# Properties

 Label 6930.2.a.bb Level $6930$ Weight $2$ Character orbit 6930.a Self dual yes Analytic conductor $55.336$ Analytic rank $0$ 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] = [6930,2,Mod(1,6930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6930, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6930.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6930.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: $$55.3363286007$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2310) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + q^5 - q^7 + q^8 $$q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} - q^{11} - 2 q^{13} - q^{14} + q^{16} + 2 q^{17} + 4 q^{19} + q^{20} - q^{22} + 4 q^{23} + q^{25} - 2 q^{26} - q^{28} + 6 q^{29} - 4 q^{31} + q^{32} + 2 q^{34} - q^{35} - 2 q^{37} + 4 q^{38} + q^{40} - 10 q^{41} + 4 q^{43} - q^{44} + 4 q^{46} + 8 q^{47} + q^{49} + q^{50} - 2 q^{52} + 6 q^{53} - q^{55} - q^{56} + 6 q^{58} + 4 q^{59} - 2 q^{61} - 4 q^{62} + q^{64} - 2 q^{65} - 12 q^{67} + 2 q^{68} - q^{70} + 8 q^{71} + 2 q^{73} - 2 q^{74} + 4 q^{76} + q^{77} + 4 q^{79} + q^{80} - 10 q^{82} + 2 q^{85} + 4 q^{86} - q^{88} + 10 q^{89} + 2 q^{91} + 4 q^{92} + 8 q^{94} + 4 q^{95} - 2 q^{97} + q^{98}+O(q^{100})$$ q + q^2 + q^4 + q^5 - q^7 + q^8 + q^10 - q^11 - 2 * q^13 - q^14 + q^16 + 2 * q^17 + 4 * q^19 + q^20 - q^22 + 4 * q^23 + q^25 - 2 * q^26 - q^28 + 6 * q^29 - 4 * q^31 + q^32 + 2 * q^34 - q^35 - 2 * q^37 + 4 * q^38 + q^40 - 10 * q^41 + 4 * q^43 - q^44 + 4 * q^46 + 8 * q^47 + q^49 + q^50 - 2 * q^52 + 6 * q^53 - q^55 - q^56 + 6 * q^58 + 4 * q^59 - 2 * q^61 - 4 * q^62 + q^64 - 2 * q^65 - 12 * q^67 + 2 * q^68 - q^70 + 8 * q^71 + 2 * q^73 - 2 * q^74 + 4 * q^76 + q^77 + 4 * q^79 + q^80 - 10 * q^82 + 2 * q^85 + 4 * q^86 - q^88 + 10 * q^89 + 2 * q^91 + 4 * q^92 + 8 * q^94 + 4 * q^95 - 2 * q^97 + 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 1.00000 0 −1.00000 1.00000 0 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$1$$

## 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 6930.2.a.bb 1
3.b odd 2 1 2310.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2310.2.a.f 1 3.b odd 2 1
6930.2.a.bb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6930))$$:

 $$T_{13} + 2$$ T13 + 2 $$T_{17} - 2$$ T17 - 2 $$T_{19} - 4$$ T19 - 4 $$T_{23} - 4$$ T23 - 4 $$T_{29} - 6$$ T29 - 6 $$T_{31} + 4$$ T31 + 4

## Hecke characteristic polynomials

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