Newform orbit 475.2.bg.a

• Label: 475.2.bg.a
• Level: $475$
• Weight: $2$
• Character orbit: 475.bg
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $1152$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.bg (of order \(90\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(1152\)
Relative dimension:	\(48\) over \(\Q(\zeta_{90})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{90}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1152 q - 30 q^{2} - 30 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 15 q^{8} - 18 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−2.17775	−	1.70145i	2.36405	+	1.15302i	1.36384	+	5.47007i	−2.13897	−	0.651764i	−3.18651	−	6.53331i	−0.343090	−	0.198083i	4.08880	−	9.18359i	2.41228	+	3.08758i	3.54921	+	5.05873i
4.2	−2.14701	−	1.67743i	−1.15430	−	0.562988i	1.31204	+	5.26231i	0.0477449	+	2.23556i	1.53391	+	3.14499i	−0.910389	−	0.525613i	3.79379	−	8.52100i	−0.831538	−	1.06432i	3.64748	−	4.87985i
4.3	−2.08635	−	1.63004i	−2.34849	−	1.14543i	1.21200	+	4.86107i	−1.47610	−	1.67962i	3.03268	+	6.21791i	−2.03462	−	1.17469i	3.24130	−	7.28008i	2.35640	+	3.01605i	0.341822	+	5.91038i
4.4	−2.05032	−	1.60188i	−0.0417711	−	0.0203731i	1.15393	+	4.62815i	2.07433	−	0.834966i	0.0530087	+	0.108684i	3.36351	+	1.94192i	2.93127	−	6.58373i	−1.84565	−	2.36233i	−5.59055	−	1.61088i
4.5	−1.97733	−	1.54486i	1.69347	+	0.825962i	1.03939	+	4.16878i	2.20114	−	0.393670i	−2.07255	−	4.24937i	−4.41205	−	2.54730i	2.34373	−	5.26409i	0.338651	+	0.433454i	−4.96054	−	2.62203i
4.6	−1.75867	−	1.37402i	0.396537	+	0.193404i	0.721134	+	2.89231i	−1.65863	+	1.49964i	−0.431636	−	0.884985i	1.20019	+	0.692929i	0.890369	−	1.99980i	−1.72715	−	2.21065i	4.97753	−	0.358376i
4.7	−1.73192	−	1.35312i	−2.31623	−	1.12970i	0.684759	+	2.74642i	1.98146	−	1.03625i	2.48290	+	5.09071i	0.350552	+	0.202391i	0.742416	−	1.66749i	2.24172	+	2.86927i	−4.83391	−	0.886455i
4.8	−1.59580	−	1.24677i	0.954886	+	0.465729i	0.508283	+	2.03861i	−0.548667	−	2.16771i	−0.943146	−	1.93374i	2.76874	+	1.59853i	0.0832081	−	0.186888i	−1.15208	−	1.47460i	−1.82708	+	4.14329i
4.9	−1.51409	−	1.18294i	−2.07207	−	1.01062i	0.409287	+	1.64156i	−2.22133	−	0.256277i	1.94181	+	3.98130i	1.90310	+	1.09875i	−0.240849	+	0.540957i	1.42516	+	1.82412i	3.06014	+	3.01573i
4.10	−1.47740	−	1.15427i	0.734804	+	0.358388i	0.366519	+	1.47003i	−1.67862	−	1.47724i	−0.671921	−	1.37764i	−2.36945	−	1.36800i	−0.369821	+	0.830632i	−1.43549	−	1.83734i	0.774845	+	4.12004i
4.11	−1.45968	−	1.14043i	−0.925186	−	0.451243i	0.346250	+	1.38873i	1.40580	+	1.73888i	0.835866	+	1.71378i	−1.96031	−	1.13179i	−0.428512	+	0.962453i	−1.19464	−	1.52906i	−0.0689501	−	4.14143i
4.12	−1.35810	−	1.06107i	2.76297	+	1.34759i	0.234739	+	0.941488i	0.856960	−	2.06534i	−2.32252	−	4.76187i	0.750840	+	0.433497i	−0.721806	+	1.62120i	3.97102	+	5.08267i	−3.35530	+	1.89565i
4.13	−1.26816	−	0.990795i	2.84571	+	1.38795i	0.142711	+	0.572383i	−1.35135	+	1.78153i	−2.23365	−	4.57965i	−1.42757	−	0.824207i	−0.923004	+	2.07310i	4.32469	+	5.53535i	3.47886	−	0.920361i
4.14	−1.17042	−	0.914429i	1.49793	+	0.730589i	0.0498483	+	0.199931i	2.21518	+	0.304932i	−1.08513	−	2.22484i	0.124678	+	0.0719829i	−1.08376	+	2.43415i	−0.136954	−	0.175293i	−2.31384	−	2.38252i
4.15	−1.02997	−	0.804701i	−0.940841	−	0.458879i	−0.0705486	−	0.282955i	1.49718	+	1.66085i	0.599778	+	1.22973i	1.88469	+	1.08813i	−1.21828	+	2.73631i	−1.17237	−	1.50057i	−0.205565	−	2.91542i
4.16	−0.989684	−	0.773226i	−1.10200	−	0.537480i	−0.102248	−	0.410094i	0.678052	−	2.13079i	0.675035	+	1.38403i	−3.49201	−	2.01611i	−1.23756	+	2.77962i	−0.921472	−	1.17943i	−2.31863	+	1.58452i
4.17	−0.968759	−	0.756878i	−2.97441	−	1.45071i	−0.118213	−	0.474126i	−0.961506	+	2.01879i	1.78347	+	3.65666i	−3.87700	−	2.23839i	−1.24440	+	2.79496i	4.89554	+	6.26600i	2.45944	−	1.22798i
4.18	−0.702068	−	0.548516i	0.672801	+	0.328147i	−0.291814	−	1.17040i	−1.03473	+	1.98225i	−0.292358	−	0.599423i	3.67058	+	2.11921i	−1.16186	+	2.60959i	−1.50200	−	1.92248i	1.81375	−	0.824110i
4.19	−0.608668	−	0.475544i	1.41905	+	0.692115i	−0.339509	−	1.36169i	−1.25916	+	1.84784i	−0.534598	−	1.09609i	−3.04521	−	1.75815i	−1.06923	+	2.40154i	−0.312314	−	0.399744i	1.64514	−	0.525936i
4.20	−0.561298	−	0.438534i	−1.64099	−	0.800365i	−0.361100	−	1.44829i	−2.14371	−	0.636019i	0.570098	+	1.16887i	0.979278	+	0.565387i	−1.01188	+	2.27271i	0.205286	+	0.262754i	0.924342	+	1.29709i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.e	even	9	1	inner
25.e	even	10	1	inner
475.bg	even	90	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	475.2.bg.a	✓	1152
19.e	even	9	1	inner	475.2.bg.a	✓	1152
25.e	even	10	1	inner	475.2.bg.a	✓	1152
475.bg	even	90	1	inner	475.2.bg.a	✓	1152

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
475.2.bg.a	✓	1152	1.a	even	1	1	trivial
475.2.bg.a	✓	1152	19.e	even	9	1	inner
475.2.bg.a	✓	1152	25.e	even	10	1	inner
475.2.bg.a	✓	1152	475.bg	even	90	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(475, [\chi])\).