LMFDB - Embedded newform 409.4.a.b.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [409,4,Mod(1,409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("409.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 409 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	409.a (trivial)

Newform invariants

comment:select newform

sage:traces = [55] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$24.1317811923$
Analytic rank:	$0$
Dimension:	$55$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	409.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.12250 q^{2} +1.94641 q^{3} +18.2400 q^{4} +18.8054 q^{5} -9.97048 q^{6} +21.6967 q^{7} -52.4542 q^{8} -23.2115 q^{9} -96.3307 q^{10} +62.6075 q^{11} +35.5025 q^{12} +36.7490 q^{13} -111.141 q^{14} +36.6031 q^{15} +122.777 q^{16} +128.125 q^{17} +118.901 q^{18} +117.205 q^{19} +343.010 q^{20} +42.2306 q^{21} -320.707 q^{22} -146.331 q^{23} -102.097 q^{24} +228.644 q^{25} -188.246 q^{26} -97.7322 q^{27} +395.746 q^{28} -5.75419 q^{29} -187.499 q^{30} -124.600 q^{31} -209.289 q^{32} +121.860 q^{33} -656.320 q^{34} +408.015 q^{35} -423.377 q^{36} -38.7969 q^{37} -600.381 q^{38} +71.5286 q^{39} -986.423 q^{40} -377.297 q^{41} -216.326 q^{42} -288.174 q^{43} +1141.96 q^{44} -436.502 q^{45} +749.578 q^{46} +257.894 q^{47} +238.974 q^{48} +127.745 q^{49} -1171.23 q^{50} +249.384 q^{51} +670.300 q^{52} -58.4100 q^{53} +500.633 q^{54} +1177.36 q^{55} -1138.08 q^{56} +228.129 q^{57} +29.4758 q^{58} +26.3658 q^{59} +667.639 q^{60} -878.809 q^{61} +638.264 q^{62} -503.611 q^{63} +89.8703 q^{64} +691.080 q^{65} -624.227 q^{66} -783.749 q^{67} +2337.00 q^{68} -284.820 q^{69} -2090.05 q^{70} +347.770 q^{71} +1217.54 q^{72} -722.580 q^{73} +198.737 q^{74} +445.035 q^{75} +2137.81 q^{76} +1358.37 q^{77} -366.405 q^{78} -340.400 q^{79} +2308.87 q^{80} +436.483 q^{81} +1932.70 q^{82} -500.823 q^{83} +770.285 q^{84} +2409.45 q^{85} +1476.17 q^{86} -11.2000 q^{87} -3284.03 q^{88} -693.112 q^{89} +2235.98 q^{90} +797.330 q^{91} -2669.07 q^{92} -242.523 q^{93} -1321.06 q^{94} +2204.09 q^{95} -407.363 q^{96} -387.479 q^{97} -654.371 q^{98} -1453.21 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 55 q + 13 q^{2} + 10 q^{3} + 257 q^{4} + 60 q^{5} + 48 q^{6} + 72 q^{7} + 156 q^{8} + 639 q^{9} + 21 q^{10} + 362 q^{11} + 21 q^{12} + 48 q^{13} + 279 q^{14} + 512 q^{15} + 1237 q^{16} + 270 q^{17} + 360 q^{18}+ \cdots + 9542 q^{99}+O(q^{100}) $ 55 * q + 13 * q^2 + 10 * q^3 + 257 * q^4 + 60 * q^5 + 48 * q^6 + 72 * q^7 + 156 * q^8 + 639 * q^9 + 21 * q^10 + 362 * q^11 + 21 * q^12 + 48 * q^13 + 279 * q^14 + 512 * q^15 + 1237 * q^16 + 270 * q^17 + 360 * q^18 + 294 * q^19 + 732 * q^20 + 488 * q^21 + 123 * q^22 + 1212 * q^23 + 427 * q^24 + 1723 * q^25 + 659 * q^26 + 376 * q^27 + 622 * q^28 + 730 * q^29 + 106 * q^30 + 972 * q^31 + 1506 * q^32 + 358 * q^33 + 505 * q^34 + 1352 * q^35 + 4215 * q^36 + 692 * q^37 + 781 * q^38 + 1920 * q^39 + 792 * q^40 + 1050 * q^41 + 1491 * q^42 + 480 * q^43 + 3920 * q^44 + 1524 * q^45 + 752 * q^46 + 3432 * q^47 + 152 * q^48 + 3729 * q^49 + 3225 * q^50 + 2088 * q^51 + 984 * q^52 + 1558 * q^53 + 2433 * q^54 + 1662 * q^55 + 2465 * q^56 + 1188 * q^57 + 2048 * q^58 + 3468 * q^59 + 4144 * q^60 + 392 * q^61 + 1966 * q^62 + 2534 * q^63 + 8120 * q^64 + 1452 * q^65 + 1593 * q^66 + 1814 * q^67 + 3496 * q^68 + 2302 * q^69 + 1744 * q^70 + 8798 * q^71 + 3545 * q^72 + 918 * q^73 + 2759 * q^74 + 1898 * q^75 + 2719 * q^76 + 1830 * q^77 + 2899 * q^78 + 2918 * q^79 + 9155 * q^80 + 9571 * q^81 + 1581 * q^82 + 4192 * q^83 + 6815 * q^84 + 590 * q^85 + 7177 * q^86 + 2360 * q^87 + 1826 * q^88 + 4878 * q^89 + 175 * q^90 + 2926 * q^91 + 10822 * q^92 + 4040 * q^93 + 1886 * q^94 + 8912 * q^95 + 4614 * q^96 + 2264 * q^97 + 2870 * q^98 + 9542 * q^99

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 $

$2$	−5.12250	−1.81108	−0.905538	−	0.424265i	$-0.860532\pi$
$2$	−5.12250	−1.81108	−0.905538	+	0.424265i	$0.860532\pi$
$3$	1.94641	0.374587	0.187294	−	0.982304i	$-0.440028\pi$
$3$	1.94641	0.374587	0.187294	+	0.982304i	$0.440028\pi$
$4$	18.2400	2.28000
$5$	18.8054	1.68201	0.841004	−	0.541029i	$-0.181965\pi$
$5$	18.8054	1.68201	0.841004	+	0.541029i	$0.181965\pi$
$6$	−9.97048	−0.678405
$7$	21.6967	1.17151	0.585755	−	0.810489i	$-0.300798\pi$
$7$	21.6967	1.17151	0.585755	+	0.810489i	$0.300798\pi$
$8$	−52.4542	−2.31817
$9$	−23.2115	−0.859685
$10$	−96.3307	−3.04624
$11$	62.6075	1.71608	0.858040	−	0.513583i	$-0.171682\pi$
$11$	62.6075	1.71608	0.858040	+	0.513583i	$0.171682\pi$
$12$	35.5025	0.854057
$13$	36.7490	0.784025	0.392013	−	0.919960i	$-0.371779\pi$
$13$	36.7490	0.784025	0.392013	+	0.919960i	$0.371779\pi$
$14$	−111.141	−2.12169
$15$	36.6031	0.630058
$16$	122.777	1.91838
$17$	128.125	1.82794	0.913968	−	0.405787i	$-0.133002\pi$
$17$	128.125	1.82794	0.913968	+	0.405787i	$0.133002\pi$
$18$	118.901	1.55695
$19$	117.205	1.41519	0.707595	−	0.706618i	$-0.249780\pi$
$19$	117.205	1.41519	0.707595	+	0.706618i	$0.249780\pi$
$20$	343.010	3.83497
$21$	42.2306	0.438832
$22$	−320.707	−3.10795
$23$	−146.331	−1.32661	−0.663305	−	0.748349i	$-0.730847\pi$
$23$	−146.331	−1.32661	−0.663305	+	0.748349i	$0.730847\pi$
$24$	−102.097	−0.868356
$25$	228.644	1.82915
$26$	−188.246	−1.41993
$27$	−97.7322	−0.696614
$28$	395.746	2.67104
$29$	−5.75419	−0.0368457	−0.0184229	−	0.999830i	$-0.505865\pi$
$29$	−5.75419	−0.0368457	−0.0184229	+	0.999830i	$0.505865\pi$
$30$	−187.499	−1.14108
$31$	−124.600	−0.721898	−0.360949	−	0.932586i	$-0.617547\pi$
$31$	−124.600	−0.721898	−0.360949	+	0.932586i	$0.617547\pi$
$32$	−209.289	−1.15617
$33$	121.860	0.642821
$34$	−656.320	−3.31053
$35$	408.015	1.97049
$36$	−423.377	−1.96008
$37$	−38.7969	−0.172383	−0.0861914	−	0.996279i	$-0.527470\pi$
$37$	−38.7969	−0.172383	−0.0861914	+	0.996279i	$0.527470\pi$
$38$	−600.381	−2.56302
$39$	71.5286	0.293686
$40$	−986.423	−3.89918
$41$	−377.297	−1.43717	−0.718585	−	0.695439i	$-0.755210\pi$
$41$	−377.297	−1.43717	−0.718585	+	0.695439i	$0.755210\pi$
$42$	−216.326	−0.794758
$43$	−288.174	−1.02200	−0.511002	−	0.859579i	$-0.670726\pi$
$43$	−288.174	−1.02200	−0.511002	+	0.859579i	$0.670726\pi$
$44$	1141.96	3.91265
$45$	−436.502	−1.44600
$46$	749.578	2.40259
$47$	257.894	0.800376	0.400188	−	0.916433i	$-0.368945\pi$
$47$	257.894	0.800376	0.400188	+	0.916433i	$0.368945\pi$
$48$	238.974	0.718602
$49$	127.745	0.372433
$50$	−1171.23	−3.31273
$51$	249.384	0.684721
$52$	670.300	1.78757
$53$	−58.4100	−0.151382	−0.0756908	−	0.997131i	$-0.524116\pi$
$53$	−58.4100	−0.151382	−0.0756908	+	0.997131i	$0.524116\pi$
$54$	500.633	1.26162
$55$	1177.36	2.88646
$56$	−1138.08	−2.71576
$57$	228.129	0.530112
$58$	29.4758	0.0667304
$59$	26.3658	0.0581786	0.0290893	−	0.999577i	$-0.490739\pi$
$59$	26.3658	0.0581786	0.0290893	+	0.999577i	$0.490739\pi$
$60$	667.639	1.43653
$61$	−878.809	−1.84459	−0.922294	−	0.386488i	$-0.873688\pi$
$61$	−878.809	−1.84459	−0.922294	+	0.386488i	$0.873688\pi$
$62$	638.264	1.30741
$63$	−503.611	−1.00713
$64$	89.8703	0.175528
$65$	691.080	1.31874
$66$	−624.227	−1.16420
$67$	−783.749	−1.42911	−0.714554	−	0.699580i	$-0.753370\pi$
$67$	−783.749	−1.42911	−0.714554	+	0.699580i	$0.753370\pi$
$68$	2337.00	4.16769
$69$	−284.820	−0.496931
$70$	−2090.05	−3.56870
$71$	347.770	0.581306	0.290653	−	0.956828i	$-0.406127\pi$
$71$	347.770	0.581306	0.290653	+	0.956828i	$0.406127\pi$
$72$	1217.54	1.99289
$73$	−722.580	−1.15851	−0.579257	−	0.815145i	$-0.696657\pi$
$73$	−722.580	−1.15851	−0.579257	+	0.815145i	$0.696657\pi$
$74$	198.737	0.312198
$75$	445.035	0.685177
$76$	2137.81	3.22663
$77$	1358.37	2.01040
$78$	−366.405	−0.531887
$79$	−340.400	−0.484785	−0.242392	−	0.970178i	$-0.577932\pi$
$79$	−340.400	−0.484785	−0.242392	+	0.970178i	$0.577932\pi$
$80$	2308.87	3.22674
$81$	436.483	0.598742
$82$	1932.70	2.60282
$83$	−500.823	−0.662319	−0.331159	−	0.943575i	$-0.607440\pi$
$83$	−500.823	−0.662319	−0.331159	+	0.943575i	$0.607440\pi$
$84$	770.285	1.00054
$85$	2409.45	3.07460
$86$	1476.17	1.85093
$87$	−11.2000	−0.0138019
$88$	−3284.03	−3.97816
$89$	−693.112	−0.825503	−0.412751	−	0.910844i	$-0.635432\pi$
$89$	−693.112	−0.825503	−0.412751	+	0.910844i	$0.635432\pi$
$90$	2235.98	2.61881
$91$	797.330	0.918493
$92$	−2669.07	−3.02467
$93$	−242.523	−0.270414
$94$	−1321.06	−1.44954
$95$	2204.09	2.38036
$96$	−407.363	−0.433086
$97$	−387.479	−0.405593	−0.202796	−	0.979221i	$-0.565003\pi$
$97$	−387.479	−0.405593	−0.202796	+	0.979221i	$0.565003\pi$
$98$	−654.371	−0.674505
$99$	−1453.21	−1.47529
$100$	4170.46	4.17046
$101$	941.265	0.927320	0.463660	−	0.886013i	$-0.346536\pi$
$101$	941.265	0.927320	0.463660	+	0.886013i	$0.346536\pi$
$102$	−1277.47	−1.24008
$103$	−551.594	−0.527672	−0.263836	−	0.964568i	$-0.584988\pi$
$103$	−551.594	−0.527672	−0.263836	+	0.964568i	$0.584988\pi$
$104$	−1927.64	−1.81750
$105$	794.164	0.738119
$106$	299.205	0.274164
$107$	549.783	0.496724	0.248362	−	0.968667i	$-0.420108\pi$
$107$	549.783	0.496724	0.248362	+	0.968667i	$0.420108\pi$
$108$	−1782.63	−1.58828
$109$	1334.04	1.17227	0.586135	−	0.810213i	$-0.300649\pi$
$109$	1334.04	1.17227	0.586135	+	0.810213i	$0.300649\pi$
$110$	−6031.03	−5.22760
$111$	−75.5146	−0.0645724
$112$	2663.84	2.24740
$113$	−224.426	−0.186833	−0.0934167	−	0.995627i	$-0.529779\pi$
$113$	−224.426	−0.186833	−0.0934167	+	0.995627i	$0.529779\pi$
$114$	−1168.59	−0.960073
$115$	−2751.81	−2.23137
$116$	−104.956	−0.0840081
$117$	−852.998	−0.674015
$118$	−135.059	−0.105366
$119$	2779.89	2.14144
$120$	−1919.99	−1.46058
$121$	2588.70	1.94493
$122$	4501.69	3.34069
$123$	−734.376	−0.538345
$124$	−2272.70	−1.64592
$125$	1949.07	1.39464
$126$	2579.75	1.82399
$127$	−2407.26	−1.68196	−0.840982	−	0.541064i	$-0.818022\pi$
$127$	−2407.26	−1.68196	−0.840982	+	0.541064i	$0.818022\pi$
$128$	1213.95	0.838276
$129$	−560.906	−0.382830
$130$	−3540.05	−2.38833
$131$	−527.365	−0.351726	−0.175863	−	0.984415i	$-0.556272\pi$
$131$	−527.365	−0.351726	−0.175863	+	0.984415i	$0.556272\pi$
$132$	2222.72	1.46563
$133$	2542.95	1.65791
$134$	4014.75	2.58822
$135$	−1837.90	−1.17171
$136$	−6720.70	−4.23746
$137$	1955.60	1.21955	0.609775	−	0.792574i	$-0.291260\pi$
$137$	1955.60	1.21955	0.609775	+	0.792574i	$0.291260\pi$
$138$	1458.99	0.899980
$139$	77.2170	0.0471184	0.0235592	−	0.999722i	$-0.492500\pi$
$139$	77.2170	0.0471184	0.0235592	+	0.999722i	$0.492500\pi$
$140$	7442.17	4.49270
$141$	501.968	0.299811
$142$	−1781.45	−1.05279
$143$	2300.76	1.34545
$144$	−2849.83	−1.64921
$145$	−108.210	−0.0619748
$146$	3701.41	2.09816
$147$	248.644	0.139509
$148$	−707.653	−0.393032
$149$	−464.056	−0.255147	−0.127574	−	0.991829i	$-0.540719\pi$
$149$	−464.056	−0.255147	−0.127574	+	0.991829i	$0.540719\pi$
$150$	−2279.69	−1.24091
$151$	−1905.89	−1.02714	−0.513572	−	0.858046i	$-0.671678\pi$
$151$	−1905.89	−1.02714	−0.513572	+	0.858046i	$0.671678\pi$
$152$	−6147.88	−3.28065
$153$	−2973.97	−1.57145
$154$	−6958.26	−3.64099
$155$	−2343.16	−1.21424
$156$	1304.68	0.669602
$157$	−87.4100	−0.0444336	−0.0222168	−	0.999753i	$-0.507072\pi$
$157$	−87.4100	−0.0444336	−0.0222168	+	0.999753i	$0.507072\pi$
$158$	1743.70	0.877982
$159$	−113.690	−0.0567056
$160$	−3935.77	−1.94469
$161$	−3174.88	−1.55414
$162$	−2235.88	−1.08437
$163$	1250.22	0.600765	0.300383	−	0.953819i	$-0.402886\pi$
$163$	1250.22	0.600765	0.300383	+	0.953819i	$0.402886\pi$
$164$	−6881.89	−3.27674
$165$	2291.63	1.08123
$166$	2565.46	1.19951
$167$	−1155.59	−0.535462	−0.267731	−	0.963494i	$-0.586274\pi$
$167$	−1155.59	−0.535462	−0.267731	+	0.963494i	$0.586274\pi$
$168$	−2215.17	−1.01729
$169$	−846.513	−0.385304
$170$	−12342.4	−5.56834
$171$	−2720.50	−1.21662
$172$	−5256.29	−2.33017
$173$	3206.24	1.40905	0.704526	−	0.709679i	$-0.251160\pi$
$173$	3206.24	1.40905	0.704526	+	0.709679i	$0.251160\pi$
$174$	57.3720	0.0249963
$175$	4960.81	2.14287
$176$	7686.74	3.29210
$177$	51.3187	0.0217929
$178$	3550.46	1.49505
$179$	3971.93	1.65853	0.829263	−	0.558858i	$-0.188760\pi$
$179$	3971.93	1.65853	0.829263	+	0.558858i	$0.188760\pi$
$180$	−7961.78	−3.29687
$181$	865.035	0.355235	0.177617	−	0.984100i	$-0.443161\pi$
$181$	865.035	0.355235	0.177617	+	0.984100i	$0.443161\pi$
$182$	−4084.32	−1.66346
$183$	−1710.52	−0.690959
$184$	7675.65	3.07531
$185$	−729.591	−0.289949
$186$	1242.32	0.489740
$187$	8021.59	3.13688
$188$	4703.98	1.82485
$189$	−2120.46	−0.816089
$190$	−11290.4	−4.31102
$191$	4109.58	1.55685	0.778426	−	0.627736i	$-0.216018\pi$
$191$	4109.58	1.55685	0.778426	+	0.627736i	$0.216018\pi$
$192$	174.925	0.0657505
$193$	−3650.52	−1.36150	−0.680751	−	0.732515i	$-0.738346\pi$
$193$	−3650.52	−1.36150	−0.680751	+	0.732515i	$0.738346\pi$
$194$	1984.86	0.734560
$195$	1345.13	0.493982
$196$	2330.06	0.849146
$197$	−3463.14	−1.25248	−0.626240	−	0.779631i	$-0.715407\pi$
$197$	−3463.14	−1.25248	−0.626240	+	0.779631i	$0.715407\pi$
$198$	7444.08	2.67186
$199$	−836.797	−0.298085	−0.149043	−	0.988831i	$-0.547619\pi$
$199$	−836.797	−0.298085	−0.149043	+	0.988831i	$0.547619\pi$
$200$	−11993.3	−4.24028
$201$	−1525.50	−0.535325
$202$	−4821.63	−1.67945
$203$	−124.847	−0.0431651
$204$	4548.76	1.56116
$205$	−7095.24	−2.41733
$206$	2825.54	0.955654
$207$	3396.55	1.14047
$208$	4511.91	1.50406
$209$	7337.90	2.42858
$210$	−4068.10	−1.33679
$211$	−116.001	−0.0378474	−0.0189237	−	0.999821i	$-0.506024\pi$
$211$	−116.001	−0.0378474	−0.0189237	+	0.999821i	$0.506024\pi$
$212$	−1065.40	−0.345149
$213$	676.904	0.217750
$214$	−2816.26	−0.899605
$215$	−5419.24	−1.71902
$216$	5126.46	1.61487
$217$	−2703.41	−0.845710
$218$	−6833.59	−2.12307
$219$	−1406.44	−0.433965
$220$	21475.0	6.58112
$221$	4708.47	1.43315
$222$	386.823	0.116945
$223$	−2188.95	−0.657322	−0.328661	−	0.944448i	$-0.606597\pi$
$223$	−2188.95	−0.657322	−0.328661	+	0.944448i	$0.606597\pi$
$224$	−4540.87	−1.35446
$225$	−5307.17	−1.57249
$226$	1149.62	0.338370
$227$	−123.136	−0.0360037	−0.0180018	−	0.999838i	$-0.505730\pi$
$227$	−123.136	−0.0360037	−0.0180018	+	0.999838i	$0.505730\pi$
$228$	4161.06	1.20865
$229$	2272.16	0.655672	0.327836	−	0.944735i	$-0.393681\pi$
$229$	2272.16	0.655672	0.327836	+	0.944735i	$0.393681\pi$
$230$	14096.1	4.04118
$231$	2643.95	0.753071
$232$	301.831	0.0854146
$233$	−1831.90	−0.515071	−0.257535	−	0.966269i	$-0.582910\pi$
$233$	−1831.90	−0.515071	−0.257535	+	0.966269i	$0.582910\pi$
$234$	4369.48	1.22069
$235$	4849.80	1.34624
$236$	480.911	0.132647
$237$	−662.559	−0.181594
$238$	−14240.0	−3.87832
$239$	−2646.63	−0.716302	−0.358151	−	0.933664i	$-0.616593\pi$
$239$	−2646.63	−0.716302	−0.358151	+	0.933664i	$0.616593\pi$
$240$	4494.00	1.20869
$241$	2462.13	0.658090	0.329045	−	0.944314i	$-0.393273\pi$
$241$	2462.13	0.658090	0.329045	+	0.944314i	$0.393273\pi$
$242$	−13260.6	−3.52241
$243$	3488.34	0.920895
$244$	−16029.4	−4.20565
$245$	2402.29	0.626436
$246$	3761.84	0.974984
$247$	4307.15	1.10955
$248$	6535.80	1.67348
$249$	−974.807	−0.248096
$250$	−9984.10	−2.52580
$251$	411.889	0.103579	0.0517893	−	0.998658i	$-0.483508\pi$
$251$	411.889	0.103579	0.0517893	+	0.998658i	$0.483508\pi$
$252$	−9185.86	−2.29625
$253$	−9161.39	−2.27657
$254$	12331.2	3.04616
$255$	4689.78	1.15171
$256$	−6937.43	−1.69371
$257$	−3990.68	−0.968607	−0.484303	−	0.874900i	$-0.660927\pi$
$257$	−3990.68	−0.968607	−0.484303	+	0.874900i	$0.660927\pi$
$258$	2873.24	0.693333
$259$	−841.762	−0.201948
$260$	12605.3	3.00672
$261$	133.563	0.0316757
$262$	2701.42	0.637002
$263$	−956.868	−0.224346	−0.112173	−	0.993689i	$-0.535781\pi$
$263$	−956.868	−0.224346	−0.112173	+	0.993689i	$0.535781\pi$
$264$	−6392.06	−1.49017
$265$	−1098.42	−0.254625
$266$	−13026.3	−3.00260
$267$	−1349.08	−0.309223
$268$	−14295.6	−3.25836
$269$	5754.01	1.30419	0.652097	−	0.758136i	$-0.273890\pi$
$269$	5754.01	1.30419	0.652097	+	0.758136i	$0.273890\pi$
$270$	9414.61	2.12206
$271$	3500.95	0.784752	0.392376	−	0.919805i	$-0.371653\pi$
$271$	3500.95	0.784752	0.392376	+	0.919805i	$0.371653\pi$
$272$	15730.8	3.50668
$273$	1551.93	0.344055
$274$	−10017.6	−2.20870
$275$	14314.8	3.13897
$276$	−5195.10	−1.13300
$277$	7881.07	1.70949	0.854743	−	0.519051i	$-0.173715\pi$
$277$	7881.07	1.70949	0.854743	+	0.519051i	$0.173715\pi$
$278$	−395.544	−0.0853350
$279$	2892.15	0.620605
$280$	−21402.1	−4.56792
$281$	1883.35	0.399827	0.199913	−	0.979814i	$-0.435934\pi$
$281$	1883.35	0.399827	0.199913	+	0.979814i	$0.435934\pi$
$282$	−2571.33	−0.542980
$283$	3761.63	0.790126	0.395063	−	0.918654i	$-0.370723\pi$
$283$	3761.63	0.790126	0.395063	+	0.918654i	$0.370723\pi$
$284$	6343.32	1.32538
$285$	4290.06	0.891653
$286$	−11785.6	−2.43671
$287$	−8186.09	−1.68366
$288$	4857.91	0.993942
$289$	11503.0	2.34135
$290$	554.305	0.112241
$291$	−754.193	−0.151930
$292$	−13179.8	−2.64141
$293$	−300.672	−0.0599503	−0.0299751	−	0.999551i	$-0.509543\pi$
$293$	−300.672	−0.0599503	−0.0299751	+	0.999551i	$0.509543\pi$
$294$	−1273.68	−0.252661
$295$	495.820	0.0978568
$296$	2035.06	0.399613
$297$	−6118.77	−1.19544
$298$	2377.13	0.462091
$299$	−5377.50	−1.04010
$300$	8117.43	1.56220
$301$	−6252.42	−1.19729
$302$	9762.90	1.86024
$303$	1832.09	0.347362
$304$	14390.0	2.71488
$305$	−16526.4	−3.10261
$306$	15234.2	2.84601
$307$	9446.42	1.75614	0.878071	−	0.478530i	$-0.158830\pi$
$307$	9446.42	1.75614	0.878071	+	0.478530i	$0.158830\pi$
$308$	24776.7	4.58371
$309$	−1073.63	−0.197659
$310$	12002.8	2.19908
$311$	−3652.98	−0.666050	−0.333025	−	0.942918i	$-0.608069\pi$
$311$	−3652.98	−0.666050	−0.333025	+	0.942918i	$0.608069\pi$
$312$	−3751.97	−0.680813
$313$	1245.20	0.224865	0.112433	−	0.993659i	$-0.464136\pi$
$313$	1245.20	0.224865	0.112433	+	0.993659i	$0.464136\pi$
$314$	447.757	0.0804726
$315$	−9470.63	−1.69400
$316$	−6208.89	−1.10531
$317$	−10083.5	−1.78657	−0.893286	−	0.449488i	$-0.851606\pi$
$317$	−10083.5	−1.78657	−0.893286	+	0.449488i	$0.851606\pi$
$318$	582.376	0.102698
$319$	−360.255	−0.0632302
$320$	1690.05	0.295239
$321$	1070.10	0.186066
$322$	16263.3	2.81466
$323$	15016.9	2.58688
$324$	7961.44	1.36513
$325$	8402.43	1.43410
$326$	−6404.24	−1.08803
$327$	2596.58	0.439117
$328$	19790.8	3.33160
$329$	5595.43	0.937648
$330$	−11738.9	−1.95819
$331$	−2550.14	−0.423469	−0.211734	−	0.977327i	$-0.567911\pi$
$331$	−2550.14	−0.423469	−0.211734	+	0.977327i	$0.567911\pi$
$332$	−9134.99	−1.51008
$333$	900.533	0.148195
$334$	5919.50	0.969762
$335$	−14738.7	−2.40377
$336$	5184.93	0.841849
$337$	−11928.5	−1.92816	−0.964079	−	0.265616i	$-0.914425\pi$
$337$	−11928.5	−1.92816	−0.964079	+	0.265616i	$0.914425\pi$
$338$	4336.26	0.697815
$339$	−436.824	−0.0699854
$340$	43948.2	7.01008
$341$	−7800.90	−1.23883
$342$	13935.7	2.20339
$343$	−4670.32	−0.735200
$344$	15116.0	2.36918
$345$	−5356.15	−0.835842
$346$	−16423.9	−2.55190
$347$	−5778.52	−0.893970	−0.446985	−	0.894542i	$-0.647502\pi$
$347$	−5778.52	−0.893970	−0.446985	+	0.894542i	$0.647502\pi$
$348$	−204.288	−0.0314683
$349$	−2629.49	−0.403305	−0.201652	−	0.979457i	$-0.564631\pi$
$349$	−2629.49	−0.403305	−0.201652	+	0.979457i	$0.564631\pi$
$350$	−25411.7	−3.88090
$351$	−3591.56	−0.546163
$352$	−13103.1	−1.98408
$353$	−11040.9	−1.66472	−0.832360	−	0.554235i	$-0.813011\pi$
$353$	−11040.9	−1.66472	−0.832360	+	0.554235i	$0.813011\pi$
$354$	−262.880	−0.0394687
$355$	6539.97	0.977762
$356$	−12642.3	−1.88214
$357$	5410.80	0.802157
$358$	−20346.2	−3.00372
$359$	12954.3	1.90446	0.952229	−	0.305384i	$-0.0987849\pi$
$359$	12954.3	1.90446	0.952229	+	0.305384i	$0.0987849\pi$
$360$	22896.3	3.35206
$361$	6877.95	1.00276
$362$	−4431.14	−0.643357
$363$	5038.68	0.728545
$364$	14543.3	2.09416
$365$	−13588.4	−1.94863
$366$	8762.15	1.25138
$367$	1763.13	0.250776	0.125388	−	0.992108i	$-0.459982\pi$
$367$	1763.13	0.250776	0.125388	+	0.992108i	$0.459982\pi$
$368$	−17966.0	−2.54495
$369$	8757.63	1.23551
$370$	3737.33	0.525120
$371$	−1267.30	−0.177345
$372$	−4423.61	−0.616542
$373$	−3490.28	−0.484503	−0.242252	−	0.970213i	$-0.577886\pi$
$373$	−3490.28	−0.484503	−0.242252	+	0.970213i	$0.577886\pi$
$374$	−41090.6	−5.68113
$375$	3793.69	0.522414
$376$	−13527.6	−1.85541
$377$	−211.460	−0.0288880
$378$	10862.1	1.47800
$379$	−3148.23	−0.426686	−0.213343	−	0.976977i	$-0.568435\pi$
$379$	−3148.23	−0.426686	−0.213343	+	0.976977i	$0.568435\pi$
$380$	40202.4	5.42721
$381$	−4685.51	−0.630042
$382$	−21051.3	−2.81958
$383$	3915.75	0.522417	0.261208	−	0.965282i	$-0.415879\pi$
$383$	3915.75	0.522417	0.261208	+	0.965282i	$0.415879\pi$
$384$	2362.85	0.314007
$385$	25544.8	3.38151
$386$	18699.7	2.46578
$387$	6688.96	0.878601
$388$	−7067.60	−0.924750
$389$	11321.5	1.47563	0.737817	−	0.675000i	$-0.235857\pi$
$389$	11321.5	1.47563	0.737817	+	0.675000i	$0.235857\pi$
$390$	−6890.40	−0.894639
$391$	−18748.6	−2.42496
$392$	−6700.74	−0.863364
$393$	−1026.47	−0.131752
$394$	17739.9	2.26834
$395$	−6401.37	−0.815412
$396$	−26506.6	−3.36365
$397$	−6039.93	−0.763565	−0.381782	−	0.924252i	$-0.624690\pi$
$397$	−6039.93	−0.763565	−0.381782	+	0.924252i	$0.624690\pi$
$398$	4286.49	0.539855
$399$	4949.63	0.621031
$400$	28072.1	3.50902
$401$	−10201.6	−1.27043	−0.635215	−	0.772336i	$-0.719088\pi$
$401$	−10201.6	−1.27043	−0.635215	+	0.772336i	$0.719088\pi$
$402$	7814.36	0.969515
$403$	−4578.93	−0.565987
$404$	17168.6	2.11429
$405$	8208.25	1.00709
$406$	639.526	0.0781752
$407$	−2428.97	−0.295823
$408$	−13081.2	−1.58730
$409$	−409.000	−0.0494468
$409$	−409.000	−0.0494468
$410$	36345.3	4.37797
$411$	3806.41	0.456828
$412$	−10061.1	−1.20309
$413$	572.050	0.0681567
$414$	−17398.8	−2.06547
$415$	−9418.19	−1.11403
$416$	−7691.16	−0.906467
$417$	150.296	0.0176499
$418$	−37588.4	−4.39834
$419$	9702.59	1.13127	0.565636	−	0.824655i	$-0.308631\pi$
$419$	9702.59	1.13127	0.565636	+	0.824655i	$0.308631\pi$
$420$	14485.5	1.68291
$421$	4479.12	0.518525	0.259263	−	0.965807i	$-0.416520\pi$
$421$	4479.12	0.518525	0.259263	+	0.965807i	$0.416520\pi$
$422$	594.212	0.0685446
$423$	−5986.10	−0.688071
$424$	3063.85	0.350928
$425$	29295.0	3.34357
$426$	−3467.44	−0.394361
$427$	−19067.2	−2.16095
$428$	10028.0	1.13253
$429$	4478.23	0.503988
$430$	27760.0	3.11328
$431$	−2298.84	−0.256918	−0.128459	−	0.991715i	$-0.541003\pi$
$431$	−2298.84	−0.256918	−0.128459	+	0.991715i	$0.541003\pi$
$432$	−11999.2	−1.33637
$433$	−14212.5	−1.57739	−0.788696	−	0.614783i	$-0.789244\pi$
$433$	−14212.5	−1.57739	−0.788696	+	0.614783i	$0.789244\pi$
$434$	13848.2	1.53165
$435$	−210.621	−0.0232150
$436$	24332.8	2.67277
$437$	−17150.6	−1.87741
$438$	7204.47	0.785943
$439$	8425.30	0.915986	0.457993	−	0.888956i	$-0.348568\pi$
$439$	8425.30	0.915986	0.457993	+	0.888956i	$0.348568\pi$
$440$	−61757.5	−6.69130
$441$	−2965.14	−0.320175
$442$	−24119.1	−2.59554
$443$	8072.63	0.865784	0.432892	−	0.901446i	$-0.357493\pi$
$443$	8072.63	0.865784	0.432892	+	0.901446i	$0.357493\pi$
$444$	−1377.38	−0.147225
$445$	−13034.3	−1.38850
$446$	11212.9	1.19046
$447$	−903.244	−0.0955749
$448$	1949.88	0.205633
$449$	−4594.25	−0.482887	−0.241443	−	0.970415i	$-0.577621\pi$
$449$	−4594.25	−0.482887	−0.241443	+	0.970415i	$0.577621\pi$
$450$	27185.9	2.84791
$451$	−23621.6	−2.46630
$452$	−4093.51	−0.425980
$453$	−3709.64	−0.384755
$454$	630.764	0.0652054
$455$	14994.1	1.54491
$456$	−11966.3	−1.22889
$457$	−5202.90	−0.532563	−0.266281	−	0.963895i	$-0.585795\pi$
$457$	−5202.90	−0.532563	−0.266281	+	0.963895i	$0.585795\pi$
$458$	−11639.1	−1.18747
$459$	−12521.9	−1.27336
$460$	−50192.9	−5.08751
$461$	−11224.0	−1.13395	−0.566976	−	0.823734i	$-0.691887\pi$
$461$	−11224.0	−1.13395	−0.566976	+	0.823734i	$0.691887\pi$
$462$	−13543.6	−1.36387
$463$	−837.261	−0.0840406	−0.0420203	−	0.999117i	$-0.513379\pi$
$463$	−837.261	−0.0840406	−0.0420203	+	0.999117i	$0.513379\pi$
$464$	−706.479	−0.0706842
$465$	−4560.75	−0.454838
$466$	9383.88	0.932833
$467$	−16798.5	−1.66455	−0.832273	−	0.554365i	$-0.812961\pi$
$467$	−16798.5	−1.66455	−0.832273	+	0.554365i	$0.812961\pi$
$468$	−15558.7	−1.53675
$469$	−17004.7	−1.67421
$470$	−24843.1	−2.43814
$471$	−170.136	−0.0166443
$472$	−1383.00	−0.134868
$473$	−18041.9	−1.75384
$474$	3393.95	0.328881
$475$	26798.2	2.58860
$476$	50705.0	4.88248
$477$	1355.78	0.130140
$478$	13557.3	1.29728
$479$	6836.41	0.652116	0.326058	−	0.945350i	$-0.394279\pi$
$479$	6836.41	0.652116	0.326058	+	0.945350i	$0.394279\pi$
$480$	−7660.63	−0.728455
$481$	−1425.74	−0.135153
$482$	−12612.2	−1.19185
$483$	−6179.63	−0.582159
$484$	47217.8	4.43443
$485$	−7286.70	−0.682211
$486$	−17869.0	−1.66781
$487$	1350.28	0.125641	0.0628204	−	0.998025i	$-0.479990\pi$
$487$	1350.28	0.125641	0.0628204	+	0.998025i	$0.479990\pi$
$488$	46097.2	4.27607
$489$	2433.44	0.225039
$490$	−12305.7	−1.13452
$491$	17872.3	1.64270	0.821349	−	0.570427i	$-0.193222\pi$
$491$	17872.3	1.64270	0.821349	+	0.570427i	$0.193222\pi$
$492$	−13395.0	−1.22742
$493$	−737.256	−0.0673516
$494$	−22063.4	−2.00947
$495$	−27328.3	−2.48145
$496$	−15298.0	−1.38488
$497$	7545.45	0.681006
$498$	4993.45	0.449321
$499$	21653.3	1.94255	0.971276	−	0.237954i	$-0.0764768\pi$
$499$	21653.3	1.94255	0.971276	+	0.237954i	$0.0764768\pi$
$500$	35551.0	3.17978
$501$	−2249.25	−0.200577
$502$	−2109.90	−0.187589
$503$	4216.10	0.373731	0.186866	−	0.982385i	$-0.440167\pi$
$503$	4216.10	0.373731	0.186866	+	0.982385i	$0.440167\pi$
$504$	26416.5	2.33469
$505$	17700.9	1.55976
$506$	46929.2	4.12304
$507$	−1647.66	−0.144330
$508$	−43908.2	−3.83487
$509$	453.733	0.0395115	0.0197557	−	0.999805i	$-0.493711\pi$
$509$	453.733	0.0395115	0.0197557	+	0.999805i	$0.493711\pi$
$510$	−24023.4	−2.08583
$511$	−15677.6	−1.35721
$512$	25825.3	2.22916
$513$	−11454.7	−0.985841
$514$	20442.3	1.75422
$515$	−10373.0	−0.887548
$516$	−10230.9	−0.872850
$517$	16146.1	1.37351
$518$	4311.92	0.365743
$519$	6240.66	0.527812
$520$	−36250.0	−3.05706
$521$	16723.0	1.40623	0.703115	−	0.711076i	$-0.251792\pi$
$521$	16723.0	1.40623	0.703115	+	0.711076i	$0.251792\pi$
$522$	−684.177	−0.0573671
$523$	−1422.00	−0.118890	−0.0594452	−	0.998232i	$-0.518933\pi$
$523$	−1422.00	−0.118890	−0.0594452	+	0.998232i	$0.518933\pi$
$524$	−9619.12	−0.801933
$525$	9655.77	0.802691
$526$	4901.55	0.406308
$527$	−15964.4	−1.31958
$528$	14961.6	1.23318
$529$	9245.65	0.759895
$530$	5626.67	0.461145
$531$	−611.989	−0.0500152
$532$	46383.3	3.78002
$533$	−13865.3	−1.12678
$534$	6910.66	0.560025
$535$	10338.9	0.835495
$536$	41110.9	3.31291
$537$	7731.02	0.621263
$538$	−29474.9	−2.36199
$539$	7997.77	0.639125
$540$	−33523.1	−2.67149
$541$	−14167.5	−1.12589	−0.562947	−	0.826493i	$-0.690332\pi$
$541$	−14167.5	−1.12589	−0.562947	+	0.826493i	$0.690332\pi$
$542$	−17933.6	−1.42125
$543$	1683.71	0.133066
$544$	−26815.2	−2.11341
$545$	25087.1	1.97177
$546$	−7949.76	−0.623111
$547$	11677.9	0.912815	0.456408	−	0.889771i	$-0.349136\pi$
$547$	11677.9	0.912815	0.456408	+	0.889771i	$0.349136\pi$
$548$	35670.1	2.78057
$549$	20398.5	1.58576
$550$	−73327.7	−5.68491
$551$	−674.418	−0.0521437
$552$	14940.0	1.15197
$553$	−7385.54	−0.567930
$554$	−40370.8	−3.09601
$555$	−1420.08	−0.108611
$556$	1408.44	0.107430
$557$	−6675.38	−0.507801	−0.253900	−	0.967230i	$-0.581714\pi$
$557$	−6675.38	−0.507801	−0.253900	+	0.967230i	$0.581714\pi$
$558$	−14815.0	−1.12396
$559$	−10590.1	−0.801277
$560$	50094.7	3.78015
$561$	15613.3	1.17504
$562$	−9647.47	−0.724117
$563$	22255.4	1.66599	0.832995	−	0.553280i	$-0.186624\pi$
$563$	22255.4	1.66599	0.832995	+	0.553280i	$0.186624\pi$
$564$	9155.87	0.683567
$565$	−4220.42	−0.314255
$566$	−19268.9	−1.43098
$567$	9470.22	0.701432
$568$	−18242.0	−1.34757
$569$	−17972.5	−1.32416	−0.662079	−	0.749434i	$-0.730326\pi$
$569$	−17972.5	−1.32416	−0.662079	+	0.749434i	$0.730326\pi$
$570$	−21975.8	−1.61485
$571$	−11889.4	−0.871374	−0.435687	−	0.900098i	$-0.643495\pi$
$571$	−11889.4	−0.871374	−0.435687	+	0.900098i	$0.643495\pi$
$572$	41965.8	3.06762
$573$	7998.93	0.583176
$574$	41933.2	3.04923
$575$	−33457.6	−2.42657
$576$	−2086.02	−0.150899
$577$	−21393.6	−1.54355	−0.771773	−	0.635899i	$-0.780630\pi$
$577$	−21393.6	−1.54355	−0.771773	+	0.635899i	$0.780630\pi$
$578$	−58924.3	−4.24036
$579$	−7105.40	−0.510001
$580$	−1973.75	−0.141302
$581$	−10866.2	−0.775912
$582$	3863.35	0.275156
$583$	−3656.90	−0.259783
$584$	37902.3	2.68563
$585$	−16041.0	−1.13370
$586$	1540.19	0.108575
$587$	13429.3	0.944270	0.472135	−	0.881526i	$-0.343483\pi$
$587$	13429.3	0.944270	0.472135	+	0.881526i	$0.343483\pi$
$588$	4535.25	0.318079
$589$	−14603.7	−1.02162
$590$	−2539.84	−0.177226
$591$	−6740.69	−0.469162
$592$	−4763.35	−0.330697
$593$	23675.3	1.63951	0.819753	−	0.572717i	$-0.194110\pi$
$593$	23675.3	1.63951	0.819753	+	0.572717i	$0.194110\pi$
$594$	31343.4	2.16504
$595$	52276.9	3.60193
$596$	−8464.37	−0.581735
$597$	−1628.75	−0.111659
$598$	27546.2	1.88369
$599$	12978.5	0.885288	0.442644	−	0.896697i	$-0.354040\pi$
$599$	12978.5	0.885288	0.442644	+	0.896697i	$0.354040\pi$
$600$	−23344.0	−1.58836
$601$	−9750.49	−0.661781	−0.330891	−	0.943669i	$-0.607349\pi$
$601$	−9750.49	−0.661781	−0.330891	+	0.943669i	$0.607349\pi$
$602$	32028.0	2.16838
$603$	18192.0	1.22858
$604$	−34763.3	−2.34189
$605$	48681.6	3.27139
$606$	−9384.87	−0.629099
$607$	7100.69	0.474808	0.237404	−	0.971411i	$-0.423704\pi$
$607$	7100.69	0.474808	0.237404	+	0.971411i	$0.423704\pi$
$608$	−24529.7	−1.63620
$609$	−243.003	−0.0161691
$610$	84656.3	5.61907
$611$	9477.33	0.627515
$612$	−54245.2	−3.58289
$613$	7619.49	0.502037	0.251018	−	0.967982i	$-0.419235\pi$
$613$	7619.49	0.502037	0.251018	+	0.967982i	$0.419235\pi$
$614$	−48389.2	−3.18051
$615$	−13810.2	−0.905501
$616$	−71252.4	−4.66045
$617$	14786.7	0.964813	0.482406	−	0.875947i	$-0.339763\pi$
$617$	14786.7	0.964813	0.482406	+	0.875947i	$0.339763\pi$
$618$	5499.66	0.357975
$619$	22104.7	1.43532	0.717659	−	0.696395i	$-0.245214\pi$
$619$	22104.7	1.43532	0.717659	+	0.696395i	$0.245214\pi$
$620$	−42739.1	−2.76846
$621$	14301.2	0.924135
$622$	18712.4	1.20627
$623$	−15038.2	−0.967084
$624$	8782.04	0.563402
$625$	8072.59	0.516645
$626$	−6378.53	−0.407248
$627$	14282.6	0.909714
$628$	−1594.36	−0.101308
$629$	−4970.85	−0.315105
$630$	48513.3	3.06796
$631$	−1915.48	−0.120846	−0.0604232	−	0.998173i	$-0.519245\pi$
$631$	−1915.48	−0.120846	−0.0604232	+	0.998173i	$0.519245\pi$
$632$	17855.4	1.12381
$633$	−225.785	−0.0141772
$634$	51652.5	3.23562
$635$	−45269.5	−2.82908
$636$	−2073.70	−0.129288
$637$	4694.48	0.291997
$638$	1845.41	0.114515
$639$	−8072.27	−0.499740
$640$	22828.9	1.40999
$641$	−9516.78	−0.586412	−0.293206	−	0.956049i	$-0.594722\pi$
$641$	−9516.78	−0.586412	−0.293206	+	0.956049i	$0.594722\pi$
$642$	−5481.60	−0.336981
$643$	13206.9	0.809998	0.404999	−	0.914317i	$-0.367272\pi$
$643$	13206.9	0.809998	0.404999	+	0.914317i	$0.367272\pi$
$644$	−57909.8	−3.54342
$645$	−10548.1	−0.643922
$646$	−76923.9	−4.68503
$647$	−15539.6	−0.944241	−0.472120	−	0.881534i	$-0.656511\pi$
$647$	−15539.6	−0.944241	−0.472120	+	0.881534i	$0.656511\pi$
$648$	−22895.4	−1.38799
$649$	1650.70	0.0998391
$650$	−43041.4	−2.59727
$651$	−5261.94	−0.316792
$652$	22803.9	1.36974
$653$	−14350.5	−0.859996	−0.429998	−	0.902830i	$-0.641486\pi$
$653$	−14350.5	−0.859996	−0.429998	+	0.902830i	$0.641486\pi$
$654$	−13301.0	−0.795275
$655$	−9917.32	−0.591606
$656$	−46323.3	−2.75704
$657$	16772.2	0.995957
$658$	−28662.6	−1.69815
$659$	13987.0	0.826791	0.413396	−	0.910552i	$-0.364343\pi$
$659$	13987.0	0.826791	0.413396	+	0.910552i	$0.364343\pi$
$660$	41799.2	2.46520
$661$	19009.2	1.11857	0.559284	−	0.828976i	$-0.311076\pi$
$661$	19009.2	1.11857	0.559284	+	0.828976i	$0.311076\pi$
$662$	13063.1	0.766934
$663$	9164.61	0.536839
$664$	26270.2	1.53537
$665$	47821.3	2.78862
$666$	−4612.98	−0.268392
$667$	842.014	0.0488799
$668$	−21077.9	−1.22085
$669$	−4260.59	−0.246224
$670$	75499.1	4.35341
$671$	−55020.0	−3.16546
$672$	−8838.41	−0.507365
$673$	4720.78	0.270390	0.135195	−	0.990819i	$-0.456834\pi$
$673$	4720.78	0.270390	0.135195	+	0.990819i	$0.456834\pi$
$674$	61103.9	3.49204
$675$	−22345.9	−1.27421
$676$	−15440.4	−0.878492
$677$	−1929.40	−0.109532	−0.0547658	−	0.998499i	$-0.517441\pi$
$677$	−1929.40	−0.109532	−0.0547658	+	0.998499i	$0.517441\pi$
$678$	2237.63	0.126749
$679$	−8406.99	−0.475156
$680$	−126386.	−7.12745
$681$	−239.674	−0.0134865
$682$	39960.1	2.24362
$683$	12286.8	0.688346	0.344173	−	0.938906i	$-0.388159\pi$
$683$	12286.8	0.688346	0.344173	+	0.938906i	$0.388159\pi$
$684$	−49621.8	−2.77388
$685$	36775.9	2.05129
$686$	23923.7	1.33150
$687$	4422.57	0.245606
$688$	−35381.1	−1.96060
$689$	−2146.51	−0.118687
$690$	27436.9	1.51377
$691$	28263.3	1.55599	0.777993	−	0.628273i	$-0.216238\pi$
$691$	28263.3	1.55599	0.777993	+	0.628273i	$0.216238\pi$
$692$	58481.7	3.21263
$693$	−31529.9	−1.72831
$694$	29600.5	1.61905
$695$	1452.10	0.0792536
$696$	587.487	0.0319952
$697$	−48341.3	−2.62705
$698$	13469.6	0.730416
$699$	−3565.62	−0.192939
$700$	90485.0	4.88573
$701$	9008.52	0.485374	0.242687	−	0.970105i	$-0.421971\pi$
$701$	9008.52	0.485374	0.242687	+	0.970105i	$0.421971\pi$
$702$	18397.7	0.989142
$703$	−4547.18	−0.243954
$704$	5626.56	0.301220
$705$	9439.71	0.504284
$706$	56556.8	3.01494
$707$	20422.3	1.08636
$708$	936.051	0.0496878
$709$	29002.8	1.53628	0.768141	−	0.640281i	$-0.221182\pi$
$709$	29002.8	1.53628	0.768141	+	0.640281i	$0.221182\pi$
$710$	−33501.0	−1.77080
$711$	7901.19	0.416762
$712$	36356.6	1.91365
$713$	18232.8	0.957678
$714$	−27716.8	−1.45277
$715$	43266.8	2.26306
$716$	72447.9	3.78143
$717$	−5151.42	−0.268317
$718$	−66358.2	−3.44912
$719$	−37976.9	−1.96982	−0.984909	−	0.173070i	$-0.944631\pi$
$719$	−37976.9	−1.96982	−0.984909	+	0.173070i	$0.944631\pi$
$720$	−53592.2	−2.77398
$721$	−11967.7	−0.618172
$722$	−35232.3	−1.81608
$723$	4792.31	0.246512
$724$	15778.2	0.809934
$725$	−1315.66	−0.0673964
$726$	−25810.6	−1.31945
$727$	1300.77	0.0663590	0.0331795	−	0.999449i	$-0.489437\pi$
$727$	1300.77	0.0663590	0.0331795	+	0.999449i	$0.489437\pi$
$728$	−41823.3	−2.12922
$729$	−4995.29	−0.253787
$730$	69606.6	3.52912
$731$	−36922.4	−1.86816
$732$	−31199.9	−1.57538
$733$	14050.9	0.708024	0.354012	−	0.935241i	$-0.384817\pi$
$733$	14050.9	0.708024	0.354012	+	0.935241i	$0.384817\pi$
$734$	−9031.65	−0.454175
$735$	4675.85	0.234655
$736$	30625.4	1.53379
$737$	−49068.6	−2.45246
$738$	−44860.9	−2.23761
$739$	−14712.4	−0.732348	−0.366174	−	0.930546i	$-0.619333\pi$
$739$	−14712.4	−0.732348	−0.366174	+	0.930546i	$0.619333\pi$
$740$	−13307.7	−0.661083
$741$	8383.49	0.415621
$742$	6491.74	0.321185
$743$	15063.6	0.743783	0.371892	−	0.928276i	$-0.378709\pi$
$743$	15063.6	0.743783	0.371892	+	0.928276i	$0.378709\pi$
$744$	12721.4	0.626865
$745$	−8726.77	−0.429160
$746$	17878.9	0.877473
$747$	11624.8	0.569385
$748$	146314.	7.15208
$749$	11928.4	0.581917
$750$	−19433.2	−0.946132
$751$	15183.3	0.737746	0.368873	−	0.929480i	$-0.379744\pi$
$751$	15183.3	0.737746	0.368873	+	0.929480i	$0.379744\pi$
$752$	31663.3	1.53543
$753$	801.705	0.0387992
$754$	1083.21	0.0523183
$755$	−35841.0	−1.72767
$756$	−38677.1	−1.86068
$757$	28873.6	1.38630	0.693149	−	0.720794i	$-0.256223\pi$
$757$	28873.6	1.38630	0.693149	+	0.720794i	$0.256223\pi$
$758$	16126.8	0.772760
$759$	−17831.8	−0.852773
$760$	−115613.	−5.51808
$761$	16374.1	0.779973	0.389986	−	0.920821i	$-0.372480\pi$
$761$	16374.1	0.779973	0.389986	+	0.920821i	$0.372480\pi$
$762$	24001.5	1.14105
$763$	28944.1	1.37333
$764$	74958.6	3.54962
$765$	−55926.8	−2.64319
$766$	−20058.4	−0.946136
$767$	968.916	0.0456135
$768$	−13503.1	−0.634442
$769$	−15513.0	−0.727454	−0.363727	−	0.931506i	$-0.618496\pi$
$769$	−15513.0	−0.727454	−0.363727	+	0.931506i	$0.618496\pi$
$770$	−130853.	−6.12418
$771$	−7767.51	−0.362827
$772$	−66585.3	−3.10422
$773$	−27277.0	−1.26919	−0.634595	−	0.772845i	$-0.718833\pi$
$773$	−27277.0	−1.26919	−0.634595	+	0.772845i	$0.718833\pi$
$774$	−34264.1	−1.59121
$775$	−28489.1	−1.32046
$776$	20324.9	0.940233
$777$	−1638.41	−0.0756471
$778$	−57994.2	−2.67249
$779$	−44221.0	−2.03387
$780$	24535.0	1.12628
$781$	21773.0	0.997568
$782$	96039.8	4.39178
$783$	562.369	0.0256672
$784$	15684.1	0.714470
$785$	−1643.78	−0.0747377
$786$	5258.08	0.238613
$787$	18599.9	0.842457	0.421229	−	0.906955i	$-0.361599\pi$
$787$	18599.9	0.842457	0.421229	+	0.906955i	$0.361599\pi$
$788$	−63167.5	−2.85565
$789$	−1862.46	−0.0840371
$790$	32791.0	1.47677
$791$	−4869.28	−0.218877
$792$	76227.1	3.41996
$793$	−32295.3	−1.44620
$794$	30939.5	1.38287
$795$	−2137.99	−0.0953793
$796$	−15263.2	−0.679633
$797$	9041.95	0.401860	0.200930	−	0.979606i	$-0.435604\pi$
$797$	9041.95	0.401860	0.200930	+	0.979606i	$0.435604\pi$
$798$	−25354.4	−1.12473
$799$	33042.7	1.46304
$800$	−47852.7	−2.11481
$801$	16088.2	0.709672
$802$	52257.5	2.30084
$803$	−45238.9	−1.98810
$804$	−27825.0	−1.22054
$805$	−59705.0	−2.61407
$806$	23455.5	1.02504
$807$	11199.7	0.488534
$808$	−49373.3	−2.14969
$809$	−28216.4	−1.22625	−0.613124	−	0.789987i	$-0.710087\pi$
$809$	−28216.4	−1.22625	−0.613124	+	0.789987i	$0.710087\pi$
$810$	−42046.7	−1.82392
$811$	−26630.2	−1.15304	−0.576519	−	0.817084i	$-0.695589\pi$
$811$	−26630.2	−1.15304	−0.576519	+	0.817084i	$0.695589\pi$
$812$	−2277.20	−0.0984162
$813$	6814.30	0.293958
$814$	12442.4	0.535757
$815$	23510.9	1.01049
$816$	30618.5	1.31356
$817$	−33775.4	−1.44633
$818$	2095.10	0.0895519
$819$	−18507.2	−0.789614
$820$	−129417.	−5.51150
$821$	−19708.1	−0.837782	−0.418891	−	0.908036i	$-0.637581\pi$
$821$	−19708.1	−0.837782	−0.418891	+	0.908036i	$0.637581\pi$
$822$	−19498.3	−0.827350
$823$	22800.9	0.965722	0.482861	−	0.875697i	$-0.339598\pi$
$823$	22800.9	0.965722	0.482861	+	0.875697i	$0.339598\pi$
$824$	28933.4	1.22323
$825$	27862.6	1.17582
$826$	−2930.32	−0.123437
$827$	−41147.2	−1.73014	−0.865071	−	0.501649i	$-0.832727\pi$
$827$	−41147.2	−1.73014	−0.865071	+	0.501649i	$0.832727\pi$
$828$	61953.0	2.60026
$829$	18461.8	0.773470	0.386735	−	0.922191i	$-0.373603\pi$
$829$	18461.8	0.773470	0.386735	+	0.922191i	$0.373603\pi$
$830$	48244.6	2.01758
$831$	15339.8	0.640351
$832$	3302.64	0.137618
$833$	16367.3	0.680784
$834$	−769.891	−0.0319654
$835$	−21731.3	−0.900651
$836$	133843.	5.53715
$837$	12177.4	0.502884
$838$	−49701.5	−2.04882
$839$	32667.9	1.34424	0.672121	−	0.740441i	$-0.265383\pi$
$839$	32667.9	1.34424	0.672121	+	0.740441i	$0.265383\pi$
$840$	−41657.2	−1.71109
$841$	−24355.9	−0.998642
$842$	−22944.3	−0.939089
$843$	3665.78	0.149770
$844$	−2115.85	−0.0862920
$845$	−15919.0	−0.648085
$846$	30663.8	1.24615
$847$	56166.1	2.27850
$848$	−7171.38	−0.290408
$849$	7321.68	0.295971
$850$	−150064.	−6.05546
$851$	5677.17	0.228685
$852$	12346.7	0.496469
$853$	44576.2	1.78929	0.894643	−	0.446782i	$-0.147430\pi$
$853$	44576.2	1.78929	0.894643	+	0.446782i	$0.147430\pi$
$854$	97671.7	3.91365
$855$	−51160.1	−2.04636
$856$	−28838.4	−1.15149
$857$	189.774	0.00756424	0.00378212	−	0.999993i	$-0.498796\pi$
$857$	189.774	0.00756424	0.00378212	+	0.999993i	$0.498796\pi$
$858$	−22939.7	−0.912761
$859$	39891.5	1.58449	0.792247	−	0.610201i	$-0.208911\pi$
$859$	39891.5	1.58449	0.792247	+	0.610201i	$0.208911\pi$
$860$	−98846.8	−3.91936
$861$	−15933.5	−0.630676
$862$	11775.8	0.465297
$863$	−32233.8	−1.27144	−0.635719	−	0.771920i	$-0.719296\pi$
$863$	−32233.8	−1.27144	−0.635719	+	0.771920i	$0.719296\pi$
$864$	20454.3	0.805404
$865$	60294.7	2.37004
$866$	72803.7	2.85678
$867$	22389.7	0.877039
$868$	−49310.0	−1.92822
$869$	−21311.6	−0.831929
$870$	1078.91	0.0420440
$871$	−28802.0	−1.12046
$872$	−69975.8	−2.71752
$873$	8993.96	0.348682
$874$	87854.1	3.40012
$875$	42288.3	1.63383
$876$	−25653.4	−0.989438
$877$	−41072.1	−1.58142	−0.790711	−	0.612189i	$-0.790289\pi$
$877$	−41072.1	−1.58142	−0.790711	+	0.612189i	$0.790289\pi$
$878$	−43158.6	−1.65892
$879$	−585.231	−0.0224566
$880$	144552.	5.53734
$881$	−16834.3	−0.643772	−0.321886	−	0.946778i	$-0.604317\pi$
$881$	−16834.3	−0.643772	−0.321886	+	0.946778i	$0.604317\pi$
$882$	15188.9	0.579862
$883$	−34548.0	−1.31669	−0.658343	−	0.752718i	$-0.728743\pi$
$883$	−34548.0	−1.31669	−0.658343	+	0.752718i	$0.728743\pi$
$884$	85882.3	3.26757
$885$	965.070	0.0366559
$886$	−41352.0	−1.56800
$887$	−23722.1	−0.897981	−0.448990	−	0.893537i	$-0.648216\pi$
$887$	−23722.1	−0.897981	−0.448990	+	0.893537i	$0.648216\pi$
$888$	3961.06	0.149690
$889$	−52229.4	−1.97044
$890$	66768.0	2.51468
$891$	27327.1	1.02749
$892$	−39926.3	−1.49869
$893$	30226.4	1.13268
$894$	4626.86	0.173093
$895$	74693.9	2.78966
$896$	26338.7	0.982048
$897$	−10466.8	−0.389607
$898$	23534.0	0.874545
$899$	716.972	0.0265989
$900$	−96802.5	−3.58528
$901$	−7483.78	−0.276716
$902$	121002.	4.46665
$903$	−12169.8	−0.448488
$904$	11772.1	0.433112
$905$	16267.3	0.597508
$906$	19002.6	0.696821
$907$	12450.3	0.455793	0.227896	−	0.973685i	$-0.426815\pi$
$907$	12450.3	0.455793	0.227896	+	0.973685i	$0.426815\pi$
$908$	−2246.00	−0.0820882
$909$	−21848.2	−0.797203
$910$	−76807.3	−2.79795
$911$	9668.40	0.351623	0.175811	−	0.984424i	$-0.443745\pi$
$911$	9668.40	0.351623	0.175811	+	0.984424i	$0.443745\pi$
$912$	28008.9	1.01696
$913$	−31355.3	−1.13659
$914$	26651.8	0.964512
$915$	−32167.1	−1.16220
$916$	41444.2	1.49493
$917$	−11442.0	−0.412050
$918$	64143.6	2.30616
$919$	−24179.1	−0.867894	−0.433947	−	0.900938i	$-0.642879\pi$
$919$	−24179.1	−0.867894	−0.433947	+	0.900938i	$0.642879\pi$
$920$	144344.	5.17269
$921$	18386.6	0.657828
$922$	57494.7	2.05367
$923$	12780.2	0.455759
$924$	48225.6	1.71700
$925$	−8870.67	−0.315314
$926$	4288.86	0.152204
$927$	12803.3	0.453631
$928$	1204.29	0.0425999
$929$	13587.5	0.479862	0.239931	−	0.970790i	$-0.422875\pi$
$929$	13587.5	0.479862	0.239931	+	0.970790i	$0.422875\pi$
$930$	23362.4	0.823746
$931$	14972.3	0.527064
$932$	−33413.7	−1.17436
$933$	−7110.21	−0.249494
$934$	86050.4	3.01462
$935$	150849.	5.27626
$936$	44743.3	1.56248
$937$	12034.9	0.419596	0.209798	−	0.977745i	$-0.432719\pi$
$937$	12034.9	0.419596	0.209798	+	0.977745i	$0.432719\pi$
$938$	87106.7	3.03213
$939$	2423.67	0.0842316
$940$	88460.3	3.06942
$941$	−48732.0	−1.68822	−0.844112	−	0.536167i	$-0.819872\pi$
$941$	−48732.0	−1.68822	−0.844112	+	0.536167i	$0.819872\pi$
$942$	871.520	0.0301440
$943$	55210.2	1.90656
$944$	3237.10	0.111609
$945$	−39876.2	−1.37267
$946$	92419.5	3.17634
$947$	37740.5	1.29504	0.647519	−	0.762049i	$-0.275807\pi$
$947$	37740.5	1.29504	0.647519	+	0.762049i	$0.275807\pi$
$948$	−12085.0	−0.414034
$949$	−26554.1	−0.908305
$950$	−137273.	−4.68815
$951$	−19626.6	−0.669227
$952$	−145817.	−4.96423
$953$	11053.3	0.375709	0.187855	−	0.982197i	$-0.439847\pi$
$953$	11053.3	0.375709	0.187855	+	0.982197i	$0.439847\pi$
$954$	−6944.99	−0.235694
$955$	77282.4	2.61864
$956$	−48274.4	−1.63316
$957$	−701.205	−0.0236852
$958$	−35019.5	−1.18103
$959$	42430.0	1.42871
$960$	3289.53	0.110593
$961$	−14265.8	−0.478863
$962$	7303.37	0.244771
$963$	−12761.3	−0.427026
$964$	44909.1	1.50044
$965$	−68649.5	−2.29006
$966$	31655.1	1.05433
$967$	29535.2	0.982202	0.491101	−	0.871103i	$-0.336595\pi$
$967$	29535.2	0.982202	0.491101	+	0.871103i	$0.336595\pi$
$968$	−135788.	−4.50867
$969$	29229.0	0.969010
$970$	37326.1	1.23554
$971$	39995.9	1.32186	0.660931	−	0.750446i	$-0.270161\pi$
$971$	39995.9	1.32186	0.660931	+	0.750446i	$0.270161\pi$
$972$	63627.3	2.09964
$973$	1675.35	0.0551997
$974$	−6916.81	−0.227545
$975$	16354.6	0.537196
$976$	−107897.	−3.53863
$977$	40534.8	1.32735	0.663676	−	0.748020i	$-0.268995\pi$
$977$	40534.8	1.32735	0.663676	+	0.748020i	$0.268995\pi$
$978$	−12465.3	−0.407562
$979$	−43394.0	−1.41663
$980$	43817.7	1.42827
$981$	−30965.0	−1.00778
$982$	−91550.6	−2.97505
$983$	−17308.3	−0.561595	−0.280798	−	0.959767i	$-0.590599\pi$
$983$	−17308.3	−0.561595	−0.280798	+	0.959767i	$0.590599\pi$
$984$	38521.1	1.24797
$985$	−65125.8	−2.10668
$986$	3776.59	0.121979
$987$	10891.0	0.351231
$988$	78562.3	2.52976
$989$	42168.7	1.35580
$990$	139989.	4.49409
$991$	2565.18	0.0822257	0.0411129	−	0.999155i	$-0.486910\pi$
$991$	2565.18	0.0822257	0.0411129	+	0.999155i	$0.486910\pi$
$992$	26077.5	0.834637
$993$	−4963.61	−0.158626
$994$	−38651.6	−1.23335
$995$	−15736.3	−0.501382
$996$	−17780.4	−0.565658
$997$	−27164.9	−0.862911	−0.431455	−	0.902134i	$-0.642000\pi$
$997$	−27164.9	−0.862911	−0.431455	+	0.902134i	$0.642000\pi$
$998$	−110919.	−3.51811
$999$	3791.70	0.120084

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	409.4.a.b.1.3	✓	55

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
409.4.a.b.1.3	✓	55	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:	\( N \)	\(=\)	\( 409 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	409.a (trivial)

\(f(q)\)	\(=\)	\(q-5.12250 q^{2} +1.94641 q^{3} +18.2400 q^{4} +18.8054 q^{5} -9.97048 q^{6} +21.6967 q^{7} -52.4542 q^{8} -23.2115 q^{9} -96.3307 q^{10} +62.6075 q^{11} +35.5025 q^{12} +36.7490 q^{13} -111.141 q^{14} +36.6031 q^{15} +122.777 q^{16} +128.125 q^{17} +118.901 q^{18} +117.205 q^{19} +343.010 q^{20} +42.2306 q^{21} -320.707 q^{22} -146.331 q^{23} -102.097 q^{24} +228.644 q^{25} -188.246 q^{26} -97.7322 q^{27} +395.746 q^{28} -5.75419 q^{29} -187.499 q^{30} -124.600 q^{31} -209.289 q^{32} +121.860 q^{33} -656.320 q^{34} +408.015 q^{35} -423.377 q^{36} -38.7969 q^{37} -600.381 q^{38} +71.5286 q^{39} -986.423 q^{40} -377.297 q^{41} -216.326 q^{42} -288.174 q^{43} +1141.96 q^{44} -436.502 q^{45} +749.578 q^{46} +257.894 q^{47} +238.974 q^{48} +127.745 q^{49} -1171.23 q^{50} +249.384 q^{51} +670.300 q^{52} -58.4100 q^{53} +500.633 q^{54} +1177.36 q^{55} -1138.08 q^{56} +228.129 q^{57} +29.4758 q^{58} +26.3658 q^{59} +667.639 q^{60} -878.809 q^{61} +638.264 q^{62} -503.611 q^{63} +89.8703 q^{64} +691.080 q^{65} -624.227 q^{66} -783.749 q^{67} +2337.00 q^{68} -284.820 q^{69} -2090.05 q^{70} +347.770 q^{71} +1217.54 q^{72} -722.580 q^{73} +198.737 q^{74} +445.035 q^{75} +2137.81 q^{76} +1358.37 q^{77} -366.405 q^{78} -340.400 q^{79} +2308.87 q^{80} +436.483 q^{81} +1932.70 q^{82} -500.823 q^{83} +770.285 q^{84} +2409.45 q^{85} +1476.17 q^{86} -11.2000 q^{87} -3284.03 q^{88} -693.112 q^{89} +2235.98 q^{90} +797.330 q^{91} -2669.07 q^{92} -242.523 q^{93} -1321.06 q^{94} +2204.09 q^{95} -407.363 q^{96} -387.479 q^{97} -654.371 q^{98} -1453.21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 55 q + 13 q^{2} + 10 q^{3} + 257 q^{4} + 60 q^{5} + 48 q^{6} + 72 q^{7} + 156 q^{8} + 639 q^{9} + 21 q^{10} + 362 q^{11} + 21 q^{12} + 48 q^{13} + 279 q^{14} + 512 q^{15} + 1237 q^{16} + 270 q^{17} + 360 q^{18}+ \cdots + 9542 q^{99}+O(q^{100}) \) 55 * q + 13 * q^2 + 10 * q^3 + 257 * q^4 + 60 * q^5 + 48 * q^6 + 72 * q^7 + 156 * q^8 + 639 * q^9 + 21 * q^10 + 362 * q^11 + 21 * q^12 + 48 * q^13 + 279 * q^14 + 512 * q^15 + 1237 * q^16 + 270 * q^17 + 360 * q^18 + 294 * q^19 + 732 * q^20 + 488 * q^21 + 123 * q^22 + 1212 * q^23 + 427 * q^24 + 1723 * q^25 + 659 * q^26 + 376 * q^27 + 622 * q^28 + 730 * q^29 + 106 * q^30 + 972 * q^31 + 1506 * q^32 + 358 * q^33 + 505 * q^34 + 1352 * q^35 + 4215 * q^36 + 692 * q^37 + 781 * q^38 + 1920 * q^39 + 792 * q^40 + 1050 * q^41 + 1491 * q^42 + 480 * q^43 + 3920 * q^44 + 1524 * q^45 + 752 * q^46 + 3432 * q^47 + 152 * q^48 + 3729 * q^49 + 3225 * q^50 + 2088 * q^51 + 984 * q^52 + 1558 * q^53 + 2433 * q^54 + 1662 * q^55 + 2465 * q^56 + 1188 * q^57 + 2048 * q^58 + 3468 * q^59 + 4144 * q^60 + 392 * q^61 + 1966 * q^62 + 2534 * q^63 + 8120 * q^64 + 1452 * q^65 + 1593 * q^66 + 1814 * q^67 + 3496 * q^68 + 2302 * q^69 + 1744 * q^70 + 8798 * q^71 + 3545 * q^72 + 918 * q^73 + 2759 * q^74 + 1898 * q^75 + 2719 * q^76 + 1830 * q^77 + 2899 * q^78 + 2918 * q^79 + 9155 * q^80 + 9571 * q^81 + 1581 * q^82 + 4192 * q^83 + 6815 * q^84 + 590 * q^85 + 7177 * q^86 + 2360 * q^87 + 1826 * q^88 + 4878 * q^89 + 175 * q^90 + 2926 * q^91 + 10822 * q^92 + 4040 * q^93 + 1886 * q^94 + 8912 * q^95 + 4614 * q^96 + 2264 * q^97 + 2870 * q^98 + 9542 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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