# Properties

 Label 63.4.e.b Level $63$ Weight $4$ Character orbit 63.e Analytic conductor $3.717$ 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] = [63,4,Mod(37,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("63.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 63.e (of order $$3$$, degree $$2$$, 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: $$3.71712033036$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} + 4 \zeta_{6} q^{4} + ( - 7 \zeta_{6} + 7) q^{5} + ( - 14 \zeta_{6} + 21) q^{7} + 24 q^{8}+O(q^{10})$$ q + (-2*z + 2) * q^2 + 4*z * q^4 + (-7*z + 7) * q^5 + (-14*z + 21) * q^7 + 24 * q^8 $$q + ( - 2 \zeta_{6} + 2) q^{2} + 4 \zeta_{6} q^{4} + ( - 7 \zeta_{6} + 7) q^{5} + ( - 14 \zeta_{6} + 21) q^{7} + 24 q^{8} - 14 \zeta_{6} q^{10} - 5 \zeta_{6} q^{11} - 14 q^{13} + ( - 42 \zeta_{6} + 14) q^{14} + ( - 16 \zeta_{6} + 16) q^{16} - 21 \zeta_{6} q^{17} + (49 \zeta_{6} - 49) q^{19} + 28 q^{20} - 10 q^{22} + (159 \zeta_{6} - 159) q^{23} + 76 \zeta_{6} q^{25} + (28 \zeta_{6} - 28) q^{26} + (28 \zeta_{6} + 56) q^{28} - 58 q^{29} - 147 \zeta_{6} q^{31} + 160 \zeta_{6} q^{32} - 42 q^{34} + ( - 147 \zeta_{6} + 49) q^{35} + (219 \zeta_{6} - 219) q^{37} + 98 \zeta_{6} q^{38} + ( - 168 \zeta_{6} + 168) q^{40} - 350 q^{41} - 124 q^{43} + ( - 20 \zeta_{6} + 20) q^{44} + 318 \zeta_{6} q^{46} + ( - 525 \zeta_{6} + 525) q^{47} + ( - 392 \zeta_{6} + 245) q^{49} + 152 q^{50} - 56 \zeta_{6} q^{52} + 303 \zeta_{6} q^{53} - 35 q^{55} + ( - 336 \zeta_{6} + 504) q^{56} + (116 \zeta_{6} - 116) q^{58} - 105 \zeta_{6} q^{59} + ( - 413 \zeta_{6} + 413) q^{61} - 294 q^{62} + 448 q^{64} + (98 \zeta_{6} - 98) q^{65} - 415 \zeta_{6} q^{67} + ( - 84 \zeta_{6} + 84) q^{68} + ( - 98 \zeta_{6} - 196) q^{70} + 432 q^{71} + 1113 \zeta_{6} q^{73} + 438 \zeta_{6} q^{74} - 196 q^{76} + ( - 35 \zeta_{6} - 70) q^{77} + ( - 103 \zeta_{6} + 103) q^{79} - 112 \zeta_{6} q^{80} + (700 \zeta_{6} - 700) q^{82} - 1092 q^{83} - 147 q^{85} + (248 \zeta_{6} - 248) q^{86} - 120 \zeta_{6} q^{88} + (329 \zeta_{6} - 329) q^{89} + (196 \zeta_{6} - 294) q^{91} - 636 q^{92} - 1050 \zeta_{6} q^{94} + 343 \zeta_{6} q^{95} - 882 q^{97} + ( - 490 \zeta_{6} - 294) q^{98} +O(q^{100})$$ q + (-2*z + 2) * q^2 + 4*z * q^4 + (-7*z + 7) * q^5 + (-14*z + 21) * q^7 + 24 * q^8 - 14*z * q^10 - 5*z * q^11 - 14 * q^13 + (-42*z + 14) * q^14 + (-16*z + 16) * q^16 - 21*z * q^17 + (49*z - 49) * q^19 + 28 * q^20 - 10 * q^22 + (159*z - 159) * q^23 + 76*z * q^25 + (28*z - 28) * q^26 + (28*z + 56) * q^28 - 58 * q^29 - 147*z * q^31 + 160*z * q^32 - 42 * q^34 + (-147*z + 49) * q^35 + (219*z - 219) * q^37 + 98*z * q^38 + (-168*z + 168) * q^40 - 350 * q^41 - 124 * q^43 + (-20*z + 20) * q^44 + 318*z * q^46 + (-525*z + 525) * q^47 + (-392*z + 245) * q^49 + 152 * q^50 - 56*z * q^52 + 303*z * q^53 - 35 * q^55 + (-336*z + 504) * q^56 + (116*z - 116) * q^58 - 105*z * q^59 + (-413*z + 413) * q^61 - 294 * q^62 + 448 * q^64 + (98*z - 98) * q^65 - 415*z * q^67 + (-84*z + 84) * q^68 + (-98*z - 196) * q^70 + 432 * q^71 + 1113*z * q^73 + 438*z * q^74 - 196 * q^76 + (-35*z - 70) * q^77 + (-103*z + 103) * q^79 - 112*z * q^80 + (700*z - 700) * q^82 - 1092 * q^83 - 147 * q^85 + (248*z - 248) * q^86 - 120*z * q^88 + (329*z - 329) * q^89 + (196*z - 294) * q^91 - 636 * q^92 - 1050*z * q^94 + 343*z * q^95 - 882 * q^97 + (-490*z - 294) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 4 q^{4} + 7 q^{5} + 28 q^{7} + 48 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 4 * q^4 + 7 * q^5 + 28 * q^7 + 48 * q^8 $$2 q + 2 q^{2} + 4 q^{4} + 7 q^{5} + 28 q^{7} + 48 q^{8} - 14 q^{10} - 5 q^{11} - 28 q^{13} - 14 q^{14} + 16 q^{16} - 21 q^{17} - 49 q^{19} + 56 q^{20} - 20 q^{22} - 159 q^{23} + 76 q^{25} - 28 q^{26} + 140 q^{28} - 116 q^{29} - 147 q^{31} + 160 q^{32} - 84 q^{34} - 49 q^{35} - 219 q^{37} + 98 q^{38} + 168 q^{40} - 700 q^{41} - 248 q^{43} + 20 q^{44} + 318 q^{46} + 525 q^{47} + 98 q^{49} + 304 q^{50} - 56 q^{52} + 303 q^{53} - 70 q^{55} + 672 q^{56} - 116 q^{58} - 105 q^{59} + 413 q^{61} - 588 q^{62} + 896 q^{64} - 98 q^{65} - 415 q^{67} + 84 q^{68} - 490 q^{70} + 864 q^{71} + 1113 q^{73} + 438 q^{74} - 392 q^{76} - 175 q^{77} + 103 q^{79} - 112 q^{80} - 700 q^{82} - 2184 q^{83} - 294 q^{85} - 248 q^{86} - 120 q^{88} - 329 q^{89} - 392 q^{91} - 1272 q^{92} - 1050 q^{94} + 343 q^{95} - 1764 q^{97} - 1078 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 4 * q^4 + 7 * q^5 + 28 * q^7 + 48 * q^8 - 14 * q^10 - 5 * q^11 - 28 * q^13 - 14 * q^14 + 16 * q^16 - 21 * q^17 - 49 * q^19 + 56 * q^20 - 20 * q^22 - 159 * q^23 + 76 * q^25 - 28 * q^26 + 140 * q^28 - 116 * q^29 - 147 * q^31 + 160 * q^32 - 84 * q^34 - 49 * q^35 - 219 * q^37 + 98 * q^38 + 168 * q^40 - 700 * q^41 - 248 * q^43 + 20 * q^44 + 318 * q^46 + 525 * q^47 + 98 * q^49 + 304 * q^50 - 56 * q^52 + 303 * q^53 - 70 * q^55 + 672 * q^56 - 116 * q^58 - 105 * q^59 + 413 * q^61 - 588 * q^62 + 896 * q^64 - 98 * q^65 - 415 * q^67 + 84 * q^68 - 490 * q^70 + 864 * q^71 + 1113 * q^73 + 438 * q^74 - 392 * q^76 - 175 * q^77 + 103 * q^79 - 112 * q^80 - 700 * q^82 - 2184 * q^83 - 294 * q^85 - 248 * q^86 - 120 * q^88 - 329 * q^89 - 392 * q^91 - 1272 * q^92 - 1050 * q^94 + 343 * q^95 - 1764 * q^97 - 1078 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$$.

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 1.73205i 0 2.00000 + 3.46410i 3.50000 6.06218i 0 14.0000 12.1244i 24.0000 0 −7.00000 12.1244i
46.1 1.00000 + 1.73205i 0 2.00000 3.46410i 3.50000 + 6.06218i 0 14.0000 + 12.1244i 24.0000 0 −7.00000 + 12.1244i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.4.e.b 2
3.b odd 2 1 7.4.c.a 2
7.b odd 2 1 441.4.e.k 2
7.c even 3 1 inner 63.4.e.b 2
7.c even 3 1 441.4.a.d 1
7.d odd 6 1 441.4.a.e 1
7.d odd 6 1 441.4.e.k 2
12.b even 2 1 112.4.i.c 2
15.d odd 2 1 175.4.e.a 2
15.e even 4 2 175.4.k.a 4
21.c even 2 1 49.4.c.a 2
21.g even 6 1 49.4.a.c 1
21.g even 6 1 49.4.c.a 2
21.h odd 6 1 7.4.c.a 2
21.h odd 6 1 49.4.a.d 1
24.f even 2 1 448.4.i.a 2
24.h odd 2 1 448.4.i.f 2
84.j odd 6 1 784.4.a.r 1
84.n even 6 1 112.4.i.c 2
84.n even 6 1 784.4.a.b 1
105.o odd 6 1 175.4.e.a 2
105.o odd 6 1 1225.4.a.c 1
105.p even 6 1 1225.4.a.d 1
105.x even 12 2 175.4.k.a 4
168.s odd 6 1 448.4.i.f 2
168.v even 6 1 448.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 3.b odd 2 1
7.4.c.a 2 21.h odd 6 1
49.4.a.c 1 21.g even 6 1
49.4.a.d 1 21.h odd 6 1
49.4.c.a 2 21.c even 2 1
49.4.c.a 2 21.g even 6 1
63.4.e.b 2 1.a even 1 1 trivial
63.4.e.b 2 7.c even 3 1 inner
112.4.i.c 2 12.b even 2 1
112.4.i.c 2 84.n even 6 1
175.4.e.a 2 15.d odd 2 1
175.4.e.a 2 105.o odd 6 1
175.4.k.a 4 15.e even 4 2
175.4.k.a 4 105.x even 12 2
441.4.a.d 1 7.c even 3 1
441.4.a.e 1 7.d odd 6 1
441.4.e.k 2 7.b odd 2 1
441.4.e.k 2 7.d odd 6 1
448.4.i.a 2 24.f even 2 1
448.4.i.a 2 168.v even 6 1
448.4.i.f 2 24.h odd 2 1
448.4.i.f 2 168.s odd 6 1
784.4.a.b 1 84.n even 6 1
784.4.a.r 1 84.j odd 6 1
1225.4.a.c 1 105.o odd 6 1
1225.4.a.d 1 105.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2T_{2} + 4$$ acting on $$S_{4}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 7T + 49$$
$7$ $$T^{2} - 28T + 343$$
$11$ $$T^{2} + 5T + 25$$
$13$ $$(T + 14)^{2}$$
$17$ $$T^{2} + 21T + 441$$
$19$ $$T^{2} + 49T + 2401$$
$23$ $$T^{2} + 159T + 25281$$
$29$ $$(T + 58)^{2}$$
$31$ $$T^{2} + 147T + 21609$$
$37$ $$T^{2} + 219T + 47961$$
$41$ $$(T + 350)^{2}$$
$43$ $$(T + 124)^{2}$$
$47$ $$T^{2} - 525T + 275625$$
$53$ $$T^{2} - 303T + 91809$$
$59$ $$T^{2} + 105T + 11025$$
$61$ $$T^{2} - 413T + 170569$$
$67$ $$T^{2} + 415T + 172225$$
$71$ $$(T - 432)^{2}$$
$73$ $$T^{2} - 1113 T + 1238769$$
$79$ $$T^{2} - 103T + 10609$$
$83$ $$(T + 1092)^{2}$$
$89$ $$T^{2} + 329T + 108241$$
$97$ $$(T + 882)^{2}$$