Newform orbit 330.2.w.a

• Label: 330.2.w.a
• Level: $330$
• Weight: $2$
• Character orbit: 330.w
• Analytic conductor: $2.635$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	330.w (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.63506326670\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(24\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 4 q^{3} + 8 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
47.1	−0.156434	+	0.987688i	−1.66802	+	0.466602i	−0.951057	−	0.309017i	1.42604	−	1.72232i	−0.199922	−	1.72047i	−1.97139	+	1.00447i	0.453990	−	0.891007i	2.56457	−	1.55660i	1.47804	+	1.67792i
47.2	−0.156434	+	0.987688i	−1.61459	+	0.626972i	−0.951057	−	0.309017i	1.61159	+	1.55009i	−0.366675	−	1.69279i	3.94629	−	2.01074i	0.453990	−	0.891007i	2.21381	−	2.02461i	−1.78311	+	1.34926i
47.3	−0.156434	+	0.987688i	−1.32529	−	1.11517i	−0.951057	−	0.309017i	−0.756005	−	2.10439i	1.30876	−	1.13452i	−0.601396	+	0.306427i	0.453990	−	0.891007i	0.512798	+	2.95585i	2.19675	−	0.417498i
47.4	−0.156434	+	0.987688i	−1.17991	−	1.26799i	−0.951057	−	0.309017i	−1.88104	+	1.20900i	1.43696	−	0.967028i	1.70796	−	0.870250i	0.453990	−	0.891007i	−0.215611	+	2.99224i	−0.899851	−	2.04701i
47.5	−0.156434	+	0.987688i	−0.642835	+	1.60834i	−0.951057	−	0.309017i	−1.85818	−	1.24385i	−1.48798	−	0.886520i	−0.310922	+	0.158423i	0.453990	−	0.891007i	−2.17353	−	2.06780i	1.51922	−	1.64072i
47.6	−0.156434	+	0.987688i	−0.587373	+	1.62941i	−0.951057	−	0.309017i	0.305347	+	2.21512i	−1.51747	−	0.835038i	−2.94617	+	1.50115i	0.453990	−	0.891007i	−2.30998	−	1.91415i	−2.23562	−	0.0449340i
47.7	−0.156434	+	0.987688i	0.137895	−	1.72655i	−0.951057	−	0.309017i	−0.281340	+	2.21830i	1.68372	+	0.406290i	−3.75100	+	1.91123i	0.453990	−	0.891007i	−2.96197	−	0.476167i	−2.14698	−	0.624894i
47.8	−0.156434	+	0.987688i	0.255726	−	1.71307i	−0.951057	−	0.309017i	2.18764	−	0.462852i	1.65197	+	0.520561i	2.26459	−	1.15387i	0.453990	−	0.891007i	−2.86921	−	0.876154i	0.114931	+	2.23311i
47.9	−0.156434	+	0.987688i	0.764925	+	1.55399i	−0.951057	−	0.309017i	0.935010	−	2.03120i	−1.65452	+	0.512409i	4.23098	−	2.15579i	0.453990	−	0.891007i	−1.82978	+	2.37737i	1.85992	+	1.24125i
47.10	−0.156434	+	0.987688i	1.40640	−	1.01096i	−0.951057	−	0.309017i	−2.05226	−	0.887817i	0.778502	+	1.54723i	1.88440	−	0.960151i	0.453990	−	0.891007i	0.955928	−	2.84362i	1.19793	−	1.88811i
47.11	−0.156434	+	0.987688i	1.53403	+	0.804202i	−0.951057	−	0.309017i	−1.90976	+	1.16310i	−1.03428	+	1.38934i	0.0369078	−	0.0188055i	0.453990	−	0.891007i	1.70652	+	2.46734i	−0.850027	−	2.06820i
47.12	−0.156434	+	0.987688i	1.65897	−	0.497823i	−0.951057	−	0.309017i	2.17628	−	0.513603i	0.232174	+	1.71642i	−2.45142	+	1.24906i	0.453990	−	0.891007i	2.50435	−	1.65174i	0.166834	+	2.22984i
47.13	0.156434	−	0.987688i	−1.71936	−	0.209262i	−0.951057	−	0.309017i	1.88104	−	1.20900i	−0.475653	+	1.66546i	1.70796	−	0.870250i	−0.453990	+	0.891007i	2.91242	+	0.719595i	−0.899851	−	2.04701i
47.14	0.156434	−	0.987688i	−1.68118	−	0.416704i	−0.951057	−	0.309017i	0.756005	+	2.10439i	−0.674568	+	1.59529i	−0.601396	+	0.306427i	−0.453990	+	0.891007i	2.65272	+	1.40111i	2.19675	−	0.417498i
47.15	0.156434	−	0.987688i	−1.31576	+	1.12640i	−0.951057	−	0.309017i	0.281340	−	2.21830i	0.906704	+	1.47577i	−3.75100	+	1.91123i	−0.453990	+	0.891007i	0.462437	−	2.96414i	−2.14698	−	0.624894i
47.16	0.156434	−	0.987688i	−1.23559	+	1.21380i	−0.951057	−	0.309017i	−2.18764	+	0.462852i	1.00557	+	1.41026i	2.26459	−	1.15387i	−0.453990	+	0.891007i	0.0533625	−	2.99953i	0.114931	+	2.23311i
47.17	0.156434	−	0.987688i	−0.602947	−	1.62372i	−0.951057	−	0.309017i	−1.42604	+	1.72232i	−1.69805	+	0.341519i	−1.97139	+	1.00447i	−0.453990	+	0.891007i	−2.27291	+	1.95803i	1.47804	+	1.67792i
47.18	0.156434	−	0.987688i	−0.441802	−	1.67476i	−0.951057	−	0.309017i	−1.61159	−	1.55009i	−1.72325	+	0.174373i	3.94629	−	2.01074i	−0.453990	+	0.891007i	−2.60962	+	1.47982i	−1.78311	+	1.34926i
47.19	0.156434	−	0.987688i	0.00877955	+	1.73203i	−0.951057	−	0.309017i	2.05226	+	0.887817i	1.71208	+	0.262277i	1.88440	−	0.960151i	−0.453990	+	0.891007i	−2.99985	+	0.0304129i	1.19793	−	1.88811i
47.20	0.156434	−	0.987688i	0.572370	+	1.63475i	−0.951057	−	0.309017i	−2.17628	+	0.513603i	1.70416	−	0.309592i	−2.45142	+	1.24906i	−0.453990	+	0.891007i	−2.34479	+	1.87136i	0.166834	+	2.22984i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
11.c	even	5	1	inner
15.e	even	4	1	inner
33.h	odd	10	1	inner
55.k	odd	20	1	inner
165.v	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	330.2.w.a	✓	192
3.b	odd	2	1	inner	330.2.w.a	✓	192
5.c	odd	4	1	inner	330.2.w.a	✓	192
11.c	even	5	1	inner	330.2.w.a	✓	192
15.e	even	4	1	inner	330.2.w.a	✓	192
33.h	odd	10	1	inner	330.2.w.a	✓	192
55.k	odd	20	1	inner	330.2.w.a	✓	192
165.v	even	20	1	inner	330.2.w.a	✓	192

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
330.2.w.a	✓	192	1.a	even	1	1	trivial
330.2.w.a	✓	192	3.b	odd	2	1	inner
330.2.w.a	✓	192	5.c	odd	4	1	inner
330.2.w.a	✓	192	11.c	even	5	1	inner
330.2.w.a	✓	192	15.e	even	4	1	inner
330.2.w.a	✓	192	33.h	odd	10	1	inner
330.2.w.a	✓	192	55.k	odd	20	1	inner
330.2.w.a	✓	192	165.v	even	20	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(330, [\chi])\).