Newform orbit 99.4.a.c

• Label: 99.4.a.c
• 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:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta - 1) q^{2} + ( - 2 \beta - 4) q^{4} + (8 \beta - 1) q^{5} + (4 \beta + 10) q^{7} + ( - 10 \beta + 6) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 8 q^{4} - 2 q^{5} + 20 q^{7} + 12 q^{8}+O(q^{10}) \)

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

−1.73205
1.73205

−2.73205

0

−0.535898

−14.8564

0

3.07180

23.3205

0

40.5885

1.2

0.732051

0

−7.46410

12.8564

0

16.9282

−11.3205

0

9.41154

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.c		2
3.b	odd	2	1		11.4.a.a	✓	2
4.b	odd	2	1		1584.4.a.bc		2
5.b	even	2	1		2475.4.a.q		2
11.b	odd	2	1		1089.4.a.v		2
12.b	even	2	1		176.4.a.i		2
15.d	odd	2	1		275.4.a.b		2
15.e	even	4	2		275.4.b.c		4
21.c	even	2	1		539.4.a.e		2
24.f	even	2	1		704.4.a.n		2
24.h	odd	2	1		704.4.a.p		2
33.d	even	2	1		121.4.a.c		2
33.f	even	10	4		121.4.c.f		8
33.h	odd	10	4		121.4.c.c		8
39.d	odd	2	1		1859.4.a.a		2
132.d	odd	2	1		1936.4.a.w		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
11.4.a.a	✓	2	3.b	odd	2	1
99.4.a.c		2	1.a	even	1	1	trivial
121.4.a.c		2	33.d	even	2	1
121.4.c.c		8	33.h	odd	10	4
121.4.c.f		8	33.f	even	10	4
176.4.a.i		2	12.b	even	2	1
275.4.a.b		2	15.d	odd	2	1
275.4.b.c		4	15.e	even	4	2
539.4.a.e		2	21.c	even	2	1
704.4.a.n		2	24.f	even	2	1
704.4.a.p		2	24.h	odd	2	1
1089.4.a.v		2	11.b	odd	2	1
1584.4.a.bc		2	4.b	odd	2	1
1859.4.a.a		2	39.d	odd	2	1
1936.4.a.w		2	132.d	odd	2	1
2475.4.a.q		2	5.b	even	2	1

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T - 2 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 2T - 191 \)
$7$	\( T^{2} - 20T + 52 \)
$11$	\( (T - 11)^{2} \)