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

Related objects

Downloads

Learn more about

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: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q\) \( -\beta q^{3} \) \( + ( -1 + \beta ) q^{5} \) \( + ( 2 + \beta ) q^{7} \) \( + 2 q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta q^{3} \) \( + ( -1 + \beta ) q^{5} \) \( + ( 2 + \beta ) q^{7} \) \( + 2 q^{9} \) \( + ( 2 + \beta ) q^{11} \) \( + ( -1 - 2 \beta ) q^{13} \) \( + ( -5 + \beta ) q^{15} \) \( + 2 q^{17} \) \( + ( 1 + 3 \beta ) q^{19} \) \( + ( -5 - 2 \beta ) q^{21} \) \( -\beta q^{23} \) \( + ( 1 - 2 \beta ) q^{25} \) \( + \beta q^{27} \) \( + ( -5 + \beta ) q^{29} \) \( + ( -1 - 3 \beta ) q^{31} \) \( + ( -5 - 2 \beta ) q^{33} \) \( + ( 3 + \beta ) q^{35} \) \( -2 \beta q^{37} \) \( + ( 10 + \beta ) q^{39} \) \( + ( -3 - \beta ) q^{41} \) \( + ( -8 - \beta ) q^{43} \) \( + ( -2 + 2 \beta ) q^{45} \) \( + ( -2 - \beta ) q^{47} \) \( + ( 2 + 4 \beta ) q^{49} \) \( -2 \beta q^{51} \) \( + ( 4 - 2 \beta ) q^{53} \) \( + ( 3 + \beta ) q^{55} \) \( + ( -15 - \beta ) q^{57} \) \( -4 \beta q^{59} \) \( + ( 3 + 2 \beta ) q^{61} \) \( + ( 4 + 2 \beta ) q^{63} \) \( + ( -9 + \beta ) q^{65} \) \( + ( 2 - 3 \beta ) q^{67} \) \( + 5 q^{69} \) \( + ( -3 + \beta ) q^{71} \) \( + ( -12 + 2 \beta ) q^{73} \) \( + ( 10 - \beta ) q^{75} \) \( + ( 9 + 4 \beta ) q^{77} \) \( + ( -4 - 4 \beta ) q^{79} \) \( -11 q^{81} \) \( + ( -10 - 3 \beta ) q^{83} \) \( + ( -2 + 2 \beta ) q^{85} \) \( + ( -5 + 5 \beta ) q^{87} \) \( + ( -10 - 2 \beta ) q^{89} \) \( + ( -12 - 5 \beta ) q^{91} \) \( + ( 15 + \beta ) q^{93} \) \( + ( 14 - 2 \beta ) q^{95} \) \( -10 q^{97} \) \( + ( 4 + 2 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut 4q^{11} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut 10q^{15} \) \(\mathstrut +\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 10q^{21} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 10q^{29} \) \(\mathstrut -\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 10q^{33} \) \(\mathstrut +\mathstrut 6q^{35} \) \(\mathstrut +\mathstrut 20q^{39} \) \(\mathstrut -\mathstrut 6q^{41} \) \(\mathstrut -\mathstrut 16q^{43} \) \(\mathstrut -\mathstrut 4q^{45} \) \(\mathstrut -\mathstrut 4q^{47} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 8q^{53} \) \(\mathstrut +\mathstrut 6q^{55} \) \(\mathstrut -\mathstrut 30q^{57} \) \(\mathstrut +\mathstrut 6q^{61} \) \(\mathstrut +\mathstrut 8q^{63} \) \(\mathstrut -\mathstrut 18q^{65} \) \(\mathstrut +\mathstrut 4q^{67} \) \(\mathstrut +\mathstrut 10q^{69} \) \(\mathstrut -\mathstrut 6q^{71} \) \(\mathstrut -\mathstrut 24q^{73} \) \(\mathstrut +\mathstrut 20q^{75} \) \(\mathstrut +\mathstrut 18q^{77} \) \(\mathstrut -\mathstrut 8q^{79} \) \(\mathstrut -\mathstrut 22q^{81} \) \(\mathstrut -\mathstrut 20q^{83} \) \(\mathstrut -\mathstrut 4q^{85} \) \(\mathstrut -\mathstrut 10q^{87} \) \(\mathstrut -\mathstrut 20q^{89} \) \(\mathstrut -\mathstrut 24q^{91} \) \(\mathstrut +\mathstrut 30q^{93} \) \(\mathstrut +\mathstrut 28q^{95} \) \(\mathstrut -\mathstrut 20q^{97} \) \(\mathstrut +\mathstrut 8q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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.

Atkin-Lehner signs

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

Hecke kernels

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

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