LMFDB - Embedded newform 9360.2.a.cp.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2340)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} -3.37228 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -3.37228 q^{7} -1.37228 q^{11} +1.00000 q^{13} +7.37228 q^{17} +4.00000 q^{19} -7.37228 q^{23} +1.00000 q^{25} -2.74456 q^{29} +6.74456 q^{31} -3.37228 q^{35} -8.11684 q^{37} -1.37228 q^{41} -10.7446 q^{43} +11.4891 q^{47} +4.37228 q^{49} +7.37228 q^{53} -1.37228 q^{55} -8.74456 q^{59} +6.62772 q^{61} +1.00000 q^{65} +4.00000 q^{67} -1.37228 q^{71} +7.48913 q^{73} +4.62772 q^{77} +16.8614 q^{79} -3.25544 q^{83} +7.37228 q^{85} -4.11684 q^{89} -3.37228 q^{91} +4.00000 q^{95} -17.3723 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - q^7	$ 2 q + 2 q^{5} - q^{7} + 3 q^{11} + 2 q^{13} + 9 q^{17} + 8 q^{19} - 9 q^{23} + 2 q^{25} + 6 q^{29} + 2 q^{31} - q^{35} + q^{37} + 3 q^{41} - 10 q^{43} + 3 q^{49} + 9 q^{53} + 3 q^{55} - 6 q^{59} + 19 q^{61} + 2 q^{65} + 8 q^{67} + 3 q^{71} - 8 q^{73} + 15 q^{77} + 5 q^{79} - 18 q^{83} + 9 q^{85} + 9 q^{89} - q^{91} + 8 q^{95} - 29 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - q^7 + 3 * q^11 + 2 * q^13 + 9 * q^17 + 8 * q^19 - 9 * q^23 + 2 * q^25 + 6 * q^29 + 2 * q^31 - q^35 + q^37 + 3 * q^41 - 10 * q^43 + 3 * q^49 + 9 * q^53 + 3 * q^55 - 6 * q^59 + 19 * q^61 + 2 * q^65 + 8 * q^67 + 3 * q^71 - 8 * q^73 + 15 * q^77 + 5 * q^79 - 18 * q^83 + 9 * q^85 + 9 * q^89 - q^91 + 8 * q^95 - 29 * 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
$7$	−3.37228	−1.27460	−0.637301	−	0.770615i	$-0.719949\pi$
$7$	−3.37228	−1.27460	−0.637301	+	0.770615i	$0.719949\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.37228	−0.413758	−0.206879	−	0.978366i	$-0.566331\pi$
$11$	−1.37228	−0.413758	−0.206879	+	0.978366i	$0.566331\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.37228	1.78804	0.894020	−	0.448026i	$-0.147873\pi$
$17$	7.37228	1.78804	0.894020	+	0.448026i	$0.147873\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.37228	−1.53723	−0.768613	−	0.639713i	$-0.779053\pi$
$23$	−7.37228	−1.53723	−0.768613	+	0.639713i	$0.779053\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.74456	−0.509652	−0.254826	−	0.966987i	$-0.582018\pi$
$29$	−2.74456	−0.509652	−0.254826	+	0.966987i	$0.582018\pi$
$30$	0	0
$31$	6.74456	1.21136	0.605680	−	0.795709i	$-0.292901\pi$
$31$	6.74456	1.21136	0.605680	+	0.795709i	$0.292901\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.37228	−0.570020
$36$	0	0
$37$	−8.11684	−1.33440	−0.667200	−	0.744878i	$-0.732508\pi$
$37$	−8.11684	−1.33440	−0.667200	+	0.744878i	$0.732508\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.37228	−0.214314	−0.107157	−	0.994242i	$-0.534175\pi$
$41$	−1.37228	−0.214314	−0.107157	+	0.994242i	$0.534175\pi$
$42$	0	0
$43$	−10.7446	−1.63853	−0.819265	−	0.573415i	$-0.805618\pi$
$43$	−10.7446	−1.63853	−0.819265	+	0.573415i	$0.805618\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.4891	1.67586	0.837931	−	0.545777i	$-0.183765\pi$
$47$	11.4891	1.67586	0.837931	+	0.545777i	$0.183765\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.37228	1.01266	0.506330	−	0.862340i	$-0.331002\pi$
$53$	7.37228	1.01266	0.506330	+	0.862340i	$0.331002\pi$
$54$	0	0
$55$	−1.37228	−0.185038
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.74456	−1.13845	−0.569223	−	0.822183i	$-0.692756\pi$
$59$	−8.74456	−1.13845	−0.569223	+	0.822183i	$0.692756\pi$
$60$	0	0
$61$	6.62772	0.848592	0.424296	−	0.905523i	$-0.360522\pi$
$61$	6.62772	0.848592	0.424296	+	0.905523i	$0.360522\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.37228	−0.162860	−0.0814299	−	0.996679i	$-0.525949\pi$
$71$	−1.37228	−0.162860	−0.0814299	+	0.996679i	$0.525949\pi$
$72$	0	0
$73$	7.48913	0.876536	0.438268	−	0.898844i	$-0.355592\pi$
$73$	7.48913	0.876536	0.438268	+	0.898844i	$0.355592\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.62772	0.527377
$78$	0	0
$79$	16.8614	1.89706	0.948528	−	0.316693i	$-0.102572\pi$
$79$	16.8614	1.89706	0.948528	+	0.316693i	$0.102572\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.25544	−0.357331	−0.178665	−	0.983910i	$-0.557178\pi$
$83$	−3.25544	−0.357331	−0.178665	+	0.983910i	$0.557178\pi$
$84$	0	0
$85$	7.37228	0.799636
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.11684	−0.436385	−0.218192	−	0.975906i	$-0.570016\pi$
$89$	−4.11684	−0.436385	−0.218192	+	0.975906i	$0.570016\pi$
$90$	0	0
$91$	−3.37228	−0.353511
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−17.3723	−1.76389	−0.881944	−	0.471354i	$-0.843765\pi$
$97$	−17.3723	−1.76389	−0.881944	+	0.471354i	$0.843765\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	1.25544	0.123702	0.0618510	−	0.998085i	$-0.480300\pi$
$103$	1.25544	0.123702	0.0618510	+	0.998085i	$0.480300\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.37228	−0.712705	−0.356353	−	0.934352i	$-0.615980\pi$
$107$	−7.37228	−0.712705	−0.356353	+	0.934352i	$0.615980\pi$
$108$	0	0
$109$	7.48913	0.717328	0.358664	−	0.933467i	$-0.383232\pi$
$109$	7.48913	0.717328	0.358664	+	0.933467i	$0.383232\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−20.2337	−1.90343	−0.951713	−	0.306990i	$-0.900678\pi$
$113$	−20.2337	−1.90343	−0.951713	+	0.306990i	$0.900678\pi$
$114$	0	0
$115$	−7.37228	−0.687469
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−24.8614	−2.27904
$120$	0	0
$121$	−9.11684	−0.828804
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	−13.4891	−1.16966
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.74456	0.747098	0.373549	−	0.927610i	$-0.378141\pi$
$137$	8.74456	0.747098	0.373549	+	0.927610i	$0.378141\pi$
$138$	0	0
$139$	−3.37228	−0.286033	−0.143017	−	0.989720i	$-0.545680\pi$
$139$	−3.37228	−0.286033	−0.143017	+	0.989720i	$0.545680\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.37228	−0.114756
$144$	0	0
$145$	−2.74456	−0.227924
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.8614	1.54519	0.772593	−	0.634901i	$-0.218959\pi$
$149$	18.8614	1.54519	0.772593	+	0.634901i	$0.218959\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.74456	0.541736
$156$	0	0
$157$	4.74456	0.378657	0.189329	−	0.981914i	$-0.439369\pi$
$157$	4.74456	0.378657	0.189329	+	0.981914i	$0.439369\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	24.8614	1.95935
$162$	0	0
$163$	14.1168	1.10572	0.552858	−	0.833275i	$-0.313537\pi$
$163$	14.1168	1.10572	0.552858	+	0.833275i	$0.313537\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.510875	0.0395327	0.0197663	−	0.999805i	$-0.493708\pi$
$167$	0.510875	0.0395327	0.0197663	+	0.999805i	$0.493708\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	20.2337	1.53834	0.769169	−	0.639045i	$-0.220670\pi$
$173$	20.2337	1.53834	0.769169	+	0.639045i	$0.220670\pi$
$174$	0	0
$175$	−3.37228	−0.254921
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.2337	1.51234	0.756168	−	0.654377i	$-0.227069\pi$
$179$	20.2337	1.51234	0.756168	+	0.654377i	$0.227069\pi$
$180$	0	0
$181$	12.1168	0.900638	0.450319	−	0.892868i	$-0.351310\pi$
$181$	12.1168	0.900638	0.450319	+	0.892868i	$0.351310\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.11684	−0.596762
$186$	0	0
$187$	−10.1168	−0.739817
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.2337	1.46406	0.732029	−	0.681273i	$-0.238573\pi$
$191$	20.2337	1.46406	0.732029	+	0.681273i	$0.238573\pi$
$192$	0	0
$193$	9.37228	0.674632	0.337316	−	0.941392i	$-0.390481\pi$
$193$	9.37228	0.674632	0.337316	+	0.941392i	$0.390481\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.7446	1.47799	0.738994	−	0.673712i	$-0.235301\pi$
$197$	20.7446	1.47799	0.738994	+	0.673712i	$0.235301\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.25544	0.649604
$204$	0	0
$205$	−1.37228	−0.0958443
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.48913	−0.379691
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.7446	−0.732773
$216$	0	0
$217$	−22.7446	−1.54400
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.37228	0.495913
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.74456	0.580397	0.290199	−	0.956966i	$-0.406279\pi$
$227$	8.74456	0.580397	0.290199	+	0.956966i	$0.406279\pi$
$228$	0	0
$229$	−18.2337	−1.20492	−0.602458	−	0.798151i	$-0.705812\pi$
$229$	−18.2337	−1.20492	−0.602458	+	0.798151i	$0.705812\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.1168	0.662776	0.331388	−	0.943494i	$-0.392483\pi$
$233$	10.1168	0.662776	0.331388	+	0.943494i	$0.392483\pi$
$234$	0	0
$235$	11.4891	0.749468
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.6277	1.46367	0.731833	−	0.681484i	$-0.238665\pi$
$239$	22.6277	1.46367	0.731833	+	0.681484i	$0.238665\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.37228	0.279335
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	10.1168	0.636041
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.7446	0.919741	0.459870	−	0.887986i	$-0.347896\pi$
$257$	14.7446	0.919741	0.459870	+	0.887986i	$0.347896\pi$
$258$	0	0
$259$	27.3723	1.70083
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.51087	−0.401478	−0.200739	−	0.979645i	$-0.564334\pi$
$263$	−6.51087	−0.401478	−0.200739	+	0.979645i	$0.564334\pi$
$264$	0	0
$265$	7.37228	0.452876
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.7446	1.63064	0.815322	−	0.579007i	$-0.196560\pi$
$269$	26.7446	1.63064	0.815322	+	0.579007i	$0.196560\pi$
$270$	0	0
$271$	−2.51087	−0.152525	−0.0762624	−	0.997088i	$-0.524299\pi$
$271$	−2.51087	−0.152525	−0.0762624	+	0.997088i	$0.524299\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.37228	−0.0827517
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−16.9783	−1.01284	−0.506419	−	0.862288i	$-0.669031\pi$
$281$	−16.9783	−1.01284	−0.506419	+	0.862288i	$0.669031\pi$
$282$	0	0
$283$	26.9783	1.60369	0.801845	−	0.597532i	$-0.203852\pi$
$283$	26.9783	1.60369	0.801845	+	0.597532i	$0.203852\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.62772	0.273166
$288$	0	0
$289$	37.3505	2.19709
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−8.74456	−0.509128
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.37228	−0.426350
$300$	0	0
$301$	36.2337	2.08848
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.62772	0.379502
$306$	0	0
$307$	8.62772	0.492410	0.246205	−	0.969218i	$-0.420816\pi$
$307$	8.62772	0.492410	0.246205	+	0.969218i	$0.420816\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	32.2337	1.82780	0.913902	−	0.405934i	$-0.133054\pi$
$311$	32.2337	1.82780	0.913902	+	0.405934i	$0.133054\pi$
$312$	0	0
$313$	10.2337	0.578442	0.289221	−	0.957262i	$-0.406604\pi$
$313$	10.2337	0.578442	0.289221	+	0.957262i	$0.406604\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.4891	−1.31928	−0.659640	−	0.751581i	$-0.729291\pi$
$317$	−23.4891	−1.31928	−0.659640	+	0.751581i	$0.729291\pi$
$318$	0	0
$319$	3.76631	0.210873
$320$	0	0
$321$	0	0
$322$	0	0
$323$	29.4891	1.64082
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−38.7446	−2.13606
$330$	0	0
$331$	−25.4891	−1.40101	−0.700505	−	0.713648i	$-0.747042\pi$
$331$	−25.4891	−1.40101	−0.700505	+	0.713648i	$0.747042\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9.25544	−0.501210
$342$	0	0
$343$	8.86141	0.478471
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.62772	0.248429	0.124214	−	0.992255i	$-0.460359\pi$
$347$	4.62772	0.248429	0.124214	+	0.992255i	$0.460359\pi$
$348$	0	0
$349$	−24.7446	−1.32455	−0.662273	−	0.749263i	$-0.730408\pi$
$349$	−24.7446	−1.32455	−0.662273	+	0.749263i	$0.730408\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	−1.37228	−0.0728331
$356$	0	0
$357$	0	0
$358$	0	0
$359$	32.7446	1.72819	0.864096	−	0.503327i	$-0.167891\pi$
$359$	32.7446	1.72819	0.864096	+	0.503327i	$0.167891\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.48913	0.391999
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−24.8614	−1.29074
$372$	0	0
$373$	−32.9783	−1.70755	−0.853775	−	0.520643i	$-0.825692\pi$
$373$	−32.9783	−1.70755	−0.853775	+	0.520643i	$0.825692\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.74456	−0.141352
$378$	0	0
$379$	24.2337	1.24480	0.622400	−	0.782699i	$-0.286158\pi$
$379$	24.2337	1.24480	0.622400	+	0.782699i	$0.286158\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	38.2337	1.95365	0.976825	−	0.214039i	$-0.0686620\pi$
$383$	38.2337	1.95365	0.976825	+	0.214039i	$0.0686620\pi$
$384$	0	0
$385$	4.62772	0.235850
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14.7446	0.747579	0.373790	−	0.927514i	$-0.378058\pi$
$389$	14.7446	0.747579	0.373790	+	0.927514i	$0.378058\pi$
$390$	0	0
$391$	−54.3505	−2.74862
$392$	0	0
$393$	0	0
$394$	0	0
$395$	16.8614	0.848389
$396$	0	0
$397$	−13.6060	−0.682864	−0.341432	−	0.939906i	$-0.610912\pi$
$397$	−13.6060	−0.682864	−0.341432	+	0.939906i	$0.610912\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	6.74456	0.335971
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.1386	0.552120
$408$	0	0
$409$	−6.23369	−0.308236	−0.154118	−	0.988052i	$-0.549254\pi$
$409$	−6.23369	−0.308236	−0.154118	+	0.988052i	$0.549254\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	29.4891	1.45106
$414$	0	0
$415$	−3.25544	−0.159803
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.74456	−0.134081	−0.0670403	−	0.997750i	$-0.521356\pi$
$419$	−2.74456	−0.134081	−0.0670403	+	0.997750i	$0.521356\pi$
$420$	0	0
$421$	16.7446	0.816080	0.408040	−	0.912964i	$-0.366212\pi$
$421$	16.7446	0.816080	0.408040	+	0.912964i	$0.366212\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.37228	0.357608
$426$	0	0
$427$	−22.3505	−1.08162
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.7446	0.999230	0.499615	−	0.866247i	$-0.333475\pi$
$431$	20.7446	0.999230	0.499615	+	0.866247i	$0.333475\pi$
$432$	0	0
$433$	−24.7446	−1.18915	−0.594574	−	0.804041i	$-0.702679\pi$
$433$	−24.7446	−1.18915	−0.594574	+	0.804041i	$0.702679\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−29.4891	−1.41066
$438$	0	0
$439$	3.13859	0.149797	0.0748984	−	0.997191i	$-0.476137\pi$
$439$	3.13859	0.149797	0.0748984	+	0.997191i	$0.476137\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.8614	−0.611064	−0.305532	−	0.952182i	$-0.598834\pi$
$443$	−12.8614	−0.611064	−0.305532	+	0.952182i	$0.598834\pi$
$444$	0	0
$445$	−4.11684	−0.195157
$446$	0	0
$447$	0	0
$448$	0	0
$449$	16.1168	0.760601	0.380300	−	0.924863i	$-0.375821\pi$
$449$	16.1168	0.760601	0.380300	+	0.924863i	$0.375821\pi$
$450$	0	0
$451$	1.88316	0.0886744
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.37228	−0.158095
$456$	0	0
$457$	12.1168	0.566802	0.283401	−	0.959001i	$-0.408537\pi$
$457$	12.1168	0.566802	0.283401	+	0.959001i	$0.408537\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.39403	−0.111501	−0.0557506	−	0.998445i	$-0.517755\pi$
$461$	−2.39403	−0.111501	−0.0557506	+	0.998445i	$0.517755\pi$
$462$	0	0
$463$	−27.3723	−1.27210	−0.636049	−	0.771649i	$-0.719432\pi$
$463$	−27.3723	−1.27210	−0.636049	+	0.771649i	$0.719432\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.88316	−0.0871421	−0.0435710	−	0.999050i	$-0.513873\pi$
$467$	−1.88316	−0.0871421	−0.0435710	+	0.999050i	$0.513873\pi$
$468$	0	0
$469$	−13.4891	−0.622870
$470$	0	0
$471$	0	0
$472$	0	0
$473$	14.7446	0.677956
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.60597	−0.438908	−0.219454	−	0.975623i	$-0.570428\pi$
$479$	−9.60597	−0.438908	−0.219454	+	0.975623i	$0.570428\pi$
$480$	0	0
$481$	−8.11684	−0.370096
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−17.3723	−0.788835
$486$	0	0
$487$	5.88316	0.266591	0.133296	−	0.991076i	$-0.457444\pi$
$487$	5.88316	0.266591	0.133296	+	0.991076i	$0.457444\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.74456	−0.123860	−0.0619302	−	0.998080i	$-0.519726\pi$
$491$	−2.74456	−0.123860	−0.0619302	+	0.998080i	$0.519726\pi$
$492$	0	0
$493$	−20.2337	−0.911279
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.62772	0.207582
$498$	0	0
$499$	24.2337	1.08485	0.542424	−	0.840105i	$-0.317506\pi$
$499$	24.2337	1.08485	0.542424	+	0.840105i	$0.317506\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.51087	0.290306	0.145153	−	0.989409i	$-0.453633\pi$
$503$	6.51087	0.290306	0.145153	+	0.989409i	$0.453633\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.60597	−0.425777	−0.212889	−	0.977076i	$-0.568287\pi$
$509$	−9.60597	−0.425777	−0.212889	+	0.977076i	$0.568287\pi$
$510$	0	0
$511$	−25.2554	−1.11723
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.25544	0.0553212
$516$	0	0
$517$	−15.7663	−0.693402
$518$	0	0
$519$	0	0
$520$	0	0
$521$	21.2554	0.931218	0.465609	−	0.884991i	$-0.345835\pi$
$521$	21.2554	0.931218	0.465609	+	0.884991i	$0.345835\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	49.7228	2.16596
$528$	0	0
$529$	31.3505	1.36307
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.37228	−0.0594401
$534$	0	0
$535$	−7.37228	−0.318732
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.00000	−0.258438
$540$	0	0
$541$	16.7446	0.719905	0.359952	−	0.932971i	$-0.382793\pi$
$541$	16.7446	0.719905	0.359952	+	0.932971i	$0.382793\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.48913	0.320799
$546$	0	0
$547$	18.7446	0.801460	0.400730	−	0.916196i	$-0.368757\pi$
$547$	18.7446	0.801460	0.400730	+	0.916196i	$0.368757\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10.9783	−0.467689
$552$	0	0
$553$	−56.8614	−2.41799
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	−10.7446	−0.454447
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.6277	1.20651	0.603257	−	0.797547i	$-0.293869\pi$
$563$	28.6277	1.20651	0.603257	+	0.797547i	$0.293869\pi$
$564$	0	0
$565$	−20.2337	−0.851238
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.2337	1.35131	0.675653	−	0.737220i	$-0.263862\pi$
$569$	32.2337	1.35131	0.675653	+	0.737220i	$0.263862\pi$
$570$	0	0
$571$	10.3505	0.433156	0.216578	−	0.976265i	$-0.430510\pi$
$571$	10.3505	0.433156	0.216578	+	0.976265i	$0.430510\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.37228	−0.307445
$576$	0	0
$577$	−40.3505	−1.67981	−0.839907	−	0.542730i	$-0.817391\pi$
$577$	−40.3505	−1.67981	−0.839907	+	0.542730i	$0.817391\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.9783	0.455455
$582$	0	0
$583$	−10.1168	−0.418997
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−32.7446	−1.35151	−0.675756	−	0.737125i	$-0.736183\pi$
$587$	−32.7446	−1.35151	−0.675756	+	0.737125i	$0.736183\pi$
$588$	0	0
$589$	26.9783	1.11162
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12.5109	−0.513760	−0.256880	−	0.966443i	$-0.582695\pi$
$593$	−12.5109	−0.513760	−0.256880	+	0.966443i	$0.582695\pi$
$594$	0	0
$595$	−24.8614	−1.01922
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.2337	1.31703	0.658516	−	0.752566i	$-0.271184\pi$
$599$	32.2337	1.31703	0.658516	+	0.752566i	$0.271184\pi$
$600$	0	0
$601$	3.88316	0.158397	0.0791986	−	0.996859i	$-0.474764\pi$
$601$	3.88316	0.158397	0.0791986	+	0.996859i	$0.474764\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.11684	−0.370652
$606$	0	0
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11.4891	0.464800
$612$	0	0
$613$	17.6060	0.711098	0.355549	−	0.934658i	$-0.384294\pi$
$613$	17.6060	0.711098	0.355549	+	0.934658i	$0.384294\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.97825	0.200417	0.100208	−	0.994966i	$-0.468049\pi$
$617$	4.97825	0.200417	0.100208	+	0.994966i	$0.468049\pi$
$618$	0	0
$619$	−1.48913	−0.0598530	−0.0299265	−	0.999552i	$-0.509527\pi$
$619$	−1.48913	−0.0598530	−0.0299265	+	0.999552i	$0.509527\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	13.8832	0.556217
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−59.8397	−2.38596
$630$	0	0
$631$	−11.7663	−0.468409	−0.234205	−	0.972187i	$-0.575249\pi$
$631$	−11.7663	−0.468409	−0.234205	+	0.972187i	$0.575249\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.00000	0.158735
$636$	0	0
$637$	4.37228	0.173236
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−34.9783	−1.38156	−0.690779	−	0.723066i	$-0.742732\pi$
$641$	−34.9783	−1.38156	−0.690779	+	0.723066i	$0.742732\pi$
$642$	0	0
$643$	−26.3505	−1.03916	−0.519582	−	0.854421i	$-0.673912\pi$
$643$	−26.3505	−1.03916	−0.519582	+	0.854421i	$0.673912\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.861407	0.0338654	0.0169327	−	0.999857i	$-0.494610\pi$
$647$	0.861407	0.0338654	0.0169327	+	0.999857i	$0.494610\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−31.2119	−1.22142	−0.610709	−	0.791855i	$-0.709115\pi$
$653$	−31.2119	−1.22142	−0.610709	+	0.791855i	$0.709115\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20.2337	0.788193	0.394096	−	0.919069i	$-0.371058\pi$
$659$	20.2337	0.788193	0.394096	+	0.919069i	$0.371058\pi$
$660$	0	0
$661$	0.978251	0.0380495	0.0190248	−	0.999819i	$-0.493944\pi$
$661$	0.978251	0.0380495	0.0190248	+	0.999819i	$0.493944\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−13.4891	−0.523086
$666$	0	0
$667$	20.2337	0.783452
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.09509	−0.351112
$672$	0	0
$673$	31.4891	1.21382	0.606908	−	0.794772i	$-0.292410\pi$
$673$	31.4891	1.21382	0.606908	+	0.794772i	$0.292410\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25.8832	0.994770	0.497385	−	0.867530i	$-0.334294\pi$
$677$	25.8832	0.994770	0.497385	+	0.867530i	$0.334294\pi$
$678$	0	0
$679$	58.5842	2.24826
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20.7446	0.793769	0.396884	−	0.917869i	$-0.370091\pi$
$683$	20.7446	0.793769	0.396884	+	0.917869i	$0.370091\pi$
$684$	0	0
$685$	8.74456	0.334113
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.37228	0.280862
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.37228	−0.127918
$696$	0	0
$697$	−10.1168	−0.383203
$698$	0	0
$699$	0	0
$700$	0	0
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	0	0
$703$	−32.4674	−1.22453
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−40.4674	−1.52193
$708$	0	0
$709$	51.7228	1.94249	0.971246	−	0.238080i	$-0.0765181\pi$
$709$	51.7228	1.94249	0.971246	+	0.238080i	$0.0765181\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−49.7228	−1.86213
$714$	0	0
$715$	−1.37228	−0.0513204
$716$	0	0
$717$	0	0
$718$	0	0
$719$	41.4891	1.54728	0.773642	−	0.633623i	$-0.218433\pi$
$719$	41.4891	1.54728	0.773642	+	0.633623i	$0.218433\pi$
$720$	0	0
$721$	−4.23369	−0.157671
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.74456	−0.101930
$726$	0	0
$727$	−10.7446	−0.398494	−0.199247	−	0.979949i	$-0.563850\pi$
$727$	−10.7446	−0.398494	−0.199247	+	0.979949i	$0.563850\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−79.2119	−2.92976
$732$	0	0
$733$	20.3505	0.751664	0.375832	−	0.926688i	$-0.377357\pi$
$733$	20.3505	0.751664	0.375832	+	0.926688i	$0.377357\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.48913	−0.202195
$738$	0	0
$739$	−16.2337	−0.597166	−0.298583	−	0.954384i	$-0.596514\pi$
$739$	−16.2337	−0.597166	−0.298583	+	0.954384i	$0.596514\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.2337	−0.522183	−0.261092	−	0.965314i	$-0.584082\pi$
$743$	−14.2337	−0.522183	−0.261092	+	0.965314i	$0.584082\pi$
$744$	0	0
$745$	18.8614	0.691028
$746$	0	0
$747$	0	0
$748$	0	0
$749$	24.8614	0.908416
$750$	0	0
$751$	−9.88316	−0.360641	−0.180321	−	0.983608i	$-0.557714\pi$
$751$	−9.88316	−0.360641	−0.180321	+	0.983608i	$0.557714\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−40.9783	−1.48546	−0.742730	−	0.669591i	$-0.766469\pi$
$761$	−40.9783	−1.48546	−0.742730	+	0.669591i	$0.766469\pi$
$762$	0	0
$763$	−25.2554	−0.914308
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.74456	−0.315748
$768$	0	0
$769$	−38.4674	−1.38717	−0.693585	−	0.720375i	$-0.743970\pi$
$769$	−38.4674	−1.38717	−0.693585	+	0.720375i	$0.743970\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−31.7228	−1.14099	−0.570495	−	0.821301i	$-0.693249\pi$
$773$	−31.7228	−1.14099	−0.570495	+	0.821301i	$0.693249\pi$
$774$	0	0
$775$	6.74456	0.242272
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.48913	−0.196668
$780$	0	0
$781$	1.88316	0.0673846
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.74456	0.169341
$786$	0	0
$787$	−30.9783	−1.10426	−0.552128	−	0.833760i	$-0.686184\pi$
$787$	−30.9783	−1.10426	−0.552128	+	0.833760i	$0.686184\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	68.2337	2.42611
$792$	0	0
$793$	6.62772	0.235357
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−53.3288	−1.88900	−0.944501	−	0.328508i	$-0.893454\pi$
$797$	−53.3288	−1.88900	−0.944501	+	0.328508i	$0.893454\pi$
$798$	0	0
$799$	84.7011	2.99651
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.2772	−0.362674
$804$	0	0
$805$	24.8614	0.876249
$806$	0	0
$807$	0	0
$808$	0	0
$809$	52.4674	1.84465	0.922327	−	0.386409i	$-0.126285\pi$
$809$	52.4674	1.84465	0.922327	+	0.386409i	$0.126285\pi$
$810$	0	0
$811$	−10.7446	−0.377293	−0.188646	−	0.982045i	$-0.560410\pi$
$811$	−10.7446	−0.377293	−0.188646	+	0.982045i	$0.560410\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	14.1168	0.494491
$816$	0	0
$817$	−42.9783	−1.50362
$818$	0	0
$819$	0	0
$820$	0	0
$821$	46.6277	1.62732	0.813659	−	0.581342i	$-0.197472\pi$
$821$	46.6277	1.62732	0.813659	+	0.581342i	$0.197472\pi$
$822$	0	0
$823$	32.4674	1.13174	0.565871	−	0.824494i	$-0.308540\pi$
$823$	32.4674	1.13174	0.565871	+	0.824494i	$0.308540\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.97825	−0.173111	−0.0865554	−	0.996247i	$-0.527586\pi$
$827$	−4.97825	−0.173111	−0.0865554	+	0.996247i	$0.527586\pi$
$828$	0	0
$829$	19.4891	0.676885	0.338443	−	0.940987i	$-0.390100\pi$
$829$	19.4891	0.676885	0.338443	+	0.940987i	$0.390100\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	32.2337	1.11683
$834$	0	0
$835$	0.510875	0.0176795
$836$	0	0
$837$	0	0
$838$	0	0
$839$	21.6060	0.745921	0.372960	−	0.927847i	$-0.378343\pi$
$839$	21.6060	0.745921	0.372960	+	0.927847i	$0.378343\pi$
$840$	0	0
$841$	−21.4674	−0.740254
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	30.7446	1.05640
$848$	0	0
$849$	0	0
$850$	0	0
$851$	59.8397	2.05128
$852$	0	0
$853$	−28.3505	−0.970704	−0.485352	−	0.874319i	$-0.661308\pi$
$853$	−28.3505	−0.970704	−0.485352	+	0.874319i	$0.661308\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.6060	0.533090	0.266545	−	0.963823i	$-0.414118\pi$
$857$	15.6060	0.533090	0.266545	+	0.963823i	$0.414118\pi$
$858$	0	0
$859$	43.6060	1.48782	0.743908	−	0.668282i	$-0.232970\pi$
$859$	43.6060	1.48782	0.743908	+	0.668282i	$0.232970\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.5109	−0.834360	−0.417180	−	0.908824i	$-0.636982\pi$
$863$	−24.5109	−0.834360	−0.417180	+	0.908824i	$0.636982\pi$
$864$	0	0
$865$	20.2337	0.687966
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−23.1386	−0.784923
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.37228	−0.114004
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	45.2554	1.52469	0.762347	−	0.647168i	$-0.224047\pi$
$881$	45.2554	1.52469	0.762347	+	0.647168i	$0.224047\pi$
$882$	0	0
$883$	1.25544	0.0422488	0.0211244	−	0.999777i	$-0.493275\pi$
$883$	1.25544	0.0422488	0.0211244	+	0.999777i	$0.493275\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−21.0951	−0.708304	−0.354152	−	0.935188i	$-0.615230\pi$
$887$	−21.0951	−0.708304	−0.354152	+	0.935188i	$0.615230\pi$
$888$	0	0
$889$	−13.4891	−0.452411
$890$	0	0
$891$	0	0
$892$	0	0
$893$	45.9565	1.53788
$894$	0	0
$895$	20.2337	0.676338
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−18.5109	−0.617372
$900$	0	0
$901$	54.3505	1.81068
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.1168	0.402778
$906$	0	0
$907$	52.7011	1.74991	0.874955	−	0.484204i	$-0.160891\pi$
$907$	52.7011	1.74991	0.874955	+	0.484204i	$0.160891\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10.9783	0.363726	0.181863	−	0.983324i	$-0.441787\pi$
$911$	10.9783	0.363726	0.181863	+	0.983324i	$0.441787\pi$
$912$	0	0
$913$	4.46738	0.147849
$914$	0	0
$915$	0	0
$916$	0	0
$917$	40.4674	1.33635
$918$	0	0
$919$	−34.5842	−1.14083	−0.570414	−	0.821357i	$-0.693217\pi$
$919$	−34.5842	−1.14083	−0.570414	+	0.821357i	$0.693217\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.37228	−0.0451692
$924$	0	0
$925$	−8.11684	−0.266880
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−40.1168	−1.31619	−0.658095	−	0.752934i	$-0.728638\pi$
$929$	−40.1168	−1.31619	−0.658095	+	0.752934i	$0.728638\pi$
$930$	0	0
$931$	17.4891	0.573183
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.1168	−0.330856
$936$	0	0
$937$	10.2337	0.334320	0.167160	−	0.985930i	$-0.446540\pi$
$937$	10.2337	0.334320	0.167160	+	0.985930i	$0.446540\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−16.1168	−0.525394	−0.262697	−	0.964878i	$-0.584612\pi$
$941$	−16.1168	−0.525394	−0.262697	+	0.964878i	$0.584612\pi$
$942$	0	0
$943$	10.1168	0.329450
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.97825	0.161771	0.0808857	−	0.996723i	$-0.474225\pi$
$947$	4.97825	0.161771	0.0808857	+	0.996723i	$0.474225\pi$
$948$	0	0
$949$	7.48913	0.243107
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.35053	0.205714	0.102857	−	0.994696i	$-0.467202\pi$
$953$	6.35053	0.205714	0.102857	+	0.994696i	$0.467202\pi$
$954$	0	0
$955$	20.2337	0.654747
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−29.4891	−0.952254
$960$	0	0
$961$	14.4891	0.467391
$962$	0	0
$963$	0	0
$964$	0	0
$965$	9.37228	0.301704
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	11.3723	0.364579
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23.4891	−0.751484	−0.375742	−	0.926724i	$-0.622612\pi$
$977$	−23.4891	−0.751484	−0.375742	+	0.926724i	$0.622612\pi$
$978$	0	0
$979$	5.64947	0.180558
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−37.2119	−1.18688	−0.593438	−	0.804880i	$-0.702230\pi$
$983$	−37.2119	−1.18688	−0.593438	+	0.804880i	$0.702230\pi$
$984$	0	0
$985$	20.7446	0.660977
$986$	0	0
$987$	0	0
$988$	0	0
$989$	79.2119	2.51879
$990$	0	0
$991$	−50.3505	−1.59944	−0.799719	−	0.600375i	$-0.795018\pi$
$991$	−50.3505	−1.59944	−0.799719	+	0.600375i	$0.795018\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	54.4674	1.72500	0.862500	−	0.506058i	$-0.168898\pi$
$997$	54.4674	1.72500	0.862500	+	0.506058i	$0.168898\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9360.2.a.cp.1.1		2
3.2	odd	2		9360.2.a.cg.1.1		2
4.3	odd	2		2340.2.a.m.1.2	yes	2
12.11	even	2		2340.2.a.j.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.a.j.1.2	✓	2	12.11	even	2
2340.2.a.m.1.2	yes	2	4.3	odd	2
9360.2.a.cg.1.1		2	3.2	odd	2
9360.2.a.cp.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.37228 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -3.37228 q^{7} -1.37228 q^{11} +1.00000 q^{13} +7.37228 q^{17} +4.00000 q^{19} -7.37228 q^{23} +1.00000 q^{25} -2.74456 q^{29} +6.74456 q^{31} -3.37228 q^{35} -8.11684 q^{37} -1.37228 q^{41} -10.7446 q^{43} +11.4891 q^{47} +4.37228 q^{49} +7.37228 q^{53} -1.37228 q^{55} -8.74456 q^{59} +6.62772 q^{61} +1.00000 q^{65} +4.00000 q^{67} -1.37228 q^{71} +7.48913 q^{73} +4.62772 q^{77} +16.8614 q^{79} -3.25544 q^{83} +7.37228 q^{85} -4.11684 q^{89} -3.37228 q^{91} +4.00000 q^{95} -17.3723 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - q^7	\( 2 q + 2 q^{5} - q^{7} + 3 q^{11} + 2 q^{13} + 9 q^{17} + 8 q^{19} - 9 q^{23} + 2 q^{25} + 6 q^{29} + 2 q^{31} - q^{35} + q^{37} + 3 q^{41} - 10 q^{43} + 3 q^{49} + 9 q^{53} + 3 q^{55} - 6 q^{59} + 19 q^{61} + 2 q^{65} + 8 q^{67} + 3 q^{71} - 8 q^{73} + 15 q^{77} + 5 q^{79} - 18 q^{83} + 9 q^{85} + 9 q^{89} - q^{91} + 8 q^{95} - 29 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - q^7 + 3 * q^11 + 2 * q^13 + 9 * q^17 + 8 * q^19 - 9 * q^23 + 2 * q^25 + 6 * q^29 + 2 * q^31 - q^35 + q^37 + 3 * q^41 - 10 * q^43 + 3 * q^49 + 9 * q^53 + 3 * q^55 - 6 * q^59 + 19 * q^61 + 2 * q^65 + 8 * q^67 + 3 * q^71 - 8 * q^73 + 15 * q^77 + 5 * q^79 - 18 * q^83 + 9 * q^85 + 9 * q^89 - q^91 + 8 * q^95 - 29 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more