# Properties

 Label 256.2.g.d Level $256$ Weight $2$ Character orbit 256.g Analytic conductor $2.044$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.04417029174$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{8})$$ Coefficient field: 8.0.18939904.2 Defining polynomial: $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 32) Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{6} - \beta_{7} ) q^{3} + ( \beta_{6} + \beta_{7} ) q^{5} + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{6} - \beta_{7} ) q^{3} + ( \beta_{6} + \beta_{7} ) q^{5} + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{9} + ( -1 + \beta_{1} - \beta_{6} + \beta_{7} ) q^{11} + ( 2 + 2 \beta_{3} - \beta_{4} + \beta_{7} ) q^{13} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{15} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{17} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{19} + ( -1 + 2 \beta_{1} + \beta_{4} + 3 \beta_{6} - 3 \beta_{7} ) q^{21} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{23} + ( -1 + 3 \beta_{4} + \beta_{7} ) q^{25} + ( -\beta_{2} + \beta_{3} + 2 \beta_{6} + 3 \beta_{7} ) q^{27} + ( -2 \beta_{1} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{29} + ( 4 - 2 \beta_{4} - 2 \beta_{6} ) q^{31} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{33} + ( -2 - \beta_{1} - \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{35} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{37} + ( 3 - \beta_{1} + \beta_{3} + 4 \beta_{4} - \beta_{7} ) q^{39} + ( 1 - 2 \beta_{1} - 2 \beta_{3} - 4 \beta_{6} + \beta_{7} ) q^{41} + ( 1 + \beta_{1} + 2 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} ) q^{43} + ( -1 - 2 \beta_{3} - \beta_{6} ) q^{45} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{47} + ( -2 - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{49} + ( 2 + 4 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{51} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 5 \beta_{6} + 3 \beta_{7} ) q^{53} + ( -1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{55} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{57} + ( 3 - \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{59} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{61} + ( -5 + \beta_{1} - \beta_{2} + 4 \beta_{4} + 4 \beta_{6} ) q^{63} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{6} ) q^{65} + ( 6 - 3 \beta_{2} + 3 \beta_{6} + 3 \beta_{7} ) q^{67} + ( -3 - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{69} + ( -3 - 3 \beta_{2} - 5 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{71} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{73} + ( 3 - 3 \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{6} + 4 \beta_{7} ) q^{75} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{77} + ( 4 - \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 6 \beta_{7} ) q^{79} + ( \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{81} + ( -4 + 3 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{83} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} ) q^{85} + ( 7 - 3 \beta_{1} - 3 \beta_{3} - 2 \beta_{6} + 7 \beta_{7} ) q^{87} + ( -1 - \beta_{1} - 3 \beta_{2} + \beta_{3} - 5 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{89} + ( -8 + 3 \beta_{2} - 3 \beta_{3} + 8 \beta_{4} - 3 \beta_{7} ) q^{91} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{93} + ( -1 + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{95} + ( 4 + \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{97} + ( -4 + 3 \beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{3} - 8q^{7} + O(q^{10})$$ $$8q + 4q^{3} - 8q^{7} - 4q^{11} + 8q^{13} - 4q^{19} - 8q^{23} - 8q^{25} - 8q^{27} + 32q^{31} - 16q^{33} - 16q^{35} + 8q^{37} + 16q^{39} + 8q^{41} + 12q^{43} + 16q^{51} - 8q^{53} - 16q^{55} + 16q^{57} + 20q^{59} - 24q^{61} - 40q^{63} + 36q^{67} - 32q^{69} - 24q^{71} - 32q^{73} + 12q^{75} - 16q^{77} - 20q^{83} - 8q^{85} + 56q^{87} - 16q^{89} - 40q^{91} + 16q^{93} - 8q^{95} + 32q^{97} - 28q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{7} - 4 \nu^{6} + 14 \nu^{5} - 27 \nu^{4} + 41 \nu^{3} - 37 \nu^{2} + 24 \nu - 5$$ $$\beta_{2}$$ $$=$$ $$\nu^{7} - 4 \nu^{6} + 14 \nu^{5} - 28 \nu^{4} + 43 \nu^{3} - 44 \nu^{2} + 30 \nu - 10$$ $$\beta_{3}$$ $$=$$ $$-5 \nu^{7} + 17 \nu^{6} - 59 \nu^{5} + 102 \nu^{4} - 146 \nu^{3} + 121 \nu^{2} - 66 \nu + 15$$ $$\beta_{4}$$ $$=$$ $$5 \nu^{7} - 17 \nu^{6} + 60 \nu^{5} - 105 \nu^{4} + 155 \nu^{3} - 133 \nu^{2} + 77 \nu - 19$$ $$\beta_{5}$$ $$=$$ $$-5 \nu^{7} + 18 \nu^{6} - 62 \nu^{5} + 113 \nu^{4} - 163 \nu^{3} + 145 \nu^{2} - 82 \nu + 20$$ $$\beta_{6}$$ $$=$$ $$-5 \nu^{7} + 18 \nu^{6} - 63 \nu^{5} + 115 \nu^{4} - 170 \nu^{3} + 152 \nu^{2} - 89 \nu + 23$$ $$\beta_{7}$$ $$=$$ $$-8 \nu^{7} + 28 \nu^{6} - 98 \nu^{5} + 175 \nu^{4} - 256 \nu^{3} + 223 \nu^{2} - 126 \nu + 31$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + 2 \beta_{6} + \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + 3 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{1} - 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{7} - 5 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + \beta_{3} - 4 \beta_{2} - \beta_{1} - 3$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$11 \beta_{7} - 19 \beta_{6} + 3 \beta_{5} - \beta_{4} - 5 \beta_{3} - 4 \beta_{2} - 8 \beta_{1} + 12$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-13 \beta_{7} + 2 \beta_{6} + 15 \beta_{5} - 16 \beta_{4} - 10 \beta_{3} + 13 \beta_{2} - 2 \beta_{1} + 23$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-67 \beta_{7} + 90 \beta_{6} + 4 \beta_{5} - 10 \beta_{4} + 16 \beta_{3} + 31 \beta_{2} + 33 \beta_{1} - 28$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-7 \beta_{7} + 87 \beta_{6} - 68 \beta_{5} + 71 \beta_{4} + 65 \beta_{3} - 26 \beta_{2} + 37 \beta_{1} - 125$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$255$$ $$\chi(n)$$ $$-\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 0.5 + 0.691860i 0.5 − 2.10607i 0.5 + 0.0297061i 0.5 − 1.44392i 0.5 − 0.0297061i 0.5 + 1.44392i 0.5 − 0.691860i 0.5 + 2.10607i
0 −0.0794708 + 0.191860i 0 −0.707107 + 0.292893i 0 −2.27133 + 2.27133i 0 2.09083 + 2.09083i 0
33.2 0 1.07947 2.60607i 0 −0.707107 + 0.292893i 0 1.68554 1.68554i 0 −3.50504 3.50504i 0
97.1 0 −1.27882 + 0.529706i 0 0.707107 1.70711i 0 −2.74912 2.74912i 0 −0.766519 + 0.766519i 0
97.2 0 2.27882 0.943920i 0 0.707107 1.70711i 0 −0.665096 0.665096i 0 2.18073 2.18073i 0
161.1 0 −1.27882 0.529706i 0 0.707107 + 1.70711i 0 −2.74912 + 2.74912i 0 −0.766519 0.766519i 0
161.2 0 2.27882 + 0.943920i 0 0.707107 + 1.70711i 0 −0.665096 + 0.665096i 0 2.18073 + 2.18073i 0
225.1 0 −0.0794708 0.191860i 0 −0.707107 0.292893i 0 −2.27133 2.27133i 0 2.09083 2.09083i 0
225.2 0 1.07947 + 2.60607i 0 −0.707107 0.292893i 0 1.68554 + 1.68554i 0 −3.50504 + 3.50504i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 225.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.2.g.d 8
4.b odd 2 1 256.2.g.c 8
8.b even 2 1 32.2.g.b 8
8.d odd 2 1 128.2.g.b 8
16.e even 4 1 512.2.g.e 8
16.e even 4 1 512.2.g.h 8
16.f odd 4 1 512.2.g.f 8
16.f odd 4 1 512.2.g.g 8
24.f even 2 1 1152.2.v.b 8
24.h odd 2 1 288.2.v.b 8
32.g even 8 1 32.2.g.b 8
32.g even 8 1 inner 256.2.g.d 8
32.g even 8 1 512.2.g.e 8
32.g even 8 1 512.2.g.h 8
32.h odd 8 1 128.2.g.b 8
32.h odd 8 1 256.2.g.c 8
32.h odd 8 1 512.2.g.f 8
32.h odd 8 1 512.2.g.g 8
40.f even 2 1 800.2.y.b 8
40.i odd 4 1 800.2.ba.c 8
40.i odd 4 1 800.2.ba.d 8
64.i even 16 2 4096.2.a.k 8
64.j odd 16 2 4096.2.a.q 8
96.o even 8 1 1152.2.v.b 8
96.p odd 8 1 288.2.v.b 8
160.v odd 8 1 800.2.ba.c 8
160.z even 8 1 800.2.y.b 8
160.bb odd 8 1 800.2.ba.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.b 8 8.b even 2 1
32.2.g.b 8 32.g even 8 1
128.2.g.b 8 8.d odd 2 1
128.2.g.b 8 32.h odd 8 1
256.2.g.c 8 4.b odd 2 1
256.2.g.c 8 32.h odd 8 1
256.2.g.d 8 1.a even 1 1 trivial
256.2.g.d 8 32.g even 8 1 inner
288.2.v.b 8 24.h odd 2 1
288.2.v.b 8 96.p odd 8 1
512.2.g.e 8 16.e even 4 1
512.2.g.e 8 32.g even 8 1
512.2.g.f 8 16.f odd 4 1
512.2.g.f 8 32.h odd 8 1
512.2.g.g 8 16.f odd 4 1
512.2.g.g 8 32.h odd 8 1
512.2.g.h 8 16.e even 4 1
512.2.g.h 8 32.g even 8 1
800.2.y.b 8 40.f even 2 1
800.2.y.b 8 160.z even 8 1
800.2.ba.c 8 40.i odd 4 1
800.2.ba.c 8 160.v odd 8 1
800.2.ba.d 8 40.i odd 4 1
800.2.ba.d 8 160.bb odd 8 1
1152.2.v.b 8 24.f even 2 1
1152.2.v.b 8 96.o even 8 1
4096.2.a.k 8 64.i even 16 2
4096.2.a.q 8 64.j odd 16 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 4 T_{3}^{7} + 8 T_{3}^{6} - 32 T_{3}^{4} + 24 T_{3}^{3} + 96 T_{3}^{2} + 16 T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(256, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$4 + 16 T + 96 T^{2} + 24 T^{3} - 32 T^{4} + 8 T^{6} - 4 T^{7} + T^{8}$$
$5$ $$( 2 + 4 T + 2 T^{2} + T^{4} )^{2}$$
$7$ $$784 + 1344 T + 1152 T^{2} + 224 T^{3} + 56 T^{4} + 48 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$11$ $$4 + 48 T + 160 T^{2} - 56 T^{3} + 224 T^{4} + 64 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$13$ $$6724 + 8528 T + 2520 T^{2} - 448 T^{3} + 200 T^{4} - 104 T^{5} + 36 T^{6} - 8 T^{7} + T^{8}$$
$17$ $$256 + 5120 T^{2} + 1056 T^{4} + 64 T^{6} + T^{8}$$
$19$ $$196 + 336 T + 832 T^{2} + 168 T^{3} + 32 T^{4} - 48 T^{5} - 8 T^{6} + 4 T^{7} + T^{8}$$
$23$ $$16 - 64 T + 128 T^{2} - 32 T^{3} - 8 T^{4} + 16 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$29$ $$188356 - 17360 T + 17272 T^{2} + 6256 T^{3} + 72 T^{4} - 168 T^{5} - 12 T^{6} + T^{8}$$
$31$ $$( 8 - 8 T + T^{2} )^{4}$$
$37$ $$64516 + 67056 T + 59320 T^{2} + 20640 T^{3} + 3464 T^{4} + 168 T^{5} - 44 T^{6} - 8 T^{7} + T^{8}$$
$41$ $$26896 + 7872 T + 1152 T^{2} - 416 T^{3} + 968 T^{4} + 240 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$43$ $$31684 - 12816 T + 5888 T^{2} - 4856 T^{3} + 1760 T^{4} - 256 T^{5} + 56 T^{6} - 12 T^{7} + T^{8}$$
$47$ $$256 + 1024 T^{2} + 544 T^{4} + 64 T^{6} + T^{8}$$
$53$ $$158404 + 238800 T + 147160 T^{2} + 47328 T^{3} + 9800 T^{4} + 1272 T^{5} + 100 T^{6} + 8 T^{7} + T^{8}$$
$59$ $$643204 - 211728 T + 44896 T^{2} - 20712 T^{3} + 5408 T^{4} - 528 T^{5} + 136 T^{6} - 20 T^{7} + T^{8}$$
$61$ $$42436 + 24720 T + 34968 T^{2} - 6336 T^{3} + 72 T^{4} - 648 T^{5} + 132 T^{6} + 24 T^{7} + T^{8}$$
$67$ $$1285956 - 734832 T + 233280 T^{2} - 68040 T^{3} + 18144 T^{4} - 3456 T^{5} + 504 T^{6} - 36 T^{7} + T^{8}$$
$71$ $$21196816 + 9060672 T + 1936512 T^{2} + 173472 T^{3} + 10232 T^{4} + 1200 T^{5} + 288 T^{6} + 24 T^{7} + T^{8}$$
$73$ $$38416 + 112896 T + 165888 T^{2} + 88192 T^{3} + 26504 T^{4} + 4672 T^{5} + 512 T^{6} + 32 T^{7} + T^{8}$$
$79$ $$99361024 + 4775936 T^{2} + 78880 T^{4} + 512 T^{6} + T^{8}$$
$83$ $$138250564 + 16837456 T + 1548544 T^{2} - 131864 T^{3} - 22752 T^{4} - 304 T^{5} + 184 T^{6} + 20 T^{7} + T^{8}$$
$89$ $$17007376 + 7786112 T + 1782272 T^{2} + 209472 T^{3} + 14024 T^{4} + 672 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$97$ $$( -992 + 288 T + 40 T^{2} - 16 T^{3} + T^{4} )^{2}$$