# Properties

 Label 1900.2.c.e Level $1900$ Weight $2$ Character orbit 1900.c Analytic conductor $15.172$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1749,1900)]

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, 1, 0]))

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

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

chi := DirichletCharacter("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.c (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: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$4$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 380) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + (\beta_{2} - \beta_1) q^{7} + (\beta_{3} - 1) q^{9}+O(q^{10})$$ q - b2 * q^3 + (b2 - b1) * q^7 + (b3 - 1) * q^9 $$q - \beta_{2} q^{3} + (\beta_{2} - \beta_1) q^{7} + (\beta_{3} - 1) q^{9} + \beta_{3} q^{11} + \beta_{2} q^{13} + ( - \beta_{2} - \beta_1) q^{17} - q^{19} + ( - \beta_{3} + 2) q^{21} + (\beta_{2} + \beta_1) q^{23} + (2 \beta_{2} - 2 \beta_1) q^{27} + \beta_{3} q^{29} + (\beta_{3} + 2) q^{31} + (4 \beta_{2} - 2 \beta_1) q^{33} + (3 \beta_{2} - 2 \beta_1) q^{37} + ( - \beta_{3} + 4) q^{39} - 6 q^{41} + (\beta_{2} + 3 \beta_1) q^{43} + (5 \beta_{2} - \beta_1) q^{47} + 3 q^{49} + (\beta_{3} - 6) q^{51} + (5 \beta_{2} - 4 \beta_1) q^{53} + \beta_{2} q^{57} + 2 \beta_{3} q^{59} + ( - 3 \beta_{3} + 2) q^{61} + ( - 3 \beta_{2} - \beta_1) q^{63} + (3 \beta_{2} - 2 \beta_1) q^{67} + ( - \beta_{3} + 6) q^{69} + ( - \beta_{3} - 6) q^{71} + (\beta_{2} - 3 \beta_1) q^{73} + ( - 2 \beta_{2} - 2 \beta_1) q^{77} + (2 \beta_{3} + 4) q^{79} + (\beta_{3} + 1) q^{81} + ( - \beta_{2} - \beta_1) q^{83} + (4 \beta_{2} - 2 \beta_1) q^{87} + ( - \beta_{3} + 12) q^{89} + (\beta_{3} - 2) q^{91} + (2 \beta_{2} - 2 \beta_1) q^{93} + ( - 5 \beta_{2} - 4 \beta_1) q^{97} + ( - \beta_{3} + 12) q^{99}+O(q^{100})$$ q - b2 * q^3 + (b2 - b1) * q^7 + (b3 - 1) * q^9 + b3 * q^11 + b2 * q^13 + (-b2 - b1) * q^17 - q^19 + (-b3 + 2) * q^21 + (b2 + b1) * q^23 + (2*b2 - 2*b1) * q^27 + b3 * q^29 + (b3 + 2) * q^31 + (4*b2 - 2*b1) * q^33 + (3*b2 - 2*b1) * q^37 + (-b3 + 4) * q^39 - 6 * q^41 + (b2 + 3*b1) * q^43 + (5*b2 - b1) * q^47 + 3 * q^49 + (b3 - 6) * q^51 + (5*b2 - 4*b1) * q^53 + b2 * q^57 + 2*b3 * q^59 + (-3*b3 + 2) * q^61 + (-3*b2 - b1) * q^63 + (3*b2 - 2*b1) * q^67 + (-b3 + 6) * q^69 + (-b3 - 6) * q^71 + (b2 - 3*b1) * q^73 + (-2*b2 - 2*b1) * q^77 + (2*b3 + 4) * q^79 + (b3 + 1) * q^81 + (-b2 - b1) * q^83 + (4*b2 - 2*b1) * q^87 + (-b3 + 12) * q^89 + (b3 - 2) * q^91 + (2*b2 - 2*b1) * q^93 + (-5*b2 - 4*b1) * q^97 + (-b3 + 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} - 4 q^{19} + 8 q^{21} + 8 q^{31} + 16 q^{39} - 24 q^{41} + 12 q^{49} - 24 q^{51} + 8 q^{61} + 24 q^{69} - 24 q^{71} + 16 q^{79} + 4 q^{81} + 48 q^{89} - 8 q^{91} + 48 q^{99}+O(q^{100})$$ 4 * q - 4 * q^9 - 4 * q^19 + 8 * q^21 + 8 * q^31 + 16 * q^39 - 24 * q^41 + 12 * q^49 - 24 * q^51 + 8 * q^61 + 24 * q^69 - 24 * q^71 + 16 * q^79 + 4 * q^81 + 48 * q^89 - 8 * q^91 + 48 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1$$ v^3 + 2*v^2 - 1 $$\beta_{2}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1$$ -v^3 + 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-2\zeta_{12}^{3} + 4\zeta_{12}$$ -2*v^3 + 4*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (b3 - b2 + b1) / 4 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + \beta _1 + 2 ) / 4$$ (b2 + b1 + 2) / 4 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{2} + \beta_1 ) / 2$$ (-b2 + b1) / 2

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1749.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 2.73205i 0 0 0 2.00000i 0 −4.46410 0
1749.2 0 0.732051i 0 0 0 2.00000i 0 2.46410 0
1749.3 0 0.732051i 0 0 0 2.00000i 0 2.46410 0
1749.4 0 2.73205i 0 0 0 2.00000i 0 −4.46410 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.c.e 4
5.b even 2 1 inner 1900.2.c.e 4
5.c odd 4 1 380.2.a.d 2
5.c odd 4 1 1900.2.a.d 2
15.e even 4 1 3420.2.a.h 2
20.e even 4 1 1520.2.a.l 2
20.e even 4 1 7600.2.a.bf 2
40.i odd 4 1 6080.2.a.z 2
40.k even 4 1 6080.2.a.bj 2
95.g even 4 1 7220.2.a.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.d 2 5.c odd 4 1
1520.2.a.l 2 20.e even 4 1
1900.2.a.d 2 5.c odd 4 1
1900.2.c.e 4 1.a even 1 1 trivial
1900.2.c.e 4 5.b even 2 1 inner
3420.2.a.h 2 15.e even 4 1
6080.2.a.z 2 40.i odd 4 1
6080.2.a.bj 2 40.k even 4 1
7220.2.a.h 2 95.g even 4 1
7600.2.a.bf 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1900, [\chi])$$:

 $$T_{3}^{4} + 8T_{3}^{2} + 4$$ T3^4 + 8*T3^2 + 4 $$T_{7}^{2} + 4$$ T7^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 8T^{2} + 4$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} - 12)^{2}$$
$13$ $$T^{4} + 8T^{2} + 4$$
$17$ $$(T^{2} + 12)^{2}$$
$19$ $$(T + 1)^{4}$$
$23$ $$(T^{2} + 12)^{2}$$
$29$ $$(T^{2} - 12)^{2}$$
$31$ $$(T^{2} - 4 T - 8)^{2}$$
$37$ $$T^{4} + 56T^{2} + 484$$
$41$ $$(T + 6)^{4}$$
$43$ $$T^{4} + 104T^{2} + 1936$$
$47$ $$T^{4} + 168T^{2} + 144$$
$53$ $$T^{4} + 168T^{2} + 6084$$
$59$ $$(T^{2} - 48)^{2}$$
$61$ $$(T^{2} - 4 T - 104)^{2}$$
$67$ $$T^{4} + 56T^{2} + 484$$
$71$ $$(T^{2} + 12 T + 24)^{2}$$
$73$ $$T^{4} + 56T^{2} + 16$$
$79$ $$(T^{2} - 8 T - 32)^{2}$$
$83$ $$(T^{2} + 12)^{2}$$
$89$ $$(T^{2} - 24 T + 132)^{2}$$
$97$ $$T^{4} + 488 T^{2} + 58564$$