Properties

Label 3006.2.a.s
Level $3006$
Weight $2$
Character orbit 3006.a
Self dual yes
Analytic conductor $24.003$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3006 = 2 \cdot 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3006.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.0030308476\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2777.1
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} + x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1002)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( -1 + \beta_{1} ) q^{5} + ( \beta_{2} - \beta_{3} ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( -1 + \beta_{1} ) q^{5} + ( \beta_{2} - \beta_{3} ) q^{7} - q^{8} + ( 1 - \beta_{1} ) q^{10} + ( 2 + \beta_{3} ) q^{13} + ( -\beta_{2} + \beta_{3} ) q^{14} + q^{16} + ( -3 - \beta_{1} + \beta_{2} ) q^{17} -2 \beta_{2} q^{19} + ( -1 + \beta_{1} ) q^{20} -2 \beta_{2} q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} + ( -2 - \beta_{3} ) q^{26} + ( \beta_{2} - \beta_{3} ) q^{28} + ( -4 - 2 \beta_{1} ) q^{29} + ( \beta_{2} + \beta_{3} ) q^{31} - q^{32} + ( 3 + \beta_{1} - \beta_{2} ) q^{34} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{35} + ( 2 - 3 \beta_{2} ) q^{37} + 2 \beta_{2} q^{38} + ( 1 - \beta_{1} ) q^{40} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{41} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + 2 \beta_{2} q^{46} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{47} + ( 3 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{49} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{50} + ( 2 + \beta_{3} ) q^{52} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{53} + ( -\beta_{2} + \beta_{3} ) q^{56} + ( 4 + 2 \beta_{1} ) q^{58} + ( 2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{59} -2 \beta_{1} q^{61} + ( -\beta_{2} - \beta_{3} ) q^{62} + q^{64} + ( 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -3 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{67} + ( -3 - \beta_{1} + \beta_{2} ) q^{68} + ( 2 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{70} + ( -2 - 2 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{73} + ( -2 + 3 \beta_{2} ) q^{74} -2 \beta_{2} q^{76} + ( -3 - 3 \beta_{1} - \beta_{2} ) q^{79} + ( -1 + \beta_{1} ) q^{80} + ( 3 + \beta_{1} + 3 \beta_{2} ) q^{82} + ( -4 - 4 \beta_{1} - \beta_{2} ) q^{83} + ( -2 - 2 \beta_{1} - 2 \beta_{3} ) q^{85} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{86} + ( -4 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{89} + ( -8 + 2 \beta_{2} - 2 \beta_{3} ) q^{91} -2 \beta_{2} q^{92} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{94} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{95} + ( 2 + 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{97} + ( -3 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{4} - 5q^{5} + q^{7} - 4q^{8} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{4} - 5q^{5} + q^{7} - 4q^{8} + 5q^{10} + 8q^{13} - q^{14} + 4q^{16} - 10q^{17} - 2q^{19} - 5q^{20} - 2q^{23} + 5q^{25} - 8q^{26} + q^{28} - 14q^{29} + q^{31} - 4q^{32} + 10q^{34} - 5q^{35} + 5q^{37} + 2q^{38} + 5q^{40} - 14q^{41} + 2q^{43} + 2q^{46} - 7q^{47} + 13q^{49} - 5q^{50} + 8q^{52} - 3q^{53} - q^{56} + 14q^{58} + 5q^{59} + 2q^{61} - q^{62} + 4q^{64} - 6q^{65} - 7q^{67} - 10q^{68} + 5q^{70} - 2q^{71} + 2q^{73} - 5q^{74} - 2q^{76} - 10q^{79} - 5q^{80} + 14q^{82} - 13q^{83} - 6q^{85} - 2q^{86} - 13q^{89} - 30q^{91} - 2q^{92} + 7q^{94} + 2q^{95} + 5q^{97} - 13q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - \nu^{2} - 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - \beta_{2} + \beta_{1} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} - 3 \beta_{2} + 5 \beta_{1} + 7\)\()/2\)

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
−1.50848
0.825785
−0.679643
2.36234
−1.00000 0 1.00000 −3.69113 0 −2.24216 −1.00000 0 3.69113
1.2 −1.00000 0 1.00000 −2.77037 0 1.86579 −1.00000 0 2.77037
1.3 −1.00000 0 1.00000 −0.416566 0 4.65960 −1.00000 0 0.416566
1.4 −1.00000 0 1.00000 1.87806 0 −3.28324 −1.00000 0 −1.87806
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(167\) \(-1\)

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 3006.2.a.s 4
3.b odd 2 1 1002.2.a.i 4
12.b even 2 1 8016.2.a.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.i 4 3.b odd 2 1
3006.2.a.s 4 1.a even 1 1 trivial
8016.2.a.o 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3006))\):

\( T_{5}^{4} + 5 T_{5}^{3} - 20 T_{5} - 8 \)
\( T_{7}^{4} - T_{7}^{3} - 20 T_{7}^{2} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( T^{4} \)
$5$ \( -8 - 20 T + 5 T^{3} + T^{4} \)
$7$ \( 64 - 20 T^{2} - T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( -16 + 56 T + 4 T^{2} - 8 T^{3} + T^{4} \)
$17$ \( -16 - 16 T + 20 T^{2} + 10 T^{3} + T^{4} \)
$19$ \( 128 - 32 T - 32 T^{2} + 2 T^{3} + T^{4} \)
$23$ \( 128 - 32 T - 32 T^{2} + 2 T^{3} + T^{4} \)
$29$ \( -32 - 56 T + 36 T^{2} + 14 T^{3} + T^{4} \)
$31$ \( -64 + 96 T - 36 T^{2} - T^{3} + T^{4} \)
$37$ \( 568 + 184 T - 66 T^{2} - 5 T^{3} + T^{4} \)
$41$ \( -1072 - 480 T - 12 T^{2} + 14 T^{3} + T^{4} \)
$43$ \( 32 - 40 T - 32 T^{2} - 2 T^{3} + T^{4} \)
$47$ \( -64 - 208 T - 32 T^{2} + 7 T^{3} + T^{4} \)
$53$ \( 8 - 36 T - 40 T^{2} + 3 T^{3} + T^{4} \)
$59$ \( 808 - 36 T - 116 T^{2} - 5 T^{3} + T^{4} \)
$61$ \( 128 + 72 T - 36 T^{2} - 2 T^{3} + T^{4} \)
$67$ \( -3208 - 1848 T - 186 T^{2} + 7 T^{3} + T^{4} \)
$71$ \( 6784 - 128 T - 168 T^{2} + 2 T^{3} + T^{4} \)
$73$ \( 512 + 72 T - 132 T^{2} - 2 T^{3} + T^{4} \)
$79$ \( 1472 - 376 T - 56 T^{2} + 10 T^{3} + T^{4} \)
$83$ \( 4072 - 772 T - 96 T^{2} + 13 T^{3} + T^{4} \)
$89$ \( -1448 - 516 T - 2 T^{2} + 13 T^{3} + T^{4} \)
$97$ \( 3256 + 628 T - 146 T^{2} - 5 T^{3} + T^{4} \)
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