LMFDB - Embedded newform 8046.2.a.o.1.12

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 29 x^{10} + 76 x^{9} + 320 x^{8} - 724 x^{7} - 1643 x^{6} + 3265 x^{5} + 3921 x^{4} + \cdots + 423 $ x^12 - 3x^11 - 29x^10 + 76x^9 + 320x^8 - 724x^7 - 1643x^6 + 3265x^5 + 3921x^4 - 6927x^3 - 3639x^2 + 5508*x + 423
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			1.12
Root			$4.06809$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +4.06809 q^{5} +2.63250 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +4.06809 q^{5} +2.63250 q^{7} +1.00000 q^{8} +4.06809 q^{10} +1.07210 q^{11} +2.84696 q^{13} +2.63250 q^{14} +1.00000 q^{16} -0.331985 q^{17} +0.959153 q^{19} +4.06809 q^{20} +1.07210 q^{22} +5.80459 q^{23} +11.5494 q^{25} +2.84696 q^{26} +2.63250 q^{28} -4.87419 q^{29} -5.37845 q^{31} +1.00000 q^{32} -0.331985 q^{34} +10.7093 q^{35} -3.06600 q^{37} +0.959153 q^{38} +4.06809 q^{40} +2.68806 q^{41} -2.54945 q^{43} +1.07210 q^{44} +5.80459 q^{46} -2.19087 q^{47} -0.0699233 q^{49} +11.5494 q^{50} +2.84696 q^{52} +2.65038 q^{53} +4.36141 q^{55} +2.63250 q^{56} -4.87419 q^{58} -0.987970 q^{59} -6.17092 q^{61} -5.37845 q^{62} +1.00000 q^{64} +11.5817 q^{65} +7.95555 q^{67} -0.331985 q^{68} +10.7093 q^{70} -11.5304 q^{71} -3.22607 q^{73} -3.06600 q^{74} +0.959153 q^{76} +2.82232 q^{77} -2.48939 q^{79} +4.06809 q^{80} +2.68806 q^{82} -4.40459 q^{83} -1.35055 q^{85} -2.54945 q^{86} +1.07210 q^{88} -5.09083 q^{89} +7.49464 q^{91} +5.80459 q^{92} -2.19087 q^{94} +3.90192 q^{95} -3.11689 q^{97} -0.0699233 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8} + 3 q^{10} + 10 q^{11} + 5 q^{13} + 6 q^{14} + 12 q^{16} + 8 q^{17} + 2 q^{19} + 3 q^{20} + 10 q^{22} + 9 q^{23} + 7 q^{25} + 5 q^{26} + 6 q^{28} + 19 q^{29} + 10 q^{31} + 12 q^{32} + 8 q^{34} + 20 q^{35} + 11 q^{37} + 2 q^{38} + 3 q^{40} + 8 q^{41} + 13 q^{43} + 10 q^{44} + 9 q^{46} + 11 q^{47} + 2 q^{49} + 7 q^{50} + 5 q^{52} + 24 q^{53} + 3 q^{55} + 6 q^{56} + 19 q^{58} + 10 q^{59} + 10 q^{62} + 12 q^{64} + 28 q^{65} + 21 q^{67} + 8 q^{68} + 20 q^{70} + 37 q^{71} - 2 q^{73} + 11 q^{74} + 2 q^{76} + 2 q^{77} + 7 q^{79} + 3 q^{80} + 8 q^{82} + 22 q^{83} + 15 q^{85} + 13 q^{86} + 10 q^{88} + 40 q^{89} + q^{91} + 9 q^{92} + 11 q^{94} + 11 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8 + 3 * q^10 + 10 * q^11 + 5 * q^13 + 6 * q^14 + 12 * q^16 + 8 * q^17 + 2 * q^19 + 3 * q^20 + 10 * q^22 + 9 * q^23 + 7 * q^25 + 5 * q^26 + 6 * q^28 + 19 * q^29 + 10 * q^31 + 12 * q^32 + 8 * q^34 + 20 * q^35 + 11 * q^37 + 2 * q^38 + 3 * q^40 + 8 * q^41 + 13 * q^43 + 10 * q^44 + 9 * q^46 + 11 * q^47 + 2 * q^49 + 7 * q^50 + 5 * q^52 + 24 * q^53 + 3 * q^55 + 6 * q^56 + 19 * q^58 + 10 * q^59 + 10 * q^62 + 12 * q^64 + 28 * q^65 + 21 * q^67 + 8 * q^68 + 20 * q^70 + 37 * q^71 - 2 * q^73 + 11 * q^74 + 2 * q^76 + 2 * q^77 + 7 * q^79 + 3 * q^80 + 8 * q^82 + 22 * q^83 + 15 * q^85 + 13 * q^86 + 10 * q^88 + 40 * q^89 + q^91 + 9 * q^92 + 11 * q^94 + 11 * q^95 + 7 * q^97 + 2 * 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
$5$	4.06809	1.81931	0.909653	−	0.415370i	$-0.136348\pi$
$5$	4.06809	1.81931	0.909653	+	0.415370i	$0.136348\pi$
$6$	0	0
$7$	2.63250	0.994993	0.497496	−	0.867466i	$-0.334253\pi$
$7$	2.63250	0.994993	0.497496	+	0.867466i	$0.334253\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	4.06809	1.28644
$11$	1.07210	0.323251	0.161626	−	0.986852i	$-0.448326\pi$
$11$	1.07210	0.323251	0.161626	+	0.986852i	$0.448326\pi$
$12$	0	0
$13$	2.84696	0.789605	0.394803	−	0.918766i	$-0.370813\pi$
$13$	2.84696	0.789605	0.394803	+	0.918766i	$0.370813\pi$
$14$	2.63250	0.703566
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.331985	−0.0805182	−0.0402591	−	0.999189i	$-0.512818\pi$
$17$	−0.331985	−0.0805182	−0.0402591	+	0.999189i	$0.512818\pi$
$18$	0	0
$19$	0.959153	0.220045	0.110022	−	0.993929i	$-0.464908\pi$
$19$	0.959153	0.220045	0.110022	+	0.993929i	$0.464908\pi$
$20$	4.06809	0.909653
$21$	0	0
$22$	1.07210	0.228573
$23$	5.80459	1.21034	0.605170	−	0.796096i	$-0.293105\pi$
$23$	5.80459	1.21034	0.605170	+	0.796096i	$0.293105\pi$
$24$	0	0
$25$	11.5494	2.30987
$26$	2.84696	0.558335
$27$	0	0
$28$	2.63250	0.497496
$29$	−4.87419	−0.905114	−0.452557	−	0.891735i	$-0.649488\pi$
$29$	−4.87419	−0.905114	−0.452557	+	0.891735i	$0.649488\pi$
$30$	0	0
$31$	−5.37845	−0.965999	−0.482999	−	0.875621i	$-0.660453\pi$
$31$	−5.37845	−0.965999	−0.482999	+	0.875621i	$0.660453\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.331985	−0.0569350
$35$	10.7093	1.81020
$36$	0	0
$37$	−3.06600	−0.504047	−0.252023	−	0.967721i	$-0.581096\pi$
$37$	−3.06600	−0.504047	−0.252023	+	0.967721i	$0.581096\pi$
$38$	0.959153	0.155595
$39$	0	0
$40$	4.06809	0.643222
$41$	2.68806	0.419804	0.209902	−	0.977722i	$-0.432686\pi$
$41$	2.68806	0.419804	0.209902	+	0.977722i	$0.432686\pi$
$42$	0	0
$43$	−2.54945	−0.388788	−0.194394	−	0.980924i	$-0.562274\pi$
$43$	−2.54945	−0.388788	−0.194394	+	0.980924i	$0.562274\pi$
$44$	1.07210	0.161626
$45$	0	0
$46$	5.80459	0.855840
$47$	−2.19087	−0.319572	−0.159786	−	0.987152i	$-0.551080\pi$
$47$	−2.19087	−0.319572	−0.159786	+	0.987152i	$0.551080\pi$
$48$	0	0
$49$	−0.0699233	−0.00998905
$50$	11.5494	1.63333
$51$	0	0
$52$	2.84696	0.394803
$53$	2.65038	0.364057	0.182029	−	0.983293i	$-0.441734\pi$
$53$	2.65038	0.364057	0.182029	+	0.983293i	$0.441734\pi$
$54$	0	0
$55$	4.36141	0.588093
$56$	2.63250	0.351783
$57$	0	0
$58$	−4.87419	−0.640012
$59$	−0.987970	−0.128623	−0.0643114	−	0.997930i	$-0.520485\pi$
$59$	−0.987970	−0.128623	−0.0643114	+	0.997930i	$0.520485\pi$
$60$	0	0
$61$	−6.17092	−0.790105	−0.395053	−	0.918658i	$-0.629274\pi$
$61$	−6.17092	−0.790105	−0.395053	+	0.918658i	$0.629274\pi$
$62$	−5.37845	−0.683064
$63$	0	0
$64$	1.00000	0.125000
$65$	11.5817	1.43653
$66$	0	0
$67$	7.95555	0.971925	0.485963	−	0.873980i	$-0.338469\pi$
$67$	7.95555	0.971925	0.485963	+	0.873980i	$0.338469\pi$
$68$	−0.331985	−0.0402591
$69$	0	0
$70$	10.7093	1.28000
$71$	−11.5304	−1.36841	−0.684204	−	0.729291i	$-0.739850\pi$
$71$	−11.5304	−1.36841	−0.684204	+	0.729291i	$0.739850\pi$
$72$	0	0
$73$	−3.22607	−0.377583	−0.188791	−	0.982017i	$-0.560457\pi$
$73$	−3.22607	−0.377583	−0.188791	+	0.982017i	$0.560457\pi$
$74$	−3.06600	−0.356415
$75$	0	0
$76$	0.959153	0.110022
$77$	2.82232	0.321633
$78$	0	0
$79$	−2.48939	−0.280078	−0.140039	−	0.990146i	$-0.544723\pi$
$79$	−2.48939	−0.280078	−0.140039	+	0.990146i	$0.544723\pi$
$80$	4.06809	0.454826
$81$	0	0
$82$	2.68806	0.296846
$83$	−4.40459	−0.483467	−0.241733	−	0.970343i	$-0.577716\pi$
$83$	−4.40459	−0.483467	−0.241733	+	0.970343i	$0.577716\pi$
$84$	0	0
$85$	−1.35055	−0.146487
$86$	−2.54945	−0.274915
$87$	0	0
$88$	1.07210	0.114287
$89$	−5.09083	−0.539627	−0.269813	−	0.962913i	$-0.586962\pi$
$89$	−5.09083	−0.539627	−0.269813	+	0.962913i	$0.586962\pi$
$90$	0	0
$91$	7.49464	0.785652
$92$	5.80459	0.605170
$93$	0	0
$94$	−2.19087	−0.225971
$95$	3.90192	0.400328
$96$	0	0
$97$	−3.11689	−0.316473	−0.158236	−	0.987401i	$-0.550581\pi$
$97$	−3.11689	−0.316473	−0.158236	+	0.987401i	$0.550581\pi$
$98$	−0.0699233	−0.00706332
$99$	0	0
$100$	11.5494	1.15494
$101$	−11.1240	−1.10688	−0.553440	−	0.832889i	$-0.686685\pi$
$101$	−11.1240	−1.10688	−0.553440	+	0.832889i	$0.686685\pi$
$102$	0	0
$103$	11.6650	1.14939	0.574693	−	0.818369i	$-0.305121\pi$
$103$	11.6650	1.14939	0.574693	+	0.818369i	$0.305121\pi$
$104$	2.84696	0.279168
$105$	0	0
$106$	2.65038	0.257427
$107$	0.353561	0.0341800	0.0170900	−	0.999854i	$-0.494560\pi$
$107$	0.353561	0.0341800	0.0170900	+	0.999854i	$0.494560\pi$
$108$	0	0
$109$	−2.60687	−0.249692	−0.124846	−	0.992176i	$-0.539844\pi$
$109$	−2.60687	−0.249692	−0.124846	+	0.992176i	$0.539844\pi$
$110$	4.36141	0.415844
$111$	0	0
$112$	2.63250	0.248748
$113$	8.85203	0.832729	0.416365	−	0.909198i	$-0.363304\pi$
$113$	8.85203	0.832729	0.416365	+	0.909198i	$0.363304\pi$
$114$	0	0
$115$	23.6136	2.20198
$116$	−4.87419	−0.452557
$117$	0	0
$118$	−0.987970	−0.0909500
$119$	−0.873952	−0.0801150
$120$	0	0
$121$	−9.85059	−0.895509
$122$	−6.17092	−0.558689
$123$	0	0
$124$	−5.37845	−0.482999
$125$	26.6434	2.38306
$126$	0	0
$127$	−3.51949	−0.312304	−0.156152	−	0.987733i	$-0.549909\pi$
$127$	−3.51949	−0.312304	−0.156152	+	0.987733i	$0.549909\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	11.5817	1.01578
$131$	−5.39749	−0.471581	−0.235790	−	0.971804i	$-0.575768\pi$
$131$	−5.39749	−0.471581	−0.235790	+	0.971804i	$0.575768\pi$
$132$	0	0
$133$	2.52497	0.218943
$134$	7.95555	0.687255
$135$	0	0
$136$	−0.331985	−0.0284675
$137$	−18.7677	−1.60343	−0.801716	−	0.597705i	$-0.796079\pi$
$137$	−18.7677	−1.60343	−0.801716	+	0.597705i	$0.796079\pi$
$138$	0	0
$139$	−11.7601	−0.997477	−0.498739	−	0.866752i	$-0.666203\pi$
$139$	−11.7601	−0.997477	−0.498739	+	0.866752i	$0.666203\pi$
$140$	10.7093	0.905098
$141$	0	0
$142$	−11.5304	−0.967610
$143$	3.05224	0.255241
$144$	0	0
$145$	−19.8286	−1.64668
$146$	−3.22607	−0.266991
$147$	0	0
$148$	−3.06600	−0.252023
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	11.1478	0.907198	0.453599	−	0.891206i	$-0.350140\pi$
$151$	11.1478	0.907198	0.453599	+	0.891206i	$0.350140\pi$
$152$	0.959153	0.0777975
$153$	0	0
$154$	2.82232	0.227429
$155$	−21.8800	−1.75745
$156$	0	0
$157$	3.29831	0.263234	0.131617	−	0.991301i	$-0.457983\pi$
$157$	3.29831	0.263234	0.131617	+	0.991301i	$0.457983\pi$
$158$	−2.48939	−0.198045
$159$	0	0
$160$	4.06809	0.321611
$161$	15.2806	1.20428
$162$	0	0
$163$	−3.10966	−0.243567	−0.121784	−	0.992557i	$-0.538861\pi$
$163$	−3.10966	−0.243567	−0.121784	+	0.992557i	$0.538861\pi$
$164$	2.68806	0.209902
$165$	0	0
$166$	−4.40459	−0.341863
$167$	10.3945	0.804349	0.402175	−	0.915563i	$-0.368254\pi$
$167$	10.3945	0.804349	0.402175	+	0.915563i	$0.368254\pi$
$168$	0	0
$169$	−4.89480	−0.376523
$170$	−1.35055	−0.103582
$171$	0	0
$172$	−2.54945	−0.194394
$173$	−17.8165	−1.35456	−0.677281	−	0.735724i	$-0.736842\pi$
$173$	−17.8165	−1.35456	−0.677281	+	0.735724i	$0.736842\pi$
$174$	0	0
$175$	30.4037	2.29831
$176$	1.07210	0.0808128
$177$	0	0
$178$	−5.09083	−0.381574
$179$	3.52763	0.263668	0.131834	−	0.991272i	$-0.457913\pi$
$179$	3.52763	0.263668	0.131834	+	0.991272i	$0.457913\pi$
$180$	0	0
$181$	6.77841	0.503835	0.251918	−	0.967749i	$-0.418939\pi$
$181$	6.77841	0.503835	0.251918	+	0.967749i	$0.418939\pi$
$182$	7.49464	0.555540
$183$	0	0
$184$	5.80459	0.427920
$185$	−12.4728	−0.917015
$186$	0	0
$187$	−0.355922	−0.0260276
$188$	−2.19087	−0.159786
$189$	0	0
$190$	3.90192	0.283075
$191$	3.68811	0.266862	0.133431	−	0.991058i	$-0.457401\pi$
$191$	3.68811	0.266862	0.133431	+	0.991058i	$0.457401\pi$
$192$	0	0
$193$	9.00315	0.648061	0.324030	−	0.946047i	$-0.394962\pi$
$193$	9.00315	0.648061	0.324030	+	0.946047i	$0.394962\pi$
$194$	−3.11689	−0.223780
$195$	0	0
$196$	−0.0699233	−0.00499452
$197$	17.0613	1.21557	0.607785	−	0.794102i	$-0.292058\pi$
$197$	17.0613	1.21557	0.607785	+	0.794102i	$0.292058\pi$
$198$	0	0
$199$	−18.6964	−1.32536	−0.662678	−	0.748904i	$-0.730580\pi$
$199$	−18.6964	−1.32536	−0.662678	+	0.748904i	$0.730580\pi$
$200$	11.5494	0.816663
$201$	0	0
$202$	−11.1240	−0.782682
$203$	−12.8313	−0.900582
$204$	0	0
$205$	10.9353	0.763751
$206$	11.6650	0.812739
$207$	0	0
$208$	2.84696	0.197401
$209$	1.02831	0.0711297
$210$	0	0
$211$	−9.43981	−0.649863	−0.324932	−	0.945738i	$-0.605341\pi$
$211$	−9.43981	−0.649863	−0.324932	+	0.945738i	$0.605341\pi$
$212$	2.65038	0.182029
$213$	0	0
$214$	0.353561	0.0241689
$215$	−10.3714	−0.707324
$216$	0	0
$217$	−14.1588	−0.961162
$218$	−2.60687	−0.176559
$219$	0	0
$220$	4.36141	0.294046
$221$	−0.945149	−0.0635776
$222$	0	0
$223$	13.7381	0.919970	0.459985	−	0.887927i	$-0.347855\pi$
$223$	13.7381	0.919970	0.459985	+	0.887927i	$0.347855\pi$
$224$	2.63250	0.175892
$225$	0	0
$226$	8.85203	0.588828
$227$	18.4887	1.22714	0.613570	−	0.789640i	$-0.289733\pi$
$227$	18.4887	1.22714	0.613570	+	0.789640i	$0.289733\pi$
$228$	0	0
$229$	6.92015	0.457297	0.228648	−	0.973509i	$-0.426569\pi$
$229$	6.92015	0.457297	0.228648	+	0.973509i	$0.426569\pi$
$230$	23.6136	1.55703
$231$	0	0
$232$	−4.87419	−0.320006
$233$	11.1460	0.730199	0.365099	−	0.930969i	$-0.381035\pi$
$233$	11.1460	0.730199	0.365099	+	0.930969i	$0.381035\pi$
$234$	0	0
$235$	−8.91267	−0.581399
$236$	−0.987970	−0.0643114
$237$	0	0
$238$	−0.873952	−0.0566499
$239$	20.8940	1.35152	0.675761	−	0.737120i	$-0.263815\pi$
$239$	20.8940	1.35152	0.675761	+	0.737120i	$0.263815\pi$
$240$	0	0
$241$	−16.7467	−1.07875	−0.539373	−	0.842067i	$-0.681339\pi$
$241$	−16.7467	−1.07875	−0.539373	+	0.842067i	$0.681339\pi$
$242$	−9.85059	−0.633220
$243$	0	0
$244$	−6.17092	−0.395053
$245$	−0.284454	−0.0181731
$246$	0	0
$247$	2.73067	0.173748
$248$	−5.37845	−0.341532
$249$	0	0
$250$	26.6434	1.68508
$251$	−10.7867	−0.680852	−0.340426	−	0.940271i	$-0.610571\pi$
$251$	−10.7867	−0.680852	−0.340426	+	0.940271i	$0.610571\pi$
$252$	0	0
$253$	6.22312	0.391244
$254$	−3.51949	−0.220832
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.98108	0.248333	0.124166	−	0.992261i	$-0.460374\pi$
$257$	3.98108	0.248333	0.124166	+	0.992261i	$0.460374\pi$
$258$	0	0
$259$	−8.07125	−0.501523
$260$	11.5817	0.718267
$261$	0	0
$262$	−5.39749	−0.333458
$263$	−10.9363	−0.674362	−0.337181	−	0.941440i	$-0.609473\pi$
$263$	−10.9363	−0.674362	−0.337181	+	0.941440i	$0.609473\pi$
$264$	0	0
$265$	10.7820	0.662331
$266$	2.52497	0.154816
$267$	0	0
$268$	7.95555	0.485963
$269$	−27.9488	−1.70407	−0.852035	−	0.523485i	$-0.824632\pi$
$269$	−27.9488	−1.70407	−0.852035	+	0.523485i	$0.824632\pi$
$270$	0	0
$271$	0.0551793	0.00335190	0.00167595	−	0.999999i	$-0.499467\pi$
$271$	0.0551793	0.00335190	0.00167595	+	0.999999i	$0.499467\pi$
$272$	−0.331985	−0.0201295
$273$	0	0
$274$	−18.7677	−1.13380
$275$	12.3821	0.746669
$276$	0	0
$277$	−17.7850	−1.06859	−0.534297	−	0.845297i	$-0.679424\pi$
$277$	−17.7850	−1.06859	−0.534297	+	0.845297i	$0.679424\pi$
$278$	−11.7601	−0.705323
$279$	0	0
$280$	10.7093	0.640001
$281$	−16.4633	−0.982120	−0.491060	−	0.871126i	$-0.663390\pi$
$281$	−16.4633	−0.982120	−0.491060	+	0.871126i	$0.663390\pi$
$282$	0	0
$283$	9.34388	0.555436	0.277718	−	0.960663i	$-0.410422\pi$
$283$	9.34388	0.555436	0.277718	+	0.960663i	$0.410422\pi$
$284$	−11.5304	−0.684204
$285$	0	0
$286$	3.05224	0.180483
$287$	7.07632	0.417702
$288$	0	0
$289$	−16.8898	−0.993517
$290$	−19.8286	−1.16438
$291$	0	0
$292$	−3.22607	−0.188791
$293$	17.2017	1.00493	0.502466	−	0.864597i	$-0.332426\pi$
$293$	17.2017	1.00493	0.502466	+	0.864597i	$0.332426\pi$
$294$	0	0
$295$	−4.01915	−0.234004
$296$	−3.06600	−0.178207
$297$	0	0
$298$	1.00000	0.0579284
$299$	16.5254	0.955691
$300$	0	0
$301$	−6.71144	−0.386841
$302$	11.1478	0.641486
$303$	0	0
$304$	0.959153	0.0550112
$305$	−25.1039	−1.43744
$306$	0	0
$307$	−26.7313	−1.52564	−0.762818	−	0.646613i	$-0.776185\pi$
$307$	−26.7313	−1.52564	−0.762818	+	0.646613i	$0.776185\pi$
$308$	2.82232	0.160816
$309$	0	0
$310$	−21.8800	−1.24270
$311$	−13.0996	−0.742809	−0.371404	−	0.928471i	$-0.621124\pi$
$311$	−13.0996	−0.742809	−0.371404	+	0.928471i	$0.621124\pi$
$312$	0	0
$313$	−19.9256	−1.12626	−0.563131	−	0.826367i	$-0.690404\pi$
$313$	−19.9256	−1.12626	−0.563131	+	0.826367i	$0.690404\pi$
$314$	3.29831	0.186134
$315$	0	0
$316$	−2.48939	−0.140039
$317$	20.4727	1.14986	0.574930	−	0.818203i	$-0.305029\pi$
$317$	20.4727	1.14986	0.574930	+	0.818203i	$0.305029\pi$
$318$	0	0
$319$	−5.22563	−0.292579
$320$	4.06809	0.227413
$321$	0	0
$322$	15.2806	0.851554
$323$	−0.318424	−0.0177176
$324$	0	0
$325$	32.8806	1.82389
$326$	−3.10966	−0.172228
$327$	0	0
$328$	2.68806	0.148423
$329$	−5.76748	−0.317972
$330$	0	0
$331$	25.4933	1.40124	0.700619	−	0.713536i	$-0.252907\pi$
$331$	25.4933	1.40124	0.700619	+	0.713536i	$0.252907\pi$
$332$	−4.40459	−0.241733
$333$	0	0
$334$	10.3945	0.568761
$335$	32.3639	1.76823
$336$	0	0
$337$	21.0185	1.14495	0.572476	−	0.819922i	$-0.305983\pi$
$337$	21.0185	1.14495	0.572476	+	0.819922i	$0.305983\pi$
$338$	−4.89480	−0.266242
$339$	0	0
$340$	−1.35055	−0.0732436
$341$	−5.76626	−0.312260
$342$	0	0
$343$	−18.6116	−1.00493
$344$	−2.54945	−0.137457
$345$	0	0
$346$	−17.8165	−0.957820
$347$	−11.2620	−0.604574	−0.302287	−	0.953217i	$-0.597750\pi$
$347$	−11.2620	−0.604574	−0.302287	+	0.953217i	$0.597750\pi$
$348$	0	0
$349$	−7.15639	−0.383073	−0.191536	−	0.981486i	$-0.561347\pi$
$349$	−7.15639	−0.383073	−0.191536	+	0.981486i	$0.561347\pi$
$350$	30.4037	1.62515
$351$	0	0
$352$	1.07210	0.0571433
$353$	20.2426	1.07740	0.538702	−	0.842496i	$-0.318915\pi$
$353$	20.2426	1.07740	0.538702	+	0.842496i	$0.318915\pi$
$354$	0	0
$355$	−46.9067	−2.48955
$356$	−5.09083	−0.269813
$357$	0	0
$358$	3.52763	0.186441
$359$	29.1262	1.53722	0.768611	−	0.639717i	$-0.220948\pi$
$359$	29.1262	1.53722	0.768611	+	0.639717i	$0.220948\pi$
$360$	0	0
$361$	−18.0800	−0.951580
$362$	6.77841	0.356265
$363$	0	0
$364$	7.49464	0.392826
$365$	−13.1239	−0.686939
$366$	0	0
$367$	−25.7576	−1.34454	−0.672269	−	0.740307i	$-0.734680\pi$
$367$	−25.7576	−1.34454	−0.672269	+	0.740307i	$0.734680\pi$
$368$	5.80459	0.302585
$369$	0	0
$370$	−12.4728	−0.648428
$371$	6.97712	0.362234
$372$	0	0
$373$	11.3193	0.586091	0.293046	−	0.956098i	$-0.405331\pi$
$373$	11.3193	0.586091	0.293046	+	0.956098i	$0.405331\pi$
$374$	−0.355922	−0.0184043
$375$	0	0
$376$	−2.19087	−0.112986
$377$	−13.8766	−0.714683
$378$	0	0
$379$	−3.01634	−0.154939	−0.0774695	−	0.996995i	$-0.524684\pi$
$379$	−3.01634	−0.154939	−0.0774695	+	0.996995i	$0.524684\pi$
$380$	3.90192	0.200164
$381$	0	0
$382$	3.68811	0.188700
$383$	7.32011	0.374040	0.187020	−	0.982356i	$-0.440117\pi$
$383$	7.32011	0.374040	0.187020	+	0.982356i	$0.440117\pi$
$384$	0	0
$385$	11.4814	0.585148
$386$	9.00315	0.458248
$387$	0	0
$388$	−3.11689	−0.158236
$389$	−15.5719	−0.789528	−0.394764	−	0.918783i	$-0.629174\pi$
$389$	−15.5719	−0.789528	−0.394764	+	0.918783i	$0.629174\pi$
$390$	0	0
$391$	−1.92704	−0.0974544
$392$	−0.0699233	−0.00353166
$393$	0	0
$394$	17.0613	0.859537
$395$	−10.1271	−0.509548
$396$	0	0
$397$	21.6642	1.08730	0.543649	−	0.839313i	$-0.317042\pi$
$397$	21.6642	1.08730	0.543649	+	0.839313i	$0.317042\pi$
$398$	−18.6964	−0.937168
$399$	0	0
$400$	11.5494	0.577468
$401$	19.0124	0.949434	0.474717	−	0.880138i	$-0.342550\pi$
$401$	19.0124	0.949434	0.474717	+	0.880138i	$0.342550\pi$
$402$	0	0
$403$	−15.3123	−0.762758
$404$	−11.1240	−0.553440
$405$	0	0
$406$	−12.8313	−0.636808
$407$	−3.28707	−0.162934
$408$	0	0
$409$	−7.19250	−0.355646	−0.177823	−	0.984062i	$-0.556905\pi$
$409$	−7.19250	−0.355646	−0.177823	+	0.984062i	$0.556905\pi$
$410$	10.9353	0.540054
$411$	0	0
$412$	11.6650	0.574693
$413$	−2.60084	−0.127979
$414$	0	0
$415$	−17.9183	−0.879574
$416$	2.84696	0.139584
$417$	0	0
$418$	1.02831	0.0502963
$419$	16.0977	0.786424	0.393212	−	0.919448i	$-0.371364\pi$
$419$	16.0977	0.786424	0.393212	+	0.919448i	$0.371364\pi$
$420$	0	0
$421$	−12.6851	−0.618233	−0.309117	−	0.951024i	$-0.600033\pi$
$421$	−12.6851	−0.618233	−0.309117	+	0.951024i	$0.600033\pi$
$422$	−9.43981	−0.459523
$423$	0	0
$424$	2.65038	0.128714
$425$	−3.83422	−0.185987
$426$	0	0
$427$	−16.2450	−0.786149
$428$	0.353561	0.0170900
$429$	0	0
$430$	−10.3714	−0.500153
$431$	12.6652	0.610061	0.305030	−	0.952343i	$-0.401333\pi$
$431$	12.6652	0.610061	0.305030	+	0.952343i	$0.401333\pi$
$432$	0	0
$433$	−26.3929	−1.26836	−0.634181	−	0.773185i	$-0.718663\pi$
$433$	−26.3929	−1.26836	−0.634181	+	0.773185i	$0.718663\pi$
$434$	−14.1588	−0.679644
$435$	0	0
$436$	−2.60687	−0.124846
$437$	5.56748	0.266329
$438$	0	0
$439$	−12.1111	−0.578032	−0.289016	−	0.957324i	$-0.593328\pi$
$439$	−12.1111	−0.578032	−0.289016	+	0.957324i	$0.593328\pi$
$440$	4.36141	0.207922
$441$	0	0
$442$	−0.945149	−0.0449562
$443$	29.0224	1.37890	0.689449	−	0.724334i	$-0.257853\pi$
$443$	29.0224	1.37890	0.689449	+	0.724334i	$0.257853\pi$
$444$	0	0
$445$	−20.7100	−0.981746
$446$	13.7381	0.650517
$447$	0	0
$448$	2.63250	0.124374
$449$	3.86088	0.182206	0.0911031	−	0.995841i	$-0.470961\pi$
$449$	3.86088	0.182206	0.0911031	+	0.995841i	$0.470961\pi$
$450$	0	0
$451$	2.88187	0.135702
$452$	8.85203	0.416365
$453$	0	0
$454$	18.4887	0.867719
$455$	30.4889	1.42934
$456$	0	0
$457$	9.46069	0.442552	0.221276	−	0.975211i	$-0.428978\pi$
$457$	9.46069	0.442552	0.221276	+	0.975211i	$0.428978\pi$
$458$	6.92015	0.323358
$459$	0	0
$460$	23.6136	1.10099
$461$	12.0723	0.562262	0.281131	−	0.959669i	$-0.409290\pi$
$461$	12.0723	0.562262	0.281131	+	0.959669i	$0.409290\pi$
$462$	0	0
$463$	1.25868	0.0584958	0.0292479	−	0.999572i	$-0.490689\pi$
$463$	1.25868	0.0584958	0.0292479	+	0.999572i	$0.490689\pi$
$464$	−4.87419	−0.226278
$465$	0	0
$466$	11.1460	0.516328
$467$	10.0999	0.467365	0.233683	−	0.972313i	$-0.424922\pi$
$467$	10.0999	0.467365	0.233683	+	0.972313i	$0.424922\pi$
$468$	0	0
$469$	20.9430	0.967059
$470$	−8.91267	−0.411111
$471$	0	0
$472$	−0.987970	−0.0454750
$473$	−2.73328	−0.125676
$474$	0	0
$475$	11.0776	0.508275
$476$	−0.873952	−0.0400575
$477$	0	0
$478$	20.8940	0.955671
$479$	25.7524	1.17666	0.588328	−	0.808622i	$-0.299786\pi$
$479$	25.7524	1.17666	0.588328	+	0.808622i	$0.299786\pi$
$480$	0	0
$481$	−8.72878	−0.397998
$482$	−16.7467	−0.762789
$483$	0	0
$484$	−9.85059	−0.447754
$485$	−12.6798	−0.575760
$486$	0	0
$487$	6.47037	0.293200	0.146600	−	0.989196i	$-0.453167\pi$
$487$	6.47037	0.293200	0.146600	+	0.989196i	$0.453167\pi$
$488$	−6.17092	−0.279344
$489$	0	0
$490$	−0.284454	−0.0128503
$491$	13.0686	0.589778	0.294889	−	0.955531i	$-0.404717\pi$
$491$	13.0686	0.589778	0.294889	+	0.955531i	$0.404717\pi$
$492$	0	0
$493$	1.61816	0.0728781
$494$	2.73067	0.122859
$495$	0	0
$496$	−5.37845	−0.241500
$497$	−30.3538	−1.36156
$498$	0	0
$499$	4.34402	0.194465	0.0972326	−	0.995262i	$-0.469001\pi$
$499$	4.34402	0.194465	0.0972326	+	0.995262i	$0.469001\pi$
$500$	26.6434	1.19153
$501$	0	0
$502$	−10.7867	−0.481435
$503$	40.3423	1.79877	0.899386	−	0.437155i	$-0.144014\pi$
$503$	40.3423	1.79877	0.899386	+	0.437155i	$0.144014\pi$
$504$	0	0
$505$	−45.2535	−2.01375
$506$	6.22312	0.276651
$507$	0	0
$508$	−3.51949	−0.156152
$509$	−30.6332	−1.35779	−0.678897	−	0.734233i	$-0.737542\pi$
$509$	−30.6332	−1.35779	−0.678897	+	0.734233i	$0.737542\pi$
$510$	0	0
$511$	−8.49264	−0.375692
$512$	1.00000	0.0441942
$513$	0	0
$514$	3.98108	0.175598
$515$	47.4543	2.09108
$516$	0	0
$517$	−2.34884	−0.103302
$518$	−8.07125	−0.354630
$519$	0	0
$520$	11.5817	0.507891
$521$	37.0048	1.62121	0.810606	−	0.585592i	$-0.199138\pi$
$521$	37.0048	1.62121	0.810606	+	0.585592i	$0.199138\pi$
$522$	0	0
$523$	43.6361	1.90807	0.954036	−	0.299693i	$-0.0968843\pi$
$523$	43.6361	1.90807	0.954036	+	0.299693i	$0.0968843\pi$
$524$	−5.39749	−0.235790
$525$	0	0
$526$	−10.9363	−0.476846
$527$	1.78557	0.0777805
$528$	0	0
$529$	10.6932	0.464923
$530$	10.7820	0.468339
$531$	0	0
$532$	2.52497	0.109471
$533$	7.65280	0.331479
$534$	0	0
$535$	1.43832	0.0621838
$536$	7.95555	0.343627
$537$	0	0
$538$	−27.9488	−1.20496
$539$	−0.0749650	−0.00322897
$540$	0	0
$541$	−5.08430	−0.218591	−0.109296	−	0.994009i	$-0.534860\pi$
$541$	−5.08430	−0.218591	−0.109296	+	0.994009i	$0.534860\pi$
$542$	0.0551793	0.00237015
$543$	0	0
$544$	−0.331985	−0.0142337
$545$	−10.6050	−0.454267
$546$	0	0
$547$	3.81481	0.163109	0.0815547	−	0.996669i	$-0.474011\pi$
$547$	3.81481	0.163109	0.0815547	+	0.996669i	$0.474011\pi$
$548$	−18.7677	−0.801716
$549$	0	0
$550$	12.3821	0.527975
$551$	−4.67509	−0.199166
$552$	0	0
$553$	−6.55333	−0.278676
$554$	−17.7850	−0.755610
$555$	0	0
$556$	−11.7601	−0.498739
$557$	34.7406	1.47201	0.736003	−	0.676978i	$-0.236711\pi$
$557$	34.7406	1.47201	0.736003	+	0.676978i	$0.236711\pi$
$558$	0	0
$559$	−7.25820	−0.306989
$560$	10.7093	0.452549
$561$	0	0
$562$	−16.4633	−0.694464
$563$	11.6146	0.489495	0.244748	−	0.969587i	$-0.421295\pi$
$563$	11.6146	0.489495	0.244748	+	0.969587i	$0.421295\pi$
$564$	0	0
$565$	36.0109	1.51499
$566$	9.34388	0.392752
$567$	0	0
$568$	−11.5304	−0.483805
$569$	19.8936	0.833984	0.416992	−	0.908910i	$-0.363084\pi$
$569$	19.8936	0.833984	0.416992	+	0.908910i	$0.363084\pi$
$570$	0	0
$571$	−14.0985	−0.590005	−0.295002	−	0.955496i	$-0.595320\pi$
$571$	−14.0985	−0.590005	−0.295002	+	0.955496i	$0.595320\pi$
$572$	3.05224	0.127620
$573$	0	0
$574$	7.07632	0.295360
$575$	67.0393	2.79573
$576$	0	0
$577$	−12.1498	−0.505803	−0.252901	−	0.967492i	$-0.581385\pi$
$577$	−12.1498	−0.505803	−0.252901	+	0.967492i	$0.581385\pi$
$578$	−16.8898	−0.702522
$579$	0	0
$580$	−19.8286	−0.823339
$581$	−11.5951	−0.481046
$582$	0	0
$583$	2.84148	0.117682
$584$	−3.22607	−0.133496
$585$	0	0
$586$	17.2017	0.710595
$587$	3.77704	0.155895	0.0779475	−	0.996957i	$-0.475163\pi$
$587$	3.77704	0.155895	0.0779475	+	0.996957i	$0.475163\pi$
$588$	0	0
$589$	−5.15876	−0.212563
$590$	−4.01915	−0.165466
$591$	0	0
$592$	−3.06600	−0.126012
$593$	1.74278	0.0715674	0.0357837	−	0.999360i	$-0.488607\pi$
$593$	1.74278	0.0715674	0.0357837	+	0.999360i	$0.488607\pi$
$594$	0	0
$595$	−3.55532	−0.145754
$596$	1.00000	0.0409616
$597$	0	0
$598$	16.5254	0.675776
$599$	−15.5961	−0.637241	−0.318620	−	0.947882i	$-0.603219\pi$
$599$	−15.5961	−0.637241	−0.318620	+	0.947882i	$0.603219\pi$
$600$	0	0
$601$	33.3589	1.36074	0.680370	−	0.732869i	$-0.261819\pi$
$601$	33.3589	1.36074	0.680370	+	0.732869i	$0.261819\pi$
$602$	−6.71144	−0.273538
$603$	0	0
$604$	11.1478	0.453599
$605$	−40.0731	−1.62920
$606$	0	0
$607$	−24.5279	−0.995557	−0.497778	−	0.867304i	$-0.665851\pi$
$607$	−24.5279	−0.995557	−0.497778	+	0.867304i	$0.665851\pi$
$608$	0.959153	0.0388988
$609$	0	0
$610$	−25.1039	−1.01643
$611$	−6.23734	−0.252336
$612$	0	0
$613$	20.6479	0.833960	0.416980	−	0.908916i	$-0.363088\pi$
$613$	20.6479	0.833960	0.416980	+	0.908916i	$0.363088\pi$
$614$	−26.7313	−1.07879
$615$	0	0
$616$	2.82232	0.113714
$617$	−15.0013	−0.603929	−0.301965	−	0.953319i	$-0.597642\pi$
$617$	−15.0013	−0.603929	−0.301965	+	0.953319i	$0.597642\pi$
$618$	0	0
$619$	16.2352	0.652548	0.326274	−	0.945275i	$-0.394207\pi$
$619$	16.2352	0.652548	0.326274	+	0.945275i	$0.394207\pi$
$620$	−21.8800	−0.878723
$621$	0	0
$622$	−13.0996	−0.525245
$623$	−13.4016	−0.536925
$624$	0	0
$625$	50.6410	2.02564
$626$	−19.9256	−0.796388
$627$	0	0
$628$	3.29831	0.131617
$629$	1.01787	0.0405849
$630$	0	0
$631$	0.157658	0.00627628	0.00313814	−	0.999995i	$-0.499001\pi$
$631$	0.157658	0.00627628	0.00313814	+	0.999995i	$0.499001\pi$
$632$	−2.48939	−0.0990227
$633$	0	0
$634$	20.4727	0.813073
$635$	−14.3176	−0.568177
$636$	0	0
$637$	−0.199069	−0.00788741
$638$	−5.22563	−0.206885
$639$	0	0
$640$	4.06809	0.160805
$641$	−28.6965	−1.13344	−0.566722	−	0.823909i	$-0.691789\pi$
$641$	−28.6965	−1.13344	−0.566722	+	0.823909i	$0.691789\pi$
$642$	0	0
$643$	−11.3240	−0.446576	−0.223288	−	0.974752i	$-0.571679\pi$
$643$	−11.3240	−0.446576	−0.223288	+	0.974752i	$0.571679\pi$
$644$	15.2806	0.602140
$645$	0	0
$646$	−0.318424	−0.0125282
$647$	19.2464	0.756654	0.378327	−	0.925672i	$-0.376499\pi$
$647$	19.2464	0.756654	0.378327	+	0.925672i	$0.376499\pi$
$648$	0	0
$649$	−1.05921	−0.0415775
$650$	32.8806	1.28968
$651$	0	0
$652$	−3.10966	−0.121784
$653$	−17.2851	−0.676420	−0.338210	−	0.941071i	$-0.609821\pi$
$653$	−17.2851	−0.676420	−0.338210	+	0.941071i	$0.609821\pi$
$654$	0	0
$655$	−21.9575	−0.857949
$656$	2.68806	0.104951
$657$	0	0
$658$	−5.76748	−0.224840
$659$	7.47309	0.291110	0.145555	−	0.989350i	$-0.453503\pi$
$659$	7.47309	0.291110	0.145555	+	0.989350i	$0.453503\pi$
$660$	0	0
$661$	44.1755	1.71823	0.859113	−	0.511786i	$-0.171016\pi$
$661$	44.1755	1.71823	0.859113	+	0.511786i	$0.171016\pi$
$662$	25.4933	0.990825
$663$	0	0
$664$	−4.40459	−0.170931
$665$	10.2718	0.398324
$666$	0	0
$667$	−28.2926	−1.09550
$668$	10.3945	0.402175
$669$	0	0
$670$	32.3639	1.25033
$671$	−6.61586	−0.255403
$672$	0	0
$673$	−10.9337	−0.421464	−0.210732	−	0.977544i	$-0.567585\pi$
$673$	−10.9337	−0.421464	−0.210732	+	0.977544i	$0.567585\pi$
$674$	21.0185	0.809603
$675$	0	0
$676$	−4.89480	−0.188262
$677$	19.2755	0.740817	0.370409	−	0.928869i	$-0.379218\pi$
$677$	19.2755	0.740817	0.370409	+	0.928869i	$0.379218\pi$
$678$	0	0
$679$	−8.20523	−0.314888
$680$	−1.35055	−0.0517910
$681$	0	0
$682$	−5.76626	−0.220801
$683$	46.5414	1.78086	0.890428	−	0.455124i	$-0.150405\pi$
$683$	46.5414	1.78086	0.890428	+	0.455124i	$0.150405\pi$
$684$	0	0
$685$	−76.3487	−2.91713
$686$	−18.6116	−0.710594
$687$	0	0
$688$	−2.54945	−0.0971970
$689$	7.54552	0.287461
$690$	0	0
$691$	21.0171	0.799529	0.399765	−	0.916618i	$-0.369092\pi$
$691$	21.0171	0.799529	0.399765	+	0.916618i	$0.369092\pi$
$692$	−17.8165	−0.677281
$693$	0	0
$694$	−11.2620	−0.427498
$695$	−47.8411	−1.81472
$696$	0	0
$697$	−0.892394	−0.0338018
$698$	−7.15639	−0.270873
$699$	0	0
$700$	30.4037	1.14915
$701$	−30.0950	−1.13667	−0.568337	−	0.822796i	$-0.692413\pi$
$701$	−30.0950	−1.13667	−0.568337	+	0.822796i	$0.692413\pi$
$702$	0	0
$703$	−2.94076	−0.110913
$704$	1.07210	0.0404064
$705$	0	0
$706$	20.2426	0.761840
$707$	−29.2840	−1.10134
$708$	0	0
$709$	28.8337	1.08287	0.541436	−	0.840742i	$-0.317881\pi$
$709$	28.8337	1.08287	0.541436	+	0.840742i	$0.317881\pi$
$710$	−46.9067	−1.76038
$711$	0	0
$712$	−5.09083	−0.190787
$713$	−31.2197	−1.16919
$714$	0	0
$715$	12.4168	0.464361
$716$	3.52763	0.131834
$717$	0	0
$718$	29.1262	1.08698
$719$	−36.7108	−1.36908	−0.684540	−	0.728975i	$-0.739997\pi$
$719$	−36.7108	−1.36908	−0.684540	+	0.728975i	$0.739997\pi$
$720$	0	0
$721$	30.7081	1.14363
$722$	−18.0800	−0.672869
$723$	0	0
$724$	6.77841	0.251918
$725$	−56.2938	−2.09070
$726$	0	0
$727$	33.6172	1.24679	0.623397	−	0.781905i	$-0.285752\pi$
$727$	33.6172	1.24679	0.623397	+	0.781905i	$0.285752\pi$
$728$	7.49464	0.277770
$729$	0	0
$730$	−13.1239	−0.485739
$731$	0.846380	0.0313045
$732$	0	0
$733$	−20.0920	−0.742114	−0.371057	−	0.928610i	$-0.621005\pi$
$733$	−20.0920	−0.742114	−0.371057	+	0.928610i	$0.621005\pi$
$734$	−25.7576	−0.950732
$735$	0	0
$736$	5.80459	0.213960
$737$	8.52917	0.314176
$738$	0	0
$739$	13.3516	0.491146	0.245573	−	0.969378i	$-0.421024\pi$
$739$	13.3516	0.491146	0.245573	+	0.969378i	$0.421024\pi$
$740$	−12.4728	−0.458508
$741$	0	0
$742$	6.97712	0.256138
$743$	16.4901	0.604964	0.302482	−	0.953155i	$-0.402185\pi$
$743$	16.4901	0.604964	0.302482	+	0.953155i	$0.402185\pi$
$744$	0	0
$745$	4.06809	0.149043
$746$	11.3193	0.414429
$747$	0	0
$748$	−0.355922	−0.0130138
$749$	0.930749	0.0340088
$750$	0	0
$751$	−39.9137	−1.45647	−0.728235	−	0.685327i	$-0.759659\pi$
$751$	−39.9137	−1.45647	−0.728235	+	0.685327i	$0.759659\pi$
$752$	−2.19087	−0.0798929
$753$	0	0
$754$	−13.8766	−0.505357
$755$	45.3504	1.65047
$756$	0	0
$757$	−21.4241	−0.778674	−0.389337	−	0.921095i	$-0.627296\pi$
$757$	−21.4241	−0.778674	−0.389337	+	0.921095i	$0.627296\pi$
$758$	−3.01634	−0.109558
$759$	0	0
$760$	3.90192	0.141537
$761$	19.4620	0.705495	0.352748	−	0.935718i	$-0.385247\pi$
$761$	19.4620	0.705495	0.352748	+	0.935718i	$0.385247\pi$
$762$	0	0
$763$	−6.86258	−0.248442
$764$	3.68811	0.133431
$765$	0	0
$766$	7.32011	0.264486
$767$	−2.81272	−0.101561
$768$	0	0
$769$	−11.5576	−0.416779	−0.208390	−	0.978046i	$-0.566822\pi$
$769$	−11.5576	−0.416779	−0.208390	+	0.978046i	$0.566822\pi$
$770$	11.4814	0.413762
$771$	0	0
$772$	9.00315	0.324030
$773$	−30.7332	−1.10540	−0.552698	−	0.833382i	$-0.686402\pi$
$773$	−30.7332	−1.10540	−0.552698	+	0.833382i	$0.686402\pi$
$774$	0	0
$775$	−62.1177	−2.23133
$776$	−3.11689	−0.111890
$777$	0	0
$778$	−15.5719	−0.558281
$779$	2.57826	0.0923756
$780$	0	0
$781$	−12.3618	−0.442340
$782$	−1.92704	−0.0689107
$783$	0	0
$784$	−0.0699233	−0.00249726
$785$	13.4178	0.478902
$786$	0	0
$787$	31.6922	1.12970	0.564852	−	0.825192i	$-0.308933\pi$
$787$	31.6922	1.12970	0.564852	+	0.825192i	$0.308933\pi$
$788$	17.0613	0.607785
$789$	0	0
$790$	−10.1271	−0.360305
$791$	23.3030	0.828560
$792$	0	0
$793$	−17.5684	−0.623872
$794$	21.6642	0.768835
$795$	0	0
$796$	−18.6964	−0.662678
$797$	36.2155	1.28282	0.641409	−	0.767199i	$-0.278350\pi$
$797$	36.2155	1.28282	0.641409	+	0.767199i	$0.278350\pi$
$798$	0	0
$799$	0.727337	0.0257313
$800$	11.5494	0.408332
$801$	0	0
$802$	19.0124	0.671351
$803$	−3.45868	−0.122054
$804$	0	0
$805$	62.1629	2.19095
$806$	−15.3123	−0.539351
$807$	0	0
$808$	−11.1240	−0.391341
$809$	18.0049	0.633019	0.316510	−	0.948589i	$-0.397489\pi$
$809$	18.0049	0.633019	0.316510	+	0.948589i	$0.397489\pi$
$810$	0	0
$811$	−14.0694	−0.494043	−0.247022	−	0.969010i	$-0.579452\pi$
$811$	−14.0694	−0.494043	−0.247022	+	0.969010i	$0.579452\pi$
$812$	−12.8313	−0.450291
$813$	0	0
$814$	−3.28707	−0.115212
$815$	−12.6504	−0.443123
$816$	0	0
$817$	−2.44531	−0.0855507
$818$	−7.19250	−0.251480
$819$	0	0
$820$	10.9353	0.381876
$821$	41.3787	1.44413	0.722063	−	0.691827i	$-0.243194\pi$
$821$	41.3787	1.44413	0.722063	+	0.691827i	$0.243194\pi$
$822$	0	0
$823$	42.1953	1.47084	0.735418	−	0.677614i	$-0.236986\pi$
$823$	42.1953	1.47084	0.735418	+	0.677614i	$0.236986\pi$
$824$	11.6650	0.406369
$825$	0	0
$826$	−2.60084	−0.0904947
$827$	6.91034	0.240296	0.120148	−	0.992756i	$-0.461663\pi$
$827$	6.91034	0.240296	0.120148	+	0.992756i	$0.461663\pi$
$828$	0	0
$829$	−46.3631	−1.61026	−0.805128	−	0.593101i	$-0.797903\pi$
$829$	−46.3631	−1.61026	−0.805128	+	0.593101i	$0.797903\pi$
$830$	−17.9183	−0.621952
$831$	0	0
$832$	2.84696	0.0987007
$833$	0.0232135	0.000804300 0
$834$	0	0
$835$	42.2857	1.46336
$836$	1.02831	0.0355649
$837$	0	0
$838$	16.0977	0.556086
$839$	17.0494	0.588610	0.294305	−	0.955711i	$-0.404912\pi$
$839$	17.0494	0.588610	0.294305	+	0.955711i	$0.404912\pi$
$840$	0	0
$841$	−5.24229	−0.180769
$842$	−12.6851	−0.437157
$843$	0	0
$844$	−9.43981	−0.324932
$845$	−19.9125	−0.685011
$846$	0	0
$847$	−25.9317	−0.891025
$848$	2.65038	0.0910143
$849$	0	0
$850$	−3.83422	−0.131513
$851$	−17.7968	−0.610068
$852$	0	0
$853$	−20.3850	−0.697969	−0.348984	−	0.937129i	$-0.613473\pi$
$853$	−20.3850	−0.697969	−0.348984	+	0.937129i	$0.613473\pi$
$854$	−16.2450	−0.555891
$855$	0	0
$856$	0.353561	0.0120844
$857$	12.7320	0.434918	0.217459	−	0.976069i	$-0.430223\pi$
$857$	12.7320	0.434918	0.217459	+	0.976069i	$0.430223\pi$
$858$	0	0
$859$	−45.3033	−1.54573	−0.772864	−	0.634571i	$-0.781177\pi$
$859$	−45.3033	−1.54573	−0.772864	+	0.634571i	$0.781177\pi$
$860$	−10.3714	−0.353662
$861$	0	0
$862$	12.6652	0.431378
$863$	15.5647	0.529829	0.264914	−	0.964272i	$-0.414656\pi$
$863$	15.5647	0.529829	0.264914	+	0.964272i	$0.414656\pi$
$864$	0	0
$865$	−72.4791	−2.46436
$866$	−26.3929	−0.896867
$867$	0	0
$868$	−14.1588	−0.480581
$869$	−2.66888	−0.0905357
$870$	0	0
$871$	22.6492	0.767437
$872$	−2.60687	−0.0882796
$873$	0	0
$874$	5.56748	0.188323
$875$	70.1389	2.37113
$876$	0	0
$877$	−1.00925	−0.0340799	−0.0170399	−	0.999855i	$-0.505424\pi$
$877$	−1.00925	−0.0340799	−0.0170399	+	0.999855i	$0.505424\pi$
$878$	−12.1111	−0.408730
$879$	0	0
$880$	4.36141	0.147023
$881$	−10.8860	−0.366758	−0.183379	−	0.983042i	$-0.558704\pi$
$881$	−10.8860	−0.366758	−0.183379	+	0.983042i	$0.558704\pi$
$882$	0	0
$883$	−10.2256	−0.344117	−0.172059	−	0.985087i	$-0.555042\pi$
$883$	−10.2256	−0.344117	−0.172059	+	0.985087i	$0.555042\pi$
$884$	−0.945149	−0.0317888
$885$	0	0
$886$	29.0224	0.975028
$887$	−13.2506	−0.444911	−0.222456	−	0.974943i	$-0.571407\pi$
$887$	−13.2506	−0.444911	−0.222456	+	0.974943i	$0.571407\pi$
$888$	0	0
$889$	−9.26507	−0.310740
$890$	−20.7100	−0.694199
$891$	0	0
$892$	13.7381	0.459985
$893$	−2.10138	−0.0703201
$894$	0	0
$895$	14.3507	0.479692
$896$	2.63250	0.0879458
$897$	0	0
$898$	3.86088	0.128839
$899$	26.2156	0.874339
$900$	0	0
$901$	−0.879885	−0.0293132
$902$	2.88187	0.0959559
$903$	0	0
$904$	8.85203	0.294414
$905$	27.5752	0.916630
$906$	0	0
$907$	36.6320	1.21634	0.608172	−	0.793805i	$-0.291903\pi$
$907$	36.6320	1.21634	0.608172	+	0.793805i	$0.291903\pi$
$908$	18.4887	0.613570
$909$	0	0
$910$	30.4889	1.01070
$911$	−12.3801	−0.410171	−0.205085	−	0.978744i	$-0.565747\pi$
$911$	−12.3801	−0.410171	−0.205085	+	0.978744i	$0.565747\pi$
$912$	0	0
$913$	−4.72218	−0.156281
$914$	9.46069	0.312932
$915$	0	0
$916$	6.92015	0.228648
$917$	−14.2089	−0.469219
$918$	0	0
$919$	−1.59582	−0.0526412	−0.0263206	−	0.999654i	$-0.508379\pi$
$919$	−1.59582	−0.0526412	−0.0263206	+	0.999654i	$0.508379\pi$
$920$	23.6136	0.778517
$921$	0	0
$922$	12.0723	0.397579
$923$	−32.8266	−1.08050
$924$	0	0
$925$	−35.4103	−1.16428
$926$	1.25868	0.0413628
$927$	0	0
$928$	−4.87419	−0.160003
$929$	54.8284	1.79886	0.899430	−	0.437064i	$-0.143982\pi$
$929$	54.8284	1.79886	0.899430	+	0.437064i	$0.143982\pi$
$930$	0	0
$931$	−0.0670671	−0.00219804
$932$	11.1460	0.365099
$933$	0	0
$934$	10.0999	0.330477
$935$	−1.44792	−0.0473522
$936$	0	0
$937$	12.0163	0.392556	0.196278	−	0.980548i	$-0.437114\pi$
$937$	12.0163	0.392556	0.196278	+	0.980548i	$0.437114\pi$
$938$	20.9430	0.683814
$939$	0	0
$940$	−8.91267	−0.290699
$941$	−28.3069	−0.922780	−0.461390	−	0.887197i	$-0.652649\pi$
$941$	−28.3069	−0.922780	−0.461390	+	0.887197i	$0.652649\pi$
$942$	0	0
$943$	15.6031	0.508105
$944$	−0.987970	−0.0321557
$945$	0	0
$946$	−2.73328	−0.0888665
$947$	49.7961	1.61816	0.809078	−	0.587701i	$-0.199967\pi$
$947$	49.7961	1.61816	0.809078	+	0.587701i	$0.199967\pi$
$948$	0	0
$949$	−9.18450	−0.298141
$950$	11.0776	0.359405
$951$	0	0
$952$	−0.873952	−0.0283249
$953$	50.7339	1.64343	0.821716	−	0.569897i	$-0.193017\pi$
$953$	50.7339	1.64343	0.821716	+	0.569897i	$0.193017\pi$
$954$	0	0
$955$	15.0036	0.485504
$956$	20.8940	0.675761
$957$	0	0
$958$	25.7524	0.832022
$959$	−49.4060	−1.59540
$960$	0	0
$961$	−2.07224	−0.0668464
$962$	−8.72878	−0.281427
$963$	0	0
$964$	−16.7467	−0.539373
$965$	36.6256	1.17902
$966$	0	0
$967$	−8.56265	−0.275356	−0.137678	−	0.990477i	$-0.543964\pi$
$967$	−8.56265	−0.275356	−0.137678	+	0.990477i	$0.543964\pi$
$968$	−9.85059	−0.316610
$969$	0	0
$970$	−12.6798	−0.407124
$971$	9.84085	0.315808	0.157904	−	0.987454i	$-0.449526\pi$
$971$	9.84085	0.315808	0.157904	+	0.987454i	$0.449526\pi$
$972$	0	0
$973$	−30.9585	−0.992483
$974$	6.47037	0.207324
$975$	0	0
$976$	−6.17092	−0.197526
$977$	−52.6489	−1.68439	−0.842194	−	0.539175i	$-0.818736\pi$
$977$	−52.6489	−1.68439	−0.842194	+	0.539175i	$0.818736\pi$
$978$	0	0
$979$	−5.45790	−0.174435
$980$	−0.284454	−0.00908656
$981$	0	0
$982$	13.0686	0.417036
$983$	−28.5668	−0.911140	−0.455570	−	0.890200i	$-0.650564\pi$
$983$	−28.5668	−0.911140	−0.455570	+	0.890200i	$0.650564\pi$
$984$	0	0
$985$	69.4070	2.21149
$986$	1.61816	0.0515326
$987$	0	0
$988$	2.73067	0.0868742
$989$	−14.7985	−0.470565
$990$	0	0
$991$	−45.0466	−1.43095	−0.715476	−	0.698637i	$-0.753790\pi$
$991$	−45.0466	−1.43095	−0.715476	+	0.698637i	$0.753790\pi$
$992$	−5.37845	−0.170766
$993$	0	0
$994$	−30.3538	−0.962765
$995$	−76.0588	−2.41123
$996$	0	0
$997$	0.146618	0.00464344	0.00232172	−	0.999997i	$-0.499261\pi$
$997$	0.146618	0.00464344	0.00232172	+	0.999997i	$0.499261\pi$
$998$	4.34402	0.137508
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.o.1.12	yes	12
3.2	odd	2		8046.2.a.j.1.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.j.1.1	✓	12	3.2	odd	2
8046.2.a.o.1.12	yes	12	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.06809 q^{5} +2.63250 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.06809 q^{5} +2.63250 q^{7} +1.00000 q^{8} +4.06809 q^{10} +1.07210 q^{11} +2.84696 q^{13} +2.63250 q^{14} +1.00000 q^{16} -0.331985 q^{17} +0.959153 q^{19} +4.06809 q^{20} +1.07210 q^{22} +5.80459 q^{23} +11.5494 q^{25} +2.84696 q^{26} +2.63250 q^{28} -4.87419 q^{29} -5.37845 q^{31} +1.00000 q^{32} -0.331985 q^{34} +10.7093 q^{35} -3.06600 q^{37} +0.959153 q^{38} +4.06809 q^{40} +2.68806 q^{41} -2.54945 q^{43} +1.07210 q^{44} +5.80459 q^{46} -2.19087 q^{47} -0.0699233 q^{49} +11.5494 q^{50} +2.84696 q^{52} +2.65038 q^{53} +4.36141 q^{55} +2.63250 q^{56} -4.87419 q^{58} -0.987970 q^{59} -6.17092 q^{61} -5.37845 q^{62} +1.00000 q^{64} +11.5817 q^{65} +7.95555 q^{67} -0.331985 q^{68} +10.7093 q^{70} -11.5304 q^{71} -3.22607 q^{73} -3.06600 q^{74} +0.959153 q^{76} +2.82232 q^{77} -2.48939 q^{79} +4.06809 q^{80} +2.68806 q^{82} -4.40459 q^{83} -1.35055 q^{85} -2.54945 q^{86} +1.07210 q^{88} -5.09083 q^{89} +7.49464 q^{91} +5.80459 q^{92} -2.19087 q^{94} +3.90192 q^{95} -3.11689 q^{97} -0.0699233 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8} + 3 q^{10} + 10 q^{11} + 5 q^{13} + 6 q^{14} + 12 q^{16} + 8 q^{17} + 2 q^{19} + 3 q^{20} + 10 q^{22} + 9 q^{23} + 7 q^{25} + 5 q^{26} + 6 q^{28} + 19 q^{29} + 10 q^{31} + 12 q^{32} + 8 q^{34} + 20 q^{35} + 11 q^{37} + 2 q^{38} + 3 q^{40} + 8 q^{41} + 13 q^{43} + 10 q^{44} + 9 q^{46} + 11 q^{47} + 2 q^{49} + 7 q^{50} + 5 q^{52} + 24 q^{53} + 3 q^{55} + 6 q^{56} + 19 q^{58} + 10 q^{59} + 10 q^{62} + 12 q^{64} + 28 q^{65} + 21 q^{67} + 8 q^{68} + 20 q^{70} + 37 q^{71} - 2 q^{73} + 11 q^{74} + 2 q^{76} + 2 q^{77} + 7 q^{79} + 3 q^{80} + 8 q^{82} + 22 q^{83} + 15 q^{85} + 13 q^{86} + 10 q^{88} + 40 q^{89} + q^{91} + 9 q^{92} + 11 q^{94} + 11 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8 + 3 * q^10 + 10 * q^11 + 5 * q^13 + 6 * q^14 + 12 * q^16 + 8 * q^17 + 2 * q^19 + 3 * q^20 + 10 * q^22 + 9 * q^23 + 7 * q^25 + 5 * q^26 + 6 * q^28 + 19 * q^29 + 10 * q^31 + 12 * q^32 + 8 * q^34 + 20 * q^35 + 11 * q^37 + 2 * q^38 + 3 * q^40 + 8 * q^41 + 13 * q^43 + 10 * q^44 + 9 * q^46 + 11 * q^47 + 2 * q^49 + 7 * q^50 + 5 * q^52 + 24 * q^53 + 3 * q^55 + 6 * q^56 + 19 * q^58 + 10 * q^59 + 10 * q^62 + 12 * q^64 + 28 * q^65 + 21 * q^67 + 8 * q^68 + 20 * q^70 + 37 * q^71 - 2 * q^73 + 11 * q^74 + 2 * q^76 + 2 * q^77 + 7 * q^79 + 3 * q^80 + 8 * q^82 + 22 * q^83 + 15 * q^85 + 13 * q^86 + 10 * q^88 + 40 * q^89 + q^91 + 9 * q^92 + 11 * q^94 + 11 * q^95 + 7 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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