Properties

Label 546.2.a.h
Level $546$
Weight $2$
Character orbit 546.a
Self dual yes
Analytic conductor $4.360$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

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

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
4.27492
−3.27492
−1.00000 −1.00000 1.00000 −4.27492 1.00000 1.00000 −1.00000 1.00000 4.27492
1.2 −1.00000 −1.00000 1.00000 3.27492 1.00000 1.00000 −1.00000 1.00000 −3.27492
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(1\)

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 546.2.a.h 2
3.b odd 2 1 1638.2.a.y 2
4.b odd 2 1 4368.2.a.bh 2
7.b odd 2 1 3822.2.a.bm 2
13.b even 2 1 7098.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.h 2 1.a even 1 1 trivial
1638.2.a.y 2 3.b odd 2 1
3822.2.a.bm 2 7.b odd 2 1
4368.2.a.bh 2 4.b odd 2 1
7098.2.a.bu 2 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(546))\):

\( T_{5}^{2} + T_{5} - 14 \)
\( T_{11}^{2} - 3 T_{11} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -14 + T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -12 - 3 T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -2 + 7 T + T^{2} \)
$19$ \( -12 - 3 T + T^{2} \)
$23$ \( -12 + 3 T + T^{2} \)
$29$ \( 6 - 9 T + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( -14 + T + T^{2} \)
$41$ \( -56 + 2 T + T^{2} \)
$43$ \( -12 - 3 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( -10 + T )^{2} \)
$59$ \( ( -8 + T )^{2} \)
$61$ \( 58 + 17 T + T^{2} \)
$67$ \( -32 - 10 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( -126 + 3 T + T^{2} \)
$79$ \( -32 - 10 T + T^{2} \)
$83$ \( -48 - 6 T + T^{2} \)
$89$ \( ( 14 + T )^{2} \)
$97$ \( -228 + T^{2} \)
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