LMFDB - Embedded newform 2151.4.a.b.1.8

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			1.8
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)

$f(q)$	$=$	$q-3.15491 q^{2} +1.95344 q^{4} -4.03944 q^{5} +6.03808 q^{7} +19.0763 q^{8} +O(q^{10})$	\(q-3.15491 q^{2} +1.95344 q^{4} -4.03944 q^{5} +6.03808 q^{7} +19.0763 q^{8} +12.7440 q^{10} +39.3940 q^{11} +58.0257 q^{13} -19.0496 q^{14} -75.8116 q^{16} -34.5475 q^{17} +61.7709 q^{19} -7.89078 q^{20} -124.284 q^{22} +169.687 q^{23} -108.683 q^{25} -183.066 q^{26} +11.7950 q^{28} -234.372 q^{29} -330.766 q^{31} +86.5677 q^{32} +108.994 q^{34} -24.3904 q^{35} -350.759 q^{37} -194.882 q^{38} -77.0577 q^{40} +127.904 q^{41} +168.116 q^{43} +76.9537 q^{44} -535.347 q^{46} +543.673 q^{47} -306.542 q^{49} +342.885 q^{50} +113.349 q^{52} -564.508 q^{53} -159.130 q^{55} +115.184 q^{56} +739.421 q^{58} +306.438 q^{59} +112.536 q^{61} +1043.54 q^{62} +333.380 q^{64} -234.391 q^{65} -953.913 q^{67} -67.4863 q^{68} +76.9496 q^{70} -357.946 q^{71} +69.4737 q^{73} +1106.61 q^{74} +120.666 q^{76} +237.864 q^{77} -487.442 q^{79} +306.236 q^{80} -403.526 q^{82} -221.865 q^{83} +139.553 q^{85} -530.392 q^{86} +751.494 q^{88} +873.031 q^{89} +350.364 q^{91} +331.473 q^{92} -1715.24 q^{94} -249.520 q^{95} +144.732 q^{97} +967.110 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.15491	−1.11543	−0.557714	−	0.830033i	$-0.688321\pi$
$2$	−3.15491	−1.11543	−0.557714	+	0.830033i	$0.688321\pi$
$3$	0	0
$3$	0	0
$4$	1.95344	0.244179
$5$	−4.03944	−0.361298	−0.180649	−	0.983548i	$-0.557820\pi$
$5$	−4.03944	−0.361298	−0.180649	+	0.983548i	$0.557820\pi$
$6$	0	0
$7$	6.03808	0.326026	0.163013	−	0.986624i	$-0.447879\pi$
$7$	6.03808	0.326026	0.163013	+	0.986624i	$0.447879\pi$
$8$	19.0763	0.843063
$9$	0	0
$10$	12.7440	0.403002
$11$	39.3940	1.07979	0.539897	−	0.841731i	$-0.318463\pi$
$11$	39.3940	1.07979	0.539897	+	0.841731i	$0.318463\pi$
$12$	0	0
$13$	58.0257	1.23796	0.618978	−	0.785408i	$-0.287547\pi$
$13$	58.0257	1.23796	0.618978	+	0.785408i	$0.287547\pi$
$14$	−19.0496	−0.363658
$15$	0	0
$16$	−75.8116	−1.18456
$17$	−34.5475	−0.492883	−0.246441	−	0.969158i	$-0.579261\pi$
$17$	−34.5475	−0.492883	−0.246441	+	0.969158i	$0.579261\pi$
$18$	0	0
$19$	61.7709	0.745854	0.372927	−	0.927861i	$-0.378354\pi$
$19$	61.7709	0.745854	0.372927	+	0.927861i	$0.378354\pi$
$20$	−7.89078	−0.0882216
$21$	0	0
$22$	−124.284	−1.20443
$23$	169.687	1.53836	0.769178	−	0.639034i	$-0.220666\pi$
$23$	169.687	1.53836	0.769178	+	0.639034i	$0.220666\pi$
$24$	0	0
$25$	−108.683	−0.869464
$26$	−183.066	−1.38085
$27$	0	0
$28$	11.7950	0.0796087
$29$	−234.372	−1.50075	−0.750375	−	0.661012i	$-0.770127\pi$
$29$	−234.372	−1.50075	−0.750375	+	0.661012i	$0.770127\pi$
$30$	0	0
$31$	−330.766	−1.91637	−0.958184	−	0.286152i	$-0.907624\pi$
$31$	−330.766	−1.91637	−0.958184	+	0.286152i	$0.907624\pi$
$32$	86.5677	0.478223
$33$	0	0
$34$	108.994	0.549775
$35$	−24.3904	−0.117793
$36$	0	0
$37$	−350.759	−1.55850	−0.779249	−	0.626714i	$-0.784399\pi$
$37$	−350.759	−1.55850	−0.779249	+	0.626714i	$0.784399\pi$
$38$	−194.882	−0.831946
$39$	0	0
$40$	−77.0577	−0.304597
$41$	127.904	0.487202	0.243601	−	0.969876i	$-0.421671\pi$
$41$	127.904	0.487202	0.243601	+	0.969876i	$0.421671\pi$
$42$	0	0
$43$	168.116	0.596221	0.298111	−	0.954531i	$-0.403644\pi$
$43$	168.116	0.596221	0.298111	+	0.954531i	$0.403644\pi$
$44$	76.9537	0.263664
$45$	0	0
$46$	−535.347	−1.71593
$47$	543.673	1.68730	0.843648	−	0.536897i	$-0.180404\pi$
$47$	543.673	1.68730	0.843648	+	0.536897i	$0.180404\pi$
$48$	0	0
$49$	−306.542	−0.893707
$50$	342.885	0.969824
$51$	0	0
$52$	113.349	0.302284
$53$	−564.508	−1.46304	−0.731520	−	0.681820i	$-0.761189\pi$
$53$	−564.508	−1.46304	−0.731520	+	0.681820i	$0.761189\pi$
$54$	0	0
$55$	−159.130	−0.390128
$56$	115.184	0.274860
$57$	0	0
$58$	739.421	1.67398
$59$	306.438	0.676183	0.338092	−	0.941113i	$-0.390219\pi$
$59$	306.438	0.676183	0.338092	+	0.941113i	$0.390219\pi$
$60$	0	0
$61$	112.536	0.236210	0.118105	−	0.993001i	$-0.462318\pi$
$61$	112.536	0.236210	0.118105	+	0.993001i	$0.462318\pi$
$62$	1043.54	2.13757
$63$	0	0
$64$	333.380	0.651132
$65$	−234.391	−0.447272
$66$	0	0
$67$	−953.913	−1.73939	−0.869694	−	0.493591i	$-0.835684\pi$
$67$	−953.913	−1.73939	−0.869694	+	0.493591i	$0.835684\pi$
$68$	−67.4863	−0.120352
$69$	0	0
$70$	76.9496	0.131389
$71$	−357.946	−0.598315	−0.299157	−	0.954204i	$-0.596706\pi$
$71$	−357.946	−0.598315	−0.299157	+	0.954204i	$0.596706\pi$
$72$	0	0
$73$	69.4737	0.111387	0.0556937	−	0.998448i	$-0.482263\pi$
$73$	69.4737	0.111387	0.0556937	+	0.998448i	$0.482263\pi$
$74$	1106.61	1.73839
$75$	0	0
$76$	120.666	0.182122
$77$	237.864	0.352041
$78$	0	0
$79$	−487.442	−0.694197	−0.347098	−	0.937829i	$-0.612833\pi$
$79$	−487.442	−0.694197	−0.347098	+	0.937829i	$0.612833\pi$
$80$	306.236	0.427978
$81$	0	0
$82$	−403.526	−0.543438
$83$	−221.865	−0.293408	−0.146704	−	0.989180i	$-0.546867\pi$
$83$	−221.865	−0.293408	−0.146704	+	0.989180i	$0.546867\pi$
$84$	0	0
$85$	139.553	0.178078
$86$	−530.392	−0.665042
$87$	0	0
$88$	751.494	0.910336
$89$	873.031	1.03979	0.519894	−	0.854231i	$-0.325972\pi$
$89$	873.031	1.03979	0.519894	+	0.854231i	$0.325972\pi$
$90$	0	0
$91$	350.364	0.403606
$92$	331.473	0.375635
$93$	0	0
$94$	−1715.24	−1.88206
$95$	−249.520	−0.269476
$96$	0	0
$97$	144.732	0.151498	0.0757492	−	0.997127i	$-0.475865\pi$
$97$	144.732	0.151498	0.0757492	+	0.997127i	$0.475865\pi$
$98$	967.110	0.996866
$99$	0	0
$100$	−212.305	−0.212305
$101$	1244.50	1.22606	0.613032	−	0.790058i	$-0.289950\pi$
$101$	1244.50	1.22606	0.613032	+	0.790058i	$0.289950\pi$
$102$	0	0
$103$	−1493.44	−1.42867	−0.714336	−	0.699803i	$-0.753271\pi$
$103$	−1493.44	−1.42867	−0.714336	+	0.699803i	$0.753271\pi$
$104$	1106.92	1.04368
$105$	0	0
$106$	1780.97	1.63192
$107$	−1208.86	−1.09220	−0.546098	−	0.837721i	$-0.683887\pi$
$107$	−1208.86	−1.09220	−0.546098	+	0.837721i	$0.683887\pi$
$108$	0	0
$109$	1941.49	1.70606	0.853030	−	0.521862i	$-0.174762\pi$
$109$	1941.49	1.70606	0.853030	+	0.521862i	$0.174762\pi$
$110$	502.039	0.435160
$111$	0	0
$112$	−457.756	−0.386196
$113$	1967.55	1.63798	0.818988	−	0.573810i	$-0.194535\pi$
$113$	1967.55	1.63798	0.818988	+	0.573810i	$0.194535\pi$
$114$	0	0
$115$	−685.440	−0.555805
$116$	−457.830	−0.366452
$117$	0	0
$118$	−966.783	−0.754234
$119$	−208.601	−0.160692
$120$	0	0
$121$	220.889	0.165957
$122$	−355.042	−0.263475
$123$	0	0
$124$	−646.131	−0.467938
$125$	943.948	0.675434
$126$	0	0
$127$	−1114.32	−0.778580	−0.389290	−	0.921115i	$-0.627280\pi$
$127$	−1114.32	−0.778580	−0.389290	+	0.921115i	$0.627280\pi$
$128$	−1744.32	−1.20451
$129$	0	0
$130$	739.482	0.498899
$131$	−151.660	−0.101150	−0.0505749	−	0.998720i	$-0.516105\pi$
$131$	−151.660	−0.101150	−0.0505749	+	0.998720i	$0.516105\pi$
$132$	0	0
$133$	372.978	0.243167
$134$	3009.51	1.94016
$135$	0	0
$136$	−659.040	−0.415531
$137$	−1690.41	−1.05417	−0.527085	−	0.849813i	$-0.676715\pi$
$137$	−1690.41	−1.05417	−0.527085	+	0.849813i	$0.676715\pi$
$138$	0	0
$139$	−250.946	−0.153129	−0.0765645	−	0.997065i	$-0.524395\pi$
$139$	−250.946	−0.153129	−0.0765645	+	0.997065i	$0.524395\pi$
$140$	−47.6452	−0.0287625
$141$	0	0
$142$	1129.29	0.667377
$143$	2285.87	1.33674
$144$	0	0
$145$	946.731	0.542219
$146$	−219.183	−0.124245
$147$	0	0
$148$	−685.185	−0.380553
$149$	−345.081	−0.189732	−0.0948662	−	0.995490i	$-0.530242\pi$
$149$	−345.081	−0.189732	−0.0948662	+	0.995490i	$0.530242\pi$
$150$	0	0
$151$	−1625.93	−0.876269	−0.438135	−	0.898909i	$-0.644361\pi$
$151$	−1625.93	−0.876269	−0.438135	+	0.898909i	$0.644361\pi$
$152$	1178.36	0.628802
$153$	0	0
$154$	−750.439	−0.392676
$155$	1336.11	0.692381
$156$	0	0
$157$	−459.778	−0.233722	−0.116861	−	0.993148i	$-0.537283\pi$
$157$	−459.778	−0.233722	−0.116861	+	0.993148i	$0.537283\pi$
$158$	1537.84	0.774327
$159$	0	0
$160$	−349.685	−0.172781
$161$	1024.58	0.501544
$162$	0	0
$163$	−1308.31	−0.628680	−0.314340	−	0.949310i	$-0.601783\pi$
$163$	−1308.31	−0.628680	−0.314340	+	0.949310i	$0.601783\pi$
$164$	249.852	0.118965
$165$	0	0
$166$	699.964	0.327276
$167$	−1556.98	−0.721455	−0.360727	−	0.932671i	$-0.617471\pi$
$167$	−1556.98	−0.721455	−0.360727	+	0.932671i	$0.617471\pi$
$168$	0	0
$169$	1169.98	0.532537
$170$	−440.275	−0.198633
$171$	0	0
$172$	328.405	0.145585
$173$	−907.420	−0.398786	−0.199393	−	0.979920i	$-0.563897\pi$
$173$	−907.420	−0.398786	−0.199393	+	0.979920i	$0.563897\pi$
$174$	0	0
$175$	−656.236	−0.283467
$176$	−2986.52	−1.27908
$177$	0	0
$178$	−2754.33	−1.15981
$179$	−4230.52	−1.76650	−0.883250	−	0.468902i	$-0.844650\pi$
$179$	−4230.52	−1.76650	−0.883250	+	0.468902i	$0.844650\pi$
$180$	0	0
$181$	3255.94	1.33709	0.668543	−	0.743674i	$-0.266918\pi$
$181$	3255.94	1.33709	0.668543	+	0.743674i	$0.266918\pi$
$182$	−1105.37	−0.450193
$183$	0	0
$184$	3237.01	1.29693
$185$	1416.87	0.563083
$186$	0	0
$187$	−1360.97	−0.532212
$188$	1062.03	0.412003
$189$	0	0
$190$	787.212	0.300581
$191$	−3738.95	−1.41644	−0.708222	−	0.705990i	$-0.750502\pi$
$191$	−3738.95	−1.41644	−0.708222	+	0.705990i	$0.750502\pi$
$192$	0	0
$193$	2123.80	0.792095	0.396048	−	0.918230i	$-0.370381\pi$
$193$	2123.80	0.792095	0.396048	+	0.918230i	$0.370381\pi$
$194$	−456.617	−0.168986
$195$	0	0
$196$	−598.809	−0.218225
$197$	−1657.08	−0.599299	−0.299650	−	0.954049i	$-0.596870\pi$
$197$	−1657.08	−0.599299	−0.299650	+	0.954049i	$0.596870\pi$
$198$	0	0
$199$	156.120	0.0556133	0.0278067	−	0.999613i	$-0.491148\pi$
$199$	156.120	0.0556133	0.0278067	+	0.999613i	$0.491148\pi$
$200$	−2073.27	−0.733013
$201$	0	0
$202$	−3926.28	−1.36759
$203$	−1415.16	−0.489283
$204$	0	0
$205$	−516.661	−0.176025
$206$	4711.67	1.59358
$207$	0	0
$208$	−4399.02	−1.46643
$209$	2433.41	0.805369
$210$	0	0
$211$	1318.88	0.430311	0.215156	−	0.976580i	$-0.430974\pi$
$211$	1318.88	0.430311	0.215156	+	0.976580i	$0.430974\pi$
$212$	−1102.73	−0.357244
$213$	0	0
$214$	3813.84	1.21827
$215$	−679.096	−0.215414
$216$	0	0
$217$	−1997.19	−0.624785
$218$	−6125.20	−1.90299
$219$	0	0
$220$	−310.850	−0.0952612
$221$	−2004.64	−0.610167
$222$	0	0
$223$	4044.56	1.21454	0.607272	−	0.794494i	$-0.292264\pi$
$223$	4044.56	1.21454	0.607272	+	0.794494i	$0.292264\pi$
$224$	522.702	0.155913
$225$	0	0
$226$	−6207.43	−1.82704
$227$	−2195.50	−0.641939	−0.320970	−	0.947090i	$-0.604009\pi$
$227$	−2195.50	−0.641939	−0.320970	+	0.947090i	$0.604009\pi$
$228$	0	0
$229$	1934.88	0.558342	0.279171	−	0.960241i	$-0.409940\pi$
$229$	1934.88	0.558342	0.279171	+	0.960241i	$0.409940\pi$
$230$	2162.50	0.619961
$231$	0	0
$232$	−4470.96	−1.26523
$233$	3012.48	0.847013	0.423507	−	0.905893i	$-0.360799\pi$
$233$	3012.48	0.847013	0.423507	+	0.905893i	$0.360799\pi$
$234$	0	0
$235$	−2196.13	−0.609617
$236$	598.607	0.165110
$237$	0	0
$238$	658.115	0.179241
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−3484.32	−0.931307	−0.465653	−	0.884967i	$-0.654181\pi$
$241$	−3484.32	−0.931307	−0.465653	+	0.884967i	$0.654181\pi$
$242$	−696.884	−0.185113
$243$	0	0
$244$	219.833	0.0576777
$245$	1238.26	0.322895
$246$	0	0
$247$	3584.30	0.923335
$248$	−6309.82	−1.61562
$249$	0	0
$250$	−2978.07	−0.753398
$251$	5123.33	1.28837	0.644187	−	0.764868i	$-0.277196\pi$
$251$	5123.33	1.28837	0.644187	+	0.764868i	$0.277196\pi$
$252$	0	0
$253$	6684.65	1.66111
$254$	3515.56	0.868449
$255$	0	0
$256$	2836.14	0.692417
$257$	−6797.77	−1.64994	−0.824968	−	0.565179i	$-0.808807\pi$
$257$	−6797.77	−1.64994	−0.824968	+	0.565179i	$0.808807\pi$
$258$	0	0
$259$	−2117.91	−0.508111
$260$	−457.868	−0.109215
$261$	0	0
$262$	478.474	0.112825
$263$	−717.341	−0.168187	−0.0840935	−	0.996458i	$-0.526799\pi$
$263$	−717.341	−0.168187	−0.0840935	+	0.996458i	$0.526799\pi$
$264$	0	0
$265$	2280.29	0.528594
$266$	−1176.71	−0.271236
$267$	0	0
$268$	−1863.41	−0.424723
$269$	7043.73	1.59652	0.798260	−	0.602313i	$-0.205754\pi$
$269$	7043.73	1.59652	0.798260	+	0.602313i	$0.205754\pi$
$270$	0	0
$271$	4463.76	1.00057	0.500284	−	0.865861i	$-0.333229\pi$
$271$	4463.76	1.00057	0.500284	+	0.865861i	$0.333229\pi$
$272$	2619.10	0.583847
$273$	0	0
$274$	5333.08	1.17585
$275$	−4281.46	−0.938842
$276$	0	0
$277$	6248.44	1.35535	0.677676	−	0.735360i	$-0.262987\pi$
$277$	6248.44	1.35535	0.677676	+	0.735360i	$0.262987\pi$
$278$	791.710	0.170804
$279$	0	0
$280$	−465.281	−0.0993066
$281$	1538.60	0.326638	0.163319	−	0.986573i	$-0.447780\pi$
$281$	1538.60	0.326638	0.163319	+	0.986573i	$0.447780\pi$
$282$	0	0
$283$	7430.22	1.56071	0.780354	−	0.625338i	$-0.215039\pi$
$283$	7430.22	1.56071	0.780354	+	0.625338i	$0.215039\pi$
$284$	−699.224	−0.146096
$285$	0	0
$286$	−7211.69	−1.49104
$287$	772.295	0.158840
$288$	0	0
$289$	−3719.47	−0.757067
$290$	−2986.85	−0.604806
$291$	0	0
$292$	135.712	0.0271985
$293$	−8996.17	−1.79373	−0.896863	−	0.442307i	$-0.854160\pi$
$293$	−8996.17	−1.79373	−0.896863	+	0.442307i	$0.854160\pi$
$294$	0	0
$295$	−1237.84	−0.244304
$296$	−6691.20	−1.31391
$297$	0	0
$298$	1088.70	0.211633
$299$	9846.21	1.90442
$300$	0	0
$301$	1015.10	0.194383
$302$	5129.67	0.977415
$303$	0	0
$304$	−4682.95	−0.883506
$305$	−454.584	−0.0853423
$306$	0	0
$307$	427.768	0.0795244	0.0397622	−	0.999209i	$-0.487340\pi$
$307$	427.768	0.0795244	0.0397622	+	0.999209i	$0.487340\pi$
$308$	464.652	0.0859611
$309$	0	0
$310$	−4215.30	−0.772301
$311$	6758.28	1.23224	0.616121	−	0.787652i	$-0.288703\pi$
$311$	6758.28	1.23224	0.616121	+	0.787652i	$0.288703\pi$
$312$	0	0
$313$	−5890.83	−1.06380	−0.531900	−	0.846807i	$-0.678522\pi$
$313$	−5890.83	−1.06380	−0.531900	+	0.846807i	$0.678522\pi$
$314$	1450.56	0.260700
$315$	0	0
$316$	−952.187	−0.169509
$317$	7600.58	1.34666	0.673330	−	0.739342i	$-0.264863\pi$
$317$	7600.58	1.34666	0.673330	+	0.739342i	$0.264863\pi$
$318$	0	0
$319$	−9232.85	−1.62050
$320$	−1346.67	−0.235253
$321$	0	0
$322$	−3232.47	−0.559436
$323$	−2134.03	−0.367618
$324$	0	0
$325$	−6306.40	−1.07636
$326$	4127.60	0.701247
$327$	0	0
$328$	2439.94	0.410742
$329$	3282.74	0.550102
$330$	0	0
$331$	−11667.4	−1.93746	−0.968729	−	0.248122i	$-0.920187\pi$
$331$	−11667.4	−1.93746	−0.968729	+	0.248122i	$0.920187\pi$
$332$	−433.399	−0.0716442
$333$	0	0
$334$	4912.13	0.804731
$335$	3853.27	0.628438
$336$	0	0
$337$	−11033.5	−1.78349	−0.891744	−	0.452540i	$-0.850518\pi$
$337$	−11033.5	−1.78349	−0.891744	+	0.452540i	$0.850518\pi$
$338$	−3691.19	−0.594006
$339$	0	0
$340$	272.607	0.0434829
$341$	−13030.2	−2.06928
$342$	0	0
$343$	−3921.98	−0.617397
$344$	3207.05	0.502652
$345$	0	0
$346$	2862.83	0.444816
$347$	1593.90	0.246585	0.123293	−	0.992370i	$-0.460655\pi$
$347$	1593.90	0.246585	0.123293	+	0.992370i	$0.460655\pi$
$348$	0	0
$349$	2272.32	0.348522	0.174261	−	0.984699i	$-0.444246\pi$
$349$	2272.32	0.348522	0.174261	+	0.984699i	$0.444246\pi$
$350$	2070.36	0.316187
$351$	0	0
$352$	3410.25	0.516383
$353$	6178.52	0.931584	0.465792	−	0.884894i	$-0.345769\pi$
$353$	6178.52	0.931584	0.465792	+	0.884894i	$0.345769\pi$
$354$	0	0
$355$	1445.90	0.216170
$356$	1705.41	0.253895
$357$	0	0
$358$	13346.9	1.97040
$359$	6583.14	0.967814	0.483907	−	0.875120i	$-0.339217\pi$
$359$	6583.14	0.967814	0.483907	+	0.875120i	$0.339217\pi$
$360$	0	0
$361$	−3043.35	−0.443702
$362$	−10272.2	−1.49142
$363$	0	0
$364$	684.413	0.0985522
$365$	−280.635	−0.0402441
$366$	0	0
$367$	1730.54	0.246140	0.123070	−	0.992398i	$-0.460726\pi$
$367$	1730.54	0.246140	0.123070	+	0.992398i	$0.460726\pi$
$368$	−12864.2	−1.82227
$369$	0	0
$370$	−4470.09	−0.628078
$371$	−3408.54	−0.476989
$372$	0	0
$373$	−11120.9	−1.54375	−0.771875	−	0.635775i	$-0.780681\pi$
$373$	−11120.9	−1.54375	−0.771875	+	0.635775i	$0.780681\pi$
$374$	4293.72	0.593644
$375$	0	0
$376$	10371.3	1.42250
$377$	−13599.6	−1.85786
$378$	0	0
$379$	−316.130	−0.0428457	−0.0214228	−	0.999771i	$-0.506820\pi$
$379$	−316.130	−0.0428457	−0.0214228	+	0.999771i	$0.506820\pi$
$380$	−487.421	−0.0658004
$381$	0	0
$382$	11796.0	1.57994
$383$	−5335.79	−0.711870	−0.355935	−	0.934511i	$-0.615838\pi$
$383$	−5335.79	−0.711870	−0.355935	+	0.934511i	$0.615838\pi$
$384$	0	0
$385$	−960.838	−0.127192
$386$	−6700.39	−0.883525
$387$	0	0
$388$	282.725	0.0369928
$389$	−8642.11	−1.12641	−0.563204	−	0.826318i	$-0.690431\pi$
$389$	−8642.11	−1.12641	−0.563204	+	0.826318i	$0.690431\pi$
$390$	0	0
$391$	−5862.27	−0.758229
$392$	−5847.69	−0.753452
$393$	0	0
$394$	5227.93	0.668475
$395$	1968.99	0.250812
$396$	0	0
$397$	−13769.4	−1.74072	−0.870359	−	0.492418i	$-0.836113\pi$
$397$	−13769.4	−1.74072	−0.870359	+	0.492418i	$0.836113\pi$
$398$	−492.544	−0.0620326
$399$	0	0
$400$	8239.42	1.02993
$401$	−7575.69	−0.943421	−0.471710	−	0.881754i	$-0.656363\pi$
$401$	−7575.69	−0.943421	−0.471710	+	0.881754i	$0.656363\pi$
$402$	0	0
$403$	−19193.0	−2.37238
$404$	2431.05	0.299379
$405$	0	0
$406$	4464.68	0.545760
$407$	−13817.8	−1.68286
$408$	0	0
$409$	10254.2	1.23970	0.619850	−	0.784721i	$-0.287194\pi$
$409$	10254.2	1.23970	0.619850	+	0.784721i	$0.287194\pi$
$410$	1630.02	0.196343
$411$	0	0
$412$	−2917.34	−0.348852
$413$	1850.30	0.220453
$414$	0	0
$415$	896.211	0.106008
$416$	5023.15	0.592020
$417$	0	0
$418$	−7677.17	−0.898331
$419$	−4005.23	−0.466989	−0.233494	−	0.972358i	$-0.575016\pi$
$419$	−4005.23	−0.466989	−0.233494	+	0.972358i	$0.575016\pi$
$420$	0	0
$421$	−14855.0	−1.71969	−0.859846	−	0.510553i	$-0.829441\pi$
$421$	−14855.0	−1.71969	−0.859846	+	0.510553i	$0.829441\pi$
$422$	−4160.95	−0.479981
$423$	0	0
$424$	−10768.7	−1.23344
$425$	3754.73	0.428543
$426$	0	0
$427$	679.504	0.0770106
$428$	−2361.43	−0.266692
$429$	0	0
$430$	2142.48	0.240279
$431$	−3546.55	−0.396360	−0.198180	−	0.980166i	$-0.563503\pi$
$431$	−3546.55	−0.396360	−0.198180	+	0.980166i	$0.563503\pi$
$432$	0	0
$433$	3061.24	0.339755	0.169877	−	0.985465i	$-0.445663\pi$
$433$	3061.24	0.339755	0.169877	+	0.985465i	$0.445663\pi$
$434$	6300.96	0.696903
$435$	0	0
$436$	3792.57	0.416585
$437$	10481.7	1.14739
$438$	0	0
$439$	−15117.1	−1.64351	−0.821756	−	0.569840i	$-0.807005\pi$
$439$	−15117.1	−1.64351	−0.821756	+	0.569840i	$0.807005\pi$
$440$	−3035.61	−0.328903
$441$	0	0
$442$	6324.46	0.680598
$443$	−11505.1	−1.23392	−0.616959	−	0.786995i	$-0.711636\pi$
$443$	−11505.1	−1.23392	−0.616959	+	0.786995i	$0.711636\pi$
$444$	0	0
$445$	−3526.55	−0.375673
$446$	−12760.2	−1.35474
$447$	0	0
$448$	2012.97	0.212286
$449$	−14749.0	−1.55022	−0.775110	−	0.631826i	$-0.782306\pi$
$449$	−14749.0	−1.55022	−0.775110	+	0.631826i	$0.782306\pi$
$450$	0	0
$451$	5038.66	0.526078
$452$	3843.48	0.399960
$453$	0	0
$454$	6926.58	0.716037
$455$	−1415.27	−0.145822
$456$	0	0
$457$	10793.2	1.10478	0.552390	−	0.833586i	$-0.313716\pi$
$457$	10793.2	1.10478	0.552390	+	0.833586i	$0.313716\pi$
$458$	−6104.36	−0.622790
$459$	0	0
$460$	−1338.96	−0.135716
$461$	667.769	0.0674645	0.0337322	−	0.999431i	$-0.489261\pi$
$461$	667.769	0.0674645	0.0337322	+	0.999431i	$0.489261\pi$
$462$	0	0
$463$	1135.62	0.113989	0.0569943	−	0.998375i	$-0.481848\pi$
$463$	1135.62	0.113989	0.0569943	+	0.998375i	$0.481848\pi$
$464$	17768.1	1.77772
$465$	0	0
$466$	−9504.09	−0.944782
$467$	6904.34	0.684143	0.342071	−	0.939674i	$-0.388872\pi$
$467$	6904.34	0.684143	0.342071	+	0.939674i	$0.388872\pi$
$468$	0	0
$469$	−5759.80	−0.567085
$470$	6928.60	0.679984
$471$	0	0
$472$	5845.72	0.570065
$473$	6622.78	0.643797
$474$	0	0
$475$	−6713.45	−0.648493
$476$	−407.488	−0.0392378
$477$	0	0
$478$	754.023	0.0721510
$479$	2697.68	0.257328	0.128664	−	0.991688i	$-0.458931\pi$
$479$	2697.68	0.257328	0.128664	+	0.991688i	$0.458931\pi$
$480$	0	0
$481$	−20353.1	−1.92935
$482$	10992.7	1.03881
$483$	0	0
$484$	431.492	0.0405233
$485$	−584.637	−0.0547361
$486$	0	0
$487$	4510.66	0.419708	0.209854	−	0.977733i	$-0.432701\pi$
$487$	4510.66	0.419708	0.209854	+	0.977733i	$0.432701\pi$
$488$	2146.78	0.199140
$489$	0	0
$490$	−3906.58	−0.360166
$491$	−12337.1	−1.13394	−0.566972	−	0.823737i	$-0.691885\pi$
$491$	−12337.1	−1.13394	−0.566972	+	0.823737i	$0.691885\pi$
$492$	0	0
$493$	8096.97	0.739694
$494$	−11308.1	−1.02991
$495$	0	0
$496$	25075.9	2.27005
$497$	−2161.31	−0.195066
$498$	0	0
$499$	4153.41	0.372609	0.186305	−	0.982492i	$-0.440349\pi$
$499$	4153.41	0.372609	0.186305	+	0.982492i	$0.440349\pi$
$500$	1843.94	0.164927
$501$	0	0
$502$	−16163.6	−1.43709
$503$	442.109	0.0391902	0.0195951	−	0.999808i	$-0.493762\pi$
$503$	442.109	0.0391902	0.0195951	+	0.999808i	$0.493762\pi$
$504$	0	0
$505$	−5027.08	−0.442975
$506$	−21089.5	−1.85285
$507$	0	0
$508$	−2176.75	−0.190113
$509$	21920.8	1.90888	0.954441	−	0.298399i	$-0.0964527\pi$
$509$	21920.8	1.90888	0.954441	+	0.298399i	$0.0964527\pi$
$510$	0	0
$511$	419.487	0.0363151
$512$	5006.83	0.432174
$513$	0	0
$514$	21446.3	1.84038
$515$	6032.66	0.516176
$516$	0	0
$517$	21417.5	1.82193
$518$	6681.81	0.566761
$519$	0	0
$520$	−4471.33	−0.377078
$521$	−3438.04	−0.289104	−0.144552	−	0.989497i	$-0.546174\pi$
$521$	−3438.04	−0.289104	−0.144552	+	0.989497i	$0.546174\pi$
$522$	0	0
$523$	8386.75	0.701199	0.350599	−	0.936526i	$-0.385978\pi$
$523$	8386.75	0.701199	0.350599	+	0.936526i	$0.385978\pi$
$524$	−296.259	−0.0246987
$525$	0	0
$526$	2263.15	0.187600
$527$	11427.2	0.944544
$528$	0	0
$529$	16626.7	1.36654
$530$	−7194.12	−0.589608
$531$	0	0
$532$	728.588	0.0593765
$533$	7421.73	0.603134
$534$	0	0
$535$	4883.12	0.394608
$536$	−18197.2	−1.46641
$537$	0	0
$538$	−22222.3	−1.78080
$539$	−12075.9	−0.965021
$540$	0	0
$541$	−4891.24	−0.388708	−0.194354	−	0.980931i	$-0.562261\pi$
$541$	−4891.24	−0.388708	−0.194354	+	0.980931i	$0.562261\pi$
$542$	−14082.7	−1.11606
$543$	0	0
$544$	−2990.70	−0.235708
$545$	−7842.51	−0.616397
$546$	0	0
$547$	3384.46	0.264551	0.132275	−	0.991213i	$-0.457772\pi$
$547$	3384.46	0.264551	0.132275	+	0.991213i	$0.457772\pi$
$548$	−3302.10	−0.257407
$549$	0	0
$550$	13507.6	1.04721
$551$	−14477.4	−1.11934
$552$	0	0
$553$	−2943.22	−0.226326
$554$	−19713.3	−1.51180
$555$	0	0
$556$	−490.206	−0.0373909
$557$	6137.41	0.466877	0.233438	−	0.972372i	$-0.425002\pi$
$557$	6137.41	0.466877	0.233438	+	0.972372i	$0.425002\pi$
$558$	0	0
$559$	9755.08	0.738096
$560$	1849.08	0.139532
$561$	0	0
$562$	−4854.15	−0.364342
$563$	−24756.5	−1.85322	−0.926611	−	0.376021i	$-0.877292\pi$
$563$	−24756.5	−1.85322	−0.926611	+	0.376021i	$0.877292\pi$
$564$	0	0
$565$	−7947.79	−0.591798
$566$	−23441.6	−1.74086
$567$	0	0
$568$	−6828.30	−0.504417
$569$	−9576.37	−0.705558	−0.352779	−	0.935707i	$-0.614763\pi$
$569$	−9576.37	−0.705558	−0.352779	+	0.935707i	$0.614763\pi$
$570$	0	0
$571$	−25871.4	−1.89612	−0.948059	−	0.318095i	$-0.896957\pi$
$571$	−25871.4	−1.89612	−0.948059	+	0.318095i	$0.896957\pi$
$572$	4465.29	0.326404
$573$	0	0
$574$	−2436.52	−0.177175
$575$	−18442.1	−1.33754
$576$	0	0
$577$	11492.6	0.829189	0.414594	−	0.910006i	$-0.363923\pi$
$577$	11492.6	0.829189	0.414594	+	0.910006i	$0.363923\pi$
$578$	11734.6	0.844453
$579$	0	0
$580$	1849.38	0.132399
$581$	−1339.64	−0.0956586
$582$	0	0
$583$	−22238.2	−1.57978
$584$	1325.30	0.0939066
$585$	0	0
$586$	28382.1	2.00077
$587$	−2641.83	−0.185758	−0.0928791	−	0.995677i	$-0.529607\pi$
$587$	−2641.83	−0.185758	−0.0928791	+	0.995677i	$0.529607\pi$
$588$	0	0
$589$	−20431.8	−1.42933
$590$	3905.26	0.272503
$591$	0	0
$592$	26591.6	1.84613
$593$	24981.6	1.72997	0.864985	−	0.501797i	$-0.167328\pi$
$593$	24981.6	1.72997	0.864985	+	0.501797i	$0.167328\pi$
$594$	0	0
$595$	842.629	0.0580579
$596$	−674.093	−0.0463287
$597$	0	0
$598$	−31063.9	−2.12424
$599$	2241.17	0.152874	0.0764372	−	0.997074i	$-0.475646\pi$
$599$	2241.17	0.152874	0.0764372	+	0.997074i	$0.475646\pi$
$600$	0	0
$601$	−21204.0	−1.43915	−0.719574	−	0.694415i	$-0.755663\pi$
$601$	−21204.0	−1.43915	−0.719574	+	0.694415i	$0.755663\pi$
$602$	−3202.55	−0.216821
$603$	0	0
$604$	−3176.16	−0.213967
$605$	−892.267	−0.0599600
$606$	0	0
$607$	−29614.1	−1.98023	−0.990114	−	0.140262i	$-0.955206\pi$
$607$	−29614.1	−1.98023	−0.990114	+	0.140262i	$0.955206\pi$
$608$	5347.36	0.356685
$609$	0	0
$610$	1434.17	0.0951932
$611$	31547.0	2.08880
$612$	0	0
$613$	−21394.5	−1.40965	−0.704826	−	0.709380i	$-0.748975\pi$
$613$	−21394.5	−1.40965	−0.704826	+	0.709380i	$0.748975\pi$
$614$	−1349.57	−0.0887037
$615$	0	0
$616$	4537.58	0.296793
$617$	6969.51	0.454752	0.227376	−	0.973807i	$-0.426985\pi$
$617$	6969.51	0.454752	0.227376	+	0.973807i	$0.426985\pi$
$618$	0	0
$619$	16011.2	1.03965	0.519826	−	0.854272i	$-0.325997\pi$
$619$	16011.2	1.03965	0.519826	+	0.854272i	$0.325997\pi$
$620$	2610.01	0.169065
$621$	0	0
$622$	−21321.7	−1.37448
$623$	5271.43	0.338997
$624$	0	0
$625$	9772.35	0.625430
$626$	18585.0	1.18659
$627$	0	0
$628$	−898.147	−0.0570700
$629$	12117.9	0.768157
$630$	0	0
$631$	17426.4	1.09942	0.549709	−	0.835356i	$-0.314738\pi$
$631$	17426.4	1.09942	0.549709	+	0.835356i	$0.314738\pi$
$632$	−9298.62	−0.585252
$633$	0	0
$634$	−23979.1	−1.50210
$635$	4501.21	0.281299
$636$	0	0
$637$	−17787.3	−1.10637
$638$	29128.8	1.80755
$639$	0	0
$640$	7046.08	0.435189
$641$	−29873.2	−1.84075	−0.920376	−	0.391034i	$-0.872117\pi$
$641$	−29873.2	−1.84075	−0.920376	+	0.391034i	$0.872117\pi$
$642$	0	0
$643$	2721.41	0.166908	0.0834540	−	0.996512i	$-0.473405\pi$
$643$	2721.41	0.166908	0.0834540	+	0.996512i	$0.473405\pi$
$644$	2001.46	0.122467
$645$	0	0
$646$	6732.67	0.410052
$647$	−11458.4	−0.696252	−0.348126	−	0.937448i	$-0.613182\pi$
$647$	−11458.4	−0.696252	−0.348126	+	0.937448i	$0.613182\pi$
$648$	0	0
$649$	12071.8	0.730139
$650$	19896.1	1.20060
$651$	0	0
$652$	−2555.70	−0.153511
$653$	21649.5	1.29741	0.648706	−	0.761039i	$-0.275310\pi$
$653$	21649.5	1.29741	0.648706	+	0.761039i	$0.275310\pi$
$654$	0	0
$655$	612.622	0.0365453
$656$	−9696.61	−0.577118
$657$	0	0
$658$	−10356.7	−0.613599
$659$	6686.12	0.395227	0.197613	−	0.980280i	$-0.436681\pi$
$659$	6686.12	0.395227	0.197613	+	0.980280i	$0.436681\pi$
$660$	0	0
$661$	9519.87	0.560182	0.280091	−	0.959974i	$-0.409635\pi$
$661$	9519.87	0.560182	0.280091	+	0.959974i	$0.409635\pi$
$662$	36809.6	2.16109
$663$	0	0
$664$	−4232.38	−0.247362
$665$	−1506.62	−0.0878560
$666$	0	0
$667$	−39769.9	−2.30869
$668$	−3041.46	−0.176164
$669$	0	0
$670$	−12156.7	−0.700977
$671$	4433.26	0.255059
$672$	0	0
$673$	2332.42	0.133593	0.0667967	−	0.997767i	$-0.478722\pi$
$673$	2332.42	0.133593	0.0667967	+	0.997767i	$0.478722\pi$
$674$	34809.8	1.98935
$675$	0	0
$676$	2285.49	0.130034
$677$	−32303.1	−1.83384	−0.916919	−	0.399073i	$-0.869332\pi$
$677$	−32303.1	−1.83384	−0.916919	+	0.399073i	$0.869332\pi$
$678$	0	0
$679$	873.905	0.0493924
$680$	2662.15	0.150131
$681$	0	0
$682$	41109.1	2.30814
$683$	23772.7	1.33183	0.665913	−	0.746030i	$-0.268042\pi$
$683$	23772.7	1.33183	0.665913	+	0.746030i	$0.268042\pi$
$684$	0	0
$685$	6828.30	0.380870
$686$	12373.5	0.688662
$687$	0	0
$688$	−12745.2	−0.706257
$689$	−32756.0	−1.81118
$690$	0	0
$691$	1757.83	0.0967745	0.0483872	−	0.998829i	$-0.484592\pi$
$691$	1757.83	0.0967745	0.0483872	+	0.998829i	$0.484592\pi$
$692$	−1772.59	−0.0973752
$693$	0	0
$694$	−5028.61	−0.275048
$695$	1013.68	0.0553253
$696$	0	0
$697$	−4418.77	−0.240133
$698$	−7168.95	−0.388752
$699$	0	0
$700$	−1281.91	−0.0692169
$701$	8349.75	0.449880	0.224940	−	0.974373i	$-0.427781\pi$
$701$	8349.75	0.449880	0.224940	+	0.974373i	$0.427781\pi$
$702$	0	0
$703$	−21666.7	−1.16241
$704$	13133.2	0.703089
$705$	0	0
$706$	−19492.6	−1.03912
$707$	7514.39	0.399728
$708$	0	0
$709$	2194.46	0.116241	0.0581204	−	0.998310i	$-0.481489\pi$
$709$	2194.46	0.116241	0.0581204	+	0.998310i	$0.481489\pi$
$710$	−4561.68	−0.241122
$711$	0	0
$712$	16654.2	0.876606
$713$	−56126.8	−2.94806
$714$	0	0
$715$	−9233.61	−0.482962
$716$	−8264.04	−0.431343
$717$	0	0
$718$	−20769.2	−1.07953
$719$	10955.9	0.568271	0.284136	−	0.958784i	$-0.408293\pi$
$719$	10955.9	0.568271	0.284136	+	0.958784i	$0.408293\pi$
$720$	0	0
$721$	−9017.52	−0.465783
$722$	9601.49	0.494918
$723$	0	0
$724$	6360.28	0.326489
$725$	25472.2	1.30485
$726$	0	0
$727$	3292.73	0.167979	0.0839893	−	0.996467i	$-0.473234\pi$
$727$	3292.73	0.167979	0.0839893	+	0.996467i	$0.473234\pi$
$728$	6683.66	0.340265
$729$	0	0
$730$	885.376	0.0448893
$731$	−5808.01	−0.293867
$732$	0	0
$733$	−9067.28	−0.456900	−0.228450	−	0.973556i	$-0.573366\pi$
$733$	−9067.28	−0.456900	−0.228450	+	0.973556i	$0.573366\pi$
$734$	−5459.69	−0.274551
$735$	0	0
$736$	14689.4	0.735678
$737$	−37578.5	−1.87818
$738$	0	0
$739$	−39353.1	−1.95890	−0.979451	−	0.201684i	$-0.935358\pi$
$739$	−39353.1	−1.95890	−0.979451	+	0.201684i	$0.935358\pi$
$740$	2767.76	0.137493
$741$	0	0
$742$	10753.6	0.532046
$743$	30197.9	1.49106	0.745528	−	0.666474i	$-0.232197\pi$
$743$	30197.9	1.49106	0.745528	+	0.666474i	$0.232197\pi$
$744$	0	0
$745$	1393.93	0.0685500
$746$	35085.4	1.72194
$747$	0	0
$748$	−2658.56	−0.129955
$749$	−7299.19	−0.356084
$750$	0	0
$751$	−6990.69	−0.339672	−0.169836	−	0.985472i	$-0.554324\pi$
$751$	−6990.69	−0.339672	−0.169836	+	0.985472i	$0.554324\pi$
$752$	−41216.7	−1.99870
$753$	0	0
$754$	42905.4	2.07231
$755$	6567.86	0.316595
$756$	0	0
$757$	23976.1	1.15116	0.575579	−	0.817746i	$-0.304777\pi$
$757$	23976.1	1.15116	0.575579	+	0.817746i	$0.304777\pi$
$758$	997.361	0.0477913
$759$	0	0
$760$	−4759.93	−0.227185
$761$	19587.7	0.933055	0.466527	−	0.884507i	$-0.345505\pi$
$761$	19587.7	0.933055	0.466527	+	0.884507i	$0.345505\pi$
$762$	0	0
$763$	11722.8	0.556219
$764$	−7303.79	−0.345866
$765$	0	0
$766$	16833.9	0.794040
$767$	17781.3	0.837086
$768$	0	0
$769$	−7517.64	−0.352527	−0.176263	−	0.984343i	$-0.556401\pi$
$769$	−7517.64	−0.352527	−0.176263	+	0.984343i	$0.556401\pi$
$770$	3031.35	0.141873
$771$	0	0
$772$	4148.70	0.193413
$773$	−16442.5	−0.765065	−0.382533	−	0.923942i	$-0.624948\pi$
$773$	−16442.5	−0.765065	−0.382533	+	0.923942i	$0.624948\pi$
$774$	0	0
$775$	35948.7	1.66621
$776$	2760.96	0.127723
$777$	0	0
$778$	27265.1	1.25643
$779$	7900.76	0.363381
$780$	0	0
$781$	−14100.9	−0.646057
$782$	18494.9	0.845750
$783$	0	0
$784$	23239.4	1.05865
$785$	1857.25	0.0844433
$786$	0	0
$787$	−8747.34	−0.396200	−0.198100	−	0.980182i	$-0.563477\pi$
$787$	−8747.34	−0.396200	−0.198100	+	0.980182i	$0.563477\pi$
$788$	−3237.00	−0.146337
$789$	0	0
$790$	−6211.99	−0.279763
$791$	11880.2	0.534022
$792$	0	0
$793$	6530.01	0.292418
$794$	43441.1	1.94165
$795$	0	0
$796$	304.970	0.0135796
$797$	21481.8	0.954736	0.477368	−	0.878703i	$-0.341591\pi$
$797$	21481.8	0.954736	0.477368	+	0.878703i	$0.341591\pi$
$798$	0	0
$799$	−18782.6	−0.831639
$800$	−9408.43	−0.415798
$801$	0	0
$802$	23900.6	1.05232
$803$	2736.85	0.120275
$804$	0	0
$805$	−4138.74	−0.181207
$806$	60552.0	2.64622
$807$	0	0
$808$	23740.5	1.03365
$809$	−32879.8	−1.42892	−0.714458	−	0.699678i	$-0.753327\pi$
$809$	−32879.8	−1.42892	−0.714458	+	0.699678i	$0.753327\pi$
$810$	0	0
$811$	19728.0	0.854185	0.427092	−	0.904208i	$-0.359538\pi$
$811$	19728.0	0.854185	0.427092	+	0.904208i	$0.359538\pi$
$812$	−2764.42	−0.119473
$813$	0	0
$814$	43593.9	1.87711
$815$	5284.84	0.227141
$816$	0	0
$817$	10384.7	0.444694
$818$	−32351.0	−1.38280
$819$	0	0
$820$	−1009.26	−0.0429817
$821$	−6331.20	−0.269136	−0.134568	−	0.990904i	$-0.542965\pi$
$821$	−6331.20	−0.269136	−0.134568	+	0.990904i	$0.542965\pi$
$822$	0	0
$823$	−37045.0	−1.56902	−0.784512	−	0.620114i	$-0.787086\pi$
$823$	−37045.0	−1.56902	−0.784512	+	0.620114i	$0.787086\pi$
$824$	−28489.4	−1.20446
$825$	0	0
$826$	−5837.51	−0.245900
$827$	−10054.3	−0.422759	−0.211379	−	0.977404i	$-0.567796\pi$
$827$	−10054.3	−0.422759	−0.211379	+	0.977404i	$0.567796\pi$
$828$	0	0
$829$	12046.4	0.504691	0.252346	−	0.967637i	$-0.418798\pi$
$829$	12046.4	0.504691	0.252346	+	0.967637i	$0.418798\pi$
$830$	−2827.46	−0.118244
$831$	0	0
$832$	19344.6	0.806074
$833$	10590.3	0.440493
$834$	0	0
$835$	6289.33	0.260660
$836$	4753.50	0.196655
$837$	0	0
$838$	12636.1	0.520892
$839$	−4917.51	−0.202350	−0.101175	−	0.994869i	$-0.532260\pi$
$839$	−4917.51	−0.202350	−0.101175	+	0.994869i	$0.532260\pi$
$840$	0	0
$841$	30541.2	1.25225
$842$	46866.3	1.91819
$843$	0	0
$844$	2576.35	0.105073
$845$	−4726.07	−0.192405
$846$	0	0
$847$	1333.75	0.0541063
$848$	42796.2	1.73305
$849$	0	0
$850$	−11845.8	−0.478009
$851$	−59519.3	−2.39753
$852$	0	0
$853$	16607.2	0.666612	0.333306	−	0.942819i	$-0.391836\pi$
$853$	16607.2	0.666612	0.333306	+	0.942819i	$0.391836\pi$
$854$	−2143.77	−0.0858997
$855$	0	0
$856$	−23060.6	−0.920790
$857$	−8132.36	−0.324150	−0.162075	−	0.986778i	$-0.551819\pi$
$857$	−8132.36	−0.324150	−0.162075	+	0.986778i	$0.551819\pi$
$858$	0	0
$859$	−19809.6	−0.786839	−0.393420	−	0.919359i	$-0.628708\pi$
$859$	−19809.6	−0.786839	−0.393420	+	0.919359i	$0.628708\pi$
$860$	−1326.57	−0.0525996
$861$	0	0
$862$	11189.0	0.442111
$863$	36260.2	1.43026	0.715129	−	0.698993i	$-0.246368\pi$
$863$	36260.2	1.43026	0.715129	+	0.698993i	$0.246368\pi$
$864$	0	0
$865$	3665.47	0.144081
$866$	−9657.93	−0.378972
$867$	0	0
$868$	−3901.39	−0.152560
$869$	−19202.3	−0.749590
$870$	0	0
$871$	−55351.5	−2.15329
$872$	37036.4	1.43832
$873$	0	0
$874$	−33068.9	−1.27983
$875$	5699.63	0.220209
$876$	0	0
$877$	−1325.48	−0.0510358	−0.0255179	−	0.999674i	$-0.508123\pi$
$877$	−1325.48	−0.0510358	−0.0255179	+	0.999674i	$0.508123\pi$
$878$	47693.1	1.83322
$879$	0	0
$880$	12063.9	0.462129
$881$	28346.4	1.08401	0.542007	−	0.840374i	$-0.317665\pi$
$881$	28346.4	1.08401	0.542007	+	0.840374i	$0.317665\pi$
$882$	0	0
$883$	17073.1	0.650686	0.325343	−	0.945596i	$-0.394520\pi$
$883$	17073.1	0.650686	0.325343	+	0.945596i	$0.394520\pi$
$884$	−3915.94	−0.148990
$885$	0	0
$886$	36297.6	1.37635
$887$	−17138.8	−0.648776	−0.324388	−	0.945924i	$-0.605158\pi$
$887$	−17138.8	−0.648776	−0.324388	+	0.945924i	$0.605158\pi$
$888$	0	0
$889$	−6728.33	−0.253837
$890$	11125.9	0.419036
$891$	0	0
$892$	7900.78	0.296567
$893$	33583.2	1.25848
$894$	0	0
$895$	17088.9	0.638234
$896$	−10532.4	−0.392703
$897$	0	0
$898$	46531.8	1.72916
$899$	77522.4	2.87599
$900$	0	0
$901$	19502.3	0.721107
$902$	−15896.5	−0.586802
$903$	0	0
$904$	37533.6	1.38092
$905$	−13152.2	−0.483087
$906$	0	0
$907$	−53242.1	−1.94915	−0.974573	−	0.224072i	$-0.928065\pi$
$907$	−53242.1	−1.94915	−0.974573	+	0.224072i	$0.928065\pi$
$908$	−4288.76	−0.156748
$909$	0	0
$910$	4465.05	0.162654
$911$	22379.7	0.813909	0.406955	−	0.913448i	$-0.366591\pi$
$911$	22379.7	0.813909	0.406955	+	0.913448i	$0.366591\pi$
$912$	0	0
$913$	−8740.17	−0.316821
$914$	−34051.5	−1.23230
$915$	0	0
$916$	3779.66	0.136336
$917$	−915.737	−0.0329774
$918$	0	0
$919$	47154.3	1.69258	0.846289	−	0.532725i	$-0.178832\pi$
$919$	47154.3	1.69258	0.846289	+	0.532725i	$0.178832\pi$
$920$	−13075.7	−0.468579
$921$	0	0
$922$	−2106.75	−0.0752518
$923$	−20770.1	−0.740688
$924$	0	0
$925$	38121.5	1.35506
$926$	−3582.77	−0.127146
$927$	0	0
$928$	−20289.0	−0.717694
$929$	38119.0	1.34623	0.673113	−	0.739539i	$-0.264957\pi$
$929$	38119.0	1.34623	0.673113	+	0.739539i	$0.264957\pi$
$930$	0	0
$931$	−18935.4	−0.666575
$932$	5884.68	0.206823
$933$	0	0
$934$	−21782.5	−0.763112
$935$	5497.54	0.192287
$936$	0	0
$937$	54300.2	1.89318	0.946590	−	0.322440i	$-0.104503\pi$
$937$	54300.2	1.89318	0.946590	+	0.322440i	$0.104503\pi$
$938$	18171.6	0.632543
$939$	0	0
$940$	−4290.01	−0.148856
$941$	−48459.9	−1.67880	−0.839398	−	0.543517i	$-0.817092\pi$
$941$	−48459.9	−1.67880	−0.839398	+	0.543517i	$0.817092\pi$
$942$	0	0
$943$	21703.7	0.749490
$944$	−23231.5	−0.800977
$945$	0	0
$946$	−20894.3	−0.718109
$947$	−265.339	−0.00910492	−0.00455246	−	0.999990i	$-0.501449\pi$
$947$	−265.339	−0.00910492	−0.00455246	+	0.999990i	$0.501449\pi$
$948$	0	0
$949$	4031.26	0.137893
$950$	21180.3	0.723347
$951$	0	0
$952$	−3979.34	−0.135474
$953$	−31191.8	−1.06023	−0.530116	−	0.847925i	$-0.677852\pi$
$953$	−31191.8	−1.06023	−0.530116	+	0.847925i	$0.677852\pi$
$954$	0	0
$955$	15103.2	0.511758
$956$	−466.871	−0.0157947
$957$	0	0
$958$	−8510.93	−0.287031
$959$	−10206.8	−0.343686
$960$	0	0
$961$	79615.5	2.67247
$962$	64212.0	2.15206
$963$	0	0
$964$	−6806.40	−0.227406
$965$	−8578.95	−0.286183
$966$	0	0
$967$	48436.8	1.61078	0.805389	−	0.592746i	$-0.201956\pi$
$967$	48436.8	1.61078	0.805389	+	0.592746i	$0.201956\pi$
$968$	4213.75	0.139912
$969$	0	0
$970$	1844.48	0.0610542
$971$	44011.8	1.45459	0.727294	−	0.686326i	$-0.240778\pi$
$971$	44011.8	1.45459	0.727294	+	0.686326i	$0.240778\pi$
$972$	0	0
$973$	−1515.23	−0.0499240
$974$	−14230.7	−0.468154
$975$	0	0
$976$	−8531.57	−0.279804
$977$	−15371.9	−0.503370	−0.251685	−	0.967809i	$-0.580985\pi$
$977$	−15371.9	−0.503370	−0.251685	+	0.967809i	$0.580985\pi$
$978$	0	0
$979$	34392.2	1.12276
$980$	2418.85	0.0788443
$981$	0	0
$982$	38922.4	1.26483
$983$	−16218.4	−0.526233	−0.263117	−	0.964764i	$-0.584750\pi$
$983$	−16218.4	−0.526233	−0.263117	+	0.964764i	$0.584750\pi$
$984$	0	0
$985$	6693.67	0.216526
$986$	−25545.2	−0.825075
$987$	0	0
$988$	7001.70	0.225459
$989$	28527.2	0.917201
$990$	0	0
$991$	13148.0	0.421454	0.210727	−	0.977545i	$-0.432417\pi$
$991$	13148.0	0.421454	0.210727	+	0.977545i	$0.432417\pi$
$992$	−28633.7	−0.916452
$993$	0	0
$994$	6818.72	0.217582
$995$	−630.637	−0.0200930
$996$	0	0
$997$	−26736.8	−0.849311	−0.424656	−	0.905355i	$-0.639605\pi$
$997$	−26736.8	−0.849311	−0.424656	+	0.905355i	$0.639605\pi$
$998$	−13103.6	−0.415619
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.b.1.8		28
3.2	odd	2		717.4.a.b.1.21	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.b.1.21	✓	28	3.2	odd	2
2151.4.a.b.1.8		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)

\(f(q)\)	\(=\)	\(q-3.15491 q^{2} +1.95344 q^{4} -4.03944 q^{5} +6.03808 q^{7} +19.0763 q^{8} +O(q^{10})\)	\(q-3.15491 q^{2} +1.95344 q^{4} -4.03944 q^{5} +6.03808 q^{7} +19.0763 q^{8} +12.7440 q^{10} +39.3940 q^{11} +58.0257 q^{13} -19.0496 q^{14} -75.8116 q^{16} -34.5475 q^{17} +61.7709 q^{19} -7.89078 q^{20} -124.284 q^{22} +169.687 q^{23} -108.683 q^{25} -183.066 q^{26} +11.7950 q^{28} -234.372 q^{29} -330.766 q^{31} +86.5677 q^{32} +108.994 q^{34} -24.3904 q^{35} -350.759 q^{37} -194.882 q^{38} -77.0577 q^{40} +127.904 q^{41} +168.116 q^{43} +76.9537 q^{44} -535.347 q^{46} +543.673 q^{47} -306.542 q^{49} +342.885 q^{50} +113.349 q^{52} -564.508 q^{53} -159.130 q^{55} +115.184 q^{56} +739.421 q^{58} +306.438 q^{59} +112.536 q^{61} +1043.54 q^{62} +333.380 q^{64} -234.391 q^{65} -953.913 q^{67} -67.4863 q^{68} +76.9496 q^{70} -357.946 q^{71} +69.4737 q^{73} +1106.61 q^{74} +120.666 q^{76} +237.864 q^{77} -487.442 q^{79} +306.236 q^{80} -403.526 q^{82} -221.865 q^{83} +139.553 q^{85} -530.392 q^{86} +751.494 q^{88} +873.031 q^{89} +350.364 q^{91} +331.473 q^{92} -1715.24 q^{94} -249.520 q^{95} +144.732 q^{97} +967.110 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) \) 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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