Properties

Label 258.2.a.f
Level 258
Weight 2
Character orbit 258.a
Self dual yes
Analytic conductor 2.060
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 258 = 2 \cdot 3 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 258.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.06014037215\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} + 5q^{11} + q^{12} - 7q^{13} + q^{14} - q^{15} + q^{16} + 4q^{17} + q^{18} - q^{19} - q^{20} + q^{21} + 5q^{22} - 4q^{23} + q^{24} - 4q^{25} - 7q^{26} + q^{27} + q^{28} - 5q^{29} - q^{30} - 10q^{31} + q^{32} + 5q^{33} + 4q^{34} - q^{35} + q^{36} + 10q^{37} - q^{38} - 7q^{39} - q^{40} + q^{42} + q^{43} + 5q^{44} - q^{45} - 4q^{46} - q^{47} + q^{48} - 6q^{49} - 4q^{50} + 4q^{51} - 7q^{52} + 12q^{53} + q^{54} - 5q^{55} + q^{56} - q^{57} - 5q^{58} + 4q^{59} - q^{60} - 8q^{61} - 10q^{62} + q^{63} + q^{64} + 7q^{65} + 5q^{66} - 2q^{67} + 4q^{68} - 4q^{69} - q^{70} - 12q^{71} + q^{72} + 4q^{73} + 10q^{74} - 4q^{75} - q^{76} + 5q^{77} - 7q^{78} + 10q^{79} - q^{80} + q^{81} - 7q^{83} + q^{84} - 4q^{85} + q^{86} - 5q^{87} + 5q^{88} + 6q^{89} - q^{90} - 7q^{91} - 4q^{92} - 10q^{93} - q^{94} + q^{95} + q^{96} - 7q^{97} - 6q^{98} + 5q^{99} + O(q^{100}) \)

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

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 258.2.a.f 1
3.b odd 2 1 774.2.a.c 1
4.b odd 2 1 2064.2.a.c 1
5.b even 2 1 6450.2.a.e 1
8.b even 2 1 8256.2.a.n 1
8.d odd 2 1 8256.2.a.bl 1
12.b even 2 1 6192.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
258.2.a.f 1 1.a even 1 1 trivial
774.2.a.c 1 3.b odd 2 1
2064.2.a.c 1 4.b odd 2 1
6192.2.a.n 1 12.b even 2 1
6450.2.a.e 1 5.b even 2 1
8256.2.a.n 1 8.b even 2 1
8256.2.a.bl 1 8.d odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(43\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(258))\):

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

Hecke Characteristic Polynomials

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