Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta q^{5} - q^{8} -3 q^{9} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta q^{5} - q^{8} -3 q^{9} -\beta q^{10} + q^{11} -2 \beta q^{13} + q^{16} -\beta q^{17} + 3 q^{18} -3 \beta q^{19} + \beta q^{20} - q^{22} -8 q^{23} + 3 q^{25} + 2 \beta q^{26} -6 q^{29} + 3 \beta q^{31} - q^{32} + \beta q^{34} -3 q^{36} -6 q^{37} + 3 \beta q^{38} -\beta q^{40} + 3 \beta q^{41} -4 q^{43} + q^{44} -3 \beta q^{45} + 8 q^{46} + \beta q^{47} -3 q^{50} -2 \beta q^{52} + 6 q^{53} + \beta q^{55} + 6 q^{58} + 2 \beta q^{59} + 2 \beta q^{61} -3 \beta q^{62} + q^{64} -16 q^{65} -4 q^{67} -\beta q^{68} + 3 q^{72} + 3 \beta q^{73} + 6 q^{74} -3 \beta q^{76} + \beta q^{80} + 9 q^{81} -3 \beta q^{82} + \beta q^{83} -8 q^{85} + 4 q^{86} - q^{88} -4 \beta q^{89} + 3 \beta q^{90} -8 q^{92} -\beta q^{94} -24 q^{95} -4 \beta q^{97} -3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} - 6q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} - 6q^{9} + 2q^{11} + 2q^{16} + 6q^{18} - 2q^{22} - 16q^{23} + 6q^{25} - 12q^{29} - 2q^{32} - 6q^{36} - 12q^{37} - 8q^{43} + 2q^{44} + 16q^{46} - 6q^{50} + 12q^{53} + 12q^{58} + 2q^{64} - 32q^{65} - 8q^{67} + 6q^{72} + 12q^{74} + 18q^{81} - 16q^{85} + 8q^{86} - 2q^{88} - 16q^{92} - 48q^{95} - 6q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
−1.00000 0 1.00000 −2.82843 0 0 −1.00000 −3.00000 2.82843
1.2 −1.00000 0 1.00000 2.82843 0 0 −1.00000 −3.00000 −2.82843
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.a.o 2
3.b odd 2 1 9702.2.a.dk 2
4.b odd 2 1 8624.2.a.bo 2
7.b odd 2 1 inner 1078.2.a.o 2
7.c even 3 2 1078.2.e.u 4
7.d odd 6 2 1078.2.e.u 4
21.c even 2 1 9702.2.a.dk 2
28.d even 2 1 8624.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.o 2 1.a even 1 1 trivial
1078.2.a.o 2 7.b odd 2 1 inner
1078.2.e.u 4 7.c even 3 2
1078.2.e.u 4 7.d odd 6 2
8624.2.a.bo 2 4.b odd 2 1
8624.2.a.bo 2 28.d even 2 1
9702.2.a.dk 2 3.b odd 2 1
9702.2.a.dk 2 21.c even 2 1

Hecke kernels

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

\( T_{3} \)
\( T_{5}^{2} - 8 \)
\( T_{13}^{2} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -8 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -32 + T^{2} \)
$17$ \( -8 + T^{2} \)
$19$ \( -72 + T^{2} \)
$23$ \( ( 8 + T )^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( -72 + T^{2} \)
$37$ \( ( 6 + T )^{2} \)
$41$ \( -72 + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( -8 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( -32 + T^{2} \)
$61$ \( -32 + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( -72 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( -8 + T^{2} \)
$89$ \( -128 + T^{2} \)
$97$ \( -128 + T^{2} \)
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