Properties

Label 6909.2.a.k
Level $6909$
Weight $2$
Character orbit 6909.a
Self dual yes
Analytic conductor $55.169$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6909 = 3 \cdot 7^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6909.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.1686427565\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 141)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
2.00000 −1.00000 2.00000 1.00000 −2.00000 0 0 1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(47\) \(-1\)

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 6909.2.a.k 1
7.b odd 2 1 141.2.a.e 1
21.c even 2 1 423.2.a.b 1
28.d even 2 1 2256.2.a.e 1
35.c odd 2 1 3525.2.a.c 1
56.e even 2 1 9024.2.a.bq 1
56.h odd 2 1 9024.2.a.n 1
84.h odd 2 1 6768.2.a.n 1
329.c even 2 1 6627.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.2.a.e 1 7.b odd 2 1
423.2.a.b 1 21.c even 2 1
2256.2.a.e 1 28.d even 2 1
3525.2.a.c 1 35.c odd 2 1
6627.2.a.i 1 329.c even 2 1
6768.2.a.n 1 84.h odd 2 1
6909.2.a.k 1 1.a even 1 1 trivial
9024.2.a.n 1 56.h odd 2 1
9024.2.a.bq 1 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6909))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( 1 + T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( -1 + T \)
$13$ \( -2 + T \)
$17$ \( 2 + T \)
$19$ \( 6 + T \)
$23$ \( -3 + T \)
$29$ \( -3 + T \)
$31$ \( 2 + T \)
$37$ \( 7 + T \)
$41$ \( 10 + T \)
$43$ \( 10 + T \)
$47$ \( -1 + T \)
$53$ \( -4 + T \)
$59$ \( 8 + T \)
$61$ \( -10 + T \)
$67$ \( -10 + T \)
$71$ \( 14 + T \)
$73$ \( -10 + T \)
$79$ \( -17 + T \)
$83$ \( 8 + T \)
$89$ \( 6 + T \)
$97$ \( 1 + T \)
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