# Properties

 Label 3800.1.o.b Level $3800$ Weight $1$ Character orbit 3800.o Self dual yes Analytic conductor $1.896$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -152 Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,1,Mod(1101,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3800.1101");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3800.o (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: yes Analytic conductor: $$1.89644704801$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.152.1 Artin image: $D_6$ Artin field: Galois closure of 6.2.23104000.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8}+O(q^{10})$$ q + q^2 - q^3 + q^4 - q^6 + q^7 + q^8 $$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} - q^{12} - q^{13} + q^{14} + q^{16} + q^{17} - q^{19} - q^{21} + q^{23} - q^{24} - q^{26} + q^{27} + q^{28} + q^{29} + q^{32} + q^{34} + 2 q^{37} - q^{38} + q^{39} - q^{42} + q^{46} - 2 q^{47} - q^{48} - q^{51} - q^{52} - q^{53} + q^{54} + q^{56} + q^{57} + q^{58} + q^{59} + q^{64} - q^{67} + q^{68} - q^{69} + q^{73} + 2 q^{74} - q^{76} + q^{78} - q^{81} - q^{84} - q^{87} - q^{91} + q^{92} - 2 q^{94} - q^{96}+O(q^{100})$$ q + q^2 - q^3 + q^4 - q^6 + q^7 + q^8 - q^12 - q^13 + q^14 + q^16 + q^17 - q^19 - q^21 + q^23 - q^24 - q^26 + q^27 + q^28 + q^29 + q^32 + q^34 + 2 * q^37 - q^38 + q^39 - q^42 + q^46 - 2 * q^47 - q^48 - q^51 - q^52 - q^53 + q^54 + q^56 + q^57 + q^58 + q^59 + q^64 - q^67 + q^68 - q^69 + q^73 + 2 * q^74 - q^76 + q^78 - q^81 - q^84 - q^87 - q^91 + q^92 - 2 * q^94 - q^96

## Character values

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

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$1$$ $$0$$ $$1$$ $$0$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1101.1
 0
1.00000 −1.00000 1.00000 0 −1.00000 1.00000 1.00000 0 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
152.g odd 2 1 CM by $$\Q(\sqrt{-38})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.o.b 1
5.b even 2 1 152.1.g.a 1
5.c odd 4 2 3800.1.b.a 2
8.b even 2 1 3800.1.o.a 1
15.d odd 2 1 1368.1.i.b 1
19.b odd 2 1 3800.1.o.a 1
20.d odd 2 1 608.1.g.a 1
40.e odd 2 1 608.1.g.b 1
40.f even 2 1 152.1.g.b yes 1
40.i odd 4 2 3800.1.b.b 2
95.d odd 2 1 152.1.g.b yes 1
95.g even 4 2 3800.1.b.b 2
95.h odd 6 2 2888.1.l.a 2
95.i even 6 2 2888.1.l.b 2
95.o odd 18 6 2888.1.s.a 6
95.p even 18 6 2888.1.s.b 6
120.i odd 2 1 1368.1.i.a 1
152.g odd 2 1 CM 3800.1.o.b 1
285.b even 2 1 1368.1.i.a 1
380.d even 2 1 608.1.g.b 1
760.b odd 2 1 152.1.g.a 1
760.p even 2 1 608.1.g.a 1
760.t even 4 2 3800.1.b.a 2
760.z even 6 2 2888.1.l.a 2
760.bh odd 6 2 2888.1.l.b 2
760.cj even 18 6 2888.1.s.a 6
760.ck odd 18 6 2888.1.s.b 6
2280.m even 2 1 1368.1.i.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.g.a 1 5.b even 2 1
152.1.g.a 1 760.b odd 2 1
152.1.g.b yes 1 40.f even 2 1
152.1.g.b yes 1 95.d odd 2 1
608.1.g.a 1 20.d odd 2 1
608.1.g.a 1 760.p even 2 1
608.1.g.b 1 40.e odd 2 1
608.1.g.b 1 380.d even 2 1
1368.1.i.a 1 120.i odd 2 1
1368.1.i.a 1 285.b even 2 1
1368.1.i.b 1 15.d odd 2 1
1368.1.i.b 1 2280.m even 2 1
2888.1.l.a 2 95.h odd 6 2
2888.1.l.a 2 760.z even 6 2
2888.1.l.b 2 95.i even 6 2
2888.1.l.b 2 760.bh odd 6 2
2888.1.s.a 6 95.o odd 18 6
2888.1.s.a 6 760.cj even 18 6
2888.1.s.b 6 95.p even 18 6
2888.1.s.b 6 760.ck odd 18 6
3800.1.b.a 2 5.c odd 4 2
3800.1.b.a 2 760.t even 4 2
3800.1.b.b 2 40.i odd 4 2
3800.1.b.b 2 95.g even 4 2
3800.1.o.a 1 8.b even 2 1
3800.1.o.a 1 19.b odd 2 1
3800.1.o.b 1 1.a even 1 1 trivial
3800.1.o.b 1 152.g odd 2 1 CM

## Hecke kernels

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

 $$T_{3} + 1$$ T3 + 1 $$T_{7} - 1$$ T7 - 1

## Hecke characteristic polynomials

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