# Properties

 Label 49.6.c.a Level $49$ Weight $6$ Character orbit 49.c Analytic conductor $7.859$ Analytic rank $0$ Dimension $2$ CM discriminant -7 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,6,Mod(18,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("49.18");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 49.c (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: $$7.85880717084$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - 11 \zeta_{6} q^{2} + (89 \zeta_{6} - 89) q^{4} + 627 q^{8} + 243 \zeta_{6} q^{9} +O(q^{10})$$ q - 11*z * q^2 + (89*z - 89) * q^4 + 627 * q^8 + 243*z * q^9 $$q - 11 \zeta_{6} q^{2} + (89 \zeta_{6} - 89) q^{4} + 627 q^{8} + 243 \zeta_{6} q^{9} + ( - 76 \zeta_{6} + 76) q^{11} - 4049 \zeta_{6} q^{16} + ( - 2673 \zeta_{6} + 2673) q^{18} - 836 q^{22} + 4952 \zeta_{6} q^{23} + ( - 3125 \zeta_{6} + 3125) q^{25} + 7282 q^{29} + (24475 \zeta_{6} - 24475) q^{32} - 21627 q^{36} + 8886 \zeta_{6} q^{37} + 11748 q^{43} + 6764 \zeta_{6} q^{44} + ( - 54472 \zeta_{6} + 54472) q^{46} - 34375 q^{50} + (24550 \zeta_{6} - 24550) q^{53} - 80102 \zeta_{6} q^{58} + 139657 q^{64} + (69364 \zeta_{6} - 69364) q^{67} - 2224 q^{71} + 152361 \zeta_{6} q^{72} + ( - 97746 \zeta_{6} + 97746) q^{74} - 80168 \zeta_{6} q^{79} + (59049 \zeta_{6} - 59049) q^{81} - 129228 \zeta_{6} q^{86} + ( - 47652 \zeta_{6} + 47652) q^{88} - 440728 q^{92} + 18468 q^{99} +O(q^{100})$$ q - 11*z * q^2 + (89*z - 89) * q^4 + 627 * q^8 + 243*z * q^9 + (-76*z + 76) * q^11 - 4049*z * q^16 + (-2673*z + 2673) * q^18 - 836 * q^22 + 4952*z * q^23 + (-3125*z + 3125) * q^25 + 7282 * q^29 + (24475*z - 24475) * q^32 - 21627 * q^36 + 8886*z * q^37 + 11748 * q^43 + 6764*z * q^44 + (-54472*z + 54472) * q^46 - 34375 * q^50 + (24550*z - 24550) * q^53 - 80102*z * q^58 + 139657 * q^64 + (69364*z - 69364) * q^67 - 2224 * q^71 + 152361*z * q^72 + (-97746*z + 97746) * q^74 - 80168*z * q^79 + (59049*z - 59049) * q^81 - 129228*z * q^86 + (-47652*z + 47652) * q^88 - 440728 * q^92 + 18468 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 11 q^{2} - 89 q^{4} + 1254 q^{8} + 243 q^{9}+O(q^{10})$$ 2 * q - 11 * q^2 - 89 * q^4 + 1254 * q^8 + 243 * q^9 $$2 q - 11 q^{2} - 89 q^{4} + 1254 q^{8} + 243 q^{9} + 76 q^{11} - 4049 q^{16} + 2673 q^{18} - 1672 q^{22} + 4952 q^{23} + 3125 q^{25} + 14564 q^{29} - 24475 q^{32} - 43254 q^{36} + 8886 q^{37} + 23496 q^{43} + 6764 q^{44} + 54472 q^{46} - 68750 q^{50} - 24550 q^{53} - 80102 q^{58} + 279314 q^{64} - 69364 q^{67} - 4448 q^{71} + 152361 q^{72} + 97746 q^{74} - 80168 q^{79} - 59049 q^{81} - 129228 q^{86} + 47652 q^{88} - 881456 q^{92} + 36936 q^{99}+O(q^{100})$$ 2 * q - 11 * q^2 - 89 * q^4 + 1254 * q^8 + 243 * q^9 + 76 * q^11 - 4049 * q^16 + 2673 * q^18 - 1672 * q^22 + 4952 * q^23 + 3125 * q^25 + 14564 * q^29 - 24475 * q^32 - 43254 * q^36 + 8886 * q^37 + 23496 * q^43 + 6764 * q^44 + 54472 * q^46 - 68750 * q^50 - 24550 * q^53 - 80102 * q^58 + 279314 * q^64 - 69364 * q^67 - 4448 * q^71 + 152361 * q^72 + 97746 * q^74 - 80168 * q^79 - 59049 * q^81 - 129228 * q^86 + 47652 * q^88 - 881456 * q^92 + 36936 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/49\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 0.5 + 0.866025i 0.5 − 0.866025i
−5.50000 9.52628i 0 −44.5000 + 77.0763i 0 0 0 627.000 121.500 + 210.444i 0
30.1 −5.50000 + 9.52628i 0 −44.5000 77.0763i 0 0 0 627.000 121.500 210.444i 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.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.6.c.a 2
7.b odd 2 1 CM 49.6.c.a 2
7.c even 3 1 49.6.a.b 1
7.c even 3 1 inner 49.6.c.a 2
7.d odd 6 1 49.6.a.b 1
7.d odd 6 1 inner 49.6.c.a 2
21.g even 6 1 441.6.a.a 1
21.h odd 6 1 441.6.a.a 1
28.f even 6 1 784.6.a.g 1
28.g odd 6 1 784.6.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.b 1 7.c even 3 1
49.6.a.b 1 7.d odd 6 1
49.6.c.a 2 1.a even 1 1 trivial
49.6.c.a 2 7.b odd 2 1 CM
49.6.c.a 2 7.c even 3 1 inner
49.6.c.a 2 7.d odd 6 1 inner
441.6.a.a 1 21.g even 6 1
441.6.a.a 1 21.h odd 6 1
784.6.a.g 1 28.f even 6 1
784.6.a.g 1 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(49, [\chi])$$:

 $$T_{2}^{2} + 11T_{2} + 121$$ T2^2 + 11*T2 + 121 $$T_{3}$$ T3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 11T + 121$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 76T + 5776$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 4952 T + 24522304$$
$29$ $$(T - 7282)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 8886 T + 78960996$$
$41$ $$T^{2}$$
$43$ $$(T - 11748)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 24550 T + 602702500$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2} + \cdots + 4811364496$$
$71$ $$(T + 2224)^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2} + \cdots + 6426908224$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$