# Properties

 Label 1815.1.v.a Level $1815$ Weight $1$ Character orbit 1815.v Analytic conductor $0.906$ Analytic rank $0$ Dimension $8$ Projective image $D_{4}$ CM discriminant -11 Inner twists $16$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,1,Mod(233,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([10, 15, 2]))

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

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

chi := DirichletCharacter("1815.233");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1815.v (of order $$20$$, degree $$8$$, 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: $$0.905802997929$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{20})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + 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 165) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.12375.1 Artin image: $C_4\wr C_2\times C_{10}$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{80} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{20}^{7} q^{3} + \zeta_{20}^{3} q^{4} + \zeta_{20} q^{5} - \zeta_{20}^{4} q^{9} +O(q^{10})$$ q - z^7 * q^3 + z^3 * q^4 + z * q^5 - z^4 * q^9 $$q - \zeta_{20}^{7} q^{3} + \zeta_{20}^{3} q^{4} + \zeta_{20} q^{5} - \zeta_{20}^{4} q^{9} + q^{12} - \zeta_{20}^{8} q^{15} + \zeta_{20}^{6} q^{16} + \zeta_{20}^{4} q^{20} + ( - \zeta_{20}^{5} - 1) q^{23} + \zeta_{20}^{2} q^{25} - \zeta_{20} q^{27} - \zeta_{20}^{7} q^{36} + (\zeta_{20}^{8} - \zeta_{20}^{3}) q^{37} - \zeta_{20}^{5} q^{45} + ( - \zeta_{20}^{7} + \zeta_{20}^{2}) q^{47} + \zeta_{20}^{3} q^{48} + \zeta_{20} q^{49} + (\zeta_{20}^{9} + \zeta_{20}^{4}) q^{53} + \zeta_{20}^{8} q^{59} + \zeta_{20} q^{60} + \zeta_{20}^{9} q^{64} + ( - \zeta_{20}^{5} - 1) q^{67} + (\zeta_{20}^{7} - \zeta_{20}^{2}) q^{69} - \zeta_{20}^{9} q^{75} + \zeta_{20}^{7} q^{80} + \zeta_{20}^{8} q^{81} + ( - \zeta_{20}^{8} - \zeta_{20}^{3}) q^{92} + ( - \zeta_{20}^{9} - \zeta_{20}^{4}) q^{97} +O(q^{100})$$ q - z^7 * q^3 + z^3 * q^4 + z * q^5 - z^4 * q^9 + q^12 - z^8 * q^15 + z^6 * q^16 + z^4 * q^20 + (-z^5 - 1) * q^23 + z^2 * q^25 - z * q^27 - z^7 * q^36 + (z^8 - z^3) * q^37 - z^5 * q^45 + (-z^7 + z^2) * q^47 + z^3 * q^48 + z * q^49 + (z^9 + z^4) * q^53 + z^8 * q^59 + z * q^60 + z^9 * q^64 + (-z^5 - 1) * q^67 + (z^7 - z^2) * q^69 - z^9 * q^75 + z^7 * q^80 + z^8 * q^81 + (-z^8 - z^3) * q^92 + (-z^9 - z^4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{9}+O(q^{10})$$ 8 * q + 2 * q^9 $$8 q + 2 q^{9} + 8 q^{12} + 2 q^{15} + 2 q^{16} - 2 q^{20} - 8 q^{23} + 2 q^{25} - 2 q^{37} + 2 q^{47} - 2 q^{53} - 4 q^{59} - 8 q^{67} - 2 q^{69} - 2 q^{81} + 2 q^{92} + 2 q^{97}+O(q^{100})$$ 8 * q + 2 * q^9 + 8 * q^12 + 2 * q^15 + 2 * q^16 - 2 * q^20 - 8 * q^23 + 2 * q^25 - 2 * q^37 + 2 * q^47 - 2 * q^53 - 4 * q^59 - 8 * q^67 - 2 * q^69 - 2 * q^81 + 2 * q^92 + 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times$$.

 $$n$$ $$727$$ $$1211$$ $$1696$$ $$\chi(n)$$ $$\zeta_{20}^{5}$$ $$-1$$ $$-\zeta_{20}^{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
233.1
 0.951057 − 0.309017i −0.951057 − 0.309017i 0.587785 − 0.809017i −0.587785 − 0.809017i −0.951057 + 0.309017i −0.587785 + 0.809017i 0.587785 + 0.809017i 0.951057 + 0.309017i
0 0.587785 + 0.809017i 0.587785 0.809017i 0.951057 0.309017i 0 0 0 −0.309017 + 0.951057i 0
578.1 0 −0.587785 + 0.809017i −0.587785 0.809017i −0.951057 0.309017i 0 0 0 −0.309017 0.951057i 0
602.1 0 −0.951057 + 0.309017i −0.951057 0.309017i 0.587785 0.809017i 0 0 0 0.809017 0.587785i 0
887.1 0 0.951057 + 0.309017i 0.951057 0.309017i −0.587785 0.809017i 0 0 0 0.809017 + 0.587785i 0
1322.1 0 −0.587785 0.809017i −0.587785 + 0.809017i −0.951057 + 0.309017i 0 0 0 −0.309017 + 0.951057i 0
1328.1 0 0.951057 0.309017i 0.951057 + 0.309017i −0.587785 + 0.809017i 0 0 0 0.809017 0.587785i 0
1613.1 0 −0.951057 0.309017i −0.951057 + 0.309017i 0.587785 + 0.809017i 0 0 0 0.809017 + 0.587785i 0
1667.1 0 0.587785 0.809017i 0.587785 + 0.809017i 0.951057 + 0.309017i 0 0 0 −0.309017 0.951057i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 233.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
11.c even 5 3 inner
11.d odd 10 3 inner
15.e even 4 1 inner
165.l odd 4 1 inner
165.u odd 20 3 inner
165.v even 20 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.1.v.a 8
3.b odd 2 1 1815.1.v.b 8
5.c odd 4 1 1815.1.v.b 8
11.b odd 2 1 CM 1815.1.v.a 8
11.c even 5 1 165.1.l.b yes 2
11.c even 5 3 inner 1815.1.v.a 8
11.d odd 10 1 165.1.l.b yes 2
11.d odd 10 3 inner 1815.1.v.a 8
15.e even 4 1 inner 1815.1.v.a 8
33.d even 2 1 1815.1.v.b 8
33.f even 10 1 165.1.l.a 2
33.f even 10 3 1815.1.v.b 8
33.h odd 10 1 165.1.l.a 2
33.h odd 10 3 1815.1.v.b 8
44.g even 10 1 2640.1.ch.a 2
44.h odd 10 1 2640.1.ch.a 2
55.e even 4 1 1815.1.v.b 8
55.h odd 10 1 825.1.l.a 2
55.j even 10 1 825.1.l.a 2
55.k odd 20 1 165.1.l.a 2
55.k odd 20 1 825.1.l.b 2
55.k odd 20 3 1815.1.v.b 8
55.l even 20 1 165.1.l.a 2
55.l even 20 1 825.1.l.b 2
55.l even 20 3 1815.1.v.b 8
132.n odd 10 1 2640.1.ch.b 2
132.o even 10 1 2640.1.ch.b 2
165.l odd 4 1 inner 1815.1.v.a 8
165.o odd 10 1 825.1.l.b 2
165.r even 10 1 825.1.l.b 2
165.u odd 20 1 165.1.l.b yes 2
165.u odd 20 1 825.1.l.a 2
165.u odd 20 3 inner 1815.1.v.a 8
165.v even 20 1 165.1.l.b yes 2
165.v even 20 1 825.1.l.a 2
165.v even 20 3 inner 1815.1.v.a 8
220.v even 20 1 2640.1.ch.b 2
220.w odd 20 1 2640.1.ch.b 2
660.bp odd 20 1 2640.1.ch.a 2
660.bv even 20 1 2640.1.ch.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.1.l.a 2 33.f even 10 1
165.1.l.a 2 33.h odd 10 1
165.1.l.a 2 55.k odd 20 1
165.1.l.a 2 55.l even 20 1
165.1.l.b yes 2 11.c even 5 1
165.1.l.b yes 2 11.d odd 10 1
165.1.l.b yes 2 165.u odd 20 1
165.1.l.b yes 2 165.v even 20 1
825.1.l.a 2 55.h odd 10 1
825.1.l.a 2 55.j even 10 1
825.1.l.a 2 165.u odd 20 1
825.1.l.a 2 165.v even 20 1
825.1.l.b 2 55.k odd 20 1
825.1.l.b 2 55.l even 20 1
825.1.l.b 2 165.o odd 10 1
825.1.l.b 2 165.r even 10 1
1815.1.v.a 8 1.a even 1 1 trivial
1815.1.v.a 8 11.b odd 2 1 CM
1815.1.v.a 8 11.c even 5 3 inner
1815.1.v.a 8 11.d odd 10 3 inner
1815.1.v.a 8 15.e even 4 1 inner
1815.1.v.a 8 165.l odd 4 1 inner
1815.1.v.a 8 165.u odd 20 3 inner
1815.1.v.a 8 165.v even 20 3 inner
1815.1.v.b 8 3.b odd 2 1
1815.1.v.b 8 5.c odd 4 1
1815.1.v.b 8 33.d even 2 1
1815.1.v.b 8 33.f even 10 3
1815.1.v.b 8 33.h odd 10 3
1815.1.v.b 8 55.e even 4 1
1815.1.v.b 8 55.k odd 20 3
1815.1.v.b 8 55.l even 20 3
2640.1.ch.a 2 44.g even 10 1
2640.1.ch.a 2 44.h odd 10 1
2640.1.ch.a 2 660.bp odd 20 1
2640.1.ch.a 2 660.bv even 20 1
2640.1.ch.b 2 132.n odd 10 1
2640.1.ch.b 2 132.o even 10 1
2640.1.ch.b 2 220.v even 20 1
2640.1.ch.b 2 220.w odd 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{23}^{2} + 2T_{23} + 2$$ acting on $$S_{1}^{\mathrm{new}}(1815, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$5$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$(T^{2} + 2 T + 2)^{4}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8} + 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 16$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8} - 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 16$$
$53$ $$T^{8} + 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 16$$
$59$ $$(T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16)^{2}$$
$61$ $$T^{8}$$
$67$ $$(T^{2} + 2 T + 2)^{4}$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8} - 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 16$$