Properties

Label 3200.2.a.bu
Level $3200$
Weight $2$
Character orbit 3200.a
Self dual yes
Analytic conductor $25.552$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 640)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + (\beta_{2} + 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + (\beta_{2} + 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} - 2 \beta_1 q^{11} + ( - \beta_{2} - \beta_1 - 2) q^{13} + ( - 2 \beta_1 + 2) q^{17} - 2 \beta_{2} q^{19} + (\beta_{2} - 3 \beta_1 + 2) q^{21} + ( - \beta_{2} + 3) q^{23} + (2 \beta_{2} + 2) q^{27} + 2 q^{29} + (2 \beta_{2} + 2 \beta_1 - 4) q^{31} + (2 \beta_{2} + 6) q^{33} + (\beta_{2} + 3 \beta_1 - 4) q^{37} + 4 \beta_1 q^{39} + ( - \beta_{2} - 3 \beta_1 + 2) q^{41} + ( - 2 \beta_{2} + \beta_1 + 5) q^{43} + ( - \beta_{2} + 2 \beta_1 + 1) q^{47} + (\beta_{2} - \beta_1 + 1) q^{49} + (2 \beta_{2} - 2 \beta_1 + 8) q^{51} + (\beta_{2} + \beta_1 - 6) q^{53} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{57} + 2 \beta_{2} q^{59} + (\beta_{2} - 3 \beta_1 + 4) q^{61} + (\beta_{2} - 4 \beta_1 + 9) q^{63} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{67} + ( - \beta_{2} - \beta_1 + 2) q^{69} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{71} + ( - 2 \beta_{2} + 4 \beta_1 + 6) q^{73} + ( - 6 \beta_1 + 2) q^{77} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{79} + ( - \beta_{2} - 3 \beta_1 + 1) q^{81} + (2 \beta_{2} + \beta_1 + 9) q^{83} + ( - 2 \beta_1 + 2) q^{87} + ( - 4 \beta_{2} - 2) q^{89} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{91} - 8 q^{93} + ( - 2 \beta_1 + 10) q^{97} + (2 \beta_{2} - 4 \beta_1 + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} + 4 q^{7} + 3 q^{9} - 2 q^{11} - 8 q^{13} + 4 q^{17} - 2 q^{19} + 4 q^{21} + 8 q^{23} + 8 q^{27} + 6 q^{29} - 8 q^{31} + 20 q^{33} - 8 q^{37} + 4 q^{39} + 2 q^{41} + 14 q^{43} + 4 q^{47} + 3 q^{49} + 24 q^{51} - 16 q^{53} - 4 q^{57} + 2 q^{59} + 10 q^{61} + 24 q^{63} + 10 q^{67} + 4 q^{69} + 12 q^{71} + 20 q^{73} - 16 q^{79} - q^{81} + 30 q^{83} + 4 q^{87} - 10 q^{89} - 28 q^{91} - 24 q^{93} + 28 q^{97} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 2 \) 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
2.17009
−1.48119
0.311108
0 −1.70928 0 0 0 2.63090 0 −0.0783777 0
1.2 0 0.806063 0 0 0 −2.15633 0 −2.35026 0
1.3 0 2.90321 0 0 0 3.52543 0 5.42864 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)

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 3200.2.a.bu 3
4.b odd 2 1 3200.2.a.bp 3
5.b even 2 1 3200.2.a.bo 3
5.c odd 4 2 640.2.c.c yes 6
8.b even 2 1 3200.2.a.br 3
8.d odd 2 1 3200.2.a.bs 3
20.d odd 2 1 3200.2.a.bv 3
20.e even 4 2 640.2.c.d yes 6
40.e odd 2 1 3200.2.a.bq 3
40.f even 2 1 3200.2.a.bt 3
40.i odd 4 2 640.2.c.b yes 6
40.k even 4 2 640.2.c.a 6
80.i odd 4 2 1280.2.f.j 6
80.j even 4 2 1280.2.f.i 6
80.s even 4 2 1280.2.f.l 6
80.t odd 4 2 1280.2.f.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.c.a 6 40.k even 4 2
640.2.c.b yes 6 40.i odd 4 2
640.2.c.c yes 6 5.c odd 4 2
640.2.c.d yes 6 20.e even 4 2
1280.2.f.i 6 80.j even 4 2
1280.2.f.j 6 80.i odd 4 2
1280.2.f.k 6 80.t odd 4 2
1280.2.f.l 6 80.s even 4 2
3200.2.a.bo 3 5.b even 2 1
3200.2.a.bp 3 4.b odd 2 1
3200.2.a.bq 3 40.e odd 2 1
3200.2.a.br 3 8.b even 2 1
3200.2.a.bs 3 8.d odd 2 1
3200.2.a.bt 3 40.f even 2 1
3200.2.a.bu 3 1.a even 1 1 trivial
3200.2.a.bv 3 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3200))\):

\( T_{3}^{3} - 2T_{3}^{2} - 4T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{3} - 4T_{7}^{2} - 4T_{7} + 20 \) Copy content Toggle raw display
\( T_{11}^{3} + 2T_{11}^{2} - 20T_{11} - 8 \) Copy content Toggle raw display
\( T_{13}^{3} + 8T_{13}^{2} + 8T_{13} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 4 T^{2} - 4 T + 20 \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} - 20 T - 8 \) Copy content Toggle raw display
$13$ \( T^{3} + 8 T^{2} + 8 T - 16 \) Copy content Toggle raw display
$17$ \( T^{3} - 4 T^{2} - 16 T + 32 \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} - 36 T - 104 \) Copy content Toggle raw display
$23$ \( T^{3} - 8 T^{2} + 12 T - 4 \) Copy content Toggle raw display
$29$ \( (T - 2)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} + 8 T^{2} - 32 T - 128 \) Copy content Toggle raw display
$37$ \( T^{3} + 8 T^{2} - 32 T - 272 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} - 52 T + 184 \) Copy content Toggle raw display
$43$ \( T^{3} - 14 T^{2} + 20 T + 100 \) Copy content Toggle raw display
$47$ \( T^{3} - 4 T^{2} - 28 T + 116 \) Copy content Toggle raw display
$53$ \( T^{3} + 16 T^{2} + 72 T + 80 \) Copy content Toggle raw display
$59$ \( T^{3} - 2 T^{2} - 36 T + 104 \) Copy content Toggle raw display
$61$ \( T^{3} - 10 T^{2} - 28 T + 8 \) Copy content Toggle raw display
$67$ \( T^{3} - 10 T^{2} - 60 T + 604 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} - 16 T + 320 \) Copy content Toggle raw display
$73$ \( T^{3} - 20T^{2} + 1184 \) Copy content Toggle raw display
$79$ \( T^{3} + 16 T^{2} + 32 T - 128 \) Copy content Toggle raw display
$83$ \( T^{3} - 30 T^{2} + 260 T - 524 \) Copy content Toggle raw display
$89$ \( T^{3} + 10 T^{2} - 116 T - 1096 \) Copy content Toggle raw display
$97$ \( T^{3} - 28 T^{2} + 240 T - 608 \) Copy content Toggle raw display
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