Properties

Label 6045.2.a.o
Level 6045
Weight 2
Character orbit 6045.a
Self dual yes
Analytic conductor 48.270
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2695680219\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

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

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
−1.73205 −1.00000 1.00000 −1.00000 1.73205 2.00000 1.73205 1.00000 1.73205
1.2 1.73205 −1.00000 1.00000 −1.00000 −1.73205 2.00000 −1.73205 1.00000 −1.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(13\) \(-1\)
\(31\) \(1\)

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 6045.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.o 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2}^{2} - 3 \)
\( T_{7} - 2 \)
\( T_{11}^{2} + 8 T_{11} + 13 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} + 4 T^{4} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{2} \)
$11$ \( 1 + 8 T + 35 T^{2} + 88 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - T )^{2} \)
$17$ \( 1 + 22 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 3 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 + 3 T + 23 T^{2} )^{2} \)
$29$ \( 1 + 55 T^{2} + 841 T^{4} \)
$31$ \( ( 1 + T )^{2} \)
$37$ \( 1 + 4 T + 30 T^{2} + 148 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - T + 41 T^{2} )^{2} \)
$43$ \( 1 - 16 T + 138 T^{2} - 688 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 91 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 8 T + 110 T^{2} - 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 8 T + 122 T^{2} + 472 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 4 T + 114 T^{2} + 244 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 10 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 2 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 12 T + 170 T^{2} - 876 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 146 T^{2} + 6241 T^{4} \)
$83$ \( 1 + 4 T - 22 T^{2} + 332 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 14 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 146 T^{2} + 9409 T^{4} \)
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