Properties

Label 414.2.a.c
Level $414$
Weight $2$
Character orbit 414.a
Self dual yes
Analytic conductor $3.306$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.30580664368\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + 2 q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + 2 q^{7} + q^{8} + 2 q^{13} + 2 q^{14} + q^{16} + 2 q^{19} + q^{23} - 5 q^{25} + 2 q^{26} + 2 q^{28} + 6 q^{29} - 4 q^{31} + q^{32} - 10 q^{37} + 2 q^{38} + 6 q^{41} + 2 q^{43} + q^{46} - 3 q^{49} - 5 q^{50} + 2 q^{52} - 12 q^{53} + 2 q^{56} + 6 q^{58} - 12 q^{59} - 10 q^{61} - 4 q^{62} + q^{64} + 14 q^{67} + 2 q^{73} - 10 q^{74} + 2 q^{76} - 10 q^{79} + 6 q^{82} + 2 q^{86} - 12 q^{89} + 4 q^{91} + q^{92} - 10 q^{97} - 3 q^{98}+O(q^{100}) \) 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 0 1.00000 0 0 2.00000 1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(-1\)

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 414.2.a.c 1
3.b odd 2 1 138.2.a.b 1
4.b odd 2 1 3312.2.a.h 1
12.b even 2 1 1104.2.a.b 1
15.d odd 2 1 3450.2.a.o 1
15.e even 4 2 3450.2.d.g 2
21.c even 2 1 6762.2.a.g 1
23.b odd 2 1 9522.2.a.k 1
24.f even 2 1 4416.2.a.t 1
24.h odd 2 1 4416.2.a.i 1
69.c even 2 1 3174.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.b 1 3.b odd 2 1
414.2.a.c 1 1.a even 1 1 trivial
1104.2.a.b 1 12.b even 2 1
3174.2.a.d 1 69.c even 2 1
3312.2.a.h 1 4.b odd 2 1
3450.2.a.o 1 15.d odd 2 1
3450.2.d.g 2 15.e even 4 2
4416.2.a.i 1 24.h odd 2 1
4416.2.a.t 1 24.f even 2 1
6762.2.a.g 1 21.c even 2 1
9522.2.a.k 1 23.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(414))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T - 1 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 2 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 12 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T - 14 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T + 10 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 12 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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