Properties

Label 3330.2.a.bb
Level $3330$
Weight $2$
Character orbit 3330.a
Self dual yes
Analytic conductor $26.590$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + ( -1 - \beta ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} - q^{5} + ( -1 - \beta ) q^{7} - q^{8} + q^{10} + ( 1 - \beta ) q^{11} + 2 \beta q^{13} + ( 1 + \beta ) q^{14} + q^{16} + ( 3 - \beta ) q^{17} -2 q^{19} - q^{20} + ( -1 + \beta ) q^{22} + ( -2 + 2 \beta ) q^{23} + q^{25} -2 \beta q^{26} + ( -1 - \beta ) q^{28} + ( -1 + 3 \beta ) q^{29} + ( -5 - \beta ) q^{31} - q^{32} + ( -3 + \beta ) q^{34} + ( 1 + \beta ) q^{35} + q^{37} + 2 q^{38} + q^{40} + ( -3 + \beta ) q^{41} + ( 5 - \beta ) q^{43} + ( 1 - \beta ) q^{44} + ( 2 - 2 \beta ) q^{46} + ( 4 - 2 \beta ) q^{47} + ( 2 + 3 \beta ) q^{49} - q^{50} + 2 \beta q^{52} + ( 1 + \beta ) q^{53} + ( -1 + \beta ) q^{55} + ( 1 + \beta ) q^{56} + ( 1 - 3 \beta ) q^{58} + ( -8 + 2 \beta ) q^{59} + ( -3 + \beta ) q^{61} + ( 5 + \beta ) q^{62} + q^{64} -2 \beta q^{65} -2 \beta q^{67} + ( 3 - \beta ) q^{68} + ( -1 - \beta ) q^{70} + ( 2 - 2 \beta ) q^{71} + ( 4 - 2 \beta ) q^{73} - q^{74} -2 q^{76} + ( 7 + \beta ) q^{77} + 2 \beta q^{79} - q^{80} + ( 3 - \beta ) q^{82} + ( -4 - 2 \beta ) q^{83} + ( -3 + \beta ) q^{85} + ( -5 + \beta ) q^{86} + ( -1 + \beta ) q^{88} -10 q^{89} + ( -16 - 4 \beta ) q^{91} + ( -2 + 2 \beta ) q^{92} + ( -4 + 2 \beta ) q^{94} + 2 q^{95} + ( 7 + 3 \beta ) q^{97} + ( -2 - 3 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} - 3q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} - 3q^{7} - 2q^{8} + 2q^{10} + q^{11} + 2q^{13} + 3q^{14} + 2q^{16} + 5q^{17} - 4q^{19} - 2q^{20} - q^{22} - 2q^{23} + 2q^{25} - 2q^{26} - 3q^{28} + q^{29} - 11q^{31} - 2q^{32} - 5q^{34} + 3q^{35} + 2q^{37} + 4q^{38} + 2q^{40} - 5q^{41} + 9q^{43} + q^{44} + 2q^{46} + 6q^{47} + 7q^{49} - 2q^{50} + 2q^{52} + 3q^{53} - q^{55} + 3q^{56} - q^{58} - 14q^{59} - 5q^{61} + 11q^{62} + 2q^{64} - 2q^{65} - 2q^{67} + 5q^{68} - 3q^{70} + 2q^{71} + 6q^{73} - 2q^{74} - 4q^{76} + 15q^{77} + 2q^{79} - 2q^{80} + 5q^{82} - 10q^{83} - 5q^{85} - 9q^{86} - q^{88} - 20q^{89} - 36q^{91} - 2q^{92} - 6q^{94} + 4q^{95} + 17q^{97} - 7q^{98} + O(q^{100}) \)

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
3.37228
−2.37228
−1.00000 0 1.00000 −1.00000 0 −4.37228 −1.00000 0 1.00000
1.2 −1.00000 0 1.00000 −1.00000 0 1.37228 −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.bb 2
3.b odd 2 1 370.2.a.f 2
12.b even 2 1 2960.2.a.o 2
15.d odd 2 1 1850.2.a.q 2
15.e even 4 2 1850.2.b.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.f 2 3.b odd 2 1
1850.2.a.q 2 15.d odd 2 1
1850.2.b.m 4 15.e even 4 2
2960.2.a.o 2 12.b even 2 1
3330.2.a.bb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3330))\):

\( T_{7}^{2} + 3 T_{7} - 6 \)
\( T_{11}^{2} - T_{11} - 8 \)
\( T_{13}^{2} - 2 T_{13} - 32 \)
\( T_{17}^{2} - 5 T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -6 + 3 T + T^{2} \)
$11$ \( -8 - T + T^{2} \)
$13$ \( -32 - 2 T + T^{2} \)
$17$ \( -2 - 5 T + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( -32 + 2 T + T^{2} \)
$29$ \( -74 - T + T^{2} \)
$31$ \( 22 + 11 T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( -2 + 5 T + T^{2} \)
$43$ \( 12 - 9 T + T^{2} \)
$47$ \( -24 - 6 T + T^{2} \)
$53$ \( -6 - 3 T + T^{2} \)
$59$ \( 16 + 14 T + T^{2} \)
$61$ \( -2 + 5 T + T^{2} \)
$67$ \( -32 + 2 T + T^{2} \)
$71$ \( -32 - 2 T + T^{2} \)
$73$ \( -24 - 6 T + T^{2} \)
$79$ \( -32 - 2 T + T^{2} \)
$83$ \( -8 + 10 T + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( -2 - 17 T + T^{2} \)
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