LMFDB - Embedded newform 3136.2.a.bm.1.1

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:	$0$
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 98)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-1.41421 q^{3} -2.82843 q^{5} -1.00000 q^{9} -2.00000 q^{11} +4.00000 q^{15} -1.41421 q^{17} -7.07107 q^{19} +4.00000 q^{23} +3.00000 q^{25} +5.65685 q^{27} -2.00000 q^{29} -8.48528 q^{31} +2.82843 q^{33} -10.0000 q^{37} -9.89949 q^{41} +2.00000 q^{43} +2.82843 q^{45} -2.82843 q^{47} +2.00000 q^{51} +2.00000 q^{53} +5.65685 q^{55} +10.0000 q^{57} -1.41421 q^{59} -2.82843 q^{61} +12.0000 q^{67} -5.65685 q^{69} +12.0000 q^{71} -1.41421 q^{73} -4.24264 q^{75} +4.00000 q^{79} -5.00000 q^{81} +9.89949 q^{83} +4.00000 q^{85} +2.82843 q^{87} -7.07107 q^{89} +12.0000 q^{93} +20.0000 q^{95} +9.89949 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} - 4 q^{11} + 8 q^{15} + 8 q^{23} + 6 q^{25} - 4 q^{29} - 20 q^{37} + 4 q^{43} + 4 q^{51} + 4 q^{53} + 20 q^{57} + 24 q^{67} + 24 q^{71} + 8 q^{79} - 10 q^{81} + 8 q^{85} + 24 q^{93} + 40 q^{95}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 4 * q^11 + 8 * q^15 + 8 * q^23 + 6 * q^25 - 4 * q^29 - 20 * q^37 + 4 * q^43 + 4 * q^51 + 4 * q^53 + 20 * q^57 + 24 * q^67 + 24 * q^71 + 8 * q^79 - 10 * q^81 + 8 * q^85 + 24 * q^93 + 40 * q^95 + 4 * 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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	0	0
$17$	−1.41421	−0.342997	−0.171499	−	0.985184i	$-0.554861\pi$
$17$	−1.41421	−0.342997	−0.171499	+	0.985184i	$0.554861\pi$
$18$	0	0
$19$	−7.07107	−1.62221	−0.811107	−	0.584898i	$-0.801135\pi$
$19$	−7.07107	−1.62221	−0.811107	+	0.584898i	$0.801135\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	0	0
$33$	2.82843	0.492366
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.89949	−1.54604	−0.773021	−	0.634381i	$-0.781255\pi$
$41$	−9.89949	−1.54604	−0.773021	+	0.634381i	$0.781255\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	2.82843	0.421637
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	5.65685	0.762770
$56$	0	0
$57$	10.0000	1.32453
$58$	0	0
$59$	−1.41421	−0.184115	−0.0920575	−	0.995754i	$-0.529344\pi$
$59$	−1.41421	−0.184115	−0.0920575	+	0.995754i	$0.529344\pi$
$60$	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
$65$	0	0
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	−5.65685	−0.681005
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−1.41421	−0.165521	−0.0827606	−	0.996569i	$-0.526374\pi$
$73$	−1.41421	−0.165521	−0.0827606	+	0.996569i	$0.526374\pi$
$74$	0	0
$75$	−4.24264	−0.489898
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	9.89949	1.08661	0.543305	−	0.839535i	$-0.317173\pi$
$83$	9.89949	1.08661	0.543305	+	0.839535i	$0.317173\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	2.82843	0.303239
$88$	0	0
$89$	−7.07107	−0.749532	−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107	−0.749532	−0.374766	+	0.927119i	$0.622277\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	12.0000	1.24434
$94$	0	0
$95$	20.0000	2.05196
$96$	0	0
$97$	9.89949	1.00514	0.502571	−	0.864536i	$-0.332388\pi$
$97$	9.89949	1.00514	0.502571	+	0.864536i	$0.332388\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	−8.48528	−0.844317	−0.422159	−	0.906522i	$-0.638727\pi$
$101$	−8.48528	−0.844317	−0.422159	+	0.906522i	$0.638727\pi$
$102$	0	0
$103$	−2.82843	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	14.1421	1.34231
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	−11.3137	−1.05501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	14.0000	1.26234
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	−2.82843	−0.249029
$130$	0	0
$131$	12.7279	1.11204	0.556022	−	0.831168i	$-0.312327\pi$
$131$	12.7279	1.11204	0.556022	+	0.831168i	$0.312327\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−16.0000	−1.37706
$136$	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
$139$	9.89949	0.839664	0.419832	−	0.907602i	$-0.362089\pi$
$139$	9.89949	0.839664	0.419832	+	0.907602i	$0.362089\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.65685	0.469776
$146$	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
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	1.41421	0.114332
$154$	0	0
$155$	24.0000	1.92773
$156$	0	0
$157$	11.3137	0.902932	0.451466	−	0.892288i	$-0.350901\pi$
$157$	11.3137	0.902932	0.451466	+	0.892288i	$0.350901\pi$
$158$	0	0
$159$	−2.82843	−0.224309
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	0	0
$165$	−8.00000	−0.622799
$166$	0	0
$167$	19.7990	1.53209	0.766046	−	0.642786i	$-0.222221\pi$
$167$	19.7990	1.53209	0.766046	+	0.642786i	$0.222221\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	7.07107	0.540738
$172$	0	0
$173$	16.9706	1.29025	0.645124	−	0.764078i	$-0.276806\pi$
$173$	16.9706	1.29025	0.645124	+	0.764078i	$0.276806\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.00000	0.150329
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	28.2843	2.07950
$186$	0	0
$187$	2.82843	0.206835
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	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
$195$	0	0
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−8.48528	−0.601506	−0.300753	−	0.953702i	$-0.597238\pi$
$199$	−8.48528	−0.601506	−0.300753	+	0.953702i	$0.597238\pi$
$200$	0	0
$201$	−16.9706	−1.19701
$202$	0	0
$203$	0	0
$204$	0	0
$205$	28.0000	1.95560
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	14.1421	0.978232
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	−16.9706	−1.16280
$214$	0	0
$215$	−5.65685	−0.385794
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	−21.2132	−1.40797	−0.703985	−	0.710215i	$-0.748598\pi$
$227$	−21.2132	−1.40797	−0.703985	+	0.710215i	$0.748598\pi$
$228$	0	0
$229$	16.9706	1.12145	0.560723	−	0.828003i	$-0.310523\pi$
$229$	16.9706	1.12145	0.560723	+	0.828003i	$0.310523\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	−5.65685	−0.367452
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−21.2132	−1.36646	−0.683231	−	0.730202i	$-0.739426\pi$
$241$	−21.2132	−1.36646	−0.683231	+	0.730202i	$0.739426\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−14.0000	−0.887214
$250$	0	0
$251$	9.89949	0.624851	0.312425	−	0.949942i	$-0.398859\pi$
$251$	9.89949	0.624851	0.312425	+	0.949942i	$0.398859\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	−5.65685	−0.354246
$256$	0	0
$257$	12.7279	0.793946	0.396973	−	0.917830i	$-0.370061\pi$
$257$	12.7279	0.793946	0.396973	+	0.917830i	$0.370061\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−5.65685	−0.347498
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	11.3137	0.689809	0.344904	−	0.938638i	$-0.387911\pi$
$269$	11.3137	0.689809	0.344904	+	0.938638i	$0.387911\pi$
$270$	0	0
$271$	−22.6274	−1.37452	−0.687259	−	0.726413i	$-0.741186\pi$
$271$	−22.6274	−1.37452	−0.687259	+	0.726413i	$0.741186\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.00000	−0.361814
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	8.48528	0.508001
$280$	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
$283$	−1.41421	−0.0840663	−0.0420331	−	0.999116i	$-0.513384\pi$
$283$	−1.41421	−0.0840663	−0.0420331	+	0.999116i	$0.513384\pi$
$284$	0	0
$285$	−28.2843	−1.67542
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.0000	−0.882353
$290$	0	0
$291$	−14.0000	−0.820695
$292$	0	0
$293$	19.7990	1.15667	0.578335	−	0.815800i	$-0.303703\pi$
$293$	19.7990	1.15667	0.578335	+	0.815800i	$0.303703\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−11.3137	−0.656488
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	12.0000	0.689382
$304$	0	0
$305$	8.00000	0.458079
$306$	0	0
$307$	−9.89949	−0.564994	−0.282497	−	0.959268i	$-0.591163\pi$
$307$	−9.89949	−0.564994	−0.282497	+	0.959268i	$0.591163\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	11.3137	0.641542	0.320771	−	0.947157i	$-0.396058\pi$
$311$	11.3137	0.641542	0.320771	+	0.947157i	$0.396058\pi$
$312$	0	0
$313$	12.7279	0.719425	0.359712	−	0.933063i	$-0.382875\pi$
$313$	12.7279	0.719425	0.359712	+	0.933063i	$0.382875\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	5.65685	0.315735
$322$	0	0
$323$	10.0000	0.556415
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.82843	−0.156412
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	0	0
$335$	−33.9411	−1.85440
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	16.9706	0.921714
$340$	0	0
$341$	16.9706	0.919007
$342$	0	0
$343$	0	0
$344$	0	0
$345$	16.0000	0.861411
$346$	0	0
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.41421	−0.0752710	−0.0376355	−	0.999292i	$-0.511983\pi$
$353$	−1.41421	−0.0752710	−0.0376355	+	0.999292i	$0.511983\pi$
$354$	0	0
$355$	−33.9411	−1.80141
$356$	0	0
$357$	0	0
$358$	0	0
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	31.0000	1.63158
$362$	0	0
$363$	9.89949	0.519589
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	−28.2843	−1.47643	−0.738213	−	0.674567i	$-0.764330\pi$
$367$	−28.2843	−1.47643	−0.738213	+	0.674567i	$0.764330\pi$
$368$	0	0
$369$	9.89949	0.515347
$370$	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
$375$	−8.00000	−0.413118
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	0	0
$381$	22.6274	1.15924
$382$	0	0
$383$	36.7696	1.87884	0.939418	−	0.342773i	$-0.111366\pi$
$383$	36.7696	1.87884	0.939418	+	0.342773i	$0.111366\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	−5.65685	−0.286079
$392$	0	0
$393$	−18.0000	−0.907980
$394$	0	0
$395$	−11.3137	−0.569254
$396$	0	0
$397$	−22.6274	−1.13564	−0.567819	−	0.823154i	$-0.692213\pi$
$397$	−22.6274	−1.13564	−0.567819	+	0.823154i	$0.692213\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	14.1421	0.702728
$406$	0	0
$407$	20.0000	0.991363
$408$	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
$411$	−16.9706	−0.837096
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−28.0000	−1.37447
$416$	0	0
$417$	−14.0000	−0.685583
$418$	0	0
$419$	−9.89949	−0.483622	−0.241811	−	0.970323i	$-0.577741\pi$
$419$	−9.89949	−0.483622	−0.241811	+	0.970323i	$0.577741\pi$
$420$	0	0
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	0	0
$423$	2.82843	0.137523
$424$	0	0
$425$	−4.24264	−0.205798
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−29.6985	−1.42722	−0.713609	−	0.700544i	$-0.752941\pi$
$433$	−29.6985	−1.42722	−0.713609	+	0.700544i	$0.752941\pi$
$434$	0	0
$435$	−8.00000	−0.383571
$436$	0	0
$437$	−28.2843	−1.35302
$438$	0	0
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	20.0000	0.948091
$446$	0	0
$447$	14.1421	0.668900
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	19.7990	0.932298
$452$	0	0
$453$	−22.6274	−1.06313
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.0000	1.12267	0.561336	−	0.827588i	$-0.310287\pi$
$457$	24.0000	1.12267	0.561336	+	0.827588i	$0.310287\pi$
$458$	0	0
$459$	−8.00000	−0.373408
$460$	0	0
$461$	−39.5980	−1.84426	−0.922131	−	0.386878i	$-0.873553\pi$
$461$	−39.5980	−1.84426	−0.922131	+	0.386878i	$0.873553\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	−33.9411	−1.57398
$466$	0	0
$467$	32.5269	1.50517	0.752583	−	0.658497i	$-0.228808\pi$
$467$	32.5269	1.50517	0.752583	+	0.658497i	$0.228808\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−16.0000	−0.737241
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−21.2132	−0.973329
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	31.1127	1.42158	0.710788	−	0.703407i	$-0.248339\pi$
$479$	31.1127	1.42158	0.710788	+	0.703407i	$0.248339\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−28.0000	−1.27141
$486$	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
$489$	−14.1421	−0.639529
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	2.82843	0.127386
$494$	0	0
$495$	−5.65685	−0.254257
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	−28.0000	−1.25095
$502$	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
$505$	24.0000	1.06799
$506$	0	0
$507$	18.3848	0.816497
$508$	0	0
$509$	−22.6274	−1.00294	−0.501471	−	0.865174i	$-0.667208\pi$
$509$	−22.6274	−1.00294	−0.501471	+	0.865174i	$0.667208\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−40.0000	−1.76604
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	5.65685	0.248788
$518$	0	0
$519$	−24.0000	−1.05348
$520$	0	0
$521$	−1.41421	−0.0619578	−0.0309789	−	0.999520i	$-0.509862\pi$
$521$	−1.41421	−0.0619578	−0.0309789	+	0.999520i	$0.509862\pi$
$522$	0	0
$523$	12.7279	0.556553	0.278277	−	0.960501i	$-0.410237\pi$
$523$	12.7279	0.556553	0.278277	+	0.960501i	$0.410237\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	1.41421	0.0613716
$532$	0	0
$533$	0	0
$534$	0	0
$535$	11.3137	0.489134
$536$	0	0
$537$	−16.9706	−0.732334
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.65685	−0.242313
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	0	0
$549$	2.82843	0.120714
$550$	0	0
$551$	14.1421	0.602475
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−40.0000	−1.69791
$556$	0	0
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	−1.41421	−0.0596020	−0.0298010	−	0.999556i	$-0.509487\pi$
$563$	−1.41421	−0.0596020	−0.0298010	+	0.999556i	$0.509487\pi$
$564$	0	0
$565$	33.9411	1.42791
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−2.00000	−0.0836974	−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000	−0.0836974	−0.0418487	+	0.999124i	$0.513325\pi$
$572$	0	0
$573$	−5.65685	−0.236318
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	−21.2132	−0.883117	−0.441559	−	0.897232i	$-0.645574\pi$
$577$	−21.2132	−0.883117	−0.441559	+	0.897232i	$0.645574\pi$
$578$	0	0
$579$	22.6274	0.940363
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4.00000	−0.165663
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−29.6985	−1.22579	−0.612894	−	0.790165i	$-0.709995\pi$
$587$	−29.6985	−1.22579	−0.612894	+	0.790165i	$0.709995\pi$
$588$	0	0
$589$	60.0000	2.47226
$590$	0	0
$591$	2.82843	0.116346
$592$	0	0
$593$	−7.07107	−0.290374	−0.145187	−	0.989404i	$-0.546378\pi$
$593$	−7.07107	−0.290374	−0.145187	+	0.989404i	$0.546378\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.0000	0.491127
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	29.6985	1.21143	0.605713	−	0.795683i	$-0.292888\pi$
$601$	29.6985	1.21143	0.605713	+	0.795683i	$0.292888\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	19.7990	0.804943
$606$	0	0
$607$	16.9706	0.688814	0.344407	−	0.938820i	$-0.388080\pi$
$607$	16.9706	0.688814	0.344407	+	0.938820i	$0.388080\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	30.0000	1.21169	0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000	1.21169	0.605844	+	0.795583i	$0.292835\pi$
$614$	0	0
$615$	−39.5980	−1.59674
$616$	0	0
$617$	−26.0000	−1.04672	−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000	−1.04672	−0.523360	+	0.852111i	$0.675322\pi$
$618$	0	0
$619$	18.3848	0.738947	0.369473	−	0.929241i	$-0.379538\pi$
$619$	18.3848	0.738947	0.369473	+	0.929241i	$0.379538\pi$
$620$	0	0
$621$	22.6274	0.908007
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	−20.0000	−0.798723
$628$	0	0
$629$	14.1421	0.563884
$630$	0	0
$631$	−44.0000	−1.75161	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000	−1.75161	−0.875806	+	0.482663i	$0.839670\pi$
$632$	0	0
$633$	16.9706	0.674519
$634$	0	0
$635$	45.2548	1.79588
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	9.89949	0.390398	0.195199	−	0.980764i	$-0.437465\pi$
$643$	9.89949	0.390398	0.195199	+	0.980764i	$0.437465\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−8.48528	−0.333591	−0.166795	−	0.985992i	$-0.553342\pi$
$647$	−8.48528	−0.333591	−0.166795	+	0.985992i	$0.553342\pi$
$648$	0	0
$649$	2.82843	0.111025
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	−36.0000	−1.40664
$656$	0	0
$657$	1.41421	0.0551737
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	−8.48528	−0.330039	−0.165020	−	0.986290i	$-0.552769\pi$
$661$	−8.48528	−0.330039	−0.165020	+	0.986290i	$0.552769\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.00000	−0.309761
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.65685	0.218380
$672$	0	0
$673$	−12.0000	−0.462566	−0.231283	−	0.972887i	$-0.574292\pi$
$673$	−12.0000	−0.462566	−0.231283	+	0.972887i	$0.574292\pi$
$674$	0	0
$675$	16.9706	0.653197
$676$	0	0
$677$	16.9706	0.652232	0.326116	−	0.945330i	$-0.394260\pi$
$677$	16.9706	0.652232	0.326116	+	0.945330i	$0.394260\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	30.0000	1.14960
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	−33.9411	−1.29682
$686$	0	0
$687$	−24.0000	−0.915657
$688$	0	0
$689$	0	0
$690$	0	0
$691$	12.7279	0.484193	0.242096	−	0.970252i	$-0.422165\pi$
$691$	12.7279	0.484193	0.242096	+	0.970252i	$0.422165\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−28.0000	−1.06210
$696$	0	0
$697$	14.0000	0.530288
$698$	0	0
$699$	−33.9411	−1.28377
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	70.7107	2.66690
$704$	0	0
$705$	−11.3137	−0.426099
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	−33.9411	−1.27111
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−16.9706	−0.633777
$718$	0	0
$719$	−2.82843	−0.105483	−0.0527413	−	0.998608i	$-0.516796\pi$
$719$	−2.82843	−0.105483	−0.0527413	+	0.998608i	$0.516796\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	30.0000	1.11571
$724$	0	0
$725$	−6.00000	−0.222834
$726$	0	0
$727$	19.7990	0.734304	0.367152	−	0.930161i	$-0.380333\pi$
$727$	19.7990	0.734304	0.367152	+	0.930161i	$0.380333\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−2.82843	−0.104613
$732$	0	0
$733$	−42.4264	−1.56706	−0.783528	−	0.621357i	$-0.786582\pi$
$733$	−42.4264	−1.56706	−0.783528	+	0.621357i	$0.786582\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	28.2843	1.03626
$746$	0	0
$747$	−9.89949	−0.362204
$748$	0	0
$749$	0	0
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	−14.0000	−0.510188
$754$	0	0
$755$	−45.2548	−1.64699
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	11.3137	0.410662
$760$	0	0
$761$	−7.07107	−0.256326	−0.128163	−	0.991753i	$-0.540908\pi$
$761$	−7.07107	−0.256326	−0.128163	+	0.991753i	$0.540908\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.00000	−0.144620
$766$	0	0
$767$	0	0
$768$	0	0
$769$	29.6985	1.07095	0.535477	−	0.844550i	$-0.320132\pi$
$769$	29.6985	1.07095	0.535477	+	0.844550i	$0.320132\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	0	0
$773$	−48.0833	−1.72943	−0.864717	−	0.502259i	$-0.832502\pi$
$773$	−48.0833	−1.72943	−0.864717	+	0.502259i	$0.832502\pi$
$774$	0	0
$775$	−25.4558	−0.914401
$776$	0	0
$777$	0	0
$778$	0	0
$779$	70.0000	2.50801
$780$	0	0
$781$	−24.0000	−0.858788
$782$	0	0
$783$	−11.3137	−0.404319
$784$	0	0
$785$	−32.0000	−1.14213
$786$	0	0
$787$	−1.41421	−0.0504113	−0.0252056	−	0.999682i	$-0.508024\pi$
$787$	−1.41421	−0.0504113	−0.0252056	+	0.999682i	$0.508024\pi$
$788$	0	0
$789$	16.9706	0.604168
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	8.00000	0.283731
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	0	0
$801$	7.07107	0.249844
$802$	0	0
$803$	2.82843	0.0998130
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−16.0000	−0.563227
$808$	0	0
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	0	0
$811$	−29.6985	−1.04285	−0.521427	−	0.853296i	$-0.674600\pi$
$811$	−29.6985	−1.04285	−0.521427	+	0.853296i	$0.674600\pi$
$812$	0	0
$813$	32.0000	1.12229
$814$	0	0
$815$	−28.2843	−0.990755
$816$	0	0
$817$	−14.1421	−0.494771
$818$	0	0
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	8.48528	0.295420
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	31.1127	1.08059	0.540294	−	0.841476i	$-0.318313\pi$
$829$	31.1127	1.08059	0.540294	+	0.841476i	$0.318313\pi$
$830$	0	0
$831$	−2.82843	−0.0981170
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−56.0000	−1.93796
$836$	0	0
$837$	−48.0000	−1.65912
$838$	0	0
$839$	−19.7990	−0.683537	−0.341769	−	0.939784i	$-0.611026\pi$
$839$	−19.7990	−0.683537	−0.341769	+	0.939784i	$0.611026\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−22.6274	−0.779330
$844$	0	0
$845$	36.7696	1.26491
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.00000	0.0686398
$850$	0	0
$851$	−40.0000	−1.37118
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	−20.0000	−0.683986
$856$	0	0
$857$	18.3848	0.628012	0.314006	−	0.949421i	$-0.398329\pi$
$857$	18.3848	0.628012	0.314006	+	0.949421i	$0.398329\pi$
$858$	0	0
$859$	−26.8701	−0.916795	−0.458397	−	0.888747i	$-0.651576\pi$
$859$	−26.8701	−0.916795	−0.458397	+	0.888747i	$0.651576\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	−48.0000	−1.63205
$866$	0	0
$867$	21.2132	0.720438
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−9.89949	−0.335047
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.0000	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$878$	0	0
$879$	−28.0000	−0.944417
$880$	0	0
$881$	29.6985	1.00057	0.500284	−	0.865862i	$-0.333229\pi$
$881$	29.6985	1.00057	0.500284	+	0.865862i	$0.333229\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	−5.65685	−0.190153
$886$	0	0
$887$	36.7696	1.23460	0.617300	−	0.786728i	$-0.288226\pi$
$887$	36.7696	1.23460	0.617300	+	0.786728i	$0.288226\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	10.0000	0.335013
$892$	0	0
$893$	20.0000	0.669274
$894$	0	0
$895$	−33.9411	−1.13453
$896$	0	0
$897$	0	0
$898$	0	0
$899$	16.9706	0.566000
$900$	0	0
$901$	−2.82843	−0.0942286
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	0	0
$909$	8.48528	0.281439
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	−19.7990	−0.655251
$914$	0	0
$915$	−11.3137	−0.374020
$916$	0	0
$917$	0	0
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	14.0000	0.461316
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−30.0000	−0.986394
$926$	0	0
$927$	2.82843	0.0928977
$928$	0	0
$929$	32.5269	1.06717	0.533587	−	0.845745i	$-0.320844\pi$
$929$	32.5269	1.06717	0.533587	+	0.845745i	$0.320844\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−16.0000	−0.523816
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	9.89949	0.323402	0.161701	−	0.986840i	$-0.448302\pi$
$937$	9.89949	0.323402	0.161701	+	0.986840i	$0.448302\pi$
$938$	0	0
$939$	−18.0000	−0.587408
$940$	0	0
$941$	31.1127	1.01424	0.507122	−	0.861874i	$-0.330709\pi$
$941$	31.1127	1.01424	0.507122	+	0.861874i	$0.330709\pi$
$942$	0	0
$943$	−39.5980	−1.28949
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	14.1421	0.458590
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	−11.3137	−0.366103
$956$	0	0
$957$	−5.65685	−0.182860
$958$	0	0
$959$	0	0
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	45.2548	1.45680
$966$	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
$969$	−14.1421	−0.454311
$970$	0	0
$971$	32.5269	1.04384	0.521919	−	0.852995i	$-0.325216\pi$
$971$	32.5269	1.04384	0.521919	+	0.852995i	$0.325216\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	0	0
$979$	14.1421	0.451985
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	−48.0833	−1.53362	−0.766809	−	0.641875i	$-0.778157\pi$
$983$	−48.0833	−1.53362	−0.766809	+	0.641875i	$0.778157\pi$
$984$	0	0
$985$	5.65685	0.180242
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	−14.1421	−0.448787
$994$	0	0
$995$	24.0000	0.760851
$996$	0	0
$997$	31.1127	0.985349	0.492675	−	0.870214i	$-0.336019\pi$
$997$	31.1127	0.985349	0.492675	+	0.870214i	$0.336019\pi$
$998$	0	0
$999$	−56.5685	−1.78975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.2.a.bm.1.1		2
4.3	odd	2		3136.2.a.bn.1.2		2
7.6	odd	2	inner	3136.2.a.bm.1.2		2
8.3	odd	2		98.2.a.b.1.1	✓	2
8.5	even	2		784.2.a.l.1.2		2
24.5	odd	2		7056.2.a.cl.1.1		2
24.11	even	2		882.2.a.n.1.1		2
28.27	even	2		3136.2.a.bn.1.1		2
40.3	even	4		2450.2.c.v.99.1		4
40.19	odd	2		2450.2.a.bj.1.2		2
40.27	even	4		2450.2.c.v.99.4		4
56.3	even	6		98.2.c.c.79.1		4
56.5	odd	6		784.2.i.m.753.2		4
56.11	odd	6		98.2.c.c.79.2		4
56.13	odd	2		784.2.a.l.1.1		2
56.19	even	6		98.2.c.c.67.1		4
56.27	even	2		98.2.a.b.1.2	yes	2
56.37	even	6		784.2.i.m.753.1		4
56.45	odd	6		784.2.i.m.177.2		4
56.51	odd	6		98.2.c.c.67.2		4
56.53	even	6		784.2.i.m.177.1		4
168.11	even	6		882.2.g.l.667.2		4
168.59	odd	6		882.2.g.l.667.1		4
168.83	odd	2		882.2.a.n.1.2		2
168.107	even	6		882.2.g.l.361.2		4
168.125	even	2		7056.2.a.cl.1.2		2
168.131	odd	6		882.2.g.l.361.1		4
280.27	odd	4		2450.2.c.v.99.3		4
280.83	odd	4		2450.2.c.v.99.2		4
280.139	even	2		2450.2.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.2.a.b.1.1	✓	2	8.3	odd	2
98.2.a.b.1.2	yes	2	56.27	even	2
98.2.c.c.67.1		4	56.19	even	6
98.2.c.c.67.2		4	56.51	odd	6
98.2.c.c.79.1		4	56.3	even	6
98.2.c.c.79.2		4	56.11	odd	6
784.2.a.l.1.1		2	56.13	odd	2
784.2.a.l.1.2		2	8.5	even	2
784.2.i.m.177.1		4	56.53	even	6
784.2.i.m.177.2		4	56.45	odd	6
784.2.i.m.753.1		4	56.37	even	6
784.2.i.m.753.2		4	56.5	odd	6
882.2.a.n.1.1		2	24.11	even	2
882.2.a.n.1.2		2	168.83	odd	2
882.2.g.l.361.1		4	168.131	odd	6
882.2.g.l.361.2		4	168.107	even	6
882.2.g.l.667.1		4	168.59	odd	6
882.2.g.l.667.2		4	168.11	even	6
2450.2.a.bj.1.1		2	280.139	even	2
2450.2.a.bj.1.2		2	40.19	odd	2
2450.2.c.v.99.1		4	40.3	even	4
2450.2.c.v.99.2		4	280.83	odd	4
2450.2.c.v.99.3		4	280.27	odd	4
2450.2.c.v.99.4		4	40.27	even	4
3136.2.a.bm.1.1		2	1.1	even	1	trivial
3136.2.a.bm.1.2		2	7.6	odd	2	inner
3136.2.a.bn.1.1		2	28.27	even	2
3136.2.a.bn.1.2		2	4.3	odd	2
7056.2.a.cl.1.1		2	24.5	odd	2
7056.2.a.cl.1.2		2	168.125	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3136.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} -2.82843 q^{5} -1.00000 q^{9} -2.00000 q^{11} +4.00000 q^{15} -1.41421 q^{17} -7.07107 q^{19} +4.00000 q^{23} +3.00000 q^{25} +5.65685 q^{27} -2.00000 q^{29} -8.48528 q^{31} +2.82843 q^{33} -10.0000 q^{37} -9.89949 q^{41} +2.00000 q^{43} +2.82843 q^{45} -2.82843 q^{47} +2.00000 q^{51} +2.00000 q^{53} +5.65685 q^{55} +10.0000 q^{57} -1.41421 q^{59} -2.82843 q^{61} +12.0000 q^{67} -5.65685 q^{69} +12.0000 q^{71} -1.41421 q^{73} -4.24264 q^{75} +4.00000 q^{79} -5.00000 q^{81} +9.89949 q^{83} +4.00000 q^{85} +2.82843 q^{87} -7.07107 q^{89} +12.0000 q^{93} +20.0000 q^{95} +9.89949 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} - 4 q^{11} + 8 q^{15} + 8 q^{23} + 6 q^{25} - 4 q^{29} - 20 q^{37} + 4 q^{43} + 4 q^{51} + 4 q^{53} + 20 q^{57} + 24 q^{67} + 24 q^{71} + 8 q^{79} - 10 q^{81} + 8 q^{85} + 24 q^{93} + 40 q^{95}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 4 * q^11 + 8 * q^15 + 8 * q^23 + 6 * q^25 - 4 * q^29 - 20 * q^37 + 4 * q^43 + 4 * q^51 + 4 * q^53 + 20 * q^57 + 24 * q^67 + 24 * q^71 + 8 * q^79 - 10 * q^81 + 8 * q^85 + 24 * q^93 + 40 * q^95 + 4 * q^99

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