# Properties

 Label 1305.1.o.a Level $1305$ Weight $1$ Character orbit 1305.o Analytic conductor $0.651$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ RM discriminant 29 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,1,Mod(28,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("1305.28");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1305.o (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: $$0.651279841486$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.3625.1 Artin image: $C_4^2:C_2^2$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - i q^{4} + i q^{5} + (i - 1) q^{7} +O(q^{10})$$ q - z * q^4 + z * q^5 + (z - 1) * q^7 $$q - i q^{4} + i q^{5} + (i - 1) q^{7} + (i + 1) q^{13} - q^{16} + q^{20} + (i + 1) q^{23} - q^{25} + (i + 1) q^{28} + i q^{29} + ( - i - 1) q^{35} - i q^{49} + ( - i + 1) q^{52} + (i + 1) q^{53} - 2 i q^{59} + i q^{64} + (i - 1) q^{65} + ( - i + 1) q^{67} - i q^{80} + ( - i - 1) q^{83} - 2 q^{91} + ( - i + 1) q^{92} +O(q^{100})$$ q - z * q^4 + z * q^5 + (z - 1) * q^7 + (z + 1) * q^13 - q^16 + q^20 + (z + 1) * q^23 - q^25 + (z + 1) * q^28 + z * q^29 + (-z - 1) * q^35 - z * q^49 + (-z + 1) * q^52 + (z + 1) * q^53 - 2*z * q^59 + z * q^64 + (z - 1) * q^65 + (-z + 1) * q^67 - z * q^80 + (-z - 1) * q^83 - 2 * q^91 + (-z + 1) * q^92 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^7 $$2 q - 2 q^{7} + 2 q^{13} - 2 q^{16} + 2 q^{20} + 2 q^{23} - 2 q^{25} + 2 q^{28} - 2 q^{35} + 2 q^{52} + 2 q^{53} - 2 q^{65} + 2 q^{67} - 2 q^{83} - 4 q^{91} + 2 q^{92}+O(q^{100})$$ 2 * q - 2 * q^7 + 2 * q^13 - 2 * q^16 + 2 * q^20 + 2 * q^23 - 2 * q^25 + 2 * q^28 - 2 * q^35 + 2 * q^52 + 2 * q^53 - 2 * q^65 + 2 * q^67 - 2 * q^83 - 4 * q^91 + 2 * q^92

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$$.

 $$n$$ $$146$$ $$262$$ $$901$$ $$\chi(n)$$ $$1$$ $$-i$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1
 1.00000i − 1.00000i
0 0 1.00000i 1.00000i 0 −1.00000 + 1.00000i 0 0 0
1072.1 0 0 1.00000i 1.00000i 0 −1.00000 1.00000i 0 0 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
29.b even 2 1 RM by $$\Q(\sqrt{29})$$
5.c odd 4 1 inner
145.h odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1305.1.o.a 2
3.b odd 2 1 145.1.h.a 2
5.c odd 4 1 inner 1305.1.o.a 2
12.b even 2 1 2320.1.bm.a 2
15.d odd 2 1 725.1.h.a 2
15.e even 4 1 145.1.h.a 2
15.e even 4 1 725.1.h.a 2
29.b even 2 1 RM 1305.1.o.a 2
60.l odd 4 1 2320.1.bm.a 2
87.d odd 2 1 145.1.h.a 2
145.h odd 4 1 inner 1305.1.o.a 2
348.b even 2 1 2320.1.bm.a 2
435.b odd 2 1 725.1.h.a 2
435.p even 4 1 145.1.h.a 2
435.p even 4 1 725.1.h.a 2
1740.v odd 4 1 2320.1.bm.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.1.h.a 2 3.b odd 2 1
145.1.h.a 2 15.e even 4 1
145.1.h.a 2 87.d odd 2 1
145.1.h.a 2 435.p even 4 1
725.1.h.a 2 15.d odd 2 1
725.1.h.a 2 15.e even 4 1
725.1.h.a 2 435.b odd 2 1
725.1.h.a 2 435.p even 4 1
1305.1.o.a 2 1.a even 1 1 trivial
1305.1.o.a 2 5.c odd 4 1 inner
1305.1.o.a 2 29.b even 2 1 RM
1305.1.o.a 2 145.h odd 4 1 inner
2320.1.bm.a 2 12.b even 2 1
2320.1.bm.a 2 60.l odd 4 1
2320.1.bm.a 2 348.b even 2 1
2320.1.bm.a 2 1740.v odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1305, [\chi])$$.

## Hecke characteristic polynomials

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