Properties

Label 1216.2.a.b
Level $1216$
Weight $2$
Character orbit 1216.a
Self dual yes
Analytic conductor $9.710$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.70980888579\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{3} - 3q^{5} + q^{7} + q^{9} + O(q^{10}) \) \( q - 2q^{3} - 3q^{5} + q^{7} + q^{9} + 3q^{11} + 4q^{13} + 6q^{15} - 3q^{17} + q^{19} - 2q^{21} + 4q^{25} + 4q^{27} - 6q^{29} + 4q^{31} - 6q^{33} - 3q^{35} - 2q^{37} - 8q^{39} - 6q^{41} - q^{43} - 3q^{45} + 3q^{47} - 6q^{49} + 6q^{51} - 12q^{53} - 9q^{55} - 2q^{57} - 6q^{59} + q^{61} + q^{63} - 12q^{65} - 4q^{67} - 6q^{71} - 7q^{73} - 8q^{75} + 3q^{77} - 8q^{79} - 11q^{81} + 12q^{83} + 9q^{85} + 12q^{87} + 12q^{89} + 4q^{91} - 8q^{93} - 3q^{95} + 8q^{97} + 3q^{99} + O(q^{100}) \)

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
0
0 −2.00000 0 −3.00000 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

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 1216.2.a.b 1
4.b odd 2 1 1216.2.a.o 1
8.b even 2 1 304.2.a.f 1
8.d odd 2 1 19.2.a.a 1
24.f even 2 1 171.2.a.b 1
24.h odd 2 1 2736.2.a.c 1
40.e odd 2 1 475.2.a.b 1
40.f even 2 1 7600.2.a.c 1
40.k even 4 2 475.2.b.a 2
56.e even 2 1 931.2.a.a 1
56.k odd 6 2 931.2.f.c 2
56.m even 6 2 931.2.f.b 2
88.g even 2 1 2299.2.a.b 1
104.h odd 2 1 3211.2.a.a 1
120.m even 2 1 4275.2.a.i 1
136.e odd 2 1 5491.2.a.b 1
152.b even 2 1 361.2.a.b 1
152.g odd 2 1 5776.2.a.c 1
152.k odd 6 2 361.2.c.c 2
152.o even 6 2 361.2.c.a 2
152.u odd 18 6 361.2.e.d 6
152.v even 18 6 361.2.e.e 6
168.e odd 2 1 8379.2.a.j 1
456.l odd 2 1 3249.2.a.d 1
760.p even 2 1 9025.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.a.a 1 8.d odd 2 1
171.2.a.b 1 24.f even 2 1
304.2.a.f 1 8.b even 2 1
361.2.a.b 1 152.b even 2 1
361.2.c.a 2 152.o even 6 2
361.2.c.c 2 152.k odd 6 2
361.2.e.d 6 152.u odd 18 6
361.2.e.e 6 152.v even 18 6
475.2.a.b 1 40.e odd 2 1
475.2.b.a 2 40.k even 4 2
931.2.a.a 1 56.e even 2 1
931.2.f.b 2 56.m even 6 2
931.2.f.c 2 56.k odd 6 2
1216.2.a.b 1 1.a even 1 1 trivial
1216.2.a.o 1 4.b odd 2 1
2299.2.a.b 1 88.g even 2 1
2736.2.a.c 1 24.h odd 2 1
3211.2.a.a 1 104.h odd 2 1
3249.2.a.d 1 456.l odd 2 1
4275.2.a.i 1 120.m even 2 1
5491.2.a.b 1 136.e odd 2 1
5776.2.a.c 1 152.g odd 2 1
7600.2.a.c 1 40.f even 2 1
8379.2.a.j 1 168.e odd 2 1
9025.2.a.d 1 760.p even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1216))\):

\( T_{3} + 2 \)
\( T_{5} + 3 \)
\( T_{7} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 2 + T \)
$5$ \( 3 + T \)
$7$ \( -1 + T \)
$11$ \( -3 + T \)
$13$ \( -4 + T \)
$17$ \( 3 + T \)
$19$ \( -1 + T \)
$23$ \( T \)
$29$ \( 6 + T \)
$31$ \( -4 + T \)
$37$ \( 2 + T \)
$41$ \( 6 + T \)
$43$ \( 1 + T \)
$47$ \( -3 + T \)
$53$ \( 12 + T \)
$59$ \( 6 + T \)
$61$ \( -1 + T \)
$67$ \( 4 + T \)
$71$ \( 6 + T \)
$73$ \( 7 + T \)
$79$ \( 8 + T \)
$83$ \( -12 + T \)
$89$ \( -12 + T \)
$97$ \( -8 + T \)
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