Properties

Label 1078.2.a.b
Level $1078$
Weight $2$
Character orbit 1078.a
Self dual yes
Analytic conductor $8.608$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.60787333789\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - 2 q^{3} + q^{4} - 2 q^{5} + 2 q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - 2 q^{3} + q^{4} - 2 q^{5} + 2 q^{6} - q^{8} + q^{9} + 2 q^{10} + q^{11} - 2 q^{12} + 4 q^{13} + 4 q^{15} + q^{16} - q^{18} - 4 q^{19} - 2 q^{20} - q^{22} + 4 q^{23} + 2 q^{24} - q^{25} - 4 q^{26} + 4 q^{27} + 2 q^{29} - 4 q^{30} + 10 q^{31} - q^{32} - 2 q^{33} + q^{36} - 6 q^{37} + 4 q^{38} - 8 q^{39} + 2 q^{40} - 4 q^{43} + q^{44} - 2 q^{45} - 4 q^{46} - 10 q^{47} - 2 q^{48} + q^{50} + 4 q^{52} - 14 q^{53} - 4 q^{54} - 2 q^{55} + 8 q^{57} - 2 q^{58} - 10 q^{59} + 4 q^{60} + 8 q^{61} - 10 q^{62} + q^{64} - 8 q^{65} + 2 q^{66} + 8 q^{67} - 8 q^{69} - 4 q^{71} - q^{72} - 4 q^{73} + 6 q^{74} + 2 q^{75} - 4 q^{76} + 8 q^{78} + 16 q^{79} - 2 q^{80} - 11 q^{81} - 4 q^{83} + 4 q^{86} - 4 q^{87} - q^{88} - 10 q^{89} + 2 q^{90} + 4 q^{92} - 20 q^{93} + 10 q^{94} + 8 q^{95} + 2 q^{96} - 6 q^{97} + q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 1078.2.a.b 1
3.b odd 2 1 9702.2.a.bz 1
4.b odd 2 1 8624.2.a.z 1
7.b odd 2 1 154.2.a.b 1
7.c even 3 2 1078.2.e.l 2
7.d odd 6 2 1078.2.e.h 2
21.c even 2 1 1386.2.a.f 1
28.d even 2 1 1232.2.a.c 1
35.c odd 2 1 3850.2.a.o 1
35.f even 4 2 3850.2.c.d 2
56.e even 2 1 4928.2.a.bf 1
56.h odd 2 1 4928.2.a.d 1
77.b even 2 1 1694.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.b 1 7.b odd 2 1
1078.2.a.b 1 1.a even 1 1 trivial
1078.2.e.h 2 7.d odd 6 2
1078.2.e.l 2 7.c even 3 2
1232.2.a.c 1 28.d even 2 1
1386.2.a.f 1 21.c even 2 1
1694.2.a.i 1 77.b even 2 1
3850.2.a.o 1 35.c odd 2 1
3850.2.c.d 2 35.f even 4 2
4928.2.a.d 1 56.h odd 2 1
4928.2.a.bf 1 56.e even 2 1
8624.2.a.z 1 4.b odd 2 1
9702.2.a.bz 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1078))\):

\( T_{3} + 2 \)
\( T_{5} + 2 \)
\( T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 2 + T \)
$5$ \( 2 + T \)
$7$ \( T \)
$11$ \( -1 + T \)
$13$ \( -4 + T \)
$17$ \( T \)
$19$ \( 4 + T \)
$23$ \( -4 + T \)
$29$ \( -2 + T \)
$31$ \( -10 + T \)
$37$ \( 6 + T \)
$41$ \( T \)
$43$ \( 4 + T \)
$47$ \( 10 + T \)
$53$ \( 14 + T \)
$59$ \( 10 + T \)
$61$ \( -8 + T \)
$67$ \( -8 + T \)
$71$ \( 4 + T \)
$73$ \( 4 + T \)
$79$ \( -16 + T \)
$83$ \( 4 + T \)
$89$ \( 10 + T \)
$97$ \( 6 + T \)
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