Properties

Label 1078.2.a.s
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: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta q^{3} + q^{4} + ( 1 - \beta ) q^{5} -\beta q^{6} - q^{8} + 4 q^{9} +O(q^{10})\) \( q - q^{2} + \beta q^{3} + q^{4} + ( 1 - \beta ) q^{5} -\beta q^{6} - q^{8} + 4 q^{9} + ( -1 + \beta ) q^{10} + q^{11} + \beta q^{12} + 5 q^{13} + ( -7 + \beta ) q^{15} + q^{16} + 6 q^{17} -4 q^{18} + ( -3 - \beta ) q^{19} + ( 1 - \beta ) q^{20} - q^{22} + ( -1 + \beta ) q^{23} -\beta q^{24} + ( 3 - 2 \beta ) q^{25} -5 q^{26} + \beta q^{27} + ( 1 + 2 \beta ) q^{29} + ( 7 - \beta ) q^{30} -4 q^{31} - q^{32} + \beta q^{33} -6 q^{34} + 4 q^{36} + ( 1 + \beta ) q^{37} + ( 3 + \beta ) q^{38} + 5 \beta q^{39} + ( -1 + \beta ) q^{40} + ( 3 + 3 \beta ) q^{41} -4 q^{43} + q^{44} + ( 4 - 4 \beta ) q^{45} + ( 1 - \beta ) q^{46} + ( 8 - 2 \beta ) q^{47} + \beta q^{48} + ( -3 + 2 \beta ) q^{50} + 6 \beta q^{51} + 5 q^{52} + ( -1 + \beta ) q^{53} -\beta q^{54} + ( 1 - \beta ) q^{55} + ( -7 - 3 \beta ) q^{57} + ( -1 - 2 \beta ) q^{58} + ( 2 + \beta ) q^{59} + ( -7 + \beta ) q^{60} + ( 9 + 2 \beta ) q^{61} + 4 q^{62} + q^{64} + ( 5 - 5 \beta ) q^{65} -\beta q^{66} + ( -4 - 3 \beta ) q^{67} + 6 q^{68} + ( 7 - \beta ) q^{69} + ( 7 - \beta ) q^{71} -4 q^{72} + ( 3 - \beta ) q^{73} + ( -1 - \beta ) q^{74} + ( -14 + 3 \beta ) q^{75} + ( -3 - \beta ) q^{76} -5 \beta q^{78} -\beta q^{79} + ( 1 - \beta ) q^{80} -5 q^{81} + ( -3 - 3 \beta ) q^{82} + ( 8 - 2 \beta ) q^{83} + ( 6 - 6 \beta ) q^{85} + 4 q^{86} + ( 14 + \beta ) q^{87} - q^{88} + ( -4 + 4 \beta ) q^{89} + ( -4 + 4 \beta ) q^{90} + ( -1 + \beta ) q^{92} -4 \beta q^{93} + ( -8 + 2 \beta ) q^{94} + ( 4 + 2 \beta ) q^{95} -\beta q^{96} + ( -11 - 2 \beta ) q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} - 2q^{8} + 8q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} - 2q^{8} + 8q^{9} - 2q^{10} + 2q^{11} + 10q^{13} - 14q^{15} + 2q^{16} + 12q^{17} - 8q^{18} - 6q^{19} + 2q^{20} - 2q^{22} - 2q^{23} + 6q^{25} - 10q^{26} + 2q^{29} + 14q^{30} - 8q^{31} - 2q^{32} - 12q^{34} + 8q^{36} + 2q^{37} + 6q^{38} - 2q^{40} + 6q^{41} - 8q^{43} + 2q^{44} + 8q^{45} + 2q^{46} + 16q^{47} - 6q^{50} + 10q^{52} - 2q^{53} + 2q^{55} - 14q^{57} - 2q^{58} + 4q^{59} - 14q^{60} + 18q^{61} + 8q^{62} + 2q^{64} + 10q^{65} - 8q^{67} + 12q^{68} + 14q^{69} + 14q^{71} - 8q^{72} + 6q^{73} - 2q^{74} - 28q^{75} - 6q^{76} + 2q^{80} - 10q^{81} - 6q^{82} + 16q^{83} + 12q^{85} + 8q^{86} + 28q^{87} - 2q^{88} - 8q^{89} - 8q^{90} - 2q^{92} - 16q^{94} + 8q^{95} - 22q^{97} + 8q^{99} + O(q^{100}) \)

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
−2.64575
2.64575
−1.00000 −2.64575 1.00000 3.64575 2.64575 0 −1.00000 4.00000 −3.64575
1.2 −1.00000 2.64575 1.00000 −1.64575 −2.64575 0 −1.00000 4.00000 1.64575
\(n\): 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.s 2
3.b odd 2 1 9702.2.a.cz 2
4.b odd 2 1 8624.2.a.ca 2
7.b odd 2 1 1078.2.a.n 2
7.c even 3 2 154.2.e.f 4
7.d odd 6 2 1078.2.e.v 4
21.c even 2 1 9702.2.a.dr 2
21.h odd 6 2 1386.2.k.s 4
28.d even 2 1 8624.2.a.bk 2
28.g odd 6 2 1232.2.q.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.f 4 7.c even 3 2
1078.2.a.n 2 7.b odd 2 1
1078.2.a.s 2 1.a even 1 1 trivial
1078.2.e.v 4 7.d odd 6 2
1232.2.q.g 4 28.g odd 6 2
1386.2.k.s 4 21.h odd 6 2
8624.2.a.bk 2 28.d even 2 1
8624.2.a.ca 2 4.b odd 2 1
9702.2.a.cz 2 3.b odd 2 1
9702.2.a.dr 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1078))\):

\( T_{3}^{2} - 7 \)
\( T_{5}^{2} - 2 T_{5} - 6 \)
\( T_{13} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -7 + T^{2} \)
$5$ \( -6 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( ( -5 + T )^{2} \)
$17$ \( ( -6 + T )^{2} \)
$19$ \( 2 + 6 T + T^{2} \)
$23$ \( -6 + 2 T + T^{2} \)
$29$ \( -27 - 2 T + T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( -6 - 2 T + T^{2} \)
$41$ \( -54 - 6 T + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( 36 - 16 T + T^{2} \)
$53$ \( -6 + 2 T + T^{2} \)
$59$ \( -3 - 4 T + T^{2} \)
$61$ \( 53 - 18 T + T^{2} \)
$67$ \( -47 + 8 T + T^{2} \)
$71$ \( 42 - 14 T + T^{2} \)
$73$ \( 2 - 6 T + T^{2} \)
$79$ \( -7 + T^{2} \)
$83$ \( 36 - 16 T + T^{2} \)
$89$ \( -96 + 8 T + T^{2} \)
$97$ \( 93 + 22 T + T^{2} \)
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