Properties

Label 418.2.a.a
Level $418$
Weight $2$
Character orbit 418.a
Self dual yes
Analytic conductor $3.338$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.33774680449\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} - 3 q^{7} + q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} - 3 q^{7} + q^{8} - 2 q^{9} - 2 q^{10} - q^{11} - q^{12} + q^{13} - 3 q^{14} + 2 q^{15} + q^{16} - 7 q^{17} - 2 q^{18} + q^{19} - 2 q^{20} + 3 q^{21} - q^{22} - 5 q^{23} - q^{24} - q^{25} + q^{26} + 5 q^{27} - 3 q^{28} + q^{29} + 2 q^{30} + 10 q^{31} + q^{32} + q^{33} - 7 q^{34} + 6 q^{35} - 2 q^{36} - 6 q^{37} + q^{38} - q^{39} - 2 q^{40} + 6 q^{41} + 3 q^{42} - 4 q^{43} - q^{44} + 4 q^{45} - 5 q^{46} - q^{48} + 2 q^{49} - q^{50} + 7 q^{51} + q^{52} - q^{53} + 5 q^{54} + 2 q^{55} - 3 q^{56} - q^{57} + q^{58} + 3 q^{59} + 2 q^{60} - 12 q^{61} + 10 q^{62} + 6 q^{63} + q^{64} - 2 q^{65} + q^{66} + 3 q^{67} - 7 q^{68} + 5 q^{69} + 6 q^{70} - 10 q^{71} - 2 q^{72} + 3 q^{73} - 6 q^{74} + q^{75} + q^{76} + 3 q^{77} - q^{78} + 8 q^{79} - 2 q^{80} + q^{81} + 6 q^{82} + 8 q^{83} + 3 q^{84} + 14 q^{85} - 4 q^{86} - q^{87} - q^{88} - 8 q^{89} + 4 q^{90} - 3 q^{91} - 5 q^{92} - 10 q^{93} - 2 q^{95} - q^{96} + 8 q^{97} + 2 q^{98} + 2 q^{99}+O(q^{100}) \) 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 1.00000 −2.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(1\)
\(19\) \(-1\)

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 418.2.a.a 1
3.b odd 2 1 3762.2.a.g 1
4.b odd 2 1 3344.2.a.h 1
11.b odd 2 1 4598.2.a.b 1
19.b odd 2 1 7942.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.a 1 1.a even 1 1 trivial
3344.2.a.h 1 4.b odd 2 1
3762.2.a.g 1 3.b odd 2 1
4598.2.a.b 1 11.b odd 2 1
7942.2.a.i 1 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(418))\). 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
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T + 7 \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T + 5 \) Copy content Toggle raw display
$29$ \( T - 1 \) Copy content Toggle raw display
$31$ \( T - 10 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 1 \) Copy content Toggle raw display
$59$ \( T - 3 \) Copy content Toggle raw display
$61$ \( T + 12 \) Copy content Toggle raw display
$67$ \( T - 3 \) Copy content Toggle raw display
$71$ \( T + 10 \) Copy content Toggle raw display
$73$ \( T - 3 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T - 8 \) Copy content Toggle raw display
$89$ \( T + 8 \) Copy content Toggle raw display
$97$ \( T - 8 \) Copy content Toggle raw display
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