Properties

Label 1840.2.a.q
Level $1840$
Weight $2$
Character orbit 1840.a
Self dual yes
Analytic conductor $14.692$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.6924739719\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{3} + q^{5} + ( -2 + \beta_{2} ) q^{7} + ( 1 + \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{3} + q^{5} + ( -2 + \beta_{2} ) q^{7} + ( 1 + \beta_{1} ) q^{9} + ( -1 - \beta_{1} ) q^{11} + ( 2 + \beta_{1} ) q^{13} + ( -1 - \beta_{2} ) q^{15} + ( -1 - 3 \beta_{1} + 2 \beta_{2} ) q^{17} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{19} + ( -1 - \beta_{1} + 3 \beta_{2} ) q^{21} + q^{23} + q^{25} + ( 1 - 2 \beta_{1} + 2 \beta_{2} ) q^{27} + ( -5 \beta_{1} + \beta_{2} ) q^{29} + ( -4 + \beta_{1} - 2 \beta_{2} ) q^{31} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{33} + ( -2 + \beta_{2} ) q^{35} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{37} + ( -3 - 2 \beta_{1} - 2 \beta_{2} ) q^{39} + ( -3 + 4 \beta_{1} + \beta_{2} ) q^{41} + ( -4 + 4 \beta_{1} ) q^{43} + ( 1 + \beta_{1} ) q^{45} + ( -\beta_{1} + \beta_{2} ) q^{47} + ( \beta_{1} - 6 \beta_{2} ) q^{49} + ( -2 + 4 \beta_{1} + 3 \beta_{2} ) q^{51} + ( 2 - 4 \beta_{2} ) q^{53} + ( -1 - \beta_{1} ) q^{55} + ( -3 - 5 \beta_{1} + 3 \beta_{2} ) q^{57} + ( -2 - 6 \beta_{1} + 4 \beta_{2} ) q^{59} + ( -3 - 3 \beta_{1} + 4 \beta_{2} ) q^{61} + ( -1 - \beta_{1} + \beta_{2} ) q^{63} + ( 2 + \beta_{1} ) q^{65} + ( 4 - 2 \beta_{1} + 6 \beta_{2} ) q^{67} + ( -1 - \beta_{2} ) q^{69} + ( -5 - 5 \beta_{2} ) q^{71} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{73} + ( -1 - \beta_{2} ) q^{75} + ( 1 + \beta_{1} - \beta_{2} ) q^{77} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -8 - \beta_{1} + \beta_{2} ) q^{81} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -1 - 3 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 2 + 9 \beta_{1} + \beta_{2} ) q^{87} + ( -8 + 4 \beta_{1} - 4 \beta_{2} ) q^{89} + ( -3 - \beta_{1} + 2 \beta_{2} ) q^{91} + ( 9 + 2 \beta_{2} ) q^{93} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{95} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{97} + ( -4 - 2 \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{3} + 3q^{5} - 7q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 2q^{3} + 3q^{5} - 7q^{7} + 3q^{9} - 3q^{11} + 6q^{13} - 2q^{15} - 5q^{17} - 7q^{19} - 6q^{21} + 3q^{23} + 3q^{25} + q^{27} - q^{29} - 10q^{31} + 5q^{33} - 7q^{35} + 2q^{37} - 7q^{39} - 10q^{41} - 12q^{43} + 3q^{45} - q^{47} + 6q^{49} - 9q^{51} + 10q^{53} - 3q^{55} - 12q^{57} - 10q^{59} - 13q^{61} - 4q^{63} + 6q^{65} + 6q^{67} - 2q^{69} - 10q^{71} + 7q^{73} - 2q^{75} + 4q^{77} - 14q^{79} - 25q^{81} - 16q^{83} - 5q^{85} + 5q^{87} - 20q^{89} - 11q^{91} + 25q^{93} - 7q^{95} + 5q^{97} - 11q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 4 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
2.11491
−1.86081
−0.254102
0 −2.47283 0 1.00000 0 −0.527166 0 3.11491 0
1.2 0 −1.46260 0 1.00000 0 −1.53740 0 −0.860806 0
1.3 0 1.93543 0 1.00000 0 −4.93543 0 0.745898 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 1840.2.a.q 3
4.b odd 2 1 920.2.a.i 3
5.b even 2 1 9200.2.a.ci 3
8.b even 2 1 7360.2.a.cf 3
8.d odd 2 1 7360.2.a.bw 3
12.b even 2 1 8280.2.a.bl 3
20.d odd 2 1 4600.2.a.v 3
20.e even 4 2 4600.2.e.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.i 3 4.b odd 2 1
1840.2.a.q 3 1.a even 1 1 trivial
4600.2.a.v 3 20.d odd 2 1
4600.2.e.q 6 20.e even 4 2
7360.2.a.bw 3 8.d odd 2 1
7360.2.a.cf 3 8.b even 2 1
8280.2.a.bl 3 12.b even 2 1
9200.2.a.ci 3 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{3} + 2 T_{3}^{2} - 4 T_{3} - 7 \)
\( T_{7}^{3} + 7 T_{7}^{2} + 11 T_{7} + 4 \)
\( T_{11}^{3} + 3 T_{11}^{2} - T_{11} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -7 - 4 T + 2 T^{2} + T^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( 4 + 11 T + 7 T^{2} + T^{3} \)
$11$ \( -2 - T + 3 T^{2} + T^{3} \)
$13$ \( -1 + 8 T - 6 T^{2} + T^{3} \)
$17$ \( -148 - 31 T + 5 T^{2} + T^{3} \)
$19$ \( -106 - 11 T + 7 T^{2} + T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( -148 - 90 T + T^{2} + T^{3} \)
$31$ \( -53 + 14 T + 10 T^{2} + T^{3} \)
$37$ \( 512 - 128 T - 2 T^{2} + T^{3} \)
$41$ \( -481 - 48 T + 10 T^{2} + T^{3} \)
$43$ \( -256 - 16 T + 12 T^{2} + T^{3} \)
$47$ \( -4 - 6 T + T^{2} + T^{3} \)
$53$ \( 8 - 52 T - 10 T^{2} + T^{3} \)
$59$ \( -1184 - 124 T + 10 T^{2} + T^{3} \)
$61$ \( -214 - 29 T + 13 T^{2} + T^{3} \)
$67$ \( 1184 - 160 T - 6 T^{2} + T^{3} \)
$71$ \( -875 - 100 T + 10 T^{2} + T^{3} \)
$73$ \( 236 - 34 T - 7 T^{2} + T^{3} \)
$79$ \( -224 + 16 T + 14 T^{2} + T^{3} \)
$83$ \( 32 + 60 T + 16 T^{2} + T^{3} \)
$89$ \( -256 + 32 T + 20 T^{2} + T^{3} \)
$97$ \( 56 - 23 T - 5 T^{2} + T^{3} \)
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