# Properties

 Label 63.6.a.d Level $63$ Weight $6$ Character orbit 63.a Self dual yes Analytic conductor $10.104$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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)

 $$f(q)$$ $$=$$ $$q + 6 q^{2} + 4 q^{4} - 78 q^{5} + 49 q^{7} - 168 q^{8}+O(q^{10})$$ q + 6 * q^2 + 4 * q^4 - 78 * q^5 + 49 * q^7 - 168 * q^8 $$q + 6 q^{2} + 4 q^{4} - 78 q^{5} + 49 q^{7} - 168 q^{8} - 468 q^{10} - 444 q^{11} - 442 q^{13} + 294 q^{14} - 1136 q^{16} + 126 q^{17} + 2684 q^{19} - 312 q^{20} - 2664 q^{22} - 4200 q^{23} + 2959 q^{25} - 2652 q^{26} + 196 q^{28} + 5442 q^{29} + 80 q^{31} - 1440 q^{32} + 756 q^{34} - 3822 q^{35} - 5434 q^{37} + 16104 q^{38} + 13104 q^{40} - 7962 q^{41} - 11524 q^{43} - 1776 q^{44} - 25200 q^{46} + 13920 q^{47} + 2401 q^{49} + 17754 q^{50} - 1768 q^{52} + 9594 q^{53} + 34632 q^{55} - 8232 q^{56} + 32652 q^{58} - 27492 q^{59} + 49478 q^{61} + 480 q^{62} + 27712 q^{64} + 34476 q^{65} - 59356 q^{67} + 504 q^{68} - 22932 q^{70} - 32040 q^{71} - 61846 q^{73} - 32604 q^{74} + 10736 q^{76} - 21756 q^{77} - 65776 q^{79} + 88608 q^{80} - 47772 q^{82} - 40188 q^{83} - 9828 q^{85} - 69144 q^{86} + 74592 q^{88} + 7974 q^{89} - 21658 q^{91} - 16800 q^{92} + 83520 q^{94} - 209352 q^{95} - 143662 q^{97} + 14406 q^{98}+O(q^{100})$$ q + 6 * q^2 + 4 * q^4 - 78 * q^5 + 49 * q^7 - 168 * q^8 - 468 * q^10 - 444 * q^11 - 442 * q^13 + 294 * q^14 - 1136 * q^16 + 126 * q^17 + 2684 * q^19 - 312 * q^20 - 2664 * q^22 - 4200 * q^23 + 2959 * q^25 - 2652 * q^26 + 196 * q^28 + 5442 * q^29 + 80 * q^31 - 1440 * q^32 + 756 * q^34 - 3822 * q^35 - 5434 * q^37 + 16104 * q^38 + 13104 * q^40 - 7962 * q^41 - 11524 * q^43 - 1776 * q^44 - 25200 * q^46 + 13920 * q^47 + 2401 * q^49 + 17754 * q^50 - 1768 * q^52 + 9594 * q^53 + 34632 * q^55 - 8232 * q^56 + 32652 * q^58 - 27492 * q^59 + 49478 * q^61 + 480 * q^62 + 27712 * q^64 + 34476 * q^65 - 59356 * q^67 + 504 * q^68 - 22932 * q^70 - 32040 * q^71 - 61846 * q^73 - 32604 * q^74 + 10736 * q^76 - 21756 * q^77 - 65776 * q^79 + 88608 * q^80 - 47772 * q^82 - 40188 * q^83 - 9828 * q^85 - 69144 * q^86 + 74592 * q^88 + 7974 * q^89 - 21658 * q^91 - 16800 * q^92 + 83520 * q^94 - 209352 * q^95 - 143662 * q^97 + 14406 * 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
6.00000 0 4.00000 −78.0000 0 49.0000 −168.000 0 −468.000
 $$n$$: 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.d 1
3.b odd 2 1 21.6.a.a 1
4.b odd 2 1 1008.6.a.c 1
7.b odd 2 1 441.6.a.j 1
12.b even 2 1 336.6.a.r 1
15.d odd 2 1 525.6.a.d 1
15.e even 4 2 525.6.d.b 2
21.c even 2 1 147.6.a.b 1
21.g even 6 2 147.6.e.i 2
21.h odd 6 2 147.6.e.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.a 1 3.b odd 2 1
63.6.a.d 1 1.a even 1 1 trivial
147.6.a.b 1 21.c even 2 1
147.6.e.i 2 21.g even 6 2
147.6.e.j 2 21.h odd 6 2
336.6.a.r 1 12.b even 2 1
441.6.a.j 1 7.b odd 2 1
525.6.a.d 1 15.d odd 2 1
525.6.d.b 2 15.e even 4 2
1008.6.a.c 1 4.b odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 6$$
$3$ $$T$$
$5$ $$T + 78$$
$7$ $$T - 49$$
$11$ $$T + 444$$
$13$ $$T + 442$$
$17$ $$T - 126$$
$19$ $$T - 2684$$
$23$ $$T + 4200$$
$29$ $$T - 5442$$
$31$ $$T - 80$$
$37$ $$T + 5434$$
$41$ $$T + 7962$$
$43$ $$T + 11524$$
$47$ $$T - 13920$$
$53$ $$T - 9594$$
$59$ $$T + 27492$$
$61$ $$T - 49478$$
$67$ $$T + 59356$$
$71$ $$T + 32040$$
$73$ $$T + 61846$$
$79$ $$T + 65776$$
$83$ $$T + 40188$$
$89$ $$T - 7974$$
$97$ $$T + 143662$$