# Properties

 Label 256.7.c.d Level $256$ Weight $7$ Character orbit 256.c Analytic conductor $58.894$ Analytic rank $0$ Dimension $2$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 256.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$58.8938454067$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 23 \beta q^{3} - 1387 q^{9}+O(q^{10})$$ q + 23*b * q^3 - 1387 * q^9 $$q + 23 \beta q^{3} - 1387 q^{9} + 1169 \beta q^{11} - 1726 q^{17} - 1241 \beta q^{19} - 15625 q^{25} - 15134 \beta q^{27} - 107548 q^{33} - 134642 q^{41} + 37457 \beta q^{43} + 117649 q^{49} - 39698 \beta q^{51} + 114172 q^{57} - 152479 \beta q^{59} - 298313 \beta q^{67} + 593134 q^{73} - 359375 \beta q^{75} + 381205 q^{81} + 339463 \beta q^{83} + 357262 q^{89} + 1822754 q^{97} - 1621403 \beta q^{99} +O(q^{100})$$ q + 23*b * q^3 - 1387 * q^9 + 1169*b * q^11 - 1726 * q^17 - 1241*b * q^19 - 15625 * q^25 - 15134*b * q^27 - 107548 * q^33 - 134642 * q^41 + 37457*b * q^43 + 117649 * q^49 - 39698*b * q^51 + 114172 * q^57 - 152479*b * q^59 - 298313*b * q^67 + 593134 * q^73 - 359375*b * q^75 + 381205 * q^81 + 339463*b * q^83 + 357262 * q^89 + 1822754 * q^97 - 1621403*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2774 q^{9}+O(q^{10})$$ 2 * q - 2774 * q^9 $$2 q - 2774 q^{9} - 3452 q^{17} - 31250 q^{25} - 215096 q^{33} - 269284 q^{41} + 235298 q^{49} + 228344 q^{57} + 1186268 q^{73} + 762410 q^{81} + 714524 q^{89} + 3645508 q^{97}+O(q^{100})$$ 2 * q - 2774 * q^9 - 3452 * q^17 - 31250 * q^25 - 215096 * q^33 - 269284 * q^41 + 235298 * q^49 + 228344 * q^57 + 1186268 * q^73 + 762410 * q^81 + 714524 * q^89 + 3645508 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
255.1
 − 1.00000i 1.00000i
0 46.0000i 0 0 0 0 0 −1387.00 0
255.2 0 46.0000i 0 0 0 0 0 −1387.00 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.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
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 256.7.c.d 2
4.b odd 2 1 inner 256.7.c.d 2
8.b even 2 1 inner 256.7.c.d 2
8.d odd 2 1 CM 256.7.c.d 2
16.e even 4 1 8.7.d.a 1
16.e even 4 1 32.7.d.a 1
16.f odd 4 1 8.7.d.a 1
16.f odd 4 1 32.7.d.a 1
48.i odd 4 1 72.7.b.a 1
48.i odd 4 1 288.7.b.a 1
48.k even 4 1 72.7.b.a 1
48.k even 4 1 288.7.b.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.7.d.a 1 16.e even 4 1
8.7.d.a 1 16.f odd 4 1
32.7.d.a 1 16.e even 4 1
32.7.d.a 1 16.f odd 4 1
72.7.b.a 1 48.i odd 4 1
72.7.b.a 1 48.k even 4 1
256.7.c.d 2 1.a even 1 1 trivial
256.7.c.d 2 4.b odd 2 1 inner
256.7.c.d 2 8.b even 2 1 inner
256.7.c.d 2 8.d odd 2 1 CM
288.7.b.a 1 48.i odd 4 1
288.7.b.a 1 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{7}^{\mathrm{new}}(256, [\chi])$$:

 $$T_{3}^{2} + 2116$$ T3^2 + 2116 $$T_{5}$$ T5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2116$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 5466244$$
$13$ $$T^{2}$$
$17$ $$(T + 1726)^{2}$$
$19$ $$T^{2} + 6160324$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T + 134642)^{2}$$
$43$ $$T^{2} + 5612107396$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 92999381764$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 355962583876$$
$71$ $$T^{2}$$
$73$ $$(T - 593134)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 460940513476$$
$89$ $$(T - 357262)^{2}$$
$97$ $$(T - 1822754)^{2}$$