LMFDB - Embedded newform 8575.2.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8575,2,Mod(1,8575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8575 = 5^{2} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8575.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:	$68.4717197332$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 343)
Fricke sign:	$+1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.3
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	8575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.04892 q^{2} +2.19806 q^{4} +0.405813 q^{8} -3.00000 q^{9} +O(q^{10})$	$q+2.04892 q^{2} +2.19806 q^{4} +0.405813 q^{8} -3.00000 q^{9} +4.26875 q^{11} -3.56465 q^{16} -6.14675 q^{18} +8.74632 q^{22} -9.12498 q^{23} +9.52111 q^{29} -8.11529 q^{32} -6.59419 q^{36} -11.6528 q^{37} -8.51573 q^{43} +9.38298 q^{44} -18.6963 q^{46} -4.41789 q^{53} +19.5080 q^{58} -9.49827 q^{64} +9.91723 q^{67} -16.7114 q^{71} -1.21744 q^{72} -23.8756 q^{74} -0.320060 q^{79} +9.00000 q^{81} -17.4480 q^{86} +1.73232 q^{88} -20.0573 q^{92} -12.8062 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 11 q^{4} - 12 q^{8} - 9 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 11 * q^4 - 12 * q^8 - 9 * q^9	$ 3 q - 3 q^{2} + 11 q^{4} - 12 q^{8} - 9 q^{9} + 5 q^{11} + 11 q^{16} + 9 q^{18} + 9 q^{22} - 3 q^{23} + 13 q^{29} - 22 q^{32} - 33 q^{36} - 17 q^{37} - 13 q^{43} + 2 q^{44} - 32 q^{46} - 19 q^{53} + 8 q^{58} + 23 q^{67} - q^{71} + 36 q^{72} - 11 q^{74} - 25 q^{79} + 27 q^{81} - 8 q^{86} + 29 q^{88} - 4 q^{92} - 15 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 11 * q^4 - 12 * q^8 - 9 * q^9 + 5 * q^11 + 11 * q^16 + 9 * q^18 + 9 * q^22 - 3 * q^23 + 13 * q^29 - 22 * q^32 - 33 * q^36 - 17 * q^37 - 13 * q^43 + 2 * q^44 - 32 * q^46 - 19 * q^53 + 8 * q^58 + 23 * q^67 - q^71 + 36 * q^72 - 11 * q^74 - 25 * q^79 + 27 * q^81 - 8 * q^86 + 29 * q^88 - 4 * q^92 - 15 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.04892	1.44880	0.724402	−	0.689378i	$-0.242116\pi$
$2$	2.04892	1.44880	0.724402	+	0.689378i	$0.242116\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	2.19806	1.09903
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0.405813	0.143477
$9$	−3.00000	−1.00000
$10$	0	0
$11$	4.26875	1.28708	0.643538	−	0.765414i	$-0.277466\pi$
$11$	4.26875	1.28708	0.643538	+	0.765414i	$0.277466\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	−3.56465	−0.891162
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−6.14675	−1.44880
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	8.74632	1.86472
$23$	−9.12498	−1.90269	−0.951345	−	0.308127i	$-0.900298\pi$
$23$	−9.12498	−1.90269	−0.951345	+	0.308127i	$0.900298\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.52111	1.76803	0.884013	−	0.467463i	$-0.154832\pi$
$29$	9.52111	1.76803	0.884013	+	0.467463i	$0.154832\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−8.11529	−1.43459
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−6.59419	−1.09903
$37$	−11.6528	−1.91571	−0.957854	−	0.287257i	$-0.907257\pi$
$37$	−11.6528	−1.91571	−0.957854	+	0.287257i	$0.907257\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−8.51573	−1.29864	−0.649318	−	0.760517i	$-0.724946\pi$
$43$	−8.51573	−1.29864	−0.649318	+	0.760517i	$0.724946\pi$
$44$	9.38298	1.41454
$45$	0	0
$46$	−18.6963	−2.75662
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.41789	−0.606845	−0.303422	−	0.952856i	$-0.598129\pi$
$53$	−4.41789	−0.606845	−0.303422	+	0.952856i	$0.598129\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	19.5080	2.56152
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	−9.49827	−1.18728
$65$	0	0
$66$	0	0
$67$	9.91723	1.21158	0.605791	−	0.795624i	$-0.292857\pi$
$67$	9.91723	1.21158	0.605791	+	0.795624i	$0.292857\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−16.7114	−1.98328	−0.991639	−	0.129042i	$-0.958810\pi$
$71$	−16.7114	−1.98328	−0.991639	+	0.129042i	$0.958810\pi$
$72$	−1.21744	−0.143477
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−23.8756	−2.77548
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.320060	−0.0360096	−0.0180048	−	0.999838i	$-0.505731\pi$
$79$	−0.320060	−0.0360096	−0.0180048	+	0.999838i	$0.505731\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	−17.4480	−1.88147
$87$	0	0
$88$	1.73232	0.184665
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	−20.0573	−2.09112
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	−12.8062	−1.28708
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	−9.05190	−0.879198
$107$	8.33273	0.805556	0.402778	−	0.915298i	$-0.368045\pi$
$107$	8.33273	0.805556	0.402778	+	0.915298i	$0.368045\pi$
$108$	0	0
$109$	−20.8092	−1.99316	−0.996582	−	0.0826150i	$-0.973673\pi$
$109$	−20.8092	−1.99316	−0.996582	+	0.0826150i	$0.973673\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−20.1903	−1.89934	−0.949671	−	0.313248i	$-0.898583\pi$
$113$	−20.1903	−1.89934	−0.949671	+	0.313248i	$0.898583\pi$
$114$	0	0
$115$	0	0
$116$	20.9280	1.94311
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.22223	0.656566
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	19.0368	1.68925	0.844623	−	0.535362i	$-0.179825\pi$
$127$	19.0368	1.68925	0.844623	+	0.535362i	$0.179825\pi$
$128$	−3.23059	−0.285546
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	20.3196	1.75534
$135$	0	0
$136$	0	0
$137$	−10.3134	−0.881129	−0.440565	−	0.897721i	$-0.645222\pi$
$137$	−10.3134	−0.881129	−0.440565	+	0.897721i	$0.645222\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	−34.2403	−2.87338
$143$	0	0
$144$	10.6939	0.891162
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−25.6136	−2.10542
$149$	5.42221	0.444204	0.222102	−	0.975023i	$-0.428708\pi$
$149$	5.42221	0.444204	0.222102	+	0.975023i	$0.428708\pi$
$150$	0	0
$151$	10.4993	0.854424	0.427212	−	0.904152i	$-0.359496\pi$
$151$	10.4993	0.854424	0.427212	+	0.904152i	$0.359496\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	−0.655777	−0.0521708
$159$	0	0
$160$	0	0
$161$	0	0
$162$	18.4403	1.44880
$163$	−24.9071	−1.95087	−0.975436	−	0.220283i	$-0.929302\pi$
$163$	−24.9071	−1.95087	−0.975436	+	0.220283i	$0.929302\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	−18.7181	−1.42724
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	−15.2166	−1.14699
$177$	0	0
$178$	0	0
$179$	7.87561	0.588651	0.294325	−	0.955705i	$-0.404905\pi$
$179$	7.87561	0.588651	0.294325	+	0.955705i	$0.404905\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	−3.70304	−0.272992
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−27.5743	−1.99521	−0.997604	−	0.0691764i	$-0.977963\pi$
$191$	−27.5743	−1.99521	−0.997604	+	0.0691764i	$0.977963\pi$
$192$	0	0
$193$	−27.7711	−1.99901	−0.999503	−	0.0315342i	$-0.989961\pi$
$193$	−27.7711	−1.99901	−0.999503	+	0.0315342i	$0.989961\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.93661	0.565460	0.282730	−	0.959200i	$-0.408760\pi$
$197$	7.93661	0.565460	0.282730	+	0.959200i	$0.408760\pi$
$198$	−26.2389	−1.86472
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	27.3749	1.90269
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−28.1672	−1.93911	−0.969555	−	0.244874i	$-0.921253\pi$
$211$	−28.1672	−1.93911	−0.969555	+	0.244874i	$0.921253\pi$
$212$	−9.71081	−0.666941
$213$	0	0
$214$	17.0731	1.16709
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−42.6364	−2.88770
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	−41.3682	−2.75177
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	3.86379	0.253670
$233$	29.0049	1.90017	0.950087	−	0.311985i	$-0.100994\pi$
$233$	29.0049	1.90017	0.950087	+	0.311985i	$0.100994\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.7095	−0.692739	−0.346369	−	0.938098i	$-0.612586\pi$
$239$	−10.7095	−0.692739	−0.346369	+	0.938098i	$0.612586\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	14.7977	0.951235
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−38.9523	−2.44891
$254$	39.0049	2.44739
$255$	0	0
$256$	12.3773	0.773584
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−28.5633	−1.76803
$262$	0	0
$263$	1.96184	0.120972	0.0604860	−	0.998169i	$-0.480735\pi$
$263$	1.96184	0.120972	0.0604860	+	0.998169i	$0.480735\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	21.7987	1.33157
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	−21.1312	−1.27658
$275$	0	0
$276$	0	0
$277$	28.7278	1.72609	0.863043	−	0.505131i	$-0.168556\pi$
$277$	28.7278	1.72609	0.863043	+	0.505131i	$0.168556\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.4209	1.57614	0.788069	−	0.615587i	$-0.211081\pi$
$281$	26.4209	1.57614	0.788069	+	0.615587i	$0.211081\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−36.7327	−2.17968
$285$	0	0
$286$	0	0
$287$	0	0
$288$	24.3459	1.43459
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	−4.72886	−0.274859
$297$	0	0
$298$	11.1097	0.643565
$299$	0	0
$300$	0	0
$301$	0	0
$302$	21.5123	1.23789
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	−0.703512	−0.0395756
$317$	−17.8834	−1.00443	−0.502215	−	0.864743i	$-0.667482\pi$
$317$	−17.8834	−1.00443	−0.502215	+	0.864743i	$0.667482\pi$
$318$	0	0
$319$	40.6432	2.27558
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	19.7826	1.09903
$325$	0	0
$326$	−51.0325	−2.82643
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	26.5827	1.46112	0.730559	−	0.682850i	$-0.239260\pi$
$331$	26.5827	1.46112	0.730559	+	0.682850i	$0.239260\pi$
$332$	0	0
$333$	34.9584	1.91571
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−13.9597	−0.760434	−0.380217	−	0.924897i	$-0.624151\pi$
$337$	−13.9597	−0.760434	−0.380217	+	0.924897i	$0.624151\pi$
$338$	−26.6359	−1.44880
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−3.45580	−0.186324
$345$	0	0
$346$	0	0
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	−34.6422	−1.84643
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	16.1365	0.852839
$359$	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
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	32.5273	1.69560
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11.1056	0.575026	0.287513	−	0.957777i	$-0.407171\pi$
$373$	11.1056	0.575026	0.287513	+	0.957777i	$0.407171\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	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
$382$	−56.4975	−2.89067
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	−56.9006	−2.89617
$387$	25.5472	1.29864
$388$	0	0
$389$	29.6450	1.50306	0.751531	−	0.659698i	$-0.229316\pi$
$389$	29.6450	1.50306	0.751531	+	0.659698i	$0.229316\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	16.2615	0.819240
$395$	0	0
$396$	−28.1489	−1.41454
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.4494	1.07113	0.535565	−	0.844494i	$-0.320099\pi$
$401$	21.4494	1.07113	0.535565	+	0.844494i	$0.320099\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−49.7429	−2.46566
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	56.0890	2.75662
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	37.2006	1.81304	0.906522	−	0.422158i	$-0.138727\pi$
$421$	37.2006	1.81304	0.906522	+	0.422158i	$0.138727\pi$
$422$	−57.7123	−2.80939
$423$	0	0
$424$	−1.79284	−0.0870680
$425$	0	0
$426$	0	0
$427$	0	0
$428$	18.3159	0.885331
$429$	0	0
$430$	0	0
$431$	−17.3515	−0.835793	−0.417897	−	0.908495i	$-0.637233\pi$
$431$	−17.3515	−0.835793	−0.417897	+	0.908495i	$0.637233\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−45.7400	−2.19055
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−20.1691	−0.951839	−0.475920	−	0.879489i	$-0.657885\pi$
$449$	−20.1691	−0.951839	−0.475920	+	0.879489i	$0.657885\pi$
$450$	0	0
$451$	0	0
$452$	−44.3795	−2.08744
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−29.3556	−1.37320	−0.686598	−	0.727037i	$-0.740897\pi$
$457$	−29.3556	−1.37320	−0.686598	+	0.727037i	$0.740897\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	6.57566	0.305597	0.152798	−	0.988257i	$-0.451171\pi$
$463$	6.57566	0.305597	0.152798	+	0.988257i	$0.451171\pi$
$464$	−33.9394	−1.57560
$465$	0	0
$466$	59.4286	2.75298
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−36.3515	−1.67144
$474$	0	0
$475$	0	0
$476$	0	0
$477$	13.2537	0.606845
$478$	−21.9428	−1.00364
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	15.8749	0.721586
$485$	0	0
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−41.9385	−1.89266	−0.946330	−	0.323203i	$-0.895240\pi$
$491$	−41.9385	−1.89266	−0.946330	+	0.323203i	$0.895240\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−33.8049	−1.51332	−0.756658	−	0.653811i	$-0.773169\pi$
$499$	−33.8049	−1.51332	−0.756658	+	0.653811i	$0.773169\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	−79.8100	−3.54799
$507$	0	0
$508$	41.8442	1.85653
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	31.8213	1.40632
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	−58.5239	−2.56152
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	4.01964	0.175265
$527$	0	0
$528$	0	0
$529$	60.2653	2.62023
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	4.02454	0.173834
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−46.0210	−1.97860	−0.989299	−	0.145900i	$-0.953392\pi$
$541$	−46.0210	−1.97860	−0.989299	+	0.145900i	$0.953392\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	25.2674	1.08036	0.540178	−	0.841550i	$-0.318357\pi$
$547$	25.2674	1.08036	0.540178	+	0.841550i	$0.318357\pi$
$548$	−22.6694	−0.968389
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	58.8609	2.50076
$555$	0	0
$556$	0	0
$557$	46.0364	1.95062	0.975312	−	0.220833i	$-0.0708776\pi$
$557$	46.0364	1.95062	0.975312	+	0.220833i	$0.0708776\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	54.1342	2.28351
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−6.78171	−0.284554
$569$	46.8133	1.96251	0.981257	−	0.192701i	$-0.0617249\pi$
$569$	46.8133	1.96251	0.981257	+	0.192701i	$0.0617249\pi$
$570$	0	0
$571$	−24.2669	−1.01554	−0.507770	−	0.861493i	$-0.669530\pi$
$571$	−24.2669	−1.01554	−0.507770	+	0.861493i	$0.669530\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	28.4948	1.18728
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	−34.8316	−1.44880
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−18.8589	−0.781055
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	41.5381	1.70721
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	11.9183	0.488195
$597$	0	0
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	−29.7517	−1.21158
$604$	23.0782	0.939038
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	22.4972	0.908654	0.454327	−	0.890835i	$-0.349880\pi$
$613$	22.4972	0.908654	0.454327	+	0.890835i	$0.349880\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.05802	−0.203628	−0.101814	−	0.994803i	$-0.532465\pi$
$617$	−5.05802	−0.203628	−0.101814	+	0.994803i	$0.532465\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	47.2094	1.87938	0.939688	−	0.342032i	$-0.111115\pi$
$631$	47.2094	1.87938	0.939688	+	0.342032i	$0.111115\pi$
$632$	−0.129885	−0.00516653
$633$	0	0
$634$	−36.6416	−1.45522
$635$	0	0
$636$	0	0
$637$	0	0
$638$	83.2746	3.29687
$639$	50.1342	1.98328
$640$	0	0
$641$	45.2288	1.78643	0.893215	−	0.449630i	$-0.148444\pi$
$641$	45.2288	1.78643	0.893215	+	0.449630i	$0.148444\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	3.65232	0.143477
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−54.7473	−2.14407
$653$	0.808379	0.0316343	0.0158172	−	0.999875i	$-0.494965\pi$
$653$	0.808379	0.0316343	0.0158172	+	0.999875i	$0.494965\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.74823	0.262874	0.131437	−	0.991325i	$-0.458041\pi$
$659$	6.74823	0.262874	0.131437	+	0.991325i	$0.458041\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	54.4657	2.11687
$663$	0	0
$664$	0	0
$665$	0	0
$666$	71.6268	2.77548
$667$	−86.8799	−3.36400
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	45.3962	1.74990	0.874948	−	0.484216i	$-0.160895\pi$
$673$	45.3962	1.74990	0.874948	+	0.484216i	$0.160895\pi$
$674$	−28.6023	−1.10172
$675$	0	0
$676$	−28.5748	−1.09903
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−16.7299	−0.640153	−0.320076	−	0.947392i	$-0.603709\pi$
$683$	−16.7299	−0.640153	−0.320076	+	0.947392i	$0.603709\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	30.3556	1.15730
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	−8.19567	−0.311103
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−52.0334	−1.96527	−0.982637	−	0.185540i	$-0.940597\pi$
$701$	−52.0334	−1.96527	−0.982637	+	0.185540i	$0.940597\pi$
$702$	0	0
$703$	0	0
$704$	−40.5457	−1.52813
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	28.3648	1.06526	0.532631	−	0.846348i	$-0.321203\pi$
$709$	28.3648	1.06526	0.532631	+	0.846348i	$0.321203\pi$
$710$	0	0
$711$	0.960180	0.0360096
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	17.3111	0.646945
$717$	0	0
$718$	16.3913	0.611719
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	−38.9294	−1.44880
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	74.0519	2.72959
$737$	42.3342	1.55940
$738$	0	0
$739$	−44.8327	−1.64920	−0.824598	−	0.565719i	$-0.808599\pi$
$739$	−44.8327	−1.64920	−0.824598	+	0.565719i	$0.808599\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	0	0
$745$	0	0
$746$	22.7545	0.833100
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−15.1132	−0.551487	−0.275744	−	0.961231i	$-0.588924\pi$
$751$	−15.1132	−0.551487	−0.275744	+	0.961231i	$0.588924\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.3943	0.922972	0.461486	−	0.887147i	$-0.347316\pi$
$757$	25.3943	0.922972	0.461486	+	0.887147i	$0.347316\pi$
$758$	−24.5870	−0.893040
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	−60.6101	−2.19280
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	−61.0425	−2.19697
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	52.3441	1.88147
$775$	0	0
$776$	0	0
$777$	0	0
$778$	60.7402	2.17764
$779$	0	0
$780$	0	0
$781$	−71.3368	−2.55263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	17.4452	0.621458
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−5.19695	−0.184665
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	43.9480	1.55186
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	49.7265	1.74829	0.874145	−	0.485666i	$-0.161423\pi$
$809$	49.7265	1.74829	0.874145	+	0.485666i	$0.161423\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	−101.919	−3.57226
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.13765	0.109505	0.0547524	−	0.998500i	$-0.482563\pi$
$821$	3.13765	0.109505	0.0547524	+	0.998500i	$0.482563\pi$
$822$	0	0
$823$	49.4941	1.72526	0.862628	−	0.505840i	$-0.168817\pi$
$823$	49.4941	1.72526	0.862628	+	0.505840i	$0.168817\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−44.0000	−1.53003	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000	−1.53003	−0.765015	+	0.644013i	$0.777268\pi$
$828$	60.1718	2.09112
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	61.6515	2.12591
$842$	76.2209	2.62675
$843$	0	0
$844$	−61.9132	−2.13114
$845$	0	0
$846$	0	0
$847$	0	0
$848$	15.7482	0.540797
$849$	0	0
$850$	0	0
$851$	106.332	3.64500
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	3.38153	0.115578
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	−35.5518	−1.21090
$863$	32.4626	1.10504	0.552520	−	0.833500i	$-0.313666\pi$
$863$	32.4626	1.10504	0.552520	+	0.833500i	$0.313666\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.36626	−0.0463471
$870$	0	0
$871$	0	0
$872$	−8.44466	−0.285972
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6.35211	−0.214496	−0.107248	−	0.994232i	$-0.534204\pi$
$877$	−6.35211	−0.214496	−0.107248	+	0.994232i	$0.534204\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−59.4174	−1.99956	−0.999778	−	0.0210840i	$-0.993288\pi$
$883$	−59.4174	−1.99956	−0.999778	+	0.0210840i	$0.993288\pi$
$884$	0	0
$885$	0	0
$886$	40.9783	1.37669
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	38.4187	1.28708
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−41.3248	−1.37903
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−8.19349	−0.272511
$905$	0	0
$906$	0	0
$907$	8.19242	0.272025	0.136012	−	0.990707i	$-0.456571\pi$
$907$	8.19242	0.272025	0.136012	+	0.990707i	$0.456571\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	53.1868	1.76216	0.881079	−	0.472969i	$-0.156818\pi$
$911$	53.1868	1.76216	0.881079	+	0.472969i	$0.156818\pi$
$912$	0	0
$913$	0	0
$914$	−60.1471	−1.98949
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	13.4730	0.442750
$927$	0	0
$928$	−77.2666	−2.53640
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	63.7546	2.08835
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	−74.4813	−2.42159
$947$	7.23549	0.235122	0.117561	−	0.993066i	$-0.462492\pi$
$947$	7.23549	0.235122	0.117561	+	0.993066i	$0.462492\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−7.72912	−0.250371	−0.125185	−	0.992133i	$-0.539953\pi$
$953$	−7.72912	−0.250371	−0.125185	+	0.992133i	$0.539953\pi$
$954$	27.1557	0.879198
$955$	0	0
$956$	−23.5401	−0.761341
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−24.9982	−0.805556
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−12.2940	−0.395348	−0.197674	−	0.980268i	$-0.563339\pi$
$967$	−12.2940	−0.395348	−0.197674	+	0.980268i	$0.563339\pi$
$968$	2.93087	0.0942019
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	49.1740	1.57564
$975$	0	0
$976$	0	0
$977$	−31.0347	−0.992888	−0.496444	−	0.868069i	$-0.665361\pi$
$977$	−31.0347	−0.992888	−0.496444	+	0.868069i	$0.665361\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	62.4277	1.99316
$982$	−85.9286	−2.74209
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	77.7059	2.47090
$990$	0	0
$991$	30.5439	0.970260	0.485130	−	0.874442i	$-0.338772\pi$
$991$	30.5439	0.970260	0.485130	+	0.874442i	$0.338772\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	−69.2635	−2.19250
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8575.2.a.a.1.3		3
5.4	even	2		343.2.a.b.1.1	✓	3
7.6	odd	2	CM	8575.2.a.a.1.3		3
15.14	odd	2		3087.2.a.a.1.3		3
20.19	odd	2		5488.2.a.c.1.2		3
35.4	even	6		343.2.c.a.324.3		6
35.9	even	6		343.2.c.a.18.3		6
35.19	odd	6		343.2.c.a.18.3		6
35.24	odd	6		343.2.c.a.324.3		6
35.34	odd	2		343.2.a.b.1.1	✓	3
105.104	even	2		3087.2.a.a.1.3		3
140.139	even	2		5488.2.a.c.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
343.2.a.b.1.1	✓	3	5.4	even	2
343.2.a.b.1.1	✓	3	35.34	odd	2
343.2.c.a.18.3		6	35.9	even	6
343.2.c.a.18.3		6	35.19	odd	6
343.2.c.a.324.3		6	35.4	even	6
343.2.c.a.324.3		6	35.24	odd	6
3087.2.a.a.1.3		3	15.14	odd	2
3087.2.a.a.1.3		3	105.104	even	2
5488.2.a.c.1.2		3	20.19	odd	2
5488.2.a.c.1.2		3	140.139	even	2
8575.2.a.a.1.3		3	1.1	even	1	trivial
8575.2.a.a.1.3		3	7.6	odd	2	CM

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:	\( N \)	\(=\)	\( 8575 = 5^{2} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8575.a (trivial)

\(f(q)\)	\(=\)	\(q+2.04892 q^{2} +2.19806 q^{4} +0.405813 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+2.04892 q^{2} +2.19806 q^{4} +0.405813 q^{8} -3.00000 q^{9} +4.26875 q^{11} -3.56465 q^{16} -6.14675 q^{18} +8.74632 q^{22} -9.12498 q^{23} +9.52111 q^{29} -8.11529 q^{32} -6.59419 q^{36} -11.6528 q^{37} -8.51573 q^{43} +9.38298 q^{44} -18.6963 q^{46} -4.41789 q^{53} +19.5080 q^{58} -9.49827 q^{64} +9.91723 q^{67} -16.7114 q^{71} -1.21744 q^{72} -23.8756 q^{74} -0.320060 q^{79} +9.00000 q^{81} -17.4480 q^{86} +1.73232 q^{88} -20.0573 q^{92} -12.8062 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 11 q^{4} - 12 q^{8} - 9 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 11 * q^4 - 12 * q^8 - 9 * q^9	\( 3 q - 3 q^{2} + 11 q^{4} - 12 q^{8} - 9 q^{9} + 5 q^{11} + 11 q^{16} + 9 q^{18} + 9 q^{22} - 3 q^{23} + 13 q^{29} - 22 q^{32} - 33 q^{36} - 17 q^{37} - 13 q^{43} + 2 q^{44} - 32 q^{46} - 19 q^{53} + 8 q^{58} + 23 q^{67} - q^{71} + 36 q^{72} - 11 q^{74} - 25 q^{79} + 27 q^{81} - 8 q^{86} + 29 q^{88} - 4 q^{92} - 15 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 11 * q^4 - 12 * q^8 - 9 * q^9 + 5 * q^11 + 11 * q^16 + 9 * q^18 + 9 * q^22 - 3 * q^23 + 13 * q^29 - 22 * q^32 - 33 * q^36 - 17 * q^37 - 13 * q^43 + 2 * q^44 - 32 * q^46 - 19 * q^53 + 8 * q^58 + 23 * q^67 - q^71 + 36 * q^72 - 11 * q^74 - 25 * q^79 + 27 * q^81 - 8 * q^86 + 29 * q^88 - 4 * q^92 - 15 * q^99

Introduction

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