Properties

Label 9680.2.a.bs
Level $9680$
Weight $2$
Character orbit 9680.a
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.2951891566\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
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.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{5} - \beta q^{7} + (\beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + q^{5} - \beta q^{7} + (\beta + 1) q^{9} - 2 q^{13} + \beta q^{15} + (\beta - 2) q^{17} + \beta q^{19} + ( - \beta - 4) q^{21} - 2 \beta q^{23} + q^{25} + ( - \beta + 4) q^{27} + ( - 3 \beta - 2) q^{29} + ( - \beta - 4) q^{31} - \beta q^{35} + ( - 3 \beta + 2) q^{37} - 2 \beta q^{39} - 2 q^{41} + 4 \beta q^{43} + (\beta + 1) q^{45} + (2 \beta + 8) q^{47} + (\beta - 3) q^{49} + ( - \beta + 4) q^{51} + (\beta + 2) q^{53} + (\beta + 4) q^{57} + ( - 2 \beta + 4) q^{59} + (3 \beta - 10) q^{61} + ( - 2 \beta - 4) q^{63} - 2 q^{65} + ( - 4 \beta + 4) q^{67} + ( - 2 \beta - 8) q^{69} + ( - 3 \beta + 4) q^{71} + 2 q^{73} + \beta q^{75} - 6 \beta q^{79} - 7 q^{81} + 2 \beta q^{83} + (\beta - 2) q^{85} + ( - 5 \beta - 12) q^{87} + ( - \beta - 10) q^{89} + 2 \beta q^{91} + ( - 5 \beta - 4) q^{93} + \beta q^{95} + (2 \beta + 2) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 2 q^{5} - q^{7} + 3 q^{9} - 4 q^{13} + q^{15} - 3 q^{17} + q^{19} - 9 q^{21} - 2 q^{23} + 2 q^{25} + 7 q^{27} - 7 q^{29} - 9 q^{31} - q^{35} + q^{37} - 2 q^{39} - 4 q^{41} + 4 q^{43} + 3 q^{45} + 18 q^{47} - 5 q^{49} + 7 q^{51} + 5 q^{53} + 9 q^{57} + 6 q^{59} - 17 q^{61} - 10 q^{63} - 4 q^{65} + 4 q^{67} - 18 q^{69} + 5 q^{71} + 4 q^{73} + q^{75} - 6 q^{79} - 14 q^{81} + 2 q^{83} - 3 q^{85} - 29 q^{87} - 21 q^{89} + 2 q^{91} - 13 q^{93} + q^{95} + 6 q^{97}+O(q^{100}) \) 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
−1.56155
2.56155
0 −1.56155 0 1.00000 0 1.56155 0 −0.561553 0
1.2 0 2.56155 0 1.00000 0 −2.56155 0 3.56155 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(11\) \(-1\)

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 9680.2.a.bs 2
4.b odd 2 1 4840.2.a.j 2
11.b odd 2 1 880.2.a.o 2
33.d even 2 1 7920.2.a.bu 2
44.c even 2 1 440.2.a.e 2
55.d odd 2 1 4400.2.a.bj 2
55.e even 4 2 4400.2.b.t 4
88.b odd 2 1 3520.2.a.bk 2
88.g even 2 1 3520.2.a.bp 2
132.d odd 2 1 3960.2.a.w 2
220.g even 2 1 2200.2.a.s 2
220.i odd 4 2 2200.2.b.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.e 2 44.c even 2 1
880.2.a.o 2 11.b odd 2 1
2200.2.a.s 2 220.g even 2 1
2200.2.b.i 4 220.i odd 4 2
3520.2.a.bk 2 88.b odd 2 1
3520.2.a.bp 2 88.g even 2 1
3960.2.a.w 2 132.d odd 2 1
4400.2.a.bj 2 55.d odd 2 1
4400.2.b.t 4 55.e even 4 2
4840.2.a.j 2 4.b odd 2 1
7920.2.a.bu 2 33.d even 2 1
9680.2.a.bs 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9680))\):

\( T_{3}^{2} - T_{3} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 4 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 3T_{17} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$19$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 7T - 26 \) Copy content Toggle raw display
$31$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$47$ \( T^{2} - 18T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$61$ \( T^{2} + 17T + 34 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$71$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 6T - 144 \) Copy content Toggle raw display
$83$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$89$ \( T^{2} + 21T + 106 \) Copy content Toggle raw display
$97$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
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