Properties

Label 256.5.c.f
Level $256$
Weight $5$
Character orbit 256.c
Analytic conductor $26.463$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 256.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.4627105495\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
Defining polynomial: \(x^{2} + 6\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 8 q^{5} -8 \beta q^{7} -15 q^{9} +O(q^{10})\) \( q + \beta q^{3} + 8 q^{5} -8 \beta q^{7} -15 q^{9} + 11 \beta q^{11} -216 q^{13} + 8 \beta q^{15} -162 q^{17} + 45 \beta q^{19} + 768 q^{21} + 72 \beta q^{23} -561 q^{25} + 66 \beta q^{27} -1304 q^{29} -64 \beta q^{31} -1056 q^{33} -64 \beta q^{35} + 1512 q^{37} -216 \beta q^{39} -1890 q^{41} -297 \beta q^{43} -120 q^{45} + 144 \beta q^{47} -3743 q^{49} -162 \beta q^{51} -1976 q^{53} + 88 \beta q^{55} -4320 q^{57} + 231 \beta q^{59} + 2376 q^{61} + 120 \beta q^{63} -1728 q^{65} -171 \beta q^{67} -6912 q^{69} + 792 \beta q^{71} -2750 q^{73} -561 \beta q^{75} + 8448 q^{77} -816 \beta q^{79} -7551 q^{81} + 953 \beta q^{83} -1296 q^{85} -1304 \beta q^{87} -2430 q^{89} + 1728 \beta q^{91} + 6144 q^{93} + 360 \beta q^{95} + 7454 q^{97} -165 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 16q^{5} - 30q^{9} + O(q^{10}) \) \( 2q + 16q^{5} - 30q^{9} - 432q^{13} - 324q^{17} + 1536q^{21} - 1122q^{25} - 2608q^{29} - 2112q^{33} + 3024q^{37} - 3780q^{41} - 240q^{45} - 7486q^{49} - 3952q^{53} - 8640q^{57} + 4752q^{61} - 3456q^{65} - 13824q^{69} - 5500q^{73} + 16896q^{77} - 15102q^{81} - 2592q^{85} - 4860q^{89} + 12288q^{93} + 14908q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
2.44949i
2.44949i
0 9.79796i 0 8.00000 0 78.3837i 0 −15.0000 0
255.2 0 9.79796i 0 8.00000 0 78.3837i 0 −15.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.5.c.f 2
4.b odd 2 1 inner 256.5.c.f 2
8.b even 2 1 256.5.c.c 2
8.d odd 2 1 256.5.c.c 2
16.e even 4 2 128.5.d.c 4
16.f odd 4 2 128.5.d.c 4
48.i odd 4 2 1152.5.b.i 4
48.k even 4 2 1152.5.b.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.5.d.c 4 16.e even 4 2
128.5.d.c 4 16.f odd 4 2
256.5.c.c 2 8.b even 2 1
256.5.c.c 2 8.d odd 2 1
256.5.c.f 2 1.a even 1 1 trivial
256.5.c.f 2 4.b odd 2 1 inner
1152.5.b.i 4 48.i odd 4 2
1152.5.b.i 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{2} + 96 \)
\( T_{5} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 66 T^{2} + 6561 T^{4} \)
$5$ \( ( 1 - 8 T + 625 T^{2} )^{2} \)
$7$ \( 1 + 1342 T^{2} + 5764801 T^{4} \)
$11$ \( 1 - 17666 T^{2} + 214358881 T^{4} \)
$13$ \( ( 1 + 216 T + 28561 T^{2} )^{2} \)
$17$ \( ( 1 + 162 T + 83521 T^{2} )^{2} \)
$19$ \( 1 - 66242 T^{2} + 16983563041 T^{4} \)
$23$ \( 1 - 62018 T^{2} + 78310985281 T^{4} \)
$29$ \( ( 1 + 1304 T + 707281 T^{2} )^{2} \)
$31$ \( 1 - 1453826 T^{2} + 852891037441 T^{4} \)
$37$ \( ( 1 - 1512 T + 1874161 T^{2} )^{2} \)
$41$ \( ( 1 + 1890 T + 2825761 T^{2} )^{2} \)
$43$ \( 1 + 1630462 T^{2} + 11688200277601 T^{4} \)
$47$ \( 1 - 7768706 T^{2} + 23811286661761 T^{4} \)
$53$ \( ( 1 + 1976 T + 7890481 T^{2} )^{2} \)
$59$ \( 1 - 19112066 T^{2} + 146830437604321 T^{4} \)
$61$ \( ( 1 - 2376 T + 13845841 T^{2} )^{2} \)
$67$ \( 1 - 37495106 T^{2} + 406067677556641 T^{4} \)
$71$ \( 1 + 9393982 T^{2} + 645753531245761 T^{4} \)
$73$ \( ( 1 + 2750 T + 28398241 T^{2} )^{2} \)
$79$ \( 1 - 13977986 T^{2} + 1517108809906561 T^{4} \)
$83$ \( 1 - 7728578 T^{2} + 2252292232139041 T^{4} \)
$89$ \( ( 1 + 2430 T + 62742241 T^{2} )^{2} \)
$97$ \( ( 1 - 7454 T + 88529281 T^{2} )^{2} \)
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