# Properties

 Label 1296.2.i.e Level $1296$ Weight $2$ Character orbit 1296.i Analytic conductor $10.349$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.3486121020$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{5} +O(q^{10})$$ q - 2*z * q^5 $$q - 2 \zeta_{6} q^{5} + (4 \zeta_{6} - 4) q^{11} + 2 \zeta_{6} q^{13} - 2 q^{17} + 4 q^{19} + 8 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} + ( - 6 \zeta_{6} + 6) q^{29} + 8 \zeta_{6} q^{31} + 6 q^{37} - 6 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} + 7 \zeta_{6} q^{49} + 2 q^{53} + 8 q^{55} - 4 \zeta_{6} q^{59} + ( - 2 \zeta_{6} + 2) q^{61} + ( - 4 \zeta_{6} + 4) q^{65} - 4 \zeta_{6} q^{67} + 8 q^{71} + 10 q^{73} + (8 \zeta_{6} - 8) q^{79} + ( - 4 \zeta_{6} + 4) q^{83} + 4 \zeta_{6} q^{85} + 6 q^{89} - 8 \zeta_{6} q^{95} + (2 \zeta_{6} - 2) q^{97} +O(q^{100})$$ q - 2*z * q^5 + (4*z - 4) * q^11 + 2*z * q^13 - 2 * q^17 + 4 * q^19 + 8*z * q^23 + (-z + 1) * q^25 + (-6*z + 6) * q^29 + 8*z * q^31 + 6 * q^37 - 6*z * q^41 + (-4*z + 4) * q^43 + 7*z * q^49 + 2 * q^53 + 8 * q^55 - 4*z * q^59 + (-2*z + 2) * q^61 + (-4*z + 4) * q^65 - 4*z * q^67 + 8 * q^71 + 10 * q^73 + (8*z - 8) * q^79 + (-4*z + 4) * q^83 + 4*z * q^85 + 6 * q^89 - 8*z * q^95 + (2*z - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^5 $$2 q - 2 q^{5} - 4 q^{11} + 2 q^{13} - 4 q^{17} + 8 q^{19} + 8 q^{23} + q^{25} + 6 q^{29} + 8 q^{31} + 12 q^{37} - 6 q^{41} + 4 q^{43} + 7 q^{49} + 4 q^{53} + 16 q^{55} - 4 q^{59} + 2 q^{61} + 4 q^{65} - 4 q^{67} + 16 q^{71} + 20 q^{73} - 8 q^{79} + 4 q^{83} + 4 q^{85} + 12 q^{89} - 8 q^{95} - 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 4 * q^11 + 2 * q^13 - 4 * q^17 + 8 * q^19 + 8 * q^23 + q^25 + 6 * q^29 + 8 * q^31 + 12 * q^37 - 6 * q^41 + 4 * q^43 + 7 * q^49 + 4 * q^53 + 16 * q^55 - 4 * q^59 + 2 * q^61 + 4 * q^65 - 4 * q^67 + 16 * q^71 + 20 * q^73 - 8 * q^79 + 4 * q^83 + 4 * q^85 + 12 * q^89 - 8 * q^95 - 2 * q^97

## Character values

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

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
433.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −1.00000 + 1.73205i 0 0 0 0 0
865.1 0 0 0 −1.00000 1.73205i 0 0 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.i.e 2
3.b odd 2 1 1296.2.i.m 2
4.b odd 2 1 648.2.i.b 2
9.c even 3 1 144.2.a.b 1
9.c even 3 1 inner 1296.2.i.e 2
9.d odd 6 1 48.2.a.a 1
9.d odd 6 1 1296.2.i.m 2
12.b even 2 1 648.2.i.g 2
36.f odd 6 1 72.2.a.a 1
36.f odd 6 1 648.2.i.b 2
36.h even 6 1 24.2.a.a 1
36.h even 6 1 648.2.i.g 2
45.h odd 6 1 1200.2.a.d 1
45.j even 6 1 3600.2.a.v 1
45.k odd 12 2 3600.2.f.r 2
45.l even 12 2 1200.2.f.b 2
63.i even 6 1 2352.2.q.r 2
63.j odd 6 1 2352.2.q.l 2
63.l odd 6 1 7056.2.a.q 1
63.n odd 6 1 2352.2.q.l 2
63.o even 6 1 2352.2.a.i 1
63.s even 6 1 2352.2.q.r 2
72.j odd 6 1 192.2.a.b 1
72.l even 6 1 192.2.a.d 1
72.n even 6 1 576.2.a.b 1
72.p odd 6 1 576.2.a.d 1
99.g even 6 1 5808.2.a.s 1
117.n odd 6 1 8112.2.a.be 1
144.u even 12 2 768.2.d.e 2
144.v odd 12 2 2304.2.d.i 2
144.w odd 12 2 768.2.d.d 2
144.x even 12 2 2304.2.d.k 2
180.n even 6 1 600.2.a.h 1
180.p odd 6 1 1800.2.a.m 1
180.v odd 12 2 600.2.f.e 2
180.x even 12 2 1800.2.f.c 2
252.n even 6 1 3528.2.s.y 2
252.o even 6 1 1176.2.q.i 2
252.r odd 6 1 1176.2.q.a 2
252.s odd 6 1 1176.2.a.i 1
252.u odd 6 1 3528.2.s.j 2
252.bb even 6 1 1176.2.q.i 2
252.bi even 6 1 3528.2.a.d 1
252.bj even 6 1 3528.2.s.y 2
252.bl odd 6 1 3528.2.s.j 2
252.bn odd 6 1 1176.2.q.a 2
360.bd even 6 1 4800.2.a.q 1
360.bh odd 6 1 4800.2.a.cc 1
360.br even 12 2 4800.2.f.bg 2
360.bt odd 12 2 4800.2.f.d 2
396.k even 6 1 8712.2.a.u 1
396.o odd 6 1 2904.2.a.c 1
468.x even 6 1 4056.2.a.i 1
468.ch odd 12 2 4056.2.c.e 2
504.cc even 6 1 9408.2.a.cc 1
504.co odd 6 1 9408.2.a.h 1
612.n even 6 1 6936.2.a.p 1
684.bh odd 6 1 8664.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 36.h even 6 1
48.2.a.a 1 9.d odd 6 1
72.2.a.a 1 36.f odd 6 1
144.2.a.b 1 9.c even 3 1
192.2.a.b 1 72.j odd 6 1
192.2.a.d 1 72.l even 6 1
576.2.a.b 1 72.n even 6 1
576.2.a.d 1 72.p odd 6 1
600.2.a.h 1 180.n even 6 1
600.2.f.e 2 180.v odd 12 2
648.2.i.b 2 4.b odd 2 1
648.2.i.b 2 36.f odd 6 1
648.2.i.g 2 12.b even 2 1
648.2.i.g 2 36.h even 6 1
768.2.d.d 2 144.w odd 12 2
768.2.d.e 2 144.u even 12 2
1176.2.a.i 1 252.s odd 6 1
1176.2.q.a 2 252.r odd 6 1
1176.2.q.a 2 252.bn odd 6 1
1176.2.q.i 2 252.o even 6 1
1176.2.q.i 2 252.bb even 6 1
1200.2.a.d 1 45.h odd 6 1
1200.2.f.b 2 45.l even 12 2
1296.2.i.e 2 1.a even 1 1 trivial
1296.2.i.e 2 9.c even 3 1 inner
1296.2.i.m 2 3.b odd 2 1
1296.2.i.m 2 9.d odd 6 1
1800.2.a.m 1 180.p odd 6 1
1800.2.f.c 2 180.x even 12 2
2304.2.d.i 2 144.v odd 12 2
2304.2.d.k 2 144.x even 12 2
2352.2.a.i 1 63.o even 6 1
2352.2.q.l 2 63.j odd 6 1
2352.2.q.l 2 63.n odd 6 1
2352.2.q.r 2 63.i even 6 1
2352.2.q.r 2 63.s even 6 1
2904.2.a.c 1 396.o odd 6 1
3528.2.a.d 1 252.bi even 6 1
3528.2.s.j 2 252.u odd 6 1
3528.2.s.j 2 252.bl odd 6 1
3528.2.s.y 2 252.n even 6 1
3528.2.s.y 2 252.bj even 6 1
3600.2.a.v 1 45.j even 6 1
3600.2.f.r 2 45.k odd 12 2
4056.2.a.i 1 468.x even 6 1
4056.2.c.e 2 468.ch odd 12 2
4800.2.a.q 1 360.bd even 6 1
4800.2.a.cc 1 360.bh odd 6 1
4800.2.f.d 2 360.bt odd 12 2
4800.2.f.bg 2 360.br even 12 2
5808.2.a.s 1 99.g even 6 1
6936.2.a.p 1 612.n even 6 1
7056.2.a.q 1 63.l odd 6 1
8112.2.a.be 1 117.n odd 6 1
8664.2.a.j 1 684.bh odd 6 1
8712.2.a.u 1 396.k even 6 1
9408.2.a.h 1 504.co odd 6 1
9408.2.a.cc 1 504.cc even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1296, [\chi])$$:

 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$(T + 2)^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} - 8T + 64$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$T^{2} - 8T + 64$$
$37$ $$(T - 6)^{2}$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2}$$
$53$ $$(T - 2)^{2}$$
$59$ $$T^{2} + 4T + 16$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$(T - 8)^{2}$$
$73$ $$(T - 10)^{2}$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$T^{2} - 4T + 16$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 2T + 4$$