# Properties

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

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## 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: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 18) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} + (\beta_{2} + 1) q^{7}+O(q^{10})$$ q + b3 * q^5 + (b2 + 1) * q^7 $$q + \beta_{3} q^{5} + (\beta_{2} + 1) q^{7} + ( - \beta_{3} + \beta_1) q^{11} + ( - 2 \beta_{2} + 5) q^{13} + (2 \beta_{3} + 2 \beta_1) q^{17} + (2 \beta_{2} + 10) q^{19} + (\beta_{3} - \beta_1) q^{23} - 2 q^{25} + (\beta_{3} + 2 \beta_1) q^{29} + ( - 3 \beta_{2} + 19) q^{31} + (\beta_{3} + 9 \beta_1) q^{35} + (2 \beta_{2} + 32) q^{37} + (7 \beta_{3} - 6 \beta_1) q^{41} + ( - 3 \beta_{2} - 23) q^{43} + ( - 3 \beta_{3} - 7 \beta_1) q^{47} + (2 \beta_{2} + 6) q^{49} + (10 \beta_{3} + 10 \beta_1) q^{53} + ( - 3 \beta_{2} + 27) q^{55} + (7 \beta_{3} - 13 \beta_1) q^{59} + (6 \beta_{2} - 31) q^{61} + (5 \beta_{3} - 18 \beta_1) q^{65} + ( - 3 \beta_{2} - 53) q^{67} + (10 \beta_{3} - 8 \beta_1) q^{71} + (6 \beta_{2} - 52) q^{73} + (5 \beta_{3} - 8 \beta_1) q^{77} + (5 \beta_{2} + 7) q^{79} + (21 \beta_{3} + 5 \beta_1) q^{83} + ( - 6 \beta_{2} - 54) q^{85} + ( - 10 \beta_{3} - 22 \beta_1) q^{89} + (3 \beta_{2} - 103) q^{91} + (10 \beta_{3} + 18 \beta_1) q^{95} + ( - 14 \beta_{2} - 7) q^{97}+O(q^{100})$$ q + b3 * q^5 + (b2 + 1) * q^7 + (-b3 + b1) * q^11 + (-2*b2 + 5) * q^13 + (2*b3 + 2*b1) * q^17 + (2*b2 + 10) * q^19 + (b3 - b1) * q^23 - 2 * q^25 + (b3 + 2*b1) * q^29 + (-3*b2 + 19) * q^31 + (b3 + 9*b1) * q^35 + (2*b2 + 32) * q^37 + (7*b3 - 6*b1) * q^41 + (-3*b2 - 23) * q^43 + (-3*b3 - 7*b1) * q^47 + (2*b2 + 6) * q^49 + (10*b3 + 10*b1) * q^53 + (-3*b2 + 27) * q^55 + (7*b3 - 13*b1) * q^59 + (6*b2 - 31) * q^61 + (5*b3 - 18*b1) * q^65 + (-3*b2 - 53) * q^67 + (10*b3 - 8*b1) * q^71 + (6*b2 - 52) * q^73 + (5*b3 - 8*b1) * q^77 + (5*b2 + 7) * q^79 + (21*b3 + 5*b1) * q^83 + (-6*b2 - 54) * q^85 + (-10*b3 - 22*b1) * q^89 + (3*b2 - 103) * q^91 + (10*b3 + 18*b1) * q^95 + (-14*b2 - 7) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{7}+O(q^{10})$$ 4 * q + 4 * q^7 $$4 q + 4 q^{7} + 20 q^{13} + 40 q^{19} - 8 q^{25} + 76 q^{31} + 128 q^{37} - 92 q^{43} + 24 q^{49} + 108 q^{55} - 124 q^{61} - 212 q^{67} - 208 q^{73} + 28 q^{79} - 216 q^{85} - 412 q^{91} - 28 q^{97}+O(q^{100})$$ 4 * q + 4 * q^7 + 20 * q^13 + 40 * q^19 - 8 * q^25 + 76 * q^31 + 128 * q^37 - 92 * q^43 + 24 * q^49 + 108 * q^55 - 124 * q^61 - 212 * q^67 - 208 * q^73 + 28 * q^79 - 216 * q^85 - 412 * q^91 - 28 * q^97

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

 $$\beta_{1}$$ $$=$$ $$( 3\nu^{3} ) / 2$$ (3*v^3) / 2 $$\beta_{2}$$ $$=$$ $$( -3\nu^{3} + 12\nu ) / 2$$ (-3*v^3 + 12*v) / 2 $$\beta_{3}$$ $$=$$ $$3\nu^{2} - 3$$ 3*v^2 - 3
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 6$$ (b2 + b1) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 3 ) / 3$$ (b3 + 3) / 3 $$\nu^{3}$$ $$=$$ $$( 2\beta_1 ) / 3$$ (2*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
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
0 0 0 5.19615i 0 −6.34847 0 0 0
161.2 0 0 0 5.19615i 0 8.34847 0 0 0
161.3 0 0 0 5.19615i 0 −6.34847 0 0 0
161.4 0 0 0 5.19615i 0 8.34847 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.g 4
3.b odd 2 1 inner 1296.3.e.g 4
4.b odd 2 1 162.3.b.a 4
9.c even 3 1 144.3.q.c 4
9.c even 3 1 432.3.q.d 4
9.d odd 6 1 144.3.q.c 4
9.d odd 6 1 432.3.q.d 4
12.b even 2 1 162.3.b.a 4
36.f odd 6 1 18.3.d.a 4
36.f odd 6 1 54.3.d.a 4
36.h even 6 1 18.3.d.a 4
36.h even 6 1 54.3.d.a 4
72.j odd 6 1 576.3.q.e 4
72.j odd 6 1 1728.3.q.c 4
72.l even 6 1 576.3.q.f 4
72.l even 6 1 1728.3.q.d 4
72.n even 6 1 576.3.q.e 4
72.n even 6 1 1728.3.q.c 4
72.p odd 6 1 576.3.q.f 4
72.p odd 6 1 1728.3.q.d 4
180.n even 6 1 450.3.i.b 4
180.n even 6 1 1350.3.i.b 4
180.p odd 6 1 450.3.i.b 4
180.p odd 6 1 1350.3.i.b 4
180.v odd 12 2 450.3.k.a 8
180.v odd 12 2 1350.3.k.a 8
180.x even 12 2 450.3.k.a 8
180.x even 12 2 1350.3.k.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 36.f odd 6 1
18.3.d.a 4 36.h even 6 1
54.3.d.a 4 36.f odd 6 1
54.3.d.a 4 36.h even 6 1
144.3.q.c 4 9.c even 3 1
144.3.q.c 4 9.d odd 6 1
162.3.b.a 4 4.b odd 2 1
162.3.b.a 4 12.b even 2 1
432.3.q.d 4 9.c even 3 1
432.3.q.d 4 9.d odd 6 1
450.3.i.b 4 180.n even 6 1
450.3.i.b 4 180.p odd 6 1
450.3.k.a 8 180.v odd 12 2
450.3.k.a 8 180.x even 12 2
576.3.q.e 4 72.j odd 6 1
576.3.q.e 4 72.n even 6 1
576.3.q.f 4 72.l even 6 1
576.3.q.f 4 72.p odd 6 1
1296.3.e.g 4 1.a even 1 1 trivial
1296.3.e.g 4 3.b odd 2 1 inner
1350.3.i.b 4 180.n even 6 1
1350.3.i.b 4 180.p odd 6 1
1350.3.k.a 8 180.v odd 12 2
1350.3.k.a 8 180.x even 12 2
1728.3.q.c 4 72.j odd 6 1
1728.3.q.c 4 72.n even 6 1
1728.3.q.d 4 72.l even 6 1
1728.3.q.d 4 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 $$S_{3}^{\mathrm{new}}(1296, [\chi])$$:

 $$T_{5}^{2} + 27$$ T5^2 + 27 $$T_{7}^{2} - 2T_{7} - 53$$ T7^2 - 2*T7 - 53

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 27)^{2}$$
$7$ $$(T^{2} - 2 T - 53)^{2}$$
$11$ $$T^{4} + 90T^{2} + 81$$
$13$ $$(T^{2} - 10 T - 191)^{2}$$
$17$ $$T^{4} + 360T^{2} + 1296$$
$19$ $$(T^{2} - 20 T - 116)^{2}$$
$23$ $$T^{4} + 90T^{2} + 81$$
$29$ $$T^{4} + 198T^{2} + 2025$$
$31$ $$(T^{2} - 38 T - 125)^{2}$$
$37$ $$(T^{2} - 64 T + 808)^{2}$$
$41$ $$T^{4} + 3942 T^{2} + 455625$$
$43$ $$(T^{2} + 46 T + 43)^{2}$$
$47$ $$T^{4} + 2250 T^{2} + 408321$$
$53$ $$T^{4} + 9000 T^{2} + 810000$$
$59$ $$T^{4} + 8730 T^{2} + \cdots + 2954961$$
$61$ $$(T^{2} + 62 T - 983)^{2}$$
$67$ $$(T^{2} + 106 T + 2323)^{2}$$
$71$ $$T^{4} + 7704 T^{2} + \cdots + 2396304$$
$73$ $$(T^{2} + 104 T + 760)^{2}$$
$79$ $$(T^{2} - 14 T - 1301)^{2}$$
$83$ $$T^{4} + 24714 T^{2} + \cdots + 131262849$$
$89$ $$T^{4} + 22824 T^{2} + \cdots + 36144144$$
$97$ $$(T^{2} + 14 T - 10535)^{2}$$
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