# Properties

 Label 930.2.a.o Level $930$ Weight $2$ Character orbit 930.a Self dual yes Analytic conductor $7.426$ 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] = [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: $$0$$ 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 $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} - 4 q^{11} + q^{12} + 6 q^{13} + q^{15} + q^{16} + 2 q^{17} + q^{18} + 4 q^{19} + q^{20} - 4 q^{22} - 8 q^{23} + q^{24} + q^{25} + 6 q^{26} + q^{27} + 6 q^{29} + q^{30} - q^{31} + q^{32} - 4 q^{33} + 2 q^{34} + q^{36} - 2 q^{37} + 4 q^{38} + 6 q^{39} + q^{40} + 10 q^{41} - 4 q^{43} - 4 q^{44} + q^{45} - 8 q^{46} + q^{48} - 7 q^{49} + q^{50} + 2 q^{51} + 6 q^{52} - 10 q^{53} + q^{54} - 4 q^{55} + 4 q^{57} + 6 q^{58} - 12 q^{59} + q^{60} - 2 q^{61} - q^{62} + q^{64} + 6 q^{65} - 4 q^{66} - 4 q^{67} + 2 q^{68} - 8 q^{69} + q^{72} + 2 q^{73} - 2 q^{74} + q^{75} + 4 q^{76} + 6 q^{78} + q^{80} + q^{81} + 10 q^{82} + 4 q^{83} + 2 q^{85} - 4 q^{86} + 6 q^{87} - 4 q^{88} - 14 q^{89} + q^{90} - 8 q^{92} - q^{93} + 4 q^{95} + q^{96} + 18 q^{97} - 7 q^{98} - 4 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^5 + q^6 + q^8 + q^9 + q^10 - 4 * q^11 + q^12 + 6 * q^13 + q^15 + q^16 + 2 * q^17 + q^18 + 4 * q^19 + q^20 - 4 * q^22 - 8 * q^23 + q^24 + q^25 + 6 * q^26 + q^27 + 6 * q^29 + q^30 - q^31 + q^32 - 4 * q^33 + 2 * q^34 + q^36 - 2 * q^37 + 4 * q^38 + 6 * q^39 + q^40 + 10 * q^41 - 4 * q^43 - 4 * q^44 + q^45 - 8 * q^46 + q^48 - 7 * q^49 + q^50 + 2 * q^51 + 6 * q^52 - 10 * q^53 + q^54 - 4 * q^55 + 4 * q^57 + 6 * q^58 - 12 * q^59 + q^60 - 2 * q^61 - q^62 + q^64 + 6 * q^65 - 4 * q^66 - 4 * q^67 + 2 * q^68 - 8 * q^69 + q^72 + 2 * q^73 - 2 * q^74 + q^75 + 4 * q^76 + 6 * q^78 + q^80 + q^81 + 10 * q^82 + 4 * q^83 + 2 * q^85 - 4 * q^86 + 6 * q^87 - 4 * q^88 - 14 * q^89 + q^90 - 8 * q^92 - q^93 + 4 * q^95 + q^96 + 18 * q^97 - 7 * q^98 - 4 * 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.o 1
3.b odd 2 1 2790.2.a.c 1
4.b odd 2 1 7440.2.a.j 1
5.b even 2 1 4650.2.a.h 1
5.c odd 4 2 4650.2.d.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.o 1 1.a even 1 1 trivial
2790.2.a.c 1 3.b odd 2 1
4650.2.a.h 1 5.b even 2 1
4650.2.d.n 2 5.c odd 4 2
7440.2.a.j 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 $$T_{11} + 4$$ T11 + 4 $$T_{19} - 4$$ T19 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T - 6$$
$17$ $$T - 2$$
$19$ $$T - 4$$
$23$ $$T + 8$$
$29$ $$T - 6$$
$31$ $$T + 1$$
$37$ $$T + 2$$
$41$ $$T - 10$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T + 10$$
$59$ $$T + 12$$
$61$ $$T + 2$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T$$
$83$ $$T - 4$$
$89$ $$T + 14$$
$97$ $$T - 18$$