Newform orbit 9200.2.a.ca

• Label: 9200.2.a.ca
• Level: $9200$
• Weight: $2$
• Character orbit: 9200.a
• Self dual: yes
• Analytic conductor: $73.462$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(73.4623698596\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{13}) \)
Defining polynomial:	\( x^{2} - x - 3 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b + 1) * q^3 + (-b + 2) * q^7 + (3*b + 1) * q^9 + (b + 3) * q^11 + (b - 2) * q^13 + (3*b - 3) * q^17 + (3*b - 2) * q^19 - q^21 - q^23 + (4*b + 7) * q^27 + 2*b * q^29 + (-3*b + 4) * q^31 + (5*b + 6) * q^33 - 8 * q^37 + q^39 + (-3*b - 3) * q^41 + (4*b - 4) * q^43 + 2*b * q^47 - 3*b * q^49 + (3*b + 6) * q^51 + (-4*b + 6) * q^53 + (4*b + 7) * q^57 + (2*b + 6) * q^59 + (-5*b + 5) * q^61 + (2*b - 7) * q^63 - 4 * q^67 + (-b - 1) * q^69 + (-b + 15) * q^71 + (-6*b - 2) * q^73 + (-2*b + 3) * q^77 + (-8*b + 4) * q^79 + (6*b + 16) * q^81 + (-4*b + 6) * q^83 + (4*b + 6) * q^87 + (3*b - 7) * q^91 + (-2*b - 5) * q^93 + (b - 5) * q^97 + (13*b + 12) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 3 * q^3 + 3 * q^7 + 5 * q^9 + 7 * q^11 - 3 * q^13 - 3 * q^17 - q^19 - 2 * q^21 - 2 * q^23 + 18 * q^27 + 2 * q^29 + 5 * q^31 + 17 * q^33 - 16 * q^37 + 2 * q^39 - 9 * q^41 - 4 * q^43 + 2 * q^47 - 3 * q^49 + 15 * q^51 + 8 * q^53 + 18 * q^57 + 14 * q^59 + 5 * q^61 - 12 * q^63 - 8 * q^67 - 3 * q^69 + 29 * q^71 - 10 * q^73 + 4 * q^77 + 38 * q^81 + 8 * q^83 + 16 * q^87 - 11 * q^91 - 12 * q^93 - 9 * q^97 + 37 * q^99

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−1.30278
2.30278

0

−0.302776

0

0

0

3.30278

0

−2.90833

0

1.2

0

3.30278

0

0

0

−0.302776

0

7.90833

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(5\)	\( +1 \)
\(23\)	\( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	9200.2.a.ca		2
4.b	odd	2	1		1150.2.a.m		2
5.b	even	2	1		1840.2.a.j		2
20.d	odd	2	1		230.2.a.b	✓	2
20.e	even	4	2		1150.2.b.f		4
40.e	odd	2	1		7360.2.a.bc		2
40.f	even	2	1		7360.2.a.bu		2
60.h	even	2	1		2070.2.a.w		2
460.g	even	2	1		5290.2.a.j		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
230.2.a.b	✓	2	20.d	odd	2	1
1150.2.a.m		2	4.b	odd	2	1
1150.2.b.f		4	20.e	even	4	2
1840.2.a.j		2	5.b	even	2	1
2070.2.a.w		2	60.h	even	2	1
5290.2.a.j		2	460.g	even	2	1
7360.2.a.bc		2	40.e	odd	2	1
7360.2.a.bu		2	40.f	even	2	1
9200.2.a.ca		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9200))\):

\( T_{3}^{2} - 3T_{3} - 1 \)

\( T_{7}^{2} - 3T_{7} - 1 \)

\( T_{11}^{2} - 7T_{11} + 9 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 3T - 1 \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 3T - 1 \)
$11$	\( T^{2} - 7T + 9 \)