LMFDB - Embedded newform 495.6.a.m.1.2

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:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 120x^{4} + 70x^{3} + 2825x^{2} - 2101x - 2690 $ x^6 - x^5 - 120x^4 + 70x^3 + 2825x^2 - 2101x - 2690
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.25379$ 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)

$f(q)$	$=$	$q-4.25379 q^{2} -13.9052 q^{4} -25.0000 q^{5} +30.2542 q^{7} +195.271 q^{8} +O(q^{10})$	\(q-4.25379 q^{2} -13.9052 q^{4} -25.0000 q^{5} +30.2542 q^{7} +195.271 q^{8} +106.345 q^{10} -121.000 q^{11} +176.845 q^{13} -128.695 q^{14} -385.676 q^{16} -597.161 q^{17} -2035.28 q^{19} +347.631 q^{20} +514.709 q^{22} -794.556 q^{23} +625.000 q^{25} -752.262 q^{26} -420.692 q^{28} +5836.23 q^{29} +5210.05 q^{31} -4608.10 q^{32} +2540.20 q^{34} -756.356 q^{35} +2916.39 q^{37} +8657.64 q^{38} -4881.79 q^{40} +12900.7 q^{41} +6585.65 q^{43} +1682.54 q^{44} +3379.88 q^{46} +15109.4 q^{47} -15891.7 q^{49} -2658.62 q^{50} -2459.07 q^{52} -5493.88 q^{53} +3025.00 q^{55} +5907.78 q^{56} -24826.1 q^{58} +34314.8 q^{59} +2669.68 q^{61} -22162.5 q^{62} +31943.5 q^{64} -4421.12 q^{65} -56618.3 q^{67} +8303.67 q^{68} +3217.38 q^{70} +30791.1 q^{71} +3435.22 q^{73} -12405.7 q^{74} +28301.0 q^{76} -3660.76 q^{77} -46626.8 q^{79} +9641.90 q^{80} -54877.1 q^{82} -41073.4 q^{83} +14929.0 q^{85} -28014.0 q^{86} -23627.8 q^{88} -37286.2 q^{89} +5350.31 q^{91} +11048.5 q^{92} -64272.3 q^{94} +50881.9 q^{95} -90018.1 q^{97} +67599.9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{2} + 53 q^{4} - 150 q^{5} - 80 q^{7} + 255 q^{8}+O(q^{10}) $ 6 * q + 5 * q^2 + 53 * q^4 - 150 * q^5 - 80 * q^7 + 255 * q^8	$ 6 q + 5 q^{2} + 53 q^{4} - 150 q^{5} - 80 q^{7} + 255 q^{8} - 125 q^{10} - 726 q^{11} - 420 q^{13} + 776 q^{14} + 1169 q^{16} + 2820 q^{17} - 220 q^{19} - 1325 q^{20} - 605 q^{22} + 2680 q^{23} + 3750 q^{25} + 1896 q^{26} - 11760 q^{28} + 1092 q^{29} - 7688 q^{31} + 4535 q^{32} - 354 q^{34} + 2000 q^{35} - 14020 q^{37} + 1570 q^{38} - 6375 q^{40} - 8196 q^{41} - 17340 q^{43} - 6413 q^{44} - 9982 q^{46} + 4200 q^{47} - 16890 q^{49} + 3125 q^{50} - 13440 q^{52} + 32900 q^{53} + 18150 q^{55} - 41824 q^{56} - 98010 q^{58} + 44512 q^{59} + 26636 q^{61} - 50680 q^{62} - 74607 q^{64} + 10500 q^{65} - 9920 q^{67} - 34810 q^{68} - 19400 q^{70} + 27344 q^{71} - 106620 q^{73} - 244014 q^{74} - 5638 q^{76} + 9680 q^{77} - 7168 q^{79} - 29225 q^{80} - 90250 q^{82} - 113480 q^{83} - 70500 q^{85} - 96314 q^{86} - 30855 q^{88} - 38352 q^{89} - 115232 q^{91} - 116910 q^{92} - 161846 q^{94} + 5500 q^{95} - 299100 q^{97} - 70715 q^{98}+O(q^{100}) $ 6 * q + 5 * q^2 + 53 * q^4 - 150 * q^5 - 80 * q^7 + 255 * q^8 - 125 * q^10 - 726 * q^11 - 420 * q^13 + 776 * q^14 + 1169 * q^16 + 2820 * q^17 - 220 * q^19 - 1325 * q^20 - 605 * q^22 + 2680 * q^23 + 3750 * q^25 + 1896 * q^26 - 11760 * q^28 + 1092 * q^29 - 7688 * q^31 + 4535 * q^32 - 354 * q^34 + 2000 * q^35 - 14020 * q^37 + 1570 * q^38 - 6375 * q^40 - 8196 * q^41 - 17340 * q^43 - 6413 * q^44 - 9982 * q^46 + 4200 * q^47 - 16890 * q^49 + 3125 * q^50 - 13440 * q^52 + 32900 * q^53 + 18150 * q^55 - 41824 * q^56 - 98010 * q^58 + 44512 * q^59 + 26636 * q^61 - 50680 * q^62 - 74607 * q^64 + 10500 * q^65 - 9920 * q^67 - 34810 * q^68 - 19400 * q^70 + 27344 * q^71 - 106620 * q^73 - 244014 * q^74 - 5638 * q^76 + 9680 * q^77 - 7168 * q^79 - 29225 * q^80 - 90250 * q^82 - 113480 * q^83 - 70500 * q^85 - 96314 * q^86 - 30855 * q^88 - 38352 * q^89 - 115232 * q^91 - 116910 * q^92 - 161846 * q^94 + 5500 * q^95 - 299100 * q^97 - 70715 * 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 $

$2$	−4.25379	−0.751971	−0.375986	−	0.926625i	$-0.622696\pi$
$2$	−4.25379	−0.751971	−0.375986	+	0.926625i	$0.622696\pi$
$3$	0	0
$3$	0	0
$4$	−13.9052	−0.434539
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	30.2542	0.233368	0.116684	−	0.993169i	$-0.462774\pi$
$7$	30.2542	0.233368	0.116684	+	0.993169i	$0.462774\pi$
$8$	195.271	1.07873
$9$	0	0
$10$	106.345	0.336292
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	176.845	0.290225	0.145112	−	0.989415i	$-0.453646\pi$
$13$	176.845	0.290225	0.145112	+	0.989415i	$0.453646\pi$
$14$	−128.695	−0.175486
$15$	0	0
$16$	−385.676	−0.376637
$17$	−597.161	−0.501152	−0.250576	−	0.968097i	$-0.580620\pi$
$17$	−597.161	−0.501152	−0.250576	+	0.968097i	$0.580620\pi$
$18$	0	0
$19$	−2035.28	−1.29342	−0.646710	−	0.762736i	$-0.723856\pi$
$19$	−2035.28	−1.29342	−0.646710	+	0.762736i	$0.723856\pi$
$20$	347.631	0.194332
$21$	0	0
$22$	514.709	0.226728
$23$	−794.556	−0.313188	−0.156594	−	0.987663i	$-0.550051\pi$
$23$	−794.556	−0.313188	−0.156594	+	0.987663i	$0.550051\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−752.262	−0.218241
$27$	0	0
$28$	−420.692	−0.101407
$29$	5836.23	1.28866	0.644329	−	0.764749i	$-0.277137\pi$
$29$	5836.23	1.28866	0.644329	+	0.764749i	$0.277137\pi$
$30$	0	0
$31$	5210.05	0.973728	0.486864	−	0.873478i	$-0.338141\pi$
$31$	5210.05	0.973728	0.486864	+	0.873478i	$0.338141\pi$
$32$	−4608.10	−0.795512
$33$	0	0
$34$	2540.20	0.376852
$35$	−756.356	−0.104365
$36$	0	0
$37$	2916.39	0.350220	0.175110	−	0.984549i	$-0.443972\pi$
$37$	2916.39	0.350220	0.175110	+	0.984549i	$0.443972\pi$
$38$	8657.64	0.972615
$39$	0	0
$40$	−4881.79	−0.482424
$41$	12900.7	1.19855	0.599274	−	0.800544i	$-0.295456\pi$
$41$	12900.7	1.19855	0.599274	+	0.800544i	$0.295456\pi$
$42$	0	0
$43$	6585.65	0.543160	0.271580	−	0.962416i	$-0.412454\pi$
$43$	6585.65	0.543160	0.271580	+	0.962416i	$0.412454\pi$
$44$	1682.54	0.131018
$45$	0	0
$46$	3379.88	0.235508
$47$	15109.4	0.997707	0.498853	−	0.866686i	$-0.333755\pi$
$47$	15109.4	0.997707	0.498853	+	0.866686i	$0.333755\pi$
$48$	0	0
$49$	−15891.7	−0.945539
$50$	−2658.62	−0.150394
$51$	0	0
$52$	−2459.07	−0.126114
$53$	−5493.88	−0.268651	−0.134326	−	0.990937i	$-0.542887\pi$
$53$	−5493.88	−0.268651	−0.134326	+	0.990937i	$0.542887\pi$
$54$	0	0
$55$	3025.00	0.134840
$56$	5907.78	0.251741
$57$	0	0
$58$	−24826.1	−0.969034
$59$	34314.8	1.28337	0.641685	−	0.766969i	$-0.278236\pi$
$59$	34314.8	1.28337	0.641685	+	0.766969i	$0.278236\pi$
$60$	0	0
$61$	2669.68	0.0918617	0.0459309	−	0.998945i	$-0.485375\pi$
$61$	2669.68	0.0918617	0.0459309	+	0.998945i	$0.485375\pi$
$62$	−22162.5	−0.732215
$63$	0	0
$64$	31943.5	0.974839
$65$	−4421.12	−0.129792
$66$	0	0
$67$	−56618.3	−1.54088	−0.770442	−	0.637510i	$-0.779964\pi$
$67$	−56618.3	−1.54088	−0.770442	+	0.637510i	$0.779964\pi$
$68$	8303.67	0.217770
$69$	0	0
$70$	3217.38	0.0784797
$71$	30791.1	0.724901	0.362451	−	0.932003i	$-0.381940\pi$
$71$	30791.1	0.724901	0.362451	+	0.932003i	$0.381940\pi$
$72$	0	0
$73$	3435.22	0.0754480	0.0377240	−	0.999288i	$-0.487989\pi$
$73$	3435.22	0.0754480	0.0377240	+	0.999288i	$0.487989\pi$
$74$	−12405.7	−0.263356
$75$	0	0
$76$	28301.0	0.562041
$77$	−3660.76	−0.0703630
$78$	0	0
$79$	−46626.8	−0.840559	−0.420279	−	0.907395i	$-0.638068\pi$
$79$	−46626.8	−0.840559	−0.420279	+	0.907395i	$0.638068\pi$
$80$	9641.90	0.168437
$81$	0	0
$82$	−54877.1	−0.901273
$83$	−41073.4	−0.654433	−0.327217	−	0.944949i	$-0.606111\pi$
$83$	−41073.4	−0.654433	−0.327217	+	0.944949i	$0.606111\pi$
$84$	0	0
$85$	14929.0	0.224122
$86$	−28014.0	−0.408441
$87$	0	0
$88$	−23627.8	−0.325250
$89$	−37286.2	−0.498968	−0.249484	−	0.968379i	$-0.580261\pi$
$89$	−37286.2	−0.498968	−0.249484	+	0.968379i	$0.580261\pi$
$90$	0	0
$91$	5350.31	0.0677291
$92$	11048.5	0.136092
$93$	0	0
$94$	−64272.3	−0.750247
$95$	50881.9	0.578435
$96$	0	0
$97$	−90018.1	−0.971406	−0.485703	−	0.874124i	$-0.661436\pi$
$97$	−90018.1	−0.971406	−0.485703	+	0.874124i	$0.661436\pi$
$98$	67599.9	0.711019
$99$	0	0
$100$	−8690.78	−0.0869078
$101$	−4515.65	−0.0440471	−0.0220235	−	0.999757i	$-0.507011\pi$
$101$	−4515.65	−0.0440471	−0.0220235	+	0.999757i	$0.507011\pi$
$102$	0	0
$103$	−59383.4	−0.551533	−0.275767	−	0.961225i	$-0.588932\pi$
$103$	−59383.4	−0.551533	−0.275767	+	0.961225i	$0.588932\pi$
$104$	34532.8	0.313075
$105$	0	0
$106$	23369.8	0.202018
$107$	162134.	1.36904	0.684518	−	0.728996i	$-0.260013\pi$
$107$	162134.	1.36904	0.684518	+	0.728996i	$0.260013\pi$
$108$	0	0
$109$	−230982.	−1.86214	−0.931071	−	0.364839i	$-0.881124\pi$
$109$	−230982.	−1.86214	−0.931071	+	0.364839i	$0.881124\pi$
$110$	−12867.7	−0.101396
$111$	0	0
$112$	−11668.3	−0.0878949
$113$	80157.4	0.590538	0.295269	−	0.955414i	$-0.404591\pi$
$113$	80157.4	0.590538	0.295269	+	0.955414i	$0.404591\pi$
$114$	0	0
$115$	19863.9	0.140062
$116$	−81154.3	−0.559972
$117$	0	0
$118$	−145968.	−0.965057
$119$	−18066.6	−0.116953
$120$	0	0
$121$	14641.0	0.0909091
$122$	−11356.3	−0.0690774
$123$	0	0
$124$	−72447.0	−0.423123
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	151788.	0.835080	0.417540	−	0.908658i	$-0.362892\pi$
$127$	151788.	0.835080	0.417540	+	0.908658i	$0.362892\pi$
$128$	11578.0	0.0624609
$129$	0	0
$130$	18806.5	0.0976002
$131$	56755.1	0.288953	0.144476	−	0.989508i	$-0.453850\pi$
$131$	56755.1	0.288953	0.144476	+	0.989508i	$0.453850\pi$
$132$	0	0
$133$	−61575.7	−0.301842
$134$	240843.	1.15870
$135$	0	0
$136$	−116608.	−0.540608
$137$	−92136.0	−0.419400	−0.209700	−	0.977766i	$-0.567249\pi$
$137$	−92136.0	−0.419400	−0.209700	+	0.977766i	$0.567249\pi$
$138$	0	0
$139$	162221.	0.712148	0.356074	−	0.934458i	$-0.384115\pi$
$139$	162221.	0.712148	0.356074	+	0.934458i	$0.384115\pi$
$140$	10517.3	0.0453508
$141$	0	0
$142$	−130979.	−0.545105
$143$	−21398.2	−0.0875060
$144$	0	0
$145$	−145906.	−0.576305
$146$	−14612.7	−0.0567348
$147$	0	0
$148$	−40553.1	−0.152184
$149$	322513.	1.19010	0.595048	−	0.803690i	$-0.297133\pi$
$149$	322513.	1.19010	0.595048	+	0.803690i	$0.297133\pi$
$150$	0	0
$151$	−337206.	−1.20352	−0.601760	−	0.798677i	$-0.705534\pi$
$151$	−337206.	−1.20352	−0.601760	+	0.798677i	$0.705534\pi$
$152$	−397431.	−1.39525
$153$	0	0
$154$	15572.1	0.0529110
$155$	−130251.	−0.435464
$156$	0	0
$157$	−518335.	−1.67827	−0.839135	−	0.543923i	$-0.816938\pi$
$157$	−518335.	−1.67827	−0.839135	+	0.543923i	$0.816938\pi$
$158$	198341.	0.632076
$159$	0	0
$160$	115202.	0.355764
$161$	−24038.7	−0.0730879
$162$	0	0
$163$	98714.0	0.291011	0.145506	−	0.989357i	$-0.453519\pi$
$163$	98714.0	0.291011	0.145506	+	0.989357i	$0.453519\pi$
$164$	−179388.	−0.520816
$165$	0	0
$166$	174718.	0.492115
$167$	−453087.	−1.25716	−0.628579	−	0.777746i	$-0.716363\pi$
$167$	−453087.	−1.25716	−0.628579	+	0.777746i	$0.716363\pi$
$168$	0	0
$169$	−340019.	−0.915770
$170$	−63505.0	−0.168533
$171$	0	0
$172$	−91575.1	−0.236024
$173$	−539696.	−1.37099	−0.685494	−	0.728078i	$-0.740414\pi$
$173$	−539696.	−1.37099	−0.685494	+	0.728078i	$0.740414\pi$
$174$	0	0
$175$	18908.9	0.0466736
$176$	46666.8	0.113560
$177$	0	0
$178$	158608.	0.375210
$179$	−617234.	−1.43985	−0.719926	−	0.694051i	$-0.755824\pi$
$179$	−617234.	−1.43985	−0.719926	+	0.694051i	$0.755824\pi$
$180$	0	0
$181$	−317984.	−0.721455	−0.360728	−	0.932671i	$-0.617472\pi$
$181$	−317984.	−0.721455	−0.360728	+	0.932671i	$0.617472\pi$
$182$	−22759.1	−0.0509303
$183$	0	0
$184$	−155154.	−0.337846
$185$	−72909.7	−0.156623
$186$	0	0
$187$	72256.5	0.151103
$188$	−210100.	−0.433543
$189$	0	0
$190$	−216441.	−0.434966
$191$	−685556.	−1.35975	−0.679875	−	0.733328i	$-0.737966\pi$
$191$	−685556.	−1.35975	−0.679875	+	0.733328i	$0.737966\pi$
$192$	0	0
$193$	543262.	1.04982	0.524911	−	0.851157i	$-0.324098\pi$
$193$	543262.	1.04982	0.524911	+	0.851157i	$0.324098\pi$
$194$	382918.	0.730469
$195$	0	0
$196$	220978.	0.410874
$197$	114090.	0.209451	0.104725	−	0.994501i	$-0.466604\pi$
$197$	114090.	0.209451	0.104725	+	0.994501i	$0.466604\pi$
$198$	0	0
$199$	−690624.	−1.23626	−0.618129	−	0.786077i	$-0.712109\pi$
$199$	−690624.	−1.23626	−0.618129	+	0.786077i	$0.712109\pi$
$200$	122045.	0.215746
$201$	0	0
$202$	19208.6	0.0331221
$203$	176571.	0.300731
$204$	0	0
$205$	−322519.	−0.536007
$206$	252605.	0.414737
$207$	0	0
$208$	−68204.9	−0.109309
$209$	246268.	0.389981
$210$	0	0
$211$	−360253.	−0.557059	−0.278529	−	0.960428i	$-0.589847\pi$
$211$	−360253.	−0.557059	−0.278529	+	0.960428i	$0.589847\pi$
$212$	76393.7	0.116740
$213$	0	0
$214$	−689685.	−1.02948
$215$	−164641.	−0.242908
$216$	0	0
$217$	157626.	0.227237
$218$	982551.	1.40028
$219$	0	0
$220$	−42063.4	−0.0585932
$221$	−105605.	−0.145447
$222$	0	0
$223$	−672994.	−0.906252	−0.453126	−	0.891446i	$-0.649691\pi$
$223$	−672994.	−0.906252	−0.453126	+	0.891446i	$0.649691\pi$
$224$	−139414.	−0.185647
$225$	0	0
$226$	−340973.	−0.444067
$227$	−816735.	−1.05200	−0.526001	−	0.850484i	$-0.676309\pi$
$227$	−816735.	−1.05200	−0.526001	+	0.850484i	$0.676309\pi$
$228$	0	0
$229$	328572.	0.414040	0.207020	−	0.978337i	$-0.433623\pi$
$229$	328572.	0.414040	0.207020	+	0.978337i	$0.433623\pi$
$230$	−84496.9	−0.105323
$231$	0	0
$232$	1.13965e6	1.39012
$233$	−649323.	−0.783558	−0.391779	−	0.920059i	$-0.628140\pi$
$233$	−649323.	−0.783558	−0.391779	+	0.920059i	$0.628140\pi$
$234$	0	0
$235$	−377735.	−0.446188
$236$	−477156.	−0.557674
$237$	0	0
$238$	76851.7	0.0879450
$239$	154243.	0.174667	0.0873336	−	0.996179i	$-0.472165\pi$
$239$	154243.	0.174667	0.0873336	+	0.996179i	$0.472165\pi$
$240$	0	0
$241$	−347372.	−0.385258	−0.192629	−	0.981272i	$-0.561701\pi$
$241$	−347372.	−0.385258	−0.192629	+	0.981272i	$0.561701\pi$
$242$	−62279.8	−0.0683610
$243$	0	0
$244$	−37122.6	−0.0399175
$245$	397292.	0.422858
$246$	0	0
$247$	−359928.	−0.375382
$248$	1.01737e6	1.05039
$249$	0	0
$250$	66465.5	0.0672584
$251$	968833.	0.970655	0.485327	−	0.874333i	$-0.338700\pi$
$251$	968833.	0.970655	0.485327	+	0.874333i	$0.338700\pi$
$252$	0	0
$253$	96141.3	0.0944297
$254$	−645675.	−0.627957
$255$	0	0
$256$	−1.07144e6	−1.02181
$257$	311524.	0.294211	0.147106	−	0.989121i	$-0.453004\pi$
$257$	311524.	0.294211	0.147106	+	0.989121i	$0.453004\pi$
$258$	0	0
$259$	88233.1	0.0817301
$260$	61476.8	0.0563999
$261$	0	0
$262$	−241425.	−0.217284
$263$	872210.	0.777556	0.388778	−	0.921331i	$-0.372897\pi$
$263$	872210.	0.777556	0.388778	+	0.921331i	$0.372897\pi$
$264$	0	0
$265$	137347.	0.120145
$266$	261930.	0.226977
$267$	0	0
$268$	787292.	0.669574
$269$	−501519.	−0.422578	−0.211289	−	0.977424i	$-0.567766\pi$
$269$	−501519.	−0.422578	−0.211289	+	0.977424i	$0.567766\pi$
$270$	0	0
$271$	1.48671e6	1.22971	0.614855	−	0.788640i	$-0.289215\pi$
$271$	1.48671e6	1.22971	0.614855	+	0.788640i	$0.289215\pi$
$272$	230311.	0.188752
$273$	0	0
$274$	391928.	0.315377
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	1.19949e6	0.939281	0.469640	−	0.882858i	$-0.344384\pi$
$277$	1.19949e6	0.939281	0.469640	+	0.882858i	$0.344384\pi$
$278$	−690055.	−0.535515
$279$	0	0
$280$	−147695.	−0.112582
$281$	−619137.	−0.467757	−0.233879	−	0.972266i	$-0.575142\pi$
$281$	−619137.	−0.467757	−0.233879	+	0.972266i	$0.575142\pi$
$282$	0	0
$283$	913180.	0.677782	0.338891	−	0.940826i	$-0.389948\pi$
$283$	913180.	0.677782	0.338891	+	0.940826i	$0.389948\pi$
$284$	−428158.	−0.314998
$285$	0	0
$286$	91023.7	0.0658020
$287$	390302.	0.279702
$288$	0	0
$289$	−1.06326e6	−0.748847
$290$	620653.	0.433365
$291$	0	0
$292$	−47767.6	−0.0327851
$293$	52328.8	0.0356100	0.0178050	−	0.999841i	$-0.494332\pi$
$293$	52328.8	0.0356100	0.0178050	+	0.999841i	$0.494332\pi$
$294$	0	0
$295$	−857871.	−0.573940
$296$	569487.	0.377794
$297$	0	0
$298$	−1.37190e6	−0.894918
$299$	−140513.	−0.0908949
$300$	0	0
$301$	199244.	0.126756
$302$	1.43441e6	0.905013
$303$	0	0
$304$	784958.	0.487149
$305$	−66742.0	−0.0410818
$306$	0	0
$307$	199403.	0.120750	0.0603749	−	0.998176i	$-0.480770\pi$
$307$	199403.	0.120750	0.0603749	+	0.998176i	$0.480770\pi$
$308$	50903.8	0.0305755
$309$	0	0
$310$	554062.	0.327457
$311$	−1.10889e6	−0.650109	−0.325054	−	0.945695i	$-0.605383\pi$
$311$	−1.10889e6	−0.650109	−0.325054	+	0.945695i	$0.605383\pi$
$312$	0	0
$313$	2.39830e6	1.38370	0.691850	−	0.722041i	$-0.256796\pi$
$313$	2.39830e6	1.38370	0.691850	+	0.722041i	$0.256796\pi$
$314$	2.20489e6	1.26201
$315$	0	0
$316$	648357.	0.365255
$317$	−1.83569e6	−1.02601	−0.513005	−	0.858386i	$-0.671468\pi$
$317$	−1.83569e6	−1.02601	−0.513005	+	0.858386i	$0.671468\pi$
$318$	0	0
$319$	−706184.	−0.388545
$320$	−798588.	−0.435961
$321$	0	0
$322$	102256.	0.0549600
$323$	1.21539e6	0.648199
$324$	0	0
$325$	110528.	0.0580449
$326$	−419909.	−0.218832
$327$	0	0
$328$	2.51915e6	1.29291
$329$	457124.	0.232833
$330$	0	0
$331$	1.72773e6	0.866773	0.433387	−	0.901208i	$-0.357318\pi$
$331$	1.72773e6	0.866773	0.433387	+	0.901208i	$0.357318\pi$
$332$	571136.	0.284377
$333$	0	0
$334$	1.92734e6	0.945347
$335$	1.41546e6	0.689104
$336$	0	0
$337$	931624.	0.446854	0.223427	−	0.974721i	$-0.428276\pi$
$337$	931624.	0.446854	0.223427	+	0.974721i	$0.428276\pi$
$338$	1.44637e6	0.688633
$339$	0	0
$340$	−207592.	−0.0973897
$341$	−630416.	−0.293590
$342$	0	0
$343$	−989273.	−0.454026
$344$	1.28599e6	0.585924
$345$	0	0
$346$	2.29575e6	1.03094
$347$	−2.25108e6	−1.00362	−0.501809	−	0.864979i	$-0.667332\pi$
$347$	−2.25108e6	−1.00362	−0.501809	+	0.864979i	$0.667332\pi$
$348$	0	0
$349$	−2.85894e6	−1.25644	−0.628220	−	0.778036i	$-0.716216\pi$
$349$	−2.85894e6	−1.25644	−0.628220	+	0.778036i	$0.716216\pi$
$350$	−80434.5	−0.0350972
$351$	0	0
$352$	557580.	0.239856
$353$	2.44462e6	1.04418	0.522089	−	0.852891i	$-0.325153\pi$
$353$	2.44462e6	1.04418	0.522089	+	0.852891i	$0.325153\pi$
$354$	0	0
$355$	−769777.	−0.324186
$356$	518474.	0.216821
$357$	0	0
$358$	2.62559e6	1.08273
$359$	−406390.	−0.166420	−0.0832102	−	0.996532i	$-0.526517\pi$
$359$	−406390.	−0.166420	−0.0832102	+	0.996532i	$0.526517\pi$
$360$	0	0
$361$	1.66625e6	0.672934
$362$	1.35264e6	0.542514
$363$	0	0
$364$	−74397.4	−0.0294309
$365$	−85880.6	−0.0337414
$366$	0	0
$367$	−3.22852e6	−1.25123	−0.625617	−	0.780130i	$-0.715153\pi$
$367$	−3.22852e6	−1.25123	−0.625617	+	0.780130i	$0.715153\pi$
$368$	306441.	0.117958
$369$	0	0
$370$	310143.	0.117776
$371$	−166213.	−0.0626946
$372$	0	0
$373$	1.27684e6	0.475187	0.237594	−	0.971365i	$-0.423641\pi$
$373$	1.27684e6	0.475187	0.237594	+	0.971365i	$0.423641\pi$
$374$	−307364.	−0.113625
$375$	0	0
$376$	2.95044e6	1.07626
$377$	1.03211e6	0.374000
$378$	0	0
$379$	465673.	0.166526	0.0832631	−	0.996528i	$-0.473466\pi$
$379$	465673.	0.166526	0.0832631	+	0.996528i	$0.473466\pi$
$380$	−707526.	−0.251353
$381$	0	0
$382$	2.91621e6	1.02249
$383$	1.09265e6	0.380613	0.190307	−	0.981725i	$-0.439052\pi$
$383$	1.09265e6	0.380613	0.190307	+	0.981725i	$0.439052\pi$
$384$	0	0
$385$	91519.0	0.0314673
$386$	−2.31092e6	−0.789437
$387$	0	0
$388$	1.25172e6	0.422114
$389$	−1.25244e6	−0.419646	−0.209823	−	0.977739i	$-0.567289\pi$
$389$	−1.25244e6	−0.419646	−0.209823	+	0.977739i	$0.567289\pi$
$390$	0	0
$391$	474478.	0.156955
$392$	−3.10319e6	−1.01998
$393$	0	0
$394$	−485315.	−0.157501
$395$	1.16567e6	0.375909
$396$	0	0
$397$	3.04295e6	0.968989	0.484494	−	0.874794i	$-0.339004\pi$
$397$	3.04295e6	0.968989	0.484494	+	0.874794i	$0.339004\pi$
$398$	2.93777e6	0.929630
$399$	0	0
$400$	−241048.	−0.0753274
$401$	178677.	0.0554892	0.0277446	−	0.999615i	$-0.491167\pi$
$401$	178677.	0.0554892	0.0277446	+	0.999615i	$0.491167\pi$
$402$	0	0
$403$	921371.	0.282600
$404$	62791.2	0.0191402
$405$	0	0
$406$	−751095.	−0.226141
$407$	−352883.	−0.105595
$408$	0	0
$409$	−5.72008e6	−1.69081	−0.845403	−	0.534129i	$-0.820640\pi$
$409$	−5.72008e6	−1.69081	−0.845403	+	0.534129i	$0.820640\pi$
$410$	1.37193e6	0.403062
$411$	0	0
$412$	825741.	0.239663
$413$	1.03817e6	0.299497
$414$	0	0
$415$	1.02684e6	0.292672
$416$	−814919.	−0.230877
$417$	0	0
$418$	−1.04758e6	−0.293254
$419$	−790896.	−0.220082	−0.110041	−	0.993927i	$-0.535098\pi$
$419$	−790896.	−0.220082	−0.110041	+	0.993927i	$0.535098\pi$
$420$	0	0
$421$	3.27556e6	0.900699	0.450350	−	0.892852i	$-0.351299\pi$
$421$	3.27556e6	0.900699	0.450350	+	0.892852i	$0.351299\pi$
$422$	1.53244e6	0.418892
$423$	0	0
$424$	−1.07280e6	−0.289803
$425$	−373226.	−0.100230
$426$	0	0
$427$	80769.1	0.0214376
$428$	−2.25452e6	−0.594900
$429$	0	0
$430$	700350.	0.182660
$431$	−5.97944e6	−1.55048	−0.775242	−	0.631664i	$-0.782372\pi$
$431$	−5.97944e6	−1.55048	−0.775242	+	0.631664i	$0.782372\pi$
$432$	0	0
$433$	5.66358e6	1.45168	0.725841	−	0.687863i	$-0.241451\pi$
$433$	5.66358e6	1.45168	0.725841	+	0.687863i	$0.241451\pi$
$434$	−670508.	−0.170875
$435$	0	0
$436$	3.21187e6	0.809173
$437$	1.61714e6	0.405083
$438$	0	0
$439$	−3.60949e6	−0.893890	−0.446945	−	0.894561i	$-0.647488\pi$
$439$	−3.60949e6	−0.893890	−0.446945	+	0.894561i	$0.647488\pi$
$440$	590696.	0.145456
$441$	0	0
$442$	449221.	0.109372
$443$	−2.24416e6	−0.543307	−0.271653	−	0.962395i	$-0.587570\pi$
$443$	−2.24416e6	−0.543307	−0.271653	+	0.962395i	$0.587570\pi$
$444$	0	0
$445$	932155.	0.223145
$446$	2.86278e6	0.681476
$447$	0	0
$448$	966427.	0.227496
$449$	−1.98207e6	−0.463985	−0.231993	−	0.972718i	$-0.574524\pi$
$449$	−1.98207e6	−0.463985	−0.231993	+	0.972718i	$0.574524\pi$
$450$	0	0
$451$	−1.56099e6	−0.361376
$452$	−1.11461e6	−0.256612
$453$	0	0
$454$	3.47422e6	0.791075
$455$	−133758.	−0.0302894
$456$	0	0
$457$	−7.42071e6	−1.66209	−0.831045	−	0.556204i	$-0.812257\pi$
$457$	−7.42071e6	−1.66209	−0.831045	+	0.556204i	$0.812257\pi$
$458$	−1.39768e6	−0.311346
$459$	0	0
$460$	−276212.	−0.0608623
$461$	−6.20338e6	−1.35949	−0.679745	−	0.733448i	$-0.737910\pi$
$461$	−6.20338e6	−1.35949	−0.679745	+	0.733448i	$0.737910\pi$
$462$	0	0
$463$	5.46550e6	1.18489	0.592444	−	0.805611i	$-0.298163\pi$
$463$	5.46550e6	1.18489	0.592444	+	0.805611i	$0.298163\pi$
$464$	−2.25090e6	−0.485356
$465$	0	0
$466$	2.76209e6	0.589213
$467$	−1.38116e6	−0.293057	−0.146528	−	0.989206i	$-0.546810\pi$
$467$	−1.38116e6	−0.293057	−0.146528	+	0.989206i	$0.546810\pi$
$468$	0	0
$469$	−1.71294e6	−0.359593
$470$	1.60681e6	0.335521
$471$	0	0
$472$	6.70070e6	1.38441
$473$	−796864.	−0.163769
$474$	0	0
$475$	−1.27205e6	−0.258684
$476$	251221.	0.0508205
$477$	0	0
$478$	−656119.	−0.131345
$479$	−1.47152e6	−0.293040	−0.146520	−	0.989208i	$-0.546807\pi$
$479$	−1.47152e6	−0.293040	−0.146520	+	0.989208i	$0.546807\pi$
$480$	0	0
$481$	515749.	0.101643
$482$	1.47765e6	0.289703
$483$	0	0
$484$	−203587.	−0.0395035
$485$	2.25045e6	0.434426
$486$	0	0
$487$	−8.12247e6	−1.55191	−0.775954	−	0.630790i	$-0.782731\pi$
$487$	−8.12247e6	−1.55191	−0.775954	+	0.630790i	$0.782731\pi$
$488$	521312.	0.0990942
$489$	0	0
$490$	−1.69000e6	−0.317977
$491$	−2.82155e6	−0.528182	−0.264091	−	0.964498i	$-0.585072\pi$
$491$	−2.82155e6	−0.528182	−0.264091	+	0.964498i	$0.585072\pi$
$492$	0	0
$493$	−3.48517e6	−0.645813
$494$	1.53106e6	0.282277
$495$	0	0
$496$	−2.00939e6	−0.366742
$497$	931560.	0.169169
$498$	0	0
$499$	5.66317e6	1.01814	0.509071	−	0.860725i	$-0.329989\pi$
$499$	5.66317e6	1.01814	0.509071	+	0.860725i	$0.329989\pi$
$500$	217270.	0.0388664
$501$	0	0
$502$	−4.12122e6	−0.729905
$503$	6.63876e6	1.16995	0.584974	−	0.811052i	$-0.301105\pi$
$503$	6.63876e6	1.16995	0.584974	+	0.811052i	$0.301105\pi$
$504$	0	0
$505$	112891.	0.0196984
$506$	−408965.	−0.0710084
$507$	0	0
$508$	−2.11065e6	−0.362875
$509$	−5.77939e6	−0.988752	−0.494376	−	0.869248i	$-0.664603\pi$
$509$	−5.77939e6	−0.988752	−0.494376	+	0.869248i	$0.664603\pi$
$510$	0	0
$511$	103930.	0.0176071
$512$	4.18720e6	0.705910
$513$	0	0
$514$	−1.32516e6	−0.221238
$515$	1.48458e6	0.246653
$516$	0	0
$517$	−1.82824e6	−0.300820
$518$	−375325.	−0.0614587
$519$	0	0
$520$	−863319.	−0.140011
$521$	9.08159e6	1.46578	0.732888	−	0.680349i	$-0.238172\pi$
$521$	9.08159e6	1.46578	0.732888	+	0.680349i	$0.238172\pi$
$522$	0	0
$523$	−4.30713e6	−0.688547	−0.344273	−	0.938869i	$-0.611875\pi$
$523$	−4.30713e6	−0.688547	−0.344273	+	0.938869i	$0.611875\pi$
$524$	−789194.	−0.125561
$525$	0	0
$526$	−3.71020e6	−0.584700
$527$	−3.11124e6	−0.487985
$528$	0	0
$529$	−5.80502e6	−0.901913
$530$	−584245.	−0.0903453
$531$	0	0
$532$	856226.	0.131162
$533$	2.28143e6	0.347848
$534$	0	0
$535$	−4.05335e6	−0.612252
$536$	−1.10559e7	−1.66220
$537$	0	0
$538$	2.13336e6	0.317766
$539$	1.92289e6	0.285091
$540$	0	0
$541$	−5.07155e6	−0.744984	−0.372492	−	0.928035i	$-0.621497\pi$
$541$	−5.07155e6	−0.744984	−0.372492	+	0.928035i	$0.621497\pi$
$542$	−6.32415e6	−0.924706
$543$	0	0
$544$	2.75178e6	0.398672
$545$	5.77456e6	0.832775
$546$	0	0
$547$	1.00442e7	1.43532	0.717660	−	0.696394i	$-0.245213\pi$
$547$	1.00442e7	1.43532	0.717660	+	0.696394i	$0.245213\pi$
$548$	1.28117e6	0.182246
$549$	0	0
$550$	321693.	0.0453456
$551$	−1.18783e7	−1.66678
$552$	0	0
$553$	−1.41066e6	−0.196159
$554$	−5.10236e6	−0.706312
$555$	0	0
$556$	−2.25572e6	−0.309456
$557$	306075.	0.0418013	0.0209006	−	0.999782i	$-0.493347\pi$
$557$	306075.	0.0418013	0.0209006	+	0.999782i	$0.493347\pi$
$558$	0	0
$559$	1.16464e6	0.157638
$560$	291708.	0.0393078
$561$	0	0
$562$	2.63368e6	0.351740
$563$	8.74487e6	1.16274	0.581370	−	0.813640i	$-0.302517\pi$
$563$	8.74487e6	1.16274	0.581370	+	0.813640i	$0.302517\pi$
$564$	0	0
$565$	−2.00394e6	−0.264096
$566$	−3.88448e6	−0.509673
$567$	0	0
$568$	6.01262e6	0.781975
$569$	−2.01648e6	−0.261104	−0.130552	−	0.991441i	$-0.541675\pi$
$569$	−2.01648e6	−0.261104	−0.130552	+	0.991441i	$0.541675\pi$
$570$	0	0
$571$	4.75332e6	0.610108	0.305054	−	0.952335i	$-0.401326\pi$
$571$	4.75332e6	0.610108	0.305054	+	0.952335i	$0.401326\pi$
$572$	297548.	0.0380248
$573$	0	0
$574$	−1.66026e6	−0.210328
$575$	−496597.	−0.0626376
$576$	0	0
$577$	5.81137e6	0.726673	0.363336	−	0.931658i	$-0.381638\pi$
$577$	5.81137e6	0.726673	0.363336	+	0.931658i	$0.381638\pi$
$578$	4.52287e6	0.563112
$579$	0	0
$580$	2.02886e6	0.250427
$581$	−1.24264e6	−0.152724
$582$	0	0
$583$	664759.	0.0810015
$584$	670801.	0.0813882
$585$	0	0
$586$	−222596.	−0.0267777
$587$	−1.52224e7	−1.82342	−0.911711	−	0.410833i	$-0.865238\pi$
$587$	−1.52224e7	−1.82342	−0.911711	+	0.410833i	$0.865238\pi$
$588$	0	0
$589$	−1.06039e7	−1.25944
$590$	3.64920e6	0.431587
$591$	0	0
$592$	−1.12478e6	−0.131906
$593$	7.23587e6	0.844995	0.422498	−	0.906364i	$-0.361153\pi$
$593$	7.23587e6	0.844995	0.422498	+	0.906364i	$0.361153\pi$
$594$	0	0
$595$	451666.	0.0523028
$596$	−4.48463e6	−0.517143
$597$	0	0
$598$	597714.	0.0683503
$599$	1.07455e7	1.22366	0.611829	−	0.790990i	$-0.290434\pi$
$599$	1.07455e7	1.22366	0.611829	+	0.790990i	$0.290434\pi$
$600$	0	0
$601$	−1.01708e7	−1.14860	−0.574302	−	0.818643i	$-0.694727\pi$
$601$	−1.01708e7	−1.14860	−0.574302	+	0.818643i	$0.694727\pi$
$602$	−847541.	−0.0953169
$603$	0	0
$604$	4.68894e6	0.522976
$605$	−366025.	−0.0406558
$606$	0	0
$607$	−4.87810e6	−0.537377	−0.268689	−	0.963227i	$-0.586590\pi$
$607$	−4.87810e6	−0.537377	−0.268689	+	0.963227i	$0.586590\pi$
$608$	9.37876e6	1.02893
$609$	0	0
$610$	283907.	0.0308924
$611$	2.67202e6	0.289559
$612$	0	0
$613$	−1.13744e7	−1.22258	−0.611291	−	0.791406i	$-0.709349\pi$
$613$	−1.13744e7	−1.22258	−0.611291	+	0.791406i	$0.709349\pi$
$614$	−848221.	−0.0908004
$615$	0	0
$616$	−714842.	−0.0759029
$617$	5.32888e6	0.563538	0.281769	−	0.959482i	$-0.409079\pi$
$617$	5.32888e6	0.563538	0.281769	+	0.959482i	$0.409079\pi$
$618$	0	0
$619$	−8.99716e6	−0.943798	−0.471899	−	0.881653i	$-0.656431\pi$
$619$	−8.99716e6	−0.943798	−0.471899	+	0.881653i	$0.656431\pi$
$620$	1.81118e6	0.189226
$621$	0	0
$622$	4.71697e6	0.488863
$623$	−1.12806e6	−0.116443
$624$	0	0
$625$	390625.	0.0400000
$626$	−1.02019e7	−1.04050
$627$	0	0
$628$	7.20758e6	0.729274
$629$	−1.74155e6	−0.175513
$630$	0	0
$631$	−1.83252e7	−1.83221	−0.916103	−	0.400943i	$-0.868683\pi$
$631$	−1.83252e7	−1.83221	−0.916103	+	0.400943i	$0.868683\pi$
$632$	−9.10488e6	−0.906738
$633$	0	0
$634$	7.80864e6	0.771529
$635$	−3.79470e6	−0.373459
$636$	0	0
$637$	−2.81036e6	−0.274419
$638$	3.00396e6	0.292175
$639$	0	0
$640$	−289450.	−0.0279334
$641$	4.11533e6	0.395603	0.197802	−	0.980242i	$-0.436620\pi$
$641$	4.11533e6	0.395603	0.197802	+	0.980242i	$0.436620\pi$
$642$	0	0
$643$	−4.30054e6	−0.410200	−0.205100	−	0.978741i	$-0.565752\pi$
$643$	−4.30054e6	−0.410200	−0.205100	+	0.978741i	$0.565752\pi$
$644$	334264.	0.0317596
$645$	0	0
$646$	−5.17001e6	−0.487427
$647$	−4.19706e6	−0.394171	−0.197086	−	0.980386i	$-0.563148\pi$
$647$	−4.19706e6	−0.394171	−0.197086	+	0.980386i	$0.563148\pi$
$648$	0	0
$649$	−4.15209e6	−0.386950
$650$	−470164.	−0.0436481
$651$	0	0
$652$	−1.37264e6	−0.126456
$653$	9.02497e6	0.828253	0.414126	−	0.910219i	$-0.364087\pi$
$653$	9.02497e6	0.828253	0.414126	+	0.910219i	$0.364087\pi$
$654$	0	0
$655$	−1.41888e6	−0.129224
$656$	−4.97551e6	−0.451417
$657$	0	0
$658$	−1.94451e6	−0.175083
$659$	747098.	0.0670138	0.0335069	−	0.999438i	$-0.489332\pi$
$659$	747098.	0.0670138	0.0335069	+	0.999438i	$0.489332\pi$
$660$	0	0
$661$	6.18905e6	0.550960	0.275480	−	0.961307i	$-0.411163\pi$
$661$	6.18905e6	0.550960	0.275480	+	0.961307i	$0.411163\pi$
$662$	−7.34940e6	−0.651789
$663$	0	0
$664$	−8.02046e6	−0.705958
$665$	1.53939e6	0.134988
$666$	0	0
$667$	−4.63721e6	−0.403592
$668$	6.30028e6	0.546284
$669$	0	0
$670$	−6.02107e6	−0.518187
$671$	−323031.	−0.0276974
$672$	0	0
$673$	−1.74383e7	−1.48411	−0.742054	−	0.670340i	$-0.766148\pi$
$673$	−1.74383e7	−1.48411	−0.742054	+	0.670340i	$0.766148\pi$
$674$	−3.96293e6	−0.336021
$675$	0	0
$676$	4.72805e6	0.397938
$677$	1.08778e7	0.912160	0.456080	−	0.889939i	$-0.349253\pi$
$677$	1.08778e7	0.912160	0.456080	+	0.889939i	$0.349253\pi$
$678$	0	0
$679$	−2.72343e6	−0.226695
$680$	2.91521e6	0.241767
$681$	0	0
$682$	2.68166e6	0.220771
$683$	1.57505e7	1.29194	0.645972	−	0.763361i	$-0.276452\pi$
$683$	1.57505e7	1.29194	0.645972	+	0.763361i	$0.276452\pi$
$684$	0	0
$685$	2.30340e6	0.187561
$686$	4.20816e6	0.341415
$687$	0	0
$688$	−2.53993e6	−0.204574
$689$	−971564.	−0.0779693
$690$	0	0
$691$	9.28838e6	0.740023	0.370011	−	0.929027i	$-0.379354\pi$
$691$	9.28838e6	0.740023	0.370011	+	0.929027i	$0.379354\pi$
$692$	7.50460e6	0.595748
$693$	0	0
$694$	9.57565e6	0.754691
$695$	−4.05553e6	−0.318482
$696$	0	0
$697$	−7.70382e6	−0.600654
$698$	1.21613e7	0.944806
$699$	0	0
$700$	−262933.	−0.0202815
$701$	−9.92501e6	−0.762845	−0.381422	−	0.924401i	$-0.624566\pi$
$701$	−9.92501e6	−0.762845	−0.381422	+	0.924401i	$0.624566\pi$
$702$	0	0
$703$	−5.93566e6	−0.452982
$704$	−3.86517e6	−0.293925
$705$	0	0
$706$	−1.03989e7	−0.785192
$707$	−136618.	−0.0102792
$708$	0	0
$709$	−635698.	−0.0474936	−0.0237468	−	0.999718i	$-0.507560\pi$
$709$	−635698.	−0.0474936	−0.0237468	+	0.999718i	$0.507560\pi$
$710$	3.27447e6	0.243778
$711$	0	0
$712$	−7.28093e6	−0.538253
$713$	−4.13968e6	−0.304960
$714$	0	0
$715$	534956.	0.0391339
$716$	8.58280e6	0.625671
$717$	0	0
$718$	1.72870e6	0.125143
$719$	1.69438e7	1.22233	0.611164	−	0.791504i	$-0.290702\pi$
$719$	1.69438e7	1.22233	0.611164	+	0.791504i	$0.290702\pi$
$720$	0	0
$721$	−1.79660e6	−0.128710
$722$	−7.08789e6	−0.506027
$723$	0	0
$724$	4.42165e6	0.313500
$725$	3.64765e6	0.257732
$726$	0	0
$727$	−2.98090e6	−0.209176	−0.104588	−	0.994516i	$-0.533352\pi$
$727$	−2.98090e6	−0.209176	−0.104588	+	0.994516i	$0.533352\pi$
$728$	1.04476e6	0.0730616
$729$	0	0
$730$	365318.	0.0253726
$731$	−3.93269e6	−0.272205
$732$	0	0
$733$	−9.10996e6	−0.626262	−0.313131	−	0.949710i	$-0.601378\pi$
$733$	−9.10996e6	−0.626262	−0.313131	+	0.949710i	$0.601378\pi$
$734$	1.37335e7	0.940892
$735$	0	0
$736$	3.66139e6	0.249145
$737$	6.85082e6	0.464594
$738$	0	0
$739$	−9.56174e6	−0.644059	−0.322030	−	0.946730i	$-0.604365\pi$
$739$	−9.56174e6	−0.644059	−0.322030	+	0.946730i	$0.604365\pi$
$740$	1.01383e6	0.0680589
$741$	0	0
$742$	707035.	0.0471445
$743$	1.64823e7	1.09533	0.547665	−	0.836698i	$-0.315517\pi$
$743$	1.64823e7	1.09533	0.547665	+	0.836698i	$0.315517\pi$
$744$	0	0
$745$	−8.06283e6	−0.532227
$746$	−5.43142e6	−0.357327
$747$	0	0
$748$	−1.00474e6	−0.0656601
$749$	4.90524e6	0.319489
$750$	0	0
$751$	−1.92250e7	−1.24385	−0.621923	−	0.783078i	$-0.713648\pi$
$751$	−1.92250e7	−1.24385	−0.621923	+	0.783078i	$0.713648\pi$
$752$	−5.82734e6	−0.375773
$753$	0	0
$754$	−4.39038e6	−0.281238
$755$	8.43016e6	0.538230
$756$	0	0
$757$	3.84681e6	0.243984	0.121992	−	0.992531i	$-0.461072\pi$
$757$	3.84681e6	0.243984	0.121992	+	0.992531i	$0.461072\pi$
$758$	−1.98088e6	−0.125223
$759$	0	0
$760$	9.93578e6	0.623976
$761$	−1.25696e7	−0.786794	−0.393397	−	0.919369i	$-0.628700\pi$
$761$	−1.25696e7	−0.786794	−0.393397	+	0.919369i	$0.628700\pi$
$762$	0	0
$763$	−6.98819e6	−0.434564
$764$	9.53282e6	0.590865
$765$	0	0
$766$	−4.64790e6	−0.286210
$767$	6.06840e6	0.372466
$768$	0	0
$769$	2.47362e7	1.50840	0.754201	−	0.656643i	$-0.228024\pi$
$769$	2.47362e7	1.50840	0.754201	+	0.656643i	$0.228024\pi$
$770$	−389303.	−0.0236625
$771$	0	0
$772$	−7.55419e6	−0.456189
$773$	2.27422e7	1.36894	0.684469	−	0.729042i	$-0.260034\pi$
$773$	2.27422e7	1.36894	0.684469	+	0.729042i	$0.260034\pi$
$774$	0	0
$775$	3.25628e6	0.194746
$776$	−1.75780e7	−1.04789
$777$	0	0
$778$	5.32762e6	0.315562
$779$	−2.62566e7	−1.55022
$780$	0	0
$781$	−3.72572e6	−0.218566
$782$	−2.01833e6	−0.118025
$783$	0	0
$784$	6.12904e6	0.356125
$785$	1.29584e7	0.750545
$786$	0	0
$787$	−6.40223e6	−0.368463	−0.184232	−	0.982883i	$-0.558980\pi$
$787$	−6.40223e6	−0.368463	−0.184232	+	0.982883i	$0.558980\pi$
$788$	−1.58645e6	−0.0910145
$789$	0	0
$790$	−4.95852e6	−0.282673
$791$	2.42510e6	0.137812
$792$	0	0
$793$	472120.	0.0266605
$794$	−1.29441e7	−0.728652
$795$	0	0
$796$	9.60330e6	0.537202
$797$	5.85339e6	0.326408	0.163204	−	0.986592i	$-0.447817\pi$
$797$	5.85339e6	0.326408	0.163204	+	0.986592i	$0.447817\pi$
$798$	0	0
$799$	−9.02275e6	−0.500002
$800$	−2.88006e6	−0.159102
$801$	0	0
$802$	−760056.	−0.0417263
$803$	−415662.	−0.0227484
$804$	0	0
$805$	600967.	0.0326859
$806$	−3.91932e6	−0.212507
$807$	0	0
$808$	−881778.	−0.0475150
$809$	−2.05685e7	−1.10492	−0.552459	−	0.833540i	$-0.686311\pi$
$809$	−2.05685e7	−1.10492	−0.552459	+	0.833540i	$0.686311\pi$
$810$	0	0
$811$	−3.13935e6	−0.167605	−0.0838027	−	0.996482i	$-0.526707\pi$
$811$	−3.13935e6	−0.167605	−0.0838027	+	0.996482i	$0.526707\pi$
$812$	−2.45526e6	−0.130679
$813$	0	0
$814$	1.50109e6	0.0794047
$815$	−2.46785e6	−0.130144
$816$	0	0
$817$	−1.34036e7	−0.702533
$818$	2.43320e7	1.27144
$819$	0	0
$820$	4.48470e6	0.232916
$821$	−1.19851e6	−0.0620557	−0.0310279	−	0.999519i	$-0.509878\pi$
$821$	−1.19851e6	−0.0620557	−0.0310279	+	0.999519i	$0.509878\pi$
$822$	0	0
$823$	2.63904e7	1.35815	0.679073	−	0.734071i	$-0.262382\pi$
$823$	2.63904e7	1.35815	0.679073	+	0.734071i	$0.262382\pi$
$824$	−1.15959e7	−0.594957
$825$	0	0
$826$	−4.41615e6	−0.225213
$827$	2.14515e6	0.109067	0.0545335	−	0.998512i	$-0.482633\pi$
$827$	2.14515e6	0.109067	0.0545335	+	0.998512i	$0.482633\pi$
$828$	0	0
$829$	254431.	0.0128583	0.00642916	−	0.999979i	$-0.497954\pi$
$829$	254431.	0.0128583	0.00642916	+	0.999979i	$0.497954\pi$
$830$	−4.36794e6	−0.220081
$831$	0	0
$832$	5.64905e6	0.282922
$833$	9.48989e6	0.473859
$834$	0	0
$835$	1.13272e7	0.562218
$836$	−3.42442e6	−0.169462
$837$	0	0
$838$	3.36431e6	0.165495
$839$	5.05327e6	0.247838	0.123919	−	0.992292i	$-0.460454\pi$
$839$	5.05327e6	0.247838	0.123919	+	0.992292i	$0.460454\pi$
$840$	0	0
$841$	1.35505e7	0.660639
$842$	−1.39335e7	−0.677300
$843$	0	0
$844$	5.00940e6	0.242064
$845$	8.50047e6	0.409545
$846$	0	0
$847$	442952.	0.0212153
$848$	2.11886e6	0.101184
$849$	0	0
$850$	1.58762e6	0.0753703
$851$	−2.31723e6	−0.109685
$852$	0	0
$853$	−3.94059e6	−0.185434	−0.0927169	−	0.995693i	$-0.529555\pi$
$853$	−3.94059e6	−0.185434	−0.0927169	+	0.995693i	$0.529555\pi$
$854$	−343575.	−0.0161204
$855$	0	0
$856$	3.16602e7	1.47682
$857$	−3.77045e7	−1.75364	−0.876822	−	0.480816i	$-0.840341\pi$
$857$	−3.77045e7	−1.75364	−0.876822	+	0.480816i	$0.840341\pi$
$858$	0	0
$859$	2.07610e7	0.959988	0.479994	−	0.877272i	$-0.340639\pi$
$859$	2.07610e7	0.959988	0.479994	+	0.877272i	$0.340639\pi$
$860$	2.28938e6	0.105553
$861$	0	0
$862$	2.54353e7	1.16592
$863$	1.65018e6	0.0754233	0.0377116	−	0.999289i	$-0.487993\pi$
$863$	1.65018e6	0.0754233	0.0377116	+	0.999289i	$0.487993\pi$
$864$	0	0
$865$	1.34924e7	0.613124
$866$	−2.40917e7	−1.09162
$867$	0	0
$868$	−2.19183e6	−0.0987432
$869$	5.64184e6	0.253438
$870$	0	0
$871$	−1.00127e7	−0.447203
$872$	−4.51043e7	−2.00875
$873$	0	0
$874$	−6.87898e6	−0.304611
$875$	−472722.	−0.0208730
$876$	0	0
$877$	−2.97115e7	−1.30444	−0.652222	−	0.758028i	$-0.726163\pi$
$877$	−2.97115e7	−1.30444	−0.652222	+	0.758028i	$0.726163\pi$
$878$	1.53540e7	0.672180
$879$	0	0
$880$	−1.16667e6	−0.0507857
$881$	−4.20269e7	−1.82426	−0.912132	−	0.409897i	$-0.865565\pi$
$881$	−4.20269e7	−1.82426	−0.912132	+	0.409897i	$0.865565\pi$
$882$	0	0
$883$	−4.02308e7	−1.73643	−0.868215	−	0.496189i	$-0.834732\pi$
$883$	−4.02308e7	−1.73643	−0.868215	+	0.496189i	$0.834732\pi$
$884$	1.46846e6	0.0632022
$885$	0	0
$886$	9.54620e6	0.408551
$887$	−4.03389e7	−1.72153	−0.860767	−	0.509000i	$-0.830015\pi$
$887$	−4.03389e7	−1.72153	−0.860767	+	0.509000i	$0.830015\pi$
$888$	0	0
$889$	4.59223e6	0.194881
$890$	−3.96519e6	−0.167799
$891$	0	0
$892$	9.35815e6	0.393802
$893$	−3.07518e7	−1.29045
$894$	0	0
$895$	1.54309e7	0.643921
$896$	350283.	0.0145764
$897$	0	0
$898$	8.43133e6	0.348903
$899$	3.04071e7	1.25480
$900$	0	0
$901$	3.28073e6	0.134635
$902$	6.64013e6	0.271744
$903$	0	0
$904$	1.56525e7	0.637032
$905$	7.94961e6	0.322645
$906$	0	0
$907$	−8.71646e6	−0.351821	−0.175911	−	0.984406i	$-0.556287\pi$
$907$	−8.71646e6	−0.351821	−0.175911	+	0.984406i	$0.556287\pi$
$908$	1.13569e7	0.457136
$909$	0	0
$910$	568977.	0.0227767
$911$	−1.35604e7	−0.541350	−0.270675	−	0.962671i	$-0.587247\pi$
$911$	−1.35604e7	−0.541350	−0.270675	+	0.962671i	$0.587247\pi$
$912$	0	0
$913$	4.96988e6	0.197319
$914$	3.15661e7	1.24984
$915$	0	0
$916$	−4.56888e6	−0.179917
$917$	1.71708e6	0.0674323
$918$	0	0
$919$	4.52932e7	1.76907	0.884534	−	0.466476i	$-0.154477\pi$
$919$	4.52932e7	1.76907	0.884534	+	0.466476i	$0.154477\pi$
$920$	3.87885e6	0.151089
$921$	0	0
$922$	2.63879e7	1.02230
$923$	5.44525e6	0.210384
$924$	0	0
$925$	1.82274e6	0.0700440
$926$	−2.32491e7	−0.891003
$927$	0	0
$928$	−2.68939e7	−1.02514
$929$	−2.52580e7	−0.960196	−0.480098	−	0.877215i	$-0.659399\pi$
$929$	−2.52580e7	−0.960196	−0.480098	+	0.877215i	$0.659399\pi$
$930$	0	0
$931$	3.23440e7	1.22298
$932$	9.02900e6	0.340486
$933$	0	0
$934$	5.87516e6	0.220370
$935$	−1.80641e6	−0.0675753
$936$	0	0
$937$	−3.36741e7	−1.25299	−0.626495	−	0.779426i	$-0.715511\pi$
$937$	−3.36741e7	−1.25299	−0.626495	+	0.779426i	$0.715511\pi$
$938$	7.28651e6	0.270403
$939$	0	0
$940$	5.25250e6	0.193886
$941$	1.71731e7	0.632228	0.316114	−	0.948721i	$-0.397622\pi$
$941$	1.71731e7	0.632228	0.316114	+	0.948721i	$0.397622\pi$
$942$	0	0
$943$	−1.02504e7	−0.375370
$944$	−1.32344e7	−0.483364
$945$	0	0
$946$	3.38969e6	0.123149
$947$	4.75076e7	1.72142	0.860712	−	0.509093i	$-0.170019\pi$
$947$	4.75076e7	1.72142	0.860712	+	0.509093i	$0.170019\pi$
$948$	0	0
$949$	607502.	0.0218969
$950$	5.41103e6	0.194523
$951$	0	0
$952$	−3.52790e6	−0.126161
$953$	3.94175e7	1.40591	0.702954	−	0.711235i	$-0.251864\pi$
$953$	3.94175e7	1.40591	0.702954	+	0.711235i	$0.251864\pi$
$954$	0	0
$955$	1.71389e7	0.608099
$956$	−2.14479e6	−0.0758998
$957$	0	0
$958$	6.25953e6	0.220358
$959$	−2.78750e6	−0.0978744
$960$	0	0
$961$	−1.48454e6	−0.0518541
$962$	−2.19389e6	−0.0764323
$963$	0	0
$964$	4.83029e6	0.167410
$965$	−1.35815e7	−0.469495
$966$	0	0
$967$	4.50428e7	1.54903	0.774514	−	0.632557i	$-0.217995\pi$
$967$	4.50428e7	1.54903	0.774514	+	0.632557i	$0.217995\pi$
$968$	2.85897e6	0.0980666
$969$	0	0
$970$	−9.57296e6	−0.326676
$971$	2.75360e6	0.0937243	0.0468621	−	0.998901i	$-0.485078\pi$
$971$	2.75360e6	0.0937243	0.0468621	+	0.998901i	$0.485078\pi$
$972$	0	0
$973$	4.90787e6	0.166192
$974$	3.45513e7	1.16699
$975$	0	0
$976$	−1.02963e6	−0.0345985
$977$	−4.46217e7	−1.49558	−0.747791	−	0.663935i	$-0.768885\pi$
$977$	−4.46217e7	−1.49558	−0.747791	+	0.663935i	$0.768885\pi$
$978$	0	0
$979$	4.51163e6	0.150445
$980$	−5.52444e6	−0.183748
$981$	0	0
$982$	1.20023e7	0.397178
$983$	2.47656e7	0.817457	0.408728	−	0.912656i	$-0.365972\pi$
$983$	2.47656e7	0.817457	0.408728	+	0.912656i	$0.365972\pi$
$984$	0	0
$985$	−2.85225e6	−0.0936692
$986$	1.48252e7	0.485633
$987$	0	0
$988$	5.00490e6	0.163118
$989$	−5.23267e6	−0.170111
$990$	0	0
$991$	4.24080e7	1.37171	0.685857	−	0.727736i	$-0.259427\pi$
$991$	4.24080e7	1.37171	0.685857	+	0.727736i	$0.259427\pi$
$992$	−2.40084e7	−0.774612
$993$	0	0
$994$	−3.96266e6	−0.127210
$995$	1.72656e7	0.552871
$996$	0	0
$997$	1.90471e7	0.606864	0.303432	−	0.952853i	$-0.401867\pi$
$997$	1.90471e7	0.606864	0.303432	+	0.952853i	$0.401867\pi$
$998$	−2.40899e7	−0.765613
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.m.1.2	yes	6
3.2	odd	2		495.6.a.k.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.6.a.k.1.5	✓	6	3.2	odd	2
495.6.a.m.1.2	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-4.25379 q^{2} -13.9052 q^{4} -25.0000 q^{5} +30.2542 q^{7} +195.271 q^{8} +O(q^{10})\)	\(q-4.25379 q^{2} -13.9052 q^{4} -25.0000 q^{5} +30.2542 q^{7} +195.271 q^{8} +106.345 q^{10} -121.000 q^{11} +176.845 q^{13} -128.695 q^{14} -385.676 q^{16} -597.161 q^{17} -2035.28 q^{19} +347.631 q^{20} +514.709 q^{22} -794.556 q^{23} +625.000 q^{25} -752.262 q^{26} -420.692 q^{28} +5836.23 q^{29} +5210.05 q^{31} -4608.10 q^{32} +2540.20 q^{34} -756.356 q^{35} +2916.39 q^{37} +8657.64 q^{38} -4881.79 q^{40} +12900.7 q^{41} +6585.65 q^{43} +1682.54 q^{44} +3379.88 q^{46} +15109.4 q^{47} -15891.7 q^{49} -2658.62 q^{50} -2459.07 q^{52} -5493.88 q^{53} +3025.00 q^{55} +5907.78 q^{56} -24826.1 q^{58} +34314.8 q^{59} +2669.68 q^{61} -22162.5 q^{62} +31943.5 q^{64} -4421.12 q^{65} -56618.3 q^{67} +8303.67 q^{68} +3217.38 q^{70} +30791.1 q^{71} +3435.22 q^{73} -12405.7 q^{74} +28301.0 q^{76} -3660.76 q^{77} -46626.8 q^{79} +9641.90 q^{80} -54877.1 q^{82} -41073.4 q^{83} +14929.0 q^{85} -28014.0 q^{86} -23627.8 q^{88} -37286.2 q^{89} +5350.31 q^{91} +11048.5 q^{92} -64272.3 q^{94} +50881.9 q^{95} -90018.1 q^{97} +67599.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{2} + 53 q^{4} - 150 q^{5} - 80 q^{7} + 255 q^{8}+O(q^{10}) \) 6 * q + 5 * q^2 + 53 * q^4 - 150 * q^5 - 80 * q^7 + 255 * q^8	\( 6 q + 5 q^{2} + 53 q^{4} - 150 q^{5} - 80 q^{7} + 255 q^{8} - 125 q^{10} - 726 q^{11} - 420 q^{13} + 776 q^{14} + 1169 q^{16} + 2820 q^{17} - 220 q^{19} - 1325 q^{20} - 605 q^{22} + 2680 q^{23} + 3750 q^{25} + 1896 q^{26} - 11760 q^{28} + 1092 q^{29} - 7688 q^{31} + 4535 q^{32} - 354 q^{34} + 2000 q^{35} - 14020 q^{37} + 1570 q^{38} - 6375 q^{40} - 8196 q^{41} - 17340 q^{43} - 6413 q^{44} - 9982 q^{46} + 4200 q^{47} - 16890 q^{49} + 3125 q^{50} - 13440 q^{52} + 32900 q^{53} + 18150 q^{55} - 41824 q^{56} - 98010 q^{58} + 44512 q^{59} + 26636 q^{61} - 50680 q^{62} - 74607 q^{64} + 10500 q^{65} - 9920 q^{67} - 34810 q^{68} - 19400 q^{70} + 27344 q^{71} - 106620 q^{73} - 244014 q^{74} - 5638 q^{76} + 9680 q^{77} - 7168 q^{79} - 29225 q^{80} - 90250 q^{82} - 113480 q^{83} - 70500 q^{85} - 96314 q^{86} - 30855 q^{88} - 38352 q^{89} - 115232 q^{91} - 116910 q^{92} - 161846 q^{94} + 5500 q^{95} - 299100 q^{97} - 70715 q^{98}+O(q^{100}) \) 6 * q + 5 * q^2 + 53 * q^4 - 150 * q^5 - 80 * q^7 + 255 * q^8 - 125 * q^10 - 726 * q^11 - 420 * q^13 + 776 * q^14 + 1169 * q^16 + 2820 * q^17 - 220 * q^19 - 1325 * q^20 - 605 * q^22 + 2680 * q^23 + 3750 * q^25 + 1896 * q^26 - 11760 * q^28 + 1092 * q^29 - 7688 * q^31 + 4535 * q^32 - 354 * q^34 + 2000 * q^35 - 14020 * q^37 + 1570 * q^38 - 6375 * q^40 - 8196 * q^41 - 17340 * q^43 - 6413 * q^44 - 9982 * q^46 + 4200 * q^47 - 16890 * q^49 + 3125 * q^50 - 13440 * q^52 + 32900 * q^53 + 18150 * q^55 - 41824 * q^56 - 98010 * q^58 + 44512 * q^59 + 26636 * q^61 - 50680 * q^62 - 74607 * q^64 + 10500 * q^65 - 9920 * q^67 - 34810 * q^68 - 19400 * q^70 + 27344 * q^71 - 106620 * q^73 - 244014 * q^74 - 5638 * q^76 + 9680 * q^77 - 7168 * q^79 - 29225 * q^80 - 90250 * q^82 - 113480 * q^83 - 70500 * q^85 - 96314 * q^86 - 30855 * q^88 - 38352 * q^89 - 115232 * q^91 - 116910 * q^92 - 161846 * q^94 + 5500 * q^95 - 299100 * q^97 - 70715 * q^98

Introduction

L-functions

Modular forms

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Database

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