Properties

Label 8048.2.a.l
Level 8048
Weight 2
Character orbit 8048.a
Self dual yes
Analytic conductor 64.264
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.2636035467\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 1006)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \beta ) q^{3} + ( -2 + 2 \beta ) q^{5} + ( 1 + 2 \beta ) q^{7} + 2 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \beta ) q^{3} + ( -2 + 2 \beta ) q^{5} + ( 1 + 2 \beta ) q^{7} + 2 q^{9} + ( 1 + 2 \beta ) q^{11} + ( 1 - 4 \beta ) q^{13} + ( -6 + 2 \beta ) q^{15} + 2 q^{17} + ( -2 + 6 \beta ) q^{19} + ( -3 - 4 \beta ) q^{21} + ( 1 - 2 \beta ) q^{23} + ( 3 - 4 \beta ) q^{25} + ( -1 + 2 \beta ) q^{27} + ( -6 + 2 \beta ) q^{29} + ( 2 - 6 \beta ) q^{31} + ( -3 - 4 \beta ) q^{33} + ( 2 + 2 \beta ) q^{35} + ( 2 - 4 \beta ) q^{37} + ( 9 + 2 \beta ) q^{39} + ( -2 - 2 \beta ) q^{41} + ( -7 - 2 \beta ) q^{43} + ( -4 + 4 \beta ) q^{45} + ( -1 - 2 \beta ) q^{47} + ( -2 + 8 \beta ) q^{49} + ( 2 - 4 \beta ) q^{51} + ( 6 - 4 \beta ) q^{53} + ( 2 + 2 \beta ) q^{55} + ( -14 - 2 \beta ) q^{57} + ( 4 - 8 \beta ) q^{59} + ( 1 + 4 \beta ) q^{61} + ( 2 + 4 \beta ) q^{63} + ( -10 + 2 \beta ) q^{65} + ( 5 - 6 \beta ) q^{67} + 5 q^{69} + ( -4 + 2 \beta ) q^{71} + ( -14 + 4 \beta ) q^{73} + ( 11 - 2 \beta ) q^{75} + ( 5 + 8 \beta ) q^{77} -8 \beta q^{79} -11 q^{81} + ( -7 - 6 \beta ) q^{83} + ( -4 + 4 \beta ) q^{85} + ( -10 + 10 \beta ) q^{87} + ( -8 - 4 \beta ) q^{89} + ( -7 - 10 \beta ) q^{91} + ( 14 + 2 \beta ) q^{93} + ( 16 - 4 \beta ) q^{95} -10 q^{97} + ( 2 + 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + 4q^{7} + 4q^{9} + O(q^{10}) \) \( 2q - 2q^{5} + 4q^{7} + 4q^{9} + 4q^{11} - 2q^{13} - 10q^{15} + 4q^{17} + 2q^{19} - 10q^{21} + 2q^{25} - 10q^{29} - 2q^{31} - 10q^{33} + 6q^{35} + 20q^{39} - 6q^{41} - 16q^{43} - 4q^{45} - 4q^{47} + 4q^{49} + 8q^{53} + 6q^{55} - 30q^{57} + 6q^{61} + 8q^{63} - 18q^{65} + 4q^{67} + 10q^{69} - 6q^{71} - 24q^{73} + 20q^{75} + 18q^{77} - 8q^{79} - 22q^{81} - 20q^{83} - 4q^{85} - 10q^{87} - 20q^{89} - 24q^{91} + 30q^{93} + 28q^{95} - 20q^{97} + 8q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
0 −2.23607 0 1.23607 0 4.23607 0 2.00000 0
1.2 0 2.23607 0 −3.23607 0 −0.236068 0 2.00000 0
\(n\): 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 8048.2.a.l 2
4.b odd 2 1 1006.2.a.f 2
12.b even 2 1 9054.2.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1006.2.a.f 2 4.b odd 2 1
8048.2.a.l 2 1.a even 1 1 trivial
9054.2.a.y 2 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(503\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\( T_{3}^{2} - 5 \)
\( T_{5}^{2} + 2 T_{5} - 4 \)
\( T_{7}^{2} - 4 T_{7} - 1 \)
\( T_{13}^{2} + 2 T_{13} - 19 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T^{2} + 9 T^{4} \)
$5$ \( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( 1 - 4 T + 13 T^{2} - 28 T^{3} + 49 T^{4} \)
$11$ \( 1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( 1 + 2 T + 7 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 2 T - 6 T^{2} - 38 T^{3} + 361 T^{4} \)
$23$ \( 1 + 41 T^{2} + 529 T^{4} \)
$29$ \( 1 + 10 T + 78 T^{2} + 290 T^{3} + 841 T^{4} \)
$31$ \( 1 + 2 T + 18 T^{2} + 62 T^{3} + 961 T^{4} \)
$37$ \( 1 + 54 T^{2} + 1369 T^{4} \)
$41$ \( 1 + 6 T + 86 T^{2} + 246 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 16 T + 145 T^{2} + 688 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 4 T + 93 T^{2} + 188 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 38 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 6 T + 111 T^{2} - 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 4 T + 93 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 6 T + 146 T^{2} + 426 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 24 T + 270 T^{2} + 1752 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 8 T + 94 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 20 T + 221 T^{2} + 1660 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 20 T + 258 T^{2} + 1780 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 10 T + 97 T^{2} )^{2} \)
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