Newform orbit 429.2.t.a

• Label: 429.2.t.a
• Level: $429$
• Weight: $2$
• Character orbit: 429.t
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(52\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 2 q^{3} + 44 q^{4} - 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
296.1	−2.31547	+	1.33684i	0.0244717	+	1.73188i	2.57426	−	4.45876i	2.76446			−2.37190	−	3.97739i	0.565134	−	0.978841i			8.41814i	−2.99880	+	0.0847639i	−6.40102	+	3.69563i
296.2	−2.31547	+	1.33684i	1.48761	+	0.887132i	2.57426	−	4.45876i	−2.76446			−4.63048	−	0.0654292i	−0.565134	+	0.978841i			8.41814i	1.42599	+	2.63942i	6.40102	−	3.69563i
296.3	−2.17254	+	1.25431i	−0.913417	−	1.47162i	2.14661	−	3.71803i	2.90929			3.83031	+	2.05144i	0.464602	−	0.804714i			5.75282i	−1.33134	+	2.68841i	−6.32053	+	3.64916i
296.4	−2.17254	+	1.25431i	−0.817753	−	1.52685i	2.14661	−	3.71803i	−2.90929			3.69175	+	2.29142i	−0.464602	+	0.804714i			5.75282i	−1.66256	+	2.49718i	6.32053	−	3.64916i
296.5	−2.13554	+	1.23295i	−1.72013	+	0.202872i	2.04035	−	3.53400i	−0.635450			3.42327	−	2.55408i	−2.03776	+	3.52950i			5.13084i	2.91769	−	0.697932i	1.35703	−	0.783481i
296.6	−2.13554	+	1.23295i	1.03576	−	1.38824i	2.04035	−	3.53400i	0.635450			−0.500264	+	4.24168i	2.03776	−	3.52950i			5.13084i	−0.854416	−	2.87576i	−1.35703	+	0.783481i
296.7	−1.92182	+	1.10956i	−1.59672	+	0.671171i	1.46227	−	2.53272i	1.97176			2.32391	−	3.06154i	0.477966	−	0.827861i			2.05166i	2.09906	−	2.14335i	−3.78937	+	2.18779i
296.8	−1.92182	+	1.10956i	1.37961	−	1.04722i	1.46227	−	2.53272i	−1.97176			−1.48942	+	3.54334i	−0.477966	+	0.827861i			2.05166i	0.806667	−	2.88951i	3.78937	−	2.18779i
296.9	−1.63490	+	0.943908i	0.664837	+	1.59937i	0.781925	−	1.35433i	0.196251			−2.59660	−	1.98726i	1.83210	−	3.17329i		−	0.823370i	−2.11598	+	2.12665i	−0.320851	+	0.185243i
296.10	−1.63490	+	0.943908i	1.05268	+	1.37545i	0.781925	−	1.35433i	−0.196251			−3.01932	−	1.25509i	−1.83210	+	3.17329i		−	0.823370i	−0.783738	+	2.89582i	0.320851	−	0.185243i
296.11	−1.41960	+	0.819608i	−0.961092	+	1.44094i	0.343515	−	0.594985i	−1.26601			0.183365	−	2.83328i	0.208022	−	0.360304i		−	2.15224i	−1.15260	−	2.76975i	1.79724	−	1.03764i
296.12	−1.41960	+	0.819608i	1.72843	−	0.111861i	0.343515	−	0.594985i	1.26601			−2.36201	+	1.57544i	−0.208022	+	0.360304i		−	2.15224i	2.97497	−	0.386690i	−1.79724	+	1.03764i
296.13	−1.25571	+	0.724987i	−1.55144	−	0.770078i	0.0512119	−	0.0887017i	−0.906362			2.50647	−	0.157780i	1.12801	−	1.95377i		−	2.75144i	1.81396	+	2.38947i	1.13813	−	0.657100i
296.14	−1.25571	+	0.724987i	0.108816	−	1.72863i	0.0512119	−	0.0887017i	0.906362			1.11659	+	2.24955i	−1.12801	+	1.95377i		−	2.75144i	−2.97632	−	0.376204i	−1.13813	+	0.657100i
296.15	−1.09482	+	0.632095i	−1.58106	−	0.707293i	−0.200911	+	0.347987i	4.13662			2.17805	−	0.225019i	−0.620239	+	1.07429i		−	3.03636i	1.99947	+	2.23654i	−4.52886	+	2.61474i
296.16	−1.09482	+	0.632095i	0.177995	−	1.72288i	−0.200911	+	0.347987i	−4.13662			0.894153	+	1.99876i	0.620239	−	1.07429i		−	3.03636i	−2.93664	−	0.613327i	4.52886	−	2.61474i
296.17	−0.949310	+	0.548084i	−0.500493	+	1.65816i	−0.399207	+	0.691447i	3.63865			−0.433691	−	1.84842i	−2.32754	+	4.03142i		−	3.06753i	−2.49901	−	1.65980i	−3.45421	+	1.99429i
296.18	−0.949310	+	0.548084i	1.68626	+	0.395642i	−0.399207	+	0.691447i	−3.63865			−1.81763	+	0.548625i	2.32754	−	4.03142i		−	3.06753i	2.68693	+	1.33431i	3.45421	−	1.99429i
296.19	−0.716153	+	0.413471i	−1.68064	+	0.418881i	−0.658083	+	1.13983i	−3.60945			1.03040	−	0.994878i	−1.45351	+	2.51755i		−	2.74228i	2.64908	−	1.40797i	2.58492	−	1.49241i
296.20	−0.716153	+	0.413471i	1.20308	−	1.24603i	−0.658083	+	1.13983i	3.60945			−0.346391	+	1.38979i	1.45351	−	2.51755i		−	2.74228i	−0.105197	−	2.99816i	−2.58492	+	1.49241i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
11.b	odd	2	1	inner
13.e	even	6	1	inner
33.d	even	2	1	inner
39.h	odd	6	1	inner
143.i	odd	6	1	inner
429.t	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	429.2.t.a	✓	104
3.b	odd	2	1	inner	429.2.t.a	✓	104
11.b	odd	2	1	inner	429.2.t.a	✓	104
13.e	even	6	1	inner	429.2.t.a	✓	104
33.d	even	2	1	inner	429.2.t.a	✓	104
39.h	odd	6	1	inner	429.2.t.a	✓	104
143.i	odd	6	1	inner	429.2.t.a	✓	104
429.t	even	6	1	inner	429.2.t.a	✓	104

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
429.2.t.a	✓	104	1.a	even	1	1	trivial
429.2.t.a	✓	104	3.b	odd	2	1	inner
429.2.t.a	✓	104	11.b	odd	2	1	inner
429.2.t.a	✓	104	13.e	even	6	1	inner
429.2.t.a	✓	104	33.d	even	2	1	inner
429.2.t.a	✓	104	39.h	odd	6	1	inner
429.2.t.a	✓	104	143.i	odd	6	1	inner
429.2.t.a	✓	104	429.t	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(429, [\chi])\).