LMFDB - Embedded newform 6422.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, 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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +3.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +3.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +3.00000 q^{10} +3.00000 q^{12} -3.00000 q^{14} +9.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} +6.00000 q^{18} -1.00000 q^{19} +3.00000 q^{20} -9.00000 q^{21} +6.00000 q^{23} +3.00000 q^{24} +4.00000 q^{25} +9.00000 q^{27} -3.00000 q^{28} -8.00000 q^{29} +9.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} +5.00000 q^{34} -9.00000 q^{35} +6.00000 q^{36} +5.00000 q^{37} -1.00000 q^{38} +3.00000 q^{40} +10.0000 q^{41} -9.00000 q^{42} +7.00000 q^{43} +18.0000 q^{45} +6.00000 q^{46} +1.00000 q^{47} +3.00000 q^{48} +2.00000 q^{49} +4.00000 q^{50} +15.0000 q^{51} -10.0000 q^{53} +9.00000 q^{54} -3.00000 q^{56} -3.00000 q^{57} -8.00000 q^{58} -6.00000 q^{59} +9.00000 q^{60} -6.00000 q^{61} -8.00000 q^{62} -18.0000 q^{63} +1.00000 q^{64} +4.00000 q^{67} +5.00000 q^{68} +18.0000 q^{69} -9.00000 q^{70} +5.00000 q^{71} +6.00000 q^{72} -8.00000 q^{73} +5.00000 q^{74} +12.0000 q^{75} -1.00000 q^{76} +12.0000 q^{79} +3.00000 q^{80} +9.00000 q^{81} +10.0000 q^{82} +8.00000 q^{83} -9.00000 q^{84} +15.0000 q^{85} +7.00000 q^{86} -24.0000 q^{87} +18.0000 q^{90} +6.00000 q^{92} -24.0000 q^{93} +1.00000 q^{94} -3.00000 q^{95} +3.00000 q^{96} +2.00000 q^{97} +2.00000 q^{98} +O(q^{100})\)

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
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	1.00000	0.353553
$9$	6.00000	2.00000
$10$	3.00000	0.948683
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	3.00000	0.866025
$13$	0	0
$13$	0	0
$14$	−3.00000	−0.801784
$15$	9.00000	2.32379
$16$	1.00000	0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	6.00000	1.41421
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	3.00000	0.670820
$21$	−9.00000	−1.96396
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	3.00000	0.612372
$25$	4.00000	0.800000
$26$	0	0
$27$	9.00000	1.73205
$28$	−3.00000	−0.566947
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	9.00000	1.64317
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.00000	0.857493
$35$	−9.00000	−1.52128
$36$	6.00000	1.00000
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	3.00000	0.474342
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	−9.00000	−1.38873
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	0	0
$45$	18.0000	2.68328
$46$	6.00000	0.884652
$47$	1.00000	0.145865	0.0729325	−	0.997337i	$-0.476764\pi$
$47$	1.00000	0.145865	0.0729325	+	0.997337i	$0.476764\pi$
$48$	3.00000	0.433013
$49$	2.00000	0.285714
$50$	4.00000	0.565685
$51$	15.0000	2.10042
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	9.00000	1.22474
$55$	0	0
$56$	−3.00000	−0.400892
$57$	−3.00000	−0.397360
$58$	−8.00000	−1.05045
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	9.00000	1.16190
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−8.00000	−1.01600
$63$	−18.0000	−2.26779
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	5.00000	0.606339
$69$	18.0000	2.16695
$70$	−9.00000	−1.07571
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	6.00000	0.707107
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	5.00000	0.581238
$75$	12.0000	1.38564
$76$	−1.00000	−0.114708
$77$	0	0
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	3.00000	0.335410
$81$	9.00000	1.00000
$82$	10.0000	1.10432
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	−9.00000	−0.981981
$85$	15.0000	1.62698
$86$	7.00000	0.754829
$87$	−24.0000	−2.57307
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	18.0000	1.89737
$91$	0	0
$92$	6.00000	0.625543
$93$	−24.0000	−2.48868
$94$	1.00000	0.103142
$95$	−3.00000	−0.307794
$96$	3.00000	0.306186
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	2.00000	0.202031
$99$	0	0
$100$	4.00000	0.400000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	15.0000	1.48522
$103$	−18.0000	−1.77359	−0.886796	−	0.462160i	$-0.847074\pi$
$103$	−18.0000	−1.77359	−0.886796	+	0.462160i	$0.847074\pi$
$104$	0	0
$105$	−27.0000	−2.63493
$106$	−10.0000	−0.971286
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	9.00000	0.866025
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	15.0000	1.42374
$112$	−3.00000	−0.283473
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−3.00000	−0.280976
$115$	18.0000	1.67851
$116$	−8.00000	−0.742781
$117$	0	0
$118$	−6.00000	−0.552345
$119$	−15.0000	−1.37505
$120$	9.00000	0.821584
$121$	−11.0000	−1.00000
$122$	−6.00000	−0.543214
$123$	30.0000	2.70501
$124$	−8.00000	−0.718421
$125$	−3.00000	−0.268328
$126$	−18.0000	−1.60357
$127$	−10.0000	−0.887357	−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000	−0.887357	−0.443678	+	0.896186i	$0.646327\pi$
$128$	1.00000	0.0883883
$129$	21.0000	1.84895
$130$	0	0
$131$	−5.00000	−0.436852	−0.218426	−	0.975854i	$-0.570092\pi$
$131$	−5.00000	−0.436852	−0.218426	+	0.975854i	$0.570092\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	4.00000	0.345547
$135$	27.0000	2.32379
$136$	5.00000	0.428746
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	18.0000	1.53226
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	−9.00000	−0.760639
$141$	3.00000	0.252646
$142$	5.00000	0.419591
$143$	0	0
$144$	6.00000	0.500000
$145$	−24.0000	−1.99309
$146$	−8.00000	−0.662085
$147$	6.00000	0.494872
$148$	5.00000	0.410997
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	12.0000	0.979796
$151$	−3.00000	−0.244137	−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000	−0.244137	−0.122068	+	0.992522i	$0.538953\pi$
$152$	−1.00000	−0.0811107
$153$	30.0000	2.42536
$154$	0	0
$155$	−24.0000	−1.92773
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	12.0000	0.954669
$159$	−30.0000	−2.37915
$160$	3.00000	0.237171
$161$	−18.0000	−1.41860
$162$	9.00000	0.707107
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	8.00000	0.620920
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−9.00000	−0.694365
$169$	0	0
$170$	15.0000	1.15045
$171$	−6.00000	−0.458831
$172$	7.00000	0.533745
$173$	26.0000	1.97674	0.988372	−	0.152057i	$-0.0485898\pi$
$173$	26.0000	1.97674	0.988372	+	0.152057i	$0.0485898\pi$
$174$	−24.0000	−1.81944
$175$	−12.0000	−0.907115
$176$	0	0
$177$	−18.0000	−1.35296
$178$	0	0
$179$	5.00000	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$180$	18.0000	1.34164
$181$	−24.0000	−1.78391	−0.891953	−	0.452128i	$-0.850665\pi$
$181$	−24.0000	−1.78391	−0.891953	+	0.452128i	$0.850665\pi$
$182$	0	0
$183$	−18.0000	−1.33060
$184$	6.00000	0.442326
$185$	15.0000	1.10282
$186$	−24.0000	−1.75977
$187$	0	0
$188$	1.00000	0.0729325
$189$	−27.0000	−1.96396
$190$	−3.00000	−0.217643
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	3.00000	0.216506
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	2.00000	0.142857
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	4.00000	0.282843
$201$	12.0000	0.846415
$202$	−18.0000	−1.26648
$203$	24.0000	1.68447
$204$	15.0000	1.05021
$205$	30.0000	2.09529
$206$	−18.0000	−1.25412
$207$	36.0000	2.50217
$208$	0	0
$209$	0	0
$210$	−27.0000	−1.86318
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	−10.0000	−0.686803
$213$	15.0000	1.02778
$214$	16.0000	1.09374
$215$	21.0000	1.43219
$216$	9.00000	0.612372
$217$	24.0000	1.62923
$218$	5.00000	0.338643
$219$	−24.0000	−1.62177
$220$	0	0
$221$	0	0
$222$	15.0000	1.00673
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	−3.00000	−0.200446
$225$	24.0000	1.60000
$226$	6.00000	0.399114
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	−3.00000	−0.198680
$229$	−23.0000	−1.51988	−0.759941	−	0.649992i	$-0.774772\pi$
$229$	−23.0000	−1.51988	−0.759941	+	0.649992i	$0.774772\pi$
$230$	18.0000	1.18688
$231$	0	0
$232$	−8.00000	−0.525226
$233$	−11.0000	−0.720634	−0.360317	−	0.932830i	$-0.617331\pi$
$233$	−11.0000	−0.720634	−0.360317	+	0.932830i	$0.617331\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	−6.00000	−0.390567
$237$	36.0000	2.33845
$238$	−15.0000	−0.972306
$239$	5.00000	0.323423	0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000	0.323423	0.161712	+	0.986838i	$0.448299\pi$
$240$	9.00000	0.580948
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	−6.00000	−0.384111
$245$	6.00000	0.383326
$246$	30.0000	1.91273
$247$	0	0
$248$	−8.00000	−0.508001
$249$	24.0000	1.52094
$250$	−3.00000	−0.189737
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	−18.0000	−1.13389
$253$	0	0
$254$	−10.0000	−0.627456
$255$	45.0000	2.81801
$256$	1.00000	0.0625000
$257$	−21.0000	−1.30994	−0.654972	−	0.755653i	$-0.727320\pi$
$257$	−21.0000	−1.30994	−0.654972	+	0.755653i	$0.727320\pi$
$258$	21.0000	1.30740
$259$	−15.0000	−0.932055
$260$	0	0
$261$	−48.0000	−2.97113
$262$	−5.00000	−0.308901
$263$	14.0000	0.863277	0.431638	−	0.902047i	$-0.357936\pi$
$263$	14.0000	0.863277	0.431638	+	0.902047i	$0.357936\pi$
$264$	0	0
$265$	−30.0000	−1.84289
$266$	3.00000	0.183942
$267$	0	0
$268$	4.00000	0.244339
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	27.0000	1.64317
$271$	−7.00000	−0.425220	−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000	−0.425220	−0.212610	+	0.977137i	$0.568196\pi$
$272$	5.00000	0.303170
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	18.0000	1.08347
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−5.00000	−0.299880
$279$	−48.0000	−2.87368
$280$	−9.00000	−0.537853
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	3.00000	0.178647
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	5.00000	0.296695
$285$	−9.00000	−0.533114
$286$	0	0
$287$	−30.0000	−1.77084
$288$	6.00000	0.353553
$289$	8.00000	0.470588
$290$	−24.0000	−1.40933
$291$	6.00000	0.351726
$292$	−8.00000	−0.468165
$293$	−19.0000	−1.10999	−0.554996	−	0.831853i	$-0.687280\pi$
$293$	−19.0000	−1.10999	−0.554996	+	0.831853i	$0.687280\pi$
$294$	6.00000	0.349927
$295$	−18.0000	−1.04800
$296$	5.00000	0.290619
$297$	0	0
$298$	22.0000	1.27443
$299$	0	0
$300$	12.0000	0.692820
$301$	−21.0000	−1.21042
$302$	−3.00000	−0.172631
$303$	−54.0000	−3.10222
$304$	−1.00000	−0.0573539
$305$	−18.0000	−1.03068
$306$	30.0000	1.71499
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	−54.0000	−3.07195
$310$	−24.0000	−1.36311
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	7.00000	0.395663	0.197832	−	0.980236i	$-0.436610\pi$
$313$	7.00000	0.395663	0.197832	+	0.980236i	$0.436610\pi$
$314$	−22.0000	−1.24153
$315$	−54.0000	−3.04256
$316$	12.0000	0.675053
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	−30.0000	−1.68232
$319$	0	0
$320$	3.00000	0.167705
$321$	48.0000	2.67910
$322$	−18.0000	−1.00310
$323$	−5.00000	−0.278207
$324$	9.00000	0.500000
$325$	0	0
$326$	−10.0000	−0.553849
$327$	15.0000	0.829502
$328$	10.0000	0.552158
$329$	−3.00000	−0.165395
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	8.00000	0.439057
$333$	30.0000	1.64399
$334$	0	0
$335$	12.0000	0.655630
$336$	−9.00000	−0.490990
$337$	−11.0000	−0.599208	−0.299604	−	0.954064i	$-0.596855\pi$
$337$	−11.0000	−0.599208	−0.299604	+	0.954064i	$0.596855\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	15.0000	0.813489
$341$	0	0
$342$	−6.00000	−0.324443
$343$	15.0000	0.809924
$344$	7.00000	0.377415
$345$	54.0000	2.90726
$346$	26.0000	1.39777
$347$	7.00000	0.375780	0.187890	−	0.982190i	$-0.439835\pi$
$347$	7.00000	0.375780	0.187890	+	0.982190i	$0.439835\pi$
$348$	−24.0000	−1.28654
$349$	−33.0000	−1.76645	−0.883225	−	0.468950i	$-0.844632\pi$
$349$	−33.0000	−1.76645	−0.883225	+	0.468950i	$0.844632\pi$
$350$	−12.0000	−0.641427
$351$	0	0
$352$	0	0
$353$	−16.0000	−0.851594	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	−16.0000	−0.851594	−0.425797	+	0.904819i	$0.640006\pi$
$354$	−18.0000	−0.956689
$355$	15.0000	0.796117
$356$	0	0
$357$	−45.0000	−2.38165
$358$	5.00000	0.264258
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	18.0000	0.948683
$361$	1.00000	0.0526316
$362$	−24.0000	−1.26141
$363$	−33.0000	−1.73205
$364$	0	0
$365$	−24.0000	−1.25622
$366$	−18.0000	−0.940875
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	6.00000	0.312772
$369$	60.0000	3.12348
$370$	15.0000	0.779813
$371$	30.0000	1.55752
$372$	−24.0000	−1.24434
$373$	−38.0000	−1.96757	−0.983783	−	0.179364i	$-0.942596\pi$
$373$	−38.0000	−1.96757	−0.983783	+	0.179364i	$0.942596\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	1.00000	0.0515711
$377$	0	0
$378$	−27.0000	−1.38873
$379$	−38.0000	−1.95193	−0.975964	−	0.217930i	$-0.930070\pi$
$379$	−38.0000	−1.95193	−0.975964	+	0.217930i	$0.930070\pi$
$380$	−3.00000	−0.153897
$381$	−30.0000	−1.53695
$382$	6.00000	0.306987
$383$	−23.0000	−1.17525	−0.587623	−	0.809135i	$-0.699936\pi$
$383$	−23.0000	−1.17525	−0.587623	+	0.809135i	$0.699936\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	24.0000	1.22157
$387$	42.0000	2.13498
$388$	2.00000	0.101535
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	30.0000	1.51717
$392$	2.00000	0.101015
$393$	−15.0000	−0.756650
$394$	−3.00000	−0.151138
$395$	36.0000	1.81136
$396$	0	0
$397$	−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
$399$	9.00000	0.450564
$400$	4.00000	0.200000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	12.0000	0.598506
$403$	0	0
$404$	−18.0000	−0.895533
$405$	27.0000	1.34164
$406$	24.0000	1.19110
$407$	0	0
$408$	15.0000	0.742611
$409$	−16.0000	−0.791149	−0.395575	−	0.918434i	$-0.629455\pi$
$409$	−16.0000	−0.791149	−0.395575	+	0.918434i	$0.629455\pi$
$410$	30.0000	1.48159
$411$	−36.0000	−1.77575
$412$	−18.0000	−0.886796
$413$	18.0000	0.885722
$414$	36.0000	1.76930
$415$	24.0000	1.17811
$416$	0	0
$417$	−15.0000	−0.734553
$418$	0	0
$419$	33.0000	1.61216	0.806078	−	0.591810i	$-0.201586\pi$
$419$	33.0000	1.61216	0.806078	+	0.591810i	$0.201586\pi$
$420$	−27.0000	−1.31747
$421$	−11.0000	−0.536107	−0.268054	−	0.963404i	$-0.586380\pi$
$421$	−11.0000	−0.536107	−0.268054	+	0.963404i	$0.586380\pi$
$422$	5.00000	0.243396
$423$	6.00000	0.291730
$424$	−10.0000	−0.485643
$425$	20.0000	0.970143
$426$	15.0000	0.726752
$427$	18.0000	0.871081
$428$	16.0000	0.773389
$429$	0	0
$430$	21.0000	1.01271
$431$	35.0000	1.68589	0.842945	−	0.537999i	$-0.180820\pi$
$431$	35.0000	1.68589	0.842945	+	0.537999i	$0.180820\pi$
$432$	9.00000	0.433013
$433$	−39.0000	−1.87422	−0.937110	−	0.349034i	$-0.886510\pi$
$433$	−39.0000	−1.87422	−0.937110	+	0.349034i	$0.886510\pi$
$434$	24.0000	1.15204
$435$	−72.0000	−3.45214
$436$	5.00000	0.239457
$437$	−6.00000	−0.287019
$438$	−24.0000	−1.14676
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	−15.0000	−0.712672	−0.356336	−	0.934358i	$-0.615974\pi$
$443$	−15.0000	−0.712672	−0.356336	+	0.934358i	$0.615974\pi$
$444$	15.0000	0.711868
$445$	0	0
$446$	−7.00000	−0.331460
$447$	66.0000	3.12169
$448$	−3.00000	−0.141737
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	24.0000	1.13137
$451$	0	0
$452$	6.00000	0.282216
$453$	−9.00000	−0.422857
$454$	6.00000	0.281594
$455$	0	0
$456$	−3.00000	−0.140488
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−23.0000	−1.07472
$459$	45.0000	2.10042
$460$	18.0000	0.839254
$461$	−17.0000	−0.791769	−0.395884	−	0.918300i	$-0.629562\pi$
$461$	−17.0000	−0.791769	−0.395884	+	0.918300i	$0.629562\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	−8.00000	−0.371391
$465$	−72.0000	−3.33892
$466$	−11.0000	−0.509565
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	3.00000	0.138380
$471$	−66.0000	−3.04112
$472$	−6.00000	−0.276172
$473$	0	0
$474$	36.0000	1.65353
$475$	−4.00000	−0.183533
$476$	−15.0000	−0.687524
$477$	−60.0000	−2.74721
$478$	5.00000	0.228695
$479$	25.0000	1.14228	0.571140	−	0.820853i	$-0.306501\pi$
$479$	25.0000	1.14228	0.571140	+	0.820853i	$0.306501\pi$
$480$	9.00000	0.410792
$481$	0	0
$482$	−2.00000	−0.0910975
$483$	−54.0000	−2.45709
$484$	−11.0000	−0.500000
$485$	6.00000	0.272446
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	−6.00000	−0.271607
$489$	−30.0000	−1.35665
$490$	6.00000	0.271052
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	30.0000	1.35250
$493$	−40.0000	−1.80151
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−15.0000	−0.672842
$498$	24.0000	1.07547
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	20.0000	0.892644
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	−18.0000	−0.801784
$505$	−54.0000	−2.40297
$506$	0	0
$507$	0	0
$508$	−10.0000	−0.443678
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	45.0000	1.99263
$511$	24.0000	1.06170
$512$	1.00000	0.0441942
$513$	−9.00000	−0.397360
$514$	−21.0000	−0.926270
$515$	−54.0000	−2.37952
$516$	21.0000	0.924473
$517$	0	0
$518$	−15.0000	−0.659062
$519$	78.0000	3.42382
$520$	0	0
$521$	17.0000	0.744784	0.372392	−	0.928076i	$-0.378538\pi$
$521$	17.0000	0.744784	0.372392	+	0.928076i	$0.378538\pi$
$522$	−48.0000	−2.10090
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−5.00000	−0.218426
$525$	−36.0000	−1.57117
$526$	14.0000	0.610429
$527$	−40.0000	−1.74243
$528$	0	0
$529$	13.0000	0.565217
$530$	−30.0000	−1.30312
$531$	−36.0000	−1.56227
$532$	3.00000	0.130066
$533$	0	0
$534$	0	0
$535$	48.0000	2.07522
$536$	4.00000	0.172774
$537$	15.0000	0.647298
$538$	18.0000	0.776035
$539$	0	0
$540$	27.0000	1.16190
$541$	33.0000	1.41878	0.709390	−	0.704816i	$-0.248970\pi$
$541$	33.0000	1.41878	0.709390	+	0.704816i	$0.248970\pi$
$542$	−7.00000	−0.300676
$543$	−72.0000	−3.08982
$544$	5.00000	0.214373
$545$	15.0000	0.642529
$546$	0	0
$547$	35.0000	1.49649	0.748246	−	0.663421i	$-0.230896\pi$
$547$	35.0000	1.49649	0.748246	+	0.663421i	$0.230896\pi$
$548$	−12.0000	−0.512615
$549$	−36.0000	−1.53644
$550$	0	0
$551$	8.00000	0.340811
$552$	18.0000	0.766131
$553$	−36.0000	−1.53088
$554$	10.0000	0.424859
$555$	45.0000	1.91014
$556$	−5.00000	−0.212047
$557$	−31.0000	−1.31351	−0.656756	−	0.754103i	$-0.728072\pi$
$557$	−31.0000	−1.31351	−0.656756	+	0.754103i	$0.728072\pi$
$558$	−48.0000	−2.03200
$559$	0	0
$560$	−9.00000	−0.380319
$561$	0	0
$562$	6.00000	0.253095
$563$	−31.0000	−1.30649	−0.653247	−	0.757145i	$-0.726594\pi$
$563$	−31.0000	−1.30649	−0.653247	+	0.757145i	$0.726594\pi$
$564$	3.00000	0.126323
$565$	18.0000	0.757266
$566$	−16.0000	−0.672530
$567$	−27.0000	−1.13389
$568$	5.00000	0.209795
$569$	21.0000	0.880366	0.440183	−	0.897908i	$-0.354914\pi$
$569$	21.0000	0.880366	0.440183	+	0.897908i	$0.354914\pi$
$570$	−9.00000	−0.376969
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	−30.0000	−1.25218
$575$	24.0000	1.00087
$576$	6.00000	0.250000
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	8.00000	0.332756
$579$	72.0000	2.99222
$580$	−24.0000	−0.996546
$581$	−24.0000	−0.995688
$582$	6.00000	0.248708
$583$	0	0
$584$	−8.00000	−0.331042
$585$	0	0
$586$	−19.0000	−0.784883
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	6.00000	0.247436
$589$	8.00000	0.329634
$590$	−18.0000	−0.741048
$591$	−9.00000	−0.370211
$592$	5.00000	0.205499
$593$	32.0000	1.31408	0.657041	−	0.753855i	$-0.271808\pi$
$593$	32.0000	1.31408	0.657041	+	0.753855i	$0.271808\pi$
$594$	0	0
$595$	−45.0000	−1.84482
$596$	22.0000	0.901155
$597$	42.0000	1.71895
$598$	0	0
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	12.0000	0.489898
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	−21.0000	−0.855896
$603$	24.0000	0.977356
$604$	−3.00000	−0.122068
$605$	−33.0000	−1.34164
$606$	−54.0000	−2.19360
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	−1.00000	−0.0405554
$609$	72.0000	2.91759
$610$	−18.0000	−0.728799
$611$	0	0
$612$	30.0000	1.21268
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	−4.00000	−0.161427
$615$	90.0000	3.62915
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	−54.0000	−2.17220
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	−24.0000	−0.963863
$621$	54.0000	2.16695
$622$	18.0000	0.721734
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	7.00000	0.279776
$627$	0	0
$628$	−22.0000	−0.877896
$629$	25.0000	0.996815
$630$	−54.0000	−2.15141
$631$	43.0000	1.71180	0.855901	−	0.517139i	$-0.173003\pi$
$631$	43.0000	1.71180	0.855901	+	0.517139i	$0.173003\pi$
$632$	12.0000	0.477334
$633$	15.0000	0.596196
$634$	−22.0000	−0.873732
$635$	−30.0000	−1.19051
$636$	−30.0000	−1.18958
$637$	0	0
$638$	0	0
$639$	30.0000	1.18678
$640$	3.00000	0.118585
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	48.0000	1.89441
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	−18.0000	−0.709299
$645$	63.0000	2.48062
$646$	−5.00000	−0.196722
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	9.00000	0.353553
$649$	0	0
$650$	0	0
$651$	72.0000	2.82190
$652$	−10.0000	−0.391630
$653$	−12.0000	−0.469596	−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000	−0.469596	−0.234798	+	0.972044i	$0.575443\pi$
$654$	15.0000	0.586546
$655$	−15.0000	−0.586098
$656$	10.0000	0.390434
$657$	−48.0000	−1.87266
$658$	−3.00000	−0.116952
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	8.00000	0.310460
$665$	9.00000	0.349005
$666$	30.0000	1.16248
$667$	−48.0000	−1.85857
$668$	0	0
$669$	−21.0000	−0.811907
$670$	12.0000	0.463600
$671$	0	0
$672$	−9.00000	−0.347183
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	−11.0000	−0.423704
$675$	36.0000	1.38564
$676$	0	0
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	18.0000	0.691286
$679$	−6.00000	−0.230259
$680$	15.0000	0.575224
$681$	18.0000	0.689761
$682$	0	0
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	−6.00000	−0.229416
$685$	−36.0000	−1.37549
$686$	15.0000	0.572703
$687$	−69.0000	−2.63251
$688$	7.00000	0.266872
$689$	0	0
$690$	54.0000	2.05574
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	26.0000	0.988372
$693$	0	0
$694$	7.00000	0.265716
$695$	−15.0000	−0.568982
$696$	−24.0000	−0.909718
$697$	50.0000	1.89389
$698$	−33.0000	−1.24907
$699$	−33.0000	−1.24817
$700$	−12.0000	−0.453557
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	−5.00000	−0.188579
$704$	0	0
$705$	9.00000	0.338960
$706$	−16.0000	−0.602168
$707$	54.0000	2.03088
$708$	−18.0000	−0.676481
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	15.0000	0.562940
$711$	72.0000	2.70021
$712$	0	0
$713$	−48.0000	−1.79761
$714$	−45.0000	−1.68408
$715$	0	0
$716$	5.00000	0.186859
$717$	15.0000	0.560185
$718$	16.0000	0.597115
$719$	26.0000	0.969636	0.484818	−	0.874615i	$-0.338886\pi$
$719$	26.0000	0.969636	0.484818	+	0.874615i	$0.338886\pi$
$720$	18.0000	0.670820
$721$	54.0000	2.01107
$722$	1.00000	0.0372161
$723$	−6.00000	−0.223142
$724$	−24.0000	−0.891953
$725$	−32.0000	−1.18845
$726$	−33.0000	−1.22474
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	−24.0000	−0.888280
$731$	35.0000	1.29452
$732$	−18.0000	−0.665299
$733$	−11.0000	−0.406294	−0.203147	−	0.979148i	$-0.565117\pi$
$733$	−11.0000	−0.406294	−0.203147	+	0.979148i	$0.565117\pi$
$734$	0	0
$735$	18.0000	0.663940
$736$	6.00000	0.221163
$737$	0	0
$738$	60.0000	2.20863
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	15.0000	0.551411
$741$	0	0
$742$	30.0000	1.10133
$743$	7.00000	0.256805	0.128403	−	0.991722i	$-0.459015\pi$
$743$	7.00000	0.256805	0.128403	+	0.991722i	$0.459015\pi$
$744$	−24.0000	−0.879883
$745$	66.0000	2.41805
$746$	−38.0000	−1.39128
$747$	48.0000	1.75623
$748$	0	0
$749$	−48.0000	−1.75388
$750$	−9.00000	−0.328634
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	1.00000	0.0364662
$753$	60.0000	2.18652
$754$	0	0
$755$	−9.00000	−0.327544
$756$	−27.0000	−0.981981
$757$	14.0000	0.508839	0.254419	−	0.967094i	$-0.418116\pi$
$757$	14.0000	0.508839	0.254419	+	0.967094i	$0.418116\pi$
$758$	−38.0000	−1.38022
$759$	0	0
$760$	−3.00000	−0.108821
$761$	−2.00000	−0.0724999	−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000	−0.0724999	−0.0362500	+	0.999343i	$0.511541\pi$
$762$	−30.0000	−1.08679
$763$	−15.0000	−0.543036
$764$	6.00000	0.217072
$765$	90.0000	3.25396
$766$	−23.0000	−0.831024
$767$	0	0
$768$	3.00000	0.108253
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	−63.0000	−2.26889
$772$	24.0000	0.863779
$773$	−27.0000	−0.971123	−0.485561	−	0.874203i	$-0.661385\pi$
$773$	−27.0000	−0.971123	−0.485561	+	0.874203i	$0.661385\pi$
$774$	42.0000	1.50966
$775$	−32.0000	−1.14947
$776$	2.00000	0.0717958
$777$	−45.0000	−1.61437
$778$	36.0000	1.29066
$779$	−10.0000	−0.358287
$780$	0	0
$781$	0	0
$782$	30.0000	1.07280
$783$	−72.0000	−2.57307
$784$	2.00000	0.0714286
$785$	−66.0000	−2.35564
$786$	−15.0000	−0.535032
$787$	−18.0000	−0.641631	−0.320815	−	0.947142i	$-0.603957\pi$
$787$	−18.0000	−0.641631	−0.320815	+	0.947142i	$0.603957\pi$
$788$	−3.00000	−0.106871
$789$	42.0000	1.49524
$790$	36.0000	1.28082
$791$	−18.0000	−0.640006
$792$	0	0
$793$	0	0
$794$	−18.0000	−0.638796
$795$	−90.0000	−3.19197
$796$	14.0000	0.496217
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	9.00000	0.318597
$799$	5.00000	0.176887
$800$	4.00000	0.141421
$801$	0	0
$802$	−6.00000	−0.211867
$803$	0	0
$804$	12.0000	0.423207
$805$	−54.0000	−1.90325
$806$	0	0
$807$	54.0000	1.90089
$808$	−18.0000	−0.633238
$809$	51.0000	1.79306	0.896532	−	0.442978i	$-0.146078\pi$
$809$	51.0000	1.79306	0.896532	+	0.442978i	$0.146078\pi$
$810$	27.0000	0.948683
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	24.0000	0.842235
$813$	−21.0000	−0.736502
$814$	0	0
$815$	−30.0000	−1.05085
$816$	15.0000	0.525105
$817$	−7.00000	−0.244899
$818$	−16.0000	−0.559427
$819$	0	0
$820$	30.0000	1.04765
$821$	−13.0000	−0.453703	−0.226852	−	0.973929i	$-0.572843\pi$
$821$	−13.0000	−0.453703	−0.226852	+	0.973929i	$0.572843\pi$
$822$	−36.0000	−1.25564
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	−18.0000	−0.627060
$825$	0	0
$826$	18.0000	0.626300
$827$	26.0000	0.904109	0.452054	−	0.891990i	$-0.350691\pi$
$827$	26.0000	0.904109	0.452054	+	0.891990i	$0.350691\pi$
$828$	36.0000	1.25109
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	24.0000	0.833052
$831$	30.0000	1.04069
$832$	0	0
$833$	10.0000	0.346479
$834$	−15.0000	−0.519408
$835$	0	0
$836$	0	0
$837$	−72.0000	−2.48868
$838$	33.0000	1.13997
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	−27.0000	−0.931589
$841$	35.0000	1.20690
$842$	−11.0000	−0.379085
$843$	18.0000	0.619953
$844$	5.00000	0.172107
$845$	0	0
$846$	6.00000	0.206284
$847$	33.0000	1.13389
$848$	−10.0000	−0.343401
$849$	−48.0000	−1.64736
$850$	20.0000	0.685994
$851$	30.0000	1.02839
$852$	15.0000	0.513892
$853$	17.0000	0.582069	0.291034	−	0.956713i	$-0.406001\pi$
$853$	17.0000	0.582069	0.291034	+	0.956713i	$0.406001\pi$
$854$	18.0000	0.615947
$855$	−18.0000	−0.615587
$856$	16.0000	0.546869
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	21.0000	0.716094
$861$	−90.0000	−3.06719
$862$	35.0000	1.19210
$863$	−45.0000	−1.53182	−0.765909	−	0.642949i	$-0.777711\pi$
$863$	−45.0000	−1.53182	−0.765909	+	0.642949i	$0.777711\pi$
$864$	9.00000	0.306186
$865$	78.0000	2.65208
$866$	−39.0000	−1.32527
$867$	24.0000	0.815083
$868$	24.0000	0.814613
$869$	0	0
$870$	−72.0000	−2.44103
$871$	0	0
$872$	5.00000	0.169321
$873$	12.0000	0.406138
$874$	−6.00000	−0.202953
$875$	9.00000	0.304256
$876$	−24.0000	−0.810885
$877$	11.0000	0.371444	0.185722	−	0.982602i	$-0.440538\pi$
$877$	11.0000	0.371444	0.185722	+	0.982602i	$0.440538\pi$
$878$	14.0000	0.472477
$879$	−57.0000	−1.92256
$880$	0	0
$881$	25.0000	0.842271	0.421136	−	0.906998i	$-0.361632\pi$
$881$	25.0000	0.842271	0.421136	+	0.906998i	$0.361632\pi$
$882$	12.0000	0.404061
$883$	9.00000	0.302874	0.151437	−	0.988467i	$-0.451610\pi$
$883$	9.00000	0.302874	0.151437	+	0.988467i	$0.451610\pi$
$884$	0	0
$885$	−54.0000	−1.81519
$886$	−15.0000	−0.503935
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	15.0000	0.503367
$889$	30.0000	1.00617
$890$	0	0
$891$	0	0
$892$	−7.00000	−0.234377
$893$	−1.00000	−0.0334637
$894$	66.0000	2.20737
$895$	15.0000	0.501395
$896$	−3.00000	−0.100223
$897$	0	0
$898$	−2.00000	−0.0667409
$899$	64.0000	2.13452
$900$	24.0000	0.800000
$901$	−50.0000	−1.66574
$902$	0	0
$903$	−63.0000	−2.09651
$904$	6.00000	0.199557
$905$	−72.0000	−2.39336
$906$	−9.00000	−0.299005
$907$	−15.0000	−0.498067	−0.249033	−	0.968495i	$-0.580113\pi$
$907$	−15.0000	−0.498067	−0.249033	+	0.968495i	$0.580113\pi$
$908$	6.00000	0.199117
$909$	−108.000	−3.58213
$910$	0	0
$911$	50.0000	1.65657	0.828287	−	0.560304i	$-0.189316\pi$
$911$	50.0000	1.65657	0.828287	+	0.560304i	$0.189316\pi$
$912$	−3.00000	−0.0993399
$913$	0	0
$914$	22.0000	0.727695
$915$	−54.0000	−1.78518
$916$	−23.0000	−0.759941
$917$	15.0000	0.495344
$918$	45.0000	1.48522
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	18.0000	0.593442
$921$	−12.0000	−0.395413
$922$	−17.0000	−0.559865
$923$	0	0
$924$	0	0
$925$	20.0000	0.657596
$926$	−24.0000	−0.788689
$927$	−108.000	−3.54719
$928$	−8.00000	−0.262613
$929$	−4.00000	−0.131236	−0.0656179	−	0.997845i	$-0.520902\pi$
$929$	−4.00000	−0.131236	−0.0656179	+	0.997845i	$0.520902\pi$
$930$	−72.0000	−2.36097
$931$	−2.00000	−0.0655474
$932$	−11.0000	−0.360317
$933$	54.0000	1.76788
$934$	−8.00000	−0.261768
$935$	0	0
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	−12.0000	−0.391814
$939$	21.0000	0.685309
$940$	3.00000	0.0978492
$941$	51.0000	1.66255	0.831276	−	0.555860i	$-0.187611\pi$
$941$	51.0000	1.66255	0.831276	+	0.555860i	$0.187611\pi$
$942$	−66.0000	−2.15040
$943$	60.0000	1.95387
$944$	−6.00000	−0.195283
$945$	−81.0000	−2.63493
$946$	0	0
$947$	42.0000	1.36482	0.682408	−	0.730971i	$-0.260933\pi$
$947$	42.0000	1.36482	0.682408	+	0.730971i	$0.260933\pi$
$948$	36.0000	1.16923
$949$	0	0
$950$	−4.00000	−0.129777
$951$	−66.0000	−2.14020
$952$	−15.0000	−0.486153
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	−60.0000	−1.94257
$955$	18.0000	0.582466
$956$	5.00000	0.161712
$957$	0	0
$958$	25.0000	0.807713
$959$	36.0000	1.16250
$960$	9.00000	0.290474
$961$	33.0000	1.06452
$962$	0	0
$963$	96.0000	3.09356
$964$	−2.00000	−0.0644157
$965$	72.0000	2.31776
$966$	−54.0000	−1.73742
$967$	−37.0000	−1.18984	−0.594920	−	0.803785i	$-0.702816\pi$
$967$	−37.0000	−1.18984	−0.594920	+	0.803785i	$0.702816\pi$
$968$	−11.0000	−0.353553
$969$	−15.0000	−0.481869
$970$	6.00000	0.192648
$971$	3.00000	0.0962746	0.0481373	−	0.998841i	$-0.484672\pi$
$971$	3.00000	0.0962746	0.0481373	+	0.998841i	$0.484672\pi$
$972$	0	0
$973$	15.0000	0.480878
$974$	16.0000	0.512673
$975$	0	0
$976$	−6.00000	−0.192055
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	−30.0000	−0.959294
$979$	0	0
$980$	6.00000	0.191663
$981$	30.0000	0.957826
$982$	15.0000	0.478669
$983$	−17.0000	−0.542216	−0.271108	−	0.962549i	$-0.587390\pi$
$983$	−17.0000	−0.542216	−0.271108	+	0.962549i	$0.587390\pi$
$984$	30.0000	0.956365
$985$	−9.00000	−0.286764
$986$	−40.0000	−1.27386
$987$	−9.00000	−0.286473
$988$	0	0
$989$	42.0000	1.33552
$990$	0	0
$991$	22.0000	0.698853	0.349427	−	0.936964i	$-0.386376\pi$
$991$	22.0000	0.698853	0.349427	+	0.936964i	$0.386376\pi$
$992$	−8.00000	−0.254000
$993$	24.0000	0.761617
$994$	−15.0000	−0.475771
$995$	42.0000	1.33149
$996$	24.0000	0.760469
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	−10.0000	−0.316544
$999$	45.0000	1.42374

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.j.1.1		1
13.12	even	2		494.2.a.c.1.1	✓	1
39.38	odd	2		4446.2.a.u.1.1		1
52.51	odd	2		3952.2.a.a.1.1		1
247.246	odd	2		9386.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.a.c.1.1	✓	1	13.12	even	2
3952.2.a.a.1.1		1	52.51	odd	2
4446.2.a.u.1.1		1	39.38	odd	2
6422.2.a.j.1.1		1	1.1	even	1	trivial
9386.2.a.h.1.1		1	247.246	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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