LMFDB - Embedded newform 8046.2.a.t.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 46 x^{14} + 192 x^{13} + 752 x^{12} - 3378 x^{11} - 5277 x^{10} + 27132 x^{9} + \cdots - 4260 $ x^16 - 4x^15 - 46x^14 + 192x^13 + 752x^12 - 3378x^11 - 5277x^10 + 27132x^9 + 15173x^8 - 102466x^7 - 16612x^6 + 186120x^5 + 117x^4 - 156722x^3 + 10545x^2 + 48580*x - 4260
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.23039$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.23039 q^{5} +0.0132875 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.23039 q^{5} +0.0132875 q^{7} +1.00000 q^{8} -1.23039 q^{10} +4.94865 q^{11} +6.21005 q^{13} +0.0132875 q^{14} +1.00000 q^{16} -0.0222600 q^{17} +0.0813053 q^{19} -1.23039 q^{20} +4.94865 q^{22} +6.99731 q^{23} -3.48613 q^{25} +6.21005 q^{26} +0.0132875 q^{28} +3.85576 q^{29} -2.81064 q^{31} +1.00000 q^{32} -0.0222600 q^{34} -0.0163488 q^{35} +8.13950 q^{37} +0.0813053 q^{38} -1.23039 q^{40} -1.31437 q^{41} +2.94775 q^{43} +4.94865 q^{44} +6.99731 q^{46} -1.16943 q^{47} -6.99982 q^{49} -3.48613 q^{50} +6.21005 q^{52} -8.32188 q^{53} -6.08879 q^{55} +0.0132875 q^{56} +3.85576 q^{58} -8.08065 q^{59} -9.76901 q^{61} -2.81064 q^{62} +1.00000 q^{64} -7.64081 q^{65} +11.7226 q^{67} -0.0222600 q^{68} -0.0163488 q^{70} -1.68449 q^{71} +16.2334 q^{73} +8.13950 q^{74} +0.0813053 q^{76} +0.0657552 q^{77} -13.0834 q^{79} -1.23039 q^{80} -1.31437 q^{82} +3.37384 q^{83} +0.0273886 q^{85} +2.94775 q^{86} +4.94865 q^{88} -12.2061 q^{89} +0.0825160 q^{91} +6.99731 q^{92} -1.16943 q^{94} -0.100038 q^{95} +9.34623 q^{97} -6.99982 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8	$ 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8} + 4 q^{10} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 10 q^{19} + 4 q^{20} + 6 q^{22} + 10 q^{23} + 28 q^{25} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 21 q^{31} + 16 q^{32} + q^{34} + 16 q^{35} + 17 q^{37} + 10 q^{38} + 4 q^{40} - 4 q^{41} + 16 q^{43} + 6 q^{44} + 10 q^{46} + 25 q^{47} + 36 q^{49} + 28 q^{50} + 6 q^{52} + 14 q^{53} + 19 q^{55} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 23 q^{61} + 21 q^{62} + 16 q^{64} + 20 q^{65} + 22 q^{67} + q^{68} + 16 q^{70} + 10 q^{71} + 16 q^{73} + 17 q^{74} + 10 q^{76} - 2 q^{77} + 37 q^{79} + 4 q^{80} - 4 q^{82} + 33 q^{83} + 43 q^{85} + 16 q^{86} + 6 q^{88} - 3 q^{89} + 28 q^{91} + 10 q^{92} + 25 q^{94} + 14 q^{95} - 3 q^{97} + 36 q^{98}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8 + 4 * q^10 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 10 * q^19 + 4 * q^20 + 6 * q^22 + 10 * q^23 + 28 * q^25 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 21 * q^31 + 16 * q^32 + q^34 + 16 * q^35 + 17 * q^37 + 10 * q^38 + 4 * q^40 - 4 * q^41 + 16 * q^43 + 6 * q^44 + 10 * q^46 + 25 * q^47 + 36 * q^49 + 28 * q^50 + 6 * q^52 + 14 * q^53 + 19 * q^55 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 23 * q^61 + 21 * q^62 + 16 * q^64 + 20 * q^65 + 22 * q^67 + q^68 + 16 * q^70 + 10 * q^71 + 16 * q^73 + 17 * q^74 + 10 * q^76 - 2 * q^77 + 37 * q^79 + 4 * q^80 - 4 * q^82 + 33 * q^83 + 43 * q^85 + 16 * q^86 + 6 * q^88 - 3 * q^89 + 28 * q^91 + 10 * q^92 + 25 * q^94 + 14 * q^95 - 3 * q^97 + 36 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.23039	−0.550249	−0.275124	−	0.961409i	$-0.588719\pi$
$5$	−1.23039	−0.550249	−0.275124	+	0.961409i	$0.588719\pi$
$6$	0	0
$7$	0.0132875	0.00502220	0.00251110	−	0.999997i	$-0.499201\pi$
$7$	0.0132875	0.00502220	0.00251110	+	0.999997i	$0.499201\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.23039	−0.389085
$11$	4.94865	1.49208	0.746038	−	0.665904i	$-0.231954\pi$
$11$	4.94865	1.49208	0.746038	+	0.665904i	$0.231954\pi$
$12$	0	0
$13$	6.21005	1.72236	0.861179	−	0.508302i	$-0.169726\pi$
$13$	6.21005	1.72236	0.861179	+	0.508302i	$0.169726\pi$
$14$	0.0132875	0.00355123
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.0222600	−0.00539885	−0.00269942	−	0.999996i	$-0.500859\pi$
$17$	−0.0222600	−0.00539885	−0.00269942	+	0.999996i	$0.500859\pi$
$18$	0	0
$19$	0.0813053	0.0186527	0.00932636	−	0.999957i	$-0.497031\pi$
$19$	0.0813053	0.0186527	0.00932636	+	0.999957i	$0.497031\pi$
$20$	−1.23039	−0.275124
$21$	0	0
$22$	4.94865	1.05506
$23$	6.99731	1.45904	0.729520	−	0.683959i	$-0.239743\pi$
$23$	6.99731	1.45904	0.729520	+	0.683959i	$0.239743\pi$
$24$	0	0
$25$	−3.48613	−0.697226
$26$	6.21005	1.21789
$27$	0	0
$28$	0.0132875	0.00251110
$29$	3.85576	0.715996	0.357998	−	0.933722i	$-0.383459\pi$
$29$	3.85576	0.715996	0.357998	+	0.933722i	$0.383459\pi$
$30$	0	0
$31$	−2.81064	−0.504806	−0.252403	−	0.967622i	$-0.581221\pi$
$31$	−2.81064	−0.504806	−0.252403	+	0.967622i	$0.581221\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.0222600	−0.00381756
$35$	−0.0163488	−0.00276346
$36$	0	0
$37$	8.13950	1.33813	0.669063	−	0.743206i	$-0.266696\pi$
$37$	8.13950	1.33813	0.669063	+	0.743206i	$0.266696\pi$
$38$	0.0813053	0.0131895
$39$	0	0
$40$	−1.23039	−0.194542
$41$	−1.31437	−0.205270	−0.102635	−	0.994719i	$-0.532727\pi$
$41$	−1.31437	−0.205270	−0.102635	+	0.994719i	$0.532727\pi$
$42$	0	0
$43$	2.94775	0.449528	0.224764	−	0.974413i	$-0.427839\pi$
$43$	2.94775	0.449528	0.224764	+	0.974413i	$0.427839\pi$
$44$	4.94865	0.746038
$45$	0	0
$46$	6.99731	1.03170
$47$	−1.16943	−0.170578	−0.0852891	−	0.996356i	$-0.527181\pi$
$47$	−1.16943	−0.170578	−0.0852891	+	0.996356i	$0.527181\pi$
$48$	0	0
$49$	−6.99982	−0.999975
$50$	−3.48613	−0.493013
$51$	0	0
$52$	6.21005	0.861179
$53$	−8.32188	−1.14310	−0.571549	−	0.820568i	$-0.693657\pi$
$53$	−8.32188	−1.14310	−0.571549	+	0.820568i	$0.693657\pi$
$54$	0	0
$55$	−6.08879	−0.821013
$56$	0.0132875	0.00177562
$57$	0	0
$58$	3.85576	0.506286
$59$	−8.08065	−1.05201	−0.526005	−	0.850481i	$-0.676311\pi$
$59$	−8.08065	−1.05201	−0.526005	+	0.850481i	$0.676311\pi$
$60$	0	0
$61$	−9.76901	−1.25079	−0.625397	−	0.780307i	$-0.715063\pi$
$61$	−9.76901	−1.25079	−0.625397	+	0.780307i	$0.715063\pi$
$62$	−2.81064	−0.356952
$63$	0	0
$64$	1.00000	0.125000
$65$	−7.64081	−0.947726
$66$	0	0
$67$	11.7226	1.43214	0.716069	−	0.698029i	$-0.245940\pi$
$67$	11.7226	1.43214	0.716069	+	0.698029i	$0.245940\pi$
$68$	−0.0222600	−0.00269942
$69$	0	0
$70$	−0.0163488	−0.00195406
$71$	−1.68449	−0.199913	−0.0999563	−	0.994992i	$-0.531870\pi$
$71$	−1.68449	−0.199913	−0.0999563	+	0.994992i	$0.531870\pi$
$72$	0	0
$73$	16.2334	1.89997	0.949985	−	0.312295i	$-0.101098\pi$
$73$	16.2334	1.89997	0.949985	+	0.312295i	$0.101098\pi$
$74$	8.13950	0.946198
$75$	0	0
$76$	0.0813053	0.00932636
$77$	0.0657552	0.00749350
$78$	0	0
$79$	−13.0834	−1.47200	−0.736001	−	0.676981i	$-0.763288\pi$
$79$	−13.0834	−1.47200	−0.736001	+	0.676981i	$0.763288\pi$
$80$	−1.23039	−0.137562
$81$	0	0
$82$	−1.31437	−0.145148
$83$	3.37384	0.370327	0.185164	−	0.982708i	$-0.440718\pi$
$83$	3.37384	0.370327	0.185164	+	0.982708i	$0.440718\pi$
$84$	0	0
$85$	0.0273886	0.00297071
$86$	2.94775	0.317864
$87$	0	0
$88$	4.94865	0.527528
$89$	−12.2061	−1.29384	−0.646920	−	0.762558i	$-0.723943\pi$
$89$	−12.2061	−1.29384	−0.646920	+	0.762558i	$0.723943\pi$
$90$	0	0
$91$	0.0825160	0.00865003
$92$	6.99731	0.729520
$93$	0	0
$94$	−1.16943	−0.120617
$95$	−0.100038	−0.0102636
$96$	0	0
$97$	9.34623	0.948966	0.474483	−	0.880265i	$-0.342635\pi$
$97$	9.34623	0.948966	0.474483	+	0.880265i	$0.342635\pi$
$98$	−6.99982	−0.707089
$99$	0	0
$100$	−3.48613	−0.348613
$101$	4.44282	0.442077	0.221038	−	0.975265i	$-0.429055\pi$
$101$	4.44282	0.442077	0.221038	+	0.975265i	$0.429055\pi$
$102$	0	0
$103$	−6.62444	−0.652725	−0.326363	−	0.945245i	$-0.605823\pi$
$103$	−6.62444	−0.652725	−0.326363	+	0.945245i	$0.605823\pi$
$104$	6.21005	0.608946
$105$	0	0
$106$	−8.32188	−0.808292
$107$	−5.57220	−0.538685	−0.269342	−	0.963044i	$-0.586806\pi$
$107$	−5.57220	−0.538685	−0.269342	+	0.963044i	$0.586806\pi$
$108$	0	0
$109$	14.1181	1.35226	0.676132	−	0.736780i	$-0.263655\pi$
$109$	14.1181	1.35226	0.676132	+	0.736780i	$0.263655\pi$
$110$	−6.08879	−0.580544
$111$	0	0
$112$	0.0132875	0.00125555
$113$	1.91638	0.180278	0.0901388	−	0.995929i	$-0.471269\pi$
$113$	1.91638	0.180278	0.0901388	+	0.995929i	$0.471269\pi$
$114$	0	0
$115$	−8.60945	−0.802836
$116$	3.85576	0.357998
$117$	0	0
$118$	−8.08065	−0.743884
$119$	−0.000295780 0	−2.71141e−5 0
$120$	0	0
$121$	13.4892	1.22629
$122$	−9.76901	−0.884444
$123$	0	0
$124$	−2.81064	−0.252403
$125$	10.4413	0.933897
$126$	0	0
$127$	15.9605	1.41627	0.708134	−	0.706078i	$-0.249537\pi$
$127$	15.9605	1.41627	0.708134	+	0.706078i	$0.249537\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−7.64081	−0.670143
$131$	−19.4835	−1.70228	−0.851139	−	0.524941i	$-0.824087\pi$
$131$	−19.4835	−1.70228	−0.851139	+	0.524941i	$0.824087\pi$
$132$	0	0
$133$	0.00108034	9.36777e−5 0
$134$	11.7226	1.01267
$135$	0	0
$136$	−0.0222600	−0.00190878
$137$	9.46101	0.808309	0.404155	−	0.914691i	$-0.367566\pi$
$137$	9.46101	0.808309	0.404155	+	0.914691i	$0.367566\pi$
$138$	0	0
$139$	4.25411	0.360829	0.180415	−	0.983591i	$-0.442256\pi$
$139$	4.25411	0.360829	0.180415	+	0.983591i	$0.442256\pi$
$140$	−0.0163488	−0.00138173
$141$	0	0
$142$	−1.68449	−0.141360
$143$	30.7314	2.56989
$144$	0	0
$145$	−4.74410	−0.393976
$146$	16.2334	1.34348
$147$	0	0
$148$	8.13950	0.669063
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−5.66234	−0.460795	−0.230397	−	0.973097i	$-0.574003\pi$
$151$	−5.66234	−0.460795	−0.230397	+	0.973097i	$0.574003\pi$
$152$	0.0813053	0.00659473
$153$	0	0
$154$	0.0657552	0.00529870
$155$	3.45819	0.277769
$156$	0	0
$157$	−9.53252	−0.760778	−0.380389	−	0.924827i	$-0.624210\pi$
$157$	−9.53252	−0.760778	−0.380389	+	0.924827i	$0.624210\pi$
$158$	−13.0834	−1.04086
$159$	0	0
$160$	−1.23039	−0.0972712
$161$	0.0929767	0.00732759
$162$	0	0
$163$	11.4090	0.893623	0.446811	−	0.894628i	$-0.352559\pi$
$163$	11.4090	0.893623	0.446811	+	0.894628i	$0.352559\pi$
$164$	−1.31437	−0.102635
$165$	0	0
$166$	3.37384	0.261861
$167$	2.88238	0.223045	0.111523	−	0.993762i	$-0.464427\pi$
$167$	2.88238	0.223045	0.111523	+	0.993762i	$0.464427\pi$
$168$	0	0
$169$	25.5647	1.96652
$170$	0.0273886	0.00210061
$171$	0	0
$172$	2.94775	0.224764
$173$	−3.22628	−0.245289	−0.122645	−	0.992451i	$-0.539138\pi$
$173$	−3.22628	−0.245289	−0.122645	+	0.992451i	$0.539138\pi$
$174$	0	0
$175$	−0.0463219	−0.00350161
$176$	4.94865	0.373019
$177$	0	0
$178$	−12.2061	−0.914883
$179$	7.84987	0.586726	0.293363	−	0.956001i	$-0.405225\pi$
$179$	7.84987	0.586726	0.293363	+	0.956001i	$0.405225\pi$
$180$	0	0
$181$	−11.4292	−0.849524	−0.424762	−	0.905305i	$-0.639642\pi$
$181$	−11.4292	−0.849524	−0.424762	+	0.905305i	$0.639642\pi$
$182$	0.0825160	0.00611649
$183$	0	0
$184$	6.99731	0.515849
$185$	−10.0148	−0.736302
$186$	0	0
$187$	−0.110157	−0.00805548
$188$	−1.16943	−0.0852891
$189$	0	0
$190$	−0.100038	−0.00725749
$191$	11.7177	0.847866	0.423933	−	0.905693i	$-0.360649\pi$
$191$	11.7177	0.847866	0.423933	+	0.905693i	$0.360649\pi$
$192$	0	0
$193$	21.6129	1.55573	0.777866	−	0.628430i	$-0.216302\pi$
$193$	21.6129	1.55573	0.777866	+	0.628430i	$0.216302\pi$
$194$	9.34623	0.671020
$195$	0	0
$196$	−6.99982	−0.499987
$197$	−27.7107	−1.97431	−0.987154	−	0.159769i	$-0.948925\pi$
$197$	−27.7107	−1.97431	−0.987154	+	0.159769i	$0.948925\pi$
$198$	0	0
$199$	1.93424	0.137115	0.0685575	−	0.997647i	$-0.478160\pi$
$199$	1.93424	0.137115	0.0685575	+	0.997647i	$0.478160\pi$
$200$	−3.48613	−0.246507
$201$	0	0
$202$	4.44282	0.312595
$203$	0.0512334	0.00359588
$204$	0	0
$205$	1.61719	0.112950
$206$	−6.62444	−0.461546
$207$	0	0
$208$	6.21005	0.430590
$209$	0.402352	0.0278313
$210$	0	0
$211$	−17.3992	−1.19781	−0.598905	−	0.800820i	$-0.704397\pi$
$211$	−17.3992	−1.19781	−0.598905	+	0.800820i	$0.704397\pi$
$212$	−8.32188	−0.571549
$213$	0	0
$214$	−5.57220	−0.380908
$215$	−3.62690	−0.247352
$216$	0	0
$217$	−0.0373463	−0.00253524
$218$	14.1181	0.956195
$219$	0	0
$220$	−6.08879	−0.410506
$221$	−0.138236	−0.00929875
$222$	0	0
$223$	21.4489	1.43632	0.718162	−	0.695875i	$-0.244983\pi$
$223$	21.4489	1.43632	0.718162	+	0.695875i	$0.244983\pi$
$224$	0.0132875	0.000887808 0
$225$	0	0
$226$	1.91638	0.127475
$227$	−7.79276	−0.517224	−0.258612	−	0.965981i	$-0.583265\pi$
$227$	−7.79276	−0.517224	−0.258612	+	0.965981i	$0.583265\pi$
$228$	0	0
$229$	15.4567	1.02141	0.510703	−	0.859757i	$-0.329385\pi$
$229$	15.4567	1.02141	0.510703	+	0.859757i	$0.329385\pi$
$230$	−8.60945	−0.567690
$231$	0	0
$232$	3.85576	0.253143
$233$	−15.6601	−1.02592	−0.512962	−	0.858411i	$-0.671452\pi$
$233$	−15.6601	−1.02592	−0.512962	+	0.858411i	$0.671452\pi$
$234$	0	0
$235$	1.43885	0.0938605
$236$	−8.08065	−0.526005
$237$	0	0
$238$	−0.000295780 0	−1.91725e−5 0
$239$	−4.96908	−0.321423	−0.160712	−	0.987001i	$-0.551379\pi$
$239$	−4.96908	−0.321423	−0.160712	+	0.987001i	$0.551379\pi$
$240$	0	0
$241$	6.87321	0.442743	0.221371	−	0.975190i	$-0.428947\pi$
$241$	6.87321	0.442743	0.221371	+	0.975190i	$0.428947\pi$
$242$	13.4892	0.867117
$243$	0	0
$244$	−9.76901	−0.625397
$245$	8.61254	0.550235
$246$	0	0
$247$	0.504910	0.0321267
$248$	−2.81064	−0.178476
$249$	0	0
$250$	10.4413	0.660365
$251$	−15.8669	−1.00151	−0.500756	−	0.865589i	$-0.666945\pi$
$251$	−15.8669	−1.00151	−0.500756	+	0.865589i	$0.666945\pi$
$252$	0	0
$253$	34.6273	2.17700
$254$	15.9605	1.00145
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.2497	1.88693	0.943464	−	0.331476i	$-0.107547\pi$
$257$	30.2497	1.88693	0.943464	+	0.331476i	$0.107547\pi$
$258$	0	0
$259$	0.108154	0.00672034
$260$	−7.64081	−0.473863
$261$	0	0
$262$	−19.4835	−1.20369
$263$	20.8380	1.28493	0.642463	−	0.766316i	$-0.277912\pi$
$263$	20.8380	1.28493	0.642463	+	0.766316i	$0.277912\pi$
$264$	0	0
$265$	10.2392	0.628989
$266$	0.00108034	6.62401e−5 0
$267$	0	0
$268$	11.7226	0.716069
$269$	−21.5901	−1.31637	−0.658185	−	0.752856i	$-0.728676\pi$
$269$	−21.5901	−1.31637	−0.658185	+	0.752856i	$0.728676\pi$
$270$	0	0
$271$	−21.2284	−1.28954	−0.644768	−	0.764378i	$-0.723046\pi$
$271$	−21.2284	−1.28954	−0.644768	+	0.764378i	$0.723046\pi$
$272$	−0.0222600	−0.00134971
$273$	0	0
$274$	9.46101	0.571561
$275$	−17.2517	−1.04031
$276$	0	0
$277$	0.0377590	0.00226872	0.00113436	−	0.999999i	$-0.499639\pi$
$277$	0.0377590	0.00226872	0.00113436	+	0.999999i	$0.499639\pi$
$278$	4.25411	0.255145
$279$	0	0
$280$	−0.0163488	−0.000977031 0
$281$	12.0257	0.717394	0.358697	−	0.933454i	$-0.383221\pi$
$281$	12.0257	0.717394	0.358697	+	0.933454i	$0.383221\pi$
$282$	0	0
$283$	−2.44057	−0.145077	−0.0725383	−	0.997366i	$-0.523110\pi$
$283$	−2.44057	−0.145077	−0.0725383	+	0.997366i	$0.523110\pi$
$284$	−1.68449	−0.0999563
$285$	0	0
$286$	30.7314	1.81719
$287$	−0.0174647	−0.00103091
$288$	0	0
$289$	−16.9995	−0.999971
$290$	−4.74410	−0.278583
$291$	0	0
$292$	16.2334	0.949985
$293$	11.8317	0.691217	0.345608	−	0.938379i	$-0.387673\pi$
$293$	11.8317	0.691217	0.345608	+	0.938379i	$0.387673\pi$
$294$	0	0
$295$	9.94238	0.578868
$296$	8.13950	0.473099
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	43.4537	2.51299
$300$	0	0
$301$	0.0391682	0.00225762
$302$	−5.66234	−0.325831
$303$	0	0
$304$	0.0813053	0.00466318
$305$	12.0197	0.688248
$306$	0	0
$307$	26.2111	1.49595	0.747973	−	0.663730i	$-0.231027\pi$
$307$	26.2111	1.49595	0.747973	+	0.663730i	$0.231027\pi$
$308$	0.0657552	0.00374675
$309$	0	0
$310$	3.45819	0.196412
$311$	9.19816	0.521580	0.260790	−	0.965396i	$-0.416017\pi$
$311$	9.19816	0.521580	0.260790	+	0.965396i	$0.416017\pi$
$312$	0	0
$313$	19.1407	1.08190	0.540950	−	0.841055i	$-0.318065\pi$
$313$	19.1407	1.08190	0.540950	+	0.841055i	$0.318065\pi$
$314$	−9.53252	−0.537951
$315$	0	0
$316$	−13.0834	−0.736001
$317$	−32.0692	−1.80118	−0.900592	−	0.434665i	$-0.856867\pi$
$317$	−32.0692	−1.80118	−0.900592	+	0.434665i	$0.856867\pi$
$318$	0	0
$319$	19.0808	1.06832
$320$	−1.23039	−0.0687811
$321$	0	0
$322$	0.0929767	0.00518139
$323$	−0.00180986	−0.000100703 0
$324$	0	0
$325$	−21.6490	−1.20087
$326$	11.4090	0.631887
$327$	0	0
$328$	−1.31437	−0.0725738
$329$	−0.0155387	−0.000856678 0
$330$	0	0
$331$	9.69446	0.532856	0.266428	−	0.963855i	$-0.414157\pi$
$331$	9.69446	0.532856	0.266428	+	0.963855i	$0.414157\pi$
$332$	3.37384	0.185164
$333$	0	0
$334$	2.88238	0.157717
$335$	−14.4234	−0.788033
$336$	0	0
$337$	12.1879	0.663915	0.331958	−	0.943294i	$-0.392291\pi$
$337$	12.1879	0.663915	0.331958	+	0.943294i	$0.392291\pi$
$338$	25.5647	1.39054
$339$	0	0
$340$	0.0273886	0.00148535
$341$	−13.9089	−0.753208
$342$	0	0
$343$	−0.186023	−0.0100443
$344$	2.94775	0.158932
$345$	0	0
$346$	−3.22628	−0.173446
$347$	22.1524	1.18921	0.594603	−	0.804020i	$-0.297309\pi$
$347$	22.1524	1.18921	0.594603	+	0.804020i	$0.297309\pi$
$348$	0	0
$349$	18.0167	0.964412	0.482206	−	0.876058i	$-0.339836\pi$
$349$	18.0167	0.964412	0.482206	+	0.876058i	$0.339836\pi$
$350$	−0.0463219	−0.00247601
$351$	0	0
$352$	4.94865	0.263764
$353$	−10.1474	−0.540093	−0.270047	−	0.962847i	$-0.587039\pi$
$353$	−10.1474	−0.540093	−0.270047	+	0.962847i	$0.587039\pi$
$354$	0	0
$355$	2.07259	0.110002
$356$	−12.2061	−0.646920
$357$	0	0
$358$	7.84987	0.414878
$359$	29.4270	1.55310	0.776549	−	0.630056i	$-0.216968\pi$
$359$	29.4270	1.55310	0.776549	+	0.630056i	$0.216968\pi$
$360$	0	0
$361$	−18.9934	−0.999652
$362$	−11.4292	−0.600704
$363$	0	0
$364$	0.0825160	0.00432501
$365$	−19.9734	−1.04546
$366$	0	0
$367$	22.1630	1.15690	0.578449	−	0.815719i	$-0.303658\pi$
$367$	22.1630	1.15690	0.578449	+	0.815719i	$0.303658\pi$
$368$	6.99731	0.364760
$369$	0	0
$370$	−10.0148	−0.520644
$371$	−0.110577	−0.00574087
$372$	0	0
$373$	12.1239	0.627752	0.313876	−	0.949464i	$-0.398372\pi$
$373$	12.1239	0.627752	0.313876	+	0.949464i	$0.398372\pi$
$374$	−0.110157	−0.00569609
$375$	0	0
$376$	−1.16943	−0.0603085
$377$	23.9445	1.23320
$378$	0	0
$379$	−21.0325	−1.08037	−0.540184	−	0.841547i	$-0.681645\pi$
$379$	−21.0325	−1.08037	−0.540184	+	0.841547i	$0.681645\pi$
$380$	−0.100038	−0.00513182
$381$	0	0
$382$	11.7177	0.599532
$383$	−14.3893	−0.735258	−0.367629	−	0.929972i	$-0.619830\pi$
$383$	−14.3893	−0.735258	−0.367629	+	0.929972i	$0.619830\pi$
$384$	0	0
$385$	−0.0809048	−0.00412329
$386$	21.6129	1.10007
$387$	0	0
$388$	9.34623	0.474483
$389$	−34.1218	−1.73004	−0.865021	−	0.501736i	$-0.832695\pi$
$389$	−34.1218	−1.73004	−0.865021	+	0.501736i	$0.832695\pi$
$390$	0	0
$391$	−0.155760	−0.00787714
$392$	−6.99982	−0.353544
$393$	0	0
$394$	−27.7107	−1.39605
$395$	16.0978	0.809967
$396$	0	0
$397$	3.97908	0.199704	0.0998522	−	0.995002i	$-0.468163\pi$
$397$	3.97908	0.199704	0.0998522	+	0.995002i	$0.468163\pi$
$398$	1.93424	0.0969549
$399$	0	0
$400$	−3.48613	−0.174307
$401$	6.14843	0.307038	0.153519	−	0.988146i	$-0.450939\pi$
$401$	6.14843	0.307038	0.153519	+	0.988146i	$0.450939\pi$
$402$	0	0
$403$	−17.4542	−0.869456
$404$	4.44282	0.221038
$405$	0	0
$406$	0.0512334	0.00254267
$407$	40.2796	1.99658
$408$	0	0
$409$	−34.7647	−1.71900	−0.859502	−	0.511133i	$-0.829226\pi$
$409$	−34.7647	−1.71900	−0.859502	+	0.511133i	$0.829226\pi$
$410$	1.61719	0.0798674
$411$	0	0
$412$	−6.62444	−0.326363
$413$	−0.107372	−0.00528341
$414$	0	0
$415$	−4.15116	−0.203772
$416$	6.21005	0.304473
$417$	0	0
$418$	0.402352	0.0196797
$419$	−2.50658	−0.122454	−0.0612272	−	0.998124i	$-0.519501\pi$
$419$	−2.50658	−0.122454	−0.0612272	+	0.998124i	$0.519501\pi$
$420$	0	0
$421$	28.1397	1.37145	0.685723	−	0.727863i	$-0.259486\pi$
$421$	28.1397	1.37145	0.685723	+	0.727863i	$0.259486\pi$
$422$	−17.3992	−0.846979
$423$	0	0
$424$	−8.32188	−0.404146
$425$	0.0776013	0.00376422
$426$	0	0
$427$	−0.129806	−0.00628173
$428$	−5.57220	−0.269342
$429$	0	0
$430$	−3.62690	−0.174904
$431$	6.95736	0.335124	0.167562	−	0.985862i	$-0.446411\pi$
$431$	6.95736	0.335124	0.167562	+	0.985862i	$0.446411\pi$
$432$	0	0
$433$	−25.6108	−1.23078	−0.615388	−	0.788224i	$-0.711001\pi$
$433$	−25.6108	−1.23078	−0.615388	+	0.788224i	$0.711001\pi$
$434$	−0.0373463	−0.00179268
$435$	0	0
$436$	14.1181	0.676132
$437$	0.568919	0.0272151
$438$	0	0
$439$	−10.8651	−0.518563	−0.259281	−	0.965802i	$-0.583486\pi$
$439$	−10.8651	−0.518563	−0.259281	+	0.965802i	$0.583486\pi$
$440$	−6.08879	−0.290272
$441$	0	0
$442$	−0.138236	−0.00657521
$443$	27.5696	1.30987	0.654935	−	0.755685i	$-0.272696\pi$
$443$	27.5696	1.30987	0.654935	+	0.755685i	$0.272696\pi$
$444$	0	0
$445$	15.0183	0.711934
$446$	21.4489	1.01564
$447$	0	0
$448$	0.0132875	0.000627775 0
$449$	26.3072	1.24152	0.620758	−	0.784002i	$-0.286825\pi$
$449$	26.3072	1.24152	0.620758	+	0.784002i	$0.286825\pi$
$450$	0	0
$451$	−6.50435	−0.306278
$452$	1.91638	0.0901388
$453$	0	0
$454$	−7.79276	−0.365732
$455$	−0.101527	−0.00475967
$456$	0	0
$457$	−6.50176	−0.304139	−0.152070	−	0.988370i	$-0.548594\pi$
$457$	−6.50176	−0.304139	−0.152070	+	0.988370i	$0.548594\pi$
$458$	15.4567	0.722243
$459$	0	0
$460$	−8.60945	−0.401418
$461$	−13.6734	−0.636835	−0.318418	−	0.947951i	$-0.603151\pi$
$461$	−13.6734	−0.636835	−0.318418	+	0.947951i	$0.603151\pi$
$462$	0	0
$463$	8.96653	0.416710	0.208355	−	0.978053i	$-0.433189\pi$
$463$	8.96653	0.416710	0.208355	+	0.978053i	$0.433189\pi$
$464$	3.85576	0.178999
$465$	0	0
$466$	−15.6601	−0.725438
$467$	−35.3968	−1.63797	−0.818985	−	0.573815i	$-0.805463\pi$
$467$	−35.3968	−1.63797	−0.818985	+	0.573815i	$0.805463\pi$
$468$	0	0
$469$	0.155763	0.00719248
$470$	1.43885	0.0663694
$471$	0	0
$472$	−8.08065	−0.371942
$473$	14.5874	0.670729
$474$	0	0
$475$	−0.283441	−0.0130052
$476$	−0.000295780 0	−1.35570e−5 0
$477$	0	0
$478$	−4.96908	−0.227280
$479$	−6.70359	−0.306295	−0.153147	−	0.988203i	$-0.548941\pi$
$479$	−6.70359	−0.306295	−0.153147	+	0.988203i	$0.548941\pi$
$480$	0	0
$481$	50.5467	2.30473
$482$	6.87321	0.313066
$483$	0	0
$484$	13.4892	0.613144
$485$	−11.4995	−0.522168
$486$	0	0
$487$	−10.1850	−0.461527	−0.230764	−	0.973010i	$-0.574122\pi$
$487$	−10.1850	−0.461527	−0.230764	+	0.973010i	$0.574122\pi$
$488$	−9.76901	−0.442222
$489$	0	0
$490$	8.61254	0.389075
$491$	−2.46753	−0.111358	−0.0556791	−	0.998449i	$-0.517732\pi$
$491$	−2.46753	−0.111358	−0.0556791	+	0.998449i	$0.517732\pi$
$492$	0	0
$493$	−0.0858292	−0.00386555
$494$	0.504910	0.0227170
$495$	0	0
$496$	−2.81064	−0.126201
$497$	−0.0223827	−0.00100400
$498$	0	0
$499$	−30.2863	−1.35580	−0.677900	−	0.735155i	$-0.737110\pi$
$499$	−30.2863	−1.35580	−0.677900	+	0.735155i	$0.737110\pi$
$500$	10.4413	0.466948
$501$	0	0
$502$	−15.8669	−0.708176
$503$	−1.86012	−0.0829384	−0.0414692	−	0.999140i	$-0.513204\pi$
$503$	−1.86012	−0.0829384	−0.0414692	+	0.999140i	$0.513204\pi$
$504$	0	0
$505$	−5.46641	−0.243252
$506$	34.6273	1.53937
$507$	0	0
$508$	15.9605	0.708134
$509$	8.33701	0.369531	0.184766	−	0.982783i	$-0.440847\pi$
$509$	8.33701	0.369531	0.184766	+	0.982783i	$0.440847\pi$
$510$	0	0
$511$	0.215701	0.00954203
$512$	1.00000	0.0441942
$513$	0	0
$514$	30.2497	1.33426
$515$	8.15067	0.359161
$516$	0	0
$517$	−5.78708	−0.254516
$518$	0.108154	0.00475199
$519$	0	0
$520$	−7.64081	−0.335072
$521$	3.54896	0.155483	0.0777413	−	0.996974i	$-0.475229\pi$
$521$	3.54896	0.155483	0.0777413	+	0.996974i	$0.475229\pi$
$522$	0	0
$523$	−27.4107	−1.19859	−0.599293	−	0.800530i	$-0.704551\pi$
$523$	−27.4107	−1.19859	−0.599293	+	0.800530i	$0.704551\pi$
$524$	−19.4835	−0.851139
$525$	0	0
$526$	20.8380	0.908580
$527$	0.0625649	0.00272537
$528$	0	0
$529$	25.9624	1.12880
$530$	10.2392	0.444762
$531$	0	0
$532$	0.00108034	4.68388e−5 0
$533$	−8.16229	−0.353548
$534$	0	0
$535$	6.85600	0.296411
$536$	11.7226	0.506337
$537$	0	0
$538$	−21.5901	−0.930814
$539$	−34.6397	−1.49204
$540$	0	0
$541$	23.1583	0.995655	0.497827	−	0.867276i	$-0.334131\pi$
$541$	23.1583	0.995655	0.497827	+	0.867276i	$0.334131\pi$
$542$	−21.2284	−0.911839
$543$	0	0
$544$	−0.0222600	−0.000954390 0
$545$	−17.3708	−0.744082
$546$	0	0
$547$	−43.0677	−1.84144	−0.920722	−	0.390220i	$-0.872399\pi$
$547$	−43.0677	−1.84144	−0.920722	+	0.390220i	$0.872399\pi$
$548$	9.46101	0.404155
$549$	0	0
$550$	−17.2517	−0.735613
$551$	0.313494	0.0133553
$552$	0	0
$553$	−0.173846	−0.00739269
$554$	0.0377590	0.00160422
$555$	0	0
$556$	4.25411	0.180415
$557$	16.8389	0.713486	0.356743	−	0.934203i	$-0.383887\pi$
$557$	16.8389	0.713486	0.356743	+	0.934203i	$0.383887\pi$
$558$	0	0
$559$	18.3057	0.774248
$560$	−0.0163488	−0.000690865 0
$561$	0	0
$562$	12.0257	0.507275
$563$	11.6855	0.492484	0.246242	−	0.969208i	$-0.420804\pi$
$563$	11.6855	0.492484	0.246242	+	0.969208i	$0.420804\pi$
$564$	0	0
$565$	−2.35790	−0.0991975
$566$	−2.44057	−0.102585
$567$	0	0
$568$	−1.68449	−0.0706798
$569$	42.1826	1.76839	0.884193	−	0.467121i	$-0.154709\pi$
$569$	42.1826	1.76839	0.884193	+	0.467121i	$0.154709\pi$
$570$	0	0
$571$	27.4768	1.14987	0.574935	−	0.818199i	$-0.305027\pi$
$571$	27.4768	1.14987	0.574935	+	0.818199i	$0.305027\pi$
$572$	30.7314	1.28494
$573$	0	0
$574$	−0.0174647	−0.000728961 0
$575$	−24.3935	−1.01728
$576$	0	0
$577$	1.68764	0.0702575	0.0351288	−	0.999383i	$-0.488816\pi$
$577$	1.68764	0.0702575	0.0351288	+	0.999383i	$0.488816\pi$
$578$	−16.9995	−0.707086
$579$	0	0
$580$	−4.74410	−0.196988
$581$	0.0448299	0.00185986
$582$	0	0
$583$	−41.1821	−1.70559
$584$	16.2334	0.671741
$585$	0	0
$586$	11.8317	0.488764
$587$	−26.4988	−1.09372	−0.546860	−	0.837224i	$-0.684177\pi$
$587$	−26.4988	−1.09372	−0.546860	+	0.837224i	$0.684177\pi$
$588$	0	0
$589$	−0.228520	−0.00941600
$590$	9.94238	0.409321
$591$	0	0
$592$	8.13950	0.334531
$593$	0.800243	0.0328621	0.0164310	−	0.999865i	$-0.494770\pi$
$593$	0.800243	0.0328621	0.0164310	+	0.999865i	$0.494770\pi$
$594$	0	0
$595$	0.000363926 0	1.49195e−5 0
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	43.4537	1.77695
$599$	−36.5595	−1.49378	−0.746890	−	0.664947i	$-0.768454\pi$
$599$	−36.5595	−1.49378	−0.746890	+	0.664947i	$0.768454\pi$
$600$	0	0
$601$	25.2003	1.02794	0.513970	−	0.857808i	$-0.328174\pi$
$601$	25.2003	1.02794	0.513970	+	0.857808i	$0.328174\pi$
$602$	0.0391682	0.00159638
$603$	0	0
$604$	−5.66234	−0.230397
$605$	−16.5970	−0.674764
$606$	0	0
$607$	4.80462	0.195014	0.0975068	−	0.995235i	$-0.468913\pi$
$607$	4.80462	0.195014	0.0975068	+	0.995235i	$0.468913\pi$
$608$	0.0813053	0.00329737
$609$	0	0
$610$	12.0197	0.486665
$611$	−7.26219	−0.293797
$612$	0	0
$613$	18.2681	0.737841	0.368920	−	0.929461i	$-0.379727\pi$
$613$	18.2681	0.737841	0.368920	+	0.929461i	$0.379727\pi$
$614$	26.2111	1.05779
$615$	0	0
$616$	0.0657552	0.00264935
$617$	−35.2561	−1.41936	−0.709678	−	0.704526i	$-0.751160\pi$
$617$	−35.2561	−1.41936	−0.709678	+	0.704526i	$0.751160\pi$
$618$	0	0
$619$	39.6127	1.59217	0.796084	−	0.605186i	$-0.206901\pi$
$619$	39.6127	1.59217	0.796084	+	0.605186i	$0.206901\pi$
$620$	3.45819	0.138884
$621$	0	0
$622$	9.19816	0.368813
$623$	−0.162188	−0.00649792
$624$	0	0
$625$	4.58376	0.183350
$626$	19.1407	0.765018
$627$	0	0
$628$	−9.53252	−0.380389
$629$	−0.181185	−0.00722434
$630$	0	0
$631$	−4.40495	−0.175358	−0.0876791	−	0.996149i	$-0.527945\pi$
$631$	−4.40495	−0.175358	−0.0876791	+	0.996149i	$0.527945\pi$
$632$	−13.0834	−0.520431
$633$	0	0
$634$	−32.0692	−1.27363
$635$	−19.6377	−0.779300
$636$	0	0
$637$	−43.4693	−1.72231
$638$	19.0808	0.755417
$639$	0	0
$640$	−1.23039	−0.0486356
$641$	−37.9958	−1.50074	−0.750372	−	0.661016i	$-0.770125\pi$
$641$	−37.9958	−1.50074	−0.750372	+	0.661016i	$0.770125\pi$
$642$	0	0
$643$	16.1619	0.637363	0.318681	−	0.947862i	$-0.396760\pi$
$643$	16.1619	0.637363	0.318681	+	0.947862i	$0.396760\pi$
$644$	0.0929767	0.00366380
$645$	0	0
$646$	−0.00180986	−7.12079e−5 0
$647$	−44.5642	−1.75200	−0.876000	−	0.482311i	$-0.839797\pi$
$647$	−44.5642	−1.75200	−0.876000	+	0.482311i	$0.839797\pi$
$648$	0	0
$649$	−39.9883	−1.56968
$650$	−21.6490	−0.849145
$651$	0	0
$652$	11.4090	0.446811
$653$	−26.3430	−1.03088	−0.515440	−	0.856926i	$-0.672371\pi$
$653$	−26.3430	−1.03088	−0.515440	+	0.856926i	$0.672371\pi$
$654$	0	0
$655$	23.9723	0.936676
$656$	−1.31437	−0.0513175
$657$	0	0
$658$	−0.0155387	−0.000605763 0
$659$	−24.9337	−0.971278	−0.485639	−	0.874159i	$-0.661413\pi$
$659$	−24.9337	−0.971278	−0.485639	+	0.874159i	$0.661413\pi$
$660$	0	0
$661$	−4.48024	−0.174261	−0.0871306	−	0.996197i	$-0.527770\pi$
$661$	−4.48024	−0.174261	−0.0871306	+	0.996197i	$0.527770\pi$
$662$	9.69446	0.376786
$663$	0	0
$664$	3.37384	0.130930
$665$	−0.00132925	−5.15461e−5 0
$666$	0	0
$667$	26.9799	1.04467
$668$	2.88238	0.111523
$669$	0	0
$670$	−14.4234	−0.557223
$671$	−48.3434	−1.86628
$672$	0	0
$673$	−32.2827	−1.24441	−0.622204	−	0.782855i	$-0.713763\pi$
$673$	−32.2827	−1.24441	−0.622204	+	0.782855i	$0.713763\pi$
$674$	12.1879	0.469459
$675$	0	0
$676$	25.5647	0.983259
$677$	−38.2496	−1.47005	−0.735026	−	0.678039i	$-0.762830\pi$
$677$	−38.2496	−1.47005	−0.735026	+	0.678039i	$0.762830\pi$
$678$	0	0
$679$	0.124188	0.00476590
$680$	0.0273886	0.00105030
$681$	0	0
$682$	−13.9089	−0.532599
$683$	−12.5710	−0.481016	−0.240508	−	0.970647i	$-0.577314\pi$
$683$	−12.5710	−0.481016	−0.240508	+	0.970647i	$0.577314\pi$
$684$	0	0
$685$	−11.6408	−0.444771
$686$	−0.186023	−0.00710237
$687$	0	0
$688$	2.94775	0.112382
$689$	−51.6793	−1.96882
$690$	0	0
$691$	−7.46631	−0.284032	−0.142016	−	0.989864i	$-0.545358\pi$
$691$	−7.46631	−0.284032	−0.142016	+	0.989864i	$0.545358\pi$
$692$	−3.22628	−0.122645
$693$	0	0
$694$	22.1524	0.840895
$695$	−5.23424	−0.198546
$696$	0	0
$697$	0.0292579	0.00110822
$698$	18.0167	0.681942
$699$	0	0
$700$	−0.0463219	−0.00175080
$701$	−27.1451	−1.02526	−0.512629	−	0.858610i	$-0.671328\pi$
$701$	−27.1451	−1.02526	−0.512629	+	0.858610i	$0.671328\pi$
$702$	0	0
$703$	0.661785	0.0249597
$704$	4.94865	0.186509
$705$	0	0
$706$	−10.1474	−0.381904
$707$	0.0590339	0.00222020
$708$	0	0
$709$	33.2194	1.24758	0.623791	−	0.781591i	$-0.285592\pi$
$709$	33.2194	1.24758	0.623791	+	0.781591i	$0.285592\pi$
$710$	2.07259	0.0777829
$711$	0	0
$712$	−12.2061	−0.457442
$713$	−19.6669	−0.736532
$714$	0	0
$715$	−37.8117	−1.41408
$716$	7.84987	0.293363
$717$	0	0
$718$	29.4270	1.09821
$719$	−34.3815	−1.28221	−0.641107	−	0.767451i	$-0.721525\pi$
$719$	−34.3815	−1.28221	−0.641107	+	0.767451i	$0.721525\pi$
$720$	0	0
$721$	−0.0880221	−0.00327812
$722$	−18.9934	−0.706861
$723$	0	0
$724$	−11.4292	−0.424762
$725$	−13.4417	−0.499211
$726$	0	0
$727$	11.6064	0.430457	0.215229	−	0.976564i	$-0.430950\pi$
$727$	11.6064	0.430457	0.215229	+	0.976564i	$0.430950\pi$
$728$	0.0825160	0.00305825
$729$	0	0
$730$	−19.9734	−0.739250
$731$	−0.0656170	−0.00242693
$732$	0	0
$733$	−16.0040	−0.591121	−0.295561	−	0.955324i	$-0.595506\pi$
$733$	−16.0040	−0.591121	−0.295561	+	0.955324i	$0.595506\pi$
$734$	22.1630	0.818050
$735$	0	0
$736$	6.99731	0.257924
$737$	58.0109	2.13686
$738$	0	0
$739$	9.93953	0.365632	0.182816	−	0.983147i	$-0.441479\pi$
$739$	9.93953	0.365632	0.182816	+	0.983147i	$0.441479\pi$
$740$	−10.0148	−0.368151
$741$	0	0
$742$	−0.110577	−0.00405941
$743$	2.90735	0.106660	0.0533302	−	0.998577i	$-0.483016\pi$
$743$	2.90735	0.106660	0.0533302	+	0.998577i	$0.483016\pi$
$744$	0	0
$745$	1.23039	0.0450782
$746$	12.1239	0.443888
$747$	0	0
$748$	−0.110157	−0.00402774
$749$	−0.0740405	−0.00270538
$750$	0	0
$751$	−24.8677	−0.907435	−0.453717	−	0.891146i	$-0.649902\pi$
$751$	−24.8677	−0.907435	−0.453717	+	0.891146i	$0.649902\pi$
$752$	−1.16943	−0.0426446
$753$	0	0
$754$	23.9445	0.872006
$755$	6.96691	0.253552
$756$	0	0
$757$	−45.7132	−1.66147	−0.830737	−	0.556665i	$-0.812081\pi$
$757$	−45.7132	−1.66147	−0.830737	+	0.556665i	$0.812081\pi$
$758$	−21.0325	−0.763935
$759$	0	0
$760$	−0.100038	−0.00362875
$761$	−47.1571	−1.70944	−0.854722	−	0.519085i	$-0.826273\pi$
$761$	−47.1571	−1.70944	−0.854722	+	0.519085i	$0.826273\pi$
$762$	0	0
$763$	0.187593	0.00679134
$764$	11.7177	0.423933
$765$	0	0
$766$	−14.3893	−0.519906
$767$	−50.1812	−1.81194
$768$	0	0
$769$	16.9976	0.612948	0.306474	−	0.951879i	$-0.400851\pi$
$769$	16.9976	0.612948	0.306474	+	0.951879i	$0.400851\pi$
$770$	−0.0809048	−0.00291561
$771$	0	0
$772$	21.6129	0.777866
$773$	29.3643	1.05616	0.528080	−	0.849195i	$-0.322912\pi$
$773$	29.3643	1.05616	0.528080	+	0.849195i	$0.322912\pi$
$774$	0	0
$775$	9.79826	0.351964
$776$	9.34623	0.335510
$777$	0	0
$778$	−34.1218	−1.22332
$779$	−0.106865	−0.00382884
$780$	0	0
$781$	−8.33597	−0.298285
$782$	−0.155760	−0.00556998
$783$	0	0
$784$	−6.99982	−0.249994
$785$	11.7288	0.418617
$786$	0	0
$787$	43.2924	1.54321	0.771604	−	0.636104i	$-0.219455\pi$
$787$	43.2924	1.54321	0.771604	+	0.636104i	$0.219455\pi$
$788$	−27.7107	−0.987154
$789$	0	0
$790$	16.0978	0.572733
$791$	0.0254638	0.000905390 0
$792$	0	0
$793$	−60.6660	−2.15431
$794$	3.97908	0.141212
$795$	0	0
$796$	1.93424	0.0685575
$797$	−17.4867	−0.619412	−0.309706	−	0.950832i	$-0.600231\pi$
$797$	−17.4867	−0.619412	−0.309706	+	0.950832i	$0.600231\pi$
$798$	0	0
$799$	0.0260314	0.000920926 0
$800$	−3.48613	−0.123253
$801$	0	0
$802$	6.14843	0.217109
$803$	80.3333	2.83490
$804$	0	0
$805$	−0.114398	−0.00403200
$806$	−17.4542	−0.614799
$807$	0	0
$808$	4.44282	0.156298
$809$	5.88250	0.206818	0.103409	−	0.994639i	$-0.467025\pi$
$809$	5.88250	0.206818	0.103409	+	0.994639i	$0.467025\pi$
$810$	0	0
$811$	7.14956	0.251055	0.125527	−	0.992090i	$-0.459938\pi$
$811$	7.14956	0.251055	0.125527	+	0.992090i	$0.459938\pi$
$812$	0.0512334	0.00179794
$813$	0	0
$814$	40.2796	1.41180
$815$	−14.0376	−0.491715
$816$	0	0
$817$	0.239668	0.00838492
$818$	−34.7647	−1.21552
$819$	0	0
$820$	1.61719	0.0564748
$821$	36.7677	1.28320	0.641601	−	0.767039i	$-0.278271\pi$
$821$	36.7677	1.28320	0.641601	+	0.767039i	$0.278271\pi$
$822$	0	0
$823$	0.493333	0.0171965	0.00859825	−	0.999963i	$-0.497263\pi$
$823$	0.493333	0.0171965	0.00859825	+	0.999963i	$0.497263\pi$
$824$	−6.62444	−0.230773
$825$	0	0
$826$	−0.107372	−0.00373593
$827$	−39.1701	−1.36208	−0.681039	−	0.732247i	$-0.738472\pi$
$827$	−39.1701	−1.36208	−0.681039	+	0.732247i	$0.738472\pi$
$828$	0	0
$829$	7.22829	0.251049	0.125524	−	0.992091i	$-0.459939\pi$
$829$	7.22829	0.251049	0.125524	+	0.992091i	$0.459939\pi$
$830$	−4.15116	−0.144089
$831$	0	0
$832$	6.21005	0.215295
$833$	0.155816	0.00539871
$834$	0	0
$835$	−3.54647	−0.122731
$836$	0.402352	0.0139156
$837$	0	0
$838$	−2.50658	−0.0865883
$839$	23.9071	0.825366	0.412683	−	0.910875i	$-0.364592\pi$
$839$	23.9071	0.825366	0.412683	+	0.910875i	$0.364592\pi$
$840$	0	0
$841$	−14.1331	−0.487349
$842$	28.1397	0.969759
$843$	0	0
$844$	−17.3992	−0.598905
$845$	−31.4547	−1.08207
$846$	0	0
$847$	0.179237	0.00615867
$848$	−8.32188	−0.285774
$849$	0	0
$850$	0.0776013	0.00266170
$851$	56.9546	1.95238
$852$	0	0
$853$	16.0219	0.548578	0.274289	−	0.961647i	$-0.411557\pi$
$853$	16.0219	0.548578	0.274289	+	0.961647i	$0.411557\pi$
$854$	−0.129806	−0.00444186
$855$	0	0
$856$	−5.57220	−0.190454
$857$	30.3041	1.03517	0.517584	−	0.855632i	$-0.326831\pi$
$857$	30.3041	1.03517	0.517584	+	0.855632i	$0.326831\pi$
$858$	0	0
$859$	37.4753	1.27864	0.639320	−	0.768941i	$-0.279216\pi$
$859$	37.4753	1.27864	0.639320	+	0.768941i	$0.279216\pi$
$860$	−3.62690	−0.123676
$861$	0	0
$862$	6.95736	0.236968
$863$	33.6153	1.14428	0.572139	−	0.820157i	$-0.306114\pi$
$863$	33.6153	1.14428	0.572139	+	0.820157i	$0.306114\pi$
$864$	0	0
$865$	3.96959	0.134970
$866$	−25.6108	−0.870290
$867$	0	0
$868$	−0.0373463	−0.00126762
$869$	−64.7454	−2.19634
$870$	0	0
$871$	72.7977	2.46666
$872$	14.1181	0.478098
$873$	0	0
$874$	0.568919	0.0192440
$875$	0.138738	0.00469022
$876$	0	0
$877$	−0.845610	−0.0285542	−0.0142771	−	0.999898i	$-0.504545\pi$
$877$	−0.845610	−0.0285542	−0.0142771	+	0.999898i	$0.504545\pi$
$878$	−10.8651	−0.366679
$879$	0	0
$880$	−6.08879	−0.205253
$881$	−46.3821	−1.56265	−0.781327	−	0.624122i	$-0.785457\pi$
$881$	−46.3821	−1.56265	−0.781327	+	0.624122i	$0.785457\pi$
$882$	0	0
$883$	25.8132	0.868684	0.434342	−	0.900748i	$-0.356981\pi$
$883$	25.8132	0.868684	0.434342	+	0.900748i	$0.356981\pi$
$884$	−0.138236	−0.00464937
$885$	0	0
$886$	27.5696	0.926218
$887$	−6.58446	−0.221084	−0.110542	−	0.993871i	$-0.535259\pi$
$887$	−6.58446	−0.221084	−0.110542	+	0.993871i	$0.535259\pi$
$888$	0	0
$889$	0.212075	0.00711278
$890$	15.0183	0.503414
$891$	0	0
$892$	21.4489	0.718162
$893$	−0.0950805	−0.00318175
$894$	0	0
$895$	−9.65843	−0.322846
$896$	0.0132875	0.000443904 0
$897$	0	0
$898$	26.3072	0.877884
$899$	−10.8371	−0.361439
$900$	0	0
$901$	0.185245	0.00617141
$902$	−6.50435	−0.216571
$903$	0	0
$904$	1.91638	0.0637377
$905$	14.0624	0.467450
$906$	0	0
$907$	3.46400	0.115020	0.0575101	−	0.998345i	$-0.481684\pi$
$907$	3.46400	0.115020	0.0575101	+	0.998345i	$0.481684\pi$
$908$	−7.79276	−0.258612
$909$	0	0
$910$	−0.101527	−0.00336559
$911$	37.9740	1.25814	0.629068	−	0.777350i	$-0.283437\pi$
$911$	37.9740	1.25814	0.629068	+	0.777350i	$0.283437\pi$
$912$	0	0
$913$	16.6960	0.552556
$914$	−6.50176	−0.215059
$915$	0	0
$916$	15.4567	0.510703
$917$	−0.258886	−0.00854917
$918$	0	0
$919$	−50.0980	−1.65258	−0.826290	−	0.563245i	$-0.809553\pi$
$919$	−50.0980	−1.65258	−0.826290	+	0.563245i	$0.809553\pi$
$920$	−8.60945	−0.283845
$921$	0	0
$922$	−13.6734	−0.450311
$923$	−10.4608	−0.344321
$924$	0	0
$925$	−28.3754	−0.932976
$926$	8.96653	0.294659
$927$	0	0
$928$	3.85576	0.126571
$929$	13.8674	0.454976	0.227488	−	0.973781i	$-0.426949\pi$
$929$	13.8674	0.454976	0.227488	+	0.973781i	$0.426949\pi$
$930$	0	0
$931$	−0.569123	−0.0186523
$932$	−15.6601	−0.512962
$933$	0	0
$934$	−35.3968	−1.15822
$935$	0.135537	0.00443252
$936$	0	0
$937$	−7.40116	−0.241785	−0.120893	−	0.992666i	$-0.538576\pi$
$937$	−7.40116	−0.241785	−0.120893	+	0.992666i	$0.538576\pi$
$938$	0.155763	0.00508585
$939$	0	0
$940$	1.43885	0.0469302
$941$	−14.2713	−0.465230	−0.232615	−	0.972569i	$-0.574728\pi$
$941$	−14.2713	−0.465230	−0.232615	+	0.972569i	$0.574728\pi$
$942$	0	0
$943$	−9.19705	−0.299497
$944$	−8.08065	−0.263003
$945$	0	0
$946$	14.5874	0.474277
$947$	15.4148	0.500914	0.250457	−	0.968128i	$-0.419419\pi$
$947$	15.4148	0.500914	0.250457	+	0.968128i	$0.419419\pi$
$948$	0	0
$949$	100.810	3.27243
$950$	−0.283441	−0.00919604
$951$	0	0
$952$	−0.000295780 0	−9.58627e−6 0
$953$	−28.1256	−0.911077	−0.455538	−	0.890216i	$-0.650553\pi$
$953$	−28.1256	−0.911077	−0.455538	+	0.890216i	$0.650553\pi$
$954$	0	0
$955$	−14.4174	−0.466538
$956$	−4.96908	−0.160712
$957$	0	0
$958$	−6.70359	−0.216583
$959$	0.125713	0.00405949
$960$	0	0
$961$	−23.1003	−0.745171
$962$	50.5467	1.62969
$963$	0	0
$964$	6.87321	0.221371
$965$	−26.5924	−0.856040
$966$	0	0
$967$	6.31009	0.202919	0.101459	−	0.994840i	$-0.467649\pi$
$967$	6.31009	0.202919	0.101459	+	0.994840i	$0.467649\pi$
$968$	13.4892	0.433558
$969$	0	0
$970$	−11.4995	−0.369228
$971$	28.8500	0.925840	0.462920	−	0.886400i	$-0.346802\pi$
$971$	28.8500	0.925840	0.462920	+	0.886400i	$0.346802\pi$
$972$	0	0
$973$	0.0565265	0.00181216
$974$	−10.1850	−0.326349
$975$	0	0
$976$	−9.76901	−0.312698
$977$	−9.09724	−0.291047	−0.145523	−	0.989355i	$-0.546487\pi$
$977$	−9.09724	−0.291047	−0.145523	+	0.989355i	$0.546487\pi$
$978$	0	0
$979$	−60.4036	−1.93051
$980$	8.61254	0.275118
$981$	0	0
$982$	−2.46753	−0.0787421
$983$	0.289539	0.00923486	0.00461743	−	0.999989i	$-0.498530\pi$
$983$	0.289539	0.00923486	0.00461743	+	0.999989i	$0.498530\pi$
$984$	0	0
$985$	34.0951	1.08636
$986$	−0.0858292	−0.00273336
$987$	0	0
$988$	0.504910	0.0160633
$989$	20.6263	0.655879
$990$	0	0
$991$	−2.81375	−0.0893816	−0.0446908	−	0.999001i	$-0.514230\pi$
$991$	−2.81375	−0.0893816	−0.0446908	+	0.999001i	$0.514230\pi$
$992$	−2.81064	−0.0892379
$993$	0	0
$994$	−0.0223827	−0.000709936 0
$995$	−2.37988	−0.0754474
$996$	0	0
$997$	−25.2654	−0.800162	−0.400081	−	0.916480i	$-0.631018\pi$
$997$	−25.2654	−0.800162	−0.400081	+	0.916480i	$0.631018\pi$
$998$	−30.2863	−0.958695
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.t.1.5	yes	16
3.2	odd	2		8046.2.a.s.1.12	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.s.1.12	✓	16	3.2	odd	2
8046.2.a.t.1.5	yes	16	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.23039 q^{5} +0.0132875 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.23039 q^{5} +0.0132875 q^{7} +1.00000 q^{8} -1.23039 q^{10} +4.94865 q^{11} +6.21005 q^{13} +0.0132875 q^{14} +1.00000 q^{16} -0.0222600 q^{17} +0.0813053 q^{19} -1.23039 q^{20} +4.94865 q^{22} +6.99731 q^{23} -3.48613 q^{25} +6.21005 q^{26} +0.0132875 q^{28} +3.85576 q^{29} -2.81064 q^{31} +1.00000 q^{32} -0.0222600 q^{34} -0.0163488 q^{35} +8.13950 q^{37} +0.0813053 q^{38} -1.23039 q^{40} -1.31437 q^{41} +2.94775 q^{43} +4.94865 q^{44} +6.99731 q^{46} -1.16943 q^{47} -6.99982 q^{49} -3.48613 q^{50} +6.21005 q^{52} -8.32188 q^{53} -6.08879 q^{55} +0.0132875 q^{56} +3.85576 q^{58} -8.08065 q^{59} -9.76901 q^{61} -2.81064 q^{62} +1.00000 q^{64} -7.64081 q^{65} +11.7226 q^{67} -0.0222600 q^{68} -0.0163488 q^{70} -1.68449 q^{71} +16.2334 q^{73} +8.13950 q^{74} +0.0813053 q^{76} +0.0657552 q^{77} -13.0834 q^{79} -1.23039 q^{80} -1.31437 q^{82} +3.37384 q^{83} +0.0273886 q^{85} +2.94775 q^{86} +4.94865 q^{88} -12.2061 q^{89} +0.0825160 q^{91} +6.99731 q^{92} -1.16943 q^{94} -0.100038 q^{95} +9.34623 q^{97} -6.99982 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8} + 4 q^{10} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 10 q^{19} + 4 q^{20} + 6 q^{22} + 10 q^{23} + 28 q^{25} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 21 q^{31} + 16 q^{32} + q^{34} + 16 q^{35} + 17 q^{37} + 10 q^{38} + 4 q^{40} - 4 q^{41} + 16 q^{43} + 6 q^{44} + 10 q^{46} + 25 q^{47} + 36 q^{49} + 28 q^{50} + 6 q^{52} + 14 q^{53} + 19 q^{55} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 23 q^{61} + 21 q^{62} + 16 q^{64} + 20 q^{65} + 22 q^{67} + q^{68} + 16 q^{70} + 10 q^{71} + 16 q^{73} + 17 q^{74} + 10 q^{76} - 2 q^{77} + 37 q^{79} + 4 q^{80} - 4 q^{82} + 33 q^{83} + 43 q^{85} + 16 q^{86} + 6 q^{88} - 3 q^{89} + 28 q^{91} + 10 q^{92} + 25 q^{94} + 14 q^{95} - 3 q^{97} + 36 q^{98}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8 + 4 * q^10 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 10 * q^19 + 4 * q^20 + 6 * q^22 + 10 * q^23 + 28 * q^25 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 21 * q^31 + 16 * q^32 + q^34 + 16 * q^35 + 17 * q^37 + 10 * q^38 + 4 * q^40 - 4 * q^41 + 16 * q^43 + 6 * q^44 + 10 * q^46 + 25 * q^47 + 36 * q^49 + 28 * q^50 + 6 * q^52 + 14 * q^53 + 19 * q^55 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 23 * q^61 + 21 * q^62 + 16 * q^64 + 20 * q^65 + 22 * q^67 + q^68 + 16 * q^70 + 10 * q^71 + 16 * q^73 + 17 * q^74 + 10 * q^76 - 2 * q^77 + 37 * q^79 + 4 * q^80 - 4 * q^82 + 33 * q^83 + 43 * q^85 + 16 * q^86 + 6 * q^88 - 3 * q^89 + 28 * q^91 + 10 * q^92 + 25 * q^94 + 14 * q^95 - 3 * q^97 + 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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