Properties

Label 3330.2.a.bc
Level $3330$
Weight $2$
Character orbit 3330.a
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - q^{10} + ( 1 + 2 \beta ) q^{11} + ( -2 + 3 \beta ) q^{13} + q^{14} + q^{16} + 5 q^{17} + ( -2 + \beta ) q^{19} + q^{20} + ( -1 - 2 \beta ) q^{22} + ( 4 + \beta ) q^{23} + q^{25} + ( 2 - 3 \beta ) q^{26} - q^{28} + ( -1 + 3 \beta ) q^{29} + ( -1 - \beta ) q^{31} - q^{32} -5 q^{34} - q^{35} + q^{37} + ( 2 - \beta ) q^{38} - q^{40} + ( -1 - 3 \beta ) q^{41} + ( -5 - 3 \beta ) q^{43} + ( 1 + 2 \beta ) q^{44} + ( -4 - \beta ) q^{46} + 8 q^{47} -6 q^{49} - q^{50} + ( -2 + 3 \beta ) q^{52} + ( 5 - 4 \beta ) q^{53} + ( 1 + 2 \beta ) q^{55} + q^{56} + ( 1 - 3 \beta ) q^{58} + ( 6 - 4 \beta ) q^{59} + ( -1 - 5 \beta ) q^{61} + ( 1 + \beta ) q^{62} + q^{64} + ( -2 + 3 \beta ) q^{65} + ( 8 + 2 \beta ) q^{67} + 5 q^{68} + q^{70} + ( 4 - 2 \beta ) q^{71} + ( -4 - \beta ) q^{73} - q^{74} + ( -2 + \beta ) q^{76} + ( -1 - 2 \beta ) q^{77} + ( 4 + 4 \beta ) q^{79} + q^{80} + ( 1 + 3 \beta ) q^{82} + ( 10 - \beta ) q^{83} + 5 q^{85} + ( 5 + 3 \beta ) q^{86} + ( -1 - 2 \beta ) q^{88} + ( 2 - 3 \beta ) q^{89} + ( 2 - 3 \beta ) q^{91} + ( 4 + \beta ) q^{92} -8 q^{94} + ( -2 + \beta ) q^{95} + ( 11 - 3 \beta ) q^{97} + 6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} - 2q^{8} - 2q^{10} + 4q^{11} - q^{13} + 2q^{14} + 2q^{16} + 10q^{17} - 3q^{19} + 2q^{20} - 4q^{22} + 9q^{23} + 2q^{25} + q^{26} - 2q^{28} + q^{29} - 3q^{31} - 2q^{32} - 10q^{34} - 2q^{35} + 2q^{37} + 3q^{38} - 2q^{40} - 5q^{41} - 13q^{43} + 4q^{44} - 9q^{46} + 16q^{47} - 12q^{49} - 2q^{50} - q^{52} + 6q^{53} + 4q^{55} + 2q^{56} - q^{58} + 8q^{59} - 7q^{61} + 3q^{62} + 2q^{64} - q^{65} + 18q^{67} + 10q^{68} + 2q^{70} + 6q^{71} - 9q^{73} - 2q^{74} - 3q^{76} - 4q^{77} + 12q^{79} + 2q^{80} + 5q^{82} + 19q^{83} + 10q^{85} + 13q^{86} - 4q^{88} + q^{89} + q^{91} + 9q^{92} - 16q^{94} - 3q^{95} + 19q^{97} + 12q^{98} + O(q^{100}) \)

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
−1.00000 0 1.00000 1.00000 0 −1.00000 −1.00000 0 −1.00000
1.2 −1.00000 0 1.00000 1.00000 0 −1.00000 −1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
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.bc 2
3.b odd 2 1 1110.2.a.r 2
12.b even 2 1 8880.2.a.bp 2
15.d odd 2 1 5550.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.r 2 3.b odd 2 1
3330.2.a.bc 2 1.a even 1 1 trivial
5550.2.a.bw 2 15.d odd 2 1
8880.2.a.bp 2 12.b even 2 1

Hecke kernels

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

\( T_{7} + 1 \)
\( T_{11}^{2} - 4 T_{11} - 13 \)
\( T_{13}^{2} + T_{13} - 38 \)
\( T_{17} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -13 - 4 T + T^{2} \)
$13$ \( -38 + T + T^{2} \)
$17$ \( ( -5 + T )^{2} \)
$19$ \( -2 + 3 T + T^{2} \)
$23$ \( 16 - 9 T + T^{2} \)
$29$ \( -38 - T + T^{2} \)
$31$ \( -2 + 3 T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( -32 + 5 T + T^{2} \)
$43$ \( 4 + 13 T + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( -59 - 6 T + T^{2} \)
$59$ \( -52 - 8 T + T^{2} \)
$61$ \( -94 + 7 T + T^{2} \)
$67$ \( 64 - 18 T + T^{2} \)
$71$ \( -8 - 6 T + T^{2} \)
$73$ \( 16 + 9 T + T^{2} \)
$79$ \( -32 - 12 T + T^{2} \)
$83$ \( 86 - 19 T + T^{2} \)
$89$ \( -38 - T + T^{2} \)
$97$ \( 52 - 19 T + T^{2} \)
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