# Properties

 Label 63.7.d.d Level $63$ Weight $7$ Character orbit 63.d Analytic conductor $14.493$ 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] = [63,7,Mod(55,63)]

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, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("63.55");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 63.d (of order $$2$$, degree $$1$$, 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: $$14.4934072681$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-510})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 510$$ x^2 + 510 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = 2\sqrt{-510}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 8 q^{2} - \beta q^{5} + ( - 7 \beta + 133) q^{7} - 512 q^{8} +O(q^{10})$$ q + 8 * q^2 - b * q^5 + (-7*b + 133) * q^7 - 512 * q^8 $$q + 8 q^{2} - \beta q^{5} + ( - 7 \beta + 133) q^{7} - 512 q^{8} - 8 \beta q^{10} - 874 q^{11} - 49 \beta q^{13} + ( - 56 \beta + 1064) q^{14} - 4096 q^{16} - 132 \beta q^{17} - 69 \beta q^{19} - 6992 q^{22} - 4738 q^{23} + 13585 q^{25} - 392 \beta q^{26} - 11146 q^{29} + 608 \beta q^{31} - 1056 \beta q^{34} + ( - 133 \beta - 14280) q^{35} + 3002 q^{37} - 552 \beta q^{38} + 512 \beta q^{40} + 1274 \beta q^{41} + 31418 q^{43} - 37904 q^{46} - 1604 \beta q^{47} + ( - 1862 \beta - 82271) q^{49} + 108680 q^{50} + 76406 q^{53} + 874 \beta q^{55} + (3584 \beta - 68096) q^{56} - 89168 q^{58} + 2507 \beta q^{59} + 6091 \beta q^{61} + 4864 \beta q^{62} + 262144 q^{64} - 99960 q^{65} + 495242 q^{67} + ( - 1064 \beta - 114240) q^{70} + 184406 q^{71} + 1350 \beta q^{73} + 24016 q^{74} + (6118 \beta - 116242) q^{77} - 534934 q^{79} + 4096 \beta q^{80} + 10192 \beta q^{82} - 15827 \beta q^{83} - 269280 q^{85} + 251344 q^{86} + 447488 q^{88} - 13938 \beta q^{89} + ( - 6517 \beta - 699720) q^{91} - 12832 \beta q^{94} - 140760 q^{95} - 18032 \beta q^{97} + ( - 14896 \beta - 658168) q^{98} +O(q^{100})$$ q + 8 * q^2 - b * q^5 + (-7*b + 133) * q^7 - 512 * q^8 - 8*b * q^10 - 874 * q^11 - 49*b * q^13 + (-56*b + 1064) * q^14 - 4096 * q^16 - 132*b * q^17 - 69*b * q^19 - 6992 * q^22 - 4738 * q^23 + 13585 * q^25 - 392*b * q^26 - 11146 * q^29 + 608*b * q^31 - 1056*b * q^34 + (-133*b - 14280) * q^35 + 3002 * q^37 - 552*b * q^38 + 512*b * q^40 + 1274*b * q^41 + 31418 * q^43 - 37904 * q^46 - 1604*b * q^47 + (-1862*b - 82271) * q^49 + 108680 * q^50 + 76406 * q^53 + 874*b * q^55 + (3584*b - 68096) * q^56 - 89168 * q^58 + 2507*b * q^59 + 6091*b * q^61 + 4864*b * q^62 + 262144 * q^64 - 99960 * q^65 + 495242 * q^67 + (-1064*b - 114240) * q^70 + 184406 * q^71 + 1350*b * q^73 + 24016 * q^74 + (6118*b - 116242) * q^77 - 534934 * q^79 + 4096*b * q^80 + 10192*b * q^82 - 15827*b * q^83 - 269280 * q^85 + 251344 * q^86 + 447488 * q^88 - 13938*b * q^89 + (-6517*b - 699720) * q^91 - 12832*b * q^94 - 140760 * q^95 - 18032*b * q^97 + (-14896*b - 658168) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 16 q^{2} + 266 q^{7} - 1024 q^{8}+O(q^{10})$$ 2 * q + 16 * q^2 + 266 * q^7 - 1024 * q^8 $$2 q + 16 q^{2} + 266 q^{7} - 1024 q^{8} - 1748 q^{11} + 2128 q^{14} - 8192 q^{16} - 13984 q^{22} - 9476 q^{23} + 27170 q^{25} - 22292 q^{29} - 28560 q^{35} + 6004 q^{37} + 62836 q^{43} - 75808 q^{46} - 164542 q^{49} + 217360 q^{50} + 152812 q^{53} - 136192 q^{56} - 178336 q^{58} + 524288 q^{64} - 199920 q^{65} + 990484 q^{67} - 228480 q^{70} + 368812 q^{71} + 48032 q^{74} - 232484 q^{77} - 1069868 q^{79} - 538560 q^{85} + 502688 q^{86} + 894976 q^{88} - 1399440 q^{91} - 281520 q^{95} - 1316336 q^{98}+O(q^{100})$$ 2 * q + 16 * q^2 + 266 * q^7 - 1024 * q^8 - 1748 * q^11 + 2128 * q^14 - 8192 * q^16 - 13984 * q^22 - 9476 * q^23 + 27170 * q^25 - 22292 * q^29 - 28560 * q^35 + 6004 * q^37 + 62836 * q^43 - 75808 * q^46 - 164542 * q^49 + 217360 * q^50 + 152812 * q^53 - 136192 * q^56 - 178336 * q^58 + 524288 * q^64 - 199920 * q^65 + 990484 * q^67 - 228480 * q^70 + 368812 * q^71 + 48032 * q^74 - 232484 * q^77 - 1069868 * q^79 - 538560 * q^85 + 502688 * q^86 + 894976 * q^88 - 1399440 * q^91 - 281520 * q^95 - 1316336 * q^98

## Character values

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

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 22.5832i − 22.5832i
8.00000 0 0 45.1664i 0 133.000 316.165i −512.000 0 361.331i
55.2 8.00000 0 0 45.1664i 0 133.000 + 316.165i −512.000 0 361.331i
 $$n$$: 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.7.d.d 2
3.b odd 2 1 7.7.b.b 2
7.b odd 2 1 inner 63.7.d.d 2
12.b even 2 1 112.7.c.b 2
15.d odd 2 1 175.7.d.e 2
15.e even 4 2 175.7.c.c 4
21.c even 2 1 7.7.b.b 2
21.g even 6 2 49.7.d.d 4
21.h odd 6 2 49.7.d.d 4
24.f even 2 1 448.7.c.c 2
24.h odd 2 1 448.7.c.d 2
84.h odd 2 1 112.7.c.b 2
105.g even 2 1 175.7.d.e 2
105.k odd 4 2 175.7.c.c 4
168.e odd 2 1 448.7.c.c 2
168.i even 2 1 448.7.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.b.b 2 3.b odd 2 1
7.7.b.b 2 21.c even 2 1
49.7.d.d 4 21.g even 6 2
49.7.d.d 4 21.h odd 6 2
63.7.d.d 2 1.a even 1 1 trivial
63.7.d.d 2 7.b odd 2 1 inner
112.7.c.b 2 12.b even 2 1
112.7.c.b 2 84.h odd 2 1
175.7.c.c 4 15.e even 4 2
175.7.c.c 4 105.k odd 4 2
175.7.d.e 2 15.d odd 2 1
175.7.d.e 2 105.g even 2 1
448.7.c.c 2 24.f even 2 1
448.7.c.c 2 168.e odd 2 1
448.7.c.d 2 24.h odd 2 1
448.7.c.d 2 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 8$$ acting on $$S_{7}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 8)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2040$$
$7$ $$T^{2} - 266T + 117649$$
$11$ $$(T + 874)^{2}$$
$13$ $$T^{2} + 4898040$$
$17$ $$T^{2} + 35544960$$
$19$ $$T^{2} + 9712440$$
$23$ $$(T + 4738)^{2}$$
$29$ $$(T + 11146)^{2}$$
$31$ $$T^{2} + 754114560$$
$37$ $$(T - 3002)^{2}$$
$41$ $$T^{2} + 3311075040$$
$43$ $$(T - 31418)^{2}$$
$47$ $$T^{2} + 5248544640$$
$53$ $$(T - 76406)^{2}$$
$59$ $$T^{2} + 12821499960$$
$61$ $$T^{2} + 75684573240$$
$67$ $$(T - 495242)^{2}$$
$71$ $$(T - 184406)^{2}$$
$73$ $$T^{2} + 3717900000$$
$79$ $$(T + 534934)^{2}$$
$83$ $$T^{2} + 511007615160$$
$89$ $$T^{2} + 396306401760$$
$97$ $$T^{2} + 663312168960$$