Newform orbit 310.3.i.a

• Label: 310.3.i.a
• Level: $310$
• Weight: $3$
• Character orbit: 310.i
• Analytic conductor: $8.447$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	310.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.44688819517\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
99.1		−	1.41421i	−2.49546	−	4.32227i	−2.00000			−1.93356	−	4.61100i	−6.11261	+	3.52912i	−7.12616	+	4.11429i			2.82843i	−7.95467	+	13.7779i	−6.52094	+	2.73447i
99.2		−	1.41421i	−2.49294	−	4.31790i	−2.00000			−3.07623	+	3.94168i	−6.10643	+	3.52555i	1.56148	−	0.901524i			2.82843i	−7.92949	+	13.7343i	5.57438	+	4.35044i
99.3		−	1.41421i	−1.63133	−	2.82555i	−2.00000			4.11851	+	2.83511i	−3.99593	+	2.30705i	6.79002	−	3.92022i			2.82843i	−0.822498	+	1.42461i	4.00945	−	5.82446i
99.4		−	1.41421i	−1.60457	−	2.77920i	−2.00000			−2.86911	−	4.09490i	−3.93038	+	2.26920i	11.5265	−	6.65484i			2.82843i	−0.649284	+	1.12459i	−5.79106	+	4.05754i
99.5		−	1.41421i	−1.14812	−	1.98861i	−2.00000			3.83759	−	3.20514i	−2.81232	+	1.62369i	−6.29732	+	3.63576i			2.82843i	1.86363	−	3.22790i	−4.53275	−	5.42717i
99.6		−	1.41421i	−0.779809	−	1.35067i	−2.00000			3.55129	+	3.51971i	−1.91013	+	1.10282i	−7.43315	+	4.29153i			2.82843i	3.28380	−	5.68770i	4.97762	−	5.02228i
99.7		−	1.41421i	−0.624440	−	1.08156i	−2.00000			2.78986	−	4.14930i	−1.52956	+	0.883091i	5.40475	−	3.12043i			2.82843i	3.72015	−	6.44349i	−5.86799	−	3.94546i
99.8		−	1.41421i	−0.404044	−	0.699825i	−2.00000			0.104343	+	4.99891i	−0.989702	+	0.571405i	0.505559	−	0.291885i			2.82843i	4.17350	−	7.22871i	7.06953	−	0.147563i
99.9		−	1.41421i	−0.136432	−	0.236307i	−2.00000			−4.56478	−	2.04028i	−0.334188	+	0.192944i	−6.27915	+	3.62527i			2.82843i	4.46277	−	7.72975i	−2.88540	+	6.45558i
99.10		−	1.41421i	0.246583	+	0.427094i	−2.00000			−3.79180	+	3.25918i	0.604003	−	0.348721i	−4.59664	+	2.65387i			2.82843i	4.37839	−	7.58360i	4.60917	+	5.36242i
99.11		−	1.41421i	1.17913	+	2.04232i	−2.00000			−1.84871	−	4.64567i	2.88827	−	1.66754i	1.35646	−	0.783150i			2.82843i	1.71930	−	2.97791i	−6.56997	+	2.61447i
99.12		−	1.41421i	1.19046	+	2.06194i	−2.00000			−4.71572	+	1.66192i	2.91602	−	1.68357i	9.16680	−	5.29245i			2.82843i	1.66560	−	2.88491i	2.35031	+	6.66904i
99.13		−	1.41421i	1.41600	+	2.45259i	−2.00000			4.35084	−	2.46378i	3.46848	−	2.00253i	−0.711149	+	0.410582i			2.82843i	0.489879	−	0.848494i	−3.48431	−	6.15301i
99.14		−	1.41421i	2.02117	+	3.50078i	−2.00000			3.49508	+	3.57553i	4.95084	−	2.85837i	5.44792	−	3.14536i			2.82843i	−3.67029	+	6.35712i	5.05656	−	4.94279i
99.15		−	1.41421i	2.55575	+	4.42669i	−2.00000			−4.97323	+	0.516724i	6.26029	−	3.61438i	−1.46156	+	0.843833i			2.82843i	−8.56373	+	14.8328i	0.730759	+	7.03321i
99.16		−	1.41421i	2.70805	+	4.69048i	−2.00000			2.12665	−	4.52519i	6.63334	−	3.82976i	−7.85437	+	4.53472i			2.82843i	−10.1671	+	17.6099i	−6.39959	−	3.00754i
99.17			1.41421i	−2.70805	−	4.69048i	−2.00000			2.85561	−	4.10433i	6.63334	−	3.82976i	7.85437	−	4.53472i		−	2.82843i	−10.1671	+	17.6099i	5.80440	+	4.03844i
99.18			1.41421i	−2.55575	−	4.42669i	−2.00000			2.03912	+	4.56530i	6.26029	−	3.61438i	1.46156	−	0.843833i		−	2.82843i	−8.56373	+	14.8328i	−6.45631	+	2.88375i
99.19			1.41421i	−2.02117	−	3.50078i	−2.00000			−4.84404	−	1.23907i	4.95084	−	2.85837i	−5.44792	+	3.14536i		−	2.82843i	−3.67029	+	6.35712i	1.75231	−	6.85051i
99.20			1.41421i	−1.41600	−	2.45259i	−2.00000			−0.0417218	−	4.99983i	3.46848	−	2.00253i	0.711149	−	0.410582i		−	2.82843i	0.489879	−	0.848494i	7.07082	−	0.0590035i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
31.e	odd	6	1	inner
155.i	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	310.3.i.a	✓	64
5.b	even	2	1	inner	310.3.i.a	✓	64
31.e	odd	6	1	inner	310.3.i.a	✓	64
155.i	odd	6	1	inner	310.3.i.a	✓	64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
310.3.i.a	✓	64	1.a	even	1	1	trivial
310.3.i.a	✓	64	5.b	even	2	1	inner
310.3.i.a	✓	64	31.e	odd	6	1	inner
310.3.i.a	✓	64	155.i	odd	6	1	inner

Hecke kernels

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