Newform orbit 975.2.cn.a

• Label: 975.2.cn.a
• Level: $975$
• Weight: $2$
• Character orbit: 975.cn
• Analytic conductor: $7.785$
• Analytic rank: $0$
• Dimension: $576$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.cn (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.78541419707\)
Analytic rank:	\(0\)
Dimension:	\(576\)
Relative dimension:	\(72\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 576 q + 76 q^{4} - 72 q^{9}+O(q^{10}) \)

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	−1.85126	−	2.05603i	−0.406737	−	0.913545i	−0.591049	+	5.62345i	1.36565	+	1.77059i	−1.12530	+	2.52747i	2.28746	+	3.96199i	8.17963	−	5.94285i	−0.669131	+	0.743145i	1.11222	−	6.08565i
4.2	−1.81009	−	2.01031i	−0.406737	−	0.913545i	−0.555859	+	5.28865i	−0.0824796	−	2.23455i	−1.10028	+	2.47127i	−0.821248	−	1.42244i	7.26098	−	5.27541i	−0.669131	+	0.743145i	−4.34284	+	4.21054i
4.3	−1.80947	−	2.00962i	0.406737	+	0.913545i	−0.555332	+	5.28363i	1.92879	−	1.13128i	1.09990	−	2.47042i	−0.182109	−	0.315421i	7.24744	−	5.26557i	−0.669131	+	0.743145i	−5.76351	−	1.82912i
4.4	−1.78307	−	1.98030i	0.406737	+	0.913545i	−0.533193	+	5.07299i	0.243025	+	2.22282i	1.08385	−	2.43438i	−1.67027	−	2.89299i	6.68510	−	4.85701i	−0.669131	+	0.743145i	3.96853	−	4.44471i
4.5	−1.76557	−	1.96086i	−0.406737	−	0.913545i	−0.518692	+	4.93502i	−1.78737	+	1.34362i	−1.07321	+	2.41048i	−2.59981	−	4.50300i	6.32333	−	4.59417i	−0.669131	+	0.743145i	5.79037	+	1.13254i
4.6	−1.61279	−	1.79118i	0.406737	+	0.913545i	−0.398193	+	3.78855i	−1.73892	+	1.40576i	0.980346	−	2.20189i	2.03112	+	3.51800i	3.52828	−	2.56344i	−0.669131	+	0.743145i	5.32248	+	0.847535i
4.7	−1.60217	−	1.77939i	0.406737	+	0.913545i	−0.390220	+	3.71270i	2.16830	+	0.546325i	0.973890	−	2.18739i	1.65332	+	2.86363i	3.35729	−	2.43922i	−0.669131	+	0.743145i	−2.50186	−	4.73355i
4.8	−1.59054	−	1.76648i	0.406737	+	0.913545i	−0.381557	+	3.63028i	−1.42171	−	1.72590i	0.966825	−	2.17153i	−1.93096	−	3.34453i	3.17357	−	2.30573i	−0.669131	+	0.743145i	−0.787469	+	5.25654i
4.9	−1.58496	−	1.76028i	−0.406737	−	0.913545i	−0.377421	+	3.59092i	2.13607	+	0.661211i	−0.963434	+	2.16391i	−0.998592	−	1.72961i	3.08661	−	2.24255i	−0.669131	+	0.743145i	−2.22168	−	4.80808i
4.10	−1.52521	−	1.69392i	−0.406737	−	0.913545i	−0.334035	+	3.17813i	−1.57843	−	1.58384i	−0.927113	+	2.08233i	0.843822	+	1.46154i	2.20484	−	1.60191i	−0.669131	+	0.743145i	−0.275447	+	5.08943i
4.11	−1.49815	−	1.66387i	0.406737	+	0.913545i	−0.314936	+	2.99642i	−1.09065	−	1.95205i	0.910665	−	2.04539i	1.80493	+	3.12622i	1.83476	−	1.33303i	−0.669131	+	0.743145i	−1.61398	+	4.73916i
4.12	−1.40380	−	1.55908i	−0.406737	−	0.913545i	−0.251015	+	2.38825i	−2.22879	−	0.180250i	−0.853314	+	1.91658i	0.693105	+	1.20049i	0.681297	−	0.494992i	−0.669131	+	0.743145i	2.84776	+	3.72791i
4.13	−1.38754	−	1.54102i	−0.406737	−	0.913545i	−0.240416	+	2.28740i	−0.181841	+	2.22866i	−0.843427	+	1.89437i	0.986663	+	1.70895i	0.503291	−	0.365663i	−0.669131	+	0.743145i	3.68672	−	2.81214i
4.14	−1.26041	−	1.39983i	0.406737	+	0.913545i	−0.161828	+	1.53969i	2.01245	+	0.974712i	0.766153	−	1.72081i	−0.944975	−	1.63674i	−0.688549	+	0.500260i	−0.669131	+	0.743145i	−1.17208	−	4.04562i
4.15	−1.19725	−	1.32968i	−0.406737	−	0.913545i	−0.125588	+	1.19489i	1.06410	−	1.96665i	−0.727760	+	1.63458i	2.25877	+	3.91231i	−1.15590	+	0.839813i	−0.669131	+	0.743145i	−3.88901	+	0.939662i
4.16	−1.17768	−	1.30795i	0.406737	+	0.913545i	−0.114737	+	1.09165i	−1.95899	+	1.07812i	0.715864	−	1.60786i	−0.348602	−	0.603796i	−1.28482	+	0.933479i	−0.669131	+	0.743145i	3.71720	+	1.29258i
4.17	−1.10181	−	1.22369i	0.406737	+	0.913545i	−0.0743620	+	0.707507i	1.65935	−	1.49885i	0.669746	−	1.50428i	−0.281784	−	0.488065i	−1.71661	+	1.24719i	−0.669131	+	0.743145i	−3.66242	−	0.379077i
4.18	−1.08693	−	1.20716i	−0.406737	−	0.913545i	−0.0667562	+	0.635143i	1.06836	−	1.96433i	−0.660699	+	1.48395i	−0.695415	−	1.20449i	−1.78904	+	1.29981i	−0.669131	+	0.743145i	−3.53249	+	0.845415i
4.19	−1.08614	−	1.20628i	−0.406737	−	0.913545i	−0.0663549	+	0.631325i	2.06546	+	0.856666i	−0.660218	+	1.48287i	0.0352314	+	0.0610226i	−1.79278	+	1.30253i	−0.669131	+	0.743145i	−1.21000	−	3.42198i
4.20	−0.957683	−	1.06361i	−0.406737	−	0.913545i	−0.00506271	+	0.0481685i	0.323081	+	2.21260i	−0.582136	+	1.30750i	−2.48205	−	4.29903i	−2.25971	+	1.64177i	−0.669131	+	0.743145i	2.04395	−	2.46261i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	even	6	1	inner
25.e	even	10	1	inner
325.bh	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	975.2.cn.a	✓	576
13.e	even	6	1	inner	975.2.cn.a	✓	576
25.e	even	10	1	inner	975.2.cn.a	✓	576
325.bh	even	30	1	inner	975.2.cn.a	✓	576

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
975.2.cn.a	✓	576	1.a	even	1	1	trivial
975.2.cn.a	✓	576	13.e	even	6	1	inner
975.2.cn.a	✓	576	25.e	even	10	1	inner
975.2.cn.a	✓	576	325.bh	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(975, [\chi])\).