Properties

Label 5550.2.a.cj
Level $5550$
Weight $2$
Character orbit 5550.a
Self dual yes
Analytic conductor $44.317$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.3169731218\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 3 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1110)
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.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{6} + ( -1 - \beta_{2} ) q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} - q^{3} + q^{4} + q^{6} + ( -1 - \beta_{2} ) q^{7} - q^{8} + q^{9} + ( 1 - \beta_{1} ) q^{11} - q^{12} + ( 1 - \beta_{1} + \beta_{3} ) q^{13} + ( 1 + \beta_{2} ) q^{14} + q^{16} + ( -1 + \beta_{2} - 2 \beta_{3} ) q^{17} - q^{18} + ( 1 - \beta_{2} - \beta_{3} ) q^{19} + ( 1 + \beta_{2} ) q^{21} + ( -1 + \beta_{1} ) q^{22} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{23} + q^{24} + ( -1 + \beta_{1} - \beta_{3} ) q^{26} - q^{27} + ( -1 - \beta_{2} ) q^{28} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{29} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} - q^{32} + ( -1 + \beta_{1} ) q^{33} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{34} + q^{36} + q^{37} + ( -1 + \beta_{2} + \beta_{3} ) q^{38} + ( -1 + \beta_{1} - \beta_{3} ) q^{39} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{41} + ( -1 - \beta_{2} ) q^{42} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{43} + ( 1 - \beta_{1} ) q^{44} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{46} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{47} - q^{48} + ( 4 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{49} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{51} + ( 1 - \beta_{1} + \beta_{3} ) q^{52} + ( 5 - \beta_{2} + 2 \beta_{3} ) q^{53} + q^{54} + ( 1 + \beta_{2} ) q^{56} + ( -1 + \beta_{2} + \beta_{3} ) q^{57} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{58} + ( -2 - 2 \beta_{2} ) q^{59} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{61} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{62} + ( -1 - \beta_{2} ) q^{63} + q^{64} + ( 1 - \beta_{1} ) q^{66} + ( -2 - 2 \beta_{2} + 2 \beta_{3} ) q^{67} + ( -1 + \beta_{2} - 2 \beta_{3} ) q^{68} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{69} + ( -6 + 2 \beta_{1} + 2 \beta_{3} ) q^{71} - q^{72} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{73} - q^{74} + ( 1 - \beta_{2} - \beta_{3} ) q^{76} + ( 1 - 3 \beta_{2} ) q^{77} + ( 1 - \beta_{1} + \beta_{3} ) q^{78} + 4 q^{79} + q^{81} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{82} + ( -1 + 3 \beta_{1} - \beta_{3} ) q^{83} + ( 1 + \beta_{2} ) q^{84} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{86} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{87} + ( -1 + \beta_{1} ) q^{88} + ( -3 + 3 \beta_{1} + \beta_{3} ) q^{89} + ( -3 + \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{91} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{92} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{94} + q^{96} + ( -2 + 2 \beta_{1} - 3 \beta_{3} ) q^{97} + ( -4 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{98} + ( 1 - \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} - 4q^{7} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} - 4q^{7} - 4q^{8} + 4q^{9} + 2q^{11} - 4q^{12} + 3q^{13} + 4q^{14} + 4q^{16} - 6q^{17} - 4q^{18} + 3q^{19} + 4q^{21} - 2q^{22} + q^{23} + 4q^{24} - 3q^{26} - 4q^{27} - 4q^{28} - 3q^{29} + 3q^{31} - 4q^{32} - 2q^{33} + 6q^{34} + 4q^{36} + 4q^{37} - 3q^{38} - 3q^{39} - 11q^{41} - 4q^{42} + 5q^{43} + 2q^{44} - q^{46} - 4q^{48} + 14q^{49} + 6q^{51} + 3q^{52} + 22q^{53} + 4q^{54} + 4q^{56} - 3q^{57} + 3q^{58} - 8q^{59} - 5q^{61} - 3q^{62} - 4q^{63} + 4q^{64} + 2q^{66} - 6q^{67} - 6q^{68} - q^{69} - 18q^{71} - 4q^{72} + 5q^{73} - 4q^{74} + 3q^{76} + 4q^{77} + 3q^{78} + 16q^{79} + 4q^{81} + 11q^{82} + q^{83} + 4q^{84} - 5q^{86} + 3q^{87} - 2q^{88} - 5q^{89} - 13q^{91} + q^{92} - 3q^{93} + 4q^{96} - 7q^{97} - 14q^{98} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 9 x^{2} + 3 x + 2\):

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

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
−2.67673
−0.339102
3.36007
0.655762
−1.00000 −1.00000 1.00000 0 1.00000 −5.13277 −1.00000 1.00000 0
1.2 −1.00000 −1.00000 1.00000 0 1.00000 −1.84556 −1.00000 1.00000 0
1.3 −1.00000 −1.00000 1.00000 0 1.00000 −0.487359 −1.00000 1.00000 0
1.4 −1.00000 −1.00000 1.00000 0 1.00000 3.46569 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(1\)
\(37\) \(-1\)

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 5550.2.a.cj 4
5.b even 2 1 1110.2.a.s 4
15.d odd 2 1 3330.2.a.bj 4
20.d odd 2 1 8880.2.a.cg 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.s 4 5.b even 2 1
3330.2.a.bj 4 15.d odd 2 1
5550.2.a.cj 4 1.a even 1 1 trivial
8880.2.a.cg 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5550))\):

\( T_{7}^{4} + 4 T_{7}^{3} - 13 T_{7}^{2} - 40 T_{7} - 16 \)
\( T_{11}^{4} - 2 T_{11}^{3} - 23 T_{11}^{2} + 68 T_{11} - 40 \)
\( T_{13}^{4} - 3 T_{13}^{3} - 38 T_{13}^{2} + 44 T_{13} + 328 \)
\( T_{17}^{4} + 6 T_{17}^{3} - 47 T_{17}^{2} - 412 T_{17} - 764 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( -16 - 40 T - 13 T^{2} + 4 T^{3} + T^{4} \)
$11$ \( -40 + 68 T - 23 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( 328 + 44 T - 38 T^{2} - 3 T^{3} + T^{4} \)
$17$ \( -764 - 412 T - 47 T^{2} + 6 T^{3} + T^{4} \)
$19$ \( -80 + 208 T - 50 T^{2} - 3 T^{3} + T^{4} \)
$23$ \( 1280 + 64 T - 76 T^{2} - T^{3} + T^{4} \)
$29$ \( 1208 - 204 T - 82 T^{2} + 3 T^{3} + T^{4} \)
$31$ \( 32 - 38 T^{2} - 3 T^{3} + T^{4} \)
$37$ \( ( -1 + T )^{4} \)
$41$ \( -80 - 84 T + 4 T^{2} + 11 T^{3} + T^{4} \)
$43$ \( 1280 + 128 T - 76 T^{2} - 5 T^{3} + T^{4} \)
$47$ \( 1280 + 96 T - 88 T^{2} + T^{4} \)
$53$ \( -2524 + 244 T + 121 T^{2} - 22 T^{3} + T^{4} \)
$59$ \( -256 - 320 T - 52 T^{2} + 8 T^{3} + T^{4} \)
$61$ \( 1096 + 196 T - 130 T^{2} + 5 T^{3} + T^{4} \)
$67$ \( 128 - 32 T - 72 T^{2} + 6 T^{3} + T^{4} \)
$71$ \( 512 - 1536 T - 56 T^{2} + 18 T^{3} + T^{4} \)
$73$ \( -2336 + 1132 T - 128 T^{2} - 5 T^{3} + T^{4} \)
$79$ \( ( -4 + T )^{4} \)
$83$ \( 5344 - 704 T - 234 T^{2} - T^{3} + T^{4} \)
$89$ \( -3208 - 1956 T - 234 T^{2} + 5 T^{3} + T^{4} \)
$97$ \( 9344 - 340 T - 236 T^{2} + 7 T^{3} + T^{4} \)
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