LMFDB - Embedded newform 9360.2.a.cw.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9360,2,Mod(1,9360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

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

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

chi := DirichletCharacter("9360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9360.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:	$74.7399762919$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	9360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +0.438447 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +0.438447 q^{7} +1.56155 q^{11} -1.00000 q^{13} +1.56155 q^{17} +5.12311 q^{19} -2.43845 q^{23} +1.00000 q^{25} -7.12311 q^{29} -6.00000 q^{31} +0.438447 q^{35} -10.6847 q^{37} +3.56155 q^{41} -3.12311 q^{43} -11.1231 q^{47} -6.80776 q^{49} -4.68466 q^{53} +1.56155 q^{55} -12.0000 q^{59} -6.68466 q^{61} -1.00000 q^{65} +11.3693 q^{67} -10.4384 q^{71} -6.00000 q^{73} +0.684658 q^{77} -4.68466 q^{79} +16.4924 q^{83} +1.56155 q^{85} +10.6847 q^{89} -0.438447 q^{91} +5.12311 q^{95} +16.9309 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 5 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 + 5 * q^7	$ 2 q + 2 q^{5} + 5 q^{7} - q^{11} - 2 q^{13} - q^{17} + 2 q^{19} - 9 q^{23} + 2 q^{25} - 6 q^{29} - 12 q^{31} + 5 q^{35} - 9 q^{37} + 3 q^{41} + 2 q^{43} - 14 q^{47} + 7 q^{49} + 3 q^{53} - q^{55} - 24 q^{59} - q^{61} - 2 q^{65} - 2 q^{67} - 25 q^{71} - 12 q^{73} - 11 q^{77} + 3 q^{79} - q^{85} + 9 q^{89} - 5 q^{91} + 2 q^{95} + 5 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 5 * q^7 - q^11 - 2 * q^13 - q^17 + 2 * q^19 - 9 * q^23 + 2 * q^25 - 6 * q^29 - 12 * q^31 + 5 * q^35 - 9 * q^37 + 3 * q^41 + 2 * q^43 - 14 * q^47 + 7 * q^49 + 3 * q^53 - q^55 - 24 * q^59 - q^61 - 2 * q^65 - 2 * q^67 - 25 * q^71 - 12 * q^73 - 11 * q^77 + 3 * q^79 - q^85 + 9 * q^89 - 5 * q^91 + 2 * q^95 + 5 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.438447	0.165717	0.0828587	−	0.996561i	$-0.473595\pi$
$7$	0.438447	0.165717	0.0828587	+	0.996561i	$0.473595\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.56155	0.470826	0.235413	−	0.971895i	$-0.424356\pi$
$11$	1.56155	0.470826	0.235413	+	0.971895i	$0.424356\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.56155	0.378732	0.189366	−	0.981907i	$-0.439357\pi$
$17$	1.56155	0.378732	0.189366	+	0.981907i	$0.439357\pi$
$18$	0	0
$19$	5.12311	1.17532	0.587661	−	0.809108i	$-0.300049\pi$
$19$	5.12311	1.17532	0.587661	+	0.809108i	$0.300049\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.43845	−0.508451	−0.254226	−	0.967145i	$-0.581821\pi$
$23$	−2.43845	−0.508451	−0.254226	+	0.967145i	$0.581821\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.12311	−1.32273	−0.661364	−	0.750065i	$-0.730022\pi$
$29$	−7.12311	−1.32273	−0.661364	+	0.750065i	$0.730022\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.438447	0.0741111
$36$	0	0
$37$	−10.6847	−1.75655	−0.878274	−	0.478159i	$-0.841304\pi$
$37$	−10.6847	−1.75655	−0.878274	+	0.478159i	$0.841304\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.56155	0.556221	0.278111	−	0.960549i	$-0.410292\pi$
$41$	3.56155	0.556221	0.278111	+	0.960549i	$0.410292\pi$
$42$	0	0
$43$	−3.12311	−0.476269	−0.238135	−	0.971232i	$-0.576536\pi$
$43$	−3.12311	−0.476269	−0.238135	+	0.971232i	$0.576536\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.1231	−1.62247	−0.811236	−	0.584719i	$-0.801205\pi$
$47$	−11.1231	−1.62247	−0.811236	+	0.584719i	$0.801205\pi$
$48$	0	0
$49$	−6.80776	−0.972538
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.68466	−0.643487	−0.321744	−	0.946827i	$-0.604269\pi$
$53$	−4.68466	−0.643487	−0.321744	+	0.946827i	$0.604269\pi$
$54$	0	0
$55$	1.56155	0.210560
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−6.68466	−0.855883	−0.427941	−	0.903806i	$-0.640761\pi$
$61$	−6.68466	−0.855883	−0.427941	+	0.903806i	$0.640761\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	11.3693	1.38898	0.694492	−	0.719501i	$-0.255629\pi$
$67$	11.3693	1.38898	0.694492	+	0.719501i	$0.255629\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.4384	−1.23882	−0.619408	−	0.785069i	$-0.712627\pi$
$71$	−10.4384	−1.23882	−0.619408	+	0.785069i	$0.712627\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.684658	0.0780241
$78$	0	0
$79$	−4.68466	−0.527065	−0.263533	−	0.964650i	$-0.584888\pi$
$79$	−4.68466	−0.527065	−0.263533	+	0.964650i	$0.584888\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.4924	1.81028	0.905139	−	0.425115i	$-0.139766\pi$
$83$	16.4924	1.81028	0.905139	+	0.425115i	$0.139766\pi$
$84$	0	0
$85$	1.56155	0.169374
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.6847	1.13257	0.566286	−	0.824209i	$-0.308380\pi$
$89$	10.6847	1.13257	0.566286	+	0.824209i	$0.308380\pi$
$90$	0	0
$91$	−0.438447	−0.0459618
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.12311	0.525620
$96$	0	0
$97$	16.9309	1.71907	0.859535	−	0.511077i	$-0.170753\pi$
$97$	16.9309	1.71907	0.859535	+	0.511077i	$0.170753\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.2462	1.01954	0.509768	−	0.860312i	$-0.329731\pi$
$101$	10.2462	1.01954	0.509768	+	0.860312i	$0.329731\pi$
$102$	0	0
$103$	15.1231	1.49012	0.745062	−	0.666995i	$-0.232420\pi$
$103$	15.1231	1.49012	0.745062	+	0.666995i	$0.232420\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.9309	−1.05673	−0.528364	−	0.849018i	$-0.677194\pi$
$107$	−10.9309	−1.05673	−0.528364	+	0.849018i	$0.677194\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.87689	−0.458780	−0.229390	−	0.973335i	$-0.573673\pi$
$113$	−4.87689	−0.458780	−0.229390	+	0.973335i	$0.573673\pi$
$114$	0	0
$115$	−2.43845	−0.227386
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.684658	0.0627625
$120$	0	0
$121$	−8.56155	−0.778323
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.75379	−0.155624	−0.0778118	−	0.996968i	$-0.524793\pi$
$127$	−1.75379	−0.155624	−0.0778118	+	0.996968i	$0.524793\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	2.24621	0.194771
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.12311	0.0959534	0.0479767	−	0.998848i	$-0.484723\pi$
$137$	1.12311	0.0959534	0.0479767	+	0.998848i	$0.484723\pi$
$138$	0	0
$139$	−3.31534	−0.281204	−0.140602	−	0.990066i	$-0.544904\pi$
$139$	−3.31534	−0.281204	−0.140602	+	0.990066i	$0.544904\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.56155	−0.130584
$144$	0	0
$145$	−7.12311	−0.591542
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.8078	−1.45887	−0.729434	−	0.684051i	$-0.760217\pi$
$149$	−17.8078	−1.45887	−0.729434	+	0.684051i	$0.760217\pi$
$150$	0	0
$151$	−11.3693	−0.925222	−0.462611	−	0.886561i	$-0.653087\pi$
$151$	−11.3693	−0.925222	−0.462611	+	0.886561i	$0.653087\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.00000	−0.481932
$156$	0	0
$157$	3.36932	0.268901	0.134450	−	0.990920i	$-0.457073\pi$
$157$	3.36932	0.268901	0.134450	+	0.990920i	$0.457073\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.06913	−0.0842593
$162$	0	0
$163$	−16.0540	−1.25744	−0.628722	−	0.777630i	$-0.716422\pi$
$163$	−16.0540	−1.25744	−0.628722	+	0.777630i	$0.716422\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.87689	−0.377385	−0.188693	−	0.982036i	$-0.560425\pi$
$167$	−4.87689	−0.377385	−0.188693	+	0.982036i	$0.560425\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.8769	0.979012	0.489506	−	0.872000i	$-0.337177\pi$
$173$	12.8769	0.979012	0.489506	+	0.872000i	$0.337177\pi$
$174$	0	0
$175$	0.438447	0.0331435
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.87689	0.364516	0.182258	−	0.983251i	$-0.441659\pi$
$179$	4.87689	0.364516	0.182258	+	0.983251i	$0.441659\pi$
$180$	0	0
$181$	13.3153	0.989722	0.494861	−	0.868972i	$-0.335219\pi$
$181$	13.3153	0.989722	0.494861	+	0.868972i	$0.335219\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−10.6847	−0.785552
$186$	0	0
$187$	2.43845	0.178317
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.6155	−1.41933	−0.709665	−	0.704539i	$-0.751154\pi$
$191$	−19.6155	−1.41933	−0.709665	+	0.704539i	$0.751154\pi$
$192$	0	0
$193$	19.5616	1.40807	0.704036	−	0.710165i	$-0.251379\pi$
$193$	19.5616	1.40807	0.704036	+	0.710165i	$0.251379\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.36932	−0.240054	−0.120027	−	0.992771i	$-0.538298\pi$
$197$	−3.36932	−0.240054	−0.120027	+	0.992771i	$0.538298\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.12311	−0.219199
$204$	0	0
$205$	3.56155	0.248750
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	−6.24621	−0.430007	−0.215003	−	0.976613i	$-0.568976\pi$
$211$	−6.24621	−0.430007	−0.215003	+	0.976613i	$0.568976\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.12311	−0.212994
$216$	0	0
$217$	−2.63068	−0.178582
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.56155	−0.105041
$222$	0	0
$223$	15.3693	1.02921	0.514603	−	0.857429i	$-0.327939\pi$
$223$	15.3693	1.02921	0.514603	+	0.857429i	$0.327939\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.75379	−0.381892	−0.190946	−	0.981601i	$-0.561156\pi$
$227$	−5.75379	−0.381892	−0.190946	+	0.981601i	$0.561156\pi$
$228$	0	0
$229$	17.1231	1.13153	0.565763	−	0.824568i	$-0.308582\pi$
$229$	17.1231	1.13153	0.565763	+	0.824568i	$0.308582\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−27.8078	−1.82175	−0.910874	−	0.412685i	$-0.864591\pi$
$233$	−27.8078	−1.82175	−0.910874	+	0.412685i	$0.864591\pi$
$234$	0	0
$235$	−11.1231	−0.725591
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−22.9309	−1.48327	−0.741637	−	0.670801i	$-0.765950\pi$
$239$	−22.9309	−1.48327	−0.741637	+	0.670801i	$0.765950\pi$
$240$	0	0
$241$	24.7386	1.59356	0.796778	−	0.604272i	$-0.206536\pi$
$241$	24.7386	1.59356	0.796778	+	0.604272i	$0.206536\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.80776	−0.434932
$246$	0	0
$247$	−5.12311	−0.325975
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−26.2462	−1.65665	−0.828323	−	0.560251i	$-0.810705\pi$
$251$	−26.2462	−1.65665	−0.828323	+	0.560251i	$0.810705\pi$
$252$	0	0
$253$	−3.80776	−0.239392
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.8769	−0.803239	−0.401619	−	0.915807i	$-0.631552\pi$
$257$	−12.8769	−0.803239	−0.401619	+	0.915807i	$0.631552\pi$
$258$	0	0
$259$	−4.68466	−0.291091
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.24621	0.138507	0.0692537	−	0.997599i	$-0.477938\pi$
$263$	2.24621	0.138507	0.0692537	+	0.997599i	$0.477938\pi$
$264$	0	0
$265$	−4.68466	−0.287776
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−0.876894	−0.0534652	−0.0267326	−	0.999643i	$-0.508510\pi$
$269$	−0.876894	−0.0534652	−0.0267326	+	0.999643i	$0.508510\pi$
$270$	0	0
$271$	19.3693	1.17660	0.588301	−	0.808642i	$-0.299797\pi$
$271$	19.3693	1.17660	0.588301	+	0.808642i	$0.299797\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.56155	0.0941652
$276$	0	0
$277$	−12.2462	−0.735804	−0.367902	−	0.929865i	$-0.619924\pi$
$277$	−12.2462	−0.735804	−0.367902	+	0.929865i	$0.619924\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.24621	0.253308	0.126654	−	0.991947i	$-0.459576\pi$
$281$	4.24621	0.253308	0.126654	+	0.991947i	$0.459576\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.56155	0.0921755
$288$	0	0
$289$	−14.5616	−0.856562
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−20.2462	−1.18280	−0.591398	−	0.806380i	$-0.701424\pi$
$293$	−20.2462	−1.18280	−0.591398	+	0.806380i	$0.701424\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.43845	0.141019
$300$	0	0
$301$	−1.36932	−0.0789261
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.68466	−0.382762
$306$	0	0
$307$	−7.56155	−0.431561	−0.215780	−	0.976442i	$-0.569230\pi$
$307$	−7.56155	−0.431561	−0.215780	+	0.976442i	$0.569230\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.63068	−0.149172	−0.0745862	−	0.997215i	$-0.523764\pi$
$311$	−2.63068	−0.149172	−0.0745862	+	0.997215i	$0.523764\pi$
$312$	0	0
$313$	29.1231	1.64614	0.823068	−	0.567943i	$-0.192261\pi$
$313$	29.1231	1.64614	0.823068	+	0.567943i	$0.192261\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.50758	0.0846740	0.0423370	−	0.999103i	$-0.486520\pi$
$317$	1.50758	0.0846740	0.0423370	+	0.999103i	$0.486520\pi$
$318$	0	0
$319$	−11.1231	−0.622774
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.87689	−0.268872
$330$	0	0
$331$	−29.1231	−1.60075	−0.800375	−	0.599499i	$-0.795366\pi$
$331$	−29.1231	−1.60075	−0.800375	+	0.599499i	$0.795366\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.3693	0.621172
$336$	0	0
$337$	−30.4924	−1.66103	−0.830514	−	0.556998i	$-0.811953\pi$
$337$	−30.4924	−1.66103	−0.830514	+	0.556998i	$0.811953\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9.36932	−0.507377
$342$	0	0
$343$	−6.05398	−0.326884
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.0540	1.39865	0.699325	−	0.714804i	$-0.253484\pi$
$347$	26.0540	1.39865	0.699325	+	0.714804i	$0.253484\pi$
$348$	0	0
$349$	−23.3693	−1.25093	−0.625465	−	0.780252i	$-0.715091\pi$
$349$	−23.3693	−1.25093	−0.625465	+	0.780252i	$0.715091\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−22.4924	−1.19715	−0.598575	−	0.801066i	$-0.704266\pi$
$353$	−22.4924	−1.19715	−0.598575	+	0.801066i	$0.704266\pi$
$354$	0	0
$355$	−10.4384	−0.554015
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.2462	0.751886	0.375943	−	0.926643i	$-0.377319\pi$
$359$	14.2462	0.751886	0.375943	+	0.926643i	$0.377319\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	1.75379	0.0915470	0.0457735	−	0.998952i	$-0.485425\pi$
$367$	1.75379	0.0915470	0.0457735	+	0.998952i	$0.485425\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.05398	−0.106637
$372$	0	0
$373$	12.2462	0.634085	0.317042	−	0.948411i	$-0.397310\pi$
$373$	12.2462	0.634085	0.317042	+	0.948411i	$0.397310\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.12311	0.366859
$378$	0	0
$379$	30.4924	1.56629	0.783145	−	0.621839i	$-0.213614\pi$
$379$	30.4924	1.56629	0.783145	+	0.621839i	$0.213614\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.50758	−0.179229	−0.0896144	−	0.995977i	$-0.528563\pi$
$383$	−3.50758	−0.179229	−0.0896144	+	0.995977i	$0.528563\pi$
$384$	0	0
$385$	0.684658	0.0348934
$386$	0	0
$387$	0	0
$388$	0	0
$389$	37.8617	1.91967	0.959833	−	0.280571i	$-0.0905239\pi$
$389$	37.8617	1.91967	0.959833	+	0.280571i	$0.0905239\pi$
$390$	0	0
$391$	−3.80776	−0.192567
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.68466	−0.235711
$396$	0	0
$397$	−4.43845	−0.222759	−0.111380	−	0.993778i	$-0.535527\pi$
$397$	−4.43845	−0.222759	−0.111380	+	0.993778i	$0.535527\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.75379	0.187455	0.0937276	−	0.995598i	$-0.470122\pi$
$401$	3.75379	0.187455	0.0937276	+	0.995598i	$0.470122\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.6847	−0.827028
$408$	0	0
$409$	−6.87689	−0.340041	−0.170020	−	0.985441i	$-0.554383\pi$
$409$	−6.87689	−0.340041	−0.170020	+	0.985441i	$0.554383\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.26137	−0.258895
$414$	0	0
$415$	16.4924	0.809581
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7.61553	−0.372043	−0.186021	−	0.982546i	$-0.559559\pi$
$419$	−7.61553	−0.372043	−0.186021	+	0.982546i	$0.559559\pi$
$420$	0	0
$421$	39.3693	1.91874	0.959372	−	0.282146i	$-0.0910462\pi$
$421$	39.3693	1.91874	0.959372	+	0.282146i	$0.0910462\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.56155	0.0757464
$426$	0	0
$427$	−2.93087	−0.141835
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.50758	0.168954	0.0844770	−	0.996425i	$-0.473078\pi$
$431$	3.50758	0.168954	0.0844770	+	0.996425i	$0.473078\pi$
$432$	0	0
$433$	−9.61553	−0.462093	−0.231046	−	0.972943i	$-0.574215\pi$
$433$	−9.61553	−0.462093	−0.231046	+	0.972943i	$0.574215\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.4924	−0.597594
$438$	0	0
$439$	−22.0540	−1.05258	−0.526289	−	0.850306i	$-0.676417\pi$
$439$	−22.0540	−1.05258	−0.526289	+	0.850306i	$0.676417\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.80776	0.370958	0.185479	−	0.982648i	$-0.440616\pi$
$443$	7.80776	0.370958	0.185479	+	0.982648i	$0.440616\pi$
$444$	0	0
$445$	10.6847	0.506501
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−35.1771	−1.66011	−0.830055	−	0.557682i	$-0.811691\pi$
$449$	−35.1771	−1.66011	−0.830055	+	0.557682i	$0.811691\pi$
$450$	0	0
$451$	5.56155	0.261883
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.438447	−0.0205547
$456$	0	0
$457$	−13.3153	−0.622865	−0.311433	−	0.950268i	$-0.600809\pi$
$457$	−13.3153	−0.622865	−0.311433	+	0.950268i	$0.600809\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.05398	0.375111	0.187556	−	0.982254i	$-0.439944\pi$
$461$	8.05398	0.375111	0.187556	+	0.982254i	$0.439944\pi$
$462$	0	0
$463$	28.9309	1.34453	0.672266	−	0.740310i	$-0.265321\pi$
$463$	28.9309	1.34453	0.672266	+	0.740310i	$0.265321\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.93087	−0.320722	−0.160361	−	0.987058i	$-0.551266\pi$
$467$	−6.93087	−0.320722	−0.160361	+	0.987058i	$0.551266\pi$
$468$	0	0
$469$	4.98485	0.230179
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.87689	−0.224240
$474$	0	0
$475$	5.12311	0.235064
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−26.0540	−1.19044	−0.595218	−	0.803564i	$-0.702934\pi$
$479$	−26.0540	−1.19044	−0.595218	+	0.803564i	$0.702934\pi$
$480$	0	0
$481$	10.6847	0.487178
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.9309	0.768791
$486$	0	0
$487$	32.0540	1.45250	0.726252	−	0.687428i	$-0.241260\pi$
$487$	32.0540	1.45250	0.726252	+	0.687428i	$0.241260\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	27.6155	1.24627	0.623136	−	0.782114i	$-0.285858\pi$
$491$	27.6155	1.24627	0.623136	+	0.782114i	$0.285858\pi$
$492$	0	0
$493$	−11.1231	−0.500959
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.57671	−0.205293
$498$	0	0
$499$	−34.9848	−1.56614	−0.783068	−	0.621936i	$-0.786347\pi$
$499$	−34.9848	−1.56614	−0.783068	+	0.621936i	$0.786347\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−13.7538	−0.613251	−0.306626	−	0.951830i	$-0.599200\pi$
$503$	−13.7538	−0.613251	−0.306626	+	0.951830i	$0.599200\pi$
$504$	0	0
$505$	10.2462	0.455950
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−23.5616	−1.04435	−0.522174	−	0.852839i	$-0.674879\pi$
$509$	−23.5616	−1.04435	−0.522174	+	0.852839i	$0.674879\pi$
$510$	0	0
$511$	−2.63068	−0.116375
$512$	0	0
$513$	0	0
$514$	0	0
$515$	15.1231	0.666404
$516$	0	0
$517$	−17.3693	−0.763902
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−25.8617	−1.13302	−0.566512	−	0.824054i	$-0.691707\pi$
$521$	−25.8617	−1.13302	−0.566512	+	0.824054i	$0.691707\pi$
$522$	0	0
$523$	30.7386	1.34411	0.672053	−	0.740503i	$-0.265413\pi$
$523$	30.7386	1.34411	0.672053	+	0.740503i	$0.265413\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.36932	−0.408134
$528$	0	0
$529$	−17.0540	−0.741477
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.56155	−0.154268
$534$	0	0
$535$	−10.9309	−0.472583
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−10.6307	−0.457896
$540$	0	0
$541$	18.8769	0.811581	0.405791	−	0.913966i	$-0.366996\pi$
$541$	18.8769	0.811581	0.405791	+	0.913966i	$0.366996\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	5.36932	0.229575	0.114788	−	0.993390i	$-0.463381\pi$
$547$	5.36932	0.229575	0.114788	+	0.993390i	$0.463381\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−36.4924	−1.55463
$552$	0	0
$553$	−2.05398	−0.0873439
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.49242	0.275093	0.137546	−	0.990495i	$-0.456078\pi$
$557$	6.49242	0.275093	0.137546	+	0.990495i	$0.456078\pi$
$558$	0	0
$559$	3.12311	0.132093
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.3153	0.814045	0.407022	−	0.913418i	$-0.366567\pi$
$563$	19.3153	0.814045	0.407022	+	0.913418i	$0.366567\pi$
$564$	0	0
$565$	−4.87689	−0.205172
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−32.8769	−1.37827	−0.689136	−	0.724632i	$-0.742010\pi$
$569$	−32.8769	−1.37827	−0.689136	+	0.724632i	$0.742010\pi$
$570$	0	0
$571$	22.0540	0.922930	0.461465	−	0.887158i	$-0.347324\pi$
$571$	22.0540	0.922930	0.461465	+	0.887158i	$0.347324\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.43845	−0.101690
$576$	0	0
$577$	24.4384	1.01739	0.508693	−	0.860948i	$-0.330129\pi$
$577$	24.4384	1.01739	0.508693	+	0.860948i	$0.330129\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.23106	0.299995
$582$	0	0
$583$	−7.31534	−0.302970
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.4924	1.34111	0.670553	−	0.741862i	$-0.266057\pi$
$587$	32.4924	1.34111	0.670553	+	0.741862i	$0.266057\pi$
$588$	0	0
$589$	−30.7386	−1.26656
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.2462	0.995673	0.497836	−	0.867271i	$-0.334128\pi$
$593$	24.2462	0.995673	0.497836	+	0.867271i	$0.334128\pi$
$594$	0	0
$595$	0.684658	0.0280683
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.36932	0.382820	0.191410	−	0.981510i	$-0.438694\pi$
$599$	9.36932	0.382820	0.191410	+	0.981510i	$0.438694\pi$
$600$	0	0
$601$	−28.5464	−1.16443	−0.582216	−	0.813034i	$-0.697814\pi$
$601$	−28.5464	−1.16443	−0.582216	+	0.813034i	$0.697814\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.56155	−0.348077
$606$	0	0
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11.1231	0.449993
$612$	0	0
$613$	−32.5464	−1.31454	−0.657268	−	0.753657i	$-0.728288\pi$
$613$	−32.5464	−1.31454	−0.657268	+	0.753657i	$0.728288\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.738634	0.0297363	0.0148681	−	0.999889i	$-0.495267\pi$
$617$	0.738634	0.0297363	0.0148681	+	0.999889i	$0.495267\pi$
$618$	0	0
$619$	8.63068	0.346896	0.173448	−	0.984843i	$-0.444509\pi$
$619$	8.63068	0.346896	0.173448	+	0.984843i	$0.444509\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.68466	0.187687
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16.6847	−0.665261
$630$	0	0
$631$	32.7386	1.30330	0.651652	−	0.758518i	$-0.274076\pi$
$631$	32.7386	1.30330	0.651652	+	0.758518i	$0.274076\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.75379	−0.0695970
$636$	0	0
$637$	6.80776	0.269733
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−32.9848	−1.30282	−0.651412	−	0.758725i	$-0.725823\pi$
$641$	−32.9848	−1.30282	−0.651412	+	0.758725i	$0.725823\pi$
$642$	0	0
$643$	10.6847	0.421362	0.210681	−	0.977555i	$-0.432432\pi$
$643$	10.6847	0.421362	0.210681	+	0.977555i	$0.432432\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−31.8078	−1.25049	−0.625246	−	0.780428i	$-0.715001\pi$
$647$	−31.8078	−1.25049	−0.625246	+	0.780428i	$0.715001\pi$
$648$	0	0
$649$	−18.7386	−0.735556
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−33.3693	−1.30584	−0.652921	−	0.757426i	$-0.726457\pi$
$653$	−33.3693	−1.30584	−0.652921	+	0.757426i	$0.726457\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−0.876894	−0.0341590	−0.0170795	−	0.999854i	$-0.505437\pi$
$659$	−0.876894	−0.0341590	−0.0170795	+	0.999854i	$0.505437\pi$
$660$	0	0
$661$	6.49242	0.252526	0.126263	−	0.991997i	$-0.459702\pi$
$661$	6.49242	0.252526	0.126263	+	0.991997i	$0.459702\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.24621	0.0871043
$666$	0	0
$667$	17.3693	0.672543
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.4384	−0.402972
$672$	0	0
$673$	16.7386	0.645227	0.322613	−	0.946531i	$-0.395439\pi$
$673$	16.7386	0.645227	0.322613	+	0.946531i	$0.395439\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14.4384	−0.554915	−0.277457	−	0.960738i	$-0.589492\pi$
$677$	−14.4384	−0.554915	−0.277457	+	0.960738i	$0.589492\pi$
$678$	0	0
$679$	7.42329	0.284880
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−32.4924	−1.24329	−0.621644	−	0.783300i	$-0.713535\pi$
$683$	−32.4924	−1.24329	−0.621644	+	0.783300i	$0.713535\pi$
$684$	0	0
$685$	1.12311	0.0429117
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.68466	0.178471
$690$	0	0
$691$	21.6155	0.822293	0.411147	−	0.911569i	$-0.365128\pi$
$691$	21.6155	0.822293	0.411147	+	0.911569i	$0.365128\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.31534	−0.125758
$696$	0	0
$697$	5.56155	0.210659
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−48.9848	−1.85013	−0.925066	−	0.379806i	$-0.875991\pi$
$701$	−48.9848	−1.85013	−0.925066	+	0.379806i	$0.875991\pi$
$702$	0	0
$703$	−54.7386	−2.06451
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.49242	0.168955
$708$	0	0
$709$	9.12311	0.342625	0.171313	−	0.985217i	$-0.445199\pi$
$709$	9.12311	0.342625	0.171313	+	0.985217i	$0.445199\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	14.6307	0.547923
$714$	0	0
$715$	−1.56155	−0.0583988
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	6.63068	0.246940
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.12311	−0.264546
$726$	0	0
$727$	−12.8769	−0.477578	−0.238789	−	0.971072i	$-0.576750\pi$
$727$	−12.8769	−0.477578	−0.238789	+	0.971072i	$0.576750\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.87689	−0.180378
$732$	0	0
$733$	−16.4384	−0.607168	−0.303584	−	0.952805i	$-0.598183\pi$
$733$	−16.4384	−0.607168	−0.303584	+	0.952805i	$0.598183\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	17.7538	0.653969
$738$	0	0
$739$	−48.7386	−1.79288	−0.896440	−	0.443166i	$-0.853855\pi$
$739$	−48.7386	−1.79288	−0.896440	+	0.443166i	$0.853855\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−18.7386	−0.687454	−0.343727	−	0.939070i	$-0.611689\pi$
$743$	−18.7386	−0.687454	−0.343727	+	0.939070i	$0.611689\pi$
$744$	0	0
$745$	−17.8078	−0.652426
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.79261	−0.175118
$750$	0	0
$751$	−14.0540	−0.512837	−0.256418	−	0.966566i	$-0.582542\pi$
$751$	−14.0540	−0.512837	−0.256418	+	0.966566i	$0.582542\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11.3693	−0.413772
$756$	0	0
$757$	−14.4924	−0.526736	−0.263368	−	0.964695i	$-0.584833\pi$
$757$	−14.4924	−0.526736	−0.263368	+	0.964695i	$0.584833\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	45.2311	1.63962	0.819812	−	0.572632i	$-0.194078\pi$
$761$	45.2311	1.63962	0.819812	+	0.572632i	$0.194078\pi$
$762$	0	0
$763$	−0.876894	−0.0317457
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.0000	0.433295
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.12311	−0.328135	−0.164068	−	0.986449i	$-0.552462\pi$
$773$	−9.12311	−0.328135	−0.164068	+	0.986449i	$0.552462\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18.2462	0.653738
$780$	0	0
$781$	−16.3002	−0.583267
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.36932	0.120256
$786$	0	0
$787$	−11.3693	−0.405272	−0.202636	−	0.979254i	$-0.564951\pi$
$787$	−11.3693	−0.405272	−0.202636	+	0.979254i	$0.564951\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.13826	−0.0760278
$792$	0	0
$793$	6.68466	0.237379
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.68466	0.165939	0.0829696	−	0.996552i	$-0.473560\pi$
$797$	4.68466	0.165939	0.0829696	+	0.996552i	$0.473560\pi$
$798$	0	0
$799$	−17.3693	−0.614482
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−9.36932	−0.330636
$804$	0	0
$805$	−1.06913	−0.0376819
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.50758	0.263952	0.131976	−	0.991253i	$-0.457868\pi$
$809$	7.50758	0.263952	0.131976	+	0.991253i	$0.457868\pi$
$810$	0	0
$811$	−4.24621	−0.149105	−0.0745523	−	0.997217i	$-0.523753\pi$
$811$	−4.24621	−0.149105	−0.0745523	+	0.997217i	$0.523753\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16.0540	−0.562346
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.5464	1.69428	0.847140	−	0.531369i	$-0.178322\pi$
$821$	48.5464	1.69428	0.847140	+	0.531369i	$0.178322\pi$
$822$	0	0
$823$	29.7538	1.03715	0.518576	−	0.855032i	$-0.326462\pi$
$823$	29.7538	1.03715	0.518576	+	0.855032i	$0.326462\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.12311	−0.247695	−0.123847	−	0.992301i	$-0.539523\pi$
$827$	−7.12311	−0.247695	−0.123847	+	0.992301i	$0.539523\pi$
$828$	0	0
$829$	0.738634	0.0256538	0.0128269	−	0.999918i	$-0.495917\pi$
$829$	0.738634	0.0256538	0.0128269	+	0.999918i	$0.495917\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−10.6307	−0.368331
$834$	0	0
$835$	−4.87689	−0.168772
$836$	0	0
$837$	0	0
$838$	0	0
$839$	44.7926	1.54641	0.773206	−	0.634155i	$-0.218652\pi$
$839$	44.7926	1.54641	0.773206	+	0.634155i	$0.218652\pi$
$840$	0	0
$841$	21.7386	0.749608
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−3.75379	−0.128982
$848$	0	0
$849$	0	0
$850$	0	0
$851$	26.0540	0.893119
$852$	0	0
$853$	41.4233	1.41831	0.709153	−	0.705054i	$-0.249077\pi$
$853$	41.4233	1.41831	0.709153	+	0.705054i	$0.249077\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.43845	0.0832958	0.0416479	−	0.999132i	$-0.486739\pi$
$857$	2.43845	0.0832958	0.0416479	+	0.999132i	$0.486739\pi$
$858$	0	0
$859$	−3.80776	−0.129919	−0.0649596	−	0.997888i	$-0.520692\pi$
$859$	−3.80776	−0.129919	−0.0649596	+	0.997888i	$0.520692\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.36932	0.318935	0.159468	−	0.987203i	$-0.449022\pi$
$863$	9.36932	0.318935	0.159468	+	0.987203i	$0.449022\pi$
$864$	0	0
$865$	12.8769	0.437828
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7.31534	−0.248156
$870$	0	0
$871$	−11.3693	−0.385235
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.438447	0.0148222
$876$	0	0
$877$	46.9848	1.58657	0.793283	−	0.608853i	$-0.208370\pi$
$877$	46.9848	1.58657	0.793283	+	0.608853i	$0.208370\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−21.3693	−0.719951	−0.359975	−	0.932962i	$-0.617215\pi$
$881$	−21.3693	−0.719951	−0.359975	+	0.932962i	$0.617215\pi$
$882$	0	0
$883$	56.1080	1.88818	0.944091	−	0.329684i	$-0.106942\pi$
$883$	56.1080	1.88818	0.944091	+	0.329684i	$0.106942\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−5.56155	−0.186739	−0.0933693	−	0.995632i	$-0.529764\pi$
$887$	−5.56155	−0.186739	−0.0933693	+	0.995632i	$0.529764\pi$
$888$	0	0
$889$	−0.768944	−0.0257895
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−56.9848	−1.90693
$894$	0	0
$895$	4.87689	0.163017
$896$	0	0
$897$	0	0
$898$	0	0
$899$	42.7386	1.42541
$900$	0	0
$901$	−7.31534	−0.243709
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.3153	0.442617
$906$	0	0
$907$	2.63068	0.0873504	0.0436752	−	0.999046i	$-0.486093\pi$
$907$	2.63068	0.0873504	0.0436752	+	0.999046i	$0.486093\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	25.7538	0.852326
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.94602	−0.0641934	−0.0320967	−	0.999485i	$-0.510218\pi$
$919$	−1.94602	−0.0641934	−0.0320967	+	0.999485i	$0.510218\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10.4384	0.343586
$924$	0	0
$925$	−10.6847	−0.351309
$926$	0	0
$927$	0	0
$928$	0	0
$929$	39.6695	1.30151	0.650757	−	0.759286i	$-0.274452\pi$
$929$	39.6695	1.30151	0.650757	+	0.759286i	$0.274452\pi$
$930$	0	0
$931$	−34.8769	−1.14304
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.43845	0.0797458
$936$	0	0
$937$	−27.3693	−0.894117	−0.447058	−	0.894505i	$-0.647528\pi$
$937$	−27.3693	−0.894117	−0.447058	+	0.894505i	$0.647528\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−41.8078	−1.36289	−0.681447	−	0.731867i	$-0.738649\pi$
$941$	−41.8078	−1.36289	−0.681447	+	0.731867i	$0.738649\pi$
$942$	0	0
$943$	−8.68466	−0.282811
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−60.6004	−1.96925	−0.984624	−	0.174688i	$-0.944108\pi$
$947$	−60.6004	−1.96925	−0.984624	+	0.174688i	$0.944108\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	29.5616	0.957593	0.478796	−	0.877926i	$-0.341073\pi$
$953$	29.5616	0.957593	0.478796	+	0.877926i	$0.341073\pi$
$954$	0	0
$955$	−19.6155	−0.634744
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.492423	0.0159012
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	0	0
$965$	19.5616	0.629709
$966$	0	0
$967$	7.36932	0.236981	0.118491	−	0.992955i	$-0.462194\pi$
$967$	7.36932	0.236981	0.118491	+	0.992955i	$0.462194\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24.4924	0.785999	0.393000	−	0.919539i	$-0.371437\pi$
$971$	24.4924	0.785999	0.393000	+	0.919539i	$0.371437\pi$
$972$	0	0
$973$	−1.45360	−0.0466003
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.4773	−1.39096	−0.695481	−	0.718545i	$-0.744808\pi$
$977$	−43.4773	−1.39096	−0.695481	+	0.718545i	$0.744808\pi$
$978$	0	0
$979$	16.6847	0.533244
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−42.7386	−1.36315	−0.681575	−	0.731748i	$-0.738705\pi$
$983$	−42.7386	−1.36315	−0.681575	+	0.731748i	$0.738705\pi$
$984$	0	0
$985$	−3.36932	−0.107355
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.61553	0.242160
$990$	0	0
$991$	−10.0540	−0.319375	−0.159688	−	0.987168i	$-0.551049\pi$
$991$	−10.0540	−0.319375	−0.159688	+	0.987168i	$0.551049\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	28.2462	0.894566	0.447283	−	0.894392i	$-0.352392\pi$
$997$	28.2462	0.894566	0.447283	+	0.894392i	$0.352392\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9360.2.a.cw.1.1		2
3.2	odd	2		9360.2.a.cl.1.1		2
4.3	odd	2		585.2.a.l.1.2	yes	2
12.11	even	2		585.2.a.j.1.1	✓	2
20.3	even	4		2925.2.c.p.2224.1		4
20.7	even	4		2925.2.c.p.2224.4		4
20.19	odd	2		2925.2.a.x.1.1		2
52.51	odd	2		7605.2.a.bd.1.1		2
60.23	odd	4		2925.2.c.o.2224.4		4
60.47	odd	4		2925.2.c.o.2224.1		4
60.59	even	2		2925.2.a.bc.1.2		2
156.155	even	2		7605.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.a.j.1.1	✓	2	12.11	even	2
585.2.a.l.1.2	yes	2	4.3	odd	2
2925.2.a.x.1.1		2	20.19	odd	2
2925.2.a.bc.1.2		2	60.59	even	2
2925.2.c.o.2224.1		4	60.47	odd	4
2925.2.c.o.2224.4		4	60.23	odd	4
2925.2.c.p.2224.1		4	20.3	even	4
2925.2.c.p.2224.4		4	20.7	even	4
7605.2.a.bd.1.1		2	52.51	odd	2
7605.2.a.bi.1.2		2	156.155	even	2
9360.2.a.cl.1.1		2	3.2	odd	2
9360.2.a.cw.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9360.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +0.438447 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +0.438447 q^{7} +1.56155 q^{11} -1.00000 q^{13} +1.56155 q^{17} +5.12311 q^{19} -2.43845 q^{23} +1.00000 q^{25} -7.12311 q^{29} -6.00000 q^{31} +0.438447 q^{35} -10.6847 q^{37} +3.56155 q^{41} -3.12311 q^{43} -11.1231 q^{47} -6.80776 q^{49} -4.68466 q^{53} +1.56155 q^{55} -12.0000 q^{59} -6.68466 q^{61} -1.00000 q^{65} +11.3693 q^{67} -10.4384 q^{71} -6.00000 q^{73} +0.684658 q^{77} -4.68466 q^{79} +16.4924 q^{83} +1.56155 q^{85} +10.6847 q^{89} -0.438447 q^{91} +5.12311 q^{95} +16.9309 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 5 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 5 * q^7	\( 2 q + 2 q^{5} + 5 q^{7} - q^{11} - 2 q^{13} - q^{17} + 2 q^{19} - 9 q^{23} + 2 q^{25} - 6 q^{29} - 12 q^{31} + 5 q^{35} - 9 q^{37} + 3 q^{41} + 2 q^{43} - 14 q^{47} + 7 q^{49} + 3 q^{53} - q^{55} - 24 q^{59} - q^{61} - 2 q^{65} - 2 q^{67} - 25 q^{71} - 12 q^{73} - 11 q^{77} + 3 q^{79} - q^{85} + 9 q^{89} - 5 q^{91} + 2 q^{95} + 5 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 5 * q^7 - q^11 - 2 * q^13 - q^17 + 2 * q^19 - 9 * q^23 + 2 * q^25 - 6 * q^29 - 12 * q^31 + 5 * q^35 - 9 * q^37 + 3 * q^41 + 2 * q^43 - 14 * q^47 + 7 * q^49 + 3 * q^53 - q^55 - 24 * q^59 - q^61 - 2 * q^65 - 2 * q^67 - 25 * q^71 - 12 * q^73 - 11 * q^77 + 3 * q^79 - q^85 + 9 * q^89 - 5 * q^91 + 2 * q^95 + 5 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more