LMFDB - Embedded newform 2151.4.a.g.1.59

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:	$59$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.59
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)

$f(q)$	$=$	$q+5.26507 q^{2} +19.7210 q^{4} -20.1774 q^{5} -13.2314 q^{7} +61.7120 q^{8} +O(q^{10})$	\(q+5.26507 q^{2} +19.7210 q^{4} -20.1774 q^{5} -13.2314 q^{7} +61.7120 q^{8} -106.236 q^{10} +68.0955 q^{11} +33.3968 q^{13} -69.6642 q^{14} +167.150 q^{16} -135.241 q^{17} -37.9908 q^{19} -397.920 q^{20} +358.528 q^{22} -177.852 q^{23} +282.129 q^{25} +175.837 q^{26} -260.936 q^{28} +141.413 q^{29} +105.071 q^{31} +386.362 q^{32} -712.054 q^{34} +266.975 q^{35} +271.247 q^{37} -200.024 q^{38} -1245.19 q^{40} -460.303 q^{41} -187.115 q^{43} +1342.91 q^{44} -936.406 q^{46} -150.225 q^{47} -167.931 q^{49} +1485.43 q^{50} +658.618 q^{52} -92.8507 q^{53} -1373.99 q^{55} -816.534 q^{56} +744.551 q^{58} +171.792 q^{59} -835.237 q^{61} +553.207 q^{62} +697.024 q^{64} -673.862 q^{65} -311.402 q^{67} -2667.09 q^{68} +1405.64 q^{70} -525.740 q^{71} -90.5209 q^{73} +1428.14 q^{74} -749.217 q^{76} -900.996 q^{77} -333.645 q^{79} -3372.66 q^{80} -2423.53 q^{82} -734.292 q^{83} +2728.82 q^{85} -985.175 q^{86} +4202.31 q^{88} -741.275 q^{89} -441.885 q^{91} -3507.43 q^{92} -790.945 q^{94} +766.557 q^{95} -597.767 q^{97} -884.168 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * 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 $

$2$	5.26507	1.86148	0.930742	−	0.365675i	$-0.119162\pi$
$2$	5.26507	1.86148	0.930742	+	0.365675i	$0.119162\pi$
$3$	0	0
$3$	0	0
$4$	19.7210	2.46513
$5$	−20.1774	−1.80473	−0.902363	−	0.430977i	$-0.858169\pi$
$5$	−20.1774	−1.80473	−0.902363	+	0.430977i	$0.858169\pi$
$6$	0	0
$7$	−13.2314	−0.714427	−0.357213	−	0.934023i	$-0.616273\pi$
$7$	−13.2314	−0.714427	−0.357213	+	0.934023i	$0.616273\pi$
$8$	61.7120	2.72731
$9$	0	0
$10$	−106.236	−3.35947
$11$	68.0955	1.86651	0.933253	−	0.359221i	$-0.116958\pi$
$11$	68.0955	1.86651	0.933253	+	0.359221i	$0.116958\pi$
$12$	0	0
$13$	33.3968	0.712508	0.356254	−	0.934389i	$-0.384054\pi$
$13$	33.3968	0.712508	0.356254	+	0.934389i	$0.384054\pi$
$14$	−69.6642	−1.32989
$15$	0	0
$16$	167.150	2.61172
$17$	−135.241	−1.92946	−0.964729	−	0.263246i	$-0.915207\pi$
$17$	−135.241	−1.92946	−0.964729	+	0.263246i	$0.915207\pi$
$18$	0	0
$19$	−37.9908	−0.458720	−0.229360	−	0.973342i	$-0.573663\pi$
$19$	−37.9908	−0.458720	−0.229360	+	0.973342i	$0.573663\pi$
$20$	−397.920	−4.44888
$21$	0	0
$22$	358.528	3.47447
$23$	−177.852	−1.61238	−0.806191	−	0.591655i	$-0.798475\pi$
$23$	−177.852	−1.61238	−0.806191	+	0.591655i	$0.798475\pi$
$24$	0	0
$25$	282.129	2.25703
$26$	175.837	1.32632
$27$	0	0
$28$	−260.936	−1.76115
$29$	141.413	0.905510	0.452755	−	0.891635i	$-0.350441\pi$
$29$	141.413	0.905510	0.452755	+	0.891635i	$0.350441\pi$
$30$	0	0
$31$	105.071	0.608752	0.304376	−	0.952552i	$-0.401552\pi$
$31$	105.071	0.608752	0.304376	+	0.952552i	$0.401552\pi$
$32$	386.362	2.13437
$33$	0	0
$34$	−712.054	−3.59166
$35$	266.975	1.28934
$36$	0	0
$37$	271.247	1.20521	0.602604	−	0.798040i	$-0.294130\pi$
$37$	271.247	1.20521	0.602604	+	0.798040i	$0.294130\pi$
$38$	−200.024	−0.853901
$39$	0	0
$40$	−1245.19	−4.92205
$41$	−460.303	−1.75335	−0.876673	−	0.481086i	$-0.840242\pi$
$41$	−460.303	−1.75335	−0.876673	+	0.481086i	$0.840242\pi$
$42$	0	0
$43$	−187.115	−0.663600	−0.331800	−	0.943350i	$-0.607656\pi$
$43$	−187.115	−0.663600	−0.331800	+	0.943350i	$0.607656\pi$
$44$	1342.91	4.60117
$45$	0	0
$46$	−936.406	−3.00143
$47$	−150.225	−0.466224	−0.233112	−	0.972450i	$-0.574891\pi$
$47$	−150.225	−0.466224	−0.233112	+	0.972450i	$0.574891\pi$
$48$	0	0
$49$	−167.931	−0.489594
$50$	1485.43	4.20144
$51$	0	0
$52$	658.618	1.75642
$53$	−92.8507	−0.240642	−0.120321	−	0.992735i	$-0.538392\pi$
$53$	−92.8507	−0.240642	−0.120321	+	0.992735i	$0.538392\pi$
$54$	0	0
$55$	−1373.99	−3.36853
$56$	−816.534	−1.94846
$57$	0	0
$58$	744.551	1.68559
$59$	171.792	0.379075	0.189537	−	0.981874i	$-0.439301\pi$
$59$	171.792	0.379075	0.189537	+	0.981874i	$0.439301\pi$
$60$	0	0
$61$	−835.237	−1.75313	−0.876567	−	0.481281i	$-0.840172\pi$
$61$	−835.237	−1.75313	−0.876567	+	0.481281i	$0.840172\pi$
$62$	553.207	1.13318
$63$	0	0
$64$	697.024	1.36137
$65$	−673.862	−1.28588
$66$	0	0
$67$	−311.402	−0.567819	−0.283909	−	0.958851i	$-0.591631\pi$
$67$	−311.402	−0.567819	−0.283909	+	0.958851i	$0.591631\pi$
$68$	−2667.09	−4.75636
$69$	0	0
$70$	1405.64	2.40010
$71$	−525.740	−0.878786	−0.439393	−	0.898295i	$-0.644807\pi$
$71$	−525.740	−0.878786	−0.439393	+	0.898295i	$0.644807\pi$
$72$	0	0
$73$	−90.5209	−0.145132	−0.0725662	−	0.997364i	$-0.523119\pi$
$73$	−90.5209	−0.145132	−0.0725662	+	0.997364i	$0.523119\pi$
$74$	1428.14	2.24348
$75$	0	0
$76$	−749.217	−1.13080
$77$	−900.996	−1.33348
$78$	0	0
$79$	−333.645	−0.475164	−0.237582	−	0.971367i	$-0.576355\pi$
$79$	−333.645	−0.475164	−0.237582	+	0.971367i	$0.576355\pi$
$80$	−3372.66	−4.71344
$81$	0	0
$82$	−2423.53	−3.26383
$83$	−734.292	−0.971072	−0.485536	−	0.874217i	$-0.661376\pi$
$83$	−734.292	−0.971072	−0.485536	+	0.874217i	$0.661376\pi$
$84$	0	0
$85$	2728.82	3.48214
$86$	−985.175	−1.23528
$87$	0	0
$88$	4202.31	5.09054
$89$	−741.275	−0.882865	−0.441432	−	0.897295i	$-0.645529\pi$
$89$	−741.275	−0.882865	−0.441432	+	0.897295i	$0.645529\pi$
$90$	0	0
$91$	−441.885	−0.509035
$92$	−3507.43	−3.97473
$93$	0	0
$94$	−790.945	−0.867869
$95$	766.557	0.827864
$96$	0	0
$97$	−597.767	−0.625712	−0.312856	−	0.949801i	$-0.601286\pi$
$97$	−597.767	−0.625712	−0.312856	+	0.949801i	$0.601286\pi$
$98$	−884.168	−0.911372
$99$	0	0
$100$	5563.87	5.56387
$101$	−333.618	−0.328675	−0.164338	−	0.986404i	$-0.552549\pi$
$101$	−333.618	−0.328675	−0.164338	+	0.986404i	$0.552549\pi$
$102$	0	0
$103$	580.062	0.554905	0.277453	−	0.960739i	$-0.410510\pi$
$103$	580.062	0.554905	0.277453	+	0.960739i	$0.410510\pi$
$104$	2060.98	1.94323
$105$	0	0
$106$	−488.866	−0.447951
$107$	1481.03	1.33810	0.669050	−	0.743217i	$-0.266701\pi$
$107$	1481.03	1.33810	0.669050	+	0.743217i	$0.266701\pi$
$108$	0	0
$109$	−1885.54	−1.65690	−0.828451	−	0.560061i	$-0.810778\pi$
$109$	−1885.54	−1.65690	−0.828451	+	0.560061i	$0.810778\pi$
$110$	−7234.17	−6.27047
$111$	0	0
$112$	−2211.63	−1.86588
$113$	1100.64	0.916276	0.458138	−	0.888881i	$-0.348517\pi$
$113$	1100.64	0.916276	0.458138	+	0.888881i	$0.348517\pi$
$114$	0	0
$115$	3588.61	2.90991
$116$	2788.81	2.23220
$117$	0	0
$118$	904.498	0.705642
$119$	1789.42	1.37846
$120$	0	0
$121$	3305.99	2.48384
$122$	−4397.58	−3.26343
$123$	0	0
$124$	2072.11	1.50065
$125$	−3170.47	−2.26860
$126$	0	0
$127$	608.505	0.425166	0.212583	−	0.977143i	$-0.431812\pi$
$127$	608.505	0.425166	0.212583	+	0.977143i	$0.431812\pi$
$128$	578.986	0.399809
$129$	0	0
$130$	−3547.93	−2.39365
$131$	741.290	0.494403	0.247201	−	0.968964i	$-0.420489\pi$
$131$	741.290	0.494403	0.247201	+	0.968964i	$0.420489\pi$
$132$	0	0
$133$	502.670	0.327722
$134$	−1639.56	−1.05699
$135$	0	0
$136$	−8345.99	−5.26223
$137$	−46.8439	−0.0292127	−0.0146064	−	0.999893i	$-0.504650\pi$
$137$	−46.8439	−0.0292127	−0.0146064	+	0.999893i	$0.504650\pi$
$138$	0	0
$139$	−1210.76	−0.738815	−0.369408	−	0.929267i	$-0.620439\pi$
$139$	−1210.76	−0.738815	−0.369408	+	0.929267i	$0.620439\pi$
$140$	5265.02	3.17840
$141$	0	0
$142$	−2768.06	−1.63585
$143$	2274.17	1.32990
$144$	0	0
$145$	−2853.36	−1.63420
$146$	−476.599	−0.270162
$147$	0	0
$148$	5349.26	2.97099
$149$	−2779.16	−1.52804	−0.764019	−	0.645193i	$-0.776777\pi$
$149$	−2779.16	−1.52804	−0.764019	+	0.645193i	$0.776777\pi$
$150$	0	0
$151$	−1613.32	−0.869471	−0.434736	−	0.900558i	$-0.643158\pi$
$151$	−1613.32	−0.869471	−0.434736	+	0.900558i	$0.643158\pi$
$152$	−2344.49	−1.25107
$153$	0	0
$154$	−4743.81	−2.48226
$155$	−2120.06	−1.09863
$156$	0	0
$157$	−990.570	−0.503542	−0.251771	−	0.967787i	$-0.581013\pi$
$157$	−990.570	−0.503542	−0.251771	+	0.967787i	$0.581013\pi$
$158$	−1756.67	−0.884511
$159$	0	0
$160$	−7795.80	−3.85195
$161$	2353.23	1.15193
$162$	0	0
$163$	1548.67	0.744180	0.372090	−	0.928197i	$-0.378641\pi$
$163$	1548.67	0.744180	0.372090	+	0.928197i	$0.378641\pi$
$164$	−9077.63	−4.32222
$165$	0	0
$166$	−3866.10	−1.80764
$167$	1299.02	0.601923	0.300961	−	0.953636i	$-0.402692\pi$
$167$	1299.02	0.601923	0.300961	+	0.953636i	$0.402692\pi$
$168$	0	0
$169$	−1081.65	−0.492333
$170$	14367.4	6.48195
$171$	0	0
$172$	−3690.10	−1.63586
$173$	363.132	0.159586	0.0797930	−	0.996811i	$-0.474574\pi$
$173$	363.132	0.159586	0.0797930	+	0.996811i	$0.474574\pi$
$174$	0	0
$175$	−3732.96	−1.61249
$176$	11382.2	4.87479
$177$	0	0
$178$	−3902.87	−1.64344
$179$	−600.492	−0.250742	−0.125371	−	0.992110i	$-0.540012\pi$
$179$	−600.492	−0.250742	−0.125371	+	0.992110i	$0.540012\pi$
$180$	0	0
$181$	3425.37	1.40666	0.703331	−	0.710863i	$-0.251695\pi$
$181$	3425.37	1.40666	0.703331	+	0.710863i	$0.251695\pi$
$182$	−2326.56	−0.947560
$183$	0	0
$184$	−10975.6	−4.39747
$185$	−5473.07	−2.17507
$186$	0	0
$187$	−9209.30	−3.60134
$188$	−2962.58	−1.14930
$189$	0	0
$190$	4035.98	1.54106
$191$	−885.654	−0.335517	−0.167758	−	0.985828i	$-0.553653\pi$
$191$	−885.654	−0.335517	−0.167758	+	0.985828i	$0.553653\pi$
$192$	0	0
$193$	−1354.18	−0.505056	−0.252528	−	0.967590i	$-0.581262\pi$
$193$	−1354.18	−0.505056	−0.252528	+	0.967590i	$0.581262\pi$
$194$	−3147.29	−1.16475
$195$	0	0
$196$	−3311.77	−1.20691
$197$	1095.00	0.396019	0.198009	−	0.980200i	$-0.436552\pi$
$197$	1095.00	0.396019	0.198009	+	0.980200i	$0.436552\pi$
$198$	0	0
$199$	3997.52	1.42400	0.712002	−	0.702178i	$-0.247789\pi$
$199$	3997.52	1.42400	0.712002	+	0.702178i	$0.247789\pi$
$200$	17410.8	6.15563
$201$	0	0
$202$	−1756.52	−0.611824
$203$	−1871.09	−0.646921
$204$	0	0
$205$	9287.73	3.16431
$206$	3054.07	1.03295
$207$	0	0
$208$	5582.28	1.86087
$209$	−2587.00	−0.856204
$210$	0	0
$211$	2251.18	0.734492	0.367246	−	0.930124i	$-0.380301\pi$
$211$	2251.18	0.734492	0.367246	+	0.930124i	$0.380301\pi$
$212$	−1831.11	−0.593213
$213$	0	0
$214$	7797.74	2.49085
$215$	3775.51	1.19762
$216$	0	0
$217$	−1390.23	−0.434909
$218$	−9927.53	−3.08430
$219$	0	0
$220$	−27096.5	−8.30385
$221$	−4516.62	−1.37475
$222$	0	0
$223$	2793.48	0.838858	0.419429	−	0.907788i	$-0.362230\pi$
$223$	2793.48	0.838858	0.419429	+	0.907788i	$0.362230\pi$
$224$	−5112.10	−1.52485
$225$	0	0
$226$	5794.93	1.70563
$227$	−6450.38	−1.88602	−0.943010	−	0.332764i	$-0.892019\pi$
$227$	−6450.38	−1.88602	−0.943010	+	0.332764i	$0.892019\pi$
$228$	0	0
$229$	722.752	0.208562	0.104281	−	0.994548i	$-0.466746\pi$
$229$	722.752	0.208562	0.104281	+	0.994548i	$0.466746\pi$
$230$	18894.3	5.41675
$231$	0	0
$232$	8726.89	2.46961
$233$	3388.50	0.952738	0.476369	−	0.879245i	$-0.341953\pi$
$233$	3388.50	0.952738	0.476369	+	0.879245i	$0.341953\pi$
$234$	0	0
$235$	3031.15	0.841407
$236$	3387.91	0.934467
$237$	0	0
$238$	9421.45	2.56598
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	6482.01	1.73254	0.866272	−	0.499573i	$-0.166510\pi$
$241$	6482.01	1.73254	0.866272	+	0.499573i	$0.166510\pi$
$242$	17406.3	4.62363
$243$	0	0
$244$	−16471.7	−4.32169
$245$	3388.41	0.883583
$246$	0	0
$247$	−1268.77	−0.326842
$248$	6484.14	1.66026
$249$	0	0
$250$	−16692.7	−4.22297
$251$	−2013.62	−0.506370	−0.253185	−	0.967418i	$-0.581478\pi$
$251$	−2013.62	−0.506370	−0.253185	+	0.967418i	$0.581478\pi$
$252$	0	0
$253$	−12110.9	−3.00952
$254$	3203.82	0.791440
$255$	0	0
$256$	−2527.79	−0.617136
$257$	−1280.22	−0.310731	−0.155365	−	0.987857i	$-0.549655\pi$
$257$	−1280.22	−0.310731	−0.155365	+	0.987857i	$0.549655\pi$
$258$	0	0
$259$	−3588.97	−0.861034
$260$	−13289.2	−3.16986
$261$	0	0
$262$	3902.95	0.920324
$263$	−7629.58	−1.78882	−0.894411	−	0.447245i	$-0.852405\pi$
$263$	−7629.58	−1.78882	−0.894411	+	0.447245i	$0.852405\pi$
$264$	0	0
$265$	1873.49	0.434293
$266$	2646.60	0.610050
$267$	0	0
$268$	−6141.17	−1.39974
$269$	−3449.88	−0.781944	−0.390972	−	0.920403i	$-0.627861\pi$
$269$	−3449.88	−0.781944	−0.390972	+	0.920403i	$0.627861\pi$
$270$	0	0
$271$	409.007	0.0916805	0.0458403	−	0.998949i	$-0.485403\pi$
$271$	409.007	0.0916805	0.0458403	+	0.998949i	$0.485403\pi$
$272$	−22605.6	−5.03920
$273$	0	0
$274$	−246.636	−0.0543790
$275$	19211.7	4.21277
$276$	0	0
$277$	−2697.20	−0.585052	−0.292526	−	0.956258i	$-0.594496\pi$
$277$	−2697.20	−0.585052	−0.292526	+	0.956258i	$0.594496\pi$
$278$	−6374.74	−1.37529
$279$	0	0
$280$	16475.6	3.51644
$281$	−2747.53	−0.583288	−0.291644	−	0.956527i	$-0.594202\pi$
$281$	−2747.53	−0.583288	−0.291644	+	0.956527i	$0.594202\pi$
$282$	0	0
$283$	−5880.65	−1.23522	−0.617612	−	0.786483i	$-0.711900\pi$
$283$	−5880.65	−1.23522	−0.617612	+	0.786483i	$0.711900\pi$
$284$	−10368.1	−2.16632
$285$	0	0
$286$	11973.7	2.47559
$287$	6090.44	1.25264
$288$	0	0
$289$	13377.1	2.72281
$290$	−15023.1	−3.04203
$291$	0	0
$292$	−1785.16	−0.357770
$293$	8311.20	1.65715	0.828576	−	0.559877i	$-0.189151\pi$
$293$	8311.20	1.65715	0.828576	+	0.559877i	$0.189151\pi$
$294$	0	0
$295$	−3466.32	−0.684126
$296$	16739.2	3.28698
$297$	0	0
$298$	−14632.5	−2.84442
$299$	−5939.70	−1.14884
$300$	0	0
$301$	2475.79	0.474094
$302$	−8494.25	−1.61851
$303$	0	0
$304$	−6350.17	−1.19805
$305$	16852.9	3.16392
$306$	0	0
$307$	2796.23	0.519834	0.259917	−	0.965631i	$-0.416305\pi$
$307$	2796.23	0.519834	0.259917	+	0.965631i	$0.416305\pi$
$308$	−17768.6	−3.28720
$309$	0	0
$310$	−11162.3	−2.04508
$311$	7690.57	1.40222	0.701112	−	0.713051i	$-0.252687\pi$
$311$	7690.57	1.40222	0.701112	+	0.713051i	$0.252687\pi$
$312$	0	0
$313$	9300.22	1.67949	0.839743	−	0.542983i	$-0.182705\pi$
$313$	9300.22	1.67949	0.839743	+	0.542983i	$0.182705\pi$
$314$	−5215.42	−0.937336
$315$	0	0
$316$	−6579.81	−1.17134
$317$	−428.900	−0.0759919	−0.0379959	−	0.999278i	$-0.512097\pi$
$317$	−428.900	−0.0759919	−0.0379959	+	0.999278i	$0.512097\pi$
$318$	0	0
$319$	9629.60	1.69014
$320$	−14064.2	−2.45691
$321$	0	0
$322$	12389.9	2.14430
$323$	5137.91	0.885081
$324$	0	0
$325$	9422.21	1.60815
$326$	8153.87	1.38528
$327$	0	0
$328$	−28406.2	−4.78192
$329$	1987.68	0.333083
$330$	0	0
$331$	−8728.76	−1.44947	−0.724737	−	0.689025i	$-0.758039\pi$
$331$	−8728.76	−1.44947	−0.724737	+	0.689025i	$0.758039\pi$
$332$	−14481.0	−2.39382
$333$	0	0
$334$	6839.43	1.12047
$335$	6283.30	1.02476
$336$	0	0
$337$	11015.3	1.78053	0.890266	−	0.455442i	$-0.150519\pi$
$337$	11015.3	1.78053	0.890266	+	0.455442i	$0.150519\pi$
$338$	−5694.99	−0.916470
$339$	0	0
$340$	53815.1	8.58392
$341$	7154.86	1.13624
$342$	0	0
$343$	6760.32	1.06421
$344$	−11547.2	−1.80984
$345$	0	0
$346$	1911.92	0.297067
$347$	−7148.46	−1.10591	−0.552953	−	0.833212i	$-0.686499\pi$
$347$	−7148.46	−1.10591	−0.552953	+	0.833212i	$0.686499\pi$
$348$	0	0
$349$	3630.38	0.556818	0.278409	−	0.960463i	$-0.410193\pi$
$349$	3630.38	0.556818	0.278409	+	0.960463i	$0.410193\pi$
$350$	−19654.3	−3.00162
$351$	0	0
$352$	26309.5	3.98381
$353$	−3706.20	−0.558814	−0.279407	−	0.960173i	$-0.590138\pi$
$353$	−3706.20	−0.558814	−0.279407	+	0.960173i	$0.590138\pi$
$354$	0	0
$355$	10608.1	1.58597
$356$	−14618.7	−2.17637
$357$	0	0
$358$	−3161.63	−0.466753
$359$	5691.48	0.836727	0.418363	−	0.908280i	$-0.362604\pi$
$359$	5691.48	0.836727	0.418363	+	0.908280i	$0.362604\pi$
$360$	0	0
$361$	−5415.70	−0.789576
$362$	18034.8	2.61848
$363$	0	0
$364$	−8714.42	−1.25483
$365$	1826.48	0.261924
$366$	0	0
$367$	1962.63	0.279151	0.139575	−	0.990211i	$-0.455426\pi$
$367$	1962.63	0.279151	0.139575	+	0.990211i	$0.455426\pi$
$368$	−29728.1	−4.21109
$369$	0	0
$370$	−28816.1	−4.04886
$371$	1228.54	0.171921
$372$	0	0
$373$	1605.24	0.222832	0.111416	−	0.993774i	$-0.464461\pi$
$373$	1605.24	0.222832	0.111416	+	0.993774i	$0.464461\pi$
$374$	−48487.7	−6.70384
$375$	0	0
$376$	−9270.67	−1.27154
$377$	4722.75	0.645183
$378$	0	0
$379$	−3963.46	−0.537175	−0.268588	−	0.963255i	$-0.586557\pi$
$379$	−3963.46	−0.537175	−0.268588	+	0.963255i	$0.586557\pi$
$380$	15117.3	2.04079
$381$	0	0
$382$	−4663.03	−0.624559
$383$	582.785	0.0777518	0.0388759	−	0.999244i	$-0.487622\pi$
$383$	582.785	0.0777518	0.0388759	+	0.999244i	$0.487622\pi$
$384$	0	0
$385$	18179.8	2.40657
$386$	−7129.85	−0.940155
$387$	0	0
$388$	−11788.6	−1.54246
$389$	−6856.77	−0.893707	−0.446853	−	0.894607i	$-0.647455\pi$
$389$	−6856.77	−0.893707	−0.446853	+	0.894607i	$0.647455\pi$
$390$	0	0
$391$	24053.0	3.11102
$392$	−10363.3	−1.33528
$393$	0	0
$394$	5765.27	0.737183
$395$	6732.10	0.857541
$396$	0	0
$397$	8288.31	1.04780	0.523902	−	0.851778i	$-0.324476\pi$
$397$	8288.31	1.04780	0.523902	+	0.851778i	$0.324476\pi$
$398$	21047.2	2.65076
$399$	0	0
$400$	47157.9	5.89474
$401$	−14917.7	−1.85774	−0.928871	−	0.370403i	$-0.879219\pi$
$401$	−14917.7	−1.85774	−0.928871	+	0.370403i	$0.879219\pi$
$402$	0	0
$403$	3509.03	0.433741
$404$	−6579.28	−0.810226
$405$	0	0
$406$	−9851.44	−1.20423
$407$	18470.7	2.24953
$408$	0	0
$409$	1787.00	0.216042	0.108021	−	0.994149i	$-0.465549\pi$
$409$	1787.00	0.216042	0.108021	+	0.994149i	$0.465549\pi$
$410$	48900.6	5.89031
$411$	0	0
$412$	11439.4	1.36791
$413$	−2273.04	−0.270821
$414$	0	0
$415$	14816.1	1.75252
$416$	12903.2	1.52075
$417$	0	0
$418$	−13620.8	−1.59381
$419$	12421.9	1.44833	0.724163	−	0.689628i	$-0.242226\pi$
$419$	12421.9	1.44833	0.724163	+	0.689628i	$0.242226\pi$
$420$	0	0
$421$	−7526.06	−0.871254	−0.435627	−	0.900127i	$-0.643473\pi$
$421$	−7526.06	−0.871254	−0.435627	+	0.900127i	$0.643473\pi$
$422$	11852.7	1.36725
$423$	0	0
$424$	−5730.00	−0.656305
$425$	−38155.5	−4.35485
$426$	0	0
$427$	11051.3	1.25249
$428$	29207.4	3.29859
$429$	0	0
$430$	19878.3	2.22934
$431$	−17221.9	−1.92471	−0.962357	−	0.271790i	$-0.912385\pi$
$431$	−17221.9	−1.92471	−0.962357	+	0.271790i	$0.912385\pi$
$432$	0	0
$433$	9694.62	1.07597	0.537984	−	0.842955i	$-0.319186\pi$
$433$	9694.62	1.07597	0.537984	+	0.842955i	$0.319186\pi$
$434$	−7319.68	−0.809576
$435$	0	0
$436$	−37184.8	−4.08447
$437$	6756.75	0.739633
$438$	0	0
$439$	666.978	0.0725128	0.0362564	−	0.999343i	$-0.488457\pi$
$439$	666.978	0.0725128	0.0362564	+	0.999343i	$0.488457\pi$
$440$	−84791.8	−9.18703
$441$	0	0
$442$	−23780.3	−2.55908
$443$	5208.79	0.558639	0.279319	−	0.960198i	$-0.409891\pi$
$443$	5208.79	0.558639	0.279319	+	0.960198i	$0.409891\pi$
$444$	0	0
$445$	14957.0	1.59333
$446$	14707.9	1.56152
$447$	0	0
$448$	−9222.58	−0.972602
$449$	10865.0	1.14198	0.570990	−	0.820957i	$-0.306559\pi$
$449$	10865.0	1.14198	0.570990	+	0.820957i	$0.306559\pi$
$450$	0	0
$451$	−31344.5	−3.27263
$452$	21705.7	2.25874
$453$	0	0
$454$	−33961.7	−3.51080
$455$	8916.12	0.918668
$456$	0	0
$457$	7334.58	0.750759	0.375380	−	0.926871i	$-0.377512\pi$
$457$	7334.58	0.750759	0.375380	+	0.926871i	$0.377512\pi$
$458$	3805.34	0.388236
$459$	0	0
$460$	70771.0	7.17329
$461$	3797.57	0.383667	0.191833	−	0.981428i	$-0.438557\pi$
$461$	3797.57	0.383667	0.191833	+	0.981428i	$0.438557\pi$
$462$	0	0
$463$	−12486.0	−1.25329	−0.626646	−	0.779304i	$-0.715573\pi$
$463$	−12486.0	−1.25329	−0.626646	+	0.779304i	$0.715573\pi$
$464$	23637.2	2.36494
$465$	0	0
$466$	17840.7	1.77351
$467$	−5250.91	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$467$	−5250.91	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$468$	0	0
$469$	4120.28	0.405665
$470$	15959.2	1.56627
$471$	0	0
$472$	10601.6	1.03385
$473$	−12741.7	−1.23861
$474$	0	0
$475$	−10718.3	−1.03535
$476$	35289.3	3.39807
$477$	0	0
$478$	1258.35	0.120409
$479$	−7322.21	−0.698456	−0.349228	−	0.937038i	$-0.613556\pi$
$479$	−7322.21	−0.698456	−0.349228	+	0.937038i	$0.613556\pi$
$480$	0	0
$481$	9058.78	0.858721
$482$	34128.3	3.22510
$483$	0	0
$484$	65197.5	6.12298
$485$	12061.4	1.12924
$486$	0	0
$487$	2735.05	0.254491	0.127246	−	0.991871i	$-0.459386\pi$
$487$	2735.05	0.254491	0.127246	+	0.991871i	$0.459386\pi$
$488$	−51544.1	−4.78134
$489$	0	0
$490$	17840.3	1.64478
$491$	6821.94	0.627026	0.313513	−	0.949584i	$-0.398494\pi$
$491$	6821.94	0.627026	0.313513	+	0.949584i	$0.398494\pi$
$492$	0	0
$493$	−19124.9	−1.74714
$494$	−6680.17	−0.608411
$495$	0	0
$496$	17562.6	1.58989
$497$	6956.26	0.627829
$498$	0	0
$499$	−15541.3	−1.39424	−0.697120	−	0.716954i	$-0.745536\pi$
$499$	−15541.3	−1.39424	−0.697120	+	0.716954i	$0.745536\pi$
$500$	−62524.8	−5.59239
$501$	0	0
$502$	−10601.9	−0.942600
$503$	−2490.11	−0.220733	−0.110366	−	0.993891i	$-0.535202\pi$
$503$	−2490.11	−0.220733	−0.110366	+	0.993891i	$0.535202\pi$
$504$	0	0
$505$	6731.55	0.593169
$506$	−63765.0	−5.60218
$507$	0	0
$508$	12000.3	1.04809
$509$	−4863.51	−0.423519	−0.211759	−	0.977322i	$-0.567919\pi$
$509$	−4863.51	−0.423519	−0.211759	+	0.977322i	$0.567919\pi$
$510$	0	0
$511$	1197.72	0.103687
$512$	−17940.9	−1.54860
$513$	0	0
$514$	−6740.44	−0.578420
$515$	−11704.2	−1.00145
$516$	0	0
$517$	−10229.6	−0.870210
$518$	−18896.2	−1.60280
$519$	0	0
$520$	−41585.3	−3.50700
$521$	280.964	0.0236262	0.0118131	−	0.999930i	$-0.496240\pi$
$521$	280.964	0.0236262	0.0118131	+	0.999930i	$0.496240\pi$
$522$	0	0
$523$	−16027.2	−1.34000	−0.669999	−	0.742362i	$-0.733706\pi$
$523$	−16027.2	−1.34000	−0.669999	+	0.742362i	$0.733706\pi$
$524$	14619.0	1.21877
$525$	0	0
$526$	−40170.3	−3.32987
$527$	−14209.9	−1.17456
$528$	0	0
$529$	19464.5	1.59978
$530$	9864.06	0.808429
$531$	0	0
$532$	9913.16	0.807876
$533$	−15372.6	−1.24927
$534$	0	0
$535$	−29883.4	−2.41490
$536$	−19217.3	−1.54862
$537$	0	0
$538$	−18163.9	−1.45558
$539$	−11435.3	−0.913830
$540$	0	0
$541$	16591.9	1.31856	0.659282	−	0.751896i	$-0.270860\pi$
$541$	16591.9	1.31856	0.659282	+	0.751896i	$0.270860\pi$
$542$	2153.45	0.170662
$543$	0	0
$544$	−52252.0	−4.11817
$545$	38045.5	2.99025
$546$	0	0
$547$	1767.79	0.138181	0.0690906	−	0.997610i	$-0.477990\pi$
$547$	1767.79	0.138181	0.0690906	+	0.997610i	$0.477990\pi$
$548$	−923.808	−0.0720130
$549$	0	0
$550$	101151.	7.84200
$551$	−5372.40	−0.415376
$552$	0	0
$553$	4414.58	0.339470
$554$	−14201.0	−1.08907
$555$	0	0
$556$	−23877.4	−1.82127
$557$	13963.8	1.06223	0.531117	−	0.847298i	$-0.321772\pi$
$557$	13963.8	1.06223	0.531117	+	0.847298i	$0.321772\pi$
$558$	0	0
$559$	−6249.04	−0.472820
$560$	44625.0	3.36741
$561$	0	0
$562$	−14466.0	−1.08578
$563$	−11204.3	−0.838727	−0.419364	−	0.907818i	$-0.637747\pi$
$563$	−11204.3	−0.838727	−0.419364	+	0.907818i	$0.637747\pi$
$564$	0	0
$565$	−22208.0	−1.65363
$566$	−30962.1	−2.29935
$567$	0	0
$568$	−32444.4	−2.39672
$569$	−17933.8	−1.32131	−0.660653	−	0.750691i	$-0.729721\pi$
$569$	−17933.8	−1.32131	−0.660653	+	0.750691i	$0.729721\pi$
$570$	0	0
$571$	3649.96	0.267506	0.133753	−	0.991015i	$-0.457297\pi$
$571$	3649.96	0.267506	0.133753	+	0.991015i	$0.457297\pi$
$572$	44848.9	3.27837
$573$	0	0
$574$	32066.6	2.33177
$575$	−50177.4	−3.63920
$576$	0	0
$577$	−10571.6	−0.762737	−0.381369	−	0.924423i	$-0.624547\pi$
$577$	−10571.6	−0.762737	−0.381369	+	0.924423i	$0.624547\pi$
$578$	70431.7	5.06846
$579$	0	0
$580$	−56271.1	−4.02850
$581$	9715.69	0.693760
$582$	0	0
$583$	−6322.71	−0.449159
$584$	−5586.22	−0.395821
$585$	0	0
$586$	43759.1	3.08476
$587$	−9237.12	−0.649500	−0.324750	−	0.945800i	$-0.605280\pi$
$587$	−9237.12	−0.649500	−0.324750	+	0.945800i	$0.605280\pi$
$588$	0	0
$589$	−3991.73	−0.279247
$590$	−18250.4	−1.27349
$591$	0	0
$592$	45339.0	3.14767
$593$	−15168.6	−1.05042	−0.525210	−	0.850973i	$-0.676013\pi$
$593$	−15168.6	−1.05042	−0.525210	+	0.850973i	$0.676013\pi$
$594$	0	0
$595$	−36106.0	−2.48773
$596$	−54807.9	−3.76681
$597$	0	0
$598$	−31273.0	−2.13854
$599$	21957.6	1.49777	0.748884	−	0.662701i	$-0.230590\pi$
$599$	21957.6	1.49777	0.748884	+	0.662701i	$0.230590\pi$
$600$	0	0
$601$	22860.6	1.55158	0.775792	−	0.630988i	$-0.217350\pi$
$601$	22860.6	1.55158	0.775792	+	0.630988i	$0.217350\pi$
$602$	13035.2	0.882518
$603$	0	0
$604$	−31816.3	−2.14336
$605$	−66706.5	−4.48265
$606$	0	0
$607$	17516.0	1.17126	0.585629	−	0.810580i	$-0.300848\pi$
$607$	17516.0	1.17126	0.585629	+	0.810580i	$0.300848\pi$
$608$	−14678.2	−0.979078
$609$	0	0
$610$	88732.0	5.88960
$611$	−5017.02	−0.332188
$612$	0	0
$613$	2369.01	0.156090	0.0780452	−	0.996950i	$-0.475132\pi$
$613$	2369.01	0.156090	0.0780452	+	0.996950i	$0.475132\pi$
$614$	14722.3	0.967664
$615$	0	0
$616$	−55602.3	−3.63682
$617$	18863.8	1.23084	0.615420	−	0.788199i	$-0.288986\pi$
$617$	18863.8	1.23084	0.615420	+	0.788199i	$0.288986\pi$
$618$	0	0
$619$	−25535.7	−1.65810	−0.829051	−	0.559173i	$-0.811119\pi$
$619$	−25535.7	−1.65810	−0.829051	+	0.559173i	$0.811119\pi$
$620$	−41809.8	−2.70826
$621$	0	0
$622$	40491.4	2.61022
$623$	9808.08	0.630742
$624$	0	0
$625$	28705.8	1.83717
$626$	48966.3	3.12634
$627$	0	0
$628$	−19535.0	−1.24129
$629$	−36683.7	−2.32540
$630$	0	0
$631$	26395.5	1.66528	0.832638	−	0.553818i	$-0.186829\pi$
$631$	26395.5	1.66528	0.832638	+	0.553818i	$0.186829\pi$
$632$	−20589.9	−1.29592
$633$	0	0
$634$	−2258.19	−0.141458
$635$	−12278.1	−0.767308
$636$	0	0
$637$	−5608.35	−0.348840
$638$	50700.6	3.14617
$639$	0	0
$640$	−11682.4	−0.721546
$641$	796.235	0.0490630	0.0245315	−	0.999699i	$-0.492191\pi$
$641$	796.235	0.0490630	0.0245315	+	0.999699i	$0.492191\pi$
$642$	0	0
$643$	28684.2	1.75924	0.879622	−	0.475674i	$-0.157796\pi$
$643$	28684.2	1.75924	0.879622	+	0.475674i	$0.157796\pi$
$644$	46408.1	2.83965
$645$	0	0
$646$	27051.5	1.64757
$647$	19433.6	1.18086	0.590429	−	0.807090i	$-0.298959\pi$
$647$	19433.6	1.18086	0.590429	+	0.807090i	$0.298959\pi$
$648$	0	0
$649$	11698.3	0.707545
$650$	49608.6	2.99356
$651$	0	0
$652$	30541.4	1.83450
$653$	12293.8	0.736744	0.368372	−	0.929678i	$-0.379915\pi$
$653$	12293.8	0.736744	0.368372	+	0.929678i	$0.379915\pi$
$654$	0	0
$655$	−14957.3	−0.892262
$656$	−76939.7	−4.57925
$657$	0	0
$658$	10465.3	0.620029
$659$	−4962.98	−0.293369	−0.146685	−	0.989183i	$-0.546860\pi$
$659$	−4962.98	−0.293369	−0.146685	+	0.989183i	$0.546860\pi$
$660$	0	0
$661$	2665.38	0.156840	0.0784200	−	0.996920i	$-0.475012\pi$
$661$	2665.38	0.156840	0.0784200	+	0.996920i	$0.475012\pi$
$662$	−45957.6	−2.69817
$663$	0	0
$664$	−45314.6	−2.64842
$665$	−10142.6	−0.591448
$666$	0	0
$667$	−25150.7	−1.46003
$668$	25618.0	1.48382
$669$	0	0
$670$	33082.1	1.90757
$671$	−56875.8	−3.27223
$672$	0	0
$673$	−14281.1	−0.817972	−0.408986	−	0.912541i	$-0.634118\pi$
$673$	−14281.1	−0.817972	−0.408986	+	0.912541i	$0.634118\pi$
$674$	57996.1	3.31443
$675$	0	0
$676$	−21331.3	−1.21366
$677$	10640.4	0.604054	0.302027	−	0.953299i	$-0.402337\pi$
$677$	10640.4	0.604054	0.302027	+	0.953299i	$0.402337\pi$
$678$	0	0
$679$	7909.28	0.447025
$680$	168401.	9.49688
$681$	0	0
$682$	37670.9	2.11509
$683$	5985.39	0.335322	0.167661	−	0.985845i	$-0.446379\pi$
$683$	5985.39	0.335322	0.167661	+	0.985845i	$0.446379\pi$
$684$	0	0
$685$	945.189	0.0527209
$686$	35593.6	1.98100
$687$	0	0
$688$	−31276.3	−1.73314
$689$	−3100.91	−0.171459
$690$	0	0
$691$	6136.31	0.337824	0.168912	−	0.985631i	$-0.445975\pi$
$691$	6136.31	0.337824	0.168912	+	0.985631i	$0.445975\pi$
$692$	7161.32	0.393400
$693$	0	0
$694$	−37637.2	−2.05863
$695$	24430.0	1.33336
$696$	0	0
$697$	62251.8	3.38301
$698$	19114.2	1.03651
$699$	0	0
$700$	−73617.7	−3.97498
$701$	−3702.99	−0.199515	−0.0997576	−	0.995012i	$-0.531807\pi$
$701$	−3702.99	−0.199515	−0.0997576	+	0.995012i	$0.531807\pi$
$702$	0	0
$703$	−10304.9	−0.552854
$704$	47464.2	2.54101
$705$	0	0
$706$	−19513.4	−1.04022
$707$	4414.22	0.234814
$708$	0	0
$709$	22752.4	1.20520	0.602598	−	0.798045i	$-0.294132\pi$
$709$	22752.4	1.20520	0.602598	+	0.798045i	$0.294132\pi$
$710$	55852.4	2.95226
$711$	0	0
$712$	−45745.5	−2.40785
$713$	−18687.1	−0.981541
$714$	0	0
$715$	−45886.9	−2.40010
$716$	−11842.3	−0.618111
$717$	0	0
$718$	29966.1	1.55755
$719$	−31170.0	−1.61675	−0.808376	−	0.588666i	$-0.799653\pi$
$719$	−31170.0	−1.61675	−0.808376	+	0.588666i	$0.799653\pi$
$720$	0	0
$721$	−7675.02	−0.396439
$722$	−28514.1	−1.46978
$723$	0	0
$724$	67551.7	3.46760
$725$	39896.8	2.04377
$726$	0	0
$727$	−9753.18	−0.497559	−0.248780	−	0.968560i	$-0.580029\pi$
$727$	−9753.18	−0.497559	−0.248780	+	0.968560i	$0.580029\pi$
$728$	−27269.6	−1.38830
$729$	0	0
$730$	9616.55	0.487568
$731$	25305.6	1.28039
$732$	0	0
$733$	24162.4	1.21754	0.608771	−	0.793346i	$-0.291663\pi$
$733$	24162.4	1.21754	0.608771	+	0.793346i	$0.291663\pi$
$734$	10333.4	0.519635
$735$	0	0
$736$	−68715.4	−3.44142
$737$	−21205.1	−1.05984
$738$	0	0
$739$	−32378.7	−1.61173	−0.805866	−	0.592099i	$-0.798300\pi$
$739$	−32378.7	−1.61173	−0.805866	+	0.592099i	$0.798300\pi$
$740$	−107934.	−5.36182
$741$	0	0
$742$	6468.37	0.320029
$743$	30746.0	1.51812	0.759060	−	0.651021i	$-0.225659\pi$
$743$	30746.0	1.51812	0.759060	+	0.651021i	$0.225659\pi$
$744$	0	0
$745$	56076.4	2.75769
$746$	8451.73	0.414799
$747$	0	0
$748$	−181617.	−8.87776
$749$	−19596.1	−0.955975
$750$	0	0
$751$	−3552.72	−0.172624	−0.0863121	−	0.996268i	$-0.527508\pi$
$751$	−3552.72	−0.172624	−0.0863121	+	0.996268i	$0.527508\pi$
$752$	−25110.1	−1.21765
$753$	0	0
$754$	24865.6	1.20100
$755$	32552.7	1.56916
$756$	0	0
$757$	−10905.3	−0.523594	−0.261797	−	0.965123i	$-0.584315\pi$
$757$	−10905.3	−0.523594	−0.261797	+	0.965123i	$0.584315\pi$
$758$	−20867.9	−0.999944
$759$	0	0
$760$	47305.8	2.25784
$761$	−1097.56	−0.0522818	−0.0261409	−	0.999658i	$-0.508322\pi$
$761$	−1097.56	−0.0522818	−0.0261409	+	0.999658i	$0.508322\pi$
$762$	0	0
$763$	24948.3	1.18374
$764$	−17466.0	−0.827091
$765$	0	0
$766$	3068.41	0.144734
$767$	5737.30	0.270094
$768$	0	0
$769$	6373.24	0.298862	0.149431	−	0.988772i	$-0.452256\pi$
$769$	6373.24	0.298862	0.149431	+	0.988772i	$0.452256\pi$
$770$	95718.0	4.47979
$771$	0	0
$772$	−26705.8	−1.24503
$773$	30799.4	1.43309	0.716545	−	0.697541i	$-0.245722\pi$
$773$	30799.4	1.43309	0.716545	+	0.697541i	$0.245722\pi$
$774$	0	0
$775$	29643.6	1.37397
$776$	−36889.4	−1.70651
$777$	0	0
$778$	−36101.4	−1.66362
$779$	17487.3	0.804296
$780$	0	0
$781$	−35800.5	−1.64026
$782$	126641.	5.79112
$783$	0	0
$784$	−28069.7	−1.27868
$785$	19987.2	0.908755
$786$	0	0
$787$	28238.3	1.27902	0.639509	−	0.768783i	$-0.279138\pi$
$787$	28238.3	1.27902	0.639509	+	0.768783i	$0.279138\pi$
$788$	21594.6	0.976236
$789$	0	0
$790$	35445.0	1.59630
$791$	−14562.9	−0.654612
$792$	0	0
$793$	−27894.2	−1.24912
$794$	43638.6	1.95047
$795$	0	0
$796$	78835.1	3.51035
$797$	−30650.8	−1.36224	−0.681122	−	0.732170i	$-0.738508\pi$
$797$	−30650.8	−1.36224	−0.681122	+	0.732170i	$0.738508\pi$
$798$	0	0
$799$	20316.6	0.899559
$800$	109004.	4.81734
$801$	0	0
$802$	−78542.8	−3.45816
$803$	−6164.06	−0.270890
$804$	0	0
$805$	−47482.2	−2.07892
$806$	18475.3	0.807401
$807$	0	0
$808$	−20588.2	−0.896399
$809$	−40477.0	−1.75908	−0.879541	−	0.475824i	$-0.842150\pi$
$809$	−40477.0	−1.75908	−0.879541	+	0.475824i	$0.842150\pi$
$810$	0	0
$811$	12850.5	0.556404	0.278202	−	0.960523i	$-0.410262\pi$
$811$	12850.5	0.556404	0.278202	+	0.960523i	$0.410262\pi$
$812$	−36899.8	−1.59474
$813$	0	0
$814$	97249.6	4.18746
$815$	−31248.2	−1.34304
$816$	0	0
$817$	7108.65	0.304407
$818$	9408.68	0.402160
$819$	0	0
$820$	183163.	7.80042
$821$	−34174.1	−1.45272	−0.726360	−	0.687314i	$-0.758790\pi$
$821$	−34174.1	−1.45272	−0.726360	+	0.687314i	$0.758790\pi$
$822$	0	0
$823$	−18526.9	−0.784700	−0.392350	−	0.919816i	$-0.628338\pi$
$823$	−18526.9	−0.784700	−0.392350	+	0.919816i	$0.628338\pi$
$824$	35796.8	1.51340
$825$	0	0
$826$	−11967.7	−0.504130
$827$	−13750.8	−0.578188	−0.289094	−	0.957301i	$-0.593354\pi$
$827$	−13750.8	−0.578188	−0.289094	+	0.957301i	$0.593354\pi$
$828$	0	0
$829$	36741.2	1.53930	0.769648	−	0.638469i	$-0.220432\pi$
$829$	36741.2	1.53930	0.769648	+	0.638469i	$0.220432\pi$
$830$	78008.0	3.26229
$831$	0	0
$832$	23278.4	0.969990
$833$	22711.1	0.944651
$834$	0	0
$835$	−26210.9	−1.08631
$836$	−51018.3	−2.11065
$837$	0	0
$838$	65402.2	2.69604
$839$	−29148.3	−1.19942	−0.599709	−	0.800218i	$-0.704717\pi$
$839$	−29148.3	−1.19942	−0.599709	+	0.800218i	$0.704717\pi$
$840$	0	0
$841$	−4391.29	−0.180052
$842$	−39625.3	−1.62183
$843$	0	0
$844$	44395.6	1.81062
$845$	21825.0	0.888525
$846$	0	0
$847$	−43742.8	−1.77452
$848$	−15520.0	−0.628490
$849$	0	0
$850$	−200891.	−8.10649
$851$	−48241.9	−1.94326
$852$	0	0
$853$	−12165.1	−0.488306	−0.244153	−	0.969737i	$-0.578510\pi$
$853$	−12165.1	−0.488306	−0.244153	+	0.969737i	$0.578510\pi$
$854$	58186.1	2.33148
$855$	0	0
$856$	91397.4	3.64941
$857$	39605.6	1.57865	0.789324	−	0.613977i	$-0.210431\pi$
$857$	39605.6	1.57865	0.789324	+	0.613977i	$0.210431\pi$
$858$	0	0
$859$	−1672.71	−0.0664401	−0.0332200	−	0.999448i	$-0.510576\pi$
$859$	−1672.71	−0.0664401	−0.0332200	+	0.999448i	$0.510576\pi$
$860$	74456.8	2.95227
$861$	0	0
$862$	−90674.8	−3.58283
$863$	23847.6	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$863$	23847.6	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$864$	0	0
$865$	−7327.07	−0.288009
$866$	51042.9	2.00290
$867$	0	0
$868$	−27416.8	−1.07211
$869$	−22719.7	−0.886897
$870$	0	0
$871$	−10399.8	−0.404575
$872$	−116361.	−4.51889
$873$	0	0
$874$	35574.8	1.37681
$875$	41949.6	1.62075
$876$	0	0
$877$	5550.23	0.213704	0.106852	−	0.994275i	$-0.465923\pi$
$877$	5550.23	0.213704	0.106852	+	0.994275i	$0.465923\pi$
$878$	3511.69	0.134981
$879$	0	0
$880$	−229663.	−8.79766
$881$	39226.6	1.50009	0.750044	−	0.661388i	$-0.230033\pi$
$881$	39226.6	1.50009	0.750044	+	0.661388i	$0.230033\pi$
$882$	0	0
$883$	44129.7	1.68186	0.840930	−	0.541144i	$-0.182008\pi$
$883$	44129.7	1.68186	0.840930	+	0.541144i	$0.182008\pi$
$884$	−89072.2	−3.38894
$885$	0	0
$886$	27424.6	1.03990
$887$	−34242.6	−1.29623	−0.648113	−	0.761544i	$-0.724442\pi$
$887$	−34242.6	−1.29623	−0.648113	+	0.761544i	$0.724442\pi$
$888$	0	0
$889$	−8051.35	−0.303750
$890$	78749.9	2.96596
$891$	0	0
$892$	55090.3	2.06789
$893$	5707.16	0.213866
$894$	0	0
$895$	12116.4	0.452521
$896$	−7660.77	−0.285634
$897$	0	0
$898$	57204.8	2.12578
$899$	14858.4	0.551231
$900$	0	0
$901$	12557.2	0.464308
$902$	−165031.	−6.09195
$903$	0	0
$904$	67922.5	2.49897
$905$	−69115.2	−2.53864
$906$	0	0
$907$	−52054.3	−1.90566	−0.952830	−	0.303504i	$-0.901843\pi$
$907$	−52054.3	−1.90566	−0.952830	+	0.303504i	$0.901843\pi$
$908$	−127208.	−4.64928
$909$	0	0
$910$	46944.0	1.71009
$911$	−49109.3	−1.78602	−0.893010	−	0.450036i	$-0.851411\pi$
$911$	−49109.3	−1.78602	−0.893010	+	0.450036i	$0.851411\pi$
$912$	0	0
$913$	−50001.9	−1.81251
$914$	38617.1	1.39753
$915$	0	0
$916$	14253.4	0.514133
$917$	−9808.28	−0.353215
$918$	0	0
$919$	−39316.6	−1.41125	−0.705623	−	0.708587i	$-0.749333\pi$
$919$	−39316.6	−1.41125	−0.705623	+	0.708587i	$0.749333\pi$
$920$	221460.	7.93622
$921$	0	0
$922$	19994.5	0.714190
$923$	−17558.0	−0.626142
$924$	0	0
$925$	76526.7	2.72020
$926$	−65739.7	−2.33298
$927$	0	0
$928$	54636.7	1.93269
$929$	−43407.2	−1.53299	−0.766493	−	0.642253i	$-0.778000\pi$
$929$	−43407.2	−1.53299	−0.766493	+	0.642253i	$0.778000\pi$
$930$	0	0
$931$	6379.82	0.224587
$932$	66824.6	2.34862
$933$	0	0
$934$	−27646.4	−0.968542
$935$	185820.	6.49943
$936$	0	0
$937$	−27279.5	−0.951102	−0.475551	−	0.879688i	$-0.657751\pi$
$937$	−27279.5	−0.951102	−0.475551	+	0.879688i	$0.657751\pi$
$938$	21693.6	0.755139
$939$	0	0
$940$	59777.4	2.07417
$941$	24741.4	0.857116	0.428558	−	0.903514i	$-0.359022\pi$
$941$	24741.4	0.857116	0.428558	+	0.903514i	$0.359022\pi$
$942$	0	0
$943$	81866.0	2.82707
$944$	28715.1	0.990037
$945$	0	0
$946$	−67086.0	−2.30566
$947$	44553.0	1.52880	0.764402	−	0.644740i	$-0.223034\pi$
$947$	44553.0	1.52880	0.764402	+	0.644740i	$0.223034\pi$
$948$	0	0
$949$	−3023.11	−0.103408
$950$	−56432.7	−1.92728
$951$	0	0
$952$	110429.	3.75948
$953$	52298.9	1.77768	0.888839	−	0.458220i	$-0.151513\pi$
$953$	52298.9	1.77768	0.888839	+	0.458220i	$0.151513\pi$
$954$	0	0
$955$	17870.2	0.605515
$956$	4713.32	0.159456
$957$	0	0
$958$	−38552.0	−1.30016
$959$	619.808	0.0208703
$960$	0	0
$961$	−18751.1	−0.629421
$962$	47695.1	1.59850
$963$	0	0
$964$	127832.	4.27094
$965$	27323.9	0.911488
$966$	0	0
$967$	21652.6	0.720064	0.360032	−	0.932940i	$-0.382766\pi$
$967$	21652.6	0.720064	0.360032	+	0.932940i	$0.382766\pi$
$968$	204019.	6.77421
$969$	0	0
$970$	63504.2	2.10206
$971$	−40757.1	−1.34702	−0.673510	−	0.739178i	$-0.735214\pi$
$971$	−40757.1	−1.34702	−0.673510	+	0.739178i	$0.735214\pi$
$972$	0	0
$973$	16020.0	0.527830
$974$	14400.3	0.473731
$975$	0	0
$976$	−139610.	−4.57869
$977$	−57315.1	−1.87684	−0.938420	−	0.345497i	$-0.887710\pi$
$977$	−57315.1	−1.87684	−0.938420	+	0.345497i	$0.887710\pi$
$978$	0	0
$979$	−50477.4	−1.64787
$980$	66823.0	2.17814
$981$	0	0
$982$	35918.0	1.16720
$983$	4287.35	0.139110	0.0695551	−	0.997578i	$-0.477842\pi$
$983$	4287.35	0.139110	0.0695551	+	0.997578i	$0.477842\pi$
$984$	0	0
$985$	−22094.4	−0.714705
$986$	−100694.	−3.25228
$987$	0	0
$988$	−25021.4	−0.805706
$989$	33278.9	1.06998
$990$	0	0
$991$	5168.68	0.165680	0.0828398	−	0.996563i	$-0.473601\pi$
$991$	5168.68	0.165680	0.0828398	+	0.996563i	$0.473601\pi$
$992$	40595.4	1.29930
$993$	0	0
$994$	36625.2	1.16869
$995$	−80659.7	−2.56994
$996$	0	0
$997$	36154.6	1.14847	0.574237	−	0.818689i	$-0.305299\pi$
$997$	36154.6	1.14847	0.574237	+	0.818689i	$0.305299\pi$
$998$	−81826.3	−2.59536
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.59	✓	59
3.2	odd	2		2151.4.a.h.1.1	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.59	✓	59	1.1	even	1	trivial
2151.4.a.h.1.1	yes	59	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+5.26507 q^{2} +19.7210 q^{4} -20.1774 q^{5} -13.2314 q^{7} +61.7120 q^{8} +O(q^{10})\)	\(q+5.26507 q^{2} +19.7210 q^{4} -20.1774 q^{5} -13.2314 q^{7} +61.7120 q^{8} -106.236 q^{10} +68.0955 q^{11} +33.3968 q^{13} -69.6642 q^{14} +167.150 q^{16} -135.241 q^{17} -37.9908 q^{19} -397.920 q^{20} +358.528 q^{22} -177.852 q^{23} +282.129 q^{25} +175.837 q^{26} -260.936 q^{28} +141.413 q^{29} +105.071 q^{31} +386.362 q^{32} -712.054 q^{34} +266.975 q^{35} +271.247 q^{37} -200.024 q^{38} -1245.19 q^{40} -460.303 q^{41} -187.115 q^{43} +1342.91 q^{44} -936.406 q^{46} -150.225 q^{47} -167.931 q^{49} +1485.43 q^{50} +658.618 q^{52} -92.8507 q^{53} -1373.99 q^{55} -816.534 q^{56} +744.551 q^{58} +171.792 q^{59} -835.237 q^{61} +553.207 q^{62} +697.024 q^{64} -673.862 q^{65} -311.402 q^{67} -2667.09 q^{68} +1405.64 q^{70} -525.740 q^{71} -90.5209 q^{73} +1428.14 q^{74} -749.217 q^{76} -900.996 q^{77} -333.645 q^{79} -3372.66 q^{80} -2423.53 q^{82} -734.292 q^{83} +2728.82 q^{85} -985.175 q^{86} +4202.31 q^{88} -741.275 q^{89} -441.885 q^{91} -3507.43 q^{92} -790.945 q^{94} +766.557 q^{95} -597.767 q^{97} -884.168 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Introduction

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