# Properties

 Label 930.2.a.l Level $930$ Weight $2$ Character orbit 930.a Self dual yes Analytic conductor $7.426$ 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] = [930,2,Mod(1,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, 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("930.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.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: $$7.42608738798$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 - q^5 - q^6 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - 6 q^{11} - q^{12} - 2 q^{13} + q^{15} + q^{16} - 4 q^{17} + q^{18} - q^{20} - 6 q^{22} + 2 q^{23} - q^{24} + q^{25} - 2 q^{26} - q^{27} - 8 q^{29} + q^{30} + q^{31} + q^{32} + 6 q^{33} - 4 q^{34} + q^{36} - 6 q^{37} + 2 q^{39} - q^{40} - 2 q^{41} + 4 q^{43} - 6 q^{44} - q^{45} + 2 q^{46} + 4 q^{47} - q^{48} - 7 q^{49} + q^{50} + 4 q^{51} - 2 q^{52} - 6 q^{53} - q^{54} + 6 q^{55} - 8 q^{58} + q^{60} + 4 q^{61} + q^{62} + q^{64} + 2 q^{65} + 6 q^{66} - 4 q^{67} - 4 q^{68} - 2 q^{69} - 8 q^{71} + q^{72} - 4 q^{73} - 6 q^{74} - q^{75} + 2 q^{78} - 4 q^{79} - q^{80} + q^{81} - 2 q^{82} + 4 q^{85} + 4 q^{86} + 8 q^{87} - 6 q^{88} - 2 q^{89} - q^{90} + 2 q^{92} - q^{93} + 4 q^{94} - q^{96} + 14 q^{97} - 7 q^{98} - 6 q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 - q^5 - q^6 + q^8 + q^9 - q^10 - 6 * q^11 - q^12 - 2 * q^13 + q^15 + q^16 - 4 * q^17 + q^18 - q^20 - 6 * q^22 + 2 * q^23 - q^24 + q^25 - 2 * q^26 - q^27 - 8 * q^29 + q^30 + q^31 + q^32 + 6 * q^33 - 4 * q^34 + q^36 - 6 * q^37 + 2 * q^39 - q^40 - 2 * q^41 + 4 * q^43 - 6 * q^44 - q^45 + 2 * q^46 + 4 * q^47 - q^48 - 7 * q^49 + q^50 + 4 * q^51 - 2 * q^52 - 6 * q^53 - q^54 + 6 * q^55 - 8 * q^58 + q^60 + 4 * q^61 + q^62 + q^64 + 2 * q^65 + 6 * q^66 - 4 * q^67 - 4 * q^68 - 2 * q^69 - 8 * q^71 + q^72 - 4 * q^73 - 6 * q^74 - q^75 + 2 * q^78 - 4 * q^79 - q^80 + q^81 - 2 * q^82 + 4 * q^85 + 4 * q^86 + 8 * q^87 - 6 * q^88 - 2 * q^89 - q^90 + 2 * q^92 - q^93 + 4 * q^94 - q^96 + 14 * q^97 - 7 * q^98 - 6 * 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 −1.00000 1.00000 −1.00000 −1.00000 0 1.00000 1.00000 −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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$31$$ $$-1$$

## 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 930.2.a.l 1
3.b odd 2 1 2790.2.a.j 1
4.b odd 2 1 7440.2.a.r 1
5.b even 2 1 4650.2.a.r 1
5.c odd 4 2 4650.2.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.l 1 1.a even 1 1 trivial
2790.2.a.j 1 3.b odd 2 1
4650.2.a.r 1 5.b even 2 1
4650.2.d.a 2 5.c odd 4 2
7440.2.a.r 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(930))$$:

 $$T_{7}$$ T7 $$T_{11} + 6$$ T11 + 6 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T + 6$$
$13$ $$T + 2$$
$17$ $$T + 4$$
$19$ $$T$$
$23$ $$T - 2$$
$29$ $$T + 8$$
$31$ $$T - 1$$
$37$ $$T + 6$$
$41$ $$T + 2$$
$43$ $$T - 4$$
$47$ $$T - 4$$
$53$ $$T + 6$$
$59$ $$T$$
$61$ $$T - 4$$
$67$ $$T + 4$$
$71$ $$T + 8$$
$73$ $$T + 4$$
$79$ $$T + 4$$
$83$ $$T$$
$89$ $$T + 2$$
$97$ $$T - 14$$