Properties

Label 546.2.a.i
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: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

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

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
3.70156
−2.70156
1.00000 −1.00000 1.00000 −3.70156 −1.00000 1.00000 1.00000 1.00000 −3.70156
1.2 1.00000 −1.00000 1.00000 2.70156 −1.00000 1.00000 1.00000 1.00000 2.70156
\(n\): 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.i 2
3.b odd 2 1 1638.2.a.w 2
4.b odd 2 1 4368.2.a.bg 2
7.b odd 2 1 3822.2.a.bt 2
13.b even 2 1 7098.2.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.i 2 1.a even 1 1 trivial
1638.2.a.w 2 3.b odd 2 1
3822.2.a.bt 2 7.b odd 2 1
4368.2.a.bg 2 4.b odd 2 1
7098.2.a.bh 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(546))\):

\( T_{5}^{2} + T_{5} - 10 \)
\( T_{11}^{2} - 5 T_{11} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -10 + T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -4 - 5 T + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -10 - T + T^{2} \)
$19$ \( -4 - 5 T + T^{2} \)
$23$ \( -8 - 3 T + T^{2} \)
$29$ \( -10 + T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 46 - 15 T + T^{2} \)
$41$ \( -32 + 6 T + T^{2} \)
$43$ \( -92 + T + T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( ( 2 + T )^{2} \)
$59$ \( -160 + 4 T + T^{2} \)
$61$ \( 10 - 9 T + T^{2} \)
$67$ \( -40 + 2 T + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 10 + 9 T + T^{2} \)
$79$ \( -32 - 6 T + T^{2} \)
$83$ \( 8 + 14 T + T^{2} \)
$89$ \( -148 - 8 T + T^{2} \)
$97$ \( -148 - 8 T + T^{2} \)
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