LMFDB - Embedded newform 1078.2.a.l.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:	$0$
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	$\chi$	$=$	1078.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} +4.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +2.00000 q^{19} +2.00000 q^{20} +1.00000 q^{22} +2.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -4.00000 q^{27} +6.00000 q^{29} +4.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} -4.00000 q^{39} +2.00000 q^{40} -8.00000 q^{41} +12.0000 q^{43} +1.00000 q^{44} +2.00000 q^{45} -12.0000 q^{47} +2.00000 q^{48} -1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} -4.00000 q^{54} +2.00000 q^{55} +4.00000 q^{57} +6.00000 q^{58} -10.0000 q^{59} +4.00000 q^{60} +10.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +2.00000 q^{66} -12.0000 q^{67} +4.00000 q^{71} +1.00000 q^{72} -12.0000 q^{73} +2.00000 q^{74} -2.00000 q^{75} +2.00000 q^{76} -4.00000 q^{78} +2.00000 q^{80} -11.0000 q^{81} -8.00000 q^{82} +18.0000 q^{83} +12.0000 q^{86} +12.0000 q^{87} +1.00000 q^{88} +2.00000 q^{90} -8.00000 q^{93} -12.0000 q^{94} +4.00000 q^{95} +2.00000 q^{96} +12.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	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
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	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
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	1.00000	0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	2.00000	0.408248
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	−4.00000	−0.769800
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	4.00000	0.730297
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	2.00000	0.348155
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	2.00000	0.324443
$39$	−4.00000	−0.640513
$40$	2.00000	0.316228
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	1.00000	0.150756
$45$	2.00000	0.298142
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	2.00000	0.288675
$49$	0	0
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−4.00000	−0.544331
$55$	2.00000	0.269680
$56$	0	0
$57$	4.00000	0.529813
$58$	6.00000	0.787839
$59$	−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
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	2.00000	0.246183
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	1.00000	0.117851
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	2.00000	0.232495
$75$	−2.00000	−0.230940
$76$	2.00000	0.229416
$77$	0	0
$78$	−4.00000	−0.452911
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	2.00000	0.223607
$81$	−11.0000	−1.22222
$82$	−8.00000	−0.883452
$83$	18.0000	1.97576	0.987878	−	0.155230i	$-0.0496119\pi$
$83$	18.0000	1.97576	0.987878	+	0.155230i	$0.0496119\pi$
$84$	0	0
$85$	0	0
$86$	12.0000	1.29399
$87$	12.0000	1.28654
$88$	1.00000	0.106600
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	0	0
$93$	−8.00000	−0.829561
$94$	−12.0000	−1.23771
$95$	4.00000	0.410391
$96$	2.00000	0.204124
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	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	$0.596490\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−2.00000	−0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−4.00000	−0.384900
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	2.00000	0.190693
$111$	4.00000	0.379663
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	−2.00000	−0.184900
$118$	−10.0000	−0.920575
$119$	0	0
$120$	4.00000	0.365148
$121$	1.00000	0.0909091
$122$	10.0000	0.905357
$123$	−16.0000	−1.44267
$124$	−4.00000	−0.359211
$125$	−12.0000	−1.07331
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	1.00000	0.0883883
$129$	24.0000	2.11308
$130$	−4.00000	−0.350823
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	−12.0000	−1.03664
$135$	−8.00000	−0.688530
$136$	0	0
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	−24.0000	−2.02116
$142$	4.00000	0.335673
$143$	−2.00000	−0.167248
$144$	1.00000	0.0833333
$145$	12.0000	0.996546
$146$	−12.0000	−0.993127
$147$	0	0
$148$	2.00000	0.164399
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	−2.00000	−0.163299
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	0	0
$155$	−8.00000	−0.642575
$156$	−4.00000	−0.320256
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	2.00000	0.158114
$161$	0	0
$162$	−11.0000	−0.864242
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	−8.00000	−0.624695
$165$	4.00000	0.311400
$166$	18.0000	1.39707
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	2.00000	0.152944
$172$	12.0000	0.914991
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	12.0000	0.909718
$175$	0	0
$176$	1.00000	0.0753778
$177$	−20.0000	−1.50329
$178$	0	0
$179$	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
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	20.0000	1.47844
$184$	0	0
$185$	4.00000	0.294086
$186$	−8.00000	−0.586588
$187$	0	0
$188$	−12.0000	−0.875190
$189$	0	0
$190$	4.00000	0.290191
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	2.00000	0.144338
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	12.0000	0.861550
$195$	−8.00000	−0.572892
$196$	0	0
$197$	−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
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	−1.00000	−0.0707107
$201$	−24.0000	−1.69283
$202$	−6.00000	−0.422159
$203$	0	0
$204$	0	0
$205$	−16.0000	−1.11749
$206$	−12.0000	−0.836080
$207$	0	0
$208$	−2.00000	−0.138675
$209$	2.00000	0.138343
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−2.00000	−0.137361
$213$	8.00000	0.548151
$214$	12.0000	0.820303
$215$	24.0000	1.63679
$216$	−4.00000	−0.272166
$217$	0	0
$218$	−10.0000	−0.677285
$219$	−24.0000	−1.62177
$220$	2.00000	0.134840
$221$	0	0
$222$	4.00000	0.268462
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−10.0000	−0.665190
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	4.00000	0.264906
$229$	30.0000	1.98246	0.991228	−	0.132164i	$-0.0421925\pi$
$229$	30.0000	1.98246	0.991228	+	0.132164i	$0.0421925\pi$
$230$	0	0
$231$	0	0
$232$	6.00000	0.393919
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	−2.00000	−0.130744
$235$	−24.0000	−1.56559
$236$	−10.0000	−0.650945
$237$	0	0
$238$	0	0
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	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
$244$	10.0000	0.640184
$245$	0	0
$246$	−16.0000	−1.02012
$247$	−4.00000	−0.254514
$248$	−4.00000	−0.254000
$249$	36.0000	2.28141
$250$	−12.0000	−0.758947
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	0	0
$254$	4.00000	0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.0000	−0.998053	−0.499026	−	0.866587i	$-0.666309\pi$
$257$	−16.0000	−0.998053	−0.499026	+	0.866587i	$0.666309\pi$
$258$	24.0000	1.49417
$259$	0	0
$260$	−4.00000	−0.248069
$261$	6.00000	0.371391
$262$	2.00000	0.123560
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	2.00000	0.123091
$265$	−4.00000	−0.245718
$266$	0	0
$267$	0	0
$268$	−12.0000	−0.733017
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	−8.00000	−0.486864
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	0	0
$274$	22.0000	1.32907
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	14.0000	0.839664
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	−24.0000	−1.42918
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	4.00000	0.237356
$285$	8.00000	0.473879
$286$	−2.00000	−0.118262
$287$	0	0
$288$	1.00000	0.0589256
$289$	−17.0000	−1.00000
$290$	12.0000	0.704664
$291$	24.0000	1.40690
$292$	−12.0000	−0.702247
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	0	0
$295$	−20.0000	−1.16445
$296$	2.00000	0.116248
$297$	−4.00000	−0.232104
$298$	−14.0000	−0.810998
$299$	0	0
$300$	−2.00000	−0.115470
$301$	0	0
$302$	−4.00000	−0.230174
$303$	−12.0000	−0.689382
$304$	2.00000	0.114708
$305$	20.0000	1.14520
$306$	0	0
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	0	0
$309$	−24.0000	−1.36531
$310$	−8.00000	−0.454369
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	−4.00000	−0.226455
$313$	4.00000	0.226093	0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000	0.226093	0.113047	+	0.993590i	$0.463939\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	0	0
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	−4.00000	−0.224309
$319$	6.00000	0.335936
$320$	2.00000	0.111803
$321$	24.0000	1.33955
$322$	0	0
$323$	0	0
$324$	−11.0000	−0.611111
$325$	2.00000	0.110940
$326$	12.0000	0.664619
$327$	−20.0000	−1.10600
$328$	−8.00000	−0.441726
$329$	0	0
$330$	4.00000	0.220193
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	18.0000	0.987878
$333$	2.00000	0.109599
$334$	−12.0000	−0.656611
$335$	−24.0000	−1.31126
$336$	0	0
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	−9.00000	−0.489535
$339$	−20.0000	−1.08625
$340$	0	0
$341$	−4.00000	−0.216612
$342$	2.00000	0.108148
$343$	0	0
$344$	12.0000	0.646997
$345$	0	0
$346$	6.00000	0.322562
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	12.0000	0.643268
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	1.00000	0.0533002
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	−20.0000	−1.06299
$355$	8.00000	0.424596
$356$	0	0
$357$	0	0
$358$	12.0000	0.634220
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	2.00000	0.105409
$361$	−15.0000	−0.789474
$362$	−18.0000	−0.946059
$363$	2.00000	0.104973
$364$	0	0
$365$	−24.0000	−1.25622
$366$	20.0000	1.04542
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	−8.00000	−0.416463
$370$	4.00000	0.207950
$371$	0	0
$372$	−8.00000	−0.414781
$373$	−30.0000	−1.55334	−0.776671	−	0.629907i	$-0.783093\pi$
$373$	−30.0000	−1.55334	−0.776671	+	0.629907i	$0.783093\pi$
$374$	0	0
$375$	−24.0000	−1.23935
$376$	−12.0000	−0.618853
$377$	−12.0000	−0.618031
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	4.00000	0.205196
$381$	8.00000	0.409852
$382$	12.0000	0.613973
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	−2.00000	−0.101797
$387$	12.0000	0.609994
$388$	12.0000	0.609208
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	−8.00000	−0.405096
$391$	0	0
$392$	0	0
$393$	4.00000	0.201773
$394$	−6.00000	−0.302276
$395$	0	0
$396$	1.00000	0.0502519
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	−24.0000	−1.19701
$403$	8.00000	0.398508
$404$	−6.00000	−0.298511
$405$	−22.0000	−1.09319
$406$	0	0
$407$	2.00000	0.0991363
$408$	0	0
$409$	−16.0000	−0.791149	−0.395575	−	0.918434i	$-0.629455\pi$
$409$	−16.0000	−0.791149	−0.395575	+	0.918434i	$0.629455\pi$
$410$	−16.0000	−0.790184
$411$	44.0000	2.17036
$412$	−12.0000	−0.591198
$413$	0	0
$414$	0	0
$415$	36.0000	1.76717
$416$	−2.00000	−0.0980581
$417$	28.0000	1.37117
$418$	2.00000	0.0978232
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	−4.00000	−0.194717
$423$	−12.0000	−0.583460
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	8.00000	0.387601
$427$	0	0
$428$	12.0000	0.580042
$429$	−4.00000	−0.193122
$430$	24.0000	1.15738
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	−4.00000	−0.192450
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	0	0
$435$	24.0000	1.15071
$436$	−10.0000	−0.478913
$437$	0	0
$438$	−24.0000	−1.14676
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	2.00000	0.0953463
$441$	0	0
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−8.00000	−0.378811
$447$	−28.0000	−1.32435
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	−1.00000	−0.0471405
$451$	−8.00000	−0.376705
$452$	−10.0000	−0.470360
$453$	−8.00000	−0.375873
$454$	−6.00000	−0.281594
$455$	0	0
$456$	4.00000	0.187317
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	30.0000	1.40181
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	6.00000	0.278543
$465$	−16.0000	−0.741982
$466$	−10.0000	−0.463241
$467$	22.0000	1.01804	0.509019	−	0.860755i	$-0.330008\pi$
$467$	22.0000	1.01804	0.509019	+	0.860755i	$0.330008\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	−24.0000	−1.10704
$471$	28.0000	1.29017
$472$	−10.0000	−0.460287
$473$	12.0000	0.551761
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	−4.00000	−0.182956
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	4.00000	0.182574
$481$	−4.00000	−0.182384
$482$	−8.00000	−0.364390
$483$	0	0
$484$	1.00000	0.0454545
$485$	24.0000	1.08978
$486$	−10.0000	−0.453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	10.0000	0.452679
$489$	24.0000	1.08532
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	−16.0000	−0.721336
$493$	0	0
$494$	−4.00000	−0.179969
$495$	2.00000	0.0898933
$496$	−4.00000	−0.179605
$497$	0	0
$498$	36.0000	1.61320
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−12.0000	−0.536656
$501$	−24.0000	−1.07224
$502$	18.0000	0.803379
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	−18.0000	−0.799408
$508$	4.00000	0.177471
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	−8.00000	−0.353209
$514$	−16.0000	−0.705730
$515$	−24.0000	−1.05757
$516$	24.0000	1.05654
$517$	−12.0000	−0.527759
$518$	0	0
$519$	12.0000	0.526742
$520$	−4.00000	−0.175412
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	6.00000	0.262613
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	2.00000	0.0873704
$525$	0	0
$526$	16.0000	0.697633
$527$	0	0
$528$	2.00000	0.0870388
$529$	−23.0000	−1.00000
$530$	−4.00000	−0.173749
$531$	−10.0000	−0.433963
$532$	0	0
$533$	16.0000	0.693037
$534$	0	0
$535$	24.0000	1.03761
$536$	−12.0000	−0.518321
$537$	24.0000	1.03568
$538$	6.00000	0.258678
$539$	0	0
$540$	−8.00000	−0.344265
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	−16.0000	−0.687259
$543$	−36.0000	−1.54491
$544$	0	0
$545$	−20.0000	−0.856706
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	22.0000	0.939793
$549$	10.0000	0.426790
$550$	−1.00000	−0.0426401
$551$	12.0000	0.511217
$552$	0	0
$553$	0	0
$554$	−14.0000	−0.594803
$555$	8.00000	0.339581
$556$	14.0000	0.593732
$557$	34.0000	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$558$	−4.00000	−0.169334
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	−10.0000	−0.421825
$563$	22.0000	0.927189	0.463595	−	0.886047i	$-0.346559\pi$
$563$	22.0000	0.927189	0.463595	+	0.886047i	$0.346559\pi$
$564$	−24.0000	−1.01058
$565$	−20.0000	−0.841406
$566$	2.00000	0.0840663
$567$	0	0
$568$	4.00000	0.167836
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	8.00000	0.335083
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−2.00000	−0.0836242
$573$	24.0000	1.00261
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	−17.0000	−0.707107
$579$	−4.00000	−0.166234
$580$	12.0000	0.498273
$581$	0	0
$582$	24.0000	0.994832
$583$	−2.00000	−0.0828315
$584$	−12.0000	−0.496564
$585$	−4.00000	−0.165380
$586$	26.0000	1.07405
$587$	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
$589$	−8.00000	−0.329634
$590$	−20.0000	−0.823387
$591$	−12.0000	−0.493614
$592$	2.00000	0.0821995
$593$	−20.0000	−0.821302	−0.410651	−	0.911793i	$-0.634698\pi$
$593$	−20.0000	−0.821302	−0.410651	+	0.911793i	$0.634698\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−14.0000	−0.573462
$597$	−8.00000	−0.327418
$598$	0	0
$599$	28.0000	1.14405	0.572024	−	0.820237i	$-0.306158\pi$
$599$	28.0000	1.14405	0.572024	+	0.820237i	$0.306158\pi$
$600$	−2.00000	−0.0816497
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	−4.00000	−0.162758
$605$	2.00000	0.0813116
$606$	−12.0000	−0.487467
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	20.0000	0.809776
$611$	24.0000	0.970936
$612$	0	0
$613$	−42.0000	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$614$	−14.0000	−0.564994
$615$	−32.0000	−1.29036
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	−24.0000	−0.965422
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	−16.0000	−0.641542
$623$	0	0
$624$	−4.00000	−0.160128
$625$	−19.0000	−0.760000
$626$	4.00000	0.159872
$627$	4.00000	0.159745
$628$	14.0000	0.558661
$629$	0	0
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	30.0000	1.19145
$635$	8.00000	0.317470
$636$	−4.00000	−0.158610
$637$	0	0
$638$	6.00000	0.237542
$639$	4.00000	0.158238
$640$	2.00000	0.0790569
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	24.0000	0.947204
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	0	0
$645$	48.0000	1.89000
$646$	0	0
$647$	44.0000	1.72982	0.864909	−	0.501928i	$-0.167376\pi$
$647$	44.0000	1.72982	0.864909	+	0.501928i	$0.167376\pi$
$648$	−11.0000	−0.432121
$649$	−10.0000	−0.392534
$650$	2.00000	0.0784465
$651$	0	0
$652$	12.0000	0.469956
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	−20.0000	−0.782062
$655$	4.00000	0.156293
$656$	−8.00000	−0.312348
$657$	−12.0000	−0.468165
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	4.00000	0.155700
$661$	−6.00000	−0.233373	−0.116686	−	0.993169i	$-0.537227\pi$
$661$	−6.00000	−0.233373	−0.116686	+	0.993169i	$0.537227\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	18.0000	0.698535
$665$	0	0
$666$	2.00000	0.0774984
$667$	0	0
$668$	−12.0000	−0.464294
$669$	−16.0000	−0.618596
$670$	−24.0000	−0.927201
$671$	10.0000	0.386046
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	34.0000	1.30963
$675$	4.00000	0.153960
$676$	−9.00000	−0.346154
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	−20.0000	−0.768095
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	−4.00000	−0.153168
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	2.00000	0.0764719
$685$	44.0000	1.68115
$686$	0	0
$687$	60.0000	2.28914
$688$	12.0000	0.457496
$689$	4.00000	0.152388
$690$	0	0
$691$	−50.0000	−1.90209	−0.951045	−	0.309053i	$-0.899988\pi$
$691$	−50.0000	−1.90209	−0.951045	+	0.309053i	$0.899988\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	20.0000	0.759190
$695$	28.0000	1.06210
$696$	12.0000	0.454859
$697$	0	0
$698$	14.0000	0.529908
$699$	−20.0000	−0.756469
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	8.00000	0.301941
$703$	4.00000	0.150863
$704$	1.00000	0.0376889
$705$	−48.0000	−1.80778
$706$	−24.0000	−0.903252
$707$	0	0
$708$	−20.0000	−0.751646
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	8.00000	0.300235
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−4.00000	−0.149592
$716$	12.0000	0.448461
$717$	−8.00000	−0.298765
$718$	20.0000	0.746393
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	−15.0000	−0.558242
$723$	−16.0000	−0.595046
$724$	−18.0000	−0.668965
$725$	−6.00000	−0.222834
$726$	2.00000	0.0742270
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−24.0000	−0.888280
$731$	0	0
$732$	20.0000	0.739221
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	0	0
$737$	−12.0000	−0.442026
$738$	−8.00000	−0.294484
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	4.00000	0.147043
$741$	−8.00000	−0.293887
$742$	0	0
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	−8.00000	−0.293294
$745$	−28.0000	−1.02584
$746$	−30.0000	−1.09838
$747$	18.0000	0.658586
$748$	0	0
$749$	0	0
$750$	−24.0000	−0.876356
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	−12.0000	−0.437595
$753$	36.0000	1.31191
$754$	−12.0000	−0.437014
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−30.0000	−1.09037	−0.545184	−	0.838316i	$-0.683540\pi$
$757$	−30.0000	−1.09037	−0.545184	+	0.838316i	$0.683540\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	4.00000	0.145095
$761$	−8.00000	−0.290000	−0.145000	−	0.989432i	$-0.546318\pi$
$761$	−8.00000	−0.290000	−0.145000	+	0.989432i	$0.546318\pi$
$762$	8.00000	0.289809
$763$	0	0
$764$	12.0000	0.434145
$765$	0	0
$766$	−12.0000	−0.433578
$767$	20.0000	0.722158
$768$	2.00000	0.0721688
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	−32.0000	−1.15245
$772$	−2.00000	−0.0719816
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	12.0000	0.431331
$775$	4.00000	0.143684
$776$	12.0000	0.430775
$777$	0	0
$778$	10.0000	0.358517
$779$	−16.0000	−0.573259
$780$	−8.00000	−0.286446
$781$	4.00000	0.143131
$782$	0	0
$783$	−24.0000	−0.857690
$784$	0	0
$785$	28.0000	0.999363
$786$	4.00000	0.142675
$787$	−14.0000	−0.499046	−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000	−0.499046	−0.249523	+	0.968369i	$0.580274\pi$
$788$	−6.00000	−0.213741
$789$	32.0000	1.13923
$790$	0	0
$791$	0	0
$792$	1.00000	0.0355335
$793$	−20.0000	−0.710221
$794$	−34.0000	−1.20661
$795$	−8.00000	−0.283731
$796$	−4.00000	−0.141776
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	18.0000	0.635602
$803$	−12.0000	−0.423471
$804$	−24.0000	−0.846415
$805$	0	0
$806$	8.00000	0.281788
$807$	12.0000	0.422420
$808$	−6.00000	−0.211079
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	−22.0000	−0.773001
$811$	50.0000	1.75574	0.877869	−	0.478901i	$-0.158965\pi$
$811$	50.0000	1.75574	0.877869	+	0.478901i	$0.158965\pi$
$812$	0	0
$813$	−32.0000	−1.12229
$814$	2.00000	0.0701000
$815$	24.0000	0.840683
$816$	0	0
$817$	24.0000	0.839654
$818$	−16.0000	−0.559427
$819$	0	0
$820$	−16.0000	−0.558744
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	44.0000	1.53468
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	−12.0000	−0.418040
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	36.0000	1.24958
$831$	−28.0000	−0.971309
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	28.0000	0.969561
$835$	−24.0000	−0.830554
$836$	2.00000	0.0691714
$837$	16.0000	0.553041
$838$	18.0000	0.621800
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−34.0000	−1.17172
$843$	−20.0000	−0.688837
$844$	−4.00000	−0.137686
$845$	−18.0000	−0.619219
$846$	−12.0000	−0.412568
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	4.00000	0.137280
$850$	0	0
$851$	0	0
$852$	8.00000	0.274075
$853$	−42.0000	−1.43805	−0.719026	−	0.694983i	$-0.755412\pi$
$853$	−42.0000	−1.43805	−0.719026	+	0.694983i	$0.755412\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	12.0000	0.410152
$857$	−48.0000	−1.63965	−0.819824	−	0.572615i	$-0.805929\pi$
$857$	−48.0000	−1.63965	−0.819824	+	0.572615i	$0.805929\pi$
$858$	−4.00000	−0.136558
$859$	34.0000	1.16007	0.580033	−	0.814593i	$-0.303040\pi$
$859$	34.0000	1.16007	0.580033	+	0.814593i	$0.303040\pi$
$860$	24.0000	0.818393
$861$	0	0
$862$	−36.0000	−1.22616
$863$	−52.0000	−1.77010	−0.885050	−	0.465495i	$-0.845876\pi$
$863$	−52.0000	−1.77010	−0.885050	+	0.465495i	$0.845876\pi$
$864$	−4.00000	−0.136083
$865$	12.0000	0.408012
$866$	−4.00000	−0.135926
$867$	−34.0000	−1.15470
$868$	0	0
$869$	0	0
$870$	24.0000	0.813676
$871$	24.0000	0.813209
$872$	−10.0000	−0.338643
$873$	12.0000	0.406138
$874$	0	0
$875$	0	0
$876$	−24.0000	−0.810885
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	16.0000	0.539974
$879$	52.0000	1.75392
$880$	2.00000	0.0674200
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	0	0
$885$	−40.0000	−1.34459
$886$	12.0000	0.403148
$887$	−20.0000	−0.671534	−0.335767	−	0.941945i	$-0.608996\pi$
$887$	−20.0000	−0.671534	−0.335767	+	0.941945i	$0.608996\pi$
$888$	4.00000	0.134231
$889$	0	0
$890$	0	0
$891$	−11.0000	−0.368514
$892$	−8.00000	−0.267860
$893$	−24.0000	−0.803129
$894$	−28.0000	−0.936460
$895$	24.0000	0.802232
$896$	0	0
$897$	0	0
$898$	−30.0000	−1.00111
$899$	−24.0000	−0.800445
$900$	−1.00000	−0.0333333
$901$	0	0
$902$	−8.00000	−0.266371
$903$	0	0
$904$	−10.0000	−0.332595
$905$	−36.0000	−1.19668
$906$	−8.00000	−0.265782
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	−6.00000	−0.199117
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	4.00000	0.132453
$913$	18.0000	0.595713
$914$	6.00000	0.198462
$915$	40.0000	1.32236
$916$	30.0000	0.991228
$917$	0	0
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	6.00000	0.197599
$923$	−8.00000	−0.263323
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	4.00000	0.131448
$927$	−12.0000	−0.394132
$928$	6.00000	0.196960
$929$	4.00000	0.131236	0.0656179	−	0.997845i	$-0.479098\pi$
$929$	4.00000	0.131236	0.0656179	+	0.997845i	$0.479098\pi$
$930$	−16.0000	−0.524661
$931$	0	0
$932$	−10.0000	−0.327561
$933$	−32.0000	−1.04763
$934$	22.0000	0.719862
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	0	0
$939$	8.00000	0.261070
$940$	−24.0000	−0.782794
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	28.0000	0.912289
$943$	0	0
$944$	−10.0000	−0.325472
$945$	0	0
$946$	12.0000	0.390154
$947$	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
$949$	24.0000	0.779073
$950$	−2.00000	−0.0648886
$951$	60.0000	1.94563
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	−2.00000	−0.0647524
$955$	24.0000	0.776622
$956$	−4.00000	−0.129369
$957$	12.0000	0.387905
$958$	36.0000	1.16311
$959$	0	0
$960$	4.00000	0.129099
$961$	−15.0000	−0.483871
$962$	−4.00000	−0.128965
$963$	12.0000	0.386695
$964$	−8.00000	−0.257663
$965$	−4.00000	−0.128765
$966$	0	0
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	24.0000	0.770594
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	−10.0000	−0.320750
$973$	0	0
$974$	8.00000	0.256337
$975$	4.00000	0.128103
$976$	10.0000	0.320092
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	24.0000	0.767435
$979$	0	0
$980$	0	0
$981$	−10.0000	−0.319275
$982$	28.0000	0.893516
$983$	−44.0000	−1.40338	−0.701691	−	0.712481i	$-0.747571\pi$
$983$	−44.0000	−1.40338	−0.701691	+	0.712481i	$0.747571\pi$
$984$	−16.0000	−0.510061
$985$	−12.0000	−0.382352
$986$	0	0
$987$	0	0
$988$	−4.00000	−0.127257
$989$	0	0
$990$	2.00000	0.0635642
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	−4.00000	−0.127000
$993$	24.0000	0.761617
$994$	0	0
$995$	−8.00000	−0.253617
$996$	36.0000	1.14070
$997$	−62.0000	−1.96356	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−62.0000	−1.96356	−0.981780	+	0.190022i	$0.939144\pi$
$998$	20.0000	0.633089
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1078.2.a.l.1.1	yes	1
3.2	odd	2		9702.2.a.e.1.1		1
4.3	odd	2		8624.2.a.g.1.1		1
7.2	even	3		1078.2.e.a.67.1		2
7.3	odd	6		1078.2.e.e.177.1		2
7.4	even	3		1078.2.e.a.177.1		2
7.5	odd	6		1078.2.e.e.67.1		2
7.6	odd	2		1078.2.a.h.1.1	✓	1
21.20	even	2		9702.2.a.t.1.1		1
28.27	even	2		8624.2.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1078.2.a.h.1.1	✓	1	7.6	odd	2
1078.2.a.l.1.1	yes	1	1.1	even	1	trivial
1078.2.e.a.67.1		2	7.2	even	3
1078.2.e.a.177.1		2	7.4	even	3
1078.2.e.e.67.1		2	7.5	odd	6
1078.2.e.e.177.1		2	7.3	odd	6
8624.2.a.g.1.1		1	4.3	odd	2
8624.2.a.y.1.1		1	28.27	even	2
9702.2.a.e.1.1		1	3.2	odd	2
9702.2.a.t.1.1		1	21.20	even	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

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