Properties

Label 7098.2.a.a
Level $7098$
Weight $2$
Character orbit 7098.a
Self dual yes
Analytic conductor $56.678$
Analytic rank $2$
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: \(2\)
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

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

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 −3.00000 1.00000 1.00000 −1.00000 1.00000 3.00000
\(n\): 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.a 1
13.b even 2 1 7098.2.a.v 1
13.e even 6 2 546.2.l.b 2
39.h odd 6 2 1638.2.r.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.b 2 13.e even 6 2
1638.2.r.o 2 39.h odd 6 2
7098.2.a.a 1 1.a even 1 1 trivial
7098.2.a.v 1 13.b even 2 1

Hecke kernels

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

\( T_{5} + 3 \)
\( T_{11} + 4 \)
\( T_{17} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( 3 + T \)
$7$ \( -1 + T \)
$11$ \( 4 + T \)
$13$ \( T \)
$17$ \( 5 + T \)
$19$ \( 4 + T \)
$23$ \( 4 + T \)
$29$ \( 9 + T \)
$31$ \( T \)
$37$ \( 7 + T \)
$41$ \( 7 + T \)
$43$ \( -8 + T \)
$47$ \( 12 + T \)
$53$ \( -7 + T \)
$59$ \( -8 + T \)
$61$ \( -7 + T \)
$67$ \( 12 + T \)
$71$ \( 12 + T \)
$73$ \( -1 + T \)
$79$ \( 16 + T \)
$83$ \( -8 + T \)
$89$ \( -6 + T \)
$97$ \( 18 + T \)
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