Properties

Label 574.2.a.l
Level $574$
Weight $2$
Character orbit 574.a
Self dual yes
Analytic conductor $4.583$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.58341307602\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} -\beta_{2} q^{3} + q^{4} + \beta_{1} q^{5} -\beta_{2} q^{6} - q^{7} + q^{8} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q + q^{2} -\beta_{2} q^{3} + q^{4} + \beta_{1} q^{5} -\beta_{2} q^{6} - q^{7} + q^{8} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{9} + \beta_{1} q^{10} + ( 2 + \beta_{1} + \beta_{2} ) q^{11} -\beta_{2} q^{12} - q^{14} + ( 2 + \beta_{2} ) q^{15} + q^{16} + ( -2 + \beta_{2} ) q^{17} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{18} + ( 2 + 2 \beta_{2} ) q^{19} + \beta_{1} q^{20} + \beta_{2} q^{21} + ( 2 + \beta_{1} + \beta_{2} ) q^{22} + ( 2 - 2 \beta_{1} ) q^{23} -\beta_{2} q^{24} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{25} + ( 2 - 2 \beta_{1} - 3 \beta_{2} ) q^{27} - q^{28} + ( 2 - \beta_{1} ) q^{29} + ( 2 + \beta_{2} ) q^{30} + ( 2 - \beta_{2} ) q^{31} + q^{32} + ( -4 + 2 \beta_{1} ) q^{33} + ( -2 + \beta_{2} ) q^{34} -\beta_{1} q^{35} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{36} + 2 q^{37} + ( 2 + 2 \beta_{2} ) q^{38} + \beta_{1} q^{40} - q^{41} + \beta_{2} q^{42} + ( -2 + 3 \beta_{2} ) q^{43} + ( 2 + \beta_{1} + \beta_{2} ) q^{44} + ( -6 - \beta_{1} - \beta_{2} ) q^{45} + ( 2 - 2 \beta_{1} ) q^{46} + 2 q^{47} -\beta_{2} q^{48} + q^{49} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{50} + ( -6 + 2 \beta_{1} + 3 \beta_{2} ) q^{51} + ( -2 - 3 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 2 - 2 \beta_{1} - 3 \beta_{2} ) q^{54} + ( 2 + 4 \beta_{1} ) q^{55} - q^{56} + ( -12 + 4 \beta_{1} ) q^{57} + ( 2 - \beta_{1} ) q^{58} + ( -3 \beta_{1} - \beta_{2} ) q^{59} + ( 2 + \beta_{2} ) q^{60} + ( -4 + \beta_{1} + 2 \beta_{2} ) q^{61} + ( 2 - \beta_{2} ) q^{62} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{63} + q^{64} + ( -4 + 2 \beta_{1} ) q^{66} + ( 6 + \beta_{1} - 3 \beta_{2} ) q^{67} + ( -2 + \beta_{2} ) q^{68} + ( -4 - 4 \beta_{2} ) q^{69} -\beta_{1} q^{70} + ( 2 + 4 \beta_{1} + 3 \beta_{2} ) q^{71} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{72} + ( -10 - 2 \beta_{2} ) q^{73} + 2 q^{74} + ( -2 + 2 \beta_{1} + 4 \beta_{2} ) q^{75} + ( 2 + 2 \beta_{2} ) q^{76} + ( -2 - \beta_{1} - \beta_{2} ) q^{77} + ( 6 + 2 \beta_{1} + \beta_{2} ) q^{79} + \beta_{1} q^{80} + ( 5 - 4 \beta_{2} ) q^{81} - q^{82} + ( -4 - \beta_{1} + \beta_{2} ) q^{83} + \beta_{2} q^{84} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{85} + ( -2 + 3 \beta_{2} ) q^{86} + ( -2 - 3 \beta_{2} ) q^{87} + ( 2 + \beta_{1} + \beta_{2} ) q^{88} + ( 8 - 2 \beta_{1} - \beta_{2} ) q^{89} + ( -6 - \beta_{1} - \beta_{2} ) q^{90} + ( 2 - 2 \beta_{1} ) q^{92} + ( 6 - 2 \beta_{1} - 3 \beta_{2} ) q^{93} + 2 q^{94} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{95} -\beta_{2} q^{96} + ( -8 - 2 \beta_{1} - \beta_{2} ) q^{97} + q^{98} + ( -2 - 3 \beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + q^{3} + 3q^{4} + q^{5} + q^{6} - 3q^{7} + 3q^{8} + 8q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + q^{3} + 3q^{4} + q^{5} + q^{6} - 3q^{7} + 3q^{8} + 8q^{9} + q^{10} + 6q^{11} + q^{12} - 3q^{14} + 5q^{15} + 3q^{16} - 7q^{17} + 8q^{18} + 4q^{19} + q^{20} - q^{21} + 6q^{22} + 4q^{23} + q^{24} - 2q^{25} + 7q^{27} - 3q^{28} + 5q^{29} + 5q^{30} + 7q^{31} + 3q^{32} - 10q^{33} - 7q^{34} - q^{35} + 8q^{36} + 6q^{37} + 4q^{38} + q^{40} - 3q^{41} - q^{42} - 9q^{43} + 6q^{44} - 18q^{45} + 4q^{46} + 6q^{47} + q^{48} + 3q^{49} - 2q^{50} - 19q^{51} - 7q^{53} + 7q^{54} + 10q^{55} - 3q^{56} - 32q^{57} + 5q^{58} - 2q^{59} + 5q^{60} - 13q^{61} + 7q^{62} - 8q^{63} + 3q^{64} - 10q^{66} + 22q^{67} - 7q^{68} - 8q^{69} - q^{70} + 7q^{71} + 8q^{72} - 28q^{73} + 6q^{74} - 8q^{75} + 4q^{76} - 6q^{77} + 19q^{79} + q^{80} + 19q^{81} - 3q^{82} - 14q^{83} - q^{84} - 7q^{85} - 9q^{86} - 3q^{87} + 6q^{88} + 23q^{89} - 18q^{90} + 4q^{92} + 19q^{93} + 6q^{94} - 8q^{95} + q^{96} - 25q^{97} + 3q^{98} - 12q^{99} + O(q^{100}) \)

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

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

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
−1.76156
3.12489
−0.363328
1.00000 −2.62620 1.00000 −1.76156 −2.62620 −1.00000 1.00000 3.89692 −1.76156
1.2 1.00000 0.484862 1.00000 3.12489 0.484862 −1.00000 1.00000 −2.76491 3.12489
1.3 1.00000 3.14134 1.00000 −0.363328 3.14134 −1.00000 1.00000 6.86799 −0.363328
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(41\) \(1\)

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 574.2.a.l 3
3.b odd 2 1 5166.2.a.bt 3
4.b odd 2 1 4592.2.a.s 3
7.b odd 2 1 4018.2.a.bg 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.a.l 3 1.a even 1 1 trivial
4018.2.a.bg 3 7.b odd 2 1
4592.2.a.s 3 4.b odd 2 1
5166.2.a.bt 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(574))\):

\( T_{3}^{3} - T_{3}^{2} - 8 T_{3} + 4 \)
\( T_{5}^{3} - T_{5}^{2} - 6 T_{5} - 2 \)
\( T_{11}^{3} - 6 T_{11}^{2} + 2 T_{11} + 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( 4 - 8 T - T^{2} + T^{3} \)
$5$ \( -2 - 6 T - T^{2} + T^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( 20 + 2 T - 6 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( -8 + 8 T + 7 T^{2} + T^{3} \)
$19$ \( 32 - 28 T - 4 T^{2} + T^{3} \)
$23$ \( 64 - 20 T - 4 T^{2} + T^{3} \)
$29$ \( 10 + 2 T - 5 T^{2} + T^{3} \)
$31$ \( 8 + 8 T - 7 T^{2} + T^{3} \)
$37$ \( ( -2 + T )^{3} \)
$41$ \( ( 1 + T )^{3} \)
$43$ \( -232 - 48 T + 9 T^{2} + T^{3} \)
$47$ \( ( -2 + T )^{3} \)
$53$ \( -110 - 46 T + 7 T^{2} + T^{3} \)
$59$ \( 100 - 50 T + 2 T^{2} + T^{3} \)
$61$ \( 10 + 26 T + 13 T^{2} + T^{3} \)
$67$ \( 580 + 66 T - 22 T^{2} + T^{3} \)
$71$ \( 328 - 104 T - 7 T^{2} + T^{3} \)
$73$ \( 512 + 228 T + 28 T^{2} + T^{3} \)
$79$ \( -128 + 96 T - 19 T^{2} + T^{3} \)
$83$ \( -20 + 46 T + 14 T^{2} + T^{3} \)
$89$ \( -236 + 152 T - 23 T^{2} + T^{3} \)
$97$ \( 404 + 184 T + 25 T^{2} + T^{3} \)
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