Properties

Label 7098.2.a.r
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} + q^{8} + q^{9} - 2q^{10} + q^{11} - q^{12} + q^{14} + 2q^{15} + q^{16} - q^{17} + q^{18} - q^{19} - 2q^{20} - q^{21} + q^{22} - 6q^{23} - q^{24} - q^{25} - q^{27} + q^{28} + 9q^{29} + 2q^{30} - 2q^{31} + q^{32} - q^{33} - q^{34} - 2q^{35} + q^{36} + 2q^{37} - q^{38} - 2q^{40} - 5q^{41} - q^{42} - 2q^{43} + q^{44} - 2q^{45} - 6q^{46} + 7q^{47} - q^{48} + q^{49} - q^{50} + q^{51} - q^{53} - q^{54} - 2q^{55} + q^{56} + q^{57} + 9q^{58} - 10q^{59} + 2q^{60} - 11q^{61} - 2q^{62} + q^{63} + q^{64} - q^{66} - 2q^{67} - q^{68} + 6q^{69} - 2q^{70} + 8q^{71} + q^{72} - 8q^{73} + 2q^{74} + q^{75} - q^{76} + q^{77} + 11q^{79} - 2q^{80} + q^{81} - 5q^{82} + 12q^{83} - q^{84} + 2q^{85} - 2q^{86} - 9q^{87} + q^{88} - 11q^{89} - 2q^{90} - 6q^{92} + 2q^{93} + 7q^{94} + 2q^{95} - q^{96} + 14q^{97} + q^{98} + q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(1\)

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 7098.2.a.r 1
13.b even 2 1 7098.2.a.e 1
13.e even 6 2 546.2.l.g 2
39.h odd 6 2 1638.2.r.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.g 2 13.e even 6 2
1638.2.r.d 2 39.h odd 6 2
7098.2.a.e 1 13.b even 2 1
7098.2.a.r 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

\( T_{5} + 2 \)
\( T_{11} - 1 \)
\( T_{17} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( 2 + T \)
$7$ \( -1 + T \)
$11$ \( -1 + T \)
$13$ \( T \)
$17$ \( 1 + T \)
$19$ \( 1 + T \)
$23$ \( 6 + T \)
$29$ \( -9 + T \)
$31$ \( 2 + T \)
$37$ \( -2 + T \)
$41$ \( 5 + T \)
$43$ \( 2 + T \)
$47$ \( -7 + T \)
$53$ \( 1 + T \)
$59$ \( 10 + T \)
$61$ \( 11 + T \)
$67$ \( 2 + T \)
$71$ \( -8 + T \)
$73$ \( 8 + T \)
$79$ \( -11 + T \)
$83$ \( -12 + T \)
$89$ \( 11 + T \)
$97$ \( -14 + T \)
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