LMFDB - Embedded newform 2124.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2124,2,Mod(1,2124)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2124, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2124.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2124 = 2^{2} \cdot 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2124.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:	$16.9602253893$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.22896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - 4x + 3 $ x^4 - 9x^2 - 4x + 3
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			1.3
Root			$0.398683$ of defining polynomial
Character	$\chi$	$=$	2124.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.852003 q^{5} +3.13705 q^{7} +O(q^{10})$	$q+0.852003 q^{5} +3.13705 q^{7} -6.38773 q^{11} +2.53573 q^{13} -6.72741 q^{17} -5.84105 q^{19} -3.39868 q^{23} -4.27409 q^{25} -1.60132 q^{29} -11.0261 q^{31} +2.67277 q^{35} -1.19168 q^{37} +8.65087 q^{41} +7.71646 q^{43} -0.213586 q^{47} +2.84105 q^{49} -8.72083 q^{53} -5.44237 q^{55} +1.00000 q^{59} +7.43142 q^{61} +2.16045 q^{65} +0.261638 q^{67} -2.35063 q^{71} +4.93004 q^{73} -20.0386 q^{77} +6.29874 q^{79} -8.08241 q^{83} -5.73178 q^{85} -1.28504 q^{89} +7.95469 q^{91} -4.97660 q^{95} +9.64775 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{7}+O(q^{10}) $ 4 * q + 2 * q^7	$ 4 q + 2 q^{7} - 10 q^{11} - 2 q^{13} - 4 q^{17} - 6 q^{19} - 12 q^{23} + 4 q^{25} - 8 q^{29} - 8 q^{31} - 12 q^{35} + 6 q^{37} - 16 q^{41} - 6 q^{43} - 18 q^{47} - 6 q^{49} - 4 q^{53} - 6 q^{55} + 4 q^{59} - 18 q^{65} + 10 q^{67} - 16 q^{71} - 14 q^{77} + 12 q^{79} - 22 q^{83} - 6 q^{85} + 2 q^{89} + 20 q^{91} - 36 q^{95} + 6 q^{97}+O(q^{100}) $ 4 * q + 2 * q^7 - 10 * q^11 - 2 * q^13 - 4 * q^17 - 6 * q^19 - 12 * q^23 + 4 * q^25 - 8 * q^29 - 8 * q^31 - 12 * q^35 + 6 * q^37 - 16 * q^41 - 6 * q^43 - 18 * q^47 - 6 * q^49 - 4 * q^53 - 6 * q^55 + 4 * q^59 - 18 * q^65 + 10 * q^67 - 16 * q^71 - 14 * q^77 + 12 * q^79 - 22 * q^83 - 6 * q^85 + 2 * q^89 + 20 * q^91 - 36 * q^95 + 6 * 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
$5$	0.852003	0.381028	0.190514	−	0.981685i	$-0.438985\pi$
$5$	0.852003	0.381028	0.190514	+	0.981685i	$0.438985\pi$
$6$	0	0
$7$	3.13705	1.18569	0.592846	−	0.805316i	$-0.298004\pi$
$7$	3.13705	1.18569	0.592846	+	0.805316i	$0.298004\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.38773	−1.92597	−0.962987	−	0.269549i	$-0.913125\pi$
$11$	−6.38773	−1.92597	−0.962987	+	0.269549i	$0.913125\pi$
$12$	0	0
$13$	2.53573	0.703284	0.351642	−	0.936134i	$-0.385623\pi$
$13$	2.53573	0.703284	0.351642	+	0.936134i	$0.385623\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.72741	−1.63164	−0.815818	−	0.578308i	$-0.803713\pi$
$17$	−6.72741	−1.63164	−0.815818	+	0.578308i	$0.803713\pi$
$18$	0	0
$19$	−5.84105	−1.34003	−0.670015	−	0.742348i	$-0.733712\pi$
$19$	−5.84105	−1.34003	−0.670015	+	0.742348i	$0.733712\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.39868	−0.708674	−0.354337	−	0.935118i	$-0.615293\pi$
$23$	−3.39868	−0.708674	−0.354337	+	0.935118i	$0.615293\pi$
$24$	0	0
$25$	−4.27409	−0.854818
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.60132	−0.297357	−0.148679	−	0.988886i	$-0.547502\pi$
$29$	−1.60132	−0.297357	−0.148679	+	0.988886i	$0.547502\pi$
$30$	0	0
$31$	−11.0261	−1.98035	−0.990177	−	0.139817i	$-0.955349\pi$
$31$	−11.0261	−1.98035	−0.990177	+	0.139817i	$0.955349\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.67277	0.451781
$36$	0	0
$37$	−1.19168	−0.195911	−0.0979557	−	0.995191i	$-0.531230\pi$
$37$	−1.19168	−0.195911	−0.0979557	+	0.995191i	$0.531230\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.65087	1.35104	0.675519	−	0.737342i	$-0.263919\pi$
$41$	8.65087	1.35104	0.675519	+	0.737342i	$0.263919\pi$
$42$	0	0
$43$	7.71646	1.17675	0.588374	−	0.808589i	$-0.299768\pi$
$43$	7.71646	1.17675	0.588374	+	0.808589i	$0.299768\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.213586	−0.0311547	−0.0155773	−	0.999879i	$-0.504959\pi$
$47$	−0.213586	−0.0311547	−0.0155773	+	0.999879i	$0.504959\pi$
$48$	0	0
$49$	2.84105	0.405865
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.72083	−1.19790	−0.598949	−	0.800787i	$-0.704415\pi$
$53$	−8.72083	−1.19790	−0.598949	+	0.800787i	$0.704415\pi$
$54$	0	0
$55$	−5.44237	−0.733849
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	7.43142	0.951496	0.475748	−	0.879582i	$-0.342178\pi$
$61$	7.43142	0.951496	0.475748	+	0.879582i	$0.342178\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.16045	0.267971
$66$	0	0
$67$	0.261638	0.0319641	0.0159821	−	0.999872i	$-0.494913\pi$
$67$	0.261638	0.0319641	0.0159821	+	0.999872i	$0.494913\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.35063	−0.278969	−0.139484	−	0.990224i	$-0.544544\pi$
$71$	−2.35063	−0.278969	−0.139484	+	0.990224i	$0.544544\pi$
$72$	0	0
$73$	4.93004	0.577018	0.288509	−	0.957477i	$-0.406840\pi$
$73$	4.93004	0.577018	0.288509	+	0.957477i	$0.406840\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−20.0386	−2.28361
$78$	0	0
$79$	6.29874	0.708663	0.354332	−	0.935120i	$-0.384708\pi$
$79$	6.29874	0.708663	0.354332	+	0.935120i	$0.384708\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.08241	−0.887159	−0.443580	−	0.896235i	$-0.646292\pi$
$83$	−8.08241	−0.887159	−0.443580	+	0.896235i	$0.646292\pi$
$84$	0	0
$85$	−5.73178	−0.621698
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.28504	−0.136214	−0.0681071	−	0.997678i	$-0.521696\pi$
$89$	−1.28504	−0.136214	−0.0681071	+	0.997678i	$0.521696\pi$
$90$	0	0
$91$	7.95469	0.833878
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.97660	−0.510588
$96$	0	0
$97$	9.64775	0.979580	0.489790	−	0.871840i	$-0.337073\pi$
$97$	9.64775	0.979580	0.489790	+	0.871840i	$0.337073\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.2768	1.12209	0.561044	−	0.827786i	$-0.310400\pi$
$101$	11.2768	1.12209	0.561044	+	0.827786i	$0.310400\pi$
$102$	0	0
$103$	−12.7755	−1.25880	−0.629402	−	0.777080i	$-0.716700\pi$
$103$	−12.7755	−1.25880	−0.629402	+	0.777080i	$0.716700\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.20413	0.406429	0.203215	−	0.979134i	$-0.434861\pi$
$107$	4.20413	0.406429	0.203215	+	0.979134i	$0.434861\pi$
$108$	0	0
$109$	−6.66841	−0.638718	−0.319359	−	0.947634i	$-0.603468\pi$
$109$	−6.66841	−0.638718	−0.319359	+	0.947634i	$0.603468\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.69964	−0.724321	−0.362161	−	0.932116i	$-0.617961\pi$
$113$	−7.69964	−0.724321	−0.362161	+	0.932116i	$0.617961\pi$
$114$	0	0
$115$	−2.89569	−0.270024
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−21.1042	−1.93462
$120$	0	0
$121$	29.8031	2.70937
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.90156	−0.706737
$126$	0	0
$127$	16.3236	1.44849	0.724245	−	0.689543i	$-0.242189\pi$
$127$	16.3236	1.44849	0.724245	+	0.689543i	$0.242189\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.0277	1.13823	0.569116	−	0.822258i	$-0.307286\pi$
$131$	13.0277	1.13823	0.569116	+	0.822258i	$0.307286\pi$
$132$	0	0
$133$	−18.3236	−1.58886
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.1975	−1.38385	−0.691925	−	0.721969i	$-0.743237\pi$
$137$	−16.1975	−1.38385	−0.691925	+	0.721969i	$0.743237\pi$
$138$	0	0
$139$	−3.77546	−0.320231	−0.160115	−	0.987098i	$-0.551187\pi$
$139$	−3.77546	−0.320231	−0.160115	+	0.987098i	$0.551187\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−16.1975	−1.35451
$144$	0	0
$145$	−1.36433	−0.113301
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.9250	1.46847	0.734235	−	0.678895i	$-0.237541\pi$
$149$	17.9250	1.46847	0.734235	+	0.678895i	$0.237541\pi$
$150$	0	0
$151$	−5.09486	−0.414614	−0.207307	−	0.978276i	$-0.566470\pi$
$151$	−5.09486	−0.414614	−0.207307	+	0.978276i	$0.566470\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.39432	−0.754570
$156$	0	0
$157$	−20.1959	−1.61181	−0.805905	−	0.592044i	$-0.798321\pi$
$157$	−20.1959	−1.61181	−0.805905	+	0.592044i	$0.798321\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−10.6618	−0.840269
$162$	0	0
$163$	−7.63842	−0.598287	−0.299144	−	0.954208i	$-0.596701\pi$
$163$	−7.63842	−0.598287	−0.299144	+	0.954208i	$0.596701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.6509	−1.44325	−0.721624	−	0.692285i	$-0.756604\pi$
$167$	−18.6509	−1.44325	−0.721624	+	0.692285i	$0.756604\pi$
$168$	0	0
$169$	−6.57008	−0.505391
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.4190	0.792139	0.396070	−	0.918220i	$-0.370374\pi$
$173$	10.4190	0.792139	0.396070	+	0.918220i	$0.370374\pi$
$174$	0	0
$175$	−13.4080	−1.01355
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.6180	1.69055	0.845275	−	0.534332i	$-0.179437\pi$
$179$	22.6180	1.69055	0.845275	+	0.534332i	$0.179437\pi$
$180$	0	0
$181$	−7.06833	−0.525385	−0.262693	−	0.964880i	$-0.584611\pi$
$181$	−7.06833	−0.525385	−0.262693	+	0.964880i	$0.584611\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.01532	−0.0746476
$186$	0	0
$187$	42.9729	3.14249
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−25.6153	−1.85346	−0.926728	−	0.375733i	$-0.877391\pi$
$191$	−25.6153	−1.85346	−0.926728	+	0.375733i	$0.877391\pi$
$192$	0	0
$193$	−0.523276	−0.0376662	−0.0188331	−	0.999823i	$-0.505995\pi$
$193$	−0.523276	−0.0376662	−0.0188331	+	0.999823i	$0.505995\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.9781	1.13839	0.569196	−	0.822202i	$-0.307255\pi$
$197$	15.9781	1.13839	0.569196	+	0.822202i	$0.307255\pi$
$198$	0	0
$199$	−11.5072	−0.815727	−0.407863	−	0.913043i	$-0.633726\pi$
$199$	−11.5072	−0.815727	−0.407863	+	0.913043i	$0.633726\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.02340	−0.352574
$204$	0	0
$205$	7.37057	0.514783
$206$	0	0
$207$	0	0
$208$	0	0
$209$	37.3111	2.58086
$210$	0	0
$211$	13.4526	0.926115	0.463058	−	0.886328i	$-0.346752\pi$
$211$	13.4526	0.926115	0.463058	+	0.886328i	$0.346752\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.57445	0.448374
$216$	0	0
$217$	−34.5895	−2.34809
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−17.0589	−1.14750
$222$	0	0
$223$	0.109275	0.00731760	0.00365880	−	0.999993i	$-0.498835\pi$
$223$	0.109275	0.00731760	0.00365880	+	0.999993i	$0.498835\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−23.8562	−1.58339	−0.791697	−	0.610913i	$-0.790802\pi$
$227$	−23.8562	−1.58339	−0.791697	+	0.610913i	$0.790802\pi$
$228$	0	0
$229$	3.74931	0.247762	0.123881	−	0.992297i	$-0.460466\pi$
$229$	3.74931	0.247762	0.123881	+	0.992297i	$0.460466\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.32064	0.610616	0.305308	−	0.952254i	$-0.401241\pi$
$233$	9.32064	0.610616	0.305308	+	0.952254i	$0.401241\pi$
$234$	0	0
$235$	−0.181976	−0.0118708
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21.4495	−1.38745	−0.693726	−	0.720239i	$-0.744032\pi$
$239$	−21.4495	−1.38745	−0.693726	+	0.720239i	$0.744032\pi$
$240$	0	0
$241$	−6.67936	−0.430255	−0.215128	−	0.976586i	$-0.569017\pi$
$241$	−6.67936	−0.430255	−0.215128	+	0.976586i	$0.569017\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.42059	0.154646
$246$	0	0
$247$	−14.8113	−0.942422
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−21.9889	−1.38793	−0.693964	−	0.720009i	$-0.744138\pi$
$251$	−21.9889	−1.38793	−0.693964	+	0.720009i	$0.744138\pi$
$252$	0	0
$253$	21.7099	1.36489
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.8223	0.799831	0.399916	−	0.916552i	$-0.369039\pi$
$257$	12.8223	0.799831	0.399916	+	0.916552i	$0.369039\pi$
$258$	0	0
$259$	−3.73836	−0.232291
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.41460	0.0872279	0.0436140	−	0.999048i	$-0.486113\pi$
$263$	1.41460	0.0872279	0.0436140	+	0.999048i	$0.486113\pi$
$264$	0	0
$265$	−7.43017	−0.456432
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.9737	1.33976	0.669881	−	0.742468i	$-0.266345\pi$
$269$	21.9737	1.33976	0.669881	+	0.742468i	$0.266345\pi$
$270$	0	0
$271$	16.6196	1.00957	0.504786	−	0.863245i	$-0.331572\pi$
$271$	16.6196	1.00957	0.504786	+	0.863245i	$0.331572\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	27.3017	1.64636
$276$	0	0
$277$	−16.6821	−1.00233	−0.501165	−	0.865352i	$-0.667095\pi$
$277$	−16.6821	−1.00233	−0.501165	+	0.865352i	$0.667095\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8.02739	−0.478874	−0.239437	−	0.970912i	$-0.576963\pi$
$281$	−8.02739	−0.478874	−0.239437	+	0.970912i	$0.576963\pi$
$282$	0	0
$283$	−9.87104	−0.586772	−0.293386	−	0.955994i	$-0.594782\pi$
$283$	−9.87104	−0.586772	−0.293386	+	0.955994i	$0.594782\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	27.1382	1.60192
$288$	0	0
$289$	28.2581	1.66224
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.8922	1.33738	0.668689	−	0.743542i	$-0.266856\pi$
$293$	22.8922	1.33738	0.668689	+	0.743542i	$0.266856\pi$
$294$	0	0
$295$	0.852003	0.0496056
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.61814	−0.498400
$300$	0	0
$301$	24.2069	1.39526
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.33159	0.362546
$306$	0	0
$307$	27.8656	1.59037	0.795186	−	0.606365i	$-0.207373\pi$
$307$	27.8656	1.59037	0.795186	+	0.606365i	$0.207373\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10.1168	−0.573669	−0.286834	−	0.957980i	$-0.592603\pi$
$311$	−10.1168	−0.573669	−0.286834	+	0.957980i	$0.592603\pi$
$312$	0	0
$313$	−20.6333	−1.16626	−0.583132	−	0.812377i	$-0.698173\pi$
$313$	−20.6333	−1.16626	−0.583132	+	0.812377i	$0.698173\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.46540	−0.250802	−0.125401	−	0.992106i	$-0.540022\pi$
$317$	−4.46540	−0.250802	−0.125401	+	0.992106i	$0.540022\pi$
$318$	0	0
$319$	10.2288	0.572702
$320$	0	0
$321$	0	0
$322$	0	0
$323$	39.2952	2.18644
$324$	0	0
$325$	−10.8379	−0.601180
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.670028	−0.0369398
$330$	0	0
$331$	−2.46081	−0.135258	−0.0676291	−	0.997711i	$-0.521543\pi$
$331$	−2.46081	−0.135258	−0.0676291	+	0.997711i	$0.521543\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.222916	0.0121792
$336$	0	0
$337$	17.0050	0.926319	0.463160	−	0.886275i	$-0.346716\pi$
$337$	17.0050	0.926319	0.463160	+	0.886275i	$0.346716\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	70.4321	3.81411
$342$	0	0
$343$	−13.0468	−0.704461
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.7152	−0.736271	−0.368136	−	0.929772i	$-0.620004\pi$
$347$	−13.7152	−0.736271	−0.368136	+	0.929772i	$0.620004\pi$
$348$	0	0
$349$	2.01245	0.107724	0.0538621	−	0.998548i	$-0.482847\pi$
$349$	2.01245	0.107724	0.0538621	+	0.998548i	$0.482847\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	16.9934	0.904468	0.452234	−	0.891899i	$-0.350627\pi$
$353$	16.9934	0.904468	0.452234	+	0.891899i	$0.350627\pi$
$354$	0	0
$355$	−2.00275	−0.106295
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.1183	1.11458	0.557290	−	0.830318i	$-0.311841\pi$
$359$	21.1183	1.11458	0.557290	+	0.830318i	$0.311841\pi$
$360$	0	0
$361$	15.1179	0.795678
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.20041	0.219860
$366$	0	0
$367$	4.69181	0.244911	0.122455	−	0.992474i	$-0.460923\pi$
$367$	4.69181	0.244911	0.122455	+	0.992474i	$0.460923\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−27.3576	−1.42034
$372$	0	0
$373$	−5.84105	−0.302438	−0.151219	−	0.988500i	$-0.548320\pi$
$373$	−5.84105	−0.302438	−0.151219	+	0.988500i	$0.548320\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.06050	−0.209127
$378$	0	0
$379$	−29.5818	−1.51952	−0.759758	−	0.650206i	$-0.774683\pi$
$379$	−29.5818	−1.51952	−0.759758	+	0.650206i	$0.774683\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−34.9979	−1.78831	−0.894155	−	0.447758i	$-0.852223\pi$
$383$	−34.9979	−1.78831	−0.894155	+	0.447758i	$0.852223\pi$
$384$	0	0
$385$	−17.0730	−0.870118
$386$	0	0
$387$	0	0
$388$	0	0
$389$	20.8661	1.05795	0.528976	−	0.848637i	$-0.322576\pi$
$389$	20.8661	1.05795	0.528976	+	0.848637i	$0.322576\pi$
$390$	0	0
$391$	22.8643	1.15630
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.36655	0.270020
$396$	0	0
$397$	25.7282	1.29126	0.645631	−	0.763650i	$-0.276595\pi$
$397$	25.7282	1.29126	0.645631	+	0.763650i	$0.276595\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.9469	−0.596598	−0.298299	−	0.954472i	$-0.596419\pi$
$401$	−11.9469	−0.596598	−0.298299	+	0.954472i	$0.596419\pi$
$402$	0	0
$403$	−27.9593	−1.39275
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.61215	0.377320
$408$	0	0
$409$	−7.72591	−0.382022	−0.191011	−	0.981588i	$-0.561177\pi$
$409$	−7.72591	−0.382022	−0.191011	+	0.981588i	$0.561177\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.13705	0.154364
$414$	0	0
$415$	−6.88624	−0.338032
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−23.8656	−1.16591	−0.582955	−	0.812504i	$-0.698104\pi$
$419$	−23.8656	−1.16591	−0.582955	+	0.812504i	$0.698104\pi$
$420$	0	0
$421$	21.8038	1.06265	0.531327	−	0.847167i	$-0.321694\pi$
$421$	21.8038	1.06265	0.531327	+	0.847167i	$0.321694\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	28.7536	1.39475
$426$	0	0
$427$	23.3127	1.12818
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.8051	−0.953977	−0.476988	−	0.878910i	$-0.658272\pi$
$431$	−19.8051	−0.953977	−0.476988	+	0.878910i	$0.658272\pi$
$432$	0	0
$433$	11.8191	0.567992	0.283996	−	0.958825i	$-0.408340\pi$
$433$	11.8191	0.567992	0.283996	+	0.958825i	$0.408340\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.8519	0.949644
$438$	0	0
$439$	−29.6168	−1.41353	−0.706766	−	0.707448i	$-0.749846\pi$
$439$	−29.6168	−1.41353	−0.706766	+	0.707448i	$0.749846\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	33.0370	1.56963	0.784817	−	0.619727i	$-0.212757\pi$
$443$	33.0370	1.56963	0.784817	+	0.619727i	$0.212757\pi$
$444$	0	0
$445$	−1.09486	−0.0519013
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14.9043	−0.703377	−0.351689	−	0.936117i	$-0.614392\pi$
$449$	−14.9043	−0.703377	−0.351689	+	0.936117i	$0.614392\pi$
$450$	0	0
$451$	−55.2594	−2.60207
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.77743	0.317731
$456$	0	0
$457$	−36.5059	−1.70767	−0.853836	−	0.520541i	$-0.825730\pi$
$457$	−36.5059	−1.70767	−0.853836	+	0.520541i	$0.825730\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.89223	−0.134704	−0.0673522	−	0.997729i	$-0.521455\pi$
$461$	−2.89223	−0.134704	−0.0673522	+	0.997729i	$0.521455\pi$
$462$	0	0
$463$	−8.80832	−0.409357	−0.204679	−	0.978829i	$-0.565615\pi$
$463$	−8.80832	−0.409357	−0.204679	+	0.978829i	$0.565615\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.2607	−1.67794	−0.838972	−	0.544174i	$-0.816843\pi$
$467$	−36.2607	−1.67794	−0.838972	+	0.544174i	$0.816843\pi$
$468$	0	0
$469$	0.820770	0.0378996
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−49.2907	−2.26639
$474$	0	0
$475$	24.9652	1.14548
$476$	0	0
$477$	0	0
$478$	0	0
$479$	38.4081	1.75491	0.877455	−	0.479659i	$-0.159240\pi$
$479$	38.4081	1.75491	0.877455	+	0.479659i	$0.159240\pi$
$480$	0	0
$481$	−3.02178	−0.137781
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.21991	0.373247
$486$	0	0
$487$	14.9093	0.675603	0.337802	−	0.941217i	$-0.390317\pi$
$487$	14.9093	0.675603	0.337802	+	0.941217i	$0.390317\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−18.3987	−0.830321	−0.415160	−	0.909748i	$-0.636275\pi$
$491$	−18.3987	−0.830321	−0.415160	+	0.909748i	$0.636275\pi$
$492$	0	0
$493$	10.7727	0.485179
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.37403	−0.330771
$498$	0	0
$499$	26.8250	1.20085	0.600426	−	0.799680i	$-0.294998\pi$
$499$	26.8250	1.20085	0.600426	+	0.799680i	$0.294998\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.2136	0.901279	0.450640	−	0.892706i	$-0.351196\pi$
$503$	20.2136	0.901279	0.450640	+	0.892706i	$0.351196\pi$
$504$	0	0
$505$	9.60790	0.427546
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.40605	−0.416916	−0.208458	−	0.978031i	$-0.566844\pi$
$509$	−9.40605	−0.416916	−0.208458	+	0.978031i	$0.566844\pi$
$510$	0	0
$511$	15.4658	0.684166
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.8847	−0.479639
$516$	0	0
$517$	1.36433	0.0600031
$518$	0	0
$519$	0	0
$520$	0	0
$521$	23.0835	1.01131	0.505654	−	0.862736i	$-0.331251\pi$
$521$	23.0835	1.01131	0.505654	+	0.862736i	$0.331251\pi$
$522$	0	0
$523$	−36.1925	−1.58259	−0.791293	−	0.611437i	$-0.790592\pi$
$523$	−36.1925	−1.58259	−0.791293	+	0.611437i	$0.790592\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	74.1774	3.23122
$528$	0	0
$529$	−11.4490	−0.497781
$530$	0	0
$531$	0	0
$532$	0	0
$533$	21.9363	0.950165
$534$	0	0
$535$	3.58194	0.154861
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−18.1479	−0.781684
$540$	0	0
$541$	−18.9495	−0.814702	−0.407351	−	0.913272i	$-0.633547\pi$
$541$	−18.9495	−0.814702	−0.407351	+	0.913272i	$0.633547\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.68151	−0.243369
$546$	0	0
$547$	−43.7315	−1.86983	−0.934913	−	0.354878i	$-0.884523\pi$
$547$	−43.7315	−1.86983	−0.934913	+	0.354878i	$0.884523\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.35338	0.398467
$552$	0	0
$553$	19.7594	0.840256
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.8688	1.18084	0.590420	−	0.807096i	$-0.298962\pi$
$557$	27.8688	1.18084	0.590420	+	0.807096i	$0.298962\pi$
$558$	0	0
$559$	19.5668	0.827589
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.0479	1.26637	0.633185	−	0.774001i	$-0.281747\pi$
$563$	30.0479	1.26637	0.633185	+	0.774001i	$0.281747\pi$
$564$	0	0
$565$	−6.56012	−0.275986
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14.0792	−0.590229	−0.295115	−	0.955462i	$-0.595358\pi$
$569$	−14.0792	−0.590229	−0.295115	+	0.955462i	$0.595358\pi$
$570$	0	0
$571$	13.8750	0.580652	0.290326	−	0.956928i	$-0.406236\pi$
$571$	13.8750	0.580652	0.290326	+	0.956928i	$0.406236\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	14.5263	0.605788
$576$	0	0
$577$	37.7653	1.57219	0.786095	−	0.618106i	$-0.212100\pi$
$577$	37.7653	1.57219	0.786095	+	0.618106i	$0.212100\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−25.3549	−1.05190
$582$	0	0
$583$	55.7063	2.30712
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.61252	0.355477	0.177738	−	0.984078i	$-0.443122\pi$
$587$	8.61252	0.355477	0.177738	+	0.984078i	$0.443122\pi$
$588$	0	0
$589$	64.4043	2.65373
$590$	0	0
$591$	0	0
$592$	0	0
$593$	8.03473	0.329947	0.164973	−	0.986298i	$-0.447246\pi$
$593$	8.03473	0.329947	0.164973	+	0.986298i	$0.447246\pi$
$594$	0	0
$595$	−17.9808	−0.737143
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.40827	0.343553	0.171776	−	0.985136i	$-0.445049\pi$
$599$	8.40827	0.343553	0.171776	+	0.985136i	$0.445049\pi$
$600$	0	0
$601$	−5.39706	−0.220151	−0.110075	−	0.993923i	$-0.535109\pi$
$601$	−5.39706	−0.220151	−0.110075	+	0.993923i	$0.535109\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	25.3924	1.03235
$606$	0	0
$607$	−28.8773	−1.17209	−0.586047	−	0.810277i	$-0.699317\pi$
$607$	−28.8773	−1.17209	−0.586047	+	0.810277i	$0.699317\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.541595	−0.0219106
$612$	0	0
$613$	−22.2084	−0.896988	−0.448494	−	0.893786i	$-0.648039\pi$
$613$	−22.2084	−0.896988	−0.448494	+	0.893786i	$0.648039\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−33.7583	−1.35906	−0.679529	−	0.733649i	$-0.737816\pi$
$617$	−33.7583	−1.35906	−0.679529	+	0.733649i	$0.737816\pi$
$618$	0	0
$619$	14.3368	0.576245	0.288123	−	0.957594i	$-0.406969\pi$
$619$	14.3368	0.576245	0.288123	+	0.957594i	$0.406969\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.03123	−0.161508
$624$	0	0
$625$	14.6383	0.585532
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.01694	0.319656
$630$	0	0
$631$	−17.4989	−0.696620	−0.348310	−	0.937379i	$-0.613244\pi$
$631$	−17.4989	−0.696620	−0.348310	+	0.937379i	$0.613244\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	13.9078	0.551914
$636$	0	0
$637$	7.20413	0.285438
$638$	0	0
$639$	0	0
$640$	0	0
$641$	44.3840	1.75306	0.876532	−	0.481344i	$-0.159851\pi$
$641$	44.3840	1.75306	0.876532	+	0.481344i	$0.159851\pi$
$642$	0	0
$643$	14.3618	0.566376	0.283188	−	0.959064i	$-0.408608\pi$
$643$	14.3618	0.566376	0.283188	+	0.959064i	$0.408608\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.04331	−0.0410167	−0.0205084	−	0.999790i	$-0.506528\pi$
$647$	−1.04331	−0.0410167	−0.0205084	+	0.999790i	$0.506528\pi$
$648$	0	0
$649$	−6.38773	−0.250740
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15.5881	0.610011	0.305006	−	0.952351i	$-0.401342\pi$
$653$	15.5881	0.610011	0.305006	+	0.952351i	$0.401342\pi$
$654$	0	0
$655$	11.0996	0.433697
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.8740	0.968952	0.484476	−	0.874804i	$-0.339010\pi$
$659$	24.8740	0.968952	0.484476	+	0.874804i	$0.339010\pi$
$660$	0	0
$661$	−48.3482	−1.88053	−0.940263	−	0.340449i	$-0.889421\pi$
$661$	−48.3482	−1.88053	−0.940263	+	0.340449i	$0.889421\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−15.6118	−0.605400
$666$	0	0
$667$	5.44237	0.210729
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−47.4699	−1.83256
$672$	0	0
$673$	8.21411	0.316631	0.158315	−	0.987389i	$-0.449394\pi$
$673$	8.21411	0.316631	0.158315	+	0.987389i	$0.449394\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−23.2798	−0.894717	−0.447358	−	0.894355i	$-0.647635\pi$
$677$	−23.2798	−0.894717	−0.447358	+	0.894355i	$0.647635\pi$
$678$	0	0
$679$	30.2654	1.16148
$680$	0	0
$681$	0	0
$682$	0	0
$683$	35.1600	1.34536	0.672679	−	0.739934i	$-0.265143\pi$
$683$	35.1600	1.34536	0.672679	+	0.739934i	$0.265143\pi$
$684$	0	0
$685$	−13.8004	−0.527285
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−22.1136	−0.842463
$690$	0	0
$691$	−3.72267	−0.141617	−0.0708085	−	0.997490i	$-0.522558\pi$
$691$	−3.72267	−0.141617	−0.0708085	+	0.997490i	$0.522558\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.21671	−0.122017
$696$	0	0
$697$	−58.1980	−2.20440
$698$	0	0
$699$	0	0
$700$	0	0
$701$	32.6273	1.23232	0.616159	−	0.787622i	$-0.288688\pi$
$701$	32.6273	1.23232	0.616159	+	0.787622i	$0.288688\pi$
$702$	0	0
$703$	6.96068	0.262527
$704$	0	0
$705$	0	0
$706$	0	0
$707$	35.3759	1.33045
$708$	0	0
$709$	13.9781	0.524958	0.262479	−	0.964938i	$-0.415460\pi$
$709$	13.9781	0.524958	0.262479	+	0.964938i	$0.415460\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	37.4744	1.40343
$714$	0	0
$715$	−13.8004	−0.516104
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12.6076	0.470183	0.235091	−	0.971973i	$-0.424461\pi$
$719$	12.6076	0.470183	0.235091	+	0.971973i	$0.424461\pi$
$720$	0	0
$721$	−40.0772	−1.49255
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.84417	0.254186
$726$	0	0
$727$	3.47921	0.129037	0.0645184	−	0.997917i	$-0.479449\pi$
$727$	3.47921	0.129037	0.0645184	+	0.997917i	$0.479449\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−51.9118	−1.92003
$732$	0	0
$733$	−18.4830	−0.682684	−0.341342	−	0.939939i	$-0.610881\pi$
$733$	−18.4830	−0.682684	−0.341342	+	0.939939i	$0.610881\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.67127	−0.0615621
$738$	0	0
$739$	27.4469	1.00965	0.504825	−	0.863222i	$-0.331557\pi$
$739$	27.4469	1.00965	0.504825	+	0.863222i	$0.331557\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−45.2778	−1.66108	−0.830541	−	0.556957i	$-0.811969\pi$
$743$	−45.2778	−1.66108	−0.830541	+	0.556957i	$0.811969\pi$
$744$	0	0
$745$	15.2721	0.559527
$746$	0	0
$747$	0	0
$748$	0	0
$749$	13.1886	0.481900
$750$	0	0
$751$	−46.5567	−1.69888	−0.849439	−	0.527688i	$-0.823059\pi$
$751$	−46.5567	−1.69888	−0.849439	+	0.527688i	$0.823059\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.34084	−0.157979
$756$	0	0
$757$	−41.3427	−1.50263	−0.751313	−	0.659946i	$-0.770579\pi$
$757$	−41.3427	−1.50263	−0.751313	+	0.659946i	$0.770579\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−23.1699	−0.839908	−0.419954	−	0.907545i	$-0.637954\pi$
$761$	−23.1699	−0.839908	−0.419954	+	0.907545i	$0.637954\pi$
$762$	0	0
$763$	−20.9191	−0.757322
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.53573	0.0915598
$768$	0	0
$769$	−33.7297	−1.21632	−0.608162	−	0.793813i	$-0.708093\pi$
$769$	−33.7297	−1.21632	−0.608162	+	0.793813i	$0.708093\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−19.3697	−0.696680	−0.348340	−	0.937368i	$-0.613254\pi$
$773$	−19.3697	−0.696680	−0.348340	+	0.937368i	$0.613254\pi$
$774$	0	0
$775$	47.1268	1.69284
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−50.5302	−1.81043
$780$	0	0
$781$	15.0152	0.537286
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−17.2070	−0.614144
$786$	0	0
$787$	−30.7810	−1.09722	−0.548611	−	0.836078i	$-0.684843\pi$
$787$	−30.7810	−1.09722	−0.548611	+	0.836078i	$0.684843\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−24.1541	−0.858822
$792$	0	0
$793$	18.8441	0.669172
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.70950	0.202241	0.101120	−	0.994874i	$-0.467757\pi$
$797$	5.70950	0.202241	0.101120	+	0.994874i	$0.467757\pi$
$798$	0	0
$799$	1.43688	0.0508331
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−31.4918	−1.11132
$804$	0	0
$805$	−9.08391	−0.320166
$806$	0	0
$807$	0	0
$808$	0	0
$809$	52.2013	1.83530	0.917650	−	0.397390i	$-0.130084\pi$
$809$	52.2013	1.83530	0.917650	+	0.397390i	$0.130084\pi$
$810$	0	0
$811$	6.84966	0.240524	0.120262	−	0.992742i	$-0.461626\pi$
$811$	6.84966	0.240524	0.120262	+	0.992742i	$0.461626\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.50796	−0.227964
$816$	0	0
$817$	−45.0722	−1.57688
$818$	0	0
$819$	0	0
$820$	0	0
$821$	14.9734	0.522575	0.261287	−	0.965261i	$-0.415853\pi$
$821$	14.9734	0.522575	0.261287	+	0.965261i	$0.415853\pi$
$822$	0	0
$823$	28.0224	0.976799	0.488399	−	0.872620i	$-0.337581\pi$
$823$	28.0224	0.976799	0.488399	+	0.872620i	$0.337581\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.4527	1.33713	0.668566	−	0.743652i	$-0.266908\pi$
$827$	38.4527	1.33713	0.668566	+	0.743652i	$0.266908\pi$
$828$	0	0
$829$	−54.8364	−1.90455	−0.952273	−	0.305248i	$-0.901261\pi$
$829$	−54.8364	−1.90455	−0.952273	+	0.305248i	$0.901261\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−19.1129	−0.662224
$834$	0	0
$835$	−15.8906	−0.549917
$836$	0	0
$837$	0	0
$838$	0	0
$839$	11.3598	0.392185	0.196093	−	0.980585i	$-0.437175\pi$
$839$	11.3598	0.392185	0.196093	+	0.980585i	$0.437175\pi$
$840$	0	0
$841$	−26.4358	−0.911579
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−5.59773	−0.192568
$846$	0	0
$847$	93.4937	3.21248
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.05015	0.138837
$852$	0	0
$853$	2.73152	0.0935255	0.0467627	−	0.998906i	$-0.485110\pi$
$853$	2.73152	0.0935255	0.0467627	+	0.998906i	$0.485110\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−56.0728	−1.91541	−0.957706	−	0.287749i	$-0.907093\pi$
$857$	−56.0728	−1.91541	−0.957706	+	0.287749i	$0.907093\pi$
$858$	0	0
$859$	6.13991	0.209491	0.104746	−	0.994499i	$-0.466597\pi$
$859$	6.13991	0.209491	0.104746	+	0.994499i	$0.466597\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38.7746	−1.31990	−0.659952	−	0.751308i	$-0.729423\pi$
$863$	−38.7746	−1.31990	−0.659952	+	0.751308i	$0.729423\pi$
$864$	0	0
$865$	8.87699	0.301827
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−40.2347	−1.36487
$870$	0	0
$871$	0.663442	0.0224799
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−24.7875	−0.837972
$876$	0	0
$877$	−13.4025	−0.452571	−0.226286	−	0.974061i	$-0.572658\pi$
$877$	−13.4025	−0.452571	−0.226286	+	0.974061i	$0.572658\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−20.4389	−0.688604	−0.344302	−	0.938859i	$-0.611884\pi$
$881$	−20.4389	−0.688604	−0.344302	+	0.938859i	$0.611884\pi$
$882$	0	0
$883$	10.8059	0.363646	0.181823	−	0.983331i	$-0.441800\pi$
$883$	10.8059	0.363646	0.181823	+	0.983331i	$0.441800\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17.1139	−0.574628	−0.287314	−	0.957836i	$-0.592762\pi$
$887$	−17.1139	−0.574628	−0.287314	+	0.957836i	$0.592762\pi$
$888$	0	0
$889$	51.2080	1.71746
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.24756	0.0417482
$894$	0	0
$895$	19.2706	0.644146
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.6564	0.588873
$900$	0	0
$901$	58.6686	1.95453
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.02225	−0.200186
$906$	0	0
$907$	7.04930	0.234068	0.117034	−	0.993128i	$-0.462661\pi$
$907$	7.04930	0.234068	0.117034	+	0.993128i	$0.462661\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6.05039	−0.200458	−0.100229	−	0.994964i	$-0.531958\pi$
$911$	−6.05039	−0.200458	−0.100229	+	0.994964i	$0.531958\pi$
$912$	0	0
$913$	51.6282	1.70865
$914$	0	0
$915$	0	0
$916$	0	0
$917$	40.8683	1.34959
$918$	0	0
$919$	−7.04655	−0.232444	−0.116222	−	0.993223i	$-0.537078\pi$
$919$	−7.04655	−0.232444	−0.116222	+	0.993223i	$0.537078\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.96056	−0.196194
$924$	0	0
$925$	5.09336	0.167469
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−17.7183	−0.581319	−0.290659	−	0.956827i	$-0.593875\pi$
$929$	−17.7183	−0.581319	−0.290659	+	0.956827i	$0.593875\pi$
$930$	0	0
$931$	−16.5947	−0.543870
$932$	0	0
$933$	0	0
$934$	0	0
$935$	36.6131	1.19737
$936$	0	0
$937$	22.1685	0.724214	0.362107	−	0.932136i	$-0.382057\pi$
$937$	22.1685	0.724214	0.362107	+	0.932136i	$0.382057\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.75662	−0.0572641	−0.0286320	−	0.999590i	$-0.509115\pi$
$941$	−1.75662	−0.0572641	−0.0286320	+	0.999590i	$0.509115\pi$
$942$	0	0
$943$	−29.4016	−0.957447
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−55.3738	−1.79941	−0.899703	−	0.436502i	$-0.856217\pi$
$947$	−55.3738	−1.79941	−0.899703	+	0.436502i	$0.856217\pi$
$948$	0	0
$949$	12.5013	0.405808
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−25.1588	−0.814974	−0.407487	−	0.913211i	$-0.633595\pi$
$953$	−25.1588	−0.814974	−0.407487	+	0.913211i	$0.633595\pi$
$954$	0	0
$955$	−21.8243	−0.706218
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−50.8124	−1.64082
$960$	0	0
$961$	90.5760	2.92181
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.445833	−0.0143519
$966$	0	0
$967$	0.0734189	0.00236099	0.00118050	−	0.999999i	$-0.499624\pi$
$967$	0.0734189	0.00236099	0.00118050	+	0.999999i	$0.499624\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.97610	0.255965	0.127983	−	0.991776i	$-0.459150\pi$
$971$	7.97610	0.255965	0.127983	+	0.991776i	$0.459150\pi$
$972$	0	0
$973$	−11.8438	−0.379695
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.94166	0.126105	0.0630524	−	0.998010i	$-0.479916\pi$
$977$	3.94166	0.126105	0.0630524	+	0.998010i	$0.479916\pi$
$978$	0	0
$979$	8.20850	0.262345
$980$	0	0
$981$	0	0
$982$	0	0
$983$	14.1049	0.449877	0.224938	−	0.974373i	$-0.427782\pi$
$983$	14.1049	0.449877	0.224938	+	0.974373i	$0.427782\pi$
$984$	0	0
$985$	13.6134	0.433759
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−26.2258	−0.833932
$990$	0	0
$991$	−57.0752	−1.81305	−0.906527	−	0.422147i	$-0.861277\pi$
$991$	−57.0752	−1.81305	−0.906527	+	0.422147i	$0.861277\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9.80421	−0.310814
$996$	0	0
$997$	31.2175	0.988668	0.494334	−	0.869272i	$-0.335412\pi$
$997$	31.2175	0.988668	0.494334	+	0.869272i	$0.335412\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2124.2.a.i.1.3	✓	4
3.2	odd	2		2124.2.a.j.1.2	yes	4
4.3	odd	2		8496.2.a.bs.1.3		4
12.11	even	2		8496.2.a.bp.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2124.2.a.i.1.3	✓	4	1.1	even	1	trivial
2124.2.a.j.1.2	yes	4	3.2	odd	2
8496.2.a.bp.1.2		4	12.11	even	2
8496.2.a.bs.1.3		4	4.3	odd	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:	\( N \)	\(=\)	\( 2124 = 2^{2} \cdot 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2124.a (trivial)

\(f(q)\)	\(=\)	\(q+0.852003 q^{5} +3.13705 q^{7} +O(q^{10})\)	\(q+0.852003 q^{5} +3.13705 q^{7} -6.38773 q^{11} +2.53573 q^{13} -6.72741 q^{17} -5.84105 q^{19} -3.39868 q^{23} -4.27409 q^{25} -1.60132 q^{29} -11.0261 q^{31} +2.67277 q^{35} -1.19168 q^{37} +8.65087 q^{41} +7.71646 q^{43} -0.213586 q^{47} +2.84105 q^{49} -8.72083 q^{53} -5.44237 q^{55} +1.00000 q^{59} +7.43142 q^{61} +2.16045 q^{65} +0.261638 q^{67} -2.35063 q^{71} +4.93004 q^{73} -20.0386 q^{77} +6.29874 q^{79} -8.08241 q^{83} -5.73178 q^{85} -1.28504 q^{89} +7.95469 q^{91} -4.97660 q^{95} +9.64775 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{7}+O(q^{10}) \) 4 * q + 2 * q^7	\( 4 q + 2 q^{7} - 10 q^{11} - 2 q^{13} - 4 q^{17} - 6 q^{19} - 12 q^{23} + 4 q^{25} - 8 q^{29} - 8 q^{31} - 12 q^{35} + 6 q^{37} - 16 q^{41} - 6 q^{43} - 18 q^{47} - 6 q^{49} - 4 q^{53} - 6 q^{55} + 4 q^{59} - 18 q^{65} + 10 q^{67} - 16 q^{71} - 14 q^{77} + 12 q^{79} - 22 q^{83} - 6 q^{85} + 2 q^{89} + 20 q^{91} - 36 q^{95} + 6 q^{97}+O(q^{100}) \) 4 * q + 2 * q^7 - 10 * q^11 - 2 * q^13 - 4 * q^17 - 6 * q^19 - 12 * q^23 + 4 * q^25 - 8 * q^29 - 8 * q^31 - 12 * q^35 + 6 * q^37 - 16 * q^41 - 6 * q^43 - 18 * q^47 - 6 * q^49 - 4 * q^53 - 6 * q^55 + 4 * q^59 - 18 * q^65 + 10 * q^67 - 16 * q^71 - 14 * q^77 + 12 * q^79 - 22 * q^83 - 6 * q^85 + 2 * q^89 + 20 * q^91 - 36 * q^95 + 6 * q^97

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