# Properties

 Label 256.12.b.e Level $256$ Weight $12$ Character orbit 256.b Analytic conductor $196.696$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 256.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$196.695854223$$ 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 1) 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.

 $$f(q)$$ $$=$$ $$q + 252 i q^{3} -4830 i q^{5} + 16744 q^{7} + 113643 q^{9} +O(q^{10})$$ $$q + 252 i q^{3} -4830 i q^{5} + 16744 q^{7} + 113643 q^{9} -534612 i q^{11} -577738 i q^{13} + 1217160 q^{15} -6905934 q^{17} + 10661420 i q^{19} + 4219488 i q^{21} -18643272 q^{23} + 25499225 q^{25} + 73279080 i q^{27} + 128406630 i q^{29} -52843168 q^{31} + 134722224 q^{33} -80873520 i q^{35} + 182213314 i q^{37} + 145589976 q^{39} -308120442 q^{41} + 17125708 i q^{43} -548895690 i q^{45} + 2687348496 q^{47} -1696965207 q^{49} -1740295368 i q^{51} + 1596055698 i q^{53} -2582175960 q^{55} -2686677840 q^{57} + 5189203740 i q^{59} + 6956478662 i q^{61} + 1902838392 q^{63} -2790474540 q^{65} -15481826884 i q^{67} -4698104544 i q^{69} -9791485272 q^{71} -1463791322 q^{73} + 6425804700 i q^{75} -8951543328 i q^{77} + 38116845680 q^{79} + 1665188361 q^{81} -29335099668 i q^{83} + 33355661220 i q^{85} -32358470760 q^{87} + 24992917110 q^{89} -9673645072 i q^{91} -13316478336 i q^{93} + 51494658600 q^{95} + 75013568546 q^{97} -60754911516 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 33488q^{7} + 227286q^{9} + O(q^{10})$$ $$2q + 33488q^{7} + 227286q^{9} + 2434320q^{15} - 13811868q^{17} - 37286544q^{23} + 50998450q^{25} - 105686336q^{31} + 269444448q^{33} + 291179952q^{39} - 616240884q^{41} + 5374696992q^{47} - 3393930414q^{49} - 5164351920q^{55} - 5373355680q^{57} + 3805676784q^{63} - 5580949080q^{65} - 19582970544q^{71} - 2927582644q^{73} + 76233691360q^{79} + 3330376722q^{81} - 64716941520q^{87} + 49985834220q^{89} + 102989317200q^{95} + 150027137092q^{97} + O(q^{100})$$

## 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
0 252.000i 0 4830.00i 0 16744.0 0 113643. 0
129.2 0 252.000i 0 4830.00i 0 16744.0 0 113643. 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 256.12.b.e 2
4.b odd 2 1 256.12.b.c 2
8.b even 2 1 inner 256.12.b.e 2
8.d odd 2 1 256.12.b.c 2
16.e even 4 1 1.12.a.a 1
16.e even 4 1 64.12.a.b 1
16.f odd 4 1 16.12.a.a 1
16.f odd 4 1 64.12.a.f 1
48.i odd 4 1 9.12.a.b 1
48.k even 4 1 144.12.a.d 1
80.i odd 4 1 25.12.b.b 2
80.q even 4 1 25.12.a.b 1
80.t odd 4 1 25.12.b.b 2
112.l odd 4 1 49.12.a.a 1
112.w even 12 2 49.12.c.b 2
112.x odd 12 2 49.12.c.c 2
144.w odd 12 2 81.12.c.b 2
144.x even 12 2 81.12.c.d 2
176.l odd 4 1 121.12.a.b 1
208.p even 4 1 169.12.a.a 1
240.bb even 4 1 225.12.b.d 2
240.bf even 4 1 225.12.b.d 2
240.bm odd 4 1 225.12.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 16.e even 4 1
9.12.a.b 1 48.i odd 4 1
16.12.a.a 1 16.f odd 4 1
25.12.a.b 1 80.q even 4 1
25.12.b.b 2 80.i odd 4 1
25.12.b.b 2 80.t odd 4 1
49.12.a.a 1 112.l odd 4 1
49.12.c.b 2 112.w even 12 2
49.12.c.c 2 112.x odd 12 2
64.12.a.b 1 16.e even 4 1
64.12.a.f 1 16.f odd 4 1
81.12.c.b 2 144.w odd 12 2
81.12.c.d 2 144.x even 12 2
121.12.a.b 1 176.l odd 4 1
144.12.a.d 1 48.k even 4 1
169.12.a.a 1 208.p even 4 1
225.12.a.b 1 240.bm odd 4 1
225.12.b.d 2 240.bb even 4 1
225.12.b.d 2 240.bf even 4 1
256.12.b.c 2 4.b odd 2 1
256.12.b.c 2 8.d odd 2 1
256.12.b.e 2 1.a even 1 1 trivial
256.12.b.e 2 8.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{12}^{\mathrm{new}}(256, [\chi])$$:

 $$T_{3}^{2} + 63504$$ $$T_{7} - 16744$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$63504 + T^{2}$$
$5$ $$23328900 + T^{2}$$
$7$ $$( -16744 + T )^{2}$$
$11$ $$285809990544 + T^{2}$$
$13$ $$333781196644 + T^{2}$$
$17$ $$( 6905934 + T )^{2}$$
$19$ $$113665876416400 + T^{2}$$
$23$ $$( 18643272 + T )^{2}$$
$29$ $$16488262627956900 + T^{2}$$
$31$ $$( 52843168 + T )^{2}$$
$37$ $$33201691798862596 + T^{2}$$
$41$ $$( 308120442 + T )^{2}$$
$43$ $$293289874501264 + T^{2}$$
$47$ $$( -2687348496 + T )^{2}$$
$53$ $$2547393791118267204 + T^{2}$$
$59$ $$26927835455229987600 + T^{2}$$
$61$ $$48392595374861310244 + T^{2}$$
$67$ $$23\!\cdots\!56$$$$+ T^{2}$$
$71$ $$( 9791485272 + T )^{2}$$
$73$ $$( 1463791322 + T )^{2}$$
$79$ $$( -38116845680 + T )^{2}$$
$83$ $$86\!\cdots\!24$$$$+ T^{2}$$
$89$ $$( -24992917110 + T )^{2}$$
$97$ $$( -75013568546 + T )^{2}$$