# Properties

 Label 58.2.a.a Level $58$ Weight $2$ Character orbit 58.a Self dual yes Analytic conductor $0.463$ 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] = [58,2,Mod(1,58)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(58, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("58.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$58 = 2 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 58.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: $$0.463132331723$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - 3 q^{3} + q^{4} - 3 q^{5} + 3 q^{6} - 2 q^{7} - q^{8} + 6 q^{9}+O(q^{10})$$ q - q^2 - 3 * q^3 + q^4 - 3 * q^5 + 3 * q^6 - 2 * q^7 - q^8 + 6 * q^9 $$q - q^{2} - 3 q^{3} + q^{4} - 3 q^{5} + 3 q^{6} - 2 q^{7} - q^{8} + 6 q^{9} + 3 q^{10} - q^{11} - 3 q^{12} + 3 q^{13} + 2 q^{14} + 9 q^{15} + q^{16} - 4 q^{17} - 6 q^{18} - 8 q^{19} - 3 q^{20} + 6 q^{21} + q^{22} + 3 q^{24} + 4 q^{25} - 3 q^{26} - 9 q^{27} - 2 q^{28} - q^{29} - 9 q^{30} + 3 q^{31} - q^{32} + 3 q^{33} + 4 q^{34} + 6 q^{35} + 6 q^{36} - 8 q^{37} + 8 q^{38} - 9 q^{39} + 3 q^{40} - 2 q^{41} - 6 q^{42} + 7 q^{43} - q^{44} - 18 q^{45} + 11 q^{47} - 3 q^{48} - 3 q^{49} - 4 q^{50} + 12 q^{51} + 3 q^{52} + q^{53} + 9 q^{54} + 3 q^{55} + 2 q^{56} + 24 q^{57} + q^{58} - 4 q^{59} + 9 q^{60} + 4 q^{61} - 3 q^{62} - 12 q^{63} + q^{64} - 9 q^{65} - 3 q^{66} - 4 q^{67} - 4 q^{68} - 6 q^{70} - 2 q^{71} - 6 q^{72} - 12 q^{73} + 8 q^{74} - 12 q^{75} - 8 q^{76} + 2 q^{77} + 9 q^{78} - 7 q^{79} - 3 q^{80} + 9 q^{81} + 2 q^{82} + 6 q^{84} + 12 q^{85} - 7 q^{86} + 3 q^{87} + q^{88} - 6 q^{89} + 18 q^{90} - 6 q^{91} - 9 q^{93} - 11 q^{94} + 24 q^{95} + 3 q^{96} - 6 q^{97} + 3 q^{98} - 6 q^{99}+O(q^{100})$$ q - q^2 - 3 * q^3 + q^4 - 3 * q^5 + 3 * q^6 - 2 * q^7 - q^8 + 6 * q^9 + 3 * q^10 - q^11 - 3 * q^12 + 3 * q^13 + 2 * q^14 + 9 * q^15 + q^16 - 4 * q^17 - 6 * q^18 - 8 * q^19 - 3 * q^20 + 6 * q^21 + q^22 + 3 * q^24 + 4 * q^25 - 3 * q^26 - 9 * q^27 - 2 * q^28 - q^29 - 9 * q^30 + 3 * q^31 - q^32 + 3 * q^33 + 4 * q^34 + 6 * q^35 + 6 * q^36 - 8 * q^37 + 8 * q^38 - 9 * q^39 + 3 * q^40 - 2 * q^41 - 6 * q^42 + 7 * q^43 - q^44 - 18 * q^45 + 11 * q^47 - 3 * q^48 - 3 * q^49 - 4 * q^50 + 12 * q^51 + 3 * q^52 + q^53 + 9 * q^54 + 3 * q^55 + 2 * q^56 + 24 * q^57 + q^58 - 4 * q^59 + 9 * q^60 + 4 * q^61 - 3 * q^62 - 12 * q^63 + q^64 - 9 * q^65 - 3 * q^66 - 4 * q^67 - 4 * q^68 - 6 * q^70 - 2 * q^71 - 6 * q^72 - 12 * q^73 + 8 * q^74 - 12 * q^75 - 8 * q^76 + 2 * q^77 + 9 * q^78 - 7 * q^79 - 3 * q^80 + 9 * q^81 + 2 * q^82 + 6 * q^84 + 12 * q^85 - 7 * q^86 + 3 * q^87 + q^88 - 6 * q^89 + 18 * q^90 - 6 * q^91 - 9 * q^93 - 11 * q^94 + 24 * q^95 + 3 * q^96 - 6 * q^97 + 3 * 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 −3.00000 1.00000 −3.00000 3.00000 −2.00000 −1.00000 6.00000 3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$29$$ $$1$$

## 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 58.2.a.a 1
3.b odd 2 1 522.2.a.k 1
4.b odd 2 1 464.2.a.f 1
5.b even 2 1 1450.2.a.i 1
5.c odd 4 2 1450.2.b.f 2
7.b odd 2 1 2842.2.a.d 1
8.b even 2 1 1856.2.a.p 1
8.d odd 2 1 1856.2.a.b 1
11.b odd 2 1 7018.2.a.c 1
12.b even 2 1 4176.2.a.bh 1
13.b even 2 1 9802.2.a.d 1
29.b even 2 1 1682.2.a.j 1
29.c odd 4 2 1682.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.a.a 1 1.a even 1 1 trivial
464.2.a.f 1 4.b odd 2 1
522.2.a.k 1 3.b odd 2 1
1450.2.a.i 1 5.b even 2 1
1450.2.b.f 2 5.c odd 4 2
1682.2.a.j 1 29.b even 2 1
1682.2.b.e 2 29.c odd 4 2
1856.2.a.b 1 8.d odd 2 1
1856.2.a.p 1 8.b even 2 1
2842.2.a.d 1 7.b odd 2 1
4176.2.a.bh 1 12.b even 2 1
7018.2.a.c 1 11.b odd 2 1
9802.2.a.d 1 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(58))$$.

## Hecke characteristic polynomials

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