Properties

Label 574.2.a.k
Level $574$
Weight $2$
Character orbit 574.a
Self dual yes
Analytic conductor $4.583$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
1.00000 2.00000 1.00000 −0.732051 2.00000 1.00000 1.00000 1.00000 −0.732051
1.2 1.00000 2.00000 1.00000 2.73205 2.00000 1.00000 1.00000 1.00000 2.73205
\(n\): 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.k 2
3.b odd 2 1 5166.2.a.bo 2
4.b odd 2 1 4592.2.a.m 2
7.b odd 2 1 4018.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.a.k 2 1.a even 1 1 trivial
4018.2.a.x 2 7.b odd 2 1
4592.2.a.m 2 4.b odd 2 1
5166.2.a.bo 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(574))\):

\( T_{3} - 2 \)
\( T_{5}^{2} - 2 T_{5} - 2 \)
\( T_{11}^{2} + 2 T_{11} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( -2 + T )^{2} \)
$5$ \( -2 - 2 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -2 + 2 T + T^{2} \)
$13$ \( -8 + 4 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( -12 + T^{2} \)
$23$ \( ( 2 + T )^{2} \)
$29$ \( -74 - 2 T + T^{2} \)
$31$ \( -8 + 4 T + T^{2} \)
$37$ \( -44 + 4 T + T^{2} \)
$41$ \( ( -1 + T )^{2} \)
$43$ \( -104 - 4 T + T^{2} \)
$47$ \( 4 - 8 T + T^{2} \)
$53$ \( -74 + 2 T + T^{2} \)
$59$ \( -146 - 2 T + T^{2} \)
$61$ \( 22 - 10 T + T^{2} \)
$67$ \( -2 + 10 T + T^{2} \)
$71$ \( 24 - 12 T + T^{2} \)
$73$ \( -44 + 4 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 6 - 6 T + T^{2} \)
$89$ \( -12 - 12 T + T^{2} \)
$97$ \( ( -2 + T )^{2} \)
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