Properties

Label 7098.2.a.w
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

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

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.w 1
13.b even 2 1 546.2.a.d 1
39.d odd 2 1 1638.2.a.l 1
52.b odd 2 1 4368.2.a.l 1
91.b odd 2 1 3822.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.d 1 13.b even 2 1
1638.2.a.l 1 39.d odd 2 1
3822.2.a.a 1 91.b odd 2 1
4368.2.a.l 1 52.b odd 2 1
7098.2.a.w 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -1 + T \)
$5$ \( 3 + T \)
$7$ \( 1 + T \)
$11$ \( 3 + T \)
$13$ \( T \)
$17$ \( 3 + T \)
$19$ \( -7 + T \)
$23$ \( -9 + T \)
$29$ \( 9 + T \)
$31$ \( -4 + T \)
$37$ \( -7 + T \)
$41$ \( 12 + T \)
$43$ \( 1 + T \)
$47$ \( T \)
$53$ \( 6 + T \)
$59$ \( 12 + T \)
$61$ \( 1 + T \)
$67$ \( 14 + T \)
$71$ \( 12 + T \)
$73$ \( -7 + T \)
$79$ \( 10 + T \)
$83$ \( -6 + T \)
$89$ \( -6 + T \)
$97$ \( -10 + T \)
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