LMFDB - Embedded newform 8037.2.a.n.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$1$
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} - 13 x^{14} + 65 x^{13} + 47 x^{12} - 390 x^{11} + 4 x^{10} + 1115 x^{9} - 320 x^{8} + \cdots - 25 $ x^16 - 4x^15 - 13x^14 + 65x^13 + 47x^12 - 390x^11 + 4x^10 + 1115x^9 - 320x^8 - 1639x^7 + 618x^6 + 1250x^5 - 487x^4 - 456x^3 + 179x^2 + 62*x - 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.38504$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.38504 q^{2} -0.0816767 q^{4} +2.56770 q^{5} +0.393019 q^{7} -2.88320 q^{8} +O(q^{10})$	\(q+1.38504 q^{2} -0.0816767 q^{4} +2.56770 q^{5} +0.393019 q^{7} -2.88320 q^{8} +3.55636 q^{10} -0.0100745 q^{11} -1.56900 q^{13} +0.544345 q^{14} -3.82998 q^{16} -0.120713 q^{17} -1.00000 q^{19} -0.209721 q^{20} -0.0139535 q^{22} +3.27458 q^{23} +1.59309 q^{25} -2.17312 q^{26} -0.0321005 q^{28} -9.33471 q^{29} -4.52673 q^{31} +0.461740 q^{32} -0.167192 q^{34} +1.00916 q^{35} -2.88028 q^{37} -1.38504 q^{38} -7.40319 q^{40} +7.99554 q^{41} -11.2676 q^{43} +0.000822852 q^{44} +4.53541 q^{46} +1.00000 q^{47} -6.84554 q^{49} +2.20649 q^{50} +0.128151 q^{52} -1.80501 q^{53} -0.0258683 q^{55} -1.13315 q^{56} -12.9289 q^{58} +8.28210 q^{59} -7.98700 q^{61} -6.26968 q^{62} +8.29948 q^{64} -4.02871 q^{65} +16.2318 q^{67} +0.00985943 q^{68} +1.39772 q^{70} -14.5282 q^{71} -0.336076 q^{73} -3.98929 q^{74} +0.0816767 q^{76} -0.00395947 q^{77} -16.0103 q^{79} -9.83423 q^{80} +11.0741 q^{82} +14.2035 q^{83} -0.309955 q^{85} -15.6060 q^{86} +0.0290468 q^{88} -9.82812 q^{89} -0.616646 q^{91} -0.267457 q^{92} +1.38504 q^{94} -2.56770 q^{95} -17.1465 q^{97} -9.48131 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8}+O(q^{10}) $ 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8	$ 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8} - 15 q^{10} - 19 q^{13} + 6 q^{14} + 10 q^{16} + 8 q^{17} - 16 q^{19} + 11 q^{20} - 12 q^{22} + 5 q^{23} - 3 q^{25} - 9 q^{26} - 17 q^{28} + 2 q^{29} - 18 q^{31} - 3 q^{32} - 14 q^{34} + 11 q^{35} - 24 q^{37} - 4 q^{38} - 50 q^{40} + 6 q^{41} - 34 q^{43} + 4 q^{44} - 3 q^{46} + 16 q^{47} + 5 q^{49} - 26 q^{50} - 44 q^{52} + 23 q^{53} - 48 q^{55} + 3 q^{56} - 26 q^{58} + 32 q^{59} - 16 q^{61} - 32 q^{62} + 7 q^{64} + 18 q^{65} - 67 q^{67} + 19 q^{68} + 24 q^{70} - 19 q^{71} - 2 q^{73} + 29 q^{74} - 10 q^{76} - 14 q^{77} - 27 q^{79} - 15 q^{80} - 56 q^{82} + 17 q^{83} + 15 q^{85} + q^{86} - 13 q^{88} - 20 q^{89} - 42 q^{91} - 45 q^{92} + 4 q^{94} - q^{95} - 50 q^{97} - 13 q^{98}+O(q^{100}) $ 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8 - 15 * q^10 - 19 * q^13 + 6 * q^14 + 10 * q^16 + 8 * q^17 - 16 * q^19 + 11 * q^20 - 12 * q^22 + 5 * q^23 - 3 * q^25 - 9 * q^26 - 17 * q^28 + 2 * q^29 - 18 * q^31 - 3 * q^32 - 14 * q^34 + 11 * q^35 - 24 * q^37 - 4 * q^38 - 50 * q^40 + 6 * q^41 - 34 * q^43 + 4 * q^44 - 3 * q^46 + 16 * q^47 + 5 * q^49 - 26 * q^50 - 44 * q^52 + 23 * q^53 - 48 * q^55 + 3 * q^56 - 26 * q^58 + 32 * q^59 - 16 * q^61 - 32 * q^62 + 7 * q^64 + 18 * q^65 - 67 * q^67 + 19 * q^68 + 24 * q^70 - 19 * q^71 - 2 * q^73 + 29 * q^74 - 10 * q^76 - 14 * q^77 - 27 * q^79 - 15 * q^80 - 56 * q^82 + 17 * q^83 + 15 * q^85 + q^86 - 13 * q^88 - 20 * q^89 - 42 * q^91 - 45 * q^92 + 4 * q^94 - q^95 - 50 * q^97 - 13 * 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 $

$2$	1.38504	0.979368	0.489684	−	0.871900i	$-0.337112\pi$
$2$	1.38504	0.979368	0.489684	+	0.871900i	$0.337112\pi$
$3$	0	0
$3$	0	0
$4$	−0.0816767	−0.0408384
$5$	2.56770	1.14831	0.574156	−	0.818746i	$-0.305330\pi$
$5$	2.56770	1.14831	0.574156	+	0.818746i	$0.305330\pi$
$6$	0	0
$7$	0.393019	0.148547	0.0742736	−	0.997238i	$-0.476336\pi$
$7$	0.393019	0.148547	0.0742736	+	0.997238i	$0.476336\pi$
$8$	−2.88320	−1.01936
$9$	0	0
$10$	3.55636	1.12462
$11$	−0.0100745	−0.00303758	−0.00151879	−	0.999999i	$-0.500483\pi$
$11$	−0.0100745	−0.00303758	−0.00151879	+	0.999999i	$0.500483\pi$
$12$	0	0
$13$	−1.56900	−0.435161	−0.217581	−	0.976042i	$-0.569817\pi$
$13$	−1.56900	−0.435161	−0.217581	+	0.976042i	$0.569817\pi$
$14$	0.544345	0.145482
$15$	0	0
$16$	−3.82998	−0.957494
$17$	−0.120713	−0.0292772	−0.0146386	−	0.999893i	$-0.504660\pi$
$17$	−0.120713	−0.0292772	−0.0146386	+	0.999893i	$0.504660\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.209721	−0.0468951
$21$	0	0
$22$	−0.0139535	−0.00297490
$23$	3.27458	0.682798	0.341399	−	0.939918i	$-0.389099\pi$
$23$	3.27458	0.682798	0.341399	+	0.939918i	$0.389099\pi$
$24$	0	0
$25$	1.59309	0.318618
$26$	−2.17312	−0.426183
$27$	0	0
$28$	−0.0321005	−0.00606643
$29$	−9.33471	−1.73341	−0.866706	−	0.498819i	$-0.833767\pi$
$29$	−9.33471	−1.73341	−0.866706	+	0.498819i	$0.833767\pi$
$30$	0	0
$31$	−4.52673	−0.813025	−0.406513	−	0.913645i	$-0.633255\pi$
$31$	−4.52673	−0.813025	−0.406513	+	0.913645i	$0.633255\pi$
$32$	0.461740	0.0816249
$33$	0	0
$34$	−0.167192	−0.0286731
$35$	1.00916	0.170578
$36$	0	0
$37$	−2.88028	−0.473515	−0.236757	−	0.971569i	$-0.576085\pi$
$37$	−2.88028	−0.473515	−0.236757	+	0.971569i	$0.576085\pi$
$38$	−1.38504	−0.224682
$39$	0	0
$40$	−7.40319	−1.17055
$41$	7.99554	1.24869	0.624346	−	0.781148i	$-0.285365\pi$
$41$	7.99554	1.24869	0.624346	+	0.781148i	$0.285365\pi$
$42$	0	0
$43$	−11.2676	−1.71829	−0.859145	−	0.511731i	$-0.829004\pi$
$43$	−11.2676	−1.71829	−0.859145	+	0.511731i	$0.829004\pi$
$44$	0.000822852 0	0.000124050 0
$45$	0	0
$46$	4.53541	0.668710
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−6.84554	−0.977934
$50$	2.20649	0.312044
$51$	0	0
$52$	0.128151	0.0177713
$53$	−1.80501	−0.247937	−0.123969	−	0.992286i	$-0.539562\pi$
$53$	−1.80501	−0.247937	−0.123969	+	0.992286i	$0.539562\pi$
$54$	0	0
$55$	−0.0258683	−0.00348808
$56$	−1.13315	−0.151424
$57$	0	0
$58$	−12.9289	−1.69765
$59$	8.28210	1.07824	0.539118	−	0.842230i	$-0.318757\pi$
$59$	8.28210	1.07824	0.539118	+	0.842230i	$0.318757\pi$
$60$	0	0
$61$	−7.98700	−1.02263	−0.511316	−	0.859393i	$-0.670842\pi$
$61$	−7.98700	−1.02263	−0.511316	+	0.859393i	$0.670842\pi$
$62$	−6.26968	−0.796251
$63$	0	0
$64$	8.29948	1.03743
$65$	−4.02871	−0.499701
$66$	0	0
$67$	16.2318	1.98303	0.991515	−	0.129992i	$-0.0414951\pi$
$67$	16.2318	1.98303	0.991515	+	0.129992i	$0.0414951\pi$
$68$	0.00985943	0.00119563
$69$	0	0
$70$	1.39772	0.167059
$71$	−14.5282	−1.72418	−0.862088	−	0.506758i	$-0.830844\pi$
$71$	−14.5282	−1.72418	−0.862088	+	0.506758i	$0.830844\pi$
$72$	0	0
$73$	−0.336076	−0.0393347	−0.0196673	−	0.999807i	$-0.506261\pi$
$73$	−0.336076	−0.0393347	−0.0196673	+	0.999807i	$0.506261\pi$
$74$	−3.98929	−0.463745
$75$	0	0
$76$	0.0816767	0.00936896
$77$	−0.00395947	−0.000451223 0
$78$	0	0
$79$	−16.0103	−1.80130	−0.900649	−	0.434548i	$-0.856908\pi$
$79$	−16.0103	−1.80130	−0.900649	+	0.434548i	$0.856908\pi$
$80$	−9.83423	−1.09950
$81$	0	0
$82$	11.0741	1.22293
$83$	14.2035	1.55904	0.779519	−	0.626379i	$-0.215464\pi$
$83$	14.2035	1.55904	0.779519	+	0.626379i	$0.215464\pi$
$84$	0	0
$85$	−0.309955	−0.0336193
$86$	−15.6060	−1.68284
$87$	0	0
$88$	0.0290468	0.00309639
$89$	−9.82812	−1.04178	−0.520889	−	0.853624i	$-0.674400\pi$
$89$	−9.82812	−1.04178	−0.520889	+	0.853624i	$0.674400\pi$
$90$	0	0
$91$	−0.616646	−0.0646420
$92$	−0.267457	−0.0278844
$93$	0	0
$94$	1.38504	0.142856
$95$	−2.56770	−0.263441
$96$	0	0
$97$	−17.1465	−1.74096	−0.870480	−	0.492205i	$-0.836191\pi$
$97$	−17.1465	−1.74096	−0.870480	+	0.492205i	$0.836191\pi$
$98$	−9.48131	−0.957757
$99$	0	0
$100$	−0.130118	−0.0130118
$101$	7.36977	0.733319	0.366660	−	0.930355i	$-0.380501\pi$
$101$	7.36977	0.733319	0.366660	+	0.930355i	$0.380501\pi$
$102$	0	0
$103$	5.79202	0.570705	0.285352	−	0.958423i	$-0.407889\pi$
$103$	5.79202	0.570705	0.285352	+	0.958423i	$0.407889\pi$
$104$	4.52372	0.443588
$105$	0	0
$106$	−2.50000	−0.242822
$107$	−13.4870	−1.30384	−0.651919	−	0.758288i	$-0.726036\pi$
$107$	−13.4870	−1.30384	−0.651919	+	0.758288i	$0.726036\pi$
$108$	0	0
$109$	2.86778	0.274683	0.137342	−	0.990524i	$-0.456144\pi$
$109$	2.86778	0.274683	0.137342	+	0.990524i	$0.456144\pi$
$110$	−0.0358285	−0.00341611
$111$	0	0
$112$	−1.50525	−0.142233
$113$	−4.45856	−0.419426	−0.209713	−	0.977763i	$-0.567253\pi$
$113$	−4.45856	−0.419426	−0.209713	+	0.977763i	$0.567253\pi$
$114$	0	0
$115$	8.40815	0.784064
$116$	0.762429	0.0707897
$117$	0	0
$118$	11.4710	1.05599
$119$	−0.0474425	−0.00434904
$120$	0	0
$121$	−10.9999	−0.999991
$122$	−11.0623	−1.00153
$123$	0	0
$124$	0.369729	0.0332026
$125$	−8.74793	−0.782438
$126$	0	0
$127$	2.05854	0.182666	0.0913331	−	0.995820i	$-0.470887\pi$
$127$	2.05854	0.182666	0.0913331	+	0.995820i	$0.470887\pi$
$128$	10.5716	0.934405
$129$	0	0
$130$	−5.57991	−0.489391
$131$	13.0862	1.14335	0.571674	−	0.820481i	$-0.306294\pi$
$131$	13.0862	1.14335	0.571674	+	0.820481i	$0.306294\pi$
$132$	0	0
$133$	−0.393019	−0.0340791
$134$	22.4816	1.94212
$135$	0	0
$136$	0.348039	0.0298441
$137$	−2.31844	−0.198077	−0.0990386	−	0.995084i	$-0.531577\pi$
$137$	−2.31844	−0.198077	−0.0990386	+	0.995084i	$0.531577\pi$
$138$	0	0
$139$	8.26237	0.700805	0.350403	−	0.936599i	$-0.386045\pi$
$139$	8.26237	0.700805	0.350403	+	0.936599i	$0.386045\pi$
$140$	−0.0824246	−0.00696615
$141$	0	0
$142$	−20.1220	−1.68860
$143$	0.0158069	0.00132184
$144$	0	0
$145$	−23.9687	−1.99050
$146$	−0.465477	−0.0385231
$147$	0	0
$148$	0.235252	0.0193376
$149$	2.41447	0.197801	0.0989004	−	0.995097i	$-0.468467\pi$
$149$	2.41447	0.197801	0.0989004	+	0.995097i	$0.468467\pi$
$150$	0	0
$151$	3.14544	0.255972	0.127986	−	0.991776i	$-0.459149\pi$
$151$	3.14544	0.255972	0.127986	+	0.991776i	$0.459149\pi$
$152$	2.88320	0.233858
$153$	0	0
$154$	−0.00548401	−0.000441914 0
$155$	−11.6233	−0.933606
$156$	0	0
$157$	11.3637	0.906925	0.453463	−	0.891275i	$-0.350189\pi$
$157$	11.3637	0.906925	0.453463	+	0.891275i	$0.350189\pi$
$158$	−22.1748	−1.76413
$159$	0	0
$160$	1.18561	0.0937308
$161$	1.28697	0.101428
$162$	0	0
$163$	−1.66318	−0.130271	−0.0651353	−	0.997876i	$-0.520748\pi$
$163$	−1.66318	−0.130271	−0.0651353	+	0.997876i	$0.520748\pi$
$164$	−0.653049	−0.0509946
$165$	0	0
$166$	19.6724	1.52687
$167$	−1.80823	−0.139925	−0.0699627	−	0.997550i	$-0.522288\pi$
$167$	−1.80823	−0.139925	−0.0699627	+	0.997550i	$0.522288\pi$
$168$	0	0
$169$	−10.5382	−0.810635
$170$	−0.429298	−0.0329257
$171$	0	0
$172$	0.920300	0.0701722
$173$	−0.787679	−0.0598861	−0.0299430	−	0.999552i	$-0.509533\pi$
$173$	−0.787679	−0.0598861	−0.0299430	+	0.999552i	$0.509533\pi$
$174$	0	0
$175$	0.626115	0.0473299
$176$	0.0385851	0.00290846
$177$	0	0
$178$	−13.6123	−1.02028
$179$	−18.1728	−1.35830	−0.679151	−	0.733999i	$-0.737652\pi$
$179$	−18.1728	−1.35830	−0.679151	+	0.733999i	$0.737652\pi$
$180$	0	0
$181$	−0.960394	−0.0713855	−0.0356928	−	0.999363i	$-0.511364\pi$
$181$	−0.960394	−0.0713855	−0.0356928	+	0.999363i	$0.511364\pi$
$182$	−0.854076	−0.0633083
$183$	0	0
$184$	−9.44127	−0.696019
$185$	−7.39569	−0.543742
$186$	0	0
$187$	0.00121612	8.89316e−5 0
$188$	−0.0816767	−0.00595689
$189$	0	0
$190$	−3.55636	−0.258005
$191$	−6.18554	−0.447570	−0.223785	−	0.974639i	$-0.571841\pi$
$191$	−6.18554	−0.447570	−0.223785	+	0.974639i	$0.571841\pi$
$192$	0	0
$193$	12.2223	0.879784	0.439892	−	0.898051i	$-0.355017\pi$
$193$	12.2223	0.879784	0.439892	+	0.898051i	$0.355017\pi$
$194$	−23.7485	−1.70504
$195$	0	0
$196$	0.559121	0.0399372
$197$	4.28552	0.305330	0.152665	−	0.988278i	$-0.451214\pi$
$197$	4.28552	0.305330	0.152665	+	0.988278i	$0.451214\pi$
$198$	0	0
$199$	−8.18679	−0.580346	−0.290173	−	0.956974i	$-0.593713\pi$
$199$	−8.18679	−0.580346	−0.290173	+	0.956974i	$0.593713\pi$
$200$	−4.59319	−0.324788
$201$	0	0
$202$	10.2074	0.718189
$203$	−3.66872	−0.257494
$204$	0	0
$205$	20.5301	1.43389
$206$	8.02215	0.558930
$207$	0	0
$208$	6.00922	0.416664
$209$	0.0100745	0.000696868 0
$210$	0	0
$211$	−3.34188	−0.230065	−0.115032	−	0.993362i	$-0.536697\pi$
$211$	−3.34188	−0.230065	−0.115032	+	0.993362i	$0.536697\pi$
$212$	0.147427	0.0101253
$213$	0	0
$214$	−18.6800	−1.27694
$215$	−28.9318	−1.97313
$216$	0	0
$217$	−1.77909	−0.120773
$218$	3.97197	0.269016
$219$	0	0
$220$	0.00211284	0.000142448 0
$221$	0.189398	0.0127403
$222$	0	0
$223$	−13.2332	−0.886159	−0.443079	−	0.896482i	$-0.646114\pi$
$223$	−13.2332	−0.886159	−0.443079	+	0.896482i	$0.646114\pi$
$224$	0.181473	0.0121252
$225$	0	0
$226$	−6.17526	−0.410772
$227$	−10.7255	−0.711878	−0.355939	−	0.934509i	$-0.615839\pi$
$227$	−10.7255	−0.711878	−0.355939	+	0.934509i	$0.615839\pi$
$228$	0	0
$229$	9.22981	0.609923	0.304961	−	0.952365i	$-0.401356\pi$
$229$	9.22981	0.609923	0.304961	+	0.952365i	$0.401356\pi$
$230$	11.6456	0.767888
$231$	0	0
$232$	26.9138	1.76698
$233$	3.98965	0.261371	0.130685	−	0.991424i	$-0.458282\pi$
$233$	3.98965	0.261371	0.130685	+	0.991424i	$0.458282\pi$
$234$	0	0
$235$	2.56770	0.167498
$236$	−0.676455	−0.0440334
$237$	0	0
$238$	−0.0657095	−0.00425931
$239$	−23.2267	−1.50241	−0.751205	−	0.660068i	$-0.770527\pi$
$239$	−23.2267	−1.50241	−0.751205	+	0.660068i	$0.770527\pi$
$240$	0	0
$241$	25.0645	1.61455	0.807274	−	0.590177i	$-0.200942\pi$
$241$	25.0645	1.61455	0.807274	+	0.590177i	$0.200942\pi$
$242$	−15.2352	−0.979359
$243$	0	0
$244$	0.652352	0.0417626
$245$	−17.5773	−1.12297
$246$	0	0
$247$	1.56900	0.0998329
$248$	13.0515	0.828768
$249$	0	0
$250$	−12.1162	−0.766295
$251$	11.1648	0.704716	0.352358	−	0.935865i	$-0.385380\pi$
$251$	11.1648	0.704716	0.352358	+	0.935865i	$0.385380\pi$
$252$	0	0
$253$	−0.0329898	−0.00207405
$254$	2.85116	0.178897
$255$	0	0
$256$	−1.95693	−0.122308
$257$	−8.16947	−0.509598	−0.254799	−	0.966994i	$-0.582009\pi$
$257$	−8.16947	−0.509598	−0.254799	+	0.966994i	$0.582009\pi$
$258$	0	0
$259$	−1.13200	−0.0703393
$260$	0.329052	0.0204070
$261$	0	0
$262$	18.1249	1.11976
$263$	−20.0485	−1.23624	−0.618120	−	0.786083i	$-0.712106\pi$
$263$	−20.0485	−1.23624	−0.618120	+	0.786083i	$0.712106\pi$
$264$	0	0
$265$	−4.63472	−0.284709
$266$	−0.544345	−0.0333760
$267$	0	0
$268$	−1.32576	−0.0809837
$269$	−8.07553	−0.492374	−0.246187	−	0.969222i	$-0.579178\pi$
$269$	−8.07553	−0.492374	−0.246187	+	0.969222i	$0.579178\pi$
$270$	0	0
$271$	18.3820	1.11663	0.558314	−	0.829630i	$-0.311449\pi$
$271$	18.3820	1.11663	0.558314	+	0.829630i	$0.311449\pi$
$272$	0.462327	0.0280327
$273$	0	0
$274$	−3.21111	−0.193991
$275$	−0.0160496	−0.000967827 0
$276$	0	0
$277$	24.6885	1.48339	0.741694	−	0.670738i	$-0.234022\pi$
$277$	24.6885	1.48339	0.741694	+	0.670738i	$0.234022\pi$
$278$	11.4437	0.686346
$279$	0	0
$280$	−2.90959	−0.173882
$281$	−7.39613	−0.441216	−0.220608	−	0.975363i	$-0.570804\pi$
$281$	−7.39613	−0.441216	−0.220608	+	0.975363i	$0.570804\pi$
$282$	0	0
$283$	−32.3699	−1.92419	−0.962094	−	0.272717i	$-0.912078\pi$
$283$	−32.3699	−1.92419	−0.962094	+	0.272717i	$0.912078\pi$
$284$	1.18661	0.0704125
$285$	0	0
$286$	0.0218930	0.00129456
$287$	3.14240	0.185490
$288$	0	0
$289$	−16.9854	−0.999143
$290$	−33.1976	−1.94943
$291$	0	0
$292$	0.0274496	0.00160636
$293$	22.3967	1.30843	0.654215	−	0.756308i	$-0.272999\pi$
$293$	22.3967	1.30843	0.654215	+	0.756308i	$0.272999\pi$
$294$	0	0
$295$	21.2659	1.23815
$296$	8.30440	0.482684
$297$	0	0
$298$	3.34412	0.193720
$299$	−5.13781	−0.297127
$300$	0	0
$301$	−4.42838	−0.255247
$302$	4.35654	0.250691
$303$	0	0
$304$	3.82998	0.219664
$305$	−20.5082	−1.17430
$306$	0	0
$307$	14.8593	0.848064	0.424032	−	0.905647i	$-0.360614\pi$
$307$	14.8593	0.848064	0.424032	+	0.905647i	$0.360614\pi$
$308$	0.000323397 0	1.84272e−5 0
$309$	0	0
$310$	−16.0987	−0.914343
$311$	−24.3847	−1.38273	−0.691364	−	0.722506i	$-0.742990\pi$
$311$	−24.3847	−1.38273	−0.691364	+	0.722506i	$0.742990\pi$
$312$	0	0
$313$	−6.35368	−0.359131	−0.179566	−	0.983746i	$-0.557469\pi$
$313$	−6.35368	−0.359131	−0.179566	+	0.983746i	$0.557469\pi$
$314$	15.7392	0.888214
$315$	0	0
$316$	1.30767	0.0735620
$317$	13.6198	0.764965	0.382483	−	0.923963i	$-0.375069\pi$
$317$	13.6198	0.764965	0.382483	+	0.923963i	$0.375069\pi$
$318$	0	0
$319$	0.0940425	0.00526537
$320$	21.3106	1.19130
$321$	0	0
$322$	1.78250	0.0993351
$323$	0.120713	0.00671664
$324$	0	0
$325$	−2.49955	−0.138650
$326$	−2.30357	−0.127583
$327$	0	0
$328$	−23.0527	−1.27287
$329$	0.393019	0.0216678
$330$	0	0
$331$	−28.7848	−1.58215	−0.791076	−	0.611717i	$-0.790479\pi$
$331$	−28.7848	−1.58215	−0.791076	+	0.611717i	$0.790479\pi$
$332$	−1.16010	−0.0636686
$333$	0	0
$334$	−2.50447	−0.137038
$335$	41.6784	2.27714
$336$	0	0
$337$	−1.72477	−0.0939540	−0.0469770	−	0.998896i	$-0.514959\pi$
$337$	−1.72477	−0.0939540	−0.0469770	+	0.998896i	$0.514959\pi$
$338$	−14.5958	−0.793910
$339$	0	0
$340$	0.0253161	0.00137296
$341$	0.0456045	0.00246962
$342$	0	0
$343$	−5.44156	−0.293817
$344$	32.4867	1.75156
$345$	0	0
$346$	−1.09096	−0.0586505
$347$	12.9585	0.695651	0.347826	−	0.937559i	$-0.386920\pi$
$347$	12.9585	0.695651	0.347826	+	0.937559i	$0.386920\pi$
$348$	0	0
$349$	6.13117	0.328194	0.164097	−	0.986444i	$-0.447529\pi$
$349$	6.13117	0.328194	0.164097	+	0.986444i	$0.447529\pi$
$350$	0.867192	0.0463533
$351$	0	0
$352$	−0.00465180	−0.000247942 0
$353$	14.1346	0.752308	0.376154	−	0.926557i	$-0.377246\pi$
$353$	14.1346	0.752308	0.376154	+	0.926557i	$0.377246\pi$
$354$	0	0
$355$	−37.3040	−1.97989
$356$	0.802729	0.0425445
$357$	0	0
$358$	−25.1700	−1.33028
$359$	−24.0513	−1.26938	−0.634691	−	0.772766i	$-0.718872\pi$
$359$	−24.0513	−1.26938	−0.634691	+	0.772766i	$0.718872\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−1.33018	−0.0699127
$363$	0	0
$364$	0.0503656	0.00263987
$365$	−0.862942	−0.0451684
$366$	0	0
$367$	8.74642	0.456559	0.228280	−	0.973596i	$-0.426690\pi$
$367$	8.74642	0.456559	0.228280	+	0.973596i	$0.426690\pi$
$368$	−12.5416	−0.653775
$369$	0	0
$370$	−10.2433	−0.532524
$371$	−0.709403	−0.0368304
$372$	0	0
$373$	−6.49202	−0.336144	−0.168072	−	0.985775i	$-0.553754\pi$
$373$	−6.49202	−0.336144	−0.168072	+	0.985775i	$0.553754\pi$
$374$	0.00168437	8.70968e−5 0
$375$	0	0
$376$	−2.88320	−0.148689
$377$	14.6461	0.754314
$378$	0	0
$379$	−17.0312	−0.874833	−0.437417	−	0.899259i	$-0.644106\pi$
$379$	−17.0312	−0.874833	−0.437417	+	0.899259i	$0.644106\pi$
$380$	0.209721	0.0107585
$381$	0	0
$382$	−8.56719	−0.438336
$383$	32.3471	1.65286	0.826430	−	0.563039i	$-0.190368\pi$
$383$	32.3471	1.65286	0.826430	+	0.563039i	$0.190368\pi$
$384$	0	0
$385$	−0.0101667	−0.000518145 0
$386$	16.9284	0.861632
$387$	0	0
$388$	1.40047	0.0710979
$389$	28.2671	1.43320	0.716600	−	0.697484i	$-0.245697\pi$
$389$	28.2671	1.43320	0.716600	+	0.697484i	$0.245697\pi$
$390$	0	0
$391$	−0.395284	−0.0199904
$392$	19.7370	0.996870
$393$	0	0
$394$	5.93559	0.299031
$395$	−41.1096	−2.06845
$396$	0	0
$397$	18.0693	0.906873	0.453437	−	0.891289i	$-0.350198\pi$
$397$	18.0693	0.906873	0.453437	+	0.891289i	$0.350198\pi$
$398$	−11.3390	−0.568372
$399$	0	0
$400$	−6.10150	−0.305075
$401$	−22.3836	−1.11779	−0.558893	−	0.829240i	$-0.688774\pi$
$401$	−22.3836	−1.11779	−0.558893	+	0.829240i	$0.688774\pi$
$402$	0	0
$403$	7.10243	0.353797
$404$	−0.601938	−0.0299476
$405$	0	0
$406$	−5.08131	−0.252181
$407$	0.0290173	0.00143834
$408$	0	0
$409$	−5.78201	−0.285902	−0.142951	−	0.989730i	$-0.545659\pi$
$409$	−5.78201	−0.285902	−0.142951	+	0.989730i	$0.545659\pi$
$410$	28.4350	1.40430
$411$	0	0
$412$	−0.473073	−0.0233066
$413$	3.25502	0.160169
$414$	0	0
$415$	36.4704	1.79026
$416$	−0.724469	−0.0355200
$417$	0	0
$418$	0.0139535	0.000682490 0
$419$	19.5188	0.953554	0.476777	−	0.879024i	$-0.341805\pi$
$419$	19.5188	0.953554	0.476777	+	0.879024i	$0.341805\pi$
$420$	0	0
$421$	19.2266	0.937049	0.468524	−	0.883451i	$-0.344786\pi$
$421$	19.2266	0.937049	0.468524	+	0.883451i	$0.344786\pi$
$422$	−4.62862	−0.225318
$423$	0	0
$424$	5.20420	0.252738
$425$	−0.192307	−0.00932824
$426$	0	0
$427$	−3.13905	−0.151909
$428$	1.10158	0.0532466
$429$	0	0
$430$	−40.0716	−1.93242
$431$	−11.3818	−0.548243	−0.274121	−	0.961695i	$-0.588387\pi$
$431$	−11.3818	−0.548243	−0.274121	+	0.961695i	$0.588387\pi$
$432$	0	0
$433$	−13.7345	−0.660036	−0.330018	−	0.943975i	$-0.607055\pi$
$433$	−13.7345	−0.660036	−0.330018	+	0.943975i	$0.607055\pi$
$434$	−2.46411	−0.118281
$435$	0	0
$436$	−0.234231	−0.0112176
$437$	−3.27458	−0.156645
$438$	0	0
$439$	−6.45666	−0.308159	−0.154080	−	0.988058i	$-0.549241\pi$
$439$	−6.45666	−0.308159	−0.154080	+	0.988058i	$0.549241\pi$
$440$	0.0745834	0.00355562
$441$	0	0
$442$	0.262323	0.0124774
$443$	−18.9321	−0.899492	−0.449746	−	0.893157i	$-0.648485\pi$
$443$	−18.9321	−0.899492	−0.449746	+	0.893157i	$0.648485\pi$
$444$	0	0
$445$	−25.2357	−1.19629
$446$	−18.3284	−0.867876
$447$	0	0
$448$	3.26185	0.154108
$449$	29.3584	1.38551	0.692755	−	0.721173i	$-0.256397\pi$
$449$	29.3584	1.38551	0.692755	+	0.721173i	$0.256397\pi$
$450$	0	0
$451$	−0.0805510	−0.00379300
$452$	0.364160	0.0171287
$453$	0	0
$454$	−14.8552	−0.697190
$455$	−1.58336	−0.0742292
$456$	0	0
$457$	−20.3630	−0.952539	−0.476270	−	0.879299i	$-0.658011\pi$
$457$	−20.3630	−0.952539	−0.476270	+	0.879299i	$0.658011\pi$
$458$	12.7836	0.597339
$459$	0	0
$460$	−0.686751	−0.0320199
$461$	−8.40744	−0.391574	−0.195787	−	0.980646i	$-0.562726\pi$
$461$	−8.40744	−0.391574	−0.195787	+	0.980646i	$0.562726\pi$
$462$	0	0
$463$	−3.88323	−0.180469	−0.0902346	−	0.995921i	$-0.528762\pi$
$463$	−3.88323	−0.180469	−0.0902346	+	0.995921i	$0.528762\pi$
$464$	35.7517	1.65973
$465$	0	0
$466$	5.52581	0.255978
$467$	20.9183	0.967984	0.483992	−	0.875072i	$-0.339186\pi$
$467$	20.9183	0.967984	0.483992	+	0.875072i	$0.339186\pi$
$468$	0	0
$469$	6.37941	0.294574
$470$	3.55636	0.164043
$471$	0	0
$472$	−23.8789	−1.09912
$473$	0.113515	0.00521944
$474$	0	0
$475$	−1.59309	−0.0730960
$476$	0.00387495	0.000177608 0
$477$	0	0
$478$	−32.1698	−1.47141
$479$	−28.4796	−1.30127	−0.650633	−	0.759392i	$-0.725497\pi$
$479$	−28.4796	−1.30127	−0.650633	+	0.759392i	$0.725497\pi$
$480$	0	0
$481$	4.51914	0.206055
$482$	34.7153	1.58124
$483$	0	0
$484$	0.898436	0.0408380
$485$	−44.0270	−1.99916
$486$	0	0
$487$	29.2716	1.32642	0.663211	−	0.748432i	$-0.269193\pi$
$487$	29.2716	1.32642	0.663211	+	0.748432i	$0.269193\pi$
$488$	23.0281	1.04243
$489$	0	0
$490$	−24.3452	−1.09980
$491$	13.4068	0.605042	0.302521	−	0.953143i	$-0.402172\pi$
$491$	13.4068	0.605042	0.302521	+	0.953143i	$0.402172\pi$
$492$	0	0
$493$	1.12682	0.0507494
$494$	2.17312	0.0977731
$495$	0	0
$496$	17.3373	0.778467
$497$	−5.70985	−0.256122
$498$	0	0
$499$	−6.87528	−0.307780	−0.153890	−	0.988088i	$-0.549180\pi$
$499$	−6.87528	−0.307780	−0.153890	+	0.988088i	$0.549180\pi$
$500$	0.714502	0.0319535
$501$	0	0
$502$	15.4636	0.690176
$503$	25.8002	1.15038	0.575188	−	0.818021i	$-0.304929\pi$
$503$	25.8002	1.15038	0.575188	+	0.818021i	$0.304929\pi$
$504$	0	0
$505$	18.9234	0.842078
$506$	−0.0456920	−0.00203126
$507$	0	0
$508$	−0.168135	−0.00745979
$509$	−12.9475	−0.573888	−0.286944	−	0.957947i	$-0.592639\pi$
$509$	−12.9475	−0.573888	−0.286944	+	0.957947i	$0.592639\pi$
$510$	0	0
$511$	−0.132084	−0.00584306
$512$	−23.8536	−1.05419
$513$	0	0
$514$	−11.3150	−0.499084
$515$	14.8722	0.655346
$516$	0	0
$517$	−0.0100745	−0.000443076 0
$518$	−1.56787	−0.0688881
$519$	0	0
$520$	11.6156	0.509377
$521$	38.0219	1.66577	0.832885	−	0.553446i	$-0.186687\pi$
$521$	38.0219	1.66577	0.832885	+	0.553446i	$0.186687\pi$
$522$	0	0
$523$	−35.1345	−1.53632	−0.768162	−	0.640256i	$-0.778828\pi$
$523$	−35.1345	−1.53632	−0.768162	+	0.640256i	$0.778828\pi$
$524$	−1.06884	−0.0466924
$525$	0	0
$526$	−27.7678	−1.21073
$527$	0.546435	0.0238031
$528$	0	0
$529$	−12.2771	−0.533787
$530$	−6.41926	−0.278835
$531$	0	0
$532$	0.0321005	0.00139173
$533$	−12.5450	−0.543383
$534$	0	0
$535$	−34.6306	−1.49721
$536$	−46.7995	−2.02143
$537$	0	0
$538$	−11.1849	−0.482215
$539$	0.0689653	0.00297055
$540$	0	0
$541$	−24.1873	−1.03989	−0.519946	−	0.854199i	$-0.674048\pi$
$541$	−24.1873	−1.03989	−0.519946	+	0.854199i	$0.674048\pi$
$542$	25.4597	1.09359
$543$	0	0
$544$	−0.0557380	−0.00238975
$545$	7.36359	0.315422
$546$	0	0
$547$	−26.8676	−1.14877	−0.574387	−	0.818584i	$-0.694760\pi$
$547$	−26.8676	−1.14877	−0.574387	+	0.818584i	$0.694760\pi$
$548$	0.189362	0.00808915
$549$	0	0
$550$	−0.0222293	−0.000947858 0
$551$	9.33471	0.397672
$552$	0	0
$553$	−6.29235	−0.267578
$554$	34.1945	1.45278
$555$	0	0
$556$	−0.674844	−0.0286197
$557$	21.5001	0.910988	0.455494	−	0.890239i	$-0.349463\pi$
$557$	21.5001	0.910988	0.455494	+	0.890239i	$0.349463\pi$
$558$	0	0
$559$	17.6788	0.747734
$560$	−3.86504	−0.163328
$561$	0	0
$562$	−10.2439	−0.432113
$563$	4.83209	0.203648	0.101824	−	0.994802i	$-0.467532\pi$
$563$	4.83209	0.203648	0.101824	+	0.994802i	$0.467532\pi$
$564$	0	0
$565$	−11.4482	−0.481631
$566$	−44.8334	−1.88449
$567$	0	0
$568$	41.8876	1.75756
$569$	−44.8011	−1.87816	−0.939080	−	0.343700i	$-0.888320\pi$
$569$	−44.8011	−1.87816	−0.939080	+	0.343700i	$0.888320\pi$
$570$	0	0
$571$	23.5482	0.985460	0.492730	−	0.870182i	$-0.335999\pi$
$571$	23.5482	0.985460	0.492730	+	0.870182i	$0.335999\pi$
$572$	−0.00129105	−5.39816e−5 0
$573$	0	0
$574$	4.35233	0.181663
$575$	5.21671	0.217552
$576$	0	0
$577$	34.0837	1.41892	0.709462	−	0.704744i	$-0.248938\pi$
$577$	34.0837	1.41892	0.709462	+	0.704744i	$0.248938\pi$
$578$	−23.5254	−0.978529
$579$	0	0
$580$	1.95769	0.0812886
$581$	5.58225	0.231591
$582$	0	0
$583$	0.0181846	0.000753127 0
$584$	0.968972	0.0400964
$585$	0	0
$586$	31.0203	1.28143
$587$	−12.1360	−0.500908	−0.250454	−	0.968129i	$-0.580580\pi$
$587$	−12.1360	−0.500908	−0.250454	+	0.968129i	$0.580580\pi$
$588$	0	0
$589$	4.52673	0.186521
$590$	29.4541	1.21261
$591$	0	0
$592$	11.0314	0.453387
$593$	−9.65600	−0.396525	−0.198262	−	0.980149i	$-0.563530\pi$
$593$	−9.65600	−0.396525	−0.198262	+	0.980149i	$0.563530\pi$
$594$	0	0
$595$	−0.121818	−0.00499406
$596$	−0.197206	−0.00807786
$597$	0	0
$598$	−7.11605	−0.290997
$599$	40.3861	1.65013	0.825066	−	0.565036i	$-0.191138\pi$
$599$	40.3861	1.65013	0.825066	+	0.565036i	$0.191138\pi$
$600$	0	0
$601$	18.2571	0.744723	0.372361	−	0.928088i	$-0.378548\pi$
$601$	18.2571	0.744723	0.372361	+	0.928088i	$0.378548\pi$
$602$	−6.13346	−0.249981
$603$	0	0
$604$	−0.256909	−0.0104535
$605$	−28.2445	−1.14830
$606$	0	0
$607$	−41.5095	−1.68482	−0.842408	−	0.538840i	$-0.818863\pi$
$607$	−41.5095	−1.68482	−0.842408	+	0.538840i	$0.818863\pi$
$608$	−0.461740	−0.0187260
$609$	0	0
$610$	−28.4046	−1.15007
$611$	−1.56900	−0.0634748
$612$	0	0
$613$	−4.66642	−0.188475	−0.0942374	−	0.995550i	$-0.530041\pi$
$613$	−4.66642	−0.188475	−0.0942374	+	0.995550i	$0.530041\pi$
$614$	20.5806	0.830567
$615$	0	0
$616$	0.0114159	0.000459961 0
$617$	−19.0215	−0.765776	−0.382888	−	0.923795i	$-0.625070\pi$
$617$	−19.0215	−0.765776	−0.382888	+	0.923795i	$0.625070\pi$
$618$	0	0
$619$	45.8139	1.84142	0.920708	−	0.390253i	$-0.127613\pi$
$619$	45.8139	1.84142	0.920708	+	0.390253i	$0.127613\pi$
$620$	0.949353	0.0381269
$621$	0	0
$622$	−33.7737	−1.35420
$623$	−3.86264	−0.154753
$624$	0	0
$625$	−30.4275	−1.21710
$626$	−8.80008	−0.351722
$627$	0	0
$628$	−0.928153	−0.0370374
$629$	0.347687	0.0138632
$630$	0	0
$631$	−11.7673	−0.468447	−0.234223	−	0.972183i	$-0.575255\pi$
$631$	−11.7673	−0.468447	−0.234223	+	0.972183i	$0.575255\pi$
$632$	46.1608	1.83618
$633$	0	0
$634$	18.8639	0.749182
$635$	5.28573	0.209758
$636$	0	0
$637$	10.7406	0.425559
$638$	0.130252	0.00515673
$639$	0	0
$640$	27.1447	1.07299
$641$	−3.17004	−0.125209	−0.0626046	−	0.998038i	$-0.519941\pi$
$641$	−3.17004	−0.125209	−0.0626046	+	0.998038i	$0.519941\pi$
$642$	0	0
$643$	−30.4215	−1.19971	−0.599854	−	0.800110i	$-0.704775\pi$
$643$	−30.4215	−1.19971	−0.599854	+	0.800110i	$0.704775\pi$
$644$	−0.105116	−0.00414214
$645$	0	0
$646$	0.167192	0.00657807
$647$	11.7665	0.462591	0.231295	−	0.972884i	$-0.425704\pi$
$647$	11.7665	0.462591	0.231295	+	0.972884i	$0.425704\pi$
$648$	0	0
$649$	−0.0834379	−0.00327523
$650$	−3.46197	−0.135790
$651$	0	0
$652$	0.135843	0.00532004
$653$	12.1513	0.475517	0.237758	−	0.971324i	$-0.423587\pi$
$653$	12.1513	0.475517	0.237758	+	0.971324i	$0.423587\pi$
$654$	0	0
$655$	33.6015	1.31292
$656$	−30.6227	−1.19562
$657$	0	0
$658$	0.544345	0.0212208
$659$	−17.8420	−0.695026	−0.347513	−	0.937675i	$-0.612974\pi$
$659$	−17.8420	−0.695026	−0.347513	+	0.937675i	$0.612974\pi$
$660$	0	0
$661$	−16.3142	−0.634549	−0.317274	−	0.948334i	$-0.602768\pi$
$661$	−16.3142	−0.634549	−0.317274	+	0.948334i	$0.602768\pi$
$662$	−39.8679	−1.54951
$663$	0	0
$664$	−40.9515	−1.58923
$665$	−1.00916	−0.0391334
$666$	0	0
$667$	−30.5673	−1.18357
$668$	0.147691	0.00571432
$669$	0	0
$670$	57.7261	2.23015
$671$	0.0804651	0.00310632
$672$	0	0
$673$	12.7703	0.492257	0.246129	−	0.969237i	$-0.420841\pi$
$673$	12.7703	0.492257	0.246129	+	0.969237i	$0.420841\pi$
$674$	−2.38886	−0.0920155
$675$	0	0
$676$	0.860730	0.0331050
$677$	38.2417	1.46975	0.734874	−	0.678203i	$-0.237241\pi$
$677$	38.2417	1.46975	0.734874	+	0.678203i	$0.237241\pi$
$678$	0	0
$679$	−6.73889	−0.258615
$680$	0.893660	0.0342703
$681$	0	0
$682$	0.0631639	0.00241867
$683$	8.11601	0.310550	0.155275	−	0.987871i	$-0.450374\pi$
$683$	8.11601	0.310550	0.155275	+	0.987871i	$0.450374\pi$
$684$	0	0
$685$	−5.95305	−0.227454
$686$	−7.53675	−0.287755
$687$	0	0
$688$	43.1546	1.64525
$689$	2.83205	0.107893
$690$	0	0
$691$	11.2703	0.428741	0.214371	−	0.976752i	$-0.431230\pi$
$691$	11.2703	0.428741	0.214371	+	0.976752i	$0.431230\pi$
$692$	0.0643350	0.00244565
$693$	0	0
$694$	17.9480	0.681299
$695$	21.2153	0.804742
$696$	0	0
$697$	−0.965164	−0.0365582
$698$	8.49189	0.321423
$699$	0	0
$700$	−0.0511390	−0.00193287
$701$	2.68493	0.101408	0.0507042	−	0.998714i	$-0.483853\pi$
$701$	2.68493	0.101408	0.0507042	+	0.998714i	$0.483853\pi$
$702$	0	0
$703$	2.88028	0.108632
$704$	−0.0836131	−0.00315129
$705$	0	0
$706$	19.5769	0.736787
$707$	2.89646	0.108933
$708$	0	0
$709$	26.2471	0.985729	0.492864	−	0.870106i	$-0.335950\pi$
$709$	26.2471	0.985729	0.492864	+	0.870106i	$0.335950\pi$
$710$	−51.6674	−1.93904
$711$	0	0
$712$	28.3364	1.06195
$713$	−14.8232	−0.555132
$714$	0	0
$715$	0.0405873	0.00151788
$716$	1.48430	0.0554708
$717$	0	0
$718$	−33.3120	−1.24319
$719$	43.4851	1.62172	0.810860	−	0.585241i	$-0.199000\pi$
$719$	43.4851	1.62172	0.810860	+	0.585241i	$0.199000\pi$
$720$	0	0
$721$	2.27637	0.0847766
$722$	1.38504	0.0515457
$723$	0	0
$724$	0.0784419	0.00291527
$725$	−14.8710	−0.552296
$726$	0	0
$727$	−33.6590	−1.24834	−0.624171	−	0.781288i	$-0.714563\pi$
$727$	−33.6590	−1.24834	−0.624171	+	0.781288i	$0.714563\pi$
$728$	1.77791	0.0658937
$729$	0	0
$730$	−1.19521	−0.0442365
$731$	1.36014	0.0503067
$732$	0	0
$733$	10.8553	0.400950	0.200475	−	0.979699i	$-0.435752\pi$
$733$	10.8553	0.400950	0.200475	+	0.979699i	$0.435752\pi$
$734$	12.1141	0.447140
$735$	0	0
$736$	1.51201	0.0557333
$737$	−0.163527	−0.00602360
$738$	0	0
$739$	−45.3715	−1.66902	−0.834509	−	0.550994i	$-0.814249\pi$
$739$	−45.3715	−1.66902	−0.834509	+	0.550994i	$0.814249\pi$
$740$	0.604056	0.0222055
$741$	0	0
$742$	−0.982548	−0.0360705
$743$	36.2810	1.33102	0.665511	−	0.746388i	$-0.268214\pi$
$743$	36.2810	1.33102	0.665511	+	0.746388i	$0.268214\pi$
$744$	0	0
$745$	6.19963	0.227137
$746$	−8.99168	−0.329209
$747$	0	0
$748$	−9.93288e−5 0	−3.63182e−6 0
$749$	−5.30065	−0.193682
$750$	0	0
$751$	20.4249	0.745317	0.372658	−	0.927969i	$-0.378446\pi$
$751$	20.4249	0.745317	0.372658	+	0.927969i	$0.378446\pi$
$752$	−3.82998	−0.139665
$753$	0	0
$754$	20.2854	0.738751
$755$	8.07655	0.293936
$756$	0	0
$757$	9.61077	0.349309	0.174655	−	0.984630i	$-0.444119\pi$
$757$	9.61077	0.349309	0.174655	+	0.984630i	$0.444119\pi$
$758$	−23.5888	−0.856783
$759$	0	0
$760$	7.40319	0.268542
$761$	1.71522	0.0621768	0.0310884	−	0.999517i	$-0.490103\pi$
$761$	1.71522	0.0621768	0.0310884	+	0.999517i	$0.490103\pi$
$762$	0	0
$763$	1.12709	0.0408034
$764$	0.505215	0.0182780
$765$	0	0
$766$	44.8019	1.61876
$767$	−12.9946	−0.469207
$768$	0	0
$769$	49.2410	1.77568	0.887838	−	0.460156i	$-0.152207\pi$
$769$	49.2410	1.77568	0.887838	+	0.460156i	$0.152207\pi$
$770$	−0.0140813	−0.000507455 0
$771$	0	0
$772$	−0.998281	−0.0359289
$773$	−6.18578	−0.222487	−0.111244	−	0.993793i	$-0.535483\pi$
$773$	−6.18578	−0.222487	−0.111244	+	0.993793i	$0.535483\pi$
$774$	0	0
$775$	−7.21150	−0.259045
$776$	49.4366	1.77467
$777$	0	0
$778$	39.1510	1.40363
$779$	−7.99554	−0.286470
$780$	0	0
$781$	0.146364	0.00523731
$782$	−0.547483	−0.0195780
$783$	0	0
$784$	26.2182	0.936366
$785$	29.1787	1.04143
$786$	0	0
$787$	−23.8364	−0.849677	−0.424838	−	0.905269i	$-0.639669\pi$
$787$	−23.8364	−0.849677	−0.424838	+	0.905269i	$0.639669\pi$
$788$	−0.350027	−0.0124692
$789$	0	0
$790$	−56.9383	−2.02577
$791$	−1.75230	−0.0623046
$792$	0	0
$793$	12.5316	0.445010
$794$	25.0266	0.888162
$795$	0	0
$796$	0.668670	0.0237004
$797$	−30.8248	−1.09187	−0.545935	−	0.837827i	$-0.683826\pi$
$797$	−30.8248	−1.09187	−0.545935	+	0.837827i	$0.683826\pi$
$798$	0	0
$799$	−0.120713	−0.00427051
$800$	0.735594	0.0260072
$801$	0	0
$802$	−31.0021	−1.09472
$803$	0.00338579	0.000119482 0
$804$	0	0
$805$	3.30457	0.116471
$806$	9.83711	0.346498
$807$	0	0
$808$	−21.2485	−0.747519
$809$	−41.6031	−1.46269	−0.731343	−	0.682009i	$-0.761106\pi$
$809$	−41.6031	−1.46269	−0.731343	+	0.682009i	$0.761106\pi$
$810$	0	0
$811$	−27.1683	−0.954010	−0.477005	−	0.878901i	$-0.658278\pi$
$811$	−27.1683	−0.954010	−0.477005	+	0.878901i	$0.658278\pi$
$812$	0.299649	0.0105156
$813$	0	0
$814$	0.0401900	0.00140866
$815$	−4.27056	−0.149591
$816$	0	0
$817$	11.2676	0.394203
$818$	−8.00829	−0.280003
$819$	0	0
$820$	−1.67684	−0.0585576
$821$	28.9175	1.00923	0.504614	−	0.863345i	$-0.331635\pi$
$821$	28.9175	1.00923	0.504614	+	0.863345i	$0.331635\pi$
$822$	0	0
$823$	45.8587	1.59853	0.799266	−	0.600977i	$-0.205222\pi$
$823$	45.8587	1.59853	0.799266	+	0.600977i	$0.205222\pi$
$824$	−16.6995	−0.581756
$825$	0	0
$826$	4.50832	0.156865
$827$	7.51274	0.261243	0.130622	−	0.991432i	$-0.458303\pi$
$827$	7.51274	0.261243	0.130622	+	0.991432i	$0.458303\pi$
$828$	0	0
$829$	−50.0796	−1.73934	−0.869668	−	0.493636i	$-0.835667\pi$
$829$	−50.0796	−1.73934	−0.869668	+	0.493636i	$0.835667\pi$
$830$	50.5128	1.75332
$831$	0	0
$832$	−13.0219	−0.451451
$833$	0.826344	0.0286311
$834$	0	0
$835$	−4.64300	−0.160678
$836$	−0.000822852 0	−2.84589e−5 0
$837$	0	0
$838$	27.0342	0.933880
$839$	−1.23516	−0.0426424	−0.0213212	−	0.999773i	$-0.506787\pi$
$839$	−1.23516	−0.0426424	−0.0213212	+	0.999773i	$0.506787\pi$
$840$	0	0
$841$	58.1368	2.00472
$842$	26.6296	0.917716
$843$	0	0
$844$	0.272954	0.00939546
$845$	−27.0591	−0.930861
$846$	0	0
$847$	−4.32317	−0.148546
$848$	6.91314	0.237398
$849$	0	0
$850$	−0.266351	−0.00913578
$851$	−9.43171	−0.323315
$852$	0	0
$853$	−6.51561	−0.223090	−0.111545	−	0.993759i	$-0.535580\pi$
$853$	−6.51561	−0.223090	−0.111545	+	0.993759i	$0.535580\pi$
$854$	−4.34769	−0.148775
$855$	0	0
$856$	38.8857	1.32909
$857$	48.3722	1.65236	0.826181	−	0.563404i	$-0.190509\pi$
$857$	48.3722	1.65236	0.826181	+	0.563404i	$0.190509\pi$
$858$	0	0
$859$	−54.0385	−1.84377	−0.921884	−	0.387465i	$-0.873351\pi$
$859$	−54.0385	−1.84377	−0.921884	+	0.387465i	$0.873351\pi$
$860$	2.36306	0.0805795
$861$	0	0
$862$	−15.7642	−0.536931
$863$	0.171163	0.00582644	0.00291322	−	0.999996i	$-0.499073\pi$
$863$	0.171163	0.00582644	0.00291322	+	0.999996i	$0.499073\pi$
$864$	0	0
$865$	−2.02252	−0.0687679
$866$	−19.0227	−0.646418
$867$	0	0
$868$	0.145310	0.00493216
$869$	0.161296	0.00547158
$870$	0	0
$871$	−25.4676	−0.862938
$872$	−8.26836	−0.280002
$873$	0	0
$874$	−4.53541	−0.153413
$875$	−3.43810	−0.116229
$876$	0	0
$877$	−6.98768	−0.235957	−0.117979	−	0.993016i	$-0.537641\pi$
$877$	−6.98768	−0.235957	−0.117979	+	0.993016i	$0.537641\pi$
$878$	−8.94270	−0.301802
$879$	0	0
$880$	0.0990750	0.00333982
$881$	13.1058	0.441547	0.220774	−	0.975325i	$-0.429142\pi$
$881$	13.1058	0.441547	0.220774	+	0.975325i	$0.429142\pi$
$882$	0	0
$883$	−8.25097	−0.277667	−0.138834	−	0.990316i	$-0.544335\pi$
$883$	−8.25097	−0.277667	−0.138834	+	0.990316i	$0.544335\pi$
$884$	−0.0154694	−0.000520293 0
$885$	0	0
$886$	−26.2216	−0.880933
$887$	−20.8380	−0.699671	−0.349835	−	0.936811i	$-0.613762\pi$
$887$	−20.8380	−0.699671	−0.349835	+	0.936811i	$0.613762\pi$
$888$	0	0
$889$	0.809047	0.0271346
$890$	−34.9523	−1.17160
$891$	0	0
$892$	1.08084	0.0361893
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	−46.6624	−1.55975
$896$	4.15484	0.138803
$897$	0	0
$898$	40.6625	1.35692
$899$	42.2557	1.40931
$900$	0	0
$901$	0.217888	0.00725890
$902$	−0.111566	−0.00371474
$903$	0	0
$904$	12.8549	0.427547
$905$	−2.46601	−0.0819728
$906$	0	0
$907$	−42.8363	−1.42236	−0.711178	−	0.703012i	$-0.751838\pi$
$907$	−42.8363	−1.42236	−0.711178	+	0.703012i	$0.751838\pi$
$908$	0.876025	0.0290719
$909$	0	0
$910$	−2.19301	−0.0726977
$911$	39.3818	1.30478	0.652389	−	0.757884i	$-0.273767\pi$
$911$	39.3818	1.30478	0.652389	+	0.757884i	$0.273767\pi$
$912$	0	0
$913$	−0.143093	−0.00473569
$914$	−28.2034	−0.932887
$915$	0	0
$916$	−0.753861	−0.0249083
$917$	5.14313	0.169841
$918$	0	0
$919$	7.15812	0.236125	0.118062	−	0.993006i	$-0.462332\pi$
$919$	7.15812	0.236125	0.118062	+	0.993006i	$0.462332\pi$
$920$	−24.2424	−0.799247
$921$	0	0
$922$	−11.6446	−0.383495
$923$	22.7946	0.750295
$924$	0	0
$925$	−4.58854	−0.150870
$926$	−5.37842	−0.176746
$927$	0	0
$928$	−4.31021	−0.141490
$929$	−6.38262	−0.209407	−0.104704	−	0.994503i	$-0.533389\pi$
$929$	−6.38262	−0.209407	−0.104704	+	0.994503i	$0.533389\pi$
$930$	0	0
$931$	6.84554	0.224353
$932$	−0.325862	−0.0106740
$933$	0	0
$934$	28.9726	0.948013
$935$	0.00312264	0.000102121 0
$936$	0	0
$937$	35.4247	1.15727	0.578636	−	0.815586i	$-0.303585\pi$
$937$	35.4247	1.15727	0.578636	+	0.815586i	$0.303585\pi$
$938$	8.83571	0.288496
$939$	0	0
$940$	−0.209721	−0.00684036
$941$	26.3533	0.859093	0.429546	−	0.903045i	$-0.358674\pi$
$941$	26.3533	0.859093	0.429546	+	0.903045i	$0.358674\pi$
$942$	0	0
$943$	26.1821	0.852605
$944$	−31.7202	−1.03241
$945$	0	0
$946$	0.157223	0.00511175
$947$	−12.3587	−0.401603	−0.200802	−	0.979632i	$-0.564355\pi$
$947$	−12.3587	−0.401603	−0.200802	+	0.979632i	$0.564355\pi$
$948$	0	0
$949$	0.527302	0.0171169
$950$	−2.20649	−0.0715879
$951$	0	0
$952$	0.136786	0.00443326
$953$	21.0268	0.681123	0.340562	−	0.940222i	$-0.389383\pi$
$953$	21.0268	0.681123	0.340562	+	0.940222i	$0.389383\pi$
$954$	0	0
$955$	−15.8826	−0.513949
$956$	1.89708	0.0613560
$957$	0	0
$958$	−39.4453	−1.27442
$959$	−0.911189	−0.0294238
$960$	0	0
$961$	−10.5087	−0.338990
$962$	6.25918	0.201804
$963$	0	0
$964$	−2.04719	−0.0659355
$965$	31.3833	1.01027
$966$	0	0
$967$	−53.5624	−1.72245	−0.861225	−	0.508223i	$-0.830303\pi$
$967$	−53.5624	−1.72245	−0.861225	+	0.508223i	$0.830303\pi$
$968$	31.7149	1.01935
$969$	0	0
$970$	−60.9789	−1.95792
$971$	8.55319	0.274485	0.137243	−	0.990537i	$-0.456176\pi$
$971$	8.55319	0.274485	0.137243	+	0.990537i	$0.456176\pi$
$972$	0	0
$973$	3.24727	0.104103
$974$	40.5422	1.29906
$975$	0	0
$976$	30.5900	0.979163
$977$	−4.10390	−0.131295	−0.0656477	−	0.997843i	$-0.520911\pi$
$977$	−4.10390	−0.131295	−0.0656477	+	0.997843i	$0.520911\pi$
$978$	0	0
$979$	0.0990133	0.00316448
$980$	1.43566	0.0458603
$981$	0	0
$982$	18.5690	0.592559
$983$	−45.9452	−1.46542	−0.732712	−	0.680539i	$-0.761746\pi$
$983$	−45.9452	−1.46542	−0.732712	+	0.680539i	$0.761746\pi$
$984$	0	0
$985$	11.0039	0.350614
$986$	1.56069	0.0497023
$987$	0	0
$988$	−0.128151	−0.00407701
$989$	−36.8967	−1.17325
$990$	0	0
$991$	19.3069	0.613304	0.306652	−	0.951822i	$-0.400791\pi$
$991$	19.3069	0.613304	0.306652	+	0.951822i	$0.400791\pi$
$992$	−2.09018	−0.0663631
$993$	0	0
$994$	−7.90834	−0.250837
$995$	−21.0212	−0.666418
$996$	0	0
$997$	53.3673	1.69016	0.845080	−	0.534639i	$-0.179553\pi$
$997$	53.3673	1.69016	0.845080	+	0.534639i	$0.179553\pi$
$998$	−9.52250	−0.301429
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.n.1.12		16
3.2	odd	2		893.2.a.b.1.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.b.1.5	✓	16	3.2	odd	2
8037.2.a.n.1.12		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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+1.38504 q^{2} -0.0816767 q^{4} +2.56770 q^{5} +0.393019 q^{7} -2.88320 q^{8} +O(q^{10})\)	\(q+1.38504 q^{2} -0.0816767 q^{4} +2.56770 q^{5} +0.393019 q^{7} -2.88320 q^{8} +3.55636 q^{10} -0.0100745 q^{11} -1.56900 q^{13} +0.544345 q^{14} -3.82998 q^{16} -0.120713 q^{17} -1.00000 q^{19} -0.209721 q^{20} -0.0139535 q^{22} +3.27458 q^{23} +1.59309 q^{25} -2.17312 q^{26} -0.0321005 q^{28} -9.33471 q^{29} -4.52673 q^{31} +0.461740 q^{32} -0.167192 q^{34} +1.00916 q^{35} -2.88028 q^{37} -1.38504 q^{38} -7.40319 q^{40} +7.99554 q^{41} -11.2676 q^{43} +0.000822852 q^{44} +4.53541 q^{46} +1.00000 q^{47} -6.84554 q^{49} +2.20649 q^{50} +0.128151 q^{52} -1.80501 q^{53} -0.0258683 q^{55} -1.13315 q^{56} -12.9289 q^{58} +8.28210 q^{59} -7.98700 q^{61} -6.26968 q^{62} +8.29948 q^{64} -4.02871 q^{65} +16.2318 q^{67} +0.00985943 q^{68} +1.39772 q^{70} -14.5282 q^{71} -0.336076 q^{73} -3.98929 q^{74} +0.0816767 q^{76} -0.00395947 q^{77} -16.0103 q^{79} -9.83423 q^{80} +11.0741 q^{82} +14.2035 q^{83} -0.309955 q^{85} -15.6060 q^{86} +0.0290468 q^{88} -9.82812 q^{89} -0.616646 q^{91} -0.267457 q^{92} +1.38504 q^{94} -2.56770 q^{95} -17.1465 q^{97} -9.48131 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8}+O(q^{10}) \) 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8	\( 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8} - 15 q^{10} - 19 q^{13} + 6 q^{14} + 10 q^{16} + 8 q^{17} - 16 q^{19} + 11 q^{20} - 12 q^{22} + 5 q^{23} - 3 q^{25} - 9 q^{26} - 17 q^{28} + 2 q^{29} - 18 q^{31} - 3 q^{32} - 14 q^{34} + 11 q^{35} - 24 q^{37} - 4 q^{38} - 50 q^{40} + 6 q^{41} - 34 q^{43} + 4 q^{44} - 3 q^{46} + 16 q^{47} + 5 q^{49} - 26 q^{50} - 44 q^{52} + 23 q^{53} - 48 q^{55} + 3 q^{56} - 26 q^{58} + 32 q^{59} - 16 q^{61} - 32 q^{62} + 7 q^{64} + 18 q^{65} - 67 q^{67} + 19 q^{68} + 24 q^{70} - 19 q^{71} - 2 q^{73} + 29 q^{74} - 10 q^{76} - 14 q^{77} - 27 q^{79} - 15 q^{80} - 56 q^{82} + 17 q^{83} + 15 q^{85} + q^{86} - 13 q^{88} - 20 q^{89} - 42 q^{91} - 45 q^{92} + 4 q^{94} - q^{95} - 50 q^{97} - 13 q^{98}+O(q^{100}) \) 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8 - 15 * q^10 - 19 * q^13 + 6 * q^14 + 10 * q^16 + 8 * q^17 - 16 * q^19 + 11 * q^20 - 12 * q^22 + 5 * q^23 - 3 * q^25 - 9 * q^26 - 17 * q^28 + 2 * q^29 - 18 * q^31 - 3 * q^32 - 14 * q^34 + 11 * q^35 - 24 * q^37 - 4 * q^38 - 50 * q^40 + 6 * q^41 - 34 * q^43 + 4 * q^44 - 3 * q^46 + 16 * q^47 + 5 * q^49 - 26 * q^50 - 44 * q^52 + 23 * q^53 - 48 * q^55 + 3 * q^56 - 26 * q^58 + 32 * q^59 - 16 * q^61 - 32 * q^62 + 7 * q^64 + 18 * q^65 - 67 * q^67 + 19 * q^68 + 24 * q^70 - 19 * q^71 - 2 * q^73 + 29 * q^74 - 10 * q^76 - 14 * q^77 - 27 * q^79 - 15 * q^80 - 56 * q^82 + 17 * q^83 + 15 * q^85 + q^86 - 13 * q^88 - 20 * q^89 - 42 * q^91 - 45 * q^92 + 4 * q^94 - q^95 - 50 * q^97 - 13 * q^98

Introduction

L-functions

Modular forms

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Database

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