Properties

Label 256.6.b.h
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: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 32)
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.

\(f(q)\) \(=\) \( q + 8 i q^{3} + 14 i q^{5} + 208 q^{7} + 179 q^{9} +O(q^{10})\) \( q + 8 i q^{3} + 14 i q^{5} + 208 q^{7} + 179 q^{9} -536 i q^{11} -694 i q^{13} -112 q^{15} -1278 q^{17} -1112 i q^{19} + 1664 i q^{21} -3216 q^{23} + 2929 q^{25} + 3376 i q^{27} -2918 i q^{29} -2624 q^{31} + 4288 q^{33} + 2912 i q^{35} -9458 i q^{37} + 5552 q^{39} -170 q^{41} -19928 i q^{43} + 2506 i q^{45} + 32 q^{47} + 26457 q^{49} -10224 i q^{51} -22178 i q^{53} + 7504 q^{55} + 8896 q^{57} + 41480 i q^{59} -15462 i q^{61} + 37232 q^{63} + 9716 q^{65} + 20744 i q^{67} -25728 i q^{69} -28592 q^{71} + 53670 q^{73} + 23432 i q^{75} -111488 i q^{77} -69152 q^{79} + 16489 q^{81} + 37800 i q^{83} -17892 i q^{85} + 23344 q^{87} + 126806 q^{89} -144352 i q^{91} -20992 i q^{93} + 15568 q^{95} + 62290 q^{97} -95944 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 416q^{7} + 358q^{9} + O(q^{10}) \) \( 2q + 416q^{7} + 358q^{9} - 224q^{15} - 2556q^{17} - 6432q^{23} + 5858q^{25} - 5248q^{31} + 8576q^{33} + 11104q^{39} - 340q^{41} + 64q^{47} + 52914q^{49} + 15008q^{55} + 17792q^{57} + 74464q^{63} + 19432q^{65} - 57184q^{71} + 107340q^{73} - 138304q^{79} + 32978q^{81} + 46688q^{87} + 253612q^{89} + 31136q^{95} + 124580q^{97} + O(q^{100}) \)

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 8.00000i 0 14.0000i 0 208.000 0 179.000 0
129.2 0 8.00000i 0 14.0000i 0 208.000 0 179.000 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.h 2
4.b odd 2 1 256.6.b.b 2
8.b even 2 1 inner 256.6.b.h 2
8.d odd 2 1 256.6.b.b 2
16.e even 4 1 32.6.a.a 1
16.e even 4 1 64.6.a.e 1
16.f odd 4 1 32.6.a.c yes 1
16.f odd 4 1 64.6.a.c 1
48.i odd 4 1 288.6.a.d 1
48.i odd 4 1 576.6.a.u 1
48.k even 4 1 288.6.a.e 1
48.k even 4 1 576.6.a.v 1
80.i odd 4 1 800.6.c.a 2
80.j even 4 1 800.6.c.b 2
80.k odd 4 1 800.6.a.a 1
80.q even 4 1 800.6.a.e 1
80.s even 4 1 800.6.c.b 2
80.t odd 4 1 800.6.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.6.a.a 1 16.e even 4 1
32.6.a.c yes 1 16.f odd 4 1
64.6.a.c 1 16.f odd 4 1
64.6.a.e 1 16.e even 4 1
256.6.b.b 2 4.b odd 2 1
256.6.b.b 2 8.d odd 2 1
256.6.b.h 2 1.a even 1 1 trivial
256.6.b.h 2 8.b even 2 1 inner
288.6.a.d 1 48.i odd 4 1
288.6.a.e 1 48.k even 4 1
576.6.a.u 1 48.i odd 4 1
576.6.a.v 1 48.k even 4 1
800.6.a.a 1 80.k odd 4 1
800.6.a.e 1 80.q even 4 1
800.6.c.a 2 80.i odd 4 1
800.6.c.a 2 80.t odd 4 1
800.6.c.b 2 80.j even 4 1
800.6.c.b 2 80.s even 4 1

Hecke kernels

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

\( T_{3}^{2} + 64 \)
\( T_{7} - 208 \)