# Properties

 Label 3751.1.t.a Level $3751$ Weight $1$ Character orbit 3751.t Analytic conductor $1.872$ Analytic rank $0$ Dimension $4$ Projective image $D_{3}$ CM discriminant -31 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3751,1,Mod(2138,3751)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3751, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([2, 5]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3751.2138");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3751 = 11^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3751.t (of order $$10$$, degree $$4$$, 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: $$1.87199286239$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 31) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.31.1 Artin image: $C_{10}\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{30} - \cdots)$$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{10}^{3} q^{2} - \zeta_{10}^{2} q^{5} - \zeta_{10} q^{7} - \zeta_{10}^{4} q^{8} - \zeta_{10}^{3} q^{9} +O(q^{10})$$ q - z^3 * q^2 - z^2 * q^5 - z * q^7 - z^4 * q^8 - z^3 * q^9 $$q - \zeta_{10}^{3} q^{2} - \zeta_{10}^{2} q^{5} - \zeta_{10} q^{7} - \zeta_{10}^{4} q^{8} - \zeta_{10}^{3} q^{9} - q^{10} + \zeta_{10}^{4} q^{14} - \zeta_{10}^{2} q^{16} - \zeta_{10} q^{18} + \zeta_{10}^{4} q^{19} - \zeta_{10}^{3} q^{31} + q^{32} + \zeta_{10}^{3} q^{35} + \zeta_{10}^{2} q^{38} - \zeta_{10} q^{40} + \zeta_{10}^{4} q^{41} - q^{45} + \zeta_{10}^{4} q^{47} - q^{56} + \zeta_{10} q^{59} - \zeta_{10} q^{62} + \zeta_{10}^{4} q^{63} - \zeta_{10}^{3} q^{64} + q^{67} + \zeta_{10} q^{70} - \zeta_{10}^{2} q^{71} - \zeta_{10}^{2} q^{72} + \zeta_{10}^{4} q^{80} - \zeta_{10} q^{81} + \zeta_{10}^{2} q^{82} + \zeta_{10}^{3} q^{90} + 2 \zeta_{10}^{2} q^{94} + \zeta_{10} q^{95} + \zeta_{10}^{3} q^{97} +O(q^{100})$$ q - z^3 * q^2 - z^2 * q^5 - z * q^7 - z^4 * q^8 - z^3 * q^9 - q^10 + z^4 * q^14 - z^2 * q^16 - z * q^18 + z^4 * q^19 - z^3 * q^31 + q^32 + z^3 * q^35 + z^2 * q^38 - z * q^40 + z^4 * q^41 - q^45 + z^4 * q^47 - q^56 + z * q^59 - z * q^62 + z^4 * q^63 - z^3 * q^64 + q^67 + z * q^70 - z^2 * q^71 - z^2 * q^72 + z^4 * q^80 - z * q^81 + z^2 * q^82 + z^3 * q^90 + 2*z^2 * q^94 + z * q^95 + z^3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} + q^{5} - q^{7} + q^{8} - q^{9}+O(q^{10})$$ 4 * q - q^2 + q^5 - q^7 + q^8 - q^9 $$4 q - q^{2} + q^{5} - q^{7} + q^{8} - q^{9} - 4 q^{10} - q^{14} + q^{16} - q^{18} - q^{19} - q^{31} + q^{35} - q^{38} - q^{40} - q^{41} - 4 q^{45} - 2 q^{47} - 4 q^{56} + q^{59} - q^{62} - q^{63} - q^{64} + 8 q^{67} + q^{70} + q^{71} + q^{72} - q^{80} - q^{81} - q^{82} + q^{90} - 2 q^{94} + q^{95} + q^{97}+O(q^{100})$$ 4 * q - q^2 + q^5 - q^7 + q^8 - q^9 - 4 * q^10 - q^14 + q^16 - q^18 - q^19 - q^31 + q^35 - q^38 - q^40 - q^41 - 4 * q^45 - 2 * q^47 - 4 * q^56 + q^59 - q^62 - q^63 - q^64 + 8 * q^67 + q^70 + q^71 + q^72 - q^80 - q^81 - q^82 + q^90 - 2 * q^94 + q^95 + q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3751\mathbb{Z}\right)^\times$$.

 $$n$$ $$2421$$ $$2543$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2138.1
 −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i
−0.809017 0.587785i 0 0 0.809017 0.587785i 0 0.309017 + 0.951057i −0.309017 + 0.951057i −0.809017 0.587785i −1.00000
2665.1 −0.809017 + 0.587785i 0 0 0.809017 + 0.587785i 0 0.309017 0.951057i −0.309017 0.951057i −0.809017 + 0.587785i −1.00000
2913.1 0.309017 0.951057i 0 0 −0.309017 0.951057i 0 −0.809017 0.587785i 0.809017 0.587785i 0.309017 0.951057i −1.00000
3657.1 0.309017 + 0.951057i 0 0 −0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000
 $$n$$: 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
31.b odd 2 1 CM by $$\Q(\sqrt{-31})$$
11.c even 5 3 inner
341.t odd 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3751.1.t.a 4
11.b odd 2 1 3751.1.t.c 4
11.c even 5 1 3751.1.d.b 1
11.c even 5 3 inner 3751.1.t.a 4
11.d odd 10 1 31.1.b.a 1
11.d odd 10 3 3751.1.t.c 4
31.b odd 2 1 CM 3751.1.t.a 4
33.f even 10 1 279.1.d.b 1
44.g even 10 1 496.1.e.a 1
55.h odd 10 1 775.1.d.b 1
55.l even 20 2 775.1.c.a 2
77.l even 10 1 1519.1.c.a 1
77.n even 30 2 1519.1.n.a 2
77.o odd 30 2 1519.1.n.b 2
88.k even 10 1 1984.1.e.b 1
88.p odd 10 1 1984.1.e.a 1
99.o odd 30 2 2511.1.m.e 2
99.p even 30 2 2511.1.m.a 2
341.b even 2 1 3751.1.t.c 4
341.o odd 10 1 961.1.f.a 4
341.p even 10 1 961.1.f.a 4
341.t odd 10 1 3751.1.d.b 1
341.t odd 10 3 inner 3751.1.t.a 4
341.v even 10 1 961.1.f.a 4
341.x odd 10 1 961.1.f.a 4
341.y odd 10 1 961.1.f.a 4
341.ba even 10 1 31.1.b.a 1
341.ba even 10 3 3751.1.t.c 4
341.bb even 10 1 961.1.f.a 4
341.bc even 10 1 961.1.f.a 4
341.be odd 10 1 961.1.f.a 4
341.bm even 30 2 961.1.h.a 8
341.bn odd 30 2 961.1.h.a 8
341.bt odd 30 2 961.1.h.a 8
341.bv even 30 2 961.1.h.a 8
341.bw even 30 2 961.1.h.a 8
341.bx even 30 2 961.1.e.a 2
341.bz odd 30 2 961.1.h.a 8
341.ca odd 30 2 961.1.h.a 8
341.cb odd 30 2 961.1.e.a 2
341.cc even 30 2 961.1.h.a 8
1023.bg odd 10 1 279.1.d.b 1
1364.bc odd 10 1 496.1.e.a 1
1705.bi even 10 1 775.1.d.b 1
1705.dk odd 20 2 775.1.c.a 2
2387.cv odd 10 1 1519.1.c.a 1
2387.gi even 30 2 1519.1.n.b 2
2387.gt odd 30 2 1519.1.n.a 2
2728.cw even 10 1 1984.1.e.a 1
2728.eb odd 10 1 1984.1.e.b 1
3069.gp even 30 2 2511.1.m.e 2
3069.ib odd 30 2 2511.1.m.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 11.d odd 10 1
31.1.b.a 1 341.ba even 10 1
279.1.d.b 1 33.f even 10 1
279.1.d.b 1 1023.bg odd 10 1
496.1.e.a 1 44.g even 10 1
496.1.e.a 1 1364.bc odd 10 1
775.1.c.a 2 55.l even 20 2
775.1.c.a 2 1705.dk odd 20 2
775.1.d.b 1 55.h odd 10 1
775.1.d.b 1 1705.bi even 10 1
961.1.e.a 2 341.bx even 30 2
961.1.e.a 2 341.cb odd 30 2
961.1.f.a 4 341.o odd 10 1
961.1.f.a 4 341.p even 10 1
961.1.f.a 4 341.v even 10 1
961.1.f.a 4 341.x odd 10 1
961.1.f.a 4 341.y odd 10 1
961.1.f.a 4 341.bb even 10 1
961.1.f.a 4 341.bc even 10 1
961.1.f.a 4 341.be odd 10 1
961.1.h.a 8 341.bm even 30 2
961.1.h.a 8 341.bn odd 30 2
961.1.h.a 8 341.bt odd 30 2
961.1.h.a 8 341.bv even 30 2
961.1.h.a 8 341.bw even 30 2
961.1.h.a 8 341.bz odd 30 2
961.1.h.a 8 341.ca odd 30 2
961.1.h.a 8 341.cc even 30 2
1519.1.c.a 1 77.l even 10 1
1519.1.c.a 1 2387.cv odd 10 1
1519.1.n.a 2 77.n even 30 2
1519.1.n.a 2 2387.gt odd 30 2
1519.1.n.b 2 77.o odd 30 2
1519.1.n.b 2 2387.gi even 30 2
1984.1.e.a 1 88.p odd 10 1
1984.1.e.a 1 2728.cw even 10 1
1984.1.e.b 1 88.k even 10 1
1984.1.e.b 1 2728.eb odd 10 1
2511.1.m.a 2 99.p even 30 2
2511.1.m.a 2 3069.ib odd 30 2
2511.1.m.e 2 99.o odd 30 2
2511.1.m.e 2 3069.gp even 30 2
3751.1.d.b 1 11.c even 5 1
3751.1.d.b 1 341.t odd 10 1
3751.1.t.a 4 1.a even 1 1 trivial
3751.1.t.a 4 11.c even 5 3 inner
3751.1.t.a 4 31.b odd 2 1 CM
3751.1.t.a 4 341.t odd 10 3 inner
3751.1.t.c 4 11.b odd 2 1
3751.1.t.c 4 11.d odd 10 3
3751.1.t.c 4 341.b even 2 1
3751.1.t.c 4 341.ba even 10 3

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + T_{2}^{3} + T_{2}^{2} + T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3751, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$7$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$37$ $$T^{4}$$
$41$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$43$ $$T^{4}$$
$47$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$53$ $$T^{4}$$
$59$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$61$ $$T^{4}$$
$67$ $$(T - 2)^{4}$$
$71$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} - T^{3} + T^{2} - T + 1$$