# Properties

 Label 63.6.e.c Level $63$ Weight $6$ Character orbit 63.e Analytic conductor $10.104$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-83})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 20x^{2} - 21x + 441$$ x^4 - x^3 - 20*x^2 - 21*x + 441 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - 2 \beta_1 - 2) q^{2} + (3 \beta_{3} - 3 \beta_{2} + 34 \beta_1) q^{4} + ( - 7 \beta_{3} - 20 \beta_1 - 20) q^{5} + ( - 14 \beta_{3} + 7 \beta_{2} + \cdots - 91) q^{7}+ \cdots + (5 \beta_{2} + 190) q^{8}+O(q^{10})$$ q + (-b3 - 2*b1 - 2) * q^2 + (3*b3 - 3*b2 + 34*b1) * q^4 + (-7*b3 - 20*b1 - 20) * q^5 + (-14*b3 + 7*b2 - 7*b1 - 91) * q^7 + (5*b2 + 190) * q^8 $$q + ( - \beta_{3} - 2 \beta_1 - 2) q^{2} + (3 \beta_{3} - 3 \beta_{2} + 34 \beta_1) q^{4} + ( - 7 \beta_{3} - 20 \beta_1 - 20) q^{5} + ( - 14 \beta_{3} + 7 \beta_{2} + \cdots - 91) q^{7}+ \cdots + ( - 343 \beta_{3} + 3675 \beta_{2} + \cdots + 80164) q^{98}+O(q^{100})$$ q + (-b3 - 2*b1 - 2) * q^2 + (3*b3 - 3*b2 + 34*b1) * q^4 + (-7*b3 - 20*b1 - 20) * q^5 + (-14*b3 + 7*b2 - 7*b1 - 91) * q^7 + (5*b2 + 190) * q^8 + (27*b3 - 27*b2 + 474*b1) * q^10 + (-b3 + b2 + 568*b1) * q^11 + (9*b2 + 467) * q^13 + (98*b3 - 21*b2 + 616*b1 - 266) * q^14 + (-99*b3 + 398*b1 + 398) * q^16 + (-148*b3 + 148*b2 + 88*b1) * q^17 + (-27*b3 - 1169*b1 - 1169) * q^19 + (277*b2 + 1982) * q^20 + (567*b2 + 1074) * q^22 + (308*b3 + 952*b1 + 952) * q^23 + (231*b3 - 231*b2 + 313*b1) * q^25 + (-476*b3 - 1492*b1 - 1492) * q^26 + (-35*b3 + 490*b2 - 1554*b1 + 2842) * q^28 + (-45*b2 + 1086) * q^29 + (768*b3 - 768*b2 + 2531*b1) * q^31 + (-459*b3 + 459*b2 - 738*b1) * q^32 + (-60*b2 - 9000) * q^34 + (728*b3 - 231*b2 + 4858*b1 - 1358) * q^35 + (-855*b3 + 9127*b1 + 9127) * q^37 + (1196*b3 - 1196*b2 + 4012*b1) * q^38 + (-1395*b3 - 5970*b1 - 5970) * q^40 + (-846*b2 + 6006) * q^41 + (-2043*b2 - 2407) * q^43 + (-1673*b3 - 19126*b1 - 19126) * q^44 + (-1260*b3 + 1260*b2 - 21000*b1) * q^46 + (-604*b3 - 11882*b1 - 11882) * q^47 + (2450*b3 - 1225*b2 + 1225*b1 - 882) * q^49 + (544*b2 + 14948) * q^50 + (1680*b3 - 1680*b2 + 17552*b1) * q^52 + (1751*b3 - 1751*b2 + 16702*b1) * q^53 + (3963*b2 + 10926) * q^55 + (-2625*b3 + 875*b2 - 5670*b1 - 19460) * q^56 + (-1041*b3 + 618*b1 + 618) * q^58 + (3917*b3 - 3917*b2 - 18590*b1) * q^59 + (2544*b3 - 19754*b1 - 19754) * q^61 + (3299*b2 + 52678) * q^62 + (-4365*b2 - 17198) * q^64 + (-3386*b3 - 13246*b1 - 13246) * q^65 + (-4461*b3 + 4461*b2 + 13151*b1) * q^67 + (4324*b3 + 24536*b1 + 24536) * q^68 + (861*b3 + 5586*b2 - 28098*b1 + 26754) * q^70 + (-1404*b2 - 51750) * q^71 + (5247*b3 - 5247*b2 - 11665*b1) * q^73 + (-8272*b3 + 8272*b2 + 34756*b1) * q^74 + (4344*b2 + 44768) * q^76 + (4067*b3 + 3892*b2 - 48146*b1 + 3108) * q^77 + (-6834*b3 + 5815*b1 + 5815) * q^79 + (-1499*b3 + 1499*b2 + 35006*b1) * q^80 + (-5160*b3 + 40440*b1 + 40440) * q^82 + (1899*b2 - 29640) * q^83 + (-1308*b2 - 62472) * q^85 + (4450*b3 + 131480*b1 + 131480) * q^86 + (2655*b3 - 2655*b2 + 107610*b1) * q^88 + (-130*b3 + 14596*b1 + 14596) * q^89 + (-6475*b3 + 2450*b2 - 11081*b1 - 46403) * q^91 + (-12404*b2 - 89656) * q^92 + (12486*b3 - 12486*b2 + 61212*b1) * q^94 + (8534*b3 - 8534*b2 + 35098*b1) * q^95 + (-1017*b2 - 5404) * q^97 + (-343*b3 + 3675*b2 - 74186*b1 + 80164) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{2} - 65 q^{4} - 33 q^{5} - 350 q^{7} + 750 q^{8}+O(q^{10})$$ 4 * q - 3 * q^2 - 65 * q^4 - 33 * q^5 - 350 * q^7 + 750 * q^8 $$4 q - 3 q^{2} - 65 q^{4} - 33 q^{5} - 350 q^{7} + 750 q^{8} - 921 q^{10} - 1137 q^{11} + 1850 q^{13} - 2352 q^{14} + 895 q^{16} - 324 q^{17} - 2311 q^{19} + 7374 q^{20} + 3162 q^{22} + 1596 q^{23} - 395 q^{25} - 2508 q^{26} + 13531 q^{28} + 4434 q^{29} - 4294 q^{31} + 1017 q^{32} - 35880 q^{34} - 15414 q^{35} + 19109 q^{37} - 6828 q^{38} - 10545 q^{40} + 25716 q^{41} - 5542 q^{43} - 36579 q^{44} + 40740 q^{46} - 23160 q^{47} - 5978 q^{49} + 58704 q^{50} - 33424 q^{52} - 31653 q^{53} + 35778 q^{55} - 65625 q^{56} + 2277 q^{58} + 41097 q^{59} - 42052 q^{61} + 204114 q^{62} - 60062 q^{64} - 23106 q^{65} - 30763 q^{67} + 44748 q^{68} + 151179 q^{70} - 204192 q^{71} + 28577 q^{73} - 77784 q^{74} + 170384 q^{76} + 96873 q^{77} + 18464 q^{79} - 71511 q^{80} + 86040 q^{82} - 122358 q^{83} - 247272 q^{85} + 258510 q^{86} - 212565 q^{88} + 29322 q^{89} - 161875 q^{91} - 333816 q^{92} - 109938 q^{94} - 61662 q^{95} - 19582 q^{97} + 462021 q^{98}+O(q^{100})$$ 4 * q - 3 * q^2 - 65 * q^4 - 33 * q^5 - 350 * q^7 + 750 * q^8 - 921 * q^10 - 1137 * q^11 + 1850 * q^13 - 2352 * q^14 + 895 * q^16 - 324 * q^17 - 2311 * q^19 + 7374 * q^20 + 3162 * q^22 + 1596 * q^23 - 395 * q^25 - 2508 * q^26 + 13531 * q^28 + 4434 * q^29 - 4294 * q^31 + 1017 * q^32 - 35880 * q^34 - 15414 * q^35 + 19109 * q^37 - 6828 * q^38 - 10545 * q^40 + 25716 * q^41 - 5542 * q^43 - 36579 * q^44 + 40740 * q^46 - 23160 * q^47 - 5978 * q^49 + 58704 * q^50 - 33424 * q^52 - 31653 * q^53 + 35778 * q^55 - 65625 * q^56 + 2277 * q^58 + 41097 * q^59 - 42052 * q^61 + 204114 * q^62 - 60062 * q^64 - 23106 * q^65 - 30763 * q^67 + 44748 * q^68 + 151179 * q^70 - 204192 * q^71 + 28577 * q^73 - 77784 * q^74 + 170384 * q^76 + 96873 * q^77 + 18464 * q^79 - 71511 * q^80 + 86040 * q^82 - 122358 * q^83 - 247272 * q^85 + 258510 * q^86 - 212565 * q^88 + 29322 * q^89 - 161875 * q^91 - 333816 * q^92 - 109938 * q^94 - 61662 * q^95 - 19582 * q^97 + 462021 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 20x^{2} - 21x + 441$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 20\nu^{2} - 20\nu - 441 ) / 420$$ (v^3 + 20*v^2 - 20*v - 441) / 420 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 41\nu ) / 21$$ (-v^3 + v^2 + 41*v) / 21 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 20\nu - 41 ) / 20$$ (v^3 + 20*v - 41) / 20
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3$$ (b3 + b2 - b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 61\beta _1 + 62 ) / 3$$ (-b3 + 2*b2 + 61*b1 + 62) / 3 $$\nu^{3}$$ $$=$$ $$( 40\beta_{3} - 20\beta_{2} + 20\beta _1 + 103 ) / 3$$ (40*b3 - 20*b2 + 20*b1 + 103) / 3

## Character values

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

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1 - \beta_{1}$$ $$1$$

## 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
 4.19493 − 1.84460i −3.69493 + 2.71062i 4.19493 + 1.84460i −3.69493 − 2.71062i
−4.69493 + 8.13186i 0 −28.0848 48.6443i −35.8645 + 62.1192i 0 −87.5000 + 95.6596i 226.949 0 −336.763 583.291i
37.2 3.19493 5.53379i 0 −4.41520 7.64735i 19.3645 33.5404i 0 −87.5000 95.6596i 148.051 0 −123.737 214.318i
46.1 −4.69493 8.13186i 0 −28.0848 + 48.6443i −35.8645 62.1192i 0 −87.5000 95.6596i 226.949 0 −336.763 + 583.291i
46.2 3.19493 + 5.53379i 0 −4.41520 + 7.64735i 19.3645 + 33.5404i 0 −87.5000 + 95.6596i 148.051 0 −123.737 + 214.318i
 $$n$$: 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.c 4
3.b odd 2 1 21.6.e.b 4
7.c even 3 1 inner 63.6.e.c 4
7.c even 3 1 441.6.a.t 2
7.d odd 6 1 441.6.a.s 2
12.b even 2 1 336.6.q.e 4
21.c even 2 1 147.6.e.l 4
21.g even 6 1 147.6.a.k 2
21.g even 6 1 147.6.e.l 4
21.h odd 6 1 21.6.e.b 4
21.h odd 6 1 147.6.a.i 2
84.n even 6 1 336.6.q.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.b 4 3.b odd 2 1
21.6.e.b 4 21.h odd 6 1
63.6.e.c 4 1.a even 1 1 trivial
63.6.e.c 4 7.c even 3 1 inner
147.6.a.i 2 21.h odd 6 1
147.6.a.k 2 21.g even 6 1
147.6.e.l 4 21.c even 2 1
147.6.e.l 4 21.g even 6 1
336.6.q.e 4 12.b even 2 1
336.6.q.e 4 84.n even 6 1
441.6.a.s 2 7.d odd 6 1
441.6.a.t 2 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3T_{2}^{3} + 69T_{2}^{2} - 180T_{2} + 3600$$ acting on $$S_{6}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3 T^{3} + \cdots + 3600$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 33 T^{3} + \cdots + 7717284$$
$7$ $$(T^{2} + 175 T + 16807)^{2}$$
$11$ $$T^{4} + \cdots + 104412996900$$
$13$ $$(T^{2} - 925 T + 208864)^{2}$$
$17$ $$T^{4} + \cdots + 1788317798400$$
$19$ $$T^{4} + \cdots + 1663584040000$$
$23$ $$T^{4} + \cdots + 27756881510400$$
$29$ $$(T^{2} - 2217 T + 1102716)^{2}$$
$31$ $$T^{4} + \cdots + 10\!\cdots\!25$$
$37$ $$T^{4} + \cdots + 20\!\cdots\!96$$
$41$ $$(T^{2} - 12858 T - 3221280)^{2}$$
$43$ $$(T^{2} + 2771 T - 257902490)^{2}$$
$47$ $$T^{4} + \cdots + 12\!\cdots\!16$$
$53$ $$T^{4} + \cdots + 35\!\cdots\!00$$
$59$ $$T^{4} + \cdots + 28\!\cdots\!44$$
$61$ $$T^{4} + \cdots + 15\!\cdots\!00$$
$67$ $$T^{4} + \cdots + 10\!\cdots\!00$$
$71$ $$(T^{2} + 102096 T + 2483190108)^{2}$$
$73$ $$T^{4} + \cdots + 22\!\cdots\!84$$
$79$ $$T^{4} + \cdots + 79\!\cdots\!69$$
$83$ $$(T^{2} + 61179 T + 711231498)^{2}$$
$89$ $$T^{4} + \cdots + 45\!\cdots\!16$$
$97$ $$(T^{2} + 9791 T - 40418570)^{2}$$