Properties

Label 256.6.b.g
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4)
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.

\(f(q)\) \(=\) \( q + 12 i q^{3} + 54 i q^{5} + 88 q^{7} + 99 q^{9} +O(q^{10})\) \( q + 12 i q^{3} + 54 i q^{5} + 88 q^{7} + 99 q^{9} + 540 i q^{11} + 418 i q^{13} -648 q^{15} + 594 q^{17} -836 i q^{19} + 1056 i q^{21} + 4104 q^{23} + 209 q^{25} + 4104 i q^{27} + 594 i q^{29} + 4256 q^{31} -6480 q^{33} + 4752 i q^{35} -298 i q^{37} -5016 q^{39} -17226 q^{41} -12100 i q^{43} + 5346 i q^{45} -1296 q^{47} -9063 q^{49} + 7128 i q^{51} + 19494 i q^{53} -29160 q^{55} + 10032 q^{57} -7668 i q^{59} + 34738 i q^{61} + 8712 q^{63} -22572 q^{65} -21812 i q^{67} + 49248 i q^{69} + 46872 q^{71} -67562 q^{73} + 2508 i q^{75} + 47520 i q^{77} -76912 q^{79} -25191 q^{81} -67716 i q^{83} + 32076 i q^{85} -7128 q^{87} -29754 q^{89} + 36784 i q^{91} + 51072 i q^{93} + 45144 q^{95} -122398 q^{97} + 53460 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 176 q^{7} + 198 q^{9} + O(q^{10}) \) \( 2 q + 176 q^{7} + 198 q^{9} - 1296 q^{15} + 1188 q^{17} + 8208 q^{23} + 418 q^{25} + 8512 q^{31} - 12960 q^{33} - 10032 q^{39} - 34452 q^{41} - 2592 q^{47} - 18126 q^{49} - 58320 q^{55} + 20064 q^{57} + 17424 q^{63} - 45144 q^{65} + 93744 q^{71} - 135124 q^{73} - 153824 q^{79} - 50382 q^{81} - 14256 q^{87} - 59508 q^{89} + 90288 q^{95} - 244796 q^{97} + O(q^{100}) \)

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 12.0000i 0 54.0000i 0 88.0000 0 99.0000 0
129.2 0 12.0000i 0 54.0000i 0 88.0000 0 99.0000 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
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.g 2
4.b odd 2 1 256.6.b.c 2
8.b even 2 1 inner 256.6.b.g 2
8.d odd 2 1 256.6.b.c 2
16.e even 4 1 4.6.a.a 1
16.e even 4 1 64.6.a.f 1
16.f odd 4 1 16.6.a.b 1
16.f odd 4 1 64.6.a.b 1
48.i odd 4 1 36.6.a.a 1
48.i odd 4 1 576.6.a.bc 1
48.k even 4 1 144.6.a.c 1
48.k even 4 1 576.6.a.bd 1
80.i odd 4 1 100.6.c.b 2
80.j even 4 1 400.6.c.f 2
80.k odd 4 1 400.6.a.d 1
80.q even 4 1 100.6.a.b 1
80.s even 4 1 400.6.c.f 2
80.t odd 4 1 100.6.c.b 2
112.j even 4 1 784.6.a.d 1
112.l odd 4 1 196.6.a.e 1
112.w even 12 2 196.6.e.g 2
112.x odd 12 2 196.6.e.d 2
144.w odd 12 2 324.6.e.d 2
144.x even 12 2 324.6.e.a 2
176.l odd 4 1 484.6.a.a 1
208.m odd 4 1 676.6.d.a 2
208.p even 4 1 676.6.a.a 1
208.r odd 4 1 676.6.d.a 2
240.bb even 4 1 900.6.d.a 2
240.bf even 4 1 900.6.d.a 2
240.bm odd 4 1 900.6.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 16.e even 4 1
16.6.a.b 1 16.f odd 4 1
36.6.a.a 1 48.i odd 4 1
64.6.a.b 1 16.f odd 4 1
64.6.a.f 1 16.e even 4 1
100.6.a.b 1 80.q even 4 1
100.6.c.b 2 80.i odd 4 1
100.6.c.b 2 80.t odd 4 1
144.6.a.c 1 48.k even 4 1
196.6.a.e 1 112.l odd 4 1
196.6.e.d 2 112.x odd 12 2
196.6.e.g 2 112.w even 12 2
256.6.b.c 2 4.b odd 2 1
256.6.b.c 2 8.d odd 2 1
256.6.b.g 2 1.a even 1 1 trivial
256.6.b.g 2 8.b even 2 1 inner
324.6.e.a 2 144.x even 12 2
324.6.e.d 2 144.w odd 12 2
400.6.a.d 1 80.k odd 4 1
400.6.c.f 2 80.j even 4 1
400.6.c.f 2 80.s even 4 1
484.6.a.a 1 176.l odd 4 1
576.6.a.bc 1 48.i odd 4 1
576.6.a.bd 1 48.k even 4 1
676.6.a.a 1 208.p even 4 1
676.6.d.a 2 208.m odd 4 1
676.6.d.a 2 208.r odd 4 1
784.6.a.d 1 112.j even 4 1
900.6.a.h 1 240.bm odd 4 1
900.6.d.a 2 240.bb even 4 1
900.6.d.a 2 240.bf even 4 1

Hecke kernels

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

\( T_{3}^{2} + 144 \)
\( T_{7} - 88 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 144 + T^{2} \)
$5$ \( 2916 + T^{2} \)
$7$ \( ( -88 + T )^{2} \)
$11$ \( 291600 + T^{2} \)
$13$ \( 174724 + T^{2} \)
$17$ \( ( -594 + T )^{2} \)
$19$ \( 698896 + T^{2} \)
$23$ \( ( -4104 + T )^{2} \)
$29$ \( 352836 + T^{2} \)
$31$ \( ( -4256 + T )^{2} \)
$37$ \( 88804 + T^{2} \)
$41$ \( ( 17226 + T )^{2} \)
$43$ \( 146410000 + T^{2} \)
$47$ \( ( 1296 + T )^{2} \)
$53$ \( 380016036 + T^{2} \)
$59$ \( 58798224 + T^{2} \)
$61$ \( 1206728644 + T^{2} \)
$67$ \( 475763344 + T^{2} \)
$71$ \( ( -46872 + T )^{2} \)
$73$ \( ( 67562 + T )^{2} \)
$79$ \( ( 76912 + T )^{2} \)
$83$ \( 4585456656 + T^{2} \)
$89$ \( ( 29754 + T )^{2} \)
$97$ \( ( 122398 + T )^{2} \)
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