Properties

Label 5520.2.a.cc
Level $5520$
Weight $2$
Character orbit 5520.a
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.20087896.1
Defining polynomial: \(x^{5} - x^{4} - 21 x^{3} + 5 x^{2} + 84 x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2760)
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.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + ( 1 - \beta_{1} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + q^{5} + ( 1 - \beta_{1} ) q^{7} + q^{9} + ( 1 - \beta_{2} ) q^{11} + ( 1 - \beta_{4} ) q^{13} + q^{15} + ( 2 + \beta_{1} + \beta_{3} ) q^{17} + ( 1 + \beta_{3} ) q^{19} + ( 1 - \beta_{1} ) q^{21} + q^{23} + q^{25} + q^{27} + ( 2 - \beta_{1} + \beta_{2} ) q^{29} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} ) q^{31} + ( 1 - \beta_{2} ) q^{33} + ( 1 - \beta_{1} ) q^{35} + ( 1 + \beta_{1} ) q^{37} + ( 1 - \beta_{4} ) q^{39} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{41} + ( -\beta_{3} + \beta_{4} ) q^{43} + q^{45} + ( -1 - \beta_{3} ) q^{47} + ( 4 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{49} + ( 2 + \beta_{1} + \beta_{3} ) q^{51} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{53} + ( 1 - \beta_{2} ) q^{55} + ( 1 + \beta_{3} ) q^{57} + ( -2 + 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{59} + ( 3 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{61} + ( 1 - \beta_{1} ) q^{63} + ( 1 - \beta_{4} ) q^{65} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{67} + q^{69} + ( -2 + \beta_{1} + \beta_{2} ) q^{71} + ( 3 - 2 \beta_{1} - \beta_{3} ) q^{73} + q^{75} + ( 3 - 2 \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{77} + ( -4 - \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{79} + q^{81} + ( 4 + \beta_{1} + \beta_{3} ) q^{83} + ( 2 + \beta_{1} + \beta_{3} ) q^{85} + ( 2 - \beta_{1} + \beta_{2} ) q^{87} + ( 4 + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{89} + ( -5 + 2 \beta_{2} - 3 \beta_{3} ) q^{91} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} ) q^{93} + ( 1 + \beta_{3} ) q^{95} + ( 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{97} + ( 1 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} + 5q^{5} + 4q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} + 5q^{5} + 4q^{7} + 5q^{9} + 4q^{11} + 4q^{13} + 5q^{15} + 10q^{17} + 4q^{19} + 4q^{21} + 5q^{23} + 5q^{25} + 5q^{27} + 10q^{29} - 6q^{31} + 4q^{33} + 4q^{35} + 6q^{37} + 4q^{39} + 12q^{41} + 2q^{43} + 5q^{45} - 4q^{47} + 19q^{49} + 10q^{51} + 4q^{55} + 4q^{57} - 6q^{59} + 16q^{61} + 4q^{63} + 4q^{65} + 4q^{67} + 5q^{69} - 8q^{71} + 14q^{73} + 5q^{75} + 16q^{77} - 18q^{79} + 5q^{81} + 20q^{83} + 10q^{85} + 10q^{87} + 18q^{89} - 20q^{91} - 6q^{93} + 4q^{95} + 4q^{97} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 21 x^{3} + 5 x^{2} + 84 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{2} - \nu - 8 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} + 4 \nu^{3} + 13 \nu^{2} - 36 \nu - 20 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - 21 \nu^{2} - 16 \nu + 60 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 21 \nu^{2} + 60 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} - \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} - \beta_{3} + 4 \beta_{1} + 16\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(15 \beta_{4} - 11 \beta_{3} + 4 \beta_{2} + 8 \beta_{1} + 12\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(37 \beta_{4} - 21 \beta_{3} + 84 \beta_{1} + 216\)\()/2\)

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
4.50582
−3.43588
−2.39144
2.27399
0.0475116
0 1.00000 0 1.00000 0 −2.89831 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 −2.62057 0 1.00000 0
1.3 0 1.00000 0 1.00000 0 0.944775 0 1.00000 0
1.4 0 1.00000 0 1.00000 0 3.55148 0 1.00000 0
1.5 0 1.00000 0 1.00000 0 5.02263 0 1.00000 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
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 5520.2.a.cc 5
4.b odd 2 1 2760.2.a.w 5
12.b even 2 1 8280.2.a.br 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.w 5 4.b odd 2 1
5520.2.a.cc 5 1.a even 1 1 trivial
8280.2.a.br 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5520))\):

\( T_{7}^{5} - 4 T_{7}^{4} - 19 T_{7}^{3} + 54 T_{7}^{2} + 104 T_{7} - 128 \)
\( T_{11}^{5} - 4 T_{11}^{4} - 46 T_{11}^{3} + 160 T_{11}^{2} + 408 T_{11} - 1312 \)
\( T_{13}^{5} - 4 T_{13}^{4} - 66 T_{13}^{3} + 248 T_{13}^{2} + 968 T_{13} - 3584 \)
\( T_{17}^{5} - 10 T_{17}^{4} - 25 T_{17}^{3} + 584 T_{17}^{2} - 2148 T_{17} + 2416 \)
\( T_{19}^{5} - 4 T_{19}^{4} - 54 T_{19}^{3} + 120 T_{19}^{2} + 256 T_{19} - 512 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( ( -1 + T )^{5} \)
$5$ \( ( -1 + T )^{5} \)
$7$ \( -128 + 104 T + 54 T^{2} - 19 T^{3} - 4 T^{4} + T^{5} \)
$11$ \( -1312 + 408 T + 160 T^{2} - 46 T^{3} - 4 T^{4} + T^{5} \)
$13$ \( -3584 + 968 T + 248 T^{2} - 66 T^{3} - 4 T^{4} + T^{5} \)
$17$ \( 2416 - 2148 T + 584 T^{2} - 25 T^{3} - 10 T^{4} + T^{5} \)
$19$ \( -512 + 256 T + 120 T^{2} - 54 T^{3} - 4 T^{4} + T^{5} \)
$23$ \( ( -1 + T )^{5} \)
$29$ \( 1424 - 1476 T + 472 T^{2} - 25 T^{3} - 10 T^{4} + T^{5} \)
$31$ \( 4096 + 128 T - 568 T^{2} - 79 T^{3} + 6 T^{4} + T^{5} \)
$37$ \( -112 + 44 T + 76 T^{2} - 11 T^{3} - 6 T^{4} + T^{5} \)
$41$ \( 1912 - 3196 T + 1010 T^{2} - 53 T^{3} - 12 T^{4} + T^{5} \)
$43$ \( -128 + 1344 T + 40 T^{2} - 84 T^{3} - 2 T^{4} + T^{5} \)
$47$ \( 512 + 256 T - 120 T^{2} - 54 T^{3} + 4 T^{4} + T^{5} \)
$53$ \( -2344 + 3364 T + 202 T^{2} - 141 T^{3} + T^{5} \)
$59$ \( 53248 + 14032 T - 1536 T^{2} - 269 T^{3} + 6 T^{4} + T^{5} \)
$61$ \( 25856 - 17480 T + 3368 T^{2} - 126 T^{3} - 16 T^{4} + T^{5} \)
$67$ \( 6176 + 4160 T + 162 T^{2} - 123 T^{3} - 4 T^{4} + T^{5} \)
$71$ \( 6464 + 1000 T - 482 T^{2} - 65 T^{3} + 8 T^{4} + T^{5} \)
$73$ \( -656 + 88 T + 316 T^{2} - 42 T^{3} - 14 T^{4} + T^{5} \)
$79$ \( 32768 - 768 T - 1984 T^{2} - 88 T^{3} + 18 T^{4} + T^{5} \)
$83$ \( 9056 - 4384 T + 414 T^{2} + 95 T^{3} - 20 T^{4} + T^{5} \)
$89$ \( -3328 + 7936 T + 1552 T^{2} - 120 T^{3} - 18 T^{4} + T^{5} \)
$97$ \( -13952 + 13536 T + 1360 T^{2} - 364 T^{3} - 4 T^{4} + T^{5} \)
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