Newform orbit 310.3.j.a

• Label: 310.3.j.a
• Level: $310$
• Weight: $3$
• Character orbit: 310.j
• Analytic conductor: $8.447$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(8.44688819517\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 12 q^{3} + 80 q^{4} - 12 q^{7} + 48 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
161.1	−1.41421			−3.75129	+	2.16581i	2.00000			−1.11803	+	1.93649i	5.30512	−	3.06292i	2.93590	+	5.08513i	−2.82843			4.88145	−	8.45492i	1.58114	−	2.73861i
161.2	−1.41421			−3.02394	+	1.74587i	2.00000			1.11803	−	1.93649i	4.27650	−	2.46904i	1.43526	+	2.48593i	−2.82843			1.59615	−	2.76462i	−1.58114	+	2.73861i
161.3	−1.41421			−2.15057	+	1.24163i	2.00000			−1.11803	+	1.93649i	3.04136	−	1.75593i	−0.891381	−	1.54392i	−2.82843			−1.41670	+	2.45380i	1.58114	−	2.73861i
161.4	−1.41421			−0.984057	+	0.568146i	2.00000			1.11803	−	1.93649i	1.39167	−	0.803479i	−6.82150	−	11.8152i	−2.82843			−3.85442	+	6.67605i	−1.58114	+	2.73861i
161.5	−1.41421			−0.707949	+	0.408735i	2.00000			1.11803	−	1.93649i	1.00119	−	0.578038i	2.94544	+	5.10165i	−2.82843			−4.16587	+	7.21550i	−1.58114	+	2.73861i
161.6	−1.41421			1.06079	−	0.612450i	2.00000			−1.11803	+	1.93649i	−1.50019	+	0.866135i	−3.70069	−	6.40978i	−2.82843			−3.74981	+	6.49486i	1.58114	−	2.73861i
161.7	−1.41421			1.34325	−	0.775525i	2.00000			−1.11803	+	1.93649i	−1.89964	+	1.09676i	−0.798717	−	1.38342i	−2.82843			−3.29712	+	5.71078i	1.58114	−	2.73861i
161.8	−1.41421			2.61959	−	1.51242i	2.00000			1.11803	−	1.93649i	−3.70466	+	2.13888i	2.32621	+	4.02911i	−2.82843			0.0748230	−	0.129597i	−1.58114	+	2.73861i
161.9	−1.41421			3.59636	−	2.07636i	2.00000			1.11803	−	1.93649i	−5.08602	+	2.93642i	−2.91437	−	5.04783i	−2.82843			4.12255	−	7.14046i	−1.58114	+	2.73861i
161.10	−1.41421			4.99781	−	2.88549i	2.00000			−1.11803	+	1.93649i	−7.06798	+	4.08070i	3.89806	+	6.75164i	−2.82843			12.1521	−	21.0481i	1.58114	−	2.73861i
161.11	1.41421			−4.23886	+	2.44730i	2.00000			1.11803	−	1.93649i	−5.99465	+	3.46101i	−3.23478	−	5.60281i	2.82843			7.47860	−	12.9533i	1.58114	−	2.73861i
161.12	1.41421			−3.51722	+	2.03067i	2.00000			−1.11803	+	1.93649i	−4.97411	+	2.87180i	−4.12357	−	7.14223i	2.82843			3.74724	−	6.49041i	−1.58114	+	2.73861i
161.13	1.41421			−2.09880	+	1.21174i	2.00000			−1.11803	+	1.93649i	−2.96815	+	1.71366i	1.93625	+	3.35368i	2.82843			−1.56335	+	2.70781i	−1.58114	+	2.73861i
161.14	1.41421			−1.20321	+	0.694672i	2.00000			1.11803	−	1.93649i	−1.70159	+	0.982414i	−6.41765	−	11.1157i	2.82843			−3.53486	+	6.12256i	1.58114	−	2.73861i
161.15	1.41421			−0.517474	+	0.298764i	2.00000			1.11803	−	1.93649i	−0.731819	+	0.422516i	4.24656	+	7.35526i	2.82843			−4.32148	+	7.48502i	1.58114	−	2.73861i
161.16	1.41421			−0.238579	+	0.137744i	2.00000			−1.11803	+	1.93649i	−0.337402	+	0.194799i	−1.48169	−	2.56637i	2.82843			−4.46205	+	7.72850i	−1.58114	+	2.73861i
161.17	1.41421			2.74847	−	1.58683i	2.00000			−1.11803	+	1.93649i	3.88692	−	2.24411i	5.76711	+	9.98892i	2.82843			0.536042	−	0.928452i	−1.58114	+	2.73861i
161.18	1.41421			3.06786	−	1.77123i	2.00000			1.11803	−	1.93649i	4.33862	−	2.50490i	−2.36899	−	4.10322i	2.82843			1.77453	−	3.07358i	1.58114	−	2.73861i
161.19	1.41421			4.39167	−	2.53553i	2.00000			1.11803	−	1.93649i	6.21076	−	3.58578i	3.33170	+	5.77067i	2.82843			8.35785	−	14.4762i	1.58114	−	2.73861i
161.20	1.41421			4.60614	−	2.65936i	2.00000			−1.11803	+	1.93649i	6.51406	−	3.76090i	−2.06913	−	3.58384i	2.82843			9.64434	−	16.7045i	−1.58114	+	2.73861i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.e	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.j.a	✓	40
31.e	odd	6	1	inner	310.3.j.a	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
310.3.j.a	✓	40	1.a	even	1	1	trivial
310.3.j.a	✓	40	31.e	odd	6	1	inner

Hecke kernels

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