LMFDB - Embedded newform 5220.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5220,2,Mod(1,5220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5220, 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("5220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5220.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:	$41.6819098551$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 580)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	5220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -1.52543 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -1.52543 q^{7} +1.52543 q^{11} +0.622216 q^{13} -5.52543 q^{17} +2.90321 q^{19} +2.90321 q^{23} +1.00000 q^{25} -1.00000 q^{29} -2.90321 q^{31} +1.52543 q^{35} +2.47457 q^{37} +8.23506 q^{41} -5.65878 q^{43} +4.70964 q^{47} -4.67307 q^{49} +11.2859 q^{53} -1.52543 q^{55} -10.2351 q^{59} -7.93978 q^{61} -0.622216 q^{65} -2.90321 q^{67} -7.47949 q^{71} +2.90321 q^{73} -2.32693 q^{77} -16.5763 q^{79} +6.76986 q^{83} +5.52543 q^{85} -13.9081 q^{89} -0.949145 q^{91} -2.90321 q^{95} +17.5669 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 2 q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 + 2 * q^7	$ 3 q - 3 q^{5} + 2 q^{7} - 2 q^{11} + 2 q^{13} - 10 q^{17} + 2 q^{19} + 2 q^{23} + 3 q^{25} - 3 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} - 2 q^{41} - 10 q^{43} - 6 q^{47} - q^{49} - 6 q^{53} + 2 q^{55} - 4 q^{59} - 10 q^{61} - 2 q^{65} - 2 q^{67} + 4 q^{71} + 2 q^{73} - 20 q^{77} - 30 q^{79} + 14 q^{83} + 10 q^{85} - 2 q^{89} - 16 q^{91} - 2 q^{95} + 6 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 2 * q^7 - 2 * q^11 + 2 * q^13 - 10 * q^17 + 2 * q^19 + 2 * q^23 + 3 * q^25 - 3 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 - 2 * q^41 - 10 * q^43 - 6 * q^47 - q^49 - 6 * q^53 + 2 * q^55 - 4 * q^59 - 10 * q^61 - 2 * q^65 - 2 * q^67 + 4 * q^71 + 2 * q^73 - 20 * q^77 - 30 * q^79 + 14 * q^83 + 10 * q^85 - 2 * q^89 - 16 * q^91 - 2 * q^95 + 6 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.52543	−0.576557	−0.288279	−	0.957547i	$-0.593083\pi$
$7$	−1.52543	−0.576557	−0.288279	+	0.957547i	$0.593083\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.52543	0.459934	0.229967	−	0.973198i	$-0.426138\pi$
$11$	1.52543	0.459934	0.229967	+	0.973198i	$0.426138\pi$
$12$	0	0
$13$	0.622216	0.172572	0.0862858	−	0.996270i	$-0.472500\pi$
$13$	0.622216	0.172572	0.0862858	+	0.996270i	$0.472500\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.52543	−1.34011	−0.670057	−	0.742310i	$-0.733730\pi$
$17$	−5.52543	−1.34011	−0.670057	+	0.742310i	$0.733730\pi$
$18$	0	0
$19$	2.90321	0.666042	0.333021	−	0.942919i	$-0.391932\pi$
$19$	2.90321	0.666042	0.333021	+	0.942919i	$0.391932\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.90321	0.605362	0.302681	−	0.953092i	$-0.402118\pi$
$23$	2.90321	0.605362	0.302681	+	0.953092i	$0.402118\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−2.90321	−0.521432	−0.260716	−	0.965416i	$-0.583959\pi$
$31$	−2.90321	−0.521432	−0.260716	+	0.965416i	$0.583959\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.52543	0.257844
$36$	0	0
$37$	2.47457	0.406817	0.203409	−	0.979094i	$-0.434798\pi$
$37$	2.47457	0.406817	0.203409	+	0.979094i	$0.434798\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.23506	1.28610	0.643050	−	0.765824i	$-0.277669\pi$
$41$	8.23506	1.28610	0.643050	+	0.765824i	$0.277669\pi$
$42$	0	0
$43$	−5.65878	−0.862956	−0.431478	−	0.902123i	$-0.642008\pi$
$43$	−5.65878	−0.862956	−0.431478	+	0.902123i	$0.642008\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.70964	0.686971	0.343485	−	0.939158i	$-0.388392\pi$
$47$	4.70964	0.686971	0.343485	+	0.939158i	$0.388392\pi$
$48$	0	0
$49$	−4.67307	−0.667582
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.2859	1.55024	0.775120	−	0.631814i	$-0.217689\pi$
$53$	11.2859	1.55024	0.775120	+	0.631814i	$0.217689\pi$
$54$	0	0
$55$	−1.52543	−0.205689
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.2351	−1.33249	−0.666246	−	0.745732i	$-0.732100\pi$
$59$	−10.2351	−1.33249	−0.666246	+	0.745732i	$0.732100\pi$
$60$	0	0
$61$	−7.93978	−1.01658	−0.508292	−	0.861185i	$-0.669723\pi$
$61$	−7.93978	−1.01658	−0.508292	+	0.861185i	$0.669723\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.622216	−0.0771764
$66$	0	0
$67$	−2.90321	−0.354684	−0.177342	−	0.984149i	$-0.556750\pi$
$67$	−2.90321	−0.354684	−0.177342	+	0.984149i	$0.556750\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.47949	−0.887653	−0.443826	−	0.896113i	$-0.646379\pi$
$71$	−7.47949	−0.887653	−0.443826	+	0.896113i	$0.646379\pi$
$72$	0	0
$73$	2.90321	0.339795	0.169898	−	0.985462i	$-0.445656\pi$
$73$	2.90321	0.339795	0.169898	+	0.985462i	$0.445656\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.32693	−0.265178
$78$	0	0
$79$	−16.5763	−1.86498	−0.932489	−	0.361199i	$-0.882368\pi$
$79$	−16.5763	−1.86498	−0.932489	+	0.361199i	$0.882368\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.76986	0.743089	0.371544	−	0.928415i	$-0.378828\pi$
$83$	6.76986	0.743089	0.371544	+	0.928415i	$0.378828\pi$
$84$	0	0
$85$	5.52543	0.599317
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.9081	−1.47426	−0.737130	−	0.675751i	$-0.763819\pi$
$89$	−13.9081	−1.47426	−0.737130	+	0.675751i	$0.763819\pi$
$90$	0	0
$91$	−0.949145	−0.0994974
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.90321	−0.297863
$96$	0	0
$97$	17.5669	1.78365	0.891825	−	0.452381i	$-0.149425\pi$
$97$	17.5669	1.78365	0.891825	+	0.452381i	$0.149425\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.05086	−0.502579	−0.251289	−	0.967912i	$-0.580855\pi$
$101$	−5.05086	−0.502579	−0.251289	+	0.967912i	$0.580855\pi$
$102$	0	0
$103$	9.56691	0.942656	0.471328	−	0.881958i	$-0.343775\pi$
$103$	9.56691	0.942656	0.471328	+	0.881958i	$0.343775\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.576283	0.0557113	0.0278557	−	0.999612i	$-0.491132\pi$
$107$	0.576283	0.0557113	0.0278557	+	0.999612i	$0.491132\pi$
$108$	0	0
$109$	7.28592	0.697864	0.348932	−	0.937148i	$-0.386544\pi$
$109$	7.28592	0.697864	0.348932	+	0.937148i	$0.386544\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−11.7605	−1.10633	−0.553167	−	0.833070i	$-0.686581\pi$
$113$	−11.7605	−1.10633	−0.553167	+	0.833070i	$0.686581\pi$
$114$	0	0
$115$	−2.90321	−0.270726
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.42864	0.772652
$120$	0	0
$121$	−8.67307	−0.788461
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.5161	−1.28809	−0.644046	−	0.764987i	$-0.722745\pi$
$127$	−14.5161	−1.28809	−0.644046	+	0.764987i	$0.722745\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.7605	1.37700	0.688500	−	0.725236i	$-0.258269\pi$
$131$	15.7605	1.37700	0.688500	+	0.725236i	$0.258269\pi$
$132$	0	0
$133$	−4.42864	−0.384012
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.52543	−0.472069	−0.236035	−	0.971745i	$-0.575848\pi$
$137$	−5.52543	−0.472069	−0.236035	+	0.971745i	$0.575848\pi$
$138$	0	0
$139$	−17.4193	−1.47748	−0.738742	−	0.673989i	$-0.764580\pi$
$139$	−17.4193	−1.47748	−0.738742	+	0.673989i	$0.764580\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.949145	0.0793715
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−5.37778	−0.437638	−0.218819	−	0.975765i	$-0.570220\pi$
$151$	−5.37778	−0.437638	−0.218819	+	0.975765i	$0.570220\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.90321	0.233192
$156$	0	0
$157$	−1.39207	−0.111100	−0.0555498	−	0.998456i	$-0.517691\pi$
$157$	−1.39207	−0.111100	−0.0555498	+	0.998456i	$0.517691\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.42864	−0.349026
$162$	0	0
$163$	7.19850	0.563830	0.281915	−	0.959439i	$-0.409030\pi$
$163$	7.19850	0.563830	0.281915	+	0.959439i	$0.409030\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.7605	−1.21958	−0.609792	−	0.792562i	$-0.708747\pi$
$167$	−15.7605	−1.21958	−0.609792	+	0.792562i	$0.708747\pi$
$168$	0	0
$169$	−12.6128	−0.970219
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−13.4795	−1.02483	−0.512413	−	0.858739i	$-0.671248\pi$
$173$	−13.4795	−1.02483	−0.512413	+	0.858739i	$0.671248\pi$
$174$	0	0
$175$	−1.52543	−0.115311
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0.266706	0.0199346	0.00996728	−	0.999950i	$-0.496827\pi$
$179$	0.266706	0.0199346	0.00996728	+	0.999950i	$0.496827\pi$
$180$	0	0
$181$	−20.7971	−1.54583	−0.772916	−	0.634508i	$-0.781203\pi$
$181$	−20.7971	−1.54583	−0.772916	+	0.634508i	$0.781203\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.47457	−0.181934
$186$	0	0
$187$	−8.42864	−0.616363
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.43356	0.682589	0.341294	−	0.939956i	$-0.389135\pi$
$191$	9.43356	0.682589	0.341294	+	0.939956i	$0.389135\pi$
$192$	0	0
$193$	5.23014	0.376474	0.188237	−	0.982124i	$-0.439723\pi$
$193$	5.23014	0.376474	0.188237	+	0.982124i	$0.439723\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.5397	−0.822171	−0.411085	−	0.911597i	$-0.634850\pi$
$197$	−11.5397	−0.822171	−0.411085	+	0.911597i	$0.634850\pi$
$198$	0	0
$199$	8.33677	0.590978	0.295489	−	0.955346i	$-0.404517\pi$
$199$	8.33677	0.590978	0.295489	+	0.955346i	$0.404517\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.52543	0.107064
$204$	0	0
$205$	−8.23506	−0.575162
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.42864	0.306335
$210$	0	0
$211$	9.79213	0.674118	0.337059	−	0.941483i	$-0.390568\pi$
$211$	9.79213	0.674118	0.337059	+	0.941483i	$0.390568\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.65878	0.385926
$216$	0	0
$217$	4.42864	0.300636
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.43801	−0.231265
$222$	0	0
$223$	16.0558	1.07517	0.537587	−	0.843208i	$-0.319336\pi$
$223$	16.0558	1.07517	0.537587	+	0.843208i	$0.319336\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−19.1985	−1.27425	−0.637125	−	0.770761i	$-0.719876\pi$
$227$	−19.1985	−1.27425	−0.637125	+	0.770761i	$0.719876\pi$
$228$	0	0
$229$	−10.5906	−0.699845	−0.349922	−	0.936779i	$-0.613792\pi$
$229$	−10.5906	−0.699845	−0.349922	+	0.936779i	$0.613792\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−23.1240	−1.51490	−0.757451	−	0.652892i	$-0.773556\pi$
$233$	−23.1240	−1.51490	−0.757451	+	0.652892i	$0.773556\pi$
$234$	0	0
$235$	−4.70964	−0.307223
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.9906	1.09903	0.549516	−	0.835483i	$-0.314812\pi$
$239$	16.9906	1.09903	0.549516	+	0.835483i	$0.314812\pi$
$240$	0	0
$241$	0.368416	0.0237318	0.0118659	−	0.999930i	$-0.496223\pi$
$241$	0.368416	0.0237318	0.0118659	+	0.999930i	$0.496223\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.67307	0.298552
$246$	0	0
$247$	1.80642	0.114940
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.7190	−0.992175	−0.496087	−	0.868273i	$-0.665230\pi$
$251$	−15.7190	−0.992175	−0.496087	+	0.868273i	$0.665230\pi$
$252$	0	0
$253$	4.42864	0.278426
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.6035	−1.16045	−0.580227	−	0.814455i	$-0.697036\pi$
$257$	−18.6035	−1.16045	−0.580227	+	0.814455i	$0.697036\pi$
$258$	0	0
$259$	−3.77478	−0.234553
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.7096	0.783710	0.391855	−	0.920027i	$-0.371834\pi$
$263$	12.7096	0.783710	0.391855	+	0.920027i	$0.371834\pi$
$264$	0	0
$265$	−11.2859	−0.693288
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−21.7462	−1.32589	−0.662945	−	0.748668i	$-0.730694\pi$
$269$	−21.7462	−1.32589	−0.662945	+	0.748668i	$0.730694\pi$
$270$	0	0
$271$	−18.3827	−1.11667	−0.558335	−	0.829616i	$-0.688560\pi$
$271$	−18.3827	−1.11667	−0.558335	+	0.829616i	$0.688560\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.52543	0.0919867
$276$	0	0
$277$	12.7971	0.768901	0.384450	−	0.923146i	$-0.374391\pi$
$277$	12.7971	0.768901	0.384450	+	0.923146i	$0.374391\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−21.1842	−1.26374	−0.631872	−	0.775073i	$-0.717713\pi$
$281$	−21.1842	−1.26374	−0.631872	+	0.775073i	$0.717713\pi$
$282$	0	0
$283$	−5.68736	−0.338079	−0.169039	−	0.985609i	$-0.554066\pi$
$283$	−5.68736	−0.338079	−0.169039	+	0.985609i	$0.554066\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.5620	−0.741511
$288$	0	0
$289$	13.5303	0.795903
$290$	0	0
$291$	0	0
$292$	0	0
$293$	25.0879	1.46565	0.732825	−	0.680417i	$-0.238201\pi$
$293$	25.0879	1.46565	0.732825	+	0.680417i	$0.238201\pi$
$294$	0	0
$295$	10.2351	0.595908
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.80642	0.104468
$300$	0	0
$301$	8.63206	0.497544
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.93978	0.454630
$306$	0	0
$307$	30.8113	1.75850	0.879248	−	0.476364i	$-0.158046\pi$
$307$	30.8113	1.75850	0.879248	+	0.476364i	$0.158046\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−30.0370	−1.70324	−0.851622	−	0.524156i	$-0.824381\pi$
$311$	−30.0370	−1.70324	−0.851622	+	0.524156i	$0.824381\pi$
$312$	0	0
$313$	−18.3368	−1.03646	−0.518228	−	0.855243i	$-0.673408\pi$
$313$	−18.3368	−1.03646	−0.518228	+	0.855243i	$0.673408\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.4558	−1.14891	−0.574457	−	0.818535i	$-0.694787\pi$
$317$	−20.4558	−1.14891	−0.574457	+	0.818535i	$0.694787\pi$
$318$	0	0
$319$	−1.52543	−0.0854075
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−16.0415	−0.892572
$324$	0	0
$325$	0.622216	0.0345143
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.18421	−0.396078
$330$	0	0
$331$	−12.0272	−0.661075	−0.330537	−	0.943793i	$-0.607230\pi$
$331$	−12.0272	−0.661075	−0.330537	+	0.943793i	$0.607230\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.90321	0.158619
$336$	0	0
$337$	−32.2306	−1.75571	−0.877857	−	0.478923i	$-0.841027\pi$
$337$	−32.2306	−1.75571	−0.877857	+	0.478923i	$0.841027\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.42864	−0.239824
$342$	0	0
$343$	17.8064	0.961457
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.9037	1.60531	0.802657	−	0.596441i	$-0.203419\pi$
$347$	29.9037	1.60531	0.802657	+	0.596441i	$0.203419\pi$
$348$	0	0
$349$	29.8163	1.59603	0.798014	−	0.602639i	$-0.205884\pi$
$349$	29.8163	1.59603	0.798014	+	0.602639i	$0.205884\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.1842	−1.12752	−0.563761	−	0.825938i	$-0.690646\pi$
$353$	−21.1842	−1.12752	−0.563761	+	0.825938i	$0.690646\pi$
$354$	0	0
$355$	7.47949	0.396970
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.1985	−0.591034	−0.295517	−	0.955337i	$-0.595492\pi$
$359$	−11.1985	−0.591034	−0.295517	+	0.955337i	$0.595492\pi$
$360$	0	0
$361$	−10.5714	−0.556387
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.90321	−0.151961
$366$	0	0
$367$	−9.95407	−0.519598	−0.259799	−	0.965663i	$-0.583656\pi$
$367$	−9.95407	−0.519598	−0.259799	+	0.965663i	$0.583656\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−17.2159	−0.893802
$372$	0	0
$373$	2.29529	0.118845	0.0594227	−	0.998233i	$-0.481074\pi$
$373$	2.29529	0.118845	0.0594227	+	0.998233i	$0.481074\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.622216	−0.0320457
$378$	0	0
$379$	9.61729	0.494007	0.247004	−	0.969015i	$-0.420554\pi$
$379$	9.61729	0.494007	0.247004	+	0.969015i	$0.420554\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.86172	0.350618	0.175309	−	0.984513i	$-0.443908\pi$
$383$	6.86172	0.350618	0.175309	+	0.984513i	$0.443908\pi$
$384$	0	0
$385$	2.32693	0.118591
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−25.3590	−1.28575	−0.642877	−	0.765969i	$-0.722260\pi$
$389$	−25.3590	−1.28575	−0.642877	+	0.765969i	$0.722260\pi$
$390$	0	0
$391$	−16.0415	−0.811253
$392$	0	0
$393$	0	0
$394$	0	0
$395$	16.5763	0.834043
$396$	0	0
$397$	24.2766	1.21841	0.609203	−	0.793015i	$-0.291490\pi$
$397$	24.2766	1.21841	0.609203	+	0.793015i	$0.291490\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.30465	−0.264902	−0.132451	−	0.991190i	$-0.542285\pi$
$401$	−5.30465	−0.264902	−0.132451	+	0.991190i	$0.542285\pi$
$402$	0	0
$403$	−1.80642	−0.0899844
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.77478	0.187109
$408$	0	0
$409$	0.714082	0.0353091	0.0176545	−	0.999844i	$-0.494380\pi$
$409$	0.714082	0.0353091	0.0176545	+	0.999844i	$0.494380\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	15.6128	0.768258
$414$	0	0
$415$	−6.76986	−0.332319
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16.6321	0.812529	0.406265	−	0.913755i	$-0.366831\pi$
$419$	16.6321	0.812529	0.406265	+	0.913755i	$0.366831\pi$
$420$	0	0
$421$	−30.4286	−1.48300	−0.741501	−	0.670952i	$-0.765886\pi$
$421$	−30.4286	−1.48300	−0.741501	+	0.670952i	$0.765886\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.52543	−0.268023
$426$	0	0
$427$	12.1116	0.586119
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.2257	1.50409	0.752044	−	0.659112i	$-0.229068\pi$
$431$	31.2257	1.50409	0.752044	+	0.659112i	$0.229068\pi$
$432$	0	0
$433$	35.2400	1.69353	0.846763	−	0.531971i	$-0.178548\pi$
$433$	35.2400	1.69353	0.846763	+	0.531971i	$0.178548\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.42864	0.403197
$438$	0	0
$439$	30.6133	1.46109	0.730547	−	0.682862i	$-0.239265\pi$
$439$	30.6133	1.46109	0.730547	+	0.682862i	$0.239265\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−31.4938	−1.49631	−0.748157	−	0.663521i	$-0.769061\pi$
$443$	−31.4938	−1.49631	−0.748157	+	0.663521i	$0.769061\pi$
$444$	0	0
$445$	13.9081	0.659309
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18.6321	0.879301	0.439651	−	0.898169i	$-0.355102\pi$
$449$	18.6321	0.879301	0.439651	+	0.898169i	$0.355102\pi$
$450$	0	0
$451$	12.5620	0.591521
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.949145	0.0444966
$456$	0	0
$457$	−31.2859	−1.46349	−0.731747	−	0.681577i	$-0.761295\pi$
$457$	−31.2859	−1.46349	−0.731747	+	0.681577i	$0.761295\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.9304	1.34742	0.673712	−	0.738994i	$-0.264699\pi$
$461$	28.9304	1.34742	0.673712	+	0.738994i	$0.264699\pi$
$462$	0	0
$463$	−14.1289	−0.656626	−0.328313	−	0.944569i	$-0.606480\pi$
$463$	−14.1289	−0.656626	−0.328313	+	0.944569i	$0.606480\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−30.6909	−1.42021	−0.710103	−	0.704098i	$-0.751352\pi$
$467$	−30.6909	−1.42021	−0.710103	+	0.704098i	$0.751352\pi$
$468$	0	0
$469$	4.42864	0.204496
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.63206	−0.396903
$474$	0	0
$475$	2.90321	0.133208
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.04641	−0.413341	−0.206670	−	0.978411i	$-0.566263\pi$
$479$	−9.04641	−0.413341	−0.206670	+	0.978411i	$0.566263\pi$
$480$	0	0
$481$	1.53972	0.0702051
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−17.5669	−0.797673
$486$	0	0
$487$	−14.6494	−0.663828	−0.331914	−	0.943310i	$-0.607694\pi$
$487$	−14.6494	−0.663828	−0.331914	+	0.943310i	$0.607694\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	22.1289	0.998664	0.499332	−	0.866411i	$-0.333579\pi$
$491$	22.1289	0.998664	0.499332	+	0.866411i	$0.333579\pi$
$492$	0	0
$493$	5.52543	0.248853
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.4094	0.511783
$498$	0	0
$499$	29.9813	1.34215	0.671073	−	0.741391i	$-0.265834\pi$
$499$	29.9813	1.34215	0.671073	+	0.741391i	$0.265834\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17.6874	0.788640	0.394320	−	0.918973i	$-0.370980\pi$
$503$	17.6874	0.788640	0.394320	+	0.918973i	$0.370980\pi$
$504$	0	0
$505$	5.05086	0.224760
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.45091	−0.0643107	−0.0321553	−	0.999483i	$-0.510237\pi$
$509$	−1.45091	−0.0643107	−0.0321553	+	0.999483i	$0.510237\pi$
$510$	0	0
$511$	−4.42864	−0.195911
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.56691	−0.421569
$516$	0	0
$517$	7.18421	0.315961
$518$	0	0
$519$	0	0
$520$	0	0
$521$	31.8765	1.39653	0.698267	−	0.715837i	$-0.253955\pi$
$521$	31.8765	1.39653	0.698267	+	0.715837i	$0.253955\pi$
$522$	0	0
$523$	14.9032	0.651672	0.325836	−	0.945426i	$-0.394354\pi$
$523$	14.9032	0.651672	0.325836	+	0.945426i	$0.394354\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.0415	0.698778
$528$	0	0
$529$	−14.5714	−0.633537
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.12399	0.221944
$534$	0	0
$535$	−0.576283	−0.0249149
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7.12843	−0.307043
$540$	0	0
$541$	−12.7841	−0.549633	−0.274817	−	0.961497i	$-0.588617\pi$
$541$	−12.7841	−0.549633	−0.274817	+	0.961497i	$0.588617\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.28592	−0.312094
$546$	0	0
$547$	−18.8617	−0.806469	−0.403235	−	0.915097i	$-0.632114\pi$
$547$	−18.8617	−0.806469	−0.403235	+	0.915097i	$0.632114\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.90321	−0.123681
$552$	0	0
$553$	25.2859	1.07527
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15.6316	0.662331	0.331166	−	0.943573i	$-0.392558\pi$
$557$	15.6316	0.662331	0.331166	+	0.943573i	$0.392558\pi$
$558$	0	0
$559$	−3.52098	−0.148922
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.18865	0.0500958	0.0250479	−	0.999686i	$-0.492026\pi$
$563$	1.18865	0.0500958	0.0250479	+	0.999686i	$0.492026\pi$
$564$	0	0
$565$	11.7605	0.494768
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−28.5303	−1.19605	−0.598027	−	0.801476i	$-0.704049\pi$
$569$	−28.5303	−1.19605	−0.598027	+	0.801476i	$0.704049\pi$
$570$	0	0
$571$	−22.8385	−0.955763	−0.477882	−	0.878424i	$-0.658595\pi$
$571$	−22.8385	−0.955763	−0.477882	+	0.878424i	$0.658595\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.90321	0.121072
$576$	0	0
$577$	4.18913	0.174396	0.0871979	−	0.996191i	$-0.472209\pi$
$577$	4.18913	0.174396	0.0871979	+	0.996191i	$0.472209\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−10.3269	−0.428433
$582$	0	0
$583$	17.2159	0.713008
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−39.2815	−1.62132	−0.810660	−	0.585517i	$-0.800891\pi$
$587$	−39.2815	−1.62132	−0.810660	+	0.585517i	$0.800891\pi$
$588$	0	0
$589$	−8.42864	−0.347296
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.4509	0.880883	0.440442	−	0.897781i	$-0.354822\pi$
$593$	21.4509	0.880883	0.440442	+	0.897781i	$0.354822\pi$
$594$	0	0
$595$	−8.42864	−0.345541
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18.3827	−0.751097	−0.375549	−	0.926803i	$-0.622546\pi$
$599$	−18.3827	−0.751097	−0.375549	+	0.926803i	$0.622546\pi$
$600$	0	0
$601$	−12.3555	−0.503992	−0.251996	−	0.967728i	$-0.581087\pi$
$601$	−12.3555	−0.503992	−0.251996	+	0.967728i	$0.581087\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.67307	0.352610
$606$	0	0
$607$	48.4340	1.96588	0.982938	−	0.183935i	$-0.0588835\pi$
$607$	48.4340	1.96588	0.982938	+	0.183935i	$0.0588835\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.93041	0.118552
$612$	0	0
$613$	−0.958989	−0.0387332	−0.0193666	−	0.999812i	$-0.506165\pi$
$613$	−0.958989	−0.0387332	−0.0193666	+	0.999812i	$0.506165\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.09679	−0.205189	−0.102594	−	0.994723i	$-0.532714\pi$
$617$	−5.09679	−0.205189	−0.102594	+	0.994723i	$0.532714\pi$
$618$	0	0
$619$	0.0142901	0.000574368 0	0.000287184	−	1.00000i	$-0.499909\pi$
$619$	0.0142901	0.000574368 0	0.000287184		1.00000i	$0.499909\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	21.2159	0.849995
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.6731	−0.545181
$630$	0	0
$631$	−12.8859	−0.512978	−0.256489	−	0.966547i	$-0.582566\pi$
$631$	−12.8859	−0.512978	−0.256489	+	0.966547i	$0.582566\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14.5161	0.576052
$636$	0	0
$637$	−2.90766	−0.115206
$638$	0	0
$639$	0	0
$640$	0	0
$641$	46.6865	1.84400	0.922002	−	0.387185i	$-0.126553\pi$
$641$	46.6865	1.84400	0.922002	+	0.387185i	$0.126553\pi$
$642$	0	0
$643$	−28.1388	−1.10968	−0.554842	−	0.831956i	$-0.687221\pi$
$643$	−28.1388	−1.10968	−0.554842	+	0.831956i	$0.687221\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	47.5067	1.86768	0.933840	−	0.357690i	$-0.116436\pi$
$647$	47.5067	1.86768	0.933840	+	0.357690i	$0.116436\pi$
$648$	0	0
$649$	−15.6128	−0.612858
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25.5383	−0.999392	−0.499696	−	0.866201i	$-0.666555\pi$
$653$	−25.5383	−0.999392	−0.499696	+	0.866201i	$0.666555\pi$
$654$	0	0
$655$	−15.7605	−0.615813
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−11.4938	−0.447734	−0.223867	−	0.974620i	$-0.571868\pi$
$659$	−11.4938	−0.447734	−0.223867	+	0.974620i	$0.571868\pi$
$660$	0	0
$661$	48.4929	1.88615	0.943077	−	0.332574i	$-0.107917\pi$
$661$	48.4929	1.88615	0.943077	+	0.332574i	$0.107917\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.42864	0.171735
$666$	0	0
$667$	−2.90321	−0.112413
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.1116	−0.467561
$672$	0	0
$673$	−15.1526	−0.584088	−0.292044	−	0.956405i	$-0.594335\pi$
$673$	−15.1526	−0.584088	−0.292044	+	0.956405i	$0.594335\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−38.8943	−1.49483	−0.747415	−	0.664357i	$-0.768705\pi$
$677$	−38.8943	−1.49483	−0.747415	+	0.664357i	$0.768705\pi$
$678$	0	0
$679$	−26.7971	−1.02838
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11.3733	−0.435189	−0.217594	−	0.976039i	$-0.569821\pi$
$683$	−11.3733	−0.435189	−0.217594	+	0.976039i	$0.569821\pi$
$684$	0	0
$685$	5.52543	0.211116
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.02227	0.267527
$690$	0	0
$691$	−29.4893	−1.12183	−0.560914	−	0.827874i	$-0.689550\pi$
$691$	−29.4893	−1.12183	−0.560914	+	0.827874i	$0.689550\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.4193	0.660751
$696$	0	0
$697$	−45.5022	−1.72352
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−52.2895	−1.97495	−0.987473	−	0.157789i	$-0.949563\pi$
$701$	−52.2895	−1.97495	−0.987473	+	0.157789i	$0.949563\pi$
$702$	0	0
$703$	7.18421	0.270958
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.70471	0.289766
$708$	0	0
$709$	9.18421	0.344920	0.172460	−	0.985017i	$-0.444828\pi$
$709$	9.18421	0.344920	0.172460	+	0.985017i	$0.444828\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.42864	−0.315655
$714$	0	0
$715$	−0.949145	−0.0354960
$716$	0	0
$717$	0	0
$718$	0	0
$719$	7.74620	0.288885	0.144442	−	0.989513i	$-0.453861\pi$
$719$	7.74620	0.288885	0.144442	+	0.989513i	$0.453861\pi$
$720$	0	0
$721$	−14.5936	−0.543495
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	5.06821	0.187969	0.0939847	−	0.995574i	$-0.470040\pi$
$727$	5.06821	0.187969	0.0939847	+	0.995574i	$0.470040\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	31.2672	1.15646
$732$	0	0
$733$	−0.659257	−0.0243502	−0.0121751	−	0.999926i	$-0.503876\pi$
$733$	−0.659257	−0.0243502	−0.0121751	+	0.999926i	$0.503876\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.42864	−0.163131
$738$	0	0
$739$	−7.07805	−0.260370	−0.130185	−	0.991490i	$-0.541557\pi$
$739$	−7.07805	−0.260370	−0.130185	+	0.991490i	$0.541557\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.3412	0.966366	0.483183	−	0.875519i	$-0.339481\pi$
$743$	26.3412	0.966366	0.483183	+	0.875519i	$0.339481\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−0.879077	−0.0321208
$750$	0	0
$751$	4.92195	0.179604	0.0898022	−	0.995960i	$-0.471377\pi$
$751$	4.92195	0.179604	0.0898022	+	0.995960i	$0.471377\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.37778	0.195718
$756$	0	0
$757$	30.9032	1.12320	0.561598	−	0.827410i	$-0.310187\pi$
$757$	30.9032	1.12320	0.561598	+	0.827410i	$0.310187\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−31.3274	−1.13562	−0.567809	−	0.823160i	$-0.692209\pi$
$761$	−31.3274	−1.13562	−0.567809	+	0.823160i	$0.692209\pi$
$762$	0	0
$763$	−11.1141	−0.402359
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.36842	−0.229950
$768$	0	0
$769$	−26.5620	−0.957850	−0.478925	−	0.877856i	$-0.658973\pi$
$769$	−26.5620	−0.957850	−0.478925	+	0.877856i	$0.658973\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14.8528	0.534219	0.267110	−	0.963666i	$-0.413931\pi$
$773$	14.8528	0.534219	0.267110	+	0.963666i	$0.413931\pi$
$774$	0	0
$775$	−2.90321	−0.104286
$776$	0	0
$777$	0	0
$778$	0	0
$779$	23.9081	0.856598
$780$	0	0
$781$	−11.4094	−0.408261
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.39207	0.0496853
$786$	0	0
$787$	−25.5540	−0.910902	−0.455451	−	0.890261i	$-0.650522\pi$
$787$	−25.5540	−0.910902	−0.455451	+	0.890261i	$0.650522\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	17.9398	0.637865
$792$	0	0
$793$	−4.94025	−0.175434
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.2208	0.362038	0.181019	−	0.983480i	$-0.442060\pi$
$797$	10.2208	0.362038	0.181019	+	0.983480i	$0.442060\pi$
$798$	0	0
$799$	−26.0228	−0.920619
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.42864	0.156283
$804$	0	0
$805$	4.42864	0.156089
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.44155	0.0858402	0.0429201	−	0.999079i	$-0.486334\pi$
$809$	2.44155	0.0858402	0.0429201	+	0.999079i	$0.486334\pi$
$810$	0	0
$811$	−2.33984	−0.0821628	−0.0410814	−	0.999156i	$-0.513080\pi$
$811$	−2.33984	−0.0821628	−0.0410814	+	0.999156i	$0.513080\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.19850	−0.252152
$816$	0	0
$817$	−16.4286	−0.574765
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14.6539	−0.511423	−0.255712	−	0.966753i	$-0.582310\pi$
$821$	−14.6539	−0.511423	−0.255712	+	0.966753i	$0.582310\pi$
$822$	0	0
$823$	47.3733	1.65133	0.825665	−	0.564160i	$-0.190800\pi$
$823$	47.3733	1.65133	0.825665	+	0.564160i	$0.190800\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18.3412	0.637787	0.318893	−	0.947791i	$-0.396689\pi$
$827$	18.3412	0.637787	0.318893	+	0.947791i	$0.396689\pi$
$828$	0	0
$829$	45.5625	1.58245	0.791225	−	0.611525i	$-0.209444\pi$
$829$	45.5625	1.58245	0.791225	+	0.611525i	$0.209444\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	25.8207	0.894635
$834$	0	0
$835$	15.7605	0.545414
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30.9733	1.06932	0.534658	−	0.845068i	$-0.320440\pi$
$839$	30.9733	1.06932	0.534658	+	0.845068i	$0.320440\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.6128	0.433895
$846$	0	0
$847$	13.2301	0.454593
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.18421	0.246271
$852$	0	0
$853$	25.4050	0.869850	0.434925	−	0.900467i	$-0.356775\pi$
$853$	25.4050	0.869850	0.434925	+	0.900467i	$0.356775\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.2573	0.657818	0.328909	−	0.944362i	$-0.393319\pi$
$857$	19.2573	0.657818	0.328909	+	0.944362i	$0.393319\pi$
$858$	0	0
$859$	42.2623	1.44197	0.720985	−	0.692951i	$-0.243690\pi$
$859$	42.2623	1.44197	0.720985	+	0.692951i	$0.243690\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.88400	0.336455	0.168228	−	0.985748i	$-0.446196\pi$
$863$	9.88400	0.336455	0.168228	+	0.985748i	$0.446196\pi$
$864$	0	0
$865$	13.4795	0.458317
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−25.2859	−0.857766
$870$	0	0
$871$	−1.80642	−0.0612083
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.52543	0.0515689
$876$	0	0
$877$	−12.6508	−0.427187	−0.213594	−	0.976923i	$-0.568517\pi$
$877$	−12.6508	−0.427187	−0.213594	+	0.976923i	$0.568517\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0.0888033	0.00299186	0.00149593	−	0.999999i	$-0.499524\pi$
$881$	0.0888033	0.00299186	0.00149593	+	0.999999i	$0.499524\pi$
$882$	0	0
$883$	57.3131	1.92874	0.964370	−	0.264557i	$-0.0852258\pi$
$883$	57.3131	1.92874	0.964370	+	0.264557i	$0.0852258\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−27.0781	−0.909192	−0.454596	−	0.890698i	$-0.650216\pi$
$887$	−27.0781	−0.909192	−0.454596	+	0.890698i	$0.650216\pi$
$888$	0	0
$889$	22.1432	0.742659
$890$	0	0
$891$	0	0
$892$	0	0
$893$	13.6731	0.457552
$894$	0	0
$895$	−0.266706	−0.00891500
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.90321	0.0968275
$900$	0	0
$901$	−62.3595	−2.07750
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.7971	0.691318
$906$	0	0
$907$	−38.6276	−1.28261	−0.641304	−	0.767287i	$-0.721606\pi$
$907$	−38.6276	−1.28261	−0.641304	+	0.767287i	$0.721606\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.27163	−0.174657	−0.0873284	−	0.996180i	$-0.527833\pi$
$911$	−5.27163	−0.174657	−0.0873284	+	0.996180i	$0.527833\pi$
$912$	0	0
$913$	10.3269	0.341771
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−24.0415	−0.793920
$918$	0	0
$919$	11.6632	0.384734	0.192367	−	0.981323i	$-0.438384\pi$
$919$	11.6632	0.384734	0.192367	+	0.981323i	$0.438384\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4.65386	−0.153184
$924$	0	0
$925$	2.47457	0.0813634
$926$	0	0
$927$	0	0
$928$	0	0
$929$	46.8198	1.53611	0.768054	−	0.640385i	$-0.221225\pi$
$929$	46.8198	1.53611	0.768054	+	0.640385i	$0.221225\pi$
$930$	0	0
$931$	−13.5669	−0.444638
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.42864	0.275646
$936$	0	0
$937$	4.44738	0.145289	0.0726447	−	0.997358i	$-0.476856\pi$
$937$	4.44738	0.145289	0.0726447	+	0.997358i	$0.476856\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	12.6726	0.413115	0.206557	−	0.978434i	$-0.433774\pi$
$941$	12.6726	0.413115	0.206557	+	0.978434i	$0.433774\pi$
$942$	0	0
$943$	23.9081	0.778556
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.2306	0.657407	0.328703	−	0.944433i	$-0.393388\pi$
$947$	20.2306	0.657407	0.328703	+	0.944433i	$0.393388\pi$
$948$	0	0
$949$	1.80642	0.0586390
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24.5718	0.795960	0.397980	−	0.917394i	$-0.369711\pi$
$953$	24.5718	0.795960	0.397980	+	0.917394i	$0.369711\pi$
$954$	0	0
$955$	−9.43356	−0.305263
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.42864	0.272175
$960$	0	0
$961$	−22.5714	−0.728108
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.23014	−0.168364
$966$	0	0
$967$	52.9960	1.70424	0.852119	−	0.523349i	$-0.175317\pi$
$967$	52.9960	1.70424	0.852119	+	0.523349i	$0.175317\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.02720	−0.129239	−0.0646195	−	0.997910i	$-0.520583\pi$
$971$	−4.02720	−0.129239	−0.0646195	+	0.997910i	$0.520583\pi$
$972$	0	0
$973$	26.5718	0.851854
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.2761	−0.424739	−0.212370	−	0.977189i	$-0.568118\pi$
$977$	−13.2761	−0.424739	−0.212370	+	0.977189i	$0.568118\pi$
$978$	0	0
$979$	−21.2159	−0.678062
$980$	0	0
$981$	0	0
$982$	0	0
$983$	23.5857	0.752265	0.376133	−	0.926566i	$-0.377254\pi$
$983$	23.5857	0.752265	0.376133	+	0.926566i	$0.377254\pi$
$984$	0	0
$985$	11.5397	0.367686
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.4286	−0.522400
$990$	0	0
$991$	50.7882	1.61334	0.806670	−	0.591003i	$-0.201268\pi$
$991$	50.7882	1.61334	0.806670	+	0.591003i	$0.201268\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−8.33677	−0.264293
$996$	0	0
$997$	−54.5705	−1.72826	−0.864132	−	0.503266i	$-0.832132\pi$
$997$	−54.5705	−1.72826	−0.864132	+	0.503266i	$0.832132\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5220.2.a.w.1.1		3
3.2	odd	2		580.2.a.d.1.3	✓	3
12.11	even	2		2320.2.a.o.1.1		3
15.2	even	4		2900.2.c.g.349.1		6
15.8	even	4		2900.2.c.g.349.6		6
15.14	odd	2		2900.2.a.f.1.1		3
24.5	odd	2		9280.2.a.bh.1.1		3
24.11	even	2		9280.2.a.bt.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.2.a.d.1.3	✓	3	3.2	odd	2
2320.2.a.o.1.1		3	12.11	even	2
2900.2.a.f.1.1		3	15.14	odd	2
2900.2.c.g.349.1		6	15.2	even	4
2900.2.c.g.349.6		6	15.8	even	4
5220.2.a.w.1.1		3	1.1	even	1	trivial
9280.2.a.bh.1.1		3	24.5	odd	2
9280.2.a.bt.1.3		3	24.11	even	2

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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:	\( N \)	\(=\)	\( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5220.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -1.52543 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -1.52543 q^{7} +1.52543 q^{11} +0.622216 q^{13} -5.52543 q^{17} +2.90321 q^{19} +2.90321 q^{23} +1.00000 q^{25} -1.00000 q^{29} -2.90321 q^{31} +1.52543 q^{35} +2.47457 q^{37} +8.23506 q^{41} -5.65878 q^{43} +4.70964 q^{47} -4.67307 q^{49} +11.2859 q^{53} -1.52543 q^{55} -10.2351 q^{59} -7.93978 q^{61} -0.622216 q^{65} -2.90321 q^{67} -7.47949 q^{71} +2.90321 q^{73} -2.32693 q^{77} -16.5763 q^{79} +6.76986 q^{83} +5.52543 q^{85} -13.9081 q^{89} -0.949145 q^{91} -2.90321 q^{95} +17.5669 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 2 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 + 2 * q^7	\( 3 q - 3 q^{5} + 2 q^{7} - 2 q^{11} + 2 q^{13} - 10 q^{17} + 2 q^{19} + 2 q^{23} + 3 q^{25} - 3 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} - 2 q^{41} - 10 q^{43} - 6 q^{47} - q^{49} - 6 q^{53} + 2 q^{55} - 4 q^{59} - 10 q^{61} - 2 q^{65} - 2 q^{67} + 4 q^{71} + 2 q^{73} - 20 q^{77} - 30 q^{79} + 14 q^{83} + 10 q^{85} - 2 q^{89} - 16 q^{91} - 2 q^{95} + 6 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 2 * q^7 - 2 * q^11 + 2 * q^13 - 10 * q^17 + 2 * q^19 + 2 * q^23 + 3 * q^25 - 3 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 - 2 * q^41 - 10 * q^43 - 6 * q^47 - q^49 - 6 * q^53 + 2 * q^55 - 4 * q^59 - 10 * q^61 - 2 * q^65 - 2 * q^67 + 4 * q^71 + 2 * q^73 - 20 * q^77 - 30 * q^79 + 14 * q^83 + 10 * q^85 - 2 * q^89 - 16 * q^91 - 2 * q^95 + 6 * q^97

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