Properties

Label 4592.2.a.bb
Level $4592$
Weight $2$
Character orbit 4592.a
Self dual yes
Analytic conductor $36.667$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4592 = 2^{4} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4592.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.6673046082\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.633117.1
Defining polynomial: \(x^{5} - x^{4} - 6 x^{3} + 4 x^{2} + 6 x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 287)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{3} + ( -1 - \beta_{2} + \beta_{3} ) q^{5} - q^{7} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{3} + ( -1 - \beta_{2} + \beta_{3} ) q^{5} - q^{7} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{9} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{11} + ( 1 + \beta_{2} + \beta_{4} ) q^{13} + ( 2 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{15} + ( 3 - \beta_{3} - 2 \beta_{4} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{19} + ( 1 - \beta_{1} ) q^{21} + ( -1 - \beta_{1} - 2 \beta_{3} ) q^{23} + ( 5 - \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{25} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{27} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{29} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} + ( -1 + 4 \beta_{1} - \beta_{2} + 3 \beta_{3} + 4 \beta_{4} ) q^{33} + ( 1 + \beta_{2} - \beta_{3} ) q^{35} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{37} + ( -1 + 2 \beta_{1} - \beta_{4} ) q^{39} - q^{41} + ( 1 + 3 \beta_{3} ) q^{43} + ( -8 + 7 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{45} + ( -3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{47} + q^{49} + ( -3 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{51} + ( 3 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{53} + ( -11 + 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{55} + ( \beta_{1} + \beta_{2} - \beta_{4} ) q^{57} + ( -3 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{59} + ( 5 - \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{61} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{63} + ( -7 + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{65} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{67} + ( -6 + 2 \beta_{1} - 3 \beta_{2} ) q^{69} + ( 5 + \beta_{1} + 4 \beta_{3} + 3 \beta_{4} ) q^{71} + ( 9 - 3 \beta_{1} - \beta_{4} ) q^{73} + ( -8 + 6 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} ) q^{75} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{77} + ( 8 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{79} + ( 2 - \beta_{1} - \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{81} + ( 5 - 3 \beta_{1} - 5 \beta_{4} ) q^{83} + ( -5 + 2 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{85} + ( 7 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} ) q^{87} + ( 1 + \beta_{1} - \beta_{3} ) q^{89} + ( -1 - \beta_{2} - \beta_{4} ) q^{91} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{93} + ( 3 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} - \beta_{4} ) q^{95} + ( 3 + 5 \beta_{1} + 2 \beta_{3} ) q^{97} + ( 11 - 6 \beta_{1} + \beta_{2} - 5 \beta_{3} - 6 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 4q^{3} - 5q^{5} - 5q^{7} + q^{9} + O(q^{10}) \) \( 5q - 4q^{3} - 5q^{5} - 5q^{7} + q^{9} - 2q^{11} + 5q^{13} + 5q^{15} + 13q^{17} + 4q^{21} - 2q^{23} + 22q^{25} - 10q^{27} - 5q^{29} - 17q^{31} + 3q^{33} + 5q^{35} - 7q^{37} - 5q^{39} - 5q^{41} - q^{43} - 23q^{45} - 9q^{47} + 5q^{49} - 5q^{51} + 5q^{53} - 33q^{55} - 3q^{57} - 7q^{59} + 22q^{61} - q^{63} - 31q^{65} + 3q^{67} - 22q^{69} + 24q^{71} + 40q^{73} - 24q^{75} + 2q^{77} + 42q^{79} + 9q^{81} + 12q^{83} - 23q^{85} + 32q^{87} + 8q^{89} - 5q^{91} - 11q^{93} + 17q^{95} + 16q^{97} + 45q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 6 \nu^{2} - \nu + 4 \)
\(\beta_{4}\)\(=\)\( -\nu^{4} + \nu^{3} + 5 \nu^{2} - 3 \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{3} + 6 \beta_{2} + \beta_{1} + 14\)

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
−2.03121
−1.08727
0.460315
1.20098
2.45719
0 −3.03121 0 −3.82713 0 −1.00000 0 6.18825 0
1.2 0 −2.08727 0 0.209668 0 −1.00000 0 1.35670 0
1.3 0 −0.539685 0 4.10136 0 −1.00000 0 −2.70874 0
1.4 0 0.200978 0 −3.21704 0 −1.00000 0 −2.95961 0
1.5 0 1.45719 0 −2.26685 0 −1.00000 0 −0.876597 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
\(2\) \(-1\)
\(7\) \(1\)
\(41\) \(1\)

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 4592.2.a.bb 5
4.b odd 2 1 287.2.a.e 5
12.b even 2 1 2583.2.a.r 5
20.d odd 2 1 7175.2.a.n 5
28.d even 2 1 2009.2.a.n 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.a.e 5 4.b odd 2 1
2009.2.a.n 5 28.d even 2 1
2583.2.a.r 5 12.b even 2 1
4592.2.a.bb 5 1.a even 1 1 trivial
7175.2.a.n 5 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4592))\):

\( T_{3}^{5} + 4 T_{3}^{4} - 10 T_{3}^{2} - 3 T_{3} + 1 \)
\( T_{5}^{5} + 5 T_{5}^{4} - 11 T_{5}^{3} - 86 T_{5}^{2} - 96 T_{5} + 24 \)
\( T_{11}^{5} + 2 T_{11}^{4} - 63 T_{11}^{3} - 140 T_{11}^{2} + 972 T_{11} + 2472 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( 1 - 3 T - 10 T^{2} + 4 T^{4} + T^{5} \)
$5$ \( 24 - 96 T - 86 T^{2} - 11 T^{3} + 5 T^{4} + T^{5} \)
$7$ \( ( 1 + T )^{5} \)
$11$ \( 2472 + 972 T - 140 T^{2} - 63 T^{3} + 2 T^{4} + T^{5} \)
$13$ \( 49 - 120 T + 80 T^{2} - 9 T^{3} - 5 T^{4} + T^{5} \)
$17$ \( 2049 - 1554 T + 304 T^{2} + 25 T^{3} - 13 T^{4} + T^{5} \)
$19$ \( 1 - 23 T + 132 T^{2} - 48 T^{3} + T^{5} \)
$23$ \( -1317 + 1149 T - 26 T^{2} - 66 T^{3} + 2 T^{4} + T^{5} \)
$29$ \( 1512 + 396 T - 350 T^{2} - 71 T^{3} + 5 T^{4} + T^{5} \)
$31$ \( 56 - 148 T - 24 T^{2} + 71 T^{3} + 17 T^{4} + T^{5} \)
$37$ \( 4 + 78 T - 157 T^{2} - 36 T^{3} + 7 T^{4} + T^{5} \)
$41$ \( ( 1 + T )^{5} \)
$43$ \( 1751 + 2222 T + 26 T^{2} - 113 T^{3} + T^{4} + T^{5} \)
$47$ \( 10092 + 318 T - 1039 T^{2} - 120 T^{3} + 9 T^{4} + T^{5} \)
$53$ \( 2328 - 2820 T + 1024 T^{2} - 109 T^{3} - 5 T^{4} + T^{5} \)
$59$ \( 21000 + 7500 T - 880 T^{2} - 171 T^{3} + 7 T^{4} + T^{5} \)
$61$ \( 2504 - 948 T - 252 T^{2} + 153 T^{3} - 22 T^{4} + T^{5} \)
$67$ \( 472 - 932 T + 576 T^{2} - 105 T^{3} - 3 T^{4} + T^{5} \)
$71$ \( -43128 - 3876 T + 1862 T^{2} + 41 T^{3} - 24 T^{4} + T^{5} \)
$73$ \( -12184 + 11532 T - 3882 T^{2} + 585 T^{3} - 40 T^{4} + T^{5} \)
$79$ \( 75008 - 13760 T - 1920 T^{2} + 572 T^{3} - 42 T^{4} + T^{5} \)
$83$ \( -24696 + 11676 T + 2542 T^{2} - 259 T^{3} - 12 T^{4} + T^{5} \)
$89$ \( 3 - 9 T + 2 T^{2} + 12 T^{3} - 8 T^{4} + T^{5} \)
$97$ \( -10493 + 4957 T + 1814 T^{2} - 162 T^{3} - 16 T^{4} + T^{5} \)
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