LMFDB - Embedded newform 1764.4.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(1,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1764.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.136768.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 23x^{2} + 18x + 119 $ x^4 - 2x^3 - 23x^2 + 18*x + 119
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 7 $
Twist minimal:	no (minimal twist has level 588)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.31012$ of defining polynomial
Character	$\chi$	$=$	1764.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.6547 q^{5} +O(q^{10})$	$q-16.6547 q^{5} +71.7600 q^{11} +65.3878 q^{13} +90.4316 q^{17} -163.818 q^{19} -79.2288 q^{23} +152.379 q^{25} +43.2995 q^{29} -135.636 q^{31} +270.740 q^{37} +152.241 q^{41} -177.641 q^{43} -45.6331 q^{47} +158.438 q^{53} -1195.14 q^{55} -391.769 q^{59} -551.295 q^{61} -1089.01 q^{65} +458.630 q^{67} +486.786 q^{71} -574.892 q^{73} -668.319 q^{79} +76.2450 q^{83} -1506.11 q^{85} +1366.80 q^{89} +2728.34 q^{95} -242.655 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 48 q^{17} - 192 q^{19} - 192 q^{23} + 324 q^{25} - 96 q^{29} - 48 q^{31} + 256 q^{37} + 1008 q^{41} - 112 q^{43} + 864 q^{47} + 648 q^{53} - 2352 q^{55} + 336 q^{59} - 960 q^{61} + 360 q^{65} + 720 q^{67} + 1344 q^{71} - 672 q^{73} - 1984 q^{79} + 3120 q^{83} + 680 q^{85} + 2160 q^{89} + 3744 q^{95} - 2016 q^{97}+O(q^{100}) $ 4 * q + 48 * q^17 - 192 * q^19 - 192 * q^23 + 324 * q^25 - 96 * q^29 - 48 * q^31 + 256 * q^37 + 1008 * q^41 - 112 * q^43 + 864 * q^47 + 648 * q^53 - 2352 * q^55 + 336 * q^59 - 960 * q^61 + 360 * q^65 + 720 * q^67 + 1344 * q^71 - 672 * q^73 - 1984 * q^79 + 3120 * q^83 + 680 * q^85 + 2160 * q^89 + 3744 * q^95 - 2016 * 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
$5$	−16.6547	−1.48964	−0.744820	−	0.667265i	$-0.767465\pi$
$5$	−16.6547	−1.48964	−0.744820	+	0.667265i	$0.767465\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	71.7600	1.96695	0.983475	−	0.181044i	$-0.0579477\pi$
$11$	71.7600	1.96695	0.983475	+	0.181044i	$0.0579477\pi$
$12$	0	0
$13$	65.3878	1.39502	0.697512	−	0.716573i	$-0.254291\pi$
$13$	65.3878	1.39502	0.697512	+	0.716573i	$0.254291\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	90.4316	1.29017	0.645085	−	0.764111i	$-0.276822\pi$
$17$	90.4316	1.29017	0.645085	+	0.764111i	$0.276822\pi$
$18$	0	0
$19$	−163.818	−1.97803	−0.989014	−	0.147824i	$-0.952773\pi$
$19$	−163.818	−1.97803	−0.989014	+	0.147824i	$0.952773\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−79.2288	−0.718276	−0.359138	−	0.933284i	$-0.616929\pi$
$23$	−79.2288	−0.718276	−0.359138	+	0.933284i	$0.616929\pi$
$24$	0	0
$25$	152.379	1.21903
$26$	0	0
$27$	0	0
$28$	0	0
$29$	43.2995	0.277259	0.138630	−	0.990344i	$-0.455730\pi$
$29$	43.2995	0.277259	0.138630	+	0.990344i	$0.455730\pi$
$30$	0	0
$31$	−135.636	−0.785836	−0.392918	−	0.919574i	$-0.628534\pi$
$31$	−135.636	−0.785836	−0.392918	+	0.919574i	$0.628534\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	270.740	1.20296	0.601479	−	0.798889i	$-0.294578\pi$
$37$	270.740	1.20296	0.601479	+	0.798889i	$0.294578\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	152.241	0.579902	0.289951	−	0.957041i	$-0.406361\pi$
$41$	152.241	0.579902	0.289951	+	0.957041i	$0.406361\pi$
$42$	0	0
$43$	−177.641	−0.630001	−0.315001	−	0.949091i	$-0.602005\pi$
$43$	−177.641	−0.630001	−0.315001	+	0.949091i	$0.602005\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−45.6331	−0.141623	−0.0708114	−	0.997490i	$-0.522559\pi$
$47$	−45.6331	−0.141623	−0.0708114	+	0.997490i	$0.522559\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	158.438	0.410624	0.205312	−	0.978697i	$-0.434179\pi$
$53$	158.438	0.410624	0.205312	+	0.978697i	$0.434179\pi$
$54$	0	0
$55$	−1195.14	−2.93005
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−391.769	−0.864474	−0.432237	−	0.901760i	$-0.642276\pi$
$59$	−391.769	−0.864474	−0.432237	+	0.901760i	$0.642276\pi$
$60$	0	0
$61$	−551.295	−1.15715	−0.578574	−	0.815630i	$-0.696391\pi$
$61$	−551.295	−1.15715	−0.578574	+	0.815630i	$0.696391\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1089.01	−2.07808
$66$	0	0
$67$	458.630	0.836277	0.418139	−	0.908383i	$-0.362683\pi$
$67$	458.630	0.836277	0.418139	+	0.908383i	$0.362683\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	486.786	0.813674	0.406837	−	0.913501i	$-0.366632\pi$
$71$	486.786	0.813674	0.406837	+	0.913501i	$0.366632\pi$
$72$	0	0
$73$	−574.892	−0.921726	−0.460863	−	0.887471i	$-0.652460\pi$
$73$	−574.892	−0.921726	−0.460863	+	0.887471i	$0.652460\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−668.319	−0.951795	−0.475897	−	0.879501i	$-0.657877\pi$
$79$	−668.319	−0.951795	−0.475897	+	0.879501i	$0.657877\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	76.2450	0.100831	0.0504155	−	0.998728i	$-0.483945\pi$
$83$	76.2450	0.100831	0.0504155	+	0.998728i	$0.483945\pi$
$84$	0	0
$85$	−1506.11	−1.92189
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1366.80	1.62787	0.813937	−	0.580953i	$-0.197320\pi$
$89$	1366.80	1.62787	0.813937	+	0.580953i	$0.197320\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2728.34	2.94655
$96$	0	0
$97$	−242.655	−0.253999	−0.127000	−	0.991903i	$-0.540535\pi$
$97$	−242.655	−0.253999	−0.127000	+	0.991903i	$0.540535\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	694.743	0.684450	0.342225	−	0.939618i	$-0.388819\pi$
$101$	694.743	0.684450	0.342225	+	0.939618i	$0.388819\pi$
$102$	0	0
$103$	174.218	0.166662	0.0833310	−	0.996522i	$-0.473444\pi$
$103$	174.218	0.166662	0.0833310	+	0.996522i	$0.473444\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−47.6051	−0.0430108	−0.0215054	−	0.999769i	$-0.506846\pi$
$107$	−47.6051	−0.0430108	−0.0215054	+	0.999769i	$0.506846\pi$
$108$	0	0
$109$	−1633.10	−1.43507	−0.717534	−	0.696523i	$-0.754729\pi$
$109$	−1633.10	−1.43507	−0.717534	+	0.696523i	$0.754729\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1127.29	0.938464	0.469232	−	0.883075i	$-0.344531\pi$
$113$	1127.29	0.938464	0.469232	+	0.883075i	$0.344531\pi$
$114$	0	0
$115$	1319.53	1.06997
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3818.50	2.86889
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−455.982	−0.326274
$126$	0	0
$127$	2398.66	1.67596	0.837979	−	0.545702i	$-0.183737\pi$
$127$	2398.66	1.67596	0.837979	+	0.545702i	$0.183737\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1548.96	1.03308	0.516541	−	0.856263i	$-0.327219\pi$
$131$	1548.96	1.03308	0.516541	+	0.856263i	$0.327219\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1523.27	−0.949939	−0.474969	−	0.880002i	$-0.657541\pi$
$137$	−1523.27	−0.949939	−0.474969	+	0.880002i	$0.657541\pi$
$138$	0	0
$139$	−551.445	−0.336496	−0.168248	−	0.985745i	$-0.553811\pi$
$139$	−551.445	−0.336496	−0.168248	+	0.985745i	$0.553811\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4692.23	2.74394
$144$	0	0
$145$	−721.140	−0.413017
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2400.24	1.31970	0.659851	−	0.751397i	$-0.270619\pi$
$149$	2400.24	1.31970	0.659851	+	0.751397i	$0.270619\pi$
$150$	0	0
$151$	621.560	0.334979	0.167490	−	0.985874i	$-0.446434\pi$
$151$	621.560	0.334979	0.167490	+	0.985874i	$0.446434\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2258.97	1.17061
$156$	0	0
$157$	2691.58	1.36822	0.684112	−	0.729377i	$-0.260190\pi$
$157$	2691.58	1.36822	0.684112	+	0.729377i	$0.260190\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1057.31	0.508065	0.254032	−	0.967196i	$-0.418243\pi$
$163$	1057.31	0.508065	0.254032	+	0.967196i	$0.418243\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−243.840	−0.112987	−0.0564937	−	0.998403i	$-0.517992\pi$
$167$	−243.840	−0.112987	−0.0564937	+	0.998403i	$0.517992\pi$
$168$	0	0
$169$	2078.56	0.946090
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1687.26	−0.741505	−0.370753	−	0.928732i	$-0.620900\pi$
$173$	−1687.26	−0.741505	−0.370753	+	0.928732i	$0.620900\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	359.139	0.149963	0.0749814	−	0.997185i	$-0.476110\pi$
$179$	359.139	0.149963	0.0749814	+	0.997185i	$0.476110\pi$
$180$	0	0
$181$	−2982.58	−1.22483	−0.612413	−	0.790538i	$-0.709801\pi$
$181$	−2982.58	−1.22483	−0.612413	+	0.790538i	$0.709801\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4509.10	−1.79198
$186$	0	0
$187$	6489.37	2.53770
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3490.77	−1.32243	−0.661213	−	0.750198i	$-0.729958\pi$
$191$	−3490.77	−1.32243	−0.661213	+	0.750198i	$0.729958\pi$
$192$	0	0
$193$	1604.23	0.598315	0.299158	−	0.954204i	$-0.403294\pi$
$193$	1604.23	0.598315	0.299158	+	0.954204i	$0.403294\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	455.935	0.164893	0.0824467	−	0.996595i	$-0.473727\pi$
$197$	455.935	0.164893	0.0824467	+	0.996595i	$0.473727\pi$
$198$	0	0
$199$	−2084.16	−0.742423	−0.371212	−	0.928548i	$-0.621058\pi$
$199$	−2084.16	−0.742423	−0.371212	+	0.928548i	$0.621058\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2535.52	−0.863846
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−11755.6	−3.89068
$210$	0	0
$211$	2568.06	0.837880	0.418940	−	0.908014i	$-0.362402\pi$
$211$	2568.06	0.837880	0.418940	+	0.908014i	$0.362402\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2958.56	0.938475
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5913.12	1.79982
$222$	0	0
$223$	4659.55	1.39922	0.699612	−	0.714523i	$-0.253356\pi$
$223$	4659.55	1.39922	0.699612	+	0.714523i	$0.253356\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4828.77	1.41188	0.705940	−	0.708271i	$-0.250525\pi$
$227$	4828.77	1.41188	0.705940	+	0.708271i	$0.250525\pi$
$228$	0	0
$229$	3222.59	0.929932	0.464966	−	0.885328i	$-0.346066\pi$
$229$	3222.59	0.929932	0.464966	+	0.885328i	$0.346066\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5702.42	1.60334	0.801670	−	0.597767i	$-0.203945\pi$
$233$	5702.42	1.60334	0.801670	+	0.597767i	$0.203945\pi$
$234$	0	0
$235$	760.005	0.210967
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1994.47	−0.539796	−0.269898	−	0.962889i	$-0.586990\pi$
$239$	−1994.47	−0.539796	−0.269898	+	0.962889i	$0.586990\pi$
$240$	0	0
$241$	1420.89	0.379781	0.189891	−	0.981805i	$-0.439187\pi$
$241$	1420.89	0.379781	0.189891	+	0.981805i	$0.439187\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10711.7	−2.75939
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3855.44	−0.969534	−0.484767	−	0.874643i	$-0.661096\pi$
$251$	−3855.44	−0.969534	−0.484767	+	0.874643i	$0.661096\pi$
$252$	0	0
$253$	−5685.46	−1.41281
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4053.28	0.983800	0.491900	−	0.870652i	$-0.336303\pi$
$257$	4053.28	0.983800	0.491900	+	0.870652i	$0.336303\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	500.787	0.117414	0.0587069	−	0.998275i	$-0.481302\pi$
$263$	500.787	0.117414	0.0587069	+	0.998275i	$0.481302\pi$
$264$	0	0
$265$	−2638.73	−0.611683
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3281.74	0.743834	0.371917	−	0.928266i	$-0.378701\pi$
$269$	3281.74	0.743834	0.371917	+	0.928266i	$0.378701\pi$
$270$	0	0
$271$	−3324.36	−0.745168	−0.372584	−	0.927999i	$-0.621528\pi$
$271$	−3324.36	−0.745168	−0.372584	+	0.927999i	$0.621528\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10934.7	2.39777
$276$	0	0
$277$	8619.75	1.86971	0.934856	−	0.355026i	$-0.115528\pi$
$277$	8619.75	1.86971	0.934856	+	0.355026i	$0.115528\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7710.80	1.63697	0.818484	−	0.574529i	$-0.194815\pi$
$281$	7710.80	1.63697	0.818484	+	0.574529i	$0.194815\pi$
$282$	0	0
$283$	139.271	0.0292537	0.0146268	−	0.999893i	$-0.495344\pi$
$283$	139.271	0.0292537	0.0146268	+	0.999893i	$0.495344\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3264.87	0.664537
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4437.09	−0.884702	−0.442351	−	0.896842i	$-0.645855\pi$
$293$	−4437.09	−0.884702	−0.442351	+	0.896842i	$0.645855\pi$
$294$	0	0
$295$	6524.79	1.28776
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5180.60	−1.00201
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9181.64	1.72373
$306$	0	0
$307$	299.480	0.0556749	0.0278375	−	0.999612i	$-0.491138\pi$
$307$	299.480	0.0556749	0.0278375	+	0.999612i	$0.491138\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	913.944	0.166640	0.0833199	−	0.996523i	$-0.473448\pi$
$311$	913.944	0.166640	0.0833199	+	0.996523i	$0.473448\pi$
$312$	0	0
$313$	−3887.38	−0.702005	−0.351002	−	0.936375i	$-0.614159\pi$
$313$	−3887.38	−0.702005	−0.351002	+	0.936375i	$0.614159\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2879.30	−0.510151	−0.255075	−	0.966921i	$-0.582100\pi$
$317$	−2879.30	−0.510151	−0.255075	+	0.966921i	$0.582100\pi$
$318$	0	0
$319$	3107.17	0.545355
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−14814.4	−2.55199
$324$	0	0
$325$	9963.70	1.70057
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2810.79	−0.466752	−0.233376	−	0.972387i	$-0.574977\pi$
$331$	−2810.79	−0.466752	−0.233376	+	0.972387i	$0.574977\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7638.34	−1.24575
$336$	0	0
$337$	2960.90	0.478607	0.239303	−	0.970945i	$-0.423081\pi$
$337$	2960.90	0.478607	0.239303	+	0.970945i	$0.423081\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9733.22	−1.54570
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4284.51	0.662838	0.331419	−	0.943484i	$-0.392473\pi$
$347$	4284.51	0.662838	0.331419	+	0.943484i	$0.392473\pi$
$348$	0	0
$349$	−3295.36	−0.505434	−0.252717	−	0.967540i	$-0.581324\pi$
$349$	−3295.36	−0.505434	−0.252717	+	0.967540i	$0.581324\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10629.7	−1.60272	−0.801360	−	0.598183i	$-0.795890\pi$
$353$	−10629.7	−1.60272	−0.801360	+	0.598183i	$0.795890\pi$
$354$	0	0
$355$	−8107.26	−1.21208
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3185.01	−0.468241	−0.234121	−	0.972208i	$-0.575221\pi$
$359$	−3185.01	−0.468241	−0.234121	+	0.972208i	$0.575221\pi$
$360$	0	0
$361$	19977.5	2.91259
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9574.65	1.37304
$366$	0	0
$367$	−7603.03	−1.08140	−0.540702	−	0.841214i	$-0.681841\pi$
$367$	−7603.03	−1.08140	−0.540702	+	0.841214i	$0.681841\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1511.96	0.209882	0.104941	−	0.994478i	$-0.466535\pi$
$373$	1511.96	0.209882	0.104941	+	0.994478i	$0.466535\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2831.26	0.386783
$378$	0	0
$379$	8848.36	1.19923	0.599617	−	0.800287i	$-0.295320\pi$
$379$	8848.36	1.19923	0.599617	+	0.800287i	$0.295320\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4134.10	0.551548	0.275774	−	0.961223i	$-0.411066\pi$
$383$	4134.10	0.551548	0.275774	+	0.961223i	$0.411066\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7630.61	−0.994569	−0.497284	−	0.867588i	$-0.665669\pi$
$389$	−7630.61	−0.994569	−0.497284	+	0.867588i	$0.665669\pi$
$390$	0	0
$391$	−7164.79	−0.926698
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11130.6	1.41783
$396$	0	0
$397$	11618.5	1.46881	0.734403	−	0.678714i	$-0.237462\pi$
$397$	11618.5	1.46881	0.734403	+	0.678714i	$0.237462\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3047.01	−0.379453	−0.189726	−	0.981837i	$-0.560760\pi$
$401$	−3047.01	−0.379453	−0.189726	+	0.981837i	$0.560760\pi$
$402$	0	0
$403$	−8868.92	−1.09626
$404$	0	0
$405$	0	0
$406$	0	0
$407$	19428.3	2.36616
$408$	0	0
$409$	−11582.8	−1.40033	−0.700164	−	0.713982i	$-0.746890\pi$
$409$	−11582.8	−1.40033	−0.700164	+	0.713982i	$0.746890\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1269.84	−0.150202
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1318.83	−0.153768	−0.0768841	−	0.997040i	$-0.524497\pi$
$419$	−1318.83	−0.153768	−0.0768841	+	0.997040i	$0.524497\pi$
$420$	0	0
$421$	−9733.56	−1.12680	−0.563402	−	0.826183i	$-0.690508\pi$
$421$	−9733.56	−1.12680	−0.563402	+	0.826183i	$0.690508\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	13779.8	1.57275
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14428.9	1.61257	0.806284	−	0.591528i	$-0.201475\pi$
$431$	14428.9	1.61257	0.806284	+	0.591528i	$0.201475\pi$
$432$	0	0
$433$	16620.3	1.84463	0.922313	−	0.386444i	$-0.126297\pi$
$433$	16620.3	1.84463	0.922313	+	0.386444i	$0.126297\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12979.1	1.42077
$438$	0	0
$439$	515.565	0.0560515	0.0280257	−	0.999607i	$-0.491078\pi$
$439$	515.565	0.0560515	0.0280257	+	0.999607i	$0.491078\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8976.09	−0.962679	−0.481340	−	0.876534i	$-0.659850\pi$
$443$	−8976.09	−0.962679	−0.481340	+	0.876534i	$0.659850\pi$
$444$	0	0
$445$	−22763.7	−2.42495
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2106.83	−0.221442	−0.110721	−	0.993852i	$-0.535316\pi$
$449$	−2106.83	−0.221442	−0.110721	+	0.993852i	$0.535316\pi$
$450$	0	0
$451$	10924.8	1.14064
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15726.2	1.60971	0.804857	−	0.593469i	$-0.202242\pi$
$457$	15726.2	1.60971	0.804857	+	0.593469i	$0.202242\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7921.16	−0.800272	−0.400136	−	0.916456i	$-0.631037\pi$
$461$	−7921.16	−0.800272	−0.400136	+	0.916456i	$0.631037\pi$
$462$	0	0
$463$	−5821.20	−0.584306	−0.292153	−	0.956372i	$-0.594372\pi$
$463$	−5821.20	−0.584306	−0.292153	+	0.956372i	$0.594372\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11787.9	1.16805	0.584024	−	0.811737i	$-0.301477\pi$
$467$	11787.9	1.16805	0.584024	+	0.811737i	$0.301477\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−12747.5	−1.23918
$474$	0	0
$475$	−24962.4	−2.41127
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5314.43	−0.506936	−0.253468	−	0.967344i	$-0.581571\pi$
$479$	−5314.43	−0.506936	−0.253468	+	0.967344i	$0.581571\pi$
$480$	0	0
$481$	17703.1	1.67816
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4041.35	0.378367
$486$	0	0
$487$	−18662.7	−1.73652	−0.868261	−	0.496107i	$-0.834762\pi$
$487$	−18662.7	−1.73652	−0.868261	+	0.496107i	$0.834762\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−13519.1	−1.24259	−0.621293	−	0.783578i	$-0.713392\pi$
$491$	−13519.1	−1.24259	−0.621293	+	0.783578i	$0.713392\pi$
$492$	0	0
$493$	3915.64	0.357711
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	13832.3	1.24092	0.620462	−	0.784237i	$-0.286945\pi$
$499$	13832.3	1.24092	0.620462	+	0.784237i	$0.286945\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	13469.0	1.19394	0.596970	−	0.802264i	$-0.296371\pi$
$503$	13469.0	1.19394	0.596970	+	0.802264i	$0.296371\pi$
$504$	0	0
$505$	−11570.7	−1.01958
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8775.25	0.764157	0.382079	−	0.924130i	$-0.375208\pi$
$509$	8775.25	0.764157	0.382079	+	0.924130i	$0.375208\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2901.54	−0.248267
$516$	0	0
$517$	−3274.63	−0.278565
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1660.76	−0.139653	−0.0698267	−	0.997559i	$-0.522245\pi$
$521$	−1660.76	−0.139653	−0.0698267	+	0.997559i	$0.522245\pi$
$522$	0	0
$523$	−6324.82	−0.528805	−0.264402	−	0.964412i	$-0.585175\pi$
$523$	−6324.82	−0.528805	−0.264402	+	0.964412i	$0.585175\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12265.8	−1.01386
$528$	0	0
$529$	−5889.79	−0.484079
$530$	0	0
$531$	0	0
$532$	0	0
$533$	9954.68	0.808977
$534$	0	0
$535$	792.848	0.0640707
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	4467.88	0.355063	0.177532	−	0.984115i	$-0.443189\pi$
$541$	4467.88	0.355063	0.177532	+	0.984115i	$0.443189\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	27198.7	2.13774
$546$	0	0
$547$	−10939.0	−0.855063	−0.427531	−	0.904000i	$-0.640617\pi$
$547$	−10939.0	−0.855063	−0.427531	+	0.904000i	$0.640617\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7093.26	−0.548427
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14325.1	1.08972	0.544862	−	0.838526i	$-0.316582\pi$
$557$	14325.1	1.08972	0.544862	+	0.838526i	$0.316582\pi$
$558$	0	0
$559$	−11615.6	−0.878867
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20402.8	1.52731	0.763656	−	0.645624i	$-0.223402\pi$
$563$	20402.8	1.52731	0.763656	+	0.645624i	$0.223402\pi$
$564$	0	0
$565$	−18774.6	−1.39797
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7250.43	0.534189	0.267095	−	0.963670i	$-0.413936\pi$
$569$	7250.43	0.534189	0.267095	+	0.963670i	$0.413936\pi$
$570$	0	0
$571$	24894.8	1.82454	0.912272	−	0.409586i	$-0.134327\pi$
$571$	24894.8	1.82454	0.912272	+	0.409586i	$0.134327\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12072.8	−0.875599
$576$	0	0
$577$	−1363.45	−0.0983726	−0.0491863	−	0.998790i	$-0.515663\pi$
$577$	−1363.45	−0.0983726	−0.0491863	+	0.998790i	$0.515663\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	11369.5	0.807677
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23854.8	1.67733	0.838666	−	0.544647i	$-0.183336\pi$
$587$	23854.8	1.67733	0.838666	+	0.544647i	$0.183336\pi$
$588$	0	0
$589$	22219.6	1.55440
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3437.20	0.238025	0.119013	−	0.992893i	$-0.462027\pi$
$593$	3437.20	0.238025	0.119013	+	0.992893i	$0.462027\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16511.7	−1.12630	−0.563148	−	0.826356i	$-0.690410\pi$
$599$	−16511.7	−1.12630	−0.563148	+	0.826356i	$0.690410\pi$
$600$	0	0
$601$	11148.3	0.756656	0.378328	−	0.925672i	$-0.376499\pi$
$601$	11148.3	0.756656	0.378328	+	0.925672i	$0.376499\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−63595.8	−4.27362
$606$	0	0
$607$	391.583	0.0261843	0.0130921	−	0.999914i	$-0.495833\pi$
$607$	391.583	0.0261843	0.0130921	+	0.999914i	$0.495833\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2983.85	−0.197567
$612$	0	0
$613$	−11740.5	−0.773563	−0.386782	−	0.922171i	$-0.626413\pi$
$613$	−11740.5	−0.773563	−0.386782	+	0.922171i	$0.626413\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8818.95	0.575426	0.287713	−	0.957717i	$-0.407105\pi$
$617$	8818.95	0.575426	0.287713	+	0.957717i	$0.407105\pi$
$618$	0	0
$619$	15924.7	1.03404	0.517018	−	0.855975i	$-0.327042\pi$
$619$	15924.7	1.03404	0.517018	+	0.855975i	$0.327042\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11453.1	−0.732998
$626$	0	0
$627$	0	0
$628$	0	0
$629$	24483.5	1.55202
$630$	0	0
$631$	29206.4	1.84261	0.921307	−	0.388836i	$-0.127123\pi$
$631$	29206.4	1.84261	0.921307	+	0.388836i	$0.127123\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−39948.9	−2.49658
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5613.01	−0.345867	−0.172933	−	0.984934i	$-0.555325\pi$
$641$	−5613.01	−0.345867	−0.172933	+	0.984934i	$0.555325\pi$
$642$	0	0
$643$	18995.0	1.16499	0.582496	−	0.812834i	$-0.302076\pi$
$643$	18995.0	1.16499	0.582496	+	0.812834i	$0.302076\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22665.0	1.37721	0.688604	−	0.725138i	$-0.258224\pi$
$647$	22665.0	1.37721	0.688604	+	0.725138i	$0.258224\pi$
$648$	0	0
$649$	−28113.3	−1.70038
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−26527.8	−1.58976	−0.794879	−	0.606768i	$-0.792466\pi$
$653$	−26527.8	−1.58976	−0.794879	+	0.606768i	$0.792466\pi$
$654$	0	0
$655$	−25797.5	−1.53892
$656$	0	0
$657$	0	0
$658$	0	0
$659$	17114.8	1.01168	0.505839	−	0.862628i	$-0.331183\pi$
$659$	17114.8	1.01168	0.505839	+	0.862628i	$0.331183\pi$
$660$	0	0
$661$	1191.82	0.0701305	0.0350653	−	0.999385i	$-0.488836\pi$
$661$	1191.82	0.0701305	0.0350653	+	0.999385i	$0.488836\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3430.57	−0.199149
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−39560.9	−2.27605
$672$	0	0
$673$	−9415.20	−0.539271	−0.269636	−	0.962962i	$-0.586903\pi$
$673$	−9415.20	−0.539271	−0.269636	+	0.962962i	$0.586903\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7917.13	0.449453	0.224727	−	0.974422i	$-0.427851\pi$
$677$	7917.13	0.449453	0.224727	+	0.974422i	$0.427851\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−29234.4	−1.63781	−0.818904	−	0.573930i	$-0.805418\pi$
$683$	−29234.4	−1.63781	−0.818904	+	0.573930i	$0.805418\pi$
$684$	0	0
$685$	25369.6	1.41507
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10359.9	0.572831
$690$	0	0
$691$	1222.46	0.0673001	0.0336501	−	0.999434i	$-0.489287\pi$
$691$	1222.46	0.0673001	0.0336501	+	0.999434i	$0.489287\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9184.14	0.501258
$696$	0	0
$697$	13767.4	0.748172
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13054.6	−0.703372	−0.351686	−	0.936118i	$-0.614392\pi$
$701$	−13054.6	−0.703372	−0.351686	+	0.936118i	$0.614392\pi$
$702$	0	0
$703$	−44352.3	−2.37948
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−19975.7	−1.05812	−0.529058	−	0.848585i	$-0.677455\pi$
$709$	−19975.7	−1.05812	−0.529058	+	0.848585i	$0.677455\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10746.3	0.564447
$714$	0	0
$715$	−78147.5	−4.08749
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6512.26	0.337783	0.168892	−	0.985635i	$-0.445981\pi$
$719$	6512.26	0.337783	0.168892	+	0.985635i	$0.445981\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6597.92	0.337987
$726$	0	0
$727$	−11437.1	−0.583467	−0.291733	−	0.956500i	$-0.594232\pi$
$727$	−11437.1	−0.583467	−0.291733	+	0.956500i	$0.594232\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−16064.4	−0.812808
$732$	0	0
$733$	12276.0	0.618589	0.309295	−	0.950966i	$-0.399907\pi$
$733$	12276.0	0.618589	0.309295	+	0.950966i	$0.399907\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32911.3	1.64492
$738$	0	0
$739$	12093.8	0.602000	0.301000	−	0.953624i	$-0.402680\pi$
$739$	12093.8	0.602000	0.301000	+	0.953624i	$0.402680\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−34331.6	−1.69516	−0.847580	−	0.530668i	$-0.821941\pi$
$743$	−34331.6	−1.69516	−0.847580	+	0.530668i	$0.821941\pi$
$744$	0	0
$745$	−39975.3	−1.96588
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−14557.9	−0.707358	−0.353679	−	0.935367i	$-0.615070\pi$
$751$	−14557.9	−0.707358	−0.353679	+	0.935367i	$0.615070\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10351.9	−0.498998
$756$	0	0
$757$	27115.5	1.30189	0.650944	−	0.759126i	$-0.274373\pi$
$757$	27115.5	1.30189	0.650944	+	0.759126i	$0.274373\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−10964.0	−0.522264	−0.261132	−	0.965303i	$-0.584096\pi$
$761$	−10964.0	−0.522264	−0.261132	+	0.965303i	$0.584096\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−25616.9	−1.20596
$768$	0	0
$769$	7835.53	0.367434	0.183717	−	0.982979i	$-0.441187\pi$
$769$	7835.53	0.367434	0.183717	+	0.982979i	$0.441187\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−28767.7	−1.33856	−0.669278	−	0.743012i	$-0.733396\pi$
$773$	−28767.7	−1.33856	−0.669278	+	0.743012i	$0.733396\pi$
$774$	0	0
$775$	−20668.0	−0.957956
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24939.8	−1.14706
$780$	0	0
$781$	34931.7	1.60046
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−44827.3	−2.03816
$786$	0	0
$787$	2695.83	0.122104	0.0610520	−	0.998135i	$-0.480554\pi$
$787$	2695.83	0.122104	0.0610520	+	0.998135i	$0.480554\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−36047.9	−1.61425
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12592.1	0.559643	0.279822	−	0.960052i	$-0.409725\pi$
$797$	12592.1	0.559643	0.279822	+	0.960052i	$0.409725\pi$
$798$	0	0
$799$	−4126.67	−0.182717
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−41254.2	−1.81299
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7799.49	0.338956	0.169478	−	0.985534i	$-0.445792\pi$
$809$	7799.49	0.338956	0.169478	+	0.985534i	$0.445792\pi$
$810$	0	0
$811$	19323.5	0.836669	0.418335	−	0.908293i	$-0.362614\pi$
$811$	19323.5	0.836669	0.418335	+	0.908293i	$0.362614\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−17609.1	−0.756834
$816$	0	0
$817$	29100.9	1.24616
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1085.30	−0.0461354	−0.0230677	−	0.999734i	$-0.507343\pi$
$821$	−1085.30	−0.0461354	−0.0230677	+	0.999734i	$0.507343\pi$
$822$	0	0
$823$	−24350.8	−1.03137	−0.515683	−	0.856779i	$-0.672462\pi$
$823$	−24350.8	−1.03137	−0.515683	+	0.856779i	$0.672462\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−31664.9	−1.33143	−0.665716	−	0.746205i	$-0.731874\pi$
$827$	−31664.9	−1.33143	−0.665716	+	0.746205i	$0.731874\pi$
$828$	0	0
$829$	3418.02	0.143200	0.0716000	−	0.997433i	$-0.477190\pi$
$829$	3418.02	0.143200	0.0716000	+	0.997433i	$0.477190\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	4061.08	0.168311
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9114.25	0.375040	0.187520	−	0.982261i	$-0.439955\pi$
$839$	9114.25	0.375040	0.187520	+	0.982261i	$0.439955\pi$
$840$	0	0
$841$	−22514.2	−0.923127
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−34617.8	−1.40933
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−21450.5	−0.864056
$852$	0	0
$853$	36229.5	1.45425	0.727124	−	0.686506i	$-0.240857\pi$
$853$	36229.5	1.45425	0.727124	+	0.686506i	$0.240857\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8645.18	0.344590	0.172295	−	0.985045i	$-0.444882\pi$
$857$	8645.18	0.344590	0.172295	+	0.985045i	$0.444882\pi$
$858$	0	0
$859$	8212.58	0.326205	0.163102	−	0.986609i	$-0.447850\pi$
$859$	8212.58	0.326205	0.163102	+	0.986609i	$0.447850\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2611.61	0.103013	0.0515065	−	0.998673i	$-0.483598\pi$
$863$	2611.61	0.103013	0.0515065	+	0.998673i	$0.483598\pi$
$864$	0	0
$865$	28100.9	1.10458
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−47958.6	−1.87213
$870$	0	0
$871$	29988.8	1.16663
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6446.58	−0.248216	−0.124108	−	0.992269i	$-0.539607\pi$
$877$	−6446.58	−0.248216	−0.124108	+	0.992269i	$0.539607\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	37166.3	1.42130	0.710649	−	0.703547i	$-0.248401\pi$
$881$	37166.3	1.42130	0.710649	+	0.703547i	$0.248401\pi$
$882$	0	0
$883$	24377.3	0.929060	0.464530	−	0.885557i	$-0.346223\pi$
$883$	24377.3	0.929060	0.464530	+	0.885557i	$0.346223\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36344.5	−1.37579	−0.687896	−	0.725809i	$-0.741466\pi$
$887$	−36344.5	−1.37579	−0.687896	+	0.725809i	$0.741466\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7475.55	0.280134
$894$	0	0
$895$	−5981.35	−0.223391
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5872.97	−0.217880
$900$	0	0
$901$	14327.8	0.529775
$902$	0	0
$903$	0	0
$904$	0	0
$905$	49673.9	1.82455
$906$	0	0
$907$	326.875	0.0119666	0.00598329	−	0.999982i	$-0.498095\pi$
$907$	326.875	0.0119666	0.00598329	+	0.999982i	$0.498095\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	38230.7	1.39038	0.695191	−	0.718825i	$-0.255320\pi$
$911$	38230.7	1.39038	0.695191	+	0.718825i	$0.255320\pi$
$912$	0	0
$913$	5471.34	0.198330
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−20049.3	−0.719659	−0.359829	−	0.933018i	$-0.617165\pi$
$919$	−20049.3	−0.719659	−0.359829	+	0.933018i	$0.617165\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	31829.8	1.13509
$924$	0	0
$925$	41255.0	1.46644
$926$	0	0
$927$	0	0
$928$	0	0
$929$	43341.5	1.53066	0.765332	−	0.643635i	$-0.222575\pi$
$929$	43341.5	1.53066	0.765332	+	0.643635i	$0.222575\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−108078.	−3.78026
$936$	0	0
$937$	−17530.0	−0.611185	−0.305592	−	0.952162i	$-0.598854\pi$
$937$	−17530.0	−0.611185	−0.305592	+	0.952162i	$0.598854\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−32625.0	−1.13023	−0.565115	−	0.825012i	$-0.691168\pi$
$941$	−32625.0	−1.13023	−0.565115	+	0.825012i	$0.691168\pi$
$942$	0	0
$943$	−12061.9	−0.416530
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19394.2	0.665497	0.332749	−	0.943016i	$-0.392024\pi$
$947$	19394.2	0.665497	0.332749	+	0.943016i	$0.392024\pi$
$948$	0	0
$949$	−37590.9	−1.28583
$950$	0	0
$951$	0	0
$952$	0	0
$953$	49618.5	1.68657	0.843285	−	0.537467i	$-0.180619\pi$
$953$	49618.5	1.68657	0.843285	+	0.537467i	$0.180619\pi$
$954$	0	0
$955$	58137.7	1.96994
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−11393.9	−0.382462
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−26717.9	−0.891274
$966$	0	0
$967$	28432.8	0.945538	0.472769	−	0.881186i	$-0.343254\pi$
$967$	28432.8	0.945538	0.472769	+	0.881186i	$0.343254\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	42690.2	1.41091	0.705455	−	0.708755i	$-0.250743\pi$
$971$	42690.2	1.41091	0.705455	+	0.708755i	$0.250743\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−22735.1	−0.744482	−0.372241	−	0.928136i	$-0.621411\pi$
$977$	−22735.1	−0.744482	−0.372241	+	0.928136i	$0.621411\pi$
$978$	0	0
$979$	98081.7	3.20195
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5643.83	0.183123	0.0915616	−	0.995799i	$-0.470814\pi$
$983$	5643.83	0.183123	0.0915616	+	0.995799i	$0.470814\pi$
$984$	0	0
$985$	−7593.45	−0.245632
$986$	0	0
$987$	0	0
$988$	0	0
$989$	14074.3	0.452515
$990$	0	0
$991$	−54844.6	−1.75802	−0.879009	−	0.476806i	$-0.841794\pi$
$991$	−54844.6	−1.75802	−0.879009	+	0.476806i	$0.841794\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	34711.0	1.10594
$996$	0	0
$997$	−44881.1	−1.42568	−0.712838	−	0.701328i	$-0.752591\pi$
$997$	−44881.1	−1.42568	−0.712838	+	0.701328i	$0.752591\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.a.bc.1.1		4
3.2	odd	2		588.4.a.j.1.4	✓	4
7.2	even	3		1764.4.k.bb.361.4		8
7.3	odd	6		1764.4.k.bd.1549.1		8
7.4	even	3		1764.4.k.bb.1549.4		8
7.5	odd	6		1764.4.k.bd.361.1		8
7.6	odd	2		1764.4.a.ba.1.4		4
12.11	even	2		2352.4.a.cq.1.4		4
21.2	odd	6		588.4.i.l.361.1		8
21.5	even	6		588.4.i.k.361.4		8
21.11	odd	6		588.4.i.l.373.1		8
21.17	even	6		588.4.i.k.373.4		8
21.20	even	2		588.4.a.k.1.1	yes	4
84.83	odd	2		2352.4.a.cl.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.4.a.j.1.4	✓	4	3.2	odd	2
588.4.a.k.1.1	yes	4	21.20	even	2
588.4.i.k.361.4		8	21.5	even	6
588.4.i.k.373.4		8	21.17	even	6
588.4.i.l.361.1		8	21.2	odd	6
588.4.i.l.373.1		8	21.11	odd	6
1764.4.a.ba.1.4		4	7.6	odd	2
1764.4.a.bc.1.1		4	1.1	even	1	trivial
1764.4.k.bb.361.4		8	7.2	even	3
1764.4.k.bb.1549.4		8	7.4	even	3
1764.4.k.bd.361.1		8	7.5	odd	6
1764.4.k.bd.1549.1		8	7.3	odd	6
2352.4.a.cl.1.1		4	84.83	odd	2
2352.4.a.cq.1.4		4	12.11	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.a (trivial)

\(f(q)\)	\(=\)	\(q-16.6547 q^{5} +O(q^{10})\)	\(q-16.6547 q^{5} +71.7600 q^{11} +65.3878 q^{13} +90.4316 q^{17} -163.818 q^{19} -79.2288 q^{23} +152.379 q^{25} +43.2995 q^{29} -135.636 q^{31} +270.740 q^{37} +152.241 q^{41} -177.641 q^{43} -45.6331 q^{47} +158.438 q^{53} -1195.14 q^{55} -391.769 q^{59} -551.295 q^{61} -1089.01 q^{65} +458.630 q^{67} +486.786 q^{71} -574.892 q^{73} -668.319 q^{79} +76.2450 q^{83} -1506.11 q^{85} +1366.80 q^{89} +2728.34 q^{95} -242.655 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 48 q^{17} - 192 q^{19} - 192 q^{23} + 324 q^{25} - 96 q^{29} - 48 q^{31} + 256 q^{37} + 1008 q^{41} - 112 q^{43} + 864 q^{47} + 648 q^{53} - 2352 q^{55} + 336 q^{59} - 960 q^{61} + 360 q^{65} + 720 q^{67} + 1344 q^{71} - 672 q^{73} - 1984 q^{79} + 3120 q^{83} + 680 q^{85} + 2160 q^{89} + 3744 q^{95} - 2016 q^{97}+O(q^{100}) \) 4 * q + 48 * q^17 - 192 * q^19 - 192 * q^23 + 324 * q^25 - 96 * q^29 - 48 * q^31 + 256 * q^37 + 1008 * q^41 - 112 * q^43 + 864 * q^47 + 648 * q^53 - 2352 * q^55 + 336 * q^59 - 960 * q^61 + 360 * q^65 + 720 * q^67 + 1344 * q^71 - 672 * q^73 - 1984 * q^79 + 3120 * q^83 + 680 * q^85 + 2160 * q^89 + 3744 * q^95 - 2016 * q^97

Introduction

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