# Properties

 Label 784.2.i.e Level $784$ Weight $2$ Character orbit 784.i Analytic conductor $6.260$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,2,Mod(177,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("784.177");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.i (of order $$3$$, 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: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} +O(q^{10})$$ q - 2*z * q^5 + 3*z * q^9 $$q - 2 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} + (4 \zeta_{6} - 4) q^{11} + 2 q^{13} + ( - 6 \zeta_{6} + 6) q^{17} + 8 \zeta_{6} q^{19} + ( - \zeta_{6} + 1) q^{25} + 6 q^{29} + ( - 8 \zeta_{6} + 8) q^{31} + 2 \zeta_{6} q^{37} + 2 q^{41} + 4 q^{43} + ( - 6 \zeta_{6} + 6) q^{45} - 8 \zeta_{6} q^{47} + (6 \zeta_{6} - 6) q^{53} + 8 q^{55} + 6 \zeta_{6} q^{61} - 4 \zeta_{6} q^{65} + (4 \zeta_{6} - 4) q^{67} + 8 q^{71} + (10 \zeta_{6} - 10) q^{73} + 16 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} - 8 q^{83} - 12 q^{85} + 6 \zeta_{6} q^{89} + ( - 16 \zeta_{6} + 16) q^{95} - 6 q^{97} - 12 q^{99} +O(q^{100})$$ q - 2*z * q^5 + 3*z * q^9 + (4*z - 4) * q^11 + 2 * q^13 + (-6*z + 6) * q^17 + 8*z * q^19 + (-z + 1) * q^25 + 6 * q^29 + (-8*z + 8) * q^31 + 2*z * q^37 + 2 * q^41 + 4 * q^43 + (-6*z + 6) * q^45 - 8*z * q^47 + (6*z - 6) * q^53 + 8 * q^55 + 6*z * q^61 - 4*z * q^65 + (4*z - 4) * q^67 + 8 * q^71 + (10*z - 10) * q^73 + 16*z * q^79 + (9*z - 9) * q^81 - 8 * q^83 - 12 * q^85 + 6*z * q^89 + (-16*z + 16) * q^95 - 6 * q^97 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 3 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 + 3 * q^9 $$2 q - 2 q^{5} + 3 q^{9} - 4 q^{11} + 4 q^{13} + 6 q^{17} + 8 q^{19} + q^{25} + 12 q^{29} + 8 q^{31} + 2 q^{37} + 4 q^{41} + 8 q^{43} + 6 q^{45} - 8 q^{47} - 6 q^{53} + 16 q^{55} + 6 q^{61} - 4 q^{65} - 4 q^{67} + 16 q^{71} - 10 q^{73} + 16 q^{79} - 9 q^{81} - 16 q^{83} - 24 q^{85} + 6 q^{89} + 16 q^{95} - 12 q^{97} - 24 q^{99}+O(q^{100})$$ 2 * q - 2 * q^5 + 3 * q^9 - 4 * q^11 + 4 * q^13 + 6 * q^17 + 8 * q^19 + q^25 + 12 * q^29 + 8 * q^31 + 2 * q^37 + 4 * q^41 + 8 * q^43 + 6 * q^45 - 8 * q^47 - 6 * q^53 + 16 * q^55 + 6 * q^61 - 4 * q^65 - 4 * q^67 + 16 * q^71 - 10 * q^73 + 16 * q^79 - 9 * q^81 - 16 * q^83 - 24 * q^85 + 6 * q^89 + 16 * q^95 - 12 * q^97 - 24 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$687$$ $$689$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −1.00000 + 1.73205i 0 0 0 1.50000 2.59808i 0
753.1 0 0 0 −1.00000 1.73205i 0 0 0 1.50000 + 2.59808i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.i.e 2
4.b odd 2 1 392.2.i.c 2
7.b odd 2 1 784.2.i.g 2
7.c even 3 1 112.2.a.b 1
7.c even 3 1 inner 784.2.i.e 2
7.d odd 6 1 784.2.a.e 1
7.d odd 6 1 784.2.i.g 2
12.b even 2 1 3528.2.s.t 2
21.g even 6 1 7056.2.a.bo 1
21.h odd 6 1 1008.2.a.d 1
28.d even 2 1 392.2.i.d 2
28.f even 6 1 392.2.a.d 1
28.f even 6 1 392.2.i.d 2
28.g odd 6 1 56.2.a.a 1
28.g odd 6 1 392.2.i.c 2
35.j even 6 1 2800.2.a.p 1
35.l odd 12 2 2800.2.g.p 2
56.j odd 6 1 3136.2.a.p 1
56.k odd 6 1 448.2.a.d 1
56.m even 6 1 3136.2.a.q 1
56.p even 6 1 448.2.a.e 1
84.h odd 2 1 3528.2.s.e 2
84.j odd 6 1 3528.2.a.x 1
84.j odd 6 1 3528.2.s.e 2
84.n even 6 1 504.2.a.c 1
84.n even 6 1 3528.2.s.t 2
112.u odd 12 2 1792.2.b.i 2
112.w even 12 2 1792.2.b.d 2
140.p odd 6 1 1400.2.a.g 1
140.s even 6 1 9800.2.a.u 1
140.w even 12 2 1400.2.g.g 2
168.s odd 6 1 4032.2.a.bk 1
168.v even 6 1 4032.2.a.bb 1
308.n even 6 1 6776.2.a.g 1
364.bl odd 6 1 9464.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.a 1 28.g odd 6 1
112.2.a.b 1 7.c even 3 1
392.2.a.d 1 28.f even 6 1
392.2.i.c 2 4.b odd 2 1
392.2.i.c 2 28.g odd 6 1
392.2.i.d 2 28.d even 2 1
392.2.i.d 2 28.f even 6 1
448.2.a.d 1 56.k odd 6 1
448.2.a.e 1 56.p even 6 1
504.2.a.c 1 84.n even 6 1
784.2.a.e 1 7.d odd 6 1
784.2.i.e 2 1.a even 1 1 trivial
784.2.i.e 2 7.c even 3 1 inner
784.2.i.g 2 7.b odd 2 1
784.2.i.g 2 7.d odd 6 1
1008.2.a.d 1 21.h odd 6 1
1400.2.a.g 1 140.p odd 6 1
1400.2.g.g 2 140.w even 12 2
1792.2.b.d 2 112.w even 12 2
1792.2.b.i 2 112.u odd 12 2
2800.2.a.p 1 35.j even 6 1
2800.2.g.p 2 35.l odd 12 2
3136.2.a.p 1 56.j odd 6 1
3136.2.a.q 1 56.m even 6 1
3528.2.a.x 1 84.j odd 6 1
3528.2.s.e 2 84.h odd 2 1
3528.2.s.e 2 84.j odd 6 1
3528.2.s.t 2 12.b even 2 1
3528.2.s.t 2 84.n even 6 1
4032.2.a.bb 1 168.v even 6 1
4032.2.a.bk 1 168.s odd 6 1
6776.2.a.g 1 308.n even 6 1
7056.2.a.bo 1 21.g even 6 1
9464.2.a.c 1 364.bl odd 6 1
9800.2.a.u 1 140.s even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(784, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{11}^{2} + 4T_{11} + 16$$ T11^2 + 4*T11 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} - 8T + 64$$
$23$ $$T^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} - 8T + 64$$
$37$ $$T^{2} - 2T + 4$$
$41$ $$(T - 2)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 8T + 64$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 6T + 36$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} + 10T + 100$$
$79$ $$T^{2} - 16T + 256$$
$83$ $$(T + 8)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T + 6)^{2}$$