Properties

Label 3300.2.a.w
Level $3300$
Weight $2$
Character orbit 3300.a
Self dual yes
Analytic conductor $26.351$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3300.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.3506326670\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 660)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 - \beta ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 - \beta ) q^{7} + q^{9} + q^{11} + ( 1 + \beta ) q^{13} + ( -3 - \beta ) q^{17} -2 \beta q^{19} + ( -1 - \beta ) q^{21} + q^{27} + 8 q^{29} + ( 2 + 2 \beta ) q^{31} + q^{33} + ( -4 + 2 \beta ) q^{37} + ( 1 + \beta ) q^{39} + 8 q^{41} + ( 7 - \beta ) q^{43} + ( -2 + 2 \beta ) q^{47} + ( 7 + 2 \beta ) q^{49} + ( -3 - \beta ) q^{51} -2 q^{53} -2 \beta q^{57} + 8 q^{59} + 2 \beta q^{61} + ( -1 - \beta ) q^{63} + 4 q^{67} -4 \beta q^{71} + ( 3 - \beta ) q^{73} + ( -1 - \beta ) q^{77} + ( -4 - 2 \beta ) q^{79} + q^{81} + ( 7 + \beta ) q^{83} + 8 q^{87} + 6 q^{89} + ( -14 - 2 \beta ) q^{91} + ( 2 + 2 \beta ) q^{93} + ( -2 + 4 \beta ) q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 6 q^{17} - 2 q^{21} + 2 q^{27} + 16 q^{29} + 4 q^{31} + 2 q^{33} - 8 q^{37} + 2 q^{39} + 16 q^{41} + 14 q^{43} - 4 q^{47} + 14 q^{49} - 6 q^{51} - 4 q^{53} + 16 q^{59} - 2 q^{63} + 8 q^{67} + 6 q^{73} - 2 q^{77} - 8 q^{79} + 2 q^{81} + 14 q^{83} + 16 q^{87} + 12 q^{89} - 28 q^{91} + 4 q^{93} - 4 q^{97} + 2 q^{99} + O(q^{100}) \)

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
2.30278
−1.30278
0 1.00000 0 0 0 −4.60555 0 1.00000 0
1.2 0 1.00000 0 0 0 2.60555 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)

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 3300.2.a.w 2
3.b odd 2 1 9900.2.a.bl 2
5.b even 2 1 660.2.a.e 2
5.c odd 4 2 3300.2.c.l 4
15.d odd 2 1 1980.2.a.h 2
15.e even 4 2 9900.2.c.q 4
20.d odd 2 1 2640.2.a.bc 2
55.d odd 2 1 7260.2.a.w 2
60.h even 2 1 7920.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.a.e 2 5.b even 2 1
1980.2.a.h 2 15.d odd 2 1
2640.2.a.bc 2 20.d odd 2 1
3300.2.a.w 2 1.a even 1 1 trivial
3300.2.c.l 4 5.c odd 4 2
7260.2.a.w 2 55.d odd 2 1
7920.2.a.bo 2 60.h even 2 1
9900.2.a.bl 2 3.b odd 2 1
9900.2.c.q 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3300))\):

\( T_{7}^{2} + 2 T_{7} - 12 \)
\( T_{13}^{2} - 2 T_{13} - 12 \)
\( T_{17}^{2} + 6 T_{17} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -12 + 2 T + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -12 - 2 T + T^{2} \)
$17$ \( -4 + 6 T + T^{2} \)
$19$ \( -52 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -8 + T )^{2} \)
$31$ \( -48 - 4 T + T^{2} \)
$37$ \( -36 + 8 T + T^{2} \)
$41$ \( ( -8 + T )^{2} \)
$43$ \( 36 - 14 T + T^{2} \)
$47$ \( -48 + 4 T + T^{2} \)
$53$ \( ( 2 + T )^{2} \)
$59$ \( ( -8 + T )^{2} \)
$61$ \( -52 + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( -208 + T^{2} \)
$73$ \( -4 - 6 T + T^{2} \)
$79$ \( -36 + 8 T + T^{2} \)
$83$ \( 36 - 14 T + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( -204 + 4 T + T^{2} \)
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