# Properties

 Label 208.4.i.a Level $208$ Weight $4$ Character orbit 208.i Analytic conductor $12.272$ 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] = [208,4,Mod(81,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("208.81");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 208.i (of order $$3$$, degree $$2$$, 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: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (3 \zeta_{6} - 3) q^{3} + 2 q^{5} - 5 \zeta_{6} q^{7} + 18 \zeta_{6} q^{9} +O(q^{10})$$ q + (3*z - 3) * q^3 + 2 * q^5 - 5*z * q^7 + 18*z * q^9 $$q + (3 \zeta_{6} - 3) q^{3} + 2 q^{5} - 5 \zeta_{6} q^{7} + 18 \zeta_{6} q^{9} + ( - 13 \zeta_{6} + 13) q^{11} + (52 \zeta_{6} - 39) q^{13} + (6 \zeta_{6} - 6) q^{15} - 27 \zeta_{6} q^{17} + 75 \zeta_{6} q^{19} + 15 q^{21} + (187 \zeta_{6} - 187) q^{23} - 121 q^{25} - 135 q^{27} + ( - 13 \zeta_{6} + 13) q^{29} + 104 q^{31} + 39 \zeta_{6} q^{33} - 10 \zeta_{6} q^{35} + (423 \zeta_{6} - 423) q^{37} + ( - 117 \zeta_{6} - 39) q^{39} + (195 \zeta_{6} - 195) q^{41} + 199 \zeta_{6} q^{43} + 36 \zeta_{6} q^{45} - 388 q^{47} + ( - 318 \zeta_{6} + 318) q^{49} + 81 q^{51} + 618 q^{53} + ( - 26 \zeta_{6} + 26) q^{55} - 225 q^{57} + 491 \zeta_{6} q^{59} - 175 \zeta_{6} q^{61} + ( - 90 \zeta_{6} + 90) q^{63} + (104 \zeta_{6} - 78) q^{65} + ( - 817 \zeta_{6} + 817) q^{67} - 561 \zeta_{6} q^{69} + 79 \zeta_{6} q^{71} + 230 q^{73} + ( - 363 \zeta_{6} + 363) q^{75} - 65 q^{77} - 764 q^{79} + (81 \zeta_{6} - 81) q^{81} + 732 q^{83} - 54 \zeta_{6} q^{85} + 39 \zeta_{6} q^{87} + ( - 1041 \zeta_{6} + 1041) q^{89} + ( - 65 \zeta_{6} + 260) q^{91} + (312 \zeta_{6} - 312) q^{93} + 150 \zeta_{6} q^{95} + 97 \zeta_{6} q^{97} + 234 q^{99} +O(q^{100})$$ q + (3*z - 3) * q^3 + 2 * q^5 - 5*z * q^7 + 18*z * q^9 + (-13*z + 13) * q^11 + (52*z - 39) * q^13 + (6*z - 6) * q^15 - 27*z * q^17 + 75*z * q^19 + 15 * q^21 + (187*z - 187) * q^23 - 121 * q^25 - 135 * q^27 + (-13*z + 13) * q^29 + 104 * q^31 + 39*z * q^33 - 10*z * q^35 + (423*z - 423) * q^37 + (-117*z - 39) * q^39 + (195*z - 195) * q^41 + 199*z * q^43 + 36*z * q^45 - 388 * q^47 + (-318*z + 318) * q^49 + 81 * q^51 + 618 * q^53 + (-26*z + 26) * q^55 - 225 * q^57 + 491*z * q^59 - 175*z * q^61 + (-90*z + 90) * q^63 + (104*z - 78) * q^65 + (-817*z + 817) * q^67 - 561*z * q^69 + 79*z * q^71 + 230 * q^73 + (-363*z + 363) * q^75 - 65 * q^77 - 764 * q^79 + (81*z - 81) * q^81 + 732 * q^83 - 54*z * q^85 + 39*z * q^87 + (-1041*z + 1041) * q^89 + (-65*z + 260) * q^91 + (312*z - 312) * q^93 + 150*z * q^95 + 97*z * q^97 + 234 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + 4 q^{5} - 5 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 + 4 * q^5 - 5 * q^7 + 18 * q^9 $$2 q - 3 q^{3} + 4 q^{5} - 5 q^{7} + 18 q^{9} + 13 q^{11} - 26 q^{13} - 6 q^{15} - 27 q^{17} + 75 q^{19} + 30 q^{21} - 187 q^{23} - 242 q^{25} - 270 q^{27} + 13 q^{29} + 208 q^{31} + 39 q^{33} - 10 q^{35} - 423 q^{37} - 195 q^{39} - 195 q^{41} + 199 q^{43} + 36 q^{45} - 776 q^{47} + 318 q^{49} + 162 q^{51} + 1236 q^{53} + 26 q^{55} - 450 q^{57} + 491 q^{59} - 175 q^{61} + 90 q^{63} - 52 q^{65} + 817 q^{67} - 561 q^{69} + 79 q^{71} + 460 q^{73} + 363 q^{75} - 130 q^{77} - 1528 q^{79} - 81 q^{81} + 1464 q^{83} - 54 q^{85} + 39 q^{87} + 1041 q^{89} + 455 q^{91} - 312 q^{93} + 150 q^{95} + 97 q^{97} + 468 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 + 4 * q^5 - 5 * q^7 + 18 * q^9 + 13 * q^11 - 26 * q^13 - 6 * q^15 - 27 * q^17 + 75 * q^19 + 30 * q^21 - 187 * q^23 - 242 * q^25 - 270 * q^27 + 13 * q^29 + 208 * q^31 + 39 * q^33 - 10 * q^35 - 423 * q^37 - 195 * q^39 - 195 * q^41 + 199 * q^43 + 36 * q^45 - 776 * q^47 + 318 * q^49 + 162 * q^51 + 1236 * q^53 + 26 * q^55 - 450 * q^57 + 491 * q^59 - 175 * q^61 + 90 * q^63 - 52 * q^65 + 817 * q^67 - 561 * q^69 + 79 * q^71 + 460 * q^73 + 363 * q^75 - 130 * q^77 - 1528 * q^79 - 81 * q^81 + 1464 * q^83 - 54 * q^85 + 39 * q^87 + 1041 * q^89 + 455 * q^91 - 312 * q^93 + 150 * q^95 + 97 * q^97 + 468 * q^99

## Character values

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

 $$n$$ $$53$$ $$79$$ $$145$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.50000 2.59808i 0 2.00000 0 −2.50000 + 4.33013i 0 9.00000 15.5885i 0
113.1 0 −1.50000 + 2.59808i 0 2.00000 0 −2.50000 4.33013i 0 9.00000 + 15.5885i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.4.i.a 2
4.b odd 2 1 26.4.c.a 2
12.b even 2 1 234.4.h.b 2
13.c even 3 1 inner 208.4.i.a 2
52.b odd 2 1 338.4.c.d 2
52.f even 4 2 338.4.e.d 4
52.i odd 6 1 338.4.a.d 1
52.i odd 6 1 338.4.c.d 2
52.j odd 6 1 26.4.c.a 2
52.j odd 6 1 338.4.a.a 1
52.l even 12 2 338.4.b.a 2
52.l even 12 2 338.4.e.d 4
156.p even 6 1 234.4.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.c.a 2 4.b odd 2 1
26.4.c.a 2 52.j odd 6 1
208.4.i.a 2 1.a even 1 1 trivial
208.4.i.a 2 13.c even 3 1 inner
234.4.h.b 2 12.b even 2 1
234.4.h.b 2 156.p even 6 1
338.4.a.a 1 52.j odd 6 1
338.4.a.d 1 52.i odd 6 1
338.4.b.a 2 52.l even 12 2
338.4.c.d 2 52.b odd 2 1
338.4.c.d 2 52.i odd 6 1
338.4.e.d 4 52.f even 4 2
338.4.e.d 4 52.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 3T_{3} + 9$$ acting on $$S_{4}^{\mathrm{new}}(208, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$(T - 2)^{2}$$
$7$ $$T^{2} + 5T + 25$$
$11$ $$T^{2} - 13T + 169$$
$13$ $$T^{2} + 26T + 2197$$
$17$ $$T^{2} + 27T + 729$$
$19$ $$T^{2} - 75T + 5625$$
$23$ $$T^{2} + 187T + 34969$$
$29$ $$T^{2} - 13T + 169$$
$31$ $$(T - 104)^{2}$$
$37$ $$T^{2} + 423T + 178929$$
$41$ $$T^{2} + 195T + 38025$$
$43$ $$T^{2} - 199T + 39601$$
$47$ $$(T + 388)^{2}$$
$53$ $$(T - 618)^{2}$$
$59$ $$T^{2} - 491T + 241081$$
$61$ $$T^{2} + 175T + 30625$$
$67$ $$T^{2} - 817T + 667489$$
$71$ $$T^{2} - 79T + 6241$$
$73$ $$(T - 230)^{2}$$
$79$ $$(T + 764)^{2}$$
$83$ $$(T - 732)^{2}$$
$89$ $$T^{2} - 1041 T + 1083681$$
$97$ $$T^{2} - 97T + 9409$$