Properties

Label 4026.2.a.h
Level 4026
Weight 2
Character orbit 4026.a
Self dual yes
Analytic conductor 32.148
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4026.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1477718538\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + 3q^{5} - q^{6} + 2q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} + 3q^{5} - q^{6} + 2q^{7} + q^{8} + q^{9} + 3q^{10} - q^{11} - q^{12} - 3q^{13} + 2q^{14} - 3q^{15} + q^{16} + q^{18} - 4q^{19} + 3q^{20} - 2q^{21} - q^{22} - 4q^{23} - q^{24} + 4q^{25} - 3q^{26} - q^{27} + 2q^{28} + 7q^{29} - 3q^{30} + 9q^{31} + q^{32} + q^{33} + 6q^{35} + q^{36} + 6q^{37} - 4q^{38} + 3q^{39} + 3q^{40} + 7q^{41} - 2q^{42} + 12q^{43} - q^{44} + 3q^{45} - 4q^{46} + 8q^{47} - q^{48} - 3q^{49} + 4q^{50} - 3q^{52} - q^{54} - 3q^{55} + 2q^{56} + 4q^{57} + 7q^{58} - 3q^{59} - 3q^{60} - q^{61} + 9q^{62} + 2q^{63} + q^{64} - 9q^{65} + q^{66} - 4q^{67} + 4q^{69} + 6q^{70} + 6q^{71} + q^{72} + 12q^{73} + 6q^{74} - 4q^{75} - 4q^{76} - 2q^{77} + 3q^{78} - 4q^{79} + 3q^{80} + q^{81} + 7q^{82} - 4q^{83} - 2q^{84} + 12q^{86} - 7q^{87} - q^{88} + 7q^{89} + 3q^{90} - 6q^{91} - 4q^{92} - 9q^{93} + 8q^{94} - 12q^{95} - q^{96} - 3q^{97} - 3q^{98} - q^{99} + O(q^{100}) \)

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
1.00000 −1.00000 1.00000 3.00000 −1.00000 2.00000 1.00000 1.00000 3.00000
\(n\): 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 4026.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.h 1 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(1\)
\(61\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5} - 3 \)
\( T_{7} - 2 \)
\( T_{13} + 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 + T \)
$5$ \( 1 - 3 T + 5 T^{2} \)
$7$ \( 1 - 2 T + 7 T^{2} \)
$11$ \( 1 + T \)
$13$ \( 1 + 3 T + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 4 T + 23 T^{2} \)
$29$ \( 1 - 7 T + 29 T^{2} \)
$31$ \( 1 - 9 T + 31 T^{2} \)
$37$ \( 1 - 6 T + 37 T^{2} \)
$41$ \( 1 - 7 T + 41 T^{2} \)
$43$ \( 1 - 12 T + 43 T^{2} \)
$47$ \( 1 - 8 T + 47 T^{2} \)
$53$ \( 1 + 53 T^{2} \)
$59$ \( 1 + 3 T + 59 T^{2} \)
$61$ \( 1 + T \)
$67$ \( 1 + 4 T + 67 T^{2} \)
$71$ \( 1 - 6 T + 71 T^{2} \)
$73$ \( 1 - 12 T + 73 T^{2} \)
$79$ \( 1 + 4 T + 79 T^{2} \)
$83$ \( 1 + 4 T + 83 T^{2} \)
$89$ \( 1 - 7 T + 89 T^{2} \)
$97$ \( 1 + 3 T + 97 T^{2} \)
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