# Properties

 Label 58.2.b.a Level $58$ Weight $2$ Character orbit 58.b Analytic conductor $0.463$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [58,2,Mod(57,58)]

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([1]))

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

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

chi := DirichletCharacter("58.57");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$58 = 2 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 58.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$0.463132331723$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} - q^{4} + q^{5} - q^{6} - 2 q^{7} - i q^{8} + 2 q^{9} +O(q^{10})$$ q + i * q^2 + i * q^3 - q^4 + q^5 - q^6 - 2 * q^7 - i * q^8 + 2 * q^9 $$q + i q^{2} + i q^{3} - q^{4} + q^{5} - q^{6} - 2 q^{7} - i q^{8} + 2 q^{9} + i q^{10} - 5 i q^{11} - i q^{12} - q^{13} - 2 i q^{14} + i q^{15} + q^{16} - 2 i q^{17} + 2 i q^{18} + 4 i q^{19} - q^{20} - 2 i q^{21} + 5 q^{22} - 6 q^{23} + q^{24} - 4 q^{25} - i q^{26} + 5 i q^{27} + 2 q^{28} + (2 i + 5) q^{29} - q^{30} + 5 i q^{31} + i q^{32} + 5 q^{33} + 2 q^{34} - 2 q^{35} - 2 q^{36} + 8 i q^{37} - 4 q^{38} - i q^{39} - i q^{40} - 10 i q^{41} + 2 q^{42} - 9 i q^{43} + 5 i q^{44} + 2 q^{45} - 6 i q^{46} + 3 i q^{47} + i q^{48} - 3 q^{49} - 4 i q^{50} + 2 q^{51} + q^{52} - q^{53} - 5 q^{54} - 5 i q^{55} + 2 i q^{56} - 4 q^{57} + (5 i - 2) q^{58} + 10 q^{59} - i q^{60} - 10 i q^{61} - 5 q^{62} - 4 q^{63} - q^{64} - q^{65} + 5 i q^{66} + 8 q^{67} + 2 i q^{68} - 6 i q^{69} - 2 i q^{70} - 8 q^{71} - 2 i q^{72} + 16 i q^{73} - 8 q^{74} - 4 i q^{75} - 4 i q^{76} + 10 i q^{77} + q^{78} - i q^{79} + q^{80} + q^{81} + 10 q^{82} + 14 q^{83} + 2 i q^{84} - 2 i q^{85} + 9 q^{86} + (5 i - 2) q^{87} - 5 q^{88} + 14 i q^{89} + 2 i q^{90} + 2 q^{91} + 6 q^{92} - 5 q^{93} - 3 q^{94} + 4 i q^{95} - q^{96} - 2 i q^{97} - 3 i q^{98} - 10 i q^{99} +O(q^{100})$$ q + i * q^2 + i * q^3 - q^4 + q^5 - q^6 - 2 * q^7 - i * q^8 + 2 * q^9 + i * q^10 - 5*i * q^11 - i * q^12 - q^13 - 2*i * q^14 + i * q^15 + q^16 - 2*i * q^17 + 2*i * q^18 + 4*i * q^19 - q^20 - 2*i * q^21 + 5 * q^22 - 6 * q^23 + q^24 - 4 * q^25 - i * q^26 + 5*i * q^27 + 2 * q^28 + (2*i + 5) * q^29 - q^30 + 5*i * q^31 + i * q^32 + 5 * q^33 + 2 * q^34 - 2 * q^35 - 2 * q^36 + 8*i * q^37 - 4 * q^38 - i * q^39 - i * q^40 - 10*i * q^41 + 2 * q^42 - 9*i * q^43 + 5*i * q^44 + 2 * q^45 - 6*i * q^46 + 3*i * q^47 + i * q^48 - 3 * q^49 - 4*i * q^50 + 2 * q^51 + q^52 - q^53 - 5 * q^54 - 5*i * q^55 + 2*i * q^56 - 4 * q^57 + (5*i - 2) * q^58 + 10 * q^59 - i * q^60 - 10*i * q^61 - 5 * q^62 - 4 * q^63 - q^64 - q^65 + 5*i * q^66 + 8 * q^67 + 2*i * q^68 - 6*i * q^69 - 2*i * q^70 - 8 * q^71 - 2*i * q^72 + 16*i * q^73 - 8 * q^74 - 4*i * q^75 - 4*i * q^76 + 10*i * q^77 + q^78 - i * q^79 + q^80 + q^81 + 10 * q^82 + 14 * q^83 + 2*i * q^84 - 2*i * q^85 + 9 * q^86 + (5*i - 2) * q^87 - 5 * q^88 + 14*i * q^89 + 2*i * q^90 + 2 * q^91 + 6 * q^92 - 5 * q^93 - 3 * q^94 + 4*i * q^95 - q^96 - 2*i * q^97 - 3*i * q^98 - 10*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{5} - 2 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^5 - 2 * q^6 - 4 * q^7 + 4 * q^9 $$2 q - 2 q^{4} + 2 q^{5} - 2 q^{6} - 4 q^{7} + 4 q^{9} - 2 q^{13} + 2 q^{16} - 2 q^{20} + 10 q^{22} - 12 q^{23} + 2 q^{24} - 8 q^{25} + 4 q^{28} + 10 q^{29} - 2 q^{30} + 10 q^{33} + 4 q^{34} - 4 q^{35} - 4 q^{36} - 8 q^{38} + 4 q^{42} + 4 q^{45} - 6 q^{49} + 4 q^{51} + 2 q^{52} - 2 q^{53} - 10 q^{54} - 8 q^{57} - 4 q^{58} + 20 q^{59} - 10 q^{62} - 8 q^{63} - 2 q^{64} - 2 q^{65} + 16 q^{67} - 16 q^{71} - 16 q^{74} + 2 q^{78} + 2 q^{80} + 2 q^{81} + 20 q^{82} + 28 q^{83} + 18 q^{86} - 4 q^{87} - 10 q^{88} + 4 q^{91} + 12 q^{92} - 10 q^{93} - 6 q^{94} - 2 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^5 - 2 * q^6 - 4 * q^7 + 4 * q^9 - 2 * q^13 + 2 * q^16 - 2 * q^20 + 10 * q^22 - 12 * q^23 + 2 * q^24 - 8 * q^25 + 4 * q^28 + 10 * q^29 - 2 * q^30 + 10 * q^33 + 4 * q^34 - 4 * q^35 - 4 * q^36 - 8 * q^38 + 4 * q^42 + 4 * q^45 - 6 * q^49 + 4 * q^51 + 2 * q^52 - 2 * q^53 - 10 * q^54 - 8 * q^57 - 4 * q^58 + 20 * q^59 - 10 * q^62 - 8 * q^63 - 2 * q^64 - 2 * q^65 + 16 * q^67 - 16 * q^71 - 16 * q^74 + 2 * q^78 + 2 * q^80 + 2 * q^81 + 20 * q^82 + 28 * q^83 + 18 * q^86 - 4 * q^87 - 10 * q^88 + 4 * q^91 + 12 * q^92 - 10 * q^93 - 6 * q^94 - 2 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/58\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 1.00000 −1.00000 −2.00000 1.00000i 2.00000 1.00000i
57.2 1.00000i 1.00000i −1.00000 1.00000 −1.00000 −2.00000 1.00000i 2.00000 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 58.2.b.a 2
3.b odd 2 1 522.2.d.a 2
4.b odd 2 1 464.2.e.c 2
5.b even 2 1 1450.2.c.a 2
5.c odd 4 1 1450.2.d.b 2
5.c odd 4 1 1450.2.d.c 2
8.b even 2 1 1856.2.e.b 2
8.d odd 2 1 1856.2.e.d 2
12.b even 2 1 4176.2.o.d 2
29.b even 2 1 inner 58.2.b.a 2
29.c odd 4 1 1682.2.a.c 1
29.c odd 4 1 1682.2.a.g 1
87.d odd 2 1 522.2.d.a 2
116.d odd 2 1 464.2.e.c 2
145.d even 2 1 1450.2.c.a 2
145.h odd 4 1 1450.2.d.b 2
145.h odd 4 1 1450.2.d.c 2
232.b odd 2 1 1856.2.e.d 2
232.g even 2 1 1856.2.e.b 2
348.b even 2 1 4176.2.o.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.b.a 2 1.a even 1 1 trivial
58.2.b.a 2 29.b even 2 1 inner
464.2.e.c 2 4.b odd 2 1
464.2.e.c 2 116.d odd 2 1
522.2.d.a 2 3.b odd 2 1
522.2.d.a 2 87.d odd 2 1
1450.2.c.a 2 5.b even 2 1
1450.2.c.a 2 145.d even 2 1
1450.2.d.b 2 5.c odd 4 1
1450.2.d.b 2 145.h odd 4 1
1450.2.d.c 2 5.c odd 4 1
1450.2.d.c 2 145.h odd 4 1
1682.2.a.c 1 29.c odd 4 1
1682.2.a.g 1 29.c odd 4 1
1856.2.e.b 2 8.b even 2 1
1856.2.e.b 2 232.g even 2 1
1856.2.e.d 2 8.d odd 2 1
1856.2.e.d 2 232.b odd 2 1
4176.2.o.d 2 12.b even 2 1
4176.2.o.d 2 348.b even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(58, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} + 25$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2} + 16$$
$23$ $$(T + 6)^{2}$$
$29$ $$T^{2} - 10T + 29$$
$31$ $$T^{2} + 25$$
$37$ $$T^{2} + 64$$
$41$ $$T^{2} + 100$$
$43$ $$T^{2} + 81$$
$47$ $$T^{2} + 9$$
$53$ $$(T + 1)^{2}$$
$59$ $$(T - 10)^{2}$$
$61$ $$T^{2} + 100$$
$67$ $$(T - 8)^{2}$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 256$$
$79$ $$T^{2} + 1$$
$83$ $$(T - 14)^{2}$$
$89$ $$T^{2} + 196$$
$97$ $$T^{2} + 4$$