Properties

Label 1078.2.a.x
Level $1078$
Weight $2$
Character orbit 1078.a
Self dual yes
Analytic conductor $8.608$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{2} + ( 1 + \beta ) q^{3} + q^{4} + ( 2 - \beta ) q^{5} + ( 1 + \beta ) q^{6} + q^{8} + 2 \beta q^{9} +O(q^{10})\) \( q + q^{2} + ( 1 + \beta ) q^{3} + q^{4} + ( 2 - \beta ) q^{5} + ( 1 + \beta ) q^{6} + q^{8} + 2 \beta q^{9} + ( 2 - \beta ) q^{10} + q^{11} + ( 1 + \beta ) q^{12} + ( 1 + 2 \beta ) q^{13} + \beta q^{15} + q^{16} + ( 2 - 4 \beta ) q^{17} + 2 \beta q^{18} + ( 2 - \beta ) q^{19} + ( 2 - \beta ) q^{20} + q^{22} + ( -2 - 3 \beta ) q^{23} + ( 1 + \beta ) q^{24} + ( 1 - 4 \beta ) q^{25} + ( 1 + 2 \beta ) q^{26} + ( 1 - \beta ) q^{27} + ( -3 + 4 \beta ) q^{29} + \beta q^{30} + 4 q^{31} + q^{32} + ( 1 + \beta ) q^{33} + ( 2 - 4 \beta ) q^{34} + 2 \beta q^{36} + ( -8 - \beta ) q^{37} + ( 2 - \beta ) q^{38} + ( 5 + 3 \beta ) q^{39} + ( 2 - \beta ) q^{40} + ( 4 + \beta ) q^{41} -4 \beta q^{43} + q^{44} + ( -4 + 4 \beta ) q^{45} + ( -2 - 3 \beta ) q^{46} + ( 2 + 6 \beta ) q^{47} + ( 1 + \beta ) q^{48} + ( 1 - 4 \beta ) q^{50} + ( -6 - 2 \beta ) q^{51} + ( 1 + 2 \beta ) q^{52} + ( -2 + 7 \beta ) q^{53} + ( 1 - \beta ) q^{54} + ( 2 - \beta ) q^{55} + \beta q^{57} + ( -3 + 4 \beta ) q^{58} + ( 7 - \beta ) q^{59} + \beta q^{60} + ( -9 - 2 \beta ) q^{61} + 4 q^{62} + q^{64} + ( -2 + 3 \beta ) q^{65} + ( 1 + \beta ) q^{66} + ( 7 - 3 \beta ) q^{67} + ( 2 - 4 \beta ) q^{68} + ( -8 - 5 \beta ) q^{69} + ( -4 - 5 \beta ) q^{71} + 2 \beta q^{72} + ( 8 + \beta ) q^{73} + ( -8 - \beta ) q^{74} + ( -7 - 3 \beta ) q^{75} + ( 2 - \beta ) q^{76} + ( 5 + 3 \beta ) q^{78} + ( -9 - 3 \beta ) q^{79} + ( 2 - \beta ) q^{80} + ( -1 - 6 \beta ) q^{81} + ( 4 + \beta ) q^{82} + ( -2 + 10 \beta ) q^{83} + ( 12 - 10 \beta ) q^{85} -4 \beta q^{86} + ( 5 + \beta ) q^{87} + q^{88} + ( -4 - 6 \beta ) q^{89} + ( -4 + 4 \beta ) q^{90} + ( -2 - 3 \beta ) q^{92} + ( 4 + 4 \beta ) q^{93} + ( 2 + 6 \beta ) q^{94} + ( 6 - 4 \beta ) q^{95} + ( 1 + \beta ) q^{96} + ( 1 + 2 \beta ) q^{97} + 2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + 4 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{16} + 4 q^{17} + 4 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{26} + 2 q^{27} - 6 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{33} + 4 q^{34} - 16 q^{37} + 4 q^{38} + 10 q^{39} + 4 q^{40} + 8 q^{41} + 2 q^{44} - 8 q^{45} - 4 q^{46} + 4 q^{47} + 2 q^{48} + 2 q^{50} - 12 q^{51} + 2 q^{52} - 4 q^{53} + 2 q^{54} + 4 q^{55} - 6 q^{58} + 14 q^{59} - 18 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{65} + 2 q^{66} + 14 q^{67} + 4 q^{68} - 16 q^{69} - 8 q^{71} + 16 q^{73} - 16 q^{74} - 14 q^{75} + 4 q^{76} + 10 q^{78} - 18 q^{79} + 4 q^{80} - 2 q^{81} + 8 q^{82} - 4 q^{83} + 24 q^{85} + 10 q^{87} + 2 q^{88} - 8 q^{89} - 8 q^{90} - 4 q^{92} + 8 q^{93} + 4 q^{94} + 12 q^{95} + 2 q^{96} + 2 q^{97} + O(q^{100}) \)

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 −0.414214 1.00000 3.41421 −0.414214 0 1.00000 −2.82843 3.41421
1.2 1.00000 2.41421 1.00000 0.585786 2.41421 0 1.00000 2.82843 0.585786
\(n\): 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.x 2
3.b odd 2 1 9702.2.a.ch 2
4.b odd 2 1 8624.2.a.bh 2
7.b odd 2 1 1078.2.a.t 2
7.c even 3 2 1078.2.e.m 4
7.d odd 6 2 154.2.e.e 4
21.c even 2 1 9702.2.a.cx 2
21.g even 6 2 1386.2.k.t 4
28.d even 2 1 8624.2.a.cc 2
28.f even 6 2 1232.2.q.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.e 4 7.d odd 6 2
1078.2.a.t 2 7.b odd 2 1
1078.2.a.x 2 1.a even 1 1 trivial
1078.2.e.m 4 7.c even 3 2
1232.2.q.f 4 28.f even 6 2
1386.2.k.t 4 21.g even 6 2
8624.2.a.bh 2 4.b odd 2 1
8624.2.a.cc 2 28.d even 2 1
9702.2.a.ch 2 3.b odd 2 1
9702.2.a.cx 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1078))\):

\( T_{3}^{2} - 2 T_{3} - 1 \)
\( T_{5}^{2} - 4 T_{5} + 2 \)
\( T_{13}^{2} - 2 T_{13} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -1 - 2 T + T^{2} \)
$5$ \( 2 - 4 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -7 - 2 T + T^{2} \)
$17$ \( -28 - 4 T + T^{2} \)
$19$ \( 2 - 4 T + T^{2} \)
$23$ \( -14 + 4 T + T^{2} \)
$29$ \( -23 + 6 T + T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 62 + 16 T + T^{2} \)
$41$ \( 14 - 8 T + T^{2} \)
$43$ \( -32 + T^{2} \)
$47$ \( -68 - 4 T + T^{2} \)
$53$ \( -94 + 4 T + T^{2} \)
$59$ \( 47 - 14 T + T^{2} \)
$61$ \( 73 + 18 T + T^{2} \)
$67$ \( 31 - 14 T + T^{2} \)
$71$ \( -34 + 8 T + T^{2} \)
$73$ \( 62 - 16 T + T^{2} \)
$79$ \( 63 + 18 T + T^{2} \)
$83$ \( -196 + 4 T + T^{2} \)
$89$ \( -56 + 8 T + T^{2} \)
$97$ \( -7 - 2 T + T^{2} \)
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