LMFDB - Embedded newform 3920.2.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(1,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.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:	$31.3013575923$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} +O(q^{10})$	$q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} +0.171573 q^{11} +4.41421 q^{13} +0.414214 q^{15} +3.24264 q^{17} -6.00000 q^{19} -7.41421 q^{23} +1.00000 q^{25} -2.41421 q^{27} -8.65685 q^{29} -10.2426 q^{31} +0.0710678 q^{33} +2.24264 q^{37} +1.82843 q^{39} -6.24264 q^{41} -2.00000 q^{43} -2.82843 q^{45} +7.24264 q^{47} +1.34315 q^{51} +4.24264 q^{53} +0.171573 q^{55} -2.48528 q^{57} +2.24264 q^{59} -2.82843 q^{61} +4.41421 q^{65} +8.24264 q^{67} -3.07107 q^{69} +3.17157 q^{71} -8.48528 q^{73} +0.414214 q^{75} -1.48528 q^{79} +7.48528 q^{81} +3.24264 q^{85} -3.58579 q^{87} -8.00000 q^{89} -4.24264 q^{93} -6.00000 q^{95} +13.2426 q^{97} -0.485281 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5	$ 2 q - 2 q^{3} + 2 q^{5} + 6 q^{11} + 6 q^{13} - 2 q^{15} - 2 q^{17} - 12 q^{19} - 12 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} - 12 q^{31} - 14 q^{33} - 4 q^{37} - 2 q^{39} - 4 q^{41} - 4 q^{43} + 6 q^{47} + 14 q^{51} + 6 q^{55} + 12 q^{57} - 4 q^{59} + 6 q^{65} + 8 q^{67} + 8 q^{69} + 12 q^{71} - 2 q^{75} + 14 q^{79} - 2 q^{81} - 2 q^{85} - 10 q^{87} - 16 q^{89} - 12 q^{95} + 18 q^{97} + 16 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 6 * q^11 + 6 * q^13 - 2 * q^15 - 2 * q^17 - 12 * q^19 - 12 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 - 12 * q^31 - 14 * q^33 - 4 * q^37 - 2 * q^39 - 4 * q^41 - 4 * q^43 + 6 * q^47 + 14 * q^51 + 6 * q^55 + 12 * q^57 - 4 * q^59 + 6 * q^65 + 8 * q^67 + 8 * q^69 + 12 * q^71 - 2 * q^75 + 14 * q^79 - 2 * q^81 - 2 * q^85 - 10 * q^87 - 16 * q^89 - 12 * q^95 + 18 * q^97 + 16 * q^99

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
$3$	0.414214	0.239146	0.119573	−	0.992825i	$-0.461847\pi$
$3$	0.414214	0.239146	0.119573	+	0.992825i	$0.461847\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	0.171573	0.0517312	0.0258656	−	0.999665i	$-0.491766\pi$
$11$	0.171573	0.0517312	0.0258656	+	0.999665i	$0.491766\pi$
$12$	0	0
$13$	4.41421	1.22428	0.612141	−	0.790748i	$-0.290308\pi$
$13$	4.41421	1.22428	0.612141	+	0.790748i	$0.290308\pi$
$14$	0	0
$15$	0.414214	0.106949
$16$	0	0
$17$	3.24264	0.786456	0.393228	−	0.919441i	$-0.371358\pi$
$17$	3.24264	0.786456	0.393228	+	0.919441i	$0.371358\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.41421	−1.54597	−0.772985	−	0.634424i	$-0.781237\pi$
$23$	−7.41421	−1.54597	−0.772985	+	0.634424i	$0.781237\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.41421	−0.464616
$28$	0	0
$29$	−8.65685	−1.60754	−0.803769	−	0.594942i	$-0.797175\pi$
$29$	−8.65685	−1.60754	−0.803769	+	0.594942i	$0.797175\pi$
$30$	0	0
$31$	−10.2426	−1.83963	−0.919816	−	0.392349i	$-0.871662\pi$
$31$	−10.2426	−1.83963	−0.919816	+	0.392349i	$0.871662\pi$
$32$	0	0
$33$	0.0710678	0.0123713
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.24264	0.368688	0.184344	−	0.982862i	$-0.440984\pi$
$37$	2.24264	0.368688	0.184344	+	0.982862i	$0.440984\pi$
$38$	0	0
$39$	1.82843	0.292783
$40$	0	0
$41$	−6.24264	−0.974937	−0.487468	−	0.873141i	$-0.662080\pi$
$41$	−6.24264	−0.974937	−0.487468	+	0.873141i	$0.662080\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0	0
$47$	7.24264	1.05645	0.528224	−	0.849105i	$-0.322858\pi$
$47$	7.24264	1.05645	0.528224	+	0.849105i	$0.322858\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.34315	0.188078
$52$	0	0
$53$	4.24264	0.582772	0.291386	−	0.956606i	$-0.405884\pi$
$53$	4.24264	0.582772	0.291386	+	0.956606i	$0.405884\pi$
$54$	0	0
$55$	0.171573	0.0231349
$56$	0	0
$57$	−2.48528	−0.329184
$58$	0	0
$59$	2.24264	0.291967	0.145983	−	0.989287i	$-0.453365\pi$
$59$	2.24264	0.291967	0.145983	+	0.989287i	$0.453365\pi$
$60$	0	0
$61$	−2.82843	−0.362143	−0.181071	−	0.983470i	$-0.557957\pi$
$61$	−2.82843	−0.362143	−0.181071	+	0.983470i	$0.557957\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.41421	0.547516
$66$	0	0
$67$	8.24264	1.00700	0.503499	−	0.863996i	$-0.332046\pi$
$67$	8.24264	1.00700	0.503499	+	0.863996i	$0.332046\pi$
$68$	0	0
$69$	−3.07107	−0.369713
$70$	0	0
$71$	3.17157	0.376396	0.188198	−	0.982131i	$-0.439735\pi$
$71$	3.17157	0.376396	0.188198	+	0.982131i	$0.439735\pi$
$72$	0	0
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	0	0
$75$	0.414214	0.0478293
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.48528	−0.167107	−0.0835536	−	0.996503i	$-0.526627\pi$
$79$	−1.48528	−0.167107	−0.0835536	+	0.996503i	$0.526627\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	3.24264	0.351714
$86$	0	0
$87$	−3.58579	−0.384437
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.24264	−0.439941
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	13.2426	1.34459	0.672293	−	0.740285i	$-0.265309\pi$
$97$	13.2426	1.34459	0.672293	+	0.740285i	$0.265309\pi$
$98$	0	0
$99$	−0.485281	−0.0487726
$100$	0	0
$101$	2.48528	0.247295	0.123647	−	0.992326i	$-0.460541\pi$
$101$	2.48528	0.247295	0.123647	+	0.992326i	$0.460541\pi$
$102$	0	0
$103$	−19.2426	−1.89603	−0.948017	−	0.318220i	$-0.896915\pi$
$103$	−19.2426	−1.89603	−0.948017	+	0.318220i	$0.896915\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.48528	0.240261	0.120131	−	0.992758i	$-0.461669\pi$
$107$	2.48528	0.240261	0.120131	+	0.992758i	$0.461669\pi$
$108$	0	0
$109$	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
$111$	0.928932	0.0881703
$112$	0	0
$113$	−13.0711	−1.22962	−0.614811	−	0.788674i	$-0.710768\pi$
$113$	−13.0711	−1.22962	−0.614811	+	0.788674i	$0.710768\pi$
$114$	0	0
$115$	−7.41421	−0.691379
$116$	0	0
$117$	−12.4853	−1.15426
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.9706	−0.997324
$122$	0	0
$123$	−2.58579	−0.233153
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.24264	−0.731416	−0.365708	−	0.930730i	$-0.619173\pi$
$127$	−8.24264	−0.731416	−0.365708	+	0.930730i	$0.619173\pi$
$128$	0	0
$129$	−0.828427	−0.0729389
$130$	0	0
$131$	−12.2426	−1.06964	−0.534822	−	0.844965i	$-0.679621\pi$
$131$	−12.2426	−1.06964	−0.534822	+	0.844965i	$0.679621\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.41421	−0.207782
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−7.75736	−0.657971	−0.328985	−	0.944335i	$-0.606707\pi$
$139$	−7.75736	−0.657971	−0.328985	+	0.944335i	$0.606707\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	0	0
$143$	0.757359	0.0633336
$144$	0	0
$145$	−8.65685	−0.718913
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.17157	−0.751365	−0.375682	−	0.926749i	$-0.622592\pi$
$149$	−9.17157	−0.751365	−0.375682	+	0.926749i	$0.622592\pi$
$150$	0	0
$151$	7.48528	0.609144	0.304572	−	0.952489i	$-0.401487\pi$
$151$	7.48528	0.609144	0.304572	+	0.952489i	$0.401487\pi$
$152$	0	0
$153$	−9.17157	−0.741478
$154$	0	0
$155$	−10.2426	−0.822709
$156$	0	0
$157$	−15.1716	−1.21082	−0.605412	−	0.795913i	$-0.706992\pi$
$157$	−15.1716	−1.21082	−0.605412	+	0.795913i	$0.706992\pi$
$158$	0	0
$159$	1.75736	0.139368
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.2426	0.802266	0.401133	−	0.916020i	$-0.368617\pi$
$163$	10.2426	0.802266	0.401133	+	0.916020i	$0.368617\pi$
$164$	0	0
$165$	0.0710678	0.00553262
$166$	0	0
$167$	−0.757359	−0.0586062	−0.0293031	−	0.999571i	$-0.509329\pi$
$167$	−0.757359	−0.0586062	−0.0293031	+	0.999571i	$0.509329\pi$
$168$	0	0
$169$	6.48528	0.498868
$170$	0	0
$171$	16.9706	1.29777
$172$	0	0
$173$	−7.24264	−0.550648	−0.275324	−	0.961352i	$-0.588785\pi$
$173$	−7.24264	−0.550648	−0.275324	+	0.961352i	$0.588785\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.928932	0.0698228
$178$	0	0
$179$	14.4853	1.08268	0.541340	−	0.840804i	$-0.317917\pi$
$179$	14.4853	1.08268	0.541340	+	0.840804i	$0.317917\pi$
$180$	0	0
$181$	−18.7279	−1.39204	−0.696018	−	0.718025i	$-0.745047\pi$
$181$	−18.7279	−1.39204	−0.696018	+	0.718025i	$0.745047\pi$
$182$	0	0
$183$	−1.17157	−0.0866052
$184$	0	0
$185$	2.24264	0.164882
$186$	0	0
$187$	0.556349	0.0406843
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.9706	−1.01087	−0.505437	−	0.862863i	$-0.668669\pi$
$191$	−13.9706	−1.01087	−0.505437	+	0.862863i	$0.668669\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	1.82843	0.130936
$196$	0	0
$197$	13.4142	0.955723	0.477862	−	0.878435i	$-0.341412\pi$
$197$	13.4142	0.955723	0.477862	+	0.878435i	$0.341412\pi$
$198$	0	0
$199$	7.41421	0.525580	0.262790	−	0.964853i	$-0.415357\pi$
$199$	7.41421	0.525580	0.262790	+	0.964853i	$0.415357\pi$
$200$	0	0
$201$	3.41421	0.240820
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.24264	−0.436005
$206$	0	0
$207$	20.9706	1.45755
$208$	0	0
$209$	−1.02944	−0.0712077
$210$	0	0
$211$	−9.00000	−0.619586	−0.309793	−	0.950804i	$-0.600260\pi$
$211$	−9.00000	−0.619586	−0.309793	+	0.950804i	$0.600260\pi$
$212$	0	0
$213$	1.31371	0.0900138
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−3.51472	−0.237503
$220$	0	0
$221$	14.3137	0.962844
$222$	0	0
$223$	24.2132	1.62144	0.810718	−	0.585437i	$-0.199077\pi$
$223$	24.2132	1.62144	0.810718	+	0.585437i	$0.199077\pi$
$224$	0	0
$225$	−2.82843	−0.188562
$226$	0	0
$227$	−15.7279	−1.04390	−0.521949	−	0.852976i	$-0.674795\pi$
$227$	−15.7279	−1.04390	−0.521949	+	0.852976i	$0.674795\pi$
$228$	0	0
$229$	18.0416	1.19222	0.596112	−	0.802901i	$-0.296711\pi$
$229$	18.0416	1.19222	0.596112	+	0.802901i	$0.296711\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.17157	−0.600850	−0.300425	−	0.953805i	$-0.597128\pi$
$233$	−9.17157	−0.600850	−0.300425	+	0.953805i	$0.597128\pi$
$234$	0	0
$235$	7.24264	0.472458
$236$	0	0
$237$	−0.615224	−0.0399631
$238$	0	0
$239$	17.4853	1.13103	0.565514	−	0.824738i	$-0.308678\pi$
$239$	17.4853	1.13103	0.565514	+	0.824738i	$0.308678\pi$
$240$	0	0
$241$	−0.727922	−0.0468896	−0.0234448	−	0.999725i	$-0.507463\pi$
$241$	−0.727922	−0.0468896	−0.0234448	+	0.999725i	$0.507463\pi$
$242$	0	0
$243$	10.3431	0.663513
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−26.4853	−1.68522
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−17.2132	−1.08649	−0.543244	−	0.839575i	$-0.682804\pi$
$251$	−17.2132	−1.08649	−0.543244	+	0.839575i	$0.682804\pi$
$252$	0	0
$253$	−1.27208	−0.0799749
$254$	0	0
$255$	1.34315	0.0841110
$256$	0	0
$257$	−9.51472	−0.593512	−0.296756	−	0.954953i	$-0.595905\pi$
$257$	−9.51472	−0.593512	−0.296756	+	0.954953i	$0.595905\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	24.4853	1.51560
$262$	0	0
$263$	16.6274	1.02529	0.512645	−	0.858601i	$-0.328666\pi$
$263$	16.6274	1.02529	0.512645	+	0.858601i	$0.328666\pi$
$264$	0	0
$265$	4.24264	0.260623
$266$	0	0
$267$	−3.31371	−0.202796
$268$	0	0
$269$	−16.2426	−0.990331	−0.495166	−	0.868799i	$-0.664893\pi$
$269$	−16.2426	−0.990331	−0.495166	+	0.868799i	$0.664893\pi$
$270$	0	0
$271$	0.686292	0.0416892	0.0208446	−	0.999783i	$-0.493364\pi$
$271$	0.686292	0.0416892	0.0208446	+	0.999783i	$0.493364\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.171573	0.0103462
$276$	0	0
$277$	15.2132	0.914073	0.457036	−	0.889448i	$-0.348911\pi$
$277$	15.2132	0.914073	0.457036	+	0.889448i	$0.348911\pi$
$278$	0	0
$279$	28.9706	1.73442
$280$	0	0
$281$	2.31371	0.138024	0.0690121	−	0.997616i	$-0.478015\pi$
$281$	2.31371	0.138024	0.0690121	+	0.997616i	$0.478015\pi$
$282$	0	0
$283$	−24.5563	−1.45972	−0.729862	−	0.683595i	$-0.760416\pi$
$283$	−24.5563	−1.45972	−0.729862	+	0.683595i	$0.760416\pi$
$284$	0	0
$285$	−2.48528	−0.147215
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−6.48528	−0.381487
$290$	0	0
$291$	5.48528	0.321553
$292$	0	0
$293$	25.7279	1.50304	0.751521	−	0.659710i	$-0.229321\pi$
$293$	25.7279	1.50304	0.751521	+	0.659710i	$0.229321\pi$
$294$	0	0
$295$	2.24264	0.130572
$296$	0	0
$297$	−0.414214	−0.0240351
$298$	0	0
$299$	−32.7279	−1.89270
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.02944	0.0591396
$304$	0	0
$305$	−2.82843	−0.161955
$306$	0	0
$307$	−30.8995	−1.76353	−0.881764	−	0.471691i	$-0.843644\pi$
$307$	−30.8995	−1.76353	−0.881764	+	0.471691i	$0.843644\pi$
$308$	0	0
$309$	−7.97056	−0.453429
$310$	0	0
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	−18.2132	−1.02947	−0.514736	−	0.857349i	$-0.672110\pi$
$313$	−18.2132	−1.02947	−0.514736	+	0.857349i	$0.672110\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.343146	−0.0192730	−0.00963649	−	0.999954i	$-0.503067\pi$
$317$	−0.343146	−0.0192730	−0.00963649	+	0.999954i	$0.503067\pi$
$318$	0	0
$319$	−1.48528	−0.0831598
$320$	0	0
$321$	1.02944	0.0574576
$322$	0	0
$323$	−19.4558	−1.08255
$324$	0	0
$325$	4.41421	0.244857
$326$	0	0
$327$	2.07107	0.114530
$328$	0	0
$329$	0	0
$330$	0	0
$331$	27.4558	1.50911	0.754555	−	0.656237i	$-0.227853\pi$
$331$	27.4558	1.50911	0.754555	+	0.656237i	$0.227853\pi$
$332$	0	0
$333$	−6.34315	−0.347602
$334$	0	0
$335$	8.24264	0.450344
$336$	0	0
$337$	22.2426	1.21163	0.605817	−	0.795604i	$-0.292846\pi$
$337$	22.2426	1.21163	0.605817	+	0.795604i	$0.292846\pi$
$338$	0	0
$339$	−5.41421	−0.294060
$340$	0	0
$341$	−1.75736	−0.0951663
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.07107	−0.165341
$346$	0	0
$347$	−13.0711	−0.701692	−0.350846	−	0.936433i	$-0.614106\pi$
$347$	−13.0711	−0.701692	−0.350846	+	0.936433i	$0.614106\pi$
$348$	0	0
$349$	−10.9706	−0.587241	−0.293620	−	0.955922i	$-0.594860\pi$
$349$	−10.9706	−0.587241	−0.293620	+	0.955922i	$0.594860\pi$
$350$	0	0
$351$	−10.6569	−0.568821
$352$	0	0
$353$	−6.21320	−0.330695	−0.165348	−	0.986235i	$-0.552875\pi$
$353$	−6.21320	−0.330695	−0.165348	+	0.986235i	$0.552875\pi$
$354$	0	0
$355$	3.17157	0.168330
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.3137	0.597115	0.298557	−	0.954392i	$-0.403495\pi$
$359$	11.3137	0.597115	0.298557	+	0.954392i	$0.403495\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−4.54416	−0.238506
$364$	0	0
$365$	−8.48528	−0.444140
$366$	0	0
$367$	23.8701	1.24601	0.623003	−	0.782219i	$-0.285912\pi$
$367$	23.8701	1.24601	0.623003	+	0.782219i	$0.285912\pi$
$368$	0	0
$369$	17.6569	0.919179
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−16.4853	−0.853576	−0.426788	−	0.904352i	$-0.640355\pi$
$373$	−16.4853	−0.853576	−0.426788	+	0.904352i	$0.640355\pi$
$374$	0	0
$375$	0.414214	0.0213899
$376$	0	0
$377$	−38.2132	−1.96808
$378$	0	0
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	−3.41421	−0.174915
$382$	0	0
$383$	−4.48528	−0.229187	−0.114594	−	0.993412i	$-0.536557\pi$
$383$	−4.48528	−0.229187	−0.114594	+	0.993412i	$0.536557\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.65685	0.287554
$388$	0	0
$389$	−6.85786	−0.347708	−0.173854	−	0.984771i	$-0.555622\pi$
$389$	−6.85786	−0.347708	−0.173854	+	0.984771i	$0.555622\pi$
$390$	0	0
$391$	−24.0416	−1.21584
$392$	0	0
$393$	−5.07107	−0.255802
$394$	0	0
$395$	−1.48528	−0.0747326
$396$	0	0
$397$	1.58579	0.0795883	0.0397942	−	0.999208i	$-0.487330\pi$
$397$	1.58579	0.0795883	0.0397942	+	0.999208i	$0.487330\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.8284	0.590683	0.295342	−	0.955392i	$-0.404566\pi$
$401$	11.8284	0.590683	0.295342	+	0.955392i	$0.404566\pi$
$402$	0	0
$403$	−45.2132	−2.25223
$404$	0	0
$405$	7.48528	0.371947
$406$	0	0
$407$	0.384776	0.0190727
$408$	0	0
$409$	2.48528	0.122889	0.0614446	−	0.998110i	$-0.480429\pi$
$409$	2.48528	0.122889	0.0614446	+	0.998110i	$0.480429\pi$
$410$	0	0
$411$	4.97056	0.245180
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3.21320	−0.157351
$418$	0	0
$419$	−18.7279	−0.914919	−0.457459	−	0.889230i	$-0.651241\pi$
$419$	−18.7279	−0.914919	−0.457459	+	0.889230i	$0.651241\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	0	0
$423$	−20.4853	−0.996028
$424$	0	0
$425$	3.24264	0.157291
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.313708	0.0151460
$430$	0	0
$431$	28.7990	1.38720	0.693599	−	0.720361i	$-0.256024\pi$
$431$	28.7990	1.38720	0.693599	+	0.720361i	$0.256024\pi$
$432$	0	0
$433$	10.9706	0.527212	0.263606	−	0.964630i	$-0.415088\pi$
$433$	10.9706	0.527212	0.263606	+	0.964630i	$0.415088\pi$
$434$	0	0
$435$	−3.58579	−0.171925
$436$	0	0
$437$	44.4853	2.12802
$438$	0	0
$439$	30.3848	1.45019	0.725093	−	0.688651i	$-0.241797\pi$
$439$	30.3848	1.45019	0.725093	+	0.688651i	$0.241797\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.1716	−0.720823	−0.360412	−	0.932793i	$-0.617364\pi$
$443$	−15.1716	−0.720823	−0.360412	+	0.932793i	$0.617364\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	0	0
$447$	−3.79899	−0.179686
$448$	0	0
$449$	−24.1716	−1.14073	−0.570364	−	0.821392i	$-0.693198\pi$
$449$	−24.1716	−1.14073	−0.570364	+	0.821392i	$0.693198\pi$
$450$	0	0
$451$	−1.07107	−0.0504346
$452$	0	0
$453$	3.10051	0.145674
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.7574	−0.549986	−0.274993	−	0.961446i	$-0.588676\pi$
$457$	−11.7574	−0.549986	−0.274993	+	0.961446i	$0.588676\pi$
$458$	0	0
$459$	−7.82843	−0.365400
$460$	0	0
$461$	3.02944	0.141095	0.0705475	−	0.997508i	$-0.477525\pi$
$461$	3.02944	0.141095	0.0705475	+	0.997508i	$0.477525\pi$
$462$	0	0
$463$	21.4558	0.997138	0.498569	−	0.866850i	$-0.333859\pi$
$463$	21.4558	0.997138	0.498569	+	0.866850i	$0.333859\pi$
$464$	0	0
$465$	−4.24264	−0.196748
$466$	0	0
$467$	5.72792	0.265057	0.132528	−	0.991179i	$-0.457690\pi$
$467$	5.72792	0.265057	0.132528	+	0.991179i	$0.457690\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6.28427	−0.289564
$472$	0	0
$473$	−0.343146	−0.0157779
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	−12.0000	−0.549442
$478$	0	0
$479$	−11.7574	−0.537207	−0.268604	−	0.963251i	$-0.586562\pi$
$479$	−11.7574	−0.537207	−0.268604	+	0.963251i	$0.586562\pi$
$480$	0	0
$481$	9.89949	0.451378
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.2426	0.601317
$486$	0	0
$487$	−27.6985	−1.25514	−0.627569	−	0.778561i	$-0.715950\pi$
$487$	−27.6985	−1.25514	−0.627569	+	0.778561i	$0.715950\pi$
$488$	0	0
$489$	4.24264	0.191859
$490$	0	0
$491$	37.2843	1.68262	0.841308	−	0.540556i	$-0.181786\pi$
$491$	37.2843	1.68262	0.841308	+	0.540556i	$0.181786\pi$
$492$	0	0
$493$	−28.0711	−1.26426
$494$	0	0
$495$	−0.485281	−0.0218118
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3.00000	−0.134298	−0.0671492	−	0.997743i	$-0.521390\pi$
$499$	−3.00000	−0.134298	−0.0671492	+	0.997743i	$0.521390\pi$
$500$	0	0
$501$	−0.313708	−0.0140155
$502$	0	0
$503$	41.2426	1.83892	0.919459	−	0.393185i	$-0.128627\pi$
$503$	41.2426	1.83892	0.919459	+	0.393185i	$0.128627\pi$
$504$	0	0
$505$	2.48528	0.110594
$506$	0	0
$507$	2.68629	0.119302
$508$	0	0
$509$	−25.2132	−1.11756	−0.558778	−	0.829317i	$-0.688730\pi$
$509$	−25.2132	−1.11756	−0.558778	+	0.829317i	$0.688730\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	14.4853	0.639541
$514$	0	0
$515$	−19.2426	−0.847932
$516$	0	0
$517$	1.24264	0.0546513
$518$	0	0
$519$	−3.00000	−0.131685
$520$	0	0
$521$	14.9706	0.655872	0.327936	−	0.944700i	$-0.393647\pi$
$521$	14.9706	0.655872	0.327936	+	0.944700i	$0.393647\pi$
$522$	0	0
$523$	−32.4853	−1.42048	−0.710241	−	0.703959i	$-0.751414\pi$
$523$	−32.4853	−1.42048	−0.710241	+	0.703959i	$0.751414\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−33.2132	−1.44679
$528$	0	0
$529$	31.9706	1.39002
$530$	0	0
$531$	−6.34315	−0.275269
$532$	0	0
$533$	−27.5563	−1.19360
$534$	0	0
$535$	2.48528	0.107448
$536$	0	0
$537$	6.00000	0.258919
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−21.9706	−0.944588	−0.472294	−	0.881441i	$-0.656574\pi$
$541$	−21.9706	−0.944588	−0.472294	+	0.881441i	$0.656574\pi$
$542$	0	0
$543$	−7.75736	−0.332900
$544$	0	0
$545$	5.00000	0.214176
$546$	0	0
$547$	24.4853	1.04692	0.523458	−	0.852052i	$-0.324642\pi$
$547$	24.4853	1.04692	0.523458	+	0.852052i	$0.324642\pi$
$548$	0	0
$549$	8.00000	0.341432
$550$	0	0
$551$	51.9411	2.21277
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.928932	0.0394310
$556$	0	0
$557$	−7.79899	−0.330454	−0.165227	−	0.986256i	$-0.552836\pi$
$557$	−7.79899	−0.330454	−0.165227	+	0.986256i	$0.552836\pi$
$558$	0	0
$559$	−8.82843	−0.373403
$560$	0	0
$561$	0.230447	0.00972950
$562$	0	0
$563$	31.9411	1.34616	0.673079	−	0.739571i	$-0.264971\pi$
$563$	31.9411	1.34616	0.673079	+	0.739571i	$0.264971\pi$
$564$	0	0
$565$	−13.0711	−0.549904
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.1421	1.09594	0.547968	−	0.836500i	$-0.315402\pi$
$569$	26.1421	1.09594	0.547968	+	0.836500i	$0.315402\pi$
$570$	0	0
$571$	17.5147	0.732968	0.366484	−	0.930424i	$-0.380561\pi$
$571$	17.5147	0.732968	0.366484	+	0.930424i	$0.380561\pi$
$572$	0	0
$573$	−5.78680	−0.241747
$574$	0	0
$575$	−7.41421	−0.309194
$576$	0	0
$577$	15.7279	0.654762	0.327381	−	0.944892i	$-0.393834\pi$
$577$	15.7279	0.654762	0.327381	+	0.944892i	$0.393834\pi$
$578$	0	0
$579$	−6.62742	−0.275426
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0.727922	0.0301475
$584$	0	0
$585$	−12.4853	−0.516203
$586$	0	0
$587$	37.4558	1.54597	0.772984	−	0.634425i	$-0.218763\pi$
$587$	37.4558	1.54597	0.772984	+	0.634425i	$0.218763\pi$
$588$	0	0
$589$	61.4558	2.53224
$590$	0	0
$591$	5.55635	0.228558
$592$	0	0
$593$	19.2426	0.790201	0.395100	−	0.918638i	$-0.370710\pi$
$593$	19.2426	0.790201	0.395100	+	0.918638i	$0.370710\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.07107	0.125690
$598$	0	0
$599$	17.8284	0.728450	0.364225	−	0.931311i	$-0.381334\pi$
$599$	17.8284	0.728450	0.364225	+	0.931311i	$0.381334\pi$
$600$	0	0
$601$	−10.9706	−0.447499	−0.223749	−	0.974647i	$-0.571830\pi$
$601$	−10.9706	−0.447499	−0.223749	+	0.974647i	$0.571830\pi$
$602$	0	0
$603$	−23.3137	−0.949408
$604$	0	0
$605$	−10.9706	−0.446017
$606$	0	0
$607$	5.10051	0.207023	0.103512	−	0.994628i	$-0.466992\pi$
$607$	5.10051	0.207023	0.103512	+	0.994628i	$0.466992\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	31.9706	1.29339
$612$	0	0
$613$	−23.9411	−0.966973	−0.483486	−	0.875352i	$-0.660630\pi$
$613$	−23.9411	−0.966973	−0.483486	+	0.875352i	$0.660630\pi$
$614$	0	0
$615$	−2.58579	−0.104269
$616$	0	0
$617$	4.58579	0.184617	0.0923084	−	0.995730i	$-0.470575\pi$
$617$	4.58579	0.184617	0.0923084	+	0.995730i	$0.470575\pi$
$618$	0	0
$619$	−16.9289	−0.680431	−0.340216	−	0.940347i	$-0.610500\pi$
$619$	−16.9289	−0.680431	−0.340216	+	0.940347i	$0.610500\pi$
$620$	0	0
$621$	17.8995	0.718282
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.426407	−0.0170291
$628$	0	0
$629$	7.27208	0.289957
$630$	0	0
$631$	42.4558	1.69014	0.845070	−	0.534655i	$-0.179559\pi$
$631$	42.4558	1.69014	0.845070	+	0.534655i	$0.179559\pi$
$632$	0	0
$633$	−3.72792	−0.148172
$634$	0	0
$635$	−8.24264	−0.327099
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8.97056	−0.354870
$640$	0	0
$641$	−23.3137	−0.920836	−0.460418	−	0.887702i	$-0.652300\pi$
$641$	−23.3137	−0.920836	−0.460418	+	0.887702i	$0.652300\pi$
$642$	0	0
$643$	−2.27208	−0.0896020	−0.0448010	−	0.998996i	$-0.514265\pi$
$643$	−2.27208	−0.0896020	−0.0448010	+	0.998996i	$0.514265\pi$
$644$	0	0
$645$	−0.828427	−0.0326193
$646$	0	0
$647$	11.5147	0.452690	0.226345	−	0.974047i	$-0.427322\pi$
$647$	11.5147	0.452690	0.226345	+	0.974047i	$0.427322\pi$
$648$	0	0
$649$	0.384776	0.0151038
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.9706	1.36850	0.684252	−	0.729246i	$-0.260129\pi$
$653$	34.9706	1.36850	0.684252	+	0.729246i	$0.260129\pi$
$654$	0	0
$655$	−12.2426	−0.478360
$656$	0	0
$657$	24.0000	0.936329
$658$	0	0
$659$	−19.9706	−0.777943	−0.388971	−	0.921250i	$-0.627169\pi$
$659$	−19.9706	−0.777943	−0.388971	+	0.921250i	$0.627169\pi$
$660$	0	0
$661$	13.4558	0.523372	0.261686	−	0.965153i	$-0.415722\pi$
$661$	13.4558	0.523372	0.261686	+	0.965153i	$0.415722\pi$
$662$	0	0
$663$	5.92893	0.230261
$664$	0	0
$665$	0	0
$666$	0	0
$667$	64.1838	2.48521
$668$	0	0
$669$	10.0294	0.387760
$670$	0	0
$671$	−0.485281	−0.0187341
$672$	0	0
$673$	3.51472	0.135482	0.0677412	−	0.997703i	$-0.478421\pi$
$673$	3.51472	0.135482	0.0677412	+	0.997703i	$0.478421\pi$
$674$	0	0
$675$	−2.41421	−0.0929231
$676$	0	0
$677$	44.2132	1.69925	0.849626	−	0.527386i	$-0.176828\pi$
$677$	44.2132	1.69925	0.849626	+	0.527386i	$0.176828\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−6.51472	−0.249645
$682$	0	0
$683$	−31.7990	−1.21675	−0.608377	−	0.793648i	$-0.708179\pi$
$683$	−31.7990	−1.21675	−0.608377	+	0.793648i	$0.708179\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	7.47309	0.285116
$688$	0	0
$689$	18.7279	0.713477
$690$	0	0
$691$	−14.8284	−0.564100	−0.282050	−	0.959400i	$-0.591014\pi$
$691$	−14.8284	−0.564100	−0.282050	+	0.959400i	$0.591014\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.75736	−0.294253
$696$	0	0
$697$	−20.2426	−0.766745
$698$	0	0
$699$	−3.79899	−0.143691
$700$	0	0
$701$	46.4558	1.75461	0.877307	−	0.479931i	$-0.159338\pi$
$701$	46.4558	1.75461	0.877307	+	0.479931i	$0.159338\pi$
$702$	0	0
$703$	−13.4558	−0.507497
$704$	0	0
$705$	3.00000	0.112987
$706$	0	0
$707$	0	0
$708$	0	0
$709$	17.0000	0.638448	0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000	0.638448	0.319224	+	0.947679i	$0.396578\pi$
$710$	0	0
$711$	4.20101	0.157550
$712$	0	0
$713$	75.9411	2.84402
$714$	0	0
$715$	0.757359	0.0283236
$716$	0	0
$717$	7.24264	0.270481
$718$	0	0
$719$	25.2132	0.940294	0.470147	−	0.882588i	$-0.344201\pi$
$719$	25.2132	0.940294	0.470147	+	0.882588i	$0.344201\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−0.301515	−0.0112135
$724$	0	0
$725$	−8.65685	−0.321507
$726$	0	0
$727$	6.68629	0.247981	0.123990	−	0.992283i	$-0.460431\pi$
$727$	6.68629	0.247981	0.123990	+	0.992283i	$0.460431\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	−6.48528	−0.239867
$732$	0	0
$733$	14.6985	0.542901	0.271450	−	0.962452i	$-0.412497\pi$
$733$	14.6985	0.542901	0.271450	+	0.962452i	$0.412497\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.41421	0.0520932
$738$	0	0
$739$	−29.9706	−1.10248	−0.551242	−	0.834345i	$-0.685846\pi$
$739$	−29.9706	−1.10248	−0.551242	+	0.834345i	$0.685846\pi$
$740$	0	0
$741$	−10.9706	−0.403014
$742$	0	0
$743$	−11.2721	−0.413532	−0.206766	−	0.978390i	$-0.566294\pi$
$743$	−11.2721	−0.413532	−0.206766	+	0.978390i	$0.566294\pi$
$744$	0	0
$745$	−9.17157	−0.336020
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−29.4853	−1.07593	−0.537967	−	0.842966i	$-0.680807\pi$
$751$	−29.4853	−1.07593	−0.537967	+	0.842966i	$0.680807\pi$
$752$	0	0
$753$	−7.12994	−0.259830
$754$	0	0
$755$	7.48528	0.272417
$756$	0	0
$757$	0.485281	0.0176379	0.00881893	−	0.999961i	$-0.497193\pi$
$757$	0.485281	0.0176379	0.00881893	+	0.999961i	$0.497193\pi$
$758$	0	0
$759$	−0.526912	−0.0191257
$760$	0	0
$761$	−12.7279	−0.461387	−0.230693	−	0.973026i	$-0.574099\pi$
$761$	−12.7279	−0.461387	−0.230693	+	0.973026i	$0.574099\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−9.17157	−0.331599
$766$	0	0
$767$	9.89949	0.357450
$768$	0	0
$769$	−8.82843	−0.318361	−0.159181	−	0.987249i	$-0.550885\pi$
$769$	−8.82843	−0.318361	−0.159181	+	0.987249i	$0.550885\pi$
$770$	0	0
$771$	−3.94113	−0.141936
$772$	0	0
$773$	−6.21320	−0.223473	−0.111737	−	0.993738i	$-0.535641\pi$
$773$	−6.21320	−0.223473	−0.111737	+	0.993738i	$0.535641\pi$
$774$	0	0
$775$	−10.2426	−0.367927
$776$	0	0
$777$	0	0
$778$	0	0
$779$	37.4558	1.34199
$780$	0	0
$781$	0.544156	0.0194714
$782$	0	0
$783$	20.8995	0.746887
$784$	0	0
$785$	−15.1716	−0.541497
$786$	0	0
$787$	−20.2721	−0.722622	−0.361311	−	0.932445i	$-0.617671\pi$
$787$	−20.2721	−0.722622	−0.361311	+	0.932445i	$0.617671\pi$
$788$	0	0
$789$	6.88730	0.245194
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−12.4853	−0.443365
$794$	0	0
$795$	1.75736	0.0623271
$796$	0	0
$797$	−21.1838	−0.750367	−0.375184	−	0.926950i	$-0.622420\pi$
$797$	−21.1838	−0.750367	−0.375184	+	0.926950i	$0.622420\pi$
$798$	0	0
$799$	23.4853	0.830850
$800$	0	0
$801$	22.6274	0.799500
$802$	0	0
$803$	−1.45584	−0.0513756
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.72792	−0.236834
$808$	0	0
$809$	−48.5980	−1.70861	−0.854307	−	0.519769i	$-0.826018\pi$
$809$	−48.5980	−1.70861	−0.854307	+	0.519769i	$0.826018\pi$
$810$	0	0
$811$	−47.3553	−1.66287	−0.831435	−	0.555621i	$-0.812480\pi$
$811$	−47.3553	−1.66287	−0.831435	+	0.555621i	$0.812480\pi$
$812$	0	0
$813$	0.284271	0.00996983
$814$	0	0
$815$	10.2426	0.358784
$816$	0	0
$817$	12.0000	0.419827
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23.4853	−0.819642	−0.409821	−	0.912166i	$-0.634409\pi$
$821$	−23.4853	−0.819642	−0.409821	+	0.912166i	$0.634409\pi$
$822$	0	0
$823$	−16.7279	−0.583099	−0.291549	−	0.956556i	$-0.594171\pi$
$823$	−16.7279	−0.583099	−0.291549	+	0.956556i	$0.594171\pi$
$824$	0	0
$825$	0.0710678	0.00247426
$826$	0	0
$827$	−42.0416	−1.46193	−0.730965	−	0.682415i	$-0.760930\pi$
$827$	−42.0416	−1.46193	−0.730965	+	0.682415i	$0.760930\pi$
$828$	0	0
$829$	5.95837	0.206943	0.103471	−	0.994632i	$-0.467005\pi$
$829$	5.95837	0.206943	0.103471	+	0.994632i	$0.467005\pi$
$830$	0	0
$831$	6.30152	0.218597
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−0.757359	−0.0262095
$836$	0	0
$837$	24.7279	0.854722
$838$	0	0
$839$	23.2721	0.803441	0.401721	−	0.915762i	$-0.368412\pi$
$839$	23.2721	0.803441	0.401721	+	0.915762i	$0.368412\pi$
$840$	0	0
$841$	45.9411	1.58418
$842$	0	0
$843$	0.958369	0.0330080
$844$	0	0
$845$	6.48528	0.223100
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−10.1716	−0.349087
$850$	0	0
$851$	−16.6274	−0.569981
$852$	0	0
$853$	10.9706	0.375625	0.187812	−	0.982205i	$-0.439860\pi$
$853$	10.9706	0.375625	0.187812	+	0.982205i	$0.439860\pi$
$854$	0	0
$855$	16.9706	0.580381
$856$	0	0
$857$	−22.4853	−0.768083	−0.384041	−	0.923316i	$-0.625468\pi$
$857$	−22.4853	−0.768083	−0.384041	+	0.923316i	$0.625468\pi$
$858$	0	0
$859$	24.7696	0.845126	0.422563	−	0.906334i	$-0.361130\pi$
$859$	24.7696	0.845126	0.422563	+	0.906334i	$0.361130\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.3848	−0.625825	−0.312913	−	0.949782i	$-0.601305\pi$
$863$	−18.3848	−0.625825	−0.312913	+	0.949782i	$0.601305\pi$
$864$	0	0
$865$	−7.24264	−0.246257
$866$	0	0
$867$	−2.68629	−0.0912312
$868$	0	0
$869$	−0.254834	−0.00864465
$870$	0	0
$871$	36.3848	1.23285
$872$	0	0
$873$	−37.4558	−1.26769
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.0294	0.439973	0.219986	−	0.975503i	$-0.429399\pi$
$877$	13.0294	0.439973	0.219986	+	0.975503i	$0.429399\pi$
$878$	0	0
$879$	10.6569	0.359447
$880$	0	0
$881$	−52.9706	−1.78462	−0.892312	−	0.451420i	$-0.850918\pi$
$881$	−52.9706	−1.78462	−0.892312	+	0.451420i	$0.850918\pi$
$882$	0	0
$883$	−31.5147	−1.06055	−0.530277	−	0.847824i	$-0.677912\pi$
$883$	−31.5147	−1.06055	−0.530277	+	0.847824i	$0.677912\pi$
$884$	0	0
$885$	0.928932	0.0312257
$886$	0	0
$887$	−2.97056	−0.0997417	−0.0498709	−	0.998756i	$-0.515881\pi$
$887$	−2.97056	−0.0997417	−0.0498709	+	0.998756i	$0.515881\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.28427	0.0430247
$892$	0	0
$893$	−43.4558	−1.45419
$894$	0	0
$895$	14.4853	0.484190
$896$	0	0
$897$	−13.5563	−0.452633
$898$	0	0
$899$	88.6690	2.95728
$900$	0	0
$901$	13.7574	0.458324
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18.7279	−0.622537
$906$	0	0
$907$	44.1838	1.46710	0.733549	−	0.679637i	$-0.237863\pi$
$907$	44.1838	1.46710	0.733549	+	0.679637i	$0.237863\pi$
$908$	0	0
$909$	−7.02944	−0.233152
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−1.17157	−0.0387310
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−12.4558	−0.410880	−0.205440	−	0.978670i	$-0.565863\pi$
$919$	−12.4558	−0.410880	−0.205440	+	0.978670i	$0.565863\pi$
$920$	0	0
$921$	−12.7990	−0.421741
$922$	0	0
$923$	14.0000	0.460816
$924$	0	0
$925$	2.24264	0.0737376
$926$	0	0
$927$	54.4264	1.78760
$928$	0	0
$929$	29.2721	0.960386	0.480193	−	0.877163i	$-0.340567\pi$
$929$	29.2721	0.960386	0.480193	+	0.877163i	$0.340567\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	4.14214	0.135607
$934$	0	0
$935$	0.556349	0.0181946
$936$	0	0
$937$	−48.5563	−1.58627	−0.793133	−	0.609048i	$-0.791552\pi$
$937$	−48.5563	−1.58627	−0.793133	+	0.609048i	$0.791552\pi$
$938$	0	0
$939$	−7.54416	−0.246194
$940$	0	0
$941$	28.9706	0.944413	0.472207	−	0.881488i	$-0.343458\pi$
$941$	28.9706	0.944413	0.472207	+	0.881488i	$0.343458\pi$
$942$	0	0
$943$	46.2843	1.50722
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.2426	1.69766	0.848829	−	0.528668i	$-0.177308\pi$
$947$	52.2426	1.69766	0.848829	+	0.528668i	$0.177308\pi$
$948$	0	0
$949$	−37.4558	−1.21587
$950$	0	0
$951$	−0.142136	−0.00460906
$952$	0	0
$953$	53.0122	1.71723	0.858617	−	0.512618i	$-0.171324\pi$
$953$	53.0122	1.71723	0.858617	+	0.512618i	$0.171324\pi$
$954$	0	0
$955$	−13.9706	−0.452077
$956$	0	0
$957$	−0.615224	−0.0198874
$958$	0	0
$959$	0	0
$960$	0	0
$961$	73.9117	2.38425
$962$	0	0
$963$	−7.02944	−0.226520
$964$	0	0
$965$	−16.0000	−0.515058
$966$	0	0
$967$	60.4264	1.94318	0.971591	−	0.236666i	$-0.0760546\pi$
$967$	60.4264	1.94318	0.971591	+	0.236666i	$0.0760546\pi$
$968$	0	0
$969$	−8.05887	−0.258888
$970$	0	0
$971$	17.2721	0.554287	0.277144	−	0.960828i	$-0.410612\pi$
$971$	17.2721	0.554287	0.277144	+	0.960828i	$0.410612\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.82843	0.0585565
$976$	0	0
$977$	42.7696	1.36832	0.684160	−	0.729332i	$-0.260169\pi$
$977$	42.7696	1.36832	0.684160	+	0.729332i	$0.260169\pi$
$978$	0	0
$979$	−1.37258	−0.0438679
$980$	0	0
$981$	−14.1421	−0.451524
$982$	0	0
$983$	−0.213203	−0.00680013	−0.00340007	−	0.999994i	$-0.501082\pi$
$983$	−0.213203	−0.00680013	−0.00340007	+	0.999994i	$0.501082\pi$
$984$	0	0
$985$	13.4142	0.427412
$986$	0	0
$987$	0	0
$988$	0	0
$989$	14.8284	0.471517
$990$	0	0
$991$	−19.9411	−0.633451	−0.316725	−	0.948517i	$-0.602583\pi$
$991$	−19.9411	−0.633451	−0.316725	+	0.948517i	$0.602583\pi$
$992$	0	0
$993$	11.3726	0.360898
$994$	0	0
$995$	7.41421	0.235046
$996$	0	0
$997$	−21.7279	−0.688130	−0.344065	−	0.938946i	$-0.611804\pi$
$997$	−21.7279	−0.688130	−0.344065	+	0.938946i	$0.611804\pi$
$998$	0	0
$999$	−5.41421	−0.171298

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.a.br.1.2		2
4.3	odd	2		245.2.a.f.1.1	yes	2
7.6	odd	2		3920.2.a.bw.1.1		2
12.11	even	2		2205.2.a.t.1.2		2
20.3	even	4		1225.2.b.j.99.4		4
20.7	even	4		1225.2.b.j.99.1		4
20.19	odd	2		1225.2.a.p.1.2		2
28.3	even	6		245.2.e.g.226.2		4
28.11	odd	6		245.2.e.f.226.2		4
28.19	even	6		245.2.e.g.116.2		4
28.23	odd	6		245.2.e.f.116.2		4
28.27	even	2		245.2.a.e.1.1	✓	2
84.83	odd	2		2205.2.a.v.1.2		2
140.27	odd	4		1225.2.b.i.99.2		4
140.83	odd	4		1225.2.b.i.99.3		4
140.139	even	2		1225.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.a.e.1.1	✓	2	28.27	even	2
245.2.a.f.1.1	yes	2	4.3	odd	2
245.2.e.f.116.2		4	28.23	odd	6
245.2.e.f.226.2		4	28.11	odd	6
245.2.e.g.116.2		4	28.19	even	6
245.2.e.g.226.2		4	28.3	even	6
1225.2.a.p.1.2		2	20.19	odd	2
1225.2.a.r.1.2		2	140.139	even	2
1225.2.b.i.99.2		4	140.27	odd	4
1225.2.b.i.99.3		4	140.83	odd	4
1225.2.b.j.99.1		4	20.7	even	4
1225.2.b.j.99.4		4	20.3	even	4
2205.2.a.t.1.2		2	12.11	even	2
2205.2.a.v.1.2		2	84.83	odd	2
3920.2.a.br.1.2		2	1.1	even	1	trivial
3920.2.a.bw.1.1		2	7.6	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:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} +O(q^{10})\)	\(q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} +0.171573 q^{11} +4.41421 q^{13} +0.414214 q^{15} +3.24264 q^{17} -6.00000 q^{19} -7.41421 q^{23} +1.00000 q^{25} -2.41421 q^{27} -8.65685 q^{29} -10.2426 q^{31} +0.0710678 q^{33} +2.24264 q^{37} +1.82843 q^{39} -6.24264 q^{41} -2.00000 q^{43} -2.82843 q^{45} +7.24264 q^{47} +1.34315 q^{51} +4.24264 q^{53} +0.171573 q^{55} -2.48528 q^{57} +2.24264 q^{59} -2.82843 q^{61} +4.41421 q^{65} +8.24264 q^{67} -3.07107 q^{69} +3.17157 q^{71} -8.48528 q^{73} +0.414214 q^{75} -1.48528 q^{79} +7.48528 q^{81} +3.24264 q^{85} -3.58579 q^{87} -8.00000 q^{89} -4.24264 q^{93} -6.00000 q^{95} +13.2426 q^{97} -0.485281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5	\( 2 q - 2 q^{3} + 2 q^{5} + 6 q^{11} + 6 q^{13} - 2 q^{15} - 2 q^{17} - 12 q^{19} - 12 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} - 12 q^{31} - 14 q^{33} - 4 q^{37} - 2 q^{39} - 4 q^{41} - 4 q^{43} + 6 q^{47} + 14 q^{51} + 6 q^{55} + 12 q^{57} - 4 q^{59} + 6 q^{65} + 8 q^{67} + 8 q^{69} + 12 q^{71} - 2 q^{75} + 14 q^{79} - 2 q^{81} - 2 q^{85} - 10 q^{87} - 16 q^{89} - 12 q^{95} + 18 q^{97} + 16 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 6 * q^11 + 6 * q^13 - 2 * q^15 - 2 * q^17 - 12 * q^19 - 12 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 - 12 * q^31 - 14 * q^33 - 4 * q^37 - 2 * q^39 - 4 * q^41 - 4 * q^43 + 6 * q^47 + 14 * q^51 + 6 * q^55 + 12 * q^57 - 4 * q^59 + 6 * q^65 + 8 * q^67 + 8 * q^69 + 12 * q^71 - 2 * q^75 + 14 * q^79 - 2 * q^81 - 2 * q^85 - 10 * q^87 - 16 * q^89 - 12 * q^95 + 18 * q^97 + 16 * q^99

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