Properties

Label 405.2.a.g
Level $405$
Weight $2$
Character orbit 405.a
Self dual yes
Analytic conductor $3.234$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.23394128186\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + ( - 2 \beta + 2) q^{4} + q^{5} + ( - \beta - 3) q^{7} + (2 \beta - 6) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + ( - 2 \beta + 2) q^{4} + q^{5} + ( - \beta - 3) q^{7} + (2 \beta - 6) q^{8} + (\beta - 1) q^{10} + ( - \beta - 4) q^{11} + (2 \beta - 2) q^{13} - 2 \beta q^{14} + ( - 4 \beta + 8) q^{16} + ( - \beta - 1) q^{17} + (2 \beta + 1) q^{19} + ( - 2 \beta + 2) q^{20} + ( - 3 \beta + 1) q^{22} - 2 \beta q^{23} + q^{25} + ( - 4 \beta + 8) q^{26} + 4 \beta q^{28} + (3 \beta - 2) q^{29} - 3 q^{31} + (8 \beta - 8) q^{32} - 2 q^{34} + ( - \beta - 3) q^{35} + ( - \beta - 1) q^{37} + ( - \beta + 5) q^{38} + (2 \beta - 6) q^{40} + ( - 3 \beta - 2) q^{41} + (3 \beta - 5) q^{43} + (6 \beta - 2) q^{44} + (2 \beta - 6) q^{46} + ( - \beta - 7) q^{47} + (6 \beta + 5) q^{49} + (\beta - 1) q^{50} + (8 \beta - 16) q^{52} + (\beta + 5) q^{53} + ( - \beta - 4) q^{55} + 12 q^{56} + ( - 5 \beta + 11) q^{58} + (\beta - 10) q^{59} + 4 q^{61} + ( - 3 \beta + 3) q^{62} + ( - 8 \beta + 16) q^{64} + (2 \beta - 2) q^{65} + 2 \beta q^{67} + 4 q^{68} - 2 \beta q^{70} + ( - \beta - 2) q^{71} + ( - 5 \beta + 1) q^{73} - 2 q^{74} + (2 \beta - 10) q^{76} + (7 \beta + 15) q^{77} + (2 \beta + 12) q^{79} + ( - 4 \beta + 8) q^{80} + (\beta - 7) q^{82} + (3 \beta - 3) q^{83} + ( - \beta - 1) q^{85} + ( - 8 \beta + 14) q^{86} + ( - 2 \beta + 18) q^{88} + 3 \beta q^{89} - 4 \beta q^{91} + ( - 4 \beta + 12) q^{92} + ( - 6 \beta + 4) q^{94} + (2 \beta + 1) q^{95} + ( - 5 \beta - 1) q^{97} + ( - \beta + 13) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} - 6 q^{7} - 12 q^{8} - 2 q^{10} - 8 q^{11} - 4 q^{13} + 16 q^{16} - 2 q^{17} + 2 q^{19} + 4 q^{20} + 2 q^{22} + 2 q^{25} + 16 q^{26} - 4 q^{29} - 6 q^{31} - 16 q^{32} - 4 q^{34} - 6 q^{35} - 2 q^{37} + 10 q^{38} - 12 q^{40} - 4 q^{41} - 10 q^{43} - 4 q^{44} - 12 q^{46} - 14 q^{47} + 10 q^{49} - 2 q^{50} - 32 q^{52} + 10 q^{53} - 8 q^{55} + 24 q^{56} + 22 q^{58} - 20 q^{59} + 8 q^{61} + 6 q^{62} + 32 q^{64} - 4 q^{65} + 8 q^{68} - 4 q^{71} + 2 q^{73} - 4 q^{74} - 20 q^{76} + 30 q^{77} + 24 q^{79} + 16 q^{80} - 14 q^{82} - 6 q^{83} - 2 q^{85} + 28 q^{86} + 36 q^{88} + 24 q^{92} + 8 q^{94} + 2 q^{95} - 2 q^{97} + 26 q^{98}+O(q^{100}) \) 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
−1.73205
1.73205
−2.73205 0 5.46410 1.00000 0 −1.26795 −9.46410 0 −2.73205
1.2 0.732051 0 −1.46410 1.00000 0 −4.73205 −2.53590 0 0.732051
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)

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 405.2.a.g 2
3.b odd 2 1 405.2.a.h yes 2
4.b odd 2 1 6480.2.a.br 2
5.b even 2 1 2025.2.a.m 2
5.c odd 4 2 2025.2.b.g 4
9.c even 3 2 405.2.e.l 4
9.d odd 6 2 405.2.e.i 4
12.b even 2 1 6480.2.a.bi 2
15.d odd 2 1 2025.2.a.g 2
15.e even 4 2 2025.2.b.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.g 2 1.a even 1 1 trivial
405.2.a.h yes 2 3.b odd 2 1
405.2.e.i 4 9.d odd 6 2
405.2.e.l 4 9.c even 3 2
2025.2.a.g 2 15.d odd 2 1
2025.2.a.m 2 5.b even 2 1
2025.2.b.g 4 5.c odd 4 2
2025.2.b.h 4 15.e even 4 2
6480.2.a.bi 2 12.b even 2 1
6480.2.a.br 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(405))\):

\( T_{2}^{2} + 2T_{2} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 8T_{11} + 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$11$ \( T^{2} + 8T + 13 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T - 11 \) Copy content Toggle raw display
$23$ \( T^{2} - 12 \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 23 \) Copy content Toggle raw display
$31$ \( (T + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 23 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T - 2 \) Copy content Toggle raw display
$47$ \( T^{2} + 14T + 46 \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 22 \) Copy content Toggle raw display
$59$ \( T^{2} + 20T + 97 \) Copy content Toggle raw display
$61$ \( (T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 12 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T + 1 \) Copy content Toggle raw display
$73$ \( T^{2} - 2T - 74 \) Copy content Toggle raw display
$79$ \( T^{2} - 24T + 132 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T - 18 \) Copy content Toggle raw display
$89$ \( T^{2} - 27 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T - 74 \) Copy content Toggle raw display
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