LMFDB - Embedded newform 2151.4.a.g.1.20

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			1.20
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)

$f(q)$	$=$	$q-2.81951 q^{2} -0.0503577 q^{4} -16.9496 q^{5} +31.0725 q^{7} +22.6981 q^{8} +O(q^{10})$	\(q-2.81951 q^{2} -0.0503577 q^{4} -16.9496 q^{5} +31.0725 q^{7} +22.6981 q^{8} +47.7895 q^{10} +36.1823 q^{11} +62.3183 q^{13} -87.6093 q^{14} -63.5946 q^{16} +110.776 q^{17} -88.4291 q^{19} +0.853542 q^{20} -102.017 q^{22} -162.116 q^{23} +162.288 q^{25} -175.707 q^{26} -1.56474 q^{28} -266.852 q^{29} -96.2067 q^{31} -2.27890 q^{32} -312.334 q^{34} -526.666 q^{35} +330.295 q^{37} +249.327 q^{38} -384.723 q^{40} -365.779 q^{41} -277.298 q^{43} -1.82206 q^{44} +457.088 q^{46} +402.448 q^{47} +622.501 q^{49} -457.573 q^{50} -3.13821 q^{52} -599.178 q^{53} -613.276 q^{55} +705.286 q^{56} +752.393 q^{58} -217.315 q^{59} +443.210 q^{61} +271.256 q^{62} +515.182 q^{64} -1056.27 q^{65} -167.419 q^{67} -5.57842 q^{68} +1484.94 q^{70} -165.185 q^{71} +669.691 q^{73} -931.270 q^{74} +4.45309 q^{76} +1124.28 q^{77} -342.722 q^{79} +1077.90 q^{80} +1031.32 q^{82} -12.9071 q^{83} -1877.61 q^{85} +781.844 q^{86} +821.270 q^{88} -684.380 q^{89} +1936.39 q^{91} +8.16380 q^{92} -1134.71 q^{94} +1498.84 q^{95} +428.930 q^{97} -1755.15 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.81951	−0.996848	−0.498424	−	0.866933i	$-0.666088\pi$
$2$	−2.81951	−0.996848	−0.498424	+	0.866933i	$0.666088\pi$
$3$	0	0
$3$	0	0
$4$	−0.0503577	−0.00629472
$5$	−16.9496	−1.51602	−0.758008	−	0.652245i	$-0.773827\pi$
$5$	−16.9496	−1.51602	−0.758008	+	0.652245i	$0.773827\pi$
$6$	0	0
$7$	31.0725	1.67776	0.838879	−	0.544318i	$-0.183211\pi$
$7$	31.0725	1.67776	0.838879	+	0.544318i	$0.183211\pi$
$8$	22.6981	1.00312
$9$	0	0
$10$	47.7895	1.51124
$11$	36.1823	0.991763	0.495881	−	0.868390i	$-0.334845\pi$
$11$	36.1823	0.991763	0.495881	+	0.868390i	$0.334845\pi$
$12$	0	0
$13$	62.3183	1.32954	0.664769	−	0.747049i	$-0.268530\pi$
$13$	62.3183	1.32954	0.664769	+	0.747049i	$0.268530\pi$
$14$	−87.6093	−1.67247
$15$	0	0
$16$	−63.5946	−0.993666
$17$	110.776	1.58042	0.790209	−	0.612837i	$-0.209972\pi$
$17$	110.776	1.58042	0.790209	+	0.612837i	$0.209972\pi$
$18$	0	0
$19$	−88.4291	−1.06774	−0.533869	−	0.845567i	$-0.679262\pi$
$19$	−88.4291	−1.06774	−0.533869	+	0.845567i	$0.679262\pi$
$20$	0.853542	0.00954289
$21$	0	0
$22$	−102.017	−0.988636
$23$	−162.116	−1.46972	−0.734860	−	0.678219i	$-0.762752\pi$
$23$	−162.116	−1.46972	−0.734860	+	0.678219i	$0.762752\pi$
$24$	0	0
$25$	162.288	1.29831
$26$	−175.707	−1.32535
$27$	0	0
$28$	−1.56474	−0.0105610
$29$	−266.852	−1.70873	−0.854366	−	0.519672i	$-0.826054\pi$
$29$	−266.852	−1.70873	−0.854366	+	0.519672i	$0.826054\pi$
$30$	0	0
$31$	−96.2067	−0.557394	−0.278697	−	0.960379i	$-0.589903\pi$
$31$	−96.2067	−0.557394	−0.278697	+	0.960379i	$0.589903\pi$
$32$	−2.27890	−0.0125892
$33$	0	0
$34$	−312.334	−1.57544
$35$	−526.666	−2.54351
$36$	0	0
$37$	330.295	1.46757	0.733785	−	0.679381i	$-0.237752\pi$
$37$	330.295	1.46757	0.733785	+	0.679381i	$0.237752\pi$
$38$	249.327	1.06437
$39$	0	0
$40$	−384.723	−1.52075
$41$	−365.779	−1.39330	−0.696648	−	0.717413i	$-0.745326\pi$
$41$	−365.779	−1.39330	−0.696648	+	0.717413i	$0.745326\pi$
$42$	0	0
$43$	−277.298	−0.983430	−0.491715	−	0.870756i	$-0.663630\pi$
$43$	−277.298	−0.983430	−0.491715	+	0.870756i	$0.663630\pi$
$44$	−1.82206	−0.00624286
$45$	0	0
$46$	457.088	1.46509
$47$	402.448	1.24900	0.624501	−	0.781024i	$-0.285303\pi$
$47$	402.448	1.24900	0.624501	+	0.781024i	$0.285303\pi$
$48$	0	0
$49$	622.501	1.81487
$50$	−457.573	−1.29421
$51$	0	0
$52$	−3.13821	−0.00836906
$53$	−599.178	−1.55289	−0.776447	−	0.630182i	$-0.782980\pi$
$53$	−599.178	−1.55289	−0.776447	+	0.630182i	$0.782980\pi$
$54$	0	0
$55$	−613.276	−1.50353
$56$	705.286	1.68300
$57$	0	0
$58$	752.393	1.70334
$59$	−217.315	−0.479527	−0.239763	−	0.970831i	$-0.577070\pi$
$59$	−217.315	−0.479527	−0.239763	+	0.970831i	$0.577070\pi$
$60$	0	0
$61$	443.210	0.930283	0.465141	−	0.885236i	$-0.346003\pi$
$61$	443.210	0.930283	0.465141	+	0.885236i	$0.346003\pi$
$62$	271.256	0.555637
$63$	0	0
$64$	515.182	1.00622
$65$	−1056.27	−2.01560
$66$	0	0
$67$	−167.419	−0.305275	−0.152638	−	0.988282i	$-0.548777\pi$
$67$	−167.419	−0.305275	−0.152638	+	0.988282i	$0.548777\pi$
$68$	−5.57842	−0.00994828
$69$	0	0
$70$	1484.94	2.53549
$71$	−165.185	−0.276111	−0.138055	−	0.990425i	$-0.544085\pi$
$71$	−165.185	−0.276111	−0.138055	+	0.990425i	$0.544085\pi$
$72$	0	0
$73$	669.691	1.07372	0.536859	−	0.843672i	$-0.319611\pi$
$73$	669.691	1.07372	0.536859	+	0.843672i	$0.319611\pi$
$74$	−931.270	−1.46294
$75$	0	0
$76$	4.45309	0.00672111
$77$	1124.28	1.66394
$78$	0	0
$79$	−342.722	−0.488091	−0.244045	−	0.969764i	$-0.578475\pi$
$79$	−342.722	−0.488091	−0.244045	+	0.969764i	$0.578475\pi$
$80$	1077.90	1.50641
$81$	0	0
$82$	1031.32	1.38890
$83$	−12.9071	−0.0170691	−0.00853454	−	0.999964i	$-0.502717\pi$
$83$	−12.9071	−0.0170691	−0.00853454	+	0.999964i	$0.502717\pi$
$84$	0	0
$85$	−1877.61	−2.39594
$86$	781.844	0.980330
$87$	0	0
$88$	821.270	0.994859
$89$	−684.380	−0.815102	−0.407551	−	0.913182i	$-0.633617\pi$
$89$	−684.380	−0.815102	−0.407551	+	0.913182i	$0.633617\pi$
$90$	0	0
$91$	1936.39	2.23064
$92$	8.16380	0.00925146
$93$	0	0
$94$	−1134.71	−1.24506
$95$	1498.84	1.61871
$96$	0	0
$97$	428.930	0.448982	0.224491	−	0.974476i	$-0.427928\pi$
$97$	428.930	0.448982	0.224491	+	0.974476i	$0.427928\pi$
$98$	−1755.15	−1.80915
$99$	0	0
$100$	−8.17247	−0.00817247
$101$	−56.3950	−0.0555596	−0.0277798	−	0.999614i	$-0.508844\pi$
$101$	−56.3950	−0.0555596	−0.0277798	+	0.999614i	$0.508844\pi$
$102$	0	0
$103$	−517.489	−0.495046	−0.247523	−	0.968882i	$-0.579617\pi$
$103$	−517.489	−0.495046	−0.247523	+	0.968882i	$0.579617\pi$
$104$	1414.51	1.33369
$105$	0	0
$106$	1689.39	1.54800
$107$	1725.02	1.55854	0.779271	−	0.626688i	$-0.215590\pi$
$107$	1725.02	1.55854	0.779271	+	0.626688i	$0.215590\pi$
$108$	0	0
$109$	−1279.49	−1.12434	−0.562170	−	0.827022i	$-0.690033\pi$
$109$	−1279.49	−1.12434	−0.562170	+	0.827022i	$0.690033\pi$
$110$	1729.14	1.49879
$111$	0	0
$112$	−1976.04	−1.66713
$113$	−1808.38	−1.50547	−0.752733	−	0.658326i	$-0.771265\pi$
$113$	−1808.38	−1.50547	−0.752733	+	0.658326i	$0.771265\pi$
$114$	0	0
$115$	2747.80	2.22812
$116$	13.4381	0.0107560
$117$	0	0
$118$	612.723	0.478015
$119$	3442.09	2.65156
$120$	0	0
$121$	−21.8378	−0.0164071
$122$	−1249.64	−0.927350
$123$	0	0
$124$	4.84475	0.00350864
$125$	−632.020	−0.452236
$126$	0	0
$127$	−1303.72	−0.910920	−0.455460	−	0.890256i	$-0.650525\pi$
$127$	−1303.72	−0.910920	−0.455460	+	0.890256i	$0.650525\pi$
$128$	−1434.33	−0.990454
$129$	0	0
$130$	2978.16	2.00925
$131$	1647.45	1.09876	0.549382	−	0.835571i	$-0.314863\pi$
$131$	1647.45	1.09876	0.549382	+	0.835571i	$0.314863\pi$
$132$	0	0
$133$	−2747.71	−1.79141
$134$	472.038	0.304313
$135$	0	0
$136$	2514.40	1.58535
$137$	−2311.23	−1.44132	−0.720662	−	0.693287i	$-0.756162\pi$
$137$	−2311.23	−1.44132	−0.720662	+	0.693287i	$0.756162\pi$
$138$	0	0
$139$	1689.77	1.03111	0.515555	−	0.856856i	$-0.327586\pi$
$139$	1689.77	1.03111	0.515555	+	0.856856i	$0.327586\pi$
$140$	26.5217	0.0160107
$141$	0	0
$142$	465.741	0.275240
$143$	2254.82	1.31859
$144$	0	0
$145$	4523.03	2.59046
$146$	−1888.20	−1.07033
$147$	0	0
$148$	−16.6329	−0.00923794
$149$	−1466.10	−0.806090	−0.403045	−	0.915180i	$-0.632048\pi$
$149$	−1466.10	−0.806090	−0.403045	+	0.915180i	$0.632048\pi$
$150$	0	0
$151$	863.538	0.465389	0.232694	−	0.972550i	$-0.425246\pi$
$151$	863.538	0.465389	0.232694	+	0.972550i	$0.425246\pi$
$152$	−2007.17	−1.07107
$153$	0	0
$154$	−3169.91	−1.65869
$155$	1630.66	0.845019
$156$	0	0
$157$	−2610.08	−1.32680	−0.663398	−	0.748267i	$-0.730886\pi$
$157$	−2610.08	−1.32680	−0.663398	+	0.748267i	$0.730886\pi$
$158$	966.307	0.486552
$159$	0	0
$160$	38.6263	0.0190855
$161$	−5037.35	−2.46583
$162$	0	0
$163$	−4023.36	−1.93334	−0.966669	−	0.256028i	$-0.917586\pi$
$163$	−4023.36	−1.93334	−0.966669	+	0.256028i	$0.917586\pi$
$164$	18.4198	0.00877040
$165$	0	0
$166$	36.3916	0.0170153
$167$	−2991.47	−1.38615	−0.693074	−	0.720867i	$-0.743744\pi$
$167$	−2991.47	−1.38615	−0.693074	+	0.720867i	$0.743744\pi$
$168$	0	0
$169$	1686.57	0.767672
$170$	5293.93	2.38839
$171$	0	0
$172$	13.9641	0.00619041
$173$	751.064	0.330071	0.165036	−	0.986288i	$-0.447226\pi$
$173$	751.064	0.330071	0.165036	+	0.986288i	$0.447226\pi$
$174$	0	0
$175$	5042.70	2.17824
$176$	−2301.00	−0.985480
$177$	0	0
$178$	1929.62	0.812533
$179$	−3650.49	−1.52430	−0.762152	−	0.647398i	$-0.775857\pi$
$179$	−3650.49	−1.52430	−0.762152	+	0.647398i	$0.775857\pi$
$180$	0	0
$181$	2115.07	0.868573	0.434286	−	0.900775i	$-0.357001\pi$
$181$	2115.07	0.868573	0.434286	+	0.900775i	$0.357001\pi$
$182$	−5459.67	−2.22361
$183$	0	0
$184$	−3679.72	−1.47431
$185$	−5598.36	−2.22486
$186$	0	0
$187$	4008.13	1.56740
$188$	−20.2664	−0.00786211
$189$	0	0
$190$	−4225.98	−1.61361
$191$	−464.311	−0.175897	−0.0879486	−	0.996125i	$-0.528031\pi$
$191$	−464.311	−0.175897	−0.0879486	+	0.996125i	$0.528031\pi$
$192$	0	0
$193$	−3926.90	−1.46458	−0.732291	−	0.680992i	$-0.761549\pi$
$193$	−3926.90	−1.46458	−0.732291	+	0.680992i	$0.761549\pi$
$194$	−1209.37	−0.447567
$195$	0	0
$196$	−31.3477	−0.0114241
$197$	420.061	0.151919	0.0759597	−	0.997111i	$-0.475798\pi$
$197$	420.061	0.151919	0.0759597	+	0.997111i	$0.475798\pi$
$198$	0	0
$199$	−1722.80	−0.613698	−0.306849	−	0.951758i	$-0.599275\pi$
$199$	−1722.80	−0.613698	−0.306849	+	0.951758i	$0.599275\pi$
$200$	3683.63	1.30236
$201$	0	0
$202$	159.006	0.0553844
$203$	−8291.77	−2.86684
$204$	0	0
$205$	6199.81	2.11226
$206$	1459.07	0.493485
$207$	0	0
$208$	−3963.11	−1.32112
$209$	−3199.57	−1.05894
$210$	0	0
$211$	−537.623	−0.175410	−0.0877049	−	0.996147i	$-0.527953\pi$
$211$	−537.623	−0.175410	−0.0877049	+	0.996147i	$0.527953\pi$
$212$	30.1732	0.00977503
$213$	0	0
$214$	−4863.71	−1.55363
$215$	4700.08	1.49090
$216$	0	0
$217$	−2989.38	−0.935173
$218$	3607.54	1.12080
$219$	0	0
$220$	30.8832	0.00946428
$221$	6903.37	2.10123
$222$	0	0
$223$	2430.96	0.729996	0.364998	−	0.931008i	$-0.381070\pi$
$223$	2430.96	0.729996	0.364998	+	0.931008i	$0.381070\pi$
$224$	−70.8110	−0.0211217
$225$	0	0
$226$	5098.73	1.50072
$227$	4770.27	1.39477	0.697387	−	0.716695i	$-0.254346\pi$
$227$	4770.27	1.39477	0.697387	+	0.716695i	$0.254346\pi$
$228$	0	0
$229$	−68.9090	−0.0198849	−0.00994243	−	0.999951i	$-0.503165\pi$
$229$	−68.9090	−0.0198849	−0.00994243	+	0.999951i	$0.503165\pi$
$230$	−7747.45	−2.22109
$231$	0	0
$232$	−6057.03	−1.71407
$233$	281.435	0.0791306	0.0395653	−	0.999217i	$-0.487403\pi$
$233$	281.435	0.0791306	0.0395653	+	0.999217i	$0.487403\pi$
$234$	0	0
$235$	−6821.32	−1.89351
$236$	10.9435	0.00301848
$237$	0	0
$238$	−9705.00	−2.64320
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	3132.42	0.837249	0.418625	−	0.908159i	$-0.362512\pi$
$241$	3132.42	0.837249	0.418625	+	0.908159i	$0.362512\pi$
$242$	61.5719	0.0163553
$243$	0	0
$244$	−22.3191	−0.00585586
$245$	−10551.1	−2.75138
$246$	0	0
$247$	−5510.75	−1.41960
$248$	−2183.71	−0.559135
$249$	0	0
$250$	1781.99	0.450811
$251$	−3561.11	−0.895520	−0.447760	−	0.894154i	$-0.647778\pi$
$251$	−3561.11	−0.895520	−0.447760	+	0.894154i	$0.647778\pi$
$252$	0	0
$253$	−5865.74	−1.45761
$254$	3675.87	0.908049
$255$	0	0
$256$	−77.3464	−0.0188834
$257$	6080.12	1.47575	0.737874	−	0.674939i	$-0.235830\pi$
$257$	6080.12	1.47575	0.737874	+	0.674939i	$0.235830\pi$
$258$	0	0
$259$	10263.1	2.46223
$260$	53.1913	0.0126876
$261$	0	0
$262$	−4645.00	−1.09530
$263$	−2065.83	−0.484352	−0.242176	−	0.970232i	$-0.577861\pi$
$263$	−2065.83	−0.484352	−0.242176	+	0.970232i	$0.577861\pi$
$264$	0	0
$265$	10155.8	2.35421
$266$	7747.21	1.78576
$267$	0	0
$268$	8.43082	0.00192162
$269$	2495.45	0.565615	0.282807	−	0.959177i	$-0.408734\pi$
$269$	2495.45	0.565615	0.282807	+	0.959177i	$0.408734\pi$
$270$	0	0
$271$	2805.76	0.628921	0.314460	−	0.949271i	$-0.398176\pi$
$271$	2805.76	0.628921	0.314460	+	0.949271i	$0.398176\pi$
$272$	−7044.75	−1.57041
$273$	0	0
$274$	6516.53	1.43678
$275$	5871.97	1.28761
$276$	0	0
$277$	3612.67	0.783626	0.391813	−	0.920045i	$-0.371848\pi$
$277$	3612.67	0.783626	0.391813	+	0.920045i	$0.371848\pi$
$278$	−4764.32	−1.02786
$279$	0	0
$280$	−11954.3	−2.55145
$281$	−6326.51	−1.34309	−0.671544	−	0.740965i	$-0.734369\pi$
$281$	−6326.51	−1.34309	−0.671544	+	0.740965i	$0.734369\pi$
$282$	0	0
$283$	3208.97	0.674041	0.337021	−	0.941497i	$-0.390581\pi$
$283$	3208.97	0.674041	0.337021	+	0.941497i	$0.390581\pi$
$284$	8.31834	0.00173804
$285$	0	0
$286$	−6357.50	−1.31443
$287$	−11365.7	−2.33761
$288$	0	0
$289$	7358.30	1.49772
$290$	−12752.7	−2.58230
$291$	0	0
$292$	−33.7241	−0.00675875
$293$	6159.46	1.22812	0.614060	−	0.789259i	$-0.289535\pi$
$293$	6159.46	1.22812	0.614060	+	0.789259i	$0.289535\pi$
$294$	0	0
$295$	3683.41	0.726970
$296$	7497.05	1.47215
$297$	0	0
$298$	4133.68	0.803549
$299$	−10102.8	−1.95405
$300$	0	0
$301$	−8616.34	−1.64996
$302$	−2434.75	−0.463922
$303$	0	0
$304$	5623.61	1.06097
$305$	−7512.23	−1.41032
$306$	0	0
$307$	3775.22	0.701835	0.350917	−	0.936406i	$-0.385870\pi$
$307$	3775.22	0.701835	0.350917	+	0.936406i	$0.385870\pi$
$308$	−56.6160	−0.0104740
$309$	0	0
$310$	−4597.67	−0.842355
$311$	−1781.74	−0.324865	−0.162432	−	0.986720i	$-0.551934\pi$
$311$	−1781.74	−0.324865	−0.162432	+	0.986720i	$0.551934\pi$
$312$	0	0
$313$	−3065.92	−0.553661	−0.276830	−	0.960919i	$-0.589284\pi$
$313$	−3065.92	−0.553661	−0.276830	+	0.960919i	$0.589284\pi$
$314$	7359.14	1.32261
$315$	0	0
$316$	17.2587	0.00307239
$317$	6701.47	1.18736	0.593679	−	0.804702i	$-0.297675\pi$
$317$	6701.47	1.18736	0.593679	+	0.804702i	$0.297675\pi$
$318$	0	0
$319$	−9655.34	−1.69466
$320$	−8732.12	−1.52544
$321$	0	0
$322$	14202.9	2.45806
$323$	−9795.81	−1.68747
$324$	0	0
$325$	10113.5	1.72615
$326$	11343.9	1.92724
$327$	0	0
$328$	−8302.49	−1.39765
$329$	12505.1	2.09552
$330$	0	0
$331$	8836.84	1.46742	0.733711	−	0.679462i	$-0.237787\pi$
$331$	8836.84	1.46742	0.733711	+	0.679462i	$0.237787\pi$
$332$	0.649970	0.000107445 0
$333$	0	0
$334$	8434.47	1.38178
$335$	2837.67	0.462802
$336$	0	0
$337$	−11580.8	−1.87195	−0.935975	−	0.352067i	$-0.885479\pi$
$337$	−11580.8	−1.87195	−0.935975	+	0.352067i	$0.885479\pi$
$338$	−4755.32	−0.765252
$339$	0	0
$340$	94.5519	0.0150818
$341$	−3480.98	−0.552803
$342$	0	0
$343$	8684.80	1.36716
$344$	−6294.12	−0.986501
$345$	0	0
$346$	−2117.63	−0.329031
$347$	−3101.56	−0.479829	−0.239914	−	0.970794i	$-0.577119\pi$
$347$	−3101.56	−0.479829	−0.239914	+	0.970794i	$0.577119\pi$
$348$	0	0
$349$	8190.63	1.25626	0.628130	−	0.778109i	$-0.283821\pi$
$349$	8190.63	1.25626	0.628130	+	0.778109i	$0.283821\pi$
$350$	−14218.0	−2.17138
$351$	0	0
$352$	−82.4558	−0.0124855
$353$	5656.11	0.852817	0.426409	−	0.904531i	$-0.359779\pi$
$353$	5656.11	0.852817	0.426409	+	0.904531i	$0.359779\pi$
$354$	0	0
$355$	2799.82	0.418588
$356$	34.4638	0.00513084
$357$	0	0
$358$	10292.6	1.51950
$359$	8527.76	1.25370	0.626850	−	0.779140i	$-0.284344\pi$
$359$	8527.76	1.25370	0.626850	+	0.779140i	$0.284344\pi$
$360$	0	0
$361$	960.705	0.140065
$362$	−5963.45	−0.865835
$363$	0	0
$364$	−97.5121	−0.0140413
$365$	−11351.0	−1.62777
$366$	0	0
$367$	4242.70	0.603453	0.301726	−	0.953395i	$-0.402437\pi$
$367$	4242.70	0.603453	0.301726	+	0.953395i	$0.402437\pi$
$368$	10309.7	1.46041
$369$	0	0
$370$	15784.6	2.21785
$371$	−18618.0	−2.60538
$372$	0	0
$373$	5781.60	0.802573	0.401287	−	0.915953i	$-0.368563\pi$
$373$	5781.60	0.802573	0.401287	+	0.915953i	$0.368563\pi$
$374$	−11301.0	−1.56246
$375$	0	0
$376$	9134.79	1.25290
$377$	−16629.8	−2.27182
$378$	0	0
$379$	5380.51	0.729230	0.364615	−	0.931158i	$-0.381201\pi$
$379$	5380.51	0.729230	0.364615	+	0.931158i	$0.381201\pi$
$380$	−75.4780	−0.0101893
$381$	0	0
$382$	1309.13	0.175343
$383$	−2066.18	−0.275658	−0.137829	−	0.990456i	$-0.544012\pi$
$383$	−2066.18	−0.275658	−0.137829	+	0.990456i	$0.544012\pi$
$384$	0	0
$385$	−19056.0	−2.52256
$386$	11071.9	1.45997
$387$	0	0
$388$	−21.5999	−0.00282621
$389$	205.552	0.0267915	0.0133958	−	0.999910i	$-0.495736\pi$
$389$	205.552	0.0267915	0.0133958	+	0.999910i	$0.495736\pi$
$390$	0	0
$391$	−17958.6	−2.32277
$392$	14129.6	1.82054
$393$	0	0
$394$	−1184.37	−0.151441
$395$	5808.99	0.739954
$396$	0	0
$397$	−2751.58	−0.347853	−0.173927	−	0.984759i	$-0.555646\pi$
$397$	−2751.58	−0.347853	−0.173927	+	0.984759i	$0.555646\pi$
$398$	4857.44	0.611763
$399$	0	0
$400$	−10320.7	−1.29008
$401$	5733.40	0.713996	0.356998	−	0.934105i	$-0.383800\pi$
$401$	5733.40	0.713996	0.356998	+	0.934105i	$0.383800\pi$
$402$	0	0
$403$	−5995.44	−0.741077
$404$	2.83993	0.000349732 0
$405$	0	0
$406$	23378.7	2.85780
$407$	11950.8	1.45548
$408$	0	0
$409$	−11138.6	−1.34662	−0.673310	−	0.739361i	$-0.735128\pi$
$409$	−11138.6	−1.34662	−0.673310	+	0.739361i	$0.735128\pi$
$410$	−17480.4	−2.10560
$411$	0	0
$412$	26.0596	0.00311617
$413$	−6752.54	−0.804530
$414$	0	0
$415$	218.769	0.0258770
$416$	−142.017	−0.0167379
$417$	0	0
$418$	9021.23	1.05560
$419$	320.730	0.0373954	0.0186977	−	0.999825i	$-0.494048\pi$
$419$	320.730	0.0373954	0.0186977	+	0.999825i	$0.494048\pi$
$420$	0	0
$421$	−12357.2	−1.43053	−0.715265	−	0.698854i	$-0.753694\pi$
$421$	−12357.2	−1.43053	−0.715265	+	0.698854i	$0.753694\pi$
$422$	1515.83	0.174857
$423$	0	0
$424$	−13600.2	−1.55774
$425$	17977.6	2.05187
$426$	0	0
$427$	13771.7	1.56079
$428$	−86.8680	−0.00981057
$429$	0	0
$430$	−13251.9	−1.48620
$431$	3781.20	0.422585	0.211292	−	0.977423i	$-0.432233\pi$
$431$	3781.20	0.422585	0.211292	+	0.977423i	$0.432233\pi$
$432$	0	0
$433$	−6014.55	−0.667531	−0.333766	−	0.942656i	$-0.608319\pi$
$433$	−6014.55	−0.667531	−0.333766	+	0.942656i	$0.608319\pi$
$434$	8428.60	0.932225
$435$	0	0
$436$	64.4322	0.00707740
$437$	14335.8	1.56928
$438$	0	0
$439$	−10883.5	−1.18323	−0.591617	−	0.806219i	$-0.701510\pi$
$439$	−10883.5	−1.18323	−0.591617	+	0.806219i	$0.701510\pi$
$440$	−13920.2	−1.50822
$441$	0	0
$442$	−19464.1	−2.09460
$443$	−1048.37	−0.112437	−0.0562186	−	0.998418i	$-0.517904\pi$
$443$	−1048.37	−0.112437	−0.0562186	+	0.998418i	$0.517904\pi$
$444$	0	0
$445$	11599.9	1.23571
$446$	−6854.12	−0.727695
$447$	0	0
$448$	16008.0	1.68819
$449$	−2304.99	−0.242270	−0.121135	−	0.992636i	$-0.538653\pi$
$449$	−2304.99	−0.242270	−0.121135	+	0.992636i	$0.538653\pi$
$450$	0	0
$451$	−13234.8	−1.38182
$452$	91.0657	0.00947648
$453$	0	0
$454$	−13449.8	−1.39038
$455$	−32820.9	−3.38169
$456$	0	0
$457$	−13364.0	−1.36792	−0.683961	−	0.729519i	$-0.739744\pi$
$457$	−13364.0	−1.36792	−0.683961	+	0.729519i	$0.739744\pi$
$458$	194.290	0.0198222
$459$	0	0
$460$	−138.373	−0.0140254
$461$	5105.41	0.515798	0.257899	−	0.966172i	$-0.416970\pi$
$461$	5105.41	0.515798	0.257899	+	0.966172i	$0.416970\pi$
$462$	0	0
$463$	−6617.88	−0.664274	−0.332137	−	0.943231i	$-0.607770\pi$
$463$	−6617.88	−0.664274	−0.332137	+	0.943231i	$0.607770\pi$
$464$	16970.4	1.69791
$465$	0	0
$466$	−793.510	−0.0788812
$467$	2186.16	0.216624	0.108312	−	0.994117i	$-0.465455\pi$
$467$	2186.16	0.216624	0.108312	+	0.994117i	$0.465455\pi$
$468$	0	0
$469$	−5202.12	−0.512178
$470$	19232.8	1.88754
$471$	0	0
$472$	−4932.64	−0.481024
$473$	−10033.3	−0.975329
$474$	0	0
$475$	−14351.0	−1.38625
$476$	−173.336	−0.0166908
$477$	0	0
$478$	−673.863	−0.0644807
$479$	9850.30	0.939607	0.469803	−	0.882771i	$-0.344325\pi$
$479$	9850.30	0.939607	0.469803	+	0.882771i	$0.344325\pi$
$480$	0	0
$481$	20583.4	1.95119
$482$	−8831.90	−0.834610
$483$	0	0
$484$	1.09970	0.000103278 0
$485$	−7270.18	−0.680664
$486$	0	0
$487$	3207.50	0.298451	0.149226	−	0.988803i	$-0.452322\pi$
$487$	3207.50	0.298451	0.149226	+	0.988803i	$0.452322\pi$
$488$	10060.0	0.933188
$489$	0	0
$490$	29749.0	2.74270
$491$	14987.9	1.37758	0.688792	−	0.724959i	$-0.258141\pi$
$491$	14987.9	1.37758	0.688792	+	0.724959i	$0.258141\pi$
$492$	0	0
$493$	−29560.8	−2.70051
$494$	15537.6	1.41512
$495$	0	0
$496$	6118.22	0.553864
$497$	−5132.71	−0.463247
$498$	0	0
$499$	−15010.4	−1.34661	−0.673303	−	0.739367i	$-0.735125\pi$
$499$	−15010.4	−1.34661	−0.673303	+	0.739367i	$0.735125\pi$
$500$	31.8271	0.00284670
$501$	0	0
$502$	10040.6	0.892697
$503$	−20308.2	−1.80019	−0.900097	−	0.435690i	$-0.856504\pi$
$503$	−20308.2	−1.80019	−0.900097	+	0.435690i	$0.856504\pi$
$504$	0	0
$505$	955.872	0.0842292
$506$	16538.5	1.45302
$507$	0	0
$508$	65.6526	0.00573398
$509$	−18004.6	−1.56786	−0.783930	−	0.620849i	$-0.786788\pi$
$509$	−18004.6	−1.56786	−0.783930	+	0.620849i	$0.786788\pi$
$510$	0	0
$511$	20809.0	1.80144
$512$	11692.7	1.00928
$513$	0	0
$514$	−17143.0	−1.47110
$515$	8771.23	0.750498
$516$	0	0
$517$	14561.5	1.23871
$518$	−28936.9	−2.45447
$519$	0	0
$520$	−23975.3	−2.02190
$521$	−20054.4	−1.68637	−0.843186	−	0.537622i	$-0.819323\pi$
$521$	−20054.4	−1.68637	−0.843186	+	0.537622i	$0.819323\pi$
$522$	0	0
$523$	−18623.7	−1.55709	−0.778546	−	0.627588i	$-0.784042\pi$
$523$	−18623.7	−1.55709	−0.778546	+	0.627588i	$0.784042\pi$
$524$	−82.9617	−0.00691641
$525$	0	0
$526$	5824.63	0.482825
$527$	−10657.4	−0.880916
$528$	0	0
$529$	14114.6	1.16007
$530$	−28634.4	−2.34679
$531$	0	0
$532$	138.369	0.0112764
$533$	−22794.8	−1.85244
$534$	0	0
$535$	−29238.4	−2.36277
$536$	−3800.08	−0.306228
$537$	0	0
$538$	−7035.95	−0.563832
$539$	22523.6	1.79992
$540$	0	0
$541$	21975.2	1.74637	0.873187	−	0.487386i	$-0.162049\pi$
$541$	21975.2	1.74637	0.873187	+	0.487386i	$0.162049\pi$
$542$	−7910.86	−0.626938
$543$	0	0
$544$	−252.447	−0.0198963
$545$	21686.8	1.70452
$546$	0	0
$547$	−22353.0	−1.74725	−0.873625	−	0.486600i	$-0.838237\pi$
$547$	−22353.0	−1.74725	−0.873625	+	0.486600i	$0.838237\pi$
$548$	116.388	0.00907272
$549$	0	0
$550$	−16556.1	−1.28355
$551$	23597.5	1.82448
$552$	0	0
$553$	−10649.2	−0.818898
$554$	−10186.0	−0.781156
$555$	0	0
$556$	−85.0929	−0.00649055
$557$	−7399.47	−0.562883	−0.281441	−	0.959578i	$-0.590812\pi$
$557$	−7399.47	−0.562883	−0.281441	+	0.959578i	$0.590812\pi$
$558$	0	0
$559$	−17280.7	−1.30751
$560$	33493.1	2.52740
$561$	0	0
$562$	17837.7	1.33885
$563$	−918.430	−0.0687517	−0.0343758	−	0.999409i	$-0.510944\pi$
$563$	−918.430	−0.0687517	−0.0343758	+	0.999409i	$0.510944\pi$
$564$	0	0
$565$	30651.2	2.28231
$566$	−9047.74	−0.671916
$567$	0	0
$568$	−3749.38	−0.276973
$569$	6118.79	0.450814	0.225407	−	0.974265i	$-0.427629\pi$
$569$	6118.79	0.450814	0.225407	+	0.974265i	$0.427629\pi$
$570$	0	0
$571$	14405.4	1.05577	0.527886	−	0.849315i	$-0.322985\pi$
$571$	14405.4	1.05577	0.527886	+	0.849315i	$0.322985\pi$
$572$	−113.548	−0.00830012
$573$	0	0
$574$	32045.7	2.33025
$575$	−26309.5	−1.90814
$576$	0	0
$577$	15609.7	1.12624	0.563119	−	0.826376i	$-0.309601\pi$
$577$	15609.7	1.12624	0.563119	+	0.826376i	$0.309601\pi$
$578$	−20746.8	−1.49300
$579$	0	0
$580$	−227.770	−0.0163062
$581$	−401.054	−0.0286378
$582$	0	0
$583$	−21679.7	−1.54010
$584$	15200.7	1.07707
$585$	0	0
$586$	−17366.7	−1.22425
$587$	−6664.51	−0.468609	−0.234305	−	0.972163i	$-0.575281\pi$
$587$	−6664.51	−0.468609	−0.234305	+	0.972163i	$0.575281\pi$
$588$	0	0
$589$	8507.47	0.595151
$590$	−10385.4	−0.724679
$591$	0	0
$592$	−21005.0	−1.45827
$593$	22217.7	1.53857	0.769285	−	0.638906i	$-0.220613\pi$
$593$	22217.7	1.53857	0.769285	+	0.638906i	$0.220613\pi$
$594$	0	0
$595$	−58341.9	−4.01981
$596$	73.8293	0.00507411
$597$	0	0
$598$	28485.0	1.94789
$599$	−14137.6	−0.964353	−0.482177	−	0.876074i	$-0.660154\pi$
$599$	−14137.6	−0.964353	−0.482177	+	0.876074i	$0.660154\pi$
$600$	0	0
$601$	−7507.68	−0.509558	−0.254779	−	0.966999i	$-0.582003\pi$
$601$	−7507.68	−0.509558	−0.254779	+	0.966999i	$0.582003\pi$
$602$	24293.9	1.64476
$603$	0	0
$604$	−43.4858	−0.00292949
$605$	370.142	0.0248734
$606$	0	0
$607$	−9633.18	−0.644149	−0.322075	−	0.946714i	$-0.604380\pi$
$607$	−9633.18	−0.644149	−0.322075	+	0.946714i	$0.604380\pi$
$608$	201.521	0.0134420
$609$	0	0
$610$	21180.8	1.40588
$611$	25079.9	1.66059
$612$	0	0
$613$	−16668.2	−1.09824	−0.549120	−	0.835743i	$-0.685037\pi$
$613$	−16668.2	−1.09824	−0.549120	+	0.835743i	$0.685037\pi$
$614$	−10644.3	−0.699622
$615$	0	0
$616$	25518.9	1.66913
$617$	5629.36	0.367309	0.183654	−	0.982991i	$-0.441207\pi$
$617$	5629.36	0.367309	0.183654	+	0.982991i	$0.441207\pi$
$618$	0	0
$619$	−8435.98	−0.547771	−0.273886	−	0.961762i	$-0.588309\pi$
$619$	−8435.98	−0.547771	−0.273886	+	0.961762i	$0.588309\pi$
$620$	−82.1164	−0.00531915
$621$	0	0
$622$	5023.62	0.323841
$623$	−21265.4	−1.36754
$624$	0	0
$625$	−9573.56	−0.612708
$626$	8644.38	0.551915
$627$	0	0
$628$	131.438	0.00835180
$629$	36588.7	2.31938
$630$	0	0
$631$	−9651.33	−0.608896	−0.304448	−	0.952529i	$-0.598472\pi$
$631$	−9651.33	−0.608896	−0.304448	+	0.952529i	$0.598472\pi$
$632$	−7779.12	−0.489615
$633$	0	0
$634$	−18894.9	−1.18361
$635$	22097.6	1.38097
$636$	0	0
$637$	38793.2	2.41294
$638$	27223.3	1.68931
$639$	0	0
$640$	24311.3	1.50154
$641$	−15430.0	−0.950778	−0.475389	−	0.879776i	$-0.657693\pi$
$641$	−15430.0	−0.950778	−0.475389	+	0.879776i	$0.657693\pi$
$642$	0	0
$643$	8343.38	0.511712	0.255856	−	0.966715i	$-0.417643\pi$
$643$	8343.38	0.511712	0.255856	+	0.966715i	$0.417643\pi$
$644$	253.670	0.0155217
$645$	0	0
$646$	27619.4	1.68215
$647$	−28265.0	−1.71748	−0.858741	−	0.512411i	$-0.828753\pi$
$647$	−28265.0	−1.71748	−0.858741	+	0.512411i	$0.828753\pi$
$648$	0	0
$649$	−7862.98	−0.475576
$650$	−28515.2	−1.72071
$651$	0	0
$652$	202.607	0.0121698
$653$	5937.98	0.355852	0.177926	−	0.984044i	$-0.443061\pi$
$653$	5937.98	0.355852	0.177926	+	0.984044i	$0.443061\pi$
$654$	0	0
$655$	−27923.6	−1.66575
$656$	23261.6	1.38447
$657$	0	0
$658$	−35258.2	−2.08892
$659$	3558.05	0.210322	0.105161	−	0.994455i	$-0.466464\pi$
$659$	3558.05	0.210322	0.105161	+	0.994455i	$0.466464\pi$
$660$	0	0
$661$	−28349.6	−1.66819	−0.834095	−	0.551621i	$-0.814009\pi$
$661$	−28349.6	−1.66819	−0.834095	+	0.551621i	$0.814009\pi$
$662$	−24915.6	−1.46280
$663$	0	0
$664$	−292.965	−0.0171224
$665$	46572.6	2.71580
$666$	0	0
$667$	43261.0	2.51136
$668$	150.643	0.00872540
$669$	0	0
$670$	−8000.85	−0.461343
$671$	16036.4	0.922620
$672$	0	0
$673$	−22066.0	−1.26387	−0.631934	−	0.775022i	$-0.717739\pi$
$673$	−22066.0	−1.26387	−0.631934	+	0.775022i	$0.717739\pi$
$674$	32652.2	1.86605
$675$	0	0
$676$	−84.9321	−0.00483227
$677$	−29434.1	−1.67097	−0.835484	−	0.549514i	$-0.814813\pi$
$677$	−29434.1	−1.67097	−0.835484	+	0.549514i	$0.814813\pi$
$678$	0	0
$679$	13327.9	0.753283
$680$	−42618.0	−2.40342
$681$	0	0
$682$	9814.67	0.551060
$683$	−9249.41	−0.518183	−0.259092	−	0.965853i	$-0.583423\pi$
$683$	−9249.41	−0.518183	−0.259092	+	0.965853i	$0.583423\pi$
$684$	0	0
$685$	39174.3	2.18507
$686$	−24486.9	−1.36285
$687$	0	0
$688$	17634.6	0.977201
$689$	−37339.8	−2.06463
$690$	0	0
$691$	−6905.74	−0.380183	−0.190092	−	0.981766i	$-0.560879\pi$
$691$	−6905.74	−0.380183	−0.190092	+	0.981766i	$0.560879\pi$
$692$	−37.8219	−0.00207770
$693$	0	0
$694$	8744.89	0.478316
$695$	−28640.9	−1.56318
$696$	0	0
$697$	−40519.6	−2.20199
$698$	−23093.6	−1.25230
$699$	0	0
$700$	−253.939	−0.0137114
$701$	9953.58	0.536293	0.268147	−	0.963378i	$-0.413589\pi$
$701$	9953.58	0.536293	0.268147	+	0.963378i	$0.413589\pi$
$702$	0	0
$703$	−29207.7	−1.56698
$704$	18640.5	0.997927
$705$	0	0
$706$	−15947.5	−0.850129
$707$	−1752.34	−0.0932155
$708$	0	0
$709$	−23631.9	−1.25179	−0.625893	−	0.779909i	$-0.715265\pi$
$709$	−23631.9	−1.25179	−0.625893	+	0.779909i	$0.715265\pi$
$710$	−7894.11	−0.417269
$711$	0	0
$712$	−15534.1	−0.817647
$713$	15596.6	0.819213
$714$	0	0
$715$	−38218.3	−1.99900
$716$	183.830	0.00959506
$717$	0	0
$718$	−24044.1	−1.24975
$719$	−2341.61	−0.121457	−0.0607283	−	0.998154i	$-0.519342\pi$
$719$	−2341.61	−0.121457	−0.0607283	+	0.998154i	$0.519342\pi$
$720$	0	0
$721$	−16079.7	−0.830567
$722$	−2708.72	−0.139623
$723$	0	0
$724$	−106.510	−0.00546742
$725$	−43307.0	−2.21846
$726$	0	0
$727$	−12821.0	−0.654066	−0.327033	−	0.945013i	$-0.606049\pi$
$727$	−12821.0	−0.654066	−0.327033	+	0.945013i	$0.606049\pi$
$728$	43952.3	2.23761
$729$	0	0
$730$	32004.2	1.62264
$731$	−30717.9	−1.55423
$732$	0	0
$733$	−16577.6	−0.835344	−0.417672	−	0.908598i	$-0.637154\pi$
$733$	−16577.6	−0.835344	−0.417672	+	0.908598i	$0.637154\pi$
$734$	−11962.3	−0.601551
$735$	0	0
$736$	369.446	0.0185027
$737$	−6057.60	−0.302760
$738$	0	0
$739$	25251.7	1.25697	0.628483	−	0.777823i	$-0.283676\pi$
$739$	25251.7	1.25697	0.628483	+	0.777823i	$0.283676\pi$
$740$	281.920	0.0140049
$741$	0	0
$742$	52493.5	2.59717
$743$	2859.85	0.141208	0.0706042	−	0.997504i	$-0.477507\pi$
$743$	2859.85	0.141208	0.0706042	+	0.997504i	$0.477507\pi$
$744$	0	0
$745$	24849.7	1.22205
$746$	−16301.3	−0.800043
$747$	0	0
$748$	−201.840	−0.00986633
$749$	53600.7	2.61486
$750$	0	0
$751$	−29020.5	−1.41008	−0.705042	−	0.709166i	$-0.749072\pi$
$751$	−29020.5	−1.41008	−0.705042	+	0.709166i	$0.749072\pi$
$752$	−25593.5	−1.24109
$753$	0	0
$754$	46887.9	2.26466
$755$	−14636.6	−0.705537
$756$	0	0
$757$	−6714.23	−0.322368	−0.161184	−	0.986924i	$-0.551531\pi$
$757$	−6714.23	−0.322368	−0.161184	+	0.986924i	$0.551531\pi$
$758$	−15170.4	−0.726931
$759$	0	0
$760$	34020.7	1.62376
$761$	28880.5	1.37571	0.687855	−	0.725848i	$-0.258552\pi$
$761$	28880.5	1.37571	0.687855	+	0.725848i	$0.258552\pi$
$762$	0	0
$763$	−39757.0	−1.88637
$764$	23.3817	0.00110722
$765$	0	0
$766$	5825.62	0.274789
$767$	−13542.7	−0.637549
$768$	0	0
$769$	−6141.52	−0.287996	−0.143998	−	0.989578i	$-0.545996\pi$
$769$	−6141.52	−0.287996	−0.143998	+	0.989578i	$0.545996\pi$
$770$	53728.6	2.51460
$771$	0	0
$772$	197.750	0.00921913
$773$	−13359.9	−0.621634	−0.310817	−	0.950470i	$-0.600603\pi$
$773$	−13359.9	−0.621634	−0.310817	+	0.950470i	$0.600603\pi$
$774$	0	0
$775$	−15613.2	−0.723668
$776$	9735.89	0.450384
$777$	0	0
$778$	−579.556	−0.0267071
$779$	32345.5	1.48768
$780$	0	0
$781$	−5976.78	−0.273836
$782$	50634.4	2.31545
$783$	0	0
$784$	−39587.7	−1.80338
$785$	44239.7	2.01144
$786$	0	0
$787$	−29950.5	−1.35657	−0.678285	−	0.734799i	$-0.737277\pi$
$787$	−29950.5	−1.35657	−0.678285	+	0.734799i	$0.737277\pi$
$788$	−21.1533	−0.000956290 0
$789$	0	0
$790$	−16378.5	−0.737621
$791$	−56190.8	−2.52581
$792$	0	0
$793$	27620.1	1.23685
$794$	7758.11	0.346757
$795$	0	0
$796$	86.7561	0.00386305
$797$	15365.8	0.682916	0.341458	−	0.939897i	$-0.389079\pi$
$797$	15365.8	0.682916	0.341458	+	0.939897i	$0.389079\pi$
$798$	0	0
$799$	44581.5	1.97394
$800$	−369.838	−0.0163447
$801$	0	0
$802$	−16165.4	−0.711745
$803$	24231.0	1.06487
$804$	0	0
$805$	85381.0	3.73824
$806$	16904.2	0.738741
$807$	0	0
$808$	−1280.06	−0.0557331
$809$	34436.6	1.49657	0.748286	−	0.663376i	$-0.230877\pi$
$809$	34436.6	1.49657	0.748286	+	0.663376i	$0.230877\pi$
$810$	0	0
$811$	−17874.3	−0.773925	−0.386963	−	0.922095i	$-0.626476\pi$
$811$	−17874.3	−0.773925	−0.386963	+	0.922095i	$0.626476\pi$
$812$	417.555	0.0180459
$813$	0	0
$814$	−33695.5	−1.45089
$815$	68194.3	2.93097
$816$	0	0
$817$	24521.2	1.05005
$818$	31405.3	1.34237
$819$	0	0
$820$	−312.208	−0.0132961
$821$	−1645.58	−0.0699526	−0.0349763	−	0.999388i	$-0.511136\pi$
$821$	−1645.58	−0.0699526	−0.0349763	+	0.999388i	$0.511136\pi$
$822$	0	0
$823$	−5442.41	−0.230511	−0.115256	−	0.993336i	$-0.536769\pi$
$823$	−5442.41	−0.230511	−0.115256	+	0.993336i	$0.536769\pi$
$824$	−11746.0	−0.496592
$825$	0	0
$826$	19038.9	0.801993
$827$	20191.8	0.849018	0.424509	−	0.905424i	$-0.360447\pi$
$827$	20191.8	0.849018	0.424509	+	0.905424i	$0.360447\pi$
$828$	0	0
$829$	26526.1	1.11133	0.555664	−	0.831407i	$-0.312464\pi$
$829$	26526.1	1.11133	0.555664	+	0.831407i	$0.312464\pi$
$830$	−616.822	−0.0257954
$831$	0	0
$832$	32105.3	1.33780
$833$	68958.1	2.86826
$834$	0	0
$835$	50704.1	2.10142
$836$	161.123	0.00666574
$837$	0	0
$838$	−904.301	−0.0372775
$839$	−35566.6	−1.46352	−0.731761	−	0.681562i	$-0.761301\pi$
$839$	−35566.6	−1.46352	−0.731761	+	0.681562i	$0.761301\pi$
$840$	0	0
$841$	46821.1	1.91976
$842$	34841.2	1.42602
$843$	0	0
$844$	27.0734	0.00110415
$845$	−28586.7	−1.16380
$846$	0	0
$847$	−678.556	−0.0275271
$848$	38104.5	1.54306
$849$	0	0
$850$	−50688.1	−2.04540
$851$	−53546.1	−2.15692
$852$	0	0
$853$	15735.3	0.631613	0.315807	−	0.948824i	$-0.397725\pi$
$853$	15735.3	0.631613	0.315807	+	0.948824i	$0.397725\pi$
$854$	−38829.3	−1.55587
$855$	0	0
$856$	39154.6	1.56341
$857$	−15441.3	−0.615478	−0.307739	−	0.951471i	$-0.599572\pi$
$857$	−15441.3	−0.615478	−0.307739	+	0.951471i	$0.599572\pi$
$858$	0	0
$859$	38408.7	1.52560	0.762799	−	0.646635i	$-0.223824\pi$
$859$	38408.7	1.52560	0.762799	+	0.646635i	$0.223824\pi$
$860$	−236.685	−0.00938477
$861$	0	0
$862$	−10661.1	−0.421252
$863$	25846.0	1.01948	0.509739	−	0.860329i	$-0.329742\pi$
$863$	25846.0	1.01948	0.509739	+	0.860329i	$0.329742\pi$
$864$	0	0
$865$	−12730.2	−0.500393
$866$	16958.1	0.665427
$867$	0	0
$868$	150.539	0.00588665
$869$	−12400.5	−0.484070
$870$	0	0
$871$	−10433.2	−0.405875
$872$	−29042.0	−1.12785
$873$	0	0
$874$	−40419.9	−1.56433
$875$	−19638.4	−0.758743
$876$	0	0
$877$	42710.7	1.64451	0.822257	−	0.569117i	$-0.192715\pi$
$877$	42710.7	1.64451	0.822257	+	0.569117i	$0.192715\pi$
$878$	30686.1	1.17950
$879$	0	0
$880$	39001.0	1.49400
$881$	−1004.95	−0.0384311	−0.0192155	−	0.999815i	$-0.506117\pi$
$881$	−1004.95	−0.0384311	−0.0192155	+	0.999815i	$0.506117\pi$
$882$	0	0
$883$	28806.3	1.09786	0.548929	−	0.835869i	$-0.315036\pi$
$883$	28806.3	1.09786	0.548929	+	0.835869i	$0.315036\pi$
$884$	−347.638	−0.0132266
$885$	0	0
$886$	2955.90	0.112083
$887$	19516.9	0.738797	0.369399	−	0.929271i	$-0.379564\pi$
$887$	19516.9	0.738797	0.369399	+	0.929271i	$0.379564\pi$
$888$	0	0
$889$	−40510.0	−1.52830
$890$	−32706.2	−1.23181
$891$	0	0
$892$	−122.418	−0.00459512
$893$	−35588.1	−1.33361
$894$	0	0
$895$	61874.2	2.31087
$896$	−44568.3	−1.66174
$897$	0	0
$898$	6498.96	0.241507
$899$	25673.0	0.952437
$900$	0	0
$901$	−66374.5	−2.45422
$902$	37315.5	1.37746
$903$	0	0
$904$	−41046.6	−1.51017
$905$	−35849.5	−1.31677
$906$	0	0
$907$	35990.3	1.31757	0.658786	−	0.752331i	$-0.271070\pi$
$907$	35990.3	1.31757	0.658786	+	0.752331i	$0.271070\pi$
$908$	−240.220	−0.00877970
$909$	0	0
$910$	92539.0	3.37103
$911$	15213.2	0.553276	0.276638	−	0.960974i	$-0.410780\pi$
$911$	15213.2	0.553276	0.276638	+	0.960974i	$0.410780\pi$
$912$	0	0
$913$	−467.007	−0.0169285
$914$	37679.9	1.36361
$915$	0	0
$916$	3.47010	0.000125170 0
$917$	51190.4	1.84346
$918$	0	0
$919$	18369.5	0.659364	0.329682	−	0.944092i	$-0.393059\pi$
$919$	18369.5	0.659364	0.329682	+	0.944092i	$0.393059\pi$
$920$	62369.7	2.23508
$921$	0	0
$922$	−14394.8	−0.514172
$923$	−10294.1	−0.367100
$924$	0	0
$925$	53602.9	1.90536
$926$	18659.2	0.662180
$927$	0	0
$928$	608.129	0.0215116
$929$	−33838.0	−1.19504	−0.597518	−	0.801855i	$-0.703846\pi$
$929$	−33838.0	−1.19504	−0.597518	+	0.801855i	$0.703846\pi$
$930$	0	0
$931$	−55047.2	−1.93781
$932$	−14.1724	−0.000498105 0
$933$	0	0
$934$	−6163.90	−0.215941
$935$	−67936.2	−2.37620
$936$	0	0
$937$	4517.50	0.157503	0.0787515	−	0.996894i	$-0.474907\pi$
$937$	4517.50	0.157503	0.0787515	+	0.996894i	$0.474907\pi$
$938$	14667.4	0.510563
$939$	0	0
$940$	343.506	0.0119191
$941$	35152.6	1.21779	0.608896	−	0.793250i	$-0.291613\pi$
$941$	35152.6	1.21779	0.608896	+	0.793250i	$0.291613\pi$
$942$	0	0
$943$	59298.7	2.04775
$944$	13820.1	0.476489
$945$	0	0
$946$	28288.9	0.972255
$947$	6858.34	0.235339	0.117670	−	0.993053i	$-0.462458\pi$
$947$	6858.34	0.235339	0.117670	+	0.993053i	$0.462458\pi$
$948$	0	0
$949$	41734.0	1.42755
$950$	40462.8	1.38188
$951$	0	0
$952$	78128.7	2.65984
$953$	−47948.0	−1.62979	−0.814894	−	0.579610i	$-0.803205\pi$
$953$	−47948.0	−1.62979	−0.814894	+	0.579610i	$0.803205\pi$
$954$	0	0
$955$	7869.88	0.266663
$956$	−12.0355	−0.000407171 0
$957$	0	0
$958$	−27773.0	−0.936645
$959$	−71815.6	−2.41819
$960$	0	0
$961$	−20535.3	−0.689311
$962$	−58035.2	−1.94504
$963$	0	0
$964$	−157.742	−0.00527025
$965$	66559.3	2.22033
$966$	0	0
$967$	17688.3	0.588228	0.294114	−	0.955770i	$-0.404976\pi$
$967$	17688.3	0.588228	0.294114	+	0.955770i	$0.404976\pi$
$968$	−495.676	−0.0164583
$969$	0	0
$970$	20498.4	0.678518
$971$	17942.1	0.592986	0.296493	−	0.955035i	$-0.404183\pi$
$971$	17942.1	0.592986	0.296493	+	0.955035i	$0.404183\pi$
$972$	0	0
$973$	52505.4	1.72995
$974$	−9043.58	−0.297510
$975$	0	0
$976$	−28185.8	−0.924390
$977$	1135.50	0.0371832	0.0185916	−	0.999827i	$-0.494082\pi$
$977$	1135.50	0.0371832	0.0185916	+	0.999827i	$0.494082\pi$
$978$	0	0
$979$	−24762.5	−0.808388
$980$	531.331	0.0173191
$981$	0	0
$982$	−42258.5	−1.37324
$983$	30099.9	0.976642	0.488321	−	0.872664i	$-0.337609\pi$
$983$	30099.9	0.976642	0.488321	+	0.872664i	$0.337609\pi$
$984$	0	0
$985$	−7119.86	−0.230312
$986$	83347.0	2.69200
$987$	0	0
$988$	277.509	0.00893597
$989$	44954.4	1.44537
$990$	0	0
$991$	−58791.0	−1.88452	−0.942259	−	0.334886i	$-0.891302\pi$
$991$	−58791.0	−1.88452	−0.942259	+	0.334886i	$0.891302\pi$
$992$	219.245	0.00701717
$993$	0	0
$994$	14471.7	0.461787
$995$	29200.7	0.930376
$996$	0	0
$997$	−40230.9	−1.27796	−0.638979	−	0.769224i	$-0.720643\pi$
$997$	−40230.9	−1.27796	−0.638979	+	0.769224i	$0.720643\pi$
$998$	42321.9	1.34236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.20	✓	59
3.2	odd	2		2151.4.a.h.1.40	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.20	✓	59	1.1	even	1	trivial
2151.4.a.h.1.40	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)

\(f(q)\)	\(=\)	\(q-2.81951 q^{2} -0.0503577 q^{4} -16.9496 q^{5} +31.0725 q^{7} +22.6981 q^{8} +O(q^{10})\)	\(q-2.81951 q^{2} -0.0503577 q^{4} -16.9496 q^{5} +31.0725 q^{7} +22.6981 q^{8} +47.7895 q^{10} +36.1823 q^{11} +62.3183 q^{13} -87.6093 q^{14} -63.5946 q^{16} +110.776 q^{17} -88.4291 q^{19} +0.853542 q^{20} -102.017 q^{22} -162.116 q^{23} +162.288 q^{25} -175.707 q^{26} -1.56474 q^{28} -266.852 q^{29} -96.2067 q^{31} -2.27890 q^{32} -312.334 q^{34} -526.666 q^{35} +330.295 q^{37} +249.327 q^{38} -384.723 q^{40} -365.779 q^{41} -277.298 q^{43} -1.82206 q^{44} +457.088 q^{46} +402.448 q^{47} +622.501 q^{49} -457.573 q^{50} -3.13821 q^{52} -599.178 q^{53} -613.276 q^{55} +705.286 q^{56} +752.393 q^{58} -217.315 q^{59} +443.210 q^{61} +271.256 q^{62} +515.182 q^{64} -1056.27 q^{65} -167.419 q^{67} -5.57842 q^{68} +1484.94 q^{70} -165.185 q^{71} +669.691 q^{73} -931.270 q^{74} +4.45309 q^{76} +1124.28 q^{77} -342.722 q^{79} +1077.90 q^{80} +1031.32 q^{82} -12.9071 q^{83} -1877.61 q^{85} +781.844 q^{86} +821.270 q^{88} -684.380 q^{89} +1936.39 q^{91} +8.16380 q^{92} -1134.71 q^{94} +1498.84 q^{95} +428.930 q^{97} -1755.15 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 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

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