Properties

Label 5775.2.a.bp
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: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} - q^{3} + (\beta_1 + 2) q^{4} + (\beta_{2} + 1) q^{6} + q^{7} + ( - 2 \beta_1 - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} - q^{3} + (\beta_1 + 2) q^{4} + (\beta_{2} + 1) q^{6} + q^{7} + ( - 2 \beta_1 - 1) q^{8} + q^{9} - q^{11} + ( - \beta_1 - 2) q^{12} + ( - \beta_{2} + 3 \beta_1 + 1) q^{13} + ( - \beta_{2} - 1) q^{14} + (\beta_{2} + 2 \beta_1 - 1) q^{16} + (2 \beta_{2} - 4 \beta_1 - 2) q^{17} + ( - \beta_{2} - 1) q^{18} + ( - \beta_{2} + \beta_1 - 3) q^{19} - q^{21} + (\beta_{2} + 1) q^{22} + ( - 2 \beta_{2} - 4) q^{23} + (2 \beta_1 + 1) q^{24} + ( - 2 \beta_{2} - 5 \beta_1 - 1) q^{26} - q^{27} + (\beta_1 + 2) q^{28} + (\beta_{2} - 3 \beta_1 - 1) q^{29} + ( - 4 \beta_{2} + 2 \beta_1 - 2) q^{31} + (2 \beta_{2} - \beta_1 - 2) q^{32} + q^{33} + (4 \beta_{2} + 6 \beta_1) q^{34} + (\beta_1 + 2) q^{36} + ( - 3 \beta_{2} + \beta_1 - 1) q^{37} + (2 \beta_{2} - \beta_1 + 5) q^{38} + (\beta_{2} - 3 \beta_1 - 1) q^{39} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{41} + (\beta_{2} + 1) q^{42} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{43} + ( - \beta_1 - 2) q^{44} + (2 \beta_{2} + 2 \beta_1 + 10) q^{46} + ( - 3 \beta_{2} - \beta_1 - 1) q^{47} + ( - \beta_{2} - 2 \beta_1 + 1) q^{48} + q^{49} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{51} + (\beta_{2} + 6 \beta_1 + 10) q^{52} + 2 \beta_1 q^{53} + (\beta_{2} + 1) q^{54} + ( - 2 \beta_1 - 1) q^{56} + (\beta_{2} - \beta_1 + 3) q^{57} + (2 \beta_{2} + 5 \beta_1 + 1) q^{58} + (3 \beta_{2} - 3 \beta_1 + 1) q^{59} - 2 q^{61} + ( - 2 \beta_{2} + 12) q^{62} + q^{63} + (2 \beta_{2} - 4 \beta_1 - 1) q^{64} + ( - \beta_{2} - 1) q^{66} + ( - \beta_{2} + 5 \beta_1 + 1) q^{67} + ( - 8 \beta_1 - 14) q^{68} + (2 \beta_{2} + 4) q^{69} + (4 \beta_1 - 4) q^{71} + ( - 2 \beta_1 - 1) q^{72} + (\beta_{2} - 3 \beta_1 + 7) q^{73} + ( - 2 \beta_{2} + \beta_1 + 9) q^{74} + ( - \beta_{2} - 2 \beta_1 - 4) q^{76} - q^{77} + (2 \beta_{2} + 5 \beta_1 + 1) q^{78} + ( - 4 \beta_1 + 4) q^{79} + q^{81} + ( - 6 \beta_{2} - 2 \beta_1) q^{82} + (6 \beta_1 - 2) q^{83} + ( - \beta_1 - 2) q^{84} + ( - 6 \beta_{2} + 10 \beta_1 + 6) q^{86} + ( - \beta_{2} + 3 \beta_1 + 1) q^{87} + (2 \beta_1 + 1) q^{88} + (4 \beta_{2} + 10) q^{89} + ( - \beta_{2} + 3 \beta_1 + 1) q^{91} + ( - 4 \beta_{2} - 6 \beta_1 - 10) q^{92} + (4 \beta_{2} - 2 \beta_1 + 2) q^{93} + ( - 2 \beta_{2} + 5 \beta_1 + 11) q^{94} + ( - 2 \beta_{2} + \beta_1 + 2) q^{96} + ( - 4 \beta_{2} - 2 \beta_1) q^{97} + ( - \beta_{2} - 1) q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 3 q^{3} + 6 q^{4} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} - 3 q^{3} + 6 q^{4} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{11} - 6 q^{12} + 4 q^{13} - 2 q^{14} - 4 q^{16} - 8 q^{17} - 2 q^{18} - 8 q^{19} - 3 q^{21} + 2 q^{22} - 10 q^{23} + 3 q^{24} - q^{26} - 3 q^{27} + 6 q^{28} - 4 q^{29} - 2 q^{31} - 8 q^{32} + 3 q^{33} - 4 q^{34} + 6 q^{36} + 13 q^{38} - 4 q^{39} + 14 q^{41} + 2 q^{42} + 14 q^{43} - 6 q^{44} + 28 q^{46} + 4 q^{48} + 3 q^{49} + 8 q^{51} + 29 q^{52} + 2 q^{54} - 3 q^{56} + 8 q^{57} + q^{58} - 6 q^{61} + 38 q^{62} + 3 q^{63} - 5 q^{64} - 2 q^{66} + 4 q^{67} - 42 q^{68} + 10 q^{69} - 12 q^{71} - 3 q^{72} + 20 q^{73} + 29 q^{74} - 11 q^{76} - 3 q^{77} + q^{78} + 12 q^{79} + 3 q^{81} + 6 q^{82} - 6 q^{83} - 6 q^{84} + 24 q^{86} + 4 q^{87} + 3 q^{88} + 26 q^{89} + 4 q^{91} - 26 q^{92} + 2 q^{93} + 35 q^{94} + 8 q^{96} + 4 q^{97} - 2 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display

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
2.11491
−1.86081
−0.254102
−2.47283 −1.00000 4.11491 0 2.47283 1.00000 −5.22982 1.00000 0
1.2 −1.46260 −1.00000 0.139194 0 1.46260 1.00000 2.72161 1.00000 0
1.3 1.93543 −1.00000 1.74590 0 −1.93543 1.00000 −0.491797 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.bp 3
5.b even 2 1 231.2.a.e 3
15.d odd 2 1 693.2.a.l 3
20.d odd 2 1 3696.2.a.bo 3
35.c odd 2 1 1617.2.a.t 3
55.d odd 2 1 2541.2.a.bg 3
105.g even 2 1 4851.2.a.bi 3
165.d even 2 1 7623.2.a.cd 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.e 3 5.b even 2 1
693.2.a.l 3 15.d odd 2 1
1617.2.a.t 3 35.c odd 2 1
2541.2.a.bg 3 55.d odd 2 1
3696.2.a.bo 3 20.d odd 2 1
4851.2.a.bi 3 105.g even 2 1
5775.2.a.bp 3 1.a even 1 1 trivial
7623.2.a.cd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5775))\):

\( T_{2}^{3} + 2T_{2}^{2} - 4T_{2} - 7 \) Copy content Toggle raw display
\( T_{13}^{3} - 4T_{13}^{2} - 27T_{13} + 94 \) Copy content Toggle raw display
\( T_{17}^{3} + 8T_{17}^{2} - 40T_{17} - 328 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 2 T^{2} - 4 T - 7 \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 4 T^{2} - 27 T + 94 \) Copy content Toggle raw display
$17$ \( T^{3} + 8 T^{2} - 40 T - 328 \) Copy content Toggle raw display
$19$ \( T^{3} + 8 T^{2} + 15 T + 4 \) Copy content Toggle raw display
$23$ \( T^{3} + 10 T^{2} + 12 T - 64 \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} - 27 T - 94 \) Copy content Toggle raw display
$31$ \( T^{3} + 2 T^{2} - 76 T - 256 \) Copy content Toggle raw display
$37$ \( T^{3} - 43T - 106 \) Copy content Toggle raw display
$41$ \( T^{3} - 14 T^{2} + 40 T + 32 \) Copy content Toggle raw display
$43$ \( T^{3} - 14 T^{2} - 44 T + 848 \) Copy content Toggle raw display
$47$ \( T^{3} - 61T - 32 \) Copy content Toggle raw display
$53$ \( T^{3} - 16T - 8 \) Copy content Toggle raw display
$59$ \( T^{3} - 57T - 52 \) Copy content Toggle raw display
$61$ \( (T + 2)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} - 4 T^{2} - 85 T + 236 \) Copy content Toggle raw display
$71$ \( T^{3} + 12 T^{2} - 16 T - 256 \) Copy content Toggle raw display
$73$ \( T^{3} - 20 T^{2} + 101 T - 134 \) Copy content Toggle raw display
$79$ \( T^{3} - 12 T^{2} - 16 T + 256 \) Copy content Toggle raw display
$83$ \( T^{3} + 6 T^{2} - 132 T - 496 \) Copy content Toggle raw display
$89$ \( T^{3} - 26 T^{2} + 140 T + 328 \) Copy content Toggle raw display
$97$ \( T^{3} - 4 T^{2} - 120 T + 232 \) Copy content Toggle raw display
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