LMFDB - Embedded newform 31.6.a.b.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [31,6,Mod(1,31)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, 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("31.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 31 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	31.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8] 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:	$4.97189841420$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 199x^{6} + 256x^{5} + 12633x^{4} - 18583x^{3} - 260319x^{2} + 410640x + 275908 $ x^8 - x^7 - 199x^6 + 256x^5 + 12633x^4 - 18583x^3 - 260319x^2 + 410640x + 275908
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 5\cdot 13 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-7.89102$ of defining polynomial
Character	$\chi$	$=$	31.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.89102 q^{2} +18.9313 q^{3} +47.0502 q^{4} -33.8630 q^{5} +168.319 q^{6} -134.850 q^{7} +133.811 q^{8} +115.395 q^{9} -301.077 q^{10} -191.415 q^{11} +890.722 q^{12} +766.219 q^{13} -1198.95 q^{14} -641.071 q^{15} -315.888 q^{16} +1357.39 q^{17} +1025.98 q^{18} -164.832 q^{19} -1593.26 q^{20} -2552.89 q^{21} -1701.88 q^{22} +3313.68 q^{23} +2533.22 q^{24} -1978.30 q^{25} +6812.46 q^{26} -2415.73 q^{27} -6344.71 q^{28} -6675.50 q^{29} -5699.78 q^{30} +961.000 q^{31} -7090.52 q^{32} -3623.74 q^{33} +12068.6 q^{34} +4566.42 q^{35} +5429.35 q^{36} +10334.6 q^{37} -1465.53 q^{38} +14505.5 q^{39} -4531.25 q^{40} +13434.9 q^{41} -22697.7 q^{42} +16907.4 q^{43} -9006.12 q^{44} -3907.62 q^{45} +29462.0 q^{46} -18714.2 q^{47} -5980.17 q^{48} +1377.48 q^{49} -17589.1 q^{50} +25697.2 q^{51} +36050.7 q^{52} +11410.8 q^{53} -21478.3 q^{54} +6481.90 q^{55} -18044.4 q^{56} -3120.50 q^{57} -59351.9 q^{58} +7475.61 q^{59} -30162.5 q^{60} -55634.9 q^{61} +8544.27 q^{62} -15561.0 q^{63} -52933.5 q^{64} -25946.5 q^{65} -32218.8 q^{66} +26388.2 q^{67} +63865.5 q^{68} +62732.4 q^{69} +40600.1 q^{70} -14258.2 q^{71} +15441.1 q^{72} +6721.72 q^{73} +91885.2 q^{74} -37451.8 q^{75} -7755.39 q^{76} +25812.3 q^{77} +128969. q^{78} -29059.9 q^{79} +10696.9 q^{80} -73774.0 q^{81} +119450. q^{82} -37568.9 q^{83} -120114. q^{84} -45965.3 q^{85} +150324. q^{86} -126376. q^{87} -25613.5 q^{88} +91193.1 q^{89} -34742.7 q^{90} -103325. q^{91} +155909. q^{92} +18193.0 q^{93} -166388. q^{94} +5581.72 q^{95} -134233. q^{96} +95596.4 q^{97} +12247.2 q^{98} -22088.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 7 q^{2} - 2 q^{3} + 149 q^{4} + 128 q^{5} + 72 q^{6} + 88 q^{7} + 924 q^{8} + 1512 q^{9} + 1581 q^{10} + 574 q^{11} - 46 q^{12} - 122 q^{13} - 309 q^{14} - 524 q^{15} + 833 q^{16} + 1932 q^{17} - 6845 q^{18}+ \cdots - 180150 q^{99}+O(q^{100}) $ 8 * q + 7 * q^2 - 2 * q^3 + 149 * q^4 + 128 * q^5 + 72 * q^6 + 88 * q^7 + 924 * q^8 + 1512 * q^9 + 1581 * q^10 + 574 * q^11 - 46 * q^12 - 122 * q^13 - 309 * q^14 - 524 * q^15 + 833 * q^16 + 1932 * q^17 - 6845 * q^18 - 1796 * q^19 - 37 * q^20 + 996 * q^21 - 9000 * q^22 - 4136 * q^23 - 17468 * q^24 + 3308 * q^25 + 1524 * q^26 - 2624 * q^27 - 21245 * q^28 + 5050 * q^29 - 30450 * q^30 + 7688 * q^31 + 21045 * q^32 + 27920 * q^33 - 24738 * q^34 + 25700 * q^35 + 21637 * q^36 + 23674 * q^37 - 4289 * q^38 + 38652 * q^39 + 36606 * q^40 + 44828 * q^41 - 9994 * q^42 + 21058 * q^43 + 13168 * q^44 - 1344 * q^45 + 6198 * q^46 + 5348 * q^47 - 69588 * q^48 + 19356 * q^49 + 33242 * q^50 + 22300 * q^51 - 60386 * q^52 + 4926 * q^53 - 24476 * q^54 - 56892 * q^55 - 87652 * q^56 + 22072 * q^57 - 25002 * q^58 + 16944 * q^59 - 138556 * q^60 - 73682 * q^61 + 6727 * q^62 - 138784 * q^63 - 1300 * q^64 + 63316 * q^65 - 128796 * q^66 - 134768 * q^67 - 9524 * q^68 + 3992 * q^69 - 142737 * q^70 + 123724 * q^71 + 93232 * q^72 + 70792 * q^73 + 307468 * q^74 - 302902 * q^75 - 65405 * q^76 + 107932 * q^77 + 36820 * q^78 - 26036 * q^79 + 576783 * q^80 + 82528 * q^81 + 142335 * q^82 + 21882 * q^83 + 299144 * q^84 - 130464 * q^85 + 480478 * q^86 - 440244 * q^87 - 71982 * q^88 + 164392 * q^89 + 397969 * q^90 - 60028 * q^91 + 31012 * q^92 - 1922 * q^93 - 170388 * q^94 - 7404 * q^95 - 331010 * q^96 - 131948 * q^97 + 288768 * q^98 - 180150 * 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 $

$2$	8.89102	1.57172	0.785862	−	0.618402i	$-0.212220\pi$
$2$	8.89102	1.57172	0.785862	+	0.618402i	$0.212220\pi$
$3$	18.9313	1.21444	0.607222	−	0.794532i	$-0.292284\pi$
$3$	18.9313	1.21444	0.607222	+	0.794532i	$0.292284\pi$
$4$	47.0502	1.47032
$5$	−33.8630	−0.605760	−0.302880	−	0.953029i	$-0.597948\pi$
$5$	−33.8630	−0.605760	−0.302880	+	0.953029i	$0.597948\pi$
$6$	168.319	1.90877
$7$	−134.850	−1.04017	−0.520086	−	0.854114i	$-0.674100\pi$
$7$	−134.850	−1.04017	−0.520086	+	0.854114i	$0.674100\pi$
$8$	133.811	0.739209
$9$	115.395	0.474876
$10$	−301.077	−0.952088
$11$	−191.415	−0.476974	−0.238487	−	0.971146i	$-0.576651\pi$
$11$	−191.415	−0.476974	−0.238487	+	0.971146i	$0.576651\pi$
$12$	890.722	1.78562
$13$	766.219	1.25746	0.628730	−	0.777623i	$-0.283575\pi$
$13$	766.219	1.25746	0.628730	+	0.777623i	$0.283575\pi$
$14$	−1198.95	−1.63486
$15$	−641.071	−0.735662
$16$	−315.888	−0.308484
$17$	1357.39	1.13915	0.569577	−	0.821938i	$-0.307107\pi$
$17$	1357.39	1.13915	0.569577	+	0.821938i	$0.307107\pi$
$18$	1025.98	0.746375
$19$	−164.832	−0.104751	−0.0523755	−	0.998627i	$-0.516679\pi$
$19$	−164.832	−0.104751	−0.0523755	+	0.998627i	$0.516679\pi$
$20$	−1593.26	−0.890659
$21$	−2552.89	−1.26323
$22$	−1701.88	−0.749672
$23$	3313.68	1.30615	0.653073	−	0.757295i	$-0.273480\pi$
$23$	3313.68	1.30615	0.653073	+	0.757295i	$0.273480\pi$
$24$	2533.22	0.897729
$25$	−1978.30	−0.633055
$26$	6812.46	1.97638
$27$	−2415.73	−0.637734
$28$	−6344.71	−1.52938
$29$	−6675.50	−1.47397	−0.736985	−	0.675909i	$-0.763751\pi$
$29$	−6675.50	−1.47397	−0.736985	+	0.675909i	$0.763751\pi$
$30$	−5699.78	−1.15626
$31$	961.000	0.179605
$31$	961.000	0.179605
$32$	−7090.52	−1.22406
$33$	−3623.74	−0.579259
$34$	12068.6	1.79044
$35$	4566.42	0.630095
$36$	5429.35	0.698219
$37$	10334.6	1.24105	0.620526	−	0.784186i	$-0.286919\pi$
$37$	10334.6	1.24105	0.620526	+	0.784186i	$0.286919\pi$
$38$	−1465.53	−0.164640
$39$	14505.5	1.52712
$40$	−4531.25	−0.447783
$41$	13434.9	1.24817	0.624085	−	0.781356i	$-0.285472\pi$
$41$	13434.9	1.24817	0.624085	+	0.781356i	$0.285472\pi$
$42$	−22697.7	−1.98545
$43$	16907.4	1.39446	0.697228	−	0.716850i	$-0.254417\pi$
$43$	16907.4	1.39446	0.697228	+	0.716850i	$0.254417\pi$
$44$	−9006.12	−0.701303
$45$	−3907.62	−0.287661
$46$	29462.0	2.05290
$47$	−18714.2	−1.23574	−0.617869	−	0.786281i	$-0.712004\pi$
$47$	−18714.2	−1.23574	−0.617869	+	0.786281i	$0.712004\pi$
$48$	−5980.17	−0.374637
$49$	1377.48	0.0819589
$50$	−17589.1	−0.994988
$51$	25697.2	1.38344
$52$	36050.7	1.84887
$53$	11410.8	0.557990	0.278995	−	0.960293i	$-0.409999\pi$
$53$	11410.8	0.557990	0.278995	+	0.960293i	$0.409999\pi$
$54$	−21478.3	−1.00234
$55$	6481.90	0.288932
$56$	−18044.4	−0.768905
$57$	−3120.50	−0.127214
$58$	−59351.9	−2.31667
$59$	7475.61	0.279587	0.139793	−	0.990181i	$-0.455356\pi$
$59$	7475.61	0.279587	0.139793	+	0.990181i	$0.455356\pi$
$60$	−30162.5	−1.08166
$61$	−55634.9	−1.91436	−0.957179	−	0.289498i	$-0.906512\pi$
$61$	−55634.9	−1.91436	−0.957179	+	0.289498i	$0.906512\pi$
$62$	8544.27	0.282290
$63$	−15561.0	−0.493953
$64$	−52933.5	−1.61540
$65$	−25946.5	−0.761719
$66$	−32218.8	−0.910435
$67$	26388.2	0.718161	0.359081	−	0.933307i	$-0.383090\pi$
$67$	26388.2	0.718161	0.359081	+	0.933307i	$0.383090\pi$
$68$	63865.5	1.67492
$69$	62732.4	1.58624
$70$	40600.1	0.990335
$71$	−14258.2	−0.335674	−0.167837	−	0.985815i	$-0.553678\pi$
$71$	−14258.2	−0.335674	−0.167837	+	0.985815i	$0.553678\pi$
$72$	15441.1	0.351033
$73$	6721.72	0.147630	0.0738148	−	0.997272i	$-0.476483\pi$
$73$	6721.72	0.147630	0.0738148	+	0.997272i	$0.476483\pi$
$74$	91885.2	1.95059
$75$	−37451.8	−0.768810
$76$	−7755.39	−0.154017
$77$	25812.3	0.496135
$78$	128969.	2.40021
$79$	−29059.9	−0.523874	−0.261937	−	0.965085i	$-0.584361\pi$
$79$	−29059.9	−0.523874	−0.261937	+	0.965085i	$0.584361\pi$
$80$	10696.9	0.186867
$81$	−73774.0	−1.24937
$82$	119450.	1.96178
$83$	−37568.9	−0.598595	−0.299298	−	0.954160i	$-0.596752\pi$
$83$	−37568.9	−0.598595	−0.299298	+	0.954160i	$0.596752\pi$
$84$	−120114.	−1.85735
$85$	−45965.3	−0.690054
$86$	150324.	2.19170
$87$	−126376.	−1.79005
$88$	−25613.5	−0.352584
$89$	91193.1	1.22036	0.610179	−	0.792264i	$-0.291098\pi$
$89$	91193.1	1.22036	0.610179	+	0.792264i	$0.291098\pi$
$90$	−34742.7	−0.452124
$91$	−103325.	−1.30798
$92$	155909.	1.92045
$93$	18193.0	0.218121
$94$	−166388.	−1.94224
$95$	5581.72	0.0634540
$96$	−134233.	−1.48655
$97$	95596.4	1.03160	0.515801	−	0.856708i	$-0.327494\pi$
$97$	95596.4	1.03160	0.515801	+	0.856708i	$0.327494\pi$
$98$	12247.2	0.128817
$99$	−22088.4	−0.226504
$100$	−93079.2	−0.930792
$101$	−71056.1	−0.693103	−0.346551	−	0.938031i	$-0.612647\pi$
$101$	−71056.1	−0.693103	−0.346551	+	0.938031i	$0.612647\pi$
$102$	228474.	2.17439
$103$	−176351.	−1.63789	−0.818946	−	0.573870i	$-0.805441\pi$
$103$	−176351.	−1.63789	−0.818946	+	0.573870i	$0.805441\pi$
$104$	102529.	0.929527
$105$	86448.4	0.765215
$106$	101454.	0.877006
$107$	141241.	1.19262	0.596308	−	0.802756i	$-0.296634\pi$
$107$	141241.	1.19262	0.596308	+	0.802756i	$0.296634\pi$
$108$	−113661.	−0.937671
$109$	−70513.7	−0.568470	−0.284235	−	0.958755i	$-0.591740\pi$
$109$	−70513.7	−0.568470	−0.284235	+	0.958755i	$0.591740\pi$
$110$	57630.6	0.454121
$111$	195648.	1.50719
$112$	42597.4	0.320877
$113$	170060.	1.25287	0.626436	−	0.779473i	$-0.284513\pi$
$113$	170060.	1.25287	0.626436	+	0.779473i	$0.284513\pi$
$114$	−27744.4	−0.199946
$115$	−112211.	−0.791211
$116$	−314083.	−2.16720
$117$	88417.8	0.597138
$118$	66465.7	0.439433
$119$	−183044.	−1.18492
$120$	−85782.5	−0.543808
$121$	−124411.	−0.772496
$122$	−494651.	−3.00884
$123$	254340.	1.51583
$124$	45215.2	0.264077
$125$	172813.	0.989239
$126$	−138353.	−0.776358
$127$	−198550.	−1.09235	−0.546175	−	0.837671i	$-0.683916\pi$
$127$	−198550.	−1.09235	−0.546175	+	0.837671i	$0.683916\pi$
$128$	−243736.	−1.31491
$129$	320079.	1.69349
$130$	−230691.	−1.19721
$131$	−78021.3	−0.397223	−0.198612	−	0.980078i	$-0.563643\pi$
$131$	−78021.3	−0.397223	−0.198612	+	0.980078i	$0.563643\pi$
$132$	−170498.	−0.851694
$133$	22227.6	0.108959
$134$	234618.	1.12875
$135$	81804.0	0.386314
$136$	181634.	0.842074
$137$	131437.	0.598298	0.299149	−	0.954206i	$-0.403297\pi$
$137$	131437.	0.598298	0.299149	+	0.954206i	$0.403297\pi$
$138$	557755.	2.49313
$139$	−42476.0	−0.186469	−0.0932345	−	0.995644i	$-0.529721\pi$
$139$	−42476.0	−0.186469	−0.0932345	+	0.995644i	$0.529721\pi$
$140$	214851.	0.926439
$141$	−354284.	−1.50074
$142$	−126770.	−0.527587
$143$	−146666.	−0.599776
$144$	−36451.8	−0.146492
$145$	226052.	0.892872
$146$	59762.9	0.232033
$147$	26077.6	0.0995346
$148$	486245.	1.82474
$149$	292210.	1.07827	0.539137	−	0.842218i	$-0.318751\pi$
$149$	292210.	1.07827	0.539137	+	0.842218i	$0.318751\pi$
$150$	−332984.	−1.20836
$151$	−425103.	−1.51723	−0.758615	−	0.651539i	$-0.774124\pi$
$151$	−425103.	−1.51723	−0.758615	+	0.651539i	$0.774124\pi$
$152$	−22056.4	−0.0774330
$153$	156636.	0.540957
$154$	229498.	0.779788
$155$	−32542.4	−0.108798
$156$	682488.	2.24535
$157$	−239463.	−0.775336	−0.387668	−	0.921799i	$-0.626719\pi$
$157$	−239463.	−0.775336	−0.387668	+	0.921799i	$0.626719\pi$
$158$	−258372.	−0.823385
$159$	216022.	0.677648
$160$	240106.	0.741487
$161$	−446850.	−1.35862
$162$	−655926.	−1.96366
$163$	388547.	1.14544	0.572722	−	0.819749i	$-0.305887\pi$
$163$	388547.	1.14544	0.572722	+	0.819749i	$0.305887\pi$
$164$	632113.	1.83521
$165$	122711.	0.350892
$166$	−334026.	−0.940827
$167$	−306915.	−0.851583	−0.425792	−	0.904821i	$-0.640004\pi$
$167$	−306915.	−0.851583	−0.425792	+	0.904821i	$0.640004\pi$
$168$	−341605.	−0.933793
$169$	215798.	0.581208
$170$	−408679.	−1.08457
$171$	−19020.8	−0.0497438
$172$	795494.	2.05029
$173$	655244.	1.66451	0.832257	−	0.554389i	$-0.187048\pi$
$173$	655244.	1.66451	0.832257	+	0.554389i	$0.187048\pi$
$174$	−1.12361e6	−2.81347
$175$	266773.	0.658486
$176$	60465.7	0.147139
$177$	141523.	0.339543
$178$	810799.	1.91807
$179$	50816.0	0.118541	0.0592705	−	0.998242i	$-0.481123\pi$
$179$	50816.0	0.118541	0.0592705	+	0.998242i	$0.481123\pi$
$180$	−183854.	−0.422953
$181$	−549255.	−1.24617	−0.623085	−	0.782154i	$-0.714121\pi$
$181$	−549255.	−1.24617	−0.623085	+	0.782154i	$0.714121\pi$
$182$	−918660.	−2.05578
$183$	−1.05324e6	−2.32488
$184$	443408.	0.965515
$185$	−349961.	−0.751779
$186$	161754.	0.342826
$187$	−259825.	−0.543347
$188$	−880505.	−1.81693
$189$	325761.	0.663353
$190$	49627.2	0.0997322
$191$	−554457.	−1.09973	−0.549863	−	0.835255i	$-0.685320\pi$
$191$	−554457.	−1.09973	−0.549863	+	0.835255i	$0.685320\pi$
$192$	−1.00210e6	−1.96182
$193$	47999.3	0.0927559	0.0463780	−	0.998924i	$-0.485232\pi$
$193$	47999.3	0.0927559	0.0463780	+	0.998924i	$0.485232\pi$
$194$	849949.	1.62139
$195$	−491201.	−0.925066
$196$	64810.8	0.120506
$197$	1.00912e6	1.85258	0.926292	−	0.376807i	$-0.122978\pi$
$197$	1.00912e6	1.85258	0.926292	+	0.376807i	$0.122978\pi$
$198$	−196388.	−0.356001
$199$	504306.	0.902738	0.451369	−	0.892337i	$-0.350936\pi$
$199$	504306.	0.902738	0.451369	+	0.892337i	$0.350936\pi$
$200$	−264718.	−0.467960
$201$	499563.	0.872167
$202$	−631761.	−1.08937
$203$	900190.	1.53318
$204$	1.20906e6	2.03410
$205$	−454945.	−0.756092
$206$	−1.56794e6	−2.57432
$207$	382382.	0.620258
$208$	−242039.	−0.387907
$209$	31551.4	0.0499636
$210$	768614.	1.20271
$211$	403332.	0.623673	0.311836	−	0.950136i	$-0.399056\pi$
$211$	403332.	0.623673	0.311836	+	0.950136i	$0.399056\pi$
$212$	536880.	0.820422
$213$	−269926.	−0.407658
$214$	1.25577e6	1.87446
$215$	−572534.	−0.844705
$216$	−323252.	−0.471419
$217$	−129591.	−0.186821
$218$	−626939.	−0.893478
$219$	127251.	0.179288
$220$	304974.	0.424822
$221$	1.04006e6	1.43244
$222$	1.73951e6	2.36889
$223$	381052.	0.513124	0.256562	−	0.966528i	$-0.417410\pi$
$223$	381052.	0.513124	0.256562	+	0.966528i	$0.417410\pi$
$224$	956156.	1.27323
$225$	−228285.	−0.300623
$226$	1.51201e6	1.96917
$227$	−1.00818e6	−1.29860	−0.649298	−	0.760534i	$-0.724937\pi$
$227$	−1.00818e6	−1.29860	−0.649298	+	0.760534i	$0.724937\pi$
$228$	−146820.	−0.187046
$229$	−840771.	−1.05947	−0.529735	−	0.848163i	$-0.677709\pi$
$229$	−840771.	−1.05947	−0.529735	+	0.848163i	$0.677709\pi$
$230$	−997673.	−1.24357
$231$	488661.	0.602529
$232$	−893256.	−1.08957
$233$	−1.18940e6	−1.43528	−0.717640	−	0.696415i	$-0.754778\pi$
$233$	−1.18940e6	−1.43528	−0.717640	+	0.696415i	$0.754778\pi$
$234$	786124.	0.938537
$235$	633719.	0.748560
$236$	351729.	0.411081
$237$	−550143.	−0.636216
$238$	−1.62745e6	−1.86236
$239$	1.06364e6	1.20448	0.602240	−	0.798315i	$-0.294275\pi$
$239$	1.06364e6	1.20448	0.602240	+	0.798315i	$0.294275\pi$
$240$	202507.	0.226940
$241$	1.10322e6	1.22354	0.611772	−	0.791034i	$-0.290457\pi$
$241$	1.10322e6	1.22354	0.611772	+	0.791034i	$0.290457\pi$
$242$	−1.10614e6	−1.21415
$243$	−809616.	−0.879556
$244$	−2.61763e6	−2.81471
$245$	−46645.7	−0.0496474
$246$	2.26134e6	2.38247
$247$	−126298.	−0.131720
$248$	128593.	0.132766
$249$	−711229.	−0.726961
$250$	1.53648e6	1.55481
$251$	−103694.	−0.103889	−0.0519445	−	0.998650i	$-0.516542\pi$
$251$	−103694.	−0.103889	−0.0519445	+	0.998650i	$0.516542\pi$
$252$	−732147.	−0.726268
$253$	−634290.	−0.622998
$254$	−1.76531e6	−1.71687
$255$	−870185.	−0.838033
$256$	−473189.	−0.451268
$257$	423080.	0.399567	0.199783	−	0.979840i	$-0.435976\pi$
$257$	423080.	0.399567	0.199783	+	0.979840i	$0.435976\pi$
$258$	2.84582e6	2.66170
$259$	−1.39362e6	−1.29091
$260$	−1.22079e6	−1.11997
$261$	−770318.	−0.699953
$262$	−693688.	−0.624326
$263$	1.79143e6	1.59702	0.798511	−	0.601980i	$-0.205621\pi$
$263$	1.79143e6	1.59702	0.798511	+	0.601980i	$0.205621\pi$
$264$	−484897.	−0.428194
$265$	−386404.	−0.338008
$266$	197626.	0.171254
$267$	1.72641e6	1.48206
$268$	1.24157e6	1.05593
$269$	−67703.6	−0.0570468	−0.0285234	−	0.999593i	$-0.509081\pi$
$269$	−67703.6	−0.0570468	−0.0285234	+	0.999593i	$0.509081\pi$
$270$	727320.	0.607178
$271$	−936407.	−0.774536	−0.387268	−	0.921967i	$-0.626581\pi$
$271$	−936407.	−0.774536	−0.387268	+	0.921967i	$0.626581\pi$
$272$	−428783.	−0.351411
$273$	−1.95607e6	−1.58846
$274$	1.16861e6	0.940360
$275$	378676.	0.301951
$276$	2.95157e6	2.33228
$277$	713290.	0.558556	0.279278	−	0.960210i	$-0.409905\pi$
$277$	713290.	0.558556	0.279278	+	0.960210i	$0.409905\pi$
$278$	−377655.	−0.293078
$279$	110895.	0.0852903
$280$	611038.	0.465772
$281$	908575.	0.686428	0.343214	−	0.939257i	$-0.388484\pi$
$281$	908575.	0.686428	0.343214	+	0.939257i	$0.388484\pi$
$282$	−3.14995e6	−2.35874
$283$	−1.96084e6	−1.45538	−0.727691	−	0.685905i	$-0.759406\pi$
$283$	−1.96084e6	−1.45538	−0.727691	+	0.685905i	$0.759406\pi$
$284$	−670849.	−0.493547
$285$	105669.	0.0770614
$286$	−1.30401e6	−0.942683
$287$	−1.81169e6	−1.29831
$288$	−818210.	−0.581278
$289$	422653.	0.297673
$290$	2.00984e6	1.40335
$291$	1.80977e6	1.25282
$292$	316258.	0.217062
$293$	816856.	0.555874	0.277937	−	0.960599i	$-0.410349\pi$
$293$	816856.	0.555874	0.277937	+	0.960599i	$0.410349\pi$
$294$	231856.	0.156441
$295$	−253147.	−0.169362
$296$	1.38289e6	0.917397
$297$	462408.	0.304183
$298$	2.59804e6	1.69475
$299$	2.53901e6	1.64243
$300$	−1.76211e6	−1.13040
$301$	−2.27995e6	−1.45047
$302$	−3.77960e6	−2.38467
$303$	−1.34519e6	−0.841735
$304$	52068.5	0.0323141
$305$	1.88397e6	1.15964
$306$	1.39265e6	0.850236
$307$	842.455	0.000510153 0	0.000255077	−	1.00000i	$-0.499919\pi$
$307$	842.455	0.000510153 0	0.000255077		1.00000i	$0.499919\pi$
$308$	1.21447e6	0.729477
$309$	−3.33856e6	−1.98913
$310$	−289335.	−0.171000
$311$	−275493.	−0.161514	−0.0807568	−	0.996734i	$-0.525734\pi$
$311$	−275493.	−0.161514	−0.0807568	+	0.996734i	$0.525734\pi$
$312$	1.94100e6	1.12886
$313$	−919046.	−0.530245	−0.265122	−	0.964215i	$-0.585412\pi$
$313$	−919046.	−0.530245	−0.265122	+	0.964215i	$0.585412\pi$
$314$	−2.12907e6	−1.21861
$315$	526942.	0.299217
$316$	−1.36727e6	−0.770261
$317$	−297343.	−0.166192	−0.0830960	−	0.996542i	$-0.526481\pi$
$317$	−297343.	−0.166192	−0.0830960	+	0.996542i	$0.526481\pi$
$318$	1.92065e6	1.06508
$319$	1.27779e6	0.703045
$320$	1.79249e6	0.978546
$321$	2.67387e6	1.44837
$322$	−3.97295e6	−2.13537
$323$	−223742.	−0.119328
$324$	−3.47108e6	−1.83697
$325$	−1.51581e6	−0.796042
$326$	3.45457e6	1.80032
$327$	−1.33492e6	−0.690375
$328$	1.79774e6	0.922660
$329$	2.52361e6	1.28538
$330$	1.09102e6	0.551505
$331$	−808752.	−0.405738	−0.202869	−	0.979206i	$-0.565026\pi$
$331$	−808752.	−0.405738	−0.202869	+	0.979206i	$0.565026\pi$
$332$	−1.76762e6	−0.880125
$333$	1.19256e6	0.589346
$334$	−2.72879e6	−1.33845
$335$	−893582.	−0.435033
$336$	806425.	0.389687
$337$	3.71052e6	1.77975	0.889876	−	0.456203i	$-0.150791\pi$
$337$	3.71052e6	1.77975	0.889876	+	0.456203i	$0.150791\pi$
$338$	1.91867e6	0.913498
$339$	3.21947e6	1.52154
$340$	−2.16268e6	−1.01460
$341$	−183950.	−0.0856671
$342$	−169114.	−0.0781836
$343$	2.08067e6	0.954921
$344$	2.26239e6	1.03079
$345$	−2.12431e6	−0.960882
$346$	5.82578e6	2.61616
$347$	1.52902e6	0.681694	0.340847	−	0.940119i	$-0.389286\pi$
$347$	1.52902e6	0.681694	0.340847	+	0.940119i	$0.389286\pi$
$348$	−5.94601e6	−2.63195
$349$	3.11316e6	1.36816	0.684082	−	0.729405i	$-0.260203\pi$
$349$	3.11316e6	1.36816	0.684082	+	0.729405i	$0.260203\pi$
$350$	2.37188e6	1.03496
$351$	−1.85098e6	−0.801925
$352$	1.35723e6	0.583846
$353$	1.90044e6	0.811741	0.405871	−	0.913931i	$-0.366968\pi$
$353$	1.90044e6	0.811741	0.405871	+	0.913931i	$0.366968\pi$
$354$	1.25828e6	0.533667
$355$	482824.	0.203338
$356$	4.29065e6	1.79431
$357$	−3.46526e6	−1.43902
$358$	451806.	0.186314
$359$	6638.27	0.00271844	0.00135922	−	0.999999i	$-0.499567\pi$
$359$	6638.27	0.00271844	0.00135922	+	0.999999i	$0.499567\pi$
$360$	−522883.	−0.212642
$361$	−2.44893e6	−0.989027
$362$	−4.88343e6	−1.95864
$363$	−2.35527e6	−0.938153
$364$	−4.86143e6	−1.92314
$365$	−227618.	−0.0894281
$366$	−9.36440e6	−3.65407
$367$	−154555.	−0.0598989	−0.0299494	−	0.999551i	$-0.509535\pi$
$367$	−154555.	−0.0598989	−0.0299494	+	0.999551i	$0.509535\pi$
$368$	−1.04675e6	−0.402925
$369$	1.55032e6	0.592727
$370$	−3.11151e6	−1.18159
$371$	−1.53874e6	−0.580406
$372$	855983.	0.320707
$373$	2.82330e6	1.05072	0.525358	−	0.850881i	$-0.323931\pi$
$373$	2.82330e6	1.05072	0.525358	+	0.850881i	$0.323931\pi$
$374$	−2.31011e6	−0.853992
$375$	3.27158e6	1.20138
$376$	−2.50417e6	−0.913469
$377$	−5.11489e6	−1.85346
$378$	2.89635e6	1.04261
$379$	−2.30151e6	−0.823028	−0.411514	−	0.911403i	$-0.635000\pi$
$379$	−2.30151e6	−0.823028	−0.411514	+	0.911403i	$0.635000\pi$
$380$	262621.	0.0932975
$381$	−3.75882e6	−1.32660
$382$	−4.92969e6	−1.72847
$383$	−416401.	−0.145049	−0.0725245	−	0.997367i	$-0.523106\pi$
$383$	−416401.	−0.145049	−0.0725245	+	0.997367i	$0.523106\pi$
$384$	−4.61425e6	−1.59688
$385$	−874083.	−0.300539
$386$	426763.	0.145787
$387$	1.95102e6	0.662194
$388$	4.49783e6	1.51678
$389$	3.82595e6	1.28193	0.640967	−	0.767568i	$-0.278533\pi$
$389$	3.82595e6	1.28193	0.640967	+	0.767568i	$0.278533\pi$
$390$	−4.36728e6	−1.45395
$391$	4.49797e6	1.48790
$392$	184323.	0.0605848
$393$	−1.47705e6	−0.482406
$394$	8.97211e6	2.91175
$395$	984056.	0.317342
$396$	−1.03926e6	−0.333032
$397$	3.10740e6	0.989513	0.494756	−	0.869032i	$-0.335257\pi$
$397$	3.10740e6	0.989513	0.494756	+	0.869032i	$0.335257\pi$
$398$	4.48380e6	1.41886
$399$	420798.	0.132325
$400$	624920.	0.195287
$401$	339771.	0.105518	0.0527588	−	0.998607i	$-0.483199\pi$
$401$	339771.	0.105518	0.0527588	+	0.998607i	$0.483199\pi$
$402$	4.44162e6	1.37081
$403$	736336.	0.225847
$404$	−3.34320e6	−1.01908
$405$	2.49821e6	0.756818
$406$	8.00360e6	2.40974
$407$	−1.97820e6	−0.591950
$408$	3.43857e6	1.02265
$409$	1.56215e6	0.461759	0.230880	−	0.972982i	$-0.425840\pi$
$409$	1.56215e6	0.461759	0.230880	+	0.972982i	$0.425840\pi$
$410$	−4.04493e6	−1.18837
$411$	2.48829e6	0.726600
$412$	−8.29735e6	−2.40822
$413$	−1.00808e6	−0.290818
$414$	3.39977e6	0.974874
$415$	1.27220e6	0.362605
$416$	−5.43289e6	−1.53921
$417$	−804127.	−0.226456
$418$	280524.	0.0785290
$419$	−430682.	−0.119846	−0.0599228	−	0.998203i	$-0.519085\pi$
$419$	−430682.	−0.119846	−0.0599228	+	0.998203i	$0.519085\pi$
$420$	4.06741e6	1.12511
$421$	−3.15827e6	−0.868448	−0.434224	−	0.900805i	$-0.642977\pi$
$421$	−3.15827e6	−0.868448	−0.434224	+	0.900805i	$0.642977\pi$
$422$	3.58603e6	0.980241
$423$	−2.15952e6	−0.586822
$424$	1.52689e6	0.412471
$425$	−2.68532e6	−0.721147
$426$	−2.39992e6	−0.640725
$427$	7.50236e6	1.99126
$428$	6.64540e6	1.75352
$429$	−2.77658e6	−0.728395
$430$	−5.09041e6	−1.32764
$431$	2.85837e6	0.741182	0.370591	−	0.928796i	$-0.379155\pi$
$431$	2.85837e6	0.741182	0.370591	+	0.928796i	$0.379155\pi$
$432$	763100.	0.196731
$433$	−4.33104e6	−1.11013	−0.555063	−	0.831808i	$-0.687306\pi$
$433$	−4.33104e6	−1.11013	−0.555063	+	0.831808i	$0.687306\pi$
$434$	−1.15219e6	−0.293630
$435$	4.27947e6	1.08434
$436$	−3.31768e6	−0.835831
$437$	−546203.	−0.136820
$438$	1.13139e6	0.281791
$439$	−7.01460e6	−1.73717	−0.868584	−	0.495542i	$-0.834969\pi$
$439$	−7.01460e6	−1.73717	−0.868584	+	0.495542i	$0.834969\pi$
$440$	867350.	0.213581
$441$	158955.	0.0389203
$442$	9.24718e6	2.25140
$443$	−5.04828e6	−1.22218	−0.611088	−	0.791563i	$-0.709268\pi$
$443$	−5.04828e6	−1.22218	−0.611088	+	0.791563i	$0.709268\pi$
$444$	9.20526e6	2.21605
$445$	−3.08807e6	−0.739244
$446$	3.38794e6	0.806490
$447$	5.53191e6	1.30950
$448$	7.13808e6	1.68030
$449$	−108991.	−0.0255138	−0.0127569	−	0.999919i	$-0.504061\pi$
$449$	−108991.	−0.0255138	−0.0127569	+	0.999919i	$0.504061\pi$
$450$	−2.02969e6	−0.472496
$451$	−2.57164e6	−0.595345
$452$	8.00136e6	1.84212
$453$	−8.04776e6	−1.84259
$454$	−8.96375e6	−2.04103
$455$	3.49888e6	0.792320
$456$	−417557.	−0.0940381
$457$	−813040.	−0.182105	−0.0910524	−	0.995846i	$-0.529023\pi$
$457$	−813040.	−0.182105	−0.0910524	+	0.995846i	$0.529023\pi$
$458$	−7.47531e6	−1.66520
$459$	−3.27909e6	−0.726477
$460$	−5.27956e6	−1.16333
$461$	−8.69658e6	−1.90588	−0.952941	−	0.303155i	$-0.901960\pi$
$461$	−8.69658e6	−1.90588	−0.952941	+	0.303155i	$0.901960\pi$
$462$	4.34470e6	0.947010
$463$	6.93409e6	1.50327	0.751635	−	0.659579i	$-0.229265\pi$
$463$	6.93409e6	1.50327	0.751635	+	0.659579i	$0.229265\pi$
$464$	2.10871e6	0.454696
$465$	−616070.	−0.132129
$466$	−1.05749e7	−2.25586
$467$	−6.34554e6	−1.34641	−0.673204	−	0.739457i	$-0.735082\pi$
$467$	−6.34554e6	−1.34641	−0.673204	+	0.739457i	$0.735082\pi$
$468$	4.16007e6	0.877983
$469$	−3.55844e6	−0.747012
$470$	5.63440e6	1.17653
$471$	−4.53336e6	−0.941603
$472$	1.00032e6	0.206673
$473$	−3.23633e6	−0.665119
$474$	−4.89133e6	−0.999956
$475$	326087.	0.0663132
$476$	−8.61225e6	−1.74220
$477$	1.31675e6	0.264976
$478$	9.45683e6	1.89311
$479$	5.97957e6	1.19078	0.595390	−	0.803437i	$-0.296998\pi$
$479$	5.97957e6	1.19078	0.595390	+	0.803437i	$0.296998\pi$
$480$	4.54553e6	0.900495
$481$	7.91857e6	1.56057
$482$	9.80875e6	1.92307
$483$	−8.45946e6	−1.64997
$484$	−5.85357e6	−1.13581
$485$	−3.23718e6	−0.624903
$486$	−7.19831e6	−1.38242
$487$	4.60436e6	0.879724	0.439862	−	0.898065i	$-0.355027\pi$
$487$	4.60436e6	0.879724	0.439862	+	0.898065i	$0.355027\pi$
$488$	−7.44458e6	−1.41511
$489$	7.35570e6	1.39108
$490$	−414728.	−0.0780321
$491$	1.79128e6	0.335321	0.167660	−	0.985845i	$-0.446379\pi$
$491$	1.79128e6	0.335321	0.167660	+	0.985845i	$0.446379\pi$
$492$	1.19667e7	2.22876
$493$	−9.06126e6	−1.67908
$494$	−1.12291e6	−0.207028
$495$	747978.	0.137207
$496$	−303568.	−0.0554054
$497$	1.92271e6	0.349159
$498$	−6.32355e6	−1.14258
$499$	−6.85216e6	−1.23190	−0.615951	−	0.787785i	$-0.711228\pi$
$499$	−6.85216e6	−1.23190	−0.615951	+	0.787785i	$0.711228\pi$
$500$	8.13088e6	1.45450
$501$	−5.81031e6	−1.03420
$502$	−921945.	−0.163285
$503$	−2.80195e6	−0.493787	−0.246894	−	0.969043i	$-0.579410\pi$
$503$	−2.80195e6	−0.493787	−0.246894	+	0.969043i	$0.579410\pi$
$504$	−2.08223e6	−0.365135
$505$	2.40617e6	0.419854
$506$	−5.63948e6	−0.979181
$507$	4.08535e6	0.705845
$508$	−9.34183e6	−1.60610
$509$	2.09295e6	0.358066	0.179033	−	0.983843i	$-0.442703\pi$
$509$	2.09295e6	0.358066	0.179033	+	0.983843i	$0.442703\pi$
$510$	−7.73682e6	−1.31716
$511$	−906423.	−0.153560
$512$	3.59243e6	0.605638
$513$	398191.	0.0668033
$514$	3.76161e6	0.628009
$515$	5.97178e6	0.992170
$516$	1.50597e7	2.48997
$517$	3.58218e6	0.589415
$518$	−1.23907e7	−2.02895
$519$	1.24046e7	2.02146
$520$	−3.47193e6	−0.563070
$521$	8.12070e6	1.31069	0.655344	−	0.755331i	$-0.272524\pi$
$521$	8.12070e6	1.31069	0.655344	+	0.755331i	$0.272524\pi$
$522$	−6.84891e6	−1.10013
$523$	−7.70062e6	−1.23104	−0.615519	−	0.788122i	$-0.711053\pi$
$523$	−7.70062e6	−1.23104	−0.615519	+	0.788122i	$0.711053\pi$
$524$	−3.67091e6	−0.584044
$525$	5.05037e6	0.799695
$526$	1.59277e7	2.51008
$527$	1.30445e6	0.204598
$528$	1.14470e6	0.178692
$529$	4.54416e6	0.706016
$530$	−3.43552e6	−0.531255
$531$	862647.	0.132769
$532$	1.04581e6	0.160205
$533$	1.02941e7	1.56953
$534$	1.53495e7	2.32938
$535$	−4.78283e6	−0.722438
$536$	3.53103e6	0.530872
$537$	962015.	0.143961
$538$	−601954.	−0.0896618
$539$	−263671.	−0.0390923
$540$	3.84889e6	0.568004
$541$	5.20442e6	0.764504	0.382252	−	0.924058i	$-0.375149\pi$
$541$	5.20442e6	0.764504	0.382252	+	0.924058i	$0.375149\pi$
$542$	−8.32561e6	−1.21736
$543$	−1.03981e7	−1.51341
$544$	−9.62461e6	−1.39439
$545$	2.38781e6	0.344356
$546$	−1.73914e7	−2.49663
$547$	−8.48906e6	−1.21309	−0.606543	−	0.795051i	$-0.707444\pi$
$547$	−8.48906e6	−1.21309	−0.606543	+	0.795051i	$0.707444\pi$
$548$	6.18415e6	0.879688
$549$	−6.41999e6	−0.909083
$550$	3.36682e6	0.474584
$551$	1.10034e6	0.154400
$552$	8.39430e6	1.17256
$553$	3.91873e6	0.544919
$554$	6.34187e6	0.877895
$555$	−6.62523e6	−0.912995
$556$	−1.99850e6	−0.274169
$557$	−4.24828e6	−0.580197	−0.290098	−	0.956997i	$-0.593688\pi$
$557$	−4.24828e6	−0.580197	−0.290098	+	0.956997i	$0.593688\pi$
$558$	985965.	0.134053
$559$	1.29547e7	1.75347
$560$	−1.44248e6	−0.194374
$561$	−4.91884e6	−0.659865
$562$	8.07816e6	1.07888
$563$	7.61230e6	1.01215	0.506075	−	0.862489i	$-0.331096\pi$
$563$	7.61230e6	1.01215	0.506075	+	0.862489i	$0.331096\pi$
$564$	−1.66691e7	−2.20656
$565$	−5.75875e6	−0.758940
$566$	−1.74339e7	−2.28746
$567$	9.94841e6	1.29956
$568$	−1.90790e6	−0.248133
$569$	1.33947e7	1.73441	0.867204	−	0.497953i	$-0.165915\pi$
$569$	1.33947e7	1.73441	0.867204	+	0.497953i	$0.165915\pi$
$570$	939508.	0.121119
$571$	−8.74767e6	−1.12280	−0.561400	−	0.827545i	$-0.689737\pi$
$571$	−8.74767e6	−1.12280	−0.561400	+	0.827545i	$0.689737\pi$
$572$	−6.90066e6	−0.881862
$573$	−1.04966e7	−1.33556
$574$	−1.61078e7	−2.04059
$575$	−6.55545e6	−0.826862
$576$	−6.10826e6	−0.767116
$577$	1.01902e7	1.27421	0.637106	−	0.770776i	$-0.280131\pi$
$577$	1.01902e7	1.27421	0.637106	+	0.770776i	$0.280131\pi$
$578$	3.75782e6	0.467860
$579$	908690.	0.112647
$580$	1.06358e7	1.31280
$581$	5.06616e6	0.622642
$582$	1.60907e7	1.96909
$583$	−2.18420e6	−0.266147
$584$	899441.	0.109129
$585$	−2.99409e6	−0.361722
$586$	7.26268e6	0.873681
$587$	−1.15490e7	−1.38340	−0.691702	−	0.722183i	$-0.743139\pi$
$587$	−1.15490e7	−1.38340	−0.691702	+	0.722183i	$0.743139\pi$
$588$	1.22695e6	0.146347
$589$	−158404.	−0.0188139
$590$	−2.25073e6	−0.266191
$591$	1.91040e7	2.24986
$592$	−3.26458e6	−0.382845
$593$	5.09360e6	0.594824	0.297412	−	0.954749i	$-0.403877\pi$
$593$	5.09360e6	0.594824	0.297412	+	0.954749i	$0.403877\pi$
$594$	4.11128e6	0.478091
$595$	6.19842e6	0.717775
$596$	1.37485e7	1.58540
$597$	9.54719e6	1.09633
$598$	2.25744e7	2.58144
$599$	−1.18765e7	−1.35245	−0.676227	−	0.736694i	$-0.736386\pi$
$599$	−1.18765e7	−1.35245	−0.676227	+	0.736694i	$0.736386\pi$
$600$	−5.01147e6	−0.568312
$601$	1.19316e7	1.34745	0.673726	−	0.738981i	$-0.264693\pi$
$601$	1.19316e7	1.34745	0.673726	+	0.738981i	$0.264693\pi$
$602$	−2.02711e7	−2.27975
$603$	3.04506e6	0.341038
$604$	−2.00012e7	−2.23081
$605$	4.21294e6	0.467947
$606$	−1.19601e7	−1.32298
$607$	−5.04218e6	−0.555452	−0.277726	−	0.960660i	$-0.589581\pi$
$607$	−5.04218e6	−0.555452	−0.277726	+	0.960660i	$0.589581\pi$
$608$	1.16875e6	0.128222
$609$	1.70418e7	1.86197
$610$	1.67504e7	1.82264
$611$	−1.43392e7	−1.55389
$612$	7.36975e6	0.795379
$613$	−4.40032e6	−0.472969	−0.236485	−	0.971635i	$-0.575995\pi$
$613$	−4.40032e6	−0.472969	−0.236485	+	0.971635i	$0.575995\pi$
$614$	7490.28	0.000801821 0
$615$	−8.61272e6	−0.918232
$616$	3.45398e6	0.366748
$617$	−1.19518e7	−1.26393	−0.631964	−	0.774998i	$-0.717751\pi$
$617$	−1.19518e7	−1.26393	−0.631964	+	0.774998i	$0.717751\pi$
$618$	−2.96832e7	−3.12636
$619$	1.26938e7	1.33158	0.665789	−	0.746140i	$-0.268095\pi$
$619$	1.26938e7	1.33158	0.665789	+	0.746140i	$0.268095\pi$
$620$	−1.53112e6	−0.159967
$621$	−8.00498e6	−0.832973
$622$	−2.44941e6	−0.253855
$623$	−1.22974e7	−1.26938
$624$	−4.58212e6	−0.471091
$625$	330210.	0.0338135
$626$	−8.17125e6	−0.833399
$627$	597310.	0.0606780
$628$	−1.12668e7	−1.13999
$629$	1.40281e7	1.41375
$630$	4.68505e6	0.470287
$631$	−1.48083e7	−1.48058	−0.740290	−	0.672287i	$-0.765312\pi$
$631$	−1.48083e7	−1.48058	−0.740290	+	0.672287i	$0.765312\pi$
$632$	−3.88854e6	−0.387252
$633$	7.63561e6	0.757416
$634$	−2.64368e6	−0.261208
$635$	6.72351e6	0.661701
$636$	1.01638e7	0.996358
$637$	1.05545e6	0.103060
$638$	1.13609e7	1.10499
$639$	−1.64532e6	−0.159404
$640$	8.25364e6	0.796518
$641$	1.32101e6	0.126987	0.0634937	−	0.997982i	$-0.479776\pi$
$641$	1.32101e6	0.126987	0.0634937	+	0.997982i	$0.479776\pi$
$642$	2.37734e7	2.27643
$643$	−5.16446e6	−0.492603	−0.246302	−	0.969193i	$-0.579215\pi$
$643$	−5.16446e6	−0.492603	−0.246302	+	0.969193i	$0.579215\pi$
$644$	−2.10244e7	−1.99760
$645$	−1.08388e7	−1.02585
$646$	−1.98929e6	−0.187550
$647$	−4.00756e6	−0.376374	−0.188187	−	0.982133i	$-0.560261\pi$
$647$	−4.00756e6	−0.376374	−0.188187	+	0.982133i	$0.560261\pi$
$648$	−9.87178e6	−0.923545
$649$	−1.43095e6	−0.133356
$650$	−1.34771e7	−1.25116
$651$	−2.45332e6	−0.226883
$652$	1.82812e7	1.68417
$653$	−1.96976e7	−1.80772	−0.903858	−	0.427832i	$-0.859277\pi$
$653$	−1.96976e7	−1.80772	−0.903858	+	0.427832i	$0.859277\pi$
$654$	−1.18688e7	−1.08508
$655$	2.64203e6	0.240622
$656$	−4.24391e6	−0.385041
$657$	775653.	0.0701058
$658$	2.24374e7	2.02026
$659$	−1.19800e7	−1.07460	−0.537298	−	0.843393i	$-0.680555\pi$
$659$	−1.19800e7	−1.07460	−0.537298	+	0.843393i	$0.680555\pi$
$660$	5.77357e6	0.515922
$661$	−7.62846e6	−0.679099	−0.339550	−	0.940588i	$-0.610275\pi$
$661$	−7.62846e6	−0.679099	−0.339550	+	0.940588i	$0.610275\pi$
$662$	−7.19062e6	−0.637708
$663$	1.96897e7	1.73962
$664$	−5.02714e6	−0.442487
$665$	−752694.	−0.0660031
$666$	1.06031e7	0.926289
$667$	−2.21205e7	−1.92522
$668$	−1.44404e7	−1.25210
$669$	7.21382e6	0.623161
$670$	−7.94486e6	−0.683753
$671$	1.06494e7	0.913099
$672$	1.81013e7	1.54627
$673$	3.71037e6	0.315776	0.157888	−	0.987457i	$-0.449531\pi$
$673$	3.71037e6	0.315776	0.157888	+	0.987457i	$0.449531\pi$
$674$	3.29902e7	2.79728
$675$	4.77904e6	0.403721
$676$	1.01533e7	0.854560
$677$	6.66373e6	0.558786	0.279393	−	0.960177i	$-0.409867\pi$
$677$	6.66373e6	0.558786	0.279393	+	0.960177i	$0.409867\pi$
$678$	2.86243e7	2.39145
$679$	−1.28912e7	−1.07304
$680$	−6.15068e6	−0.510094
$681$	−1.90862e7	−1.57707
$682$	−1.63550e6	−0.134645
$683$	5.32022e6	0.436393	0.218196	−	0.975905i	$-0.429983\pi$
$683$	5.32022e6	0.436393	0.218196	+	0.975905i	$0.429983\pi$
$684$	−894933.	−0.0731392
$685$	−4.45087e6	−0.362425
$686$	1.84993e7	1.50087
$687$	−1.59169e7	−1.28667
$688$	−5.34083e6	−0.430167
$689$	8.74317e6	0.701651
$690$	−1.88873e7	−1.51024
$691$	2.44212e6	0.194568	0.0972842	−	0.995257i	$-0.468984\pi$
$691$	2.44212e6	0.194568	0.0972842	+	0.995257i	$0.468984\pi$
$692$	3.08293e7	2.44737
$693$	2.97861e6	0.235603
$694$	1.35945e7	1.07144
$695$	1.43836e6	0.112955
$696$	−1.69105e7	−1.32323
$697$	1.82364e7	1.42186
$698$	2.76792e7	2.15038
$699$	−2.25168e7	−1.74307
$700$	1.25517e7	0.968184
$701$	−9.95456e6	−0.765116	−0.382558	−	0.923932i	$-0.624957\pi$
$701$	−9.95456e6	−0.765116	−0.382558	+	0.923932i	$0.624957\pi$
$702$	−1.64571e7	−1.26041
$703$	−1.70348e6	−0.130002
$704$	1.01323e7	0.770506
$705$	1.19971e7	0.909085
$706$	1.68968e7	1.27583
$707$	9.58190e6	0.720947
$708$	6.65869e6	0.499235
$709$	−4.30998e6	−0.322003	−0.161002	−	0.986954i	$-0.551472\pi$
$709$	−4.30998e6	−0.322003	−0.161002	+	0.986954i	$0.551472\pi$
$710$	4.29280e6	0.319591
$711$	−3.35337e6	−0.248775
$712$	1.22027e7	0.902100
$713$	3.18445e6	0.234591
$714$	−3.08097e7	−2.26174
$715$	4.96655e6	0.363320
$716$	2.39090e6	0.174293
$717$	2.01361e7	1.46277
$718$	59021.0	0.00427263
$719$	−8.19873e6	−0.591459	−0.295729	−	0.955272i	$-0.595563\pi$
$719$	−8.19873e6	−0.591459	−0.295729	+	0.955272i	$0.595563\pi$
$720$	1.23437e6	0.0887389
$721$	2.37809e7	1.70369
$722$	−2.17735e7	−1.55448
$723$	2.08854e7	1.48593
$724$	−2.58425e7	−1.83227
$725$	1.32061e7	0.933104
$726$	−2.09407e7	−1.47452
$727$	−698821.	−0.0490377	−0.0245188	−	0.999699i	$-0.507805\pi$
$727$	−698821.	−0.0490377	−0.0245188	+	0.999699i	$0.507805\pi$
$728$	−1.38260e7	−0.966868
$729$	2.59998e6	0.181197
$730$	−2.02375e6	−0.140556
$731$	2.29499e7	1.58850
$732$	−4.95552e7	−3.41831
$733$	−4.55058e6	−0.312829	−0.156414	−	0.987692i	$-0.549994\pi$
$733$	−4.55058e6	−0.312829	−0.156414	+	0.987692i	$0.549994\pi$
$734$	−1.37415e6	−0.0941445
$735$	−883065.	−0.0602941
$736$	−2.34957e7	−1.59880
$737$	−5.05110e6	−0.342544
$738$	1.37839e7	0.931603
$739$	2.71324e6	0.182758	0.0913791	−	0.995816i	$-0.470873\pi$
$739$	2.71324e6	0.182758	0.0913791	+	0.995816i	$0.470873\pi$
$740$	−1.64657e7	−1.10535
$741$	−2.39098e6	−0.159967
$742$	−1.36810e7	−0.912238
$743$	−7.49985e6	−0.498403	−0.249201	−	0.968452i	$-0.580168\pi$
$743$	−7.49985e6	−0.498403	−0.249201	+	0.968452i	$0.580168\pi$
$744$	2.43443e6	0.161237
$745$	−9.89510e6	−0.653175
$746$	2.51020e7	1.65144
$747$	−4.33526e6	−0.284259
$748$	−1.22248e7	−0.798893
$749$	−1.90463e7	−1.24053
$750$	2.90877e7	1.88823
$751$	1.28965e7	0.834398	0.417199	−	0.908815i	$-0.363012\pi$
$751$	1.28965e7	0.834398	0.417199	+	0.908815i	$0.363012\pi$
$752$	5.91158e6	0.381205
$753$	−1.96306e6	−0.126167
$754$	−4.54766e7	−2.91313
$755$	1.43953e7	0.919078
$756$	1.53271e7	0.975340
$757$	3.93920e6	0.249844	0.124922	−	0.992167i	$-0.460132\pi$
$757$	3.93920e6	0.249844	0.124922	+	0.992167i	$0.460132\pi$
$758$	−2.04628e7	−1.29357
$759$	−1.20079e7	−0.756596
$760$	746897.	0.0469058
$761$	−6.39591e6	−0.400351	−0.200175	−	0.979760i	$-0.564151\pi$
$761$	−6.39591e6	−0.400351	−0.200175	+	0.979760i	$0.564151\pi$
$762$	−3.34197e7	−2.08505
$763$	9.50877e6	0.591307
$764$	−2.60873e7	−1.61695
$765$	−5.30417e6	−0.327690
$766$	−3.70223e6	−0.227977
$767$	5.72795e6	0.351569
$768$	−8.95809e6	−0.548040
$769$	−2.87887e7	−1.75552	−0.877760	−	0.479101i	$-0.840963\pi$
$769$	−2.87887e7	−1.75552	−0.877760	+	0.479101i	$0.840963\pi$
$770$	−7.77148e6	−0.472364
$771$	8.00946e6	0.485252
$772$	2.25838e6	0.136381
$773$	−1.67733e6	−0.100965	−0.0504824	−	0.998725i	$-0.516076\pi$
$773$	−1.67733e6	−0.100965	−0.0504824	+	0.998725i	$0.516076\pi$
$774$	1.73466e7	1.04079
$775$	−1.90114e6	−0.113700
$776$	1.27919e7	0.762570
$777$	−2.63831e7	−1.56774
$778$	3.40166e7	2.01485
$779$	−2.21450e6	−0.130747
$780$	−2.31111e7	−1.36014
$781$	2.72923e6	0.160108
$782$	3.99915e7	2.33857
$783$	1.61262e7	0.940000
$784$	−435130.	−0.0252830
$785$	8.10895e6	0.469668
$786$	−1.31324e7	−0.758209
$787$	−1.64671e7	−0.947719	−0.473859	−	0.880601i	$-0.657140\pi$
$787$	−1.64671e7	−0.947719	−0.473859	+	0.880601i	$0.657140\pi$
$788$	4.74793e7	2.72389
$789$	3.39142e7	1.93950
$790$	8.74926e6	0.498774
$791$	−2.29326e7	−1.30320
$792$	−2.95567e6	−0.167434
$793$	−4.26285e7	−2.40723
$794$	2.76280e7	1.55524
$795$	−7.31514e6	−0.410492
$796$	2.37277e7	1.32731
$797$	−2.06976e7	−1.15418	−0.577092	−	0.816679i	$-0.695813\pi$
$797$	−2.06976e7	−1.15418	−0.577092	+	0.816679i	$0.695813\pi$
$798$	3.74132e6	0.207978
$799$	−2.54025e7	−1.40770
$800$	1.40272e7	0.774898
$801$	1.05232e7	0.579519
$802$	3.02091e6	0.165845
$803$	−1.28664e6	−0.0704155
$804$	2.35045e7	1.28236
$805$	1.51317e7	0.822996
$806$	6.54678e6	0.354969
$807$	−1.28172e6	−0.0692802
$808$	−9.50810e6	−0.512348
$809$	1.76250e7	0.946797	0.473399	−	0.880848i	$-0.343027\pi$
$809$	1.76250e7	0.946797	0.473399	+	0.880848i	$0.343027\pi$
$810$	2.22116e7	1.18951
$811$	3.58516e7	1.91406	0.957031	−	0.289987i	$-0.0936509\pi$
$811$	3.58516e7	1.91406	0.957031	+	0.289987i	$0.0936509\pi$
$812$	4.23541e7	2.25427
$813$	−1.77274e7	−0.940631
$814$	−1.75882e7	−0.930382
$815$	−1.31574e7	−0.693864
$816$	−8.11743e6	−0.426769
$817$	−2.78688e6	−0.146071
$818$	1.38891e7	0.725758
$819$	−1.19231e7	−0.621127
$820$	−2.14053e7	−1.11170
$821$	1.26101e7	0.652922	0.326461	−	0.945211i	$-0.394144\pi$
$821$	1.26101e7	0.652922	0.326461	+	0.945211i	$0.394144\pi$
$822$	2.21234e7	1.14202
$823$	−412984.	−0.0212537	−0.0106268	−	0.999944i	$-0.503383\pi$
$823$	−412984.	−0.0212537	−0.0106268	+	0.999944i	$0.503383\pi$
$824$	−2.35978e7	−1.21075
$825$	7.16884e6	0.366703
$826$	−8.96290e6	−0.457086
$827$	−1.39510e7	−0.709318	−0.354659	−	0.934996i	$-0.615403\pi$
$827$	−1.39510e7	−0.709318	−0.354659	+	0.934996i	$0.615403\pi$
$828$	1.79912e7	0.911975
$829$	1.37175e7	0.693247	0.346623	−	0.938004i	$-0.387328\pi$
$829$	1.37175e7	0.693247	0.346623	+	0.938004i	$0.387328\pi$
$830$	1.13111e7	0.569915
$831$	1.35035e7	0.678335
$832$	−4.05587e7	−2.03131
$833$	1.86978e6	0.0933639
$834$	−7.14950e6	−0.355927
$835$	1.03931e7	0.515855
$836$	1.48450e6	0.0734623
$837$	−2.32152e6	−0.114540
$838$	−3.82920e6	−0.188364
$839$	−1.98766e7	−0.974849	−0.487424	−	0.873165i	$-0.662063\pi$
$839$	−1.98766e7	−0.974849	−0.487424	+	0.873165i	$0.662063\pi$
$840$	1.15678e7	0.565654
$841$	2.40511e7	1.17259
$842$	−2.80802e7	−1.36496
$843$	1.72005e7	0.833629
$844$	1.89768e7	0.916997
$845$	−7.30758e6	−0.352072
$846$	−1.92003e7	−0.922323
$847$	1.67768e7	0.803529
$848$	−3.60453e6	−0.172131
$849$	−3.71214e7	−1.76748
$850$	−2.38752e7	−1.13344
$851$	3.42456e7	1.62099
$852$	−1.27001e7	−0.599386
$853$	−3.14168e7	−1.47839	−0.739196	−	0.673490i	$-0.764794\pi$
$853$	−3.14168e7	−1.47839	−0.739196	+	0.673490i	$0.764794\pi$
$854$	6.67036e7	3.12971
$855$	644102.	0.0301328
$856$	1.88996e7	0.881592
$857$	−1.20160e7	−0.558865	−0.279432	−	0.960165i	$-0.590146\pi$
$857$	−1.20160e7	−0.558865	−0.279432	+	0.960165i	$0.590146\pi$
$858$	−2.46866e7	−1.14484
$859$	3.96288e6	0.183243	0.0916216	−	0.995794i	$-0.470795\pi$
$859$	3.96288e6	0.183243	0.0916216	+	0.995794i	$0.470795\pi$
$860$	−2.69378e7	−1.24198
$861$	−3.42977e7	−1.57673
$862$	2.54138e7	1.16493
$863$	−1.59005e7	−0.726750	−0.363375	−	0.931643i	$-0.618376\pi$
$863$	−1.59005e7	−0.726750	−0.363375	+	0.931643i	$0.618376\pi$
$864$	1.71288e7	0.780625
$865$	−2.21885e7	−1.00830
$866$	−3.85073e7	−1.74481
$867$	8.00138e6	0.361507
$868$	−6.09726e6	−0.274685
$869$	5.56251e6	0.249874
$870$	3.80488e7	1.70429
$871$	2.02191e7	0.903060
$872$	−9.43553e6	−0.420218
$873$	1.10313e7	0.489883
$874$	−4.85630e6	−0.215044
$875$	−2.33038e7	−1.02898
$876$	5.98718e6	0.263610
$877$	−3.99748e7	−1.75504	−0.877520	−	0.479540i	$-0.840803\pi$
$877$	−3.99748e7	−1.75504	−0.877520	+	0.479540i	$0.840803\pi$
$878$	−6.23669e7	−2.73035
$879$	1.54642e7	0.675079
$880$	−2.04755e6	−0.0891309
$881$	3.99093e7	1.73235	0.866173	−	0.499745i	$-0.166573\pi$
$881$	3.99093e7	1.73235	0.866173	+	0.499745i	$0.166573\pi$
$882$	1.41327e6	0.0611721
$883$	−1.13118e7	−0.488235	−0.244118	−	0.969746i	$-0.578498\pi$
$883$	−1.13118e7	−0.488235	−0.244118	+	0.969746i	$0.578498\pi$
$884$	4.89349e7	2.10614
$885$	−4.79240e6	−0.205681
$886$	−4.48843e7	−1.92092
$887$	3.72644e7	1.59032	0.795161	−	0.606398i	$-0.207386\pi$
$887$	3.72644e7	1.59032	0.795161	+	0.606398i	$0.207386\pi$
$888$	2.61799e7	1.11413
$889$	2.67745e7	1.13623
$890$	−2.74561e7	−1.16189
$891$	1.41215e7	0.595917
$892$	1.79286e7	0.754455
$893$	3.08470e6	0.129445
$894$	4.91843e7	2.05818
$895$	−1.72078e6	−0.0718073
$896$	3.28678e7	1.36773
$897$	4.80668e7	1.99464
$898$	−969041.	−0.0401006
$899$	−6.41515e6	−0.264733
$900$	−1.07409e7	−0.442011
$901$	1.54889e7	0.635637
$902$	−2.28645e7	−0.935719
$903$	−4.31626e7	−1.76152
$904$	2.27560e7	0.926135
$905$	1.85994e7	0.754880
$906$	−7.15528e7	−2.89605
$907$	1.92037e7	0.775114	0.387557	−	0.921846i	$-0.373319\pi$
$907$	1.92037e7	0.775114	0.387557	+	0.921846i	$0.373319\pi$
$908$	−4.74351e7	−1.90935
$909$	−8.19951e6	−0.329138
$910$	3.11086e7	1.24531
$911$	−1.78298e7	−0.711788	−0.355894	−	0.934526i	$-0.615824\pi$
$911$	−1.78298e7	−0.711788	−0.355894	+	0.934526i	$0.615824\pi$
$912$	985726.	0.0392436
$913$	7.19126e6	0.285514
$914$	−7.22875e6	−0.286219
$915$	3.56660e7	1.40832
$916$	−3.95584e7	−1.55776
$917$	1.05212e7	0.413181
$918$	−2.91545e7	−1.14182
$919$	5.10041e7	1.99212	0.996061	−	0.0886662i	$-0.0282604\pi$
$919$	5.10041e7	1.99212	0.996061	+	0.0886662i	$0.0282604\pi$
$920$	−1.50151e7	−0.584870
$921$	15948.8	0.000619553 0
$922$	−7.73214e7	−2.99552
$923$	−1.09249e7	−0.422097
$924$	2.29916e7	0.885909
$925$	−2.04449e7	−0.785654
$926$	6.16511e7	2.36273
$927$	−2.03500e7	−0.777796
$928$	4.73327e7	1.80423
$929$	−1.75776e7	−0.668223	−0.334111	−	0.942534i	$-0.608436\pi$
$929$	−1.75776e7	−0.668223	−0.334111	+	0.942534i	$0.608436\pi$
$930$	−5.47749e6	−0.207670
$931$	−227054.	−0.00858529
$932$	−5.59612e7	−2.11032
$933$	−5.21544e6	−0.196149
$934$	−5.64183e7	−2.11618
$935$	8.79847e6	0.329138
$936$	1.18313e7	0.441410
$937$	−2.52501e7	−0.939537	−0.469768	−	0.882790i	$-0.655663\pi$
$937$	−2.52501e7	−0.939537	−0.469768	+	0.882790i	$0.655663\pi$
$938$	−3.16381e7	−1.17410
$939$	−1.73988e7	−0.643953
$940$	2.98166e7	1.10062
$941$	−8.38617e6	−0.308738	−0.154369	−	0.988013i	$-0.549334\pi$
$941$	−8.38617e6	−0.308738	−0.154369	+	0.988013i	$0.549334\pi$
$942$	−4.03062e7	−1.47994
$943$	4.45189e7	1.63029
$944$	−2.36145e6	−0.0862481
$945$	−1.10313e7	−0.401833
$946$	−2.87742e7	−1.04538
$947$	−3.78162e7	−1.37026	−0.685129	−	0.728422i	$-0.740254\pi$
$947$	−3.78162e7	−1.37026	−0.685129	+	0.728422i	$0.740254\pi$
$948$	−2.58843e7	−0.935439
$949$	5.15031e6	0.185638
$950$	2.89925e6	0.104226
$951$	−5.62910e6	−0.201831
$952$	−2.44933e7	−0.875902
$953$	3.78762e7	1.35093	0.675467	−	0.737390i	$-0.263942\pi$
$953$	3.78762e7	1.35093	0.675467	+	0.737390i	$0.263942\pi$
$954$	1.17072e7	0.416470
$955$	1.87756e7	0.666170
$956$	5.00444e7	1.77097
$957$	2.41903e7	0.853810
$958$	5.31645e7	1.87158
$959$	−1.77243e7	−0.622333
$960$	3.39342e7	1.18839
$961$	923521.	0.0322581
$962$	7.04042e7	2.45279
$963$	1.62985e7	0.566345
$964$	5.19067e7	1.79900
$965$	−1.62540e6	−0.0561878
$966$	−7.52132e7	−2.59329
$967$	−2.07069e7	−0.712113	−0.356056	−	0.934465i	$-0.615879\pi$
$967$	−2.07069e7	−0.712113	−0.356056	+	0.934465i	$0.615879\pi$
$968$	−1.66476e7	−0.571036
$969$	−4.23573e6	−0.144917
$970$	−2.87818e7	−0.982176
$971$	1.71800e7	0.584757	0.292379	−	0.956303i	$-0.405553\pi$
$971$	1.71800e7	0.584757	0.292379	+	0.956303i	$0.405553\pi$
$972$	−3.80926e7	−1.29323
$973$	5.72788e6	0.193960
$974$	4.09374e7	1.38268
$975$	−2.86963e7	−0.966749
$976$	1.75744e7	0.590549
$977$	2.05109e7	0.687462	0.343731	−	0.939068i	$-0.388309\pi$
$977$	2.05109e7	0.687462	0.343731	+	0.939068i	$0.388309\pi$
$978$	6.53996e7	2.18639
$979$	−1.74558e7	−0.582079
$980$	−2.19469e6	−0.0729975
$981$	−8.13693e6	−0.269953
$982$	1.59263e7	0.527032
$983$	−1.21189e7	−0.400019	−0.200010	−	0.979794i	$-0.564097\pi$
$983$	−1.21189e7	−0.400019	−0.200010	+	0.979794i	$0.564097\pi$
$984$	3.40335e7	1.12052
$985$	−3.41719e7	−1.12222
$986$	−8.05638e7	−2.63905
$987$	4.77752e7	1.56102
$988$	−5.94233e6	−0.193671
$989$	5.60256e7	1.82136
$990$	6.65028e6	0.215651
$991$	3.37573e7	1.09190	0.545951	−	0.837817i	$-0.316169\pi$
$991$	3.37573e7	1.09190	0.545951	+	0.837817i	$0.316169\pi$
$992$	−6.81399e6	−0.219848
$993$	−1.53107e7	−0.492746
$994$	1.70949e7	0.548782
$995$	−1.70773e7	−0.546843
$996$	−3.34634e7	−1.06886
$997$	1.84529e7	0.587932	0.293966	−	0.955816i	$-0.405025\pi$
$997$	1.84529e7	0.587932	0.293966	+	0.955816i	$0.405025\pi$
$998$	−6.09226e7	−1.93621
$999$	−2.49657e7	−0.791461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	31.6.a.b.1.7	✓	8
3.2	odd	2		279.6.a.f.1.2		8
4.3	odd	2		496.6.a.h.1.3		8
5.4	even	2		775.6.a.b.1.2		8
31.30	odd	2		961.6.a.c.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.6.a.b.1.7	✓	8	1.1	even	1	trivial
279.6.a.f.1.2		8	3.2	odd	2
496.6.a.h.1.3		8	4.3	odd	2
775.6.a.b.1.2		8	5.4	even	2
961.6.a.c.1.7		8	31.30	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:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	31.a (trivial)

\(f(q)\)	\(=\)	\(q+8.89102 q^{2} +18.9313 q^{3} +47.0502 q^{4} -33.8630 q^{5} +168.319 q^{6} -134.850 q^{7} +133.811 q^{8} +115.395 q^{9} -301.077 q^{10} -191.415 q^{11} +890.722 q^{12} +766.219 q^{13} -1198.95 q^{14} -641.071 q^{15} -315.888 q^{16} +1357.39 q^{17} +1025.98 q^{18} -164.832 q^{19} -1593.26 q^{20} -2552.89 q^{21} -1701.88 q^{22} +3313.68 q^{23} +2533.22 q^{24} -1978.30 q^{25} +6812.46 q^{26} -2415.73 q^{27} -6344.71 q^{28} -6675.50 q^{29} -5699.78 q^{30} +961.000 q^{31} -7090.52 q^{32} -3623.74 q^{33} +12068.6 q^{34} +4566.42 q^{35} +5429.35 q^{36} +10334.6 q^{37} -1465.53 q^{38} +14505.5 q^{39} -4531.25 q^{40} +13434.9 q^{41} -22697.7 q^{42} +16907.4 q^{43} -9006.12 q^{44} -3907.62 q^{45} +29462.0 q^{46} -18714.2 q^{47} -5980.17 q^{48} +1377.48 q^{49} -17589.1 q^{50} +25697.2 q^{51} +36050.7 q^{52} +11410.8 q^{53} -21478.3 q^{54} +6481.90 q^{55} -18044.4 q^{56} -3120.50 q^{57} -59351.9 q^{58} +7475.61 q^{59} -30162.5 q^{60} -55634.9 q^{61} +8544.27 q^{62} -15561.0 q^{63} -52933.5 q^{64} -25946.5 q^{65} -32218.8 q^{66} +26388.2 q^{67} +63865.5 q^{68} +62732.4 q^{69} +40600.1 q^{70} -14258.2 q^{71} +15441.1 q^{72} +6721.72 q^{73} +91885.2 q^{74} -37451.8 q^{75} -7755.39 q^{76} +25812.3 q^{77} +128969. q^{78} -29059.9 q^{79} +10696.9 q^{80} -73774.0 q^{81} +119450. q^{82} -37568.9 q^{83} -120114. q^{84} -45965.3 q^{85} +150324. q^{86} -126376. q^{87} -25613.5 q^{88} +91193.1 q^{89} -34742.7 q^{90} -103325. q^{91} +155909. q^{92} +18193.0 q^{93} -166388. q^{94} +5581.72 q^{95} -134233. q^{96} +95596.4 q^{97} +12247.2 q^{98} -22088.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 7 q^{2} - 2 q^{3} + 149 q^{4} + 128 q^{5} + 72 q^{6} + 88 q^{7} + 924 q^{8} + 1512 q^{9} + 1581 q^{10} + 574 q^{11} - 46 q^{12} - 122 q^{13} - 309 q^{14} - 524 q^{15} + 833 q^{16} + 1932 q^{17} - 6845 q^{18}+ \cdots - 180150 q^{99}+O(q^{100}) \) 8 * q + 7 * q^2 - 2 * q^3 + 149 * q^4 + 128 * q^5 + 72 * q^6 + 88 * q^7 + 924 * q^8 + 1512 * q^9 + 1581 * q^10 + 574 * q^11 - 46 * q^12 - 122 * q^13 - 309 * q^14 - 524 * q^15 + 833 * q^16 + 1932 * q^17 - 6845 * q^18 - 1796 * q^19 - 37 * q^20 + 996 * q^21 - 9000 * q^22 - 4136 * q^23 - 17468 * q^24 + 3308 * q^25 + 1524 * q^26 - 2624 * q^27 - 21245 * q^28 + 5050 * q^29 - 30450 * q^30 + 7688 * q^31 + 21045 * q^32 + 27920 * q^33 - 24738 * q^34 + 25700 * q^35 + 21637 * q^36 + 23674 * q^37 - 4289 * q^38 + 38652 * q^39 + 36606 * q^40 + 44828 * q^41 - 9994 * q^42 + 21058 * q^43 + 13168 * q^44 - 1344 * q^45 + 6198 * q^46 + 5348 * q^47 - 69588 * q^48 + 19356 * q^49 + 33242 * q^50 + 22300 * q^51 - 60386 * q^52 + 4926 * q^53 - 24476 * q^54 - 56892 * q^55 - 87652 * q^56 + 22072 * q^57 - 25002 * q^58 + 16944 * q^59 - 138556 * q^60 - 73682 * q^61 + 6727 * q^62 - 138784 * q^63 - 1300 * q^64 + 63316 * q^65 - 128796 * q^66 - 134768 * q^67 - 9524 * q^68 + 3992 * q^69 - 142737 * q^70 + 123724 * q^71 + 93232 * q^72 + 70792 * q^73 + 307468 * q^74 - 302902 * q^75 - 65405 * q^76 + 107932 * q^77 + 36820 * q^78 - 26036 * q^79 + 576783 * q^80 + 82528 * q^81 + 142335 * q^82 + 21882 * q^83 + 299144 * q^84 - 130464 * q^85 + 480478 * q^86 - 440244 * q^87 - 71982 * q^88 + 164392 * q^89 + 397969 * q^90 - 60028 * q^91 + 31012 * q^92 - 1922 * q^93 - 170388 * q^94 - 7404 * q^95 - 331010 * q^96 - 131948 * q^97 + 288768 * q^98 - 180150 * q^99

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