LMFDB - Embedded newform 678.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [678,2,Mod(1,678)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(678, 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("678.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 678 = 2 \cdot 3 \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	678.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:	$5.41385725704$
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	$\chi$	$=$	678.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} +7.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -2.00000 q^{22} +3.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} +7.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} -3.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} -3.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -4.00000 q^{37} +6.00000 q^{38} +7.00000 q^{39} -1.00000 q^{40} +1.00000 q^{42} +2.00000 q^{43} -2.00000 q^{44} -1.00000 q^{45} +3.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -4.00000 q^{50} -3.00000 q^{51} +7.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} +1.00000 q^{56} +6.00000 q^{57} +2.00000 q^{58} -3.00000 q^{59} -1.00000 q^{60} -1.00000 q^{61} -3.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -7.00000 q^{65} -2.00000 q^{66} -2.00000 q^{67} -3.00000 q^{68} +3.00000 q^{69} -1.00000 q^{70} -5.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} -4.00000 q^{74} -4.00000 q^{75} +6.00000 q^{76} -2.00000 q^{77} +7.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +1.00000 q^{84} +3.00000 q^{85} +2.00000 q^{86} +2.00000 q^{87} -2.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} +7.00000 q^{91} +3.00000 q^{92} -3.00000 q^{93} -8.00000 q^{94} -6.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} -6.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	1.00000	0.408248
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	1.00000	0.288675
$13$	7.00000	1.94145	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	7.00000	1.94145	0.970725	+	0.240192i	$0.0772105\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	1.00000	0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−1.00000	−0.223607
$21$	1.00000	0.218218
$22$	−2.00000	−0.426401
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	7.00000	1.37281
$27$	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
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	1.00000	0.176777
$33$	−2.00000	−0.348155
$34$	−3.00000	−0.514496
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	6.00000	0.973329
$39$	7.00000	1.12090
$40$	−1.00000	−0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	1.00000	0.154303
$43$	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
$46$	3.00000	0.442326
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	−4.00000	−0.565685
$51$	−3.00000	−0.420084
$52$	7.00000	0.970725
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	2.00000	0.269680
$56$	1.00000	0.133631
$57$	6.00000	0.794719
$58$	2.00000	0.262613
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	−1.00000	−0.129099
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−3.00000	−0.381000
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	−7.00000	−0.868243
$66$	−2.00000	−0.246183
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−3.00000	−0.363803
$69$	3.00000	0.361158
$70$	−1.00000	−0.119523
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	1.00000	0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−4.00000	−0.464991
$75$	−4.00000	−0.461880
$76$	6.00000	0.688247
$77$	−2.00000	−0.227921
$78$	7.00000	0.792594
$79$	−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
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	1.00000	0.109109
$85$	3.00000	0.325396
$86$	2.00000	0.215666
$87$	2.00000	0.214423
$88$	−2.00000	−0.213201
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−0.105409
$91$	7.00000	0.733799
$92$	3.00000	0.312772
$93$	−3.00000	−0.311086
$94$	−8.00000	−0.825137
$95$	−6.00000	−0.615587
$96$	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	$0.751641\pi$
$98$	−6.00000	−0.606092
$99$	−2.00000	−0.201008
$100$	−4.00000	−0.400000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	−3.00000	−0.297044
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	7.00000	0.686406
$105$	−1.00000	−0.0975900
$106$	−2.00000	−0.194257
$107$	−11.0000	−1.06341	−0.531705	−	0.846930i	$-0.678449\pi$
$107$	−11.0000	−1.06341	−0.531705	+	0.846930i	$0.678449\pi$
$108$	1.00000	0.0962250
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	2.00000	0.190693
$111$	−4.00000	−0.379663
$112$	1.00000	0.0944911
$113$	1.00000	0.0940721
$113$	1.00000	0.0940721
$114$	6.00000	0.561951
$115$	−3.00000	−0.279751
$116$	2.00000	0.185695
$117$	7.00000	0.647150
$118$	−3.00000	−0.276172
$119$	−3.00000	−0.275010
$120$	−1.00000	−0.0912871
$121$	−7.00000	−0.636364
$122$	−1.00000	−0.0905357
$123$	0	0
$124$	−3.00000	−0.269408
$125$	9.00000	0.804984
$126$	1.00000	0.0890871
$127$	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
$129$	2.00000	0.176090
$130$	−7.00000	−0.613941
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	−2.00000	−0.174078
$133$	6.00000	0.520266
$134$	−2.00000	−0.172774
$135$	−1.00000	−0.0860663
$136$	−3.00000	−0.257248
$137$	5.00000	0.427179	0.213589	−	0.976924i	$-0.431485\pi$
$137$	5.00000	0.427179	0.213589	+	0.976924i	$0.431485\pi$
$138$	3.00000	0.255377
$139$	21.0000	1.78120	0.890598	−	0.454791i	$-0.150286\pi$
$139$	21.0000	1.78120	0.890598	+	0.454791i	$0.150286\pi$
$140$	−1.00000	−0.0845154
$141$	−8.00000	−0.673722
$142$	−5.00000	−0.419591
$143$	−14.0000	−1.17074
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	4.00000	0.331042
$147$	−6.00000	−0.494872
$148$	−4.00000	−0.328798
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	−4.00000	−0.326599
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	6.00000	0.486664
$153$	−3.00000	−0.242536
$154$	−2.00000	−0.161165
$155$	3.00000	0.240966
$156$	7.00000	0.560449
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−4.00000	−0.318223
$159$	−2.00000	−0.158610
$160$	−1.00000	−0.0790569
$161$	3.00000	0.236433
$162$	1.00000	0.0785674
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	21.0000	1.62503	0.812514	−	0.582941i	$-0.198098\pi$
$167$	21.0000	1.62503	0.812514	+	0.582941i	$0.198098\pi$
$168$	1.00000	0.0771517
$169$	36.0000	2.76923
$170$	3.00000	0.230089
$171$	6.00000	0.458831
$172$	2.00000	0.152499
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	2.00000	0.151620
$175$	−4.00000	−0.302372
$176$	−2.00000	−0.150756
$177$	−3.00000	−0.225494
$178$	6.00000	0.449719
$179$	−23.0000	−1.71910	−0.859550	−	0.511051i	$-0.829256\pi$
$179$	−23.0000	−1.71910	−0.859550	+	0.511051i	$0.829256\pi$
$180$	−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	$0.674182\pi$
$182$	7.00000	0.518875
$183$	−1.00000	−0.0739221
$184$	3.00000	0.221163
$185$	4.00000	0.294086
$186$	−3.00000	−0.219971
$187$	6.00000	0.438763
$188$	−8.00000	−0.583460
$189$	1.00000	0.0727393
$190$	−6.00000	−0.435286
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	1.00000	0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−14.0000	−1.00514
$195$	−7.00000	−0.501280
$196$	−6.00000	−0.428571
$197$	−26.0000	−1.85242	−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000	−1.85242	−0.926212	+	0.377004i	$0.876954\pi$
$198$	−2.00000	−0.142134
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	−4.00000	−0.282843
$201$	−2.00000	−0.141069
$202$	−3.00000	−0.211079
$203$	2.00000	0.140372
$204$	−3.00000	−0.210042
$205$	0	0
$206$	−8.00000	−0.557386
$207$	3.00000	0.208514
$208$	7.00000	0.485363
$209$	−12.0000	−0.830057
$210$	−1.00000	−0.0690066
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	−2.00000	−0.137361
$213$	−5.00000	−0.342594
$214$	−11.0000	−0.751945
$215$	−2.00000	−0.136399
$216$	1.00000	0.0680414
$217$	−3.00000	−0.203653
$218$	5.00000	0.338643
$219$	4.00000	0.270295
$220$	2.00000	0.134840
$221$	−21.0000	−1.41261
$222$	−4.00000	−0.268462
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	1.00000	0.0668153
$225$	−4.00000	−0.266667
$226$	1.00000	0.0665190
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	6.00000	0.397360
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	−3.00000	−0.197814
$231$	−2.00000	−0.131590
$232$	2.00000	0.131306
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	7.00000	0.457604
$235$	8.00000	0.521862
$236$	−3.00000	−0.195283
$237$	−4.00000	−0.259828
$238$	−3.00000	−0.194461
$239$	30.0000	1.94054	0.970269	−	0.242028i	$-0.0778125\pi$
$239$	30.0000	1.94054	0.970269	+	0.242028i	$0.0778125\pi$
$240$	−1.00000	−0.0645497
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−7.00000	−0.449977
$243$	1.00000	0.0641500
$244$	−1.00000	−0.0640184
$245$	6.00000	0.383326
$246$	0	0
$247$	42.0000	2.67240
$248$	−3.00000	−0.190500
$249$	0	0
$250$	9.00000	0.569210
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	1.00000	0.0629941
$253$	−6.00000	−0.377217
$254$	16.0000	1.00393
$255$	3.00000	0.187867
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	2.00000	0.124515
$259$	−4.00000	−0.248548
$260$	−7.00000	−0.434122
$261$	2.00000	0.123797
$262$	6.00000	0.370681
$263$	5.00000	0.308313	0.154157	−	0.988046i	$-0.450734\pi$
$263$	5.00000	0.308313	0.154157	+	0.988046i	$0.450734\pi$
$264$	−2.00000	−0.123091
$265$	2.00000	0.122859
$266$	6.00000	0.367884
$267$	6.00000	0.367194
$268$	−2.00000	−0.122169
$269$	−3.00000	−0.182913	−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000	−0.182913	−0.0914566	+	0.995809i	$0.529152\pi$
$270$	−1.00000	−0.0608581
$271$	6.00000	0.364474	0.182237	−	0.983255i	$-0.441666\pi$
$271$	6.00000	0.364474	0.182237	+	0.983255i	$0.441666\pi$
$272$	−3.00000	−0.181902
$273$	7.00000	0.423659
$274$	5.00000	0.302061
$275$	8.00000	0.482418
$276$	3.00000	0.180579
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	21.0000	1.25950
$279$	−3.00000	−0.179605
$280$	−1.00000	−0.0597614
$281$	9.00000	0.536895	0.268447	−	0.963294i	$-0.413489\pi$
$281$	9.00000	0.536895	0.268447	+	0.963294i	$0.413489\pi$
$282$	−8.00000	−0.476393
$283$	11.0000	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$283$	11.0000	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$284$	−5.00000	−0.296695
$285$	−6.00000	−0.355409
$286$	−14.0000	−0.827837
$287$	0	0
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	−2.00000	−0.117444
$291$	−14.0000	−0.820695
$292$	4.00000	0.234082
$293$	−7.00000	−0.408944	−0.204472	−	0.978872i	$-0.565548\pi$
$293$	−7.00000	−0.408944	−0.204472	+	0.978872i	$0.565548\pi$
$294$	−6.00000	−0.349927
$295$	3.00000	0.174667
$296$	−4.00000	−0.232495
$297$	−2.00000	−0.116052
$298$	−18.0000	−1.04271
$299$	21.0000	1.21446
$300$	−4.00000	−0.230940
$301$	2.00000	0.115278
$302$	12.0000	0.690522
$303$	−3.00000	−0.172345
$304$	6.00000	0.344124
$305$	1.00000	0.0572598
$306$	−3.00000	−0.171499
$307$	21.0000	1.19853	0.599267	−	0.800549i	$-0.295459\pi$
$307$	21.0000	1.19853	0.599267	+	0.800549i	$0.295459\pi$
$308$	−2.00000	−0.113961
$309$	−8.00000	−0.455104
$310$	3.00000	0.170389
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	7.00000	0.396297
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	−10.0000	−0.564333
$315$	−1.00000	−0.0563436
$316$	−4.00000	−0.225018
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	−2.00000	−0.112154
$319$	−4.00000	−0.223957
$320$	−1.00000	−0.0559017
$321$	−11.0000	−0.613960
$322$	3.00000	0.167183
$323$	−18.0000	−1.00155
$324$	1.00000	0.0555556
$325$	−28.0000	−1.55316
$326$	−11.0000	−0.609234
$327$	5.00000	0.276501
$328$	0	0
$329$	−8.00000	−0.441054
$330$	2.00000	0.110096
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	21.0000	1.14907
$335$	2.00000	0.109272
$336$	1.00000	0.0545545
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	36.0000	1.95814
$339$	1.00000	0.0543125
$340$	3.00000	0.162698
$341$	6.00000	0.324918
$342$	6.00000	0.324443
$343$	−13.0000	−0.701934
$344$	2.00000	0.107833
$345$	−3.00000	−0.161515
$346$	−22.0000	−1.18273
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	2.00000	0.107211
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	−4.00000	−0.213809
$351$	7.00000	0.373632
$352$	−2.00000	−0.106600
$353$	32.0000	1.70319	0.851594	−	0.524202i	$-0.175636\pi$
$353$	32.0000	1.70319	0.851594	+	0.524202i	$0.175636\pi$
$354$	−3.00000	−0.159448
$355$	5.00000	0.265372
$356$	6.00000	0.317999
$357$	−3.00000	−0.158777
$358$	−23.0000	−1.21559
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−1.00000	−0.0527046
$361$	17.0000	0.894737
$362$	−14.0000	−0.735824
$363$	−7.00000	−0.367405
$364$	7.00000	0.366900
$365$	−4.00000	−0.209370
$366$	−1.00000	−0.0522708
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	3.00000	0.156386
$369$	0	0
$370$	4.00000	0.207950
$371$	−2.00000	−0.103835
$372$	−3.00000	−0.155543
$373$	24.0000	1.24267	0.621336	−	0.783544i	$-0.286590\pi$
$373$	24.0000	1.24267	0.621336	+	0.783544i	$0.286590\pi$
$374$	6.00000	0.310253
$375$	9.00000	0.464758
$376$	−8.00000	−0.412568
$377$	14.0000	0.721037
$378$	1.00000	0.0514344
$379$	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
$381$	16.0000	0.819705
$382$	3.00000	0.153493
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	1.00000	0.0510310
$385$	2.00000	0.101929
$386$	−2.00000	−0.101797
$387$	2.00000	0.101666
$388$	−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	$0.270926\pi$
$390$	−7.00000	−0.354459
$391$	−9.00000	−0.455150
$392$	−6.00000	−0.303046
$393$	6.00000	0.302660
$394$	−26.0000	−1.30986
$395$	4.00000	0.201262
$396$	−2.00000	−0.100504
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	4.00000	0.200502
$399$	6.00000	0.300376
$400$	−4.00000	−0.200000
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	−2.00000	−0.0997509
$403$	−21.0000	−1.04608
$404$	−3.00000	−0.149256
$405$	−1.00000	−0.0496904
$406$	2.00000	0.0992583
$407$	8.00000	0.396545
$408$	−3.00000	−0.148522
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	5.00000	0.246632
$412$	−8.00000	−0.394132
$413$	−3.00000	−0.147620
$414$	3.00000	0.147442
$415$	0	0
$416$	7.00000	0.343203
$417$	21.0000	1.02837
$418$	−12.0000	−0.586939
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	−1.00000	−0.0487950
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	23.0000	1.11962
$423$	−8.00000	−0.388973
$424$	−2.00000	−0.0971286
$425$	12.0000	0.582086
$426$	−5.00000	−0.242251
$427$	−1.00000	−0.0483934
$428$	−11.0000	−0.531705
$429$	−14.0000	−0.675926
$430$	−2.00000	−0.0964486
$431$	−23.0000	−1.10787	−0.553936	−	0.832560i	$-0.686875\pi$
$431$	−23.0000	−1.10787	−0.553936	+	0.832560i	$0.686875\pi$
$432$	1.00000	0.0481125
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	−3.00000	−0.144005
$435$	−2.00000	−0.0958927
$436$	5.00000	0.239457
$437$	18.0000	0.861057
$438$	4.00000	0.191127
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	2.00000	0.0953463
$441$	−6.00000	−0.285714
$442$	−21.0000	−0.998868
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−4.00000	−0.189832
$445$	−6.00000	−0.284427
$446$	−14.0000	−0.662919
$447$	−18.0000	−0.851371
$448$	1.00000	0.0472456
$449$	−5.00000	−0.235965	−0.117982	−	0.993016i	$-0.537643\pi$
$449$	−5.00000	−0.235965	−0.117982	+	0.993016i	$0.537643\pi$
$450$	−4.00000	−0.188562
$451$	0	0
$452$	1.00000	0.0470360
$453$	12.0000	0.563809
$454$	18.0000	0.844782
$455$	−7.00000	−0.328165
$456$	6.00000	0.280976
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−8.00000	−0.373815
$459$	−3.00000	−0.140028
$460$	−3.00000	−0.139876
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	−2.00000	−0.0930484
$463$	23.0000	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$464$	2.00000	0.0928477
$465$	3.00000	0.139122
$466$	10.0000	0.463241
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	7.00000	0.323575
$469$	−2.00000	−0.0923514
$470$	8.00000	0.369012
$471$	−10.0000	−0.460776
$472$	−3.00000	−0.138086
$473$	−4.00000	−0.183920
$474$	−4.00000	−0.183726
$475$	−24.0000	−1.10120
$476$	−3.00000	−0.137505
$477$	−2.00000	−0.0915737
$478$	30.0000	1.37217
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	−1.00000	−0.0456435
$481$	−28.0000	−1.27669
$482$	−10.0000	−0.455488
$483$	3.00000	0.136505
$484$	−7.00000	−0.318182
$485$	14.0000	0.635707
$486$	1.00000	0.0453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−1.00000	−0.0452679
$489$	−11.0000	−0.497437
$490$	6.00000	0.271052
$491$	−5.00000	−0.225647	−0.112823	−	0.993615i	$-0.535989\pi$
$491$	−5.00000	−0.225647	−0.112823	+	0.993615i	$0.535989\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	42.0000	1.88967
$495$	2.00000	0.0898933
$496$	−3.00000	−0.134704
$497$	−5.00000	−0.224281
$498$	0	0
$499$	38.0000	1.70111	0.850557	−	0.525883i	$-0.176265\pi$
$499$	38.0000	1.70111	0.850557	+	0.525883i	$0.176265\pi$
$500$	9.00000	0.402492
$501$	21.0000	0.938211
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	1.00000	0.0445435
$505$	3.00000	0.133498
$506$	−6.00000	−0.266733
$507$	36.0000	1.59882
$508$	16.0000	0.709885
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	3.00000	0.132842
$511$	4.00000	0.176950
$512$	1.00000	0.0441942
$513$	6.00000	0.264906
$514$	6.00000	0.264649
$515$	8.00000	0.352522
$516$	2.00000	0.0880451
$517$	16.0000	0.703679
$518$	−4.00000	−0.175750
$519$	−22.0000	−0.965693
$520$	−7.00000	−0.306970
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	2.00000	0.0875376
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	6.00000	0.262111
$525$	−4.00000	−0.174574
$526$	5.00000	0.218010
$527$	9.00000	0.392046
$528$	−2.00000	−0.0870388
$529$	−14.0000	−0.608696
$530$	2.00000	0.0868744
$531$	−3.00000	−0.130189
$532$	6.00000	0.260133
$533$	0	0
$534$	6.00000	0.259645
$535$	11.0000	0.475571
$536$	−2.00000	−0.0863868
$537$	−23.0000	−0.992523
$538$	−3.00000	−0.129339
$539$	12.0000	0.516877
$540$	−1.00000	−0.0430331
$541$	−32.0000	−1.37579	−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000	−1.37579	−0.687894	+	0.725811i	$0.741464\pi$
$542$	6.00000	0.257722
$543$	−14.0000	−0.600798
$544$	−3.00000	−0.128624
$545$	−5.00000	−0.214176
$546$	7.00000	0.299572
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	5.00000	0.213589
$549$	−1.00000	−0.0426790
$550$	8.00000	0.341121
$551$	12.0000	0.511217
$552$	3.00000	0.127688
$553$	−4.00000	−0.170097
$554$	5.00000	0.212430
$555$	4.00000	0.169791
$556$	21.0000	0.890598
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	−3.00000	−0.127000
$559$	14.0000	0.592137
$560$	−1.00000	−0.0422577
$561$	6.00000	0.253320
$562$	9.00000	0.379642
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	−8.00000	−0.336861
$565$	−1.00000	−0.0420703
$566$	11.0000	0.462364
$567$	1.00000	0.0419961
$568$	−5.00000	−0.209795
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	−6.00000	−0.251312
$571$	−30.0000	−1.25546	−0.627730	−	0.778431i	$-0.716016\pi$
$571$	−30.0000	−1.25546	−0.627730	+	0.778431i	$0.716016\pi$
$572$	−14.0000	−0.585369
$573$	3.00000	0.125327
$574$	0	0
$575$	−12.0000	−0.500435
$576$	1.00000	0.0416667
$577$	−24.0000	−0.999133	−0.499567	−	0.866276i	$-0.666507\pi$
$577$	−24.0000	−0.999133	−0.499567	+	0.866276i	$0.666507\pi$
$578$	−8.00000	−0.332756
$579$	−2.00000	−0.0831172
$580$	−2.00000	−0.0830455
$581$	0	0
$582$	−14.0000	−0.580319
$583$	4.00000	0.165663
$584$	4.00000	0.165521
$585$	−7.00000	−0.289414
$586$	−7.00000	−0.289167
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	−6.00000	−0.247436
$589$	−18.0000	−0.741677
$590$	3.00000	0.123508
$591$	−26.0000	−1.06950
$592$	−4.00000	−0.164399
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	−2.00000	−0.0820610
$595$	3.00000	0.122988
$596$	−18.0000	−0.737309
$597$	4.00000	0.163709
$598$	21.0000	0.858754
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	−4.00000	−0.163299
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	2.00000	0.0815139
$603$	−2.00000	−0.0814463
$604$	12.0000	0.488273
$605$	7.00000	0.284590
$606$	−3.00000	−0.121867
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	6.00000	0.243332
$609$	2.00000	0.0810441
$610$	1.00000	0.0404888
$611$	−56.0000	−2.26552
$612$	−3.00000	−0.121268
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	21.0000	0.847491
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	−40.0000	−1.61034	−0.805170	−	0.593045i	$-0.797926\pi$
$617$	−40.0000	−1.61034	−0.805170	+	0.593045i	$0.797926\pi$
$618$	−8.00000	−0.321807
$619$	−38.0000	−1.52735	−0.763674	−	0.645601i	$-0.776607\pi$
$619$	−38.0000	−1.52735	−0.763674	+	0.645601i	$0.776607\pi$
$620$	3.00000	0.120483
$621$	3.00000	0.120386
$622$	−24.0000	−0.962312
$623$	6.00000	0.240385
$624$	7.00000	0.280224
$625$	11.0000	0.440000
$626$	−1.00000	−0.0399680
$627$	−12.0000	−0.479234
$628$	−10.0000	−0.399043
$629$	12.0000	0.478471
$630$	−1.00000	−0.0398410
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	−4.00000	−0.159111
$633$	23.0000	0.914168
$634$	10.0000	0.397151
$635$	−16.0000	−0.634941
$636$	−2.00000	−0.0793052
$637$	−42.0000	−1.66410
$638$	−4.00000	−0.158362
$639$	−5.00000	−0.197797
$640$	−1.00000	−0.0395285
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	−11.0000	−0.434135
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	3.00000	0.118217
$645$	−2.00000	−0.0787499
$646$	−18.0000	−0.708201
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	1.00000	0.0392837
$649$	6.00000	0.235521
$650$	−28.0000	−1.09825
$651$	−3.00000	−0.117579
$652$	−11.0000	−0.430793
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	5.00000	0.195515
$655$	−6.00000	−0.234439
$656$	0	0
$657$	4.00000	0.156055
$658$	−8.00000	−0.311872
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	2.00000	0.0778499
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	17.0000	0.660724
$663$	−21.0000	−0.815572
$664$	0	0
$665$	−6.00000	−0.232670
$666$	−4.00000	−0.154997
$667$	6.00000	0.232321
$668$	21.0000	0.812514
$669$	−14.0000	−0.541271
$670$	2.00000	0.0772667
$671$	2.00000	0.0772091
$672$	1.00000	0.0385758
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	23.0000	0.885927
$675$	−4.00000	−0.153960
$676$	36.0000	1.38462
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	1.00000	0.0384048
$679$	−14.0000	−0.537271
$680$	3.00000	0.115045
$681$	18.0000	0.689761
$682$	6.00000	0.229752
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	6.00000	0.229416
$685$	−5.00000	−0.191040
$686$	−13.0000	−0.496342
$687$	−8.00000	−0.305219
$688$	2.00000	0.0762493
$689$	−14.0000	−0.533358
$690$	−3.00000	−0.114208
$691$	13.0000	0.494543	0.247272	−	0.968946i	$-0.420466\pi$
$691$	13.0000	0.494543	0.247272	+	0.968946i	$0.420466\pi$
$692$	−22.0000	−0.836315
$693$	−2.00000	−0.0759737
$694$	12.0000	0.455514
$695$	−21.0000	−0.796575
$696$	2.00000	0.0758098
$697$	0	0
$698$	−28.0000	−1.05982
$699$	10.0000	0.378235
$700$	−4.00000	−0.151186
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	7.00000	0.264198
$703$	−24.0000	−0.905177
$704$	−2.00000	−0.0753778
$705$	8.00000	0.301297
$706$	32.0000	1.20434
$707$	−3.00000	−0.112827
$708$	−3.00000	−0.112747
$709$	−53.0000	−1.99046	−0.995228	−	0.0975728i	$-0.968892\pi$
$709$	−53.0000	−1.99046	−0.995228	+	0.0975728i	$0.968892\pi$
$710$	5.00000	0.187647
$711$	−4.00000	−0.150012
$712$	6.00000	0.224860
$713$	−9.00000	−0.337053
$714$	−3.00000	−0.112272
$715$	14.0000	0.523570
$716$	−23.0000	−0.859550
$717$	30.0000	1.12037
$718$	24.0000	0.895672
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	−1.00000	−0.0372678
$721$	−8.00000	−0.297936
$722$	17.0000	0.632674
$723$	−10.0000	−0.371904
$724$	−14.0000	−0.520306
$725$	−8.00000	−0.297113
$726$	−7.00000	−0.259794
$727$	7.00000	0.259616	0.129808	−	0.991539i	$-0.458564\pi$
$727$	7.00000	0.259616	0.129808	+	0.991539i	$0.458564\pi$
$728$	7.00000	0.259437
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	−6.00000	−0.221918
$732$	−1.00000	−0.0369611
$733$	−36.0000	−1.32969	−0.664845	−	0.746981i	$-0.731502\pi$
$733$	−36.0000	−1.32969	−0.664845	+	0.746981i	$0.731502\pi$
$734$	−17.0000	−0.627481
$735$	6.00000	0.221313
$736$	3.00000	0.110581
$737$	4.00000	0.147342
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	4.00000	0.147043
$741$	42.0000	1.54291
$742$	−2.00000	−0.0734223
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	−3.00000	−0.109985
$745$	18.0000	0.659469
$746$	24.0000	0.878702
$747$	0	0
$748$	6.00000	0.219382
$749$	−11.0000	−0.401931
$750$	9.00000	0.328634
$751$	−18.0000	−0.656829	−0.328415	−	0.944534i	$-0.606514\pi$
$751$	−18.0000	−0.656829	−0.328415	+	0.944534i	$0.606514\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	14.0000	0.509850
$755$	−12.0000	−0.436725
$756$	1.00000	0.0363696
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	16.0000	0.581146
$759$	−6.00000	−0.217786
$760$	−6.00000	−0.217643
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	16.0000	0.579619
$763$	5.00000	0.181012
$764$	3.00000	0.108536
$765$	3.00000	0.108465
$766$	20.0000	0.722629
$767$	−21.0000	−0.758266
$768$	1.00000	0.0360844
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	2.00000	0.0720750
$771$	6.00000	0.216085
$772$	−2.00000	−0.0719816
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	2.00000	0.0718885
$775$	12.0000	0.431053
$776$	−14.0000	−0.502571
$777$	−4.00000	−0.143499
$778$	26.0000	0.932145
$779$	0	0
$780$	−7.00000	−0.250640
$781$	10.0000	0.357828
$782$	−9.00000	−0.321839
$783$	2.00000	0.0714742
$784$	−6.00000	−0.214286
$785$	10.0000	0.356915
$786$	6.00000	0.214013
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−26.0000	−0.926212
$789$	5.00000	0.178005
$790$	4.00000	0.142314
$791$	1.00000	0.0355559
$792$	−2.00000	−0.0710669
$793$	−7.00000	−0.248577
$794$	−8.00000	−0.283909
$795$	2.00000	0.0709327
$796$	4.00000	0.141776
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	6.00000	0.212398
$799$	24.0000	0.849059
$800$	−4.00000	−0.141421
$801$	6.00000	0.212000
$802$	38.0000	1.34183
$803$	−8.00000	−0.282314
$804$	−2.00000	−0.0705346
$805$	−3.00000	−0.105736
$806$	−21.0000	−0.739693
$807$	−3.00000	−0.105605
$808$	−3.00000	−0.105540
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	−1.00000	−0.0351364
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	2.00000	0.0701862
$813$	6.00000	0.210429
$814$	8.00000	0.280400
$815$	11.0000	0.385313
$816$	−3.00000	−0.105021
$817$	12.0000	0.419827
$818$	32.0000	1.11885
$819$	7.00000	0.244600
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	5.00000	0.174395
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	−8.00000	−0.278693
$825$	8.00000	0.278524
$826$	−3.00000	−0.104383
$827$	−26.0000	−0.904109	−0.452054	−	0.891990i	$-0.649309\pi$
$827$	−26.0000	−0.904109	−0.452054	+	0.891990i	$0.649309\pi$
$828$	3.00000	0.104257
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	5.00000	0.173448
$832$	7.00000	0.242681
$833$	18.0000	0.623663
$834$	21.0000	0.727171
$835$	−21.0000	−0.726735
$836$	−12.0000	−0.415029
$837$	−3.00000	−0.103695
$838$	−21.0000	−0.725433
$839$	−35.0000	−1.20833	−0.604167	−	0.796858i	$-0.706494\pi$
$839$	−35.0000	−1.20833	−0.604167	+	0.796858i	$0.706494\pi$
$840$	−1.00000	−0.0345033
$841$	−25.0000	−0.862069
$842$	−19.0000	−0.654783
$843$	9.00000	0.309976
$844$	23.0000	0.791693
$845$	−36.0000	−1.23844
$846$	−8.00000	−0.275046
$847$	−7.00000	−0.240523
$848$	−2.00000	−0.0686803
$849$	11.0000	0.377519
$850$	12.0000	0.411597
$851$	−12.0000	−0.411355
$852$	−5.00000	−0.171297
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	−1.00000	−0.0342193
$855$	−6.00000	−0.205196
$856$	−11.0000	−0.375972
$857$	39.0000	1.33221	0.666107	−	0.745856i	$-0.267959\pi$
$857$	39.0000	1.33221	0.666107	+	0.745856i	$0.267959\pi$
$858$	−14.0000	−0.477952
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	−2.00000	−0.0681994
$861$	0	0
$862$	−23.0000	−0.783383
$863$	38.0000	1.29354	0.646768	−	0.762687i	$-0.276120\pi$
$863$	38.0000	1.29354	0.646768	+	0.762687i	$0.276120\pi$
$864$	1.00000	0.0340207
$865$	22.0000	0.748022
$866$	−28.0000	−0.951479
$867$	−8.00000	−0.271694
$868$	−3.00000	−0.101827
$869$	8.00000	0.271381
$870$	−2.00000	−0.0678064
$871$	−14.0000	−0.474372
$872$	5.00000	0.169321
$873$	−14.0000	−0.473828
$874$	18.0000	0.608859
$875$	9.00000	0.304256
$876$	4.00000	0.135147
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	−8.00000	−0.269987
$879$	−7.00000	−0.236104
$880$	2.00000	0.0674200
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	−6.00000	−0.202031
$883$	−40.0000	−1.34611	−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000	−1.34611	−0.673054	+	0.739594i	$0.735018\pi$
$884$	−21.0000	−0.706306
$885$	3.00000	0.100844
$886$	24.0000	0.806296
$887$	55.0000	1.84672	0.923360	−	0.383936i	$-0.125432\pi$
$887$	55.0000	1.84672	0.923360	+	0.383936i	$0.125432\pi$
$888$	−4.00000	−0.134231
$889$	16.0000	0.536623
$890$	−6.00000	−0.201120
$891$	−2.00000	−0.0670025
$892$	−14.0000	−0.468755
$893$	−48.0000	−1.60626
$894$	−18.0000	−0.602010
$895$	23.0000	0.768805
$896$	1.00000	0.0334077
$897$	21.0000	0.701170
$898$	−5.00000	−0.166852
$899$	−6.00000	−0.200111
$900$	−4.00000	−0.133333
$901$	6.00000	0.199889
$902$	0	0
$903$	2.00000	0.0665558
$904$	1.00000	0.0332595
$905$	14.0000	0.465376
$906$	12.0000	0.398673
$907$	−16.0000	−0.531271	−0.265636	−	0.964073i	$-0.585582\pi$
$907$	−16.0000	−0.531271	−0.265636	+	0.964073i	$0.585582\pi$
$908$	18.0000	0.597351
$909$	−3.00000	−0.0995037
$910$	−7.00000	−0.232048
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	6.00000	0.198680
$913$	0	0
$914$	10.0000	0.330771
$915$	1.00000	0.0330590
$916$	−8.00000	−0.264327
$917$	6.00000	0.198137
$918$	−3.00000	−0.0990148
$919$	52.0000	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$920$	−3.00000	−0.0989071
$921$	21.0000	0.691974
$922$	0	0
$923$	−35.0000	−1.15204
$924$	−2.00000	−0.0657952
$925$	16.0000	0.526077
$926$	23.0000	0.755827
$927$	−8.00000	−0.262754
$928$	2.00000	0.0656532
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	3.00000	0.0983739
$931$	−36.0000	−1.17985
$932$	10.0000	0.327561
$933$	−24.0000	−0.785725
$934$	6.00000	0.196326
$935$	−6.00000	−0.196221
$936$	7.00000	0.228802
$937$	56.0000	1.82944	0.914720	−	0.404088i	$-0.132411\pi$
$937$	56.0000	1.82944	0.914720	+	0.404088i	$0.132411\pi$
$938$	−2.00000	−0.0653023
$939$	−1.00000	−0.0326338
$940$	8.00000	0.260931
$941$	53.0000	1.72775	0.863875	−	0.503706i	$-0.168030\pi$
$941$	53.0000	1.72775	0.863875	+	0.503706i	$0.168030\pi$
$942$	−10.0000	−0.325818
$943$	0	0
$944$	−3.00000	−0.0976417
$945$	−1.00000	−0.0325300
$946$	−4.00000	−0.130051
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	−4.00000	−0.129914
$949$	28.0000	0.908918
$950$	−24.0000	−0.778663
$951$	10.0000	0.324272
$952$	−3.00000	−0.0972306
$953$	16.0000	0.518291	0.259145	−	0.965838i	$-0.416559\pi$
$953$	16.0000	0.518291	0.259145	+	0.965838i	$0.416559\pi$
$954$	−2.00000	−0.0647524
$955$	−3.00000	−0.0970777
$956$	30.0000	0.970269
$957$	−4.00000	−0.129302
$958$	−24.0000	−0.775405
$959$	5.00000	0.161458
$960$	−1.00000	−0.0322749
$961$	−22.0000	−0.709677
$962$	−28.0000	−0.902756
$963$	−11.0000	−0.354470
$964$	−10.0000	−0.322078
$965$	2.00000	0.0643823
$966$	3.00000	0.0965234
$967$	37.0000	1.18984	0.594920	−	0.803785i	$-0.297184\pi$
$967$	37.0000	1.18984	0.594920	+	0.803785i	$0.297184\pi$
$968$	−7.00000	−0.224989
$969$	−18.0000	−0.578243
$970$	14.0000	0.449513
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	1.00000	0.0320750
$973$	21.0000	0.673229
$974$	−16.0000	−0.512673
$975$	−28.0000	−0.896718
$976$	−1.00000	−0.0320092
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	−11.0000	−0.351741
$979$	−12.0000	−0.383522
$980$	6.00000	0.191663
$981$	5.00000	0.159638
$982$	−5.00000	−0.159556
$983$	−17.0000	−0.542216	−0.271108	−	0.962549i	$-0.587390\pi$
$983$	−17.0000	−0.542216	−0.271108	+	0.962549i	$0.587390\pi$
$984$	0	0
$985$	26.0000	0.828429
$986$	−6.00000	−0.191079
$987$	−8.00000	−0.254643
$988$	42.0000	1.33620
$989$	6.00000	0.190789
$990$	2.00000	0.0635642
$991$	19.0000	0.603555	0.301777	−	0.953378i	$-0.402420\pi$
$991$	19.0000	0.603555	0.301777	+	0.953378i	$0.402420\pi$
$992$	−3.00000	−0.0952501
$993$	17.0000	0.539479
$994$	−5.00000	−0.158590
$995$	−4.00000	−0.126809
$996$	0	0
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	38.0000	1.20287
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	678.2.a.e.1.1	✓	1
3.2	odd	2		2034.2.a.d.1.1		1
4.3	odd	2		5424.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
678.2.a.e.1.1	✓	1	1.1	even	1	trivial
2034.2.a.d.1.1		1	3.2	odd	2
5424.2.a.e.1.1		1	4.3	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:	\( N \)	\(=\)	\( 678 = 2 \cdot 3 \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	678.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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