Newform orbit 234.2.z.a

• Label: 234.2.z.a
• Level: $234$
• Weight: $2$
• Character orbit: 234.z
• Analytic conductor: $1.868$
• Analytic rank: $0$
• Dimension: $56$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	234.z (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.86849940730\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
41.1	−0.965926	+	0.258819i	−1.71677	+	0.229570i	0.866025	−	0.500000i	−2.88241	+	0.772339i	1.59885	−	0.666081i	2.05205	+	2.05205i	−0.707107	+	0.707107i	2.89459	−	0.788239i	2.58430	−	1.49204i
41.2	−0.965926	+	0.258819i	−0.958654	−	1.44256i	0.866025	−	0.500000i	3.32591	−	0.891175i	1.29935	+	1.14529i	−0.0201488	−	0.0201488i	−0.707107	+	0.707107i	−1.16197	+	2.76583i	−2.98193	+	1.72162i
41.3	−0.965926	+	0.258819i	−0.840145	−	1.51465i	0.866025	−	0.500000i	−1.83272	+	0.491075i	1.20354	+	1.24559i	−1.44971	−	1.44971i	−0.707107	+	0.707107i	−1.58831	+	2.54505i	1.64317	−	0.948684i
41.4	−0.965926	+	0.258819i	−0.679534	+	1.59318i	0.866025	−	0.500000i	−1.16856	+	0.313114i	0.244033	−	1.71477i	−3.18466	−	3.18466i	−0.707107	+	0.707107i	−2.07647	−	2.16525i	1.04770	−	0.604889i
41.5	−0.965926	+	0.258819i	0.880877	+	1.49133i	0.866025	−	0.500000i	1.93052	−	0.517281i	−1.23685	−	1.21252i	1.87601	+	1.87601i	−0.707107	+	0.707107i	−1.44811	+	2.62735i	−1.73086	+	0.999310i
41.6	−0.965926	+	0.258819i	1.60665	−	0.647043i	0.866025	−	0.500000i	2.84756	−	0.763001i	−1.38444	+	1.04083i	−1.37958	−	1.37958i	−0.707107	+	0.707107i	2.16267	−	2.07915i	−2.55305	+	1.47400i
41.7	−0.965926	+	0.258819i	1.70757	+	0.290170i	0.866025	−	0.500000i	−2.22031	+	0.594929i	−1.72449	+	0.161669i	1.10604	+	1.10604i	−0.707107	+	0.707107i	2.83160	+	0.990974i	1.99067	−	1.14931i
41.8	0.965926	−	0.258819i	−1.70264	+	0.317829i	0.866025	−	0.500000i	1.48652	−	0.398313i	−1.56236	+	0.747675i	1.01155	+	1.01155i	0.707107	−	0.707107i	2.79797	−	1.08230i	1.33278	−	0.769482i
41.9	0.965926	−	0.258819i	−1.09566	−	1.34146i	0.866025	−	0.500000i	−3.60132	+	0.964970i	−1.40553	−	1.01217i	−2.35233	−	2.35233i	0.707107	−	0.707107i	−0.599043	+	2.93958i	−3.22885	+	1.86418i
41.10	0.965926	−	0.258819i	−0.125978	−	1.72746i	0.866025	−	0.500000i	1.13912	−	0.305227i	−0.568786	−	1.63600i	2.54027	+	2.54027i	0.707107	−	0.707107i	−2.96826	+	0.435246i	1.02131	−	0.589653i
41.11	0.965926	−	0.258819i	0.0473290	+	1.73140i	0.866025	−	0.500000i	3.94593	−	1.05731i	0.493837	+	1.66016i	−2.29775	−	2.29775i	0.707107	−	0.707107i	−2.99552	+	0.163891i	3.53782	−	2.04256i
41.12	0.965926	−	0.258819i	0.161445	+	1.72451i	0.866025	−	0.500000i	−2.21510	+	0.593533i	0.602280	+	1.62396i	2.94637	+	2.94637i	0.707107	−	0.707107i	−2.94787	+	0.556826i	−1.98600	+	1.14662i
41.13	0.965926	−	0.258819i	1.10250	−	1.33585i	0.866025	−	0.500000i	−0.00707659	+	0.00189617i	0.719190	−	1.57568i	−2.39172	−	2.39172i	0.707107	−	0.707107i	−0.568989	−	2.94555i	−0.00634470	+	0.00366311i
41.14	0.965926	−	0.258819i	1.61301	+	0.631032i	0.866025	−	0.500000i	−0.748082	+	0.200448i	1.72137	+	0.192053i	−0.456400	−	0.456400i	0.707107	−	0.707107i	2.20360	+	2.03572i	−0.670712	+	0.387236i
137.1	−0.965926	−	0.258819i	−1.71677	−	0.229570i	0.866025	+	0.500000i	−2.88241	−	0.772339i	1.59885	+	0.666081i	2.05205	−	2.05205i	−0.707107	−	0.707107i	2.89459	+	0.788239i	2.58430	+	1.49204i
137.2	−0.965926	−	0.258819i	−0.958654	+	1.44256i	0.866025	+	0.500000i	3.32591	+	0.891175i	1.29935	−	1.14529i	−0.0201488	+	0.0201488i	−0.707107	−	0.707107i	−1.16197	−	2.76583i	−2.98193	−	1.72162i
137.3	−0.965926	−	0.258819i	−0.840145	+	1.51465i	0.866025	+	0.500000i	−1.83272	−	0.491075i	1.20354	−	1.24559i	−1.44971	+	1.44971i	−0.707107	−	0.707107i	−1.58831	−	2.54505i	1.64317	+	0.948684i
137.4	−0.965926	−	0.258819i	−0.679534	−	1.59318i	0.866025	+	0.500000i	−1.16856	−	0.313114i	0.244033	+	1.71477i	−3.18466	+	3.18466i	−0.707107	−	0.707107i	−2.07647	+	2.16525i	1.04770	+	0.604889i
137.5	−0.965926	−	0.258819i	0.880877	−	1.49133i	0.866025	+	0.500000i	1.93052	+	0.517281i	−1.23685	+	1.21252i	1.87601	−	1.87601i	−0.707107	−	0.707107i	−1.44811	−	2.62735i	−1.73086	−	0.999310i
137.6	−0.965926	−	0.258819i	1.60665	+	0.647043i	0.866025	+	0.500000i	2.84756	+	0.763001i	−1.38444	−	1.04083i	−1.37958	+	1.37958i	−0.707107	−	0.707107i	2.16267	+	2.07915i	−2.55305	−	1.47400i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
117.bc	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	234.2.z.a	yes	56
3.b	odd	2	1		702.2.bc.a		56
9.c	even	3	1		702.2.bb.a		56
9.d	odd	6	1		234.2.y.a	✓	56
13.f	odd	12	1		234.2.y.a	✓	56
39.k	even	12	1		702.2.bb.a		56
117.bb	odd	12	1		702.2.bc.a		56
117.bc	even	12	1	inner	234.2.z.a	yes	56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
234.2.y.a	✓	56	9.d	odd	6	1
234.2.y.a	✓	56	13.f	odd	12	1
234.2.z.a	yes	56	1.a	even	1	1	trivial
234.2.z.a	yes	56	117.bc	even	12	1	inner
702.2.bb.a		56	9.c	even	3	1
702.2.bb.a		56	39.k	even	12	1
702.2.bc.a		56	3.b	odd	2	1
702.2.bc.a		56	117.bb	odd	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(234, [\chi])\).