Properties

Label 3330.2.a.bk
Level $3330$
Weight $2$
Character orbit 3330.a
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.23544108.1
Defining polynomial: \( x^{5} - x^{4} - 20x^{3} + 39x^{2} + 9x - 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + (\beta_{4} + 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - q^{5} + (\beta_{4} + 1) q^{7} - q^{8} + q^{10} + ( - \beta_{3} - 1) q^{11} + (\beta_{4} - \beta_{2} + 1) q^{13} + ( - \beta_{4} - 1) q^{14} + q^{16} + (\beta_{4} - 1) q^{17} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{19} - q^{20} + (\beta_{3} + 1) q^{22} + (\beta_{4} - \beta_{2} - 1) q^{23} + q^{25} + ( - \beta_{4} + \beta_{2} - 1) q^{26} + (\beta_{4} + 1) q^{28} + \beta_{2} q^{29} + ( - \beta_{2} - \beta_1 + 3) q^{31} - q^{32} + ( - \beta_{4} + 1) q^{34} + ( - \beta_{4} - 1) q^{35} + q^{37} + (\beta_{3} - \beta_{2} - \beta_1) q^{38} + q^{40} + ( - \beta_{2} + \beta_1 - 1) q^{41} + (\beta_{2} - \beta_1 + 3) q^{43} + ( - \beta_{3} - 1) q^{44} + ( - \beta_{4} + \beta_{2} + 1) q^{46} + (\beta_{4} + \beta_{3} + \beta_1 - 1) q^{47} + (\beta_{4} - 2 \beta_1 + 4) q^{49} - q^{50} + (\beta_{4} - \beta_{2} + 1) q^{52} + (\beta_{4} - \beta_1) q^{53} + (\beta_{3} + 1) q^{55} + ( - \beta_{4} - 1) q^{56} - \beta_{2} q^{58} + (2 \beta_{3} + 2) q^{59} + ( - 3 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{61} + (\beta_{2} + \beta_1 - 3) q^{62} + q^{64} + ( - \beta_{4} + \beta_{2} - 1) q^{65} + ( - 2 \beta_{4} + 2 \beta_{3}) q^{67} + (\beta_{4} - 1) q^{68} + (\beta_{4} + 1) q^{70} + ( - 2 \beta_1 + 2) q^{71} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{73} - q^{74} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{76} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{77} + 2 \beta_1 q^{79} - q^{80} + (\beta_{2} - \beta_1 + 1) q^{82} + (\beta_{4} + 2 \beta_{3} - \beta_{2} + 1) q^{83} + ( - \beta_{4} + 1) q^{85} + ( - \beta_{2} + \beta_1 - 3) q^{86} + (\beta_{3} + 1) q^{88} + (3 \beta_{4} + \beta_{2} + 3) q^{89} + (\beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 7) q^{91} + (\beta_{4} - \beta_{2} - 1) q^{92} + ( - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{94} + (\beta_{3} - \beta_{2} - \beta_1) q^{95} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 2) q^{97} + ( - \beta_{4} + 2 \beta_1 - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8} + 5 q^{10} - 5 q^{11} + 5 q^{13} - 3 q^{14} + 5 q^{16} - 7 q^{17} - q^{19} - 5 q^{20} + 5 q^{22} - 5 q^{23} + 5 q^{25} - 5 q^{26} + 3 q^{28} - 2 q^{29} + 16 q^{31} - 5 q^{32} + 7 q^{34} - 3 q^{35} + 5 q^{37} + q^{38} + 5 q^{40} - 2 q^{41} + 12 q^{43} - 5 q^{44} + 5 q^{46} - 6 q^{47} + 16 q^{49} - 5 q^{50} + 5 q^{52} - 3 q^{53} + 5 q^{55} - 3 q^{56} + 2 q^{58} + 10 q^{59} + 16 q^{61} - 16 q^{62} + 5 q^{64} - 5 q^{65} + 4 q^{67} - 7 q^{68} + 3 q^{70} + 8 q^{71} - 3 q^{73} - 5 q^{74} - q^{76} - 3 q^{77} + 2 q^{79} - 5 q^{80} + 2 q^{82} + 5 q^{83} + 7 q^{85} - 12 q^{86} + 5 q^{88} + 7 q^{89} + 33 q^{91} - 5 q^{92} + 6 q^{94} + q^{95} + 14 q^{97} - 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 20x^{3} + 39x^{2} + 9x - 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 20\nu^{2} - 19\nu - 22 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{4} - \nu^{3} + 37\nu^{2} - 21\nu - 36 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{4} + 4\nu^{3} - 94\nu^{2} + 33\nu + 120 ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{4} - 2\nu^{3} + 77\nu^{2} - 39\nu - 96 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - \beta_{3} - 4\beta_{2} + \beta _1 + 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{4} + 10\beta_{3} + 3\beta_{2} - 6\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{4} - 39\beta_{3} - 80\beta_{2} + 35\beta _1 + 237 ) / 2 \) 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
1.43959
−4.76376
−0.892002
3.56575
1.65043
−1.00000 0 1.00000 −1.00000 0 −4.23800 −1.00000 0 1.00000
1.2 −1.00000 0 1.00000 −1.00000 0 −1.19175 −1.00000 0 1.00000
1.3 −1.00000 0 1.00000 −1.00000 0 0.647226 −1.00000 0 1.00000
1.4 −1.00000 0 1.00000 −1.00000 0 3.21441 −1.00000 0 1.00000
1.5 −1.00000 0 1.00000 −1.00000 0 4.56812 −1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(1\)
\(37\) \(-1\)

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 3330.2.a.bk 5
3.b odd 2 1 3330.2.a.bl yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3330.2.a.bk 5 1.a even 1 1 trivial
3330.2.a.bl yes 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3330))\):

\( T_{7}^{5} - 3T_{7}^{4} - 21T_{7}^{3} + 55T_{7}^{2} + 48T_{7} - 48 \) Copy content Toggle raw display
\( T_{11}^{5} + 5T_{11}^{4} - 29T_{11}^{3} - 111T_{11}^{2} + 210T_{11} + 588 \) Copy content Toggle raw display
\( T_{13}^{5} - 5T_{13}^{4} - 32T_{13}^{3} + 112T_{13}^{2} + 304T_{13} - 368 \) Copy content Toggle raw display
\( T_{17}^{5} + 7T_{17}^{4} - 5T_{17}^{3} - 63T_{17}^{2} + 84 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( (T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 3 T^{4} - 21 T^{3} + 55 T^{2} + \cdots - 48 \) Copy content Toggle raw display
$11$ \( T^{5} + 5 T^{4} - 29 T^{3} - 111 T^{2} + \cdots + 588 \) Copy content Toggle raw display
$13$ \( T^{5} - 5 T^{4} - 32 T^{3} + 112 T^{2} + \cdots - 368 \) Copy content Toggle raw display
$17$ \( T^{5} + 7 T^{4} - 5 T^{3} - 63 T^{2} + \cdots + 84 \) Copy content Toggle raw display
$19$ \( T^{5} + T^{4} - 86 T^{3} + 40 T^{2} + \cdots - 3632 \) Copy content Toggle raw display
$23$ \( T^{5} + 5 T^{4} - 32 T^{3} - 120 T^{2} + \cdots + 384 \) Copy content Toggle raw display
$29$ \( T^{5} + 2 T^{4} - 35 T^{3} - 24 T^{2} + \cdots + 96 \) Copy content Toggle raw display
$31$ \( T^{5} - 16 T^{4} + 37 T^{3} + \cdots + 3568 \) Copy content Toggle raw display
$37$ \( (T - 1)^{5} \) Copy content Toggle raw display
$41$ \( T^{5} + 2 T^{4} - 95 T^{3} - 174 T^{2} + \cdots - 384 \) Copy content Toggle raw display
$43$ \( T^{5} - 12 T^{4} - 39 T^{3} + \cdots - 1488 \) Copy content Toggle raw display
$47$ \( T^{5} + 6 T^{4} - 120 T^{3} + \cdots + 12096 \) Copy content Toggle raw display
$53$ \( T^{5} + 3 T^{4} - 51 T^{3} - 141 T^{2} + \cdots - 144 \) Copy content Toggle raw display
$59$ \( T^{5} - 10 T^{4} - 116 T^{3} + \cdots - 18816 \) Copy content Toggle raw display
$61$ \( T^{5} - 16 T^{4} - 167 T^{3} + \cdots - 61592 \) Copy content Toggle raw display
$67$ \( T^{5} - 4 T^{4} - 248 T^{3} + \cdots - 37376 \) Copy content Toggle raw display
$71$ \( T^{5} - 8 T^{4} - 152 T^{3} + \cdots + 4608 \) Copy content Toggle raw display
$73$ \( T^{5} + 3 T^{4} - 252 T^{3} + \cdots - 21792 \) Copy content Toggle raw display
$79$ \( T^{5} - 2 T^{4} - 176 T^{3} + \cdots - 16736 \) Copy content Toggle raw display
$83$ \( T^{5} - 5 T^{4} - 176 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$89$ \( T^{5} - 7 T^{4} - 296 T^{3} + \cdots - 83856 \) Copy content Toggle raw display
$97$ \( T^{5} - 14 T^{4} - 47 T^{3} + \cdots + 784 \) Copy content Toggle raw display
show more
show less