LMFDB - Embedded newform 82.6.a.c.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [82,6,Mod(1,82)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(82, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("82.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 82 = 2 \cdot 41 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	82.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5] 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:	$13.1514732247$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 730x^{3} - 4674x^{2} + 68790x + 487116 $ x^5 - 730x^3 - 4674x^2 + 68790*x + 487116
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-19.2082$ of defining polynomial
Character	$\chi$	$=$	82.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} +17.2082 q^{3} +16.0000 q^{4} -11.6574 q^{5} -68.8328 q^{6} -121.255 q^{7} -64.0000 q^{8} +53.1223 q^{9} +46.6298 q^{10} -379.908 q^{11} +275.331 q^{12} -771.276 q^{13} +485.021 q^{14} -200.604 q^{15} +256.000 q^{16} +2288.12 q^{17} -212.489 q^{18} +401.355 q^{19} -186.519 q^{20} -2086.58 q^{21} +1519.63 q^{22} -4787.55 q^{23} -1101.33 q^{24} -2989.10 q^{25} +3085.11 q^{26} -3267.45 q^{27} -1940.08 q^{28} -5724.34 q^{29} +802.415 q^{30} +3444.52 q^{31} -1024.00 q^{32} -6537.54 q^{33} -9152.49 q^{34} +1413.53 q^{35} +849.957 q^{36} +8660.85 q^{37} -1605.42 q^{38} -13272.3 q^{39} +746.076 q^{40} -1681.00 q^{41} +8346.34 q^{42} +8179.43 q^{43} -6078.53 q^{44} -619.270 q^{45} +19150.2 q^{46} -18780.7 q^{47} +4405.30 q^{48} -2104.18 q^{49} +11956.4 q^{50} +39374.5 q^{51} -12340.4 q^{52} +2021.20 q^{53} +13069.8 q^{54} +4428.76 q^{55} +7760.33 q^{56} +6906.60 q^{57} +22897.4 q^{58} +45753.6 q^{59} -3209.66 q^{60} -25094.8 q^{61} -13778.1 q^{62} -6441.35 q^{63} +4096.00 q^{64} +8991.11 q^{65} +26150.2 q^{66} +27844.3 q^{67} +36609.9 q^{68} -82385.1 q^{69} -5654.10 q^{70} -23526.6 q^{71} -3399.83 q^{72} +19764.3 q^{73} -34643.4 q^{74} -51437.1 q^{75} +6421.68 q^{76} +46065.8 q^{77} +53089.1 q^{78} +92632.2 q^{79} -2984.31 q^{80} -69135.7 q^{81} +6724.00 q^{82} -98359.5 q^{83} -33385.3 q^{84} -26673.6 q^{85} -32717.7 q^{86} -98505.6 q^{87} +24314.1 q^{88} +5411.37 q^{89} +2477.08 q^{90} +93521.3 q^{91} -76600.7 q^{92} +59274.0 q^{93} +75122.9 q^{94} -4678.78 q^{95} -17621.2 q^{96} -37923.2 q^{97} +8416.72 q^{98} -20181.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 20 q^{2} - 10 q^{3} + 80 q^{4} - 38 q^{5} + 40 q^{6} - 38 q^{7} - 320 q^{8} + 265 q^{9} + 152 q^{10} + 416 q^{11} - 160 q^{12} + 268 q^{13} + 152 q^{14} + 22 q^{15} + 1280 q^{16} - 2198 q^{17} - 1060 q^{18}+ \cdots - 139442 q^{99}+O(q^{100}) $ 5 * q - 20 * q^2 - 10 * q^3 + 80 * q^4 - 38 * q^5 + 40 * q^6 - 38 * q^7 - 320 * q^8 + 265 * q^9 + 152 * q^10 + 416 * q^11 - 160 * q^12 + 268 * q^13 + 152 * q^14 + 22 * q^15 + 1280 * q^16 - 2198 * q^17 - 1060 * q^18 - 2250 * q^19 - 608 * q^20 - 6344 * q^21 - 1664 * q^22 - 12008 * q^23 + 640 * q^24 - 9861 * q^25 - 1072 * q^26 - 17962 * q^27 - 608 * q^28 - 11684 * q^29 - 88 * q^30 - 6780 * q^31 - 5120 * q^32 - 15876 * q^33 + 8792 * q^34 - 23158 * q^35 + 4240 * q^36 - 5822 * q^37 + 9000 * q^38 - 14440 * q^39 + 2432 * q^40 - 8405 * q^41 + 25376 * q^42 + 13248 * q^43 + 6656 * q^44 + 5058 * q^45 + 48032 * q^46 - 2000 * q^47 - 2560 * q^48 + 51789 * q^49 + 39444 * q^50 + 36628 * q^51 + 4288 * q^52 - 32036 * q^53 + 71848 * q^54 + 28670 * q^55 + 2432 * q^56 + 5476 * q^57 + 46736 * q^58 - 41376 * q^59 + 352 * q^60 + 45314 * q^61 + 27120 * q^62 + 25568 * q^63 + 20480 * q^64 + 20012 * q^65 + 63504 * q^66 + 47244 * q^67 - 35168 * q^68 + 77140 * q^69 + 92632 * q^70 - 87348 * q^71 - 16960 * q^72 + 44550 * q^73 + 23288 * q^74 + 23370 * q^75 - 36000 * q^76 - 19040 * q^77 + 57760 * q^78 - 6864 * q^79 - 9728 * q^80 + 154261 * q^81 + 33620 * q^82 - 119888 * q^83 - 101504 * q^84 - 22700 * q^85 - 52992 * q^86 - 242980 * q^87 - 26624 * q^88 - 136626 * q^89 - 20232 * q^90 - 30964 * q^91 - 192128 * q^92 - 180684 * q^93 + 8000 * q^94 - 153770 * q^95 + 10240 * q^96 + 7086 * q^97 - 207156 * q^98 - 139442 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	17.2082	1.10391	0.551953	−	0.833875i	$-0.313883\pi$
$3$	17.2082	1.10391	0.551953	+	0.833875i	$0.313883\pi$
$4$	16.0000	0.500000
$5$	−11.6574	−0.208535	−0.104267	−	0.994549i	$-0.533250\pi$
$5$	−11.6574	−0.208535	−0.104267	+	0.994549i	$0.533250\pi$
$6$	−68.8328	−0.780580
$7$	−121.255	−0.935309	−0.467655	−	0.883911i	$-0.654901\pi$
$7$	−121.255	−0.935309	−0.467655	+	0.883911i	$0.654901\pi$
$8$	−64.0000	−0.353553
$9$	53.1223	0.218610
$10$	46.6298	0.147456
$11$	−379.908	−0.946666	−0.473333	−	0.880883i	$-0.656949\pi$
$11$	−379.908	−0.946666	−0.473333	+	0.880883i	$0.656949\pi$
$12$	275.331	0.551953
$13$	−771.276	−1.26576	−0.632880	−	0.774250i	$-0.718128\pi$
$13$	−771.276	−1.26576	−0.632880	+	0.774250i	$0.718128\pi$
$14$	485.021	0.661364
$15$	−200.604	−0.230203
$16$	256.000	0.250000
$17$	2288.12	1.92025	0.960123	−	0.279579i	$-0.0901948\pi$
$17$	2288.12	1.92025	0.960123	+	0.279579i	$0.0901948\pi$
$18$	−212.489	−0.154581
$19$	401.355	0.255061	0.127531	−	0.991835i	$-0.459295\pi$
$19$	401.355	0.255061	0.127531	+	0.991835i	$0.459295\pi$
$20$	−186.519	−0.104267
$21$	−2086.58	−1.03249
$22$	1519.63	0.669394
$23$	−4787.55	−1.88709	−0.943547	−	0.331240i	$-0.892533\pi$
$23$	−4787.55	−1.88709	−0.943547	+	0.331240i	$0.892533\pi$
$24$	−1101.33	−0.390290
$25$	−2989.10	−0.956513
$26$	3085.11	0.895028
$27$	−3267.45	−0.862581
$28$	−1940.08	−0.467655
$29$	−5724.34	−1.26395	−0.631976	−	0.774988i	$-0.717756\pi$
$29$	−5724.34	−1.26395	−0.631976	+	0.774988i	$0.717756\pi$
$30$	802.415	0.162778
$31$	3444.52	0.643761	0.321881	−	0.946780i	$-0.395685\pi$
$31$	3444.52	0.643761	0.321881	+	0.946780i	$0.395685\pi$
$32$	−1024.00	−0.176777
$33$	−6537.54	−1.04503
$34$	−9152.49	−1.35782
$35$	1413.53	0.195044
$36$	849.957	0.109305
$37$	8660.85	1.04005	0.520027	−	0.854150i	$-0.325922\pi$
$37$	8660.85	1.04005	0.520027	+	0.854150i	$0.325922\pi$
$38$	−1605.42	−0.180356
$39$	−13272.3	−1.39728
$40$	746.076	0.0737282
$41$	−1681.00	−0.156174
$41$	−1681.00	−0.156174
$42$	8346.34	0.730084
$43$	8179.43	0.674608	0.337304	−	0.941396i	$-0.390485\pi$
$43$	8179.43	0.674608	0.337304	+	0.941396i	$0.390485\pi$
$44$	−6078.53	−0.473333
$45$	−619.270	−0.0455878
$46$	19150.2	1.33438
$47$	−18780.7	−1.24013	−0.620066	−	0.784550i	$-0.712894\pi$
$47$	−18780.7	−1.24013	−0.620066	+	0.784550i	$0.712894\pi$
$48$	4405.30	0.275977
$49$	−2104.18	−0.125197
$50$	11956.4	0.676357
$51$	39374.5	2.11977
$52$	−12340.4	−0.632880
$53$	2021.20	0.0988370	0.0494185	−	0.998778i	$-0.484263\pi$
$53$	2021.20	0.0988370	0.0494185	+	0.998778i	$0.484263\pi$
$54$	13069.8	0.609937
$55$	4428.76	0.197413
$56$	7760.33	0.330682
$57$	6906.60	0.281564
$58$	22897.4	0.893748
$59$	45753.6	1.71118	0.855590	−	0.517655i	$-0.173195\pi$
$59$	45753.6	1.71118	0.855590	+	0.517655i	$0.173195\pi$
$60$	−3209.66	−0.115101
$61$	−25094.8	−0.863492	−0.431746	−	0.901995i	$-0.642102\pi$
$61$	−25094.8	−0.863492	−0.431746	+	0.901995i	$0.642102\pi$
$62$	−13778.1	−0.455208
$63$	−6441.35	−0.204468
$64$	4096.00	0.125000
$65$	8991.11	0.263955
$66$	26150.2	0.738949
$67$	27844.3	0.757790	0.378895	−	0.925440i	$-0.376304\pi$
$67$	27844.3	0.757790	0.378895	+	0.925440i	$0.376304\pi$
$68$	36609.9	0.960123
$69$	−82385.1	−2.08318
$70$	−5654.10	−0.137917
$71$	−23526.6	−0.553877	−0.276939	−	0.960888i	$-0.589320\pi$
$71$	−23526.6	−0.553877	−0.276939	+	0.960888i	$0.589320\pi$
$72$	−3399.83	−0.0772904
$73$	19764.3	0.434086	0.217043	−	0.976162i	$-0.430359\pi$
$73$	19764.3	0.434086	0.217043	+	0.976162i	$0.430359\pi$
$74$	−34643.4	−0.735430
$75$	−51437.1	−1.05590
$76$	6421.68	0.127531
$77$	46065.8	0.885426
$78$	53089.1	0.988028
$79$	92632.2	1.66991	0.834957	−	0.550315i	$-0.185492\pi$
$79$	92632.2	1.66991	0.834957	+	0.550315i	$0.185492\pi$
$80$	−2984.31	−0.0521337
$81$	−69135.7	−1.17082
$82$	6724.00	0.110432
$83$	−98359.5	−1.56719	−0.783594	−	0.621274i	$-0.786616\pi$
$83$	−98359.5	−1.56719	−0.783594	+	0.621274i	$0.786616\pi$
$84$	−33385.3	−0.516247
$85$	−26673.6	−0.400438
$86$	−32717.7	−0.477020
$87$	−98505.6	−1.39528
$88$	24314.1	0.334697
$89$	5411.37	0.0724157	0.0362078	−	0.999344i	$-0.488472\pi$
$89$	5411.37	0.0724157	0.0362078	+	0.999344i	$0.488472\pi$
$90$	2477.08	0.0322355
$91$	93521.3	1.18388
$92$	−76600.7	−0.943547
$93$	59274.0	0.710652
$94$	75122.9	0.876906
$95$	−4678.78	−0.0531892
$96$	−17621.2	−0.195145
$97$	−37923.2	−0.409238	−0.204619	−	0.978842i	$-0.565596\pi$
$97$	−37923.2	−0.409238	−0.204619	+	0.978842i	$0.565596\pi$
$98$	8416.72	0.0885274
$99$	−20181.6	−0.206951
$100$	−47825.7	−0.478257
$101$	−11287.7	−0.110103	−0.0550517	−	0.998484i	$-0.517532\pi$
$101$	−11287.7	−0.110103	−0.0550517	+	0.998484i	$0.517532\pi$
$102$	−157498.	−1.49891
$103$	85113.2	0.790504	0.395252	−	0.918573i	$-0.370657\pi$
$103$	85113.2	0.790504	0.395252	+	0.918573i	$0.370657\pi$
$104$	49361.7	0.447514
$105$	24324.2	0.215311
$106$	−8084.80	−0.0698883
$107$	−172432.	−1.45599	−0.727996	−	0.685582i	$-0.759548\pi$
$107$	−172432.	−1.45599	−0.727996	+	0.685582i	$0.759548\pi$
$108$	−52279.3	−0.431291
$109$	220320.	1.77618	0.888092	−	0.459666i	$-0.152031\pi$
$109$	220320.	1.77618	0.888092	+	0.459666i	$0.152031\pi$
$110$	−17715.0	−0.139592
$111$	149038.	1.14812
$112$	−31041.3	−0.233827
$113$	201882.	1.48731	0.743654	−	0.668565i	$-0.233091\pi$
$113$	201882.	1.48731	0.743654	+	0.668565i	$0.233091\pi$
$114$	−27626.4	−0.199096
$115$	55810.6	0.393524
$116$	−91589.4	−0.631976
$117$	−40972.0	−0.276708
$118$	−183015.	−1.20999
$119$	−277447.	−1.79602
$120$	12838.6	0.0813890
$121$	−16720.8	−0.103823
$122$	100379.	0.610581
$123$	−28927.0	−0.172401
$124$	55112.4	0.321881
$125$	71274.8	0.408001
$126$	25765.4	0.144581
$127$	−130701.	−0.719070	−0.359535	−	0.933132i	$-0.617065\pi$
$127$	−130701.	−0.719070	−0.359535	+	0.933132i	$0.617065\pi$
$128$	−16384.0	−0.0883883
$129$	140753.	0.744705
$130$	−35964.4	−0.186644
$131$	−69625.8	−0.354480	−0.177240	−	0.984168i	$-0.556717\pi$
$131$	−69625.8	−0.354480	−0.177240	+	0.984168i	$0.556717\pi$
$132$	−104601.	−0.522516
$133$	−48666.4	−0.238561
$134$	−111377.	−0.535839
$135$	38090.2	0.179878
$136$	−146440.	−0.678909
$137$	234560.	1.06771	0.533855	−	0.845576i	$-0.320743\pi$
$137$	234560.	1.06771	0.533855	+	0.845576i	$0.320743\pi$
$138$	329540.	1.47303
$139$	−232569.	−1.02097	−0.510487	−	0.859885i	$-0.670535\pi$
$139$	−232569.	−1.02097	−0.510487	+	0.859885i	$0.670535\pi$
$140$	22616.4	0.0975222
$141$	−323183.	−1.36899
$142$	94106.5	0.391650
$143$	293014.	1.19825
$144$	13599.3	0.0546526
$145$	66731.2	0.263578
$146$	−79057.4	−0.306945
$147$	−36209.2	−0.138205
$148$	138574.	0.520027
$149$	−256426.	−0.946230	−0.473115	−	0.881001i	$-0.656870\pi$
$149$	−256426.	−0.946230	−0.473115	+	0.881001i	$0.656870\pi$
$150$	205748.	0.746635
$151$	36721.0	0.131061	0.0655303	−	0.997851i	$-0.479126\pi$
$151$	36721.0	0.131061	0.0655303	+	0.997851i	$0.479126\pi$
$152$	−25686.7	−0.0901778
$153$	121550.	0.419785
$154$	−184263.	−0.626091
$155$	−40154.3	−0.134247
$156$	−212357.	−0.698641
$157$	123544.	0.400012	0.200006	−	0.979795i	$-0.435904\pi$
$157$	123544.	0.400012	0.200006	+	0.979795i	$0.435904\pi$
$158$	−370529.	−1.18081
$159$	34781.2	0.109107
$160$	11937.2	0.0368641
$161$	580515.	1.76502
$162$	276543.	0.827895
$163$	306138.	0.902502	0.451251	−	0.892397i	$-0.350978\pi$
$163$	306138.	0.902502	0.451251	+	0.892397i	$0.350978\pi$
$164$	−26896.0	−0.0780869
$165$	76211.0	0.217925
$166$	393438.	1.10817
$167$	−596701.	−1.65564	−0.827819	−	0.560995i	$-0.810419\pi$
$167$	−596701.	−1.65564	−0.827819	+	0.560995i	$0.810419\pi$
$168$	133541.	0.365042
$169$	223574.	0.602151
$170$	106695.	0.283152
$171$	21320.9	0.0557590
$172$	130871.	0.337304
$173$	55163.9	0.140133	0.0700664	−	0.997542i	$-0.477679\pi$
$173$	55163.9	0.140133	0.0700664	+	0.997542i	$0.477679\pi$
$174$	394022.	0.986615
$175$	362444.	0.894636
$176$	−97256.5	−0.236667
$177$	787338.	1.88898
$178$	−21645.5	−0.0512056
$179$	−90282.8	−0.210607	−0.105303	−	0.994440i	$-0.533581\pi$
$179$	−90282.8	−0.210607	−0.105303	+	0.994440i	$0.533581\pi$
$180$	−9908.32	−0.0227939
$181$	106707.	0.242100	0.121050	−	0.992646i	$-0.461374\pi$
$181$	106707.	0.242100	0.121050	+	0.992646i	$0.461374\pi$
$182$	−374085.	−0.837128
$183$	−431836.	−0.953215
$184$	306403.	0.667188
$185$	−100963.	−0.216887
$186$	−237096.	−0.502507
$187$	−869276.	−1.81783
$188$	−300492.	−0.620066
$189$	396196.	0.806780
$190$	18715.1	0.0376104
$191$	−646690.	−1.28266	−0.641331	−	0.767264i	$-0.721618\pi$
$191$	−646690.	−1.28266	−0.641331	+	0.767264i	$0.721618\pi$
$192$	70484.8	0.137988
$193$	457039.	0.883201	0.441601	−	0.897212i	$-0.354411\pi$
$193$	457039.	0.883201	0.441601	+	0.897212i	$0.354411\pi$
$194$	151693.	0.289375
$195$	154721.	0.291382
$196$	−33666.9	−0.0625983
$197$	−513012.	−0.941808	−0.470904	−	0.882184i	$-0.656072\pi$
$197$	−513012.	−0.941808	−0.470904	+	0.882184i	$0.656072\pi$
$198$	80726.4	0.146336
$199$	−615301.	−1.10143	−0.550713	−	0.834695i	$-0.685644\pi$
$199$	−615301.	−1.10143	−0.550713	+	0.834695i	$0.685644\pi$
$200$	191303.	0.338179
$201$	479150.	0.836530
$202$	45150.6	0.0778548
$203$	694106.	1.18219
$204$	629991.	1.05989
$205$	19596.2	0.0325676
$206$	−340453.	−0.558971
$207$	−254325.	−0.412538
$208$	−197447.	−0.316440
$209$	−152478.	−0.241458
$210$	−97296.9	−0.152248
$211$	−483353.	−0.747409	−0.373705	−	0.927548i	$-0.621913\pi$
$211$	−483353.	−0.747409	−0.373705	+	0.927548i	$0.621913\pi$
$212$	32339.2	0.0494185
$213$	−404851.	−0.611429
$214$	689728.	1.02954
$215$	−95351.2	−0.140679
$216$	209117.	0.304969
$217$	−417666.	−0.602116
$218$	−881281.	−1.25595
$219$	340109.	0.479190
$220$	70860.1	0.0987064
$221$	−1.76477e6	−2.43057
$222$	−596150.	−0.811846
$223$	−26435.3	−0.0355978	−0.0177989	−	0.999842i	$-0.505666\pi$
$223$	−26435.3	−0.0355978	−0.0177989	+	0.999842i	$0.505666\pi$
$224$	124165.	0.165341
$225$	−158788.	−0.209104
$226$	−807526.	−1.05168
$227$	−812709.	−1.04682	−0.523408	−	0.852082i	$-0.675340\pi$
$227$	−812709.	−1.04682	−0.523408	+	0.852082i	$0.675340\pi$
$228$	110506.	0.140782
$229$	159021.	0.200385	0.100193	−	0.994968i	$-0.468054\pi$
$229$	159021.	0.200385	0.100193	+	0.994968i	$0.468054\pi$
$230$	−223242.	−0.278264
$231$	792710.	0.977428
$232$	366358.	0.446874
$233$	−59625.4	−0.0719517	−0.0359759	−	0.999353i	$-0.511454\pi$
$233$	−59625.4	−0.0719517	−0.0359759	+	0.999353i	$0.511454\pi$
$234$	163888.	0.195662
$235$	218935.	0.258611
$236$	732058.	0.855590
$237$	1.59403e6	1.84343
$238$	1.10979e6	1.26998
$239$	−742389.	−0.840691	−0.420346	−	0.907364i	$-0.638091\pi$
$239$	−742389.	−0.840691	−0.420346	+	0.907364i	$0.638091\pi$
$240$	−51354.5	−0.0575507
$241$	36559.2	0.0405465	0.0202733	−	0.999794i	$-0.493546\pi$
$241$	36559.2	0.0405465	0.0202733	+	0.999794i	$0.493546\pi$
$242$	66883.1	0.0734138
$243$	−395710.	−0.429894
$244$	−401516.	−0.431746
$245$	24529.4	0.0261078
$246$	115708.	0.121906
$247$	−309556.	−0.322847
$248$	−220449.	−0.227604
$249$	−1.69259e6	−1.73003
$250$	−285099.	−0.288500
$251$	1.18970e6	1.19193	0.595966	−	0.803010i	$-0.296769\pi$
$251$	1.18970e6	1.19193	0.595966	+	0.803010i	$0.296769\pi$
$252$	−103062.	−0.102234
$253$	1.81883e6	1.78645
$254$	522806.	0.508459
$255$	−459006.	−0.442046
$256$	65536.0	0.0625000
$257$	−1.99616e6	−1.88522	−0.942612	−	0.333891i	$-0.891638\pi$
$257$	−1.99616e6	−1.88522	−0.942612	+	0.333891i	$0.891638\pi$
$258$	−563013.	−0.526586
$259$	−1.05017e6	−0.972772
$260$	143858.	0.131978
$261$	−304090.	−0.276313
$262$	278503.	0.250655
$263$	−1.75336e6	−1.56308	−0.781541	−	0.623854i	$-0.785566\pi$
$263$	−1.75336e6	−1.56308	−0.781541	+	0.623854i	$0.785566\pi$
$264$	418402.	0.369474
$265$	−23562.0	−0.0206109
$266$	194666.	0.168688
$267$	93120.0	0.0799401
$268$	445508.	0.378895
$269$	−58101.7	−0.0489563	−0.0244781	−	0.999700i	$-0.507792\pi$
$269$	−58101.7	−0.0489563	−0.0244781	+	0.999700i	$0.507792\pi$
$270$	−152361.	−0.127193
$271$	191066.	0.158038	0.0790189	−	0.996873i	$-0.474821\pi$
$271$	191066.	0.158038	0.0790189	+	0.996873i	$0.474821\pi$
$272$	585759.	0.480061
$273$	1.60933e6	1.30689
$274$	−938241.	−0.754984
$275$	1.13559e6	0.905499
$276$	−1.31816e6	−1.04159
$277$	−1.61158e6	−1.26198	−0.630989	−	0.775792i	$-0.717351\pi$
$277$	−1.61158e6	−1.26198	−0.630989	+	0.775792i	$0.717351\pi$
$278$	930276.	0.721938
$279$	182981.	0.140733
$280$	−90465.6	−0.0689586
$281$	1.10967e6	0.838355	0.419177	−	0.907904i	$-0.362319\pi$
$281$	1.10967e6	0.838355	0.419177	+	0.907904i	$0.362319\pi$
$282$	1.29273e6	0.968022
$283$	−1.85363e6	−1.37581	−0.687903	−	0.725802i	$-0.741469\pi$
$283$	−1.85363e6	−1.37581	−0.687903	+	0.725802i	$0.741469\pi$
$284$	−376426.	−0.276939
$285$	−80513.3	−0.0587159
$286$	−1.17206e6	−0.847293
$287$	203830.	0.146071
$288$	−54397.2	−0.0386452
$289$	3.81564e6	2.68734
$290$	−266925.	−0.186378
$291$	−652590.	−0.451760
$292$	316230.	0.217043
$293$	468790.	0.319014	0.159507	−	0.987197i	$-0.449010\pi$
$293$	468790.	0.319014	0.159507	+	0.987197i	$0.449010\pi$
$294$	144837.	0.0977260
$295$	−533370.	−0.356840
$296$	−554294.	−0.367715
$297$	1.24133e6	0.816577
$298$	1.02570e6	0.669085
$299$	3.69252e6	2.38861
$300$	−822994.	−0.527951
$301$	−991798.	−0.630967
$302$	−146884.	−0.0926738
$303$	−194240.	−0.121544
$304$	102747.	0.0637654
$305$	292541.	0.180068
$306$	−486201.	−0.296833
$307$	−1.15839e6	−0.701472	−0.350736	−	0.936474i	$-0.614069\pi$
$307$	−1.15839e6	−0.701472	−0.350736	+	0.936474i	$0.614069\pi$
$308$	737053.	0.442713
$309$	1.46465e6	0.872643
$310$	160617.	0.0949266
$311$	1.31726e6	0.772271	0.386136	−	0.922442i	$-0.373810\pi$
$311$	1.31726e6	0.772271	0.386136	+	0.922442i	$0.373810\pi$
$312$	849426.	0.494014
$313$	1.51337e6	0.873140	0.436570	−	0.899670i	$-0.356193\pi$
$313$	1.51337e6	0.873140	0.436570	+	0.899670i	$0.356193\pi$
$314$	−494177.	−0.282851
$315$	75089.7	0.0426387
$316$	1.48212e6	0.834957
$317$	793249.	0.443365	0.221682	−	0.975119i	$-0.428845\pi$
$317$	793249.	0.443365	0.221682	+	0.975119i	$0.428845\pi$
$318$	−139125.	−0.0771502
$319$	2.17472e6	1.19654
$320$	−47748.9	−0.0260668
$321$	−2.96725e6	−1.60728
$322$	−2.32206e6	−1.24805
$323$	918349.	0.489781
$324$	−1.10617e6	−0.585410
$325$	2.30543e6	1.21072
$326$	−1.22455e6	−0.638165
$327$	3.79131e6	1.96074
$328$	107584.	0.0552158
$329$	2.27726e6	1.15991
$330$	−304844.	−0.154096
$331$	1.27130e6	0.637791	0.318896	−	0.947790i	$-0.396688\pi$
$331$	1.27130e6	0.637791	0.318896	+	0.947790i	$0.396688\pi$
$332$	−1.57375e6	−0.783594
$333$	460084.	0.227367
$334$	2.38680e6	1.17071
$335$	−324593.	−0.158026
$336$	−534166.	−0.258124
$337$	−3.22914e6	−1.54886	−0.774429	−	0.632661i	$-0.781963\pi$
$337$	−3.22914e6	−1.54886	−0.774429	+	0.632661i	$0.781963\pi$
$338$	−894297.	−0.425785
$339$	3.47402e6	1.64185
$340$	−426778.	−0.200219
$341$	−1.30860e6	−0.609427
$342$	−85283.6	−0.0394276
$343$	2.29308e6	1.05241
$344$	−523483.	−0.238510
$345$	960399.	0.434414
$346$	−220656.	−0.0990889
$347$	−3.58896e6	−1.60009	−0.800046	−	0.599939i	$-0.795191\pi$
$347$	−3.58896e6	−1.60009	−0.800046	+	0.599939i	$0.795191\pi$
$348$	−1.57609e6	−0.697642
$349$	−2.16986e6	−0.953606	−0.476803	−	0.879010i	$-0.658205\pi$
$349$	−2.16986e6	−0.953606	−0.476803	+	0.879010i	$0.658205\pi$
$350$	−1.44978e6	−0.632603
$351$	2.52011e6	1.09182
$352$	389026.	0.167349
$353$	−3.83421e6	−1.63772	−0.818858	−	0.573996i	$-0.805393\pi$
$353$	−3.83421e6	−1.63772	−0.818858	+	0.573996i	$0.805393\pi$
$354$	−3.14935e6	−1.33571
$355$	274260.	0.115503
$356$	86582.0	0.0362078
$357$	−4.77436e6	−1.98264
$358$	361131.	0.148922
$359$	−2.82546e6	−1.15705	−0.578526	−	0.815664i	$-0.696372\pi$
$359$	−2.82546e6	−1.15705	−0.578526	+	0.815664i	$0.696372\pi$
$360$	39633.3	0.0161177
$361$	−2.31501e6	−0.934944
$362$	−426827.	−0.171191
$363$	−287734.	−0.114611
$364$	1.49634e6	0.591939
$365$	−230402.	−0.0905219
$366$	1.72734e6	0.674025
$367$	−2.72488e6	−1.05604	−0.528022	−	0.849231i	$-0.677066\pi$
$367$	−2.72488e6	−1.05604	−0.528022	+	0.849231i	$0.677066\pi$
$368$	−1.22561e6	−0.471773
$369$	−89298.6	−0.0341412
$370$	403853.	0.153363
$371$	−245081.	−0.0924431
$372$	948385.	0.355326
$373$	−2.06470e6	−0.768396	−0.384198	−	0.923251i	$-0.625522\pi$
$373$	−2.06470e6	−0.768396	−0.384198	+	0.923251i	$0.625522\pi$
$374$	3.47710e6	1.28540
$375$	1.22651e6	0.450395
$376$	1.20197e6	0.438453
$377$	4.41505e6	1.59986
$378$	−1.58478e6	−0.570480
$379$	−879611.	−0.314552	−0.157276	−	0.987555i	$-0.550271\pi$
$379$	−879611.	−0.314552	−0.157276	+	0.987555i	$0.550271\pi$
$380$	−74860.4	−0.0265946
$381$	−2.24914e6	−0.793786
$382$	2.58676e6	0.906979
$383$	1.08234e6	0.377021	0.188511	−	0.982071i	$-0.439634\pi$
$383$	1.08234e6	0.377021	0.188511	+	0.982071i	$0.439634\pi$
$384$	−281939.	−0.0975725
$385$	−537010.	−0.184642
$386$	−1.82816e6	−0.624518
$387$	434510.	0.147476
$388$	−606771.	−0.204619
$389$	4.48669e6	1.50332	0.751661	−	0.659550i	$-0.229253\pi$
$389$	4.48669e6	1.50332	0.751661	+	0.659550i	$0.229253\pi$
$390$	−618884.	−0.206038
$391$	−1.09545e7	−3.62368
$392$	134668.	0.0442637
$393$	−1.19814e6	−0.391313
$394$	2.05205e6	0.665959
$395$	−1.07985e6	−0.348235
$396$	−322905.	−0.103475
$397$	−1.44653e6	−0.460629	−0.230314	−	0.973116i	$-0.573975\pi$
$397$	−1.44653e6	−0.460629	−0.230314	+	0.973116i	$0.573975\pi$
$398$	2.46120e6	0.778825
$399$	−837461.	−0.263349
$400$	−765211.	−0.239128
$401$	5.12175e6	1.59059	0.795294	−	0.606224i	$-0.207316\pi$
$401$	5.12175e6	1.59059	0.795294	+	0.606224i	$0.207316\pi$
$402$	−1.91660e6	−0.591516
$403$	−2.65668e6	−0.814848
$404$	−180603.	−0.0550517
$405$	805946.	0.244157
$406$	−2.77642e6	−0.835931
$407$	−3.29033e6	−0.984584
$408$	−2.51997e6	−0.749453
$409$	4.30793e6	1.27339	0.636693	−	0.771117i	$-0.280302\pi$
$409$	4.30793e6	1.27339	0.636693	+	0.771117i	$0.280302\pi$
$410$	−78384.7	−0.0230288
$411$	4.03636e6	1.17865
$412$	1.36181e6	0.395252
$413$	−5.54786e6	−1.60048
$414$	1.01730e6	0.291708
$415$	1.14662e6	0.326813
$416$	789787.	0.223757
$417$	−4.00210e6	−1.12706
$418$	609912.	0.170737
$419$	4.20512e6	1.17015	0.585077	−	0.810978i	$-0.301064\pi$
$419$	4.20512e6	1.17015	0.585077	+	0.810978i	$0.301064\pi$
$420$	389188.	0.107655
$421$	4.33022e6	1.19071	0.595353	−	0.803464i	$-0.297012\pi$
$421$	4.33022e6	1.19071	0.595353	+	0.803464i	$0.297012\pi$
$422$	1.93341e6	0.528498
$423$	−997676.	−0.271106
$424$	−129357.	−0.0349441
$425$	−6.83943e6	−1.83674
$426$	1.61940e6	0.432346
$427$	3.04287e6	0.807632
$428$	−2.75891e6	−0.727996
$429$	5.04225e6	1.32276
$430$	381405.	0.0994753
$431$	5.17175e6	1.34105	0.670524	−	0.741888i	$-0.266069\pi$
$431$	5.17175e6	1.34105	0.670524	+	0.741888i	$0.266069\pi$
$432$	−836468.	−0.215645
$433$	−233332.	−0.0598074	−0.0299037	−	0.999553i	$-0.509520\pi$
$433$	−233332.	−0.0598074	−0.0299037	+	0.999553i	$0.509520\pi$
$434$	1.67066e6	0.425760
$435$	1.14832e6	0.290965
$436$	3.52512e6	0.888092
$437$	−1.92151e6	−0.481325
$438$	−1.36044e6	−0.338838
$439$	−3.29986e6	−0.817211	−0.408606	−	0.912711i	$-0.633985\pi$
$439$	−3.29986e6	−0.817211	−0.408606	+	0.912711i	$0.633985\pi$
$440$	−283441.	−0.0697960
$441$	−111779.	−0.0273693
$442$	7.05910e6	1.71867
$443$	1.37780e6	0.333562	0.166781	−	0.985994i	$-0.446663\pi$
$443$	1.37780e6	0.333562	0.166781	+	0.985994i	$0.446663\pi$
$444$	2.38460e6	0.574062
$445$	−63082.8	−0.0151012
$446$	105741.	0.0251714
$447$	−4.41263e6	−1.04455
$448$	−496661.	−0.116914
$449$	866827.	0.202916	0.101458	−	0.994840i	$-0.467649\pi$
$449$	866827.	0.202916	0.101458	+	0.994840i	$0.467649\pi$
$450$	635152.	0.147859
$451$	638626.	0.147844
$452$	3.23011e6	0.743654
$453$	631902.	0.144679
$454$	3.25083e6	0.740210
$455$	−1.09022e6	−0.246880
$456$	−442022.	−0.0995479
$457$	5.48827e6	1.22926	0.614631	−	0.788815i	$-0.289305\pi$
$457$	5.48827e6	1.22926	0.614631	+	0.788815i	$0.289305\pi$
$458$	−636083.	−0.141694
$459$	−7.47633e6	−1.65637
$460$	892969.	0.196762
$461$	6.52468e6	1.42991	0.714953	−	0.699173i	$-0.246448\pi$
$461$	6.52468e6	1.42991	0.714953	+	0.699173i	$0.246448\pi$
$462$	−3.17084e6	−0.691146
$463$	847051.	0.183636	0.0918178	−	0.995776i	$-0.470732\pi$
$463$	847051.	0.183636	0.0918178	+	0.995776i	$0.470732\pi$
$464$	−1.46543e6	−0.315988
$465$	−690984.	−0.148196
$466$	238501.	0.0508776
$467$	−4.69693e6	−0.996603	−0.498302	−	0.867004i	$-0.666043\pi$
$467$	−4.69693e6	−0.996603	−0.498302	+	0.867004i	$0.666043\pi$
$468$	−655552.	−0.138354
$469$	−3.37626e6	−0.708768
$470$	−875742.	−0.182865
$471$	2.12597e6	0.441576
$472$	−2.92823e6	−0.604993
$473$	−3.10743e6	−0.638629
$474$	−6.37614e6	−1.30350
$475$	−1.19969e6	−0.243970
$476$	−4.43915e6	−0.898012
$477$	107371.	0.0216068
$478$	2.96955e6	0.594458
$479$	2.95657e6	0.588776	0.294388	−	0.955686i	$-0.404884\pi$
$479$	2.95657e6	0.588776	0.294388	+	0.955686i	$0.404884\pi$
$480$	205418.	0.0406945
$481$	−6.67991e6	−1.31646
$482$	−146237.	−0.0286707
$483$	9.98962e6	1.94841
$484$	−267532.	−0.0519114
$485$	442088.	0.0853403
$486$	1.58284e6	0.303981
$487$	2.50852e6	0.479286	0.239643	−	0.970861i	$-0.422970\pi$
$487$	2.50852e6	0.479286	0.239643	+	0.970861i	$0.422970\pi$
$488$	1.60606e6	0.305291
$489$	5.26808e6	0.996278
$490$	−98117.4	−0.0184610
$491$	9.20277e6	1.72272	0.861361	−	0.507994i	$-0.169613\pi$
$491$	9.20277e6	1.72272	0.861361	+	0.507994i	$0.169613\pi$
$492$	−462832.	−0.0862006
$493$	−1.30980e7	−2.42710
$494$	1.23822e6	0.228287
$495$	235266.	0.0431565
$496$	881798.	0.160940
$497$	2.85272e6	0.518047
$498$	6.77036e6	1.22332
$499$	−4.58684e6	−0.824635	−0.412318	−	0.911040i	$-0.635281\pi$
$499$	−4.58684e6	−0.824635	−0.412318	+	0.911040i	$0.635281\pi$
$500$	1.14040e6	0.204000
$501$	−1.02681e7	−1.82767
$502$	−4.75878e6	−0.842823
$503$	−5.00732e6	−0.882440	−0.441220	−	0.897399i	$-0.645454\pi$
$503$	−5.00732e6	−0.882440	−0.441220	+	0.897399i	$0.645454\pi$
$504$	412247.	0.0722904
$505$	131585.	0.0229604
$506$	−7.27531e6	−1.26321
$507$	3.84731e6	0.664718
$508$	−2.09122e6	−0.359535
$509$	−6.76573e6	−1.15750	−0.578749	−	0.815505i	$-0.696459\pi$
$509$	−6.76573e6	−1.15750	−0.578749	+	0.815505i	$0.696459\pi$
$510$	1.83602e6	0.312574
$511$	−2.39653e6	−0.406004
$512$	−262144.	−0.0441942
$513$	−1.31141e6	−0.220011
$514$	7.98465e6	1.33305
$515$	−992203.	−0.164847
$516$	2.25205e6	0.372352
$517$	7.13496e6	1.17399
$518$	4.20069e6	0.687854
$519$	949272.	0.154694
$520$	−575431.	−0.0933222
$521$	6.65292e6	1.07379	0.536893	−	0.843650i	$-0.319598\pi$
$521$	6.65292e6	1.07379	0.536893	+	0.843650i	$0.319598\pi$
$522$	1.21636e6	0.195383
$523$	1.61858e6	0.258749	0.129374	−	0.991596i	$-0.458703\pi$
$523$	1.61858e6	0.258749	0.129374	+	0.991596i	$0.458703\pi$
$524$	−1.11401e6	−0.177240
$525$	6.23702e6	0.987594
$526$	7.01344e6	1.10527
$527$	7.88149e6	1.23618
$528$	−1.67361e6	−0.261258
$529$	1.64843e7	2.56112
$530$	94248.1	0.0145741
$531$	2.43054e6	0.374081
$532$	−778662.	−0.119281
$533$	1.29652e6	0.197679
$534$	−372480.	−0.0565262
$535$	2.01012e6	0.303625
$536$	−1.78203e6	−0.267919
$537$	−1.55361e6	−0.232490
$538$	232407.	0.0346173
$539$	799395.	0.118519
$540$	609443.	0.0899391
$541$	−5.54011e6	−0.813814	−0.406907	−	0.913470i	$-0.633393\pi$
$541$	−5.54011e6	−0.813814	−0.406907	+	0.913470i	$0.633393\pi$
$542$	−764265.	−0.111750
$543$	1.83623e6	0.267256
$544$	−2.34304e6	−0.339455
$545$	−2.56837e6	−0.370396
$546$	−6.43733e6	−0.924111
$547$	5.08966e6	0.727312	0.363656	−	0.931533i	$-0.381528\pi$
$547$	5.08966e6	0.727312	0.363656	+	0.931533i	$0.381528\pi$
$548$	3.75296e6	0.533855
$549$	−1.33309e6	−0.188768
$550$	−4.54234e6	−0.640284
$551$	−2.29749e6	−0.322385
$552$	5.27264e6	0.736514
$553$	−1.12321e7	−1.56189
$554$	6.44631e6	0.892353
$555$	−1.73740e6	−0.239424
$556$	−3.72111e6	−0.510487
$557$	−2.56423e6	−0.350202	−0.175101	−	0.984551i	$-0.556025\pi$
$557$	−2.56423e6	−0.350202	−0.175101	+	0.984551i	$0.556025\pi$
$558$	−731924.	−0.0995131
$559$	−6.30860e6	−0.853893
$560$	361863.	0.0487611
$561$	−1.49587e7	−2.00672
$562$	−4.43868e6	−0.592806
$563$	−2.17607e6	−0.289336	−0.144668	−	0.989480i	$-0.546211\pi$
$563$	−2.17607e6	−0.289336	−0.144668	+	0.989480i	$0.546211\pi$
$564$	−5.17092e6	−0.684495
$565$	−2.35342e6	−0.310155
$566$	7.41453e6	0.972842
$567$	8.38307e6	1.09508
$568$	1.50570e6	0.195825
$569$	1.01231e6	0.131079	0.0655396	−	0.997850i	$-0.479123\pi$
$569$	1.01231e6	0.131079	0.0655396	+	0.997850i	$0.479123\pi$
$570$	322053.	0.0415184
$571$	4.89623e6	0.628451	0.314225	−	0.949348i	$-0.398255\pi$
$571$	4.89623e6	0.628451	0.314225	+	0.949348i	$0.398255\pi$
$572$	4.68823e6	0.599127
$573$	−1.11284e7	−1.41594
$574$	−815320.	−0.103288
$575$	1.43105e7	1.80503
$576$	217589.	0.0273263
$577$	−9.71524e6	−1.21483	−0.607413	−	0.794386i	$-0.707793\pi$
$577$	−9.71524e6	−1.21483	−0.607413	+	0.794386i	$0.707793\pi$
$578$	−1.52626e7	−1.90024
$579$	7.86482e6	0.974972
$580$	1.06770e6	0.131789
$581$	1.19266e7	1.46581
$582$	2.61036e6	0.319443
$583$	−767870.	−0.0935656
$584$	−1.26492e6	−0.153472
$585$	477628.	0.0577033
$586$	−1.87516e6	−0.225577
$587$	−3.69221e6	−0.442274	−0.221137	−	0.975243i	$-0.570977\pi$
$587$	−3.69221e6	−0.442274	−0.221137	+	0.975243i	$0.570977\pi$
$588$	−579346.	−0.0691027
$589$	1.38248e6	0.164199
$590$	2.13348e6	0.252324
$591$	−8.82802e6	−1.03967
$592$	2.21718e6	0.260014
$593$	−2.13984e6	−0.249887	−0.124944	−	0.992164i	$-0.539875\pi$
$593$	−2.13984e6	−0.249887	−0.124944	+	0.992164i	$0.539875\pi$
$594$	−4.96533e6	−0.577407
$595$	3.23432e6	0.374533
$596$	−4.10282e6	−0.473115
$597$	−1.05882e7	−1.21587
$598$	−1.47701e7	−1.68900
$599$	−1.50623e7	−1.71524	−0.857619	−	0.514286i	$-0.828057\pi$
$599$	−1.50623e7	−1.71524	−0.857619	+	0.514286i	$0.828057\pi$
$600$	3.29198e6	0.373318
$601$	−1.34923e7	−1.52370	−0.761849	−	0.647755i	$-0.775708\pi$
$601$	−1.34923e7	−1.52370	−0.761849	+	0.647755i	$0.775708\pi$
$602$	3.96719e6	0.446161
$603$	1.47915e6	0.165661
$604$	587536.	0.0655303
$605$	194921.	0.0216507
$606$	776961.	0.0859444
$607$	−222180.	−0.0244756	−0.0122378	−	0.999925i	$-0.503896\pi$
$607$	−222180.	−0.0244756	−0.0122378	+	0.999925i	$0.503896\pi$
$608$	−410988.	−0.0450889
$609$	1.19443e7	1.30502
$610$	−1.17016e6	−0.127327
$611$	1.44851e7	1.56971
$612$	1.94480e6	0.209893
$613$	−3.66739e6	−0.394190	−0.197095	−	0.980384i	$-0.563151\pi$
$613$	−3.66739e6	−0.394190	−0.197095	+	0.980384i	$0.563151\pi$
$614$	4.63358e6	0.496016
$615$	337215.	0.0359517
$616$	−2.94821e6	−0.313045
$617$	−1.35201e7	−1.42977	−0.714886	−	0.699241i	$-0.753521\pi$
$617$	−1.35201e7	−1.42977	−0.714886	+	0.699241i	$0.753521\pi$
$618$	−5.85858e6	−0.617052
$619$	5.14426e6	0.539630	0.269815	−	0.962912i	$-0.413037\pi$
$619$	5.14426e6	0.539630	0.269815	+	0.962912i	$0.413037\pi$
$620$	−642469.	−0.0671233
$621$	1.56431e7	1.62777
$622$	−5.26903e6	−0.546078
$623$	−656157.	−0.0677310
$624$	−3.39770e6	−0.349321
$625$	8.51007e6	0.871431
$626$	−6.05347e6	−0.617403
$627$	−2.62387e6	−0.266547
$628$	1.97671e6	0.200006
$629$	1.98171e7	1.99716
$630$	−300359.	−0.0301501
$631$	−1.10859e7	−1.10841	−0.554204	−	0.832381i	$-0.686977\pi$
$631$	−1.10859e7	−1.10841	−0.554204	+	0.832381i	$0.686977\pi$
$632$	−5.92846e6	−0.590404
$633$	−8.31764e6	−0.825070
$634$	−3.17299e6	−0.313506
$635$	1.52365e6	0.149951
$636$	556499.	0.0545534
$637$	1.62290e6	0.158469
$638$	−8.69889e6	−0.846082
$639$	−1.24979e6	−0.121083
$640$	190996.	0.0184320
$641$	2.04308e6	0.196400	0.0981998	−	0.995167i	$-0.468692\pi$
$641$	2.04308e6	0.196400	0.0981998	+	0.995167i	$0.468692\pi$
$642$	1.18690e7	1.13652
$643$	3.32184e6	0.316848	0.158424	−	0.987371i	$-0.449359\pi$
$643$	3.32184e6	0.316848	0.158424	+	0.987371i	$0.449359\pi$
$644$	9.28824e6	0.882508
$645$	−1.64082e6	−0.155297
$646$	−3.67340e6	−0.346327
$647$	−4.74219e6	−0.445367	−0.222684	−	0.974891i	$-0.571482\pi$
$647$	−4.74219e6	−0.445367	−0.222684	+	0.974891i	$0.571482\pi$
$648$	4.42469e6	0.413947
$649$	−1.73822e7	−1.61992
$650$	−9.22170e6	−0.856106
$651$	−7.18728e6	−0.664680
$652$	4.89821e6	0.451251
$653$	5.28532e6	0.485052	0.242526	−	0.970145i	$-0.422024\pi$
$653$	5.28532e6	0.485052	0.242526	+	0.970145i	$0.422024\pi$
$654$	−1.51653e7	−1.38645
$655$	811659.	0.0739215
$656$	−430336.	−0.0390434
$657$	1.04993e6	0.0948955
$658$	−9.10905e6	−0.820178
$659$	6.67742e6	0.598957	0.299478	−	0.954103i	$-0.403187\pi$
$659$	6.67742e6	0.598957	0.299478	+	0.954103i	$0.403187\pi$
$660$	1.21938e6	0.108963
$661$	1.01972e7	0.907771	0.453885	−	0.891060i	$-0.350038\pi$
$661$	1.01972e7	0.907771	0.453885	+	0.891060i	$0.350038\pi$
$662$	−5.08520e6	−0.450986
$663$	−3.03686e7	−2.68312
$664$	6.29501e6	0.554084
$665$	567326.	0.0497483
$666$	−1.84034e6	−0.160772
$667$	2.74055e7	2.38519
$668$	−9.54721e6	−0.827819
$669$	−454905.	−0.0392966
$670$	1.29837e6	0.111741
$671$	9.53370e6	0.817439
$672$	2.13666e6	0.182521
$673$	1.42122e7	1.20955	0.604774	−	0.796397i	$-0.293264\pi$
$673$	1.42122e7	1.20955	0.604774	+	0.796397i	$0.293264\pi$
$674$	1.29165e7	1.09521
$675$	9.76676e6	0.825071
$676$	3.57719e6	0.301075
$677$	1.46664e7	1.22985	0.614924	−	0.788586i	$-0.289187\pi$
$677$	1.46664e7	1.22985	0.614924	+	0.788586i	$0.289187\pi$
$678$	−1.38961e7	−1.16096
$679$	4.59839e6	0.382764
$680$	1.70711e6	0.141576
$681$	−1.39853e7	−1.15559
$682$	5.23441e6	0.430930
$683$	692122.	0.0567716	0.0283858	−	0.999597i	$-0.490963\pi$
$683$	692122.	0.0567716	0.0283858	+	0.999597i	$0.490963\pi$
$684$	341134.	0.0278795
$685$	−2.73437e6	−0.222654
$686$	−9.17231e6	−0.744164
$687$	2.73646e6	0.221206
$688$	2.09393e6	0.168652
$689$	−1.55890e6	−0.125104
$690$	−3.84160e6	−0.307177
$691$	3.64423e6	0.290342	0.145171	−	0.989407i	$-0.453627\pi$
$691$	3.64423e6	0.290342	0.145171	+	0.989407i	$0.453627\pi$
$692$	882623.	0.0700664
$693$	2.44712e6	0.193563
$694$	1.43558e7	1.13144
$695$	2.71116e6	0.212909
$696$	6.30436e6	0.493307
$697$	−3.84633e6	−0.299892
$698$	8.67946e6	0.674301
$699$	−1.02605e6	−0.0794280
$700$	5.79911e6	0.447318
$701$	2.14432e7	1.64814	0.824070	−	0.566488i	$-0.191698\pi$
$701$	2.14432e7	1.64814	0.824070	+	0.566488i	$0.191698\pi$
$702$	−1.00804e7	−0.772035
$703$	3.47608e6	0.265278
$704$	−1.55610e6	−0.118333
$705$	3.76748e6	0.285482
$706$	1.53368e7	1.15804
$707$	1.36869e6	0.102981
$708$	1.25974e7	0.944491
$709$	9.52389e6	0.711539	0.355769	−	0.934574i	$-0.384219\pi$
$709$	9.52389e6	0.711539	0.355769	+	0.934574i	$0.384219\pi$
$710$	−1.09704e6	−0.0816727
$711$	4.92084e6	0.365060
$712$	−346328.	−0.0256028
$713$	−1.64908e7	−1.21484
$714$	1.90974e7	1.40194
$715$	−3.41580e6	−0.249877
$716$	−1.44453e6	−0.105303
$717$	−1.27752e7	−0.928045
$718$	1.13018e7	0.818159
$719$	−1.15568e6	−0.0833711	−0.0416855	−	0.999131i	$-0.513273\pi$
$719$	−1.15568e6	−0.0833711	−0.0416855	+	0.999131i	$0.513273\pi$
$720$	−158533.	−0.0113970
$721$	−1.03204e7	−0.739366
$722$	9.26005e6	0.661105
$723$	629117.	0.0447596
$724$	1.70731e6	0.121050
$725$	1.71106e7	1.20899
$726$	1.15094e6	0.0810420
$727$	−1.99629e7	−1.40084	−0.700419	−	0.713732i	$-0.747003\pi$
$727$	−1.99629e7	−1.40084	−0.700419	+	0.713732i	$0.747003\pi$
$728$	−5.98536e6	−0.418564
$729$	9.99052e6	0.696256
$730$	921607.	0.0640087
$731$	1.87155e7	1.29541
$732$	−6.90937e6	−0.476607
$733$	−1.87823e7	−1.29119	−0.645593	−	0.763682i	$-0.723390\pi$
$733$	−1.87823e7	−1.29119	−0.645593	+	0.763682i	$0.723390\pi$
$734$	1.08995e7	0.746735
$735$	422106.	0.0288206
$736$	4.90245e6	0.333594
$737$	−1.05783e7	−0.717374
$738$	357194.	0.0241415
$739$	−1.38934e7	−0.935830	−0.467915	−	0.883773i	$-0.654995\pi$
$739$	−1.38934e7	−0.935830	−0.467915	+	0.883773i	$0.654995\pi$
$740$	−1.61541e6	−0.108444
$741$	−5.32690e6	−0.356393
$742$	980323.	0.0653672
$743$	5.72049e6	0.380156	0.190078	−	0.981769i	$-0.439126\pi$
$743$	5.72049e6	0.380156	0.190078	+	0.981769i	$0.439126\pi$
$744$	−3.79354e6	−0.251254
$745$	2.98927e6	0.197322
$746$	8.25881e6	0.543338
$747$	−5.22508e6	−0.342603
$748$	−1.39084e7	−0.908916
$749$	2.09083e7	1.36180
$750$	−4.90605e6	−0.318477
$751$	−1.12332e7	−0.726781	−0.363390	−	0.931637i	$-0.618381\pi$
$751$	−1.12332e7	−0.726781	−0.363390	+	0.931637i	$0.618381\pi$
$752$	−4.80787e6	−0.310033
$753$	2.04725e7	1.31578
$754$	−1.76602e7	−1.13127
$755$	−428073.	−0.0273307
$756$	6.33913e6	0.403390
$757$	−885804.	−0.0561821	−0.0280911	−	0.999605i	$-0.508943\pi$
$757$	−885804.	−0.0561821	−0.0280911	+	0.999605i	$0.508943\pi$
$758$	3.51845e6	0.222422
$759$	3.12988e7	1.97207
$760$	299442.	0.0188052
$761$	−2.39502e7	−1.49916	−0.749580	−	0.661914i	$-0.769745\pi$
$761$	−2.39502e7	−1.49916	−0.749580	+	0.661914i	$0.769745\pi$
$762$	8.99655e6	0.561292
$763$	−2.67150e7	−1.66128
$764$	−1.03470e7	−0.641331
$765$	−1.41697e6	−0.0875398
$766$	−4.32935e6	−0.266594
$767$	−3.52887e7	−2.16594
$768$	1.12776e6	0.0689942
$769$	2.21567e7	1.35110	0.675552	−	0.737312i	$-0.263905\pi$
$769$	2.21567e7	1.35110	0.675552	+	0.737312i	$0.263905\pi$
$770$	2.14804e6	0.130562
$771$	−3.43504e7	−2.08111
$772$	7.31262e6	0.441601
$773$	2.67193e7	1.60834	0.804168	−	0.594402i	$-0.202611\pi$
$773$	2.67193e7	1.60834	0.804168	+	0.594402i	$0.202611\pi$
$774$	−1.73804e6	−0.104281
$775$	−1.02960e7	−0.615766
$776$	2.42709e6	0.144687
$777$	−1.80716e7	−1.07385
$778$	−1.79468e7	−1.06301
$779$	−674678.	−0.0398339
$780$	2.47553e6	0.145691
$781$	8.93795e6	0.524337
$782$	4.38179e7	2.56233
$783$	1.87040e7	1.09026
$784$	−538670.	−0.0312992
$785$	−1.44021e6	−0.0834164
$786$	4.79254e6	0.276700
$787$	1.13376e7	0.652505	0.326252	−	0.945283i	$-0.394214\pi$
$787$	1.13376e7	0.652505	0.326252	+	0.945283i	$0.394214\pi$
$788$	−8.20820e6	−0.470904
$789$	−3.01722e7	−1.72550
$790$	4.31942e6	0.246239
$791$	−2.44792e7	−1.39109
$792$	1.29162e6	0.0731682
$793$	1.93550e7	1.09297
$794$	5.78612e6	0.325714
$795$	−405460.	−0.0227526
$796$	−9.84482e6	−0.550713
$797$	3.02752e7	1.68827	0.844135	−	0.536131i	$-0.180115\pi$
$797$	3.02752e7	1.68827	0.844135	+	0.536131i	$0.180115\pi$
$798$	3.34984e6	0.186216
$799$	−4.29726e7	−2.38136
$800$	3.06084e6	0.169089
$801$	287465.	0.0158308
$802$	−2.04870e7	−1.12472
$803$	−7.50864e6	−0.410934
$804$	7.66640e6	0.418265
$805$	−6.76732e6	−0.368067
$806$	1.06267e7	0.576184
$807$	−999827.	−0.0540432
$808$	722410.	0.0389274
$809$	1.48319e7	0.796754	0.398377	−	0.917222i	$-0.369574\pi$
$809$	1.48319e7	0.796754	0.398377	+	0.917222i	$0.369574\pi$
$810$	−3.22378e6	−0.172645
$811$	2.51247e6	0.134137	0.0670686	−	0.997748i	$-0.478635\pi$
$811$	2.51247e6	0.134137	0.0670686	+	0.997748i	$0.478635\pi$
$812$	1.11057e7	0.591093
$813$	3.28791e6	0.174459
$814$	1.31613e7	0.696206
$815$	−3.56879e6	−0.188203
$816$	1.00799e7	0.529943
$817$	3.28285e6	0.172067
$818$	−1.72317e7	−0.900420
$819$	4.96806e6	0.258808
$820$	313539.	0.0162838
$821$	−2.86543e7	−1.48365	−0.741825	−	0.670593i	$-0.766040\pi$
$821$	−2.86543e7	−1.48365	−0.741825	+	0.670593i	$0.766040\pi$
$822$	−1.61454e7	−0.833432
$823$	−2.29169e7	−1.17939	−0.589693	−	0.807627i	$-0.700751\pi$
$823$	−2.29169e7	−1.17939	−0.589693	+	0.807627i	$0.700751\pi$
$824$	−5.44725e6	−0.279485
$825$	1.95414e7	0.999586
$826$	2.21915e7	1.13171
$827$	−2.54816e7	−1.29558	−0.647789	−	0.761820i	$-0.724306\pi$
$827$	−2.54816e7	−1.29558	−0.647789	+	0.761820i	$0.724306\pi$
$828$	−4.06921e6	−0.206269
$829$	−7.67593e6	−0.387922	−0.193961	−	0.981009i	$-0.562134\pi$
$829$	−7.67593e6	−0.387922	−0.193961	+	0.981009i	$0.562134\pi$
$830$	−4.58648e6	−0.231092
$831$	−2.77323e7	−1.39311
$832$	−3.15915e6	−0.158220
$833$	−4.81462e6	−0.240408
$834$	1.60084e7	0.796952
$835$	6.95601e6	0.345258
$836$	−2.43965e6	−0.120729
$837$	−1.12548e7	−0.555296
$838$	−1.68205e7	−0.827424
$839$	2.05736e7	1.00903	0.504517	−	0.863402i	$-0.331671\pi$
$839$	2.05736e7	1.00903	0.504517	+	0.863402i	$0.331671\pi$
$840$	−1.55675e6	−0.0761239
$841$	1.22569e7	0.597572
$842$	−1.73209e7	−0.841957
$843$	1.90954e7	0.925466
$844$	−7.73365e6	−0.373705
$845$	−2.60631e6	−0.125569
$846$	3.99070e6	0.191701
$847$	2.02748e6	0.0971065
$848$	517427.	0.0247092
$849$	−3.18977e7	−1.51876
$850$	2.73577e7	1.29877
$851$	−4.14642e7	−1.96268
$852$	−6.47761e6	−0.305715
$853$	2.83130e7	1.33233	0.666167	−	0.745803i	$-0.267934\pi$
$853$	2.83130e7	1.33233	0.666167	+	0.745803i	$0.267934\pi$
$854$	−1.21715e7	−0.571082
$855$	−248547.	−0.0116277
$856$	1.10357e7	0.514771
$857$	8.95870e6	0.416671	0.208335	−	0.978057i	$-0.433195\pi$
$857$	8.95870e6	0.416671	0.208335	+	0.978057i	$0.433195\pi$
$858$	−2.01690e7	−0.935333
$859$	1.67431e7	0.774199	0.387099	−	0.922038i	$-0.373477\pi$
$859$	1.67431e7	0.774199	0.387099	+	0.922038i	$0.373477\pi$
$860$	−1.52562e6	−0.0703396
$861$	3.50755e6	0.161249
$862$	−2.06870e7	−0.948264
$863$	−1.28254e7	−0.586196	−0.293098	−	0.956082i	$-0.594686\pi$
$863$	−1.28254e7	−0.586196	−0.293098	+	0.956082i	$0.594686\pi$
$864$	3.34587e6	0.152484
$865$	−643070.	−0.0292226
$866$	933329.	0.0422902
$867$	6.56604e7	2.96658
$868$	−6.68266e6	−0.301058
$869$	−3.51917e7	−1.58085
$870$	−4.59329e6	−0.205743
$871$	−2.14756e7	−0.959181
$872$	−1.41005e7	−0.627976
$873$	−2.01457e6	−0.0894636
$874$	7.68602e6	0.340348
$875$	−8.64244e6	−0.381607
$876$	5.44174e6	0.239595
$877$	3.69190e7	1.62088	0.810439	−	0.585823i	$-0.199228\pi$
$877$	3.69190e7	1.62088	0.810439	+	0.585823i	$0.199228\pi$
$878$	1.31994e7	0.577856
$879$	8.06703e6	0.352161
$880$	1.13376e6	0.0493532
$881$	−3.95146e7	−1.71521	−0.857605	−	0.514308i	$-0.828049\pi$
$881$	−3.95146e7	−1.71521	−0.857605	+	0.514308i	$0.828049\pi$
$882$	447115.	0.0193530
$883$	1.73805e7	0.750172	0.375086	−	0.926990i	$-0.377613\pi$
$883$	1.73805e7	0.750172	0.375086	+	0.926990i	$0.377613\pi$
$884$	−2.82364e7	−1.21529
$885$	−9.17835e6	−0.393918
$886$	−5.51120e6	−0.235864
$887$	−5.95831e6	−0.254281	−0.127140	−	0.991885i	$-0.540580\pi$
$887$	−5.95831e6	−0.254281	−0.127140	+	0.991885i	$0.540580\pi$
$888$	−9.53841e6	−0.405923
$889$	1.58482e7	0.672553
$890$	252331.	0.0106781
$891$	2.62652e7	1.10838
$892$	−422965.	−0.0177989
$893$	−7.53774e6	−0.316310
$894$	1.76505e7	0.738608
$895$	1.05247e6	0.0439188
$896$	1.98664e6	0.0826704
$897$	6.35417e7	2.63680
$898$	−3.46731e6	−0.143483
$899$	−1.97176e7	−0.813683
$900$	−2.54061e6	−0.104552
$901$	4.62475e6	0.189791
$902$	−2.55450e6	−0.104542
$903$	−1.70671e7	−0.696529
$904$	−1.29204e7	−0.525842
$905$	−1.24393e6	−0.0504863
$906$	−2.52761e6	−0.102303
$907$	−3.06111e7	−1.23555	−0.617776	−	0.786354i	$-0.711966\pi$
$907$	−3.06111e7	−1.23555	−0.617776	+	0.786354i	$0.711966\pi$
$908$	−1.30033e7	−0.523408
$909$	−599626.	−0.0240697
$910$	4.36088e6	0.174570
$911$	−2.42170e7	−0.966774	−0.483387	−	0.875407i	$-0.660594\pi$
$911$	−2.42170e7	−0.966774	−0.483387	+	0.875407i	$0.660594\pi$
$912$	1.76809e6	0.0703910
$913$	3.73676e7	1.48360
$914$	−2.19531e7	−0.869220
$915$	5.03410e6	0.198778
$916$	2.54433e6	0.100193
$917$	8.44249e6	0.331549
$918$	2.99053e7	1.17123
$919$	9.71885e6	0.379600	0.189800	−	0.981823i	$-0.439216\pi$
$919$	9.71885e6	0.379600	0.189800	+	0.981823i	$0.439216\pi$
$920$	−3.57188e6	−0.139132
$921$	−1.99339e7	−0.774360
$922$	−2.60987e7	−1.01110
$923$	1.81455e7	0.701076
$924$	1.26834e7	0.488714
$925$	−2.58882e7	−0.994826
$926$	−3.38820e6	−0.129850
$927$	4.52141e6	0.172812
$928$	5.86172e6	0.223437
$929$	−1.64495e7	−0.625336	−0.312668	−	0.949862i	$-0.601223\pi$
$929$	−1.64495e7	−0.625336	−0.312668	+	0.949862i	$0.601223\pi$
$930$	2.76394e6	0.104790
$931$	−844523.	−0.0319328
$932$	−954006.	−0.0359759
$933$	2.26676e7	0.852515
$934$	1.87877e7	0.704705
$935$	1.01335e7	0.379081
$936$	2.62221e6	0.0978312
$937$	4.00964e7	1.49196	0.745978	−	0.665971i	$-0.231982\pi$
$937$	4.00964e7	1.49196	0.745978	+	0.665971i	$0.231982\pi$
$938$	1.35051e7	0.501175
$939$	2.60423e7	0.963865
$940$	3.50297e6	0.129305
$941$	−2.03772e6	−0.0750190	−0.0375095	−	0.999296i	$-0.511942\pi$
$941$	−2.03772e6	−0.0750190	−0.0375095	+	0.999296i	$0.511942\pi$
$942$	−8.50390e6	−0.312242
$943$	8.04787e6	0.294714
$944$	1.17129e7	0.427795
$945$	−4.61863e6	−0.168242
$946$	1.24297e7	0.451579
$947$	−2.60027e7	−0.942201	−0.471100	−	0.882080i	$-0.656143\pi$
$947$	−2.60027e7	−0.942201	−0.471100	+	0.882080i	$0.656143\pi$
$948$	2.55045e7	0.921715
$949$	−1.52438e7	−0.549449
$950$	4.79877e6	0.172513
$951$	1.36504e7	0.489433
$952$	1.77566e7	0.634990
$953$	3.15712e7	1.12605	0.563027	−	0.826439i	$-0.309637\pi$
$953$	3.15712e7	1.12605	0.563027	+	0.826439i	$0.309637\pi$
$954$	−429483.	−0.0152783
$955$	7.53875e6	0.267480
$956$	−1.18782e7	−0.420346
$957$	3.74231e7	1.32087
$958$	−1.18263e7	−0.416327
$959$	−2.84416e7	−0.998638
$960$	−821673.	−0.0287754
$961$	−1.67644e7	−0.585572
$962$	2.67196e7	0.930878
$963$	−9.15999e6	−0.318295
$964$	584947.	0.0202733
$965$	−5.32790e6	−0.184178
$966$	−3.99585e7	−1.37774
$967$	−4.38150e7	−1.50680	−0.753402	−	0.657560i	$-0.771588\pi$
$967$	−4.38150e7	−1.50680	−0.753402	+	0.657560i	$0.771588\pi$
$968$	1.07013e6	0.0367069
$969$	1.58031e7	0.540672
$970$	−1.76835e6	−0.0603447
$971$	3.52313e7	1.19917	0.599584	−	0.800312i	$-0.295333\pi$
$971$	3.52313e7	1.19917	0.599584	+	0.800312i	$0.295333\pi$
$972$	−6.33137e6	−0.214947
$973$	2.82002e7	0.954927
$974$	−1.00341e7	−0.338907
$975$	3.96722e7	1.33652
$976$	−6.42426e6	−0.215873
$977$	2.70560e7	0.906833	0.453417	−	0.891299i	$-0.350205\pi$
$977$	2.70560e7	0.906833	0.453417	+	0.891299i	$0.350205\pi$
$978$	−2.10723e7	−0.704475
$979$	−2.05583e6	−0.0685535
$980$	392470.	0.0130539
$981$	1.17039e7	0.388292
$982$	−3.68111e7	−1.21815
$983$	−1.15654e7	−0.381747	−0.190874	−	0.981615i	$-0.561132\pi$
$983$	−1.15654e7	−0.381747	−0.190874	+	0.981615i	$0.561132\pi$
$984$	1.85133e6	0.0609531
$985$	5.98041e6	0.196400
$986$	5.23919e7	1.71622
$987$	3.91876e7	1.28043
$988$	−4.95289e6	−0.161423
$989$	−3.91594e7	−1.27305
$990$	−941063.	−0.0305162
$991$	2.45423e7	0.793835	0.396918	−	0.917854i	$-0.370080\pi$
$991$	2.45423e7	0.793835	0.396918	+	0.917854i	$0.370080\pi$
$992$	−3.52719e6	−0.113802
$993$	2.18768e7	0.704062
$994$	−1.14109e7	−0.366314
$995$	7.17284e6	0.229685
$996$	−2.70814e7	−0.865015
$997$	−3.67032e7	−1.16941	−0.584704	−	0.811247i	$-0.698789\pi$
$997$	−3.67032e7	−1.16941	−0.584704	+	0.811247i	$0.698789\pi$
$998$	1.83474e7	0.583105
$999$	−2.82989e7	−0.897132

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	82.6.a.c.1.5	✓	5
3.2	odd	2		738.6.a.k.1.3		5
4.3	odd	2		656.6.a.c.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
82.6.a.c.1.5	✓	5	1.1	even	1	trivial
656.6.a.c.1.1		5	4.3	odd	2
738.6.a.k.1.3		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 82 = 2 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	82.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +17.2082 q^{3} +16.0000 q^{4} -11.6574 q^{5} -68.8328 q^{6} -121.255 q^{7} -64.0000 q^{8} +53.1223 q^{9} +46.6298 q^{10} -379.908 q^{11} +275.331 q^{12} -771.276 q^{13} +485.021 q^{14} -200.604 q^{15} +256.000 q^{16} +2288.12 q^{17} -212.489 q^{18} +401.355 q^{19} -186.519 q^{20} -2086.58 q^{21} +1519.63 q^{22} -4787.55 q^{23} -1101.33 q^{24} -2989.10 q^{25} +3085.11 q^{26} -3267.45 q^{27} -1940.08 q^{28} -5724.34 q^{29} +802.415 q^{30} +3444.52 q^{31} -1024.00 q^{32} -6537.54 q^{33} -9152.49 q^{34} +1413.53 q^{35} +849.957 q^{36} +8660.85 q^{37} -1605.42 q^{38} -13272.3 q^{39} +746.076 q^{40} -1681.00 q^{41} +8346.34 q^{42} +8179.43 q^{43} -6078.53 q^{44} -619.270 q^{45} +19150.2 q^{46} -18780.7 q^{47} +4405.30 q^{48} -2104.18 q^{49} +11956.4 q^{50} +39374.5 q^{51} -12340.4 q^{52} +2021.20 q^{53} +13069.8 q^{54} +4428.76 q^{55} +7760.33 q^{56} +6906.60 q^{57} +22897.4 q^{58} +45753.6 q^{59} -3209.66 q^{60} -25094.8 q^{61} -13778.1 q^{62} -6441.35 q^{63} +4096.00 q^{64} +8991.11 q^{65} +26150.2 q^{66} +27844.3 q^{67} +36609.9 q^{68} -82385.1 q^{69} -5654.10 q^{70} -23526.6 q^{71} -3399.83 q^{72} +19764.3 q^{73} -34643.4 q^{74} -51437.1 q^{75} +6421.68 q^{76} +46065.8 q^{77} +53089.1 q^{78} +92632.2 q^{79} -2984.31 q^{80} -69135.7 q^{81} +6724.00 q^{82} -98359.5 q^{83} -33385.3 q^{84} -26673.6 q^{85} -32717.7 q^{86} -98505.6 q^{87} +24314.1 q^{88} +5411.37 q^{89} +2477.08 q^{90} +93521.3 q^{91} -76600.7 q^{92} +59274.0 q^{93} +75122.9 q^{94} -4678.78 q^{95} -17621.2 q^{96} -37923.2 q^{97} +8416.72 q^{98} -20181.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 20 q^{2} - 10 q^{3} + 80 q^{4} - 38 q^{5} + 40 q^{6} - 38 q^{7} - 320 q^{8} + 265 q^{9} + 152 q^{10} + 416 q^{11} - 160 q^{12} + 268 q^{13} + 152 q^{14} + 22 q^{15} + 1280 q^{16} - 2198 q^{17} - 1060 q^{18}+ \cdots - 139442 q^{99}+O(q^{100}) \) 5 * q - 20 * q^2 - 10 * q^3 + 80 * q^4 - 38 * q^5 + 40 * q^6 - 38 * q^7 - 320 * q^8 + 265 * q^9 + 152 * q^10 + 416 * q^11 - 160 * q^12 + 268 * q^13 + 152 * q^14 + 22 * q^15 + 1280 * q^16 - 2198 * q^17 - 1060 * q^18 - 2250 * q^19 - 608 * q^20 - 6344 * q^21 - 1664 * q^22 - 12008 * q^23 + 640 * q^24 - 9861 * q^25 - 1072 * q^26 - 17962 * q^27 - 608 * q^28 - 11684 * q^29 - 88 * q^30 - 6780 * q^31 - 5120 * q^32 - 15876 * q^33 + 8792 * q^34 - 23158 * q^35 + 4240 * q^36 - 5822 * q^37 + 9000 * q^38 - 14440 * q^39 + 2432 * q^40 - 8405 * q^41 + 25376 * q^42 + 13248 * q^43 + 6656 * q^44 + 5058 * q^45 + 48032 * q^46 - 2000 * q^47 - 2560 * q^48 + 51789 * q^49 + 39444 * q^50 + 36628 * q^51 + 4288 * q^52 - 32036 * q^53 + 71848 * q^54 + 28670 * q^55 + 2432 * q^56 + 5476 * q^57 + 46736 * q^58 - 41376 * q^59 + 352 * q^60 + 45314 * q^61 + 27120 * q^62 + 25568 * q^63 + 20480 * q^64 + 20012 * q^65 + 63504 * q^66 + 47244 * q^67 - 35168 * q^68 + 77140 * q^69 + 92632 * q^70 - 87348 * q^71 - 16960 * q^72 + 44550 * q^73 + 23288 * q^74 + 23370 * q^75 - 36000 * q^76 - 19040 * q^77 + 57760 * q^78 - 6864 * q^79 - 9728 * q^80 + 154261 * q^81 + 33620 * q^82 - 119888 * q^83 - 101504 * q^84 - 22700 * q^85 - 52992 * q^86 - 242980 * q^87 - 26624 * q^88 - 136626 * q^89 - 20232 * q^90 - 30964 * q^91 - 192128 * q^92 - 180684 * q^93 + 8000 * q^94 - 153770 * q^95 + 10240 * q^96 + 7086 * q^97 - 207156 * q^98 - 139442 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more