Properties

Label 3960.2.a.w
Level $3960$
Weight $2$
Character orbit 3960.a
Self dual yes
Analytic conductor $31.621$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.6207592004\)
Analytic rank: \(0\)
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 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.

\(f(q)\) \(=\) \( q - q^{5} - \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} - \beta q^{7} - q^{11} + 2 q^{13} + (\beta - 2) q^{17} + \beta q^{19} - 2 \beta q^{23} + q^{25} + ( - 3 \beta - 2) q^{29} + (\beta + 4) q^{31} + \beta q^{35} + ( - 3 \beta + 2) q^{37} - 2 q^{41} + 4 \beta q^{43} + (2 \beta + 8) q^{47} + (\beta - 3) q^{49} + ( - \beta - 2) q^{53} + q^{55} + ( - 2 \beta + 4) q^{59} + ( - 3 \beta + 10) q^{61} - 2 q^{65} + (4 \beta - 4) q^{67} + ( - 3 \beta + 4) q^{71} - 2 q^{73} + \beta q^{77} - 6 \beta q^{79} - 2 \beta q^{83} + ( - \beta + 2) q^{85} + (\beta + 10) q^{89} - 2 \beta q^{91} - \beta q^{95} + (2 \beta + 2) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - q^{7} - 2 q^{11} + 4 q^{13} - 3 q^{17} + q^{19} - 2 q^{23} + 2 q^{25} - 7 q^{29} + 9 q^{31} + q^{35} + q^{37} - 4 q^{41} + 4 q^{43} + 18 q^{47} - 5 q^{49} - 5 q^{53} + 2 q^{55} + 6 q^{59} + 17 q^{61} - 4 q^{65} - 4 q^{67} + 5 q^{71} - 4 q^{73} + q^{77} - 6 q^{79} - 2 q^{83} + 3 q^{85} + 21 q^{89} - 2 q^{91} - q^{95} + 6 q^{97}+O(q^{100}) \) 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
2.56155
−1.56155
0 0 0 −1.00000 0 −2.56155 0 0 0
1.2 0 0 0 −1.00000 0 1.56155 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)

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 3960.2.a.w 2
3.b odd 2 1 440.2.a.e 2
4.b odd 2 1 7920.2.a.bu 2
12.b even 2 1 880.2.a.o 2
15.d odd 2 1 2200.2.a.s 2
15.e even 4 2 2200.2.b.i 4
24.f even 2 1 3520.2.a.bk 2
24.h odd 2 1 3520.2.a.bp 2
33.d even 2 1 4840.2.a.j 2
60.h even 2 1 4400.2.a.bj 2
60.l odd 4 2 4400.2.b.t 4
132.d odd 2 1 9680.2.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.e 2 3.b odd 2 1
880.2.a.o 2 12.b even 2 1
2200.2.a.s 2 15.d odd 2 1
2200.2.b.i 4 15.e even 4 2
3520.2.a.bk 2 24.f even 2 1
3520.2.a.bp 2 24.h odd 2 1
3960.2.a.w 2 1.a even 1 1 trivial
4400.2.a.bj 2 60.h even 2 1
4400.2.b.t 4 60.l odd 4 2
4840.2.a.j 2 33.d even 2 1
7920.2.a.bu 2 4.b odd 2 1
9680.2.a.bs 2 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3960))\):

\( T_{7}^{2} + T_{7} - 4 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 3T_{17} - 2 \) Copy content Toggle raw display
\( T_{19}^{2} - T_{19} - 4 \) Copy content Toggle raw display
\( T_{23}^{2} + 2T_{23} - 16 \) Copy content Toggle raw display
\( T_{29}^{2} + 7T_{29} - 26 \) 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
$11$ \( (T + 1)^{2} \) 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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