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 $1$

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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: \(x^{5} - x^{4} - 20 x^{3} + 39 x^{2} + 9 x - 36\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{4} + 20 \nu^{2} - 19 \nu - 22 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{4} - \nu^{3} + 37 \nu^{2} - 21 \nu - 36 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 5 \nu^{4} + 4 \nu^{3} - 94 \nu^{2} + 33 \nu + 120 \)\()/6\)
\(\beta_{4}\)\(=\)\((\)\( -4 \nu^{4} - 2 \nu^{3} + 77 \nu^{2} - 39 \nu - 96 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} - \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} - \beta_{3} - 4 \beta_{2} + \beta_{1} + 15\)\()/2\)
\(\nu^{3}\)\(=\)\(7 \beta_{4} + 10 \beta_{3} + 3 \beta_{2} - 6 \beta_{1} - 6\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{4} - 39 \beta_{3} - 80 \beta_{2} + 35 \beta_{1} + 237\)\()/2\)

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3330))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{5} \)
$3$ \( T^{5} \)
$5$ \( ( 1 + T )^{5} \)
$7$ \( -48 + 48 T + 55 T^{2} - 21 T^{3} - 3 T^{4} + T^{5} \)
$11$ \( 588 + 210 T - 111 T^{2} - 29 T^{3} + 5 T^{4} + T^{5} \)
$13$ \( -368 + 304 T + 112 T^{2} - 32 T^{3} - 5 T^{4} + T^{5} \)
$17$ \( 84 - 63 T^{2} - 5 T^{3} + 7 T^{4} + T^{5} \)
$19$ \( -3632 + 1912 T + 40 T^{2} - 86 T^{3} + T^{4} + T^{5} \)
$23$ \( 384 + 288 T - 120 T^{2} - 32 T^{3} + 5 T^{4} + T^{5} \)
$29$ \( 96 + 252 T - 24 T^{2} - 35 T^{3} + 2 T^{4} + T^{5} \)
$31$ \( 3568 - 2620 T + 472 T^{2} + 37 T^{3} - 16 T^{4} + T^{5} \)
$37$ \( ( -1 + T )^{5} \)
$41$ \( -384 + 1632 T - 174 T^{2} - 95 T^{3} + 2 T^{4} + T^{5} \)
$43$ \( -1488 - 60 T + 616 T^{2} - 39 T^{3} - 12 T^{4} + T^{5} \)
$47$ \( 12096 + 3024 T - 504 T^{2} - 120 T^{3} + 6 T^{4} + T^{5} \)
$53$ \( -144 + 324 T - 141 T^{2} - 51 T^{3} + 3 T^{4} + T^{5} \)
$59$ \( -18816 + 3360 T + 888 T^{2} - 116 T^{3} - 10 T^{4} + T^{5} \)
$61$ \( -61592 + 8552 T + 2440 T^{2} - 167 T^{3} - 16 T^{4} + T^{5} \)
$67$ \( -37376 + 5120 T + 1600 T^{2} - 248 T^{3} - 4 T^{4} + T^{5} \)
$71$ \( 4608 + 5760 T + 480 T^{2} - 152 T^{3} - 8 T^{4} + T^{5} \)
$73$ \( -21792 + 14832 T - 188 T^{2} - 252 T^{3} + 3 T^{4} + T^{5} \)
$79$ \( -16736 + 5680 T + 544 T^{2} - 176 T^{3} - 2 T^{4} + T^{5} \)
$83$ \( 2304 + 864 T - 516 T^{2} - 176 T^{3} - 5 T^{4} + T^{5} \)
$89$ \( -83856 + 16848 T + 1344 T^{2} - 296 T^{3} - 7 T^{4} + T^{5} \)
$97$ \( 784 + 1372 T + 406 T^{2} - 47 T^{3} - 14 T^{4} + T^{5} \)
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