Properties

Label 256.4.b.b
Level $256$
Weight $4$
Character orbit 256.b
Analytic conductor $15.104$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 256.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.1044889615\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + 2 i q^{3} -6 i q^{5} -20 q^{7} + 23 q^{9} +O(q^{10})\) \( q + 2 i q^{3} -6 i q^{5} -20 q^{7} + 23 q^{9} -14 i q^{11} + 54 i q^{13} + 12 q^{15} -66 q^{17} + 162 i q^{19} -40 i q^{21} -172 q^{23} + 89 q^{25} + 100 i q^{27} -2 i q^{29} -128 q^{31} + 28 q^{33} + 120 i q^{35} -158 i q^{37} -108 q^{39} -202 q^{41} + 298 i q^{43} -138 i q^{45} -408 q^{47} + 57 q^{49} -132 i q^{51} + 690 i q^{53} -84 q^{55} -324 q^{57} + 322 i q^{59} -298 i q^{61} -460 q^{63} + 324 q^{65} + 202 i q^{67} -344 i q^{69} + 700 q^{71} + 418 q^{73} + 178 i q^{75} + 280 i q^{77} + 744 q^{79} + 421 q^{81} -678 i q^{83} + 396 i q^{85} + 4 q^{87} + 82 q^{89} -1080 i q^{91} -256 i q^{93} + 972 q^{95} -1122 q^{97} -322 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 40q^{7} + 46q^{9} + O(q^{10}) \) \( 2q - 40q^{7} + 46q^{9} + 24q^{15} - 132q^{17} - 344q^{23} + 178q^{25} - 256q^{31} + 56q^{33} - 216q^{39} - 404q^{41} - 816q^{47} + 114q^{49} - 168q^{55} - 648q^{57} - 920q^{63} + 648q^{65} + 1400q^{71} + 836q^{73} + 1488q^{79} + 842q^{81} + 8q^{87} + 164q^{89} + 1944q^{95} - 2244q^{97} + O(q^{100}) \)

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 2.00000i 0 6.00000i 0 −20.0000 0 23.0000 0
129.2 0 2.00000i 0 6.00000i 0 −20.0000 0 23.0000 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 256.4.b.b 2
4.b odd 2 1 256.4.b.f 2
8.b even 2 1 inner 256.4.b.b 2
8.d odd 2 1 256.4.b.f 2
16.e even 4 1 128.4.a.a 1
16.e even 4 1 128.4.a.d yes 1
16.f odd 4 1 128.4.a.b yes 1
16.f odd 4 1 128.4.a.c yes 1
48.i odd 4 1 1152.4.a.d 1
48.i odd 4 1 1152.4.a.j 1
48.k even 4 1 1152.4.a.c 1
48.k even 4 1 1152.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.a.a 1 16.e even 4 1
128.4.a.b yes 1 16.f odd 4 1
128.4.a.c yes 1 16.f odd 4 1
128.4.a.d yes 1 16.e even 4 1
256.4.b.b 2 1.a even 1 1 trivial
256.4.b.b 2 8.b even 2 1 inner
256.4.b.f 2 4.b odd 2 1
256.4.b.f 2 8.d odd 2 1
1152.4.a.c 1 48.k even 4 1
1152.4.a.d 1 48.i odd 4 1
1152.4.a.i 1 48.k even 4 1
1152.4.a.j 1 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{7} + 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( 36 + T^{2} \)
$7$ \( ( 20 + T )^{2} \)
$11$ \( 196 + T^{2} \)
$13$ \( 2916 + T^{2} \)
$17$ \( ( 66 + T )^{2} \)
$19$ \( 26244 + T^{2} \)
$23$ \( ( 172 + T )^{2} \)
$29$ \( 4 + T^{2} \)
$31$ \( ( 128 + T )^{2} \)
$37$ \( 24964 + T^{2} \)
$41$ \( ( 202 + T )^{2} \)
$43$ \( 88804 + T^{2} \)
$47$ \( ( 408 + T )^{2} \)
$53$ \( 476100 + T^{2} \)
$59$ \( 103684 + T^{2} \)
$61$ \( 88804 + T^{2} \)
$67$ \( 40804 + T^{2} \)
$71$ \( ( -700 + T )^{2} \)
$73$ \( ( -418 + T )^{2} \)
$79$ \( ( -744 + T )^{2} \)
$83$ \( 459684 + T^{2} \)
$89$ \( ( -82 + T )^{2} \)
$97$ \( ( 1122 + T )^{2} \)
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