LMFDB - Embedded newform 6020.2.a.j.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6020,2,Mod(1,6020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6020, 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("6020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6020.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:	$48.0699420168$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - x^{12} - 28 x^{11} + 26 x^{10} + 286 x^{9} - 235 x^{8} - 1298 x^{7} + 895 x^{6} + 2571 x^{5} + \cdots - 216 $ x^13 - x^12 - 28x^11 + 26x^10 + 286x^9 - 235x^8 - 1298x^7 + 895x^6 + 2571x^5 - 1548x^4 - 1833x^3 + 1119x^2 + 342*x - 216
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-1.87398$ of defining polynomial
Character	$\chi$	$=$	6020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.87398 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.511819 q^{9} +O(q^{10})$	$q+1.87398 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.511819 q^{9} -1.79941 q^{11} -4.71817 q^{13} -1.87398 q^{15} +2.34845 q^{17} +0.828630 q^{19} -1.87398 q^{21} +0.819083 q^{23} +1.00000 q^{25} -4.66281 q^{27} +4.13031 q^{29} +9.56274 q^{31} -3.37206 q^{33} +1.00000 q^{35} -0.321229 q^{37} -8.84178 q^{39} +10.2372 q^{41} -1.00000 q^{43} -0.511819 q^{45} -5.71350 q^{47} +1.00000 q^{49} +4.40097 q^{51} -9.41701 q^{53} +1.79941 q^{55} +1.55284 q^{57} +11.4162 q^{59} +5.47171 q^{61} -0.511819 q^{63} +4.71817 q^{65} +5.63361 q^{67} +1.53495 q^{69} -3.27795 q^{71} -1.82738 q^{73} +1.87398 q^{75} +1.79941 q^{77} +2.59519 q^{79} -10.2735 q^{81} +0.296736 q^{83} -2.34845 q^{85} +7.74014 q^{87} +8.36859 q^{89} +4.71817 q^{91} +17.9204 q^{93} -0.828630 q^{95} +0.240040 q^{97} -0.920971 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9}+O(q^{10}) $ 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9	$ 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9} + 6 q^{11} + q^{13} + q^{15} - 6 q^{17} + 2 q^{19} + q^{21} - 6 q^{23} + 13 q^{25} - q^{27} + 19 q^{29} - 24 q^{31} + 17 q^{33} + 13 q^{35} + 15 q^{39} + 4 q^{41} - 13 q^{43} - 18 q^{45} - 3 q^{47} + 13 q^{49} + 7 q^{51} - 5 q^{53} - 6 q^{55} + 6 q^{57} - 6 q^{59} + 3 q^{61} - 18 q^{63} - q^{65} - 2 q^{67} + 20 q^{69} + 18 q^{71} + 14 q^{73} - q^{75} - 6 q^{77} + 12 q^{79} + 37 q^{81} + 2 q^{83} + 6 q^{85} - 2 q^{87} + 17 q^{89} - q^{91} + 15 q^{93} - 2 q^{95} + 17 q^{97} + 80 q^{99}+O(q^{100}) $ 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9 + 6 * q^11 + q^13 + q^15 - 6 * q^17 + 2 * q^19 + q^21 - 6 * q^23 + 13 * q^25 - q^27 + 19 * q^29 - 24 * q^31 + 17 * q^33 + 13 * q^35 + 15 * q^39 + 4 * q^41 - 13 * q^43 - 18 * q^45 - 3 * q^47 + 13 * q^49 + 7 * q^51 - 5 * q^53 - 6 * q^55 + 6 * q^57 - 6 * q^59 + 3 * q^61 - 18 * q^63 - q^65 - 2 * q^67 + 20 * q^69 + 18 * q^71 + 14 * q^73 - q^75 - 6 * q^77 + 12 * q^79 + 37 * q^81 + 2 * q^83 + 6 * q^85 - 2 * q^87 + 17 * q^89 - q^91 + 15 * q^93 - 2 * q^95 + 17 * q^97 + 80 * 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
$3$	1.87398	1.08195	0.540973	−	0.841040i	$-0.318056\pi$
$3$	1.87398	1.08195	0.540973	+	0.841040i	$0.318056\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.511819	0.170606
$10$	0	0
$11$	−1.79941	−0.542542	−0.271271	−	0.962503i	$-0.587444\pi$
$11$	−1.79941	−0.542542	−0.271271	+	0.962503i	$0.587444\pi$
$12$	0	0
$13$	−4.71817	−1.30859	−0.654293	−	0.756242i	$-0.727033\pi$
$13$	−4.71817	−1.30859	−0.654293	+	0.756242i	$0.727033\pi$
$14$	0	0
$15$	−1.87398	−0.483861
$16$	0	0
$17$	2.34845	0.569584	0.284792	−	0.958589i	$-0.408076\pi$
$17$	2.34845	0.569584	0.284792	+	0.958589i	$0.408076\pi$
$18$	0	0
$19$	0.828630	0.190101	0.0950503	−	0.995472i	$-0.469699\pi$
$19$	0.828630	0.190101	0.0950503	+	0.995472i	$0.469699\pi$
$20$	0	0
$21$	−1.87398	−0.408937
$22$	0	0
$23$	0.819083	0.170791	0.0853953	−	0.996347i	$-0.472785\pi$
$23$	0.819083	0.170791	0.0853953	+	0.996347i	$0.472785\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.66281	−0.897359
$28$	0	0
$29$	4.13031	0.766979	0.383490	−	0.923545i	$-0.374722\pi$
$29$	4.13031	0.766979	0.383490	+	0.923545i	$0.374722\pi$
$30$	0	0
$31$	9.56274	1.71752	0.858759	−	0.512380i	$-0.171236\pi$
$31$	9.56274	1.71752	0.858759	+	0.512380i	$0.171236\pi$
$32$	0	0
$33$	−3.37206	−0.587001
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−0.321229	−0.0528097	−0.0264048	−	0.999651i	$-0.508406\pi$
$37$	−0.321229	−0.0528097	−0.0264048	+	0.999651i	$0.508406\pi$
$38$	0	0
$39$	−8.84178	−1.41582
$40$	0	0
$41$	10.2372	1.59879	0.799394	−	0.600807i	$-0.205154\pi$
$41$	10.2372	1.59879	0.799394	+	0.600807i	$0.205154\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	−0.511819	−0.0762975
$46$	0	0
$47$	−5.71350	−0.833399	−0.416699	−	0.909044i	$-0.636813\pi$
$47$	−5.71350	−0.833399	−0.416699	+	0.909044i	$0.636813\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.40097	0.616258
$52$	0	0
$53$	−9.41701	−1.29353	−0.646763	−	0.762691i	$-0.723878\pi$
$53$	−9.41701	−1.29353	−0.646763	+	0.762691i	$0.723878\pi$
$54$	0	0
$55$	1.79941	0.242632
$56$	0	0
$57$	1.55284	0.205679
$58$	0	0
$59$	11.4162	1.48627	0.743134	−	0.669142i	$-0.233338\pi$
$59$	11.4162	1.48627	0.743134	+	0.669142i	$0.233338\pi$
$60$	0	0
$61$	5.47171	0.700580	0.350290	−	0.936641i	$-0.386083\pi$
$61$	5.47171	0.700580	0.350290	+	0.936641i	$0.386083\pi$
$62$	0	0
$63$	−0.511819	−0.0644832
$64$	0	0
$65$	4.71817	0.585217
$66$	0	0
$67$	5.63361	0.688254	0.344127	−	0.938923i	$-0.388175\pi$
$67$	5.63361	0.688254	0.344127	+	0.938923i	$0.388175\pi$
$68$	0	0
$69$	1.53495	0.184786
$70$	0	0
$71$	−3.27795	−0.389021	−0.194511	−	0.980900i	$-0.562312\pi$
$71$	−3.27795	−0.389021	−0.194511	+	0.980900i	$0.562312\pi$
$72$	0	0
$73$	−1.82738	−0.213879	−0.106939	−	0.994266i	$-0.534105\pi$
$73$	−1.82738	−0.213879	−0.106939	+	0.994266i	$0.534105\pi$
$74$	0	0
$75$	1.87398	0.216389
$76$	0	0
$77$	1.79941	0.205061
$78$	0	0
$79$	2.59519	0.291982	0.145991	−	0.989286i	$-0.453363\pi$
$79$	2.59519	0.291982	0.145991	+	0.989286i	$0.453363\pi$
$80$	0	0
$81$	−10.2735	−1.14150
$82$	0	0
$83$	0.296736	0.0325710	0.0162855	−	0.999867i	$-0.494816\pi$
$83$	0.296736	0.0325710	0.0162855	+	0.999867i	$0.494816\pi$
$84$	0	0
$85$	−2.34845	−0.254725
$86$	0	0
$87$	7.74014	0.829830
$88$	0	0
$89$	8.36859	0.887069	0.443534	−	0.896257i	$-0.353724\pi$
$89$	8.36859	0.887069	0.443534	+	0.896257i	$0.353724\pi$
$90$	0	0
$91$	4.71817	0.494599
$92$	0	0
$93$	17.9204	1.85826
$94$	0	0
$95$	−0.828630	−0.0850156
$96$	0	0
$97$	0.240040	0.0243723	0.0121862	−	0.999926i	$-0.496121\pi$
$97$	0.240040	0.0243723	0.0121862	+	0.999926i	$0.496121\pi$
$98$	0	0
$99$	−0.920971	−0.0925611
$100$	0	0
$101$	9.36187	0.931541	0.465771	−	0.884905i	$-0.345777\pi$
$101$	9.36187	0.931541	0.465771	+	0.884905i	$0.345777\pi$
$102$	0	0
$103$	2.91098	0.286827	0.143414	−	0.989663i	$-0.454192\pi$
$103$	2.91098	0.286827	0.143414	+	0.989663i	$0.454192\pi$
$104$	0	0
$105$	1.87398	0.182882
$106$	0	0
$107$	8.25101	0.797655	0.398828	−	0.917026i	$-0.369417\pi$
$107$	8.25101	0.797655	0.398828	+	0.917026i	$0.369417\pi$
$108$	0	0
$109$	7.71564	0.739024	0.369512	−	0.929226i	$-0.379525\pi$
$109$	7.71564	0.739024	0.369512	+	0.929226i	$0.379525\pi$
$110$	0	0
$111$	−0.601978	−0.0571372
$112$	0	0
$113$	6.33716	0.596150	0.298075	−	0.954542i	$-0.403655\pi$
$113$	6.33716	0.596150	0.298075	+	0.954542i	$0.403655\pi$
$114$	0	0
$115$	−0.819083	−0.0763798
$116$	0	0
$117$	−2.41485	−0.223253
$118$	0	0
$119$	−2.34845	−0.215282
$120$	0	0
$121$	−7.76213	−0.705649
$122$	0	0
$123$	19.1844	1.72980
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	10.6667	0.946520	0.473260	−	0.880923i	$-0.343077\pi$
$127$	10.6667	0.946520	0.473260	+	0.880923i	$0.343077\pi$
$128$	0	0
$129$	−1.87398	−0.164995
$130$	0	0
$131$	−9.50129	−0.830132	−0.415066	−	0.909791i	$-0.636242\pi$
$131$	−9.50129	−0.830132	−0.415066	+	0.909791i	$0.636242\pi$
$132$	0	0
$133$	−0.828630	−0.0718513
$134$	0	0
$135$	4.66281	0.401311
$136$	0	0
$137$	13.1516	1.12362	0.561808	−	0.827268i	$-0.310106\pi$
$137$	13.1516	1.12362	0.561808	+	0.827268i	$0.310106\pi$
$138$	0	0
$139$	6.73830	0.571535	0.285768	−	0.958299i	$-0.407751\pi$
$139$	6.73830	0.571535	0.285768	+	0.958299i	$0.407751\pi$
$140$	0	0
$141$	−10.7070	−0.901692
$142$	0	0
$143$	8.48991	0.709962
$144$	0	0
$145$	−4.13031	−0.343003
$146$	0	0
$147$	1.87398	0.154564
$148$	0	0
$149$	7.09349	0.581121	0.290561	−	0.956857i	$-0.406158\pi$
$149$	7.09349	0.581121	0.290561	+	0.956857i	$0.406158\pi$
$150$	0	0
$151$	10.1435	0.825467	0.412734	−	0.910852i	$-0.364574\pi$
$151$	10.1435	0.825467	0.412734	+	0.910852i	$0.364574\pi$
$152$	0	0
$153$	1.20198	0.0971746
$154$	0	0
$155$	−9.56274	−0.768097
$156$	0	0
$157$	2.97083	0.237098	0.118549	−	0.992948i	$-0.462176\pi$
$157$	2.97083	0.237098	0.118549	+	0.992948i	$0.462176\pi$
$158$	0	0
$159$	−17.6473	−1.39952
$160$	0	0
$161$	−0.819083	−0.0645527
$162$	0	0
$163$	21.0718	1.65047	0.825234	−	0.564791i	$-0.191043\pi$
$163$	21.0718	1.65047	0.825234	+	0.564791i	$0.191043\pi$
$164$	0	0
$165$	3.37206	0.262515
$166$	0	0
$167$	−12.4096	−0.960286	−0.480143	−	0.877190i	$-0.659415\pi$
$167$	−12.4096	−0.960286	−0.480143	+	0.877190i	$0.659415\pi$
$168$	0	0
$169$	9.26113	0.712395
$170$	0	0
$171$	0.424109	0.0324324
$172$	0	0
$173$	18.1690	1.38136	0.690680	−	0.723160i	$-0.257311\pi$
$173$	18.1690	1.38136	0.690680	+	0.723160i	$0.257311\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	21.3939	1.60806
$178$	0	0
$179$	13.5444	1.01235	0.506177	−	0.862430i	$-0.331058\pi$
$179$	13.5444	1.01235	0.506177	+	0.862430i	$0.331058\pi$
$180$	0	0
$181$	−18.6260	−1.38446	−0.692230	−	0.721677i	$-0.743372\pi$
$181$	−18.6260	−1.38446	−0.692230	+	0.721677i	$0.743372\pi$
$182$	0	0
$183$	10.2539	0.757990
$184$	0	0
$185$	0.321229	0.0236172
$186$	0	0
$187$	−4.22582	−0.309023
$188$	0	0
$189$	4.66281	0.339170
$190$	0	0
$191$	4.21785	0.305193	0.152597	−	0.988289i	$-0.451236\pi$
$191$	4.21785	0.305193	0.152597	+	0.988289i	$0.451236\pi$
$192$	0	0
$193$	−15.7353	−1.13265	−0.566325	−	0.824182i	$-0.691635\pi$
$193$	−15.7353	−1.13265	−0.566325	+	0.824182i	$0.691635\pi$
$194$	0	0
$195$	8.84178	0.633173
$196$	0	0
$197$	−23.1220	−1.64738	−0.823689	−	0.567042i	$-0.808088\pi$
$197$	−23.1220	−1.64738	−0.823689	+	0.567042i	$0.808088\pi$
$198$	0	0
$199$	−4.91166	−0.348179	−0.174089	−	0.984730i	$-0.555698\pi$
$199$	−4.91166	−0.348179	−0.174089	+	0.984730i	$0.555698\pi$
$200$	0	0
$201$	10.5573	0.744654
$202$	0	0
$203$	−4.13031	−0.289891
$204$	0	0
$205$	−10.2372	−0.715000
$206$	0	0
$207$	0.419222	0.0291380
$208$	0	0
$209$	−1.49104	−0.103138
$210$	0	0
$211$	11.3821	0.783577	0.391789	−	0.920055i	$-0.371856\pi$
$211$	11.3821	0.783577	0.391789	+	0.920055i	$0.371856\pi$
$212$	0	0
$213$	−6.14283	−0.420900
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	−9.56274	−0.649161
$218$	0	0
$219$	−3.42449	−0.231405
$220$	0	0
$221$	−11.0804	−0.745348
$222$	0	0
$223$	−19.8964	−1.33236	−0.666179	−	0.745792i	$-0.732071\pi$
$223$	−19.8964	−1.33236	−0.666179	+	0.745792i	$0.732071\pi$
$224$	0	0
$225$	0.511819	0.0341213
$226$	0	0
$227$	−1.61724	−0.107340	−0.0536701	−	0.998559i	$-0.517092\pi$
$227$	−1.61724	−0.107340	−0.0536701	+	0.998559i	$0.517092\pi$
$228$	0	0
$229$	12.9221	0.853920	0.426960	−	0.904271i	$-0.359585\pi$
$229$	12.9221	0.853920	0.426960	+	0.904271i	$0.359585\pi$
$230$	0	0
$231$	3.37206	0.221865
$232$	0	0
$233$	−13.4063	−0.878276	−0.439138	−	0.898420i	$-0.644716\pi$
$233$	−13.4063	−0.878276	−0.439138	+	0.898420i	$0.644716\pi$
$234$	0	0
$235$	5.71350	0.372707
$236$	0	0
$237$	4.86335	0.315908
$238$	0	0
$239$	−6.97546	−0.451205	−0.225603	−	0.974219i	$-0.572435\pi$
$239$	−6.97546	−0.451205	−0.225603	+	0.974219i	$0.572435\pi$
$240$	0	0
$241$	15.8884	1.02346	0.511730	−	0.859147i	$-0.329005\pi$
$241$	15.8884	1.02346	0.511730	+	0.859147i	$0.329005\pi$
$242$	0	0
$243$	−5.26394	−0.337682
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−3.90962	−0.248763
$248$	0	0
$249$	0.556079	0.0352401
$250$	0	0
$251$	27.8489	1.75781	0.878905	−	0.476997i	$-0.158275\pi$
$251$	27.8489	1.75781	0.878905	+	0.476997i	$0.158275\pi$
$252$	0	0
$253$	−1.47386	−0.0926610
$254$	0	0
$255$	−4.40097	−0.275599
$256$	0	0
$257$	5.78918	0.361119	0.180559	−	0.983564i	$-0.442209\pi$
$257$	5.78918	0.361119	0.180559	+	0.983564i	$0.442209\pi$
$258$	0	0
$259$	0.321229	0.0199602
$260$	0	0
$261$	2.11397	0.130852
$262$	0	0
$263$	1.37480	0.0847735	0.0423868	−	0.999101i	$-0.486504\pi$
$263$	1.37480	0.0847735	0.0423868	+	0.999101i	$0.486504\pi$
$264$	0	0
$265$	9.41701	0.578482
$266$	0	0
$267$	15.6826	0.959760
$268$	0	0
$269$	23.7949	1.45080	0.725401	−	0.688327i	$-0.241654\pi$
$269$	23.7949	1.45080	0.725401	+	0.688327i	$0.241654\pi$
$270$	0	0
$271$	−29.8190	−1.81138	−0.905688	−	0.423945i	$-0.860645\pi$
$271$	−29.8190	−1.81138	−0.905688	+	0.423945i	$0.860645\pi$
$272$	0	0
$273$	8.84178	0.535129
$274$	0	0
$275$	−1.79941	−0.108508
$276$	0	0
$277$	10.0712	0.605118	0.302559	−	0.953131i	$-0.402159\pi$
$277$	10.0712	0.605118	0.302559	+	0.953131i	$0.402159\pi$
$278$	0	0
$279$	4.89439	0.293020
$280$	0	0
$281$	−24.9969	−1.49119	−0.745595	−	0.666400i	$-0.767834\pi$
$281$	−24.9969	−1.49119	−0.745595	+	0.666400i	$0.767834\pi$
$282$	0	0
$283$	12.7664	0.758883	0.379442	−	0.925216i	$-0.376116\pi$
$283$	12.7664	0.758883	0.379442	+	0.925216i	$0.376116\pi$
$284$	0	0
$285$	−1.55284	−0.0919823
$286$	0	0
$287$	−10.2372	−0.604285
$288$	0	0
$289$	−11.4848	−0.675575
$290$	0	0
$291$	0.449831	0.0263696
$292$	0	0
$293$	5.46819	0.319455	0.159727	−	0.987161i	$-0.448938\pi$
$293$	5.46819	0.319455	0.159727	+	0.987161i	$0.448938\pi$
$294$	0	0
$295$	−11.4162	−0.664679
$296$	0	0
$297$	8.39030	0.486855
$298$	0	0
$299$	−3.86457	−0.223494
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	17.5440	1.00788
$304$	0	0
$305$	−5.47171	−0.313309
$306$	0	0
$307$	11.0727	0.631951	0.315976	−	0.948767i	$-0.397668\pi$
$307$	11.0727	0.631951	0.315976	+	0.948767i	$0.397668\pi$
$308$	0	0
$309$	5.45513	0.310332
$310$	0	0
$311$	−5.58073	−0.316454	−0.158227	−	0.987403i	$-0.550578\pi$
$311$	−5.58073	−0.316454	−0.158227	+	0.987403i	$0.550578\pi$
$312$	0	0
$313$	−11.1613	−0.630871	−0.315436	−	0.948947i	$-0.602151\pi$
$313$	−11.1613	−0.630871	−0.315436	+	0.948947i	$0.602151\pi$
$314$	0	0
$315$	0.511819	0.0288377
$316$	0	0
$317$	23.9852	1.34714	0.673572	−	0.739121i	$-0.264759\pi$
$317$	23.9852	1.34714	0.673572	+	0.739121i	$0.264759\pi$
$318$	0	0
$319$	−7.43211	−0.416118
$320$	0	0
$321$	15.4623	0.863020
$322$	0	0
$323$	1.94600	0.108278
$324$	0	0
$325$	−4.71817	−0.261717
$326$	0	0
$327$	14.4590	0.799584
$328$	0	0
$329$	5.71350	0.314995
$330$	0	0
$331$	−1.46186	−0.0803509	−0.0401755	−	0.999193i	$-0.512792\pi$
$331$	−1.46186	−0.0803509	−0.0401755	+	0.999193i	$0.512792\pi$
$332$	0	0
$333$	−0.164411	−0.00900966
$334$	0	0
$335$	−5.63361	−0.307797
$336$	0	0
$337$	29.4134	1.60225	0.801126	−	0.598496i	$-0.204235\pi$
$337$	29.4134	1.60225	0.801126	+	0.598496i	$0.204235\pi$
$338$	0	0
$339$	11.8757	0.645002
$340$	0	0
$341$	−17.2073	−0.931825
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.53495	−0.0826388
$346$	0	0
$347$	−4.34773	−0.233398	−0.116699	−	0.993167i	$-0.537231\pi$
$347$	−4.34773	−0.233398	−0.116699	+	0.993167i	$0.537231\pi$
$348$	0	0
$349$	0.848332	0.0454102	0.0227051	−	0.999742i	$-0.492772\pi$
$349$	0.848332	0.0454102	0.0227051	+	0.999742i	$0.492772\pi$
$350$	0	0
$351$	21.9999	1.17427
$352$	0	0
$353$	−28.6543	−1.52512	−0.762558	−	0.646919i	$-0.776057\pi$
$353$	−28.6543	−1.52512	−0.762558	+	0.646919i	$0.776057\pi$
$354$	0	0
$355$	3.27795	0.173976
$356$	0	0
$357$	−4.40097	−0.232924
$358$	0	0
$359$	−6.79226	−0.358482	−0.179241	−	0.983805i	$-0.557364\pi$
$359$	−6.79226	−0.358482	−0.179241	+	0.983805i	$0.557364\pi$
$360$	0	0
$361$	−18.3134	−0.963862
$362$	0	0
$363$	−14.5461	−0.763473
$364$	0	0
$365$	1.82738	0.0956496
$366$	0	0
$367$	0.292260	0.0152558	0.00762792	−	0.999971i	$-0.497572\pi$
$367$	0.292260	0.0152558	0.00762792	+	0.999971i	$0.497572\pi$
$368$	0	0
$369$	5.23961	0.272763
$370$	0	0
$371$	9.41701	0.488907
$372$	0	0
$373$	−35.2361	−1.82446	−0.912228	−	0.409683i	$-0.865639\pi$
$373$	−35.2361	−1.82446	−0.912228	+	0.409683i	$0.865639\pi$
$374$	0	0
$375$	−1.87398	−0.0967722
$376$	0	0
$377$	−19.4875	−1.00366
$378$	0	0
$379$	−5.79456	−0.297647	−0.148823	−	0.988864i	$-0.547549\pi$
$379$	−5.79456	−0.297647	−0.148823	+	0.988864i	$0.547549\pi$
$380$	0	0
$381$	19.9893	1.02408
$382$	0	0
$383$	−29.2398	−1.49408	−0.747041	−	0.664778i	$-0.768526\pi$
$383$	−29.2398	−1.49408	−0.747041	+	0.664778i	$0.768526\pi$
$384$	0	0
$385$	−1.79941	−0.0917063
$386$	0	0
$387$	−0.511819	−0.0260172
$388$	0	0
$389$	15.1281	0.767025	0.383512	−	0.923536i	$-0.374714\pi$
$389$	15.1281	0.767025	0.383512	+	0.923536i	$0.374714\pi$
$390$	0	0
$391$	1.92358	0.0972795
$392$	0	0
$393$	−17.8053	−0.898158
$394$	0	0
$395$	−2.59519	−0.130578
$396$	0	0
$397$	21.7497	1.09159	0.545793	−	0.837920i	$-0.316228\pi$
$397$	21.7497	1.09159	0.545793	+	0.837920i	$0.316228\pi$
$398$	0	0
$399$	−1.55284	−0.0777392
$400$	0	0
$401$	15.4497	0.771523	0.385762	−	0.922599i	$-0.373939\pi$
$401$	15.4497	0.771523	0.385762	+	0.922599i	$0.373939\pi$
$402$	0	0
$403$	−45.1186	−2.24752
$404$	0	0
$405$	10.2735	0.510494
$406$	0	0
$407$	0.578021	0.0286514
$408$	0	0
$409$	0.303181	0.0149914	0.00749568	−	0.999972i	$-0.497614\pi$
$409$	0.303181	0.0149914	0.00749568	+	0.999972i	$0.497614\pi$
$410$	0	0
$411$	24.6459	1.21569
$412$	0	0
$413$	−11.4162	−0.561757
$414$	0	0
$415$	−0.296736	−0.0145662
$416$	0	0
$417$	12.6275	0.618370
$418$	0	0
$419$	−21.3841	−1.04468	−0.522340	−	0.852737i	$-0.674941\pi$
$419$	−21.3841	−1.04468	−0.522340	+	0.852737i	$0.674941\pi$
$420$	0	0
$421$	22.7487	1.10870	0.554352	−	0.832282i	$-0.312966\pi$
$421$	22.7487	1.10870	0.554352	+	0.832282i	$0.312966\pi$
$422$	0	0
$423$	−2.92428	−0.142183
$424$	0	0
$425$	2.34845	0.113917
$426$	0	0
$427$	−5.47171	−0.264795
$428$	0	0
$429$	15.9100	0.768140
$430$	0	0
$431$	−21.0195	−1.01247	−0.506236	−	0.862395i	$-0.668964\pi$
$431$	−21.0195	−1.01247	−0.506236	+	0.862395i	$0.668964\pi$
$432$	0	0
$433$	−21.2742	−1.02237	−0.511187	−	0.859470i	$-0.670794\pi$
$433$	−21.2742	−1.02237	−0.511187	+	0.859470i	$0.670794\pi$
$434$	0	0
$435$	−7.74014	−0.371111
$436$	0	0
$437$	0.678716	0.0324674
$438$	0	0
$439$	16.1335	0.770008	0.385004	−	0.922915i	$-0.374200\pi$
$439$	16.1335	0.770008	0.385004	+	0.922915i	$0.374200\pi$
$440$	0	0
$441$	0.511819	0.0243723
$442$	0	0
$443$	16.7888	0.797659	0.398829	−	0.917025i	$-0.369417\pi$
$443$	16.7888	0.797659	0.398829	+	0.917025i	$0.369417\pi$
$444$	0	0
$445$	−8.36859	−0.396709
$446$	0	0
$447$	13.2931	0.628742
$448$	0	0
$449$	24.0153	1.13335	0.566676	−	0.823941i	$-0.308229\pi$
$449$	24.0153	1.13335	0.566676	+	0.823941i	$0.308229\pi$
$450$	0	0
$451$	−18.4210	−0.867409
$452$	0	0
$453$	19.0088	0.893110
$454$	0	0
$455$	−4.71817	−0.221191
$456$	0	0
$457$	32.0333	1.49845	0.749227	−	0.662313i	$-0.230425\pi$
$457$	32.0333	1.49845	0.749227	+	0.662313i	$0.230425\pi$
$458$	0	0
$459$	−10.9504	−0.511121
$460$	0	0
$461$	−36.6785	−1.70829	−0.854144	−	0.520037i	$-0.825918\pi$
$461$	−36.6785	−1.70829	−0.854144	+	0.520037i	$0.825918\pi$
$462$	0	0
$463$	27.6767	1.28625	0.643123	−	0.765763i	$-0.277639\pi$
$463$	27.6767	1.28625	0.643123	+	0.765763i	$0.277639\pi$
$464$	0	0
$465$	−17.9204	−0.831040
$466$	0	0
$467$	32.9517	1.52482	0.762411	−	0.647093i	$-0.224015\pi$
$467$	32.9517	1.52482	0.762411	+	0.647093i	$0.224015\pi$
$468$	0	0
$469$	−5.63361	−0.260136
$470$	0	0
$471$	5.56729	0.256527
$472$	0	0
$473$	1.79941	0.0827368
$474$	0	0
$475$	0.828630	0.0380201
$476$	0	0
$477$	−4.81981	−0.220684
$478$	0	0
$479$	−6.68909	−0.305632	−0.152816	−	0.988255i	$-0.548834\pi$
$479$	−6.68909	−0.305632	−0.152816	+	0.988255i	$0.548834\pi$
$480$	0	0
$481$	1.51561	0.0691059
$482$	0	0
$483$	−1.53495	−0.0698426
$484$	0	0
$485$	−0.240040	−0.0108996
$486$	0	0
$487$	17.1657	0.777851	0.388926	−	0.921269i	$-0.372846\pi$
$487$	17.1657	0.777851	0.388926	+	0.921269i	$0.372846\pi$
$488$	0	0
$489$	39.4882	1.78572
$490$	0	0
$491$	−16.6792	−0.752723	−0.376362	−	0.926473i	$-0.622825\pi$
$491$	−16.6792	−0.752723	−0.376362	+	0.926473i	$0.622825\pi$
$492$	0	0
$493$	9.69984	0.436859
$494$	0	0
$495$	0.920971	0.0413946
$496$	0	0
$497$	3.27795	0.147036
$498$	0	0
$499$	17.0943	0.765245	0.382622	−	0.923905i	$-0.375021\pi$
$499$	17.0943	0.765245	0.382622	+	0.923905i	$0.375021\pi$
$500$	0	0
$501$	−23.2555	−1.03898
$502$	0	0
$503$	−11.9043	−0.530789	−0.265394	−	0.964140i	$-0.585502\pi$
$503$	−11.9043	−0.530789	−0.265394	+	0.964140i	$0.585502\pi$
$504$	0	0
$505$	−9.36187	−0.416598
$506$	0	0
$507$	17.3552	0.770773
$508$	0	0
$509$	−10.7162	−0.474987	−0.237493	−	0.971389i	$-0.576326\pi$
$509$	−10.7162	−0.474987	−0.237493	+	0.971389i	$0.576326\pi$
$510$	0	0
$511$	1.82738	0.0808387
$512$	0	0
$513$	−3.86375	−0.170589
$514$	0	0
$515$	−2.91098	−0.128273
$516$	0	0
$517$	10.2809	0.452154
$518$	0	0
$519$	34.0483	1.49456
$520$	0	0
$521$	−15.1966	−0.665774	−0.332887	−	0.942967i	$-0.608023\pi$
$521$	−15.1966	−0.665774	−0.332887	+	0.942967i	$0.608023\pi$
$522$	0	0
$523$	14.7441	0.644713	0.322356	−	0.946618i	$-0.395525\pi$
$523$	14.7441	0.644713	0.322356	+	0.946618i	$0.395525\pi$
$524$	0	0
$525$	−1.87398	−0.0817874
$526$	0	0
$527$	22.4576	0.978270
$528$	0	0
$529$	−22.3291	−0.970831
$530$	0	0
$531$	5.84305	0.253567
$532$	0	0
$533$	−48.3010	−2.09215
$534$	0	0
$535$	−8.25101	−0.356722
$536$	0	0
$537$	25.3819	1.09531
$538$	0	0
$539$	−1.79941	−0.0775060
$540$	0	0
$541$	−12.7195	−0.546853	−0.273426	−	0.961893i	$-0.588157\pi$
$541$	−12.7195	−0.546853	−0.273426	+	0.961893i	$0.588157\pi$
$542$	0	0
$543$	−34.9049	−1.49791
$544$	0	0
$545$	−7.71564	−0.330502
$546$	0	0
$547$	−14.1735	−0.606014	−0.303007	−	0.952988i	$-0.597991\pi$
$547$	−14.1735	−0.606014	−0.303007	+	0.952988i	$0.597991\pi$
$548$	0	0
$549$	2.80053	0.119523
$550$	0	0
$551$	3.42250	0.145803
$552$	0	0
$553$	−2.59519	−0.110359
$554$	0	0
$555$	0.601978	0.0255525
$556$	0	0
$557$	−32.7090	−1.38593	−0.692963	−	0.720973i	$-0.743695\pi$
$557$	−32.7090	−1.38593	−0.692963	+	0.720973i	$0.743695\pi$
$558$	0	0
$559$	4.71817	0.199557
$560$	0	0
$561$	−7.91913	−0.334346
$562$	0	0
$563$	−2.89230	−0.121896	−0.0609480	−	0.998141i	$-0.519412\pi$
$563$	−2.89230	−0.121896	−0.0609480	+	0.998141i	$0.519412\pi$
$564$	0	0
$565$	−6.33716	−0.266606
$566$	0	0
$567$	10.2735	0.431446
$568$	0	0
$569$	10.0265	0.420332	0.210166	−	0.977666i	$-0.432600\pi$
$569$	10.0265	0.420332	0.210166	+	0.977666i	$0.432600\pi$
$570$	0	0
$571$	13.0418	0.545780	0.272890	−	0.962045i	$-0.412020\pi$
$571$	13.0418	0.545780	0.272890	+	0.962045i	$0.412020\pi$
$572$	0	0
$573$	7.90420	0.330203
$574$	0	0
$575$	0.819083	0.0341581
$576$	0	0
$577$	−22.9291	−0.954551	−0.477276	−	0.878754i	$-0.658376\pi$
$577$	−22.9291	−0.954551	−0.477276	+	0.878754i	$0.658376\pi$
$578$	0	0
$579$	−29.4876	−1.22546
$580$	0	0
$581$	−0.296736	−0.0123107
$582$	0	0
$583$	16.9450	0.701792
$584$	0	0
$585$	2.41485	0.0998418
$586$	0	0
$587$	6.18783	0.255399	0.127699	−	0.991813i	$-0.459241\pi$
$587$	6.18783	0.255399	0.127699	+	0.991813i	$0.459241\pi$
$588$	0	0
$589$	7.92397	0.326501
$590$	0	0
$591$	−43.3304	−1.78237
$592$	0	0
$593$	9.14437	0.375514	0.187757	−	0.982215i	$-0.439878\pi$
$593$	9.14437	0.375514	0.187757	+	0.982215i	$0.439878\pi$
$594$	0	0
$595$	2.34845	0.0962772
$596$	0	0
$597$	−9.20438	−0.376710
$598$	0	0
$599$	−11.3121	−0.462199	−0.231100	−	0.972930i	$-0.574232\pi$
$599$	−11.3121	−0.462199	−0.231100	+	0.972930i	$0.574232\pi$
$600$	0	0
$601$	34.3746	1.40217	0.701084	−	0.713079i	$-0.252700\pi$
$601$	34.3746	1.40217	0.701084	+	0.713079i	$0.252700\pi$
$602$	0	0
$603$	2.88339	0.117421
$604$	0	0
$605$	7.76213	0.315576
$606$	0	0
$607$	−0.407271	−0.0165306	−0.00826531	−	0.999966i	$-0.502631\pi$
$607$	−0.407271	−0.0165306	−0.00826531	+	0.999966i	$0.502631\pi$
$608$	0	0
$609$	−7.74014	−0.313646
$610$	0	0
$611$	26.9572	1.09057
$612$	0	0
$613$	−4.65424	−0.187983	−0.0939916	−	0.995573i	$-0.529963\pi$
$613$	−4.65424	−0.187983	−0.0939916	+	0.995573i	$0.529963\pi$
$614$	0	0
$615$	−19.1844	−0.773591
$616$	0	0
$617$	−21.9702	−0.884489	−0.442244	−	0.896895i	$-0.645818\pi$
$617$	−21.9702	−0.884489	−0.442244	+	0.896895i	$0.645818\pi$
$618$	0	0
$619$	1.49723	0.0601786	0.0300893	−	0.999547i	$-0.490421\pi$
$619$	1.49723	0.0601786	0.0300893	+	0.999547i	$0.490421\pi$
$620$	0	0
$621$	−3.81923	−0.153260
$622$	0	0
$623$	−8.36859	−0.335280
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.79419	−0.111589
$628$	0	0
$629$	−0.754390	−0.0300795
$630$	0	0
$631$	16.2583	0.647234	0.323617	−	0.946188i	$-0.395101\pi$
$631$	16.2583	0.647234	0.323617	+	0.946188i	$0.395101\pi$
$632$	0	0
$633$	21.3299	0.847788
$634$	0	0
$635$	−10.6667	−0.423297
$636$	0	0
$637$	−4.71817	−0.186941
$638$	0	0
$639$	−1.67772	−0.0663695
$640$	0	0
$641$	−7.70312	−0.304255	−0.152127	−	0.988361i	$-0.548612\pi$
$641$	−7.70312	−0.304255	−0.152127	+	0.988361i	$0.548612\pi$
$642$	0	0
$643$	13.4157	0.529062	0.264531	−	0.964377i	$-0.414783\pi$
$643$	13.4157	0.529062	0.264531	+	0.964377i	$0.414783\pi$
$644$	0	0
$645$	1.87398	0.0737881
$646$	0	0
$647$	−5.47198	−0.215126	−0.107563	−	0.994198i	$-0.534305\pi$
$647$	−5.47198	−0.215126	−0.107563	+	0.994198i	$0.534305\pi$
$648$	0	0
$649$	−20.5425	−0.806363
$650$	0	0
$651$	−17.9204	−0.702357
$652$	0	0
$653$	10.7254	0.419718	0.209859	−	0.977732i	$-0.432699\pi$
$653$	10.7254	0.419718	0.209859	+	0.977732i	$0.432699\pi$
$654$	0	0
$655$	9.50129	0.371246
$656$	0	0
$657$	−0.935290	−0.0364891
$658$	0	0
$659$	28.4624	1.10874	0.554370	−	0.832271i	$-0.312959\pi$
$659$	28.4624	1.10874	0.554370	+	0.832271i	$0.312959\pi$
$660$	0	0
$661$	13.8615	0.539150	0.269575	−	0.962979i	$-0.413117\pi$
$661$	13.8615	0.539150	0.269575	+	0.962979i	$0.413117\pi$
$662$	0	0
$663$	−20.7645	−0.806427
$664$	0	0
$665$	0.828630	0.0321329
$666$	0	0
$667$	3.38306	0.130993
$668$	0	0
$669$	−37.2855	−1.44154
$670$	0	0
$671$	−9.84583	−0.380094
$672$	0	0
$673$	−10.2941	−0.396808	−0.198404	−	0.980120i	$-0.563576\pi$
$673$	−10.2941	−0.396808	−0.198404	+	0.980120i	$0.563576\pi$
$674$	0	0
$675$	−4.66281	−0.179472
$676$	0	0
$677$	−16.2883	−0.626008	−0.313004	−	0.949752i	$-0.601335\pi$
$677$	−16.2883	−0.626008	−0.313004	+	0.949752i	$0.601335\pi$
$678$	0	0
$679$	−0.240040	−0.00921188
$680$	0	0
$681$	−3.03069	−0.116136
$682$	0	0
$683$	−34.2526	−1.31064	−0.655321	−	0.755351i	$-0.727466\pi$
$683$	−34.2526	−1.31064	−0.655321	+	0.755351i	$0.727466\pi$
$684$	0	0
$685$	−13.1516	−0.502496
$686$	0	0
$687$	24.2159	0.923895
$688$	0	0
$689$	44.4311	1.69269
$690$	0	0
$691$	−16.6645	−0.633946	−0.316973	−	0.948435i	$-0.602666\pi$
$691$	−16.6645	−0.633946	−0.316973	+	0.948435i	$0.602666\pi$
$692$	0	0
$693$	0.920971	0.0349848
$694$	0	0
$695$	−6.73830	−0.255598
$696$	0	0
$697$	24.0417	0.910643
$698$	0	0
$699$	−25.1232	−0.950247
$700$	0	0
$701$	0.795428	0.0300429	0.0150215	−	0.999887i	$-0.495218\pi$
$701$	0.795428	0.0300429	0.0150215	+	0.999887i	$0.495218\pi$
$702$	0	0
$703$	−0.266180	−0.0100392
$704$	0	0
$705$	10.7070	0.403249
$706$	0	0
$707$	−9.36187	−0.352090
$708$	0	0
$709$	15.3053	0.574803	0.287401	−	0.957810i	$-0.407209\pi$
$709$	15.3053	0.574803	0.287401	+	0.957810i	$0.407209\pi$
$710$	0	0
$711$	1.32827	0.0498139
$712$	0	0
$713$	7.83267	0.293336
$714$	0	0
$715$	−8.48991	−0.317505
$716$	0	0
$717$	−13.0719	−0.488179
$718$	0	0
$719$	−37.6053	−1.40244	−0.701220	−	0.712945i	$-0.747361\pi$
$719$	−37.6053	−1.40244	−0.701220	+	0.712945i	$0.747361\pi$
$720$	0	0
$721$	−2.91098	−0.108411
$722$	0	0
$723$	29.7745	1.10733
$724$	0	0
$725$	4.13031	0.153396
$726$	0	0
$727$	5.28896	0.196157	0.0980784	−	0.995179i	$-0.468730\pi$
$727$	5.28896	0.196157	0.0980784	+	0.995179i	$0.468730\pi$
$728$	0	0
$729$	20.9560	0.776146
$730$	0	0
$731$	−2.34845	−0.0868607
$732$	0	0
$733$	−0.460469	−0.0170078	−0.00850391	−	0.999964i	$-0.502707\pi$
$733$	−0.460469	−0.0170078	−0.00850391	+	0.999964i	$0.502707\pi$
$734$	0	0
$735$	−1.87398	−0.0691230
$736$	0	0
$737$	−10.1371	−0.373407
$738$	0	0
$739$	−4.26836	−0.157014	−0.0785071	−	0.996914i	$-0.525015\pi$
$739$	−4.26836	−0.157014	−0.0785071	+	0.996914i	$0.525015\pi$
$740$	0	0
$741$	−7.32656	−0.269148
$742$	0	0
$743$	−30.7372	−1.12764	−0.563819	−	0.825898i	$-0.690669\pi$
$743$	−30.7372	−1.12764	−0.563819	+	0.825898i	$0.690669\pi$
$744$	0	0
$745$	−7.09349	−0.259885
$746$	0	0
$747$	0.151875	0.00555682
$748$	0	0
$749$	−8.25101	−0.301485
$750$	0	0
$751$	−45.8339	−1.67250	−0.836252	−	0.548346i	$-0.815258\pi$
$751$	−45.8339	−1.67250	−0.836252	+	0.548346i	$0.815258\pi$
$752$	0	0
$753$	52.1885	1.90185
$754$	0	0
$755$	−10.1435	−0.369160
$756$	0	0
$757$	1.84286	0.0669799	0.0334899	−	0.999439i	$-0.489338\pi$
$757$	1.84286	0.0669799	0.0334899	+	0.999439i	$0.489338\pi$
$758$	0	0
$759$	−2.76200	−0.100254
$760$	0	0
$761$	34.5242	1.25150	0.625751	−	0.780023i	$-0.284793\pi$
$761$	34.5242	1.25150	0.625751	+	0.780023i	$0.284793\pi$
$762$	0	0
$763$	−7.71564	−0.279325
$764$	0	0
$765$	−1.20198	−0.0434578
$766$	0	0
$767$	−53.8638	−1.94491
$768$	0	0
$769$	5.65368	0.203877	0.101938	−	0.994791i	$-0.467496\pi$
$769$	5.65368	0.203877	0.101938	+	0.994791i	$0.467496\pi$
$770$	0	0
$771$	10.8488	0.390711
$772$	0	0
$773$	29.5823	1.06400	0.532001	−	0.846744i	$-0.321440\pi$
$773$	29.5823	1.06400	0.532001	+	0.846744i	$0.321440\pi$
$774$	0	0
$775$	9.56274	0.343504
$776$	0	0
$777$	0.601978	0.0215958
$778$	0	0
$779$	8.48288	0.303931
$780$	0	0
$781$	5.89837	0.211060
$782$	0	0
$783$	−19.2589	−0.688255
$784$	0	0
$785$	−2.97083	−0.106034
$786$	0	0
$787$	31.3100	1.11608	0.558041	−	0.829813i	$-0.311553\pi$
$787$	31.3100	1.11608	0.558041	+	0.829813i	$0.311553\pi$
$788$	0	0
$789$	2.57635	0.0917203
$790$	0	0
$791$	−6.33716	−0.225324
$792$	0	0
$793$	−25.8165	−0.916769
$794$	0	0
$795$	17.6473	0.625886
$796$	0	0
$797$	−31.6943	−1.12267	−0.561335	−	0.827589i	$-0.689712\pi$
$797$	−31.6943	−1.12267	−0.561335	+	0.827589i	$0.689712\pi$
$798$	0	0
$799$	−13.4179	−0.474690
$800$	0	0
$801$	4.28320	0.151340
$802$	0	0
$803$	3.28821	0.116038
$804$	0	0
$805$	0.819083	0.0288689
$806$	0	0
$807$	44.5913	1.56969
$808$	0	0
$809$	−1.54220	−0.0542209	−0.0271104	−	0.999632i	$-0.508631\pi$
$809$	−1.54220	−0.0542209	−0.0271104	+	0.999632i	$0.508631\pi$
$810$	0	0
$811$	−12.1348	−0.426109	−0.213054	−	0.977040i	$-0.568341\pi$
$811$	−12.1348	−0.426109	−0.213054	+	0.977040i	$0.568341\pi$
$812$	0	0
$813$	−55.8804	−1.95981
$814$	0	0
$815$	−21.0718	−0.738111
$816$	0	0
$817$	−0.828630	−0.0289901
$818$	0	0
$819$	2.41485	0.0843817
$820$	0	0
$821$	44.7087	1.56035	0.780173	−	0.625564i	$-0.215131\pi$
$821$	44.7087	1.56035	0.780173	+	0.625564i	$0.215131\pi$
$822$	0	0
$823$	−4.24978	−0.148138	−0.0740691	−	0.997253i	$-0.523599\pi$
$823$	−4.24978	−0.148138	−0.0740691	+	0.997253i	$0.523599\pi$
$824$	0	0
$825$	−3.37206	−0.117400
$826$	0	0
$827$	−45.8424	−1.59410	−0.797048	−	0.603916i	$-0.793606\pi$
$827$	−45.8424	−1.59410	−0.797048	+	0.603916i	$0.793606\pi$
$828$	0	0
$829$	−0.236134	−0.00820128	−0.00410064	−	0.999992i	$-0.501305\pi$
$829$	−0.236134	−0.00820128	−0.00410064	+	0.999992i	$0.501305\pi$
$830$	0	0
$831$	18.8732	0.654705
$832$	0	0
$833$	2.34845	0.0813691
$834$	0	0
$835$	12.4096	0.429453
$836$	0	0
$837$	−44.5892	−1.54123
$838$	0	0
$839$	−6.34854	−0.219176	−0.109588	−	0.993977i	$-0.534953\pi$
$839$	−6.34854	−0.219176	−0.109588	+	0.993977i	$0.534953\pi$
$840$	0	0
$841$	−11.9405	−0.411743
$842$	0	0
$843$	−46.8438	−1.61339
$844$	0	0
$845$	−9.26113	−0.318593
$846$	0	0
$847$	7.76213	0.266710
$848$	0	0
$849$	23.9240	0.821071
$850$	0	0
$851$	−0.263113	−0.00901939
$852$	0	0
$853$	43.0982	1.47565	0.737827	−	0.674990i	$-0.235852\pi$
$853$	43.0982	1.47565	0.737827	+	0.674990i	$0.235852\pi$
$854$	0	0
$855$	−0.424109	−0.0145042
$856$	0	0
$857$	19.0936	0.652224	0.326112	−	0.945331i	$-0.394261\pi$
$857$	19.0936	0.652224	0.326112	+	0.945331i	$0.394261\pi$
$858$	0	0
$859$	−10.4925	−0.358000	−0.179000	−	0.983849i	$-0.557286\pi$
$859$	−10.4925	−0.358000	−0.179000	+	0.983849i	$0.557286\pi$
$860$	0	0
$861$	−19.1844	−0.653804
$862$	0	0
$863$	−38.4020	−1.30722	−0.653610	−	0.756832i	$-0.726746\pi$
$863$	−38.4020	−1.30722	−0.653610	+	0.756832i	$0.726746\pi$
$864$	0	0
$865$	−18.1690	−0.617763
$866$	0	0
$867$	−21.5223	−0.730935
$868$	0	0
$869$	−4.66980	−0.158412
$870$	0	0
$871$	−26.5803	−0.900639
$872$	0	0
$873$	0.122857	0.00415808
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	48.1698	1.62658	0.813289	−	0.581861i	$-0.197675\pi$
$877$	48.1698	1.62658	0.813289	+	0.581861i	$0.197675\pi$
$878$	0	0
$879$	10.2473	0.345633
$880$	0	0
$881$	14.5677	0.490798	0.245399	−	0.969422i	$-0.421081\pi$
$881$	14.5677	0.490798	0.245399	+	0.969422i	$0.421081\pi$
$882$	0	0
$883$	56.1185	1.88854	0.944269	−	0.329174i	$-0.106770\pi$
$883$	56.1185	1.88854	0.944269	+	0.329174i	$0.106770\pi$
$884$	0	0
$885$	−21.3939	−0.719147
$886$	0	0
$887$	41.1964	1.38324	0.691620	−	0.722262i	$-0.256897\pi$
$887$	41.1964	1.38324	0.691620	+	0.722262i	$0.256897\pi$
$888$	0	0
$889$	−10.6667	−0.357751
$890$	0	0
$891$	18.4862	0.619311
$892$	0	0
$893$	−4.73437	−0.158430
$894$	0	0
$895$	−13.5444	−0.452738
$896$	0	0
$897$	−7.24215	−0.241808
$898$	0	0
$899$	39.4970	1.31730
$900$	0	0
$901$	−22.1154	−0.736771
$902$	0	0
$903$	1.87398	0.0623623
$904$	0	0
$905$	18.6260	0.619150
$906$	0	0
$907$	38.7285	1.28596	0.642979	−	0.765884i	$-0.277698\pi$
$907$	38.7285	1.28596	0.642979	+	0.765884i	$0.277698\pi$
$908$	0	0
$909$	4.79159	0.158927
$910$	0	0
$911$	−8.27712	−0.274233	−0.137117	−	0.990555i	$-0.543784\pi$
$911$	−8.27712	−0.274233	−0.137117	+	0.990555i	$0.543784\pi$
$912$	0	0
$913$	−0.533949	−0.0176711
$914$	0	0
$915$	−10.2539	−0.338983
$916$	0	0
$917$	9.50129	0.313760
$918$	0	0
$919$	−12.8243	−0.423034	−0.211517	−	0.977374i	$-0.567840\pi$
$919$	−12.8243	−0.423034	−0.211517	+	0.977374i	$0.567840\pi$
$920$	0	0
$921$	20.7500	0.683737
$922$	0	0
$923$	15.4659	0.509068
$924$	0	0
$925$	−0.321229	−0.0105619
$926$	0	0
$927$	1.48990	0.0489346
$928$	0	0
$929$	10.8393	0.355627	0.177814	−	0.984064i	$-0.443098\pi$
$929$	10.8393	0.355627	0.177814	+	0.984064i	$0.443098\pi$
$930$	0	0
$931$	0.828630	0.0271572
$932$	0	0
$933$	−10.4582	−0.342386
$934$	0	0
$935$	4.22582	0.138199
$936$	0	0
$937$	15.7730	0.515283	0.257641	−	0.966241i	$-0.417055\pi$
$937$	15.7730	0.515283	0.257641	+	0.966241i	$0.417055\pi$
$938$	0	0
$939$	−20.9160	−0.682568
$940$	0	0
$941$	−20.6190	−0.672159	−0.336080	−	0.941834i	$-0.609101\pi$
$941$	−20.6190	−0.672159	−0.336080	+	0.941834i	$0.609101\pi$
$942$	0	0
$943$	8.38514	0.273058
$944$	0	0
$945$	−4.66281	−0.151681
$946$	0	0
$947$	41.7447	1.35652	0.678260	−	0.734822i	$-0.262734\pi$
$947$	41.7447	1.35652	0.678260	+	0.734822i	$0.262734\pi$
$948$	0	0
$949$	8.62190	0.279879
$950$	0	0
$951$	44.9480	1.45754
$952$	0	0
$953$	5.05036	0.163597	0.0817986	−	0.996649i	$-0.473934\pi$
$953$	5.05036	0.163597	0.0817986	+	0.996649i	$0.473934\pi$
$954$	0	0
$955$	−4.21785	−0.136487
$956$	0	0
$957$	−13.9277	−0.450217
$958$	0	0
$959$	−13.1516	−0.424687
$960$	0	0
$961$	60.4459	1.94987
$962$	0	0
$963$	4.22303	0.136085
$964$	0	0
$965$	15.7353	0.506536
$966$	0	0
$967$	−19.0702	−0.613255	−0.306628	−	0.951830i	$-0.599201\pi$
$967$	−19.0702	−0.613255	−0.306628	+	0.951830i	$0.599201\pi$
$968$	0	0
$969$	3.64677	0.117151
$970$	0	0
$971$	22.1651	0.711313	0.355657	−	0.934617i	$-0.384257\pi$
$971$	22.1651	0.711313	0.355657	+	0.934617i	$0.384257\pi$
$972$	0	0
$973$	−6.73830	−0.216020
$974$	0	0
$975$	−8.84178	−0.283164
$976$	0	0
$977$	55.6148	1.77927	0.889637	−	0.456668i	$-0.150957\pi$
$977$	55.6148	1.77927	0.889637	+	0.456668i	$0.150957\pi$
$978$	0	0
$979$	−15.0585	−0.481272
$980$	0	0
$981$	3.94901	0.126082
$982$	0	0
$983$	−57.6149	−1.83763	−0.918815	−	0.394688i	$-0.870853\pi$
$983$	−57.6149	−1.83763	−0.918815	+	0.394688i	$0.870853\pi$
$984$	0	0
$985$	23.1220	0.736730
$986$	0	0
$987$	10.7070	0.340808
$988$	0	0
$989$	−0.819083	−0.0260453
$990$	0	0
$991$	48.4350	1.53859	0.769295	−	0.638894i	$-0.220608\pi$
$991$	48.4350	1.53859	0.769295	+	0.638894i	$0.220608\pi$
$992$	0	0
$993$	−2.73950	−0.0869353
$994$	0	0
$995$	4.91166	0.155710
$996$	0	0
$997$	−23.6963	−0.750469	−0.375234	−	0.926930i	$-0.622438\pi$
$997$	−23.6963	−0.750469	−0.375234	+	0.926930i	$0.622438\pi$
$998$	0	0
$999$	1.49783	0.0473892

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.j.1.11	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.j.1.11	✓	13	1.1	even	1	trivial

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.87398 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.511819 q^{9} +O(q^{10})\)	\(q+1.87398 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.511819 q^{9} -1.79941 q^{11} -4.71817 q^{13} -1.87398 q^{15} +2.34845 q^{17} +0.828630 q^{19} -1.87398 q^{21} +0.819083 q^{23} +1.00000 q^{25} -4.66281 q^{27} +4.13031 q^{29} +9.56274 q^{31} -3.37206 q^{33} +1.00000 q^{35} -0.321229 q^{37} -8.84178 q^{39} +10.2372 q^{41} -1.00000 q^{43} -0.511819 q^{45} -5.71350 q^{47} +1.00000 q^{49} +4.40097 q^{51} -9.41701 q^{53} +1.79941 q^{55} +1.55284 q^{57} +11.4162 q^{59} +5.47171 q^{61} -0.511819 q^{63} +4.71817 q^{65} +5.63361 q^{67} +1.53495 q^{69} -3.27795 q^{71} -1.82738 q^{73} +1.87398 q^{75} +1.79941 q^{77} +2.59519 q^{79} -10.2735 q^{81} +0.296736 q^{83} -2.34845 q^{85} +7.74014 q^{87} +8.36859 q^{89} +4.71817 q^{91} +17.9204 q^{93} -0.828630 q^{95} +0.240040 q^{97} -0.920971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9}+O(q^{10}) \) 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9	\( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9} + 6 q^{11} + q^{13} + q^{15} - 6 q^{17} + 2 q^{19} + q^{21} - 6 q^{23} + 13 q^{25} - q^{27} + 19 q^{29} - 24 q^{31} + 17 q^{33} + 13 q^{35} + 15 q^{39} + 4 q^{41} - 13 q^{43} - 18 q^{45} - 3 q^{47} + 13 q^{49} + 7 q^{51} - 5 q^{53} - 6 q^{55} + 6 q^{57} - 6 q^{59} + 3 q^{61} - 18 q^{63} - q^{65} - 2 q^{67} + 20 q^{69} + 18 q^{71} + 14 q^{73} - q^{75} - 6 q^{77} + 12 q^{79} + 37 q^{81} + 2 q^{83} + 6 q^{85} - 2 q^{87} + 17 q^{89} - q^{91} + 15 q^{93} - 2 q^{95} + 17 q^{97} + 80 q^{99}+O(q^{100}) \) 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9 + 6 * q^11 + q^13 + q^15 - 6 * q^17 + 2 * q^19 + q^21 - 6 * q^23 + 13 * q^25 - q^27 + 19 * q^29 - 24 * q^31 + 17 * q^33 + 13 * q^35 + 15 * q^39 + 4 * q^41 - 13 * q^43 - 18 * q^45 - 3 * q^47 + 13 * q^49 + 7 * q^51 - 5 * q^53 - 6 * q^55 + 6 * q^57 - 6 * q^59 + 3 * q^61 - 18 * q^63 - q^65 - 2 * q^67 + 20 * q^69 + 18 * q^71 + 14 * q^73 - q^75 - 6 * q^77 + 12 * q^79 + 37 * q^81 + 2 * q^83 + 6 * q^85 - 2 * q^87 + 17 * q^89 - q^91 + 15 * q^93 - 2 * q^95 + 17 * q^97 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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