# Properties

 Label 462.2.a.f Level $462$ Weight $2$ Character orbit 462.a Self dual yes Analytic conductor $3.689$ 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] = [462,2,Mod(1,462)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, 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("462.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.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: $$3.68908857338$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + q^6 + q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} + q^{9} - q^{11} + q^{12} + 2 q^{13} + q^{14} + q^{16} + q^{18} + 2 q^{19} + q^{21} - q^{22} + q^{24} - 5 q^{25} + 2 q^{26} + q^{27} + q^{28} - 6 q^{29} + 2 q^{31} + q^{32} - q^{33} + q^{36} + 2 q^{37} + 2 q^{38} + 2 q^{39} + q^{42} - 4 q^{43} - q^{44} - 6 q^{47} + q^{48} + q^{49} - 5 q^{50} + 2 q^{52} - 6 q^{53} + q^{54} + q^{56} + 2 q^{57} - 6 q^{58} + 2 q^{61} + 2 q^{62} + q^{63} + q^{64} - q^{66} - 4 q^{67} - 12 q^{71} + q^{72} - 4 q^{73} + 2 q^{74} - 5 q^{75} + 2 q^{76} - q^{77} + 2 q^{78} + 8 q^{79} + q^{81} + 6 q^{83} + q^{84} - 4 q^{86} - 6 q^{87} - q^{88} - 6 q^{89} + 2 q^{91} + 2 q^{93} - 6 q^{94} + q^{96} + 2 q^{97} + q^{98} - q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 + q^7 + q^8 + q^9 - q^11 + q^12 + 2 * q^13 + q^14 + q^16 + q^18 + 2 * q^19 + q^21 - q^22 + q^24 - 5 * q^25 + 2 * q^26 + q^27 + q^28 - 6 * q^29 + 2 * q^31 + q^32 - q^33 + q^36 + 2 * q^37 + 2 * q^38 + 2 * q^39 + q^42 - 4 * q^43 - q^44 - 6 * q^47 + q^48 + q^49 - 5 * q^50 + 2 * q^52 - 6 * q^53 + q^54 + q^56 + 2 * q^57 - 6 * q^58 + 2 * q^61 + 2 * q^62 + q^63 + q^64 - q^66 - 4 * q^67 - 12 * q^71 + q^72 - 4 * q^73 + 2 * q^74 - 5 * q^75 + 2 * q^76 - q^77 + 2 * q^78 + 8 * q^79 + q^81 + 6 * q^83 + q^84 - 4 * q^86 - 6 * q^87 - q^88 - 6 * q^89 + 2 * q^91 + 2 * q^93 - 6 * q^94 + q^96 + 2 * q^97 + q^98 - 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 0 1.00000 1.00000 1.00000 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$+1$$

## 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 462.2.a.f 1
3.b odd 2 1 1386.2.a.c 1
4.b odd 2 1 3696.2.a.i 1
7.b odd 2 1 3234.2.a.q 1
11.b odd 2 1 5082.2.a.l 1
21.c even 2 1 9702.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.f 1 1.a even 1 1 trivial
1386.2.a.c 1 3.b odd 2 1
3234.2.a.q 1 7.b odd 2 1
3696.2.a.i 1 4.b odd 2 1
5082.2.a.l 1 11.b odd 2 1
9702.2.a.n 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(462))$$:

 $$T_{5}$$ T5 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

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