Properties

Label 414.2.a.e
Level $414$
Weight $2$
Character orbit 414.a
Self dual yes
Analytic conductor $3.306$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta - 1) q^{5} + 2 q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta - 1) q^{5} + 2 q^{7} - q^{8} + ( - \beta + 1) q^{10} + ( - \beta + 1) q^{11} + 2 \beta q^{13} - 2 q^{14} + q^{16} + ( - 2 \beta + 2) q^{17} + ( - \beta + 3) q^{19} + (\beta - 1) q^{20} + (\beta - 1) q^{22} + q^{23} + ( - 2 \beta + 3) q^{25} - 2 \beta q^{26} + 2 q^{28} + (2 \beta + 4) q^{29} + ( - 2 \beta + 4) q^{31} - q^{32} + (2 \beta - 2) q^{34} + (2 \beta - 2) q^{35} + (3 \beta - 1) q^{37} + (\beta - 3) q^{38} + ( - \beta + 1) q^{40} + 6 q^{41} + ( - \beta + 3) q^{43} + ( - \beta + 1) q^{44} - q^{46} + 6 q^{47} - 3 q^{49} + (2 \beta - 3) q^{50} + 2 \beta q^{52} + ( - \beta + 1) q^{53} + (2 \beta - 8) q^{55} - 2 q^{56} + ( - 2 \beta - 4) q^{58} + ( - \beta + 3) q^{61} + (2 \beta - 4) q^{62} + q^{64} + ( - 2 \beta + 14) q^{65} + ( - 3 \beta - 7) q^{67} + ( - 2 \beta + 2) q^{68} + ( - 2 \beta + 2) q^{70} - 6 q^{71} + ( - 2 \beta - 2) q^{73} + ( - 3 \beta + 1) q^{74} + ( - \beta + 3) q^{76} + ( - 2 \beta + 2) q^{77} + (4 \beta - 2) q^{79} + (\beta - 1) q^{80} - 6 q^{82} + ( - \beta - 11) q^{83} + (4 \beta - 16) q^{85} + (\beta - 3) q^{86} + (\beta - 1) q^{88} + 4 \beta q^{91} + q^{92} - 6 q^{94} + (4 \beta - 10) q^{95} + ( - 2 \beta + 4) q^{97} + 3 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} + 2 q^{11} - 4 q^{14} + 2 q^{16} + 4 q^{17} + 6 q^{19} - 2 q^{20} - 2 q^{22} + 2 q^{23} + 6 q^{25} + 4 q^{28} + 8 q^{29} + 8 q^{31} - 2 q^{32} - 4 q^{34} - 4 q^{35} - 2 q^{37} - 6 q^{38} + 2 q^{40} + 12 q^{41} + 6 q^{43} + 2 q^{44} - 2 q^{46} + 12 q^{47} - 6 q^{49} - 6 q^{50} + 2 q^{53} - 16 q^{55} - 4 q^{56} - 8 q^{58} + 6 q^{61} - 8 q^{62} + 2 q^{64} + 28 q^{65} - 14 q^{67} + 4 q^{68} + 4 q^{70} - 12 q^{71} - 4 q^{73} + 2 q^{74} + 6 q^{76} + 4 q^{77} - 4 q^{79} - 2 q^{80} - 12 q^{82} - 22 q^{83} - 32 q^{85} - 6 q^{86} - 2 q^{88} + 2 q^{92} - 12 q^{94} - 20 q^{95} + 8 q^{97} + 6 q^{98}+O(q^{100}) \) 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
−2.64575
2.64575
−1.00000 0 1.00000 −3.64575 0 2.00000 −1.00000 0 3.64575
1.2 −1.00000 0 1.00000 1.64575 0 2.00000 −1.00000 0 −1.64575
\(n\): 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.e 2
3.b odd 2 1 414.2.a.g yes 2
4.b odd 2 1 3312.2.a.v 2
12.b even 2 1 3312.2.a.z 2
23.b odd 2 1 9522.2.a.bb 2
69.c even 2 1 9522.2.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.a.e 2 1.a even 1 1 trivial
414.2.a.g yes 2 3.b odd 2 1
3312.2.a.v 2 4.b odd 2 1
3312.2.a.z 2 12.b even 2 1
9522.2.a.bb 2 23.b odd 2 1
9522.2.a.bc 2 69.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2T_{5} - 6 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(414))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 6 \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 6 \) Copy content Toggle raw display
$13$ \( T^{2} - 28 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 24 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 2 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 8T - 12 \) Copy content Toggle raw display
$31$ \( T^{2} - 8T - 12 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 62 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 6T + 2 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 2T - 6 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 6T + 2 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T - 14 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 24 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 108 \) Copy content Toggle raw display
$83$ \( T^{2} + 22T + 114 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 12 \) Copy content Toggle raw display
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