# Properties

 Label 256.6.b.d Level $256$ Weight $6$ Character orbit 256.b Analytic conductor $41.058$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 256.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$41.0582578721$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 8) 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.

 $$f(q)$$ $$=$$ $$q + 10 \beta q^{3} - 37 \beta q^{5} - 24 q^{7} - 157 q^{9} +O(q^{10})$$ q + 10*b * q^3 - 37*b * q^5 - 24 * q^7 - 157 * q^9 $$q + 10 \beta q^{3} - 37 \beta q^{5} - 24 q^{7} - 157 q^{9} - 62 \beta q^{11} - 239 \beta q^{13} + 1480 q^{15} - 1198 q^{17} + 1522 \beta q^{19} - 240 \beta q^{21} + 184 q^{23} - 2351 q^{25} + 860 \beta q^{27} + 1641 \beta q^{29} + 5728 q^{31} + 2480 q^{33} + 888 \beta q^{35} + 5163 \beta q^{37} + 9560 q^{39} + 8886 q^{41} + 4594 \beta q^{43} + 5809 \beta q^{45} - 23664 q^{47} - 16231 q^{49} - 11980 \beta q^{51} + 5843 \beta q^{53} - 9176 q^{55} - 60880 q^{57} - 8438 \beta q^{59} + 9241 \beta q^{61} + 3768 q^{63} - 35372 q^{65} - 7766 \beta q^{67} + 1840 \beta q^{69} - 31960 q^{71} + 4886 q^{73} - 23510 \beta q^{75} + 1488 \beta q^{77} - 44560 q^{79} - 72551 q^{81} + 33682 \beta q^{83} + 44326 \beta q^{85} - 65640 q^{87} - 71994 q^{89} + 5736 \beta q^{91} + 57280 \beta q^{93} + 225256 q^{95} + 48866 q^{97} + 9734 \beta q^{99} +O(q^{100})$$ q + 10*b * q^3 - 37*b * q^5 - 24 * q^7 - 157 * q^9 - 62*b * q^11 - 239*b * q^13 + 1480 * q^15 - 1198 * q^17 + 1522*b * q^19 - 240*b * q^21 + 184 * q^23 - 2351 * q^25 + 860*b * q^27 + 1641*b * q^29 + 5728 * q^31 + 2480 * q^33 + 888*b * q^35 + 5163*b * q^37 + 9560 * q^39 + 8886 * q^41 + 4594*b * q^43 + 5809*b * q^45 - 23664 * q^47 - 16231 * q^49 - 11980*b * q^51 + 5843*b * q^53 - 9176 * q^55 - 60880 * q^57 - 8438*b * q^59 + 9241*b * q^61 + 3768 * q^63 - 35372 * q^65 - 7766*b * q^67 + 1840*b * q^69 - 31960 * q^71 + 4886 * q^73 - 23510*b * q^75 + 1488*b * q^77 - 44560 * q^79 - 72551 * q^81 + 33682*b * q^83 + 44326*b * q^85 - 65640 * q^87 - 71994 * q^89 + 5736*b * q^91 + 57280*b * q^93 + 225256 * q^95 + 48866 * q^97 + 9734*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 48 q^{7} - 314 q^{9}+O(q^{10})$$ 2 * q - 48 * q^7 - 314 * q^9 $$2 q - 48 q^{7} - 314 q^{9} + 2960 q^{15} - 2396 q^{17} + 368 q^{23} - 4702 q^{25} + 11456 q^{31} + 4960 q^{33} + 19120 q^{39} + 17772 q^{41} - 47328 q^{47} - 32462 q^{49} - 18352 q^{55} - 121760 q^{57} + 7536 q^{63} - 70744 q^{65} - 63920 q^{71} + 9772 q^{73} - 89120 q^{79} - 145102 q^{81} - 131280 q^{87} - 143988 q^{89} + 450512 q^{95} + 97732 q^{97}+O(q^{100})$$ 2 * q - 48 * q^7 - 314 * q^9 + 2960 * q^15 - 2396 * q^17 + 368 * q^23 - 4702 * q^25 + 11456 * q^31 + 4960 * q^33 + 19120 * q^39 + 17772 * q^41 - 47328 * q^47 - 32462 * q^49 - 18352 * q^55 - 121760 * q^57 + 7536 * q^63 - 70744 * q^65 - 63920 * q^71 + 9772 * q^73 - 89120 * q^79 - 145102 * q^81 - 131280 * q^87 - 143988 * q^89 + 450512 * q^95 + 97732 * q^97

## 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 20.0000i 0 74.0000i 0 −24.0000 0 −157.000 0
129.2 0 20.0000i 0 74.0000i 0 −24.0000 0 −157.000 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 256.6.b.d 2
4.b odd 2 1 256.6.b.f 2
8.b even 2 1 inner 256.6.b.d 2
8.d odd 2 1 256.6.b.f 2
16.e even 4 1 16.6.a.a 1
16.e even 4 1 64.6.a.g 1
16.f odd 4 1 8.6.a.a 1
16.f odd 4 1 64.6.a.a 1
48.i odd 4 1 144.6.a.k 1
48.i odd 4 1 576.6.a.h 1
48.k even 4 1 72.6.a.f 1
48.k even 4 1 576.6.a.g 1
80.i odd 4 1 400.6.c.d 2
80.j even 4 1 200.6.c.a 2
80.k odd 4 1 200.6.a.a 1
80.q even 4 1 400.6.a.l 1
80.s even 4 1 200.6.c.a 2
80.t odd 4 1 400.6.c.d 2
112.j even 4 1 392.6.a.b 1
112.l odd 4 1 784.6.a.l 1
112.u odd 12 2 392.6.i.b 2
112.v even 12 2 392.6.i.e 2
176.i even 4 1 968.6.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.a.a 1 16.f odd 4 1
16.6.a.a 1 16.e even 4 1
64.6.a.a 1 16.f odd 4 1
64.6.a.g 1 16.e even 4 1
72.6.a.f 1 48.k even 4 1
144.6.a.k 1 48.i odd 4 1
200.6.a.a 1 80.k odd 4 1
200.6.c.a 2 80.j even 4 1
200.6.c.a 2 80.s even 4 1
256.6.b.d 2 1.a even 1 1 trivial
256.6.b.d 2 8.b even 2 1 inner
256.6.b.f 2 4.b odd 2 1
256.6.b.f 2 8.d odd 2 1
392.6.a.b 1 112.j even 4 1
392.6.i.b 2 112.u odd 12 2
392.6.i.e 2 112.v even 12 2
400.6.a.l 1 80.q even 4 1
400.6.c.d 2 80.i odd 4 1
400.6.c.d 2 80.t odd 4 1
576.6.a.g 1 48.k even 4 1
576.6.a.h 1 48.i odd 4 1
784.6.a.l 1 112.l odd 4 1
968.6.a.a 1 176.i even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(256, [\chi])$$:

 $$T_{3}^{2} + 400$$ T3^2 + 400 $$T_{7} + 24$$ T7 + 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 400$$
$5$ $$T^{2} + 5476$$
$7$ $$(T + 24)^{2}$$
$11$ $$T^{2} + 15376$$
$13$ $$T^{2} + 228484$$
$17$ $$(T + 1198)^{2}$$
$19$ $$T^{2} + 9265936$$
$23$ $$(T - 184)^{2}$$
$29$ $$T^{2} + 10771524$$
$31$ $$(T - 5728)^{2}$$
$37$ $$T^{2} + 106626276$$
$41$ $$(T - 8886)^{2}$$
$43$ $$T^{2} + 84419344$$
$47$ $$(T + 23664)^{2}$$
$53$ $$T^{2} + 136562596$$
$59$ $$T^{2} + 284799376$$
$61$ $$T^{2} + 341584324$$
$67$ $$T^{2} + 241243024$$
$71$ $$(T + 31960)^{2}$$
$73$ $$(T - 4886)^{2}$$
$79$ $$(T + 44560)^{2}$$
$83$ $$T^{2} + 4537908496$$
$89$ $$(T + 71994)^{2}$$
$97$ $$(T - 48866)^{2}$$