# Properties

 Label 1296.3.q.o Level $1296$ Weight $3$ Character orbit 1296.q Analytic conductor $35.313$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1296.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$35.3134422611$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3^{8}$$ Twist minimal: no (minimal twist has level 162) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{5} + (2 \beta_{3} + 2 \beta_{2}) q^{7}+O(q^{10})$$ q + b6 * q^5 + (2*b3 + 2*b2) * q^7 $$q + \beta_{6} q^{5} + (2 \beta_{3} + 2 \beta_{2}) q^{7} + (2 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + 2 \beta_1) q^{11} + ( - \beta_{4} + \beta_{3} - 16 \beta_{2} + 16) q^{13} + (5 \beta_{7} - 5 \beta_{6} - 5 \beta_{5}) q^{17} + (2 \beta_{4} - 14) q^{19} + (2 \beta_{6} + 2 \beta_1) q^{23} + ( - 3 \beta_{3} - 7 \beta_{2}) q^{25} + (7 \beta_{7} - 7 \beta_{6} + \beta_{5} + 7 \beta_1) q^{29} + ( - 8 \beta_{2} + 8) q^{31} + ( - 6 \beta_{7} - 4 \beta_{6} - 4 \beta_{5}) q^{35} + (8 \beta_{4} - 19) q^{37} + ( - 5 \beta_{6} - 3 \beta_1) q^{41} + ( - 6 \beta_{3} - 22 \beta_{2}) q^{43} + (4 \beta_{7} - 4 \beta_{6} + 4 \beta_1) q^{47} + ( - 8 \beta_{4} + 8 \beta_{3} + 63 \beta_{2} - 63) q^{49} + ( - 17 \beta_{7} + 8 \beta_{6} + 8 \beta_{5}) q^{53} + ( - 6 \beta_{4} + 54) q^{55} + ( - 16 \beta_{6} + 20 \beta_1) q^{59} + 13 \beta_{2} q^{61} + ( - 3 \beta_{7} + 3 \beta_{6} - 19 \beta_{5} - 3 \beta_1) q^{65} + ( - 6 \beta_{4} + 6 \beta_{3} + 10 \beta_{2} - 10) q^{67} + ( - 2 \beta_{7} - 16 \beta_{6} - 16 \beta_{5}) q^{71} + (3 \beta_{4} + 56) q^{73} + ( - 32 \beta_{6} + 40 \beta_1) q^{77} + ( - 14 \beta_{3} + 26 \beta_{2}) q^{79} + ( - 8 \beta_{7} + 8 \beta_{6} - 12 \beta_{5} - 8 \beta_1) q^{83} + ( - 45 \beta_{2} + 45) q^{85} + (8 \beta_{7} - 29 \beta_{6} - 29 \beta_{5}) q^{89} + (30 \beta_{4} - 22) q^{91} + ( - 26 \beta_{6} + 6 \beta_1) q^{95} + ( - 16 \beta_{3} - 8 \beta_{2}) q^{97}+O(q^{100})$$ q + b6 * q^5 + (2*b3 + 2*b2) * q^7 + (2*b7 - 2*b6 - 4*b5 + 2*b1) * q^11 + (-b4 + b3 - 16*b2 + 16) * q^13 + (5*b7 - 5*b6 - 5*b5) * q^17 + (2*b4 - 14) * q^19 + (2*b6 + 2*b1) * q^23 + (-3*b3 - 7*b2) * q^25 + (7*b7 - 7*b6 + b5 + 7*b1) * q^29 + (-8*b2 + 8) * q^31 + (-6*b7 - 4*b6 - 4*b5) * q^35 + (8*b4 - 19) * q^37 + (-5*b6 - 3*b1) * q^41 + (-6*b3 - 22*b2) * q^43 + (4*b7 - 4*b6 + 4*b1) * q^47 + (-8*b4 + 8*b3 + 63*b2 - 63) * q^49 + (-17*b7 + 8*b6 + 8*b5) * q^53 + (-6*b4 + 54) * q^55 + (-16*b6 + 20*b1) * q^59 + 13*b2 * q^61 + (-3*b7 + 3*b6 - 19*b5 - 3*b1) * q^65 + (-6*b4 + 6*b3 + 10*b2 - 10) * q^67 + (-2*b7 - 16*b6 - 16*b5) * q^71 + (3*b4 + 56) * q^73 + (-32*b6 + 40*b1) * q^77 + (-14*b3 + 26*b2) * q^79 + (-8*b7 + 8*b6 - 12*b5 - 8*b1) * q^83 + (-45*b2 + 45) * q^85 + (8*b7 - 29*b6 - 29*b5) * q^89 + (30*b4 - 22) * q^91 + (-26*b6 + 6*b1) * q^95 + (-16*b3 - 8*b2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{7}+O(q^{10})$$ 8 * q + 8 * q^7 $$8 q + 8 q^{7} + 64 q^{13} - 112 q^{19} - 28 q^{25} + 32 q^{31} - 152 q^{37} - 88 q^{43} - 252 q^{49} + 432 q^{55} + 52 q^{61} - 40 q^{67} + 448 q^{73} + 104 q^{79} + 180 q^{85} - 176 q^{91} - 32 q^{97}+O(q^{100})$$ 8 * q + 8 * q^7 + 64 * q^13 - 112 * q^19 - 28 * q^25 + 32 * q^31 - 152 * q^37 - 88 * q^43 - 252 * q^49 + 432 * q^55 + 52 * q^61 - 40 * q^67 + 448 * q^73 + 104 * q^79 + 180 * q^85 - 176 * q^91 - 32 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$3\zeta_{24}^{3} + 3\zeta_{24}$$ 3*v^3 + 3*v $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$3\zeta_{24}^{6} + 3\zeta_{24}^{2}$$ 3*v^6 + 3*v^2 $$\beta_{4}$$ $$=$$ $$-3\zeta_{24}^{6} + 6\zeta_{24}^{2}$$ -3*v^6 + 6*v^2 $$\beta_{5}$$ $$=$$ $$3\zeta_{24}^{5} - 3\zeta_{24}^{3}$$ 3*v^5 - 3*v^3 $$\beta_{6}$$ $$=$$ $$3\zeta_{24}^{7} - 3\zeta_{24}^{5} + 3\zeta_{24}$$ 3*v^7 - 3*v^5 + 3*v $$\beta_{7}$$ $$=$$ $$-3\zeta_{24}^{5} - 3\zeta_{24}^{3} + 3\zeta_{24}$$ -3*v^5 - 3*v^3 + 3*v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + \beta_{5} + 2\beta_1 ) / 9$$ (b7 + b5 + 2*b1) / 9 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{4} + \beta_{3} ) / 9$$ (b4 + b3) / 9 $$\zeta_{24}^{3}$$ $$=$$ $$( -\beta_{7} - \beta_{5} + \beta_1 ) / 9$$ (-b7 - b5 + b1) / 9 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{7} + 2\beta_{5} + \beta_1 ) / 9$$ (-b7 + 2*b5 + b1) / 9 $$\zeta_{24}^{6}$$ $$=$$ $$( -\beta_{4} + 2\beta_{3} ) / 9$$ (-b4 + 2*b3) / 9 $$\zeta_{24}^{7}$$ $$=$$ $$( -2\beta_{7} + 3\beta_{6} + \beta_{5} - \beta_1 ) / 9$$ (-2*b7 + 3*b6 + b5 - b1) / 9

## Character values

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

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
593.1
 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 − 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 + 0.965926i
0 0 0 −5.01910 + 2.89778i 0 −4.19615 + 7.26795i 0 0 0
593.2 0 0 0 −1.34486 + 0.776457i 0 6.19615 10.7321i 0 0 0
593.3 0 0 0 1.34486 0.776457i 0 6.19615 10.7321i 0 0 0
593.4 0 0 0 5.01910 2.89778i 0 −4.19615 + 7.26795i 0 0 0
1025.1 0 0 0 −5.01910 2.89778i 0 −4.19615 7.26795i 0 0 0
1025.2 0 0 0 −1.34486 0.776457i 0 6.19615 + 10.7321i 0 0 0
1025.3 0 0 0 1.34486 + 0.776457i 0 6.19615 + 10.7321i 0 0 0
1025.4 0 0 0 5.01910 + 2.89778i 0 −4.19615 7.26795i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1025.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.3.q.o 8
3.b odd 2 1 inner 1296.3.q.o 8
4.b odd 2 1 162.3.d.c 8
9.c even 3 1 1296.3.e.d 4
9.c even 3 1 inner 1296.3.q.o 8
9.d odd 6 1 1296.3.e.d 4
9.d odd 6 1 inner 1296.3.q.o 8
12.b even 2 1 162.3.d.c 8
36.f odd 6 1 162.3.b.b 4
36.f odd 6 1 162.3.d.c 8
36.h even 6 1 162.3.b.b 4
36.h even 6 1 162.3.d.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.3.b.b 4 36.f odd 6 1
162.3.b.b 4 36.h even 6 1
162.3.d.c 8 4.b odd 2 1
162.3.d.c 8 12.b even 2 1
162.3.d.c 8 36.f odd 6 1
162.3.d.c 8 36.h even 6 1
1296.3.e.d 4 9.c even 3 1
1296.3.e.d 4 9.d odd 6 1
1296.3.q.o 8 1.a even 1 1 trivial
1296.3.q.o 8 3.b odd 2 1 inner
1296.3.q.o 8 9.c even 3 1 inner
1296.3.q.o 8 9.d odd 6 1 inner

## Hecke kernels

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

 $$T_{5}^{8} - 36T_{5}^{6} + 1215T_{5}^{4} - 2916T_{5}^{2} + 6561$$ T5^8 - 36*T5^6 + 1215*T5^4 - 2916*T5^2 + 6561 $$T_{7}^{4} - 4T_{7}^{3} + 120T_{7}^{2} + 416T_{7} + 10816$$ T7^4 - 4*T7^3 + 120*T7^2 + 416*T7 + 10816

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 36 T^{6} + 1215 T^{4} + \cdots + 6561$$
$7$ $$(T^{4} - 4 T^{3} + 120 T^{2} + 416 T + 10816)^{2}$$
$11$ $$(T^{4} - 216 T^{2} + 46656)^{2}$$
$13$ $$(T^{4} - 32 T^{3} + 795 T^{2} + \cdots + 52441)^{2}$$
$17$ $$(T^{4} + 900 T^{2} + 50625)^{2}$$
$19$ $$(T^{2} + 28 T + 88)^{4}$$
$23$ $$(T^{4} - 216 T^{2} + 46656)^{2}$$
$29$ $$T^{8} - 2052 T^{6} + \cdots + 996005996001$$
$31$ $$(T^{2} - 8 T + 64)^{4}$$
$37$ $$(T^{2} + 38 T - 1367)^{4}$$
$41$ $$T^{8} - 1764 T^{6} + \cdots + 512249392656$$
$43$ $$(T^{4} + 44 T^{3} + 2424 T^{2} + \cdots + 238144)^{2}$$
$47$ $$(T^{4} - 288 T^{2} + 82944)^{2}$$
$53$ $$(T^{4} + 7812 T^{2} + 4743684)^{2}$$
$59$ $$T^{8} + \cdots + 994737284775936$$
$61$ $$(T^{2} - 13 T + 169)^{4}$$
$67$ $$(T^{4} + 20 T^{3} + 1272 T^{2} + \cdots + 760384)^{2}$$
$71$ $$(T^{4} + 10512 T^{2} + 2742336)^{2}$$
$73$ $$(T^{2} - 112 T + 2893)^{4}$$
$79$ $$(T^{4} - 52 T^{3} + 7320 T^{2} + \cdots + 21307456)^{2}$$
$83$ $$T^{8} - 10944 T^{6} + \cdots + 6295362011136$$
$89$ $$(T^{4} + 24228 T^{2} + \cdots + 112211649)^{2}$$
$97$ $$(T^{4} + 16 T^{3} + 7104 T^{2} + \cdots + 46895104)^{2}$$