Properties

Label 5600.2.a.bw
Level $5600$
Weight $2$
Character orbit 5600.a
Self dual yes
Analytic conductor $44.716$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5600 = 2^{5} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.7162251319\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.504568.1
Defining polynomial: \(x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 7 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1120)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - q^{7} + ( 1 + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} - q^{7} + ( 1 + \beta_{2} + \beta_{3} ) q^{9} + ( -1 - \beta_{1} - \beta_{3} ) q^{11} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{13} + ( 1 - \beta_{2} + \beta_{4} ) q^{17} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{19} -\beta_{2} q^{21} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{23} + ( 2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{27} + ( -2 + \beta_{1} - \beta_{4} ) q^{29} + ( -2 - 2 \beta_{4} ) q^{31} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{33} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{37} + ( -5 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{39} + 2 \beta_{1} q^{41} + ( 2 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{43} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{47} + q^{49} + ( -3 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{51} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{53} + ( \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{57} + ( -4 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{59} + ( -2 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{61} + ( -1 - \beta_{2} - \beta_{3} ) q^{63} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{67} + ( 2 - 4 \beta_{2} + 2 \beta_{4} ) q^{69} + ( -\beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{71} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{73} + ( 1 + \beta_{1} + \beta_{3} ) q^{77} + ( -7 + \beta_{2} + \beta_{4} ) q^{79} + ( -3 - 3 \beta_{1} + 7 \beta_{2} + \beta_{3} - \beta_{4} ) q^{81} + ( 6 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{83} + ( 2 + 3 \beta_{1} - 6 \beta_{2} + 3 \beta_{4} ) q^{87} + ( -2 \beta_{3} + 2 \beta_{4} ) q^{89} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{91} + ( -2 + 2 \beta_{1} - 6 \beta_{2} + 4 \beta_{4} ) q^{93} + ( -3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{97} + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 2q^{3} - 5q^{7} + 7q^{9} + O(q^{10}) \) \( 5q + 2q^{3} - 5q^{7} + 7q^{9} - 4q^{11} + 4q^{17} - 12q^{19} - 2q^{21} - 8q^{23} + 14q^{27} - 12q^{29} - 12q^{31} - 8q^{33} + 12q^{37} - 32q^{39} - 2q^{41} + 8q^{43} - 14q^{47} + 5q^{49} - 12q^{51} + 8q^{53} - 8q^{57} - 16q^{59} - 10q^{61} - 7q^{63} + 4q^{67} + 4q^{69} - 4q^{71} - 12q^{73} + 4q^{77} - 32q^{79} + q^{81} + 24q^{83} - 2q^{87} + 2q^{89} - 20q^{93} - 12q^{97} - 40q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 7 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 3 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 5 \nu + 2 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 6 \nu^{2} - 3 \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{4} - \beta_{3} - 3 \beta_{2} + 5 \beta_{1} + 6\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{4} + 3 \beta_{2} + 9 \beta_{1} + 23\)\()/2\)

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
0.271831
1.28447
−1.23118
2.52064
−1.84576
0 −2.19794 0 0 0 −1.00000 0 1.83094 0
1.2 0 −1.63460 0 0 0 −1.00000 0 −0.328072 0
1.3 0 0.746976 0 0 0 −1.00000 0 −2.44203 0
1.4 0 1.83297 0 0 0 −1.00000 0 0.359777 0
1.5 0 3.25260 0 0 0 −1.00000 0 7.57939 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
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

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 5600.2.a.bw 5
4.b odd 2 1 5600.2.a.bv 5
5.b even 2 1 5600.2.a.bu 5
5.c odd 4 2 1120.2.g.b 10
20.d odd 2 1 5600.2.a.bx 5
20.e even 4 2 1120.2.g.c yes 10
40.i odd 4 2 2240.2.g.o 10
40.k even 4 2 2240.2.g.n 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.g.b 10 5.c odd 4 2
1120.2.g.c yes 10 20.e even 4 2
2240.2.g.n 10 40.k even 4 2
2240.2.g.o 10 40.i odd 4 2
5600.2.a.bu 5 5.b even 2 1
5600.2.a.bv 5 4.b odd 2 1
5600.2.a.bw 5 1.a even 1 1 trivial
5600.2.a.bx 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5600))\):

\( T_{3}^{5} - 2 T_{3}^{4} - 9 T_{3}^{3} + 12 T_{3}^{2} + 18 T_{3} - 16 \)
\( T_{11}^{5} + 4 T_{11}^{4} - 25 T_{11}^{3} - 152 T_{11}^{2} - 248 T_{11} - 128 \)
\( T_{13}^{5} - 47 T_{13}^{3} + 80 T_{13}^{2} + 378 T_{13} - 824 \)
\( T_{19}^{5} + 12 T_{19}^{4} + 30 T_{19}^{3} - 72 T_{19}^{2} - 184 T_{19} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( -16 + 18 T + 12 T^{2} - 9 T^{3} - 2 T^{4} + T^{5} \)
$5$ \( T^{5} \)
$7$ \( ( 1 + T )^{5} \)
$11$ \( -128 - 248 T - 152 T^{2} - 25 T^{3} + 4 T^{4} + T^{5} \)
$13$ \( -824 + 378 T + 80 T^{2} - 47 T^{3} + T^{5} \)
$17$ \( 32 + 76 T + 20 T^{2} - 25 T^{3} - 4 T^{4} + T^{5} \)
$19$ \( 256 - 184 T - 72 T^{2} + 30 T^{3} + 12 T^{4} + T^{5} \)
$23$ \( 512 + 64 T - 144 T^{2} - 20 T^{3} + 8 T^{4} + T^{5} \)
$29$ \( 4744 - 124 T - 462 T^{2} - 15 T^{3} + 12 T^{4} + T^{5} \)
$31$ \( -4096 - 4480 T - 1184 T^{2} - 56 T^{3} + 12 T^{4} + T^{5} \)
$37$ \( -256 - 256 T + 256 T^{2} + 4 T^{3} - 12 T^{4} + T^{5} \)
$41$ \( -128 - 1056 T - 632 T^{2} - 100 T^{3} + 2 T^{4} + T^{5} \)
$43$ \( -15104 + 4832 T + 736 T^{2} - 140 T^{3} - 8 T^{4} + T^{5} \)
$47$ \( -64 - 616 T - 160 T^{2} + 37 T^{3} + 14 T^{4} + T^{5} \)
$53$ \( -1856 + 400 T + 496 T^{2} - 72 T^{3} - 8 T^{4} + T^{5} \)
$59$ \( 58112 + 2232 T - 2048 T^{2} - 106 T^{3} + 16 T^{4} + T^{5} \)
$61$ \( 1648 + 72 T - 468 T^{2} - 46 T^{3} + 10 T^{4} + T^{5} \)
$67$ \( 512 + 320 T - 192 T^{2} - 108 T^{3} - 4 T^{4} + T^{5} \)
$71$ \( -1024 - 2240 T - 1344 T^{2} - 228 T^{3} + 4 T^{4} + T^{5} \)
$73$ \( 2048 + 144 T - 496 T^{2} - 40 T^{3} + 12 T^{4} + T^{5} \)
$79$ \( 3904 + 4500 T + 1884 T^{2} + 363 T^{3} + 32 T^{4} + T^{5} \)
$83$ \( 27392 - 20320 T + 4032 T^{2} - 54 T^{3} - 24 T^{4} + T^{5} \)
$89$ \( -3104 + 1936 T + 400 T^{2} - 168 T^{3} - 2 T^{4} + T^{5} \)
$97$ \( 3424 - 2644 T - 1156 T^{2} - 65 T^{3} + 12 T^{4} + T^{5} \)
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