# Properties

 Label 1900.2.l.a Level $1900$ Weight $2$ Character orbit 1900.l Analytic conductor $15.172$ Analytic rank $0$ Dimension $8$ CM discriminant -19 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(493,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("1900.493");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.l (of order $$4$$, 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: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.2702336256.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625$$ x^8 + 9*x^6 + 56*x^4 + 225*x^2 + 625 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{3}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 380) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} + \beta_1 + 1) q^{7} - 3 \beta_1 q^{9}+O(q^{10})$$ q + (b5 + b1 + 1) * q^7 - 3*b1 * q^9 $$q + (\beta_{5} + \beta_1 + 1) q^{7} - 3 \beta_1 q^{9} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 1) q^{11}+ \cdots + ( - 3 \beta_{6} + 3 \beta_{5} + \cdots + 3 \beta_1) q^{99}+O(q^{100})$$ q + (b5 + b1 + 1) * q^7 - 3*b1 * q^9 + (-b7 - b6 - b5 + b4 - 1) * q^11 + (b5 - b4 - b2 - b1 - 1) * q^17 - b3 * q^19 + (-b7 - b3 + 2*b1 - 2) * q^23 + (-b7 - b6 - b4 - b3 + b2) * q^43 + (-b7 - 3*b5 + b4 + b3 + b2 + 2*b1 + 2) * q^47 + (b6 - b5 - b3 + b2 + 6*b1) * q^49 + (b7 + 3*b6 + 3*b5 - 3*b4 + 3) * q^61 + (3*b6 - 3*b1 + 3) * q^63 + (-b7 - b6 + 2*b4 - b3 - 2*b2 + 4*b1 - 4) * q^73 + (-3*b7 - 3*b5 + 2*b4 + 3*b3 + 2*b2 - 5*b1 - 5) * q^77 - 9 * q^81 + (-b7 - b3 - 8*b1 + 8) * q^83 + (-3*b6 + 3*b5 - 3*b3 - 3*b2 + 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 6 q^{7}+O(q^{10})$$ 8 * q + 6 * q^7 $$8 q + 6 q^{7} - 14 q^{17} - 16 q^{23} - 2 q^{43} + 26 q^{47} + 18 q^{63} - 22 q^{73} - 26 q^{77} - 72 q^{81} + 64 q^{83}+O(q^{100})$$ 8 * q + 6 * q^7 - 14 * q^17 - 16 * q^23 - 2 * q^43 + 26 * q^47 + 18 * q^63 - 22 * q^73 - 26 * q^77 - 72 * q^81 + 64 * q^83

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625$$ :

 $$\beta_{1}$$ $$=$$ $$( 9\nu^{7} + 56\nu^{5} + 154\nu^{3} + 625\nu ) / 1750$$ (9*v^7 + 56*v^5 + 154*v^3 + 625*v) / 1750 $$\beta_{2}$$ $$=$$ $$( -2\nu^{7} - 9\nu^{6} - 28\nu^{5} - 56\nu^{4} - 252\nu^{3} - 504\nu^{2} - 1710\nu - 1325 ) / 700$$ (-2*v^7 - 9*v^6 - 28*v^5 - 56*v^4 - 252*v^3 - 504*v^2 - 1710*v - 1325) / 700 $$\beta_{3}$$ $$=$$ $$( \nu^{6} - 27 ) / 28$$ (v^6 - 27) / 28 $$\beta_{4}$$ $$=$$ $$( \nu^{7} - 20\nu^{6} - 16\nu^{5} - 180\nu^{4} - 44\nu^{3} - 620\nu^{2} - 175\nu - 2000 ) / 500$$ (v^7 - 20*v^6 - 16*v^5 - 180*v^4 - 44*v^3 - 620*v^2 - 175*v - 2000) / 500 $$\beta_{5}$$ $$=$$ $$( -7\nu^{7} - \nu^{6} - 28\nu^{5} - 84\nu^{4} - 252\nu^{3} - 56\nu^{2} - 1015\nu - 925 ) / 700$$ (-7*v^7 - v^6 - 28*v^5 - 84*v^4 - 252*v^3 - 56*v^2 - 1015*v - 925) / 700 $$\beta_{6}$$ $$=$$ $$( -7\nu^{7} - 135\nu^{6} + 112\nu^{5} - 840\nu^{4} + 308\nu^{3} - 4060\nu^{2} + 4725\nu - 12875 ) / 3500$$ (-7*v^7 - 135*v^6 + 112*v^5 - 840*v^4 + 308*v^3 - 4060*v^2 + 4725*v - 12875) / 3500 $$\beta_{7}$$ $$=$$ $$( 9\nu^{7} + 56\nu^{5} + 404\nu^{3} + 625\nu ) / 500$$ (9*v^7 + 56*v^5 + 404*v^3 + 625*v) / 500
 $$\nu$$ $$=$$ $$( -3\beta_{7} + \beta_{6} - 3\beta_{5} + \beta_{4} + \beta_{3} - 3\beta_{2} + 3\beta _1 - 1 ) / 10$$ (-3*b7 + b6 - 3*b5 + b4 + b3 - 3*b2 + 3*b1 - 1) / 10 $$\nu^{2}$$ $$=$$ $$( \beta_{7} - 7\beta_{6} + 11\beta_{5} + 3\beta_{4} - 7\beta_{3} - 9\beta_{2} + 9\beta _1 - 23 ) / 10$$ (b7 - 7*b6 + 11*b5 + 3*b4 - 7*b3 - 9*b2 + 9*b1 - 23) / 10 $$\nu^{3}$$ $$=$$ $$2\beta_{7} - 7\beta_1$$ 2*b7 - 7*b1 $$\nu^{4}$$ $$=$$ $$( -9\beta_{7} + 13\beta_{6} - 49\beta_{5} - 27\beta_{4} - 7\beta_{3} + 31\beta_{2} - 31\beta _1 - 73 ) / 10$$ (-9*b7 + 13*b6 - 49*b5 - 27*b4 - 7*b3 + 31*b2 - 31*b1 - 73) / 10 $$\nu^{5}$$ $$=$$ $$( 23\beta_{7} + 79\beta_{6} + 123\beta_{5} - 101\beta_{4} - 11\beta_{3} + 33\beta_{2} + 247\beta _1 + 101 ) / 10$$ (23*b7 + 79*b6 + 123*b5 - 101*b4 - 11*b3 + 33*b2 + 247*b1 + 101) / 10 $$\nu^{6}$$ $$=$$ $$28\beta_{3} + 27$$ 28*b3 + 27 $$\nu^{7}$$ $$=$$ $$( -277\beta_{7} - 561\beta_{6} - 557\beta_{5} + 559\beta_{4} - \beta_{3} + 3\beta_{2} + 1397\beta _1 - 559 ) / 10$$ (-277*b7 - 561*b6 - 557*b5 + 559*b4 - b3 + 3*b2 + 1397*b1 - 559) / 10

## Character values

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

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$\beta_{1}$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
493.1
 −0.656712 − 2.13746i 0.656712 − 2.13746i 1.52274 + 1.63746i −1.52274 + 1.63746i −0.656712 + 2.13746i 0.656712 + 2.13746i 1.52274 − 1.63746i −1.52274 − 1.63746i
0 0 0 0 0 −3.52622 + 3.52622i 0 3.00000i 0
493.2 0 0 0 0 0 1.25130 1.25130i 0 3.00000i 0
493.3 0 0 0 0 0 2.42815 2.42815i 0 3.00000i 0
493.4 0 0 0 0 0 2.84677 2.84677i 0 3.00000i 0
1557.1 0 0 0 0 0 −3.52622 3.52622i 0 3.00000i 0
1557.2 0 0 0 0 0 1.25130 + 1.25130i 0 3.00000i 0
1557.3 0 0 0 0 0 2.42815 + 2.42815i 0 3.00000i 0
1557.4 0 0 0 0 0 2.84677 + 2.84677i 0 3.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 493.4 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 CM by $$\Q(\sqrt{-19})$$
5.c odd 4 1 inner
95.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.l.a 8
5.b even 2 1 380.2.l.a 8
5.c odd 4 1 380.2.l.a 8
5.c odd 4 1 inner 1900.2.l.a 8
15.d odd 2 1 3420.2.bb.c 8
15.e even 4 1 3420.2.bb.c 8
19.b odd 2 1 CM 1900.2.l.a 8
95.d odd 2 1 380.2.l.a 8
95.g even 4 1 380.2.l.a 8
95.g even 4 1 inner 1900.2.l.a 8
285.b even 2 1 3420.2.bb.c 8
285.j odd 4 1 3420.2.bb.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.l.a 8 5.b even 2 1
380.2.l.a 8 5.c odd 4 1
380.2.l.a 8 95.d odd 2 1
380.2.l.a 8 95.g even 4 1
1900.2.l.a 8 1.a even 1 1 trivial
1900.2.l.a 8 5.c odd 4 1 inner
1900.2.l.a 8 19.b odd 2 1 CM
1900.2.l.a 8 95.g even 4 1 inner
3420.2.bb.c 8 15.d odd 2 1
3420.2.bb.c 8 15.e even 4 1
3420.2.bb.c 8 285.b even 2 1
3420.2.bb.c 8 285.j odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 6 T^{7} + \cdots + 14884$$
$11$ $$(T^{4} - 47 T^{2} + 196)^{2}$$
$13$ $$T^{8}$$
$17$ $$T^{8} + 14 T^{7} + \cdots + 412164$$
$19$ $$(T^{2} + 19)^{4}$$
$23$ $$(T^{4} + 8 T^{3} + \cdots + 900)^{2}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8} + 2 T^{7} + \cdots + 2815684$$
$47$ $$T^{8} - 26 T^{7} + \cdots + 1004004$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$(T^{4} - 347 T^{2} + 26896)^{2}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$T^{8} + 22 T^{7} + \cdots + 235991044$$
$79$ $$T^{8}$$
$83$ $$(T^{4} - 32 T^{3} + \cdots + 8100)^{2}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$