LMFDB - Embedded newform 2151.4.a.g.1.48

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			1.48
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)

$f(q)$	$=$	$q+3.78499 q^{2} +6.32612 q^{4} +5.39134 q^{5} +9.96233 q^{7} -6.33562 q^{8} +O(q^{10})$	\(q+3.78499 q^{2} +6.32612 q^{4} +5.39134 q^{5} +9.96233 q^{7} -6.33562 q^{8} +20.4061 q^{10} -23.9107 q^{11} -68.8672 q^{13} +37.7073 q^{14} -74.5892 q^{16} +101.415 q^{17} -4.75784 q^{19} +34.1062 q^{20} -90.5017 q^{22} +138.385 q^{23} -95.9335 q^{25} -260.661 q^{26} +63.0229 q^{28} +147.063 q^{29} -284.339 q^{31} -231.634 q^{32} +383.854 q^{34} +53.7103 q^{35} +299.816 q^{37} -18.0083 q^{38} -34.1575 q^{40} -128.369 q^{41} -253.017 q^{43} -151.262 q^{44} +523.784 q^{46} -113.991 q^{47} -243.752 q^{49} -363.107 q^{50} -435.662 q^{52} -472.814 q^{53} -128.911 q^{55} -63.1175 q^{56} +556.632 q^{58} -756.581 q^{59} -476.984 q^{61} -1076.22 q^{62} -280.018 q^{64} -371.286 q^{65} -50.4211 q^{67} +641.563 q^{68} +203.293 q^{70} +722.838 q^{71} +145.341 q^{73} +1134.80 q^{74} -30.0986 q^{76} -238.206 q^{77} -632.943 q^{79} -402.135 q^{80} -485.877 q^{82} +401.146 q^{83} +546.762 q^{85} -957.667 q^{86} +151.489 q^{88} -1337.22 q^{89} -686.078 q^{91} +875.438 q^{92} -431.455 q^{94} -25.6511 q^{95} -367.601 q^{97} -922.598 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.78499	1.33819	0.669097	−	0.743175i	$-0.266681\pi$
$2$	3.78499	1.33819	0.669097	+	0.743175i	$0.266681\pi$
$3$	0	0
$3$	0	0
$4$	6.32612	0.790765
$5$	5.39134	0.482216	0.241108	−	0.970498i	$-0.422489\pi$
$5$	5.39134	0.482216	0.241108	+	0.970498i	$0.422489\pi$
$6$	0	0
$7$	9.96233	0.537915	0.268958	−	0.963152i	$-0.413321\pi$
$7$	9.96233	0.537915	0.268958	+	0.963152i	$0.413321\pi$
$8$	−6.33562	−0.279997
$9$	0	0
$10$	20.4061	0.645299
$11$	−23.9107	−0.655395	−0.327698	−	0.944783i	$-0.606273\pi$
$11$	−23.9107	−0.655395	−0.327698	+	0.944783i	$0.606273\pi$
$12$	0	0
$13$	−68.8672	−1.46926	−0.734628	−	0.678470i	$-0.762643\pi$
$13$	−68.8672	−1.46926	−0.734628	+	0.678470i	$0.762643\pi$
$14$	37.7073	0.719835
$15$	0	0
$16$	−74.5892	−1.16546
$17$	101.415	1.44687	0.723433	−	0.690395i	$-0.242563\pi$
$17$	101.415	1.44687	0.723433	+	0.690395i	$0.242563\pi$
$18$	0	0
$19$	−4.75784	−0.0574486	−0.0287243	−	0.999587i	$-0.509144\pi$
$19$	−4.75784	−0.0574486	−0.0287243	+	0.999587i	$0.509144\pi$
$20$	34.1062	0.381319
$21$	0	0
$22$	−90.5017	−0.877046
$23$	138.385	1.25457	0.627287	−	0.778788i	$-0.284165\pi$
$23$	138.385	1.25457	0.627287	+	0.778788i	$0.284165\pi$
$24$	0	0
$25$	−95.9335	−0.767468
$26$	−260.661	−1.96615
$27$	0	0
$28$	63.0229	0.425365
$29$	147.063	0.941687	0.470843	−	0.882217i	$-0.343950\pi$
$29$	147.063	0.941687	0.470843	+	0.882217i	$0.343950\pi$
$30$	0	0
$31$	−284.339	−1.64738	−0.823689	−	0.567042i	$-0.808088\pi$
$31$	−284.339	−1.64738	−0.823689	+	0.567042i	$0.808088\pi$
$32$	−231.634	−1.27961
$33$	0	0
$34$	383.854	1.93619
$35$	53.7103	0.259391
$36$	0	0
$37$	299.816	1.33215	0.666073	−	0.745887i	$-0.267974\pi$
$37$	299.816	1.33215	0.666073	+	0.745887i	$0.267974\pi$
$38$	−18.0083	−0.0768774
$39$	0	0
$40$	−34.1575	−0.135019
$41$	−128.369	−0.488974	−0.244487	−	0.969653i	$-0.578620\pi$
$41$	−128.369	−0.488974	−0.244487	+	0.969653i	$0.578620\pi$
$42$	0	0
$43$	−253.017	−0.897321	−0.448660	−	0.893702i	$-0.648099\pi$
$43$	−253.017	−0.897321	−0.448660	+	0.893702i	$0.648099\pi$
$44$	−151.262	−0.518263
$45$	0	0
$46$	523.784	1.67886
$47$	−113.991	−0.353773	−0.176886	−	0.984231i	$-0.556603\pi$
$47$	−113.991	−0.353773	−0.176886	+	0.984231i	$0.556603\pi$
$48$	0	0
$49$	−243.752	−0.710647
$50$	−363.107	−1.02702
$51$	0	0
$52$	−435.662	−1.16184
$53$	−472.814	−1.22540	−0.612698	−	0.790317i	$-0.709916\pi$
$53$	−472.814	−1.22540	−0.612698	+	0.790317i	$0.709916\pi$
$54$	0	0
$55$	−128.911	−0.316042
$56$	−63.1175	−0.150615
$57$	0	0
$58$	556.632	1.26016
$59$	−756.581	−1.66947	−0.834733	−	0.550655i	$-0.814378\pi$
$59$	−756.581	−1.66947	−0.834733	+	0.550655i	$0.814378\pi$
$60$	0	0
$61$	−476.984	−1.00117	−0.500587	−	0.865686i	$-0.666882\pi$
$61$	−476.984	−1.00117	−0.500587	+	0.865686i	$0.666882\pi$
$62$	−1076.22	−2.20451
$63$	0	0
$64$	−280.018	−0.546910
$65$	−371.286	−0.708498
$66$	0	0
$67$	−50.4211	−0.0919390	−0.0459695	−	0.998943i	$-0.514638\pi$
$67$	−50.4211	−0.0919390	−0.0459695	+	0.998943i	$0.514638\pi$
$68$	641.563	1.14413
$69$	0	0
$70$	203.293	0.347116
$71$	722.838	1.20824	0.604121	−	0.796893i	$-0.293525\pi$
$71$	722.838	1.20824	0.604121	+	0.796893i	$0.293525\pi$
$72$	0	0
$73$	145.341	0.233026	0.116513	−	0.993189i	$-0.462828\pi$
$73$	145.341	0.233026	0.116513	+	0.993189i	$0.462828\pi$
$74$	1134.80	1.78267
$75$	0	0
$76$	−30.0986	−0.0454283
$77$	−238.206	−0.352547
$78$	0	0
$79$	−632.943	−0.901413	−0.450707	−	0.892672i	$-0.648828\pi$
$79$	−632.943	−0.901413	−0.450707	+	0.892672i	$0.648828\pi$
$80$	−402.135	−0.562001
$81$	0	0
$82$	−485.877	−0.654343
$83$	401.146	0.530500	0.265250	−	0.964180i	$-0.414546\pi$
$83$	401.146	0.530500	0.265250	+	0.964180i	$0.414546\pi$
$84$	0	0
$85$	546.762	0.697702
$86$	−957.667	−1.20079
$87$	0	0
$88$	151.489	0.183509
$89$	−1337.22	−1.59264	−0.796320	−	0.604876i	$-0.793223\pi$
$89$	−1337.22	−1.59264	−0.796320	+	0.604876i	$0.793223\pi$
$90$	0	0
$91$	−686.078	−0.790335
$92$	875.438	0.992073
$93$	0	0
$94$	−431.455	−0.473417
$95$	−25.6511	−0.0277026
$96$	0	0
$97$	−367.601	−0.384786	−0.192393	−	0.981318i	$-0.561625\pi$
$97$	−367.601	−0.384786	−0.192393	+	0.981318i	$0.561625\pi$
$98$	−922.598	−0.950984
$99$	0	0
$100$	−606.887	−0.606887
$101$	934.506	0.920662	0.460331	−	0.887747i	$-0.347731\pi$
$101$	934.506	0.920662	0.460331	+	0.887747i	$0.347731\pi$
$102$	0	0
$103$	−1470.29	−1.40653	−0.703264	−	0.710929i	$-0.748275\pi$
$103$	−1470.29	−1.40653	−0.703264	+	0.710929i	$0.748275\pi$
$104$	436.316	0.411388
$105$	0	0
$106$	−1789.59	−1.63982
$107$	1090.90	0.985618	0.492809	−	0.870138i	$-0.335970\pi$
$107$	1090.90	0.985618	0.492809	+	0.870138i	$0.335970\pi$
$108$	0	0
$109$	−169.075	−0.148573	−0.0742864	−	0.997237i	$-0.523668\pi$
$109$	−169.075	−0.148573	−0.0742864	+	0.997237i	$0.523668\pi$
$110$	−487.925	−0.422926
$111$	0	0
$112$	−743.082	−0.626917
$113$	−1473.23	−1.22646	−0.613230	−	0.789904i	$-0.710130\pi$
$113$	−1473.23	−1.22646	−0.613230	+	0.789904i	$0.710130\pi$
$114$	0	0
$115$	746.078	0.604975
$116$	930.338	0.744653
$117$	0	0
$118$	−2863.65	−2.23407
$119$	1010.33	0.778292
$120$	0	0
$121$	−759.278	−0.570457
$122$	−1805.38	−1.33977
$123$	0	0
$124$	−1798.76	−1.30269
$125$	−1191.13	−0.852301
$126$	0	0
$127$	146.269	0.102199	0.0510995	−	0.998694i	$-0.483727\pi$
$127$	146.269	0.102199	0.0510995	+	0.998694i	$0.483727\pi$
$128$	793.208	0.547737
$129$	0	0
$130$	−1405.31	−0.948108
$131$	−2900.64	−1.93458	−0.967290	−	0.253675i	$-0.918361\pi$
$131$	−2900.64	−1.93458	−0.967290	+	0.253675i	$0.918361\pi$
$132$	0	0
$133$	−47.3992	−0.0309025
$134$	−190.843	−0.123032
$135$	0	0
$136$	−642.526	−0.405119
$137$	924.185	0.576339	0.288170	−	0.957579i	$-0.406953\pi$
$137$	924.185	0.576339	0.288170	+	0.957579i	$0.406953\pi$
$138$	0	0
$139$	2798.11	1.70743	0.853714	−	0.520743i	$-0.174345\pi$
$139$	2798.11	1.70743	0.853714	+	0.520743i	$0.174345\pi$
$140$	339.778	0.205117
$141$	0	0
$142$	2735.93	1.61686
$143$	1646.66	0.962943
$144$	0	0
$145$	792.866	0.454096
$146$	550.113	0.311833
$147$	0	0
$148$	1896.67	1.05341
$149$	2584.90	1.42123	0.710615	−	0.703581i	$-0.248417\pi$
$149$	2584.90	1.42123	0.710615	+	0.703581i	$0.248417\pi$
$150$	0	0
$151$	101.170	0.0545240	0.0272620	−	0.999628i	$-0.491321\pi$
$151$	101.170	0.0545240	0.0272620	+	0.999628i	$0.491321\pi$
$152$	30.1438	0.0160855
$153$	0	0
$154$	−901.608	−0.471777
$155$	−1532.96	−0.794391
$156$	0	0
$157$	−2156.84	−1.09640	−0.548200	−	0.836348i	$-0.684687\pi$
$157$	−2156.84	−1.09640	−0.548200	+	0.836348i	$0.684687\pi$
$158$	−2395.68	−1.20627
$159$	0	0
$160$	−1248.82	−0.617048
$161$	1378.63	0.674854
$162$	0	0
$163$	2839.46	1.36444	0.682219	−	0.731148i	$-0.261015\pi$
$163$	2839.46	1.36444	0.682219	+	0.731148i	$0.261015\pi$
$164$	−812.080	−0.386663
$165$	0	0
$166$	1518.33	0.709912
$167$	−1018.41	−0.471900	−0.235950	−	0.971765i	$-0.575820\pi$
$167$	−1018.41	−0.471900	−0.235950	+	0.971765i	$0.575820\pi$
$168$	0	0
$169$	2545.69	1.15871
$170$	2069.49	0.933661
$171$	0	0
$172$	−1600.62	−0.709569
$173$	−110.834	−0.0487085	−0.0243542	−	0.999703i	$-0.507753\pi$
$173$	−110.834	−0.0487085	−0.0243542	+	0.999703i	$0.507753\pi$
$174$	0	0
$175$	−955.721	−0.412833
$176$	1783.48	0.763834
$177$	0	0
$178$	−5061.35	−2.13126
$179$	−1730.63	−0.722647	−0.361323	−	0.932441i	$-0.617675\pi$
$179$	−1730.63	−0.722647	−0.361323	+	0.932441i	$0.617675\pi$
$180$	0	0
$181$	−2118.36	−0.869926	−0.434963	−	0.900448i	$-0.643239\pi$
$181$	−2118.36	−0.869926	−0.434963	+	0.900448i	$0.643239\pi$
$182$	−2596.79	−1.05762
$183$	0	0
$184$	−876.753	−0.351277
$185$	1616.41	0.642381
$186$	0	0
$187$	−2424.90	−0.948269
$188$	−721.122	−0.279751
$189$	0	0
$190$	−97.0891	−0.0370715
$191$	−259.696	−0.0983821	−0.0491910	−	0.998789i	$-0.515664\pi$
$191$	−259.696	−0.0983821	−0.0491910	+	0.998789i	$0.515664\pi$
$192$	0	0
$193$	−1595.13	−0.594923	−0.297461	−	0.954734i	$-0.596140\pi$
$193$	−1595.13	−0.594923	−0.297461	+	0.954734i	$0.596140\pi$
$194$	−1391.37	−0.514919
$195$	0	0
$196$	−1542.00	−0.561955
$197$	−415.809	−0.150381	−0.0751907	−	0.997169i	$-0.523957\pi$
$197$	−415.809	−0.150381	−0.0751907	+	0.997169i	$0.523957\pi$
$198$	0	0
$199$	5337.78	1.90143	0.950717	−	0.310061i	$-0.100349\pi$
$199$	5337.78	1.90143	0.950717	+	0.310061i	$0.100349\pi$
$200$	607.798	0.214889
$201$	0	0
$202$	3537.09	1.23202
$203$	1465.09	0.506548
$204$	0	0
$205$	−692.083	−0.235791
$206$	−5565.04	−1.88221
$207$	0	0
$208$	5136.75	1.71235
$209$	113.763	0.0376515
$210$	0	0
$211$	2425.65	0.791414	0.395707	−	0.918377i	$-0.370500\pi$
$211$	2425.65	0.791414	0.395707	+	0.918377i	$0.370500\pi$
$212$	−2991.08	−0.969000
$213$	0	0
$214$	4129.03	1.31895
$215$	−1364.10	−0.432702
$216$	0	0
$217$	−2832.67	−0.886150
$218$	−639.946	−0.198819
$219$	0	0
$220$	−815.504	−0.249915
$221$	−6984.16	−2.12582
$222$	0	0
$223$	−2394.31	−0.718990	−0.359495	−	0.933147i	$-0.617051\pi$
$223$	−2394.31	−0.718990	−0.359495	+	0.933147i	$0.617051\pi$
$224$	−2307.61	−0.688321
$225$	0	0
$226$	−5576.16	−1.64124
$227$	3203.94	0.936796	0.468398	−	0.883518i	$-0.344831\pi$
$227$	3203.94	0.936796	0.468398	+	0.883518i	$0.344831\pi$
$228$	0	0
$229$	−1598.36	−0.461235	−0.230618	−	0.973044i	$-0.574075\pi$
$229$	−1598.36	−0.461235	−0.230618	+	0.973044i	$0.574075\pi$
$230$	2823.90	0.809575
$231$	0	0
$232$	−931.735	−0.263670
$233$	1658.01	0.466178	0.233089	−	0.972455i	$-0.425117\pi$
$233$	1658.01	0.466178	0.233089	+	0.972455i	$0.425117\pi$
$234$	0	0
$235$	−614.565	−0.170595
$236$	−4786.22	−1.32015
$237$	0	0
$238$	3824.08	1.04151
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−1185.09	−0.316756	−0.158378	−	0.987379i	$-0.550626\pi$
$241$	−1185.09	−0.316756	−0.158378	+	0.987379i	$0.550626\pi$
$242$	−2873.86	−0.763383
$243$	0	0
$244$	−3017.46	−0.791693
$245$	−1314.15	−0.342685
$246$	0	0
$247$	327.659	0.0844066
$248$	1801.46	0.461262
$249$	0	0
$250$	−4508.40	−1.14054
$251$	−3398.24	−0.854563	−0.427281	−	0.904119i	$-0.640529\pi$
$251$	−3398.24	−0.854563	−0.427281	+	0.904119i	$0.640529\pi$
$252$	0	0
$253$	−3308.87	−0.822242
$254$	553.626	0.136762
$255$	0	0
$256$	5242.42	1.27989
$257$	554.979	0.134703	0.0673514	−	0.997729i	$-0.478545\pi$
$257$	554.979	0.134703	0.0673514	+	0.997729i	$0.478545\pi$
$258$	0	0
$259$	2986.86	0.716581
$260$	−2348.80	−0.560255
$261$	0	0
$262$	−10978.9	−2.58884
$263$	−4424.11	−1.03727	−0.518635	−	0.854996i	$-0.673560\pi$
$263$	−4424.11	−1.03727	−0.518635	+	0.854996i	$0.673560\pi$
$264$	0	0
$265$	−2549.10	−0.590905
$266$	−179.405	−0.0413535
$267$	0	0
$268$	−318.970	−0.0727021
$269$	475.646	0.107809	0.0539045	−	0.998546i	$-0.482833\pi$
$269$	475.646	0.107809	0.0539045	+	0.998546i	$0.482833\pi$
$270$	0	0
$271$	−3360.51	−0.753270	−0.376635	−	0.926362i	$-0.622919\pi$
$271$	−3360.51	−0.753270	−0.376635	+	0.926362i	$0.622919\pi$
$272$	−7564.45	−1.68626
$273$	0	0
$274$	3498.03	0.771254
$275$	2293.84	0.502995
$276$	0	0
$277$	−5384.17	−1.16788	−0.583941	−	0.811796i	$-0.698490\pi$
$277$	−5384.17	−1.16788	−0.583941	+	0.811796i	$0.698490\pi$
$278$	10590.8	2.28487
$279$	0	0
$280$	−340.288	−0.0726289
$281$	5548.30	1.17788	0.588939	−	0.808178i	$-0.299546\pi$
$281$	5548.30	1.17788	0.588939	+	0.808178i	$0.299546\pi$
$282$	0	0
$283$	960.377	0.201726	0.100863	−	0.994900i	$-0.467840\pi$
$283$	960.377	0.201726	0.100863	+	0.994900i	$0.467840\pi$
$284$	4572.76	0.955434
$285$	0	0
$286$	6232.59	1.28861
$287$	−1278.86	−0.263027
$288$	0	0
$289$	5371.98	1.09342
$290$	3000.99	0.607669
$291$	0	0
$292$	919.444	0.184268
$293$	1616.62	0.322334	0.161167	−	0.986927i	$-0.448474\pi$
$293$	1616.62	0.322334	0.161167	+	0.986927i	$0.448474\pi$
$294$	0	0
$295$	−4078.98	−0.805043
$296$	−1899.52	−0.372997
$297$	0	0
$298$	9783.81	1.90188
$299$	−9530.16	−1.84329
$300$	0	0
$301$	−2520.64	−0.482683
$302$	382.928	0.0729637
$303$	0	0
$304$	354.883	0.0669538
$305$	−2571.58	−0.482782
$306$	0	0
$307$	−3044.79	−0.566043	−0.283021	−	0.959114i	$-0.591337\pi$
$307$	−3044.79	−0.566043	−0.283021	+	0.959114i	$0.591337\pi$
$308$	−1506.92	−0.278782
$309$	0	0
$310$	−5802.25	−1.06305
$311$	3266.24	0.595535	0.297767	−	0.954638i	$-0.403758\pi$
$311$	3266.24	0.595535	0.297767	+	0.954638i	$0.403758\pi$
$312$	0	0
$313$	9926.21	1.79253	0.896266	−	0.443517i	$-0.146269\pi$
$313$	9926.21	1.79253	0.896266	+	0.443517i	$0.146269\pi$
$314$	−8163.61	−1.46720
$315$	0	0
$316$	−4004.07	−0.712806
$317$	−4692.00	−0.831322	−0.415661	−	0.909520i	$-0.636450\pi$
$317$	−4692.00	−0.831322	−0.415661	+	0.909520i	$0.636450\pi$
$318$	0	0
$319$	−3516.38	−0.617177
$320$	−1509.67	−0.263729
$321$	0	0
$322$	5218.11	0.903087
$323$	−482.516	−0.0831204
$324$	0	0
$325$	6606.67	1.12761
$326$	10747.3	1.82588
$327$	0	0
$328$	813.300	0.136912
$329$	−1135.62	−0.190300
$330$	0	0
$331$	287.433	0.0477304	0.0238652	−	0.999715i	$-0.492403\pi$
$331$	287.433	0.0477304	0.0238652	+	0.999715i	$0.492403\pi$
$332$	2537.70	0.419500
$333$	0	0
$334$	−3854.68	−0.631494
$335$	−271.837	−0.0443345
$336$	0	0
$337$	818.172	0.132251	0.0661256	−	0.997811i	$-0.478936\pi$
$337$	818.172	0.132251	0.0661256	+	0.997811i	$0.478936\pi$
$338$	9635.39	1.55058
$339$	0	0
$340$	3458.88	0.551718
$341$	6798.73	1.07968
$342$	0	0
$343$	−5845.42	−0.920183
$344$	1603.02	0.251247
$345$	0	0
$346$	−419.506	−0.0651814
$347$	−6501.16	−1.00577	−0.502883	−	0.864355i	$-0.667727\pi$
$347$	−6501.16	−1.00577	−0.502883	+	0.864355i	$0.667727\pi$
$348$	0	0
$349$	301.109	0.0461833	0.0230917	−	0.999733i	$-0.492649\pi$
$349$	301.109	0.0461833	0.0230917	+	0.999733i	$0.492649\pi$
$350$	−3617.39	−0.552451
$351$	0	0
$352$	5538.53	0.838650
$353$	8116.43	1.22378	0.611890	−	0.790943i	$-0.290410\pi$
$353$	8116.43	1.22378	0.611890	+	0.790943i	$0.290410\pi$
$354$	0	0
$355$	3897.06	0.582633
$356$	−8459.40	−1.25940
$357$	0	0
$358$	−6550.43	−0.967042
$359$	3735.04	0.549103	0.274551	−	0.961572i	$-0.411471\pi$
$359$	3735.04	0.549103	0.274551	+	0.961572i	$0.411471\pi$
$360$	0	0
$361$	−6836.36	−0.996700
$362$	−8017.97	−1.16413
$363$	0	0
$364$	−4340.21	−0.624969
$365$	783.582	0.112369
$366$	0	0
$367$	−6786.94	−0.965328	−0.482664	−	0.875806i	$-0.660331\pi$
$367$	−6786.94	−0.965328	−0.482664	+	0.875806i	$0.660331\pi$
$368$	−10322.0	−1.46215
$369$	0	0
$370$	6118.08	0.859631
$371$	−4710.33	−0.659159
$372$	0	0
$373$	−8021.78	−1.11354	−0.556772	−	0.830665i	$-0.687960\pi$
$373$	−8021.78	−1.11354	−0.556772	+	0.830665i	$0.687960\pi$
$374$	−9178.22	−1.26897
$375$	0	0
$376$	722.205	0.0990555
$377$	−10127.8	−1.38358
$378$	0	0
$379$	5810.90	0.787561	0.393781	−	0.919204i	$-0.371167\pi$
$379$	5810.90	0.787561	0.393781	+	0.919204i	$0.371167\pi$
$380$	−162.272	−0.0219062
$381$	0	0
$382$	−982.947	−0.131654
$383$	−7403.68	−0.987755	−0.493878	−	0.869531i	$-0.664421\pi$
$383$	−7403.68	−0.987755	−0.493878	+	0.869531i	$0.664421\pi$
$384$	0	0
$385$	−1284.25	−0.170004
$386$	−6037.55	−0.796122
$387$	0	0
$388$	−2325.49	−0.304275
$389$	−457.630	−0.0596472	−0.0298236	−	0.999555i	$-0.509495\pi$
$389$	−457.630	−0.0596472	−0.0298236	+	0.999555i	$0.509495\pi$
$390$	0	0
$391$	14034.3	1.81520
$392$	1544.32	0.198979
$393$	0	0
$394$	−1573.83	−0.201240
$395$	−3412.41	−0.434676
$396$	0	0
$397$	−291.335	−0.0368305	−0.0184152	−	0.999830i	$-0.505862\pi$
$397$	−291.335	−0.0368305	−0.0184152	+	0.999830i	$0.505862\pi$
$398$	20203.4	2.54449
$399$	0	0
$400$	7155.60	0.894450
$401$	−3567.33	−0.444249	−0.222125	−	0.975018i	$-0.571299\pi$
$401$	−3567.33	−0.444249	−0.222125	+	0.975018i	$0.571299\pi$
$402$	0	0
$403$	19581.6	2.42042
$404$	5911.80	0.728027
$405$	0	0
$406$	5545.35	0.677860
$407$	−7168.80	−0.873082
$408$	0	0
$409$	3528.46	0.426580	0.213290	−	0.976989i	$-0.431582\pi$
$409$	3528.46	0.426580	0.213290	+	0.976989i	$0.431582\pi$
$410$	−2619.52	−0.315534
$411$	0	0
$412$	−9301.25	−1.11223
$413$	−7537.31	−0.898031
$414$	0	0
$415$	2162.71	0.255815
$416$	15952.0	1.88007
$417$	0	0
$418$	430.592	0.0503851
$419$	−3459.28	−0.403334	−0.201667	−	0.979454i	$-0.564636\pi$
$419$	−3459.28	−0.403334	−0.201667	+	0.979454i	$0.564636\pi$
$420$	0	0
$421$	15493.6	1.79361	0.896805	−	0.442426i	$-0.145882\pi$
$421$	15493.6	1.79361	0.896805	+	0.442426i	$0.145882\pi$
$422$	9181.04	1.05907
$423$	0	0
$424$	2995.57	0.343108
$425$	−9729.08	−1.11042
$426$	0	0
$427$	−4751.88	−0.538547
$428$	6901.15	0.779392
$429$	0	0
$430$	−5163.11	−0.579040
$431$	−8338.54	−0.931910	−0.465955	−	0.884808i	$-0.654289\pi$
$431$	−8338.54	−0.931910	−0.465955	+	0.884808i	$0.654289\pi$
$432$	0	0
$433$	16692.7	1.85266	0.926330	−	0.376713i	$-0.122946\pi$
$433$	16692.7	1.85266	0.926330	+	0.376713i	$0.122946\pi$
$434$	−10721.6	−1.18584
$435$	0	0
$436$	−1069.59	−0.117486
$437$	−658.412	−0.0720735
$438$	0	0
$439$	12115.7	1.31720	0.658601	−	0.752493i	$-0.271149\pi$
$439$	12115.7	1.31720	0.658601	+	0.752493i	$0.271149\pi$
$440$	816.729	0.0884909
$441$	0	0
$442$	−26434.9	−2.84476
$443$	−4961.23	−0.532088	−0.266044	−	0.963961i	$-0.585717\pi$
$443$	−4961.23	−0.532088	−0.266044	+	0.963961i	$0.585717\pi$
$444$	0	0
$445$	−7209.40	−0.767996
$446$	−9062.42	−0.962149
$447$	0	0
$448$	−2789.63	−0.294191
$449$	12845.1	1.35011	0.675056	−	0.737767i	$-0.264120\pi$
$449$	12845.1	1.35011	0.675056	+	0.737767i	$0.264120\pi$
$450$	0	0
$451$	3069.40	0.320471
$452$	−9319.84	−0.969841
$453$	0	0
$454$	12126.9	1.25362
$455$	−3698.88	−0.381112
$456$	0	0
$457$	−17334.8	−1.77437	−0.887186	−	0.461413i	$-0.847343\pi$
$457$	−17334.8	−1.77437	−0.887186	+	0.461413i	$0.847343\pi$
$458$	−6049.79	−0.617223
$459$	0	0
$460$	4719.78	0.478393
$461$	−11143.0	−1.12577	−0.562886	−	0.826535i	$-0.690309\pi$
$461$	−11143.0	−1.12577	−0.562886	+	0.826535i	$0.690309\pi$
$462$	0	0
$463$	11700.8	1.17447	0.587235	−	0.809416i	$-0.300216\pi$
$463$	11700.8	1.17447	0.587235	+	0.809416i	$0.300216\pi$
$464$	−10969.3	−1.09749
$465$	0	0
$466$	6275.53	0.623837
$467$	−10130.2	−1.00379	−0.501894	−	0.864929i	$-0.667363\pi$
$467$	−10130.2	−1.00379	−0.501894	+	0.864929i	$0.667363\pi$
$468$	0	0
$469$	−502.312	−0.0494554
$470$	−2326.12	−0.228289
$471$	0	0
$472$	4793.41	0.467446
$473$	6049.82	0.588100
$474$	0	0
$475$	456.436	0.0440899
$476$	6391.46	0.615446
$477$	0	0
$478$	904.612	0.0865606
$479$	4302.76	0.410434	0.205217	−	0.978716i	$-0.434210\pi$
$479$	4302.76	0.410434	0.205217	+	0.978716i	$0.434210\pi$
$480$	0	0
$481$	−20647.5	−1.95726
$482$	−4485.54	−0.423881
$483$	0	0
$484$	−4803.28	−0.451097
$485$	−1981.86	−0.185550
$486$	0	0
$487$	12589.1	1.17139	0.585696	−	0.810531i	$-0.300821\pi$
$487$	12589.1	1.17139	0.585696	+	0.810531i	$0.300821\pi$
$488$	3021.99	0.280326
$489$	0	0
$490$	−4974.03	−0.458580
$491$	−4301.47	−0.395362	−0.197681	−	0.980266i	$-0.563341\pi$
$491$	−4301.47	−0.395362	−0.197681	+	0.980266i	$0.563341\pi$
$492$	0	0
$493$	14914.4	1.36249
$494$	1240.18	0.112952
$495$	0	0
$496$	21208.6	1.91995
$497$	7201.15	0.649931
$498$	0	0
$499$	10328.7	0.926603	0.463302	−	0.886201i	$-0.346665\pi$
$499$	10328.7	0.926603	0.463302	+	0.886201i	$0.346665\pi$
$500$	−7535.21	−0.673970
$501$	0	0
$502$	−12862.3	−1.14357
$503$	−6664.46	−0.590763	−0.295381	−	0.955379i	$-0.595447\pi$
$503$	−6664.46	−0.590763	−0.295381	+	0.955379i	$0.595447\pi$
$504$	0	0
$505$	5038.24	0.443958
$506$	−12524.0	−1.10032
$507$	0	0
$508$	925.314	0.0808153
$509$	9576.76	0.833953	0.416977	−	0.908917i	$-0.363090\pi$
$509$	9576.76	0.833953	0.416977	+	0.908917i	$0.363090\pi$
$510$	0	0
$511$	1447.93	0.125348
$512$	13496.8	1.16500
$513$	0	0
$514$	2100.59	0.180259
$515$	−7926.85	−0.678250
$516$	0	0
$517$	2725.61	0.231861
$518$	11305.2	0.958925
$519$	0	0
$520$	2352.33	0.198378
$521$	20897.5	1.75727	0.878636	−	0.477493i	$-0.158454\pi$
$521$	20897.5	1.75727	0.878636	+	0.477493i	$0.158454\pi$
$522$	0	0
$523$	1660.23	0.138808	0.0694041	−	0.997589i	$-0.477890\pi$
$523$	1660.23	0.138808	0.0694041	+	0.997589i	$0.477890\pi$
$524$	−18349.8	−1.52980
$525$	0	0
$526$	−16745.2	−1.38807
$527$	−28836.2	−2.38353
$528$	0	0
$529$	6983.31	0.573955
$530$	−9648.30	−0.790746
$531$	0	0
$532$	−299.853	−0.0244366
$533$	8840.44	0.718428
$534$	0	0
$535$	5881.40	0.475280
$536$	319.449	0.0257427
$537$	0	0
$538$	1800.31	0.144270
$539$	5828.28	0.465755
$540$	0	0
$541$	23034.4	1.83055	0.915274	−	0.402831i	$-0.131974\pi$
$541$	23034.4	1.83055	0.915274	+	0.402831i	$0.131974\pi$
$542$	−12719.5	−1.00802
$543$	0	0
$544$	−23491.1	−1.85142
$545$	−911.539	−0.0716441
$546$	0	0
$547$	−15457.2	−1.20823	−0.604117	−	0.796896i	$-0.706474\pi$
$547$	−15457.2	−1.20823	−0.604117	+	0.796896i	$0.706474\pi$
$548$	5846.51	0.455749
$549$	0	0
$550$	8682.14	0.673105
$551$	−699.702	−0.0540986
$552$	0	0
$553$	−6305.59	−0.484884
$554$	−20379.0	−1.56285
$555$	0	0
$556$	17701.2	1.35017
$557$	13685.6	1.04107	0.520535	−	0.853840i	$-0.325733\pi$
$557$	13685.6	1.04107	0.520535	+	0.853840i	$0.325733\pi$
$558$	0	0
$559$	17424.6	1.31839
$560$	−4006.21	−0.302309
$561$	0	0
$562$	21000.2	1.57623
$563$	−180.930	−0.0135440	−0.00677202	−	0.999977i	$-0.502156\pi$
$563$	−180.930	−0.0135440	−0.00677202	+	0.999977i	$0.502156\pi$
$564$	0	0
$565$	−7942.69	−0.591418
$566$	3635.01	0.269949
$567$	0	0
$568$	−4579.63	−0.338304
$569$	990.907	0.0730070	0.0365035	−	0.999334i	$-0.488378\pi$
$569$	990.907	0.0730070	0.0365035	+	0.999334i	$0.488378\pi$
$570$	0	0
$571$	14592.9	1.06952	0.534759	−	0.845004i	$-0.320402\pi$
$571$	14592.9	1.06952	0.534759	+	0.845004i	$0.320402\pi$
$572$	10417.0	0.761461
$573$	0	0
$574$	−4840.46	−0.351981
$575$	−13275.7	−0.962845
$576$	0	0
$577$	−71.1062	−0.00513031	−0.00256516	−	0.999997i	$-0.500817\pi$
$577$	−71.1062	−0.00513031	−0.00256516	+	0.999997i	$0.500817\pi$
$578$	20332.9	1.46321
$579$	0	0
$580$	5015.77	0.359083
$581$	3996.35	0.285364
$582$	0	0
$583$	11305.3	0.803118
$584$	−920.824	−0.0652466
$585$	0	0
$586$	6118.87	0.431345
$587$	9855.47	0.692979	0.346490	−	0.938054i	$-0.387374\pi$
$587$	9855.47	0.692979	0.346490	+	0.938054i	$0.387374\pi$
$588$	0	0
$589$	1352.84	0.0946395
$590$	−15438.9	−1.07730
$591$	0	0
$592$	−22363.0	−1.55256
$593$	−3528.47	−0.244345	−0.122173	−	0.992509i	$-0.538986\pi$
$593$	−3528.47	−0.244345	−0.122173	+	0.992509i	$0.538986\pi$
$594$	0	0
$595$	5447.02	0.375304
$596$	16352.4	1.12386
$597$	0	0
$598$	−36071.5	−2.46668
$599$	18538.1	1.26452	0.632259	−	0.774757i	$-0.282128\pi$
$599$	18538.1	1.26452	0.632259	+	0.774757i	$0.282128\pi$
$600$	0	0
$601$	−4577.09	−0.310654	−0.155327	−	0.987863i	$-0.549643\pi$
$601$	−4577.09	−0.310654	−0.155327	+	0.987863i	$0.549643\pi$
$602$	−9540.60	−0.645923
$603$	0	0
$604$	640.015	0.0431156
$605$	−4093.52	−0.275083
$606$	0	0
$607$	−6748.64	−0.451267	−0.225633	−	0.974212i	$-0.572445\pi$
$607$	−6748.64	−0.451267	−0.225633	+	0.974212i	$0.572445\pi$
$608$	1102.08	0.0735117
$609$	0	0
$610$	−9733.41	−0.646056
$611$	7850.25	0.519782
$612$	0	0
$613$	3105.57	0.204621	0.102311	−	0.994753i	$-0.467376\pi$
$613$	3105.57	0.204621	0.102311	+	0.994753i	$0.467376\pi$
$614$	−11524.5	−0.757475
$615$	0	0
$616$	1509.18	0.0987123
$617$	17559.7	1.14575	0.572873	−	0.819644i	$-0.305829\pi$
$617$	17559.7	1.14575	0.572873	+	0.819644i	$0.305829\pi$
$618$	0	0
$619$	6603.88	0.428808	0.214404	−	0.976745i	$-0.431219\pi$
$619$	6603.88	0.428808	0.214404	+	0.976745i	$0.431219\pi$
$620$	−9697.71	−0.628177
$621$	0	0
$622$	12362.7	0.796942
$623$	−13321.8	−0.856705
$624$	0	0
$625$	5569.92	0.356475
$626$	37570.6	2.39876
$627$	0	0
$628$	−13644.4	−0.866994
$629$	30405.8	1.92744
$630$	0	0
$631$	−20581.4	−1.29847	−0.649234	−	0.760589i	$-0.724910\pi$
$631$	−20581.4	−1.29847	−0.649234	+	0.760589i	$0.724910\pi$
$632$	4010.09	0.252393
$633$	0	0
$634$	−17759.2	−1.11247
$635$	788.585	0.0492819
$636$	0	0
$637$	16786.5	1.04412
$638$	−13309.5	−0.825903
$639$	0	0
$640$	4276.45	0.264127
$641$	−17570.3	−1.08266	−0.541332	−	0.840809i	$-0.682080\pi$
$641$	−17570.3	−1.08266	−0.541332	+	0.840809i	$0.682080\pi$
$642$	0	0
$643$	13570.7	0.832311	0.416156	−	0.909293i	$-0.363377\pi$
$643$	13570.7	0.832311	0.416156	+	0.909293i	$0.363377\pi$
$644$	8721.40	0.533651
$645$	0	0
$646$	−1826.31	−0.111231
$647$	8755.42	0.532011	0.266006	−	0.963971i	$-0.414296\pi$
$647$	8755.42	0.532011	0.266006	+	0.963971i	$0.414296\pi$
$648$	0	0
$649$	18090.4	1.09416
$650$	25006.1	1.50896
$651$	0	0
$652$	17962.7	1.07895
$653$	−3949.95	−0.236713	−0.118356	−	0.992971i	$-0.537763\pi$
$653$	−3949.95	−0.236713	−0.118356	+	0.992971i	$0.537763\pi$
$654$	0	0
$655$	−15638.3	−0.932885
$656$	9574.97	0.569878
$657$	0	0
$658$	−4298.30	−0.254658
$659$	9579.76	0.566274	0.283137	−	0.959080i	$-0.408625\pi$
$659$	9579.76	0.566274	0.283137	+	0.959080i	$0.408625\pi$
$660$	0	0
$661$	10579.5	0.622533	0.311267	−	0.950323i	$-0.399247\pi$
$661$	10579.5	0.622533	0.311267	+	0.950323i	$0.399247\pi$
$662$	1087.93	0.0638726
$663$	0	0
$664$	−2541.51	−0.148539
$665$	−255.545	−0.0149017
$666$	0	0
$667$	20351.3	1.18142
$668$	−6442.61	−0.373162
$669$	0	0
$670$	−1028.90	−0.0593281
$671$	11405.0	0.656165
$672$	0	0
$673$	−4256.15	−0.243778	−0.121889	−	0.992544i	$-0.538895\pi$
$673$	−4256.15	−0.243778	−0.121889	+	0.992544i	$0.538895\pi$
$674$	3096.77	0.176978
$675$	0	0
$676$	16104.3	0.916268
$677$	−18996.7	−1.07844	−0.539219	−	0.842165i	$-0.681281\pi$
$677$	−18996.7	−1.07844	−0.539219	+	0.842165i	$0.681281\pi$
$678$	0	0
$679$	−3662.17	−0.206982
$680$	−3464.07	−0.195355
$681$	0	0
$682$	25733.1	1.44483
$683$	28376.8	1.58977	0.794883	−	0.606763i	$-0.207532\pi$
$683$	28376.8	1.58977	0.794883	+	0.606763i	$0.207532\pi$
$684$	0	0
$685$	4982.59	0.277920
$686$	−22124.8	−1.23138
$687$	0	0
$688$	18872.4	1.04579
$689$	32561.3	1.80042
$690$	0	0
$691$	−23570.2	−1.29761	−0.648807	−	0.760953i	$-0.724732\pi$
$691$	−23570.2	−1.29761	−0.648807	+	0.760953i	$0.724732\pi$
$692$	−701.150	−0.0385170
$693$	0	0
$694$	−24606.8	−1.34591
$695$	15085.5	0.823348
$696$	0	0
$697$	−13018.6	−0.707480
$698$	1139.69	0.0618023
$699$	0	0
$700$	−6046.01	−0.326454
$701$	−25244.6	−1.36016	−0.680082	−	0.733136i	$-0.738056\pi$
$701$	−25244.6	−1.36016	−0.680082	+	0.733136i	$0.738056\pi$
$702$	0	0
$703$	−1426.47	−0.0765298
$704$	6695.43	0.358442
$705$	0	0
$706$	30720.6	1.63765
$707$	9309.86	0.495238
$708$	0	0
$709$	14943.8	0.791576	0.395788	−	0.918342i	$-0.370472\pi$
$709$	14943.8	0.791576	0.395788	+	0.918342i	$0.370472\pi$
$710$	14750.3	0.779676
$711$	0	0
$712$	8472.11	0.445935
$713$	−39348.1	−2.06676
$714$	0	0
$715$	8877.71	0.464346
$716$	−10948.2	−0.571443
$717$	0	0
$718$	14137.1	0.734806
$719$	−25609.1	−1.32832	−0.664158	−	0.747592i	$-0.731210\pi$
$719$	−25609.1	−1.32832	−0.664158	+	0.747592i	$0.731210\pi$
$720$	0	0
$721$	−14647.6	−0.756593
$722$	−25875.5	−1.33378
$723$	0	0
$724$	−13401.0	−0.687907
$725$	−14108.3	−0.722715
$726$	0	0
$727$	−2306.17	−0.117649	−0.0588247	−	0.998268i	$-0.518735\pi$
$727$	−2306.17	−0.117649	−0.0588247	+	0.998268i	$0.518735\pi$
$728$	4346.73	0.221292
$729$	0	0
$730$	2965.85	0.150371
$731$	−25659.7	−1.29830
$732$	0	0
$733$	−14863.6	−0.748977	−0.374488	−	0.927232i	$-0.622182\pi$
$733$	−14863.6	−0.748977	−0.374488	+	0.927232i	$0.622182\pi$
$734$	−25688.5	−1.29180
$735$	0	0
$736$	−32054.6	−1.60536
$737$	1205.60	0.0602564
$738$	0	0
$739$	35857.8	1.78491	0.892457	−	0.451133i	$-0.148980\pi$
$739$	35857.8	1.78491	0.892457	+	0.451133i	$0.148980\pi$
$740$	10225.6	0.507973
$741$	0	0
$742$	−17828.5	−0.882083
$743$	−4247.42	−0.209721	−0.104861	−	0.994487i	$-0.533440\pi$
$743$	−4247.42	−0.209721	−0.104861	+	0.994487i	$0.533440\pi$
$744$	0	0
$745$	13936.1	0.685340
$746$	−30362.3	−1.49014
$747$	0	0
$748$	−15340.2	−0.749858
$749$	10867.9	0.530179
$750$	0	0
$751$	22971.0	1.11614	0.558072	−	0.829792i	$-0.311541\pi$
$751$	22971.0	1.11614	0.558072	+	0.829792i	$0.311541\pi$
$752$	8502.51	0.412307
$753$	0	0
$754$	−38333.6	−1.85150
$755$	545.443	0.0262923
$756$	0	0
$757$	12971.9	0.622818	0.311409	−	0.950276i	$-0.399199\pi$
$757$	12971.9	0.622818	0.311409	+	0.950276i	$0.399199\pi$
$758$	21994.2	1.05391
$759$	0	0
$760$	162.516	0.00775666
$761$	25835.3	1.23066	0.615328	−	0.788271i	$-0.289024\pi$
$761$	25835.3	1.23066	0.615328	+	0.788271i	$0.289024\pi$
$762$	0	0
$763$	−1684.38	−0.0799196
$764$	−1642.87	−0.0777971
$765$	0	0
$766$	−28022.8	−1.32181
$767$	52103.6	2.45287
$768$	0	0
$769$	−39553.5	−1.85479	−0.927395	−	0.374082i	$-0.877958\pi$
$769$	−39553.5	−1.85479	−0.927395	+	0.374082i	$0.877958\pi$
$770$	−4860.87	−0.227498
$771$	0	0
$772$	−10091.0	−0.470444
$773$	−28781.1	−1.33918	−0.669588	−	0.742732i	$-0.733529\pi$
$773$	−28781.1	−1.33918	−0.669588	+	0.742732i	$0.733529\pi$
$774$	0	0
$775$	27277.6	1.26431
$776$	2328.98	0.107739
$777$	0	0
$778$	−1732.12	−0.0798195
$779$	610.761	0.0280909
$780$	0	0
$781$	−17283.6	−0.791876
$782$	53119.5	2.42909
$783$	0	0
$784$	18181.3	0.828228
$785$	−11628.3	−0.528701
$786$	0	0
$787$	−17177.2	−0.778020	−0.389010	−	0.921234i	$-0.627183\pi$
$787$	−17177.2	−0.778020	−0.389010	+	0.921234i	$0.627183\pi$
$788$	−2630.46	−0.118916
$789$	0	0
$790$	−12915.9	−0.581681
$791$	−14676.8	−0.659732
$792$	0	0
$793$	32848.6	1.47098
$794$	−1102.70	−0.0492863
$795$	0	0
$796$	33767.4	1.50359
$797$	28084.7	1.24820	0.624098	−	0.781346i	$-0.285467\pi$
$797$	28084.7	1.24820	0.624098	+	0.781346i	$0.285467\pi$
$798$	0	0
$799$	−11560.4	−0.511862
$800$	22221.5	0.982059
$801$	0	0
$802$	−13502.3	−0.594492
$803$	−3475.20	−0.152724
$804$	0	0
$805$	7432.68	0.325425
$806$	74116.0	3.23899
$807$	0	0
$808$	−5920.68	−0.257783
$809$	−15309.4	−0.665329	−0.332664	−	0.943045i	$-0.607948\pi$
$809$	−15309.4	−0.665329	−0.332664	+	0.943045i	$0.607948\pi$
$810$	0	0
$811$	27565.6	1.19354	0.596768	−	0.802414i	$-0.296451\pi$
$811$	27565.6	1.19354	0.596768	+	0.802414i	$0.296451\pi$
$812$	9268.34	0.400560
$813$	0	0
$814$	−27133.8	−1.16835
$815$	15308.5	0.657954
$816$	0	0
$817$	1203.82	0.0515498
$818$	13355.2	0.570847
$819$	0	0
$820$	−4378.20	−0.186455
$821$	−15159.1	−0.644404	−0.322202	−	0.946671i	$-0.604423\pi$
$821$	−15159.1	−0.644404	−0.322202	+	0.946671i	$0.604423\pi$
$822$	0	0
$823$	−37930.0	−1.60651	−0.803253	−	0.595637i	$-0.796900\pi$
$823$	−37930.0	−1.60651	−0.803253	+	0.595637i	$0.796900\pi$
$824$	9315.22	0.393824
$825$	0	0
$826$	−28528.6	−1.20174
$827$	−30447.1	−1.28023	−0.640115	−	0.768279i	$-0.721113\pi$
$827$	−30447.1	−1.28023	−0.640115	+	0.768279i	$0.721113\pi$
$828$	0	0
$829$	−6923.16	−0.290050	−0.145025	−	0.989428i	$-0.546326\pi$
$829$	−6923.16	−0.290050	−0.145025	+	0.989428i	$0.546326\pi$
$830$	8185.83	0.342331
$831$	0	0
$832$	19284.1	0.803551
$833$	−24720.1	−1.02821
$834$	0	0
$835$	−5490.62	−0.227558
$836$	719.680	0.0297735
$837$	0	0
$838$	−13093.3	−0.539739
$839$	45271.4	1.86286	0.931432	−	0.363916i	$-0.118560\pi$
$839$	45271.4	1.86286	0.931432	+	0.363916i	$0.118560\pi$
$840$	0	0
$841$	−2761.46	−0.113226
$842$	58642.9	2.40020
$843$	0	0
$844$	15344.9	0.625822
$845$	13724.7	0.558749
$846$	0	0
$847$	−7564.18	−0.306858
$848$	35266.8	1.42814
$849$	0	0
$850$	−36824.4	−1.48596
$851$	41489.9	1.67127
$852$	0	0
$853$	6399.49	0.256875	0.128438	−	0.991718i	$-0.459004\pi$
$853$	6399.49	0.256875	0.128438	+	0.991718i	$0.459004\pi$
$854$	−17985.8	−0.720680
$855$	0	0
$856$	−6911.52	−0.275971
$857$	−39912.0	−1.59086	−0.795430	−	0.606045i	$-0.792755\pi$
$857$	−39912.0	−1.59086	−0.795430	+	0.606045i	$0.792755\pi$
$858$	0	0
$859$	2195.35	0.0871994	0.0435997	−	0.999049i	$-0.486117\pi$
$859$	2195.35	0.0871994	0.0435997	+	0.999049i	$0.486117\pi$
$860$	−8629.47	−0.342166
$861$	0	0
$862$	−31561.2	−1.24708
$863$	29140.2	1.14942	0.574708	−	0.818359i	$-0.305116\pi$
$863$	29140.2	1.14942	0.574708	+	0.818359i	$0.305116\pi$
$864$	0	0
$865$	−597.544	−0.0234880
$866$	63181.8	2.47922
$867$	0	0
$868$	−17919.8	−0.700736
$869$	15134.1	0.590782
$870$	0	0
$871$	3472.36	0.135082
$872$	1071.19	0.0416000
$873$	0	0
$874$	−2492.08	−0.0964483
$875$	−11866.4	−0.458466
$876$	0	0
$877$	−13975.7	−0.538114	−0.269057	−	0.963124i	$-0.586712\pi$
$877$	−13975.7	−0.538114	−0.269057	+	0.963124i	$0.586712\pi$
$878$	45857.8	1.76267
$879$	0	0
$880$	9615.34	0.368333
$881$	−4596.29	−0.175770	−0.0878848	−	0.996131i	$-0.528011\pi$
$881$	−4596.29	−0.175770	−0.0878848	+	0.996131i	$0.528011\pi$
$882$	0	0
$883$	9476.47	0.361165	0.180582	−	0.983560i	$-0.442202\pi$
$883$	9476.47	0.361165	0.180582	+	0.983560i	$0.442202\pi$
$884$	−44182.6	−1.68102
$885$	0	0
$886$	−18778.2	−0.712037
$887$	35132.1	1.32990	0.664949	−	0.746889i	$-0.268453\pi$
$887$	35132.1	1.32990	0.664949	+	0.746889i	$0.268453\pi$
$888$	0	0
$889$	1457.18	0.0549744
$890$	−27287.5	−1.02773
$891$	0	0
$892$	−15146.7	−0.568552
$893$	542.351	0.0203237
$894$	0	0
$895$	−9330.44	−0.348472
$896$	7902.20	0.294636
$897$	0	0
$898$	48618.7	1.80671
$899$	−41815.7	−1.55131
$900$	0	0
$901$	−47950.4	−1.77298
$902$	11617.6	0.428853
$903$	0	0
$904$	9333.84	0.343406
$905$	−11420.8	−0.419492
$906$	0	0
$907$	−17329.8	−0.634430	−0.317215	−	0.948354i	$-0.602748\pi$
$907$	−17329.8	−0.634430	−0.317215	+	0.948354i	$0.602748\pi$
$908$	20268.5	0.740785
$909$	0	0
$910$	−14000.2	−0.510002
$911$	−42270.8	−1.53731	−0.768657	−	0.639661i	$-0.779075\pi$
$911$	−42270.8	−1.53731	−0.768657	+	0.639661i	$0.779075\pi$
$912$	0	0
$913$	−9591.68	−0.347687
$914$	−65612.0	−2.37445
$915$	0	0
$916$	−10111.4	−0.364729
$917$	−28897.1	−1.04064
$918$	0	0
$919$	−18991.4	−0.681686	−0.340843	−	0.940120i	$-0.610713\pi$
$919$	−18991.4	−0.681686	−0.340843	+	0.940120i	$0.610713\pi$
$920$	−4726.87	−0.169392
$921$	0	0
$922$	−42176.1	−1.50650
$923$	−49779.8	−1.77521
$924$	0	0
$925$	−28762.4	−1.02238
$926$	44287.2	1.57167
$927$	0	0
$928$	−34064.8	−1.20499
$929$	−19692.9	−0.695483	−0.347741	−	0.937590i	$-0.613051\pi$
$929$	−19692.9	−0.695483	−0.347741	+	0.937590i	$0.613051\pi$
$930$	0	0
$931$	1159.73	0.0408257
$932$	10488.7	0.368637
$933$	0	0
$934$	−38342.6	−1.34326
$935$	−13073.5	−0.457270
$936$	0	0
$937$	41739.0	1.45523	0.727616	−	0.685984i	$-0.240628\pi$
$937$	41739.0	1.45523	0.727616	+	0.685984i	$0.240628\pi$
$938$	−1901.24	−0.0661810
$939$	0	0
$940$	−3887.81	−0.134900
$941$	8040.14	0.278535	0.139267	−	0.990255i	$-0.455525\pi$
$941$	8040.14	0.278535	0.139267	+	0.990255i	$0.455525\pi$
$942$	0	0
$943$	−17764.4	−0.613454
$944$	56432.8	1.94569
$945$	0	0
$946$	22898.5	0.786992
$947$	−14958.3	−0.513284	−0.256642	−	0.966506i	$-0.582616\pi$
$947$	−14958.3	−0.513284	−0.256642	+	0.966506i	$0.582616\pi$
$948$	0	0
$949$	−10009.2	−0.342374
$950$	1727.60	0.0590009
$951$	0	0
$952$	−6401.06	−0.217920
$953$	−21420.2	−0.728089	−0.364044	−	0.931382i	$-0.618604\pi$
$953$	−21420.2	−0.728089	−0.364044	+	0.931382i	$0.618604\pi$
$954$	0	0
$955$	−1400.11	−0.0474414
$956$	1511.94	0.0511503
$957$	0	0
$958$	16285.9	0.549241
$959$	9207.04	0.310022
$960$	0	0
$961$	51057.4	1.71385
$962$	−78150.3	−2.61920
$963$	0	0
$964$	−7497.00	−0.250479
$965$	−8599.89	−0.286881
$966$	0	0
$967$	−31297.0	−1.04079	−0.520395	−	0.853925i	$-0.674215\pi$
$967$	−31297.0	−1.04079	−0.520395	+	0.853925i	$0.674215\pi$
$968$	4810.50	0.159727
$969$	0	0
$970$	−7501.32	−0.248302
$971$	47837.1	1.58101	0.790507	−	0.612452i	$-0.209817\pi$
$971$	47837.1	1.58101	0.790507	+	0.612452i	$0.209817\pi$
$972$	0	0
$973$	27875.7	0.918451
$974$	47649.7	1.56755
$975$	0	0
$976$	35577.9	1.16682
$977$	−37480.5	−1.22734	−0.613668	−	0.789564i	$-0.710307\pi$
$977$	−37480.5	−1.22734	−0.613668	+	0.789564i	$0.710307\pi$
$978$	0	0
$979$	31973.8	1.04381
$980$	−8313.46	−0.270983
$981$	0	0
$982$	−16281.0	−0.529071
$983$	22314.5	0.724030	0.362015	−	0.932172i	$-0.382089\pi$
$983$	22314.5	0.724030	0.362015	+	0.932172i	$0.382089\pi$
$984$	0	0
$985$	−2241.77	−0.0725163
$986$	56450.7	1.82328
$987$	0	0
$988$	2072.81	0.0667458
$989$	−35013.7	−1.12575
$990$	0	0
$991$	−18810.4	−0.602957	−0.301479	−	0.953473i	$-0.597480\pi$
$991$	−18810.4	−0.602957	−0.301479	+	0.953473i	$0.597480\pi$
$992$	65862.5	2.10800
$993$	0	0
$994$	27256.3	0.869735
$995$	28777.8	0.916901
$996$	0	0
$997$	33209.9	1.05493	0.527466	−	0.849576i	$-0.323142\pi$
$997$	33209.9	1.05493	0.527466	+	0.849576i	$0.323142\pi$
$998$	39093.9	1.23998
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.48	✓	59
3.2	odd	2		2151.4.a.h.1.12	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.48	✓	59	1.1	even	1	trivial
2151.4.a.h.1.12	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)

\(f(q)\)	\(=\)	\(q+3.78499 q^{2} +6.32612 q^{4} +5.39134 q^{5} +9.96233 q^{7} -6.33562 q^{8} +O(q^{10})\)	\(q+3.78499 q^{2} +6.32612 q^{4} +5.39134 q^{5} +9.96233 q^{7} -6.33562 q^{8} +20.4061 q^{10} -23.9107 q^{11} -68.8672 q^{13} +37.7073 q^{14} -74.5892 q^{16} +101.415 q^{17} -4.75784 q^{19} +34.1062 q^{20} -90.5017 q^{22} +138.385 q^{23} -95.9335 q^{25} -260.661 q^{26} +63.0229 q^{28} +147.063 q^{29} -284.339 q^{31} -231.634 q^{32} +383.854 q^{34} +53.7103 q^{35} +299.816 q^{37} -18.0083 q^{38} -34.1575 q^{40} -128.369 q^{41} -253.017 q^{43} -151.262 q^{44} +523.784 q^{46} -113.991 q^{47} -243.752 q^{49} -363.107 q^{50} -435.662 q^{52} -472.814 q^{53} -128.911 q^{55} -63.1175 q^{56} +556.632 q^{58} -756.581 q^{59} -476.984 q^{61} -1076.22 q^{62} -280.018 q^{64} -371.286 q^{65} -50.4211 q^{67} +641.563 q^{68} +203.293 q^{70} +722.838 q^{71} +145.341 q^{73} +1134.80 q^{74} -30.0986 q^{76} -238.206 q^{77} -632.943 q^{79} -402.135 q^{80} -485.877 q^{82} +401.146 q^{83} +546.762 q^{85} -957.667 q^{86} +151.489 q^{88} -1337.22 q^{89} -686.078 q^{91} +875.438 q^{92} -431.455 q^{94} -25.6511 q^{95} -367.601 q^{97} -922.598 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 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

L-functions

Modular forms

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Database

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