# Properties

 Label 1900.3.e.b Level $1900$ Weight $3$ Character orbit 1900.e Analytic conductor $51.771$ 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] = [1900,3,Mod(1101,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1900.1101");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1900.e (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: $$51.7712502285$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-29})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 29$$ x^2 + 29 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) 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 $$\beta = \sqrt{-29}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + q^{7} - 20 q^{9}+O(q^{10})$$ q + b * q^3 + q^7 - 20 * q^9 $$q + \beta q^{3} + q^{7} - 20 q^{9} + 14 q^{11} + 3 \beta q^{13} - 23 q^{17} + ( - 3 \beta + 10) q^{19} + \beta q^{21} + q^{23} - 11 \beta q^{27} + 9 \beta q^{29} + 6 \beta q^{31} + 14 \beta q^{33} + 6 \beta q^{37} - 87 q^{39} - 6 \beta q^{41} - 68 q^{43} - 26 q^{47} - 48 q^{49} - 23 \beta q^{51} - 15 \beta q^{53} + (10 \beta + 87) q^{57} - 3 \beta q^{59} - 40 q^{61} - 20 q^{63} - 3 \beta q^{67} + \beta q^{69} + 6 \beta q^{71} + 7 q^{73} + 14 q^{77} - 18 \beta q^{79} + 139 q^{81} - 32 q^{83} - 261 q^{87} + 24 \beta q^{89} + 3 \beta q^{91} - 174 q^{93} - 18 \beta q^{97} - 280 q^{99} +O(q^{100})$$ q + b * q^3 + q^7 - 20 * q^9 + 14 * q^11 + 3*b * q^13 - 23 * q^17 + (-3*b + 10) * q^19 + b * q^21 + q^23 - 11*b * q^27 + 9*b * q^29 + 6*b * q^31 + 14*b * q^33 + 6*b * q^37 - 87 * q^39 - 6*b * q^41 - 68 * q^43 - 26 * q^47 - 48 * q^49 - 23*b * q^51 - 15*b * q^53 + (10*b + 87) * q^57 - 3*b * q^59 - 40 * q^61 - 20 * q^63 - 3*b * q^67 + b * q^69 + 6*b * q^71 + 7 * q^73 + 14 * q^77 - 18*b * q^79 + 139 * q^81 - 32 * q^83 - 261 * q^87 + 24*b * q^89 + 3*b * q^91 - 174 * q^93 - 18*b * q^97 - 280 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{7} - 40 q^{9}+O(q^{10})$$ 2 * q + 2 * q^7 - 40 * q^9 $$2 q + 2 q^{7} - 40 q^{9} + 28 q^{11} - 46 q^{17} + 20 q^{19} + 2 q^{23} - 174 q^{39} - 136 q^{43} - 52 q^{47} - 96 q^{49} + 174 q^{57} - 80 q^{61} - 40 q^{63} + 14 q^{73} + 28 q^{77} + 278 q^{81} - 64 q^{83} - 522 q^{87} - 348 q^{93} - 560 q^{99}+O(q^{100})$$ 2 * q + 2 * q^7 - 40 * q^9 + 28 * q^11 - 46 * q^17 + 20 * q^19 + 2 * q^23 - 174 * q^39 - 136 * q^43 - 52 * q^47 - 96 * q^49 + 174 * q^57 - 80 * q^61 - 40 * q^63 + 14 * q^73 + 28 * q^77 + 278 * q^81 - 64 * q^83 - 522 * q^87 - 348 * q^93 - 560 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1101.1
 − 5.38516i 5.38516i
0 5.38516i 0 0 0 1.00000 0 −20.0000 0
1101.2 0 5.38516i 0 0 0 1.00000 0 −20.0000 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
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.3.e.b 2
5.b even 2 1 76.3.c.a 2
5.c odd 4 2 1900.3.g.b 4
15.d odd 2 1 684.3.h.c 2
19.b odd 2 1 inner 1900.3.e.b 2
20.d odd 2 1 304.3.e.b 2
40.e odd 2 1 1216.3.e.l 2
40.f even 2 1 1216.3.e.k 2
60.h even 2 1 2736.3.o.i 2
95.d odd 2 1 76.3.c.a 2
95.g even 4 2 1900.3.g.b 4
285.b even 2 1 684.3.h.c 2
380.d even 2 1 304.3.e.b 2
760.b odd 2 1 1216.3.e.k 2
760.p even 2 1 1216.3.e.l 2
1140.p odd 2 1 2736.3.o.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.c.a 2 5.b even 2 1
76.3.c.a 2 95.d odd 2 1
304.3.e.b 2 20.d odd 2 1
304.3.e.b 2 380.d even 2 1
684.3.h.c 2 15.d odd 2 1
684.3.h.c 2 285.b even 2 1
1216.3.e.k 2 40.f even 2 1
1216.3.e.k 2 760.b odd 2 1
1216.3.e.l 2 40.e odd 2 1
1216.3.e.l 2 760.p even 2 1
1900.3.e.b 2 1.a even 1 1 trivial
1900.3.e.b 2 19.b odd 2 1 inner
1900.3.g.b 4 5.c odd 4 2
1900.3.g.b 4 95.g even 4 2
2736.3.o.i 2 60.h even 2 1
2736.3.o.i 2 1140.p odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1900, [\chi])$$:

 $$T_{3}^{2} + 29$$ T3^2 + 29 $$T_{7} - 1$$ T7 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 29$$
$5$ $$T^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T - 14)^{2}$$
$13$ $$T^{2} + 261$$
$17$ $$(T + 23)^{2}$$
$19$ $$T^{2} - 20T + 361$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} + 2349$$
$31$ $$T^{2} + 1044$$
$37$ $$T^{2} + 1044$$
$41$ $$T^{2} + 1044$$
$43$ $$(T + 68)^{2}$$
$47$ $$(T + 26)^{2}$$
$53$ $$T^{2} + 6525$$
$59$ $$T^{2} + 261$$
$61$ $$(T + 40)^{2}$$
$67$ $$T^{2} + 261$$
$71$ $$T^{2} + 1044$$
$73$ $$(T - 7)^{2}$$
$79$ $$T^{2} + 9396$$
$83$ $$(T + 32)^{2}$$
$89$ $$T^{2} + 16704$$
$97$ $$T^{2} + 9396$$