LMFDB - Embedded newform 1225.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, 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("1225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.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:	$9.78167424761$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Root			$-1.79129$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.79129 q^{2} +1.20871 q^{4} +1.41742 q^{8} -3.00000 q^{9} +O(q^{10})$	$q-1.79129 q^{2} +1.20871 q^{4} +1.41742 q^{8} -3.00000 q^{9} -6.58258 q^{11} -4.95644 q^{16} +5.37386 q^{18} +11.7913 q^{22} +8.58258 q^{23} +8.16515 q^{29} +6.04356 q^{32} -3.62614 q^{36} -12.1652 q^{37} -1.41742 q^{43} -7.95644 q^{44} -15.3739 q^{46} +10.0000 q^{53} -14.6261 q^{58} -0.912878 q^{64} +15.7477 q^{67} -3.41742 q^{71} -4.25227 q^{72} +21.7913 q^{74} +9.74773 q^{79} +9.00000 q^{81} +2.53901 q^{86} -9.33030 q^{88} +10.3739 q^{92} +19.7477 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 7 q^{4} + 12 q^{8} - 6 q^{9}+O(q^{10}) $ 2 * q + q^2 + 7 * q^4 + 12 * q^8 - 6 * q^9	$ 2 q + q^{2} + 7 q^{4} + 12 q^{8} - 6 q^{9} - 4 q^{11} + 13 q^{16} - 3 q^{18} + 19 q^{22} + 8 q^{23} - 2 q^{29} + 35 q^{32} - 21 q^{36} - 6 q^{37} - 12 q^{43} + 7 q^{44} - 17 q^{46} + 20 q^{53} - 43 q^{58} + 44 q^{64} + 4 q^{67} - 16 q^{71} - 36 q^{72} + 39 q^{74} - 8 q^{79} + 18 q^{81} - 27 q^{86} + 18 q^{88} + 7 q^{92} + 12 q^{99}+O(q^{100}) $ 2 * q + q^2 + 7 * q^4 + 12 * q^8 - 6 * q^9 - 4 * q^11 + 13 * q^16 - 3 * q^18 + 19 * q^22 + 8 * q^23 - 2 * q^29 + 35 * q^32 - 21 * q^36 - 6 * q^37 - 12 * q^43 + 7 * q^44 - 17 * q^46 + 20 * q^53 - 43 * q^58 + 44 * q^64 + 4 * q^67 - 16 * q^71 - 36 * q^72 + 39 * q^74 - 8 * q^79 + 18 * q^81 - 27 * q^86 + 18 * q^88 + 7 * q^92 + 12 * 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 $

$2$	−1.79129	−1.26663	−0.633316	−	0.773893i	$-0.718307\pi$
$2$	−1.79129	−1.26663	−0.633316	+	0.773893i	$0.718307\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	1.20871	0.604356
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.41742	0.501135
$9$	−3.00000	−1.00000
$10$	0	0
$11$	−6.58258	−1.98472	−0.992361	−	0.123371i	$-0.960630\pi$
$11$	−6.58258	−1.98472	−0.992361	+	0.123371i	$0.960630\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	−4.95644	−1.23911
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	5.37386	1.26663
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	11.7913	2.51391
$23$	8.58258	1.78959	0.894795	−	0.446476i	$-0.147321\pi$
$23$	8.58258	1.78959	0.894795	+	0.446476i	$0.147321\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.16515	1.51623	0.758115	−	0.652121i	$-0.226120\pi$
$29$	8.16515	1.51623	0.758115	+	0.652121i	$0.226120\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	6.04356	1.06836
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−3.62614	−0.604356
$37$	−12.1652	−1.99994	−0.999969	−	0.00783774i	$-0.997505\pi$
$37$	−12.1652	−1.99994	−0.999969	+	0.00783774i	$0.997505\pi$
$38$	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
$43$	−1.41742	−0.216155	−0.108078	−	0.994142i	$-0.534469\pi$
$43$	−1.41742	−0.216155	−0.108078	+	0.994142i	$0.534469\pi$
$44$	−7.95644	−1.19948
$45$	0	0
$46$	−15.3739	−2.26675
$47$	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
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−14.6261	−1.92051
$59$	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
$64$	−0.912878	−0.114110
$65$	0	0
$66$	0	0
$67$	15.7477	1.92389	0.961946	−	0.273241i	$-0.0880957\pi$
$67$	15.7477	1.92389	0.961946	+	0.273241i	$0.0880957\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.41742	−0.405574	−0.202787	−	0.979223i	$-0.565000\pi$
$71$	−3.41742	−0.405574	−0.202787	+	0.979223i	$0.565000\pi$
$72$	−4.25227	−0.501135
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	21.7913	2.53319
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	9.74773	1.09670	0.548352	−	0.836247i	$-0.315255\pi$
$79$	9.74773	1.09670	0.548352	+	0.836247i	$0.315255\pi$
$80$	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
$86$	2.53901	0.273789
$87$	0	0
$88$	−9.33030	−0.994614
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	10.3739	1.08155
$93$	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
$99$	19.7477	1.98472
$100$	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
$106$	−17.9129	−1.73985
$107$	20.0000	1.93347	0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000	1.93347	0.966736	+	0.255774i	$0.0823304\pi$
$108$	0	0
$109$	−18.1652	−1.73991	−0.869953	−	0.493135i	$-0.835851\pi$
$109$	−18.1652	−1.73991	−0.869953	+	0.493135i	$0.835851\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	19.3303	1.81844	0.909221	−	0.416314i	$-0.136678\pi$
$113$	19.3303	1.81844	0.909221	+	0.416314i	$0.136678\pi$
$114$	0	0
$115$	0	0
$116$	9.86932	0.916343
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	32.3303	2.93912
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−5.74773	−0.510028	−0.255014	−	0.966937i	$-0.582080\pi$
$127$	−5.74773	−0.510028	−0.255014	+	0.966937i	$0.582080\pi$
$128$	−10.4519	−0.923826
$129$	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
$134$	−28.2087	−2.43686
$135$	0	0
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	6.12159	0.513712
$143$	0	0
$144$	14.8693	1.23911
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−14.7042	−1.20868
$149$	−1.83485	−0.150317	−0.0751583	−	0.997172i	$-0.523946\pi$
$149$	−1.83485	−0.150317	−0.0751583	+	0.997172i	$0.523946\pi$
$150$	0	0
$151$	16.5826	1.34947	0.674735	−	0.738060i	$-0.264258\pi$
$151$	16.5826	1.34947	0.674735	+	0.738060i	$0.264258\pi$
$152$	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	$0.5\pi$
$158$	−17.4610	−1.38912
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−16.1216	−1.26663
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	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
$172$	−1.71326	−0.130635
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	32.6261	2.45929
$177$	0	0
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	12.1652	0.896827
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−9.33030	−0.671610	−0.335805	−	0.941932i	$-0.609008\pi$
$193$	−9.33030	−0.671610	−0.335805	+	0.941932i	$0.609008\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−22.1652	−1.57920	−0.789601	−	0.613621i	$-0.789712\pi$
$197$	−22.1652	−1.57920	−0.789601	+	0.613621i	$0.789712\pi$
$198$	−35.3739	−2.51391
$199$	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
$207$	−25.7477	−1.78959
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	12.0871	0.830147
$213$	0	0
$214$	−35.8258	−2.44900
$215$	0	0
$216$	0	0
$217$	0	0
$218$	32.5390	2.20382
$219$	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
$226$	−34.6261	−2.30330
$227$	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
$232$	11.5735	0.759836
$233$	29.3303	1.92149	0.960746	−	0.277429i	$-0.0894825\pi$
$233$	29.3303	1.92149	0.960746	+	0.277429i	$0.0894825\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	−57.9129	−3.72278
$243$	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
$253$	−56.4955	−3.55184
$254$	10.2958	0.646018
$255$	0	0
$256$	20.5481	1.28426
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−24.4955	−1.51623
$262$	0	0
$263$	11.4174	0.704029	0.352014	−	0.935995i	$-0.385497\pi$
$263$	11.4174	0.704029	0.352014	+	0.935995i	$0.385497\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	19.0345	1.16272
$269$	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
$274$	−17.9129	−1.08216
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	31.3303	1.86901	0.934505	−	0.355951i	$-0.115843\pi$
$281$	31.3303	1.86901	0.934505	+	0.355951i	$0.115843\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−4.13068	−0.245111
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−18.1307	−1.06836
$289$	−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
$296$	−17.2432	−1.00224
$297$	0	0
$298$	3.28674	0.190396
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−29.7042	−1.70928
$303$	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
$316$	11.7822	0.662800
$317$	−7.83485	−0.440049	−0.220024	−	0.975494i	$-0.570614\pi$
$317$	−7.83485	−0.440049	−0.220024	+	0.975494i	$0.570614\pi$
$318$	0	0
$319$	−53.7477	−3.00929
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	10.8784	0.604356
$325$	0	0
$326$	−35.8258	−1.98421
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−13.4174	−0.737488	−0.368744	−	0.929531i	$-0.620212\pi$
$331$	−13.4174	−0.737488	−0.368744	+	0.929531i	$0.620212\pi$
$332$	0	0
$333$	36.4955	1.99994
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	23.2867	1.26663
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−2.00909	−0.108323
$345$	0	0
$346$	0	0
$347$	−30.0780	−1.61467	−0.807337	−	0.590091i	$-0.799092\pi$
$347$	−30.0780	−1.61467	−0.807337	+	0.590091i	$0.799092\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	−39.7822	−2.12040
$353$	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
$358$	−7.16515	−0.378690
$359$	−36.0780	−1.90413	−0.952063	−	0.305903i	$-0.901042\pi$
$359$	−36.0780	−1.90413	−0.952063	+	0.305903i	$0.901042\pi$
$360$	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	$0.5\pi$
$368$	−42.5390	−2.21750
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−16.4955	−0.854102	−0.427051	−	0.904227i	$-0.640448\pi$
$373$	−16.4955	−0.854102	−0.427051	+	0.904227i	$0.640448\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−26.0780	−1.33954	−0.669769	−	0.742569i	$-0.733607\pi$
$379$	−26.0780	−1.33954	−0.669769	+	0.742569i	$0.733607\pi$
$380$	0	0
$381$	0	0
$382$	−14.3303	−0.733202
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	16.7133	0.850682
$387$	4.25227	0.216155
$388$	0	0
$389$	28.1652	1.42803	0.714015	−	0.700130i	$-0.246875\pi$
$389$	28.1652	1.42803	0.714015	+	0.700130i	$0.246875\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	39.7042	2.00027
$395$	0	0
$396$	23.8693	1.19948
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.33030	−0.0664322	−0.0332161	−	0.999448i	$-0.510575\pi$
$401$	−1.33030	−0.0664322	−0.0332161	+	0.999448i	$0.510575\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	80.0780	3.96932
$408$	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
$414$	46.1216	2.26675
$415$	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
$421$	−14.4955	−0.706465	−0.353233	−	0.935536i	$-0.614918\pi$
$421$	−14.4955	−0.706465	−0.353233	+	0.935536i	$0.614918\pi$
$422$	21.4955	1.04638
$423$	0	0
$424$	14.1742	0.688362
$425$	0	0
$426$	0	0
$427$	0	0
$428$	24.1742	1.16851
$429$	0	0
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−21.9564	−1.05152
$437$	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
$449$	−37.6606	−1.77731	−0.888657	−	0.458573i	$-0.848361\pi$
$449$	−37.6606	−1.77731	−0.888657	+	0.458573i	$0.848361\pi$
$450$	0	0
$451$	0	0
$452$	23.3648	1.09899
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.6606	1.57458	0.787288	−	0.616585i	$-0.211484\pi$
$457$	33.6606	1.57458	0.787288	+	0.616585i	$0.211484\pi$
$458$	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
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	−40.4701	−1.87878
$465$	0	0
$466$	−52.5390	−2.43382
$467$	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
$473$	9.33030	0.429008
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−30.0000	−1.37361
$478$	−28.6606	−1.31091
$479$	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
$484$	39.0780	1.77627
$485$	0	0
$486$	0	0
$487$	20.0780	0.909822	0.454911	−	0.890537i	$-0.349671\pi$
$487$	20.0780	0.909822	0.454911	+	0.890537i	$0.349671\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−26.5826	−1.19965	−0.599827	−	0.800129i	$-0.704764\pi$
$491$	−26.5826	−1.19965	−0.599827	+	0.800129i	$0.704764\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	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
$506$	101.200	4.49887
$507$	0	0
$508$	−6.94735	−0.308239
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	−15.9038	−0.702855
$513$	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	$0.5\pi$
$522$	43.8784	1.92051
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	−20.4519	−0.891745
$527$	0	0
$528$	0	0
$529$	50.6606	2.20264
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	22.3212	0.964129
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	44.4955	1.91301	0.956504	−	0.291718i	$-0.0942267\pi$
$541$	44.4955	1.91301	0.956504	+	0.291718i	$0.0942267\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	35.7477	1.52846	0.764231	−	0.644942i	$-0.223119\pi$
$547$	35.7477	1.52846	0.764231	+	0.644942i	$0.223119\pi$
$548$	12.0871	0.516336
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−17.9129	−0.761045
$555$	0	0
$556$	0	0
$557$	32.1652	1.36288	0.681441	−	0.731873i	$-0.261354\pi$
$557$	32.1652	1.36288	0.681441	+	0.731873i	$0.261354\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−56.1216	−2.36735
$563$	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
$568$	−4.84394	−0.203247
$569$	−47.6606	−1.99804	−0.999018	−	0.0443003i	$-0.985894\pi$
$569$	−47.6606	−1.99804	−0.999018	+	0.0443003i	$0.985894\pi$
$570$	0	0
$571$	39.2432	1.64228	0.821138	−	0.570730i	$-0.193340\pi$
$571$	39.2432	1.64228	0.821138	+	0.570730i	$0.193340\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	2.73864	0.114110
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	30.4519	1.26663
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−65.8258	−2.72622
$584$	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
$592$	60.2958	2.47814
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	−2.21780	−0.0908448
$597$	0	0
$598$	0	0
$599$	16.0780	0.656930	0.328465	−	0.944516i	$-0.393469\pi$
$599$	16.0780	0.656930	0.328465	+	0.944516i	$0.393469\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	−47.2432	−1.92389
$604$	20.0436	0.815561
$605$	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
$613$	−46.4955	−1.87793	−0.938967	−	0.344008i	$-0.888215\pi$
$613$	−46.4955	−1.87793	−0.938967	+	0.344008i	$0.888215\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23.6606	0.952540	0.476270	−	0.879299i	$-0.341988\pi$
$617$	23.6606	0.952540	0.476270	+	0.879299i	$0.341988\pi$
$618$	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
$631$	−49.2432	−1.96034	−0.980170	−	0.198158i	$-0.936504\pi$
$631$	−49.2432	−1.96034	−0.980170	+	0.198158i	$0.936504\pi$
$632$	13.8167	0.549597
$633$	0	0
$634$	14.0345	0.557380
$635$	0	0
$636$	0	0
$637$	0	0
$638$	96.2777	3.81167
$639$	10.2523	0.405574
$640$	0	0
$641$	−41.3303	−1.63245	−0.816224	−	0.577735i	$-0.803937\pi$
$641$	−41.3303	−1.63245	−0.816224	+	0.577735i	$0.803937\pi$
$642$	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	$0.5\pi$
$648$	12.7568	0.501135
$649$	0	0
$650$	0	0
$651$	0	0
$652$	24.1742	0.946736
$653$	−50.0000	−1.95665	−0.978326	−	0.207072i	$-0.933606\pi$
$653$	−50.0000	−1.95665	−0.978326	+	0.207072i	$0.933606\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	24.0345	0.934126
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−65.3739	−2.53319
$667$	70.0780	2.71343
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	53.7386	2.06993
$675$	0	0
$676$	−15.7133	−0.604356
$677$	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
$683$	−21.4174	−0.819515	−0.409757	−	0.912194i	$-0.634387\pi$
$683$	−21.4174	−0.819515	−0.409757	+	0.912194i	$0.634387\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	7.02538	0.267840
$689$	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
$694$	53.8784	2.04520
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	0	0
$704$	6.00909	0.226476
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	−29.2432	−1.09670
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	4.83485	0.180687
$717$	0	0
$718$	64.6261	2.41183
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	34.0345	1.26663
$723$	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
$736$	51.8693	1.91193
$737$	−103.661	−3.81839
$738$	0	0
$739$	39.7477	1.46214	0.731072	−	0.682300i	$-0.239020\pi$
$739$	39.7477	1.46214	0.731072	+	0.682300i	$0.239020\pi$
$740$	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
$746$	29.5481	1.08183
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−17.8348	−0.648219	−0.324109	−	0.946020i	$-0.605065\pi$
$757$	−17.8348	−0.648219	−0.324109	+	0.946020i	$0.605065\pi$
$758$	46.7133	1.69670
$759$	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
$764$	9.66970	0.349837
$765$	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
$772$	−11.2777	−0.405892
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	−7.61704	−0.273789
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−50.4519	−1.80879
$779$	0	0
$780$	0	0
$781$	22.4955	0.804951
$782$	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	$0.5\pi$
$788$	−26.7913	−0.954400
$789$	0	0
$790$	0	0
$791$	0	0
$792$	27.9909	0.994614
$793$	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
$802$	2.38296	0.0841451
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−17.6606	−0.620914	−0.310457	−	0.950587i	$-0.600482\pi$
$809$	−17.6606	−0.620914	−0.310457	+	0.950587i	$0.600482\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	−143.443	−5.02767
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	0	0
$823$	57.2432	1.99537	0.997686	−	0.0679910i	$-0.0216589\pi$
$823$	57.2432	1.99537	0.997686	+	0.0679910i	$0.0216589\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.0780	−0.350447	−0.175224	−	0.984529i	$-0.556065\pi$
$827$	−10.0780	−0.350447	−0.175224	+	0.984529i	$0.556065\pi$
$828$	−31.1216	−1.08155
$829$	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
$841$	37.6697	1.29896
$842$	25.9655	0.894831
$843$	0	0
$844$	−14.5045	−0.499267
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−49.5644	−1.70205
$849$	0	0
$850$	0	0
$851$	−104.408	−3.57907
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	28.3485	0.968931
$857$	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
$862$	−57.3212	−1.95237
$863$	54.4083	1.85208	0.926041	−	0.377424i	$-0.123190\pi$
$863$	54.4083	1.85208	0.926041	+	0.377424i	$0.123190\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−64.1652	−2.17665
$870$	0	0
$871$	0	0
$872$	−25.7477	−0.871928
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	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
$883$	44.4083	1.49446	0.747230	−	0.664566i	$-0.231383\pi$
$883$	44.4083	1.49446	0.747230	+	0.664566i	$0.231383\pi$
$884$	0	0
$885$	0	0
$886$	−35.8258	−1.20359
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−59.2432	−1.98472
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	67.4610	2.25120
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	27.3992	0.911285
$905$	0	0
$906$	0	0
$907$	−60.0000	−1.99227	−0.996134	−	0.0878507i	$-0.972000\pi$
$907$	−60.0000	−1.99227	−0.996134	+	0.0878507i	$0.972000\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	42.4083	1.40505	0.702525	−	0.711659i	$-0.252056\pi$
$911$	42.4083	1.40505	0.702525	+	0.711659i	$0.252056\pi$
$912$	0	0
$913$	0	0
$914$	−60.2958	−1.99441
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	56.0780	1.84984	0.924922	−	0.380158i	$-0.124130\pi$
$919$	56.0780	1.84984	0.924922	+	0.380158i	$0.124130\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−71.6515	−2.35461
$927$	0	0
$928$	49.3466	1.61988
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	35.4519	1.16127
$933$	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
$946$	−16.7133	−0.543395
$947$	20.0000	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$947$	20.0000	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.6697	0.345625	0.172813	−	0.984955i	$-0.444714\pi$
$953$	10.6697	0.345625	0.172813	+	0.984955i	$0.444714\pi$
$954$	53.7386	1.73985
$955$	0	0
$956$	19.3394	0.625481
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−60.0000	−1.93347
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	45.8258	1.47290
$969$	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
$974$	−35.9655	−1.15241
$975$	0	0
$976$	0	0
$977$	−13.6606	−0.437041	−0.218521	−	0.975832i	$-0.570123\pi$
$977$	−13.6606	−0.437041	−0.218521	+	0.975832i	$0.570123\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	54.4955	1.73991
$982$	47.6170	1.51952
$983$	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
$989$	−12.1652	−0.386829
$990$	0	0
$991$	62.4083	1.98247	0.991233	−	0.132125i	$-0.0421802\pi$
$991$	62.4083	1.98247	0.991233	+	0.132125i	$0.0421802\pi$
$992$	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	$0.5\pi$
$998$	−64.4864	−2.04128
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.a.t.1.1	yes	2
5.2	odd	4		1225.2.b.e.99.2		4
5.3	odd	4		1225.2.b.e.99.3		4
5.4	even	2		1225.2.a.o.1.2	✓	2
7.6	odd	2	CM	1225.2.a.t.1.1	yes	2
35.13	even	4		1225.2.b.e.99.3		4
35.27	even	4		1225.2.b.e.99.2		4
35.34	odd	2		1225.2.a.o.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1225.2.a.o.1.2	✓	2	5.4	even	2
1225.2.a.o.1.2	✓	2	35.34	odd	2
1225.2.a.t.1.1	yes	2	1.1	even	1	trivial
1225.2.a.t.1.1	yes	2	7.6	odd	2	CM
1225.2.b.e.99.2		4	5.2	odd	4
1225.2.b.e.99.2		4	35.27	even	4
1225.2.b.e.99.3		4	5.3	odd	4
1225.2.b.e.99.3		4	35.13	even	4

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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79129 q^{2} +1.20871 q^{4} +1.41742 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q-1.79129 q^{2} +1.20871 q^{4} +1.41742 q^{8} -3.00000 q^{9} -6.58258 q^{11} -4.95644 q^{16} +5.37386 q^{18} +11.7913 q^{22} +8.58258 q^{23} +8.16515 q^{29} +6.04356 q^{32} -3.62614 q^{36} -12.1652 q^{37} -1.41742 q^{43} -7.95644 q^{44} -15.3739 q^{46} +10.0000 q^{53} -14.6261 q^{58} -0.912878 q^{64} +15.7477 q^{67} -3.41742 q^{71} -4.25227 q^{72} +21.7913 q^{74} +9.74773 q^{79} +9.00000 q^{81} +2.53901 q^{86} -9.33030 q^{88} +10.3739 q^{92} +19.7477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 7 q^{4} + 12 q^{8} - 6 q^{9}+O(q^{10}) \) 2 * q + q^2 + 7 * q^4 + 12 * q^8 - 6 * q^9	\( 2 q + q^{2} + 7 q^{4} + 12 q^{8} - 6 q^{9} - 4 q^{11} + 13 q^{16} - 3 q^{18} + 19 q^{22} + 8 q^{23} - 2 q^{29} + 35 q^{32} - 21 q^{36} - 6 q^{37} - 12 q^{43} + 7 q^{44} - 17 q^{46} + 20 q^{53} - 43 q^{58} + 44 q^{64} + 4 q^{67} - 16 q^{71} - 36 q^{72} + 39 q^{74} - 8 q^{79} + 18 q^{81} - 27 q^{86} + 18 q^{88} + 7 q^{92} + 12 q^{99}+O(q^{100}) \) 2 * q + q^2 + 7 * q^4 + 12 * q^8 - 6 * q^9 - 4 * q^11 + 13 * q^16 - 3 * q^18 + 19 * q^22 + 8 * q^23 - 2 * q^29 + 35 * q^32 - 21 * q^36 - 6 * q^37 - 12 * q^43 + 7 * q^44 - 17 * q^46 + 20 * q^53 - 43 * q^58 + 44 * q^64 + 4 * q^67 - 16 * q^71 - 36 * q^72 + 39 * q^74 - 8 * q^79 + 18 * q^81 - 27 * q^86 + 18 * q^88 + 7 * q^92 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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