# Properties

 Label 2178.3.c.d Level $2178$ Weight $3$ Character orbit 2178.c Analytic conductor $59.346$ 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] = [2178,3,Mod(485,2178)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2178, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("2178.485");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2178 = 2 \cdot 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2178.c (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: $$59.3462015777$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) 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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 2 q^{4} + 3 \beta q^{5} + 4 q^{7} - 2 \beta q^{8} +O(q^{10})$$ q + b * q^2 - 2 * q^4 + 3*b * q^5 + 4 * q^7 - 2*b * q^8 $$q + \beta q^{2} - 2 q^{4} + 3 \beta q^{5} + 4 q^{7} - 2 \beta q^{8} - 6 q^{10} - 8 q^{13} + 4 \beta q^{14} + 4 q^{16} - 9 \beta q^{17} + 16 q^{19} - 6 \beta q^{20} + 12 \beta q^{23} + 7 q^{25} - 8 \beta q^{26} - 8 q^{28} + 3 \beta q^{29} + 44 q^{31} + 4 \beta q^{32} + 18 q^{34} + 12 \beta q^{35} - 34 q^{37} + 16 \beta q^{38} + 12 q^{40} + 33 \beta q^{41} + 40 q^{43} - 24 q^{46} + 60 \beta q^{47} - 33 q^{49} + 7 \beta q^{50} + 16 q^{52} - 27 \beta q^{53} - 8 \beta q^{56} - 6 q^{58} - 24 \beta q^{59} - 50 q^{61} + 44 \beta q^{62} - 8 q^{64} - 24 \beta q^{65} + 8 q^{67} + 18 \beta q^{68} - 24 q^{70} + 36 \beta q^{71} + 16 q^{73} - 34 \beta q^{74} - 32 q^{76} + 76 q^{79} + 12 \beta q^{80} - 66 q^{82} + 84 \beta q^{83} + 54 q^{85} + 40 \beta q^{86} - 9 \beta q^{89} - 32 q^{91} - 24 \beta q^{92} - 120 q^{94} + 48 \beta q^{95} + 176 q^{97} - 33 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 2 * q^4 + 3*b * q^5 + 4 * q^7 - 2*b * q^8 - 6 * q^10 - 8 * q^13 + 4*b * q^14 + 4 * q^16 - 9*b * q^17 + 16 * q^19 - 6*b * q^20 + 12*b * q^23 + 7 * q^25 - 8*b * q^26 - 8 * q^28 + 3*b * q^29 + 44 * q^31 + 4*b * q^32 + 18 * q^34 + 12*b * q^35 - 34 * q^37 + 16*b * q^38 + 12 * q^40 + 33*b * q^41 + 40 * q^43 - 24 * q^46 + 60*b * q^47 - 33 * q^49 + 7*b * q^50 + 16 * q^52 - 27*b * q^53 - 8*b * q^56 - 6 * q^58 - 24*b * q^59 - 50 * q^61 + 44*b * q^62 - 8 * q^64 - 24*b * q^65 + 8 * q^67 + 18*b * q^68 - 24 * q^70 + 36*b * q^71 + 16 * q^73 - 34*b * q^74 - 32 * q^76 + 76 * q^79 + 12*b * q^80 - 66 * q^82 + 84*b * q^83 + 54 * q^85 + 40*b * q^86 - 9*b * q^89 - 32 * q^91 - 24*b * q^92 - 120 * q^94 + 48*b * q^95 + 176 * q^97 - 33*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} + 8 q^{7}+O(q^{10})$$ 2 * q - 4 * q^4 + 8 * q^7 $$2 q - 4 q^{4} + 8 q^{7} - 12 q^{10} - 16 q^{13} + 8 q^{16} + 32 q^{19} + 14 q^{25} - 16 q^{28} + 88 q^{31} + 36 q^{34} - 68 q^{37} + 24 q^{40} + 80 q^{43} - 48 q^{46} - 66 q^{49} + 32 q^{52} - 12 q^{58} - 100 q^{61} - 16 q^{64} + 16 q^{67} - 48 q^{70} + 32 q^{73} - 64 q^{76} + 152 q^{79} - 132 q^{82} + 108 q^{85} - 64 q^{91} - 240 q^{94} + 352 q^{97}+O(q^{100})$$ 2 * q - 4 * q^4 + 8 * q^7 - 12 * q^10 - 16 * q^13 + 8 * q^16 + 32 * q^19 + 14 * q^25 - 16 * q^28 + 88 * q^31 + 36 * q^34 - 68 * q^37 + 24 * q^40 + 80 * q^43 - 48 * q^46 - 66 * q^49 + 32 * q^52 - 12 * q^58 - 100 * q^61 - 16 * q^64 + 16 * q^67 - 48 * q^70 + 32 * q^73 - 64 * q^76 + 152 * q^79 - 132 * q^82 + 108 * q^85 - 64 * q^91 - 240 * q^94 + 352 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2178\mathbb{Z}\right)^\times$$.

 $$n$$ $$1333$$ $$1937$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
485.1
 − 1.41421i 1.41421i
1.41421i 0 −2.00000 4.24264i 0 4.00000 2.82843i 0 −6.00000
485.2 1.41421i 0 −2.00000 4.24264i 0 4.00000 2.82843i 0 −6.00000
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2178.3.c.d 2
3.b odd 2 1 inner 2178.3.c.d 2
11.b odd 2 1 18.3.b.a 2
33.d even 2 1 18.3.b.a 2
44.c even 2 1 144.3.e.b 2
55.d odd 2 1 450.3.d.f 2
55.e even 4 2 450.3.b.b 4
77.b even 2 1 882.3.b.a 2
77.h odd 6 2 882.3.s.b 4
77.i even 6 2 882.3.s.d 4
88.b odd 2 1 576.3.e.c 2
88.g even 2 1 576.3.e.f 2
99.g even 6 2 162.3.d.b 4
99.h odd 6 2 162.3.d.b 4
132.d odd 2 1 144.3.e.b 2
143.d odd 2 1 3042.3.c.e 2
143.g even 4 2 3042.3.d.a 4
165.d even 2 1 450.3.d.f 2
165.l odd 4 2 450.3.b.b 4
176.i even 4 2 2304.3.h.c 4
176.l odd 4 2 2304.3.h.f 4
220.g even 2 1 3600.3.l.d 2
220.i odd 4 2 3600.3.c.b 4
231.h odd 2 1 882.3.b.a 2
231.k odd 6 2 882.3.s.d 4
231.l even 6 2 882.3.s.b 4
264.m even 2 1 576.3.e.c 2
264.p odd 2 1 576.3.e.f 2
396.k even 6 2 1296.3.q.f 4
396.o odd 6 2 1296.3.q.f 4
429.e even 2 1 3042.3.c.e 2
429.l odd 4 2 3042.3.d.a 4
528.s odd 4 2 2304.3.h.c 4
528.x even 4 2 2304.3.h.f 4
660.g odd 2 1 3600.3.l.d 2
660.q even 4 2 3600.3.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 11.b odd 2 1
18.3.b.a 2 33.d even 2 1
144.3.e.b 2 44.c even 2 1
144.3.e.b 2 132.d odd 2 1
162.3.d.b 4 99.g even 6 2
162.3.d.b 4 99.h odd 6 2
450.3.b.b 4 55.e even 4 2
450.3.b.b 4 165.l odd 4 2
450.3.d.f 2 55.d odd 2 1
450.3.d.f 2 165.d even 2 1
576.3.e.c 2 88.b odd 2 1
576.3.e.c 2 264.m even 2 1
576.3.e.f 2 88.g even 2 1
576.3.e.f 2 264.p odd 2 1
882.3.b.a 2 77.b even 2 1
882.3.b.a 2 231.h odd 2 1
882.3.s.b 4 77.h odd 6 2
882.3.s.b 4 231.l even 6 2
882.3.s.d 4 77.i even 6 2
882.3.s.d 4 231.k odd 6 2
1296.3.q.f 4 396.k even 6 2
1296.3.q.f 4 396.o odd 6 2
2178.3.c.d 2 1.a even 1 1 trivial
2178.3.c.d 2 3.b odd 2 1 inner
2304.3.h.c 4 176.i even 4 2
2304.3.h.c 4 528.s odd 4 2
2304.3.h.f 4 176.l odd 4 2
2304.3.h.f 4 528.x even 4 2
3042.3.c.e 2 143.d odd 2 1
3042.3.c.e 2 429.e even 2 1
3042.3.d.a 4 143.g even 4 2
3042.3.d.a 4 429.l odd 4 2
3600.3.c.b 4 220.i odd 4 2
3600.3.c.b 4 660.q even 4 2
3600.3.l.d 2 220.g even 2 1
3600.3.l.d 2 660.g odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(2178, [\chi])$$:

 $$T_{5}^{2} + 18$$ T5^2 + 18 $$T_{7} - 4$$ T7 - 4 $$T_{13} + 8$$ T13 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 18$$
$7$ $$(T - 4)^{2}$$
$11$ $$T^{2}$$
$13$ $$(T + 8)^{2}$$
$17$ $$T^{2} + 162$$
$19$ $$(T - 16)^{2}$$
$23$ $$T^{2} + 288$$
$29$ $$T^{2} + 18$$
$31$ $$(T - 44)^{2}$$
$37$ $$(T + 34)^{2}$$
$41$ $$T^{2} + 2178$$
$43$ $$(T - 40)^{2}$$
$47$ $$T^{2} + 7200$$
$53$ $$T^{2} + 1458$$
$59$ $$T^{2} + 1152$$
$61$ $$(T + 50)^{2}$$
$67$ $$(T - 8)^{2}$$
$71$ $$T^{2} + 2592$$
$73$ $$(T - 16)^{2}$$
$79$ $$(T - 76)^{2}$$
$83$ $$T^{2} + 14112$$
$89$ $$T^{2} + 162$$
$97$ $$(T - 176)^{2}$$