# Properties

 Label 256.6.b.g 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: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 4) 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.

 $$f(q)$$ $$=$$ $$q + 12 i q^{3} + 54 i q^{5} + 88 q^{7} + 99 q^{9} +O(q^{10})$$ $$q + 12 i q^{3} + 54 i q^{5} + 88 q^{7} + 99 q^{9} + 540 i q^{11} + 418 i q^{13} -648 q^{15} + 594 q^{17} -836 i q^{19} + 1056 i q^{21} + 4104 q^{23} + 209 q^{25} + 4104 i q^{27} + 594 i q^{29} + 4256 q^{31} -6480 q^{33} + 4752 i q^{35} -298 i q^{37} -5016 q^{39} -17226 q^{41} -12100 i q^{43} + 5346 i q^{45} -1296 q^{47} -9063 q^{49} + 7128 i q^{51} + 19494 i q^{53} -29160 q^{55} + 10032 q^{57} -7668 i q^{59} + 34738 i q^{61} + 8712 q^{63} -22572 q^{65} -21812 i q^{67} + 49248 i q^{69} + 46872 q^{71} -67562 q^{73} + 2508 i q^{75} + 47520 i q^{77} -76912 q^{79} -25191 q^{81} -67716 i q^{83} + 32076 i q^{85} -7128 q^{87} -29754 q^{89} + 36784 i q^{91} + 51072 i q^{93} + 45144 q^{95} -122398 q^{97} + 53460 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 176 q^{7} + 198 q^{9} + O(q^{10})$$ $$2 q + 176 q^{7} + 198 q^{9} - 1296 q^{15} + 1188 q^{17} + 8208 q^{23} + 418 q^{25} + 8512 q^{31} - 12960 q^{33} - 10032 q^{39} - 34452 q^{41} - 2592 q^{47} - 18126 q^{49} - 58320 q^{55} + 20064 q^{57} + 17424 q^{63} - 45144 q^{65} + 93744 q^{71} - 135124 q^{73} - 153824 q^{79} - 50382 q^{81} - 14256 q^{87} - 59508 q^{89} + 90288 q^{95} - 244796 q^{97} + O(q^{100})$$

## 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 12.0000i 0 54.0000i 0 88.0000 0 99.0000 0
129.2 0 12.0000i 0 54.0000i 0 88.0000 0 99.0000 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.g 2
4.b odd 2 1 256.6.b.c 2
8.b even 2 1 inner 256.6.b.g 2
8.d odd 2 1 256.6.b.c 2
16.e even 4 1 4.6.a.a 1
16.e even 4 1 64.6.a.f 1
16.f odd 4 1 16.6.a.b 1
16.f odd 4 1 64.6.a.b 1
48.i odd 4 1 36.6.a.a 1
48.i odd 4 1 576.6.a.bc 1
48.k even 4 1 144.6.a.c 1
48.k even 4 1 576.6.a.bd 1
80.i odd 4 1 100.6.c.b 2
80.j even 4 1 400.6.c.f 2
80.k odd 4 1 400.6.a.d 1
80.q even 4 1 100.6.a.b 1
80.s even 4 1 400.6.c.f 2
80.t odd 4 1 100.6.c.b 2
112.j even 4 1 784.6.a.d 1
112.l odd 4 1 196.6.a.e 1
112.w even 12 2 196.6.e.g 2
112.x odd 12 2 196.6.e.d 2
144.w odd 12 2 324.6.e.d 2
144.x even 12 2 324.6.e.a 2
176.l odd 4 1 484.6.a.a 1
208.m odd 4 1 676.6.d.a 2
208.p even 4 1 676.6.a.a 1
208.r odd 4 1 676.6.d.a 2
240.bb even 4 1 900.6.d.a 2
240.bf even 4 1 900.6.d.a 2
240.bm odd 4 1 900.6.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 16.e even 4 1
16.6.a.b 1 16.f odd 4 1
36.6.a.a 1 48.i odd 4 1
64.6.a.b 1 16.f odd 4 1
64.6.a.f 1 16.e even 4 1
100.6.a.b 1 80.q even 4 1
100.6.c.b 2 80.i odd 4 1
100.6.c.b 2 80.t odd 4 1
144.6.a.c 1 48.k even 4 1
196.6.a.e 1 112.l odd 4 1
196.6.e.d 2 112.x odd 12 2
196.6.e.g 2 112.w even 12 2
256.6.b.c 2 4.b odd 2 1
256.6.b.c 2 8.d odd 2 1
256.6.b.g 2 1.a even 1 1 trivial
256.6.b.g 2 8.b even 2 1 inner
324.6.e.a 2 144.x even 12 2
324.6.e.d 2 144.w odd 12 2
400.6.a.d 1 80.k odd 4 1
400.6.c.f 2 80.j even 4 1
400.6.c.f 2 80.s even 4 1
484.6.a.a 1 176.l odd 4 1
576.6.a.bc 1 48.i odd 4 1
576.6.a.bd 1 48.k even 4 1
676.6.a.a 1 208.p even 4 1
676.6.d.a 2 208.m odd 4 1
676.6.d.a 2 208.r odd 4 1
784.6.a.d 1 112.j even 4 1
900.6.a.h 1 240.bm odd 4 1
900.6.d.a 2 240.bb even 4 1
900.6.d.a 2 240.bf even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 144$$ $$T_{7} - 88$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$144 + T^{2}$$
$5$ $$2916 + T^{2}$$
$7$ $$( -88 + T )^{2}$$
$11$ $$291600 + T^{2}$$
$13$ $$174724 + T^{2}$$
$17$ $$( -594 + T )^{2}$$
$19$ $$698896 + T^{2}$$
$23$ $$( -4104 + T )^{2}$$
$29$ $$352836 + T^{2}$$
$31$ $$( -4256 + T )^{2}$$
$37$ $$88804 + T^{2}$$
$41$ $$( 17226 + T )^{2}$$
$43$ $$146410000 + T^{2}$$
$47$ $$( 1296 + T )^{2}$$
$53$ $$380016036 + T^{2}$$
$59$ $$58798224 + T^{2}$$
$61$ $$1206728644 + T^{2}$$
$67$ $$475763344 + T^{2}$$
$71$ $$( -46872 + T )^{2}$$
$73$ $$( 67562 + T )^{2}$$
$79$ $$( 76912 + T )^{2}$$
$83$ $$4585456656 + T^{2}$$
$89$ $$( 29754 + T )^{2}$$
$97$ $$( 122398 + T )^{2}$$