LMFDB - Embedded newform 495.6.a.p.1.5

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3 x^{9} - 246 x^{8} + 640 x^{7} + 20433 x^{6} - 44595 x^{5} - 667026 x^{4} + 1173648 x^{3} + \cdots - 30445728 $ x^10 - 3x^9 - 246x^8 + 640x^7 + 20433x^6 - 44595x^5 - 667026x^4 + 1173648x^3 + 7949136x^2 - 8226800*x - 30445728
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.97152$ 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)

$f(q)$	$=$	$q-1.97152 q^{2} -28.1131 q^{4} +25.0000 q^{5} +213.116 q^{7} +118.514 q^{8} +O(q^{10})$	\(q-1.97152 q^{2} -28.1131 q^{4} +25.0000 q^{5} +213.116 q^{7} +118.514 q^{8} -49.2880 q^{10} -121.000 q^{11} +941.794 q^{13} -420.162 q^{14} +665.967 q^{16} +1228.09 q^{17} -649.243 q^{19} -702.828 q^{20} +238.554 q^{22} +2372.53 q^{23} +625.000 q^{25} -1856.76 q^{26} -5991.35 q^{28} +5138.20 q^{29} +4673.09 q^{31} -5105.42 q^{32} -2421.20 q^{34} +5327.90 q^{35} -5345.80 q^{37} +1279.99 q^{38} +2962.85 q^{40} +648.735 q^{41} -9140.04 q^{43} +3401.69 q^{44} -4677.49 q^{46} -27692.2 q^{47} +28611.4 q^{49} -1232.20 q^{50} -26476.8 q^{52} -23416.5 q^{53} -3025.00 q^{55} +25257.2 q^{56} -10130.0 q^{58} -13615.6 q^{59} -7673.40 q^{61} -9213.08 q^{62} -11245.5 q^{64} +23544.9 q^{65} +8759.02 q^{67} -34525.4 q^{68} -10504.0 q^{70} -10812.2 q^{71} +65657.0 q^{73} +10539.3 q^{74} +18252.2 q^{76} -25787.0 q^{77} +15899.7 q^{79} +16649.2 q^{80} -1278.99 q^{82} +9330.13 q^{83} +30702.2 q^{85} +18019.7 q^{86} -14340.2 q^{88} +92668.6 q^{89} +200711. q^{91} -66699.3 q^{92} +54595.6 q^{94} -16231.1 q^{95} +156410. q^{97} -56407.9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{2} + 181 q^{4} + 250 q^{5} + 116 q^{7} + 129 q^{8}+O(q^{10}) $ 10 * q + 3 * q^2 + 181 * q^4 + 250 * q^5 + 116 * q^7 + 129 * q^8	$ 10 q + 3 q^{2} + 181 q^{4} + 250 q^{5} + 116 q^{7} + 129 q^{8} + 75 q^{10} - 1210 q^{11} + 932 q^{13} - 1332 q^{14} + 2701 q^{16} + 96 q^{17} + 1664 q^{19} + 4525 q^{20} - 363 q^{22} + 6288 q^{23} + 6250 q^{25} + 13380 q^{26} + 13868 q^{28} + 11208 q^{29} + 9032 q^{31} + 9801 q^{32} + 14610 q^{34} + 2900 q^{35} + 21572 q^{37} + 15870 q^{38} + 3225 q^{40} + 10800 q^{41} + 21128 q^{43} - 21901 q^{44} + 83982 q^{46} - 17400 q^{47} + 71610 q^{49} + 1875 q^{50} + 40640 q^{52} + 5004 q^{53} - 30250 q^{55} - 54012 q^{56} - 9786 q^{58} - 25272 q^{59} + 52004 q^{61} + 34740 q^{62} + 56953 q^{64} + 23300 q^{65} + 4160 q^{67} - 87978 q^{68} - 33300 q^{70} - 65232 q^{71} + 44252 q^{73} - 49842 q^{74} + 233246 q^{76} - 14036 q^{77} + 112604 q^{79} + 67525 q^{80} + 167910 q^{82} + 70032 q^{83} + 2400 q^{85} - 72978 q^{86} - 15609 q^{88} - 46848 q^{89} + 130672 q^{91} + 121302 q^{92} + 252294 q^{94} + 41600 q^{95} + 129932 q^{97} - 316137 q^{98}+O(q^{100}) $ 10 * q + 3 * q^2 + 181 * q^4 + 250 * q^5 + 116 * q^7 + 129 * q^8 + 75 * q^10 - 1210 * q^11 + 932 * q^13 - 1332 * q^14 + 2701 * q^16 + 96 * q^17 + 1664 * q^19 + 4525 * q^20 - 363 * q^22 + 6288 * q^23 + 6250 * q^25 + 13380 * q^26 + 13868 * q^28 + 11208 * q^29 + 9032 * q^31 + 9801 * q^32 + 14610 * q^34 + 2900 * q^35 + 21572 * q^37 + 15870 * q^38 + 3225 * q^40 + 10800 * q^41 + 21128 * q^43 - 21901 * q^44 + 83982 * q^46 - 17400 * q^47 + 71610 * q^49 + 1875 * q^50 + 40640 * q^52 + 5004 * q^53 - 30250 * q^55 - 54012 * q^56 - 9786 * q^58 - 25272 * q^59 + 52004 * q^61 + 34740 * q^62 + 56953 * q^64 + 23300 * q^65 + 4160 * q^67 - 87978 * q^68 - 33300 * q^70 - 65232 * q^71 + 44252 * q^73 - 49842 * q^74 + 233246 * q^76 - 14036 * q^77 + 112604 * q^79 + 67525 * q^80 + 167910 * q^82 + 70032 * q^83 + 2400 * q^85 - 72978 * q^86 - 15609 * q^88 - 46848 * q^89 + 130672 * q^91 + 121302 * q^92 + 252294 * q^94 + 41600 * q^95 + 129932 * q^97 - 316137 * 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 $

$2$	−1.97152	−0.348518	−0.174259	−	0.984700i	$-0.555753\pi$
$2$	−1.97152	−0.348518	−0.174259	+	0.984700i	$0.555753\pi$
$3$	0	0
$3$	0	0
$4$	−28.1131	−0.878535
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	213.116	1.64388	0.821941	−	0.569572i	$-0.192891\pi$
$7$	213.116	1.64388	0.821941	+	0.569572i	$0.192891\pi$
$8$	118.514	0.654704
$9$	0	0
$10$	−49.2880	−0.155862
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	941.794	1.54560	0.772801	−	0.634648i	$-0.218855\pi$
$13$	941.794	1.54560	0.772801	+	0.634648i	$0.218855\pi$
$14$	−420.162	−0.572924
$15$	0	0
$16$	665.967	0.650358
$17$	1228.09	1.03064	0.515321	−	0.856998i	$-0.327673\pi$
$17$	1228.09	1.03064	0.515321	+	0.856998i	$0.327673\pi$
$18$	0	0
$19$	−649.243	−0.412594	−0.206297	−	0.978489i	$-0.566141\pi$
$19$	−649.243	−0.412594	−0.206297	+	0.978489i	$0.566141\pi$
$20$	−702.828	−0.392893
$21$	0	0
$22$	238.554	0.105082
$23$	2372.53	0.935175	0.467588	−	0.883947i	$-0.345123\pi$
$23$	2372.53	0.935175	0.467588	+	0.883947i	$0.345123\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−1856.76	−0.538671
$27$	0	0
$28$	−5991.35	−1.44421
$29$	5138.20	1.13453	0.567265	−	0.823536i	$-0.308002\pi$
$29$	5138.20	1.13453	0.567265	+	0.823536i	$0.308002\pi$
$30$	0	0
$31$	4673.09	0.873374	0.436687	−	0.899614i	$-0.356152\pi$
$31$	4673.09	0.873374	0.436687	+	0.899614i	$0.356152\pi$
$32$	−5105.42	−0.881366
$33$	0	0
$34$	−2421.20	−0.359197
$35$	5327.90	0.735167
$36$	0	0
$37$	−5345.80	−0.641961	−0.320980	−	0.947086i	$-0.604012\pi$
$37$	−5345.80	−0.641961	−0.320980	+	0.947086i	$0.604012\pi$
$38$	1279.99	0.143797
$39$	0	0
$40$	2962.85	0.292793
$41$	648.735	0.0602709	0.0301355	−	0.999546i	$-0.490406\pi$
$41$	648.735	0.0602709	0.0301355	+	0.999546i	$0.490406\pi$
$42$	0	0
$43$	−9140.04	−0.753836	−0.376918	−	0.926247i	$-0.623016\pi$
$43$	−9140.04	−0.753836	−0.376918	+	0.926247i	$0.623016\pi$
$44$	3401.69	0.264888
$45$	0	0
$46$	−4677.49	−0.325926
$47$	−27692.2	−1.82857	−0.914287	−	0.405068i	$-0.867248\pi$
$47$	−27692.2	−1.82857	−0.914287	+	0.405068i	$0.867248\pi$
$48$	0	0
$49$	28611.4	1.70235
$50$	−1232.20	−0.0697037
$51$	0	0
$52$	−26476.8	−1.35787
$53$	−23416.5	−1.14507	−0.572536	−	0.819879i	$-0.694040\pi$
$53$	−23416.5	−1.14507	−0.572536	+	0.819879i	$0.694040\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	25257.2	1.07626
$57$	0	0
$58$	−10130.0	−0.395404
$59$	−13615.6	−0.509223	−0.254612	−	0.967043i	$-0.581948\pi$
$59$	−13615.6	−0.509223	−0.254612	+	0.967043i	$0.581948\pi$
$60$	0	0
$61$	−7673.40	−0.264036	−0.132018	−	0.991247i	$-0.542146\pi$
$61$	−7673.40	−0.264036	−0.132018	+	0.991247i	$0.542146\pi$
$62$	−9213.08	−0.304387
$63$	0	0
$64$	−11245.5	−0.343186
$65$	23544.9	0.691214
$66$	0	0
$67$	8759.02	0.238379	0.119190	−	0.992871i	$-0.461970\pi$
$67$	8759.02	0.238379	0.119190	+	0.992871i	$0.461970\pi$
$68$	−34525.4	−0.905454
$69$	0	0
$70$	−10504.0	−0.256219
$71$	−10812.2	−0.254547	−0.127274	−	0.991868i	$-0.540623\pi$
$71$	−10812.2	−0.254547	−0.127274	+	0.991868i	$0.540623\pi$
$72$	0	0
$73$	65657.0	1.44203	0.721014	−	0.692921i	$-0.243676\pi$
$73$	65657.0	1.44203	0.721014	+	0.692921i	$0.243676\pi$
$74$	10539.3	0.223735
$75$	0	0
$76$	18252.2	0.362478
$77$	−25787.0	−0.495649
$78$	0	0
$79$	15899.7	0.286629	0.143315	−	0.989677i	$-0.454224\pi$
$79$	15899.7	0.286629	0.143315	+	0.989677i	$0.454224\pi$
$80$	16649.2	0.290849
$81$	0	0
$82$	−1278.99	−0.0210055
$83$	9330.13	0.148659	0.0743297	−	0.997234i	$-0.476318\pi$
$83$	9330.13	0.148659	0.0743297	+	0.997234i	$0.476318\pi$
$84$	0	0
$85$	30702.2	0.460917
$86$	18019.7	0.262726
$87$	0	0
$88$	−14340.2	−0.197401
$89$	92668.6	1.24010	0.620052	−	0.784561i	$-0.287112\pi$
$89$	92668.6	1.24010	0.620052	+	0.784561i	$0.287112\pi$
$90$	0	0
$91$	200711.	2.54079
$92$	−66699.3	−0.821584
$93$	0	0
$94$	54595.6	0.637291
$95$	−16231.1	−0.184518
$96$	0	0
$97$	156410.	1.68785	0.843927	−	0.536458i	$-0.180238\pi$
$97$	156410.	1.68785	0.843927	+	0.536458i	$0.180238\pi$
$98$	−56407.9	−0.593301
$99$	0	0
$100$	−17570.7	−0.175707
$101$	−72747.9	−0.709606	−0.354803	−	0.934941i	$-0.615452\pi$
$101$	−72747.9	−0.709606	−0.354803	+	0.934941i	$0.615452\pi$
$102$	0	0
$103$	183784.	1.70692	0.853461	−	0.521157i	$-0.174499\pi$
$103$	183784.	1.70692	0.853461	+	0.521157i	$0.174499\pi$
$104$	111616.	1.01191
$105$	0	0
$106$	46166.1	0.399079
$107$	−50408.3	−0.425641	−0.212820	−	0.977091i	$-0.568265\pi$
$107$	−50408.3	−0.425641	−0.212820	+	0.977091i	$0.568265\pi$
$108$	0	0
$109$	28506.1	0.229811	0.114906	−	0.993376i	$-0.463343\pi$
$109$	28506.1	0.229811	0.114906	+	0.993376i	$0.463343\pi$
$110$	5963.84	0.0469942
$111$	0	0
$112$	141928.	1.06911
$113$	−213346.	−1.57177	−0.785883	−	0.618375i	$-0.787791\pi$
$113$	−213346.	−1.57177	−0.785883	+	0.618375i	$0.787791\pi$
$114$	0	0
$115$	59313.4	0.418223
$116$	−144451.	−0.996724
$117$	0	0
$118$	26843.5	0.177474
$119$	261725.	1.69425
$120$	0	0
$121$	14641.0	0.0909091
$122$	15128.3	0.0920215
$123$	0	0
$124$	−131375.	−0.767289
$125$	15625.0	0.0894427
$126$	0	0
$127$	−181811.	−1.00026	−0.500129	−	0.865951i	$-0.666714\pi$
$127$	−181811.	−1.00026	−0.500129	+	0.865951i	$0.666714\pi$
$128$	185544.	1.00097
$129$	0	0
$130$	−46419.1	−0.240901
$131$	−201366.	−1.02520	−0.512600	−	0.858627i	$-0.671318\pi$
$131$	−201366.	−1.02520	−0.512600	+	0.858627i	$0.671318\pi$
$132$	0	0
$133$	−138364.	−0.678256
$134$	−17268.6	−0.0830796
$135$	0	0
$136$	145546.	0.674765
$137$	351321.	1.59920	0.799600	−	0.600533i	$-0.205045\pi$
$137$	351321.	1.59920	0.799600	+	0.600533i	$0.205045\pi$
$138$	0	0
$139$	−241689.	−1.06101	−0.530505	−	0.847682i	$-0.677998\pi$
$139$	−241689.	−1.06101	−0.530505	+	0.847682i	$0.677998\pi$
$140$	−149784.	−0.645870
$141$	0	0
$142$	21316.5	0.0887144
$143$	−113957.	−0.466017
$144$	0	0
$145$	128455.	0.507377
$146$	−129444.	−0.502573
$147$	0	0
$148$	150287.	0.563985
$149$	429711.	1.58566	0.792831	−	0.609441i	$-0.208606\pi$
$149$	429711.	1.58566	0.792831	+	0.609441i	$0.208606\pi$
$150$	0	0
$151$	−28846.5	−0.102956	−0.0514780	−	0.998674i	$-0.516393\pi$
$151$	−28846.5	−0.102956	−0.0514780	+	0.998674i	$0.516393\pi$
$152$	−76944.4	−0.270127
$153$	0	0
$154$	50839.6	0.172743
$155$	116827.	0.390585
$156$	0	0
$157$	−7907.42	−0.0256027	−0.0128014	−	0.999918i	$-0.504075\pi$
$157$	−7907.42	−0.0256027	−0.0128014	+	0.999918i	$0.504075\pi$
$158$	−31346.5	−0.0998957
$159$	0	0
$160$	−127635.	−0.394159
$161$	505625.	1.53732
$162$	0	0
$163$	111642.	0.329125	0.164562	−	0.986367i	$-0.447379\pi$
$163$	111642.	0.329125	0.164562	+	0.986367i	$0.447379\pi$
$164$	−18238.0	−0.0529501
$165$	0	0
$166$	−18394.5	−0.0518106
$167$	−410631.	−1.13936	−0.569680	−	0.821867i	$-0.692933\pi$
$167$	−410631.	−1.13936	−0.569680	+	0.821867i	$0.692933\pi$
$168$	0	0
$169$	515684.	1.38889
$170$	−60530.0	−0.160638
$171$	0	0
$172$	256955.	0.662271
$173$	−235594.	−0.598478	−0.299239	−	0.954178i	$-0.596733\pi$
$173$	−235594.	−0.598478	−0.299239	+	0.954178i	$0.596733\pi$
$174$	0	0
$175$	133197.	0.328777
$176$	−80582.0	−0.196090
$177$	0	0
$178$	−182698.	−0.432199
$179$	723484.	1.68770	0.843852	−	0.536576i	$-0.180283\pi$
$179$	723484.	1.68770	0.843852	+	0.536576i	$0.180283\pi$
$180$	0	0
$181$	537812.	1.22021	0.610104	−	0.792321i	$-0.291128\pi$
$181$	537812.	1.22021	0.610104	+	0.792321i	$0.291128\pi$
$182$	−395706.	−0.885512
$183$	0	0
$184$	281179.	0.612263
$185$	−133645.	−0.287094
$186$	0	0
$187$	−148599.	−0.310750
$188$	778513.	1.60647
$189$	0	0
$190$	31999.8	0.0643078
$191$	−304289.	−0.603536	−0.301768	−	0.953381i	$-0.597577\pi$
$191$	−304289.	−0.603536	−0.301768	+	0.953381i	$0.597577\pi$
$192$	0	0
$193$	594720.	1.14926	0.574631	−	0.818412i	$-0.305145\pi$
$193$	594720.	1.14926	0.574631	+	0.818412i	$0.305145\pi$
$194$	−308365.	−0.588248
$195$	0	0
$196$	−804356.	−1.49557
$197$	−892310.	−1.63814	−0.819068	−	0.573696i	$-0.805509\pi$
$197$	−892310.	−1.63814	−0.819068	+	0.573696i	$0.805509\pi$
$198$	0	0
$199$	823790.	1.47463	0.737316	−	0.675548i	$-0.236093\pi$
$199$	823790.	1.47463	0.737316	+	0.675548i	$0.236093\pi$
$200$	74071.3	0.130941
$201$	0	0
$202$	143424.	0.247311
$203$	1.09503e6	1.86503
$204$	0	0
$205$	16218.4	0.0269540
$206$	−362333.	−0.594894
$207$	0	0
$208$	627204.	1.00520
$209$	78558.4	0.124402
$210$	0	0
$211$	107251.	0.165842	0.0829211	−	0.996556i	$-0.473575\pi$
$211$	107251.	0.165842	0.0829211	+	0.996556i	$0.473575\pi$
$212$	658312.	1.00599
$213$	0	0
$214$	99381.0	0.148344
$215$	−228501.	−0.337126
$216$	0	0
$217$	995910.	1.43572
$218$	−56200.2	−0.0800934
$219$	0	0
$220$	85042.2	0.118462
$221$	1.15661e6	1.59296
$222$	0	0
$223$	510012.	0.686781	0.343391	−	0.939193i	$-0.388425\pi$
$223$	510012.	0.686781	0.343391	+	0.939193i	$0.388425\pi$
$224$	−1.08805e6	−1.44886
$225$	0	0
$226$	420615.	0.547789
$227$	−794682.	−1.02360	−0.511798	−	0.859106i	$-0.671020\pi$
$227$	−794682.	−1.02360	−0.511798	+	0.859106i	$0.671020\pi$
$228$	0	0
$229$	−748115.	−0.942713	−0.471357	−	0.881943i	$-0.656236\pi$
$229$	−748115.	−0.942713	−0.471357	+	0.881943i	$0.656236\pi$
$230$	−116937.	−0.145758
$231$	0	0
$232$	608949.	0.742781
$233$	698413.	0.842796	0.421398	−	0.906876i	$-0.361540\pi$
$233$	698413.	0.842796	0.421398	+	0.906876i	$0.361540\pi$
$234$	0	0
$235$	−692304.	−0.817763
$236$	382778.	0.447370
$237$	0	0
$238$	−515996.	−0.590478
$239$	−829121.	−0.938908	−0.469454	−	0.882957i	$-0.655549\pi$
$239$	−829121.	−0.938908	−0.469454	+	0.882957i	$0.655549\pi$
$240$	0	0
$241$	−1.14674e6	−1.27181	−0.635906	−	0.771766i	$-0.719374\pi$
$241$	−1.14674e6	−1.27181	−0.635906	+	0.771766i	$0.719374\pi$
$242$	−28865.0	−0.0316835
$243$	0	0
$244$	215723.	0.231965
$245$	715285.	0.761315
$246$	0	0
$247$	−611453.	−0.637706
$248$	553827.	0.571801
$249$	0	0
$250$	−30805.0	−0.0311724
$251$	−891736.	−0.893413	−0.446706	−	0.894681i	$-0.647403\pi$
$251$	−891736.	−0.893413	−0.446706	+	0.894681i	$0.647403\pi$
$252$	0	0
$253$	−287077.	−0.281966
$254$	358444.	0.348608
$255$	0	0
$256$	−5946.77	−0.00567128
$257$	−886431.	−0.837167	−0.418584	−	0.908178i	$-0.637473\pi$
$257$	−886431.	−0.837167	−0.418584	+	0.908178i	$0.637473\pi$
$258$	0	0
$259$	−1.13928e6	−1.05531
$260$	−661919.	−0.607256
$261$	0	0
$262$	396997.	0.357301
$263$	−792157.	−0.706190	−0.353095	−	0.935587i	$-0.614871\pi$
$263$	−792157.	−0.706190	−0.353095	+	0.935587i	$0.614871\pi$
$264$	0	0
$265$	−585413.	−0.512092
$266$	272787.	0.236385
$267$	0	0
$268$	−246243.	−0.209425
$269$	−1.52456e6	−1.28459	−0.642293	−	0.766459i	$-0.722017\pi$
$269$	−1.52456e6	−1.28459	−0.642293	+	0.766459i	$0.722017\pi$
$270$	0	0
$271$	2.39201e6	1.97852	0.989258	−	0.146180i	$-0.0466978\pi$
$271$	2.39201e6	1.97852	0.989258	+	0.146180i	$0.0466978\pi$
$272$	817867.	0.670286
$273$	0	0
$274$	−692636.	−0.557351
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	500540.	0.391958	0.195979	−	0.980608i	$-0.437212\pi$
$277$	500540.	0.391958	0.195979	+	0.980608i	$0.437212\pi$
$278$	476493.	0.369781
$279$	0	0
$280$	631431.	0.481317
$281$	1.24897e6	0.943593	0.471797	−	0.881707i	$-0.343606\pi$
$281$	1.24897e6	0.943593	0.471797	+	0.881707i	$0.343606\pi$
$282$	0	0
$283$	1.31259e6	0.974230	0.487115	−	0.873338i	$-0.338049\pi$
$283$	1.31259e6	0.974230	0.487115	+	0.873338i	$0.338049\pi$
$284$	303965.	0.223629
$285$	0	0
$286$	224669.	0.162415
$287$	138256.	0.0990784
$288$	0	0
$289$	88344.8	0.0622209
$290$	−253251.	−0.176830
$291$	0	0
$292$	−1.84582e6	−1.26687
$293$	2.39294e6	1.62841	0.814204	−	0.580578i	$-0.197173\pi$
$293$	2.39294e6	1.62841	0.814204	+	0.580578i	$0.197173\pi$
$294$	0	0
$295$	−340391.	−0.227731
$296$	−633553.	−0.420294
$297$	0	0
$298$	−847183.	−0.552633
$299$	2.23444e6	1.44541
$300$	0	0
$301$	−1.94789e6	−1.23922
$302$	56871.5	0.0358820
$303$	0	0
$304$	−432374.	−0.268334
$305$	−191835.	−0.118081
$306$	0	0
$307$	−60908.9	−0.0368838	−0.0184419	−	0.999830i	$-0.505871\pi$
$307$	−60908.9	−0.0368838	−0.0184419	+	0.999830i	$0.505871\pi$
$308$	724954.	0.435445
$309$	0	0
$310$	−230327.	−0.136126
$311$	2.10266e6	1.23273	0.616366	−	0.787460i	$-0.288604\pi$
$311$	2.10266e6	1.23273	0.616366	+	0.787460i	$0.288604\pi$
$312$	0	0
$313$	−1.46355e6	−0.844398	−0.422199	−	0.906503i	$-0.638742\pi$
$313$	−1.46355e6	−0.844398	−0.422199	+	0.906503i	$0.638742\pi$
$314$	15589.6	0.00892302
$315$	0	0
$316$	−446990.	−0.251814
$317$	−802172.	−0.448352	−0.224176	−	0.974549i	$-0.571969\pi$
$317$	−802172.	−0.448352	−0.224176	+	0.974549i	$0.571969\pi$
$318$	0	0
$319$	−621722.	−0.342073
$320$	−281138.	−0.153478
$321$	0	0
$322$	−996849.	−0.535784
$323$	−797328.	−0.425236
$324$	0	0
$325$	588622.	0.309120
$326$	−220105.	−0.114706
$327$	0	0
$328$	76884.3	0.0394596
$329$	−5.90164e6	−3.00596
$330$	0	0
$331$	1.03390e6	0.518692	0.259346	−	0.965784i	$-0.416493\pi$
$331$	1.03390e6	0.518692	0.259346	+	0.965784i	$0.416493\pi$
$332$	−262299.	−0.130603
$333$	0	0
$334$	809567.	0.397088
$335$	218976.	0.106607
$336$	0	0
$337$	−3.55089e6	−1.70319	−0.851594	−	0.524203i	$-0.824363\pi$
$337$	−3.55089e6	−1.70319	−0.851594	+	0.524203i	$0.824363\pi$
$338$	−1.01668e6	−0.484053
$339$	0	0
$340$	−863135.	−0.404931
$341$	−565444.	−0.263332
$342$	0	0
$343$	2.51571e6	1.15458
$344$	−1.08322e6	−0.493539
$345$	0	0
$346$	464477.	0.208581
$347$	−2.63585e6	−1.17516	−0.587580	−	0.809166i	$-0.699919\pi$
$347$	−2.63585e6	−1.17516	−0.587580	+	0.809166i	$0.699919\pi$
$348$	0	0
$349$	−1.80180e6	−0.791850	−0.395925	−	0.918283i	$-0.629576\pi$
$349$	−1.80180e6	−0.791850	−0.395925	+	0.918283i	$0.629576\pi$
$350$	−262601.	−0.114585
$351$	0	0
$352$	617755.	0.265742
$353$	218218.	0.0932080	0.0466040	−	0.998913i	$-0.485160\pi$
$353$	218218.	0.0932080	0.0466040	+	0.998913i	$0.485160\pi$
$354$	0	0
$355$	−270305.	−0.113837
$356$	−2.60520e6	−1.08947
$357$	0	0
$358$	−1.42636e6	−0.588196
$359$	595543.	0.243881	0.121940	−	0.992537i	$-0.461088\pi$
$359$	595543.	0.243881	0.121940	+	0.992537i	$0.461088\pi$
$360$	0	0
$361$	−2.05458e6	−0.829766
$362$	−1.06031e6	−0.425265
$363$	0	0
$364$	−5.64262e6	−2.23217
$365$	1.64142e6	0.644894
$366$	0	0
$367$	4.03997e6	1.56572	0.782858	−	0.622200i	$-0.213761\pi$
$367$	4.03997e6	1.56572	0.782858	+	0.622200i	$0.213761\pi$
$368$	1.58003e6	0.608199
$369$	0	0
$370$	263484.	0.100057
$371$	−4.99044e6	−1.88236
$372$	0	0
$373$	−1.19455e6	−0.444561	−0.222280	−	0.974983i	$-0.571350\pi$
$373$	−1.19455e6	−0.444561	−0.222280	+	0.974983i	$0.571350\pi$
$374$	292965.	0.108302
$375$	0	0
$376$	−3.28191e6	−1.19717
$377$	4.83913e6	1.75353
$378$	0	0
$379$	−2.51994e6	−0.901141	−0.450570	−	0.892741i	$-0.648779\pi$
$379$	−2.51994e6	−0.901141	−0.450570	+	0.892741i	$0.648779\pi$
$380$	456306.	0.162105
$381$	0	0
$382$	599912.	0.210344
$383$	3.13792e6	1.09306	0.546531	−	0.837439i	$-0.315948\pi$
$383$	3.13792e6	1.09306	0.546531	+	0.837439i	$0.315948\pi$
$384$	0	0
$385$	−644676.	−0.221661
$386$	−1.17250e6	−0.400539
$387$	0	0
$388$	−4.39717e6	−1.48284
$389$	4.31716e6	1.44652	0.723259	−	0.690576i	$-0.242643\pi$
$389$	4.31716e6	1.44652	0.723259	+	0.690576i	$0.242643\pi$
$390$	0	0
$391$	2.91368e6	0.963830
$392$	3.39086e6	1.11454
$393$	0	0
$394$	1.75920e6	0.570921
$395$	397492.	0.128185
$396$	0	0
$397$	−1.26065e6	−0.401438	−0.200719	−	0.979649i	$-0.564328\pi$
$397$	−1.26065e6	−0.401438	−0.200719	+	0.979649i	$0.564328\pi$
$398$	−1.62412e6	−0.513937
$399$	0	0
$400$	416229.	0.130072
$401$	−4.76332e6	−1.47927	−0.739637	−	0.673006i	$-0.765003\pi$
$401$	−4.76332e6	−1.47927	−0.739637	+	0.673006i	$0.765003\pi$
$402$	0	0
$403$	4.40109e6	1.34989
$404$	2.04517e6	0.623414
$405$	0	0
$406$	−2.15887e6	−0.649999
$407$	646842.	0.193558
$408$	0	0
$409$	−856253.	−0.253101	−0.126550	−	0.991960i	$-0.540391\pi$
$409$	−856253.	−0.253101	−0.126550	+	0.991960i	$0.540391\pi$
$410$	−31974.8	−0.00939396
$411$	0	0
$412$	−5.16673e6	−1.49959
$413$	−2.90171e6	−0.837103
$414$	0	0
$415$	233253.	0.0664825
$416$	−4.80825e6	−1.36224
$417$	0	0
$418$	−154879.	−0.0433563
$419$	1.49598e6	0.416286	0.208143	−	0.978098i	$-0.433258\pi$
$419$	1.49598e6	0.416286	0.208143	+	0.978098i	$0.433258\pi$
$420$	0	0
$421$	1.67483e6	0.460537	0.230269	−	0.973127i	$-0.426040\pi$
$421$	1.67483e6	0.460537	0.230269	+	0.973127i	$0.426040\pi$
$422$	−211447.	−0.0577990
$423$	0	0
$424$	−2.77519e6	−0.749684
$425$	767555.	0.206128
$426$	0	0
$427$	−1.63532e6	−0.434045
$428$	1.41714e6	0.373940
$429$	0	0
$430$	450494.	0.117495
$431$	−810701.	−0.210217	−0.105108	−	0.994461i	$-0.533519\pi$
$431$	−810701.	−0.210217	−0.105108	+	0.994461i	$0.533519\pi$
$432$	0	0
$433$	−3.57278e6	−0.915771	−0.457886	−	0.889011i	$-0.651393\pi$
$433$	−3.57278e6	−0.915771	−0.457886	+	0.889011i	$0.651393\pi$
$434$	−1.96346e6	−0.500376
$435$	0	0
$436$	−801394.	−0.201897
$437$	−1.54035e6	−0.385848
$438$	0	0
$439$	1.55659e6	0.385489	0.192745	−	0.981249i	$-0.438261\pi$
$439$	1.55659e6	0.385489	0.192745	+	0.981249i	$0.438261\pi$
$440$	−358505.	−0.0882803
$441$	0	0
$442$	−2.28027e6	−0.555176
$443$	1.50549e6	0.364475	0.182238	−	0.983255i	$-0.441666\pi$
$443$	1.50549e6	0.364475	0.182238	+	0.983255i	$0.441666\pi$
$444$	0	0
$445$	2.31672e6	0.554591
$446$	−1.00550e6	−0.239356
$447$	0	0
$448$	−2.39660e6	−0.564158
$449$	3.75288e6	0.878515	0.439258	−	0.898361i	$-0.355242\pi$
$449$	3.75288e6	0.878515	0.439258	+	0.898361i	$0.355242\pi$
$450$	0	0
$451$	−78497.0	−0.0181724
$452$	5.99781e6	1.38085
$453$	0	0
$454$	1.56673e6	0.356742
$455$	5.01779e6	1.13628
$456$	0	0
$457$	5.49561e6	1.23091	0.615454	−	0.788173i	$-0.288973\pi$
$457$	5.49561e6	1.23091	0.615454	+	0.788173i	$0.288973\pi$
$458$	1.47492e6	0.328553
$459$	0	0
$460$	−1.66748e6	−0.367424
$461$	3.65677e6	0.801393	0.400697	−	0.916211i	$-0.368768\pi$
$461$	3.65677e6	0.801393	0.400697	+	0.916211i	$0.368768\pi$
$462$	0	0
$463$	−1.66187e6	−0.360283	−0.180142	−	0.983641i	$-0.557656\pi$
$463$	−1.66187e6	−0.360283	−0.180142	+	0.983641i	$0.557656\pi$
$464$	3.42187e6	0.737851
$465$	0	0
$466$	−1.37693e6	−0.293730
$467$	−3.58971e6	−0.761671	−0.380835	−	0.924643i	$-0.624364\pi$
$467$	−3.58971e6	−0.761671	−0.380835	+	0.924643i	$0.624364\pi$
$468$	0	0
$469$	1.86669e6	0.391868
$470$	1.36489e6	0.285005
$471$	0	0
$472$	−1.61365e6	−0.333390
$473$	1.10594e6	0.227290
$474$	0	0
$475$	−405777.	−0.0825188
$476$	−7.35791e6	−1.48846
$477$	0	0
$478$	1.63463e6	0.327227
$479$	−6.84466e6	−1.36305	−0.681527	−	0.731793i	$-0.738684\pi$
$479$	−6.84466e6	−1.36305	−0.681527	+	0.731793i	$0.738684\pi$
$480$	0	0
$481$	−5.03464e6	−0.992216
$482$	2.26082e6	0.443250
$483$	0	0
$484$	−411604.	−0.0798668
$485$	3.91025e6	0.754831
$486$	0	0
$487$	−6.17722e6	−1.18024	−0.590120	−	0.807315i	$-0.700920\pi$
$487$	−6.17722e6	−1.18024	−0.590120	+	0.807315i	$0.700920\pi$
$488$	−909406.	−0.172866
$489$	0	0
$490$	−1.41020e6	−0.265332
$491$	4.97445e6	0.931196	0.465598	−	0.884996i	$-0.345839\pi$
$491$	4.97445e6	0.931196	0.465598	+	0.884996i	$0.345839\pi$
$492$	0	0
$493$	6.31016e6	1.16929
$494$	1.20549e6	0.222252
$495$	0	0
$496$	3.11213e6	0.568006
$497$	−2.30425e6	−0.418446
$498$	0	0
$499$	4.64342e6	0.834808	0.417404	−	0.908721i	$-0.362940\pi$
$499$	4.64342e6	0.834808	0.417404	+	0.908721i	$0.362940\pi$
$500$	−439267.	−0.0785785
$501$	0	0
$502$	1.75807e6	0.311371
$503$	1.63001e6	0.287257	0.143628	−	0.989632i	$-0.454123\pi$
$503$	1.63001e6	0.287257	0.143628	+	0.989632i	$0.454123\pi$
$504$	0	0
$505$	−1.81870e6	−0.317345
$506$	565977.	0.0982703
$507$	0	0
$508$	5.11128e6	0.878761
$509$	−1.04687e7	−1.79100	−0.895502	−	0.445058i	$-0.853183\pi$
$509$	−1.04687e7	−1.79100	−0.895502	+	0.445058i	$0.853183\pi$
$510$	0	0
$511$	1.39925e7	2.37052
$512$	−5.92569e6	−0.998996
$513$	0	0
$514$	1.74761e6	0.291768
$515$	4.59459e6	0.763359
$516$	0	0
$517$	3.35075e6	0.551336
$518$	2.24610e6	0.367794
$519$	0	0
$520$	2.79040e6	0.452541
$521$	5.58267e6	0.901048	0.450524	−	0.892764i	$-0.351237\pi$
$521$	5.58267e6	0.901048	0.450524	+	0.892764i	$0.351237\pi$
$522$	0	0
$523$	6.94785e6	1.11070	0.555349	−	0.831617i	$-0.312585\pi$
$523$	6.94785e6	1.11070	0.555349	+	0.831617i	$0.312585\pi$
$524$	5.66104e6	0.900674
$525$	0	0
$526$	1.56175e6	0.246120
$527$	5.73897e6	0.900135
$528$	0	0
$529$	−807423.	−0.125448
$530$	1.15415e6	0.178473
$531$	0	0
$532$	3.88984e6	0.595872
$533$	610975.	0.0931549
$534$	0	0
$535$	−1.26021e6	−0.190352
$536$	1.03807e6	0.156068
$537$	0	0
$538$	3.00569e6	0.447702
$539$	−3.46198e6	−0.513278
$540$	0	0
$541$	8.09420e6	1.18900	0.594499	−	0.804097i	$-0.297351\pi$
$541$	8.09420e6	1.18900	0.594499	+	0.804097i	$0.297351\pi$
$542$	−4.71589e6	−0.689549
$543$	0	0
$544$	−6.26990e6	−0.908372
$545$	712651.	0.102775
$546$	0	0
$547$	1.50039e6	0.214406	0.107203	−	0.994237i	$-0.465810\pi$
$547$	1.50039e6	0.214406	0.107203	+	0.994237i	$0.465810\pi$
$548$	−9.87673e6	−1.40495
$549$	0	0
$550$	149096.	0.0210165
$551$	−3.33594e6	−0.468100
$552$	0	0
$553$	3.38848e6	0.471185
$554$	−986823.	−0.136604
$555$	0	0
$556$	6.79462e6	0.932134
$557$	−7.46926e6	−1.02009	−0.510046	−	0.860147i	$-0.670372\pi$
$557$	−7.46926e6	−1.02009	−0.510046	+	0.860147i	$0.670372\pi$
$558$	0	0
$559$	−8.60804e6	−1.16513
$560$	3.54821e6	0.478122
$561$	0	0
$562$	−2.46236e6	−0.328860
$563$	3.85619e6	0.512729	0.256364	−	0.966580i	$-0.417475\pi$
$563$	3.85619e6	0.512729	0.256364	+	0.966580i	$0.417475\pi$
$564$	0	0
$565$	−5.33364e6	−0.702915
$566$	−2.58779e6	−0.339537
$567$	0	0
$568$	−1.28140e6	−0.166653
$569$	6.66587e6	0.863130	0.431565	−	0.902082i	$-0.357962\pi$
$569$	6.66587e6	0.863130	0.431565	+	0.902082i	$0.357962\pi$
$570$	0	0
$571$	−1.54310e7	−1.98063	−0.990315	−	0.138840i	$-0.955663\pi$
$571$	−1.54310e7	−1.98063	−0.990315	+	0.138840i	$0.955663\pi$
$572$	3.20369e6	0.409412
$573$	0	0
$574$	−272574.	−0.0345306
$575$	1.48283e6	0.187035
$576$	0	0
$577$	−3.51132e6	−0.439067	−0.219533	−	0.975605i	$-0.570454\pi$
$577$	−3.51132e6	−0.439067	−0.219533	+	0.975605i	$0.570454\pi$
$578$	−174173.	−0.0216851
$579$	0	0
$580$	−3.61127e6	−0.445748
$581$	1.98840e6	0.244379
$582$	0	0
$583$	2.83340e6	0.345252
$584$	7.78127e6	0.944101
$585$	0	0
$586$	−4.71773e6	−0.567531
$587$	−7.55149e6	−0.904560	−0.452280	−	0.891876i	$-0.649389\pi$
$587$	−7.55149e6	−0.904560	−0.452280	+	0.891876i	$0.649389\pi$
$588$	0	0
$589$	−3.03397e6	−0.360349
$590$	671087.	0.0793686
$591$	0	0
$592$	−3.56013e6	−0.417505
$593$	1.51673e7	1.77122	0.885609	−	0.464432i	$-0.153741\pi$
$593$	1.51673e7	1.77122	0.885609	+	0.464432i	$0.153741\pi$
$594$	0	0
$595$	6.54313e6	0.757693
$596$	−1.20805e7	−1.39306
$597$	0	0
$598$	−4.40524e6	−0.503752
$599$	−1.25331e7	−1.42722	−0.713611	−	0.700542i	$-0.752942\pi$
$599$	−1.25331e7	−1.42722	−0.713611	+	0.700542i	$0.752942\pi$
$600$	0	0
$601$	−1.07520e7	−1.21423	−0.607117	−	0.794612i	$-0.707674\pi$
$601$	−1.07520e7	−1.21423	−0.607117	+	0.794612i	$0.707674\pi$
$602$	3.84030e6	0.431890
$603$	0	0
$604$	810966.	0.0904504
$605$	366025.	0.0406558
$606$	0	0
$607$	1.48918e7	1.64049	0.820246	−	0.572011i	$-0.193836\pi$
$607$	1.48918e7	1.64049	0.820246	+	0.572011i	$0.193836\pi$
$608$	3.31465e6	0.363646
$609$	0	0
$610$	378206.	0.0411533
$611$	−2.60803e7	−2.82625
$612$	0	0
$613$	−7.31044e6	−0.785764	−0.392882	−	0.919589i	$-0.628522\pi$
$613$	−7.31044e6	−0.785764	−0.392882	+	0.919589i	$0.628522\pi$
$614$	120083.	0.0128547
$615$	0	0
$616$	−3.05613e6	−0.324504
$617$	−1.23461e7	−1.30562	−0.652810	−	0.757521i	$-0.726410\pi$
$617$	−1.23461e7	−1.30562	−0.652810	+	0.757521i	$0.726410\pi$
$618$	0	0
$619$	9.02622e6	0.946846	0.473423	−	0.880835i	$-0.343018\pi$
$619$	9.02622e6	0.946846	0.473423	+	0.880835i	$0.343018\pi$
$620$	−3.28438e6	−0.343142
$621$	0	0
$622$	−4.14544e6	−0.429630
$623$	1.97492e7	2.03858
$624$	0	0
$625$	390625.	0.0400000
$626$	2.88542e6	0.294288
$627$	0	0
$628$	222302.	0.0224929
$629$	−6.56512e6	−0.661631
$630$	0	0
$631$	−1.71741e6	−0.171712	−0.0858562	−	0.996308i	$-0.527363\pi$
$631$	−1.71741e6	−0.171712	−0.0858562	+	0.996308i	$0.527363\pi$
$632$	1.88434e6	0.187657
$633$	0	0
$634$	1.58150e6	0.156259
$635$	−4.54528e6	−0.447329
$636$	0	0
$637$	2.69461e7	2.63116
$638$	1.22574e6	0.119219
$639$	0	0
$640$	4.63860e6	0.447649
$641$	8.87450e6	0.853098	0.426549	−	0.904465i	$-0.359729\pi$
$641$	8.87450e6	0.853098	0.426549	+	0.904465i	$0.359729\pi$
$642$	0	0
$643$	−1.52538e7	−1.45496	−0.727478	−	0.686131i	$-0.759308\pi$
$643$	−1.52538e7	−1.45496	−0.727478	+	0.686131i	$0.759308\pi$
$644$	−1.42147e7	−1.35059
$645$	0	0
$646$	1.57195e6	0.148203
$647$	1.19504e7	1.12233	0.561167	−	0.827703i	$-0.310353\pi$
$647$	1.19504e7	1.12233	0.561167	+	0.827703i	$0.310353\pi$
$648$	0	0
$649$	1.64749e6	0.153537
$650$	−1.16048e6	−0.107734
$651$	0	0
$652$	−3.13862e6	−0.289148
$653$	1.29575e7	1.18916	0.594579	−	0.804037i	$-0.297319\pi$
$653$	1.29575e7	1.18916	0.594579	+	0.804037i	$0.297319\pi$
$654$	0	0
$655$	−5.03416e6	−0.458484
$656$	432036.	0.0391977
$657$	0	0
$658$	1.16352e7	1.04763
$659$	6.43535e6	0.577243	0.288622	−	0.957443i	$-0.406803\pi$
$659$	6.43535e6	0.577243	0.288622	+	0.957443i	$0.406803\pi$
$660$	0	0
$661$	7.44870e6	0.663097	0.331548	−	0.943438i	$-0.392429\pi$
$661$	7.44870e6	0.663097	0.331548	+	0.943438i	$0.392429\pi$
$662$	−2.03836e6	−0.180774
$663$	0	0
$664$	1.10575e6	0.0973279
$665$	−3.45910e6	−0.303326
$666$	0	0
$667$	1.21905e7	1.06098
$668$	1.15441e7	1.00097
$669$	0	0
$670$	−431714.	−0.0371543
$671$	928482.	0.0796099
$672$	0	0
$673$	2.07295e6	0.176422	0.0882109	−	0.996102i	$-0.471885\pi$
$673$	2.07295e6	0.176422	0.0882109	+	0.996102i	$0.471885\pi$
$674$	7.00064e6	0.593592
$675$	0	0
$676$	−1.44975e7	−1.22019
$677$	3.91189e6	0.328031	0.164016	−	0.986458i	$-0.447555\pi$
$677$	3.91189e6	0.328031	0.164016	+	0.986458i	$0.447555\pi$
$678$	0	0
$679$	3.33335e7	2.77464
$680$	3.63865e6	0.301764
$681$	0	0
$682$	1.11478e6	0.0917761
$683$	1.88835e6	0.154893	0.0774463	−	0.996997i	$-0.475323\pi$
$683$	1.88835e6	0.154893	0.0774463	+	0.996997i	$0.475323\pi$
$684$	0	0
$685$	8.78303e6	0.715184
$686$	−4.95977e6	−0.402393
$687$	0	0
$688$	−6.08696e6	−0.490264
$689$	−2.20536e7	−1.76983
$690$	0	0
$691$	1.92159e7	1.53097	0.765483	−	0.643456i	$-0.222500\pi$
$691$	1.92159e7	1.53097	0.765483	+	0.643456i	$0.222500\pi$
$692$	6.62327e6	0.525784
$693$	0	0
$694$	5.19662e6	0.409565
$695$	−6.04222e6	−0.474498
$696$	0	0
$697$	796705.	0.0621177
$698$	3.55228e6	0.275974
$699$	0	0
$700$	−3.74460e6	−0.288842
$701$	−1.99106e7	−1.53035	−0.765173	−	0.643825i	$-0.777346\pi$
$701$	−1.99106e7	−1.53035	−0.765173	+	0.643825i	$0.777346\pi$
$702$	0	0
$703$	3.47072e6	0.264869
$704$	1.36071e6	0.103475
$705$	0	0
$706$	−430220.	−0.0324847
$707$	−1.55037e7	−1.16651
$708$	0	0
$709$	1.85755e7	1.38780	0.693898	−	0.720073i	$-0.255892\pi$
$709$	1.85755e7	1.38780	0.693898	+	0.720073i	$0.255892\pi$
$710$	532912.	0.0396743
$711$	0	0
$712$	1.09825e7	0.811900
$713$	1.10871e7	0.816757
$714$	0	0
$715$	−2.84893e6	−0.208409
$716$	−2.03394e7	−1.48271
$717$	0	0
$718$	−1.17412e6	−0.0849969
$719$	6.46878e6	0.466659	0.233330	−	0.972398i	$-0.425038\pi$
$719$	6.46878e6	0.466659	0.233330	+	0.972398i	$0.425038\pi$
$720$	0	0
$721$	3.91672e7	2.80598
$722$	4.05065e6	0.289189
$723$	0	0
$724$	−1.51196e7	−1.07200
$725$	3.21137e6	0.226906
$726$	0	0
$727$	−1.92281e7	−1.34927	−0.674636	−	0.738151i	$-0.735699\pi$
$727$	−1.92281e7	−1.34927	−0.674636	+	0.738151i	$0.735699\pi$
$728$	2.37871e7	1.66347
$729$	0	0
$730$	−3.23610e6	−0.224758
$731$	−1.12248e7	−0.776934
$732$	0	0
$733$	2.74521e7	1.88719	0.943594	−	0.331105i	$-0.107422\pi$
$733$	2.74521e7	1.88719	0.943594	+	0.331105i	$0.107422\pi$
$734$	−7.96487e6	−0.545681
$735$	0	0
$736$	−1.21128e7	−0.824232
$737$	−1.05984e6	−0.0718741
$738$	0	0
$739$	−9.04336e6	−0.609142	−0.304571	−	0.952490i	$-0.598513\pi$
$739$	−9.04336e6	−0.609142	−0.304571	+	0.952490i	$0.598513\pi$
$740$	3.75718e6	0.252222
$741$	0	0
$742$	9.83874e6	0.656039
$743$	9.47688e6	0.629786	0.314893	−	0.949127i	$-0.398031\pi$
$743$	9.47688e6	0.629786	0.314893	+	0.949127i	$0.398031\pi$
$744$	0	0
$745$	1.07428e7	0.709130
$746$	2.35507e6	0.154938
$747$	0	0
$748$	4.17757e6	0.273005
$749$	−1.07428e7	−0.699703
$750$	0	0
$751$	2.01407e7	1.30309	0.651545	−	0.758610i	$-0.274121\pi$
$751$	2.01407e7	1.30309	0.651545	+	0.758610i	$0.274121\pi$
$752$	−1.84421e7	−1.18923
$753$	0	0
$754$	−9.54042e6	−0.611138
$755$	−721163.	−0.0460433
$756$	0	0
$757$	2.89933e6	0.183890	0.0919448	−	0.995764i	$-0.470692\pi$
$757$	2.89933e6	0.183890	0.0919448	+	0.995764i	$0.470692\pi$
$758$	4.96811e6	0.314064
$759$	0	0
$760$	−1.92361e6	−0.120804
$761$	1.45187e7	0.908798	0.454399	−	0.890798i	$-0.349854\pi$
$761$	1.45187e7	0.908798	0.454399	+	0.890798i	$0.349854\pi$
$762$	0	0
$763$	6.07510e6	0.377782
$764$	8.55453e6	0.530228
$765$	0	0
$766$	−6.18646e6	−0.380952
$767$	−1.28231e7	−0.787056
$768$	0	0
$769$	1.71918e7	1.04835	0.524175	−	0.851610i	$-0.324374\pi$
$769$	1.71918e7	1.04835	0.524175	+	0.851610i	$0.324374\pi$
$770$	1.27099e6	0.0772530
$771$	0	0
$772$	−1.67194e7	−1.00967
$773$	1.49463e7	0.899675	0.449838	−	0.893110i	$-0.351482\pi$
$773$	1.49463e7	0.899675	0.449838	+	0.893110i	$0.351482\pi$
$774$	0	0
$775$	2.92068e6	0.174675
$776$	1.85368e7	1.10505
$777$	0	0
$778$	−8.51136e6	−0.504138
$779$	−421187.	−0.0248674
$780$	0	0
$781$	1.30828e6	0.0767489
$782$	−5.74438e6	−0.335912
$783$	0	0
$784$	1.90543e7	1.10714
$785$	−197686.	−0.0114499
$786$	0	0
$787$	−2.81222e7	−1.61850	−0.809249	−	0.587466i	$-0.800126\pi$
$787$	−2.81222e7	−1.61850	−0.809249	+	0.587466i	$0.800126\pi$
$788$	2.50856e7	1.43916
$789$	0	0
$790$	−783663.	−0.0446747
$791$	−4.54674e7	−2.58380
$792$	0	0
$793$	−7.22677e6	−0.408095
$794$	2.48540e6	0.139909
$795$	0	0
$796$	−2.31593e7	−1.29552
$797$	−3.03756e7	−1.69387	−0.846933	−	0.531699i	$-0.821554\pi$
$797$	−3.03756e7	−1.69387	−0.846933	+	0.531699i	$0.821554\pi$
$798$	0	0
$799$	−3.40084e7	−1.88460
$800$	−3.19089e6	−0.176273
$801$	0	0
$802$	9.39097e6	0.515554
$803$	−7.94449e6	−0.434788
$804$	0	0
$805$	1.26406e7	0.687510
$806$	−8.67683e6	−0.470461
$807$	0	0
$808$	−8.62166e6	−0.464582
$809$	2.13456e6	0.114667	0.0573334	−	0.998355i	$-0.481740\pi$
$809$	2.13456e6	0.114667	0.0573334	+	0.998355i	$0.481740\pi$
$810$	0	0
$811$	−2.04424e7	−1.09139	−0.545695	−	0.837984i	$-0.683734\pi$
$811$	−2.04424e7	−1.09139	−0.545695	+	0.837984i	$0.683734\pi$
$812$	−3.07848e7	−1.63850
$813$	0	0
$814$	−1.27526e6	−0.0674587
$815$	2.79106e6	0.147189
$816$	0	0
$817$	5.93410e6	0.311028
$818$	1.68812e6	0.0882103
$819$	0	0
$820$	−455949.	−0.0236800
$821$	8.61986e6	0.446316	0.223158	−	0.974782i	$-0.428363\pi$
$821$	8.61986e6	0.446316	0.223158	+	0.974782i	$0.428363\pi$
$822$	0	0
$823$	−9.07362e6	−0.466962	−0.233481	−	0.972361i	$-0.575012\pi$
$823$	−9.07362e6	−0.466962	−0.233481	+	0.972361i	$0.575012\pi$
$824$	2.17809e7	1.11753
$825$	0	0
$826$	5.72077e6	0.291746
$827$	3.37873e7	1.71787	0.858933	−	0.512088i	$-0.171128\pi$
$827$	3.37873e7	1.71787	0.858933	+	0.512088i	$0.171128\pi$
$828$	0	0
$829$	2.97193e7	1.50194	0.750970	−	0.660336i	$-0.229586\pi$
$829$	2.97193e7	1.50194	0.750970	+	0.660336i	$0.229586\pi$
$830$	−459863.	−0.0231704
$831$	0	0
$832$	−1.05910e7	−0.530429
$833$	3.51374e7	1.75451
$834$	0	0
$835$	−1.02658e7	−0.509537
$836$	−2.20852e6	−0.109291
$837$	0	0
$838$	−2.94936e6	−0.145083
$839$	1.83058e6	0.0897807	0.0448904	−	0.998992i	$-0.485706\pi$
$839$	1.83058e6	0.0897807	0.0448904	+	0.998992i	$0.485706\pi$
$840$	0	0
$841$	5.88991e6	0.287157
$842$	−3.30195e6	−0.160506
$843$	0	0
$844$	−3.01516e6	−0.145698
$845$	1.28921e7	0.621129
$846$	0	0
$847$	3.12023e6	0.149444
$848$	−1.55946e7	−0.744708
$849$	0	0
$850$	−1.51325e6	−0.0718395
$851$	−1.26831e7	−0.600346
$852$	0	0
$853$	3.89435e7	1.83258	0.916290	−	0.400516i	$-0.131169\pi$
$853$	3.89435e7	1.83258	0.916290	+	0.400516i	$0.131169\pi$
$854$	3.22407e6	0.151273
$855$	0	0
$856$	−5.97410e6	−0.278669
$857$	1.25624e7	0.584278	0.292139	−	0.956376i	$-0.405633\pi$
$857$	1.25624e7	0.584278	0.292139	+	0.956376i	$0.405633\pi$
$858$	0	0
$859$	−2.24542e7	−1.03828	−0.519139	−	0.854690i	$-0.673747\pi$
$859$	−2.24542e7	−1.03828	−0.519139	+	0.854690i	$0.673747\pi$
$860$	6.42387e6	0.296177
$861$	0	0
$862$	1.59831e6	0.0732645
$863$	1.04465e7	0.477470	0.238735	−	0.971085i	$-0.423267\pi$
$863$	1.04465e7	0.477470	0.238735	+	0.971085i	$0.423267\pi$
$864$	0	0
$865$	−5.88984e6	−0.267648
$866$	7.04381e6	0.319163
$867$	0	0
$868$	−2.79981e7	−1.26133
$869$	−1.92386e6	−0.0864220
$870$	0	0
$871$	8.24920e6	0.368440
$872$	3.37837e6	0.150458
$873$	0	0
$874$	3.03683e6	0.134475
$875$	3.32994e6	0.147033
$876$	0	0
$877$	−3.37236e7	−1.48059	−0.740294	−	0.672283i	$-0.765314\pi$
$877$	−3.37236e7	−1.48059	−0.740294	+	0.672283i	$0.765314\pi$
$878$	−3.06884e6	−0.134350
$879$	0	0
$880$	−2.01455e6	−0.0876943
$881$	−3.80226e7	−1.65045	−0.825225	−	0.564804i	$-0.808952\pi$
$881$	−3.80226e7	−1.65045	−0.825225	+	0.564804i	$0.808952\pi$
$882$	0	0
$883$	−2.27127e7	−0.980319	−0.490160	−	0.871633i	$-0.663061\pi$
$883$	−2.27127e7	−0.980319	−0.490160	+	0.871633i	$0.663061\pi$
$884$	−3.25158e7	−1.39947
$885$	0	0
$886$	−2.96810e6	−0.127026
$887$	−3.59474e7	−1.53412	−0.767059	−	0.641576i	$-0.778281\pi$
$887$	−3.59474e7	−1.53412	−0.767059	+	0.641576i	$0.778281\pi$
$888$	0	0
$889$	−3.87469e7	−1.64431
$890$	−4.56745e6	−0.193285
$891$	0	0
$892$	−1.43380e7	−0.603361
$893$	1.79789e7	0.754459
$894$	0	0
$895$	1.80871e7	0.754764
$896$	3.95424e7	1.64548
$897$	0	0
$898$	−7.39888e6	−0.306179
$899$	2.40113e7	0.990868
$900$	0	0
$901$	−2.87576e7	−1.18016
$902$	154758.	0.00633341
$903$	0	0
$904$	−2.52845e7	−1.02904
$905$	1.34453e7	0.545694
$906$	0	0
$907$	−1.50358e7	−0.606886	−0.303443	−	0.952850i	$-0.598136\pi$
$907$	−1.50358e7	−0.606886	−0.303443	+	0.952850i	$0.598136\pi$
$908$	2.23410e7	0.899265
$909$	0	0
$910$	−9.89266e6	−0.396013
$911$	9.15595e6	0.365517	0.182758	−	0.983158i	$-0.441497\pi$
$911$	9.15595e6	0.365517	0.182758	+	0.983158i	$0.441497\pi$
$912$	0	0
$913$	−1.12895e6	−0.0448225
$914$	−1.08347e7	−0.428994
$915$	0	0
$916$	2.10318e7	0.828207
$917$	−4.29144e7	−1.68531
$918$	0	0
$919$	−4.54678e7	−1.77589	−0.887944	−	0.459953i	$-0.847866\pi$
$919$	−4.54678e7	−1.77589	−0.887944	+	0.459953i	$0.847866\pi$
$920$	7.02947e6	0.273812
$921$	0	0
$922$	−7.20939e6	−0.279300
$923$	−1.01829e7	−0.393429
$924$	0	0
$925$	−3.34113e6	−0.128392
$926$	3.27640e6	0.125565
$927$	0	0
$928$	−2.62326e7	−0.999936
$929$	3.35662e7	1.27604	0.638019	−	0.770021i	$-0.279754\pi$
$929$	3.35662e7	1.27604	0.638019	+	0.770021i	$0.279754\pi$
$930$	0	0
$931$	−1.85758e7	−0.702380
$932$	−1.96346e7	−0.740426
$933$	0	0
$934$	7.07718e6	0.265456
$935$	−3.71497e6	−0.138972
$936$	0	0
$937$	4.49720e6	0.167338	0.0836688	−	0.996494i	$-0.473336\pi$
$937$	4.49720e6	0.167338	0.0836688	+	0.996494i	$0.473336\pi$
$938$	−3.68021e6	−0.136573
$939$	0	0
$940$	1.94628e7	0.718433
$941$	−2.67878e7	−0.986197	−0.493099	−	0.869973i	$-0.664136\pi$
$941$	−2.67878e7	−0.986197	−0.493099	+	0.869973i	$0.664136\pi$
$942$	0	0
$943$	1.53915e6	0.0563639
$944$	−9.06757e6	−0.331178
$945$	0	0
$946$	−2.18039e6	−0.0792148
$947$	3.16458e7	1.14668	0.573338	−	0.819319i	$-0.305648\pi$
$947$	3.16458e7	1.14668	0.573338	+	0.819319i	$0.305648\pi$
$948$	0	0
$949$	6.18354e7	2.22880
$950$	799996.	0.0287593
$951$	0	0
$952$	3.10181e7	1.10923
$953$	−4.02571e7	−1.43585	−0.717926	−	0.696119i	$-0.754909\pi$
$953$	−4.02571e7	−1.43585	−0.717926	+	0.696119i	$0.754909\pi$
$954$	0	0
$955$	−7.60724e6	−0.269910
$956$	2.33092e7	0.824863
$957$	0	0
$958$	1.34944e7	0.475050
$959$	7.48721e7	2.62890
$960$	0	0
$961$	−6.79136e6	−0.237219
$962$	9.92589e6	0.345806
$963$	0	0
$964$	3.22385e7	1.11733
$965$	1.48680e7	0.513966
$966$	0	0
$967$	−8.48380e6	−0.291759	−0.145880	−	0.989302i	$-0.546601\pi$
$967$	−8.48380e6	−0.291759	−0.145880	+	0.989302i	$0.546601\pi$
$968$	1.73516e6	0.0595186
$969$	0	0
$970$	−7.70912e6	−0.263073
$971$	−903262.	−0.0307444	−0.0153722	−	0.999882i	$-0.504893\pi$
$971$	−903262.	−0.0307444	−0.0153722	+	0.999882i	$0.504893\pi$
$972$	0	0
$973$	−5.15077e7	−1.74418
$974$	1.21785e7	0.411336
$975$	0	0
$976$	−5.11023e6	−0.171718
$977$	−1.17609e7	−0.394189	−0.197095	−	0.980384i	$-0.563151\pi$
$977$	−1.17609e7	−0.394189	−0.197095	+	0.980384i	$0.563151\pi$
$978$	0	0
$979$	−1.12129e7	−0.373905
$980$	−2.01089e7	−0.668841
$981$	0	0
$982$	−9.80721e6	−0.324539
$983$	−5.25115e7	−1.73329	−0.866644	−	0.498927i	$-0.833728\pi$
$983$	−5.25115e7	−1.73329	−0.866644	+	0.498927i	$0.833728\pi$
$984$	0	0
$985$	−2.23077e7	−0.732597
$986$	−1.24406e7	−0.407520
$987$	0	0
$988$	1.71899e7	0.560247
$989$	−2.16851e7	−0.704969
$990$	0	0
$991$	−1.68764e7	−0.545877	−0.272938	−	0.962031i	$-0.587996\pi$
$991$	−1.68764e7	−0.545877	−0.272938	+	0.962031i	$0.587996\pi$
$992$	−2.38581e7	−0.769762
$993$	0	0
$994$	4.54288e6	0.145836
$995$	2.05947e7	0.659476
$996$	0	0
$997$	−8.04669e6	−0.256377	−0.128189	−	0.991750i	$-0.540916\pi$
$997$	−8.04669e6	−0.256377	−0.128189	+	0.991750i	$0.540916\pi$
$998$	−9.15458e6	−0.290946
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.p.1.5	yes	10
3.2	odd	2		495.6.a.o.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.6.a.o.1.6	✓	10	3.2	odd	2
495.6.a.p.1.5	yes	10	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)

\(f(q)\)	\(=\)	\(q-1.97152 q^{2} -28.1131 q^{4} +25.0000 q^{5} +213.116 q^{7} +118.514 q^{8} +O(q^{10})\)	\(q-1.97152 q^{2} -28.1131 q^{4} +25.0000 q^{5} +213.116 q^{7} +118.514 q^{8} -49.2880 q^{10} -121.000 q^{11} +941.794 q^{13} -420.162 q^{14} +665.967 q^{16} +1228.09 q^{17} -649.243 q^{19} -702.828 q^{20} +238.554 q^{22} +2372.53 q^{23} +625.000 q^{25} -1856.76 q^{26} -5991.35 q^{28} +5138.20 q^{29} +4673.09 q^{31} -5105.42 q^{32} -2421.20 q^{34} +5327.90 q^{35} -5345.80 q^{37} +1279.99 q^{38} +2962.85 q^{40} +648.735 q^{41} -9140.04 q^{43} +3401.69 q^{44} -4677.49 q^{46} -27692.2 q^{47} +28611.4 q^{49} -1232.20 q^{50} -26476.8 q^{52} -23416.5 q^{53} -3025.00 q^{55} +25257.2 q^{56} -10130.0 q^{58} -13615.6 q^{59} -7673.40 q^{61} -9213.08 q^{62} -11245.5 q^{64} +23544.9 q^{65} +8759.02 q^{67} -34525.4 q^{68} -10504.0 q^{70} -10812.2 q^{71} +65657.0 q^{73} +10539.3 q^{74} +18252.2 q^{76} -25787.0 q^{77} +15899.7 q^{79} +16649.2 q^{80} -1278.99 q^{82} +9330.13 q^{83} +30702.2 q^{85} +18019.7 q^{86} -14340.2 q^{88} +92668.6 q^{89} +200711. q^{91} -66699.3 q^{92} +54595.6 q^{94} -16231.1 q^{95} +156410. q^{97} -56407.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{2} + 181 q^{4} + 250 q^{5} + 116 q^{7} + 129 q^{8}+O(q^{10}) \) 10 * q + 3 * q^2 + 181 * q^4 + 250 * q^5 + 116 * q^7 + 129 * q^8	\( 10 q + 3 q^{2} + 181 q^{4} + 250 q^{5} + 116 q^{7} + 129 q^{8} + 75 q^{10} - 1210 q^{11} + 932 q^{13} - 1332 q^{14} + 2701 q^{16} + 96 q^{17} + 1664 q^{19} + 4525 q^{20} - 363 q^{22} + 6288 q^{23} + 6250 q^{25} + 13380 q^{26} + 13868 q^{28} + 11208 q^{29} + 9032 q^{31} + 9801 q^{32} + 14610 q^{34} + 2900 q^{35} + 21572 q^{37} + 15870 q^{38} + 3225 q^{40} + 10800 q^{41} + 21128 q^{43} - 21901 q^{44} + 83982 q^{46} - 17400 q^{47} + 71610 q^{49} + 1875 q^{50} + 40640 q^{52} + 5004 q^{53} - 30250 q^{55} - 54012 q^{56} - 9786 q^{58} - 25272 q^{59} + 52004 q^{61} + 34740 q^{62} + 56953 q^{64} + 23300 q^{65} + 4160 q^{67} - 87978 q^{68} - 33300 q^{70} - 65232 q^{71} + 44252 q^{73} - 49842 q^{74} + 233246 q^{76} - 14036 q^{77} + 112604 q^{79} + 67525 q^{80} + 167910 q^{82} + 70032 q^{83} + 2400 q^{85} - 72978 q^{86} - 15609 q^{88} - 46848 q^{89} + 130672 q^{91} + 121302 q^{92} + 252294 q^{94} + 41600 q^{95} + 129932 q^{97} - 316137 q^{98}+O(q^{100}) \) 10 * q + 3 * q^2 + 181 * q^4 + 250 * q^5 + 116 * q^7 + 129 * q^8 + 75 * q^10 - 1210 * q^11 + 932 * q^13 - 1332 * q^14 + 2701 * q^16 + 96 * q^17 + 1664 * q^19 + 4525 * q^20 - 363 * q^22 + 6288 * q^23 + 6250 * q^25 + 13380 * q^26 + 13868 * q^28 + 11208 * q^29 + 9032 * q^31 + 9801 * q^32 + 14610 * q^34 + 2900 * q^35 + 21572 * q^37 + 15870 * q^38 + 3225 * q^40 + 10800 * q^41 + 21128 * q^43 - 21901 * q^44 + 83982 * q^46 - 17400 * q^47 + 71610 * q^49 + 1875 * q^50 + 40640 * q^52 + 5004 * q^53 - 30250 * q^55 - 54012 * q^56 - 9786 * q^58 - 25272 * q^59 + 52004 * q^61 + 34740 * q^62 + 56953 * q^64 + 23300 * q^65 + 4160 * q^67 - 87978 * q^68 - 33300 * q^70 - 65232 * q^71 + 44252 * q^73 - 49842 * q^74 + 233246 * q^76 - 14036 * q^77 + 112604 * q^79 + 67525 * q^80 + 167910 * q^82 + 70032 * q^83 + 2400 * q^85 - 72978 * q^86 - 15609 * q^88 - 46848 * q^89 + 130672 * q^91 + 121302 * q^92 + 252294 * q^94 + 41600 * q^95 + 129932 * q^97 - 316137 * q^98

Introduction

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