Properties

Label 405.2.a.i
Level $405$
Weight $2$
Character orbit 405.a
Self dual yes
Analytic conductor $3.234$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - q^{5} + ( - \beta_1 + 2) q^{7} + ( - \beta_{2} - \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - q^{5} + ( - \beta_1 + 2) q^{7} + ( - \beta_{2} - \beta_1 - 1) q^{8} + \beta_1 q^{10} + (\beta_{2} + 1) q^{11} + ( - \beta_{2} + 1) q^{13} + (\beta_{2} - 2 \beta_1 + 4) q^{14} + (2 \beta_1 + 1) q^{16} + (\beta_{2} + 1) q^{17} + ( - \beta_{2} + 1) q^{19} + ( - \beta_{2} - 2) q^{20} + ( - \beta_{2} - 2 \beta_1 - 1) q^{22} + ( - 2 \beta_{2} + \beta_1 - 2) q^{23} + q^{25} + (\beta_{2} + 1) q^{26} + (\beta_{2} - 3 \beta_1 + 3) q^{28} + ( - 2 \beta_{2} + 2 \beta_1 + 1) q^{29} + ( - \beta_{2} + 4 \beta_1 + 1) q^{31} + (\beta_1 - 6) q^{32} + ( - \beta_{2} - 2 \beta_1 - 1) q^{34} + (\beta_1 - 2) q^{35} + (\beta_{2} - 2 \beta_1 + 3) q^{37} + (\beta_{2} + 1) q^{38} + (\beta_{2} + \beta_1 + 1) q^{40} + (\beta_{2} + 2 \beta_1 + 4) q^{41} + (\beta_{2} + 2 \beta_1 + 3) q^{43} + (\beta_{2} + 2 \beta_1 + 7) q^{44} + (\beta_{2} + 4 \beta_1 - 2) q^{46} + (\beta_{2} + 3 \beta_1 - 5) q^{47} + (\beta_{2} - 4 \beta_1 + 1) q^{49} - \beta_1 q^{50} + (\beta_{2} - 2 \beta_1 - 3) q^{52} + 2 \beta_1 q^{53} + ( - \beta_{2} - 1) q^{55} + 3 q^{56} + (\beta_1 - 6) q^{58} + 2 \beta_1 q^{59} + (\beta_{2} + 2 \beta_1) q^{61} + ( - 3 \beta_{2} - 15) q^{62} + ( - \beta_{2} + 2 \beta_1 - 6) q^{64} + (\beta_{2} - 1) q^{65} + (3 \beta_{2} - \beta_1 + 5) q^{67} + (\beta_{2} + 2 \beta_1 + 7) q^{68} + ( - \beta_{2} + 2 \beta_1 - 4) q^{70} + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{71} + (4 \beta_1 - 4) q^{73} + (\beta_{2} - 4 \beta_1 + 7) q^{74} + (\beta_{2} - 2 \beta_1 - 3) q^{76} + (\beta_{2} - 2 \beta_1 + 1) q^{77} + ( - 4 \beta_1 + 2) q^{79} + ( - 2 \beta_1 - 1) q^{80} + ( - 3 \beta_{2} - 5 \beta_1 - 9) q^{82} + ( - 3 \beta_1 + 6) q^{83} + ( - \beta_{2} - 1) q^{85} + ( - 3 \beta_{2} - 4 \beta_1 - 9) q^{86} + ( - \beta_{2} - 4 \beta_1 - 7) q^{88} + 3 q^{89} + ( - \beta_{2} + 3) q^{91} + ( - \beta_{2} - \beta_1 - 13) q^{92} + ( - 4 \beta_{2} + 4 \beta_1 - 13) q^{94} + (\beta_{2} - 1) q^{95} + ( - 4 \beta_{2} + 2 \beta_1 - 8) q^{97} + (3 \beta_{2} - 2 \beta_1 + 15) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 5 q^{4} - 3 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 5 q^{4} - 3 q^{5} + 5 q^{7} - 3 q^{8} + q^{10} + 2 q^{11} + 4 q^{13} + 9 q^{14} + 5 q^{16} + 2 q^{17} + 4 q^{19} - 5 q^{20} - 4 q^{22} - 3 q^{23} + 3 q^{25} + 2 q^{26} + 5 q^{28} + 7 q^{29} + 8 q^{31} - 17 q^{32} - 4 q^{34} - 5 q^{35} + 6 q^{37} + 2 q^{38} + 3 q^{40} + 13 q^{41} + 10 q^{43} + 22 q^{44} - 3 q^{46} - 13 q^{47} - 2 q^{49} - q^{50} - 12 q^{52} + 2 q^{53} - 2 q^{55} + 9 q^{56} - 17 q^{58} + 2 q^{59} + q^{61} - 42 q^{62} - 15 q^{64} - 4 q^{65} + 11 q^{67} + 22 q^{68} - 9 q^{70} + 10 q^{71} - 8 q^{73} + 16 q^{74} - 12 q^{76} + 2 q^{79} - 5 q^{80} - 29 q^{82} + 15 q^{83} - 2 q^{85} - 28 q^{86} - 24 q^{88} + 9 q^{89} + 10 q^{91} - 39 q^{92} - 31 q^{94} - 4 q^{95} - 18 q^{97} + 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
2.51414
0.571993
−2.08613
−2.51414 0 4.32088 −1.00000 0 −0.514137 −5.83502 0 2.51414
1.2 −0.571993 0 −1.67282 −1.00000 0 1.42801 2.10083 0 0.571993
1.3 2.08613 0 2.35194 −1.00000 0 4.08613 0.734191 0 −2.08613
\(n\): 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.i 3
3.b odd 2 1 405.2.a.j 3
4.b odd 2 1 6480.2.a.bs 3
5.b even 2 1 2025.2.a.o 3
5.c odd 4 2 2025.2.b.m 6
9.c even 3 2 135.2.e.b 6
9.d odd 6 2 45.2.e.b 6
12.b even 2 1 6480.2.a.bv 3
15.d odd 2 1 2025.2.a.n 3
15.e even 4 2 2025.2.b.l 6
36.f odd 6 2 2160.2.q.k 6
36.h even 6 2 720.2.q.i 6
45.h odd 6 2 225.2.e.b 6
45.j even 6 2 675.2.e.b 6
45.k odd 12 4 675.2.k.b 12
45.l even 12 4 225.2.k.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.b 6 9.d odd 6 2
135.2.e.b 6 9.c even 3 2
225.2.e.b 6 45.h odd 6 2
225.2.k.b 12 45.l even 12 4
405.2.a.i 3 1.a even 1 1 trivial
405.2.a.j 3 3.b odd 2 1
675.2.e.b 6 45.j even 6 2
675.2.k.b 12 45.k odd 12 4
720.2.q.i 6 36.h even 6 2
2025.2.a.n 3 15.d odd 2 1
2025.2.a.o 3 5.b even 2 1
2025.2.b.l 6 15.e even 4 2
2025.2.b.m 6 5.c odd 4 2
2160.2.q.k 6 36.f odd 6 2
6480.2.a.bs 3 4.b odd 2 1
6480.2.a.bv 3 12.b even 2 1

Hecke kernels

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

\( T_{2}^{3} + T_{2}^{2} - 5T_{2} - 3 \) Copy content Toggle raw display
\( T_{11}^{3} - 2T_{11}^{2} - 8T_{11} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 5T - 3 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 5 T^{2} + 3 T + 3 \) Copy content Toggle raw display
$11$ \( T^{3} - 2 T^{2} - 8 T + 12 \) Copy content Toggle raw display
$13$ \( T^{3} - 4 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} - 8 T + 12 \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$23$ \( T^{3} + 3 T^{2} - 33 T - 117 \) Copy content Toggle raw display
$29$ \( T^{3} - 7 T^{2} - 29 T + 51 \) Copy content Toggle raw display
$31$ \( T^{3} - 8 T^{2} - 60 T + 468 \) Copy content Toggle raw display
$37$ \( T^{3} - 6 T^{2} - 12 T + 4 \) Copy content Toggle raw display
$41$ \( T^{3} - 13 T^{2} + 19 T - 3 \) Copy content Toggle raw display
$43$ \( T^{3} - 10 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$47$ \( T^{3} + 13 T^{2} - 11 T - 369 \) Copy content Toggle raw display
$53$ \( T^{3} - 2 T^{2} - 20 T + 24 \) Copy content Toggle raw display
$59$ \( T^{3} - 2 T^{2} - 20 T + 24 \) Copy content Toggle raw display
$61$ \( T^{3} - T^{2} - 37 T - 71 \) Copy content Toggle raw display
$67$ \( T^{3} - 11 T^{2} - 39 T + 507 \) Copy content Toggle raw display
$71$ \( T^{3} - 10 T^{2} - 92 T + 708 \) Copy content Toggle raw display
$73$ \( T^{3} + 8 T^{2} - 64 T - 128 \) Copy content Toggle raw display
$79$ \( T^{3} - 2 T^{2} - 84 T - 24 \) Copy content Toggle raw display
$83$ \( T^{3} - 15 T^{2} + 27 T + 81 \) Copy content Toggle raw display
$89$ \( (T - 3)^{3} \) Copy content Toggle raw display
$97$ \( T^{3} + 18 T^{2} - 36 T - 1304 \) Copy content Toggle raw display
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