# Properties

 Label 1900.1.bb.a Level $1900$ Weight $1$ Character orbit 1900.bb Analytic conductor $0.948$ Analytic rank $0$ Dimension $8$ Projective image $D_{15}$ CM discriminant -19 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,1,Mod(341,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 2, 5]))

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

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

chi := DirichletCharacter("1900.341");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1900.bb (of order $$10$$, degree $$4$$, 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: $$0.948223524003$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{15})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1$$ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{15}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{15} + \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{30}^{7} q^{5} + ( - \zeta_{30}^{13} + \zeta_{30}^{2}) q^{7} + \zeta_{30}^{12} q^{9} +O(q^{10})$$ q - z^7 * q^5 + (-z^13 + z^2) * q^7 + z^12 * q^9 $$q - \zeta_{30}^{7} q^{5} + ( - \zeta_{30}^{13} + \zeta_{30}^{2}) q^{7} + \zeta_{30}^{12} q^{9} + ( - \zeta_{30}^{5} - \zeta_{30}) q^{11} + ( - \zeta_{30}^{13} - \zeta_{30}^{5}) q^{17} - \zeta_{30}^{9} q^{19} + (\zeta_{30}^{6} + 1) q^{23} + \zeta_{30}^{14} q^{25} + ( - \zeta_{30}^{9} - \zeta_{30}^{5}) q^{35} + (\zeta_{30}^{14} - \zeta_{30}) q^{43} + \zeta_{30}^{4} q^{45} - \zeta_{30}^{6} q^{47} + ( - \zeta_{30}^{11} + \zeta_{30}^{4} + 1) q^{49} + (\zeta_{30}^{12} + \zeta_{30}^{8}) q^{55} + (\zeta_{30}^{4} + \zeta_{30}^{2}) q^{61} + (\zeta_{30}^{14} + \zeta_{30}^{10}) q^{63} + \zeta_{30}^{3} q^{73} + (\zeta_{30}^{14} - \zeta_{30}^{7} - \zeta_{30}^{3}) q^{77} - \zeta_{30}^{9} q^{81} + ( - \zeta_{30}^{3} + 1) q^{83} + (\zeta_{30}^{12} - \zeta_{30}^{5}) q^{85} - \zeta_{30} q^{95} + ( - \zeta_{30}^{13} + \zeta_{30}^{2}) q^{99} +O(q^{100})$$ q - z^7 * q^5 + (-z^13 + z^2) * q^7 + z^12 * q^9 + (-z^5 - z) * q^11 + (-z^13 - z^5) * q^17 - z^9 * q^19 + (z^6 + 1) * q^23 + z^14 * q^25 + (-z^9 - z^5) * q^35 + (z^14 - z) * q^43 + z^4 * q^45 - z^6 * q^47 + (-z^11 + z^4 + 1) * q^49 + (z^12 + z^8) * q^55 + (z^4 + z^2) * q^61 + (z^14 + z^10) * q^63 + z^3 * q^73 + (z^14 - z^7 - z^3) * q^77 - z^9 * q^81 + (-z^3 + 1) * q^83 + (z^12 - z^5) * q^85 - z * q^95 + (-z^13 + z^2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10})$$ 8 * q + q^5 + 2 * q^7 - 2 * q^9 $$8 q + q^{5} + 2 q^{7} - 2 q^{9} - 3 q^{11} - 3 q^{17} - 2 q^{19} + 6 q^{23} + q^{25} - 6 q^{35} + 2 q^{43} + q^{45} + 2 q^{47} + 10 q^{49} - q^{55} + 2 q^{61} - 3 q^{63} + 2 q^{73} - 2 q^{77} - 2 q^{81} + 6 q^{83} - 6 q^{85} + q^{95} + 2 q^{99}+O(q^{100})$$ 8 * q + q^5 + 2 * q^7 - 2 * q^9 - 3 * q^11 - 3 * q^17 - 2 * q^19 + 6 * q^23 + q^25 - 6 * q^35 + 2 * q^43 + q^45 + 2 * q^47 + 10 * q^49 - q^55 + 2 * q^61 - 3 * q^63 + 2 * q^73 - 2 * q^77 - 2 * q^81 + 6 * q^83 - 6 * q^85 + q^95 + 2 * q^99

## Character values

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

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$\zeta_{30}^{6}$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
341.1
 −0.978148 − 0.207912i 0.669131 − 0.743145i 0.913545 − 0.406737i −0.104528 + 0.994522i 0.913545 + 0.406737i −0.104528 − 0.994522i −0.978148 + 0.207912i 0.669131 + 0.743145i
0 0 0 −0.104528 0.994522i 0 1.82709 0 −0.809017 + 0.587785i 0
341.2 0 0 0 0.913545 + 0.406737i 0 −0.209057 0 −0.809017 + 0.587785i 0
721.1 0 0 0 −0.978148 0.207912i 0 1.33826 0 0.309017 + 0.951057i 0
721.2 0 0 0 0.669131 0.743145i 0 −1.95630 0 0.309017 + 0.951057i 0
1481.1 0 0 0 −0.978148 + 0.207912i 0 1.33826 0 0.309017 0.951057i 0
1481.2 0 0 0 0.669131 + 0.743145i 0 −1.95630 0 0.309017 0.951057i 0
1861.1 0 0 0 −0.104528 + 0.994522i 0 1.82709 0 −0.809017 0.587785i 0
1861.2 0 0 0 0.913545 0.406737i 0 −0.209057 0 −0.809017 0.587785i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 341.2 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 $$\Q(\sqrt{-19})$$
25.d even 5 1 inner
475.o odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.1.bb.a 8
19.b odd 2 1 CM 1900.1.bb.a 8
25.d even 5 1 inner 1900.1.bb.a 8
475.o odd 10 1 inner 1900.1.bb.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.1.bb.a 8 1.a even 1 1 trivial
1900.1.bb.a 8 19.b odd 2 1 CM
1900.1.bb.a 8 25.d even 5 1 inner
1900.1.bb.a 8 475.o odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1$$
$7$ $$(T^{4} - T^{3} - 4 T^{2} + 4 T + 1)^{2}$$
$11$ $$T^{8} + 3 T^{7} + 8 T^{6} + 11 T^{5} + \cdots + 1$$
$13$ $$T^{8}$$
$17$ $$T^{8} + 3 T^{7} + 8 T^{6} + 11 T^{5} + \cdots + 1$$
$19$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$23$ $$(T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1)^{2}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$(T^{4} - T^{3} - 4 T^{2} + 4 T + 1)^{2}$$
$47$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$T^{8} - 2 T^{7} + 3 T^{6} + T^{5} + T^{3} + \cdots + 1$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$79$ $$T^{8}$$
$83$ $$(T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1)^{2}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$