Newform orbit 375.2.r.a

• Label: 375.2.r.a
• Level: $375$
• Weight: $2$
• Character orbit: 375.r
• Analytic conductor: $2.994$
• Analytic rank: $0$
• Dimension: $1920$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 375 = 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	375.r (of order \(100\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.99439007580\)
Analytic rank:	\(0\)
Dimension:	\(1920\)
Relative dimension:	\(48\) over \(\Q(\zeta_{100})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{100}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1920 q - 40 q^{3} - 80 q^{4} - 40 q^{6} - 80 q^{7} - 40 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−0.0833682	+	2.65282i	−1.49941	+	0.867052i	−5.03445	−	0.316741i	−1.98014	+	1.03876i	−2.17513	−	4.04994i	−2.17312	+	1.10726i	0.760419	−	8.04439i	1.49644	−	2.60013i	−2.59057	−	5.33957i
2.2	−0.0832329	+	2.64851i	−1.35138	−	1.08341i	−5.01164	−	0.315306i	1.85430	−	1.24963i	2.98190	−	3.48897i	0.881691	−	0.449244i	0.753484	−	7.97103i	0.652457	+	2.92819i	3.15533	+	5.01514i
2.3	−0.0813481	+	2.58854i	1.04616	−	1.38042i	−4.69787	−	0.295565i	−1.46650	−	1.68801i	3.48817	+	2.82031i	−2.09137	+	1.06561i	0.659797	−	6.97993i	−0.811110	−	2.88827i	4.48878	−	3.65877i
2.4	−0.0804451	+	2.55980i	1.37365	−	1.05503i	−4.55007	−	0.286266i	1.39972	+	1.74378i	2.59016	+	3.60115i	3.88418	−	1.97909i	0.616781	−	6.52486i	0.773835	−	2.89848i	−4.57634	+	3.44274i
2.5	−0.0793452	+	2.52481i	1.10996	+	1.32966i	−4.37229	−	0.275081i	2.17029	−	0.538365i	−3.44521	+	2.69692i	−3.89320	+	1.98369i	0.566002	−	5.98768i	−0.535998	+	2.95173i	1.18706	+	5.52228i
2.6	−0.0728134	+	2.31696i	0.437284	+	1.67594i	−3.36695	−	0.211831i	−0.141448	+	2.23159i	−3.91493	+	0.891140i	2.19624	−	1.11904i	0.299657	−	3.17004i	−2.61756	+	1.46573i	−5.16021	−	0.490219i
2.7	−0.0691810	+	2.20137i	−0.930805	−	1.46069i	−2.84521	−	0.179005i	−1.69478	+	1.45867i	3.27991	−	1.94800i	0.833266	−	0.424570i	0.176353	−	1.86562i	−1.26720	+	2.71923i	−3.09383	−	3.83176i
2.8	−0.0649612	+	2.06710i	1.66727	+	0.469265i	−2.27262	−	0.142981i	0.227208	−	2.22449i	−1.07832	+	3.41593i	2.51020	−	1.27901i	0.0539337	−	0.570559i	2.55958	+	1.56478i	4.58349	+	0.614168i
2.9	−0.0593083	+	1.88722i	−1.71348	+	0.252959i	−1.56204	−	0.0982750i	−1.36948	−	1.76763i	−0.375767	−	3.24872i	3.97273	−	2.02421i	−0.0772733	+	0.817466i	2.87202	−	0.866882i	3.41714	−	2.47968i
2.10	−0.0579752	+	1.84480i	0.100787	+	1.72912i	−1.40388	−	0.0883245i	−2.05055	−	0.891770i	−3.19572	+	0.0856866i	−1.21790	+	0.620552i	−0.103063	+	1.09029i	−2.97968	+	0.348546i	1.76402	−	3.73115i
2.11	−0.0570608	+	1.81570i	0.474850	−	1.66569i	−1.29747	−	0.0816302i	1.27471	+	1.83715i	2.99730	+	0.957232i	−3.85379	+	1.96360i	−0.119663	+	1.26591i	−2.54904	−	1.58190i	−3.40846	+	2.20966i
2.12	−0.0526908	+	1.67665i	−1.65374	+	0.514936i	−0.812324	−	0.0511071i	0.692419	−	2.12616i	−0.776230	−	2.79987i	−4.27447	+	2.17795i	−0.187238	+	1.98078i	2.46968	−	1.70314i	3.52834	+	1.27297i
2.13	−0.0525597	+	1.67248i	1.72817	−	0.115954i	−0.798364	−	0.0502288i	−1.94046	+	1.11113i	0.103099	+	2.89641i	−1.30754	+	0.666225i	−0.188975	+	1.99915i	2.97311	−	0.400776i	−1.75636	−	3.30377i
2.14	−0.0488416	+	1.55417i	−0.735267	+	1.56824i	−0.416993	−	0.0262350i	2.19836	−	0.408900i	−2.40140	−	1.21932i	1.67655	−	0.854243i	−0.231524	+	2.44927i	−1.91876	−	2.30615i	0.528126	+	3.43659i
2.15	−0.0409171	+	1.30200i	0.164439	−	1.72423i	0.302515	+	0.0190326i	2.12752	−	0.688244i	2.23822	+	0.284650i	2.83629	−	1.44516i	−0.282338	+	2.98683i	−2.94592	−	0.567059i	0.809044	+	2.79819i
2.16	−0.0376488	+	1.19800i	1.65356	+	0.515512i	0.562255	+	0.0353741i	1.75820	+	1.38157i	−0.679841	+	1.96156i	−0.0441764	+	0.0225090i	−0.289142	+	3.05881i	2.46849	+	1.70486i	−1.72132	+	2.05432i
2.17	−0.0296812	+	0.944471i	−1.20446	−	1.24470i	1.10491	+	0.0695149i	−2.06893	−	0.848257i	1.21134	−	1.10063i	−1.87078	+	0.953212i	−0.276303	+	2.92298i	−0.0985716	+	2.99838i	0.862563	−	1.92887i
2.18	−0.0292253	+	0.929963i	−1.71195	+	0.263095i	1.13208	+	0.0712242i	−0.425879	+	2.19514i	−0.194637	−	1.59974i	3.41195	−	1.73848i	−0.274442	+	2.90329i	2.86156	−	0.900813i	−2.02895	−	0.460205i
2.19	−0.0222028	+	0.706505i	−0.924175	+	1.46489i	1.49740	+	0.0942083i	−0.227524	+	2.22446i	−1.01443	−	0.685459i	−2.69759	+	1.37449i	−0.232847	+	2.46326i	−1.29180	−	2.70763i	−1.56654	−	0.210136i
2.20	−0.0216723	+	0.689623i	1.10344	−	1.33508i	1.52094	+	0.0956897i	−1.68390	−	1.47121i	0.896786	+	0.789888i	1.72763	−	0.880274i	−0.228815	+	2.42061i	−0.564862	−	2.94634i	1.05108	−	1.12937i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
125.i	odd	100	1	inner
375.r	even	100	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	375.2.r.a	✓	1920
3.b	odd	2	1	inner	375.2.r.a	✓	1920
125.i	odd	100	1	inner	375.2.r.a	✓	1920
375.r	even	100	1	inner	375.2.r.a	✓	1920

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
375.2.r.a	✓	1920	1.a	even	1	1	trivial
375.2.r.a	✓	1920	3.b	odd	2	1	inner
375.2.r.a	✓	1920	125.i	odd	100	1	inner
375.2.r.a	✓	1920	375.r	even	100	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(375, [\chi])\).