LMFDB - Embedded newform 6024.2.a.n.1.3

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} - 7 x^{13} - 22 x^{12} + 214 x^{11} + 91 x^{10} - 2481 x^{9} + 1285 x^{8} + 13253 x^{7} + \cdots - 5832 $ x^14 - 7x^13 - 22x^12 + 214x^11 + 91x^10 - 2481x^9 + 1285x^8 + 13253x^7 - 14287x^6 - 29907x^5 + 49572x^4 + 10044x^3 - 53784x^2 + 32076*x - 5832
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.71742$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.71742 q^{5} +3.96243 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.71742 q^{5} +3.96243 q^{7} +1.00000 q^{9} +3.43134 q^{11} +2.15782 q^{13} +3.71742 q^{15} -1.67107 q^{17} -1.67421 q^{19} -3.96243 q^{21} -7.04857 q^{23} +8.81922 q^{25} -1.00000 q^{27} -8.05220 q^{29} -6.74835 q^{31} -3.43134 q^{33} -14.7300 q^{35} +7.53454 q^{37} -2.15782 q^{39} -7.68958 q^{41} +11.0648 q^{43} -3.71742 q^{45} -9.22133 q^{47} +8.70085 q^{49} +1.67107 q^{51} +10.1710 q^{53} -12.7557 q^{55} +1.67421 q^{57} +0.718991 q^{59} +3.40447 q^{61} +3.96243 q^{63} -8.02151 q^{65} -5.28579 q^{67} +7.04857 q^{69} +14.4930 q^{71} -6.67617 q^{73} -8.81922 q^{75} +13.5964 q^{77} +2.08862 q^{79} +1.00000 q^{81} -13.2722 q^{83} +6.21209 q^{85} +8.05220 q^{87} +1.78507 q^{89} +8.55019 q^{91} +6.74835 q^{93} +6.22373 q^{95} +8.98957 q^{97} +3.43134 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9	$ 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9} + 10 q^{11} - 9 q^{13} + 7 q^{15} - 22 q^{17} - 6 q^{19} - q^{21} + 3 q^{23} + 23 q^{25} - 14 q^{27} - 12 q^{29} - 13 q^{31} - 10 q^{33} + 23 q^{35} + 5 q^{37} + 9 q^{39} - 52 q^{41} + 16 q^{43} - 7 q^{45} + q^{47} + 9 q^{49} + 22 q^{51} - 13 q^{53} - 12 q^{55} + 6 q^{57} + 12 q^{59} - 20 q^{61} + q^{63} - 40 q^{65} + 21 q^{67} - 3 q^{69} - 5 q^{71} - 14 q^{73} - 23 q^{75} - 14 q^{77} - 23 q^{79} + 14 q^{81} + 25 q^{83} - 11 q^{85} + 12 q^{87} - 79 q^{89} + 6 q^{91} + 13 q^{93} + 3 q^{95} - 17 q^{97} + 10 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9 + 10 * q^11 - 9 * q^13 + 7 * q^15 - 22 * q^17 - 6 * q^19 - q^21 + 3 * q^23 + 23 * q^25 - 14 * q^27 - 12 * q^29 - 13 * q^31 - 10 * q^33 + 23 * q^35 + 5 * q^37 + 9 * q^39 - 52 * q^41 + 16 * q^43 - 7 * q^45 + q^47 + 9 * q^49 + 22 * q^51 - 13 * q^53 - 12 * q^55 + 6 * q^57 + 12 * q^59 - 20 * q^61 + q^63 - 40 * q^65 + 21 * q^67 - 3 * q^69 - 5 * q^71 - 14 * q^73 - 23 * q^75 - 14 * q^77 - 23 * q^79 + 14 * q^81 + 25 * q^83 - 11 * q^85 + 12 * q^87 - 79 * q^89 + 6 * q^91 + 13 * q^93 + 3 * q^95 - 17 * q^97 + 10 * 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
$5$	−3.71742	−1.66248	−0.831241	−	0.555913i	$-0.812369\pi$
$5$	−3.71742	−1.66248	−0.831241	+	0.555913i	$0.812369\pi$
$6$	0	0
$7$	3.96243	1.49766	0.748829	−	0.662763i	$-0.230617\pi$
$7$	3.96243	1.49766	0.748829	+	0.662763i	$0.230617\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.43134	1.03459	0.517294	−	0.855808i	$-0.326939\pi$
$11$	3.43134	1.03459	0.517294	+	0.855808i	$0.326939\pi$
$12$	0	0
$13$	2.15782	0.598470	0.299235	−	0.954179i	$-0.403268\pi$
$13$	2.15782	0.598470	0.299235	+	0.954179i	$0.403268\pi$
$14$	0	0
$15$	3.71742	0.959834
$16$	0	0
$17$	−1.67107	−0.405295	−0.202648	−	0.979252i	$-0.564955\pi$
$17$	−1.67107	−0.405295	−0.202648	+	0.979252i	$0.564955\pi$
$18$	0	0
$19$	−1.67421	−0.384089	−0.192045	−	0.981386i	$-0.561512\pi$
$19$	−1.67421	−0.384089	−0.192045	+	0.981386i	$0.561512\pi$
$20$	0	0
$21$	−3.96243	−0.864673
$22$	0	0
$23$	−7.04857	−1.46973	−0.734864	−	0.678214i	$-0.762754\pi$
$23$	−7.04857	−1.46973	−0.734864	+	0.678214i	$0.762754\pi$
$24$	0	0
$25$	8.81922	1.76384
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.05220	−1.49526	−0.747628	−	0.664118i	$-0.768807\pi$
$29$	−8.05220	−1.49526	−0.747628	+	0.664118i	$0.768807\pi$
$30$	0	0
$31$	−6.74835	−1.21204	−0.606020	−	0.795450i	$-0.707235\pi$
$31$	−6.74835	−1.21204	−0.606020	+	0.795450i	$0.707235\pi$
$32$	0	0
$33$	−3.43134	−0.597320
$34$	0	0
$35$	−14.7300	−2.48983
$36$	0	0
$37$	7.53454	1.23867	0.619335	−	0.785126i	$-0.287402\pi$
$37$	7.53454	1.23867	0.619335	+	0.785126i	$0.287402\pi$
$38$	0	0
$39$	−2.15782	−0.345527
$40$	0	0
$41$	−7.68958	−1.20091	−0.600455	−	0.799658i	$-0.705014\pi$
$41$	−7.68958	−1.20091	−0.600455	+	0.799658i	$0.705014\pi$
$42$	0	0
$43$	11.0648	1.68736	0.843681	−	0.536845i	$-0.180384\pi$
$43$	11.0648	1.68736	0.843681	+	0.536845i	$0.180384\pi$
$44$	0	0
$45$	−3.71742	−0.554161
$46$	0	0
$47$	−9.22133	−1.34507	−0.672534	−	0.740066i	$-0.734794\pi$
$47$	−9.22133	−1.34507	−0.672534	+	0.740066i	$0.734794\pi$
$48$	0	0
$49$	8.70085	1.24298
$50$	0	0
$51$	1.67107	0.233997
$52$	0	0
$53$	10.1710	1.39709	0.698546	−	0.715566i	$-0.253831\pi$
$53$	10.1710	1.39709	0.698546	+	0.715566i	$0.253831\pi$
$54$	0	0
$55$	−12.7557	−1.71998
$56$	0	0
$57$	1.67421	0.221754
$58$	0	0
$59$	0.718991	0.0936046	0.0468023	−	0.998904i	$-0.485097\pi$
$59$	0.718991	0.0936046	0.0468023	+	0.998904i	$0.485097\pi$
$60$	0	0
$61$	3.40447	0.435898	0.217949	−	0.975960i	$-0.430063\pi$
$61$	3.40447	0.435898	0.217949	+	0.975960i	$0.430063\pi$
$62$	0	0
$63$	3.96243	0.499219
$64$	0	0
$65$	−8.02151	−0.994946
$66$	0	0
$67$	−5.28579	−0.645762	−0.322881	−	0.946440i	$-0.604651\pi$
$67$	−5.28579	−0.645762	−0.322881	+	0.946440i	$0.604651\pi$
$68$	0	0
$69$	7.04857	0.848548
$70$	0	0
$71$	14.4930	1.72001	0.860004	−	0.510287i	$-0.170461\pi$
$71$	14.4930	1.72001	0.860004	+	0.510287i	$0.170461\pi$
$72$	0	0
$73$	−6.67617	−0.781386	−0.390693	−	0.920521i	$-0.627765\pi$
$73$	−6.67617	−0.781386	−0.390693	+	0.920521i	$0.627765\pi$
$74$	0	0
$75$	−8.81922	−1.01836
$76$	0	0
$77$	13.5964	1.54946
$78$	0	0
$79$	2.08862	0.234988	0.117494	−	0.993074i	$-0.462514\pi$
$79$	2.08862	0.234988	0.117494	+	0.993074i	$0.462514\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.2722	−1.45681	−0.728405	−	0.685146i	$-0.759738\pi$
$83$	−13.2722	−1.45681	−0.728405	+	0.685146i	$0.759738\pi$
$84$	0	0
$85$	6.21209	0.673796
$86$	0	0
$87$	8.05220	0.863286
$88$	0	0
$89$	1.78507	0.189217	0.0946085	−	0.995515i	$-0.469840\pi$
$89$	1.78507	0.189217	0.0946085	+	0.995515i	$0.469840\pi$
$90$	0	0
$91$	8.55019	0.896304
$92$	0	0
$93$	6.74835	0.699771
$94$	0	0
$95$	6.22373	0.638541
$96$	0	0
$97$	8.98957	0.912753	0.456376	−	0.889787i	$-0.349147\pi$
$97$	8.98957	0.912753	0.456376	+	0.889787i	$0.349147\pi$
$98$	0	0
$99$	3.43134	0.344863
$100$	0	0
$101$	−4.78506	−0.476132	−0.238066	−	0.971249i	$-0.576513\pi$
$101$	−4.78506	−0.476132	−0.238066	+	0.971249i	$0.576513\pi$
$102$	0	0
$103$	−17.7578	−1.74973	−0.874863	−	0.484370i	$-0.839049\pi$
$103$	−17.7578	−1.74973	−0.874863	+	0.484370i	$0.839049\pi$
$104$	0	0
$105$	14.7300	1.43750
$106$	0	0
$107$	12.6620	1.22408	0.612040	−	0.790827i	$-0.290349\pi$
$107$	12.6620	1.22408	0.612040	+	0.790827i	$0.290349\pi$
$108$	0	0
$109$	−2.99197	−0.286579	−0.143289	−	0.989681i	$-0.545768\pi$
$109$	−2.99197	−0.286579	−0.143289	+	0.989681i	$0.545768\pi$
$110$	0	0
$111$	−7.53454	−0.715147
$112$	0	0
$113$	10.8987	1.02527	0.512633	−	0.858608i	$-0.328670\pi$
$113$	10.8987	1.02527	0.512633	+	0.858608i	$0.328670\pi$
$114$	0	0
$115$	26.2025	2.44340
$116$	0	0
$117$	2.15782	0.199490
$118$	0	0
$119$	−6.62152	−0.606993
$120$	0	0
$121$	0.774104	0.0703731
$122$	0	0
$123$	7.68958	0.693346
$124$	0	0
$125$	−14.1977	−1.26988
$126$	0	0
$127$	−16.9584	−1.50482	−0.752408	−	0.658697i	$-0.771108\pi$
$127$	−16.9584	−1.50482	−0.752408	+	0.658697i	$0.771108\pi$
$128$	0	0
$129$	−11.0648	−0.974199
$130$	0	0
$131$	12.1409	1.06076	0.530378	−	0.847761i	$-0.322050\pi$
$131$	12.1409	1.06076	0.530378	+	0.847761i	$0.322050\pi$
$132$	0	0
$133$	−6.63393	−0.575234
$134$	0	0
$135$	3.71742	0.319945
$136$	0	0
$137$	−10.3467	−0.883979	−0.441989	−	0.897020i	$-0.645727\pi$
$137$	−10.3467	−0.883979	−0.441989	+	0.897020i	$0.645727\pi$
$138$	0	0
$139$	11.1918	0.949274	0.474637	−	0.880182i	$-0.342579\pi$
$139$	11.1918	0.949274	0.474637	+	0.880182i	$0.342579\pi$
$140$	0	0
$141$	9.22133	0.776576
$142$	0	0
$143$	7.40420	0.619170
$144$	0	0
$145$	29.9334	2.48583
$146$	0	0
$147$	−8.70085	−0.717634
$148$	0	0
$149$	16.5425	1.35521	0.677606	−	0.735425i	$-0.263017\pi$
$149$	16.5425	1.35521	0.677606	+	0.735425i	$0.263017\pi$
$150$	0	0
$151$	−21.3366	−1.73635	−0.868174	−	0.496260i	$-0.834706\pi$
$151$	−21.3366	−1.73635	−0.868174	+	0.496260i	$0.834706\pi$
$152$	0	0
$153$	−1.67107	−0.135098
$154$	0	0
$155$	25.0865	2.01499
$156$	0	0
$157$	−10.7310	−0.856424	−0.428212	−	0.903678i	$-0.640856\pi$
$157$	−10.7310	−0.856424	−0.428212	+	0.903678i	$0.640856\pi$
$158$	0	0
$159$	−10.1710	−0.806611
$160$	0	0
$161$	−27.9295	−2.20115
$162$	0	0
$163$	−24.7637	−1.93964	−0.969820	−	0.243822i	$-0.921599\pi$
$163$	−24.7637	−1.93964	−0.969820	+	0.243822i	$0.921599\pi$
$164$	0	0
$165$	12.7557	0.993033
$166$	0	0
$167$	−20.5528	−1.59043	−0.795213	−	0.606330i	$-0.792641\pi$
$167$	−20.5528	−1.59043	−0.795213	+	0.606330i	$0.792641\pi$
$168$	0	0
$169$	−8.34383	−0.641833
$170$	0	0
$171$	−1.67421	−0.128030
$172$	0	0
$173$	−6.02293	−0.457915	−0.228957	−	0.973436i	$-0.573532\pi$
$173$	−6.02293	−0.457915	−0.228957	+	0.973436i	$0.573532\pi$
$174$	0	0
$175$	34.9456	2.64164
$176$	0	0
$177$	−0.718991	−0.0540426
$178$	0	0
$179$	14.5152	1.08491	0.542457	−	0.840084i	$-0.317494\pi$
$179$	14.5152	1.08491	0.542457	+	0.840084i	$0.317494\pi$
$180$	0	0
$181$	−20.8173	−1.54734	−0.773671	−	0.633588i	$-0.781581\pi$
$181$	−20.8173	−1.54734	−0.773671	+	0.633588i	$0.781581\pi$
$182$	0	0
$183$	−3.40447	−0.251666
$184$	0	0
$185$	−28.0091	−2.05927
$186$	0	0
$187$	−5.73403	−0.419314
$188$	0	0
$189$	−3.96243	−0.288224
$190$	0	0
$191$	19.9085	1.44053	0.720264	−	0.693700i	$-0.244020\pi$
$191$	19.9085	1.44053	0.720264	+	0.693700i	$0.244020\pi$
$192$	0	0
$193$	−12.3467	−0.888738	−0.444369	−	0.895844i	$-0.646572\pi$
$193$	−12.3467	−0.888738	−0.444369	+	0.895844i	$0.646572\pi$
$194$	0	0
$195$	8.02151	0.574432
$196$	0	0
$197$	−24.7288	−1.76185	−0.880927	−	0.473252i	$-0.843080\pi$
$197$	−24.7288	−1.76185	−0.880927	+	0.473252i	$0.843080\pi$
$198$	0	0
$199$	0.986116	0.0699039	0.0349520	−	0.999389i	$-0.488872\pi$
$199$	0.986116	0.0699039	0.0349520	+	0.999389i	$0.488872\pi$
$200$	0	0
$201$	5.28579	0.372831
$202$	0	0
$203$	−31.9063	−2.23938
$204$	0	0
$205$	28.5854	1.99649
$206$	0	0
$207$	−7.04857	−0.489909
$208$	0	0
$209$	−5.74477	−0.397374
$210$	0	0
$211$	0.167739	0.0115476	0.00577381	−	0.999983i	$-0.498162\pi$
$211$	0.167739	0.0115476	0.00577381	+	0.999983i	$0.498162\pi$
$212$	0	0
$213$	−14.4930	−0.993047
$214$	0	0
$215$	−41.1324	−2.80521
$216$	0	0
$217$	−26.7399	−1.81522
$218$	0	0
$219$	6.67617	0.451134
$220$	0	0
$221$	−3.60587	−0.242557
$222$	0	0
$223$	24.8611	1.66482	0.832410	−	0.554161i	$-0.186961\pi$
$223$	24.8611	1.66482	0.832410	+	0.554161i	$0.186961\pi$
$224$	0	0
$225$	8.81922	0.587948
$226$	0	0
$227$	−6.95633	−0.461708	−0.230854	−	0.972988i	$-0.574152\pi$
$227$	−6.95633	−0.461708	−0.230854	+	0.972988i	$0.574152\pi$
$228$	0	0
$229$	13.4132	0.886371	0.443185	−	0.896430i	$-0.353848\pi$
$229$	13.4132	0.886371	0.443185	+	0.896430i	$0.353848\pi$
$230$	0	0
$231$	−13.5964	−0.894581
$232$	0	0
$233$	8.36594	0.548071	0.274036	−	0.961720i	$-0.411641\pi$
$233$	8.36594	0.548071	0.274036	+	0.961720i	$0.411641\pi$
$234$	0	0
$235$	34.2796	2.23615
$236$	0	0
$237$	−2.08862	−0.135671
$238$	0	0
$239$	−5.26498	−0.340563	−0.170282	−	0.985395i	$-0.554468\pi$
$239$	−5.26498	−0.340563	−0.170282	+	0.985395i	$0.554468\pi$
$240$	0	0
$241$	−10.4485	−0.673044	−0.336522	−	0.941676i	$-0.609251\pi$
$241$	−10.4485	−0.673044	−0.336522	+	0.941676i	$0.609251\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−32.3447	−2.06643
$246$	0	0
$247$	−3.61263	−0.229866
$248$	0	0
$249$	13.2722	0.841090
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−24.1860	−1.52056
$254$	0	0
$255$	−6.21209	−0.389016
$256$	0	0
$257$	18.3818	1.14662	0.573312	−	0.819337i	$-0.305658\pi$
$257$	18.3818	1.14662	0.573312	+	0.819337i	$0.305658\pi$
$258$	0	0
$259$	29.8551	1.85510
$260$	0	0
$261$	−8.05220	−0.498418
$262$	0	0
$263$	−12.5294	−0.772597	−0.386298	−	0.922374i	$-0.626246\pi$
$263$	−12.5294	−0.772597	−0.386298	+	0.922374i	$0.626246\pi$
$264$	0	0
$265$	−37.8098	−2.32264
$266$	0	0
$267$	−1.78507	−0.109245
$268$	0	0
$269$	−23.2894	−1.41998	−0.709990	−	0.704212i	$-0.751300\pi$
$269$	−23.2894	−1.41998	−0.709990	+	0.704212i	$0.751300\pi$
$270$	0	0
$271$	12.9691	0.787817	0.393909	−	0.919150i	$-0.371123\pi$
$271$	12.9691	0.787817	0.393909	+	0.919150i	$0.371123\pi$
$272$	0	0
$273$	−8.55019	−0.517481
$274$	0	0
$275$	30.2618	1.82485
$276$	0	0
$277$	22.4080	1.34637	0.673184	−	0.739475i	$-0.264926\pi$
$277$	22.4080	1.34637	0.673184	+	0.739475i	$0.264926\pi$
$278$	0	0
$279$	−6.74835	−0.404013
$280$	0	0
$281$	−29.7976	−1.77757	−0.888787	−	0.458321i	$-0.848451\pi$
$281$	−29.7976	−1.77757	−0.888787	+	0.458321i	$0.848451\pi$
$282$	0	0
$283$	23.0117	1.36790	0.683950	−	0.729529i	$-0.260260\pi$
$283$	23.0117	1.36790	0.683950	+	0.729529i	$0.260260\pi$
$284$	0	0
$285$	−6.22373	−0.368662
$286$	0	0
$287$	−30.4694	−1.79855
$288$	0	0
$289$	−14.2075	−0.835736
$290$	0	0
$291$	−8.98957	−0.526978
$292$	0	0
$293$	−33.1080	−1.93419	−0.967096	−	0.254413i	$-0.918118\pi$
$293$	−33.1080	−1.93419	−0.967096	+	0.254413i	$0.918118\pi$
$294$	0	0
$295$	−2.67279	−0.155616
$296$	0	0
$297$	−3.43134	−0.199107
$298$	0	0
$299$	−15.2095	−0.879589
$300$	0	0
$301$	43.8434	2.52709
$302$	0	0
$303$	4.78506	0.274895
$304$	0	0
$305$	−12.6558	−0.724672
$306$	0	0
$307$	−17.2349	−0.983648	−0.491824	−	0.870695i	$-0.663670\pi$
$307$	−17.2349	−0.983648	−0.491824	+	0.870695i	$0.663670\pi$
$308$	0	0
$309$	17.7578	1.01020
$310$	0	0
$311$	7.37380	0.418130	0.209065	−	0.977902i	$-0.432958\pi$
$311$	7.37380	0.418130	0.209065	+	0.977902i	$0.432958\pi$
$312$	0	0
$313$	14.3940	0.813599	0.406800	−	0.913517i	$-0.366645\pi$
$313$	14.3940	0.813599	0.406800	+	0.913517i	$0.366645\pi$
$314$	0	0
$315$	−14.7300	−0.829943
$316$	0	0
$317$	−2.11785	−0.118950	−0.0594751	−	0.998230i	$-0.518943\pi$
$317$	−2.11785	−0.118950	−0.0594751	+	0.998230i	$0.518943\pi$
$318$	0	0
$319$	−27.6298	−1.54697
$320$	0	0
$321$	−12.6620	−0.706722
$322$	0	0
$323$	2.79772	0.155670
$324$	0	0
$325$	19.0303	1.05561
$326$	0	0
$327$	2.99197	0.165456
$328$	0	0
$329$	−36.5389	−2.01445
$330$	0	0
$331$	−25.5896	−1.40653	−0.703265	−	0.710928i	$-0.748275\pi$
$331$	−25.5896	−1.40653	−0.703265	+	0.710928i	$0.748275\pi$
$332$	0	0
$333$	7.53454	0.412890
$334$	0	0
$335$	19.6495	1.07357
$336$	0	0
$337$	−12.5031	−0.681089	−0.340544	−	0.940228i	$-0.610611\pi$
$337$	−12.5031	−0.681089	−0.340544	+	0.940228i	$0.610611\pi$
$338$	0	0
$339$	−10.8987	−0.591938
$340$	0	0
$341$	−23.1559	−1.25396
$342$	0	0
$343$	6.73949	0.363898
$344$	0	0
$345$	−26.2025	−1.41070
$346$	0	0
$347$	26.4314	1.41891	0.709457	−	0.704749i	$-0.248940\pi$
$347$	26.4314	1.41891	0.709457	+	0.704749i	$0.248940\pi$
$348$	0	0
$349$	−15.9345	−0.852952	−0.426476	−	0.904499i	$-0.640245\pi$
$349$	−15.9345	−0.852952	−0.426476	+	0.904499i	$0.640245\pi$
$350$	0	0
$351$	−2.15782	−0.115176
$352$	0	0
$353$	−9.54727	−0.508150	−0.254075	−	0.967185i	$-0.581771\pi$
$353$	−9.54727	−0.508150	−0.254075	+	0.967185i	$0.581771\pi$
$354$	0	0
$355$	−53.8768	−2.85948
$356$	0	0
$357$	6.62152	0.350448
$358$	0	0
$359$	−4.57134	−0.241266	−0.120633	−	0.992697i	$-0.538492\pi$
$359$	−4.57134	−0.241266	−0.120633	+	0.992697i	$0.538492\pi$
$360$	0	0
$361$	−16.1970	−0.852475
$362$	0	0
$363$	−0.774104	−0.0406299
$364$	0	0
$365$	24.8181	1.29904
$366$	0	0
$367$	28.8150	1.50413	0.752066	−	0.659088i	$-0.229058\pi$
$367$	28.8150	1.50413	0.752066	+	0.659088i	$0.229058\pi$
$368$	0	0
$369$	−7.68958	−0.400303
$370$	0	0
$371$	40.3018	2.09236
$372$	0	0
$373$	8.82992	0.457196	0.228598	−	0.973521i	$-0.426586\pi$
$373$	8.82992	0.457196	0.228598	+	0.973521i	$0.426586\pi$
$374$	0	0
$375$	14.1977	0.733164
$376$	0	0
$377$	−17.3752	−0.894866
$378$	0	0
$379$	−7.44512	−0.382430	−0.191215	−	0.981548i	$-0.561243\pi$
$379$	−7.44512	−0.382430	−0.191215	+	0.981548i	$0.561243\pi$
$380$	0	0
$381$	16.9584	0.868806
$382$	0	0
$383$	−19.0959	−0.975753	−0.487876	−	0.872913i	$-0.662228\pi$
$383$	−19.0959	−0.975753	−0.487876	+	0.872913i	$0.662228\pi$
$384$	0	0
$385$	−50.5437	−2.57595
$386$	0	0
$387$	11.0648	0.562454
$388$	0	0
$389$	11.6690	0.591642	0.295821	−	0.955243i	$-0.404407\pi$
$389$	11.6690	0.591642	0.295821	+	0.955243i	$0.404407\pi$
$390$	0	0
$391$	11.7787	0.595674
$392$	0	0
$393$	−12.1409	−0.612428
$394$	0	0
$395$	−7.76429	−0.390664
$396$	0	0
$397$	−24.8045	−1.24490	−0.622452	−	0.782658i	$-0.713863\pi$
$397$	−24.8045	−1.24490	−0.622452	+	0.782658i	$0.713863\pi$
$398$	0	0
$399$	6.63393	0.332112
$400$	0	0
$401$	−19.7329	−0.985416	−0.492708	−	0.870195i	$-0.663993\pi$
$401$	−19.7329	−0.985416	−0.492708	+	0.870195i	$0.663993\pi$
$402$	0	0
$403$	−14.5617	−0.725370
$404$	0	0
$405$	−3.71742	−0.184720
$406$	0	0
$407$	25.8536	1.28151
$408$	0	0
$409$	13.6415	0.674529	0.337265	−	0.941410i	$-0.390498\pi$
$409$	13.6415	0.674529	0.337265	+	0.941410i	$0.390498\pi$
$410$	0	0
$411$	10.3467	0.510365
$412$	0	0
$413$	2.84895	0.140188
$414$	0	0
$415$	49.3383	2.42192
$416$	0	0
$417$	−11.1918	−0.548063
$418$	0	0
$419$	−35.3313	−1.72605	−0.863024	−	0.505164i	$-0.831432\pi$
$419$	−35.3313	−1.72605	−0.863024	+	0.505164i	$0.831432\pi$
$420$	0	0
$421$	−17.8128	−0.868140	−0.434070	−	0.900879i	$-0.642923\pi$
$421$	−17.8128	−0.868140	−0.434070	+	0.900879i	$0.642923\pi$
$422$	0	0
$423$	−9.22133	−0.448356
$424$	0	0
$425$	−14.7376	−0.714878
$426$	0	0
$427$	13.4900	0.652825
$428$	0	0
$429$	−7.40420	−0.357478
$430$	0	0
$431$	−24.7231	−1.19087	−0.595435	−	0.803403i	$-0.703020\pi$
$431$	−24.7231	−1.19087	−0.595435	+	0.803403i	$0.703020\pi$
$432$	0	0
$433$	−26.7541	−1.28572	−0.642859	−	0.765985i	$-0.722252\pi$
$433$	−26.7541	−1.28572	−0.642859	+	0.765985i	$0.722252\pi$
$434$	0	0
$435$	−29.9334	−1.43520
$436$	0	0
$437$	11.8008	0.564507
$438$	0	0
$439$	37.2247	1.77664	0.888320	−	0.459225i	$-0.151873\pi$
$439$	37.2247	1.77664	0.888320	+	0.459225i	$0.151873\pi$
$440$	0	0
$441$	8.70085	0.414326
$442$	0	0
$443$	35.8595	1.70374	0.851869	−	0.523755i	$-0.175469\pi$
$443$	35.8595	1.70374	0.851869	+	0.523755i	$0.175469\pi$
$444$	0	0
$445$	−6.63586	−0.314570
$446$	0	0
$447$	−16.5425	−0.782432
$448$	0	0
$449$	−9.56448	−0.451376	−0.225688	−	0.974200i	$-0.572463\pi$
$449$	−9.56448	−0.451376	−0.225688	+	0.974200i	$0.572463\pi$
$450$	0	0
$451$	−26.3856	−1.24245
$452$	0	0
$453$	21.3366	1.00248
$454$	0	0
$455$	−31.7847	−1.49009
$456$	0	0
$457$	32.9919	1.54330	0.771648	−	0.636050i	$-0.219433\pi$
$457$	32.9919	1.54330	0.771648	+	0.636050i	$0.219433\pi$
$458$	0	0
$459$	1.67107	0.0779991
$460$	0	0
$461$	24.5912	1.14533	0.572663	−	0.819791i	$-0.305910\pi$
$461$	24.5912	1.14533	0.572663	+	0.819791i	$0.305910\pi$
$462$	0	0
$463$	−25.0378	−1.16361	−0.581803	−	0.813330i	$-0.697653\pi$
$463$	−25.0378	−1.16361	−0.581803	+	0.813330i	$0.697653\pi$
$464$	0	0
$465$	−25.0865	−1.16336
$466$	0	0
$467$	−29.8397	−1.38081	−0.690407	−	0.723421i	$-0.742569\pi$
$467$	−29.8397	−1.38081	−0.690407	+	0.723421i	$0.742569\pi$
$468$	0	0
$469$	−20.9446	−0.967130
$470$	0	0
$471$	10.7310	0.494456
$472$	0	0
$473$	37.9670	1.74573
$474$	0	0
$475$	−14.7652	−0.677474
$476$	0	0
$477$	10.1710	0.465697
$478$	0	0
$479$	15.4562	0.706213	0.353107	−	0.935583i	$-0.385125\pi$
$479$	15.4562	0.706213	0.353107	+	0.935583i	$0.385125\pi$
$480$	0	0
$481$	16.2581	0.741308
$482$	0	0
$483$	27.9295	1.27083
$484$	0	0
$485$	−33.4180	−1.51744
$486$	0	0
$487$	13.7002	0.620814	0.310407	−	0.950604i	$-0.399535\pi$
$487$	13.7002	0.620814	0.310407	+	0.950604i	$0.399535\pi$
$488$	0	0
$489$	24.7637	1.11985
$490$	0	0
$491$	−0.861072	−0.0388596	−0.0194298	−	0.999811i	$-0.506185\pi$
$491$	−0.861072	−0.0388596	−0.0194298	+	0.999811i	$0.506185\pi$
$492$	0	0
$493$	13.4558	0.606020
$494$	0	0
$495$	−12.7557	−0.573328
$496$	0	0
$497$	57.4277	2.57598
$498$	0	0
$499$	16.9431	0.758477	0.379238	−	0.925299i	$-0.376186\pi$
$499$	16.9431	0.758477	0.379238	+	0.925299i	$0.376186\pi$
$500$	0	0
$501$	20.5528	0.918233
$502$	0	0
$503$	−2.61455	−0.116577	−0.0582885	−	0.998300i	$-0.518564\pi$
$503$	−2.61455	−0.116577	−0.0582885	+	0.998300i	$0.518564\pi$
$504$	0	0
$505$	17.7881	0.791560
$506$	0	0
$507$	8.34383	0.370563
$508$	0	0
$509$	21.9549	0.973132	0.486566	−	0.873644i	$-0.338249\pi$
$509$	21.9549	0.973132	0.486566	+	0.873644i	$0.338249\pi$
$510$	0	0
$511$	−26.4538	−1.17025
$512$	0	0
$513$	1.67421	0.0739180
$514$	0	0
$515$	66.0132	2.90889
$516$	0	0
$517$	−31.6415	−1.39159
$518$	0	0
$519$	6.02293	0.264377
$520$	0	0
$521$	1.40938	0.0617459	0.0308729	−	0.999523i	$-0.490171\pi$
$521$	1.40938	0.0617459	0.0308729	+	0.999523i	$0.490171\pi$
$522$	0	0
$523$	−2.11457	−0.0924636	−0.0462318	−	0.998931i	$-0.514721\pi$
$523$	−2.11457	−0.0924636	−0.0462318	+	0.998931i	$0.514721\pi$
$524$	0	0
$525$	−34.9456	−1.52515
$526$	0	0
$527$	11.2770	0.491234
$528$	0	0
$529$	26.6823	1.16010
$530$	0	0
$531$	0.718991	0.0312015
$532$	0	0
$533$	−16.5927	−0.718709
$534$	0	0
$535$	−47.0699	−2.03501
$536$	0	0
$537$	−14.5152	−0.626375
$538$	0	0
$539$	29.8556	1.28597
$540$	0	0
$541$	−17.4114	−0.748573	−0.374286	−	0.927313i	$-0.622112\pi$
$541$	−17.4114	−0.748573	−0.374286	+	0.927313i	$0.622112\pi$
$542$	0	0
$543$	20.8173	0.893358
$544$	0	0
$545$	11.1224	0.476431
$546$	0	0
$547$	−6.73444	−0.287944	−0.143972	−	0.989582i	$-0.545987\pi$
$547$	−6.73444	−0.287944	−0.143972	+	0.989582i	$0.545987\pi$
$548$	0	0
$549$	3.40447	0.145299
$550$	0	0
$551$	13.4810	0.574312
$552$	0	0
$553$	8.27602	0.351932
$554$	0	0
$555$	28.0091	1.18892
$556$	0	0
$557$	−15.4199	−0.653364	−0.326682	−	0.945134i	$-0.605931\pi$
$557$	−15.4199	−0.653364	−0.326682	+	0.945134i	$0.605931\pi$
$558$	0	0
$559$	23.8757	1.00984
$560$	0	0
$561$	5.73403	0.242091
$562$	0	0
$563$	14.6745	0.618458	0.309229	−	0.950988i	$-0.399929\pi$
$563$	14.6745	0.618458	0.309229	+	0.950988i	$0.399929\pi$
$564$	0	0
$565$	−40.5152	−1.70449
$566$	0	0
$567$	3.96243	0.166406
$568$	0	0
$569$	−30.2948	−1.27002	−0.635012	−	0.772503i	$-0.719005\pi$
$569$	−30.2948	−1.27002	−0.635012	+	0.772503i	$0.719005\pi$
$570$	0	0
$571$	16.2404	0.679640	0.339820	−	0.940491i	$-0.389634\pi$
$571$	16.2404	0.679640	0.339820	+	0.940491i	$0.389634\pi$
$572$	0	0
$573$	−19.9085	−0.831690
$574$	0	0
$575$	−62.1629	−2.59237
$576$	0	0
$577$	3.52877	0.146905	0.0734523	−	0.997299i	$-0.476598\pi$
$577$	3.52877	0.146905	0.0734523	+	0.997299i	$0.476598\pi$
$578$	0	0
$579$	12.3467	0.513113
$580$	0	0
$581$	−52.5901	−2.18180
$582$	0	0
$583$	34.9001	1.44541
$584$	0	0
$585$	−8.02151	−0.331649
$586$	0	0
$587$	30.4365	1.25625	0.628125	−	0.778112i	$-0.283823\pi$
$587$	30.4365	1.25625	0.628125	+	0.778112i	$0.283823\pi$
$588$	0	0
$589$	11.2981	0.465531
$590$	0	0
$591$	24.7288	1.01721
$592$	0	0
$593$	−11.5821	−0.475619	−0.237809	−	0.971312i	$-0.576429\pi$
$593$	−11.5821	−0.475619	−0.237809	+	0.971312i	$0.576429\pi$
$594$	0	0
$595$	24.6150	1.00912
$596$	0	0
$597$	−0.986116	−0.0403591
$598$	0	0
$599$	0.798070	0.0326082	0.0163041	−	0.999867i	$-0.494810\pi$
$599$	0.798070	0.0326082	0.0163041	+	0.999867i	$0.494810\pi$
$600$	0	0
$601$	17.6000	0.717919	0.358959	−	0.933353i	$-0.383132\pi$
$601$	17.6000	0.717919	0.358959	+	0.933353i	$0.383132\pi$
$602$	0	0
$603$	−5.28579	−0.215254
$604$	0	0
$605$	−2.87767	−0.116994
$606$	0	0
$607$	−3.24571	−0.131739	−0.0658697	−	0.997828i	$-0.520982\pi$
$607$	−3.24571	−0.131739	−0.0658697	+	0.997828i	$0.520982\pi$
$608$	0	0
$609$	31.9063	1.29291
$610$	0	0
$611$	−19.8979	−0.804984
$612$	0	0
$613$	−30.8460	−1.24586	−0.622930	−	0.782278i	$-0.714058\pi$
$613$	−30.8460	−1.24586	−0.622930	+	0.782278i	$0.714058\pi$
$614$	0	0
$615$	−28.5854	−1.15267
$616$	0	0
$617$	−5.71142	−0.229933	−0.114966	−	0.993369i	$-0.536676\pi$
$617$	−5.71142	−0.229933	−0.114966	+	0.993369i	$0.536676\pi$
$618$	0	0
$619$	−46.3300	−1.86216	−0.931081	−	0.364814i	$-0.881133\pi$
$619$	−46.3300	−1.86216	−0.931081	+	0.364814i	$0.881133\pi$
$620$	0	0
$621$	7.04857	0.282849
$622$	0	0
$623$	7.07322	0.283382
$624$	0	0
$625$	8.68260	0.347304
$626$	0	0
$627$	5.74477	0.229424
$628$	0	0
$629$	−12.5908	−0.502027
$630$	0	0
$631$	3.46587	0.137974	0.0689871	−	0.997618i	$-0.478023\pi$
$631$	3.46587	0.137974	0.0689871	+	0.997618i	$0.478023\pi$
$632$	0	0
$633$	−0.167739	−0.00666702
$634$	0	0
$635$	63.0416	2.50173
$636$	0	0
$637$	18.7748	0.743886
$638$	0	0
$639$	14.4930	0.573336
$640$	0	0
$641$	−18.7329	−0.739905	−0.369952	−	0.929051i	$-0.620626\pi$
$641$	−18.7329	−0.739905	−0.369952	+	0.929051i	$0.620626\pi$
$642$	0	0
$643$	24.8927	0.981671	0.490835	−	0.871252i	$-0.336692\pi$
$643$	24.8927	0.981671	0.490835	+	0.871252i	$0.336692\pi$
$644$	0	0
$645$	41.1324	1.61959
$646$	0	0
$647$	−44.0960	−1.73359	−0.866796	−	0.498664i	$-0.833824\pi$
$647$	−44.0960	−1.73359	−0.866796	+	0.498664i	$0.833824\pi$
$648$	0	0
$649$	2.46710	0.0968422
$650$	0	0
$651$	26.7399	1.04802
$652$	0	0
$653$	5.68913	0.222633	0.111316	−	0.993785i	$-0.464493\pi$
$653$	5.68913	0.222633	0.111316	+	0.993785i	$0.464493\pi$
$654$	0	0
$655$	−45.1329	−1.76349
$656$	0	0
$657$	−6.67617	−0.260462
$658$	0	0
$659$	7.78173	0.303133	0.151567	−	0.988447i	$-0.451568\pi$
$659$	7.78173	0.303133	0.151567	+	0.988447i	$0.451568\pi$
$660$	0	0
$661$	−13.1389	−0.511045	−0.255523	−	0.966803i	$-0.582248\pi$
$661$	−13.1389	−0.511045	−0.255523	+	0.966803i	$0.582248\pi$
$662$	0	0
$663$	3.60587	0.140040
$664$	0	0
$665$	24.6611	0.956316
$666$	0	0
$667$	56.7565	2.19762
$668$	0	0
$669$	−24.8611	−0.961184
$670$	0	0
$671$	11.6819	0.450975
$672$	0	0
$673$	−16.2192	−0.625203	−0.312602	−	0.949884i	$-0.601200\pi$
$673$	−16.2192	−0.625203	−0.312602	+	0.949884i	$0.601200\pi$
$674$	0	0
$675$	−8.81922	−0.339452
$676$	0	0
$677$	−18.6277	−0.715920	−0.357960	−	0.933737i	$-0.616528\pi$
$677$	−18.6277	−0.715920	−0.357960	+	0.933737i	$0.616528\pi$
$678$	0	0
$679$	35.6206	1.36699
$680$	0	0
$681$	6.95633	0.266567
$682$	0	0
$683$	34.2789	1.31165	0.655823	−	0.754915i	$-0.272322\pi$
$683$	34.2789	1.31165	0.655823	+	0.754915i	$0.272322\pi$
$684$	0	0
$685$	38.4631	1.46960
$686$	0	0
$687$	−13.4132	−0.511746
$688$	0	0
$689$	21.9471	0.836118
$690$	0	0
$691$	−18.7648	−0.713845	−0.356923	−	0.934134i	$-0.616174\pi$
$691$	−18.7648	−0.713845	−0.356923	+	0.934134i	$0.616174\pi$
$692$	0	0
$693$	13.5964	0.516486
$694$	0	0
$695$	−41.6045	−1.57815
$696$	0	0
$697$	12.8499	0.486723
$698$	0	0
$699$	−8.36594	−0.316429
$700$	0	0
$701$	−22.0295	−0.832042	−0.416021	−	0.909355i	$-0.636576\pi$
$701$	−22.0295	−0.832042	−0.416021	+	0.909355i	$0.636576\pi$
$702$	0	0
$703$	−12.6144	−0.475760
$704$	0	0
$705$	−34.2796	−1.29104
$706$	0	0
$707$	−18.9605	−0.713082
$708$	0	0
$709$	46.3820	1.74191	0.870956	−	0.491360i	$-0.163500\pi$
$709$	46.3820	1.74191	0.870956	+	0.491360i	$0.163500\pi$
$710$	0	0
$711$	2.08862	0.0783295
$712$	0	0
$713$	47.5662	1.78137
$714$	0	0
$715$	−27.5245	−1.02936
$716$	0	0
$717$	5.26498	0.196624
$718$	0	0
$719$	−13.1508	−0.490442	−0.245221	−	0.969467i	$-0.578861\pi$
$719$	−13.1508	−0.490442	−0.245221	+	0.969467i	$0.578861\pi$
$720$	0	0
$721$	−70.3640	−2.62049
$722$	0	0
$723$	10.4485	0.388582
$724$	0	0
$725$	−71.0141	−2.63740
$726$	0	0
$727$	−20.9936	−0.778608	−0.389304	−	0.921109i	$-0.627284\pi$
$727$	−20.9936	−0.778608	−0.389304	+	0.921109i	$0.627284\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−18.4901	−0.683880
$732$	0	0
$733$	−19.2201	−0.709909	−0.354954	−	0.934884i	$-0.615504\pi$
$733$	−19.2201	−0.709909	−0.354954	+	0.934884i	$0.615504\pi$
$734$	0	0
$735$	32.3447	1.19305
$736$	0	0
$737$	−18.1373	−0.668097
$738$	0	0
$739$	22.8652	0.841109	0.420554	−	0.907267i	$-0.361836\pi$
$739$	22.8652	0.841109	0.420554	+	0.907267i	$0.361836\pi$
$740$	0	0
$741$	3.61263	0.132713
$742$	0	0
$743$	−15.1488	−0.555755	−0.277877	−	0.960617i	$-0.589631\pi$
$743$	−15.1488	−0.555755	−0.277877	+	0.960617i	$0.589631\pi$
$744$	0	0
$745$	−61.4954	−2.25302
$746$	0	0
$747$	−13.2722	−0.485604
$748$	0	0
$749$	50.1722	1.83325
$750$	0	0
$751$	−17.0138	−0.620842	−0.310421	−	0.950599i	$-0.600470\pi$
$751$	−17.0138	−0.620842	−0.310421	+	0.950599i	$0.600470\pi$
$752$	0	0
$753$	1.00000	0.0364420
$754$	0	0
$755$	79.3171	2.88665
$756$	0	0
$757$	−32.2154	−1.17089	−0.585444	−	0.810713i	$-0.699080\pi$
$757$	−32.2154	−1.17089	−0.585444	+	0.810713i	$0.699080\pi$
$758$	0	0
$759$	24.1860	0.877898
$760$	0	0
$761$	−22.7256	−0.823801	−0.411900	−	0.911229i	$-0.635135\pi$
$761$	−22.7256	−0.823801	−0.411900	+	0.911229i	$0.635135\pi$
$762$	0	0
$763$	−11.8555	−0.429196
$764$	0	0
$765$	6.21209	0.224599
$766$	0	0
$767$	1.55145	0.0560196
$768$	0	0
$769$	43.0406	1.55208	0.776042	−	0.630681i	$-0.217224\pi$
$769$	43.0406	1.55208	0.776042	+	0.630681i	$0.217224\pi$
$770$	0	0
$771$	−18.3818	−0.662004
$772$	0	0
$773$	32.8730	1.18236	0.591180	−	0.806540i	$-0.298662\pi$
$773$	32.8730	1.18236	0.591180	+	0.806540i	$0.298662\pi$
$774$	0	0
$775$	−59.5152	−2.13785
$776$	0	0
$777$	−29.8551	−1.07105
$778$	0	0
$779$	12.8739	0.461257
$780$	0	0
$781$	49.7306	1.77950
$782$	0	0
$783$	8.05220	0.287762
$784$	0	0
$785$	39.8915	1.42379
$786$	0	0
$787$	−19.6439	−0.700228	−0.350114	−	0.936707i	$-0.613857\pi$
$787$	−19.6439	−0.700228	−0.350114	+	0.936707i	$0.613857\pi$
$788$	0	0
$789$	12.5294	0.446059
$790$	0	0
$791$	43.1854	1.53550
$792$	0	0
$793$	7.34622	0.260872
$794$	0	0
$795$	37.8098	1.34098
$796$	0	0
$797$	19.4128	0.687635	0.343818	−	0.939036i	$-0.388280\pi$
$797$	19.4128	0.687635	0.343818	+	0.939036i	$0.388280\pi$
$798$	0	0
$799$	15.4095	0.545150
$800$	0	0
$801$	1.78507	0.0630724
$802$	0	0
$803$	−22.9082	−0.808413
$804$	0	0
$805$	103.826	3.65937
$806$	0	0
$807$	23.2894	0.819825
$808$	0	0
$809$	0.153209	0.00538655	0.00269327	−	0.999996i	$-0.499143\pi$
$809$	0.153209	0.00538655	0.00269327	+	0.999996i	$0.499143\pi$
$810$	0	0
$811$	−20.7451	−0.728460	−0.364230	−	0.931309i	$-0.618668\pi$
$811$	−20.7451	−0.728460	−0.364230	+	0.931309i	$0.618668\pi$
$812$	0	0
$813$	−12.9691	−0.454847
$814$	0	0
$815$	92.0570	3.22462
$816$	0	0
$817$	−18.5247	−0.648098
$818$	0	0
$819$	8.55019	0.298768
$820$	0	0
$821$	−40.1472	−1.40115	−0.700573	−	0.713581i	$-0.747072\pi$
$821$	−40.1472	−1.40115	−0.700573	+	0.713581i	$0.747072\pi$
$822$	0	0
$823$	−39.2648	−1.36869	−0.684343	−	0.729160i	$-0.739911\pi$
$823$	−39.2648	−1.36869	−0.684343	+	0.729160i	$0.739911\pi$
$824$	0	0
$825$	−30.2618	−1.05358
$826$	0	0
$827$	19.3880	0.674188	0.337094	−	0.941471i	$-0.390556\pi$
$827$	19.3880	0.674188	0.337094	+	0.941471i	$0.390556\pi$
$828$	0	0
$829$	17.0357	0.591676	0.295838	−	0.955238i	$-0.404401\pi$
$829$	17.0357	0.591676	0.295838	+	0.955238i	$0.404401\pi$
$830$	0	0
$831$	−22.4080	−0.777325
$832$	0	0
$833$	−14.5398	−0.503773
$834$	0	0
$835$	76.4036	2.64405
$836$	0	0
$837$	6.74835	0.233257
$838$	0	0
$839$	−19.2125	−0.663289	−0.331645	−	0.943404i	$-0.607603\pi$
$839$	−19.2125	−0.663289	−0.331645	+	0.943404i	$0.607603\pi$
$840$	0	0
$841$	35.8379	1.23579
$842$	0	0
$843$	29.7976	1.02628
$844$	0	0
$845$	31.0175	1.06704
$846$	0	0
$847$	3.06733	0.105395
$848$	0	0
$849$	−23.0117	−0.789758
$850$	0	0
$851$	−53.1077	−1.82051
$852$	0	0
$853$	5.91469	0.202515	0.101258	−	0.994860i	$-0.467713\pi$
$853$	5.91469	0.202515	0.101258	+	0.994860i	$0.467713\pi$
$854$	0	0
$855$	6.22373	0.212847
$856$	0	0
$857$	2.80081	0.0956739	0.0478370	−	0.998855i	$-0.484767\pi$
$857$	2.80081	0.0956739	0.0478370	+	0.998855i	$0.484767\pi$
$858$	0	0
$859$	2.68402	0.0915775	0.0457888	−	0.998951i	$-0.485420\pi$
$859$	2.68402	0.0915775	0.0457888	+	0.998951i	$0.485420\pi$
$860$	0	0
$861$	30.4694	1.03839
$862$	0	0
$863$	34.3343	1.16875	0.584376	−	0.811483i	$-0.301339\pi$
$863$	34.3343	1.16875	0.584376	+	0.811483i	$0.301339\pi$
$864$	0	0
$865$	22.3898	0.761275
$866$	0	0
$867$	14.2075	0.482512
$868$	0	0
$869$	7.16678	0.243116
$870$	0	0
$871$	−11.4058	−0.386469
$872$	0	0
$873$	8.98957	0.304251
$874$	0	0
$875$	−56.2573	−1.90184
$876$	0	0
$877$	46.1706	1.55907	0.779534	−	0.626359i	$-0.215456\pi$
$877$	46.1706	1.55907	0.779534	+	0.626359i	$0.215456\pi$
$878$	0	0
$879$	33.1080	1.11671
$880$	0	0
$881$	−4.23774	−0.142773	−0.0713865	−	0.997449i	$-0.522742\pi$
$881$	−4.23774	−0.142773	−0.0713865	+	0.997449i	$0.522742\pi$
$882$	0	0
$883$	−10.0333	−0.337648	−0.168824	−	0.985646i	$-0.553997\pi$
$883$	−10.0333	−0.337648	−0.168824	+	0.985646i	$0.553997\pi$
$884$	0	0
$885$	2.67279	0.0898449
$886$	0	0
$887$	3.35766	0.112739	0.0563697	−	0.998410i	$-0.482047\pi$
$887$	3.35766	0.112739	0.0563697	+	0.998410i	$0.482047\pi$
$888$	0	0
$889$	−67.1965	−2.25370
$890$	0	0
$891$	3.43134	0.114954
$892$	0	0
$893$	15.4384	0.516627
$894$	0	0
$895$	−53.9589	−1.80365
$896$	0	0
$897$	15.2095	0.507831
$898$	0	0
$899$	54.3390	1.81231
$900$	0	0
$901$	−16.9965	−0.566234
$902$	0	0
$903$	−43.8434	−1.45902
$904$	0	0
$905$	77.3869	2.57243
$906$	0	0
$907$	−8.73676	−0.290099	−0.145050	−	0.989424i	$-0.546334\pi$
$907$	−8.73676	−0.290099	−0.145050	+	0.989424i	$0.546334\pi$
$908$	0	0
$909$	−4.78506	−0.158711
$910$	0	0
$911$	−42.3965	−1.40466	−0.702330	−	0.711852i	$-0.747857\pi$
$911$	−42.3965	−1.40466	−0.702330	+	0.711852i	$0.747857\pi$
$912$	0	0
$913$	−45.5414	−1.50720
$914$	0	0
$915$	12.6558	0.418389
$916$	0	0
$917$	48.1075	1.58865
$918$	0	0
$919$	−4.89310	−0.161408	−0.0807042	−	0.996738i	$-0.525717\pi$
$919$	−4.89310	−0.161408	−0.0807042	+	0.996738i	$0.525717\pi$
$920$	0	0
$921$	17.2349	0.567909
$922$	0	0
$923$	31.2733	1.02937
$924$	0	0
$925$	66.4488	2.18482
$926$	0	0
$927$	−17.7578	−0.583242
$928$	0	0
$929$	−17.0987	−0.560989	−0.280494	−	0.959856i	$-0.590498\pi$
$929$	−17.0987	−0.560989	−0.280494	+	0.959856i	$0.590498\pi$
$930$	0	0
$931$	−14.5670	−0.477415
$932$	0	0
$933$	−7.37380	−0.241407
$934$	0	0
$935$	21.3158	0.697101
$936$	0	0
$937$	6.61397	0.216069	0.108034	−	0.994147i	$-0.465544\pi$
$937$	6.61397	0.216069	0.108034	+	0.994147i	$0.465544\pi$
$938$	0	0
$939$	−14.3940	−0.469732
$940$	0	0
$941$	−34.2101	−1.11522	−0.557608	−	0.830104i	$-0.688281\pi$
$941$	−34.2101	−1.11522	−0.557608	+	0.830104i	$0.688281\pi$
$942$	0	0
$943$	54.2005	1.76501
$944$	0	0
$945$	14.7300	0.479168
$946$	0	0
$947$	16.3518	0.531363	0.265681	−	0.964061i	$-0.414403\pi$
$947$	16.3518	0.531363	0.265681	+	0.964061i	$0.414403\pi$
$948$	0	0
$949$	−14.4059	−0.467637
$950$	0	0
$951$	2.11785	0.0686760
$952$	0	0
$953$	−4.80069	−0.155510	−0.0777548	−	0.996973i	$-0.524775\pi$
$953$	−4.80069	−0.155510	−0.0777548	+	0.996973i	$0.524775\pi$
$954$	0	0
$955$	−74.0083	−2.39485
$956$	0	0
$957$	27.6298	0.893146
$958$	0	0
$959$	−40.9981	−1.32390
$960$	0	0
$961$	14.5402	0.469040
$962$	0	0
$963$	12.6620	0.408026
$964$	0	0
$965$	45.8980	1.47751
$966$	0	0
$967$	−46.0011	−1.47930	−0.739648	−	0.672994i	$-0.765008\pi$
$967$	−46.0011	−1.47930	−0.739648	+	0.672994i	$0.765008\pi$
$968$	0	0
$969$	−2.79772	−0.0898759
$970$	0	0
$971$	−56.7808	−1.82218	−0.911092	−	0.412204i	$-0.864759\pi$
$971$	−56.7808	−1.82218	−0.911092	+	0.412204i	$0.864759\pi$
$972$	0	0
$973$	44.3466	1.42169
$974$	0	0
$975$	−19.0303	−0.609456
$976$	0	0
$977$	59.6094	1.90707	0.953537	−	0.301275i	$-0.0974122\pi$
$977$	59.6094	1.90707	0.953537	+	0.301275i	$0.0974122\pi$
$978$	0	0
$979$	6.12519	0.195762
$980$	0	0
$981$	−2.99197	−0.0955262
$982$	0	0
$983$	−21.1854	−0.675708	−0.337854	−	0.941199i	$-0.609701\pi$
$983$	−21.1854	−0.675708	−0.337854	+	0.941199i	$0.609701\pi$
$984$	0	0
$985$	91.9274	2.92905
$986$	0	0
$987$	36.5389	1.16304
$988$	0	0
$989$	−77.9908	−2.47996
$990$	0	0
$991$	−51.0386	−1.62129	−0.810647	−	0.585535i	$-0.800885\pi$
$991$	−51.0386	−1.62129	−0.810647	+	0.585535i	$0.800885\pi$
$992$	0	0
$993$	25.5896	0.812060
$994$	0	0
$995$	−3.66581	−0.116214
$996$	0	0
$997$	9.84792	0.311887	0.155943	−	0.987766i	$-0.450158\pi$
$997$	9.84792	0.311887	0.155943	+	0.987766i	$0.450158\pi$
$998$	0	0
$999$	−7.53454	−0.238382

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.n.1.3	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.n.1.3	✓	14	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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.71742 q^{5} +3.96243 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.71742 q^{5} +3.96243 q^{7} +1.00000 q^{9} +3.43134 q^{11} +2.15782 q^{13} +3.71742 q^{15} -1.67107 q^{17} -1.67421 q^{19} -3.96243 q^{21} -7.04857 q^{23} +8.81922 q^{25} -1.00000 q^{27} -8.05220 q^{29} -6.74835 q^{31} -3.43134 q^{33} -14.7300 q^{35} +7.53454 q^{37} -2.15782 q^{39} -7.68958 q^{41} +11.0648 q^{43} -3.71742 q^{45} -9.22133 q^{47} +8.70085 q^{49} +1.67107 q^{51} +10.1710 q^{53} -12.7557 q^{55} +1.67421 q^{57} +0.718991 q^{59} +3.40447 q^{61} +3.96243 q^{63} -8.02151 q^{65} -5.28579 q^{67} +7.04857 q^{69} +14.4930 q^{71} -6.67617 q^{73} -8.81922 q^{75} +13.5964 q^{77} +2.08862 q^{79} +1.00000 q^{81} -13.2722 q^{83} +6.21209 q^{85} +8.05220 q^{87} +1.78507 q^{89} +8.55019 q^{91} +6.74835 q^{93} +6.22373 q^{95} +8.98957 q^{97} +3.43134 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9	\( 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9} + 10 q^{11} - 9 q^{13} + 7 q^{15} - 22 q^{17} - 6 q^{19} - q^{21} + 3 q^{23} + 23 q^{25} - 14 q^{27} - 12 q^{29} - 13 q^{31} - 10 q^{33} + 23 q^{35} + 5 q^{37} + 9 q^{39} - 52 q^{41} + 16 q^{43} - 7 q^{45} + q^{47} + 9 q^{49} + 22 q^{51} - 13 q^{53} - 12 q^{55} + 6 q^{57} + 12 q^{59} - 20 q^{61} + q^{63} - 40 q^{65} + 21 q^{67} - 3 q^{69} - 5 q^{71} - 14 q^{73} - 23 q^{75} - 14 q^{77} - 23 q^{79} + 14 q^{81} + 25 q^{83} - 11 q^{85} + 12 q^{87} - 79 q^{89} + 6 q^{91} + 13 q^{93} + 3 q^{95} - 17 q^{97} + 10 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9 + 10 * q^11 - 9 * q^13 + 7 * q^15 - 22 * q^17 - 6 * q^19 - q^21 + 3 * q^23 + 23 * q^25 - 14 * q^27 - 12 * q^29 - 13 * q^31 - 10 * q^33 + 23 * q^35 + 5 * q^37 + 9 * q^39 - 52 * q^41 + 16 * q^43 - 7 * q^45 + q^47 + 9 * q^49 + 22 * q^51 - 13 * q^53 - 12 * q^55 + 6 * q^57 + 12 * q^59 - 20 * q^61 + q^63 - 40 * q^65 + 21 * q^67 - 3 * q^69 - 5 * q^71 - 14 * q^73 - 23 * q^75 - 14 * q^77 - 23 * q^79 + 14 * q^81 + 25 * q^83 - 11 * q^85 + 12 * q^87 - 79 * q^89 + 6 * q^91 + 13 * q^93 + 3 * q^95 - 17 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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