# Properties

 Label 1296.3.e.e 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: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 36) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} + (\beta_{2} - 1) q^{7}+O(q^{10})$$ q + b1 * q^5 + (b2 - 1) * q^7 $$q + \beta_1 q^{5} + (\beta_{2} - 1) q^{7} + ( - \beta_{3} + \beta_1) q^{11} + (\beta_{2} - 3) q^{13} + \beta_{3} q^{17} - \beta_{2} q^{19} + ( - 2 \beta_{3} - \beta_1) q^{23} + (3 \beta_{2} - 8) q^{25} + 7 \beta_1 q^{29} + (3 \beta_{2} - 5) q^{31} + ( - 2 \beta_{3} - 7 \beta_1) q^{35} + ( - 4 \beta_{2} - 14) q^{37} + (\beta_{3} - 7 \beta_1) q^{41} - 23 q^{43} + ( - 2 \beta_{3} + \beta_1) q^{47} + ( - \beta_{2} + 26) q^{49} - 8 \beta_1 q^{53} + ( - 3 \beta_{2} - 21) q^{55} + ( - \beta_{3} - 9 \beta_1) q^{59} + ( - 3 \beta_{2} - 19) q^{61} + ( - 2 \beta_{3} - 9 \beta_1) q^{65} + ( - 6 \beta_{2} + 61) q^{67} + 2 \beta_{3} q^{71} + (3 \beta_{2} + 20) q^{73} + ( - 8 \beta_{3} + 9 \beta_1) q^{77} + (5 \beta_{2} + 39) q^{79} + ( - 2 \beta_{3} + \beta_1) q^{83} + (6 \beta_{2} - 12) q^{85} - 8 \beta_{3} q^{89} + ( - 3 \beta_{2} + 77) q^{91} + (2 \beta_{3} + 6 \beta_1) q^{95} + ( - 2 \beta_{2} + 99) q^{97}+O(q^{100})$$ q + b1 * q^5 + (b2 - 1) * q^7 + (-b3 + b1) * q^11 + (b2 - 3) * q^13 + b3 * q^17 - b2 * q^19 + (-2*b3 - b1) * q^23 + (3*b2 - 8) * q^25 + 7*b1 * q^29 + (3*b2 - 5) * q^31 + (-2*b3 - 7*b1) * q^35 + (-4*b2 - 14) * q^37 + (b3 - 7*b1) * q^41 - 23 * q^43 + (-2*b3 + b1) * q^47 + (-b2 + 26) * q^49 - 8*b1 * q^53 + (-3*b2 - 21) * q^55 + (-b3 - 9*b1) * q^59 + (-3*b2 - 19) * q^61 + (-2*b3 - 9*b1) * q^65 + (-6*b2 + 61) * q^67 + 2*b3 * q^71 + (3*b2 + 20) * q^73 + (-8*b3 + 9*b1) * q^77 + (5*b2 + 39) * q^79 + (-2*b3 + b1) * q^83 + (6*b2 - 12) * q^85 - 8*b3 * q^89 + (-3*b2 + 77) * q^91 + (2*b3 + 6*b1) * q^95 + (-2*b2 + 99) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{7}+O(q^{10})$$ 4 * q - 2 * q^7 $$4 q - 2 q^{7} - 10 q^{13} - 2 q^{19} - 26 q^{25} - 14 q^{31} - 64 q^{37} - 92 q^{43} + 102 q^{49} - 90 q^{55} - 82 q^{61} + 232 q^{67} + 86 q^{73} + 166 q^{79} - 36 q^{85} + 302 q^{91} + 392 q^{97}+O(q^{100})$$ 4 * q - 2 * q^7 - 10 * q^13 - 2 * q^19 - 26 * q^25 - 14 * q^31 - 64 * q^37 - 92 * q^43 + 102 * q^49 - 90 * q^55 - 82 * q^61 + 232 * q^67 + 86 * q^73 + 166 * q^79 - 36 * q^85 + 302 * q^91 + 392 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$-\nu^{3} + \nu^{2} - \nu + 3$$ -v^3 + v^2 - v + 3 $$\beta_{2}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 5\nu + 2$$ -v^3 + v^2 + 5*v + 2 $$\beta_{3}$$ $$=$$ $$4\nu^{3} + 5\nu^{2} - 5\nu - 21$$ 4*v^3 + 5*v^2 - 5*v - 21
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta _1 + 1 ) / 6$$ (b2 - b1 + 1) / 6 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} + 3\beta_{2} + 5\beta _1 + 21 ) / 18$$ (2*b3 + 3*b2 + 5*b1 + 21) / 18 $$\nu^{3}$$ $$=$$ $$( \beta_{3} - 5\beta _1 + 36 ) / 9$$ (b3 - 5*b1 + 36) / 9

## Character values

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

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

## 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
 −1.18614 + 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 − 1.26217i
0 0 0 7.57301i 0 −9.11684 0 0 0
161.2 0 0 0 2.37686i 0 8.11684 0 0 0
161.3 0 0 0 2.37686i 0 8.11684 0 0 0
161.4 0 0 0 7.57301i 0 −9.11684 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.e 4
3.b odd 2 1 inner 1296.3.e.e 4
4.b odd 2 1 324.3.c.b 4
9.c even 3 1 144.3.q.b 4
9.c even 3 1 432.3.q.b 4
9.d odd 6 1 144.3.q.b 4
9.d odd 6 1 432.3.q.b 4
12.b even 2 1 324.3.c.b 4
36.f odd 6 1 36.3.g.a 4
36.f odd 6 1 108.3.g.a 4
36.h even 6 1 36.3.g.a 4
36.h even 6 1 108.3.g.a 4
72.j odd 6 1 576.3.q.g 4
72.j odd 6 1 1728.3.q.h 4
72.l even 6 1 576.3.q.d 4
72.l even 6 1 1728.3.q.g 4
72.n even 6 1 576.3.q.g 4
72.n even 6 1 1728.3.q.h 4
72.p odd 6 1 576.3.q.d 4
72.p odd 6 1 1728.3.q.g 4
180.n even 6 1 900.3.p.a 4
180.n even 6 1 2700.3.p.b 4
180.p odd 6 1 900.3.p.a 4
180.p odd 6 1 2700.3.p.b 4
180.v odd 12 2 900.3.u.a 8
180.v odd 12 2 2700.3.u.b 8
180.x even 12 2 900.3.u.a 8
180.x even 12 2 2700.3.u.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 36.f odd 6 1
36.3.g.a 4 36.h even 6 1
108.3.g.a 4 36.f odd 6 1
108.3.g.a 4 36.h even 6 1
144.3.q.b 4 9.c even 3 1
144.3.q.b 4 9.d odd 6 1
324.3.c.b 4 4.b odd 2 1
324.3.c.b 4 12.b even 2 1
432.3.q.b 4 9.c even 3 1
432.3.q.b 4 9.d odd 6 1
576.3.q.d 4 72.l even 6 1
576.3.q.d 4 72.p odd 6 1
576.3.q.g 4 72.j odd 6 1
576.3.q.g 4 72.n even 6 1
900.3.p.a 4 180.n even 6 1
900.3.p.a 4 180.p odd 6 1
900.3.u.a 8 180.v odd 12 2
900.3.u.a 8 180.x even 12 2
1296.3.e.e 4 1.a even 1 1 trivial
1296.3.e.e 4 3.b odd 2 1 inner
1728.3.q.g 4 72.l even 6 1
1728.3.q.g 4 72.p odd 6 1
1728.3.q.h 4 72.j odd 6 1
1728.3.q.h 4 72.n even 6 1
2700.3.p.b 4 180.n even 6 1
2700.3.p.b 4 180.p odd 6 1
2700.3.u.b 8 180.v odd 12 2
2700.3.u.b 8 180.x even 12 2

## Hecke kernels

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

 $$T_{5}^{4} + 63T_{5}^{2} + 324$$ T5^4 + 63*T5^2 + 324 $$T_{7}^{2} + T_{7} - 74$$ T7^2 + T7 - 74

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 63T^{2} + 324$$
$7$ $$(T^{2} + T - 74)^{2}$$
$11$ $$T^{4} + 414T^{2} + 81$$
$13$ $$(T^{2} + 5 T - 68)^{2}$$
$17$ $$T^{4} + 387 T^{2} + 20736$$
$19$ $$(T^{2} + T - 74)^{2}$$
$23$ $$T^{4} + 1683 T^{2} + 627264$$
$29$ $$T^{4} + 3087 T^{2} + 777924$$
$31$ $$(T^{2} + 7 T - 656)^{2}$$
$37$ $$(T^{2} + 32 T - 932)^{2}$$
$41$ $$T^{4} + 3222 T^{2} + \cdots + 2424249$$
$43$ $$(T + 23)^{4}$$
$47$ $$T^{4} + 1539 T^{2} + 104976$$
$53$ $$T^{4} + 4032 T^{2} + \cdots + 1327104$$
$59$ $$T^{4} + 5814 T^{2} + 68121$$
$61$ $$(T^{2} + 41 T - 248)^{2}$$
$67$ $$(T^{2} - 116 T + 691)^{2}$$
$71$ $$T^{4} + 1548 T^{2} + 331776$$
$73$ $$(T^{2} - 43 T - 206)^{2}$$
$79$ $$(T^{2} - 83 T - 134)^{2}$$
$83$ $$T^{4} + 1539 T^{2} + 104976$$
$89$ $$T^{4} + 24768 T^{2} + \cdots + 84934656$$
$97$ $$(T^{2} - 196 T + 9307)^{2}$$