# Properties

 Label 256.7.c.l Level 256 Weight 7 Character orbit 256.c Analytic conductor 58.894 Analytic rank 0 Dimension 8 CM no 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 11 x^{6} + 516 x^{4} - 2816 x^{2} + 65536$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{42}$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{5} q^{5} + \beta_{4} q^{7} + ( 165 + 3 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{5} q^{5} + \beta_{4} q^{7} + ( 165 + 3 \beta_{3} ) q^{9} + ( -28 \beta_{1} - 29 \beta_{2} ) q^{11} + ( 2 \beta_{5} - \beta_{6} ) q^{13} + ( -2 \beta_{4} - \beta_{7} ) q^{15} + ( 1042 - 25 \beta_{3} ) q^{17} + ( 52 \beta_{1} + 13 \beta_{2} ) q^{19} + ( -9 \beta_{5} - \beta_{6} ) q^{21} + ( 4 \beta_{4} - 3 \beta_{7} ) q^{23} + ( 5975 - 110 \beta_{3} ) q^{25} + ( 399 \beta_{1} + 207 \beta_{2} ) q^{27} + ( -37 \beta_{5} + 8 \beta_{6} ) q^{29} + ( 57 \beta_{4} + \beta_{7} ) q^{31} + ( 21012 - 55 \beta_{3} ) q^{33} + ( -110 \beta_{1} - 690 \beta_{2} ) q^{35} + ( 186 \beta_{5} + 7 \beta_{6} ) q^{37} + ( -53 \beta_{4} + 8 \beta_{7} ) q^{39} + ( 29486 - 138 \beta_{3} ) q^{41} + ( 217 \beta_{1} - 2338 \beta_{2} ) q^{43} + ( 240 \beta_{5} - 15 \beta_{6} ) q^{45} + ( -319 \beta_{4} + 11 \beta_{7} ) q^{47} + ( 529 - 712 \beta_{3} ) q^{49} + ( 5167 \beta_{1} - 1725 \beta_{2} ) q^{51} + ( -1032 \beta_{5} - 7 \beta_{6} ) q^{53} + ( 201 \beta_{4} + 28 \beta_{7} ) q^{55} + ( -31668 + 143 \beta_{3} ) q^{57} + ( 4439 \beta_{1} - 4112 \beta_{2} ) q^{59} + ( -722 \beta_{5} - 41 \beta_{6} ) q^{61} + ( 654 \beta_{4} - 3 \beta_{7} ) q^{63} + ( -51360 + 1570 \beta_{3} ) q^{65} + ( 2444 \beta_{1} - 8411 \beta_{2} ) q^{67} + ( 2817 \beta_{5} - 55 \beta_{6} ) q^{69} + ( -488 \beta_{4} - 85 \beta_{7} ) q^{71} + ( -110978 + 1067 \beta_{3} ) q^{73} + ( 24125 \beta_{1} - 7590 \beta_{2} ) q^{75} + ( 1093 \beta_{5} + 173 \beta_{6} ) q^{77} + ( -390 \beta_{4} - 52 \beta_{7} ) q^{79} + ( -142011 + 3177 \beta_{3} ) q^{81} + ( 29001 \beta_{1} - 25912 \beta_{2} ) q^{83} + ( -4417 \beta_{5} + 125 \beta_{6} ) q^{85} + ( 382 \beta_{4} - 85 \beta_{7} ) q^{87} + ( -190306 + 1491 \beta_{3} ) q^{89} + ( 1570 \beta_{1} - 41538 \beta_{2} ) q^{91} + ( -1464 \beta_{5} - 40 \beta_{6} ) q^{93} + ( -169 \beta_{4} - 52 \beta_{7} ) q^{95} + ( -231694 - 5337 \beta_{3} ) q^{97} + ( 9675 \beta_{1} - 24936 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 1320q^{9} + O(q^{10})$$ $$8q + 1320q^{9} + 8336q^{17} + 47800q^{25} + 168096q^{33} + 235888q^{41} + 4232q^{49} - 253344q^{57} - 410880q^{65} - 887824q^{73} - 1136088q^{81} - 1522448q^{89} - 1853552q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 11 x^{6} + 516 x^{4} - 2816 x^{2} + 65536$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-9 \nu^{7} + 995 \nu^{5} - 164 \nu^{3} + 293120 \nu$$$$)/26624$$ $$\beta_{2}$$ $$=$$ $$($$$$71 \nu^{7} - 1933 \nu^{5} + 30876 \nu^{3} - 300800 \nu$$$$)/26624$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - 11 \nu^{4} + 260 \nu^{2} - 1408$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} - 629 \nu^{4} + 6268 \nu^{2} - 162304$$$$)/208$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{7} - \nu^{5} + 684 \nu^{3} + 1408 \nu$$$$)/128$$ $$\beta_{6}$$ $$=$$ $$($$$$17 \nu^{7} + 165 \nu^{5} - 732 \nu^{3} + 126080 \nu$$$$)/128$$ $$\beta_{7}$$ $$=$$ $$($$$$-179 \nu^{6} - 6095 \nu^{4} - 49484 \nu^{2} - 1576448$$$$)/208$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - 11 \beta_{5} + 56 \beta_{2} + 72 \beta_{1}$$$$)/1024$$ $$\nu^{2}$$ $$=$$ $$($$$$-5 \beta_{7} + 63 \beta_{4} - 32 \beta_{3} + 5632$$$$)/2048$$ $$\nu^{3}$$ $$=$$ $$($$$$-13 \beta_{6} + 15 \beta_{5} + 760 \beta_{2} + 1928 \beta_{1}$$$$)/1024$$ $$\nu^{4}$$ $$=$$ $$($$$$-51 \beta_{7} - 23 \beta_{4} - 352 \beta_{3} - 466432$$$$)/2048$$ $$\nu^{5}$$ $$=$$ $$($$$$-275 \beta_{6} + 3665 \beta_{5} - 18232 \beta_{2} + 2232 \beta_{1}$$$$)/1024$$ $$\nu^{6}$$ $$=$$ $$($$$$739 \beta_{7} - 16633 \beta_{4} + 20832 \beta_{3} - 3711488$$$$)/2048$$ $$\nu^{7}$$ $$=$$ $$($$$$2403 \beta_{6} + 46655 \beta_{5} - 205640 \beta_{2} - 472632 \beta_{1}$$$$)/1024$$

## 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
 −3.26433 − 2.31174i 3.26433 − 2.31174i 2.84502 − 2.81174i −2.84502 − 2.81174i 2.84502 + 2.81174i −2.84502 + 2.81174i −3.26433 + 2.31174i 3.26433 + 2.31174i
0 32.4939i 0 −199.084 0 19.6656i 0 −326.854 0
255.2 0 32.4939i 0 199.084 0 19.6656i 0 −326.854 0
255.3 0 8.49390i 0 −59.7107 0 483.584i 0 656.854 0
255.4 0 8.49390i 0 59.7107 0 483.584i 0 656.854 0
255.5 0 8.49390i 0 −59.7107 0 483.584i 0 656.854 0
255.6 0 8.49390i 0 59.7107 0 483.584i 0 656.854 0
255.7 0 32.4939i 0 −199.084 0 19.6656i 0 −326.854 0
255.8 0 32.4939i 0 199.084 0 19.6656i 0 −326.854 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 255.8 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.c.l 8
4.b odd 2 1 inner 256.7.c.l 8
8.b even 2 1 inner 256.7.c.l 8
8.d odd 2 1 inner 256.7.c.l 8
16.e even 4 1 8.7.d.b 4
16.e even 4 1 32.7.d.b 4
16.f odd 4 1 8.7.d.b 4
16.f odd 4 1 32.7.d.b 4
48.i odd 4 1 72.7.b.b 4
48.i odd 4 1 288.7.b.b 4
48.k even 4 1 72.7.b.b 4
48.k even 4 1 288.7.b.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.7.d.b 4 16.e even 4 1
8.7.d.b 4 16.f odd 4 1
32.7.d.b 4 16.e even 4 1
32.7.d.b 4 16.f odd 4 1
72.7.b.b 4 48.i odd 4 1
72.7.b.b 4 48.k even 4 1
256.7.c.l 8 1.a even 1 1 trivial
256.7.c.l 8 4.b odd 2 1 inner
256.7.c.l 8 8.b even 2 1 inner
256.7.c.l 8 8.d odd 2 1 inner
288.7.b.b 4 48.i odd 4 1
288.7.b.b 4 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 $$S_{7}^{\mathrm{new}}(256, [\chi])$$:

 $$T_{3}^{4} + 1128 T_{3}^{2} + 76176$$ $$T_{5}^{4} - 43200 T_{5}^{2} + 141312000$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 1788 T^{2} + 1620198 T^{4} - 950216508 T^{6} + 282429536481 T^{8} )^{2}$$
$5$ $$( 1 + 19300 T^{2} + 256155750 T^{4} + 4711914062500 T^{6} + 59604644775390625 T^{8} )^{2}$$
$7$ $$( 1 - 236356 T^{2} + 28021959366 T^{4} - 3271471277679556 T^{6} +$$$$19\!\cdots\!01$$$$T^{8} )^{2}$$
$11$ $$( 1 - 4238012 T^{2} + 10442076641958 T^{4} - 13300697121684118652 T^{6} +$$$$98\!\cdots\!41$$$$T^{8} )^{2}$$
$13$ $$( 1 + 4994596 T^{2} + 30260873415846 T^{4} +$$$$11\!\cdots\!76$$$$T^{6} +$$$$54\!\cdots\!61$$$$T^{8} )^{2}$$
$17$ $$( 1 - 2084 T + 32560902 T^{2} - 50302693796 T^{3} + 582622237229761 T^{4} )^{4}$$
$19$ $$( 1 - 184369532 T^{2} + 12923720463999078 T^{4} -$$$$40\!\cdots\!52$$$$T^{6} +$$$$48\!\cdots\!21$$$$T^{8} )^{2}$$
$23$ $$( 1 - 212117956 T^{2} + 23026671552237126 T^{4} -$$$$46\!\cdots\!76$$$$T^{6} +$$$$48\!\cdots\!41$$$$T^{8} )^{2}$$
$29$ $$( 1 + 1409719204 T^{2} + 1160515330165289766 T^{4} +$$$$49\!\cdots\!64$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8} )^{2}$$
$31$ $$( 1 - 2758360324 T^{2} + 3362667870952277766 T^{4} -$$$$21\!\cdots\!64$$$$T^{6} +$$$$62\!\cdots\!21$$$$T^{8} )^{2}$$
$37$ $$( 1 + 8121202276 T^{2} + 29600495645847907686 T^{4} +$$$$53\!\cdots\!56$$$$T^{6} +$$$$43\!\cdots\!61$$$$T^{8} )^{2}$$
$41$ $$( 1 - 58972 T + 9857729958 T^{2} - 280123147300252 T^{3} + 22563490300366186081 T^{4} )^{4}$$
$43$ $$( 1 - 16632984764 T^{2} +$$$$13\!\cdots\!46$$$$T^{4} -$$$$66\!\cdots\!64$$$$T^{6} +$$$$15\!\cdots\!01$$$$T^{8} )^{2}$$
$47$ $$( 1 - 13578767236 T^{2} +$$$$16\!\cdots\!86$$$$T^{4} -$$$$15\!\cdots\!76$$$$T^{6} +$$$$13\!\cdots\!81$$$$T^{8} )^{2}$$
$53$ $$( 1 + 42194545636 T^{2} +$$$$11\!\cdots\!86$$$$T^{4} +$$$$20\!\cdots\!76$$$$T^{6} +$$$$24\!\cdots\!81$$$$T^{8} )^{2}$$
$59$ $$( 1 - 131907478076 T^{2} +$$$$79\!\cdots\!26$$$$T^{4} -$$$$23\!\cdots\!56$$$$T^{6} +$$$$31\!\cdots\!61$$$$T^{8} )^{2}$$
$61$ $$( 1 + 160868902564 T^{2} +$$$$11\!\cdots\!46$$$$T^{4} +$$$$42\!\cdots\!44$$$$T^{6} +$$$$70\!\cdots\!41$$$$T^{8} )^{2}$$
$67$ $$( 1 - 253874829308 T^{2} +$$$$30\!\cdots\!58$$$$T^{4} -$$$$20\!\cdots\!88$$$$T^{6} +$$$$66\!\cdots\!21$$$$T^{8} )^{2}$$
$71$ $$( 1 - 164364621124 T^{2} +$$$$21\!\cdots\!06$$$$T^{4} -$$$$26\!\cdots\!84$$$$T^{6} +$$$$26\!\cdots\!81$$$$T^{8} )^{2}$$
$73$ $$( 1 + 221956 T + 284381984742 T^{2} + 33589539530201284 T^{3} +$$$$22\!\cdots\!21$$$$T^{4} )^{4}$$
$79$ $$( 1 - 828257339524 T^{2} +$$$$28\!\cdots\!06$$$$T^{4} -$$$$48\!\cdots\!84$$$$T^{6} +$$$$34\!\cdots\!81$$$$T^{8} )^{2}$$
$83$ $$( 1 + 201038775172 T^{2} +$$$$21\!\cdots\!98$$$$T^{4} +$$$$21\!\cdots\!92$$$$T^{6} +$$$$11\!\cdots\!21$$$$T^{8} )^{2}$$
$89$ $$( 1 + 380612 T + 970422538278 T^{2} + 189157043115248132 T^{3} +$$$$24\!\cdots\!21$$$$T^{4} )^{4}$$
$97$ $$( 1 + 463388 T + 953987784774 T^{2} + 385989231420039452 T^{3} +$$$$69\!\cdots\!41$$$$T^{4} )^{4}$$