LMFDB - Embedded newform 2151.4.a.d.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.10380 q^{2} -6.78162 q^{4} -20.8736 q^{5} -33.6250 q^{7} +16.3160 q^{8} +O(q^{10})$	\(q-1.10380 q^{2} -6.78162 q^{4} -20.8736 q^{5} -33.6250 q^{7} +16.3160 q^{8} +23.0403 q^{10} -69.0452 q^{11} -37.7160 q^{13} +37.1153 q^{14} +36.2434 q^{16} -83.4196 q^{17} -80.0717 q^{19} +141.557 q^{20} +76.2121 q^{22} -1.95956 q^{23} +310.707 q^{25} +41.6310 q^{26} +228.032 q^{28} +169.396 q^{29} +251.447 q^{31} -170.533 q^{32} +92.0787 q^{34} +701.874 q^{35} +13.1079 q^{37} +88.3832 q^{38} -340.573 q^{40} -150.847 q^{41} +335.988 q^{43} +468.238 q^{44} +2.16297 q^{46} -457.072 q^{47} +787.639 q^{49} -342.958 q^{50} +255.776 q^{52} -207.825 q^{53} +1441.22 q^{55} -548.624 q^{56} -186.979 q^{58} -273.288 q^{59} -729.276 q^{61} -277.548 q^{62} -101.712 q^{64} +787.269 q^{65} -385.910 q^{67} +565.720 q^{68} -774.729 q^{70} -430.141 q^{71} -351.322 q^{73} -14.4685 q^{74} +543.016 q^{76} +2321.64 q^{77} -537.394 q^{79} -756.530 q^{80} +166.505 q^{82} -486.003 q^{83} +1741.27 q^{85} -370.864 q^{86} -1126.54 q^{88} +17.5701 q^{89} +1268.20 q^{91} +13.2890 q^{92} +504.516 q^{94} +1671.38 q^{95} -1030.76 q^{97} -869.397 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

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 $

$2$	−1.10380	−0.390253	−0.195126	−	0.980778i	$-0.562512\pi$
$2$	−1.10380	−0.390253	−0.195126	+	0.980778i	$0.562512\pi$
$3$	0	0
$3$	0	0
$4$	−6.78162	−0.847703
$5$	−20.8736	−1.86699	−0.933495	−	0.358590i	$-0.883258\pi$
$5$	−20.8736	−1.86699	−0.933495	+	0.358590i	$0.883258\pi$
$6$	0	0
$7$	−33.6250	−1.81558	−0.907789	−	0.419427i	$-0.862231\pi$
$7$	−33.6250	−1.81558	−0.907789	+	0.419427i	$0.862231\pi$
$8$	16.3160	0.721071
$9$	0	0
$10$	23.0403	0.728598
$11$	−69.0452	−1.89254	−0.946268	−	0.323383i	$-0.895180\pi$
$11$	−69.0452	−1.89254	−0.946268	+	0.323383i	$0.895180\pi$
$12$	0	0
$13$	−37.7160	−0.804657	−0.402328	−	0.915495i	$-0.631799\pi$
$13$	−37.7160	−0.804657	−0.402328	+	0.915495i	$0.631799\pi$
$14$	37.1153	0.708534
$15$	0	0
$16$	36.2434	0.566303
$17$	−83.4196	−1.19013	−0.595066	−	0.803677i	$-0.702874\pi$
$17$	−83.4196	−1.19013	−0.595066	+	0.803677i	$0.702874\pi$
$18$	0	0
$19$	−80.0717	−0.966826	−0.483413	−	0.875392i	$-0.660603\pi$
$19$	−80.0717	−0.966826	−0.483413	+	0.875392i	$0.660603\pi$
$20$	141.557	1.58265
$21$	0	0
$22$	76.2121	0.738567
$23$	−1.95956	−0.0177651	−0.00888255	−	0.999961i	$-0.502827\pi$
$23$	−1.95956	−0.0177651	−0.00888255	+	0.999961i	$0.502827\pi$
$24$	0	0
$25$	310.707	2.48565
$26$	41.6310	0.314019
$27$	0	0
$28$	228.032	1.53907
$29$	169.396	1.08469	0.542345	−	0.840156i	$-0.317536\pi$
$29$	169.396	1.08469	0.542345	+	0.840156i	$0.317536\pi$
$30$	0	0
$31$	251.447	1.45681	0.728407	−	0.685145i	$-0.240261\pi$
$31$	251.447	1.45681	0.728407	+	0.685145i	$0.240261\pi$
$32$	−170.533	−0.942072
$33$	0	0
$34$	92.0787	0.464452
$35$	701.874	3.38967
$36$	0	0
$37$	13.1079	0.0582413	0.0291206	−	0.999576i	$-0.490729\pi$
$37$	13.1079	0.0582413	0.0291206	+	0.999576i	$0.490729\pi$
$38$	88.3832	0.377307
$39$	0	0
$40$	−340.573	−1.34623
$41$	−150.847	−0.574594	−0.287297	−	0.957842i	$-0.592757\pi$
$41$	−150.847	−0.574594	−0.287297	+	0.957842i	$0.592757\pi$
$42$	0	0
$43$	335.988	1.19158	0.595788	−	0.803142i	$-0.296840\pi$
$43$	335.988	1.19158	0.595788	+	0.803142i	$0.296840\pi$
$44$	468.238	1.60431
$45$	0	0
$46$	2.16297	0.00693288
$47$	−457.072	−1.41853	−0.709263	−	0.704944i	$-0.750972\pi$
$47$	−457.072	−1.41853	−0.709263	+	0.704944i	$0.750972\pi$
$48$	0	0
$49$	787.639	2.29632
$50$	−342.958	−0.970033
$51$	0	0
$52$	255.776	0.682110
$53$	−207.825	−0.538623	−0.269311	−	0.963053i	$-0.586796\pi$
$53$	−207.825	−0.538623	−0.269311	+	0.963053i	$0.586796\pi$
$54$	0	0
$55$	1441.22	3.53335
$56$	−548.624	−1.30916
$57$	0	0
$58$	−186.979	−0.423303
$59$	−273.288	−0.603035	−0.301518	−	0.953461i	$-0.597493\pi$
$59$	−273.288	−0.603035	−0.301518	+	0.953461i	$0.597493\pi$
$60$	0	0
$61$	−729.276	−1.53073	−0.765363	−	0.643599i	$-0.777440\pi$
$61$	−729.276	−1.53073	−0.765363	+	0.643599i	$0.777440\pi$
$62$	−277.548	−0.568525
$63$	0	0
$64$	−101.712	−0.198657
$65$	787.269	1.50229
$66$	0	0
$67$	−385.910	−0.703677	−0.351839	−	0.936061i	$-0.614443\pi$
$67$	−385.910	−0.703677	−0.351839	+	0.936061i	$0.614443\pi$
$68$	565.720	1.00888
$69$	0	0
$70$	−774.729	−1.32283
$71$	−430.141	−0.718992	−0.359496	−	0.933147i	$-0.617051\pi$
$71$	−430.141	−0.718992	−0.359496	+	0.933147i	$0.617051\pi$
$72$	0	0
$73$	−351.322	−0.563276	−0.281638	−	0.959521i	$-0.590878\pi$
$73$	−351.322	−0.563276	−0.281638	+	0.959521i	$0.590878\pi$
$74$	−14.4685	−0.0227288
$75$	0	0
$76$	543.016	0.819581
$77$	2321.64	3.43605
$78$	0	0
$79$	−537.394	−0.765336	−0.382668	−	0.923886i	$-0.624995\pi$
$79$	−537.394	−0.765336	−0.382668	+	0.923886i	$0.624995\pi$
$80$	−756.530	−1.05728
$81$	0	0
$82$	166.505	0.224237
$83$	−486.003	−0.642720	−0.321360	−	0.946957i	$-0.604140\pi$
$83$	−486.003	−0.642720	−0.321360	+	0.946957i	$0.604140\pi$
$84$	0	0
$85$	1741.27	2.22196
$86$	−370.864	−0.465016
$87$	0	0
$88$	−1126.54	−1.36465
$89$	17.5701	0.0209261	0.0104631	−	0.999945i	$-0.496669\pi$
$89$	17.5701	0.0209261	0.0104631	+	0.999945i	$0.496669\pi$
$90$	0	0
$91$	1268.20	1.46092
$92$	13.2890	0.0150595
$93$	0	0
$94$	504.516	0.553584
$95$	1671.38	1.80506
$96$	0	0
$97$	−1030.76	−1.07895	−0.539473	−	0.842003i	$-0.681376\pi$
$97$	−1030.76	−1.07895	−0.539473	+	0.842003i	$0.681376\pi$
$98$	−869.397	−0.896146
$99$	0	0
$100$	−2107.10	−2.10710
$101$	−349.752	−0.344571	−0.172285	−	0.985047i	$-0.555115\pi$
$101$	−349.752	−0.344571	−0.172285	+	0.985047i	$0.555115\pi$
$102$	0	0
$103$	1766.95	1.69032	0.845161	−	0.534511i	$-0.179504\pi$
$103$	1766.95	1.69032	0.845161	+	0.534511i	$0.179504\pi$
$104$	−615.373	−0.580215
$105$	0	0
$106$	229.398	0.210199
$107$	−158.197	−0.142929	−0.0714647	−	0.997443i	$-0.522767\pi$
$107$	−158.197	−0.142929	−0.0714647	+	0.997443i	$0.522767\pi$
$108$	0	0
$109$	2076.11	1.82436	0.912181	−	0.409788i	$-0.134397\pi$
$109$	2076.11	1.82436	0.912181	+	0.409788i	$0.134397\pi$
$110$	−1590.82	−1.37890
$111$	0	0
$112$	−1218.68	−1.02817
$113$	1216.86	1.01303	0.506515	−	0.862231i	$-0.330934\pi$
$113$	1216.86	1.01303	0.506515	+	0.862231i	$0.330934\pi$
$114$	0	0
$115$	40.9031	0.0331673
$116$	−1148.78	−0.919495
$117$	0	0
$118$	301.656	0.235336
$119$	2804.98	2.16078
$120$	0	0
$121$	3436.24	2.58169
$122$	804.976	0.597370
$123$	0	0
$124$	−1705.22	−1.23495
$125$	−3876.37	−2.77370
$126$	0	0
$127$	2348.04	1.64059	0.820295	−	0.571940i	$-0.193809\pi$
$127$	2348.04	1.64059	0.820295	+	0.571940i	$0.193809\pi$
$128$	1476.54	1.01960
$129$	0	0
$130$	−868.988	−0.586271
$131$	653.161	0.435625	0.217813	−	0.975991i	$-0.430108\pi$
$131$	653.161	0.435625	0.217813	+	0.975991i	$0.430108\pi$
$132$	0	0
$133$	2692.41	1.75535
$134$	425.967	0.274612
$135$	0	0
$136$	−1361.07	−0.858169
$137$	−416.681	−0.259850	−0.129925	−	0.991524i	$-0.541474\pi$
$137$	−416.681	−0.259850	−0.129925	+	0.991524i	$0.541474\pi$
$138$	0	0
$139$	262.976	0.160470	0.0802350	−	0.996776i	$-0.474433\pi$
$139$	262.976	0.160470	0.0802350	+	0.996776i	$0.474433\pi$
$140$	−4759.84	−2.87343
$141$	0	0
$142$	474.791	0.280588
$143$	2604.11	1.52284
$144$	0	0
$145$	−3535.90	−2.02511
$146$	387.790	0.219820
$147$	0	0
$148$	−88.8929	−0.0493713
$149$	−721.231	−0.396547	−0.198274	−	0.980147i	$-0.563533\pi$
$149$	−721.231	−0.396547	−0.198274	+	0.980147i	$0.563533\pi$
$150$	0	0
$151$	−2930.42	−1.57930	−0.789649	−	0.613559i	$-0.789737\pi$
$151$	−2930.42	−1.57930	−0.789649	+	0.613559i	$0.789737\pi$
$152$	−1306.45	−0.697150
$153$	0	0
$154$	−2562.63	−1.34093
$155$	−5248.60	−2.71986
$156$	0	0
$157$	−1721.01	−0.874848	−0.437424	−	0.899255i	$-0.644109\pi$
$157$	−1721.01	−0.874848	−0.437424	+	0.899255i	$0.644109\pi$
$158$	593.176	0.298675
$159$	0	0
$160$	3559.64	1.75884
$161$	65.8903	0.0322539
$162$	0	0
$163$	2855.12	1.37197	0.685983	−	0.727618i	$-0.259372\pi$
$163$	2855.12	1.37197	0.685983	+	0.727618i	$0.259372\pi$
$164$	1022.99	0.487085
$165$	0	0
$166$	536.451	0.250823
$167$	21.7513	0.0100788	0.00503942	−	0.999987i	$-0.498396\pi$
$167$	21.7513	0.0100788	0.00503942	+	0.999987i	$0.498396\pi$
$168$	0	0
$169$	−774.502	−0.352527
$170$	−1922.01	−0.867127
$171$	0	0
$172$	−2278.55	−1.01010
$173$	−225.799	−0.0992322	−0.0496161	−	0.998768i	$-0.515800\pi$
$173$	−225.799	−0.0992322	−0.0496161	+	0.998768i	$0.515800\pi$
$174$	0	0
$175$	−10447.5	−4.51290
$176$	−2502.43	−1.07175
$177$	0	0
$178$	−19.3939	−0.00816647
$179$	2750.37	1.14845	0.574224	−	0.818698i	$-0.305304\pi$
$179$	2750.37	1.14845	0.574224	+	0.818698i	$0.305304\pi$
$180$	0	0
$181$	15.7495	0.00646769	0.00323385	−	0.999995i	$-0.498971\pi$
$181$	15.7495	0.00646769	0.00323385	+	0.999995i	$0.498971\pi$
$182$	−1399.84	−0.570127
$183$	0	0
$184$	−31.9722	−0.0128099
$185$	−273.609	−0.108736
$186$	0	0
$187$	5759.72	2.25237
$188$	3099.69	1.20249
$189$	0	0
$190$	−1844.87	−0.704428
$191$	−1477.88	−0.559871	−0.279935	−	0.960019i	$-0.590313\pi$
$191$	−1477.88	−0.559871	−0.279935	+	0.960019i	$0.590313\pi$
$192$	0	0
$193$	557.268	0.207840	0.103920	−	0.994586i	$-0.466861\pi$
$193$	557.268	0.207840	0.103920	+	0.994586i	$0.466861\pi$
$194$	1137.75	0.421061
$195$	0	0
$196$	−5341.47	−1.94660
$197$	3765.99	1.36201	0.681005	−	0.732279i	$-0.261543\pi$
$197$	3765.99	1.36201	0.681005	+	0.732279i	$0.261543\pi$
$198$	0	0
$199$	−4958.72	−1.76640	−0.883202	−	0.468992i	$-0.844617\pi$
$199$	−4958.72	−1.76640	−0.883202	+	0.468992i	$0.844617\pi$
$200$	5069.48	1.79233
$201$	0	0
$202$	386.057	0.134470
$203$	−5695.93	−1.96934
$204$	0	0
$205$	3148.72	1.07276
$206$	−1950.37	−0.659653
$207$	0	0
$208$	−1366.96	−0.455680
$209$	5528.56	1.82975
$210$	0	0
$211$	929.701	0.303333	0.151667	−	0.988432i	$-0.451536\pi$
$211$	929.701	0.303333	0.151667	+	0.988432i	$0.451536\pi$
$212$	1409.39	0.456592
$213$	0	0
$214$	174.618	0.0557786
$215$	−7013.28	−2.22466
$216$	0	0
$217$	−8454.90	−2.64496
$218$	−2291.61	−0.711962
$219$	0	0
$220$	−9773.81	−2.99523
$221$	3146.26	0.957647
$222$	0	0
$223$	87.7520	0.0263512	0.0131756	−	0.999913i	$-0.495806\pi$
$223$	87.7520	0.0263512	0.0131756	+	0.999913i	$0.495806\pi$
$224$	5734.18	1.71041
$225$	0	0
$226$	−1343.17	−0.395338
$227$	1392.98	0.407291	0.203646	−	0.979045i	$-0.434721\pi$
$227$	1392.98	0.407291	0.203646	+	0.979045i	$0.434721\pi$
$228$	0	0
$229$	−4622.09	−1.33378	−0.666892	−	0.745155i	$-0.732376\pi$
$229$	−4622.09	−1.33378	−0.666892	+	0.745155i	$0.732376\pi$
$230$	−45.1489	−0.0129436
$231$	0	0
$232$	2763.86	0.782139
$233$	2665.47	0.749444	0.374722	−	0.927137i	$-0.377738\pi$
$233$	2665.47	0.749444	0.374722	+	0.927137i	$0.377738\pi$
$234$	0	0
$235$	9540.73	2.64838
$236$	1853.34	0.511195
$237$	0	0
$238$	−3096.14	−0.843249
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−4955.60	−1.32456	−0.662278	−	0.749258i	$-0.730410\pi$
$241$	−4955.60	−1.32456	−0.662278	+	0.749258i	$0.730410\pi$
$242$	−3792.92	−1.00751
$243$	0	0
$244$	4945.68	1.29760
$245$	−16440.9	−4.28721
$246$	0	0
$247$	3019.98	0.777963
$248$	4102.60	1.05047
$249$	0	0
$250$	4278.74	1.08244
$251$	2860.82	0.719416	0.359708	−	0.933065i	$-0.382876\pi$
$251$	2860.82	0.719416	0.359708	+	0.933065i	$0.382876\pi$
$252$	0	0
$253$	135.298	0.0336211
$254$	−2591.77	−0.640245
$255$	0	0
$256$	−816.104	−0.199244
$257$	−1740.88	−0.422542	−0.211271	−	0.977428i	$-0.567760\pi$
$257$	−1740.88	−0.422542	−0.211271	+	0.977428i	$0.567760\pi$
$258$	0	0
$259$	−440.753	−0.105742
$260$	−5338.96	−1.27349
$261$	0	0
$262$	−720.959	−0.170004
$263$	−4901.88	−1.14929	−0.574644	−	0.818403i	$-0.694860\pi$
$263$	−4901.88	−1.14929	−0.574644	+	0.818403i	$0.694860\pi$
$264$	0	0
$265$	4338.06	1.00560
$266$	−2971.88	−0.685029
$267$	0	0
$268$	2617.09	0.596509
$269$	2621.83	0.594260	0.297130	−	0.954837i	$-0.403971\pi$
$269$	2621.83	0.594260	0.297130	+	0.954837i	$0.403971\pi$
$270$	0	0
$271$	−980.038	−0.219679	−0.109840	−	0.993949i	$-0.535034\pi$
$271$	−980.038	−0.219679	−0.109840	+	0.993949i	$0.535034\pi$
$272$	−3023.41	−0.673975
$273$	0	0
$274$	459.933	0.101407
$275$	−21452.8	−4.70419
$276$	0	0
$277$	−7088.82	−1.53764	−0.768819	−	0.639466i	$-0.779155\pi$
$277$	−7088.82	−1.53764	−0.768819	+	0.639466i	$0.779155\pi$
$278$	−290.273	−0.0626239
$279$	0	0
$280$	11451.8	2.44419
$281$	−3247.88	−0.689509	−0.344754	−	0.938693i	$-0.612038\pi$
$281$	−3247.88	−0.689509	−0.344754	+	0.938693i	$0.612038\pi$
$282$	0	0
$283$	4831.16	1.01478	0.507389	−	0.861717i	$-0.330611\pi$
$283$	4831.16	1.01478	0.507389	+	0.861717i	$0.330611\pi$
$284$	2917.06	0.609491
$285$	0	0
$286$	−2874.42	−0.594293
$287$	5072.23	1.04322
$288$	0	0
$289$	2045.83	0.416412
$290$	3902.93	0.790303
$291$	0	0
$292$	2382.54	0.477491
$293$	8585.06	1.71176	0.855878	−	0.517178i	$-0.173017\pi$
$293$	8585.06	1.71176	0.855878	+	0.517178i	$0.173017\pi$
$294$	0	0
$295$	5704.50	1.12586
$296$	213.868	0.0419961
$297$	0	0
$298$	796.096	0.154754
$299$	73.9069	0.0142948
$300$	0	0
$301$	−11297.6	−2.16340
$302$	3234.60	0.616325
$303$	0	0
$304$	−2902.07	−0.547517
$305$	15222.6	2.85785
$306$	0	0
$307$	−4536.46	−0.843353	−0.421677	−	0.906746i	$-0.638558\pi$
$307$	−4536.46	−0.843353	−0.421677	+	0.906746i	$0.638558\pi$
$308$	−15744.5	−2.91275
$309$	0	0
$310$	5793.41	1.06143
$311$	−1058.23	−0.192947	−0.0964736	−	0.995336i	$-0.530756\pi$
$311$	−1058.23	−0.192947	−0.0964736	+	0.995336i	$0.530756\pi$
$312$	0	0
$313$	−5570.71	−1.00599	−0.502995	−	0.864289i	$-0.667769\pi$
$313$	−5570.71	−1.00599	−0.502995	+	0.864289i	$0.667769\pi$
$314$	1899.65	0.341412
$315$	0	0
$316$	3644.40	0.648778
$317$	4833.36	0.856367	0.428183	−	0.903692i	$-0.359154\pi$
$317$	4833.36	0.856367	0.428183	+	0.903692i	$0.359154\pi$
$318$	0	0
$319$	−11696.0	−2.05282
$320$	2123.10	0.370891
$321$	0	0
$322$	−72.7298	−0.0125872
$323$	6679.55	1.15065
$324$	0	0
$325$	−11718.6	−2.00010
$326$	−3151.49	−0.535413
$327$	0	0
$328$	−2461.22	−0.414323
$329$	15369.0	2.57545
$330$	0	0
$331$	2909.13	0.483082	0.241541	−	0.970391i	$-0.422347\pi$
$331$	2909.13	0.483082	0.241541	+	0.970391i	$0.422347\pi$
$332$	3295.89	0.544836
$333$	0	0
$334$	−24.0091	−0.00393329
$335$	8055.32	1.31376
$336$	0	0
$337$	3839.70	0.620659	0.310329	−	0.950629i	$-0.399561\pi$
$337$	3839.70	0.620659	0.310329	+	0.950629i	$0.399561\pi$
$338$	854.897	0.137575
$339$	0	0
$340$	−11808.6	−1.88357
$341$	−17361.2	−2.75707
$342$	0	0
$343$	−14951.0	−2.35358
$344$	5481.98	0.859210
$345$	0	0
$346$	249.237	0.0387256
$347$	−10490.8	−1.62299	−0.811493	−	0.584362i	$-0.801345\pi$
$347$	−10490.8	−1.62299	−0.811493	+	0.584362i	$0.801345\pi$
$348$	0	0
$349$	1444.77	0.221596	0.110798	−	0.993843i	$-0.464659\pi$
$349$	1444.77	0.221596	0.110798	+	0.993843i	$0.464659\pi$
$350$	11532.0	1.76117
$351$	0	0
$352$	11774.5	1.78291
$353$	−9433.10	−1.42230	−0.711152	−	0.703038i	$-0.751826\pi$
$353$	−9433.10	−1.42230	−0.711152	+	0.703038i	$0.751826\pi$
$354$	0	0
$355$	8978.60	1.34235
$356$	−119.154	−0.0177391
$357$	0	0
$358$	−3035.86	−0.448185
$359$	11186.4	1.64455	0.822276	−	0.569089i	$-0.192704\pi$
$359$	11186.4	1.64455	0.822276	+	0.569089i	$0.192704\pi$
$360$	0	0
$361$	−447.529	−0.0652470
$362$	−17.3843	−0.00252403
$363$	0	0
$364$	−8600.45	−1.23842
$365$	7333.36	1.05163
$366$	0	0
$367$	10862.0	1.54493	0.772467	−	0.635055i	$-0.219023\pi$
$367$	10862.0	1.54493	0.772467	+	0.635055i	$0.219023\pi$
$368$	−71.0212	−0.0100604
$369$	0	0
$370$	302.010	0.0424345
$371$	6988.12	0.977911
$372$	0	0
$373$	−24.5069	−0.00340193	−0.00170096	−	0.999999i	$-0.500541\pi$
$373$	−24.5069	−0.00340193	−0.00170096	+	0.999999i	$0.500541\pi$
$374$	−6357.59	−0.878992
$375$	0	0
$376$	−7457.57	−1.02286
$377$	−6388.94	−0.872804
$378$	0	0
$379$	12349.7	1.67378	0.836890	−	0.547371i	$-0.184372\pi$
$379$	12349.7	1.67378	0.836890	+	0.547371i	$0.184372\pi$
$380$	−11334.7	−1.53015
$381$	0	0
$382$	1631.28	0.218491
$383$	4372.01	0.583288	0.291644	−	0.956527i	$-0.405798\pi$
$383$	4372.01	0.583288	0.291644	+	0.956527i	$0.405798\pi$
$384$	0	0
$385$	−48461.0	−6.41507
$386$	−615.114	−0.0811100
$387$	0	0
$388$	6990.22	0.914625
$389$	6788.40	0.884795	0.442398	−	0.896819i	$-0.354128\pi$
$389$	6788.40	0.884795	0.442398	+	0.896819i	$0.354128\pi$
$390$	0	0
$391$	163.466	0.0211428
$392$	12851.1	1.65581
$393$	0	0
$394$	−4156.91	−0.531528
$395$	11217.3	1.42888
$396$	0	0
$397$	7187.80	0.908678	0.454339	−	0.890829i	$-0.349876\pi$
$397$	7187.80	0.908678	0.454339	+	0.890829i	$0.349876\pi$
$398$	5473.44	0.689344
$399$	0	0
$400$	11261.1	1.40763
$401$	−2833.45	−0.352857	−0.176428	−	0.984313i	$-0.556454\pi$
$401$	−2833.45	−0.352857	−0.176428	+	0.984313i	$0.556454\pi$
$402$	0	0
$403$	−9483.58	−1.17224
$404$	2371.89	0.292094
$405$	0	0
$406$	6287.18	0.768540
$407$	−905.038	−0.110224
$408$	0	0
$409$	−11104.0	−1.34244	−0.671220	−	0.741259i	$-0.734229\pi$
$409$	−11104.0	−1.34244	−0.671220	+	0.741259i	$0.734229\pi$
$410$	−3475.56	−0.418648
$411$	0	0
$412$	−11982.8	−1.43289
$413$	9189.30	1.09486
$414$	0	0
$415$	10144.6	1.19995
$416$	6431.84	0.758045
$417$	0	0
$418$	−6102.43	−0.714066
$419$	7777.73	0.906843	0.453421	−	0.891296i	$-0.350203\pi$
$419$	7777.73	0.906843	0.453421	+	0.891296i	$0.350203\pi$
$420$	0	0
$421$	−3704.31	−0.428829	−0.214415	−	0.976743i	$-0.568784\pi$
$421$	−3704.31	−0.428829	−0.214415	+	0.976743i	$0.568784\pi$
$422$	−1026.21	−0.118377
$423$	0	0
$424$	−3390.87	−0.388385
$425$	−25919.0	−2.95825
$426$	0	0
$427$	24521.9	2.77915
$428$	1072.83	0.121162
$429$	0	0
$430$	7741.27	0.868180
$431$	15207.6	1.69960	0.849799	−	0.527108i	$-0.176724\pi$
$431$	15207.6	1.69960	0.849799	+	0.527108i	$0.176724\pi$
$432$	0	0
$433$	7133.00	0.791663	0.395831	−	0.918323i	$-0.370456\pi$
$433$	7133.00	0.791663	0.395831	+	0.918323i	$0.370456\pi$
$434$	9332.53	1.03220
$435$	0	0
$436$	−14079.4	−1.54652
$437$	156.905	0.0171758
$438$	0	0
$439$	−4408.33	−0.479267	−0.239633	−	0.970863i	$-0.577027\pi$
$439$	−4408.33	−0.479267	−0.239633	+	0.970863i	$0.577027\pi$
$440$	23514.9	2.54779
$441$	0	0
$442$	−3472.84	−0.373724
$443$	14143.9	1.51692	0.758461	−	0.651718i	$-0.225952\pi$
$443$	14143.9	1.51692	0.758461	+	0.651718i	$0.225952\pi$
$444$	0	0
$445$	−366.751	−0.0390689
$446$	−96.8608	−0.0102836
$447$	0	0
$448$	3420.07	0.360677
$449$	−17191.3	−1.80692	−0.903460	−	0.428673i	$-0.858981\pi$
$449$	−17191.3	−1.80692	−0.903460	+	0.428673i	$0.858981\pi$
$450$	0	0
$451$	10415.3	1.08744
$452$	−8252.27	−0.858748
$453$	0	0
$454$	−1537.57	−0.158947
$455$	−26471.9	−2.72752
$456$	0	0
$457$	2014.56	0.206208	0.103104	−	0.994671i	$-0.467123\pi$
$457$	2014.56	0.206208	0.103104	+	0.994671i	$0.467123\pi$
$458$	5101.87	0.520512
$459$	0	0
$460$	−277.390	−0.0281160
$461$	−1340.97	−0.135478	−0.0677389	−	0.997703i	$-0.521578\pi$
$461$	−1340.97	−0.135478	−0.0677389	+	0.997703i	$0.521578\pi$
$462$	0	0
$463$	−2327.30	−0.233604	−0.116802	−	0.993155i	$-0.537264\pi$
$463$	−2327.30	−0.233604	−0.116802	+	0.993155i	$0.537264\pi$
$464$	6139.48	0.614264
$465$	0	0
$466$	−2942.14	−0.292472
$467$	2298.37	0.227743	0.113872	−	0.993495i	$-0.463675\pi$
$467$	2298.37	0.227743	0.113872	+	0.993495i	$0.463675\pi$
$468$	0	0
$469$	12976.2	1.27758
$470$	−10531.1	−1.03354
$471$	0	0
$472$	−4458.96	−0.434831
$473$	−23198.4	−2.25510
$474$	0	0
$475$	−24878.8	−2.40320
$476$	−19022.3	−1.83170
$477$	0	0
$478$	263.808	0.0252433
$479$	4242.02	0.404640	0.202320	−	0.979319i	$-0.435152\pi$
$479$	4242.02	0.404640	0.202320	+	0.979319i	$0.435152\pi$
$480$	0	0
$481$	−494.378	−0.0468643
$482$	5470.00	0.516912
$483$	0	0
$484$	−23303.3	−2.18851
$485$	21515.6	2.01438
$486$	0	0
$487$	5741.79	0.534261	0.267131	−	0.963660i	$-0.413924\pi$
$487$	5741.79	0.534261	0.267131	+	0.963660i	$0.413924\pi$
$488$	−11898.8	−1.10376
$489$	0	0
$490$	18147.4	1.67310
$491$	−5108.45	−0.469534	−0.234767	−	0.972052i	$-0.575433\pi$
$491$	−5108.45	−0.469534	−0.234767	+	0.972052i	$0.575433\pi$
$492$	0	0
$493$	−14130.9	−1.29092
$494$	−3333.46	−0.303602
$495$	0	0
$496$	9113.30	0.824998
$497$	14463.5	1.30539
$498$	0	0
$499$	−14924.4	−1.33889	−0.669446	−	0.742860i	$-0.733469\pi$
$499$	−14924.4	−1.33889	−0.669446	+	0.742860i	$0.733469\pi$
$500$	26288.1	2.35127
$501$	0	0
$502$	−3157.78	−0.280754
$503$	3270.67	0.289924	0.144962	−	0.989437i	$-0.453694\pi$
$503$	3270.67	0.289924	0.144962	+	0.989437i	$0.453694\pi$
$504$	0	0
$505$	7300.59	0.643311
$506$	−149.343	−0.0131207
$507$	0	0
$508$	−15923.5	−1.39073
$509$	11640.6	1.01367	0.506836	−	0.862042i	$-0.330815\pi$
$509$	11640.6	1.01367	0.506836	+	0.862042i	$0.330815\pi$
$510$	0	0
$511$	11813.2	1.02267
$512$	−10911.5	−0.941843
$513$	0	0
$514$	1921.59	0.164898
$515$	−36882.7	−3.15582
$516$	0	0
$517$	31558.6	2.68461
$518$	486.504	0.0412659
$519$	0	0
$520$	12845.1	1.08326
$521$	−8694.56	−0.731124	−0.365562	−	0.930787i	$-0.619123\pi$
$521$	−8694.56	−0.731124	−0.365562	+	0.930787i	$0.619123\pi$
$522$	0	0
$523$	−5227.06	−0.437023	−0.218512	−	0.975834i	$-0.570120\pi$
$523$	−5227.06	−0.437023	−0.218512	+	0.975834i	$0.570120\pi$
$524$	−4429.49	−0.369281
$525$	0	0
$526$	5410.70	0.448513
$527$	−20975.6	−1.73380
$528$	0	0
$529$	−12163.2	−0.999684
$530$	−4788.36	−0.392439
$531$	0	0
$532$	−18258.9	−1.48801
$533$	5689.35	0.462351
$534$	0	0
$535$	3302.13	0.266848
$536$	−6296.49	−0.507401
$537$	0	0
$538$	−2893.98	−0.231911
$539$	−54382.7	−4.34588
$540$	0	0
$541$	−13129.3	−1.04339	−0.521695	−	0.853132i	$-0.674700\pi$
$541$	−13129.3	−1.04339	−0.521695	+	0.853132i	$0.674700\pi$
$542$	1081.77	0.0857304
$543$	0	0
$544$	14225.8	1.12119
$545$	−43335.9	−3.40607
$546$	0	0
$547$	24572.9	1.92077	0.960387	−	0.278670i	$-0.0898936\pi$
$547$	24572.9	1.92077	0.960387	+	0.278670i	$0.0898936\pi$
$548$	2825.77	0.220275
$549$	0	0
$550$	23679.6	1.83582
$551$	−13563.8	−1.04871
$552$	0	0
$553$	18069.9	1.38953
$554$	7824.65	0.600068
$555$	0	0
$556$	−1783.40	−0.136031
$557$	421.134	0.0320359	0.0160180	−	0.999872i	$-0.494901\pi$
$557$	421.134	0.0320359	0.0160180	+	0.999872i	$0.494901\pi$
$558$	0	0
$559$	−12672.1	−0.958810
$560$	25438.3	1.91958
$561$	0	0
$562$	3585.01	0.269083
$563$	−21329.2	−1.59666	−0.798329	−	0.602222i	$-0.794282\pi$
$563$	−21329.2	−1.59666	−0.798329	+	0.602222i	$0.794282\pi$
$564$	0	0
$565$	−25400.2	−1.89132
$566$	−5332.64	−0.396020
$567$	0	0
$568$	−7018.18	−0.518444
$569$	4614.68	0.339996	0.169998	−	0.985444i	$-0.445624\pi$
$569$	4614.68	0.339996	0.169998	+	0.985444i	$0.445624\pi$
$570$	0	0
$571$	13898.2	1.01860	0.509301	−	0.860588i	$-0.329904\pi$
$571$	13898.2	1.01860	0.509301	+	0.860588i	$0.329904\pi$
$572$	−17660.1	−1.29092
$573$	0	0
$574$	−5598.73	−0.407119
$575$	−608.850	−0.0441579
$576$	0	0
$577$	−13545.0	−0.977273	−0.488636	−	0.872488i	$-0.662506\pi$
$577$	−13545.0	−0.977273	−0.488636	+	0.872488i	$0.662506\pi$
$578$	−2258.19	−0.162506
$579$	0	0
$580$	23979.1	1.71669
$581$	16341.8	1.16691
$582$	0	0
$583$	14349.3	1.01936
$584$	−5732.16	−0.406162
$585$	0	0
$586$	−9476.20	−0.668017
$587$	−18263.2	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$587$	−18263.2	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$588$	0	0
$589$	−20133.8	−1.40849
$590$	−6296.64	−0.439370
$591$	0	0
$592$	475.075	0.0329822
$593$	26172.3	1.81243	0.906213	−	0.422822i	$-0.138960\pi$
$593$	26172.3	1.81243	0.906213	+	0.422822i	$0.138960\pi$
$594$	0	0
$595$	−58550.1	−4.03415
$596$	4891.12	0.336154
$597$	0	0
$598$	−81.5785	−0.00557859
$599$	7040.07	0.480216	0.240108	−	0.970746i	$-0.422817\pi$
$599$	7040.07	0.480216	0.240108	+	0.970746i	$0.422817\pi$
$600$	0	0
$601$	25935.8	1.76031	0.880153	−	0.474691i	$-0.157440\pi$
$601$	25935.8	1.76031	0.880153	+	0.474691i	$0.157440\pi$
$602$	12470.3	0.844272
$603$	0	0
$604$	19873.0	1.33877
$605$	−71726.6	−4.82000
$606$	0	0
$607$	−6181.70	−0.413356	−0.206678	−	0.978409i	$-0.566265\pi$
$607$	−6181.70	−0.413356	−0.206678	+	0.978409i	$0.566265\pi$
$608$	13654.9	0.910820
$609$	0	0
$610$	−16802.7	−1.11528
$611$	17238.9	1.14143
$612$	0	0
$613$	8606.74	0.567085	0.283542	−	0.958960i	$-0.408490\pi$
$613$	8606.74	0.567085	0.283542	+	0.958960i	$0.408490\pi$
$614$	5007.35	0.329121
$615$	0	0
$616$	37879.9	2.47763
$617$	−143.326	−0.00935181	−0.00467591	−	0.999989i	$-0.501488\pi$
$617$	−143.326	−0.00935181	−0.00467591	+	0.999989i	$0.501488\pi$
$618$	0	0
$619$	15572.1	1.01114	0.505570	−	0.862786i	$-0.331282\pi$
$619$	15572.1	1.01114	0.505570	+	0.862786i	$0.331282\pi$
$620$	35594.0	2.30563
$621$	0	0
$622$	1168.07	0.0752981
$623$	−590.793	−0.0379930
$624$	0	0
$625$	42075.3	2.69282
$626$	6148.96	0.392591
$627$	0	0
$628$	11671.2	0.741611
$629$	−1093.46	−0.0693148
$630$	0	0
$631$	7951.40	0.501649	0.250824	−	0.968033i	$-0.419298\pi$
$631$	7951.40	0.501649	0.250824	+	0.968033i	$0.419298\pi$
$632$	−8768.11	−0.551862
$633$	0	0
$634$	−5335.06	−0.334199
$635$	−49012.1	−3.06297
$636$	0	0
$637$	−29706.6	−1.84775
$638$	12910.0	0.801117
$639$	0	0
$640$	−30820.6	−1.90358
$641$	10550.1	0.650084	0.325042	−	0.945700i	$-0.394622\pi$
$641$	10550.1	0.650084	0.325042	+	0.945700i	$0.394622\pi$
$642$	0	0
$643$	−6002.98	−0.368172	−0.184086	−	0.982910i	$-0.558932\pi$
$643$	−6002.98	−0.368172	−0.184086	+	0.982910i	$0.558932\pi$
$644$	−446.843	−0.0273417
$645$	0	0
$646$	−7372.89	−0.449044
$647$	9593.94	0.582962	0.291481	−	0.956577i	$-0.405852\pi$
$647$	9593.94	0.582962	0.291481	+	0.956577i	$0.405852\pi$
$648$	0	0
$649$	18869.2	1.14127
$650$	12935.0	0.780544
$651$	0	0
$652$	−19362.4	−1.16302
$653$	3306.73	0.198166	0.0990828	−	0.995079i	$-0.468409\pi$
$653$	3306.73	0.198166	0.0990828	+	0.995079i	$0.468409\pi$
$654$	0	0
$655$	−13633.8	−0.813308
$656$	−5467.21	−0.325394
$657$	0	0
$658$	−16964.3	−1.00507
$659$	−2682.14	−0.158545	−0.0792727	−	0.996853i	$-0.525260\pi$
$659$	−2682.14	−0.158545	−0.0792727	+	0.996853i	$0.525260\pi$
$660$	0	0
$661$	22301.9	1.31232	0.656159	−	0.754623i	$-0.272180\pi$
$661$	22301.9	1.31232	0.656159	+	0.754623i	$0.272180\pi$
$662$	−3211.10	−0.188524
$663$	0	0
$664$	−7929.61	−0.463447
$665$	−56200.2	−3.27722
$666$	0	0
$667$	−331.942	−0.0192696
$668$	−147.509	−0.00854385
$669$	0	0
$670$	−8891.47	−0.512698
$671$	50353.0	2.89695
$672$	0	0
$673$	−6359.53	−0.364253	−0.182126	−	0.983275i	$-0.558298\pi$
$673$	−6359.53	−0.364253	−0.182126	+	0.983275i	$0.558298\pi$
$674$	−4238.27	−0.242214
$675$	0	0
$676$	5252.38	0.298838
$677$	−16592.9	−0.941974	−0.470987	−	0.882140i	$-0.656102\pi$
$677$	−16592.9	−0.941974	−0.470987	+	0.882140i	$0.656102\pi$
$678$	0	0
$679$	34659.2	1.95891
$680$	28410.5	1.60219
$681$	0	0
$682$	19163.3	1.07596
$683$	14138.9	0.792106	0.396053	−	0.918228i	$-0.370380\pi$
$683$	14138.9	0.792106	0.396053	+	0.918228i	$0.370380\pi$
$684$	0	0
$685$	8697.62	0.485137
$686$	16502.9	0.918489
$687$	0	0
$688$	12177.4	0.674793
$689$	7838.34	0.433406
$690$	0	0
$691$	1170.47	0.0644383	0.0322192	−	0.999481i	$-0.489743\pi$
$691$	1170.47	0.0644383	0.0322192	+	0.999481i	$0.489743\pi$
$692$	1531.28	0.0841194
$693$	0	0
$694$	11579.8	0.633375
$695$	−5489.25	−0.299596
$696$	0	0
$697$	12583.6	0.683842
$698$	−1594.74	−0.0864783
$699$	0	0
$700$	70851.0	3.82560
$701$	−7956.08	−0.428669	−0.214335	−	0.976760i	$-0.568758\pi$
$701$	−7956.08	−0.428669	−0.214335	+	0.976760i	$0.568758\pi$
$702$	0	0
$703$	−1049.57	−0.0563092
$704$	7022.74	0.375965
$705$	0	0
$706$	10412.3	0.555058
$707$	11760.4	0.625595
$708$	0	0
$709$	−61.9799	−0.00328308	−0.00164154	−	0.999999i	$-0.500523\pi$
$709$	−61.9799	−0.00328308	−0.00164154	+	0.999999i	$0.500523\pi$
$710$	−9910.59	−0.523856
$711$	0	0
$712$	286.673	0.0150892
$713$	−492.727	−0.0258804
$714$	0	0
$715$	−54357.1	−2.84313
$716$	−18652.0	−0.973543
$717$	0	0
$718$	−12347.5	−0.641791
$719$	−6419.43	−0.332969	−0.166484	−	0.986044i	$-0.553242\pi$
$719$	−6419.43	−0.332969	−0.166484	+	0.986044i	$0.553242\pi$
$720$	0	0
$721$	−59413.8	−3.06891
$722$	493.983	0.0254628
$723$	0	0
$724$	−106.807	−0.00548268
$725$	52632.4	2.69617
$726$	0	0
$727$	−992.212	−0.0506177	−0.0253089	−	0.999680i	$-0.508057\pi$
$727$	−992.212	−0.0506177	−0.0253089	+	0.999680i	$0.508057\pi$
$728$	20691.9	1.05343
$729$	0	0
$730$	−8094.57	−0.410402
$731$	−28028.0	−1.41813
$732$	0	0
$733$	6546.67	0.329887	0.164943	−	0.986303i	$-0.447256\pi$
$733$	6546.67	0.329887	0.164943	+	0.986303i	$0.447256\pi$
$734$	−11989.5	−0.602915
$735$	0	0
$736$	334.171	0.0167360
$737$	26645.2	1.33173
$738$	0	0
$739$	21474.5	1.06895	0.534474	−	0.845185i	$-0.320510\pi$
$739$	21474.5	1.06895	0.534474	+	0.845185i	$0.320510\pi$
$740$	1855.51	0.0921758
$741$	0	0
$742$	−7713.50	−0.381633
$743$	−4134.48	−0.204145	−0.102072	−	0.994777i	$-0.532547\pi$
$743$	−4134.48	−0.204145	−0.102072	+	0.994777i	$0.532547\pi$
$744$	0	0
$745$	15054.7	0.740350
$746$	27.0507	0.00132761
$747$	0	0
$748$	−39060.3	−1.90934
$749$	5319.36	0.259499
$750$	0	0
$751$	−20294.1	−0.986073	−0.493037	−	0.870009i	$-0.664113\pi$
$751$	−20294.1	−0.986073	−0.493037	+	0.870009i	$0.664113\pi$
$752$	−16565.8	−0.803316
$753$	0	0
$754$	7052.12	0.340614
$755$	61168.3	2.94853
$756$	0	0
$757$	11348.8	0.544885	0.272442	−	0.962172i	$-0.412169\pi$
$757$	11348.8	0.544885	0.272442	+	0.962172i	$0.412169\pi$
$758$	−13631.6	−0.653197
$759$	0	0
$760$	27270.2	1.30157
$761$	−20113.0	−0.958074	−0.479037	−	0.877795i	$-0.659014\pi$
$761$	−20113.0	−0.958074	−0.479037	+	0.877795i	$0.659014\pi$
$762$	0	0
$763$	−69809.2	−3.31227
$764$	10022.4	0.474604
$765$	0	0
$766$	−4825.83	−0.227630
$767$	10307.3	0.485236
$768$	0	0
$769$	33669.4	1.57887	0.789435	−	0.613835i	$-0.210374\pi$
$769$	33669.4	1.57887	0.789435	+	0.613835i	$0.210374\pi$
$770$	53491.3	2.50350
$771$	0	0
$772$	−3779.18	−0.176186
$773$	−15922.7	−0.740878	−0.370439	−	0.928857i	$-0.620793\pi$
$773$	−15922.7	−0.740878	−0.370439	+	0.928857i	$0.620793\pi$
$774$	0	0
$775$	78126.3	3.62114
$776$	−16817.8	−0.777996
$777$	0	0
$778$	−7493.04	−0.345294
$779$	12078.6	0.555532
$780$	0	0
$781$	29699.2	1.36072
$782$	−180.434	−0.00825103
$783$	0	0
$784$	28546.7	1.30041
$785$	35923.6	1.63333
$786$	0	0
$787$	−10831.5	−0.490597	−0.245298	−	0.969448i	$-0.578886\pi$
$787$	−10831.5	−0.490597	−0.245298	+	0.969448i	$0.578886\pi$
$788$	−25539.6	−1.15458
$789$	0	0
$790$	−12381.7	−0.557623
$791$	−40916.8	−1.83923
$792$	0	0
$793$	27505.4	1.23171
$794$	−7933.90	−0.354614
$795$	0	0
$796$	33628.2	1.49739
$797$	−6424.29	−0.285521	−0.142760	−	0.989757i	$-0.545598\pi$
$797$	−6424.29	−0.285521	−0.142760	+	0.989757i	$0.545598\pi$
$798$	0	0
$799$	38128.7	1.68823
$800$	−52985.8	−2.34167
$801$	0	0
$802$	3127.56	0.137703
$803$	24257.1	1.06602
$804$	0	0
$805$	−1375.37	−0.0602178
$806$	10468.0	0.457468
$807$	0	0
$808$	−5706.55	−0.248460
$809$	20406.8	0.886854	0.443427	−	0.896310i	$-0.353762\pi$
$809$	20406.8	0.886854	0.443427	+	0.896310i	$0.353762\pi$
$810$	0	0
$811$	−25576.8	−1.10743	−0.553713	−	0.832707i	$-0.686790\pi$
$811$	−25576.8	−1.10743	−0.553713	+	0.832707i	$0.686790\pi$
$812$	38627.7	1.66942
$813$	0	0
$814$	998.982	0.0430151
$815$	−59596.6	−2.56145
$816$	0	0
$817$	−26903.1	−1.15205
$818$	12256.6	0.523890
$819$	0	0
$820$	−21353.4	−0.909383
$821$	30481.2	1.29574	0.647869	−	0.761752i	$-0.275660\pi$
$821$	30481.2	1.29574	0.647869	+	0.761752i	$0.275660\pi$
$822$	0	0
$823$	24750.6	1.04830	0.524151	−	0.851625i	$-0.324383\pi$
$823$	24750.6	1.04830	0.524151	+	0.851625i	$0.324383\pi$
$824$	28829.6	1.21884
$825$	0	0
$826$	−10143.2	−0.427271
$827$	−13668.2	−0.574714	−0.287357	−	0.957824i	$-0.592777\pi$
$827$	−13668.2	−0.574714	−0.287357	+	0.957824i	$0.592777\pi$
$828$	0	0
$829$	−26187.5	−1.09714	−0.548570	−	0.836104i	$-0.684828\pi$
$829$	−26187.5	−1.09714	−0.548570	+	0.836104i	$0.684828\pi$
$830$	−11197.7	−0.468285
$831$	0	0
$832$	3836.18	0.159851
$833$	−65704.5	−2.73293
$834$	0	0
$835$	−454.027	−0.0188171
$836$	−37492.6	−1.55109
$837$	0	0
$838$	−8585.07	−0.353898
$839$	−4864.48	−0.200168	−0.100084	−	0.994979i	$-0.531911\pi$
$839$	−4864.48	−0.200168	−0.100084	+	0.994979i	$0.531911\pi$
$840$	0	0
$841$	4305.97	0.176554
$842$	4088.82	0.167352
$843$	0	0
$844$	−6304.88	−0.257136
$845$	16166.6	0.658165
$846$	0	0
$847$	−115543.	−4.68727
$848$	−7532.29	−0.305024
$849$	0	0
$850$	28609.5	1.15447
$851$	−25.6858	−0.00103466
$852$	0	0
$853$	27915.7	1.12053	0.560267	−	0.828312i	$-0.310698\pi$
$853$	27915.7	1.12053	0.560267	+	0.828312i	$0.310698\pi$
$854$	−27067.3	−1.08457
$855$	0	0
$856$	−2581.13	−0.103062
$857$	4158.31	0.165747	0.0828736	−	0.996560i	$-0.473590\pi$
$857$	4158.31	0.165747	0.0828736	+	0.996560i	$0.473590\pi$
$858$	0	0
$859$	12504.1	0.496665	0.248332	−	0.968675i	$-0.420117\pi$
$859$	12504.1	0.496665	0.248332	+	0.968675i	$0.420117\pi$
$860$	47561.4	1.88585
$861$	0	0
$862$	−16786.2	−0.663272
$863$	12964.0	0.511354	0.255677	−	0.966762i	$-0.417702\pi$
$863$	12964.0	0.511354	0.255677	+	0.966762i	$0.417702\pi$
$864$	0	0
$865$	4713.23	0.185266
$866$	−7873.41	−0.308949
$867$	0	0
$868$	57338.0	2.24214
$869$	37104.5	1.44843
$870$	0	0
$871$	14555.0	0.566219
$872$	33873.8	1.31549
$873$	0	0
$874$	−173.192	−0.00670289
$875$	130343.	5.03587
$876$	0	0
$877$	37069.9	1.42732	0.713662	−	0.700491i	$-0.247035\pi$
$877$	37069.9	1.42732	0.713662	+	0.700491i	$0.247035\pi$
$878$	4865.92	0.187035
$879$	0	0
$880$	52234.7	2.00095
$881$	−28841.6	−1.10295	−0.551475	−	0.834191i	$-0.685935\pi$
$881$	−28841.6	−1.10295	−0.551475	+	0.834191i	$0.685935\pi$
$882$	0	0
$883$	4692.31	0.178832	0.0894161	−	0.995994i	$-0.471500\pi$
$883$	4692.31	0.178832	0.0894161	+	0.995994i	$0.471500\pi$
$884$	−21336.7	−0.811800
$885$	0	0
$886$	−15612.0	−0.591983
$887$	43493.1	1.64640	0.823199	−	0.567753i	$-0.192187\pi$
$887$	43493.1	1.64640	0.823199	+	0.567753i	$0.192187\pi$
$888$	0	0
$889$	−78952.9	−2.97862
$890$	404.820	0.0152467
$891$	0	0
$892$	−595.101	−0.0223380
$893$	36598.5	1.37147
$894$	0	0
$895$	−57410.1	−2.14414
$896$	−49648.5	−1.85116
$897$	0	0
$898$	18975.8	0.705155
$899$	42594.1	1.58019
$900$	0	0
$901$	17336.7	0.641032
$902$	−11496.4	−0.424376
$903$	0	0
$904$	19854.2	0.730466
$905$	−328.749	−0.0120751
$906$	0	0
$907$	−3069.80	−0.112383	−0.0561913	−	0.998420i	$-0.517896\pi$
$907$	−3069.80	−0.112383	−0.0561913	+	0.998420i	$0.517896\pi$
$908$	−9446.65	−0.345262
$909$	0	0
$910$	29219.7	1.06442
$911$	20060.7	0.729573	0.364786	−	0.931091i	$-0.381142\pi$
$911$	20060.7	0.729573	0.364786	+	0.931091i	$0.381142\pi$
$912$	0	0
$913$	33556.2	1.21637
$914$	−2223.67	−0.0804733
$915$	0	0
$916$	31345.3	1.13065
$917$	−21962.5	−0.790912
$918$	0	0
$919$	−815.158	−0.0292596	−0.0146298	−	0.999893i	$-0.504657\pi$
$919$	−815.158	−0.0292596	−0.0146298	+	0.999893i	$0.504657\pi$
$920$	667.374	0.0239160
$921$	0	0
$922$	1480.17	0.0528706
$923$	16223.2	0.578542
$924$	0	0
$925$	4072.72	0.144768
$926$	2568.88	0.0911647
$927$	0	0
$928$	−28887.6	−1.02186
$929$	−19561.6	−0.690847	−0.345423	−	0.938447i	$-0.612265\pi$
$929$	−19561.6	−0.690847	−0.345423	+	0.938447i	$0.612265\pi$
$930$	0	0
$931$	−63067.6	−2.22015
$932$	−18076.2	−0.635306
$933$	0	0
$934$	−2536.95	−0.0888774
$935$	−120226.	−4.20515
$936$	0	0
$937$	−12608.8	−0.439607	−0.219804	−	0.975544i	$-0.570542\pi$
$937$	−12608.8	−0.439607	−0.219804	+	0.975544i	$0.570542\pi$
$938$	−14323.1	−0.498579
$939$	0	0
$940$	−64701.6	−2.24504
$941$	−44721.1	−1.54927	−0.774636	−	0.632407i	$-0.782067\pi$
$941$	−44721.1	−1.54927	−0.774636	+	0.632407i	$0.782067\pi$
$942$	0	0
$943$	295.594	0.0102077
$944$	−9904.88	−0.341501
$945$	0	0
$946$	25606.4	0.880059
$947$	−50550.7	−1.73461	−0.867306	−	0.497776i	$-0.834150\pi$
$947$	−50550.7	−1.73461	−0.867306	+	0.497776i	$0.834150\pi$
$948$	0	0
$949$	13250.5	0.453244
$950$	27461.3	0.937853
$951$	0	0
$952$	45766.0	1.55807
$953$	−24886.9	−0.845923	−0.422962	−	0.906148i	$-0.639009\pi$
$953$	−24886.9	−0.845923	−0.422962	+	0.906148i	$0.639009\pi$
$954$	0	0
$955$	30848.6	1.04527
$956$	1620.81	0.0548333
$957$	0	0
$958$	−4682.35	−0.157912
$959$	14010.9	0.471778
$960$	0	0
$961$	33434.6	1.12231
$962$	545.695	0.0182889
$963$	0	0
$964$	33607.0	1.12283
$965$	−11632.2	−0.388035
$966$	0	0
$967$	−41736.0	−1.38794	−0.693970	−	0.720004i	$-0.744140\pi$
$967$	−41736.0	−1.38794	−0.693970	+	0.720004i	$0.744140\pi$
$968$	56065.5	1.86159
$969$	0	0
$970$	−23749.0	−0.786117
$971$	−46628.6	−1.54107	−0.770537	−	0.637396i	$-0.780012\pi$
$971$	−46628.6	−1.54107	−0.770537	+	0.637396i	$0.780012\pi$
$972$	0	0
$973$	−8842.56	−0.291346
$974$	−6337.79	−0.208497
$975$	0	0
$976$	−26431.4	−0.866854
$977$	−27915.4	−0.914117	−0.457058	−	0.889437i	$-0.651097\pi$
$977$	−27915.4	−0.914117	−0.457058	+	0.889437i	$0.651097\pi$
$978$	0	0
$979$	−1213.13	−0.0396034
$980$	111496.	3.63428
$981$	0	0
$982$	5638.71	0.183237
$983$	−21254.5	−0.689638	−0.344819	−	0.938669i	$-0.612060\pi$
$983$	−21254.5	−0.689638	−0.344819	+	0.938669i	$0.612060\pi$
$984$	0	0
$985$	−78609.8	−2.54286
$986$	15597.8	0.503787
$987$	0	0
$988$	−20480.4	−0.659482
$989$	−658.391	−0.0211685
$990$	0	0
$991$	−12573.9	−0.403049	−0.201525	−	0.979483i	$-0.564590\pi$
$991$	−12573.9	−0.403049	−0.201525	+	0.979483i	$0.564590\pi$
$992$	−42880.1	−1.37242
$993$	0	0
$994$	−15964.8	−0.509430
$995$	103506.	3.29786
$996$	0	0
$997$	2548.66	0.0809597	0.0404799	−	0.999180i	$-0.487111\pi$
$997$	2548.66	0.0809597	0.0404799	+	0.999180i	$0.487111\pi$
$998$	16473.6	0.522506
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.d.1.15		32
3.2	odd	2		717.4.a.d.1.18	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.d.1.18	✓	32	3.2	odd	2
2151.4.a.d.1.15		32	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-1.10380 q^{2} -6.78162 q^{4} -20.8736 q^{5} -33.6250 q^{7} +16.3160 q^{8} +O(q^{10})\)	\(q-1.10380 q^{2} -6.78162 q^{4} -20.8736 q^{5} -33.6250 q^{7} +16.3160 q^{8} +23.0403 q^{10} -69.0452 q^{11} -37.7160 q^{13} +37.1153 q^{14} +36.2434 q^{16} -83.4196 q^{17} -80.0717 q^{19} +141.557 q^{20} +76.2121 q^{22} -1.95956 q^{23} +310.707 q^{25} +41.6310 q^{26} +228.032 q^{28} +169.396 q^{29} +251.447 q^{31} -170.533 q^{32} +92.0787 q^{34} +701.874 q^{35} +13.1079 q^{37} +88.3832 q^{38} -340.573 q^{40} -150.847 q^{41} +335.988 q^{43} +468.238 q^{44} +2.16297 q^{46} -457.072 q^{47} +787.639 q^{49} -342.958 q^{50} +255.776 q^{52} -207.825 q^{53} +1441.22 q^{55} -548.624 q^{56} -186.979 q^{58} -273.288 q^{59} -729.276 q^{61} -277.548 q^{62} -101.712 q^{64} +787.269 q^{65} -385.910 q^{67} +565.720 q^{68} -774.729 q^{70} -430.141 q^{71} -351.322 q^{73} -14.4685 q^{74} +543.016 q^{76} +2321.64 q^{77} -537.394 q^{79} -756.530 q^{80} +166.505 q^{82} -486.003 q^{83} +1741.27 q^{85} -370.864 q^{86} -1126.54 q^{88} +17.5701 q^{89} +1268.20 q^{91} +13.2890 q^{92} +504.516 q^{94} +1671.38 q^{95} -1030.76 q^{97} -869.397 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

Introduction

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