Newform orbit 308.2.f.a

• Label: 308.2.f.a
• Level: $308$
• Weight: $2$
• Character orbit: 308.f
• Analytic conductor: $2.459$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	308.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.45939238226\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 12 q^{8} + 40 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
111.1	−1.40495	−	0.161602i	−2.32114			1.94777	+	0.454086i			0.864179i	3.26109	+	0.375101i	−1.35593	−	2.27188i	−2.66314	−	0.952731i	2.38769			0.139653	−	1.21413i
111.2	−1.40495	−	0.161602i	2.32114			1.94777	+	0.454086i		−	0.864179i	−3.26109	−	0.375101i	1.35593	−	2.27188i	−2.66314	−	0.952731i	2.38769			−0.139653	+	1.21413i
111.3	−1.40495	+	0.161602i	−2.32114			1.94777	−	0.454086i		−	0.864179i	3.26109	−	0.375101i	−1.35593	+	2.27188i	−2.66314	+	0.952731i	2.38769			0.139653	+	1.21413i
111.4	−1.40495	+	0.161602i	2.32114			1.94777	−	0.454086i			0.864179i	−3.26109	+	0.375101i	1.35593	+	2.27188i	−2.66314	+	0.952731i	2.38769			−0.139653	−	1.21413i
111.5	−1.22297	−	0.710168i	−0.278505			0.991322	+	1.73703i			1.27829i	0.340603	+	0.197785i	2.61856	−	0.378345i	0.0212252	−	2.82835i	−2.92244			0.907803	−	1.56332i
111.6	−1.22297	−	0.710168i	0.278505			0.991322	+	1.73703i		−	1.27829i	−0.340603	−	0.197785i	−2.61856	−	0.378345i	0.0212252	−	2.82835i	−2.92244			−0.907803	+	1.56332i
111.7	−1.22297	+	0.710168i	−0.278505			0.991322	−	1.73703i		−	1.27829i	0.340603	−	0.197785i	2.61856	+	0.378345i	0.0212252	+	2.82835i	−2.92244			0.907803	+	1.56332i
111.8	−1.22297	+	0.710168i	0.278505			0.991322	−	1.73703i			1.27829i	−0.340603	+	0.197785i	−2.61856	+	0.378345i	0.0212252	+	2.82835i	−2.92244			−0.907803	−	1.56332i
111.9	−1.12716	−	0.854113i	−2.74291			0.540982	+	1.92544i		−	3.46756i	3.09170	+	2.34275i	2.56168	+	0.661643i	1.03477	−	2.63235i	4.52355			−2.96169	+	3.90850i
111.10	−1.12716	−	0.854113i	2.74291			0.540982	+	1.92544i			3.46756i	−3.09170	−	2.34275i	−2.56168	+	0.661643i	1.03477	−	2.63235i	4.52355			2.96169	−	3.90850i
111.11	−1.12716	+	0.854113i	−2.74291			0.540982	−	1.92544i			3.46756i	3.09170	−	2.34275i	2.56168	−	0.661643i	1.03477	+	2.63235i	4.52355			−2.96169	−	3.90850i
111.12	−1.12716	+	0.854113i	2.74291			0.540982	−	1.92544i		−	3.46756i	−3.09170	+	2.34275i	−2.56168	−	0.661643i	1.03477	+	2.63235i	4.52355			2.96169	+	3.90850i
111.13	−0.660659	−	1.25041i	−1.82041			−1.12706	+	1.65219i			3.16747i	1.20267	+	2.27626i	0.0454780	−	2.64536i	2.81052	+	0.317755i	0.313893			3.96064	−	2.09262i
111.14	−0.660659	−	1.25041i	1.82041			−1.12706	+	1.65219i		−	3.16747i	−1.20267	−	2.27626i	−0.0454780	−	2.64536i	2.81052	+	0.317755i	0.313893			−3.96064	+	2.09262i
111.15	−0.660659	+	1.25041i	−1.82041			−1.12706	−	1.65219i		−	3.16747i	1.20267	−	2.27626i	0.0454780	+	2.64536i	2.81052	−	0.317755i	0.313893			3.96064	+	2.09262i
111.16	−0.660659	+	1.25041i	1.82041			−1.12706	−	1.65219i			3.16747i	−1.20267	+	2.27626i	−0.0454780	+	2.64536i	2.81052	−	0.317755i	0.313893			−3.96064	−	2.09262i
111.17	−0.127910	−	1.40842i	−2.47071			−1.96728	+	0.360301i		−	2.31056i	0.316028	+	3.47979i	−2.23820	−	1.41084i	0.759089	+	2.72466i	3.10440			−3.25424	+	0.295544i
111.18	−0.127910	−	1.40842i	2.47071			−1.96728	+	0.360301i			2.31056i	−0.316028	−	3.47979i	2.23820	−	1.41084i	0.759089	+	2.72466i	3.10440			3.25424	−	0.295544i
111.19	−0.127910	+	1.40842i	−2.47071			−1.96728	−	0.360301i			2.31056i	0.316028	−	3.47979i	−2.23820	+	1.41084i	0.759089	−	2.72466i	3.10440			−3.25424	−	0.295544i
111.20	−0.127910	+	1.40842i	2.47071			−1.96728	−	0.360301i		−	2.31056i	−0.316028	+	3.47979i	2.23820	+	1.41084i	0.759089	−	2.72466i	3.10440			3.25424	+	0.295544i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.b	odd	2	1	inner
28.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	308.2.f.a	✓	40
4.b	odd	2	1	inner	308.2.f.a	✓	40
7.b	odd	2	1	inner	308.2.f.a	✓	40
28.d	even	2	1	inner	308.2.f.a	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
308.2.f.a	✓	40	1.a	even	1	1	trivial
308.2.f.a	✓	40	4.b	odd	2	1	inner
308.2.f.a	✓	40	7.b	odd	2	1	inner
308.2.f.a	✓	40	28.d	even	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(308, [\chi])\).