Newform orbit 403.3.n.a

• Label: 403.3.n.a
• Level: $403$
• Weight: $3$
• Character orbit: 403.n
• Analytic conductor: $10.981$
• Analytic rank: $0$
• Dimension: $146$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	403.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9809546537\)
Analytic rank:	\(0\)
Dimension:	\(146\)
Relative dimension:	\(73\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 146 q - 2 q^{2} - 146 q^{4} - 2 q^{5} + 12 q^{6} + 16 q^{7} - 10 q^{8} - 422 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
347.1	−1.97752	+	3.42517i			1.54677i	−5.82118	−	10.0826i	0.998261	−	1.72904i	−5.29794	−	3.05877i	−0.729334	+	1.26324i	30.2259			6.60750			3.94816	+	6.83842i
347.2	−1.92005	+	3.32562i		−	5.37975i	−5.37318	−	9.30663i	−2.88729	+	5.00093i	17.8910	+	10.3294i	−0.0475240	+	0.0823140i	25.9067			−19.9417			−11.0875	−	19.2041i
347.3	−1.86650	+	3.23288i		−	3.12227i	−4.96766	−	8.60425i	4.34179	−	7.52019i	10.0939	+	5.82772i	1.33349	−	2.30968i	22.1566			−0.748556			16.2079	+	28.0729i
347.4	−1.84398	+	3.19387i			5.59424i	−4.80054	−	8.31477i	−1.08765	+	1.88386i	−17.8673	−	10.3157i	5.37982	−	9.31812i	20.6566			−22.2955			−4.01120	−	6.94760i
347.5	−1.81096	+	3.13668i			2.45052i	−4.55919	−	7.89674i	−4.07389	+	7.05619i	−7.68649	−	4.43780i	−4.77851	+	8.27662i	18.5384			2.99497			−14.7553	−	25.5570i
347.6	−1.75986	+	3.04816i		−	2.24463i	−4.19420	−	7.26456i	−0.674199	+	1.16775i	6.84200	+	3.95023i	1.49834	−	2.59521i	15.4459			3.96163			−2.37299	−	4.11014i
347.7	−1.69117	+	2.92920i			3.46315i	−3.72015	−	6.44348i	2.46517	−	4.26980i	−10.1443	−	5.85679i	0.564458	−	0.977671i	11.6363			−2.99338			8.33807	+	14.4420i
347.8	−1.66215	+	2.87893i			0.165598i	−3.52550	−	6.10634i	−4.28528	+	7.42232i	−0.476746	−	0.275249i	4.53849	−	7.86089i	10.1424			8.97258			−14.2456	−	24.6741i
347.9	−1.62345	+	2.81191i			0.725475i	−3.27121	−	5.66590i	0.797924	−	1.38204i	−2.03997	−	1.17778i	−3.41461	+	5.91427i	8.25502			8.47369			2.59079	+	4.48737i
347.10	−1.59322	+	2.75953i		−	3.92061i	−3.07668	−	5.32897i	2.37035	−	4.10557i	10.8191	+	6.24639i	−5.63496	+	9.76005i	6.86156			−6.37121			7.55298	+	13.0821i
347.11	−1.47699	+	2.55822i			1.72252i	−2.36298	−	4.09281i	0.496131	−	0.859324i	−4.40657	−	2.54414i	5.78997	−	10.0285i	2.14448			6.03294			1.46556	+	2.53842i
347.12	−1.47301	+	2.55133i			4.70617i	−2.33953	−	4.05218i	4.83353	−	8.37192i	−12.0070	−	6.93225i	−3.52736	+	6.10956i	2.00050			−13.1481			14.2397	+	24.6639i
347.13	−1.45014	+	2.51172i		−	3.44080i	−2.20584	−	3.82062i	−1.72058	+	2.98013i	8.64234	+	4.98966i	2.00775	−	3.47753i	1.19398			−2.83911			−4.99017	−	8.64323i
347.14	−1.37601	+	2.38331i		−	1.54831i	−1.78679	−	3.09481i	3.95192	−	6.84492i	3.69010	+	2.13048i	5.36410	−	9.29089i	−1.17351			6.60275			10.8757	+	18.8373i
347.15	−1.35832	+	2.35267i			4.93639i	−1.69005	−	2.92725i	−2.00883	+	3.47939i	−11.6137	−	6.70518i	−3.14336	+	5.44446i	−1.68404			−15.3679			−5.45724	−	9.45222i
347.16	−1.34658	+	2.33234i		−	5.56766i	−1.62654	−	2.81725i	0.910039	−	1.57623i	12.9857	+	7.49728i	4.58756	−	7.94589i	−2.01158			−21.9988			2.45087	+	4.24504i
347.17	−1.32954	+	2.30284i		−	2.37170i	−1.53538	−	2.65935i	−1.97690	+	3.42409i	5.46165	+	3.15329i	−3.57586	+	6.19358i	−2.47094			3.37502			−5.25675	−	9.10496i
347.18	−1.14547	+	1.98401i			4.31273i	−0.624190	−	1.08113i	−0.475266	+	0.823186i	−8.55648	−	4.94009i	0.0855553	−	0.148186i	−6.30378			−9.59962			−1.08880	−	1.88586i
347.19	−1.07116	+	1.85531i			1.99962i	−0.294772	−	0.510560i	1.03220	−	1.78782i	−3.70991	−	2.14192i	3.31453	−	5.74093i	−7.30630			5.00152			2.21130	+	3.83009i
347.20	−1.03950	+	1.80047i			0.492905i	−0.161126	−	0.279078i	2.47696	−	4.29022i	−0.887461	−	0.512376i	−6.00491	+	10.4008i	−7.64605			8.75704			5.14961	+	8.91938i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
403.n	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	403.3.n.a	✓	146
13.c	even	3	1		403.3.o.a	yes	146
31.e	odd	6	1		403.3.o.a	yes	146
403.n	odd	6	1	inner	403.3.n.a	✓	146

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
403.3.n.a	✓	146	1.a	even	1	1	trivial
403.3.n.a	✓	146	403.n	odd	6	1	inner
403.3.o.a	yes	146	13.c	even	3	1
403.3.o.a	yes	146	31.e	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(403, [\chi])\).