# Properties

 Label 3800.1.bd.f Level $3800$ Weight $1$ Character orbit 3800.bd Analytic conductor $1.896$ Analytic rank $0$ Dimension $4$ Projective image $D_{3}$ CM discriminant -40 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,1,Mod(1451,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 0, 2]))

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

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

chi := DirichletCharacter("3800.1451");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3800.bd (of order $$6$$, degree $$2$$, 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: $$1.89644704801$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.14440.1 Artin image: $S_3\times C_{12}$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{2} - \zeta_{12}^{4} q^{4} + \zeta_{12}^{3} q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} +O(q^{10})$$ q + z^5 * q^2 - z^4 * q^4 + z^3 * q^7 + z^3 * q^8 - z^4 * q^9 $$q + \zeta_{12}^{5} q^{2} - \zeta_{12}^{4} q^{4} + \zeta_{12}^{3} q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} - q^{11} - 2 \zeta_{12} q^{13} - \zeta_{12}^{2} q^{14} - \zeta_{12}^{2} q^{16} + \zeta_{12}^{3} q^{18} + \zeta_{12}^{2} q^{19} - \zeta_{12}^{5} q^{22} + \zeta_{12} q^{23} + 2 q^{26} + \zeta_{12} q^{28} + \zeta_{12} q^{32} - \zeta_{12}^{2} q^{36} + \zeta_{12}^{3} q^{37} - \zeta_{12} q^{38} + \zeta_{12}^{2} q^{41} + \zeta_{12}^{4} q^{44} - q^{46} + 2 \zeta_{12} q^{47} + 2 \zeta_{12}^{5} q^{52} + \zeta_{12} q^{53} - q^{56} + 2 \zeta_{12}^{2} q^{59} + \zeta_{12} q^{63} - q^{64} + \zeta_{12} q^{72} - \zeta_{12}^{2} q^{74} + q^{76} - \zeta_{12}^{3} q^{77} - \zeta_{12}^{2} q^{81} - \zeta_{12} q^{82} - \zeta_{12}^{3} q^{88} + \zeta_{12}^{4} q^{89} - 2 \zeta_{12}^{4} q^{91} - \zeta_{12}^{5} q^{92} - 2 q^{94} + \zeta_{12}^{4} q^{99} +O(q^{100})$$ q + z^5 * q^2 - z^4 * q^4 + z^3 * q^7 + z^3 * q^8 - z^4 * q^9 - q^11 - 2*z * q^13 - z^2 * q^14 - z^2 * q^16 + z^3 * q^18 + z^2 * q^19 - z^5 * q^22 + z * q^23 + 2 * q^26 + z * q^28 + z * q^32 - z^2 * q^36 + z^3 * q^37 - z * q^38 + z^2 * q^41 + z^4 * q^44 - q^46 + 2*z * q^47 + 2*z^5 * q^52 + z * q^53 - q^56 + 2*z^2 * q^59 + z * q^63 - q^64 + z * q^72 - z^2 * q^74 + q^76 - z^3 * q^77 - z^2 * q^81 - z * q^82 - z^3 * q^88 + z^4 * q^89 - 2*z^4 * q^91 - z^5 * q^92 - 2 * q^94 + z^4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 + 2 * q^9 $$4 q + 2 q^{4} + 2 q^{9} - 4 q^{11} - 2 q^{14} - 2 q^{16} + 2 q^{19} + 8 q^{26} - 2 q^{36} + 2 q^{41} - 2 q^{44} - 4 q^{46} - 4 q^{56} + 4 q^{59} - 4 q^{64} - 2 q^{74} + 4 q^{76} - 2 q^{81} - 2 q^{89} + 4 q^{91} - 8 q^{94} - 2 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 + 2 * q^9 - 4 * q^11 - 2 * q^14 - 2 * q^16 + 2 * q^19 + 8 * q^26 - 2 * q^36 + 2 * q^41 - 2 * q^44 - 4 * q^46 - 4 * q^56 + 4 * q^59 - 4 * q^64 - 2 * q^74 + 4 * q^76 - 2 * q^81 - 2 * q^89 + 4 * q^91 - 8 * q^94 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$\zeta_{12}^{4}$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1451.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 1.00000i 1.00000i 0.500000 0.866025i 0
1451.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 1.00000i 1.00000i 0.500000 0.866025i 0
3051.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 1.00000i 1.00000i 0.500000 + 0.866025i 0
3051.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 1.00000i 1.00000i 0.500000 + 0.866025i 0
 $$n$$: 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
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
5.b even 2 1 inner
8.d odd 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner
152.k odd 6 1 inner
760.bm odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.bd.f 4
5.b even 2 1 inner 3800.1.bd.f 4
5.c odd 4 1 760.1.bm.a 2
5.c odd 4 1 760.1.bm.b yes 2
8.d odd 2 1 inner 3800.1.bd.f 4
19.c even 3 1 inner 3800.1.bd.f 4
20.e even 4 1 3040.1.cc.a 2
20.e even 4 1 3040.1.cc.b 2
40.e odd 2 1 CM 3800.1.bd.f 4
40.i odd 4 1 3040.1.cc.a 2
40.i odd 4 1 3040.1.cc.b 2
40.k even 4 1 760.1.bm.a 2
40.k even 4 1 760.1.bm.b yes 2
95.i even 6 1 inner 3800.1.bd.f 4
95.m odd 12 1 760.1.bm.a 2
95.m odd 12 1 760.1.bm.b yes 2
152.k odd 6 1 inner 3800.1.bd.f 4
380.v even 12 1 3040.1.cc.a 2
380.v even 12 1 3040.1.cc.b 2
760.bm odd 6 1 inner 3800.1.bd.f 4
760.br odd 12 1 3040.1.cc.a 2
760.br odd 12 1 3040.1.cc.b 2
760.bw even 12 1 760.1.bm.a 2
760.bw even 12 1 760.1.bm.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.1.bm.a 2 5.c odd 4 1
760.1.bm.a 2 40.k even 4 1
760.1.bm.a 2 95.m odd 12 1
760.1.bm.a 2 760.bw even 12 1
760.1.bm.b yes 2 5.c odd 4 1
760.1.bm.b yes 2 40.k even 4 1
760.1.bm.b yes 2 95.m odd 12 1
760.1.bm.b yes 2 760.bw even 12 1
3040.1.cc.a 2 20.e even 4 1
3040.1.cc.a 2 40.i odd 4 1
3040.1.cc.a 2 380.v even 12 1
3040.1.cc.a 2 760.br odd 12 1
3040.1.cc.b 2 20.e even 4 1
3040.1.cc.b 2 40.i odd 4 1
3040.1.cc.b 2 380.v even 12 1
3040.1.cc.b 2 760.br odd 12 1
3800.1.bd.f 4 1.a even 1 1 trivial
3800.1.bd.f 4 5.b even 2 1 inner
3800.1.bd.f 4 8.d odd 2 1 inner
3800.1.bd.f 4 19.c even 3 1 inner
3800.1.bd.f 4 40.e odd 2 1 CM
3800.1.bd.f 4 95.i even 6 1 inner
3800.1.bd.f 4 152.k odd 6 1 inner
3800.1.bd.f 4 760.bm odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3800, [\chi])$$:

 $$T_{3}$$ T3 $$T_{11} + 1$$ T11 + 1

## Hecke characteristic polynomials

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