Properties

Label 3249.2.a.y
Level $3249$
Weight $2$
Character orbit 3249.a
Self dual yes
Analytic conductor $25.943$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3249 = 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3249.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.9433956167\)
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 57)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} - 1) q^{5} + \beta_{2} q^{7} + (\beta_{2} + \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} - 1) q^{5} + \beta_{2} q^{7} + (\beta_{2} + \beta_1 + 1) q^{8} + ( - \beta_{2} - 2 \beta_1 - 1) q^{10} + ( - \beta_{2} + 2 \beta_1 - 1) q^{11} + (2 \beta_1 - 1) q^{13} + (\beta_{2} + \beta_1 + 1) q^{14} + (2 \beta_1 + 1) q^{16} + ( - \beta_{2} - 2 \beta_1 - 7) q^{20} + (\beta_{2} - 2 \beta_1 + 7) q^{22} + (\beta_{2} + 2 \beta_1 - 5) q^{23} + (2 \beta_1 + 1) q^{25} + (2 \beta_{2} - \beta_1 + 8) q^{26} + (2 \beta_1 + 5) q^{28} + ( - 2 \beta_{2} - 2) q^{29} + ( - \beta_{2} + 2 \beta_1 + 4) q^{31} + ( - \beta_1 + 6) q^{32} + (\beta_{2} - 2 \beta_1 - 5) q^{35} - q^{37} + ( - \beta_{2} - 4 \beta_1 - 7) q^{40} + (2 \beta_{2} + 2) q^{41} + ( - \beta_{2} + 2 \beta_1 - 2) q^{43} + (\beta_{2} + 4 \beta_1 - 5) q^{44} + (3 \beta_{2} - 4 \beta_1 + 9) q^{46} + 6 q^{47} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{49} + (2 \beta_{2} + \beta_1 + 8) q^{50} + (\beta_{2} + 6 \beta_1) q^{52} + (3 \beta_{2} + 3) q^{53} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{55} + (3 \beta_1 + 6) q^{56} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{58} + (\beta_{2} - 2 \beta_1 + 1) q^{59} + ( - 2 \beta_{2} + 2 \beta_1 + 3) q^{61} + (\beta_{2} + 3 \beta_1 + 7) q^{62} + ( - \beta_{2} + 2 \beta_1 - 6) q^{64} + ( - \beta_{2} - 4 \beta_1 - 1) q^{65} + ( - \beta_{2} - 4 \beta_1 + 4) q^{67} + ( - \beta_{2} - 4 \beta_1 - 7) q^{70} + ( - 2 \beta_{2} + 4 \beta_1 + 4) q^{71} + (2 \beta_{2} + 7) q^{73} - \beta_1 q^{74} + (3 \beta_{2} - 3) q^{77} + ( - \beta_{2} - 2 \beta_1 + 4) q^{79} + ( - 3 \beta_{2} - 4 \beta_1 - 3) q^{80} + (2 \beta_{2} + 4 \beta_1 + 2) q^{82} + 4 \beta_1 q^{83} + (\beta_{2} - 3 \beta_1 + 7) q^{86} + (3 \beta_{2} + 3) q^{88} + ( - \beta_{2} + 5) q^{89} + (\beta_{2} + 2 \beta_1 + 2) q^{91} + ( - 3 \beta_{2} + 8 \beta_1 - 3) q^{92} + 6 \beta_1 q^{94} + (2 \beta_{2} + 6 \beta_1 - 2) q^{97} + ( - 4 \beta_1 + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 5 q^{4} - 2 q^{5} - q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 5 q^{4} - 2 q^{5} - q^{7} + 3 q^{8} - 4 q^{10} - q^{13} + 3 q^{14} + 5 q^{16} - 22 q^{20} + 18 q^{22} - 14 q^{23} + 5 q^{25} + 21 q^{26} + 17 q^{28} - 4 q^{29} + 15 q^{31} + 17 q^{32} - 18 q^{35} - 3 q^{37} - 24 q^{40} + 4 q^{41} - 3 q^{43} - 12 q^{44} + 20 q^{46} + 18 q^{47} - 2 q^{49} + 23 q^{50} + 5 q^{52} + 6 q^{53} + 12 q^{55} + 21 q^{56} - 8 q^{58} + 13 q^{61} + 23 q^{62} - 15 q^{64} - 6 q^{65} + 9 q^{67} - 24 q^{70} + 18 q^{71} + 19 q^{73} - q^{74} - 12 q^{77} + 11 q^{79} - 10 q^{80} + 8 q^{82} + 4 q^{83} + 17 q^{86} + 6 q^{88} + 16 q^{89} + 7 q^{91} + 2 q^{92} + 6 q^{94} - 2 q^{97} + 14 q^{98}+O(q^{100}) \) 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
−2.08613
0.571993
2.51414
−2.08613 0 2.35194 −1.35194 0 0.351939 −0.734191 0 2.82032
1.2 0.571993 0 −1.67282 2.67282 0 −3.67282 −2.10083 0 1.52884
1.3 2.51414 0 4.32088 −3.32088 0 2.32088 5.83502 0 −8.34916
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(19\) \(1\)

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 3249.2.a.y 3
3.b odd 2 1 1083.2.a.l 3
19.b odd 2 1 3249.2.a.t 3
19.c even 3 2 171.2.f.b 6
57.d even 2 1 1083.2.a.o 3
57.h odd 6 2 57.2.e.b 6
76.g odd 6 2 2736.2.s.z 6
228.m even 6 2 912.2.q.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.e.b 6 57.h odd 6 2
171.2.f.b 6 19.c even 3 2
912.2.q.l 6 228.m even 6 2
1083.2.a.l 3 3.b odd 2 1
1083.2.a.o 3 57.d even 2 1
2736.2.s.z 6 76.g odd 6 2
3249.2.a.t 3 19.b odd 2 1
3249.2.a.y 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3249))\):

\( T_{2}^{3} - T_{2}^{2} - 5T_{2} + 3 \) Copy content Toggle raw display
\( T_{5}^{3} + 2T_{5}^{2} - 8T_{5} - 12 \) Copy content Toggle raw display
\( T_{13}^{3} + T_{13}^{2} - 21T_{13} + 3 \) 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
$5$ \( T^{3} + 2 T^{2} - 8 T - 12 \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 9T + 3 \) Copy content Toggle raw display
$11$ \( T^{3} - 24T + 36 \) Copy content Toggle raw display
$13$ \( T^{3} + T^{2} - 21T + 3 \) Copy content Toggle raw display
$17$ \( T^{3} \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 14 T^{2} + 28 T - 156 \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} - 32 T - 96 \) Copy content Toggle raw display
$31$ \( T^{3} - 15 T^{2} + 51 T + 31 \) Copy content Toggle raw display
$37$ \( (T + 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - 4 T^{2} - 32 T + 96 \) Copy content Toggle raw display
$43$ \( T^{3} + 3 T^{2} - 21 T + 13 \) Copy content Toggle raw display
$47$ \( (T - 6)^{3} \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} - 72 T + 324 \) Copy content Toggle raw display
$59$ \( T^{3} - 24T - 36 \) Copy content Toggle raw display
$61$ \( T^{3} - 13 T^{2} + 11 T + 73 \) Copy content Toggle raw display
$67$ \( T^{3} - 9 T^{2} - 81 T + 541 \) Copy content Toggle raw display
$71$ \( T^{3} - 18 T^{2} + 12 T + 648 \) Copy content Toggle raw display
$73$ \( T^{3} - 19 T^{2} + 83 T + 31 \) Copy content Toggle raw display
$79$ \( T^{3} - 11 T^{2} + 3 T + 171 \) Copy content Toggle raw display
$83$ \( T^{3} - 4 T^{2} - 80 T + 192 \) Copy content Toggle raw display
$89$ \( T^{3} - 16 T^{2} + 76 T - 108 \) Copy content Toggle raw display
$97$ \( T^{3} + 2 T^{2} - 268 T - 1448 \) Copy content Toggle raw display
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