Newform orbit 403.2.bg.a

• Label: 403.2.bg.a
• Level: $403$
• Weight: $2$
• Character orbit: 403.bg
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bg (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 8 q^{2} - 12 q^{4} - 8 q^{5} - 16 q^{7} - 28 q^{8} + 64 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
123.1	−0.705571	−	2.63323i	−1.33423	−	0.770317i	−4.70400	+	2.71585i	−0.754353	+	0.754353i	−1.08703	+	4.05684i	0.397529	−	1.48360i	6.61515	+	6.61515i	−0.313224	−	0.542519i	2.51863	+	1.45413i
123.2	−0.705571	−	2.63323i	1.33423	+	0.770317i	−4.70400	+	2.71585i	−0.754353	+	0.754353i	1.08703	−	4.05684i	0.397529	−	1.48360i	6.61515	+	6.61515i	−0.313224	−	0.542519i	2.51863	+	1.45413i
123.3	−0.623506	−	2.32695i	−1.53051	−	0.883643i	−3.29391	+	1.90174i	2.78278	−	2.78278i	−1.10191	+	4.11240i	0.171705	−	0.640810i	3.07213	+	3.07213i	0.0616504	+	0.106782i	−8.21047	−	4.74032i
123.4	−0.623506	−	2.32695i	1.53051	+	0.883643i	−3.29391	+	1.90174i	2.78278	−	2.78278i	1.10191	−	4.11240i	0.171705	−	0.640810i	3.07213	+	3.07213i	0.0616504	+	0.106782i	−8.21047	−	4.74032i
123.5	−0.598354	−	2.23309i	−1.62337	−	0.937250i	−2.89661	+	1.67236i	−1.13336	+	1.13336i	−1.12162	+	4.18593i	−0.972184	+	3.62824i	2.19825	+	2.19825i	0.256877	+	0.444923i	3.20903	+	1.85274i
123.6	−0.598354	−	2.23309i	1.62337	+	0.937250i	−2.89661	+	1.67236i	−1.13336	+	1.13336i	1.12162	−	4.18593i	−0.972184	+	3.62824i	2.19825	+	2.19825i	0.256877	+	0.444923i	3.20903	+	1.85274i
123.7	−0.436335	−	1.62842i	−0.707551	−	0.408505i	−0.729326	+	0.421077i	−2.22177	+	2.22177i	−0.356490	+	1.33044i	0.484835	−	1.80943i	−1.38026	−	1.38026i	−1.16625	−	2.02000i	4.58742	+	2.64855i
123.8	−0.436335	−	1.62842i	0.707551	+	0.408505i	−0.729326	+	0.421077i	−2.22177	+	2.22177i	0.356490	−	1.33044i	0.484835	−	1.80943i	−1.38026	−	1.38026i	−1.16625	−	2.02000i	4.58742	+	2.64855i
123.9	−0.434186	−	1.62041i	−1.90487	−	1.09978i	−0.705145	+	0.407116i	0.649841	−	0.649841i	−0.955016	+	3.56417i	0.921552	−	3.43928i	−1.40658	−	1.40658i	0.919017	+	1.59178i	−1.33516	−	0.770854i
123.10	−0.434186	−	1.62041i	1.90487	+	1.09978i	−0.705145	+	0.407116i	0.649841	−	0.649841i	0.955016	−	3.56417i	0.921552	−	3.43928i	−1.40658	−	1.40658i	0.919017	+	1.59178i	−1.33516	−	0.770854i
123.11	−0.359643	−	1.34220i	−2.72479	−	1.57316i	0.0598795	−	0.0345714i	1.15466	−	1.15466i	−1.13155	+	4.22301i	−1.21090	+	4.51915i	−2.03306	−	2.03306i	3.44967	+	5.97500i	−1.96506	−	1.13453i
123.12	−0.359643	−	1.34220i	2.72479	+	1.57316i	0.0598795	−	0.0345714i	1.15466	−	1.15466i	1.13155	−	4.22301i	−1.21090	+	4.51915i	−2.03306	−	2.03306i	3.44967	+	5.97500i	−1.96506	−	1.13453i
123.13	−0.248585	−	0.927732i	−0.327048	−	0.188821i	0.933159	−	0.538760i	1.52097	−	1.52097i	−0.0938763	+	0.350351i	−0.204081	+	0.761641i	−2.09009	−	2.09009i	−1.42869	−	2.47457i	−1.78914	−	1.03296i
123.14	−0.248585	−	0.927732i	0.327048	+	0.188821i	0.933159	−	0.538760i	1.52097	−	1.52097i	0.0938763	−	0.350351i	−0.204081	+	0.761641i	−2.09009	−	2.09009i	−1.42869	−	2.47457i	−1.78914	−	1.03296i
123.15	−0.137428	−	0.512890i	−2.33567	−	1.34850i	1.48788	−	0.859029i	0.158725	−	0.158725i	−0.370644	+	1.38326i	0.491600	−	1.83468i	−1.39599	−	1.39599i	2.13689	+	3.70121i	−0.103222	−	0.0595950i
123.16	−0.137428	−	0.512890i	2.33567	+	1.34850i	1.48788	−	0.859029i	0.158725	−	0.158725i	0.370644	−	1.38326i	0.491600	−	1.83468i	−1.39599	−	1.39599i	2.13689	+	3.70121i	−0.103222	−	0.0595950i
123.17	−0.111049	−	0.414442i	−0.896531	−	0.517612i	1.57262	−	0.907953i	−1.95188	+	1.95188i	−0.114961	+	0.429040i	−0.976537	+	3.64449i	−1.15772	−	1.15772i	−0.964155	−	1.66997i	1.02569	+	0.592184i
123.18	−0.111049	−	0.414442i	0.896531	+	0.517612i	1.57262	−	0.907953i	−1.95188	+	1.95188i	0.114961	−	0.429040i	−0.976537	+	3.64449i	−1.15772	−	1.15772i	−0.964155	−	1.66997i	1.02569	+	0.592184i
123.19	0.0151761	+	0.0566379i	−2.67732	−	1.54575i	1.72907	−	0.998281i	−2.77132	+	2.77132i	0.0469169	−	0.175096i	0.222097	−	0.828876i	0.165705	+	0.165705i	3.27869	+	5.67887i	−0.199020	−	0.114904i
123.20	0.0151761	+	0.0566379i	2.67732	+	1.54575i	1.72907	−	0.998281i	−2.77132	+	2.77132i	−0.0469169	+	0.175096i	0.222097	−	0.828876i	0.165705	+	0.165705i	3.27869	+	5.67887i	−0.199020	−	0.114904i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.f	odd	12	1	inner
31.b	odd	2	1	inner
403.bg	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	403.2.bg.a	✓	144
13.f	odd	12	1	inner	403.2.bg.a	✓	144
31.b	odd	2	1	inner	403.2.bg.a	✓	144
403.bg	even	12	1	inner	403.2.bg.a	✓	144

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
403.2.bg.a	✓	144	1.a	even	1	1	trivial
403.2.bg.a	✓	144	13.f	odd	12	1	inner
403.2.bg.a	✓	144	31.b	odd	2	1	inner
403.2.bg.a	✓	144	403.bg	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(403, [\chi])\).