Newform orbit 310.3.v.a

• Label: 310.3.v.a
• Level: $310$
• Weight: $3$
• Character orbit: 310.v
• Analytic conductor: $8.447$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	310.v (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.44688819517\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 256 q + 128 q^{4} - 6 q^{5} + 56 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
79.1	−1.34500	−	0.437016i	−3.56978	+	3.96465i	1.61803	+	1.17557i	−4.92898	−	0.839725i	6.53396	−	3.77238i	2.83866	+	0.298355i	−1.66251	−	2.28825i	−2.03431	−	19.3552i	6.26249	+	3.28347i
79.2	−1.34500	−	0.437016i	−3.26474	+	3.62586i	1.61803	+	1.17557i	1.52541	−	4.76163i	5.97562	−	3.45002i	−3.28580	−	0.345352i	−1.66251	−	2.28825i	−1.54758	−	14.7242i	−4.13258	+	5.73775i
79.3	−1.34500	−	0.437016i	−2.95182	+	3.27833i	1.61803	+	1.17557i	3.88275	+	3.15027i	5.40288	−	3.11935i	−12.6732	−	1.33201i	−1.66251	−	2.28825i	−1.09344	−	10.4034i	−3.84557	−	5.93394i
79.4	−1.34500	−	0.437016i	−2.65310	+	2.94656i	1.61803	+	1.17557i	4.77513	−	1.48259i	4.85611	−	2.80367i	10.4983	+	1.10342i	−1.66251	−	2.28825i	−0.702554	−	6.68436i	−7.07046	−	0.0927253i
79.5	−1.34500	−	0.437016i	−1.42517	+	1.58281i	1.61803	+	1.17557i	1.34691	−	4.81517i	2.60856	−	1.50605i	−1.51254	−	0.158975i	−1.66251	−	2.28825i	0.466573	+	4.43915i	−3.91589	+	5.88777i
79.6	−1.34500	−	0.437016i	−1.35994	+	1.51037i	1.61803	+	1.17557i	−3.25883	+	3.79210i	2.48917	−	1.43712i	−1.03052	−	0.108313i	−1.66251	−	2.28825i	0.508986	+	4.84268i	6.04033	−	3.67620i
79.7	−1.34500	−	0.437016i	−0.395199	+	0.438913i	1.61803	+	1.17557i	−3.90472	−	3.12301i	0.723353	−	0.417628i	−7.74856	−	0.814406i	−1.66251	−	2.28825i	0.904294	+	8.60378i	3.88703	+	5.90686i
79.8	−1.34500	−	0.437016i	−0.318506	+	0.353737i	1.61803	+	1.17557i	1.15032	+	4.86588i	0.582978	−	0.336583i	9.57121	+	1.00597i	−1.66251	−	2.28825i	0.917073	+	8.72536i	0.579292	−	7.04730i
79.9	−1.34500	−	0.437016i	0.177516	−	0.197151i	1.61803	+	1.17557i	4.64948	+	1.83911i	−0.324917	+	0.187591i	−1.90786	−	0.200524i	−1.66251	−	2.28825i	0.933399	+	8.88070i	−5.44982	−	4.50549i
79.10	−1.34500	−	0.437016i	0.704588	−	0.782524i	1.61803	+	1.17557i	−0.960877	−	4.90680i	−1.28964	+	0.744577i	12.5117	+	1.31503i	−1.66251	−	2.28825i	0.824856	+	7.84798i	−0.851975	+	7.01955i
79.11	−1.34500	−	0.437016i	1.97566	−	2.19419i	1.61803	+	1.17557i	4.77498	−	1.48311i	−3.61615	+	2.08779i	5.32896	+	0.560097i	−1.66251	−	2.28825i	0.0295087	+	0.280756i	−7.07047	−	0.0919685i
79.12	−1.34500	−	0.437016i	2.13183	−	2.36764i	1.61803	+	1.17557i	−4.53766	+	2.09992i	−3.90201	+	2.25283i	8.17295	+	0.859012i	−1.66251	−	2.28825i	−0.120255	−	1.14415i	7.02083	−	0.841361i
79.13	−1.34500	−	0.437016i	2.33417	−	2.59236i	1.61803	+	1.17557i	3.59887	−	3.47104i	−4.27236	+	2.46665i	−8.82519	−	0.927565i	−1.66251	−	2.28825i	−0.331220	−	3.15134i	−6.35737	+	3.09578i
79.14	−1.34500	−	0.437016i	2.45503	−	2.72659i	1.61803	+	1.17557i	−4.67373	−	1.77659i	−4.49358	+	2.59437i	−3.01690	−	0.317089i	−1.66251	−	2.28825i	−0.466353	−	4.43705i	5.50975	+	4.43201i
79.15	−1.34500	−	0.437016i	2.58728	−	2.87347i	1.61803	+	1.17557i	−0.434769	+	4.98106i	−4.73564	+	2.73412i	−13.2458	−	1.39219i	−1.66251	−	2.28825i	−0.622032	−	5.91824i	2.76157	−	6.50951i
79.16	−1.34500	−	0.437016i	3.57217	−	3.96730i	1.61803	+	1.17557i	3.89351	+	3.13697i	−6.53833	+	3.77491i	4.32457	+	0.454530i	−1.66251	−	2.28825i	−2.03829	−	19.3930i	−3.86585	−	5.92074i
79.17	1.34500	+	0.437016i	−3.57217	+	3.96730i	1.61803	+	1.17557i	−4.66345	−	1.80339i	−6.53833	+	3.77491i	−4.32457	−	0.454530i	1.66251	+	2.28825i	−2.03829	−	19.3930i	−5.48422	−	4.46356i
79.18	1.34500	+	0.437016i	−2.58728	+	2.87347i	1.61803	+	1.17557i	−4.09634	+	2.86705i	−4.73564	+	2.73412i	13.2458	+	1.39219i	1.66251	+	2.28825i	−0.622032	−	5.91824i	−6.76251	+	2.06601i
79.19	1.34500	+	0.437016i	−2.45503	+	2.72659i	1.61803	+	1.17557i	3.87544	+	3.15927i	−4.49358	+	2.59437i	3.01690	+	0.317089i	1.66251	+	2.28825i	−0.466353	−	4.43705i	3.83180	+	5.94284i
79.20	1.34500	+	0.437016i	−2.33417	+	2.59236i	1.61803	+	1.17557i	1.20658	−	4.85223i	−4.27236	+	2.46665i	8.82519	+	0.927565i	1.66251	+	2.28825i	−0.331220	−	3.15134i	3.74335	−	5.99895i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
31.h	odd	30	1	inner
155.v	odd	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	310.3.v.a	✓	256
5.b	even	2	1	inner	310.3.v.a	✓	256
31.h	odd	30	1	inner	310.3.v.a	✓	256
155.v	odd	30	1	inner	310.3.v.a	✓	256

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
310.3.v.a	✓	256	1.a	even	1	1	trivial
310.3.v.a	✓	256	5.b	even	2	1	inner
310.3.v.a	✓	256	31.h	odd	30	1	inner
310.3.v.a	✓	256	155.v	odd	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(310, [\chi])\).