Properties

Label 920.2.a.f
Level $920$
Weight $2$
Character orbit 920.a
Self dual yes
Analytic conductor $7.346$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{5} + 2 \beta q^{7} + ( 1 + \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{3} + q^{5} + 2 \beta q^{7} + ( 1 + \beta ) q^{9} -4 q^{11} + ( 2 - \beta ) q^{13} + \beta q^{15} + ( 2 - 2 \beta ) q^{17} + 4 q^{19} + ( 8 + 2 \beta ) q^{21} + q^{23} + q^{25} + ( 4 - \beta ) q^{27} + ( -6 - \beta ) q^{29} + ( 4 - \beta ) q^{31} -4 \beta q^{33} + 2 \beta q^{35} + ( -2 - 2 \beta ) q^{37} + ( -4 + \beta ) q^{39} + ( -2 + \beta ) q^{41} + ( -4 - 2 \beta ) q^{43} + ( 1 + \beta ) q^{45} + ( 4 - 3 \beta ) q^{47} + ( 9 + 4 \beta ) q^{49} -8 q^{51} + ( 6 - 4 \beta ) q^{53} -4 q^{55} + 4 \beta q^{57} + ( 4 - 4 \beta ) q^{59} + ( 6 + 2 \beta ) q^{61} + ( 8 + 4 \beta ) q^{63} + ( 2 - \beta ) q^{65} + ( -4 + 4 \beta ) q^{67} + \beta q^{69} + ( -4 + 3 \beta ) q^{71} + ( 14 + \beta ) q^{73} + \beta q^{75} -8 \beta q^{77} -4 \beta q^{79} -7 q^{81} + 12 q^{83} + ( 2 - 2 \beta ) q^{85} + ( -4 - 7 \beta ) q^{87} + 10 q^{89} + ( -8 + 2 \beta ) q^{91} + ( -4 + 3 \beta ) q^{93} + 4 q^{95} + ( -6 - 4 \beta ) q^{97} + ( -4 - 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 2q^{5} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{5} + 2q^{7} + 3q^{9} - 8q^{11} + 3q^{13} + q^{15} + 2q^{17} + 8q^{19} + 18q^{21} + 2q^{23} + 2q^{25} + 7q^{27} - 13q^{29} + 7q^{31} - 4q^{33} + 2q^{35} - 6q^{37} - 7q^{39} - 3q^{41} - 10q^{43} + 3q^{45} + 5q^{47} + 22q^{49} - 16q^{51} + 8q^{53} - 8q^{55} + 4q^{57} + 4q^{59} + 14q^{61} + 20q^{63} + 3q^{65} - 4q^{67} + q^{69} - 5q^{71} + 29q^{73} + q^{75} - 8q^{77} - 4q^{79} - 14q^{81} + 24q^{83} + 2q^{85} - 15q^{87} + 20q^{89} - 14q^{91} - 5q^{93} + 8q^{95} - 16q^{97} - 12q^{99} + O(q^{100}) \)

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
−1.56155
2.56155
0 −1.56155 0 1.00000 0 −3.12311 0 −0.561553 0
1.2 0 2.56155 0 1.00000 0 5.12311 0 3.56155 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 920.2.a.f 2
3.b odd 2 1 8280.2.a.bb 2
4.b odd 2 1 1840.2.a.k 2
5.b even 2 1 4600.2.a.r 2
5.c odd 4 2 4600.2.e.m 4
8.b even 2 1 7360.2.a.bj 2
8.d odd 2 1 7360.2.a.bm 2
20.d odd 2 1 9200.2.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.f 2 1.a even 1 1 trivial
1840.2.a.k 2 4.b odd 2 1
4600.2.a.r 2 5.b even 2 1
4600.2.e.m 4 5.c odd 4 2
7360.2.a.bj 2 8.b even 2 1
7360.2.a.bm 2 8.d odd 2 1
8280.2.a.bb 2 3.b odd 2 1
9200.2.a.bx 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(920))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 - T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -16 - 2 T + T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( -2 - 3 T + T^{2} \)
$17$ \( -16 - 2 T + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 38 + 13 T + T^{2} \)
$31$ \( 8 - 7 T + T^{2} \)
$37$ \( -8 + 6 T + T^{2} \)
$41$ \( -2 + 3 T + T^{2} \)
$43$ \( 8 + 10 T + T^{2} \)
$47$ \( -32 - 5 T + T^{2} \)
$53$ \( -52 - 8 T + T^{2} \)
$59$ \( -64 - 4 T + T^{2} \)
$61$ \( 32 - 14 T + T^{2} \)
$67$ \( -64 + 4 T + T^{2} \)
$71$ \( -32 + 5 T + T^{2} \)
$73$ \( 206 - 29 T + T^{2} \)
$79$ \( -64 + 4 T + T^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( -4 + 16 T + T^{2} \)
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