Properties

Label 6045.2.a.p
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} + q^{5} + \beta q^{6} + ( 1 + \beta ) q^{7} -2 \beta q^{8} + q^{9} +O(q^{10})\) \( q + \beta q^{2} + q^{3} + q^{5} + \beta q^{6} + ( 1 + \beta ) q^{7} -2 \beta q^{8} + q^{9} + \beta q^{10} + ( -2 - 2 \beta ) q^{11} + q^{13} + ( 2 + \beta ) q^{14} + q^{15} -4 q^{16} + ( 2 - 4 \beta ) q^{17} + \beta q^{18} + ( -6 + \beta ) q^{19} + ( 1 + \beta ) q^{21} + ( -4 - 2 \beta ) q^{22} + ( -6 + 2 \beta ) q^{23} -2 \beta q^{24} + q^{25} + \beta q^{26} + q^{27} + ( 3 - \beta ) q^{29} + \beta q^{30} + q^{31} + ( -2 - 2 \beta ) q^{33} + ( -8 + 2 \beta ) q^{34} + ( 1 + \beta ) q^{35} + ( -6 + 2 \beta ) q^{37} + ( 2 - 6 \beta ) q^{38} + q^{39} -2 \beta q^{40} + ( -7 + \beta ) q^{41} + ( 2 + \beta ) q^{42} + ( -1 + \beta ) q^{43} + q^{45} + ( 4 - 6 \beta ) q^{46} + ( -2 - 5 \beta ) q^{47} -4 q^{48} + ( -4 + 2 \beta ) q^{49} + \beta q^{50} + ( 2 - 4 \beta ) q^{51} + ( 4 + 3 \beta ) q^{53} + \beta q^{54} + ( -2 - 2 \beta ) q^{55} + ( -4 - 2 \beta ) q^{56} + ( -6 + \beta ) q^{57} + ( -2 + 3 \beta ) q^{58} + ( -1 - 5 \beta ) q^{59} + ( -4 + 3 \beta ) q^{61} + \beta q^{62} + ( 1 + \beta ) q^{63} + 8 q^{64} + q^{65} + ( -4 - 2 \beta ) q^{66} + ( -7 + \beta ) q^{67} + ( -6 + 2 \beta ) q^{69} + ( 2 + \beta ) q^{70} + ( -4 + 8 \beta ) q^{71} -2 \beta q^{72} + ( -6 - 5 \beta ) q^{73} + ( 4 - 6 \beta ) q^{74} + q^{75} + ( -6 - 4 \beta ) q^{77} + \beta q^{78} + 5 \beta q^{79} -4 q^{80} + q^{81} + ( 2 - 7 \beta ) q^{82} + ( -9 - 2 \beta ) q^{83} + ( 2 - 4 \beta ) q^{85} + ( 2 - \beta ) q^{86} + ( 3 - \beta ) q^{87} + ( 8 + 4 \beta ) q^{88} + ( -6 + 2 \beta ) q^{89} + \beta q^{90} + ( 1 + \beta ) q^{91} + q^{93} + ( -10 - 2 \beta ) q^{94} + ( -6 + \beta ) q^{95} + ( 9 + \beta ) q^{97} + ( 4 - 4 \beta ) q^{98} + ( -2 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{5} + 2q^{7} + 2q^{9} - 4q^{11} + 2q^{13} + 4q^{14} + 2q^{15} - 8q^{16} + 4q^{17} - 12q^{19} + 2q^{21} - 8q^{22} - 12q^{23} + 2q^{25} + 2q^{27} + 6q^{29} + 2q^{31} - 4q^{33} - 16q^{34} + 2q^{35} - 12q^{37} + 4q^{38} + 2q^{39} - 14q^{41} + 4q^{42} - 2q^{43} + 2q^{45} + 8q^{46} - 4q^{47} - 8q^{48} - 8q^{49} + 4q^{51} + 8q^{53} - 4q^{55} - 8q^{56} - 12q^{57} - 4q^{58} - 2q^{59} - 8q^{61} + 2q^{63} + 16q^{64} + 2q^{65} - 8q^{66} - 14q^{67} - 12q^{69} + 4q^{70} - 8q^{71} - 12q^{73} + 8q^{74} + 2q^{75} - 12q^{77} - 8q^{80} + 2q^{81} + 4q^{82} - 18q^{83} + 4q^{85} + 4q^{86} + 6q^{87} + 16q^{88} - 12q^{89} + 2q^{91} + 2q^{93} - 20q^{94} - 12q^{95} + 18q^{97} + 8q^{98} - 4q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
−1.41421 1.00000 0 1.00000 −1.41421 −0.414214 2.82843 1.00000 −1.41421
1.2 1.41421 1.00000 0 1.00000 1.41421 2.41421 −2.82843 1.00000 1.41421
\(n\): 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.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.p 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2}^{2} - 2 \)
\( T_{7}^{2} - 2 T_{7} - 1 \)
\( T_{11}^{2} + 4 T_{11} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 4 T^{4} \)
$3$ \( ( 1 - T )^{2} \)
$5$ \( ( 1 - T )^{2} \)
$7$ \( 1 - 2 T + 13 T^{2} - 14 T^{3} + 49 T^{4} \)
$11$ \( 1 + 4 T + 18 T^{2} + 44 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - T )^{2} \)
$17$ \( 1 - 4 T + 6 T^{2} - 68 T^{3} + 289 T^{4} \)
$19$ \( 1 + 12 T + 72 T^{2} + 228 T^{3} + 361 T^{4} \)
$23$ \( 1 + 12 T + 74 T^{2} + 276 T^{3} + 529 T^{4} \)
$29$ \( 1 - 6 T + 65 T^{2} - 174 T^{3} + 841 T^{4} \)
$31$ \( ( 1 - T )^{2} \)
$37$ \( 1 + 12 T + 102 T^{2} + 444 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 14 T + 129 T^{2} + 574 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 2 T + 85 T^{2} + 86 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 4 T + 48 T^{2} + 188 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 8 T + 104 T^{2} - 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 2 T + 69 T^{2} + 118 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 8 T + 120 T^{2} + 488 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 14 T + 181 T^{2} + 938 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 8 T + 30 T^{2} + 568 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 12 T + 132 T^{2} + 876 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 108 T^{2} + 6241 T^{4} \)
$83$ \( 1 + 18 T + 239 T^{2} + 1494 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 12 T + 206 T^{2} + 1068 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 18 T + 273 T^{2} - 1746 T^{3} + 9409 T^{4} \)
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