Properties

Label 6041.2.a.b
Level 6041
Weight 2
Character orbit 6041.a
Self dual yes
Analytic conductor 48.238
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6041 = 7 \cdot 863 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6041.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2376278611\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( 1 + \beta ) q^{3} - q^{4} + ( 1 - \beta ) q^{5} + ( -1 - \beta ) q^{6} - q^{7} + 3 q^{8} + ( -1 + 3 \beta ) q^{9} +O(q^{10})\) \( q - q^{2} + ( 1 + \beta ) q^{3} - q^{4} + ( 1 - \beta ) q^{5} + ( -1 - \beta ) q^{6} - q^{7} + 3 q^{8} + ( -1 + 3 \beta ) q^{9} + ( -1 + \beta ) q^{10} + ( -1 - \beta ) q^{11} + ( -1 - \beta ) q^{12} + ( 4 + \beta ) q^{13} + q^{14} -\beta q^{15} - q^{16} -2 \beta q^{17} + ( 1 - 3 \beta ) q^{18} + ( 4 - 5 \beta ) q^{19} + ( -1 + \beta ) q^{20} + ( -1 - \beta ) q^{21} + ( 1 + \beta ) q^{22} + ( -3 + 5 \beta ) q^{23} + ( 3 + 3 \beta ) q^{24} + ( -3 - \beta ) q^{25} + ( -4 - \beta ) q^{26} + ( -1 + 2 \beta ) q^{27} + q^{28} + ( 2 - 4 \beta ) q^{29} + \beta q^{30} + ( -4 + 4 \beta ) q^{31} -5 q^{32} + ( -2 - 3 \beta ) q^{33} + 2 \beta q^{34} + ( -1 + \beta ) q^{35} + ( 1 - 3 \beta ) q^{36} + ( -7 + 3 \beta ) q^{37} + ( -4 + 5 \beta ) q^{38} + ( 5 + 6 \beta ) q^{39} + ( 3 - 3 \beta ) q^{40} + ( -4 + 8 \beta ) q^{41} + ( 1 + \beta ) q^{42} + ( 4 - 6 \beta ) q^{43} + ( 1 + \beta ) q^{44} + ( -4 + \beta ) q^{45} + ( 3 - 5 \beta ) q^{46} + ( -2 + 2 \beta ) q^{47} + ( -1 - \beta ) q^{48} + q^{49} + ( 3 + \beta ) q^{50} + ( -2 - 4 \beta ) q^{51} + ( -4 - \beta ) q^{52} + ( 7 - 7 \beta ) q^{53} + ( 1 - 2 \beta ) q^{54} + \beta q^{55} -3 q^{56} + ( -1 - 6 \beta ) q^{57} + ( -2 + 4 \beta ) q^{58} + ( -9 - \beta ) q^{59} + \beta q^{60} + ( 4 + 6 \beta ) q^{61} + ( 4 - 4 \beta ) q^{62} + ( 1 - 3 \beta ) q^{63} + 7 q^{64} + ( 3 - 4 \beta ) q^{65} + ( 2 + 3 \beta ) q^{66} + ( -12 + 3 \beta ) q^{67} + 2 \beta q^{68} + ( 2 + 7 \beta ) q^{69} + ( 1 - \beta ) q^{70} + ( 4 - 4 \beta ) q^{71} + ( -3 + 9 \beta ) q^{72} + ( 3 - 3 \beta ) q^{73} + ( 7 - 3 \beta ) q^{74} + ( -4 - 5 \beta ) q^{75} + ( -4 + 5 \beta ) q^{76} + ( 1 + \beta ) q^{77} + ( -5 - 6 \beta ) q^{78} + ( -12 - \beta ) q^{79} + ( -1 + \beta ) q^{80} + ( 4 - 6 \beta ) q^{81} + ( 4 - 8 \beta ) q^{82} + ( 12 - 2 \beta ) q^{83} + ( 1 + \beta ) q^{84} + 2 q^{85} + ( -4 + 6 \beta ) q^{86} + ( -2 - 6 \beta ) q^{87} + ( -3 - 3 \beta ) q^{88} + ( 6 - 8 \beta ) q^{89} + ( 4 - \beta ) q^{90} + ( -4 - \beta ) q^{91} + ( 3 - 5 \beta ) q^{92} + 4 \beta q^{93} + ( 2 - 2 \beta ) q^{94} + ( 9 - 4 \beta ) q^{95} + ( -5 - 5 \beta ) q^{96} -3 \beta q^{97} - q^{98} + ( -2 - 5 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 3q^{3} - 2q^{4} + q^{5} - 3q^{6} - 2q^{7} + 6q^{8} + q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 3q^{3} - 2q^{4} + q^{5} - 3q^{6} - 2q^{7} + 6q^{8} + q^{9} - q^{10} - 3q^{11} - 3q^{12} + 9q^{13} + 2q^{14} - q^{15} - 2q^{16} - 2q^{17} - q^{18} + 3q^{19} - q^{20} - 3q^{21} + 3q^{22} - q^{23} + 9q^{24} - 7q^{25} - 9q^{26} + 2q^{28} + q^{30} - 4q^{31} - 10q^{32} - 7q^{33} + 2q^{34} - q^{35} - q^{36} - 11q^{37} - 3q^{38} + 16q^{39} + 3q^{40} + 3q^{42} + 2q^{43} + 3q^{44} - 7q^{45} + q^{46} - 2q^{47} - 3q^{48} + 2q^{49} + 7q^{50} - 8q^{51} - 9q^{52} + 7q^{53} + q^{55} - 6q^{56} - 8q^{57} - 19q^{59} + q^{60} + 14q^{61} + 4q^{62} - q^{63} + 14q^{64} + 2q^{65} + 7q^{66} - 21q^{67} + 2q^{68} + 11q^{69} + q^{70} + 4q^{71} + 3q^{72} + 3q^{73} + 11q^{74} - 13q^{75} - 3q^{76} + 3q^{77} - 16q^{78} - 25q^{79} - q^{80} + 2q^{81} + 22q^{83} + 3q^{84} + 4q^{85} - 2q^{86} - 10q^{87} - 9q^{88} + 4q^{89} + 7q^{90} - 9q^{91} + q^{92} + 4q^{93} + 2q^{94} + 14q^{95} - 15q^{96} - 3q^{97} - 2q^{98} - 9q^{99} + O(q^{100}) \)

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
−0.618034
1.61803
−1.00000 0.381966 −1.00000 1.61803 −0.381966 −1.00000 3.00000 −2.85410 −1.61803
1.2 −1.00000 2.61803 −1.00000 −0.618034 −2.61803 −1.00000 3.00000 3.85410 0.618034
\(n\): 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 6041.2.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6041.2.a.b 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(863\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6041))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + 2 T^{2} )^{2} \)
$3$ \( 1 - 3 T + 7 T^{2} - 9 T^{3} + 9 T^{4} \)
$5$ \( 1 - T + 9 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 + 3 T + 23 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( 1 - 9 T + 45 T^{2} - 117 T^{3} + 169 T^{4} \)
$17$ \( 1 + 2 T + 30 T^{2} + 34 T^{3} + 289 T^{4} \)
$19$ \( 1 - 3 T + 9 T^{2} - 57 T^{3} + 361 T^{4} \)
$23$ \( 1 + T + 15 T^{2} + 23 T^{3} + 529 T^{4} \)
$29$ \( 1 + 38 T^{2} + 841 T^{4} \)
$31$ \( 1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4} \)
$37$ \( 1 + 11 T + 93 T^{2} + 407 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 2 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 2 T + 42 T^{2} - 86 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 2 T + 90 T^{2} + 94 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 7 T + 57 T^{2} - 371 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 19 T + 207 T^{2} + 1121 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 14 T + 126 T^{2} - 854 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 21 T + 233 T^{2} + 1407 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 4 T + 126 T^{2} - 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 3 T + 137 T^{2} - 219 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 25 T + 313 T^{2} + 1975 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 22 T + 282 T^{2} - 1826 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 4 T + 102 T^{2} - 356 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 3 T + 185 T^{2} + 291 T^{3} + 9409 T^{4} \)
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