Newform orbit 403.2.h.a

• Label: 403.2.h.a
• Level: $403$
• Weight: $2$
• Character orbit: 403.h
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(15\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 6 q^{2} - 2 q^{3} + 22 q^{4} + 7 q^{5} + 6 q^{7} - 24 q^{8} - 13 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
118.1	−2.75405			1.04794	+	1.81508i	5.58479			1.03612	−	1.79462i	−2.88606	−	4.99881i	2.05805	+	3.56464i	−9.87267			−0.696336	+	1.20609i	−2.85353	+	4.94246i
118.2	−2.54177			−0.358627	−	0.621161i	4.46058			−1.54812	+	2.68142i	0.911547	+	1.57885i	−1.09448	−	1.89569i	−6.25421			1.24277	−	2.15255i	3.93495	−	6.81554i
118.3	−2.19969			−0.332345	−	0.575638i	2.83864			2.12692	−	3.68393i	0.731057	+	1.26623i	−0.0876538	−	0.151821i	−1.84476			1.27909	−	2.21546i	−4.67856	+	8.10350i
118.4	−1.54088			−0.945700	−	1.63800i	0.374297			0.246770	−	0.427419i	1.45721	+	2.52395i	−2.41271	−	4.17893i	2.50501			−0.288697	+	0.500038i	−0.380242	+	0.658599i
118.5	−1.20534			−1.24518	−	2.15671i	−0.547152			0.667125	−	1.15549i	1.50087	+	2.59957i	1.94340	+	3.36606i	3.07019			−1.60094	+	2.77291i	−0.804114	+	1.39277i
118.6	−1.11604			0.521580	+	0.903404i	−0.754455			−0.0866186	+	0.150028i	−0.582104	−	1.00823i	−0.0906774	−	0.157058i	3.07408			0.955908	−	1.65568i	0.0966698	−	0.167437i
118.7	−0.899619			1.46679	+	2.54055i	−1.19069			−1.03352	+	1.79011i	−1.31955	−	2.28553i	1.98012	+	3.42966i	2.87040			−2.80293	+	4.85482i	0.929773	−	1.61041i
118.8	0.0439466			0.820034	+	1.42034i	−1.99807			0.865000	−	1.49822i	0.0360377	+	0.0624192i	−2.17574	−	3.76849i	−0.175702			0.155089	−	0.268621i	0.0380138	−	0.0658419i
118.9	0.165462			−0.350644	−	0.607334i	−1.97262			−1.35983	+	2.35530i	−0.0580182	−	0.100491i	0.231883	+	0.401634i	−0.657317			1.25410	−	2.17216i	−0.225001	+	0.389713i
118.10	0.352077			−1.65894	−	2.87336i	−1.87604			1.09161	−	1.89072i	−0.584073	−	1.01164i	−1.14275	−	1.97930i	−1.36466			−4.00415	+	6.93538i	0.384330	−	0.665679i
118.11	1.21548			−0.671133	−	1.16244i	−0.522608			1.76937	−	3.06463i	−0.815749	−	1.41292i	0.724247	+	1.25443i	−3.06618			0.599162	−	1.03778i	2.15063	−	3.72500i
118.12	1.41262			−1.31623	−	2.27977i	−0.00449076			−1.72422	+	2.98644i	−1.85934	−	3.22047i	2.33584	+	4.04580i	−2.83159			−1.96492	+	3.40333i	−2.43568	+	4.21871i
118.13	1.58004			1.43561	+	2.48655i	0.496524			1.71834	−	2.97625i	2.26832	+	3.92885i	0.908007	+	1.57271i	−2.37555			−2.62196	+	4.54137i	2.71504	−	4.70259i
118.14	2.10012			0.700181	+	1.21275i	2.41051			−0.938510	+	1.62555i	1.47047	+	2.54692i	0.810515	+	1.40385i	0.862110			0.519492	−	0.899787i	−1.97098	+	3.41385i
118.15	2.38763			−0.113336	−	0.196303i	3.70079			0.669573	−	1.15973i	−0.270604	−	0.468700i	−0.988047	−	1.71135i	4.06087			1.47431	−	2.55358i	1.59869	−	2.76902i
222.1	−2.75405			1.04794	−	1.81508i	5.58479			1.03612	+	1.79462i	−2.88606	+	4.99881i	2.05805	−	3.56464i	−9.87267			−0.696336	−	1.20609i	−2.85353	−	4.94246i
222.2	−2.54177			−0.358627	+	0.621161i	4.46058			−1.54812	−	2.68142i	0.911547	−	1.57885i	−1.09448	+	1.89569i	−6.25421			1.24277	+	2.15255i	3.93495	+	6.81554i
222.3	−2.19969			−0.332345	+	0.575638i	2.83864			2.12692	+	3.68393i	0.731057	−	1.26623i	−0.0876538	+	0.151821i	−1.84476			1.27909	+	2.21546i	−4.67856	−	8.10350i
222.4	−1.54088			−0.945700	+	1.63800i	0.374297			0.246770	+	0.427419i	1.45721	−	2.52395i	−2.41271	+	4.17893i	2.50501			−0.288697	−	0.500038i	−0.380242	−	0.658599i
222.5	−1.20534			−1.24518	+	2.15671i	−0.547152			0.667125	+	1.15549i	1.50087	−	2.59957i	1.94340	−	3.36606i	3.07019			−1.60094	−	2.77291i	−0.804114	−	1.39277i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	403.2.h.a	✓	30
31.c	even	3	1	inner	403.2.h.a	✓	30

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
403.2.h.a	✓	30	1.a	even	1	1	trivial
403.2.h.a	✓	30	31.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^15 + 3*T2^14 - 16*T2^13 - 49*T2^12 + 96*T2^11 + 303*T2^10 - 268*T2^9 - 895*T2^8 + 344*T2^7 + 1309*T2^6 - 158*T2^5 - 858*T2^4 + 12*T2^3 + 166*T2^2 - 30*T2 + 1