Properties

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

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 422 T^{2} + 59049 T^{4} \)
$5$ \( 1 - 6054 T^{2} + 9765625 T^{4} \)
$7$ \( ( 1 + 208 T + 16807 T^{2} )^{2} \)
$11$ \( 1 - 34806 T^{2} + 25937424601 T^{4} \)
$13$ \( 1 - 260950 T^{2} + 137858491849 T^{4} \)
$17$ \( ( 1 + 1278 T + 1419857 T^{2} )^{2} \)
$19$ \( 1 - 3715654 T^{2} + 6131066257801 T^{4} \)
$23$ \( ( 1 - 3216 T + 6436343 T^{2} )^{2} \)
$29$ \( 1 - 32507574 T^{2} + 420707233300201 T^{4} \)
$31$ \( ( 1 - 2624 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 - 49234150 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 + 170 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 + 103108298 T^{2} + 21611482313284249 T^{4} \)
$47$ \( ( 1 + 32 T + 229345007 T^{2} )^{2} \)
$53$ \( 1 - 344527302 T^{2} + 174887470365513049 T^{4} \)
$59$ \( 1 + 290741802 T^{2} + 511116753300641401 T^{4} \)
$61$ \( 1 - 1450119158 T^{2} + 713342911662882601 T^{4} \)
$67$ \( 1 - 2269936678 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 - 28592 T + 1804229351 T^{2} )^{2} \)
$73$ \( ( 1 - 53670 T + 2073071593 T^{2} )^{2} \)
$79$ \( ( 1 - 69152 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 - 6449241286 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 - 126806 T + 5584059449 T^{2} )^{2} \)
$97$ \( ( 1 - 62290 T + 8587340257 T^{2} )^{2} \)
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