Newform orbit 403.2.ba.a

• Label: 403.2.ba.a
• Level: $403$
• Weight: $2$
• Character orbit: 403.ba
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(35\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 140 q - 8 q^{2} - 12 q^{4} - 2 q^{5} + 6 q^{6} + 12 q^{7} - 10 q^{8} - 124 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
6.1	−0.725626	−	2.70807i		−	1.15921i	−5.07507	+	2.93009i	0.758661	+	2.83136i	−3.13923	+	0.841153i	2.03109	−	0.544229i	7.65262	+	7.65262i	1.65623			7.11703	−	4.10902i
6.2	−0.648505	−	2.42025i			0.854246i	−3.70501	+	2.13909i	−0.846300	−	3.15843i	2.06749	−	0.553983i	4.17770	−	1.11941i	4.03636	+	4.03636i	2.27026			−7.09538	+	4.09652i
6.3	−0.626284	−	2.33732i			1.39283i	−3.33879	+	1.92765i	−0.0315189	−	0.117630i	3.25550	−	0.872308i	−1.74827	+	0.468447i	3.17450	+	3.17450i	1.06002			−0.255199	+	0.147339i
6.4	−0.616245	−	2.29986i		−	2.87995i	−3.17753	+	1.83455i	−0.235608	−	0.879300i	−6.62347	+	1.77475i	−1.87834	+	0.503300i	2.81012	+	2.81012i	−5.29412			−1.87707	+	1.08373i
6.5	−0.610803	−	2.27955i			2.86872i	−3.09121	+	1.78471i	−0.322912	−	1.20512i	6.53938	−	1.75222i	0.235099	−	0.0629946i	2.61896	+	2.61896i	−5.22953			−2.54990	+	1.47219i
6.6	−0.529121	−	1.97471i			0.0635184i	−1.88744	+	1.08972i	0.363448	+	1.35641i	0.125430	−	0.0336089i	−3.84966	+	1.03151i	0.259387	+	0.259387i	2.99597			2.48620	−	1.43541i
6.7	−0.505205	−	1.88545i		−	2.78647i	−1.56764	+	0.905077i	−0.0340828	−	0.127199i	−5.25374	+	1.40774i	4.21322	−	1.12893i	−0.262034	−	0.262034i	−4.76440			−0.222608	+	0.128523i
6.8	−0.504814	−	1.88399i			2.73123i	−1.56254	+	0.902134i	1.05873	+	3.95125i	5.14562	−	1.37876i	1.60020	−	0.428773i	−0.269949	−	0.269949i	−4.45962			6.90966	−	3.98929i
6.9	−0.495867	−	1.85060i		−	0.831703i	−1.44679	+	0.835304i	0.740052	+	2.76191i	−1.53915	+	0.412414i	1.04383	−	0.279694i	−0.446239	−	0.446239i	2.30827			4.74423	−	2.73908i
6.10	−0.379919	−	1.41788i		−	0.797458i	−0.133982	+	0.0773543i	−0.876737	−	3.27203i	−1.13070	+	0.302969i	2.28277	−	0.611667i	−1.91533	−	1.91533i	2.36406			−4.30624	+	2.48621i
6.11	−0.361997	−	1.35099i			2.34712i	0.0379121	−	0.0218885i	−0.843165	−	3.14674i	3.17094	−	0.849651i	−1.42675	+	0.382295i	−2.02129	−	2.02129i	−2.50897			−3.94599	+	2.27822i
6.12	−0.330760	−	1.23441i		−	1.61222i	0.317680	−	0.183412i	−0.646845	−	2.41406i	−1.99014	+	0.533256i	−2.14247	+	0.574074i	−2.13879	−	2.13879i	0.400758			−2.76599	+	1.59695i
6.13	−0.216378	−	0.807533i			1.94866i	1.12676	−	0.650535i	−0.0513533	−	0.191653i	1.57361	−	0.421647i	1.59671	−	0.427837i	−1.95145	−	1.95145i	−0.797286			−0.143655	+	0.0829390i
6.14	−0.167077	−	0.623539i			0.591585i	1.37116	−	0.791642i	0.549606	+	2.05116i	0.368876	−	0.0988400i	−4.24319	+	1.13696i	−1.63563	−	1.63563i	2.65003			1.18715	−	0.685401i
6.15	−0.137083	−	0.511599i		−	1.84118i	1.48911	−	0.859737i	0.868165	+	3.24004i	−0.941949	+	0.252395i	1.36093	−	0.364660i	−1.39301	−	1.39301i	−0.389962			1.53859	−	0.888306i
6.16	−0.135765	−	0.506680i		−	2.82155i	1.49376	−	0.862422i	0.266274	+	0.993746i	−1.42962	+	0.383067i	−2.70478	+	0.724745i	−1.38160	−	1.38160i	−4.96115			0.467361	−	0.269831i
6.17	−0.133986	−	0.500044i			1.10231i	1.49996	−	0.866002i	0.299955	+	1.11945i	0.551206	−	0.147695i	3.94703	−	1.05760i	−1.36613	−	1.36613i	1.78490			0.519583	−	0.299981i
6.18	0.0225085	+	0.0840028i			0.498964i	1.72550	−	0.996218i	−0.755041	−	2.81785i	−0.0419144	+	0.0112309i	−2.56590	+	0.687531i	0.245512	+	0.245512i	2.75103			0.219712	−	0.126851i
6.19	0.0759990	+	0.283632i		−	1.83195i	1.65738	−	0.956888i	−0.350763	−	1.30906i	0.519599	−	0.139226i	2.10731	−	0.564653i	0.812630	+	0.812630i	−0.356028			0.344635	−	0.198975i
6.20	0.107992	+	0.403030i			2.83596i	1.58128	−	0.912952i	−0.961540	−	3.58852i	−1.14298	+	0.306260i	1.51235	−	0.405234i	1.12879	+	1.12879i	−5.04265			1.34244	−	0.775060i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
403.ba	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.ba.a	✓	140
13.f	odd	12	1		403.2.bf.a	yes	140
31.e	odd	6	1		403.2.bf.a	yes	140
403.ba	even	12	1	inner	403.2.ba.a	✓	140

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
403.2.ba.a	✓	140	1.a	even	1	1	trivial
403.2.ba.a	✓	140	403.ba	even	12	1	inner
403.2.bf.a	yes	140	13.f	odd	12	1
403.2.bf.a	yes	140	31.e	odd	6	1

Hecke kernels

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