LMFDB - Embedded newform 3136.2.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3136,2,Mod(1,3136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3136, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3136.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3136 = 2^{6} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3136.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:	$25.0410860739$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 196)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3136.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{3} -1.41421 q^{5} +5.00000 q^{9} -4.00000 q^{11} -4.24264 q^{13} -4.00000 q^{15} +1.41421 q^{17} -2.82843 q^{19} -4.00000 q^{23} -3.00000 q^{25} +5.65685 q^{27} -8.00000 q^{29} -11.3137 q^{33} +8.00000 q^{37} -12.0000 q^{39} -7.07107 q^{41} +4.00000 q^{43} -7.07107 q^{45} +5.65685 q^{47} +4.00000 q^{51} -10.0000 q^{53} +5.65685 q^{55} -8.00000 q^{57} -14.1421 q^{59} +7.07107 q^{61} +6.00000 q^{65} -11.3137 q^{69} -7.07107 q^{73} -8.48528 q^{75} +8.00000 q^{79} +1.00000 q^{81} +14.1421 q^{83} -2.00000 q^{85} -22.6274 q^{87} +7.07107 q^{89} +4.00000 q^{95} -1.41421 q^{97} -20.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{9} - 8 q^{11} - 8 q^{15} - 8 q^{23} - 6 q^{25} - 16 q^{29} + 16 q^{37} - 24 q^{39} + 8 q^{43} + 8 q^{51} - 20 q^{53} - 16 q^{57} + 12 q^{65} + 16 q^{79} + 2 q^{81} - 4 q^{85} + 8 q^{95} - 40 q^{99}+O(q^{100}) $ 2 * q + 10 * q^9 - 8 * q^11 - 8 * q^15 - 8 * q^23 - 6 * q^25 - 16 * q^29 + 16 * q^37 - 24 * q^39 + 8 * q^43 + 8 * q^51 - 20 * q^53 - 16 * q^57 + 12 * q^65 + 16 * q^79 + 2 * q^81 - 4 * q^85 + 8 * q^95 - 40 * q^99

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
$3$	2.82843	1.63299	0.816497	−	0.577350i	$-0.195913\pi$
$3$	2.82843	1.63299	0.816497	+	0.577350i	$0.195913\pi$
$4$	0	0
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.00000	1.66667
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−4.24264	−1.17670	−0.588348	−	0.808608i	$-0.700222\pi$
$13$	−4.24264	−1.17670	−0.588348	+	0.808608i	$0.700222\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	0	0
$17$	1.41421	0.342997	0.171499	−	0.985184i	$-0.445139\pi$
$17$	1.41421	0.342997	0.171499	+	0.985184i	$0.445139\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−11.3137	−1.96946
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	−12.0000	−1.92154
$40$	0	0
$41$	−7.07107	−1.10432	−0.552158	−	0.833740i	$-0.686195\pi$
$41$	−7.07107	−1.10432	−0.552158	+	0.833740i	$0.686195\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	−7.07107	−1.05409
$46$	0	0
$47$	5.65685	0.825137	0.412568	−	0.910927i	$-0.364632\pi$
$47$	5.65685	0.825137	0.412568	+	0.910927i	$0.364632\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	5.65685	0.762770
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	−14.1421	−1.84115	−0.920575	−	0.390567i	$-0.872279\pi$
$59$	−14.1421	−1.84115	−0.920575	+	0.390567i	$0.872279\pi$
$60$	0	0
$61$	7.07107	0.905357	0.452679	−	0.891674i	$-0.350468\pi$
$61$	7.07107	0.905357	0.452679	+	0.891674i	$0.350468\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.00000	0.744208
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−11.3137	−1.36201
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−7.07107	−0.827606	−0.413803	−	0.910366i	$-0.635800\pi$
$73$	−7.07107	−0.827606	−0.413803	+	0.910366i	$0.635800\pi$
$74$	0	0
$75$	−8.48528	−0.979796
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.1421	1.55230	0.776151	−	0.630548i	$-0.217170\pi$
$83$	14.1421	1.55230	0.776151	+	0.630548i	$0.217170\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−22.6274	−2.42591
$88$	0	0
$89$	7.07107	0.749532	0.374766	−	0.927119i	$-0.377723\pi$
$89$	7.07107	0.749532	0.374766	+	0.927119i	$0.377723\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−1.41421	−0.143592	−0.0717958	−	0.997419i	$-0.522873\pi$
$97$	−1.41421	−0.143592	−0.0717958	+	0.997419i	$0.522873\pi$
$98$	0	0
$99$	−20.0000	−2.01008
$100$	0	0
$101$	12.7279	1.26648	0.633238	−	0.773957i	$-0.281726\pi$
$101$	12.7279	1.26648	0.633238	+	0.773957i	$0.281726\pi$
$102$	0	0
$103$	−11.3137	−1.11477	−0.557386	−	0.830253i	$-0.688196\pi$
$103$	−11.3137	−1.11477	−0.557386	+	0.830253i	$0.688196\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	0	0
$111$	22.6274	2.14770
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	5.65685	0.527504
$116$	0	0
$117$	−21.2132	−1.96116
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−20.0000	−1.80334
$124$	0	0
$125$	11.3137	1.01193
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	11.3137	0.996116
$130$	0	0
$131$	8.48528	0.741362	0.370681	−	0.928760i	$-0.379124\pi$
$131$	8.48528	0.741362	0.370681	+	0.928760i	$0.379124\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−8.00000	−0.688530
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−2.82843	−0.239904	−0.119952	−	0.992780i	$-0.538274\pi$
$139$	−2.82843	−0.239904	−0.119952	+	0.992780i	$0.538274\pi$
$140$	0	0
$141$	16.0000	1.34744
$142$	0	0
$143$	16.9706	1.41915
$144$	0	0
$145$	11.3137	0.939552
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	7.07107	0.571662
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−7.07107	−0.564333	−0.282166	−	0.959366i	$-0.591053\pi$
$157$	−7.07107	−0.564333	−0.282166	+	0.959366i	$0.591053\pi$
$158$	0	0
$159$	−28.2843	−2.24309
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	16.0000	1.24560
$166$	0	0
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	−14.1421	−1.08148
$172$	0	0
$173$	4.24264	0.322562	0.161281	−	0.986909i	$-0.448437\pi$
$173$	4.24264	0.322562	0.161281	+	0.986909i	$0.448437\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−40.0000	−3.00658
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	21.2132	1.57676	0.788382	−	0.615185i	$-0.210919\pi$
$181$	21.2132	1.57676	0.788382	+	0.615185i	$0.210919\pi$
$182$	0	0
$183$	20.0000	1.47844
$184$	0	0
$185$	−11.3137	−0.831800
$186$	0	0
$187$	−5.65685	−0.413670
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	16.9706	1.21529
$196$	0	0
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	10.0000	0.698430
$206$	0	0
$207$	−20.0000	−1.39010
$208$	0	0
$209$	11.3137	0.782586
$210$	0	0
$211$	−24.0000	−1.65223	−0.826114	−	0.563503i	$-0.809453\pi$
$211$	−24.0000	−1.65223	−0.826114	+	0.563503i	$0.809453\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.65685	−0.385794
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−20.0000	−1.35147
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	16.9706	1.13643	0.568216	−	0.822879i	$-0.307634\pi$
$223$	16.9706	1.13643	0.568216	+	0.822879i	$0.307634\pi$
$224$	0	0
$225$	−15.0000	−1.00000
$226$	0	0
$227$	−8.48528	−0.563188	−0.281594	−	0.959534i	$-0.590863\pi$
$227$	−8.48528	−0.563188	−0.281594	+	0.959534i	$0.590863\pi$
$228$	0	0
$229$	21.2132	1.40181	0.700904	−	0.713256i	$-0.252780\pi$
$229$	21.2132	1.40181	0.700904	+	0.713256i	$0.252780\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	22.6274	1.46981
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	12.7279	0.819878	0.409939	−	0.912113i	$-0.365550\pi$
$241$	12.7279	0.819878	0.409939	+	0.912113i	$0.365550\pi$
$242$	0	0
$243$	−14.1421	−0.907218
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	40.0000	2.53490
$250$	0	0
$251$	−19.7990	−1.24970	−0.624851	−	0.780744i	$-0.714840\pi$
$251$	−19.7990	−1.24970	−0.624851	+	0.780744i	$0.714840\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	0	0
$255$	−5.65685	−0.354246
$256$	0	0
$257$	21.2132	1.32324	0.661622	−	0.749838i	$-0.269869\pi$
$257$	21.2132	1.32324	0.661622	+	0.749838i	$0.269869\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−40.0000	−2.47594
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	14.1421	0.868744
$266$	0	0
$267$	20.0000	1.22398
$268$	0	0
$269$	18.3848	1.12094	0.560470	−	0.828175i	$-0.310621\pi$
$269$	18.3848	1.12094	0.560470	+	0.828175i	$0.310621\pi$
$270$	0	0
$271$	28.2843	1.71815	0.859074	−	0.511852i	$-0.171040\pi$
$271$	28.2843	1.71815	0.859074	+	0.511852i	$0.171040\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	2.82843	0.168133	0.0840663	−	0.996460i	$-0.473209\pi$
$283$	2.82843	0.168133	0.0840663	+	0.996460i	$0.473209\pi$
$284$	0	0
$285$	11.3137	0.670166
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.0000	−0.882353
$290$	0	0
$291$	−4.00000	−0.234484
$292$	0	0
$293$	−32.5269	−1.90024	−0.950121	−	0.311881i	$-0.899041\pi$
$293$	−32.5269	−1.90024	−0.950121	+	0.311881i	$0.899041\pi$
$294$	0	0
$295$	20.0000	1.16445
$296$	0	0
$297$	−22.6274	−1.31298
$298$	0	0
$299$	16.9706	0.981433
$300$	0	0
$301$	0	0
$302$	0	0
$303$	36.0000	2.06815
$304$	0	0
$305$	−10.0000	−0.572598
$306$	0	0
$307$	19.7990	1.12999	0.564994	−	0.825095i	$-0.308878\pi$
$307$	19.7990	1.12999	0.564994	+	0.825095i	$0.308878\pi$
$308$	0	0
$309$	−32.0000	−1.82042
$310$	0	0
$311$	−22.6274	−1.28308	−0.641542	−	0.767088i	$-0.721705\pi$
$311$	−22.6274	−1.28308	−0.641542	+	0.767088i	$0.721705\pi$
$312$	0	0
$313$	4.24264	0.239808	0.119904	−	0.992785i	$-0.461741\pi$
$313$	4.24264	0.239808	0.119904	+	0.992785i	$0.461741\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	32.0000	1.79166
$320$	0	0
$321$	−22.6274	−1.26294
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	12.7279	0.706018
$326$	0	0
$327$	22.6274	1.25130
$328$	0	0
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	40.0000	2.19199
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	16.9706	0.921714
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	16.0000	0.861411
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	4.24264	0.227103	0.113552	−	0.993532i	$-0.463777\pi$
$349$	4.24264	0.227103	0.113552	+	0.993532i	$0.463777\pi$
$350$	0	0
$351$	−24.0000	−1.28103
$352$	0	0
$353$	9.89949	0.526897	0.263448	−	0.964673i	$-0.415140\pi$
$353$	9.89949	0.526897	0.263448	+	0.964673i	$0.415140\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	14.1421	0.742270
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	5.65685	0.295285	0.147643	−	0.989041i	$-0.452831\pi$
$367$	5.65685	0.295285	0.147643	+	0.989041i	$0.452831\pi$
$368$	0	0
$369$	−35.3553	−1.84053
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	32.0000	1.65247
$376$	0	0
$377$	33.9411	1.74806
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	−56.5685	−2.89809
$382$	0	0
$383$	−5.65685	−0.289052	−0.144526	−	0.989501i	$-0.546166\pi$
$383$	−5.65685	−0.289052	−0.144526	+	0.989501i	$0.546166\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	20.0000	1.01666
$388$	0	0
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	−5.65685	−0.286079
$392$	0	0
$393$	24.0000	1.21064
$394$	0	0
$395$	−11.3137	−0.569254
$396$	0	0
$397$	−15.5563	−0.780751	−0.390375	−	0.920656i	$-0.627655\pi$
$397$	−15.5563	−0.780751	−0.390375	+	0.920656i	$0.627655\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.41421	−0.0702728
$406$	0	0
$407$	−32.0000	−1.58618
$408$	0	0
$409$	38.1838	1.88807	0.944033	−	0.329851i	$-0.106999\pi$
$409$	38.1838	1.88807	0.944033	+	0.329851i	$0.106999\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−20.0000	−0.981761
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	−14.1421	−0.690889	−0.345444	−	0.938439i	$-0.612272\pi$
$419$	−14.1421	−0.690889	−0.345444	+	0.938439i	$0.612272\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	28.2843	1.37523
$424$	0	0
$425$	−4.24264	−0.205798
$426$	0	0
$427$	0	0
$428$	0	0
$429$	48.0000	2.31746
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	−21.2132	−1.01944	−0.509721	−	0.860340i	$-0.670251\pi$
$433$	−21.2132	−1.01944	−0.509721	+	0.860340i	$0.670251\pi$
$434$	0	0
$435$	32.0000	1.53428
$436$	0	0
$437$	11.3137	0.541208
$438$	0	0
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	0	0
$447$	−28.2843	−1.33780
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	28.2843	1.33185
$452$	0	0
$453$	−11.3137	−0.531564
$454$	0	0
$455$	0	0
$456$	0	0
$457$	30.0000	1.40334	0.701670	−	0.712502i	$-0.252438\pi$
$457$	30.0000	1.40334	0.701670	+	0.712502i	$0.252438\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	−7.07107	−0.329332	−0.164666	−	0.986349i	$-0.552655\pi$
$461$	−7.07107	−0.329332	−0.164666	+	0.986349i	$0.552655\pi$
$462$	0	0
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.7990	0.916188	0.458094	−	0.888904i	$-0.348532\pi$
$467$	19.7990	0.916188	0.458094	+	0.888904i	$0.348532\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−20.0000	−0.921551
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	8.48528	0.389331
$476$	0	0
$477$	−50.0000	−2.28934
$478$	0	0
$479$	−11.3137	−0.516937	−0.258468	−	0.966020i	$-0.583218\pi$
$479$	−11.3137	−0.516937	−0.258468	+	0.966020i	$0.583218\pi$
$480$	0	0
$481$	−33.9411	−1.54758
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.00000	0.0908153
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	0	0
$489$	−11.3137	−0.511624
$490$	0	0
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	0	0
$493$	−11.3137	−0.509544
$494$	0	0
$495$	28.2843	1.27128
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	−39.5980	−1.76559	−0.882793	−	0.469762i	$-0.844340\pi$
$503$	−39.5980	−1.76559	−0.882793	+	0.469762i	$0.844340\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	0	0
$507$	14.1421	0.628074
$508$	0	0
$509$	18.3848	0.814891	0.407445	−	0.913230i	$-0.366420\pi$
$509$	18.3848	0.814891	0.407445	+	0.913230i	$0.366420\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−16.0000	−0.706417
$514$	0	0
$515$	16.0000	0.705044
$516$	0	0
$517$	−22.6274	−0.995153
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	−41.0122	−1.79678	−0.898388	−	0.439202i	$-0.855261\pi$
$521$	−41.0122	−1.79678	−0.898388	+	0.439202i	$0.855261\pi$
$522$	0	0
$523$	−42.4264	−1.85518	−0.927589	−	0.373603i	$-0.878122\pi$
$523$	−42.4264	−1.85518	−0.927589	+	0.373603i	$0.878122\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−70.7107	−3.06858
$532$	0	0
$533$	30.0000	1.29944
$534$	0	0
$535$	11.3137	0.489134
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	0	0
$543$	60.0000	2.57485
$544$	0	0
$545$	−11.3137	−0.484626
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	0	0
$549$	35.3553	1.50893
$550$	0	0
$551$	22.6274	0.963960
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−32.0000	−1.35832
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	−16.9706	−0.717778
$560$	0	0
$561$	−16.0000	−0.675521
$562$	0	0
$563$	−14.1421	−0.596020	−0.298010	−	0.954563i	$-0.596323\pi$
$563$	−14.1421	−0.596020	−0.298010	+	0.954563i	$0.596323\pi$
$564$	0	0
$565$	−8.48528	−0.356978
$566$	0	0
$567$	0	0
$568$	0	0
$569$	40.0000	1.67689	0.838444	−	0.544988i	$-0.183466\pi$
$569$	40.0000	1.67689	0.838444	+	0.544988i	$0.183466\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	−45.2548	−1.89055
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	12.7279	0.529870	0.264935	−	0.964266i	$-0.414649\pi$
$577$	12.7279	0.529870	0.264935	+	0.964266i	$0.414649\pi$
$578$	0	0
$579$	−28.2843	−1.17545
$580$	0	0
$581$	0	0
$582$	0	0
$583$	40.0000	1.65663
$584$	0	0
$585$	30.0000	1.24035
$586$	0	0
$587$	−25.4558	−1.05068	−0.525338	−	0.850894i	$-0.676061\pi$
$587$	−25.4558	−1.05068	−0.525338	+	0.850894i	$0.676061\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	28.2843	1.16346
$592$	0	0
$593$	−9.89949	−0.406524	−0.203262	−	0.979124i	$-0.565154\pi$
$593$	−9.89949	−0.406524	−0.203262	+	0.979124i	$0.565154\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	−29.6985	−1.21143	−0.605713	−	0.795683i	$-0.707112\pi$
$601$	−29.6985	−1.21143	−0.605713	+	0.795683i	$0.707112\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−7.07107	−0.287480
$606$	0	0
$607$	−33.9411	−1.37763	−0.688814	−	0.724938i	$-0.741868\pi$
$607$	−33.9411	−1.37763	−0.688814	+	0.724938i	$0.741868\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−24.0000	−0.970936
$612$	0	0
$613$	−24.0000	−0.969351	−0.484675	−	0.874694i	$-0.661062\pi$
$613$	−24.0000	−0.969351	−0.484675	+	0.874694i	$0.661062\pi$
$614$	0	0
$615$	28.2843	1.14053
$616$	0	0
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	0	0
$619$	31.1127	1.25052	0.625262	−	0.780415i	$-0.284992\pi$
$619$	31.1127	1.25052	0.625262	+	0.780415i	$0.284992\pi$
$620$	0	0
$621$	−22.6274	−0.908007
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	32.0000	1.27796
$628$	0	0
$629$	11.3137	0.451107
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	−67.8823	−2.69808
$634$	0	0
$635$	28.2843	1.12243
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.00000	0.315981	0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000	0.315981	0.157991	+	0.987441i	$0.449498\pi$
$642$	0	0
$643$	−2.82843	−0.111542	−0.0557711	−	0.998444i	$-0.517762\pi$
$643$	−2.82843	−0.111542	−0.0557711	+	0.998444i	$0.517762\pi$
$644$	0	0
$645$	−16.0000	−0.629999
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	56.5685	2.22051
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	−35.3553	−1.37934
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	21.2132	0.825098	0.412549	−	0.910935i	$-0.364639\pi$
$661$	21.2132	0.825098	0.412549	+	0.910935i	$0.364639\pi$
$662$	0	0
$663$	−16.9706	−0.659082
$664$	0	0
$665$	0	0
$666$	0	0
$667$	32.0000	1.23904
$668$	0	0
$669$	48.0000	1.85579
$670$	0	0
$671$	−28.2843	−1.09190
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	−16.9706	−0.653197
$676$	0	0
$677$	−12.7279	−0.489174	−0.244587	−	0.969627i	$-0.578652\pi$
$677$	−12.7279	−0.489174	−0.244587	+	0.969627i	$0.578652\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	60.0000	2.28914
$688$	0	0
$689$	42.4264	1.61632
$690$	0	0
$691$	42.4264	1.61398	0.806988	−	0.590567i	$-0.201096\pi$
$691$	42.4264	1.61398	0.806988	+	0.590567i	$0.201096\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	−10.0000	−0.378777
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	0	0
$703$	−22.6274	−0.853409
$704$	0	0
$705$	−22.6274	−0.852198
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8.00000	0.300446	0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000	0.300446	0.150223	+	0.988652i	$0.452001\pi$
$710$	0	0
$711$	40.0000	1.50012
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−24.0000	−0.897549
$716$	0	0
$717$	−33.9411	−1.26755
$718$	0	0
$719$	39.5980	1.47676	0.738378	−	0.674387i	$-0.235592\pi$
$719$	39.5980	1.47676	0.738378	+	0.674387i	$0.235592\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	36.0000	1.33885
$724$	0	0
$725$	24.0000	0.891338
$726$	0	0
$727$	28.2843	1.04901	0.524503	−	0.851409i	$-0.324251\pi$
$727$	28.2843	1.04901	0.524503	+	0.851409i	$0.324251\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	5.65685	0.209226
$732$	0	0
$733$	38.1838	1.41035	0.705175	−	0.709034i	$-0.250869\pi$
$733$	38.1838	1.41035	0.705175	+	0.709034i	$0.250869\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	33.9411	1.24686
$742$	0	0
$743$	−20.0000	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$744$	0	0
$745$	14.1421	0.518128
$746$	0	0
$747$	70.7107	2.58717
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	−56.0000	−2.04075
$754$	0	0
$755$	5.65685	0.205874
$756$	0	0
$757$	40.0000	1.45382	0.726912	−	0.686730i	$-0.240955\pi$
$757$	40.0000	1.45382	0.726912	+	0.686730i	$0.240955\pi$
$758$	0	0
$759$	45.2548	1.64265
$760$	0	0
$761$	41.0122	1.48669	0.743345	−	0.668908i	$-0.233238\pi$
$761$	41.0122	1.48669	0.743345	+	0.668908i	$0.233238\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−10.0000	−0.361551
$766$	0	0
$767$	60.0000	2.16647
$768$	0	0
$769$	46.6690	1.68293	0.841464	−	0.540312i	$-0.181694\pi$
$769$	46.6690	1.68293	0.841464	+	0.540312i	$0.181694\pi$
$770$	0	0
$771$	60.0000	2.16085
$772$	0	0
$773$	−24.0416	−0.864717	−0.432359	−	0.901702i	$-0.642319\pi$
$773$	−24.0416	−0.864717	−0.432359	+	0.901702i	$0.642319\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	20.0000	0.716574
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−45.2548	−1.61728
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	−48.0833	−1.71398	−0.856992	−	0.515330i	$-0.827669\pi$
$787$	−48.0833	−1.71398	−0.856992	+	0.515330i	$0.827669\pi$
$788$	0	0
$789$	−67.8823	−2.41667
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−30.0000	−1.06533
$794$	0	0
$795$	40.0000	1.41865
$796$	0	0
$797$	29.6985	1.05197	0.525987	−	0.850493i	$-0.323696\pi$
$797$	29.6985	1.05197	0.525987	+	0.850493i	$0.323696\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	35.3553	1.24922
$802$	0	0
$803$	28.2843	0.998130
$804$	0	0
$805$	0	0
$806$	0	0
$807$	52.0000	1.83049
$808$	0	0
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	−8.48528	−0.297959	−0.148979	−	0.988840i	$-0.547599\pi$
$811$	−8.48528	−0.297959	−0.148979	+	0.988840i	$0.547599\pi$
$812$	0	0
$813$	80.0000	2.80572
$814$	0	0
$815$	5.65685	0.198151
$816$	0	0
$817$	−11.3137	−0.395817
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	0	0
$825$	33.9411	1.18168
$826$	0	0
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	0	0
$829$	7.07107	0.245588	0.122794	−	0.992432i	$-0.460815\pi$
$829$	7.07107	0.245588	0.122794	+	0.992432i	$0.460815\pi$
$830$	0	0
$831$	−62.2254	−2.15858
$832$	0	0
$833$	0	0
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−11.3137	−0.390593	−0.195296	−	0.980744i	$-0.562567\pi$
$839$	−11.3137	−0.390593	−0.195296	+	0.980744i	$0.562567\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	45.2548	1.55866
$844$	0	0
$845$	−7.07107	−0.243252
$846$	0	0
$847$	0	0
$848$	0	0
$849$	8.00000	0.274559
$850$	0	0
$851$	−32.0000	−1.09695
$852$	0	0
$853$	−12.7279	−0.435796	−0.217898	−	0.975972i	$-0.569920\pi$
$853$	−12.7279	−0.435796	−0.217898	+	0.975972i	$0.569920\pi$
$854$	0	0
$855$	20.0000	0.683986
$856$	0	0
$857$	7.07107	0.241543	0.120772	−	0.992680i	$-0.461463\pi$
$857$	7.07107	0.241543	0.120772	+	0.992680i	$0.461463\pi$
$858$	0	0
$859$	−14.1421	−0.482523	−0.241262	−	0.970460i	$-0.577561\pi$
$859$	−14.1421	−0.482523	−0.241262	+	0.970460i	$0.577561\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−40.0000	−1.36162	−0.680808	−	0.732462i	$-0.738371\pi$
$863$	−40.0000	−1.36162	−0.680808	+	0.732462i	$0.738371\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	−42.4264	−1.44088
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−7.07107	−0.239319
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−8.00000	−0.270141	−0.135070	−	0.990836i	$-0.543126\pi$
$877$	−8.00000	−0.270141	−0.135070	+	0.990836i	$0.543126\pi$
$878$	0	0
$879$	−92.0000	−3.10308
$880$	0	0
$881$	−21.2132	−0.714691	−0.357345	−	0.933972i	$-0.616318\pi$
$881$	−21.2132	−0.714691	−0.357345	+	0.933972i	$0.616318\pi$
$882$	0	0
$883$	40.0000	1.34611	0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000	1.34611	0.673054	+	0.739594i	$0.264982\pi$
$884$	0	0
$885$	56.5685	1.90153
$886$	0	0
$887$	28.2843	0.949693	0.474846	−	0.880069i	$-0.342504\pi$
$887$	28.2843	0.949693	0.474846	+	0.880069i	$0.342504\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	−16.0000	−0.535420
$894$	0	0
$895$	0	0
$896$	0	0
$897$	48.0000	1.60267
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−14.1421	−0.471143
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−30.0000	−0.997234
$906$	0	0
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	0	0
$909$	63.6396	2.11079
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	0	0
$913$	−56.5685	−1.87215
$914$	0	0
$915$	−28.2843	−0.935049
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	56.0000	1.84526
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−24.0000	−0.789115
$926$	0	0
$927$	−56.5685	−1.85795
$928$	0	0
$929$	35.3553	1.15997	0.579986	−	0.814627i	$-0.303058\pi$
$929$	35.3553	1.15997	0.579986	+	0.814627i	$0.303058\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−64.0000	−2.09527
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	15.5563	0.508204	0.254102	−	0.967177i	$-0.418220\pi$
$937$	15.5563	0.508204	0.254102	+	0.967177i	$0.418220\pi$
$938$	0	0
$939$	12.0000	0.391605
$940$	0	0
$941$	7.07107	0.230510	0.115255	−	0.993336i	$-0.463231\pi$
$941$	7.07107	0.230510	0.115255	+	0.993336i	$0.463231\pi$
$942$	0	0
$943$	28.2843	0.921063
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−60.0000	−1.94974	−0.974869	−	0.222779i	$-0.928487\pi$
$947$	−60.0000	−1.94974	−0.974869	+	0.222779i	$0.928487\pi$
$948$	0	0
$949$	30.0000	0.973841
$950$	0	0
$951$	5.65685	0.183436
$952$	0	0
$953$	10.0000	0.323932	0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000	0.323932	0.161966	+	0.986796i	$0.448217\pi$
$954$	0	0
$955$	22.6274	0.732206
$956$	0	0
$957$	90.5097	2.92576
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−40.0000	−1.28898
$964$	0	0
$965$	14.1421	0.455251
$966$	0	0
$967$	12.0000	0.385894	0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000	0.385894	0.192947	+	0.981209i	$0.438195\pi$
$968$	0	0
$969$	−11.3137	−0.363449
$970$	0	0
$971$	−14.1421	−0.453843	−0.226921	−	0.973913i	$-0.572866\pi$
$971$	−14.1421	−0.453843	−0.226921	+	0.973913i	$0.572866\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	36.0000	1.15292
$976$	0	0
$977$	48.0000	1.53566	0.767828	−	0.640656i	$-0.221338\pi$
$977$	48.0000	1.53566	0.767828	+	0.640656i	$0.221338\pi$
$978$	0	0
$979$	−28.2843	−0.903969
$980$	0	0
$981$	40.0000	1.27710
$982$	0	0
$983$	45.2548	1.44341	0.721703	−	0.692203i	$-0.243360\pi$
$983$	45.2548	1.44341	0.721703	+	0.692203i	$0.243360\pi$
$984$	0	0
$985$	−14.1421	−0.450606
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	56.5685	1.79515
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.41421	−0.0447886	−0.0223943	−	0.999749i	$-0.507129\pi$
$997$	−1.41421	−0.0447886	−0.0223943	+	0.999749i	$0.507129\pi$
$998$	0	0
$999$	45.2548	1.43180

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.2.a.br.1.2		2
4.3	odd	2		3136.2.a.bs.1.1		2
7.6	odd	2	inner	3136.2.a.br.1.1		2
8.3	odd	2		784.2.a.m.1.2		2
8.5	even	2		196.2.a.c.1.1	✓	2
24.5	odd	2		1764.2.a.l.1.1		2
24.11	even	2		7056.2.a.cr.1.1		2
28.27	even	2		3136.2.a.bs.1.2		2
40.13	odd	4		4900.2.e.p.2549.1		4
40.29	even	2		4900.2.a.y.1.2		2
40.37	odd	4		4900.2.e.p.2549.3		4
56.3	even	6		784.2.i.l.177.2		4
56.5	odd	6		196.2.e.b.165.1		4
56.11	odd	6		784.2.i.l.177.1		4
56.13	odd	2		196.2.a.c.1.2	yes	2
56.19	even	6		784.2.i.l.753.2		4
56.27	even	2		784.2.a.m.1.1		2
56.37	even	6		196.2.e.b.165.2		4
56.45	odd	6		196.2.e.b.177.1		4
56.51	odd	6		784.2.i.l.753.1		4
56.53	even	6		196.2.e.b.177.2		4
168.5	even	6		1764.2.k.l.361.1		4
168.53	odd	6		1764.2.k.l.1549.2		4
168.83	odd	2		7056.2.a.cr.1.2		2
168.101	even	6		1764.2.k.l.1549.1		4
168.125	even	2		1764.2.a.l.1.2		2
168.149	odd	6		1764.2.k.l.361.2		4
280.13	even	4		4900.2.e.p.2549.4		4
280.69	odd	2		4900.2.a.y.1.1		2
280.237	even	4		4900.2.e.p.2549.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.a.c.1.1	✓	2	8.5	even	2
196.2.a.c.1.2	yes	2	56.13	odd	2
196.2.e.b.165.1		4	56.5	odd	6
196.2.e.b.165.2		4	56.37	even	6
196.2.e.b.177.1		4	56.45	odd	6
196.2.e.b.177.2		4	56.53	even	6
784.2.a.m.1.1		2	56.27	even	2
784.2.a.m.1.2		2	8.3	odd	2
784.2.i.l.177.1		4	56.11	odd	6
784.2.i.l.177.2		4	56.3	even	6
784.2.i.l.753.1		4	56.51	odd	6
784.2.i.l.753.2		4	56.19	even	6
1764.2.a.l.1.1		2	24.5	odd	2
1764.2.a.l.1.2		2	168.125	even	2
1764.2.k.l.361.1		4	168.5	even	6
1764.2.k.l.361.2		4	168.149	odd	6
1764.2.k.l.1549.1		4	168.101	even	6
1764.2.k.l.1549.2		4	168.53	odd	6
3136.2.a.br.1.1		2	7.6	odd	2	inner
3136.2.a.br.1.2		2	1.1	even	1	trivial
3136.2.a.bs.1.1		2	4.3	odd	2
3136.2.a.bs.1.2		2	28.27	even	2
4900.2.a.y.1.1		2	280.69	odd	2
4900.2.a.y.1.2		2	40.29	even	2
4900.2.e.p.2549.1		4	40.13	odd	4
4900.2.e.p.2549.2		4	280.237	even	4
4900.2.e.p.2549.3		4	40.37	odd	4
4900.2.e.p.2549.4		4	280.13	even	4
7056.2.a.cr.1.1		2	24.11	even	2
7056.2.a.cr.1.2		2	168.83	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:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3136.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82843 q^{3} -1.41421 q^{5} +5.00000 q^{9} -4.00000 q^{11} -4.24264 q^{13} -4.00000 q^{15} +1.41421 q^{17} -2.82843 q^{19} -4.00000 q^{23} -3.00000 q^{25} +5.65685 q^{27} -8.00000 q^{29} -11.3137 q^{33} +8.00000 q^{37} -12.0000 q^{39} -7.07107 q^{41} +4.00000 q^{43} -7.07107 q^{45} +5.65685 q^{47} +4.00000 q^{51} -10.0000 q^{53} +5.65685 q^{55} -8.00000 q^{57} -14.1421 q^{59} +7.07107 q^{61} +6.00000 q^{65} -11.3137 q^{69} -7.07107 q^{73} -8.48528 q^{75} +8.00000 q^{79} +1.00000 q^{81} +14.1421 q^{83} -2.00000 q^{85} -22.6274 q^{87} +7.07107 q^{89} +4.00000 q^{95} -1.41421 q^{97} -20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{9} - 8 q^{11} - 8 q^{15} - 8 q^{23} - 6 q^{25} - 16 q^{29} + 16 q^{37} - 24 q^{39} + 8 q^{43} + 8 q^{51} - 20 q^{53} - 16 q^{57} + 12 q^{65} + 16 q^{79} + 2 q^{81} - 4 q^{85} + 8 q^{95} - 40 q^{99}+O(q^{100}) \) 2 * q + 10 * q^9 - 8 * q^11 - 8 * q^15 - 8 * q^23 - 6 * q^25 - 16 * q^29 + 16 * q^37 - 24 * q^39 + 8 * q^43 + 8 * q^51 - 20 * q^53 - 16 * q^57 + 12 * q^65 + 16 * q^79 + 2 * q^81 - 4 * q^85 + 8 * q^95 - 40 * q^99

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