Properties

Label 256.2.b.a
Level $256$
Weight $2$
Character orbit 256.b
Analytic conductor $2.044$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.04417029174\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 128)
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.

\(f(q)\) \(=\) \( q + \beta q^{3} + \beta q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + \beta q^{5} - 4 q^{7} - q^{9} + \beta q^{11} - \beta q^{13} - 4 q^{15} - 2 q^{17} + \beta q^{19} - 4 \beta q^{21} + 4 q^{23} + q^{25} + 2 \beta q^{27} + 3 \beta q^{29} - 4 q^{33} - 4 \beta q^{35} + 5 \beta q^{37} + 4 q^{39} + 6 q^{41} - 3 \beta q^{43} - \beta q^{45} + 8 q^{47} + 9 q^{49} - 2 \beta q^{51} - 3 \beta q^{53} - 4 q^{55} - 4 q^{57} - 7 \beta q^{59} - \beta q^{61} + 4 q^{63} + 4 q^{65} + 5 \beta q^{67} + 4 \beta q^{69} + 12 q^{71} - 14 q^{73} + \beta q^{75} - 4 \beta q^{77} + 8 q^{79} - 11 q^{81} - 3 \beta q^{83} - 2 \beta q^{85} - 12 q^{87} + 2 q^{89} + 4 \beta q^{91} - 4 q^{95} - 2 q^{97} - \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{7} - 2 q^{9} - 8 q^{15} - 4 q^{17} + 8 q^{23} + 2 q^{25} - 8 q^{33} + 8 q^{39} + 12 q^{41} + 16 q^{47} + 18 q^{49} - 8 q^{55} - 8 q^{57} + 8 q^{63} + 8 q^{65} + 24 q^{71} - 28 q^{73} + 16 q^{79} - 22 q^{81} - 24 q^{87} + 4 q^{89} - 8 q^{95} - 4 q^{97}+O(q^{100}) \) 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 2.00000i 0 2.00000i 0 −4.00000 0 −1.00000 0
129.2 0 2.00000i 0 2.00000i 0 −4.00000 0 −1.00000 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 256.2.b.a 2
3.b odd 2 1 2304.2.d.b 2
4.b odd 2 1 256.2.b.c 2
8.b even 2 1 inner 256.2.b.a 2
8.d odd 2 1 256.2.b.c 2
12.b even 2 1 2304.2.d.r 2
16.e even 4 1 128.2.a.b yes 1
16.e even 4 1 128.2.a.c yes 1
16.f odd 4 1 128.2.a.a 1
16.f odd 4 1 128.2.a.d yes 1
24.f even 2 1 2304.2.d.r 2
24.h odd 2 1 2304.2.d.b 2
32.g even 8 4 1024.2.e.m 4
32.h odd 8 4 1024.2.e.i 4
48.i odd 4 1 1152.2.a.h 1
48.i odd 4 1 1152.2.a.r 1
48.k even 4 1 1152.2.a.c 1
48.k even 4 1 1152.2.a.m 1
80.i odd 4 1 3200.2.c.f 2
80.i odd 4 1 3200.2.c.k 2
80.j even 4 1 3200.2.c.e 2
80.j even 4 1 3200.2.c.l 2
80.k odd 4 1 3200.2.a.h 1
80.k odd 4 1 3200.2.a.x 1
80.q even 4 1 3200.2.a.e 1
80.q even 4 1 3200.2.a.u 1
80.s even 4 1 3200.2.c.e 2
80.s even 4 1 3200.2.c.l 2
80.t odd 4 1 3200.2.c.f 2
80.t odd 4 1 3200.2.c.k 2
112.j even 4 1 6272.2.a.a 1
112.j even 4 1 6272.2.a.h 1
112.l odd 4 1 6272.2.a.b 1
112.l odd 4 1 6272.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 16.f odd 4 1
128.2.a.b yes 1 16.e even 4 1
128.2.a.c yes 1 16.e even 4 1
128.2.a.d yes 1 16.f odd 4 1
256.2.b.a 2 1.a even 1 1 trivial
256.2.b.a 2 8.b even 2 1 inner
256.2.b.c 2 4.b odd 2 1
256.2.b.c 2 8.d odd 2 1
1024.2.e.i 4 32.h odd 8 4
1024.2.e.m 4 32.g even 8 4
1152.2.a.c 1 48.k even 4 1
1152.2.a.h 1 48.i odd 4 1
1152.2.a.m 1 48.k even 4 1
1152.2.a.r 1 48.i odd 4 1
2304.2.d.b 2 3.b odd 2 1
2304.2.d.b 2 24.h odd 2 1
2304.2.d.r 2 12.b even 2 1
2304.2.d.r 2 24.f even 2 1
3200.2.a.e 1 80.q even 4 1
3200.2.a.h 1 80.k odd 4 1
3200.2.a.u 1 80.q even 4 1
3200.2.a.x 1 80.k odd 4 1
3200.2.c.e 2 80.j even 4 1
3200.2.c.e 2 80.s even 4 1
3200.2.c.f 2 80.i odd 4 1
3200.2.c.f 2 80.t odd 4 1
3200.2.c.k 2 80.i odd 4 1
3200.2.c.k 2 80.t odd 4 1
3200.2.c.l 2 80.j even 4 1
3200.2.c.l 2 80.s even 4 1
6272.2.a.a 1 112.j even 4 1
6272.2.a.b 1 112.l odd 4 1
6272.2.a.g 1 112.l odd 4 1
6272.2.a.h 1 112.j even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 196 \) Copy content Toggle raw display
$61$ \( T^{2} + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 100 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 14)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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