# Properties

 Label 58.2.a.b Level $58$ Weight $2$ Character orbit 58.a Self dual yes Analytic conductor $0.463$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 2 q^{7} + q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 - 2 * q^7 + q^8 - 2 * q^9 $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 2 q^{7} + q^{8} - 2 q^{9} + q^{10} - 3 q^{11} - q^{12} - q^{13} - 2 q^{14} - q^{15} + q^{16} + 8 q^{17} - 2 q^{18} + q^{20} + 2 q^{21} - 3 q^{22} + 4 q^{23} - q^{24} - 4 q^{25} - q^{26} + 5 q^{27} - 2 q^{28} - q^{29} - q^{30} - 3 q^{31} + q^{32} + 3 q^{33} + 8 q^{34} - 2 q^{35} - 2 q^{36} + 8 q^{37} + q^{39} + q^{40} + 2 q^{41} + 2 q^{42} - 11 q^{43} - 3 q^{44} - 2 q^{45} + 4 q^{46} + 13 q^{47} - q^{48} - 3 q^{49} - 4 q^{50} - 8 q^{51} - q^{52} - 11 q^{53} + 5 q^{54} - 3 q^{55} - 2 q^{56} - q^{58} - q^{60} - 8 q^{61} - 3 q^{62} + 4 q^{63} + q^{64} - q^{65} + 3 q^{66} - 12 q^{67} + 8 q^{68} - 4 q^{69} - 2 q^{70} + 2 q^{71} - 2 q^{72} + 4 q^{73} + 8 q^{74} + 4 q^{75} + 6 q^{77} + q^{78} + 15 q^{79} + q^{80} + q^{81} + 2 q^{82} + 4 q^{83} + 2 q^{84} + 8 q^{85} - 11 q^{86} + q^{87} - 3 q^{88} - 10 q^{89} - 2 q^{90} + 2 q^{91} + 4 q^{92} + 3 q^{93} + 13 q^{94} - q^{96} - 2 q^{97} - 3 q^{98} + 6 q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 - 2 * q^7 + q^8 - 2 * q^9 + q^10 - 3 * q^11 - q^12 - q^13 - 2 * q^14 - q^15 + q^16 + 8 * q^17 - 2 * q^18 + q^20 + 2 * q^21 - 3 * q^22 + 4 * q^23 - q^24 - 4 * q^25 - q^26 + 5 * q^27 - 2 * q^28 - q^29 - q^30 - 3 * q^31 + q^32 + 3 * q^33 + 8 * q^34 - 2 * q^35 - 2 * q^36 + 8 * q^37 + q^39 + q^40 + 2 * q^41 + 2 * q^42 - 11 * q^43 - 3 * q^44 - 2 * q^45 + 4 * q^46 + 13 * q^47 - q^48 - 3 * q^49 - 4 * q^50 - 8 * q^51 - q^52 - 11 * q^53 + 5 * q^54 - 3 * q^55 - 2 * q^56 - q^58 - q^60 - 8 * q^61 - 3 * q^62 + 4 * q^63 + q^64 - q^65 + 3 * q^66 - 12 * q^67 + 8 * q^68 - 4 * q^69 - 2 * q^70 + 2 * q^71 - 2 * q^72 + 4 * q^73 + 8 * q^74 + 4 * q^75 + 6 * q^77 + q^78 + 15 * q^79 + q^80 + q^81 + 2 * q^82 + 4 * q^83 + 2 * q^84 + 8 * q^85 - 11 * q^86 + q^87 - 3 * q^88 - 10 * q^89 - 2 * q^90 + 2 * q^91 + 4 * q^92 + 3 * q^93 + 13 * q^94 - q^96 - 2 * 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 −1.00000 1.00000 1.00000 −1.00000 −2.00000 1.00000 −2.00000 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
$$2$$ $$-1$$
$$29$$ $$+1$$

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(58))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T + 2$$
$11$ $$T + 3$$
$13$ $$T + 1$$
$17$ $$T - 8$$
$19$ $$T$$
$23$ $$T - 4$$
$29$ $$T + 1$$
$31$ $$T + 3$$
$37$ $$T - 8$$
$41$ $$T - 2$$
$43$ $$T + 11$$
$47$ $$T - 13$$
$53$ $$T + 11$$
$59$ $$T$$
$61$ $$T + 8$$
$67$ $$T + 12$$
$71$ $$T - 2$$
$73$ $$T - 4$$
$79$ $$T - 15$$
$83$ $$T - 4$$
$89$ $$T + 10$$
$97$ $$T + 2$$