Properties

Label 4332.2.a.o
Level $4332$
Weight $2$
Character orbit 4332.a
Self dual yes
Analytic conductor $34.591$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4332 = 2^{2} \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4332.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.5911941556\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 228)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{7} + q^{9} + ( -1 - 2 \beta_{1} + 3 \beta_{2} ) q^{11} + ( -3 + 2 \beta_{1} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{17} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{21} + ( -4 + 2 \beta_{1} + \beta_{2} ) q^{23} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{25} + q^{27} + ( 1 - 4 \beta_{1} ) q^{29} + ( 2 - 3 \beta_{1} + 3 \beta_{2} ) q^{31} + ( -1 - 2 \beta_{1} + 3 \beta_{2} ) q^{33} + ( -1 + 4 \beta_{1} - 3 \beta_{2} ) q^{35} + ( -2 - 2 \beta_{1} + 3 \beta_{2} ) q^{37} + ( -3 + 2 \beta_{1} ) q^{39} + ( -6 - 3 \beta_{2} ) q^{41} + ( -7 + 2 \beta_{1} + 2 \beta_{2} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( -5 + \beta_{2} ) q^{47} + ( -2 \beta_{1} + \beta_{2} ) q^{49} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{51} + ( 2 + 5 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -4 \beta_{1} + 5 \beta_{2} ) q^{55} + ( 3 + \beta_{1} - 5 \beta_{2} ) q^{59} + ( 5 + 2 \beta_{1} - \beta_{2} ) q^{61} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{63} + ( -7 + 5 \beta_{1} - 2 \beta_{2} ) q^{65} + ( -2 + 2 \beta_{2} ) q^{67} + ( -4 + 2 \beta_{1} + \beta_{2} ) q^{69} + ( 3 + 6 \beta_{1} - 6 \beta_{2} ) q^{71} + ( -2 - \beta_{1} + 4 \beta_{2} ) q^{73} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{75} + ( -8 + 2 \beta_{1} + 3 \beta_{2} ) q^{77} + ( -9 + 2 \beta_{1} ) q^{79} + q^{81} + ( -5 - 4 \beta_{1} - 3 \beta_{2} ) q^{83} + ( -6 + 5 \beta_{1} - 4 \beta_{2} ) q^{85} + ( 1 - 4 \beta_{1} ) q^{87} + ( 4 + 7 \beta_{1} - \beta_{2} ) q^{89} + ( 3 - 9 \beta_{1} + 8 \beta_{2} ) q^{91} + ( 2 - 3 \beta_{1} + 3 \beta_{2} ) q^{93} + ( 2 - 2 \beta_{1} - 5 \beta_{2} ) q^{97} + ( -1 - 2 \beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} - 9 q^{13} + 3 q^{15} - 3 q^{17} - 3 q^{21} - 12 q^{23} - 6 q^{25} + 3 q^{27} + 3 q^{29} + 6 q^{31} - 3 q^{33} - 3 q^{35} - 6 q^{37} - 9 q^{39} - 18 q^{41} - 21 q^{43} + 3 q^{45} - 15 q^{47} - 3 q^{51} + 6 q^{53} + 9 q^{59} + 15 q^{61} - 3 q^{63} - 21 q^{65} - 6 q^{67} - 12 q^{69} + 9 q^{71} - 6 q^{73} - 6 q^{75} - 24 q^{77} - 27 q^{79} + 3 q^{81} - 15 q^{83} - 18 q^{85} + 3 q^{87} + 12 q^{89} + 9 q^{91} + 6 q^{93} + 6 q^{97} - 3 q^{99} + O(q^{100}) \)

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
1.87939
−0.347296
−1.53209
0 1.00000 0 −0.879385 0 −2.18479 0 1.00000 0
1.2 0 1.00000 0 1.34730 0 2.41147 0 1.00000 0
1.3 0 1.00000 0 2.53209 0 −3.22668 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(19\) \(1\)

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 4332.2.a.o 3
19.b odd 2 1 4332.2.a.n 3
19.e even 9 2 228.2.q.a 6
57.l odd 18 2 684.2.bo.a 6
76.l odd 18 2 912.2.bo.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.q.a 6 19.e even 9 2
684.2.bo.a 6 57.l odd 18 2
912.2.bo.e 6 76.l odd 18 2
4332.2.a.n 3 19.b odd 2 1
4332.2.a.o 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4332))\):

\( T_{5}^{3} - 3 T_{5}^{2} + 3 \)
\( T_{7}^{3} + 3 T_{7}^{2} - 6 T_{7} - 17 \)
\( T_{13}^{3} + 9 T_{13}^{2} + 15 T_{13} - 17 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( 3 - 3 T^{2} + T^{3} \)
$7$ \( -17 - 6 T + 3 T^{2} + T^{3} \)
$11$ \( -3 - 18 T + 3 T^{2} + T^{3} \)
$13$ \( -17 + 15 T + 9 T^{2} + T^{3} \)
$17$ \( -3 - 18 T + 3 T^{2} + T^{3} \)
$19$ \( T^{3} \)
$23$ \( -57 + 27 T + 12 T^{2} + T^{3} \)
$29$ \( 111 - 45 T - 3 T^{2} + T^{3} \)
$31$ \( 19 - 15 T - 6 T^{2} + T^{3} \)
$37$ \( -17 - 9 T + 6 T^{2} + T^{3} \)
$41$ \( 27 + 81 T + 18 T^{2} + T^{3} \)
$43$ \( 19 + 111 T + 21 T^{2} + T^{3} \)
$47$ \( 111 + 72 T + 15 T^{2} + T^{3} \)
$53$ \( 213 - 45 T - 6 T^{2} + T^{3} \)
$59$ \( -9 - 36 T - 9 T^{2} + T^{3} \)
$61$ \( -71 + 66 T - 15 T^{2} + T^{3} \)
$67$ \( -8 + 6 T^{2} + T^{3} \)
$71$ \( 513 - 81 T - 9 T^{2} + T^{3} \)
$73$ \( 19 - 27 T + 6 T^{2} + T^{3} \)
$79$ \( 613 + 231 T + 27 T^{2} + T^{3} \)
$83$ \( 3 - 36 T + 15 T^{2} + T^{3} \)
$89$ \( 381 - 81 T - 12 T^{2} + T^{3} \)
$97$ \( 379 - 105 T - 6 T^{2} + T^{3} \)
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