Properties

Label 6889.2.a.a
Level $6889$
Weight $2$
Character orbit 6889.a
Self dual yes
Analytic conductor $55.009$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 6889 = 83^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6889.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.0089419525\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 83)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} - q^{4} + 2 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} - q^{4} + 2 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} - 2 q^{9} + 2 q^{10} + 3 q^{11} + q^{12} + 6 q^{13} - 3 q^{14} - 2 q^{15} - q^{16} + 5 q^{17} - 2 q^{18} - 2 q^{19} - 2 q^{20} + 3 q^{21} + 3 q^{22} - 4 q^{23} + 3 q^{24} - q^{25} + 6 q^{26} + 5 q^{27} + 3 q^{28} - 7 q^{29} - 2 q^{30} + 5 q^{31} + 5 q^{32} - 3 q^{33} + 5 q^{34} - 6 q^{35} + 2 q^{36} - 11 q^{37} - 2 q^{38} - 6 q^{39} - 6 q^{40} - 2 q^{41} + 3 q^{42} + 8 q^{43} - 3 q^{44} - 4 q^{45} - 4 q^{46} + q^{48} + 2 q^{49} - q^{50} - 5 q^{51} - 6 q^{52} - 6 q^{53} + 5 q^{54} + 6 q^{55} + 9 q^{56} + 2 q^{57} - 7 q^{58} + 5 q^{59} + 2 q^{60} + 5 q^{61} + 5 q^{62} + 6 q^{63} + 7 q^{64} + 12 q^{65} - 3 q^{66} + 2 q^{67} - 5 q^{68} + 4 q^{69} - 6 q^{70} - 2 q^{71} + 6 q^{72} - 11 q^{74} + q^{75} + 2 q^{76} - 9 q^{77} - 6 q^{78} - 14 q^{79} - 2 q^{80} + q^{81} - 2 q^{82} - 3 q^{84} + 10 q^{85} + 8 q^{86} + 7 q^{87} - 9 q^{88} - 4 q^{90} - 18 q^{91} + 4 q^{92} - 5 q^{93} - 4 q^{95} - 5 q^{96} + 8 q^{97} + 2 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −3.00000 −3.00000 −2.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(83\) \(-1\)

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 6889.2.a.a 1
83.b odd 2 1 83.2.a.a 1
249.d even 2 1 747.2.a.d 1
332.b even 2 1 1328.2.a.c 1
415.d odd 2 1 2075.2.a.d 1
581.b even 2 1 4067.2.a.a 1
664.e odd 2 1 5312.2.a.l 1
664.h even 2 1 5312.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.2.a.a 1 83.b odd 2 1
747.2.a.d 1 249.d even 2 1
1328.2.a.c 1 332.b even 2 1
2075.2.a.d 1 415.d odd 2 1
4067.2.a.a 1 581.b even 2 1
5312.2.a.h 1 664.h even 2 1
5312.2.a.l 1 664.e odd 2 1
6889.2.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6889))\). Copy content Toggle raw display

Hecke characteristic polynomials

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