Newform orbit 330.3.n.a

• Label: 330.3.n.a
• Level: $330$
• Weight: $3$
• Character orbit: 330.n
• Analytic conductor: $8.992$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	330.n (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.99184872389\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(24\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 48 q^{4} + 8 q^{5} + 72 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−1.14412	+	0.831254i	−1.64728	+	0.535233i	0.618034	−	1.90211i	−4.91042	+	0.942239i	1.43977	−	1.98168i	−0.966454	+	2.97444i	0.874032	+	2.68999i	2.42705	−	1.76336i	4.83488	−	5.15984i
19.2	−1.14412	+	0.831254i	−1.64728	+	0.535233i	0.618034	−	1.90211i	−0.953709	+	4.90820i	1.43977	−	1.98168i	2.42147	−	7.45251i	0.874032	+	2.68999i	2.42705	−	1.76336i	−2.98880	−	6.40836i
19.3	−1.14412	+	0.831254i	−1.64728	+	0.535233i	0.618034	−	1.90211i	−0.790159	−	4.93717i	1.43977	−	1.98168i	−1.42745	+	4.39324i	0.874032	+	2.68999i	2.42705	−	1.76336i	5.00808	+	4.99191i
19.4	−1.14412	+	0.831254i	−1.64728	+	0.535233i	0.618034	−	1.90211i	0.571612	+	4.96722i	1.43977	−	1.98168i	−3.59272	+	11.0573i	0.874032	+	2.68999i	2.42705	−	1.76336i	−4.78301	−	5.20795i
19.5	−1.14412	+	0.831254i	−1.64728	+	0.535233i	0.618034	−	1.90211i	3.29702	−	3.75894i	1.43977	−	1.98168i	4.30541	−	13.2507i	0.874032	+	2.68999i	2.42705	−	1.76336i	−0.647563	+	7.04135i
19.6	−1.14412	+	0.831254i	−1.64728	+	0.535233i	0.618034	−	1.90211i	4.91991	+	0.891339i	1.43977	−	1.98168i	−0.200067	+	0.615743i	0.874032	+	2.68999i	2.42705	−	1.76336i	−6.36991	+	3.06989i
19.7	−1.14412	+	0.831254i	1.64728	−	0.535233i	0.618034	−	1.90211i	−4.38379	−	2.40465i	−1.43977	+	1.98168i	−1.18878	+	3.65869i	0.874032	+	2.68999i	2.42705	−	1.76336i	7.01447	−	0.892829i
19.8	−1.14412	+	0.831254i	1.64728	−	0.535233i	0.618034	−	1.90211i	−3.75909	+	3.29685i	−1.43977	+	1.98168i	0.732438	−	2.25421i	0.874032	+	2.68999i	2.42705	−	1.76336i	1.56035	−	6.89676i
19.9	−1.14412	+	0.831254i	1.64728	−	0.535233i	0.618034	−	1.90211i	−0.932937	−	4.91219i	−1.43977	+	1.98168i	1.81423	−	5.58362i	0.874032	+	2.68999i	2.42705	−	1.76336i	5.15067	+	4.84464i
19.10	−1.14412	+	0.831254i	1.64728	−	0.535233i	0.618034	−	1.90211i	3.36456	+	3.69862i	−1.43977	+	1.98168i	2.06300	−	6.34927i	0.874032	+	2.68999i	2.42705	−	1.76336i	−6.92395	−	1.43487i
19.11	−1.14412	+	0.831254i	1.64728	−	0.535233i	0.618034	−	1.90211i	3.73073	−	3.32891i	−1.43977	+	1.98168i	0.386762	−	1.19033i	0.874032	+	2.68999i	2.42705	−	1.76336i	−1.50125	+	6.90987i
19.12	−1.14412	+	0.831254i	1.64728	−	0.535233i	0.618034	−	1.90211i	4.00856	−	2.98856i	−1.43977	+	1.98168i	−3.26747	+	10.0562i	0.874032	+	2.68999i	2.42705	−	1.76336i	−2.10203	+	6.75141i
19.13	1.14412	−	0.831254i	−1.64728	+	0.535233i	0.618034	−	1.90211i	−4.96007	+	0.630675i	−1.43977	+	1.98168i	−1.81423	+	5.58362i	−0.874032	−	2.68999i	2.42705	−	1.76336i	−5.15067	+	4.84464i
19.14	1.14412	−	0.831254i	−1.64728	+	0.535233i	0.618034	−	1.90211i	−3.64163	−	3.42616i	−1.43977	+	1.98168i	1.18878	−	3.65869i	−0.874032	−	2.68999i	2.42705	−	1.76336i	−7.01447	−	0.892829i
19.15	1.14412	−	0.831254i	−1.64728	+	0.535233i	0.618034	−	1.90211i	−2.01312	+	4.57683i	−1.43977	+	1.98168i	−0.386762	+	1.19033i	−0.874032	−	2.68999i	2.42705	−	1.76336i	1.50125	+	6.90987i
19.16	1.14412	−	0.831254i	−1.64728	+	0.535233i	0.618034	−	1.90211i	−1.60358	+	4.73588i	−1.43977	+	1.98168i	3.26747	−	10.0562i	−0.874032	−	2.68999i	2.42705	−	1.76336i	2.10203	+	6.75141i
19.17	1.14412	−	0.831254i	−1.64728	+	0.535233i	0.618034	−	1.90211i	1.97387	−	4.59389i	−1.43977	+	1.98168i	−0.732438	+	2.25421i	−0.874032	−	2.68999i	2.42705	−	1.76336i	−1.56035	−	6.89676i
19.18	1.14412	−	0.831254i	−1.64728	+	0.535233i	0.618034	−	1.90211i	4.55730	+	2.05695i	−1.43977	+	1.98168i	−2.06300	+	6.34927i	−0.874032	−	2.68999i	2.42705	−	1.76336i	6.92395	−	1.43487i
19.19	1.14412	−	0.831254i	1.64728	−	0.535233i	0.618034	−	1.90211i	−4.93970	+	0.774183i	1.43977	−	1.98168i	1.42745	−	4.39324i	−0.874032	−	2.68999i	2.42705	−	1.76336i	−5.00808	+	4.99191i
19.20	1.14412	−	0.831254i	1.64728	−	0.535233i	0.618034	−	1.90211i	−2.55613	+	4.29723i	1.43977	−	1.98168i	−4.30541	+	13.2507i	−0.874032	−	2.68999i	2.42705	−	1.76336i	0.647563	+	7.04135i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
11.d	odd	10	1	inner
55.h	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	330.3.n.a	✓	96
5.b	even	2	1	inner	330.3.n.a	✓	96
11.d	odd	10	1	inner	330.3.n.a	✓	96
55.h	odd	10	1	inner	330.3.n.a	✓	96

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
330.3.n.a	✓	96	1.a	even	1	1	trivial
330.3.n.a	✓	96	5.b	even	2	1	inner
330.3.n.a	✓	96	11.d	odd	10	1	inner
330.3.n.a	✓	96	55.h	odd	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(330, [\chi])\).