Properties

Label 4176.2.a.bu
Level $4176$
Weight $2$
Character orbit 4176.a
Self dual yes
Analytic conductor $33.346$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4176 = 2^{4} \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4176.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.3455278841\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 232)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{5} + (2 \beta_{2} + \beta_1 + 1) q^{11} + (\beta_{2} + 2 \beta_1 + 1) q^{13} - 2 q^{17} + (2 \beta_{2} + 2) q^{19} + (2 \beta_1 - 2) q^{23} + ( - 3 \beta_{2} - 2 \beta_1 + 2) q^{25} - q^{29} + ( - \beta_1 + 5) q^{31} + 2 \beta_{2} q^{37} + (2 \beta_{2} + 4 \beta_1 - 4) q^{41} + (2 \beta_{2} - \beta_1 + 3) q^{43} + (2 \beta_{2} - 3 \beta_1 + 1) q^{47} - 7 q^{49} + (\beta_{2} - 2 \beta_1 + 1) q^{53} + ( - 4 \beta_{2} - 5 \beta_1 + 9) q^{55} + (2 \beta_1 + 2) q^{59} + ( - 4 \beta_1 + 2) q^{61} + ( - 3 \beta_{2} - 4 \beta_1 + 1) q^{65} + (4 \beta_1 - 8) q^{67} + ( - 4 \beta_{2} + 2 \beta_1 + 2) q^{71} + (2 \beta_{2} + 4 \beta_1) q^{73} + ( - 2 \beta_{2} + \beta_1 + 9) q^{79} + (2 \beta_1 + 10) q^{83} + ( - 2 \beta_{2} + 2) q^{85} + ( - 6 \beta_{2} - 4 \beta_1 - 4) q^{89} + ( - 2 \beta_{2} - 4 \beta_1 + 10) q^{95} + (2 \beta_{2} - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{5} + 2 q^{11} + 4 q^{13} - 6 q^{17} + 4 q^{19} - 4 q^{23} + 7 q^{25} - 3 q^{29} + 14 q^{31} - 2 q^{37} - 10 q^{41} + 6 q^{43} - 2 q^{47} - 21 q^{49} + 26 q^{55} + 8 q^{59} + 2 q^{61} + 2 q^{65} - 20 q^{67} + 12 q^{71} + 2 q^{73} + 30 q^{79} + 32 q^{83} + 8 q^{85} - 10 q^{89} + 28 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) 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
−0.363328
3.12489
−1.76156
0 0 0 −4.14134 0 0 0 0 0
1.2 0 0 0 −1.48486 0 0 0 0 0
1.3 0 0 0 1.62620 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(29\) \(1\)

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 4176.2.a.bu 3
3.b odd 2 1 464.2.a.j 3
4.b odd 2 1 2088.2.a.s 3
12.b even 2 1 232.2.a.d 3
24.f even 2 1 1856.2.a.x 3
24.h odd 2 1 1856.2.a.y 3
60.h even 2 1 5800.2.a.p 3
348.b even 2 1 6728.2.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.2.a.d 3 12.b even 2 1
464.2.a.j 3 3.b odd 2 1
1856.2.a.x 3 24.f even 2 1
1856.2.a.y 3 24.h odd 2 1
2088.2.a.s 3 4.b odd 2 1
4176.2.a.bu 3 1.a even 1 1 trivial
5800.2.a.p 3 60.h even 2 1
6728.2.a.j 3 348.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4176))\):

\( T_{5}^{3} + 4T_{5}^{2} - 3T_{5} - 10 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{3} - 2T_{11}^{2} - 29T_{11} + 80 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 4 T^{2} - 3 T - 10 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 2 T^{2} - 29 T + 80 \) Copy content Toggle raw display
$13$ \( T^{3} - 4 T^{2} - 19 T + 2 \) Copy content Toggle raw display
$17$ \( (T + 2)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} - 28 T + 32 \) Copy content Toggle raw display
$23$ \( T^{3} + 4 T^{2} - 20 T - 64 \) Copy content Toggle raw display
$29$ \( (T + 1)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} - 14 T^{2} + 59 T - 68 \) Copy content Toggle raw display
$37$ \( T^{3} + 2 T^{2} - 32 T - 32 \) Copy content Toggle raw display
$41$ \( T^{3} + 10 T^{2} - 64 T - 512 \) Copy content Toggle raw display
$43$ \( T^{3} - 6 T^{2} - 37 T - 32 \) Copy content Toggle raw display
$47$ \( T^{3} + 2 T^{2} - 117 T - 452 \) Copy content Toggle raw display
$53$ \( T^{3} - 43T - 58 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 4 T + 16 \) Copy content Toggle raw display
$61$ \( T^{3} - 2 T^{2} - 100 T + 328 \) Copy content Toggle raw display
$67$ \( T^{3} + 20 T^{2} + 32 T - 640 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} - 148 T + 1696 \) Copy content Toggle raw display
$73$ \( T^{3} - 2 T^{2} - 96 T - 160 \) Copy content Toggle raw display
$79$ \( T^{3} - 30 T^{2} + 251 T - 388 \) Copy content Toggle raw display
$83$ \( T^{3} - 32 T^{2} + 316 T - 976 \) Copy content Toggle raw display
$89$ \( T^{3} + 10 T^{2} - 256 T - 2816 \) Copy content Toggle raw display
$97$ \( T^{3} + 14 T^{2} + 32 T - 64 \) Copy content Toggle raw display
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