LMFDB - Embedded newform 2352.4.a.ca.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1345}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 336 $ x^2 - x - 336
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$18.8371$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -20.8371 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -20.8371 q^{5} +9.00000 q^{9} -15.1629 q^{11} -2.16288 q^{13} -62.5114 q^{15} +119.348 q^{17} -33.5114 q^{19} -0.651517 q^{23} +309.186 q^{25} +27.0000 q^{27} -163.208 q^{29} -223.326 q^{31} -45.4886 q^{33} +168.534 q^{37} -6.48864 q^{39} +323.023 q^{41} -221.557 q^{43} -187.534 q^{45} +508.045 q^{47} +358.045 q^{51} -176.511 q^{53} +315.951 q^{55} -100.534 q^{57} +454.928 q^{59} -38.6515 q^{61} +45.0682 q^{65} -141.792 q^{67} -1.95455 q^{69} -602.742 q^{71} +1102.30 q^{73} +927.557 q^{75} +116.303 q^{79} +81.0000 q^{81} -568.928 q^{83} -2486.88 q^{85} -489.625 q^{87} +383.159 q^{89} -669.977 q^{93} +698.280 q^{95} -334.701 q^{97} -136.466 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 5 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 5 * q^5 + 18 * q^9	$ 2 q + 6 q^{3} - 5 q^{5} + 18 q^{9} - 67 q^{11} - 41 q^{13} - 15 q^{15} + 92 q^{17} + 43 q^{19} - 148 q^{23} + 435 q^{25} + 54 q^{27} + 77 q^{29} - 520 q^{31} - 201 q^{33} + 7 q^{37} - 123 q^{39} + 426 q^{41} + 107 q^{43} - 45 q^{45} + 576 q^{47} + 276 q^{51} - 243 q^{53} - 505 q^{55} + 129 q^{57} - 7 q^{59} - 224 q^{61} - 570 q^{65} - 687 q^{67} - 444 q^{69} - 472 q^{71} + 921 q^{73} + 1305 q^{75} + 526 q^{79} + 162 q^{81} - 221 q^{83} - 2920 q^{85} + 231 q^{87} - 774 q^{89} - 1560 q^{93} + 1910 q^{95} - 1953 q^{97} - 603 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 5 * q^5 + 18 * q^9 - 67 * q^11 - 41 * q^13 - 15 * q^15 + 92 * q^17 + 43 * q^19 - 148 * q^23 + 435 * q^25 + 54 * q^27 + 77 * q^29 - 520 * q^31 - 201 * q^33 + 7 * q^37 - 123 * q^39 + 426 * q^41 + 107 * q^43 - 45 * q^45 + 576 * q^47 + 276 * q^51 - 243 * q^53 - 505 * q^55 + 129 * q^57 - 7 * q^59 - 224 * q^61 - 570 * q^65 - 687 * q^67 - 444 * q^69 - 472 * q^71 + 921 * q^73 + 1305 * q^75 + 526 * q^79 + 162 * q^81 - 221 * q^83 - 2920 * q^85 + 231 * q^87 - 774 * q^89 - 1560 * q^93 + 1910 * q^95 - 1953 * q^97 - 603 * q^99

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−20.8371	−1.86373	−0.931864	−	0.362807i	$-0.881818\pi$
$5$	−20.8371	−1.86373	−0.931864	+	0.362807i	$0.881818\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−15.1629	−0.415616	−0.207808	−	0.978170i	$-0.566633\pi$
$11$	−15.1629	−0.415616	−0.207808	+	0.978170i	$0.566633\pi$
$12$	0	0
$13$	−2.16288	−0.0461442	−0.0230721	−	0.999734i	$-0.507345\pi$
$13$	−2.16288	−0.0461442	−0.0230721	+	0.999734i	$0.507345\pi$
$14$	0	0
$15$	−62.5114	−1.07602
$16$	0	0
$17$	119.348	1.70272	0.851361	−	0.524581i	$-0.175778\pi$
$17$	119.348	1.70272	0.851361	+	0.524581i	$0.175778\pi$
$18$	0	0
$19$	−33.5114	−0.404633	−0.202317	−	0.979320i	$-0.564847\pi$
$19$	−33.5114	−0.404633	−0.202317	+	0.979320i	$0.564847\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.651517	−0.00590655	−0.00295327	−	0.999996i	$-0.500940\pi$
$23$	−0.651517	−0.00590655	−0.00295327	+	0.999996i	$0.500940\pi$
$24$	0	0
$25$	309.186	2.47348
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−163.208	−1.04507	−0.522535	−	0.852618i	$-0.675014\pi$
$29$	−163.208	−1.04507	−0.522535	+	0.852618i	$0.675014\pi$
$30$	0	0
$31$	−223.326	−1.29389	−0.646943	−	0.762538i	$-0.723953\pi$
$31$	−223.326	−1.29389	−0.646943	+	0.762538i	$0.723953\pi$
$32$	0	0
$33$	−45.4886	−0.239956
$34$	0	0
$35$	0	0
$36$	0	0
$37$	168.534	0.748833	0.374417	−	0.927261i	$-0.377843\pi$
$37$	168.534	0.748833	0.374417	+	0.927261i	$0.377843\pi$
$38$	0	0
$39$	−6.48864	−0.0266414
$40$	0	0
$41$	323.023	1.23043	0.615216	−	0.788359i	$-0.289069\pi$
$41$	323.023	1.23043	0.615216	+	0.788359i	$0.289069\pi$
$42$	0	0
$43$	−221.557	−0.785746	−0.392873	−	0.919593i	$-0.628519\pi$
$43$	−221.557	−0.785746	−0.392873	+	0.919593i	$0.628519\pi$
$44$	0	0
$45$	−187.534	−0.621243
$46$	0	0
$47$	508.045	1.57672	0.788362	−	0.615211i	$-0.210929\pi$
$47$	508.045	1.57672	0.788362	+	0.615211i	$0.210929\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	358.045	0.983066
$52$	0	0
$53$	−176.511	−0.457466	−0.228733	−	0.973489i	$-0.573458\pi$
$53$	−176.511	−0.457466	−0.228733	+	0.973489i	$0.573458\pi$
$54$	0	0
$55$	315.951	0.774596
$56$	0	0
$57$	−100.534	−0.233615
$58$	0	0
$59$	454.928	1.00384	0.501920	−	0.864914i	$-0.332627\pi$
$59$	454.928	1.00384	0.501920	+	0.864914i	$0.332627\pi$
$60$	0	0
$61$	−38.6515	−0.0811282	−0.0405641	−	0.999177i	$-0.512915\pi$
$61$	−38.6515	−0.0811282	−0.0405641	+	0.999177i	$0.512915\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	45.0682	0.0860003
$66$	0	0
$67$	−141.792	−0.258546	−0.129273	−	0.991609i	$-0.541264\pi$
$67$	−141.792	−0.258546	−0.129273	+	0.991609i	$0.541264\pi$
$68$	0	0
$69$	−1.95455	−0.00341015
$70$	0	0
$71$	−602.742	−1.00750	−0.503749	−	0.863850i	$-0.668046\pi$
$71$	−602.742	−1.00750	−0.503749	+	0.863850i	$0.668046\pi$
$72$	0	0
$73$	1102.30	1.76732	0.883660	−	0.468129i	$-0.155072\pi$
$73$	1102.30	1.76732	0.883660	+	0.468129i	$0.155072\pi$
$74$	0	0
$75$	927.557	1.42807
$76$	0	0
$77$	0	0
$78$	0	0
$79$	116.303	0.165634	0.0828172	−	0.996565i	$-0.473608\pi$
$79$	116.303	0.165634	0.0828172	+	0.996565i	$0.473608\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−568.928	−0.752385	−0.376193	−	0.926542i	$-0.622767\pi$
$83$	−568.928	−0.752385	−0.376193	+	0.926542i	$0.622767\pi$
$84$	0	0
$85$	−2486.88	−3.17341
$86$	0	0
$87$	−489.625	−0.603371
$88$	0	0
$89$	383.159	0.456346	0.228173	−	0.973621i	$-0.426725\pi$
$89$	383.159	0.456346	0.228173	+	0.973621i	$0.426725\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−669.977	−0.747026
$94$	0	0
$95$	698.280	0.754127
$96$	0	0
$97$	−334.701	−0.350348	−0.175174	−	0.984538i	$-0.556049\pi$
$97$	−334.701	−0.350348	−0.175174	+	0.984538i	$0.556049\pi$
$98$	0	0
$99$	−136.466	−0.138539
$100$	0	0
$101$	14.7424	0.0145240	0.00726201	−	0.999974i	$-0.497688\pi$
$101$	14.7424	0.0145240	0.00726201	+	0.999974i	$0.497688\pi$
$102$	0	0
$103$	−841.420	−0.804928	−0.402464	−	0.915436i	$-0.631846\pi$
$103$	−841.420	−0.804928	−0.402464	+	0.915436i	$0.631846\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	715.670	0.646603	0.323301	−	0.946296i	$-0.395207\pi$
$107$	715.670	0.646603	0.323301	+	0.946296i	$0.395207\pi$
$108$	0	0
$109$	600.019	0.527260	0.263630	−	0.964624i	$-0.415080\pi$
$109$	600.019	0.527260	0.263630	+	0.964624i	$0.415080\pi$
$110$	0	0
$111$	505.602	0.432339
$112$	0	0
$113$	622.644	0.518349	0.259174	−	0.965831i	$-0.416550\pi$
$113$	622.644	0.518349	0.259174	+	0.965831i	$0.416550\pi$
$114$	0	0
$115$	13.5757	0.0110082
$116$	0	0
$117$	−19.4659	−0.0153814
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1101.09	−0.827263
$122$	0	0
$123$	969.068	0.710390
$124$	0	0
$125$	−3837.90	−2.74618
$126$	0	0
$127$	180.076	0.125820	0.0629100	−	0.998019i	$-0.479962\pi$
$127$	180.076	0.125820	0.0629100	+	0.998019i	$0.479962\pi$
$128$	0	0
$129$	−664.670	−0.453651
$130$	0	0
$131$	217.860	0.145302	0.0726508	−	0.997357i	$-0.476854\pi$
$131$	217.860	0.145302	0.0726508	+	0.997357i	$0.476854\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−562.602	−0.358675
$136$	0	0
$137$	−2601.86	−1.62257	−0.811283	−	0.584654i	$-0.801230\pi$
$137$	−2601.86	−1.62257	−0.811283	+	0.584654i	$0.801230\pi$
$138$	0	0
$139$	−2651.55	−1.61800	−0.808998	−	0.587811i	$-0.799990\pi$
$139$	−2651.55	−1.61800	−0.808998	+	0.587811i	$0.799990\pi$
$140$	0	0
$141$	1524.14	0.910322
$142$	0	0
$143$	32.7955	0.0191783
$144$	0	0
$145$	3400.79	1.94773
$146$	0	0
$147$	0	0
$148$	0	0
$149$	581.023	0.319458	0.159729	−	0.987161i	$-0.448938\pi$
$149$	581.023	0.319458	0.159729	+	0.987161i	$0.448938\pi$
$150$	0	0
$151$	615.390	0.331654	0.165827	−	0.986155i	$-0.446971\pi$
$151$	615.390	0.331654	0.165827	+	0.986155i	$0.446971\pi$
$152$	0	0
$153$	1074.14	0.567574
$154$	0	0
$155$	4653.47	2.41145
$156$	0	0
$157$	306.932	0.156024	0.0780122	−	0.996952i	$-0.475143\pi$
$157$	306.932	0.156024	0.0780122	+	0.996952i	$0.475143\pi$
$158$	0	0
$159$	−529.534	−0.264118
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3514.50	−1.68882	−0.844408	−	0.535701i	$-0.820047\pi$
$163$	−3514.50	−1.68882	−0.844408	+	0.535701i	$0.820047\pi$
$164$	0	0
$165$	947.852	0.447213
$166$	0	0
$167$	1123.30	0.520502	0.260251	−	0.965541i	$-0.416195\pi$
$167$	1123.30	0.520502	0.260251	+	0.965541i	$0.416195\pi$
$168$	0	0
$169$	−2192.32	−0.997871
$170$	0	0
$171$	−301.602	−0.134878
$172$	0	0
$173$	−1530.60	−0.672655	−0.336327	−	0.941745i	$-0.609185\pi$
$173$	−1530.60	−0.672655	−0.336327	+	0.941745i	$0.609185\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1364.78	0.579568
$178$	0	0
$179$	−3413.43	−1.42532	−0.712659	−	0.701511i	$-0.752509\pi$
$179$	−3413.43	−1.42532	−0.712659	+	0.701511i	$0.752509\pi$
$180$	0	0
$181$	−1286.71	−0.528399	−0.264200	−	0.964468i	$-0.585108\pi$
$181$	−1286.71	−0.528399	−0.264200	+	0.964468i	$0.585108\pi$
$182$	0	0
$183$	−115.955	−0.0468394
$184$	0	0
$185$	−3511.77	−1.39562
$186$	0	0
$187$	−1809.67	−0.707679
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1055.30	−0.399783	−0.199891	−	0.979818i	$-0.564059\pi$
$191$	−1055.30	−0.399783	−0.199891	+	0.979818i	$0.564059\pi$
$192$	0	0
$193$	−4770.84	−1.77934	−0.889670	−	0.456604i	$-0.849066\pi$
$193$	−4770.84	−1.77934	−0.889670	+	0.456604i	$0.849066\pi$
$194$	0	0
$195$	135.205	0.0496523
$196$	0	0
$197$	1622.31	0.586725	0.293363	−	0.956001i	$-0.405226\pi$
$197$	1622.31	0.586725	0.293363	+	0.956001i	$0.405226\pi$
$198$	0	0
$199$	−3550.14	−1.26464	−0.632318	−	0.774709i	$-0.717896\pi$
$199$	−3550.14	−1.26464	−0.632318	+	0.774709i	$0.717896\pi$
$200$	0	0
$201$	−425.375	−0.149272
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6730.86	−2.29319
$206$	0	0
$207$	−5.86365	−0.00196885
$208$	0	0
$209$	508.129	0.168172
$210$	0	0
$211$	−4653.39	−1.51826	−0.759129	−	0.650941i	$-0.774375\pi$
$211$	−4653.39	−1.51826	−0.759129	+	0.650941i	$0.774375\pi$
$212$	0	0
$213$	−1808.23	−0.581679
$214$	0	0
$215$	4616.61	1.46442
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3306.90	1.02036
$220$	0	0
$221$	−258.136	−0.0785707
$222$	0	0
$223$	−4649.53	−1.39621	−0.698107	−	0.715993i	$-0.745974\pi$
$223$	−4649.53	−1.39621	−0.698107	+	0.715993i	$0.745974\pi$
$224$	0	0
$225$	2782.67	0.824495
$226$	0	0
$227$	4151.72	1.21392	0.606958	−	0.794734i	$-0.292389\pi$
$227$	4151.72	1.21392	0.606958	+	0.794734i	$0.292389\pi$
$228$	0	0
$229$	−4263.63	−1.23034	−0.615172	−	0.788393i	$-0.710913\pi$
$229$	−4263.63	−1.23034	−0.615172	+	0.788393i	$0.710913\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3049.90	0.857535	0.428768	−	0.903415i	$-0.358948\pi$
$233$	3049.90	0.857535	0.428768	+	0.903415i	$0.358948\pi$
$234$	0	0
$235$	−10586.2	−2.93859
$236$	0	0
$237$	348.909	0.0956290
$238$	0	0
$239$	−3987.20	−1.07912	−0.539562	−	0.841946i	$-0.681410\pi$
$239$	−3987.20	−1.07912	−0.539562	+	0.841946i	$0.681410\pi$
$240$	0	0
$241$	624.648	0.166959	0.0834795	−	0.996509i	$-0.473397\pi$
$241$	624.648	0.166959	0.0834795	+	0.996509i	$0.473397\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	72.4810	0.0186715
$248$	0	0
$249$	−1706.78	−0.434390
$250$	0	0
$251$	−1328.78	−0.334152	−0.167076	−	0.985944i	$-0.553432\pi$
$251$	−1328.78	−0.334152	−0.167076	+	0.985944i	$0.553432\pi$
$252$	0	0
$253$	9.87887	0.00245486
$254$	0	0
$255$	−7460.64	−1.83217
$256$	0	0
$257$	3226.18	0.783049	0.391525	−	0.920168i	$-0.371948\pi$
$257$	3226.18	0.783049	0.391525	+	0.920168i	$0.371948\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1468.87	−0.348357
$262$	0	0
$263$	−3250.61	−0.762135	−0.381067	−	0.924547i	$-0.624443\pi$
$263$	−3250.61	−0.762135	−0.381067	+	0.924547i	$0.624443\pi$
$264$	0	0
$265$	3677.99	0.852593
$266$	0	0
$267$	1149.48	0.263471
$268$	0	0
$269$	−2826.04	−0.640546	−0.320273	−	0.947325i	$-0.603775\pi$
$269$	−2826.04	−0.640546	−0.320273	+	0.947325i	$0.603775\pi$
$270$	0	0
$271$	−2396.77	−0.537245	−0.268622	−	0.963246i	$-0.586568\pi$
$271$	−2396.77	−0.537245	−0.268622	+	0.963246i	$0.586568\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4688.14	−1.02802
$276$	0	0
$277$	1820.47	0.394878	0.197439	−	0.980315i	$-0.436738\pi$
$277$	1820.47	0.394878	0.197439	+	0.980315i	$0.436738\pi$
$278$	0	0
$279$	−2009.93	−0.431296
$280$	0	0
$281$	3083.81	0.654679	0.327339	−	0.944907i	$-0.393848\pi$
$281$	3083.81	0.654679	0.327339	+	0.944907i	$0.393848\pi$
$282$	0	0
$283$	2554.77	0.536626	0.268313	−	0.963332i	$-0.413534\pi$
$283$	2554.77	0.536626	0.268313	+	0.963332i	$0.413534\pi$
$284$	0	0
$285$	2094.84	0.435395
$286$	0	0
$287$	0	0
$288$	0	0
$289$	9331.06	1.89926
$290$	0	0
$291$	−1004.10	−0.202273
$292$	0	0
$293$	1846.47	0.368163	0.184081	−	0.982911i	$-0.441069\pi$
$293$	1846.47	0.368163	0.184081	+	0.982911i	$0.441069\pi$
$294$	0	0
$295$	−9479.39	−1.87089
$296$	0	0
$297$	−409.398	−0.0799854
$298$	0	0
$299$	1.40915	0.000272553 0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	44.2272	0.00838544
$304$	0	0
$305$	805.386	0.151201
$306$	0	0
$307$	7041.50	1.30905	0.654527	−	0.756039i	$-0.272868\pi$
$307$	7041.50	1.30905	0.654527	+	0.756039i	$0.272868\pi$
$308$	0	0
$309$	−2524.26	−0.464726
$310$	0	0
$311$	−2685.99	−0.489738	−0.244869	−	0.969556i	$-0.578745\pi$
$311$	−2685.99	−0.489738	−0.244869	+	0.969556i	$0.578745\pi$
$312$	0	0
$313$	−2219.19	−0.400754	−0.200377	−	0.979719i	$-0.564217\pi$
$313$	−2219.19	−0.400754	−0.200377	+	0.979719i	$0.564217\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2221.26	0.393560	0.196780	−	0.980448i	$-0.436952\pi$
$317$	2221.26	0.393560	0.196780	+	0.980448i	$0.436952\pi$
$318$	0	0
$319$	2474.71	0.434348
$320$	0	0
$321$	2147.01	0.373316
$322$	0	0
$323$	−3999.53	−0.688978
$324$	0	0
$325$	−668.731	−0.114137
$326$	0	0
$327$	1800.06	0.304414
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4154.06	−0.689812	−0.344906	−	0.938637i	$-0.612089\pi$
$331$	−4154.06	−0.689812	−0.344906	+	0.938637i	$0.612089\pi$
$332$	0	0
$333$	1516.81	0.249611
$334$	0	0
$335$	2954.53	0.481860
$336$	0	0
$337$	−254.167	−0.0410841	−0.0205420	−	0.999789i	$-0.506539\pi$
$337$	−254.167	−0.0410841	−0.0205420	+	0.999789i	$0.506539\pi$
$338$	0	0
$339$	1867.93	0.299269
$340$	0	0
$341$	3386.26	0.537761
$342$	0	0
$343$	0	0
$344$	0	0
$345$	40.7272	0.00635559
$346$	0	0
$347$	6224.64	0.962986	0.481493	−	0.876450i	$-0.340095\pi$
$347$	6224.64	0.962986	0.481493	+	0.876450i	$0.340095\pi$
$348$	0	0
$349$	−9732.21	−1.49270	−0.746352	−	0.665552i	$-0.768196\pi$
$349$	−9732.21	−1.49270	−0.746352	+	0.665552i	$0.768196\pi$
$350$	0	0
$351$	−58.3977	−0.00888046
$352$	0	0
$353$	−1425.61	−0.214951	−0.107476	−	0.994208i	$-0.534277\pi$
$353$	−1425.61	−0.214951	−0.107476	+	0.994208i	$0.534277\pi$
$354$	0	0
$355$	12559.4	1.87770
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5766.49	0.847754	0.423877	−	0.905720i	$-0.360669\pi$
$359$	5766.49	0.847754	0.423877	+	0.905720i	$0.360669\pi$
$360$	0	0
$361$	−5735.99	−0.836272
$362$	0	0
$363$	−3303.26	−0.477621
$364$	0	0
$365$	−22968.7	−3.29381
$366$	0	0
$367$	11545.3	1.64213	0.821065	−	0.570834i	$-0.193380\pi$
$367$	11545.3	1.64213	0.821065	+	0.570834i	$0.193380\pi$
$368$	0	0
$369$	2907.20	0.410144
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6479.57	−0.899463	−0.449731	−	0.893164i	$-0.648480\pi$
$373$	−6479.57	−0.899463	−0.449731	+	0.893164i	$0.648480\pi$
$374$	0	0
$375$	−11513.7	−1.58551
$376$	0	0
$377$	353.000	0.0482239
$378$	0	0
$379$	−611.996	−0.0829449	−0.0414725	−	0.999140i	$-0.513205\pi$
$379$	−611.996	−0.0829449	−0.0414725	+	0.999140i	$0.513205\pi$
$380$	0	0
$381$	540.227	0.0726422
$382$	0	0
$383$	4360.81	0.581794	0.290897	−	0.956754i	$-0.406046\pi$
$383$	4360.81	0.581794	0.290897	+	0.956754i	$0.406046\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1994.01	−0.261915
$388$	0	0
$389$	−13146.9	−1.71356	−0.856781	−	0.515681i	$-0.827539\pi$
$389$	−13146.9	−1.71356	−0.856781	+	0.515681i	$0.827539\pi$
$390$	0	0
$391$	−77.7575	−0.0100572
$392$	0	0
$393$	653.580	0.0838899
$394$	0	0
$395$	−2423.42	−0.308697
$396$	0	0
$397$	8478.04	1.07179	0.535895	−	0.844285i	$-0.319974\pi$
$397$	8478.04	1.07179	0.535895	+	0.844285i	$0.319974\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2803.00	0.349065	0.174533	−	0.984651i	$-0.444159\pi$
$401$	2803.00	0.349065	0.174533	+	0.984651i	$0.444159\pi$
$402$	0	0
$403$	483.027	0.0597054
$404$	0	0
$405$	−1687.81	−0.207081
$406$	0	0
$407$	−2555.46	−0.311227
$408$	0	0
$409$	6385.39	0.771973	0.385987	−	0.922504i	$-0.373861\pi$
$409$	6385.39	0.771973	0.385987	+	0.922504i	$0.373861\pi$
$410$	0	0
$411$	−7805.57	−0.936789
$412$	0	0
$413$	0	0
$414$	0	0
$415$	11854.8	1.40224
$416$	0	0
$417$	−7954.65	−0.934151
$418$	0	0
$419$	4831.66	0.563346	0.281673	−	0.959510i	$-0.409111\pi$
$419$	4831.66	0.563346	0.281673	+	0.959510i	$0.409111\pi$
$420$	0	0
$421$	7475.37	0.865385	0.432693	−	0.901542i	$-0.357564\pi$
$421$	7475.37	0.865385	0.432693	+	0.901542i	$0.357564\pi$
$422$	0	0
$423$	4572.41	0.525575
$424$	0	0
$425$	36900.8	4.21165
$426$	0	0
$427$	0	0
$428$	0	0
$429$	98.3864	0.0110726
$430$	0	0
$431$	−6991.93	−0.781414	−0.390707	−	0.920515i	$-0.627769\pi$
$431$	−6991.93	−0.781414	−0.390707	+	0.920515i	$0.627769\pi$
$432$	0	0
$433$	7699.26	0.854510	0.427255	−	0.904131i	$-0.359481\pi$
$433$	7699.26	0.854510	0.427255	+	0.904131i	$0.359481\pi$
$434$	0	0
$435$	10202.4	1.12452
$436$	0	0
$437$	21.8332	0.00238999
$438$	0	0
$439$	9412.32	1.02329	0.511646	−	0.859196i	$-0.329036\pi$
$439$	9412.32	1.02329	0.511646	+	0.859196i	$0.329036\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6258.18	0.671185	0.335593	−	0.942007i	$-0.391063\pi$
$443$	6258.18	0.671185	0.335593	+	0.942007i	$0.391063\pi$
$444$	0	0
$445$	−7983.93	−0.850505
$446$	0	0
$447$	1743.07	0.184439
$448$	0	0
$449$	−11633.8	−1.22279	−0.611396	−	0.791325i	$-0.709392\pi$
$449$	−11633.8	−1.22279	−0.611396	+	0.791325i	$0.709392\pi$
$450$	0	0
$451$	−4897.95	−0.511387
$452$	0	0
$453$	1846.17	0.191480
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13104.6	−1.34138	−0.670688	−	0.741740i	$-0.734001\pi$
$457$	−13104.6	−1.34138	−0.670688	+	0.741740i	$0.734001\pi$
$458$	0	0
$459$	3222.41	0.327689
$460$	0	0
$461$	2594.63	0.262134	0.131067	−	0.991373i	$-0.458160\pi$
$461$	2594.63	0.262134	0.131067	+	0.991373i	$0.458160\pi$
$462$	0	0
$463$	14136.2	1.41893	0.709465	−	0.704741i	$-0.248937\pi$
$463$	14136.2	1.41893	0.709465	+	0.704741i	$0.248937\pi$
$464$	0	0
$465$	13960.4	1.39225
$466$	0	0
$467$	15590.2	1.54482	0.772409	−	0.635125i	$-0.219052\pi$
$467$	15590.2	1.54482	0.772409	+	0.635125i	$0.219052\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	920.795	0.0900807
$472$	0	0
$473$	3359.44	0.326569
$474$	0	0
$475$	−10361.2	−1.00085
$476$	0	0
$477$	−1588.60	−0.152489
$478$	0	0
$479$	−8453.51	−0.806369	−0.403184	−	0.915119i	$-0.632097\pi$
$479$	−8453.51	−0.806369	−0.403184	+	0.915119i	$0.632097\pi$
$480$	0	0
$481$	−364.519	−0.0345543
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6974.20	0.652953
$486$	0	0
$487$	4011.07	0.373221	0.186611	−	0.982434i	$-0.440250\pi$
$487$	4011.07	0.373221	0.186611	+	0.982434i	$0.440250\pi$
$488$	0	0
$489$	−10543.5	−0.975038
$490$	0	0
$491$	−13927.9	−1.28016	−0.640079	−	0.768309i	$-0.721098\pi$
$491$	−13927.9	−1.28016	−0.640079	+	0.768309i	$0.721098\pi$
$492$	0	0
$493$	−19478.7	−1.77946
$494$	0	0
$495$	2843.56	0.258199
$496$	0	0
$497$	0	0
$498$	0	0
$499$	3947.55	0.354141	0.177071	−	0.984198i	$-0.443338\pi$
$499$	3947.55	0.354141	0.177071	+	0.984198i	$0.443338\pi$
$500$	0	0
$501$	3369.91	0.300512
$502$	0	0
$503$	−13725.3	−1.21666	−0.608331	−	0.793684i	$-0.708161\pi$
$503$	−13725.3	−1.21666	−0.608331	+	0.793684i	$0.708161\pi$
$504$	0	0
$505$	−307.190	−0.0270688
$506$	0	0
$507$	−6576.97	−0.576121
$508$	0	0
$509$	−7830.10	−0.681853	−0.340926	−	0.940090i	$-0.610741\pi$
$509$	−7830.10	−0.681853	−0.340926	+	0.940090i	$0.610741\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−904.807	−0.0778717
$514$	0	0
$515$	17532.8	1.50017
$516$	0	0
$517$	−7703.43	−0.655312
$518$	0	0
$519$	−4591.80	−0.388357
$520$	0	0
$521$	5907.39	0.496751	0.248376	−	0.968664i	$-0.420103\pi$
$521$	5907.39	0.496751	0.248376	+	0.968664i	$0.420103\pi$
$522$	0	0
$523$	7908.06	0.661176	0.330588	−	0.943775i	$-0.392753\pi$
$523$	7908.06	0.661176	0.330588	+	0.943775i	$0.392753\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−26653.6	−2.20313
$528$	0	0
$529$	−12166.6	−0.999965
$530$	0	0
$531$	4094.35	0.334613
$532$	0	0
$533$	−698.659	−0.0567773
$534$	0	0
$535$	−14912.5	−1.20509
$536$	0	0
$537$	−10240.3	−0.822908
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3941.04	−0.313195	−0.156598	−	0.987662i	$-0.550053\pi$
$541$	−3941.04	−0.313195	−0.156598	+	0.987662i	$0.550053\pi$
$542$	0	0
$543$	−3860.12	−0.305071
$544$	0	0
$545$	−12502.7	−0.982670
$546$	0	0
$547$	1828.71	0.142943	0.0714717	−	0.997443i	$-0.477230\pi$
$547$	1828.71	0.142943	0.0714717	+	0.997443i	$0.477230\pi$
$548$	0	0
$549$	−347.864	−0.0270427
$550$	0	0
$551$	5469.33	0.422870
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−10535.3	−0.805763
$556$	0	0
$557$	22532.0	1.71402	0.857011	−	0.515298i	$-0.172319\pi$
$557$	22532.0	1.71402	0.857011	+	0.515298i	$0.172319\pi$
$558$	0	0
$559$	479.201	0.0362577
$560$	0	0
$561$	−5429.00	−0.408579
$562$	0	0
$563$	23355.7	1.74836	0.874179	−	0.485604i	$-0.161400\pi$
$563$	23355.7	1.74836	0.874179	+	0.485604i	$0.161400\pi$
$564$	0	0
$565$	−12974.1	−0.966062
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20887.6	−1.53894	−0.769468	−	0.638686i	$-0.779478\pi$
$569$	−20887.6	−1.53894	−0.769468	+	0.638686i	$0.779478\pi$
$570$	0	0
$571$	−23745.3	−1.74029	−0.870147	−	0.492792i	$-0.835976\pi$
$571$	−23745.3	−1.74029	−0.870147	+	0.492792i	$0.835976\pi$
$572$	0	0
$573$	−3165.89	−0.230815
$574$	0	0
$575$	−201.440	−0.0146098
$576$	0	0
$577$	−2454.39	−0.177084	−0.0885422	−	0.996072i	$-0.528221\pi$
$577$	−2454.39	−0.177084	−0.0885422	+	0.996072i	$0.528221\pi$
$578$	0	0
$579$	−14312.5	−1.02730
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2676.42	0.190130
$584$	0	0
$585$	405.614	0.0286668
$586$	0	0
$587$	18567.5	1.30556	0.652780	−	0.757547i	$-0.273603\pi$
$587$	18567.5	1.30556	0.652780	+	0.757547i	$0.273603\pi$
$588$	0	0
$589$	7483.95	0.523550
$590$	0	0
$591$	4866.93	0.338746
$592$	0	0
$593$	−17112.9	−1.18507	−0.592533	−	0.805546i	$-0.701872\pi$
$593$	−17112.9	−1.18507	−0.592533	+	0.805546i	$0.701872\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10650.4	−0.730138
$598$	0	0
$599$	−23264.8	−1.58694	−0.793469	−	0.608611i	$-0.791727\pi$
$599$	−23264.8	−1.58694	−0.793469	+	0.608611i	$0.791727\pi$
$600$	0	0
$601$	−25322.3	−1.71867	−0.859334	−	0.511416i	$-0.829121\pi$
$601$	−25322.3	−1.71867	−0.859334	+	0.511416i	$0.829121\pi$
$602$	0	0
$603$	−1276.13	−0.0861821
$604$	0	0
$605$	22943.5	1.54179
$606$	0	0
$607$	−21734.4	−1.45333	−0.726665	−	0.686992i	$-0.758931\pi$
$607$	−21734.4	−1.45333	−0.726665	+	0.686992i	$0.758931\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1098.84	−0.0727567
$612$	0	0
$613$	−13572.4	−0.894262	−0.447131	−	0.894468i	$-0.647554\pi$
$613$	−13572.4	−0.894262	−0.447131	+	0.894468i	$0.647554\pi$
$614$	0	0
$615$	−20192.6	−1.32397
$616$	0	0
$617$	−8497.12	−0.554427	−0.277213	−	0.960808i	$-0.589411\pi$
$617$	−8497.12	−0.554427	−0.277213	+	0.960808i	$0.589411\pi$
$618$	0	0
$619$	−22982.9	−1.49235	−0.746173	−	0.665752i	$-0.768111\pi$
$619$	−22982.9	−1.49235	−0.746173	+	0.665752i	$0.768111\pi$
$620$	0	0
$621$	−17.5910	−0.00113672
$622$	0	0
$623$	0	0
$624$	0	0
$625$	41322.5	2.64464
$626$	0	0
$627$	1524.39	0.0970943
$628$	0	0
$629$	20114.3	1.27505
$630$	0	0
$631$	15717.9	0.991635	0.495817	−	0.868427i	$-0.334869\pi$
$631$	15717.9	0.991635	0.495817	+	0.868427i	$0.334869\pi$
$632$	0	0
$633$	−13960.2	−0.876566
$634$	0	0
$635$	−3752.26	−0.234494
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−5424.68	−0.335833
$640$	0	0
$641$	29107.4	1.79356	0.896780	−	0.442478i	$-0.145900\pi$
$641$	29107.4	1.79356	0.896780	+	0.442478i	$0.145900\pi$
$642$	0	0
$643$	−3112.26	−0.190880	−0.0954398	−	0.995435i	$-0.530426\pi$
$643$	−3112.26	−0.190880	−0.0954398	+	0.995435i	$0.530426\pi$
$644$	0	0
$645$	13849.8	0.845482
$646$	0	0
$647$	−7857.59	−0.477456	−0.238728	−	0.971087i	$-0.576730\pi$
$647$	−7857.59	−0.477456	−0.238728	+	0.971087i	$0.576730\pi$
$648$	0	0
$649$	−6898.02	−0.417213
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19522.0	−1.16992	−0.584958	−	0.811063i	$-0.698889\pi$
$653$	−19522.0	−1.16992	−0.584958	+	0.811063i	$0.698889\pi$
$654$	0	0
$655$	−4539.57	−0.270803
$656$	0	0
$657$	9920.69	0.589107
$658$	0	0
$659$	−664.061	−0.0392536	−0.0196268	−	0.999807i	$-0.506248\pi$
$659$	−664.061	−0.0392536	−0.0196268	+	0.999807i	$0.506248\pi$
$660$	0	0
$661$	−15921.6	−0.936883	−0.468442	−	0.883494i	$-0.655184\pi$
$661$	−15921.6	−0.936883	−0.468442	+	0.883494i	$0.655184\pi$
$662$	0	0
$663$	−774.409	−0.0453628
$664$	0	0
$665$	0	0
$666$	0	0
$667$	106.333	0.00617276
$668$	0	0
$669$	−13948.6	−0.806105
$670$	0	0
$671$	586.068	0.0337182
$672$	0	0
$673$	24631.0	1.41078	0.705391	−	0.708819i	$-0.250771\pi$
$673$	24631.0	1.41078	0.705391	+	0.708819i	$0.250771\pi$
$674$	0	0
$675$	8348.01	0.476022
$676$	0	0
$677$	−17092.8	−0.970353	−0.485177	−	0.874416i	$-0.661245\pi$
$677$	−17092.8	−0.970353	−0.485177	+	0.874416i	$0.661245\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	12455.1	0.700855
$682$	0	0
$683$	19163.6	1.07361	0.536804	−	0.843707i	$-0.319632\pi$
$683$	19163.6	1.07361	0.536804	+	0.843707i	$0.319632\pi$
$684$	0	0
$685$	54215.2	3.02402
$686$	0	0
$687$	−12790.9	−0.710339
$688$	0	0
$689$	381.773	0.0211094
$690$	0	0
$691$	8095.87	0.445704	0.222852	−	0.974852i	$-0.428463\pi$
$691$	8095.87	0.445704	0.222852	+	0.974852i	$0.428463\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	55250.7	3.01551
$696$	0	0
$697$	38552.3	2.09508
$698$	0	0
$699$	9149.70	0.495098
$700$	0	0
$701$	12354.7	0.665664	0.332832	−	0.942986i	$-0.391996\pi$
$701$	12354.7	0.665664	0.332832	+	0.942986i	$0.391996\pi$
$702$	0	0
$703$	−5647.81	−0.303003
$704$	0	0
$705$	−31758.6	−1.69659
$706$	0	0
$707$	0	0
$708$	0	0
$709$	3828.82	0.202813	0.101406	−	0.994845i	$-0.467666\pi$
$709$	3828.82	0.202813	0.101406	+	0.994845i	$0.467666\pi$
$710$	0	0
$711$	1046.73	0.0552114
$712$	0	0
$713$	145.500	0.00764241
$714$	0	0
$715$	−683.363	−0.0357431
$716$	0	0
$717$	−11961.6	−0.623033
$718$	0	0
$719$	1223.00	0.0634356	0.0317178	−	0.999497i	$-0.489902\pi$
$719$	1223.00	0.0634356	0.0317178	+	0.999497i	$0.489902\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1873.94	0.0963938
$724$	0	0
$725$	−50461.7	−2.58496
$726$	0	0
$727$	6368.21	0.324875	0.162437	−	0.986719i	$-0.448064\pi$
$727$	6368.21	0.324875	0.162437	+	0.986719i	$0.448064\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−26442.5	−1.33791
$732$	0	0
$733$	25154.0	1.26751	0.633753	−	0.773535i	$-0.281513\pi$
$733$	25154.0	1.26751	0.633753	+	0.773535i	$0.281513\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2149.97	0.107456
$738$	0	0
$739$	10739.1	0.534566	0.267283	−	0.963618i	$-0.413874\pi$
$739$	10739.1	0.534566	0.267283	+	0.963618i	$0.413874\pi$
$740$	0	0
$741$	217.443	0.0107800
$742$	0	0
$743$	−28166.3	−1.39074	−0.695370	−	0.718652i	$-0.744760\pi$
$743$	−28166.3	−1.39074	−0.695370	+	0.718652i	$0.744760\pi$
$744$	0	0
$745$	−12106.8	−0.595383
$746$	0	0
$747$	−5120.35	−0.250795
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−28657.0	−1.39242	−0.696211	−	0.717837i	$-0.745132\pi$
$751$	−28657.0	−1.39242	−0.696211	+	0.717837i	$0.745132\pi$
$752$	0	0
$753$	−3986.35	−0.192923
$754$	0	0
$755$	−12823.0	−0.618113
$756$	0	0
$757$	−23604.1	−1.13330	−0.566648	−	0.823960i	$-0.691760\pi$
$757$	−23604.1	−1.13330	−0.566648	+	0.823960i	$0.691760\pi$
$758$	0	0
$759$	29.6366	0.00141731
$760$	0	0
$761$	−4630.97	−0.220595	−0.110297	−	0.993899i	$-0.535180\pi$
$761$	−4630.97	−0.220595	−0.110297	+	0.993899i	$0.535180\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−22381.9	−1.05780
$766$	0	0
$767$	−983.954	−0.0463214
$768$	0	0
$769$	−33276.8	−1.56046	−0.780228	−	0.625495i	$-0.784897\pi$
$769$	−33276.8	−1.56046	−0.780228	+	0.625495i	$0.784897\pi$
$770$	0	0
$771$	9678.55	0.452094
$772$	0	0
$773$	22938.8	1.06734	0.533668	−	0.845694i	$-0.320813\pi$
$773$	22938.8	1.06734	0.533668	+	0.845694i	$0.320813\pi$
$774$	0	0
$775$	−69049.1	−3.20041
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10824.9	−0.497873
$780$	0	0
$781$	9139.31	0.418733
$782$	0	0
$783$	−4406.62	−0.201124
$784$	0	0
$785$	−6395.58	−0.290787
$786$	0	0
$787$	−13514.5	−0.612120	−0.306060	−	0.952012i	$-0.599011\pi$
$787$	−13514.5	−0.612120	−0.306060	+	0.952012i	$0.599011\pi$
$788$	0	0
$789$	−9751.84	−0.440019
$790$	0	0
$791$	0	0
$792$	0	0
$793$	83.5986	0.00374360
$794$	0	0
$795$	11034.0	0.492245
$796$	0	0
$797$	−10473.4	−0.465480	−0.232740	−	0.972539i	$-0.574769\pi$
$797$	−10473.4	−0.465480	−0.232740	+	0.972539i	$0.574769\pi$
$798$	0	0
$799$	60634.5	2.68472
$800$	0	0
$801$	3448.43	0.152115
$802$	0	0
$803$	−16714.0	−0.734527
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8478.12	−0.369819
$808$	0	0
$809$	23568.0	1.02423	0.512117	−	0.858916i	$-0.328861\pi$
$809$	23568.0	1.02423	0.512117	+	0.858916i	$0.328861\pi$
$810$	0	0
$811$	−6704.22	−0.290280	−0.145140	−	0.989411i	$-0.546363\pi$
$811$	−6704.22	−0.290280	−0.145140	+	0.989411i	$0.546363\pi$
$812$	0	0
$813$	−7190.31	−0.310178
$814$	0	0
$815$	73232.1	3.14749
$816$	0	0
$817$	7424.67	0.317939
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24539.4	1.04316	0.521579	−	0.853203i	$-0.325343\pi$
$821$	24539.4	1.04316	0.521579	+	0.853203i	$0.325343\pi$
$822$	0	0
$823$	31117.0	1.31795	0.658973	−	0.752167i	$-0.270991\pi$
$823$	31117.0	1.31795	0.658973	+	0.752167i	$0.270991\pi$
$824$	0	0
$825$	−14064.4	−0.593528
$826$	0	0
$827$	−31244.9	−1.31377	−0.656887	−	0.753989i	$-0.728127\pi$
$827$	−31244.9	−1.31377	−0.656887	+	0.753989i	$0.728127\pi$
$828$	0	0
$829$	−4231.50	−0.177281	−0.0886405	−	0.996064i	$-0.528252\pi$
$829$	−4231.50	−0.177281	−0.0886405	+	0.996064i	$0.528252\pi$
$830$	0	0
$831$	5461.40	0.227983
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−23406.4	−0.970074
$836$	0	0
$837$	−6029.80	−0.249009
$838$	0	0
$839$	−38670.4	−1.59124	−0.795621	−	0.605795i	$-0.792855\pi$
$839$	−38670.4	−1.59124	−0.795621	+	0.605795i	$0.792855\pi$
$840$	0	0
$841$	2247.96	0.0921710
$842$	0	0
$843$	9251.43	0.377979
$844$	0	0
$845$	45681.7	1.85976
$846$	0	0
$847$	0	0
$848$	0	0
$849$	7664.31	0.309821
$850$	0	0
$851$	−109.803	−0.00442302
$852$	0	0
$853$	19944.4	0.800565	0.400282	−	0.916392i	$-0.368912\pi$
$853$	19944.4	0.800565	0.400282	+	0.916392i	$0.368912\pi$
$854$	0	0
$855$	6284.52	0.251376
$856$	0	0
$857$	−13882.2	−0.553334	−0.276667	−	0.960966i	$-0.589230\pi$
$857$	−13882.2	−0.553334	−0.276667	+	0.960966i	$0.589230\pi$
$858$	0	0
$859$	−4157.16	−0.165123	−0.0825614	−	0.996586i	$-0.526310\pi$
$859$	−4157.16	−0.165123	−0.0825614	+	0.996586i	$0.526310\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16237.9	0.640493	0.320246	−	0.947334i	$-0.396234\pi$
$863$	16237.9	0.640493	0.320246	+	0.947334i	$0.396234\pi$
$864$	0	0
$865$	31893.3	1.25365
$866$	0	0
$867$	27993.2	1.09654
$868$	0	0
$869$	−1763.49	−0.0688403
$870$	0	0
$871$	306.678	0.0119304
$872$	0	0
$873$	−3012.31	−0.116783
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−16489.4	−0.634900	−0.317450	−	0.948275i	$-0.602827\pi$
$877$	−16489.4	−0.634900	−0.317450	+	0.948275i	$0.602827\pi$
$878$	0	0
$879$	5539.40	0.212559
$880$	0	0
$881$	−45411.7	−1.73662	−0.868309	−	0.496023i	$-0.834793\pi$
$881$	−45411.7	−1.73662	−0.868309	+	0.496023i	$0.834793\pi$
$882$	0	0
$883$	2206.85	0.0841070	0.0420535	−	0.999115i	$-0.486610\pi$
$883$	2206.85	0.0841070	0.0420535	+	0.999115i	$0.486610\pi$
$884$	0	0
$885$	−28438.2	−1.08016
$886$	0	0
$887$	28146.2	1.06545	0.532727	−	0.846287i	$-0.321167\pi$
$887$	28146.2	1.06545	0.532727	+	0.846287i	$0.321167\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1228.19	−0.0461796
$892$	0	0
$893$	−17025.3	−0.637995
$894$	0	0
$895$	71126.1	2.65641
$896$	0	0
$897$	4.22746	0.000157359 0
$898$	0	0
$899$	36448.6	1.35220
$900$	0	0
$901$	−21066.4	−0.778937
$902$	0	0
$903$	0	0
$904$	0	0
$905$	26811.3	0.984793
$906$	0	0
$907$	5042.25	0.184592	0.0922960	−	0.995732i	$-0.470579\pi$
$907$	5042.25	0.184592	0.0922960	+	0.995732i	$0.470579\pi$
$908$	0	0
$909$	132.682	0.00484134
$910$	0	0
$911$	−29647.3	−1.07822	−0.539110	−	0.842235i	$-0.681239\pi$
$911$	−29647.3	−1.07822	−0.539110	+	0.842235i	$0.681239\pi$
$912$	0	0
$913$	8626.59	0.312704
$914$	0	0
$915$	2416.16	0.0872959
$916$	0	0
$917$	0	0
$918$	0	0
$919$	11891.3	0.426830	0.213415	−	0.976962i	$-0.431541\pi$
$919$	11891.3	0.426830	0.213415	+	0.976962i	$0.431541\pi$
$920$	0	0
$921$	21124.5	0.755782
$922$	0	0
$923$	1303.66	0.0464902
$924$	0	0
$925$	52108.3	1.85223
$926$	0	0
$927$	−7572.78	−0.268309
$928$	0	0
$929$	39188.5	1.38400	0.691999	−	0.721898i	$-0.256730\pi$
$929$	39188.5	1.38400	0.691999	+	0.721898i	$0.256730\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−8057.98	−0.282751
$934$	0	0
$935$	37708.2	1.31892
$936$	0	0
$937$	−9716.23	−0.338757	−0.169379	−	0.985551i	$-0.554176\pi$
$937$	−9716.23	−0.338757	−0.169379	+	0.985551i	$0.554176\pi$
$938$	0	0
$939$	−6657.57	−0.231375
$940$	0	0
$941$	6995.87	0.242358	0.121179	−	0.992631i	$-0.461333\pi$
$941$	6995.87	0.242358	0.121179	+	0.992631i	$0.461333\pi$
$942$	0	0
$943$	−210.455	−0.00726760
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−14979.2	−0.514002	−0.257001	−	0.966411i	$-0.582734\pi$
$947$	−14979.2	−0.514002	−0.257001	+	0.966411i	$0.582734\pi$
$948$	0	0
$949$	−2384.14	−0.0815516
$950$	0	0
$951$	6663.78	0.227222
$952$	0	0
$953$	29393.3	0.999100	0.499550	−	0.866285i	$-0.333499\pi$
$953$	29393.3	0.999100	0.499550	+	0.866285i	$0.333499\pi$
$954$	0	0
$955$	21989.3	0.745087
$956$	0	0
$957$	7424.12	0.250771
$958$	0	0
$959$	0	0
$960$	0	0
$961$	20083.4	0.674143
$962$	0	0
$963$	6441.03	0.215534
$964$	0	0
$965$	99410.6	3.31621
$966$	0	0
$967$	7133.95	0.237241	0.118621	−	0.992940i	$-0.462153\pi$
$967$	7133.95	0.237241	0.118621	+	0.992940i	$0.462153\pi$
$968$	0	0
$969$	−11998.6	−0.397781
$970$	0	0
$971$	−9688.13	−0.320192	−0.160096	−	0.987101i	$-0.551180\pi$
$971$	−9688.13	−0.320192	−0.160096	+	0.987101i	$0.551180\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2006.19	−0.0658970
$976$	0	0
$977$	21305.7	0.697676	0.348838	−	0.937183i	$-0.386576\pi$
$977$	21305.7	0.697676	0.348838	+	0.937183i	$0.386576\pi$
$978$	0	0
$979$	−5809.79	−0.189665
$980$	0	0
$981$	5400.17	0.175753
$982$	0	0
$983$	−37280.8	−1.20964	−0.604818	−	0.796364i	$-0.706754\pi$
$983$	−37280.8	−1.20964	−0.604818	+	0.796364i	$0.706754\pi$
$984$	0	0
$985$	−33804.3	−1.09350
$986$	0	0
$987$	0	0
$988$	0	0
$989$	144.348	0.00464105
$990$	0	0
$991$	51397.1	1.64751	0.823755	−	0.566946i	$-0.191875\pi$
$991$	51397.1	1.64751	0.823755	+	0.566946i	$0.191875\pi$
$992$	0	0
$993$	−12462.2	−0.398263
$994$	0	0
$995$	73974.6	2.35694
$996$	0	0
$997$	34373.8	1.09191	0.545953	−	0.837816i	$-0.316168\pi$
$997$	34373.8	1.09191	0.545953	+	0.837816i	$0.316168\pi$
$998$	0	0
$999$	4550.42	0.144113

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.ca.1.1		2
4.3	odd	2		294.4.a.m.1.1		2
7.3	odd	6		336.4.q.j.289.1		4
7.5	odd	6		336.4.q.j.193.1		4
7.6	odd	2		2352.4.a.bq.1.2		2
12.11	even	2		882.4.a.z.1.2		2
28.3	even	6		42.4.e.c.37.1	yes	4
28.11	odd	6		294.4.e.l.79.2		4
28.19	even	6		42.4.e.c.25.1	✓	4
28.23	odd	6		294.4.e.l.67.2		4
28.27	even	2		294.4.a.n.1.2		2
84.11	even	6		882.4.g.bf.667.1		4
84.23	even	6		882.4.g.bf.361.1		4
84.47	odd	6		126.4.g.g.109.2		4
84.59	odd	6		126.4.g.g.37.2		4
84.83	odd	2		882.4.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.e.c.25.1	✓	4	28.19	even	6
42.4.e.c.37.1	yes	4	28.3	even	6
126.4.g.g.37.2		4	84.59	odd	6
126.4.g.g.109.2		4	84.47	odd	6
294.4.a.m.1.1		2	4.3	odd	2
294.4.a.n.1.2		2	28.27	even	2
294.4.e.l.67.2		4	28.23	odd	6
294.4.e.l.79.2		4	28.11	odd	6
336.4.q.j.193.1		4	7.5	odd	6
336.4.q.j.289.1		4	7.3	odd	6
882.4.a.v.1.1		2	84.83	odd	2
882.4.a.z.1.2		2	12.11	even	2
882.4.g.bf.361.1		4	84.23	even	6
882.4.g.bf.667.1		4	84.11	even	6
2352.4.a.bq.1.2		2	7.6	odd	2
2352.4.a.ca.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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -20.8371 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -20.8371 q^{5} +9.00000 q^{9} -15.1629 q^{11} -2.16288 q^{13} -62.5114 q^{15} +119.348 q^{17} -33.5114 q^{19} -0.651517 q^{23} +309.186 q^{25} +27.0000 q^{27} -163.208 q^{29} -223.326 q^{31} -45.4886 q^{33} +168.534 q^{37} -6.48864 q^{39} +323.023 q^{41} -221.557 q^{43} -187.534 q^{45} +508.045 q^{47} +358.045 q^{51} -176.511 q^{53} +315.951 q^{55} -100.534 q^{57} +454.928 q^{59} -38.6515 q^{61} +45.0682 q^{65} -141.792 q^{67} -1.95455 q^{69} -602.742 q^{71} +1102.30 q^{73} +927.557 q^{75} +116.303 q^{79} +81.0000 q^{81} -568.928 q^{83} -2486.88 q^{85} -489.625 q^{87} +383.159 q^{89} -669.977 q^{93} +698.280 q^{95} -334.701 q^{97} -136.466 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 5 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 5 * q^5 + 18 * q^9	\( 2 q + 6 q^{3} - 5 q^{5} + 18 q^{9} - 67 q^{11} - 41 q^{13} - 15 q^{15} + 92 q^{17} + 43 q^{19} - 148 q^{23} + 435 q^{25} + 54 q^{27} + 77 q^{29} - 520 q^{31} - 201 q^{33} + 7 q^{37} - 123 q^{39} + 426 q^{41} + 107 q^{43} - 45 q^{45} + 576 q^{47} + 276 q^{51} - 243 q^{53} - 505 q^{55} + 129 q^{57} - 7 q^{59} - 224 q^{61} - 570 q^{65} - 687 q^{67} - 444 q^{69} - 472 q^{71} + 921 q^{73} + 1305 q^{75} + 526 q^{79} + 162 q^{81} - 221 q^{83} - 2920 q^{85} + 231 q^{87} - 774 q^{89} - 1560 q^{93} + 1910 q^{95} - 1953 q^{97} - 603 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 5 * q^5 + 18 * q^9 - 67 * q^11 - 41 * q^13 - 15 * q^15 + 92 * q^17 + 43 * q^19 - 148 * q^23 + 435 * q^25 + 54 * q^27 + 77 * q^29 - 520 * q^31 - 201 * q^33 + 7 * q^37 - 123 * q^39 + 426 * q^41 + 107 * q^43 - 45 * q^45 + 576 * q^47 + 276 * q^51 - 243 * q^53 - 505 * q^55 + 129 * q^57 - 7 * q^59 - 224 * q^61 - 570 * q^65 - 687 * q^67 - 444 * q^69 - 472 * q^71 + 921 * q^73 + 1305 * q^75 + 526 * q^79 + 162 * q^81 - 221 * q^83 - 2920 * q^85 + 231 * q^87 - 774 * q^89 - 1560 * q^93 + 1910 * q^95 - 1953 * q^97 - 603 * q^99

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