# Properties

 Label 784.2.i.c Level $784$ Weight $2$ Character orbit 784.i Analytic conductor $6.260$ Analytic rank $0$ Dimension $2$ 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 14) 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{3} - \zeta_{6} q^{9} +O(q^{10})$$ q + (2*z - 2) * q^3 - z * q^9 $$q + (2 \zeta_{6} - 2) q^{3} - \zeta_{6} q^{9} - 4 q^{13} + (6 \zeta_{6} - 6) q^{17} + 2 \zeta_{6} q^{19} + ( - 5 \zeta_{6} + 5) q^{25} - 4 q^{27} - 6 q^{29} + (4 \zeta_{6} - 4) q^{31} - 2 \zeta_{6} q^{37} + ( - 8 \zeta_{6} + 8) q^{39} + 6 q^{41} - 8 q^{43} - 12 \zeta_{6} q^{47} - 12 \zeta_{6} q^{51} + (6 \zeta_{6} - 6) q^{53} - 4 q^{57} + (6 \zeta_{6} - 6) q^{59} - 8 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{67} + (2 \zeta_{6} - 2) q^{73} + 10 \zeta_{6} q^{75} + 8 \zeta_{6} q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 6 q^{83} + ( - 12 \zeta_{6} + 12) q^{87} + 6 \zeta_{6} q^{89} - 8 \zeta_{6} q^{93} - 10 q^{97} +O(q^{100})$$ q + (2*z - 2) * q^3 - z * q^9 - 4 * q^13 + (6*z - 6) * q^17 + 2*z * q^19 + (-5*z + 5) * q^25 - 4 * q^27 - 6 * q^29 + (4*z - 4) * q^31 - 2*z * q^37 + (-8*z + 8) * q^39 + 6 * q^41 - 8 * q^43 - 12*z * q^47 - 12*z * q^51 + (6*z - 6) * q^53 - 4 * q^57 + (6*z - 6) * q^59 - 8*z * q^61 + (4*z - 4) * q^67 + (2*z - 2) * q^73 + 10*z * q^75 + 8*z * q^79 + (-11*z + 11) * q^81 + 6 * q^83 + (-12*z + 12) * q^87 + 6*z * q^89 - 8*z * q^93 - 10 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - q^9 $$2 q - 2 q^{3} - q^{9} - 8 q^{13} - 6 q^{17} + 2 q^{19} + 5 q^{25} - 8 q^{27} - 12 q^{29} - 4 q^{31} - 2 q^{37} + 8 q^{39} + 12 q^{41} - 16 q^{43} - 12 q^{47} - 12 q^{51} - 6 q^{53} - 8 q^{57} - 6 q^{59} - 8 q^{61} - 4 q^{67} - 2 q^{73} + 10 q^{75} + 8 q^{79} + 11 q^{81} + 12 q^{83} + 12 q^{87} + 6 q^{89} - 8 q^{93} - 20 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 - q^9 - 8 * q^13 - 6 * q^17 + 2 * q^19 + 5 * q^25 - 8 * q^27 - 12 * q^29 - 4 * q^31 - 2 * q^37 + 8 * q^39 + 12 * q^41 - 16 * q^43 - 12 * q^47 - 12 * q^51 - 6 * q^53 - 8 * q^57 - 6 * q^59 - 8 * q^61 - 4 * q^67 - 2 * q^73 + 10 * q^75 + 8 * q^79 + 11 * q^81 + 12 * q^83 + 12 * q^87 + 6 * q^89 - 8 * q^93 - 20 * q^97

## 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 −1.00000 1.73205i 0 0 0 0 0 −0.500000 + 0.866025i 0
753.1 0 −1.00000 + 1.73205i 0 0 0 0 0 −0.500000 0.866025i 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.c 2
4.b odd 2 1 98.2.c.b 2
7.b odd 2 1 784.2.i.i 2
7.c even 3 1 112.2.a.c 1
7.c even 3 1 inner 784.2.i.c 2
7.d odd 6 1 784.2.a.b 1
7.d odd 6 1 784.2.i.i 2
12.b even 2 1 882.2.g.c 2
21.g even 6 1 7056.2.a.bd 1
21.h odd 6 1 1008.2.a.h 1
28.d even 2 1 98.2.c.a 2
28.f even 6 1 98.2.a.a 1
28.f even 6 1 98.2.c.a 2
28.g odd 6 1 14.2.a.a 1
28.g odd 6 1 98.2.c.b 2
35.j even 6 1 2800.2.a.g 1
35.l odd 12 2 2800.2.g.h 2
56.j odd 6 1 3136.2.a.z 1
56.k odd 6 1 448.2.a.g 1
56.m even 6 1 3136.2.a.e 1
56.p even 6 1 448.2.a.a 1
84.h odd 2 1 882.2.g.d 2
84.j odd 6 1 882.2.a.i 1
84.j odd 6 1 882.2.g.d 2
84.n even 6 1 126.2.a.b 1
84.n even 6 1 882.2.g.c 2
112.u odd 12 2 1792.2.b.c 2
112.w even 12 2 1792.2.b.g 2
140.p odd 6 1 350.2.a.f 1
140.s even 6 1 2450.2.a.t 1
140.w even 12 2 350.2.c.d 2
140.x odd 12 2 2450.2.c.c 2
168.s odd 6 1 4032.2.a.r 1
168.v even 6 1 4032.2.a.w 1
252.o even 6 1 1134.2.f.f 2
252.u odd 6 1 1134.2.f.l 2
252.bb even 6 1 1134.2.f.f 2
252.bl odd 6 1 1134.2.f.l 2
308.n even 6 1 1694.2.a.e 1
364.bl odd 6 1 2366.2.a.j 1
364.ce even 12 2 2366.2.d.b 2
420.ba even 6 1 3150.2.a.i 1
420.bp odd 12 2 3150.2.g.j 2
476.o odd 6 1 4046.2.a.f 1
532.t even 6 1 5054.2.a.c 1
644.p even 6 1 7406.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 28.g odd 6 1
98.2.a.a 1 28.f even 6 1
98.2.c.a 2 28.d even 2 1
98.2.c.a 2 28.f even 6 1
98.2.c.b 2 4.b odd 2 1
98.2.c.b 2 28.g odd 6 1
112.2.a.c 1 7.c even 3 1
126.2.a.b 1 84.n even 6 1
350.2.a.f 1 140.p odd 6 1
350.2.c.d 2 140.w even 12 2
448.2.a.a 1 56.p even 6 1
448.2.a.g 1 56.k odd 6 1
784.2.a.b 1 7.d odd 6 1
784.2.i.c 2 1.a even 1 1 trivial
784.2.i.c 2 7.c even 3 1 inner
784.2.i.i 2 7.b odd 2 1
784.2.i.i 2 7.d odd 6 1
882.2.a.i 1 84.j odd 6 1
882.2.g.c 2 12.b even 2 1
882.2.g.c 2 84.n even 6 1
882.2.g.d 2 84.h odd 2 1
882.2.g.d 2 84.j odd 6 1
1008.2.a.h 1 21.h odd 6 1
1134.2.f.f 2 252.o even 6 1
1134.2.f.f 2 252.bb even 6 1
1134.2.f.l 2 252.u odd 6 1
1134.2.f.l 2 252.bl odd 6 1
1694.2.a.e 1 308.n even 6 1
1792.2.b.c 2 112.u odd 12 2
1792.2.b.g 2 112.w even 12 2
2366.2.a.j 1 364.bl odd 6 1
2366.2.d.b 2 364.ce even 12 2
2450.2.a.t 1 140.s even 6 1
2450.2.c.c 2 140.x odd 12 2
2800.2.a.g 1 35.j even 6 1
2800.2.g.h 2 35.l odd 12 2
3136.2.a.e 1 56.m even 6 1
3136.2.a.z 1 56.j odd 6 1
3150.2.a.i 1 420.ba even 6 1
3150.2.g.j 2 420.bp odd 12 2
4032.2.a.r 1 168.s odd 6 1
4032.2.a.w 1 168.v even 6 1
4046.2.a.f 1 476.o odd 6 1
5054.2.a.c 1 532.t even 6 1
7056.2.a.bd 1 21.g even 6 1
7406.2.a.a 1 644.p even 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2} + 6T + 36$$
$19$ $$T^{2} - 2T + 4$$
$23$ $$T^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2} + 12T + 144$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} + 6T + 36$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 2T + 4$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T + 10)^{2}$$