Properties

Label 256.4.b.i
Level $256$
Weight $4$
Character orbit 256.b
Analytic conductor $15.104$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
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 a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 - 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{3} + ( 8 - 16 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + ( 4 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{7} + ( -25 + 32 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{9} +O(q^{10})\) \( q + ( 4 - 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{3} + ( 8 - 16 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + ( 4 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{7} + ( -25 + 32 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{9} + ( 4 - 8 \zeta_{12}^{2} - 46 \zeta_{12}^{3} ) q^{11} + ( 24 - 48 \zeta_{12}^{2} + 50 \zeta_{12}^{3} ) q^{13} + ( -92 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{15} + ( 46 + 96 \zeta_{12} - 48 \zeta_{12}^{3} ) q^{17} + ( -28 + 56 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{19} -88 \zeta_{12}^{3} q^{21} + ( -4 - 144 \zeta_{12} + 72 \zeta_{12}^{3} ) q^{23} + ( -71 - 64 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{25} + ( -24 + 48 \zeta_{12}^{2} - 188 \zeta_{12}^{3} ) q^{27} + ( -40 + 80 \zeta_{12}^{2} + 42 \zeta_{12}^{3} ) q^{29} + ( -192 - 128 \zeta_{12} + 64 \zeta_{12}^{3} ) q^{31} + ( 44 - 352 \zeta_{12} + 176 \zeta_{12}^{3} ) q^{33} + ( 48 - 96 \zeta_{12}^{2} - 200 \zeta_{12}^{3} ) q^{35} + ( -56 + 112 \zeta_{12}^{2} + 86 \zeta_{12}^{3} ) q^{37} + ( -388 + 496 \zeta_{12} - 248 \zeta_{12}^{3} ) q^{39} + ( 150 - 64 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{41} + ( 20 - 40 \zeta_{12}^{2} - 150 \zeta_{12}^{3} ) q^{43} + ( -168 + 336 \zeta_{12}^{2} - 334 \zeta_{12}^{3} ) q^{45} + ( -8 + 352 \zeta_{12} - 176 \zeta_{12}^{3} ) q^{47} + ( -135 + 128 \zeta_{12} - 64 \zeta_{12}^{3} ) q^{49} + ( 88 - 176 \zeta_{12}^{2} - 484 \zeta_{12}^{3} ) q^{51} + ( -56 + 112 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{53} + ( -188 - 752 \zeta_{12} + 376 \zeta_{12}^{3} ) q^{55} + ( 332 - 96 \zeta_{12} + 48 \zeta_{12}^{3} ) q^{57} + ( 132 - 264 \zeta_{12}^{2} + 322 \zeta_{12}^{3} ) q^{59} + ( 280 - 560 \zeta_{12}^{2} + 146 \zeta_{12}^{3} ) q^{61} + ( 284 - 272 \zeta_{12} + 136 \zeta_{12}^{3} ) q^{63} + ( -476 + 704 \zeta_{12} - 352 \zeta_{12}^{3} ) q^{65} + ( -332 + 664 \zeta_{12}^{2} - 86 \zeta_{12}^{3} ) q^{67} + ( 128 - 256 \zeta_{12}^{2} + 856 \zeta_{12}^{3} ) q^{69} + ( -204 + 336 \zeta_{12} - 168 \zeta_{12}^{3} ) q^{71} + ( -206 + 416 \zeta_{12} - 208 \zeta_{12}^{3} ) q^{73} + ( -220 + 440 \zeta_{12}^{2} + 242 \zeta_{12}^{3} ) q^{75} + ( 384 - 768 \zeta_{12}^{2} - 280 \zeta_{12}^{3} ) q^{77} + ( -200 - 288 \zeta_{12} + 144 \zeta_{12}^{3} ) q^{79} + ( -11 - 736 \zeta_{12} + 368 \zeta_{12}^{3} ) q^{81} + ( 52 - 104 \zeta_{12}^{2} + 474 \zeta_{12}^{3} ) q^{83} + ( 464 - 928 \zeta_{12}^{2} - 1244 \zeta_{12}^{3} ) q^{85} + ( 396 + 176 \zeta_{12} - 88 \zeta_{12}^{3} ) q^{87} + ( -286 + 928 \zeta_{12} - 464 \zeta_{12}^{3} ) q^{89} + ( -304 + 608 \zeta_{12}^{2} - 376 \zeta_{12}^{3} ) q^{91} + ( -640 + 1280 \zeta_{12}^{2} + 384 \zeta_{12}^{3} ) q^{93} + ( 676 + 144 \zeta_{12} - 72 \zeta_{12}^{3} ) q^{95} + ( 1102 + 736 \zeta_{12} - 368 \zeta_{12}^{3} ) q^{97} + ( 636 - 1272 \zeta_{12}^{2} + 958 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 16q^{7} - 100q^{9} + O(q^{10}) \) \( 4q + 16q^{7} - 100q^{9} - 368q^{15} + 184q^{17} - 16q^{23} - 284q^{25} - 768q^{31} + 176q^{33} - 1552q^{39} + 600q^{41} - 32q^{47} - 540q^{49} - 752q^{55} + 1328q^{57} + 1136q^{63} - 1904q^{65} - 816q^{71} - 824q^{73} - 800q^{79} - 44q^{81} + 1584q^{87} - 1144q^{89} + 2704q^{95} + 4408q^{97} + O(q^{100}) \)

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
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 8.92820i 0 11.8564i 0 −9.85641 0 −52.7128 0
129.2 0 4.92820i 0 15.8564i 0 17.8564 0 2.71281 0
129.3 0 4.92820i 0 15.8564i 0 17.8564 0 2.71281 0
129.4 0 8.92820i 0 11.8564i 0 −9.85641 0 −52.7128 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.i 4
4.b odd 2 1 256.4.b.h 4
8.b even 2 1 inner 256.4.b.i 4
8.d odd 2 1 256.4.b.h 4
16.e even 4 1 128.4.a.e 2
16.e even 4 1 128.4.a.h yes 2
16.f odd 4 1 128.4.a.f yes 2
16.f odd 4 1 128.4.a.g yes 2
48.i odd 4 1 1152.4.a.q 2
48.i odd 4 1 1152.4.a.s 2
48.k even 4 1 1152.4.a.r 2
48.k even 4 1 1152.4.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.a.e 2 16.e even 4 1
128.4.a.f yes 2 16.f odd 4 1
128.4.a.g yes 2 16.f odd 4 1
128.4.a.h yes 2 16.e even 4 1
256.4.b.h 4 4.b odd 2 1
256.4.b.h 4 8.d odd 2 1
256.4.b.i 4 1.a even 1 1 trivial
256.4.b.i 4 8.b even 2 1 inner
1152.4.a.q 2 48.i odd 4 1
1152.4.a.r 2 48.k even 4 1
1152.4.a.s 2 48.i odd 4 1
1152.4.a.t 2 48.k even 4 1

Hecke kernels

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

\( T_{3}^{4} + 104 T_{3}^{2} + 1936 \)
\( T_{7}^{2} - 8 T_{7} - 176 \)