# Properties

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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.h 4
4.b odd 2 1 256.4.b.i 4
8.b even 2 1 inner 256.4.b.h 4
8.d odd 2 1 256.4.b.i 4
16.e even 4 1 128.4.a.f yes 2
16.e even 4 1 128.4.a.g yes 2
16.f odd 4 1 128.4.a.e 2
16.f odd 4 1 128.4.a.h yes 2
48.i odd 4 1 1152.4.a.r 2
48.i odd 4 1 1152.4.a.t 2
48.k even 4 1 1152.4.a.q 2
48.k even 4 1 1152.4.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.a.e 2 16.f odd 4 1
128.4.a.f yes 2 16.e even 4 1
128.4.a.g yes 2 16.e even 4 1
128.4.a.h yes 2 16.f odd 4 1
256.4.b.h 4 1.a even 1 1 trivial
256.4.b.h 4 8.b even 2 1 inner
256.4.b.i 4 4.b odd 2 1
256.4.b.i 4 8.d odd 2 1
1152.4.a.q 2 48.k even 4 1
1152.4.a.r 2 48.i odd 4 1
1152.4.a.s 2 48.k even 4 1
1152.4.a.t 2 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 $$S_{4}^{\mathrm{new}}(256, [\chi])$$:

 $$T_{3}^{4} + 104 T_{3}^{2} + 1936$$ $$T_{7}^{2} + 8 T_{7} - 176$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 4 T^{2} + 694 T^{4} - 2916 T^{6} + 531441 T^{8}$$
$5$ $$1 - 108 T^{2} + 31094 T^{4} - 1687500 T^{6} + 244140625 T^{8}$$
$7$ $$( 1 + 8 T + 510 T^{2} + 2744 T^{3} + 117649 T^{4} )^{2}$$
$11$ $$1 - 996 T^{2} + 3384854 T^{4} - 1764474756 T^{6} + 3138428376721 T^{8}$$
$13$ $$1 - 332 T^{2} - 7598826 T^{4} - 1602500588 T^{6} + 23298085122481 T^{8}$$
$17$ $$( 1 - 92 T + 5030 T^{2} - 451996 T^{3} + 24137569 T^{4} )^{2}$$
$19$ $$1 - 22724 T^{2} + 223149174 T^{4} - 1069070599844 T^{6} + 2213314919066161 T^{8}$$
$23$ $$( 1 - 8 T + 8798 T^{2} - 97336 T^{3} + 148035889 T^{4} )^{2}$$
$29$ $$1 - 84428 T^{2} + 2937799638 T^{4} - 50219743345388 T^{6} + 353814783205469041 T^{8}$$
$31$ $$( 1 - 384 T + 84158 T^{2} - 11439744 T^{3} + 887503681 T^{4} )^{2}$$
$37$ $$1 - 169004 T^{2} + 11993714550 T^{4} - 433618026026636 T^{6} + 6582952005840035281 T^{8}$$
$41$ $$( 1 - 300 T + 157270 T^{2} - 20676300 T^{3} + 4750104241 T^{4} )^{2}$$
$43$ $$1 - 270628 T^{2} + 30844604694 T^{4} - 1710737839224772 T^{6} + 39959630797262576401 T^{8}$$
$47$ $$( 1 - 16 T + 114782 T^{2} - 1661168 T^{3} + 10779215329 T^{4} )^{2}$$
$53$ $$1 - 576620 T^{2} + 127450023606 T^{4} - 12780413914203980 T^{6} +$$$$49\!\cdots\!41$$$$T^{8}$$
$59$ $$1 - 509604 T^{2} + 127606046294 T^{4} - 21495368665588164 T^{6} +$$$$17\!\cdots\!81$$$$T^{8}$$
$61$ $$1 - 394892 T^{2} + 121971578838 T^{4} - 20344983672164012 T^{6} +$$$$26\!\cdots\!21$$$$T^{8}$$
$67$ $$1 - 526916 T^{2} + 240544281654 T^{4} - 47663968898960804 T^{6} +$$$$81\!\cdots\!61$$$$T^{8}$$
$71$ $$( 1 - 408 T + 672766 T^{2} - 146027688 T^{3} + 128100283921 T^{4} )^{2}$$
$73$ $$( 1 + 412 T + 690678 T^{2} + 160275004 T^{3} + 151334226289 T^{4} )^{2}$$
$79$ $$( 1 - 400 T + 963870 T^{2} - 197215600 T^{3} + 243087455521 T^{4} )^{2}$$
$83$ $$1 - 1821572 T^{2} + 1476121597686 T^{4} - 595545429798516068 T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$89$ $$( 1 + 572 T + 845846 T^{2} + 403242268 T^{3} + 496981290961 T^{4} )^{2}$$
$97$ $$( 1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4} )^{2}$$