LMFDB - Embedded newform 7800.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 15x - 20 $ x^3 - 15*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.61323$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.61323 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.61323 q^{7} +1.00000 q^{9} +1.61323 q^{11} -1.00000 q^{13} -6.17103 q^{17} +5.17103 q^{19} +1.61323 q^{21} -8.34206 q^{23} -1.00000 q^{27} +1.00000 q^{29} +1.61323 q^{31} -1.61323 q^{33} +1.17103 q^{37} +1.00000 q^{39} -3.17103 q^{41} +2.00000 q^{43} +1.55780 q^{47} -4.39749 q^{49} +6.17103 q^{51} +7.34206 q^{53} -5.17103 q^{57} -4.72883 q^{59} +10.5131 q^{61} -1.61323 q^{63} -4.78426 q^{67} +8.34206 q^{69} -1.17103 q^{71} +0.773540 q^{73} -2.60251 q^{77} -2.39749 q^{79} +1.00000 q^{81} +8.72883 q^{83} -1.00000 q^{87} -6.34206 q^{89} +1.61323 q^{91} -1.61323 q^{93} +4.34206 q^{97} +1.61323 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{9} - 3 q^{13} - 6 q^{17} + 3 q^{19} - 3 q^{27} + 3 q^{29} - 9 q^{37} + 3 q^{39} + 3 q^{41} + 6 q^{43} - 3 q^{47} + 9 q^{49} + 6 q^{51} - 3 q^{53} - 3 q^{57} + 6 q^{59} - 6 q^{61} + 3 q^{67} + 9 q^{71} + 12 q^{73} - 30 q^{77} + 15 q^{79} + 3 q^{81} + 6 q^{83} - 3 q^{87} + 6 q^{89} - 12 q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^9 - 3 * q^13 - 6 * q^17 + 3 * q^19 - 3 * q^27 + 3 * q^29 - 9 * q^37 + 3 * q^39 + 3 * q^41 + 6 * q^43 - 3 * q^47 + 9 * q^49 + 6 * q^51 - 3 * q^53 - 3 * q^57 + 6 * q^59 - 6 * q^61 + 3 * q^67 + 9 * q^71 + 12 * q^73 - 30 * q^77 + 15 * q^79 + 3 * q^81 + 6 * q^83 - 3 * q^87 + 6 * q^89 - 12 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.61323	−0.609744	−0.304872	−	0.952393i	$-0.598614\pi$
$7$	−1.61323	−0.609744	−0.304872	+	0.952393i	$0.598614\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.61323	0.486407	0.243204	−	0.969975i	$-0.421802\pi$
$11$	1.61323	0.486407	0.243204	+	0.969975i	$0.421802\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.17103	−1.49669	−0.748347	−	0.663307i	$-0.769152\pi$
$17$	−6.17103	−1.49669	−0.748347	+	0.663307i	$0.769152\pi$
$18$	0	0
$19$	5.17103	1.18632	0.593158	−	0.805086i	$-0.297881\pi$
$19$	5.17103	1.18632	0.593158	+	0.805086i	$0.297881\pi$
$20$	0	0
$21$	1.61323	0.352036
$22$	0	0
$23$	−8.34206	−1.73944	−0.869720	−	0.493546i	$-0.835701\pi$
$23$	−8.34206	−1.73944	−0.869720	+	0.493546i	$0.835701\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	1.61323	0.289745	0.144872	−	0.989450i	$-0.453723\pi$
$31$	1.61323	0.289745	0.144872	+	0.989450i	$0.453723\pi$
$32$	0	0
$33$	−1.61323	−0.280827
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.17103	0.192516	0.0962581	−	0.995356i	$-0.469313\pi$
$37$	1.17103	0.192516	0.0962581	+	0.995356i	$0.469313\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−3.17103	−0.495232	−0.247616	−	0.968858i	$-0.579647\pi$
$41$	−3.17103	−0.495232	−0.247616	+	0.968858i	$0.579647\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.55780	0.227228	0.113614	−	0.993525i	$-0.463757\pi$
$47$	1.55780	0.227228	0.113614	+	0.993525i	$0.463757\pi$
$48$	0	0
$49$	−4.39749	−0.628213
$50$	0	0
$51$	6.17103	0.864117
$52$	0	0
$53$	7.34206	1.00851	0.504255	−	0.863555i	$-0.331767\pi$
$53$	7.34206	1.00851	0.504255	+	0.863555i	$0.331767\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.17103	−0.684920
$58$	0	0
$59$	−4.72883	−0.615641	−0.307821	−	0.951444i	$-0.599600\pi$
$59$	−4.72883	−0.615641	−0.307821	+	0.951444i	$0.599600\pi$
$60$	0	0
$61$	10.5131	1.34606	0.673032	−	0.739614i	$-0.264992\pi$
$61$	10.5131	1.34606	0.673032	+	0.739614i	$0.264992\pi$
$62$	0	0
$63$	−1.61323	−0.203248
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.78426	−0.584490	−0.292245	−	0.956343i	$-0.594402\pi$
$67$	−4.78426	−0.584490	−0.292245	+	0.956343i	$0.594402\pi$
$68$	0	0
$69$	8.34206	1.00427
$70$	0	0
$71$	−1.17103	−0.138976	−0.0694878	−	0.997583i	$-0.522137\pi$
$71$	−1.17103	−0.138976	−0.0694878	+	0.997583i	$0.522137\pi$
$72$	0	0
$73$	0.773540	0.0905360	0.0452680	−	0.998975i	$-0.485586\pi$
$73$	0.773540	0.0905360	0.0452680	+	0.998975i	$0.485586\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.60251	−0.296584
$78$	0	0
$79$	−2.39749	−0.269739	−0.134869	−	0.990863i	$-0.543061\pi$
$79$	−2.39749	−0.269739	−0.134869	+	0.990863i	$0.543061\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.72883	0.958114	0.479057	−	0.877784i	$-0.340979\pi$
$83$	8.72883	0.958114	0.479057	+	0.877784i	$0.340979\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−6.34206	−0.672257	−0.336128	−	0.941816i	$-0.609118\pi$
$89$	−6.34206	−0.672257	−0.336128	+	0.941816i	$0.609118\pi$
$90$	0	0
$91$	1.61323	0.169112
$92$	0	0
$93$	−1.61323	−0.167284
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.34206	0.440869	0.220435	−	0.975402i	$-0.429252\pi$
$97$	4.34206	0.440869	0.220435	+	0.975402i	$0.429252\pi$
$98$	0	0
$99$	1.61323	0.162136
$100$	0	0
$101$	−11.7395	−1.16813	−0.584064	−	0.811707i	$-0.698538\pi$
$101$	−11.7395	−1.16813	−0.584064	+	0.811707i	$0.698538\pi$
$102$	0	0
$103$	15.5685	1.53401	0.767006	−	0.641640i	$-0.221746\pi$
$103$	15.5685	1.53401	0.767006	+	0.641640i	$0.221746\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.17103	0.499902	0.249951	−	0.968258i	$-0.419585\pi$
$107$	5.17103	0.499902	0.249951	+	0.968258i	$0.419585\pi$
$108$	0	0
$109$	0.718110	0.0687825	0.0343913	−	0.999408i	$-0.489051\pi$
$109$	0.718110	0.0687825	0.0343913	+	0.999408i	$0.489051\pi$
$110$	0	0
$111$	−1.17103	−0.111149
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	9.95529	0.912600
$120$	0	0
$121$	−8.39749	−0.763408
$122$	0	0
$123$	3.17103	0.285922
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−0.0554299	−0.00491861	−0.00245930	−	0.999997i	$-0.500783\pi$
$127$	−0.0554299	−0.00491861	−0.00245930	+	0.999997i	$0.500783\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	11.1710	0.976017	0.488009	−	0.872839i	$-0.337723\pi$
$131$	11.1710	0.976017	0.488009	+	0.872839i	$0.337723\pi$
$132$	0	0
$133$	−8.34206	−0.723348
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−20.2866	−1.73320	−0.866602	−	0.499000i	$-0.833701\pi$
$137$	−20.2866	−1.73320	−0.866602	+	0.499000i	$0.833701\pi$
$138$	0	0
$139$	7.22646	0.612940	0.306470	−	0.951880i	$-0.400852\pi$
$139$	7.22646	0.612940	0.306470	+	0.951880i	$0.400852\pi$
$140$	0	0
$141$	−1.55780	−0.131190
$142$	0	0
$143$	−1.61323	−0.134905
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.39749	0.362699
$148$	0	0
$149$	14.7950	1.21205	0.606026	−	0.795445i	$-0.292763\pi$
$149$	14.7950	1.21205	0.606026	+	0.795445i	$0.292763\pi$
$150$	0	0
$151$	−10.2973	−0.837986	−0.418993	−	0.907989i	$-0.637617\pi$
$151$	−10.2973	−0.837986	−0.418993	+	0.907989i	$0.637617\pi$
$152$	0	0
$153$	−6.17103	−0.498898
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−16.5131	−1.31789	−0.658944	−	0.752192i	$-0.728997\pi$
$157$	−16.5131	−1.31789	−0.658944	+	0.752192i	$0.728997\pi$
$158$	0	0
$159$	−7.34206	−0.582263
$160$	0	0
$161$	13.4577	1.06061
$162$	0	0
$163$	11.2265	0.879324	0.439662	−	0.898163i	$-0.355098\pi$
$163$	11.2265	0.879324	0.439662	+	0.898163i	$0.355098\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.5131	−1.20044	−0.600219	−	0.799835i	$-0.704920\pi$
$167$	−15.5131	−1.20044	−0.600219	+	0.799835i	$0.704920\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	5.17103	0.395439
$172$	0	0
$173$	−11.2312	−0.853892	−0.426946	−	0.904277i	$-0.640411\pi$
$173$	−11.2312	−0.853892	−0.426946	+	0.904277i	$0.640411\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.72883	0.355441
$178$	0	0
$179$	11.5685	0.864672	0.432336	−	0.901713i	$-0.357689\pi$
$179$	11.5685	0.864672	0.432336	+	0.901713i	$0.357689\pi$
$180$	0	0
$181$	−0.281890	−0.0209527	−0.0104763	−	0.999945i	$-0.503335\pi$
$181$	−0.281890	−0.0209527	−0.0104763	+	0.999945i	$0.503335\pi$
$182$	0	0
$183$	−10.5131	−0.777150
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−9.95529	−0.728003
$188$	0	0
$189$	1.61323	0.117345
$190$	0	0
$191$	17.5685	1.27121	0.635607	−	0.772013i	$-0.280750\pi$
$191$	17.5685	1.27121	0.635607	+	0.772013i	$0.280750\pi$
$192$	0	0
$193$	13.1156	0.944082	0.472041	−	0.881577i	$-0.343518\pi$
$193$	13.1156	0.944082	0.472041	+	0.881577i	$0.343518\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.431481	−0.0307417	−0.0153709	−	0.999882i	$-0.504893\pi$
$197$	−0.431481	−0.0307417	−0.0153709	+	0.999882i	$0.504893\pi$
$198$	0	0
$199$	17.0602	1.20936	0.604682	−	0.796467i	$-0.293300\pi$
$199$	17.0602	1.20936	0.604682	+	0.796467i	$0.293300\pi$
$200$	0	0
$201$	4.78426	0.337456
$202$	0	0
$203$	−1.61323	−0.113227
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−8.34206	−0.579813
$208$	0	0
$209$	8.34206	0.577032
$210$	0	0
$211$	12.3421	0.849662	0.424831	−	0.905273i	$-0.360333\pi$
$211$	12.3421	0.849662	0.424831	+	0.905273i	$0.360333\pi$
$212$	0	0
$213$	1.17103	0.0802376
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.60251	−0.176670
$218$	0	0
$219$	−0.773540	−0.0522710
$220$	0	0
$221$	6.17103	0.415108
$222$	0	0
$223$	8.88440	0.594943	0.297472	−	0.954731i	$-0.403857\pi$
$223$	8.88440	0.594943	0.297472	+	0.954731i	$0.403857\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.49763	0.431263	0.215631	−	0.976475i	$-0.430819\pi$
$227$	6.49763	0.431263	0.215631	+	0.976475i	$0.430819\pi$
$228$	0	0
$229$	−15.0816	−0.996621	−0.498310	−	0.866999i	$-0.666046\pi$
$229$	−15.0816	−0.996621	−0.498310	+	0.866999i	$0.666046\pi$
$230$	0	0
$231$	2.60251	0.171233
$232$	0	0
$233$	18.7950	1.23130	0.615650	−	0.788020i	$-0.288894\pi$
$233$	18.7950	1.23130	0.615650	+	0.788020i	$0.288894\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.39749	0.155734
$238$	0	0
$239$	−16.4082	−1.06136	−0.530679	−	0.847573i	$-0.678063\pi$
$239$	−16.4082	−1.06136	−0.530679	+	0.847573i	$0.678063\pi$
$240$	0	0
$241$	−25.7997	−1.66191	−0.830953	−	0.556343i	$-0.812204\pi$
$241$	−25.7997	−1.66191	−0.830953	+	0.556343i	$0.812204\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.17103	−0.329025
$248$	0	0
$249$	−8.72883	−0.553167
$250$	0	0
$251$	19.0602	1.20307	0.601534	−	0.798847i	$-0.294557\pi$
$251$	19.0602	1.20307	0.601534	+	0.798847i	$0.294557\pi$
$252$	0	0
$253$	−13.4577	−0.846076
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.48691	−0.0927509	−0.0463755	−	0.998924i	$-0.514767\pi$
$257$	−1.48691	−0.0927509	−0.0463755	+	0.998924i	$0.514767\pi$
$258$	0	0
$259$	−1.88914	−0.117385
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	13.9106	0.857763	0.428882	−	0.903361i	$-0.358908\pi$
$263$	13.9106	0.857763	0.428882	+	0.903361i	$0.358908\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.34206	0.388128
$268$	0	0
$269$	15.4529	0.942181	0.471091	−	0.882085i	$-0.343860\pi$
$269$	15.4529	0.942181	0.471091	+	0.882085i	$0.343860\pi$
$270$	0	0
$271$	10.3868	0.630951	0.315476	−	0.948934i	$-0.397836\pi$
$271$	10.3868	0.630951	0.315476	+	0.948934i	$0.397836\pi$
$272$	0	0
$273$	−1.61323	−0.0976371
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−16.4529	−0.988560	−0.494280	−	0.869303i	$-0.664568\pi$
$277$	−16.4529	−0.988560	−0.494280	+	0.869303i	$0.664568\pi$
$278$	0	0
$279$	1.61323	0.0965815
$280$	0	0
$281$	18.3975	1.09750	0.548751	−	0.835986i	$-0.315103\pi$
$281$	18.3975	1.09750	0.548751	+	0.835986i	$0.315103\pi$
$282$	0	0
$283$	17.6794	1.05093	0.525465	−	0.850815i	$-0.323891\pi$
$283$	17.6794	1.05093	0.525465	+	0.850815i	$0.323891\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.11560	0.301964
$288$	0	0
$289$	21.0816	1.24009
$290$	0	0
$291$	−4.34206	−0.254536
$292$	0	0
$293$	−14.7950	−0.864332	−0.432166	−	0.901794i	$-0.642251\pi$
$293$	−14.7950	−0.864332	−0.432166	+	0.901794i	$0.642251\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.61323	−0.0936091
$298$	0	0
$299$	8.34206	0.482434
$300$	0	0
$301$	−3.22646	−0.185970
$302$	0	0
$303$	11.7395	0.674419
$304$	0	0
$305$	0	0
$306$	0	0
$307$	32.3081	1.84392	0.921959	−	0.387286i	$-0.126588\pi$
$307$	32.3081	1.84392	0.921959	+	0.387286i	$0.126588\pi$
$308$	0	0
$309$	−15.5685	−0.885662
$310$	0	0
$311$	25.5685	1.44986	0.724929	−	0.688824i	$-0.241873\pi$
$311$	25.5685	1.44986	0.724929	+	0.688824i	$0.241873\pi$
$312$	0	0
$313$	15.0000	0.847850	0.423925	−	0.905697i	$-0.360652\pi$
$313$	15.0000	0.847850	0.423925	+	0.905697i	$0.360652\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.6794	−0.768310	−0.384155	−	0.923269i	$-0.625507\pi$
$317$	−13.6794	−0.768310	−0.384155	+	0.923269i	$0.625507\pi$
$318$	0	0
$319$	1.61323	0.0903235
$320$	0	0
$321$	−5.17103	−0.288619
$322$	0	0
$323$	−31.9106	−1.77555
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.718110	−0.0397116
$328$	0	0
$329$	−2.51309	−0.138551
$330$	0	0
$331$	0.773540	0.0425176	0.0212588	−	0.999774i	$-0.493233\pi$
$331$	0.773540	0.0425176	0.0212588	+	0.999774i	$0.493233\pi$
$332$	0	0
$333$	1.17103	0.0641720
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−36.0816	−1.96549	−0.982745	−	0.184964i	$-0.940783\pi$
$337$	−36.0816	−1.96549	−0.982745	+	0.184964i	$0.940783\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	2.60251	0.140934
$342$	0	0
$343$	18.3868	0.992792
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.9660	1.39393	0.696964	−	0.717106i	$-0.254534\pi$
$347$	25.9660	1.39393	0.696964	+	0.717106i	$0.254534\pi$
$348$	0	0
$349$	13.5685	0.726306	0.363153	−	0.931729i	$-0.381700\pi$
$349$	13.5685	0.726306	0.363153	+	0.931729i	$0.381700\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	25.6239	1.36383	0.681913	−	0.731434i	$-0.261148\pi$
$353$	25.6239	1.36383	0.681913	+	0.731434i	$0.261148\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−9.95529	−0.526890
$358$	0	0
$359$	−0.895119	−0.0472426	−0.0236213	−	0.999721i	$-0.507520\pi$
$359$	−0.895119	−0.0472426	−0.0236213	+	0.999721i	$0.507520\pi$
$360$	0	0
$361$	7.73955	0.407345
$362$	0	0
$363$	8.39749	0.440754
$364$	0	0
$365$	0	0
$366$	0	0
$367$	16.2866	0.850155	0.425078	−	0.905157i	$-0.360247\pi$
$367$	16.2866	0.850155	0.425078	+	0.905157i	$0.360247\pi$
$368$	0	0
$369$	−3.17103	−0.165077
$370$	0	0
$371$	−11.8444	−0.614932
$372$	0	0
$373$	36.8551	1.90829	0.954144	−	0.299349i	$-0.0967695\pi$
$373$	36.8551	1.90829	0.954144	+	0.299349i	$0.0967695\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	11.7241	0.602226	0.301113	−	0.953588i	$-0.402642\pi$
$379$	11.7241	0.602226	0.301113	+	0.953588i	$0.402642\pi$
$380$	0	0
$381$	0.0554299	0.00283976
$382$	0	0
$383$	−17.9660	−0.918020	−0.459010	−	0.888431i	$-0.651796\pi$
$383$	−17.9660	−0.918020	−0.459010	+	0.888431i	$0.651796\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.00000	0.101666
$388$	0	0
$389$	−11.8337	−0.599993	−0.299996	−	0.953940i	$-0.596986\pi$
$389$	−11.8337	−0.599993	−0.299996	+	0.953940i	$0.596986\pi$
$390$	0	0
$391$	51.4791	2.60341
$392$	0	0
$393$	−11.1710	−0.563504
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−7.94457	−0.398727	−0.199363	−	0.979926i	$-0.563887\pi$
$397$	−7.94457	−0.398727	−0.199363	+	0.979926i	$0.563887\pi$
$398$	0	0
$399$	8.34206	0.417625
$400$	0	0
$401$	17.5685	0.877330	0.438665	−	0.898651i	$-0.355451\pi$
$401$	17.5685	0.877330	0.438665	+	0.898651i	$0.355451\pi$
$402$	0	0
$403$	−1.61323	−0.0803607
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.88914	0.0936412
$408$	0	0
$409$	14.8844	0.735986	0.367993	−	0.929829i	$-0.380045\pi$
$409$	14.8844	0.735986	0.367993	+	0.929829i	$0.380045\pi$
$410$	0	0
$411$	20.2866	1.00067
$412$	0	0
$413$	7.62869	0.375383
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−7.22646	−0.353881
$418$	0	0
$419$	−17.6239	−0.860986	−0.430493	−	0.902594i	$-0.641660\pi$
$419$	−17.6239	−0.860986	−0.430493	+	0.902594i	$0.641660\pi$
$420$	0	0
$421$	−22.3421	−1.08889	−0.544443	−	0.838798i	$-0.683259\pi$
$421$	−22.3421	−1.08889	−0.544443	+	0.838798i	$0.683259\pi$
$422$	0	0
$423$	1.55780	0.0757428
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−16.9600	−0.820753
$428$	0	0
$429$	1.61323	0.0778875
$430$	0	0
$431$	−20.1972	−0.972865	−0.486433	−	0.873718i	$-0.661702\pi$
$431$	−20.1972	−0.972865	−0.486433	+	0.873718i	$0.661702\pi$
$432$	0	0
$433$	35.1925	1.69124	0.845621	−	0.533784i	$-0.179230\pi$
$433$	35.1925	1.69124	0.845621	+	0.533784i	$0.179230\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−43.1370	−2.06352
$438$	0	0
$439$	−12.7395	−0.608026	−0.304013	−	0.952668i	$-0.598327\pi$
$439$	−12.7395	−0.608026	−0.304013	+	0.952668i	$0.598327\pi$
$440$	0	0
$441$	−4.39749	−0.209404
$442$	0	0
$443$	11.8337	0.562237	0.281118	−	0.959673i	$-0.409295\pi$
$443$	11.8337	0.562237	0.281118	+	0.959673i	$0.409295\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−14.7950	−0.699778
$448$	0	0
$449$	20.2866	0.957385	0.478693	−	0.877983i	$-0.341111\pi$
$449$	20.2866	0.957385	0.478693	+	0.877983i	$0.341111\pi$
$450$	0	0
$451$	−5.11560	−0.240884
$452$	0	0
$453$	10.2973	0.483812
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.43148	0.207296	0.103648	−	0.994614i	$-0.466948\pi$
$457$	4.43148	0.207296	0.103648	+	0.994614i	$0.466948\pi$
$458$	0	0
$459$	6.17103	0.288039
$460$	0	0
$461$	33.2479	1.54851	0.774255	−	0.632874i	$-0.218125\pi$
$461$	33.2479	1.54851	0.774255	+	0.632874i	$0.218125\pi$
$462$	0	0
$463$	16.0447	0.745661	0.372830	−	0.927899i	$-0.378387\pi$
$463$	16.0447	0.745661	0.372830	+	0.927899i	$0.378387\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.0554	0.650408	0.325204	−	0.945644i	$-0.394567\pi$
$467$	14.0554	0.650408	0.325204	+	0.945644i	$0.394567\pi$
$468$	0	0
$469$	7.71811	0.356389
$470$	0	0
$471$	16.5131	0.760883
$472$	0	0
$473$	3.22646	0.148353
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.34206	0.336170
$478$	0	0
$479$	−24.7843	−1.13242	−0.566211	−	0.824260i	$-0.691591\pi$
$479$	−24.7843	−1.13242	−0.566211	+	0.824260i	$0.691591\pi$
$480$	0	0
$481$	−1.17103	−0.0533944
$482$	0	0
$483$	−13.4577	−0.612345
$484$	0	0
$485$	0	0
$486$	0	0
$487$	34.4082	1.55918	0.779592	−	0.626287i	$-0.215426\pi$
$487$	34.4082	1.55918	0.779592	+	0.626287i	$0.215426\pi$
$488$	0	0
$489$	−11.2265	−0.507678
$490$	0	0
$491$	−5.45766	−0.246301	−0.123150	−	0.992388i	$-0.539300\pi$
$491$	−5.45766	−0.246301	−0.123150	+	0.992388i	$0.539300\pi$
$492$	0	0
$493$	−6.17103	−0.277929
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.88914	0.0847395
$498$	0	0
$499$	−11.3480	−0.508008	−0.254004	−	0.967203i	$-0.581748\pi$
$499$	−11.3480	−0.508008	−0.254004	+	0.967203i	$0.581748\pi$
$500$	0	0
$501$	15.5131	0.693074
$502$	0	0
$503$	−10.0214	−0.446834	−0.223417	−	0.974723i	$-0.571721\pi$
$503$	−10.0214	−0.446834	−0.223417	+	0.974723i	$0.571721\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	6.66268	0.295318	0.147659	−	0.989038i	$-0.452826\pi$
$509$	6.66268	0.295318	0.147659	+	0.989038i	$0.452826\pi$
$510$	0	0
$511$	−1.24790	−0.0552038
$512$	0	0
$513$	−5.17103	−0.228307
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.51309	0.110526
$518$	0	0
$519$	11.2312	0.492995
$520$	0	0
$521$	−15.4791	−0.678152	−0.339076	−	0.940759i	$-0.610114\pi$
$521$	−15.4791	−0.678152	−0.339076	+	0.940759i	$0.610114\pi$
$522$	0	0
$523$	−9.56852	−0.418402	−0.209201	−	0.977873i	$-0.567086\pi$
$523$	−9.56852	−0.418402	−0.209201	+	0.977873i	$0.567086\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.95529	−0.433659
$528$	0	0
$529$	46.5900	2.02565
$530$	0	0
$531$	−4.72883	−0.205214
$532$	0	0
$533$	3.17103	0.137353
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−11.5685	−0.499218
$538$	0	0
$539$	−7.09416	−0.305567
$540$	0	0
$541$	30.4744	1.31019	0.655097	−	0.755544i	$-0.272628\pi$
$541$	30.4744	1.31019	0.655097	+	0.755544i	$0.272628\pi$
$542$	0	0
$543$	0.281890	0.0120970
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.77354	0.289616	0.144808	−	0.989460i	$-0.453744\pi$
$547$	6.77354	0.289616	0.144808	+	0.989460i	$0.453744\pi$
$548$	0	0
$549$	10.5131	0.448688
$550$	0	0
$551$	5.17103	0.220293
$552$	0	0
$553$	3.86770	0.164471
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.91058	0.419925	0.209962	−	0.977709i	$-0.432666\pi$
$557$	9.91058	0.419925	0.209962	+	0.977709i	$0.432666\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	9.95529	0.420313
$562$	0	0
$563$	−26.9707	−1.13668	−0.568341	−	0.822793i	$-0.692414\pi$
$563$	−26.9707	−1.13668	−0.568341	+	0.822793i	$0.692414\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.61323	−0.0677493
$568$	0	0
$569$	9.60725	0.402757	0.201378	−	0.979514i	$-0.435458\pi$
$569$	9.60725	0.402757	0.201378	+	0.979514i	$0.435458\pi$
$570$	0	0
$571$	−17.0262	−0.712523	−0.356262	−	0.934386i	$-0.615949\pi$
$571$	−17.0262	−0.712523	−0.356262	+	0.934386i	$0.615949\pi$
$572$	0	0
$573$	−17.5685	−0.733935
$574$	0	0
$575$	0	0
$576$	0	0
$577$	37.0262	1.54142	0.770710	−	0.637186i	$-0.219902\pi$
$577$	37.0262	1.54142	0.770710	+	0.637186i	$0.219902\pi$
$578$	0	0
$579$	−13.1156	−0.545066
$580$	0	0
$581$	−14.0816	−0.584204
$582$	0	0
$583$	11.8444	0.490546
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.3868	0.593806	0.296903	−	0.954908i	$-0.404046\pi$
$587$	14.3868	0.593806	0.296903	+	0.954908i	$0.404046\pi$
$588$	0	0
$589$	8.34206	0.343729
$590$	0	0
$591$	0.431481	0.0177487
$592$	0	0
$593$	−31.7657	−1.30446	−0.652231	−	0.758020i	$-0.726167\pi$
$593$	−31.7657	−1.30446	−0.652231	+	0.758020i	$0.726167\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−17.0602	−0.698226
$598$	0	0
$599$	−34.0214	−1.39008	−0.695039	−	0.718972i	$-0.744613\pi$
$599$	−34.0214	−1.39008	−0.695039	+	0.718972i	$0.744613\pi$
$600$	0	0
$601$	−6.65794	−0.271583	−0.135792	−	0.990737i	$-0.543358\pi$
$601$	−6.65794	−0.271583	−0.135792	+	0.990737i	$0.543358\pi$
$602$	0	0
$603$	−4.78426	−0.194830
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0.486911	0.0197631	0.00988155	−	0.999951i	$-0.496855\pi$
$607$	0.486911	0.0197631	0.00988155	+	0.999951i	$0.496855\pi$
$608$	0	0
$609$	1.61323	0.0653714
$610$	0	0
$611$	−1.55780	−0.0630218
$612$	0	0
$613$	7.54708	0.304824	0.152412	−	0.988317i	$-0.451296\pi$
$613$	7.54708	0.304824	0.152412	+	0.988317i	$0.451296\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−23.8337	−0.959509	−0.479755	−	0.877403i	$-0.659274\pi$
$617$	−23.8337	−0.959509	−0.479755	+	0.877403i	$0.659274\pi$
$618$	0	0
$619$	26.3635	1.05964	0.529819	−	0.848111i	$-0.322260\pi$
$619$	26.3635	1.05964	0.529819	+	0.848111i	$0.322260\pi$
$620$	0	0
$621$	8.34206	0.334755
$622$	0	0
$623$	10.2312	0.409904
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−8.34206	−0.333150
$628$	0	0
$629$	−7.22646	−0.288138
$630$	0	0
$631$	45.2788	1.80252	0.901261	−	0.433277i	$-0.142643\pi$
$631$	45.2788	1.80252	0.901261	+	0.433277i	$0.142643\pi$
$632$	0	0
$633$	−12.3421	−0.490553
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.39749	0.174235
$638$	0	0
$639$	−1.17103	−0.0463252
$640$	0	0
$641$	37.7610	1.49147	0.745735	−	0.666243i	$-0.232099\pi$
$641$	37.7610	1.49147	0.745735	+	0.666243i	$0.232099\pi$
$642$	0	0
$643$	27.2914	1.07627	0.538133	−	0.842860i	$-0.319130\pi$
$643$	27.2914	1.07627	0.538133	+	0.842860i	$0.319130\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29.3682	−1.15458	−0.577292	−	0.816538i	$-0.695891\pi$
$647$	−29.3682	−1.15458	−0.577292	+	0.816538i	$0.695891\pi$
$648$	0	0
$649$	−7.62869	−0.299452
$650$	0	0
$651$	2.60251	0.102000
$652$	0	0
$653$	−28.9446	−1.13269	−0.566344	−	0.824169i	$-0.691643\pi$
$653$	−28.9446	−1.13269	−0.566344	+	0.824169i	$0.691643\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.773540	0.0301787
$658$	0	0
$659$	−10.2866	−0.400710	−0.200355	−	0.979723i	$-0.564210\pi$
$659$	−10.2866	−0.400710	−0.200355	+	0.979723i	$0.564210\pi$
$660$	0	0
$661$	9.62395	0.374328	0.187164	−	0.982329i	$-0.440070\pi$
$661$	9.62395	0.374328	0.187164	+	0.982329i	$0.440070\pi$
$662$	0	0
$663$	−6.17103	−0.239663
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.34206	−0.323006
$668$	0	0
$669$	−8.88440	−0.343491
$670$	0	0
$671$	16.9600	0.654735
$672$	0	0
$673$	−1.45292	−0.0560059	−0.0280030	−	0.999608i	$-0.508915\pi$
$673$	−1.45292	−0.0560059	−0.0280030	+	0.999608i	$0.508915\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.231200	0.00888573	0.00444286	−	0.999990i	$-0.498586\pi$
$677$	0.231200	0.00888573	0.00444286	+	0.999990i	$0.498586\pi$
$678$	0	0
$679$	−7.00474	−0.268817
$680$	0	0
$681$	−6.49763	−0.248990
$682$	0	0
$683$	7.84443	0.300159	0.150079	−	0.988674i	$-0.452047\pi$
$683$	7.84443	0.300159	0.150079	+	0.988674i	$0.452047\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15.0816	0.575399
$688$	0	0
$689$	−7.34206	−0.279710
$690$	0	0
$691$	1.44694	0.0550442	0.0275221	−	0.999621i	$-0.491238\pi$
$691$	1.44694	0.0550442	0.0275221	+	0.999621i	$0.491238\pi$
$692$	0	0
$693$	−2.60251	−0.0988612
$694$	0	0
$695$	0	0
$696$	0	0
$697$	19.5685	0.741211
$698$	0	0
$699$	−18.7950	−0.710891
$700$	0	0
$701$	18.4917	0.698420	0.349210	−	0.937044i	$-0.386450\pi$
$701$	18.4917	0.698420	0.349210	+	0.937044i	$0.386450\pi$
$702$	0	0
$703$	6.05543	0.228385
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.9386	0.712259
$708$	0	0
$709$	−16.3635	−0.614544	−0.307272	−	0.951622i	$-0.599416\pi$
$709$	−16.3635	−0.614544	−0.307272	+	0.951622i	$0.599416\pi$
$710$	0	0
$711$	−2.39749	−0.0899129
$712$	0	0
$713$	−13.4577	−0.503993
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.4082	0.612776
$718$	0	0
$719$	−32.4744	−1.21109	−0.605545	−	0.795811i	$-0.707045\pi$
$719$	−32.4744	−1.21109	−0.605545	+	0.795811i	$0.707045\pi$
$720$	0	0
$721$	−25.1156	−0.935354
$722$	0	0
$723$	25.7997	0.959502
$724$	0	0
$725$	0	0
$726$	0	0
$727$	32.0309	1.18796	0.593981	−	0.804479i	$-0.297556\pi$
$727$	32.0309	1.18796	0.593981	+	0.804479i	$0.297556\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.3421	−0.456488
$732$	0	0
$733$	−20.0649	−0.741114	−0.370557	−	0.928810i	$-0.620833\pi$
$733$	−20.0649	−0.741114	−0.370557	+	0.928810i	$0.620833\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.71811	−0.284300
$738$	0	0
$739$	−7.10488	−0.261357	−0.130679	−	0.991425i	$-0.541716\pi$
$739$	−7.10488	−0.261357	−0.130679	+	0.991425i	$0.541716\pi$
$740$	0	0
$741$	5.17103	0.189963
$742$	0	0
$743$	51.0583	1.87315	0.936574	−	0.350469i	$-0.113978\pi$
$743$	51.0583	1.87315	0.936574	+	0.350469i	$0.113978\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.72883	0.319371
$748$	0	0
$749$	−8.34206	−0.304812
$750$	0	0
$751$	38.1758	1.39305	0.696527	−	0.717531i	$-0.254728\pi$
$751$	38.1758	1.39305	0.696527	+	0.717531i	$0.254728\pi$
$752$	0	0
$753$	−19.0602	−0.694591
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−29.0863	−1.05716	−0.528581	−	0.848883i	$-0.677276\pi$
$757$	−29.0863	−1.05716	−0.528581	+	0.848883i	$0.677276\pi$
$758$	0	0
$759$	13.4577	0.488482
$760$	0	0
$761$	9.17103	0.332450	0.166225	−	0.986088i	$-0.446842\pi$
$761$	9.17103	0.332450	0.166225	+	0.986088i	$0.446842\pi$
$762$	0	0
$763$	−1.15848	−0.0419397
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.72883	0.170748
$768$	0	0
$769$	35.3588	1.27507	0.637535	−	0.770422i	$-0.279954\pi$
$769$	35.3588	1.27507	0.637535	+	0.770422i	$0.279954\pi$
$770$	0	0
$771$	1.48691	0.0535498
$772$	0	0
$773$	−23.0167	−0.827853	−0.413927	−	0.910310i	$-0.635843\pi$
$773$	−23.0167	−0.827853	−0.413927	+	0.910310i	$0.635843\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.88914	0.0677725
$778$	0	0
$779$	−16.3975	−0.587501
$780$	0	0
$781$	−1.88914	−0.0675988
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	0	0
$786$	0	0
$787$	50.7717	1.80982	0.904908	−	0.425607i	$-0.139940\pi$
$787$	50.7717	1.80982	0.904908	+	0.425607i	$0.139940\pi$
$788$	0	0
$789$	−13.9106	−0.495230
$790$	0	0
$791$	−9.67938	−0.344159
$792$	0	0
$793$	−10.5131	−0.373331
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−29.2866	−1.03739	−0.518693	−	0.854961i	$-0.673581\pi$
$797$	−29.2866	−1.03739	−0.518693	+	0.854961i	$0.673581\pi$
$798$	0	0
$799$	−9.61323	−0.340092
$800$	0	0
$801$	−6.34206	−0.224086
$802$	0	0
$803$	1.24790	0.0440374
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−15.4529	−0.543969
$808$	0	0
$809$	−25.3588	−0.891566	−0.445783	−	0.895141i	$-0.647075\pi$
$809$	−25.3588	−0.891566	−0.445783	+	0.895141i	$0.647075\pi$
$810$	0	0
$811$	−36.9600	−1.29784	−0.648921	−	0.760856i	$-0.724779\pi$
$811$	−36.9600	−1.29784	−0.648921	+	0.760856i	$0.724779\pi$
$812$	0	0
$813$	−10.3868	−0.364280
$814$	0	0
$815$	0	0
$816$	0	0
$817$	10.3421	0.361823
$818$	0	0
$819$	1.61323	0.0563708
$820$	0	0
$821$	−3.33732	−0.116473	−0.0582366	−	0.998303i	$-0.518548\pi$
$821$	−3.33732	−0.116473	−0.0582366	+	0.998303i	$0.518548\pi$
$822$	0	0
$823$	29.9446	1.04380	0.521901	−	0.853006i	$-0.325223\pi$
$823$	29.9446	1.04380	0.521901	+	0.853006i	$0.325223\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39.7550	−1.38242	−0.691209	−	0.722655i	$-0.742921\pi$
$827$	−39.7550	−1.38242	−0.691209	+	0.722655i	$0.742921\pi$
$828$	0	0
$829$	6.06017	0.210478	0.105239	−	0.994447i	$-0.466439\pi$
$829$	6.06017	0.210478	0.105239	+	0.994447i	$0.466439\pi$
$830$	0	0
$831$	16.4529	0.570745
$832$	0	0
$833$	27.1370	0.940243
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.61323	−0.0557614
$838$	0	0
$839$	8.13230	0.280758	0.140379	−	0.990098i	$-0.455168\pi$
$839$	8.13230	0.280758	0.140379	+	0.990098i	$0.455168\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	−18.3975	−0.633643
$844$	0	0
$845$	0	0
$846$	0	0
$847$	13.5471	0.465483
$848$	0	0
$849$	−17.6794	−0.606755
$850$	0	0
$851$	−9.76880	−0.334870
$852$	0	0
$853$	24.9707	0.854982	0.427491	−	0.904020i	$-0.359398\pi$
$853$	24.9707	0.854982	0.427491	+	0.904020i	$0.359398\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23.5900	0.805818	0.402909	−	0.915240i	$-0.367999\pi$
$857$	23.5900	0.805818	0.402909	+	0.915240i	$0.367999\pi$
$858$	0	0
$859$	32.9153	1.12306	0.561528	−	0.827458i	$-0.310214\pi$
$859$	32.9153	1.12306	0.561528	+	0.827458i	$0.310214\pi$
$860$	0	0
$861$	−5.11560	−0.174339
$862$	0	0
$863$	8.10014	0.275732	0.137866	−	0.990451i	$-0.455976\pi$
$863$	8.10014	0.275732	0.137866	+	0.990451i	$0.455976\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−21.0816	−0.715969
$868$	0	0
$869$	−3.86770	−0.131203
$870$	0	0
$871$	4.78426	0.162108
$872$	0	0
$873$	4.34206	0.146956
$874$	0	0
$875$	0	0
$876$	0	0
$877$	16.2866	0.549960	0.274980	−	0.961450i	$-0.411329\pi$
$877$	16.2866	0.549960	0.274980	+	0.961450i	$0.411329\pi$
$878$	0	0
$879$	14.7950	0.499022
$880$	0	0
$881$	6.51309	0.219432	0.109716	−	0.993963i	$-0.465006\pi$
$881$	6.51309	0.219432	0.109716	+	0.993963i	$0.465006\pi$
$882$	0	0
$883$	−48.5053	−1.63233	−0.816166	−	0.577817i	$-0.803905\pi$
$883$	−48.5053	−1.63233	−0.816166	+	0.577817i	$0.803905\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17.8891	−0.600659	−0.300329	−	0.953836i	$-0.597097\pi$
$887$	−17.8891	−0.600659	−0.300329	+	0.953836i	$0.597097\pi$
$888$	0	0
$889$	0.0894212	0.00299909
$890$	0	0
$891$	1.61323	0.0540452
$892$	0	0
$893$	8.05543	0.269565
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−8.34206	−0.278533
$898$	0	0
$899$	1.61323	0.0538042
$900$	0	0
$901$	−45.3081	−1.50943
$902$	0	0
$903$	3.22646	0.107370
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−29.4791	−0.978837	−0.489419	−	0.872049i	$-0.662791\pi$
$907$	−29.4791	−0.978837	−0.489419	+	0.872049i	$0.662791\pi$
$908$	0	0
$909$	−11.7395	−0.389376
$910$	0	0
$911$	10.2217	0.338661	0.169330	−	0.985559i	$-0.445840\pi$
$911$	10.2217	0.338661	0.169330	+	0.985559i	$0.445840\pi$
$912$	0	0
$913$	14.0816	0.466033
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−18.0214	−0.595120
$918$	0	0
$919$	−9.85515	−0.325091	−0.162546	−	0.986701i	$-0.551970\pi$
$919$	−9.85515	−0.325091	−0.162546	+	0.986701i	$0.551970\pi$
$920$	0	0
$921$	−32.3081	−1.06459
$922$	0	0
$923$	1.17103	0.0385449
$924$	0	0
$925$	0	0
$926$	0	0
$927$	15.5685	0.511337
$928$	0	0
$929$	14.5083	0.476003	0.238002	−	0.971265i	$-0.423508\pi$
$929$	14.5083	0.476003	0.238002	+	0.971265i	$0.423508\pi$
$930$	0	0
$931$	−22.7395	−0.745259
$932$	0	0
$933$	−25.5685	−0.837076
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.61921	0.248909	0.124454	−	0.992225i	$-0.460282\pi$
$937$	7.61921	0.248909	0.124454	+	0.992225i	$0.460282\pi$
$938$	0	0
$939$	−15.0000	−0.489506
$940$	0	0
$941$	−31.9106	−1.04026	−0.520128	−	0.854089i	$-0.674116\pi$
$941$	−31.9106	−1.04026	−0.520128	+	0.854089i	$0.674116\pi$
$942$	0	0
$943$	26.4529	0.861426
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.8444	1.03480	0.517402	−	0.855742i	$-0.326899\pi$
$947$	31.8444	1.03480	0.517402	+	0.855742i	$0.326899\pi$
$948$	0	0
$949$	−0.773540	−0.0251102
$950$	0	0
$951$	13.6794	0.443584
$952$	0	0
$953$	33.6501	1.09003	0.545017	−	0.838425i	$-0.316523\pi$
$953$	33.6501	1.09003	0.545017	+	0.838425i	$0.316523\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.61323	−0.0521483
$958$	0	0
$959$	32.7270	1.05681
$960$	0	0
$961$	−28.3975	−0.916048
$962$	0	0
$963$	5.17103	0.166634
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−25.3021	−0.813660	−0.406830	−	0.913504i	$-0.633366\pi$
$967$	−25.3021	−0.813660	−0.406830	+	0.913504i	$0.633366\pi$
$968$	0	0
$969$	31.9106	1.02512
$970$	0	0
$971$	12.9922	0.416939	0.208470	−	0.978029i	$-0.433152\pi$
$971$	12.9922	0.416939	0.208470	+	0.978029i	$0.433152\pi$
$972$	0	0
$973$	−11.6579	−0.373736
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24.5733	−0.786168	−0.393084	−	0.919503i	$-0.628592\pi$
$977$	−24.5733	−0.786168	−0.393084	+	0.919503i	$0.628592\pi$
$978$	0	0
$979$	−10.2312	−0.326991
$980$	0	0
$981$	0.718110	0.0229275
$982$	0	0
$983$	29.3140	0.934973	0.467486	−	0.884000i	$-0.345160\pi$
$983$	29.3140	0.934973	0.467486	+	0.884000i	$0.345160\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.51309	0.0799925
$988$	0	0
$989$	−16.6841	−0.530524
$990$	0	0
$991$	−51.6549	−1.64087	−0.820435	−	0.571739i	$-0.806269\pi$
$991$	−51.6549	−1.64087	−0.820435	+	0.571739i	$0.806269\pi$
$992$	0	0
$993$	−0.773540	−0.0245476
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−14.0602	−0.445290	−0.222645	−	0.974900i	$-0.571469\pi$
$997$	−14.0602	−0.445290	−0.222645	+	0.974900i	$0.571469\pi$
$998$	0	0
$999$	−1.17103	−0.0370497

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bk.1.2	✓	3
5.4	even	2		7800.2.a.bq.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bk.1.2	✓	3	1.1	even	1	trivial
7800.2.a.bq.1.2	yes	3	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.61323 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.61323 q^{7} +1.00000 q^{9} +1.61323 q^{11} -1.00000 q^{13} -6.17103 q^{17} +5.17103 q^{19} +1.61323 q^{21} -8.34206 q^{23} -1.00000 q^{27} +1.00000 q^{29} +1.61323 q^{31} -1.61323 q^{33} +1.17103 q^{37} +1.00000 q^{39} -3.17103 q^{41} +2.00000 q^{43} +1.55780 q^{47} -4.39749 q^{49} +6.17103 q^{51} +7.34206 q^{53} -5.17103 q^{57} -4.72883 q^{59} +10.5131 q^{61} -1.61323 q^{63} -4.78426 q^{67} +8.34206 q^{69} -1.17103 q^{71} +0.773540 q^{73} -2.60251 q^{77} -2.39749 q^{79} +1.00000 q^{81} +8.72883 q^{83} -1.00000 q^{87} -6.34206 q^{89} +1.61323 q^{91} -1.61323 q^{93} +4.34206 q^{97} +1.61323 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{9} - 3 q^{13} - 6 q^{17} + 3 q^{19} - 3 q^{27} + 3 q^{29} - 9 q^{37} + 3 q^{39} + 3 q^{41} + 6 q^{43} - 3 q^{47} + 9 q^{49} + 6 q^{51} - 3 q^{53} - 3 q^{57} + 6 q^{59} - 6 q^{61} + 3 q^{67} + 9 q^{71} + 12 q^{73} - 30 q^{77} + 15 q^{79} + 3 q^{81} + 6 q^{83} - 3 q^{87} + 6 q^{89} - 12 q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^9 - 3 * q^13 - 6 * q^17 + 3 * q^19 - 3 * q^27 + 3 * q^29 - 9 * q^37 + 3 * q^39 + 3 * q^41 + 6 * q^43 - 3 * q^47 + 9 * q^49 + 6 * q^51 - 3 * q^53 - 3 * q^57 + 6 * q^59 - 6 * q^61 + 3 * q^67 + 9 * q^71 + 12 * q^73 - 30 * q^77 + 15 * q^79 + 3 * q^81 + 6 * q^83 - 3 * q^87 + 6 * q^89 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more