LMFDB - Embedded newform 5040.2.a.by.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{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +1.00000 q^{7} +0.627719 q^{11} -1.37228 q^{13} -5.37228 q^{17} -6.74456 q^{19} +6.74456 q^{23} +1.00000 q^{25} -1.37228 q^{29} +8.00000 q^{31} +1.00000 q^{35} -2.00000 q^{37} +4.74456 q^{41} -2.74456 q^{43} +10.1168 q^{47} +1.00000 q^{49} +0.744563 q^{53} +0.627719 q^{55} +8.00000 q^{59} +8.74456 q^{61} -1.37228 q^{65} +4.00000 q^{67} +8.00000 q^{71} -6.00000 q^{73} +0.627719 q^{77} +2.11684 q^{79} +13.4891 q^{83} -5.37228 q^{85} -3.25544 q^{89} -1.37228 q^{91} -6.74456 q^{95} +18.8614 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 7 q^{11} + 3 q^{13} - 5 q^{17} - 2 q^{19} + 2 q^{23} + 2 q^{25} + 3 q^{29} + 16 q^{31} + 2 q^{35} - 4 q^{37} - 2 q^{41} + 6 q^{43} + 3 q^{47} + 2 q^{49} - 10 q^{53} + 7 q^{55} + 16 q^{59} + 6 q^{61} + 3 q^{65} + 8 q^{67} + 16 q^{71} - 12 q^{73} + 7 q^{77} - 13 q^{79} + 4 q^{83} - 5 q^{85} - 18 q^{89} + 3 q^{91} - 2 q^{95} + 9 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^7 + 7 * q^11 + 3 * q^13 - 5 * q^17 - 2 * q^19 + 2 * q^23 + 2 * q^25 + 3 * q^29 + 16 * q^31 + 2 * q^35 - 4 * q^37 - 2 * q^41 + 6 * q^43 + 3 * q^47 + 2 * q^49 - 10 * q^53 + 7 * q^55 + 16 * q^59 + 6 * q^61 + 3 * q^65 + 8 * q^67 + 16 * q^71 - 12 * q^73 + 7 * q^77 - 13 * q^79 + 4 * q^83 - 5 * q^85 - 18 * q^89 + 3 * q^91 - 2 * 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
$11$	0.627719	0.189264	0.0946322	−	0.995512i	$-0.469833\pi$
$11$	0.627719	0.189264	0.0946322	+	0.995512i	$0.469833\pi$
$12$	0	0
$13$	−1.37228	−0.380602	−0.190301	−	0.981726i	$-0.560946\pi$
$13$	−1.37228	−0.380602	−0.190301	+	0.981726i	$0.560946\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.37228	−1.30297	−0.651485	−	0.758662i	$-0.725854\pi$
$17$	−5.37228	−1.30297	−0.651485	+	0.758662i	$0.725854\pi$
$18$	0	0
$19$	−6.74456	−1.54731	−0.773654	−	0.633608i	$-0.781573\pi$
$19$	−6.74456	−1.54731	−0.773654	+	0.633608i	$0.781573\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.74456	1.40634	0.703169	−	0.711022i	$-0.251768\pi$
$23$	6.74456	1.40634	0.703169	+	0.711022i	$0.251768\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.37228	−0.254826	−0.127413	−	0.991850i	$-0.540667\pi$
$29$	−1.37228	−0.254826	−0.127413	+	0.991850i	$0.540667\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.74456	0.740976	0.370488	−	0.928837i	$-0.379190\pi$
$41$	4.74456	0.740976	0.370488	+	0.928837i	$0.379190\pi$
$42$	0	0
$43$	−2.74456	−0.418542	−0.209271	−	0.977858i	$-0.567109\pi$
$43$	−2.74456	−0.418542	−0.209271	+	0.977858i	$0.567109\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.1168	1.47569	0.737847	−	0.674968i	$-0.235843\pi$
$47$	10.1168	1.47569	0.737847	+	0.674968i	$0.235843\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.744563	0.102274	0.0511368	−	0.998692i	$-0.483716\pi$
$53$	0.744563	0.102274	0.0511368	+	0.998692i	$0.483716\pi$
$54$	0	0
$55$	0.627719	0.0846416
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	8.74456	1.11963	0.559813	−	0.828619i	$-0.310873\pi$
$61$	8.74456	1.11963	0.559813	+	0.828619i	$0.310873\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.37228	−0.170211
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	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
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.627719	0.0715352
$78$	0	0
$79$	2.11684	0.238164	0.119082	−	0.992884i	$-0.462005\pi$
$79$	2.11684	0.238164	0.119082	+	0.992884i	$0.462005\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.4891	1.48062	0.740312	−	0.672264i	$-0.234678\pi$
$83$	13.4891	1.48062	0.740312	+	0.672264i	$0.234678\pi$
$84$	0	0
$85$	−5.37228	−0.582706
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.25544	−0.345076	−0.172538	−	0.985003i	$-0.555197\pi$
$89$	−3.25544	−0.345076	−0.172538	+	0.985003i	$0.555197\pi$
$90$	0	0
$91$	−1.37228	−0.143854
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.74456	−0.691978
$96$	0	0
$97$	18.8614	1.91509	0.957543	−	0.288291i	$-0.0930870\pi$
$97$	18.8614	1.91509	0.957543	+	0.288291i	$0.0930870\pi$
$98$	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
$103$	11.3723	1.12054	0.560272	−	0.828309i	$-0.310697\pi$
$103$	11.3723	1.12054	0.560272	+	0.828309i	$0.310697\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.74456	−0.265327	−0.132663	−	0.991161i	$-0.542353\pi$
$107$	−2.74456	−0.265327	−0.132663	+	0.991161i	$0.542353\pi$
$108$	0	0
$109$	−5.37228	−0.514571	−0.257286	−	0.966335i	$-0.582828\pi$
$109$	−5.37228	−0.514571	−0.257286	+	0.966335i	$0.582828\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	6.74456	0.628934
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.37228	−0.492476
$120$	0	0
$121$	−10.6060	−0.964179
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.74456	0.589275	0.294638	−	0.955609i	$-0.404801\pi$
$131$	6.74456	0.589275	0.294638	+	0.955609i	$0.404801\pi$
$132$	0	0
$133$	−6.74456	−0.584828
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.25544	−0.278131	−0.139065	−	0.990283i	$-0.544410\pi$
$137$	−3.25544	−0.278131	−0.139065	+	0.990283i	$0.544410\pi$
$138$	0	0
$139$	−6.74456	−0.572066	−0.286033	−	0.958220i	$-0.592337\pi$
$139$	−6.74456	−0.572066	−0.286033	+	0.958220i	$0.592337\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.861407	−0.0720344
$144$	0	0
$145$	−1.37228	−0.113962
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.48913	0.613533	0.306767	−	0.951785i	$-0.400753\pi$
$149$	7.48913	0.613533	0.306767	+	0.951785i	$0.400753\pi$
$150$	0	0
$151$	2.11684	0.172266	0.0861332	−	0.996284i	$-0.472549\pi$
$151$	2.11684	0.172266	0.0861332	+	0.996284i	$0.472549\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	7.48913	0.597697	0.298849	−	0.954301i	$-0.403397\pi$
$157$	7.48913	0.597697	0.298849	+	0.954301i	$0.403397\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.74456	0.531546
$162$	0	0
$163$	5.25544	0.411638	0.205819	−	0.978590i	$-0.434014\pi$
$163$	5.25544	0.411638	0.205819	+	0.978590i	$0.434014\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.3723	−0.880014	−0.440007	−	0.897994i	$-0.645024\pi$
$167$	−11.3723	−0.880014	−0.440007	+	0.897994i	$0.645024\pi$
$168$	0	0
$169$	−11.1168	−0.855142
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.37228	−0.408447	−0.204223	−	0.978924i	$-0.565467\pi$
$173$	−5.37228	−0.408447	−0.204223	+	0.978924i	$0.565467\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.9783	1.71748	0.858738	−	0.512416i	$-0.171249\pi$
$179$	22.9783	1.71748	0.858738	+	0.512416i	$0.171249\pi$
$180$	0	0
$181$	−18.2337	−1.35530	−0.677650	−	0.735385i	$-0.737001\pi$
$181$	−18.2337	−1.35530	−0.677650	+	0.735385i	$0.737001\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−3.37228	−0.246606
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.8614	−1.79891	−0.899454	−	0.437015i	$-0.856036\pi$
$191$	−24.8614	−1.79891	−0.899454	+	0.437015i	$0.856036\pi$
$192$	0	0
$193$	−4.74456	−0.341521	−0.170761	−	0.985313i	$-0.554622\pi$
$193$	−4.74456	−0.341521	−0.170761	+	0.985313i	$0.554622\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.2337	−1.86907	−0.934536	−	0.355867i	$-0.884185\pi$
$197$	−26.2337	−1.86907	−0.934536	+	0.355867i	$0.884185\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.37228	−0.0963153
$204$	0	0
$205$	4.74456	0.331375
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.23369	−0.292850
$210$	0	0
$211$	8.62772	0.593957	0.296978	−	0.954884i	$-0.404021\pi$
$211$	8.62772	0.593957	0.296978	+	0.954884i	$0.404021\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.74456	−0.187178
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.37228	0.495913
$222$	0	0
$223$	11.3723	0.761544	0.380772	−	0.924669i	$-0.375658\pi$
$223$	11.3723	0.761544	0.380772	+	0.924669i	$0.375658\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.86141	−0.588152	−0.294076	−	0.955782i	$-0.595012\pi$
$227$	−8.86141	−0.588152	−0.294076	+	0.955782i	$0.595012\pi$
$228$	0	0
$229$	22.2337	1.46924	0.734622	−	0.678477i	$-0.237360\pi$
$229$	22.2337	1.46924	0.734622	+	0.678477i	$0.237360\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.7446	0.834924	0.417462	−	0.908694i	$-0.362920\pi$
$233$	12.7446	0.834924	0.417462	+	0.908694i	$0.362920\pi$
$234$	0	0
$235$	10.1168	0.659950
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.3723	1.25309	0.626544	−	0.779386i	$-0.284469\pi$
$239$	19.3723	1.25309	0.626544	+	0.779386i	$0.284469\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	9.25544	0.588909
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.74456	−0.425713	−0.212857	−	0.977083i	$-0.568277\pi$
$251$	−6.74456	−0.425713	−0.212857	+	0.977083i	$0.568277\pi$
$252$	0	0
$253$	4.23369	0.266170
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.510875	0.0318675	0.0159337	−	0.999873i	$-0.494928\pi$
$257$	0.510875	0.0318675	0.0159337	+	0.999873i	$0.494928\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.2337	−0.754362	−0.377181	−	0.926140i	$-0.623106\pi$
$263$	−12.2337	−0.754362	−0.377181	+	0.926140i	$0.623106\pi$
$264$	0	0
$265$	0.744563	0.0457381
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	13.4891	0.819406	0.409703	−	0.912219i	$-0.365632\pi$
$271$	13.4891	0.819406	0.409703	+	0.912219i	$0.365632\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.627719	0.0378529
$276$	0	0
$277$	−20.9783	−1.26046	−0.630230	−	0.776408i	$-0.717040\pi$
$277$	−20.9783	−1.26046	−0.630230	+	0.776408i	$0.717040\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.6060	1.28890	0.644452	−	0.764645i	$-0.277086\pi$
$281$	21.6060	1.28890	0.644452	+	0.764645i	$0.277086\pi$
$282$	0	0
$283$	26.1168	1.55249	0.776243	−	0.630434i	$-0.217123\pi$
$283$	26.1168	1.55249	0.776243	+	0.630434i	$0.217123\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.74456	0.280063
$288$	0	0
$289$	11.8614	0.697730
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7.88316	−0.460539	−0.230269	−	0.973127i	$-0.573961\pi$
$293$	−7.88316	−0.460539	−0.230269	+	0.973127i	$0.573961\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9.25544	−0.535256
$300$	0	0
$301$	−2.74456	−0.158194
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.74456	0.500712
$306$	0	0
$307$	−13.8832	−0.792354	−0.396177	−	0.918174i	$-0.629663\pi$
$307$	−13.8832	−0.792354	−0.396177	+	0.918174i	$0.629663\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.25544	−0.0711893	−0.0355947	−	0.999366i	$-0.511333\pi$
$311$	−1.25544	−0.0711893	−0.0355947	+	0.999366i	$0.511333\pi$
$312$	0	0
$313$	20.1168	1.13707	0.568536	−	0.822659i	$-0.307510\pi$
$313$	20.1168	1.13707	0.568536	+	0.822659i	$0.307510\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	−0.861407	−0.0482295
$320$	0	0
$321$	0	0
$322$	0	0
$323$	36.2337	2.01610
$324$	0	0
$325$	−1.37228	−0.0761205
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10.1168	0.557760
$330$	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
$335$	4.00000	0.218543
$336$	0	0
$337$	15.4891	0.843746	0.421873	−	0.906655i	$-0.361373\pi$
$337$	15.4891	0.843746	0.421873	+	0.906655i	$0.361373\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.02175	0.271943
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.2554	−0.711589	−0.355795	−	0.934564i	$-0.615790\pi$
$347$	−13.2554	−0.711589	−0.355795	+	0.934564i	$0.615790\pi$
$348$	0	0
$349$	−3.48913	−0.186769	−0.0933843	−	0.995630i	$-0.529769\pi$
$349$	−3.48913	−0.186769	−0.0933843	+	0.995630i	$0.529769\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−26.8614	−1.42969	−0.714844	−	0.699284i	$-0.753502\pi$
$353$	−26.8614	−1.42969	−0.714844	+	0.699284i	$0.753502\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	26.4891	1.39416
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	8.86141	0.462562	0.231281	−	0.972887i	$-0.425708\pi$
$367$	8.86141	0.462562	0.231281	+	0.972887i	$0.425708\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.744563	0.0386558
$372$	0	0
$373$	−19.2554	−0.997009	−0.498504	−	0.866887i	$-0.666117\pi$
$373$	−19.2554	−0.997009	−0.498504	+	0.866887i	$0.666117\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.88316	0.0969875
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.48913	0.280481	0.140241	−	0.990117i	$-0.455212\pi$
$383$	5.48913	0.280481	0.140241	+	0.990117i	$0.455212\pi$
$384$	0	0
$385$	0.627719	0.0319915
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.8614	0.550695	0.275348	−	0.961345i	$-0.411207\pi$
$389$	10.8614	0.550695	0.275348	+	0.961345i	$0.411207\pi$
$390$	0	0
$391$	−36.2337	−1.83242
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.11684	0.106510
$396$	0	0
$397$	37.3723	1.87566	0.937831	−	0.347094i	$-0.112831\pi$
$397$	37.3723	1.87566	0.937831	+	0.347094i	$0.112831\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.60597	−0.0801983	−0.0400991	−	0.999196i	$-0.512767\pi$
$401$	−1.60597	−0.0801983	−0.0400991	+	0.999196i	$0.512767\pi$
$402$	0	0
$403$	−10.9783	−0.546866
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.25544	−0.0622297
$408$	0	0
$409$	−11.4891	−0.568101	−0.284050	−	0.958809i	$-0.591678\pi$
$409$	−11.4891	−0.568101	−0.284050	+	0.958809i	$0.591678\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	13.4891	0.662155
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−37.4891	−1.83146	−0.915732	−	0.401790i	$-0.868388\pi$
$419$	−37.4891	−1.83146	−0.915732	+	0.401790i	$0.868388\pi$
$420$	0	0
$421$	21.6060	1.05301	0.526505	−	0.850172i	$-0.323502\pi$
$421$	21.6060	1.05301	0.526505	+	0.850172i	$0.323502\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.37228	−0.260594
$426$	0	0
$427$	8.74456	0.423179
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.6277	0.608256	0.304128	−	0.952631i	$-0.401635\pi$
$431$	12.6277	0.608256	0.304128	+	0.952631i	$0.401635\pi$
$432$	0	0
$433$	−16.9783	−0.815923	−0.407961	−	0.912999i	$-0.633760\pi$
$433$	−16.9783	−0.815923	−0.407961	+	0.912999i	$0.633760\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−45.4891	−2.17604
$438$	0	0
$439$	−28.2337	−1.34752	−0.673760	−	0.738950i	$-0.735322\pi$
$439$	−28.2337	−1.34752	−0.673760	+	0.738950i	$0.735322\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.25544	−0.249693	−0.124847	−	0.992176i	$-0.539844\pi$
$443$	−5.25544	−0.249693	−0.124847	+	0.992176i	$0.539844\pi$
$444$	0	0
$445$	−3.25544	−0.154323
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0.116844	0.00551421	0.00275710	−	0.999996i	$-0.499122\pi$
$449$	0.116844	0.00551421	0.00275710	+	0.999996i	$0.499122\pi$
$450$	0	0
$451$	2.97825	0.140240
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.37228	−0.0643335
$456$	0	0
$457$	16.7446	0.783278	0.391639	−	0.920119i	$-0.371908\pi$
$457$	16.7446	0.783278	0.391639	+	0.920119i	$0.371908\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.7446	0.593573	0.296787	−	0.954944i	$-0.404085\pi$
$461$	12.7446	0.593573	0.296787	+	0.954944i	$0.404085\pi$
$462$	0	0
$463$	29.4891	1.37048	0.685238	−	0.728319i	$-0.259698\pi$
$463$	29.4891	1.37048	0.685238	+	0.728319i	$0.259698\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−31.6060	−1.46255	−0.731275	−	0.682083i	$-0.761074\pi$
$467$	−31.6060	−1.46255	−0.731275	+	0.682083i	$0.761074\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.72281	−0.0792150
$474$	0	0
$475$	−6.74456	−0.309462
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12.2337	−0.558971	−0.279486	−	0.960150i	$-0.590164\pi$
$479$	−12.2337	−0.558971	−0.279486	+	0.960150i	$0.590164\pi$
$480$	0	0
$481$	2.74456	0.125141
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18.8614	0.856452
$486$	0	0
$487$	4.23369	0.191847	0.0959234	−	0.995389i	$-0.469420\pi$
$487$	4.23369	0.191847	0.0959234	+	0.995389i	$0.469420\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.8832	0.807056	0.403528	−	0.914967i	$-0.367784\pi$
$491$	17.8832	0.807056	0.403528	+	0.914967i	$0.367784\pi$
$492$	0	0
$493$	7.37228	0.332031
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	−3.13859	−0.140503	−0.0702514	−	0.997529i	$-0.522380\pi$
$499$	−3.13859	−0.140503	−0.0702514	+	0.997529i	$0.522380\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.6277	−0.563042	−0.281521	−	0.959555i	$-0.590839\pi$
$503$	−12.6277	−0.563042	−0.281521	+	0.959555i	$0.590839\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4.97825	−0.220657	−0.110329	−	0.993895i	$-0.535190\pi$
$509$	−4.97825	−0.220657	−0.110329	+	0.993895i	$0.535190\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11.3723	0.501123
$516$	0	0
$517$	6.35053	0.279296
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	−32.4674	−1.41970	−0.709850	−	0.704353i	$-0.751237\pi$
$523$	−32.4674	−1.41970	−0.709850	+	0.704353i	$0.751237\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−42.9783	−1.87216
$528$	0	0
$529$	22.4891	0.977788
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.51087	−0.282017
$534$	0	0
$535$	−2.74456	−0.118658
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.627719	0.0270378
$540$	0	0
$541$	20.3505	0.874938	0.437469	−	0.899234i	$-0.355875\pi$
$541$	20.3505	0.874938	0.437469	+	0.899234i	$0.355875\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.37228	−0.230123
$546$	0	0
$547$	−14.9783	−0.640424	−0.320212	−	0.947346i	$-0.603754\pi$
$547$	−14.9783	−0.640424	−0.320212	+	0.947346i	$0.603754\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.25544	0.394295
$552$	0	0
$553$	2.11684	0.0900174
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.2337	1.62001	0.810007	−	0.586421i	$-0.199463\pi$
$557$	38.2337	1.62001	0.810007	+	0.586421i	$0.199463\pi$
$558$	0	0
$559$	3.76631	0.159298
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.48913	−0.231339	−0.115670	−	0.993288i	$-0.536901\pi$
$563$	−5.48913	−0.231339	−0.115670	+	0.993288i	$0.536901\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.9783	−0.879454	−0.439727	−	0.898131i	$-0.644925\pi$
$569$	−20.9783	−0.879454	−0.439727	+	0.898131i	$0.644925\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.74456	0.281268
$576$	0	0
$577$	17.6060	0.732946	0.366473	−	0.930429i	$-0.380565\pi$
$577$	17.6060	0.732946	0.366473	+	0.930429i	$0.380565\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.4891	0.559623
$582$	0	0
$583$	0.467376	0.0193567
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.9783	0.453121	0.226560	−	0.973997i	$-0.427252\pi$
$587$	10.9783	0.453121	0.226560	+	0.973997i	$0.427252\pi$
$588$	0	0
$589$	−53.9565	−2.22324
$590$	0	0
$591$	0	0
$592$	0	0
$593$	25.3723	1.04191	0.520957	−	0.853583i	$-0.325575\pi$
$593$	25.3723	1.04191	0.520957	+	0.853583i	$0.325575\pi$
$594$	0	0
$595$	−5.37228	−0.220242
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7.60597	0.310771	0.155386	−	0.987854i	$-0.450338\pi$
$599$	7.60597	0.310771	0.155386	+	0.987854i	$0.450338\pi$
$600$	0	0
$601$	39.4891	1.61080	0.805398	−	0.592735i	$-0.201952\pi$
$601$	39.4891	1.61080	0.805398	+	0.592735i	$0.201952\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.6060	−0.431194
$606$	0	0
$607$	15.6060	0.633427	0.316713	−	0.948521i	$-0.397421\pi$
$607$	15.6060	0.633427	0.316713	+	0.948521i	$0.397421\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13.8832	−0.561652
$612$	0	0
$613$	−31.4891	−1.27183	−0.635917	−	0.771758i	$-0.719378\pi$
$613$	−31.4891	−1.27183	−0.635917	+	0.771758i	$0.719378\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	1.25544	0.0504603	0.0252301	−	0.999682i	$-0.491968\pi$
$619$	1.25544	0.0504603	0.0252301	+	0.999682i	$0.491968\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.25544	−0.130426
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.7446	0.428414
$630$	0	0
$631$	3.37228	0.134248	0.0671242	−	0.997745i	$-0.478618\pi$
$631$	3.37228	0.134248	0.0671242	+	0.997745i	$0.478618\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	−1.37228	−0.0543718
$638$	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
$643$	−12.6277	−0.497989	−0.248994	−	0.968505i	$-0.580100\pi$
$643$	−12.6277	−0.497989	−0.248994	+	0.968505i	$0.580100\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	0	0
$649$	5.02175	0.197121
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	6.74456	0.263532
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.11684	0.238278	0.119139	−	0.992878i	$-0.461987\pi$
$659$	6.11684	0.238278	0.119139	+	0.992878i	$0.461987\pi$
$660$	0	0
$661$	3.25544	0.126622	0.0633109	−	0.997994i	$-0.479834\pi$
$661$	3.25544	0.126622	0.0633109	+	0.997994i	$0.479834\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.74456	−0.261543
$666$	0	0
$667$	−9.25544	−0.358372
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.48913	0.211905
$672$	0	0
$673$	−31.7228	−1.22282	−0.611412	−	0.791312i	$-0.709398\pi$
$673$	−31.7228	−1.22282	−0.611412	+	0.791312i	$0.709398\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.3505	1.39706	0.698532	−	0.715579i	$-0.253837\pi$
$677$	36.3505	1.39706	0.698532	+	0.715579i	$0.253837\pi$
$678$	0	0
$679$	18.8614	0.723834
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	−3.25544	−0.124384
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.02175	−0.0389256
$690$	0	0
$691$	−13.4891	−0.513151	−0.256575	−	0.966524i	$-0.582594\pi$
$691$	−13.4891	−0.513151	−0.256575	+	0.966524i	$0.582594\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.74456	−0.255836
$696$	0	0
$697$	−25.4891	−0.965469
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30.8614	−1.16562	−0.582810	−	0.812609i	$-0.698047\pi$
$701$	−30.8614	−1.16562	−0.582810	+	0.812609i	$0.698047\pi$
$702$	0	0
$703$	13.4891	0.508752
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	−33.6060	−1.26210	−0.631049	−	0.775743i	$-0.717375\pi$
$709$	−33.6060	−1.26210	−0.631049	+	0.775743i	$0.717375\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	53.9565	2.02069
$714$	0	0
$715$	−0.861407	−0.0322148
$716$	0	0
$717$	0	0
$718$	0	0
$719$	14.7446	0.549879	0.274940	−	0.961461i	$-0.411342\pi$
$719$	14.7446	0.549879	0.274940	+	0.961461i	$0.411342\pi$
$720$	0	0
$721$	11.3723	0.423526
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.37228	−0.0509652
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14.7446	0.545347
$732$	0	0
$733$	−38.8614	−1.43538	−0.717689	−	0.696363i	$-0.754800\pi$
$733$	−38.8614	−1.43538	−0.717689	+	0.696363i	$0.754800\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.51087	0.0924893
$738$	0	0
$739$	−19.6060	−0.721217	−0.360609	−	0.932717i	$-0.617431\pi$
$739$	−19.6060	−0.721217	−0.360609	+	0.932717i	$0.617431\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29.4891	−1.08185	−0.540926	−	0.841070i	$-0.681926\pi$
$743$	−29.4891	−1.08185	−0.540926	+	0.841070i	$0.681926\pi$
$744$	0	0
$745$	7.48913	0.274380
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.74456	−0.100284
$750$	0	0
$751$	−48.8614	−1.78298	−0.891489	−	0.453042i	$-0.850339\pi$
$751$	−48.8614	−1.78298	−0.891489	+	0.453042i	$0.850339\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.11684	0.0770398
$756$	0	0
$757$	−38.2337	−1.38963	−0.694814	−	0.719190i	$-0.744513\pi$
$757$	−38.2337	−1.38963	−0.694814	+	0.719190i	$0.744513\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.9783	1.48546	0.742730	−	0.669591i	$-0.233531\pi$
$761$	40.9783	1.48546	0.742730	+	0.669591i	$0.233531\pi$
$762$	0	0
$763$	−5.37228	−0.194490
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.9783	−0.396402
$768$	0	0
$769$	−19.4891	−0.702796	−0.351398	−	0.936226i	$-0.614294\pi$
$769$	−19.4891	−0.702796	−0.351398	+	0.936226i	$0.614294\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.37228	0.0493575	0.0246788	−	0.999695i	$-0.492144\pi$
$773$	1.37228	0.0493575	0.0246788	+	0.999695i	$0.492144\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−32.0000	−1.14652
$780$	0	0
$781$	5.02175	0.179692
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.48913	0.267298
$786$	0	0
$787$	13.8832	0.494881	0.247441	−	0.968903i	$-0.420411\pi$
$787$	13.8832	0.494881	0.247441	+	0.968903i	$0.420411\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.3723	−0.757045	−0.378523	−	0.925592i	$-0.623568\pi$
$797$	−21.3723	−0.757045	−0.378523	+	0.925592i	$0.623568\pi$
$798$	0	0
$799$	−54.3505	−1.92278
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.76631	−0.132910
$804$	0	0
$805$	6.74456	0.237715
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−14.6277	−0.514283	−0.257142	−	0.966374i	$-0.582781\pi$
$809$	−14.6277	−0.514283	−0.257142	+	0.966374i	$0.582781\pi$
$810$	0	0
$811$	−1.25544	−0.0440844	−0.0220422	−	0.999757i	$-0.507017\pi$
$811$	−1.25544	−0.0440844	−0.0220422	+	0.999757i	$0.507017\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.25544	0.184090
$816$	0	0
$817$	18.5109	0.647614
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.88316	0.275124	0.137562	−	0.990493i	$-0.456073\pi$
$821$	7.88316	0.275124	0.137562	+	0.990493i	$0.456073\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.2554	0.460937	0.230468	−	0.973080i	$-0.425974\pi$
$827$	13.2554	0.460937	0.230468	+	0.973080i	$0.425974\pi$
$828$	0	0
$829$	22.2337	0.772208	0.386104	−	0.922455i	$-0.373821\pi$
$829$	22.2337	0.772208	0.386104	+	0.922455i	$0.373821\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.37228	−0.186139
$834$	0	0
$835$	−11.3723	−0.393554
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−22.7446	−0.785230	−0.392615	−	0.919703i	$-0.628429\pi$
$839$	−22.7446	−0.785230	−0.392615	+	0.919703i	$0.628429\pi$
$840$	0	0
$841$	−27.1168	−0.935064
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−11.1168	−0.382431
$846$	0	0
$847$	−10.6060	−0.364425
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−13.4891	−0.462401
$852$	0	0
$853$	−16.5109	−0.565322	−0.282661	−	0.959220i	$-0.591217\pi$
$853$	−16.5109	−0.565322	−0.282661	+	0.959220i	$0.591217\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	24.4674	0.834816	0.417408	−	0.908719i	$-0.362939\pi$
$859$	24.4674	0.834816	0.417408	+	0.908719i	$0.362939\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.51087	0.0854712	0.0427356	−	0.999086i	$-0.486393\pi$
$863$	2.51087	0.0854712	0.0427356	+	0.999086i	$0.486393\pi$
$864$	0	0
$865$	−5.37228	−0.182663
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.32878	0.0450759
$870$	0	0
$871$	−5.48913	−0.185992
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−52.9783	−1.78895	−0.894474	−	0.447120i	$-0.852450\pi$
$877$	−52.9783	−1.78895	−0.894474	+	0.447120i	$0.852450\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	36.7446	1.23796	0.618978	−	0.785408i	$-0.287547\pi$
$881$	36.7446	1.23796	0.618978	+	0.785408i	$0.287547\pi$
$882$	0	0
$883$	33.4891	1.12700	0.563499	−	0.826116i	$-0.309455\pi$
$883$	33.4891	1.12700	0.563499	+	0.826116i	$0.309455\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−58.9783	−1.98030	−0.990148	−	0.140025i	$-0.955282\pi$
$887$	−58.9783	−1.98030	−0.990148	+	0.140025i	$0.955282\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−68.2337	−2.28335
$894$	0	0
$895$	22.9783	0.768078
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10.9783	−0.366145
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18.2337	−0.606108
$906$	0	0
$907$	45.7228	1.51820	0.759101	−	0.650973i	$-0.225639\pi$
$907$	45.7228	1.51820	0.759101	+	0.650973i	$0.225639\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	45.9565	1.52261	0.761303	−	0.648396i	$-0.224560\pi$
$911$	45.9565	1.52261	0.761303	+	0.648396i	$0.224560\pi$
$912$	0	0
$913$	8.46738	0.280229
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.74456	0.222725
$918$	0	0
$919$	−19.3723	−0.639033	−0.319516	−	0.947581i	$-0.603520\pi$
$919$	−19.3723	−0.639033	−0.319516	+	0.947581i	$0.603520\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.9783	−0.361354
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	42.2337	1.38564	0.692821	−	0.721109i	$-0.256368\pi$
$929$	42.2337	1.38564	0.692821	+	0.721109i	$0.256368\pi$
$930$	0	0
$931$	−6.74456	−0.221044
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.37228	−0.110285
$936$	0	0
$937$	−35.0951	−1.14651	−0.573253	−	0.819378i	$-0.694319\pi$
$937$	−35.0951	−1.14651	−0.573253	+	0.819378i	$0.694319\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.23369	0.0728161	0.0364081	−	0.999337i	$-0.488408\pi$
$941$	2.23369	0.0728161	0.0364081	+	0.999337i	$0.488408\pi$
$942$	0	0
$943$	32.0000	1.04206
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	0	0
$949$	8.23369	0.267277
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−48.7446	−1.57899	−0.789496	−	0.613756i	$-0.789658\pi$
$953$	−48.7446	−1.57899	−0.789496	+	0.613756i	$0.789658\pi$
$954$	0	0
$955$	−24.8614	−0.804496
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.25544	−0.105124
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.74456	−0.152733
$966$	0	0
$967$	−36.2337	−1.16520	−0.582598	−	0.812760i	$-0.697964\pi$
$967$	−36.2337	−1.16520	−0.582598	+	0.812760i	$0.697964\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10.5109	0.337310	0.168655	−	0.985675i	$-0.446058\pi$
$971$	10.5109	0.337310	0.168655	+	0.985675i	$0.446058\pi$
$972$	0	0
$973$	−6.74456	−0.216221
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.02175	−0.224646	−0.112323	−	0.993672i	$-0.535829\pi$
$977$	−7.02175	−0.224646	−0.112323	+	0.993672i	$0.535829\pi$
$978$	0	0
$979$	−2.04350	−0.0653105
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.62772	−0.147601	−0.0738007	−	0.997273i	$-0.523513\pi$
$983$	−4.62772	−0.147601	−0.0738007	+	0.997273i	$0.523513\pi$
$984$	0	0
$985$	−26.2337	−0.835875
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−18.5109	−0.588612
$990$	0	0
$991$	−53.9565	−1.71398	−0.856992	−	0.515329i	$-0.827670\pi$
$991$	−53.9565	−1.71398	−0.856992	+	0.515329i	$0.827670\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	−24.1168	−0.763788	−0.381894	−	0.924206i	$-0.624728\pi$
$997$	−24.1168	−0.763788	−0.381894	+	0.924206i	$0.624728\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.a.by.1.1		2
3.2	odd	2		560.2.a.h.1.2		2
4.3	odd	2		2520.2.a.x.1.2		2
12.11	even	2		280.2.a.c.1.1	✓	2
15.2	even	4		2800.2.g.r.449.1		4
15.8	even	4		2800.2.g.r.449.4		4
15.14	odd	2		2800.2.a.bk.1.1		2
21.20	even	2		3920.2.a.bt.1.1		2
24.5	odd	2		2240.2.a.bg.1.1		2
24.11	even	2		2240.2.a.bk.1.2		2
60.23	odd	4		1400.2.g.i.449.1		4
60.47	odd	4		1400.2.g.i.449.4		4
60.59	even	2		1400.2.a.r.1.2		2
84.11	even	6		1960.2.q.t.961.2		4
84.23	even	6		1960.2.q.t.361.2		4
84.47	odd	6		1960.2.q.r.361.1		4
84.59	odd	6		1960.2.q.r.961.1		4
84.83	odd	2		1960.2.a.s.1.2		2
420.419	odd	2		9800.2.a.bu.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.a.c.1.1	✓	2	12.11	even	2
560.2.a.h.1.2		2	3.2	odd	2
1400.2.a.r.1.2		2	60.59	even	2
1400.2.g.i.449.1		4	60.23	odd	4
1400.2.g.i.449.4		4	60.47	odd	4
1960.2.a.s.1.2		2	84.83	odd	2
1960.2.q.r.361.1		4	84.47	odd	6
1960.2.q.r.961.1		4	84.59	odd	6
1960.2.q.t.361.2		4	84.23	even	6
1960.2.q.t.961.2		4	84.11	even	6
2240.2.a.bg.1.1		2	24.5	odd	2
2240.2.a.bk.1.2		2	24.11	even	2
2520.2.a.x.1.2		2	4.3	odd	2
2800.2.a.bk.1.1		2	15.14	odd	2
2800.2.g.r.449.1		4	15.2	even	4
2800.2.g.r.449.4		4	15.8	even	4
3920.2.a.bt.1.1		2	21.20	even	2
5040.2.a.by.1.1		2	1.1	even	1	trivial
9800.2.a.bu.1.1		2	420.419	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +1.00000 q^{7} +0.627719 q^{11} -1.37228 q^{13} -5.37228 q^{17} -6.74456 q^{19} +6.74456 q^{23} +1.00000 q^{25} -1.37228 q^{29} +8.00000 q^{31} +1.00000 q^{35} -2.00000 q^{37} +4.74456 q^{41} -2.74456 q^{43} +10.1168 q^{47} +1.00000 q^{49} +0.744563 q^{53} +0.627719 q^{55} +8.00000 q^{59} +8.74456 q^{61} -1.37228 q^{65} +4.00000 q^{67} +8.00000 q^{71} -6.00000 q^{73} +0.627719 q^{77} +2.11684 q^{79} +13.4891 q^{83} -5.37228 q^{85} -3.25544 q^{89} -1.37228 q^{91} -6.74456 q^{95} +18.8614 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 2 * q^7	\( 2 q + 2 q^{5} + 2 q^{7} + 7 q^{11} + 3 q^{13} - 5 q^{17} - 2 q^{19} + 2 q^{23} + 2 q^{25} + 3 q^{29} + 16 q^{31} + 2 q^{35} - 4 q^{37} - 2 q^{41} + 6 q^{43} + 3 q^{47} + 2 q^{49} - 10 q^{53} + 7 q^{55} + 16 q^{59} + 6 q^{61} + 3 q^{65} + 8 q^{67} + 16 q^{71} - 12 q^{73} + 7 q^{77} - 13 q^{79} + 4 q^{83} - 5 q^{85} - 18 q^{89} + 3 q^{91} - 2 q^{95} + 9 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^7 + 7 * q^11 + 3 * q^13 - 5 * q^17 - 2 * q^19 + 2 * q^23 + 2 * q^25 + 3 * q^29 + 16 * q^31 + 2 * q^35 - 4 * q^37 - 2 * q^41 + 6 * q^43 + 3 * q^47 + 2 * q^49 - 10 * q^53 + 7 * q^55 + 16 * q^59 + 6 * q^61 + 3 * q^65 + 8 * q^67 + 16 * q^71 - 12 * q^73 + 7 * q^77 - 13 * q^79 + 4 * q^83 - 5 * q^85 - 18 * q^89 + 3 * q^91 - 2 * q^95 + 9 * q^97

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