Properties

 Label 208.4.f.b Level $208$ Weight $4$ Character orbit 208.f Analytic conductor $12.272$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(129,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("208.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 208.f (of order $$2$$, degree $$1$$, not 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: $$12.2723972812$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 13) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + 3 i q^{5} - 5 i q^{7} - 26 q^{9} +O(q^{10})$$ q + q^3 + 3*i * q^5 - 5*i * q^7 - 26 * q^9 $$q + q^{3} + 3 i q^{5} - 5 i q^{7} - 26 q^{9} + 16 i q^{11} + (13 i + 26) q^{13} + 3 i q^{15} - 45 q^{17} + 2 i q^{19} - 5 i q^{21} - 162 q^{23} + 44 q^{25} - 53 q^{27} - 144 q^{29} + 88 i q^{31} + 16 i q^{33} + 135 q^{35} + 101 i q^{37} + (13 i + 26) q^{39} + 64 i q^{41} + 97 q^{43} - 78 i q^{45} - 37 i q^{47} + 118 q^{49} - 45 q^{51} - 414 q^{53} - 432 q^{55} + 2 i q^{57} - 174 i q^{59} + 376 q^{61} + 130 i q^{63} + (78 i - 351) q^{65} - 12 i q^{67} - 162 q^{69} + 119 i q^{71} - 366 i q^{73} + 44 q^{75} + 720 q^{77} + 830 q^{79} + 649 q^{81} - 146 i q^{83} - 135 i q^{85} - 144 q^{87} - 146 i q^{89} + ( - 130 i + 585) q^{91} + 88 i q^{93} - 54 q^{95} + 284 i q^{97} - 416 i q^{99} +O(q^{100})$$ q + q^3 + 3*i * q^5 - 5*i * q^7 - 26 * q^9 + 16*i * q^11 + (13*i + 26) * q^13 + 3*i * q^15 - 45 * q^17 + 2*i * q^19 - 5*i * q^21 - 162 * q^23 + 44 * q^25 - 53 * q^27 - 144 * q^29 + 88*i * q^31 + 16*i * q^33 + 135 * q^35 + 101*i * q^37 + (13*i + 26) * q^39 + 64*i * q^41 + 97 * q^43 - 78*i * q^45 - 37*i * q^47 + 118 * q^49 - 45 * q^51 - 414 * q^53 - 432 * q^55 + 2*i * q^57 - 174*i * q^59 + 376 * q^61 + 130*i * q^63 + (78*i - 351) * q^65 - 12*i * q^67 - 162 * q^69 + 119*i * q^71 - 366*i * q^73 + 44 * q^75 + 720 * q^77 + 830 * q^79 + 649 * q^81 - 146*i * q^83 - 135*i * q^85 - 144 * q^87 - 146*i * q^89 + (-130*i + 585) * q^91 + 88*i * q^93 - 54 * q^95 + 284*i * q^97 - 416*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 52 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 52 * q^9 $$2 q + 2 q^{3} - 52 q^{9} + 52 q^{13} - 90 q^{17} - 324 q^{23} + 88 q^{25} - 106 q^{27} - 288 q^{29} + 270 q^{35} + 52 q^{39} + 194 q^{43} + 236 q^{49} - 90 q^{51} - 828 q^{53} - 864 q^{55} + 752 q^{61} - 702 q^{65} - 324 q^{69} + 88 q^{75} + 1440 q^{77} + 1660 q^{79} + 1298 q^{81} - 288 q^{87} + 1170 q^{91} - 108 q^{95}+O(q^{100})$$ 2 * q + 2 * q^3 - 52 * q^9 + 52 * q^13 - 90 * q^17 - 324 * q^23 + 88 * q^25 - 106 * q^27 - 288 * q^29 + 270 * q^35 + 52 * q^39 + 194 * q^43 + 236 * q^49 - 90 * q^51 - 828 * q^53 - 864 * q^55 + 752 * q^61 - 702 * q^65 - 324 * q^69 + 88 * q^75 + 1440 * q^77 + 1660 * q^79 + 1298 * q^81 - 288 * q^87 + 1170 * q^91 - 108 * q^95

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$$.

 $$n$$ $$53$$ $$79$$ $$145$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 − 1.00000i 1.00000i
0 1.00000 0 9.00000i 0 15.0000i 0 −26.0000 0
129.2 0 1.00000 0 9.00000i 0 15.0000i 0 −26.0000 0
 $$n$$: 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
13.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.4.f.b 2
4.b odd 2 1 13.4.b.a 2
8.b even 2 1 832.4.f.c 2
8.d odd 2 1 832.4.f.e 2
12.b even 2 1 117.4.b.a 2
13.b even 2 1 inner 208.4.f.b 2
20.d odd 2 1 325.4.c.b 2
20.e even 4 1 325.4.d.a 2
20.e even 4 1 325.4.d.b 2
52.b odd 2 1 13.4.b.a 2
52.f even 4 1 169.4.a.b 1
52.f even 4 1 169.4.a.c 1
52.i odd 6 2 169.4.e.d 4
52.j odd 6 2 169.4.e.d 4
52.l even 12 2 169.4.c.b 2
52.l even 12 2 169.4.c.c 2
104.e even 2 1 832.4.f.c 2
104.h odd 2 1 832.4.f.e 2
156.h even 2 1 117.4.b.a 2
156.l odd 4 1 1521.4.a.d 1
156.l odd 4 1 1521.4.a.i 1
260.g odd 2 1 325.4.c.b 2
260.p even 4 1 325.4.d.a 2
260.p even 4 1 325.4.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 4.b odd 2 1
13.4.b.a 2 52.b odd 2 1
117.4.b.a 2 12.b even 2 1
117.4.b.a 2 156.h even 2 1
169.4.a.b 1 52.f even 4 1
169.4.a.c 1 52.f even 4 1
169.4.c.b 2 52.l even 12 2
169.4.c.c 2 52.l even 12 2
169.4.e.d 4 52.i odd 6 2
169.4.e.d 4 52.j odd 6 2
208.4.f.b 2 1.a even 1 1 trivial
208.4.f.b 2 13.b even 2 1 inner
325.4.c.b 2 20.d odd 2 1
325.4.c.b 2 260.g odd 2 1
325.4.d.a 2 20.e even 4 1
325.4.d.a 2 260.p even 4 1
325.4.d.b 2 20.e even 4 1
325.4.d.b 2 260.p even 4 1
832.4.f.c 2 8.b even 2 1
832.4.f.c 2 104.e even 2 1
832.4.f.e 2 8.d odd 2 1
832.4.f.e 2 104.h odd 2 1
1521.4.a.d 1 156.l odd 4 1
1521.4.a.i 1 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 1$$ acting on $$S_{4}^{\mathrm{new}}(208, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 81$$
$7$ $$T^{2} + 225$$
$11$ $$T^{2} + 2304$$
$13$ $$T^{2} - 52T + 2197$$
$17$ $$(T + 45)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T + 162)^{2}$$
$29$ $$(T + 144)^{2}$$
$31$ $$T^{2} + 69696$$
$37$ $$T^{2} + 91809$$
$41$ $$T^{2} + 36864$$
$43$ $$(T - 97)^{2}$$
$47$ $$T^{2} + 12321$$
$53$ $$(T + 414)^{2}$$
$59$ $$T^{2} + 272484$$
$61$ $$(T - 376)^{2}$$
$67$ $$T^{2} + 1296$$
$71$ $$T^{2} + 127449$$
$73$ $$T^{2} + 1205604$$
$79$ $$(T - 830)^{2}$$
$83$ $$T^{2} + 191844$$
$89$ $$T^{2} + 191844$$
$97$ $$T^{2} + 725904$$