# Properties

 Label 7191.2.a.g Level $7191$ Weight $2$ Character orbit 7191.a Self dual yes Analytic conductor $57.420$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7191,2,Mod(1,7191)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7191, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7191.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7191 = 3^{2} \cdot 17 \cdot 47$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7191.a (trivial)

## Newform invariants

comment: select newform

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

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

 Self dual: yes Analytic conductor: $$57.4204240935$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 799) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} - 4 q^{5} - 2 q^{7} - 3 q^{8}+O(q^{10})$$ q + q^2 - q^4 - 4 * q^5 - 2 * q^7 - 3 * q^8 $$q + q^{2} - q^{4} - 4 q^{5} - 2 q^{7} - 3 q^{8} - 4 q^{10} + 2 q^{13} - 2 q^{14} - q^{16} - q^{17} + 4 q^{19} + 4 q^{20} + 4 q^{23} + 11 q^{25} + 2 q^{26} + 2 q^{28} - 8 q^{29} + 8 q^{31} + 5 q^{32} - q^{34} + 8 q^{35} - 2 q^{37} + 4 q^{38} + 12 q^{40} + 8 q^{41} - 4 q^{43} + 4 q^{46} + q^{47} - 3 q^{49} + 11 q^{50} - 2 q^{52} - 6 q^{53} + 6 q^{56} - 8 q^{58} - 4 q^{59} + 6 q^{61} + 8 q^{62} + 7 q^{64} - 8 q^{65} - 4 q^{67} + q^{68} + 8 q^{70} + 6 q^{71} + 4 q^{73} - 2 q^{74} - 4 q^{76} + 2 q^{79} + 4 q^{80} + 8 q^{82} + 4 q^{85} - 4 q^{86} - 6 q^{89} - 4 q^{91} - 4 q^{92} + q^{94} - 16 q^{95} + 10 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 - q^4 - 4 * q^5 - 2 * q^7 - 3 * q^8 - 4 * q^10 + 2 * q^13 - 2 * q^14 - q^16 - q^17 + 4 * q^19 + 4 * q^20 + 4 * q^23 + 11 * q^25 + 2 * q^26 + 2 * q^28 - 8 * q^29 + 8 * q^31 + 5 * q^32 - q^34 + 8 * q^35 - 2 * q^37 + 4 * q^38 + 12 * q^40 + 8 * q^41 - 4 * q^43 + 4 * q^46 + q^47 - 3 * q^49 + 11 * q^50 - 2 * q^52 - 6 * q^53 + 6 * q^56 - 8 * q^58 - 4 * q^59 + 6 * q^61 + 8 * q^62 + 7 * q^64 - 8 * q^65 - 4 * q^67 + q^68 + 8 * q^70 + 6 * q^71 + 4 * q^73 - 2 * q^74 - 4 * q^76 + 2 * q^79 + 4 * q^80 + 8 * q^82 + 4 * q^85 - 4 * q^86 - 6 * q^89 - 4 * q^91 - 4 * q^92 + q^94 - 16 * q^95 + 10 * q^97 - 3 * q^98

## 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 −4.00000 0 −2.00000 −3.00000 0 −4.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$17$$ $$1$$
$$47$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7191.2.a.g 1
3.b odd 2 1 799.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.2.a.b 1 3.b odd 2 1
7191.2.a.g 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7191))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{5} + 4$$ T5 + 4 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

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