LMFDB - Embedded newform 16.22.a.c.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 1)
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)

$f(q)$

$=$

$q+128844. q^{3} +2.16410e7 q^{5} +7.68079e8 q^{7} +6.14042e9 q^{9} +O(q^{10})$

$q+128844. q^{3} +2.16410e7 q^{5} +7.68079e8 q^{7} +6.14042e9 q^{9} +9.47249e10 q^{11} -8.06218e10 q^{13} +2.78831e12 q^{15} +3.05228e12 q^{17} +7.92079e12 q^{19} +9.89623e13 q^{21} +7.38454e13 q^{23} -8.50644e12 q^{25} -5.56597e14 q^{27} -4.25303e15 q^{29} -1.90054e15 q^{31} +1.22047e16 q^{33} +1.66220e16 q^{35} +2.21914e16 q^{37} -1.03876e16 q^{39} -2.06228e16 q^{41} +1.93606e17 q^{43} +1.32885e17 q^{45} -1.46961e17 q^{47} +3.13992e16 q^{49} +3.93268e17 q^{51} +2.03827e18 q^{53} +2.04994e18 q^{55} +1.02055e18 q^{57} +5.97588e18 q^{59} +6.19062e18 q^{61} +4.71633e18 q^{63} -1.74473e18 q^{65} -1.69613e19 q^{67} +9.51454e18 q^{69} +5.63276e18 q^{71} -4.32848e19 q^{73} -1.09600e18 q^{75} +7.27562e19 q^{77} +5.12649e19 q^{79} -1.35945e20 q^{81} -4.89119e19 q^{83} +6.60543e19 q^{85} -5.47978e20 q^{87} -5.04303e20 q^{89} -6.19239e19 q^{91} -2.44873e20 q^{93} +1.71413e20 q^{95} +8.08275e20 q^{97} +5.81651e20 q^{99} +O(q^{100})$

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
$3$	128844.	1.25977	0.629885	−	0.776689i	$-0.283102\pi$
$3$	128844.	1.25977	0.629885	+	0.776689i	$0.283102\pi$
$4$	0	0
$5$	2.16410e7	0.991040	0.495520	−	0.868596i	$-0.334977\pi$
$5$	2.16410e7	0.991040	0.495520	+	0.868596i	$0.334977\pi$
$6$	0	0
$7$	7.68079e8	1.02772	0.513862	−	0.857873i	$-0.328214\pi$
$7$	7.68079e8	1.02772	0.513862	+	0.857873i	$0.328214\pi$
$8$	0	0
$9$	6.14042e9	0.587019
$10$	0	0
$11$	9.47249e10	1.10114	0.550568	−	0.834790i	$-0.314411\pi$
$11$	9.47249e10	1.10114	0.550568	+	0.834790i	$0.314411\pi$
$12$	0	0
$13$	−8.06218e10	−0.162199	−0.0810993	−	0.996706i	$-0.525843\pi$
$13$	−8.06218e10	−0.162199	−0.0810993	+	0.996706i	$0.525843\pi$
$14$	0	0
$15$	2.78831e12	1.24848
$16$	0	0
$17$	3.05228e12	0.367207	0.183604	−	0.983000i	$-0.441224\pi$
$17$	3.05228e12	0.367207	0.183604	+	0.983000i	$0.441224\pi$
$18$	0	0
$19$	7.92079e12	0.296385	0.148192	−	0.988959i	$-0.452655\pi$
$19$	7.92079e12	0.296385	0.148192	+	0.988959i	$0.452655\pi$
$20$	0	0
$21$	9.89623e13	1.29469
$22$	0	0
$23$	7.38454e13	0.371690	0.185845	−	0.982579i	$-0.440498\pi$
$23$	7.38454e13	0.371690	0.185845	+	0.982579i	$0.440498\pi$
$24$	0	0
$25$	−8.50644e12	−0.0178393
$26$	0	0
$27$	−5.56597e14	−0.520261
$28$	0	0
$29$	−4.25303e15	−1.87724	−0.938620	−	0.344954i	$-0.887895\pi$
$29$	−4.25303e15	−1.87724	−0.938620	+	0.344954i	$0.887895\pi$
$30$	0	0
$31$	−1.90054e15	−0.416466	−0.208233	−	0.978079i	$-0.566771\pi$
$31$	−1.90054e15	−0.416466	−0.208233	+	0.978079i	$0.566771\pi$
$32$	0	0
$33$	1.22047e16	1.38718
$34$	0	0
$35$	1.66220e16	1.01852
$36$	0	0
$37$	2.21914e16	0.758695	0.379347	−	0.925254i	$-0.376149\pi$
$37$	2.21914e16	0.758695	0.379347	+	0.925254i	$0.376149\pi$
$38$	0	0
$39$	−1.03876e16	−0.204333
$40$	0	0
$41$	−2.06228e16	−0.239948	−0.119974	−	0.992777i	$-0.538281\pi$
$41$	−2.06228e16	−0.239948	−0.119974	+	0.992777i	$0.538281\pi$
$42$	0	0
$43$	1.93606e17	1.36615	0.683077	−	0.730346i	$-0.260641\pi$
$43$	1.93606e17	1.36615	0.683077	+	0.730346i	$0.260641\pi$
$44$	0	0
$45$	1.32885e17	0.581759
$46$	0	0
$47$	−1.46961e17	−0.407543	−0.203771	−	0.979019i	$-0.565320\pi$
$47$	−1.46961e17	−0.407543	−0.203771	+	0.979019i	$0.565320\pi$
$48$	0	0
$49$	3.13992e16	0.0562160
$50$	0	0
$51$	3.93268e17	0.462596
$52$	0	0
$53$	2.03827e18	1.60090	0.800450	−	0.599399i	$-0.204594\pi$
$53$	2.03827e18	1.60090	0.800450	+	0.599399i	$0.204594\pi$
$54$	0	0
$55$	2.04994e18	1.09127
$56$	0	0
$57$	1.02055e18	0.373376
$58$	0	0
$59$	5.97588e18	1.52214	0.761072	−	0.648667i	$-0.224673\pi$
$59$	5.97588e18	1.52214	0.761072	+	0.648667i	$0.224673\pi$
$60$	0	0
$61$	6.19062e18	1.11114	0.555572	−	0.831468i	$-0.312499\pi$
$61$	6.19062e18	1.11114	0.555572	+	0.831468i	$0.312499\pi$
$62$	0	0
$63$	4.71633e18	0.603293
$64$	0	0
$65$	−1.74473e18	−0.160745
$66$	0	0
$67$	−1.69613e19	−1.13677	−0.568387	−	0.822761i	$-0.692432\pi$
$67$	−1.69613e19	−1.13677	−0.568387	+	0.822761i	$0.692432\pi$
$68$	0	0
$69$	9.51454e18	0.468244
$70$	0	0
$71$	5.63276e18	0.205357	0.102678	−	0.994715i	$-0.467259\pi$
$71$	5.63276e18	0.205357	0.102678	+	0.994715i	$0.467259\pi$
$72$	0	0
$73$	−4.32848e19	−1.17881	−0.589407	−	0.807837i	$-0.700638\pi$
$73$	−4.32848e19	−1.17881	−0.589407	+	0.807837i	$0.700638\pi$
$74$	0	0
$75$	−1.09600e18	−0.0224734
$76$	0	0
$77$	7.27562e19	1.13166
$78$	0	0
$79$	5.12649e19	0.609166	0.304583	−	0.952486i	$-0.401483\pi$
$79$	5.12649e19	0.609166	0.304583	+	0.952486i	$0.401483\pi$
$80$	0	0
$81$	−1.35945e20	−1.24243
$82$	0	0
$83$	−4.89119e19	−0.346014	−0.173007	−	0.984921i	$-0.555348\pi$
$83$	−4.89119e19	−0.346014	−0.173007	+	0.984921i	$0.555348\pi$
$84$	0	0
$85$	6.60543e19	0.363917
$86$	0	0
$87$	−5.47978e20	−2.36489
$88$	0	0
$89$	−5.04303e20	−1.71434	−0.857170	−	0.515034i	$-0.827779\pi$
$89$	−5.04303e20	−1.71434	−0.857170	+	0.515034i	$0.827779\pi$
$90$	0	0
$91$	−6.19239e19	−0.166695
$92$	0	0
$93$	−2.44873e20	−0.524651
$94$	0	0
$95$	1.71413e20	0.293729
$96$	0	0
$97$	8.08275e20	1.11290	0.556450	−	0.830881i	$-0.312163\pi$
$97$	8.08275e20	1.11290	0.556450	+	0.830881i	$0.312163\pi$
$98$	0	0
$99$	5.81651e20	0.646388
$100$	0	0
$101$	−1.00202e21	−0.902612	−0.451306	−	0.892369i	$-0.649042\pi$
$101$	−1.00202e21	−0.902612	−0.451306	+	0.892369i	$0.649042\pi$
$102$	0	0
$103$	5.89747e20	0.432389	0.216195	−	0.976350i	$-0.430635\pi$
$103$	5.89747e20	0.432389	0.216195	+	0.976350i	$0.430635\pi$
$104$	0	0
$105$	2.14164e21	1.28309
$106$	0	0
$107$	−1.12210e21	−0.551445	−0.275723	−	0.961237i	$-0.588917\pi$
$107$	−1.12210e21	−0.551445	−0.275723	+	0.961237i	$0.588917\pi$
$108$	0	0
$109$	1.72394e21	0.697499	0.348750	−	0.937216i	$-0.386606\pi$
$109$	1.72394e21	0.697499	0.348750	+	0.937216i	$0.386606\pi$
$110$	0	0
$111$	2.85923e21	0.955781
$112$	0	0
$113$	4.95810e20	0.137402	0.0687008	−	0.997637i	$-0.478115\pi$
$113$	4.95810e20	0.137402	0.0687008	+	0.997637i	$0.478115\pi$
$114$	0	0
$115$	1.59809e21	0.368360
$116$	0	0
$117$	−4.95052e20	−0.0952136
$118$	0	0
$119$	2.34439e21	0.377387
$120$	0	0
$121$	1.57256e21	0.212501
$122$	0	0
$123$	−2.65712e21	−0.302279
$124$	0	0
$125$	−1.05033e22	−1.00872
$126$	0	0
$127$	−1.63609e21	−0.133005	−0.0665027	−	0.997786i	$-0.521184\pi$
$127$	−1.63609e21	−0.133005	−0.0665027	+	0.997786i	$0.521184\pi$
$128$	0	0
$129$	2.49450e22	1.72104
$130$	0	0
$131$	1.38650e22	0.813898	0.406949	−	0.913451i	$-0.366593\pi$
$131$	1.38650e22	0.813898	0.406949	+	0.913451i	$0.366593\pi$
$132$	0	0
$133$	6.08379e21	0.304601
$134$	0	0
$135$	−1.20453e22	−0.515600
$136$	0	0
$137$	−4.00789e22	−1.47011	−0.735055	−	0.678007i	$-0.762844\pi$
$137$	−4.00789e22	−1.47011	−0.735055	+	0.678007i	$0.762844\pi$
$138$	0	0
$139$	−4.47585e22	−1.41000	−0.705001	−	0.709206i	$-0.749054\pi$
$139$	−4.47585e22	−1.41000	−0.705001	+	0.709206i	$0.749054\pi$
$140$	0	0
$141$	−1.89350e22	−0.513410
$142$	0	0
$143$	−7.63689e21	−0.178603
$144$	0	0
$145$	−9.20396e22	−1.86042
$146$	0	0
$147$	4.04560e21	0.0708191
$148$	0	0
$149$	4.93289e22	0.749283	0.374641	−	0.927170i	$-0.377766\pi$
$149$	4.93289e22	0.749283	0.374641	+	0.927170i	$0.377766\pi$
$150$	0	0
$151$	−5.70415e22	−0.753239	−0.376620	−	0.926368i	$-0.622914\pi$
$151$	−5.70415e22	−0.753239	−0.376620	+	0.926368i	$0.622914\pi$
$152$	0	0
$153$	1.87423e22	0.215557
$154$	0	0
$155$	−4.11295e22	−0.412735
$156$	0	0
$157$	6.35623e22	0.557511	0.278756	−	0.960362i	$-0.410078\pi$
$157$	6.35623e22	0.557511	0.278756	+	0.960362i	$0.410078\pi$
$158$	0	0
$159$	2.62618e23	2.01677
$160$	0	0
$161$	5.67191e22	0.381995
$162$	0	0
$163$	−8.68484e22	−0.513797	−0.256899	−	0.966438i	$-0.582701\pi$
$163$	−8.68484e22	−0.513797	−0.256899	+	0.966438i	$0.582701\pi$
$164$	0	0
$165$	2.64122e23	1.37475
$166$	0	0
$167$	1.89411e23	0.868726	0.434363	−	0.900738i	$-0.356974\pi$
$167$	1.89411e23	0.868726	0.434363	+	0.900738i	$0.356974\pi$
$168$	0	0
$169$	−2.40565e23	−0.973692
$170$	0	0
$171$	4.86370e22	0.173983
$172$	0	0
$173$	−4.18508e23	−1.32501	−0.662506	−	0.749057i	$-0.730507\pi$
$173$	−4.18508e23	−1.32501	−0.662506	+	0.749057i	$0.730507\pi$
$174$	0	0
$175$	−6.53362e21	−0.0183339
$176$	0	0
$177$	7.69957e23	1.91755
$178$	0	0
$179$	−4.76752e23	−1.05520	−0.527601	−	0.849493i	$-0.676908\pi$
$179$	−4.76752e23	−1.05520	−0.527601	+	0.849493i	$0.676908\pi$
$180$	0	0
$181$	−2.88627e22	−0.0568476	−0.0284238	−	0.999596i	$-0.509049\pi$
$181$	−2.88627e22	−0.0568476	−0.0284238	+	0.999596i	$0.509049\pi$
$182$	0	0
$183$	7.97624e23	1.39979
$184$	0	0
$185$	4.80244e23	0.751897
$186$	0	0
$187$	2.89127e23	0.404345
$188$	0	0
$189$	−4.27510e23	−0.534685
$190$	0	0
$191$	−8.86378e23	−0.992587	−0.496293	−	0.868155i	$-0.665306\pi$
$191$	−8.86378e23	−0.992587	−0.496293	+	0.868155i	$0.665306\pi$
$192$	0	0
$193$	8.63509e22	0.0866792	0.0433396	−	0.999060i	$-0.486200\pi$
$193$	8.63509e22	0.0866792	0.0433396	+	0.999060i	$0.486200\pi$
$194$	0	0
$195$	−2.24798e23	−0.202502
$196$	0	0
$197$	−6.99008e23	−0.565701	−0.282850	−	0.959164i	$-0.591280\pi$
$197$	−6.99008e23	−0.565701	−0.282850	+	0.959164i	$0.591280\pi$
$198$	0	0
$199$	1.24542e24	0.906483	0.453242	−	0.891388i	$-0.350268\pi$
$199$	1.24542e24	0.906483	0.453242	+	0.891388i	$0.350268\pi$
$200$	0	0
$201$	−2.18536e24	−1.43207
$202$	0	0
$203$	−3.26666e24	−1.92928
$204$	0	0
$205$	−4.46297e23	−0.237798
$206$	0	0
$207$	4.53442e23	0.218189
$208$	0	0
$209$	7.50296e23	0.326360
$210$	0	0
$211$	−3.50841e24	−1.38085	−0.690423	−	0.723406i	$-0.742575\pi$
$211$	−3.50841e24	−1.38085	−0.690423	+	0.723406i	$0.742575\pi$
$212$	0	0
$213$	7.25747e23	0.258702
$214$	0	0
$215$	4.18981e24	1.35391
$216$	0	0
$217$	−1.45977e24	−0.428012
$218$	0	0
$219$	−5.57698e24	−1.48503
$220$	0	0
$221$	−2.46081e23	−0.0595605
$222$	0	0
$223$	4.72350e24	1.04007	0.520035	−	0.854145i	$-0.325919\pi$
$223$	4.72350e24	1.04007	0.520035	+	0.854145i	$0.325919\pi$
$224$	0	0
$225$	−5.22331e22	−0.0104720
$226$	0	0
$227$	5.44317e24	0.994444	0.497222	−	0.867623i	$-0.334353\pi$
$227$	5.44317e24	0.994444	0.497222	+	0.867623i	$0.334353\pi$
$228$	0	0
$229$	6.90677e24	1.15081	0.575403	−	0.817870i	$-0.304845\pi$
$229$	6.90677e24	1.15081	0.575403	+	0.817870i	$0.304845\pi$
$230$	0	0
$231$	9.37420e24	1.42564
$232$	0	0
$233$	4.53650e24	0.630208	0.315104	−	0.949057i	$-0.397961\pi$
$233$	4.53650e24	0.630208	0.315104	+	0.949057i	$0.397961\pi$
$234$	0	0
$235$	−3.18036e24	−0.403891
$236$	0	0
$237$	6.60518e24	0.767409
$238$	0	0
$239$	2.73493e24	0.290916	0.145458	−	0.989364i	$-0.453534\pi$
$239$	2.73493e24	0.290916	0.145458	+	0.989364i	$0.453534\pi$
$240$	0	0
$241$	−8.08907e24	−0.788351	−0.394175	−	0.919035i	$-0.628970\pi$
$241$	−8.08907e24	−0.788351	−0.394175	+	0.919035i	$0.628970\pi$
$242$	0	0
$243$	−1.16935e25	−1.04491
$244$	0	0
$245$	6.79508e23	0.0557123
$246$	0	0
$247$	−6.38588e23	−0.0480732
$248$	0	0
$249$	−6.30200e24	−0.435898
$250$	0	0
$251$	−6.63927e24	−0.422227	−0.211113	−	0.977462i	$-0.567709\pi$
$251$	−6.63927e24	−0.422227	−0.211113	+	0.977462i	$0.567709\pi$
$252$	0	0
$253$	6.99500e24	0.409282
$254$	0	0
$255$	8.51070e24	0.458452
$256$	0	0
$257$	1.57278e24	0.0780497	0.0390249	−	0.999238i	$-0.487575\pi$
$257$	1.57278e24	0.0780497	0.0390249	+	0.999238i	$0.487575\pi$
$258$	0	0
$259$	1.70448e25	0.779729
$260$	0	0
$261$	−2.61154e25	−1.10197
$262$	0	0
$263$	3.40077e25	1.32447	0.662235	−	0.749296i	$-0.269608\pi$
$263$	3.40077e25	1.32447	0.662235	+	0.749296i	$0.269608\pi$
$264$	0	0
$265$	4.41100e25	1.58656
$266$	0	0
$267$	−6.49765e25	−2.15967
$268$	0	0
$269$	−3.57975e25	−1.10015	−0.550077	−	0.835114i	$-0.685402\pi$
$269$	−3.57975e25	−1.10015	−0.550077	+	0.835114i	$0.685402\pi$
$270$	0	0
$271$	−2.46104e25	−0.699746	−0.349873	−	0.936797i	$-0.613775\pi$
$271$	−2.46104e25	−0.699746	−0.349873	+	0.936797i	$0.613775\pi$
$272$	0	0
$273$	−7.97852e24	−0.209998
$274$	0	0
$275$	−8.05772e23	−0.0196435
$276$	0	0
$277$	−6.11679e25	−1.38193	−0.690965	−	0.722888i	$-0.742814\pi$
$277$	−6.11679e25	−1.38193	−0.690965	+	0.722888i	$0.742814\pi$
$278$	0	0
$279$	−1.16701e25	−0.244473
$280$	0	0
$281$	1.73710e25	0.337605	0.168802	−	0.985650i	$-0.446010\pi$
$281$	1.73710e25	0.337605	0.168802	+	0.985650i	$0.446010\pi$
$282$	0	0
$283$	−7.57237e25	−1.36607	−0.683037	−	0.730383i	$-0.739341\pi$
$283$	−7.57237e25	−1.36607	−0.683037	+	0.730383i	$0.739341\pi$
$284$	0	0
$285$	2.20856e25	0.370031
$286$	0	0
$287$	−1.58399e25	−0.246600
$288$	0	0
$289$	−5.97755e25	−0.865159
$290$	0	0
$291$	1.04141e26	1.40200
$292$	0	0
$293$	4.88684e25	0.612235	0.306118	−	0.951994i	$-0.400970\pi$
$293$	4.88684e25	0.612235	0.306118	+	0.951994i	$0.400970\pi$
$294$	0	0
$295$	1.29324e26	1.50851
$296$	0	0
$297$	−5.27236e25	−0.572878
$298$	0	0
$299$	−5.95355e24	−0.0602877
$300$	0	0
$301$	1.48705e26	1.40403
$302$	0	0
$303$	−1.29104e26	−1.13708
$304$	0	0
$305$	1.33971e26	1.10119
$306$	0	0
$307$	2.17987e26	1.67293	0.836466	−	0.548019i	$-0.184618\pi$
$307$	2.17987e26	1.67293	0.836466	+	0.548019i	$0.184618\pi$
$308$	0	0
$309$	7.59854e25	0.544711
$310$	0	0
$311$	4.04644e25	0.271075	0.135538	−	0.990772i	$-0.456724\pi$
$311$	4.04644e25	0.271075	0.135538	+	0.990772i	$0.456724\pi$
$312$	0	0
$313$	−8.74174e24	−0.0547498	−0.0273749	−	0.999625i	$-0.508715\pi$
$313$	−8.74174e24	−0.0547498	−0.0273749	+	0.999625i	$0.508715\pi$
$314$	0	0
$315$	1.02066e26	0.597888
$316$	0	0
$317$	−3.19758e25	−0.175267	−0.0876334	−	0.996153i	$-0.527930\pi$
$317$	−3.19758e25	−0.175267	−0.0876334	+	0.996153i	$0.527930\pi$
$318$	0	0
$319$	−4.02868e26	−2.06710
$320$	0	0
$321$	−1.44576e26	−0.694694
$322$	0	0
$323$	2.41765e25	0.108835
$324$	0	0
$325$	6.85805e23	0.00289351
$326$	0	0
$327$	2.22119e26	0.878688
$328$	0	0
$329$	−1.12877e26	−0.418841
$330$	0	0
$331$	2.52215e26	0.878166	0.439083	−	0.898446i	$-0.355303\pi$
$331$	2.52215e26	0.878166	0.439083	+	0.898446i	$0.355303\pi$
$332$	0	0
$333$	1.36265e26	0.445368
$334$	0	0
$335$	−3.67059e26	−1.12659
$336$	0	0
$337$	−1.53127e26	−0.441507	−0.220753	−	0.975330i	$-0.570852\pi$
$337$	−1.53127e26	−0.441507	−0.220753	+	0.975330i	$0.570852\pi$
$338$	0	0
$339$	6.38821e25	0.173094
$340$	0	0
$341$	−1.80029e26	−0.458586
$342$	0	0
$343$	−4.04890e26	−0.969949
$344$	0	0
$345$	2.05904e26	0.464049
$346$	0	0
$347$	−3.23436e26	−0.686008	−0.343004	−	0.939334i	$-0.611444\pi$
$347$	−3.23436e26	−0.686008	−0.343004	+	0.939334i	$0.611444\pi$
$348$	0	0
$349$	−6.77854e26	−1.35353	−0.676767	−	0.736197i	$-0.736620\pi$
$349$	−6.77854e26	−1.35353	−0.676767	+	0.736197i	$0.736620\pi$
$350$	0	0
$351$	4.48739e25	0.0843857
$352$	0	0
$353$	6.84291e26	1.21229	0.606145	−	0.795354i	$-0.292715\pi$
$353$	6.84291e26	1.21229	0.606145	+	0.795354i	$0.292715\pi$
$354$	0	0
$355$	1.21898e26	0.203517
$356$	0	0
$357$	3.02061e26	0.475421
$358$	0	0
$359$	6.85500e25	0.101746	0.0508728	−	0.998705i	$-0.483800\pi$
$359$	6.85500e25	0.101746	0.0508728	+	0.998705i	$0.483800\pi$
$360$	0	0
$361$	−6.51471e26	−0.912156
$362$	0	0
$363$	2.02615e26	0.267703
$364$	0	0
$365$	−9.36723e26	−1.16825
$366$	0	0
$367$	−1.13575e27	−1.33749	−0.668745	−	0.743492i	$-0.733168\pi$
$367$	−1.13575e27	−1.33749	−0.668745	+	0.743492i	$0.733168\pi$
$368$	0	0
$369$	−1.26633e26	−0.140854
$370$	0	0
$371$	1.56555e27	1.64528
$372$	0	0
$373$	3.82975e26	0.380389	0.190195	−	0.981746i	$-0.439088\pi$
$373$	3.82975e26	0.380389	0.190195	+	0.981746i	$0.439088\pi$
$374$	0	0
$375$	−1.35329e27	−1.27075
$376$	0	0
$377$	3.42887e26	0.304486
$378$	0	0
$379$	−7.14767e25	−0.0600417	−0.0300208	−	0.999549i	$-0.509557\pi$
$379$	−7.14767e25	−0.0600417	−0.0300208	+	0.999549i	$0.509557\pi$
$380$	0	0
$381$	−2.10801e26	−0.167556
$382$	0	0
$383$	1.38425e27	1.04142	0.520712	−	0.853732i	$-0.325666\pi$
$383$	1.38425e27	1.04142	0.520712	+	0.853732i	$0.325666\pi$
$384$	0	0
$385$	1.57451e27	1.12152
$386$	0	0
$387$	1.18882e27	0.801958
$388$	0	0
$389$	1.63213e27	1.04300	0.521500	−	0.853251i	$-0.325373\pi$
$389$	1.63213e27	1.04300	0.521500	+	0.853251i	$0.325373\pi$
$390$	0	0
$391$	2.25397e26	0.136487
$392$	0	0
$393$	1.78642e27	1.02532
$394$	0	0
$395$	1.10942e27	0.603708
$396$	0	0
$397$	1.21029e27	0.624581	0.312291	−	0.949987i	$-0.398904\pi$
$397$	1.21029e27	0.624581	0.312291	+	0.949987i	$0.398904\pi$
$398$	0	0
$399$	7.83860e26	0.383728
$400$	0	0
$401$	−6.51358e26	−0.302555	−0.151277	−	0.988491i	$-0.548339\pi$
$401$	−6.51358e26	−0.302555	−0.151277	+	0.988491i	$0.548339\pi$
$402$	0	0
$403$	1.53225e26	0.0675502
$404$	0	0
$405$	−2.94198e27	−1.23130
$406$	0	0
$407$	2.10208e27	0.835427
$408$	0	0
$409$	−4.45251e27	−1.68078	−0.840389	−	0.541984i	$-0.817673\pi$
$409$	−4.45251e27	−1.68078	−0.840389	+	0.541984i	$0.817673\pi$
$410$	0	0
$411$	−5.16393e27	−1.85200
$412$	0	0
$413$	4.58995e27	1.56434
$414$	0	0
$415$	−1.05850e27	−0.342914
$416$	0	0
$417$	−5.76686e27	−1.77628
$418$	0	0
$419$	−4.92912e27	−1.44385	−0.721925	−	0.691972i	$-0.756742\pi$
$419$	−4.92912e27	−1.44385	−0.721925	+	0.691972i	$0.756742\pi$
$420$	0	0
$421$	1.50145e27	0.418359	0.209180	−	0.977877i	$-0.432921\pi$
$421$	1.50145e27	0.418359	0.209180	+	0.977877i	$0.432921\pi$
$422$	0	0
$423$	−9.02400e26	−0.239235
$424$	0	0
$425$	−2.59641e25	−0.00655072
$426$	0	0
$427$	4.75488e27	1.14195
$428$	0	0
$429$	−9.83968e26	−0.224998
$430$	0	0
$431$	7.19221e27	1.56621	0.783107	−	0.621887i	$-0.213634\pi$
$431$	7.19221e27	1.56621	0.783107	+	0.621887i	$0.213634\pi$
$432$	0	0
$433$	4.23104e27	0.877656	0.438828	−	0.898571i	$-0.355394\pi$
$433$	4.23104e27	0.877656	0.438828	+	0.898571i	$0.355394\pi$
$434$	0	0
$435$	−1.18588e28	−2.34370
$436$	0	0
$437$	5.84914e26	0.110163
$438$	0	0
$439$	−4.53235e27	−0.813665	−0.406832	−	0.913503i	$-0.633367\pi$
$439$	−4.53235e27	−0.813665	−0.406832	+	0.913503i	$0.633367\pi$
$440$	0	0
$441$	1.92804e26	0.0329998
$442$	0	0
$443$	4.61186e27	0.752726	0.376363	−	0.926472i	$-0.377174\pi$
$443$	4.61186e27	0.752726	0.376363	+	0.926472i	$0.377174\pi$
$444$	0	0
$445$	−1.09136e28	−1.69898
$446$	0	0
$447$	6.35573e27	0.943923
$448$	0	0
$449$	−1.25692e26	−0.0178123	−0.00890615	−	0.999960i	$-0.502835\pi$
$449$	−1.25692e26	−0.0178123	−0.00890615	+	0.999960i	$0.502835\pi$
$450$	0	0
$451$	−1.95349e27	−0.264216
$452$	0	0
$453$	−7.34945e27	−0.948907
$454$	0	0
$455$	−1.34009e27	−0.165202
$456$	0	0
$457$	1.15924e28	1.36475	0.682374	−	0.731003i	$-0.260948\pi$
$457$	1.15924e28	1.36475	0.682374	+	0.731003i	$0.260948\pi$
$458$	0	0
$459$	−1.69889e27	−0.191044
$460$	0	0
$461$	1.02514e28	1.10134	0.550671	−	0.834723i	$-0.314372\pi$
$461$	1.02514e28	1.10134	0.550671	+	0.834723i	$0.314372\pi$
$462$	0	0
$463$	5.72801e27	0.588035	0.294017	−	0.955800i	$-0.405008\pi$
$463$	5.72801e27	0.588035	0.294017	+	0.955800i	$0.405008\pi$
$464$	0	0
$465$	−5.29929e27	−0.519950
$466$	0	0
$467$	−1.12658e28	−1.05666	−0.528329	−	0.849040i	$-0.677181\pi$
$467$	−1.12658e28	−1.05666	−0.528329	+	0.849040i	$0.677181\pi$
$468$	0	0
$469$	−1.30276e28	−1.16829
$470$	0	0
$471$	8.18963e27	0.702335
$472$	0	0
$473$	1.83393e28	1.50432
$474$	0	0
$475$	−6.73777e25	−0.00528729
$476$	0	0
$477$	1.25158e28	0.939759
$478$	0	0
$479$	1.17373e28	0.843422	0.421711	−	0.906730i	$-0.361430\pi$
$479$	1.17373e28	0.843422	0.421711	+	0.906730i	$0.361430\pi$
$480$	0	0
$481$	−1.78911e27	−0.123059
$482$	0	0
$483$	7.30792e27	0.481226
$484$	0	0
$485$	1.74918e28	1.10293
$486$	0	0
$487$	−4.75272e27	−0.287004	−0.143502	−	0.989650i	$-0.545836\pi$
$487$	−4.75272e27	−0.287004	−0.143502	+	0.989650i	$0.545836\pi$
$488$	0	0
$489$	−1.11899e28	−0.647266
$490$	0	0
$491$	2.59837e28	1.43994	0.719971	−	0.694004i	$-0.244155\pi$
$491$	2.59837e28	1.43994	0.719971	+	0.694004i	$0.244155\pi$
$492$	0	0
$493$	−1.29815e28	−0.689336
$494$	0	0
$495$	1.25875e28	0.640596
$496$	0	0
$497$	4.32640e27	0.211050
$498$	0	0
$499$	−3.84508e28	−1.79825	−0.899124	−	0.437695i	$-0.855795\pi$
$499$	−3.84508e28	−1.79825	−0.899124	+	0.437695i	$0.855795\pi$
$500$	0	0
$501$	2.44045e28	1.09439
$502$	0	0
$503$	3.27446e28	1.40824	0.704118	−	0.710083i	$-0.251343\pi$
$503$	3.27446e28	1.40824	0.704118	+	0.710083i	$0.251343\pi$
$504$	0	0
$505$	−2.16846e28	−0.894525
$506$	0	0
$507$	−3.09953e28	−1.22663
$508$	0	0
$509$	−2.02472e28	−0.768826	−0.384413	−	0.923161i	$-0.625596\pi$
$509$	−2.02472e28	−0.768826	−0.384413	+	0.923161i	$0.625596\pi$
$510$	0	0
$511$	−3.32461e28	−1.21149
$512$	0	0
$513$	−4.40869e27	−0.154197
$514$	0	0
$515$	1.27627e28	0.428515
$516$	0	0
$517$	−1.39208e28	−0.448760
$518$	0	0
$519$	−5.39222e28	−1.66921
$520$	0	0
$521$	4.75207e27	0.141282	0.0706411	−	0.997502i	$-0.477496\pi$
$521$	4.75207e27	0.141282	0.0706411	+	0.997502i	$0.477496\pi$
$522$	0	0
$523$	1.28180e26	0.00366061	0.00183030	−	0.999998i	$-0.499417\pi$
$523$	1.28180e26	0.00366061	0.00183030	+	0.999998i	$0.499417\pi$
$524$	0	0
$525$	−8.41817e26	−0.0230964
$526$	0	0
$527$	−5.80099e27	−0.152929
$528$	0	0
$529$	−3.40184e28	−0.861846
$530$	0	0
$531$	3.66944e28	0.893527
$532$	0	0
$533$	1.66265e27	0.0389192
$534$	0	0
$535$	−2.42833e28	−0.546504
$536$	0	0
$537$	−6.14266e28	−1.32931
$538$	0	0
$539$	2.97429e27	0.0619014
$540$	0	0
$541$	3.42747e28	0.686123	0.343061	−	0.939313i	$-0.388536\pi$
$541$	3.42747e28	0.686123	0.343061	+	0.939313i	$0.388536\pi$
$542$	0	0
$543$	−3.71879e27	−0.0716149
$544$	0	0
$545$	3.73077e28	0.691250
$546$	0	0
$547$	7.30329e28	1.30212	0.651061	−	0.759026i	$-0.274324\pi$
$547$	7.30329e28	1.30212	0.651061	+	0.759026i	$0.274324\pi$
$548$	0	0
$549$	3.80130e28	0.652263
$550$	0	0
$551$	−3.36874e28	−0.556385
$552$	0	0
$553$	3.93755e28	0.626055
$554$	0	0
$555$	6.18765e28	0.947217
$556$	0	0
$557$	−3.32597e28	−0.490274	−0.245137	−	0.969488i	$-0.578833\pi$
$557$	−3.32597e28	−0.490274	−0.245137	+	0.969488i	$0.578833\pi$
$558$	0	0
$559$	−1.56089e28	−0.221588
$560$	0	0
$561$	3.72523e28	0.509382
$562$	0	0
$563$	8.28332e28	1.09110	0.545552	−	0.838077i	$-0.316320\pi$
$563$	8.28332e28	1.09110	0.545552	+	0.838077i	$0.316320\pi$
$564$	0	0
$565$	1.07298e28	0.136170
$566$	0	0
$567$	−1.04417e29	−1.27687
$568$	0	0
$569$	7.35414e28	0.866669	0.433335	−	0.901233i	$-0.357337\pi$
$569$	7.35414e28	0.866669	0.433335	+	0.901233i	$0.357337\pi$
$570$	0	0
$571$	−1.09131e29	−1.23956	−0.619780	−	0.784776i	$-0.712778\pi$
$571$	−1.09131e29	−1.23956	−0.619780	+	0.784776i	$0.712778\pi$
$572$	0	0
$573$	−1.14204e29	−1.25043
$574$	0	0
$575$	−6.28162e26	−0.00663070
$576$	0	0
$577$	1.30727e28	0.133051	0.0665257	−	0.997785i	$-0.478809\pi$
$577$	1.30727e28	0.133051	0.0665257	+	0.997785i	$0.478809\pi$
$578$	0	0
$579$	1.11258e28	0.109196
$580$	0	0
$581$	−3.75682e28	−0.355607
$582$	0	0
$583$	1.93075e29	1.76281
$584$	0	0
$585$	−1.07134e28	−0.0943605
$586$	0	0
$587$	1.71459e29	1.45701	0.728503	−	0.685043i	$-0.240216\pi$
$587$	1.71459e29	1.45701	0.728503	+	0.685043i	$0.240216\pi$
$588$	0	0
$589$	−1.50538e28	−0.123434
$590$	0	0
$591$	−9.00630e28	−0.712652
$592$	0	0
$593$	2.35277e29	1.79682	0.898412	−	0.439153i	$-0.144721\pi$
$593$	2.35277e29	1.79682	0.898412	+	0.439153i	$0.144721\pi$
$594$	0	0
$595$	5.07349e28	0.374006
$596$	0	0
$597$	1.60465e29	1.14196
$598$	0	0
$599$	1.33384e29	0.916481	0.458240	−	0.888828i	$-0.348480\pi$
$599$	1.33384e29	0.916481	0.458240	+	0.888828i	$0.348480\pi$
$600$	0	0
$601$	−3.64474e28	−0.241816	−0.120908	−	0.992664i	$-0.538581\pi$
$601$	−3.64474e28	−0.241816	−0.120908	+	0.992664i	$0.538581\pi$
$602$	0	0
$603$	−1.04150e29	−0.667308
$604$	0	0
$605$	3.40317e28	0.210597
$606$	0	0
$607$	−2.71228e29	−1.62126	−0.810630	−	0.585559i	$-0.800875\pi$
$607$	−2.71228e29	−1.62126	−0.810630	+	0.585559i	$0.800875\pi$
$608$	0	0
$609$	−4.20890e29	−2.43045
$610$	0	0
$611$	1.18482e28	0.0661029
$612$	0	0
$613$	−2.02699e29	−1.09274	−0.546370	−	0.837544i	$-0.683991\pi$
$613$	−2.02699e29	−1.09274	−0.546370	+	0.837544i	$0.683991\pi$
$614$	0	0
$615$	−5.75027e28	−0.299571
$616$	0	0
$617$	3.04169e29	1.53151	0.765757	−	0.643130i	$-0.222364\pi$
$617$	3.04169e29	1.53151	0.765757	+	0.643130i	$0.222364\pi$
$618$	0	0
$619$	−1.21487e29	−0.591262	−0.295631	−	0.955302i	$-0.595530\pi$
$619$	−1.21487e29	−0.591262	−0.295631	+	0.955302i	$0.595530\pi$
$620$	0	0
$621$	−4.11022e28	−0.193376
$622$	0	0
$623$	−3.87345e29	−1.76187
$624$	0	0
$625$	−2.23245e29	−0.981842
$626$	0	0
$627$	9.66712e28	0.411138
$628$	0	0
$629$	6.77345e28	0.278598
$630$	0	0
$631$	1.79118e29	0.712577	0.356288	−	0.934376i	$-0.384042\pi$
$631$	1.79118e29	0.712577	0.356288	+	0.934376i	$0.384042\pi$
$632$	0	0
$633$	−4.52038e29	−1.73955
$634$	0	0
$635$	−3.54066e28	−0.131814
$636$	0	0
$637$	−2.53146e27	−0.00911815
$638$	0	0
$639$	3.45875e28	0.120548
$640$	0	0
$641$	−3.41742e29	−1.15263	−0.576315	−	0.817228i	$-0.695510\pi$
$641$	−3.41742e29	−1.15263	−0.576315	+	0.817228i	$0.695510\pi$
$642$	0	0
$643$	−7.57582e28	−0.247294	−0.123647	−	0.992326i	$-0.539459\pi$
$643$	−7.57582e28	−0.247294	−0.123647	+	0.992326i	$0.539459\pi$
$644$	0	0
$645$	5.39832e29	1.70562
$646$	0	0
$647$	2.99088e29	0.914755	0.457377	−	0.889273i	$-0.348789\pi$
$647$	2.99088e29	0.914755	0.457377	+	0.889273i	$0.348789\pi$
$648$	0	0
$649$	5.66065e29	1.67609
$650$	0	0
$651$	−1.88082e29	−0.539196
$652$	0	0
$653$	5.40507e28	0.150042	0.0750210	−	0.997182i	$-0.476098\pi$
$653$	5.40507e28	0.150042	0.0750210	+	0.997182i	$0.476098\pi$
$654$	0	0
$655$	3.00051e29	0.806606
$656$	0	0
$657$	−2.65787e29	−0.691985
$658$	0	0
$659$	−2.26606e28	−0.0571446	−0.0285723	−	0.999592i	$-0.509096\pi$
$659$	−2.26606e28	−0.0571446	−0.0285723	+	0.999592i	$0.509096\pi$
$660$	0	0
$661$	−1.87483e29	−0.457980	−0.228990	−	0.973429i	$-0.573542\pi$
$661$	−1.87483e29	−0.457980	−0.228990	+	0.973429i	$0.573542\pi$
$662$	0	0
$663$	−3.17060e28	−0.0750325
$664$	0	0
$665$	1.31659e29	0.301872
$666$	0	0
$667$	−3.14067e29	−0.697752
$668$	0	0
$669$	6.08595e29	1.31025
$670$	0	0
$671$	5.86406e29	1.22352
$672$	0	0
$673$	7.53001e29	1.52278	0.761390	−	0.648294i	$-0.224517\pi$
$673$	7.53001e29	1.52278	0.761390	+	0.648294i	$0.224517\pi$
$674$	0	0
$675$	4.73466e27	0.00928109
$676$	0	0
$677$	−8.12052e29	−1.54313	−0.771565	−	0.636150i	$-0.780526\pi$
$677$	−8.12052e29	−1.54313	−0.771565	+	0.636150i	$0.780526\pi$
$678$	0	0
$679$	6.20819e29	1.14375
$680$	0	0
$681$	7.01320e29	1.25277
$682$	0	0
$683$	−4.69973e29	−0.814058	−0.407029	−	0.913415i	$-0.633435\pi$
$683$	−4.69973e29	−0.814058	−0.407029	+	0.913415i	$0.633435\pi$
$684$	0	0
$685$	−8.67346e29	−1.45694
$686$	0	0
$687$	8.89896e29	1.44975
$688$	0	0
$689$	−1.64329e29	−0.259664
$690$	0	0
$691$	−3.58806e29	−0.549971	−0.274985	−	0.961448i	$-0.588673\pi$
$691$	−3.58806e29	−0.549971	−0.274985	+	0.961448i	$0.588673\pi$
$692$	0	0
$693$	4.46754e29	0.664308
$694$	0	0
$695$	−9.68616e29	−1.39737
$696$	0	0
$697$	−6.29466e28	−0.0881106
$698$	0	0
$699$	5.84501e29	0.793917
$700$	0	0
$701$	−7.33110e29	−0.966339	−0.483170	−	0.875527i	$-0.660515\pi$
$701$	−7.33110e29	−0.966339	−0.483170	+	0.875527i	$0.660515\pi$
$702$	0	0
$703$	1.75774e29	0.224865
$704$	0	0
$705$	−4.09771e29	−0.508810
$706$	0	0
$707$	−7.69629e29	−0.927636
$708$	0	0
$709$	1.20909e30	1.41473	0.707365	−	0.706848i	$-0.249884\pi$
$709$	1.20909e30	1.41473	0.707365	+	0.706848i	$0.249884\pi$
$710$	0	0
$711$	3.14788e29	0.357592
$712$	0	0
$713$	−1.40346e29	−0.154796
$714$	0	0
$715$	−1.65270e29	−0.177003
$716$	0	0
$717$	3.52379e29	0.366488
$718$	0	0
$719$	5.25189e29	0.530472	0.265236	−	0.964183i	$-0.414550\pi$
$719$	5.25189e29	0.530472	0.265236	+	0.964183i	$0.414550\pi$
$720$	0	0
$721$	4.52972e29	0.444377
$722$	0	0
$723$	−1.04223e30	−0.993140
$724$	0	0
$725$	3.61782e28	0.0334886
$726$	0	0
$727$	−7.23318e29	−0.650456	−0.325228	−	0.945636i	$-0.605441\pi$
$727$	−7.23318e29	−0.650456	−0.325228	+	0.945636i	$0.605441\pi$
$728$	0	0
$729$	−8.46052e28	−0.0739193
$730$	0	0
$731$	5.90940e29	0.501662
$732$	0	0
$733$	−2.11409e30	−1.74394	−0.871969	−	0.489560i	$-0.837157\pi$
$733$	−2.11409e30	−1.74394	−0.871969	+	0.489560i	$0.837157\pi$
$734$	0	0
$735$	8.75506e28	0.0701846
$736$	0	0
$737$	−1.60666e30	−1.25174
$738$	0	0
$739$	7.42069e29	0.561924	0.280962	−	0.959719i	$-0.409347\pi$
$739$	7.42069e29	0.561924	0.280962	+	0.959719i	$0.409347\pi$
$740$	0	0
$741$	−8.22782e28	−0.0605611
$742$	0	0
$743$	−1.69968e30	−1.21614	−0.608071	−	0.793882i	$-0.708057\pi$
$743$	−1.69968e30	−1.21614	−0.608071	+	0.793882i	$0.708057\pi$
$744$	0	0
$745$	1.06752e30	0.742569
$746$	0	0
$747$	−3.00339e29	−0.203117
$748$	0	0
$749$	−8.61862e29	−0.566733
$750$	0	0
$751$	−5.90225e29	−0.377397	−0.188698	−	0.982035i	$-0.560427\pi$
$751$	−5.90225e29	−0.377397	−0.188698	+	0.982035i	$0.560427\pi$
$752$	0	0
$753$	−8.55430e29	−0.531909
$754$	0	0
$755$	−1.23443e30	−0.746490
$756$	0	0
$757$	1.33308e30	0.784058	0.392029	−	0.919953i	$-0.371773\pi$
$757$	1.33308e30	0.784058	0.392029	+	0.919953i	$0.371773\pi$
$758$	0	0
$759$	9.01264e29	0.515601
$760$	0	0
$761$	−7.86217e29	−0.437526	−0.218763	−	0.975778i	$-0.570202\pi$
$761$	−7.86217e29	−0.437526	−0.218763	+	0.975778i	$0.570202\pi$
$762$	0	0
$763$	1.32412e30	0.716837
$764$	0	0
$765$	4.05601e29	0.213626
$766$	0	0
$767$	−4.81786e29	−0.246890
$768$	0	0
$769$	−1.65083e30	−0.823142	−0.411571	−	0.911378i	$-0.635020\pi$
$769$	−1.65083e30	−0.823142	−0.411571	+	0.911378i	$0.635020\pi$
$770$	0	0
$771$	2.02644e29	0.0983247
$772$	0	0
$773$	3.77930e30	1.78454	0.892271	−	0.451499i	$-0.149111\pi$
$773$	3.77930e30	1.78454	0.892271	+	0.451499i	$0.149111\pi$
$774$	0	0
$775$	1.61668e28	0.00742946
$776$	0	0
$777$	2.19612e30	0.982278
$778$	0	0
$779$	−1.63349e29	−0.0711169
$780$	0	0
$781$	5.33563e29	0.226126
$782$	0	0
$783$	2.36723e30	0.976655
$784$	0	0
$785$	1.37555e30	0.552516
$786$	0	0
$787$	3.58219e30	1.40092	0.700461	−	0.713691i	$-0.252978\pi$
$787$	3.58219e30	1.40092	0.700461	+	0.713691i	$0.252978\pi$
$788$	0	0
$789$	4.38169e30	1.66853
$790$	0	0
$791$	3.80821e29	0.141211
$792$	0	0
$793$	−4.99099e29	−0.180226
$794$	0	0
$795$	5.68331e30	1.99870
$796$	0	0
$797$	−4.45901e30	−1.52731	−0.763654	−	0.645626i	$-0.776596\pi$
$797$	−4.45901e30	−1.52731	−0.763654	+	0.645626i	$0.776596\pi$
$798$	0	0
$799$	−4.48565e29	−0.149653
$800$	0	0
$801$	−3.09664e30	−1.00635
$802$	0	0
$803$	−4.10015e30	−1.29803
$804$	0	0
$805$	1.22746e30	0.378572
$806$	0	0
$807$	−4.61229e30	−1.38594
$808$	0	0
$809$	3.09156e30	0.905144	0.452572	−	0.891728i	$-0.350507\pi$
$809$	3.09156e30	0.905144	0.452572	+	0.891728i	$0.350507\pi$
$810$	0	0
$811$	9.85711e29	0.281210	0.140605	−	0.990066i	$-0.455095\pi$
$811$	9.85711e29	0.281210	0.140605	+	0.990066i	$0.455095\pi$
$812$	0	0
$813$	−3.17090e30	−0.881518
$814$	0	0
$815$	−1.87948e30	−0.509194
$816$	0	0
$817$	1.53351e30	0.404907
$818$	0	0
$819$	−3.80239e29	−0.0978533
$820$	0	0
$821$	−5.81930e30	−1.45971	−0.729856	−	0.683601i	$-0.760413\pi$
$821$	−5.81930e30	−1.45971	−0.729856	+	0.683601i	$0.760413\pi$
$822$	0	0
$823$	−1.82848e30	−0.447086	−0.223543	−	0.974694i	$-0.571762\pi$
$823$	−1.82848e30	−0.447086	−0.223543	+	0.974694i	$0.571762\pi$
$824$	0	0
$825$	−1.03819e29	−0.0247463
$826$	0	0
$827$	3.80997e30	0.885347	0.442673	−	0.896683i	$-0.354030\pi$
$827$	3.80997e30	0.885347	0.442673	+	0.896683i	$0.354030\pi$
$828$	0	0
$829$	9.24247e29	0.209395	0.104697	−	0.994504i	$-0.466613\pi$
$829$	9.24247e29	0.209395	0.104697	+	0.994504i	$0.466613\pi$
$830$	0	0
$831$	−7.88112e30	−1.74091
$832$	0	0
$833$	9.58392e28	0.0206429
$834$	0	0
$835$	4.09904e30	0.860942
$836$	0	0
$837$	1.05784e30	0.216671
$838$	0	0
$839$	−2.18057e30	−0.435582	−0.217791	−	0.975995i	$-0.569885\pi$
$839$	−2.18057e30	−0.435582	−0.217791	+	0.975995i	$0.569885\pi$
$840$	0	0
$841$	1.29554e31	2.52403
$842$	0	0
$843$	2.23815e30	0.425304
$844$	0	0
$845$	−5.20605e30	−0.964968
$846$	0	0
$847$	1.20785e30	0.218393
$848$	0	0
$849$	−9.75655e30	−1.72094
$850$	0	0
$851$	1.63874e30	0.282000
$852$	0	0
$853$	−4.27754e30	−0.718173	−0.359087	−	0.933304i	$-0.616912\pi$
$853$	−4.27754e30	−0.718173	−0.359087	+	0.933304i	$0.616912\pi$
$854$	0	0
$855$	1.05255e30	0.172424
$856$	0	0
$857$	−1.76322e29	−0.0281843	−0.0140922	−	0.999901i	$-0.504486\pi$
$857$	−1.76322e29	−0.0281843	−0.0140922	+	0.999901i	$0.504486\pi$
$858$	0	0
$859$	−4.74009e30	−0.739365	−0.369682	−	0.929158i	$-0.620533\pi$
$859$	−4.74009e30	−0.739365	−0.369682	+	0.929158i	$0.620533\pi$
$860$	0	0
$861$	−2.04088e30	−0.310659
$862$	0	0
$863$	−8.10824e29	−0.120452	−0.0602259	−	0.998185i	$-0.519182\pi$
$863$	−8.10824e29	−0.120452	−0.0602259	+	0.998185i	$0.519182\pi$
$864$	0	0
$865$	−9.05691e30	−1.31314
$866$	0	0
$867$	−7.70171e30	−1.08990
$868$	0	0
$869$	4.85607e30	0.670775
$870$	0	0
$871$	1.36745e30	0.184383
$872$	0	0
$873$	4.96315e30	0.653293
$874$	0	0
$875$	−8.06736e30	−1.03669
$876$	0	0
$877$	1.55255e30	0.194782	0.0973912	−	0.995246i	$-0.468950\pi$
$877$	1.55255e30	0.194782	0.0973912	+	0.995246i	$0.468950\pi$
$878$	0	0
$879$	6.29641e30	0.771275
$880$	0	0
$881$	−5.75722e30	−0.688598	−0.344299	−	0.938860i	$-0.611883\pi$
$881$	−5.75722e30	−0.688598	−0.344299	+	0.938860i	$0.611883\pi$
$882$	0	0
$883$	1.29134e31	1.50818	0.754088	−	0.656774i	$-0.228079\pi$
$883$	1.29134e31	1.50818	0.754088	+	0.656774i	$0.228079\pi$
$884$	0	0
$885$	1.66626e31	1.90037
$886$	0	0
$887$	4.22752e30	0.470856	0.235428	−	0.971892i	$-0.424351\pi$
$887$	4.22752e30	0.470856	0.235428	+	0.971892i	$0.424351\pi$
$888$	0	0
$889$	−1.25665e30	−0.136693
$890$	0	0
$891$	−1.28774e31	−1.36808
$892$	0	0
$893$	−1.16404e30	−0.120789
$894$	0	0
$895$	−1.03174e31	−1.04575
$896$	0	0
$897$	−7.67079e29	−0.0759486
$898$	0	0
$899$	8.08306e30	0.781806
$900$	0	0
$901$	6.22137e30	0.587862
$902$	0	0
$903$	1.91597e31	1.76875
$904$	0	0
$905$	−6.24617e29	−0.0563383
$906$	0	0
$907$	−1.34786e31	−1.18787	−0.593935	−	0.804513i	$-0.702426\pi$
$907$	−1.34786e31	−1.18787	−0.593935	+	0.804513i	$0.702426\pi$
$908$	0	0
$909$	−6.15281e30	−0.529850
$910$	0	0
$911$	1.28886e31	1.08458	0.542292	−	0.840190i	$-0.317557\pi$
$911$	1.28886e31	1.08458	0.542292	+	0.840190i	$0.317557\pi$
$912$	0	0
$913$	−4.63317e30	−0.381009
$914$	0	0
$915$	1.72613e31	1.38724
$916$	0	0
$917$	1.06494e31	0.836463
$918$	0	0
$919$	−1.25018e30	−0.0959753	−0.0479876	−	0.998848i	$-0.515281\pi$
$919$	−1.25018e30	−0.0959753	−0.0479876	+	0.998848i	$0.515281\pi$
$920$	0	0
$921$	2.80864e31	2.10751
$922$	0	0
$923$	−4.54123e29	−0.0333086
$924$	0	0
$925$	−1.88770e29	−0.0135346
$926$	0	0
$927$	3.62130e30	0.253821
$928$	0	0
$929$	−1.41245e31	−0.967850	−0.483925	−	0.875109i	$-0.660789\pi$
$929$	−1.41245e31	−0.967850	−0.483925	+	0.875109i	$0.660789\pi$
$930$	0	0
$931$	2.48706e29	0.0166615
$932$	0	0
$933$	5.21360e30	0.341492
$934$	0	0
$935$	6.25699e30	0.400722
$936$	0	0
$937$	−7.12851e30	−0.446409	−0.223204	−	0.974772i	$-0.571652\pi$
$937$	−7.12851e30	−0.446409	−0.223204	+	0.974772i	$0.571652\pi$
$938$	0	0
$939$	−1.12632e30	−0.0689721
$940$	0	0
$941$	−1.07254e29	−0.00642275	−0.00321138	−	0.999995i	$-0.501022\pi$
$941$	−1.07254e29	−0.00642275	−0.00321138	+	0.999995i	$0.501022\pi$
$942$	0	0
$943$	−1.52290e30	−0.0891864
$944$	0	0
$945$	−9.25173e30	−0.529894
$946$	0	0
$947$	−2.92935e31	−1.64095	−0.820477	−	0.571680i	$-0.806292\pi$
$947$	−2.92935e31	−1.64095	−0.820477	+	0.571680i	$0.806292\pi$
$948$	0	0
$949$	3.48969e30	0.191202
$950$	0	0
$951$	−4.11989e30	−0.220796
$952$	0	0
$953$	3.04320e31	1.59535	0.797675	−	0.603088i	$-0.206063\pi$
$953$	3.04320e31	1.59535	0.797675	+	0.603088i	$0.206063\pi$
$954$	0	0
$955$	−1.91821e31	−0.983694
$956$	0	0
$957$	−5.19071e31	−2.60406
$958$	0	0
$959$	−3.07838e31	−1.51087
$960$	0	0
$961$	−1.72134e31	−0.826556
$962$	0	0
$963$	−6.89017e30	−0.323709
$964$	0	0
$965$	1.86871e30	0.0859026
$966$	0	0
$967$	3.78485e31	1.70244	0.851218	−	0.524813i	$-0.175865\pi$
$967$	3.78485e31	1.70244	0.851218	+	0.524813i	$0.175865\pi$
$968$	0	0
$969$	3.11500e30	0.137106
$970$	0	0
$971$	1.62334e31	0.699212	0.349606	−	0.936897i	$-0.386316\pi$
$971$	1.62334e31	0.699212	0.349606	+	0.936897i	$0.386316\pi$
$972$	0	0
$973$	−3.43780e31	−1.44909
$974$	0	0
$975$	8.83618e28	0.00364516
$976$	0	0
$977$	2.98858e30	0.120662	0.0603312	−	0.998178i	$-0.480784\pi$
$977$	2.98858e30	0.120662	0.0603312	+	0.998178i	$0.480784\pi$
$978$	0	0
$979$	−4.77701e31	−1.88772
$980$	0	0
$981$	1.05857e31	0.409445
$982$	0	0
$983$	2.41371e31	0.913846	0.456923	−	0.889506i	$-0.348951\pi$
$983$	2.41371e31	0.913846	0.456923	+	0.889506i	$0.348951\pi$
$984$	0	0
$985$	−1.51272e31	−0.560632
$986$	0	0
$987$	−1.45436e31	−0.527643
$988$	0	0
$989$	1.42969e31	0.507786
$990$	0	0
$991$	2.29436e31	0.797790	0.398895	−	0.916997i	$-0.369394\pi$
$991$	2.29436e31	0.797790	0.398895	+	0.916997i	$0.369394\pi$
$992$	0	0
$993$	3.24964e31	1.10629
$994$	0	0
$995$	2.69521e31	0.898361
$996$	0	0
$997$	3.55036e31	1.15871	0.579354	−	0.815076i	$-0.303305\pi$
$997$	3.55036e31	1.15871	0.579354	+	0.815076i	$0.303305\pi$
$998$	0	0
$999$	−1.23517e31	−0.394719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.22.a.c.1.1		1
4.3	odd	2		1.22.a.a.1.1	✓	1
8.3	odd	2		64.22.a.g.1.1		1
8.5	even	2		64.22.a.a.1.1		1
12.11	even	2		9.22.a.c.1.1		1
20.3	even	4		25.22.b.a.24.2		2
20.7	even	4		25.22.b.a.24.1		2
20.19	odd	2		25.22.a.a.1.1		1
28.27	even	2		49.22.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.22.a.a.1.1	✓	1	4.3	odd	2
9.22.a.c.1.1		1	12.11	even	2
16.22.a.c.1.1		1	1.1	even	1	trivial
25.22.a.a.1.1		1	20.19	odd	2
25.22.b.a.24.1		2	20.7	even	4
25.22.b.a.24.2		2	20.3	even	4
49.22.a.a.1.1		1	28.27	even	2
64.22.a.a.1.1		1	8.5	even	2
64.22.a.g.1.1		1	8.3	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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
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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