Newform orbit 252.12.t.a

• Label: 252.12.t.a
• Level: $252$
• Weight: $12$
• Character orbit: 252.t
• Analytic conductor: $193.622$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	252.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(193.622481501\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 31278 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1		0			0			0		−6386.12	−	11061.1i		0		44274.2	−	4137.89i		0			0			0
17.2		0			0			0		−6313.77	−	10935.8i		0		−39874.6	−	19681.0i		0			0			0
17.3		0			0			0		−5672.42	−	9824.92i		0		44073.4	−	5904.51i		0			0			0
17.4		0			0			0		−5207.12	−	9018.99i		0		−35022.7	+	27399.6i		0			0			0
17.5		0			0			0		−4768.57	−	8259.41i		0		−1253.90	−	44449.5i		0			0			0
17.6		0			0			0		−4361.32	−	7554.02i		0		−11789.7	+	42875.7i		0			0			0
17.7		0			0			0		−3708.57	−	6423.43i		0		−40837.0	+	17597.3i		0			0			0
17.8		0			0			0		−2849.90	−	4936.17i		0		36467.2	+	25445.4i		0			0			0
17.9		0			0			0		−2700.97	−	4678.22i		0		4004.48	−	44286.5i		0			0			0
17.10		0			0			0		−2528.74	−	4379.91i		0		13916.4	−	42233.4i		0			0			0
17.11		0			0			0		−2163.02	−	3746.46i		0		41588.1	−	15740.3i		0			0			0
17.12		0			0			0		−1048.47	−	1816.01i		0		−42450.8	−	13238.5i		0			0			0
17.13		0			0			0		−687.929	−	1191.53i		0		−37416.0	−	24028.5i		0			0			0
17.14		0			0			0		−265.754	−	460.300i		0		26157.3	+	35960.0i		0			0			0
17.15		0			0			0		−11.5001	−	19.9187i		0		−9655.86	+	43406.1i		0			0			0
17.16		0			0			0		11.5001	+	19.9187i		0		−9655.86	+	43406.1i		0			0			0
17.17		0			0			0		265.754	+	460.300i		0		26157.3	+	35960.0i		0			0			0
17.18		0			0			0		687.929	+	1191.53i		0		−37416.0	−	24028.5i		0			0			0
17.19		0			0			0		1048.47	+	1816.01i		0		−42450.8	−	13238.5i		0			0			0
17.20		0			0			0		2163.02	+	3746.46i		0		41588.1	−	15740.3i		0			0			0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	252.12.t.a	✓	60
3.b	odd	2	1	inner	252.12.t.a	✓	60
7.d	odd	6	1	inner	252.12.t.a	✓	60
21.g	even	6	1	inner	252.12.t.a	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.12.t.a	✓	60	1.a	even	1	1	trivial
252.12.t.a	✓	60	3.b	odd	2	1	inner
252.12.t.a	✓	60	7.d	odd	6	1	inner
252.12.t.a	✓	60	21.g	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(252, [\chi])\).