Properties

Label 2576.2.a.s
Level 2576
Weight 2
Character orbit 2576.a
Self dual yes
Analytic conductor 20.569
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2576.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.5694635607\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 161)
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.

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

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 1.00000 0 −3.23607 0 1.00000 0 −2.00000 0
1.2 0 1.00000 0 1.23607 0 1.00000 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 2576.2.a.s 2
4.b odd 2 1 161.2.a.b 2
12.b even 2 1 1449.2.a.i 2
20.d odd 2 1 4025.2.a.i 2
28.d even 2 1 1127.2.a.d 2
92.b even 2 1 3703.2.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.a.b 2 4.b odd 2 1
1127.2.a.d 2 28.d even 2 1
1449.2.a.i 2 12.b even 2 1
2576.2.a.s 2 1.a even 1 1 trivial
3703.2.a.b 2 92.b even 2 1
4025.2.a.i 2 20.d odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(23\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2576))\):

\( T_{3} - 1 \)
\( T_{5}^{2} + 2 T_{5} - 4 \)
\( T_{11}^{2} - 20 \)
\( T_{13}^{2} + 4 T_{13} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T + 3 T^{2} )^{2} \)
$5$ \( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( 1 + 2 T^{2} + 121 T^{4} \)
$13$ \( 1 + 4 T + 25 T^{2} + 52 T^{3} + 169 T^{4} \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( 1 - 10 T + 58 T^{2} - 190 T^{3} + 361 T^{4} \)
$23$ \( ( 1 - T )^{2} \)
$29$ \( 1 - 6 T + 47 T^{2} - 174 T^{3} + 841 T^{4} \)
$31$ \( ( 1 - 9 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 2 T + 30 T^{2} - 74 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 77 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 4 T + 70 T^{2} - 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 2 T + 75 T^{2} + 94 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 18 T + 182 T^{2} - 954 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 12 T + 134 T^{2} - 708 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 58 T^{2} + 3721 T^{4} \)
$67$ \( 1 - 2 T + 10 T^{2} - 134 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 16 T + 201 T^{2} - 1136 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 101 T^{2} + 5329 T^{4} \)
$79$ \( 1 - 10 T + 178 T^{2} - 790 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 4 T + 150 T^{2} + 332 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 98 T^{2} + 7921 T^{4} \)
$97$ \( 1 + 6 T + 158 T^{2} + 582 T^{3} + 9409 T^{4} \)
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