Properties

Label 867.2.a.b
Level $867$
Weight $2$
Character orbit 867.a
Self dual yes
Analytic conductor $6.923$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.92302985525\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{3} - q^{4} - q^{6} + 4q^{7} + 3q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} - q^{4} - q^{6} + 4q^{7} + 3q^{8} + q^{9} - 4q^{11} - q^{12} + 2q^{13} - 4q^{14} - q^{16} - q^{18} + 4q^{19} + 4q^{21} + 4q^{22} + 4q^{23} + 3q^{24} - 5q^{25} - 2q^{26} + q^{27} - 4q^{28} - 4q^{31} - 5q^{32} - 4q^{33} - q^{36} + 8q^{37} - 4q^{38} + 2q^{39} + 8q^{41} - 4q^{42} + 4q^{43} + 4q^{44} - 4q^{46} - 8q^{47} - q^{48} + 9q^{49} + 5q^{50} - 2q^{52} - 6q^{53} - q^{54} + 12q^{56} + 4q^{57} + 12q^{59} + 8q^{61} + 4q^{62} + 4q^{63} + 7q^{64} + 4q^{66} + 12q^{67} + 4q^{69} + 12q^{71} + 3q^{72} - 8q^{74} - 5q^{75} - 4q^{76} - 16q^{77} - 2q^{78} + 4q^{79} + q^{81} - 8q^{82} - 12q^{83} - 4q^{84} - 4q^{86} - 12q^{88} - 10q^{89} + 8q^{91} - 4q^{92} - 4q^{93} + 8q^{94} - 5q^{96} + 16q^{97} - 9q^{98} - 4q^{99} + O(q^{100}) \)

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 0 −1.00000 4.00000 3.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(1\)

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 867.2.a.b 1
3.b odd 2 1 2601.2.a.j 1
17.b even 2 1 867.2.a.a 1
17.c even 4 2 51.2.d.b 2
17.d even 8 4 867.2.e.d 4
17.e odd 16 8 867.2.h.d 8
51.c odd 2 1 2601.2.a.i 1
51.f odd 4 2 153.2.d.a 2
68.f odd 4 2 816.2.c.c 2
85.f odd 4 2 1275.2.d.b 2
85.i odd 4 2 1275.2.d.d 2
85.j even 4 2 1275.2.g.a 2
136.i even 4 2 3264.2.c.e 2
136.j odd 4 2 3264.2.c.d 2
204.l even 4 2 2448.2.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b 2 17.c even 4 2
153.2.d.a 2 51.f odd 4 2
816.2.c.c 2 68.f odd 4 2
867.2.a.a 1 17.b even 2 1
867.2.a.b 1 1.a even 1 1 trivial
867.2.e.d 4 17.d even 8 4
867.2.h.d 8 17.e odd 16 8
1275.2.d.b 2 85.f odd 4 2
1275.2.d.d 2 85.i odd 4 2
1275.2.g.a 2 85.j even 4 2
2448.2.c.j 2 204.l even 4 2
2601.2.a.i 1 51.c odd 2 1
2601.2.a.j 1 3.b odd 2 1
3264.2.c.d 2 136.j odd 4 2
3264.2.c.e 2 136.i even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(867))\):

\( T_{2} + 1 \)
\( T_{5} \)
\( T_{7} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( -4 + T \)
$11$ \( 4 + T \)
$13$ \( -2 + T \)
$17$ \( T \)
$19$ \( -4 + T \)
$23$ \( -4 + T \)
$29$ \( T \)
$31$ \( 4 + T \)
$37$ \( -8 + T \)
$41$ \( -8 + T \)
$43$ \( -4 + T \)
$47$ \( 8 + T \)
$53$ \( 6 + T \)
$59$ \( -12 + T \)
$61$ \( -8 + T \)
$67$ \( -12 + T \)
$71$ \( -12 + T \)
$73$ \( T \)
$79$ \( -4 + T \)
$83$ \( 12 + T \)
$89$ \( 10 + T \)
$97$ \( -16 + T \)
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