Properties

Label 6028.2.a.a
Level 6028
Weight 2
Character orbit 6028.a
Self dual yes
Analytic conductor 48.134
Analytic rank 2
Dimension 2
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6028.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1338223384\)
Analytic rank: \(2\)
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.

\(f(q)\) \(=\) \( q -\beta q^{3} + ( -3 + \beta ) q^{5} + ( -3 - \beta ) q^{7} + ( -2 + \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{3} + ( -3 + \beta ) q^{5} + ( -3 - \beta ) q^{7} + ( -2 + \beta ) q^{9} + q^{11} + ( -3 + 2 \beta ) q^{13} + ( -1 + 2 \beta ) q^{15} + ( -5 + 2 \beta ) q^{17} + ( -2 + \beta ) q^{19} + ( 1 + 4 \beta ) q^{21} + ( -4 + 3 \beta ) q^{23} + ( 5 - 5 \beta ) q^{25} + ( -1 + 4 \beta ) q^{27} + ( -4 - 2 \beta ) q^{29} + ( -3 - 4 \beta ) q^{31} -\beta q^{33} + ( 8 - \beta ) q^{35} + ( -1 - 3 \beta ) q^{37} + ( -2 + \beta ) q^{39} -3 q^{41} + ( -7 + 6 \beta ) q^{43} + ( 7 - 4 \beta ) q^{45} + ( 1 - 8 \beta ) q^{47} + ( 3 + 7 \beta ) q^{49} + ( -2 + 3 \beta ) q^{51} + ( -5 - 2 \beta ) q^{53} + ( -3 + \beta ) q^{55} + ( -1 + \beta ) q^{57} + ( 2 - 5 \beta ) q^{59} + ( -13 + 2 \beta ) q^{61} + ( 5 - 2 \beta ) q^{63} + ( 11 - 7 \beta ) q^{65} + ( 1 - 7 \beta ) q^{67} + ( -3 + \beta ) q^{69} + ( -5 - 3 \beta ) q^{71} + ( 11 - 8 \beta ) q^{73} + 5 q^{75} + ( -3 - \beta ) q^{77} + ( -11 + 6 \beta ) q^{79} + ( 2 - 6 \beta ) q^{81} + ( 1 + 9 \beta ) q^{83} + ( 17 - 9 \beta ) q^{85} + ( 2 + 6 \beta ) q^{87} + ( 4 - 5 \beta ) q^{89} + ( 7 - 5 \beta ) q^{91} + ( 4 + 7 \beta ) q^{93} + ( 7 - 4 \beta ) q^{95} + ( 1 + 8 \beta ) q^{97} + ( -2 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - 5q^{5} - 7q^{7} - 3q^{9} + O(q^{10}) \) \( 2q - q^{3} - 5q^{5} - 7q^{7} - 3q^{9} + 2q^{11} - 4q^{13} - 8q^{17} - 3q^{19} + 6q^{21} - 5q^{23} + 5q^{25} + 2q^{27} - 10q^{29} - 10q^{31} - q^{33} + 15q^{35} - 5q^{37} - 3q^{39} - 6q^{41} - 8q^{43} + 10q^{45} - 6q^{47} + 13q^{49} - q^{51} - 12q^{53} - 5q^{55} - q^{57} - q^{59} - 24q^{61} + 8q^{63} + 15q^{65} - 5q^{67} - 5q^{69} - 13q^{71} + 14q^{73} + 10q^{75} - 7q^{77} - 16q^{79} - 2q^{81} + 11q^{83} + 25q^{85} + 10q^{87} + 3q^{89} + 9q^{91} + 15q^{93} + 10q^{95} + 10q^{97} - 3q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
0 −1.61803 0 −1.38197 0 −4.61803 0 −0.381966 0
1.2 0 0.618034 0 −3.61803 0 −2.38197 0 −2.61803 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)
\(137\) \(-1\)

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 6028.2.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6028.2.a.a 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6028))\):

\( T_{3}^{2} + T_{3} - 1 \)
\( T_{5}^{2} + 5 T_{5} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T + 5 T^{2} + 3 T^{3} + 9 T^{4} \)
$5$ \( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} \)
$7$ \( 1 + 7 T + 25 T^{2} + 49 T^{3} + 49 T^{4} \)
$11$ \( ( 1 - T )^{2} \)
$13$ \( 1 + 4 T + 25 T^{2} + 52 T^{3} + 169 T^{4} \)
$17$ \( 1 + 8 T + 45 T^{2} + 136 T^{3} + 289 T^{4} \)
$19$ \( 1 + 3 T + 39 T^{2} + 57 T^{3} + 361 T^{4} \)
$23$ \( 1 + 5 T + 41 T^{2} + 115 T^{3} + 529 T^{4} \)
$29$ \( 1 + 10 T + 78 T^{2} + 290 T^{3} + 841 T^{4} \)
$31$ \( 1 + 10 T + 67 T^{2} + 310 T^{3} + 961 T^{4} \)
$37$ \( 1 + 5 T + 69 T^{2} + 185 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 3 T + 41 T^{2} )^{2} \)
$43$ \( 1 + 8 T + 57 T^{2} + 344 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 6 T + 23 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 12 T + 137 T^{2} + 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 + T + 87 T^{2} + 59 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 24 T + 261 T^{2} + 1464 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 5 T + 79 T^{2} + 335 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 13 T + 173 T^{2} + 923 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 14 T + 115 T^{2} - 1022 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 16 T + 177 T^{2} + 1264 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 11 T + 95 T^{2} - 913 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 3 T + 149 T^{2} - 267 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 10 T + 139 T^{2} - 970 T^{3} + 9409 T^{4} \)
show more
show less