Properties

Label 3330.2.a.bd
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} + ( -3 + \beta ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} - q^{5} + ( -3 + \beta ) q^{7} + q^{8} - q^{10} + ( 2 - 2 \beta ) q^{11} + ( -2 + 2 \beta ) q^{13} + ( -3 + \beta ) q^{14} + q^{16} + ( -2 + 2 \beta ) q^{17} + ( 1 - 3 \beta ) q^{19} - q^{20} + ( 2 - 2 \beta ) q^{22} + 8 q^{23} + q^{25} + ( -2 + 2 \beta ) q^{26} + ( -3 + \beta ) q^{28} + ( 2 + 4 \beta ) q^{29} + ( -1 - \beta ) q^{31} + q^{32} + ( -2 + 2 \beta ) q^{34} + ( 3 - \beta ) q^{35} + q^{37} + ( 1 - 3 \beta ) q^{38} - q^{40} + 2 q^{41} -4 \beta q^{43} + ( 2 - 2 \beta ) q^{44} + 8 q^{46} + ( 3 - \beta ) q^{47} + ( 5 - 6 \beta ) q^{49} + q^{50} + ( -2 + 2 \beta ) q^{52} + 6 q^{53} + ( -2 + 2 \beta ) q^{55} + ( -3 + \beta ) q^{56} + ( 2 + 4 \beta ) q^{58} + ( 5 - 3 \beta ) q^{59} + ( 2 + 4 \beta ) q^{61} + ( -1 - \beta ) q^{62} + q^{64} + ( 2 - 2 \beta ) q^{65} + ( 5 + 5 \beta ) q^{67} + ( -2 + 2 \beta ) q^{68} + ( 3 - \beta ) q^{70} + ( 4 + 4 \beta ) q^{71} + ( 6 + 4 \beta ) q^{73} + q^{74} + ( 1 - 3 \beta ) q^{76} + ( -12 + 8 \beta ) q^{77} + ( 7 - \beta ) q^{79} - q^{80} + 2 q^{82} + ( 7 - \beta ) q^{83} + ( 2 - 2 \beta ) q^{85} -4 \beta q^{86} + ( 2 - 2 \beta ) q^{88} + 2 q^{89} + ( 12 - 8 \beta ) q^{91} + 8 q^{92} + ( 3 - \beta ) q^{94} + ( -1 + 3 \beta ) q^{95} -2 q^{97} + ( 5 - 6 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} - 6q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} - 6q^{7} + 2q^{8} - 2q^{10} + 4q^{11} - 4q^{13} - 6q^{14} + 2q^{16} - 4q^{17} + 2q^{19} - 2q^{20} + 4q^{22} + 16q^{23} + 2q^{25} - 4q^{26} - 6q^{28} + 4q^{29} - 2q^{31} + 2q^{32} - 4q^{34} + 6q^{35} + 2q^{37} + 2q^{38} - 2q^{40} + 4q^{41} + 4q^{44} + 16q^{46} + 6q^{47} + 10q^{49} + 2q^{50} - 4q^{52} + 12q^{53} - 4q^{55} - 6q^{56} + 4q^{58} + 10q^{59} + 4q^{61} - 2q^{62} + 2q^{64} + 4q^{65} + 10q^{67} - 4q^{68} + 6q^{70} + 8q^{71} + 12q^{73} + 2q^{74} + 2q^{76} - 24q^{77} + 14q^{79} - 2q^{80} + 4q^{82} + 14q^{83} + 4q^{85} + 4q^{88} + 4q^{89} + 24q^{91} + 16q^{92} + 6q^{94} - 2q^{95} - 4q^{97} + 10q^{98} + O(q^{100}) \)

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
−1.73205
1.73205
1.00000 0 1.00000 −1.00000 0 −4.73205 1.00000 0 −1.00000
1.2 1.00000 0 1.00000 −1.00000 0 −1.26795 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.bd 2
3.b odd 2 1 370.2.a.e 2
12.b even 2 1 2960.2.a.q 2
15.d odd 2 1 1850.2.a.x 2
15.e even 4 2 1850.2.b.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.e 2 3.b odd 2 1
1850.2.a.x 2 15.d odd 2 1
1850.2.b.l 4 15.e even 4 2
2960.2.a.q 2 12.b even 2 1
3330.2.a.bd 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{7}^{2} + 6 T_{7} + 6 \)
\( T_{11}^{2} - 4 T_{11} - 8 \)
\( T_{13}^{2} + 4 T_{13} - 8 \)
\( T_{17}^{2} + 4 T_{17} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( 6 + 6 T + T^{2} \)
$11$ \( -8 - 4 T + T^{2} \)
$13$ \( -8 + 4 T + T^{2} \)
$17$ \( -8 + 4 T + T^{2} \)
$19$ \( -26 - 2 T + T^{2} \)
$23$ \( ( -8 + T )^{2} \)
$29$ \( -44 - 4 T + T^{2} \)
$31$ \( -2 + 2 T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( -48 + T^{2} \)
$47$ \( 6 - 6 T + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( -2 - 10 T + T^{2} \)
$61$ \( -44 - 4 T + T^{2} \)
$67$ \( -50 - 10 T + T^{2} \)
$71$ \( -32 - 8 T + T^{2} \)
$73$ \( -12 - 12 T + T^{2} \)
$79$ \( 46 - 14 T + T^{2} \)
$83$ \( 46 - 14 T + T^{2} \)
$89$ \( ( -2 + T )^{2} \)
$97$ \( ( 2 + T )^{2} \)
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