# 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: $$x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + 126 \beta q^{3} - 2415 \beta q^{5} + 16744 q^{7} + 113643 q^{9} +O(q^{10})$$ q + 126*b * q^3 - 2415*b * q^5 + 16744 * q^7 + 113643 * q^9 $$q + 126 \beta q^{3} - 2415 \beta q^{5} + 16744 q^{7} + 113643 q^{9} - 267306 \beta q^{11} - 288869 \beta q^{13} + 1217160 q^{15} - 6905934 q^{17} + 5330710 \beta q^{19} + 2109744 \beta q^{21} - 18643272 q^{23} + 25499225 q^{25} + 36639540 \beta q^{27} + 64203315 \beta q^{29} - 52843168 q^{31} + 134722224 q^{33} - 40436760 \beta q^{35} + 91106657 \beta q^{37} + 145589976 q^{39} - 308120442 q^{41} + 8562854 \beta q^{43} - 274447845 \beta q^{45} + 2687348496 q^{47} - 1696965207 q^{49} - 870147684 \beta q^{51} + 798027849 \beta q^{53} - 2582175960 q^{55} - 2686677840 q^{57} + 2594601870 \beta q^{59} + 3478239331 \beta q^{61} + 1902838392 q^{63} - 2790474540 q^{65} - 7740913442 \beta q^{67} - 2349052272 \beta q^{69} - 9791485272 q^{71} - 1463791322 q^{73} + 3212902350 \beta q^{75} - 4475771664 \beta q^{77} + 38116845680 q^{79} + 1665188361 q^{81} - 14667549834 \beta q^{83} + 16677830610 \beta q^{85} - 32358470760 q^{87} + 24992917110 q^{89} - 4836822536 \beta q^{91} - 6658239168 \beta q^{93} + 51494658600 q^{95} + 75013568546 q^{97} - 30377455758 \beta q^{99} +O(q^{100})$$ q + 126*b * q^3 - 2415*b * q^5 + 16744 * q^7 + 113643 * q^9 - 267306*b * q^11 - 288869*b * q^13 + 1217160 * q^15 - 6905934 * q^17 + 5330710*b * q^19 + 2109744*b * q^21 - 18643272 * q^23 + 25499225 * q^25 + 36639540*b * q^27 + 64203315*b * q^29 - 52843168 * q^31 + 134722224 * q^33 - 40436760*b * q^35 + 91106657*b * q^37 + 145589976 * q^39 - 308120442 * q^41 + 8562854*b * q^43 - 274447845*b * q^45 + 2687348496 * q^47 - 1696965207 * q^49 - 870147684*b * q^51 + 798027849*b * q^53 - 2582175960 * q^55 - 2686677840 * q^57 + 2594601870*b * q^59 + 3478239331*b * q^61 + 1902838392 * q^63 - 2790474540 * q^65 - 7740913442*b * q^67 - 2349052272*b * q^69 - 9791485272 * q^71 - 1463791322 * q^73 + 3212902350*b * q^75 - 4475771664*b * q^77 + 38116845680 * q^79 + 1665188361 * q^81 - 14667549834*b * q^83 + 16677830610*b * q^85 - 32358470760 * q^87 + 24992917110 * q^89 - 4836822536*b * q^91 - 6658239168*b * q^93 + 51494658600 * q^95 + 75013568546 * q^97 - 30377455758*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 33488 q^{7} + 227286 q^{9}+O(q^{10})$$ 2 * q + 33488 * q^7 + 227286 * q^9 $$2 q + 33488 q^{7} + 227286 q^{9} + 2434320 q^{15} - 13811868 q^{17} - 37286544 q^{23} + 50998450 q^{25} - 105686336 q^{31} + 269444448 q^{33} + 291179952 q^{39} - 616240884 q^{41} + 5374696992 q^{47} - 3393930414 q^{49} - 5164351920 q^{55} - 5373355680 q^{57} + 3805676784 q^{63} - 5580949080 q^{65} - 19582970544 q^{71} - 2927582644 q^{73} + 76233691360 q^{79} + 3330376722 q^{81} - 64716941520 q^{87} + 49985834220 q^{89} + 102989317200 q^{95} + 150027137092 q^{97}+O(q^{100})$$ 2 * q + 33488 * q^7 + 227286 * q^9 + 2434320 * q^15 - 13811868 * q^17 - 37286544 * q^23 + 50998450 * q^25 - 105686336 * q^31 + 269444448 * q^33 + 291179952 * q^39 - 616240884 * q^41 + 5374696992 * q^47 - 3393930414 * q^49 - 5164351920 * q^55 - 5373355680 * q^57 + 3805676784 * q^63 - 5580949080 * q^65 - 19582970544 * q^71 - 2927582644 * q^73 + 76233691360 * q^79 + 3330376722 * q^81 - 64716941520 * q^87 + 49985834220 * q^89 + 102989317200 * q^95 + 150027137092 * 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
0 252.000i 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 $$S_{12}^{\mathrm{new}}(256, [\chi])$$:

 $$T_{3}^{2} + 63504$$ T3^2 + 63504 $$T_{7} - 16744$$ T7 - 16744

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 63504$$
$5$ $$T^{2} + 23328900$$
$7$ $$(T - 16744)^{2}$$
$11$ $$T^{2} + 285809990544$$
$13$ $$T^{2} + 333781196644$$
$17$ $$(T + 6905934)^{2}$$
$19$ $$T^{2} + \cdots + 113665876416400$$
$23$ $$(T + 18643272)^{2}$$
$29$ $$T^{2} + 16\!\cdots\!00$$
$31$ $$(T + 52843168)^{2}$$
$37$ $$T^{2} + 33\!\cdots\!96$$
$41$ $$(T + 308120442)^{2}$$
$43$ $$T^{2} + \cdots + 293289874501264$$
$47$ $$(T - 2687348496)^{2}$$
$53$ $$T^{2} + 25\!\cdots\!04$$
$59$ $$T^{2} + 26\!\cdots\!00$$
$61$ $$T^{2} + 48\!\cdots\!44$$
$67$ $$T^{2} + 23\!\cdots\!56$$
$71$ $$(T + 9791485272)^{2}$$
$73$ $$(T + 1463791322)^{2}$$
$79$ $$(T - 38116845680)^{2}$$
$83$ $$T^{2} + 86\!\cdots\!24$$
$89$ $$(T - 24992917110)^{2}$$
$97$ $$(T - 75013568546)^{2}$$