LMFDB - Embedded newform 2310.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2310,2,Mod(1,2310)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2310, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2310.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2310.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:	$18.4454428669$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2310.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} +1.00000 q^{21} +1.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} -1.00000 q^{32} +1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} +2.00000 q^{39} +1.00000 q^{40} +6.00000 q^{41} -1.00000 q^{42} -8.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} -8.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +1.00000 q^{55} +1.00000 q^{56} -2.00000 q^{58} +4.00000 q^{59} +1.00000 q^{60} -6.00000 q^{61} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -1.00000 q^{66} +8.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} -1.00000 q^{70} +12.0000 q^{71} -1.00000 q^{72} -6.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} +1.00000 q^{77} -2.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} +1.00000 q^{84} -2.00000 q^{85} +8.00000 q^{86} -2.00000 q^{87} +1.00000 q^{88} -2.00000 q^{89} +1.00000 q^{90} +2.00000 q^{91} +8.00000 q^{92} +4.00000 q^{94} +1.00000 q^{96} -14.0000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	1.00000	0.267261
$15$	1.00000	0.258199
$16$	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
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	1.00000	0.218218
$22$	1.00000	0.213201
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−1.00000	−0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	−2.00000	−0.342997
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	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	$0.344787\pi$
$42$	−1.00000	−0.154303
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−1.00000	−0.150756
$45$	−1.00000	−0.149071
$46$	−8.00000	−1.17954
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	−2.00000	−0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	1.00000	0.134840
$56$	1.00000	0.133631
$57$	0	0
$58$	−2.00000	−0.262613
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	1.00000	0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−1.00000	−0.123091
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	2.00000	0.242536
$69$	−8.00000	−0.963087
$70$	−1.00000	−0.119523
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−1.00000	−0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	6.00000	0.697486
$75$	−1.00000	−0.115470
$76$	0	0
$77$	1.00000	0.113961
$78$	−2.00000	−0.226455
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	1.00000	0.109109
$85$	−2.00000	−0.216930
$86$	8.00000	0.862662
$87$	−2.00000	−0.214423
$88$	1.00000	0.106600
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	1.00000	0.105409
$91$	2.00000	0.209657
$92$	8.00000	0.834058
$93$	0	0
$94$	4.00000	0.412568
$95$	0	0
$96$	1.00000	0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−1.00000	−0.101015
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	−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
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	2.00000	0.196116
$105$	−1.00000	−0.0975900
$106$	6.00000	0.582772
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	−1.00000	−0.0962250
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−1.00000	−0.0953463
$111$	6.00000	0.569495
$112$	−1.00000	−0.0944911
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	2.00000	0.185695
$117$	−2.00000	−0.184900
$118$	−4.00000	−0.368230
$119$	−2.00000	−0.183340
$120$	−1.00000	−0.0912871
$121$	1.00000	0.0909091
$122$	6.00000	0.543214
$123$	−6.00000	−0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	1.00000	0.0890871
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−1.00000	−0.0883883
$129$	8.00000	0.704361
$130$	−2.00000	−0.175412
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−8.00000	−0.691095
$135$	1.00000	0.0860663
$136$	−2.00000	−0.171499
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	8.00000	0.681005
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	1.00000	0.0845154
$141$	4.00000	0.336861
$142$	−12.0000	−1.00702
$143$	2.00000	0.167248
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	6.00000	0.496564
$147$	−1.00000	−0.0824786
$148$	−6.00000	−0.493197
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	1.00000	0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	2.00000	0.160128
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	1.00000	0.0790569
$161$	−8.00000	−0.630488
$162$	−1.00000	−0.0785674
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	6.00000	0.468521
$165$	−1.00000	−0.0778499
$166$	4.00000	0.310460
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	2.00000	0.153393
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	2.00000	0.151620
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	−4.00000	−0.300658
$178$	2.00000	0.149906
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−1.00000	−0.0745356
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	−2.00000	−0.148250
$183$	6.00000	0.443533
$184$	−8.00000	−0.589768
$185$	6.00000	0.441129
$186$	0	0
$187$	−2.00000	−0.146254
$188$	−4.00000	−0.291730
$189$	1.00000	0.0727393
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−1.00000	−0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	14.0000	1.00514
$195$	−2.00000	−0.143223
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	1.00000	0.0710669
$199$	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
$202$	6.00000	0.422159
$203$	−2.00000	−0.140372
$204$	−2.00000	−0.140028
$205$	−6.00000	−0.419058
$206$	8.00000	0.557386
$207$	8.00000	0.556038
$208$	−2.00000	−0.138675
$209$	0	0
$210$	1.00000	0.0690066
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−6.00000	−0.412082
$213$	−12.0000	−0.822226
$214$	4.00000	0.273434
$215$	8.00000	0.545595
$216$	1.00000	0.0680414
$217$	0	0
$218$	−6.00000	−0.406371
$219$	6.00000	0.405442
$220$	1.00000	0.0674200
$221$	−4.00000	−0.269069
$222$	−6.00000	−0.402694
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	2.00000	0.133038
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	8.00000	0.527504
$231$	−1.00000	−0.0657952
$232$	−2.00000	−0.131306
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	2.00000	0.130744
$235$	4.00000	0.260931
$236$	4.00000	0.260378
$237$	0	0
$238$	2.00000	0.129641
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	1.00000	0.0645497
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	−1.00000	−0.0638877
$246$	6.00000	0.382546
$247$	0	0
$248$	0	0
$249$	4.00000	0.253490
$250$	1.00000	0.0632456
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	−1.00000	−0.0629941
$253$	−8.00000	−0.502956
$254$	16.0000	1.00393
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	−8.00000	−0.498058
$259$	6.00000	0.372822
$260$	2.00000	0.124035
$261$	2.00000	0.123797
$262$	20.0000	1.23560
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	−1.00000	−0.0615457
$265$	6.00000	0.368577
$266$	0	0
$267$	2.00000	0.122398
$268$	8.00000	0.488678
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	−1.00000	−0.0608581
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	2.00000	0.121268
$273$	−2.00000	−0.121046
$274$	−14.0000	−0.845771
$275$	−1.00000	−0.0603023
$276$	−8.00000	−0.481543
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−16.0000	−0.959616
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−4.00000	−0.238197
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−2.00000	−0.118262
$287$	−6.00000	−0.354169
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	2.00000	0.117444
$291$	14.0000	0.820695
$292$	−6.00000	−0.351123
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	1.00000	0.0583212
$295$	−4.00000	−0.232889
$296$	6.00000	0.348743
$297$	1.00000	0.0580259
$298$	14.0000	0.810998
$299$	−16.0000	−0.925304
$300$	−1.00000	−0.0577350
$301$	8.00000	0.461112
$302$	−8.00000	−0.460348
$303$	6.00000	0.344691
$304$	0	0
$305$	6.00000	0.343559
$306$	−2.00000	−0.114332
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	1.00000	0.0569803
$309$	8.00000	0.455104
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	−2.00000	−0.113228
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	18.0000	1.01580
$315$	1.00000	0.0563436
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−6.00000	−0.336463
$319$	−2.00000	−0.111979
$320$	−1.00000	−0.0559017
$321$	4.00000	0.223258
$322$	8.00000	0.445823
$323$	0	0
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	24.0000	1.32924
$327$	−6.00000	−0.331801
$328$	−6.00000	−0.331295
$329$	4.00000	0.220527
$330$	1.00000	0.0550482
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−4.00000	−0.219529
$333$	−6.00000	−0.328798
$334$	12.0000	0.656611
$335$	−8.00000	−0.437087
$336$	1.00000	0.0545545
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	9.00000	0.489535
$339$	2.00000	0.108625
$340$	−2.00000	−0.108465
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	8.00000	0.431331
$345$	8.00000	0.430706
$346$	6.00000	0.322562
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	−2.00000	−0.107211
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	1.00000	0.0534522
$351$	2.00000	0.106752
$352$	1.00000	0.0533002
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	4.00000	0.212598
$355$	−12.0000	−0.636894
$356$	−2.00000	−0.106000
$357$	2.00000	0.105851
$358$	−20.0000	−1.05703
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	1.00000	0.0527046
$361$	−19.0000	−1.00000
$362$	14.0000	0.735824
$363$	−1.00000	−0.0524864
$364$	2.00000	0.104828
$365$	6.00000	0.314054
$366$	−6.00000	−0.313625
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	8.00000	0.417029
$369$	6.00000	0.312348
$370$	−6.00000	−0.311925
$371$	6.00000	0.311504
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	2.00000	0.103418
$375$	1.00000	0.0516398
$376$	4.00000	0.206284
$377$	−4.00000	−0.206010
$378$	−1.00000	−0.0514344
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	12.0000	0.613973
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	1.00000	0.0510310
$385$	−1.00000	−0.0509647
$386$	−2.00000	−0.101797
$387$	−8.00000	−0.406663
$388$	−14.0000	−0.710742
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	2.00000	0.101274
$391$	16.0000	0.809155
$392$	−1.00000	−0.0505076
$393$	20.0000	1.00887
$394$	6.00000	0.302276
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	8.00000	0.399004
$403$	0	0
$404$	−6.00000	−0.298511
$405$	−1.00000	−0.0496904
$406$	2.00000	0.0992583
$407$	6.00000	0.297409
$408$	2.00000	0.0990148
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	6.00000	0.296319
$411$	−14.0000	−0.690569
$412$	−8.00000	−0.394132
$413$	−4.00000	−0.196827
$414$	−8.00000	−0.393179
$415$	4.00000	0.196352
$416$	2.00000	0.0980581
$417$	−16.0000	−0.783523
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	−1.00000	−0.0487950
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	20.0000	0.973585
$423$	−4.00000	−0.194487
$424$	6.00000	0.291386
$425$	2.00000	0.0970143
$426$	12.0000	0.581402
$427$	6.00000	0.290360
$428$	−4.00000	−0.193347
$429$	−2.00000	−0.0965609
$430$	−8.00000	−0.385794
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	−1.00000	−0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	6.00000	0.287348
$437$	0	0
$438$	−6.00000	−0.286691
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	−1.00000	−0.0476731
$441$	1.00000	0.0476190
$442$	4.00000	0.190261
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	6.00000	0.284747
$445$	2.00000	0.0948091
$446$	−16.0000	−0.757622
$447$	14.0000	0.662177
$448$	−1.00000	−0.0472456
$449$	−38.0000	−1.79333	−0.896665	−	0.442709i	$-0.854018\pi$
$449$	−38.0000	−1.79333	−0.896665	+	0.442709i	$0.854018\pi$
$450$	−1.00000	−0.0471405
$451$	−6.00000	−0.282529
$452$	−2.00000	−0.0940721
$453$	−8.00000	−0.375873
$454$	−4.00000	−0.187729
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	6.00000	0.280362
$459$	−2.00000	−0.0933520
$460$	−8.00000	−0.373002
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	1.00000	0.0465242
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	2.00000	0.0926482
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	−2.00000	−0.0924500
$469$	−8.00000	−0.369406
$470$	−4.00000	−0.184506
$471$	18.0000	0.829396
$472$	−4.00000	−0.184115
$473$	8.00000	0.367840
$474$	0	0
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	−6.00000	−0.274721
$478$	−4.00000	−0.182956
$479$	40.0000	1.82765	0.913823	−	0.406112i	$-0.133116\pi$
$479$	40.0000	1.82765	0.913823	+	0.406112i	$0.133116\pi$
$480$	−1.00000	−0.0456435
$481$	12.0000	0.547153
$482$	22.0000	1.00207
$483$	8.00000	0.364013
$484$	1.00000	0.0454545
$485$	14.0000	0.635707
$486$	1.00000	0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	6.00000	0.271607
$489$	24.0000	1.08532
$490$	1.00000	0.0451754
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−6.00000	−0.270501
$493$	4.00000	0.180151
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	−12.0000	−0.538274
$498$	−4.00000	−0.179244
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	−1.00000	−0.0447214
$501$	12.0000	0.536120
$502$	4.00000	0.178529
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	1.00000	0.0445435
$505$	6.00000	0.266996
$506$	8.00000	0.355643
$507$	9.00000	0.399704
$508$	−16.0000	−0.709885
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	−2.00000	−0.0885615
$511$	6.00000	0.265424
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	8.00000	0.352522
$516$	8.00000	0.352180
$517$	4.00000	0.175920
$518$	−6.00000	−0.263625
$519$	6.00000	0.263371
$520$	−2.00000	−0.0877058
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	−2.00000	−0.0875376
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−20.0000	−0.873704
$525$	1.00000	0.0436436
$526$	−16.0000	−0.697633
$527$	0	0
$528$	1.00000	0.0435194
$529$	41.0000	1.78261
$530$	−6.00000	−0.260623
$531$	4.00000	0.173585
$532$	0	0
$533$	−12.0000	−0.519778
$534$	−2.00000	−0.0865485
$535$	4.00000	0.172935
$536$	−8.00000	−0.345547
$537$	−20.0000	−0.863064
$538$	30.0000	1.29339
$539$	−1.00000	−0.0430730
$540$	1.00000	0.0430331
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	16.0000	0.687259
$543$	14.0000	0.600798
$544$	−2.00000	−0.0857493
$545$	−6.00000	−0.257012
$546$	2.00000	0.0855921
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	14.0000	0.598050
$549$	−6.00000	−0.256074
$550$	1.00000	0.0426401
$551$	0	0
$552$	8.00000	0.340503
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	−6.00000	−0.254686
$556$	16.0000	0.678551
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	1.00000	0.0422577
$561$	2.00000	0.0844401
$562$	6.00000	0.253095
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	4.00000	0.168430
$565$	2.00000	0.0841406
$566$	4.00000	0.168133
$567$	−1.00000	−0.0419961
$568$	−12.0000	−0.503509
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	2.00000	0.0836242
$573$	12.0000	0.501307
$574$	6.00000	0.250435
$575$	8.00000	0.333623
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	13.0000	0.540729
$579$	−2.00000	−0.0831172
$580$	−2.00000	−0.0830455
$581$	4.00000	0.165948
$582$	−14.0000	−0.580319
$583$	6.00000	0.248495
$584$	6.00000	0.248282
$585$	2.00000	0.0826898
$586$	6.00000	0.247858
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	−1.00000	−0.0412393
$589$	0	0
$590$	4.00000	0.164677
$591$	6.00000	0.246807
$592$	−6.00000	−0.246598
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	−1.00000	−0.0410305
$595$	2.00000	0.0819920
$596$	−14.0000	−0.573462
$597$	−16.0000	−0.654836
$598$	16.0000	0.654289
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	1.00000	0.0408248
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	−8.00000	−0.326056
$603$	8.00000	0.325785
$604$	8.00000	0.325515
$605$	−1.00000	−0.0406558
$606$	−6.00000	−0.243733
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	−6.00000	−0.242933
$611$	8.00000	0.323645
$612$	2.00000	0.0808452
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	−4.00000	−0.161427
$615$	6.00000	0.241943
$616$	−1.00000	−0.0402911
$617$	38.0000	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$617$	38.0000	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$618$	−8.00000	−0.321807
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	16.0000	0.641542
$623$	2.00000	0.0801283
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	0	0
$628$	−18.0000	−0.718278
$629$	−12.0000	−0.478471
$630$	−1.00000	−0.0398410
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	−18.0000	−0.714871
$635$	16.0000	0.634941
$636$	6.00000	0.237915
$637$	−2.00000	−0.0792429
$638$	2.00000	0.0791808
$639$	12.0000	0.474713
$640$	1.00000	0.0395285
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	−4.00000	−0.157867
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	−8.00000	−0.315244
$645$	−8.00000	−0.315000
$646$	0	0
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	−1.00000	−0.0392837
$649$	−4.00000	−0.157014
$650$	2.00000	0.0784465
$651$	0	0
$652$	−24.0000	−0.939913
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	6.00000	0.234619
$655$	20.0000	0.781465
$656$	6.00000	0.234261
$657$	−6.00000	−0.234082
$658$	−4.00000	−0.155936
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	−1.00000	−0.0389249
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	−4.00000	−0.155464
$663$	4.00000	0.155347
$664$	4.00000	0.155230
$665$	0	0
$666$	6.00000	0.232495
$667$	16.0000	0.619522
$668$	−12.0000	−0.464294
$669$	−16.0000	−0.618596
$670$	8.00000	0.309067
$671$	6.00000	0.231627
$672$	−1.00000	−0.0385758
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	14.0000	0.539260
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	−2.00000	−0.0768095
$679$	14.0000	0.537271
$680$	2.00000	0.0766965
$681$	−4.00000	−0.153280
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	−14.0000	−0.534913
$686$	1.00000	0.0381802
$687$	6.00000	0.228914
$688$	−8.00000	−0.304997
$689$	12.0000	0.457164
$690$	−8.00000	−0.304555
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	−6.00000	−0.228086
$693$	1.00000	0.0379869
$694$	−4.00000	−0.151838
$695$	−16.0000	−0.606915
$696$	2.00000	0.0758098
$697$	12.0000	0.454532
$698$	−2.00000	−0.0757011
$699$	2.00000	0.0756469
$700$	−1.00000	−0.0377964
$701$	−46.0000	−1.73740	−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−46.0000	−1.73740	−0.868698	+	0.495342i	$0.835043\pi$
$702$	−2.00000	−0.0754851
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	−4.00000	−0.150649
$706$	−10.0000	−0.376355
$707$	6.00000	0.225653
$708$	−4.00000	−0.150329
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	12.0000	0.450352
$711$	0	0
$712$	2.00000	0.0749532
$713$	0	0
$714$	−2.00000	−0.0748481
$715$	−2.00000	−0.0747958
$716$	20.0000	0.747435
$717$	−4.00000	−0.149383
$718$	4.00000	0.149279
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	−1.00000	−0.0372678
$721$	8.00000	0.297936
$722$	19.0000	0.707107
$723$	22.0000	0.818189
$724$	−14.0000	−0.520306
$725$	2.00000	0.0742781
$726$	1.00000	0.0371135
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	−2.00000	−0.0741249
$729$	1.00000	0.0370370
$730$	−6.00000	−0.222070
$731$	−16.0000	−0.591781
$732$	6.00000	0.221766
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−8.00000	−0.295285
$735$	1.00000	0.0368856
$736$	−8.00000	−0.294884
$737$	−8.00000	−0.294684
$738$	−6.00000	−0.220863
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	6.00000	0.220564
$741$	0	0
$742$	−6.00000	−0.220267
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	14.0000	0.512920
$746$	−10.0000	−0.366126
$747$	−4.00000	−0.146352
$748$	−2.00000	−0.0731272
$749$	4.00000	0.146157
$750$	−1.00000	−0.0365148
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	−4.00000	−0.145865
$753$	4.00000	0.145768
$754$	4.00000	0.145671
$755$	−8.00000	−0.291150
$756$	1.00000	0.0363696
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	−20.0000	−0.726433
$759$	8.00000	0.290382
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	−16.0000	−0.579619
$763$	−6.00000	−0.217215
$764$	−12.0000	−0.434145
$765$	−2.00000	−0.0723102
$766$	4.00000	0.144526
$767$	−8.00000	−0.288863
$768$	−1.00000	−0.0360844
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	1.00000	0.0360375
$771$	6.00000	0.216085
$772$	2.00000	0.0719816
$773$	−46.0000	−1.65451	−0.827253	−	0.561830i	$-0.810097\pi$
$773$	−46.0000	−1.65451	−0.827253	+	0.561830i	$0.810097\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	14.0000	0.502571
$777$	−6.00000	−0.215249
$778$	−26.0000	−0.932145
$779$	0	0
$780$	−2.00000	−0.0716115
$781$	−12.0000	−0.429394
$782$	−16.0000	−0.572159
$783$	−2.00000	−0.0714742
$784$	1.00000	0.0357143
$785$	18.0000	0.642448
$786$	−20.0000	−0.713376
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−6.00000	−0.213741
$789$	−16.0000	−0.569615
$790$	0	0
$791$	2.00000	0.0711118
$792$	1.00000	0.0355335
$793$	12.0000	0.426132
$794$	10.0000	0.354887
$795$	−6.00000	−0.212798
$796$	16.0000	0.567105
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	−1.00000	−0.0353553
$801$	−2.00000	−0.0706665
$802$	−2.00000	−0.0706225
$803$	6.00000	0.211735
$804$	−8.00000	−0.282138
$805$	8.00000	0.281963
$806$	0	0
$807$	30.0000	1.05605
$808$	6.00000	0.211079
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	1.00000	0.0351364
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	−2.00000	−0.0701862
$813$	16.0000	0.561144
$814$	−6.00000	−0.210300
$815$	24.0000	0.840683
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	−18.0000	−0.629355
$819$	2.00000	0.0698857
$820$	−6.00000	−0.209529
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	14.0000	0.488306
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	8.00000	0.278693
$825$	1.00000	0.0348155
$826$	4.00000	0.139178
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	8.00000	0.278019
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	−4.00000	−0.138842
$831$	−2.00000	−0.0693792
$832$	−2.00000	−0.0693375
$833$	2.00000	0.0692959
$834$	16.0000	0.554035
$835$	12.0000	0.415277
$836$	0	0
$837$	0	0
$838$	−28.0000	−0.967244
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	1.00000	0.0345033
$841$	−25.0000	−0.862069
$842$	−6.00000	−0.206774
$843$	6.00000	0.206651
$844$	−20.0000	−0.688428
$845$	9.00000	0.309609
$846$	4.00000	0.137523
$847$	−1.00000	−0.0343604
$848$	−6.00000	−0.206041
$849$	4.00000	0.137280
$850$	−2.00000	−0.0685994
$851$	−48.0000	−1.64542
$852$	−12.0000	−0.411113
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	4.00000	0.136717
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	2.00000	0.0682789
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	8.00000	0.272798
$861$	6.00000	0.204479
$862$	4.00000	0.136241
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	1.00000	0.0340207
$865$	6.00000	0.204006
$866$	−2.00000	−0.0679628
$867$	13.0000	0.441503
$868$	0	0
$869$	0	0
$870$	−2.00000	−0.0678064
$871$	−16.0000	−0.542139
$872$	−6.00000	−0.203186
$873$	−14.0000	−0.473828
$874$	0	0
$875$	1.00000	0.0338062
$876$	6.00000	0.202721
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	8.00000	0.269987
$879$	6.00000	0.202375
$880$	1.00000	0.0337100
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	−1.00000	−0.0336718
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	−4.00000	−0.134535
$885$	4.00000	0.134459
$886$	−12.0000	−0.403148
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	−6.00000	−0.201347
$889$	16.0000	0.536623
$890$	−2.00000	−0.0670402
$891$	−1.00000	−0.0335013
$892$	16.0000	0.535720
$893$	0	0
$894$	−14.0000	−0.468230
$895$	−20.0000	−0.668526
$896$	1.00000	0.0334077
$897$	16.0000	0.534224
$898$	38.0000	1.26808
$899$	0	0
$900$	1.00000	0.0333333
$901$	−12.0000	−0.399778
$902$	6.00000	0.199778
$903$	−8.00000	−0.266223
$904$	2.00000	0.0665190
$905$	14.0000	0.465376
$906$	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	$0.678286\pi$
$908$	4.00000	0.132745
$909$	−6.00000	−0.199007
$910$	2.00000	0.0662994
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	−18.0000	−0.595387
$915$	−6.00000	−0.198354
$916$	−6.00000	−0.198246
$917$	20.0000	0.660458
$918$	2.00000	0.0660098
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	8.00000	0.263752
$921$	−4.00000	−0.131804
$922$	30.0000	0.987997
$923$	−24.0000	−0.789970
$924$	−1.00000	−0.0328976
$925$	−6.00000	−0.197279
$926$	8.00000	0.262896
$927$	−8.00000	−0.262754
$928$	−2.00000	−0.0656532
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	−2.00000	−0.0655122
$933$	16.0000	0.523816
$934$	20.0000	0.654420
$935$	2.00000	0.0654070
$936$	2.00000	0.0653720
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	8.00000	0.261209
$939$	6.00000	0.195803
$940$	4.00000	0.130466
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	−18.0000	−0.586472
$943$	48.0000	1.56310
$944$	4.00000	0.130189
$945$	−1.00000	−0.0325300
$946$	−8.00000	−0.260102
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	−18.0000	−0.583690
$952$	2.00000	0.0648204
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	6.00000	0.194257
$955$	12.0000	0.388311
$956$	4.00000	0.129369
$957$	2.00000	0.0646508
$958$	−40.0000	−1.29234
$959$	−14.0000	−0.452084
$960$	1.00000	0.0322749
$961$	−31.0000	−1.00000
$962$	−12.0000	−0.386896
$963$	−4.00000	−0.128898
$964$	−22.0000	−0.708572
$965$	−2.00000	−0.0643823
$966$	−8.00000	−0.257396
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−14.0000	−0.449513
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	−1.00000	−0.0320750
$973$	−16.0000	−0.512936
$974$	−8.00000	−0.256337
$975$	2.00000	0.0640513
$976$	−6.00000	−0.192055
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	−24.0000	−0.767435
$979$	2.00000	0.0639203
$980$	−1.00000	−0.0319438
$981$	6.00000	0.191565
$982$	−12.0000	−0.382935
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	6.00000	0.191273
$985$	6.00000	0.191176
$986$	−4.00000	−0.127386
$987$	−4.00000	−0.127321
$988$	0	0
$989$	−64.0000	−2.03508
$990$	−1.00000	−0.0317821
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	12.0000	0.380617
$995$	−16.0000	−0.507234
$996$	4.00000	0.126745
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	−28.0000	−0.886325
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2310.2.a.a.1.1	✓	1
3.2	odd	2		6930.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2310.2.a.a.1.1	✓	1	1.1	even	1	trivial
6930.2.a.bd.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:	\( N \)	\(=\)	\( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2310.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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