LMFDB - Embedded newform 5040.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5040,2,Mod(1,5040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5040.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:	$40.2446026187$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2520)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	5040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -1.00000 q^{7} -3.12311 q^{11} +2.00000 q^{13} +3.12311 q^{17} +1.12311 q^{19} +3.12311 q^{23} +1.00000 q^{25} -5.12311 q^{29} -6.24621 q^{31} +1.00000 q^{35} +7.12311 q^{37} -8.24621 q^{41} +1.12311 q^{43} +1.12311 q^{47} +1.00000 q^{49} +1.12311 q^{53} +3.12311 q^{55} -4.00000 q^{59} +11.1231 q^{61} -2.00000 q^{65} +1.12311 q^{67} -6.00000 q^{71} -4.24621 q^{73} +3.12311 q^{77} -8.00000 q^{79} -3.12311 q^{85} +4.24621 q^{89} -2.00000 q^{91} -1.12311 q^{95} +6.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^7	$ 2 q - 2 q^{5} - 2 q^{7} + 2 q^{11} + 4 q^{13} - 2 q^{17} - 6 q^{19} - 2 q^{23} + 2 q^{25} - 2 q^{29} + 4 q^{31} + 2 q^{35} + 6 q^{37} - 6 q^{43} - 6 q^{47} + 2 q^{49} - 6 q^{53} - 2 q^{55} - 8 q^{59} + 14 q^{61} - 4 q^{65} - 6 q^{67} - 12 q^{71} + 8 q^{73} - 2 q^{77} - 16 q^{79} + 2 q^{85} - 8 q^{89} - 4 q^{91} + 6 q^{95} + 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 + 2 * q^11 + 4 * q^13 - 2 * q^17 - 6 * q^19 - 2 * q^23 + 2 * q^25 - 2 * q^29 + 4 * q^31 + 2 * q^35 + 6 * q^37 - 6 * q^43 - 6 * q^47 + 2 * q^49 - 6 * q^53 - 2 * q^55 - 8 * q^59 + 14 * q^61 - 4 * q^65 - 6 * q^67 - 12 * q^71 + 8 * q^73 - 2 * q^77 - 16 * q^79 + 2 * q^85 - 8 * q^89 - 4 * q^91 + 6 * q^95 + 12 * 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
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.12311	−0.941652	−0.470826	−	0.882226i	$-0.656044\pi$
$11$	−3.12311	−0.941652	−0.470826	+	0.882226i	$0.656044\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.12311	0.757464	0.378732	−	0.925506i	$-0.376360\pi$
$17$	3.12311	0.757464	0.378732	+	0.925506i	$0.376360\pi$
$18$	0	0
$19$	1.12311	0.257658	0.128829	−	0.991667i	$-0.458878\pi$
$19$	1.12311	0.257658	0.128829	+	0.991667i	$0.458878\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.12311	0.651213	0.325606	−	0.945505i	$-0.394432\pi$
$23$	3.12311	0.651213	0.325606	+	0.945505i	$0.394432\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.12311	−0.951337	−0.475668	−	0.879625i	$-0.657794\pi$
$29$	−5.12311	−0.951337	−0.475668	+	0.879625i	$0.657794\pi$
$30$	0	0
$31$	−6.24621	−1.12185	−0.560926	−	0.827866i	$-0.689555\pi$
$31$	−6.24621	−1.12185	−0.560926	+	0.827866i	$0.689555\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	7.12311	1.17103	0.585516	−	0.810661i	$-0.300892\pi$
$37$	7.12311	1.17103	0.585516	+	0.810661i	$0.300892\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.24621	−1.28784	−0.643921	−	0.765092i	$-0.722693\pi$
$41$	−8.24621	−1.28784	−0.643921	+	0.765092i	$0.722693\pi$
$42$	0	0
$43$	1.12311	0.171272	0.0856360	−	0.996326i	$-0.472708\pi$
$43$	1.12311	0.171272	0.0856360	+	0.996326i	$0.472708\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.12311	0.163822	0.0819109	−	0.996640i	$-0.473898\pi$
$47$	1.12311	0.163822	0.0819109	+	0.996640i	$0.473898\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.12311	0.154270	0.0771352	−	0.997021i	$-0.475423\pi$
$53$	1.12311	0.154270	0.0771352	+	0.997021i	$0.475423\pi$
$54$	0	0
$55$	3.12311	0.421119
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	11.1231	1.42417	0.712084	−	0.702094i	$-0.247752\pi$
$61$	11.1231	1.42417	0.712084	+	0.702094i	$0.247752\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	1.12311	0.137209	0.0686046	−	0.997644i	$-0.478145\pi$
$67$	1.12311	0.137209	0.0686046	+	0.997644i	$0.478145\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−4.24621	−0.496981	−0.248491	−	0.968634i	$-0.579935\pi$
$73$	−4.24621	−0.496981	−0.248491	+	0.968634i	$0.579935\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.12311	0.355911
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−3.12311	−0.338748
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.24621	0.450097	0.225049	−	0.974348i	$-0.427746\pi$
$89$	4.24621	0.450097	0.225049	+	0.974348i	$0.427746\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.12311	−0.115228
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−14.2462	−1.40372	−0.701860	−	0.712314i	$-0.747647\pi$
$103$	−14.2462	−1.40372	−0.701860	+	0.712314i	$0.747647\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.87689	−0.858162	−0.429081	−	0.903266i	$-0.641162\pi$
$107$	−8.87689	−0.858162	−0.429081	+	0.903266i	$0.641162\pi$
$108$	0	0
$109$	12.2462	1.17297	0.586487	−	0.809959i	$-0.300510\pi$
$109$	12.2462	1.17297	0.586487	+	0.809959i	$0.300510\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.8769	−1.02321	−0.511606	−	0.859220i	$-0.670949\pi$
$113$	−10.8769	−1.02321	−0.511606	+	0.859220i	$0.670949\pi$
$114$	0	0
$115$	−3.12311	−0.291231
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.12311	−0.286295
$120$	0	0
$121$	−1.24621	−0.113292
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.4924	−1.10852	−0.554262	−	0.832343i	$-0.686999\pi$
$127$	−12.4924	−1.10852	−0.554262	+	0.832343i	$0.686999\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.2462	−1.59418	−0.797089	−	0.603861i	$-0.793628\pi$
$131$	−18.2462	−1.59418	−0.797089	+	0.603861i	$0.793628\pi$
$132$	0	0
$133$	−1.12311	−0.0973856
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.3693	−0.971346	−0.485673	−	0.874140i	$-0.661425\pi$
$137$	−11.3693	−0.971346	−0.485673	+	0.874140i	$0.661425\pi$
$138$	0	0
$139$	−15.3693	−1.30361	−0.651804	−	0.758387i	$-0.725988\pi$
$139$	−15.3693	−1.30361	−0.651804	+	0.758387i	$0.725988\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.24621	−0.522334
$144$	0	0
$145$	5.12311	0.425451
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.12311	−0.747394	−0.373697	−	0.927551i	$-0.621910\pi$
$149$	−9.12311	−0.747394	−0.373697	+	0.927551i	$0.621910\pi$
$150$	0	0
$151$	12.4924	1.01662	0.508309	−	0.861174i	$-0.330271\pi$
$151$	12.4924	1.01662	0.508309	+	0.861174i	$0.330271\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.24621	0.501708
$156$	0	0
$157$	−14.4924	−1.15662	−0.578311	−	0.815817i	$-0.696288\pi$
$157$	−14.4924	−1.15662	−0.578311	+	0.815817i	$0.696288\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.12311	−0.246135
$162$	0	0
$163$	−6.87689	−0.538640	−0.269320	−	0.963051i	$-0.586799\pi$
$163$	−6.87689	−0.538640	−0.269320	+	0.963051i	$0.586799\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.12311	−0.705967	−0.352984	−	0.935630i	$-0.614833\pi$
$167$	−9.12311	−0.705967	−0.352984	+	0.935630i	$0.614833\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.2462	−0.931062	−0.465531	−	0.885032i	$-0.654137\pi$
$173$	−12.2462	−0.931062	−0.465531	+	0.885032i	$0.654137\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.36932	0.401322	0.200661	−	0.979661i	$-0.435691\pi$
$179$	5.36932	0.401322	0.200661	+	0.979661i	$0.435691\pi$
$180$	0	0
$181$	12.8769	0.957132	0.478566	−	0.878052i	$-0.341157\pi$
$181$	12.8769	0.957132	0.478566	+	0.878052i	$0.341157\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−7.12311	−0.523701
$186$	0	0
$187$	−9.75379	−0.713268
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.0000	−1.01300	−0.506502	−	0.862239i	$-0.669062\pi$
$191$	−14.0000	−1.01300	−0.506502	+	0.862239i	$0.669062\pi$
$192$	0	0
$193$	10.4924	0.755261	0.377631	−	0.925956i	$-0.376739\pi$
$193$	10.4924	0.755261	0.377631	+	0.925956i	$0.376739\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.12311	0.0800180	0.0400090	−	0.999199i	$-0.487261\pi$
$197$	1.12311	0.0800180	0.0400090	+	0.999199i	$0.487261\pi$
$198$	0	0
$199$	2.24621	0.159230	0.0796148	−	0.996826i	$-0.474631\pi$
$199$	2.24621	0.159230	0.0796148	+	0.996826i	$0.474631\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.12311	0.359572
$204$	0	0
$205$	8.24621	0.575940
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.50758	−0.242624
$210$	0	0
$211$	22.2462	1.53149	0.765746	−	0.643143i	$-0.222370\pi$
$211$	22.2462	1.53149	0.765746	+	0.643143i	$0.222370\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.12311	−0.0765952
$216$	0	0
$217$	6.24621	0.424020
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.24621	0.420166
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−5.36932	−0.354814	−0.177407	−	0.984138i	$-0.556771\pi$
$229$	−5.36932	−0.354814	−0.177407	+	0.984138i	$0.556771\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−25.1231	−1.64587	−0.822935	−	0.568136i	$-0.807665\pi$
$233$	−25.1231	−1.64587	−0.822935	+	0.568136i	$0.807665\pi$
$234$	0	0
$235$	−1.12311	−0.0732633
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.0000	−0.646846	−0.323423	−	0.946254i	$-0.604834\pi$
$239$	−10.0000	−0.646846	−0.323423	+	0.946254i	$0.604834\pi$
$240$	0	0
$241$	20.7386	1.33589	0.667946	−	0.744209i	$-0.267174\pi$
$241$	20.7386	1.33589	0.667946	+	0.744209i	$0.267174\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	2.24621	0.142923
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	−9.75379	−0.613215
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.87689	−0.553725	−0.276863	−	0.960909i	$-0.589295\pi$
$257$	−8.87689	−0.553725	−0.276863	+	0.960909i	$0.589295\pi$
$258$	0	0
$259$	−7.12311	−0.442608
$260$	0	0
$261$	0	0
$262$	0	0
$263$	21.3693	1.31769	0.658844	−	0.752279i	$-0.271046\pi$
$263$	21.3693	1.31769	0.658844	+	0.752279i	$0.271046\pi$
$264$	0	0
$265$	−1.12311	−0.0689918
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.75379	0.472757	0.236378	−	0.971661i	$-0.424040\pi$
$269$	7.75379	0.472757	0.236378	+	0.971661i	$0.424040\pi$
$270$	0	0
$271$	6.24621	0.379430	0.189715	−	0.981839i	$-0.439244\pi$
$271$	6.24621	0.379430	0.189715	+	0.981839i	$0.439244\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.12311	−0.188330
$276$	0	0
$277$	−4.87689	−0.293024	−0.146512	−	0.989209i	$-0.546805\pi$
$277$	−4.87689	−0.293024	−0.146512	+	0.989209i	$0.546805\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.24621	0.372618	0.186309	−	0.982491i	$-0.440347\pi$
$281$	6.24621	0.372618	0.186309	+	0.982491i	$0.440347\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.24621	0.486758
$288$	0	0
$289$	−7.24621	−0.426248
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.24621	0.361228
$300$	0	0
$301$	−1.12311	−0.0647347
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−11.1231	−0.636907
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.49242	0.481561	0.240781	−	0.970580i	$-0.422597\pi$
$311$	8.49242	0.481561	0.240781	+	0.970580i	$0.422597\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.3693	1.08789	0.543945	−	0.839121i	$-0.316930\pi$
$317$	19.3693	1.08789	0.543945	+	0.839121i	$0.316930\pi$
$318$	0	0
$319$	16.0000	0.895828
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.50758	0.195167
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.12311	−0.0619188
$330$	0	0
$331$	−18.7386	−1.02997	−0.514984	−	0.857200i	$-0.672202\pi$
$331$	−18.7386	−1.02997	−0.514984	+	0.857200i	$0.672202\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.12311	−0.0613618
$336$	0	0
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19.5076	1.05639
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−27.6155	−1.48248	−0.741240	−	0.671241i	$-0.765762\pi$
$347$	−27.6155	−1.48248	−0.741240	+	0.671241i	$0.765762\pi$
$348$	0	0
$349$	8.87689	0.475169	0.237585	−	0.971367i	$-0.423644\pi$
$349$	8.87689	0.475169	0.237585	+	0.971367i	$0.423644\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−23.1231	−1.23072	−0.615359	−	0.788247i	$-0.710989\pi$
$353$	−23.1231	−1.23072	−0.615359	+	0.788247i	$0.710989\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.7386	−1.51677	−0.758384	−	0.651809i	$-0.774011\pi$
$359$	−28.7386	−1.51677	−0.758384	+	0.651809i	$0.774011\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.24621	0.222257
$366$	0	0
$367$	20.4924	1.06970	0.534848	−	0.844948i	$-0.320369\pi$
$367$	20.4924	1.06970	0.534848	+	0.844948i	$0.320369\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.12311	−0.0583087
$372$	0	0
$373$	−2.63068	−0.136212	−0.0681058	−	0.997678i	$-0.521696\pi$
$373$	−2.63068	−0.136212	−0.0681058	+	0.997678i	$0.521696\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.2462	−0.527707
$378$	0	0
$379$	22.2462	1.14271	0.571356	−	0.820703i	$-0.306418\pi$
$379$	22.2462	1.14271	0.571356	+	0.820703i	$0.306418\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10.8769	0.555783	0.277892	−	0.960612i	$-0.410364\pi$
$383$	10.8769	0.555783	0.277892	+	0.960612i	$0.410364\pi$
$384$	0	0
$385$	−3.12311	−0.159168
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13.1231	0.665368	0.332684	−	0.943038i	$-0.392046\pi$
$389$	13.1231	0.665368	0.332684	+	0.943038i	$0.392046\pi$
$390$	0	0
$391$	9.75379	0.493270
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	0.246211	0.0123570	0.00617849	−	0.999981i	$-0.498033\pi$
$397$	0.246211	0.0123570	0.00617849	+	0.999981i	$0.498033\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.75379	−0.487081	−0.243540	−	0.969891i	$-0.578309\pi$
$401$	−9.75379	−0.487081	−0.243540	+	0.969891i	$0.578309\pi$
$402$	0	0
$403$	−12.4924	−0.622292
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−22.2462	−1.10270
$408$	0	0
$409$	−23.7538	−1.17455	−0.587275	−	0.809388i	$-0.699799\pi$
$409$	−23.7538	−1.17455	−0.587275	+	0.809388i	$0.699799\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.492423	0.0240564	0.0120282	−	0.999928i	$-0.496171\pi$
$419$	0.492423	0.0240564	0.0120282	+	0.999928i	$0.496171\pi$
$420$	0	0
$421$	−22.4924	−1.09621	−0.548107	−	0.836408i	$-0.684651\pi$
$421$	−22.4924	−1.09621	−0.548107	+	0.836408i	$0.684651\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.12311	0.151493
$426$	0	0
$427$	−11.1231	−0.538285
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.49242	0.312729	0.156364	−	0.987699i	$-0.450023\pi$
$431$	6.49242	0.312729	0.156364	+	0.987699i	$0.450023\pi$
$432$	0	0
$433$	7.75379	0.372623	0.186312	−	0.982491i	$-0.440347\pi$
$433$	7.75379	0.372623	0.186312	+	0.982491i	$0.440347\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.50758	0.167790
$438$	0	0
$439$	−28.4924	−1.35987	−0.679935	−	0.733273i	$-0.737992\pi$
$439$	−28.4924	−1.35987	−0.679935	+	0.733273i	$0.737992\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−19.6155	−0.931962	−0.465981	−	0.884795i	$-0.654298\pi$
$443$	−19.6155	−0.931962	−0.465981	+	0.884795i	$0.654298\pi$
$444$	0	0
$445$	−4.24621	−0.201290
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−22.7386	−1.07310	−0.536551	−	0.843868i	$-0.680273\pi$
$449$	−22.7386	−1.07310	−0.536551	+	0.843868i	$0.680273\pi$
$450$	0	0
$451$	25.7538	1.21270
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	−24.7386	−1.15722	−0.578612	−	0.815603i	$-0.696406\pi$
$457$	−24.7386	−1.15722	−0.578612	+	0.815603i	$0.696406\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	26.4924	1.23388	0.616938	−	0.787012i	$-0.288373\pi$
$461$	26.4924	1.23388	0.616938	+	0.787012i	$0.288373\pi$
$462$	0	0
$463$	42.2462	1.96335	0.981674	−	0.190568i	$-0.0610330\pi$
$463$	42.2462	1.96335	0.981674	+	0.190568i	$0.0610330\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.2462	1.39963	0.699814	−	0.714325i	$-0.253266\pi$
$467$	30.2462	1.39963	0.699814	+	0.714325i	$0.253266\pi$
$468$	0	0
$469$	−1.12311	−0.0518602
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.50758	−0.161279
$474$	0	0
$475$	1.12311	0.0515316
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	14.2462	0.649571
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.00000	−0.272446
$486$	0	0
$487$	−12.4924	−0.566086	−0.283043	−	0.959107i	$-0.591344\pi$
$487$	−12.4924	−0.566086	−0.283043	+	0.959107i	$0.591344\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10.6307	−0.479756	−0.239878	−	0.970803i	$-0.577107\pi$
$491$	−10.6307	−0.479756	−0.239878	+	0.970803i	$0.577107\pi$
$492$	0	0
$493$	−16.0000	−0.720604
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.00000	0.269137
$498$	0	0
$499$	14.2462	0.637748	0.318874	−	0.947797i	$-0.396695\pi$
$499$	14.2462	0.637748	0.318874	+	0.947797i	$0.396695\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	33.6155	1.49884	0.749421	−	0.662094i	$-0.230332\pi$
$503$	33.6155	1.49884	0.749421	+	0.662094i	$0.230332\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	4.24621	0.187841
$512$	0	0
$513$	0	0
$514$	0	0
$515$	14.2462	0.627763
$516$	0	0
$517$	−3.50758	−0.154263
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.9848	1.35747	0.678735	−	0.734383i	$-0.262528\pi$
$521$	30.9848	1.35747	0.678735	+	0.734383i	$0.262528\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19.5076	−0.849763
$528$	0	0
$529$	−13.2462	−0.575922
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−16.4924	−0.714366
$534$	0	0
$535$	8.87689	0.383782
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.12311	−0.134522
$540$	0	0
$541$	36.7386	1.57952	0.789759	−	0.613418i	$-0.210206\pi$
$541$	36.7386	1.57952	0.789759	+	0.613418i	$0.210206\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12.2462	−0.524570
$546$	0	0
$547$	−3.36932	−0.144062	−0.0720308	−	0.997402i	$-0.522948\pi$
$547$	−3.36932	−0.144062	−0.0720308	+	0.997402i	$0.522948\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.75379	−0.245120
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.8769	0.969325	0.484663	−	0.874701i	$-0.338942\pi$
$557$	22.8769	0.969325	0.484663	+	0.874701i	$0.338942\pi$
$558$	0	0
$559$	2.24621	0.0950046
$560$	0	0
$561$	0	0
$562$	0	0
$563$	40.4924	1.70655	0.853276	−	0.521459i	$-0.174612\pi$
$563$	40.4924	1.70655	0.853276	+	0.521459i	$0.174612\pi$
$564$	0	0
$565$	10.8769	0.457594
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−30.2462	−1.26799	−0.633994	−	0.773338i	$-0.718585\pi$
$569$	−30.2462	−1.26799	−0.633994	+	0.773338i	$0.718585\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.12311	0.130243
$576$	0	0
$577$	−1.50758	−0.0627613	−0.0313806	−	0.999508i	$-0.509990\pi$
$577$	−1.50758	−0.0627613	−0.0313806	+	0.999508i	$0.509990\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−3.50758	−0.145269
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−32.4924	−1.34111	−0.670553	−	0.741862i	$-0.733943\pi$
$587$	−32.4924	−1.34111	−0.670553	+	0.741862i	$0.733943\pi$
$588$	0	0
$589$	−7.01515	−0.289054
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−17.3693	−0.713272	−0.356636	−	0.934243i	$-0.616076\pi$
$593$	−17.3693	−0.713272	−0.356636	+	0.934243i	$0.616076\pi$
$594$	0	0
$595$	3.12311	0.128035
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−30.9848	−1.26601	−0.633003	−	0.774149i	$-0.718178\pi$
$599$	−30.9848	−1.26601	−0.633003	+	0.774149i	$0.718178\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.24621	0.0506657
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.24621	0.0908720
$612$	0	0
$613$	−48.1080	−1.94306	−0.971531	−	0.236913i	$-0.923864\pi$
$613$	−48.1080	−1.94306	−0.971531	+	0.236913i	$0.923864\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−29.6155	−1.19228	−0.596138	−	0.802882i	$-0.703299\pi$
$617$	−29.6155	−1.19228	−0.596138	+	0.802882i	$0.703299\pi$
$618$	0	0
$619$	7.36932	0.296198	0.148099	−	0.988973i	$-0.452685\pi$
$619$	7.36932	0.296198	0.148099	+	0.988973i	$0.452685\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.24621	−0.170121
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	22.2462	0.887015
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.4924	0.495747
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.49242	−0.335431	−0.167715	−	0.985835i	$-0.553639\pi$
$641$	−8.49242	−0.335431	−0.167715	+	0.985835i	$0.553639\pi$
$642$	0	0
$643$	34.7386	1.36996	0.684979	−	0.728563i	$-0.259811\pi$
$643$	34.7386	1.36996	0.684979	+	0.728563i	$0.259811\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.63068	0.182051	0.0910255	−	0.995849i	$-0.470986\pi$
$647$	4.63068	0.182051	0.0910255	+	0.995849i	$0.470986\pi$
$648$	0	0
$649$	12.4924	0.490370
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18.8769	−0.738710	−0.369355	−	0.929288i	$-0.620421\pi$
$653$	−18.8769	−0.738710	−0.369355	+	0.929288i	$0.620421\pi$
$654$	0	0
$655$	18.2462	0.712938
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0.384472	0.0149769	0.00748845	−	0.999972i	$-0.497616\pi$
$659$	0.384472	0.0149769	0.00748845	+	0.999972i	$0.497616\pi$
$660$	0	0
$661$	−4.87689	−0.189689	−0.0948446	−	0.995492i	$-0.530235\pi$
$661$	−4.87689	−0.189689	−0.0948446	+	0.995492i	$0.530235\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.12311	0.0435522
$666$	0	0
$667$	−16.0000	−0.619522
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−34.7386	−1.34107
$672$	0	0
$673$	−22.4924	−0.867019	−0.433510	−	0.901149i	$-0.642725\pi$
$673$	−22.4924	−0.867019	−0.433510	+	0.901149i	$0.642725\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.7386	1.41198	0.705990	−	0.708222i	$-0.250502\pi$
$677$	36.7386	1.41198	0.705990	+	0.708222i	$0.250502\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16.8769	0.645776	0.322888	−	0.946437i	$-0.395346\pi$
$683$	16.8769	0.645776	0.322888	+	0.946437i	$0.395346\pi$
$684$	0	0
$685$	11.3693	0.434399
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.24621	0.0855738
$690$	0	0
$691$	−13.1231	−0.499226	−0.249613	−	0.968346i	$-0.580303\pi$
$691$	−13.1231	−0.499226	−0.249613	+	0.968346i	$0.580303\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15.3693	0.582991
$696$	0	0
$697$	−25.7538	−0.975494
$698$	0	0
$699$	0	0
$700$	0	0
$701$	47.3693	1.78911	0.894557	−	0.446953i	$-0.147491\pi$
$701$	47.3693	1.78911	0.894557	+	0.446953i	$0.147491\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.00000	−0.225653
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−19.5076	−0.730565
$714$	0	0
$715$	6.24621	0.233595
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21.7538	0.811279	0.405640	−	0.914033i	$-0.367049\pi$
$719$	21.7538	0.811279	0.405640	+	0.914033i	$0.367049\pi$
$720$	0	0
$721$	14.2462	0.530557
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.12311	−0.190267
$726$	0	0
$727$	−30.2462	−1.12177	−0.560885	−	0.827894i	$-0.689539\pi$
$727$	−30.2462	−1.12177	−0.560885	+	0.827894i	$0.689539\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.50758	0.129732
$732$	0	0
$733$	−30.9848	−1.14445	−0.572226	−	0.820096i	$-0.693920\pi$
$733$	−30.9848	−1.14445	−0.572226	+	0.820096i	$0.693920\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.50758	−0.129203
$738$	0	0
$739$	1.75379	0.0645142	0.0322571	−	0.999480i	$-0.489730\pi$
$739$	1.75379	0.0645142	0.0322571	+	0.999480i	$0.489730\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.36932	0.343727	0.171863	−	0.985121i	$-0.445021\pi$
$743$	9.36932	0.343727	0.171863	+	0.985121i	$0.445021\pi$
$744$	0	0
$745$	9.12311	0.334245
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.87689	0.324355
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12.4924	−0.454646
$756$	0	0
$757$	−45.3693	−1.64898	−0.824488	−	0.565880i	$-0.808537\pi$
$757$	−45.3693	−1.64898	−0.824488	+	0.565880i	$0.808537\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	0	0
$763$	−12.2462	−0.443343
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−11.7538	−0.423852	−0.211926	−	0.977286i	$-0.567974\pi$
$769$	−11.7538	−0.423852	−0.211926	+	0.977286i	$0.567974\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	34.4924	1.24061	0.620303	−	0.784362i	$-0.287010\pi$
$773$	34.4924	1.24061	0.620303	+	0.784362i	$0.287010\pi$
$774$	0	0
$775$	−6.24621	−0.224371
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.26137	−0.331823
$780$	0	0
$781$	18.7386	0.670521
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.4924	0.517257
$786$	0	0
$787$	26.2462	0.935576	0.467788	−	0.883841i	$-0.345051\pi$
$787$	26.2462	0.935576	0.467788	+	0.883841i	$0.345051\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10.8769	0.386738
$792$	0	0
$793$	22.2462	0.789986
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−29.2311	−1.03542	−0.517709	−	0.855557i	$-0.673215\pi$
$797$	−29.2311	−1.03542	−0.517709	+	0.855557i	$0.673215\pi$
$798$	0	0
$799$	3.50758	0.124089
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13.2614	0.467983
$804$	0	0
$805$	3.12311	0.110075
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16.0000	0.562530	0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000	0.562530	0.281265	+	0.959630i	$0.409246\pi$
$810$	0	0
$811$	9.61553	0.337647	0.168823	−	0.985646i	$-0.446003\pi$
$811$	9.61553	0.337647	0.168823	+	0.985646i	$0.446003\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.87689	0.240887
$816$	0	0
$817$	1.26137	0.0441296
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−44.3542	−1.54797	−0.773985	−	0.633203i	$-0.781740\pi$
$821$	−44.3542	−1.54797	−0.773985	+	0.633203i	$0.781740\pi$
$822$	0	0
$823$	5.75379	0.200564	0.100282	−	0.994959i	$-0.468025\pi$
$823$	5.75379	0.200564	0.100282	+	0.994959i	$0.468025\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43.6155	1.51666	0.758330	−	0.651871i	$-0.226015\pi$
$827$	43.6155	1.51666	0.758330	+	0.651871i	$0.226015\pi$
$828$	0	0
$829$	−49.8617	−1.73177	−0.865885	−	0.500243i	$-0.833244\pi$
$829$	−49.8617	−1.73177	−0.865885	+	0.500243i	$0.833244\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.12311	0.108209
$834$	0	0
$835$	9.12311	0.315718
$836$	0	0
$837$	0	0
$838$	0	0
$839$	47.2311	1.63060	0.815299	−	0.579041i	$-0.196573\pi$
$839$	47.2311	1.63060	0.815299	+	0.579041i	$0.196573\pi$
$840$	0	0
$841$	−2.75379	−0.0949582
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	1.24621	0.0428203
$848$	0	0
$849$	0	0
$850$	0	0
$851$	22.2462	0.762590
$852$	0	0
$853$	41.2311	1.41172	0.705862	−	0.708349i	$-0.250560\pi$
$853$	41.2311	1.41172	0.705862	+	0.708349i	$0.250560\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.6307	0.363137	0.181569	−	0.983378i	$-0.441883\pi$
$857$	10.6307	0.363137	0.181569	+	0.983378i	$0.441883\pi$
$858$	0	0
$859$	−14.8769	−0.507593	−0.253797	−	0.967258i	$-0.581679\pi$
$859$	−14.8769	−0.507593	−0.253797	+	0.967258i	$0.581679\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	25.8617	0.880344	0.440172	−	0.897914i	$-0.354917\pi$
$863$	25.8617	0.880344	0.440172	+	0.897914i	$0.354917\pi$
$864$	0	0
$865$	12.2462	0.416384
$866$	0	0
$867$	0	0
$868$	0	0
$869$	24.9848	0.847553
$870$	0	0
$871$	2.24621	0.0761100
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−4.87689	−0.164681	−0.0823405	−	0.996604i	$-0.526240\pi$
$877$	−4.87689	−0.164681	−0.0823405	+	0.996604i	$0.526240\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−8.73863	−0.294412	−0.147206	−	0.989106i	$-0.547028\pi$
$881$	−8.73863	−0.294412	−0.147206	+	0.989106i	$0.547028\pi$
$882$	0	0
$883$	−7.86174	−0.264569	−0.132284	−	0.991212i	$-0.542231\pi$
$883$	−7.86174	−0.264569	−0.132284	+	0.991212i	$0.542231\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38.8769	1.30536	0.652679	−	0.757634i	$-0.273645\pi$
$887$	38.8769	1.30536	0.652679	+	0.757634i	$0.273645\pi$
$888$	0	0
$889$	12.4924	0.418982
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.26137	0.0422100
$894$	0	0
$895$	−5.36932	−0.179476
$896$	0	0
$897$	0	0
$898$	0	0
$899$	32.0000	1.06726
$900$	0	0
$901$	3.50758	0.116854
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.8769	−0.428042
$906$	0	0
$907$	−43.3693	−1.44005	−0.720027	−	0.693946i	$-0.755871\pi$
$907$	−43.3693	−1.44005	−0.720027	+	0.693946i	$0.755871\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−46.0000	−1.52405	−0.762024	−	0.647549i	$-0.775794\pi$
$911$	−46.0000	−1.52405	−0.762024	+	0.647549i	$0.775794\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18.2462	0.602543
$918$	0	0
$919$	−52.4924	−1.73157	−0.865783	−	0.500420i	$-0.833179\pi$
$919$	−52.4924	−1.73157	−0.865783	+	0.500420i	$0.833179\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−12.0000	−0.394985
$924$	0	0
$925$	7.12311	0.234206
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−18.4924	−0.606717	−0.303358	−	0.952877i	$-0.598108\pi$
$929$	−18.4924	−0.606717	−0.303358	+	0.952877i	$0.598108\pi$
$930$	0	0
$931$	1.12311	0.0368083
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9.75379	0.318983
$936$	0	0
$937$	−0.246211	−0.00804337	−0.00402169	−	0.999992i	$-0.501280\pi$
$937$	−0.246211	−0.00804337	−0.00402169	+	0.999992i	$0.501280\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.26137	0.106318	0.0531588	−	0.998586i	$-0.483071\pi$
$941$	3.26137	0.106318	0.0531588	+	0.998586i	$0.483071\pi$
$942$	0	0
$943$	−25.7538	−0.838659
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−12.3845	−0.402441	−0.201221	−	0.979546i	$-0.564491\pi$
$947$	−12.3845	−0.402441	−0.201221	+	0.979546i	$0.564491\pi$
$948$	0	0
$949$	−8.49242	−0.275676
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.87689	−0.0931917	−0.0465959	−	0.998914i	$-0.514837\pi$
$953$	−2.87689	−0.0931917	−0.0465959	+	0.998914i	$0.514837\pi$
$954$	0	0
$955$	14.0000	0.453029
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.3693	0.367134
$960$	0	0
$961$	8.01515	0.258553
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10.4924	−0.337763
$966$	0	0
$967$	45.4773	1.46245	0.731225	−	0.682136i	$-0.238949\pi$
$967$	45.4773	1.46245	0.731225	+	0.682136i	$0.238949\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−46.7386	−1.49991	−0.749957	−	0.661487i	$-0.769926\pi$
$971$	−46.7386	−1.49991	−0.749957	+	0.661487i	$0.769926\pi$
$972$	0	0
$973$	15.3693	0.492718
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.8617	1.53123	0.765616	−	0.643297i	$-0.222434\pi$
$977$	47.8617	1.53123	0.765616	+	0.643297i	$0.222434\pi$
$978$	0	0
$979$	−13.2614	−0.423835
$980$	0	0
$981$	0	0
$982$	0	0
$983$	26.8769	0.857240	0.428620	−	0.903485i	$-0.359000\pi$
$983$	26.8769	0.857240	0.428620	+	0.903485i	$0.359000\pi$
$984$	0	0
$985$	−1.12311	−0.0357851
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.50758	0.111534
$990$	0	0
$991$	−16.9848	−0.539541	−0.269771	−	0.962925i	$-0.586948\pi$
$991$	−16.9848	−0.539541	−0.269771	+	0.962925i	$0.586948\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.24621	−0.0712097
$996$	0	0
$997$	−4.24621	−0.134479	−0.0672394	−	0.997737i	$-0.521419\pi$
$997$	−4.24621	−0.134479	−0.0672394	+	0.997737i	$0.521419\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.a.bs.1.1		2
3.2	odd	2		5040.2.a.bv.1.2		2
4.3	odd	2		2520.2.a.u.1.2	✓	2
12.11	even	2		2520.2.a.y.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2520.2.a.u.1.2	✓	2	4.3	odd	2
2520.2.a.y.1.1	yes	2	12.11	even	2
5040.2.a.bs.1.1		2	1.1	even	1	trivial
5040.2.a.bv.1.2		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -1.00000 q^{7} -3.12311 q^{11} +2.00000 q^{13} +3.12311 q^{17} +1.12311 q^{19} +3.12311 q^{23} +1.00000 q^{25} -5.12311 q^{29} -6.24621 q^{31} +1.00000 q^{35} +7.12311 q^{37} -8.24621 q^{41} +1.12311 q^{43} +1.12311 q^{47} +1.00000 q^{49} +1.12311 q^{53} +3.12311 q^{55} -4.00000 q^{59} +11.1231 q^{61} -2.00000 q^{65} +1.12311 q^{67} -6.00000 q^{71} -4.24621 q^{73} +3.12311 q^{77} -8.00000 q^{79} -3.12311 q^{85} +4.24621 q^{89} -2.00000 q^{91} -1.12311 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7	\( 2 q - 2 q^{5} - 2 q^{7} + 2 q^{11} + 4 q^{13} - 2 q^{17} - 6 q^{19} - 2 q^{23} + 2 q^{25} - 2 q^{29} + 4 q^{31} + 2 q^{35} + 6 q^{37} - 6 q^{43} - 6 q^{47} + 2 q^{49} - 6 q^{53} - 2 q^{55} - 8 q^{59} + 14 q^{61} - 4 q^{65} - 6 q^{67} - 12 q^{71} + 8 q^{73} - 2 q^{77} - 16 q^{79} + 2 q^{85} - 8 q^{89} - 4 q^{91} + 6 q^{95} + 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 + 2 * q^11 + 4 * q^13 - 2 * q^17 - 6 * q^19 - 2 * q^23 + 2 * q^25 - 2 * q^29 + 4 * q^31 + 2 * q^35 + 6 * q^37 - 6 * q^43 - 6 * q^47 + 2 * q^49 - 6 * q^53 - 2 * q^55 - 8 * q^59 + 14 * q^61 - 4 * q^65 - 6 * q^67 - 12 * q^71 + 8 * q^73 - 2 * q^77 - 16 * q^79 + 2 * q^85 - 8 * q^89 - 4 * q^91 + 6 * q^95 + 12 * q^97

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