LMFDB - Embedded newform 8046.2.a.j.1.2

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:	$1$
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			1.2
Root			$3.34249$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.34249 q^{5} +1.09959 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.34249 q^{5} +1.09959 q^{7} -1.00000 q^{8} +3.34249 q^{10} +0.788740 q^{11} -3.98439 q^{13} -1.09959 q^{14} +1.00000 q^{16} -3.16458 q^{17} +2.35926 q^{19} -3.34249 q^{20} -0.788740 q^{22} -2.43189 q^{23} +6.17225 q^{25} +3.98439 q^{26} +1.09959 q^{28} -5.83060 q^{29} +7.12201 q^{31} -1.00000 q^{32} +3.16458 q^{34} -3.67537 q^{35} +4.00097 q^{37} -2.35926 q^{38} +3.34249 q^{40} +9.66912 q^{41} +1.90843 q^{43} +0.788740 q^{44} +2.43189 q^{46} +10.4559 q^{47} -5.79090 q^{49} -6.17225 q^{50} -3.98439 q^{52} +4.08227 q^{53} -2.63636 q^{55} -1.09959 q^{56} +5.83060 q^{58} -3.34338 q^{59} +12.6958 q^{61} -7.12201 q^{62} +1.00000 q^{64} +13.3178 q^{65} -1.62880 q^{67} -3.16458 q^{68} +3.67537 q^{70} -8.55539 q^{71} -8.82421 q^{73} -4.00097 q^{74} +2.35926 q^{76} +0.867290 q^{77} +4.14041 q^{79} -3.34249 q^{80} -9.66912 q^{82} -3.47794 q^{83} +10.5776 q^{85} -1.90843 q^{86} -0.788740 q^{88} -17.2900 q^{89} -4.38119 q^{91} -2.43189 q^{92} -10.4559 q^{94} -7.88581 q^{95} +13.2614 q^{97} +5.79090 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.34249	−1.49481	−0.747404	−	0.664370i	$-0.768700\pi$
$5$	−3.34249	−1.49481	−0.747404	+	0.664370i	$0.768700\pi$
$6$	0	0
$7$	1.09959	0.415606	0.207803	−	0.978171i	$-0.433369\pi$
$7$	1.09959	0.415606	0.207803	+	0.978171i	$0.433369\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.34249	1.05699
$11$	0.788740	0.237814	0.118907	−	0.992905i	$-0.462061\pi$
$11$	0.788740	0.237814	0.118907	+	0.992905i	$0.462061\pi$
$12$	0	0
$13$	−3.98439	−1.10507	−0.552535	−	0.833490i	$-0.686339\pi$
$13$	−3.98439	−1.10507	−0.552535	+	0.833490i	$0.686339\pi$
$14$	−1.09959	−0.293878
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.16458	−0.767524	−0.383762	−	0.923432i	$-0.625372\pi$
$17$	−3.16458	−0.767524	−0.383762	+	0.923432i	$0.625372\pi$
$18$	0	0
$19$	2.35926	0.541251	0.270626	−	0.962685i	$-0.412769\pi$
$19$	2.35926	0.541251	0.270626	+	0.962685i	$0.412769\pi$
$20$	−3.34249	−0.747404
$21$	0	0
$22$	−0.788740	−0.168160
$23$	−2.43189	−0.507084	−0.253542	−	0.967324i	$-0.581596\pi$
$23$	−2.43189	−0.507084	−0.253542	+	0.967324i	$0.581596\pi$
$24$	0	0
$25$	6.17225	1.23445
$26$	3.98439	0.781402
$27$	0	0
$28$	1.09959	0.207803
$29$	−5.83060	−1.08272	−0.541358	−	0.840792i	$-0.682090\pi$
$29$	−5.83060	−1.08272	−0.541358	+	0.840792i	$0.682090\pi$
$30$	0	0
$31$	7.12201	1.27915	0.639575	−	0.768728i	$-0.279110\pi$
$31$	7.12201	1.27915	0.639575	+	0.768728i	$0.279110\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.16458	0.542722
$35$	−3.67537	−0.621251
$36$	0	0
$37$	4.00097	0.657755	0.328878	−	0.944373i	$-0.393330\pi$
$37$	4.00097	0.657755	0.328878	+	0.944373i	$0.393330\pi$
$38$	−2.35926	−0.382723
$39$	0	0
$40$	3.34249	0.528494
$41$	9.66912	1.51006	0.755031	−	0.655689i	$-0.227622\pi$
$41$	9.66912	1.51006	0.755031	+	0.655689i	$0.227622\pi$
$42$	0	0
$43$	1.90843	0.291033	0.145516	−	0.989356i	$-0.453516\pi$
$43$	1.90843	0.291033	0.145516	+	0.989356i	$0.453516\pi$
$44$	0.788740	0.118907
$45$	0	0
$46$	2.43189	0.358562
$47$	10.4559	1.52514	0.762572	−	0.646903i	$-0.223936\pi$
$47$	10.4559	1.52514	0.762572	+	0.646903i	$0.223936\pi$
$48$	0	0
$49$	−5.79090	−0.827272
$50$	−6.17225	−0.872888
$51$	0	0
$52$	−3.98439	−0.552535
$53$	4.08227	0.560743	0.280372	−	0.959892i	$-0.409542\pi$
$53$	4.08227	0.560743	0.280372	+	0.959892i	$0.409542\pi$
$54$	0	0
$55$	−2.63636	−0.355486
$56$	−1.09959	−0.146939
$57$	0	0
$58$	5.83060	0.765596
$59$	−3.34338	−0.435271	−0.217636	−	0.976030i	$-0.569834\pi$
$59$	−3.34338	−0.435271	−0.217636	+	0.976030i	$0.569834\pi$
$60$	0	0
$61$	12.6958	1.62553	0.812764	−	0.582593i	$-0.197962\pi$
$61$	12.6958	1.62553	0.812764	+	0.582593i	$0.197962\pi$
$62$	−7.12201	−0.904496
$63$	0	0
$64$	1.00000	0.125000
$65$	13.3178	1.65187
$66$	0	0
$67$	−1.62880	−0.198989	−0.0994945	−	0.995038i	$-0.531723\pi$
$67$	−1.62880	−0.198989	−0.0994945	+	0.995038i	$0.531723\pi$
$68$	−3.16458	−0.383762
$69$	0	0
$70$	3.67537	0.439291
$71$	−8.55539	−1.01534	−0.507669	−	0.861552i	$-0.669493\pi$
$71$	−8.55539	−1.01534	−0.507669	+	0.861552i	$0.669493\pi$
$72$	0	0
$73$	−8.82421	−1.03280	−0.516398	−	0.856349i	$-0.672727\pi$
$73$	−8.82421	−1.03280	−0.516398	+	0.856349i	$0.672727\pi$
$74$	−4.00097	−0.465103
$75$	0	0
$76$	2.35926	0.270626
$77$	0.867290	0.0988369
$78$	0	0
$79$	4.14041	0.465833	0.232916	−	0.972497i	$-0.425173\pi$
$79$	4.14041	0.465833	0.232916	+	0.972497i	$0.425173\pi$
$80$	−3.34249	−0.373702
$81$	0	0
$82$	−9.66912	−1.06778
$83$	−3.47794	−0.381753	−0.190877	−	0.981614i	$-0.561133\pi$
$83$	−3.47794	−0.381753	−0.190877	+	0.981614i	$0.561133\pi$
$84$	0	0
$85$	10.5776	1.14730
$86$	−1.90843	−0.205791
$87$	0	0
$88$	−0.788740	−0.0840800
$89$	−17.2900	−1.83273	−0.916366	−	0.400342i	$-0.868891\pi$
$89$	−17.2900	−1.83273	−0.916366	+	0.400342i	$0.868891\pi$
$90$	0	0
$91$	−4.38119	−0.459273
$92$	−2.43189	−0.253542
$93$	0	0
$94$	−10.4559	−1.07844
$95$	−7.88581	−0.809067
$96$	0	0
$97$	13.2614	1.34649	0.673245	−	0.739419i	$-0.264900\pi$
$97$	13.2614	1.34649	0.673245	+	0.739419i	$0.264900\pi$
$98$	5.79090	0.584970
$99$	0	0
$100$	6.17225	0.617225
$101$	−0.991814	−0.0986892	−0.0493446	−	0.998782i	$-0.515713\pi$
$101$	−0.991814	−0.0986892	−0.0493446	+	0.998782i	$0.515713\pi$
$102$	0	0
$103$	−16.1065	−1.58702	−0.793510	−	0.608557i	$-0.791749\pi$
$103$	−16.1065	−1.58702	−0.793510	+	0.608557i	$0.791749\pi$
$104$	3.98439	0.390701
$105$	0	0
$106$	−4.08227	−0.396505
$107$	15.4541	1.49400	0.747000	−	0.664824i	$-0.231493\pi$
$107$	15.4541	1.49400	0.747000	+	0.664824i	$0.231493\pi$
$108$	0	0
$109$	−4.00527	−0.383635	−0.191817	−	0.981431i	$-0.561438\pi$
$109$	−4.00527	−0.383635	−0.191817	+	0.981431i	$0.561438\pi$
$110$	2.63636	0.251367
$111$	0	0
$112$	1.09959	0.103901
$113$	−0.887494	−0.0834884	−0.0417442	−	0.999128i	$-0.513291\pi$
$113$	−0.887494	−0.0834884	−0.0417442	+	0.999128i	$0.513291\pi$
$114$	0	0
$115$	8.12857	0.757993
$116$	−5.83060	−0.541358
$117$	0	0
$118$	3.34338	0.307783
$119$	−3.47974	−0.318988
$120$	0	0
$121$	−10.3779	−0.943444
$122$	−12.6958	−1.14942
$123$	0	0
$124$	7.12201	0.639575
$125$	−3.91825	−0.350459
$126$	0	0
$127$	16.9596	1.50492	0.752459	−	0.658639i	$-0.228868\pi$
$127$	16.9596	1.50492	0.752459	+	0.658639i	$0.228868\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−13.3178	−1.16805
$131$	−8.56282	−0.748137	−0.374068	−	0.927401i	$-0.622037\pi$
$131$	−8.56282	−0.748137	−0.374068	+	0.927401i	$0.622037\pi$
$132$	0	0
$133$	2.59422	0.224947
$134$	1.62880	0.140706
$135$	0	0
$136$	3.16458	0.271361
$137$	−9.33690	−0.797706	−0.398853	−	0.917015i	$-0.630592\pi$
$137$	−9.33690	−0.797706	−0.398853	+	0.917015i	$0.630592\pi$
$138$	0	0
$139$	8.39966	0.712450	0.356225	−	0.934400i	$-0.384064\pi$
$139$	8.39966	0.712450	0.356225	+	0.934400i	$0.384064\pi$
$140$	−3.67537	−0.310625
$141$	0	0
$142$	8.55539	0.717952
$143$	−3.14265	−0.262801
$144$	0	0
$145$	19.4887	1.61845
$146$	8.82421	0.730297
$147$	0	0
$148$	4.00097	0.328878
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−9.42362	−0.766884	−0.383442	−	0.923565i	$-0.625261\pi$
$151$	−9.42362	−0.766884	−0.383442	+	0.923565i	$0.625261\pi$
$152$	−2.35926	−0.191361
$153$	0	0
$154$	−0.867290	−0.0698883
$155$	−23.8053	−1.91208
$156$	0	0
$157$	20.1843	1.61088	0.805442	−	0.592674i	$-0.201928\pi$
$157$	20.1843	1.61088	0.805442	+	0.592674i	$0.201928\pi$
$158$	−4.14041	−0.329394
$159$	0	0
$160$	3.34249	0.264247
$161$	−2.67408	−0.210747
$162$	0	0
$163$	−14.3923	−1.12730	−0.563648	−	0.826015i	$-0.690602\pi$
$163$	−14.3923	−1.12730	−0.563648	+	0.826015i	$0.690602\pi$
$164$	9.66912	0.755031
$165$	0	0
$166$	3.47794	0.269940
$167$	−8.31122	−0.643141	−0.321571	−	0.946886i	$-0.604211\pi$
$167$	−8.31122	−0.643141	−0.321571	+	0.946886i	$0.604211\pi$
$168$	0	0
$169$	2.87533	0.221179
$170$	−10.5776	−0.811265
$171$	0	0
$172$	1.90843	0.145516
$173$	6.81143	0.517864	0.258932	−	0.965896i	$-0.416630\pi$
$173$	6.81143	0.517864	0.258932	+	0.965896i	$0.416630\pi$
$174$	0	0
$175$	6.78694	0.513045
$176$	0.788740	0.0594535
$177$	0	0
$178$	17.2900	1.29594
$179$	−0.113515	−0.00848453	−0.00424226	−	0.999991i	$-0.501350\pi$
$179$	−0.113515	−0.00848453	−0.00424226	+	0.999991i	$0.501350\pi$
$180$	0	0
$181$	−9.66111	−0.718105	−0.359052	−	0.933317i	$-0.616900\pi$
$181$	−9.66111	−0.718105	−0.359052	+	0.933317i	$0.616900\pi$
$182$	4.38119	0.324755
$183$	0	0
$184$	2.43189	0.179281
$185$	−13.3732	−0.983218
$186$	0	0
$187$	−2.49604	−0.182528
$188$	10.4559	0.762572
$189$	0	0
$190$	7.88581	0.572097
$191$	−10.6010	−0.767059	−0.383529	−	0.923529i	$-0.625292\pi$
$191$	−10.6010	−0.767059	−0.383529	+	0.923529i	$0.625292\pi$
$192$	0	0
$193$	16.1443	1.16209	0.581047	−	0.813870i	$-0.302643\pi$
$193$	16.1443	1.16209	0.581047	+	0.813870i	$0.302643\pi$
$194$	−13.2614	−0.952113
$195$	0	0
$196$	−5.79090	−0.413636
$197$	20.2484	1.44264	0.721319	−	0.692603i	$-0.243536\pi$
$197$	20.2484	1.44264	0.721319	+	0.692603i	$0.243536\pi$
$198$	0	0
$199$	−6.75835	−0.479087	−0.239543	−	0.970886i	$-0.576998\pi$
$199$	−6.75835	−0.479087	−0.239543	+	0.970886i	$0.576998\pi$
$200$	−6.17225	−0.436444
$201$	0	0
$202$	0.991814	0.0697838
$203$	−6.41127	−0.449983
$204$	0	0
$205$	−32.3190	−2.25725
$206$	16.1065	1.12219
$207$	0	0
$208$	−3.98439	−0.276267
$209$	1.86084	0.128717
$210$	0	0
$211$	28.6202	1.97030	0.985149	−	0.171704i	$-0.0549272\pi$
$211$	28.6202	1.97030	0.985149	+	0.171704i	$0.0549272\pi$
$212$	4.08227	0.280372
$213$	0	0
$214$	−15.4541	−1.05642
$215$	−6.37891	−0.435038
$216$	0	0
$217$	7.83129	0.531622
$218$	4.00527	0.271271
$219$	0	0
$220$	−2.63636	−0.177743
$221$	12.6089	0.848168
$222$	0	0
$223$	−6.35670	−0.425676	−0.212838	−	0.977088i	$-0.568271\pi$
$223$	−6.35670	−0.425676	−0.212838	+	0.977088i	$0.568271\pi$
$224$	−1.09959	−0.0734694
$225$	0	0
$226$	0.887494	0.0590352
$227$	10.6168	0.704660	0.352330	−	0.935876i	$-0.385389\pi$
$227$	10.6168	0.704660	0.352330	+	0.935876i	$0.385389\pi$
$228$	0	0
$229$	−29.5975	−1.95586	−0.977930	−	0.208933i	$-0.933001\pi$
$229$	−29.5975	−1.95586	−0.977930	+	0.208933i	$0.933001\pi$
$230$	−8.12857	−0.535982
$231$	0	0
$232$	5.83060	0.382798
$233$	−2.56299	−0.167907	−0.0839534	−	0.996470i	$-0.526755\pi$
$233$	−2.56299	−0.167907	−0.0839534	+	0.996470i	$0.526755\pi$
$234$	0	0
$235$	−34.9486	−2.27980
$236$	−3.34338	−0.217636
$237$	0	0
$238$	3.47974	0.225558
$239$	−11.9467	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$239$	−11.9467	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$240$	0	0
$241$	22.7488	1.46538	0.732690	−	0.680563i	$-0.238265\pi$
$241$	22.7488	1.46538	0.732690	+	0.680563i	$0.238265\pi$
$242$	10.3779	0.667116
$243$	0	0
$244$	12.6958	0.812764
$245$	19.3560	1.23661
$246$	0	0
$247$	−9.40020	−0.598121
$248$	−7.12201	−0.452248
$249$	0	0
$250$	3.91825	0.247812
$251$	−15.0745	−0.951496	−0.475748	−	0.879582i	$-0.657823\pi$
$251$	−15.0745	−0.951496	−0.475748	+	0.879582i	$0.657823\pi$
$252$	0	0
$253$	−1.91813	−0.120592
$254$	−16.9596	−1.06414
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.7860	1.42135	0.710676	−	0.703520i	$-0.248389\pi$
$257$	22.7860	1.42135	0.710676	+	0.703520i	$0.248389\pi$
$258$	0	0
$259$	4.39942	0.273367
$260$	13.3178	0.825933
$261$	0	0
$262$	8.56282	0.529012
$263$	21.2554	1.31066	0.655331	−	0.755342i	$-0.272529\pi$
$263$	21.2554	1.31066	0.655331	+	0.755342i	$0.272529\pi$
$264$	0	0
$265$	−13.6450	−0.838203
$266$	−2.59422	−0.159062
$267$	0	0
$268$	−1.62880	−0.0994945
$269$	10.1503	0.618876	0.309438	−	0.950920i	$-0.399859\pi$
$269$	10.1503	0.618876	0.309438	+	0.950920i	$0.399859\pi$
$270$	0	0
$271$	−25.1511	−1.52782	−0.763912	−	0.645321i	$-0.776724\pi$
$271$	−25.1511	−1.52782	−0.763912	+	0.645321i	$0.776724\pi$
$272$	−3.16458	−0.191881
$273$	0	0
$274$	9.33690	0.564063
$275$	4.86830	0.293570
$276$	0	0
$277$	12.0265	0.722600	0.361300	−	0.932450i	$-0.382333\pi$
$277$	12.0265	0.722600	0.361300	+	0.932450i	$0.382333\pi$
$278$	−8.39966	−0.503778
$279$	0	0
$280$	3.67537	0.219645
$281$	−7.06003	−0.421166	−0.210583	−	0.977576i	$-0.567536\pi$
$281$	−7.06003	−0.421166	−0.210583	+	0.977576i	$0.567536\pi$
$282$	0	0
$283$	−8.66105	−0.514846	−0.257423	−	0.966299i	$-0.582873\pi$
$283$	−8.66105	−0.514846	−0.257423	+	0.966299i	$0.582873\pi$
$284$	−8.55539	−0.507669
$285$	0	0
$286$	3.14265	0.185829
$287$	10.6321	0.627591
$288$	0	0
$289$	−6.98541	−0.410906
$290$	−19.4887	−1.14442
$291$	0	0
$292$	−8.82421	−0.516398
$293$	22.2018	1.29704	0.648520	−	0.761197i	$-0.275388\pi$
$293$	22.2018	1.29704	0.648520	+	0.761197i	$0.275388\pi$
$294$	0	0
$295$	11.1752	0.650647
$296$	−4.00097	−0.232552
$297$	0	0
$298$	1.00000	0.0579284
$299$	9.68958	0.560363
$300$	0	0
$301$	2.09849	0.120955
$302$	9.42362	0.542269
$303$	0	0
$304$	2.35926	0.135313
$305$	−42.4356	−2.42985
$306$	0	0
$307$	−19.1927	−1.09538	−0.547691	−	0.836680i	$-0.684493\pi$
$307$	−19.1927	−1.09538	−0.547691	+	0.836680i	$0.684493\pi$
$308$	0.867290	0.0494185
$309$	0	0
$310$	23.8053	1.35205
$311$	−24.1101	−1.36716	−0.683580	−	0.729875i	$-0.739578\pi$
$311$	−24.1101	−1.36716	−0.683580	+	0.729875i	$0.739578\pi$
$312$	0	0
$313$	−9.66790	−0.546462	−0.273231	−	0.961948i	$-0.588092\pi$
$313$	−9.66790	−0.546462	−0.273231	+	0.961948i	$0.588092\pi$
$314$	−20.1843	−1.13907
$315$	0	0
$316$	4.14041	0.232916
$317$	25.0388	1.40632	0.703160	−	0.711032i	$-0.251772\pi$
$317$	25.0388	1.40632	0.703160	+	0.711032i	$0.251772\pi$
$318$	0	0
$319$	−4.59883	−0.257485
$320$	−3.34249	−0.186851
$321$	0	0
$322$	2.67408	0.149021
$323$	−7.46608	−0.415424
$324$	0	0
$325$	−24.5926	−1.36415
$326$	14.3923	0.797118
$327$	0	0
$328$	−9.66912	−0.533888
$329$	11.4972	0.633859
$330$	0	0
$331$	8.19597	0.450491	0.225245	−	0.974302i	$-0.427682\pi$
$331$	8.19597	0.450491	0.225245	+	0.974302i	$0.427682\pi$
$332$	−3.47794	−0.190877
$333$	0	0
$334$	8.31122	0.454770
$335$	5.44424	0.297450
$336$	0	0
$337$	0.918346	0.0500255	0.0250128	−	0.999687i	$-0.492037\pi$
$337$	0.918346	0.0500255	0.0250128	+	0.999687i	$0.492037\pi$
$338$	−2.87533	−0.156397
$339$	0	0
$340$	10.5776	0.573651
$341$	5.61742	0.304200
$342$	0	0
$343$	−14.0647	−0.759425
$344$	−1.90843	−0.102896
$345$	0	0
$346$	−6.81143	−0.366185
$347$	−22.3735	−1.20107	−0.600536	−	0.799598i	$-0.705046\pi$
$347$	−22.3735	−1.20107	−0.600536	+	0.799598i	$0.705046\pi$
$348$	0	0
$349$	−21.4418	−1.14775	−0.573876	−	0.818942i	$-0.694561\pi$
$349$	−21.4418	−1.14775	−0.573876	+	0.818942i	$0.694561\pi$
$350$	−6.78694	−0.362777
$351$	0	0
$352$	−0.788740	−0.0420400
$353$	−12.2422	−0.651588	−0.325794	−	0.945441i	$-0.605632\pi$
$353$	−12.2422	−0.651588	−0.325794	+	0.945441i	$0.605632\pi$
$354$	0	0
$355$	28.5963	1.51773
$356$	−17.2900	−0.916366
$357$	0	0
$358$	0.113515	0.00599947
$359$	−28.8894	−1.52472	−0.762362	−	0.647151i	$-0.775960\pi$
$359$	−28.8894	−1.52472	−0.762362	+	0.647151i	$0.775960\pi$
$360$	0	0
$361$	−13.4339	−0.707047
$362$	9.66111	0.507777
$363$	0	0
$364$	−4.38119	−0.229637
$365$	29.4949	1.54383
$366$	0	0
$367$	−11.2758	−0.588594	−0.294297	−	0.955714i	$-0.595086\pi$
$367$	−11.2758	−0.588594	−0.294297	+	0.955714i	$0.595086\pi$
$368$	−2.43189	−0.126771
$369$	0	0
$370$	13.3732	0.695240
$371$	4.48882	0.233048
$372$	0	0
$373$	−11.1644	−0.578070	−0.289035	−	0.957319i	$-0.593334\pi$
$373$	−11.1644	−0.578070	−0.289035	+	0.957319i	$0.593334\pi$
$374$	2.49604	0.129067
$375$	0	0
$376$	−10.4559	−0.539220
$377$	23.2314	1.19648
$378$	0	0
$379$	−19.6485	−1.00927	−0.504637	−	0.863332i	$-0.668374\pi$
$379$	−19.6485	−1.00927	−0.504637	+	0.863332i	$0.668374\pi$
$380$	−7.88581	−0.404533
$381$	0	0
$382$	10.6010	0.542393
$383$	−22.8674	−1.16847	−0.584234	−	0.811585i	$-0.698605\pi$
$383$	−22.8674	−1.16847	−0.584234	+	0.811585i	$0.698605\pi$
$384$	0	0
$385$	−2.89891	−0.147742
$386$	−16.1443	−0.821724
$387$	0	0
$388$	13.2614	0.673245
$389$	−23.7393	−1.20363	−0.601815	−	0.798635i	$-0.705556\pi$
$389$	−23.7393	−1.20363	−0.601815	+	0.798635i	$0.705556\pi$
$390$	0	0
$391$	7.69592	0.389199
$392$	5.79090	0.292485
$393$	0	0
$394$	−20.2484	−1.02010
$395$	−13.8393	−0.696330
$396$	0	0
$397$	12.0637	0.605459	0.302730	−	0.953076i	$-0.402102\pi$
$397$	12.0637	0.605459	0.302730	+	0.953076i	$0.402102\pi$
$398$	6.75835	0.338765
$399$	0	0
$400$	6.17225	0.308613
$401$	−2.20968	−0.110346	−0.0551730	−	0.998477i	$-0.517571\pi$
$401$	−2.20968	−0.110346	−0.0551730	+	0.998477i	$0.517571\pi$
$402$	0	0
$403$	−28.3768	−1.41355
$404$	−0.991814	−0.0493446
$405$	0	0
$406$	6.41127	0.318186
$407$	3.15573	0.156424
$408$	0	0
$409$	−14.0849	−0.696451	−0.348226	−	0.937411i	$-0.613216\pi$
$409$	−14.0849	−0.696451	−0.348226	+	0.937411i	$0.613216\pi$
$410$	32.3190	1.59612
$411$	0	0
$412$	−16.1065	−0.793510
$413$	−3.67635	−0.180901
$414$	0	0
$415$	11.6250	0.570648
$416$	3.98439	0.195351
$417$	0	0
$418$	−1.86084	−0.0910168
$419$	5.25596	0.256771	0.128385	−	0.991724i	$-0.459021\pi$
$419$	5.25596	0.256771	0.128385	+	0.991724i	$0.459021\pi$
$420$	0	0
$421$	−2.51919	−0.122778	−0.0613890	−	0.998114i	$-0.519553\pi$
$421$	−2.51919	−0.122778	−0.0613890	+	0.998114i	$0.519553\pi$
$422$	−28.6202	−1.39321
$423$	0	0
$424$	−4.08227	−0.198253
$425$	−19.5326	−0.947471
$426$	0	0
$427$	13.9601	0.675579
$428$	15.4541	0.747000
$429$	0	0
$430$	6.37891	0.307618
$431$	−38.9111	−1.87428	−0.937140	−	0.348952i	$-0.886537\pi$
$431$	−38.9111	−1.87428	−0.937140	+	0.348952i	$0.886537\pi$
$432$	0	0
$433$	−15.6017	−0.749769	−0.374885	−	0.927072i	$-0.622318\pi$
$433$	−15.6017	−0.749769	−0.374885	+	0.927072i	$0.622318\pi$
$434$	−7.83129	−0.375914
$435$	0	0
$436$	−4.00527	−0.191817
$437$	−5.73746	−0.274460
$438$	0	0
$439$	6.56144	0.313160	0.156580	−	0.987665i	$-0.449953\pi$
$439$	6.56144	0.313160	0.156580	+	0.987665i	$0.449953\pi$
$440$	2.63636	0.125683
$441$	0	0
$442$	−12.6089	−0.599745
$443$	−36.1008	−1.71520	−0.857599	−	0.514318i	$-0.828045\pi$
$443$	−36.1008	−1.71520	−0.857599	+	0.514318i	$0.828045\pi$
$444$	0	0
$445$	57.7915	2.73958
$446$	6.35670	0.300998
$447$	0	0
$448$	1.09959	0.0519507
$449$	−6.20802	−0.292974	−0.146487	−	0.989213i	$-0.546797\pi$
$449$	−6.20802	−0.292974	−0.146487	+	0.989213i	$0.546797\pi$
$450$	0	0
$451$	7.62642	0.359114
$452$	−0.887494	−0.0417442
$453$	0	0
$454$	−10.6168	−0.498270
$455$	14.6441	0.686525
$456$	0	0
$457$	−38.1208	−1.78322	−0.891608	−	0.452808i	$-0.850422\pi$
$457$	−38.1208	−1.78322	−0.891608	+	0.452808i	$0.850422\pi$
$458$	29.5975	1.38300
$459$	0	0
$460$	8.12857	0.378997
$461$	−14.4758	−0.674206	−0.337103	−	0.941468i	$-0.609447\pi$
$461$	−14.4758	−0.674206	−0.337103	+	0.941468i	$0.609447\pi$
$462$	0	0
$463$	−15.8368	−0.735999	−0.368000	−	0.929826i	$-0.619957\pi$
$463$	−15.8368	−0.735999	−0.368000	+	0.929826i	$0.619957\pi$
$464$	−5.83060	−0.270679
$465$	0	0
$466$	2.56299	0.118728
$467$	−9.50173	−0.439688	−0.219844	−	0.975535i	$-0.570555\pi$
$467$	−9.50173	−0.439688	−0.219844	+	0.975535i	$0.570555\pi$
$468$	0	0
$469$	−1.79101	−0.0827010
$470$	34.9486	1.61206
$471$	0	0
$472$	3.34338	0.153892
$473$	1.50525	0.0692117
$474$	0	0
$475$	14.5620	0.668148
$476$	−3.47974	−0.159494
$477$	0	0
$478$	11.9467	0.546431
$479$	24.7914	1.13275	0.566375	−	0.824148i	$-0.308346\pi$
$479$	24.7914	1.13275	0.566375	+	0.824148i	$0.308346\pi$
$480$	0	0
$481$	−15.9414	−0.726865
$482$	−22.7488	−1.03618
$483$	0	0
$484$	−10.3779	−0.471722
$485$	−44.3261	−2.01275
$486$	0	0
$487$	17.1260	0.776055	0.388027	−	0.921648i	$-0.373157\pi$
$487$	17.1260	0.776055	0.388027	+	0.921648i	$0.373157\pi$
$488$	−12.6958	−0.574711
$489$	0	0
$490$	−19.3560	−0.874417
$491$	21.5708	0.973478	0.486739	−	0.873547i	$-0.338186\pi$
$491$	21.5708	0.973478	0.486739	+	0.873547i	$0.338186\pi$
$492$	0	0
$493$	18.4514	0.831011
$494$	9.40020	0.422935
$495$	0	0
$496$	7.12201	0.319788
$497$	−9.40741	−0.421980
$498$	0	0
$499$	8.62413	0.386069	0.193034	−	0.981192i	$-0.438167\pi$
$499$	8.62413	0.386069	0.193034	+	0.981192i	$0.438167\pi$
$500$	−3.91825	−0.175229
$501$	0	0
$502$	15.0745	0.672809
$503$	18.5744	0.828190	0.414095	−	0.910234i	$-0.364098\pi$
$503$	18.5744	0.828190	0.414095	+	0.910234i	$0.364098\pi$
$504$	0	0
$505$	3.31513	0.147521
$506$	1.91813	0.0852712
$507$	0	0
$508$	16.9596	0.752459
$509$	−8.03053	−0.355947	−0.177973	−	0.984035i	$-0.556954\pi$
$509$	−8.03053	−0.355947	−0.177973	+	0.984035i	$0.556954\pi$
$510$	0	0
$511$	−9.70301	−0.429236
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−22.7860	−1.00505
$515$	53.8358	2.37229
$516$	0	0
$517$	8.24696	0.362701
$518$	−4.39942	−0.193300
$519$	0	0
$520$	−13.3178	−0.584023
$521$	17.5085	0.767063	0.383532	−	0.923528i	$-0.374708\pi$
$521$	17.5085	0.767063	0.383532	+	0.923528i	$0.374708\pi$
$522$	0	0
$523$	−32.4654	−1.41961	−0.709807	−	0.704396i	$-0.751218\pi$
$523$	−32.4654	−1.41961	−0.709807	+	0.704396i	$0.751218\pi$
$524$	−8.56282	−0.374068
$525$	0	0
$526$	−21.2554	−0.926778
$527$	−22.5382	−0.981779
$528$	0	0
$529$	−17.0859	−0.742866
$530$	13.6450	0.592699
$531$	0	0
$532$	2.59422	0.112474
$533$	−38.5255	−1.66872
$534$	0	0
$535$	−51.6551	−2.23324
$536$	1.62880	0.0703532
$537$	0	0
$538$	−10.1503	−0.437611
$539$	−4.56752	−0.196737
$540$	0	0
$541$	−13.2445	−0.569427	−0.284714	−	0.958613i	$-0.591899\pi$
$541$	−13.2445	−0.569427	−0.284714	+	0.958613i	$0.591899\pi$
$542$	25.1511	1.08033
$543$	0	0
$544$	3.16458	0.135680
$545$	13.3876	0.573460
$546$	0	0
$547$	21.3254	0.911808	0.455904	−	0.890029i	$-0.349316\pi$
$547$	21.3254	0.911808	0.455904	+	0.890029i	$0.349316\pi$
$548$	−9.33690	−0.398853
$549$	0	0
$550$	−4.86830	−0.207585
$551$	−13.7559	−0.586022
$552$	0	0
$553$	4.55275	0.193603
$554$	−12.0265	−0.510955
$555$	0	0
$556$	8.39966	0.356225
$557$	−15.1873	−0.643506	−0.321753	−	0.946824i	$-0.604272\pi$
$557$	−15.1873	−0.643506	−0.321753	+	0.946824i	$0.604272\pi$
$558$	0	0
$559$	−7.60392	−0.321611
$560$	−3.67537	−0.155313
$561$	0	0
$562$	7.06003	0.297809
$563$	20.1936	0.851059	0.425530	−	0.904945i	$-0.360088\pi$
$563$	20.1936	0.851059	0.425530	+	0.904945i	$0.360088\pi$
$564$	0	0
$565$	2.96644	0.124799
$566$	8.66105	0.364051
$567$	0	0
$568$	8.55539	0.358976
$569$	−28.1353	−1.17949	−0.589746	−	0.807589i	$-0.700772\pi$
$569$	−28.1353	−1.17949	−0.589746	+	0.807589i	$0.700772\pi$
$570$	0	0
$571$	−19.0227	−0.796077	−0.398039	−	0.917369i	$-0.630309\pi$
$571$	−19.0227	−0.796077	−0.398039	+	0.917369i	$0.630309\pi$
$572$	−3.14265	−0.131401
$573$	0	0
$574$	−10.6321	−0.443774
$575$	−15.0102	−0.625970
$576$	0	0
$577$	−23.4607	−0.976681	−0.488341	−	0.872653i	$-0.662398\pi$
$577$	−23.4607	−0.976681	−0.488341	+	0.872653i	$0.662398\pi$
$578$	6.98541	0.290555
$579$	0	0
$580$	19.4887	0.809226
$581$	−3.82430	−0.158659
$582$	0	0
$583$	3.21985	0.133353
$584$	8.82421	0.365149
$585$	0	0
$586$	−22.2018	−0.917146
$587$	−14.8172	−0.611572	−0.305786	−	0.952100i	$-0.598919\pi$
$587$	−14.8172	−0.611572	−0.305786	+	0.952100i	$0.598919\pi$
$588$	0	0
$589$	16.8027	0.692342
$590$	−11.1752	−0.460077
$591$	0	0
$592$	4.00097	0.164439
$593$	−15.0880	−0.619589	−0.309795	−	0.950804i	$-0.600260\pi$
$593$	−15.0880	−0.619589	−0.309795	+	0.950804i	$0.600260\pi$
$594$	0	0
$595$	11.6310	0.476825
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−9.68958	−0.396237
$599$	9.05965	0.370167	0.185084	−	0.982723i	$-0.440744\pi$
$599$	9.05965	0.370167	0.185084	+	0.982723i	$0.440744\pi$
$600$	0	0
$601$	−45.3655	−1.85050	−0.925249	−	0.379359i	$-0.876144\pi$
$601$	−45.3655	−1.85050	−0.925249	+	0.379359i	$0.876144\pi$
$602$	−2.09849	−0.0855280
$603$	0	0
$604$	−9.42362	−0.383442
$605$	34.6880	1.41027
$606$	0	0
$607$	29.7797	1.20872	0.604360	−	0.796711i	$-0.293429\pi$
$607$	29.7797	1.20872	0.604360	+	0.796711i	$0.293429\pi$
$608$	−2.35926	−0.0956806
$609$	0	0
$610$	42.4356	1.71817
$611$	−41.6602	−1.68539
$612$	0	0
$613$	2.33439	0.0942850	0.0471425	−	0.998888i	$-0.484989\pi$
$613$	2.33439	0.0942850	0.0471425	+	0.998888i	$0.484989\pi$
$614$	19.1927	0.774552
$615$	0	0
$616$	−0.867290	−0.0349441
$617$	−20.9818	−0.844697	−0.422349	−	0.906434i	$-0.638794\pi$
$617$	−20.9818	−0.844697	−0.422349	+	0.906434i	$0.638794\pi$
$618$	0	0
$619$	37.7212	1.51614	0.758071	−	0.652173i	$-0.226142\pi$
$619$	37.7212	1.51614	0.758071	+	0.652173i	$0.226142\pi$
$620$	−23.8053	−0.956042
$621$	0	0
$622$	24.1101	0.966729
$623$	−19.0119	−0.761694
$624$	0	0
$625$	−17.7646	−0.710582
$626$	9.66790	0.386407
$627$	0	0
$628$	20.1843	0.805442
$629$	−12.6614	−0.504843
$630$	0	0
$631$	−0.910277	−0.0362376	−0.0181188	−	0.999836i	$-0.505768\pi$
$631$	−0.910277	−0.0362376	−0.0181188	+	0.999836i	$0.505768\pi$
$632$	−4.14041	−0.164697
$633$	0	0
$634$	−25.0388	−0.994418
$635$	−56.6872	−2.24956
$636$	0	0
$637$	23.0732	0.914193
$638$	4.59883	0.182069
$639$	0	0
$640$	3.34249	0.132124
$641$	31.3983	1.24016	0.620079	−	0.784539i	$-0.287100\pi$
$641$	31.3983	1.24016	0.620079	+	0.784539i	$0.287100\pi$
$642$	0	0
$643$	2.17653	0.0858341	0.0429171	−	0.999079i	$-0.486335\pi$
$643$	2.17653	0.0858341	0.0429171	+	0.999079i	$0.486335\pi$
$644$	−2.67408	−0.105373
$645$	0	0
$646$	7.46608	0.293749
$647$	−25.4227	−0.999468	−0.499734	−	0.866179i	$-0.666569\pi$
$647$	−25.4227	−0.999468	−0.499734	+	0.866179i	$0.666569\pi$
$648$	0	0
$649$	−2.63706	−0.103514
$650$	24.5926	0.964602
$651$	0	0
$652$	−14.3923	−0.563648
$653$	39.6345	1.55102	0.775509	−	0.631336i	$-0.217493\pi$
$653$	39.6345	1.55102	0.775509	+	0.631336i	$0.217493\pi$
$654$	0	0
$655$	28.6211	1.11832
$656$	9.66912	0.377516
$657$	0	0
$658$	−11.4972	−0.448206
$659$	−20.1780	−0.786022	−0.393011	−	0.919534i	$-0.628567\pi$
$659$	−20.1780	−0.786022	−0.393011	+	0.919534i	$0.628567\pi$
$660$	0	0
$661$	−1.81995	−0.0707880	−0.0353940	−	0.999373i	$-0.511269\pi$
$661$	−1.81995	−0.0707880	−0.0353940	+	0.999373i	$0.511269\pi$
$662$	−8.19597	−0.318545
$663$	0	0
$664$	3.47794	0.134970
$665$	−8.67115	−0.336253
$666$	0	0
$667$	14.1794	0.549028
$668$	−8.31122	−0.321571
$669$	0	0
$670$	−5.44424	−0.210329
$671$	10.0137	0.386574
$672$	0	0
$673$	22.6446	0.872884	0.436442	−	0.899732i	$-0.356238\pi$
$673$	22.6446	0.872884	0.436442	+	0.899732i	$0.356238\pi$
$674$	−0.918346	−0.0353734
$675$	0	0
$676$	2.87533	0.110590
$677$	−2.69591	−0.103612	−0.0518060	−	0.998657i	$-0.516498\pi$
$677$	−2.69591	−0.103612	−0.0518060	+	0.998657i	$0.516498\pi$
$678$	0	0
$679$	14.5821	0.559609
$680$	−10.5776	−0.405632
$681$	0	0
$682$	−5.61742	−0.215102
$683$	28.2797	1.08209	0.541047	−	0.840993i	$-0.318028\pi$
$683$	28.2797	1.08209	0.541047	+	0.840993i	$0.318028\pi$
$684$	0	0
$685$	31.2085	1.19242
$686$	14.0647	0.536994
$687$	0	0
$688$	1.90843	0.0727582
$689$	−16.2653	−0.619660
$690$	0	0
$691$	−3.17811	−0.120901	−0.0604504	−	0.998171i	$-0.519254\pi$
$691$	−3.17811	−0.120901	−0.0604504	+	0.998171i	$0.519254\pi$
$692$	6.81143	0.258932
$693$	0	0
$694$	22.3735	0.849286
$695$	−28.0758	−1.06498
$696$	0	0
$697$	−30.5987	−1.15901
$698$	21.4418	0.811583
$699$	0	0
$700$	6.78694	0.256522
$701$	13.0251	0.491951	0.245975	−	0.969276i	$-0.420892\pi$
$701$	13.0251	0.491951	0.245975	+	0.969276i	$0.420892\pi$
$702$	0	0
$703$	9.43933	0.356011
$704$	0.788740	0.0297268
$705$	0	0
$706$	12.2422	0.460743
$707$	−1.09059	−0.0410158
$708$	0	0
$709$	30.2448	1.13587	0.567934	−	0.823074i	$-0.307743\pi$
$709$	30.2448	1.13587	0.567934	+	0.823074i	$0.307743\pi$
$710$	−28.5963	−1.07320
$711$	0	0
$712$	17.2900	0.647969
$713$	−17.3199	−0.648637
$714$	0	0
$715$	10.5043	0.392837
$716$	−0.113515	−0.00424226
$717$	0	0
$718$	28.8894	1.07814
$719$	2.90249	0.108245	0.0541224	−	0.998534i	$-0.482764\pi$
$719$	2.90249	0.108245	0.0541224	+	0.998534i	$0.482764\pi$
$720$	0	0
$721$	−17.7105	−0.659574
$722$	13.4339	0.499958
$723$	0	0
$724$	−9.66111	−0.359052
$725$	−35.9880	−1.33656
$726$	0	0
$727$	34.3373	1.27350	0.636750	−	0.771070i	$-0.280278\pi$
$727$	34.3373	1.27350	0.636750	+	0.771070i	$0.280278\pi$
$728$	4.38119	0.162378
$729$	0	0
$730$	−29.4949	−1.09165
$731$	−6.03938	−0.223375
$732$	0	0
$733$	49.4893	1.82793	0.913965	−	0.405793i	$-0.133004\pi$
$733$	49.4893	1.82793	0.913965	+	0.405793i	$0.133004\pi$
$734$	11.2758	0.416199
$735$	0	0
$736$	2.43189	0.0896406
$737$	−1.28470	−0.0473224
$738$	0	0
$739$	2.93152	0.107838	0.0539188	−	0.998545i	$-0.482829\pi$
$739$	2.93152	0.107838	0.0539188	+	0.998545i	$0.482829\pi$
$740$	−13.3732	−0.491609
$741$	0	0
$742$	−4.48882	−0.164790
$743$	13.3874	0.491136	0.245568	−	0.969379i	$-0.421026\pi$
$743$	13.3874	0.491136	0.245568	+	0.969379i	$0.421026\pi$
$744$	0	0
$745$	3.34249	0.122459
$746$	11.1644	0.408757
$747$	0	0
$748$	−2.49604	−0.0912641
$749$	16.9931	0.620915
$750$	0	0
$751$	−32.7852	−1.19635	−0.598175	−	0.801366i	$-0.704107\pi$
$751$	−32.7852	−1.19635	−0.598175	+	0.801366i	$0.704107\pi$
$752$	10.4559	0.381286
$753$	0	0
$754$	−23.2314	−0.846037
$755$	31.4984	1.14634
$756$	0	0
$757$	−14.5738	−0.529694	−0.264847	−	0.964290i	$-0.585322\pi$
$757$	−14.5738	−0.529694	−0.264847	+	0.964290i	$0.585322\pi$
$758$	19.6485	0.713664
$759$	0	0
$760$	7.88581	0.286048
$761$	−4.38868	−0.159090	−0.0795448	−	0.996831i	$-0.525347\pi$
$761$	−4.38868	−0.159090	−0.0795448	+	0.996831i	$0.525347\pi$
$762$	0	0
$763$	−4.40415	−0.159441
$764$	−10.6010	−0.383529
$765$	0	0
$766$	22.8674	0.826232
$767$	13.3213	0.481005
$768$	0	0
$769$	0.983867	0.0354792	0.0177396	−	0.999843i	$-0.494353\pi$
$769$	0.983867	0.0354792	0.0177396	+	0.999843i	$0.494353\pi$
$770$	2.89891	0.104470
$771$	0	0
$772$	16.1443	0.581047
$773$	18.3395	0.659625	0.329812	−	0.944046i	$-0.393014\pi$
$773$	18.3395	0.659625	0.329812	+	0.944046i	$0.393014\pi$
$774$	0	0
$775$	43.9588	1.57905
$776$	−13.2614	−0.476056
$777$	0	0
$778$	23.7393	0.851095
$779$	22.8120	0.817324
$780$	0	0
$781$	−6.74798	−0.241462
$782$	−7.69592	−0.275205
$783$	0	0
$784$	−5.79090	−0.206818
$785$	−67.4659	−2.40796
$786$	0	0
$787$	−26.1071	−0.930618	−0.465309	−	0.885148i	$-0.654057\pi$
$787$	−26.1071	−0.930618	−0.465309	+	0.885148i	$0.654057\pi$
$788$	20.2484	0.721319
$789$	0	0
$790$	13.8393	0.492380
$791$	−0.975879	−0.0346983
$792$	0	0
$793$	−50.5849	−1.79632
$794$	−12.0637	−0.428124
$795$	0	0
$796$	−6.75835	−0.239543
$797$	34.2187	1.21209	0.606045	−	0.795431i	$-0.292755\pi$
$797$	34.2187	1.21209	0.606045	+	0.795431i	$0.292755\pi$
$798$	0	0
$799$	−33.0885	−1.17059
$800$	−6.17225	−0.218222
$801$	0	0
$802$	2.20968	0.0780264
$803$	−6.96001	−0.245614
$804$	0	0
$805$	8.93809	0.315026
$806$	28.3768	0.999531
$807$	0	0
$808$	0.991814	0.0348919
$809$	30.8992	1.08636	0.543180	−	0.839616i	$-0.317220\pi$
$809$	30.8992	1.08636	0.543180	+	0.839616i	$0.317220\pi$
$810$	0	0
$811$	−47.5568	−1.66995	−0.834973	−	0.550292i	$-0.814517\pi$
$811$	−47.5568	−1.66995	−0.834973	+	0.550292i	$0.814517\pi$
$812$	−6.41127	−0.224991
$813$	0	0
$814$	−3.15573	−0.110608
$815$	48.1063	1.68509
$816$	0	0
$817$	4.50248	0.157522
$818$	14.0849	0.492465
$819$	0	0
$820$	−32.3190	−1.12863
$821$	39.7791	1.38830	0.694150	−	0.719830i	$-0.255780\pi$
$821$	39.7791	1.38830	0.694150	+	0.719830i	$0.255780\pi$
$822$	0	0
$823$	−29.8365	−1.04004	−0.520018	−	0.854155i	$-0.674075\pi$
$823$	−29.8365	−1.04004	−0.520018	+	0.854155i	$0.674075\pi$
$824$	16.1065	0.561096
$825$	0	0
$826$	3.67635	0.127917
$827$	−8.27649	−0.287802	−0.143901	−	0.989592i	$-0.545965\pi$
$827$	−8.27649	−0.287802	−0.143901	+	0.989592i	$0.545965\pi$
$828$	0	0
$829$	−41.4481	−1.43955	−0.719775	−	0.694207i	$-0.755755\pi$
$829$	−41.4481	−1.43955	−0.719775	+	0.694207i	$0.755755\pi$
$830$	−11.6250	−0.403509
$831$	0	0
$832$	−3.98439	−0.138134
$833$	18.3258	0.634951
$834$	0	0
$835$	27.7802	0.961373
$836$	1.86084	0.0643586
$837$	0	0
$838$	−5.25596	−0.181564
$839$	−40.6197	−1.40235	−0.701174	−	0.712991i	$-0.747340\pi$
$839$	−40.6197	−1.40235	−0.701174	+	0.712991i	$0.747340\pi$
$840$	0	0
$841$	4.99594	0.172274
$842$	2.51919	0.0868171
$843$	0	0
$844$	28.6202	0.985149
$845$	−9.61076	−0.330620
$846$	0	0
$847$	−11.4114	−0.392101
$848$	4.08227	0.140186
$849$	0	0
$850$	19.5326	0.669963
$851$	−9.72991	−0.333537
$852$	0	0
$853$	−4.26634	−0.146077	−0.0730383	−	0.997329i	$-0.523270\pi$
$853$	−4.26634	−0.146077	−0.0730383	+	0.997329i	$0.523270\pi$
$854$	−13.9601	−0.477706
$855$	0	0
$856$	−15.4541	−0.528209
$857$	−27.6263	−0.943696	−0.471848	−	0.881680i	$-0.656413\pi$
$857$	−27.6263	−0.943696	−0.471848	+	0.881680i	$0.656413\pi$
$858$	0	0
$859$	18.4094	0.628120	0.314060	−	0.949403i	$-0.398311\pi$
$859$	18.4094	0.628120	0.314060	+	0.949403i	$0.398311\pi$
$860$	−6.37891	−0.217519
$861$	0	0
$862$	38.9111	1.32532
$863$	42.3998	1.44331	0.721653	−	0.692255i	$-0.243383\pi$
$863$	42.3998	1.44331	0.721653	+	0.692255i	$0.243383\pi$
$864$	0	0
$865$	−22.7672	−0.774107
$866$	15.6017	0.530167
$867$	0	0
$868$	7.83129	0.265811
$869$	3.26571	0.110782
$870$	0	0
$871$	6.48975	0.219897
$872$	4.00527	0.135635
$873$	0	0
$874$	5.73746	0.194072
$875$	−4.30846	−0.145653
$876$	0	0
$877$	−26.4203	−0.892149	−0.446074	−	0.894996i	$-0.647178\pi$
$877$	−26.4203	−0.892149	−0.446074	+	0.894996i	$0.647178\pi$
$878$	−6.56144	−0.221438
$879$	0	0
$880$	−2.63636	−0.0888716
$881$	47.8336	1.61156	0.805778	−	0.592218i	$-0.201748\pi$
$881$	47.8336	1.61156	0.805778	+	0.592218i	$0.201748\pi$
$882$	0	0
$883$	−8.90176	−0.299568	−0.149784	−	0.988719i	$-0.547858\pi$
$883$	−8.90176	−0.299568	−0.149784	+	0.988719i	$0.547858\pi$
$884$	12.6089	0.424084
$885$	0	0
$886$	36.1008	1.21283
$887$	−29.3681	−0.986086	−0.493043	−	0.870005i	$-0.664115\pi$
$887$	−29.3681	−0.986086	−0.493043	+	0.870005i	$0.664115\pi$
$888$	0	0
$889$	18.6486	0.625453
$890$	−57.7915	−1.93718
$891$	0	0
$892$	−6.35670	−0.212838
$893$	24.6681	0.825487
$894$	0	0
$895$	0.379424	0.0126827
$896$	−1.09959	−0.0367347
$897$	0	0
$898$	6.20802	0.207164
$899$	−41.5256	−1.38496
$900$	0	0
$901$	−12.9187	−0.430384
$902$	−7.62642	−0.253932
$903$	0	0
$904$	0.887494	0.0295176
$905$	32.2922	1.07343
$906$	0	0
$907$	−14.9048	−0.494905	−0.247453	−	0.968900i	$-0.579593\pi$
$907$	−14.9048	−0.494905	−0.247453	+	0.968900i	$0.579593\pi$
$908$	10.6168	0.352330
$909$	0	0
$910$	−14.6441	−0.485447
$911$	0.409485	0.0135668	0.00678342	−	0.999977i	$-0.497841\pi$
$911$	0.409485	0.0135668	0.00678342	+	0.999977i	$0.497841\pi$
$912$	0	0
$913$	−2.74319	−0.0907863
$914$	38.1208	1.26092
$915$	0	0
$916$	−29.5975	−0.977930
$917$	−9.41558	−0.310930
$918$	0	0
$919$	−8.68617	−0.286530	−0.143265	−	0.989684i	$-0.545760\pi$
$919$	−8.68617	−0.286530	−0.143265	+	0.989684i	$0.545760\pi$
$920$	−8.12857	−0.267991
$921$	0	0
$922$	14.4758	0.476736
$923$	34.0880	1.12202
$924$	0	0
$925$	24.6950	0.811966
$926$	15.8368	0.520430
$927$	0	0
$928$	5.83060	0.191399
$929$	36.9180	1.21124	0.605620	−	0.795754i	$-0.292925\pi$
$929$	36.9180	1.21124	0.605620	+	0.795754i	$0.292925\pi$
$930$	0	0
$931$	−13.6622	−0.447762
$932$	−2.56299	−0.0839534
$933$	0	0
$934$	9.50173	0.310906
$935$	8.34298	0.272845
$936$	0	0
$937$	−30.1903	−0.986274	−0.493137	−	0.869952i	$-0.664150\pi$
$937$	−30.1903	−0.986274	−0.493137	+	0.869952i	$0.664150\pi$
$938$	1.79101	0.0584784
$939$	0	0
$940$	−34.9486	−1.13990
$941$	27.6699	0.902012	0.451006	−	0.892521i	$-0.351065\pi$
$941$	27.6699	0.902012	0.451006	+	0.892521i	$0.351065\pi$
$942$	0	0
$943$	−23.5142	−0.765729
$944$	−3.34338	−0.108818
$945$	0	0
$946$	−1.50525	−0.0489401
$947$	−55.5140	−1.80396	−0.901982	−	0.431774i	$-0.857888\pi$
$947$	−55.5140	−1.80396	−0.901982	+	0.431774i	$0.857888\pi$
$948$	0	0
$949$	35.1591	1.14131
$950$	−14.5620	−0.472452
$951$	0	0
$952$	3.47974	0.112779
$953$	−31.6847	−1.02637	−0.513185	−	0.858278i	$-0.671534\pi$
$953$	−31.6847	−1.02637	−0.513185	+	0.858278i	$0.671534\pi$
$954$	0	0
$955$	35.4336	1.14661
$956$	−11.9467	−0.386385
$957$	0	0
$958$	−24.7914	−0.800975
$959$	−10.2668	−0.331531
$960$	0	0
$961$	19.7230	0.636227
$962$	15.9414	0.513971
$963$	0	0
$964$	22.7488	0.732690
$965$	−53.9622	−1.73711
$966$	0	0
$967$	−21.8616	−0.703020	−0.351510	−	0.936184i	$-0.614332\pi$
$967$	−21.8616	−0.703020	−0.351510	+	0.936184i	$0.614332\pi$
$968$	10.3779	0.333558
$969$	0	0
$970$	44.3261	1.42323
$971$	−46.9707	−1.50736	−0.753680	−	0.657241i	$-0.771723\pi$
$971$	−46.9707	−1.50736	−0.753680	+	0.657241i	$0.771723\pi$
$972$	0	0
$973$	9.23618	0.296098
$974$	−17.1260	−0.548753
$975$	0	0
$976$	12.6958	0.406382
$977$	−7.92668	−0.253597	−0.126799	−	0.991928i	$-0.540470\pi$
$977$	−7.92668	−0.253597	−0.126799	+	0.991928i	$0.540470\pi$
$978$	0	0
$979$	−13.6373	−0.435850
$980$	19.3560	0.618306
$981$	0	0
$982$	−21.5708	−0.688353
$983$	18.3144	0.584137	0.292069	−	0.956397i	$-0.405656\pi$
$983$	18.3144	0.584137	0.292069	+	0.956397i	$0.405656\pi$
$984$	0	0
$985$	−67.6801	−2.15647
$986$	−18.4514	−0.587613
$987$	0	0
$988$	−9.40020	−0.299060
$989$	−4.64109	−0.147578
$990$	0	0
$991$	10.9296	0.347189	0.173595	−	0.984817i	$-0.444462\pi$
$991$	10.9296	0.347189	0.173595	+	0.984817i	$0.444462\pi$
$992$	−7.12201	−0.226124
$993$	0	0
$994$	9.40741	0.298385
$995$	22.5897	0.716142
$996$	0	0
$997$	−18.4478	−0.584247	−0.292124	−	0.956381i	$-0.594362\pi$
$997$	−18.4478	−0.584247	−0.292124	+	0.956381i	$0.594362\pi$
$998$	−8.62413	−0.272992
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.j.1.2	✓	12
3.2	odd	2		8046.2.a.o.1.11	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.j.1.2	✓	12	1.1	even	1	trivial
8046.2.a.o.1.11	yes	12	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.34249 q^{5} +1.09959 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.34249 q^{5} +1.09959 q^{7} -1.00000 q^{8} +3.34249 q^{10} +0.788740 q^{11} -3.98439 q^{13} -1.09959 q^{14} +1.00000 q^{16} -3.16458 q^{17} +2.35926 q^{19} -3.34249 q^{20} -0.788740 q^{22} -2.43189 q^{23} +6.17225 q^{25} +3.98439 q^{26} +1.09959 q^{28} -5.83060 q^{29} +7.12201 q^{31} -1.00000 q^{32} +3.16458 q^{34} -3.67537 q^{35} +4.00097 q^{37} -2.35926 q^{38} +3.34249 q^{40} +9.66912 q^{41} +1.90843 q^{43} +0.788740 q^{44} +2.43189 q^{46} +10.4559 q^{47} -5.79090 q^{49} -6.17225 q^{50} -3.98439 q^{52} +4.08227 q^{53} -2.63636 q^{55} -1.09959 q^{56} +5.83060 q^{58} -3.34338 q^{59} +12.6958 q^{61} -7.12201 q^{62} +1.00000 q^{64} +13.3178 q^{65} -1.62880 q^{67} -3.16458 q^{68} +3.67537 q^{70} -8.55539 q^{71} -8.82421 q^{73} -4.00097 q^{74} +2.35926 q^{76} +0.867290 q^{77} +4.14041 q^{79} -3.34249 q^{80} -9.66912 q^{82} -3.47794 q^{83} +10.5776 q^{85} -1.90843 q^{86} -0.788740 q^{88} -17.2900 q^{89} -4.38119 q^{91} -2.43189 q^{92} -10.4559 q^{94} -7.88581 q^{95} +13.2614 q^{97} +5.79090 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 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

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