Newform orbit 304.3.bf.a

• Label: 304.3.bf.a
• Level: $304$
• Weight: $3$
• Character orbit: 304.bf
• Analytic conductor: $8.283$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	304.bf (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.28340003655\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
47.1		0		−3.11543	−	3.71282i		0		2.92002	+	1.06280i		0		2.88610	+	1.66629i		0		−2.51633	+	14.2708i		0
47.2		0		−2.05195	−	2.44541i		0		−7.10822	−	2.58718i		0		3.42115	+	1.97520i		0		−0.206734	+	1.17244i		0
47.3		0		−1.12759	−	1.34381i		0		3.24851	+	1.18236i		0		−5.83630	−	3.36959i		0		1.02847	−	5.83272i		0
47.4		0		1.12759	+	1.34381i		0		3.24851	+	1.18236i		0		5.83630	+	3.36959i		0		1.02847	−	5.83272i		0
47.5		0		2.05195	+	2.44541i		0		−7.10822	−	2.58718i		0		−3.42115	−	1.97520i		0		−0.206734	+	1.17244i		0
47.6		0		3.11543	+	3.71282i		0		2.92002	+	1.06280i		0		−2.88610	−	1.66629i		0		−2.51633	+	14.2708i		0
63.1		0		−1.47783	+	4.06029i		0		−1.21801	−	6.90766i		0		2.43240	+	1.40435i		0		−7.40761	−	6.21572i		0
63.2		0		−1.14415	+	3.14353i		0		1.38949	+	7.88020i		0		8.08226	+	4.66630i		0		−1.67828	−	1.40825i		0
63.3		0		−0.534758	+	1.46923i		0		0.00216362	+	0.0122705i		0		−6.17911	−	3.56751i		0		5.02172	+	4.21372i		0
63.4		0		0.534758	−	1.46923i		0		0.00216362	+	0.0122705i		0		6.17911	+	3.56751i		0		5.02172	+	4.21372i		0
63.5		0		1.14415	−	3.14353i		0		1.38949	+	7.88020i		0		−8.08226	−	4.66630i		0		−1.67828	−	1.40825i		0
63.6		0		1.47783	−	4.06029i		0		−1.21801	−	6.90766i		0		−2.43240	−	1.40435i		0		−7.40761	−	6.21572i		0
111.1		0		−1.47783	−	4.06029i		0		−1.21801	+	6.90766i		0		2.43240	−	1.40435i		0		−7.40761	+	6.21572i		0
111.2		0		−1.14415	−	3.14353i		0		1.38949	−	7.88020i		0		8.08226	−	4.66630i		0		−1.67828	+	1.40825i		0
111.3		0		−0.534758	−	1.46923i		0		0.00216362	−	0.0122705i		0		−6.17911	+	3.56751i		0		5.02172	−	4.21372i		0
111.4		0		0.534758	+	1.46923i		0		0.00216362	−	0.0122705i		0		6.17911	−	3.56751i		0		5.02172	−	4.21372i		0
111.5		0		1.14415	+	3.14353i		0		1.38949	−	7.88020i		0		−8.08226	+	4.66630i		0		−1.67828	+	1.40825i		0
111.6		0		1.47783	+	4.06029i		0		−1.21801	+	6.90766i		0		−2.43240	+	1.40435i		0		−7.40761	+	6.21572i		0
175.1		0		−4.96062	+	0.874690i		0		1.49575	−	1.25508i		0		2.80960	+	1.62212i		0		15.3854	−	5.59982i		0
175.2		0		−2.09036	+	0.368587i		0		−4.91876	+	4.12733i		0		−3.82389	−	2.20772i		0		−4.22348	+	1.53722i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
19.e	even	9	1	inner
76.l	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	304.3.bf.a	✓	36
4.b	odd	2	1	inner	304.3.bf.a	✓	36
19.e	even	9	1	inner	304.3.bf.a	✓	36
76.l	odd	18	1	inner	304.3.bf.a	✓	36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
304.3.bf.a	✓	36	1.a	even	1	1	trivial
304.3.bf.a	✓	36	4.b	odd	2	1	inner
304.3.bf.a	✓	36	19.e	even	9	1	inner
304.3.bf.a	✓	36	76.l	odd	18	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^36 + 6*T3^34 + 51*T3^32 - 12566*T3^30 - 167955*T3^28 + 835179*T3^26 + 166507038*T3^24 + 2680142187*T3^22 + 30048180060*T3^20 + 145538793471*T3^18 + 344179659150*T3^16 - 6926523728688*T3^14 - 3178232433958*T3^12 + 9803893711758*T3^10 + 319669229323857*T3^8 + 920336054786780*T3^6 + 1747786563398274*T3^4 - 196994135456037*T3^2 + 6131066257801