Newform orbit 312.2.bb.a

• Label: 312.2.bb.a
• Level: $312$
• Weight: $2$
• Character orbit: 312.bb
• Analytic conductor: $2.491$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	312.bb (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.49133254306\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 28 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
61.1	−1.40080	+	0.194311i	0.866025	+	0.500000i	1.92449	−	0.544382i			3.36712i	−1.31028	−	0.532122i	2.12695	+	3.68399i	−2.59004	+	1.13652i	0.500000	+	0.866025i	−0.654268	−	4.71666i
61.2	−1.38060	−	0.306524i	−0.866025	−	0.500000i	1.81209	+	0.846371i			0.0787347i	1.04237	+	0.955755i	0.680796	+	1.17917i	−2.24232	−	1.72394i	0.500000	+	0.866025i	0.0241341	−	0.108701i
61.3	−1.32264	+	0.500617i	0.866025	+	0.500000i	1.49877	−	1.32427i			0.369739i	−1.39575	−	0.227775i	−1.16461	−	2.01717i	−1.31938	+	2.50185i	0.500000	+	0.866025i	−0.185098	−	0.489033i
61.4	−1.26013	+	0.641935i	−0.866025	−	0.500000i	1.17584	−	1.61784i		−	1.45280i	1.41227	+	0.0741315i	1.04137	+	1.80370i	−0.443159	+	2.79349i	0.500000	+	0.866025i	0.932603	+	1.83071i
61.5	−1.21555	−	0.722795i	0.866025	+	0.500000i	0.955135	+	1.75719i		−	2.42445i	−0.691302	−	1.23373i	−0.755347	−	1.30830i	0.109072	−	2.82632i	0.500000	+	0.866025i	−1.75238	+	2.94704i
61.6	−1.18919	+	0.765401i	−0.866025	−	0.500000i	0.828323	−	1.82041i			4.17948i	1.41257	−	0.0682638i	−1.23795	−	2.14419i	0.408311	+	2.79880i	0.500000	+	0.866025i	−3.19898	−	4.97018i
61.7	−1.07157	−	0.922897i	0.866025	+	0.500000i	0.296521	+	1.97790i			3.60324i	−0.466558	−	1.33504i	−1.60828	−	2.78562i	1.50765	−	2.39311i	0.500000	+	0.866025i	3.32542	−	3.86112i
61.8	−1.04153	−	0.956674i	−0.866025	−	0.500000i	0.169550	+	1.99280i		−	3.04873i	0.423650	+	1.34927i	−1.36275	−	2.36035i	1.72987	−	2.23776i	0.500000	+	0.866025i	−2.91664	+	3.17533i
61.9	−0.853140	+	1.12790i	0.866025	+	0.500000i	−0.544303	−	1.92451i		−	0.581590i	−1.30279	+	0.550217i	1.02479	+	1.77499i	2.63501	+	1.02796i	0.500000	+	0.866025i	0.655973	+	0.496178i
61.10	−0.613215	−	1.27435i	−0.866025	−	0.500000i	−1.24794	+	1.56290i			1.62558i	−0.106115	+	1.41023i	−0.510871	−	0.884855i	2.75693	+	0.631912i	0.500000	+	0.866025i	2.07155	−	0.996827i
61.11	−0.604578	−	1.27847i	0.866025	+	0.500000i	−1.26897	+	1.54587i		−	1.47697i	0.115655	−	1.40948i	1.94969	+	3.37697i	2.74354	+	0.687742i	0.500000	+	0.866025i	−1.88826	+	0.892942i
61.12	−0.550217	+	1.30279i	−0.866025	−	0.500000i	−1.39452	−	1.43363i			0.581590i	1.12790	−	0.853140i	1.02479	+	1.77499i	2.63501	−	1.02796i	0.500000	+	0.866025i	−0.757689	−	0.320001i
61.13	−0.0682638	+	1.41257i	0.866025	+	0.500000i	−1.99068	−	0.192854i		−	4.17948i	−0.765401	+	1.18919i	−1.23795	−	2.14419i	0.408311	−	2.79880i	0.500000	+	0.866025i	5.90379	+	0.285307i
61.14	0.0741315	+	1.41227i	0.866025	+	0.500000i	−1.98901	+	0.209387i			1.45280i	−0.641935	+	1.26013i	1.04137	+	1.80370i	−0.443159	−	2.79349i	0.500000	+	0.866025i	−2.05174	+	0.107698i
61.15	0.0901764	−	1.41134i	−0.866025	−	0.500000i	−1.98374	−	0.254538i		−	3.18381i	−0.783763	+	1.17716i	−0.0947723	−	0.164150i	−0.538125	+	2.77676i	0.500000	+	0.866025i	−4.49343	−	0.287105i
61.16	0.227775	+	1.39575i	−0.866025	−	0.500000i	−1.89624	+	0.635833i		−	0.369739i	0.500617	−	1.32264i	−1.16461	−	2.01717i	−1.31938	−	2.50185i	0.500000	+	0.866025i	0.516063	−	0.0842171i
61.17	0.379426	−	1.36236i	−0.866025	−	0.500000i	−1.71207	−	1.03383i			2.89261i	−1.00977	+	0.990129i	2.18031	+	3.77641i	−2.05806	+	1.94020i	0.500000	+	0.866025i	3.94079	+	1.09753i
61.18	0.532122	+	1.31028i	−0.866025	−	0.500000i	−1.43369	+	1.39446i		−	3.36712i	0.194311	−	1.40080i	2.12695	+	3.68399i	−2.59004	−	1.13652i	0.500000	+	0.866025i	4.41188	−	1.79172i
61.19	0.579874	−	1.28986i	0.866025	+	0.500000i	−1.32749	−	1.49592i		−	1.76603i	1.14712	−	0.827117i	−2.26933	−	3.93060i	−2.69930	+	0.844842i	0.500000	+	0.866025i	−2.27793	−	1.02407i
61.20	0.827117	−	1.14712i	−0.866025	−	0.500000i	−0.631754	−	1.89760i			1.76603i	−1.28986	+	0.579874i	−2.26933	−	3.93060i	−2.69930	−	0.844842i	0.500000	+	0.866025i	2.02584	+	1.46071i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
13.c	even	3	1	inner
104.r	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	312.2.bb.a	✓	56
3.b	odd	2	1		936.2.be.c		56
4.b	odd	2	1		1248.2.br.a		56
8.b	even	2	1	inner	312.2.bb.a	✓	56
8.d	odd	2	1		1248.2.br.a		56
13.c	even	3	1	inner	312.2.bb.a	✓	56
24.h	odd	2	1		936.2.be.c		56
39.i	odd	6	1		936.2.be.c		56
52.j	odd	6	1		1248.2.br.a		56
104.n	odd	6	1		1248.2.br.a		56
104.r	even	6	1	inner	312.2.bb.a	✓	56
312.bh	odd	6	1		936.2.be.c		56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
312.2.bb.a	✓	56	1.a	even	1	1	trivial
312.2.bb.a	✓	56	8.b	even	2	1	inner
312.2.bb.a	✓	56	13.c	even	3	1	inner
312.2.bb.a	✓	56	104.r	even	6	1	inner
936.2.be.c		56	3.b	odd	2	1
936.2.be.c		56	24.h	odd	2	1
936.2.be.c		56	39.i	odd	6	1
936.2.be.c		56	312.bh	odd	6	1
1248.2.br.a		56	4.b	odd	2	1
1248.2.br.a		56	8.d	odd	2	1
1248.2.br.a		56	52.j	odd	6	1
1248.2.br.a		56	104.n	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(312, [\chi])\).