# Properties

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

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## 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} -6 \beta_{2} q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} -6 \beta_{2} q^{7} + ( \beta_{1} - \beta_{3} ) q^{11} + ( -10 + 10 \beta_{2} ) q^{13} + 4 \beta_{3} q^{17} -2 q^{19} + 2 \beta_{1} q^{23} + 7 \beta_{2} q^{25} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{29} + ( -22 + 22 \beta_{2} ) q^{31} -6 \beta_{3} q^{35} -6 q^{37} + 6 \beta_{1} q^{41} + 82 \beta_{2} q^{43} + ( 12 \beta_{1} - 12 \beta_{3} ) q^{47} + ( 13 - 13 \beta_{2} ) q^{49} + 11 \beta_{3} q^{53} + 32 q^{55} -13 \beta_{1} q^{59} + 86 \beta_{2} q^{61} + ( -10 \beta_{1} + 10 \beta_{3} ) q^{65} + ( 2 - 2 \beta_{2} ) q^{67} -22 \beta_{3} q^{71} + 82 q^{73} -6 \beta_{1} q^{77} + 10 \beta_{2} q^{79} + ( -13 \beta_{1} + 13 \beta_{3} ) q^{83} + ( -128 + 128 \beta_{2} ) q^{85} + 6 \beta_{3} q^{89} + 60 q^{91} -2 \beta_{1} q^{95} + 94 \beta_{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{7} + O(q^{10})$$ $$4q - 12q^{7} - 20q^{13} - 8q^{19} + 14q^{25} - 44q^{31} - 24q^{37} + 164q^{43} + 26q^{49} + 128q^{55} + 172q^{61} + 4q^{67} + 328q^{73} + 20q^{79} - 256q^{85} + 240q^{91} + 188q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{3}$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/4$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}$$$$/2$$

## 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
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
0 0 0 −4.89898 + 2.82843i 0 −3.00000 + 5.19615i 0 0 0
593.2 0 0 0 4.89898 2.82843i 0 −3.00000 + 5.19615i 0 0 0
1025.1 0 0 0 −4.89898 2.82843i 0 −3.00000 5.19615i 0 0 0
1025.2 0 0 0 4.89898 + 2.82843i 0 −3.00000 5.19615i 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
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.e 4
3.b odd 2 1 inner 1296.3.q.e 4
4.b odd 2 1 648.3.m.d 4
9.c even 3 1 48.3.e.b 2
9.c even 3 1 inner 1296.3.q.e 4
9.d odd 6 1 48.3.e.b 2
9.d odd 6 1 inner 1296.3.q.e 4
12.b even 2 1 648.3.m.d 4
36.f odd 6 1 24.3.e.a 2
36.f odd 6 1 648.3.m.d 4
36.h even 6 1 24.3.e.a 2
36.h even 6 1 648.3.m.d 4
45.h odd 6 1 1200.3.l.n 2
45.j even 6 1 1200.3.l.n 2
45.k odd 12 2 1200.3.c.i 4
45.l even 12 2 1200.3.c.i 4
72.j odd 6 1 192.3.e.d 2
72.l even 6 1 192.3.e.c 2
72.n even 6 1 192.3.e.d 2
72.p odd 6 1 192.3.e.c 2
144.u even 12 2 768.3.h.d 4
144.v odd 12 2 768.3.h.d 4
144.w odd 12 2 768.3.h.c 4
144.x even 12 2 768.3.h.c 4
180.n even 6 1 600.3.l.b 2
180.p odd 6 1 600.3.l.b 2
180.v odd 12 2 600.3.c.a 4
180.x even 12 2 600.3.c.a 4
252.s odd 6 1 1176.3.d.a 2
252.bi even 6 1 1176.3.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.e.a 2 36.f odd 6 1
24.3.e.a 2 36.h even 6 1
48.3.e.b 2 9.c even 3 1
48.3.e.b 2 9.d odd 6 1
192.3.e.c 2 72.l even 6 1
192.3.e.c 2 72.p odd 6 1
192.3.e.d 2 72.j odd 6 1
192.3.e.d 2 72.n even 6 1
600.3.c.a 4 180.v odd 12 2
600.3.c.a 4 180.x even 12 2
600.3.l.b 2 180.n even 6 1
600.3.l.b 2 180.p odd 6 1
648.3.m.d 4 4.b odd 2 1
648.3.m.d 4 12.b even 2 1
648.3.m.d 4 36.f odd 6 1
648.3.m.d 4 36.h even 6 1
768.3.h.c 4 144.w odd 12 2
768.3.h.c 4 144.x even 12 2
768.3.h.d 4 144.u even 12 2
768.3.h.d 4 144.v odd 12 2
1176.3.d.a 2 252.s odd 6 1
1176.3.d.a 2 252.bi even 6 1
1200.3.c.i 4 45.k odd 12 2
1200.3.c.i 4 45.l even 12 2
1200.3.l.n 2 45.h odd 6 1
1200.3.l.n 2 45.j even 6 1
1296.3.q.e 4 1.a even 1 1 trivial
1296.3.q.e 4 3.b odd 2 1 inner
1296.3.q.e 4 9.c even 3 1 inner
1296.3.q.e 4 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 $$S_{3}^{\mathrm{new}}(1296, [\chi])$$:

 $$T_{5}^{4} - 32 T_{5}^{2} + 1024$$ $$T_{7}^{2} + 6 T_{7} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$1024 - 32 T^{2} + T^{4}$$
$7$ $$( 36 + 6 T + T^{2} )^{2}$$
$11$ $$1024 - 32 T^{2} + T^{4}$$
$13$ $$( 100 + 10 T + T^{2} )^{2}$$
$17$ $$( 512 + T^{2} )^{2}$$
$19$ $$( 2 + T )^{4}$$
$23$ $$16384 - 128 T^{2} + T^{4}$$
$29$ $$82944 - 288 T^{2} + T^{4}$$
$31$ $$( 484 + 22 T + T^{2} )^{2}$$
$37$ $$( 6 + T )^{4}$$
$41$ $$1327104 - 1152 T^{2} + T^{4}$$
$43$ $$( 6724 - 82 T + T^{2} )^{2}$$
$47$ $$21233664 - 4608 T^{2} + T^{4}$$
$53$ $$( 3872 + T^{2} )^{2}$$
$59$ $$29246464 - 5408 T^{2} + T^{4}$$
$61$ $$( 7396 - 86 T + T^{2} )^{2}$$
$67$ $$( 4 - 2 T + T^{2} )^{2}$$
$71$ $$( 15488 + T^{2} )^{2}$$
$73$ $$( -82 + T )^{4}$$
$79$ $$( 100 - 10 T + T^{2} )^{2}$$
$83$ $$29246464 - 5408 T^{2} + T^{4}$$
$89$ $$( 1152 + T^{2} )^{2}$$
$97$ $$( 8836 - 94 T + T^{2} )^{2}$$
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