Properties

Label 6012.2.a.f
Level $6012$
Weight $2$
Character orbit 6012.a
Self dual yes
Analytic conductor $48.006$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6012.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0060616952\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.161121.1
Defining polynomial: \(x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2004)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{3} + \beta_{4} ) q^{5} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{7} +O(q^{10})\) \( q + ( 1 + \beta_{3} + \beta_{4} ) q^{5} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{7} + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{13} + ( 1 + 2 \beta_{1} ) q^{17} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{19} + ( 2 + \beta_{1} - \beta_{3} + \beta_{4} ) q^{23} + ( -2 - 2 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} ) q^{25} + ( 3 - \beta_{3} - 2 \beta_{4} ) q^{29} + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{31} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{35} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{37} + ( 2 + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{41} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{43} + ( 4 - 3 \beta_{1} + 4 \beta_{3} + \beta_{4} ) q^{47} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{49} + ( 4 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{53} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{55} + ( 2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{59} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{61} + ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} ) q^{65} + ( 1 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{67} + ( 6 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{71} + ( -3 + 3 \beta_{1} - \beta_{2} + \beta_{4} ) q^{73} + ( 3 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{77} + ( -4 \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} ) q^{79} + ( 2 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{83} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{85} + ( 5 - 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{89} + ( 3 - 2 \beta_{1} + \beta_{2} - 5 \beta_{3} - 3 \beta_{4} ) q^{91} + ( -\beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} ) q^{95} + ( 3 + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 7q^{5} - 2q^{7} + O(q^{10}) \) \( 5q + 7q^{5} - 2q^{7} + 5q^{11} - 8q^{13} + 7q^{17} + 2q^{19} + 13q^{23} + 2q^{25} + 11q^{29} - 12q^{31} + 12q^{35} - 7q^{37} + 12q^{41} + 19q^{47} - 9q^{49} + 21q^{53} - q^{55} + 7q^{59} - 6q^{61} - 14q^{65} + 10q^{67} + 35q^{71} - 8q^{73} + 6q^{77} + 11q^{83} + 5q^{85} + 32q^{89} + 5q^{91} + 19q^{95} + 11q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 3 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 6 \nu^{2} + 3 \nu + 4 \)
\(\beta_{4}\)\(=\)\( -2 \nu^{4} + 3 \nu^{3} + 11 \nu^{2} - 11 \nu - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} - 4 \beta_{3} + 7 \beta_{2} + 2 \beta_{1} + 15\)

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
1.31991
−2.07823
−0.261082
2.56399
−0.544588
0 0 0 −1.28459 0 −1.96468 0 0 0
1.2 0 0 0 0.614948 0 −3.46328 0 0 0
1.3 0 0 0 1.38924 0 −0.871845 0 0 0
1.4 0 0 0 1.85181 0 2.41580 0 0 0
1.5 0 0 0 4.42860 0 1.88401 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 6012.2.a.f 5
3.b odd 2 1 2004.2.a.b 5
12.b even 2 1 8016.2.a.q 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.2.a.b 5 3.b odd 2 1
6012.2.a.f 5 1.a even 1 1 trivial
8016.2.a.q 5 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 7 T_{5}^{4} + 11 T_{5}^{3} + 6 T_{5}^{2} - 21 T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( T^{5} \)
$5$ \( 9 - 21 T + 6 T^{2} + 11 T^{3} - 7 T^{4} + T^{5} \)
$7$ \( 27 + 27 T - 15 T^{2} - 11 T^{3} + 2 T^{4} + T^{5} \)
$11$ \( -243 + 45 T + 84 T^{2} - 19 T^{3} - 5 T^{4} + T^{5} \)
$13$ \( 27 - 218 T - 124 T^{2} - T^{3} + 8 T^{4} + T^{5} \)
$17$ \( -3 - 27 T + 74 T^{2} - 6 T^{3} - 7 T^{4} + T^{5} \)
$19$ \( -601 + 344 T + 66 T^{2} - 45 T^{3} - 2 T^{4} + T^{5} \)
$23$ \( 243 - 513 T + 144 T^{2} + 29 T^{3} - 13 T^{4} + T^{5} \)
$29$ \( -27 - 279 T + 102 T^{2} + 21 T^{3} - 11 T^{4} + T^{5} \)
$31$ \( 421 - 42 T - 197 T^{2} - 2 T^{3} + 12 T^{4} + T^{5} \)
$37$ \( -31 - 161 T - 116 T^{2} - 7 T^{3} + 7 T^{4} + T^{5} \)
$41$ \( 2151 - 2274 T + 605 T^{2} - 12 T^{3} - 12 T^{4} + T^{5} \)
$43$ \( 6057 + 2196 T - 197 T^{2} - 108 T^{3} + T^{5} \)
$47$ \( -69093 + 312 T + 2259 T^{2} - 67 T^{3} - 19 T^{4} + T^{5} \)
$53$ \( -603 - 1140 T + 381 T^{2} + 85 T^{3} - 21 T^{4} + T^{5} \)
$59$ \( 573 - 1512 T + 863 T^{2} - 99 T^{3} - 7 T^{4} + T^{5} \)
$61$ \( -4133 + 3728 T - 202 T^{2} - 147 T^{3} + 6 T^{4} + T^{5} \)
$67$ \( 1611 - 6302 T + 2444 T^{2} - 193 T^{3} - 10 T^{4} + T^{5} \)
$71$ \( 36999 - 6549 T - 1295 T^{2} + 408 T^{3} - 35 T^{4} + T^{5} \)
$73$ \( 173 + 952 T - 244 T^{2} - 53 T^{3} + 8 T^{4} + T^{5} \)
$79$ \( -7569 + 2262 T + 805 T^{2} - 198 T^{3} + T^{5} \)
$83$ \( 423 - 1428 T + 1095 T^{2} - 89 T^{3} - 11 T^{4} + T^{5} \)
$89$ \( 133587 - 43788 T + 3630 T^{2} + 161 T^{3} - 32 T^{4} + T^{5} \)
$97$ \( -109 - 305 T + 115 T^{2} + 20 T^{3} - 11 T^{4} + T^{5} \)
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