# Properties

 Label 882.3.s.b Level $882$ Weight $3$ Character orbit 882.s Analytic conductor $24.033$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,3,Mod(557,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("882.557");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 882.s (of order $$6$$, degree $$2$$, not 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: $$24.0327593166$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + 2 \beta_{2} q^{4} + 3 \beta_1 q^{5} - 2 \beta_{3} q^{8}+O(q^{10})$$ q - b1 * q^2 + 2*b2 * q^4 + 3*b1 * q^5 - 2*b3 * q^8 $$q - \beta_1 q^{2} + 2 \beta_{2} q^{4} + 3 \beta_1 q^{5} - 2 \beta_{3} q^{8} - 6 \beta_{2} q^{10} + ( - 12 \beta_{3} + 12 \beta_1) q^{11} + 8 q^{13} + (4 \beta_{2} - 4) q^{16} + (9 \beta_{3} - 9 \beta_1) q^{17} + ( - 16 \beta_{2} + 16) q^{19} + 6 \beta_{3} q^{20} - 24 q^{22} + 12 \beta_1 q^{23} - 7 \beta_{2} q^{25} - 8 \beta_1 q^{26} + 3 \beta_{3} q^{29} - 44 \beta_{2} q^{31} + ( - 4 \beta_{3} + 4 \beta_1) q^{32} + 18 q^{34} + ( - 34 \beta_{2} + 34) q^{37} + (16 \beta_{3} - 16 \beta_1) q^{38} + ( - 12 \beta_{2} + 12) q^{40} + 33 \beta_{3} q^{41} - 40 q^{43} + 24 \beta_1 q^{44} - 24 \beta_{2} q^{46} + 60 \beta_1 q^{47} + 7 \beta_{3} q^{50} + 16 \beta_{2} q^{52} + ( - 27 \beta_{3} + 27 \beta_1) q^{53} + 72 q^{55} + ( - 6 \beta_{2} + 6) q^{58} + ( - 24 \beta_{3} + 24 \beta_1) q^{59} + (50 \beta_{2} - 50) q^{61} + 44 \beta_{3} q^{62} - 8 q^{64} + 24 \beta_1 q^{65} - 8 \beta_{2} q^{67} - 18 \beta_1 q^{68} - 36 \beta_{3} q^{71} + 16 \beta_{2} q^{73} + (34 \beta_{3} - 34 \beta_1) q^{74} + 32 q^{76} + ( - 76 \beta_{2} + 76) q^{79} + (12 \beta_{3} - 12 \beta_1) q^{80} + ( - 66 \beta_{2} + 66) q^{82} + 84 \beta_{3} q^{83} - 54 q^{85} + 40 \beta_1 q^{86} - 48 \beta_{2} q^{88} - 9 \beta_1 q^{89} + 24 \beta_{3} q^{92} - 120 \beta_{2} q^{94} + ( - 48 \beta_{3} + 48 \beta_1) q^{95} + 176 q^{97}+O(q^{100})$$ q - b1 * q^2 + 2*b2 * q^4 + 3*b1 * q^5 - 2*b3 * q^8 - 6*b2 * q^10 + (-12*b3 + 12*b1) * q^11 + 8 * q^13 + (4*b2 - 4) * q^16 + (9*b3 - 9*b1) * q^17 + (-16*b2 + 16) * q^19 + 6*b3 * q^20 - 24 * q^22 + 12*b1 * q^23 - 7*b2 * q^25 - 8*b1 * q^26 + 3*b3 * q^29 - 44*b2 * q^31 + (-4*b3 + 4*b1) * q^32 + 18 * q^34 + (-34*b2 + 34) * q^37 + (16*b3 - 16*b1) * q^38 + (-12*b2 + 12) * q^40 + 33*b3 * q^41 - 40 * q^43 + 24*b1 * q^44 - 24*b2 * q^46 + 60*b1 * q^47 + 7*b3 * q^50 + 16*b2 * q^52 + (-27*b3 + 27*b1) * q^53 + 72 * q^55 + (-6*b2 + 6) * q^58 + (-24*b3 + 24*b1) * q^59 + (50*b2 - 50) * q^61 + 44*b3 * q^62 - 8 * q^64 + 24*b1 * q^65 - 8*b2 * q^67 - 18*b1 * q^68 - 36*b3 * q^71 + 16*b2 * q^73 + (34*b3 - 34*b1) * q^74 + 32 * q^76 + (-76*b2 + 76) * q^79 + (12*b3 - 12*b1) * q^80 + (-66*b2 + 66) * q^82 + 84*b3 * q^83 - 54 * q^85 + 40*b1 * q^86 - 48*b2 * q^88 - 9*b1 * q^89 + 24*b3 * q^92 - 120*b2 * q^94 + (-48*b3 + 48*b1) * q^95 + 176 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4}+O(q^{10})$$ 4 * q + 4 * q^4 $$4 q + 4 q^{4} - 12 q^{10} + 32 q^{13} - 8 q^{16} + 32 q^{19} - 96 q^{22} - 14 q^{25} - 88 q^{31} + 72 q^{34} + 68 q^{37} + 24 q^{40} - 160 q^{43} - 48 q^{46} + 32 q^{52} + 288 q^{55} + 12 q^{58} - 100 q^{61} - 32 q^{64} - 16 q^{67} + 32 q^{73} + 128 q^{76} + 152 q^{79} + 132 q^{82} - 216 q^{85} - 96 q^{88} - 240 q^{94} + 704 q^{97}+O(q^{100})$$ 4 * q + 4 * q^4 - 12 * q^10 + 32 * q^13 - 8 * q^16 + 32 * q^19 - 96 * q^22 - 14 * q^25 - 88 * q^31 + 72 * q^34 + 68 * q^37 + 24 * q^40 - 160 * q^43 - 48 * q^46 + 32 * q^52 + 288 * q^55 + 12 * q^58 - 100 * q^61 - 32 * q^64 - 16 * q^67 + 32 * q^73 + 128 * q^76 + 152 * q^79 + 132 * q^82 - 216 * q^85 - 96 * q^88 - 240 * q^94 + 704 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-1 + \beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
557.1
 1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 + 0.707107i −1.22474 − 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i 3.67423 2.12132i 0 0 2.82843i 0 −3.00000 + 5.19615i
557.2 1.22474 0.707107i 0 1.00000 1.73205i −3.67423 + 2.12132i 0 0 2.82843i 0 −3.00000 + 5.19615i
863.1 −1.22474 0.707107i 0 1.00000 + 1.73205i 3.67423 + 2.12132i 0 0 2.82843i 0 −3.00000 5.19615i
863.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i −3.67423 2.12132i 0 0 2.82843i 0 −3.00000 5.19615i
 $$n$$: 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
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.s.b 4
3.b odd 2 1 inner 882.3.s.b 4
7.b odd 2 1 882.3.s.d 4
7.c even 3 1 18.3.b.a 2
7.c even 3 1 inner 882.3.s.b 4
7.d odd 6 1 882.3.b.a 2
7.d odd 6 1 882.3.s.d 4
21.c even 2 1 882.3.s.d 4
21.g even 6 1 882.3.b.a 2
21.g even 6 1 882.3.s.d 4
21.h odd 6 1 18.3.b.a 2
21.h odd 6 1 inner 882.3.s.b 4
28.g odd 6 1 144.3.e.b 2
35.j even 6 1 450.3.d.f 2
35.l odd 12 2 450.3.b.b 4
56.k odd 6 1 576.3.e.f 2
56.p even 6 1 576.3.e.c 2
63.g even 3 1 162.3.d.b 4
63.h even 3 1 162.3.d.b 4
63.j odd 6 1 162.3.d.b 4
63.n odd 6 1 162.3.d.b 4
77.h odd 6 1 2178.3.c.d 2
84.n even 6 1 144.3.e.b 2
91.r even 6 1 3042.3.c.e 2
91.z odd 12 2 3042.3.d.a 4
105.o odd 6 1 450.3.d.f 2
105.x even 12 2 450.3.b.b 4
112.u odd 12 2 2304.3.h.c 4
112.w even 12 2 2304.3.h.f 4
140.p odd 6 1 3600.3.l.d 2
140.w even 12 2 3600.3.c.b 4
168.s odd 6 1 576.3.e.c 2
168.v even 6 1 576.3.e.f 2
231.l even 6 1 2178.3.c.d 2
252.o even 6 1 1296.3.q.f 4
252.u odd 6 1 1296.3.q.f 4
252.bb even 6 1 1296.3.q.f 4
252.bl odd 6 1 1296.3.q.f 4
273.w odd 6 1 3042.3.c.e 2
273.cd even 12 2 3042.3.d.a 4
336.bt odd 12 2 2304.3.h.f 4
336.bu even 12 2 2304.3.h.c 4
420.ba even 6 1 3600.3.l.d 2
420.bp odd 12 2 3600.3.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 7.c even 3 1
18.3.b.a 2 21.h odd 6 1
144.3.e.b 2 28.g odd 6 1
144.3.e.b 2 84.n even 6 1
162.3.d.b 4 63.g even 3 1
162.3.d.b 4 63.h even 3 1
162.3.d.b 4 63.j odd 6 1
162.3.d.b 4 63.n odd 6 1
450.3.b.b 4 35.l odd 12 2
450.3.b.b 4 105.x even 12 2
450.3.d.f 2 35.j even 6 1
450.3.d.f 2 105.o odd 6 1
576.3.e.c 2 56.p even 6 1
576.3.e.c 2 168.s odd 6 1
576.3.e.f 2 56.k odd 6 1
576.3.e.f 2 168.v even 6 1
882.3.b.a 2 7.d odd 6 1
882.3.b.a 2 21.g even 6 1
882.3.s.b 4 1.a even 1 1 trivial
882.3.s.b 4 3.b odd 2 1 inner
882.3.s.b 4 7.c even 3 1 inner
882.3.s.b 4 21.h odd 6 1 inner
882.3.s.d 4 7.b odd 2 1
882.3.s.d 4 7.d odd 6 1
882.3.s.d 4 21.c even 2 1
882.3.s.d 4 21.g even 6 1
1296.3.q.f 4 252.o even 6 1
1296.3.q.f 4 252.u odd 6 1
1296.3.q.f 4 252.bb even 6 1
1296.3.q.f 4 252.bl odd 6 1
2178.3.c.d 2 77.h odd 6 1
2178.3.c.d 2 231.l even 6 1
2304.3.h.c 4 112.u odd 12 2
2304.3.h.c 4 336.bu even 12 2
2304.3.h.f 4 112.w even 12 2
2304.3.h.f 4 336.bt odd 12 2
3042.3.c.e 2 91.r even 6 1
3042.3.c.e 2 273.w odd 6 1
3042.3.d.a 4 91.z odd 12 2
3042.3.d.a 4 273.cd even 12 2
3600.3.c.b 4 140.w even 12 2
3600.3.c.b 4 420.bp odd 12 2
3600.3.l.d 2 140.p odd 6 1
3600.3.l.d 2 420.ba even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} - 18T_{5}^{2} + 324$$ T5^4 - 18*T5^2 + 324 $$T_{11}^{4} - 288T_{11}^{2} + 82944$$ T11^4 - 288*T11^2 + 82944 $$T_{13} - 8$$ T13 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2T^{2} + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 18T^{2} + 324$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 288 T^{2} + 82944$$
$13$ $$(T - 8)^{4}$$
$17$ $$T^{4} - 162 T^{2} + 26244$$
$19$ $$(T^{2} - 16 T + 256)^{2}$$
$23$ $$T^{4} - 288 T^{2} + 82944$$
$29$ $$(T^{2} + 18)^{2}$$
$31$ $$(T^{2} + 44 T + 1936)^{2}$$
$37$ $$(T^{2} - 34 T + 1156)^{2}$$
$41$ $$(T^{2} + 2178)^{2}$$
$43$ $$(T + 40)^{4}$$
$47$ $$T^{4} - 7200 T^{2} + \cdots + 51840000$$
$53$ $$T^{4} - 1458 T^{2} + \cdots + 2125764$$
$59$ $$T^{4} - 1152 T^{2} + \cdots + 1327104$$
$61$ $$(T^{2} + 50 T + 2500)^{2}$$
$67$ $$(T^{2} + 8 T + 64)^{2}$$
$71$ $$(T^{2} + 2592)^{2}$$
$73$ $$(T^{2} - 16 T + 256)^{2}$$
$79$ $$(T^{2} - 76 T + 5776)^{2}$$
$83$ $$(T^{2} + 14112)^{2}$$
$89$ $$T^{4} - 162 T^{2} + 26244$$
$97$ $$(T - 176)^{4}$$