LMFDB - Embedded newform 363.4.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,4,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("363.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	363.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:	$21.4176933321$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 1 $ x^4 - 2x^3 - 7x^2 + 8*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.35008$ of defining polynomial
Character	$\chi$	$=$	363.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.96812 q^{2} -3.00000 q^{3} +7.74597 q^{4} -11.4919 q^{5} +11.9044 q^{6} -13.5724 q^{7} +1.00803 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.96812 q^{2} -3.00000 q^{3} +7.74597 q^{4} -11.4919 q^{5} +11.9044 q^{6} -13.5724 q^{7} +1.00803 q^{8} +9.00000 q^{9} +45.6014 q^{10} -23.2379 q^{12} -67.5220 q^{13} +53.8569 q^{14} +34.4758 q^{15} -65.9677 q^{16} -87.1706 q^{17} -35.7131 q^{18} -112.271 q^{19} -89.0161 q^{20} +40.7172 q^{21} +126.952 q^{23} -3.02410 q^{24} +7.06453 q^{25} +267.935 q^{26} -27.0000 q^{27} -105.131 q^{28} -194.446 q^{29} -136.804 q^{30} -159.444 q^{31} +253.704 q^{32} +345.903 q^{34} +155.973 q^{35} +69.7137 q^{36} -321.855 q^{37} +445.506 q^{38} +202.566 q^{39} -11.5843 q^{40} -438.437 q^{41} -161.571 q^{42} +4.08802 q^{43} -103.427 q^{45} -503.759 q^{46} +57.5565 q^{47} +197.903 q^{48} -158.790 q^{49} -28.0329 q^{50} +261.512 q^{51} -523.023 q^{52} +722.952 q^{53} +107.139 q^{54} -13.6814 q^{56} +336.814 q^{57} +771.585 q^{58} -2.95160 q^{59} +267.048 q^{60} +612.326 q^{61} +632.691 q^{62} -122.151 q^{63} -478.984 q^{64} +775.959 q^{65} +424.621 q^{67} -675.220 q^{68} -380.855 q^{69} -618.919 q^{70} +205.460 q^{71} +9.07231 q^{72} +596.421 q^{73} +1277.16 q^{74} -21.1936 q^{75} -869.650 q^{76} -803.806 q^{78} -812.877 q^{79} +758.097 q^{80} +81.0000 q^{81} +1739.77 q^{82} +66.1858 q^{83} +315.394 q^{84} +1001.76 q^{85} -16.2218 q^{86} +583.338 q^{87} -430.347 q^{89} +410.412 q^{90} +916.435 q^{91} +983.363 q^{92} +478.331 q^{93} -228.391 q^{94} +1290.21 q^{95} -761.111 q^{96} +810.581 q^{97} +630.099 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 16 q^{5} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 16 * q^5 + 36 * q^9	$ 4 q - 12 q^{3} + 16 q^{5} + 36 q^{9} + 76 q^{14} - 48 q^{15} - 16 q^{16} - 480 q^{20} + 136 q^{23} + 524 q^{25} + 576 q^{26} - 108 q^{27} - 204 q^{31} + 640 q^{34} - 172 q^{37} + 868 q^{38} - 228 q^{42} + 144 q^{45} + 664 q^{47} + 48 q^{48} + 976 q^{49} + 2520 q^{53} + 472 q^{56} + 1816 q^{58} + 360 q^{59} + 1440 q^{60} - 1792 q^{64} + 2628 q^{67} - 408 q^{69} - 1856 q^{70} + 512 q^{71} - 1572 q^{75} - 1728 q^{78} + 3776 q^{80} + 324 q^{81} + 3272 q^{82} + 152 q^{86} - 544 q^{89} - 672 q^{91} + 2880 q^{92} + 612 q^{93} + 20 q^{97}+O(q^{100}) $ 4 * q - 12 * q^3 + 16 * q^5 + 36 * q^9 + 76 * q^14 - 48 * q^15 - 16 * q^16 - 480 * q^20 + 136 * q^23 + 524 * q^25 + 576 * q^26 - 108 * q^27 - 204 * q^31 + 640 * q^34 - 172 * q^37 + 868 * q^38 - 228 * q^42 + 144 * q^45 + 664 * q^47 + 48 * q^48 + 976 * q^49 + 2520 * q^53 + 472 * q^56 + 1816 * q^58 + 360 * q^59 + 1440 * q^60 - 1792 * q^64 + 2628 * q^67 - 408 * q^69 - 1856 * q^70 + 512 * q^71 - 1572 * q^75 - 1728 * q^78 + 3776 * q^80 + 324 * q^81 + 3272 * q^82 + 152 * q^86 - 544 * q^89 - 672 * q^91 + 2880 * q^92 + 612 * q^93 + 20 * 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 $

$2$	−3.96812	−1.40294	−0.701471	−	0.712698i	$-0.747473\pi$
$2$	−3.96812	−1.40294	−0.701471	+	0.712698i	$0.747473\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	7.74597	0.968246
$5$	−11.4919	−1.02787	−0.513935	−	0.857829i	$-0.671813\pi$
$5$	−11.4919	−1.02787	−0.513935	+	0.857829i	$0.671813\pi$
$6$	11.9044	0.809989
$7$	−13.5724	−0.732840	−0.366420	−	0.930450i	$-0.619417\pi$
$7$	−13.5724	−0.732840	−0.366420	+	0.930450i	$0.619417\pi$
$8$	1.00803	0.0445492
$9$	9.00000	0.333333
$10$	45.6014	1.44204
$11$	0	0
$11$	0	0
$12$	−23.2379	−0.559017
$13$	−67.5220	−1.44056	−0.720279	−	0.693685i	$-0.755986\pi$
$13$	−67.5220	−1.44056	−0.720279	+	0.693685i	$0.755986\pi$
$14$	53.8569	1.02813
$15$	34.4758	0.593441
$16$	−65.9677	−1.03075
$17$	−87.1706	−1.24365	−0.621823	−	0.783158i	$-0.713608\pi$
$17$	−87.1706	−1.24365	−0.621823	+	0.783158i	$0.713608\pi$
$18$	−35.7131	−0.467647
$19$	−112.271	−1.35562	−0.677811	−	0.735236i	$-0.737071\pi$
$19$	−112.271	−1.35562	−0.677811	+	0.735236i	$0.737071\pi$
$20$	−89.0161	−0.995231
$21$	40.7172	0.423105
$22$	0	0
$23$	126.952	1.15092	0.575462	−	0.817829i	$-0.304822\pi$
$23$	126.952	1.15092	0.575462	+	0.817829i	$0.304822\pi$
$24$	−3.02410	−0.0257205
$25$	7.06453	0.0565163
$26$	267.935	2.02102
$27$	−27.0000	−0.192450
$28$	−105.131	−0.709569
$29$	−194.446	−1.24509	−0.622547	−	0.782583i	$-0.713902\pi$
$29$	−194.446	−1.24509	−0.622547	+	0.782583i	$0.713902\pi$
$30$	−136.804	−0.832563
$31$	−159.444	−0.923771	−0.461886	−	0.886940i	$-0.652827\pi$
$31$	−159.444	−0.923771	−0.461886	+	0.886940i	$0.652827\pi$
$32$	253.704	1.40153
$33$	0	0
$34$	345.903	1.74476
$35$	155.973	0.753264
$36$	69.7137	0.322749
$37$	−321.855	−1.43007	−0.715035	−	0.699088i	$-0.753589\pi$
$37$	−321.855	−1.43007	−0.715035	+	0.699088i	$0.753589\pi$
$38$	445.506	1.90186
$39$	202.566	0.831706
$40$	−11.5843	−0.0457908
$41$	−438.437	−1.67006	−0.835029	−	0.550206i	$-0.814549\pi$
$41$	−438.437	−1.67006	−0.835029	+	0.550206i	$0.814549\pi$
$42$	−161.571	−0.593592
$43$	4.08802	0.0144981	0.00724905	−	0.999974i	$-0.497693\pi$
$43$	4.08802	0.0144981	0.00724905	+	0.999974i	$0.497693\pi$
$44$	0	0
$45$	−103.427	−0.342623
$46$	−503.759	−1.61468
$47$	57.5565	0.178627	0.0893135	−	0.996004i	$-0.471533\pi$
$47$	57.5565	0.178627	0.0893135	+	0.996004i	$0.471533\pi$
$48$	197.903	0.595101
$49$	−158.790	−0.462945
$50$	−28.0329	−0.0792890
$51$	261.512	0.718019
$52$	−523.023	−1.39481
$53$	722.952	1.87368	0.936840	−	0.349758i	$-0.113736\pi$
$53$	722.952	1.87368	0.936840	+	0.349758i	$0.113736\pi$
$54$	107.139	0.269996
$55$	0	0
$56$	−13.6814	−0.0326475
$57$	336.814	0.782669
$58$	771.585	1.74679
$59$	−2.95160	−0.00651298	−0.00325649	−	0.999995i	$-0.501037\pi$
$59$	−2.95160	−0.00651298	−0.00325649	+	0.999995i	$0.501037\pi$
$60$	267.048	0.574597
$61$	612.326	1.28525	0.642626	−	0.766180i	$-0.277845\pi$
$61$	612.326	1.28525	0.642626	+	0.766180i	$0.277845\pi$
$62$	632.691	1.29600
$63$	−122.151	−0.244280
$64$	−478.984	−0.935515
$65$	775.959	1.48071
$66$	0	0
$67$	424.621	0.774264	0.387132	−	0.922024i	$-0.373466\pi$
$67$	424.621	0.774264	0.387132	+	0.922024i	$0.373466\pi$
$68$	−675.220	−1.20415
$69$	−380.855	−0.664486
$70$	−618.919	−1.05679
$71$	205.460	0.343431	0.171715	−	0.985147i	$-0.445069\pi$
$71$	205.460	0.343431	0.171715	+	0.985147i	$0.445069\pi$
$72$	9.07231	0.0148497
$73$	596.421	0.956245	0.478122	−	0.878293i	$-0.341318\pi$
$73$	596.421	0.956245	0.478122	+	0.878293i	$0.341318\pi$
$74$	1277.16	2.00631
$75$	−21.1936	−0.0326297
$76$	−869.650	−1.31258
$77$	0	0
$78$	−803.806	−1.16684
$79$	−812.877	−1.15767	−0.578834	−	0.815445i	$-0.696492\pi$
$79$	−812.877	−1.15767	−0.578834	+	0.815445i	$0.696492\pi$
$80$	758.097	1.05947
$81$	81.0000	0.111111
$82$	1739.77	2.34299
$83$	66.1858	0.0875281	0.0437641	−	0.999042i	$-0.486065\pi$
$83$	66.1858	0.0875281	0.0437641	+	0.999042i	$0.486065\pi$
$84$	315.394	0.409670
$85$	1001.76	1.27831
$86$	−16.2218	−0.0203400
$87$	583.338	0.718855
$88$	0	0
$89$	−430.347	−0.512547	−0.256273	−	0.966604i	$-0.582495\pi$
$89$	−430.347	−0.512547	−0.256273	+	0.966604i	$0.582495\pi$
$90$	410.412	0.480681
$91$	916.435	1.05570
$92$	983.363	1.11438
$93$	478.331	0.533339
$94$	−228.391	−0.250603
$95$	1290.21	1.39340
$96$	−761.111	−0.809172
$97$	810.581	0.848474	0.424237	−	0.905551i	$-0.360542\pi$
$97$	810.581	0.848474	0.424237	+	0.905551i	$0.360542\pi$
$98$	630.099	0.649485
$99$	0	0
$100$	54.7216	0.0547216
$101$	252.688	0.248945	0.124472	−	0.992223i	$-0.460276\pi$
$101$	252.688	0.248945	0.124472	+	0.992223i	$0.460276\pi$
$102$	−1037.71	−1.00734
$103$	−1959.38	−1.87440	−0.937201	−	0.348790i	$-0.886592\pi$
$103$	−1959.38	−1.87440	−0.937201	+	0.348790i	$0.886592\pi$
$104$	−68.0645	−0.0641757
$105$	−467.919	−0.434897
$106$	−2868.76	−2.62866
$107$	259.775	0.234705	0.117352	−	0.993090i	$-0.462559\pi$
$107$	259.775	0.234705	0.117352	+	0.993090i	$0.462559\pi$
$108$	−209.141	−0.186339
$109$	−1218.89	−1.07109	−0.535543	−	0.844508i	$-0.679893\pi$
$109$	−1218.89	−1.07109	−0.535543	+	0.844508i	$0.679893\pi$
$110$	0	0
$111$	965.564	0.825652
$112$	895.340	0.755372
$113$	−323.016	−0.268910	−0.134455	−	0.990920i	$-0.542928\pi$
$113$	−323.016	−0.268910	−0.134455	+	0.990920i	$0.542928\pi$
$114$	−1336.52	−1.09804
$115$	−1458.92	−1.18300
$116$	−1506.17	−1.20556
$117$	−607.698	−0.480186
$118$	11.7123	0.00913733
$119$	1183.11	0.911393
$120$	34.7528	0.0264373
$121$	0	0
$122$	−2429.78	−1.80313
$123$	1315.31	0.964208
$124$	−1235.04	−0.894438
$125$	1355.31	0.969778
$126$	484.712	0.342711
$127$	1846.66	1.29027	0.645136	−	0.764068i	$-0.276801\pi$
$127$	1846.66	1.29027	0.645136	+	0.764068i	$0.276801\pi$
$128$	−128.963	−0.0890536
$129$	−12.2641	−0.00837048
$130$	−3079.10	−2.07734
$131$	346.196	0.230895	0.115448	−	0.993314i	$-0.463170\pi$
$131$	346.196	0.230895	0.115448	+	0.993314i	$0.463170\pi$
$132$	0	0
$133$	1523.79	0.993454
$134$	−1684.95	−1.08625
$135$	310.282	0.197814
$136$	−87.8709	−0.0554035
$137$	−2432.05	−1.51667	−0.758335	−	0.651865i	$-0.773987\pi$
$137$	−2432.05	−1.51667	−0.758335	+	0.651865i	$0.773987\pi$
$138$	1511.28	0.932235
$139$	100.347	0.0612326	0.0306163	−	0.999531i	$-0.490253\pi$
$139$	100.347	0.0612326	0.0306163	+	0.999531i	$0.490253\pi$
$140$	1208.16	0.729345
$141$	−172.669	−0.103130
$142$	−815.288	−0.481813
$143$	0	0
$144$	−593.710	−0.343582
$145$	2234.56	1.27979
$146$	−2366.67	−1.34156
$147$	476.371	0.267282
$148$	−2493.08	−1.38466
$149$	585.619	0.321985	0.160993	−	0.986956i	$-0.448530\pi$
$149$	585.619	0.321985	0.160993	+	0.986956i	$0.448530\pi$
$150$	84.0987	0.0457775
$151$	−2586.79	−1.39411	−0.697053	−	0.717020i	$-0.745506\pi$
$151$	−2586.79	−1.39411	−0.697053	+	0.717020i	$0.745506\pi$
$152$	−113.173	−0.0603919
$153$	−784.535	−0.414548
$154$	0	0
$155$	1832.31	0.949516
$156$	1569.07	0.805296
$157$	−706.935	−0.359360	−0.179680	−	0.983725i	$-0.557506\pi$
$157$	−706.935	−0.359360	−0.179680	+	0.983725i	$0.557506\pi$
$158$	3225.59	1.62414
$159$	−2168.85	−1.08177
$160$	−2915.54	−1.44059
$161$	−1723.04	−0.843443
$162$	−321.418	−0.155882
$163$	2284.30	1.09767	0.548835	−	0.835931i	$-0.315072\pi$
$163$	2284.30	1.09767	0.548835	+	0.835931i	$0.315072\pi$
$164$	−3396.12	−1.61703
$165$	0	0
$166$	−262.633	−0.122797
$167$	−3214.22	−1.48936	−0.744681	−	0.667420i	$-0.767398\pi$
$167$	−3214.22	−1.48936	−0.744681	+	0.667420i	$0.767398\pi$
$168$	41.0443	0.0188490
$169$	2362.23	1.07521
$170$	−3975.10	−1.79339
$171$	−1010.44	−0.451874
$172$	31.6657	0.0140377
$173$	832.809	0.365996	0.182998	−	0.983113i	$-0.441420\pi$
$173$	832.809	0.365996	0.182998	+	0.983113i	$0.441420\pi$
$174$	−2314.75	−1.00851
$175$	−95.8826	−0.0414174
$176$	0	0
$177$	8.85480	0.00376027
$178$	1707.67	0.719073
$179$	1063.70	0.444161	0.222080	−	0.975028i	$-0.428715\pi$
$179$	1063.70	0.444161	0.222080	+	0.975028i	$0.428715\pi$
$180$	−801.145	−0.331744
$181$	83.6776	0.0343630	0.0171815	−	0.999852i	$-0.494531\pi$
$181$	83.6776	0.0343630	0.0171815	+	0.999852i	$0.494531\pi$
$182$	−3636.52	−1.48108
$183$	−1836.98	−0.742041
$184$	127.972	0.0512728
$185$	3698.73	1.46993
$186$	−1898.07	−0.748244
$187$	0	0
$188$	445.830	0.172955
$189$	366.454	0.141035
$190$	−5119.73	−1.95486
$191$	2024.31	0.766881	0.383440	−	0.923566i	$-0.374739\pi$
$191$	2024.31	0.766881	0.383440	+	0.923566i	$0.374739\pi$
$192$	1436.95	0.540120
$193$	−411.650	−0.153530	−0.0767649	−	0.997049i	$-0.524459\pi$
$193$	−411.650	−0.153530	−0.0767649	+	0.997049i	$0.524459\pi$
$194$	−3216.48	−1.19036
$195$	−2327.88	−0.854886
$196$	−1229.98	−0.448245
$197$	−1476.28	−0.533913	−0.266957	−	0.963709i	$-0.586018\pi$
$197$	−1476.28	−0.533913	−0.266957	+	0.963709i	$0.586018\pi$
$198$	0	0
$199$	−4841.72	−1.72472	−0.862362	−	0.506291i	$-0.831016\pi$
$199$	−4841.72	−1.72472	−0.862362	+	0.506291i	$0.831016\pi$
$200$	7.12129	0.00251776
$201$	−1273.86	−0.447022
$202$	−1002.70	−0.349255
$203$	2639.10	0.912454
$204$	2025.66	0.695219
$205$	5038.49	1.71660
$206$	7775.05	2.62968
$207$	1142.56	0.383641
$208$	4454.28	1.48485
$209$	0	0
$210$	1856.76	0.610136
$211$	386.336	0.126050	0.0630248	−	0.998012i	$-0.479925\pi$
$211$	386.336	0.126050	0.0630248	+	0.998012i	$0.479925\pi$
$212$	5599.96	1.81418
$213$	−616.379	−0.198280
$214$	−1030.82	−0.329277
$215$	−46.9793	−0.0149021
$216$	−27.2169	−0.00857351
$217$	2164.03	0.676977
$218$	4836.70	1.50267
$219$	−1789.26	−0.552088
$220$	0	0
$221$	5885.93	1.79154
$222$	−3831.47	−1.15834
$223$	−4623.33	−1.38835	−0.694173	−	0.719808i	$-0.744230\pi$
$223$	−4623.33	−1.38835	−0.694173	+	0.719808i	$0.744230\pi$
$224$	−3443.36	−1.02710
$225$	63.5808	0.0188388
$226$	1281.77	0.377265
$227$	1826.28	0.533983	0.266992	−	0.963699i	$-0.413970\pi$
$227$	1826.28	0.533983	0.266992	+	0.963699i	$0.413970\pi$
$228$	2608.95	0.757816
$229$	56.2259	0.0162249	0.00811247	−	0.999967i	$-0.497418\pi$
$229$	56.2259	0.0162249	0.00811247	+	0.999967i	$0.497418\pi$
$230$	5789.17	1.65968
$231$	0	0
$232$	−196.008	−0.0554680
$233$	5718.16	1.60777	0.803883	−	0.594788i	$-0.202764\pi$
$233$	5718.16	1.60777	0.803883	+	0.594788i	$0.202764\pi$
$234$	2411.42	0.673673
$235$	−661.435	−0.183605
$236$	−22.8630	−0.00630616
$237$	2438.63	0.668380
$238$	−4694.73	−1.27863
$239$	−2069.72	−0.560164	−0.280082	−	0.959976i	$-0.590362\pi$
$239$	−2069.72	−0.560164	−0.280082	+	0.959976i	$0.590362\pi$
$240$	−2274.29	−0.611687
$241$	−4050.63	−1.08267	−0.541337	−	0.840806i	$-0.682082\pi$
$241$	−4050.63	−1.08267	−0.541337	+	0.840806i	$0.682082\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	4743.06	1.24444
$245$	1824.81	0.475848
$246$	−5219.31	−1.35273
$247$	7580.79	1.95285
$248$	−160.725	−0.0411533
$249$	−198.557	−0.0505344
$250$	−5378.02	−1.36054
$251$	−5572.20	−1.40125	−0.700626	−	0.713529i	$-0.747096\pi$
$251$	−5572.20	−1.40125	−0.700626	+	0.713529i	$0.747096\pi$
$252$	−946.181	−0.236523
$253$	0	0
$254$	−7327.76	−1.81018
$255$	−3005.28	−0.738030
$256$	4343.61	1.06045
$257$	−2727.81	−0.662085	−0.331043	−	0.943616i	$-0.607400\pi$
$257$	−2727.81	−0.662085	−0.331043	+	0.943616i	$0.607400\pi$
$258$	48.6653	0.0117433
$259$	4368.34	1.04801
$260$	6010.55	1.43369
$261$	−1750.01	−0.415031
$262$	−1373.75	−0.323932
$263$	−1746.18	−0.409406	−0.204703	−	0.978824i	$-0.565623\pi$
$263$	−1746.18	−0.409406	−0.204703	+	0.978824i	$0.565623\pi$
$264$	0	0
$265$	−8308.11	−1.92590
$266$	−6046.58	−1.39376
$267$	1291.04	0.295919
$268$	3289.10	0.749678
$269$	2679.54	0.607340	0.303670	−	0.952777i	$-0.401788\pi$
$269$	2679.54	0.607340	0.303670	+	0.952777i	$0.401788\pi$
$270$	−1231.24	−0.277521
$271$	6953.32	1.55861	0.779306	−	0.626643i	$-0.215572\pi$
$271$	6953.32	1.55861	0.779306	+	0.626643i	$0.215572\pi$
$272$	5750.45	1.28188
$273$	−2749.31	−0.609508
$274$	9650.66	2.12780
$275$	0	0
$276$	−2950.09	−0.643386
$277$	1588.49	0.344559	0.172280	−	0.985048i	$-0.444887\pi$
$277$	1588.49	0.344559	0.172280	+	0.985048i	$0.444887\pi$
$278$	−398.190	−0.0859058
$279$	−1434.99	−0.307924
$280$	157.226	0.0335574
$281$	−801.544	−0.170164	−0.0850820	−	0.996374i	$-0.527115\pi$
$281$	−801.544	−0.170164	−0.0850820	+	0.996374i	$0.527115\pi$
$282$	685.173	0.144686
$283$	1993.13	0.418655	0.209327	−	0.977846i	$-0.432873\pi$
$283$	1993.13	0.418655	0.209327	+	0.977846i	$0.432873\pi$
$284$	1591.48	0.332525
$285$	−3870.64	−0.804481
$286$	0	0
$287$	5950.64	1.22389
$288$	2283.33	0.467176
$289$	2685.71	0.546654
$290$	−8867.00	−1.79548
$291$	−2431.74	−0.489867
$292$	4619.86	0.925880
$293$	−1859.46	−0.370754	−0.185377	−	0.982667i	$-0.559351\pi$
$293$	−1859.46	−0.370754	−0.185377	+	0.982667i	$0.559351\pi$
$294$	−1890.30	−0.374981
$295$	33.9196	0.00669449
$296$	−324.441	−0.0637086
$297$	0	0
$298$	−2323.81	−0.451726
$299$	−8572.03	−1.65797
$300$	−164.165	−0.0315936
$301$	−55.4843	−0.0106248
$302$	10264.7	1.95585
$303$	−758.065	−0.143728
$304$	7406.29	1.39730
$305$	−7036.81	−1.32107
$306$	3113.13	0.581587
$307$	−1127.92	−0.209687	−0.104844	−	0.994489i	$-0.533434\pi$
$307$	−1127.92	−0.209687	−0.104844	+	0.994489i	$0.533434\pi$
$308$	0	0
$309$	5878.14	1.08219
$310$	−7270.84	−1.33212
$311$	−1656.33	−0.302000	−0.151000	−	0.988534i	$-0.548249\pi$
$311$	−1656.33	−0.302000	−0.151000	+	0.988534i	$0.548249\pi$
$312$	204.194	0.0370519
$313$	9967.66	1.80002	0.900009	−	0.435872i	$-0.143560\pi$
$313$	9967.66	1.80002	0.900009	+	0.435872i	$0.143560\pi$
$314$	2805.20	0.504162
$315$	1403.76	0.251088
$316$	−6296.51	−1.12091
$317$	−1867.37	−0.330858	−0.165429	−	0.986222i	$-0.552901\pi$
$317$	−1867.37	−0.330858	−0.165429	+	0.986222i	$0.552901\pi$
$318$	8606.27	1.51766
$319$	0	0
$320$	5504.45	0.961588
$321$	−779.325	−0.135507
$322$	6837.21	1.18330
$323$	9786.76	1.68591
$324$	627.423	0.107583
$325$	−477.012	−0.0814149
$326$	−9064.37	−1.53997
$327$	3656.67	0.618392
$328$	−441.960	−0.0743998
$329$	−781.179	−0.130905
$330$	0	0
$331$	8078.97	1.34157	0.670786	−	0.741651i	$-0.265957\pi$
$331$	8078.97	1.34157	0.670786	+	0.741651i	$0.265957\pi$
$332$	512.673	0.0847487
$333$	−2896.69	−0.476690
$334$	12754.4	2.08949
$335$	−4879.72	−0.795843
$336$	−2686.02	−0.436114
$337$	−7660.53	−1.23827	−0.619133	−	0.785286i	$-0.712516\pi$
$337$	−7660.53	−1.23827	−0.619133	+	0.785286i	$0.712516\pi$
$338$	−9373.59	−1.50845
$339$	969.048	0.155255
$340$	7759.59	1.23771
$341$	0	0
$342$	4009.55	0.633953
$343$	6810.49	1.07211
$344$	4.12087	0.000645879 0
$345$	4376.76	0.683005
$346$	−3304.69	−0.513471
$347$	−6055.85	−0.936874	−0.468437	−	0.883497i	$-0.655183\pi$
$347$	−6055.85	−0.936874	−0.468437	+	0.883497i	$0.655183\pi$
$348$	4518.52	0.696028
$349$	7051.57	1.08155	0.540776	−	0.841166i	$-0.318131\pi$
$349$	7051.57	1.08155	0.540776	+	0.841166i	$0.318131\pi$
$350$	380.473	0.0581062
$351$	1823.10	0.277235
$352$	0	0
$353$	−3426.06	−0.516574	−0.258287	−	0.966068i	$-0.583158\pi$
$353$	−3426.06	−0.516574	−0.258287	+	0.966068i	$0.583158\pi$
$354$	−35.1369	−0.00527544
$355$	−2361.13	−0.353002
$356$	−3333.45	−0.496271
$357$	−3549.34	−0.526193
$358$	−4220.89	−0.623132
$359$	11247.1	1.65348	0.826740	−	0.562584i	$-0.190193\pi$
$359$	11247.1	1.65348	0.826740	+	0.562584i	$0.190193\pi$
$360$	−104.258	−0.0152636
$361$	5745.85	0.837710
$362$	−332.043	−0.0482093
$363$	0	0
$364$	7098.68	1.02218
$365$	−6854.04	−0.982895
$366$	7289.35	1.04104
$367$	−5211.92	−0.741308	−0.370654	−	0.928771i	$-0.620866\pi$
$367$	−5211.92	−0.741308	−0.370654	+	0.928771i	$0.620866\pi$
$368$	−8374.71	−1.18631
$369$	−3945.93	−0.556686
$370$	−14677.0	−2.06222
$371$	−9812.18	−1.37311
$372$	3705.13	0.516404
$373$	−82.0495	−0.0113897	−0.00569486	−	0.999984i	$-0.501813\pi$
$373$	−82.0495	−0.0113897	−0.00569486	+	0.999984i	$0.501813\pi$
$374$	0	0
$375$	−4065.92	−0.559902
$376$	58.0189	0.00795770
$377$	13129.4	1.79363
$378$	−1454.13	−0.197864
$379$	−5821.60	−0.789011	−0.394506	−	0.918893i	$-0.629084\pi$
$379$	−5821.60	−0.789011	−0.394506	+	0.918893i	$0.629084\pi$
$380$	9993.96	1.34916
$381$	−5539.98	−0.744939
$382$	−8032.72	−1.07589
$383$	−3538.65	−0.472106	−0.236053	−	0.971740i	$-0.575854\pi$
$383$	−3538.65	−0.472106	−0.236053	+	0.971740i	$0.575854\pi$
$384$	386.890	0.0514151
$385$	0	0
$386$	1633.48	0.215393
$387$	36.7922	0.00483270
$388$	6278.73	0.821531
$389$	8142.86	1.06134	0.530668	−	0.847580i	$-0.321941\pi$
$389$	8142.86	1.06134	0.530668	+	0.847580i	$0.321941\pi$
$390$	9237.29	1.19935
$391$	−11066.4	−1.43134
$392$	−160.066	−0.0206239
$393$	−1038.59	−0.133307
$394$	5858.07	0.749049
$395$	9341.52	1.18993
$396$	0	0
$397$	13241.7	1.67401	0.837005	−	0.547195i	$-0.184304\pi$
$397$	13241.7	1.67401	0.837005	+	0.547195i	$0.184304\pi$
$398$	19212.5	2.41969
$399$	−4571.37	−0.573571
$400$	−466.031	−0.0582539
$401$	5499.97	0.684926	0.342463	−	0.939531i	$-0.388739\pi$
$401$	5499.97	0.684926	0.342463	+	0.939531i	$0.388739\pi$
$402$	5054.84	0.627145
$403$	10766.0	1.33075
$404$	1957.32	0.241040
$405$	−930.847	−0.114208
$406$	−10472.2	−1.28012
$407$	0	0
$408$	263.613	0.0319872
$409$	−11307.5	−1.36704	−0.683520	−	0.729931i	$-0.739552\pi$
$409$	−11307.5	−1.36704	−0.683520	+	0.729931i	$0.739552\pi$
$410$	−19993.3	−2.40829
$411$	7296.14	0.875650
$412$	−15177.3	−1.81488
$413$	40.0603	0.00477297
$414$	−4533.83	−0.538226
$415$	−760.603	−0.0899675
$416$	−17130.6	−2.01898
$417$	−301.042	−0.0353527
$418$	0	0
$419$	−10437.1	−1.21691	−0.608455	−	0.793588i	$-0.708210\pi$
$419$	−10437.1	−1.21691	−0.608455	+	0.793588i	$0.708210\pi$
$420$	−3624.48	−0.421087
$421$	−1981.89	−0.229433	−0.114716	−	0.993398i	$-0.536596\pi$
$421$	−1981.89	−0.229433	−0.114716	+	0.993398i	$0.536596\pi$
$422$	−1533.03	−0.176840
$423$	518.008	0.0595424
$424$	728.760	0.0834710
$425$	−615.819	−0.0702862
$426$	2445.87	0.278175
$427$	−8310.73	−0.941884
$428$	2012.21	0.227252
$429$	0	0
$430$	186.419	0.0209068
$431$	8834.65	0.987355	0.493678	−	0.869645i	$-0.335652\pi$
$431$	8834.65	0.987355	0.493678	+	0.869645i	$0.335652\pi$
$432$	1781.13	0.198367
$433$	1610.08	0.178696	0.0893482	−	0.996000i	$-0.471522\pi$
$433$	1610.08	0.178696	0.0893482	+	0.996000i	$0.471522\pi$
$434$	−8587.13	−0.949759
$435$	−6703.68	−0.738889
$436$	−9441.47	−1.03707
$437$	−14253.0	−1.56022
$438$	7100.01	0.774547
$439$	−8438.73	−0.917446	−0.458723	−	0.888579i	$-0.651693\pi$
$439$	−8438.73	−0.917446	−0.458723	+	0.888579i	$0.651693\pi$
$440$	0	0
$441$	−1429.11	−0.154315
$442$	−23356.1	−2.51343
$443$	−13158.2	−1.41120	−0.705601	−	0.708609i	$-0.749323\pi$
$443$	−13158.2	−1.41120	−0.705601	+	0.708609i	$0.749323\pi$
$444$	7479.23	0.799434
$445$	4945.52	0.526831
$446$	18345.9	1.94777
$447$	−1756.86	−0.185898
$448$	6500.96	0.685583
$449$	−15508.4	−1.63004	−0.815018	−	0.579435i	$-0.803273\pi$
$449$	−15508.4	−1.63004	−0.815018	+	0.579435i	$0.803273\pi$
$450$	−252.296	−0.0264297
$451$	0	0
$452$	−2502.07	−0.260371
$453$	7760.37	0.804887
$454$	−7246.88	−0.749147
$455$	−10531.6	−1.08512
$456$	339.520	0.0348673
$457$	−2014.57	−0.206209	−0.103105	−	0.994671i	$-0.532878\pi$
$457$	−2014.57	−0.206209	−0.103105	+	0.994671i	$0.532878\pi$
$458$	−223.111	−0.0227626
$459$	2353.61	0.239340
$460$	−11300.7	−1.14543
$461$	−4384.13	−0.442927	−0.221463	−	0.975169i	$-0.571083\pi$
$461$	−4384.13	−0.442927	−0.221463	+	0.975169i	$0.571083\pi$
$462$	0	0
$463$	−14761.3	−1.48167	−0.740836	−	0.671685i	$-0.765571\pi$
$463$	−14761.3	−1.48167	−0.740836	+	0.671685i	$0.765571\pi$
$464$	12827.2	1.28337
$465$	−5496.94	−0.548204
$466$	−22690.4	−2.25560
$467$	−8355.59	−0.827945	−0.413973	−	0.910289i	$-0.635859\pi$
$467$	−8355.59	−0.827945	−0.413973	+	0.910289i	$0.635859\pi$
$468$	−4707.21	−0.464938
$469$	−5763.12	−0.567412
$470$	2624.65	0.257588
$471$	2120.81	0.207477
$472$	−2.97531	−0.000290148 0
$473$	0	0
$474$	−9676.77	−0.937698
$475$	−793.145	−0.0766147
$476$	9164.35	0.882453
$477$	6506.56	0.624560
$478$	8212.90	0.785877
$479$	6649.18	0.634256	0.317128	−	0.948383i	$-0.397281\pi$
$479$	6649.18	0.634256	0.317128	+	0.948383i	$0.397281\pi$
$480$	8746.63	0.831724
$481$	21732.3	2.06010
$482$	16073.4	1.51893
$483$	5169.11	0.486962
$484$	0	0
$485$	−9315.14	−0.872121
$486$	964.253	0.0899988
$487$	4639.77	0.431721	0.215861	−	0.976424i	$-0.430744\pi$
$487$	4639.77	0.431721	0.215861	+	0.976424i	$0.430744\pi$
$488$	617.246	0.0572570
$489$	−6852.90	−0.633740
$490$	−7241.05	−0.667586
$491$	−16552.1	−1.52136	−0.760680	−	0.649128i	$-0.775134\pi$
$491$	−16552.1	−1.52136	−0.760680	+	0.649128i	$0.775134\pi$
$492$	10188.4	0.933590
$493$	16950.0	1.54845
$494$	−30081.5	−2.73974
$495$	0	0
$496$	10518.1	0.952173
$497$	−2788.58	−0.251680
$498$	787.899	0.0708968
$499$	−18627.9	−1.67114	−0.835572	−	0.549382i	$-0.814863\pi$
$499$	−18627.9	−1.67114	−0.835572	+	0.549382i	$0.814863\pi$
$500$	10498.2	0.938984
$501$	9642.65	0.859884
$502$	22111.2	1.96587
$503$	13449.7	1.19223	0.596115	−	0.802899i	$-0.296710\pi$
$503$	13449.7	1.19223	0.596115	+	0.802899i	$0.296710\pi$
$504$	−123.133	−0.0108825
$505$	−2903.88	−0.255883
$506$	0	0
$507$	−7086.68	−0.620770
$508$	14304.2	1.24930
$509$	2780.11	0.242095	0.121047	−	0.992647i	$-0.461375\pi$
$509$	2780.11	0.242095	0.121047	+	0.992647i	$0.461375\pi$
$510$	11925.3	1.03541
$511$	−8094.86	−0.700774
$512$	−16204.3	−1.39870
$513$	3031.33	0.260890
$514$	10824.3	0.928867
$515$	22517.1	1.92664
$516$	−94.9971	−0.00810468
$517$	0	0
$518$	−17334.1	−1.47030
$519$	−2498.43	−0.211308
$520$	782.193	0.0659643
$521$	−18428.7	−1.54966	−0.774832	−	0.632168i	$-0.782165\pi$
$521$	−18428.7	−1.54966	−0.774832	+	0.632168i	$0.782165\pi$
$522$	6944.26	0.582265
$523$	−3521.88	−0.294457	−0.147229	−	0.989103i	$-0.547035\pi$
$523$	−3521.88	−0.294457	−0.147229	+	0.989103i	$0.547035\pi$
$524$	2681.62	0.223563
$525$	287.648	0.0239123
$526$	6929.04	0.574373
$527$	13898.8	1.14884
$528$	0	0
$529$	3949.71	0.324625
$530$	32967.6	2.70192
$531$	−26.5644	−0.00217099
$532$	11803.2	0.961908
$533$	29604.2	2.40581
$534$	−5123.00	−0.415157
$535$	−2985.32	−0.241246
$536$	428.033	0.0344929
$537$	−3191.10	−0.256436
$538$	−10632.7	−0.852063
$539$	0	0
$540$	2403.44	0.191532
$541$	−3403.53	−0.270479	−0.135240	−	0.990813i	$-0.543180\pi$
$541$	−3403.53	−0.270479	−0.135240	+	0.990813i	$0.543180\pi$
$542$	−27591.6	−2.18664
$543$	−251.033	−0.0198395
$544$	−22115.5	−1.74300
$545$	14007.4	1.10094
$546$	10909.6	0.855104
$547$	13015.0	1.01733	0.508665	−	0.860964i	$-0.330139\pi$
$547$	13015.0	1.01733	0.508665	+	0.860964i	$0.330139\pi$
$548$	−18838.6	−1.46851
$549$	5510.94	0.428417
$550$	0	0
$551$	21830.7	1.68788
$552$	−383.915	−0.0296023
$553$	11032.7	0.848385
$554$	−6303.30	−0.483396
$555$	−11096.2	−0.848662
$556$	777.286	0.0592882
$557$	5077.07	0.386216	0.193108	−	0.981177i	$-0.438143\pi$
$557$	5077.07	0.386216	0.193108	+	0.981177i	$0.438143\pi$
$558$	5694.22	0.431999
$559$	−276.032	−0.0208853
$560$	−10289.2	−0.776424
$561$	0	0
$562$	3180.62	0.238730
$563$	−9844.19	−0.736915	−0.368458	−	0.929645i	$-0.620114\pi$
$563$	−9844.19	−0.736915	−0.368458	+	0.929645i	$0.620114\pi$
$564$	−1337.49	−0.0998556
$565$	3712.08	0.276404
$566$	−7908.97	−0.587348
$567$	−1099.36	−0.0814267
$568$	207.110	0.0152996
$569$	5319.05	0.391891	0.195946	−	0.980615i	$-0.437222\pi$
$569$	5319.05	0.391891	0.195946	+	0.980615i	$0.437222\pi$
$570$	15359.2	1.12864
$571$	−10691.4	−0.783573	−0.391786	−	0.920056i	$-0.628143\pi$
$571$	−10691.4	−0.783573	−0.391786	+	0.920056i	$0.628143\pi$
$572$	0	0
$573$	−6072.94	−0.442759
$574$	−23612.8	−1.71704
$575$	896.854	0.0650459
$576$	−4310.85	−0.311838
$577$	−14719.5	−1.06201	−0.531005	−	0.847368i	$-0.678186\pi$
$577$	−14719.5	−1.06201	−0.531005	+	0.847368i	$0.678186\pi$
$578$	−10657.2	−0.766923
$579$	1234.95	0.0886405
$580$	17308.8	1.23916
$581$	−898.299	−0.0641441
$582$	9649.44	0.687255
$583$	0	0
$584$	601.213	0.0426000
$585$	6983.63	0.493568
$586$	7378.57	0.520147
$587$	−9308.36	−0.654510	−0.327255	−	0.944936i	$-0.606124\pi$
$587$	−9308.36	−0.654510	−0.327255	+	0.944936i	$0.606124\pi$
$588$	3689.95	0.258794
$589$	17900.9	1.25228
$590$	−134.597	−0.00939198
$591$	4428.85	0.308255
$592$	21232.0	1.47404
$593$	23670.0	1.63914	0.819569	−	0.572981i	$-0.194213\pi$
$593$	23670.0	1.63914	0.819569	+	0.572981i	$0.194213\pi$
$594$	0	0
$595$	−13596.3	−0.936793
$596$	4536.19	0.311761
$597$	14525.2	0.995770
$598$	34014.8	2.32604
$599$	7476.65	0.509996	0.254998	−	0.966942i	$-0.417925\pi$
$599$	7476.65	0.509996	0.254998	+	0.966942i	$0.417925\pi$
$600$	−21.3639	−0.00145363
$601$	−12315.0	−0.835841	−0.417920	−	0.908484i	$-0.637241\pi$
$601$	−12315.0	−0.835841	−0.417920	+	0.908484i	$0.637241\pi$
$602$	220.168	0.0149060
$603$	3821.59	0.258088
$604$	−20037.2	−1.34984
$605$	0	0
$606$	3008.09	0.201643
$607$	7209.54	0.482086	0.241043	−	0.970514i	$-0.422511\pi$
$607$	7209.54	0.482086	0.241043	+	0.970514i	$0.422511\pi$
$608$	−28483.6	−1.89994
$609$	−7917.29	−0.526806
$610$	27922.9	1.85339
$611$	−3886.33	−0.257323
$612$	−6076.98	−0.401385
$613$	−8903.43	−0.586633	−0.293317	−	0.956015i	$-0.594759\pi$
$613$	−8903.43	−0.586633	−0.293317	+	0.956015i	$0.594759\pi$
$614$	4475.73	0.294179
$615$	−15115.5	−0.991080
$616$	0	0
$617$	−2749.78	−0.179420	−0.0897099	−	0.995968i	$-0.528594\pi$
$617$	−2749.78	−0.179420	−0.0897099	+	0.995968i	$0.528594\pi$
$618$	−23325.1	−1.51824
$619$	−26559.8	−1.72460	−0.862299	−	0.506399i	$-0.830976\pi$
$619$	−26559.8	−1.72460	−0.862299	+	0.506399i	$0.830976\pi$
$620$	14193.0	0.919365
$621$	−3427.69	−0.221495
$622$	6572.52	0.423688
$623$	5840.83	0.375615
$624$	−13362.8	−0.857278
$625$	−16458.2	−1.05332
$626$	−39552.9	−2.52532
$627$	0	0
$628$	−5475.90	−0.347949
$629$	28056.3	1.77850
$630$	−5570.27	−0.352262
$631$	−26711.2	−1.68519	−0.842596	−	0.538546i	$-0.818974\pi$
$631$	−26711.2	−1.68519	−0.842596	+	0.538546i	$0.818974\pi$
$632$	−819.407	−0.0515732
$633$	−1159.01	−0.0727747
$634$	7409.95	0.464175
$635$	−21221.7	−1.32623
$636$	−16799.9	−1.04742
$637$	10721.8	0.666899
$638$	0	0
$639$	1849.14	0.114477
$640$	1482.04	0.0915355
$641$	13617.5	0.839095	0.419548	−	0.907733i	$-0.362189\pi$
$641$	13617.5	0.839095	0.419548	+	0.907733i	$0.362189\pi$
$642$	3092.46	0.190108
$643$	−7993.20	−0.490235	−0.245117	−	0.969493i	$-0.578827\pi$
$643$	−7993.20	−0.490235	−0.245117	+	0.969493i	$0.578827\pi$
$644$	−13346.6	−0.816660
$645$	140.938	0.00860376
$646$	−38835.0	−2.36524
$647$	−27825.2	−1.69076	−0.845380	−	0.534166i	$-0.820626\pi$
$647$	−27825.2	−1.69076	−0.845380	+	0.534166i	$0.820626\pi$
$648$	81.6508	0.00494992
$649$	0	0
$650$	1892.84	0.114220
$651$	−6492.09	−0.390853
$652$	17694.1	1.06281
$653$	25007.1	1.49863	0.749313	−	0.662216i	$-0.230384\pi$
$653$	25007.1	1.49863	0.749313	+	0.662216i	$0.230384\pi$
$654$	−14510.1	−0.867568
$655$	−3978.46	−0.237330
$656$	28922.7	1.72140
$657$	5367.79	0.318748
$658$	3099.81	0.183652
$659$	21087.2	1.24650	0.623248	−	0.782025i	$-0.285813\pi$
$659$	21087.2	1.24650	0.623248	+	0.782025i	$0.285813\pi$
$660$	0	0
$661$	−14271.0	−0.839751	−0.419876	−	0.907582i	$-0.637926\pi$
$661$	−14271.0	−0.839751	−0.419876	+	0.907582i	$0.637926\pi$
$662$	−32058.3	−1.88215
$663$	−17657.8	−1.03435
$664$	66.7175	0.00389931
$665$	−17511.3	−1.02114
$666$	11494.4	0.668769
$667$	−24685.2	−1.43301
$668$	−24897.2	−1.44207
$669$	13870.0	0.801562
$670$	19363.3	1.11652
$671$	0	0
$672$	10330.1	0.592994
$673$	9388.49	0.537741	0.268870	−	0.963176i	$-0.413350\pi$
$673$	9388.49	0.537741	0.268870	+	0.963176i	$0.413350\pi$
$674$	30397.9	1.73722
$675$	−190.742	−0.0108766
$676$	18297.7	1.04106
$677$	3239.61	0.183912	0.0919561	−	0.995763i	$-0.470688\pi$
$677$	3239.61	0.183912	0.0919561	+	0.995763i	$0.470688\pi$
$678$	−3845.30	−0.217814
$679$	−11001.5	−0.621796
$680$	1009.81	0.0569475
$681$	−5478.83	−0.308295
$682$	0	0
$683$	20716.5	1.16061	0.580303	−	0.814401i	$-0.302934\pi$
$683$	20716.5	1.16061	0.580303	+	0.814401i	$0.302934\pi$
$684$	−7826.85	−0.437525
$685$	27948.9	1.55894
$686$	−27024.8	−1.50410
$687$	−168.678	−0.00936747
$688$	−269.678	−0.0149438
$689$	−48815.2	−2.69914
$690$	−17367.5	−0.958216
$691$	−27118.1	−1.49294	−0.746469	−	0.665420i	$-0.768253\pi$
$691$	−27118.1	−1.49294	−0.746469	+	0.665420i	$0.768253\pi$
$692$	6450.91	0.354374
$693$	0	0
$694$	24030.3	1.31438
$695$	−1153.18	−0.0629392
$696$	588.025	0.0320244
$697$	38218.8	2.07696
$698$	−27981.5	−1.51736
$699$	−17154.5	−0.928244
$700$	−742.703	−0.0401022
$701$	−6409.27	−0.345328	−0.172664	−	0.984981i	$-0.555237\pi$
$701$	−6409.27	−0.345328	−0.172664	+	0.984981i	$0.555237\pi$
$702$	−7234.26	−0.388945
$703$	36135.1	1.93863
$704$	0	0
$705$	1984.31	0.106005
$706$	13595.0	0.724723
$707$	−3429.59	−0.182437
$708$	68.5890	0.00364086
$709$	−17210.3	−0.911632	−0.455816	−	0.890074i	$-0.650653\pi$
$709$	−17210.3	−0.911632	−0.455816	+	0.890074i	$0.650653\pi$
$710$	9369.24	0.495241
$711$	−7315.89	−0.385889
$712$	−433.804	−0.0228336
$713$	−20241.6	−1.06319
$714$	14084.2	0.738218
$715$	0	0
$716$	8239.40	0.430057
$717$	6209.16	0.323411
$718$	−44629.8	−2.31974
$719$	−10051.0	−0.521335	−0.260667	−	0.965429i	$-0.583943\pi$
$719$	−10051.0	−0.521335	−0.260667	+	0.965429i	$0.583943\pi$
$720$	6822.87	0.353157
$721$	26593.5	1.37364
$722$	−22800.2	−1.17526
$723$	12151.9	0.625082
$724$	648.164	0.0332719
$725$	−1373.67	−0.0703680
$726$	0	0
$727$	1668.92	0.0851400	0.0425700	−	0.999093i	$-0.486445\pi$
$727$	1668.92	0.0851400	0.0425700	+	0.999093i	$0.486445\pi$
$728$	923.798	0.0470306
$729$	729.000	0.0370370
$730$	27197.6	1.37894
$731$	−356.355	−0.0180305
$732$	−14229.2	−0.718478
$733$	18242.1	0.919221	0.459610	−	0.888121i	$-0.347989\pi$
$733$	18242.1	0.919221	0.459610	+	0.888121i	$0.347989\pi$
$734$	20681.5	1.04001
$735$	−5474.42	−0.274731
$736$	32208.1	1.61305
$737$	0	0
$738$	15657.9	0.780998
$739$	9596.59	0.477694	0.238847	−	0.971057i	$-0.423230\pi$
$739$	9596.59	0.477694	0.238847	+	0.971057i	$0.423230\pi$
$740$	28650.3	1.42325
$741$	−22742.4	−1.12748
$742$	38935.9	1.92639
$743$	5085.37	0.251096	0.125548	−	0.992088i	$-0.459931\pi$
$743$	5085.37	0.251096	0.125548	+	0.992088i	$0.459931\pi$
$744$	482.174	0.0237599
$745$	−6729.90	−0.330959
$746$	325.582	0.0159791
$747$	595.672	0.0291760
$748$	0	0
$749$	−3525.77	−0.172001
$750$	16134.1	0.785510
$751$	−6152.28	−0.298935	−0.149467	−	0.988767i	$-0.547756\pi$
$751$	−6152.28	−0.298935	−0.149467	+	0.988767i	$0.547756\pi$
$752$	−3796.87	−0.184119
$753$	16716.6	0.809013
$754$	−52099.0	−2.51636
$755$	29727.2	1.43296
$756$	2838.54	0.136557
$757$	32336.0	1.55254	0.776269	−	0.630402i	$-0.217110\pi$
$757$	32336.0	1.55254	0.776269	+	0.630402i	$0.217110\pi$
$758$	23100.8	1.10694
$759$	0	0
$760$	1300.58	0.0620750
$761$	−39095.6	−1.86231	−0.931154	−	0.364627i	$-0.881197\pi$
$761$	−39095.6	−1.86231	−0.931154	+	0.364627i	$0.881197\pi$
$762$	21983.3	1.04511
$763$	16543.2	0.784935
$764$	15680.3	0.742529
$765$	9015.83	0.426102
$766$	14041.8	0.662338
$767$	199.298	0.00938232
$768$	−13030.8	−0.612252
$769$	12565.9	0.589256	0.294628	−	0.955612i	$-0.404804\pi$
$769$	12565.9	0.589256	0.294628	+	0.955612i	$0.404804\pi$
$770$	0	0
$771$	8183.42	0.382255
$772$	−3188.63	−0.148655
$773$	37205.4	1.73116	0.865578	−	0.500774i	$-0.166951\pi$
$773$	37205.4	1.73116	0.865578	+	0.500774i	$0.166951\pi$
$774$	−145.996	−0.00677999
$775$	−1126.39	−0.0522081
$776$	817.093	0.0377989
$777$	−13105.0	−0.605071
$778$	−32311.8	−1.48899
$779$	49223.9	2.26397
$780$	−18031.7	−0.827739
$781$	0	0
$782$	43913.0	2.00809
$783$	5250.04	0.239618
$784$	10475.0	0.477179
$785$	8124.06	0.369376
$786$	4121.24	0.187023
$787$	−29516.6	−1.33692	−0.668459	−	0.743749i	$-0.733046\pi$
$787$	−29516.6	−1.33692	−0.668459	+	0.743749i	$0.733046\pi$
$788$	−11435.3	−0.516959
$789$	5238.53	0.236371
$790$	−37068.3	−1.66940
$791$	4384.10	0.197068
$792$	0	0
$793$	−41345.5	−1.85148
$794$	−52544.7	−2.34854
$795$	24924.3	1.11192
$796$	−37503.8	−1.66996
$797$	11288.6	0.501709	0.250855	−	0.968025i	$-0.419288\pi$
$797$	11288.6	0.501709	0.250855	+	0.968025i	$0.419288\pi$
$798$	18139.7	0.804687
$799$	−5017.23	−0.222149
$800$	1792.30	0.0792091
$801$	−3873.12	−0.170849
$802$	−21824.5	−0.960911
$803$	0	0
$804$	−9867.30	−0.432827
$805$	19801.0	0.866949
$806$	−42720.6	−1.86696
$807$	−8038.62	−0.350648
$808$	254.719	0.0110903
$809$	10197.4	0.443168	0.221584	−	0.975141i	$-0.428877\pi$
$809$	10197.4	0.443168	0.221584	+	0.975141i	$0.428877\pi$
$810$	3693.71	0.160227
$811$	8747.22	0.378738	0.189369	−	0.981906i	$-0.439356\pi$
$811$	8747.22	0.378738	0.189369	+	0.981906i	$0.439356\pi$
$812$	20442.3	0.883480
$813$	−20860.0	−0.899865
$814$	0	0
$815$	−26251.0	−1.12826
$816$	−17251.3	−0.740095
$817$	−458.968	−0.0196539
$818$	44869.5	1.91788
$819$	8247.92	0.351899
$820$	39028.0	1.66209
$821$	−2221.32	−0.0944270	−0.0472135	−	0.998885i	$-0.515034\pi$
$821$	−2221.32	−0.0944270	−0.0472135	+	0.998885i	$0.515034\pi$
$822$	−28952.0	−1.22849
$823$	19758.1	0.836844	0.418422	−	0.908253i	$-0.362583\pi$
$823$	19758.1	0.836844	0.418422	+	0.908253i	$0.362583\pi$
$824$	−1975.12	−0.0835032
$825$	0	0
$826$	−158.964	−0.00669620
$827$	−41462.1	−1.74338	−0.871691	−	0.490056i	$-0.836976\pi$
$827$	−41462.1	−1.74338	−0.871691	+	0.490056i	$0.836976\pi$
$828$	8850.27	0.371459
$829$	20493.2	0.858576	0.429288	−	0.903168i	$-0.358764\pi$
$829$	20493.2	0.858576	0.429288	+	0.903168i	$0.358764\pi$
$830$	3018.16	0.126219
$831$	−4765.46	−0.198931
$832$	32342.0	1.34766
$833$	13841.8	0.575740
$834$	1194.57	0.0495978
$835$	36937.6	1.53087
$836$	0	0
$837$	4304.98	0.177780
$838$	41415.6	1.70725
$839$	36109.8	1.48588	0.742938	−	0.669360i	$-0.233432\pi$
$839$	36109.8	1.48588	0.742938	+	0.669360i	$0.233432\pi$
$840$	−471.678	−0.0193743
$841$	13420.2	0.550257
$842$	7864.36	0.321881
$843$	2404.63	0.0982442
$844$	2992.54	0.122047
$845$	−27146.5	−1.10517
$846$	−2055.52	−0.0835345
$847$	0	0
$848$	−47691.5	−1.93129
$849$	−5979.39	−0.241710
$850$	2443.64	0.0986074
$851$	−40860.0	−1.64590
$852$	−4774.45	−0.191984
$853$	−25284.5	−1.01492	−0.507459	−	0.861676i	$-0.669415\pi$
$853$	−25284.5	−1.01492	−0.507459	+	0.861676i	$0.669415\pi$
$854$	32978.0	1.32141
$855$	11611.9	0.464468
$856$	261.862	0.0104559
$857$	547.314	0.0218155	0.0109078	−	0.999941i	$-0.496528\pi$
$857$	547.314	0.0218155	0.0109078	+	0.999941i	$0.496528\pi$
$858$	0	0
$859$	32971.6	1.30963	0.654817	−	0.755788i	$-0.272746\pi$
$859$	32971.6	1.30963	0.654817	+	0.755788i	$0.272746\pi$
$860$	−363.900	−0.0144289
$861$	−17851.9	−0.706610
$862$	−35056.9	−1.38520
$863$	1526.26	0.0602021	0.0301010	−	0.999547i	$-0.490417\pi$
$863$	1526.26	0.0602021	0.0301010	+	0.999547i	$0.490417\pi$
$864$	−6850.00	−0.269724
$865$	−9570.59	−0.376196
$866$	−6388.99	−0.250701
$867$	−8057.13	−0.315611
$868$	16762.5	0.655480
$869$	0	0
$870$	26601.0	1.03662
$871$	−28671.3	−1.11537
$872$	−1228.68	−0.0477161
$873$	7295.22	0.282825
$874$	56557.7	2.18889
$875$	−18394.7	−0.710693
$876$	−13859.6	−0.534557
$877$	4727.50	0.182026	0.0910128	−	0.995850i	$-0.470990\pi$
$877$	4727.50	0.182026	0.0910128	+	0.995850i	$0.470990\pi$
$878$	33485.9	1.28712
$879$	5578.39	0.214055
$880$	0	0
$881$	−10481.7	−0.400838	−0.200419	−	0.979710i	$-0.564230\pi$
$881$	−10481.7	−0.400838	−0.200419	+	0.979710i	$0.564230\pi$
$882$	5670.89	0.216495
$883$	−37338.2	−1.42302	−0.711512	−	0.702674i	$-0.751989\pi$
$883$	−37338.2	−1.42302	−0.711512	+	0.702674i	$0.751989\pi$
$884$	45592.3	1.73465
$885$	−101.759	−0.00386507
$886$	52213.1	1.97984
$887$	46961.5	1.77769	0.888846	−	0.458207i	$-0.151508\pi$
$887$	46961.5	1.77769	0.888846	+	0.458207i	$0.151508\pi$
$888$	973.322	0.0367822
$889$	−25063.6	−0.945563
$890$	−19624.4	−0.739114
$891$	0	0
$892$	−35812.2	−1.34426
$893$	−6461.94	−0.242151
$894$	6971.42	0.260804
$895$	−12224.0	−0.456539
$896$	1750.34	0.0652620
$897$	25716.1	0.957230
$898$	61539.1	2.28685
$899$	31003.1	1.15018
$900$	492.495	0.0182405
$901$	−63020.1	−2.33019
$902$	0	0
$903$	166.453	0.00613422
$904$	−325.611	−0.0119797
$905$	−961.617	−0.0353207
$906$	−30794.1	−1.12921
$907$	−26343.8	−0.964423	−0.482211	−	0.876055i	$-0.660166\pi$
$907$	−26343.8	−0.964423	−0.482211	+	0.876055i	$0.660166\pi$
$908$	14146.3	0.517027
$909$	2274.20	0.0829816
$910$	41790.7	1.52236
$911$	−1824.39	−0.0663498	−0.0331749	−	0.999450i	$-0.510562\pi$
$911$	−1824.39	−0.0663498	−0.0331749	+	0.999450i	$0.510562\pi$
$912$	−22218.9	−0.806732
$913$	0	0
$914$	7994.06	0.289300
$915$	21110.4	0.762721
$916$	435.524	0.0157097
$917$	−4698.70	−0.169209
$918$	−9339.39	−0.335780
$919$	20121.9	0.722262	0.361131	−	0.932515i	$-0.382391\pi$
$919$	20121.9	0.722262	0.361131	+	0.932515i	$0.382391\pi$
$920$	−1470.64	−0.0527017
$921$	3383.77	0.121063
$922$	17396.7	0.621400
$923$	−13873.1	−0.494732
$924$	0	0
$925$	−2273.75	−0.0808222
$926$	58574.5	2.07870
$927$	−17634.4	−0.624800
$928$	−49331.6	−1.74503
$929$	34106.4	1.20452	0.602258	−	0.798301i	$-0.294268\pi$
$929$	34106.4	1.20452	0.602258	+	0.798301i	$0.294268\pi$
$930$	21812.5	0.769098
$931$	17827.6	0.627579
$932$	44292.7	1.55671
$933$	4968.99	0.174360
$934$	33156.0	1.16156
$935$	0	0
$936$	−612.581	−0.0213919
$937$	−7703.60	−0.268587	−0.134293	−	0.990942i	$-0.542876\pi$
$937$	−7703.60	−0.268587	−0.134293	+	0.990942i	$0.542876\pi$
$938$	22868.7	0.796046
$939$	−29903.0	−1.03924
$940$	−5123.45	−0.177775
$941$	−9282.36	−0.321569	−0.160784	−	0.986990i	$-0.551402\pi$
$941$	−9282.36	−0.321569	−0.160784	+	0.986990i	$0.551402\pi$
$942$	−8415.61	−0.291078
$943$	−55660.3	−1.92211
$944$	194.710	0.00671322
$945$	−4211.27	−0.144966
$946$	0	0
$947$	33215.0	1.13975	0.569874	−	0.821732i	$-0.306992\pi$
$947$	33215.0	1.13975	0.569874	+	0.821732i	$0.306992\pi$
$948$	18889.5	0.647156
$949$	−40271.6	−1.37752
$950$	3147.29	0.107486
$951$	5602.11	0.191021
$952$	1192.62	0.0406019
$953$	798.262	0.0271335	0.0135668	−	0.999908i	$-0.495681\pi$
$953$	798.262	0.0271335	0.0135668	+	0.999908i	$0.495681\pi$
$954$	−25818.8	−0.876221
$955$	−23263.3	−0.788254
$956$	−16032.0	−0.542376
$957$	0	0
$958$	−26384.7	−0.889825
$959$	33008.7	1.11148
$960$	−16513.4	−0.555173
$961$	−4368.76	−0.146647
$962$	−86236.3	−2.89020
$963$	2337.98	0.0782349
$964$	−31376.1	−1.04829
$965$	4730.66	0.157809
$966$	−20511.6	−0.683179
$967$	−33883.1	−1.12679	−0.563396	−	0.826187i	$-0.690505\pi$
$967$	−33883.1	−1.12679	−0.563396	+	0.826187i	$0.690505\pi$
$968$	0	0
$969$	−29360.3	−0.973362
$970$	36963.6	1.22353
$971$	37313.7	1.23322	0.616609	−	0.787270i	$-0.288506\pi$
$971$	37313.7	1.23322	0.616609	+	0.787270i	$0.288506\pi$
$972$	−1882.27	−0.0621130
$973$	−1361.95	−0.0448737
$974$	−18411.2	−0.605680
$975$	1431.03	0.0470049
$976$	−40393.8	−1.32477
$977$	22242.2	0.728343	0.364172	−	0.931332i	$-0.381352\pi$
$977$	22242.2	0.728343	0.364172	+	0.931332i	$0.381352\pi$
$978$	27193.1	0.889100
$979$	0	0
$980$	14134.9	0.460737
$981$	−10970.0	−0.357029
$982$	65680.8	2.13438
$983$	6313.28	0.204845	0.102422	−	0.994741i	$-0.467341\pi$
$983$	6313.28	0.204845	0.102422	+	0.994741i	$0.467341\pi$
$984$	1325.88	0.0429547
$985$	16965.4	0.548793
$986$	−67259.5	−2.17239
$987$	2343.54	0.0755781
$988$	58720.5	1.89084
$989$	518.981	0.0166862
$990$	0	0
$991$	15619.7	0.500681	0.250340	−	0.968158i	$-0.419457\pi$
$991$	15619.7	0.500681	0.250340	+	0.968158i	$0.419457\pi$
$992$	−40451.4	−1.29469
$993$	−24236.9	−0.774557
$994$	11065.4	0.353092
$995$	55640.7	1.77279
$996$	−1538.02	−0.0489297
$997$	−47936.3	−1.52273	−0.761364	−	0.648325i	$-0.775470\pi$
$997$	−47936.3	−1.52273	−0.761364	+	0.648325i	$0.775470\pi$
$998$	73917.8	2.34452
$999$	8690.08	0.275217

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.4.a.r.1.1	✓	4
3.2	odd	2		1089.4.a.bb.1.4		4
11.10	odd	2	inner	363.4.a.r.1.4	yes	4
33.32	even	2		1089.4.a.bb.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.4.a.r.1.1	✓	4	1.1	even	1	trivial
363.4.a.r.1.4	yes	4	11.10	odd	2	inner
1089.4.a.bb.1.1		4	33.32	even	2
1089.4.a.bb.1.4		4	3.2	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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q-3.96812 q^{2} -3.00000 q^{3} +7.74597 q^{4} -11.4919 q^{5} +11.9044 q^{6} -13.5724 q^{7} +1.00803 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.96812 q^{2} -3.00000 q^{3} +7.74597 q^{4} -11.4919 q^{5} +11.9044 q^{6} -13.5724 q^{7} +1.00803 q^{8} +9.00000 q^{9} +45.6014 q^{10} -23.2379 q^{12} -67.5220 q^{13} +53.8569 q^{14} +34.4758 q^{15} -65.9677 q^{16} -87.1706 q^{17} -35.7131 q^{18} -112.271 q^{19} -89.0161 q^{20} +40.7172 q^{21} +126.952 q^{23} -3.02410 q^{24} +7.06453 q^{25} +267.935 q^{26} -27.0000 q^{27} -105.131 q^{28} -194.446 q^{29} -136.804 q^{30} -159.444 q^{31} +253.704 q^{32} +345.903 q^{34} +155.973 q^{35} +69.7137 q^{36} -321.855 q^{37} +445.506 q^{38} +202.566 q^{39} -11.5843 q^{40} -438.437 q^{41} -161.571 q^{42} +4.08802 q^{43} -103.427 q^{45} -503.759 q^{46} +57.5565 q^{47} +197.903 q^{48} -158.790 q^{49} -28.0329 q^{50} +261.512 q^{51} -523.023 q^{52} +722.952 q^{53} +107.139 q^{54} -13.6814 q^{56} +336.814 q^{57} +771.585 q^{58} -2.95160 q^{59} +267.048 q^{60} +612.326 q^{61} +632.691 q^{62} -122.151 q^{63} -478.984 q^{64} +775.959 q^{65} +424.621 q^{67} -675.220 q^{68} -380.855 q^{69} -618.919 q^{70} +205.460 q^{71} +9.07231 q^{72} +596.421 q^{73} +1277.16 q^{74} -21.1936 q^{75} -869.650 q^{76} -803.806 q^{78} -812.877 q^{79} +758.097 q^{80} +81.0000 q^{81} +1739.77 q^{82} +66.1858 q^{83} +315.394 q^{84} +1001.76 q^{85} -16.2218 q^{86} +583.338 q^{87} -430.347 q^{89} +410.412 q^{90} +916.435 q^{91} +983.363 q^{92} +478.331 q^{93} -228.391 q^{94} +1290.21 q^{95} -761.111 q^{96} +810.581 q^{97} +630.099 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 16 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 16 * q^5 + 36 * q^9	\( 4 q - 12 q^{3} + 16 q^{5} + 36 q^{9} + 76 q^{14} - 48 q^{15} - 16 q^{16} - 480 q^{20} + 136 q^{23} + 524 q^{25} + 576 q^{26} - 108 q^{27} - 204 q^{31} + 640 q^{34} - 172 q^{37} + 868 q^{38} - 228 q^{42} + 144 q^{45} + 664 q^{47} + 48 q^{48} + 976 q^{49} + 2520 q^{53} + 472 q^{56} + 1816 q^{58} + 360 q^{59} + 1440 q^{60} - 1792 q^{64} + 2628 q^{67} - 408 q^{69} - 1856 q^{70} + 512 q^{71} - 1572 q^{75} - 1728 q^{78} + 3776 q^{80} + 324 q^{81} + 3272 q^{82} + 152 q^{86} - 544 q^{89} - 672 q^{91} + 2880 q^{92} + 612 q^{93} + 20 q^{97}+O(q^{100}) \) 4 * q - 12 * q^3 + 16 * q^5 + 36 * q^9 + 76 * q^14 - 48 * q^15 - 16 * q^16 - 480 * q^20 + 136 * q^23 + 524 * q^25 + 576 * q^26 - 108 * q^27 - 204 * q^31 + 640 * q^34 - 172 * q^37 + 868 * q^38 - 228 * q^42 + 144 * q^45 + 664 * q^47 + 48 * q^48 + 976 * q^49 + 2520 * q^53 + 472 * q^56 + 1816 * q^58 + 360 * q^59 + 1440 * q^60 - 1792 * q^64 + 2628 * q^67 - 408 * q^69 - 1856 * q^70 + 512 * q^71 - 1572 * q^75 - 1728 * q^78 + 3776 * q^80 + 324 * q^81 + 3272 * q^82 + 152 * q^86 - 544 * q^89 - 672 * q^91 + 2880 * q^92 + 612 * q^93 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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