# Properties

 Label 1296.3.e.d Level $1296$ Weight $3$ Character orbit 1296.e Analytic conductor $35.313$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1296.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$35.3134422611$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: no (minimal twist has level 162) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} + ( - 2 \beta_{3} - 2) q^{7}+O(q^{10})$$ q + b1 * q^5 + (-2*b3 - 2) * q^7 $$q + \beta_1 q^{5} + ( - 2 \beta_{3} - 2) q^{7} + ( - 2 \beta_{2} - 2 \beta_1) q^{11} + (\beta_{3} - 16) q^{13} + 5 \beta_{2} q^{17} + (2 \beta_{3} - 14) q^{19} + (2 \beta_{2} + 2 \beta_1) q^{23} + (3 \beta_{3} + 7) q^{25} + ( - 7 \beta_{2} + 8 \beta_1) q^{29} - 8 q^{31} + ( - 6 \beta_{2} + 10 \beta_1) q^{35} + (8 \beta_{3} - 19) q^{37} + ( - 3 \beta_{2} - 5 \beta_1) q^{41} + (6 \beta_{3} + 22) q^{43} + ( - 4 \beta_{2} + 4 \beta_1) q^{47} + (8 \beta_{3} + 63) q^{49} + ( - 17 \beta_{2} + 9 \beta_1) q^{53} + ( - 6 \beta_{3} + 54) q^{55} + (20 \beta_{2} - 16 \beta_1) q^{59} - 13 q^{61} + (3 \beta_{2} - 22 \beta_1) q^{65} + (6 \beta_{3} + 10) q^{67} + ( - 2 \beta_{2} + 18 \beta_1) q^{71} + (3 \beta_{3} + 56) q^{73} + (40 \beta_{2} - 32 \beta_1) q^{77} + (14 \beta_{3} - 26) q^{79} + (8 \beta_{2} - 20 \beta_1) q^{83} - 45 q^{85} + (8 \beta_{2} + 21 \beta_1) q^{89} + (30 \beta_{3} - 22) q^{91} + (6 \beta_{2} - 26 \beta_1) q^{95} + (16 \beta_{3} + 8) q^{97}+O(q^{100})$$ q + b1 * q^5 + (-2*b3 - 2) * q^7 + (-2*b2 - 2*b1) * q^11 + (b3 - 16) * q^13 + 5*b2 * q^17 + (2*b3 - 14) * q^19 + (2*b2 + 2*b1) * q^23 + (3*b3 + 7) * q^25 + (-7*b2 + 8*b1) * q^29 - 8 * q^31 + (-6*b2 + 10*b1) * q^35 + (8*b3 - 19) * q^37 + (-3*b2 - 5*b1) * q^41 + (6*b3 + 22) * q^43 + (-4*b2 + 4*b1) * q^47 + (8*b3 + 63) * q^49 + (-17*b2 + 9*b1) * q^53 + (-6*b3 + 54) * q^55 + (20*b2 - 16*b1) * q^59 - 13 * q^61 + (3*b2 - 22*b1) * q^65 + (6*b3 + 10) * q^67 + (-2*b2 + 18*b1) * q^71 + (3*b3 + 56) * q^73 + (40*b2 - 32*b1) * q^77 + (14*b3 - 26) * q^79 + (8*b2 - 20*b1) * q^83 - 45 * q^85 + (8*b2 + 21*b1) * q^89 + (30*b3 - 22) * q^91 + (6*b2 - 26*b1) * q^95 + (16*b3 + 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{7}+O(q^{10})$$ 4 * q - 8 * q^7 $$4 q - 8 q^{7} - 64 q^{13} - 56 q^{19} + 28 q^{25} - 32 q^{31} - 76 q^{37} + 88 q^{43} + 252 q^{49} + 216 q^{55} - 52 q^{61} + 40 q^{67} + 224 q^{73} - 104 q^{79} - 180 q^{85} - 88 q^{91} + 32 q^{97}+O(q^{100})$$ 4 * q - 8 * q^7 - 64 * q^13 - 56 * q^19 + 28 * q^25 - 32 * q^31 - 76 * q^37 + 88 * q^43 + 252 * q^49 + 216 * q^55 - 52 * q^61 + 40 * q^67 + 224 * q^73 - 104 * q^79 - 180 * q^85 - 88 * q^91 + 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$3\nu$$ 3*v $$\beta_{2}$$ $$=$$ $$3\nu^{3} + 12\nu$$ 3*v^3 + 12*v $$\beta_{3}$$ $$=$$ $$3\nu^{2} + 6$$ 3*v^2 + 6
 $$\nu$$ $$=$$ $$( \beta_1 ) / 3$$ (b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 6 ) / 3$$ (b3 - 6) / 3 $$\nu^{3}$$ $$=$$ $$( \beta_{2} - 4\beta_1 ) / 3$$ (b2 - 4*b1) / 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 − 1.93185i − 0.517638i 0.517638i 1.93185i
0 0 0 5.79555i 0 8.39230 0 0 0
161.2 0 0 0 1.55291i 0 −12.3923 0 0 0
161.3 0 0 0 1.55291i 0 −12.3923 0 0 0
161.4 0 0 0 5.79555i 0 8.39230 0 0 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
3.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.e.d 4
3.b odd 2 1 inner 1296.3.e.d 4
4.b odd 2 1 162.3.b.b 4
9.c even 3 2 1296.3.q.o 8
9.d odd 6 2 1296.3.q.o 8
12.b even 2 1 162.3.b.b 4
36.f odd 6 2 162.3.d.c 8
36.h even 6 2 162.3.d.c 8

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

## Hecke kernels

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

 $$T_{5}^{4} + 36T_{5}^{2} + 81$$ T5^4 + 36*T5^2 + 81 $$T_{7}^{2} + 4T_{7} - 104$$ T7^2 + 4*T7 - 104

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 36T^{2} + 81$$
$7$ $$(T^{2} + 4 T - 104)^{2}$$
$11$ $$(T^{2} + 216)^{2}$$
$13$ $$(T^{2} + 32 T + 229)^{2}$$
$17$ $$T^{4} + 900 T^{2} + 50625$$
$19$ $$(T^{2} + 28 T + 88)^{2}$$
$23$ $$(T^{2} + 216)^{2}$$
$29$ $$T^{4} + 2052 T^{2} + 998001$$
$31$ $$(T + 8)^{4}$$
$37$ $$(T^{2} + 38 T - 1367)^{2}$$
$41$ $$T^{4} + 1764 T^{2} + 715716$$
$43$ $$(T^{2} - 44 T - 488)^{2}$$
$47$ $$(T^{2} + 288)^{2}$$
$53$ $$T^{4} + 7812 T^{2} + \cdots + 4743684$$
$59$ $$T^{4} + 12096 T^{2} + \cdots + 31539456$$
$61$ $$(T + 13)^{4}$$
$67$ $$(T^{2} - 20 T - 872)^{2}$$
$71$ $$T^{4} + 10512 T^{2} + \cdots + 2742336$$
$73$ $$(T^{2} - 112 T + 2893)^{2}$$
$79$ $$(T^{2} + 52 T - 4616)^{2}$$
$83$ $$T^{4} + 10944 T^{2} + \cdots + 2509056$$
$89$ $$T^{4} + 24228 T^{2} + \cdots + 112211649$$
$97$ $$(T^{2} - 16 T - 6848)^{2}$$