LMFDB - Embedded newform 529.4.a.k.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [529,4,Mod(1,529)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("529.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(529, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 529 = 23^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	529.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,4,0,56,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.2120103930$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 80 x^{10} + 266 x^{9} + 2256 x^{8} - 5648 x^{7} - 28495 x^{6} + 47408 x^{5} + \cdots - 4232 $ x^12 - 4x^11 - 80x^10 + 266x^9 + 2256x^8 - 5648x^7 - 28495x^6 + 47408x^5 + 160258x^4 - 126464x^3 - 313380x^2 - 103008*x - 4232
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.78050$ of defining polynomial
Character	$\chi$	$=$	529.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.36628 q^{2} -0.504290 q^{3} -6.13327 q^{4} +0.0831947 q^{5} -0.689003 q^{6} -29.1476 q^{7} -19.3101 q^{8} -26.7457 q^{9} +0.113668 q^{10} -29.4165 q^{11} +3.09295 q^{12} +23.1585 q^{13} -39.8239 q^{14} -0.0419542 q^{15} +22.6831 q^{16} +66.2488 q^{17} -36.5422 q^{18} +40.4157 q^{19} -0.510255 q^{20} +14.6988 q^{21} -40.1913 q^{22} +9.73786 q^{24} -124.993 q^{25} +31.6411 q^{26} +27.1034 q^{27} +178.770 q^{28} +150.919 q^{29} -0.0573214 q^{30} +301.179 q^{31} +185.472 q^{32} +14.8345 q^{33} +90.5146 q^{34} -2.42493 q^{35} +164.039 q^{36} -181.892 q^{37} +55.2193 q^{38} -11.6786 q^{39} -1.60649 q^{40} +275.600 q^{41} +20.0828 q^{42} -171.429 q^{43} +180.419 q^{44} -2.22510 q^{45} -460.713 q^{47} -11.4389 q^{48} +506.583 q^{49} -170.776 q^{50} -33.4086 q^{51} -142.038 q^{52} -656.391 q^{53} +37.0309 q^{54} -2.44730 q^{55} +562.842 q^{56} -20.3812 q^{57} +206.198 q^{58} +541.040 q^{59} +0.257317 q^{60} -569.326 q^{61} +411.496 q^{62} +779.573 q^{63} +71.9423 q^{64} +1.92667 q^{65} +20.2681 q^{66} +80.5001 q^{67} -406.321 q^{68} -3.31314 q^{70} +504.470 q^{71} +516.461 q^{72} -422.104 q^{73} -248.516 q^{74} +63.0327 q^{75} -247.881 q^{76} +857.422 q^{77} -15.9563 q^{78} +995.209 q^{79} +1.88712 q^{80} +708.466 q^{81} +376.547 q^{82} -988.210 q^{83} -90.1520 q^{84} +5.51154 q^{85} -234.221 q^{86} -76.1068 q^{87} +568.035 q^{88} -1205.04 q^{89} -3.04012 q^{90} -675.016 q^{91} -151.882 q^{93} -629.464 q^{94} +3.36237 q^{95} -93.5317 q^{96} +618.663 q^{97} +692.137 q^{98} +786.765 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 56 q^{4} - 6 q^{6} + 138 q^{8} + 204 q^{9} + 30 q^{12} + 160 q^{13} + 144 q^{16} + 478 q^{18} - 1188 q^{24} + 400 q^{25} + 1554 q^{26} - 684 q^{27} - 44 q^{29} + 1076 q^{31} + 248 q^{32}+ \cdots - 3512 q^{98}+O(q^{100}) $ 12 * q + 4 * q^2 + 56 * q^4 - 6 * q^6 + 138 * q^8 + 204 * q^9 + 30 * q^12 + 160 * q^13 + 144 * q^16 + 478 * q^18 - 1188 * q^24 + 400 * q^25 + 1554 * q^26 - 684 * q^27 - 44 * q^29 + 1076 * q^31 + 248 * q^32 + 1048 * q^35 + 2006 * q^36 + 468 * q^39 + 92 * q^41 + 204 * q^47 - 2282 * q^48 + 1428 * q^49 - 1932 * q^50 + 2290 * q^52 + 6426 * q^54 + 1808 * q^55 + 2058 * q^58 + 3416 * q^59 + 2654 * q^62 - 346 * q^64 - 5116 * q^70 - 660 * q^71 + 7554 * q^72 - 5860 * q^73 + 2472 * q^75 + 2712 * q^77 + 826 * q^78 + 4548 * q^81 + 5486 * q^82 - 1436 * q^85 + 1268 * q^87 - 7544 * q^93 - 1322 * q^94 + 9688 * q^95 - 6930 * q^96 - 3512 * q^98

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 $

$2$	1.36628	0.483054	0.241527	−	0.970394i	$-0.422352\pi$
$2$	1.36628	0.483054	0.241527	+	0.970394i	$0.422352\pi$
$3$	−0.504290	−0.0970506	−0.0485253	−	0.998822i	$-0.515452\pi$
$3$	−0.504290	−0.0970506	−0.0485253	+	0.998822i	$0.515452\pi$
$4$	−6.13327	−0.766659
$5$	0.0831947	0.00744116	0.00372058	−	0.999993i	$-0.498816\pi$
$5$	0.0831947	0.00744116	0.00372058	+	0.999993i	$0.498816\pi$
$6$	−0.689003	−0.0468807
$7$	−29.1476	−1.57382	−0.786912	−	0.617066i	$-0.788321\pi$
$7$	−29.1476	−1.57382	−0.786912	+	0.617066i	$0.788321\pi$
$8$	−19.3101	−0.853392
$9$	−26.7457	−0.990581
$10$	0.113668	0.00359448
$11$	−29.4165	−0.806311	−0.403155	−	0.915132i	$-0.632087\pi$
$11$	−29.4165	−0.806311	−0.403155	+	0.915132i	$0.632087\pi$
$12$	3.09295	0.0744047
$13$	23.1585	0.494079	0.247039	−	0.969005i	$-0.420542\pi$
$13$	23.1585	0.494079	0.247039	+	0.969005i	$0.420542\pi$
$14$	−39.8239	−0.760242
$15$	−0.0419542	−0.000722169 0
$16$	22.6831	0.354424
$17$	66.2488	0.945158	0.472579	−	0.881288i	$-0.343323\pi$
$17$	66.2488	0.945158	0.472579	+	0.881288i	$0.343323\pi$
$18$	−36.5422	−0.478504
$19$	40.4157	0.488000	0.244000	−	0.969775i	$-0.421540\pi$
$19$	40.4157	0.488000	0.244000	+	0.969775i	$0.421540\pi$
$20$	−0.510255	−0.00570483
$21$	14.6988	0.152741
$22$	−40.1913	−0.389492
$23$	0	0
$23$	0	0
$24$	9.73786	0.0828222
$25$	−124.993	−0.999945
$26$	31.6411	0.238667
$27$	27.1034	0.193187
$28$	178.770	1.20659
$29$	150.919	0.966377	0.483188	−	0.875516i	$-0.339479\pi$
$29$	150.919	0.966377	0.483188	+	0.875516i	$0.339479\pi$
$30$	−0.0573214	−0.000348847 0
$31$	301.179	1.74495	0.872474	−	0.488660i	$-0.162514\pi$
$31$	301.179	1.74495	0.872474	+	0.488660i	$0.162514\pi$
$32$	185.472	1.02460
$33$	14.8345	0.0782530
$34$	90.5146	0.456562
$35$	−2.42493	−0.0117111
$36$	164.039	0.759438
$37$	−181.892	−0.808184	−0.404092	−	0.914718i	$-0.632412\pi$
$37$	−181.892	−0.808184	−0.404092	+	0.914718i	$0.632412\pi$
$38$	55.2193	0.235731
$39$	−11.6786	−0.0479507
$40$	−1.60649	−0.00635022
$41$	275.600	1.04979	0.524895	−	0.851167i	$-0.324104\pi$
$41$	275.600	1.04979	0.524895	+	0.851167i	$0.324104\pi$
$42$	20.0828	0.0737820
$43$	−171.429	−0.607971	−0.303985	−	0.952677i	$-0.598317\pi$
$43$	−171.429	−0.607971	−0.303985	+	0.952677i	$0.598317\pi$
$44$	180.419	0.618165
$45$	−2.22510	−0.00737107
$46$	0	0
$47$	−460.713	−1.42983	−0.714913	−	0.699213i	$-0.753534\pi$
$47$	−460.713	−1.42983	−0.714913	+	0.699213i	$0.753534\pi$
$48$	−11.4389	−0.0343971
$49$	506.583	1.47692
$50$	−170.776	−0.483027
$51$	−33.4086	−0.0917282
$52$	−142.038	−0.378790
$53$	−656.391	−1.70117	−0.850587	−	0.525834i	$-0.823753\pi$
$53$	−656.391	−1.70117	−0.850587	+	0.525834i	$0.823753\pi$
$54$	37.0309	0.0933199
$55$	−2.44730	−0.00599989
$56$	562.842	1.34309
$57$	−20.3812	−0.0473607
$58$	206.198	0.466812
$59$	541.040	1.19386	0.596928	−	0.802295i	$-0.296388\pi$
$59$	541.040	1.19386	0.596928	+	0.802295i	$0.296388\pi$
$60$	0.257317	0.000553657 0
$61$	−569.326	−1.19500	−0.597498	−	0.801870i	$-0.703838\pi$
$61$	−569.326	−1.19500	−0.597498	+	0.801870i	$0.703838\pi$
$62$	411.496	0.842905
$63$	779.573	1.55900
$64$	71.9423	0.140512
$65$	1.92667	0.00367652
$66$	20.2681	0.0378004
$67$	80.5001	0.146786	0.0733929	−	0.997303i	$-0.476617\pi$
$67$	80.5001	0.146786	0.0733929	+	0.997303i	$0.476617\pi$
$68$	−406.321	−0.724613
$69$	0	0
$70$	−3.31314	−0.00565708
$71$	504.470	0.843233	0.421617	−	0.906774i	$-0.361463\pi$
$71$	504.470	0.843233	0.421617	+	0.906774i	$0.361463\pi$
$72$	516.461	0.845354
$73$	−422.104	−0.676761	−0.338380	−	0.941009i	$-0.609879\pi$
$73$	−422.104	−0.676761	−0.338380	+	0.941009i	$0.609879\pi$
$74$	−248.516	−0.390397
$75$	63.0327	0.0970453
$76$	−247.881	−0.374130
$77$	857.422	1.26899
$78$	−15.9563	−0.0231628
$79$	995.209	1.41734	0.708670	−	0.705541i	$-0.249296\pi$
$79$	995.209	1.41734	0.708670	+	0.705541i	$0.249296\pi$
$80$	1.88712	0.00263733
$81$	708.466	0.971832
$82$	376.547	0.507106
$83$	−988.210	−1.30687	−0.653434	−	0.756983i	$-0.726672\pi$
$83$	−988.210	−1.30687	−0.653434	+	0.756983i	$0.726672\pi$
$84$	−90.1520	−0.117100
$85$	5.51154	0.00703307
$86$	−234.221	−0.293683
$87$	−76.1068	−0.0937874
$88$	568.035	0.688099
$89$	−1205.04	−1.43522	−0.717608	−	0.696447i	$-0.754763\pi$
$89$	−1205.04	−1.43522	−0.717608	+	0.696447i	$0.754763\pi$
$90$	−3.04012	−0.00356063
$91$	−675.016	−0.777593
$92$	0	0
$93$	−151.882	−0.169348
$94$	−629.464	−0.690684
$95$	3.36237	0.00363129
$96$	−93.5317	−0.0994379
$97$	618.663	0.647584	0.323792	−	0.946128i	$-0.395042\pi$
$97$	618.663	0.647584	0.323792	+	0.946128i	$0.395042\pi$
$98$	692.137	0.713432
$99$	786.765	0.798716
$100$	766.616	0.766616
$101$	1433.73	1.41249	0.706243	−	0.707970i	$-0.250389\pi$
$101$	1433.73	1.41249	0.706243	+	0.707970i	$0.250389\pi$
$102$	−45.6456	−0.0443097
$103$	1027.03	0.982488	0.491244	−	0.871022i	$-0.336542\pi$
$103$	1027.03	0.982488	0.491244	+	0.871022i	$0.336542\pi$
$104$	−447.193	−0.421643
$105$	1.22287	0.00113657
$106$	−896.817	−0.821760
$107$	1914.69	1.72991	0.864955	−	0.501850i	$-0.167347\pi$
$107$	1914.69	1.72991	0.864955	+	0.501850i	$0.167347\pi$
$108$	−166.232	−0.148109
$109$	857.907	0.753877	0.376938	−	0.926238i	$-0.376977\pi$
$109$	857.907	0.753877	0.376938	+	0.926238i	$0.376977\pi$
$110$	−3.34370	−0.00289827
$111$	91.7261	0.0784347
$112$	−661.159	−0.557801
$113$	924.728	0.769832	0.384916	−	0.922952i	$-0.374230\pi$
$113$	924.728	0.769832	0.384916	+	0.922952i	$0.374230\pi$
$114$	−27.8466	−0.0228778
$115$	0	0
$116$	−925.626	−0.740881
$117$	−619.391	−0.489425
$118$	739.215	0.576697
$119$	−1930.99	−1.48751
$120$	0.810139	0.000616293 0
$121$	−465.668	−0.349863
$122$	−777.861	−0.577248
$123$	−138.982	−0.101883
$124$	−1847.21	−1.33778
$125$	−20.7981	−0.0148819
$126$	1065.12	0.753081
$127$	1433.21	1.00139	0.500696	−	0.865623i	$-0.333077\pi$
$127$	1433.21	1.00139	0.500696	+	0.865623i	$0.333077\pi$
$128$	−1385.48	−0.956723
$129$	86.4501	0.0590039
$130$	2.63238	0.00177596
$131$	−861.722	−0.574725	−0.287363	−	0.957822i	$-0.592779\pi$
$131$	−861.722	−0.574725	−0.287363	+	0.957822i	$0.592779\pi$
$132$	−90.9837	−0.0599933
$133$	−1178.02	−0.768026
$134$	109.986	0.0709055
$135$	2.25486	0.00143754
$136$	−1279.27	−0.806590
$137$	46.1070	0.0287532	0.0143766	−	0.999897i	$-0.495424\pi$
$137$	46.1070	0.0287532	0.0143766	+	0.999897i	$0.495424\pi$
$138$	0	0
$139$	660.701	0.403165	0.201582	−	0.979472i	$-0.435392\pi$
$139$	660.701	0.403165	0.201582	+	0.979472i	$0.435392\pi$
$140$	14.8727	0.00897839
$141$	232.333	0.138766
$142$	689.249	0.407327
$143$	−681.244	−0.398381
$144$	−606.676	−0.351086
$145$	12.5556	0.00719096
$146$	−576.714	−0.326912
$147$	−255.465	−0.143336
$148$	1115.59	0.619601
$149$	−3394.75	−1.86650	−0.933252	−	0.359223i	$-0.883042\pi$
$149$	−3394.75	−1.86650	−0.933252	+	0.359223i	$0.883042\pi$
$150$	86.1206	0.0468781
$151$	1473.84	0.794301	0.397151	−	0.917753i	$-0.369999\pi$
$151$	1473.84	0.794301	0.397151	+	0.917753i	$0.369999\pi$
$152$	−780.430	−0.416455
$153$	−1771.87	−0.936256
$154$	1171.48	0.612991
$155$	25.0565	0.0129844
$156$	71.6281	0.0367618
$157$	26.7216	0.0135835	0.00679177	−	0.999977i	$-0.497838\pi$
$157$	26.7216	0.0135835	0.00679177	+	0.999977i	$0.497838\pi$
$158$	1359.74	0.684652
$159$	331.011	0.165100
$160$	15.4303	0.00762420
$161$	0	0
$162$	967.965	0.469448
$163$	2914.04	1.40028	0.700140	−	0.714006i	$-0.253121\pi$
$163$	2914.04	1.40028	0.700140	+	0.714006i	$0.253121\pi$
$164$	−1690.33	−0.804831
$165$	1.23415	0.000582293 0
$166$	−1350.17	−0.631288
$167$	200.029	0.0926868	0.0463434	−	0.998926i	$-0.485243\pi$
$167$	200.029	0.0926868	0.0463434	+	0.998926i	$0.485243\pi$
$168$	−283.836	−0.130348
$169$	−1660.68	−0.755886
$170$	7.53033	0.00339735
$171$	−1080.95	−0.483404
$172$	1051.42	0.466106
$173$	−1413.95	−0.621389	−0.310695	−	0.950510i	$-0.600562\pi$
$173$	−1413.95	−0.621389	−0.310695	+	0.950510i	$0.600562\pi$
$174$	−103.984	−0.0453044
$175$	3643.25	1.57374
$176$	−667.259	−0.285776
$177$	−272.841	−0.115864
$178$	−1646.43	−0.693287
$179$	94.0407	0.0392678	0.0196339	−	0.999807i	$-0.493750\pi$
$179$	94.0407	0.0392678	0.0196339	+	0.999807i	$0.493750\pi$
$180$	13.6471	0.00565110
$181$	588.921	0.241846	0.120923	−	0.992662i	$-0.461415\pi$
$181$	588.921	0.241846	0.120923	+	0.992662i	$0.461415\pi$
$182$	−922.264	−0.375620
$183$	287.105	0.115975
$184$	0	0
$185$	−15.1324	−0.00601382
$186$	−207.513	−0.0818044
$187$	−1948.81	−0.762091
$188$	2825.67	1.09619
$189$	−790.000	−0.304042
$190$	4.59396	0.00175411
$191$	1990.98	0.754252	0.377126	−	0.926162i	$-0.376912\pi$
$191$	1990.98	0.754252	0.377126	+	0.926162i	$0.376912\pi$
$192$	−36.2798	−0.0136368
$193$	2694.51	1.00495	0.502473	−	0.864593i	$-0.332423\pi$
$193$	2694.51	1.00495	0.502473	+	0.864593i	$0.332423\pi$
$194$	845.269	0.312818
$195$	−0.971599	−0.000356809 0
$196$	−3107.01	−1.13229
$197$	−2724.10	−0.985198	−0.492599	−	0.870257i	$-0.663953\pi$
$197$	−2724.10	−0.985198	−0.492599	+	0.870257i	$0.663953\pi$
$198$	1074.94	0.385823
$199$	844.492	0.300826	0.150413	−	0.988623i	$-0.451940\pi$
$199$	844.492	0.300826	0.150413	+	0.988623i	$0.451940\pi$
$200$	2413.62	0.853345
$201$	−40.5954	−0.0142457
$202$	1958.88	0.682307
$203$	−4398.92	−1.52091
$204$	204.904	0.0703242
$205$	22.9284	0.00781166
$206$	1403.21	0.474595
$207$	0	0
$208$	525.309	0.175113
$209$	−1188.89	−0.393480
$210$	1.67078	0.000549023 0
$211$	2709.96	0.884178	0.442089	−	0.896971i	$-0.354238\pi$
$211$	2709.96	0.884178	0.442089	+	0.896971i	$0.354238\pi$
$212$	4025.82	1.30422
$213$	−254.399	−0.0818363
$214$	2616.01	0.835640
$215$	−14.2620	−0.00452401
$216$	−523.368	−0.164864
$217$	−8778.66	−2.74624
$218$	1172.14	0.364163
$219$	212.863	0.0656801
$220$	15.0099	0.00459986
$221$	1534.22	0.466982
$222$	125.324	0.0378882
$223$	−6196.41	−1.86073	−0.930365	−	0.366636i	$-0.880510\pi$
$223$	−6196.41	−1.86073	−0.930365	+	0.366636i	$0.880510\pi$
$224$	−5406.07	−1.61254
$225$	3343.03	0.990526
$226$	1263.44	0.371871
$227$	5438.04	1.59002	0.795012	−	0.606594i	$-0.207464\pi$
$227$	5438.04	1.59002	0.795012	+	0.606594i	$0.207464\pi$
$228$	125.004	0.0363095
$229$	6753.89	1.94895	0.974475	−	0.224498i	$-0.0720741\pi$
$229$	6753.89	1.94895	0.974475	+	0.224498i	$0.0720741\pi$
$230$	0	0
$231$	−432.389	−0.123156
$232$	−2914.25	−0.824698
$233$	683.570	0.192198	0.0960990	−	0.995372i	$-0.469363\pi$
$233$	683.570	0.192198	0.0960990	+	0.995372i	$0.469363\pi$
$234$	−846.264	−0.236419
$235$	−38.3288	−0.0106396
$236$	−3318.35	−0.915279
$237$	−501.874	−0.137554
$238$	−2638.28	−0.718549
$239$	−4079.26	−1.10404	−0.552020	−	0.833831i	$-0.686143\pi$
$239$	−4079.26	−1.10404	−0.552020	+	0.833831i	$0.686143\pi$
$240$	−0.951654	−0.000255954 0
$241$	−1690.03	−0.451719	−0.225860	−	0.974160i	$-0.572519\pi$
$241$	−1690.03	−0.451719	−0.225860	+	0.974160i	$0.572519\pi$
$242$	−636.234	−0.169003
$243$	−1089.06	−0.287504
$244$	3491.83	0.916154
$245$	42.1451	0.0109900
$246$	−189.889	−0.0492149
$247$	935.970	0.241111
$248$	−5815.79	−1.48913
$249$	498.344	0.126832
$250$	−28.4161	−0.00718877
$251$	−641.802	−0.161395	−0.0806976	−	0.996739i	$-0.525715\pi$
$251$	−641.802	−0.161395	−0.0806976	+	0.996739i	$0.525715\pi$
$252$	−4781.33	−1.19522
$253$	0	0
$254$	1958.17	0.483727
$255$	−2.77942	−0.000682564 0
$256$	−2468.50	−0.602661
$257$	4349.92	1.05580	0.527900	−	0.849306i	$-0.322979\pi$
$257$	4349.92	1.05580	0.527900	+	0.849306i	$0.322979\pi$
$258$	118.115	0.0285021
$259$	5301.71	1.27194
$260$	−11.8168	−0.00281864
$261$	−4036.43	−0.957274
$262$	−1177.36	−0.277624
$263$	−3923.36	−0.919866	−0.459933	−	0.887954i	$-0.652127\pi$
$263$	−3923.36	−0.919866	−0.459933	+	0.887954i	$0.652127\pi$
$264$	−286.454	−0.0667804
$265$	−54.6083	−0.0126587
$266$	−1609.51	−0.370998
$267$	607.691	0.139289
$268$	−493.729	−0.112535
$269$	2251.92	0.510417	0.255209	−	0.966886i	$-0.417856\pi$
$269$	2251.92	0.510417	0.255209	+	0.966886i	$0.417856\pi$
$270$	3.08078	0.000694408 0
$271$	−397.386	−0.0890755	−0.0445378	−	0.999008i	$-0.514182\pi$
$271$	−397.386	−0.0890755	−0.0445378	+	0.999008i	$0.514182\pi$
$272$	1502.73	0.334987
$273$	340.404	0.0754659
$274$	62.9953	0.0138894
$275$	3676.86	0.806266
$276$	0	0
$277$	5493.15	1.19152	0.595761	−	0.803162i	$-0.296851\pi$
$277$	5493.15	1.19152	0.595761	+	0.803162i	$0.296851\pi$
$278$	902.704	0.194750
$279$	−8055.25	−1.72851
$280$	46.8255	0.00999413
$281$	−3449.28	−0.732265	−0.366133	−	0.930563i	$-0.619318\pi$
$281$	−3449.28	−0.732265	−0.366133	+	0.930563i	$0.619318\pi$
$282$	317.432	0.0670313
$283$	−2381.80	−0.500295	−0.250148	−	0.968208i	$-0.580479\pi$
$283$	−2381.80	−0.500295	−0.250148	+	0.968208i	$0.580479\pi$
$284$	−3094.05	−0.646472
$285$	−1.69561	−0.000352419 0
$286$	−930.773	−0.192440
$287$	−8033.07	−1.65219
$288$	−4960.58	−1.01495
$289$	−524.103	−0.106677
$290$	17.1546	0.00347362
$291$	−311.985	−0.0628485
$292$	2588.88	0.518844
$293$	5170.24	1.03088	0.515442	−	0.856925i	$-0.327628\pi$
$293$	5170.24	1.03088	0.515442	+	0.856925i	$0.327628\pi$
$294$	−349.038	−0.0692391
$295$	45.0117	0.00888367
$296$	3512.34	0.689697
$297$	−797.288	−0.155769
$298$	−4638.20	−0.901622
$299$	0	0
$300$	−386.597	−0.0744006
$301$	4996.76	0.956838
$302$	2013.69	0.383691
$303$	−723.013	−0.137083
$304$	916.756	0.172959
$305$	−47.3649	−0.00889215
$306$	−2420.88	−0.452262
$307$	−1338.58	−0.248850	−0.124425	−	0.992229i	$-0.539709\pi$
$307$	−1338.58	−0.248850	−0.124425	+	0.992229i	$0.539709\pi$
$308$	−5258.80	−0.972882
$309$	−517.921	−0.0953510
$310$	34.2343	0.00627219
$311$	3570.72	0.651052	0.325526	−	0.945533i	$-0.394459\pi$
$311$	3570.72	0.651052	0.325526	+	0.945533i	$0.394459\pi$
$312$	225.515	0.0409207
$313$	−4079.52	−0.736703	−0.368352	−	0.929687i	$-0.620078\pi$
$313$	−4079.52	−0.736703	−0.368352	+	0.929687i	$0.620078\pi$
$314$	36.5093	0.00656159
$315$	64.8563	0.0116008
$316$	−6103.89	−1.08662
$317$	−320.913	−0.0568589	−0.0284294	−	0.999596i	$-0.509051\pi$
$317$	−320.913	−0.0568589	−0.0284294	+	0.999596i	$0.509051\pi$
$318$	452.256	0.0797523
$319$	−4439.51	−0.779200
$320$	5.98522	0.00104557
$321$	−965.560	−0.167889
$322$	0	0
$323$	2677.49	0.461237
$324$	−4345.21	−0.745064
$325$	−2894.66	−0.494052
$326$	3981.41	0.676411
$327$	−432.634	−0.0731642
$328$	−5321.84	−0.895883
$329$	13428.7	2.25029
$330$	1.68620	0.000281279 0
$331$	7657.70	1.27162	0.635809	−	0.771847i	$-0.280667\pi$
$331$	7657.70	1.27162	0.635809	+	0.771847i	$0.280667\pi$
$332$	6060.96	1.00192
$333$	4864.82	0.800572
$334$	273.296	0.0447728
$335$	6.69718	0.00109226
$336$	333.416	0.0541349
$337$	1942.40	0.313974	0.156987	−	0.987601i	$-0.449822\pi$
$337$	1942.40	0.313974	0.156987	+	0.987601i	$0.449822\pi$
$338$	−2268.96	−0.365134
$339$	−466.331	−0.0747127
$340$	−33.8038	−0.00539196
$341$	−8859.65	−1.40697
$342$	−1476.88	−0.233510
$343$	−4768.07	−0.750588
$344$	3310.31	0.518837
$345$	0	0
$346$	−1931.85	−0.300165
$347$	755.100	0.116818	0.0584091	−	0.998293i	$-0.481397\pi$
$347$	755.100	0.116818	0.0584091	+	0.998293i	$0.481397\pi$
$348$	466.784	0.0719030
$349$	−11707.4	−1.79566	−0.897828	−	0.440347i	$-0.854855\pi$
$349$	−11707.4	−1.79566	−0.897828	+	0.440347i	$0.854855\pi$
$350$	4977.71	0.760200
$351$	627.676	0.0954497
$352$	−5455.94	−0.826144
$353$	5119.94	0.771974	0.385987	−	0.922504i	$-0.373861\pi$
$353$	5119.94	0.771974	0.385987	+	0.922504i	$0.373861\pi$
$354$	−372.778	−0.0559688
$355$	41.9692	0.00627463
$356$	7390.85	1.10032
$357$	973.780	0.144364
$358$	128.486	0.0189685
$359$	1168.39	0.171769	0.0858845	−	0.996305i	$-0.472628\pi$
$359$	1168.39	0.171769	0.0858845	+	0.996305i	$0.472628\pi$
$360$	42.9668	0.00629041
$361$	−5225.57	−0.761856
$362$	804.633	0.116825
$363$	234.832	0.0339544
$364$	4140.06	0.596148
$365$	−35.1168	−0.00503588
$366$	392.267	0.0560223
$367$	5324.59	0.757334	0.378667	−	0.925533i	$-0.376383\pi$
$367$	5324.59	0.757334	0.378667	+	0.925533i	$0.376383\pi$
$368$	0	0
$369$	−7371.10	−1.03990
$370$	−20.6752	−0.00290500
$371$	19132.2	2.67735
$372$	931.531	0.129832
$373$	−4800.75	−0.666416	−0.333208	−	0.942853i	$-0.608131\pi$
$373$	−4800.75	−0.666416	−0.333208	+	0.942853i	$0.608131\pi$
$374$	−2662.62	−0.368131
$375$	10.4883	0.00144430
$376$	8896.39	1.22020
$377$	3495.06	0.477466
$378$	−1079.36	−0.146869
$379$	5640.98	0.764532	0.382266	−	0.924052i	$-0.375144\pi$
$379$	5640.98	0.764532	0.382266	+	0.924052i	$0.375144\pi$
$380$	−20.6223	−0.00278396
$381$	−722.754	−0.0971858
$382$	2720.24	0.364345
$383$	6165.75	0.822598	0.411299	−	0.911500i	$-0.365075\pi$
$383$	6165.75	0.822598	0.411299	+	0.911500i	$0.365075\pi$
$384$	698.685	0.0928506
$385$	71.3329	0.00944276
$386$	3681.46	0.485444
$387$	4585.00	0.602244
$388$	−3794.42	−0.496476
$389$	11917.5	1.55332	0.776661	−	0.629919i	$-0.216912\pi$
$389$	11917.5	1.55332	0.776661	+	0.629919i	$0.216912\pi$
$390$	−1.32748	−0.000172358 0
$391$	0	0
$392$	−9782.15	−1.26039
$393$	434.558	0.0557775
$394$	−3721.89	−0.475904
$395$	82.7961	0.0105466
$396$	−4825.44	−0.612343
$397$	3209.99	0.405806	0.202903	−	0.979199i	$-0.434962\pi$
$397$	3209.99	0.405806	0.202903	+	0.979199i	$0.434962\pi$
$398$	1153.82	0.145315
$399$	594.065	0.0745374
$400$	−2835.24	−0.354404
$401$	9648.67	1.20157	0.600787	−	0.799409i	$-0.294854\pi$
$401$	9648.67	1.20157	0.600787	+	0.799409i	$0.294854\pi$
$402$	−55.4648	−0.00688142
$403$	6974.88	0.862142
$404$	−8793.42	−1.08289
$405$	58.9406	0.00723156
$406$	−6010.18	−0.734680
$407$	5350.62	0.651647
$408$	645.121	0.0782801
$409$	−971.408	−0.117440	−0.0587201	−	0.998274i	$-0.518702\pi$
$409$	−971.408	−0.117440	−0.0587201	+	0.998274i	$0.518702\pi$
$410$	31.3267	0.00377346
$411$	−23.2513	−0.00279052
$412$	−6299.05	−0.753233
$413$	−15770.0	−1.87892
$414$	0	0
$415$	−82.2138	−0.00972462
$416$	4295.26	0.506232
$417$	−333.185	−0.0391274
$418$	−1624.36	−0.190072
$419$	−4858.19	−0.566440	−0.283220	−	0.959055i	$-0.591403\pi$
$419$	−4858.19	−0.566440	−0.283220	+	0.959055i	$0.591403\pi$
$420$	−7.50017	−0.000871359 0
$421$	−6799.85	−0.787184	−0.393592	−	0.919285i	$-0.628768\pi$
$421$	−6799.85	−0.787184	−0.393592	+	0.919285i	$0.628768\pi$
$422$	3702.58	0.427106
$423$	12322.1	1.41636
$424$	12675.0	1.45177
$425$	−8280.64	−0.945105
$426$	−347.581	−0.0395314
$427$	16594.5	1.88071
$428$	−11743.3	−1.32625
$429$	343.544	0.0386631
$430$	−19.4860	−0.00218534
$431$	1345.12	0.150330	0.0751648	−	0.997171i	$-0.476052\pi$
$431$	1345.12	0.150330	0.0751648	+	0.997171i	$0.476052\pi$
$432$	614.790	0.0684702
$433$	4087.92	0.453702	0.226851	−	0.973929i	$-0.427157\pi$
$433$	4087.92	0.453702	0.226851	+	0.973929i	$0.427157\pi$
$434$	−11994.1	−1.32658
$435$	−6.33168	−0.000697887 0
$436$	−5261.77	−0.577966
$437$	0	0
$438$	290.831	0.0317270
$439$	2613.28	0.284111	0.142056	−	0.989859i	$-0.454629\pi$
$439$	2613.28	0.284111	0.142056	+	0.989859i	$0.454629\pi$
$440$	47.2575	0.00512025
$441$	−13548.9	−1.46301
$442$	2096.19	0.225578
$443$	2728.09	0.292586	0.146293	−	0.989241i	$-0.453266\pi$
$443$	2728.09	0.292586	0.146293	+	0.989241i	$0.453266\pi$
$444$	−562.581	−0.0601327
$445$	−100.253	−0.0106797
$446$	−8466.06	−0.898833
$447$	1711.94	0.181145
$448$	−2096.95	−0.221142
$449$	13237.0	1.39130	0.695649	−	0.718382i	$-0.255117\pi$
$449$	13237.0	1.39130	0.695649	+	0.718382i	$0.255117\pi$
$450$	4567.52	0.478478
$451$	−8107.18	−0.846458
$452$	−5671.60	−0.590199
$453$	−743.243	−0.0770875
$454$	7429.91	0.768068
$455$	−56.1578	−0.00578619
$456$	393.563	0.0404173
$457$	6127.89	0.627244	0.313622	−	0.949548i	$-0.398457\pi$
$457$	6127.89	0.627244	0.313622	+	0.949548i	$0.398457\pi$
$458$	9227.73	0.941448
$459$	1795.57	0.182592
$460$	0	0
$461$	5412.21	0.546793	0.273397	−	0.961901i	$-0.411853\pi$
$461$	5412.21	0.546793	0.273397	+	0.961901i	$0.411853\pi$
$462$	−590.766	−0.0594912
$463$	6916.59	0.694258	0.347129	−	0.937817i	$-0.387157\pi$
$463$	6916.59	0.694258	0.347129	+	0.937817i	$0.387157\pi$
$464$	3423.31	0.342507
$465$	−12.6358	−0.00126015
$466$	933.950	0.0928421
$467$	−14421.1	−1.42897	−0.714483	−	0.699653i	$-0.753338\pi$
$467$	−14421.1	−1.42897	−0.714483	+	0.699653i	$0.753338\pi$
$468$	3798.89	0.375222
$469$	−2346.39	−0.231015
$470$	−52.3681	−0.00513949
$471$	−13.4754	−0.00131829
$472$	−10447.5	−1.01883
$473$	5042.86	0.490213
$474$	−685.702	−0.0664459
$475$	−5051.69	−0.487973
$476$	11843.3	1.14041
$477$	17555.6	1.68515
$478$	−5573.43	−0.533311
$479$	1174.30	0.112015	0.0560074	−	0.998430i	$-0.482163\pi$
$479$	1174.30	0.112015	0.0560074	+	0.998430i	$0.482163\pi$
$480$	−7.78134	−0.000739933 0
$481$	−4212.35	−0.399307
$482$	−2309.06	−0.218205
$483$	0	0
$484$	2856.07	0.268226
$485$	51.4694	0.00481878
$486$	−1487.97	−0.138880
$487$	−3668.33	−0.341330	−0.170665	−	0.985329i	$-0.554592\pi$
$487$	−3668.33	−0.341330	−0.170665	+	0.985329i	$0.554592\pi$
$488$	10993.7	1.01980
$489$	−1469.52	−0.135898
$490$	57.5821	0.00530876
$491$	2797.46	0.257124	0.128562	−	0.991701i	$-0.458964\pi$
$491$	2797.46	0.257124	0.128562	+	0.991701i	$0.458964\pi$
$492$	852.414	0.0781094
$493$	9998.18	0.913378
$494$	1278.80	0.116469
$495$	65.4547	0.00594337
$496$	6831.69	0.618452
$497$	−14704.1	−1.32710
$498$	680.879	0.0612669
$499$	1880.58	0.168710	0.0843552	−	0.996436i	$-0.473117\pi$
$499$	1880.58	0.168710	0.0843552	+	0.996436i	$0.473117\pi$
$500$	127.560	0.0114093
$501$	−100.873	−0.00899531
$502$	−876.884	−0.0779626
$503$	−17801.6	−1.57800	−0.789000	−	0.614393i	$-0.789401\pi$
$503$	−17801.6	−1.57800	−0.789000	+	0.614393i	$0.789401\pi$
$504$	−15053.6	−1.33044
$505$	119.278	0.0105105
$506$	0	0
$507$	837.465	0.0733592
$508$	−8790.27	−0.767726
$509$	10857.3	0.945462	0.472731	−	0.881207i	$-0.343268\pi$
$509$	10857.3	0.945462	0.472731	+	0.881207i	$0.343268\pi$
$510$	−3.79747	−0.000329715 0
$511$	12303.3	1.06510
$512$	7711.19	0.665605
$513$	1095.40	0.0942754
$514$	5943.23	0.510009
$515$	85.4434	0.00731085
$516$	−530.222	−0.0452359
$517$	13552.6	1.15288
$518$	7243.64	0.614415
$519$	713.038	0.0603062
$520$	−37.2041	−0.00313751
$521$	17767.9	1.49410	0.747048	−	0.664770i	$-0.231470\pi$
$521$	17767.9	1.49410	0.747048	+	0.664770i	$0.231470\pi$
$522$	−5514.91	−0.462415
$523$	3707.29	0.309959	0.154979	−	0.987918i	$-0.450469\pi$
$523$	3707.29	0.309959	0.154979	+	0.987918i	$0.450469\pi$
$524$	5285.18	0.440618
$525$	−1837.25	−0.152732
$526$	−5360.42	−0.444345
$527$	19952.8	1.64925
$528$	336.492	0.0277347
$529$	0	0
$530$	−74.6104	−0.00611484
$531$	−14470.5	−1.18261
$532$	7225.13	0.588814
$533$	6382.49	0.518679
$534$	830.278	0.0672840
$535$	159.292	0.0128725
$536$	−1554.46	−0.125266
$537$	−47.4238	−0.00381096
$538$	3076.77	0.246559
$539$	−14901.9	−1.19086
$540$	−13.8297	−0.00110210
$541$	537.343	0.0427027	0.0213514	−	0.999772i	$-0.493203\pi$
$541$	537.343	0.0427027	0.0213514	+	0.999772i	$0.493203\pi$
$542$	−542.942	−0.0430283
$543$	−296.987	−0.0234713
$544$	12287.3	0.968407
$545$	71.3733	0.00560972
$546$	465.088	0.0364541
$547$	4242.41	0.331613	0.165806	−	0.986158i	$-0.446977\pi$
$547$	4242.41	0.331613	0.165806	+	0.986158i	$0.446977\pi$
$548$	−282.787	−0.0220439
$549$	15227.0	1.18374
$550$	5023.64	0.389470
$551$	6099.49	0.471592
$552$	0	0
$553$	−29008.0	−2.23064
$554$	7505.20	0.575569
$555$	7.63112	0.000583645 0
$556$	−4052.25	−0.309090
$557$	1133.73	0.0862433	0.0431217	−	0.999070i	$-0.486270\pi$
$557$	1133.73	0.0862433	0.0431217	+	0.999070i	$0.486270\pi$
$558$	−11005.8	−0.834966
$559$	−3970.06	−0.300385
$560$	−55.0049	−0.00415068
$561$	982.764	0.0739614
$562$	−4712.69	−0.353724
$563$	−10869.8	−0.813687	−0.406844	−	0.913498i	$-0.633371\pi$
$563$	−10869.8	−0.813687	−0.406844	+	0.913498i	$0.633371\pi$
$564$	−1424.96	−0.106386
$565$	76.9324	0.00572845
$566$	−3254.22	−0.241670
$567$	−20650.1	−1.52949
$568$	−9741.34	−0.719608
$569$	−13837.3	−1.01949	−0.509745	−	0.860325i	$-0.670260\pi$
$569$	−13837.3	−1.01949	−0.509745	+	0.860325i	$0.670260\pi$
$570$	−2.31669	−0.000170237 0
$571$	21292.4	1.56052	0.780262	−	0.625453i	$-0.215086\pi$
$571$	21292.4	1.56052	0.780262	+	0.625453i	$0.215086\pi$
$572$	4178.25	0.305422
$573$	−1004.03	−0.0732006
$574$	−10975.5	−0.798095
$575$	0	0
$576$	−1924.15	−0.139189
$577$	−19873.8	−1.43390	−0.716949	−	0.697126i	$-0.754462\pi$
$577$	−19873.8	−1.43390	−0.716949	+	0.697126i	$0.754462\pi$
$578$	−716.073	−0.0515307
$579$	−1358.81	−0.0975307
$580$	−77.0071	−0.00551301
$581$	28804.0	2.05678
$582$	−426.260	−0.0303592
$583$	19308.8	1.37168
$584$	8150.85	0.577542
$585$	−51.5301	−0.00364189
$586$	7064.02	0.497972
$587$	−6960.49	−0.489421	−0.244710	−	0.969596i	$-0.578693\pi$
$587$	−6960.49	−0.489421	−0.244710	+	0.969596i	$0.578693\pi$
$588$	1566.84	0.109890
$589$	12172.4	0.851535
$590$	61.4987	0.00429129
$591$	1373.73	0.0956140
$592$	−4125.87	−0.286440
$593$	21647.2	1.49906	0.749531	−	0.661969i	$-0.230279\pi$
$593$	21647.2	1.49906	0.749531	+	0.661969i	$0.230279\pi$
$594$	−1089.32	−0.0752448
$595$	−160.648	−0.0110688
$596$	20820.9	1.43097
$597$	−425.869	−0.0291954
$598$	0	0
$599$	17123.4	1.16802	0.584010	−	0.811747i	$-0.301483\pi$
$599$	17123.4	1.16802	0.584010	+	0.811747i	$0.301483\pi$
$600$	−1217.17	−0.0828176
$601$	−11631.2	−0.789426	−0.394713	−	0.918805i	$-0.629156\pi$
$601$	−11631.2	−0.789426	−0.394713	+	0.918805i	$0.629156\pi$
$602$	6826.99	0.462205
$603$	−2153.03	−0.145403
$604$	−9039.47	−0.608958
$605$	−38.7411	−0.00260339
$606$	−987.841	−0.0662183
$607$	861.311	0.0575940	0.0287970	−	0.999585i	$-0.490832\pi$
$607$	861.311	0.0575940	0.0287970	+	0.999585i	$0.490832\pi$
$608$	7495.99	0.500004
$609$	2218.33	0.147605
$610$	−64.7139	−0.00429539
$611$	−10669.4	−0.706447
$612$	10867.3	0.717788
$613$	−16744.9	−1.10330	−0.551649	−	0.834076i	$-0.686001\pi$
$613$	−16744.9	−1.10330	−0.551649	+	0.834076i	$0.686001\pi$
$614$	−1828.89	−0.120208
$615$	−11.5626	−0.000758127 0
$616$	−16556.9	−1.08295
$617$	−20420.2	−1.33239	−0.666197	−	0.745776i	$-0.732079\pi$
$617$	−20420.2	−1.33239	−0.666197	+	0.745776i	$0.732079\pi$
$618$	−707.626	−0.0460597
$619$	−30237.8	−1.96342	−0.981712	−	0.190374i	$-0.939030\pi$
$619$	−30237.8	−1.96342	−0.981712	+	0.190374i	$0.939030\pi$
$620$	−153.678	−0.00995463
$621$	0	0
$622$	4878.62	0.314493
$623$	35124.1	2.25878
$624$	−264.908	−0.0169949
$625$	15622.4	0.999834
$626$	−5573.78	−0.355868
$627$	599.545	0.0381875
$628$	−163.891	−0.0104139
$629$	−12050.1	−0.763861
$630$	88.6122	0.00560380
$631$	300.989	0.0189892	0.00949460	−	0.999955i	$-0.496978\pi$
$631$	300.989	0.0189892	0.00949460	+	0.999955i	$0.496978\pi$
$632$	−19217.5	−1.20955
$633$	−1366.61	−0.0858100
$634$	−438.458	−0.0274659
$635$	119.236	0.00745152
$636$	−2030.18	−0.126575
$637$	11731.7	0.729715
$638$	−6065.63	−0.376396
$639$	−13492.4	−0.835291
$640$	−115.265	−0.00711913
$641$	6834.11	0.421109	0.210555	−	0.977582i	$-0.432473\pi$
$641$	6834.11	0.421109	0.210555	+	0.977582i	$0.432473\pi$
$642$	−1319.23	−0.0810994
$643$	20286.5	1.24420	0.622101	−	0.782937i	$-0.286279\pi$
$643$	20286.5	1.24420	0.622101	+	0.782937i	$0.286279\pi$
$644$	0	0
$645$	7.19219	0.000439058 0
$646$	3658.21	0.222803
$647$	2607.28	0.158428	0.0792139	−	0.996858i	$-0.474759\pi$
$647$	2607.28	0.158428	0.0792139	+	0.996858i	$0.474759\pi$
$648$	−13680.5	−0.829354
$649$	−15915.5	−0.962618
$650$	−3954.92	−0.238654
$651$	4426.99	0.266524
$652$	−17872.6	−1.07354
$653$	−20705.4	−1.24083	−0.620417	−	0.784272i	$-0.713037\pi$
$653$	−20705.4	−1.24083	−0.620417	+	0.784272i	$0.713037\pi$
$654$	−591.100	−0.0353423
$655$	−71.6907	−0.00427662
$656$	6251.46	0.372071
$657$	11289.5	0.670387
$658$	18347.4	1.08701
$659$	10735.7	0.634604	0.317302	−	0.948325i	$-0.397223\pi$
$659$	10735.7	0.634604	0.317302	+	0.948325i	$0.397223\pi$
$660$	−7.56936	−0.000446420 0
$661$	16139.5	0.949702	0.474851	−	0.880066i	$-0.342502\pi$
$661$	16139.5	0.949702	0.474851	+	0.880066i	$0.342502\pi$
$662$	10462.6	0.614260
$663$	−773.694	−0.0453209
$664$	19082.4	1.11527
$665$	−98.0052	−0.00571500
$666$	6646.72	0.386720
$667$	0	0
$668$	−1226.83	−0.0710591
$669$	3124.79	0.180585
$670$	9.15025	0.000527619 0
$671$	16747.6	0.963538
$672$	2726.23	0.156498
$673$	−29417.9	−1.68496	−0.842478	−	0.538731i	$-0.818904\pi$
$673$	−29417.9	−1.68496	−0.842478	+	0.538731i	$0.818904\pi$
$674$	2653.87	0.151666
$675$	−3387.74	−0.193176
$676$	10185.4	0.579507
$677$	3827.11	0.217264	0.108632	−	0.994082i	$-0.465353\pi$
$677$	3827.11	0.217264	0.108632	+	0.994082i	$0.465353\pi$
$678$	−637.140	−0.0360903
$679$	−18032.5	−1.01918
$680$	−106.428	−0.00600196
$681$	−2742.35	−0.154313
$682$	−12104.8	−0.679643
$683$	7395.51	0.414321	0.207161	−	0.978307i	$-0.433578\pi$
$683$	7395.51	0.414321	0.207161	+	0.978307i	$0.433578\pi$
$684$	6629.74	0.370606
$685$	3.83586	0.000213957 0
$686$	−6514.53	−0.362574
$687$	−3405.92	−0.189147
$688$	−3888.56	−0.215479
$689$	−15201.1	−0.840515
$690$	0	0
$691$	22060.6	1.21451	0.607254	−	0.794508i	$-0.292271\pi$
$691$	22060.6	1.21451	0.607254	+	0.794508i	$0.292271\pi$
$692$	8672.11	0.476393
$693$	−22932.3	−1.25704
$694$	1031.68	0.0564295
$695$	54.9668	0.00300001
$696$	1469.63	0.0800375
$697$	18258.1	0.992218
$698$	−15995.7	−0.867399
$699$	−344.717	−0.0186529
$700$	−22345.0	−1.20652
$701$	−4722.66	−0.254454	−0.127227	−	0.991874i	$-0.540608\pi$
$701$	−4722.66	−0.254454	−0.127227	+	0.991874i	$0.540608\pi$
$702$	857.583	0.0461074
$703$	−7351.28	−0.394394
$704$	−2116.29	−0.113297
$705$	19.3288	0.00103258
$706$	6995.29	0.372905
$707$	−41789.7	−2.22300
$708$	1673.41	0.0888285
$709$	−21211.9	−1.12359	−0.561797	−	0.827275i	$-0.689890\pi$
$709$	−21211.9	−1.12359	−0.561797	+	0.827275i	$0.689890\pi$
$710$	57.3418	0.00303099
$711$	−26617.6	−1.40399
$712$	23269.4	1.22480
$713$	0	0
$714$	1330.46	0.0697356
$715$	−56.6759	−0.00296442
$716$	−576.777	−0.0301050
$717$	2057.13	0.107148
$718$	1596.35	0.0829738
$719$	−21052.7	−1.09198	−0.545989	−	0.837792i	$-0.683846\pi$
$719$	−21052.7	−1.09198	−0.545989	+	0.837792i	$0.683846\pi$
$720$	−50.4722	−0.00261249
$721$	−29935.5	−1.54626
$722$	−7139.61	−0.368018
$723$	852.264	0.0438396
$724$	−3612.01	−0.185413
$725$	−18863.8	−0.966323
$726$	320.847	0.0164018
$727$	7899.08	0.402972	0.201486	−	0.979491i	$-0.435423\pi$
$727$	7899.08	0.402972	0.201486	+	0.979491i	$0.435423\pi$
$728$	13034.6	0.663591
$729$	−18579.4	−0.943930
$730$	−47.9795	−0.00243261
$731$	−11357.0	−0.574628
$732$	−1760.89	−0.0889133
$733$	−7733.34	−0.389683	−0.194841	−	0.980835i	$-0.562419\pi$
$733$	−7733.34	−0.389683	−0.194841	+	0.980835i	$0.562419\pi$
$734$	7274.91	0.365833
$735$	−21.2533	−0.00106659
$736$	0	0
$737$	−2368.03	−0.118355
$738$	−10071.0	−0.502330
$739$	−23223.3	−1.15600	−0.577998	−	0.816038i	$-0.696166\pi$
$739$	−23223.3	−1.15600	−0.577998	+	0.816038i	$0.696166\pi$
$740$	92.8112	0.00461055
$741$	−472.000	−0.0233999
$742$	26140.1	1.29330
$743$	12610.3	0.622649	0.311324	−	0.950304i	$-0.399227\pi$
$743$	12610.3	0.622649	0.311324	+	0.950304i	$0.399227\pi$
$744$	2932.84	0.144521
$745$	−282.425	−0.0138889
$746$	−6559.18	−0.321915
$747$	26430.4	1.29456
$748$	11952.6	0.584263
$749$	−55808.7	−2.72257
$750$	14.3300	0.000697674 0
$751$	17345.2	0.842791	0.421395	−	0.906877i	$-0.361540\pi$
$751$	17345.2	0.842791	0.421395	+	0.906877i	$0.361540\pi$
$752$	−10450.4	−0.506765
$753$	323.654	0.0156635
$754$	4775.24	0.230642
$755$	122.616	0.00591052
$756$	4845.28	0.233097
$757$	14515.8	0.696941	0.348470	−	0.937320i	$-0.386701\pi$
$757$	14515.8	0.696941	0.348470	+	0.937320i	$0.386701\pi$
$758$	7707.17	0.369310
$759$	0	0
$760$	−64.9276	−0.00309891
$761$	−20402.4	−0.971861	−0.485930	−	0.873998i	$-0.661519\pi$
$761$	−20402.4	−0.971861	−0.485930	+	0.873998i	$0.661519\pi$
$762$	−987.486	−0.0469460
$763$	−25005.9	−1.18647
$764$	−12211.2	−0.578254
$765$	−147.410	−0.00696683
$766$	8424.16	0.397359
$767$	12529.7	0.589859
$768$	1244.84	0.0584887
$769$	−29625.6	−1.38924	−0.694621	−	0.719376i	$-0.744428\pi$
$769$	−29625.6	−1.38924	−0.694621	+	0.719376i	$0.744428\pi$
$770$	97.4610	0.00456136
$771$	−2193.62	−0.102466
$772$	−16526.1	−0.770451
$773$	24097.7	1.12126	0.560630	−	0.828067i	$-0.310559\pi$
$773$	24097.7	1.12126	0.560630	+	0.828067i	$0.310559\pi$
$774$	6264.41	0.290917
$775$	−37645.3	−1.74485
$776$	−11946.4	−0.552643
$777$	−2673.60	−0.123442
$778$	16282.7	0.750338
$779$	11138.6	0.512298
$780$	5.95908	0.000273550 0
$781$	−14839.8	−0.679908
$782$	0	0
$783$	4090.41	0.186692
$784$	11490.9	0.523456
$785$	2.22310	0.000101077 0
$786$	593.729	0.0269435
$787$	−4460.02	−0.202011	−0.101006	−	0.994886i	$-0.532206\pi$
$787$	−4460.02	−0.202011	−0.101006	+	0.994886i	$0.532206\pi$
$788$	16707.6	0.755310
$789$	1978.51	0.0892736
$790$	113.123	0.00509460
$791$	−26953.6	−1.21158
$792$	−15192.5	−0.681618
$793$	−13184.8	−0.590422
$794$	4385.76	0.196026
$795$	27.5384	0.00122854
$796$	−5179.50	−0.230631
$797$	11101.1	0.493377	0.246688	−	0.969095i	$-0.420658\pi$
$797$	11101.1	0.493377	0.246688	+	0.969095i	$0.420658\pi$
$798$	811.661	0.0360056
$799$	−30521.6	−1.35141
$800$	−23182.7	−1.02454
$801$	32229.7	1.42170
$802$	13182.8	0.580426
$803$	12416.8	0.545679
$804$	248.982	0.0109216
$805$	0	0
$806$	9529.66	0.416461
$807$	−1135.62	−0.0495363
$808$	−27685.3	−1.20540
$809$	45548.9	1.97950	0.989749	−	0.142816i	$-0.0456156\pi$
$809$	45548.9	1.97950	0.989749	+	0.142816i	$0.0456156\pi$
$810$	80.5296	0.00349323
$811$	2491.19	0.107864	0.0539318	−	0.998545i	$-0.482825\pi$
$811$	2491.19	0.107864	0.0539318	+	0.998545i	$0.482825\pi$
$812$	26979.8	1.16602
$813$	200.398	0.00864484
$814$	7310.47	0.314781
$815$	242.433	0.0104197
$816$	−757.811	−0.0325107
$817$	−6928.44	−0.296690
$818$	−1327.22	−0.0567300
$819$	18053.8	0.770269
$820$	−140.626	−0.00598888
$821$	31242.1	1.32808	0.664042	−	0.747696i	$-0.268840\pi$
$821$	31242.1	1.32808	0.664042	+	0.747696i	$0.268840\pi$
$822$	−31.7679	−0.00134797
$823$	−20068.9	−0.850009	−0.425004	−	0.905191i	$-0.639727\pi$
$823$	−20068.9	−0.850009	−0.425004	+	0.905191i	$0.639727\pi$
$824$	−19832.0	−0.838447
$825$	−1854.20	−0.0782486
$826$	−21546.3	−0.907619
$827$	19228.6	0.808516	0.404258	−	0.914645i	$-0.367530\pi$
$827$	19228.6	0.808516	0.404258	+	0.914645i	$0.367530\pi$
$828$	0	0
$829$	5981.95	0.250617	0.125309	−	0.992118i	$-0.460008\pi$
$829$	5981.95	0.250617	0.125309	+	0.992118i	$0.460008\pi$
$830$	−112.327	−0.00469752
$831$	−2770.14	−0.115638
$832$	1666.08	0.0694242
$833$	33560.5	1.39592
$834$	−455.225	−0.0189007
$835$	16.6413	0.000689697 0
$836$	7291.78	0.301665
$837$	8162.99	0.337102
$838$	−6637.67	−0.273621
$839$	−3811.11	−0.156822	−0.0784112	−	0.996921i	$-0.524985\pi$
$839$	−3811.11	−0.156822	−0.0784112	+	0.996921i	$0.524985\pi$
$840$	−23.6136	−0.000969937 0
$841$	−1612.51	−0.0661164
$842$	−9290.53	−0.380253
$843$	1739.43	0.0710668
$844$	−16620.9	−0.677862
$845$	−138.160	−0.00562467
$846$	16835.5	0.684178
$847$	13573.1	0.550623
$848$	−14889.0	−0.602937
$849$	1201.12	0.0485540
$850$	−11313.7	−0.456537
$851$	0	0
$852$	1560.30	0.0627405
$853$	−19918.1	−0.799509	−0.399754	−	0.916622i	$-0.630905\pi$
$853$	−19918.1	−0.799509	−0.399754	+	0.916622i	$0.630905\pi$
$854$	22672.8	0.908486
$855$	−89.9290	−0.00359708
$856$	−36972.8	−1.47629
$857$	−29929.3	−1.19296	−0.596479	−	0.802629i	$-0.703434\pi$
$857$	−29929.3	−1.19296	−0.596479	+	0.802629i	$0.703434\pi$
$858$	469.379	0.0186764
$859$	−45740.1	−1.81680	−0.908401	−	0.418101i	$-0.862696\pi$
$859$	−45740.1	−1.81680	−0.908401	+	0.418101i	$0.862696\pi$
$860$	87.4728	0.00346837
$861$	4051.00	0.160346
$862$	1837.81	0.0726174
$863$	20014.0	0.789437	0.394718	−	0.918802i	$-0.370842\pi$
$863$	20014.0	0.789437	0.394718	+	0.918802i	$0.370842\pi$
$864$	5026.92	0.197939
$865$	−117.633	−0.00462385
$866$	5585.26	0.219163
$867$	264.300	0.0103530
$868$	53841.9	2.10543
$869$	−29275.6	−1.14282
$870$	−8.65088	−0.000337117 0
$871$	1864.26	0.0725238
$872$	−16566.2	−0.643352
$873$	−16546.6	−0.641485
$874$	0	0
$875$	606.215	0.0234215
$876$	−1305.54	−0.0503542
$877$	−6910.18	−0.266066	−0.133033	−	0.991112i	$-0.542472\pi$
$877$	−6910.18	−0.266066	−0.133033	+	0.991112i	$0.542472\pi$
$878$	3570.48	0.137241
$879$	−2607.30	−0.100048
$880$	−55.5124	−0.00212650
$881$	5697.86	0.217895	0.108948	−	0.994047i	$-0.465252\pi$
$881$	5697.86	0.217895	0.108948	+	0.994047i	$0.465252\pi$
$882$	−18511.7	−0.706713
$883$	−28263.8	−1.07718	−0.538591	−	0.842568i	$-0.681043\pi$
$883$	−28263.8	−1.07718	−0.538591	+	0.842568i	$0.681043\pi$
$884$	−9409.81	−0.358016
$885$	−22.6989	−0.000862166 0
$886$	3727.35	0.141335
$887$	17988.0	0.680924	0.340462	−	0.940258i	$-0.389417\pi$
$887$	17988.0	0.680924	0.340462	+	0.940258i	$0.389417\pi$
$888$	−1771.24	−0.0669356
$889$	−41774.7	−1.57602
$890$	−136.974	−0.00515886
$891$	−20840.6	−0.783599
$892$	38004.3	1.42654
$893$	−18620.0	−0.697756
$894$	2339.00	0.0875030
$895$	7.82369	0.000292198 0
$896$	40383.5	1.50571
$897$	0	0
$898$	18085.5	0.672072
$899$	45453.6	1.68628
$900$	−20503.7	−0.759396
$901$	−43485.1	−1.60788
$902$	−11076.7	−0.408885
$903$	−2519.81	−0.0928618
$904$	−17856.5	−0.656969
$905$	48.9951	0.00179961
$906$	−1015.48	−0.0372374
$907$	29565.2	1.08236	0.541178	−	0.840908i	$-0.317979\pi$
$907$	29565.2	1.08236	0.541178	+	0.840908i	$0.317979\pi$
$908$	−33353.0	−1.21901
$909$	−38346.0	−1.39918
$910$	−76.7275	−0.00279504
$911$	−3423.03	−0.124490	−0.0622449	−	0.998061i	$-0.519826\pi$
$911$	−3423.03	−0.124490	−0.0622449	+	0.998061i	$0.519826\pi$
$912$	−462.311	−0.0167858
$913$	29069.7	1.05374
$914$	8372.44	0.302993
$915$	23.8856	0.000862989 0
$916$	−41423.4	−1.49418
$917$	25117.2	0.904516
$918$	2453.25	0.0882020
$919$	−22221.6	−0.797630	−0.398815	−	0.917031i	$-0.630578\pi$
$919$	−22221.6	−0.797630	−0.398815	+	0.917031i	$0.630578\pi$
$920$	0	0
$921$	675.035	0.0241511
$922$	7394.61	0.264131
$923$	11682.8	0.416624
$924$	2651.96	0.0944189
$925$	22735.2	0.808139
$926$	9450.03	0.335364
$927$	−27468.6	−0.973234
$928$	27991.2	0.990147
$929$	−591.253	−0.0208809	−0.0104405	−	0.999945i	$-0.503323\pi$
$929$	−591.253	−0.0208809	−0.0104405	+	0.999945i	$0.503323\pi$
$930$	−17.2640	−0.000608720 0
$931$	20473.9	0.720737
$932$	−4192.52	−0.147350
$933$	−1800.68	−0.0631850
$934$	−19703.3	−0.690268
$935$	−162.130	−0.00567084
$936$	11960.5	0.417672
$937$	1385.48	0.0483048	0.0241524	−	0.999708i	$-0.492311\pi$
$937$	1385.48	0.0483048	0.0241524	+	0.999708i	$0.492311\pi$
$938$	−3205.83	−0.111593
$939$	2057.26	0.0714975
$940$	235.081	0.00815692
$941$	8337.71	0.288843	0.144422	−	0.989516i	$-0.453868\pi$
$941$	8337.71	0.288843	0.144422	+	0.989516i	$0.453868\pi$
$942$	−18.4113	−0.000636806 0
$943$	0	0
$944$	12272.5	0.423131
$945$	−65.7238	−0.00226243
$946$	6889.97	0.236800
$947$	28090.6	0.963910	0.481955	−	0.876196i	$-0.339927\pi$
$947$	28090.6	0.963910	0.481955	+	0.876196i	$0.339927\pi$
$948$	3078.13	0.105457
$949$	−9775.32	−0.334373
$950$	−6902.04	−0.235718
$951$	161.833	0.00551819
$952$	37287.6	1.26943
$953$	31766.4	1.07976	0.539882	−	0.841741i	$-0.318469\pi$
$953$	31766.4	1.07976	0.539882	+	0.841741i	$0.318469\pi$
$954$	23986.0	0.814020
$955$	165.639	0.00561251
$956$	25019.2	0.846422
$957$	2238.80	0.0756218
$958$	1604.42	0.0541092
$959$	−1343.91	−0.0452525
$960$	−3.01828	−0.000101474 0
$961$	60918.0	2.04485
$962$	−5755.26	−0.192887
$963$	−51209.8	−1.71362
$964$	10365.4	0.346314
$965$	224.169	0.00747797
$966$	0	0
$967$	−35673.3	−1.18633	−0.593163	−	0.805082i	$-0.702121\pi$
$967$	−35673.3	−1.18633	−0.593163	+	0.805082i	$0.702121\pi$
$968$	8992.07	0.298570
$969$	−1350.23	−0.0447634
$970$	70.3218	0.00232773
$971$	34889.7	1.15310	0.576552	−	0.817061i	$-0.304398\pi$
$971$	34889.7	1.15310	0.576552	+	0.817061i	$0.304398\pi$
$972$	6679.52	0.220417
$973$	−19257.8	−0.634510
$974$	−5011.98	−0.164881
$975$	1459.75	0.0479480
$976$	−12914.1	−0.423535
$977$	−21394.9	−0.700598	−0.350299	−	0.936638i	$-0.613920\pi$
$977$	−21394.9	−0.700598	−0.350299	+	0.936638i	$0.613920\pi$
$978$	−2007.78	−0.0656461
$979$	35448.2	1.15723
$980$	−258.487	−0.00842557
$981$	−22945.3	−0.746776
$982$	3822.13	0.124205
$983$	5136.74	0.166670	0.0833350	−	0.996522i	$-0.473443\pi$
$983$	5136.74	0.166670	0.0833350	+	0.996522i	$0.473443\pi$
$984$	2683.75	0.0869460
$985$	−226.630	−0.00733101
$986$	13660.4	0.441211
$987$	−6771.94	−0.218392
$988$	−5740.55	−0.184850
$989$	0	0
$990$	89.4297	0.00287097
$991$	−22185.1	−0.711134	−0.355567	−	0.934651i	$-0.615712\pi$
$991$	−22185.1	−0.711134	−0.355567	+	0.934651i	$0.615712\pi$
$992$	55860.4	1.78787
$993$	−3861.70	−0.123411
$994$	−20090.0	−0.641061
$995$	70.2573	0.00223850
$996$	−3056.48	−0.0972372
$997$	35743.5	1.13541	0.567707	−	0.823231i	$-0.307831\pi$
$997$	35743.5	1.13541	0.567707	+	0.823231i	$0.307831\pi$
$998$	2569.41	0.0814963
$999$	−4929.88	−0.156131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	529.4.a.k.1.8	yes	12
23.22	odd	2	inner	529.4.a.k.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
529.4.a.k.1.7	✓	12	23.22	odd	2	inner
529.4.a.k.1.8	yes	12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 529 = 23^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	529.a (trivial)

\(f(q)\)	\(=\)	\(q+1.36628 q^{2} -0.504290 q^{3} -6.13327 q^{4} +0.0831947 q^{5} -0.689003 q^{6} -29.1476 q^{7} -19.3101 q^{8} -26.7457 q^{9} +0.113668 q^{10} -29.4165 q^{11} +3.09295 q^{12} +23.1585 q^{13} -39.8239 q^{14} -0.0419542 q^{15} +22.6831 q^{16} +66.2488 q^{17} -36.5422 q^{18} +40.4157 q^{19} -0.510255 q^{20} +14.6988 q^{21} -40.1913 q^{22} +9.73786 q^{24} -124.993 q^{25} +31.6411 q^{26} +27.1034 q^{27} +178.770 q^{28} +150.919 q^{29} -0.0573214 q^{30} +301.179 q^{31} +185.472 q^{32} +14.8345 q^{33} +90.5146 q^{34} -2.42493 q^{35} +164.039 q^{36} -181.892 q^{37} +55.2193 q^{38} -11.6786 q^{39} -1.60649 q^{40} +275.600 q^{41} +20.0828 q^{42} -171.429 q^{43} +180.419 q^{44} -2.22510 q^{45} -460.713 q^{47} -11.4389 q^{48} +506.583 q^{49} -170.776 q^{50} -33.4086 q^{51} -142.038 q^{52} -656.391 q^{53} +37.0309 q^{54} -2.44730 q^{55} +562.842 q^{56} -20.3812 q^{57} +206.198 q^{58} +541.040 q^{59} +0.257317 q^{60} -569.326 q^{61} +411.496 q^{62} +779.573 q^{63} +71.9423 q^{64} +1.92667 q^{65} +20.2681 q^{66} +80.5001 q^{67} -406.321 q^{68} -3.31314 q^{70} +504.470 q^{71} +516.461 q^{72} -422.104 q^{73} -248.516 q^{74} +63.0327 q^{75} -247.881 q^{76} +857.422 q^{77} -15.9563 q^{78} +995.209 q^{79} +1.88712 q^{80} +708.466 q^{81} +376.547 q^{82} -988.210 q^{83} -90.1520 q^{84} +5.51154 q^{85} -234.221 q^{86} -76.1068 q^{87} +568.035 q^{88} -1205.04 q^{89} -3.04012 q^{90} -675.016 q^{91} -151.882 q^{93} -629.464 q^{94} +3.36237 q^{95} -93.5317 q^{96} +618.663 q^{97} +692.137 q^{98} +786.765 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 56 q^{4} - 6 q^{6} + 138 q^{8} + 204 q^{9} + 30 q^{12} + 160 q^{13} + 144 q^{16} + 478 q^{18} - 1188 q^{24} + 400 q^{25} + 1554 q^{26} - 684 q^{27} - 44 q^{29} + 1076 q^{31} + 248 q^{32}+ \cdots - 3512 q^{98}+O(q^{100}) \) 12 * q + 4 * q^2 + 56 * q^4 - 6 * q^6 + 138 * q^8 + 204 * q^9 + 30 * q^12 + 160 * q^13 + 144 * q^16 + 478 * q^18 - 1188 * q^24 + 400 * q^25 + 1554 * q^26 - 684 * q^27 - 44 * q^29 + 1076 * q^31 + 248 * q^32 + 1048 * q^35 + 2006 * q^36 + 468 * q^39 + 92 * q^41 + 204 * q^47 - 2282 * q^48 + 1428 * q^49 - 1932 * q^50 + 2290 * q^52 + 6426 * q^54 + 1808 * q^55 + 2058 * q^58 + 3416 * q^59 + 2654 * q^62 - 346 * q^64 - 5116 * q^70 - 660 * q^71 + 7554 * q^72 - 5860 * q^73 + 2472 * q^75 + 2712 * q^77 + 826 * q^78 + 4548 * q^81 + 5486 * q^82 - 1436 * q^85 + 1268 * q^87 - 7544 * q^93 - 1322 * q^94 + 9688 * q^95 - 6930 * q^96 - 3512 * q^98

Introduction

L-functions

Modular forms

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Database

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