Properties

Label 9360.2.a.ci
Level $9360$
Weight $2$
Character orbit 9360.a
Self dual yes
Analytic conductor $74.740$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(74.7399762919\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1560)
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.

\(f(q)\) \(=\) \( q - q^{5} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + \beta q^{7} + (\beta - 4) q^{11} + q^{13} + ( - 3 \beta + 2) q^{17} + 2 \beta q^{19} + (\beta - 4) q^{23} + q^{25} + 2 q^{29} + ( - 2 \beta + 4) q^{31} - \beta q^{35} + ( - 5 \beta + 2) q^{37} + (3 \beta + 2) q^{41} + ( - 4 \beta + 4) q^{43} + ( - 2 \beta + 4) q^{47} + (\beta - 3) q^{49} + (\beta - 2) q^{53} + ( - \beta + 4) q^{55} - 8 q^{59} + (3 \beta - 6) q^{61} - q^{65} + ( - 4 \beta + 8) q^{67} - 3 \beta q^{71} + ( - 4 \beta + 10) q^{73} + ( - 3 \beta + 4) q^{77} + (5 \beta - 4) q^{79} - 8 q^{83} + (3 \beta - 2) q^{85} + ( - 3 \beta + 10) q^{89} + \beta q^{91} - 2 \beta q^{95} + (5 \beta - 10) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + q^{7} - 7 q^{11} + 2 q^{13} + q^{17} + 2 q^{19} - 7 q^{23} + 2 q^{25} + 4 q^{29} + 6 q^{31} - q^{35} - q^{37} + 7 q^{41} + 4 q^{43} + 6 q^{47} - 5 q^{49} - 3 q^{53} + 7 q^{55} - 16 q^{59} - 9 q^{61} - 2 q^{65} + 12 q^{67} - 3 q^{71} + 16 q^{73} + 5 q^{77} - 3 q^{79} - 16 q^{83} - q^{85} + 17 q^{89} + q^{91} - 2 q^{95} - 15 q^{97}+O(q^{100}) \) 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 0 0 −1.00000 0 −1.56155 0 0 0
1.2 0 0 0 −1.00000 0 2.56155 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)

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 9360.2.a.ci 2
3.b odd 2 1 3120.2.a.bg 2
4.b odd 2 1 4680.2.a.y 2
12.b even 2 1 1560.2.a.o 2
60.h even 2 1 7800.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.a.o 2 12.b even 2 1
3120.2.a.bg 2 3.b odd 2 1
4680.2.a.y 2 4.b odd 2 1
7800.2.a.bd 2 60.h even 2 1
9360.2.a.ci 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9360))\):

\( T_{7}^{2} - T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 7T_{11} + 8 \) Copy content Toggle raw display
\( T_{17}^{2} - T_{17} - 38 \) Copy content Toggle raw display
\( T_{19}^{2} - 2T_{19} - 16 \) Copy content Toggle raw display
\( T_{31}^{2} - 6T_{31} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 7T + 8 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 7T + 8 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$37$ \( T^{2} + T - 106 \) Copy content Toggle raw display
$41$ \( T^{2} - 7T - 26 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$59$ \( (T + 8)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 9T - 18 \) Copy content Toggle raw display
$67$ \( T^{2} - 12T - 32 \) Copy content Toggle raw display
$71$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$73$ \( T^{2} - 16T - 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 3T - 104 \) Copy content Toggle raw display
$83$ \( (T + 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 17T + 34 \) Copy content Toggle raw display
$97$ \( T^{2} + 15T - 50 \) Copy content Toggle raw display
show more
show less