LMFDB - Embedded newform 2151.4.a.c.1.7

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:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-3.00561 q^{2} +1.03369 q^{4} -10.0458 q^{5} -17.8583 q^{7} +20.9380 q^{8} +O(q^{10})$	\(q-3.00561 q^{2} +1.03369 q^{4} -10.0458 q^{5} -17.8583 q^{7} +20.9380 q^{8} +30.1936 q^{10} +34.7305 q^{11} -0.0479261 q^{13} +53.6751 q^{14} -71.2010 q^{16} -40.3279 q^{17} -119.230 q^{19} -10.3842 q^{20} -104.386 q^{22} +117.750 q^{23} -24.0827 q^{25} +0.144047 q^{26} -18.4600 q^{28} -46.1206 q^{29} -42.7678 q^{31} +46.4985 q^{32} +121.210 q^{34} +179.400 q^{35} +231.884 q^{37} +358.359 q^{38} -210.338 q^{40} -164.538 q^{41} -237.501 q^{43} +35.9007 q^{44} -353.911 q^{46} +595.359 q^{47} -24.0816 q^{49} +72.3832 q^{50} -0.0495409 q^{52} +500.970 q^{53} -348.894 q^{55} -373.917 q^{56} +138.620 q^{58} -428.191 q^{59} +904.066 q^{61} +128.543 q^{62} +429.852 q^{64} +0.481454 q^{65} -390.342 q^{67} -41.6867 q^{68} -539.207 q^{70} -712.531 q^{71} -992.306 q^{73} -696.953 q^{74} -123.247 q^{76} -620.227 q^{77} -1108.82 q^{79} +715.268 q^{80} +494.536 q^{82} -228.157 q^{83} +405.124 q^{85} +713.834 q^{86} +727.187 q^{88} -1561.98 q^{89} +0.855878 q^{91} +121.717 q^{92} -1789.42 q^{94} +1197.76 q^{95} -773.799 q^{97} +72.3798 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * 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 $

$2$	−3.00561	−1.06264	−0.531322	−	0.847170i	$-0.678305\pi$
$2$	−3.00561	−1.06264	−0.531322	+	0.847170i	$0.678305\pi$
$3$	0	0
$3$	0	0
$4$	1.03369	0.129212
$5$	−10.0458	−0.898520	−0.449260	−	0.893401i	$-0.648312\pi$
$5$	−10.0458	−0.898520	−0.449260	+	0.893401i	$0.648312\pi$
$6$	0	0
$7$	−17.8583	−0.964257	−0.482128	−	0.876101i	$-0.660136\pi$
$7$	−17.8583	−0.964257	−0.482128	+	0.876101i	$0.660136\pi$
$8$	20.9380	0.925338
$9$	0	0
$10$	30.1936	0.954807
$11$	34.7305	0.951967	0.475984	−	0.879454i	$-0.342092\pi$
$11$	34.7305	0.951967	0.475984	+	0.879454i	$0.342092\pi$
$12$	0	0
$13$	−0.0479261	−0.00102248	−0.000511242	−	1.00000i	$-0.500163\pi$
$13$	−0.0479261	−0.00102248	−0.000511242		1.00000i	$0.500163\pi$
$14$	53.6751	1.02466
$15$	0	0
$16$	−71.2010	−1.11252
$17$	−40.3279	−0.575350	−0.287675	−	0.957728i	$-0.592882\pi$
$17$	−40.3279	−0.575350	−0.287675	+	0.957728i	$0.592882\pi$
$18$	0	0
$19$	−119.230	−1.43964	−0.719822	−	0.694158i	$-0.755777\pi$
$19$	−119.230	−1.43964	−0.719822	+	0.694158i	$0.755777\pi$
$20$	−10.3842	−0.116099
$21$	0	0
$22$	−104.386	−1.01160
$23$	117.750	1.06750	0.533752	−	0.845641i	$-0.320782\pi$
$23$	117.750	1.06750	0.533752	+	0.845641i	$0.320782\pi$
$24$	0	0
$25$	−24.0827	−0.192662
$26$	0.144047	0.00108654
$27$	0	0
$28$	−18.4600	−0.124593
$29$	−46.1206	−0.295323	−0.147662	−	0.989038i	$-0.547175\pi$
$29$	−46.1206	−0.295323	−0.147662	+	0.989038i	$0.547175\pi$
$30$	0	0
$31$	−42.7678	−0.247785	−0.123892	−	0.992296i	$-0.539538\pi$
$31$	−42.7678	−0.247785	−0.123892	+	0.992296i	$0.539538\pi$
$32$	46.4985	0.256870
$33$	0	0
$34$	121.210	0.611392
$35$	179.400	0.866404
$36$	0	0
$37$	231.884	1.03031	0.515156	−	0.857097i	$-0.327734\pi$
$37$	231.884	1.03031	0.515156	+	0.857097i	$0.327734\pi$
$38$	358.359	1.52983
$39$	0	0
$40$	−210.338	−0.831435
$41$	−164.538	−0.626744	−0.313372	−	0.949631i	$-0.601459\pi$
$41$	−164.538	−0.626744	−0.313372	+	0.949631i	$0.601459\pi$
$42$	0	0
$43$	−237.501	−0.842290	−0.421145	−	0.906993i	$-0.638372\pi$
$43$	−237.501	−0.842290	−0.421145	+	0.906993i	$0.638372\pi$
$44$	35.9007	0.123005
$45$	0	0
$46$	−353.911	−1.13438
$47$	595.359	1.84770	0.923852	−	0.382750i	$-0.125023\pi$
$47$	595.359	1.84770	0.923852	+	0.382750i	$0.125023\pi$
$48$	0	0
$49$	−24.0816	−0.0702087
$50$	72.3832	0.204731
$51$	0	0
$52$	−0.0495409	−0.000132117 0
$53$	500.970	1.29837	0.649185	−	0.760631i	$-0.275110\pi$
$53$	500.970	1.29837	0.649185	+	0.760631i	$0.275110\pi$
$54$	0	0
$55$	−348.894	−0.855362
$56$	−373.917	−0.892263
$57$	0	0
$58$	138.620	0.313823
$59$	−428.191	−0.944844	−0.472422	−	0.881373i	$-0.656620\pi$
$59$	−428.191	−0.944844	−0.472422	+	0.881373i	$0.656620\pi$
$60$	0	0
$61$	904.066	1.89760	0.948802	−	0.315873i	$-0.102297\pi$
$61$	904.066	1.89760	0.948802	+	0.315873i	$0.102297\pi$
$62$	128.543	0.263307
$63$	0	0
$64$	429.852	0.839554
$65$	0.481454	0.000918723 0
$66$	0	0
$67$	−390.342	−0.711759	−0.355880	−	0.934532i	$-0.615819\pi$
$67$	−390.342	−0.711759	−0.355880	+	0.934532i	$0.615819\pi$
$68$	−41.6867	−0.0743419
$69$	0	0
$70$	−539.207	−0.920679
$71$	−712.531	−1.19101	−0.595506	−	0.803351i	$-0.703049\pi$
$71$	−712.531	−1.19101	−0.595506	+	0.803351i	$0.703049\pi$
$72$	0	0
$73$	−992.306	−1.59097	−0.795484	−	0.605975i	$-0.792783\pi$
$73$	−992.306	−1.59097	−0.795484	+	0.605975i	$0.792783\pi$
$74$	−696.953	−1.09485
$75$	0	0
$76$	−123.247	−0.186019
$77$	−620.227	−0.917941
$78$	0	0
$79$	−1108.82	−1.57914	−0.789569	−	0.613662i	$-0.789696\pi$
$79$	−1108.82	−1.57914	−0.789569	+	0.613662i	$0.789696\pi$
$80$	715.268	0.999618
$81$	0	0
$82$	494.536	0.666005
$83$	−228.157	−0.301729	−0.150865	−	0.988554i	$-0.548206\pi$
$83$	−228.157	−0.301729	−0.150865	+	0.988554i	$0.548206\pi$
$84$	0	0
$85$	405.124	0.516964
$86$	713.834	0.895055
$87$	0	0
$88$	727.187	0.880891
$89$	−1561.98	−1.86033	−0.930166	−	0.367140i	$-0.880337\pi$
$89$	−1561.98	−1.86033	−0.930166	+	0.367140i	$0.880337\pi$
$90$	0	0
$91$	0.855878	0.000985938 0
$92$	121.717	0.137934
$93$	0	0
$94$	−1789.42	−1.96345
$95$	1197.76	1.29355
$96$	0	0
$97$	−773.799	−0.809973	−0.404986	−	0.914323i	$-0.632724\pi$
$97$	−773.799	−0.809973	−0.404986	+	0.914323i	$0.632724\pi$
$98$	72.3798	0.0746068
$99$	0	0
$100$	−24.8941	−0.0248941
$101$	−1736.80	−1.71107	−0.855534	−	0.517747i	$-0.826771\pi$
$101$	−1736.80	−1.71107	−0.855534	+	0.517747i	$0.826771\pi$
$102$	0	0
$103$	−897.253	−0.858340	−0.429170	−	0.903224i	$-0.641194\pi$
$103$	−897.253	−0.858340	−0.429170	+	0.903224i	$0.641194\pi$
$104$	−1.00348	−0.000946144 0
$105$	0	0
$106$	−1505.72	−1.37970
$107$	−705.482	−0.637398	−0.318699	−	0.947856i	$-0.603246\pi$
$107$	−705.482	−0.637398	−0.318699	+	0.947856i	$0.603246\pi$
$108$	0	0
$109$	−257.278	−0.226080	−0.113040	−	0.993590i	$-0.536059\pi$
$109$	−257.278	−0.226080	−0.113040	+	0.993590i	$0.536059\pi$
$110$	1048.64	0.908945
$111$	0	0
$112$	1271.53	1.07275
$113$	−1393.54	−1.16012	−0.580060	−	0.814574i	$-0.696971\pi$
$113$	−1393.54	−1.16012	−0.580060	+	0.814574i	$0.696971\pi$
$114$	0	0
$115$	−1182.89	−0.959173
$116$	−47.6745	−0.0381592
$117$	0	0
$118$	1286.98	1.00403
$119$	720.187	0.554785
$120$	0	0
$121$	−124.793	−0.0937586
$122$	−2717.27	−2.01648
$123$	0	0
$124$	−44.2088	−0.0320167
$125$	1497.65	1.07163
$126$	0	0
$127$	−40.7453	−0.0284690	−0.0142345	−	0.999899i	$-0.504531\pi$
$127$	−40.7453	−0.0284690	−0.0142345	+	0.999899i	$0.504531\pi$
$128$	−1663.96	−1.14902
$129$	0	0
$130$	−1.44706	−0.000976275 0
$131$	783.111	0.522296	0.261148	−	0.965299i	$-0.415899\pi$
$131$	783.111	0.522296	0.261148	+	0.965299i	$0.415899\pi$
$132$	0	0
$133$	2129.24	1.38819
$134$	1173.22	0.756347
$135$	0	0
$136$	−844.386	−0.532393
$137$	2392.39	1.49194	0.745969	−	0.665981i	$-0.231987\pi$
$137$	2392.39	1.49194	0.745969	+	0.665981i	$0.231987\pi$
$138$	0	0
$139$	828.376	0.505482	0.252741	−	0.967534i	$-0.418668\pi$
$139$	828.376	0.505482	0.252741	+	0.967534i	$0.418668\pi$
$140$	185.445	0.111950
$141$	0	0
$142$	2141.59	1.26562
$143$	−1.66450	−0.000973372 0
$144$	0	0
$145$	463.316	0.265354
$146$	2982.49	1.69063
$147$	0	0
$148$	239.697	0.133128
$149$	1512.48	0.831590	0.415795	−	0.909458i	$-0.363503\pi$
$149$	1512.48	0.831590	0.415795	+	0.909458i	$0.363503\pi$
$150$	0	0
$151$	−2263.09	−1.21965	−0.609827	−	0.792534i	$-0.708761\pi$
$151$	−2263.09	−1.21965	−0.609827	+	0.792534i	$0.708761\pi$
$152$	−2496.44	−1.33216
$153$	0	0
$154$	1864.16	0.975444
$155$	429.635	0.222640
$156$	0	0
$157$	1354.26	0.688418	0.344209	−	0.938893i	$-0.388147\pi$
$157$	1354.26	0.688418	0.344209	+	0.938893i	$0.388147\pi$
$158$	3332.68	1.67806
$159$	0	0
$160$	−467.113	−0.230803
$161$	−2102.81	−1.02935
$162$	0	0
$163$	834.939	0.401211	0.200606	−	0.979672i	$-0.435709\pi$
$163$	834.939	0.401211	0.200606	+	0.979672i	$0.435709\pi$
$164$	−170.082	−0.0809826
$165$	0	0
$166$	685.752	0.320630
$167$	2830.41	1.31152	0.655760	−	0.754969i	$-0.272348\pi$
$167$	2830.41	1.31152	0.655760	+	0.754969i	$0.272348\pi$
$168$	0	0
$169$	−2197.00	−0.999999
$170$	−1217.65	−0.549348
$171$	0	0
$172$	−245.503	−0.108834
$173$	2516.68	1.10601	0.553004	−	0.833178i	$-0.313481\pi$
$173$	2516.68	1.10601	0.553004	+	0.833178i	$0.313481\pi$
$174$	0	0
$175$	430.076	0.185775
$176$	−2472.85	−1.05908
$177$	0	0
$178$	4694.70	1.97687
$179$	−2610.90	−1.09021	−0.545106	−	0.838367i	$-0.683511\pi$
$179$	−2610.90	−1.09021	−0.545106	+	0.838367i	$0.683511\pi$
$180$	0	0
$181$	−796.570	−0.327119	−0.163560	−	0.986533i	$-0.552298\pi$
$181$	−796.570	−0.327119	−0.163560	+	0.986533i	$0.552298\pi$
$182$	−2.57243	−0.00104770
$183$	0	0
$184$	2465.45	0.987801
$185$	−2329.45	−0.925755
$186$	0	0
$187$	−1400.61	−0.547714
$188$	615.419	0.238745
$189$	0	0
$190$	−3599.99	−1.37458
$191$	−1892.17	−0.716820	−0.358410	−	0.933564i	$-0.616681\pi$
$191$	−1892.17	−0.716820	−0.358410	+	0.933564i	$0.616681\pi$
$192$	0	0
$193$	1401.07	0.522547	0.261273	−	0.965265i	$-0.415858\pi$
$193$	1401.07	0.522547	0.261273	+	0.965265i	$0.415858\pi$
$194$	2325.74	0.860712
$195$	0	0
$196$	−24.8930	−0.00907178
$197$	−2363.57	−0.854809	−0.427404	−	0.904061i	$-0.640572\pi$
$197$	−2363.57	−0.854809	−0.427404	+	0.904061i	$0.640572\pi$
$198$	0	0
$199$	2716.93	0.967830	0.483915	−	0.875115i	$-0.339214\pi$
$199$	2716.93	0.967830	0.483915	+	0.875115i	$0.339214\pi$
$200$	−504.244	−0.178277
$201$	0	0
$202$	5220.14	1.81826
$203$	823.634	0.284767
$204$	0	0
$205$	1652.91	0.563142
$206$	2696.79	0.912109
$207$	0	0
$208$	3.41239	0.00113753
$209$	−4140.92	−1.37049
$210$	0	0
$211$	5488.43	1.79071	0.895354	−	0.445356i	$-0.146923\pi$
$211$	5488.43	1.79071	0.895354	+	0.445356i	$0.146923\pi$
$212$	517.850	0.167764
$213$	0	0
$214$	2120.40	0.677327
$215$	2385.87	0.756815
$216$	0	0
$217$	763.760	0.238928
$218$	773.278	0.240243
$219$	0	0
$220$	−360.650	−0.110523
$221$	1.93276	0.000588287 0
$222$	0	0
$223$	−2007.67	−0.602885	−0.301443	−	0.953484i	$-0.597468\pi$
$223$	−2007.67	−0.602885	−0.301443	+	0.953484i	$0.597468\pi$
$224$	−830.384	−0.247689
$225$	0	0
$226$	4188.45	1.23279
$227$	1420.65	0.415381	0.207691	−	0.978195i	$-0.433405\pi$
$227$	1420.65	0.415381	0.207691	+	0.978195i	$0.433405\pi$
$228$	0	0
$229$	1562.36	0.450845	0.225423	−	0.974261i	$-0.427624\pi$
$229$	1562.36	0.450845	0.225423	+	0.974261i	$0.427624\pi$
$230$	3555.30	1.01926
$231$	0	0
$232$	−965.672	−0.273274
$233$	−5874.79	−1.65180	−0.825902	−	0.563814i	$-0.809334\pi$
$233$	−5874.79	−1.65180	−0.825902	+	0.563814i	$0.809334\pi$
$234$	0	0
$235$	−5980.84	−1.66020
$236$	−442.619	−0.122085
$237$	0	0
$238$	−2164.60	−0.589539
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	141.789	0.0378979	0.0189490	−	0.999820i	$-0.493968\pi$
$241$	141.789	0.0378979	0.0189490	+	0.999820i	$0.493968\pi$
$242$	375.078	0.0996319
$243$	0	0
$244$	934.527	0.245192
$245$	241.918	0.0630839
$246$	0	0
$247$	5.71423	0.00147201
$248$	−895.473	−0.229285
$249$	0	0
$250$	−4501.35	−1.13876
$251$	6867.52	1.72699	0.863494	−	0.504359i	$-0.168271\pi$
$251$	6867.52	1.72699	0.863494	+	0.504359i	$0.168271\pi$
$252$	0	0
$253$	4089.52	1.01623
$254$	122.464	0.0302524
$255$	0	0
$256$	1562.39	0.381442
$257$	1603.59	0.389218	0.194609	−	0.980881i	$-0.437656\pi$
$257$	1603.59	0.389218	0.194609	+	0.980881i	$0.437656\pi$
$258$	0	0
$259$	−4141.05	−0.993485
$260$	0.497676	0.000118710 0
$261$	0	0
$262$	−2353.73	−0.555014
$263$	−4118.19	−0.965546	−0.482773	−	0.875746i	$-0.660370\pi$
$263$	−4118.19	−0.965546	−0.482773	+	0.875746i	$0.660370\pi$
$264$	0	0
$265$	−5032.63	−1.16661
$266$	−6399.68	−1.47515
$267$	0	0
$268$	−403.494	−0.0919676
$269$	−7641.36	−1.73198	−0.865989	−	0.500063i	$-0.833310\pi$
$269$	−7641.36	−1.73198	−0.865989	+	0.500063i	$0.833310\pi$
$270$	0	0
$271$	653.831	0.146559	0.0732794	−	0.997311i	$-0.476654\pi$
$271$	653.831	0.146559	0.0732794	+	0.997311i	$0.476654\pi$
$272$	2871.39	0.640086
$273$	0	0
$274$	−7190.59	−1.58540
$275$	−836.404	−0.183408
$276$	0	0
$277$	2122.41	0.460372	0.230186	−	0.973147i	$-0.426066\pi$
$277$	2122.41	0.460372	0.230186	+	0.973147i	$0.426066\pi$
$278$	−2489.78	−0.537147
$279$	0	0
$280$	3756.28	0.801717
$281$	4833.04	1.02603	0.513016	−	0.858379i	$-0.328528\pi$
$281$	4833.04	1.02603	0.513016	+	0.858379i	$0.328528\pi$
$282$	0	0
$283$	−7180.15	−1.50818	−0.754091	−	0.656770i	$-0.771922\pi$
$283$	−7180.15	−1.50818	−0.754091	+	0.656770i	$0.771922\pi$
$284$	−736.539	−0.153893
$285$	0	0
$286$	5.00283	0.00103435
$287$	2938.36	0.604342
$288$	0	0
$289$	−3286.66	−0.668972
$290$	−1392.55	−0.281977
$291$	0	0
$292$	−1025.74	−0.205572
$293$	3926.79	0.782954	0.391477	−	0.920188i	$-0.371964\pi$
$293$	3926.79	0.782954	0.391477	+	0.920188i	$0.371964\pi$
$294$	0	0
$295$	4301.51	0.848961
$296$	4855.19	0.953386
$297$	0	0
$298$	−4545.92	−0.883684
$299$	−5.64330	−0.00109151
$300$	0	0
$301$	4241.35	0.812184
$302$	6801.97	1.29606
$303$	0	0
$304$	8489.30	1.60163
$305$	−9082.03	−1.70503
$306$	0	0
$307$	2192.04	0.407513	0.203756	−	0.979022i	$-0.434685\pi$
$307$	2192.04	0.407513	0.203756	+	0.979022i	$0.434685\pi$
$308$	−641.125	−0.118609
$309$	0	0
$310$	−1291.32	−0.236587
$311$	9645.49	1.75867	0.879333	−	0.476207i	$-0.157989\pi$
$311$	9645.49	1.75867	0.879333	+	0.476207i	$0.157989\pi$
$312$	0	0
$313$	7652.94	1.38201	0.691006	−	0.722849i	$-0.257168\pi$
$313$	7652.94	1.38201	0.691006	+	0.722849i	$0.257168\pi$
$314$	−4070.37	−0.731543
$315$	0	0
$316$	−1146.18	−0.204043
$317$	−2109.16	−0.373698	−0.186849	−	0.982389i	$-0.559828\pi$
$317$	−2109.16	−0.373698	−0.186849	+	0.982389i	$0.559828\pi$
$318$	0	0
$319$	−1601.79	−0.281138
$320$	−4318.19	−0.754356
$321$	0	0
$322$	6320.24	1.09383
$323$	4808.30	0.828300
$324$	0	0
$325$	1.15419	0.000196994 0
$326$	−2509.50	−0.426345
$327$	0	0
$328$	−3445.09	−0.579949
$329$	−10632.1	−1.78166
$330$	0	0
$331$	7548.07	1.25341	0.626706	−	0.779256i	$-0.284403\pi$
$331$	7548.07	1.25341	0.626706	+	0.779256i	$0.284403\pi$
$332$	−235.845	−0.0389869
$333$	0	0
$334$	−8507.12	−1.39368
$335$	3921.28	0.639530
$336$	0	0
$337$	−3033.59	−0.490356	−0.245178	−	0.969478i	$-0.578846\pi$
$337$	−3033.59	−0.490356	−0.245178	+	0.969478i	$0.578846\pi$
$338$	6603.32	1.06264
$339$	0	0
$340$	418.774	0.0667977
$341$	−1485.35	−0.235883
$342$	0	0
$343$	6555.45	1.03196
$344$	−4972.79	−0.779403
$345$	0	0
$346$	−7564.15	−1.17529
$347$	−4042.24	−0.625356	−0.312678	−	0.949859i	$-0.601226\pi$
$347$	−4042.24	−0.625356	−0.312678	+	0.949859i	$0.601226\pi$
$348$	0	0
$349$	−9210.15	−1.41263	−0.706315	−	0.707898i	$-0.749644\pi$
$349$	−9210.15	−1.41263	−0.706315	+	0.707898i	$0.749644\pi$
$350$	−1292.64	−0.197413
$351$	0	0
$352$	1614.92	0.244532
$353$	8069.60	1.21672	0.608359	−	0.793662i	$-0.291828\pi$
$353$	8069.60	1.21672	0.608359	+	0.793662i	$0.291828\pi$
$354$	0	0
$355$	7157.92	1.07015
$356$	−1614.61	−0.240376
$357$	0	0
$358$	7847.35	1.15851
$359$	−540.154	−0.0794101	−0.0397051	−	0.999211i	$-0.512642\pi$
$359$	−540.154	−0.0794101	−0.0397051	+	0.999211i	$0.512642\pi$
$360$	0	0
$361$	7356.80	1.07258
$362$	2394.18	0.347611
$363$	0	0
$364$	0.884715	0.000127395 0
$365$	9968.47	1.42952
$366$	0	0
$367$	12324.4	1.75294	0.876470	−	0.481457i	$-0.159892\pi$
$367$	12324.4	1.75294	0.876470	+	0.481457i	$0.159892\pi$
$368$	−8383.92	−1.18761
$369$	0	0
$370$	7001.43	0.983748
$371$	−8946.47	−1.25196
$372$	0	0
$373$	−3928.17	−0.545289	−0.272645	−	0.962115i	$-0.587898\pi$
$373$	−3928.17	−0.545289	−0.272645	+	0.962115i	$0.587898\pi$
$374$	4209.68	0.582025
$375$	0	0
$376$	12465.6	1.70975
$377$	2.21038	0.000301963 0
$378$	0	0
$379$	13170.4	1.78501	0.892505	−	0.451038i	$-0.148946\pi$
$379$	13170.4	1.78501	0.892505	+	0.451038i	$0.148946\pi$
$380$	1238.11	0.167142
$381$	0	0
$382$	5687.12	0.761724
$383$	−1138.85	−0.151939	−0.0759695	−	0.997110i	$-0.524205\pi$
$383$	−1138.85	−0.151939	−0.0759695	+	0.997110i	$0.524205\pi$
$384$	0	0
$385$	6230.65	0.824788
$386$	−4211.08	−0.555281
$387$	0	0
$388$	−799.870	−0.104658
$389$	12139.7	1.58228	0.791141	−	0.611633i	$-0.209487\pi$
$389$	12139.7	1.58228	0.791141	+	0.611633i	$0.209487\pi$
$390$	0	0
$391$	−4748.61	−0.614188
$392$	−504.220	−0.0649668
$393$	0	0
$394$	7103.97	0.908357
$395$	11138.9	1.41889
$396$	0	0
$397$	−5245.50	−0.663134	−0.331567	−	0.943432i	$-0.607577\pi$
$397$	−5245.50	−0.663134	−0.331567	+	0.943432i	$0.607577\pi$
$398$	−8166.04	−1.02846
$399$	0	0
$400$	1714.71	0.214339
$401$	3679.75	0.458249	0.229125	−	0.973397i	$-0.426414\pi$
$401$	3679.75	0.458249	0.229125	+	0.973397i	$0.426414\pi$
$402$	0	0
$403$	2.04969	0.000253356 0
$404$	−1795.32	−0.221090
$405$	0	0
$406$	−2475.52	−0.302606
$407$	8053.45	0.980822
$408$	0	0
$409$	5172.91	0.625389	0.312694	−	0.949854i	$-0.398768\pi$
$409$	5172.91	0.625389	0.312694	+	0.949854i	$0.398768\pi$
$410$	−4967.99	−0.598419
$411$	0	0
$412$	−927.485	−0.110907
$413$	7646.77	0.911072
$414$	0	0
$415$	2292.01	0.271110
$416$	−2.22849	−0.000262646 0
$417$	0	0
$418$	12446.0	1.45635
$419$	7885.21	0.919374	0.459687	−	0.888081i	$-0.347962\pi$
$419$	7885.21	0.919374	0.459687	+	0.888081i	$0.347962\pi$
$420$	0	0
$421$	6206.90	0.718541	0.359271	−	0.933233i	$-0.383025\pi$
$421$	6206.90	0.718541	0.359271	+	0.933233i	$0.383025\pi$
$422$	−16496.1	−1.90288
$423$	0	0
$424$	10489.3	1.20143
$425$	971.205	0.110848
$426$	0	0
$427$	−16145.1	−1.82978
$428$	−729.252	−0.0823592
$429$	0	0
$430$	−7171.01	−0.804225
$431$	−2116.98	−0.236592	−0.118296	−	0.992978i	$-0.537743\pi$
$431$	−2116.98	−0.236592	−0.118296	+	0.992978i	$0.537743\pi$
$432$	0	0
$433$	−3126.06	−0.346949	−0.173474	−	0.984838i	$-0.555499\pi$
$433$	−3126.06	−0.346949	−0.173474	+	0.984838i	$0.555499\pi$
$434$	−2295.57	−0.253896
$435$	0	0
$436$	−265.947	−0.0292122
$437$	−14039.3	−1.53683
$438$	0	0
$439$	−7672.85	−0.834181	−0.417090	−	0.908865i	$-0.636950\pi$
$439$	−7672.85	−0.834181	−0.417090	+	0.908865i	$0.636950\pi$
$440$	−7305.15	−0.791498
$441$	0	0
$442$	−5.80912	−0.000625139 0
$443$	6073.86	0.651417	0.325708	−	0.945470i	$-0.394397\pi$
$443$	6073.86	0.651417	0.325708	+	0.945470i	$0.394397\pi$
$444$	0	0
$445$	15691.3	1.67154
$446$	6034.27	0.640652
$447$	0	0
$448$	−7676.42	−0.809546
$449$	−2674.08	−0.281064	−0.140532	−	0.990076i	$-0.544881\pi$
$449$	−2674.08	−0.281064	−0.140532	+	0.990076i	$0.544881\pi$
$450$	0	0
$451$	−5714.48	−0.596639
$452$	−1440.50	−0.149901
$453$	0	0
$454$	−4269.91	−0.441402
$455$	−8.59794	−0.000885885 0
$456$	0	0
$457$	−11418.2	−1.16876	−0.584379	−	0.811481i	$-0.698662\pi$
$457$	−11418.2	−1.16876	−0.584379	+	0.811481i	$0.698662\pi$
$458$	−4695.84	−0.479088
$459$	0	0
$460$	−1222.74	−0.123936
$461$	8359.13	0.844519	0.422260	−	0.906475i	$-0.361237\pi$
$461$	8359.13	0.844519	0.422260	+	0.906475i	$0.361237\pi$
$462$	0	0
$463$	−8025.52	−0.805567	−0.402784	−	0.915295i	$-0.631957\pi$
$463$	−8025.52	−0.805567	−0.402784	+	0.915295i	$0.631957\pi$
$464$	3283.83	0.328552
$465$	0	0
$466$	17657.3	1.75528
$467$	−19843.9	−1.96631	−0.983155	−	0.182773i	$-0.941493\pi$
$467$	−19843.9	−1.96631	−0.983155	+	0.182773i	$0.941493\pi$
$468$	0	0
$469$	6970.84	0.686319
$470$	17976.1	1.76420
$471$	0	0
$472$	−8965.47	−0.874300
$473$	−8248.51	−0.801833
$474$	0	0
$475$	2871.38	0.277364
$476$	744.453	0.0716847
$477$	0	0
$478$	−718.341	−0.0687367
$479$	4138.27	0.394744	0.197372	−	0.980329i	$-0.436759\pi$
$479$	4138.27	0.394744	0.197372	+	0.980329i	$0.436759\pi$
$480$	0	0
$481$	−11.1133	−0.00105348
$482$	−426.161	−0.0402720
$483$	0	0
$484$	−128.997	−0.0121147
$485$	7773.40	0.727777
$486$	0	0
$487$	6780.94	0.630953	0.315476	−	0.948933i	$-0.397836\pi$
$487$	6780.94	0.630953	0.315476	+	0.948933i	$0.397836\pi$
$488$	18929.3	1.75592
$489$	0	0
$490$	−727.111	−0.0670357
$491$	2130.61	0.195831	0.0979154	−	0.995195i	$-0.468783\pi$
$491$	2130.61	0.195831	0.0979154	+	0.995195i	$0.468783\pi$
$492$	0	0
$493$	1859.94	0.169914
$494$	−17.1747	−0.00156423
$495$	0	0
$496$	3045.11	0.275665
$497$	12724.6	1.14844
$498$	0	0
$499$	−9636.21	−0.864481	−0.432240	−	0.901758i	$-0.642277\pi$
$499$	−9636.21	−0.864481	−0.432240	+	0.901758i	$0.642277\pi$
$500$	1548.11	0.138467
$501$	0	0
$502$	−20641.1	−1.83517
$503$	−15252.8	−1.35207	−0.676033	−	0.736871i	$-0.736302\pi$
$503$	−15252.8	−1.35207	−0.676033	+	0.736871i	$0.736302\pi$
$504$	0	0
$505$	17447.5	1.53743
$506$	−12291.5	−1.07989
$507$	0	0
$508$	−42.1181	−0.00367852
$509$	17851.8	1.55455	0.777277	−	0.629159i	$-0.216600\pi$
$509$	17851.8	1.55455	0.777277	+	0.629159i	$0.216600\pi$
$510$	0	0
$511$	17720.9	1.53410
$512$	8615.72	0.743681
$513$	0	0
$514$	−4819.76	−0.413600
$515$	9013.59	0.771235
$516$	0	0
$517$	20677.1	1.75895
$518$	12446.4	1.05572
$519$	0	0
$520$	10.0807	0.000850129 0
$521$	21366.2	1.79668	0.898342	−	0.439297i	$-0.144773\pi$
$521$	21366.2	1.79668	0.898342	+	0.439297i	$0.144773\pi$
$522$	0	0
$523$	−23151.3	−1.93563	−0.967816	−	0.251660i	$-0.919024\pi$
$523$	−23151.3	−1.93563	−0.967816	+	0.251660i	$0.919024\pi$
$524$	809.497	0.0674867
$525$	0	0
$526$	12377.7	1.02603
$527$	1724.74	0.142563
$528$	0	0
$529$	1698.07	0.139564
$530$	15126.1	1.23969
$531$	0	0
$532$	2200.99	0.179370
$533$	7.88565	0.000640836 0
$534$	0	0
$535$	7087.11	0.572715
$536$	−8172.98	−0.658618
$537$	0	0
$538$	22967.0	1.84048
$539$	−836.365	−0.0668364
$540$	0	0
$541$	−1046.16	−0.0831382	−0.0415691	−	0.999136i	$-0.513236\pi$
$541$	−1046.16	−0.0831382	−0.0415691	+	0.999136i	$0.513236\pi$
$542$	−1965.16	−0.155740
$543$	0	0
$544$	−1875.19	−0.147790
$545$	2584.55	0.203138
$546$	0	0
$547$	20454.8	1.59888	0.799438	−	0.600749i	$-0.205131\pi$
$547$	20454.8	1.59888	0.799438	+	0.600749i	$0.205131\pi$
$548$	2473.00	0.192776
$549$	0	0
$550$	2513.90	0.194897
$551$	5498.95	0.425160
$552$	0	0
$553$	19801.6	1.52269
$554$	−6379.13	−0.489212
$555$	0	0
$556$	856.287	0.0653141
$557$	10349.8	0.787314	0.393657	−	0.919257i	$-0.371210\pi$
$557$	10349.8	0.787314	0.393657	+	0.919257i	$0.371210\pi$
$558$	0	0
$559$	11.3825	0.000861229 0
$560$	−12773.5	−0.963889
$561$	0	0
$562$	−14526.2	−1.09031
$563$	15471.2	1.15814	0.579071	−	0.815277i	$-0.303415\pi$
$563$	15471.2	1.15814	0.579071	+	0.815277i	$0.303415\pi$
$564$	0	0
$565$	13999.2	1.04239
$566$	21580.7	1.60266
$567$	0	0
$568$	−14919.0	−1.10209
$569$	−16022.5	−1.18049	−0.590246	−	0.807224i	$-0.700969\pi$
$569$	−16022.5	−1.18049	−0.590246	+	0.807224i	$0.700969\pi$
$570$	0	0
$571$	−14658.8	−1.07435	−0.537175	−	0.843471i	$-0.680508\pi$
$571$	−14658.8	−1.07435	−0.537175	+	0.843471i	$0.680508\pi$
$572$	−1.72058	−0.000125771 0
$573$	0	0
$574$	−8831.57	−0.642200
$575$	−2835.74	−0.205667
$576$	0	0
$577$	−14162.9	−1.02185	−0.510926	−	0.859625i	$-0.670697\pi$
$577$	−14162.9	−1.02185	−0.510926	+	0.859625i	$0.670697\pi$
$578$	9878.42	0.710879
$579$	0	0
$580$	478.927	0.0342868
$581$	4074.50	0.290944
$582$	0	0
$583$	17398.9	1.23600
$584$	−20776.9	−1.47218
$585$	0	0
$586$	−11802.4	−0.832001
$587$	−3424.28	−0.240776	−0.120388	−	0.992727i	$-0.538414\pi$
$587$	−3424.28	−0.240776	−0.120388	+	0.992727i	$0.538414\pi$
$588$	0	0
$589$	5099.21	0.356722
$590$	−12928.7	−0.902143
$591$	0	0
$592$	−16510.4	−1.14624
$593$	7698.60	0.533126	0.266563	−	0.963818i	$-0.414112\pi$
$593$	7698.60	0.533126	0.266563	+	0.963818i	$0.414112\pi$
$594$	0	0
$595$	−7234.83	−0.498486
$596$	1563.44	0.107451
$597$	0	0
$598$	16.9616	0.00115988
$599$	4763.00	0.324893	0.162446	−	0.986717i	$-0.448062\pi$
$599$	4763.00	0.324893	0.162446	+	0.986717i	$0.448062\pi$
$600$	0	0
$601$	−3349.08	−0.227308	−0.113654	−	0.993520i	$-0.536255\pi$
$601$	−3349.08	−0.227308	−0.113654	+	0.993520i	$0.536255\pi$
$602$	−12747.9	−0.863063
$603$	0	0
$604$	−2339.34	−0.157594
$605$	1253.64	0.0842439
$606$	0	0
$607$	−19747.6	−1.32048	−0.660238	−	0.751056i	$-0.729545\pi$
$607$	−19747.6	−1.32048	−0.660238	+	0.751056i	$0.729545\pi$
$608$	−5544.02	−0.369802
$609$	0	0
$610$	27297.0	1.81184
$611$	−28.5332	−0.00188925
$612$	0	0
$613$	−12735.7	−0.839133	−0.419566	−	0.907725i	$-0.637818\pi$
$613$	−12735.7	−0.839133	−0.419566	+	0.907725i	$0.637818\pi$
$614$	−6588.42	−0.433041
$615$	0	0
$616$	−12986.3	−0.849405
$617$	3870.62	0.252553	0.126277	−	0.991995i	$-0.459697\pi$
$617$	3870.62	0.252553	0.126277	+	0.991995i	$0.459697\pi$
$618$	0	0
$619$	25662.6	1.66634	0.833172	−	0.553014i	$-0.186522\pi$
$619$	25662.6	1.66634	0.833172	+	0.553014i	$0.186522\pi$
$620$	444.111	0.0287676
$621$	0	0
$622$	−28990.6	−1.86884
$623$	27894.3	1.79384
$624$	0	0
$625$	−12034.7	−0.770220
$626$	−23001.8	−1.46859
$627$	0	0
$628$	1399.89	0.0889516
$629$	−9351.40	−0.592790
$630$	0	0
$631$	−8159.44	−0.514774	−0.257387	−	0.966308i	$-0.582861\pi$
$631$	−8159.44	−0.514774	−0.257387	+	0.966308i	$0.582861\pi$
$632$	−23216.5	−1.46124
$633$	0	0
$634$	6339.32	0.397108
$635$	409.317	0.0255799
$636$	0	0
$637$	1.15414	7.17873e−5 0
$638$	4814.36	0.298749
$639$	0	0
$640$	16715.7	1.03242
$641$	26893.3	1.65713	0.828567	−	0.559891i	$-0.189157\pi$
$641$	26893.3	1.65713	0.828567	+	0.559891i	$0.189157\pi$
$642$	0	0
$643$	−20875.5	−1.28033	−0.640164	−	0.768238i	$-0.721134\pi$
$643$	−20875.5	−1.28033	−0.640164	+	0.768238i	$0.721134\pi$
$644$	−2173.66	−0.133004
$645$	0	0
$646$	−14451.9	−0.880187
$647$	29596.6	1.79840	0.899199	−	0.437539i	$-0.144150\pi$
$647$	29596.6	1.79840	0.899199	+	0.437539i	$0.144150\pi$
$648$	0	0
$649$	−14871.3	−0.899460
$650$	−3.46904	−0.000209334 0
$651$	0	0
$652$	863.070	0.0518412
$653$	8140.97	0.487873	0.243936	−	0.969791i	$-0.421561\pi$
$653$	8140.97	0.487873	0.243936	+	0.969791i	$0.421561\pi$
$654$	0	0
$655$	−7866.95	−0.469293
$656$	11715.3	0.697262
$657$	0	0
$658$	31955.9	1.89327
$659$	−9284.27	−0.548807	−0.274404	−	0.961615i	$-0.588480\pi$
$659$	−9284.27	−0.548807	−0.274404	+	0.961615i	$0.588480\pi$
$660$	0	0
$661$	3581.25	0.210733	0.105367	−	0.994433i	$-0.466398\pi$
$661$	3581.25	0.210733	0.105367	+	0.994433i	$0.466398\pi$
$662$	−22686.6	−1.33193
$663$	0	0
$664$	−4777.16	−0.279201
$665$	−21389.9	−1.24731
$666$	0	0
$667$	−5430.70	−0.315258
$668$	2925.78	0.169464
$669$	0	0
$670$	−11785.8	−0.679593
$671$	31398.7	1.80646
$672$	0	0
$673$	−16326.1	−0.935106	−0.467553	−	0.883965i	$-0.654864\pi$
$673$	−16326.1	−0.935106	−0.467553	+	0.883965i	$0.654864\pi$
$674$	9117.78	0.521074
$675$	0	0
$676$	−2271.02	−0.129212
$677$	28253.2	1.60393	0.801965	−	0.597371i	$-0.203788\pi$
$677$	28253.2	1.60393	0.801965	+	0.597371i	$0.203788\pi$
$678$	0	0
$679$	13818.7	0.781022
$680$	8482.49	0.478366
$681$	0	0
$682$	4464.38	0.250660
$683$	23002.0	1.28865	0.644326	−	0.764751i	$-0.277138\pi$
$683$	23002.0	1.28865	0.644326	+	0.764751i	$0.277138\pi$
$684$	0	0
$685$	−24033.4	−1.34054
$686$	−19703.1	−1.09660
$687$	0	0
$688$	16910.3	0.937062
$689$	−24.0095	−0.00132756
$690$	0	0
$691$	18511.3	1.01911	0.509555	−	0.860438i	$-0.329810\pi$
$691$	18511.3	1.01911	0.509555	+	0.860438i	$0.329810\pi$
$692$	2601.47	0.142909
$693$	0	0
$694$	12149.4	0.664531
$695$	−8321.67	−0.454186
$696$	0	0
$697$	6635.46	0.360597
$698$	27682.1	1.50112
$699$	0	0
$700$	444.566	0.0240043
$701$	−10589.8	−0.570571	−0.285286	−	0.958443i	$-0.592088\pi$
$701$	−10589.8	−0.570571	−0.285286	+	0.958443i	$0.592088\pi$
$702$	0	0
$703$	−27647.6	−1.48328
$704$	14929.0	0.799228
$705$	0	0
$706$	−24254.1	−1.29294
$707$	31016.2	1.64991
$708$	0	0
$709$	−34064.2	−1.80438	−0.902192	−	0.431335i	$-0.858043\pi$
$709$	−34064.2	−1.80438	−0.902192	+	0.431335i	$0.858043\pi$
$710$	−21513.9	−1.13719
$711$	0	0
$712$	−32704.7	−1.72143
$713$	−5035.91	−0.264511
$714$	0	0
$715$	16.7211	0.000874594 0
$716$	−2698.87	−0.140868
$717$	0	0
$718$	1623.49	0.0843847
$719$	−9518.75	−0.493727	−0.246863	−	0.969050i	$-0.579400\pi$
$719$	−9518.75	−0.493727	−0.246863	+	0.969050i	$0.579400\pi$
$720$	0	0
$721$	16023.4	0.827660
$722$	−22111.7	−1.13977
$723$	0	0
$724$	−823.409	−0.0422676
$725$	1110.71	0.0568974
$726$	0	0
$727$	−14252.6	−0.727097	−0.363549	−	0.931575i	$-0.618435\pi$
$727$	−14252.6	−0.727097	−0.363549	+	0.931575i	$0.618435\pi$
$728$	17.9204	0.000912326 0
$729$	0	0
$730$	−29961.3	−1.51907
$731$	9577.90	0.484612
$732$	0	0
$733$	6840.24	0.344679	0.172340	−	0.985038i	$-0.444867\pi$
$733$	6840.24	0.344679	0.172340	+	0.985038i	$0.444867\pi$
$734$	−37042.4	−1.86275
$735$	0	0
$736$	5475.20	0.274210
$737$	−13556.8	−0.677571
$738$	0	0
$739$	3514.47	0.174942	0.0874708	−	0.996167i	$-0.472122\pi$
$739$	3514.47	0.174942	0.0874708	+	0.996167i	$0.472122\pi$
$740$	−2407.94	−0.119618
$741$	0	0
$742$	26889.6	1.33039
$743$	−29995.7	−1.48107	−0.740536	−	0.672016i	$-0.765428\pi$
$743$	−29995.7	−1.48107	−0.740536	+	0.672016i	$0.765428\pi$
$744$	0	0
$745$	−15194.0	−0.747200
$746$	11806.5	0.579448
$747$	0	0
$748$	−1447.80	−0.0707711
$749$	12598.7	0.614615
$750$	0	0
$751$	35226.9	1.71165	0.855824	−	0.517267i	$-0.173051\pi$
$751$	35226.9	1.71165	0.855824	+	0.517267i	$0.173051\pi$
$752$	−42390.2	−2.05560
$753$	0	0
$754$	−6.64353	−0.000320880 0
$755$	22734.5	1.09588
$756$	0	0
$757$	−16932.9	−0.812995	−0.406498	−	0.913652i	$-0.633250\pi$
$757$	−16932.9	−0.812995	−0.406498	+	0.913652i	$0.633250\pi$
$758$	−39585.1	−1.89683
$759$	0	0
$760$	25078.6	1.19697
$761$	39429.1	1.87819	0.939095	−	0.343658i	$-0.111666\pi$
$761$	39429.1	1.87819	0.939095	+	0.343658i	$0.111666\pi$
$762$	0	0
$763$	4594.55	0.218000
$764$	−1955.92	−0.0926215
$765$	0	0
$766$	3422.95	0.161457
$767$	20.5215	0.000966088 0
$768$	0	0
$769$	−5202.49	−0.243962	−0.121981	−	0.992532i	$-0.538925\pi$
$769$	−5202.49	−0.243962	−0.121981	+	0.992532i	$0.538925\pi$
$770$	−18726.9	−0.876456
$771$	0	0
$772$	1448.28	0.0675191
$773$	−1619.26	−0.0753439	−0.0376720	−	0.999290i	$-0.511994\pi$
$773$	−1619.26	−0.0753439	−0.0376720	+	0.999290i	$0.511994\pi$
$774$	0	0
$775$	1029.97	0.0477386
$776$	−16201.8	−0.749498
$777$	0	0
$778$	−36487.2	−1.68140
$779$	19617.8	0.902288
$780$	0	0
$781$	−24746.6	−1.13380
$782$	14272.5	0.652663
$783$	0	0
$784$	1714.63	0.0781083
$785$	−13604.6	−0.618557
$786$	0	0
$787$	9658.86	0.437486	0.218743	−	0.975783i	$-0.429804\pi$
$787$	9658.86	0.437486	0.218743	+	0.975783i	$0.429804\pi$
$788$	−2443.21	−0.110451
$789$	0	0
$790$	−33479.3	−1.50777
$791$	24886.3	1.11865
$792$	0	0
$793$	−43.3283	−0.00194027
$794$	15765.9	0.704675
$795$	0	0
$796$	2808.47	0.125055
$797$	−17192.5	−0.764102	−0.382051	−	0.924141i	$-0.624782\pi$
$797$	−17192.5	−0.764102	−0.382051	+	0.924141i	$0.624782\pi$
$798$	0	0
$799$	−24009.6	−1.06308
$800$	−1119.81	−0.0494891
$801$	0	0
$802$	−11059.9	−0.486956
$803$	−34463.3	−1.51455
$804$	0	0
$805$	21124.4	0.924889
$806$	−6.16058	−0.000269227 0
$807$	0	0
$808$	−36365.1	−1.58332
$809$	12452.3	0.541162	0.270581	−	0.962697i	$-0.412784\pi$
$809$	12452.3	0.541162	0.270581	+	0.962697i	$0.412784\pi$
$810$	0	0
$811$	19356.1	0.838084	0.419042	−	0.907967i	$-0.362366\pi$
$811$	19356.1	0.838084	0.419042	+	0.907967i	$0.362366\pi$
$812$	851.385	0.0367953
$813$	0	0
$814$	−24205.5	−1.04226
$815$	−8387.59	−0.360496
$816$	0	0
$817$	28317.2	1.21260
$818$	−15547.8	−0.664565
$819$	0	0
$820$	1708.60	0.0727645
$821$	7246.23	0.308033	0.154017	−	0.988068i	$-0.450779\pi$
$821$	7246.23	0.308033	0.154017	+	0.988068i	$0.450779\pi$
$822$	0	0
$823$	39779.6	1.68485	0.842424	−	0.538816i	$-0.181128\pi$
$823$	39779.6	1.68485	0.842424	+	0.538816i	$0.181128\pi$
$824$	−18786.7	−0.794254
$825$	0	0
$826$	−22983.2	−0.968145
$827$	−46314.5	−1.94741	−0.973707	−	0.227802i	$-0.926846\pi$
$827$	−46314.5	−1.94741	−0.973707	+	0.227802i	$0.926846\pi$
$828$	0	0
$829$	−6132.09	−0.256908	−0.128454	−	0.991715i	$-0.541001\pi$
$829$	−6132.09	−0.256908	−0.128454	+	0.991715i	$0.541001\pi$
$830$	−6888.90	−0.288093
$831$	0	0
$832$	−20.6011	−0.000858432 0
$833$	971.159	0.0403946
$834$	0	0
$835$	−28433.6	−1.17843
$836$	−4280.44	−0.177084
$837$	0	0
$838$	−23699.9	−0.976967
$839$	23785.0	0.978723	0.489361	−	0.872081i	$-0.337230\pi$
$839$	23785.0	0.978723	0.489361	+	0.872081i	$0.337230\pi$
$840$	0	0
$841$	−22261.9	−0.912784
$842$	−18655.5	−0.763553
$843$	0	0
$844$	5673.36	0.231380
$845$	22070.5	0.898519
$846$	0	0
$847$	2228.58	0.0904073
$848$	−35669.6	−1.44446
$849$	0	0
$850$	−2919.06	−0.117792
$851$	27304.4	1.09986
$852$	0	0
$853$	44936.5	1.80375	0.901874	−	0.431999i	$-0.142192\pi$
$853$	44936.5	1.80375	0.901874	+	0.431999i	$0.142192\pi$
$854$	48525.8	1.94440
$855$	0	0
$856$	−14771.4	−0.589808
$857$	41784.1	1.66548	0.832741	−	0.553662i	$-0.186770\pi$
$857$	41784.1	1.66548	0.832741	+	0.553662i	$0.186770\pi$
$858$	0	0
$859$	6928.43	0.275198	0.137599	−	0.990488i	$-0.456061\pi$
$859$	6928.43	0.275198	0.137599	+	0.990488i	$0.456061\pi$
$860$	2466.26	0.0977893
$861$	0	0
$862$	6362.81	0.251413
$863$	8999.11	0.354963	0.177482	−	0.984124i	$-0.443205\pi$
$863$	8999.11	0.354963	0.177482	+	0.984124i	$0.443205\pi$
$864$	0	0
$865$	−25281.9	−0.993771
$866$	9395.72	0.368683
$867$	0	0
$868$	789.494	0.0308723
$869$	−38509.8	−1.50329
$870$	0	0
$871$	18.7076	0.000727763 0
$872$	−5386.89	−0.209201
$873$	0	0
$874$	42196.8	1.63310
$875$	−26745.4	−1.03333
$876$	0	0
$877$	−10492.4	−0.403993	−0.201997	−	0.979386i	$-0.564743\pi$
$877$	−10492.4	−0.403993	−0.201997	+	0.979386i	$0.564743\pi$
$878$	23061.6	0.886437
$879$	0	0
$880$	24841.6	0.951604
$881$	−38067.5	−1.45576	−0.727882	−	0.685702i	$-0.759495\pi$
$881$	−38067.5	−1.45576	−0.727882	+	0.685702i	$0.759495\pi$
$882$	0	0
$883$	−28033.6	−1.06841	−0.534205	−	0.845355i	$-0.679389\pi$
$883$	−28033.6	−1.06841	−0.534205	+	0.845355i	$0.679389\pi$
$884$	1.99788	7.60135e−5 0
$885$	0	0
$886$	−18255.6	−0.692224
$887$	−15674.0	−0.593328	−0.296664	−	0.954982i	$-0.595874\pi$
$887$	−15674.0	−0.593328	−0.296664	+	0.954982i	$0.595874\pi$
$888$	0	0
$889$	727.641	0.0274514
$890$	−47161.8	−1.77626
$891$	0	0
$892$	−2075.31	−0.0778998
$893$	−70984.7	−2.66004
$894$	0	0
$895$	26228.5	0.979577
$896$	29715.4	1.10795
$897$	0	0
$898$	8037.25	0.298671
$899$	1972.48	0.0731766
$900$	0	0
$901$	−20203.1	−0.747017
$902$	17175.5	0.634015
$903$	0	0
$904$	−29178.0	−1.07350
$905$	8002.15	0.293923
$906$	0	0
$907$	13824.9	0.506116	0.253058	−	0.967451i	$-0.418564\pi$
$907$	13824.9	0.506116	0.253058	+	0.967451i	$0.418564\pi$
$908$	1468.51	0.0536721
$909$	0	0
$910$	25.8421	0.000941380 0
$911$	4704.80	0.171105	0.0855527	−	0.996334i	$-0.472734\pi$
$911$	4704.80	0.171105	0.0855527	+	0.996334i	$0.472734\pi$
$912$	0	0
$913$	−7924.01	−0.287236
$914$	34318.8	1.24197
$915$	0	0
$916$	1615.00	0.0582545
$917$	−13985.0	−0.503627
$918$	0	0
$919$	−29892.6	−1.07298	−0.536488	−	0.843908i	$-0.680249\pi$
$919$	−29892.6	−1.07298	−0.536488	+	0.843908i	$0.680249\pi$
$920$	−24767.3	−0.887559
$921$	0	0
$922$	−25124.3	−0.897423
$923$	34.1488	0.00121779
$924$	0	0
$925$	−5584.40	−0.198501
$926$	24121.6	0.856031
$927$	0	0
$928$	−2144.54	−0.0758598
$929$	−20792.2	−0.734307	−0.367154	−	0.930160i	$-0.619668\pi$
$929$	−20792.2	−0.734307	−0.367154	+	0.930160i	$0.619668\pi$
$930$	0	0
$931$	2871.25	0.101076
$932$	−6072.73	−0.213432
$933$	0	0
$934$	59643.1	2.08949
$935$	14070.2	0.492132
$936$	0	0
$937$	42229.8	1.47234	0.736172	−	0.676794i	$-0.236631\pi$
$937$	42229.8	1.47234	0.736172	+	0.676794i	$0.236631\pi$
$938$	−20951.6	−0.729312
$939$	0	0
$940$	−6182.35	−0.214517
$941$	14057.5	0.486993	0.243496	−	0.969902i	$-0.421706\pi$
$941$	14057.5	0.486993	0.243496	+	0.969902i	$0.421706\pi$
$942$	0	0
$943$	−19374.3	−0.669051
$944$	30487.7	1.05115
$945$	0	0
$946$	24791.8	0.852063
$947$	−10953.9	−0.375875	−0.187937	−	0.982181i	$-0.560180\pi$
$947$	−10953.9	−0.375875	−0.187937	+	0.982181i	$0.560180\pi$
$948$	0	0
$949$	47.5573	0.00162674
$950$	−8630.25	−0.294739
$951$	0	0
$952$	15079.3	0.513364
$953$	−13364.2	−0.454260	−0.227130	−	0.973864i	$-0.572934\pi$
$953$	−13364.2	−0.454260	−0.227130	+	0.973864i	$0.572934\pi$
$954$	0	0
$955$	19008.3	0.644077
$956$	247.053	0.00835801
$957$	0	0
$958$	−12438.0	−0.419472
$959$	−42724.0	−1.43861
$960$	0	0
$961$	−27961.9	−0.938603
$962$	33.4022	0.00111947
$963$	0	0
$964$	146.566	0.00489686
$965$	−14074.9	−0.469519
$966$	0	0
$967$	49843.6	1.65756	0.828782	−	0.559572i	$-0.189035\pi$
$967$	49843.6	1.65756	0.828782	+	0.559572i	$0.189035\pi$
$968$	−2612.91	−0.0867583
$969$	0	0
$970$	−23363.8	−0.773367
$971$	−50679.0	−1.67494	−0.837469	−	0.546484i	$-0.815966\pi$
$971$	−50679.0	−1.67494	−0.837469	+	0.546484i	$0.815966\pi$
$972$	0	0
$973$	−14793.4	−0.487414
$974$	−20380.9	−0.670478
$975$	0	0
$976$	−64370.4	−2.11111
$977$	−16969.0	−0.555666	−0.277833	−	0.960629i	$-0.589616\pi$
$977$	−16969.0	−0.555666	−0.277833	+	0.960629i	$0.589616\pi$
$978$	0	0
$979$	−54248.3	−1.77097
$980$	250.069	0.00815118
$981$	0	0
$982$	−6403.77	−0.208098
$983$	31703.3	1.02867	0.514333	−	0.857591i	$-0.328040\pi$
$983$	31703.3	1.02867	0.514333	+	0.857591i	$0.328040\pi$
$984$	0	0
$985$	23743.8	0.768063
$986$	−5590.27	−0.180558
$987$	0	0
$988$	5.90676	0.000190201 0
$989$	−27965.7	−0.899148
$990$	0	0
$991$	4833.41	0.154933	0.0774664	−	0.996995i	$-0.475317\pi$
$991$	4833.41	0.154933	0.0774664	+	0.996995i	$0.475317\pi$
$992$	−1988.64	−0.0636486
$993$	0	0
$994$	−38245.1	−1.22038
$995$	−27293.7	−0.869615
$996$	0	0
$997$	31094.4	0.987734	0.493867	−	0.869538i	$-0.335583\pi$
$997$	31094.4	0.987734	0.493867	+	0.869538i	$0.335583\pi$
$998$	28962.7	0.918635
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.c.1.7		28
3.2	odd	2		717.4.a.a.1.22	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.a.1.22	✓	28	3.2	odd	2
2151.4.a.c.1.7		28	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)

\(f(q)\)	\(=\)	\(q-3.00561 q^{2} +1.03369 q^{4} -10.0458 q^{5} -17.8583 q^{7} +20.9380 q^{8} +O(q^{10})\)	\(q-3.00561 q^{2} +1.03369 q^{4} -10.0458 q^{5} -17.8583 q^{7} +20.9380 q^{8} +30.1936 q^{10} +34.7305 q^{11} -0.0479261 q^{13} +53.6751 q^{14} -71.2010 q^{16} -40.3279 q^{17} -119.230 q^{19} -10.3842 q^{20} -104.386 q^{22} +117.750 q^{23} -24.0827 q^{25} +0.144047 q^{26} -18.4600 q^{28} -46.1206 q^{29} -42.7678 q^{31} +46.4985 q^{32} +121.210 q^{34} +179.400 q^{35} +231.884 q^{37} +358.359 q^{38} -210.338 q^{40} -164.538 q^{41} -237.501 q^{43} +35.9007 q^{44} -353.911 q^{46} +595.359 q^{47} -24.0816 q^{49} +72.3832 q^{50} -0.0495409 q^{52} +500.970 q^{53} -348.894 q^{55} -373.917 q^{56} +138.620 q^{58} -428.191 q^{59} +904.066 q^{61} +128.543 q^{62} +429.852 q^{64} +0.481454 q^{65} -390.342 q^{67} -41.6867 q^{68} -539.207 q^{70} -712.531 q^{71} -992.306 q^{73} -696.953 q^{74} -123.247 q^{76} -620.227 q^{77} -1108.82 q^{79} +715.268 q^{80} +494.536 q^{82} -228.157 q^{83} +405.124 q^{85} +713.834 q^{86} +727.187 q^{88} -1561.98 q^{89} +0.855878 q^{91} +121.717 q^{92} -1789.42 q^{94} +1197.76 q^{95} -773.799 q^{97} +72.3798 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * q^98

Introduction

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