LMFDB - Embedded newform 6024.2.a.o.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 43 x^{12} + 119 x^{11} + 679 x^{10} - 1667 x^{9} - 4890 x^{8} + 9662 x^{7} + \cdots - 416 $ x^14 - 3x^13 - 43x^12 + 119x^11 + 679x^10 - 1667x^9 - 4890x^8 + 9662x^7 + 16575x^6 - 20277x^5 - 25196x^4 + 8040x^3 + 10776x^2 + 912*x - 416
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-3.91931$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.91931 q^{5} -2.30632 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.91931 q^{5} -2.30632 q^{7} +1.00000 q^{9} -0.569836 q^{11} +0.0160947 q^{13} -3.91931 q^{15} +1.39707 q^{17} +0.737373 q^{19} +2.30632 q^{21} -8.19975 q^{23} +10.3610 q^{25} -1.00000 q^{27} -3.58093 q^{29} -7.46800 q^{31} +0.569836 q^{33} -9.03918 q^{35} +3.43161 q^{37} -0.0160947 q^{39} +4.10490 q^{41} -11.0416 q^{43} +3.91931 q^{45} +1.03209 q^{47} -1.68089 q^{49} -1.39707 q^{51} +3.38840 q^{53} -2.23337 q^{55} -0.737373 q^{57} +0.600517 q^{59} -13.9166 q^{61} -2.30632 q^{63} +0.0630803 q^{65} +9.60123 q^{67} +8.19975 q^{69} +4.10931 q^{71} -13.6555 q^{73} -10.3610 q^{75} +1.31422 q^{77} +8.57075 q^{79} +1.00000 q^{81} -1.99037 q^{83} +5.47554 q^{85} +3.58093 q^{87} -18.0518 q^{89} -0.0371196 q^{91} +7.46800 q^{93} +2.88999 q^{95} -8.09889 q^{97} -0.569836 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9} - 22 q^{11} + 9 q^{13} + 3 q^{15} - 7 q^{21} - 17 q^{23} + 25 q^{25} - 14 q^{27} - 18 q^{29} - 7 q^{31} + 22 q^{33} - 27 q^{35} + 9 q^{37} - 9 q^{39} - 6 q^{41} - 14 q^{43} - 3 q^{45} - 15 q^{47} + 15 q^{49} - 11 q^{53} + 2 q^{55} - 36 q^{59} + 2 q^{61} + 7 q^{63} + 8 q^{65} + 3 q^{67} + 17 q^{69} - 29 q^{71} + 2 q^{73} - 25 q^{75} + 8 q^{77} + 23 q^{79} + 14 q^{81} - 55 q^{83} + 7 q^{85} + 18 q^{87} + 9 q^{89} - 22 q^{91} + 7 q^{93} - 27 q^{95} + 17 q^{97} - 22 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9 - 22 * q^11 + 9 * q^13 + 3 * q^15 - 7 * q^21 - 17 * q^23 + 25 * q^25 - 14 * q^27 - 18 * q^29 - 7 * q^31 + 22 * q^33 - 27 * q^35 + 9 * q^37 - 9 * q^39 - 6 * q^41 - 14 * q^43 - 3 * q^45 - 15 * q^47 + 15 * q^49 - 11 * q^53 + 2 * q^55 - 36 * q^59 + 2 * q^61 + 7 * q^63 + 8 * q^65 + 3 * q^67 + 17 * q^69 - 29 * q^71 + 2 * q^73 - 25 * q^75 + 8 * q^77 + 23 * q^79 + 14 * q^81 - 55 * q^83 + 7 * q^85 + 18 * q^87 + 9 * q^89 - 22 * q^91 + 7 * q^93 - 27 * q^95 + 17 * q^97 - 22 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.91931	1.75277	0.876385	−	0.481612i	$-0.159949\pi$
$5$	3.91931	1.75277	0.876385	+	0.481612i	$0.159949\pi$
$6$	0	0
$7$	−2.30632	−0.871707	−0.435853	−	0.900018i	$-0.643553\pi$
$7$	−2.30632	−0.871707	−0.435853	+	0.900018i	$0.643553\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.569836	−0.171812	−0.0859061	−	0.996303i	$-0.527378\pi$
$11$	−0.569836	−0.171812	−0.0859061	+	0.996303i	$0.527378\pi$
$12$	0	0
$13$	0.0160947	0.00446388	0.00223194	−	0.999998i	$-0.499290\pi$
$13$	0.0160947	0.00446388	0.00223194	+	0.999998i	$0.499290\pi$
$14$	0	0
$15$	−3.91931	−1.01196
$16$	0	0
$17$	1.39707	0.338839	0.169419	−	0.985544i	$-0.445811\pi$
$17$	1.39707	0.338839	0.169419	+	0.985544i	$0.445811\pi$
$18$	0	0
$19$	0.737373	0.169165	0.0845825	−	0.996416i	$-0.473044\pi$
$19$	0.737373	0.169165	0.0845825	+	0.996416i	$0.473044\pi$
$20$	0	0
$21$	2.30632	0.503280
$22$	0	0
$23$	−8.19975	−1.70977	−0.854883	−	0.518820i	$-0.826371\pi$
$23$	−8.19975	−1.70977	−0.854883	+	0.518820i	$0.826371\pi$
$24$	0	0
$25$	10.3610	2.07220
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.58093	−0.664961	−0.332481	−	0.943110i	$-0.607886\pi$
$29$	−3.58093	−0.664961	−0.332481	+	0.943110i	$0.607886\pi$
$30$	0	0
$31$	−7.46800	−1.34129	−0.670646	−	0.741777i	$-0.733983\pi$
$31$	−7.46800	−1.34129	−0.670646	+	0.741777i	$0.733983\pi$
$32$	0	0
$33$	0.569836	0.0991958
$34$	0	0
$35$	−9.03918	−1.52790
$36$	0	0
$37$	3.43161	0.564154	0.282077	−	0.959392i	$-0.408977\pi$
$37$	3.43161	0.564154	0.282077	+	0.959392i	$0.408977\pi$
$38$	0	0
$39$	−0.0160947	−0.00257722
$40$	0	0
$41$	4.10490	0.641078	0.320539	−	0.947235i	$-0.396136\pi$
$41$	4.10490	0.641078	0.320539	+	0.947235i	$0.396136\pi$
$42$	0	0
$43$	−11.0416	−1.68382	−0.841911	−	0.539617i	$-0.818569\pi$
$43$	−11.0416	−1.68382	−0.841911	+	0.539617i	$0.818569\pi$
$44$	0	0
$45$	3.91931	0.584256
$46$	0	0
$47$	1.03209	0.150545	0.0752727	−	0.997163i	$-0.476017\pi$
$47$	1.03209	0.150545	0.0752727	+	0.997163i	$0.476017\pi$
$48$	0	0
$49$	−1.68089	−0.240127
$50$	0	0
$51$	−1.39707	−0.195629
$52$	0	0
$53$	3.38840	0.465432	0.232716	−	0.972545i	$-0.425239\pi$
$53$	3.38840	0.465432	0.232716	+	0.972545i	$0.425239\pi$
$54$	0	0
$55$	−2.23337	−0.301147
$56$	0	0
$57$	−0.737373	−0.0976674
$58$	0	0
$59$	0.600517	0.0781806	0.0390903	−	0.999236i	$-0.487554\pi$
$59$	0.600517	0.0781806	0.0390903	+	0.999236i	$0.487554\pi$
$60$	0	0
$61$	−13.9166	−1.78183	−0.890917	−	0.454167i	$-0.849937\pi$
$61$	−13.9166	−1.78183	−0.890917	+	0.454167i	$0.849937\pi$
$62$	0	0
$63$	−2.30632	−0.290569
$64$	0	0
$65$	0.0630803	0.00782415
$66$	0	0
$67$	9.60123	1.17298	0.586488	−	0.809958i	$-0.300510\pi$
$67$	9.60123	1.17298	0.586488	+	0.809958i	$0.300510\pi$
$68$	0	0
$69$	8.19975	0.987134
$70$	0	0
$71$	4.10931	0.487685	0.243843	−	0.969815i	$-0.421592\pi$
$71$	4.10931	0.487685	0.243843	+	0.969815i	$0.421592\pi$
$72$	0	0
$73$	−13.6555	−1.59825	−0.799127	−	0.601162i	$-0.794705\pi$
$73$	−13.6555	−1.59825	−0.799127	+	0.601162i	$0.794705\pi$
$74$	0	0
$75$	−10.3610	−1.19639
$76$	0	0
$77$	1.31422	0.149770
$78$	0	0
$79$	8.57075	0.964285	0.482142	−	0.876093i	$-0.339859\pi$
$79$	8.57075	0.964285	0.482142	+	0.876093i	$0.339859\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.99037	−0.218472	−0.109236	−	0.994016i	$-0.534840\pi$
$83$	−1.99037	−0.218472	−0.109236	+	0.994016i	$0.534840\pi$
$84$	0	0
$85$	5.47554	0.593906
$86$	0	0
$87$	3.58093	0.383915
$88$	0	0
$89$	−18.0518	−1.91348	−0.956741	−	0.290941i	$-0.906032\pi$
$89$	−18.0518	−1.91348	−0.956741	+	0.290941i	$0.906032\pi$
$90$	0	0
$91$	−0.0371196	−0.00389119
$92$	0	0
$93$	7.46800	0.774396
$94$	0	0
$95$	2.88999	0.296507
$96$	0	0
$97$	−8.09889	−0.822318	−0.411159	−	0.911564i	$-0.634876\pi$
$97$	−8.09889	−0.822318	−0.411159	+	0.911564i	$0.634876\pi$
$98$	0	0
$99$	−0.569836	−0.0572707
$100$	0	0
$101$	−15.6220	−1.55445	−0.777223	−	0.629226i	$-0.783372\pi$
$101$	−15.6220	−1.55445	−0.777223	+	0.629226i	$0.783372\pi$
$102$	0	0
$103$	0.653826	0.0644234	0.0322117	−	0.999481i	$-0.489745\pi$
$103$	0.653826	0.0644234	0.0322117	+	0.999481i	$0.489745\pi$
$104$	0	0
$105$	9.03918	0.882134
$106$	0	0
$107$	13.9415	1.34777	0.673887	−	0.738834i	$-0.264623\pi$
$107$	13.9415	1.34777	0.673887	+	0.738834i	$0.264623\pi$
$108$	0	0
$109$	19.4198	1.86008	0.930039	−	0.367460i	$-0.119773\pi$
$109$	19.4198	1.86008	0.930039	+	0.367460i	$0.119773\pi$
$110$	0	0
$111$	−3.43161	−0.325714
$112$	0	0
$113$	−14.1984	−1.33567	−0.667835	−	0.744309i	$-0.732779\pi$
$113$	−14.1984	−1.33567	−0.667835	+	0.744309i	$0.732779\pi$
$114$	0	0
$115$	−32.1374	−2.99683
$116$	0	0
$117$	0.0160947	0.00148796
$118$	0	0
$119$	−3.22208	−0.295368
$120$	0	0
$121$	−10.6753	−0.970481
$122$	0	0
$123$	−4.10490	−0.370127
$124$	0	0
$125$	21.0114	1.87932
$126$	0	0
$127$	1.84005	0.163278	0.0816391	−	0.996662i	$-0.473985\pi$
$127$	1.84005	0.163278	0.0816391	+	0.996662i	$0.473985\pi$
$128$	0	0
$129$	11.0416	0.972155
$130$	0	0
$131$	−12.0789	−1.05534	−0.527670	−	0.849450i	$-0.676934\pi$
$131$	−12.0789	−1.05534	−0.527670	+	0.849450i	$0.676934\pi$
$132$	0	0
$133$	−1.70062	−0.147462
$134$	0	0
$135$	−3.91931	−0.337321
$136$	0	0
$137$	15.1009	1.29016	0.645080	−	0.764115i	$-0.276824\pi$
$137$	15.1009	1.29016	0.645080	+	0.764115i	$0.276824\pi$
$138$	0	0
$139$	−7.69187	−0.652416	−0.326208	−	0.945298i	$-0.605771\pi$
$139$	−7.69187	−0.652416	−0.326208	+	0.945298i	$0.605771\pi$
$140$	0	0
$141$	−1.03209	−0.0869174
$142$	0	0
$143$	−0.00917137	−0.000766949 0
$144$	0	0
$145$	−14.0348	−1.16552
$146$	0	0
$147$	1.68089	0.138638
$148$	0	0
$149$	14.4291	1.18208	0.591040	−	0.806642i	$-0.298717\pi$
$149$	14.4291	1.18208	0.591040	+	0.806642i	$0.298717\pi$
$150$	0	0
$151$	−6.19439	−0.504093	−0.252046	−	0.967715i	$-0.581104\pi$
$151$	−6.19439	−0.504093	−0.252046	+	0.967715i	$0.581104\pi$
$152$	0	0
$153$	1.39707	0.112946
$154$	0	0
$155$	−29.2694	−2.35098
$156$	0	0
$157$	1.65115	0.131776	0.0658881	−	0.997827i	$-0.479012\pi$
$157$	1.65115	0.131776	0.0658881	+	0.997827i	$0.479012\pi$
$158$	0	0
$159$	−3.38840	−0.268717
$160$	0	0
$161$	18.9112	1.49041
$162$	0	0
$163$	−12.5967	−0.986651	−0.493325	−	0.869845i	$-0.664219\pi$
$163$	−12.5967	−0.986651	−0.493325	+	0.869845i	$0.664219\pi$
$164$	0	0
$165$	2.23337	0.173867
$166$	0	0
$167$	10.2117	0.790207	0.395104	−	0.918637i	$-0.370709\pi$
$167$	10.2117	0.790207	0.395104	+	0.918637i	$0.370709\pi$
$168$	0	0
$169$	−12.9997	−0.999980
$170$	0	0
$171$	0.737373	0.0563883
$172$	0	0
$173$	11.6804	0.888044	0.444022	−	0.896016i	$-0.353551\pi$
$173$	11.6804	0.888044	0.444022	+	0.896016i	$0.353551\pi$
$174$	0	0
$175$	−23.8958	−1.80635
$176$	0	0
$177$	−0.600517	−0.0451376
$178$	0	0
$179$	23.0721	1.72449	0.862244	−	0.506493i	$-0.169059\pi$
$179$	23.0721	1.72449	0.862244	+	0.506493i	$0.169059\pi$
$180$	0	0
$181$	−20.9351	−1.55609	−0.778045	−	0.628208i	$-0.783789\pi$
$181$	−20.9351	−1.55609	−0.778045	+	0.628208i	$0.783789\pi$
$182$	0	0
$183$	13.9166	1.02874
$184$	0	0
$185$	13.4496	0.988832
$186$	0	0
$187$	−0.796100	−0.0582166
$188$	0	0
$189$	2.30632	0.167760
$190$	0	0
$191$	11.4013	0.824970	0.412485	−	0.910964i	$-0.364661\pi$
$191$	11.4013	0.824970	0.412485	+	0.910964i	$0.364661\pi$
$192$	0	0
$193$	−8.36460	−0.602097	−0.301049	−	0.953609i	$-0.597337\pi$
$193$	−8.36460	−0.602097	−0.301049	+	0.953609i	$0.597337\pi$
$194$	0	0
$195$	−0.0630803	−0.00451728
$196$	0	0
$197$	−24.1100	−1.71777	−0.858885	−	0.512169i	$-0.828842\pi$
$197$	−24.1100	−1.71777	−0.858885	+	0.512169i	$0.828842\pi$
$198$	0	0
$199$	−19.7156	−1.39760	−0.698801	−	0.715316i	$-0.746283\pi$
$199$	−19.7156	−1.39760	−0.698801	+	0.715316i	$0.746283\pi$
$200$	0	0
$201$	−9.60123	−0.677218
$202$	0	0
$203$	8.25876	0.579651
$204$	0	0
$205$	16.0884	1.12366
$206$	0	0
$207$	−8.19975	−0.569922
$208$	0	0
$209$	−0.420182	−0.0290646
$210$	0	0
$211$	2.26284	0.155780	0.0778901	−	0.996962i	$-0.475182\pi$
$211$	2.26284	0.155780	0.0778901	+	0.996962i	$0.475182\pi$
$212$	0	0
$213$	−4.10931	−0.281565
$214$	0	0
$215$	−43.2753	−2.95135
$216$	0	0
$217$	17.2236	1.16921
$218$	0	0
$219$	13.6555	0.922753
$220$	0	0
$221$	0.0224855	0.00151254
$222$	0	0
$223$	7.65219	0.512429	0.256214	−	0.966620i	$-0.417525\pi$
$223$	7.65219	0.512429	0.256214	+	0.966620i	$0.417525\pi$
$224$	0	0
$225$	10.3610	0.690733
$226$	0	0
$227$	20.9735	1.39206	0.696031	−	0.718011i	$-0.254947\pi$
$227$	20.9735	1.39206	0.696031	+	0.718011i	$0.254947\pi$
$228$	0	0
$229$	9.85718	0.651381	0.325690	−	0.945476i	$-0.394403\pi$
$229$	9.85718	0.651381	0.325690	+	0.945476i	$0.394403\pi$
$230$	0	0
$231$	−1.31422	−0.0864696
$232$	0	0
$233$	24.6325	1.61373	0.806866	−	0.590735i	$-0.201162\pi$
$233$	24.6325	1.61373	0.806866	+	0.590735i	$0.201162\pi$
$234$	0	0
$235$	4.04507	0.263871
$236$	0	0
$237$	−8.57075	−0.556730
$238$	0	0
$239$	−27.8295	−1.80014	−0.900069	−	0.435747i	$-0.856484\pi$
$239$	−27.8295	−1.80014	−0.900069	+	0.435747i	$0.856484\pi$
$240$	0	0
$241$	5.41442	0.348773	0.174387	−	0.984677i	$-0.444206\pi$
$241$	5.41442	0.348773	0.174387	+	0.984677i	$0.444206\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−6.58794	−0.420888
$246$	0	0
$247$	0.0118678	0.000755132 0
$248$	0	0
$249$	1.99037	0.126135
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	4.67252	0.293759
$254$	0	0
$255$	−5.47554	−0.342892
$256$	0	0
$257$	−21.8374	−1.36218	−0.681089	−	0.732200i	$-0.738493\pi$
$257$	−21.8374	−1.36218	−0.681089	+	0.732200i	$0.738493\pi$
$258$	0	0
$259$	−7.91440	−0.491777
$260$	0	0
$261$	−3.58093	−0.221654
$262$	0	0
$263$	3.56135	0.219602	0.109801	−	0.993954i	$-0.464979\pi$
$263$	3.56135	0.219602	0.109801	+	0.993954i	$0.464979\pi$
$264$	0	0
$265$	13.2802	0.815795
$266$	0	0
$267$	18.0518	1.10475
$268$	0	0
$269$	3.67560	0.224105	0.112053	−	0.993702i	$-0.464258\pi$
$269$	3.67560	0.224105	0.112053	+	0.993702i	$0.464258\pi$
$270$	0	0
$271$	−4.79408	−0.291219	−0.145610	−	0.989342i	$-0.546514\pi$
$271$	−4.79408	−0.291219	−0.145610	+	0.989342i	$0.546514\pi$
$272$	0	0
$273$	0.0371196	0.00224658
$274$	0	0
$275$	−5.90407	−0.356029
$276$	0	0
$277$	19.1481	1.15050	0.575249	−	0.817978i	$-0.304905\pi$
$277$	19.1481	1.15050	0.575249	+	0.817978i	$0.304905\pi$
$278$	0	0
$279$	−7.46800	−0.447098
$280$	0	0
$281$	17.2091	1.02661	0.513303	−	0.858207i	$-0.328422\pi$
$281$	17.2091	1.02661	0.513303	+	0.858207i	$0.328422\pi$
$282$	0	0
$283$	10.5009	0.624211	0.312106	−	0.950047i	$-0.398966\pi$
$283$	10.5009	0.624211	0.312106	+	0.950047i	$0.398966\pi$
$284$	0	0
$285$	−2.88999	−0.171188
$286$	0	0
$287$	−9.46722	−0.558832
$288$	0	0
$289$	−15.0482	−0.885188
$290$	0	0
$291$	8.09889	0.474765
$292$	0	0
$293$	8.52791	0.498206	0.249103	−	0.968477i	$-0.419864\pi$
$293$	8.52791	0.498206	0.249103	+	0.968477i	$0.419864\pi$
$294$	0	0
$295$	2.35361	0.137033
$296$	0	0
$297$	0.569836	0.0330653
$298$	0	0
$299$	−0.131973	−0.00763219
$300$	0	0
$301$	25.4653	1.46780
$302$	0	0
$303$	15.6220	0.897459
$304$	0	0
$305$	−54.5433	−3.12314
$306$	0	0
$307$	−5.91233	−0.337434	−0.168717	−	0.985665i	$-0.553962\pi$
$307$	−5.91233	−0.337434	−0.168717	+	0.985665i	$0.553962\pi$
$308$	0	0
$309$	−0.653826	−0.0371949
$310$	0	0
$311$	14.4589	0.819890	0.409945	−	0.912110i	$-0.365548\pi$
$311$	14.4589	0.819890	0.409945	+	0.912110i	$0.365548\pi$
$312$	0	0
$313$	19.7293	1.11516	0.557582	−	0.830122i	$-0.311729\pi$
$313$	19.7293	1.11516	0.557582	+	0.830122i	$0.311729\pi$
$314$	0	0
$315$	−9.03918	−0.509300
$316$	0	0
$317$	−12.5503	−0.704892	−0.352446	−	0.935832i	$-0.614650\pi$
$317$	−12.5503	−0.704892	−0.352446	+	0.935832i	$0.614650\pi$
$318$	0	0
$319$	2.04054	0.114248
$320$	0	0
$321$	−13.9415	−0.778138
$322$	0	0
$323$	1.03016	0.0573196
$324$	0	0
$325$	0.166758	0.00925005
$326$	0	0
$327$	−19.4198	−1.07392
$328$	0	0
$329$	−2.38032	−0.131231
$330$	0	0
$331$	−23.7375	−1.30473	−0.652367	−	0.757904i	$-0.726224\pi$
$331$	−23.7375	−1.30473	−0.652367	+	0.757904i	$0.726224\pi$
$332$	0	0
$333$	3.43161	0.188051
$334$	0	0
$335$	37.6302	2.05596
$336$	0	0
$337$	−22.2463	−1.21183	−0.605916	−	0.795529i	$-0.707193\pi$
$337$	−22.2463	−1.21183	−0.605916	+	0.795529i	$0.707193\pi$
$338$	0	0
$339$	14.1984	0.771149
$340$	0	0
$341$	4.25554	0.230450
$342$	0	0
$343$	20.0209	1.08103
$344$	0	0
$345$	32.1374	1.73022
$346$	0	0
$347$	−31.7152	−1.70256	−0.851280	−	0.524712i	$-0.824173\pi$
$347$	−31.7152	−1.70256	−0.851280	+	0.524712i	$0.824173\pi$
$348$	0	0
$349$	18.9889	1.01645	0.508225	−	0.861224i	$-0.330302\pi$
$349$	18.9889	1.01645	0.508225	+	0.861224i	$0.330302\pi$
$350$	0	0
$351$	−0.0160947	−0.000859074 0
$352$	0	0
$353$	−30.6412	−1.63087	−0.815433	−	0.578852i	$-0.803501\pi$
$353$	−30.6412	−1.63087	−0.815433	+	0.578852i	$0.803501\pi$
$354$	0	0
$355$	16.1057	0.854800
$356$	0	0
$357$	3.22208	0.170531
$358$	0	0
$359$	−25.8312	−1.36332	−0.681660	−	0.731670i	$-0.738741\pi$
$359$	−25.8312	−1.36332	−0.681660	+	0.731670i	$0.738741\pi$
$360$	0	0
$361$	−18.4563	−0.971383
$362$	0	0
$363$	10.6753	0.560307
$364$	0	0
$365$	−53.5201	−2.80137
$366$	0	0
$367$	−5.37268	−0.280452	−0.140226	−	0.990120i	$-0.544783\pi$
$367$	−5.37268	−0.280452	−0.140226	+	0.990120i	$0.544783\pi$
$368$	0	0
$369$	4.10490	0.213693
$370$	0	0
$371$	−7.81473	−0.405720
$372$	0	0
$373$	4.47143	0.231522	0.115761	−	0.993277i	$-0.463069\pi$
$373$	4.47143	0.231522	0.115761	+	0.993277i	$0.463069\pi$
$374$	0	0
$375$	−21.0114	−1.08503
$376$	0	0
$377$	−0.0576341	−0.00296831
$378$	0	0
$379$	10.3868	0.533536	0.266768	−	0.963761i	$-0.414044\pi$
$379$	10.3868	0.533536	0.266768	+	0.963761i	$0.414044\pi$
$380$	0	0
$381$	−1.84005	−0.0942687
$382$	0	0
$383$	−23.4034	−1.19586	−0.597929	−	0.801549i	$-0.704010\pi$
$383$	−23.4034	−1.19586	−0.597929	+	0.801549i	$0.704010\pi$
$384$	0	0
$385$	5.15086	0.262512
$386$	0	0
$387$	−11.0416	−0.561274
$388$	0	0
$389$	−17.0590	−0.864926	−0.432463	−	0.901652i	$-0.642355\pi$
$389$	−17.0590	−0.864926	−0.432463	+	0.901652i	$0.642355\pi$
$390$	0	0
$391$	−11.4556	−0.579335
$392$	0	0
$393$	12.0789	0.609300
$394$	0	0
$395$	33.5914	1.69017
$396$	0	0
$397$	16.5911	0.832685	0.416343	−	0.909208i	$-0.363312\pi$
$397$	16.5911	0.832685	0.416343	+	0.909208i	$0.363312\pi$
$398$	0	0
$399$	1.70062	0.0851373
$400$	0	0
$401$	−14.6832	−0.733242	−0.366621	−	0.930370i	$-0.619486\pi$
$401$	−14.6832	−0.733242	−0.366621	+	0.930370i	$0.619486\pi$
$402$	0	0
$403$	−0.120196	−0.00598737
$404$	0	0
$405$	3.91931	0.194752
$406$	0	0
$407$	−1.95546	−0.0969285
$408$	0	0
$409$	14.6300	0.723405	0.361702	−	0.932294i	$-0.382196\pi$
$409$	14.6300	0.723405	0.361702	+	0.932294i	$0.382196\pi$
$410$	0	0
$411$	−15.1009	−0.744874
$412$	0	0
$413$	−1.38498	−0.0681506
$414$	0	0
$415$	−7.80088	−0.382930
$416$	0	0
$417$	7.69187	0.376673
$418$	0	0
$419$	−28.9538	−1.41449	−0.707243	−	0.706970i	$-0.750061\pi$
$419$	−28.9538	−1.41449	−0.707243	+	0.706970i	$0.750061\pi$
$420$	0	0
$421$	−3.16756	−0.154378	−0.0771888	−	0.997016i	$-0.524594\pi$
$421$	−3.16756	−0.154378	−0.0771888	+	0.997016i	$0.524594\pi$
$422$	0	0
$423$	1.03209	0.0501818
$424$	0	0
$425$	14.4750	0.702141
$426$	0	0
$427$	32.0960	1.55324
$428$	0	0
$429$	0.00917137	0.000442798 0
$430$	0	0
$431$	−7.27532	−0.350440	−0.175220	−	0.984529i	$-0.556064\pi$
$431$	−7.27532	−0.350440	−0.175220	+	0.984529i	$0.556064\pi$
$432$	0	0
$433$	−19.6881	−0.946148	−0.473074	−	0.881023i	$-0.656856\pi$
$433$	−19.6881	−0.946148	−0.473074	+	0.881023i	$0.656856\pi$
$434$	0	0
$435$	14.0348	0.672915
$436$	0	0
$437$	−6.04627	−0.289232
$438$	0	0
$439$	−9.80525	−0.467979	−0.233990	−	0.972239i	$-0.575178\pi$
$439$	−9.80525	−0.467979	−0.233990	+	0.972239i	$0.575178\pi$
$440$	0	0
$441$	−1.68089	−0.0800424
$442$	0	0
$443$	−24.4651	−1.16237	−0.581186	−	0.813771i	$-0.697411\pi$
$443$	−24.4651	−1.16237	−0.581186	+	0.813771i	$0.697411\pi$
$444$	0	0
$445$	−70.7504	−3.35389
$446$	0	0
$447$	−14.4291	−0.682475
$448$	0	0
$449$	15.6339	0.737809	0.368904	−	0.929467i	$-0.379733\pi$
$449$	15.6339	0.737809	0.368904	+	0.929467i	$0.379733\pi$
$450$	0	0
$451$	−2.33912	−0.110145
$452$	0	0
$453$	6.19439	0.291038
$454$	0	0
$455$	−0.145483	−0.00682036
$456$	0	0
$457$	23.0825	1.07975	0.539877	−	0.841744i	$-0.318471\pi$
$457$	23.0825	1.07975	0.539877	+	0.841744i	$0.318471\pi$
$458$	0	0
$459$	−1.39707	−0.0652095
$460$	0	0
$461$	−34.1995	−1.59283	−0.796415	−	0.604751i	$-0.793273\pi$
$461$	−34.1995	−1.59283	−0.796415	+	0.604751i	$0.793273\pi$
$462$	0	0
$463$	24.6159	1.14400	0.571998	−	0.820255i	$-0.306168\pi$
$463$	24.6159	1.14400	0.571998	+	0.820255i	$0.306168\pi$
$464$	0	0
$465$	29.2694	1.35734
$466$	0	0
$467$	22.3773	1.03550	0.517750	−	0.855532i	$-0.326770\pi$
$467$	22.3773	1.03550	0.517750	+	0.855532i	$0.326770\pi$
$468$	0	0
$469$	−22.1435	−1.02249
$470$	0	0
$471$	−1.65115	−0.0760810
$472$	0	0
$473$	6.29188	0.289301
$474$	0	0
$475$	7.63992	0.350543
$476$	0	0
$477$	3.38840	0.155144
$478$	0	0
$479$	−41.4710	−1.89486	−0.947430	−	0.319964i	$-0.896329\pi$
$479$	−41.4710	−1.89486	−0.947430	+	0.319964i	$0.896329\pi$
$480$	0	0
$481$	0.0552310	0.00251832
$482$	0	0
$483$	−18.9112	−0.860491
$484$	0	0
$485$	−31.7421	−1.44133
$486$	0	0
$487$	−8.64415	−0.391704	−0.195852	−	0.980633i	$-0.562747\pi$
$487$	−8.64415	−0.391704	−0.195852	+	0.980633i	$0.562747\pi$
$488$	0	0
$489$	12.5967	0.569643
$490$	0	0
$491$	−7.07966	−0.319501	−0.159750	−	0.987157i	$-0.551069\pi$
$491$	−7.07966	−0.319501	−0.159750	+	0.987157i	$0.551069\pi$
$492$	0	0
$493$	−5.00279	−0.225315
$494$	0	0
$495$	−2.23337	−0.100382
$496$	0	0
$497$	−9.47738	−0.425119
$498$	0	0
$499$	3.41425	0.152843	0.0764213	−	0.997076i	$-0.475651\pi$
$499$	3.41425	0.152843	0.0764213	+	0.997076i	$0.475651\pi$
$500$	0	0
$501$	−10.2117	−0.456227
$502$	0	0
$503$	17.5123	0.780836	0.390418	−	0.920638i	$-0.372331\pi$
$503$	17.5123	0.780836	0.390418	+	0.920638i	$0.372331\pi$
$504$	0	0
$505$	−61.2274	−2.72458
$506$	0	0
$507$	12.9997	0.577339
$508$	0	0
$509$	32.3376	1.43334	0.716670	−	0.697413i	$-0.245665\pi$
$509$	32.3376	1.43334	0.716670	+	0.697413i	$0.245665\pi$
$510$	0	0
$511$	31.4939	1.39321
$512$	0	0
$513$	−0.737373	−0.0325558
$514$	0	0
$515$	2.56255	0.112919
$516$	0	0
$517$	−0.588121	−0.0258655
$518$	0	0
$519$	−11.6804	−0.512712
$520$	0	0
$521$	18.8638	0.826437	0.413218	−	0.910632i	$-0.364405\pi$
$521$	18.8638	0.826437	0.413218	+	0.910632i	$0.364405\pi$
$522$	0	0
$523$	−9.42668	−0.412200	−0.206100	−	0.978531i	$-0.566077\pi$
$523$	−9.42668	−0.412200	−0.206100	+	0.978531i	$0.566077\pi$
$524$	0	0
$525$	23.8958	1.04290
$526$	0	0
$527$	−10.4333	−0.454482
$528$	0	0
$529$	44.2359	1.92330
$530$	0	0
$531$	0.600517	0.0260602
$532$	0	0
$533$	0.0660674	0.00286170
$534$	0	0
$535$	54.6410	2.36234
$536$	0	0
$537$	−23.0721	−0.995633
$538$	0	0
$539$	0.957833	0.0412568
$540$	0	0
$541$	19.2773	0.828798	0.414399	−	0.910095i	$-0.363992\pi$
$541$	19.2773	0.828798	0.414399	+	0.910095i	$0.363992\pi$
$542$	0	0
$543$	20.9351	0.898409
$544$	0	0
$545$	76.1122	3.26029
$546$	0	0
$547$	34.2176	1.46304	0.731519	−	0.681821i	$-0.238812\pi$
$547$	34.2176	1.46304	0.731519	+	0.681821i	$0.238812\pi$
$548$	0	0
$549$	−13.9166	−0.593944
$550$	0	0
$551$	−2.64048	−0.112488
$552$	0	0
$553$	−19.7669	−0.840573
$554$	0	0
$555$	−13.4496	−0.570902
$556$	0	0
$557$	1.36806	0.0579668	0.0289834	−	0.999580i	$-0.490773\pi$
$557$	1.36806	0.0579668	0.0289834	+	0.999580i	$0.490773\pi$
$558$	0	0
$559$	−0.177711	−0.00751638
$560$	0	0
$561$	0.796100	0.0336114
$562$	0	0
$563$	−13.4381	−0.566349	−0.283175	−	0.959068i	$-0.591388\pi$
$563$	−13.4381	−0.566349	−0.283175	+	0.959068i	$0.591388\pi$
$564$	0	0
$565$	−55.6478	−2.34112
$566$	0	0
$567$	−2.30632	−0.0968563
$568$	0	0
$569$	−22.0712	−0.925273	−0.462637	−	0.886548i	$-0.653097\pi$
$569$	−22.0712	−0.925273	−0.462637	+	0.886548i	$0.653097\pi$
$570$	0	0
$571$	13.0973	0.548106	0.274053	−	0.961715i	$-0.411636\pi$
$571$	13.0973	0.548106	0.274053	+	0.961715i	$0.411636\pi$
$572$	0	0
$573$	−11.4013	−0.476297
$574$	0	0
$575$	−84.9576	−3.54298
$576$	0	0
$577$	10.6019	0.441362	0.220681	−	0.975346i	$-0.429172\pi$
$577$	10.6019	0.441362	0.220681	+	0.975346i	$0.429172\pi$
$578$	0	0
$579$	8.36460	0.347621
$580$	0	0
$581$	4.59043	0.190443
$582$	0	0
$583$	−1.93083	−0.0799669
$584$	0	0
$585$	0.0630803	0.00260805
$586$	0	0
$587$	−28.2422	−1.16568	−0.582841	−	0.812586i	$-0.698059\pi$
$587$	−28.2422	−1.16568	−0.582841	+	0.812586i	$0.698059\pi$
$588$	0	0
$589$	−5.50670	−0.226900
$590$	0	0
$591$	24.1100	0.991754
$592$	0	0
$593$	−27.4397	−1.12681	−0.563407	−	0.826180i	$-0.690510\pi$
$593$	−27.4397	−1.12681	−0.563407	+	0.826180i	$0.690510\pi$
$594$	0	0
$595$	−12.6284	−0.517712
$596$	0	0
$597$	19.7156	0.806906
$598$	0	0
$599$	−27.3276	−1.11658	−0.558288	−	0.829647i	$-0.688542\pi$
$599$	−27.3276	−1.11658	−0.558288	+	0.829647i	$0.688542\pi$
$600$	0	0
$601$	−25.3175	−1.03272	−0.516361	−	0.856371i	$-0.672714\pi$
$601$	−25.3175	−1.03272	−0.516361	+	0.856371i	$0.672714\pi$
$602$	0	0
$603$	9.60123	0.390992
$604$	0	0
$605$	−41.8398	−1.70103
$606$	0	0
$607$	18.3599	0.745204	0.372602	−	0.927991i	$-0.378466\pi$
$607$	18.3599	0.745204	0.372602	+	0.927991i	$0.378466\pi$
$608$	0	0
$609$	−8.25876	−0.334662
$610$	0	0
$611$	0.0166112	0.000672016 0
$612$	0	0
$613$	−3.69384	−0.149193	−0.0745964	−	0.997214i	$-0.523767\pi$
$613$	−3.69384	−0.149193	−0.0745964	+	0.997214i	$0.523767\pi$
$614$	0	0
$615$	−16.0884	−0.648747
$616$	0	0
$617$	−0.210959	−0.00849288	−0.00424644	−	0.999991i	$-0.501352\pi$
$617$	−0.210959	−0.00849288	−0.00424644	+	0.999991i	$0.501352\pi$
$618$	0	0
$619$	−7.94591	−0.319373	−0.159687	−	0.987168i	$-0.551048\pi$
$619$	−7.94591	−0.319373	−0.159687	+	0.987168i	$0.551048\pi$
$620$	0	0
$621$	8.19975	0.329045
$622$	0	0
$623$	41.6331	1.66800
$624$	0	0
$625$	30.5453	1.22181
$626$	0	0
$627$	0.420182	0.0167804
$628$	0	0
$629$	4.79420	0.191157
$630$	0	0
$631$	30.2346	1.20362	0.601811	−	0.798638i	$-0.294446\pi$
$631$	30.2346	1.20362	0.601811	+	0.798638i	$0.294446\pi$
$632$	0	0
$633$	−2.26284	−0.0899398
$634$	0	0
$635$	7.21174	0.286189
$636$	0	0
$637$	−0.0270535	−0.00107190
$638$	0	0
$639$	4.10931	0.162562
$640$	0	0
$641$	−29.9950	−1.18473	−0.592365	−	0.805669i	$-0.701806\pi$
$641$	−29.9950	−1.18473	−0.592365	+	0.805669i	$0.701806\pi$
$642$	0	0
$643$	8.42642	0.332306	0.166153	−	0.986100i	$-0.446865\pi$
$643$	8.42642	0.332306	0.166153	+	0.986100i	$0.446865\pi$
$644$	0	0
$645$	43.2753	1.70396
$646$	0	0
$647$	30.3898	1.19475	0.597373	−	0.801964i	$-0.296211\pi$
$647$	30.3898	1.19475	0.597373	+	0.801964i	$0.296211\pi$
$648$	0	0
$649$	−0.342196	−0.0134324
$650$	0	0
$651$	−17.2236	−0.675046
$652$	0	0
$653$	−36.3888	−1.42400	−0.712002	−	0.702178i	$-0.752211\pi$
$653$	−36.3888	−1.42400	−0.712002	+	0.702178i	$0.752211\pi$
$654$	0	0
$655$	−47.3410	−1.84977
$656$	0	0
$657$	−13.6555	−0.532752
$658$	0	0
$659$	−24.2152	−0.943291	−0.471646	−	0.881788i	$-0.656340\pi$
$659$	−24.2152	−0.943291	−0.471646	+	0.881788i	$0.656340\pi$
$660$	0	0
$661$	−0.977423	−0.0380173	−0.0190087	−	0.999819i	$-0.506051\pi$
$661$	−0.977423	−0.0380173	−0.0190087	+	0.999819i	$0.506051\pi$
$662$	0	0
$663$	−0.0224855	−0.000873263 0
$664$	0	0
$665$	−6.66525	−0.258467
$666$	0	0
$667$	29.3627	1.13693
$668$	0	0
$669$	−7.65219	−0.295851
$670$	0	0
$671$	7.93016	0.306141
$672$	0	0
$673$	−4.86603	−0.187572	−0.0937858	−	0.995592i	$-0.529897\pi$
$673$	−4.86603	−0.187572	−0.0937858	+	0.995592i	$0.529897\pi$
$674$	0	0
$675$	−10.3610	−0.398795
$676$	0	0
$677$	−19.7531	−0.759172	−0.379586	−	0.925156i	$-0.623934\pi$
$677$	−19.7531	−0.759172	−0.379586	+	0.925156i	$0.623934\pi$
$678$	0	0
$679$	18.6786	0.716820
$680$	0	0
$681$	−20.9735	−0.803708
$682$	0	0
$683$	11.4373	0.437635	0.218817	−	0.975766i	$-0.429780\pi$
$683$	11.4373	0.437635	0.218817	+	0.975766i	$0.429780\pi$
$684$	0	0
$685$	59.1853	2.26135
$686$	0	0
$687$	−9.85718	−0.376075
$688$	0	0
$689$	0.0545354	0.00207763
$690$	0	0
$691$	−18.2782	−0.695334	−0.347667	−	0.937618i	$-0.613026\pi$
$691$	−18.2782	−0.695334	−0.347667	+	0.937618i	$0.613026\pi$
$692$	0	0
$693$	1.31422	0.0499233
$694$	0	0
$695$	−30.1468	−1.14353
$696$	0	0
$697$	5.73483	0.217222
$698$	0	0
$699$	−24.6325	−0.931688
$700$	0	0
$701$	−11.2815	−0.426095	−0.213047	−	0.977042i	$-0.568339\pi$
$701$	−11.2815	−0.426095	−0.213047	+	0.977042i	$0.568339\pi$
$702$	0	0
$703$	2.53038	0.0954350
$704$	0	0
$705$	−4.04507	−0.152346
$706$	0	0
$707$	36.0293	1.35502
$708$	0	0
$709$	−27.9554	−1.04989	−0.524944	−	0.851137i	$-0.675914\pi$
$709$	−27.9554	−1.04989	−0.524944	+	0.851137i	$0.675914\pi$
$710$	0	0
$711$	8.57075	0.321428
$712$	0	0
$713$	61.2358	2.29330
$714$	0	0
$715$	−0.0359455	−0.00134428
$716$	0	0
$717$	27.8295	1.03931
$718$	0	0
$719$	16.2368	0.605531	0.302766	−	0.953065i	$-0.402090\pi$
$719$	16.2368	0.605531	0.302766	+	0.953065i	$0.402090\pi$
$720$	0	0
$721$	−1.50793	−0.0561583
$722$	0	0
$723$	−5.41442	−0.201364
$724$	0	0
$725$	−37.1020	−1.37793
$726$	0	0
$727$	−49.1339	−1.82227	−0.911137	−	0.412103i	$-0.864794\pi$
$727$	−49.1339	−1.82227	−0.911137	+	0.412103i	$0.864794\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−15.4258	−0.570544
$732$	0	0
$733$	−1.25092	−0.0462038	−0.0231019	−	0.999733i	$-0.507354\pi$
$733$	−1.25092	−0.0462038	−0.0231019	+	0.999733i	$0.507354\pi$
$734$	0	0
$735$	6.58794	0.243000
$736$	0	0
$737$	−5.47113	−0.201532
$738$	0	0
$739$	16.8155	0.618567	0.309284	−	0.950970i	$-0.399911\pi$
$739$	16.8155	0.618567	0.309284	+	0.950970i	$0.399911\pi$
$740$	0	0
$741$	−0.0118678	−0.000435976 0
$742$	0	0
$743$	12.3630	0.453553	0.226776	−	0.973947i	$-0.427181\pi$
$743$	12.3630	0.453553	0.226776	+	0.973947i	$0.427181\pi$
$744$	0	0
$745$	56.5523	2.07191
$746$	0	0
$747$	−1.99037	−0.0728238
$748$	0	0
$749$	−32.1535	−1.17486
$750$	0	0
$751$	−33.3579	−1.21725	−0.608624	−	0.793459i	$-0.708278\pi$
$751$	−33.3579	−1.21725	−0.608624	+	0.793459i	$0.708278\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−24.2778	−0.883558
$756$	0	0
$757$	−13.3109	−0.483792	−0.241896	−	0.970302i	$-0.577769\pi$
$757$	−13.3109	−0.483792	−0.241896	+	0.970302i	$0.577769\pi$
$758$	0	0
$759$	−4.67252	−0.169602
$760$	0	0
$761$	48.1762	1.74639	0.873193	−	0.487375i	$-0.162045\pi$
$761$	48.1762	1.74639	0.873193	+	0.487375i	$0.162045\pi$
$762$	0	0
$763$	−44.7882	−1.62144
$764$	0	0
$765$	5.47554	0.197969
$766$	0	0
$767$	0.00966517	0.000348989 0
$768$	0	0
$769$	6.80210	0.245290	0.122645	−	0.992451i	$-0.460862\pi$
$769$	6.80210	0.245290	0.122645	+	0.992451i	$0.460862\pi$
$770$	0	0
$771$	21.8374	0.786454
$772$	0	0
$773$	32.3072	1.16201	0.581004	−	0.813901i	$-0.302660\pi$
$773$	32.3072	1.16201	0.581004	+	0.813901i	$0.302660\pi$
$774$	0	0
$775$	−77.3760	−2.77943
$776$	0	0
$777$	7.91440	0.283927
$778$	0	0
$779$	3.02684	0.108448
$780$	0	0
$781$	−2.34163	−0.0837903
$782$	0	0
$783$	3.58093	0.127972
$784$	0	0
$785$	6.47138	0.230973
$786$	0	0
$787$	−32.7250	−1.16652	−0.583259	−	0.812286i	$-0.698223\pi$
$787$	−32.7250	−1.16652	−0.583259	+	0.812286i	$0.698223\pi$
$788$	0	0
$789$	−3.56135	−0.126787
$790$	0	0
$791$	32.7460	1.16431
$792$	0	0
$793$	−0.223984	−0.00795389
$794$	0	0
$795$	−13.2802	−0.471000
$796$	0	0
$797$	−26.8386	−0.950673	−0.475336	−	0.879804i	$-0.657674\pi$
$797$	−26.8386	−0.950673	−0.475336	+	0.879804i	$0.657674\pi$
$798$	0	0
$799$	1.44190	0.0510106
$800$	0	0
$801$	−18.0518	−0.637827
$802$	0	0
$803$	7.78140	0.274600
$804$	0	0
$805$	74.1191	2.61235
$806$	0	0
$807$	−3.67560	−0.129387
$808$	0	0
$809$	5.73926	0.201782	0.100891	−	0.994897i	$-0.467831\pi$
$809$	5.73926	0.201782	0.100891	+	0.994897i	$0.467831\pi$
$810$	0	0
$811$	33.1648	1.16457	0.582287	−	0.812983i	$-0.302158\pi$
$811$	33.1648	1.16457	0.582287	+	0.812983i	$0.302158\pi$
$812$	0	0
$813$	4.79408	0.168136
$814$	0	0
$815$	−49.3704	−1.72937
$816$	0	0
$817$	−8.14174	−0.284843
$818$	0	0
$819$	−0.0371196	−0.00129706
$820$	0	0
$821$	−6.49179	−0.226565	−0.113282	−	0.993563i	$-0.536136\pi$
$821$	−6.49179	−0.226565	−0.113282	+	0.993563i	$0.536136\pi$
$822$	0	0
$823$	48.8704	1.70351	0.851757	−	0.523938i	$-0.175537\pi$
$823$	48.8704	1.70351	0.851757	+	0.523938i	$0.175537\pi$
$824$	0	0
$825$	5.90407	0.205553
$826$	0	0
$827$	−30.9672	−1.07684	−0.538418	−	0.842678i	$-0.680978\pi$
$827$	−30.9672	−1.07684	−0.538418	+	0.842678i	$0.680978\pi$
$828$	0	0
$829$	15.0860	0.523957	0.261979	−	0.965074i	$-0.415625\pi$
$829$	15.0860	0.523957	0.261979	+	0.965074i	$0.415625\pi$
$830$	0	0
$831$	−19.1481	−0.664241
$832$	0	0
$833$	−2.34832	−0.0813644
$834$	0	0
$835$	40.0230	1.38505
$836$	0	0
$837$	7.46800	0.258132
$838$	0	0
$839$	−2.23485	−0.0771556	−0.0385778	−	0.999256i	$-0.512283\pi$
$839$	−2.23485	−0.0771556	−0.0385778	+	0.999256i	$0.512283\pi$
$840$	0	0
$841$	−16.1770	−0.557827
$842$	0	0
$843$	−17.2091	−0.592711
$844$	0	0
$845$	−50.9500	−1.75273
$846$	0	0
$847$	24.6206	0.845974
$848$	0	0
$849$	−10.5009	−0.360389
$850$	0	0
$851$	−28.1384	−0.964571
$852$	0	0
$853$	34.9109	1.19533	0.597663	−	0.801747i	$-0.296096\pi$
$853$	34.9109	1.19533	0.597663	+	0.801747i	$0.296096\pi$
$854$	0	0
$855$	2.88999	0.0988357
$856$	0	0
$857$	54.9477	1.87698	0.938488	−	0.345312i	$-0.112227\pi$
$857$	54.9477	1.87698	0.938488	+	0.345312i	$0.112227\pi$
$858$	0	0
$859$	−15.2362	−0.519851	−0.259926	−	0.965629i	$-0.583698\pi$
$859$	−15.2362	−0.519851	−0.259926	+	0.965629i	$0.583698\pi$
$860$	0	0
$861$	9.46722	0.322642
$862$	0	0
$863$	16.1701	0.550438	0.275219	−	0.961382i	$-0.411250\pi$
$863$	16.1701	0.550438	0.275219	+	0.961382i	$0.411250\pi$
$864$	0	0
$865$	45.7791	1.55654
$866$	0	0
$867$	15.0482	0.511064
$868$	0	0
$869$	−4.88393	−0.165676
$870$	0	0
$871$	0.154529	0.00523603
$872$	0	0
$873$	−8.09889	−0.274106
$874$	0	0
$875$	−48.4590	−1.63821
$876$	0	0
$877$	−35.5752	−1.20129	−0.600645	−	0.799516i	$-0.705090\pi$
$877$	−35.5752	−1.20129	−0.600645	+	0.799516i	$0.705090\pi$
$878$	0	0
$879$	−8.52791	−0.287639
$880$	0	0
$881$	−38.5284	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$881$	−38.5284	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$882$	0	0
$883$	28.4807	0.958450	0.479225	−	0.877692i	$-0.340918\pi$
$883$	28.4807	0.958450	0.479225	+	0.877692i	$0.340918\pi$
$884$	0	0
$885$	−2.35361	−0.0791158
$886$	0	0
$887$	27.1237	0.910725	0.455362	−	0.890306i	$-0.349510\pi$
$887$	27.1237	0.910725	0.455362	+	0.890306i	$0.349510\pi$
$888$	0	0
$889$	−4.24375	−0.142331
$890$	0	0
$891$	−0.569836	−0.0190902
$892$	0	0
$893$	0.761033	0.0254670
$894$	0	0
$895$	90.4266	3.02263
$896$	0	0
$897$	0.131973	0.00440645
$898$	0	0
$899$	26.7424	0.891908
$900$	0	0
$901$	4.73382	0.157706
$902$	0	0
$903$	−25.4653	−0.847434
$904$	0	0
$905$	−82.0510	−2.72747
$906$	0	0
$907$	−20.2514	−0.672436	−0.336218	−	0.941784i	$-0.609148\pi$
$907$	−20.2514	−0.672436	−0.336218	+	0.941784i	$0.609148\pi$
$908$	0	0
$909$	−15.6220	−0.518148
$910$	0	0
$911$	13.4382	0.445228	0.222614	−	0.974907i	$-0.428541\pi$
$911$	13.4382	0.445228	0.222614	+	0.974907i	$0.428541\pi$
$912$	0	0
$913$	1.13419	0.0375361
$914$	0	0
$915$	54.5433	1.80315
$916$	0	0
$917$	27.8578	0.919946
$918$	0	0
$919$	42.3165	1.39589	0.697947	−	0.716150i	$-0.254097\pi$
$919$	42.3165	1.39589	0.697947	+	0.716150i	$0.254097\pi$
$920$	0	0
$921$	5.91233	0.194818
$922$	0	0
$923$	0.0661383	0.00217697
$924$	0	0
$925$	35.5550	1.16904
$926$	0	0
$927$	0.653826	0.0214745
$928$	0	0
$929$	22.8255	0.748880	0.374440	−	0.927251i	$-0.377835\pi$
$929$	22.8255	0.748880	0.374440	+	0.927251i	$0.377835\pi$
$930$	0	0
$931$	−1.23944	−0.0406211
$932$	0	0
$933$	−14.4589	−0.473364
$934$	0	0
$935$	−3.12016	−0.102040
$936$	0	0
$937$	43.9274	1.43504	0.717522	−	0.696536i	$-0.245276\pi$
$937$	43.9274	1.43504	0.717522	+	0.696536i	$0.245276\pi$
$938$	0	0
$939$	−19.7293	−0.643840
$940$	0	0
$941$	56.1448	1.83027	0.915135	−	0.403148i	$-0.132084\pi$
$941$	56.1448	1.83027	0.915135	+	0.403148i	$0.132084\pi$
$942$	0	0
$943$	−33.6592	−1.09609
$944$	0	0
$945$	9.03918	0.294045
$946$	0	0
$947$	−25.5852	−0.831409	−0.415704	−	0.909500i	$-0.636465\pi$
$947$	−25.5852	−0.831409	−0.415704	+	0.909500i	$0.636465\pi$
$948$	0	0
$949$	−0.219782	−0.00713442
$950$	0	0
$951$	12.5503	0.406970
$952$	0	0
$953$	39.7931	1.28902	0.644512	−	0.764594i	$-0.277061\pi$
$953$	39.7931	1.28902	0.644512	+	0.764594i	$0.277061\pi$
$954$	0	0
$955$	44.6853	1.44598
$956$	0	0
$957$	−2.04054	−0.0659613
$958$	0	0
$959$	−34.8276	−1.12464
$960$	0	0
$961$	24.7711	0.799067
$962$	0	0
$963$	13.9415	0.449258
$964$	0	0
$965$	−32.7835	−1.05534
$966$	0	0
$967$	26.7658	0.860730	0.430365	−	0.902655i	$-0.358385\pi$
$967$	26.7658	0.860730	0.430365	+	0.902655i	$0.358385\pi$
$968$	0	0
$969$	−1.03016	−0.0330935
$970$	0	0
$971$	−17.7569	−0.569846	−0.284923	−	0.958550i	$-0.591968\pi$
$971$	−17.7569	−0.569846	−0.284923	+	0.958550i	$0.591968\pi$
$972$	0	0
$973$	17.7399	0.568715
$974$	0	0
$975$	−0.166758	−0.00534052
$976$	0	0
$977$	19.2501	0.615865	0.307932	−	0.951408i	$-0.400363\pi$
$977$	19.2501	0.615865	0.307932	+	0.951408i	$0.400363\pi$
$978$	0	0
$979$	10.2865	0.328759
$980$	0	0
$981$	19.4198	0.620026
$982$	0	0
$983$	36.3517	1.15944	0.579720	−	0.814816i	$-0.303162\pi$
$983$	36.3517	1.15944	0.579720	+	0.814816i	$0.303162\pi$
$984$	0	0
$985$	−94.4947	−3.01085
$986$	0	0
$987$	2.38032	0.0757665
$988$	0	0
$989$	90.5380	2.87894
$990$	0	0
$991$	0.968685	0.0307713	0.0153856	−	0.999882i	$-0.495102\pi$
$991$	0.968685	0.0307713	0.0153856	+	0.999882i	$0.495102\pi$
$992$	0	0
$993$	23.7375	0.753288
$994$	0	0
$995$	−77.2716	−2.44967
$996$	0	0
$997$	−20.7827	−0.658196	−0.329098	−	0.944296i	$-0.606745\pi$
$997$	−20.7827	−0.658196	−0.329098	+	0.944296i	$0.606745\pi$
$998$	0	0
$999$	−3.43161	−0.108571

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.o.1.14	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.o.1.14	✓	14	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.91931 q^{5} -2.30632 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.91931 q^{5} -2.30632 q^{7} +1.00000 q^{9} -0.569836 q^{11} +0.0160947 q^{13} -3.91931 q^{15} +1.39707 q^{17} +0.737373 q^{19} +2.30632 q^{21} -8.19975 q^{23} +10.3610 q^{25} -1.00000 q^{27} -3.58093 q^{29} -7.46800 q^{31} +0.569836 q^{33} -9.03918 q^{35} +3.43161 q^{37} -0.0160947 q^{39} +4.10490 q^{41} -11.0416 q^{43} +3.91931 q^{45} +1.03209 q^{47} -1.68089 q^{49} -1.39707 q^{51} +3.38840 q^{53} -2.23337 q^{55} -0.737373 q^{57} +0.600517 q^{59} -13.9166 q^{61} -2.30632 q^{63} +0.0630803 q^{65} +9.60123 q^{67} +8.19975 q^{69} +4.10931 q^{71} -13.6555 q^{73} -10.3610 q^{75} +1.31422 q^{77} +8.57075 q^{79} +1.00000 q^{81} -1.99037 q^{83} +5.47554 q^{85} +3.58093 q^{87} -18.0518 q^{89} -0.0371196 q^{91} +7.46800 q^{93} +2.88999 q^{95} -8.09889 q^{97} -0.569836 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9} - 22 q^{11} + 9 q^{13} + 3 q^{15} - 7 q^{21} - 17 q^{23} + 25 q^{25} - 14 q^{27} - 18 q^{29} - 7 q^{31} + 22 q^{33} - 27 q^{35} + 9 q^{37} - 9 q^{39} - 6 q^{41} - 14 q^{43} - 3 q^{45} - 15 q^{47} + 15 q^{49} - 11 q^{53} + 2 q^{55} - 36 q^{59} + 2 q^{61} + 7 q^{63} + 8 q^{65} + 3 q^{67} + 17 q^{69} - 29 q^{71} + 2 q^{73} - 25 q^{75} + 8 q^{77} + 23 q^{79} + 14 q^{81} - 55 q^{83} + 7 q^{85} + 18 q^{87} + 9 q^{89} - 22 q^{91} + 7 q^{93} - 27 q^{95} + 17 q^{97} - 22 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9 - 22 * q^11 + 9 * q^13 + 3 * q^15 - 7 * q^21 - 17 * q^23 + 25 * q^25 - 14 * q^27 - 18 * q^29 - 7 * q^31 + 22 * q^33 - 27 * q^35 + 9 * q^37 - 9 * q^39 - 6 * q^41 - 14 * q^43 - 3 * q^45 - 15 * q^47 + 15 * q^49 - 11 * q^53 + 2 * q^55 - 36 * q^59 + 2 * q^61 + 7 * q^63 + 8 * q^65 + 3 * q^67 + 17 * q^69 - 29 * q^71 + 2 * q^73 - 25 * q^75 + 8 * q^77 + 23 * q^79 + 14 * q^81 - 55 * q^83 + 7 * q^85 + 18 * q^87 + 9 * q^89 - 22 * q^91 + 7 * q^93 - 27 * q^95 + 17 * q^97 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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