# Properties

 Label 3800.1.b.a Level $3800$ Weight $1$ Character orbit 3800.b Analytic conductor $1.896$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -152 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,1,Mod(949,3800)]

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, 1, 1]))

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

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

chi := DirichletCharacter("3800.949");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3800.b (of order $$2$$, degree $$1$$, 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: $$1.89644704801$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ 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 152) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.152.1 Artin image: $C_4\times D_6$ 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.

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

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
949.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i 0 0
949.2 1.00000i 1.00000i −1.00000 0 −1.00000 1.00000i 1.00000i 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 $$\Q(\sqrt{-38})$$
5.b even 2 1 inner
760.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.b.a 2
5.b even 2 1 inner 3800.1.b.a 2
5.c odd 4 1 152.1.g.a 1
5.c odd 4 1 3800.1.o.b 1
8.b even 2 1 3800.1.b.b 2
15.e even 4 1 1368.1.i.b 1
19.b odd 2 1 3800.1.b.b 2
20.e even 4 1 608.1.g.a 1
40.f even 2 1 3800.1.b.b 2
40.i odd 4 1 152.1.g.b yes 1
40.i odd 4 1 3800.1.o.a 1
40.k even 4 1 608.1.g.b 1
95.d odd 2 1 3800.1.b.b 2
95.g even 4 1 152.1.g.b yes 1
95.g even 4 1 3800.1.o.a 1
95.l even 12 2 2888.1.l.a 2
95.m odd 12 2 2888.1.l.b 2
95.q odd 36 6 2888.1.s.b 6
95.r even 36 6 2888.1.s.a 6
120.w even 4 1 1368.1.i.a 1
152.g odd 2 1 CM 3800.1.b.a 2
285.j odd 4 1 1368.1.i.a 1
380.j odd 4 1 608.1.g.b 1
760.b odd 2 1 inner 3800.1.b.a 2
760.t even 4 1 152.1.g.a 1
760.t even 4 1 3800.1.o.b 1
760.y odd 4 1 608.1.g.a 1
760.bp even 12 2 2888.1.l.b 2
760.br odd 12 2 2888.1.l.a 2
760.cq odd 36 6 2888.1.s.a 6
760.cs even 36 6 2888.1.s.b 6
2280.bw odd 4 1 1368.1.i.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.g.a 1 5.c odd 4 1
152.1.g.a 1 760.t even 4 1
152.1.g.b yes 1 40.i odd 4 1
152.1.g.b yes 1 95.g even 4 1
608.1.g.a 1 20.e even 4 1
608.1.g.a 1 760.y odd 4 1
608.1.g.b 1 40.k even 4 1
608.1.g.b 1 380.j odd 4 1
1368.1.i.a 1 120.w even 4 1
1368.1.i.a 1 285.j odd 4 1
1368.1.i.b 1 15.e even 4 1
1368.1.i.b 1 2280.bw odd 4 1
2888.1.l.a 2 95.l even 12 2
2888.1.l.a 2 760.br odd 12 2
2888.1.l.b 2 95.m odd 12 2
2888.1.l.b 2 760.bp even 12 2
2888.1.s.a 6 95.r even 36 6
2888.1.s.a 6 760.cq odd 36 6
2888.1.s.b 6 95.q odd 36 6
2888.1.s.b 6 760.cs even 36 6
3800.1.b.a 2 1.a even 1 1 trivial
3800.1.b.a 2 5.b even 2 1 inner
3800.1.b.a 2 152.g odd 2 1 CM
3800.1.b.a 2 760.b odd 2 1 inner
3800.1.b.b 2 8.b even 2 1
3800.1.b.b 2 19.b odd 2 1
3800.1.b.b 2 40.f even 2 1
3800.1.b.b 2 95.d odd 2 1
3800.1.o.a 1 40.i odd 4 1
3800.1.o.a 1 95.g even 4 1
3800.1.o.b 1 5.c odd 4 1
3800.1.o.b 1 760.t even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{29} + 1$$ T29 + 1

## Hecke characteristic polynomials

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