Properties

Label 256.6.b.f
Level $256$
Weight $6$
Character orbit 256.b
Analytic conductor $41.058$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 256.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(41.0582578721\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 10 \beta q^{3} + 37 \beta q^{5} + 24 q^{7} - 157 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 10 \beta q^{3} + 37 \beta q^{5} + 24 q^{7} - 157 q^{9} - 62 \beta q^{11} + 239 \beta q^{13} - 1480 q^{15} - 1198 q^{17} + 1522 \beta q^{19} + 240 \beta q^{21} - 184 q^{23} - 2351 q^{25} + 860 \beta q^{27} - 1641 \beta q^{29} - 5728 q^{31} + 2480 q^{33} + 888 \beta q^{35} - 5163 \beta q^{37} - 9560 q^{39} + 8886 q^{41} + 4594 \beta q^{43} - 5809 \beta q^{45} + 23664 q^{47} - 16231 q^{49} - 11980 \beta q^{51} - 5843 \beta q^{53} + 9176 q^{55} - 60880 q^{57} - 8438 \beta q^{59} - 9241 \beta q^{61} - 3768 q^{63} - 35372 q^{65} - 7766 \beta q^{67} - 1840 \beta q^{69} + 31960 q^{71} + 4886 q^{73} - 23510 \beta q^{75} - 1488 \beta q^{77} + 44560 q^{79} - 72551 q^{81} + 33682 \beta q^{83} - 44326 \beta q^{85} + 65640 q^{87} - 71994 q^{89} + 5736 \beta q^{91} - 57280 \beta q^{93} - 225256 q^{95} + 48866 q^{97} + 9734 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 48 q^{7} - 314 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 48 q^{7} - 314 q^{9} - 2960 q^{15} - 2396 q^{17} - 368 q^{23} - 4702 q^{25} - 11456 q^{31} + 4960 q^{33} - 19120 q^{39} + 17772 q^{41} + 47328 q^{47} - 32462 q^{49} + 18352 q^{55} - 121760 q^{57} - 7536 q^{63} - 70744 q^{65} + 63920 q^{71} + 9772 q^{73} + 89120 q^{79} - 145102 q^{81} + 131280 q^{87} - 143988 q^{89} - 450512 q^{95} + 97732 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.00000i
1.00000i
0 20.0000i 0 74.0000i 0 24.0000 0 −157.000 0
129.2 0 20.0000i 0 74.0000i 0 24.0000 0 −157.000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.6.b.f 2
4.b odd 2 1 256.6.b.d 2
8.b even 2 1 inner 256.6.b.f 2
8.d odd 2 1 256.6.b.d 2
16.e even 4 1 8.6.a.a 1
16.e even 4 1 64.6.a.a 1
16.f odd 4 1 16.6.a.a 1
16.f odd 4 1 64.6.a.g 1
48.i odd 4 1 72.6.a.f 1
48.i odd 4 1 576.6.a.g 1
48.k even 4 1 144.6.a.k 1
48.k even 4 1 576.6.a.h 1
80.i odd 4 1 200.6.c.a 2
80.j even 4 1 400.6.c.d 2
80.k odd 4 1 400.6.a.l 1
80.q even 4 1 200.6.a.a 1
80.s even 4 1 400.6.c.d 2
80.t odd 4 1 200.6.c.a 2
112.j even 4 1 784.6.a.l 1
112.l odd 4 1 392.6.a.b 1
112.w even 12 2 392.6.i.b 2
112.x odd 12 2 392.6.i.e 2
176.l odd 4 1 968.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.a.a 1 16.e even 4 1
16.6.a.a 1 16.f odd 4 1
64.6.a.a 1 16.e even 4 1
64.6.a.g 1 16.f odd 4 1
72.6.a.f 1 48.i odd 4 1
144.6.a.k 1 48.k even 4 1
200.6.a.a 1 80.q even 4 1
200.6.c.a 2 80.i odd 4 1
200.6.c.a 2 80.t odd 4 1
256.6.b.d 2 4.b odd 2 1
256.6.b.d 2 8.d odd 2 1
256.6.b.f 2 1.a even 1 1 trivial
256.6.b.f 2 8.b even 2 1 inner
392.6.a.b 1 112.l odd 4 1
392.6.i.b 2 112.w even 12 2
392.6.i.e 2 112.x odd 12 2
400.6.a.l 1 80.k odd 4 1
400.6.c.d 2 80.j even 4 1
400.6.c.d 2 80.s even 4 1
576.6.a.g 1 48.i odd 4 1
576.6.a.h 1 48.k even 4 1
784.6.a.l 1 112.j even 4 1
968.6.a.a 1 176.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{2} + 400 \) Copy content Toggle raw display
\( T_{7} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 400 \) Copy content Toggle raw display
$5$ \( T^{2} + 5476 \) Copy content Toggle raw display
$7$ \( (T - 24)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 15376 \) Copy content Toggle raw display
$13$ \( T^{2} + 228484 \) Copy content Toggle raw display
$17$ \( (T + 1198)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 9265936 \) Copy content Toggle raw display
$23$ \( (T + 184)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 10771524 \) Copy content Toggle raw display
$31$ \( (T + 5728)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 106626276 \) Copy content Toggle raw display
$41$ \( (T - 8886)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 84419344 \) Copy content Toggle raw display
$47$ \( (T - 23664)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 136562596 \) Copy content Toggle raw display
$59$ \( T^{2} + 284799376 \) Copy content Toggle raw display
$61$ \( T^{2} + 341584324 \) Copy content Toggle raw display
$67$ \( T^{2} + 241243024 \) Copy content Toggle raw display
$71$ \( (T - 31960)^{2} \) Copy content Toggle raw display
$73$ \( (T - 4886)^{2} \) Copy content Toggle raw display
$79$ \( (T - 44560)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4537908496 \) Copy content Toggle raw display
$89$ \( (T + 71994)^{2} \) Copy content Toggle raw display
$97$ \( (T - 48866)^{2} \) Copy content Toggle raw display
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