Newform orbit 403.3.m.a

• Label: 403.3.m.a
• Level: $403$
• Weight: $3$
• Character orbit: 403.m
• Analytic conductor: $10.981$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9809546537\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(72\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 12 q^{3} - 280 q^{4} + 196 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
181.1		−	3.89295i	−3.19746	−	1.84605i	−11.1551			−5.73793	+	3.31280i	−7.18660	+	12.4475i	0.262115	+	0.151332i			27.8544i	2.31582	+	4.01112i	12.8966	+	22.3375i
181.2		−	3.81143i	−3.05475	−	1.76366i	−10.5270			5.70999	−	3.29666i	−6.72207	+	11.6430i	−7.03746	−	4.06308i			24.8772i	1.72100	+	2.98085i	−12.5650	−	21.7632i
181.3		−	3.76689i	2.14408	+	1.23788i	−10.1895			−2.62439	+	1.51519i	4.66297	−	8.07651i	7.69252	+	4.44128i			23.3151i	−1.43529	−	2.48599i	5.70756	+	9.88579i
181.4		−	3.69311i	3.87348	+	2.23635i	−9.63904			0.508542	−	0.293607i	8.25909	−	14.3052i	−11.4158	−	6.59094i			20.8256i	5.50254	+	9.53069i	−1.08432	−	1.87810i
181.5		−	3.57954i	0.0436415	+	0.0251964i	−8.81308			−5.49663	+	3.17348i	0.0901915	−	0.156216i	−7.29259	−	4.21038i			17.2286i	−4.49873	−	7.79203i	11.3596	+	19.6754i
181.6		−	3.43140i	0.696493	+	0.402121i	−7.77451			3.45071	−	1.99227i	1.37984	−	2.38995i	1.25250	+	0.723130i			12.9519i	−4.17660	−	7.23408i	−6.83626	−	11.8408i
181.7		−	3.40454i	4.45186	+	2.57028i	−7.59092			5.42047	−	3.12951i	8.75065	−	15.1566i	4.41541	+	2.54924i			12.2255i	8.71272	+	15.0909i	−10.6546	−	18.4542i
181.8		−	3.37863i	−4.22208	−	2.43762i	−7.41511			0.976405	−	0.563728i	−8.23581	+	14.2648i	8.27616	+	4.77824i			11.5384i	7.38398	+	12.7894i	−1.90462	−	3.29891i
181.9		−	3.33983i	−1.59710	−	0.922083i	−7.15444			5.20485	−	3.00502i	−3.07960	+	5.33402i	9.90854	+	5.72070i			10.5353i	−2.79952	−	4.84892i	−10.0362	−	17.3833i
181.10		−	3.25264i	3.35155	+	1.93502i	−6.57969			−6.03286	+	3.48308i	6.29393	−	10.9014i	0.591531	+	0.341521i			8.39081i	2.98861	+	5.17642i	11.3292	+	19.6228i
181.11		−	3.07069i	−2.77257	−	1.60074i	−5.42915			2.42526	−	1.40022i	−4.91538	+	8.51370i	−8.96845	−	5.17794i			4.38848i	0.624748	+	1.08210i	−4.29965	−	7.44722i
181.12		−	2.89606i	−0.639377	−	0.369144i	−4.38716			−0.963670	+	0.556375i	−1.06906	+	1.85167i	−2.21690	−	1.27993i			1.12123i	−4.22746	−	7.32218i	1.61130	+	2.79085i
181.13		−	2.72859i	−3.92810	−	2.26789i	−3.44522			−5.97896	+	3.45196i	−6.18814	+	10.7182i	−0.866628	−	0.500348i		−	1.51377i	5.78663	+	10.0227i	9.41899	+	16.3142i
181.14		−	2.69072i	−0.654840	−	0.378072i	−3.23998			−3.48936	+	2.01458i	−1.01729	+	1.76199i	1.84193	+	1.06344i		−	2.04500i	−4.21412	−	7.29908i	5.42068	+	9.38889i
181.15		−	2.60572i	2.03931	+	1.17740i	−2.78976			8.29436	−	4.78875i	3.06796	−	5.31387i	−3.21156	−	1.85420i		−	3.15355i	−1.72747	−	2.99206i	−12.4781	−	21.6128i
181.16		−	2.40233i	4.31493	+	2.49123i	−1.77117			−0.219673	+	0.126828i	5.98474	−	10.3659i	3.22753	+	1.86342i		−	5.35437i	7.91241	+	13.7047i	0.304682	+	0.527725i
181.17		−	2.28297i	−0.403967	−	0.233231i	−1.21195			−8.02670	+	4.63422i	−0.532458	+	0.922244i	10.7285	+	6.19408i		−	6.36504i	−4.39121	−	7.60579i	10.5798	+	18.3247i
181.18		−	2.22173i	1.88501	+	1.08831i	−0.936093			1.78283	−	1.02932i	2.41794	−	4.18799i	−7.09488	−	4.09623i		−	6.80718i	−2.13115	−	3.69127i	−2.28687	−	3.96098i
181.19		−	2.20078i	−4.94931	−	2.85749i	−0.843427			6.54579	−	3.77921i	−6.28869	+	10.8923i	−1.94118	−	1.12074i		−	6.94692i	11.8305	+	20.4909i	−8.31721	−	14.4058i
181.20		−	2.18654i	4.34211	+	2.50692i	−0.780963			−7.01572	+	4.05053i	5.48147	−	9.49419i	−3.43213	−	1.98154i		−	7.03856i	8.06925	+	13.9764i	8.85665	+	15.3402i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner
31.e	odd	6	1	inner
403.m	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.m.a	✓	144
13.b	even	2	1	inner	403.3.m.a	✓	144
31.e	odd	6	1	inner	403.3.m.a	✓	144
403.m	odd	6	1	inner	403.3.m.a	✓	144

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
403.3.m.a	✓	144	1.a	even	1	1	trivial
403.3.m.a	✓	144	13.b	even	2	1	inner
403.3.m.a	✓	144	31.e	odd	6	1	inner
403.3.m.a	✓	144	403.m	odd	6	1	inner

Hecke kernels

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