Newform orbit 99.4.a.f

• Label: 99.4.a.f
• Level: $99$
• Weight: $4$
• Character orbit: 99.a
• Self dual: yes
• Analytic conductor: $5.841$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	99.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(5.84118909057\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{97}) \)
Defining polynomial:	\( x^{2} - x - 24 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{97})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{2} + (\beta + 16) q^{4} + (2 \beta + 6) q^{5} + ( - 4 \beta + 14) q^{7} + ( - 9 \beta - 24) q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 33 q^{4} + 14 q^{5} + 24 q^{7} - 57 q^{8}+O(q^{10}) \)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

5.42443
−4.42443

−5.42443

0

21.4244

16.8489

0

−7.69772

−72.8199

0

−91.3954

1.2

4.42443

0

11.5756

−2.84886

0

31.6977

15.8199

0

−12.6046

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(11\)	\(-1\)

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	99.4.a.f		2
3.b	odd	2	1		33.4.a.c	✓	2
4.b	odd	2	1		1584.4.a.bj		2
5.b	even	2	1		2475.4.a.p		2
11.b	odd	2	1		1089.4.a.u		2
12.b	even	2	1		528.4.a.p		2
15.d	odd	2	1		825.4.a.l		2
15.e	even	4	2		825.4.c.h		4
21.c	even	2	1		1617.4.a.k		2
24.f	even	2	1		2112.4.a.bg		2
24.h	odd	2	1		2112.4.a.bn		2
33.d	even	2	1		363.4.a.i		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.4.a.c	✓	2	3.b	odd	2	1
99.4.a.f		2	1.a	even	1	1	trivial
363.4.a.i		2	33.d	even	2	1
528.4.a.p		2	12.b	even	2	1
825.4.a.l		2	15.d	odd	2	1
825.4.c.h		4	15.e	even	4	2
1089.4.a.u		2	11.b	odd	2	1
1584.4.a.bj		2	4.b	odd	2	1
1617.4.a.k		2	21.c	even	2	1
2112.4.a.bg		2	24.f	even	2	1
2112.4.a.bn		2	24.h	odd	2	1
2475.4.a.p		2	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 24 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(99))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T - 24 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 14T - 48 \)
$7$	\( T^{2} - 24T - 244 \)
$11$	\( (T - 11)^{2} \)