Newform orbit 8001.2.a.x

• Label: 8001.2.a.x
• Level: $8001$
• Weight: $2$
• Character orbit: 8001.a
• Self dual: yes
• Analytic conductor: $63.888$
• Analytic rank: $1$
• Dimension: $22$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$8001 = 3^{2} \cdot 7 \cdot 127$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8001.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

$\operatorname{Tr}(f)(q) =$ 22 * q + 10 * q^4 + 22 * q^7 - 4 * q^10 - 10 * q^13 - 6 * q^16 - 18 * q^19 - 34 * q^22 - 26 * q^25 + 10 * q^28 - 42 * q^31 - 16 * q^34 - 36 * q^37 - 46 * q^40 - 54 * q^43 - 12 * q^46 + 22 * q^49 - 22 * q^52 - 28 * q^55 - 26 * q^58 - 22 * q^61 - 76 * q^64 - 48 * q^67 - 4 * q^70 - 48 * q^73 - 20 * q^76 - 94 * q^79 - 8 * q^82 - 40 * q^85 - 26 * q^88 - 10 * q^91 - 26 * q^94 - 56 * q^97

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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
1.1	−2.37730				0		3.65154			2.39021				0		1.00000			−3.92621				0		−5.68223
1.2	−2.32723				0		3.41602			1.82757				0		1.00000			−3.29542				0		−4.25318
1.3	−2.21969				0		2.92701			0.739015				0		1.00000			−2.05767				0		−1.64038
1.4	−1.94436				0		1.78053			−1.34248				0		1.00000			0.426737				0		2.61025
1.5	−1.69586				0		0.875937			−2.42277				0		1.00000			1.90625				0		4.10867
1.6	−1.56609				0		0.452631			−2.71234				0		1.00000			2.42332				0		4.24776
1.7	−0.775659				0		−1.39835			−0.833166				0		1.00000			2.63596				0		0.646253
1.8	−0.691771				0		−1.52145			−0.236679				0		1.00000			2.43604				0		0.163728
1.9	−0.684852				0		−1.53098			−0.182709				0		1.00000			2.41820				0		0.125129
1.10	−0.446366				0		−1.80076			2.33773				0		1.00000			1.69653				0		−1.04348
1.11	−0.384540				0		−1.85213			3.33517				0		1.00000			1.48130				0		−1.28251
1.12	0.384540				0		−1.85213			−3.33517				0		1.00000			−1.48130				0		−1.28251
1.13	0.446366				0		−1.80076			−2.33773				0		1.00000			−1.69653				0		−1.04348
1.14	0.684852				0		−1.53098			0.182709				0		1.00000			−2.41820				0		0.125129
1.15	0.691771				0		−1.52145			0.236679				0		1.00000			−2.43604				0		0.163728
1.16	0.775659				0		−1.39835			0.833166				0		1.00000			−2.63596				0		0.646253
1.17	1.56609				0		0.452631			2.71234				0		1.00000			−2.42332				0		4.24776
1.18	1.69586				0		0.875937			2.42277				0		1.00000			−1.90625				0		4.10867
1.19	1.94436				0		1.78053			1.34248				0		1.00000			−0.426737				0		2.61025
1.20	2.21969				0		2.92701			−0.739015				0		1.00000			2.05767				0		−1.64038

Atkin-Lehner signs

$p$	Sign
$3$	$+1$
$7$	$-1$
$127$	$-1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8001.2.a.x	✓	22
3.b	odd	2	1	inner	8001.2.a.x	✓	22

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8001.2.a.x	✓	22	1.a	even	1	1	trivial
8001.2.a.x	✓	22	3.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8001))$:

T2^22 - 27*T2^20 + 307*T2^18 - 1912*T2^16 + 7111*T2^14 - 16187*T2^12 + 22362*T2^10 - 18281*T2^8 + 8680*T2^6 - 2300*T2^4 + 308*T2^2 - 16

T5^22 - 42*T5^20 + 737*T5^18 - 7048*T5^16 + 40077*T5^14 - 138490*T5^12 + 285212*T5^10 - 330463*T5^8 + 194868*T5^6 - 49728*T5^4 + 3372*T5^2 - 64