Properties

Label 8048.2.a.u
Level $8048$
Weight $2$
Character orbit 8048.a
Self dual yes
Analytic conductor $64.264$
Analytic rank $0$
Dimension $26$
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: \(0\)
Dimension: \(26\)
Twist minimal: no (minimal twist has level 503)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 26q - 4q^{3} + 9q^{5} - 11q^{7} + 42q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 26q - 4q^{3} + 9q^{5} - 11q^{7} + 42q^{9} + 17q^{11} + 14q^{13} - 18q^{15} + 17q^{17} + 22q^{19} - 16q^{21} - 27q^{23} + 93q^{25} - 31q^{27} + 13q^{29} - 26q^{31} + 6q^{33} + 22q^{35} + 55q^{37} + 15q^{39} + 24q^{41} - 20q^{43} - 8q^{45} + 25q^{47} + 65q^{49} - 7q^{51} + 30q^{53} - 25q^{55} + 9q^{57} + 26q^{59} + 15q^{61} + 19q^{63} + 20q^{65} + 20q^{67} - 27q^{69} + 35q^{71} + 38q^{73} - 2q^{75} - 6q^{77} - 21q^{79} + 70q^{81} + 48q^{83} + 6q^{85} + 9q^{87} - 5q^{89} + 24q^{91} - 8q^{93} - 43q^{95} + 142q^{97} + 50q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −3.34519 0 2.36922 0 0.269938 0 8.19027 0
1.2 0 −3.26409 0 1.15565 0 1.19216 0 7.65430 0
1.3 0 −3.12024 0 −4.14994 0 3.36863 0 6.73592 0
1.4 0 −2.94588 0 −4.09318 0 −3.85397 0 5.67824 0
1.5 0 −2.76613 0 3.34245 0 5.14301 0 4.65146 0
1.6 0 −2.32813 0 1.62138 0 −3.43484 0 2.42020 0
1.7 0 −2.27911 0 3.79116 0 −4.03115 0 2.19436 0
1.8 0 −1.74581 0 1.36534 0 0.0430977 0 0.0478420 0
1.9 0 −1.69727 0 0.763844 0 0.178001 0 −0.119277 0
1.10 0 −1.34769 0 4.23270 0 3.43664 0 −1.18373 0
1.11 0 −1.22555 0 −2.04978 0 −4.12271 0 −1.49802 0
1.12 0 −0.577924 0 2.10638 0 −1.73516 0 −2.66600 0
1.13 0 −0.249551 0 −3.41933 0 3.14640 0 −2.93772 0
1.14 0 0.205279 0 2.58453 0 2.90918 0 −2.95786 0
1.15 0 0.705033 0 −0.869095 0 −5.04945 0 −2.50293 0
1.16 0 1.04994 0 −1.98731 0 −2.61319 0 −1.89762 0
1.17 0 1.08298 0 3.81229 0 −4.19464 0 −1.82716 0
1.18 0 1.08803 0 3.67682 0 −1.62501 0 −1.81619 0
1.19 0 1.16576 0 −3.04539 0 −0.946556 0 −1.64102 0
1.20 0 1.65492 0 1.16845 0 5.22170 0 −0.261242 0
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.26
Significant digits:
Format:

Atkin-Lehner signs

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

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 8048.2.a.u 26
4.b odd 2 1 503.2.a.f 26
12.b even 2 1 4527.2.a.o 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.2.a.f 26 4.b odd 2 1
4527.2.a.o 26 12.b even 2 1
8048.2.a.u 26 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\(T_{3}^{26} + \cdots\)
\(T_{5}^{26} - \cdots\)
\(T_{7}^{26} + \cdots\)
\(T_{13}^{26} - \cdots\)

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database