Properties

Label 3432.2.a.w
Level $3432$
Weight $2$
Character orbit 3432.a
Self dual yes
Analytic conductor $27.405$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3432.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.4046579737\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.2172244.1
Defining polynomial: \(x^{5} - 2 x^{4} - 8 x^{3} + 16 x^{2} + 5 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{1} q^{5} + ( -1 - \beta_{4} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta_{1} q^{5} + ( -1 - \beta_{4} ) q^{7} + q^{9} - q^{11} + q^{13} -\beta_{1} q^{15} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{17} + ( -1 - \beta_{1} + \beta_{2} ) q^{19} + ( 1 + \beta_{4} ) q^{21} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{23} + ( \beta_{2} - \beta_{3} + \beta_{4} ) q^{25} - q^{27} + ( 1 + \beta_{2} - 2 \beta_{3} ) q^{29} + ( -2 + \beta_{2} + \beta_{4} ) q^{31} + q^{33} + ( -3 \beta_{1} + \beta_{3} + \beta_{4} ) q^{35} + ( -\beta_{2} + 3 \beta_{3} ) q^{37} - q^{39} + ( 1 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{41} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{43} + \beta_{1} q^{45} + ( -4 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{47} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{49} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{51} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{53} -\beta_{1} q^{55} + ( 1 + \beta_{1} - \beta_{2} ) q^{57} + ( -2 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{59} + ( 2 - 3 \beta_{1} - \beta_{3} - \beta_{4} ) q^{61} + ( -1 - \beta_{4} ) q^{63} + \beta_{1} q^{65} + ( 2 - \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{67} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{69} + ( -6 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{71} + ( -3 + 2 \beta_{1} - \beta_{4} ) q^{73} + ( -\beta_{2} + \beta_{3} - \beta_{4} ) q^{75} + ( 1 + \beta_{4} ) q^{77} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{79} + q^{81} + ( -4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{85} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{87} + ( -2 + 2 \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{89} + ( -1 - \beta_{4} ) q^{91} + ( 2 - \beta_{2} - \beta_{4} ) q^{93} + ( -6 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{95} + ( -4 + \beta_{2} + \beta_{3} ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{3} + q^{5} - 5q^{7} + 5q^{9} + O(q^{10}) \) \( 5q - 5q^{3} + q^{5} - 5q^{7} + 5q^{9} - 5q^{11} + 5q^{13} - q^{15} - 4q^{19} + 5q^{21} - 5q^{23} + 4q^{25} - 5q^{27} + 11q^{29} - 8q^{31} + 5q^{33} - 5q^{35} - 8q^{37} - 5q^{39} - q^{41} + q^{43} + q^{45} - 18q^{47} + 10q^{49} + 2q^{53} - q^{55} + 4q^{57} - 13q^{59} + 9q^{61} - 5q^{63} + q^{65} + 5q^{67} + 5q^{69} - 24q^{71} - 13q^{73} - 4q^{75} + 5q^{77} - 6q^{79} + 5q^{81} - 22q^{83} - 22q^{85} - 11q^{87} - 14q^{89} - 5q^{91} + 8q^{93} - 32q^{95} - 20q^{97} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 8 x^{3} + 16 x^{2} + 5 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{4} + \nu^{3} - 7 \nu^{2} - 5 \nu + 4 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} + \nu^{3} + 7 \nu^{2} - 5 \nu - 2 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{4} + \nu^{3} + 7 \nu^{2} - 9 \nu - 2 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{4} + \nu^{3} + 9 \nu^{2} - 7 \nu - 10 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{4} - \beta_{3} - \beta_{2} + 8\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{3} + 7 \beta_{2} + 2 \beta_{1} - 2\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(14 \beta_{4} - 7 \beta_{3} - 9 \beta_{2} + 2 \beta_{1} + 50\)\()/2\)

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
2.19338
0.679950
−2.71661
−0.758834
2.60211
0 −1.00000 0 −3.47314 0 −3.67591 0 1.00000 0
1.2 0 −1.00000 0 −1.05398 0 4.24902 0 1.00000 0
1.3 0 −1.00000 0 0.169303 0 −1.46187 0 1.00000 0
1.4 0 −1.00000 0 1.82899 0 −0.862883 0 1.00000 0
1.5 0 −1.00000 0 3.52882 0 −3.24835 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(1\)
\(13\) \(-1\)

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 3432.2.a.w 5
4.b odd 2 1 6864.2.a.cf 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.a.w 5 1.a even 1 1 trivial
6864.2.a.cf 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3432))\):

\( T_{5}^{5} - T_{5}^{4} - 14 T_{5}^{3} + 12 T_{5}^{2} + 22 T_{5} - 4 \)
\( T_{7}^{5} + 5 T_{7}^{4} - 10 T_{7}^{3} - 88 T_{7}^{2} - 140 T_{7} - 64 \)
\( T_{17}^{5} - 52 T_{17}^{3} - 22 T_{17}^{2} + 492 T_{17} - 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( ( 1 + T )^{5} \)
$5$ \( -4 + 22 T + 12 T^{2} - 14 T^{3} - T^{4} + T^{5} \)
$7$ \( -64 - 140 T - 88 T^{2} - 10 T^{3} + 5 T^{4} + T^{5} \)
$11$ \( ( 1 + T )^{5} \)
$13$ \( ( -1 + T )^{5} \)
$17$ \( -256 + 492 T - 22 T^{2} - 52 T^{3} + T^{5} \)
$19$ \( -32 - 152 T - 164 T^{2} - 40 T^{3} + 4 T^{4} + T^{5} \)
$23$ \( 544 + 188 T - 132 T^{2} - 40 T^{3} + 5 T^{4} + T^{5} \)
$29$ \( -4892 + 182 T + 462 T^{2} - 30 T^{3} - 11 T^{4} + T^{5} \)
$31$ \( -256 + 752 T - 234 T^{2} - 42 T^{3} + 8 T^{4} + T^{5} \)
$37$ \( 17152 + 2496 T - 880 T^{2} - 124 T^{3} + 8 T^{4} + T^{5} \)
$41$ \( -2648 + 3796 T - 68 T^{2} - 138 T^{3} + T^{4} + T^{5} \)
$43$ \( -1208 + 7638 T + 118 T^{2} - 196 T^{3} - T^{4} + T^{5} \)
$47$ \( 2048 - 1248 T - 920 T^{2} + 4 T^{3} + 18 T^{4} + T^{5} \)
$53$ \( 89888 + 21200 T - 384 T^{2} - 280 T^{3} - 2 T^{4} + T^{5} \)
$59$ \( 64 - 688 T - 200 T^{2} + 28 T^{3} + 13 T^{4} + T^{5} \)
$61$ \( -9056 + 6560 T + 840 T^{2} - 150 T^{3} - 9 T^{4} + T^{5} \)
$67$ \( -248 + 66 T + 916 T^{2} - 138 T^{3} - 5 T^{4} + T^{5} \)
$71$ \( 3968 - 1584 T - 640 T^{2} + 104 T^{3} + 24 T^{4} + T^{5} \)
$73$ \( -8 + 68 T - 80 T^{2} - 2 T^{3} + 13 T^{4} + T^{5} \)
$79$ \( -16096 + 7020 T - 242 T^{2} - 152 T^{3} + 6 T^{4} + T^{5} \)
$83$ \( 105088 - 2656 T - 3392 T^{2} - 80 T^{3} + 22 T^{4} + T^{5} \)
$89$ \( -6112 - 7108 T - 2394 T^{2} - 162 T^{3} + 14 T^{4} + T^{5} \)
$97$ \( -256 - 960 T - 16 T^{2} + 108 T^{3} + 20 T^{4} + T^{5} \)
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