Properties

Label 574.2.a.m
Level $574$
Weight $2$
Character orbit 574.a
Self dual yes
Analytic conductor $4.583$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.11348.1
Defining polynomial: \(x^{4} - x^{3} - 5 x^{2} + x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} -\beta_{2} q^{3} + q^{4} + ( 1 - \beta_{2} + \beta_{3} ) q^{5} + \beta_{2} q^{6} - q^{7} - q^{8} + ( 2 - \beta_{1} + \beta_{3} ) q^{9} +O(q^{10})\) \( q - q^{2} -\beta_{2} q^{3} + q^{4} + ( 1 - \beta_{2} + \beta_{3} ) q^{5} + \beta_{2} q^{6} - q^{7} - q^{8} + ( 2 - \beta_{1} + \beta_{3} ) q^{9} + ( -1 + \beta_{2} - \beta_{3} ) q^{10} + ( 1 - \beta_{3} ) q^{11} -\beta_{2} q^{12} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + q^{14} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{15} + q^{16} + ( 2 \beta_{1} + \beta_{2} ) q^{17} + ( -2 + \beta_{1} - \beta_{3} ) q^{18} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + ( 1 - \beta_{2} + \beta_{3} ) q^{20} + \beta_{2} q^{21} + ( -1 + \beta_{3} ) q^{22} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + \beta_{2} q^{24} + ( 3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{25} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{26} + ( -3 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{27} - q^{28} + ( 4 + \beta_{1} + 2 \beta_{2} ) q^{29} + ( -3 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{30} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{31} - q^{32} + ( 2 + 2 \beta_{3} ) q^{33} + ( -2 \beta_{1} - \beta_{2} ) q^{34} + ( -1 + \beta_{2} - \beta_{3} ) q^{35} + ( 2 - \beta_{1} + \beta_{3} ) q^{36} + ( -2 + 2 \beta_{2} ) q^{37} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{38} + ( 4 + 2 \beta_{2} ) q^{39} + ( -1 + \beta_{2} - \beta_{3} ) q^{40} + q^{41} -\beta_{2} q^{42} + ( -3 + \beta_{1} + \beta_{3} ) q^{43} + ( 1 - \beta_{3} ) q^{44} + ( 8 - 3 \beta_{1} - \beta_{2} ) q^{45} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{46} + ( 1 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{47} -\beta_{2} q^{48} + q^{49} + ( -3 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{50} + ( -3 + 3 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{51} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{52} + ( 8 + \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{53} + ( 3 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{54} + ( -3 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{55} + q^{56} + 4 q^{57} + ( -4 - \beta_{1} - 2 \beta_{2} ) q^{58} + ( 4 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{59} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{60} + ( 5 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{61} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{62} + ( -2 + \beta_{1} - \beta_{3} ) q^{63} + q^{64} + ( 4 + 2 \beta_{2} - 4 \beta_{3} ) q^{65} + ( -2 - 2 \beta_{3} ) q^{66} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{67} + ( 2 \beta_{1} + \beta_{2} ) q^{68} + ( -4 - 2 \beta_{2} ) q^{69} + ( 1 - \beta_{2} + \beta_{3} ) q^{70} + ( 3 - 5 \beta_{1} - \beta_{3} ) q^{71} + ( -2 + \beta_{1} - \beta_{3} ) q^{72} + ( 2 - 4 \beta_{1} ) q^{73} + ( 2 - 2 \beta_{2} ) q^{74} + ( 7 - 3 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{75} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{76} + ( -1 + \beta_{3} ) q^{77} + ( -4 - 2 \beta_{2} ) q^{78} + ( -7 - \beta_{1} + \beta_{3} ) q^{79} + ( 1 - \beta_{2} + \beta_{3} ) q^{80} + ( 9 + 4 \beta_{2} + 4 \beta_{3} ) q^{81} - q^{82} + ( -4 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{83} + \beta_{2} q^{84} + ( -7 + 5 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{85} + ( 3 - \beta_{1} - \beta_{3} ) q^{86} + ( -9 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{87} + ( -1 + \beta_{3} ) q^{88} + ( 9 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{89} + ( -8 + 3 \beta_{1} + \beta_{2} ) q^{90} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{91} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{92} + ( -11 + 3 \beta_{1} - 3 \beta_{3} ) q^{93} + ( -1 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{94} + ( 6 - 2 \beta_{3} ) q^{95} + \beta_{2} q^{96} + ( -1 - \beta_{1} - 3 \beta_{3} ) q^{97} - q^{98} + ( -7 - 4 \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - q^{3} + 4q^{4} + 3q^{5} + q^{6} - 4q^{7} - 4q^{8} + 9q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - q^{3} + 4q^{4} + 3q^{5} + q^{6} - 4q^{7} - 4q^{8} + 9q^{9} - 3q^{10} + 4q^{11} - q^{12} - 6q^{13} + 4q^{14} + 11q^{15} + 4q^{16} - q^{17} - 9q^{18} + 2q^{19} + 3q^{20} + q^{21} - 4q^{22} + 6q^{23} + q^{24} + 13q^{25} + 6q^{26} - 13q^{27} - 4q^{28} + 17q^{29} - 11q^{30} + 5q^{31} - 4q^{32} + 8q^{33} + q^{34} - 3q^{35} + 9q^{36} - 6q^{37} - 2q^{38} + 18q^{39} - 3q^{40} + 4q^{41} - q^{42} - 13q^{43} + 4q^{44} + 34q^{45} - 6q^{46} + 6q^{47} - q^{48} + 4q^{49} - 13q^{50} - 11q^{51} - 6q^{52} + 29q^{53} + 13q^{54} - 16q^{55} + 4q^{56} + 16q^{57} - 17q^{58} + 16q^{59} + 11q^{60} + 21q^{61} - 5q^{62} - 9q^{63} + 4q^{64} + 18q^{65} - 8q^{66} + 4q^{67} - q^{68} - 18q^{69} + 3q^{70} + 17q^{71} - 9q^{72} + 12q^{73} + 6q^{74} + 26q^{75} + 2q^{76} - 4q^{77} - 18q^{78} - 27q^{79} + 3q^{80} + 40q^{81} - 4q^{82} - 18q^{83} + q^{84} - 29q^{85} + 13q^{86} - 41q^{87} - 4q^{88} + 37q^{89} - 34q^{90} + 6q^{91} + 6q^{92} - 47q^{93} - 6q^{94} + 24q^{95} + q^{96} - 3q^{97} - 4q^{98} - 32q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 3 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1} + 3\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{3} - 3 \beta_{2} + 7 \beta_{1} + 9\)\()/2\)

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
−0.589216
−1.77571
2.64119
0.723742
−1.00000 −3.39434 1.00000 1.47439 3.39434 −1.00000 −1.00000 8.52156 −1.47439
1.2 −1.00000 −1.12631 1.00000 −3.70458 1.12631 −1.00000 −1.00000 −1.73143 3.70458
1.3 −1.00000 0.757235 1.00000 1.30651 −0.757235 −1.00000 −1.00000 −2.42659 −1.30651
1.4 −1.00000 2.76342 1.00000 3.92368 −2.76342 −1.00000 −1.00000 4.63646 −3.92368
\(n\): 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.m 4
3.b odd 2 1 5166.2.a.bx 4
4.b odd 2 1 4592.2.a.ba 4
7.b odd 2 1 4018.2.a.bj 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.a.m 4 1.a even 1 1 trivial
4018.2.a.bj 4 7.b odd 2 1
4592.2.a.ba 4 4.b odd 2 1
5166.2.a.bx 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(574))\):

\( T_{3}^{4} + T_{3}^{3} - 10 T_{3}^{2} - 4 T_{3} + 8 \)
\( T_{5}^{4} - 3 T_{5}^{3} - 12 T_{5}^{2} + 40 T_{5} - 28 \)
\( T_{11}^{4} - 4 T_{11}^{3} - 8 T_{11}^{2} + 28 T_{11} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( 8 - 4 T - 10 T^{2} + T^{3} + T^{4} \)
$5$ \( -28 + 40 T - 12 T^{2} - 3 T^{3} + T^{4} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( -16 + 28 T - 8 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( -32 - 64 T - 8 T^{2} + 6 T^{3} + T^{4} \)
$17$ \( 152 - 28 T - 58 T^{2} + T^{3} + T^{4} \)
$19$ \( 32 + 8 T - 20 T^{2} - 2 T^{3} + T^{4} \)
$23$ \( -32 + 64 T - 8 T^{2} - 6 T^{3} + T^{4} \)
$29$ \( -1348 + 248 T + 62 T^{2} - 17 T^{3} + T^{4} \)
$31$ \( 32 + 176 T - 56 T^{2} - 5 T^{3} + T^{4} \)
$37$ \( 32 - 120 T - 28 T^{2} + 6 T^{3} + T^{4} \)
$41$ \( ( -1 + T )^{4} \)
$43$ \( -32 + 32 T + 48 T^{2} + 13 T^{3} + T^{4} \)
$47$ \( -1184 + 656 T - 80 T^{2} - 6 T^{3} + T^{4} \)
$53$ \( -16132 + 1992 T + 150 T^{2} - 29 T^{3} + T^{4} \)
$59$ \( -664 + 388 T + 28 T^{2} - 16 T^{3} + T^{4} \)
$61$ \( 692 + 864 T + 32 T^{2} - 21 T^{3} + T^{4} \)
$67$ \( 752 + 420 T - 96 T^{2} - 4 T^{3} + T^{4} \)
$71$ \( -18016 + 4640 T - 200 T^{2} - 17 T^{3} + T^{4} \)
$73$ \( -2896 + 2352 T - 176 T^{2} - 12 T^{3} + T^{4} \)
$79$ \( 256 + 640 T + 232 T^{2} + 27 T^{3} + T^{4} \)
$83$ \( 16 + 108 T + 80 T^{2} + 18 T^{3} + T^{4} \)
$89$ \( 1384 - 2012 T + 458 T^{2} - 37 T^{3} + T^{4} \)
$97$ \( 344 - 252 T - 98 T^{2} + 3 T^{3} + T^{4} \)
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