676$	−13.0000	−0.500000
$677$	34.0000	1.30673	0.653363	−	0.757045i	$-0.273358\pi$
$677$	34.0000	1.30673	0.653363	+	0.757045i	$0.273358\pi$
$678$	14.0000	0.537667
$679$	−7.00000	−0.268635
$680$	−3.00000	−0.115045
$681$	26.0000	0.996322
$682$	−9.00000	−0.344628
$683$	−31.0000	−1.18618	−0.593091	−	0.805135i	$-0.702093\pi$
$683$	−31.0000	−1.18618	−0.593091	+	0.805135i	$0.702093\pi$
$684$	−6.00000	−0.229416
$685$	8.00000	0.305664
$686$	13.0000	0.496342
$687$	−12.0000	−0.457829
$688$	−1.00000	−0.0381246
$689$	0	0
$690$	−4.00000	−0.152277
$691$	5.00000	0.190209	0.0951045	−	0.995467i	$-0.469681\pi$
$691$	5.00000	0.190209	0.0951045	+	0.995467i	$0.469681\pi$
$692$	9.00000	0.342129
$693$	3.00000	0.113961
$694$	28.0000	1.06287
$695$	3.00000	0.113796
$696$	6.00000	0.227429
$697$	9.00000	0.340899
$698$	22.0000	0.832712
$699$	8.00000	0.302588
$700$	1.00000	0.0377964
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	0	0
$703$	6.00000	0.226294
$704$	3.00000	0.113067
$705$	8.00000	0.301297
$706$	−19.0000	−0.715074
$707$	−10.0000	−0.376089
$708$	0	0
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	2.00000	0.0750587
$711$	−8.00000	−0.300023
$712$	18.0000	0.674579
$713$	6.00000	0.224702
$714$	−6.00000	−0.224544
$715$	0	0
$716$	24.0000	0.896922
$717$	18.0000	0.672222
$718$	−16.0000	−0.597115
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	1.00000	0.0372678
$721$	8.00000	0.297936
$722$	−17.0000	−0.632674
$723$	48.0000	1.78514
$724$	2.00000	0.0743294
$725$	−3.00000	−0.111417
$726$	4.00000	0.148454
$727$	52.0000	1.92857	0.964287	−	0.264861i	$-0.0853260\pi$
$727$	52.0000	1.92857	0.964287	+	0.264861i	$0.0853260\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−3.00000	−0.110959
$732$	−30.0000	−1.10883
$733$	−41.0000	−1.51437	−0.757185	−	0.653201i	$-0.773426\pi$
$733$	−41.0000	−1.51437	−0.757185	+	0.653201i	$0.773426\pi$
$734$	−13.0000	−0.479839
$735$	−12.0000	−0.442627
$736$	−2.00000	−0.0737210
$737$	0	0
$738$	−3.00000	−0.110432
$739$	27.0000	0.993211	0.496606	−	0.867976i	$-0.334580\pi$
$739$	27.0000	0.993211	0.496606	+	0.867976i	$0.334580\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	−13.0000	−0.477245
$743$	3.00000	0.110059	0.0550297	−	0.998485i	$-0.482475\pi$
$743$	3.00000	0.110059	0.0550297	+	0.998485i	$0.482475\pi$
$744$	−6.00000	−0.219971
$745$	18.0000	0.659469
$746$	−22.0000	−0.805477
$747$	−4.00000	−0.146352
$748$	9.00000	0.329073
$749$	2.00000	0.0730784
$750$	−2.00000	−0.0730297
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	4.00000	0.145865
$753$	12.0000	0.437304
$754$	0	0
$755$	16.0000	0.582300
$756$	−4.00000	−0.145479
$757$	−50.0000	−1.81728	−0.908640	−	0.417579i	$-0.862879\pi$
$757$	−50.0000	−1.81728	−0.908640	+	0.417579i	$0.862879\pi$
$758$	−16.0000	−0.581146
$759$	12.0000	0.435572
$760$	6.00000	0.217643
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	8.00000	0.289809
$763$	−3.00000	−0.108607
$764$	21.0000	0.759753
$765$	3.00000	0.108465
$766$	12.0000	0.433578
$767$	0	0
$768$	2.00000	0.0721688
$769$	8.00000	0.288487	0.144244	−	0.989542i	$-0.453925\pi$
$769$	8.00000	0.288487	0.144244	+	0.989542i	$0.453925\pi$
$770$	−3.00000	−0.108112
$771$	−44.0000	−1.58462
$772$	−10.0000	−0.359908
$773$	−41.0000	−1.47467	−0.737334	−	0.675529i	$-0.763915\pi$
$773$	−41.0000	−1.47467	−0.737334	+	0.675529i	$0.763915\pi$
$774$	1.00000	0.0359443
$775$	3.00000	0.107763
$776$	7.00000	0.251285
$777$	−2.00000	−0.0717496
$778$	−15.0000	−0.537776
$779$	−18.0000	−0.644917
$780$	0	0
$781$	−6.00000	−0.214697
$782$	−6.00000	−0.214560
$783$	12.0000	0.428845
$784$	−6.00000	−0.214286
$785$	−3.00000	−0.107075
$786$	24.0000	0.856052
$787$	−10.0000	−0.356462	−0.178231	−	0.983989i	$-0.557037\pi$
$787$	−10.0000	−0.356462	−0.178231	+	0.983989i	$0.557037\pi$
$788$	−18.0000	−0.641223
$789$	−2.00000	−0.0712019
$790$	8.00000	0.284627
$791$	−7.00000	−0.248891
$792$	−3.00000	−0.106600
$793$	0	0
$794$	−18.0000	−0.638796
$795$	26.0000	0.922125
$796$	8.00000	0.283552
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	12.0000	0.424795
$799$	12.0000	0.424529
$800$	−1.00000	−0.0353553
$801$	−18.0000	−0.635999
$802$	−34.0000	−1.20058
$803$	0	0
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	−20.0000	−0.704033
$808$	10.0000	0.351799
$809$	34.0000	1.19538	0.597688	−	0.801729i	$-0.296086\pi$
$809$	34.0000	1.19538	0.597688	+	0.801729i	$0.296086\pi$
$810$	11.0000	0.386501
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	−3.00000	−0.105279
$813$	−20.0000	−0.701431
$814$	3.00000	0.105150
$815$	−5.00000	−0.175142
$816$	6.00000	0.210042
$817$	6.00000	0.209913
$818$	0	0
$819$	0	0
$820$	3.00000	0.104765
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	−16.0000	−0.558064
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	−8.00000	−0.278693
$825$	6.00000	0.208893
$826$	0	0
$827$	−11.0000	−0.382507	−0.191254	−	0.981541i	$-0.561255\pi$
$827$	−11.0000	−0.382507	−0.191254	+	0.981541i	$0.561255\pi$
$828$	2.00000	0.0695048
$829$	−39.0000	−1.35453	−0.677263	−	0.735741i	$-0.736834\pi$
$829$	−39.0000	−1.35453	−0.677263	+	0.735741i	$0.736834\pi$
$830$	4.00000	0.138842
$831$	−40.0000	−1.38758
$832$	0	0
$833$	−18.0000	−0.623663
$834$	−6.00000	−0.207763
$835$	12.0000	0.415277
$836$	−18.0000	−0.622543
$837$	−12.0000	−0.414781
$838$	−12.0000	−0.414533
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	−2.00000	−0.0690066
$841$	−20.0000	−0.689655
$842$	−34.0000	−1.17172
$843$	−48.0000	−1.65321
$844$	−3.00000	−0.103264
$845$	−13.0000	−0.447214
$846$	−4.00000	−0.137523
$847$	−2.00000	−0.0687208
$848$	13.0000	0.446422
$849$	−40.0000	−1.37280
$850$	−3.00000	−0.102899
$851$	−2.00000	−0.0685591
$852$	−4.00000	−0.137038
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	15.0000	0.513289
$855$	−6.00000	−0.205196
$856$	−2.00000	−0.0683586
$857$	33.0000	1.12726	0.563629	−	0.826028i	$-0.309405\pi$
$857$	33.0000	1.12726	0.563629	+	0.826028i	$0.309405\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	−1.00000	−0.0340997
$861$	6.00000	0.204479
$862$	31.0000	1.05586
$863$	33.0000	1.12333	0.561667	−	0.827364i	$-0.310160\pi$
$863$	33.0000	1.12333	0.561667	+	0.827364i	$0.310160\pi$
$864$	4.00000	0.136083
$865$	9.00000	0.306009
$866$	22.0000	0.747590
$867$	−16.0000	−0.543388
$868$	3.00000	0.101827
$869$	−24.0000	−0.814144
$870$	6.00000	0.203419
$871$	0	0
$872$	3.00000	0.101593
$873$	−7.00000	−0.236914
$874$	12.0000	0.405906
$875$	1.00000	0.0338062
$876$	0	0
$877$	−43.0000	−1.45201	−0.726003	−	0.687691i	$-0.758624\pi$
$877$	−43.0000	−1.45201	−0.726003	+	0.687691i	$0.758624\pi$
$878$	−37.0000	−1.24869
$879$	6.00000	0.202375
$880$	3.00000	0.101130
$881$	−9.00000	−0.303218	−0.151609	−	0.988441i	$-0.548445\pi$
$881$	−9.00000	−0.303218	−0.151609	+	0.988441i	$0.548445\pi$
$882$	6.00000	0.202031
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	0	0
$885$	0	0
$886$	34.0000	1.14225
$887$	−33.0000	−1.10803	−0.554016	−	0.832506i	$-0.686905\pi$
$887$	−33.0000	−1.10803	−0.554016	+	0.832506i	$0.686905\pi$
$888$	2.00000	0.0671156
$889$	−4.00000	−0.134156
$890$	18.0000	0.603361
$891$	−33.0000	−1.10554
$892$	23.0000	0.770097
$893$	−24.0000	−0.803129
$894$	−36.0000	−1.20402
$895$	24.0000	0.802232
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	10.0000	0.333704
$899$	−9.00000	−0.300167
$900$	1.00000	0.0333333
$901$	39.0000	1.29928
$902$	−9.00000	−0.299667
$903$	−2.00000	−0.0665558
$904$	7.00000	0.232817
$905$	2.00000	0.0664822
$906$	−32.0000	−1.06313
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	13.0000	0.431420
$909$	−10.0000	−0.331679
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	−12.0000	−0.397360
$913$	−12.0000	−0.397142
$914$	−9.00000	−0.297694
$915$	−30.0000	−0.991769
$916$	−6.00000	−0.198246
$917$	−12.0000	−0.396275
$918$	12.0000	0.396059
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	−2.00000	−0.0659380
$921$	68.0000	2.24068
$922$	1.00000	0.0329332
$923$	0	0
$924$	6.00000	0.197386
$925$	−1.00000	−0.0328798
$926$	−12.0000	−0.394344
$927$	8.00000	0.262754
$928$	3.00000	0.0984798
$929$	−3.00000	−0.0984268	−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000	−0.0984268	−0.0492134	+	0.998788i	$0.515671\pi$
$930$	−6.00000	−0.196748
$931$	36.0000	1.17985
$932$	4.00000	0.131024
$933$	50.0000	1.63693
$934$	37.0000	1.21068
$935$	9.00000	0.294331
$936$	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
$939$	52.0000	1.69696
$940$	4.00000	0.130466
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	6.00000	0.195491
$943$	6.00000	0.195387
$944$	0	0
$945$	−4.00000	−0.130120
$946$	3.00000	0.0975384
$947$	9.00000	0.292461	0.146230	−	0.989251i	$-0.453286\pi$
$947$	9.00000	0.292461	0.146230	+	0.989251i	$0.453286\pi$
$948$	−16.0000	−0.519656
$949$	0	0
$950$	6.00000	0.194666
$951$	42.0000	1.36194
$952$	−3.00000	−0.0972306
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	−13.0000	−0.420891
$955$	21.0000	0.679544
$956$	9.00000	0.291081
$957$	−18.0000	−0.581857
$958$	8.00000	0.258468
$959$	8.00000	0.258333
$960$	2.00000	0.0645497
$961$	−22.0000	−0.709677
$962$	0	0
$963$	2.00000	0.0644491
$964$	24.0000	0.772988
$965$	−10.0000	−0.321911
$966$	−4.00000	−0.128698
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	2.00000	0.0642824
$969$	−36.0000	−1.15649
$970$	7.00000	0.224756
$971$	9.00000	0.288824	0.144412	−	0.989518i	$-0.453871\pi$
$971$	9.00000	0.288824	0.144412	+	0.989518i	$0.453871\pi$
$972$	−10.0000	−0.320750
$973$	3.00000	0.0961756
$974$	−34.0000	−1.08943
$975$	0	0
$976$	−15.0000	−0.480138
$977$	35.0000	1.11975	0.559875	−	0.828577i	$-0.310849\pi$
$977$	35.0000	1.11975	0.559875	+	0.828577i	$0.310849\pi$
$978$	10.0000	0.319765
$979$	−54.0000	−1.72585
$980$	−6.00000	−0.191663
$981$	−3.00000	−0.0957826
$982$	20.0000	0.638226
$983$	−27.0000	−0.861166	−0.430583	−	0.902551i	$-0.641692\pi$
$983$	−27.0000	−0.861166	−0.430583	+	0.902551i	$0.641692\pi$
$984$	−6.00000	−0.191273
$985$	−18.0000	−0.573528
$986$	9.00000	0.286618
$987$	8.00000	0.254643
$988$	0	0
$989$	−2.00000	−0.0635963
$990$	−3.00000	−0.0953463
$991$	33.0000	1.04828	0.524140	−	0.851632i	$-0.324387\pi$
$991$	33.0000	1.04828	0.524140	+	0.851632i	$0.324387\pi$
$992$	−3.00000	−0.0952501
$993$	60.0000	1.90404
$994$	2.00000	0.0634361
$995$	8.00000	0.253617
$996$	−8.00000	−0.253490
$997$	−12.0000	−0.380044	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	−12.0000	−0.380044	−0.190022	+	0.981780i	$0.560856\pi$
$998$	22.0000	0.696398
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.a.c.1.1	✓	1
3.2	odd	2		3330.2.a.p.1.1		1
4.3	odd	2		2960.2.a.c.1.1		1
5.2	odd	4		1850.2.b.c.149.1		2
5.3	odd	4		1850.2.b.c.149.2		2
5.4	even	2		1850.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.c.1.1	✓	1	1.1	even	1	trivial
1850.2.a.i.1.1		1	5.4	even	2
1850.2.b.c.149.1		2	5.2	odd	4
1850.2.b.c.149.2		2	5.3	odd	4
2960.2.a.c.1.1		1	4.3	odd	2
3330.2.a.p.1.1		1	3.2	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:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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