LMFDB - Embedded newform 6024.2.a.r.1.17

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 9 x^{19} - 31 x^{18} + 471 x^{17} - 82 x^{16} - 9476 x^{15} + 12881 x^{14} + 94079 x^{13} + \cdots + 24832 $ x^20 - 9x^19 - 31x^18 + 471x^17 - 82x^16 - 9476x^15 + 12881x^14 + 94079x^13 - 181272x^12 - 504914x^11 + 1108012x^10 + 1591114x^9 - 3399127x^8 - 3224089x^7 + 5089722x^6 + 4287384x^5 - 2840216x^4 - 2820864x^3 - 208576x^2 + 204992*x + 24832
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$3.18186$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.18186 q^{5} +2.54652 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.18186 q^{5} +2.54652 q^{7} +1.00000 q^{9} -1.62213 q^{11} +4.61755 q^{13} +3.18186 q^{15} +1.01883 q^{17} -3.19172 q^{19} +2.54652 q^{21} -8.05733 q^{23} +5.12424 q^{25} +1.00000 q^{27} +6.44948 q^{29} -3.09823 q^{31} -1.62213 q^{33} +8.10268 q^{35} +10.9412 q^{37} +4.61755 q^{39} +0.892781 q^{41} +4.16844 q^{43} +3.18186 q^{45} -8.40522 q^{47} -0.515219 q^{49} +1.01883 q^{51} -2.66438 q^{53} -5.16139 q^{55} -3.19172 q^{57} +8.88643 q^{59} +9.85906 q^{61} +2.54652 q^{63} +14.6924 q^{65} +3.55934 q^{67} -8.05733 q^{69} -10.2193 q^{71} -2.54079 q^{73} +5.12424 q^{75} -4.13079 q^{77} +9.05853 q^{79} +1.00000 q^{81} +17.8724 q^{83} +3.24176 q^{85} +6.44948 q^{87} -9.63901 q^{89} +11.7587 q^{91} -3.09823 q^{93} -10.1556 q^{95} -2.93524 q^{97} -1.62213 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) $ 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9	$ 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9} + 4 q^{11} + 21 q^{13} + 9 q^{15} + 10 q^{17} + 8 q^{19} + 9 q^{21} + 9 q^{23} + 43 q^{25} + 20 q^{27} + 18 q^{29} + 27 q^{31} + 4 q^{33} - 7 q^{35} + 33 q^{37} + 21 q^{39} + 14 q^{41} - 6 q^{43} + 9 q^{45} + 21 q^{47} + 47 q^{49} + 10 q^{51} + 23 q^{53} + 24 q^{55} + 8 q^{57} + 10 q^{59} + 28 q^{61} + 9 q^{63} + 2 q^{65} + 15 q^{67} + 9 q^{69} + 33 q^{71} + 50 q^{73} + 43 q^{75} + 20 q^{77} + 17 q^{79} + 20 q^{81} - 19 q^{83} + 41 q^{85} + 18 q^{87} + 21 q^{89} + 30 q^{91} + 27 q^{93} + 27 q^{95} + 47 q^{97} + 4 q^{99}+O(q^{100}) $ 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9 + 4 * q^11 + 21 * q^13 + 9 * q^15 + 10 * q^17 + 8 * q^19 + 9 * q^21 + 9 * q^23 + 43 * q^25 + 20 * q^27 + 18 * q^29 + 27 * q^31 + 4 * q^33 - 7 * q^35 + 33 * q^37 + 21 * q^39 + 14 * q^41 - 6 * q^43 + 9 * q^45 + 21 * q^47 + 47 * q^49 + 10 * q^51 + 23 * q^53 + 24 * q^55 + 8 * q^57 + 10 * q^59 + 28 * q^61 + 9 * q^63 + 2 * q^65 + 15 * q^67 + 9 * q^69 + 33 * q^71 + 50 * q^73 + 43 * q^75 + 20 * q^77 + 17 * q^79 + 20 * q^81 - 19 * q^83 + 41 * q^85 + 18 * q^87 + 21 * q^89 + 30 * q^91 + 27 * q^93 + 27 * q^95 + 47 * q^97 + 4 * 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
$5$	3.18186	1.42297	0.711486	−	0.702701i	$-0.248023\pi$
$5$	3.18186	1.42297	0.711486	+	0.702701i	$0.248023\pi$
$6$	0	0
$7$	2.54652	0.962495	0.481248	−	0.876585i	$-0.340184\pi$
$7$	2.54652	0.962495	0.481248	+	0.876585i	$0.340184\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.62213	−0.489090	−0.244545	−	0.969638i	$-0.578639\pi$
$11$	−1.62213	−0.489090	−0.244545	+	0.969638i	$0.578639\pi$
$12$	0	0
$13$	4.61755	1.28068	0.640339	−	0.768092i	$-0.278794\pi$
$13$	4.61755	1.28068	0.640339	+	0.768092i	$0.278794\pi$
$14$	0	0
$15$	3.18186	0.821553
$16$	0	0
$17$	1.01883	0.247101	0.123551	−	0.992338i	$-0.460572\pi$
$17$	1.01883	0.247101	0.123551	+	0.992338i	$0.460572\pi$
$18$	0	0
$19$	−3.19172	−0.732230	−0.366115	−	0.930570i	$-0.619312\pi$
$19$	−3.19172	−0.732230	−0.366115	+	0.930570i	$0.619312\pi$
$20$	0	0
$21$	2.54652	0.555697
$22$	0	0
$23$	−8.05733	−1.68007	−0.840035	−	0.542532i	$-0.817466\pi$
$23$	−8.05733	−1.68007	−0.840035	+	0.542532i	$0.817466\pi$
$24$	0	0
$25$	5.12424	1.02485
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.44948	1.19764	0.598819	−	0.800884i	$-0.295637\pi$
$29$	6.44948	1.19764	0.598819	+	0.800884i	$0.295637\pi$
$30$	0	0
$31$	−3.09823	−0.556459	−0.278229	−	0.960515i	$-0.589748\pi$
$31$	−3.09823	−0.556459	−0.278229	+	0.960515i	$0.589748\pi$
$32$	0	0
$33$	−1.62213	−0.282376
$34$	0	0
$35$	8.10268	1.36960
$36$	0	0
$37$	10.9412	1.79873	0.899365	−	0.437198i	$-0.144029\pi$
$37$	10.9412	1.79873	0.899365	+	0.437198i	$0.144029\pi$
$38$	0	0
$39$	4.61755	0.739400
$40$	0	0
$41$	0.892781	0.139429	0.0697145	−	0.997567i	$-0.477791\pi$
$41$	0.892781	0.139429	0.0697145	+	0.997567i	$0.477791\pi$
$42$	0	0
$43$	4.16844	0.635680	0.317840	−	0.948144i	$-0.397042\pi$
$43$	4.16844	0.635680	0.317840	+	0.948144i	$0.397042\pi$
$44$	0	0
$45$	3.18186	0.474324
$46$	0	0
$47$	−8.40522	−1.22603	−0.613014	−	0.790072i	$-0.710043\pi$
$47$	−8.40522	−1.22603	−0.613014	+	0.790072i	$0.710043\pi$
$48$	0	0
$49$	−0.515219	−0.0736027
$50$	0	0
$51$	1.01883	0.142664
$52$	0	0
$53$	−2.66438	−0.365981	−0.182990	−	0.983115i	$-0.558578\pi$
$53$	−2.66438	−0.365981	−0.182990	+	0.983115i	$0.558578\pi$
$54$	0	0
$55$	−5.16139	−0.695962
$56$	0	0
$57$	−3.19172	−0.422753
$58$	0	0
$59$	8.88643	1.15691	0.578457	−	0.815713i	$-0.303655\pi$
$59$	8.88643	1.15691	0.578457	+	0.815713i	$0.303655\pi$
$60$	0	0
$61$	9.85906	1.26232	0.631162	−	0.775651i	$-0.282578\pi$
$61$	9.85906	1.26232	0.631162	+	0.775651i	$0.282578\pi$
$62$	0	0
$63$	2.54652	0.320832
$64$	0	0
$65$	14.6924	1.82237
$66$	0	0
$67$	3.55934	0.434842	0.217421	−	0.976078i	$-0.430236\pi$
$67$	3.55934	0.434842	0.217421	+	0.976078i	$0.430236\pi$
$68$	0	0
$69$	−8.05733	−0.969989
$70$	0	0
$71$	−10.2193	−1.21281	−0.606403	−	0.795158i	$-0.707388\pi$
$71$	−10.2193	−1.21281	−0.606403	+	0.795158i	$0.707388\pi$
$72$	0	0
$73$	−2.54079	−0.297377	−0.148688	−	0.988884i	$-0.547505\pi$
$73$	−2.54079	−0.297377	−0.148688	+	0.988884i	$0.547505\pi$
$74$	0	0
$75$	5.12424	0.591696
$76$	0	0
$77$	−4.13079	−0.470747
$78$	0	0
$79$	9.05853	1.01916	0.509582	−	0.860422i	$-0.329800\pi$
$79$	9.05853	1.01916	0.509582	+	0.860422i	$0.329800\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	17.8724	1.96175	0.980877	−	0.194627i	$-0.0623497\pi$
$83$	17.8724	1.96175	0.980877	+	0.194627i	$0.0623497\pi$
$84$	0	0
$85$	3.24176	0.351618
$86$	0	0
$87$	6.44948	0.691457
$88$	0	0
$89$	−9.63901	−1.02173	−0.510866	−	0.859660i	$-0.670675\pi$
$89$	−9.63901	−1.02173	−0.510866	+	0.859660i	$0.670675\pi$
$90$	0	0
$91$	11.7587	1.23265
$92$	0	0
$93$	−3.09823	−0.321272
$94$	0	0
$95$	−10.1556	−1.04194
$96$	0	0
$97$	−2.93524	−0.298029	−0.149014	−	0.988835i	$-0.547610\pi$
$97$	−2.93524	−0.298029	−0.149014	+	0.988835i	$0.547610\pi$
$98$	0	0
$99$	−1.62213	−0.163030
$100$	0	0
$101$	16.3984	1.63170	0.815852	−	0.578261i	$-0.196268\pi$
$101$	16.3984	1.63170	0.815852	+	0.578261i	$0.196268\pi$
$102$	0	0
$103$	12.9427	1.27528	0.637639	−	0.770335i	$-0.279911\pi$
$103$	12.9427	1.27528	0.637639	+	0.770335i	$0.279911\pi$
$104$	0	0
$105$	8.10268	0.790741
$106$	0	0
$107$	0.440004	0.0425368	0.0212684	−	0.999774i	$-0.493230\pi$
$107$	0.440004	0.0425368	0.0212684	+	0.999774i	$0.493230\pi$
$108$	0	0
$109$	0.580594	0.0556108	0.0278054	−	0.999613i	$-0.491148\pi$
$109$	0.580594	0.0556108	0.0278054	+	0.999613i	$0.491148\pi$
$110$	0	0
$111$	10.9412	1.03850
$112$	0	0
$113$	−4.67348	−0.439644	−0.219822	−	0.975540i	$-0.570548\pi$
$113$	−4.67348	−0.439644	−0.219822	+	0.975540i	$0.570548\pi$
$114$	0	0
$115$	−25.6373	−2.39069
$116$	0	0
$117$	4.61755	0.426893
$118$	0	0
$119$	2.59446	0.237834
$120$	0	0
$121$	−8.36870	−0.760791
$122$	0	0
$123$	0.892781	0.0804994
$124$	0	0
$125$	0.395304	0.0353571
$126$	0	0
$127$	−10.9773	−0.974080	−0.487040	−	0.873380i	$-0.661923\pi$
$127$	−10.9773	−0.974080	−0.487040	+	0.873380i	$0.661923\pi$
$128$	0	0
$129$	4.16844	0.367010
$130$	0	0
$131$	−0.204993	−0.0179103	−0.00895514	−	0.999960i	$-0.502851\pi$
$131$	−0.204993	−0.0179103	−0.00895514	+	0.999960i	$0.502851\pi$
$132$	0	0
$133$	−8.12778	−0.704768
$134$	0	0
$135$	3.18186	0.273851
$136$	0	0
$137$	10.5791	0.903832	0.451916	−	0.892060i	$-0.350741\pi$
$137$	10.5791	0.903832	0.451916	+	0.892060i	$0.350741\pi$
$138$	0	0
$139$	−5.38787	−0.456993	−0.228496	−	0.973545i	$-0.573381\pi$
$139$	−5.38787	−0.456993	−0.228496	+	0.973545i	$0.573381\pi$
$140$	0	0
$141$	−8.40522	−0.707847
$142$	0	0
$143$	−7.49027	−0.626367
$144$	0	0
$145$	20.5213	1.70421
$146$	0	0
$147$	−0.515219	−0.0424945
$148$	0	0
$149$	−1.16765	−0.0956576	−0.0478288	−	0.998856i	$-0.515230\pi$
$149$	−1.16765	−0.0956576	−0.0478288	+	0.998856i	$0.515230\pi$
$150$	0	0
$151$	12.7530	1.03782	0.518912	−	0.854828i	$-0.326337\pi$
$151$	12.7530	1.03782	0.518912	+	0.854828i	$0.326337\pi$
$152$	0	0
$153$	1.01883	0.0823672
$154$	0	0
$155$	−9.85814	−0.791825
$156$	0	0
$157$	−6.86775	−0.548106	−0.274053	−	0.961715i	$-0.588364\pi$
$157$	−6.86775	−0.548106	−0.274053	+	0.961715i	$0.588364\pi$
$158$	0	0
$159$	−2.66438	−0.211299
$160$	0	0
$161$	−20.5182	−1.61706
$162$	0	0
$163$	4.15118	0.325145	0.162573	−	0.986697i	$-0.448021\pi$
$163$	4.15118	0.325145	0.162573	+	0.986697i	$0.448021\pi$
$164$	0	0
$165$	−5.16139	−0.401814
$166$	0	0
$167$	−16.3817	−1.26765	−0.633827	−	0.773475i	$-0.718517\pi$
$167$	−16.3817	−1.26765	−0.633827	+	0.773475i	$0.718517\pi$
$168$	0	0
$169$	8.32179	0.640137
$170$	0	0
$171$	−3.19172	−0.244077
$172$	0	0
$173$	−16.5984	−1.26196	−0.630978	−	0.775801i	$-0.717346\pi$
$173$	−16.5984	−1.26196	−0.630978	+	0.775801i	$0.717346\pi$
$174$	0	0
$175$	13.0490	0.986411
$176$	0	0
$177$	8.88643	0.667945
$178$	0	0
$179$	−13.9511	−1.04276	−0.521378	−	0.853326i	$-0.674582\pi$
$179$	−13.9511	−1.04276	−0.521378	+	0.853326i	$0.674582\pi$
$180$	0	0
$181$	−18.9274	−1.40686	−0.703432	−	0.710762i	$-0.748350\pi$
$181$	−18.9274	−1.40686	−0.703432	+	0.710762i	$0.748350\pi$
$182$	0	0
$183$	9.85906	0.728803
$184$	0	0
$185$	34.8135	2.55954
$186$	0	0
$187$	−1.65267	−0.120855
$188$	0	0
$189$	2.54652	0.185232
$190$	0	0
$191$	−9.81038	−0.709854	−0.354927	−	0.934894i	$-0.615494\pi$
$191$	−9.81038	−0.709854	−0.354927	+	0.934894i	$0.615494\pi$
$192$	0	0
$193$	−2.85083	−0.205207	−0.102603	−	0.994722i	$-0.532717\pi$
$193$	−2.85083	−0.205207	−0.102603	+	0.994722i	$0.532717\pi$
$194$	0	0
$195$	14.6924	1.05215
$196$	0	0
$197$	−9.00576	−0.641634	−0.320817	−	0.947141i	$-0.603957\pi$
$197$	−9.00576	−0.641634	−0.320817	+	0.947141i	$0.603957\pi$
$198$	0	0
$199$	−12.8945	−0.914068	−0.457034	−	0.889449i	$-0.651088\pi$
$199$	−12.8945	−0.914068	−0.457034	+	0.889449i	$0.651088\pi$
$200$	0	0
$201$	3.55934	0.251056
$202$	0	0
$203$	16.4238	1.15272
$204$	0	0
$205$	2.84071	0.198403
$206$	0	0
$207$	−8.05733	−0.560023
$208$	0	0
$209$	5.17738	0.358127
$210$	0	0
$211$	27.9805	1.92626	0.963128	−	0.269042i	$-0.0867071\pi$
$211$	27.9805	1.92626	0.963128	+	0.269042i	$0.0867071\pi$
$212$	0	0
$213$	−10.2193	−0.700214
$214$	0	0
$215$	13.2634	0.904555
$216$	0	0
$217$	−7.88972	−0.535589
$218$	0	0
$219$	−2.54079	−0.171691
$220$	0	0
$221$	4.70448	0.316458
$222$	0	0
$223$	−17.9010	−1.19874	−0.599371	−	0.800471i	$-0.704583\pi$
$223$	−17.9010	−1.19874	−0.599371	+	0.800471i	$0.704583\pi$
$224$	0	0
$225$	5.12424	0.341616
$226$	0	0
$227$	19.0625	1.26523	0.632613	−	0.774468i	$-0.281983\pi$
$227$	19.0625	1.26523	0.632613	+	0.774468i	$0.281983\pi$
$228$	0	0
$229$	−23.8183	−1.57395	−0.786977	−	0.616982i	$-0.788355\pi$
$229$	−23.8183	−1.57395	−0.786977	+	0.616982i	$0.788355\pi$
$230$	0	0
$231$	−4.13079	−0.271786
$232$	0	0
$233$	20.0140	1.31116	0.655580	−	0.755126i	$-0.272424\pi$
$233$	20.0140	1.31116	0.655580	+	0.755126i	$0.272424\pi$
$234$	0	0
$235$	−26.7442	−1.74460
$236$	0	0
$237$	9.05853	0.588415
$238$	0	0
$239$	12.1391	0.785212	0.392606	−	0.919707i	$-0.371574\pi$
$239$	12.1391	0.785212	0.392606	+	0.919707i	$0.371574\pi$
$240$	0	0
$241$	8.19115	0.527639	0.263819	−	0.964572i	$-0.415018\pi$
$241$	8.19115	0.527639	0.263819	+	0.964572i	$0.415018\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.63935	−0.104735
$246$	0	0
$247$	−14.7379	−0.937751
$248$	0	0
$249$	17.8724	1.13262
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	13.0700	0.821706
$254$	0	0
$255$	3.24176	0.203007
$256$	0	0
$257$	31.2209	1.94751	0.973753	−	0.227608i	$-0.0730904\pi$
$257$	31.2209	1.94751	0.973753	+	0.227608i	$0.0730904\pi$
$258$	0	0
$259$	27.8621	1.73127
$260$	0	0
$261$	6.44948	0.399213
$262$	0	0
$263$	−29.6076	−1.82568	−0.912842	−	0.408314i	$-0.866117\pi$
$263$	−29.6076	−1.82568	−0.912842	+	0.408314i	$0.866117\pi$
$264$	0	0
$265$	−8.47769	−0.520780
$266$	0	0
$267$	−9.63901	−0.589898
$268$	0	0
$269$	5.09323	0.310539	0.155270	−	0.987872i	$-0.450375\pi$
$269$	5.09323	0.310539	0.155270	+	0.987872i	$0.450375\pi$
$270$	0	0
$271$	3.24681	0.197230	0.0986149	−	0.995126i	$-0.468559\pi$
$271$	3.24681	0.197230	0.0986149	+	0.995126i	$0.468559\pi$
$272$	0	0
$273$	11.7587	0.711669
$274$	0	0
$275$	−8.31217	−0.501243
$276$	0	0
$277$	5.54366	0.333087	0.166543	−	0.986034i	$-0.446739\pi$
$277$	5.54366	0.333087	0.166543	+	0.986034i	$0.446739\pi$
$278$	0	0
$279$	−3.09823	−0.185486
$280$	0	0
$281$	−30.6076	−1.82589	−0.912947	−	0.408078i	$-0.866199\pi$
$281$	−30.6076	−1.82589	−0.912947	+	0.408078i	$0.866199\pi$
$282$	0	0
$283$	13.2155	0.785578	0.392789	−	0.919629i	$-0.371510\pi$
$283$	13.2155	0.785578	0.392789	+	0.919629i	$0.371510\pi$
$284$	0	0
$285$	−10.1556	−0.601566
$286$	0	0
$287$	2.27349	0.134200
$288$	0	0
$289$	−15.9620	−0.938941
$290$	0	0
$291$	−2.93524	−0.172067
$292$	0	0
$293$	22.3977	1.30849	0.654244	−	0.756283i	$-0.272987\pi$
$293$	22.3977	1.30849	0.654244	+	0.756283i	$0.272987\pi$
$294$	0	0
$295$	28.2754	1.64626
$296$	0	0
$297$	−1.62213	−0.0941255
$298$	0	0
$299$	−37.2052	−2.15163
$300$	0	0
$301$	10.6150	0.611839
$302$	0	0
$303$	16.3984	0.942065
$304$	0	0
$305$	31.3702	1.79625
$306$	0	0
$307$	−21.4179	−1.22238	−0.611192	−	0.791482i	$-0.709310\pi$
$307$	−21.4179	−1.22238	−0.611192	+	0.791482i	$0.709310\pi$
$308$	0	0
$309$	12.9427	0.736282
$310$	0	0
$311$	−9.68607	−0.549246	−0.274623	−	0.961552i	$-0.588553\pi$
$311$	−9.68607	−0.549246	−0.274623	+	0.961552i	$0.588553\pi$
$312$	0	0
$313$	25.2893	1.42944	0.714719	−	0.699412i	$-0.246555\pi$
$313$	25.2893	1.42944	0.714719	+	0.699412i	$0.246555\pi$
$314$	0	0
$315$	8.10268	0.456534
$316$	0	0
$317$	−24.9181	−1.39954	−0.699769	−	0.714369i	$-0.746714\pi$
$317$	−24.9181	−1.39954	−0.699769	+	0.714369i	$0.746714\pi$
$318$	0	0
$319$	−10.4619	−0.585753
$320$	0	0
$321$	0.440004	0.0245586
$322$	0	0
$323$	−3.25180	−0.180935
$324$	0	0
$325$	23.6614	1.31250
$326$	0	0
$327$	0.580594	0.0321069
$328$	0	0
$329$	−21.4041	−1.18005
$330$	0	0
$331$	11.5728	0.636099	0.318050	−	0.948074i	$-0.396972\pi$
$331$	11.5728	0.636099	0.318050	+	0.948074i	$0.396972\pi$
$332$	0	0
$333$	10.9412	0.599577
$334$	0	0
$335$	11.3253	0.618768
$336$	0	0
$337$	−0.177114	−0.00964802	−0.00482401	−	0.999988i	$-0.501536\pi$
$337$	−0.177114	−0.00964802	−0.00482401	+	0.999988i	$0.501536\pi$
$338$	0	0
$339$	−4.67348	−0.253829
$340$	0	0
$341$	5.02573	0.272159
$342$	0	0
$343$	−19.1377	−1.03334
$344$	0	0
$345$	−25.6373	−1.38027
$346$	0	0
$347$	−15.6159	−0.838308	−0.419154	−	0.907915i	$-0.637673\pi$
$347$	−15.6159	−0.838308	−0.419154	+	0.907915i	$0.637673\pi$
$348$	0	0
$349$	25.5794	1.36923	0.684617	−	0.728903i	$-0.259969\pi$
$349$	25.5794	1.36923	0.684617	+	0.728903i	$0.259969\pi$
$350$	0	0
$351$	4.61755	0.246467
$352$	0	0
$353$	−26.0040	−1.38405	−0.692025	−	0.721873i	$-0.743281\pi$
$353$	−26.0040	−1.38405	−0.692025	+	0.721873i	$0.743281\pi$
$354$	0	0
$355$	−32.5163	−1.72579
$356$	0	0
$357$	2.59446	0.137314
$358$	0	0
$359$	−11.4722	−0.605478	−0.302739	−	0.953074i	$-0.597901\pi$
$359$	−11.4722	−0.605478	−0.302739	+	0.953074i	$0.597901\pi$
$360$	0	0
$361$	−8.81295	−0.463839
$362$	0	0
$363$	−8.36870	−0.439243
$364$	0	0
$365$	−8.08444	−0.423159
$366$	0	0
$367$	−7.02999	−0.366962	−0.183481	−	0.983023i	$-0.558737\pi$
$367$	−7.02999	−0.366962	−0.183481	+	0.983023i	$0.558737\pi$
$368$	0	0
$369$	0.892781	0.0464763
$370$	0	0
$371$	−6.78491	−0.352255
$372$	0	0
$373$	−25.9323	−1.34273	−0.671363	−	0.741129i	$-0.734291\pi$
$373$	−25.9323	−1.34273	−0.671363	+	0.741129i	$0.734291\pi$
$374$	0	0
$375$	0.395304	0.0204134
$376$	0	0
$377$	29.7808	1.53379
$378$	0	0
$379$	−11.1083	−0.570595	−0.285298	−	0.958439i	$-0.592092\pi$
$379$	−11.1083	−0.570595	−0.285298	+	0.958439i	$0.592092\pi$
$380$	0	0
$381$	−10.9773	−0.562385
$382$	0	0
$383$	11.6807	0.596854	0.298427	−	0.954432i	$-0.403538\pi$
$383$	11.6807	0.596854	0.298427	+	0.954432i	$0.403538\pi$
$384$	0	0
$385$	−13.1436	−0.669860
$386$	0	0
$387$	4.16844	0.211893
$388$	0	0
$389$	7.32296	0.371289	0.185644	−	0.982617i	$-0.440563\pi$
$389$	7.32296	0.371289	0.185644	+	0.982617i	$0.440563\pi$
$390$	0	0
$391$	−8.20902	−0.415148
$392$	0	0
$393$	−0.204993	−0.0103405
$394$	0	0
$395$	28.8230	1.45024
$396$	0	0
$397$	−33.9337	−1.70308	−0.851541	−	0.524288i	$-0.824331\pi$
$397$	−33.9337	−1.70308	−0.851541	+	0.524288i	$0.824331\pi$
$398$	0	0
$399$	−8.12778	−0.406898
$400$	0	0
$401$	−33.6825	−1.68202	−0.841012	−	0.541016i	$-0.818040\pi$
$401$	−33.6825	−1.68202	−0.841012	+	0.541016i	$0.818040\pi$
$402$	0	0
$403$	−14.3062	−0.712645
$404$	0	0
$405$	3.18186	0.158108
$406$	0	0
$407$	−17.7481	−0.879742
$408$	0	0
$409$	0.209950	0.0103814	0.00519069	−	0.999987i	$-0.498348\pi$
$409$	0.209950	0.0103814	0.00519069	+	0.999987i	$0.498348\pi$
$410$	0	0
$411$	10.5791	0.521828
$412$	0	0
$413$	22.6295	1.11353
$414$	0	0
$415$	56.8676	2.79152
$416$	0	0
$417$	−5.38787	−0.263845
$418$	0	0
$419$	−23.3446	−1.14046	−0.570230	−	0.821485i	$-0.693146\pi$
$419$	−23.3446	−1.14046	−0.570230	+	0.821485i	$0.693146\pi$
$420$	0	0
$421$	−5.58174	−0.272037	−0.136019	−	0.990706i	$-0.543431\pi$
$421$	−5.58174	−0.272037	−0.136019	+	0.990706i	$0.543431\pi$
$422$	0	0
$423$	−8.40522	−0.408676
$424$	0	0
$425$	5.22070	0.253241
$426$	0	0
$427$	25.1063	1.21498
$428$	0	0
$429$	−7.49027	−0.361633
$430$	0	0
$431$	32.3722	1.55932	0.779658	−	0.626206i	$-0.215393\pi$
$431$	32.3722	1.55932	0.779658	+	0.626206i	$0.215393\pi$
$432$	0	0
$433$	25.3860	1.21998	0.609988	−	0.792411i	$-0.291174\pi$
$433$	25.3860	1.21998	0.609988	+	0.792411i	$0.291174\pi$
$434$	0	0
$435$	20.5213	0.983923
$436$	0	0
$437$	25.7167	1.23020
$438$	0	0
$439$	−12.3593	−0.589878	−0.294939	−	0.955516i	$-0.595299\pi$
$439$	−12.3593	−0.589878	−0.294939	+	0.955516i	$0.595299\pi$
$440$	0	0
$441$	−0.515219	−0.0245342
$442$	0	0
$443$	31.5409	1.49855	0.749277	−	0.662256i	$-0.230401\pi$
$443$	31.5409	1.49855	0.749277	+	0.662256i	$0.230401\pi$
$444$	0	0
$445$	−30.6700	−1.45390
$446$	0	0
$447$	−1.16765	−0.0552279
$448$	0	0
$449$	−4.54571	−0.214525	−0.107263	−	0.994231i	$-0.534209\pi$
$449$	−4.54571	−0.214525	−0.107263	+	0.994231i	$0.534209\pi$
$450$	0	0
$451$	−1.44821	−0.0681934
$452$	0	0
$453$	12.7530	0.599188
$454$	0	0
$455$	37.4146	1.75402
$456$	0	0
$457$	−10.4704	−0.489782	−0.244891	−	0.969551i	$-0.578752\pi$
$457$	−10.4704	−0.489782	−0.244891	+	0.969551i	$0.578752\pi$
$458$	0	0
$459$	1.01883	0.0475547
$460$	0	0
$461$	25.6733	1.19573	0.597863	−	0.801598i	$-0.296017\pi$
$461$	25.6733	1.19573	0.597863	+	0.801598i	$0.296017\pi$
$462$	0	0
$463$	4.74247	0.220401	0.110201	−	0.993909i	$-0.464851\pi$
$463$	4.74247	0.220401	0.110201	+	0.993909i	$0.464851\pi$
$464$	0	0
$465$	−9.85814	−0.457160
$466$	0	0
$467$	−28.6202	−1.32439	−0.662193	−	0.749333i	$-0.730374\pi$
$467$	−28.6202	−1.32439	−0.662193	+	0.749333i	$0.730374\pi$
$468$	0	0
$469$	9.06393	0.418534
$470$	0	0
$471$	−6.86775	−0.316449
$472$	0	0
$473$	−6.76174	−0.310905
$474$	0	0
$475$	−16.3551	−0.750424
$476$	0	0
$477$	−2.66438	−0.121994
$478$	0	0
$479$	−4.60861	−0.210573	−0.105286	−	0.994442i	$-0.533576\pi$
$479$	−4.60861	−0.210573	−0.105286	+	0.994442i	$0.533576\pi$
$480$	0	0
$481$	50.5218	2.30360
$482$	0	0
$483$	−20.5182	−0.933610
$484$	0	0
$485$	−9.33953	−0.424086
$486$	0	0
$487$	17.2111	0.779910	0.389955	−	0.920834i	$-0.372491\pi$
$487$	17.2111	0.779910	0.389955	+	0.920834i	$0.372491\pi$
$488$	0	0
$489$	4.15118	0.187723
$490$	0	0
$491$	−16.1783	−0.730115	−0.365057	−	0.930985i	$-0.618951\pi$
$491$	−16.1783	−0.730115	−0.365057	+	0.930985i	$0.618951\pi$
$492$	0	0
$493$	6.57090	0.295938
$494$	0	0
$495$	−5.16139	−0.231987
$496$	0	0
$497$	−26.0236	−1.16732
$498$	0	0
$499$	4.16541	0.186469	0.0932346	−	0.995644i	$-0.470279\pi$
$499$	4.16541	0.186469	0.0932346	+	0.995644i	$0.470279\pi$
$500$	0	0
$501$	−16.3817	−0.731880
$502$	0	0
$503$	21.6308	0.964470	0.482235	−	0.876042i	$-0.339825\pi$
$503$	21.6308	0.964470	0.482235	+	0.876042i	$0.339825\pi$
$504$	0	0
$505$	52.1775	2.32187
$506$	0	0
$507$	8.32179	0.369583
$508$	0	0
$509$	−14.8268	−0.657185	−0.328592	−	0.944472i	$-0.606574\pi$
$509$	−14.8268	−0.657185	−0.328592	+	0.944472i	$0.606574\pi$
$510$	0	0
$511$	−6.47018	−0.286224
$512$	0	0
$513$	−3.19172	−0.140918
$514$	0	0
$515$	41.1817	1.81468
$516$	0	0
$517$	13.6344	0.599638
$518$	0	0
$519$	−16.5984	−0.728590
$520$	0	0
$521$	−12.1515	−0.532366	−0.266183	−	0.963923i	$-0.585763\pi$
$521$	−12.1515	−0.532366	−0.266183	+	0.963923i	$0.585763\pi$
$522$	0	0
$523$	−3.00258	−0.131294	−0.0656469	−	0.997843i	$-0.520911\pi$
$523$	−3.00258	−0.131294	−0.0656469	+	0.997843i	$0.520911\pi$
$524$	0	0
$525$	13.0490	0.569505
$526$	0	0
$527$	−3.15656	−0.137502
$528$	0	0
$529$	41.9206	1.82264
$530$	0	0
$531$	8.88643	0.385638
$532$	0	0
$533$	4.12246	0.178564
$534$	0	0
$535$	1.40003	0.0605286
$536$	0	0
$537$	−13.9511	−0.602035
$538$	0	0
$539$	0.835752	0.0359984
$540$	0	0
$541$	11.7626	0.505715	0.252857	−	0.967504i	$-0.418630\pi$
$541$	11.7626	0.505715	0.252857	+	0.967504i	$0.418630\pi$
$542$	0	0
$543$	−18.9274	−0.812254
$544$	0	0
$545$	1.84737	0.0791326
$546$	0	0
$547$	12.2984	0.525840	0.262920	−	0.964818i	$-0.415315\pi$
$547$	12.2984	0.525840	0.262920	+	0.964818i	$0.415315\pi$
$548$	0	0
$549$	9.85906	0.420774
$550$	0	0
$551$	−20.5849	−0.876947
$552$	0	0
$553$	23.0678	0.980941
$554$	0	0
$555$	34.8135	1.47775
$556$	0	0
$557$	5.54560	0.234975	0.117487	−	0.993074i	$-0.462516\pi$
$557$	5.54560	0.234975	0.117487	+	0.993074i	$0.462516\pi$
$558$	0	0
$559$	19.2480	0.814102
$560$	0	0
$561$	−1.65267	−0.0697756
$562$	0	0
$563$	−27.3567	−1.15295	−0.576474	−	0.817115i	$-0.695572\pi$
$563$	−27.3567	−1.15295	−0.576474	+	0.817115i	$0.695572\pi$
$564$	0	0
$565$	−14.8704	−0.625601
$566$	0	0
$567$	2.54652	0.106944
$568$	0	0
$569$	−12.3712	−0.518627	−0.259313	−	0.965793i	$-0.583496\pi$
$569$	−12.3712	−0.518627	−0.259313	+	0.965793i	$0.583496\pi$
$570$	0	0
$571$	−0.892374	−0.0373447	−0.0186723	−	0.999826i	$-0.505944\pi$
$571$	−0.892374	−0.0373447	−0.0186723	+	0.999826i	$0.505944\pi$
$572$	0	0
$573$	−9.81038	−0.409835
$574$	0	0
$575$	−41.2877	−1.72182
$576$	0	0
$577$	−7.69701	−0.320431	−0.160215	−	0.987082i	$-0.551219\pi$
$577$	−7.69701	−0.320431	−0.160215	+	0.987082i	$0.551219\pi$
$578$	0	0
$579$	−2.85083	−0.118476
$580$	0	0
$581$	45.5126	1.88818
$582$	0	0
$583$	4.32197	0.178998
$584$	0	0
$585$	14.6924	0.607456
$586$	0	0
$587$	40.6475	1.67770	0.838851	−	0.544361i	$-0.183228\pi$
$587$	40.6475	1.67770	0.838851	+	0.544361i	$0.183228\pi$
$588$	0	0
$589$	9.88867	0.407456
$590$	0	0
$591$	−9.00576	−0.370447
$592$	0	0
$593$	−35.7480	−1.46799	−0.733997	−	0.679153i	$-0.762347\pi$
$593$	−35.7480	−1.46799	−0.733997	+	0.679153i	$0.762347\pi$
$594$	0	0
$595$	8.25522	0.338431
$596$	0	0
$597$	−12.8945	−0.527737
$598$	0	0
$599$	−21.1766	−0.865251	−0.432626	−	0.901574i	$-0.642413\pi$
$599$	−21.1766	−0.865251	−0.432626	+	0.901574i	$0.642413\pi$
$600$	0	0
$601$	−18.3403	−0.748117	−0.374058	−	0.927405i	$-0.622034\pi$
$601$	−18.3403	−0.748117	−0.374058	+	0.927405i	$0.622034\pi$
$602$	0	0
$603$	3.55934	0.144947
$604$	0	0
$605$	−26.6280	−1.08258
$606$	0	0
$607$	42.8694	1.74002	0.870009	−	0.493037i	$-0.164113\pi$
$607$	42.8694	1.74002	0.870009	+	0.493037i	$0.164113\pi$
$608$	0	0
$609$	16.4238	0.665524
$610$	0	0
$611$	−38.8115	−1.57015
$612$	0	0
$613$	18.9476	0.765288	0.382644	−	0.923896i	$-0.375014\pi$
$613$	18.9476	0.765288	0.382644	+	0.923896i	$0.375014\pi$
$614$	0	0
$615$	2.84071	0.114548
$616$	0	0
$617$	43.1414	1.73681	0.868404	−	0.495857i	$-0.165146\pi$
$617$	43.1414	1.73681	0.868404	+	0.495857i	$0.165146\pi$
$618$	0	0
$619$	12.6131	0.506963	0.253481	−	0.967340i	$-0.418424\pi$
$619$	12.6131	0.506963	0.253481	+	0.967340i	$0.418424\pi$
$620$	0	0
$621$	−8.05733	−0.323330
$622$	0	0
$623$	−24.5460	−0.983413
$624$	0	0
$625$	−24.3634	−0.974535
$626$	0	0
$627$	5.17738	0.206764
$628$	0	0
$629$	11.1472	0.444469
$630$	0	0
$631$	18.5826	0.739763	0.369881	−	0.929079i	$-0.379398\pi$
$631$	18.5826	0.739763	0.369881	+	0.929079i	$0.379398\pi$
$632$	0	0
$633$	27.9805	1.11212
$634$	0	0
$635$	−34.9283	−1.38609
$636$	0	0
$637$	−2.37905	−0.0942614
$638$	0	0
$639$	−10.2193	−0.404269
$640$	0	0
$641$	−23.4755	−0.927226	−0.463613	−	0.886038i	$-0.653447\pi$
$641$	−23.4755	−0.927226	−0.463613	+	0.886038i	$0.653447\pi$
$642$	0	0
$643$	20.3178	0.801254	0.400627	−	0.916241i	$-0.368792\pi$
$643$	20.3178	0.801254	0.400627	+	0.916241i	$0.368792\pi$
$644$	0	0
$645$	13.2634	0.522245
$646$	0	0
$647$	25.9646	1.02078	0.510388	−	0.859944i	$-0.329502\pi$
$647$	25.9646	1.02078	0.510388	+	0.859944i	$0.329502\pi$
$648$	0	0
$649$	−14.4149	−0.565836
$650$	0	0
$651$	−7.88972	−0.309222
$652$	0	0
$653$	29.9130	1.17058	0.585292	−	0.810822i	$-0.300980\pi$
$653$	29.9130	1.17058	0.585292	+	0.810822i	$0.300980\pi$
$654$	0	0
$655$	−0.652258	−0.0254858
$656$	0	0
$657$	−2.54079	−0.0991256
$658$	0	0
$659$	−36.1774	−1.40927	−0.704635	−	0.709570i	$-0.748889\pi$
$659$	−36.1774	−1.40927	−0.704635	+	0.709570i	$0.748889\pi$
$660$	0	0
$661$	32.0345	1.24600	0.622998	−	0.782223i	$-0.285914\pi$
$661$	32.0345	1.24600	0.622998	+	0.782223i	$0.285914\pi$
$662$	0	0
$663$	4.70448	0.182707
$664$	0	0
$665$	−25.8615	−1.00286
$666$	0	0
$667$	−51.9656	−2.01212
$668$	0	0
$669$	−17.9010	−0.692094
$670$	0	0
$671$	−15.9927	−0.617390
$672$	0	0
$673$	−11.5447	−0.445016	−0.222508	−	0.974931i	$-0.571424\pi$
$673$	−11.5447	−0.445016	−0.222508	+	0.974931i	$0.571424\pi$
$674$	0	0
$675$	5.12424	0.197232
$676$	0	0
$677$	19.9909	0.768314	0.384157	−	0.923268i	$-0.374492\pi$
$677$	19.9909	0.768314	0.384157	+	0.923268i	$0.374492\pi$
$678$	0	0
$679$	−7.47466	−0.286851
$680$	0	0
$681$	19.0625	0.730478
$682$	0	0
$683$	14.2003	0.543361	0.271681	−	0.962387i	$-0.412421\pi$
$683$	14.2003	0.543361	0.271681	+	0.962387i	$0.412421\pi$
$684$	0	0
$685$	33.6612	1.28613
$686$	0	0
$687$	−23.8183	−0.908723
$688$	0	0
$689$	−12.3029	−0.468704
$690$	0	0
$691$	36.9475	1.40555	0.702774	−	0.711413i	$-0.251944\pi$
$691$	36.9475	1.40555	0.702774	+	0.711413i	$0.251944\pi$
$692$	0	0
$693$	−4.13079	−0.156916
$694$	0	0
$695$	−17.1434	−0.650288
$696$	0	0
$697$	0.909588	0.0344531
$698$	0	0
$699$	20.0140	0.756998
$700$	0	0
$701$	−6.68012	−0.252305	−0.126152	−	0.992011i	$-0.540263\pi$
$701$	−6.68012	−0.252305	−0.126152	+	0.992011i	$0.540263\pi$
$702$	0	0
$703$	−34.9214	−1.31708
$704$	0	0
$705$	−26.7442	−1.00725
$706$	0	0
$707$	41.7590	1.57051
$708$	0	0
$709$	−8.84256	−0.332089	−0.166045	−	0.986118i	$-0.553100\pi$
$709$	−8.84256	−0.332089	−0.166045	+	0.986118i	$0.553100\pi$
$710$	0	0
$711$	9.05853	0.339721
$712$	0	0
$713$	24.9635	0.934890
$714$	0	0
$715$	−23.8330	−0.891303
$716$	0	0
$717$	12.1391	0.453342
$718$	0	0
$719$	14.1888	0.529153	0.264576	−	0.964365i	$-0.414768\pi$
$719$	14.1888	0.529153	0.264576	+	0.964365i	$0.414768\pi$
$720$	0	0
$721$	32.9588	1.22745
$722$	0	0
$723$	8.19115	0.304632
$724$	0	0
$725$	33.0487	1.22740
$726$	0	0
$727$	15.3785	0.570357	0.285179	−	0.958474i	$-0.407947\pi$
$727$	15.3785	0.570357	0.285179	+	0.958474i	$0.407947\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.24691	0.157078
$732$	0	0
$733$	17.1056	0.631809	0.315905	−	0.948791i	$-0.397692\pi$
$733$	17.1056	0.631809	0.315905	+	0.948791i	$0.397692\pi$
$734$	0	0
$735$	−1.63935	−0.0604685
$736$	0	0
$737$	−5.77370	−0.212677
$738$	0	0
$739$	5.04399	0.185546	0.0927731	−	0.995687i	$-0.470427\pi$
$739$	5.04399	0.185546	0.0927731	+	0.995687i	$0.470427\pi$
$740$	0	0
$741$	−14.7379	−0.541411
$742$	0	0
$743$	−24.4802	−0.898092	−0.449046	−	0.893509i	$-0.648236\pi$
$743$	−24.4802	−0.898092	−0.449046	+	0.893509i	$0.648236\pi$
$744$	0	0
$745$	−3.71530	−0.136118
$746$	0	0
$747$	17.8724	0.653918
$748$	0	0
$749$	1.12048	0.0409415
$750$	0	0
$751$	−15.6809	−0.572206	−0.286103	−	0.958199i	$-0.592360\pi$
$751$	−15.6809	−0.572206	−0.286103	+	0.958199i	$0.592360\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	40.5782	1.47679
$756$	0	0
$757$	41.0970	1.49370	0.746848	−	0.664995i	$-0.231566\pi$
$757$	41.0970	1.49370	0.746848	+	0.664995i	$0.231566\pi$
$758$	0	0
$759$	13.0700	0.474412
$760$	0	0
$761$	−47.2573	−1.71307	−0.856537	−	0.516085i	$-0.827389\pi$
$761$	−47.2573	−1.71307	−0.856537	+	0.516085i	$0.827389\pi$
$762$	0	0
$763$	1.47850	0.0535251
$764$	0	0
$765$	3.24176	0.117206
$766$	0	0
$767$	41.0336	1.48164
$768$	0	0
$769$	33.5359	1.20934	0.604668	−	0.796477i	$-0.293306\pi$
$769$	33.5359	1.20934	0.604668	+	0.796477i	$0.293306\pi$
$770$	0	0
$771$	31.2209	1.12439
$772$	0	0
$773$	−38.4683	−1.38361	−0.691804	−	0.722085i	$-0.743184\pi$
$773$	−38.4683	−1.38361	−0.691804	+	0.722085i	$0.743184\pi$
$774$	0	0
$775$	−15.8761	−0.570285
$776$	0	0
$777$	27.8621	0.999549
$778$	0	0
$779$	−2.84950	−0.102094
$780$	0	0
$781$	16.5770	0.593172
$782$	0	0
$783$	6.44948	0.230486
$784$	0	0
$785$	−21.8522	−0.779939
$786$	0	0
$787$	−14.3174	−0.510361	−0.255180	−	0.966893i	$-0.582135\pi$
$787$	−14.3174	−0.510361	−0.255180	+	0.966893i	$0.582135\pi$
$788$	0	0
$789$	−29.6076	−1.05406
$790$	0	0
$791$	−11.9011	−0.423156
$792$	0	0
$793$	45.5247	1.61663
$794$	0	0
$795$	−8.47769	−0.300673
$796$	0	0
$797$	−22.4897	−0.796627	−0.398313	−	0.917249i	$-0.630404\pi$
$797$	−22.4897	−0.796627	−0.398313	+	0.917249i	$0.630404\pi$
$798$	0	0
$799$	−8.56345	−0.302953
$800$	0	0
$801$	−9.63901	−0.340578
$802$	0	0
$803$	4.12149	0.145444
$804$	0	0
$805$	−65.2860	−2.30103
$806$	0	0
$807$	5.09323	0.179290
$808$	0	0
$809$	26.5024	0.931773	0.465887	−	0.884844i	$-0.345735\pi$
$809$	26.5024	0.931773	0.465887	+	0.884844i	$0.345735\pi$
$810$	0	0
$811$	0.247194	0.00868016	0.00434008	−	0.999991i	$-0.498619\pi$
$811$	0.247194	0.00868016	0.00434008	+	0.999991i	$0.498619\pi$
$812$	0	0
$813$	3.24681	0.113871
$814$	0	0
$815$	13.2085	0.462673
$816$	0	0
$817$	−13.3045	−0.465464
$818$	0	0
$819$	11.7587	0.410882
$820$	0	0
$821$	−3.39546	−0.118502	−0.0592512	−	0.998243i	$-0.518871\pi$
$821$	−3.39546	−0.118502	−0.0592512	+	0.998243i	$0.518871\pi$
$822$	0	0
$823$	−4.53601	−0.158115	−0.0790577	−	0.996870i	$-0.525191\pi$
$823$	−4.53601	−0.158115	−0.0790577	+	0.996870i	$0.525191\pi$
$824$	0	0
$825$	−8.31217	−0.289393
$826$	0	0
$827$	−33.0363	−1.14879	−0.574393	−	0.818580i	$-0.694762\pi$
$827$	−33.0363	−1.14879	−0.574393	+	0.818580i	$0.694762\pi$
$828$	0	0
$829$	28.1442	0.977489	0.488745	−	0.872427i	$-0.337455\pi$
$829$	28.1442	0.977489	0.488745	+	0.872427i	$0.337455\pi$
$830$	0	0
$831$	5.54366	0.192308
$832$	0	0
$833$	−0.524918	−0.0181873
$834$	0	0
$835$	−52.1243	−1.80383
$836$	0	0
$837$	−3.09823	−0.107091
$838$	0	0
$839$	−28.8869	−0.997288	−0.498644	−	0.866807i	$-0.666168\pi$
$839$	−28.8869	−0.997288	−0.498644	+	0.866807i	$0.666168\pi$
$840$	0	0
$841$	12.5958	0.434338
$842$	0	0
$843$	−30.6076	−1.05418
$844$	0	0
$845$	26.4788	0.910897
$846$	0	0
$847$	−21.3111	−0.732257
$848$	0	0
$849$	13.2155	0.453554
$850$	0	0
$851$	−88.1573	−3.02199
$852$	0	0
$853$	−2.99486	−0.102542	−0.0512711	−	0.998685i	$-0.516327\pi$
$853$	−2.99486	−0.102542	−0.0512711	+	0.998685i	$0.516327\pi$
$854$	0	0
$855$	−10.1556	−0.347314
$856$	0	0
$857$	−25.0681	−0.856311	−0.428156	−	0.903705i	$-0.640837\pi$
$857$	−25.0681	−0.856311	−0.428156	+	0.903705i	$0.640837\pi$
$858$	0	0
$859$	−57.3838	−1.95791	−0.978955	−	0.204075i	$-0.934581\pi$
$859$	−57.3838	−1.95791	−0.978955	+	0.204075i	$0.934581\pi$
$860$	0	0
$861$	2.27349	0.0774803
$862$	0	0
$863$	−35.2953	−1.20147	−0.600733	−	0.799450i	$-0.705124\pi$
$863$	−35.2953	−1.20147	−0.600733	+	0.799450i	$0.705124\pi$
$864$	0	0
$865$	−52.8139	−1.79573
$866$	0	0
$867$	−15.9620	−0.542098
$868$	0	0
$869$	−14.6941	−0.498463
$870$	0	0
$871$	16.4354	0.556893
$872$	0	0
$873$	−2.93524	−0.0993429
$874$	0	0
$875$	1.00665	0.0340310
$876$	0	0
$877$	32.6134	1.10128	0.550638	−	0.834744i	$-0.314384\pi$
$877$	32.6134	1.10128	0.550638	+	0.834744i	$0.314384\pi$
$878$	0	0
$879$	22.3977	0.755456
$880$	0	0
$881$	−28.0953	−0.946556	−0.473278	−	0.880913i	$-0.656929\pi$
$881$	−28.0953	−0.946556	−0.473278	+	0.880913i	$0.656929\pi$
$882$	0	0
$883$	−56.7008	−1.90813	−0.954067	−	0.299594i	$-0.903149\pi$
$883$	−56.7008	−1.90813	−0.954067	+	0.299594i	$0.903149\pi$
$884$	0	0
$885$	28.2754	0.950467
$886$	0	0
$887$	4.86724	0.163426	0.0817130	−	0.996656i	$-0.473961\pi$
$887$	4.86724	0.163426	0.0817130	+	0.996656i	$0.473961\pi$
$888$	0	0
$889$	−27.9540	−0.937547
$890$	0	0
$891$	−1.62213	−0.0543434
$892$	0	0
$893$	26.8271	0.897734
$894$	0	0
$895$	−44.3905	−1.48381
$896$	0	0
$897$	−37.2052	−1.24224
$898$	0	0
$899$	−19.9820	−0.666436
$900$	0	0
$901$	−2.71454	−0.0904344
$902$	0	0
$903$	10.6150	0.353246
$904$	0	0
$905$	−60.2244	−2.00193
$906$	0	0
$907$	46.2178	1.53464	0.767319	−	0.641266i	$-0.221590\pi$
$907$	46.2178	1.53464	0.767319	+	0.641266i	$0.221590\pi$
$908$	0	0
$909$	16.3984	0.543901
$910$	0	0
$911$	−33.0739	−1.09579	−0.547894	−	0.836548i	$-0.684570\pi$
$911$	−33.0739	−1.09579	−0.547894	+	0.836548i	$0.684570\pi$
$912$	0	0
$913$	−28.9914	−0.959475
$914$	0	0
$915$	31.3702	1.03707
$916$	0	0
$917$	−0.522018	−0.0172386
$918$	0	0
$919$	8.55287	0.282133	0.141067	−	0.990000i	$-0.454947\pi$
$919$	8.55287	0.282133	0.141067	+	0.990000i	$0.454947\pi$
$920$	0	0
$921$	−21.4179	−0.705744
$922$	0	0
$923$	−47.1881	−1.55321
$924$	0	0
$925$	56.0656	1.84342
$926$	0	0
$927$	12.9427	0.425093
$928$	0	0
$929$	−37.5574	−1.23222	−0.616109	−	0.787661i	$-0.711292\pi$
$929$	−37.5574	−1.23222	−0.616109	+	0.787661i	$0.711292\pi$
$930$	0	0
$931$	1.64443	0.0538941
$932$	0	0
$933$	−9.68607	−0.317108
$934$	0	0
$935$	−5.25855	−0.171973
$936$	0	0
$937$	54.5161	1.78096	0.890482	−	0.455018i	$-0.150367\pi$
$937$	54.5161	1.78096	0.890482	+	0.455018i	$0.150367\pi$
$938$	0	0
$939$	25.2893	0.825286
$940$	0	0
$941$	5.30412	0.172909	0.0864546	−	0.996256i	$-0.472446\pi$
$941$	5.30412	0.172909	0.0864546	+	0.996256i	$0.472446\pi$
$942$	0	0
$943$	−7.19344	−0.234251
$944$	0	0
$945$	8.10268	0.263580
$946$	0	0
$947$	−44.7266	−1.45342	−0.726709	−	0.686945i	$-0.758951\pi$
$947$	−44.7266	−1.45342	−0.726709	+	0.686945i	$0.758951\pi$
$948$	0	0
$949$	−11.7322	−0.380844
$950$	0	0
$951$	−24.9181	−0.808024
$952$	0	0
$953$	4.44990	0.144147	0.0720733	−	0.997399i	$-0.477038\pi$
$953$	4.44990	0.144147	0.0720733	+	0.997399i	$0.477038\pi$
$954$	0	0
$955$	−31.2153	−1.01010
$956$	0	0
$957$	−10.4619	−0.338185
$958$	0	0
$959$	26.9399	0.869935
$960$	0	0
$961$	−21.4010	−0.690354
$962$	0	0
$963$	0.440004	0.0141789
$964$	0	0
$965$	−9.07093	−0.292004
$966$	0	0
$967$	59.5929	1.91638	0.958189	−	0.286135i	$-0.0923706\pi$
$967$	59.5929	1.91638	0.958189	+	0.286135i	$0.0923706\pi$
$968$	0	0
$969$	−3.25180	−0.104463
$970$	0	0
$971$	27.5141	0.882970	0.441485	−	0.897269i	$-0.354452\pi$
$971$	27.5141	0.882970	0.441485	+	0.897269i	$0.354452\pi$
$972$	0	0
$973$	−13.7203	−0.439854
$974$	0	0
$975$	23.6614	0.757772
$976$	0	0
$977$	13.2616	0.424277	0.212139	−	0.977240i	$-0.431957\pi$
$977$	13.2616	0.424277	0.212139	+	0.977240i	$0.431957\pi$
$978$	0	0
$979$	15.6357	0.499720
$980$	0	0
$981$	0.580594	0.0185369
$982$	0	0
$983$	−19.3890	−0.618412	−0.309206	−	0.950995i	$-0.600063\pi$
$983$	−19.3890	−0.618412	−0.309206	+	0.950995i	$0.600063\pi$
$984$	0	0
$985$	−28.6551	−0.913026
$986$	0	0
$987$	−21.4041	−0.681300
$988$	0	0
$989$	−33.5865	−1.06799
$990$	0	0
$991$	−11.9938	−0.380997	−0.190498	−	0.981688i	$-0.561010\pi$
$991$	−11.9938	−0.380997	−0.190498	+	0.981688i	$0.561010\pi$
$992$	0	0
$993$	11.5728	0.367252
$994$	0	0
$995$	−41.0285	−1.30069
$996$	0	0
$997$	−40.7712	−1.29124	−0.645619	−	0.763660i	$-0.723401\pi$
$997$	−40.7712	−1.29124	−0.645619	+	0.763660i	$0.723401\pi$
$998$	0	0
$999$	10.9412	0.346166

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.r.1.17	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.r.1.17	✓	20	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.18186 q^{5} +2.54652 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.18186 q^{5} +2.54652 q^{7} +1.00000 q^{9} -1.62213 q^{11} +4.61755 q^{13} +3.18186 q^{15} +1.01883 q^{17} -3.19172 q^{19} +2.54652 q^{21} -8.05733 q^{23} +5.12424 q^{25} +1.00000 q^{27} +6.44948 q^{29} -3.09823 q^{31} -1.62213 q^{33} +8.10268 q^{35} +10.9412 q^{37} +4.61755 q^{39} +0.892781 q^{41} +4.16844 q^{43} +3.18186 q^{45} -8.40522 q^{47} -0.515219 q^{49} +1.01883 q^{51} -2.66438 q^{53} -5.16139 q^{55} -3.19172 q^{57} +8.88643 q^{59} +9.85906 q^{61} +2.54652 q^{63} +14.6924 q^{65} +3.55934 q^{67} -8.05733 q^{69} -10.2193 q^{71} -2.54079 q^{73} +5.12424 q^{75} -4.13079 q^{77} +9.05853 q^{79} +1.00000 q^{81} +17.8724 q^{83} +3.24176 q^{85} +6.44948 q^{87} -9.63901 q^{89} +11.7587 q^{91} -3.09823 q^{93} -10.1556 q^{95} -2.93524 q^{97} -1.62213 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9} + 4 q^{11} + 21 q^{13} + 9 q^{15} + 10 q^{17} + 8 q^{19} + 9 q^{21} + 9 q^{23} + 43 q^{25} + 20 q^{27} + 18 q^{29} + 27 q^{31} + 4 q^{33} - 7 q^{35} + 33 q^{37} + 21 q^{39} + 14 q^{41} - 6 q^{43} + 9 q^{45} + 21 q^{47} + 47 q^{49} + 10 q^{51} + 23 q^{53} + 24 q^{55} + 8 q^{57} + 10 q^{59} + 28 q^{61} + 9 q^{63} + 2 q^{65} + 15 q^{67} + 9 q^{69} + 33 q^{71} + 50 q^{73} + 43 q^{75} + 20 q^{77} + 17 q^{79} + 20 q^{81} - 19 q^{83} + 41 q^{85} + 18 q^{87} + 21 q^{89} + 30 q^{91} + 27 q^{93} + 27 q^{95} + 47 q^{97} + 4 q^{99}+O(q^{100}) \) 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9 + 4 * q^11 + 21 * q^13 + 9 * q^15 + 10 * q^17 + 8 * q^19 + 9 * q^21 + 9 * q^23 + 43 * q^25 + 20 * q^27 + 18 * q^29 + 27 * q^31 + 4 * q^33 - 7 * q^35 + 33 * q^37 + 21 * q^39 + 14 * q^41 - 6 * q^43 + 9 * q^45 + 21 * q^47 + 47 * q^49 + 10 * q^51 + 23 * q^53 + 24 * q^55 + 8 * q^57 + 10 * q^59 + 28 * q^61 + 9 * q^63 + 2 * q^65 + 15 * q^67 + 9 * q^69 + 33 * q^71 + 50 * q^73 + 43 * q^75 + 20 * q^77 + 17 * q^79 + 20 * q^81 - 19 * q^83 + 41 * q^85 + 18 * q^87 + 21 * q^89 + 30 * q^91 + 27 * q^93 + 27 * q^95 + 47 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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