Properties

Label 5775.2.a.bw
Level $5775$
Weight $2$
Character orbit 5775.a
Self dual yes
Analytic conductor $46.114$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5775.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.1136071673\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \(x^{3} - 6 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} -\beta_{1} q^{6} + q^{7} + ( -1 - 2 \beta_{1} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} -\beta_{1} q^{6} + q^{7} + ( -1 - 2 \beta_{1} ) q^{8} + q^{9} + q^{11} + ( 2 + \beta_{2} ) q^{12} + ( -\beta_{1} + \beta_{2} ) q^{13} -\beta_{1} q^{14} + ( 4 + \beta_{1} ) q^{16} + 2 \beta_{1} q^{17} -\beta_{1} q^{18} + ( 4 - \beta_{1} - \beta_{2} ) q^{19} + q^{21} -\beta_{1} q^{22} + ( 2 + 2 \beta_{1} ) q^{23} + ( -1 - 2 \beta_{1} ) q^{24} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{26} + q^{27} + ( 2 + \beta_{2} ) q^{28} + ( 4 - \beta_{1} + \beta_{2} ) q^{29} + ( -2 + 2 \beta_{2} ) q^{31} + ( -2 - \beta_{2} ) q^{32} + q^{33} + ( -8 - 2 \beta_{2} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( -3 \beta_{1} - \beta_{2} ) q^{37} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{38} + ( -\beta_{1} + \beta_{2} ) q^{39} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{41} -\beta_{1} q^{42} + ( -2 + 2 \beta_{1} ) q^{43} + ( 2 + \beta_{2} ) q^{44} + ( -8 - 2 \beta_{1} - 2 \beta_{2} ) q^{46} + ( 8 - \beta_{1} - \beta_{2} ) q^{47} + ( 4 + \beta_{1} ) q^{48} + q^{49} + 2 \beta_{1} q^{51} + ( 7 - 3 \beta_{1} ) q^{52} + 2 \beta_{2} q^{53} -\beta_{1} q^{54} + ( -1 - 2 \beta_{1} ) q^{56} + ( 4 - \beta_{1} - \beta_{2} ) q^{57} + ( 3 - 6 \beta_{1} + \beta_{2} ) q^{58} + ( -8 + \beta_{1} - 3 \beta_{2} ) q^{59} + 6 q^{61} + ( -2 - 2 \beta_{1} ) q^{62} + q^{63} + ( -7 + 2 \beta_{1} ) q^{64} -\beta_{1} q^{66} + ( -4 - \beta_{1} - \beta_{2} ) q^{67} + ( 2 + 8 \beta_{1} ) q^{68} + ( 2 + 2 \beta_{1} ) q^{69} + ( 4 - 4 \beta_{1} ) q^{71} + ( -1 - 2 \beta_{1} ) q^{72} + ( -8 + \beta_{1} - \beta_{2} ) q^{73} + ( 13 + 2 \beta_{1} + 3 \beta_{2} ) q^{74} + ( -1 - 5 \beta_{1} + 4 \beta_{2} ) q^{76} + q^{77} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{78} + ( 4 + 4 \beta_{1} ) q^{79} + q^{81} + ( -10 - 6 \beta_{1} - 2 \beta_{2} ) q^{82} + ( 6 - 2 \beta_{2} ) q^{83} + ( 2 + \beta_{2} ) q^{84} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{86} + ( 4 - \beta_{1} + \beta_{2} ) q^{87} + ( -1 - 2 \beta_{1} ) q^{88} + ( 6 + 4 \beta_{2} ) q^{89} + ( -\beta_{1} + \beta_{2} ) q^{91} + ( 6 + 8 \beta_{1} + 2 \beta_{2} ) q^{92} + ( -2 + 2 \beta_{2} ) q^{93} + ( 5 - 6 \beta_{1} + \beta_{2} ) q^{94} + ( -2 - \beta_{2} ) q^{96} + ( -8 - 2 \beta_{2} ) q^{97} -\beta_{1} q^{98} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 6q^{4} + 3q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 6q^{4} + 3q^{7} - 3q^{8} + 3q^{9} + 3q^{11} + 6q^{12} + 12q^{16} + 12q^{19} + 3q^{21} + 6q^{23} - 3q^{24} + 9q^{26} + 3q^{27} + 6q^{28} + 12q^{29} - 6q^{31} - 6q^{32} + 3q^{33} - 24q^{34} + 6q^{36} + 15q^{38} + 6q^{41} - 6q^{43} + 6q^{44} - 24q^{46} + 24q^{47} + 12q^{48} + 3q^{49} + 21q^{52} - 3q^{56} + 12q^{57} + 9q^{58} - 24q^{59} + 18q^{61} - 6q^{62} + 3q^{63} - 21q^{64} - 12q^{67} + 6q^{68} + 6q^{69} + 12q^{71} - 3q^{72} - 24q^{73} + 39q^{74} - 3q^{76} + 3q^{77} + 9q^{78} + 12q^{79} + 3q^{81} - 30q^{82} + 18q^{83} + 6q^{84} - 24q^{86} + 12q^{87} - 3q^{88} + 18q^{89} + 18q^{92} - 6q^{93} + 15q^{94} - 6q^{96} - 24q^{97} + 3q^{99} + O(q^{100}) \)

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

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

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
2.52892
−0.167449
−2.36147
−2.52892 1.00000 4.39543 0 −2.52892 1.00000 −6.05784 1.00000 0
1.2 0.167449 1.00000 −1.97196 0 0.167449 1.00000 −0.665102 1.00000 0
1.3 2.36147 1.00000 3.57653 0 2.36147 1.00000 3.72294 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 5775.2.a.bw 3
5.b even 2 1 231.2.a.d 3
15.d odd 2 1 693.2.a.m 3
20.d odd 2 1 3696.2.a.bp 3
35.c odd 2 1 1617.2.a.s 3
55.d odd 2 1 2541.2.a.bi 3
105.g even 2 1 4851.2.a.bp 3
165.d even 2 1 7623.2.a.cb 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.d 3 5.b even 2 1
693.2.a.m 3 15.d odd 2 1
1617.2.a.s 3 35.c odd 2 1
2541.2.a.bi 3 55.d odd 2 1
3696.2.a.bp 3 20.d odd 2 1
4851.2.a.bp 3 105.g even 2 1
5775.2.a.bw 3 1.a even 1 1 trivial
7623.2.a.cb 3 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5775))\):

\( T_{2}^{3} - 6 T_{2} + 1 \)
\( T_{13}^{3} - 15 T_{13} - 2 \)
\( T_{17}^{3} - 24 T_{17} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T + T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( -2 - 15 T + T^{3} \)
$17$ \( -8 - 24 T + T^{3} \)
$19$ \( 36 + 27 T - 12 T^{2} + T^{3} \)
$23$ \( 32 - 12 T - 6 T^{2} + T^{3} \)
$29$ \( -6 + 33 T - 12 T^{2} + T^{3} \)
$31$ \( 32 - 36 T + 6 T^{2} + T^{3} \)
$37$ \( 246 - 75 T + T^{3} \)
$41$ \( 32 - 72 T - 6 T^{2} + T^{3} \)
$43$ \( -48 - 12 T + 6 T^{2} + T^{3} \)
$47$ \( -328 + 171 T - 24 T^{2} + T^{3} \)
$53$ \( 120 - 48 T + T^{3} \)
$59$ \( -716 + 87 T + 24 T^{2} + T^{3} \)
$61$ \( ( -6 + T )^{3} \)
$67$ \( -4 + 27 T + 12 T^{2} + T^{3} \)
$71$ \( 384 - 48 T - 12 T^{2} + T^{3} \)
$73$ \( 394 + 177 T + 24 T^{2} + T^{3} \)
$79$ \( 256 - 48 T - 12 T^{2} + T^{3} \)
$83$ \( -48 + 60 T - 18 T^{2} + T^{3} \)
$89$ \( 1896 - 84 T - 18 T^{2} + T^{3} \)
$97$ \( 8 + 144 T + 24 T^{2} + T^{3} \)
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