Properties

Label 1216.2.a.l
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 152)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - 3q^{7} - 2q^{9} + O(q^{10}) \) \( q + q^{3} - 3q^{7} - 2q^{9} + 2q^{11} - q^{13} - 5q^{17} + q^{19} - 3q^{21} + q^{23} - 5q^{25} - 5q^{27} + 3q^{29} - 4q^{31} + 2q^{33} - 2q^{37} - q^{39} - 8q^{41} - 8q^{43} + 8q^{47} + 2q^{49} - 5q^{51} - 9q^{53} + q^{57} + q^{59} - 14q^{61} + 6q^{63} + 13q^{67} + q^{69} - 10q^{71} + 9q^{73} - 5q^{75} - 6q^{77} + 10q^{79} + q^{81} + 10q^{83} + 3q^{87} - 12q^{89} + 3q^{91} - 4q^{93} + 14q^{97} - 4q^{99} + O(q^{100}) \)

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 1.00000 0 0 0 −3.00000 0 −2.00000 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.l 1
4.b odd 2 1 1216.2.a.f 1
8.b even 2 1 304.2.a.b 1
8.d odd 2 1 152.2.a.b 1
24.f even 2 1 1368.2.a.g 1
24.h odd 2 1 2736.2.a.k 1
40.e odd 2 1 3800.2.a.d 1
40.f even 2 1 7600.2.a.o 1
40.k even 4 2 3800.2.d.f 2
56.e even 2 1 7448.2.a.g 1
152.b even 2 1 2888.2.a.b 1
152.g odd 2 1 5776.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.a.b 1 8.d odd 2 1
304.2.a.b 1 8.b even 2 1
1216.2.a.f 1 4.b odd 2 1
1216.2.a.l 1 1.a even 1 1 trivial
1368.2.a.g 1 24.f even 2 1
2736.2.a.k 1 24.h odd 2 1
2888.2.a.b 1 152.b even 2 1
3800.2.a.d 1 40.e odd 2 1
3800.2.d.f 2 40.k even 4 2
5776.2.a.l 1 152.g odd 2 1
7448.2.a.g 1 56.e even 2 1
7600.2.a.o 1 40.f even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( 3 + T \)
$11$ \( -2 + T \)
$13$ \( 1 + T \)
$17$ \( 5 + T \)
$19$ \( -1 + T \)
$23$ \( -1 + T \)
$29$ \( -3 + T \)
$31$ \( 4 + T \)
$37$ \( 2 + T \)
$41$ \( 8 + T \)
$43$ \( 8 + T \)
$47$ \( -8 + T \)
$53$ \( 9 + T \)
$59$ \( -1 + T \)
$61$ \( 14 + T \)
$67$ \( -13 + T \)
$71$ \( 10 + T \)
$73$ \( -9 + T \)
$79$ \( -10 + T \)
$83$ \( -10 + T \)
$89$ \( 12 + T \)
$97$ \( -14 + T \)
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