# Properties

 Label 3822.2.c.d Level $3822$ Weight $2$ Character orbit 3822.c Analytic conductor $30.519$ 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] = [3822,2,Mod(883,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3822.883");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3822.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: $$30.5188236525$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - i q^{2} - q^{3} - q^{4} + 2 i q^{5} + i q^{6} + i q^{8} + q^{9} +O(q^{10})$$ q - i * q^2 - q^3 - q^4 + 2*i * q^5 + i * q^6 + i * q^8 + q^9 $$q - i q^{2} - q^{3} - q^{4} + 2 i q^{5} + i q^{6} + i q^{8} + q^{9} + 2 q^{10} + q^{12} + ( - 2 i + 3) q^{13} - 2 i q^{15} + q^{16} + 2 q^{17} - i q^{18} - 6 i q^{19} - 2 i q^{20} + 4 q^{23} - i q^{24} + q^{25} + ( - 3 i - 2) q^{26} - q^{27} - 10 q^{29} - 2 q^{30} + 10 i q^{31} - i q^{32} - 2 i q^{34} - q^{36} - 8 i q^{37} - 6 q^{38} + (2 i - 3) q^{39} - 2 q^{40} + 10 i q^{41} + 4 q^{43} + 2 i q^{45} - 4 i q^{46} - 12 i q^{47} - q^{48} - i q^{50} - 2 q^{51} + (2 i - 3) q^{52} - 6 q^{53} + i q^{54} + 6 i q^{57} + 10 i q^{58} + 4 i q^{59} + 2 i q^{60} - 2 q^{61} + 10 q^{62} - q^{64} + (6 i + 4) q^{65} + 2 i q^{67} - 2 q^{68} - 4 q^{69} + i q^{72} - 4 i q^{73} - 8 q^{74} - q^{75} + 6 i q^{76} + (3 i + 2) q^{78} + 2 i q^{80} + q^{81} + 10 q^{82} - 4 i q^{83} + 4 i q^{85} - 4 i q^{86} + 10 q^{87} - 6 i q^{89} + 2 q^{90} - 4 q^{92} - 10 i q^{93} - 12 q^{94} + 12 q^{95} + i q^{96} - 12 i q^{97} +O(q^{100})$$ q - i * q^2 - q^3 - q^4 + 2*i * q^5 + i * q^6 + i * q^8 + q^9 + 2 * q^10 + q^12 + (-2*i + 3) * q^13 - 2*i * q^15 + q^16 + 2 * q^17 - i * q^18 - 6*i * q^19 - 2*i * q^20 + 4 * q^23 - i * q^24 + q^25 + (-3*i - 2) * q^26 - q^27 - 10 * q^29 - 2 * q^30 + 10*i * q^31 - i * q^32 - 2*i * q^34 - q^36 - 8*i * q^37 - 6 * q^38 + (2*i - 3) * q^39 - 2 * q^40 + 10*i * q^41 + 4 * q^43 + 2*i * q^45 - 4*i * q^46 - 12*i * q^47 - q^48 - i * q^50 - 2 * q^51 + (2*i - 3) * q^52 - 6 * q^53 + i * q^54 + 6*i * q^57 + 10*i * q^58 + 4*i * q^59 + 2*i * q^60 - 2 * q^61 + 10 * q^62 - q^64 + (6*i + 4) * q^65 + 2*i * q^67 - 2 * q^68 - 4 * q^69 + i * q^72 - 4*i * q^73 - 8 * q^74 - q^75 + 6*i * q^76 + (3*i + 2) * q^78 + 2*i * q^80 + q^81 + 10 * q^82 - 4*i * q^83 + 4*i * q^85 - 4*i * q^86 + 10 * q^87 - 6*i * q^89 + 2 * q^90 - 4 * q^92 - 10*i * q^93 - 12 * q^94 + 12 * q^95 + i * q^96 - 12*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 4 q^{10} + 2 q^{12} + 6 q^{13} + 2 q^{16} + 4 q^{17} + 8 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{27} - 20 q^{29} - 4 q^{30} - 2 q^{36} - 12 q^{38} - 6 q^{39} - 4 q^{40} + 8 q^{43} - 2 q^{48} - 4 q^{51} - 6 q^{52} - 12 q^{53} - 4 q^{61} + 20 q^{62} - 2 q^{64} + 8 q^{65} - 4 q^{68} - 8 q^{69} - 16 q^{74} - 2 q^{75} + 4 q^{78} + 2 q^{81} + 20 q^{82} + 20 q^{87} + 4 q^{90} - 8 q^{92} - 24 q^{94} + 24 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 4 * q^10 + 2 * q^12 + 6 * q^13 + 2 * q^16 + 4 * q^17 + 8 * q^23 + 2 * q^25 - 4 * q^26 - 2 * q^27 - 20 * q^29 - 4 * q^30 - 2 * q^36 - 12 * q^38 - 6 * q^39 - 4 * q^40 + 8 * q^43 - 2 * q^48 - 4 * q^51 - 6 * q^52 - 12 * q^53 - 4 * q^61 + 20 * q^62 - 2 * q^64 + 8 * q^65 - 4 * q^68 - 8 * q^69 - 16 * q^74 - 2 * q^75 + 4 * q^78 + 2 * q^81 + 20 * q^82 + 20 * q^87 + 4 * q^90 - 8 * q^92 - 24 * q^94 + 24 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3822\mathbb{Z}\right)^\times$$.

 $$n$$ $$1471$$ $$2549$$ $$3433$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
883.1
 1.00000i − 1.00000i
1.00000i −1.00000 −1.00000 2.00000i 1.00000i 0 1.00000i 1.00000 2.00000
883.2 1.00000i −1.00000 −1.00000 2.00000i 1.00000i 0 1.00000i 1.00000 2.00000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3822.2.c.d 2
7.b odd 2 1 78.2.b.a 2
13.b even 2 1 inner 3822.2.c.d 2
21.c even 2 1 234.2.b.a 2
28.d even 2 1 624.2.c.a 2
35.c odd 2 1 1950.2.b.c 2
35.f even 4 1 1950.2.f.d 2
35.f even 4 1 1950.2.f.g 2
56.e even 2 1 2496.2.c.m 2
56.h odd 2 1 2496.2.c.f 2
84.h odd 2 1 1872.2.c.b 2
91.b odd 2 1 78.2.b.a 2
91.i even 4 1 1014.2.a.b 1
91.i even 4 1 1014.2.a.g 1
91.n odd 6 2 1014.2.i.c 4
91.t odd 6 2 1014.2.i.c 4
91.bc even 12 2 1014.2.e.b 2
91.bc even 12 2 1014.2.e.e 2
273.g even 2 1 234.2.b.a 2
273.o odd 4 1 3042.2.a.c 1
273.o odd 4 1 3042.2.a.n 1
364.h even 2 1 624.2.c.a 2
364.p odd 4 1 8112.2.a.g 1
364.p odd 4 1 8112.2.a.j 1
455.h odd 2 1 1950.2.b.c 2
455.s even 4 1 1950.2.f.d 2
455.s even 4 1 1950.2.f.g 2
728.b even 2 1 2496.2.c.m 2
728.l odd 2 1 2496.2.c.f 2
1092.d odd 2 1 1872.2.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.b.a 2 7.b odd 2 1
78.2.b.a 2 91.b odd 2 1
234.2.b.a 2 21.c even 2 1
234.2.b.a 2 273.g even 2 1
624.2.c.a 2 28.d even 2 1
624.2.c.a 2 364.h even 2 1
1014.2.a.b 1 91.i even 4 1
1014.2.a.g 1 91.i even 4 1
1014.2.e.b 2 91.bc even 12 2
1014.2.e.e 2 91.bc even 12 2
1014.2.i.c 4 91.n odd 6 2
1014.2.i.c 4 91.t odd 6 2
1872.2.c.b 2 84.h odd 2 1
1872.2.c.b 2 1092.d odd 2 1
1950.2.b.c 2 35.c odd 2 1
1950.2.b.c 2 455.h odd 2 1
1950.2.f.d 2 35.f even 4 1
1950.2.f.d 2 455.s even 4 1
1950.2.f.g 2 35.f even 4 1
1950.2.f.g 2 455.s even 4 1
2496.2.c.f 2 56.h odd 2 1
2496.2.c.f 2 728.l odd 2 1
2496.2.c.m 2 56.e even 2 1
2496.2.c.m 2 728.b even 2 1
3042.2.a.c 1 273.o odd 4 1
3042.2.a.n 1 273.o odd 4 1
3822.2.c.d 2 1.a even 1 1 trivial
3822.2.c.d 2 13.b even 2 1 inner
8112.2.a.g 1 364.p odd 4 1
8112.2.a.j 1 364.p odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3822, [\chi])$$:

 $$T_{5}^{2} + 4$$ T5^2 + 4 $$T_{11}$$ T11 $$T_{17} - 2$$ T17 - 2 $$T_{19}^{2} + 36$$ T19^2 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 6T + 13$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T - 4)^{2}$$
$29$ $$(T + 10)^{2}$$
$31$ $$T^{2} + 100$$
$37$ $$T^{2} + 64$$
$41$ $$T^{2} + 100$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 144$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 16$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 144$$