LMFDB - Embedded newform 16.22.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,22,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 22, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	16.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:	$44.7163750859$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-71604.0 q^{3} -2.86938e7 q^{5} +8.53202e8 q^{7} -5.33322e9 q^{9} +O(q^{10})$

$q-71604.0 q^{3} -2.86938e7 q^{5} +8.53202e8 q^{7} -5.33322e9 q^{9} -8.67312e10 q^{11} -8.95323e11 q^{13} +2.05459e12 q^{15} +3.25757e12 q^{17} -2.30325e13 q^{19} -6.10927e13 q^{21} -1.46496e14 q^{23} +3.46495e14 q^{25} +1.13088e15 q^{27} -7.34052e14 q^{29} +3.14666e15 q^{31} +6.21030e15 q^{33} -2.44816e16 q^{35} -1.29638e16 q^{37} +6.41087e16 q^{39} +4.57146e16 q^{41} +2.40736e16 q^{43} +1.53030e17 q^{45} +4.49992e17 q^{47} +1.69408e17 q^{49} -2.33255e17 q^{51} +2.06484e18 q^{53} +2.48864e18 q^{55} +1.64922e18 q^{57} +3.78050e18 q^{59} -7.61981e18 q^{61} -4.55032e18 q^{63} +2.56902e19 q^{65} +1.87912e19 q^{67} +1.04897e19 q^{69} +4.52649e18 q^{71} -2.55715e19 q^{73} -2.48104e19 q^{75} -7.39992e19 q^{77} -9.93364e19 q^{79} -2.51884e19 q^{81} -2.95818e18 q^{83} -9.34719e19 q^{85} +5.25610e19 q^{87} +1.18803e20 q^{89} -7.63892e20 q^{91} -2.25314e20 q^{93} +6.60888e20 q^{95} -5.69053e20 q^{97} +4.62556e20 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−71604.0	−0.700106	−0.350053	−	0.936730i	$-0.613837\pi$
$3$	−71604.0	−0.700106	−0.350053	+	0.936730i	$0.613837\pi$
$4$	0	0
$5$	−2.86938e7	−1.31402	−0.657011	−	0.753881i	$-0.728179\pi$
$5$	−2.86938e7	−1.31402	−0.657011	+	0.753881i	$0.728179\pi$
$6$	0	0
$7$	8.53202e8	1.14162	0.570811	−	0.821081i	$-0.306629\pi$
$7$	8.53202e8	1.14162	0.570811	+	0.821081i	$0.306629\pi$
$8$	0	0
$9$	−5.33322e9	−0.509851
$10$	0	0
$11$	−8.67312e10	−1.00821	−0.504106	−	0.863642i	$-0.668178\pi$
$11$	−8.67312e10	−1.00821	−0.504106	+	0.863642i	$0.668178\pi$
$12$	0	0
$13$	−8.95323e11	−1.80125	−0.900627	−	0.434594i	$-0.856892\pi$
$13$	−8.95323e11	−1.80125	−0.900627	+	0.434594i	$0.856892\pi$
$14$	0	0
$15$	2.05459e12	0.919955
$16$	0	0
$17$	3.25757e12	0.391904	0.195952	−	0.980613i	$-0.437220\pi$
$17$	3.25757e12	0.391904	0.195952	+	0.980613i	$0.437220\pi$
$18$	0	0
$19$	−2.30325e13	−0.861842	−0.430921	−	0.902390i	$-0.641811\pi$
$19$	−2.30325e13	−0.861842	−0.430921	+	0.902390i	$0.641811\pi$
$20$	0	0
$21$	−6.10927e13	−0.799258
$22$	0	0
$23$	−1.46496e14	−0.737365	−0.368683	−	0.929555i	$-0.620191\pi$
$23$	−1.46496e14	−0.737365	−0.368683	+	0.929555i	$0.620191\pi$
$24$	0	0
$25$	3.46495e14	0.726653
$26$	0	0
$27$	1.13088e15	1.05706
$28$	0	0
$29$	−7.34052e14	−0.324002	−0.162001	−	0.986791i	$-0.551795\pi$
$29$	−7.34052e14	−0.324002	−0.162001	+	0.986791i	$0.551795\pi$
$30$	0	0
$31$	3.14666e15	0.689529	0.344765	−	0.938689i	$-0.387959\pi$
$31$	3.14666e15	0.689529	0.344765	+	0.938689i	$0.387959\pi$
$32$	0	0
$33$	6.21030e15	0.705856
$34$	0	0
$35$	−2.44816e16	−1.50012
$36$	0	0
$37$	−1.29638e16	−0.443215	−0.221608	−	0.975136i	$-0.571130\pi$
$37$	−1.29638e16	−0.443215	−0.221608	+	0.975136i	$0.571130\pi$
$38$	0	0
$39$	6.41087e16	1.26107
$40$	0	0
$41$	4.57146e16	0.531894	0.265947	−	0.963988i	$-0.414315\pi$
$41$	4.57146e16	0.531894	0.265947	+	0.963988i	$0.414315\pi$
$42$	0	0
$43$	2.40736e16	0.169872	0.0849361	−	0.996386i	$-0.472931\pi$
$43$	2.40736e16	0.169872	0.0849361	+	0.996386i	$0.472931\pi$
$44$	0	0
$45$	1.53030e17	0.669955
$46$	0	0
$47$	4.49992e17	1.24789	0.623946	−	0.781467i	$-0.285528\pi$
$47$	4.49992e17	1.24789	0.623946	+	0.781467i	$0.285528\pi$
$48$	0	0
$49$	1.69408e17	0.303303
$50$	0	0
$51$	−2.33255e17	−0.274375
$52$	0	0
$53$	2.06484e18	1.62177	0.810885	−	0.585206i	$-0.198986\pi$
$53$	2.06484e18	1.62177	0.810885	+	0.585206i	$0.198986\pi$
$54$	0	0
$55$	2.48864e18	1.32481
$56$	0	0
$57$	1.64922e18	0.603381
$58$	0	0
$59$	3.78050e18	0.962948	0.481474	−	0.876460i	$-0.340102\pi$
$59$	3.78050e18	0.962948	0.481474	+	0.876460i	$0.340102\pi$
$60$	0	0
$61$	−7.61981e18	−1.36767	−0.683835	−	0.729637i	$-0.739689\pi$
$61$	−7.61981e18	−1.36767	−0.683835	+	0.729637i	$0.739689\pi$
$62$	0	0
$63$	−4.55032e18	−0.582057
$64$	0	0
$65$	2.56902e19	2.36689
$66$	0	0
$67$	1.87912e19	1.25941	0.629706	−	0.776833i	$-0.283175\pi$
$67$	1.87912e19	1.25941	0.629706	+	0.776833i	$0.283175\pi$
$68$	0	0
$69$	1.04897e19	0.516234
$70$	0	0
$71$	4.52649e18	0.165025	0.0825123	−	0.996590i	$-0.473706\pi$
$71$	4.52649e18	0.165025	0.0825123	+	0.996590i	$0.473706\pi$
$72$	0	0
$73$	−2.55715e19	−0.696411	−0.348205	−	0.937418i	$-0.613209\pi$
$73$	−2.55715e19	−0.696411	−0.348205	+	0.937418i	$0.613209\pi$
$74$	0	0
$75$	−2.48104e19	−0.508735
$76$	0	0
$77$	−7.39992e19	−1.15100
$78$	0	0
$79$	−9.93364e19	−1.18039	−0.590193	−	0.807262i	$-0.700948\pi$
$79$	−9.93364e19	−1.18039	−0.590193	+	0.807262i	$0.700948\pi$
$80$	0	0
$81$	−2.51884e19	−0.230201
$82$	0	0
$83$	−2.95818e18	−0.0209269	−0.0104634	−	0.999945i	$-0.503331\pi$
$83$	−2.95818e18	−0.0209269	−0.0104634	+	0.999945i	$0.503331\pi$
$84$	0	0
$85$	−9.34719e19	−0.514970
$86$	0	0
$87$	5.25610e19	0.226836
$88$	0	0
$89$	1.18803e20	0.403861	0.201931	−	0.979400i	$-0.435278\pi$
$89$	1.18803e20	0.403861	0.201931	+	0.979400i	$0.435278\pi$
$90$	0	0
$91$	−7.63892e20	−2.05635
$92$	0	0
$93$	−2.25314e20	−0.482744
$94$	0	0
$95$	6.60888e20	1.13248
$96$	0	0
$97$	−5.69053e20	−0.783519	−0.391759	−	0.920068i	$-0.628133\pi$
$97$	−5.69053e20	−0.783519	−0.391759	+	0.920068i	$0.628133\pi$
$98$	0	0
$99$	4.62556e20	0.514038
$100$	0	0
$101$	−2.65047e20	−0.238753	−0.119376	−	0.992849i	$-0.538090\pi$
$101$	−2.65047e20	−0.238753	−0.119376	+	0.992849i	$0.538090\pi$
$102$	0	0
$103$	−1.51294e21	−1.10925	−0.554625	−	0.832100i	$-0.687138\pi$
$103$	−1.51294e21	−1.10925	−0.554625	+	0.832100i	$0.687138\pi$
$104$	0	0
$105$	1.75298e21	1.05024
$106$	0	0
$107$	1.78568e21	0.877552	0.438776	−	0.898596i	$-0.355412\pi$
$107$	1.78568e21	0.877552	0.438776	+	0.898596i	$0.355412\pi$
$108$	0	0
$109$	3.33894e21	1.35092	0.675462	−	0.737395i	$-0.263944\pi$
$109$	3.33894e21	1.35092	0.675462	+	0.737395i	$0.263944\pi$
$110$	0	0
$111$	9.28261e20	0.310298
$112$	0	0
$113$	−3.04712e21	−0.844434	−0.422217	−	0.906495i	$-0.638748\pi$
$113$	−3.04712e21	−0.844434	−0.422217	+	0.906495i	$0.638748\pi$
$114$	0	0
$115$	4.20351e21	0.968914
$116$	0	0
$117$	4.77496e21	0.918371
$118$	0	0
$119$	2.77936e21	0.447407
$120$	0	0
$121$	1.22048e20	0.0164924
$122$	0	0
$123$	−3.27335e21	−0.372382
$124$	0	0
$125$	3.74000e21	0.359184
$126$	0	0
$127$	7.04680e21	0.572866	0.286433	−	0.958100i	$-0.407530\pi$
$127$	7.04680e21	0.572866	0.286433	+	0.958100i	$0.407530\pi$
$128$	0	0
$129$	−1.72377e21	−0.118929
$130$	0	0
$131$	−9.42787e21	−0.553433	−0.276716	−	0.960952i	$-0.589246\pi$
$131$	−9.42787e21	−0.553433	−0.276716	+	0.960952i	$0.589246\pi$
$132$	0	0
$133$	−1.96514e22	−0.983899
$134$	0	0
$135$	−3.24493e22	−1.38900
$136$	0	0
$137$	−2.01753e21	−0.0740039	−0.0370020	−	0.999315i	$-0.511781\pi$
$137$	−2.01753e21	−0.0740039	−0.0370020	+	0.999315i	$0.511781\pi$
$138$	0	0
$139$	−3.38653e22	−1.06684	−0.533420	−	0.845851i	$-0.679093\pi$
$139$	−3.38653e22	−1.06684	−0.533420	+	0.845851i	$0.679093\pi$
$140$	0	0
$141$	−3.22212e22	−0.873658
$142$	0	0
$143$	7.76525e22	1.81605
$144$	0	0
$145$	2.10627e22	0.425746
$146$	0	0
$147$	−1.21303e22	−0.212344
$148$	0	0
$149$	−3.40800e22	−0.517658	−0.258829	−	0.965923i	$-0.583337\pi$
$149$	−3.40800e22	−0.517658	−0.258829	+	0.965923i	$0.583337\pi$
$150$	0	0
$151$	−2.38856e22	−0.315413	−0.157706	−	0.987486i	$-0.550410\pi$
$151$	−2.38856e22	−0.315413	−0.157706	+	0.987486i	$0.550410\pi$
$152$	0	0
$153$	−1.73733e22	−0.199813
$154$	0	0
$155$	−9.02897e22	−0.906056
$156$	0	0
$157$	1.27315e23	1.11669	0.558345	−	0.829609i	$-0.311436\pi$
$157$	1.27315e23	1.11669	0.558345	+	0.829609i	$0.311436\pi$
$158$	0	0
$159$	−1.47851e23	−1.13541
$160$	0	0
$161$	−1.24990e23	−0.841793
$162$	0	0
$163$	2.36289e23	1.39789	0.698946	−	0.715175i	$-0.253653\pi$
$163$	2.36289e23	1.39789	0.698946	+	0.715175i	$0.253653\pi$
$164$	0	0
$165$	−1.78197e23	−0.927510
$166$	0	0
$167$	3.15464e23	1.44686	0.723429	−	0.690398i	$-0.242565\pi$
$167$	3.15464e23	1.44686	0.723429	+	0.690398i	$0.242565\pi$
$168$	0	0
$169$	5.54540e23	2.24451
$170$	0	0
$171$	1.22837e23	0.439411
$172$	0	0
$173$	4.08353e23	1.29286	0.646431	−	0.762972i	$-0.276261\pi$
$173$	4.08353e23	1.29286	0.646431	+	0.762972i	$0.276261\pi$
$174$	0	0
$175$	2.95631e23	0.829564
$176$	0	0
$177$	−2.70699e23	−0.674166
$178$	0	0
$179$	−7.91545e23	−1.75194	−0.875970	−	0.482366i	$-0.839777\pi$
$179$	−7.91545e23	−1.75194	−0.875970	+	0.482366i	$0.839777\pi$
$180$	0	0
$181$	−4.73055e23	−0.931722	−0.465861	−	0.884858i	$-0.654255\pi$
$181$	−4.73055e23	−0.931722	−0.465861	+	0.884858i	$0.654255\pi$
$182$	0	0
$183$	5.45609e23	0.957514
$184$	0	0
$185$	3.71981e23	0.582394
$186$	0	0
$187$	−2.82533e23	−0.395122
$188$	0	0
$189$	9.64872e23	1.20676
$190$	0	0
$191$	4.27960e23	0.479239	0.239620	−	0.970867i	$-0.422977\pi$
$191$	4.27960e23	0.479239	0.239620	+	0.970867i	$0.422977\pi$
$192$	0	0
$193$	−1.88476e24	−1.89193	−0.945964	−	0.324273i	$-0.894880\pi$
$193$	−1.88476e24	−1.89193	−0.945964	+	0.324273i	$0.894880\pi$
$194$	0	0
$195$	−1.83952e24	−1.65707
$196$	0	0
$197$	8.32416e22	0.0673666	0.0336833	−	0.999433i	$-0.489276\pi$
$197$	8.32416e22	0.0673666	0.0336833	+	0.999433i	$0.489276\pi$
$198$	0	0
$199$	8.81455e23	0.641568	0.320784	−	0.947152i	$-0.396054\pi$
$199$	8.81455e23	0.641568	0.320784	+	0.947152i	$0.396054\pi$
$200$	0	0
$201$	−1.34552e24	−0.881723
$202$	0	0
$203$	−6.26295e23	−0.369888
$204$	0	0
$205$	−1.31173e24	−0.698920
$206$	0	0
$207$	7.81294e23	0.375946
$208$	0	0
$209$	1.99763e24	0.868920
$210$	0	0
$211$	−3.79755e24	−1.49464	−0.747322	−	0.664462i	$-0.768661\pi$
$211$	−3.79755e24	−1.49464	−0.747322	+	0.664462i	$0.768661\pi$
$212$	0	0
$213$	−3.24115e23	−0.115535
$214$	0	0
$215$	−6.90763e23	−0.223216
$216$	0	0
$217$	2.68474e24	0.787182
$218$	0	0
$219$	1.83102e24	0.487562
$220$	0	0
$221$	−2.91658e24	−0.705918
$222$	0	0
$223$	3.92011e24	0.863171	0.431586	−	0.902072i	$-0.357954\pi$
$223$	3.92011e24	0.863171	0.431586	+	0.902072i	$0.357954\pi$
$224$	0	0
$225$	−1.84794e24	−0.370485
$226$	0	0
$227$	1.12173e24	0.204936	0.102468	−	0.994736i	$-0.467326\pi$
$227$	1.12173e24	0.204936	0.102468	+	0.994736i	$0.467326\pi$
$228$	0	0
$229$	3.79307e23	0.0632002	0.0316001	−	0.999501i	$-0.489940\pi$
$229$	3.79307e23	0.0632002	0.0316001	+	0.999501i	$0.489940\pi$
$230$	0	0
$231$	5.29864e24	0.805821
$232$	0	0
$233$	4.03014e24	0.559865	0.279933	−	0.960020i	$-0.409688\pi$
$233$	4.03014e24	0.559865	0.279933	+	0.960020i	$0.409688\pi$
$234$	0	0
$235$	−1.29120e25	−1.63976
$236$	0	0
$237$	7.11289e24	0.826396
$238$	0	0
$239$	1.75220e25	1.86383	0.931916	−	0.362673i	$-0.118136\pi$
$239$	1.75220e25	1.86383	0.931916	+	0.362673i	$0.118136\pi$
$240$	0	0
$241$	9.55213e24	0.930939	0.465470	−	0.885064i	$-0.345885\pi$
$241$	9.55213e24	0.930939	0.465470	+	0.885064i	$0.345885\pi$
$242$	0	0
$243$	−1.00258e25	−0.895891
$244$	0	0
$245$	−4.86097e24	−0.398546
$246$	0	0
$247$	2.06215e25	1.55240
$248$	0	0
$249$	2.11818e23	0.0146511
$250$	0	0
$251$	−6.29655e24	−0.400432	−0.200216	−	0.979752i	$-0.564164\pi$
$251$	−6.29655e24	−0.400432	−0.200216	+	0.979752i	$0.564164\pi$
$252$	0	0
$253$	1.27057e25	0.743421
$254$	0	0
$255$	6.69296e24	0.360534
$256$	0	0
$257$	−1.45248e25	−0.720794	−0.360397	−	0.932799i	$-0.617359\pi$
$257$	−1.45248e25	−0.720794	−0.360397	+	0.932799i	$0.617359\pi$
$258$	0	0
$259$	−1.10608e25	−0.505985
$260$	0	0
$261$	3.91486e24	0.165193
$262$	0	0
$263$	−4.13430e25	−1.61015	−0.805075	−	0.593173i	$-0.797875\pi$
$263$	−4.13430e25	−1.61015	−0.805075	+	0.593173i	$0.797875\pi$
$264$	0	0
$265$	−5.92480e25	−2.13104
$266$	0	0
$267$	−8.50677e24	−0.282746
$268$	0	0
$269$	−5.18786e25	−1.59437	−0.797185	−	0.603735i	$-0.793678\pi$
$269$	−5.18786e25	−1.59437	−0.797185	+	0.603735i	$0.793678\pi$
$270$	0	0
$271$	2.35538e25	0.669706	0.334853	−	0.942270i	$-0.391313\pi$
$271$	2.35538e25	0.669706	0.334853	+	0.942270i	$0.391313\pi$
$272$	0	0
$273$	5.46977e25	1.43967
$274$	0	0
$275$	−3.00519e25	−0.732621
$276$	0	0
$277$	5.29041e25	1.19523	0.597615	−	0.801783i	$-0.296115\pi$
$277$	5.29041e25	1.19523	0.597615	+	0.801783i	$0.296115\pi$
$278$	0	0
$279$	−1.67819e25	−0.351557
$280$	0	0
$281$	2.17922e25	0.423530	0.211765	−	0.977321i	$-0.432079\pi$
$281$	2.17922e25	0.423530	0.211765	+	0.977321i	$0.432079\pi$
$282$	0	0
$283$	5.39243e24	0.0972808	0.0486404	−	0.998816i	$-0.484511\pi$
$283$	5.39243e24	0.0972808	0.0486404	+	0.998816i	$0.484511\pi$
$284$	0	0
$285$	−4.73222e25	−0.792856
$286$	0	0
$287$	3.90038e25	0.607222
$288$	0	0
$289$	−5.84802e25	−0.846411
$290$	0	0
$291$	4.07465e25	0.548547
$292$	0	0
$293$	3.92947e25	0.492293	0.246147	−	0.969233i	$-0.420836\pi$
$293$	3.92947e25	0.492293	0.246147	+	0.969233i	$0.420836\pi$
$294$	0	0
$295$	−1.08477e26	−1.26533
$296$	0	0
$297$	−9.80828e25	−1.06574
$298$	0	0
$299$	1.31161e26	1.32818
$300$	0	0
$301$	2.05397e25	0.193930
$302$	0	0
$303$	1.89784e25	0.167152
$304$	0	0
$305$	2.18641e26	1.79715
$306$	0	0
$307$	−1.23827e26	−0.950306	−0.475153	−	0.879903i	$-0.657607\pi$
$307$	−1.23827e26	−0.950306	−0.475153	+	0.879903i	$0.657607\pi$
$308$	0	0
$309$	1.08332e26	0.776593
$310$	0	0
$311$	1.58511e26	1.06188	0.530941	−	0.847409i	$-0.321839\pi$
$311$	1.58511e26	1.06188	0.530941	+	0.847409i	$0.321839\pi$
$312$	0	0
$313$	1.15882e26	0.725770	0.362885	−	0.931834i	$-0.381792\pi$
$313$	1.15882e26	0.725770	0.362885	+	0.931834i	$0.381792\pi$
$314$	0	0
$315$	1.30566e26	0.764836
$316$	0	0
$317$	−5.09758e25	−0.279410	−0.139705	−	0.990193i	$-0.544615\pi$
$317$	−5.09758e25	−0.279410	−0.139705	+	0.990193i	$0.544615\pi$
$318$	0	0
$319$	6.36652e25	0.326663
$320$	0	0
$321$	−1.27862e26	−0.614380
$322$	0	0
$323$	−7.50298e25	−0.337759
$324$	0	0
$325$	−3.10225e26	−1.30889
$326$	0	0
$327$	−2.39082e26	−0.945790
$328$	0	0
$329$	3.83934e26	1.42462
$330$	0	0
$331$	4.64134e26	1.61603	0.808015	−	0.589162i	$-0.200542\pi$
$331$	4.64134e26	1.61603	0.808015	+	0.589162i	$0.200542\pi$
$332$	0	0
$333$	6.91389e25	0.225974
$334$	0	0
$335$	−5.39189e26	−1.65490
$336$	0	0
$337$	−2.39093e26	−0.689371	−0.344686	−	0.938718i	$-0.612014\pi$
$337$	−2.39093e26	−0.689371	−0.344686	+	0.938718i	$0.612014\pi$
$338$	0	0
$339$	2.18186e26	0.591194
$340$	0	0
$341$	−2.72914e26	−0.695192
$342$	0	0
$343$	−3.32013e26	−0.795366
$344$	0	0
$345$	−3.00988e26	−0.678343
$346$	0	0
$347$	3.01925e26	0.640383	0.320192	−	0.947353i	$-0.396253\pi$
$347$	3.01925e26	0.640383	0.320192	+	0.947353i	$0.396253\pi$
$348$	0	0
$349$	4.92021e26	0.982464	0.491232	−	0.871029i	$-0.336547\pi$
$349$	4.92021e26	0.982464	0.491232	+	0.871029i	$0.336547\pi$
$350$	0	0
$351$	−1.01251e27	−1.90403
$352$	0	0
$353$	−6.69643e26	−1.18634	−0.593169	−	0.805078i	$-0.702124\pi$
$353$	−6.69643e26	−1.18634	−0.593169	+	0.805078i	$0.702124\pi$
$354$	0	0
$355$	−1.29882e26	−0.216846
$356$	0	0
$357$	−1.99014e26	−0.313232
$358$	0	0
$359$	−5.89124e26	−0.874410	−0.437205	−	0.899362i	$-0.644032\pi$
$359$	−5.89124e26	−0.874410	−0.437205	+	0.899362i	$0.644032\pi$
$360$	0	0
$361$	−1.83715e26	−0.257228
$362$	0	0
$363$	−8.73909e24	−0.0115464
$364$	0	0
$365$	7.33741e26	0.915099
$366$	0	0
$367$	1.61827e27	1.90571	0.952854	−	0.303428i	$-0.0981313\pi$
$367$	1.61827e27	1.90571	0.952854	+	0.303428i	$0.0981313\pi$
$368$	0	0
$369$	−2.43806e26	−0.271187
$370$	0	0
$371$	1.76172e27	1.85145
$372$	0	0
$373$	1.12579e26	0.111819	0.0559095	−	0.998436i	$-0.482194\pi$
$373$	1.12579e26	0.111819	0.0559095	+	0.998436i	$0.482194\pi$
$374$	0	0
$375$	−2.67799e26	−0.251467
$376$	0	0
$377$	6.57214e26	0.583610
$378$	0	0
$379$	−1.61687e27	−1.35820	−0.679101	−	0.734045i	$-0.737630\pi$
$379$	−1.61687e27	−1.35820	−0.679101	+	0.734045i	$0.737630\pi$
$380$	0	0
$381$	−5.04579e26	−0.401067
$382$	0	0
$383$	7.30139e25	0.0549311	0.0274656	−	0.999623i	$-0.491256\pi$
$383$	7.30139e25	0.0549311	0.0274656	+	0.999623i	$0.491256\pi$
$384$	0	0
$385$	2.12332e27	1.51244
$386$	0	0
$387$	−1.28390e26	−0.0866095
$388$	0	0
$389$	4.38998e26	0.280538	0.140269	−	0.990113i	$-0.455203\pi$
$389$	4.38998e26	0.280538	0.140269	+	0.990113i	$0.455203\pi$
$390$	0	0
$391$	−4.77220e26	−0.288976
$392$	0	0
$393$	6.75073e26	0.387462
$394$	0	0
$395$	2.85034e27	1.55105
$396$	0	0
$397$	−1.33852e27	−0.690756	−0.345378	−	0.938464i	$-0.612249\pi$
$397$	−1.33852e27	−0.690756	−0.345378	+	0.938464i	$0.612249\pi$
$398$	0	0
$399$	1.40712e27	0.688834
$400$	0	0
$401$	−2.07292e27	−0.962870	−0.481435	−	0.876482i	$-0.659884\pi$
$401$	−2.07292e27	−0.962870	−0.481435	+	0.876482i	$0.659884\pi$
$402$	0	0
$403$	−2.81728e27	−1.24202
$404$	0	0
$405$	7.22750e26	0.302489
$406$	0	0
$407$	1.12437e27	0.446855
$408$	0	0
$409$	−4.12924e26	−0.155874	−0.0779372	−	0.996958i	$-0.524833\pi$
$409$	−4.12924e26	−0.155874	−0.0779372	+	0.996958i	$0.524833\pi$
$410$	0	0
$411$	1.44463e26	0.0518106
$412$	0	0
$413$	3.22553e27	1.09932
$414$	0	0
$415$	8.48813e25	0.0274984
$416$	0	0
$417$	2.42489e27	0.746901
$418$	0	0
$419$	3.55921e27	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$419$	3.55921e27	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$420$	0	0
$421$	−3.99663e27	−1.11361	−0.556804	−	0.830644i	$-0.687972\pi$
$421$	−3.99663e27	−1.11361	−0.556804	+	0.830644i	$0.687972\pi$
$422$	0	0
$423$	−2.39991e27	−0.636239
$424$	0	0
$425$	1.12873e27	0.284778
$426$	0	0
$427$	−6.50124e27	−1.56136
$428$	0	0
$429$	−5.56023e27	−1.27143
$430$	0	0
$431$	−5.74296e27	−1.25062	−0.625308	−	0.780378i	$-0.715027\pi$
$431$	−5.74296e27	−1.25062	−0.625308	+	0.780378i	$0.715027\pi$
$432$	0	0
$433$	−6.99214e27	−1.45040	−0.725199	−	0.688540i	$-0.758252\pi$
$433$	−6.99214e27	−1.45040	−0.725199	+	0.688540i	$0.758252\pi$
$434$	0	0
$435$	−1.50817e27	−0.298067
$436$	0	0
$437$	3.37416e27	0.635492
$438$	0	0
$439$	8.74047e27	1.56912	0.784562	−	0.620050i	$-0.212888\pi$
$439$	8.74047e27	1.56912	0.784562	+	0.620050i	$0.212888\pi$
$440$	0	0
$441$	−9.03493e26	−0.154639
$442$	0	0
$443$	−1.15967e28	−1.89276	−0.946380	−	0.323055i	$-0.895290\pi$
$443$	−1.15967e28	−1.89276	−0.946380	+	0.323055i	$0.895290\pi$
$444$	0	0
$445$	−3.40891e27	−0.530683
$446$	0	0
$447$	2.44026e27	0.362416
$448$	0	0
$449$	6.44084e27	0.912758	0.456379	−	0.889785i	$-0.349146\pi$
$449$	6.44084e27	0.912758	0.456379	+	0.889785i	$0.349146\pi$
$450$	0	0
$451$	−3.96489e27	−0.536262
$452$	0	0
$453$	1.71031e27	0.220822
$454$	0	0
$455$	2.19189e28	2.70209
$456$	0	0
$457$	−8.27692e27	−0.974425	−0.487212	−	0.873283i	$-0.661986\pi$
$457$	−8.27692e27	−0.974425	−0.487212	+	0.873283i	$0.661986\pi$
$458$	0	0
$459$	3.68393e27	0.414265
$460$	0	0
$461$	4.93315e27	0.529986	0.264993	−	0.964250i	$-0.414630\pi$
$461$	4.93315e27	0.529986	0.264993	+	0.964250i	$0.414630\pi$
$462$	0	0
$463$	4.15139e26	0.0426181	0.0213090	−	0.999773i	$-0.493217\pi$
$463$	4.15139e26	0.0426181	0.0213090	+	0.999773i	$0.493217\pi$
$464$	0	0
$465$	6.46510e27	0.634336
$466$	0	0
$467$	−7.30716e27	−0.685364	−0.342682	−	0.939452i	$-0.611335\pi$
$467$	−7.30716e27	−0.685364	−0.342682	+	0.939452i	$0.611335\pi$
$468$	0	0
$469$	1.60327e28	1.43777
$470$	0	0
$471$	−9.11626e27	−0.781802
$472$	0	0
$473$	−2.08793e27	−0.171267
$474$	0	0
$475$	−7.98064e27	−0.626260
$476$	0	0
$477$	−1.10122e28	−0.826861
$478$	0	0
$479$	1.30805e28	0.939946	0.469973	−	0.882681i	$-0.344264\pi$
$479$	1.30805e28	0.939946	0.469973	+	0.882681i	$0.344264\pi$
$480$	0	0
$481$	1.16068e28	0.798343
$482$	0	0
$483$	8.94982e27	0.589345
$484$	0	0
$485$	1.63283e28	1.02956
$486$	0	0
$487$	1.68131e28	1.01530	0.507650	−	0.861564i	$-0.330514\pi$
$487$	1.68131e28	1.01530	0.507650	+	0.861564i	$0.330514\pi$
$488$	0	0
$489$	−1.69192e28	−0.978673
$490$	0	0
$491$	2.67382e28	1.48175	0.740877	−	0.671641i	$-0.234410\pi$
$491$	2.67382e28	1.48175	0.740877	+	0.671641i	$0.234410\pi$
$492$	0	0
$493$	−2.39122e27	−0.126978
$494$	0	0
$495$	−1.32725e28	−0.675457
$496$	0	0
$497$	3.86201e27	0.188396
$498$	0	0
$499$	−1.92819e28	−0.901765	−0.450882	−	0.892583i	$-0.648891\pi$
$499$	−1.92819e28	−0.901765	−0.450882	+	0.892583i	$0.648891\pi$
$500$	0	0
$501$	−2.25884e28	−1.01296
$502$	0	0
$503$	−2.46340e27	−0.105943	−0.0529714	−	0.998596i	$-0.516869\pi$
$503$	−2.46340e27	−0.105943	−0.0529714	+	0.998596i	$0.516869\pi$
$504$	0	0
$505$	7.60520e27	0.313726
$506$	0	0
$507$	−3.97072e28	−1.57140
$508$	0	0
$509$	2.34782e28	0.891513	0.445757	−	0.895154i	$-0.352935\pi$
$509$	2.34782e28	0.891513	0.445757	+	0.895154i	$0.352935\pi$
$510$	0	0
$511$	−2.18176e28	−0.795038
$512$	0	0
$513$	−2.60470e28	−0.911016
$514$	0	0
$515$	4.34118e28	1.45758
$516$	0	0
$517$	−3.90283e28	−1.25814
$518$	0	0
$519$	−2.92397e28	−0.905141
$520$	0	0
$521$	5.65820e28	1.68222	0.841109	−	0.540865i	$-0.181903\pi$
$521$	5.65820e28	1.68222	0.841109	+	0.540865i	$0.181903\pi$
$522$	0	0
$523$	2.74538e28	0.784033	0.392016	−	0.919958i	$-0.371778\pi$
$523$	2.74538e28	0.784033	0.392016	+	0.919958i	$0.371778\pi$
$524$	0	0
$525$	−2.11683e28	−0.580783
$526$	0	0
$527$	1.02505e28	0.270229
$528$	0	0
$529$	−1.80106e28	−0.456293
$530$	0	0
$531$	−2.01622e28	−0.490960
$532$	0	0
$533$	−4.09294e28	−0.958075
$534$	0	0
$535$	−5.12378e28	−1.15312
$536$	0	0
$537$	5.66778e28	1.22654
$538$	0	0
$539$	−1.46930e28	−0.305794
$540$	0	0
$541$	−1.54302e28	−0.308888	−0.154444	−	0.988002i	$-0.549359\pi$
$541$	−1.54302e28	−0.308888	−0.154444	+	0.988002i	$0.549359\pi$
$542$	0	0
$543$	3.38726e28	0.652305
$544$	0	0
$545$	−9.58069e28	−1.77514
$546$	0	0
$547$	−8.66492e27	−0.154489	−0.0772445	−	0.997012i	$-0.524612\pi$
$547$	−8.66492e27	−0.154489	−0.0772445	+	0.997012i	$0.524612\pi$
$548$	0	0
$549$	4.06381e28	0.697307
$550$	0	0
$551$	1.69070e28	0.279238
$552$	0	0
$553$	−8.47541e28	−1.34756
$554$	0	0
$555$	−2.66353e28	−0.407738
$556$	0	0
$557$	1.12653e29	1.66060	0.830300	−	0.557317i	$-0.188169\pi$
$557$	1.12653e29	1.66060	0.830300	+	0.557317i	$0.188169\pi$
$558$	0	0
$559$	−2.15537e28	−0.305983
$560$	0	0
$561$	2.02305e28	0.276628
$562$	0	0
$563$	1.30345e29	1.71695	0.858475	−	0.512856i	$-0.171412\pi$
$563$	1.30345e29	1.71695	0.858475	+	0.512856i	$0.171412\pi$
$564$	0	0
$565$	8.74333e28	1.10960
$566$	0	0
$567$	−2.14908e28	−0.262803
$568$	0	0
$569$	1.64552e29	1.93921	0.969606	−	0.244671i	$-0.0786800\pi$
$569$	1.64552e29	1.93921	0.969606	+	0.244671i	$0.0786800\pi$
$570$	0	0
$571$	9.69589e28	1.10131	0.550654	−	0.834734i	$-0.314378\pi$
$571$	9.69589e28	1.10131	0.550654	+	0.834734i	$0.314378\pi$
$572$	0	0
$573$	−3.06436e28	−0.335519
$574$	0	0
$575$	−5.07601e28	−0.535809
$576$	0	0
$577$	1.25423e28	0.127653	0.0638263	−	0.997961i	$-0.479670\pi$
$577$	1.25423e28	0.127653	0.0638263	+	0.997961i	$0.479670\pi$
$578$	0	0
$579$	1.34956e29	1.32455
$580$	0	0
$581$	−2.52393e27	−0.0238906
$582$	0	0
$583$	−1.79086e29	−1.63509
$584$	0	0
$585$	−1.37012e29	−1.20676
$586$	0	0
$587$	2.13236e29	1.81201	0.906003	−	0.423271i	$-0.139118\pi$
$587$	2.13236e29	1.81201	0.906003	+	0.423271i	$0.139118\pi$
$588$	0	0
$589$	−7.24754e28	−0.594265
$590$	0	0
$591$	−5.96043e27	−0.0471638
$592$	0	0
$593$	5.74182e28	0.438506	0.219253	−	0.975668i	$-0.429638\pi$
$593$	5.74182e28	0.438506	0.219253	+	0.975668i	$0.429638\pi$
$594$	0	0
$595$	−7.97504e28	−0.587902
$596$	0	0
$597$	−6.31157e28	−0.449166
$598$	0	0
$599$	1.18461e28	0.0813941	0.0406970	−	0.999172i	$-0.487042\pi$
$599$	1.18461e28	0.0813941	0.0406970	+	0.999172i	$0.487042\pi$
$600$	0	0
$601$	1.60567e28	0.106531	0.0532653	−	0.998580i	$-0.483037\pi$
$601$	1.60567e28	0.106531	0.0532653	+	0.998580i	$0.483037\pi$
$602$	0	0
$603$	−1.00217e29	−0.642113
$604$	0	0
$605$	−3.50200e27	−0.0216713
$606$	0	0
$607$	−9.90701e28	−0.592190	−0.296095	−	0.955158i	$-0.595685\pi$
$607$	−9.90701e28	−0.592190	−0.296095	+	0.955158i	$0.595685\pi$
$608$	0	0
$609$	4.48452e28	0.258961
$610$	0	0
$611$	−4.02888e29	−2.24777
$612$	0	0
$613$	1.45877e29	0.786416	0.393208	−	0.919450i	$-0.371365\pi$
$613$	1.45877e29	0.786416	0.393208	+	0.919450i	$0.371365\pi$
$614$	0	0
$615$	9.39248e28	0.489318
$616$	0	0
$617$	−9.52438e28	−0.479560	−0.239780	−	0.970827i	$-0.577075\pi$
$617$	−9.52438e28	−0.479560	−0.239780	+	0.970827i	$0.577075\pi$
$618$	0	0
$619$	3.36526e29	1.63782	0.818911	−	0.573921i	$-0.194578\pi$
$619$	3.36526e29	1.63782	0.818911	+	0.573921i	$0.194578\pi$
$620$	0	0
$621$	−1.65670e29	−0.779437
$622$	0	0
$623$	1.01363e29	0.461057
$624$	0	0
$625$	−2.72537e29	−1.19863
$626$	0	0
$627$	−1.43039e29	−0.608336
$628$	0	0
$629$	−4.22305e28	−0.173698
$630$	0	0
$631$	−8.77108e27	−0.0348935	−0.0174468	−	0.999848i	$-0.505554\pi$
$631$	−8.77108e27	−0.0348935	−0.0174468	+	0.999848i	$0.505554\pi$
$632$	0	0
$633$	2.71920e29	1.04641
$634$	0	0
$635$	−2.02199e29	−0.752758
$636$	0	0
$637$	−1.51675e29	−0.546325
$638$	0	0
$639$	−2.41408e28	−0.0841379
$640$	0	0
$641$	4.29903e28	0.144998	0.0724988	−	0.997368i	$-0.476903\pi$
$641$	4.29903e28	0.144998	0.0724988	+	0.997368i	$0.476903\pi$
$642$	0	0
$643$	−4.61999e28	−0.150809	−0.0754043	−	0.997153i	$-0.524025\pi$
$643$	−4.61999e28	−0.150809	−0.0754043	+	0.997153i	$0.524025\pi$
$644$	0	0
$645$	4.94614e28	0.156275
$646$	0	0
$647$	4.54118e28	0.138891	0.0694455	−	0.997586i	$-0.477877\pi$
$647$	4.54118e28	0.138891	0.0694455	+	0.997586i	$0.477877\pi$
$648$	0	0
$649$	−3.27887e29	−0.970856
$650$	0	0
$651$	−1.92238e29	−0.551111
$652$	0	0
$653$	−1.64556e29	−0.456799	−0.228400	−	0.973567i	$-0.573349\pi$
$653$	−1.64556e29	−0.456799	−0.228400	+	0.973567i	$0.573349\pi$
$654$	0	0
$655$	2.70521e29	0.727223
$656$	0	0
$657$	1.36378e29	0.355066
$658$	0	0
$659$	−2.21390e29	−0.558291	−0.279146	−	0.960249i	$-0.590051\pi$
$659$	−2.21390e29	−0.558291	−0.279146	+	0.960249i	$0.590051\pi$
$660$	0	0
$661$	2.54395e29	0.621431	0.310716	−	0.950503i	$-0.399431\pi$
$661$	2.54395e29	0.621431	0.310716	+	0.950503i	$0.399431\pi$
$662$	0	0
$663$	2.08839e29	0.494218
$664$	0	0
$665$	5.63872e29	1.29286
$666$	0	0
$667$	1.07535e29	0.238908
$668$	0	0
$669$	−2.80695e29	−0.604312
$670$	0	0
$671$	6.60875e29	1.37890
$672$	0	0
$673$	7.99394e29	1.61660	0.808300	−	0.588770i	$-0.200388\pi$
$673$	7.99394e29	1.61660	0.808300	+	0.588770i	$0.200388\pi$
$674$	0	0
$675$	3.91846e29	0.768114
$676$	0	0
$677$	4.61947e29	0.877831	0.438916	−	0.898528i	$-0.355363\pi$
$677$	4.61947e29	0.877831	0.438916	+	0.898528i	$0.355363\pi$
$678$	0	0
$679$	−4.85517e29	−0.894483
$680$	0	0
$681$	−8.03207e28	−0.143477
$682$	0	0
$683$	−3.35857e29	−0.581751	−0.290876	−	0.956761i	$-0.593947\pi$
$683$	−3.35857e29	−0.581751	−0.290876	+	0.956761i	$0.593947\pi$
$684$	0	0
$685$	5.78906e28	0.0972428
$686$	0	0
$687$	−2.71599e28	−0.0442469
$688$	0	0
$689$	−1.84870e30	−2.92122
$690$	0	0
$691$	4.41273e29	0.676376	0.338188	−	0.941079i	$-0.390186\pi$
$691$	4.41273e29	0.676376	0.338188	+	0.941079i	$0.390186\pi$
$692$	0	0
$693$	3.94654e29	0.586838
$694$	0	0
$695$	9.71722e29	1.40185
$696$	0	0
$697$	1.48919e29	0.208451
$698$	0	0
$699$	−2.88574e29	−0.391965
$700$	0	0
$701$	1.23969e29	0.163408	0.0817040	−	0.996657i	$-0.473964\pi$
$701$	1.23969e29	0.163408	0.0817040	+	0.996657i	$0.473964\pi$
$702$	0	0
$703$	2.98589e29	0.381982
$704$	0	0
$705$	9.24548e29	1.14801
$706$	0	0
$707$	−2.26139e29	−0.272566
$708$	0	0
$709$	1.27607e30	1.49311	0.746553	−	0.665326i	$-0.231708\pi$
$709$	1.27607e30	1.49311	0.746553	+	0.665326i	$0.231708\pi$
$710$	0	0
$711$	5.29783e29	0.601821
$712$	0	0
$713$	−4.60973e29	−0.508435
$714$	0	0
$715$	−2.22814e30	−2.38632
$716$	0	0
$717$	−1.25465e30	−1.30488
$718$	0	0
$719$	−1.07039e30	−1.08116	−0.540580	−	0.841293i	$-0.681795\pi$
$719$	−1.07039e30	−1.08116	−0.540580	+	0.841293i	$0.681795\pi$
$720$	0	0
$721$	−1.29084e30	−1.26635
$722$	0	0
$723$	−6.83971e29	−0.651757
$724$	0	0
$725$	−2.54345e29	−0.235437
$726$	0	0
$727$	7.64279e29	0.687291	0.343645	−	0.939100i	$-0.388338\pi$
$727$	7.64279e29	0.687291	0.343645	+	0.939100i	$0.388338\pi$
$728$	0	0
$729$	9.81370e29	0.857420
$730$	0	0
$731$	7.84214e28	0.0665736
$732$	0	0
$733$	−1.18755e30	−0.979629	−0.489814	−	0.871827i	$-0.662935\pi$
$733$	−1.18755e30	−0.979629	−0.489814	+	0.871827i	$0.662935\pi$
$734$	0	0
$735$	3.48065e29	0.279025
$736$	0	0
$737$	−1.62978e30	−1.26976
$738$	0	0
$739$	−1.62225e29	−0.122843	−0.0614217	−	0.998112i	$-0.519563\pi$
$739$	−1.62225e29	−0.122843	−0.0614217	+	0.998112i	$0.519563\pi$
$740$	0	0
$741$	−1.47658e30	−1.08684
$742$	0	0
$743$	−1.09499e30	−0.783478	−0.391739	−	0.920076i	$-0.628126\pi$
$743$	−1.09499e30	−0.783478	−0.391739	+	0.920076i	$0.628126\pi$
$744$	0	0
$745$	9.77883e29	0.680214
$746$	0	0
$747$	1.57766e28	0.0106696
$748$	0	0
$749$	1.52354e30	1.00183
$750$	0	0
$751$	1.43876e30	0.919962	0.459981	−	0.887929i	$-0.347856\pi$
$751$	1.43876e30	0.919962	0.459981	+	0.887929i	$0.347856\pi$
$752$	0	0
$753$	4.50858e29	0.280345
$754$	0	0
$755$	6.85369e29	0.414459
$756$	0	0
$757$	2.13374e29	0.125498	0.0627488	−	0.998029i	$-0.480013\pi$
$757$	2.13374e29	0.125498	0.0627488	+	0.998029i	$0.480013\pi$
$758$	0	0
$759$	−9.09782e29	−0.520474
$760$	0	0
$761$	3.15854e30	1.75771	0.878857	−	0.477086i	$-0.158307\pi$
$761$	3.15854e30	1.75771	0.878857	+	0.477086i	$0.158307\pi$
$762$	0	0
$763$	2.84879e30	1.54225
$764$	0	0
$765$	4.98506e29	0.262558
$766$	0	0
$767$	−3.38477e30	−1.73451
$768$	0	0
$769$	−1.20266e30	−0.599675	−0.299838	−	0.953990i	$-0.596933\pi$
$769$	−1.20266e30	−0.599675	−0.299838	+	0.953990i	$0.596933\pi$
$770$	0	0
$771$	1.04003e30	0.504633
$772$	0	0
$773$	−2.33922e30	−1.10455	−0.552277	−	0.833661i	$-0.686241\pi$
$773$	−2.33922e30	−1.10455	−0.552277	+	0.833661i	$0.686241\pi$
$774$	0	0
$775$	1.09030e30	0.501049
$776$	0	0
$777$	7.91994e29	0.354243
$778$	0	0
$779$	−1.05292e30	−0.458408
$780$	0	0
$781$	−3.92588e29	−0.166380
$782$	0	0
$783$	−8.30127e29	−0.342488
$784$	0	0
$785$	−3.65314e30	−1.46736
$786$	0	0
$787$	1.29344e30	0.505838	0.252919	−	0.967487i	$-0.418609\pi$
$787$	1.29344e30	0.505838	0.252919	+	0.967487i	$0.418609\pi$
$788$	0	0
$789$	2.96032e30	1.12728
$790$	0	0
$791$	−2.59981e30	−0.964025
$792$	0	0
$793$	6.82220e30	2.46352
$794$	0	0
$795$	4.24239e30	1.49196
$796$	0	0
$797$	−4.15582e29	−0.142346	−0.0711729	−	0.997464i	$-0.522674\pi$
$797$	−4.15582e29	−0.142346	−0.0711729	+	0.997464i	$0.522674\pi$
$798$	0	0
$799$	1.46588e30	0.489054
$800$	0	0
$801$	−6.33602e29	−0.205909
$802$	0	0
$803$	2.21784e30	0.702130
$804$	0	0
$805$	3.58645e30	1.10613
$806$	0	0
$807$	3.71471e30	1.11623
$808$	0	0
$809$	−6.34707e30	−1.85829	−0.929146	−	0.369713i	$-0.879456\pi$
$809$	−6.34707e30	−1.85829	−0.929146	+	0.369713i	$0.879456\pi$
$810$	0	0
$811$	1.03289e30	0.294669	0.147335	−	0.989087i	$-0.452931\pi$
$811$	1.03289e30	0.294669	0.147335	+	0.989087i	$0.452931\pi$
$812$	0	0
$813$	−1.68655e30	−0.468865
$814$	0	0
$815$	−6.78002e30	−1.83686
$816$	0	0
$817$	−5.54475e29	−0.146403
$818$	0	0
$819$	4.07400e30	1.04843
$820$	0	0
$821$	5.48361e30	1.37551	0.687755	−	0.725943i	$-0.258596\pi$
$821$	5.48361e30	1.37551	0.687755	+	0.725943i	$0.258596\pi$
$822$	0	0
$823$	−1.50751e30	−0.368605	−0.184302	−	0.982870i	$-0.559003\pi$
$823$	−1.50751e30	−0.368605	−0.184302	+	0.982870i	$0.559003\pi$
$824$	0	0
$825$	2.15184e30	0.512913
$826$	0	0
$827$	−3.24916e30	−0.755027	−0.377514	−	0.926004i	$-0.623221\pi$
$827$	−3.24916e30	−0.755027	−0.377514	+	0.926004i	$0.623221\pi$
$828$	0	0
$829$	−3.88679e30	−0.880579	−0.440290	−	0.897856i	$-0.645124\pi$
$829$	−3.88679e30	−0.880579	−0.440290	+	0.897856i	$0.645124\pi$
$830$	0	0
$831$	−3.78815e30	−0.836789
$832$	0	0
$833$	5.51859e29	0.118866
$834$	0	0
$835$	−9.05184e30	−1.90120
$836$	0	0
$837$	3.55851e30	0.728871
$838$	0	0
$839$	4.76500e30	0.951836	0.475918	−	0.879490i	$-0.342116\pi$
$839$	4.76500e30	0.951836	0.475918	+	0.879490i	$0.342116\pi$
$840$	0	0
$841$	−4.59401e30	−0.895023
$842$	0	0
$843$	−1.56041e30	−0.296516
$844$	0	0
$845$	−1.59118e31	−2.94934
$846$	0	0
$847$	1.04131e29	0.0188281
$848$	0	0
$849$	−3.86120e29	−0.0681069
$850$	0	0
$851$	1.89914e30	0.326811
$852$	0	0
$853$	2.09705e30	0.352083	0.176041	−	0.984383i	$-0.443671\pi$
$853$	2.09705e30	0.352083	0.176041	+	0.984383i	$0.443671\pi$
$854$	0	0
$855$	−3.52466e30	−0.577396
$856$	0	0
$857$	5.93597e30	0.948840	0.474420	−	0.880299i	$-0.342658\pi$
$857$	5.93597e30	0.948840	0.474420	+	0.880299i	$0.342658\pi$
$858$	0	0
$859$	2.89046e30	0.450856	0.225428	−	0.974260i	$-0.427622\pi$
$859$	2.89046e30	0.450856	0.225428	+	0.974260i	$0.427622\pi$
$860$	0	0
$861$	−2.79283e30	−0.425120
$862$	0	0
$863$	5.57344e30	0.827961	0.413981	−	0.910286i	$-0.364138\pi$
$863$	5.57344e30	0.827961	0.413981	+	0.910286i	$0.364138\pi$
$864$	0	0
$865$	−1.17172e31	−1.69885
$866$	0	0
$867$	4.18742e30	0.592578
$868$	0	0
$869$	8.61557e30	1.19008
$870$	0	0
$871$	−1.68242e31	−2.26852
$872$	0	0
$873$	3.03489e30	0.399478
$874$	0	0
$875$	3.19098e30	0.410052
$876$	0	0
$877$	3.23623e30	0.406016	0.203008	−	0.979177i	$-0.434928\pi$
$877$	3.23623e30	0.406016	0.203008	+	0.979177i	$0.434928\pi$
$878$	0	0
$879$	−2.81366e30	−0.344658
$880$	0	0
$881$	3.38000e30	0.404268	0.202134	−	0.979358i	$-0.435212\pi$
$881$	3.38000e30	0.404268	0.202134	+	0.979358i	$0.435212\pi$
$882$	0	0
$883$	4.75979e30	0.555904	0.277952	−	0.960595i	$-0.410344\pi$
$883$	4.75979e30	0.555904	0.277952	+	0.960595i	$0.410344\pi$
$884$	0	0
$885$	7.76737e30	0.885869
$886$	0	0
$887$	−4.28065e30	−0.476773	−0.238387	−	0.971170i	$-0.576619\pi$
$887$	−4.28065e30	−0.476773	−0.238387	+	0.971170i	$0.576619\pi$
$888$	0	0
$889$	6.01235e30	0.653997
$890$	0	0
$891$	2.18462e30	0.232092
$892$	0	0
$893$	−1.03644e31	−1.07549
$894$	0	0
$895$	2.27124e31	2.30209
$896$	0	0
$897$	−9.39166e30	−0.929868
$898$	0	0
$899$	−2.30981e30	−0.223409
$900$	0	0
$901$	6.72635e30	0.635578
$902$	0	0
$903$	−1.47072e30	−0.135772
$904$	0	0
$905$	1.35737e31	1.22430
$906$	0	0
$907$	−4.03498e30	−0.355602	−0.177801	−	0.984066i	$-0.556898\pi$
$907$	−4.03498e30	−0.355602	−0.177801	+	0.984066i	$0.556898\pi$
$908$	0	0
$909$	1.41355e30	0.121728
$910$	0	0
$911$	1.00965e31	0.849623	0.424812	−	0.905282i	$-0.360340\pi$
$911$	1.00965e31	0.849623	0.424812	+	0.905282i	$0.360340\pi$
$912$	0	0
$913$	2.56566e29	0.0210987
$914$	0	0
$915$	−1.56556e31	−1.25819
$916$	0	0
$917$	−8.04388e30	−0.631811
$918$	0	0
$919$	2.04869e31	1.57277	0.786383	−	0.617739i	$-0.211951\pi$
$919$	2.04869e31	1.57277	0.786383	+	0.617739i	$0.211951\pi$
$920$	0	0
$921$	8.86653e30	0.665315
$922$	0	0
$923$	−4.05267e30	−0.297251
$924$	0	0
$925$	−4.49190e30	−0.322064
$926$	0	0
$927$	8.06882e30	0.565552
$928$	0	0
$929$	1.15155e31	0.789072	0.394536	−	0.918880i	$-0.370905\pi$
$929$	1.15155e31	0.789072	0.394536	+	0.918880i	$0.370905\pi$
$930$	0	0
$931$	−3.90189e30	−0.261399
$932$	0	0
$933$	−1.13500e31	−0.743431
$934$	0	0
$935$	8.10693e30	0.519200
$936$	0	0
$937$	1.40172e31	0.877801	0.438900	−	0.898536i	$-0.355368\pi$
$937$	1.40172e31	0.877801	0.438900	+	0.898536i	$0.355368\pi$
$938$	0	0
$939$	−8.29758e30	−0.508116
$940$	0	0
$941$	−9.63512e30	−0.576987	−0.288493	−	0.957482i	$-0.593154\pi$
$941$	−9.63512e30	−0.576987	−0.288493	+	0.957482i	$0.593154\pi$
$942$	0	0
$943$	−6.69700e30	−0.392200
$944$	0	0
$945$	−2.76858e31	−1.58571
$946$	0	0
$947$	−1.70818e31	−0.956880	−0.478440	−	0.878120i	$-0.658798\pi$
$947$	−1.70818e31	−0.956880	−0.478440	+	0.878120i	$0.658798\pi$
$948$	0	0
$949$	2.28947e31	1.25441
$950$	0	0
$951$	3.65007e30	0.195617
$952$	0	0
$953$	−1.43239e31	−0.750904	−0.375452	−	0.926842i	$-0.622513\pi$
$953$	−1.43239e31	−0.750904	−0.375452	+	0.926842i	$0.622513\pi$
$954$	0	0
$955$	−1.22798e31	−0.629731
$956$	0	0
$957$	−4.55868e30	−0.228699
$958$	0	0
$959$	−1.72136e30	−0.0844846
$960$	0	0
$961$	−1.09240e31	−0.524550
$962$	0	0
$963$	−9.52340e30	−0.447421
$964$	0	0
$965$	5.40809e31	2.48603
$966$	0	0
$967$	−1.43951e31	−0.647493	−0.323747	−	0.946144i	$-0.604943\pi$
$967$	−1.43951e31	−0.647493	−0.323747	+	0.946144i	$0.604943\pi$
$968$	0	0
$969$	5.37243e30	0.236467
$970$	0	0
$971$	2.67107e31	1.15049	0.575245	−	0.817981i	$-0.304907\pi$
$971$	2.67107e31	1.15049	0.575245	+	0.817981i	$0.304907\pi$
$972$	0	0
$973$	−2.88939e31	−1.21793
$974$	0	0
$975$	2.22134e31	0.916360
$976$	0	0
$977$	−3.63714e31	−1.46848	−0.734238	−	0.678892i	$-0.762460\pi$
$977$	−3.63714e31	−1.46848	−0.734238	+	0.678892i	$0.762460\pi$
$978$	0	0
$979$	−1.03039e31	−0.407178
$980$	0	0
$981$	−1.78073e31	−0.688770
$982$	0	0
$983$	2.81078e31	1.06418	0.532089	−	0.846688i	$-0.321407\pi$
$983$	2.81078e31	1.06418	0.532089	+	0.846688i	$0.321407\pi$
$984$	0	0
$985$	−2.38851e30	−0.0885212
$986$	0	0
$987$	−2.74912e31	−0.997388
$988$	0	0
$989$	−3.52668e30	−0.125258
$990$	0	0
$991$	−3.20038e31	−1.11283	−0.556413	−	0.830906i	$-0.687823\pi$
$991$	−3.20038e31	−1.11283	−0.556413	+	0.830906i	$0.687823\pi$
$992$	0	0
$993$	−3.32339e31	−1.13139
$994$	0	0
$995$	−2.52923e31	−0.843035
$996$	0	0
$997$	−4.21028e31	−1.37408	−0.687040	−	0.726620i	$-0.741090\pi$
$997$	−4.21028e31	−1.37408	−0.687040	+	0.726620i	$0.741090\pi$
$998$	0	0
$999$	−1.46606e31	−0.468504

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.22.a.a.1.1		1
4.3	odd	2		2.22.a.a.1.1	✓	1
8.3	odd	2		64.22.a.b.1.1		1
8.5	even	2		64.22.a.f.1.1		1
12.11	even	2		18.22.a.e.1.1		1
20.3	even	4		50.22.b.a.49.2		2
20.7	even	4		50.22.b.a.49.1		2
20.19	odd	2		50.22.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.22.a.a.1.1	✓	1	4.3	odd	2
16.22.a.a.1.1		1	1.1	even	1	trivial
18.22.a.e.1.1		1	12.11	even	2
50.22.a.c.1.1		1	20.19	odd	2
50.22.b.a.49.1		2	20.7	even	4
50.22.b.a.49.2		2	20.3	even	4
64.22.a.b.1.1		1	8.3	odd	2
64.22.a.f.1.1		1	8.5	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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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