LMFDB - Embedded newform 3136.2.f.i.3135.1

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			3135.1
Root			$-1.36145 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	3136.3135
Dual form			3136.2.f.i.3135.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.72291 q^{3} -1.08239i q^{5} +4.41421 q^{9} +O(q^{10})$	$q-2.72291 q^{3} -1.08239i q^{5} +4.41421 q^{9} -2.08402i q^{11} +2.61313i q^{13} +2.94725i q^{15} -4.46088i q^{17} +1.12786 q^{19} +7.11529i q^{23} +3.82843 q^{25} -3.85077 q^{27} +1.17157 q^{29} -7.70154 q^{31} +5.67459i q^{33} +4.00000 q^{37} -7.11529i q^{39} -5.54328i q^{41} +7.97852i q^{43} -4.77791i q^{45} -5.44581 q^{47} +12.1466i q^{51} +6.48528 q^{53} -2.25573 q^{55} -3.07107 q^{57} -8.82940 q^{59} +13.0656i q^{61} +2.82843 q^{65} -10.0625i q^{67} -19.3743i q^{69} +7.97069i q^{73} -10.4244 q^{75} -4.16804i q^{79} -2.75736 q^{81} +4.31795 q^{83} -4.82843 q^{85} -3.19008 q^{87} +4.01254i q^{89} +20.9706 q^{93} -1.22079i q^{95} -3.82683i q^{97} -9.19932i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.72291	−1.57207	−0.786035	−	0.618182i	$-0.787870\pi$
$3$	−2.72291	−1.57207	−0.786035	+	0.618182i	$0.787870\pi$
$4$	0	0
$5$	− 1.08239i	− 0.484061i	−0.970269	−	0.242030i	$-0.922187\pi$
$5$	− 1.08239i	− 0.484061i	0.970269	−	0.242030i	$-0.0778133\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	4.41421	1.47140
$10$	0	0
$11$	− 2.08402i	− 0.628356i	−0.949364	−	0.314178i	$-0.898271\pi$
$11$	− 2.08402i	− 0.628356i	0.949364	−	0.314178i	$-0.101729\pi$
$12$	0	0
$13$	2.61313i	0.724751i	0.932032	+	0.362375i	$0.118034\pi$
$13$	2.61313i	0.724751i	−0.932032	+	0.362375i	$0.881966\pi$
$14$	0	0
$15$	2.94725i	0.760977i
$16$	0	0
$17$	− 4.46088i	− 1.08192i	−0.841047	−	0.540962i	$-0.818060\pi$
$17$	− 4.46088i	− 1.08192i	0.841047	−	0.540962i	$-0.181940\pi$
$18$	0	0
$19$	1.12786	0.258750	0.129375	−	0.991596i	$-0.458703\pi$
$19$	1.12786	0.258750	0.129375	+	0.991596i	$0.458703\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.11529i	1.48364i	0.670598	+	0.741821i	$0.266037\pi$
$23$	7.11529i	1.48364i	−0.670598	+	0.741821i	$0.733963\pi$
$24$	0	0
$25$	3.82843	0.765685
$26$	0	0
$27$	−3.85077	−0.741081
$28$	0	0
$29$	1.17157	0.217556	0.108778	−	0.994066i	$-0.465306\pi$
$29$	1.17157	0.217556	0.108778	+	0.994066i	$0.465306\pi$
$30$	0	0
$31$	−7.70154	−1.38324	−0.691619	−	0.722263i	$-0.743102\pi$
$31$	−7.70154	−1.38324	−0.691619	+	0.722263i	$0.743102\pi$
$32$	0	0
$33$	5.67459i	0.987820i
$34$	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
$39$	− 7.11529i	− 1.13936i
$40$	0	0
$41$	− 5.54328i	− 0.865714i	−0.901462	−	0.432857i	$-0.857505\pi$
$41$	− 5.54328i	− 0.865714i	0.901462	−	0.432857i	$-0.142495\pi$
$42$	0	0
$43$	7.97852i	1.21671i	0.793664	+	0.608357i	$0.208171\pi$
$43$	7.97852i	1.21671i	−0.793664	+	0.608357i	$0.791829\pi$
$44$	0	0
$45$	− 4.77791i	− 0.712249i
$46$	0	0
$47$	−5.44581	−0.794353	−0.397177	−	0.917742i	$-0.630010\pi$
$47$	−5.44581	−0.794353	−0.397177	+	0.917742i	$0.630010\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	12.1466i	1.70086i
$52$	0	0
$53$	6.48528	0.890822	0.445411	−	0.895326i	$-0.353058\pi$
$53$	6.48528	0.890822	0.445411	+	0.895326i	$0.353058\pi$
$54$	0	0
$55$	−2.25573	−0.304162
$56$	0	0
$57$	−3.07107	−0.406773
$58$	0	0
$59$	−8.82940	−1.14949	−0.574745	−	0.818332i	$-0.694899\pi$
$59$	−8.82940	−1.14949	−0.574745	+	0.818332i	$0.694899\pi$
$60$	0	0
$61$	13.0656i	1.67288i	0.548057	+	0.836441i	$0.315368\pi$
$61$	13.0656i	1.67288i	−0.548057	+	0.836441i	$0.684632\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.82843	0.350823
$66$	0	0
$67$	− 10.0625i	− 1.22934i	−0.788786	−	0.614668i	$-0.789290\pi$
$67$	− 10.0625i	− 1.22934i	0.788786	−	0.614668i	$-0.210710\pi$
$68$	0	0
$69$	− 19.3743i	− 2.33239i
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	7.97069i	0.932899i	0.884548	+	0.466450i	$0.154467\pi$
$73$	7.97069i	0.932899i	−0.884548	+	0.466450i	$0.845533\pi$
$74$	0	0
$75$	−10.4244	−1.20371
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 4.16804i	− 0.468941i	−0.972123	−	0.234471i	$-0.924664\pi$
$79$	− 4.16804i	− 0.468941i	0.972123	−	0.234471i	$-0.0753357\pi$
$80$	0	0
$81$	−2.75736	−0.306373
$82$	0	0
$83$	4.31795	0.473956	0.236978	−	0.971515i	$-0.423843\pi$
$83$	4.31795	0.473956	0.236978	+	0.971515i	$0.423843\pi$
$84$	0	0
$85$	−4.82843	−0.523716
$86$	0	0
$87$	−3.19008	−0.342013
$88$	0	0
$89$	4.01254i	0.425329i	0.977125	+	0.212664i	$0.0682141\pi$
$89$	4.01254i	0.425329i	−0.977125	+	0.212664i	$0.931786\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	20.9706	2.17455
$94$	0	0
$95$	− 1.22079i	− 0.125251i
$96$	0	0
$97$	− 3.82683i	− 0.388556i	−0.980946	−	0.194278i	$-0.937764\pi$
$97$	− 3.82683i	− 0.388556i	0.980946	−	0.194278i	$-0.0622364\pi$
$98$	0	0
$99$	− 9.19932i	− 0.924566i
$100$	0	0
$101$	− 0.185709i	− 0.0184788i	−0.999957	−	0.00923938i	$-0.997059\pi$
$101$	− 0.185709i	− 0.0184788i	0.999957	−	0.00923938i	$-0.00294103\pi$
$102$	0	0
$103$	15.4031	1.51771	0.758855	−	0.651259i	$-0.225759\pi$
$103$	15.4031	1.51771	0.758855	+	0.651259i	$0.225759\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.11529i	− 0.687861i	−0.938995	−	0.343931i	$-0.888241\pi$
$107$	− 7.11529i	− 0.687861i	0.938995	−	0.343931i	$-0.111759\pi$
$108$	0	0
$109$	−5.65685	−0.541828	−0.270914	−	0.962604i	$-0.587326\pi$
$109$	−5.65685	−0.541828	−0.270914	+	0.962604i	$0.587326\pi$
$110$	0	0
$111$	−10.8916	−1.03379
$112$	0	0
$113$	4.24264	0.399114	0.199557	−	0.979886i	$-0.436050\pi$
$113$	4.24264	0.399114	0.199557	+	0.979886i	$0.436050\pi$
$114$	0	0
$115$	7.70154	0.718172
$116$	0	0
$117$	11.5349i	1.06640i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.65685	0.605169
$122$	0	0
$123$	15.0938i	1.36096i
$124$	0	0
$125$	− 9.55582i	− 0.854699i
$126$	0	0
$127$	11.2833i	1.00123i	0.865669	+	0.500617i	$0.166894\pi$
$127$	11.2833i	1.00123i	−0.865669	+	0.500617i	$0.833106\pi$
$128$	0	0
$129$	− 21.7248i	− 1.91276i
$130$	0	0
$131$	15.8703	1.38659	0.693295	−	0.720654i	$-0.256158\pi$
$131$	15.8703	1.38659	0.693295	+	0.720654i	$0.256158\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.16804i	0.358728i
$136$	0	0
$137$	0.242641	0.0207302	0.0103651	−	0.999946i	$-0.496701\pi$
$137$	0.242641	0.0207302	0.0103651	+	0.999946i	$0.496701\pi$
$138$	0	0
$139$	1.78855	0.151703	0.0758515	−	0.997119i	$-0.475833\pi$
$139$	1.78855	0.151703	0.0758515	+	0.997119i	$0.475833\pi$
$140$	0	0
$141$	14.8284	1.24878
$142$	0	0
$143$	5.44581	0.455402
$144$	0	0
$145$	− 1.26810i	− 0.105310i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.828427	0.0678674	0.0339337	−	0.999424i	$-0.489196\pi$
$149$	0.828427	0.0678674	0.0339337	+	0.999424i	$0.489196\pi$
$150$	0	0
$151$	5.38883i	0.438537i	0.975665	+	0.219269i	$0.0703671\pi$
$151$	5.38883i	0.438537i	−0.975665	+	0.219269i	$0.929633\pi$
$152$	0	0
$153$	− 19.6913i	− 1.59195i
$154$	0	0
$155$	8.33609i	0.669571i
$156$	0	0
$157$	− 20.1940i	− 1.61166i	−0.592147	−	0.805830i	$-0.701720\pi$
$157$	− 20.1940i	− 1.61166i	0.592147	−	0.805830i	$-0.298280\pi$
$158$	0	0
$159$	−17.6588	−1.40043
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 3.81048i	− 0.298460i	−0.988803	−	0.149230i	$-0.952321\pi$
$163$	− 3.81048i	− 0.298460i	0.988803	−	0.149230i	$-0.0476795\pi$
$164$	0	0
$165$	6.14214	0.478165
$166$	0	0
$167$	20.8489	1.61334	0.806668	−	0.591005i	$-0.201269\pi$
$167$	20.8489	1.61334	0.806668	+	0.591005i	$0.201269\pi$
$168$	0	0
$169$	6.17157	0.474736
$170$	0	0
$171$	4.97863	0.380726
$172$	0	0
$173$	− 21.0907i	− 1.60350i	−0.597661	−	0.801749i	$-0.703903\pi$
$173$	− 21.0907i	− 1.60350i	0.597661	−	0.801749i	$-0.296097\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	24.0416	1.80708
$178$	0	0
$179$	1.22079i	0.0912462i	0.998959	+	0.0456231i	$0.0145273\pi$
$179$	1.22079i	0.0912462i	−0.998959	+	0.0456231i	$0.985473\pi$
$180$	0	0
$181$	6.04601i	0.449397i	0.974428	+	0.224698i	$0.0721396\pi$
$181$	6.04601i	0.449397i	−0.974428	+	0.224698i	$0.927860\pi$
$182$	0	0
$183$	− 35.5765i	− 2.62989i
$184$	0	0
$185$	− 4.32957i	− 0.318316i
$186$	0	0
$187$	−9.29658	−0.679833
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 20.1251i	− 1.45620i	−0.685471	−	0.728100i	$-0.740404\pi$
$191$	− 20.1251i	− 1.45620i	0.685471	−	0.728100i	$-0.259596\pi$
$192$	0	0
$193$	17.4142	1.25350	0.626751	−	0.779219i	$-0.284384\pi$
$193$	17.4142	1.25350	0.626751	+	0.779219i	$0.284384\pi$
$194$	0	0
$195$	−7.70154	−0.551519
$196$	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
$199$	21.7832	1.54417	0.772087	−	0.635517i	$-0.219213\pi$
$199$	21.7832	1.54417	0.772087	+	0.635517i	$0.219213\pi$
$200$	0	0
$201$	27.3994i	1.93260i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	31.4084i	2.18304i
$208$	0	0
$209$	− 2.35049i	− 0.162587i
$210$	0	0
$211$	− 15.9570i	− 1.09853i	−0.835649	−	0.549264i	$-0.814908\pi$
$211$	− 15.9570i	− 1.09853i	0.835649	−	0.549264i	$-0.185092\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.63589	0.588963
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 21.7034i	− 1.46658i
$220$	0	0
$221$	11.6569	0.784125
$222$	0	0
$223$	20.8489	1.39614	0.698072	−	0.716027i	$-0.254041\pi$
$223$	20.8489	1.39614	0.698072	+	0.716027i	$0.254041\pi$
$224$	0	0
$225$	16.8995	1.12663
$226$	0	0
$227$	18.1260	1.20306	0.601532	−	0.798849i	$-0.294557\pi$
$227$	18.1260	1.20306	0.601532	+	0.798849i	$0.294557\pi$
$228$	0	0
$229$	18.9259i	1.25066i	0.780360	+	0.625330i	$0.215036\pi$
$229$	18.9259i	1.25066i	−0.780360	+	0.625330i	$0.784964\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.75736	0.508202	0.254101	−	0.967178i	$-0.418221\pi$
$233$	7.75736	0.508202	0.254101	+	0.967178i	$0.418221\pi$
$234$	0	0
$235$	5.89450i	0.384515i
$236$	0	0
$237$	11.3492i	0.737209i
$238$	0	0
$239$	11.2833i	0.729858i	0.931035	+	0.364929i	$0.118907\pi$
$239$	11.2833i	0.729858i	−0.931035	+	0.364929i	$0.881093\pi$
$240$	0	0
$241$	− 8.60474i	− 0.554280i	−0.960830	−	0.277140i	$-0.910613\pi$
$241$	− 8.60474i	− 0.554280i	0.960830	−	0.277140i	$-0.0893866\pi$
$242$	0	0
$243$	19.0603	1.22272
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.94725i	0.187529i
$248$	0	0
$249$	−11.7574	−0.745092
$250$	0	0
$251$	10.4244	0.657985	0.328993	−	0.944333i	$-0.393291\pi$
$251$	10.4244	0.657985	0.328993	+	0.944333i	$0.393291\pi$
$252$	0	0
$253$	14.8284	0.932255
$254$	0	0
$255$	13.1474	0.823319
$256$	0	0
$257$	− 9.42450i	− 0.587884i	−0.955823	−	0.293942i	$-0.905033\pi$
$257$	− 9.42450i	− 0.587884i	0.955823	−	0.293942i	$-0.0949673\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5.17157	0.320112
$262$	0	0
$263$	− 21.8516i	− 1.34742i	−0.738994	−	0.673712i	$-0.764699\pi$
$263$	− 21.8516i	− 1.34742i	0.738994	−	0.673712i	$-0.235301\pi$
$264$	0	0
$265$	− 7.01962i	− 0.431212i
$266$	0	0
$267$	− 10.9258i	− 0.668647i
$268$	0	0
$269$	17.3952i	1.06060i	0.847809	+	0.530302i	$0.177921\pi$
$269$	17.3952i	1.06060i	−0.847809	+	0.530302i	$0.822079\pi$
$270$	0	0
$271$	−5.44581	−0.330809	−0.165405	−	0.986226i	$-0.552893\pi$
$271$	−5.44581	−0.330809	−0.165405	+	0.986226i	$0.552893\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 7.97852i	− 0.481123i
$276$	0	0
$277$	−24.1421	−1.45056	−0.725280	−	0.688454i	$-0.758290\pi$
$277$	−24.1421	−1.45056	−0.725280	+	0.688454i	$0.758290\pi$
$278$	0	0
$279$	−33.9962	−2.03530
$280$	0	0
$281$	−26.3848	−1.57398	−0.786992	−	0.616963i	$-0.788363\pi$
$281$	−26.3848	−1.57398	−0.786992	+	0.616963i	$0.788363\pi$
$282$	0	0
$283$	12.0195	0.714484	0.357242	−	0.934012i	$-0.383717\pi$
$283$	12.0195	0.714484	0.357242	+	0.934012i	$0.383717\pi$
$284$	0	0
$285$	3.32410i	0.196903i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2.89949	−0.170559
$290$	0	0
$291$	10.4201i	0.610838i
$292$	0	0
$293$	− 23.2555i	− 1.35860i	−0.733860	−	0.679300i	$-0.762283\pi$
$293$	− 23.2555i	− 1.35860i	0.733860	−	0.679300i	$-0.237717\pi$
$294$	0	0
$295$	9.55688i	0.556423i
$296$	0	0
$297$	8.02509i	0.465663i
$298$	0	0
$299$	−18.5932	−1.07527
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0.505668i	0.0290499i
$304$	0	0
$305$	14.1421	0.809776
$306$	0	0
$307$	10.4244	0.594954	0.297477	−	0.954729i	$-0.403855\pi$
$307$	10.4244	0.594954	0.297477	+	0.954729i	$0.403855\pi$
$308$	0	0
$309$	−41.9411	−2.38595
$310$	0	0
$311$	4.51146	0.255821	0.127911	−	0.991786i	$-0.459173\pi$
$311$	4.51146	0.255821	0.127911	+	0.991786i	$0.459173\pi$
$312$	0	0
$313$	9.87285i	0.558046i	0.960284	+	0.279023i	$0.0900106\pi$
$313$	9.87285i	0.558046i	−0.960284	+	0.279023i	$0.909989\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.4853	1.26290	0.631450	−	0.775417i	$-0.282460\pi$
$317$	22.4853	1.26290	0.631450	+	0.775417i	$0.282460\pi$
$318$	0	0
$319$	− 2.44158i	− 0.136702i
$320$	0	0
$321$	19.3743i	1.08137i
$322$	0	0
$323$	− 5.03127i	− 0.279948i
$324$	0	0
$325$	10.0042i	0.554931i
$326$	0	0
$327$	15.4031	0.851792
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 3.81048i	− 0.209443i	−0.994502	−	0.104722i	$-0.966605\pi$
$331$	− 3.81048i	− 0.209443i	0.994502	−	0.104722i	$-0.0333951\pi$
$332$	0	0
$333$	17.6569	0.967590
$334$	0	0
$335$	−10.8916	−0.595073
$336$	0	0
$337$	5.17157	0.281714	0.140857	−	0.990030i	$-0.455014\pi$
$337$	5.17157	0.281714	0.140857	+	0.990030i	$0.455014\pi$
$338$	0	0
$339$	−11.5523	−0.627435
$340$	0	0
$341$	16.0502i	0.869166i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−20.9706	−1.12902
$346$	0	0
$347$	− 2.08402i	− 0.111876i	−0.998434	−	0.0559381i	$-0.982185\pi$
$347$	− 2.08402i	− 0.111876i	0.998434	−	0.0559381i	$-0.0178149\pi$
$348$	0	0
$349$	− 5.67459i	− 0.303754i	−0.988399	−	0.151877i	$-0.951468\pi$
$349$	− 5.67459i	− 0.303754i	0.988399	−	0.151877i	$-0.0485318\pi$
$350$	0	0
$351$	− 10.0625i	− 0.537099i
$352$	0	0
$353$	27.9790i	1.48917i	0.667526	+	0.744586i	$0.267353\pi$
$353$	27.9790i	1.48917i	−0.667526	+	0.744586i	$0.732647\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 24.7988i	− 1.30883i	−0.756135	−	0.654415i	$-0.772915\pi$
$359$	− 24.7988i	− 1.30883i	0.756135	−	0.654415i	$-0.227085\pi$
$360$	0	0
$361$	−17.7279	−0.933049
$362$	0	0
$363$	−18.1260	−0.951367
$364$	0	0
$365$	8.62742	0.451580
$366$	0	0
$367$	−16.3374	−0.852807	−0.426404	−	0.904533i	$-0.640220\pi$
$367$	−16.3374	−0.852807	−0.426404	+	0.904533i	$0.640220\pi$
$368$	0	0
$369$	− 24.4692i	− 1.27382i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−8.14214	−0.421584	−0.210792	−	0.977531i	$-0.567604\pi$
$373$	−8.14214	−0.421584	−0.210792	+	0.977531i	$0.567604\pi$
$374$	0	0
$375$	26.0196i	1.34365i
$376$	0	0
$377$	3.06147i	0.157674i
$378$	0	0
$379$	23.9356i	1.22949i	0.788727	+	0.614744i	$0.210741\pi$
$379$	23.9356i	1.22949i	−0.788727	+	0.614744i	$0.789259\pi$
$380$	0	0
$381$	− 30.7235i	− 1.57401i
$382$	0	0
$383$	6.76719	0.345787	0.172894	−	0.984941i	$-0.444688\pi$
$383$	6.76719	0.345787	0.172894	+	0.984941i	$0.444688\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	35.2189i	1.79028i
$388$	0	0
$389$	14.1421	0.717035	0.358517	−	0.933523i	$-0.383282\pi$
$389$	14.1421	0.717035	0.358517	+	0.933523i	$0.383282\pi$
$390$	0	0
$391$	31.7405	1.60519
$392$	0	0
$393$	−43.2132	−2.17982
$394$	0	0
$395$	−4.51146	−0.226996
$396$	0	0
$397$	− 12.8030i	− 0.642564i	−0.946984	−	0.321282i	$-0.895886\pi$
$397$	− 12.8030i	− 0.642564i	0.946984	−	0.321282i	$-0.104114\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7.17157	−0.358131	−0.179066	−	0.983837i	$-0.557307\pi$
$401$	−7.17157	−0.358131	−0.179066	+	0.983837i	$0.557307\pi$
$402$	0	0
$403$	− 20.1251i	− 1.00250i
$404$	0	0
$405$	2.98454i	0.148303i
$406$	0	0
$407$	− 8.33609i	− 0.413204i
$408$	0	0
$409$	7.25972i	0.358970i	0.983761	+	0.179485i	$0.0574431\pi$
$409$	7.25972i	0.358970i	−0.983761	+	0.179485i	$0.942557\pi$
$410$	0	0
$411$	−0.660688	−0.0325893
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 4.67371i	− 0.229423i
$416$	0	0
$417$	−4.87006	−0.238488
$418$	0	0
$419$	16.5309	0.807589	0.403795	−	0.914850i	$-0.367691\pi$
$419$	16.5309	0.807589	0.403795	+	0.914850i	$0.367691\pi$
$420$	0	0
$421$	−6.48528	−0.316073	−0.158037	−	0.987433i	$-0.550516\pi$
$421$	−6.48528	−0.316073	−0.158037	+	0.987433i	$0.550516\pi$
$422$	0	0
$423$	−24.0390	−1.16881
$424$	0	0
$425$	− 17.0782i	− 0.828413i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−14.8284	−0.715923
$430$	0	0
$431$	− 17.1778i	− 0.827427i	−0.910407	−	0.413714i	$-0.864231\pi$
$431$	− 17.1778i	− 0.827427i	0.910407	−	0.413714i	$-0.135769\pi$
$432$	0	0
$433$	15.1760i	0.729313i	0.931142	+	0.364657i	$0.118814\pi$
$433$	15.1760i	0.729313i	−0.931142	+	0.364657i	$0.881186\pi$
$434$	0	0
$435$	3.45292i	0.165555i
$436$	0	0
$437$	8.02509i	0.383892i
$438$	0	0
$439$	−17.6588	−0.842809	−0.421404	−	0.906873i	$-0.638463\pi$
$439$	−17.6588	−0.842809	−0.421404	+	0.906873i	$0.638463\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.7347i	1.27020i	0.772428	+	0.635102i	$0.219042\pi$
$443$	26.7347i	1.27020i	−0.772428	+	0.635102i	$0.780958\pi$
$444$	0	0
$445$	4.34315	0.205885
$446$	0	0
$447$	−2.25573	−0.106692
$448$	0	0
$449$	40.2843	1.90113	0.950566	−	0.310522i	$-0.100504\pi$
$449$	40.2843	1.90113	0.950566	+	0.310522i	$0.100504\pi$
$450$	0	0
$451$	−11.5523	−0.543977
$452$	0	0
$453$	− 14.6733i	− 0.689411i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−24.7279	−1.15672	−0.578362	−	0.815780i	$-0.696308\pi$
$457$	−24.7279	−1.15672	−0.578362	+	0.815780i	$0.696308\pi$
$458$	0	0
$459$	17.1778i	0.801793i
$460$	0	0
$461$	27.4763i	1.27970i	0.768501	+	0.639849i	$0.221003\pi$
$461$	27.4763i	1.27970i	−0.768501	+	0.639849i	$0.778997\pi$
$462$	0	0
$463$	− 6.60963i	− 0.307175i	−0.988135	−	0.153588i	$-0.950917\pi$
$463$	− 6.60963i	− 0.307175i	0.988135	−	0.153588i	$-0.0490828\pi$
$464$	0	0
$465$	− 22.6984i	− 1.05261i
$466$	0	0
$467$	−7.50803	−0.347430	−0.173715	−	0.984796i	$-0.555577\pi$
$467$	−7.50803	−0.347430	−0.173715	+	0.984796i	$0.555577\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	54.9864i	2.53364i
$472$	0	0
$473$	16.6274	0.764529
$474$	0	0
$475$	4.31795	0.198121
$476$	0	0
$477$	28.6274	1.31076
$478$	0	0
$479$	−19.9145	−0.909918	−0.454959	−	0.890512i	$-0.650346\pi$
$479$	−19.9145	−0.909918	−0.454959	+	0.890512i	$0.650346\pi$
$480$	0	0
$481$	10.4525i	0.476593i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.14214	−0.188085
$486$	0	0
$487$	− 39.7445i	− 1.80100i	−0.434860	−	0.900498i	$-0.643202\pi$
$487$	− 39.7445i	− 1.80100i	0.434860	−	0.900498i	$-0.356798\pi$
$488$	0	0
$489$	10.3756i	0.469200i
$490$	0	0
$491$	15.9570i	0.720132i	0.932927	+	0.360066i	$0.117246\pi$
$491$	15.9570i	0.720132i	−0.932927	+	0.360066i	$0.882754\pi$
$492$	0	0
$493$	− 5.22625i	− 0.235379i
$494$	0	0
$495$	−9.95727	−0.447546
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 18.3986i	− 0.823636i	−0.911266	−	0.411818i	$-0.864894\pi$
$499$	− 18.3986i	− 0.823636i	0.911266	−	0.411818i	$-0.135106\pi$
$500$	0	0
$501$	−56.7696	−2.53628
$502$	0	0
$503$	−20.8489	−0.929606	−0.464803	−	0.885414i	$-0.653875\pi$
$503$	−20.8489	−0.929606	−0.464803	+	0.885414i	$0.653875\pi$
$504$	0	0
$505$	−0.201010	−0.00894483
$506$	0	0
$507$	−16.8046	−0.746319
$508$	0	0
$509$	− 21.0907i	− 0.934830i	−0.884038	−	0.467415i	$-0.845185\pi$
$509$	− 21.0907i	− 0.934830i	0.884038	−	0.467415i	$-0.154815\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.34315	−0.191755
$514$	0	0
$515$	− 16.6722i	− 0.734664i
$516$	0	0
$517$	11.3492i	0.499137i
$518$	0	0
$519$	57.4280i	2.52081i
$520$	0	0
$521$	17.9749i	0.787493i	0.919219	+	0.393746i	$0.128821\pi$
$521$	17.9749i	0.787493i	−0.919219	+	0.393746i	$0.871179\pi$
$522$	0	0
$523$	36.7191	1.60562	0.802808	−	0.596238i	$-0.203338\pi$
$523$	36.7191	1.60562	0.802808	+	0.596238i	$0.203338\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	34.3557i	1.49656i
$528$	0	0
$529$	−27.6274	−1.20119
$530$	0	0
$531$	−38.9749	−1.69137
$532$	0	0
$533$	14.4853	0.627427
$534$	0	0
$535$	−7.70154	−0.332967
$536$	0	0
$537$	− 3.32410i	− 0.143445i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22.6863	−0.975360	−0.487680	−	0.873023i	$-0.662157\pi$
$541$	−22.6863	−0.975360	−0.487680	+	0.873023i	$0.662157\pi$
$542$	0	0
$543$	− 16.4627i	− 0.706483i
$544$	0	0
$545$	6.12293i	0.262278i
$546$	0	0
$547$	14.5882i	0.623744i	0.950124	+	0.311872i	$0.100956\pi$
$547$	14.5882i	0.623744i	−0.950124	+	0.311872i	$0.899044\pi$
$548$	0	0
$549$	57.6745i	2.46149i
$550$	0	0
$551$	1.32138	0.0562925
$552$	0	0
$553$	0	0
$554$	0	0
$555$	11.7890i	0.500415i
$556$	0	0
$557$	8.34315	0.353510	0.176755	−	0.984255i	$-0.443440\pi$
$557$	8.34315	0.353510	0.176755	+	0.984255i	$0.443440\pi$
$558$	0	0
$559$	−20.8489	−0.881814
$560$	0	0
$561$	25.3137	1.06875
$562$	0	0
$563$	−11.3588	−0.478716	−0.239358	−	0.970931i	$-0.576937\pi$
$563$	−11.3588	−0.478716	−0.239358	+	0.970931i	$0.576937\pi$
$564$	0	0
$565$	− 4.59220i	− 0.193195i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13.1716	0.552181	0.276091	−	0.961132i	$-0.410961\pi$
$569$	13.1716	0.552181	0.276091	+	0.961132i	$0.410961\pi$
$570$	0	0
$571$	− 40.6077i	− 1.69938i	−0.527282	−	0.849691i	$-0.676789\pi$
$571$	− 40.6077i	− 1.69938i	0.527282	−	0.849691i	$-0.323211\pi$
$572$	0	0
$573$	54.7987i	2.28925i
$574$	0	0
$575$	27.2404i	1.13600i
$576$	0	0
$577$	− 38.6172i	− 1.60766i	−0.594862	−	0.803828i	$-0.702793\pi$
$577$	− 38.6172i	− 1.60766i	0.594862	−	0.803828i	$-0.297207\pi$
$578$	0	0
$579$	−47.4173	−1.97059
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 13.5155i	− 0.559753i
$584$	0	0
$585$	12.4853	0.516203
$586$	0	0
$587$	−1.78855	−0.0738214	−0.0369107	−	0.999319i	$-0.511752\pi$
$587$	−1.78855	−0.0738214	−0.0369107	+	0.999319i	$0.511752\pi$
$588$	0	0
$589$	−8.68629	−0.357912
$590$	0	0
$591$	5.44581	0.224011
$592$	0	0
$593$	− 34.9986i	− 1.43722i	−0.695412	−	0.718611i	$-0.744778\pi$
$593$	− 34.9986i	− 1.43722i	0.695412	−	0.718611i	$-0.255222\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−59.3137	−2.42755
$598$	0	0
$599$	41.9766i	1.71512i	0.514385	+	0.857560i	$0.328020\pi$
$599$	41.9766i	1.71512i	−0.514385	+	0.857560i	$0.671980\pi$
$600$	0	0
$601$	− 6.25425i	− 0.255116i	−0.991831	−	0.127558i	$-0.959286\pi$
$601$	− 6.25425i	− 0.255116i	0.991831	−	0.127558i	$-0.0407139\pi$
$602$	0	0
$603$	− 44.4182i	− 1.80885i
$604$	0	0
$605$	− 7.20533i	− 0.292938i
$606$	0	0
$607$	15.4031	0.625192	0.312596	−	0.949886i	$-0.398801\pi$
$607$	15.4031	0.625192	0.312596	+	0.949886i	$0.398801\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 14.2306i	− 0.575708i
$612$	0	0
$613$	31.3137	1.26475	0.632374	−	0.774663i	$-0.282080\pi$
$613$	31.3137	1.26475	0.632374	+	0.774663i	$0.282080\pi$
$614$	0	0
$615$	16.3374	0.658789
$616$	0	0
$617$	15.4558	0.622229	0.311114	−	0.950372i	$-0.399298\pi$
$617$	15.4558	0.622229	0.311114	+	0.950372i	$0.399298\pi$
$618$	0	0
$619$	−44.4207	−1.78542	−0.892709	−	0.450634i	$-0.851198\pi$
$619$	−44.4207	−1.78542	−0.892709	+	0.450634i	$0.851198\pi$
$620$	0	0
$621$	− 27.3994i	− 1.09950i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	8.79899	0.351960
$626$	0	0
$627$	6.40017i	0.255598i
$628$	0	0
$629$	− 17.8435i	− 0.711469i
$630$	0	0
$631$	− 22.5667i	− 0.898365i	−0.893440	−	0.449183i	$-0.851715\pi$
$631$	− 22.5667i	− 0.898365i	0.893440	−	0.449183i	$-0.148285\pi$
$632$	0	0
$633$	43.4495i	1.72696i
$634$	0	0
$635$	12.2130	0.484658
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.45584	−0.0575024	−0.0287512	−	0.999587i	$-0.509153\pi$
$641$	−1.45584	−0.0575024	−0.0287512	+	0.999587i	$0.509153\pi$
$642$	0	0
$643$	6.84734	0.270033	0.135016	−	0.990843i	$-0.456891\pi$
$643$	6.84734	0.270033	0.135016	+	0.990843i	$0.456891\pi$
$644$	0	0
$645$	−23.5147	−0.925891
$646$	0	0
$647$	9.57025	0.376245	0.188123	−	0.982146i	$-0.439760\pi$
$647$	9.57025	0.376245	0.188123	+	0.982146i	$0.439760\pi$
$648$	0	0
$649$	18.4007i	0.722289i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15.7990	−0.618262	−0.309131	−	0.951019i	$-0.600038\pi$
$653$	−15.7990	−0.618262	−0.309131	+	0.951019i	$0.600038\pi$
$654$	0	0
$655$	− 17.1778i	− 0.671194i
$656$	0	0
$657$	35.1843i	1.37267i
$658$	0	0
$659$	30.5452i	1.18987i	0.803773	+	0.594936i	$0.202823\pi$
$659$	30.5452i	1.18987i	−0.803773	+	0.594936i	$0.797177\pi$
$660$	0	0
$661$	12.5404i	0.487764i	0.969805	+	0.243882i	$0.0784209\pi$
$661$	12.5404i	0.487764i	−0.969805	+	0.243882i	$0.921579\pi$
$662$	0	0
$663$	−31.7405	−1.23270
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.33609i	0.322775i
$668$	0	0
$669$	−56.7696	−2.19484
$670$	0	0
$671$	27.2291	1.05117
$672$	0	0
$673$	−26.3848	−1.01706	−0.508529	−	0.861045i	$-0.669811\pi$
$673$	−26.3848	−1.01706	−0.508529	+	0.861045i	$0.669811\pi$
$674$	0	0
$675$	−14.7424	−0.567435
$676$	0	0
$677$	− 40.0936i	− 1.54092i	−0.637488	−	0.770461i	$-0.720026\pi$
$677$	− 40.0936i	− 1.54092i	0.637488	−	0.770461i	$-0.279974\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−49.3553	−1.89130
$682$	0	0
$683$	− 41.9766i	− 1.60619i	−0.595850	−	0.803096i	$-0.703185\pi$
$683$	− 41.9766i	− 1.60619i	0.595850	−	0.803096i	$-0.296815\pi$
$684$	0	0
$685$	− 0.262632i	− 0.0100347i
$686$	0	0
$687$	− 51.5335i	− 1.96613i
$688$	0	0
$689$	16.9469i	0.645624i
$690$	0	0
$691$	−4.97863	−0.189396	−0.0946981	−	0.995506i	$-0.530189\pi$
$691$	−4.97863	−0.189396	−0.0946981	+	0.995506i	$0.530189\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.93591i	− 0.0734334i
$696$	0	0
$697$	−24.7279	−0.936637
$698$	0	0
$699$	−21.1226	−0.798928
$700$	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
$703$	4.51146	0.170153
$704$	0	0
$705$	− 16.0502i	− 0.604485i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.37258	0.0515484	0.0257742	−	0.999668i	$-0.491795\pi$
$709$	1.37258	0.0515484	0.0257742	+	0.999668i	$0.491795\pi$
$710$	0	0
$711$	− 18.3986i	− 0.690003i
$712$	0	0
$713$	− 54.7987i	− 2.05223i
$714$	0	0
$715$	− 5.89450i	− 0.220442i
$716$	0	0
$717$	− 30.7235i	− 1.14739i
$718$	0	0
$719$	11.8260	0.441034	0.220517	−	0.975383i	$-0.429225\pi$
$719$	11.8260	0.441034	0.220517	+	0.975383i	$0.429225\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	23.4299i	0.871368i
$724$	0	0
$725$	4.48528	0.166579
$726$	0	0
$727$	−38.1207	−1.41382	−0.706909	−	0.707305i	$-0.749911\pi$
$727$	−38.1207	−1.41382	−0.706909	+	0.707305i	$0.749911\pi$
$728$	0	0
$729$	−43.6274	−1.61583
$730$	0	0
$731$	35.5913	1.31639
$732$	0	0
$733$	2.35049i	0.0868175i	0.999057	+	0.0434087i	$0.0138218\pi$
$733$	2.35049i	0.0868175i	−0.999057	+	0.0434087i	$0.986178\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20.9706	−0.772461
$738$	0	0
$739$	52.3968i	1.92745i	0.266905	+	0.963723i	$0.413999\pi$
$739$	52.3968i	1.92745i	−0.266905	+	0.963723i	$0.586001\pi$
$740$	0	0
$741$	− 8.02509i	− 0.294809i
$742$	0	0
$743$	17.8930i	0.656429i	0.944603	+	0.328215i	$0.106447\pi$
$743$	17.8930i	0.656429i	−0.944603	+	0.328215i	$0.893553\pi$
$744$	0	0
$745$	− 0.896683i	− 0.0328519i
$746$	0	0
$747$	19.0603	0.697381
$748$	0	0
$749$	0	0
$750$	0	0
$751$	13.7249i	0.500829i	0.968139	+	0.250415i	$0.0805670\pi$
$751$	13.7249i	0.500829i	−0.968139	+	0.250415i	$0.919433\pi$
$752$	0	0
$753$	−28.3848	−1.03440
$754$	0	0
$755$	5.83283	0.212279
$756$	0	0
$757$	−12.2843	−0.446479	−0.223240	−	0.974764i	$-0.571663\pi$
$757$	−12.2843	−0.446479	−0.223240	+	0.974764i	$0.571663\pi$
$758$	0	0
$759$	−40.3764	−1.46557
$760$	0	0
$761$	− 25.9999i	− 0.942497i	−0.882000	−	0.471249i	$-0.843803\pi$
$761$	− 25.9999i	− 0.942497i	0.882000	−	0.471249i	$-0.156197\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−21.3137	−0.770599
$766$	0	0
$767$	− 23.0723i	− 0.833094i
$768$	0	0
$769$	46.1940i	1.66580i	0.553425	+	0.832899i	$0.313320\pi$
$769$	46.1940i	1.66580i	−0.553425	+	0.832899i	$0.686680\pi$
$770$	0	0
$771$	25.6620i	0.924196i
$772$	0	0
$773$	52.9735i	1.90532i	0.304031	+	0.952662i	$0.401667\pi$
$773$	52.9735i	1.90532i	−0.304031	+	0.952662i	$0.598333\pi$
$774$	0	0
$775$	−29.4848	−1.05912
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 6.25206i	− 0.224003i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−4.51146	−0.161226
$784$	0	0
$785$	−21.8579	−0.780141
$786$	0	0
$787$	29.2913	1.04412	0.522061	−	0.852908i	$-0.325164\pi$
$787$	29.2913	1.04412	0.522061	+	0.852908i	$0.325164\pi$
$788$	0	0
$789$	59.4997i	2.11825i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−34.1421	−1.21242
$794$	0	0
$795$	19.1138i	0.677895i
$796$	0	0
$797$	19.1886i	0.679694i	0.940481	+	0.339847i	$0.110375\pi$
$797$	19.1886i	0.679694i	−0.940481	+	0.339847i	$0.889625\pi$
$798$	0	0
$799$	24.2931i	0.859429i
$800$	0	0
$801$	17.7122i	0.625831i
$802$	0	0
$803$	16.6111	0.586193
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 47.3655i	− 1.66734i
$808$	0	0
$809$	48.0416	1.68905	0.844527	−	0.535513i	$-0.179882\pi$
$809$	48.0416	1.68905	0.844527	+	0.535513i	$0.179882\pi$
$810$	0	0
$811$	42.4386	1.49022	0.745111	−	0.666941i	$-0.232397\pi$
$811$	42.4386	1.49022	0.745111	+	0.666941i	$0.232397\pi$
$812$	0	0
$813$	14.8284	0.520056
$814$	0	0
$815$	−4.12444	−0.144473
$816$	0	0
$817$	8.99869i	0.314824i
$818$	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
$823$	32.6292i	1.13738i	0.822551	+	0.568692i	$0.192550\pi$
$823$	32.6292i	1.13738i	−0.822551	+	0.568692i	$0.807450\pi$
$824$	0	0
$825$	21.7248i	0.756359i
$826$	0	0
$827$	− 17.8930i	− 0.622199i	−0.950377	−	0.311100i	$-0.899303\pi$
$827$	− 17.8930i	− 0.622199i	0.950377	−	0.311100i	$-0.100697\pi$
$828$	0	0
$829$	4.25265i	0.147700i	0.997269	+	0.0738502i	$0.0235287\pi$
$829$	4.25265i	0.147700i	−0.997269	+	0.0738502i	$0.976471\pi$
$830$	0	0
$831$	65.7368	2.28038
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 22.5667i	− 0.780952i
$836$	0	0
$837$	29.6569	1.02509
$838$	0	0
$839$	53.9108	1.86121	0.930603	−	0.366029i	$-0.119283\pi$
$839$	53.9108	1.86121	0.930603	+	0.366029i	$0.119283\pi$
$840$	0	0
$841$	−27.6274	−0.952670
$842$	0	0
$843$	71.8432	2.47441
$844$	0	0
$845$	− 6.68006i	− 0.229801i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−32.7279	−1.12322
$850$	0	0
$851$	28.4612i	0.975637i
$852$	0	0
$853$	3.61859i	0.123898i	0.998079	+	0.0619492i	$0.0197317\pi$
$853$	3.61859i	0.123898i	−0.998079	+	0.0619492i	$0.980268\pi$
$854$	0	0
$855$	− 5.38883i	− 0.184294i
$856$	0	0
$857$	39.6996i	1.35611i	0.735010	+	0.678057i	$0.237178\pi$
$857$	39.6996i	1.35611i	−0.735010	+	0.678057i	$0.762822\pi$
$858$	0	0
$859$	−19.7210	−0.672873	−0.336436	−	0.941706i	$-0.609222\pi$
$859$	−19.7210	−0.672873	−0.336436	+	0.941706i	$0.609222\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.3986i	0.626297i	0.949704	+	0.313148i	$0.101384\pi$
$863$	18.3986i	0.626297i	−0.949704	+	0.313148i	$0.898616\pi$
$864$	0	0
$865$	−22.8284	−0.776190
$866$	0	0
$867$	7.89505	0.268130
$868$	0	0
$869$	−8.68629	−0.294662
$870$	0	0
$871$	26.2947	0.890962
$872$	0	0
$873$	− 16.8925i	− 0.571723i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	54.4264	1.83785	0.918925	−	0.394433i	$-0.129059\pi$
$877$	54.4264	1.83785	0.918925	+	0.394433i	$0.129059\pi$
$878$	0	0
$879$	63.3225i	2.13582i
$880$	0	0
$881$	15.1760i	0.511293i	0.966770	+	0.255647i	$0.0822883\pi$
$881$	15.1760i	0.511293i	−0.966770	+	0.255647i	$0.917712\pi$
$882$	0	0
$883$	− 4.67371i	− 0.157283i	−0.996903	−	0.0786415i	$-0.974942\pi$
$883$	− 4.67371i	− 0.157283i	0.996903	−	0.0786415i	$-0.0250582\pi$
$884$	0	0
$885$	− 26.0225i	− 0.874736i
$886$	0	0
$887$	−17.6588	−0.592925	−0.296462	−	0.955045i	$-0.595807\pi$
$887$	−17.6588	−0.592925	−0.296462	+	0.955045i	$0.595807\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.74640i	0.192512i
$892$	0	0
$893$	−6.14214	−0.205539
$894$	0	0
$895$	1.32138	0.0441687
$896$	0	0
$897$	50.6274	1.69040
$898$	0	0
$899$	−9.02291	−0.300931
$900$	0	0
$901$	− 28.9301i	− 0.963801i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.54416	0.217535
$906$	0	0
$907$	23.7875i	0.789850i	0.918713	+	0.394925i	$0.129229\pi$
$907$	23.7875i	0.789850i	−0.918713	+	0.394925i	$0.870771\pi$
$908$	0	0
$909$	− 0.819760i	− 0.0271897i
$910$	0	0
$911$	− 43.1974i	− 1.43119i	−0.698513	−	0.715597i	$-0.746155\pi$
$911$	− 43.1974i	− 1.43119i	0.698513	−	0.715597i	$-0.253845\pi$
$912$	0	0
$913$	− 8.99869i	− 0.297813i
$914$	0	0
$915$	−38.5077	−1.27303
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 49.3014i	− 1.62630i	−0.582052	−	0.813151i	$-0.697750\pi$
$919$	− 49.3014i	− 1.62630i	0.582052	−	0.813151i	$-0.302250\pi$
$920$	0	0
$921$	−28.3848	−0.935310
$922$	0	0
$923$	0	0
$924$	0	0
$925$	15.3137	0.503512
$926$	0	0
$927$	67.9925	2.23317
$928$	0	0
$929$	− 16.7068i	− 0.548131i	−0.961711	−	0.274065i	$-0.911632\pi$
$929$	− 16.7068i	− 0.548131i	0.961711	−	0.274065i	$-0.0883685\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−12.2843	−0.402169
$934$	0	0
$935$	10.0625i	0.329080i
$936$	0	0
$937$	53.4762i	1.74699i	0.486831	+	0.873496i	$0.338153\pi$
$937$	53.4762i	1.74699i	−0.486831	+	0.873496i	$0.661847\pi$
$938$	0	0
$939$	− 26.8828i	− 0.877288i
$940$	0	0
$941$	8.10201i	0.264118i	0.991242	+	0.132059i	$0.0421588\pi$
$941$	8.10201i	0.264118i	−0.991242	+	0.132059i	$0.957841\pi$
$942$	0	0
$943$	39.4421	1.28441
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 42.3342i	− 1.37568i	−0.725864	−	0.687838i	$-0.758560\pi$
$947$	− 42.3342i	− 1.37568i	0.725864	−	0.687838i	$-0.241440\pi$
$948$	0	0
$949$	−20.8284	−0.676119
$950$	0	0
$951$	−61.2253	−1.98537
$952$	0	0
$953$	37.6569	1.21983	0.609913	−	0.792469i	$-0.291205\pi$
$953$	37.6569	1.21983	0.609913	+	0.792469i	$0.291205\pi$
$954$	0	0
$955$	−21.7832	−0.704889
$956$	0	0
$957$	6.64820i	0.214906i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	28.3137	0.913345
$962$	0	0
$963$	− 31.4084i	− 1.01212i
$964$	0	0
$965$	− 18.8490i	− 0.606771i
$966$	0	0
$967$	17.8930i	0.575399i	0.957721	+	0.287699i	$0.0928904\pi$
$967$	17.8930i	0.575399i	−0.957721	+	0.287699i	$0.907110\pi$
$968$	0	0
$969$	13.6997i	0.440097i
$970$	0	0
$971$	21.9768	0.705268	0.352634	−	0.935761i	$-0.385286\pi$
$971$	21.9768	0.705268	0.352634	+	0.935761i	$0.385286\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 27.2404i	− 0.872391i
$976$	0	0
$977$	54.3848	1.73992	0.869962	−	0.493120i	$-0.164143\pi$
$977$	54.3848	1.73992	0.869962	+	0.493120i	$0.164143\pi$
$978$	0	0
$979$	8.36223	0.267258
$980$	0	0
$981$	−24.9706	−0.797249
$982$	0	0
$983$	33.9962	1.08431	0.542156	−	0.840278i	$-0.317608\pi$
$983$	33.9962	1.08431	0.542156	+	0.840278i	$0.317608\pi$
$984$	0	0
$985$	2.16478i	0.0689758i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−56.7696	−1.80517
$990$	0	0
$991$	− 21.8516i	− 0.694137i	−0.937840	−	0.347069i	$-0.887177\pi$
$991$	− 21.8516i	− 0.694137i	0.937840	−	0.347069i	$-0.112823\pi$
$992$	0	0
$993$	10.3756i	0.329259i
$994$	0	0
$995$	− 23.5780i	− 0.747473i
$996$	0	0
$997$	6.68006i	0.211560i	0.994390	+	0.105780i	$0.0337339\pi$
$997$	6.68006i	0.211560i	−0.994390	+	0.105780i	$0.966266\pi$
$998$	0	0
$999$	−15.4031	−0.487332

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.2.f.i.3135.1		8
4.3	odd	2	inner	3136.2.f.i.3135.7		8
7.6	odd	2	inner	3136.2.f.i.3135.8		8
8.3	odd	2		196.2.d.c.195.1	✓	8
8.5	even	2		196.2.d.c.195.4	yes	8
24.5	odd	2		1764.2.b.k.1567.5		8
24.11	even	2		1764.2.b.k.1567.7		8
28.27	even	2	inner	3136.2.f.i.3135.2		8
56.3	even	6		196.2.f.d.19.7		16
56.5	odd	6		196.2.f.d.31.8		16
56.11	odd	6		196.2.f.d.19.8		16
56.13	odd	2		196.2.d.c.195.3	yes	8
56.19	even	6		196.2.f.d.31.3		16
56.27	even	2		196.2.d.c.195.2	yes	8
56.37	even	6		196.2.f.d.31.7		16
56.45	odd	6		196.2.f.d.19.4		16
56.51	odd	6		196.2.f.d.31.4		16
56.53	even	6		196.2.f.d.19.3		16
168.83	odd	2		1764.2.b.k.1567.8		8
168.125	even	2		1764.2.b.k.1567.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.d.c.195.1	✓	8	8.3	odd	2
196.2.d.c.195.2	yes	8	56.27	even	2
196.2.d.c.195.3	yes	8	56.13	odd	2
196.2.d.c.195.4	yes	8	8.5	even	2
196.2.f.d.19.3		16	56.53	even	6
196.2.f.d.19.4		16	56.45	odd	6
196.2.f.d.19.7		16	56.3	even	6
196.2.f.d.19.8		16	56.11	odd	6
196.2.f.d.31.3		16	56.19	even	6
196.2.f.d.31.4		16	56.51	odd	6
196.2.f.d.31.7		16	56.37	even	6
196.2.f.d.31.8		16	56.5	odd	6
1764.2.b.k.1567.5		8	24.5	odd	2
1764.2.b.k.1567.6		8	168.125	even	2
1764.2.b.k.1567.7		8	24.11	even	2
1764.2.b.k.1567.8		8	168.83	odd	2
3136.2.f.i.3135.1		8	1.1	even	1	trivial
3136.2.f.i.3135.2		8	28.27	even	2	inner
3136.2.f.i.3135.7		8	4.3	odd	2	inner
3136.2.f.i.3135.8		8	7.6	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)

\(f(q)\)	\(=\)	\(q-2.72291 q^{3} -1.08239i q^{5} +4.41421 q^{9} +O(q^{10})\)	\(q-2.72291 q^{3} -1.08239i q^{5} +4.41421 q^{9} -2.08402i q^{11} +2.61313i q^{13} +2.94725i q^{15} -4.46088i q^{17} +1.12786 q^{19} +7.11529i q^{23} +3.82843 q^{25} -3.85077 q^{27} +1.17157 q^{29} -7.70154 q^{31} +5.67459i q^{33} +4.00000 q^{37} -7.11529i q^{39} -5.54328i q^{41} +7.97852i q^{43} -4.77791i q^{45} -5.44581 q^{47} +12.1466i q^{51} +6.48528 q^{53} -2.25573 q^{55} -3.07107 q^{57} -8.82940 q^{59} +13.0656i q^{61} +2.82843 q^{65} -10.0625i q^{67} -19.3743i q^{69} +7.97069i q^{73} -10.4244 q^{75} -4.16804i q^{79} -2.75736 q^{81} +4.31795 q^{83} -4.82843 q^{85} -3.19008 q^{87} +4.01254i q^{89} +20.9706 q^{93} -1.22079i q^{95} -3.82683i q^{97} -9.19932i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 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

Varieties

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Representations

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Database

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