LMFDB - Embedded newform 2151.4.a.b.1.13

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

Embedding invariants

Embedding label			1.13
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)

$f(q)$	$=$	$q-0.889333 q^{2} -7.20909 q^{4} -6.34700 q^{5} -11.3318 q^{7} +13.5259 q^{8} +O(q^{10})$	\(q-0.889333 q^{2} -7.20909 q^{4} -6.34700 q^{5} -11.3318 q^{7} +13.5259 q^{8} +5.64460 q^{10} -54.9949 q^{11} -49.5708 q^{13} +10.0778 q^{14} +45.6436 q^{16} +9.76927 q^{17} +104.174 q^{19} +45.7561 q^{20} +48.9088 q^{22} +45.7323 q^{23} -84.7156 q^{25} +44.0850 q^{26} +81.6922 q^{28} -38.7844 q^{29} +159.477 q^{31} -148.800 q^{32} -8.68814 q^{34} +71.9232 q^{35} +152.261 q^{37} -92.6455 q^{38} -85.8491 q^{40} +270.821 q^{41} -12.7420 q^{43} +396.463 q^{44} -40.6712 q^{46} +71.6627 q^{47} -214.590 q^{49} +75.3404 q^{50} +357.360 q^{52} +338.444 q^{53} +349.053 q^{55} -153.274 q^{56} +34.4922 q^{58} +558.892 q^{59} -183.619 q^{61} -141.828 q^{62} -232.816 q^{64} +314.626 q^{65} +105.246 q^{67} -70.4275 q^{68} -63.9636 q^{70} +381.383 q^{71} -117.337 q^{73} -135.411 q^{74} -751.000 q^{76} +623.194 q^{77} -644.394 q^{79} -289.700 q^{80} -240.850 q^{82} +360.281 q^{83} -62.0056 q^{85} +11.3319 q^{86} -743.858 q^{88} +213.760 q^{89} +561.728 q^{91} -329.688 q^{92} -63.7320 q^{94} -661.193 q^{95} -48.8871 q^{97} +190.842 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) $ 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8	$ 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8} - 88 q^{10} + 110 q^{11} - 82 q^{13} - 126 q^{14} + 271 q^{16} - 100 q^{17} - 292 q^{19} + 52 q^{20} - 351 q^{22} + 276 q^{23} + 386 q^{25} - 84 q^{26} - 1010 q^{28} + 38 q^{29} - 432 q^{31} + 452 q^{32} - 524 q^{34} + 166 q^{35} - 936 q^{37} + 41 q^{38} - 1183 q^{40} - 1054 q^{41} - 1804 q^{43} + 341 q^{44} - 888 q^{46} + 560 q^{47} + 1074 q^{49} + 1054 q^{50} - 632 q^{52} + 160 q^{53} - 842 q^{55} - 509 q^{56} - 1266 q^{58} - 846 q^{59} - 2220 q^{61} - 82 q^{62} - 1565 q^{64} - 296 q^{65} - 4752 q^{67} + 1719 q^{68} - 5601 q^{70} + 802 q^{71} - 2732 q^{73} + 4581 q^{74} - 5614 q^{76} + 1008 q^{77} - 3172 q^{79} + 732 q^{80} - 9709 q^{82} + 4780 q^{83} - 4624 q^{85} + 2009 q^{86} - 9331 q^{88} - 4372 q^{89} - 7398 q^{91} + 6138 q^{92} - 7068 q^{94} + 3160 q^{95} - 4846 q^{97} + 3772 q^{98}+O(q^{100}) $ 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8 - 88 * q^10 + 110 * q^11 - 82 * q^13 - 126 * q^14 + 271 * q^16 - 100 * q^17 - 292 * q^19 + 52 * q^20 - 351 * q^22 + 276 * q^23 + 386 * q^25 - 84 * q^26 - 1010 * q^28 + 38 * q^29 - 432 * q^31 + 452 * q^32 - 524 * q^34 + 166 * q^35 - 936 * q^37 + 41 * q^38 - 1183 * q^40 - 1054 * q^41 - 1804 * q^43 + 341 * q^44 - 888 * q^46 + 560 * q^47 + 1074 * q^49 + 1054 * q^50 - 632 * q^52 + 160 * q^53 - 842 * q^55 - 509 * q^56 - 1266 * q^58 - 846 * q^59 - 2220 * q^61 - 82 * q^62 - 1565 * q^64 - 296 * q^65 - 4752 * q^67 + 1719 * q^68 - 5601 * q^70 + 802 * q^71 - 2732 * q^73 + 4581 * q^74 - 5614 * q^76 + 1008 * q^77 - 3172 * q^79 + 732 * q^80 - 9709 * q^82 + 4780 * q^83 - 4624 * q^85 + 2009 * q^86 - 9331 * q^88 - 4372 * q^89 - 7398 * q^91 + 6138 * q^92 - 7068 * q^94 + 3160 * q^95 - 4846 * q^97 + 3772 * 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 $

$2$	−0.889333	−0.314427	−0.157213	−	0.987565i	$-0.550251\pi$
$2$	−0.889333	−0.314427	−0.157213	+	0.987565i	$0.550251\pi$
$3$	0	0
$3$	0	0
$4$	−7.20909	−0.901136
$5$	−6.34700	−0.567693	−0.283846	−	0.958870i	$-0.591611\pi$
$5$	−6.34700	−0.567693	−0.283846	+	0.958870i	$0.591611\pi$
$6$	0	0
$7$	−11.3318	−0.611862	−0.305931	−	0.952054i	$-0.598968\pi$
$7$	−11.3318	−0.611862	−0.305931	+	0.952054i	$0.598968\pi$
$8$	13.5259	0.597768
$9$	0	0
$10$	5.64460	0.178498
$11$	−54.9949	−1.50742	−0.753709	−	0.657208i	$-0.771737\pi$
$11$	−54.9949	−1.50742	−0.753709	+	0.657208i	$0.771737\pi$
$12$	0	0
$13$	−49.5708	−1.05757	−0.528787	−	0.848754i	$-0.677353\pi$
$13$	−49.5708	−1.05757	−0.528787	+	0.848754i	$0.677353\pi$
$14$	10.0778	0.192386
$15$	0	0
$16$	45.6436	0.713182
$17$	9.76927	0.139376	0.0696881	−	0.997569i	$-0.477800\pi$
$17$	9.76927	0.139376	0.0696881	+	0.997569i	$0.477800\pi$
$18$	0	0
$19$	104.174	1.25785	0.628926	−	0.777465i	$-0.283495\pi$
$19$	104.174	1.25785	0.628926	+	0.777465i	$0.283495\pi$
$20$	45.7561	0.511568
$21$	0	0
$22$	48.9088	0.473972
$23$	45.7323	0.414602	0.207301	−	0.978277i	$-0.433532\pi$
$23$	45.7323	0.414602	0.207301	+	0.978277i	$0.433532\pi$
$24$	0	0
$25$	−84.7156	−0.677725
$26$	44.0850	0.332530
$27$	0	0
$28$	81.6922	0.551370
$29$	−38.7844	−0.248347	−0.124174	−	0.992260i	$-0.539628\pi$
$29$	−38.7844	−0.248347	−0.124174	+	0.992260i	$0.539628\pi$
$30$	0	0
$31$	159.477	0.923962	0.461981	−	0.886890i	$-0.347139\pi$
$31$	159.477	0.923962	0.461981	+	0.886890i	$0.347139\pi$
$32$	−148.800	−0.822011
$33$	0	0
$34$	−8.68814	−0.0438236
$35$	71.9232	0.347349
$36$	0	0
$37$	152.261	0.676528	0.338264	−	0.941051i	$-0.390160\pi$
$37$	152.261	0.676528	0.338264	+	0.941051i	$0.390160\pi$
$38$	−92.6455	−0.395502
$39$	0	0
$40$	−85.8491	−0.339349
$41$	270.821	1.03159	0.515795	−	0.856712i	$-0.327497\pi$
$41$	270.821	1.03159	0.515795	+	0.856712i	$0.327497\pi$
$42$	0	0
$43$	−12.7420	−0.0451894	−0.0225947	−	0.999745i	$-0.507193\pi$
$43$	−12.7420	−0.0451894	−0.0225947	+	0.999745i	$0.507193\pi$
$44$	396.463	1.35839
$45$	0	0
$46$	−40.6712	−0.130362
$47$	71.6627	0.222406	0.111203	−	0.993798i	$-0.464530\pi$
$47$	71.6627	0.222406	0.111203	+	0.993798i	$0.464530\pi$
$48$	0	0
$49$	−214.590	−0.625625
$50$	75.3404	0.213095
$51$	0	0
$52$	357.360	0.953019
$53$	338.444	0.877148	0.438574	−	0.898695i	$-0.355484\pi$
$53$	338.444	0.877148	0.438574	+	0.898695i	$0.355484\pi$
$54$	0	0
$55$	349.053	0.855750
$56$	−153.274	−0.365751
$57$	0	0
$58$	34.4922	0.0780870
$59$	558.892	1.23325	0.616623	−	0.787258i	$-0.288500\pi$
$59$	558.892	1.23325	0.616623	+	0.787258i	$0.288500\pi$
$60$	0	0
$61$	−183.619	−0.385411	−0.192705	−	0.981257i	$-0.561726\pi$
$61$	−183.619	−0.385411	−0.192705	+	0.981257i	$0.561726\pi$
$62$	−141.828	−0.290518
$63$	0	0
$64$	−232.816	−0.454720
$65$	314.626	0.600378
$66$	0	0
$67$	105.246	0.191908	0.0959541	−	0.995386i	$-0.469410\pi$
$67$	105.246	0.191908	0.0959541	+	0.995386i	$0.469410\pi$
$68$	−70.4275	−0.125597
$69$	0	0
$70$	−63.9636	−0.109216
$71$	381.383	0.637490	0.318745	−	0.947840i	$-0.396739\pi$
$71$	381.383	0.637490	0.318745	+	0.947840i	$0.396739\pi$
$72$	0	0
$73$	−117.337	−0.188126	−0.0940630	−	0.995566i	$-0.529986\pi$
$73$	−117.337	−0.188126	−0.0940630	+	0.995566i	$0.529986\pi$
$74$	−135.411	−0.212719
$75$	0	0
$76$	−751.000	−1.13350
$77$	623.194	0.922331
$78$	0	0
$79$	−644.394	−0.917721	−0.458861	−	0.888508i	$-0.651742\pi$
$79$	−644.394	−0.917721	−0.458861	+	0.888508i	$0.651742\pi$
$80$	−289.700	−0.404868
$81$	0	0
$82$	−240.850	−0.324359
$83$	360.281	0.476458	0.238229	−	0.971209i	$-0.423433\pi$
$83$	360.281	0.476458	0.238229	+	0.971209i	$0.423433\pi$
$84$	0	0
$85$	−62.0056	−0.0791229
$86$	11.3319	0.0142087
$87$	0	0
$88$	−743.858	−0.901086
$89$	213.760	0.254590	0.127295	−	0.991865i	$-0.459371\pi$
$89$	213.760	0.254590	0.127295	+	0.991865i	$0.459371\pi$
$90$	0	0
$91$	561.728	0.647089
$92$	−329.688	−0.373612
$93$	0	0
$94$	−63.7320	−0.0699304
$95$	−661.193	−0.714074
$96$	0	0
$97$	−48.8871	−0.0511725	−0.0255862	−	0.999673i	$-0.508145\pi$
$97$	−48.8871	−0.0511725	−0.0255862	+	0.999673i	$0.508145\pi$
$98$	190.842	0.196713
$99$	0	0
$100$	610.722	0.610722
$101$	−601.286	−0.592378	−0.296189	−	0.955129i	$-0.595716\pi$
$101$	−601.286	−0.592378	−0.296189	+	0.955129i	$0.595716\pi$
$102$	0	0
$103$	−1346.25	−1.28786	−0.643930	−	0.765085i	$-0.722697\pi$
$103$	−1346.25	−1.28786	−0.643930	+	0.765085i	$0.722697\pi$
$104$	−670.492	−0.632184
$105$	0	0
$106$	−300.989	−0.275799
$107$	−10.8617	−0.00981343	−0.00490672	−	0.999988i	$-0.501562\pi$
$107$	−10.8617	−0.00981343	−0.00490672	+	0.999988i	$0.501562\pi$
$108$	0	0
$109$	1134.97	0.997347	0.498674	−	0.866790i	$-0.333821\pi$
$109$	1134.97	0.997347	0.498674	+	0.866790i	$0.333821\pi$
$110$	−310.424	−0.269071
$111$	0	0
$112$	−517.226	−0.436369
$113$	1683.77	1.40173	0.700866	−	0.713293i	$-0.252797\pi$
$113$	1683.77	1.40173	0.700866	+	0.713293i	$0.252797\pi$
$114$	0	0
$115$	−290.263	−0.235366
$116$	279.600	0.223795
$117$	0	0
$118$	−497.041	−0.387766
$119$	−110.704	−0.0852790
$120$	0	0
$121$	1693.44	1.27231
$122$	163.299	0.121183
$123$	0	0
$124$	−1149.68	−0.832616
$125$	1331.06	0.952432
$126$	0	0
$127$	−2307.80	−1.61248	−0.806238	−	0.591592i	$-0.798500\pi$
$127$	−2307.80	−1.61248	−0.806238	+	0.591592i	$0.798500\pi$
$128$	1397.45	0.964987
$129$	0	0
$130$	−279.807	−0.188775
$131$	−593.184	−0.395624	−0.197812	−	0.980240i	$-0.563383\pi$
$131$	−593.184	−0.395624	−0.197812	+	0.980240i	$0.563383\pi$
$132$	0	0
$133$	−1180.48	−0.769631
$134$	−93.5988	−0.0603411
$135$	0	0
$136$	132.139	0.0833147
$137$	1068.27	0.666196	0.333098	−	0.942892i	$-0.391906\pi$
$137$	1068.27	0.666196	0.333098	+	0.942892i	$0.391906\pi$
$138$	0	0
$139$	−69.2650	−0.0422660	−0.0211330	−	0.999777i	$-0.506727\pi$
$139$	−69.2650	−0.0422660	−0.0211330	+	0.999777i	$0.506727\pi$
$140$	−518.500	−0.313009
$141$	0	0
$142$	−339.176	−0.200444
$143$	2726.14	1.59421
$144$	0	0
$145$	246.164	0.140985
$146$	104.351	0.0591519
$147$	0	0
$148$	−1097.66	−0.609644
$149$	2167.83	1.19192	0.595958	−	0.803015i	$-0.296772\pi$
$149$	2167.83	1.19192	0.595958	+	0.803015i	$0.296772\pi$
$150$	0	0
$151$	300.312	0.161848	0.0809240	−	0.996720i	$-0.474213\pi$
$151$	300.312	0.161848	0.0809240	+	0.996720i	$0.474213\pi$
$152$	1409.05	0.751903
$153$	0	0
$154$	−554.226	−0.290005
$155$	−1012.20	−0.524527
$156$	0	0
$157$	−164.042	−0.0833886	−0.0416943	−	0.999130i	$-0.513276\pi$
$157$	−164.042	−0.0833886	−0.0416943	+	0.999130i	$0.513276\pi$
$158$	573.081	0.288556
$159$	0	0
$160$	944.433	0.466650
$161$	−518.230	−0.253679
$162$	0	0
$163$	−371.219	−0.178381	−0.0891906	−	0.996015i	$-0.528428\pi$
$163$	−371.219	−0.178381	−0.0891906	+	0.996015i	$0.528428\pi$
$164$	−1952.37	−0.929602
$165$	0	0
$166$	−320.410	−0.149811
$167$	1720.00	0.796992	0.398496	−	0.917170i	$-0.369532\pi$
$167$	1720.00	0.796992	0.398496	+	0.917170i	$0.369532\pi$
$168$	0	0
$169$	260.267	0.118465
$170$	55.1436	0.0248784
$171$	0	0
$172$	91.8585	0.0407218
$173$	−838.717	−0.368592	−0.184296	−	0.982871i	$-0.559001\pi$
$173$	−838.717	−0.368592	−0.184296	+	0.982871i	$0.559001\pi$
$174$	0	0
$175$	959.983	0.414674
$176$	−2510.17	−1.07506
$177$	0	0
$178$	−190.104	−0.0800498
$179$	−2818.72	−1.17699	−0.588495	−	0.808501i	$-0.700280\pi$
$179$	−2818.72	−1.17699	−0.588495	+	0.808501i	$0.700280\pi$
$180$	0	0
$181$	2882.05	1.18354	0.591771	−	0.806106i	$-0.298429\pi$
$181$	2882.05	1.18354	0.591771	+	0.806106i	$0.298429\pi$
$182$	−499.564	−0.203462
$183$	0	0
$184$	618.572	0.247835
$185$	−966.400	−0.384060
$186$	0	0
$187$	−537.261	−0.210098
$188$	−516.623	−0.200418
$189$	0	0
$190$	588.021	0.224524
$191$	−1357.03	−0.514091	−0.257046	−	0.966399i	$-0.582749\pi$
$191$	−1357.03	−0.514091	−0.257046	+	0.966399i	$0.582749\pi$
$192$	0	0
$193$	1629.40	0.607704	0.303852	−	0.952719i	$-0.401727\pi$
$193$	1629.40	0.607704	0.303852	+	0.952719i	$0.401727\pi$
$194$	43.4769	0.0160900
$195$	0	0
$196$	1546.99	0.563774
$197$	−463.073	−0.167475	−0.0837376	−	0.996488i	$-0.526686\pi$
$197$	−463.073	−0.167475	−0.0837376	+	0.996488i	$0.526686\pi$
$198$	0	0
$199$	2919.96	1.04015	0.520076	−	0.854120i	$-0.325904\pi$
$199$	2919.96	1.04015	0.520076	+	0.854120i	$0.325904\pi$
$200$	−1145.86	−0.405122
$201$	0	0
$202$	534.744	0.186260
$203$	439.498	0.151954
$204$	0	0
$205$	−1718.90	−0.585626
$206$	1197.26	0.404937
$207$	0	0
$208$	−2262.59	−0.754243
$209$	−5729.05	−1.89611
$210$	0	0
$211$	−4342.15	−1.41671	−0.708355	−	0.705856i	$-0.750562\pi$
$211$	−4342.15	−1.41671	−0.708355	+	0.705856i	$0.750562\pi$
$212$	−2439.87	−0.790429
$213$	0	0
$214$	9.65964	0.00308560
$215$	80.8737	0.0256537
$216$	0	0
$217$	−1807.16	−0.565337
$218$	−1009.37	−0.313593
$219$	0	0
$220$	−2516.35	−0.771147
$221$	−484.271	−0.147401
$222$	0	0
$223$	−1370.94	−0.411681	−0.205840	−	0.978586i	$-0.565993\pi$
$223$	−1370.94	−0.411681	−0.205840	+	0.978586i	$0.565993\pi$
$224$	1686.18	0.502957
$225$	0	0
$226$	−1497.43	−0.440742
$227$	−2606.23	−0.762032	−0.381016	−	0.924568i	$-0.624426\pi$
$227$	−2606.23	−0.762032	−0.381016	+	0.924568i	$0.624426\pi$
$228$	0	0
$229$	−1208.26	−0.348665	−0.174333	−	0.984687i	$-0.555777\pi$
$229$	−1208.26	−0.348665	−0.174333	+	0.984687i	$0.555777\pi$
$230$	258.140	0.0740055
$231$	0	0
$232$	−524.595	−0.148454
$233$	−1298.47	−0.365088	−0.182544	−	0.983198i	$-0.558433\pi$
$233$	−1298.47	−0.365088	−0.182544	+	0.983198i	$0.558433\pi$
$234$	0	0
$235$	−454.843	−0.126258
$236$	−4029.10	−1.11132
$237$	0	0
$238$	98.4525	0.0268140
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−2241.81	−0.599201	−0.299601	−	0.954065i	$-0.596853\pi$
$241$	−2241.81	−0.599201	−0.299601	+	0.954065i	$0.596853\pi$
$242$	−1506.03	−0.400048
$243$	0	0
$244$	1323.73	0.347307
$245$	1362.00	0.355163
$246$	0	0
$247$	−5164.00	−1.33027
$248$	2157.07	0.552315
$249$	0	0
$250$	−1183.76	−0.299470
$251$	−1204.32	−0.302854	−0.151427	−	0.988468i	$-0.548387\pi$
$251$	−1204.32	−0.302854	−0.151427	+	0.988468i	$0.548387\pi$
$252$	0	0
$253$	−2515.04	−0.624978
$254$	2052.40	0.507005
$255$	0	0
$256$	619.733	0.151302
$257$	518.797	0.125921	0.0629605	−	0.998016i	$-0.479946\pi$
$257$	518.797	0.125921	0.0629605	+	0.998016i	$0.479946\pi$
$258$	0	0
$259$	−1725.40	−0.413942
$260$	−2268.17	−0.541022
$261$	0	0
$262$	527.538	0.124395
$263$	3817.07	0.894945	0.447472	−	0.894298i	$-0.352324\pi$
$263$	3817.07	0.894945	0.447472	+	0.894298i	$0.352324\pi$
$264$	0	0
$265$	−2148.10	−0.497951
$266$	1049.84	0.241993
$267$	0	0
$268$	−758.728	−0.172935
$269$	−8414.67	−1.90725	−0.953627	−	0.300992i	$-0.902682\pi$
$269$	−8414.67	−1.90725	−0.953627	+	0.300992i	$0.902682\pi$
$270$	0	0
$271$	−4700.92	−1.05373	−0.526865	−	0.849949i	$-0.676633\pi$
$271$	−4700.92	−1.05373	−0.526865	+	0.849949i	$0.676633\pi$
$272$	445.905	0.0994006
$273$	0	0
$274$	−950.051	−0.209470
$275$	4658.93	1.02161
$276$	0	0
$277$	2351.68	0.510104	0.255052	−	0.966927i	$-0.417907\pi$
$277$	2351.68	0.510104	0.255052	+	0.966927i	$0.417907\pi$
$278$	61.5996	0.0132896
$279$	0	0
$280$	972.828	0.207634
$281$	925.797	0.196542	0.0982712	−	0.995160i	$-0.468669\pi$
$281$	925.797	0.196542	0.0982712	+	0.995160i	$0.468669\pi$
$282$	0	0
$283$	−6238.81	−1.31046	−0.655228	−	0.755431i	$-0.727427\pi$
$283$	−6238.81	−1.31046	−0.655228	+	0.755431i	$0.727427\pi$
$284$	−2749.42	−0.574466
$285$	0	0
$286$	−2424.45	−0.501261
$287$	−3068.90	−0.631190
$288$	0	0
$289$	−4817.56	−0.980574
$290$	−218.922	−0.0443295
$291$	0	0
$292$	845.889	0.169527
$293$	−1162.43	−0.231774	−0.115887	−	0.993262i	$-0.536971\pi$
$293$	−1162.43	−0.231774	−0.115887	+	0.993262i	$0.536971\pi$
$294$	0	0
$295$	−3547.29	−0.700106
$296$	2059.47	0.404407
$297$	0	0
$298$	−1927.92	−0.374770
$299$	−2266.99	−0.438472
$300$	0	0
$301$	144.391	0.0276496
$302$	−267.077	−0.0508893
$303$	0	0
$304$	4754.89	0.897077
$305$	1165.43	0.218795
$306$	0	0
$307$	8985.64	1.67048	0.835241	−	0.549885i	$-0.185328\pi$
$307$	8985.64	1.67048	0.835241	+	0.549885i	$0.185328\pi$
$308$	−4492.66	−0.831146
$309$	0	0
$310$	900.180	0.164925
$311$	2765.80	0.504289	0.252145	−	0.967690i	$-0.418864\pi$
$311$	2765.80	0.504289	0.252145	+	0.967690i	$0.418864\pi$
$312$	0	0
$313$	−1196.92	−0.216146	−0.108073	−	0.994143i	$-0.534468\pi$
$313$	−1196.92	−0.216146	−0.108073	+	0.994143i	$0.534468\pi$
$314$	145.888	0.0262196
$315$	0	0
$316$	4645.49	0.826991
$317$	−5167.55	−0.915579	−0.457789	−	0.889061i	$-0.651359\pi$
$317$	−5167.55	−0.915579	−0.457789	+	0.889061i	$0.651359\pi$
$318$	0	0
$319$	2132.94	0.374363
$320$	1477.69	0.258141
$321$	0	0
$322$	460.879	0.0797634
$323$	1017.71	0.175315
$324$	0	0
$325$	4199.42	0.716745
$326$	330.137	0.0560878
$327$	0	0
$328$	3663.11	0.616651
$329$	−812.070	−0.136082
$330$	0	0
$331$	−8224.98	−1.36582	−0.682909	−	0.730504i	$-0.739285\pi$
$331$	−8224.98	−1.36582	−0.682909	+	0.730504i	$0.739285\pi$
$332$	−2597.30	−0.429353
$333$	0	0
$334$	−1529.65	−0.250595
$335$	−667.997	−0.108945
$336$	0	0
$337$	2142.13	0.346260	0.173130	−	0.984899i	$-0.444612\pi$
$337$	2142.13	0.346260	0.173130	+	0.984899i	$0.444612\pi$
$338$	−231.464	−0.0372484
$339$	0	0
$340$	447.004	0.0713005
$341$	−8770.40	−1.39280
$342$	0	0
$343$	6318.51	0.994658
$344$	−172.348	−0.0270128
$345$	0	0
$346$	745.898	0.115895
$347$	3471.27	0.537025	0.268513	−	0.963276i	$-0.413468\pi$
$347$	3471.27	0.537025	0.268513	+	0.963276i	$0.413468\pi$
$348$	0	0
$349$	3881.63	0.595355	0.297677	−	0.954667i	$-0.403788\pi$
$349$	3881.63	0.595355	0.297677	+	0.954667i	$0.403788\pi$
$350$	−853.745	−0.130384
$351$	0	0
$352$	8183.24	1.23911
$353$	1058.67	0.159625	0.0798124	−	0.996810i	$-0.474568\pi$
$353$	1058.67	0.159625	0.0798124	+	0.996810i	$0.474568\pi$
$354$	0	0
$355$	−2420.64	−0.361899
$356$	−1541.01	−0.229420
$357$	0	0
$358$	2506.78	0.370077
$359$	6878.60	1.01125	0.505625	−	0.862754i	$-0.331262\pi$
$359$	6878.60	1.01125	0.505625	+	0.862754i	$0.331262\pi$
$360$	0	0
$361$	3993.25	0.582191
$362$	−2563.10	−0.372137
$363$	0	0
$364$	−4049.55	−0.583115
$365$	744.735	0.106798
$366$	0	0
$367$	−12682.7	−1.80391	−0.901954	−	0.431832i	$-0.857867\pi$
$367$	−12682.7	−1.80391	−0.901954	+	0.431832i	$0.857867\pi$
$368$	2087.39	0.295686
$369$	0	0
$370$	859.452	0.120759
$371$	−3835.19	−0.536693
$372$	0	0
$373$	−1793.52	−0.248968	−0.124484	−	0.992222i	$-0.539728\pi$
$373$	−1793.52	−0.248968	−0.124484	+	0.992222i	$0.539728\pi$
$374$	477.803	0.0660605
$375$	0	0
$376$	969.306	0.132947
$377$	1922.57	0.262646
$378$	0	0
$379$	−4863.75	−0.659192	−0.329596	−	0.944122i	$-0.606913\pi$
$379$	−4863.75	−0.659192	−0.329596	+	0.944122i	$0.606913\pi$
$380$	4766.60	0.643477
$381$	0	0
$382$	1206.85	0.161644
$383$	7510.66	1.00203	0.501014	−	0.865439i	$-0.332960\pi$
$383$	7510.66	1.00203	0.501014	+	0.865439i	$0.332960\pi$
$384$	0	0
$385$	−3955.41	−0.523601
$386$	−1449.08	−0.191078
$387$	0	0
$388$	352.431	0.0461134
$389$	−2326.85	−0.303280	−0.151640	−	0.988436i	$-0.548455\pi$
$389$	−2326.85	−0.303280	−0.151640	+	0.988436i	$0.548455\pi$
$390$	0	0
$391$	446.771	0.0577856
$392$	−2902.53	−0.373979
$393$	0	0
$394$	411.826	0.0526587
$395$	4089.97	0.520984
$396$	0	0
$397$	11633.8	1.47074	0.735372	−	0.677664i	$-0.237008\pi$
$397$	11633.8	1.47074	0.735372	+	0.677664i	$0.237008\pi$
$398$	−2596.81	−0.327052
$399$	0	0
$400$	−3866.73	−0.483341
$401$	2017.85	0.251288	0.125644	−	0.992075i	$-0.459900\pi$
$401$	2017.85	0.251288	0.125644	+	0.992075i	$0.459900\pi$
$402$	0	0
$403$	−7905.38	−0.977159
$404$	4334.73	0.533813
$405$	0	0
$406$	−390.860	−0.0477785
$407$	−8373.58	−1.01981
$408$	0	0
$409$	4543.26	0.549266	0.274633	−	0.961549i	$-0.411444\pi$
$409$	4543.26	0.549266	0.274633	+	0.961549i	$0.411444\pi$
$410$	1528.68	0.184136
$411$	0	0
$412$	9705.20	1.16054
$413$	−6333.27	−0.754576
$414$	0	0
$415$	−2286.70	−0.270482
$416$	7376.13	0.869338
$417$	0	0
$418$	5095.03	0.596187
$419$	−1174.18	−0.136903	−0.0684517	−	0.997654i	$-0.521806\pi$
$419$	−1174.18	−0.136903	−0.0684517	+	0.997654i	$0.521806\pi$
$420$	0	0
$421$	5922.72	0.685644	0.342822	−	0.939400i	$-0.388617\pi$
$421$	5922.72	0.685644	0.342822	+	0.939400i	$0.388617\pi$
$422$	3861.61	0.445451
$423$	0	0
$424$	4577.77	0.524331
$425$	−827.610	−0.0944588
$426$	0	0
$427$	2080.74	0.235818
$428$	78.3027	0.00884324
$429$	0	0
$430$	−71.9237	−0.00806620
$431$	−5112.58	−0.571379	−0.285689	−	0.958322i	$-0.592223\pi$
$431$	−5112.58	−0.571379	−0.285689	+	0.958322i	$0.592223\pi$
$432$	0	0
$433$	−2265.90	−0.251483	−0.125742	−	0.992063i	$-0.540131\pi$
$433$	−2265.90	−0.251483	−0.125742	+	0.992063i	$0.540131\pi$
$434$	1607.17	0.177757
$435$	0	0
$436$	−8182.13	−0.898746
$437$	4764.12	0.521507
$438$	0	0
$439$	−2872.01	−0.312240	−0.156120	−	0.987738i	$-0.549899\pi$
$439$	−2872.01	−0.312240	−0.156120	+	0.987738i	$0.549899\pi$
$440$	4721.27	0.511540
$441$	0	0
$442$	430.678	0.0463468
$443$	−12690.3	−1.36102	−0.680512	−	0.732737i	$-0.738243\pi$
$443$	−12690.3	−1.36102	−0.680512	+	0.732737i	$0.738243\pi$
$444$	0	0
$445$	−1356.73	−0.144529
$446$	1219.22	0.129443
$447$	0	0
$448$	2638.24	0.278225
$449$	−13661.8	−1.43595	−0.717973	−	0.696071i	$-0.754930\pi$
$449$	−13661.8	−1.43595	−0.717973	+	0.696071i	$0.754930\pi$
$450$	0	0
$451$	−14893.8	−1.55504
$452$	−12138.4	−1.26315
$453$	0	0
$454$	2317.80	0.239603
$455$	−3565.29	−0.367348
$456$	0	0
$457$	−12073.1	−1.23579	−0.617897	−	0.786259i	$-0.712015\pi$
$457$	−12073.1	−1.23579	−0.617897	+	0.786259i	$0.712015\pi$
$458$	1074.55	0.109630
$459$	0	0
$460$	2092.53	0.212097
$461$	−4497.29	−0.454360	−0.227180	−	0.973853i	$-0.572951\pi$
$461$	−4497.29	−0.454360	−0.227180	+	0.973853i	$0.572951\pi$
$462$	0	0
$463$	1569.33	0.157523	0.0787614	−	0.996893i	$-0.474903\pi$
$463$	1569.33	0.157523	0.0787614	+	0.996893i	$0.474903\pi$
$464$	−1770.26	−0.177117
$465$	0	0
$466$	1154.77	0.114793
$467$	8336.24	0.826028	0.413014	−	0.910725i	$-0.364476\pi$
$467$	8336.24	0.826028	0.413014	+	0.910725i	$0.364476\pi$
$468$	0	0
$469$	−1192.63	−0.117421
$470$	404.507	0.0396990
$471$	0	0
$472$	7559.54	0.737195
$473$	700.748	0.0681193
$474$	0	0
$475$	−8825.17	−0.852477
$476$	798.073	0.0768480
$477$	0	0
$478$	212.551	0.0203386
$479$	11652.6	1.11153	0.555764	−	0.831340i	$-0.312426\pi$
$479$	11652.6	1.11153	0.555764	+	0.831340i	$0.312426\pi$
$480$	0	0
$481$	−7547.70	−0.715480
$482$	1993.71	0.188405
$483$	0	0
$484$	−12208.2	−1.14652
$485$	310.286	0.0290503
$486$	0	0
$487$	−810.168	−0.0753845	−0.0376922	−	0.999289i	$-0.512001\pi$
$487$	−810.168	−0.0753845	−0.0376922	+	0.999289i	$0.512001\pi$
$488$	−2483.62	−0.230386
$489$	0	0
$490$	−1211.27	−0.111673
$491$	−6471.35	−0.594802	−0.297401	−	0.954753i	$-0.596120\pi$
$491$	−6471.35	−0.594802	−0.297401	+	0.954753i	$0.596120\pi$
$492$	0	0
$493$	−378.895	−0.0346137
$494$	4592.51	0.418273
$495$	0	0
$496$	7279.09	0.658953
$497$	−4321.77	−0.390056
$498$	0	0
$499$	−9458.86	−0.848570	−0.424285	−	0.905529i	$-0.639475\pi$
$499$	−9458.86	−0.848570	−0.424285	+	0.905529i	$0.639475\pi$
$500$	−9595.76	−0.858271
$501$	0	0
$502$	1071.04	0.0952252
$503$	3963.27	0.351319	0.175659	−	0.984451i	$-0.443794\pi$
$503$	3963.27	0.351319	0.175659	+	0.984451i	$0.443794\pi$
$504$	0	0
$505$	3816.36	0.336289
$506$	2236.71	0.196510
$507$	0	0
$508$	16637.2	1.45306
$509$	−11968.0	−1.04218	−0.521091	−	0.853501i	$-0.674475\pi$
$509$	−11968.0	−1.04218	−0.521091	+	0.853501i	$0.674475\pi$
$510$	0	0
$511$	1329.64	0.115107
$512$	−11730.8	−1.01256
$513$	0	0
$514$	−461.383	−0.0395929
$515$	8544.62	0.731109
$516$	0	0
$517$	−3941.09	−0.335259
$518$	1534.45	0.130154
$519$	0	0
$520$	4255.61	0.358886
$521$	12279.7	1.03260	0.516299	−	0.856408i	$-0.327309\pi$
$521$	12279.7	1.03260	0.516299	+	0.856408i	$0.327309\pi$
$522$	0	0
$523$	1555.96	0.130091	0.0650455	−	0.997882i	$-0.479281\pi$
$523$	1555.96	0.130091	0.0650455	+	0.997882i	$0.479281\pi$
$524$	4276.31	0.356511
$525$	0	0
$526$	−3394.64	−0.281395
$527$	1557.97	0.128778
$528$	0	0
$529$	−10075.6	−0.828106
$530$	1910.38	0.156569
$531$	0	0
$532$	8510.21	0.693542
$533$	−13424.8	−1.09098
$534$	0	0
$535$	68.9390	0.00557102
$536$	1423.55	0.114717
$537$	0	0
$538$	7483.44	0.599691
$539$	11801.3	0.943079
$540$	0	0
$541$	−20689.1	−1.64416	−0.822082	−	0.569369i	$-0.807188\pi$
$541$	−20689.1	−1.64416	−0.822082	+	0.569369i	$0.807188\pi$
$542$	4180.68	0.331321
$543$	0	0
$544$	−1453.67	−0.114569
$545$	−7203.68	−0.566187
$546$	0	0
$547$	−14459.0	−1.13021	−0.565104	−	0.825020i	$-0.691164\pi$
$547$	−14459.0	−1.13021	−0.565104	+	0.825020i	$0.691164\pi$
$548$	−7701.28	−0.600333
$549$	0	0
$550$	−4143.34	−0.321223
$551$	−4040.33	−0.312384
$552$	0	0
$553$	7302.16	0.561518
$554$	−2091.43	−0.160390
$555$	0	0
$556$	499.337	0.0380874
$557$	−2886.69	−0.219593	−0.109796	−	0.993954i	$-0.535020\pi$
$557$	−2886.69	−0.219593	−0.109796	+	0.993954i	$0.535020\pi$
$558$	0	0
$559$	631.634	0.0477912
$560$	3282.83	0.247723
$561$	0	0
$562$	−823.342	−0.0617982
$563$	22368.1	1.67443	0.837215	−	0.546874i	$-0.184182\pi$
$563$	22368.1	1.67443	0.837215	+	0.546874i	$0.184182\pi$
$564$	0	0
$565$	−10686.9	−0.795753
$566$	5548.38	0.412042
$567$	0	0
$568$	5158.56	0.381071
$569$	23462.5	1.72865	0.864323	−	0.502937i	$-0.167747\pi$
$569$	23462.5	1.72865	0.864323	+	0.502937i	$0.167747\pi$
$570$	0	0
$571$	5256.97	0.385284	0.192642	−	0.981269i	$-0.438294\pi$
$571$	5256.97	0.385284	0.192642	+	0.981269i	$0.438294\pi$
$572$	−19653.0	−1.43660
$573$	0	0
$574$	2729.28	0.198463
$575$	−3874.24	−0.280986
$576$	0	0
$577$	−8469.23	−0.611055	−0.305528	−	0.952183i	$-0.598833\pi$
$577$	−8469.23	−0.611055	−0.305528	+	0.952183i	$0.598833\pi$
$578$	4284.42	0.308319
$579$	0	0
$580$	−1774.62	−0.127047
$581$	−4082.64	−0.291526
$582$	0	0
$583$	−18612.7	−1.32223
$584$	−1587.09	−0.112456
$585$	0	0
$586$	1033.78	0.0728758
$587$	24157.4	1.69861	0.849304	−	0.527904i	$-0.177022\pi$
$587$	24157.4	1.69861	0.849304	+	0.527904i	$0.177022\pi$
$588$	0	0
$589$	16613.3	1.16221
$590$	3154.72	0.220132
$591$	0	0
$592$	6949.74	0.482488
$593$	12882.1	0.892085	0.446042	−	0.895012i	$-0.352833\pi$
$593$	12882.1	0.892085	0.446042	+	0.895012i	$0.352833\pi$
$594$	0	0
$595$	702.637	0.0484123
$596$	−15628.1	−1.07408
$597$	0	0
$598$	2016.11	0.137867
$599$	−18829.8	−1.28442	−0.642209	−	0.766530i	$-0.721982\pi$
$599$	−18829.8	−1.28442	−0.642209	+	0.766530i	$0.721982\pi$
$600$	0	0
$601$	−13460.5	−0.913588	−0.456794	−	0.889573i	$-0.651002\pi$
$601$	−13460.5	−0.913588	−0.456794	+	0.889573i	$0.651002\pi$
$602$	−128.411	−0.00869379
$603$	0	0
$604$	−2164.98	−0.145847
$605$	−10748.3	−0.722281
$606$	0	0
$607$	−24468.1	−1.63613	−0.818064	−	0.575127i	$-0.804953\pi$
$607$	−24468.1	−1.63613	−0.818064	+	0.575127i	$0.804953\pi$
$608$	−15501.1	−1.03397
$609$	0	0
$610$	−1036.46	−0.0687949
$611$	−3552.38	−0.235211
$612$	0	0
$613$	−13205.7	−0.870100	−0.435050	−	0.900406i	$-0.643269\pi$
$613$	−13205.7	−0.870100	−0.435050	+	0.900406i	$0.643269\pi$
$614$	−7991.23	−0.525244
$615$	0	0
$616$	8429.28	0.551340
$617$	−15409.8	−1.00547	−0.502735	−	0.864440i	$-0.667673\pi$
$617$	−15409.8	−1.00547	−0.502735	+	0.864440i	$0.667673\pi$
$618$	0	0
$619$	25293.0	1.64234	0.821171	−	0.570682i	$-0.193321\pi$
$619$	25293.0	1.64234	0.821171	+	0.570682i	$0.193321\pi$
$620$	7297.02	0.472670
$621$	0	0
$622$	−2459.71	−0.158562
$623$	−2422.29	−0.155774
$624$	0	0
$625$	2141.18	0.137036
$626$	1064.46	0.0679620
$627$	0	0
$628$	1182.60	0.0751444
$629$	1487.48	0.0942920
$630$	0	0
$631$	−1787.05	−0.112744	−0.0563720	−	0.998410i	$-0.517953\pi$
$631$	−1787.05	−0.112744	−0.0563720	+	0.998410i	$0.517953\pi$
$632$	−8716.03	−0.548584
$633$	0	0
$634$	4595.67	0.287882
$635$	14647.6	0.915391
$636$	0	0
$637$	10637.4	0.661646
$638$	−1896.90	−0.117710
$639$	0	0
$640$	−8869.62	−0.547816
$641$	15207.0	0.937037	0.468518	−	0.883454i	$-0.344788\pi$
$641$	15207.0	0.937037	0.468518	+	0.883454i	$0.344788\pi$
$642$	0	0
$643$	4435.03	0.272007	0.136004	−	0.990708i	$-0.456574\pi$
$643$	4435.03	0.272007	0.136004	+	0.990708i	$0.456574\pi$
$644$	3735.97	0.228599
$645$	0	0
$646$	−905.079	−0.0551236
$647$	−4675.49	−0.284100	−0.142050	−	0.989859i	$-0.545369\pi$
$647$	−4675.49	−0.284100	−0.142050	+	0.989859i	$0.545369\pi$
$648$	0	0
$649$	−30736.2	−1.85902
$650$	−3734.68	−0.225364
$651$	0	0
$652$	2676.15	0.160746
$653$	−5716.16	−0.342558	−0.171279	−	0.985223i	$-0.554790\pi$
$653$	−5716.16	−0.342558	−0.171279	+	0.985223i	$0.554790\pi$
$654$	0	0
$655$	3764.94	0.224593
$656$	12361.3	0.735711
$657$	0	0
$658$	722.200	0.0427877
$659$	21565.8	1.27479	0.637394	−	0.770538i	$-0.280012\pi$
$659$	21565.8	1.27479	0.637394	+	0.770538i	$0.280012\pi$
$660$	0	0
$661$	18984.6	1.11712	0.558558	−	0.829465i	$-0.311355\pi$
$661$	18984.6	1.11712	0.558558	+	0.829465i	$0.311355\pi$
$662$	7314.74	0.429449
$663$	0	0
$664$	4873.14	0.284811
$665$	7492.53	0.436914
$666$	0	0
$667$	−1773.70	−0.102965
$668$	−12399.6	−0.718198
$669$	0	0
$670$	594.071	0.0342552
$671$	10098.1	0.580975
$672$	0	0
$673$	−4084.82	−0.233965	−0.116982	−	0.993134i	$-0.537322\pi$
$673$	−4084.82	−0.233965	−0.116982	+	0.993134i	$0.537322\pi$
$674$	−1905.07	−0.108873
$675$	0	0
$676$	−1876.29	−0.106753
$677$	−23886.7	−1.35604	−0.678020	−	0.735044i	$-0.737162\pi$
$677$	−23886.7	−1.35604	−0.678020	+	0.735044i	$0.737162\pi$
$678$	0	0
$679$	553.980	0.0313105
$680$	−838.684	−0.0472971
$681$	0	0
$682$	7799.81	0.437933
$683$	16925.7	0.948236	0.474118	−	0.880461i	$-0.342767\pi$
$683$	16925.7	0.948236	0.474118	+	0.880461i	$0.342767\pi$
$684$	0	0
$685$	−6780.33	−0.378195
$686$	−5619.26	−0.312747
$687$	0	0
$688$	−581.593	−0.0322282
$689$	−16776.9	−0.927650
$690$	0	0
$691$	23902.5	1.31591	0.657956	−	0.753056i	$-0.271421\pi$
$691$	23902.5	1.31591	0.657956	+	0.753056i	$0.271421\pi$
$692$	6046.38	0.332152
$693$	0	0
$694$	−3087.12	−0.168855
$695$	439.625	0.0239941
$696$	0	0
$697$	2645.73	0.143779
$698$	−3452.06	−0.187195
$699$	0	0
$700$	−6920.60	−0.373677
$701$	15019.4	0.809238	0.404619	−	0.914485i	$-0.367404\pi$
$701$	15019.4	0.809238	0.404619	+	0.914485i	$0.367404\pi$
$702$	0	0
$703$	15861.7	0.850973
$704$	12803.7	0.685452
$705$	0	0
$706$	−941.514	−0.0501903
$707$	6813.68	0.362454
$708$	0	0
$709$	3905.90	0.206896	0.103448	−	0.994635i	$-0.467013\pi$
$709$	3905.90	0.206896	0.103448	+	0.994635i	$0.467013\pi$
$710$	2152.75	0.113791
$711$	0	0
$712$	2891.30	0.152186
$713$	7293.22	0.383076
$714$	0	0
$715$	−17302.8	−0.905020
$716$	20320.4	1.06063
$717$	0	0
$718$	−6117.36	−0.317964
$719$	−10847.6	−0.562652	−0.281326	−	0.959612i	$-0.590774\pi$
$719$	−10847.6	−0.562652	−0.281326	+	0.959612i	$0.590774\pi$
$720$	0	0
$721$	15255.4	0.787992
$722$	−3551.33	−0.183056
$723$	0	0
$724$	−20777.0	−1.06653
$725$	3285.64	0.168311
$726$	0	0
$727$	11086.5	0.565581	0.282790	−	0.959182i	$-0.408740\pi$
$727$	11086.5	0.565581	0.282790	+	0.959182i	$0.408740\pi$
$728$	7597.91	0.386809
$729$	0	0
$730$	−662.317	−0.0335801
$731$	−124.481	−0.00629833
$732$	0	0
$733$	−4617.67	−0.232684	−0.116342	−	0.993209i	$-0.537117\pi$
$733$	−4617.67	−0.232684	−0.116342	+	0.993209i	$0.537117\pi$
$734$	11279.2	0.567197
$735$	0	0
$736$	−6804.96	−0.340807
$737$	−5788.00	−0.289286
$738$	0	0
$739$	29916.4	1.48916	0.744582	−	0.667531i	$-0.232649\pi$
$739$	29916.4	1.48916	0.744582	+	0.667531i	$0.232649\pi$
$740$	6966.87	0.346091
$741$	0	0
$742$	3410.76	0.168751
$743$	14895.4	0.735478	0.367739	−	0.929929i	$-0.380132\pi$
$743$	14895.4	0.735478	0.367739	+	0.929929i	$0.380132\pi$
$744$	0	0
$745$	−13759.2	−0.676642
$746$	1595.04	0.0782821
$747$	0	0
$748$	3873.16	0.189327
$749$	123.083	0.00600446
$750$	0	0
$751$	27218.7	1.32254	0.661269	−	0.750149i	$-0.270018\pi$
$751$	27218.7	1.32254	0.661269	+	0.750149i	$0.270018\pi$
$752$	3270.95	0.158616
$753$	0	0
$754$	−1709.81	−0.0825829
$755$	−1906.08	−0.0918800
$756$	0	0
$757$	−17727.1	−0.851127	−0.425563	−	0.904929i	$-0.639924\pi$
$757$	−17727.1	−0.851127	−0.425563	+	0.904929i	$0.639924\pi$
$758$	4325.49	0.207268
$759$	0	0
$760$	−8943.26	−0.426850
$761$	17383.4	0.828051	0.414026	−	0.910265i	$-0.364122\pi$
$761$	17383.4	0.828051	0.414026	+	0.910265i	$0.364122\pi$
$762$	0	0
$763$	−12861.3	−0.610239
$764$	9782.97	0.463266
$765$	0	0
$766$	−6679.48	−0.315065
$767$	−27704.7	−1.30425
$768$	0	0
$769$	−5660.35	−0.265432	−0.132716	−	0.991154i	$-0.542370\pi$
$769$	−5660.35	−0.265432	−0.132716	+	0.991154i	$0.542370\pi$
$770$	3517.68	0.164634
$771$	0	0
$772$	−11746.5	−0.547624
$773$	16491.1	0.767325	0.383662	−	0.923473i	$-0.374663\pi$
$773$	16491.1	0.767325	0.383662	+	0.923473i	$0.374663\pi$
$774$	0	0
$775$	−13510.1	−0.626192
$776$	−661.244	−0.0305893
$777$	0	0
$778$	2069.34	0.0953593
$779$	28212.6	1.29759
$780$	0	0
$781$	−20974.1	−0.960965
$782$	−397.328	−0.0181693
$783$	0	0
$784$	−9794.65	−0.446185
$785$	1041.18	0.0473391
$786$	0	0
$787$	22681.6	1.02734	0.513668	−	0.857989i	$-0.328286\pi$
$787$	22681.6	1.02734	0.513668	+	0.857989i	$0.328286\pi$
$788$	3338.34	0.150918
$789$	0	0
$790$	−3637.34	−0.163811
$791$	−19080.2	−0.857666
$792$	0	0
$793$	9102.16	0.407601
$794$	−10346.3	−0.462441
$795$	0	0
$796$	−21050.2	−0.937319
$797$	15113.4	0.671698	0.335849	−	0.941916i	$-0.390977\pi$
$797$	15113.4	0.671698	0.335849	+	0.941916i	$0.390977\pi$
$798$	0	0
$799$	700.093	0.0309981
$800$	12605.7	0.557097
$801$	0	0
$802$	−1794.54	−0.0790116
$803$	6452.92	0.283585
$804$	0	0
$805$	3289.21	0.144012
$806$	7030.52	0.307245
$807$	0	0
$808$	−8132.96	−0.354105
$809$	22148.6	0.962550	0.481275	−	0.876570i	$-0.340174\pi$
$809$	22148.6	0.962550	0.481275	+	0.876570i	$0.340174\pi$
$810$	0	0
$811$	−1324.71	−0.0573576	−0.0286788	−	0.999589i	$-0.509130\pi$
$811$	−1324.71	−0.0573576	−0.0286788	+	0.999589i	$0.509130\pi$
$812$	−3168.38	−0.136931
$813$	0	0
$814$	7446.90	0.320656
$815$	2356.13	0.101266
$816$	0	0
$817$	−1327.39	−0.0568415
$818$	−4040.47	−0.172704
$819$	0	0
$820$	12391.7	0.527729
$821$	−40237.1	−1.71046	−0.855228	−	0.518252i	$-0.826583\pi$
$821$	−40237.1	−1.71046	−0.855228	+	0.518252i	$0.826583\pi$
$822$	0	0
$823$	34834.3	1.47539	0.737695	−	0.675134i	$-0.235914\pi$
$823$	34834.3	1.47539	0.737695	+	0.675134i	$0.235914\pi$
$824$	−18209.2	−0.769841
$825$	0	0
$826$	5632.39	0.237259
$827$	−20883.6	−0.878107	−0.439053	−	0.898461i	$-0.644686\pi$
$827$	−20883.6	−0.878107	−0.439053	+	0.898461i	$0.644686\pi$
$828$	0	0
$829$	13459.7	0.563902	0.281951	−	0.959429i	$-0.409018\pi$
$829$	13459.7	0.563902	0.281951	+	0.959429i	$0.409018\pi$
$830$	2033.64	0.0850466
$831$	0	0
$832$	11540.9	0.480900
$833$	−2096.38	−0.0871973
$834$	0	0
$835$	−10916.8	−0.452447
$836$	41301.2	1.70865
$837$	0	0
$838$	1044.24	0.0430461
$839$	19502.5	0.802504	0.401252	−	0.915968i	$-0.368575\pi$
$839$	19502.5	0.802504	0.401252	+	0.915968i	$0.368575\pi$
$840$	0	0
$841$	−22884.8	−0.938324
$842$	−5267.27	−0.215585
$843$	0	0
$844$	31302.9	1.27665
$845$	−1651.91	−0.0672515
$846$	0	0
$847$	−19189.8	−0.778477
$848$	15447.8	0.625566
$849$	0	0
$850$	736.021	0.0297004
$851$	6963.24	0.280490
$852$	0	0
$853$	893.918	0.0358818	0.0179409	−	0.999839i	$-0.494289\pi$
$853$	893.918	0.0358818	0.0179409	+	0.999839i	$0.494289\pi$
$854$	−1850.47	−0.0741475
$855$	0	0
$856$	−146.914	−0.00586615
$857$	−23446.5	−0.934560	−0.467280	−	0.884109i	$-0.654766\pi$
$857$	−23446.5	−0.934560	−0.467280	+	0.884109i	$0.654766\pi$
$858$	0	0
$859$	15738.5	0.625135	0.312567	−	0.949896i	$-0.398811\pi$
$859$	15738.5	0.625135	0.312567	+	0.949896i	$0.398811\pi$
$860$	−583.026	−0.0231175
$861$	0	0
$862$	4546.79	0.179657
$863$	−32628.9	−1.28702	−0.643512	−	0.765436i	$-0.722523\pi$
$863$	−32628.9	−1.28702	−0.643512	+	0.765436i	$0.722523\pi$
$864$	0	0
$865$	5323.33	0.209247
$866$	2015.14	0.0790730
$867$	0	0
$868$	13028.0	0.509445
$869$	35438.4	1.38339
$870$	0	0
$871$	−5217.13	−0.202957
$872$	15351.6	0.596182
$873$	0	0
$874$	−4236.89	−0.163976
$875$	−15083.4	−0.582757
$876$	0	0
$877$	1534.21	0.0590726	0.0295363	−	0.999564i	$-0.490597\pi$
$877$	1534.21	0.0590726	0.0295363	+	0.999564i	$0.490597\pi$
$878$	2554.17	0.0981766
$879$	0	0
$880$	15932.0	0.610306
$881$	−36791.3	−1.40696	−0.703480	−	0.710715i	$-0.748372\pi$
$881$	−36791.3	−1.40696	−0.703480	+	0.710715i	$0.748372\pi$
$882$	0	0
$883$	−49814.0	−1.89850	−0.949249	−	0.314527i	$-0.898154\pi$
$883$	−49814.0	−1.89850	−0.949249	+	0.314527i	$0.898154\pi$
$884$	3491.15	0.132828
$885$	0	0
$886$	11285.9	0.427942
$887$	−5542.98	−0.209825	−0.104913	−	0.994481i	$-0.533456\pi$
$887$	−5542.98	−0.209825	−0.104913	+	0.994481i	$0.533456\pi$
$888$	0	0
$889$	26151.6	0.986612
$890$	1206.59	0.0454437
$891$	0	0
$892$	9883.22	0.370980
$893$	7465.40	0.279754
$894$	0	0
$895$	17890.4	0.668169
$896$	−15835.7	−0.590439
$897$	0	0
$898$	12149.9	0.451500
$899$	−6185.19	−0.229464
$900$	0	0
$901$	3306.35	0.122254
$902$	13245.5	0.488945
$903$	0	0
$904$	22774.6	0.837910
$905$	−18292.4	−0.671889
$906$	0	0
$907$	31713.1	1.16099	0.580493	−	0.814265i	$-0.302860\pi$
$907$	31713.1	1.16099	0.580493	+	0.814265i	$0.302860\pi$
$908$	18788.5	0.686694
$909$	0	0
$910$	3170.73	0.115504
$911$	18994.0	0.690780	0.345390	−	0.938459i	$-0.387747\pi$
$911$	18994.0	0.690780	0.345390	+	0.938459i	$0.387747\pi$
$912$	0	0
$913$	−19813.6	−0.718221
$914$	10737.0	0.388566
$915$	0	0
$916$	8710.48	0.314195
$917$	6721.86	0.242067
$918$	0	0
$919$	−36621.2	−1.31449	−0.657247	−	0.753675i	$-0.728279\pi$
$919$	−36621.2	−1.31449	−0.657247	+	0.753675i	$0.728279\pi$
$920$	−3926.08	−0.140694
$921$	0	0
$922$	3999.59	0.142863
$923$	−18905.5	−0.674194
$924$	0	0
$925$	−12898.9	−0.458500
$926$	−1395.66	−0.0495294
$927$	0	0
$928$	5771.11	0.204144
$929$	−25240.1	−0.891388	−0.445694	−	0.895185i	$-0.647043\pi$
$929$	−25240.1	−0.891388	−0.445694	+	0.895185i	$0.647043\pi$
$930$	0	0
$931$	−22354.7	−0.786944
$932$	9360.76	0.328993
$933$	0	0
$934$	−7413.69	−0.259725
$935$	3409.99	0.119271
$936$	0	0
$937$	−24780.4	−0.863969	−0.431984	−	0.901881i	$-0.642186\pi$
$937$	−24780.4	−0.863969	−0.431984	+	0.901881i	$0.642186\pi$
$938$	1060.65	0.0369204
$939$	0	0
$940$	3279.00	0.113776
$941$	13338.5	0.462085	0.231043	−	0.972944i	$-0.425786\pi$
$941$	13338.5	0.462085	0.231043	+	0.972944i	$0.425786\pi$
$942$	0	0
$943$	12385.3	0.427699
$944$	25509.9	0.879529
$945$	0	0
$946$	−623.198	−0.0214185
$947$	−38939.5	−1.33618	−0.668090	−	0.744080i	$-0.732888\pi$
$947$	−38939.5	−1.33618	−0.668090	+	0.744080i	$0.732888\pi$
$948$	0	0
$949$	5816.47	0.198957
$950$	7848.52	0.268042
$951$	0	0
$952$	−1497.37	−0.0509770
$953$	−33165.8	−1.12733	−0.563664	−	0.826004i	$-0.690609\pi$
$953$	−33165.8	−1.12733	−0.563664	+	0.826004i	$0.690609\pi$
$954$	0	0
$955$	8613.09	0.291846
$956$	1722.97	0.0582896
$957$	0	0
$958$	−10363.1	−0.349494
$959$	−12105.5	−0.407620
$960$	0	0
$961$	−4358.24	−0.146294
$962$	6712.42	0.224966
$963$	0	0
$964$	16161.4	0.539962
$965$	−10341.8	−0.344989
$966$	0	0
$967$	43255.5	1.43847	0.719236	−	0.694765i	$-0.244492\pi$
$967$	43255.5	1.43847	0.719236	+	0.694765i	$0.244492\pi$
$968$	22905.4	0.760545
$969$	0	0
$970$	−275.948	−0.00913417
$971$	−5315.25	−0.175669	−0.0878344	−	0.996135i	$-0.527995\pi$
$971$	−5315.25	−0.175669	−0.0878344	+	0.996135i	$0.527995\pi$
$972$	0	0
$973$	784.899	0.0258610
$974$	720.509	0.0237029
$975$	0	0
$976$	−8381.05	−0.274868
$977$	3628.58	0.118821	0.0594107	−	0.998234i	$-0.481078\pi$
$977$	3628.58	0.118821	0.0594107	+	0.998234i	$0.481078\pi$
$978$	0	0
$979$	−11755.7	−0.383773
$980$	−9818.77	−0.320050
$981$	0	0
$982$	5755.18	0.187022
$983$	−4308.64	−0.139801	−0.0699004	−	0.997554i	$-0.522268\pi$
$983$	−4308.64	−0.139801	−0.0699004	+	0.997554i	$0.522268\pi$
$984$	0	0
$985$	2939.13	0.0950745
$986$	336.964	0.0108835
$987$	0	0
$988$	37227.7	1.19876
$989$	−582.722	−0.0187356
$990$	0	0
$991$	11922.0	0.382153	0.191076	−	0.981575i	$-0.438802\pi$
$991$	11922.0	0.382153	0.191076	+	0.981575i	$0.438802\pi$
$992$	−23730.1	−0.759507
$993$	0	0
$994$	3843.49	0.122644
$995$	−18533.0	−0.590487
$996$	0	0
$997$	21350.6	0.678214	0.339107	−	0.940748i	$-0.389875\pi$
$997$	21350.6	0.678214	0.339107	+	0.940748i	$0.389875\pi$
$998$	8412.07	0.266813
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.b.1.13		28
3.2	odd	2		717.4.a.b.1.16	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.b.1.16	✓	28	3.2	odd	2
2151.4.a.b.1.13		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)

\(f(q)\)	\(=\)	\(q-0.889333 q^{2} -7.20909 q^{4} -6.34700 q^{5} -11.3318 q^{7} +13.5259 q^{8} +O(q^{10})\)	\(q-0.889333 q^{2} -7.20909 q^{4} -6.34700 q^{5} -11.3318 q^{7} +13.5259 q^{8} +5.64460 q^{10} -54.9949 q^{11} -49.5708 q^{13} +10.0778 q^{14} +45.6436 q^{16} +9.76927 q^{17} +104.174 q^{19} +45.7561 q^{20} +48.9088 q^{22} +45.7323 q^{23} -84.7156 q^{25} +44.0850 q^{26} +81.6922 q^{28} -38.7844 q^{29} +159.477 q^{31} -148.800 q^{32} -8.68814 q^{34} +71.9232 q^{35} +152.261 q^{37} -92.6455 q^{38} -85.8491 q^{40} +270.821 q^{41} -12.7420 q^{43} +396.463 q^{44} -40.6712 q^{46} +71.6627 q^{47} -214.590 q^{49} +75.3404 q^{50} +357.360 q^{52} +338.444 q^{53} +349.053 q^{55} -153.274 q^{56} +34.4922 q^{58} +558.892 q^{59} -183.619 q^{61} -141.828 q^{62} -232.816 q^{64} +314.626 q^{65} +105.246 q^{67} -70.4275 q^{68} -63.9636 q^{70} +381.383 q^{71} -117.337 q^{73} -135.411 q^{74} -751.000 q^{76} +623.194 q^{77} -644.394 q^{79} -289.700 q^{80} -240.850 q^{82} +360.281 q^{83} -62.0056 q^{85} +11.3319 q^{86} -743.858 q^{88} +213.760 q^{89} +561.728 q^{91} -329.688 q^{92} -63.7320 q^{94} -661.193 q^{95} -48.8871 q^{97} +190.842 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) \) 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8} - 88 q^{10} + 110 q^{11} - 82 q^{13} - 126 q^{14} + 271 q^{16} - 100 q^{17} - 292 q^{19} + 52 q^{20} - 351 q^{22} + 276 q^{23} + 386 q^{25} - 84 q^{26} - 1010 q^{28} + 38 q^{29} - 432 q^{31} + 452 q^{32} - 524 q^{34} + 166 q^{35} - 936 q^{37} + 41 q^{38} - 1183 q^{40} - 1054 q^{41} - 1804 q^{43} + 341 q^{44} - 888 q^{46} + 560 q^{47} + 1074 q^{49} + 1054 q^{50} - 632 q^{52} + 160 q^{53} - 842 q^{55} - 509 q^{56} - 1266 q^{58} - 846 q^{59} - 2220 q^{61} - 82 q^{62} - 1565 q^{64} - 296 q^{65} - 4752 q^{67} + 1719 q^{68} - 5601 q^{70} + 802 q^{71} - 2732 q^{73} + 4581 q^{74} - 5614 q^{76} + 1008 q^{77} - 3172 q^{79} + 732 q^{80} - 9709 q^{82} + 4780 q^{83} - 4624 q^{85} + 2009 q^{86} - 9331 q^{88} - 4372 q^{89} - 7398 q^{91} + 6138 q^{92} - 7068 q^{94} + 3160 q^{95} - 4846 q^{97} + 3772 q^{98}+O(q^{100}) \) 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8 - 88 * q^10 + 110 * q^11 - 82 * q^13 - 126 * q^14 + 271 * q^16 - 100 * q^17 - 292 * q^19 + 52 * q^20 - 351 * q^22 + 276 * q^23 + 386 * q^25 - 84 * q^26 - 1010 * q^28 + 38 * q^29 - 432 * q^31 + 452 * q^32 - 524 * q^34 + 166 * q^35 - 936 * q^37 + 41 * q^38 - 1183 * q^40 - 1054 * q^41 - 1804 * q^43 + 341 * q^44 - 888 * q^46 + 560 * q^47 + 1074 * q^49 + 1054 * q^50 - 632 * q^52 + 160 * q^53 - 842 * q^55 - 509 * q^56 - 1266 * q^58 - 846 * q^59 - 2220 * q^61 - 82 * q^62 - 1565 * q^64 - 296 * q^65 - 4752 * q^67 + 1719 * q^68 - 5601 * q^70 + 802 * q^71 - 2732 * q^73 + 4581 * q^74 - 5614 * q^76 + 1008 * q^77 - 3172 * q^79 + 732 * q^80 - 9709 * q^82 + 4780 * q^83 - 4624 * q^85 + 2009 * q^86 - 9331 * q^88 - 4372 * q^89 - 7398 * q^91 + 6138 * q^92 - 7068 * q^94 + 3160 * q^95 - 4846 * q^97 + 3772 * q^98

Introduction

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