# Properties

 Label 63.6.e.a Level $63$ Weight $6$ Character orbit 63.e Analytic conductor $10.104$ 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] = [63,6,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, 6, names="a")

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

chi := DirichletCharacter("63.37");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$10.1041806482$$ 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 21) 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} + 28 \zeta_{6} q^{4} + ( - 11 \zeta_{6} + 11) q^{5} + (7 \zeta_{6} + 126) q^{7} - 120 q^{8}+O(q^{10})$$ q + (2*z - 2) * q^2 + 28*z * q^4 + (-11*z + 11) * q^5 + (7*z + 126) * q^7 - 120 * q^8 $$q + (2 \zeta_{6} - 2) q^{2} + 28 \zeta_{6} q^{4} + ( - 11 \zeta_{6} + 11) q^{5} + (7 \zeta_{6} + 126) q^{7} - 120 q^{8} + 22 \zeta_{6} q^{10} + 269 \zeta_{6} q^{11} - 308 q^{13} + (252 \zeta_{6} - 266) q^{14} + (656 \zeta_{6} - 656) q^{16} + 1896 \zeta_{6} q^{17} + ( - 164 \zeta_{6} + 164) q^{19} + 308 q^{20} - 538 q^{22} + (3264 \zeta_{6} - 3264) q^{23} + 3004 \zeta_{6} q^{25} + ( - 616 \zeta_{6} + 616) q^{26} + (3724 \zeta_{6} - 196) q^{28} - 2417 q^{29} - 2841 \zeta_{6} q^{31} - 5152 \zeta_{6} q^{32} - 3792 q^{34} + ( - 1386 \zeta_{6} + 1463) q^{35} + ( - 11328 \zeta_{6} + 11328) q^{37} + 328 \zeta_{6} q^{38} + (1320 \zeta_{6} - 1320) q^{40} + 16856 q^{41} - 7894 q^{43} + (7532 \zeta_{6} - 7532) q^{44} - 6528 \zeta_{6} q^{46} + ( - 21102 \zeta_{6} + 21102) q^{47} + (1813 \zeta_{6} + 15827) q^{49} - 6008 q^{50} - 8624 \zeta_{6} q^{52} - 29691 \zeta_{6} q^{53} + 2959 q^{55} + ( - 840 \zeta_{6} - 15120) q^{56} + ( - 4834 \zeta_{6} + 4834) q^{58} - 8163 \zeta_{6} q^{59} + (15166 \zeta_{6} - 15166) q^{61} + 5682 q^{62} - 10688 q^{64} + (3388 \zeta_{6} - 3388) q^{65} + 32078 \zeta_{6} q^{67} + (53088 \zeta_{6} - 53088) q^{68} + (2926 \zeta_{6} - 154) q^{70} + 38274 q^{71} - 34866 \zeta_{6} q^{73} + 22656 \zeta_{6} q^{74} + 4592 q^{76} + (35777 \zeta_{6} - 1883) q^{77} + (13529 \zeta_{6} - 13529) q^{79} + 7216 \zeta_{6} q^{80} + (33712 \zeta_{6} - 33712) q^{82} + 68103 q^{83} + 20856 q^{85} + ( - 15788 \zeta_{6} + 15788) q^{86} - 32280 \zeta_{6} q^{88} + (114922 \zeta_{6} - 114922) q^{89} + ( - 2156 \zeta_{6} - 38808) q^{91} - 91392 q^{92} + 42204 \zeta_{6} q^{94} - 1804 \zeta_{6} q^{95} + 154959 q^{97} + (31654 \zeta_{6} - 35280) q^{98} +O(q^{100})$$ q + (2*z - 2) * q^2 + 28*z * q^4 + (-11*z + 11) * q^5 + (7*z + 126) * q^7 - 120 * q^8 + 22*z * q^10 + 269*z * q^11 - 308 * q^13 + (252*z - 266) * q^14 + (656*z - 656) * q^16 + 1896*z * q^17 + (-164*z + 164) * q^19 + 308 * q^20 - 538 * q^22 + (3264*z - 3264) * q^23 + 3004*z * q^25 + (-616*z + 616) * q^26 + (3724*z - 196) * q^28 - 2417 * q^29 - 2841*z * q^31 - 5152*z * q^32 - 3792 * q^34 + (-1386*z + 1463) * q^35 + (-11328*z + 11328) * q^37 + 328*z * q^38 + (1320*z - 1320) * q^40 + 16856 * q^41 - 7894 * q^43 + (7532*z - 7532) * q^44 - 6528*z * q^46 + (-21102*z + 21102) * q^47 + (1813*z + 15827) * q^49 - 6008 * q^50 - 8624*z * q^52 - 29691*z * q^53 + 2959 * q^55 + (-840*z - 15120) * q^56 + (-4834*z + 4834) * q^58 - 8163*z * q^59 + (15166*z - 15166) * q^61 + 5682 * q^62 - 10688 * q^64 + (3388*z - 3388) * q^65 + 32078*z * q^67 + (53088*z - 53088) * q^68 + (2926*z - 154) * q^70 + 38274 * q^71 - 34866*z * q^73 + 22656*z * q^74 + 4592 * q^76 + (35777*z - 1883) * q^77 + (13529*z - 13529) * q^79 + 7216*z * q^80 + (33712*z - 33712) * q^82 + 68103 * q^83 + 20856 * q^85 + (-15788*z + 15788) * q^86 - 32280*z * q^88 + (114922*z - 114922) * q^89 + (-2156*z - 38808) * q^91 - 91392 * q^92 + 42204*z * q^94 - 1804*z * q^95 + 154959 * q^97 + (31654*z - 35280) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 28 q^{4} + 11 q^{5} + 259 q^{7} - 240 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 28 * q^4 + 11 * q^5 + 259 * q^7 - 240 * q^8 $$2 q - 2 q^{2} + 28 q^{4} + 11 q^{5} + 259 q^{7} - 240 q^{8} + 22 q^{10} + 269 q^{11} - 616 q^{13} - 280 q^{14} - 656 q^{16} + 1896 q^{17} + 164 q^{19} + 616 q^{20} - 1076 q^{22} - 3264 q^{23} + 3004 q^{25} + 616 q^{26} + 3332 q^{28} - 4834 q^{29} - 2841 q^{31} - 5152 q^{32} - 7584 q^{34} + 1540 q^{35} + 11328 q^{37} + 328 q^{38} - 1320 q^{40} + 33712 q^{41} - 15788 q^{43} - 7532 q^{44} - 6528 q^{46} + 21102 q^{47} + 33467 q^{49} - 12016 q^{50} - 8624 q^{52} - 29691 q^{53} + 5918 q^{55} - 31080 q^{56} + 4834 q^{58} - 8163 q^{59} - 15166 q^{61} + 11364 q^{62} - 21376 q^{64} - 3388 q^{65} + 32078 q^{67} - 53088 q^{68} + 2618 q^{70} + 76548 q^{71} - 34866 q^{73} + 22656 q^{74} + 9184 q^{76} + 32011 q^{77} - 13529 q^{79} + 7216 q^{80} - 33712 q^{82} + 136206 q^{83} + 41712 q^{85} + 15788 q^{86} - 32280 q^{88} - 114922 q^{89} - 79772 q^{91} - 182784 q^{92} + 42204 q^{94} - 1804 q^{95} + 309918 q^{97} - 38906 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 28 * q^4 + 11 * q^5 + 259 * q^7 - 240 * q^8 + 22 * q^10 + 269 * q^11 - 616 * q^13 - 280 * q^14 - 656 * q^16 + 1896 * q^17 + 164 * q^19 + 616 * q^20 - 1076 * q^22 - 3264 * q^23 + 3004 * q^25 + 616 * q^26 + 3332 * q^28 - 4834 * q^29 - 2841 * q^31 - 5152 * q^32 - 7584 * q^34 + 1540 * q^35 + 11328 * q^37 + 328 * q^38 - 1320 * q^40 + 33712 * q^41 - 15788 * q^43 - 7532 * q^44 - 6528 * q^46 + 21102 * q^47 + 33467 * q^49 - 12016 * q^50 - 8624 * q^52 - 29691 * q^53 + 5918 * q^55 - 31080 * q^56 + 4834 * q^58 - 8163 * q^59 - 15166 * q^61 + 11364 * q^62 - 21376 * q^64 - 3388 * q^65 + 32078 * q^67 - 53088 * q^68 + 2618 * q^70 + 76548 * q^71 - 34866 * q^73 + 22656 * q^74 + 9184 * q^76 + 32011 * q^77 - 13529 * q^79 + 7216 * q^80 - 33712 * q^82 + 136206 * q^83 + 41712 * q^85 + 15788 * q^86 - 32280 * q^88 - 114922 * q^89 - 79772 * q^91 - 182784 * q^92 + 42204 * q^94 - 1804 * q^95 + 309918 * q^97 - 38906 * 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 14.0000 + 24.2487i 5.50000 9.52628i 0 129.500 + 6.06218i −120.000 0 11.0000 + 19.0526i
46.1 −1.00000 1.73205i 0 14.0000 24.2487i 5.50000 + 9.52628i 0 129.500 6.06218i −120.000 0 11.0000 19.0526i
 $$n$$: 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 63.6.e.a 2
3.b odd 2 1 21.6.e.a 2
7.c even 3 1 inner 63.6.e.a 2
7.c even 3 1 441.6.a.g 1
7.d odd 6 1 441.6.a.h 1
12.b even 2 1 336.6.q.b 2
21.c even 2 1 147.6.e.g 2
21.g even 6 1 147.6.a.d 1
21.g even 6 1 147.6.e.g 2
21.h odd 6 1 21.6.e.a 2
21.h odd 6 1 147.6.a.c 1
84.n even 6 1 336.6.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.a 2 3.b odd 2 1
21.6.e.a 2 21.h odd 6 1
63.6.e.a 2 1.a even 1 1 trivial
63.6.e.a 2 7.c even 3 1 inner
147.6.a.c 1 21.h odd 6 1
147.6.a.d 1 21.g even 6 1
147.6.e.g 2 21.c even 2 1
147.6.e.g 2 21.g even 6 1
336.6.q.b 2 12.b even 2 1
336.6.q.b 2 84.n even 6 1
441.6.a.g 1 7.c even 3 1
441.6.a.h 1 7.d odd 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 $$S_{6}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 11T + 121$$
$7$ $$T^{2} - 259T + 16807$$
$11$ $$T^{2} - 269T + 72361$$
$13$ $$(T + 308)^{2}$$
$17$ $$T^{2} - 1896 T + 3594816$$
$19$ $$T^{2} - 164T + 26896$$
$23$ $$T^{2} + 3264 T + 10653696$$
$29$ $$(T + 2417)^{2}$$
$31$ $$T^{2} + 2841 T + 8071281$$
$37$ $$T^{2} - 11328 T + 128323584$$
$41$ $$(T - 16856)^{2}$$
$43$ $$(T + 7894)^{2}$$
$47$ $$T^{2} - 21102 T + 445294404$$
$53$ $$T^{2} + 29691 T + 881555481$$
$59$ $$T^{2} + 8163 T + 66634569$$
$61$ $$T^{2} + 15166 T + 230007556$$
$67$ $$T^{2} - 32078 T + 1028998084$$
$71$ $$(T - 38274)^{2}$$
$73$ $$T^{2} + 34866 T + 1215637956$$
$79$ $$T^{2} + 13529 T + 183033841$$
$83$ $$(T - 68103)^{2}$$
$89$ $$T^{2} + 114922 T + 13207066084$$
$97$ $$(T - 154959)^{2}$$