LMFDB - Embedded newform 7742.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7742,2,Mod(1,7742)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7742, 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("7742.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7742 = 2 \cdot 7^{2} \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7742.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:	$61.8201812449$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 158)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7742.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} +3.00000 q^{10} -2.00000 q^{11} +3.00000 q^{12} +5.00000 q^{13} +9.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +6.00000 q^{18} +3.00000 q^{20} -2.00000 q^{22} -2.00000 q^{23} +3.00000 q^{24} +4.00000 q^{25} +5.00000 q^{26} +9.00000 q^{27} +6.00000 q^{29} +9.00000 q^{30} +10.0000 q^{31} +1.00000 q^{32} -6.00000 q^{33} -6.00000 q^{34} +6.00000 q^{36} -10.0000 q^{37} +15.0000 q^{39} +3.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} -2.00000 q^{44} +18.0000 q^{45} -2.00000 q^{46} +3.00000 q^{47} +3.00000 q^{48} +4.00000 q^{50} -18.0000 q^{51} +5.00000 q^{52} -12.0000 q^{53} +9.00000 q^{54} -6.00000 q^{55} +6.00000 q^{58} +1.00000 q^{59} +9.00000 q^{60} -12.0000 q^{61} +10.0000 q^{62} +1.00000 q^{64} +15.0000 q^{65} -6.00000 q^{66} -8.00000 q^{67} -6.00000 q^{68} -6.00000 q^{69} -3.00000 q^{71} +6.00000 q^{72} +6.00000 q^{73} -10.0000 q^{74} +12.0000 q^{75} +15.0000 q^{78} +1.00000 q^{79} +3.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} -14.0000 q^{83} -18.0000 q^{85} +4.00000 q^{86} +18.0000 q^{87} -2.00000 q^{88} +7.00000 q^{89} +18.0000 q^{90} -2.00000 q^{92} +30.0000 q^{93} +3.00000 q^{94} +3.00000 q^{96} +11.0000 q^{97} -12.0000 q^{99} +O(q^{100})\)

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
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	3.00000	1.22474
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	6.00000	2.00000
$10$	3.00000	0.948683
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	3.00000	0.866025
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	9.00000	2.32379
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	6.00000	1.41421
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	3.00000	0.612372
$25$	4.00000	0.800000
$26$	5.00000	0.980581
$27$	9.00000	1.73205
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	9.00000	1.64317
$31$	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
$33$	−6.00000	−1.04447
$34$	−6.00000	−1.02899
$35$	0	0
$36$	6.00000	1.00000
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	15.0000	2.40192
$40$	3.00000	0.474342
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−2.00000	−0.301511
$45$	18.0000	2.68328
$46$	−2.00000	−0.294884
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	3.00000	0.433013
$49$	0	0
$50$	4.00000	0.565685
$51$	−18.0000	−2.52050
$52$	5.00000	0.693375
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	9.00000	1.22474
$55$	−6.00000	−0.809040
$56$	0	0
$57$	0	0
$58$	6.00000	0.787839
$59$	1.00000	0.130189	0.0650945	−	0.997879i	$-0.479265\pi$
$59$	1.00000	0.130189	0.0650945	+	0.997879i	$0.479265\pi$
$60$	9.00000	1.16190
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	10.0000	1.27000
$63$	0	0
$64$	1.00000	0.125000
$65$	15.0000	1.86052
$66$	−6.00000	−0.738549
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−6.00000	−0.727607
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	6.00000	0.707107
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−10.0000	−1.16248
$75$	12.0000	1.38564
$76$	0	0
$77$	0	0
$78$	15.0000	1.69842
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	3.00000	0.335410
$81$	9.00000	1.00000
$82$	−2.00000	−0.220863
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	0	0
$85$	−18.0000	−1.95237
$86$	4.00000	0.431331
$87$	18.0000	1.92980
$88$	−2.00000	−0.213201
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	18.0000	1.89737
$91$	0	0
$92$	−2.00000	−0.208514
$93$	30.0000	3.11086
$94$	3.00000	0.309426
$95$	0	0
$96$	3.00000	0.306186
$97$	11.0000	1.11688	0.558440	−	0.829545i	$-0.311400\pi$
$97$	11.0000	1.11688	0.558440	+	0.829545i	$0.311400\pi$
$98$	0	0
$99$	−12.0000	−1.20605
$100$	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	$0.547688\pi$
$102$	−18.0000	−1.78227
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−1.00000	−0.0966736	−0.0483368	−	0.998831i	$-0.515392\pi$
$107$	−1.00000	−0.0966736	−0.0483368	+	0.998831i	$0.515392\pi$
$108$	9.00000	0.866025
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−6.00000	−0.572078
$111$	−30.0000	−2.84747
$112$	0	0
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	6.00000	0.557086
$117$	30.0000	2.77350
$118$	1.00000	0.0920575
$119$	0	0
$120$	9.00000	0.821584
$121$	−7.00000	−0.636364
$122$	−12.0000	−1.08643
$123$	−6.00000	−0.541002
$124$	10.0000	0.898027
$125$	−3.00000	−0.268328
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	1.00000	0.0883883
$129$	12.0000	1.05654
$130$	15.0000	1.31559
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	−8.00000	−0.691095
$135$	27.0000	2.32379
$136$	−6.00000	−0.514496
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	−6.00000	−0.510754
$139$	7.00000	0.593732	0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000	0.593732	0.296866	+	0.954919i	$0.404058\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	−3.00000	−0.251754
$143$	−10.0000	−0.836242
$144$	6.00000	0.500000
$145$	18.0000	1.49482
$146$	6.00000	0.496564
$147$	0	0
$148$	−10.0000	−0.821995
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	12.0000	0.979796
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	0	0
$153$	−36.0000	−2.91043
$154$	0	0
$155$	30.0000	2.40966
$156$	15.0000	1.20096
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	1.00000	0.0795557
$159$	−36.0000	−2.85499
$160$	3.00000	0.237171
$161$	0	0
$162$	9.00000	0.707107
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−2.00000	−0.156174
$165$	−18.0000	−1.40130
$166$	−14.0000	−1.08661
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	−18.0000	−1.38054
$171$	0	0
$172$	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	$0.548590\pi$
$174$	18.0000	1.36458
$175$	0	0
$176$	−2.00000	−0.150756
$177$	3.00000	0.225494
$178$	7.00000	0.524672
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	18.0000	1.34164
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	−36.0000	−2.66120
$184$	−2.00000	−0.147442
$185$	−30.0000	−2.20564
$186$	30.0000	2.19971
$187$	12.0000	0.877527
$188$	3.00000	0.218797
$189$	0	0
$190$	0	0
$191$	−21.0000	−1.51951	−0.759753	−	0.650211i	$-0.774680\pi$
$191$	−21.0000	−1.51951	−0.759753	+	0.650211i	$0.774680\pi$
$192$	3.00000	0.216506
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	11.0000	0.789754
$195$	45.0000	3.22252
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−12.0000	−0.852803
$199$	−9.00000	−0.637993	−0.318997	−	0.947756i	$-0.603346\pi$
$199$	−9.00000	−0.637993	−0.318997	+	0.947756i	$0.603346\pi$
$200$	4.00000	0.282843
$201$	−24.0000	−1.69283
$202$	−3.00000	−0.211079
$203$	0	0
$204$	−18.0000	−1.26025
$205$	−6.00000	−0.419058
$206$	−13.0000	−0.905753
$207$	−12.0000	−0.834058
$208$	5.00000	0.346688
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−12.0000	−0.824163
$213$	−9.00000	−0.616670
$214$	−1.00000	−0.0683586
$215$	12.0000	0.818393
$216$	9.00000	0.612372
$217$	0	0
$218$	2.00000	0.135457
$219$	18.0000	1.21633
$220$	−6.00000	−0.404520
$221$	−30.0000	−2.01802
$222$	−30.0000	−2.01347
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	24.0000	1.60000
$226$	12.0000	0.798228
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	6.00000	0.393919
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	30.0000	1.96116
$235$	9.00000	0.587095
$236$	1.00000	0.0650945
$237$	3.00000	0.194871
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	9.00000	0.580948
$241$	23.0000	1.48156	0.740780	−	0.671748i	$-0.234456\pi$
$241$	23.0000	1.48156	0.740780	+	0.671748i	$0.234456\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	−12.0000	−0.768221
$245$	0	0
$246$	−6.00000	−0.382546
$247$	0	0
$248$	10.0000	0.635001
$249$	−42.0000	−2.66164
$250$	−3.00000	−0.189737
$251$	7.00000	0.441836	0.220918	−	0.975292i	$-0.429095\pi$
$251$	7.00000	0.441836	0.220918	+	0.975292i	$0.429095\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	7.00000	0.439219
$255$	−54.0000	−3.38161
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	12.0000	0.747087
$259$	0	0
$260$	15.0000	0.930261
$261$	36.0000	2.22834
$262$	6.00000	0.370681
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	−6.00000	−0.369274
$265$	−36.0000	−2.21146
$266$	0	0
$267$	21.0000	1.28518
$268$	−8.00000	−0.488678
$269$	−5.00000	−0.304855	−0.152428	−	0.988315i	$-0.548709\pi$
$269$	−5.00000	−0.304855	−0.152428	+	0.988315i	$0.548709\pi$
$270$	27.0000	1.64317
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−4.00000	−0.241649
$275$	−8.00000	−0.482418
$276$	−6.00000	−0.361158
$277$	−13.0000	−0.781094	−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000	−0.781094	−0.390547	+	0.920583i	$0.627714\pi$
$278$	7.00000	0.419832
$279$	60.0000	3.59211
$280$	0	0
$281$	−9.00000	−0.536895	−0.268447	−	0.963294i	$-0.586511\pi$
$281$	−9.00000	−0.536895	−0.268447	+	0.963294i	$0.586511\pi$
$282$	9.00000	0.535942
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	−3.00000	−0.178017
$285$	0	0
$286$	−10.0000	−0.591312
$287$	0	0
$288$	6.00000	0.353553
$289$	19.0000	1.11765
$290$	18.0000	1.05700
$291$	33.0000	1.93449
$292$	6.00000	0.351123
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	0	0
$295$	3.00000	0.174667
$296$	−10.0000	−0.581238
$297$	−18.0000	−1.04447
$298$	14.0000	0.810998
$299$	−10.0000	−0.578315
$300$	12.0000	0.692820
$301$	0	0
$302$	14.0000	0.805609
$303$	−9.00000	−0.517036
$304$	0	0
$305$	−36.0000	−2.06135
$306$	−36.0000	−2.05798
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−39.0000	−2.21863
$310$	30.0000	1.70389
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	15.0000	0.849208
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	1.00000	0.0562544
$317$	−9.00000	−0.505490	−0.252745	−	0.967533i	$-0.581333\pi$
$317$	−9.00000	−0.505490	−0.252745	+	0.967533i	$0.581333\pi$
$318$	−36.0000	−2.01878
$319$	−12.0000	−0.671871
$320$	3.00000	0.167705
$321$	−3.00000	−0.167444
$322$	0	0
$323$	0	0
$324$	9.00000	0.500000
$325$	20.0000	1.10940
$326$	4.00000	0.221540
$327$	6.00000	0.331801
$328$	−2.00000	−0.110432
$329$	0	0
$330$	−18.0000	−0.990867
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−14.0000	−0.768350
$333$	−60.0000	−3.28798
$334$	−18.0000	−0.984916
$335$	−24.0000	−1.31126
$336$	0	0
$337$	31.0000	1.68868	0.844339	−	0.535810i	$-0.179994\pi$
$337$	31.0000	1.68868	0.844339	+	0.535810i	$0.179994\pi$
$338$	12.0000	0.652714
$339$	36.0000	1.95525
$340$	−18.0000	−0.976187
$341$	−20.0000	−1.08306
$342$	0	0
$343$	0	0
$344$	4.00000	0.215666
$345$	−18.0000	−0.969087
$346$	−4.00000	−0.215041
$347$	14.0000	0.751559	0.375780	−	0.926709i	$-0.377375\pi$
$347$	14.0000	0.751559	0.375780	+	0.926709i	$0.377375\pi$
$348$	18.0000	0.964901
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	45.0000	2.40192
$352$	−2.00000	−0.106600
$353$	12.0000	0.638696	0.319348	−	0.947638i	$-0.396536\pi$
$353$	12.0000	0.638696	0.319348	+	0.947638i	$0.396536\pi$
$354$	3.00000	0.159448
$355$	−9.00000	−0.477670
$356$	7.00000	0.370999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−25.0000	−1.31945	−0.659725	−	0.751507i	$-0.729327\pi$
$359$	−25.0000	−1.31945	−0.659725	+	0.751507i	$0.729327\pi$
$360$	18.0000	0.948683
$361$	−19.0000	−1.00000
$362$	18.0000	0.946059
$363$	−21.0000	−1.10221
$364$	0	0
$365$	18.0000	0.942163
$366$	−36.0000	−1.88175
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	−2.00000	−0.104257
$369$	−12.0000	−0.624695
$370$	−30.0000	−1.55963
$371$	0	0
$372$	30.0000	1.55543
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	12.0000	0.620505
$375$	−9.00000	−0.464758
$376$	3.00000	0.154713
$377$	30.0000	1.54508
$378$	0	0
$379$	27.0000	1.38690	0.693448	−	0.720506i	$-0.256091\pi$
$379$	27.0000	1.38690	0.693448	+	0.720506i	$0.256091\pi$
$380$	0	0
$381$	21.0000	1.07586
$382$	−21.0000	−1.07445
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	−26.0000	−1.32337
$387$	24.0000	1.21999
$388$	11.0000	0.558440
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	45.0000	2.27866
$391$	12.0000	0.606866
$392$	0	0
$393$	18.0000	0.907980
$394$	−6.00000	−0.302276
$395$	3.00000	0.150946
$396$	−12.0000	−0.603023
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	−9.00000	−0.451129
$399$	0	0
$400$	4.00000	0.200000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	−24.0000	−1.19701
$403$	50.0000	2.49068
$404$	−3.00000	−0.149256
$405$	27.0000	1.34164
$406$	0	0
$407$	20.0000	0.991363
$408$	−18.0000	−0.891133
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	−6.00000	−0.296319
$411$	−12.0000	−0.591916
$412$	−13.0000	−0.640464
$413$	0	0
$414$	−12.0000	−0.589768
$415$	−42.0000	−2.06170
$416$	5.00000	0.245145
$417$	21.0000	1.02837
$418$	0	0
$419$	−27.0000	−1.31904	−0.659518	−	0.751689i	$-0.729240\pi$
$419$	−27.0000	−1.31904	−0.659518	+	0.751689i	$0.729240\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	4.00000	0.194717
$423$	18.0000	0.875190
$424$	−12.0000	−0.582772
$425$	−24.0000	−1.16417
$426$	−9.00000	−0.436051
$427$	0	0
$428$	−1.00000	−0.0483368
$429$	−30.0000	−1.44841
$430$	12.0000	0.578691
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	9.00000	0.433013
$433$	−21.0000	−1.00920	−0.504598	−	0.863355i	$-0.668359\pi$
$433$	−21.0000	−1.00920	−0.504598	+	0.863355i	$0.668359\pi$
$434$	0	0
$435$	54.0000	2.58910
$436$	2.00000	0.0957826
$437$	0	0
$438$	18.0000	0.860073
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	−6.00000	−0.286039
$441$	0	0
$442$	−30.0000	−1.42695
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−30.0000	−1.42374
$445$	21.0000	0.995495
$446$	−2.00000	−0.0947027
$447$	42.0000	1.98653
$448$	0	0
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	24.0000	1.13137
$451$	4.00000	0.188353
$452$	12.0000	0.564433
$453$	42.0000	1.97333
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.0000	1.07589	0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000	1.07589	0.537947	+	0.842978i	$0.319200\pi$
$458$	4.00000	0.186908
$459$	−54.0000	−2.52050
$460$	−6.00000	−0.279751
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	7.00000	0.325318	0.162659	−	0.986682i	$-0.447993\pi$
$463$	7.00000	0.325318	0.162659	+	0.986682i	$0.447993\pi$
$464$	6.00000	0.278543
$465$	90.0000	4.17365
$466$	2.00000	0.0926482
$467$	−2.00000	−0.0925490	−0.0462745	−	0.998929i	$-0.514735\pi$
$467$	−2.00000	−0.0925490	−0.0462745	+	0.998929i	$0.514735\pi$
$468$	30.0000	1.38675
$469$	0	0
$470$	9.00000	0.415139
$471$	42.0000	1.93526
$472$	1.00000	0.0460287
$473$	−8.00000	−0.367840
$474$	3.00000	0.137795
$475$	0	0
$476$	0	0
$477$	−72.0000	−3.29665
$478$	−24.0000	−1.09773
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	9.00000	0.410792
$481$	−50.0000	−2.27980
$482$	23.0000	1.04762
$483$	0	0
$484$	−7.00000	−0.318182
$485$	33.0000	1.49845
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−12.0000	−0.543214
$489$	12.0000	0.542659
$490$	0	0
$491$	23.0000	1.03798	0.518988	−	0.854782i	$-0.326309\pi$
$491$	23.0000	1.03798	0.518988	+	0.854782i	$0.326309\pi$
$492$	−6.00000	−0.270501
$493$	−36.0000	−1.62136
$494$	0	0
$495$	−36.0000	−1.61808
$496$	10.0000	0.449013
$497$	0	0
$498$	−42.0000	−1.88207
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	−3.00000	−0.134164
$501$	−54.0000	−2.41254
$502$	7.00000	0.312425
$503$	15.0000	0.668817	0.334408	−	0.942428i	$-0.391463\pi$
$503$	15.0000	0.668817	0.334408	+	0.942428i	$0.391463\pi$
$504$	0	0
$505$	−9.00000	−0.400495
$506$	4.00000	0.177822
$507$	36.0000	1.59882
$508$	7.00000	0.310575
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	−54.0000	−2.39116
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	2.00000	0.0882162
$515$	−39.0000	−1.71855
$516$	12.0000	0.528271
$517$	−6.00000	−0.263880
$518$	0	0
$519$	−12.0000	−0.526742
$520$	15.0000	0.657794
$521$	−44.0000	−1.92767	−0.963837	−	0.266491i	$-0.914136\pi$
$521$	−44.0000	−1.92767	−0.963837	+	0.266491i	$0.914136\pi$
$522$	36.0000	1.57568
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	6.00000	0.261612
$527$	−60.0000	−2.61364
$528$	−6.00000	−0.261116
$529$	−19.0000	−0.826087
$530$	−36.0000	−1.56374
$531$	6.00000	0.260378
$532$	0	0
$533$	−10.0000	−0.433148
$534$	21.0000	0.908759
$535$	−3.00000	−0.129701
$536$	−8.00000	−0.345547
$537$	−12.0000	−0.517838
$538$	−5.00000	−0.215565
$539$	0	0
$540$	27.0000	1.16190
$541$	3.00000	0.128980	0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000	0.128980	0.0644900	+	0.997918i	$0.479458\pi$
$542$	−8.00000	−0.343629
$543$	54.0000	2.31736
$544$	−6.00000	−0.257248
$545$	6.00000	0.257012
$546$	0	0
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	−4.00000	−0.170872
$549$	−72.0000	−3.07289
$550$	−8.00000	−0.341121
$551$	0	0
$552$	−6.00000	−0.255377
$553$	0	0
$554$	−13.0000	−0.552317
$555$	−90.0000	−3.82029
$556$	7.00000	0.296866
$557$	45.0000	1.90671	0.953356	−	0.301849i	$-0.0976040\pi$
$557$	45.0000	1.90671	0.953356	+	0.301849i	$0.0976040\pi$
$558$	60.0000	2.54000
$559$	20.0000	0.845910
$560$	0	0
$561$	36.0000	1.51992
$562$	−9.00000	−0.379642
$563$	14.0000	0.590030	0.295015	−	0.955493i	$-0.404675\pi$
$563$	14.0000	0.590030	0.295015	+	0.955493i	$0.404675\pi$
$564$	9.00000	0.378968
$565$	36.0000	1.51453
$566$	12.0000	0.504398
$567$	0	0
$568$	−3.00000	−0.125877
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	0	0
$571$	−14.0000	−0.585882	−0.292941	−	0.956131i	$-0.594634\pi$
$571$	−14.0000	−0.585882	−0.292941	+	0.956131i	$0.594634\pi$
$572$	−10.0000	−0.418121
$573$	−63.0000	−2.63186
$574$	0	0
$575$	−8.00000	−0.333623
$576$	6.00000	0.250000
$577$	−36.0000	−1.49870	−0.749350	−	0.662174i	$-0.769634\pi$
$577$	−36.0000	−1.49870	−0.749350	+	0.662174i	$0.769634\pi$
$578$	19.0000	0.790296
$579$	−78.0000	−3.24157
$580$	18.0000	0.747409
$581$	0	0
$582$	33.0000	1.36789
$583$	24.0000	0.993978
$584$	6.00000	0.248282
$585$	90.0000	3.72104
$586$	16.0000	0.660954
$587$	−3.00000	−0.123823	−0.0619116	−	0.998082i	$-0.519720\pi$
$587$	−3.00000	−0.123823	−0.0619116	+	0.998082i	$0.519720\pi$
$588$	0	0
$589$	0	0
$590$	3.00000	0.123508
$591$	−18.0000	−0.740421
$592$	−10.0000	−0.410997
$593$	29.0000	1.19089	0.595444	−	0.803397i	$-0.296976\pi$
$593$	29.0000	1.19089	0.595444	+	0.803397i	$0.296976\pi$
$594$	−18.0000	−0.738549
$595$	0	0
$596$	14.0000	0.573462
$597$	−27.0000	−1.10504
$598$	−10.0000	−0.408930
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	12.0000	0.489898
$601$	36.0000	1.46847	0.734235	−	0.678895i	$-0.237541\pi$
$601$	36.0000	1.46847	0.734235	+	0.678895i	$0.237541\pi$
$602$	0	0
$603$	−48.0000	−1.95471
$604$	14.0000	0.569652
$605$	−21.0000	−0.853771
$606$	−9.00000	−0.365600
$607$	41.0000	1.66414	0.832069	−	0.554672i	$-0.187156\pi$
$607$	41.0000	1.66414	0.832069	+	0.554672i	$0.187156\pi$
$608$	0	0
$609$	0	0
$610$	−36.0000	−1.45760
$611$	15.0000	0.606835
$612$	−36.0000	−1.45521
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	12.0000	0.484281
$615$	−18.0000	−0.725830
$616$	0	0
$617$	29.0000	1.16750	0.583748	−	0.811935i	$-0.301586\pi$
$617$	29.0000	1.16750	0.583748	+	0.811935i	$0.301586\pi$
$618$	−39.0000	−1.56881
$619$	−31.0000	−1.24600	−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000	−1.24600	−0.622998	+	0.782224i	$0.714085\pi$
$620$	30.0000	1.20483
$621$	−18.0000	−0.722315
$622$	28.0000	1.12270
$623$	0	0
$624$	15.0000	0.600481
$625$	−29.0000	−1.16000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	14.0000	0.558661
$629$	60.0000	2.39236
$630$	0	0
$631$	−5.00000	−0.199047	−0.0995234	−	0.995035i	$-0.531732\pi$
$631$	−5.00000	−0.199047	−0.0995234	+	0.995035i	$0.531732\pi$
$632$	1.00000	0.0397779
$633$	12.0000	0.476957
$634$	−9.00000	−0.357436
$635$	21.0000	0.833360
$636$	−36.0000	−1.42749
$637$	0	0
$638$	−12.0000	−0.475085
$639$	−18.0000	−0.712069
$640$	3.00000	0.118585
$641$	−25.0000	−0.987441	−0.493720	−	0.869621i	$-0.664363\pi$
$641$	−25.0000	−0.987441	−0.493720	+	0.869621i	$0.664363\pi$
$642$	−3.00000	−0.118401
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	36.0000	1.41750
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	9.00000	0.353553
$649$	−2.00000	−0.0785069
$650$	20.0000	0.784465
$651$	0	0
$652$	4.00000	0.156652
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	6.00000	0.234619
$655$	18.0000	0.703318
$656$	−2.00000	−0.0780869
$657$	36.0000	1.40449
$658$	0	0
$659$	−3.00000	−0.116863	−0.0584317	−	0.998291i	$-0.518610\pi$
$659$	−3.00000	−0.116863	−0.0584317	+	0.998291i	$0.518610\pi$
$660$	−18.0000	−0.700649
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	−12.0000	−0.466393
$663$	−90.0000	−3.49531
$664$	−14.0000	−0.543305
$665$	0	0
$666$	−60.0000	−2.32495
$667$	−12.0000	−0.464642
$668$	−18.0000	−0.696441
$669$	−6.00000	−0.231973
$670$	−24.0000	−0.927201
$671$	24.0000	0.926510
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	31.0000	1.19408
$675$	36.0000	1.38564
$676$	12.0000	0.461538
$677$	−17.0000	−0.653363	−0.326682	−	0.945134i	$-0.605930\pi$
$677$	−17.0000	−0.653363	−0.326682	+	0.945134i	$0.605930\pi$
$678$	36.0000	1.38257
$679$	0	0
$680$	−18.0000	−0.690268
$681$	0	0
$682$	−20.0000	−0.765840
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	12.0000	0.457829
$688$	4.00000	0.152499
$689$	−60.0000	−2.28582
$690$	−18.0000	−0.685248
$691$	5.00000	0.190209	0.0951045	−	0.995467i	$-0.469681\pi$
$691$	5.00000	0.190209	0.0951045	+	0.995467i	$0.469681\pi$
$692$	−4.00000	−0.152057
$693$	0	0
$694$	14.0000	0.531433
$695$	21.0000	0.796575
$696$	18.0000	0.682288
$697$	12.0000	0.454532
$698$	−20.0000	−0.757011
$699$	6.00000	0.226941
$700$	0	0
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	45.0000	1.69842
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	27.0000	1.01688
$706$	12.0000	0.451626
$707$	0	0
$708$	3.00000	0.112747
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	−9.00000	−0.337764
$711$	6.00000	0.225018
$712$	7.00000	0.262336
$713$	−20.0000	−0.749006
$714$	0	0
$715$	−30.0000	−1.12194
$716$	−4.00000	−0.149487
$717$	−72.0000	−2.68889
$718$	−25.0000	−0.932992
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	18.0000	0.670820
$721$	0	0
$722$	−19.0000	−0.707107
$723$	69.0000	2.56614
$724$	18.0000	0.668965
$725$	24.0000	0.891338
$726$	−21.0000	−0.779383
$727$	42.0000	1.55769	0.778847	−	0.627214i	$-0.215805\pi$
$727$	42.0000	1.55769	0.778847	+	0.627214i	$0.215805\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	18.0000	0.666210
$731$	−24.0000	−0.887672
$732$	−36.0000	−1.33060
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	2.00000	0.0738213
$735$	0	0
$736$	−2.00000	−0.0737210
$737$	16.0000	0.589368
$738$	−12.0000	−0.441726
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	−30.0000	−1.10282
$741$	0	0
$742$	0	0
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	30.0000	1.09985
$745$	42.0000	1.53876
$746$	14.0000	0.512576
$747$	−84.0000	−3.07340
$748$	12.0000	0.438763
$749$	0	0
$750$	−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	$0.606514\pi$
$752$	3.00000	0.109399
$753$	21.0000	0.765283
$754$	30.0000	1.09254
$755$	42.0000	1.52854
$756$	0	0
$757$	31.0000	1.12671	0.563357	−	0.826214i	$-0.309510\pi$
$757$	31.0000	1.12671	0.563357	+	0.826214i	$0.309510\pi$
$758$	27.0000	0.980684
$759$	12.0000	0.435572
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	21.0000	0.760750
$763$	0	0
$764$	−21.0000	−0.759753
$765$	−108.000	−3.90475
$766$	−6.00000	−0.216789
$767$	5.00000	0.180540
$768$	3.00000	0.108253
$769$	−16.0000	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−26.0000	−0.935760
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	24.0000	0.862662
$775$	40.0000	1.43684
$776$	11.0000	0.394877
$777$	0	0
$778$	−9.00000	−0.322666
$779$	0	0
$780$	45.0000	1.61126
$781$	6.00000	0.214697
$782$	12.0000	0.429119
$783$	54.0000	1.92980
$784$	0	0
$785$	42.0000	1.49904
$786$	18.0000	0.642039
$787$	−6.00000	−0.213877	−0.106938	−	0.994266i	$-0.534105\pi$
$787$	−6.00000	−0.213877	−0.106938	+	0.994266i	$0.534105\pi$
$788$	−6.00000	−0.213741
$789$	18.0000	0.640817
$790$	3.00000	0.106735
$791$	0	0
$792$	−12.0000	−0.426401
$793$	−60.0000	−2.13066
$794$	−7.00000	−0.248421
$795$	−108.000	−3.83037
$796$	−9.00000	−0.318997
$797$	26.0000	0.920967	0.460484	−	0.887668i	$-0.347676\pi$
$797$	26.0000	0.920967	0.460484	+	0.887668i	$0.347676\pi$
$798$	0	0
$799$	−18.0000	−0.636794
$800$	4.00000	0.141421
$801$	42.0000	1.48400
$802$	−30.0000	−1.05934
$803$	−12.0000	−0.423471
$804$	−24.0000	−0.846415
$805$	0	0
$806$	50.0000	1.76117
$807$	−15.0000	−0.528025
$808$	−3.00000	−0.105540
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	27.0000	0.948683
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	20.0000	0.701000
$815$	12.0000	0.420342
$816$	−18.0000	−0.630126
$817$	0	0
$818$	4.00000	0.139857
$819$	0	0
$820$	−6.00000	−0.209529
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	−12.0000	−0.418548
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	−13.0000	−0.452876
$825$	−24.0000	−0.835573
$826$	0	0
$827$	37.0000	1.28662	0.643308	−	0.765607i	$-0.277561\pi$
$827$	37.0000	1.28662	0.643308	+	0.765607i	$0.277561\pi$
$828$	−12.0000	−0.417029
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	−42.0000	−1.45784
$831$	−39.0000	−1.35290
$832$	5.00000	0.173344
$833$	0	0
$834$	21.0000	0.727171
$835$	−54.0000	−1.86875
$836$	0	0
$837$	90.0000	3.11086
$838$	−27.0000	−0.932700
$839$	52.0000	1.79524	0.897620	−	0.440771i	$-0.145295\pi$
$839$	52.0000	1.79524	0.897620	+	0.440771i	$0.145295\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−13.0000	−0.448010
$843$	−27.0000	−0.929929
$844$	4.00000	0.137686
$845$	36.0000	1.23844
$846$	18.0000	0.618853
$847$	0	0
$848$	−12.0000	−0.412082
$849$	36.0000	1.23552
$850$	−24.0000	−0.823193
$851$	20.0000	0.685591
$852$	−9.00000	−0.308335
$853$	−56.0000	−1.91740	−0.958702	−	0.284413i	$-0.908201\pi$
$853$	−56.0000	−1.91740	−0.958702	+	0.284413i	$0.908201\pi$
$854$	0	0
$855$	0	0
$856$	−1.00000	−0.0341793
$857$	7.00000	0.239115	0.119558	−	0.992827i	$-0.461852\pi$
$857$	7.00000	0.239115	0.119558	+	0.992827i	$0.461852\pi$
$858$	−30.0000	−1.02418
$859$	41.0000	1.39890	0.699451	−	0.714681i	$-0.253428\pi$
$859$	41.0000	1.39890	0.699451	+	0.714681i	$0.253428\pi$
$860$	12.0000	0.409197
$861$	0	0
$862$	24.0000	0.817443
$863$	54.0000	1.83818	0.919091	−	0.394046i	$-0.128925\pi$
$863$	54.0000	1.83818	0.919091	+	0.394046i	$0.128925\pi$
$864$	9.00000	0.306186
$865$	−12.0000	−0.408012
$866$	−21.0000	−0.713609
$867$	57.0000	1.93582
$868$	0	0
$869$	−2.00000	−0.0678454
$870$	54.0000	1.83077
$871$	−40.0000	−1.35535
$872$	2.00000	0.0677285
$873$	66.0000	2.23376
$874$	0	0
$875$	0	0
$876$	18.0000	0.608164
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	16.0000	0.539974
$879$	48.0000	1.61900
$880$	−6.00000	−0.202260
$881$	−20.0000	−0.673817	−0.336909	−	0.941537i	$-0.609381\pi$
$881$	−20.0000	−0.673817	−0.336909	+	0.941537i	$0.609381\pi$
$882$	0	0
$883$	37.0000	1.24515	0.622575	−	0.782560i	$-0.286087\pi$
$883$	37.0000	1.24515	0.622575	+	0.782560i	$0.286087\pi$
$884$	−30.0000	−1.00901
$885$	9.00000	0.302532
$886$	−12.0000	−0.403148
$887$	10.0000	0.335767	0.167884	−	0.985807i	$-0.446307\pi$
$887$	10.0000	0.335767	0.167884	+	0.985807i	$0.446307\pi$
$888$	−30.0000	−1.00673
$889$	0	0
$890$	21.0000	0.703922
$891$	−18.0000	−0.603023
$892$	−2.00000	−0.0669650
$893$	0	0
$894$	42.0000	1.40469
$895$	−12.0000	−0.401116
$896$	0	0
$897$	−30.0000	−1.00167
$898$	22.0000	0.734150
$899$	60.0000	2.00111
$900$	24.0000	0.800000
$901$	72.0000	2.39867
$902$	4.00000	0.133185
$903$	0	0
$904$	12.0000	0.399114
$905$	54.0000	1.79502
$906$	42.0000	1.39536
$907$	36.0000	1.19536	0.597680	−	0.801735i	$-0.296089\pi$
$907$	36.0000	1.19536	0.597680	+	0.801735i	$0.296089\pi$
$908$	0	0
$909$	−18.0000	−0.597022
$910$	0	0
$911$	28.0000	0.927681	0.463841	−	0.885919i	$-0.346471\pi$
$911$	28.0000	0.927681	0.463841	+	0.885919i	$0.346471\pi$
$912$	0	0
$913$	28.0000	0.926665
$914$	23.0000	0.760772
$915$	−108.000	−3.57037
$916$	4.00000	0.132164
$917$	0	0
$918$	−54.0000	−1.78227
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−6.00000	−0.197814
$921$	36.0000	1.18624
$922$	2.00000	0.0658665
$923$	−15.0000	−0.493731
$924$	0	0
$925$	−40.0000	−1.31519
$926$	7.00000	0.230034
$927$	−78.0000	−2.56186
$928$	6.00000	0.196960
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	90.0000	2.95122
$931$	0	0
$932$	2.00000	0.0655122
$933$	84.0000	2.75004
$934$	−2.00000	−0.0654420
$935$	36.0000	1.17733
$936$	30.0000	0.980581
$937$	36.0000	1.17607	0.588034	−	0.808836i	$-0.299902\pi$
$937$	36.0000	1.17607	0.588034	+	0.808836i	$0.299902\pi$
$938$	0	0
$939$	−42.0000	−1.37062
$940$	9.00000	0.293548
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	42.0000	1.36843
$943$	4.00000	0.130258
$944$	1.00000	0.0325472
$945$	0	0
$946$	−8.00000	−0.260102
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	3.00000	0.0974355
$949$	30.0000	0.973841
$950$	0	0
$951$	−27.0000	−0.875535
$952$	0	0
$953$	−23.0000	−0.745043	−0.372522	−	0.928024i	$-0.621507\pi$
$953$	−23.0000	−0.745043	−0.372522	+	0.928024i	$0.621507\pi$
$954$	−72.0000	−2.33109
$955$	−63.0000	−2.03863
$956$	−24.0000	−0.776215
$957$	−36.0000	−1.16371
$958$	−36.0000	−1.16311
$959$	0	0
$960$	9.00000	0.290474
$961$	69.0000	2.22581
$962$	−50.0000	−1.61206
$963$	−6.00000	−0.193347
$964$	23.0000	0.740780
$965$	−78.0000	−2.51091
$966$	0	0
$967$	−28.0000	−0.900419	−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000	−0.900419	−0.450210	+	0.892923i	$0.648651\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	33.0000	1.05957
$971$	54.0000	1.73294	0.866471	−	0.499227i	$-0.166383\pi$
$971$	54.0000	1.73294	0.866471	+	0.499227i	$0.166383\pi$
$972$	0	0
$973$	0	0
$974$	2.00000	0.0640841
$975$	60.0000	1.92154
$976$	−12.0000	−0.384111
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	12.0000	0.383718
$979$	−14.0000	−0.447442
$980$	0	0
$981$	12.0000	0.383131
$982$	23.0000	0.733959
$983$	−56.0000	−1.78612	−0.893061	−	0.449935i	$-0.851447\pi$
$983$	−56.0000	−1.78612	−0.893061	+	0.449935i	$0.851447\pi$
$984$	−6.00000	−0.191273
$985$	−18.0000	−0.573528
$986$	−36.0000	−1.14647
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	−36.0000	−1.14416
$991$	−35.0000	−1.11181	−0.555906	−	0.831245i	$-0.687628\pi$
$991$	−35.0000	−1.11181	−0.555906	+	0.831245i	$0.687628\pi$
$992$	10.0000	0.317500
$993$	−36.0000	−1.14243
$994$	0	0
$995$	−27.0000	−0.855958
$996$	−42.0000	−1.33082
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	10.0000	0.316544
$999$	−90.0000	−2.84747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7742.2.a.n.1.1		1
7.6	odd	2		158.2.a.c.1.1	✓	1
21.20	even	2		1422.2.a.c.1.1		1
28.27	even	2		1264.2.a.h.1.1		1
35.34	odd	2		3950.2.a.f.1.1		1
56.13	odd	2		5056.2.a.v.1.1		1
56.27	even	2		5056.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
158.2.a.c.1.1	✓	1	7.6	odd	2
1264.2.a.h.1.1		1	28.27	even	2
1422.2.a.c.1.1		1	21.20	even	2
3950.2.a.f.1.1		1	35.34	odd	2
5056.2.a.b.1.1		1	56.27	even	2
5056.2.a.v.1.1		1	56.13	odd	2
7742.2.a.n.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7742 = 2 \cdot 7^{2} \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7742.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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