# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,3,Mod(593,1296)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1296.593");

S:= CuspForms(chi, 3);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$35.3134422611$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} - 4 \beta_{2} q^{7}+O(q^{10})$$ q + b1 * q^5 - 4*b2 * q^7 $$q + \beta_1 q^{5} - 4 \beta_{2} q^{7} + (4 \beta_{3} - 4 \beta_1) q^{11} + (8 \beta_{2} - 8) q^{13} - 3 \beta_{3} q^{17} + 16 q^{19} - 4 \beta_1 q^{23} - 7 \beta_{2} q^{25} + ( - \beta_{3} + \beta_1) q^{29} + ( - 44 \beta_{2} + 44) q^{31} - 4 \beta_{3} q^{35} - 34 q^{37} - 11 \beta_1 q^{41} - 40 \beta_{2} q^{43} + ( - 20 \beta_{3} + 20 \beta_1) q^{47} + ( - 33 \beta_{2} + 33) q^{49} + 9 \beta_{3} q^{53} - 72 q^{55} + 8 \beta_1 q^{59} - 50 \beta_{2} q^{61} + (8 \beta_{3} - 8 \beta_1) q^{65} + ( - 8 \beta_{2} + 8) q^{67} + 12 \beta_{3} q^{71} - 16 q^{73} + 16 \beta_1 q^{77} - 76 \beta_{2} q^{79} + (28 \beta_{3} - 28 \beta_1) q^{83} + ( - 54 \beta_{2} + 54) q^{85} + 3 \beta_{3} q^{89} + 32 q^{91} + 16 \beta_1 q^{95} - 176 \beta_{2} q^{97}+O(q^{100})$$ q + b1 * q^5 - 4*b2 * q^7 + (4*b3 - 4*b1) * q^11 + (8*b2 - 8) * q^13 - 3*b3 * q^17 + 16 * q^19 - 4*b1 * q^23 - 7*b2 * q^25 + (-b3 + b1) * q^29 + (-44*b2 + 44) * q^31 - 4*b3 * q^35 - 34 * q^37 - 11*b1 * q^41 - 40*b2 * q^43 + (-20*b3 + 20*b1) * q^47 + (-33*b2 + 33) * q^49 + 9*b3 * q^53 - 72 * q^55 + 8*b1 * q^59 - 50*b2 * q^61 + (8*b3 - 8*b1) * q^65 + (-8*b2 + 8) * q^67 + 12*b3 * q^71 - 16 * q^73 + 16*b1 * q^77 - 76*b2 * q^79 + (28*b3 - 28*b1) * q^83 + (-54*b2 + 54) * q^85 + 3*b3 * q^89 + 32 * q^91 + 16*b1 * q^95 - 176*b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{7}+O(q^{10})$$ 4 * q - 8 * q^7 $$4 q - 8 q^{7} - 16 q^{13} + 64 q^{19} - 14 q^{25} + 88 q^{31} - 136 q^{37} - 80 q^{43} + 66 q^{49} - 288 q^{55} - 100 q^{61} + 16 q^{67} - 64 q^{73} - 152 q^{79} + 108 q^{85} + 128 q^{91} - 352 q^{97}+O(q^{100})$$ 4 * q - 8 * q^7 - 16 * q^13 + 64 * q^19 - 14 * q^25 + 88 * q^31 - 136 * q^37 - 80 * q^43 + 66 * q^49 - 288 * q^55 - 100 * q^61 + 16 * q^67 - 64 * q^73 - 152 * q^79 + 108 * q^85 + 128 * q^91 - 352 * q^97

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

 $$\beta_{1}$$ $$=$$ $$3\nu$$ 3*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( 3\nu^{3} ) / 2$$ (3*v^3) / 2
 $$\nu$$ $$=$$ $$( \beta_1 ) / 3$$ (b1) / 3 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{3} ) / 3$$ (2*b3) / 3

## Character values

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

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 - \beta_{2}$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 −3.67423 + 2.12132i 0 −2.00000 + 3.46410i 0 0 0
593.2 0 0 0 3.67423 2.12132i 0 −2.00000 + 3.46410i 0 0 0
1025.1 0 0 0 −3.67423 2.12132i 0 −2.00000 3.46410i 0 0 0
1025.2 0 0 0 3.67423 + 2.12132i 0 −2.00000 3.46410i 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.f 4
3.b odd 2 1 inner 1296.3.q.f 4
4.b odd 2 1 162.3.d.b 4
9.c even 3 1 144.3.e.b 2
9.c even 3 1 inner 1296.3.q.f 4
9.d odd 6 1 144.3.e.b 2
9.d odd 6 1 inner 1296.3.q.f 4
12.b even 2 1 162.3.d.b 4
36.f odd 6 1 18.3.b.a 2
36.f odd 6 1 162.3.d.b 4
36.h even 6 1 18.3.b.a 2
36.h even 6 1 162.3.d.b 4
45.h odd 6 1 3600.3.l.d 2
45.j even 6 1 3600.3.l.d 2
45.k odd 12 2 3600.3.c.b 4
45.l even 12 2 3600.3.c.b 4
72.j odd 6 1 576.3.e.f 2
72.l even 6 1 576.3.e.c 2
72.n even 6 1 576.3.e.f 2
72.p odd 6 1 576.3.e.c 2
144.u even 12 2 2304.3.h.f 4
144.v odd 12 2 2304.3.h.f 4
144.w odd 12 2 2304.3.h.c 4
144.x even 12 2 2304.3.h.c 4
180.n even 6 1 450.3.d.f 2
180.p odd 6 1 450.3.d.f 2
180.v odd 12 2 450.3.b.b 4
180.x even 12 2 450.3.b.b 4
252.n even 6 1 882.3.s.d 4
252.o even 6 1 882.3.s.b 4
252.r odd 6 1 882.3.s.d 4
252.s odd 6 1 882.3.b.a 2
252.u odd 6 1 882.3.s.b 4
252.bb even 6 1 882.3.s.b 4
252.bi even 6 1 882.3.b.a 2
252.bj even 6 1 882.3.s.d 4
252.bl odd 6 1 882.3.s.b 4
252.bn odd 6 1 882.3.s.d 4
396.k even 6 1 2178.3.c.d 2
396.o odd 6 1 2178.3.c.d 2
468.x even 6 1 3042.3.c.e 2
468.bg odd 6 1 3042.3.c.e 2
468.bs even 12 2 3042.3.d.a 4
468.ch odd 12 2 3042.3.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 36.f odd 6 1
18.3.b.a 2 36.h even 6 1
144.3.e.b 2 9.c even 3 1
144.3.e.b 2 9.d odd 6 1
162.3.d.b 4 4.b odd 2 1
162.3.d.b 4 12.b even 2 1
162.3.d.b 4 36.f odd 6 1
162.3.d.b 4 36.h even 6 1
450.3.b.b 4 180.v odd 12 2
450.3.b.b 4 180.x even 12 2
450.3.d.f 2 180.n even 6 1
450.3.d.f 2 180.p odd 6 1
576.3.e.c 2 72.l even 6 1
576.3.e.c 2 72.p odd 6 1
576.3.e.f 2 72.j odd 6 1
576.3.e.f 2 72.n even 6 1
882.3.b.a 2 252.s odd 6 1
882.3.b.a 2 252.bi even 6 1
882.3.s.b 4 252.o even 6 1
882.3.s.b 4 252.u odd 6 1
882.3.s.b 4 252.bb even 6 1
882.3.s.b 4 252.bl odd 6 1
882.3.s.d 4 252.n even 6 1
882.3.s.d 4 252.r odd 6 1
882.3.s.d 4 252.bj even 6 1
882.3.s.d 4 252.bn odd 6 1
1296.3.q.f 4 1.a even 1 1 trivial
1296.3.q.f 4 3.b odd 2 1 inner
1296.3.q.f 4 9.c even 3 1 inner
1296.3.q.f 4 9.d odd 6 1 inner
2178.3.c.d 2 396.k even 6 1
2178.3.c.d 2 396.o odd 6 1
2304.3.h.c 4 144.w odd 12 2
2304.3.h.c 4 144.x even 12 2
2304.3.h.f 4 144.u even 12 2
2304.3.h.f 4 144.v odd 12 2
3042.3.c.e 2 468.x even 6 1
3042.3.c.e 2 468.bg odd 6 1
3042.3.d.a 4 468.bs even 12 2
3042.3.d.a 4 468.ch odd 12 2
3600.3.c.b 4 45.k odd 12 2
3600.3.c.b 4 45.l even 12 2
3600.3.l.d 2 45.h odd 6 1
3600.3.l.d 2 45.j even 6 1

## Hecke kernels

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

 $$T_{5}^{4} - 18T_{5}^{2} + 324$$ T5^4 - 18*T5^2 + 324 $$T_{7}^{2} + 4T_{7} + 16$$ T7^2 + 4*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 18T^{2} + 324$$
$7$ $$(T^{2} + 4 T + 16)^{2}$$
$11$ $$T^{4} - 288 T^{2} + 82944$$
$13$ $$(T^{2} + 8 T + 64)^{2}$$
$17$ $$(T^{2} + 162)^{2}$$
$19$ $$(T - 16)^{4}$$
$23$ $$T^{4} - 288 T^{2} + 82944$$
$29$ $$T^{4} - 18T^{2} + 324$$
$31$ $$(T^{2} - 44 T + 1936)^{2}$$
$37$ $$(T + 34)^{4}$$
$41$ $$T^{4} - 2178 T^{2} + \cdots + 4743684$$
$43$ $$(T^{2} + 40 T + 1600)^{2}$$
$47$ $$T^{4} - 7200 T^{2} + \cdots + 51840000$$
$53$ $$(T^{2} + 1458)^{2}$$
$59$ $$T^{4} - 1152 T^{2} + \cdots + 1327104$$
$61$ $$(T^{2} + 50 T + 2500)^{2}$$
$67$ $$(T^{2} - 8 T + 64)^{2}$$
$71$ $$(T^{2} + 2592)^{2}$$
$73$ $$(T + 16)^{4}$$
$79$ $$(T^{2} + 76 T + 5776)^{2}$$
$83$ $$T^{4} - 14112 T^{2} + \cdots + 199148544$$
$89$ $$(T^{2} + 162)^{2}$$
$97$ $$(T^{2} + 176 T + 30976)^{2}$$