LMFDB - Embedded newform 6024.2.a.q.1.15

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:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 57 x^{16} + 51 x^{15} + 1328 x^{14} - 1116 x^{13} - 16275 x^{12} + 13699 x^{11} + \cdots + 44032 $ x^18 - x^17 - 57x^16 + 51x^15 + 1328x^14 - 1116x^13 - 16275x^12 + 13699x^11 + 112394x^10 - 101250x^9 - 432956x^8 + 439806x^7 + 844117x^6 - 1010785x^5 - 593552x^4 + 965980x^3 - 88040x^2 - 184992x + 44032
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$2.64131$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.64131 q^{5} +4.83855 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.64131 q^{5} +4.83855 q^{7} +1.00000 q^{9} -0.990762 q^{11} -3.25983 q^{13} +2.64131 q^{15} +1.72141 q^{17} +3.60916 q^{19} +4.83855 q^{21} +5.68178 q^{23} +1.97649 q^{25} +1.00000 q^{27} +1.57232 q^{29} -10.1788 q^{31} -0.990762 q^{33} +12.7801 q^{35} -2.88721 q^{37} -3.25983 q^{39} +7.72461 q^{41} +0.980077 q^{43} +2.64131 q^{45} -13.2657 q^{47} +16.4116 q^{49} +1.72141 q^{51} +6.87297 q^{53} -2.61690 q^{55} +3.60916 q^{57} -1.82848 q^{59} +4.09375 q^{61} +4.83855 q^{63} -8.61022 q^{65} -2.37096 q^{67} +5.68178 q^{69} +5.27223 q^{71} +1.42676 q^{73} +1.97649 q^{75} -4.79386 q^{77} +16.6060 q^{79} +1.00000 q^{81} -3.23531 q^{83} +4.54678 q^{85} +1.57232 q^{87} +1.65330 q^{89} -15.7729 q^{91} -10.1788 q^{93} +9.53290 q^{95} -2.90278 q^{97} -0.990762 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9} + 8 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} + 21 q^{19} + 7 q^{21} - 2 q^{23} + 25 q^{25} + 18 q^{27} + 5 q^{29} + 23 q^{31} + 8 q^{33} + 17 q^{35} + 15 q^{37} + 2 q^{39} + 20 q^{41} + 37 q^{43} + q^{45} - q^{47} + 33 q^{49} + 2 q^{51} + 2 q^{53} + 26 q^{55} + 21 q^{57} + 24 q^{59} + 20 q^{61} + 7 q^{63} + 26 q^{65} + 49 q^{67} - 2 q^{69} + 15 q^{71} + 15 q^{73} + 25 q^{75} + 6 q^{77} + 23 q^{79} + 18 q^{81} + 38 q^{83} + 21 q^{85} + 5 q^{87} + 31 q^{89} + 48 q^{91} + 23 q^{93} + 13 q^{95} + 25 q^{97} + 8 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9 + 8 * q^11 + 2 * q^13 + q^15 + 2 * q^17 + 21 * q^19 + 7 * q^21 - 2 * q^23 + 25 * q^25 + 18 * q^27 + 5 * q^29 + 23 * q^31 + 8 * q^33 + 17 * q^35 + 15 * q^37 + 2 * q^39 + 20 * q^41 + 37 * q^43 + q^45 - q^47 + 33 * q^49 + 2 * q^51 + 2 * q^53 + 26 * q^55 + 21 * q^57 + 24 * q^59 + 20 * q^61 + 7 * q^63 + 26 * q^65 + 49 * q^67 - 2 * q^69 + 15 * q^71 + 15 * q^73 + 25 * q^75 + 6 * q^77 + 23 * q^79 + 18 * q^81 + 38 * q^83 + 21 * q^85 + 5 * q^87 + 31 * q^89 + 48 * q^91 + 23 * q^93 + 13 * q^95 + 25 * q^97 + 8 * 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
$5$	2.64131	1.18123	0.590614	−	0.806954i	$-0.298886\pi$
$5$	2.64131	1.18123	0.590614	+	0.806954i	$0.298886\pi$
$6$	0	0
$7$	4.83855	1.82880	0.914401	−	0.404810i	$-0.132662\pi$
$7$	4.83855	1.82880	0.914401	+	0.404810i	$0.132662\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.990762	−0.298726	−0.149363	−	0.988782i	$-0.547722\pi$
$11$	−0.990762	−0.298726	−0.149363	+	0.988782i	$0.547722\pi$
$12$	0	0
$13$	−3.25983	−0.904115	−0.452058	−	0.891989i	$-0.649310\pi$
$13$	−3.25983	−0.904115	−0.452058	+	0.891989i	$0.649310\pi$
$14$	0	0
$15$	2.64131	0.681982
$16$	0	0
$17$	1.72141	0.417504	0.208752	−	0.977969i	$-0.433060\pi$
$17$	1.72141	0.417504	0.208752	+	0.977969i	$0.433060\pi$
$18$	0	0
$19$	3.60916	0.827999	0.414000	−	0.910277i	$-0.364132\pi$
$19$	3.60916	0.827999	0.414000	+	0.910277i	$0.364132\pi$
$20$	0	0
$21$	4.83855	1.05586
$22$	0	0
$23$	5.68178	1.18473	0.592366	−	0.805669i	$-0.298194\pi$
$23$	5.68178	1.18473	0.592366	+	0.805669i	$0.298194\pi$
$24$	0	0
$25$	1.97649	0.395299
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.57232	0.291972	0.145986	−	0.989287i	$-0.453364\pi$
$29$	1.57232	0.291972	0.145986	+	0.989287i	$0.453364\pi$
$30$	0	0
$31$	−10.1788	−1.82816	−0.914082	−	0.405529i	$-0.867087\pi$
$31$	−10.1788	−1.82816	−0.914082	+	0.405529i	$0.867087\pi$
$32$	0	0
$33$	−0.990762	−0.172469
$34$	0	0
$35$	12.7801	2.16023
$36$	0	0
$37$	−2.88721	−0.474654	−0.237327	−	0.971430i	$-0.576271\pi$
$37$	−2.88721	−0.474654	−0.237327	+	0.971430i	$0.576271\pi$
$38$	0	0
$39$	−3.25983	−0.521991
$40$	0	0
$41$	7.72461	1.20638	0.603190	−	0.797597i	$-0.293896\pi$
$41$	7.72461	1.20638	0.603190	+	0.797597i	$0.293896\pi$
$42$	0	0
$43$	0.980077	0.149460	0.0747302	−	0.997204i	$-0.476190\pi$
$43$	0.980077	0.149460	0.0747302	+	0.997204i	$0.476190\pi$
$44$	0	0
$45$	2.64131	0.393743
$46$	0	0
$47$	−13.2657	−1.93501	−0.967503	−	0.252860i	$-0.918629\pi$
$47$	−13.2657	−1.93501	−0.967503	+	0.252860i	$0.918629\pi$
$48$	0	0
$49$	16.4116	2.34452
$50$	0	0
$51$	1.72141	0.241046
$52$	0	0
$53$	6.87297	0.944075	0.472037	−	0.881579i	$-0.343519\pi$
$53$	6.87297	0.944075	0.472037	+	0.881579i	$0.343519\pi$
$54$	0	0
$55$	−2.61690	−0.352863
$56$	0	0
$57$	3.60916	0.478045
$58$	0	0
$59$	−1.82848	−0.238048	−0.119024	−	0.992891i	$-0.537977\pi$
$59$	−1.82848	−0.238048	−0.119024	+	0.992891i	$0.537977\pi$
$60$	0	0
$61$	4.09375	0.524150	0.262075	−	0.965047i	$-0.415593\pi$
$61$	4.09375	0.524150	0.262075	+	0.965047i	$0.415593\pi$
$62$	0	0
$63$	4.83855	0.609601
$64$	0	0
$65$	−8.61022	−1.06797
$66$	0	0
$67$	−2.37096	−0.289659	−0.144829	−	0.989457i	$-0.546263\pi$
$67$	−2.37096	−0.289659	−0.144829	+	0.989457i	$0.546263\pi$
$68$	0	0
$69$	5.68178	0.684006
$70$	0	0
$71$	5.27223	0.625698	0.312849	−	0.949803i	$-0.398717\pi$
$71$	5.27223	0.625698	0.312849	+	0.949803i	$0.398717\pi$
$72$	0	0
$73$	1.42676	0.166990	0.0834948	−	0.996508i	$-0.473392\pi$
$73$	1.42676	0.166990	0.0834948	+	0.996508i	$0.473392\pi$
$74$	0	0
$75$	1.97649	0.228226
$76$	0	0
$77$	−4.79386	−0.546310
$78$	0	0
$79$	16.6060	1.86833	0.934163	−	0.356847i	$-0.116148\pi$
$79$	16.6060	1.86833	0.934163	+	0.356847i	$0.116148\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.23531	−0.355121	−0.177561	−	0.984110i	$-0.556821\pi$
$83$	−3.23531	−0.355121	−0.177561	+	0.984110i	$0.556821\pi$
$84$	0	0
$85$	4.54678	0.493168
$86$	0	0
$87$	1.57232	0.168570
$88$	0	0
$89$	1.65330	0.175249	0.0876245	−	0.996154i	$-0.472072\pi$
$89$	1.65330	0.175249	0.0876245	+	0.996154i	$0.472072\pi$
$90$	0	0
$91$	−15.7729	−1.65345
$92$	0	0
$93$	−10.1788	−1.05549
$94$	0	0
$95$	9.53290	0.978055
$96$	0	0
$97$	−2.90278	−0.294733	−0.147366	−	0.989082i	$-0.547080\pi$
$97$	−2.90278	−0.294733	−0.147366	+	0.989082i	$0.547080\pi$
$98$	0	0
$99$	−0.990762	−0.0995753
$100$	0	0
$101$	−8.60064	−0.855796	−0.427898	−	0.903827i	$-0.640746\pi$
$101$	−8.60064	−0.855796	−0.427898	+	0.903827i	$0.640746\pi$
$102$	0	0
$103$	0.484047	0.0476946	0.0238473	−	0.999716i	$-0.492408\pi$
$103$	0.484047	0.0476946	0.0238473	+	0.999716i	$0.492408\pi$
$104$	0	0
$105$	12.7801	1.24721
$106$	0	0
$107$	7.92863	0.766490	0.383245	−	0.923647i	$-0.374806\pi$
$107$	7.92863	0.766490	0.383245	+	0.923647i	$0.374806\pi$
$108$	0	0
$109$	12.9202	1.23753	0.618765	−	0.785576i	$-0.287633\pi$
$109$	12.9202	1.23753	0.618765	+	0.785576i	$0.287633\pi$
$110$	0	0
$111$	−2.88721	−0.274041
$112$	0	0
$113$	−0.717974	−0.0675413	−0.0337706	−	0.999430i	$-0.510752\pi$
$113$	−0.717974	−0.0675413	−0.0337706	+	0.999430i	$0.510752\pi$
$114$	0	0
$115$	15.0073	1.39944
$116$	0	0
$117$	−3.25983	−0.301372
$118$	0	0
$119$	8.32916	0.763533
$120$	0	0
$121$	−10.0184	−0.910763
$122$	0	0
$123$	7.72461	0.696504
$124$	0	0
$125$	−7.98601	−0.714290
$126$	0	0
$127$	−5.30515	−0.470756	−0.235378	−	0.971904i	$-0.575633\pi$
$127$	−5.30515	−0.470756	−0.235378	+	0.971904i	$0.575633\pi$
$128$	0	0
$129$	0.980077	0.0862910
$130$	0	0
$131$	9.46869	0.827283	0.413642	−	0.910440i	$-0.364257\pi$
$131$	9.46869	0.827283	0.413642	+	0.910440i	$0.364257\pi$
$132$	0	0
$133$	17.4631	1.51425
$134$	0	0
$135$	2.64131	0.227327
$136$	0	0
$137$	−21.3721	−1.82595	−0.912973	−	0.408020i	$-0.866219\pi$
$137$	−21.3721	−1.82595	−0.912973	+	0.408020i	$0.866219\pi$
$138$	0	0
$139$	8.29655	0.703704	0.351852	−	0.936056i	$-0.385552\pi$
$139$	8.29655	0.703704	0.351852	+	0.936056i	$0.385552\pi$
$140$	0	0
$141$	−13.2657	−1.11718
$142$	0	0
$143$	3.22972	0.270083
$144$	0	0
$145$	4.15297	0.344886
$146$	0	0
$147$	16.4116	1.35361
$148$	0	0
$149$	−17.9970	−1.47437	−0.737185	−	0.675691i	$-0.763845\pi$
$149$	−17.9970	−1.47437	−0.737185	+	0.675691i	$0.763845\pi$
$150$	0	0
$151$	−2.17851	−0.177285	−0.0886423	−	0.996064i	$-0.528253\pi$
$151$	−2.17851	−0.177285	−0.0886423	+	0.996064i	$0.528253\pi$
$152$	0	0
$153$	1.72141	0.139168
$154$	0	0
$155$	−26.8853	−2.15948
$156$	0	0
$157$	20.6371	1.64702	0.823511	−	0.567301i	$-0.192012\pi$
$157$	20.6371	1.64702	0.823511	+	0.567301i	$0.192012\pi$
$158$	0	0
$159$	6.87297	0.545062
$160$	0	0
$161$	27.4916	2.16664
$162$	0	0
$163$	17.4320	1.36538	0.682689	−	0.730709i	$-0.260810\pi$
$163$	17.4320	1.36538	0.682689	+	0.730709i	$0.260810\pi$
$164$	0	0
$165$	−2.61690	−0.203726
$166$	0	0
$167$	1.62017	0.125372	0.0626861	−	0.998033i	$-0.480033\pi$
$167$	1.62017	0.125372	0.0626861	+	0.998033i	$0.480033\pi$
$168$	0	0
$169$	−2.37349	−0.182576
$170$	0	0
$171$	3.60916	0.276000
$172$	0	0
$173$	−5.10630	−0.388225	−0.194112	−	0.980979i	$-0.562183\pi$
$173$	−5.10630	−0.388225	−0.194112	+	0.980979i	$0.562183\pi$
$174$	0	0
$175$	9.56337	0.722923
$176$	0	0
$177$	−1.82848	−0.137437
$178$	0	0
$179$	−9.60452	−0.717876	−0.358938	−	0.933361i	$-0.616861\pi$
$179$	−9.60452	−0.717876	−0.358938	+	0.933361i	$0.616861\pi$
$180$	0	0
$181$	7.09425	0.527312	0.263656	−	0.964617i	$-0.415072\pi$
$181$	7.09425	0.527312	0.263656	+	0.964617i	$0.415072\pi$
$182$	0	0
$183$	4.09375	0.302618
$184$	0	0
$185$	−7.62599	−0.560674
$186$	0	0
$187$	−1.70551	−0.124719
$188$	0	0
$189$	4.83855	0.351953
$190$	0	0
$191$	−7.52941	−0.544809	−0.272405	−	0.962183i	$-0.587819\pi$
$191$	−7.52941	−0.544809	−0.272405	+	0.962183i	$0.587819\pi$
$192$	0	0
$193$	−3.65324	−0.262966	−0.131483	−	0.991318i	$-0.541974\pi$
$193$	−3.65324	−0.262966	−0.131483	+	0.991318i	$0.541974\pi$
$194$	0	0
$195$	−8.61022	−0.616590
$196$	0	0
$197$	−24.8012	−1.76701	−0.883505	−	0.468422i	$-0.844823\pi$
$197$	−24.8012	−1.76701	−0.883505	+	0.468422i	$0.844823\pi$
$198$	0	0
$199$	−16.3209	−1.15696	−0.578478	−	0.815698i	$-0.696353\pi$
$199$	−16.3209	−1.15696	−0.578478	+	0.815698i	$0.696353\pi$
$200$	0	0
$201$	−2.37096	−0.167235
$202$	0	0
$203$	7.60775	0.533960
$204$	0	0
$205$	20.4030	1.42501
$206$	0	0
$207$	5.68178	0.394911
$208$	0	0
$209$	−3.57582	−0.247345
$210$	0	0
$211$	19.7285	1.35816	0.679081	−	0.734063i	$-0.262378\pi$
$211$	19.7285	1.35816	0.679081	+	0.734063i	$0.262378\pi$
$212$	0	0
$213$	5.27223	0.361247
$214$	0	0
$215$	2.58868	0.176547
$216$	0	0
$217$	−49.2506	−3.34335
$218$	0	0
$219$	1.42676	0.0964115
$220$	0	0
$221$	−5.61152	−0.377472
$222$	0	0
$223$	−21.6132	−1.44733	−0.723664	−	0.690153i	$-0.757543\pi$
$223$	−21.6132	−1.44733	−0.723664	+	0.690153i	$0.757543\pi$
$224$	0	0
$225$	1.97649	0.131766
$226$	0	0
$227$	12.2295	0.811703	0.405852	−	0.913939i	$-0.366975\pi$
$227$	12.2295	0.811703	0.405852	+	0.913939i	$0.366975\pi$
$228$	0	0
$229$	3.88401	0.256663	0.128331	−	0.991731i	$-0.459038\pi$
$229$	3.88401	0.256663	0.128331	+	0.991731i	$0.459038\pi$
$230$	0	0
$231$	−4.79386	−0.315413
$232$	0	0
$233$	−12.4153	−0.813354	−0.406677	−	0.913572i	$-0.633313\pi$
$233$	−12.4153	−0.813354	−0.406677	+	0.913572i	$0.633313\pi$
$234$	0	0
$235$	−35.0388	−2.28568
$236$	0	0
$237$	16.6060	1.07868
$238$	0	0
$239$	17.2415	1.11526	0.557631	−	0.830089i	$-0.311711\pi$
$239$	17.2415	1.11526	0.557631	+	0.830089i	$0.311711\pi$
$240$	0	0
$241$	−14.1409	−0.910894	−0.455447	−	0.890263i	$-0.650521\pi$
$241$	−14.1409	−0.910894	−0.455447	+	0.890263i	$0.650521\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	43.3481	2.76941
$246$	0	0
$247$	−11.7653	−0.748606
$248$	0	0
$249$	−3.23531	−0.205029
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−5.62929	−0.353910
$254$	0	0
$255$	4.54678	0.284730
$256$	0	0
$257$	−21.7655	−1.35769	−0.678846	−	0.734280i	$-0.737520\pi$
$257$	−21.7655	−1.35769	−0.678846	+	0.734280i	$0.737520\pi$
$258$	0	0
$259$	−13.9699	−0.868047
$260$	0	0
$261$	1.57232	0.0973241
$262$	0	0
$263$	−16.7893	−1.03527	−0.517636	−	0.855601i	$-0.673188\pi$
$263$	−16.7893	−1.03527	−0.517636	+	0.855601i	$0.673188\pi$
$264$	0	0
$265$	18.1536	1.11517
$266$	0	0
$267$	1.65330	0.101180
$268$	0	0
$269$	−10.0455	−0.612487	−0.306243	−	0.951953i	$-0.599072\pi$
$269$	−10.0455	−0.612487	−0.306243	+	0.951953i	$0.599072\pi$
$270$	0	0
$271$	10.1464	0.616352	0.308176	−	0.951329i	$-0.400281\pi$
$271$	10.1464	0.616352	0.308176	+	0.951329i	$0.400281\pi$
$272$	0	0
$273$	−15.7729	−0.954618
$274$	0	0
$275$	−1.95823	−0.118086
$276$	0	0
$277$	23.0023	1.38208	0.691038	−	0.722818i	$-0.257154\pi$
$277$	23.0023	1.38208	0.691038	+	0.722818i	$0.257154\pi$
$278$	0	0
$279$	−10.1788	−0.609388
$280$	0	0
$281$	−9.91685	−0.591590	−0.295795	−	0.955251i	$-0.595584\pi$
$281$	−9.91685	−0.591590	−0.295795	+	0.955251i	$0.595584\pi$
$282$	0	0
$283$	−26.4769	−1.57389	−0.786945	−	0.617023i	$-0.788339\pi$
$283$	−26.4769	−1.57389	−0.786945	+	0.617023i	$0.788339\pi$
$284$	0	0
$285$	9.53290	0.564680
$286$	0	0
$287$	37.3759	2.20623
$288$	0	0
$289$	−14.0367	−0.825690
$290$	0	0
$291$	−2.90278	−0.170164
$292$	0	0
$293$	2.13762	0.124881	0.0624404	−	0.998049i	$-0.480112\pi$
$293$	2.13762	0.124881	0.0624404	+	0.998049i	$0.480112\pi$
$294$	0	0
$295$	−4.82958	−0.281189
$296$	0	0
$297$	−0.990762	−0.0574898
$298$	0	0
$299$	−18.5216	−1.07113
$300$	0	0
$301$	4.74216	0.273333
$302$	0	0
$303$	−8.60064	−0.494094
$304$	0	0
$305$	10.8128	0.619141
$306$	0	0
$307$	1.26623	0.0722674	0.0361337	−	0.999347i	$-0.488496\pi$
$307$	1.26623	0.0722674	0.0361337	+	0.999347i	$0.488496\pi$
$308$	0	0
$309$	0.484047	0.0275365
$310$	0	0
$311$	12.9899	0.736589	0.368295	−	0.929709i	$-0.379942\pi$
$311$	12.9899	0.736589	0.368295	+	0.929709i	$0.379942\pi$
$312$	0	0
$313$	−28.2727	−1.59807	−0.799035	−	0.601285i	$-0.794656\pi$
$313$	−28.2727	−1.59807	−0.799035	+	0.601285i	$0.794656\pi$
$314$	0	0
$315$	12.7801	0.720077
$316$	0	0
$317$	−32.6601	−1.83437	−0.917187	−	0.398458i	$-0.869545\pi$
$317$	−32.6601	−1.83437	−0.917187	+	0.398458i	$0.869545\pi$
$318$	0	0
$319$	−1.55779	−0.0872197
$320$	0	0
$321$	7.92863	0.442533
$322$	0	0
$323$	6.21287	0.345693
$324$	0	0
$325$	−6.44304	−0.357395
$326$	0	0
$327$	12.9202	0.714489
$328$	0	0
$329$	−64.1870	−3.53874
$330$	0	0
$331$	−18.3361	−1.00784	−0.503921	−	0.863750i	$-0.668110\pi$
$331$	−18.3361	−1.00784	−0.503921	+	0.863750i	$0.668110\pi$
$332$	0	0
$333$	−2.88721	−0.158218
$334$	0	0
$335$	−6.26242	−0.342153
$336$	0	0
$337$	−25.3984	−1.38354	−0.691769	−	0.722119i	$-0.743168\pi$
$337$	−25.3984	−1.38354	−0.691769	+	0.722119i	$0.743168\pi$
$338$	0	0
$339$	−0.717974	−0.0389950
$340$	0	0
$341$	10.0848	0.546120
$342$	0	0
$343$	45.5386	2.45885
$344$	0	0
$345$	15.0073	0.807966
$346$	0	0
$347$	32.9893	1.77096	0.885479	−	0.464679i	$-0.153830\pi$
$347$	32.9893	1.77096	0.885479	+	0.464679i	$0.153830\pi$
$348$	0	0
$349$	21.3694	1.14388	0.571938	−	0.820297i	$-0.306192\pi$
$349$	21.3694	1.14388	0.571938	+	0.820297i	$0.306192\pi$
$350$	0	0
$351$	−3.25983	−0.173997
$352$	0	0
$353$	16.5215	0.879353	0.439676	−	0.898156i	$-0.355093\pi$
$353$	16.5215	0.879353	0.439676	+	0.898156i	$0.355093\pi$
$354$	0	0
$355$	13.9256	0.739092
$356$	0	0
$357$	8.32916	0.440826
$358$	0	0
$359$	−10.8112	−0.570593	−0.285296	−	0.958439i	$-0.592092\pi$
$359$	−10.8112	−0.570593	−0.285296	+	0.958439i	$0.592092\pi$
$360$	0	0
$361$	−5.97393	−0.314418
$362$	0	0
$363$	−10.0184	−0.525829
$364$	0	0
$365$	3.76851	0.197253
$366$	0	0
$367$	6.22121	0.324744	0.162372	−	0.986730i	$-0.448085\pi$
$367$	6.22121	0.324744	0.162372	+	0.986730i	$0.448085\pi$
$368$	0	0
$369$	7.72461	0.402127
$370$	0	0
$371$	33.2552	1.72653
$372$	0	0
$373$	−0.432556	−0.0223969	−0.0111984	−	0.999937i	$-0.503565\pi$
$373$	−0.432556	−0.0223969	−0.0111984	+	0.999937i	$0.503565\pi$
$374$	0	0
$375$	−7.98601	−0.412396
$376$	0	0
$377$	−5.12550	−0.263977
$378$	0	0
$379$	21.0939	1.08352	0.541760	−	0.840533i	$-0.317758\pi$
$379$	21.0939	1.08352	0.541760	+	0.840533i	$0.317758\pi$
$380$	0	0
$381$	−5.30515	−0.271791
$382$	0	0
$383$	27.5324	1.40684	0.703421	−	0.710773i	$-0.251655\pi$
$383$	27.5324	1.40684	0.703421	+	0.710773i	$0.251655\pi$
$384$	0	0
$385$	−12.6620	−0.645317
$386$	0	0
$387$	0.980077	0.0498201
$388$	0	0
$389$	11.7021	0.593321	0.296660	−	0.954983i	$-0.404127\pi$
$389$	11.7021	0.593321	0.296660	+	0.954983i	$0.404127\pi$
$390$	0	0
$391$	9.78069	0.494631
$392$	0	0
$393$	9.46869	0.477632
$394$	0	0
$395$	43.8616	2.20692
$396$	0	0
$397$	−12.6413	−0.634449	−0.317224	−	0.948351i	$-0.602751\pi$
$397$	−12.6413	−0.634449	−0.317224	+	0.948351i	$0.602751\pi$
$398$	0	0
$399$	17.4631	0.874250
$400$	0	0
$401$	14.1525	0.706741	0.353370	−	0.935483i	$-0.385036\pi$
$401$	14.1525	0.706741	0.353370	+	0.935483i	$0.385036\pi$
$402$	0	0
$403$	33.1812	1.65287
$404$	0	0
$405$	2.64131	0.131248
$406$	0	0
$407$	2.86053	0.141791
$408$	0	0
$409$	24.0106	1.18725	0.593623	−	0.804743i	$-0.297697\pi$
$409$	24.0106	1.18725	0.593623	+	0.804743i	$0.297697\pi$
$410$	0	0
$411$	−21.3721	−1.05421
$412$	0	0
$413$	−8.84722	−0.435343
$414$	0	0
$415$	−8.54543	−0.419479
$416$	0	0
$417$	8.29655	0.406284
$418$	0	0
$419$	−30.2558	−1.47809	−0.739046	−	0.673655i	$-0.764723\pi$
$419$	−30.2558	−1.47809	−0.739046	+	0.673655i	$0.764723\pi$
$420$	0	0
$421$	15.9209	0.775938	0.387969	−	0.921672i	$-0.373177\pi$
$421$	15.9209	0.775938	0.387969	+	0.921672i	$0.373177\pi$
$422$	0	0
$423$	−13.2657	−0.645002
$424$	0	0
$425$	3.40236	0.165039
$426$	0	0
$427$	19.8078	0.958567
$428$	0	0
$429$	3.22972	0.155932
$430$	0	0
$431$	−9.39262	−0.452427	−0.226213	−	0.974078i	$-0.572635\pi$
$431$	−9.39262	−0.452427	−0.226213	+	0.974078i	$0.572635\pi$
$432$	0	0
$433$	30.1377	1.44832	0.724162	−	0.689630i	$-0.242227\pi$
$433$	30.1377	1.44832	0.724162	+	0.689630i	$0.242227\pi$
$434$	0	0
$435$	4.15297	0.199120
$436$	0	0
$437$	20.5065	0.980957
$438$	0	0
$439$	25.4860	1.21638	0.608189	−	0.793792i	$-0.291896\pi$
$439$	25.4860	1.21638	0.608189	+	0.793792i	$0.291896\pi$
$440$	0	0
$441$	16.4116	0.781505
$442$	0	0
$443$	−11.6122	−0.551712	−0.275856	−	0.961199i	$-0.588961\pi$
$443$	−11.6122	−0.551712	−0.275856	+	0.961199i	$0.588961\pi$
$444$	0	0
$445$	4.36686	0.207009
$446$	0	0
$447$	−17.9970	−0.851228
$448$	0	0
$449$	−24.6111	−1.16147	−0.580735	−	0.814093i	$-0.697235\pi$
$449$	−24.6111	−1.16147	−0.580735	+	0.814093i	$0.697235\pi$
$450$	0	0
$451$	−7.65324	−0.360377
$452$	0	0
$453$	−2.17851	−0.102355
$454$	0	0
$455$	−41.6610	−1.95310
$456$	0	0
$457$	4.30125	0.201204	0.100602	−	0.994927i	$-0.467923\pi$
$457$	4.30125	0.201204	0.100602	+	0.994927i	$0.467923\pi$
$458$	0	0
$459$	1.72141	0.0803487
$460$	0	0
$461$	36.4239	1.69643	0.848215	−	0.529652i	$-0.177677\pi$
$461$	36.4239	1.69643	0.848215	+	0.529652i	$0.177677\pi$
$462$	0	0
$463$	−0.123459	−0.00573764	−0.00286882	−	0.999996i	$-0.500913\pi$
$463$	−0.123459	−0.00573764	−0.00286882	+	0.999996i	$0.500913\pi$
$464$	0	0
$465$	−26.8853	−1.24678
$466$	0	0
$467$	21.5753	0.998386	0.499193	−	0.866491i	$-0.333630\pi$
$467$	21.5753	0.998386	0.499193	+	0.866491i	$0.333630\pi$
$468$	0	0
$469$	−11.4720	−0.529728
$470$	0	0
$471$	20.6371	0.950908
$472$	0	0
$473$	−0.971023	−0.0446477
$474$	0	0
$475$	7.13349	0.327307
$476$	0	0
$477$	6.87297	0.314692
$478$	0	0
$479$	−5.12204	−0.234032	−0.117016	−	0.993130i	$-0.537333\pi$
$479$	−5.12204	−0.234032	−0.117016	+	0.993130i	$0.537333\pi$
$480$	0	0
$481$	9.41181	0.429142
$482$	0	0
$483$	27.4916	1.25091
$484$	0	0
$485$	−7.66713	−0.348146
$486$	0	0
$487$	28.9024	1.30969	0.654846	−	0.755762i	$-0.272733\pi$
$487$	28.9024	1.30969	0.654846	+	0.755762i	$0.272733\pi$
$488$	0	0
$489$	17.4320	0.788302
$490$	0	0
$491$	−25.6109	−1.15580	−0.577902	−	0.816106i	$-0.696128\pi$
$491$	−25.6109	−1.15580	−0.577902	+	0.816106i	$0.696128\pi$
$492$	0	0
$493$	2.70661	0.121900
$494$	0	0
$495$	−2.61690	−0.117621
$496$	0	0
$497$	25.5100	1.14428
$498$	0	0
$499$	−3.81791	−0.170913	−0.0854566	−	0.996342i	$-0.527235\pi$
$499$	−3.81791	−0.170913	−0.0854566	+	0.996342i	$0.527235\pi$
$500$	0	0
$501$	1.62017	0.0723836
$502$	0	0
$503$	0.497686	0.0221907	0.0110954	−	0.999938i	$-0.496468\pi$
$503$	0.497686	0.0221907	0.0110954	+	0.999938i	$0.496468\pi$
$504$	0	0
$505$	−22.7169	−1.01089
$506$	0	0
$507$	−2.37349	−0.105410
$508$	0	0
$509$	−10.3809	−0.460127	−0.230064	−	0.973176i	$-0.573893\pi$
$509$	−10.3809	−0.460127	−0.230064	+	0.973176i	$0.573893\pi$
$510$	0	0
$511$	6.90345	0.305391
$512$	0	0
$513$	3.60916	0.159348
$514$	0	0
$515$	1.27852	0.0563382
$516$	0	0
$517$	13.1432	0.578036
$518$	0	0
$519$	−5.10630	−0.224142
$520$	0	0
$521$	16.0652	0.703831	0.351915	−	0.936032i	$-0.385530\pi$
$521$	16.0652	0.703831	0.351915	+	0.936032i	$0.385530\pi$
$522$	0	0
$523$	−21.7888	−0.952757	−0.476379	−	0.879240i	$-0.658051\pi$
$523$	−21.7888	−0.952757	−0.476379	+	0.879240i	$0.658051\pi$
$524$	0	0
$525$	9.56337	0.417380
$526$	0	0
$527$	−17.5219	−0.763266
$528$	0	0
$529$	9.28259	0.403591
$530$	0	0
$531$	−1.82848	−0.0793494
$532$	0	0
$533$	−25.1809	−1.09071
$534$	0	0
$535$	20.9419	0.905399
$536$	0	0
$537$	−9.60452	−0.414466
$538$	0	0
$539$	−16.2600	−0.700368
$540$	0	0
$541$	−34.5702	−1.48629	−0.743143	−	0.669132i	$-0.766666\pi$
$541$	−34.5702	−1.48629	−0.743143	+	0.669132i	$0.766666\pi$
$542$	0	0
$543$	7.09425	0.304444
$544$	0	0
$545$	34.1262	1.46181
$546$	0	0
$547$	38.2335	1.63475	0.817373	−	0.576109i	$-0.195430\pi$
$547$	38.2335	1.63475	0.817373	+	0.576109i	$0.195430\pi$
$548$	0	0
$549$	4.09375	0.174717
$550$	0	0
$551$	5.67476	0.241753
$552$	0	0
$553$	80.3492	3.41680
$554$	0	0
$555$	−7.62599	−0.323705
$556$	0	0
$557$	−8.60017	−0.364401	−0.182200	−	0.983261i	$-0.558322\pi$
$557$	−8.60017	−0.364401	−0.182200	+	0.983261i	$0.558322\pi$
$558$	0	0
$559$	−3.19489	−0.135129
$560$	0	0
$561$	−1.70551	−0.0720067
$562$	0	0
$563$	21.9954	0.926995	0.463497	−	0.886098i	$-0.346594\pi$
$563$	21.9954	0.926995	0.463497	+	0.886098i	$0.346594\pi$
$564$	0	0
$565$	−1.89639	−0.0797816
$566$	0	0
$567$	4.83855	0.203200
$568$	0	0
$569$	26.2219	1.09928	0.549640	−	0.835402i	$-0.314765\pi$
$569$	26.2219	1.09928	0.549640	+	0.835402i	$0.314765\pi$
$570$	0	0
$571$	−10.1874	−0.426330	−0.213165	−	0.977016i	$-0.568377\pi$
$571$	−10.1874	−0.426330	−0.213165	+	0.977016i	$0.568377\pi$
$572$	0	0
$573$	−7.52941	−0.314546
$574$	0	0
$575$	11.2300	0.468323
$576$	0	0
$577$	17.0621	0.710306	0.355153	−	0.934808i	$-0.384429\pi$
$577$	17.0621	0.710306	0.355153	+	0.934808i	$0.384429\pi$
$578$	0	0
$579$	−3.65324	−0.151823
$580$	0	0
$581$	−15.6542	−0.649446
$582$	0	0
$583$	−6.80947	−0.282020
$584$	0	0
$585$	−8.61022	−0.355989
$586$	0	0
$587$	−27.6380	−1.14074	−0.570371	−	0.821387i	$-0.693201\pi$
$587$	−27.6380	−1.14074	−0.570371	+	0.821387i	$0.693201\pi$
$588$	0	0
$589$	−36.7369	−1.51372
$590$	0	0
$591$	−24.8012	−1.02018
$592$	0	0
$593$	−17.4084	−0.714878	−0.357439	−	0.933936i	$-0.616350\pi$
$593$	−17.4084	−0.714878	−0.357439	+	0.933936i	$0.616350\pi$
$594$	0	0
$595$	21.9998	0.901906
$596$	0	0
$597$	−16.3209	−0.667969
$598$	0	0
$599$	−36.6340	−1.49682	−0.748412	−	0.663234i	$-0.769184\pi$
$599$	−36.6340	−1.49682	−0.748412	+	0.663234i	$0.769184\pi$
$600$	0	0
$601$	−13.3927	−0.546299	−0.273150	−	0.961972i	$-0.588065\pi$
$601$	−13.3927	−0.546299	−0.273150	+	0.961972i	$0.588065\pi$
$602$	0	0
$603$	−2.37096	−0.0965529
$604$	0	0
$605$	−26.4616	−1.07582
$606$	0	0
$607$	33.6441	1.36557	0.682787	−	0.730618i	$-0.260768\pi$
$607$	33.6441	1.36557	0.682787	+	0.730618i	$0.260768\pi$
$608$	0	0
$609$	7.60775	0.308282
$610$	0	0
$611$	43.2441	1.74947
$612$	0	0
$613$	−22.0765	−0.891660	−0.445830	−	0.895118i	$-0.647091\pi$
$613$	−22.0765	−0.891660	−0.445830	+	0.895118i	$0.647091\pi$
$614$	0	0
$615$	20.4030	0.822730
$616$	0	0
$617$	−2.33311	−0.0939274	−0.0469637	−	0.998897i	$-0.514955\pi$
$617$	−2.33311	−0.0939274	−0.0469637	+	0.998897i	$0.514955\pi$
$618$	0	0
$619$	4.85186	0.195013	0.0975064	−	0.995235i	$-0.468913\pi$
$619$	4.85186	0.195013	0.0975064	+	0.995235i	$0.468913\pi$
$620$	0	0
$621$	5.68178	0.228002
$622$	0	0
$623$	7.99956	0.320496
$624$	0	0
$625$	−30.9759	−1.23904
$626$	0	0
$627$	−3.57582	−0.142805
$628$	0	0
$629$	−4.97008	−0.198170
$630$	0	0
$631$	−32.4207	−1.29065	−0.645325	−	0.763909i	$-0.723278\pi$
$631$	−32.4207	−1.29065	−0.645325	+	0.763909i	$0.723278\pi$
$632$	0	0
$633$	19.7285	0.784136
$634$	0	0
$635$	−14.0125	−0.556069
$636$	0	0
$637$	−53.4991	−2.11971
$638$	0	0
$639$	5.27223	0.208566
$640$	0	0
$641$	20.4079	0.806062	0.403031	−	0.915186i	$-0.367957\pi$
$641$	20.4079	0.806062	0.403031	+	0.915186i	$0.367957\pi$
$642$	0	0
$643$	39.9108	1.57393	0.786965	−	0.616998i	$-0.211651\pi$
$643$	39.9108	1.57393	0.786965	+	0.616998i	$0.211651\pi$
$644$	0	0
$645$	2.58868	0.101929
$646$	0	0
$647$	−19.8900	−0.781955	−0.390978	−	0.920400i	$-0.627863\pi$
$647$	−19.8900	−0.781955	−0.390978	+	0.920400i	$0.627863\pi$
$648$	0	0
$649$	1.81159	0.0711112
$650$	0	0
$651$	−49.2506	−1.93028
$652$	0	0
$653$	−7.31517	−0.286265	−0.143132	−	0.989704i	$-0.545717\pi$
$653$	−7.31517	−0.286265	−0.143132	+	0.989704i	$0.545717\pi$
$654$	0	0
$655$	25.0097	0.977210
$656$	0	0
$657$	1.42676	0.0556632
$658$	0	0
$659$	−34.1795	−1.33144	−0.665721	−	0.746200i	$-0.731876\pi$
$659$	−34.1795	−1.33144	−0.665721	+	0.746200i	$0.731876\pi$
$660$	0	0
$661$	34.8216	1.35440	0.677202	−	0.735797i	$-0.263192\pi$
$661$	34.8216	1.35440	0.677202	+	0.735797i	$0.263192\pi$
$662$	0	0
$663$	−5.61152	−0.217934
$664$	0	0
$665$	46.1255	1.78867
$666$	0	0
$667$	8.93357	0.345909
$668$	0	0
$669$	−21.6132	−0.835615
$670$	0	0
$671$	−4.05593	−0.156577
$672$	0	0
$673$	16.5198	0.636791	0.318396	−	0.947958i	$-0.396856\pi$
$673$	16.5198	0.636791	0.318396	+	0.947958i	$0.396856\pi$
$674$	0	0
$675$	1.97649	0.0760752
$676$	0	0
$677$	−32.8882	−1.26399	−0.631997	−	0.774970i	$-0.717765\pi$
$677$	−32.8882	−1.26399	−0.631997	+	0.774970i	$0.717765\pi$
$678$	0	0
$679$	−14.0453	−0.539008
$680$	0	0
$681$	12.2295	0.468637
$682$	0	0
$683$	45.7191	1.74939	0.874697	−	0.484670i	$-0.161060\pi$
$683$	45.7191	1.74939	0.874697	+	0.484670i	$0.161060\pi$
$684$	0	0
$685$	−56.4504	−2.15686
$686$	0	0
$687$	3.88401	0.148184
$688$	0	0
$689$	−22.4047	−0.853552
$690$	0	0
$691$	−7.00553	−0.266503	−0.133251	−	0.991082i	$-0.542542\pi$
$691$	−7.00553	−0.266503	−0.133251	+	0.991082i	$0.542542\pi$
$692$	0	0
$693$	−4.79386	−0.182103
$694$	0	0
$695$	21.9137	0.831234
$696$	0	0
$697$	13.2972	0.503669
$698$	0	0
$699$	−12.4153	−0.469590
$700$	0	0
$701$	−36.0061	−1.35993	−0.679965	−	0.733244i	$-0.738005\pi$
$701$	−36.0061	−1.35993	−0.679965	+	0.733244i	$0.738005\pi$
$702$	0	0
$703$	−10.4204	−0.393013
$704$	0	0
$705$	−35.0388	−1.31964
$706$	0	0
$707$	−41.6147	−1.56508
$708$	0	0
$709$	−27.2929	−1.02501	−0.512503	−	0.858685i	$-0.671282\pi$
$709$	−27.2929	−1.02501	−0.512503	+	0.858685i	$0.671282\pi$
$710$	0	0
$711$	16.6060	0.622775
$712$	0	0
$713$	−57.8336	−2.16589
$714$	0	0
$715$	8.53067	0.319029
$716$	0	0
$717$	17.2415	0.643896
$718$	0	0
$719$	−47.3778	−1.76689	−0.883447	−	0.468531i	$-0.844784\pi$
$719$	−47.3778	−1.76689	−0.883447	+	0.468531i	$0.844784\pi$
$720$	0	0
$721$	2.34209	0.0872240
$722$	0	0
$723$	−14.1409	−0.525905
$724$	0	0
$725$	3.10768	0.115416
$726$	0	0
$727$	3.93293	0.145864	0.0729321	−	0.997337i	$-0.476764\pi$
$727$	3.93293	0.145864	0.0729321	+	0.997337i	$0.476764\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.68712	0.0624004
$732$	0	0
$733$	23.0542	0.851528	0.425764	−	0.904834i	$-0.360005\pi$
$733$	23.0542	0.851528	0.425764	+	0.904834i	$0.360005\pi$
$734$	0	0
$735$	43.3481	1.59892
$736$	0	0
$737$	2.34905	0.0865286
$738$	0	0
$739$	17.2712	0.635330	0.317665	−	0.948203i	$-0.397101\pi$
$739$	17.2712	0.635330	0.317665	+	0.948203i	$0.397101\pi$
$740$	0	0
$741$	−11.7653	−0.432208
$742$	0	0
$743$	−4.61549	−0.169326	−0.0846630	−	0.996410i	$-0.526981\pi$
$743$	−4.61549	−0.169326	−0.0846630	+	0.996410i	$0.526981\pi$
$744$	0	0
$745$	−47.5355	−1.74157
$746$	0	0
$747$	−3.23531	−0.118374
$748$	0	0
$749$	38.3631	1.40176
$750$	0	0
$751$	21.1366	0.771284	0.385642	−	0.922648i	$-0.373980\pi$
$751$	21.1366	0.771284	0.385642	+	0.922648i	$0.373980\pi$
$752$	0	0
$753$	1.00000	0.0364420
$754$	0	0
$755$	−5.75411	−0.209414
$756$	0	0
$757$	−38.8837	−1.41325	−0.706627	−	0.707587i	$-0.749784\pi$
$757$	−38.8837	−1.41325	−0.706627	+	0.707587i	$0.749784\pi$
$758$	0	0
$759$	−5.62929	−0.204330
$760$	0	0
$761$	30.5022	1.10570	0.552852	−	0.833280i	$-0.313540\pi$
$761$	30.5022	1.10570	0.552852	+	0.833280i	$0.313540\pi$
$762$	0	0
$763$	62.5151	2.26320
$764$	0	0
$765$	4.54678	0.164389
$766$	0	0
$767$	5.96055	0.215223
$768$	0	0
$769$	−41.2690	−1.48820	−0.744099	−	0.668069i	$-0.767121\pi$
$769$	−41.2690	−1.48820	−0.744099	+	0.668069i	$0.767121\pi$
$770$	0	0
$771$	−21.7655	−0.783864
$772$	0	0
$773$	−2.47382	−0.0889770	−0.0444885	−	0.999010i	$-0.514166\pi$
$773$	−2.47382	−0.0889770	−0.0444885	+	0.999010i	$0.514166\pi$
$774$	0	0
$775$	−20.1183	−0.722671
$776$	0	0
$777$	−13.9699	−0.501167
$778$	0	0
$779$	27.8794	0.998882
$780$	0	0
$781$	−5.22352	−0.186912
$782$	0	0
$783$	1.57232	0.0561901
$784$	0	0
$785$	54.5089	1.94551
$786$	0	0
$787$	54.3174	1.93621	0.968104	−	0.250550i	$-0.0806113\pi$
$787$	54.3174	1.93621	0.968104	+	0.250550i	$0.0806113\pi$
$788$	0	0
$789$	−16.7893	−0.597714
$790$	0	0
$791$	−3.47395	−0.123520
$792$	0	0
$793$	−13.3449	−0.473892
$794$	0	0
$795$	18.1536	0.643842
$796$	0	0
$797$	2.89882	0.102681	0.0513407	−	0.998681i	$-0.483651\pi$
$797$	2.89882	0.102681	0.0513407	+	0.998681i	$0.483651\pi$
$798$	0	0
$799$	−22.8358	−0.807873
$800$	0	0
$801$	1.65330	0.0584163
$802$	0	0
$803$	−1.41358	−0.0498841
$804$	0	0
$805$	72.6137	2.55930
$806$	0	0
$807$	−10.0455	−0.353619
$808$	0	0
$809$	24.3146	0.854854	0.427427	−	0.904050i	$-0.359420\pi$
$809$	24.3146	0.854854	0.427427	+	0.904050i	$0.359420\pi$
$810$	0	0
$811$	−2.28570	−0.0802619	−0.0401310	−	0.999194i	$-0.512778\pi$
$811$	−2.28570	−0.0802619	−0.0401310	+	0.999194i	$0.512778\pi$
$812$	0	0
$813$	10.1464	0.355851
$814$	0	0
$815$	46.0432	1.61282
$816$	0	0
$817$	3.53726	0.123753
$818$	0	0
$819$	−15.7729	−0.551149
$820$	0	0
$821$	29.6473	1.03470	0.517349	−	0.855775i	$-0.326919\pi$
$821$	29.6473	1.03470	0.517349	+	0.855775i	$0.326919\pi$
$822$	0	0
$823$	−46.8337	−1.63252	−0.816260	−	0.577685i	$-0.803956\pi$
$823$	−46.8337	−1.63252	−0.816260	+	0.577685i	$0.803956\pi$
$824$	0	0
$825$	−1.95823	−0.0681769
$826$	0	0
$827$	3.49088	0.121390	0.0606949	−	0.998156i	$-0.480668\pi$
$827$	3.49088	0.121390	0.0606949	+	0.998156i	$0.480668\pi$
$828$	0	0
$829$	−46.2312	−1.60568	−0.802838	−	0.596197i	$-0.796678\pi$
$829$	−46.2312	−1.60568	−0.802838	+	0.596197i	$0.796678\pi$
$830$	0	0
$831$	23.0023	0.797942
$832$	0	0
$833$	28.2512	0.978845
$834$	0	0
$835$	4.27935	0.148093
$836$	0	0
$837$	−10.1788	−0.351830
$838$	0	0
$839$	2.18809	0.0755411	0.0377706	−	0.999286i	$-0.487974\pi$
$839$	2.18809	0.0755411	0.0377706	+	0.999286i	$0.487974\pi$
$840$	0	0
$841$	−26.5278	−0.914752
$842$	0	0
$843$	−9.91685	−0.341555
$844$	0	0
$845$	−6.26910	−0.215664
$846$	0	0
$847$	−48.4745	−1.66560
$848$	0	0
$849$	−26.4769	−0.908686
$850$	0	0
$851$	−16.4045	−0.562337
$852$	0	0
$853$	45.0595	1.54281	0.771403	−	0.636346i	$-0.219555\pi$
$853$	45.0595	1.54281	0.771403	+	0.636346i	$0.219555\pi$
$854$	0	0
$855$	9.53290	0.326018
$856$	0	0
$857$	−18.3076	−0.625375	−0.312687	−	0.949856i	$-0.601229\pi$
$857$	−18.3076	−0.625375	−0.312687	+	0.949856i	$0.601229\pi$
$858$	0	0
$859$	5.59941	0.191049	0.0955247	−	0.995427i	$-0.469547\pi$
$859$	5.59941	0.191049	0.0955247	+	0.995427i	$0.469547\pi$
$860$	0	0
$861$	37.3759	1.27377
$862$	0	0
$863$	−1.57829	−0.0537256	−0.0268628	−	0.999639i	$-0.508552\pi$
$863$	−1.57829	−0.0537256	−0.0268628	+	0.999639i	$0.508552\pi$
$864$	0	0
$865$	−13.4873	−0.458582
$866$	0	0
$867$	−14.0367	−0.476712
$868$	0	0
$869$	−16.4526	−0.558117
$870$	0	0
$871$	7.72893	0.261885
$872$	0	0
$873$	−2.90278	−0.0982443
$874$	0	0
$875$	−38.6407	−1.30629
$876$	0	0
$877$	−45.9144	−1.55042	−0.775210	−	0.631703i	$-0.782356\pi$
$877$	−45.9144	−1.55042	−0.775210	+	0.631703i	$0.782356\pi$
$878$	0	0
$879$	2.13762	0.0721000
$880$	0	0
$881$	−2.11541	−0.0712699	−0.0356349	−	0.999365i	$-0.511345\pi$
$881$	−2.11541	−0.0712699	−0.0356349	+	0.999365i	$0.511345\pi$
$882$	0	0
$883$	17.5668	0.591170	0.295585	−	0.955316i	$-0.404485\pi$
$883$	17.5668	0.591170	0.295585	+	0.955316i	$0.404485\pi$
$884$	0	0
$885$	−4.82958	−0.162345
$886$	0	0
$887$	−34.9466	−1.17339	−0.586696	−	0.809808i	$-0.699571\pi$
$887$	−34.9466	−1.17339	−0.586696	+	0.809808i	$0.699571\pi$
$888$	0	0
$889$	−25.6692	−0.860919
$890$	0	0
$891$	−0.990762	−0.0331918
$892$	0	0
$893$	−47.8782	−1.60218
$894$	0	0
$895$	−25.3685	−0.847975
$896$	0	0
$897$	−18.5216	−0.618420
$898$	0	0
$899$	−16.0043	−0.533773
$900$	0	0
$901$	11.8312	0.394155
$902$	0	0
$903$	4.74216	0.157809
$904$	0	0
$905$	18.7381	0.622875
$906$	0	0
$907$	−11.4977	−0.381775	−0.190888	−	0.981612i	$-0.561137\pi$
$907$	−11.4977	−0.381775	−0.190888	+	0.981612i	$0.561137\pi$
$908$	0	0
$909$	−8.60064	−0.285265
$910$	0	0
$911$	−40.3621	−1.33726	−0.668629	−	0.743596i	$-0.733118\pi$
$911$	−40.3621	−1.33726	−0.668629	+	0.743596i	$0.733118\pi$
$912$	0	0
$913$	3.20542	0.106084
$914$	0	0
$915$	10.8128	0.357461
$916$	0	0
$917$	45.8148	1.51294
$918$	0	0
$919$	57.8470	1.90820	0.954098	−	0.299494i	$-0.0968177\pi$
$919$	57.8470	1.90820	0.954098	+	0.299494i	$0.0968177\pi$
$920$	0	0
$921$	1.26623	0.0417236
$922$	0	0
$923$	−17.1866	−0.565703
$924$	0	0
$925$	−5.70654	−0.187630
$926$	0	0
$927$	0.484047	0.0158982
$928$	0	0
$929$	14.8290	0.486525	0.243262	−	0.969961i	$-0.421782\pi$
$929$	14.8290	0.486525	0.243262	+	0.969961i	$0.421782\pi$
$930$	0	0
$931$	59.2322	1.94126
$932$	0	0
$933$	12.9899	0.425270
$934$	0	0
$935$	−4.50478	−0.147322
$936$	0	0
$937$	−38.8030	−1.26764	−0.633820	−	0.773480i	$-0.718514\pi$
$937$	−38.8030	−1.26764	−0.633820	+	0.773480i	$0.718514\pi$
$938$	0	0
$939$	−28.2727	−0.922646
$940$	0	0
$941$	−8.97474	−0.292568	−0.146284	−	0.989243i	$-0.546731\pi$
$941$	−8.97474	−0.292568	−0.146284	+	0.989243i	$0.546731\pi$
$942$	0	0
$943$	43.8895	1.42924
$944$	0	0
$945$	12.7801	0.415737
$946$	0	0
$947$	17.9383	0.582917	0.291458	−	0.956584i	$-0.405860\pi$
$947$	17.9383	0.582917	0.291458	+	0.956584i	$0.405860\pi$
$948$	0	0
$949$	−4.65100	−0.150978
$950$	0	0
$951$	−32.6601	−1.05908
$952$	0	0
$953$	−45.6193	−1.47775	−0.738877	−	0.673841i	$-0.764643\pi$
$953$	−45.6193	−1.47775	−0.738877	+	0.673841i	$0.764643\pi$
$954$	0	0
$955$	−19.8875	−0.643544
$956$	0	0
$957$	−1.55779	−0.0503563
$958$	0	0
$959$	−103.410	−3.33929
$960$	0	0
$961$	72.6077	2.34218
$962$	0	0
$963$	7.92863	0.255497
$964$	0	0
$965$	−9.64932	−0.310623
$966$	0	0
$967$	−4.43370	−0.142578	−0.0712891	−	0.997456i	$-0.522711\pi$
$967$	−4.43370	−0.142578	−0.0712891	+	0.997456i	$0.522711\pi$
$968$	0	0
$969$	6.21287	0.199586
$970$	0	0
$971$	−13.4554	−0.431803	−0.215901	−	0.976415i	$-0.569269\pi$
$971$	−13.4554	−0.431803	−0.215901	+	0.976415i	$0.569269\pi$
$972$	0	0
$973$	40.1433	1.28693
$974$	0	0
$975$	−6.44304	−0.206342
$976$	0	0
$977$	−20.3818	−0.652071	−0.326036	−	0.945357i	$-0.605713\pi$
$977$	−20.3818	−0.652071	−0.326036	+	0.945357i	$0.605713\pi$
$978$	0	0
$979$	−1.63802	−0.0523514
$980$	0	0
$981$	12.9202	0.412510
$982$	0	0
$983$	23.6155	0.753218	0.376609	−	0.926372i	$-0.377090\pi$
$983$	23.6155	0.753218	0.376609	+	0.926372i	$0.377090\pi$
$984$	0	0
$985$	−65.5074	−2.08724
$986$	0	0
$987$	−64.1870	−2.04309
$988$	0	0
$989$	5.56858	0.177071
$990$	0	0
$991$	27.6416	0.878064	0.439032	−	0.898471i	$-0.355321\pi$
$991$	27.6416	0.878064	0.439032	+	0.898471i	$0.355321\pi$
$992$	0	0
$993$	−18.3361	−0.581878
$994$	0	0
$995$	−43.1084	−1.36663
$996$	0	0
$997$	1.33170	0.0421755	0.0210877	−	0.999778i	$-0.493287\pi$
$997$	1.33170	0.0421755	0.0210877	+	0.999778i	$0.493287\pi$
$998$	0	0
$999$	−2.88721	−0.0913471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.q.1.15	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.q.1.15	✓	18	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.64131 q^{5} +4.83855 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.64131 q^{5} +4.83855 q^{7} +1.00000 q^{9} -0.990762 q^{11} -3.25983 q^{13} +2.64131 q^{15} +1.72141 q^{17} +3.60916 q^{19} +4.83855 q^{21} +5.68178 q^{23} +1.97649 q^{25} +1.00000 q^{27} +1.57232 q^{29} -10.1788 q^{31} -0.990762 q^{33} +12.7801 q^{35} -2.88721 q^{37} -3.25983 q^{39} +7.72461 q^{41} +0.980077 q^{43} +2.64131 q^{45} -13.2657 q^{47} +16.4116 q^{49} +1.72141 q^{51} +6.87297 q^{53} -2.61690 q^{55} +3.60916 q^{57} -1.82848 q^{59} +4.09375 q^{61} +4.83855 q^{63} -8.61022 q^{65} -2.37096 q^{67} +5.68178 q^{69} +5.27223 q^{71} +1.42676 q^{73} +1.97649 q^{75} -4.79386 q^{77} +16.6060 q^{79} +1.00000 q^{81} -3.23531 q^{83} +4.54678 q^{85} +1.57232 q^{87} +1.65330 q^{89} -15.7729 q^{91} -10.1788 q^{93} +9.53290 q^{95} -2.90278 q^{97} -0.990762 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9} + 8 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} + 21 q^{19} + 7 q^{21} - 2 q^{23} + 25 q^{25} + 18 q^{27} + 5 q^{29} + 23 q^{31} + 8 q^{33} + 17 q^{35} + 15 q^{37} + 2 q^{39} + 20 q^{41} + 37 q^{43} + q^{45} - q^{47} + 33 q^{49} + 2 q^{51} + 2 q^{53} + 26 q^{55} + 21 q^{57} + 24 q^{59} + 20 q^{61} + 7 q^{63} + 26 q^{65} + 49 q^{67} - 2 q^{69} + 15 q^{71} + 15 q^{73} + 25 q^{75} + 6 q^{77} + 23 q^{79} + 18 q^{81} + 38 q^{83} + 21 q^{85} + 5 q^{87} + 31 q^{89} + 48 q^{91} + 23 q^{93} + 13 q^{95} + 25 q^{97} + 8 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9 + 8 * q^11 + 2 * q^13 + q^15 + 2 * q^17 + 21 * q^19 + 7 * q^21 - 2 * q^23 + 25 * q^25 + 18 * q^27 + 5 * q^29 + 23 * q^31 + 8 * q^33 + 17 * q^35 + 15 * q^37 + 2 * q^39 + 20 * q^41 + 37 * q^43 + q^45 - q^47 + 33 * q^49 + 2 * q^51 + 2 * q^53 + 26 * q^55 + 21 * q^57 + 24 * q^59 + 20 * q^61 + 7 * q^63 + 26 * q^65 + 49 * q^67 - 2 * q^69 + 15 * q^71 + 15 * q^73 + 25 * q^75 + 6 * q^77 + 23 * q^79 + 18 * q^81 + 38 * q^83 + 21 * q^85 + 5 * q^87 + 31 * q^89 + 48 * q^91 + 23 * q^93 + 13 * q^95 + 25 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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