# Properties

 Label 462.2.a.b 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + q^6 - q^7 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} + q^{9} - q^{11} - q^{12} - 2 q^{13} + q^{14} + q^{16} - 4 q^{17} - q^{18} + 6 q^{19} + q^{21} + q^{22} - 4 q^{23} + q^{24} - 5 q^{25} + 2 q^{26} - q^{27} - q^{28} - 10 q^{29} + 6 q^{31} - q^{32} + q^{33} + 4 q^{34} + q^{36} - 6 q^{37} - 6 q^{38} + 2 q^{39} - 12 q^{41} - q^{42} - 8 q^{43} - q^{44} + 4 q^{46} + 2 q^{47} - q^{48} + q^{49} + 5 q^{50} + 4 q^{51} - 2 q^{52} + 6 q^{53} + q^{54} + q^{56} - 6 q^{57} + 10 q^{58} - 8 q^{59} + 6 q^{61} - 6 q^{62} - q^{63} + q^{64} - q^{66} - 4 q^{67} - 4 q^{68} + 4 q^{69} - q^{72} - 12 q^{73} + 6 q^{74} + 5 q^{75} + 6 q^{76} + q^{77} - 2 q^{78} + q^{81} + 12 q^{82} + 14 q^{83} + q^{84} + 8 q^{86} + 10 q^{87} + q^{88} + 10 q^{89} + 2 q^{91} - 4 q^{92} - 6 q^{93} - 2 q^{94} + q^{96} + 10 q^{97} - q^{98} - q^{99}+O(q^{100})$$ 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 - 4 * q^17 - q^18 + 6 * q^19 + q^21 + q^22 - 4 * q^23 + q^24 - 5 * q^25 + 2 * q^26 - q^27 - q^28 - 10 * q^29 + 6 * q^31 - q^32 + q^33 + 4 * q^34 + q^36 - 6 * q^37 - 6 * q^38 + 2 * q^39 - 12 * q^41 - q^42 - 8 * q^43 - q^44 + 4 * q^46 + 2 * q^47 - q^48 + q^49 + 5 * q^50 + 4 * q^51 - 2 * q^52 + 6 * q^53 + q^54 + q^56 - 6 * q^57 + 10 * q^58 - 8 * q^59 + 6 * q^61 - 6 * q^62 - q^63 + q^64 - q^66 - 4 * q^67 - 4 * q^68 + 4 * q^69 - q^72 - 12 * q^73 + 6 * q^74 + 5 * q^75 + 6 * q^76 + q^77 - 2 * q^78 + q^81 + 12 * q^82 + 14 * q^83 + q^84 + 8 * q^86 + 10 * q^87 + q^88 + 10 * q^89 + 2 * q^91 - 4 * q^92 - 6 * q^93 - 2 * q^94 + q^96 + 10 * 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.b 1
3.b odd 2 1 1386.2.a.i 1
4.b odd 2 1 3696.2.a.y 1
7.b odd 2 1 3234.2.a.k 1
11.b odd 2 1 5082.2.a.s 1
21.c even 2 1 9702.2.a.bt 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.b 1 1.a even 1 1 trivial
1386.2.a.i 1 3.b odd 2 1
3234.2.a.k 1 7.b odd 2 1
3696.2.a.y 1 4.b odd 2 1
5082.2.a.s 1 11.b odd 2 1
9702.2.a.bt 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

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