# Properties

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

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 - 2 * q^5 + q^6 + q^7 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + 2 q^{10} + q^{11} - q^{12} + 2 q^{13} - q^{14} + 2 q^{15} + q^{16} - 6 q^{17} - q^{18} - 4 q^{19} - 2 q^{20} - q^{21} - q^{22} - 4 q^{23} + q^{24} - q^{25} - 2 q^{26} - q^{27} + q^{28} + 2 q^{29} - 2 q^{30} - 4 q^{31} - q^{32} - q^{33} + 6 q^{34} - 2 q^{35} + q^{36} - 2 q^{37} + 4 q^{38} - 2 q^{39} + 2 q^{40} - 6 q^{41} + q^{42} + q^{44} - 2 q^{45} + 4 q^{46} - 8 q^{47} - q^{48} + q^{49} + q^{50} + 6 q^{51} + 2 q^{52} - 14 q^{53} + q^{54} - 2 q^{55} - q^{56} + 4 q^{57} - 2 q^{58} + 12 q^{59} + 2 q^{60} - 14 q^{61} + 4 q^{62} + q^{63} + q^{64} - 4 q^{65} + q^{66} + 4 q^{67} - 6 q^{68} + 4 q^{69} + 2 q^{70} + 12 q^{71} - q^{72} + 6 q^{73} + 2 q^{74} + q^{75} - 4 q^{76} + q^{77} + 2 q^{78} - 2 q^{80} + q^{81} + 6 q^{82} - q^{84} + 12 q^{85} - 2 q^{87} - q^{88} - 6 q^{89} + 2 q^{90} + 2 q^{91} - 4 q^{92} + 4 q^{93} + 8 q^{94} + 8 q^{95} + q^{96} - 14 q^{97} - q^{98} + q^{99}+O(q^{100})$$ q - q^2 - q^3 + q^4 - 2 * q^5 + q^6 + q^7 - q^8 + q^9 + 2 * q^10 + q^11 - q^12 + 2 * q^13 - q^14 + 2 * q^15 + q^16 - 6 * q^17 - q^18 - 4 * q^19 - 2 * q^20 - q^21 - q^22 - 4 * q^23 + q^24 - q^25 - 2 * q^26 - q^27 + q^28 + 2 * q^29 - 2 * q^30 - 4 * q^31 - q^32 - q^33 + 6 * q^34 - 2 * q^35 + q^36 - 2 * q^37 + 4 * q^38 - 2 * q^39 + 2 * q^40 - 6 * q^41 + q^42 + q^44 - 2 * q^45 + 4 * q^46 - 8 * q^47 - q^48 + q^49 + q^50 + 6 * q^51 + 2 * q^52 - 14 * q^53 + q^54 - 2 * q^55 - q^56 + 4 * q^57 - 2 * q^58 + 12 * q^59 + 2 * q^60 - 14 * q^61 + 4 * q^62 + q^63 + q^64 - 4 * q^65 + q^66 + 4 * q^67 - 6 * q^68 + 4 * q^69 + 2 * q^70 + 12 * q^71 - q^72 + 6 * q^73 + 2 * q^74 + q^75 - 4 * q^76 + q^77 + 2 * q^78 - 2 * q^80 + q^81 + 6 * q^82 - q^84 + 12 * q^85 - 2 * q^87 - q^88 - 6 * q^89 + 2 * q^90 + 2 * q^91 - 4 * q^92 + 4 * q^93 + 8 * q^94 + 8 * q^95 + q^96 - 14 * 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 −2.00000 1.00000 1.00000 −1.00000 1.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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.a 1
3.b odd 2 1 1386.2.a.k 1
4.b odd 2 1 3696.2.a.s 1
7.b odd 2 1 3234.2.a.n 1
11.b odd 2 1 5082.2.a.q 1
21.c even 2 1 9702.2.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.a 1 1.a even 1 1 trivial
1386.2.a.k 1 3.b odd 2 1
3234.2.a.n 1 7.b odd 2 1
3696.2.a.s 1 4.b odd 2 1
5082.2.a.q 1 11.b odd 2 1
9702.2.a.bf 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(462))$$:

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

## Hecke characteristic polynomials

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