Properties

 Label 49.2.a.a Level $49$ Weight $2$ Character orbit 49.a Self dual yes Analytic conductor $0.391$ Analytic rank $0$ Dimension $1$ CM discriminant -7 Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,2,Mod(1,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("49.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 49.a (trivial)

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: yes Analytic conductor: $$0.391266969904$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} - 3 q^{8} - 3 q^{9}+O(q^{10})$$ q + q^2 - q^4 - 3 * q^8 - 3 * q^9 $$q + q^{2} - q^{4} - 3 q^{8} - 3 q^{9} + 4 q^{11} - q^{16} - 3 q^{18} + 4 q^{22} + 8 q^{23} - 5 q^{25} + 2 q^{29} + 5 q^{32} + 3 q^{36} - 6 q^{37} - 12 q^{43} - 4 q^{44} + 8 q^{46} - 5 q^{50} - 10 q^{53} + 2 q^{58} + 7 q^{64} + 4 q^{67} + 16 q^{71} + 9 q^{72} - 6 q^{74} + 8 q^{79} + 9 q^{81} - 12 q^{86} - 12 q^{88} - 8 q^{92} - 12 q^{99}+O(q^{100})$$ q + q^2 - q^4 - 3 * q^8 - 3 * q^9 + 4 * q^11 - q^16 - 3 * q^18 + 4 * q^22 + 8 * q^23 - 5 * q^25 + 2 * q^29 + 5 * q^32 + 3 * q^36 - 6 * q^37 - 12 * q^43 - 4 * q^44 + 8 * q^46 - 5 * q^50 - 10 * q^53 + 2 * q^58 + 7 * q^64 + 4 * q^67 + 16 * q^71 + 9 * q^72 - 6 * q^74 + 8 * q^79 + 9 * q^81 - 12 * q^86 - 12 * q^88 - 8 * q^92 - 12 * q^99

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 0 −1.00000 0 0 0 −3.00000 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.2.a.a 1
3.b odd 2 1 441.2.a.c 1
4.b odd 2 1 784.2.a.f 1
5.b even 2 1 1225.2.a.c 1
5.c odd 4 2 1225.2.b.c 2
7.b odd 2 1 CM 49.2.a.a 1
7.c even 3 2 49.2.c.a 2
7.d odd 6 2 49.2.c.a 2
8.b even 2 1 3136.2.a.n 1
8.d odd 2 1 3136.2.a.o 1
11.b odd 2 1 5929.2.a.c 1
12.b even 2 1 7056.2.a.bg 1
13.b even 2 1 8281.2.a.d 1
21.c even 2 1 441.2.a.c 1
21.g even 6 2 441.2.e.d 2
21.h odd 6 2 441.2.e.d 2
28.d even 2 1 784.2.a.f 1
28.f even 6 2 784.2.i.f 2
28.g odd 6 2 784.2.i.f 2
35.c odd 2 1 1225.2.a.c 1
35.f even 4 2 1225.2.b.c 2
56.e even 2 1 3136.2.a.o 1
56.h odd 2 1 3136.2.a.n 1
77.b even 2 1 5929.2.a.c 1
84.h odd 2 1 7056.2.a.bg 1
91.b odd 2 1 8281.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.a.a 1 1.a even 1 1 trivial
49.2.a.a 1 7.b odd 2 1 CM
49.2.c.a 2 7.c even 3 2
49.2.c.a 2 7.d odd 6 2
441.2.a.c 1 3.b odd 2 1
441.2.a.c 1 21.c even 2 1
441.2.e.d 2 21.g even 6 2
441.2.e.d 2 21.h odd 6 2
784.2.a.f 1 4.b odd 2 1
784.2.a.f 1 28.d even 2 1
784.2.i.f 2 28.f even 6 2
784.2.i.f 2 28.g odd 6 2
1225.2.a.c 1 5.b even 2 1
1225.2.a.c 1 35.c odd 2 1
1225.2.b.c 2 5.c odd 4 2
1225.2.b.c 2 35.f even 4 2
3136.2.a.n 1 8.b even 2 1
3136.2.a.n 1 56.h odd 2 1
3136.2.a.o 1 8.d odd 2 1
3136.2.a.o 1 56.e even 2 1
5929.2.a.c 1 11.b odd 2 1
5929.2.a.c 1 77.b even 2 1
7056.2.a.bg 1 12.b even 2 1
7056.2.a.bg 1 84.h odd 2 1
8281.2.a.d 1 13.b even 2 1
8281.2.a.d 1 91.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(49))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 4$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 8$$
$29$ $$T - 2$$
$31$ $$T$$
$37$ $$T + 6$$
$41$ $$T$$
$43$ $$T + 12$$
$47$ $$T$$
$53$ $$T + 10$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T - 4$$
$71$ $$T - 16$$
$73$ $$T$$
$79$ $$T - 8$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$