LMFDB - Embedded newform 495.6.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$79.3899908074$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 33x^{2} - 8x + 116 $ x^4 - 2x^3 - 33x^2 - 8*x + 116
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.95665$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.82466 q^{2} +29.2253 q^{4} +25.0000 q^{5} -125.436 q^{7} +21.7111 q^{8} +O(q^{10})$	\(q-7.82466 q^{2} +29.2253 q^{4} +25.0000 q^{5} -125.436 q^{7} +21.7111 q^{8} -195.616 q^{10} +121.000 q^{11} +532.300 q^{13} +981.491 q^{14} -1105.09 q^{16} +1373.09 q^{17} -554.639 q^{19} +730.632 q^{20} -946.784 q^{22} -4250.72 q^{23} +625.000 q^{25} -4165.06 q^{26} -3665.89 q^{28} +6973.40 q^{29} +3130.03 q^{31} +7952.21 q^{32} -10744.0 q^{34} -3135.89 q^{35} +1384.70 q^{37} +4339.86 q^{38} +542.778 q^{40} -679.385 q^{41} -1721.06 q^{43} +3536.26 q^{44} +33260.4 q^{46} +15143.3 q^{47} -1072.91 q^{49} -4890.41 q^{50} +15556.6 q^{52} +9544.44 q^{53} +3025.00 q^{55} -2723.35 q^{56} -54564.5 q^{58} -27582.7 q^{59} -40527.5 q^{61} -24491.5 q^{62} -26860.4 q^{64} +13307.5 q^{65} -58726.5 q^{67} +40129.0 q^{68} +24537.3 q^{70} +42527.8 q^{71} +23753.0 q^{73} -10834.8 q^{74} -16209.5 q^{76} -15177.7 q^{77} -78690.1 q^{79} -27627.3 q^{80} +5315.96 q^{82} -52252.1 q^{83} +34327.3 q^{85} +13466.7 q^{86} +2627.05 q^{88} -8156.09 q^{89} -66769.3 q^{91} -124228. q^{92} -118491. q^{94} -13866.0 q^{95} +79010.1 q^{97} +8395.17 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{2} + 61 q^{4} + 100 q^{5} - 90 q^{7} + 135 q^{8}+O(q^{10}) $ 4 * q + 5 * q^2 + 61 * q^4 + 100 * q^5 - 90 * q^7 + 135 * q^8	$ 4 q + 5 q^{2} + 61 q^{4} + 100 q^{5} - 90 q^{7} + 135 q^{8} + 125 q^{10} + 484 q^{11} + 820 q^{13} + 1687 q^{14} - 2671 q^{16} + 3800 q^{17} - 3394 q^{19} + 1525 q^{20} + 605 q^{22} + 3020 q^{23} + 2500 q^{25} + 3650 q^{26} + 2635 q^{28} + 5248 q^{29} + 4732 q^{31} - 11505 q^{32} + 9759 q^{34} - 2250 q^{35} + 10210 q^{37} - 16945 q^{38} + 3375 q^{40} + 21068 q^{41} - 12140 q^{43} + 7381 q^{44} + 58442 q^{46} - 4720 q^{47} + 24550 q^{49} + 3125 q^{50} + 58250 q^{52} + 21670 q^{53} + 12100 q^{55} + 28985 q^{56} - 45435 q^{58} + 69068 q^{59} - 44000 q^{61} - 33375 q^{62} - 68223 q^{64} + 20500 q^{65} - 8720 q^{67} + 107335 q^{68} + 42175 q^{70} + 47516 q^{71} - 2480 q^{73} + 55717 q^{74} - 102461 q^{76} - 10890 q^{77} - 188192 q^{79} - 66775 q^{80} + 279030 q^{82} - 68620 q^{83} + 95000 q^{85} - 115328 q^{86} + 16335 q^{88} + 170266 q^{89} + 97740 q^{91} - 53950 q^{92} - 152926 q^{94} - 84850 q^{95} + 186160 q^{97} + 393590 q^{98}+O(q^{100}) $ 4 * q + 5 * q^2 + 61 * q^4 + 100 * q^5 - 90 * q^7 + 135 * q^8 + 125 * q^10 + 484 * q^11 + 820 * q^13 + 1687 * q^14 - 2671 * q^16 + 3800 * q^17 - 3394 * q^19 + 1525 * q^20 + 605 * q^22 + 3020 * q^23 + 2500 * q^25 + 3650 * q^26 + 2635 * q^28 + 5248 * q^29 + 4732 * q^31 - 11505 * q^32 + 9759 * q^34 - 2250 * q^35 + 10210 * q^37 - 16945 * q^38 + 3375 * q^40 + 21068 * q^41 - 12140 * q^43 + 7381 * q^44 + 58442 * q^46 - 4720 * q^47 + 24550 * q^49 + 3125 * q^50 + 58250 * q^52 + 21670 * q^53 + 12100 * q^55 + 28985 * q^56 - 45435 * q^58 + 69068 * q^59 - 44000 * q^61 - 33375 * q^62 - 68223 * q^64 + 20500 * q^65 - 8720 * q^67 + 107335 * q^68 + 42175 * q^70 + 47516 * q^71 - 2480 * q^73 + 55717 * q^74 - 102461 * q^76 - 10890 * q^77 - 188192 * q^79 - 66775 * q^80 + 279030 * q^82 - 68620 * q^83 + 95000 * q^85 - 115328 * q^86 + 16335 * q^88 + 170266 * q^89 + 97740 * q^91 - 53950 * q^92 - 152926 * q^94 - 84850 * q^95 + 186160 * q^97 + 393590 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−7.82466	−1.38322	−0.691609	−	0.722272i	$-0.743098\pi$
$2$	−7.82466	−1.38322	−0.691609	+	0.722272i	$0.743098\pi$
$3$	0	0
$3$	0	0
$4$	29.2253	0.913290
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−125.436	−0.967555	−0.483778	−	0.875191i	$-0.660736\pi$
$7$	−125.436	−0.967555	−0.483778	+	0.875191i	$0.660736\pi$
$8$	21.7111	0.119938
$9$	0	0
$10$	−195.616	−0.618594
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	532.300	0.873570	0.436785	−	0.899566i	$-0.356117\pi$
$13$	532.300	0.873570	0.436785	+	0.899566i	$0.356117\pi$
$14$	981.491	1.33834
$15$	0	0
$16$	−1105.09	−1.07919
$17$	1373.09	1.15233	0.576166	−	0.817333i	$-0.304548\pi$
$17$	1373.09	1.15233	0.576166	+	0.817333i	$0.304548\pi$
$18$	0	0
$19$	−554.639	−0.352474	−0.176237	−	0.984348i	$-0.556392\pi$
$19$	−554.639	−0.352474	−0.176237	+	0.984348i	$0.556392\pi$
$20$	730.632	0.408436
$21$	0	0
$22$	−946.784	−0.417056
$23$	−4250.72	−1.67549	−0.837746	−	0.546060i	$-0.816127\pi$
$23$	−4250.72	−1.67549	−0.837746	+	0.546060i	$0.816127\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−4165.06	−1.20834
$27$	0	0
$28$	−3665.89	−0.883659
$29$	6973.40	1.53975	0.769874	−	0.638196i	$-0.220319\pi$
$29$	6973.40	1.53975	0.769874	+	0.638196i	$0.220319\pi$
$30$	0	0
$31$	3130.03	0.584985	0.292493	−	0.956268i	$-0.405515\pi$
$31$	3130.03	0.584985	0.292493	+	0.956268i	$0.405515\pi$
$32$	7952.21	1.37282
$33$	0	0
$34$	−10744.0	−1.59392
$35$	−3135.89	−0.432704
$36$	0	0
$37$	1384.70	0.166284	0.0831422	−	0.996538i	$-0.473504\pi$
$37$	1384.70	0.166284	0.0831422	+	0.996538i	$0.473504\pi$
$38$	4339.86	0.487548
$39$	0	0
$40$	542.778	0.0536380
$41$	−679.385	−0.0631185	−0.0315592	−	0.999502i	$-0.510047\pi$
$41$	−679.385	−0.0631185	−0.0315592	+	0.999502i	$0.510047\pi$
$42$	0	0
$43$	−1721.06	−0.141946	−0.0709732	−	0.997478i	$-0.522610\pi$
$43$	−1721.06	−0.141946	−0.0709732	+	0.997478i	$0.522610\pi$
$44$	3536.26	0.275367
$45$	0	0
$46$	33260.4	2.31757
$47$	15143.3	0.999945	0.499972	−	0.866041i	$-0.333344\pi$
$47$	15143.3	0.999945	0.499972	+	0.866041i	$0.333344\pi$
$48$	0	0
$49$	−1072.91	−0.0638372
$50$	−4890.41	−0.276643
$51$	0	0
$52$	15556.6	0.797823
$53$	9544.44	0.466724	0.233362	−	0.972390i	$-0.425027\pi$
$53$	9544.44	0.466724	0.233362	+	0.972390i	$0.425027\pi$
$54$	0	0
$55$	3025.00	0.134840
$56$	−2723.35	−0.116047
$57$	0	0
$58$	−54564.5	−2.12981
$59$	−27582.7	−1.03159	−0.515794	−	0.856713i	$-0.672503\pi$
$59$	−27582.7	−1.03159	−0.515794	+	0.856713i	$0.672503\pi$
$60$	0	0
$61$	−40527.5	−1.39452	−0.697261	−	0.716817i	$-0.745598\pi$
$61$	−40527.5	−1.39452	−0.697261	+	0.716817i	$0.745598\pi$
$62$	−24491.5	−0.809162
$63$	0	0
$64$	−26860.4	−0.819714
$65$	13307.5	0.390672
$66$	0	0
$67$	−58726.5	−1.59826	−0.799130	−	0.601159i	$-0.794706\pi$
$67$	−58726.5	−1.59826	−0.799130	+	0.601159i	$0.794706\pi$
$68$	40129.0	1.05241
$69$	0	0
$70$	24537.3	0.598523
$71$	42527.8	1.00121	0.500607	−	0.865675i	$-0.333110\pi$
$71$	42527.8	1.00121	0.500607	+	0.865675i	$0.333110\pi$
$72$	0	0
$73$	23753.0	0.521690	0.260845	−	0.965381i	$-0.415999\pi$
$73$	23753.0	0.521690	0.260845	+	0.965381i	$0.415999\pi$
$74$	−10834.8	−0.230007
$75$	0	0
$76$	−16209.5	−0.321911
$77$	−15177.7	−0.291729
$78$	0	0
$79$	−78690.1	−1.41857	−0.709287	−	0.704919i	$-0.750983\pi$
$79$	−78690.1	−1.41857	−0.709287	+	0.704919i	$0.750983\pi$
$80$	−27627.3	−0.482629
$81$	0	0
$82$	5315.96	0.0873066
$83$	−52252.1	−0.832546	−0.416273	−	0.909240i	$-0.636664\pi$
$83$	−52252.1	−0.832546	−0.416273	+	0.909240i	$0.636664\pi$
$84$	0	0
$85$	34327.3	0.515338
$86$	13466.7	0.196343
$87$	0	0
$88$	2627.05	0.0361627
$89$	−8156.09	−0.109146	−0.0545729	−	0.998510i	$-0.517380\pi$
$89$	−8156.09	−0.109146	−0.0545729	+	0.998510i	$0.517380\pi$
$90$	0	0
$91$	−66769.3	−0.845227
$92$	−124228.	−1.53021
$93$	0	0
$94$	−118491.	−1.38314
$95$	−13866.0	−0.157631
$96$	0	0
$97$	79010.1	0.852615	0.426308	−	0.904578i	$-0.359814\pi$
$97$	79010.1	0.852615	0.426308	+	0.904578i	$0.359814\pi$
$98$	8395.17	0.0883007
$99$	0	0
$100$	18265.8	0.182658
$101$	116968.	1.14095	0.570473	−	0.821317i	$-0.306760\pi$
$101$	116968.	1.14095	0.570473	+	0.821317i	$0.306760\pi$
$102$	0	0
$103$	99313.3	0.922389	0.461195	−	0.887299i	$-0.347421\pi$
$103$	99313.3	0.922389	0.461195	+	0.887299i	$0.347421\pi$
$104$	11556.8	0.104774
$105$	0	0
$106$	−74682.0	−0.645581
$107$	−122146.	−1.03138	−0.515692	−	0.856774i	$-0.672465\pi$
$107$	−122146.	−1.03138	−0.515692	+	0.856774i	$0.672465\pi$
$108$	0	0
$109$	−68840.4	−0.554980	−0.277490	−	0.960729i	$-0.589503\pi$
$109$	−68840.4	−0.554980	−0.277490	+	0.960729i	$0.589503\pi$
$110$	−23669.6	−0.186513
$111$	0	0
$112$	138618.	1.04418
$113$	65961.5	0.485953	0.242977	−	0.970032i	$-0.421876\pi$
$113$	65961.5	0.485953	0.242977	+	0.970032i	$0.421876\pi$
$114$	0	0
$115$	−106268.	−0.749303
$116$	203800.	1.40624
$117$	0	0
$118$	215825.	1.42691
$119$	−172235.	−1.11494
$120$	0	0
$121$	14641.0	0.0909091
$122$	317114.	1.92893
$123$	0	0
$124$	91476.2	0.534261
$125$	15625.0	0.0894427
$126$	0	0
$127$	296082.	1.62893	0.814464	−	0.580214i	$-0.197031\pi$
$127$	296082.	1.62893	0.814464	+	0.580214i	$0.197031\pi$
$128$	−44297.3	−0.238975
$129$	0	0
$130$	−104127.	−0.540385
$131$	−308100.	−1.56860	−0.784302	−	0.620379i	$-0.786979\pi$
$131$	−308100.	−1.56860	−0.784302	+	0.620379i	$0.786979\pi$
$132$	0	0
$133$	69571.5	0.341038
$134$	459515.	2.21074
$135$	0	0
$136$	29811.4	0.138209
$137$	420407.	1.91368	0.956839	−	0.290617i	$-0.0938606\pi$
$137$	420407.	1.91368	0.956839	+	0.290617i	$0.0938606\pi$
$138$	0	0
$139$	−263129.	−1.15513	−0.577566	−	0.816344i	$-0.695997\pi$
$139$	−263129.	−1.15513	−0.577566	+	0.816344i	$0.695997\pi$
$140$	−91647.3	−0.395184
$141$	0	0
$142$	−332765.	−1.38490
$143$	64408.3	0.263391
$144$	0	0
$145$	174335.	0.688596
$146$	−185860.	−0.721610
$147$	0	0
$148$	40468.3	0.151866
$149$	−134194.	−0.495186	−0.247593	−	0.968864i	$-0.579640\pi$
$149$	−134194.	−0.495186	−0.247593	+	0.968864i	$0.579640\pi$
$150$	0	0
$151$	93266.4	0.332876	0.166438	−	0.986052i	$-0.446773\pi$
$151$	93266.4	0.332876	0.166438	+	0.986052i	$0.446773\pi$
$152$	−12041.9	−0.0422751
$153$	0	0
$154$	118760.	0.403524
$155$	78250.9	0.261613
$156$	0	0
$157$	427883.	1.38540	0.692701	−	0.721225i	$-0.256420\pi$
$157$	427883.	1.38540	0.692701	+	0.721225i	$0.256420\pi$
$158$	615723.	1.96220
$159$	0	0
$160$	198805.	0.613943
$161$	533191.	1.62113
$162$	0	0
$163$	655901.	1.93361	0.966806	−	0.255511i	$-0.0822437\pi$
$163$	655901.	1.93361	0.966806	+	0.255511i	$0.0822437\pi$
$164$	−19855.2	−0.0576455
$165$	0	0
$166$	408855.	1.15159
$167$	−26926.6	−0.0747120	−0.0373560	−	0.999302i	$-0.511894\pi$
$167$	−26926.6	−0.0747120	−0.0373560	+	0.999302i	$0.511894\pi$
$168$	0	0
$169$	−87950.0	−0.236875
$170$	−268599.	−0.712825
$171$	0	0
$172$	−50298.4	−0.129638
$173$	−30708.4	−0.0780084	−0.0390042	−	0.999239i	$-0.512419\pi$
$173$	−30708.4	−0.0780084	−0.0390042	+	0.999239i	$0.512419\pi$
$174$	0	0
$175$	−78397.2	−0.193511
$176$	−133716.	−0.325388
$177$	0	0
$178$	63818.6	0.150972
$179$	535046.	1.24813	0.624063	−	0.781374i	$-0.285481\pi$
$179$	535046.	1.24813	0.624063	+	0.781374i	$0.285481\pi$
$180$	0	0
$181$	774794.	1.75788	0.878941	−	0.476930i	$-0.158250\pi$
$181$	774794.	1.75788	0.878941	+	0.476930i	$0.158250\pi$
$182$	522447.	1.16913
$183$	0	0
$184$	−92287.8	−0.200956
$185$	34617.5	0.0743646
$186$	0	0
$187$	166144.	0.347441
$188$	442567.	0.913240
$189$	0	0
$190$	108497.	0.218038
$191$	662092.	1.31321	0.656606	−	0.754234i	$-0.271992\pi$
$191$	662092.	1.31321	0.656606	+	0.754234i	$0.271992\pi$
$192$	0	0
$193$	−311639.	−0.602225	−0.301112	−	0.953589i	$-0.597358\pi$
$193$	−311639.	−0.602225	−0.301112	+	0.953589i	$0.597358\pi$
$194$	−618227.	−1.17935
$195$	0	0
$196$	−31356.2	−0.0583019
$197$	1.00469e6	1.84445	0.922223	−	0.386659i	$-0.126371\pi$
$197$	1.00469e6	1.84445	0.922223	+	0.386659i	$0.126371\pi$
$198$	0	0
$199$	495472.	0.886923	0.443462	−	0.896293i	$-0.353750\pi$
$199$	495472.	0.886923	0.443462	+	0.896293i	$0.353750\pi$
$200$	13569.5	0.0239876
$201$	0	0
$202$	−915237.	−1.57818
$203$	−874712.	−1.48979
$204$	0	0
$205$	−16984.6	−0.0282274
$206$	−777093.	−1.27586
$207$	0	0
$208$	−588240.	−0.942749
$209$	−67111.4	−0.106275
$210$	0	0
$211$	113891.	0.176109	0.0880547	−	0.996116i	$-0.471935\pi$
$211$	113891.	0.176109	0.0880547	+	0.996116i	$0.471935\pi$
$212$	278939.	0.426255
$213$	0	0
$214$	955751.	1.42663
$215$	−43026.4	−0.0634803
$216$	0	0
$217$	−392618.	−0.566005
$218$	538653.	0.767658
$219$	0	0
$220$	88406.5	0.123148
$221$	730896.	1.00664
$222$	0	0
$223$	−187638.	−0.252673	−0.126337	−	0.991987i	$-0.540322\pi$
$223$	−187638.	−0.252673	−0.126337	+	0.991987i	$0.540322\pi$
$224$	−997490.	−1.32828
$225$	0	0
$226$	−516126.	−0.672179
$227$	839421.	1.08122	0.540611	−	0.841273i	$-0.318193\pi$
$227$	839421.	1.08122	0.540611	+	0.841273i	$0.318193\pi$
$228$	0	0
$229$	−464599.	−0.585450	−0.292725	−	0.956197i	$-0.594562\pi$
$229$	−464599.	−0.585450	−0.292725	+	0.956197i	$0.594562\pi$
$230$	831510.	1.03645
$231$	0	0
$232$	151400.	0.184675
$233$	−1.18305e6	−1.42763	−0.713813	−	0.700336i	$-0.753033\pi$
$233$	−1.18305e6	−1.42763	−0.713813	+	0.700336i	$0.753033\pi$
$234$	0	0
$235$	378583.	0.447189
$236$	−806112.	−0.942139
$237$	0	0
$238$	1.34768e6	1.54221
$239$	119668.	0.135514	0.0677570	−	0.997702i	$-0.478416\pi$
$239$	119668.	0.135514	0.0677570	+	0.997702i	$0.478416\pi$
$240$	0	0
$241$	103550.	0.114844	0.0574220	−	0.998350i	$-0.481712\pi$
$241$	103550.	0.114844	0.0574220	+	0.998350i	$0.481712\pi$
$242$	−114561.	−0.125747
$243$	0	0
$244$	−1.18443e6	−1.27360
$245$	−26822.8	−0.0285489
$246$	0	0
$247$	−295234.	−0.307911
$248$	67956.6	0.0701621
$249$	0	0
$250$	−122260.	−0.123719
$251$	−210204.	−0.210599	−0.105300	−	0.994441i	$-0.533580\pi$
$251$	−210204.	−0.210599	−0.105300	+	0.994441i	$0.533580\pi$
$252$	0	0
$253$	−514337.	−0.505180
$254$	−2.31674e6	−2.25316
$255$	0	0
$256$	1.20614e6	1.15027
$257$	1.20825e6	1.14110	0.570548	−	0.821264i	$-0.306731\pi$
$257$	1.20825e6	1.14110	0.570548	+	0.821264i	$0.306731\pi$
$258$	0	0
$259$	−173691.	−0.160889
$260$	388915.	0.356797
$261$	0	0
$262$	2.41078e6	2.16972
$263$	1.82802e6	1.62964	0.814819	−	0.579715i	$-0.196836\pi$
$263$	1.82802e6	1.62964	0.814819	+	0.579715i	$0.196836\pi$
$264$	0	0
$265$	238611.	0.208726
$266$	−544373.	−0.471729
$267$	0	0
$268$	−1.71630e6	−1.45968
$269$	478347.	0.403053	0.201526	−	0.979483i	$-0.435410\pi$
$269$	478347.	0.403053	0.201526	+	0.979483i	$0.435410\pi$
$270$	0	0
$271$	−1.59909e6	−1.32267	−0.661333	−	0.750093i	$-0.730009\pi$
$271$	−1.59909e6	−1.32267	−0.661333	+	0.750093i	$0.730009\pi$
$272$	−1.51739e6	−1.24359
$273$	0	0
$274$	−3.28955e6	−2.64703
$275$	75625.0	0.0603023
$276$	0	0
$277$	−515488.	−0.403663	−0.201831	−	0.979420i	$-0.564689\pi$
$277$	−515488.	−0.403663	−0.201831	+	0.979420i	$0.564689\pi$
$278$	2.05889e6	1.59780
$279$	0	0
$280$	−68083.7	−0.0518977
$281$	2.37586e6	1.79496	0.897480	−	0.441054i	$-0.145395\pi$
$281$	2.37586e6	1.79496	0.897480	+	0.441054i	$0.145395\pi$
$282$	0	0
$283$	1.32014e6	0.979836	0.489918	−	0.871769i	$-0.337027\pi$
$283$	1.32014e6	0.979836	0.489918	+	0.871769i	$0.337027\pi$
$284$	1.24289e6	0.914399
$285$	0	0
$286$	−503973.	−0.364327
$287$	85219.1	0.0610706
$288$	0	0
$289$	465524.	0.327867
$290$	−1.36411e6	−0.952478
$291$	0	0
$292$	694190.	0.476454
$293$	966751.	0.657878	0.328939	−	0.944351i	$-0.393309\pi$
$293$	966751.	0.657878	0.328939	+	0.944351i	$0.393309\pi$
$294$	0	0
$295$	−689567.	−0.461340
$296$	30063.4	0.0199439
$297$	0	0
$298$	1.05003e6	0.684950
$299$	−2.26265e6	−1.46366
$300$	0	0
$301$	215882.	0.137341
$302$	−729778.	−0.460440
$303$	0	0
$304$	612927.	0.380386
$305$	−1.01319e6	−0.623649
$306$	0	0
$307$	1.00328e6	0.607543	0.303772	−	0.952745i	$-0.401754\pi$
$307$	1.00328e6	0.607543	0.303772	+	0.952745i	$0.401754\pi$
$308$	−443573.	−0.266433
$309$	0	0
$310$	−612286.	−0.361868
$311$	−1.94850e6	−1.14235	−0.571174	−	0.820829i	$-0.693512\pi$
$311$	−1.94850e6	−1.14235	−0.571174	+	0.820829i	$0.693512\pi$
$312$	0	0
$313$	−967479.	−0.558188	−0.279094	−	0.960264i	$-0.590034\pi$
$313$	−967479.	−0.558188	−0.279094	+	0.960264i	$0.590034\pi$
$314$	−3.34804e6	−1.91631
$315$	0	0
$316$	−2.29974e6	−1.29557
$317$	−1.52395e6	−0.851772	−0.425886	−	0.904777i	$-0.640037\pi$
$317$	−1.52395e6	−0.851772	−0.425886	+	0.904777i	$0.640037\pi$
$318$	0	0
$319$	843781.	0.464251
$320$	−671510.	−0.366587
$321$	0	0
$322$	−4.17204e6	−2.24238
$323$	−761571.	−0.406166
$324$	0	0
$325$	332687.	0.174714
$326$	−5.13220e6	−2.67461
$327$	0	0
$328$	−14750.2	−0.00757032
$329$	−1.89951e6	−0.967501
$330$	0	0
$331$	3.66226e6	1.83730	0.918649	−	0.395074i	$-0.129281\pi$
$331$	3.66226e6	1.83730	0.918649	+	0.395074i	$0.129281\pi$
$332$	−1.52708e6	−0.760356
$333$	0	0
$334$	210691.	0.103343
$335$	−1.46816e6	−0.714763
$336$	0	0
$337$	−1.23032e6	−0.590125	−0.295062	−	0.955478i	$-0.595340\pi$
$337$	−1.23032e6	−0.590125	−0.295062	+	0.955478i	$0.595340\pi$
$338$	688179.	0.327650
$339$	0	0
$340$	1.00323e6	0.470653
$341$	378734.	0.176380
$342$	0	0
$343$	2.24278e6	1.02932
$344$	−37366.1	−0.0170248
$345$	0	0
$346$	240283.	0.107903
$347$	−1.93838e6	−0.864200	−0.432100	−	0.901826i	$-0.642227\pi$
$347$	−1.93838e6	−0.864200	−0.432100	+	0.901826i	$0.642227\pi$
$348$	0	0
$349$	1.13923e6	0.500664	0.250332	−	0.968160i	$-0.419460\pi$
$349$	1.13923e6	0.500664	0.250332	+	0.968160i	$0.419460\pi$
$350$	613432.	0.267668
$351$	0	0
$352$	962217.	0.413920
$353$	4.57641e6	1.95474	0.977369	−	0.211542i	$-0.0678486\pi$
$353$	4.57641e6	1.95474	0.977369	+	0.211542i	$0.0678486\pi$
$354$	0	0
$355$	1.06319e6	0.447756
$356$	−238364.	−0.0996818
$357$	0	0
$358$	−4.18655e6	−1.72643
$359$	4.22743e6	1.73117	0.865586	−	0.500760i	$-0.166946\pi$
$359$	4.22743e6	1.73117	0.865586	+	0.500760i	$0.166946\pi$
$360$	0	0
$361$	−2.16847e6	−0.875762
$362$	−6.06250e6	−2.43153
$363$	0	0
$364$	−1.95135e6	−0.771938
$365$	593826.	0.233307
$366$	0	0
$367$	−3.26625e6	−1.26585	−0.632927	−	0.774211i	$-0.718147\pi$
$367$	−3.26625e6	−1.26585	−0.632927	+	0.774211i	$0.718147\pi$
$368$	4.69743e6	1.80818
$369$	0	0
$370$	−270870.	−0.102862
$371$	−1.19721e6	−0.451582
$372$	0	0
$373$	1.27410e6	0.474166	0.237083	−	0.971489i	$-0.423809\pi$
$373$	1.27410e6	0.474166	0.237083	+	0.971489i	$0.423809\pi$
$374$	−1.30002e6	−0.480586
$375$	0	0
$376$	328778.	0.119932
$377$	3.71194e6	1.34508
$378$	0	0
$379$	−2.53361e6	−0.906028	−0.453014	−	0.891503i	$-0.649651\pi$
$379$	−2.53361e6	−0.906028	−0.453014	+	0.891503i	$0.649651\pi$
$380$	−405237.	−0.143963
$381$	0	0
$382$	−5.18064e6	−1.81646
$383$	780098.	0.271739	0.135870	−	0.990727i	$-0.456617\pi$
$383$	780098.	0.271739	0.135870	+	0.990727i	$0.456617\pi$
$384$	0	0
$385$	−379443.	−0.130465
$386$	2.43847e6	0.833008
$387$	0	0
$388$	2.30909e6	0.778685
$389$	−3.15717e6	−1.05785	−0.528924	−	0.848669i	$-0.677404\pi$
$389$	−3.15717e6	−1.05785	−0.528924	+	0.848669i	$0.677404\pi$
$390$	0	0
$391$	−5.83662e6	−1.93072
$392$	−23294.1	−0.00765652
$393$	0	0
$394$	−7.86135e6	−2.55127
$395$	−1.96725e6	−0.634406
$396$	0	0
$397$	4.05582e6	1.29152	0.645761	−	0.763540i	$-0.276540\pi$
$397$	4.05582e6	1.29152	0.645761	+	0.763540i	$0.276540\pi$
$398$	−3.87690e6	−1.22681
$399$	0	0
$400$	−690682.	−0.215838
$401$	−4.07817e6	−1.26650	−0.633249	−	0.773949i	$-0.718279\pi$
$401$	−4.07817e6	−1.26650	−0.633249	+	0.773949i	$0.718279\pi$
$402$	0	0
$403$	1.66612e6	0.511026
$404$	3.41843e6	1.04201
$405$	0	0
$406$	6.84433e6	2.06070
$407$	167549.	0.0501366
$408$	0	0
$409$	169572.	0.0501240	0.0250620	−	0.999686i	$-0.492022\pi$
$409$	169572.	0.0501240	0.0250620	+	0.999686i	$0.492022\pi$
$410$	132899.	0.0390447
$411$	0	0
$412$	2.90246e6	0.842409
$413$	3.45985e6	0.998118
$414$	0	0
$415$	−1.30630e6	−0.372326
$416$	4.23296e6	1.19925
$417$	0	0
$418$	525124.	0.147001
$419$	730962.	0.203404	0.101702	−	0.994815i	$-0.467571\pi$
$419$	730962.	0.203404	0.101702	+	0.994815i	$0.467571\pi$
$420$	0	0
$421$	−1.76443e6	−0.485175	−0.242587	−	0.970130i	$-0.577996\pi$
$421$	−1.76443e6	−0.485175	−0.242587	+	0.970130i	$0.577996\pi$
$422$	−891157.	−0.243598
$423$	0	0
$424$	207221.	0.0559781
$425$	858182.	0.230466
$426$	0	0
$427$	5.08359e6	1.34928
$428$	−3.56975e6	−0.941952
$429$	0	0
$430$	336667.	0.0878071
$431$	−2.36048e6	−0.612078	−0.306039	−	0.952019i	$-0.599004\pi$
$431$	−2.36048e6	−0.612078	−0.306039	+	0.952019i	$0.599004\pi$
$432$	0	0
$433$	729625.	0.187017	0.0935083	−	0.995619i	$-0.470192\pi$
$433$	729625.	0.187017	0.0935083	+	0.995619i	$0.470192\pi$
$434$	3.07210e6	0.782909
$435$	0	0
$436$	−2.01188e6	−0.506858
$437$	2.35761e6	0.590567
$438$	0	0
$439$	−590313.	−0.146191	−0.0730956	−	0.997325i	$-0.523288\pi$
$439$	−590313.	−0.146191	−0.0730956	+	0.997325i	$0.523288\pi$
$440$	65676.2	0.0161725
$441$	0	0
$442$	−5.71902e6	−1.39241
$443$	−4.23488e6	−1.02525	−0.512627	−	0.858611i	$-0.671328\pi$
$443$	−4.23488e6	−1.02525	−0.512627	+	0.858611i	$0.671328\pi$
$444$	0	0
$445$	−203902.	−0.0488115
$446$	1.46821e6	0.349502
$447$	0	0
$448$	3.36925e6	0.793119
$449$	−5.25315e6	−1.22971	−0.614857	−	0.788639i	$-0.710786\pi$
$449$	−5.25315e6	−1.22971	−0.614857	+	0.788639i	$0.710786\pi$
$450$	0	0
$451$	−82205.6	−0.0190309
$452$	1.92775e6	0.443817
$453$	0	0
$454$	−6.56818e6	−1.49557
$455$	−1.66923e6	−0.377997
$456$	0	0
$457$	−74256.4	−0.0166320	−0.00831598	−	0.999965i	$-0.502647\pi$
$457$	−74256.4	−0.0166320	−0.00831598	+	0.999965i	$0.502647\pi$
$458$	3.63533e6	0.809804
$459$	0	0
$460$	−3.10571e6	−0.684331
$461$	4.48265e6	0.982387	0.491193	−	0.871051i	$-0.336561\pi$
$461$	4.48265e6	0.982387	0.491193	+	0.871051i	$0.336561\pi$
$462$	0	0
$463$	6.45533e6	1.39948	0.699739	−	0.714399i	$-0.253300\pi$
$463$	6.45533e6	1.39948	0.699739	+	0.714399i	$0.253300\pi$
$464$	−7.70624e6	−1.66168
$465$	0	0
$466$	9.25699e6	1.97472
$467$	4.62392e6	0.981111	0.490556	−	0.871410i	$-0.336794\pi$
$467$	4.62392e6	0.981111	0.490556	+	0.871410i	$0.336794\pi$
$468$	0	0
$469$	7.36640e6	1.54640
$470$	−2.96228e6	−0.618559
$471$	0	0
$472$	−598851.	−0.123727
$473$	−208248.	−0.0427984
$474$	0	0
$475$	−346650.	−0.0704947
$476$	−5.03361e6	−1.01827
$477$	0	0
$478$	−936364.	−0.187445
$479$	−7.34630e6	−1.46295	−0.731476	−	0.681868i	$-0.761168\pi$
$479$	−7.34630e6	−1.46295	−0.731476	+	0.681868i	$0.761168\pi$
$480$	0	0
$481$	737076.	0.145261
$482$	−810245.	−0.158854
$483$	0	0
$484$	427888.	0.0830264
$485$	1.97525e6	0.381301
$486$	0	0
$487$	−3.52414e6	−0.673335	−0.336667	−	0.941624i	$-0.609300\pi$
$487$	−3.52414e6	−0.673335	−0.336667	+	0.941624i	$0.609300\pi$
$488$	−879898.	−0.167256
$489$	0	0
$490$	209879.	0.0394893
$491$	9.12404e6	1.70798	0.853992	−	0.520287i	$-0.174175\pi$
$491$	9.12404e6	1.70798	0.853992	+	0.520287i	$0.174175\pi$
$492$	0	0
$493$	9.57512e6	1.77430
$494$	2.31011e6	0.425907
$495$	0	0
$496$	−3.45898e6	−0.631311
$497$	−5.33450e6	−0.968729
$498$	0	0
$499$	4.97520e6	0.894457	0.447229	−	0.894420i	$-0.352411\pi$
$499$	4.97520e6	0.894457	0.447229	+	0.894420i	$0.352411\pi$
$500$	456645.	0.0816872
$501$	0	0
$502$	1.64477e6	0.291304
$503$	5.63542e6	0.993131	0.496565	−	0.867999i	$-0.334594\pi$
$503$	5.63542e6	0.993131	0.496565	+	0.867999i	$0.334594\pi$
$504$	0	0
$505$	2.92421e6	0.510246
$506$	4.02451e6	0.698774
$507$	0	0
$508$	8.65307e6	1.48768
$509$	−6.57066e6	−1.12413	−0.562063	−	0.827095i	$-0.689992\pi$
$509$	−6.57066e6	−1.12413	−0.562063	+	0.827095i	$0.689992\pi$
$510$	0	0
$511$	−2.97948e6	−0.504763
$512$	−8.02015e6	−1.35210
$513$	0	0
$514$	−9.45411e6	−1.57838
$515$	2.48283e6	0.412505
$516$	0	0
$517$	1.83234e6	0.301495
$518$	1.35907e6	0.222545
$519$	0	0
$520$	288921.	0.0468566
$521$	1.13639e7	1.83415	0.917073	−	0.398720i	$-0.130546\pi$
$521$	1.13639e7	1.83415	0.917073	+	0.398720i	$0.130546\pi$
$522$	0	0
$523$	8.29972e6	1.32681	0.663405	−	0.748260i	$-0.269110\pi$
$523$	8.29972e6	1.32681	0.663405	+	0.748260i	$0.269110\pi$
$524$	−9.00432e6	−1.43259
$525$	0	0
$526$	−1.43036e7	−2.25414
$527$	4.29783e6	0.674097
$528$	0	0
$529$	1.16322e7	1.80727
$530$	−1.86705e6	−0.288713
$531$	0	0
$532$	2.03325e6	0.311466
$533$	−361637.	−0.0551384
$534$	0	0
$535$	−3.05365e6	−0.461249
$536$	−1.27502e6	−0.191692
$537$	0	0
$538$	−3.74290e6	−0.557510
$539$	−129822.	−0.0192476
$540$	0	0
$541$	2.95545e6	0.434140	0.217070	−	0.976156i	$-0.430350\pi$
$541$	2.95545e6	0.434140	0.217070	+	0.976156i	$0.430350\pi$
$542$	1.25123e7	1.82953
$543$	0	0
$544$	1.09191e7	1.58194
$545$	−1.72101e6	−0.248195
$546$	0	0
$547$	−3.70026e6	−0.528767	−0.264384	−	0.964418i	$-0.585168\pi$
$547$	−3.70026e6	−0.528767	−0.264384	+	0.964418i	$0.585168\pi$
$548$	1.22865e7	1.74774
$549$	0	0
$550$	−591740.	−0.0834111
$551$	−3.86772e6	−0.542720
$552$	0	0
$553$	9.87054e6	1.37255
$554$	4.03352e6	0.558354
$555$	0	0
$556$	−7.69002e6	−1.05497
$557$	−1.17422e7	−1.60365	−0.801827	−	0.597557i	$-0.796138\pi$
$557$	−1.17422e7	−1.60365	−0.801827	+	0.597557i	$0.796138\pi$
$558$	0	0
$559$	−916118.	−0.124000
$560$	3.46545e6	0.466970
$561$	0	0
$562$	−1.85903e7	−2.48282
$563$	1.44514e7	1.92150	0.960749	−	0.277418i	$-0.0894787\pi$
$563$	1.44514e7	1.92150	0.960749	+	0.277418i	$0.0894787\pi$
$564$	0	0
$565$	1.64904e6	0.217325
$566$	−1.03296e7	−1.35533
$567$	0	0
$568$	923326.	0.120084
$569$	−89979.8	−0.0116510	−0.00582552	−	0.999983i	$-0.501854\pi$
$569$	−89979.8	−0.0116510	−0.00582552	+	0.999983i	$0.501854\pi$
$570$	0	0
$571$	−5.85857e6	−0.751972	−0.375986	−	0.926625i	$-0.622696\pi$
$571$	−5.85857e6	−0.751972	−0.375986	+	0.926625i	$0.622696\pi$
$572$	1.88235e6	0.240553
$573$	0	0
$574$	−666810.	−0.0844739
$575$	−2.65670e6	−0.335098
$576$	0	0
$577$	−1.36701e6	−0.170935	−0.0854675	−	0.996341i	$-0.527238\pi$
$577$	−1.36701e6	−0.170935	−0.0854675	+	0.996341i	$0.527238\pi$
$578$	−3.64257e6	−0.453511
$579$	0	0
$580$	5.09499e6	0.628888
$581$	6.55427e6	0.805534
$582$	0	0
$583$	1.15488e6	0.140723
$584$	515706.	0.0625705
$585$	0	0
$586$	−7.56450e6	−0.909989
$587$	5.83375e6	0.698799	0.349400	−	0.936974i	$-0.386386\pi$
$587$	5.83375e6	0.698799	0.349400	+	0.936974i	$0.386386\pi$
$588$	0	0
$589$	−1.73604e6	−0.206192
$590$	5.39563e6	0.638134
$591$	0	0
$592$	−1.53022e6	−0.179453
$593$	1.43090e7	1.67098	0.835490	−	0.549506i	$-0.185184\pi$
$593$	1.43090e7	1.67098	0.835490	+	0.549506i	$0.185184\pi$
$594$	0	0
$595$	−4.30587e6	−0.498618
$596$	−3.92187e6	−0.452249
$597$	0	0
$598$	1.77045e7	2.02456
$599$	2.05474e6	0.233986	0.116993	−	0.993133i	$-0.462675\pi$
$599$	2.05474e6	0.233986	0.116993	+	0.993133i	$0.462675\pi$
$600$	0	0
$601$	5.27159e6	0.595326	0.297663	−	0.954671i	$-0.403793\pi$
$601$	5.27159e6	0.595326	0.297663	+	0.954671i	$0.403793\pi$
$602$	−1.68920e6	−0.189972
$603$	0	0
$604$	2.72574e6	0.304013
$605$	366025.	0.0406558
$606$	0	0
$607$	−3.60917e6	−0.397590	−0.198795	−	0.980041i	$-0.563703\pi$
$607$	−3.60917e6	−0.397590	−0.198795	+	0.980041i	$0.563703\pi$
$608$	−4.41061e6	−0.483882
$609$	0	0
$610$	7.92785e6	0.862642
$611$	8.06077e6	0.873522
$612$	0	0
$613$	−1.72474e7	−1.85384	−0.926921	−	0.375256i	$-0.877555\pi$
$613$	−1.72474e7	−1.85384	−0.926921	+	0.375256i	$0.877555\pi$
$614$	−7.85034e6	−0.840365
$615$	0	0
$616$	−329525.	−0.0349894
$617$	1.64726e7	1.74201	0.871005	−	0.491275i	$-0.163469\pi$
$617$	1.64726e7	1.74201	0.871005	+	0.491275i	$0.163469\pi$
$618$	0	0
$619$	−1.06465e7	−1.11681	−0.558404	−	0.829569i	$-0.688586\pi$
$619$	−1.06465e7	−1.11681	−0.558404	+	0.829569i	$0.688586\pi$
$620$	2.28690e6	0.238929
$621$	0	0
$622$	1.52463e7	1.58012
$623$	1.02306e6	0.105605
$624$	0	0
$625$	390625.	0.0400000
$626$	7.57019e6	0.772095
$627$	0	0
$628$	1.25050e7	1.26527
$629$	1.90132e6	0.191615
$630$	0	0
$631$	−1.49787e6	−0.149761	−0.0748806	−	0.997193i	$-0.523858\pi$
$631$	−1.49787e6	−0.149761	−0.0748806	+	0.997193i	$0.523858\pi$
$632$	−1.70845e6	−0.170141
$633$	0	0
$634$	1.19244e7	1.17819
$635$	7.40204e6	0.728479
$636$	0	0
$637$	−571111.	−0.0557663
$638$	−6.60230e6	−0.642160
$639$	0	0
$640$	−1.10743e6	−0.106873
$641$	−1.27849e7	−1.22901	−0.614503	−	0.788915i	$-0.710643\pi$
$641$	−1.27849e7	−1.22901	−0.614503	+	0.788915i	$0.710643\pi$
$642$	0	0
$643$	−2.60701e6	−0.248665	−0.124333	−	0.992241i	$-0.539679\pi$
$643$	−2.60701e6	−0.248665	−0.124333	+	0.992241i	$0.539679\pi$
$644$	1.55827e7	1.48056
$645$	0	0
$646$	5.95903e6	0.561816
$647$	1.91733e7	1.80068	0.900338	−	0.435190i	$-0.143319\pi$
$647$	1.91733e7	1.80068	0.900338	+	0.435190i	$0.143319\pi$
$648$	0	0
$649$	−3.33750e6	−0.311035
$650$	−2.60316e6	−0.241668
$651$	0	0
$652$	1.91689e7	1.76595
$653$	−6.02156e6	−0.552619	−0.276309	−	0.961069i	$-0.589111\pi$
$653$	−6.02156e6	−0.552619	−0.276309	+	0.961069i	$0.589111\pi$
$654$	0	0
$655$	−7.70250e6	−0.701501
$656$	750783.	0.0681169
$657$	0	0
$658$	1.48630e7	1.33826
$659$	736404.	0.0660545	0.0330273	−	0.999454i	$-0.489485\pi$
$659$	736404.	0.0660545	0.0330273	+	0.999454i	$0.489485\pi$
$660$	0	0
$661$	7.40219e6	0.658956	0.329478	−	0.944163i	$-0.393127\pi$
$661$	7.40219e6	0.658956	0.329478	+	0.944163i	$0.393127\pi$
$662$	−2.86560e7	−2.54138
$663$	0	0
$664$	−1.13445e6	−0.0998541
$665$	1.73929e6	0.152517
$666$	0	0
$667$	−2.96419e7	−2.57983
$668$	−786938.	−0.0682337
$669$	0	0
$670$	1.14879e7	0.988673
$671$	−4.90383e6	−0.420464
$672$	0	0
$673$	−2.43422e6	−0.207167	−0.103584	−	0.994621i	$-0.533031\pi$
$673$	−2.43422e6	−0.207167	−0.103584	+	0.994621i	$0.533031\pi$
$674$	9.62685e6	0.816271
$675$	0	0
$676$	−2.57037e6	−0.216336
$677$	−2.27337e7	−1.90633	−0.953166	−	0.302447i	$-0.902197\pi$
$677$	−2.27337e7	−1.90633	−0.953166	+	0.302447i	$0.902197\pi$
$678$	0	0
$679$	−9.91068e6	−0.824952
$680$	745285.	0.0618087
$681$	0	0
$682$	−2.96347e6	−0.243971
$683$	−4.65948e6	−0.382196	−0.191098	−	0.981571i	$-0.561205\pi$
$683$	−4.65948e6	−0.382196	−0.191098	+	0.981571i	$0.561205\pi$
$684$	0	0
$685$	1.05102e7	0.855823
$686$	−1.75490e7	−1.42377
$687$	0	0
$688$	1.90193e6	0.153187
$689$	5.08050e6	0.407717
$690$	0	0
$691$	5.24048e6	0.417519	0.208760	−	0.977967i	$-0.433057\pi$
$691$	5.24048e6	0.417519	0.208760	+	0.977967i	$0.433057\pi$
$692$	−897461.	−0.0712443
$693$	0	0
$694$	1.51671e7	1.19538
$695$	−6.57822e6	−0.516590
$696$	0	0
$697$	−932858.	−0.0727334
$698$	−8.91405e6	−0.692527
$699$	0	0
$700$	−2.29118e6	−0.176732
$701$	2.67335e6	0.205475	0.102738	−	0.994708i	$-0.467240\pi$
$701$	2.67335e6	0.205475	0.102738	+	0.994708i	$0.467240\pi$
$702$	0	0
$703$	−768009.	−0.0586109
$704$	−3.25011e6	−0.247153
$705$	0	0
$706$	−3.58089e7	−2.70383
$707$	−1.46720e7	−1.10393
$708$	0	0
$709$	−1.03026e7	−0.769714	−0.384857	−	0.922976i	$-0.625749\pi$
$709$	−1.03026e7	−0.769714	−0.384857	+	0.922976i	$0.625749\pi$
$710$	−8.31913e6	−0.619344
$711$	0	0
$712$	−177078.	−0.0130907
$713$	−1.33049e7	−0.980138
$714$	0	0
$715$	1.61021e6	0.117792
$716$	1.56369e7	1.13990
$717$	0	0
$718$	−3.30782e7	−2.39459
$719$	3.49018e6	0.251783	0.125891	−	0.992044i	$-0.459821\pi$
$719$	3.49018e6	0.251783	0.125891	+	0.992044i	$0.459821\pi$
$720$	0	0
$721$	−1.24574e7	−0.892462
$722$	1.69676e7	1.21137
$723$	0	0
$724$	2.26436e7	1.60546
$725$	4.35837e6	0.307949
$726$	0	0
$727$	8.80514e6	0.617875	0.308937	−	0.951082i	$-0.400027\pi$
$727$	8.80514e6	0.617875	0.308937	+	0.951082i	$0.400027\pi$
$728$	−1.44964e6	−0.101375
$729$	0	0
$730$	−4.64649e6	−0.322714
$731$	−2.36317e6	−0.163569
$732$	0	0
$733$	−1.51293e7	−1.04006	−0.520030	−	0.854148i	$-0.674079\pi$
$733$	−1.51293e7	−1.04006	−0.520030	+	0.854148i	$0.674079\pi$
$734$	2.55573e7	1.75095
$735$	0	0
$736$	−3.38026e7	−2.30015
$737$	−7.10591e6	−0.481893
$738$	0	0
$739$	−1.28312e7	−0.864283	−0.432142	−	0.901806i	$-0.642242\pi$
$739$	−1.28312e7	−0.864283	−0.432142	+	0.901806i	$0.642242\pi$
$740$	1.01171e6	0.0679165
$741$	0	0
$742$	9.36778e6	0.624636
$743$	6.27463e6	0.416981	0.208490	−	0.978024i	$-0.433145\pi$
$743$	6.27463e6	0.416981	0.208490	+	0.978024i	$0.433145\pi$
$744$	0	0
$745$	−3.35486e6	−0.221454
$746$	−9.96937e6	−0.655874
$747$	0	0
$748$	4.85561e6	0.317314
$749$	1.53215e7	0.997920
$750$	0	0
$751$	−2.27348e6	−0.147093	−0.0735464	−	0.997292i	$-0.523432\pi$
$751$	−2.27348e6	−0.147093	−0.0735464	+	0.997292i	$0.523432\pi$
$752$	−1.67347e7	−1.07913
$753$	0	0
$754$	−2.90446e7	−1.86053
$755$	2.33166e6	0.148867
$756$	0	0
$757$	1.31980e7	0.837082	0.418541	−	0.908198i	$-0.362542\pi$
$757$	1.31980e7	0.837082	0.418541	+	0.908198i	$0.362542\pi$
$758$	1.98246e7	1.25323
$759$	0	0
$760$	−301046.	−0.0189060
$761$	−1.28529e7	−0.804528	−0.402264	−	0.915524i	$-0.631777\pi$
$761$	−1.28529e7	−0.804528	−0.402264	+	0.915524i	$0.631777\pi$
$762$	0	0
$763$	8.63504e6	0.536974
$764$	1.93498e7	1.19934
$765$	0	0
$766$	−6.10400e6	−0.375874
$767$	−1.46822e7	−0.901165
$768$	0	0
$769$	1.02398e7	0.624421	0.312211	−	0.950013i	$-0.398931\pi$
$769$	1.02398e7	0.624421	0.312211	+	0.950013i	$0.398931\pi$
$770$	2.96901e6	0.180462
$771$	0	0
$772$	−9.10774e6	−0.550006
$773$	−1.44400e7	−0.869199	−0.434600	−	0.900624i	$-0.643110\pi$
$773$	−1.44400e7	−0.869199	−0.434600	+	0.900624i	$0.643110\pi$
$774$	0	0
$775$	1.95627e6	0.116997
$776$	1.71540e6	0.102261
$777$	0	0
$778$	2.47038e7	1.46323
$779$	376814.	0.0222476
$780$	0	0
$781$	5.14586e6	0.301877
$782$	4.56696e7	2.67061
$783$	0	0
$784$	1.18567e6	0.0688925
$785$	1.06971e7	0.619571
$786$	0	0
$787$	7.81654e6	0.449860	0.224930	−	0.974375i	$-0.427785\pi$
$787$	7.81654e6	0.449860	0.224930	+	0.974375i	$0.427785\pi$
$788$	2.93623e7	1.68451
$789$	0	0
$790$	1.53931e7	0.877521
$791$	−8.27392e6	−0.470187
$792$	0	0
$793$	−2.15728e7	−1.21821
$794$	−3.17354e7	−1.78646
$795$	0	0
$796$	1.44803e7	0.810018
$797$	−4.41572e6	−0.246238	−0.123119	−	0.992392i	$-0.539290\pi$
$797$	−4.41572e6	−0.246238	−0.123119	+	0.992392i	$0.539290\pi$
$798$	0	0
$799$	2.07931e7	1.15227
$800$	4.97013e6	0.274564
$801$	0	0
$802$	3.19103e7	1.75184
$803$	2.87412e6	0.157295
$804$	0	0
$805$	1.33298e7	0.724992
$806$	−1.30368e7	−0.706860
$807$	0	0
$808$	2.53951e6	0.136843
$809$	2.28687e7	1.22849	0.614243	−	0.789117i	$-0.289462\pi$
$809$	2.28687e7	1.22849	0.614243	+	0.789117i	$0.289462\pi$
$810$	0	0
$811$	−1.28190e7	−0.684388	−0.342194	−	0.939629i	$-0.611170\pi$
$811$	−1.28190e7	−0.684388	−0.342194	+	0.939629i	$0.611170\pi$
$812$	−2.55637e7	−1.36061
$813$	0	0
$814$	−1.31101e6	−0.0693499
$815$	1.63975e7	0.864738
$816$	0	0
$817$	954566.	0.0500323
$818$	−1.32684e6	−0.0693323
$819$	0	0
$820$	−496381.	−0.0257799
$821$	77248.3	0.00399973	0.00199987	−	0.999998i	$-0.499363\pi$
$821$	77248.3	0.00399973	0.00199987	+	0.999998i	$0.499363\pi$
$822$	0	0
$823$	6.41258e6	0.330015	0.165007	−	0.986292i	$-0.447235\pi$
$823$	6.41258e6	0.330015	0.165007	+	0.986292i	$0.447235\pi$
$824$	2.15620e6	0.110630
$825$	0	0
$826$	−2.70721e7	−1.38061
$827$	2.68414e7	1.36471	0.682356	−	0.731020i	$-0.260956\pi$
$827$	2.68414e7	1.36471	0.682356	+	0.731020i	$0.260956\pi$
$828$	0	0
$829$	−1.93130e7	−0.976028	−0.488014	−	0.872836i	$-0.662279\pi$
$829$	−1.93130e7	−0.976028	−0.488014	+	0.872836i	$0.662279\pi$
$830$	1.02214e7	0.515008
$831$	0	0
$832$	−1.42978e7	−0.716078
$833$	−1.47321e6	−0.0735616
$834$	0	0
$835$	−673165.	−0.0334122
$836$	−1.96135e6	−0.0970598
$837$	0	0
$838$	−5.71953e6	−0.281352
$839$	−2.99842e7	−1.47058	−0.735288	−	0.677754i	$-0.762953\pi$
$839$	−2.99842e7	−1.47058	−0.735288	+	0.677754i	$0.762953\pi$
$840$	0	0
$841$	2.81171e7	1.37082
$842$	1.38060e7	0.671102
$843$	0	0
$844$	3.32849e6	0.160839
$845$	−2.19875e6	−0.105934
$846$	0	0
$847$	−1.83650e6	−0.0879596
$848$	−1.05475e7	−0.503685
$849$	0	0
$850$	−6.71499e6	−0.318785
$851$	−5.88597e6	−0.278608
$852$	0	0
$853$	−1.79675e7	−0.845504	−0.422752	−	0.906245i	$-0.638936\pi$
$853$	−1.79675e7	−0.845504	−0.422752	+	0.906245i	$0.638936\pi$
$854$	−3.97774e7	−1.86634
$855$	0	0
$856$	−2.65193e6	−0.123702
$857$	−3.57131e7	−1.66102	−0.830511	−	0.557003i	$-0.811951\pi$
$857$	−3.57131e7	−1.66102	−0.830511	+	0.557003i	$0.811951\pi$
$858$	0	0
$859$	7.29428e6	0.337287	0.168643	−	0.985677i	$-0.446061\pi$
$859$	7.29428e6	0.337287	0.168643	+	0.985677i	$0.446061\pi$
$860$	−1.25746e6	−0.0579760
$861$	0	0
$862$	1.84699e7	0.846637
$863$	7.09299e6	0.324192	0.162096	−	0.986775i	$-0.448175\pi$
$863$	7.09299e6	0.324192	0.162096	+	0.986775i	$0.448175\pi$
$864$	0	0
$865$	−767709.	−0.0348864
$866$	−5.70907e6	−0.258685
$867$	0	0
$868$	−1.14744e7	−0.516927
$869$	−9.52150e6	−0.427716
$870$	0	0
$871$	−3.12601e7	−1.39619
$872$	−1.49460e6	−0.0665633
$873$	0	0
$874$	−1.84475e7	−0.816882
$875$	−1.95993e6	−0.0865408
$876$	0	0
$877$	3.26769e7	1.43464	0.717319	−	0.696745i	$-0.245369\pi$
$877$	3.26769e7	1.43464	0.717319	+	0.696745i	$0.245369\pi$
$878$	4.61900e6	0.202214
$879$	0	0
$880$	−3.34290e6	−0.145518
$881$	3.26117e7	1.41558	0.707789	−	0.706423i	$-0.249693\pi$
$881$	3.26117e7	1.41558	0.707789	+	0.706423i	$0.249693\pi$
$882$	0	0
$883$	−1.59642e7	−0.689042	−0.344521	−	0.938779i	$-0.611959\pi$
$883$	−1.59642e7	−0.689042	−0.344521	+	0.938779i	$0.611959\pi$
$884$	2.13607e7	0.919357
$885$	0	0
$886$	3.31365e7	1.41815
$887$	−1.02195e7	−0.436136	−0.218068	−	0.975934i	$-0.569975\pi$
$887$	−1.02195e7	−0.436136	−0.218068	+	0.975934i	$0.569975\pi$
$888$	0	0
$889$	−3.71392e7	−1.57608
$890$	1.59547e6	0.0675169
$891$	0	0
$892$	−5.48379e6	−0.230764
$893$	−8.39907e6	−0.352454
$894$	0	0
$895$	1.33761e7	0.558179
$896$	5.55645e6	0.231221
$897$	0	0
$898$	4.11041e7	1.70096
$899$	2.18270e7	0.900730
$900$	0	0
$901$	1.31054e7	0.537821
$902$	643231.	0.0263239
$903$	0	0
$904$	1.43210e6	0.0582844
$905$	1.93699e7	0.786149
$906$	0	0
$907$	2.75776e7	1.11311	0.556556	−	0.830810i	$-0.312123\pi$
$907$	2.75776e7	1.11311	0.556556	+	0.830810i	$0.312123\pi$
$908$	2.45323e7	0.987470
$909$	0	0
$910$	1.30612e7	0.522852
$911$	1.19367e7	0.476526	0.238263	−	0.971201i	$-0.423422\pi$
$911$	1.19367e7	0.476526	0.238263	+	0.971201i	$0.423422\pi$
$912$	0	0
$913$	−6.32250e6	−0.251022
$914$	581031.	0.0230056
$915$	0	0
$916$	−1.35780e7	−0.534686
$917$	3.86467e7	1.51771
$918$	0	0
$919$	3.32890e7	1.30020	0.650102	−	0.759847i	$-0.274726\pi$
$919$	3.32890e7	1.30020	0.650102	+	0.759847i	$0.274726\pi$
$920$	−2.30720e6	−0.0898700
$921$	0	0
$922$	−3.50752e7	−1.35885
$923$	2.26375e7	0.874631
$924$	0	0
$925$	865438.	0.0332569
$926$	−5.05108e7	−1.93578
$927$	0	0
$928$	5.54539e7	2.11379
$929$	2.85341e7	1.08474	0.542370	−	0.840140i	$-0.317527\pi$
$929$	2.85341e7	1.08474	0.542370	+	0.840140i	$0.317527\pi$
$930$	0	0
$931$	595079.	0.0225009
$932$	−3.45751e7	−1.30384
$933$	0	0
$934$	−3.61806e7	−1.35709
$935$	4.15360e6	0.155380
$936$	0	0
$937$	−3.00251e7	−1.11721	−0.558605	−	0.829434i	$-0.688663\pi$
$937$	−3.00251e7	−1.11721	−0.558605	+	0.829434i	$0.688663\pi$
$938$	−5.76395e7	−2.13901
$939$	0	0
$940$	1.10642e7	0.408413
$941$	−2.38196e7	−0.876921	−0.438461	−	0.898750i	$-0.644476\pi$
$941$	−2.38196e7	−0.876921	−0.438461	+	0.898750i	$0.644476\pi$
$942$	0	0
$943$	2.88787e6	0.105755
$944$	3.04814e7	1.11328
$945$	0	0
$946$	1.62947e6	0.0591995
$947$	2.01566e7	0.730368	0.365184	−	0.930935i	$-0.381006\pi$
$947$	2.01566e7	0.730368	0.365184	+	0.930935i	$0.381006\pi$
$948$	0	0
$949$	1.26437e7	0.455733
$950$	2.71242e6	0.0975096
$951$	0	0
$952$	−3.73941e6	−0.133724
$953$	−2.29824e7	−0.819715	−0.409858	−	0.912150i	$-0.634422\pi$
$953$	−2.29824e7	−0.819715	−0.409858	+	0.912150i	$0.634422\pi$
$954$	0	0
$955$	1.65523e7	0.587286
$956$	3.49734e6	0.123764
$957$	0	0
$958$	5.74823e7	2.02358
$959$	−5.27341e7	−1.85159
$960$	0	0
$961$	−1.88320e7	−0.657792
$962$	−5.76737e6	−0.200928
$963$	0	0
$964$	3.02629e6	0.104886
$965$	−7.79098e6	−0.269323
$966$	0	0
$967$	3.99056e6	0.137236	0.0686179	−	0.997643i	$-0.478141\pi$
$967$	3.99056e6	0.137236	0.0686179	+	0.997643i	$0.478141\pi$
$968$	317873.	0.0109035
$969$	0	0
$970$	−1.54557e7	−0.527422
$971$	−5.32237e6	−0.181158	−0.0905789	−	0.995889i	$-0.528872\pi$
$971$	−5.32237e6	−0.181158	−0.0905789	+	0.995889i	$0.528872\pi$
$972$	0	0
$973$	3.30057e7	1.11765
$974$	2.75752e7	0.931369
$975$	0	0
$976$	4.47866e7	1.50496
$977$	−1.59829e7	−0.535696	−0.267848	−	0.963461i	$-0.586312\pi$
$977$	−1.59829e7	−0.535696	−0.267848	+	0.963461i	$0.586312\pi$
$978$	0	0
$979$	−986887.	−0.0329087
$980$	−783904.	−0.0260734
$981$	0	0
$982$	−7.13925e7	−2.36251
$983$	−3.82997e7	−1.26419	−0.632095	−	0.774891i	$-0.717805\pi$
$983$	−3.82997e7	−1.26419	−0.632095	+	0.774891i	$0.717805\pi$
$984$	0	0
$985$	2.51172e7	0.824861
$986$	−7.49220e7	−2.45424
$987$	0	0
$988$	−8.62831e6	−0.281212
$989$	7.31572e6	0.237830
$990$	0	0
$991$	−1.01405e7	−0.328000	−0.164000	−	0.986460i	$-0.552440\pi$
$991$	−1.01405e7	−0.328000	−0.164000	+	0.986460i	$0.552440\pi$
$992$	2.48907e7	0.803078
$993$	0	0
$994$	4.17406e7	1.33996
$995$	1.23868e7	0.396644
$996$	0	0
$997$	−2.79398e7	−0.890195	−0.445098	−	0.895482i	$-0.646831\pi$
$997$	−2.79398e7	−0.890195	−0.445098	+	0.895482i	$0.646831\pi$
$998$	−3.89293e7	−1.23723
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.g.1.1		4
3.2	odd	2		55.6.a.b.1.4	✓	4
12.11	even	2		880.6.a.n.1.4		4
15.2	even	4		275.6.b.d.199.7		8
15.8	even	4		275.6.b.d.199.2		8
15.14	odd	2		275.6.a.d.1.1		4
33.32	even	2		605.6.a.c.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.6.a.b.1.4	✓	4	3.2	odd	2
275.6.a.d.1.1		4	15.14	odd	2
275.6.b.d.199.2		8	15.8	even	4
275.6.b.d.199.7		8	15.2	even	4
495.6.a.g.1.1		4	1.1	even	1	trivial
605.6.a.c.1.1		4	33.32	even	2
880.6.a.n.1.4		4	12.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-7.82466 q^{2} +29.2253 q^{4} +25.0000 q^{5} -125.436 q^{7} +21.7111 q^{8} +O(q^{10})\)	\(q-7.82466 q^{2} +29.2253 q^{4} +25.0000 q^{5} -125.436 q^{7} +21.7111 q^{8} -195.616 q^{10} +121.000 q^{11} +532.300 q^{13} +981.491 q^{14} -1105.09 q^{16} +1373.09 q^{17} -554.639 q^{19} +730.632 q^{20} -946.784 q^{22} -4250.72 q^{23} +625.000 q^{25} -4165.06 q^{26} -3665.89 q^{28} +6973.40 q^{29} +3130.03 q^{31} +7952.21 q^{32} -10744.0 q^{34} -3135.89 q^{35} +1384.70 q^{37} +4339.86 q^{38} +542.778 q^{40} -679.385 q^{41} -1721.06 q^{43} +3536.26 q^{44} +33260.4 q^{46} +15143.3 q^{47} -1072.91 q^{49} -4890.41 q^{50} +15556.6 q^{52} +9544.44 q^{53} +3025.00 q^{55} -2723.35 q^{56} -54564.5 q^{58} -27582.7 q^{59} -40527.5 q^{61} -24491.5 q^{62} -26860.4 q^{64} +13307.5 q^{65} -58726.5 q^{67} +40129.0 q^{68} +24537.3 q^{70} +42527.8 q^{71} +23753.0 q^{73} -10834.8 q^{74} -16209.5 q^{76} -15177.7 q^{77} -78690.1 q^{79} -27627.3 q^{80} +5315.96 q^{82} -52252.1 q^{83} +34327.3 q^{85} +13466.7 q^{86} +2627.05 q^{88} -8156.09 q^{89} -66769.3 q^{91} -124228. q^{92} -118491. q^{94} -13866.0 q^{95} +79010.1 q^{97} +8395.17 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{2} + 61 q^{4} + 100 q^{5} - 90 q^{7} + 135 q^{8}+O(q^{10}) \) 4 * q + 5 * q^2 + 61 * q^4 + 100 * q^5 - 90 * q^7 + 135 * q^8	\( 4 q + 5 q^{2} + 61 q^{4} + 100 q^{5} - 90 q^{7} + 135 q^{8} + 125 q^{10} + 484 q^{11} + 820 q^{13} + 1687 q^{14} - 2671 q^{16} + 3800 q^{17} - 3394 q^{19} + 1525 q^{20} + 605 q^{22} + 3020 q^{23} + 2500 q^{25} + 3650 q^{26} + 2635 q^{28} + 5248 q^{29} + 4732 q^{31} - 11505 q^{32} + 9759 q^{34} - 2250 q^{35} + 10210 q^{37} - 16945 q^{38} + 3375 q^{40} + 21068 q^{41} - 12140 q^{43} + 7381 q^{44} + 58442 q^{46} - 4720 q^{47} + 24550 q^{49} + 3125 q^{50} + 58250 q^{52} + 21670 q^{53} + 12100 q^{55} + 28985 q^{56} - 45435 q^{58} + 69068 q^{59} - 44000 q^{61} - 33375 q^{62} - 68223 q^{64} + 20500 q^{65} - 8720 q^{67} + 107335 q^{68} + 42175 q^{70} + 47516 q^{71} - 2480 q^{73} + 55717 q^{74} - 102461 q^{76} - 10890 q^{77} - 188192 q^{79} - 66775 q^{80} + 279030 q^{82} - 68620 q^{83} + 95000 q^{85} - 115328 q^{86} + 16335 q^{88} + 170266 q^{89} + 97740 q^{91} - 53950 q^{92} - 152926 q^{94} - 84850 q^{95} + 186160 q^{97} + 393590 q^{98}+O(q^{100}) \) 4 * q + 5 * q^2 + 61 * q^4 + 100 * q^5 - 90 * q^7 + 135 * q^8 + 125 * q^10 + 484 * q^11 + 820 * q^13 + 1687 * q^14 - 2671 * q^16 + 3800 * q^17 - 3394 * q^19 + 1525 * q^20 + 605 * q^22 + 3020 * q^23 + 2500 * q^25 + 3650 * q^26 + 2635 * q^28 + 5248 * q^29 + 4732 * q^31 - 11505 * q^32 + 9759 * q^34 - 2250 * q^35 + 10210 * q^37 - 16945 * q^38 + 3375 * q^40 + 21068 * q^41 - 12140 * q^43 + 7381 * q^44 + 58442 * q^46 - 4720 * q^47 + 24550 * q^49 + 3125 * q^50 + 58250 * q^52 + 21670 * q^53 + 12100 * q^55 + 28985 * q^56 - 45435 * q^58 + 69068 * q^59 - 44000 * q^61 - 33375 * q^62 - 68223 * q^64 + 20500 * q^65 - 8720 * q^67 + 107335 * q^68 + 42175 * q^70 + 47516 * q^71 - 2480 * q^73 + 55717 * q^74 - 102461 * q^76 - 10890 * q^77 - 188192 * q^79 - 66775 * q^80 + 279030 * q^82 - 68620 * q^83 + 95000 * q^85 - 115328 * q^86 + 16335 * q^88 + 170266 * q^89 + 97740 * q^91 - 53950 * q^92 - 152926 * q^94 - 84850 * q^95 + 186160 * q^97 + 393590 * q^98

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