# Properties

 Label 256.7.d.f Level $256$ Weight $7$ Character orbit 256.d Analytic conductor $58.894$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$58.8938454067$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{15})$$ Defining polynomial: $$x^{4} - 7 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{10}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} -5 \beta_{1} q^{5} + 5 \beta_{3} q^{7} + 231 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} -5 \beta_{1} q^{5} + 5 \beta_{3} q^{7} + 231 q^{9} + 31 \beta_{2} q^{11} + 733 \beta_{1} q^{13} + 5 \beta_{3} q^{15} -4766 q^{17} + 243 \beta_{2} q^{19} -4800 \beta_{1} q^{21} -169 \beta_{3} q^{23} + 15525 q^{25} + 498 \beta_{2} q^{27} + 12749 \beta_{1} q^{29} -676 \beta_{3} q^{31} -29760 q^{33} + 100 \beta_{2} q^{35} -997 \beta_{1} q^{37} -733 \beta_{3} q^{39} -29362 q^{41} + 695 \beta_{2} q^{43} -1155 \beta_{1} q^{45} + 122 \beta_{3} q^{47} + 21649 q^{49} + 4766 \beta_{2} q^{51} + 96427 \beta_{1} q^{53} -155 \beta_{3} q^{55} -233280 q^{57} + 2531 \beta_{2} q^{59} -5459 \beta_{1} q^{61} + 1155 \beta_{3} q^{63} + 14660 q^{65} -12721 \beta_{2} q^{67} + 162240 \beta_{1} q^{69} + 8589 \beta_{3} q^{71} -288626 q^{73} -15525 \beta_{2} q^{75} + 148800 \beta_{1} q^{77} + 5014 \beta_{3} q^{79} -646479 q^{81} -6589 \beta_{2} q^{83} + 23830 \beta_{1} q^{85} -12749 \beta_{3} q^{87} -310738 q^{89} -14660 \beta_{2} q^{91} + 648960 \beta_{1} q^{93} -1215 \beta_{3} q^{95} -1457086 q^{97} + 7161 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 924q^{9} + O(q^{10})$$ $$4q + 924q^{9} - 19064q^{17} + 62100q^{25} - 119040q^{33} - 117448q^{41} + 86596q^{49} - 933120q^{57} + 58640q^{65} - 1154504q^{73} - 2585916q^{81} - 1242952q^{89} - 5828344q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 7 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 3 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$-2 \nu^{3} + 22 \nu$$ $$\beta_{3}$$ $$=$$ $$32 \nu^{2} - 112$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 4 \beta_{1}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 112$$$$)/32$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{2} + 44 \beta_{1}$$$$)/16$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 1.93649 + 0.500000i 1.93649 − 0.500000i −1.93649 + 0.500000i −1.93649 − 0.500000i
0 −30.9839 0 10.0000i 0 309.839i 0 231.000 0
127.2 0 −30.9839 0 10.0000i 0 309.839i 0 231.000 0
127.3 0 30.9839 0 10.0000i 0 309.839i 0 231.000 0
127.4 0 30.9839 0 10.0000i 0 309.839i 0 231.000 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.7.d.f 4
4.b odd 2 1 inner 256.7.d.f 4
8.b even 2 1 inner 256.7.d.f 4
8.d odd 2 1 inner 256.7.d.f 4
16.e even 4 1 4.7.b.a 2
16.e even 4 1 64.7.c.c 2
16.f odd 4 1 4.7.b.a 2
16.f odd 4 1 64.7.c.c 2
48.i odd 4 1 36.7.d.c 2
48.i odd 4 1 576.7.g.h 2
48.k even 4 1 36.7.d.c 2
48.k even 4 1 576.7.g.h 2
80.i odd 4 1 100.7.d.a 4
80.j even 4 1 100.7.d.a 4
80.k odd 4 1 100.7.b.c 2
80.q even 4 1 100.7.b.c 2
80.s even 4 1 100.7.d.a 4
80.t odd 4 1 100.7.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.7.b.a 2 16.e even 4 1
4.7.b.a 2 16.f odd 4 1
36.7.d.c 2 48.i odd 4 1
36.7.d.c 2 48.k even 4 1
64.7.c.c 2 16.e even 4 1
64.7.c.c 2 16.f odd 4 1
100.7.b.c 2 80.k odd 4 1
100.7.b.c 2 80.q even 4 1
100.7.d.a 4 80.i odd 4 1
100.7.d.a 4 80.j even 4 1
100.7.d.a 4 80.s even 4 1
100.7.d.a 4 80.t odd 4 1
256.7.d.f 4 1.a even 1 1 trivial
256.7.d.f 4 4.b odd 2 1 inner
256.7.d.f 4 8.b even 2 1 inner
256.7.d.f 4 8.d odd 2 1 inner
576.7.g.h 2 48.i odd 4 1
576.7.g.h 2 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 960$$ acting on $$S_{7}^{\mathrm{new}}(256, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -960 + T^{2} )^{2}$$
$5$ $$( 100 + T^{2} )^{2}$$
$7$ $$( 96000 + T^{2} )^{2}$$
$11$ $$( -922560 + T^{2} )^{2}$$
$13$ $$( 2149156 + T^{2} )^{2}$$
$17$ $$( 4766 + T )^{4}$$
$19$ $$( -56687040 + T^{2} )^{2}$$
$23$ $$( 109674240 + T^{2} )^{2}$$
$29$ $$( 650148004 + T^{2} )^{2}$$
$31$ $$( 1754787840 + T^{2} )^{2}$$
$37$ $$( 3976036 + T^{2} )^{2}$$
$41$ $$( 29362 + T )^{4}$$
$43$ $$( -463704000 + T^{2} )^{2}$$
$47$ $$( 57154560 + T^{2} )^{2}$$
$53$ $$( 37192665316 + T^{2} )^{2}$$
$59$ $$( -6149722560 + T^{2} )^{2}$$
$61$ $$( 119202724 + T^{2} )^{2}$$
$67$ $$( -155350887360 + T^{2} )^{2}$$
$71$ $$( 283280336640 + T^{2} )^{2}$$
$73$ $$( 288626 + T )^{4}$$
$79$ $$( 96538352640 + T^{2} )^{2}$$
$83$ $$( -41678324160 + T^{2} )^{2}$$
$89$ $$( 310738 + T )^{4}$$
$97$ $$( 1457086 + T )^{4}$$