LMFDB - Embedded newform 2151.4.a.c.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:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q+0.797577 q^{2} -7.36387 q^{4} +7.16277 q^{5} +0.569206 q^{7} -12.2539 q^{8} +O(q^{10})$	\(q+0.797577 q^{2} -7.36387 q^{4} +7.16277 q^{5} +0.569206 q^{7} -12.2539 q^{8} +5.71286 q^{10} +72.6144 q^{11} +76.1671 q^{13} +0.453986 q^{14} +49.1376 q^{16} -76.7287 q^{17} +149.048 q^{19} -52.7457 q^{20} +57.9156 q^{22} +94.9473 q^{23} -73.6948 q^{25} +60.7491 q^{26} -4.19156 q^{28} +148.174 q^{29} +170.687 q^{31} +137.222 q^{32} -61.1971 q^{34} +4.07709 q^{35} -288.916 q^{37} +118.877 q^{38} -87.7716 q^{40} -144.739 q^{41} -103.378 q^{43} -534.723 q^{44} +75.7278 q^{46} +242.300 q^{47} -342.676 q^{49} -58.7773 q^{50} -560.884 q^{52} +165.612 q^{53} +520.120 q^{55} -6.97498 q^{56} +118.180 q^{58} -54.7756 q^{59} +180.229 q^{61} +136.136 q^{62} -283.655 q^{64} +545.567 q^{65} -979.806 q^{67} +565.021 q^{68} +3.25179 q^{70} +170.531 q^{71} +924.830 q^{73} -230.433 q^{74} -1097.57 q^{76} +41.3326 q^{77} -197.036 q^{79} +351.961 q^{80} -115.441 q^{82} +733.506 q^{83} -549.590 q^{85} -82.4523 q^{86} -889.808 q^{88} +252.743 q^{89} +43.3548 q^{91} -699.180 q^{92} +193.253 q^{94} +1067.59 q^{95} -1614.99 q^{97} -273.311 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.797577	0.281986	0.140993	−	0.990011i	$-0.454970\pi$
$2$	0.797577	0.281986	0.140993	+	0.990011i	$0.454970\pi$
$3$	0	0
$3$	0	0
$4$	−7.36387	−0.920484
$5$	7.16277	0.640657	0.320329	−	0.947306i	$-0.396207\pi$
$5$	7.16277	0.640657	0.320329	+	0.947306i	$0.396207\pi$
$6$	0	0
$7$	0.569206	0.0307342	0.0153671	−	0.999882i	$-0.495108\pi$
$7$	0.569206	0.0307342	0.0153671	+	0.999882i	$0.495108\pi$
$8$	−12.2539	−0.541550
$9$	0	0
$10$	5.71286	0.180656
$11$	72.6144	1.99037	0.995185	−	0.0980112i	$-0.0312481\pi$
$11$	72.6144	1.99037	0.995185	+	0.0980112i	$0.0312481\pi$
$12$	0	0
$13$	76.1671	1.62500	0.812498	−	0.582964i	$-0.198107\pi$
$13$	76.1671	1.62500	0.812498	+	0.582964i	$0.198107\pi$
$14$	0.453986	0.00866663
$15$	0	0
$16$	49.1376	0.767774
$17$	−76.7287	−1.09467	−0.547337	−	0.836912i	$-0.684358\pi$
$17$	−76.7287	−1.09467	−0.547337	+	0.836912i	$0.684358\pi$
$18$	0	0
$19$	149.048	1.79968	0.899839	−	0.436223i	$-0.143684\pi$
$19$	149.048	1.79968	0.899839	+	0.436223i	$0.143684\pi$
$20$	−52.7457	−0.589715
$21$	0	0
$22$	57.9156	0.561257
$23$	94.9473	0.860778	0.430389	−	0.902644i	$-0.358376\pi$
$23$	94.9473	0.860778	0.430389	+	0.902644i	$0.358376\pi$
$24$	0	0
$25$	−73.6948	−0.589558
$26$	60.7491	0.458226
$27$	0	0
$28$	−4.19156	−0.0282904
$29$	148.174	0.948802	0.474401	−	0.880309i	$-0.342665\pi$
$29$	148.174	0.948802	0.474401	+	0.880309i	$0.342665\pi$
$30$	0	0
$31$	170.687	0.988913	0.494456	−	0.869202i	$-0.335367\pi$
$31$	170.687	0.988913	0.494456	+	0.869202i	$0.335367\pi$
$32$	137.222	0.758051
$33$	0	0
$34$	−61.1971	−0.308683
$35$	4.07709	0.0196901
$36$	0	0
$37$	−288.916	−1.28372	−0.641858	−	0.766824i	$-0.721836\pi$
$37$	−288.916	−1.28372	−0.641858	+	0.766824i	$0.721836\pi$
$38$	118.877	0.507484
$39$	0	0
$40$	−87.7716	−0.346948
$41$	−144.739	−0.551329	−0.275664	−	0.961254i	$-0.588898\pi$
$41$	−144.739	−0.551329	−0.275664	+	0.961254i	$0.588898\pi$
$42$	0	0
$43$	−103.378	−0.366630	−0.183315	−	0.983054i	$-0.558683\pi$
$43$	−103.378	−0.366630	−0.183315	+	0.983054i	$0.558683\pi$
$44$	−534.723	−1.83210
$45$	0	0
$46$	75.7278	0.242727
$47$	242.300	0.751982	0.375991	−	0.926623i	$-0.377302\pi$
$47$	242.300	0.751982	0.375991	+	0.926623i	$0.377302\pi$
$48$	0	0
$49$	−342.676	−0.999055
$50$	−58.7773	−0.166247
$51$	0	0
$52$	−560.884	−1.49578
$53$	165.612	0.429217	0.214608	−	0.976700i	$-0.431152\pi$
$53$	165.612	0.429217	0.214608	+	0.976700i	$0.431152\pi$
$54$	0	0
$55$	520.120	1.27515
$56$	−6.97498	−0.0166441
$57$	0	0
$58$	118.180	0.267549
$59$	−54.7756	−0.120867	−0.0604336	−	0.998172i	$-0.519248\pi$
$59$	−54.7756	−0.120867	−0.0604336	+	0.998172i	$0.519248\pi$
$60$	0	0
$61$	180.229	0.378295	0.189148	−	0.981949i	$-0.439428\pi$
$61$	180.229	0.378295	0.189148	+	0.981949i	$0.439428\pi$
$62$	136.136	0.278860
$63$	0	0
$64$	−283.655	−0.554014
$65$	545.567	1.04107
$66$	0	0
$67$	−979.806	−1.78660	−0.893301	−	0.449458i	$-0.851617\pi$
$67$	−979.806	−1.78660	−0.893301	+	0.449458i	$0.851617\pi$
$68$	565.021	1.00763
$69$	0	0
$70$	3.25179	0.00555234
$71$	170.531	0.285047	0.142523	−	0.989791i	$-0.454478\pi$
$71$	170.531	0.285047	0.142523	+	0.989791i	$0.454478\pi$
$72$	0	0
$73$	924.830	1.48278	0.741391	−	0.671073i	$-0.234166\pi$
$73$	924.830	1.48278	0.741391	+	0.671073i	$0.234166\pi$
$74$	−230.433	−0.361990
$75$	0	0
$76$	−1097.57	−1.65657
$77$	41.3326	0.0611725
$78$	0	0
$79$	−197.036	−0.280611	−0.140305	−	0.990108i	$-0.544808\pi$
$79$	−197.036	−0.280611	−0.140305	+	0.990108i	$0.544808\pi$
$80$	351.961	0.491880
$81$	0	0
$82$	−115.441	−0.155467
$83$	733.506	0.970033	0.485017	−	0.874505i	$-0.338814\pi$
$83$	733.506	0.970033	0.485017	+	0.874505i	$0.338814\pi$
$84$	0	0
$85$	−549.590	−0.701311
$86$	−82.4523	−0.103384
$87$	0	0
$88$	−889.808	−1.07788
$89$	252.743	0.301019	0.150510	−	0.988609i	$-0.451909\pi$
$89$	252.743	0.301019	0.150510	+	0.988609i	$0.451909\pi$
$90$	0	0
$91$	43.3548	0.0499430
$92$	−699.180	−0.792332
$93$	0	0
$94$	193.253	0.212048
$95$	1067.59	1.15298
$96$	0	0
$97$	−1614.99	−1.69049	−0.845244	−	0.534380i	$-0.820545\pi$
$97$	−1614.99	−1.69049	−0.845244	+	0.534380i	$0.820545\pi$
$98$	−273.311	−0.281720
$99$	0	0
$100$	542.679	0.542679
$101$	−1175.00	−1.15759	−0.578796	−	0.815472i	$-0.696477\pi$
$101$	−1175.00	−1.15759	−0.578796	+	0.815472i	$0.696477\pi$
$102$	0	0
$103$	482.083	0.461176	0.230588	−	0.973052i	$-0.425935\pi$
$103$	482.083	0.461176	0.230588	+	0.973052i	$0.425935\pi$
$104$	−933.342	−0.880016
$105$	0	0
$106$	132.088	0.121033
$107$	139.089	0.125665	0.0628327	−	0.998024i	$-0.479987\pi$
$107$	139.089	0.125665	0.0628327	+	0.998024i	$0.479987\pi$
$108$	0	0
$109$	−1526.87	−1.34172	−0.670860	−	0.741584i	$-0.734075\pi$
$109$	−1526.87	−1.34172	−0.670860	+	0.741584i	$0.734075\pi$
$110$	414.836	0.359573
$111$	0	0
$112$	27.9694	0.0235970
$113$	1044.24	0.869329	0.434664	−	0.900593i	$-0.356867\pi$
$113$	1044.24	0.869329	0.434664	+	0.900593i	$0.356867\pi$
$114$	0	0
$115$	680.085	0.551463
$116$	−1091.14	−0.873357
$117$	0	0
$118$	−43.6877	−0.0340829
$119$	−43.6745	−0.0336440
$120$	0	0
$121$	3941.86	2.96158
$122$	143.747	0.106674
$123$	0	0
$124$	−1256.92	−0.910278
$125$	−1423.20	−1.01836
$126$	0	0
$127$	2293.02	1.60215	0.801074	−	0.598566i	$-0.204262\pi$
$127$	2293.02	1.60215	0.801074	+	0.598566i	$0.204262\pi$
$128$	−1324.01	−0.914276
$129$	0	0
$130$	435.132	0.293566
$131$	2600.97	1.73472	0.867358	−	0.497685i	$-0.165816\pi$
$131$	2600.97	1.73472	0.867358	+	0.497685i	$0.165816\pi$
$132$	0	0
$133$	84.8388	0.0553117
$134$	−781.471	−0.503797
$135$	0	0
$136$	940.224	0.592820
$137$	−1043.82	−0.650947	−0.325474	−	0.945551i	$-0.605524\pi$
$137$	−1043.82	−0.650947	−0.325474	+	0.945551i	$0.605524\pi$
$138$	0	0
$139$	−2260.68	−1.37949	−0.689743	−	0.724054i	$-0.742277\pi$
$139$	−2260.68	−1.37949	−0.689743	+	0.724054i	$0.742277\pi$
$140$	−30.0232	−0.0181244
$141$	0	0
$142$	136.012	0.0803793
$143$	5530.83	3.23434
$144$	0	0
$145$	1061.34	0.607857
$146$	737.623	0.418124
$147$	0	0
$148$	2127.54	1.18164
$149$	2779.15	1.52803	0.764016	−	0.645197i	$-0.223225\pi$
$149$	2779.15	1.52803	0.764016	+	0.645197i	$0.223225\pi$
$150$	0	0
$151$	−2162.14	−1.16525	−0.582625	−	0.812741i	$-0.697974\pi$
$151$	−2162.14	−1.16525	−0.582625	+	0.812741i	$0.697974\pi$
$152$	−1826.41	−0.974615
$153$	0	0
$154$	32.9659	0.0172498
$155$	1222.59	0.633554
$156$	0	0
$157$	246.528	0.125319	0.0626594	−	0.998035i	$-0.480042\pi$
$157$	246.528	0.125319	0.0626594	+	0.998035i	$0.480042\pi$
$158$	−157.151	−0.0791284
$159$	0	0
$160$	982.889	0.485651
$161$	54.0446	0.0264553
$162$	0	0
$163$	1042.16	0.500788	0.250394	−	0.968144i	$-0.419440\pi$
$163$	1042.16	0.500788	0.250394	+	0.968144i	$0.419440\pi$
$164$	1065.84	0.507489
$165$	0	0
$166$	585.028	0.273536
$167$	2965.10	1.37393	0.686966	−	0.726690i	$-0.258942\pi$
$167$	2965.10	1.37393	0.686966	+	0.726690i	$0.258942\pi$
$168$	0	0
$169$	3604.42	1.64061
$170$	−438.340	−0.197760
$171$	0	0
$172$	761.266	0.337477
$173$	−2565.17	−1.12732	−0.563660	−	0.826007i	$-0.690607\pi$
$173$	−2565.17	−1.12732	−0.563660	+	0.826007i	$0.690607\pi$
$174$	0	0
$175$	−41.9475	−0.0181196
$176$	3568.10	1.52816
$177$	0	0
$178$	201.582	0.0848832
$179$	1027.83	0.429181	0.214591	−	0.976704i	$-0.431158\pi$
$179$	1027.83	0.429181	0.214591	+	0.976704i	$0.431158\pi$
$180$	0	0
$181$	−2101.85	−0.863144	−0.431572	−	0.902079i	$-0.642041\pi$
$181$	−2101.85	−0.863144	−0.431572	+	0.902079i	$0.642041\pi$
$182$	34.5788	0.0140832
$183$	0	0
$184$	−1163.47	−0.466154
$185$	−2069.44	−0.822422
$186$	0	0
$187$	−5571.61	−2.17881
$188$	−1784.27	−0.692187
$189$	0	0
$190$	851.488	0.325123
$191$	1231.96	0.466708	0.233354	−	0.972392i	$-0.425030\pi$
$191$	1231.96	0.466708	0.233354	+	0.972392i	$0.425030\pi$
$192$	0	0
$193$	−2073.78	−0.773439	−0.386720	−	0.922197i	$-0.626392\pi$
$193$	−2073.78	−0.773439	−0.386720	+	0.922197i	$0.626392\pi$
$194$	−1288.08	−0.476694
$195$	0	0
$196$	2523.42	0.919614
$197$	3831.52	1.38571	0.692854	−	0.721078i	$-0.256353\pi$
$197$	3831.52	1.38571	0.692854	+	0.721078i	$0.256353\pi$
$198$	0	0
$199$	−1788.71	−0.637178	−0.318589	−	0.947893i	$-0.603209\pi$
$199$	−1788.71	−0.637178	−0.318589	+	0.947893i	$0.603209\pi$
$200$	903.047	0.319275
$201$	0	0
$202$	−937.152	−0.326425
$203$	84.3417	0.0291607
$204$	0	0
$205$	−1036.73	−0.353213
$206$	384.499	0.130045
$207$	0	0
$208$	3742.66	1.24763
$209$	10823.0	3.58203
$210$	0	0
$211$	3228.33	1.05331	0.526653	−	0.850080i	$-0.323447\pi$
$211$	3228.33	1.05331	0.526653	+	0.850080i	$0.323447\pi$
$212$	−1219.54	−0.395087
$213$	0	0
$214$	110.934	0.0354359
$215$	−740.476	−0.234884
$216$	0	0
$217$	97.1561	0.0303935
$218$	−1217.80	−0.378346
$219$	0	0
$220$	−3830.10	−1.17375
$221$	−5844.20	−1.77884
$222$	0	0
$223$	5503.99	1.65280	0.826400	−	0.563083i	$-0.190385\pi$
$223$	5503.99	1.65280	0.826400	+	0.563083i	$0.190385\pi$
$224$	78.1076	0.0232981
$225$	0	0
$226$	832.865	0.245139
$227$	−930.852	−0.272171	−0.136086	−	0.990697i	$-0.543452\pi$
$227$	−930.852	−0.272171	−0.136086	+	0.990697i	$0.543452\pi$
$228$	0	0
$229$	−2094.91	−0.604523	−0.302261	−	0.953225i	$-0.597742\pi$
$229$	−2094.91	−0.604523	−0.302261	+	0.953225i	$0.597742\pi$
$230$	542.421	0.155505
$231$	0	0
$232$	−1815.71	−0.513824
$233$	−5635.74	−1.58459	−0.792295	−	0.610139i	$-0.791114\pi$
$233$	−5635.74	−1.58459	−0.792295	+	0.610139i	$0.791114\pi$
$234$	0	0
$235$	1735.54	0.481763
$236$	403.360	0.111256
$237$	0	0
$238$	−34.8338	−0.00948713
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−4406.54	−1.17780	−0.588901	−	0.808205i	$-0.700439\pi$
$241$	−4406.54	−1.17780	−0.588901	+	0.808205i	$0.700439\pi$
$242$	3143.94	0.835123
$243$	0	0
$244$	−1327.19	−0.348215
$245$	−2454.51	−0.640052
$246$	0	0
$247$	11352.5	2.92447
$248$	−2091.58	−0.535546
$249$	0	0
$250$	−1135.12	−0.287164
$251$	6558.78	1.64935	0.824674	−	0.565608i	$-0.191358\pi$
$251$	6558.78	1.64935	0.824674	+	0.565608i	$0.191358\pi$
$252$	0	0
$253$	6894.55	1.71327
$254$	1828.86	0.451783
$255$	0	0
$256$	1213.24	0.296201
$257$	−4595.31	−1.11536	−0.557680	−	0.830056i	$-0.688308\pi$
$257$	−4595.31	−1.11536	−0.557680	+	0.830056i	$0.688308\pi$
$258$	0	0
$259$	−164.453	−0.0394540
$260$	−4017.48	−0.958284
$261$	0	0
$262$	2074.47	0.489166
$263$	1854.65	0.434840	0.217420	−	0.976078i	$-0.430236\pi$
$263$	1854.65	0.434840	0.217420	+	0.976078i	$0.430236\pi$
$264$	0	0
$265$	1186.24	0.274981
$266$	67.6655	0.0155971
$267$	0	0
$268$	7215.17	1.64454
$269$	−6285.89	−1.42475	−0.712375	−	0.701799i	$-0.752380\pi$
$269$	−6285.89	−1.42475	−0.712375	+	0.701799i	$0.752380\pi$
$270$	0	0
$271$	1702.73	0.381674	0.190837	−	0.981622i	$-0.438880\pi$
$271$	1702.73	0.381674	0.190837	+	0.981622i	$0.438880\pi$
$272$	−3770.26	−0.840462
$273$	0	0
$274$	−832.529	−0.183558
$275$	−5351.31	−1.17344
$276$	0	0
$277$	−2734.93	−0.593234	−0.296617	−	0.954997i	$-0.595858\pi$
$277$	−2734.93	−0.593234	−0.296617	+	0.954997i	$0.595858\pi$
$278$	−1803.07	−0.388996
$279$	0	0
$280$	−49.9601	−0.0106632
$281$	−884.626	−0.187802	−0.0939010	−	0.995582i	$-0.529934\pi$
$281$	−884.626	−0.187802	−0.0939010	+	0.995582i	$0.529934\pi$
$282$	0	0
$283$	−461.744	−0.0969887	−0.0484944	−	0.998823i	$-0.515442\pi$
$283$	−461.744	−0.0969887	−0.0484944	+	0.998823i	$0.515442\pi$
$284$	−1255.77	−0.262381
$285$	0	0
$286$	4411.26	0.912040
$287$	−82.3865	−0.0169447
$288$	0	0
$289$	974.301	0.198311
$290$	846.499	0.171407
$291$	0	0
$292$	−6810.33	−1.36488
$293$	−21.4596	−0.00427879	−0.00213939	−	0.999998i	$-0.500681\pi$
$293$	−21.4596	−0.00427879	−0.00213939	+	0.999998i	$0.500681\pi$
$294$	0	0
$295$	−392.344	−0.0774345
$296$	3540.34	0.695196
$297$	0	0
$298$	2216.59	0.430884
$299$	7231.86	1.39876
$300$	0	0
$301$	−58.8437	−0.0112681
$302$	−1724.48	−0.328584
$303$	0	0
$304$	7323.84	1.38175
$305$	1290.94	0.242358
$306$	0	0
$307$	8984.15	1.67020	0.835102	−	0.550095i	$-0.185409\pi$
$307$	8984.15	1.67020	0.835102	+	0.550095i	$0.185409\pi$
$308$	−304.368	−0.0563083
$309$	0	0
$310$	975.111	0.178653
$311$	5517.84	1.00607	0.503035	−	0.864266i	$-0.332217\pi$
$311$	5517.84	1.00607	0.503035	+	0.864266i	$0.332217\pi$
$312$	0	0
$313$	−9741.44	−1.75917	−0.879583	−	0.475746i	$-0.842178\pi$
$313$	−9741.44	−1.75917	−0.879583	+	0.475746i	$0.842178\pi$
$314$	196.625	0.0353382
$315$	0	0
$316$	1450.95	0.258298
$317$	1953.82	0.346174	0.173087	−	0.984907i	$-0.444626\pi$
$317$	1953.82	0.346174	0.173087	+	0.984907i	$0.444626\pi$
$318$	0	0
$319$	10759.6	1.88847
$320$	−2031.76	−0.354933
$321$	0	0
$322$	43.1047	0.00746004
$323$	−11436.2	−1.97006
$324$	0	0
$325$	−5613.12	−0.958030
$326$	831.204	0.141215
$327$	0	0
$328$	1773.62	0.298572
$329$	137.919	0.0231116
$330$	0	0
$331$	−2154.89	−0.357836	−0.178918	−	0.983864i	$-0.557260\pi$
$331$	−2154.89	−0.357836	−0.178918	+	0.983864i	$0.557260\pi$
$332$	−5401.45	−0.892900
$333$	0	0
$334$	2364.90	0.387430
$335$	−7018.12	−1.14460
$336$	0	0
$337$	−10457.0	−1.69030	−0.845149	−	0.534531i	$-0.820488\pi$
$337$	−10457.0	−1.69030	−0.845149	+	0.534531i	$0.820488\pi$
$338$	2874.81	0.462630
$339$	0	0
$340$	4047.11	0.645545
$341$	12394.3	1.96830
$342$	0	0
$343$	−390.291	−0.0614395
$344$	1266.79	0.198548
$345$	0	0
$346$	−2045.92	−0.317889
$347$	3772.55	0.583635	0.291817	−	0.956474i	$-0.405740\pi$
$347$	3772.55	0.583635	0.291817	+	0.956474i	$0.405740\pi$
$348$	0	0
$349$	10410.4	1.59672	0.798359	−	0.602182i	$-0.205702\pi$
$349$	10410.4	1.59672	0.798359	+	0.602182i	$0.205702\pi$
$350$	−33.4564	−0.00510948
$351$	0	0
$352$	9964.30	1.50880
$353$	613.156	0.0924504	0.0462252	−	0.998931i	$-0.485281\pi$
$353$	613.156	0.0924504	0.0462252	+	0.998931i	$0.485281\pi$
$354$	0	0
$355$	1221.47	0.182617
$356$	−1861.17	−0.277083
$357$	0	0
$358$	819.772	0.121023
$359$	−4528.45	−0.665745	−0.332873	−	0.942972i	$-0.608018\pi$
$359$	−4528.45	−0.665745	−0.332873	+	0.942972i	$0.608018\pi$
$360$	0	0
$361$	15356.2	2.23884
$362$	−1676.38	−0.243395
$363$	0	0
$364$	−319.259	−0.0459717
$365$	6624.34	0.949956
$366$	0	0
$367$	−13206.5	−1.87840	−0.939199	−	0.343373i	$-0.888431\pi$
$367$	−13206.5	−1.87840	−0.939199	+	0.343373i	$0.888431\pi$
$368$	4665.48	0.660883
$369$	0	0
$370$	−1650.54	−0.231911
$371$	94.2671	0.0131917
$372$	0	0
$373$	11397.8	1.58218	0.791092	−	0.611697i	$-0.209513\pi$
$373$	11397.8	1.58218	0.791092	+	0.611697i	$0.209513\pi$
$374$	−4443.79	−0.614393
$375$	0	0
$376$	−2969.12	−0.407236
$377$	11286.0	1.54180
$378$	0	0
$379$	13484.0	1.82751	0.913755	−	0.406265i	$-0.133169\pi$
$379$	13484.0	1.82751	0.913755	+	0.406265i	$0.133169\pi$
$380$	−7861.62	−1.06130
$381$	0	0
$382$	982.579	0.131605
$383$	7644.37	1.01987	0.509933	−	0.860214i	$-0.329670\pi$
$383$	7644.37	1.01987	0.509933	+	0.860214i	$0.329670\pi$
$384$	0	0
$385$	296.056	0.0391906
$386$	−1654.00	−0.218099
$387$	0	0
$388$	11892.6	1.55607
$389$	3526.50	0.459642	0.229821	−	0.973233i	$-0.426186\pi$
$389$	3526.50	0.459642	0.229821	+	0.973233i	$0.426186\pi$
$390$	0	0
$391$	−7285.19	−0.942271
$392$	4199.11	0.541038
$393$	0	0
$394$	3055.93	0.390750
$395$	−1411.32	−0.179775
$396$	0	0
$397$	3217.94	0.406810	0.203405	−	0.979095i	$-0.434799\pi$
$397$	3217.94	0.406810	0.203405	+	0.979095i	$0.434799\pi$
$398$	−1426.64	−0.179675
$399$	0	0
$400$	−3621.18	−0.452648
$401$	−15211.7	−1.89435	−0.947176	−	0.320715i	$-0.896077\pi$
$401$	−15211.7	−1.89435	−0.947176	+	0.320715i	$0.896077\pi$
$402$	0	0
$403$	13000.7	1.60698
$404$	8652.54	1.06554
$405$	0	0
$406$	67.2690	0.00822292
$407$	−20979.5	−2.55507
$408$	0	0
$409$	−2142.25	−0.258992	−0.129496	−	0.991580i	$-0.541336\pi$
$409$	−2142.25	−0.258992	−0.129496	+	0.991580i	$0.541336\pi$
$410$	−826.875	−0.0996011
$411$	0	0
$412$	−3550.00	−0.424505
$413$	−31.1786	−0.00371476
$414$	0	0
$415$	5253.93	0.621459
$416$	10451.8	1.23183
$417$	0	0
$418$	8632.19	1.01008
$419$	−5836.82	−0.680542	−0.340271	−	0.940327i	$-0.610519\pi$
$419$	−5836.82	−0.680542	−0.340271	+	0.940327i	$0.610519\pi$
$420$	0	0
$421$	−435.097	−0.0503690	−0.0251845	−	0.999683i	$-0.508017\pi$
$421$	−435.097	−0.0503690	−0.0251845	+	0.999683i	$0.508017\pi$
$422$	2574.85	0.297018
$423$	0	0
$424$	−2029.38	−0.232442
$425$	5654.51	0.645374
$426$	0	0
$427$	102.588	0.0116266
$428$	−1024.23	−0.115673
$429$	0	0
$430$	−590.587	−0.0662340
$431$	4879.45	0.545324	0.272662	−	0.962110i	$-0.412096\pi$
$431$	4879.45	0.545324	0.272662	+	0.962110i	$0.412096\pi$
$432$	0	0
$433$	−6288.61	−0.697948	−0.348974	−	0.937132i	$-0.613470\pi$
$433$	−6288.61	−0.697948	−0.348974	+	0.937132i	$0.613470\pi$
$434$	77.4895	0.00857054
$435$	0	0
$436$	11243.7	1.23503
$437$	14151.7	1.54912
$438$	0	0
$439$	10850.4	1.17964	0.589821	−	0.807534i	$-0.299198\pi$
$439$	10850.4	1.17964	0.589821	+	0.807534i	$0.299198\pi$
$440$	−6373.49	−0.690555
$441$	0	0
$442$	−4661.20	−0.501608
$443$	1904.29	0.204233	0.102117	−	0.994772i	$-0.467438\pi$
$443$	1904.29	0.204233	0.102117	+	0.994772i	$0.467438\pi$
$444$	0	0
$445$	1810.34	0.192850
$446$	4389.86	0.466067
$447$	0	0
$448$	−161.458	−0.0170272
$449$	8515.42	0.895028	0.447514	−	0.894277i	$-0.352309\pi$
$449$	8515.42	0.895028	0.447514	+	0.894277i	$0.352309\pi$
$450$	0	0
$451$	−10510.2	−1.09735
$452$	−7689.67	−0.800203
$453$	0	0
$454$	−742.427	−0.0767485
$455$	310.540	0.0319964
$456$	0	0
$457$	9127.30	0.934260	0.467130	−	0.884189i	$-0.345288\pi$
$457$	9127.30	0.934260	0.467130	+	0.884189i	$0.345288\pi$
$458$	−1670.85	−0.170467
$459$	0	0
$460$	−5008.06	−0.507613
$461$	10749.3	1.08600	0.543000	−	0.839732i	$-0.317288\pi$
$461$	10749.3	1.08600	0.543000	+	0.839732i	$0.317288\pi$
$462$	0	0
$463$	−2847.59	−0.285829	−0.142914	−	0.989735i	$-0.545647\pi$
$463$	−2847.59	−0.285829	−0.142914	+	0.989735i	$0.545647\pi$
$464$	7280.92	0.728466
$465$	0	0
$466$	−4494.93	−0.446832
$467$	−11887.8	−1.17795	−0.588974	−	0.808152i	$-0.700468\pi$
$467$	−11887.8	−1.17795	−0.588974	+	0.808152i	$0.700468\pi$
$468$	0	0
$469$	−557.712	−0.0549099
$470$	1384.23	0.135850
$471$	0	0
$472$	671.213	0.0654557
$473$	−7506.77	−0.729729
$474$	0	0
$475$	−10984.0	−1.06101
$476$	321.613	0.0309687
$477$	0	0
$478$	190.621	0.0182402
$479$	14483.0	1.38151	0.690755	−	0.723089i	$-0.257278\pi$
$479$	14483.0	1.38151	0.690755	+	0.723089i	$0.257278\pi$
$480$	0	0
$481$	−22005.9	−2.08603
$482$	−3514.55	−0.332124
$483$	0	0
$484$	−29027.3	−2.72608
$485$	−11567.8	−1.08302
$486$	0	0
$487$	−10535.6	−0.980313	−0.490157	−	0.871634i	$-0.663060\pi$
$487$	−10535.6	−0.980313	−0.490157	+	0.871634i	$0.663060\pi$
$488$	−2208.51	−0.204866
$489$	0	0
$490$	−1957.66	−0.180486
$491$	−965.419	−0.0887347	−0.0443674	−	0.999015i	$-0.514127\pi$
$491$	−965.419	−0.0887347	−0.0443674	+	0.999015i	$0.514127\pi$
$492$	0	0
$493$	−11369.2	−1.03863
$494$	9054.51	0.824660
$495$	0	0
$496$	8387.14	0.759262
$497$	97.0674	0.00876070
$498$	0	0
$499$	−2864.98	−0.257022	−0.128511	−	0.991708i	$-0.541020\pi$
$499$	−2864.98	−0.257022	−0.128511	+	0.991708i	$0.541020\pi$
$500$	10480.3	0.937386
$501$	0	0
$502$	5231.13	0.465093
$503$	6587.92	0.583977	0.291989	−	0.956422i	$-0.405683\pi$
$503$	6587.92	0.583977	0.291989	+	0.956422i	$0.405683\pi$
$504$	0	0
$505$	−8416.24	−0.741620
$506$	5498.93	0.483117
$507$	0	0
$508$	−16885.5	−1.47475
$509$	−20142.0	−1.75398	−0.876991	−	0.480507i	$-0.840453\pi$
$509$	−20142.0	−1.75398	−0.876991	+	0.480507i	$0.840453\pi$
$510$	0	0
$511$	526.419	0.0455722
$512$	11559.8	0.997800
$513$	0	0
$514$	−3665.11	−0.314516
$515$	3453.05	0.295455
$516$	0	0
$517$	17594.5	1.49672
$518$	−131.164	−0.0111255
$519$	0	0
$520$	−6685.31	−0.563789
$521$	6329.92	0.532282	0.266141	−	0.963934i	$-0.414251\pi$
$521$	6329.92	0.532282	0.266141	+	0.963934i	$0.414251\pi$
$522$	0	0
$523$	−14099.9	−1.17887	−0.589433	−	0.807817i	$-0.700649\pi$
$523$	−14099.9	−1.17887	−0.589433	+	0.807817i	$0.700649\pi$
$524$	−19153.2	−1.59678
$525$	0	0
$526$	1479.23	0.122619
$527$	−13096.6	−1.08254
$528$	0	0
$529$	−3152.01	−0.259062
$530$	946.115	0.0775408
$531$	0	0
$532$	−624.742	−0.0509135
$533$	−11024.4	−0.895907
$534$	0	0
$535$	996.258	0.0805084
$536$	12006.4	0.967534
$537$	0	0
$538$	−5013.48	−0.401759
$539$	−24883.2	−1.98849
$540$	0	0
$541$	4421.64	0.351389	0.175694	−	0.984445i	$-0.443783\pi$
$541$	4421.64	0.351389	0.175694	+	0.984445i	$0.443783\pi$
$542$	1358.06	0.107627
$543$	0	0
$544$	−10528.9	−0.829819
$545$	−10936.6	−0.859583
$546$	0	0
$547$	−7653.86	−0.598273	−0.299137	−	0.954210i	$-0.596699\pi$
$547$	−7653.86	−0.598273	−0.299137	+	0.954210i	$0.596699\pi$
$548$	7686.57	0.599186
$549$	0	0
$550$	−4268.08	−0.330894
$551$	22085.0	1.70754
$552$	0	0
$553$	−112.154	−0.00862436
$554$	−2181.32	−0.167284
$555$	0	0
$556$	16647.4	1.26980
$557$	21063.4	1.60231	0.801153	−	0.598459i	$-0.204220\pi$
$557$	21063.4	1.60231	0.801153	+	0.598459i	$0.204220\pi$
$558$	0	0
$559$	−7874.04	−0.595772
$560$	200.338	0.0151176
$561$	0	0
$562$	−705.557	−0.0529575
$563$	−3809.28	−0.285154	−0.142577	−	0.989784i	$-0.545539\pi$
$563$	−3809.28	−0.285154	−0.142577	+	0.989784i	$0.545539\pi$
$564$	0	0
$565$	7479.67	0.556942
$566$	−368.276	−0.0273495
$567$	0	0
$568$	−2089.67	−0.154367
$569$	4844.78	0.356948	0.178474	−	0.983945i	$-0.442884\pi$
$569$	4844.78	0.356948	0.178474	+	0.983945i	$0.442884\pi$
$570$	0	0
$571$	−20776.4	−1.52270	−0.761352	−	0.648339i	$-0.775464\pi$
$571$	−20776.4	−1.52270	−0.761352	+	0.648339i	$0.775464\pi$
$572$	−40728.3	−2.97716
$573$	0	0
$574$	−65.7096	−0.00477816
$575$	−6997.12	−0.507479
$576$	0	0
$577$	2751.09	0.198491	0.0992455	−	0.995063i	$-0.468357\pi$
$577$	2751.09	0.198491	0.0992455	+	0.995063i	$0.468357\pi$
$578$	777.080	0.0559209
$579$	0	0
$580$	−7815.55	−0.559523
$581$	417.516	0.0298132
$582$	0	0
$583$	12025.8	0.854301
$584$	−11332.7	−0.803001
$585$	0	0
$586$	−17.1157	−0.00120656
$587$	−200.790	−0.0141184	−0.00705919	−	0.999975i	$-0.502247\pi$
$587$	−200.790	−0.0141184	−0.00705919	+	0.999975i	$0.502247\pi$
$588$	0	0
$589$	25440.5	1.77972
$590$	−312.925	−0.0218355
$591$	0	0
$592$	−14196.6	−0.985604
$593$	−19348.1	−1.33985	−0.669926	−	0.742428i	$-0.733674\pi$
$593$	−19348.1	−1.33985	−0.669926	+	0.742428i	$0.733674\pi$
$594$	0	0
$595$	−312.830	−0.0215543
$596$	−20465.3	−1.40653
$597$	0	0
$598$	5767.97	0.394431
$599$	5949.11	0.405800	0.202900	−	0.979199i	$-0.434963\pi$
$599$	5949.11	0.405800	0.202900	+	0.979199i	$0.434963\pi$
$600$	0	0
$601$	15088.9	1.02411	0.512054	−	0.858953i	$-0.328885\pi$
$601$	15088.9	1.02411	0.512054	+	0.858953i	$0.328885\pi$
$602$	−46.9324	−0.00317744
$603$	0	0
$604$	15921.7	1.07259
$605$	28234.6	1.89735
$606$	0	0
$607$	24315.1	1.62589	0.812947	−	0.582338i	$-0.197862\pi$
$607$	24315.1	1.62589	0.812947	+	0.582338i	$0.197862\pi$
$608$	20452.6	1.36425
$609$	0	0
$610$	1029.63	0.0683415
$611$	18455.3	1.22197
$612$	0	0
$613$	−22931.1	−1.51090	−0.755449	−	0.655208i	$-0.772581\pi$
$613$	−22931.1	−1.51090	−0.755449	+	0.655208i	$0.772581\pi$
$614$	7165.56	0.470974
$615$	0	0
$616$	−506.484	−0.0331280
$617$	−23058.2	−1.50452	−0.752259	−	0.658867i	$-0.771036\pi$
$617$	−23058.2	−1.50452	−0.752259	+	0.658867i	$0.771036\pi$
$618$	0	0
$619$	5892.07	0.382589	0.191294	−	0.981533i	$-0.438731\pi$
$619$	5892.07	0.382589	0.191294	+	0.981533i	$0.438731\pi$
$620$	−9003.00	−0.583176
$621$	0	0
$622$	4400.90	0.283698
$623$	143.863	0.00925160
$624$	0	0
$625$	−982.228	−0.0628626
$626$	−7769.55	−0.496060
$627$	0	0
$628$	−1815.40	−0.115354
$629$	22168.2	1.40525
$630$	0	0
$631$	−6185.97	−0.390269	−0.195134	−	0.980777i	$-0.562514\pi$
$631$	−6185.97	−0.390269	−0.195134	+	0.980777i	$0.562514\pi$
$632$	2414.45	0.151965
$633$	0	0
$634$	1558.32	0.0976164
$635$	16424.4	1.02643
$636$	0	0
$637$	−26100.6	−1.62346
$638$	8581.60	0.532522
$639$	0	0
$640$	−9483.59	−0.585737
$641$	18486.0	1.13909	0.569544	−	0.821961i	$-0.307120\pi$
$641$	18486.0	1.13909	0.569544	+	0.821961i	$0.307120\pi$
$642$	0	0
$643$	1694.35	0.103917	0.0519585	−	0.998649i	$-0.483454\pi$
$643$	1694.35	0.103917	0.0519585	+	0.998649i	$0.483454\pi$
$644$	−397.977	−0.0243517
$645$	0	0
$646$	−9121.28	−0.555530
$647$	−24274.3	−1.47499	−0.737496	−	0.675352i	$-0.763992\pi$
$647$	−24274.3	−1.47499	−0.737496	+	0.675352i	$0.763992\pi$
$648$	0	0
$649$	−3977.50	−0.240571
$650$	−4476.89	−0.270151
$651$	0	0
$652$	−7674.34	−0.460967
$653$	−7752.11	−0.464569	−0.232285	−	0.972648i	$-0.574620\pi$
$653$	−7752.11	−0.464569	−0.232285	+	0.972648i	$0.574620\pi$
$654$	0	0
$655$	18630.1	1.11136
$656$	−7112.13	−0.423296
$657$	0	0
$658$	110.001	0.00651715
$659$	19354.1	1.14405	0.572024	−	0.820237i	$-0.306159\pi$
$659$	19354.1	1.14405	0.572024	+	0.820237i	$0.306159\pi$
$660$	0	0
$661$	28603.0	1.68310	0.841550	−	0.540179i	$-0.181643\pi$
$661$	28603.0	1.68310	0.841550	+	0.540179i	$0.181643\pi$
$662$	−1718.69	−0.100905
$663$	0	0
$664$	−8988.29	−0.525321
$665$	607.681	0.0354359
$666$	0	0
$667$	14068.8	0.816708
$668$	−21834.6	−1.26468
$669$	0	0
$670$	−5597.49	−0.322761
$671$	13087.3	0.752948
$672$	0	0
$673$	−20019.1	−1.14663	−0.573314	−	0.819335i	$-0.694343\pi$
$673$	−20019.1	−1.14663	−0.573314	+	0.819335i	$0.694343\pi$
$674$	−8340.28	−0.476640
$675$	0	0
$676$	−26542.5	−1.51016
$677$	3795.45	0.215467	0.107733	−	0.994180i	$-0.465641\pi$
$677$	3795.45	0.215467	0.107733	+	0.994180i	$0.465641\pi$
$678$	0	0
$679$	−919.262	−0.0519559
$680$	6734.61	0.379795
$681$	0	0
$682$	9885.45	0.555034
$683$	20610.6	1.15468	0.577338	−	0.816506i	$-0.304092\pi$
$683$	20610.6	1.15468	0.577338	+	0.816506i	$0.304092\pi$
$684$	0	0
$685$	−7476.65	−0.417034
$686$	−311.287	−0.0173251
$687$	0	0
$688$	−5079.77	−0.281489
$689$	12614.1	0.697476
$690$	0	0
$691$	−3316.43	−0.182580	−0.0912902	−	0.995824i	$-0.529099\pi$
$691$	−3316.43	−0.182580	−0.0912902	+	0.995824i	$0.529099\pi$
$692$	18889.6	1.03768
$693$	0	0
$694$	3008.90	0.164577
$695$	−16192.7	−0.883778
$696$	0	0
$697$	11105.7	0.603525
$698$	8303.08	0.450252
$699$	0	0
$700$	308.896	0.0166788
$701$	27084.7	1.45931	0.729655	−	0.683815i	$-0.239681\pi$
$701$	27084.7	1.45931	0.729655	+	0.683815i	$0.239681\pi$
$702$	0	0
$703$	−43062.2	−2.31027
$704$	−20597.5	−1.10269
$705$	0	0
$706$	489.039	0.0260697
$707$	−668.817	−0.0355777
$708$	0	0
$709$	−8267.67	−0.437939	−0.218970	−	0.975732i	$-0.570270\pi$
$709$	−8267.67	−0.437939	−0.218970	+	0.975732i	$0.570270\pi$
$710$	974.220	0.0514955
$711$	0	0
$712$	−3097.08	−0.163017
$713$	16206.3	0.851234
$714$	0	0
$715$	39616.0	2.07211
$716$	−7568.79	−0.395055
$717$	0	0
$718$	−3611.79	−0.187731
$719$	−4927.60	−0.255589	−0.127795	−	0.991801i	$-0.540790\pi$
$719$	−4927.60	−0.255589	−0.127795	+	0.991801i	$0.540790\pi$
$720$	0	0
$721$	274.405	0.0141739
$722$	12247.8	0.631322
$723$	0	0
$724$	15477.7	0.794510
$725$	−10919.7	−0.559374
$726$	0	0
$727$	−2081.33	−0.106179	−0.0530896	−	0.998590i	$-0.516907\pi$
$727$	−2081.33	−0.106179	−0.0530896	+	0.998590i	$0.516907\pi$
$728$	−531.264	−0.0270466
$729$	0	0
$730$	5283.42	0.267874
$731$	7932.10	0.401340
$732$	0	0
$733$	12721.4	0.641031	0.320515	−	0.947243i	$-0.396144\pi$
$733$	12721.4	0.641031	0.320515	+	0.947243i	$0.396144\pi$
$734$	−10533.2	−0.529682
$735$	0	0
$736$	13028.9	0.652514
$737$	−71148.1	−3.55600
$738$	0	0
$739$	−14139.2	−0.703814	−0.351907	−	0.936035i	$-0.614467\pi$
$739$	−14139.2	−0.703814	−0.351907	+	0.936035i	$0.614467\pi$
$740$	15239.1	0.757026
$741$	0	0
$742$	75.1853	0.00371986
$743$	14856.2	0.733542	0.366771	−	0.930311i	$-0.380463\pi$
$743$	14856.2	0.733542	0.366771	+	0.930311i	$0.380463\pi$
$744$	0	0
$745$	19906.4	0.978945
$746$	9090.61	0.446154
$747$	0	0
$748$	41028.7	2.00556
$749$	79.1700	0.00386223
$750$	0	0
$751$	−18379.4	−0.893040	−0.446520	−	0.894774i	$-0.647337\pi$
$751$	−18379.4	−0.893040	−0.446520	+	0.894774i	$0.647337\pi$
$752$	11906.0	0.577352
$753$	0	0
$754$	9001.46	0.434766
$755$	−15486.9	−0.746525
$756$	0	0
$757$	−6223.62	−0.298813	−0.149406	−	0.988776i	$-0.547736\pi$
$757$	−6223.62	−0.298813	−0.149406	+	0.988776i	$0.547736\pi$
$758$	10754.5	0.515333
$759$	0	0
$760$	−13082.2	−0.624394
$761$	−23019.0	−1.09650	−0.548251	−	0.836314i	$-0.684706\pi$
$761$	−23019.0	−1.09650	−0.548251	+	0.836314i	$0.684706\pi$
$762$	0	0
$763$	−869.103	−0.0412368
$764$	−9071.96	−0.429597
$765$	0	0
$766$	6096.97	0.287588
$767$	−4172.09	−0.196409
$768$	0	0
$769$	−487.629	−0.0228665	−0.0114332	−	0.999935i	$-0.503639\pi$
$769$	−487.629	−0.0228665	−0.0114332	+	0.999935i	$0.503639\pi$
$770$	236.127	0.0110512
$771$	0	0
$772$	15271.0	0.711938
$773$	4971.70	0.231332	0.115666	−	0.993288i	$-0.463100\pi$
$773$	4971.70	0.231332	0.115666	+	0.993288i	$0.463100\pi$
$774$	0	0
$775$	−12578.7	−0.583022
$776$	19789.9	0.915484
$777$	0	0
$778$	2812.66	0.129613
$779$	−21573.0	−0.992214
$780$	0	0
$781$	12383.0	0.567349
$782$	−5810.50	−0.265707
$783$	0	0
$784$	−16838.3	−0.767049
$785$	1765.82	0.0802864
$786$	0	0
$787$	5173.20	0.234313	0.117157	−	0.993113i	$-0.462622\pi$
$787$	5173.20	0.234313	0.117157	+	0.993113i	$0.462622\pi$
$788$	−28214.8	−1.27552
$789$	0	0
$790$	−1125.64	−0.0506941
$791$	594.390	0.0267182
$792$	0	0
$793$	13727.6	0.614728
$794$	2566.55	0.114715
$795$	0	0
$796$	13171.9	0.586512
$797$	−8153.67	−0.362381	−0.181191	−	0.983448i	$-0.557995\pi$
$797$	−8153.67	−0.362381	−0.181191	+	0.983448i	$0.557995\pi$
$798$	0	0
$799$	−18591.4	−0.823175
$800$	−10112.5	−0.446916
$801$	0	0
$802$	−12132.5	−0.534181
$803$	67156.0	2.95129
$804$	0	0
$805$	387.109	0.0169488
$806$	10369.1	0.453146
$807$	0	0
$808$	14398.3	0.626894
$809$	−16405.6	−0.712967	−0.356484	−	0.934302i	$-0.616024\pi$
$809$	−16405.6	−0.712967	−0.356484	+	0.934302i	$0.616024\pi$
$810$	0	0
$811$	−13257.7	−0.574035	−0.287017	−	0.957925i	$-0.592664\pi$
$811$	−13257.7	−0.574035	−0.287017	+	0.957925i	$0.592664\pi$
$812$	−621.081	−0.0268420
$813$	0	0
$814$	−16732.7	−0.720494
$815$	7464.76	0.320833
$816$	0	0
$817$	−15408.3	−0.659815
$818$	−1708.61	−0.0730321
$819$	0	0
$820$	7634.37	0.325127
$821$	39513.1	1.67968	0.839841	−	0.542833i	$-0.182648\pi$
$821$	39513.1	1.67968	0.839841	+	0.542833i	$0.182648\pi$
$822$	0	0
$823$	15238.2	0.645406	0.322703	−	0.946500i	$-0.395409\pi$
$823$	15238.2	0.645406	0.322703	+	0.946500i	$0.395409\pi$
$824$	−5907.39	−0.249750
$825$	0	0
$826$	−24.8673	−0.00104751
$827$	9561.03	0.402019	0.201009	−	0.979589i	$-0.435578\pi$
$827$	9561.03	0.402019	0.201009	+	0.979589i	$0.435578\pi$
$828$	0	0
$829$	−15696.4	−0.657610	−0.328805	−	0.944398i	$-0.606646\pi$
$829$	−15696.4	−0.657610	−0.328805	+	0.944398i	$0.606646\pi$
$830$	4190.42	0.175243
$831$	0	0
$832$	−21605.2	−0.900271
$833$	26293.1	1.09364
$834$	0	0
$835$	21238.3	0.880219
$836$	−79699.2	−3.29720
$837$	0	0
$838$	−4655.31	−0.191903
$839$	30185.3	1.24209	0.621045	−	0.783775i	$-0.286708\pi$
$839$	30185.3	1.24209	0.621045	+	0.783775i	$0.286708\pi$
$840$	0	0
$841$	−2433.39	−0.0997740
$842$	−347.024	−0.0142034
$843$	0	0
$844$	−23773.0	−0.969552
$845$	25817.6	1.05107
$846$	0	0
$847$	2243.73	0.0910218
$848$	8137.75	0.329542
$849$	0	0
$850$	4509.91	0.181987
$851$	−27431.8	−1.10499
$852$	0	0
$853$	9990.04	0.400999	0.200500	−	0.979694i	$-0.435743\pi$
$853$	9990.04	0.400999	0.200500	+	0.979694i	$0.435743\pi$
$854$	81.8216	0.00327855
$855$	0	0
$856$	−1704.37	−0.0680540
$857$	−13094.2	−0.521923	−0.260962	−	0.965349i	$-0.584040\pi$
$857$	−13094.2	−0.521923	−0.260962	+	0.965349i	$0.584040\pi$
$858$	0	0
$859$	2750.52	0.109251	0.0546256	−	0.998507i	$-0.482603\pi$
$859$	2750.52	0.109251	0.0546256	+	0.998507i	$0.482603\pi$
$860$	5452.77	0.216207
$861$	0	0
$862$	3891.74	0.153774
$863$	36163.2	1.42643	0.713216	−	0.700945i	$-0.247238\pi$
$863$	36163.2	1.42643	0.713216	+	0.700945i	$0.247238\pi$
$864$	0	0
$865$	−18373.7	−0.722226
$866$	−5015.65	−0.196812
$867$	0	0
$868$	−715.445	−0.0279767
$869$	−14307.6	−0.558520
$870$	0	0
$871$	−74629.0	−2.90322
$872$	18710.1	0.726608
$873$	0	0
$874$	11287.1	0.436831
$875$	−810.097	−0.0312986
$876$	0	0
$877$	−11667.1	−0.449226	−0.224613	−	0.974448i	$-0.572112\pi$
$877$	−11667.1	−0.449226	−0.224613	+	0.974448i	$0.572112\pi$
$878$	8654.05	0.332643
$879$	0	0
$880$	25557.4	0.979024
$881$	−37363.6	−1.42885	−0.714423	−	0.699714i	$-0.753311\pi$
$881$	−37363.6	−1.42885	−0.714423	+	0.699714i	$0.753311\pi$
$882$	0	0
$883$	2230.51	0.0850086	0.0425043	−	0.999096i	$-0.486466\pi$
$883$	2230.51	0.0850086	0.0425043	+	0.999096i	$0.486466\pi$
$884$	43036.0	1.63739
$885$	0	0
$886$	1518.82	0.0575910
$887$	−40114.0	−1.51849	−0.759243	−	0.650808i	$-0.774430\pi$
$887$	−40114.0	−1.51849	−0.759243	+	0.650808i	$0.774430\pi$
$888$	0	0
$889$	1305.20	0.0492408
$890$	1443.89	0.0543811
$891$	0	0
$892$	−40530.7	−1.52138
$893$	36114.3	1.35332
$894$	0	0
$895$	7362.09	0.274958
$896$	−753.636	−0.0280996
$897$	0	0
$898$	6791.71	0.252385
$899$	25291.4	0.938283
$900$	0	0
$901$	−12707.2	−0.469853
$902$	−8382.66	−0.309437
$903$	0	0
$904$	−12796.0	−0.470785
$905$	−15055.0	−0.552979
$906$	0	0
$907$	−25611.3	−0.937606	−0.468803	−	0.883303i	$-0.655315\pi$
$907$	−25611.3	−0.937606	−0.468803	+	0.883303i	$0.655315\pi$
$908$	6854.68	0.250529
$909$	0	0
$910$	247.680	0.00902253
$911$	−5675.38	−0.206404	−0.103202	−	0.994660i	$-0.532909\pi$
$911$	−5675.38	−0.206404	−0.103202	+	0.994660i	$0.532909\pi$
$912$	0	0
$913$	53263.1	1.93073
$914$	7279.73	0.263448
$915$	0	0
$916$	15426.7	0.556454
$917$	1480.49	0.0533152
$918$	0	0
$919$	8043.82	0.288728	0.144364	−	0.989525i	$-0.453886\pi$
$919$	8043.82	0.288728	0.144364	+	0.989525i	$0.453886\pi$
$920$	−8333.68	−0.298645
$921$	0	0
$922$	8573.42	0.306237
$923$	12988.9	0.463200
$924$	0	0
$925$	21291.6	0.756825
$926$	−2271.17	−0.0805997
$927$	0	0
$928$	20332.8	0.719241
$929$	−13906.6	−0.491131	−0.245566	−	0.969380i	$-0.578974\pi$
$929$	−13906.6	−0.491131	−0.245566	+	0.969380i	$0.578974\pi$
$930$	0	0
$931$	−51075.0	−1.79798
$932$	41500.8	1.45859
$933$	0	0
$934$	−9481.44	−0.332165
$935$	−39908.2	−1.39587
$936$	0	0
$937$	−25169.6	−0.877540	−0.438770	−	0.898600i	$-0.644586\pi$
$937$	−25169.6	−0.877540	−0.438770	+	0.898600i	$0.644586\pi$
$938$	−444.818	−0.0154838
$939$	0	0
$940$	−12780.3	−0.443455
$941$	18602.8	0.644457	0.322229	−	0.946662i	$-0.395568\pi$
$941$	18602.8	0.644457	0.322229	+	0.946662i	$0.395568\pi$
$942$	0	0
$943$	−13742.6	−0.474571
$944$	−2691.54	−0.0927988
$945$	0	0
$946$	−5987.23	−0.205773
$947$	24289.7	0.833484	0.416742	−	0.909025i	$-0.363172\pi$
$947$	24289.7	0.833484	0.416742	+	0.909025i	$0.363172\pi$
$948$	0	0
$949$	70441.6	2.40952
$950$	−8760.62	−0.299191
$951$	0	0
$952$	535.181	0.0182199
$953$	−14995.7	−0.509713	−0.254857	−	0.966979i	$-0.582028\pi$
$953$	−14995.7	−0.509713	−0.254857	+	0.966979i	$0.582028\pi$
$954$	0	0
$955$	8824.21	0.299000
$956$	−1759.97	−0.0595411
$957$	0	0
$958$	11551.3	0.389567
$959$	−594.150	−0.0200064
$960$	0	0
$961$	−656.936	−0.0220515
$962$	−17551.4	−0.588232
$963$	0	0
$964$	32449.2	1.08415
$965$	−14854.0	−0.495510
$966$	0	0
$967$	−26721.9	−0.888644	−0.444322	−	0.895867i	$-0.646555\pi$
$967$	−26721.9	−0.888644	−0.444322	+	0.895867i	$0.646555\pi$
$968$	−48303.0	−1.60384
$969$	0	0
$970$	−9226.21	−0.305398
$971$	−26175.3	−0.865092	−0.432546	−	0.901612i	$-0.642385\pi$
$971$	−26175.3	−0.865092	−0.432546	+	0.901612i	$0.642385\pi$
$972$	0	0
$973$	−1286.79	−0.0423975
$974$	−8402.94	−0.276435
$975$	0	0
$976$	8856.04	0.290445
$977$	−43912.1	−1.43794	−0.718972	−	0.695039i	$-0.755387\pi$
$977$	−43912.1	−1.43794	−0.718972	+	0.695039i	$0.755387\pi$
$978$	0	0
$979$	18352.8	0.599140
$980$	18074.7	0.589158
$981$	0	0
$982$	−769.996	−0.0250220
$983$	−47646.8	−1.54598	−0.772989	−	0.634419i	$-0.781239\pi$
$983$	−47646.8	−1.54598	−0.772989	+	0.634419i	$0.781239\pi$
$984$	0	0
$985$	27444.3	0.887763
$986$	−9067.84	−0.292879
$987$	0	0
$988$	−83598.5	−2.69193
$989$	−9815.51	−0.315587
$990$	0	0
$991$	−40238.0	−1.28981	−0.644905	−	0.764263i	$-0.723103\pi$
$991$	−40238.0	−1.28981	−0.644905	+	0.764263i	$0.723103\pi$
$992$	23422.0	0.749647
$993$	0	0
$994$	77.4187	0.00247040
$995$	−12812.1	−0.408213
$996$	0	0
$997$	−18447.1	−0.585983	−0.292991	−	0.956115i	$-0.594651\pi$
$997$	−18447.1	−0.585983	−0.292991	+	0.956115i	$0.594651\pi$
$998$	−2285.04	−0.0724767
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.c.1.15		28
3.2	odd	2		717.4.a.a.1.14	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.a.1.14	✓	28	3.2	odd	2
2151.4.a.c.1.15		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)

\(f(q)\)	\(=\)	\(q+0.797577 q^{2} -7.36387 q^{4} +7.16277 q^{5} +0.569206 q^{7} -12.2539 q^{8} +O(q^{10})\)	\(q+0.797577 q^{2} -7.36387 q^{4} +7.16277 q^{5} +0.569206 q^{7} -12.2539 q^{8} +5.71286 q^{10} +72.6144 q^{11} +76.1671 q^{13} +0.453986 q^{14} +49.1376 q^{16} -76.7287 q^{17} +149.048 q^{19} -52.7457 q^{20} +57.9156 q^{22} +94.9473 q^{23} -73.6948 q^{25} +60.7491 q^{26} -4.19156 q^{28} +148.174 q^{29} +170.687 q^{31} +137.222 q^{32} -61.1971 q^{34} +4.07709 q^{35} -288.916 q^{37} +118.877 q^{38} -87.7716 q^{40} -144.739 q^{41} -103.378 q^{43} -534.723 q^{44} +75.7278 q^{46} +242.300 q^{47} -342.676 q^{49} -58.7773 q^{50} -560.884 q^{52} +165.612 q^{53} +520.120 q^{55} -6.97498 q^{56} +118.180 q^{58} -54.7756 q^{59} +180.229 q^{61} +136.136 q^{62} -283.655 q^{64} +545.567 q^{65} -979.806 q^{67} +565.021 q^{68} +3.25179 q^{70} +170.531 q^{71} +924.830 q^{73} -230.433 q^{74} -1097.57 q^{76} +41.3326 q^{77} -197.036 q^{79} +351.961 q^{80} -115.441 q^{82} +733.506 q^{83} -549.590 q^{85} -82.4523 q^{86} -889.808 q^{88} +252.743 q^{89} +43.3548 q^{91} -699.180 q^{92} +193.253 q^{94} +1067.59 q^{95} -1614.99 q^{97} -273.311 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) \) 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

L-functions

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Database

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