Properties

Label 2960.2.a.bb
Level $2960$
Weight $2$
Character orbit 2960.a
Self dual yes
Analytic conductor $23.636$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.6357189983\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.583504.1
Defining polynomial: \( x^{5} - 8x^{3} - 2x^{2} + 15x + 8 \) 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 1480)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + q^{5} + ( - \beta_{4} + \beta_{2} + 1) q^{7} + (\beta_{2} - 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + q^{5} + ( - \beta_{4} + \beta_{2} + 1) q^{7} + (\beta_{2} - 2 \beta_1 + 1) q^{9} + (\beta_{4} + \beta_{3} + 1) q^{11} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{13} + ( - \beta_1 + 1) q^{15} + (\beta_{4} + \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 1) q^{17} + (\beta_{4} + \beta_{3} + \beta_1 + 2) q^{19} + ( - 2 \beta_{4} + \beta_{2} - 2 \beta_1) q^{21} + ( - 2 \beta_{3} - 2 \beta_{2} + 2) q^{23} + q^{25} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 + 3) q^{27} + ( - 2 \beta_{3} - 2 \beta_{2} + 2) q^{29} + (\beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{31} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + 1) q^{33} + ( - \beta_{4} + \beta_{2} + 1) q^{35} - q^{37} + ( - 3 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{39} + (\beta_{4} + \beta_{3} + 1) q^{41} + (2 \beta_{2} - 2 \beta_1 + 4) q^{43} + (\beta_{2} - 2 \beta_1 + 1) q^{45} + ( - \beta_{4} - \beta_{2} + 2 \beta_1 + 3) q^{47} + (2 \beta_{3} - 3 \beta_{2} + 3) q^{49} + (\beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 2) q^{51} + ( - \beta_{4} - 5 \beta_{3} + 2 \beta_1 - 1) q^{53} + (\beta_{4} + \beta_{3} + 1) q^{55} + (\beta_{4} - \beta_{3} - 3 \beta_{2} - 1) q^{57} + (2 \beta_{4} - \beta_{3} - \beta_{2} + 2) q^{59} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{61} + ( - \beta_{4} + \beta_{3} - 3 \beta_1 + 2) q^{63} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{65} + ( - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 6) q^{67} + (2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{69} + (3 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 3) q^{71} + (\beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{73} + ( - \beta_1 + 1) q^{75} + ( - \beta_{4} + \beta_{3} + 6 \beta_{2} - 5) q^{77} + ( - 3 \beta_{3} + \beta_{2}) q^{79} + (\beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{81} + (2 \beta_{3} - 4 \beta_{2} - 3 \beta_1 + 3) q^{83} + (\beta_{4} + \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 1) q^{85} + (2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{87} + (2 \beta_{4} - 4 \beta_{2} + 2 \beta_1 + 2) q^{89} + ( - 3 \beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 2) q^{91} + ( - \beta_{4} + \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 3) q^{93} + (\beta_{4} + \beta_{3} + \beta_1 + 2) q^{95} + ( - 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 2) q^{97} - 2 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{3} + 5 q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{3} + 5 q^{5} + 5 q^{7} + 6 q^{9} + 7 q^{11} - 6 q^{13} + 5 q^{15} + 12 q^{19} - q^{21} + 6 q^{23} + 5 q^{25} + 17 q^{27} + 6 q^{29} + 10 q^{31} + 3 q^{33} + 5 q^{35} - 5 q^{37} - 4 q^{39} + 7 q^{41} + 22 q^{43} + 6 q^{45} + 13 q^{47} + 14 q^{49} + 10 q^{51} - 11 q^{53} + 7 q^{55} - 8 q^{57} + 10 q^{59} + 10 q^{63} - 6 q^{65} + 24 q^{67} + 26 q^{69} + 13 q^{71} - 13 q^{73} + 5 q^{75} - 19 q^{77} - 2 q^{79} + q^{81} + 13 q^{83} + 26 q^{87} + 8 q^{89} + 8 q^{91} - 20 q^{93} + 12 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 8x^{3} - 2x^{2} + 15x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 6\nu^{2} + 4\nu + 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 6\beta_{2} + 12 \) 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
2.34794
1.84514
−0.592644
−1.44679
−2.15365
0 −1.34794 0 1.00000 0 2.75056 0 −1.18306 0
1.2 0 −0.845141 0 1.00000 0 2.14219 0 −2.28574 0
1.3 0 1.59264 0 1.00000 0 −4.50235 0 −0.463486 0
1.4 0 2.44679 0 1.00000 0 4.02963 0 2.98676 0
1.5 0 3.15365 0 1.00000 0 0.579974 0 6.94552 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
\(2\) \(1\)
\(5\) \(-1\)
\(37\) \(1\)

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 2960.2.a.bb 5
4.b odd 2 1 1480.2.a.g 5
20.d odd 2 1 7400.2.a.r 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.a.g 5 4.b odd 2 1
2960.2.a.bb 5 1.a even 1 1 trivial
7400.2.a.r 5 20.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2960))\):

\( T_{3}^{5} - 5T_{3}^{4} + 2T_{3}^{3} + 16T_{3}^{2} - 8T_{3} - 14 \) Copy content Toggle raw display
\( T_{7}^{5} - 5T_{7}^{4} - 12T_{7}^{3} + 100T_{7}^{2} - 160T_{7} + 62 \) Copy content Toggle raw display
\( T_{13}^{5} + 6T_{13}^{4} - 28T_{13}^{3} - 172T_{13}^{2} - 196T_{13} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 5 T^{4} + 2 T^{3} + 16 T^{2} + \cdots - 14 \) Copy content Toggle raw display
$5$ \( (T - 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 5 T^{4} - 12 T^{3} + 100 T^{2} + \cdots + 62 \) Copy content Toggle raw display
$11$ \( T^{5} - 7 T^{4} + 48 T^{2} + 16 T - 32 \) Copy content Toggle raw display
$13$ \( T^{5} + 6 T^{4} - 28 T^{3} - 172 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$17$ \( T^{5} - 76 T^{3} - 100 T^{2} + \cdots + 2896 \) Copy content Toggle raw display
$19$ \( T^{5} - 12 T^{4} + 26 T^{3} + 62 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$23$ \( T^{5} - 6 T^{4} - 48 T^{3} + 432 T^{2} + \cdots + 224 \) Copy content Toggle raw display
$29$ \( T^{5} - 6 T^{4} - 48 T^{3} + 432 T^{2} + \cdots + 224 \) Copy content Toggle raw display
$31$ \( T^{5} - 10 T^{4} - 10 T^{3} + \cdots + 1112 \) Copy content Toggle raw display
$37$ \( (T + 1)^{5} \) Copy content Toggle raw display
$41$ \( T^{5} - 7 T^{4} + 48 T^{2} + 16 T - 32 \) Copy content Toggle raw display
$43$ \( T^{5} - 22 T^{4} + 152 T^{3} + \cdots + 32 \) Copy content Toggle raw display
$47$ \( T^{5} - 13 T^{4} + 24 T^{3} + \cdots + 1238 \) Copy content Toggle raw display
$53$ \( T^{5} + 11 T^{4} - 192 T^{3} + \cdots + 58544 \) Copy content Toggle raw display
$59$ \( T^{5} - 10 T^{4} - 42 T^{3} + \cdots - 1448 \) Copy content Toggle raw display
$61$ \( T^{5} - 128 T^{3} + 144 T^{2} + \cdots - 10816 \) Copy content Toggle raw display
$67$ \( T^{5} - 24 T^{4} + 50 T^{3} + \cdots - 992 \) Copy content Toggle raw display
$71$ \( T^{5} - 13 T^{4} - 212 T^{3} + \cdots - 99136 \) Copy content Toggle raw display
$73$ \( T^{5} + 13 T^{4} - 108 T^{3} + \cdots + 11632 \) Copy content Toggle raw display
$79$ \( T^{5} + 2 T^{4} - 110 T^{3} + \cdots + 11456 \) Copy content Toggle raw display
$83$ \( T^{5} - 13 T^{4} - 254 T^{3} + \cdots + 35434 \) Copy content Toggle raw display
$89$ \( T^{5} - 8 T^{4} - 144 T^{3} + \cdots + 1984 \) Copy content Toggle raw display
$97$ \( T^{5} + 20 T^{4} - 80 T^{3} + \cdots - 31744 \) Copy content Toggle raw display
show more
show less