Properties

Label 6045.2.a.m
Level 6045
Weight 2
Character orbit 6045.a
Self dual yes
Analytic conductor 48.270
Analytic rank 1
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.56155
2.56155
−2.00000 1.00000 2.00000 −1.00000 −2.00000 −3.56155 0 1.00000 2.00000
1.2 −2.00000 1.00000 2.00000 −1.00000 −2.00000 0.561553 0 1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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

Atkin-Lehner signs

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

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T + 2 T^{2} )^{2} \)
$3$ \( ( 1 - T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( 1 + 3 T + 12 T^{2} + 21 T^{3} + 49 T^{4} \)
$11$ \( 1 - 3 T + 20 T^{2} - 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - T )^{2} \)
$17$ \( 1 + 2 T + 18 T^{2} + 34 T^{3} + 289 T^{4} \)
$19$ \( 1 + 2 T + 22 T^{2} + 38 T^{3} + 361 T^{4} \)
$23$ \( ( 1 + 2 T + 23 T^{2} )^{2} \)
$29$ \( 1 + 3 T + 22 T^{2} + 87 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + T )^{2} \)
$37$ \( 1 - 13 T + 112 T^{2} - 481 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 3 T + 80 T^{2} - 123 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 5 T + 88 T^{2} + 215 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 4 T + 30 T^{2} - 188 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 6 T + 98 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 9 T + 32 T^{2} + 531 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 6 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 11 T + 126 T^{2} + 737 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 12 T + 110 T^{2} + 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 3 T + 144 T^{2} + 219 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 14 T + 190 T^{2} - 1106 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 31 T + 402 T^{2} + 2573 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 21 T + 284 T^{2} + 1869 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 27 T + 372 T^{2} - 2619 T^{3} + 9409 T^{4} \)
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