# Properties

 Label 63.6.a.e Level $63$ Weight $6$ Character orbit 63.a Self dual yes Analytic conductor $10.104$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,6,Mod(1,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("63.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 63.a (trivial)

## 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: yes Analytic conductor: $$10.1041806482$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 10 q^{2} + 68 q^{4} + 56 q^{5} - 49 q^{7} + 360 q^{8}+O(q^{10})$$ q + 10 * q^2 + 68 * q^4 + 56 * q^5 - 49 * q^7 + 360 * q^8 $$q + 10 q^{2} + 68 q^{4} + 56 q^{5} - 49 q^{7} + 360 q^{8} + 560 q^{10} - 232 q^{11} - 140 q^{13} - 490 q^{14} + 1424 q^{16} + 1722 q^{17} - 98 q^{19} + 3808 q^{20} - 2320 q^{22} - 1824 q^{23} + 11 q^{25} - 1400 q^{26} - 3332 q^{28} - 3418 q^{29} - 7644 q^{31} + 2720 q^{32} + 17220 q^{34} - 2744 q^{35} - 10398 q^{37} - 980 q^{38} + 20160 q^{40} + 17962 q^{41} + 10880 q^{43} - 15776 q^{44} - 18240 q^{46} - 9324 q^{47} + 2401 q^{49} + 110 q^{50} - 9520 q^{52} - 2262 q^{53} - 12992 q^{55} - 17640 q^{56} - 34180 q^{58} + 2730 q^{59} + 25648 q^{61} - 76440 q^{62} - 18368 q^{64} - 7840 q^{65} - 48404 q^{67} + 117096 q^{68} - 27440 q^{70} + 58560 q^{71} + 68082 q^{73} - 103980 q^{74} - 6664 q^{76} + 11368 q^{77} + 31784 q^{79} + 79744 q^{80} + 179620 q^{82} + 20538 q^{83} + 96432 q^{85} + 108800 q^{86} - 83520 q^{88} + 50582 q^{89} + 6860 q^{91} - 124032 q^{92} - 93240 q^{94} - 5488 q^{95} - 58506 q^{97} + 24010 q^{98}+O(q^{100})$$ q + 10 * q^2 + 68 * q^4 + 56 * q^5 - 49 * q^7 + 360 * q^8 + 560 * q^10 - 232 * q^11 - 140 * q^13 - 490 * q^14 + 1424 * q^16 + 1722 * q^17 - 98 * q^19 + 3808 * q^20 - 2320 * q^22 - 1824 * q^23 + 11 * q^25 - 1400 * q^26 - 3332 * q^28 - 3418 * q^29 - 7644 * q^31 + 2720 * q^32 + 17220 * q^34 - 2744 * q^35 - 10398 * q^37 - 980 * q^38 + 20160 * q^40 + 17962 * q^41 + 10880 * q^43 - 15776 * q^44 - 18240 * q^46 - 9324 * q^47 + 2401 * q^49 + 110 * q^50 - 9520 * q^52 - 2262 * q^53 - 12992 * q^55 - 17640 * q^56 - 34180 * q^58 + 2730 * q^59 + 25648 * q^61 - 76440 * q^62 - 18368 * q^64 - 7840 * q^65 - 48404 * q^67 + 117096 * q^68 - 27440 * q^70 + 58560 * q^71 + 68082 * q^73 - 103980 * q^74 - 6664 * q^76 + 11368 * q^77 + 31784 * q^79 + 79744 * q^80 + 179620 * q^82 + 20538 * q^83 + 96432 * q^85 + 108800 * q^86 - 83520 * q^88 + 50582 * q^89 + 6860 * q^91 - 124032 * q^92 - 93240 * q^94 - 5488 * q^95 - 58506 * q^97 + 24010 * q^98

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
10.0000 0 68.0000 56.0000 0 −49.0000 360.000 0 560.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.6.a.e 1
3.b odd 2 1 7.6.a.a 1
4.b odd 2 1 1008.6.a.y 1
7.b odd 2 1 441.6.a.k 1
12.b even 2 1 112.6.a.g 1
15.d odd 2 1 175.6.a.b 1
15.e even 4 2 175.6.b.a 2
21.c even 2 1 49.6.a.a 1
21.g even 6 2 49.6.c.b 2
21.h odd 6 2 49.6.c.c 2
24.f even 2 1 448.6.a.c 1
24.h odd 2 1 448.6.a.m 1
33.d even 2 1 847.6.a.b 1
84.h odd 2 1 784.6.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 3.b odd 2 1
49.6.a.a 1 21.c even 2 1
49.6.c.b 2 21.g even 6 2
49.6.c.c 2 21.h odd 6 2
63.6.a.e 1 1.a even 1 1 trivial
112.6.a.g 1 12.b even 2 1
175.6.a.b 1 15.d odd 2 1
175.6.b.a 2 15.e even 4 2
441.6.a.k 1 7.b odd 2 1
448.6.a.c 1 24.f even 2 1
448.6.a.m 1 24.h odd 2 1
784.6.a.c 1 84.h odd 2 1
847.6.a.b 1 33.d even 2 1
1008.6.a.y 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 10$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(63))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 10$$
$3$ $$T$$
$5$ $$T - 56$$
$7$ $$T + 49$$
$11$ $$T + 232$$
$13$ $$T + 140$$
$17$ $$T - 1722$$
$19$ $$T + 98$$
$23$ $$T + 1824$$
$29$ $$T + 3418$$
$31$ $$T + 7644$$
$37$ $$T + 10398$$
$41$ $$T - 17962$$
$43$ $$T - 10880$$
$47$ $$T + 9324$$
$53$ $$T + 2262$$
$59$ $$T - 2730$$
$61$ $$T - 25648$$
$67$ $$T + 48404$$
$71$ $$T - 58560$$
$73$ $$T - 68082$$
$79$ $$T - 31784$$
$83$ $$T - 20538$$
$89$ $$T - 50582$$
$97$ $$T + 58506$$