Properties

Label 920.2.a.j
Level $920$
Weight $2$
Character orbit 920.a
Self dual yes
Analytic conductor $7.346$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.34623698596\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.13955077.1
Defining polynomial: \( x^{5} - 14x^{3} - x^{2} + 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} + ( - \beta_{4} - \beta_{3}) q^{7} + (\beta_{4} + \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - q^{5} + ( - \beta_{4} - \beta_{3}) q^{7} + (\beta_{4} + \beta_{2} + 2) q^{9} + ( - \beta_{2} + \beta_1) q^{11} + ( - \beta_{4} + \beta_{2} + 1) q^{13} - \beta_1 q^{15} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{17} + (\beta_{4} + 2 \beta_1 + 1) q^{19} + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{21} - q^{23} + q^{25} + ( - \beta_{4} + 2 \beta_{3} + 3 \beta_1 + 1) q^{27} + (\beta_{3} - \beta_{2} + 1) q^{29} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{31} + (\beta_{4} - 2 \beta_{3} - 2 \beta_1 + 3) q^{33} + (\beta_{4} + \beta_{3}) q^{35} + ( - \beta_{3} + 3) q^{37} + (\beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{39} + (\beta_{3} - \beta_1 + 5) q^{41} + ( - \beta_{4} - \beta_{2} - 2) q^{45} + ( - 2 \beta_{3} - \beta_{2} - 2) q^{47} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 6) q^{49} + (\beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 3) q^{51} + ( - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{53} + (\beta_{2} - \beta_1) q^{55} + (\beta_{4} + \beta_{2} + 3 \beta_1 + 9) q^{57} + (2 \beta_{4} - \beta_{3} - 1) q^{59} + ( - 2 \beta_{4} - \beta_{2} - \beta_1) q^{61} + (\beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 - 8) q^{63} + (\beta_{4} - \beta_{2} - 1) q^{65} + (2 \beta_{4} + \beta_{3} - 2 \beta_1 + 1) q^{67} - \beta_1 q^{69} + ( - 2 \beta_{4} + \beta_{3} - 3 \beta_1 + 1) q^{71} + ( - 2 \beta_{3} - \beta_{2}) q^{73} + \beta_1 q^{75} + (\beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{77} - 2 \beta_{2} q^{79} + (\beta_{4} + 5 \beta_{2} - 3 \beta_1 + 10) q^{81} + (\beta_{3} - 9) q^{83} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{85} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{87} + ( - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 2) q^{89} + ( - 4 \beta_{4} - \beta_{2} - 5 \beta_1 + 4) q^{91} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 7) q^{93} + ( - \beta_{4} - 2 \beta_1 - 1) q^{95} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 + 6) q^{97} + ( - 3 \beta_{4} - 4 \beta_{2} + 4 \beta_1 - 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{5} - 2 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{5} - 2 q^{7} + 13 q^{9} - q^{11} + 4 q^{13} + 4 q^{17} + 7 q^{19} + 6 q^{21} - 5 q^{23} + 5 q^{25} + 3 q^{27} + 4 q^{29} + 19 q^{31} + 17 q^{33} + 2 q^{35} + 15 q^{37} + 19 q^{39} + 25 q^{41} - 13 q^{45} - 11 q^{47} + 25 q^{49} + 19 q^{51} + 3 q^{53} + q^{55} + 48 q^{57} - q^{59} - 5 q^{61} - 41 q^{63} - 4 q^{65} + 9 q^{67} + q^{71} - q^{73} + 18 q^{77} - 2 q^{79} + 57 q^{81} - 45 q^{83} - 4 q^{85} - 9 q^{87} + 6 q^{89} + 11 q^{91} - 39 q^{93} - 7 q^{95} + 25 q^{97} - 65 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 14x^{3} - x^{2} + 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 10\nu^{2} + 3\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 4\nu^{3} + 14\nu^{2} - 39\nu - 28 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{4} + 14\nu^{2} - 3\nu - 24 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + 2\beta_{3} + 9\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{4} + 14\beta_{2} - 3\beta _1 + 46 \) 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
−3.36002
−1.31091
−0.568386
1.93283
3.30649
0 −3.36002 0 −1.00000 0 −1.90754 0 8.28974 0
1.2 0 −1.31091 0 −1.00000 0 −4.66212 0 −1.28151 0
1.3 0 −0.568386 0 −1.00000 0 4.73770 0 −2.67694 0
1.4 0 1.93283 0 −1.00000 0 2.38236 0 0.735829 0
1.5 0 3.30649 0 −1.00000 0 −2.55040 0 7.93288 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
\(2\) \(-1\)
\(5\) \(1\)
\(23\) \(1\)

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 920.2.a.j 5
3.b odd 2 1 8280.2.a.bs 5
4.b odd 2 1 1840.2.a.v 5
5.b even 2 1 4600.2.a.be 5
5.c odd 4 2 4600.2.e.u 10
8.b even 2 1 7360.2.a.co 5
8.d odd 2 1 7360.2.a.cp 5
20.d odd 2 1 9200.2.a.cu 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.j 5 1.a even 1 1 trivial
1840.2.a.v 5 4.b odd 2 1
4600.2.a.be 5 5.b even 2 1
4600.2.e.u 10 5.c odd 4 2
7360.2.a.co 5 8.b even 2 1
7360.2.a.cp 5 8.d odd 2 1
8280.2.a.bs 5 3.b odd 2 1
9200.2.a.cu 5 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 14T_{3}^{3} - T_{3}^{2} + 32T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(920))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 14 T^{3} - T^{2} + 32 T + 16 \) Copy content Toggle raw display
$5$ \( (T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 2 T^{4} - 28 T^{3} - 57 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{5} + T^{4} - 35 T^{3} - 28 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{5} - 4 T^{4} - 46 T^{3} + 75 T^{2} + \cdots + 500 \) Copy content Toggle raw display
$17$ \( T^{5} - 4 T^{4} - 46 T^{3} + 157 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$19$ \( T^{5} - 7 T^{4} - 41 T^{3} + 180 T^{2} + \cdots + 512 \) Copy content Toggle raw display
$23$ \( (T + 1)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 4 T^{4} - 41 T^{3} - 36 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$31$ \( T^{5} - 19 T^{4} + 72 T^{3} + \cdots - 128 \) Copy content Toggle raw display
$37$ \( T^{5} - 15 T^{4} + 64 T^{3} - 36 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( T^{5} - 25 T^{4} + 212 T^{3} + \cdots + 2182 \) Copy content Toggle raw display
$43$ \( T^{5} \) Copy content Toggle raw display
$47$ \( T^{5} + 11 T^{4} - 110 T^{3} + \cdots + 512 \) Copy content Toggle raw display
$53$ \( T^{5} - 3 T^{4} - 160 T^{3} + \cdots - 20272 \) Copy content Toggle raw display
$59$ \( T^{5} + T^{4} - 142 T^{3} + \cdots + 13568 \) Copy content Toggle raw display
$61$ \( T^{5} + 5 T^{4} - 115 T^{3} + \cdots + 7664 \) Copy content Toggle raw display
$67$ \( T^{5} - 9 T^{4} - 126 T^{3} + \cdots + 8192 \) Copy content Toggle raw display
$71$ \( T^{5} - T^{4} - 214 T^{3} + 407 T^{2} + \cdots - 3968 \) Copy content Toggle raw display
$73$ \( T^{5} + T^{4} - 158 T^{3} - 272 T^{2} + \cdots - 1328 \) Copy content Toggle raw display
$79$ \( T^{5} + 2 T^{4} - 128 T^{3} + \cdots - 1024 \) Copy content Toggle raw display
$83$ \( T^{5} + 45 T^{4} + 784 T^{3} + \cdots + 41216 \) Copy content Toggle raw display
$89$ \( T^{5} - 6 T^{4} - 136 T^{3} + \cdots - 8192 \) Copy content Toggle raw display
$97$ \( T^{5} - 25 T^{4} - 39 T^{3} + \cdots - 49616 \) Copy content Toggle raw display
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