Properties

Label 256.8.b.c
Level $256$
Weight $8$
Character orbit 256.b
Analytic conductor $79.971$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(79.9705665239\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + 84 i q^{3} + 82 i q^{5} -456 q^{7} -4869 q^{9} +O(q^{10})\) \( q + 84 i q^{3} + 82 i q^{5} -456 q^{7} -4869 q^{9} -2524 i q^{11} -10778 i q^{13} -6888 q^{15} -11150 q^{17} -4124 i q^{19} -38304 i q^{21} + 81704 q^{23} + 71401 q^{25} -225288 i q^{27} + 99798 i q^{29} + 40480 q^{31} + 212016 q^{33} -37392 i q^{35} + 419442 i q^{37} + 905352 q^{39} -141402 q^{41} -690428 i q^{43} -399258 i q^{45} + 682032 q^{47} -615607 q^{49} -936600 i q^{51} -1813118 i q^{53} + 206968 q^{55} + 346416 q^{57} -966028 i q^{59} + 1887670 i q^{61} + 2220264 q^{63} + 883796 q^{65} -2965868 i q^{67} + 6863136 i q^{69} -2548232 q^{71} + 1680326 q^{73} + 5997684 i q^{75} + 1150944 i q^{77} -4038064 q^{79} + 8275689 q^{81} + 5385764 i q^{83} -914300 i q^{85} -8383032 q^{87} + 6473046 q^{89} + 4914768 i q^{91} + 3400320 i q^{93} + 338168 q^{95} -6065758 q^{97} + 12289356 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 912q^{7} - 9738q^{9} + O(q^{10}) \) \( 2q - 912q^{7} - 9738q^{9} - 13776q^{15} - 22300q^{17} + 163408q^{23} + 142802q^{25} + 80960q^{31} + 424032q^{33} + 1810704q^{39} - 282804q^{41} + 1364064q^{47} - 1231214q^{49} + 413936q^{55} + 692832q^{57} + 4440528q^{63} + 1767592q^{65} - 5096464q^{71} + 3360652q^{73} - 8076128q^{79} + 16551378q^{81} - 16766064q^{87} + 12946092q^{89} + 676336q^{95} - 12131516q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.00000i
1.00000i
0 84.0000i 0 82.0000i 0 −456.000 0 −4869.00 0
129.2 0 84.0000i 0 82.0000i 0 −456.000 0 −4869.00 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 256.8.b.c 2
4.b odd 2 1 256.8.b.e 2
8.b even 2 1 inner 256.8.b.c 2
8.d odd 2 1 256.8.b.e 2
16.e even 4 1 16.8.a.c 1
16.e even 4 1 64.8.a.a 1
16.f odd 4 1 8.8.a.a 1
16.f odd 4 1 64.8.a.g 1
48.i odd 4 1 144.8.a.g 1
48.i odd 4 1 576.8.a.k 1
48.k even 4 1 72.8.a.d 1
48.k even 4 1 576.8.a.j 1
80.i odd 4 1 400.8.c.b 2
80.j even 4 1 200.8.c.a 2
80.k odd 4 1 200.8.a.i 1
80.q even 4 1 400.8.a.b 1
80.s even 4 1 200.8.c.a 2
80.t odd 4 1 400.8.c.b 2
112.j even 4 1 392.8.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.a 1 16.f odd 4 1
16.8.a.c 1 16.e even 4 1
64.8.a.a 1 16.e even 4 1
64.8.a.g 1 16.f odd 4 1
72.8.a.d 1 48.k even 4 1
144.8.a.g 1 48.i odd 4 1
200.8.a.i 1 80.k odd 4 1
200.8.c.a 2 80.j even 4 1
200.8.c.a 2 80.s even 4 1
256.8.b.c 2 1.a even 1 1 trivial
256.8.b.c 2 8.b even 2 1 inner
256.8.b.e 2 4.b odd 2 1
256.8.b.e 2 8.d odd 2 1
392.8.a.d 1 112.j even 4 1
400.8.a.b 1 80.q even 4 1
400.8.c.b 2 80.i odd 4 1
400.8.c.b 2 80.t odd 4 1
576.8.a.j 1 48.k even 4 1
576.8.a.k 1 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{2} + 7056 \)
\( T_{7} + 456 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 7056 + T^{2} \)
$5$ \( 6724 + T^{2} \)
$7$ \( ( 456 + T )^{2} \)
$11$ \( 6370576 + T^{2} \)
$13$ \( 116165284 + T^{2} \)
$17$ \( ( 11150 + T )^{2} \)
$19$ \( 17007376 + T^{2} \)
$23$ \( ( -81704 + T )^{2} \)
$29$ \( 9959640804 + T^{2} \)
$31$ \( ( -40480 + T )^{2} \)
$37$ \( 175931591364 + T^{2} \)
$41$ \( ( 141402 + T )^{2} \)
$43$ \( 476690823184 + T^{2} \)
$47$ \( ( -682032 + T )^{2} \)
$53$ \( 3287396881924 + T^{2} \)
$59$ \( 933210096784 + T^{2} \)
$61$ \( 3563298028900 + T^{2} \)
$67$ \( 8796372993424 + T^{2} \)
$71$ \( ( 2548232 + T )^{2} \)
$73$ \( ( -1680326 + T )^{2} \)
$79$ \( ( 4038064 + T )^{2} \)
$83$ \( 29006453863696 + T^{2} \)
$89$ \( ( -6473046 + T )^{2} \)
$97$ \( ( 6065758 + T )^{2} \)
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