Newform orbit 930.2.bf.a

• Label: 930.2.bf.a
• Level: $930$
• Weight: $2$
• Character orbit: 930.bf
• Analytic conductor: $7.426$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.42608738798\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(64\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q + 8 q^{3} - 4 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
377.1	−0.707107	+	0.707107i	−1.72168	−	0.189281i		−	1.00000i	0.213393	+	2.22586i	1.35125	−	1.08357i	3.30534	−	0.885662i	0.707107	+	0.707107i	2.92835	+	0.651761i	−1.72481	−	1.42303i
377.2	−0.707107	+	0.707107i	−1.70669	+	0.295334i		−	1.00000i	2.22324	+	0.239148i	0.997976	−	1.41564i	−3.38718	+	0.907593i	0.707107	+	0.707107i	2.82556	−	1.00809i	−1.74117	+	1.40297i
377.3	−0.707107	+	0.707107i	−1.65429	−	0.513158i		−	1.00000i	1.83992	−	1.27071i	1.53262	−	0.806900i	3.00096	−	0.804106i	0.707107	+	0.707107i	2.47334	+	1.69782i	−0.402493	+	2.19955i
377.4	−0.707107	+	0.707107i	−1.64704	−	0.535974i		−	1.00000i	−0.601583	+	2.15362i	1.54362	−	0.785641i	−4.77476	+	1.27939i	0.707107	+	0.707107i	2.42546	+	1.76554i	−1.09746	−	1.94823i
377.5	−0.707107	+	0.707107i	−1.64517	+	0.541693i		−	1.00000i	−2.17658	−	0.512363i	0.780273	−	1.54634i	0.122813	−	0.0329078i	0.707107	+	0.707107i	2.41314	−	1.78235i	1.90137	−	1.17678i
377.6	−0.707107	+	0.707107i	−1.45410	−	0.941054i		−	1.00000i	−1.41418	−	1.73208i	1.69363	−	0.362781i	−1.47385	+	0.394917i	0.707107	+	0.707107i	1.22883	+	2.73678i	2.22474	+	0.224784i
377.7	−0.707107	+	0.707107i	−1.38753	+	1.03670i		−	1.00000i	−0.924875	+	2.03583i	0.248075	−	1.71419i	−0.101233	+	0.0271252i	0.707107	+	0.707107i	0.850495	−	2.87692i	−0.785564	−	2.09354i
377.8	−0.707107	+	0.707107i	−1.34147	+	1.09565i		−	1.00000i	2.02568	+	0.946905i	0.173818	−	1.72331i	2.29933	−	0.616104i	0.707107	+	0.707107i	0.599084	−	2.93957i	−2.10193	+	0.762808i
377.9	−0.707107	+	0.707107i	−1.27826	−	1.16878i		−	1.00000i	−2.23600	−	0.0178176i	1.73032	−	0.0774131i	4.17555	−	1.11884i	0.707107	+	0.707107i	0.267899	+	2.98801i	1.59369	−	1.56849i
377.10	−0.707107	+	0.707107i	−1.00727	+	1.40904i		−	1.00000i	−0.926393	−	2.03514i	−0.284092	−	1.70859i	3.45336	−	0.925324i	0.707107	+	0.707107i	−0.970795	−	2.83858i	2.09412	+	0.784002i
377.11	−0.707107	+	0.707107i	−1.00673	−	1.40943i		−	1.00000i	0.813777	−	2.08273i	1.70848	+	0.284755i	−1.29509	+	0.347018i	0.707107	+	0.707107i	−0.972997	+	2.83783i	0.897286	+	2.04814i
377.12	−0.707107	+	0.707107i	−0.792487	−	1.54012i		−	1.00000i	−0.947623	+	2.02534i	1.64940	+	0.528655i	−1.77478	+	0.475552i	0.707107	+	0.707107i	−1.74393	+	2.44105i	−0.762062	−	2.10220i
377.13	−0.707107	+	0.707107i	−0.537394	−	1.64657i		−	1.00000i	0.858744	+	2.06460i	1.54430	+	0.784309i	2.17271	−	0.582177i	0.707107	+	0.707107i	−2.42242	+	1.76972i	−2.06711	−	0.852667i
377.14	−0.707107	+	0.707107i	−0.390383	+	1.68748i		−	1.00000i	−1.57881	−	1.58347i	−0.917189	−	1.46927i	−4.83994	+	1.29686i	0.707107	+	0.707107i	−2.69520	−	1.31753i	2.23607	+	0.00329236i
377.15	−0.707107	+	0.707107i	−0.320052	+	1.70222i		−	1.00000i	1.97048	−	1.05698i	−0.977343	−	1.42997i	1.65949	−	0.444658i	0.707107	+	0.707107i	−2.79513	−	1.08960i	−0.645943	+	2.14074i
377.16	−0.707107	+	0.707107i	−0.136833	+	1.72664i		−	1.00000i	−2.04548	+	0.903327i	−1.12416	−	1.31767i	−0.382918	+	0.102603i	0.707107	+	0.707107i	−2.96255	−	0.472523i	0.807626	−	2.08512i
377.17	−0.707107	+	0.707107i	−0.134174	−	1.72685i		−	1.00000i	2.18104	+	0.493020i	1.31594	+	1.12619i	−3.79793	+	1.01765i	0.707107	+	0.707107i	−2.96399	+	0.463396i	−1.89085	+	1.19361i
377.18	−0.707107	+	0.707107i	0.321845	−	1.70189i		−	1.00000i	−0.575509	−	2.16074i	0.975836	+	1.43099i	2.59908	−	0.696422i	0.707107	+	0.707107i	−2.79283	−	1.09549i	1.93482	+	1.12093i
377.19	−0.707107	+	0.707107i	0.609255	+	1.62136i		−	1.00000i	1.73683	−	1.40835i	−1.57728	−	0.715666i	−3.28936	+	0.881380i	0.707107	+	0.707107i	−2.25762	+	1.97564i	−0.232271	+	2.22397i
377.20	−0.707107	+	0.707107i	0.661860	−	1.60061i		−	1.00000i	−2.12466	+	0.697001i	0.663795	+	1.59981i	1.92566	−	0.515980i	0.707107	+	0.707107i	−2.12388	−	2.11875i	1.00951	−	1.99522i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner
31.c	even	3	1	inner
93.h	odd	6	1	inner
155.o	odd	12	1	inner
465.be	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	930.2.bf.a	✓	256
3.b	odd	2	1	inner	930.2.bf.a	✓	256
5.c	odd	4	1	inner	930.2.bf.a	✓	256
15.e	even	4	1	inner	930.2.bf.a	✓	256
31.c	even	3	1	inner	930.2.bf.a	✓	256
93.h	odd	6	1	inner	930.2.bf.a	✓	256
155.o	odd	12	1	inner	930.2.bf.a	✓	256
465.be	even	12	1	inner	930.2.bf.a	✓	256

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
930.2.bf.a	✓	256	1.a	even	1	1	trivial
930.2.bf.a	✓	256	3.b	odd	2	1	inner
930.2.bf.a	✓	256	5.c	odd	4	1	inner
930.2.bf.a	✓	256	15.e	even	4	1	inner
930.2.bf.a	✓	256	31.c	even	3	1	inner
930.2.bf.a	✓	256	93.h	odd	6	1	inner
930.2.bf.a	✓	256	155.o	odd	12	1	inner
930.2.bf.a	✓	256	465.be	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(930, [\chi])\).