Properties

Label 256.2.b.b
Level $256$
Weight $2$
Character orbit 256.b
Analytic conductor $2.044$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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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: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 i q^{5} + 3 q^{9} +O(q^{10})\) \( q + 2 i q^{5} + 3 q^{9} + 6 i q^{13} + 2 q^{17} + q^{25} -10 i q^{29} + 2 i q^{37} -10 q^{41} + 6 i q^{45} -7 q^{49} -14 i q^{53} -10 i q^{61} -12 q^{65} + 6 q^{73} + 9 q^{81} + 4 i q^{85} -10 q^{89} + 18 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{9} + O(q^{10}) \) \( 2q + 6q^{9} + 4q^{17} + 2q^{25} - 20q^{41} - 14q^{49} - 24q^{65} + 12q^{73} + 18q^{81} - 20q^{89} + 36q^{97} + O(q^{100}) \)

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 0 0 2.00000i 0 0 0 3.00000 0
129.2 0 0 0 2.00000i 0 0 0 3.00000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
8.b even 2 1 inner
8.d odd 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.b 2
3.b odd 2 1 2304.2.d.j 2
4.b odd 2 1 CM 256.2.b.b 2
8.b even 2 1 inner 256.2.b.b 2
8.d odd 2 1 inner 256.2.b.b 2
12.b even 2 1 2304.2.d.j 2
16.e even 4 1 32.2.a.a 1
16.e even 4 1 64.2.a.a 1
16.f odd 4 1 32.2.a.a 1
16.f odd 4 1 64.2.a.a 1
24.f even 2 1 2304.2.d.j 2
24.h odd 2 1 2304.2.d.j 2
32.g even 8 4 1024.2.e.j 4
32.h odd 8 4 1024.2.e.j 4
48.i odd 4 1 288.2.a.d 1
48.i odd 4 1 576.2.a.c 1
48.k even 4 1 288.2.a.d 1
48.k even 4 1 576.2.a.c 1
80.i odd 4 1 800.2.c.e 2
80.i odd 4 1 1600.2.c.l 2
80.j even 4 1 800.2.c.e 2
80.j even 4 1 1600.2.c.l 2
80.k odd 4 1 800.2.a.d 1
80.k odd 4 1 1600.2.a.n 1
80.q even 4 1 800.2.a.d 1
80.q even 4 1 1600.2.a.n 1
80.s even 4 1 800.2.c.e 2
80.s even 4 1 1600.2.c.l 2
80.t odd 4 1 800.2.c.e 2
80.t odd 4 1 1600.2.c.l 2
112.j even 4 1 1568.2.a.e 1
112.j even 4 1 3136.2.a.m 1
112.l odd 4 1 1568.2.a.e 1
112.l odd 4 1 3136.2.a.m 1
112.u odd 12 2 1568.2.i.g 2
112.v even 12 2 1568.2.i.f 2
112.w even 12 2 1568.2.i.g 2
112.x odd 12 2 1568.2.i.f 2
144.u even 12 2 2592.2.i.e 2
144.v odd 12 2 2592.2.i.t 2
144.w odd 12 2 2592.2.i.e 2
144.x even 12 2 2592.2.i.t 2
176.i even 4 1 3872.2.a.f 1
176.i even 4 1 7744.2.a.v 1
176.l odd 4 1 3872.2.a.f 1
176.l odd 4 1 7744.2.a.v 1
208.o odd 4 1 5408.2.a.g 1
208.p even 4 1 5408.2.a.g 1
240.t even 4 1 7200.2.a.v 1
240.z odd 4 1 7200.2.f.m 2
240.bb even 4 1 7200.2.f.m 2
240.bd odd 4 1 7200.2.f.m 2
240.bf even 4 1 7200.2.f.m 2
240.bm odd 4 1 7200.2.a.v 1
272.k odd 4 1 9248.2.a.f 1
272.r even 4 1 9248.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 16.e even 4 1
32.2.a.a 1 16.f odd 4 1
64.2.a.a 1 16.e even 4 1
64.2.a.a 1 16.f odd 4 1
256.2.b.b 2 1.a even 1 1 trivial
256.2.b.b 2 4.b odd 2 1 CM
256.2.b.b 2 8.b even 2 1 inner
256.2.b.b 2 8.d odd 2 1 inner
288.2.a.d 1 48.i odd 4 1
288.2.a.d 1 48.k even 4 1
576.2.a.c 1 48.i odd 4 1
576.2.a.c 1 48.k even 4 1
800.2.a.d 1 80.k odd 4 1
800.2.a.d 1 80.q even 4 1
800.2.c.e 2 80.i odd 4 1
800.2.c.e 2 80.j even 4 1
800.2.c.e 2 80.s even 4 1
800.2.c.e 2 80.t odd 4 1
1024.2.e.j 4 32.g even 8 4
1024.2.e.j 4 32.h odd 8 4
1568.2.a.e 1 112.j even 4 1
1568.2.a.e 1 112.l odd 4 1
1568.2.i.f 2 112.v even 12 2
1568.2.i.f 2 112.x odd 12 2
1568.2.i.g 2 112.u odd 12 2
1568.2.i.g 2 112.w even 12 2
1600.2.a.n 1 80.k odd 4 1
1600.2.a.n 1 80.q even 4 1
1600.2.c.l 2 80.i odd 4 1
1600.2.c.l 2 80.j even 4 1
1600.2.c.l 2 80.s even 4 1
1600.2.c.l 2 80.t odd 4 1
2304.2.d.j 2 3.b odd 2 1
2304.2.d.j 2 12.b even 2 1
2304.2.d.j 2 24.f even 2 1
2304.2.d.j 2 24.h odd 2 1
2592.2.i.e 2 144.u even 12 2
2592.2.i.e 2 144.w odd 12 2
2592.2.i.t 2 144.v odd 12 2
2592.2.i.t 2 144.x even 12 2
3136.2.a.m 1 112.j even 4 1
3136.2.a.m 1 112.l odd 4 1
3872.2.a.f 1 176.i even 4 1
3872.2.a.f 1 176.l odd 4 1
5408.2.a.g 1 208.o odd 4 1
5408.2.a.g 1 208.p even 4 1
7200.2.a.v 1 240.t even 4 1
7200.2.a.v 1 240.bm odd 4 1
7200.2.f.m 2 240.z odd 4 1
7200.2.f.m 2 240.bb even 4 1
7200.2.f.m 2 240.bd odd 4 1
7200.2.f.m 2 240.bf even 4 1
7744.2.a.v 1 176.i even 4 1
7744.2.a.v 1 176.l odd 4 1
9248.2.a.f 1 272.k odd 4 1
9248.2.a.f 1 272.r even 4 1

Hecke kernels

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

\( T_{3} \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 100 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 196 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 100 + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( ( -18 + T )^{2} \)
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