# Properties

 Label 675.1.c.c Level $675$ Weight $1$ Character orbit 675.c Analytic conductor $0.337$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -15 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,1,Mod(26,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("675.26");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 675.c (of order $$2$$, degree $$1$$, 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: $$0.336868883527$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 135) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.135.1 Artin image: $C_4\times S_3$ Artin field: Galois closure of 12.0.1037970703125.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - i q^{2} - i q^{8} +O(q^{10})$$ q - z * q^2 - z * q^8 $$q - i q^{2} - i q^{8} - q^{16} - i q^{17} + q^{19} + i q^{23} - q^{31} - q^{34} - i q^{38} + q^{46} + 2 i q^{47} - q^{49} + i q^{53} - q^{61} + i q^{62} - q^{64} + q^{79} + i q^{83} + 2 q^{94} + i q^{98} +O(q^{100})$$ q - z * q^2 - z * q^8 - q^16 - z * q^17 + q^19 + z * q^23 - q^31 - q^34 - z * q^38 + q^46 + 2*z * q^47 - q^49 + z * q^53 - q^61 + z * q^62 - q^64 + q^79 + z * q^83 + 2 * q^94 + z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 2 q^{16} + 2 q^{19} - 2 q^{31} - 2 q^{34} + 2 q^{46} - 2 q^{49} - 2 q^{61} - 2 q^{64} + 2 q^{79} + 4 q^{94}+O(q^{100})$$ 2 * q - 2 * q^16 + 2 * q^19 - 2 * q^31 - 2 * q^34 + 2 * q^46 - 2 * q^49 - 2 * q^61 - 2 * q^64 + 2 * q^79 + 4 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 1.00000i − 1.00000i
1.00000i 0 0 0 0 0 1.00000i 0 0
26.2 1.00000i 0 0 0 0 0 1.00000i 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
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 675.1.c.c 2
3.b odd 2 1 inner 675.1.c.c 2
5.b even 2 1 inner 675.1.c.c 2
5.c odd 4 1 135.1.d.a 1
5.c odd 4 1 135.1.d.b yes 1
9.c even 3 2 2025.1.j.c 4
9.d odd 6 2 2025.1.j.c 4
15.d odd 2 1 CM 675.1.c.c 2
15.e even 4 1 135.1.d.a 1
15.e even 4 1 135.1.d.b yes 1
20.e even 4 1 2160.1.c.a 1
20.e even 4 1 2160.1.c.b 1
45.h odd 6 2 2025.1.j.c 4
45.j even 6 2 2025.1.j.c 4
45.k odd 12 2 405.1.h.a 2
45.k odd 12 2 405.1.h.b 2
45.l even 12 2 405.1.h.a 2
45.l even 12 2 405.1.h.b 2
60.l odd 4 1 2160.1.c.a 1
60.l odd 4 1 2160.1.c.b 1
135.q even 36 6 3645.1.n.d 6
135.q even 36 6 3645.1.n.e 6
135.r odd 36 6 3645.1.n.d 6
135.r odd 36 6 3645.1.n.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.1.d.a 1 5.c odd 4 1
135.1.d.a 1 15.e even 4 1
135.1.d.b yes 1 5.c odd 4 1
135.1.d.b yes 1 15.e even 4 1
405.1.h.a 2 45.k odd 12 2
405.1.h.a 2 45.l even 12 2
405.1.h.b 2 45.k odd 12 2
405.1.h.b 2 45.l even 12 2
675.1.c.c 2 1.a even 1 1 trivial
675.1.c.c 2 3.b odd 2 1 inner
675.1.c.c 2 5.b even 2 1 inner
675.1.c.c 2 15.d odd 2 1 CM
2025.1.j.c 4 9.c even 3 2
2025.1.j.c 4 9.d odd 6 2
2025.1.j.c 4 45.h odd 6 2
2025.1.j.c 4 45.j even 6 2
2160.1.c.a 1 20.e even 4 1
2160.1.c.a 1 60.l odd 4 1
2160.1.c.b 1 20.e even 4 1
2160.1.c.b 1 60.l odd 4 1
3645.1.n.d 6 135.q even 36 6
3645.1.n.d 6 135.r odd 36 6
3645.1.n.e 6 135.q even 36 6
3645.1.n.e 6 135.r odd 36 6

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(675, [\chi])$$:

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{7}$$ T7

## Hecke characteristic polynomials

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