# Properties

 Label 1296.3.g.k Level $1296$ Weight $3$ Character orbit 1296.g Analytic conductor $35.313$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1296.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$35.3134422611$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.856615824.2 Defining polynomial: $$x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}\cdot 3^{6}$$ Twist minimal: no (minimal twist has level 144) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{3} ) q^{5} -\beta_{6} q^{7} +O(q^{10})$$ $$q + ( 1 + \beta_{3} ) q^{5} -\beta_{6} q^{7} + ( -\beta_{1} + \beta_{2} + \beta_{6} ) q^{11} + ( -1 + \beta_{3} + \beta_{4} ) q^{13} + ( -1 + \beta_{4} + \beta_{7} ) q^{17} + ( \beta_{2} - 3 \beta_{5} - \beta_{6} ) q^{19} + ( \beta_{1} + 3 \beta_{2} + \beta_{5} ) q^{23} + ( 7 + 4 \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{25} + ( 17 - \beta_{3} - 3 \beta_{4} ) q^{29} + ( 3 \beta_{1} + \beta_{2} - 3 \beta_{5} - 2 \beta_{6} ) q^{31} + ( 4 \beta_{1} + 7 \beta_{2} - 4 \beta_{5} + \beta_{6} ) q^{35} + ( -4 - 4 \beta_{3} - \beta_{4} + 2 \beta_{7} ) q^{37} + ( 15 + 4 \beta_{3} + \beta_{4} - 2 \beta_{7} ) q^{41} + ( -6 \beta_{1} + 2 \beta_{2} + 3 \beta_{5} + 2 \beta_{6} ) q^{43} + ( -2 \beta_{1} + 10 \beta_{2} + 2 \beta_{5} - 5 \beta_{6} ) q^{47} + ( -9 + 6 \beta_{3} + 3 \beta_{4} + \beta_{7} ) q^{49} + ( 30 - 6 \beta_{3} + 3 \beta_{4} ) q^{53} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{5} ) q^{55} + ( -4 \beta_{1} + 14 \beta_{2} + 7 \beta_{5} + 2 \beta_{6} ) q^{59} + ( -3 - 7 \beta_{3} - \beta_{4} - 2 \beta_{7} ) q^{61} + ( 22 - 6 \beta_{3} - \beta_{4} - \beta_{7} ) q^{65} + ( 3 \beta_{1} - \beta_{2} + 6 \beta_{5} + 3 \beta_{6} ) q^{67} + ( -5 \beta_{1} + 19 \beta_{2} - \beta_{5} - 5 \beta_{6} ) q^{71} + ( 9 - 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{73} + ( 51 - 3 \beta_{3} ) q^{77} + ( 9 \beta_{1} - 5 \beta_{2} + 3 \beta_{5} + 2 \beta_{6} ) q^{79} + ( -2 \beta_{1} + 23 \beta_{2} - 6 \beta_{5} + 5 \beta_{6} ) q^{83} + ( 4 - 16 \beta_{3} - \beta_{4} - 2 \beta_{7} ) q^{85} + ( 22 + 4 \beta_{3} - 3 \beta_{4} ) q^{89} + ( 9 \beta_{1} - 3 \beta_{5} + 8 \beta_{6} ) q^{91} + ( 10 \beta_{1} + 26 \beta_{2} - 18 \beta_{5} - 10 \beta_{6} ) q^{95} + ( 3 + 6 \beta_{3} + 3 \beta_{4} + 4 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 6q^{5} + O(q^{10})$$ $$8q + 6q^{5} - 10q^{13} - 6q^{17} + 46q^{25} + 138q^{29} - 20q^{37} + 108q^{41} - 82q^{49} + 252q^{53} - 14q^{61} + 186q^{65} + 74q^{73} + 414q^{77} + 60q^{85} + 168q^{89} + 20q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$3 \nu^{3} + 12 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{5} + 7 \nu^{3} + 10 \nu$$ $$\beta_{3}$$ $$=$$ $$\nu^{6} + 8 \nu^{4} + 14 \nu^{2}$$ $$\beta_{4}$$ $$=$$ $$-\nu^{6} - 5 \nu^{4} - 2 \nu^{2} - 4$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{7} + 30 \nu^{5} + 87 \nu^{3} + 66 \nu$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{7} - 32 \nu^{5} - 95 \nu^{3} - 50 \nu$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$3 \nu^{6} + 30 \nu^{4} + 84 \nu^{2} + 43$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} + \beta_{2} - \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - 2 \beta_{4} - 5 \beta_{3} - 51$$$$)/18$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 3 \beta_{1}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{7} + 7 \beta_{4} + 13 \beta_{3} + 114$$$$)/9$$ $$\nu^{5}$$ $$=$$ $$($$$$9 \beta_{6} + 9 \beta_{5} + 12 \beta_{2} - 16 \beta_{1}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$3 \beta_{7} - 14 \beta_{4} - 20 \beta_{3} - 185$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-43 \beta_{6} - 41 \beta_{5} - 73 \beta_{2} + 84 \beta_{1}$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1135.1
 1.07834i − 1.07834i − 0.385731i 0.385731i 2.33086i − 2.33086i 2.06288i − 2.06288i
0 0 0 −6.03459 0 11.8163i 0 0 0
1135.2 0 0 0 −6.03459 0 11.8163i 0 0 0
1135.3 0 0 0 −0.909226 0 7.05186i 0 0 0
1135.4 0 0 0 −0.909226 0 7.05186i 0 0 0
1135.5 0 0 0 0.710609 0 3.12324i 0 0 0
1135.6 0 0 0 0.710609 0 3.12324i 0 0 0
1135.7 0 0 0 9.23321 0 6.15562i 0 0 0
1135.8 0 0 0 9.23321 0 6.15562i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1135.8 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.3.g.k 8
3.b odd 2 1 1296.3.g.j 8
4.b odd 2 1 inner 1296.3.g.k 8
9.c even 3 1 432.3.o.a 8
9.c even 3 1 432.3.o.b 8
9.d odd 6 1 144.3.o.a 8
9.d odd 6 1 144.3.o.c yes 8
12.b even 2 1 1296.3.g.j 8
36.f odd 6 1 432.3.o.a 8
36.f odd 6 1 432.3.o.b 8
36.h even 6 1 144.3.o.a 8
36.h even 6 1 144.3.o.c yes 8
72.j odd 6 1 576.3.o.d 8
72.j odd 6 1 576.3.o.f 8
72.l even 6 1 576.3.o.d 8
72.l even 6 1 576.3.o.f 8
72.n even 6 1 1728.3.o.e 8
72.n even 6 1 1728.3.o.f 8
72.p odd 6 1 1728.3.o.e 8
72.p odd 6 1 1728.3.o.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.o.a 8 9.d odd 6 1
144.3.o.a 8 36.h even 6 1
144.3.o.c yes 8 9.d odd 6 1
144.3.o.c yes 8 36.h even 6 1
432.3.o.a 8 9.c even 3 1
432.3.o.a 8 36.f odd 6 1
432.3.o.b 8 9.c even 3 1
432.3.o.b 8 36.f odd 6 1
576.3.o.d 8 72.j odd 6 1
576.3.o.d 8 72.l even 6 1
576.3.o.f 8 72.j odd 6 1
576.3.o.f 8 72.l even 6 1
1296.3.g.j 8 3.b odd 2 1
1296.3.g.j 8 12.b even 2 1
1296.3.g.k 8 1.a even 1 1 trivial
1296.3.g.k 8 4.b odd 2 1 inner
1728.3.o.e 8 72.n even 6 1
1728.3.o.e 8 72.p odd 6 1
1728.3.o.f 8 72.n even 6 1
1728.3.o.f 8 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1296, [\chi])$$:

 $$T_{5}^{4} - 3 T_{5}^{3} - 57 T_{5}^{2} - 9 T_{5} + 36$$ $$T_{17}^{4} + 3 T_{17}^{3} - 822 T_{17}^{2} - 1908 T_{17} + 84168$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 36 - 9 T - 57 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$7$ $$2566404 + 400815 T^{2} + 16335 T^{4} + 237 T^{6} + T^{8}$$
$11$ $$12131289 + 1221804 T^{2} + 37422 T^{4} + 396 T^{6} + T^{8}$$
$13$ $$( -3194 - 2353 T - 291 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$17$ $$( 84168 - 1908 T - 822 T^{2} + 3 T^{3} + T^{4} )^{2}$$
$19$ $$2931572736 + 85791744 T^{2} + 739584 T^{4} + 1731 T^{6} + T^{8}$$
$23$ $$19131876 + 5137263 T^{2} + 273375 T^{4} + 1053 T^{6} + T^{8}$$
$29$ $$( -2011626 + 119457 T - 579 T^{2} - 69 T^{3} + T^{4} )^{2}$$
$31$ $$944784 + 49200939 T^{2} + 1357155 T^{4} + 2673 T^{6} + T^{8}$$
$37$ $$( -613568 - 117320 T - 3756 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$41$ $$( -2330613 + 190998 T - 2700 T^{2} - 54 T^{3} + T^{4} )^{2}$$
$43$ $$29016737649 + 3679120116 T^{2} + 9809910 T^{4} + 6372 T^{6} + T^{8}$$
$47$ $$28643839776036 + 59032500543 T^{2} + 41801103 T^{4} + 11709 T^{6} + T^{8}$$
$53$ $$( -6508512 + 274104 T + 972 T^{2} - 126 T^{3} + T^{4} )^{2}$$
$59$ $$48359409452649 + 114842866284 T^{2} + 77288094 T^{4} + 15948 T^{6} + T^{8}$$
$61$ $$( -556736 - 222251 T - 6357 T^{2} + 7 T^{3} + T^{4} )^{2}$$
$67$ $$68036119056801 + 102586425588 T^{2} + 54676998 T^{4} + 12324 T^{6} + T^{8}$$
$71$ $$726110197530624 + 765915906816 T^{2} + 231242688 T^{4} + 26208 T^{6} + T^{8}$$
$73$ $$( 416536 + 25628 T - 1002 T^{2} - 37 T^{3} + T^{4} )^{2}$$
$79$ $$240627852449856 + 354653956251 T^{2} + 162185139 T^{4} + 23169 T^{6} + T^{8}$$
$83$ $$1517530356962064 + 2504009122227 T^{2} + 545735475 T^{4} + 40761 T^{6} + T^{8}$$
$89$ $$( -1161936 + 109152 T - 984 T^{2} - 84 T^{3} + T^{4} )^{2}$$
$97$ $$( -5516309 + 583178 T - 14988 T^{2} - 10 T^{3} + T^{4} )^{2}$$