LMFDB - Embedded newform 3136.2.f.i.3135.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3136,2,Mod(3135,3136)]

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([1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3136.3135");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3136 = 2^{6} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3136.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$25.0410860739$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1212153856.10
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 10x^{4} - 16x^{2} + 16 $ x^8 - 4x^6 + 10x^4 - 16*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 196)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3135.4
Root			$-1.07072 - 0.923880i$ of defining polynomial
Character	$\chi$	$=$	3136.3135
Dual form			3136.2.f.i.3135.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14144 q^{3} +2.61313i q^{5} +1.58579 q^{9} +O(q^{10})$	$q-2.14144 q^{3} +2.61313i q^{5} +1.58579 q^{9} -3.95687i q^{11} +1.08239i q^{13} -5.59587i q^{15} -0.317025i q^{17} -5.16991 q^{19} +2.31788i q^{23} -1.82843 q^{25} +3.02846 q^{27} +6.82843 q^{29} +6.05692 q^{31} +8.47343i q^{33} +4.00000 q^{37} -2.31788i q^{39} +2.29610i q^{41} -7.23486i q^{43} +4.14386i q^{45} -4.28289 q^{47} +0.678892i q^{51} -10.4853 q^{53} +10.3398 q^{55} +11.0711 q^{57} +11.2268 q^{59} +5.41196i q^{61} -2.82843 q^{65} +3.27798i q^{67} -4.96362i q^{69} +14.0167i q^{73} +3.91548 q^{75} -7.91375i q^{79} -11.2426 q^{81} +9.45280 q^{83} +0.828427 q^{85} -14.6227 q^{87} -5.99162i q^{89} -12.9706 q^{93} -13.5096i q^{95} -9.23880i q^{97} -6.27476i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} + 8 q^{25} + 32 q^{29} + 32 q^{37} - 16 q^{53} + 32 q^{57} - 56 q^{81} - 16 q^{85} + 32 q^{93}+O(q^{100}) $ 8 * q + 24 * q^9 + 8 * q^25 + 32 * q^29 + 32 * q^37 - 16 * q^53 + 32 * q^57 - 56 * q^81 - 16 * q^85 + 32 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−2.14144	−1.23636	−0.618182	−	0.786035i	$-0.712130\pi$
$3$	−2.14144	−1.23636	−0.618182	+	0.786035i	$0.712130\pi$
$4$	0	0
$5$	2.61313i	1.16863i	0.811529	+	0.584313i	$0.198636\pi$
$5$	2.61313i	1.16863i	−0.811529	+	0.584313i	$0.801364\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.58579	0.528595
$10$	0	0
$11$	− 3.95687i	− 1.19304i	−0.802597	−	0.596521i	$-0.796549\pi$
$11$	− 3.95687i	− 1.19304i	0.802597	−	0.596521i	$-0.203451\pi$
$12$	0	0
$13$	1.08239i	0.300202i	0.988671	+	0.150101i	$0.0479598\pi$
$13$	1.08239i	0.300202i	−0.988671	+	0.150101i	$0.952040\pi$
$14$	0	0
$15$	− 5.59587i	− 1.44485i
$16$	0	0
$17$	− 0.317025i	− 0.0768899i	−0.999261	−	0.0384450i	$-0.987760\pi$
$17$	− 0.317025i	− 0.0768899i	0.999261	−	0.0384450i	$-0.0122404\pi$
$18$	0	0
$19$	−5.16991	−1.18606	−0.593029	−	0.805181i	$-0.702068\pi$
$19$	−5.16991	−1.18606	−0.593029	+	0.805181i	$0.702068\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.31788i	0.483312i	0.970362	+	0.241656i	$0.0776906\pi$
$23$	2.31788i	0.483312i	−0.970362	+	0.241656i	$0.922309\pi$
$24$	0	0
$25$	−1.82843	−0.365685
$26$	0	0
$27$	3.02846	0.582827
$28$	0	0
$29$	6.82843	1.26801	0.634004	−	0.773330i	$-0.281410\pi$
$29$	6.82843	1.26801	0.634004	+	0.773330i	$0.281410\pi$
$30$	0	0
$31$	6.05692	1.08786	0.543928	−	0.839132i	$-0.316937\pi$
$31$	6.05692	1.08786	0.543928	+	0.839132i	$0.316937\pi$
$32$	0	0
$33$	8.47343i	1.47503i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	− 2.31788i	− 0.371158i
$40$	0	0
$41$	2.29610i	0.358591i	0.983795	+	0.179295i	$0.0573818\pi$
$41$	2.29610i	0.358591i	−0.983795	+	0.179295i	$0.942618\pi$
$42$	0	0
$43$	− 7.23486i	− 1.10331i	−0.834074	−	0.551653i	$-0.813997\pi$
$43$	− 7.23486i	− 1.10331i	0.834074	−	0.551653i	$-0.186003\pi$
$44$	0	0
$45$	4.14386i	0.617730i
$46$	0	0
$47$	−4.28289	−0.624724	−0.312362	−	0.949963i	$-0.601120\pi$
$47$	−4.28289	−0.624724	−0.312362	+	0.949963i	$0.601120\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.678892i	0.0950639i
$52$	0	0
$53$	−10.4853	−1.44026	−0.720132	−	0.693837i	$-0.755919\pi$
$53$	−10.4853	−1.44026	−0.720132	+	0.693837i	$0.755919\pi$
$54$	0	0
$55$	10.3398	1.39422
$56$	0	0
$57$	11.0711	1.46640
$58$	0	0
$59$	11.2268	1.46161	0.730804	−	0.682587i	$-0.239145\pi$
$59$	11.2268	1.46161	0.730804	+	0.682587i	$0.239145\pi$
$60$	0	0
$61$	5.41196i	0.692931i	0.938063	+	0.346465i	$0.112618\pi$
$61$	5.41196i	0.692931i	−0.938063	+	0.346465i	$0.887382\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.82843	−0.350823
$66$	0	0
$67$	3.27798i	0.400469i	0.979748	+	0.200235i	$0.0641704\pi$
$67$	3.27798i	0.400469i	−0.979748	+	0.200235i	$0.935830\pi$
$68$	0	0
$69$	− 4.96362i	− 0.597550i
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	14.0167i	1.64053i	0.571983	+	0.820266i	$0.306174\pi$
$73$	14.0167i	1.64053i	−0.571983	+	0.820266i	$0.693826\pi$
$74$	0	0
$75$	3.91548	0.452120
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 7.91375i	− 0.890366i	−0.895440	−	0.445183i	$-0.853139\pi$
$79$	− 7.91375i	− 0.890366i	0.895440	−	0.445183i	$-0.146861\pi$
$80$	0	0
$81$	−11.2426	−1.24918
$82$	0	0
$83$	9.45280	1.03758	0.518790	−	0.854902i	$-0.326383\pi$
$83$	9.45280	1.03758	0.518790	+	0.854902i	$0.326383\pi$
$84$	0	0
$85$	0.828427	0.0898555
$86$	0	0
$87$	−14.6227	−1.56772
$88$	0	0
$89$	− 5.99162i	− 0.635110i	−0.948240	−	0.317555i	$-0.897138\pi$
$89$	− 5.99162i	− 0.635110i	0.948240	−	0.317555i	$-0.102862\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−12.9706	−1.34498
$94$	0	0
$95$	− 13.5096i	− 1.38606i
$96$	0	0
$97$	− 9.23880i	− 0.938058i	−0.883183	−	0.469029i	$-0.844604\pi$
$97$	− 9.23880i	− 0.938058i	0.883183	−	0.469029i	$-0.155396\pi$
$98$	0	0
$99$	− 6.27476i	− 0.630637i
$100$	0	0
$101$	15.2304i	1.51548i	0.652555	+	0.757741i	$0.273697\pi$
$101$	15.2304i	1.51548i	−0.652555	+	0.757741i	$0.726303\pi$
$102$	0	0
$103$	−12.1138	−1.19361	−0.596806	−	0.802385i	$-0.703564\pi$
$103$	−12.1138	−1.19361	−0.596806	+	0.802385i	$0.703564\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.31788i	− 0.224078i	−0.993704	−	0.112039i	$-0.964262\pi$
$107$	− 2.31788i	− 0.224078i	0.993704	−	0.112039i	$-0.0357382\pi$
$108$	0	0
$109$	5.65685	0.541828	0.270914	−	0.962604i	$-0.412674\pi$
$109$	5.65685	0.541828	0.270914	+	0.962604i	$0.412674\pi$
$110$	0	0
$111$	−8.56578	−0.813028
$112$	0	0
$113$	−4.24264	−0.399114	−0.199557	−	0.979886i	$-0.563950\pi$
$113$	−4.24264	−0.399114	−0.199557	+	0.979886i	$0.563950\pi$
$114$	0	0
$115$	−6.05692	−0.564811
$116$	0	0
$117$	1.71644i	0.158685i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−4.65685	−0.423350
$122$	0	0
$123$	− 4.91697i	− 0.443349i
$124$	0	0
$125$	8.28772i	0.741276i
$126$	0	0
$127$	10.2316i	0.907911i	0.891024	+	0.453955i	$0.149987\pi$
$127$	10.2316i	0.907911i	−0.891024	+	0.453955i	$0.850013\pi$
$128$	0	0
$129$	15.4930i	1.36409i
$130$	0	0
$131$	0.367414	0.0321011	0.0160505	−	0.999871i	$-0.494891\pi$
$131$	0.367414	0.0321011	0.0160505	+	0.999871i	$0.494891\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	7.91375i	0.681107i
$136$	0	0
$137$	−8.24264	−0.704216	−0.352108	−	0.935959i	$-0.614535\pi$
$137$	−8.24264	−0.704216	−0.352108	+	0.935959i	$0.614535\pi$
$138$	0	0
$139$	−22.8211	−1.93566	−0.967829	−	0.251610i	$-0.919040\pi$
$139$	−22.8211	−1.93566	−0.967829	+	0.251610i	$0.919040\pi$
$140$	0	0
$141$	9.17157	0.772386
$142$	0	0
$143$	4.28289	0.358153
$144$	0	0
$145$	17.8435i	1.48183i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.82843	−0.395560	−0.197780	−	0.980246i	$-0.563373\pi$
$149$	−4.82843	−0.395560	−0.197780	+	0.980246i	$0.563373\pi$
$150$	0	0
$151$	21.4234i	1.74341i	0.490032	+	0.871704i	$0.336985\pi$
$151$	21.4234i	1.74341i	−0.490032	+	0.871704i	$0.663015\pi$
$152$	0	0
$153$	− 0.502734i	− 0.0406437i
$154$	0	0
$155$	15.8275i	1.27130i
$156$	0	0
$157$	19.1886i	1.53141i	0.643189	+	0.765707i	$0.277611\pi$
$157$	19.1886i	1.53141i	−0.643189	+	0.765707i	$0.722389\pi$
$158$	0	0
$159$	22.4537	1.78069
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.1486i	1.18653i	0.805007	+	0.593265i	$0.202161\pi$
$163$	15.1486i	1.18653i	−0.805007	+	0.593265i	$0.797839\pi$
$164$	0	0
$165$	−22.1421	−1.72376
$166$	0	0
$167$	−7.83095	−0.605977	−0.302989	−	0.952994i	$-0.597984\pi$
$167$	−7.83095	−0.605977	−0.302989	+	0.952994i	$0.597984\pi$
$168$	0	0
$169$	11.8284	0.909879
$170$	0	0
$171$	−8.19837	−0.626945
$172$	0	0
$173$	6.57128i	0.499605i	0.968297	+	0.249802i	$0.0803657\pi$
$173$	6.57128i	0.499605i	−0.968297	+	0.249802i	$0.919634\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−24.0416	−1.80708
$178$	0	0
$179$	13.5096i	1.00976i	0.863191	+	0.504878i	$0.168463\pi$
$179$	13.5096i	1.00976i	−0.863191	+	0.504878i	$0.831537\pi$
$180$	0	0
$181$	− 21.9874i	− 1.63431i	−0.576418	−	0.817155i	$-0.695550\pi$
$181$	− 21.9874i	− 1.63431i	0.576418	−	0.817155i	$-0.304450\pi$
$182$	0	0
$183$	− 11.5894i	− 0.856714i
$184$	0	0
$185$	10.4525i	0.768483i
$186$	0	0
$187$	−1.25443	−0.0917330
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.55596i	0.474373i	0.971464	+	0.237186i	$0.0762252\pi$
$191$	6.55596i	0.474373i	−0.971464	+	0.237186i	$0.923775\pi$
$192$	0	0
$193$	14.5858	1.04991	0.524954	−	0.851131i	$-0.324083\pi$
$193$	14.5858	1.04991	0.524954	+	0.851131i	$0.324083\pi$
$194$	0	0
$195$	6.05692	0.433745
$196$	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
$199$	17.1316	1.21442	0.607212	−	0.794540i	$-0.292288\pi$
$199$	17.1316	1.21442	0.607212	+	0.794540i	$0.292288\pi$
$200$	0	0
$201$	− 7.01962i	− 0.495126i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	3.67567i	0.255477i
$208$	0	0
$209$	20.4567i	1.41502i
$210$	0	0
$211$	14.4697i	0.996136i	0.867138	+	0.498068i	$0.165957\pi$
$211$	14.4697i	0.996136i	−0.867138	+	0.498068i	$0.834043\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	18.9056	1.28935
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 30.0160i	− 2.02829i
$220$	0	0
$221$	0.343146	0.0230825
$222$	0	0
$223$	−7.83095	−0.524399	−0.262200	−	0.965014i	$-0.584448\pi$
$223$	−7.83095	−0.524399	−0.262200	+	0.965014i	$0.584448\pi$
$224$	0	0
$225$	−2.89949	−0.193300
$226$	0	0
$227$	−9.97240	−0.661891	−0.330946	−	0.943650i	$-0.607368\pi$
$227$	−9.97240	−0.661891	−0.330946	+	0.943650i	$0.607368\pi$
$228$	0	0
$229$	− 1.34502i	− 0.0888817i	−0.999012	−	0.0444409i	$-0.985849\pi$
$229$	− 1.34502i	− 0.0888817i	0.999012	−	0.0444409i	$-0.0141506\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.2426	1.06409	0.532045	−	0.846716i	$-0.321424\pi$
$233$	16.2426	1.06409	0.532045	+	0.846716i	$0.321424\pi$
$234$	0	0
$235$	− 11.1917i	− 0.730068i
$236$	0	0
$237$	16.9469i	1.10082i
$238$	0	0
$239$	10.2316i	0.661829i	0.943661	+	0.330915i	$0.107357\pi$
$239$	10.2316i	0.661829i	−0.943661	+	0.330915i	$0.892643\pi$
$240$	0	0
$241$	− 5.09494i	− 0.328194i	−0.986444	−	0.164097i	$-0.947529\pi$
$241$	− 5.09494i	− 0.328194i	0.986444	−	0.164097i	$-0.0524710\pi$
$242$	0	0
$243$	14.9901	0.961616
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 5.59587i	− 0.356056i
$248$	0	0
$249$	−20.2426	−1.28283
$250$	0	0
$251$	−3.91548	−0.247143	−0.123571	−	0.992336i	$-0.539435\pi$
$251$	−3.91548	−0.247143	−0.123571	+	0.992336i	$0.539435\pi$
$252$	0	0
$253$	9.17157	0.576612
$254$	0	0
$255$	−1.77403	−0.111094
$256$	0	0
$257$	19.0572i	1.18876i	0.804185	+	0.594379i	$0.202602\pi$
$257$	19.0572i	1.18876i	−0.804185	+	0.594379i	$0.797398\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	10.8284	0.670263
$262$	0	0
$263$	25.6614i	1.58235i	0.611588	+	0.791176i	$0.290531\pi$
$263$	25.6614i	1.58235i	−0.611588	+	0.791176i	$0.709469\pi$
$264$	0	0
$265$	− 27.3994i	− 1.68313i
$266$	0	0
$267$	12.8307i	0.785227i
$268$	0	0
$269$	− 5.04054i	− 0.307327i	−0.988123	−	0.153664i	$-0.950893\pi$
$269$	− 5.04054i	− 0.307327i	0.988123	−	0.153664i	$-0.0491072\pi$
$270$	0	0
$271$	−4.28289	−0.260167	−0.130084	−	0.991503i	$-0.541525\pi$
$271$	−4.28289	−0.260167	−0.130084	+	0.991503i	$0.541525\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.23486i	0.436278i
$276$	0	0
$277$	4.14214	0.248877	0.124438	−	0.992227i	$-0.460287\pi$
$277$	4.14214	0.248877	0.124438	+	0.992227i	$0.460287\pi$
$278$	0	0
$279$	9.60498	0.575035
$280$	0	0
$281$	10.3848	0.619504	0.309752	−	0.950817i	$-0.399754\pi$
$281$	10.3848	0.619504	0.309752	+	0.950817i	$0.399754\pi$
$282$	0	0
$283$	3.39587	0.201864	0.100932	−	0.994893i	$-0.467818\pi$
$283$	3.39587	0.201864	0.100932	+	0.994893i	$0.467818\pi$
$284$	0	0
$285$	28.9301i	1.71367i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.8995	0.994088
$290$	0	0
$291$	19.7844i	1.15978i
$292$	0	0
$293$	11.7975i	0.689219i	0.938746	+	0.344609i	$0.111989\pi$
$293$	11.7975i	0.689219i	−0.938746	+	0.344609i	$0.888011\pi$
$294$	0	0
$295$	29.3371i	1.70807i
$296$	0	0
$297$	− 11.9832i	− 0.695338i
$298$	0	0
$299$	−2.50886	−0.145091
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 32.6151i	− 1.87369i
$304$	0	0
$305$	−14.1421	−0.809776
$306$	0	0
$307$	−3.91548	−0.223468	−0.111734	−	0.993738i	$-0.535640\pi$
$307$	−3.91548	−0.223468	−0.111734	+	0.993738i	$0.535640\pi$
$308$	0	0
$309$	25.9411	1.47574
$310$	0	0
$311$	−20.6796	−1.17263	−0.586317	−	0.810082i	$-0.699423\pi$
$311$	−20.6796	−1.17263	−0.586317	+	0.810082i	$0.699423\pi$
$312$	0	0
$313$	− 12.7486i	− 0.720594i	−0.932838	−	0.360297i	$-0.882675\pi$
$313$	− 12.7486i	− 0.720594i	0.932838	−	0.360297i	$-0.117325\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.51472	0.309737	0.154869	−	0.987935i	$-0.450505\pi$
$317$	5.51472	0.309737	0.154869	+	0.987935i	$0.450505\pi$
$318$	0	0
$319$	− 27.0192i	− 1.51279i
$320$	0	0
$321$	4.96362i	0.277042i
$322$	0	0
$323$	1.63899i	0.0911959i
$324$	0	0
$325$	− 1.97908i	− 0.109779i
$326$	0	0
$327$	−12.1138	−0.669897
$328$	0	0
$329$	0	0
$330$	0	0
$331$	15.1486i	0.832643i	0.909218	+	0.416321i	$0.136681\pi$
$331$	15.1486i	0.832643i	−0.909218	+	0.416321i	$0.863319\pi$
$332$	0	0
$333$	6.34315	0.347602
$334$	0	0
$335$	−8.56578	−0.467999
$336$	0	0
$337$	10.8284	0.589862	0.294931	−	0.955519i	$-0.404703\pi$
$337$	10.8284	0.589862	0.294931	+	0.955519i	$0.404703\pi$
$338$	0	0
$339$	9.08538	0.493450
$340$	0	0
$341$	− 23.9665i	− 1.29786i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	12.9706	0.698312
$346$	0	0
$347$	− 3.95687i	− 0.212416i	−0.994344	−	0.106208i	$-0.966129\pi$
$347$	− 3.95687i	− 0.212416i	0.994344	−	0.106208i	$-0.0338710\pi$
$348$	0	0
$349$	− 8.47343i	− 0.453572i	−0.973945	−	0.226786i	$-0.927178\pi$
$349$	− 8.47343i	− 0.453572i	0.973945	−	0.226786i	$-0.0728218\pi$
$350$	0	0
$351$	3.27798i	0.174966i
$352$	0	0
$353$	10.0586i	0.535363i	0.963507	+	0.267681i	$0.0862575\pi$
$353$	10.0586i	0.535363i	−0.963507	+	0.267681i	$0.913743\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.2573i	1.64970i	0.565353	+	0.824849i	$0.308740\pi$
$359$	31.2573i	1.64970i	−0.565353	+	0.824849i	$0.691260\pi$
$360$	0	0
$361$	7.72792	0.406733
$362$	0	0
$363$	9.97240	0.523415
$364$	0	0
$365$	−36.6274	−1.91717
$366$	0	0
$367$	−12.8487	−0.670695	−0.335348	−	0.942094i	$-0.608854\pi$
$367$	−12.8487	−0.670695	−0.335348	+	0.942094i	$0.608854\pi$
$368$	0	0
$369$	3.64113i	0.189549i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	20.1421	1.04292	0.521460	−	0.853276i	$-0.325388\pi$
$373$	20.1421	1.04292	0.521460	+	0.853276i	$0.325388\pi$
$374$	0	0
$375$	− 17.7477i	− 0.916487i
$376$	0	0
$377$	7.39104i	0.380658i
$378$	0	0
$379$	− 21.7046i	− 1.11489i	−0.830214	−	0.557444i	$-0.811782\pi$
$379$	− 21.7046i	− 1.11489i	0.830214	−	0.557444i	$-0.188218\pi$
$380$	0	0
$381$	− 21.9105i	− 1.12251i
$382$	0	0
$383$	−31.0194	−1.58502	−0.792509	−	0.609860i	$-0.791226\pi$
$383$	−31.0194	−1.58502	−0.792509	+	0.609860i	$0.791226\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 11.4729i	− 0.583202i
$388$	0	0
$389$	−14.1421	−0.717035	−0.358517	−	0.933523i	$-0.616718\pi$
$389$	−14.1421	−0.717035	−0.358517	+	0.933523i	$0.616718\pi$
$390$	0	0
$391$	0.734828	0.0371618
$392$	0	0
$393$	−0.786797	−0.0396886
$394$	0	0
$395$	20.6796	1.04050
$396$	0	0
$397$	16.1271i	0.809396i	0.914450	+	0.404698i	$0.132623\pi$
$397$	16.1271i	0.809396i	−0.914450	+	0.404698i	$0.867377\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.8284	−0.640621	−0.320311	−	0.947313i	$-0.603787\pi$
$401$	−12.8284	−0.640621	−0.320311	+	0.947313i	$0.603787\pi$
$402$	0	0
$403$	6.55596i	0.326576i
$404$	0	0
$405$	− 29.3784i	− 1.45983i
$406$	0	0
$407$	− 15.8275i	− 0.784540i
$408$	0	0
$409$	− 13.8310i	− 0.683899i	−0.939718	−	0.341949i	$-0.888913\pi$
$409$	− 13.8310i	− 0.683899i	0.939718	−	0.341949i	$-0.111087\pi$
$410$	0	0
$411$	17.6512	0.870668
$412$	0	0
$413$	0	0
$414$	0	0
$415$	24.7013i	1.21254i
$416$	0	0
$417$	48.8701	2.39318
$418$	0	0
$419$	−17.2837	−0.844366	−0.422183	−	0.906511i	$-0.638736\pi$
$419$	−17.2837	−0.844366	−0.422183	+	0.906511i	$0.638736\pi$
$420$	0	0
$421$	10.4853	0.511021	0.255511	−	0.966806i	$-0.417756\pi$
$421$	10.4853	0.511021	0.255511	+	0.966806i	$0.417756\pi$
$422$	0	0
$423$	−6.79175	−0.330226
$424$	0	0
$425$	0.579658i	0.0281175i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−9.17157	−0.442808
$430$	0	0
$431$	0.960099i	0.0462463i	0.999733	+	0.0231232i	$0.00736099\pi$
$431$	0.960099i	0.0462463i	−0.999733	+	0.0231232i	$0.992639\pi$
$432$	0	0
$433$	26.1857i	1.25840i	0.777243	+	0.629201i	$0.216618\pi$
$433$	26.1857i	1.25840i	−0.777243	+	0.629201i	$0.783382\pi$
$434$	0	0
$435$	− 38.2110i	− 1.83208i
$436$	0	0
$437$	− 11.9832i	− 0.573236i
$438$	0	0
$439$	22.4537	1.07165	0.535827	−	0.844328i	$-0.320000\pi$
$439$	22.4537	1.07165	0.535827	+	0.844328i	$0.320000\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	28.3770i	1.34823i	0.738625	+	0.674116i	$0.235475\pi$
$443$	28.3770i	1.34823i	−0.738625	+	0.674116i	$0.764525\pi$
$444$	0	0
$445$	15.6569	0.742206
$446$	0	0
$447$	10.3398	0.489056
$448$	0	0
$449$	−16.2843	−0.768502	−0.384251	−	0.923229i	$-0.625540\pi$
$449$	−16.2843	−0.768502	−0.384251	+	0.923229i	$0.625540\pi$
$450$	0	0
$451$	9.08538	0.427814
$452$	0	0
$453$	− 45.8770i	− 2.15549i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.727922	0.0340508	0.0170254	−	0.999855i	$-0.494580\pi$
$457$	0.727922	0.0340508	0.0170254	+	0.999855i	$0.494580\pi$
$458$	0	0
$459$	− 0.960099i	− 0.0448136i
$460$	0	0
$461$	29.7499i	1.38559i	0.721135	+	0.692794i	$0.243621\pi$
$461$	29.7499i	1.38559i	−0.721135	+	0.692794i	$0.756379\pi$
$462$	0	0
$463$	− 34.9330i	− 1.62347i	−0.584024	−	0.811737i	$-0.698523\pi$
$463$	− 34.9330i	− 1.62347i	0.584024	−	0.811737i	$-0.301477\pi$
$464$	0	0
$465$	− 33.8937i	− 1.57178i
$466$	0	0
$467$	−24.0755	−1.11408	−0.557041	−	0.830485i	$-0.688063\pi$
$467$	−24.0755	−1.11408	−0.557041	+	0.830485i	$0.688063\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 41.0913i	− 1.89339i
$472$	0	0
$473$	−28.6274	−1.31629
$474$	0	0
$475$	9.45280	0.433724
$476$	0	0
$477$	−16.6274	−0.761317
$478$	0	0
$479$	32.7935	1.49837	0.749186	−	0.662360i	$-0.230445\pi$
$479$	32.7935	1.49837	0.749186	+	0.662360i	$0.230445\pi$
$480$	0	0
$481$	4.32957i	0.197411i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.1421	1.09624
$486$	0	0
$487$	− 19.5032i	− 0.883773i	−0.897071	−	0.441886i	$-0.854309\pi$
$487$	− 19.5032i	− 0.883773i	0.897071	−	0.441886i	$-0.145691\pi$
$488$	0	0
$489$	− 32.4399i	− 1.46698i
$490$	0	0
$491$	− 14.4697i	− 0.653009i	−0.945196	−	0.326504i	$-0.894129\pi$
$491$	− 14.4697i	− 0.653009i	0.945196	−	0.326504i	$-0.105871\pi$
$492$	0	0
$493$	− 2.16478i	− 0.0974970i
$494$	0	0
$495$	16.3967	0.736978
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 12.5495i	− 0.561793i	−0.959738	−	0.280897i	$-0.909368\pi$
$499$	− 12.5495i	− 0.561793i	0.959738	−	0.280897i	$-0.0906318\pi$
$500$	0	0
$501$	16.7696	0.749208
$502$	0	0
$503$	7.83095	0.349165	0.174582	−	0.984643i	$-0.444142\pi$
$503$	7.83095	0.349165	0.174582	+	0.984643i	$0.444142\pi$
$504$	0	0
$505$	−39.7990	−1.77103
$506$	0	0
$507$	−25.3299	−1.12494
$508$	0	0
$509$	6.57128i	0.291267i	0.989339	+	0.145633i	$0.0465220\pi$
$509$	6.57128i	0.291267i	−0.989339	+	0.145633i	$0.953478\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−15.6569	−0.691267
$514$	0	0
$515$	− 31.6550i	− 1.39489i
$516$	0	0
$517$	16.9469i	0.745322i
$518$	0	0
$519$	− 14.0720i	− 0.617693i
$520$	0	0
$521$	12.0376i	0.527378i	0.964608	+	0.263689i	$0.0849393\pi$
$521$	12.0376i	0.527378i	−0.964608	+	0.263689i	$0.915061\pi$
$522$	0	0
$523$	−7.46354	−0.326358	−0.163179	−	0.986597i	$-0.552175\pi$
$523$	−7.46354	−0.326358	−0.163179	+	0.986597i	$0.552175\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.92020i	− 0.0836451i
$528$	0	0
$529$	17.6274	0.766409
$530$	0	0
$531$	17.8033	0.772600
$532$	0	0
$533$	−2.48528	−0.107649
$534$	0	0
$535$	6.05692	0.261864
$536$	0	0
$537$	− 28.9301i	− 1.24843i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−45.3137	−1.94819	−0.974094	−	0.226142i	$-0.927389\pi$
$541$	−45.3137	−1.94819	−0.974094	+	0.226142i	$0.927389\pi$
$542$	0	0
$543$	47.0848i	2.02060i
$544$	0	0
$545$	14.7821i	0.633194i
$546$	0	0
$547$	27.6981i	1.18429i	0.805833	+	0.592143i	$0.201718\pi$
$547$	27.6981i	1.18429i	−0.805833	+	0.592143i	$0.798282\pi$
$548$	0	0
$549$	8.58221i	0.366280i
$550$	0	0
$551$	−35.3023	−1.50393
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 22.3835i	− 0.950125i
$556$	0	0
$557$	19.6569	0.832888	0.416444	−	0.909161i	$-0.363276\pi$
$557$	19.6569	0.832888	0.416444	+	0.909161i	$0.363276\pi$
$558$	0	0
$559$	7.83095	0.331214
$560$	0	0
$561$	2.68629	0.113415
$562$	0	0
$563$	−21.0470	−0.887027	−0.443513	−	0.896268i	$-0.646268\pi$
$563$	−21.0470	−0.887027	−0.443513	+	0.896268i	$0.646268\pi$
$564$	0	0
$565$	− 11.0866i	− 0.466415i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.8284	0.789329	0.394664	−	0.918825i	$-0.370861\pi$
$569$	18.8284	0.789329	0.394664	+	0.918825i	$0.370861\pi$
$570$	0	0
$571$	− 9.95043i	− 0.416412i	−0.978085	−	0.208206i	$-0.933237\pi$
$571$	− 9.95043i	− 0.416412i	0.978085	−	0.208206i	$-0.0667625\pi$
$572$	0	0
$573$	− 14.0392i	− 0.586497i
$574$	0	0
$575$	− 4.23808i	− 0.176740i
$576$	0	0
$577$	0.842290i	0.0350650i	0.999846	+	0.0175325i	$0.00558105\pi$
$577$	0.842290i	0.0350650i	−0.999846	+	0.0175325i	$0.994419\pi$
$578$	0	0
$579$	−31.2347	−1.29807
$580$	0	0
$581$	0	0
$582$	0	0
$583$	41.4889i	1.71830i
$584$	0	0
$585$	−4.48528	−0.185444
$586$	0	0
$587$	22.8211	0.941926	0.470963	−	0.882153i	$-0.343906\pi$
$587$	22.8211	0.941926	0.470963	+	0.882153i	$0.343906\pi$
$588$	0	0
$589$	−31.3137	−1.29026
$590$	0	0
$591$	4.28289	0.176175
$592$	0	0
$593$	− 37.4579i	− 1.53821i	−0.639121	−	0.769106i	$-0.720702\pi$
$593$	− 37.4579i	− 1.53821i	0.639121	−	0.769106i	$-0.279298\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−36.6863	−1.50147
$598$	0	0
$599$	− 32.2174i	− 1.31637i	−0.752857	−	0.658184i	$-0.771325\pi$
$599$	− 32.2174i	− 1.31637i	0.752857	−	0.658184i	$-0.228675\pi$
$600$	0	0
$601$	− 25.5516i	− 1.04227i	−0.853474	−	0.521136i	$-0.825509\pi$
$601$	− 25.5516i	− 1.04227i	0.853474	−	0.521136i	$-0.174491\pi$
$602$	0	0
$603$	5.19818i	0.211686i
$604$	0	0
$605$	− 12.1689i	− 0.494738i
$606$	0	0
$607$	−12.1138	−0.491686	−0.245843	−	0.969310i	$-0.579065\pi$
$607$	−12.1138	−0.491686	−0.245843	+	0.969310i	$0.579065\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 4.63577i	− 0.187543i
$612$	0	0
$613$	8.68629	0.350836	0.175418	−	0.984494i	$-0.443872\pi$
$613$	8.68629	0.350836	0.175418	+	0.984494i	$0.443872\pi$
$614$	0	0
$615$	12.8487	0.518108
$616$	0	0
$617$	−35.4558	−1.42740	−0.713699	−	0.700452i	$-0.752982\pi$
$617$	−35.4558	−1.42740	−0.713699	+	0.700452i	$0.752982\pi$
$618$	0	0
$619$	13.5205	0.543433	0.271717	−	0.962377i	$-0.412409\pi$
$619$	13.5205	0.543433	0.271717	+	0.962377i	$0.412409\pi$
$620$	0	0
$621$	7.01962i	0.281688i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−30.7990	−1.23196
$626$	0	0
$627$	− 43.8068i	− 1.74948i
$628$	0	0
$629$	− 1.26810i	− 0.0505625i
$630$	0	0
$631$	− 20.4633i	− 0.814630i	−0.913288	−	0.407315i	$-0.866465\pi$
$631$	− 20.4633i	− 0.814630i	0.913288	−	0.407315i	$-0.133535\pi$
$632$	0	0
$633$	− 30.9861i	− 1.23159i
$634$	0	0
$635$	−26.7365	−1.06101
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	49.4558	1.95339	0.976694	−	0.214636i	$-0.0688564\pi$
$641$	49.4558	1.95339	0.976694	+	0.214636i	$0.0688564\pi$
$642$	0	0
$643$	41.7267	1.64554	0.822769	−	0.568375i	$-0.192428\pi$
$643$	41.7267	1.64554	0.822769	+	0.568375i	$0.192428\pi$
$644$	0	0
$645$	−40.4853	−1.59411
$646$	0	0
$647$	43.8681	1.72463	0.862317	−	0.506370i	$-0.169013\pi$
$647$	43.8681	1.72463	0.862317	+	0.506370i	$0.169013\pi$
$648$	0	0
$649$	− 44.4231i	− 1.74376i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	23.7990	0.931326	0.465663	−	0.884962i	$-0.345816\pi$
$653$	23.7990	0.931326	0.465663	+	0.884962i	$0.345816\pi$
$654$	0	0
$655$	0.960099i	0.0375142i
$656$	0	0
$657$	22.2275i	0.867177i
$658$	0	0
$659$	13.2284i	0.515306i	0.966238	+	0.257653i	$0.0829491\pi$
$659$	13.2284i	0.515306i	−0.966238	+	0.257653i	$0.917051\pi$
$660$	0	0
$661$	− 37.6662i	− 1.46504i	−0.680744	−	0.732522i	$-0.738343\pi$
$661$	− 37.6662i	− 1.46504i	0.680744	−	0.732522i	$-0.261657\pi$
$662$	0	0
$663$	−0.734828	−0.0285383
$664$	0	0
$665$	0	0
$666$	0	0
$667$	15.8275i	0.612843i
$668$	0	0
$669$	16.7696	0.648348
$670$	0	0
$671$	21.4144	0.826696
$672$	0	0
$673$	10.3848	0.400304	0.200152	−	0.979765i	$-0.435856\pi$
$673$	10.3848	0.400304	0.200152	+	0.979765i	$0.435856\pi$
$674$	0	0
$675$	−5.53732	−0.213132
$676$	0	0
$677$	− 28.8532i	− 1.10892i	−0.832211	−	0.554459i	$-0.812925\pi$
$677$	− 28.8532i	− 1.10892i	0.832211	−	0.554459i	$-0.187075\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	21.3553	0.818338
$682$	0	0
$683$	32.2174i	1.23276i	0.787447	+	0.616382i	$0.211402\pi$
$683$	32.2174i	1.23276i	−0.787447	+	0.616382i	$0.788598\pi$
$684$	0	0
$685$	− 21.5391i	− 0.822965i
$686$	0	0
$687$	2.88030i	0.109890i
$688$	0	0
$689$	− 11.3492i	− 0.432370i
$690$	0	0
$691$	8.19837	0.311881	0.155940	−	0.987766i	$-0.450159\pi$
$691$	8.19837	0.311881	0.155940	+	0.987766i	$0.450159\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 59.6343i	− 2.26206i
$696$	0	0
$697$	0.727922	0.0275720
$698$	0	0
$699$	−34.7827	−1.31560
$700$	0	0
$701$	−16.0000	−0.604312	−0.302156	−	0.953259i	$-0.597706\pi$
$701$	−16.0000	−0.604312	−0.302156	+	0.953259i	$0.597706\pi$
$702$	0	0
$703$	−20.6796	−0.779947
$704$	0	0
$705$	23.9665i	0.902630i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	46.6274	1.75113	0.875565	−	0.483101i	$-0.160490\pi$
$709$	46.6274	1.75113	0.875565	+	0.483101i	$0.160490\pi$
$710$	0	0
$711$	− 12.5495i	− 0.470644i
$712$	0	0
$713$	14.0392i	0.525774i
$714$	0	0
$715$	11.1917i	0.418547i
$716$	0	0
$717$	− 21.9105i	− 0.818262i
$718$	0	0
$719$	33.5283	1.25039	0.625197	−	0.780467i	$-0.285019\pi$
$719$	33.5283	1.25039	0.625197	+	0.780467i	$0.285019\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	10.9105i	0.405767i
$724$	0	0
$725$	−12.4853	−0.463692
$726$	0	0
$727$	−29.9802	−1.11191	−0.555953	−	0.831214i	$-0.687646\pi$
$727$	−29.9802	−1.11191	−0.555953	+	0.831214i	$0.687646\pi$
$728$	0	0
$729$	1.62742	0.0602747
$730$	0	0
$731$	−2.29363	−0.0848331
$732$	0	0
$733$	− 20.4567i	− 0.755584i	−0.925890	−	0.377792i	$-0.876683\pi$
$733$	− 20.4567i	− 0.755584i	0.925890	−	0.377792i	$-0.123317\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.9706	0.477777
$738$	0	0
$739$	− 12.4330i	− 0.457357i	−0.973502	−	0.228678i	$-0.926560\pi$
$739$	− 12.4330i	− 0.457357i	0.973502	−	0.228678i	$-0.0734404\pi$
$740$	0	0
$741$	11.9832i	0.440215i
$742$	0	0
$743$	45.1646i	1.65693i	0.560042	+	0.828464i	$0.310785\pi$
$743$	45.1646i	1.65693i	−0.560042	+	0.828464i	$0.689215\pi$
$744$	0	0
$745$	− 12.6173i	− 0.462262i
$746$	0	0
$747$	14.9901	0.548460
$748$	0	0
$749$	0	0
$750$	0	0
$751$	37.2509i	1.35930i	0.733535	+	0.679652i	$0.237869\pi$
$751$	37.2509i	1.35930i	−0.733535	+	0.679652i	$0.762131\pi$
$752$	0	0
$753$	8.38478	0.305558
$754$	0	0
$755$	−55.9819	−2.03739
$756$	0	0
$757$	44.2843	1.60954	0.804770	−	0.593587i	$-0.202289\pi$
$757$	44.2843	1.60954	0.804770	+	0.593587i	$0.202289\pi$
$758$	0	0
$759$	−19.6404	−0.712902
$760$	0	0
$761$	− 0.0543929i	− 0.00197174i	−1.00000		0.000985871i	$-0.999686\pi$
$761$	− 0.0543929i	− 0.00197174i	1.00000		0.000985871i	$-0.000313813\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.31371	0.0474972
$766$	0	0
$767$	12.1518i	0.438777i
$768$	0	0
$769$	− 19.1342i	− 0.689996i	−0.938603	−	0.344998i	$-0.887880\pi$
$769$	− 19.1342i	− 0.689996i	0.938603	−	0.344998i	$-0.112120\pi$
$770$	0	0
$771$	− 40.8100i	− 1.46974i
$772$	0	0
$773$	49.4955i	1.78023i	0.455735	+	0.890116i	$0.349376\pi$
$773$	49.4955i	1.78023i	−0.455735	+	0.890116i	$0.650624\pi$
$774$	0	0
$775$	−11.0746	−0.397813
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 11.8706i	− 0.425309i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	20.6796	0.739029
$784$	0	0
$785$	−50.1421	−1.78965
$786$	0	0
$787$	41.2071	1.46887	0.734436	−	0.678677i	$-0.237447\pi$
$787$	41.2071	1.46887	0.734436	+	0.678677i	$0.237447\pi$
$788$	0	0
$789$	− 54.9526i	− 1.95636i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5.85786	−0.208019
$794$	0	0
$795$	58.6742i	2.08096i
$796$	0	0
$797$	20.1940i	0.715309i	0.933854	+	0.357655i	$0.116424\pi$
$797$	20.1940i	0.715309i	−0.933854	+	0.357655i	$0.883576\pi$
$798$	0	0
$799$	1.35778i	0.0480350i
$800$	0	0
$801$	− 9.50143i	− 0.335716i
$802$	0	0
$803$	55.4623	1.95722
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.7940i	0.379968i
$808$	0	0
$809$	−0.0416306	−0.00146365	−0.000731826	−	1.00000i	$-0.500233\pi$
$809$	−0.0416306	−0.00146365	−0.000731826		1.00000i	$0.500233\pi$
$810$	0	0
$811$	39.4330	1.38468	0.692340	−	0.721571i	$-0.256580\pi$
$811$	39.4330	1.38468	0.692340	+	0.721571i	$0.256580\pi$
$812$	0	0
$813$	9.17157	0.321661
$814$	0	0
$815$	−39.5852	−1.38661
$816$	0	0
$817$	37.4035i	1.30858i
$818$	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
$823$	17.1853i	0.599041i	0.954090	+	0.299521i	$0.0968267\pi$
$823$	17.1853i	0.599041i	−0.954090	+	0.299521i	$0.903173\pi$
$824$	0	0
$825$	− 15.4930i	− 0.539399i
$826$	0	0
$827$	− 45.1646i	− 1.57053i	−0.619162	−	0.785264i	$-0.712527\pi$
$827$	− 45.1646i	− 1.57053i	0.619162	−	0.785264i	$-0.287473\pi$
$828$	0	0
$829$	− 47.2220i	− 1.64009i	−0.572301	−	0.820044i	$-0.693949\pi$
$829$	− 47.2220i	− 1.64009i	0.572301	−	0.820044i	$-0.306051\pi$
$830$	0	0
$831$	−8.87016	−0.307702
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 20.4633i	− 0.708160i
$836$	0	0
$837$	18.3431	0.634032
$838$	0	0
$839$	−42.3984	−1.46376	−0.731878	−	0.681435i	$-0.761356\pi$
$839$	−42.3984	−1.46376	−0.731878	+	0.681435i	$0.761356\pi$
$840$	0	0
$841$	17.6274	0.607842
$842$	0	0
$843$	−22.2384	−0.765932
$844$	0	0
$845$	30.9092i	1.06331i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−7.27208	−0.249577
$850$	0	0
$851$	9.27153i	0.317824i
$852$	0	0
$853$	− 38.3002i	− 1.31137i	−0.755033	−	0.655687i	$-0.772379\pi$
$853$	− 38.3002i	− 1.31137i	0.755033	−	0.655687i	$-0.227621\pi$
$854$	0	0
$855$	− 21.4234i	− 0.732664i
$856$	0	0
$857$	− 3.45542i	− 0.118035i	−0.998257	−	0.0590174i	$-0.981203\pi$
$857$	− 3.45542i	− 0.118035i	0.998257	−	0.0590174i	$-0.0187967\pi$
$858$	0	0
$859$	2.66105	0.0907937	0.0453969	−	0.998969i	$-0.485545\pi$
$859$	2.66105	0.0907937	0.0453969	+	0.998969i	$0.485545\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12.5495i	0.427190i	0.976922	+	0.213595i	$0.0685174\pi$
$863$	12.5495i	0.427190i	−0.976922	+	0.213595i	$0.931483\pi$
$864$	0	0
$865$	−17.1716	−0.583851
$866$	0	0
$867$	−36.1893	−1.22905
$868$	0	0
$869$	−31.3137	−1.06224
$870$	0	0
$871$	−3.54806	−0.120221
$872$	0	0
$873$	− 14.6508i	− 0.495853i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−30.4264	−1.02743	−0.513713	−	0.857962i	$-0.671731\pi$
$877$	−30.4264	−1.02743	−0.513713	+	0.857962i	$0.671731\pi$
$878$	0	0
$879$	− 25.2638i	− 0.852125i
$880$	0	0
$881$	26.1857i	0.882217i	0.897454	+	0.441109i	$0.145415\pi$
$881$	26.1857i	0.882217i	−0.897454	+	0.441109i	$0.854585\pi$
$882$	0	0
$883$	24.7013i	0.831266i	0.909532	+	0.415633i	$0.136440\pi$
$883$	24.7013i	0.831266i	−0.909532	+	0.415633i	$0.863560\pi$
$884$	0	0
$885$	− 62.8238i	− 2.11180i
$886$	0	0
$887$	22.4537	0.753920	0.376960	−	0.926230i	$-0.376969\pi$
$887$	22.4537	0.753920	0.376960	+	0.926230i	$0.376969\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	44.4857i	1.49033i
$892$	0	0
$893$	22.1421	0.740958
$894$	0	0
$895$	−35.3023	−1.18003
$896$	0	0
$897$	5.37258	0.179385
$898$	0	0
$899$	41.3592	1.37941
$900$	0	0
$901$	3.32410i	0.110742i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	57.4558	1.90990
$906$	0	0
$907$	33.9729i	1.12805i	0.825757	+	0.564025i	$0.190748\pi$
$907$	33.9729i	1.12805i	−0.825757	+	0.564025i	$0.809252\pi$
$908$	0	0
$909$	24.1522i	0.801077i
$910$	0	0
$911$	18.7078i	0.619817i	0.950766	+	0.309908i	$0.100298\pi$
$911$	18.7078i	0.619817i	−0.950766	+	0.309908i	$0.899702\pi$
$912$	0	0
$913$	− 37.4035i	− 1.23788i
$914$	0	0
$915$	30.2846	1.00118
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 48.8403i	− 1.61109i	−0.592533	−	0.805546i	$-0.701872\pi$
$919$	− 48.8403i	− 1.61109i	0.592533	−	0.805546i	$-0.298128\pi$
$920$	0	0
$921$	8.38478	0.276288
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−7.31371	−0.240473
$926$	0	0
$927$	−19.2100	−0.630938
$928$	0	0
$929$	− 29.8812i	− 0.980369i	−0.871619	−	0.490185i	$-0.836929\pi$
$929$	− 29.8812i	− 0.980369i	0.871619	−	0.490185i	$-0.163071\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	44.2843	1.44980
$934$	0	0
$935$	− 3.27798i	− 0.107201i
$936$	0	0
$937$	29.8042i	0.973662i	0.873496	+	0.486831i	$0.161847\pi$
$937$	29.8042i	0.973662i	−0.873496	+	0.486831i	$0.838153\pi$
$938$	0	0
$939$	27.3004i	0.890916i
$940$	0	0
$941$	24.7862i	0.808008i	0.914757	+	0.404004i	$0.132382\pi$
$941$	24.7862i	0.808008i	−0.914757	+	0.404004i	$0.867618\pi$
$942$	0	0
$943$	−5.32209	−0.173311
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.15505i	0.297499i	0.988875	+	0.148750i	$0.0475249\pi$
$947$	9.15505i	0.297499i	−0.988875	+	0.148750i	$0.952475\pi$
$948$	0	0
$949$	−15.1716	−0.492490
$950$	0	0
$951$	−11.8095	−0.382948
$952$	0	0
$953$	26.3431	0.853338	0.426669	−	0.904408i	$-0.359687\pi$
$953$	26.3431	0.853338	0.426669	+	0.904408i	$0.359687\pi$
$954$	0	0
$955$	−17.1316	−0.554364
$956$	0	0
$957$	57.8602i	1.87035i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	5.68629	0.183429
$962$	0	0
$963$	− 3.67567i	− 0.118447i
$964$	0	0
$965$	38.1145i	1.22695i
$966$	0	0
$967$	45.1646i	1.45240i	0.687486	+	0.726198i	$0.258714\pi$
$967$	45.1646i	1.45240i	−0.687486	+	0.726198i	$0.741286\pi$
$968$	0	0
$969$	− 3.50981i	− 0.112751i
$970$	0	0
$971$	−13.0009	−0.417217	−0.208609	−	0.977999i	$-0.566894\pi$
$971$	−13.0009	−0.417217	−0.208609	+	0.977999i	$0.566894\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4.23808i	0.135727i
$976$	0	0
$977$	17.6152	0.563561	0.281780	−	0.959479i	$-0.409075\pi$
$977$	17.6152	0.563561	0.281780	+	0.959479i	$0.409075\pi$
$978$	0	0
$979$	−23.7081	−0.757714
$980$	0	0
$981$	8.97056	0.286408
$982$	0	0
$983$	−9.60498	−0.306351	−0.153176	−	0.988199i	$-0.548950\pi$
$983$	−9.60498	−0.306351	−0.153176	+	0.988199i	$0.548950\pi$
$984$	0	0
$985$	− 5.22625i	− 0.166522i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.7696	0.533241
$990$	0	0
$991$	25.6614i	0.815163i	0.913169	+	0.407581i	$0.133628\pi$
$991$	25.6614i	0.815163i	−0.913169	+	0.407581i	$0.866372\pi$
$992$	0	0
$993$	− 32.4399i	− 1.02945i
$994$	0	0
$995$	44.7669i	1.41921i
$996$	0	0
$997$	− 30.9092i	− 0.978903i	−0.872030	−	0.489452i	$-0.837197\pi$
$997$	− 30.9092i	− 0.978903i	0.872030	−	0.489452i	$-0.162803\pi$
$998$	0	0
$999$	12.1138	0.383265

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.2.f.i.3135.4		8
4.3	odd	2	inner	3136.2.f.i.3135.6		8
7.6	odd	2	inner	3136.2.f.i.3135.5		8
8.3	odd	2		196.2.d.c.195.7	yes	8
8.5	even	2		196.2.d.c.195.6	yes	8
24.5	odd	2		1764.2.b.k.1567.4		8
24.11	even	2		1764.2.b.k.1567.2		8
28.27	even	2	inner	3136.2.f.i.3135.3		8
56.3	even	6		196.2.f.d.19.1		16
56.5	odd	6		196.2.f.d.31.2		16
56.11	odd	6		196.2.f.d.19.2		16
56.13	odd	2		196.2.d.c.195.5	✓	8
56.19	even	6		196.2.f.d.31.5		16
56.27	even	2		196.2.d.c.195.8	yes	8
56.37	even	6		196.2.f.d.31.1		16
56.45	odd	6		196.2.f.d.19.6		16
56.51	odd	6		196.2.f.d.31.6		16
56.53	even	6		196.2.f.d.19.5		16
168.83	odd	2		1764.2.b.k.1567.1		8
168.125	even	2		1764.2.b.k.1567.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.d.c.195.5	✓	8	56.13	odd	2
196.2.d.c.195.6	yes	8	8.5	even	2
196.2.d.c.195.7	yes	8	8.3	odd	2
196.2.d.c.195.8	yes	8	56.27	even	2
196.2.f.d.19.1		16	56.3	even	6
196.2.f.d.19.2		16	56.11	odd	6
196.2.f.d.19.5		16	56.53	even	6
196.2.f.d.19.6		16	56.45	odd	6
196.2.f.d.31.1		16	56.37	even	6
196.2.f.d.31.2		16	56.5	odd	6
196.2.f.d.31.5		16	56.19	even	6
196.2.f.d.31.6		16	56.51	odd	6
1764.2.b.k.1567.1		8	168.83	odd	2
1764.2.b.k.1567.2		8	24.11	even	2
1764.2.b.k.1567.3		8	168.125	even	2
1764.2.b.k.1567.4		8	24.5	odd	2
3136.2.f.i.3135.3		8	28.27	even	2	inner
3136.2.f.i.3135.4		8	1.1	even	1	trivial
3136.2.f.i.3135.5		8	7.6	odd	2	inner
3136.2.f.i.3135.6		8	4.3	odd	2	inner

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.14144 q^{3} +2.61313i q^{5} +1.58579 q^{9} +O(q^{10})\)	\(q-2.14144 q^{3} +2.61313i q^{5} +1.58579 q^{9} -3.95687i q^{11} +1.08239i q^{13} -5.59587i q^{15} -0.317025i q^{17} -5.16991 q^{19} +2.31788i q^{23} -1.82843 q^{25} +3.02846 q^{27} +6.82843 q^{29} +6.05692 q^{31} +8.47343i q^{33} +4.00000 q^{37} -2.31788i q^{39} +2.29610i q^{41} -7.23486i q^{43} +4.14386i q^{45} -4.28289 q^{47} +0.678892i q^{51} -10.4853 q^{53} +10.3398 q^{55} +11.0711 q^{57} +11.2268 q^{59} +5.41196i q^{61} -2.82843 q^{65} +3.27798i q^{67} -4.96362i q^{69} +14.0167i q^{73} +3.91548 q^{75} -7.91375i q^{79} -11.2426 q^{81} +9.45280 q^{83} +0.828427 q^{85} -14.6227 q^{87} -5.99162i q^{89} -12.9706 q^{93} -13.5096i q^{95} -9.23880i q^{97} -6.27476i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} + 8 q^{25} + 32 q^{29} + 32 q^{37} - 16 q^{53} + 32 q^{57} - 56 q^{81} - 16 q^{85} + 32 q^{93}+O(q^{100}) \) 8 * q + 24 * q^9 + 8 * q^25 + 32 * q^29 + 32 * q^37 - 16 * q^53 + 32 * q^57 - 56 * q^81 - 16 * q^85 + 32 * q^93

Introduction

L-functions

Modular forms

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