LMFDB - Embedded newform 1078.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(1,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, 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("1078.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1078.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:	$8.60787333789$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1078.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} -2.00000 q^{12} +4.00000 q^{13} +4.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} +4.00000 q^{23} +2.00000 q^{24} -1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} +2.00000 q^{29} -4.00000 q^{30} +10.0000 q^{31} -1.00000 q^{32} -2.00000 q^{33} +1.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} -8.00000 q^{39} +2.00000 q^{40} -4.00000 q^{43} +1.00000 q^{44} -2.00000 q^{45} -4.00000 q^{46} -10.0000 q^{47} -2.00000 q^{48} +1.00000 q^{50} +4.00000 q^{52} -14.0000 q^{53} -4.00000 q^{54} -2.00000 q^{55} +8.00000 q^{57} -2.00000 q^{58} -10.0000 q^{59} +4.00000 q^{60} +8.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} -8.00000 q^{65} +2.00000 q^{66} +8.00000 q^{67} -8.00000 q^{69} -4.00000 q^{71} -1.00000 q^{72} -4.00000 q^{73} +6.00000 q^{74} +2.00000 q^{75} -4.00000 q^{76} +8.00000 q^{78} +16.0000 q^{79} -2.00000 q^{80} -11.0000 q^{81} -4.00000 q^{83} +4.00000 q^{86} -4.00000 q^{87} -1.00000 q^{88} -10.0000 q^{89} +2.00000 q^{90} +4.00000 q^{92} -20.0000 q^{93} +10.0000 q^{94} +8.00000 q^{95} +2.00000 q^{96} -6.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	2.00000	0.816497
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.00000	−0.577350
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−1.00000	−0.213201
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	2.00000	0.408248
$25$	−1.00000	−0.200000
$26$	−4.00000	−0.784465
$27$	4.00000	0.769800
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−4.00000	−0.730297
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	−1.00000	−0.176777
$33$	−2.00000	−0.348155
$34$	0	0
$35$	0	0
$36$	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	$0.664172\pi$
$38$	4.00000	0.648886
$39$	−8.00000	−1.28103
$40$	2.00000	0.316228
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	1.00000	0.150756
$45$	−2.00000	−0.298142
$46$	−4.00000	−0.589768
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	−2.00000	−0.288675
$49$	0	0
$50$	1.00000	0.141421
$51$	0	0
$52$	4.00000	0.554700
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	−4.00000	−0.544331
$55$	−2.00000	−0.269680
$56$	0	0
$57$	8.00000	1.05963
$58$	−2.00000	−0.262613
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	4.00000	0.516398
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	1.00000	0.125000
$65$	−8.00000	−0.992278
$66$	2.00000	0.246183
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−1.00000	−0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	6.00000	0.697486
$75$	2.00000	0.230940
$76$	−4.00000	−0.458831
$77$	0	0
$78$	8.00000	0.905822
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	−2.00000	−0.223607
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	−4.00000	−0.428845
$88$	−1.00000	−0.106600
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	4.00000	0.417029
$93$	−20.0000	−2.07390
$94$	10.0000	1.03142
$95$	8.00000	0.820783
$96$	2.00000	0.204124
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	−1.00000	−0.100000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	14.0000	1.35980
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	4.00000	0.384900
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	2.00000	0.190693
$111$	12.0000	1.13899
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−8.00000	−0.749269
$115$	−8.00000	−0.746004
$116$	2.00000	0.185695
$117$	4.00000	0.369800
$118$	10.0000	0.920575
$119$	0	0
$120$	−4.00000	−0.365148
$121$	1.00000	0.0909091
$122$	−8.00000	−0.724286
$123$	0	0
$124$	10.0000	0.898027
$125$	12.0000	1.07331
$126$	0	0
$127$	−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
$130$	8.00000	0.701646
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	−2.00000	−0.174078
$133$	0	0
$134$	−8.00000	−0.691095
$135$	−8.00000	−0.688530
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	8.00000	0.681005
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	20.0000	1.68430
$142$	4.00000	0.335673
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	4.00000	0.331042
$147$	0	0
$148$	−6.00000	−0.493197
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	−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	$0.274337\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	0	0
$155$	−20.0000	−1.60644
$156$	−8.00000	−0.640513
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−16.0000	−1.27289
$159$	28.0000	2.22054
$160$	2.00000	0.158114
$161$	0	0
$162$	11.0000	0.864242
$163$	24.0000	1.87983	0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000	1.87983	0.939913	+	0.341415i	$0.110906\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	4.00000	0.310460
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−4.00000	−0.304997
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	1.00000	0.0753778
$177$	20.0000	1.50329
$178$	10.0000	0.749532
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	−2.00000	−0.149071
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	−16.0000	−1.18275
$184$	−4.00000	−0.294884
$185$	12.0000	0.882258
$186$	20.0000	1.46647
$187$	0	0
$188$	−10.0000	−0.729325
$189$	0	0
$190$	−8.00000	−0.580381
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	−2.00000	−0.144338
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	6.00000	0.430775
$195$	16.0000	1.14578
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	−1.00000	−0.0710669
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	1.00000	0.0707107
$201$	−16.0000	−1.12855
$202$	12.0000	0.844317
$203$	0	0
$204$	0	0
$205$	0	0
$206$	2.00000	0.139347
$207$	4.00000	0.278019
$208$	4.00000	0.277350
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−14.0000	−0.961524
$213$	8.00000	0.548151
$214$	12.0000	0.820303
$215$	8.00000	0.545595
$216$	−4.00000	−0.272166
$217$	0	0
$218$	14.0000	0.948200
$219$	8.00000	0.540590
$220$	−2.00000	−0.134840
$221$	0	0
$222$	−12.0000	−0.805387
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	14.0000	0.931266
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	8.00000	0.529813
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	−2.00000	−0.131306
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−4.00000	−0.261488
$235$	20.0000	1.30466
$236$	−10.0000	−0.650945
$237$	−32.0000	−2.07862
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	4.00000	0.258199
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	−1.00000	−0.0642824
$243$	10.0000	0.641500
$244$	8.00000	0.512148
$245$	0	0
$246$	0	0
$247$	−16.0000	−1.01806
$248$	−10.0000	−0.635001
$249$	8.00000	0.506979
$250$	−12.0000	−0.758947
$251$	26.0000	1.64111	0.820553	−	0.571571i	$-0.193666\pi$
$251$	26.0000	1.64111	0.820553	+	0.571571i	$0.193666\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	−8.00000	−0.498058
$259$	0	0
$260$	−8.00000	−0.496139
$261$	2.00000	0.123797
$262$	8.00000	0.494242
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	2.00000	0.123091
$265$	28.0000	1.72003
$266$	0	0
$267$	20.0000	1.22398
$268$	8.00000	0.488678
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	8.00000	0.486864
$271$	28.0000	1.70088	0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000	1.70088	0.850439	+	0.526073i	$0.176336\pi$
$272$	0	0
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−1.00000	−0.0603023
$276$	−8.00000	−0.481543
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	20.0000	1.19952
$279$	10.0000	0.598684
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	−20.0000	−1.19098
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−4.00000	−0.237356
$285$	−16.0000	−0.947758
$286$	−4.00000	−0.236525
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	4.00000	0.234888
$291$	12.0000	0.703452
$292$	−4.00000	−0.234082
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	20.0000	1.16445
$296$	6.00000	0.348743
$297$	4.00000	0.232104
$298$	−22.0000	−1.27443
$299$	16.0000	0.925304
$300$	2.00000	0.115470
$301$	0	0
$302$	−16.0000	−0.920697
$303$	24.0000	1.37876
$304$	−4.00000	−0.229416
$305$	−16.0000	−0.916157
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	20.0000	1.13592
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	8.00000	0.452911
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	16.0000	0.900070
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−28.0000	−1.57016
$319$	2.00000	0.111979
$320$	−2.00000	−0.111803
$321$	24.0000	1.33955
$322$	0	0
$323$	0	0
$324$	−11.0000	−0.611111
$325$	−4.00000	−0.221880
$326$	−24.0000	−1.32924
$327$	28.0000	1.54840
$328$	0	0
$329$	0	0
$330$	−4.00000	−0.220193
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−4.00000	−0.219529
$333$	−6.00000	−0.328798
$334$	−8.00000	−0.437741
$335$	−16.0000	−0.874173
$336$	0	0
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	−3.00000	−0.163178
$339$	28.0000	1.52075
$340$	0	0
$341$	10.0000	0.541530
$342$	4.00000	0.216295
$343$	0	0
$344$	4.00000	0.215666
$345$	16.0000	0.861411
$346$	4.00000	0.215041
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−4.00000	−0.214423
$349$	−32.0000	−1.71292	−0.856460	−	0.516213i	$-0.827341\pi$
$349$	−32.0000	−1.71292	−0.856460	+	0.516213i	$0.827341\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	−1.00000	−0.0533002
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	−20.0000	−1.06299
$355$	8.00000	0.424596
$356$	−10.0000	−0.529999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	2.00000	0.105409
$361$	−3.00000	−0.157895
$362$	14.0000	0.735824
$363$	−2.00000	−0.104973
$364$	0	0
$365$	8.00000	0.418739
$366$	16.0000	0.836333
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	−12.0000	−0.623850
$371$	0	0
$372$	−20.0000	−1.03695
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	0	0
$375$	−24.0000	−1.23935
$376$	10.0000	0.515711
$377$	8.00000	0.412021
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	8.00000	0.410391
$381$	32.0000	1.63941
$382$	−8.00000	−0.409316
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	6.00000	0.305392
$387$	−4.00000	−0.203331
$388$	−6.00000	−0.304604
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	−16.0000	−0.810191
$391$	0	0
$392$	0	0
$393$	16.0000	0.807093
$394$	18.0000	0.906827
$395$	−32.0000	−1.61009
$396$	1.00000	0.0502519
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−14.0000	−0.701757
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	16.0000	0.798007
$403$	40.0000	1.99254
$404$	−12.0000	−0.597022
$405$	22.0000	1.09319
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	−2.00000	−0.0985329
$413$	0	0
$414$	−4.00000	−0.196589
$415$	8.00000	0.392705
$416$	−4.00000	−0.196116
$417$	40.0000	1.95881
$418$	4.00000	0.195646
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	4.00000	0.194717
$423$	−10.0000	−0.486217
$424$	14.0000	0.679900
$425$	0	0
$426$	−8.00000	−0.387601
$427$	0	0
$428$	−12.0000	−0.580042
$429$	−8.00000	−0.386244
$430$	−8.00000	−0.385794
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	4.00000	0.192450
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	0	0
$435$	8.00000	0.383571
$436$	−14.0000	−0.670478
$437$	−16.0000	−0.765384
$438$	−8.00000	−0.382255
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	2.00000	0.0953463
$441$	0	0
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	12.0000	0.569495
$445$	20.0000	0.948091
$446$	−14.0000	−0.662919
$447$	−44.0000	−2.08113
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−14.0000	−0.658505
$453$	−32.0000	−1.50349
$454$	8.00000	0.375459
$455$	0	0
$456$	−8.00000	−0.374634
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	−8.00000	−0.373002
$461$	−32.0000	−1.49039	−0.745194	−	0.666847i	$-0.767643\pi$
$461$	−32.0000	−1.49039	−0.745194	+	0.666847i	$0.767643\pi$
$462$	0	0
$463$	12.0000	0.557687	0.278844	−	0.960337i	$-0.410049\pi$
$463$	12.0000	0.557687	0.278844	+	0.960337i	$0.410049\pi$
$464$	2.00000	0.0928477
$465$	40.0000	1.85496
$466$	−6.00000	−0.277945
$467$	−14.0000	−0.647843	−0.323921	−	0.946084i	$-0.605001\pi$
$467$	−14.0000	−0.647843	−0.323921	+	0.946084i	$0.605001\pi$
$468$	4.00000	0.184900
$469$	0	0
$470$	−20.0000	−0.922531
$471$	20.0000	0.921551
$472$	10.0000	0.460287
$473$	−4.00000	−0.183920
$474$	32.0000	1.46981
$475$	4.00000	0.183533
$476$	0	0
$477$	−14.0000	−0.641016
$478$	8.00000	0.365911
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	−4.00000	−0.182574
$481$	−24.0000	−1.09431
$482$	8.00000	0.364390
$483$	0	0
$484$	1.00000	0.0454545
$485$	12.0000	0.544892
$486$	−10.0000	−0.453609
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	−8.00000	−0.362143
$489$	−48.0000	−2.17064
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	0	0
$494$	16.0000	0.719874
$495$	−2.00000	−0.0898933
$496$	10.0000	0.449013
$497$	0	0
$498$	−8.00000	−0.358489
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	12.0000	0.536656
$501$	−16.0000	−0.714827
$502$	−26.0000	−1.16044
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	−4.00000	−0.177822
$507$	−6.00000	−0.266469
$508$	−16.0000	−0.709885
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−16.0000	−0.706417
$514$	2.00000	0.0882162
$515$	4.00000	0.176261
$516$	8.00000	0.352180
$517$	−10.0000	−0.439799
$518$	0	0
$519$	8.00000	0.351161
$520$	8.00000	0.350823
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	−2.00000	−0.0875376
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	24.0000	1.04645
$527$	0	0
$528$	−2.00000	−0.0870388
$529$	−7.00000	−0.304348
$530$	−28.0000	−1.21624
$531$	−10.0000	−0.433963
$532$	0	0
$533$	0	0
$534$	−20.0000	−0.865485
$535$	24.0000	1.03761
$536$	−8.00000	−0.345547
$537$	−24.0000	−1.03568
$538$	14.0000	0.603583
$539$	0	0
$540$	−8.00000	−0.344265
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−28.0000	−1.20270
$543$	28.0000	1.20160
$544$	0	0
$545$	28.0000	1.19939
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	6.00000	0.256307
$549$	8.00000	0.341432
$550$	1.00000	0.0426401
$551$	−8.00000	−0.340811
$552$	8.00000	0.340503
$553$	0	0
$554$	−6.00000	−0.254916
$555$	−24.0000	−1.01874
$556$	−20.0000	−0.848189
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	−10.0000	−0.423334
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	−30.0000	−1.26547
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	20.0000	0.842152
$565$	28.0000	1.17797
$566$	0	0
$567$	0	0
$568$	4.00000	0.167836
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	16.0000	0.670166
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	4.00000	0.167248
$573$	−16.0000	−0.668410
$574$	0	0
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	17.0000	0.707107
$579$	12.0000	0.498703
$580$	−4.00000	−0.166091
$581$	0	0
$582$	−12.0000	−0.497416
$583$	−14.0000	−0.579821
$584$	4.00000	0.165521
$585$	−8.00000	−0.330759
$586$	0	0
$587$	42.0000	1.73353	0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000	1.73353	0.866763	+	0.498721i	$0.166197\pi$
$588$	0	0
$589$	−40.0000	−1.64817
$590$	−20.0000	−0.823387
$591$	36.0000	1.48084
$592$	−6.00000	−0.246598
$593$	−12.0000	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$593$	−12.0000	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	22.0000	0.901155
$597$	−28.0000	−1.14596
$598$	−16.0000	−0.654289
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	−2.00000	−0.0816497
$601$	24.0000	0.978980	0.489490	−	0.872009i	$-0.337183\pi$
$601$	24.0000	0.978980	0.489490	+	0.872009i	$0.337183\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	16.0000	0.651031
$605$	−2.00000	−0.0813116
$606$	−24.0000	−0.974933
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	16.0000	0.647821
$611$	−40.0000	−1.61823
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	0	0
$617$	38.0000	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$617$	38.0000	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$618$	−4.00000	−0.160904
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	−20.0000	−0.803219
$621$	16.0000	0.642058
$622$	−6.00000	−0.240578
$623$	0	0
$624$	−8.00000	−0.320256
$625$	−19.0000	−0.760000
$626$	−6.00000	−0.239808
$627$	8.00000	0.319489
$628$	−10.0000	−0.399043
$629$	0	0
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−16.0000	−0.636446
$633$	8.00000	0.317971
$634$	18.0000	0.714871
$635$	32.0000	1.26988
$636$	28.0000	1.11027
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	−4.00000	−0.158238
$640$	2.00000	0.0790569
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	−24.0000	−0.947204
$643$	−22.0000	−0.867595	−0.433798	−	0.901010i	$-0.642827\pi$
$643$	−22.0000	−0.867595	−0.433798	+	0.901010i	$0.642827\pi$
$644$	0	0
$645$	−16.0000	−0.629999
$646$	0	0
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	11.0000	0.432121
$649$	−10.0000	−0.392534
$650$	4.00000	0.156893
$651$	0	0
$652$	24.0000	0.939913
$653$	46.0000	1.80012	0.900060	−	0.435767i	$-0.143523\pi$
$653$	46.0000	1.80012	0.900060	+	0.435767i	$0.143523\pi$
$654$	−28.0000	−1.09489
$655$	16.0000	0.625172
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	4.00000	0.155700
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	4.00000	0.155230
$665$	0	0
$666$	6.00000	0.232495
$667$	8.00000	0.309761
$668$	8.00000	0.309529
$669$	−28.0000	−1.08254
$670$	16.0000	0.618134
$671$	8.00000	0.308837
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	34.0000	1.30963
$675$	−4.00000	−0.153960
$676$	3.00000	0.115385
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	−28.0000	−1.07533
$679$	0	0
$680$	0	0
$681$	16.0000	0.613121
$682$	−10.0000	−0.382920
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	−4.00000	−0.152944
$685$	−12.0000	−0.458496
$686$	0	0
$687$	−20.0000	−0.763048
$688$	−4.00000	−0.152499
$689$	−56.0000	−2.13343
$690$	−16.0000	−0.609110
$691$	−42.0000	−1.59776	−0.798878	−	0.601494i	$-0.794573\pi$
$691$	−42.0000	−1.59776	−0.798878	+	0.601494i	$0.794573\pi$
$692$	−4.00000	−0.152057
$693$	0	0
$694$	12.0000	0.455514
$695$	40.0000	1.51729
$696$	4.00000	0.151620
$697$	0	0
$698$	32.0000	1.21122
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	−16.0000	−0.603881
$703$	24.0000	0.905177
$704$	1.00000	0.0376889
$705$	−40.0000	−1.50649
$706$	2.00000	0.0752710
$707$	0	0
$708$	20.0000	0.751646
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	−8.00000	−0.300235
$711$	16.0000	0.600047
$712$	10.0000	0.374766
$713$	40.0000	1.49801
$714$	0	0
$715$	−8.00000	−0.299183
$716$	12.0000	0.448461
$717$	16.0000	0.597531
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	3.00000	0.111648
$723$	16.0000	0.595046
$724$	−14.0000	−0.520306
$725$	−2.00000	−0.0742781
$726$	2.00000	0.0742270
$727$	−46.0000	−1.70605	−0.853023	−	0.521874i	$-0.825233\pi$
$727$	−46.0000	−1.70605	−0.853023	+	0.521874i	$0.825233\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−8.00000	−0.296093
$731$	0	0
$732$	−16.0000	−0.591377
$733$	8.00000	0.295487	0.147743	−	0.989026i	$-0.452799\pi$
$733$	8.00000	0.295487	0.147743	+	0.989026i	$0.452799\pi$
$734$	18.0000	0.664392
$735$	0	0
$736$	−4.00000	−0.147442
$737$	8.00000	0.294684
$738$	0	0
$739$	−52.0000	−1.91285	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000	−1.91285	−0.956425	+	0.291977i	$0.905687\pi$
$740$	12.0000	0.441129
$741$	32.0000	1.17555
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	20.0000	0.733236
$745$	−44.0000	−1.61204
$746$	34.0000	1.24483
$747$	−4.00000	−0.146352
$748$	0	0
$749$	0	0
$750$	24.0000	0.876356
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	−10.0000	−0.364662
$753$	−52.0000	−1.89499
$754$	−8.00000	−0.291343
$755$	−32.0000	−1.16460
$756$	0	0
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	−8.00000	−0.290573
$759$	−8.00000	−0.290382
$760$	−8.00000	−0.290191
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	−32.0000	−1.15924
$763$	0	0
$764$	8.00000	0.289430
$765$	0	0
$766$	14.0000	0.505841
$767$	−40.0000	−1.44432
$768$	−2.00000	−0.0721688
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	0	0
$771$	4.00000	0.144056
$772$	−6.00000	−0.215945
$773$	34.0000	1.22290	0.611448	−	0.791285i	$-0.290588\pi$
$773$	34.0000	1.22290	0.611448	+	0.791285i	$0.290588\pi$
$774$	4.00000	0.143777
$775$	−10.0000	−0.359211
$776$	6.00000	0.215387
$777$	0	0
$778$	18.0000	0.645331
$779$	0	0
$780$	16.0000	0.572892
$781$	−4.00000	−0.143131
$782$	0	0
$783$	8.00000	0.285897
$784$	0	0
$785$	20.0000	0.713831
$786$	−16.0000	−0.570701
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	−18.0000	−0.641223
$789$	48.0000	1.70885
$790$	32.0000	1.13851
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	32.0000	1.13635
$794$	18.0000	0.638796
$795$	−56.0000	−1.98612
$796$	14.0000	0.496217
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	−10.0000	−0.353333
$802$	−10.0000	−0.353112
$803$	−4.00000	−0.141157
$804$	−16.0000	−0.564276
$805$	0	0
$806$	−40.0000	−1.40894
$807$	28.0000	0.985647
$808$	12.0000	0.422159
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	−22.0000	−0.773001
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	0	0
$813$	−56.0000	−1.96401
$814$	6.00000	0.210300
$815$	−48.0000	−1.68137
$816$	0	0
$817$	16.0000	0.559769
$818$	−4.00000	−0.139857
$819$	0	0
$820$	0	0
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	12.0000	0.418548
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	2.00000	0.0696733
$825$	2.00000	0.0696311
$826$	0	0
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	4.00000	0.139010
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	−8.00000	−0.277684
$831$	−12.0000	−0.416275
$832$	4.00000	0.138675
$833$	0	0
$834$	−40.0000	−1.38509
$835$	−16.0000	−0.553703
$836$	−4.00000	−0.138343
$837$	40.0000	1.38260
$838$	−30.0000	−1.03633
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−10.0000	−0.344623
$843$	−60.0000	−2.06651
$844$	−4.00000	−0.137686
$845$	−6.00000	−0.206406
$846$	10.0000	0.343807
$847$	0	0
$848$	−14.0000	−0.480762
$849$	0	0
$850$	0	0
$851$	−24.0000	−0.822709
$852$	8.00000	0.274075
$853$	−4.00000	−0.136957	−0.0684787	−	0.997653i	$-0.521815\pi$
$853$	−4.00000	−0.136957	−0.0684787	+	0.997653i	$0.521815\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	12.0000	0.410152
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	8.00000	0.273115
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	16.0000	0.544962
$863$	40.0000	1.36162	0.680808	−	0.732462i	$-0.261629\pi$
$863$	40.0000	1.36162	0.680808	+	0.732462i	$0.261629\pi$
$864$	−4.00000	−0.136083
$865$	8.00000	0.272008
$866$	−10.0000	−0.339814
$867$	34.0000	1.15470
$868$	0	0
$869$	16.0000	0.542763
$870$	−8.00000	−0.271225
$871$	32.0000	1.08428
$872$	14.0000	0.474100
$873$	−6.00000	−0.203069
$874$	16.0000	0.541208
$875$	0	0
$876$	8.00000	0.270295
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−28.0000	−0.944954
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	0	0
$885$	−40.0000	−1.34459
$886$	−4.00000	−0.134383
$887$	28.0000	0.940148	0.470074	−	0.882627i	$-0.344227\pi$
$887$	28.0000	0.940148	0.470074	+	0.882627i	$0.344227\pi$
$888$	−12.0000	−0.402694
$889$	0	0
$890$	−20.0000	−0.670402
$891$	−11.0000	−0.368514
$892$	14.0000	0.468755
$893$	40.0000	1.33855
$894$	44.0000	1.47158
$895$	−24.0000	−0.802232
$896$	0	0
$897$	−32.0000	−1.06845
$898$	−6.00000	−0.200223
$899$	20.0000	0.667037
$900$	−1.00000	−0.0333333
$901$	0	0
$902$	0	0
$903$	0	0
$904$	14.0000	0.465633
$905$	28.0000	0.930751
$906$	32.0000	1.06313
$907$	48.0000	1.59381	0.796907	−	0.604102i	$-0.206468\pi$
$907$	48.0000	1.59381	0.796907	+	0.604102i	$0.206468\pi$
$908$	−8.00000	−0.265489
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−4.00000	−0.132526	−0.0662630	−	0.997802i	$-0.521108\pi$
$911$	−4.00000	−0.132526	−0.0662630	+	0.997802i	$0.521108\pi$
$912$	8.00000	0.264906
$913$	−4.00000	−0.132381
$914$	38.0000	1.25693
$915$	32.0000	1.05789
$916$	10.0000	0.330409
$917$	0	0
$918$	0	0
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	8.00000	0.263752
$921$	32.0000	1.05444
$922$	32.0000	1.05386
$923$	−16.0000	−0.526646
$924$	0	0
$925$	6.00000	0.197279
$926$	−12.0000	−0.394344
$927$	−2.00000	−0.0656886
$928$	−2.00000	−0.0656532
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	−40.0000	−1.31165
$931$	0	0
$932$	6.00000	0.196537
$933$	−12.0000	−0.392862
$934$	14.0000	0.458094
$935$	0	0
$936$	−4.00000	−0.130744
$937$	52.0000	1.69877	0.849383	−	0.527777i	$-0.176974\pi$
$937$	52.0000	1.69877	0.849383	+	0.527777i	$0.176974\pi$
$938$	0	0
$939$	−12.0000	−0.391605
$940$	20.0000	0.652328
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	−20.0000	−0.651635
$943$	0	0
$944$	−10.0000	−0.325472
$945$	0	0
$946$	4.00000	0.130051
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−32.0000	−1.03931
$949$	−16.0000	−0.519382
$950$	−4.00000	−0.129777
$951$	36.0000	1.16738
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	14.0000	0.453267
$955$	−16.0000	−0.517748
$956$	−8.00000	−0.258738
$957$	−4.00000	−0.129302
$958$	12.0000	0.387702
$959$	0	0
$960$	4.00000	0.129099
$961$	69.0000	2.22581
$962$	24.0000	0.773791
$963$	−12.0000	−0.386695
$964$	−8.00000	−0.257663
$965$	12.0000	0.386294
$966$	0	0
$967$	−24.0000	−0.771788	−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000	−0.771788	−0.385894	+	0.922543i	$0.626107\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−12.0000	−0.385297
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	10.0000	0.320750
$973$	0	0
$974$	−12.0000	−0.384505
$975$	8.00000	0.256205
$976$	8.00000	0.256074
$977$	−22.0000	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$977$	−22.0000	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$978$	48.0000	1.53487
$979$	−10.0000	−0.319601
$980$	0	0
$981$	−14.0000	−0.446986
$982$	28.0000	0.893516
$983$	−26.0000	−0.829271	−0.414636	−	0.909988i	$-0.636091\pi$
$983$	−26.0000	−0.829271	−0.414636	+	0.909988i	$0.636091\pi$
$984$	0	0
$985$	36.0000	1.14706
$986$	0	0
$987$	0	0
$988$	−16.0000	−0.509028
$989$	−16.0000	−0.508770
$990$	2.00000	0.0635642
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	−10.0000	−0.317500
$993$	40.0000	1.26936
$994$	0	0
$995$	−28.0000	−0.887660
$996$	8.00000	0.253490
$997$	−36.0000	−1.14013	−0.570066	−	0.821599i	$-0.693082\pi$
$997$	−36.0000	−1.14013	−0.570066	+	0.821599i	$0.693082\pi$
$998$	16.0000	0.506471
$999$	−24.0000	−0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1078.2.a.b.1.1		1
3.2	odd	2		9702.2.a.bz.1.1		1
4.3	odd	2		8624.2.a.z.1.1		1
7.2	even	3		1078.2.e.l.67.1		2
7.3	odd	6		1078.2.e.h.177.1		2
7.4	even	3		1078.2.e.l.177.1		2
7.5	odd	6		1078.2.e.h.67.1		2
7.6	odd	2		154.2.a.b.1.1	✓	1
21.20	even	2		1386.2.a.f.1.1		1
28.27	even	2		1232.2.a.c.1.1		1
35.13	even	4		3850.2.c.d.1849.2		2
35.27	even	4		3850.2.c.d.1849.1		2
35.34	odd	2		3850.2.a.o.1.1		1
56.13	odd	2		4928.2.a.d.1.1		1
56.27	even	2		4928.2.a.bf.1.1		1
77.76	even	2		1694.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.a.b.1.1	✓	1	7.6	odd	2
1078.2.a.b.1.1		1	1.1	even	1	trivial
1078.2.e.h.67.1		2	7.5	odd	6
1078.2.e.h.177.1		2	7.3	odd	6
1078.2.e.l.67.1		2	7.2	even	3
1078.2.e.l.177.1		2	7.4	even	3
1232.2.a.c.1.1		1	28.27	even	2
1386.2.a.f.1.1		1	21.20	even	2
1694.2.a.i.1.1		1	77.76	even	2
3850.2.a.o.1.1		1	35.34	odd	2
3850.2.c.d.1849.1		2	35.27	even	4
3850.2.c.d.1849.2		2	35.13	even	4
4928.2.a.d.1.1		1	56.13	odd	2
4928.2.a.bf.1.1		1	56.27	even	2
8624.2.a.z.1.1		1	4.3	odd	2
9702.2.a.bz.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:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more