# Properties

 Label 1900.2.s.a Level $1900$ Weight $2$ Character orbit 1900.s Analytic conductor $15.172$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(49,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1900.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.s (of order $$6$$, degree $$2$$, 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: $$15.1715763840$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{3} - 2 \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + z * q^3 - 2*z^2 * q^9 $$q + \zeta_{12} q^{3} - 2 \zeta_{12}^{2} q^{9} - 4 q^{11} + (\zeta_{12}^{3} - \zeta_{12}) q^{13} + 3 \zeta_{12} q^{17} + ( - 2 \zeta_{12}^{2} + 5) q^{19} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{23} - 5 \zeta_{12}^{3} q^{27} + 7 \zeta_{12}^{2} q^{29} + 4 q^{31} - 4 \zeta_{12} q^{33} - 10 \zeta_{12}^{3} q^{37} - q^{39} + ( - 5 \zeta_{12}^{2} + 5) q^{41} + 5 \zeta_{12} q^{43} + ( - 7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{47} + 7 q^{49} + 3 \zeta_{12}^{2} q^{51} + ( - 11 \zeta_{12}^{3} + 11 \zeta_{12}) q^{53} + ( - 2 \zeta_{12}^{3} + 5 \zeta_{12}) q^{57} + ( - 3 \zeta_{12}^{2} + 3) q^{59} - 11 \zeta_{12}^{2} q^{61} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{67} + 5 q^{69} + (11 \zeta_{12}^{2} - 11) q^{71} - 15 \zeta_{12} q^{73} + (13 \zeta_{12}^{2} - 13) q^{79} + (\zeta_{12}^{2} - 1) q^{81} + 7 \zeta_{12}^{3} q^{87} + 3 \zeta_{12}^{2} q^{89} + 4 \zeta_{12} q^{93} - 5 \zeta_{12} q^{97} + 8 \zeta_{12}^{2} q^{99} +O(q^{100})$$ q + z * q^3 - 2*z^2 * q^9 - 4 * q^11 + (z^3 - z) * q^13 + 3*z * q^17 + (-2*z^2 + 5) * q^19 + (-5*z^3 + 5*z) * q^23 - 5*z^3 * q^27 + 7*z^2 * q^29 + 4 * q^31 - 4*z * q^33 - 10*z^3 * q^37 - q^39 + (-5*z^2 + 5) * q^41 + 5*z * q^43 + (-7*z^3 + 7*z) * q^47 + 7 * q^49 + 3*z^2 * q^51 + (-11*z^3 + 11*z) * q^53 + (-2*z^3 + 5*z) * q^57 + (-3*z^2 + 3) * q^59 - 11*z^2 * q^61 + (-3*z^3 + 3*z) * q^67 + 5 * q^69 + (11*z^2 - 11) * q^71 - 15*z * q^73 + (13*z^2 - 13) * q^79 + (z^2 - 1) * q^81 + 7*z^3 * q^87 + 3*z^2 * q^89 + 4*z * q^93 - 5*z * q^97 + 8*z^2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} - 16 q^{11} + 16 q^{19} + 14 q^{29} + 16 q^{31} - 4 q^{39} + 10 q^{41} + 28 q^{49} + 6 q^{51} + 6 q^{59} - 22 q^{61} + 20 q^{69} - 22 q^{71} - 26 q^{79} - 2 q^{81} + 6 q^{89} + 16 q^{99}+O(q^{100})$$ 4 * q - 4 * q^9 - 16 * q^11 + 16 * q^19 + 14 * q^29 + 16 * q^31 - 4 * q^39 + 10 * q^41 + 28 * q^49 + 6 * q^51 + 6 * q^59 - 22 * q^61 + 20 * q^69 - 22 * q^71 - 26 * q^79 - 2 * q^81 + 6 * q^89 + 16 * q^99

## Character values

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

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 −0.866025 0.500000i 0 0 0 0 0 −1.00000 1.73205i 0
49.2 0 0.866025 + 0.500000i 0 0 0 0 0 −1.00000 1.73205i 0
349.1 0 −0.866025 + 0.500000i 0 0 0 0 0 −1.00000 + 1.73205i 0
349.2 0 0.866025 0.500000i 0 0 0 0 0 −1.00000 + 1.73205i 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
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.s.a 4
5.b even 2 1 inner 1900.2.s.a 4
5.c odd 4 1 76.2.e.a 2
5.c odd 4 1 1900.2.i.a 2
15.e even 4 1 684.2.k.b 2
19.c even 3 1 inner 1900.2.s.a 4
20.e even 4 1 304.2.i.a 2
40.i odd 4 1 1216.2.i.c 2
40.k even 4 1 1216.2.i.g 2
60.l odd 4 1 2736.2.s.g 2
95.g even 4 1 1444.2.e.b 2
95.i even 6 1 inner 1900.2.s.a 4
95.l even 12 1 1444.2.a.c 1
95.l even 12 1 1444.2.e.b 2
95.m odd 12 1 76.2.e.a 2
95.m odd 12 1 1444.2.a.b 1
95.m odd 12 1 1900.2.i.a 2
285.v even 12 1 684.2.k.b 2
380.v even 12 1 304.2.i.a 2
380.v even 12 1 5776.2.a.k 1
380.w odd 12 1 5776.2.a.f 1
760.br odd 12 1 1216.2.i.c 2
760.bw even 12 1 1216.2.i.g 2
1140.bu odd 12 1 2736.2.s.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.e.a 2 5.c odd 4 1
76.2.e.a 2 95.m odd 12 1
304.2.i.a 2 20.e even 4 1
304.2.i.a 2 380.v even 12 1
684.2.k.b 2 15.e even 4 1
684.2.k.b 2 285.v even 12 1
1216.2.i.c 2 40.i odd 4 1
1216.2.i.c 2 760.br odd 12 1
1216.2.i.g 2 40.k even 4 1
1216.2.i.g 2 760.bw even 12 1
1444.2.a.b 1 95.m odd 12 1
1444.2.a.c 1 95.l even 12 1
1444.2.e.b 2 95.g even 4 1
1444.2.e.b 2 95.l even 12 1
1900.2.i.a 2 5.c odd 4 1
1900.2.i.a 2 95.m odd 12 1
1900.2.s.a 4 1.a even 1 1 trivial
1900.2.s.a 4 5.b even 2 1 inner
1900.2.s.a 4 19.c even 3 1 inner
1900.2.s.a 4 95.i even 6 1 inner
2736.2.s.g 2 60.l odd 4 1
2736.2.s.g 2 1140.bu odd 12 1
5776.2.a.f 1 380.w odd 12 1
5776.2.a.k 1 380.v even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$(T + 4)^{4}$$
$13$ $$T^{4} - T^{2} + 1$$
$17$ $$T^{4} - 9T^{2} + 81$$
$19$ $$(T^{2} - 8 T + 19)^{2}$$
$23$ $$T^{4} - 25T^{2} + 625$$
$29$ $$(T^{2} - 7 T + 49)^{2}$$
$31$ $$(T - 4)^{4}$$
$37$ $$(T^{2} + 100)^{2}$$
$41$ $$(T^{2} - 5 T + 25)^{2}$$
$43$ $$T^{4} - 25T^{2} + 625$$
$47$ $$T^{4} - 49T^{2} + 2401$$
$53$ $$T^{4} - 121 T^{2} + 14641$$
$59$ $$(T^{2} - 3 T + 9)^{2}$$
$61$ $$(T^{2} + 11 T + 121)^{2}$$
$67$ $$T^{4} - 9T^{2} + 81$$
$71$ $$(T^{2} + 11 T + 121)^{2}$$
$73$ $$T^{4} - 225 T^{2} + 50625$$
$79$ $$(T^{2} + 13 T + 169)^{2}$$
$83$ $$T^{4}$$
$89$ $$(T^{2} - 3 T + 9)^{2}$$
$97$ $$T^{4} - 25T^{2} + 625$$