LMFDB - Embedded newform 5040.2.a.bt.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:	$0$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
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)

$f(q)$	$=$	$q-1.00000 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +1.00000 q^{7} -1.56155 q^{11} +0.438447 q^{13} +0.438447 q^{17} +7.12311 q^{19} +3.12311 q^{23} +1.00000 q^{25} -6.68466 q^{29} -1.00000 q^{35} +6.00000 q^{37} -5.12311 q^{41} -0.876894 q^{43} -8.68466 q^{47} +1.00000 q^{49} +5.12311 q^{53} +1.56155 q^{55} -4.00000 q^{59} +15.3693 q^{61} -0.438447 q^{65} -10.2462 q^{67} +8.00000 q^{71} -12.2462 q^{73} -1.56155 q^{77} +2.43845 q^{79} +4.00000 q^{83} -0.438447 q^{85} +1.12311 q^{89} +0.438447 q^{91} -7.12311 q^{95} +5.80776 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + q^{11} + 5 q^{13} + 5 q^{17} + 6 q^{19} - 2 q^{23} + 2 q^{25} - q^{29} - 2 q^{35} + 12 q^{37} - 2 q^{41} - 10 q^{43} - 5 q^{47} + 2 q^{49} + 2 q^{53} - q^{55} - 8 q^{59} + 6 q^{61} - 5 q^{65} - 4 q^{67} + 16 q^{71} - 8 q^{73} + q^{77} + 9 q^{79} + 8 q^{83} - 5 q^{85} - 6 q^{89} + 5 q^{91} - 6 q^{95} - 9 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 + q^11 + 5 * q^13 + 5 * q^17 + 6 * q^19 - 2 * q^23 + 2 * q^25 - q^29 - 2 * q^35 + 12 * q^37 - 2 * q^41 - 10 * q^43 - 5 * q^47 + 2 * q^49 + 2 * q^53 - q^55 - 8 * q^59 + 6 * q^61 - 5 * q^65 - 4 * q^67 + 16 * q^71 - 8 * q^73 + q^77 + 9 * q^79 + 8 * q^83 - 5 * q^85 - 6 * q^89 + 5 * q^91 - 6 * q^95 - 9 * 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
$11$	−1.56155	−0.470826	−0.235413	−	0.971895i	$-0.575644\pi$
$11$	−1.56155	−0.470826	−0.235413	+	0.971895i	$0.575644\pi$
$12$	0	0
$13$	0.438447	0.121603	0.0608017	−	0.998150i	$-0.480634\pi$
$13$	0.438447	0.121603	0.0608017	+	0.998150i	$0.480634\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.438447	0.106339	0.0531695	−	0.998586i	$-0.483068\pi$
$17$	0.438447	0.106339	0.0531695	+	0.998586i	$0.483068\pi$
$18$	0	0
$19$	7.12311	1.63415	0.817076	−	0.576530i	$-0.195593\pi$
$19$	7.12311	1.63415	0.817076	+	0.576530i	$0.195593\pi$
$20$	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
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.12311	−0.800095	−0.400047	−	0.916494i	$-0.631006\pi$
$41$	−5.12311	−0.800095	−0.400047	+	0.916494i	$0.631006\pi$
$42$	0	0
$43$	−0.876894	−0.133725	−0.0668626	−	0.997762i	$-0.521299\pi$
$43$	−0.876894	−0.133725	−0.0668626	+	0.997762i	$0.521299\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.68466	−1.26679	−0.633394	−	0.773830i	$-0.718339\pi$
$47$	−8.68466	−1.26679	−0.633394	+	0.773830i	$0.718339\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.12311	0.703713	0.351856	−	0.936054i	$-0.385551\pi$
$53$	5.12311	0.703713	0.351856	+	0.936054i	$0.385551\pi$
$54$	0	0
$55$	1.56155	0.210560
$56$	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
$61$	15.3693	1.96784	0.983920	−	0.178611i	$-0.0571605\pi$
$61$	15.3693	1.96784	0.983920	+	0.178611i	$0.0571605\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.438447	−0.0543827
$66$	0	0
$67$	−10.2462	−1.25177	−0.625887	−	0.779914i	$-0.715263\pi$
$67$	−10.2462	−1.25177	−0.625887	+	0.779914i	$0.715263\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−12.2462	−1.43331	−0.716655	−	0.697428i	$-0.754328\pi$
$73$	−12.2462	−1.43331	−0.716655	+	0.697428i	$0.754328\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.56155	−0.177955
$78$	0	0
$79$	2.43845	0.274347	0.137173	−	0.990547i	$-0.456198\pi$
$79$	2.43845	0.274347	0.137173	+	0.990547i	$0.456198\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−0.438447	−0.0475563
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.12311	0.119049	0.0595245	−	0.998227i	$-0.481042\pi$
$89$	1.12311	0.119049	0.0595245	+	0.998227i	$0.481042\pi$
$90$	0	0
$91$	0.438447	0.0459618
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.12311	−0.730815
$96$	0	0
$97$	5.80776	0.589689	0.294845	−	0.955545i	$-0.404732\pi$
$97$	5.80776	0.589689	0.294845	+	0.955545i	$0.404732\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.2462	1.61656	0.808279	−	0.588799i	$-0.200399\pi$
$101$	16.2462	1.61656	0.808279	+	0.588799i	$0.200399\pi$
$102$	0	0
$103$	−5.56155	−0.547996	−0.273998	−	0.961730i	$-0.588346\pi$
$103$	−5.56155	−0.547996	−0.273998	+	0.961730i	$0.588346\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.3693	1.29246	0.646230	−	0.763142i	$-0.276345\pi$
$107$	13.3693	1.29246	0.646230	+	0.763142i	$0.276345\pi$
$108$	0	0
$109$	5.31534	0.509117	0.254559	−	0.967057i	$-0.418070\pi$
$109$	5.31534	0.509117	0.254559	+	0.967057i	$0.418070\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	−3.12311	−0.291231
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.438447	0.0401924
$120$	0	0
$121$	−8.56155	−0.778323
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.24621	0.554262	0.277131	−	0.960832i	$-0.410616\pi$
$127$	6.24621	0.554262	0.277131	+	0.960832i	$0.410616\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−0.876894	−0.0766146	−0.0383073	−	0.999266i	$-0.512197\pi$
$131$	−0.876894	−0.0766146	−0.0383073	+	0.999266i	$0.512197\pi$
$132$	0	0
$133$	7.12311	0.617652
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.1231	1.46293	0.731463	−	0.681881i	$-0.238838\pi$
$137$	17.1231	1.46293	0.731463	+	0.681881i	$0.238838\pi$
$138$	0	0
$139$	15.1231	1.28273	0.641363	−	0.767238i	$-0.278369\pi$
$139$	15.1231	1.28273	0.641363	+	0.767238i	$0.278369\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.684658	−0.0572540
$144$	0	0
$145$	6.68466	0.555131
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.2462	−1.00325	−0.501624	−	0.865086i	$-0.667264\pi$
$149$	−12.2462	−1.00325	−0.501624	+	0.865086i	$0.667264\pi$
$150$	0	0
$151$	6.93087	0.564026	0.282013	−	0.959411i	$-0.408998\pi$
$151$	6.93087	0.564026	0.282013	+	0.959411i	$0.408998\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	20.2462	1.61582	0.807912	−	0.589303i	$-0.200598\pi$
$157$	20.2462	1.61582	0.807912	+	0.589303i	$0.200598\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.12311	0.246135
$162$	0	0
$163$	7.12311	0.557925	0.278962	−	0.960302i	$-0.410010\pi$
$163$	7.12311	0.557925	0.278962	+	0.960302i	$0.410010\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.93087	−0.536327	−0.268163	−	0.963373i	$-0.586417\pi$
$167$	−6.93087	−0.536327	−0.268163	+	0.963373i	$0.586417\pi$
$168$	0	0
$169$	−12.8078	−0.985213
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.43845	0.337449	0.168724	−	0.985663i	$-0.446035\pi$
$173$	4.43845	0.337449	0.168724	+	0.985663i	$0.446035\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	−17.6155	−1.30935	−0.654676	−	0.755910i	$-0.727195\pi$
$181$	−17.6155	−1.30935	−0.654676	+	0.755910i	$0.727195\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	−0.684658	−0.0500672
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.5616	−0.981280	−0.490640	−	0.871363i	$-0.663237\pi$
$191$	−13.5616	−0.981280	−0.490640	+	0.871363i	$0.663237\pi$
$192$	0	0
$193$	19.3693	1.39423	0.697117	−	0.716957i	$-0.254466\pi$
$193$	19.3693	1.39423	0.697117	+	0.716957i	$0.254466\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.12311	−0.0800180	−0.0400090	−	0.999199i	$-0.512739\pi$
$197$	−1.12311	−0.0800180	−0.0400090	+	0.999199i	$0.512739\pi$
$198$	0	0
$199$	1.75379	0.124323	0.0621614	−	0.998066i	$-0.480201\pi$
$199$	1.75379	0.124323	0.0621614	+	0.998066i	$0.480201\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.68466	−0.469171
$204$	0	0
$205$	5.12311	0.357813
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−11.1231	−0.769401
$210$	0	0
$211$	−14.0540	−0.967516	−0.483758	−	0.875202i	$-0.660728\pi$
$211$	−14.0540	−0.967516	−0.483758	+	0.875202i	$0.660728\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.876894	0.0598037
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.192236	0.0129312
$222$	0	0
$223$	2.43845	0.163291	0.0816453	−	0.996661i	$-0.473983\pi$
$223$	2.43845	0.163291	0.0816453	+	0.996661i	$0.473983\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.3153	0.751026	0.375513	−	0.926817i	$-0.377467\pi$
$227$	11.3153	0.751026	0.375513	+	0.926817i	$0.377467\pi$
$228$	0	0
$229$	10.8769	0.718765	0.359383	−	0.933190i	$-0.382987\pi$
$229$	10.8769	0.718765	0.359383	+	0.933190i	$0.382987\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.12311	−0.335626	−0.167813	−	0.985819i	$-0.553670\pi$
$233$	−5.12311	−0.335626	−0.167813	+	0.985819i	$0.553670\pi$
$234$	0	0
$235$	8.68466	0.566525
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.8078	1.28126	0.640629	−	0.767851i	$-0.278674\pi$
$239$	19.8078	1.28126	0.640629	+	0.767851i	$0.278674\pi$
$240$	0	0
$241$	−4.24621	−0.273523	−0.136761	−	0.990604i	$-0.543669\pi$
$241$	−4.24621	−0.273523	−0.136761	+	0.990604i	$0.543669\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	3.12311	0.198718
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.87689	−0.560305	−0.280152	−	0.959956i	$-0.590385\pi$
$251$	−8.87689	−0.560305	−0.280152	+	0.959956i	$0.590385\pi$
$252$	0	0
$253$	−4.87689	−0.306608
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.4924	0.654499	0.327250	−	0.944938i	$-0.393878\pi$
$257$	10.4924	0.654499	0.327250	+	0.944938i	$0.393878\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.8769	−0.794023	−0.397012	−	0.917814i	$-0.629953\pi$
$263$	−12.8769	−0.794023	−0.397012	+	0.917814i	$0.629953\pi$
$264$	0	0
$265$	−5.12311	−0.314710
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.7386	1.26446	0.632228	−	0.774782i	$-0.282140\pi$
$269$	20.7386	1.26446	0.632228	+	0.774782i	$0.282140\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.56155	−0.0941652
$276$	0	0
$277$	−0.246211	−0.0147934	−0.00739670	−	0.999973i	$-0.502354\pi$
$277$	−0.246211	−0.0147934	−0.00739670	+	0.999973i	$0.502354\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.4384	−0.742016	−0.371008	−	0.928630i	$-0.620988\pi$
$281$	−12.4384	−0.742016	−0.371008	+	0.928630i	$0.620988\pi$
$282$	0	0
$283$	11.3153	0.672627	0.336314	−	0.941750i	$-0.390820\pi$
$283$	11.3153	0.672627	0.336314	+	0.941750i	$0.390820\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.12311	−0.302407
$288$	0	0
$289$	−16.8078	−0.988692
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.68466	0.156839	0.0784197	−	0.996920i	$-0.475013\pi$
$293$	2.68466	0.156839	0.0784197	+	0.996920i	$0.475013\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.36932	0.0791896
$300$	0	0
$301$	−0.876894	−0.0505434
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−15.3693	−0.880045
$306$	0	0
$307$	19.3153	1.10238	0.551192	−	0.834378i	$-0.314173\pi$
$307$	19.3153	1.10238	0.551192	+	0.834378i	$0.314173\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	31.6155	1.79275	0.896376	−	0.443294i	$-0.146190\pi$
$311$	31.6155	1.79275	0.896376	+	0.443294i	$0.146190\pi$
$312$	0	0
$313$	−22.3002	−1.26048	−0.630241	−	0.776400i	$-0.717044\pi$
$313$	−22.3002	−1.26048	−0.630241	+	0.776400i	$0.717044\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.4924	−0.589313	−0.294657	−	0.955603i	$-0.595205\pi$
$317$	−10.4924	−0.589313	−0.294657	+	0.955603i	$0.595205\pi$
$318$	0	0
$319$	10.4384	0.584441
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.12311	0.173774
$324$	0	0
$325$	0.438447	0.0243207
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.68466	−0.478801
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.2462	0.559810
$336$	0	0
$337$	−1.50758	−0.0821230	−0.0410615	−	0.999157i	$-0.513074\pi$
$337$	−1.50758	−0.0821230	−0.0410615	+	0.999157i	$0.513074\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.12311	0.382388	0.191194	−	0.981552i	$-0.438764\pi$
$347$	7.12311	0.382388	0.191194	+	0.981552i	$0.438764\pi$
$348$	0	0
$349$	10.4924	0.561646	0.280823	−	0.959760i	$-0.409393\pi$
$349$	10.4924	0.561646	0.280823	+	0.959760i	$0.409393\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5.80776	−0.309116	−0.154558	−	0.987984i	$-0.549395\pi$
$353$	−5.80776	−0.309116	−0.154558	+	0.987984i	$0.549395\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	31.7386	1.67045
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.2462	0.640996
$366$	0	0
$367$	8.68466	0.453335	0.226668	−	0.973972i	$-0.427217\pi$
$367$	8.68466	0.453335	0.226668	+	0.973972i	$0.427217\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.12311	0.265978
$372$	0	0
$373$	4.63068	0.239768	0.119884	−	0.992788i	$-0.461748\pi$
$373$	4.63068	0.239768	0.119884	+	0.992788i	$0.461748\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.93087	−0.150947
$378$	0	0
$379$	16.4924	0.847159	0.423579	−	0.905859i	$-0.360773\pi$
$379$	16.4924	0.847159	0.423579	+	0.905859i	$0.360773\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.24621	0.319166	0.159583	−	0.987184i	$-0.448985\pi$
$383$	6.24621	0.319166	0.159583	+	0.987184i	$0.448985\pi$
$384$	0	0
$385$	1.56155	0.0795841
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.9309	1.26405	0.632023	−	0.774950i	$-0.282225\pi$
$389$	24.9309	1.26405	0.632023	+	0.774950i	$0.282225\pi$
$390$	0	0
$391$	1.36932	0.0692493
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.43845	−0.122692
$396$	0	0
$397$	27.5616	1.38327	0.691637	−	0.722245i	$-0.256890\pi$
$397$	27.5616	1.38327	0.691637	+	0.722245i	$0.256890\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−31.5616	−1.57611	−0.788054	−	0.615606i	$-0.788911\pi$
$401$	−31.5616	−1.57611	−0.788054	+	0.615606i	$0.788911\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.36932	−0.464420
$408$	0	0
$409$	6.49242	0.321030	0.160515	−	0.987033i	$-0.448685\pi$
$409$	6.49242	0.321030	0.160515	+	0.987033i	$0.448685\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	26.2462	1.28221	0.641106	−	0.767453i	$-0.278476\pi$
$419$	26.2462	1.28221	0.641106	+	0.767453i	$0.278476\pi$
$420$	0	0
$421$	−2.68466	−0.130842	−0.0654211	−	0.997858i	$-0.520839\pi$
$421$	−2.68466	−0.130842	−0.0654211	+	0.997858i	$0.520839\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.438447	0.0212678
$426$	0	0
$427$	15.3693	0.743773
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.8078	−0.954106	−0.477053	−	0.878874i	$-0.658295\pi$
$431$	−19.8078	−0.954106	−0.477053	+	0.878874i	$0.658295\pi$
$432$	0	0
$433$	8.24621	0.396288	0.198144	−	0.980173i	$-0.436509\pi$
$433$	8.24621	0.396288	0.198144	+	0.980173i	$0.436509\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	22.2462	1.06418
$438$	0	0
$439$	−9.36932	−0.447173	−0.223587	−	0.974684i	$-0.571777\pi$
$439$	−9.36932	−0.447173	−0.223587	+	0.974684i	$0.571777\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.63068	−0.124988	−0.0624938	−	0.998045i	$-0.519905\pi$
$443$	−2.63068	−0.124988	−0.0624938	+	0.998045i	$0.519905\pi$
$444$	0	0
$445$	−1.12311	−0.0532403
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.80776	0.0853137	0.0426568	−	0.999090i	$-0.486418\pi$
$449$	1.80776	0.0853137	0.0426568	+	0.999090i	$0.486418\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.438447	−0.0205547
$456$	0	0
$457$	−17.1231	−0.800985	−0.400493	−	0.916300i	$-0.631161\pi$
$457$	−17.1231	−0.800985	−0.400493	+	0.916300i	$0.631161\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13.1231	0.611204	0.305602	−	0.952159i	$-0.401142\pi$
$461$	13.1231	0.611204	0.305602	+	0.952159i	$0.401142\pi$
$462$	0	0
$463$	−12.4924	−0.580572	−0.290286	−	0.956940i	$-0.593750\pi$
$463$	−12.4924	−0.580572	−0.290286	+	0.956940i	$0.593750\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.4384	1.03833	0.519164	−	0.854675i	$-0.326243\pi$
$467$	22.4384	1.03833	0.519164	+	0.854675i	$0.326243\pi$
$468$	0	0
$469$	−10.2462	−0.473126
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.36932	0.0629613
$474$	0	0
$475$	7.12311	0.326831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.87689	0.222831	0.111415	−	0.993774i	$-0.464462\pi$
$479$	4.87689	0.222831	0.111415	+	0.993774i	$0.464462\pi$
$480$	0	0
$481$	2.63068	0.119949
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.80776	−0.263717
$486$	0	0
$487$	3.12311	0.141521	0.0707607	−	0.997493i	$-0.477457\pi$
$487$	3.12311	0.141521	0.0707607	+	0.997493i	$0.477457\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−41.1771	−1.85830	−0.929148	−	0.369708i	$-0.879458\pi$
$491$	−41.1771	−1.85830	−0.929148	+	0.369708i	$0.879458\pi$
$492$	0	0
$493$	−2.93087	−0.132000
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	−41.1771	−1.84334	−0.921670	−	0.387976i	$-0.873174\pi$
$499$	−41.1771	−1.84334	−0.921670	+	0.387976i	$0.873174\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	38.9309	1.73584	0.867921	−	0.496703i	$-0.165456\pi$
$503$	38.9309	1.73584	0.867921	+	0.496703i	$0.165456\pi$
$504$	0	0
$505$	−16.2462	−0.722947
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11.7538	0.520978	0.260489	−	0.965477i	$-0.416116\pi$
$509$	11.7538	0.520978	0.260489	+	0.965477i	$0.416116\pi$
$510$	0	0
$511$	−12.2462	−0.541740
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.56155	0.245071
$516$	0	0
$517$	13.5616	0.596436
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	−40.4924	−1.77061	−0.885305	−	0.465011i	$-0.846050\pi$
$523$	−40.4924	−1.77061	−0.885305	+	0.465011i	$0.846050\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−13.2462	−0.575922
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.24621	−0.0972942
$534$	0	0
$535$	−13.3693	−0.578006
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.56155	−0.0672608
$540$	0	0
$541$	−37.8078	−1.62548	−0.812741	−	0.582625i	$-0.802026\pi$
$541$	−37.8078	−1.62548	−0.812741	+	0.582625i	$0.802026\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.31534	−0.227684
$546$	0	0
$547$	2.24621	0.0960411	0.0480205	−	0.998846i	$-0.484709\pi$
$547$	2.24621	0.0960411	0.0480205	+	0.998846i	$0.484709\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−47.6155	−2.02849
$552$	0	0
$553$	2.43845	0.103693
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.1231	0.556044	0.278022	−	0.960575i	$-0.410321\pi$
$557$	13.1231	0.556044	0.278022	+	0.960575i	$0.410321\pi$
$558$	0	0
$559$	−0.384472	−0.0162614
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	0	0
$565$	−14.0000	−0.588984
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.9848	1.29895	0.649476	−	0.760382i	$-0.274988\pi$
$569$	30.9848	1.29895	0.649476	+	0.760382i	$0.274988\pi$
$570$	0	0
$571$	−40.4924	−1.69456	−0.847278	−	0.531150i	$-0.821760\pi$
$571$	−40.4924	−1.69456	−0.847278	+	0.531150i	$0.821760\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.12311	0.130243
$576$	0	0
$577$	−24.0540	−1.00138	−0.500690	−	0.865627i	$-0.666920\pi$
$577$	−24.0540	−1.00138	−0.500690	+	0.865627i	$0.666920\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	−8.00000	−0.331326
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26.2462	−1.08330	−0.541649	−	0.840605i	$-0.682200\pi$
$587$	−26.2462	−1.08330	−0.541649	+	0.840605i	$0.682200\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	27.5616	1.13182	0.565909	−	0.824468i	$-0.308525\pi$
$593$	27.5616	1.13182	0.565909	+	0.824468i	$0.308525\pi$
$594$	0	0
$595$	−0.438447	−0.0179746
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−11.8078	−0.482452	−0.241226	−	0.970469i	$-0.577550\pi$
$599$	−11.8078	−0.482452	−0.241226	+	0.970469i	$0.577550\pi$
$600$	0	0
$601$	6.49242	0.264831	0.132416	−	0.991194i	$-0.457727\pi$
$601$	6.49242	0.264831	0.132416	+	0.991194i	$0.457727\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.56155	0.348077
$606$	0	0
$607$	42.0540	1.70692	0.853459	−	0.521160i	$-0.174500\pi$
$607$	42.0540	1.70692	0.853459	+	0.521160i	$0.174500\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.80776	−0.154046
$612$	0	0
$613$	40.7386	1.64542	0.822709	−	0.568463i	$-0.192462\pi$
$613$	40.7386	1.64542	0.822709	+	0.568463i	$0.192462\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−32.2462	−1.29818	−0.649092	−	0.760710i	$-0.724851\pi$
$617$	−32.2462	−1.29818	−0.649092	+	0.760710i	$0.724851\pi$
$618$	0	0
$619$	−32.1080	−1.29053	−0.645264	−	0.763960i	$-0.723253\pi$
$619$	−32.1080	−1.29053	−0.645264	+	0.763960i	$0.723253\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.12311	0.0449963
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.63068	0.104892
$630$	0	0
$631$	11.8078	0.470060	0.235030	−	0.971988i	$-0.424481\pi$
$631$	11.8078	0.470060	0.235030	+	0.971988i	$0.424481\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.24621	−0.247873
$636$	0	0
$637$	0.438447	0.0173719
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	−1.56155	−0.0615816	−0.0307908	−	0.999526i	$-0.509803\pi$
$643$	−1.56155	−0.0615816	−0.0307908	+	0.999526i	$0.509803\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.4924	1.43467	0.717333	−	0.696731i	$-0.245363\pi$
$647$	36.4924	1.43467	0.717333	+	0.696731i	$0.245363\pi$
$648$	0	0
$649$	6.24621	0.245185
$650$	0	0
$651$	0	0
$652$	0	0
$653$	33.2311	1.30043	0.650216	−	0.759750i	$-0.274678\pi$
$653$	33.2311	1.30043	0.650216	+	0.759750i	$0.274678\pi$
$654$	0	0
$655$	0.876894	0.0342631
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9.17708	0.357488	0.178744	−	0.983896i	$-0.442797\pi$
$659$	9.17708	0.357488	0.178744	+	0.983896i	$0.442797\pi$
$660$	0	0
$661$	−5.12311	−0.199266	−0.0996329	−	0.995024i	$-0.531767\pi$
$661$	−5.12311	−0.199266	−0.0996329	+	0.995024i	$0.531767\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.12311	−0.276222
$666$	0	0
$667$	−20.8769	−0.808357
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	31.8617	1.22818	0.614090	−	0.789236i	$-0.289523\pi$
$673$	31.8617	1.22818	0.614090	+	0.789236i	$0.289523\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.93087	−0.189509	−0.0947544	−	0.995501i	$-0.530207\pi$
$677$	−4.93087	−0.189509	−0.0947544	+	0.995501i	$0.530207\pi$
$678$	0	0
$679$	5.80776	0.222882
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6.73863	−0.257847	−0.128923	−	0.991655i	$-0.541152\pi$
$683$	−6.73863	−0.257847	−0.128923	+	0.991655i	$0.541152\pi$
$684$	0	0
$685$	−17.1231	−0.654240
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.24621	0.0855738
$690$	0	0
$691$	24.4924	0.931736	0.465868	−	0.884854i	$-0.345742\pi$
$691$	24.4924	0.931736	0.465868	+	0.884854i	$0.345742\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.1231	−0.573652
$696$	0	0
$697$	−2.24621	−0.0850813
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−28.9309	−1.09270	−0.546352	−	0.837556i	$-0.683984\pi$
$701$	−28.9309	−1.09270	−0.546352	+	0.837556i	$0.683984\pi$
$702$	0	0
$703$	42.7386	1.61192
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.2462	0.611002
$708$	0	0
$709$	27.1771	1.02066	0.510328	−	0.859980i	$-0.329524\pi$
$709$	27.1771	1.02066	0.510328	+	0.859980i	$0.329524\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0.684658	0.0256048
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.38447	0.312688	0.156344	−	0.987703i	$-0.450029\pi$
$719$	8.38447	0.312688	0.156344	+	0.987703i	$0.450029\pi$
$720$	0	0
$721$	−5.56155	−0.207123
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.68466	−0.248262
$726$	0	0
$727$	−52.4924	−1.94684	−0.973418	−	0.229035i	$-0.926443\pi$
$727$	−52.4924	−1.94684	−0.973418	+	0.229035i	$0.926443\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.384472	−0.0142202
$732$	0	0
$733$	6.68466	0.246903	0.123452	−	0.992351i	$-0.460604\pi$
$733$	6.68466	0.246903	0.123452	+	0.992351i	$0.460604\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	−34.9309	−1.28495	−0.642476	−	0.766305i	$-0.722093\pi$
$739$	−34.9309	−1.28495	−0.642476	+	0.766305i	$0.722093\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−32.9848	−1.21010	−0.605048	−	0.796189i	$-0.706846\pi$
$743$	−32.9848	−1.21010	−0.605048	+	0.796189i	$0.706846\pi$
$744$	0	0
$745$	12.2462	0.448666
$746$	0	0
$747$	0	0
$748$	0	0
$749$	13.3693	0.488504
$750$	0	0
$751$	−17.0691	−0.622861	−0.311431	−	0.950269i	$-0.600808\pi$
$751$	−17.0691	−0.622861	−0.311431	+	0.950269i	$0.600808\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.93087	−0.252240
$756$	0	0
$757$	39.3693	1.43090	0.715451	−	0.698663i	$-0.246221\pi$
$757$	39.3693	1.43090	0.715451	+	0.698663i	$0.246221\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−48.2462	−1.74892	−0.874462	−	0.485094i	$-0.838785\pi$
$761$	−48.2462	−1.74892	−0.874462	+	0.485094i	$0.838785\pi$
$762$	0	0
$763$	5.31534	0.192428
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.75379	−0.0633256
$768$	0	0
$769$	−42.4924	−1.53232	−0.766158	−	0.642652i	$-0.777834\pi$
$769$	−42.4924	−1.53232	−0.766158	+	0.642652i	$0.777834\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−36.9309	−1.32831	−0.664156	−	0.747594i	$-0.731209\pi$
$773$	−36.9309	−1.32831	−0.664156	+	0.747594i	$0.731209\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−36.4924	−1.30748
$780$	0	0
$781$	−12.4924	−0.447014
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−20.2462	−0.722618
$786$	0	0
$787$	49.1771	1.75297	0.876487	−	0.481426i	$-0.159881\pi$
$787$	49.1771	1.75297	0.876487	+	0.481426i	$0.159881\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	6.73863	0.239296
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.0540	−0.852036	−0.426018	−	0.904715i	$-0.640084\pi$
$797$	−24.0540	−0.852036	−0.426018	+	0.904715i	$0.640084\pi$
$798$	0	0
$799$	−3.80776	−0.134709
$800$	0	0
$801$	0	0
$802$	0	0
$803$	19.1231	0.674840
$804$	0	0
$805$	−3.12311	−0.110075
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−16.5464	−0.581740	−0.290870	−	0.956763i	$-0.593945\pi$
$809$	−16.5464	−0.581740	−0.290870	+	0.956763i	$0.593945\pi$
$810$	0	0
$811$	−19.6155	−0.688794	−0.344397	−	0.938824i	$-0.611917\pi$
$811$	−19.6155	−0.688794	−0.344397	+	0.938824i	$0.611917\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.12311	−0.249512
$816$	0	0
$817$	−6.24621	−0.218527
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.4233	0.747678	0.373839	−	0.927494i	$-0.378041\pi$
$821$	21.4233	0.747678	0.373839	+	0.927494i	$0.378041\pi$
$822$	0	0
$823$	36.4924	1.27205	0.636023	−	0.771670i	$-0.280578\pi$
$823$	36.4924	1.27205	0.636023	+	0.771670i	$0.280578\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5.36932	−0.186709	−0.0933547	−	0.995633i	$-0.529759\pi$
$827$	−5.36932	−0.186709	−0.0933547	+	0.995633i	$0.529759\pi$
$828$	0	0
$829$	34.8769	1.21132	0.605662	−	0.795722i	$-0.292908\pi$
$829$	34.8769	1.21132	0.605662	+	0.795722i	$0.292908\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.438447	0.0151913
$834$	0	0
$835$	6.93087	0.239853
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−28.8769	−0.996941	−0.498471	−	0.866907i	$-0.666105\pi$
$839$	−28.8769	−0.996941	−0.498471	+	0.866907i	$0.666105\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.8078	0.440600
$846$	0	0
$847$	−8.56155	−0.294178
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18.7386	0.642352
$852$	0	0
$853$	−7.26137	−0.248624	−0.124312	−	0.992243i	$-0.539672\pi$
$853$	−7.26137	−0.248624	−0.124312	+	0.992243i	$0.539672\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.7538	0.538139	0.269070	−	0.963121i	$-0.413284\pi$
$857$	15.7538	0.538139	0.269070	+	0.963121i	$0.413284\pi$
$858$	0	0
$859$	16.4924	0.562714	0.281357	−	0.959603i	$-0.409215\pi$
$859$	16.4924	0.562714	0.281357	+	0.959603i	$0.409215\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.7538	−0.876669	−0.438335	−	0.898812i	$-0.644431\pi$
$863$	−25.7538	−0.876669	−0.438335	+	0.898812i	$0.644431\pi$
$864$	0	0
$865$	−4.43845	−0.150912
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.80776	−0.129170
$870$	0	0
$871$	−4.49242	−0.152220
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−40.2462	−1.35902	−0.679509	−	0.733667i	$-0.737807\pi$
$877$	−40.2462	−1.35902	−0.679509	+	0.733667i	$0.737807\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	11.8617	0.399632	0.199816	−	0.979833i	$-0.435966\pi$
$881$	11.8617	0.399632	0.199816	+	0.979833i	$0.435966\pi$
$882$	0	0
$883$	8.49242	0.285793	0.142896	−	0.989738i	$-0.454358\pi$
$883$	8.49242	0.285793	0.142896	+	0.989738i	$0.454358\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.4924	0.688068	0.344034	−	0.938957i	$-0.388206\pi$
$887$	20.4924	0.688068	0.344034	+	0.938957i	$0.388206\pi$
$888$	0	0
$889$	6.24621	0.209491
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−61.8617	−2.07012
$894$	0	0
$895$	−20.0000	−0.668526
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	2.24621	0.0748321
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.6155	0.585560
$906$	0	0
$907$	24.1080	0.800491	0.400246	−	0.916408i	$-0.368925\pi$
$907$	24.1080	0.800491	0.400246	+	0.916408i	$0.368925\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−28.4924	−0.943996	−0.471998	−	0.881600i	$-0.656467\pi$
$911$	−28.4924	−0.943996	−0.471998	+	0.881600i	$0.656467\pi$
$912$	0	0
$913$	−6.24621	−0.206719
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.876894	−0.0289576
$918$	0	0
$919$	−40.3002	−1.32938	−0.664690	−	0.747119i	$-0.731436\pi$
$919$	−40.3002	−1.32938	−0.664690	+	0.747119i	$0.731436\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.50758	0.115453
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−22.1080	−0.725338	−0.362669	−	0.931918i	$-0.618134\pi$
$929$	−22.1080	−0.725338	−0.362669	+	0.931918i	$0.618134\pi$
$930$	0	0
$931$	7.12311	0.233450
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.684658	0.0223907
$936$	0	0
$937$	−55.6695	−1.81864	−0.909322	−	0.416094i	$-0.863399\pi$
$937$	−55.6695	−1.81864	−0.909322	+	0.416094i	$0.863399\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−43.8617	−1.42985	−0.714926	−	0.699200i	$-0.753540\pi$
$941$	−43.8617	−1.42985	−0.714926	+	0.699200i	$0.753540\pi$
$942$	0	0
$943$	−16.0000	−0.521032
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	−5.36932	−0.174295
$950$	0	0
$951$	0	0
$952$	0	0
$953$	33.1231	1.07296	0.536481	−	0.843912i	$-0.319753\pi$
$953$	33.1231	1.07296	0.536481	+	0.843912i	$0.319753\pi$
$954$	0	0
$955$	13.5616	0.438842
$956$	0	0
$957$	0	0
$958$	0	0
$959$	17.1231	0.552934
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−19.3693	−0.623520
$966$	0	0
$967$	35.1231	1.12948	0.564741	−	0.825268i	$-0.308976\pi$
$967$	35.1231	1.12948	0.564741	+	0.825268i	$0.308976\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	49.4773	1.58780	0.793901	−	0.608048i	$-0.208047\pi$
$971$	49.4773	1.58780	0.793901	+	0.608048i	$0.208047\pi$
$972$	0	0
$973$	15.1231	0.484825
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−33.2311	−1.06316	−0.531578	−	0.847009i	$-0.678401\pi$
$977$	−33.2311	−1.06316	−0.531578	+	0.847009i	$0.678401\pi$
$978$	0	0
$979$	−1.75379	−0.0560513
$980$	0	0
$981$	0	0
$982$	0	0
$983$	51.4233	1.64015	0.820074	−	0.572257i	$-0.193932\pi$
$983$	51.4233	1.64015	0.820074	+	0.572257i	$0.193932\pi$
$984$	0	0
$985$	1.12311	0.0357851
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.73863	−0.0870835
$990$	0	0
$991$	−12.4924	−0.396835	−0.198417	−	0.980118i	$-0.563580\pi$
$991$	−12.4924	−0.396835	−0.198417	+	0.980118i	$0.563580\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.75379	−0.0555988
$996$	0	0
$997$	−2.68466	−0.0850240	−0.0425120	−	0.999096i	$-0.513536\pi$
$997$	−2.68466	−0.0850240	−0.0425120	+	0.999096i	$0.513536\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.a.bt.1.1		2
3.2	odd	2		560.2.a.i.1.1		2
4.3	odd	2		315.2.a.e.1.2		2
12.11	even	2		35.2.a.b.1.1	✓	2
15.2	even	4		2800.2.g.t.449.3		4
15.8	even	4		2800.2.g.t.449.2		4
15.14	odd	2		2800.2.a.bi.1.2		2
20.3	even	4		1575.2.d.e.1324.1		4
20.7	even	4		1575.2.d.e.1324.4		4
20.19	odd	2		1575.2.a.p.1.1		2
21.20	even	2		3920.2.a.bs.1.2		2
24.5	odd	2		2240.2.a.bd.1.2		2
24.11	even	2		2240.2.a.bh.1.1		2
28.27	even	2		2205.2.a.x.1.2		2
60.23	odd	4		175.2.b.b.99.4		4
60.47	odd	4		175.2.b.b.99.1		4
60.59	even	2		175.2.a.f.1.2		2
84.11	even	6		245.2.e.i.226.2		4
84.23	even	6		245.2.e.i.116.2		4
84.47	odd	6		245.2.e.h.116.2		4
84.59	odd	6		245.2.e.h.226.2		4
84.83	odd	2		245.2.a.d.1.1		2
132.131	odd	2		4235.2.a.m.1.2		2
156.155	even	2		5915.2.a.l.1.2		2
420.83	even	4		1225.2.b.f.99.4		4
420.167	even	4		1225.2.b.f.99.1		4
420.419	odd	2		1225.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.a.b.1.1	✓	2	12.11	even	2
175.2.a.f.1.2		2	60.59	even	2
175.2.b.b.99.1		4	60.47	odd	4
175.2.b.b.99.4		4	60.23	odd	4
245.2.a.d.1.1		2	84.83	odd	2
245.2.e.h.116.2		4	84.47	odd	6
245.2.e.h.226.2		4	84.59	odd	6
245.2.e.i.116.2		4	84.23	even	6
245.2.e.i.226.2		4	84.11	even	6
315.2.a.e.1.2		2	4.3	odd	2
560.2.a.i.1.1		2	3.2	odd	2
1225.2.a.s.1.2		2	420.419	odd	2
1225.2.b.f.99.1		4	420.167	even	4
1225.2.b.f.99.4		4	420.83	even	4
1575.2.a.p.1.1		2	20.19	odd	2
1575.2.d.e.1324.1		4	20.3	even	4
1575.2.d.e.1324.4		4	20.7	even	4
2205.2.a.x.1.2		2	28.27	even	2
2240.2.a.bd.1.2		2	24.5	odd	2
2240.2.a.bh.1.1		2	24.11	even	2
2800.2.a.bi.1.2		2	15.14	odd	2
2800.2.g.t.449.2		4	15.8	even	4
2800.2.g.t.449.3		4	15.2	even	4
3920.2.a.bs.1.2		2	21.20	even	2
4235.2.a.m.1.2		2	132.131	odd	2
5040.2.a.bt.1.1		2	1.1	even	1	trivial
5915.2.a.l.1.2		2	156.155	even	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +1.00000 q^{7} -1.56155 q^{11} +0.438447 q^{13} +0.438447 q^{17} +7.12311 q^{19} +3.12311 q^{23} +1.00000 q^{25} -6.68466 q^{29} -1.00000 q^{35} +6.00000 q^{37} -5.12311 q^{41} -0.876894 q^{43} -8.68466 q^{47} +1.00000 q^{49} +5.12311 q^{53} +1.56155 q^{55} -4.00000 q^{59} +15.3693 q^{61} -0.438447 q^{65} -10.2462 q^{67} +8.00000 q^{71} -12.2462 q^{73} -1.56155 q^{77} +2.43845 q^{79} +4.00000 q^{83} -0.438447 q^{85} +1.12311 q^{89} +0.438447 q^{91} -7.12311 q^{95} +5.80776 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7	\( 2 q - 2 q^{5} + 2 q^{7} + q^{11} + 5 q^{13} + 5 q^{17} + 6 q^{19} - 2 q^{23} + 2 q^{25} - q^{29} - 2 q^{35} + 12 q^{37} - 2 q^{41} - 10 q^{43} - 5 q^{47} + 2 q^{49} + 2 q^{53} - q^{55} - 8 q^{59} + 6 q^{61} - 5 q^{65} - 4 q^{67} + 16 q^{71} - 8 q^{73} + q^{77} + 9 q^{79} + 8 q^{83} - 5 q^{85} - 6 q^{89} + 5 q^{91} - 6 q^{95} - 9 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 + q^11 + 5 * q^13 + 5 * q^17 + 6 * q^19 - 2 * q^23 + 2 * q^25 - q^29 - 2 * q^35 + 12 * q^37 - 2 * q^41 - 10 * q^43 - 5 * q^47 + 2 * q^49 + 2 * q^53 - q^55 - 8 * q^59 + 6 * q^61 - 5 * q^65 - 4 * q^67 + 16 * q^71 - 8 * q^73 + q^77 + 9 * q^79 + 8 * q^83 - 5 * q^85 - 6 * q^89 + 5 * q^91 - 6 * q^95 - 9 * q^97

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