# Properties

 Label 3675.1.c.f Level $3675$ Weight $1$ Character orbit 3675.c Analytic conductor $1.834$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -15 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,1,Mod(1226,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3675.1226");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3675 = 3 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3675.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: $$1.83406392143$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 735) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.15435.1 Artin image: $C_4\times D_8$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{32} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{8}^{3} - \zeta_{8}) q^{2} + \zeta_{8}^{2} q^{3} - q^{4} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{6} - q^{9}+O(q^{10})$$ q + (-z^3 - z) * q^2 + z^2 * q^3 - q^4 + (-z^3 + z) * q^6 - q^9 $$q + ( - \zeta_{8}^{3} - \zeta_{8}) q^{2} + \zeta_{8}^{2} q^{3} - q^{4} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{6} - q^{9} - \zeta_{8}^{2} q^{12} - q^{16} + (\zeta_{8}^{3} + \zeta_{8}) q^{18} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{19} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{23} - \zeta_{8}^{2} q^{27} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{31} + (\zeta_{8}^{3} + \zeta_{8}) q^{32} + q^{36} - \zeta_{8}^{2} q^{38} + ( - \zeta_{8}^{2} - 2) q^{46} - \zeta_{8}^{2} q^{48} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{53} + (\zeta_{8}^{3} - \zeta_{8}) q^{54} + (\zeta_{8}^{3} + \zeta_{8}) q^{57} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{61} - \zeta_{8}^{2} q^{62} + q^{64} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{69} + (\zeta_{8}^{3} - \zeta_{8}) q^{76} + q^{81} + (\zeta_{8}^{3} + \zeta_{8}) q^{92} + (\zeta_{8}^{3} + \zeta_{8}) q^{93} + (\zeta_{8}^{3} - \zeta_{8}) q^{96} +O(q^{100})$$ q + (-z^3 - z) * q^2 + z^2 * q^3 - q^4 + (-z^3 + z) * q^6 - q^9 - z^2 * q^12 - q^16 + (z^3 + z) * q^18 + (-z^3 + z) * q^19 + (-z^3 - z) * q^23 - z^2 * q^27 + (-z^3 + z) * q^31 + (z^3 + z) * q^32 + q^36 - z^2 * q^38 + (-z^2 - 2) * q^46 - z^2 * q^48 + (-z^3 - z) * q^53 + (z^3 - z) * q^54 + (z^3 + z) * q^57 + (-z^3 + z) * q^61 - z^2 * q^62 + q^64 + (-z^3 + z) * q^69 + (z^3 - z) * q^76 + q^81 + (z^3 + z) * q^92 + (z^3 + z) * q^93 + (z^3 - z) * q^96 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^9 $$4 q - 4 q^{4} - 4 q^{9} - 4 q^{16} + 4 q^{36} - 8 q^{46} + 4 q^{64} + 4 q^{81}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^9 - 4 * q^16 + 4 * q^36 - 8 * q^46 + 4 * q^64 + 4 * q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3675\mathbb{Z}\right)^\times$$.

 $$n$$ $$1177$$ $$1226$$ $$2551$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1226.1
 −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i
1.41421i 1.00000i −1.00000 0 −1.41421 0 0 −1.00000 0
1226.2 1.41421i 1.00000i −1.00000 0 1.41421 0 0 −1.00000 0
1226.3 1.41421i 1.00000i −1.00000 0 1.41421 0 0 −1.00000 0
1226.4 1.41421i 1.00000i −1.00000 0 −1.41421 0 0 −1.00000 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
7.b odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3675.1.c.f 4
3.b odd 2 1 inner 3675.1.c.f 4
5.b even 2 1 inner 3675.1.c.f 4
5.c odd 4 1 735.1.f.c 2
5.c odd 4 1 735.1.f.d yes 2
7.b odd 2 1 inner 3675.1.c.f 4
7.c even 3 2 3675.1.u.f 8
7.d odd 6 2 3675.1.u.f 8
15.d odd 2 1 CM 3675.1.c.f 4
15.e even 4 1 735.1.f.c 2
15.e even 4 1 735.1.f.d yes 2
21.c even 2 1 inner 3675.1.c.f 4
21.g even 6 2 3675.1.u.f 8
21.h odd 6 2 3675.1.u.f 8
35.c odd 2 1 inner 3675.1.c.f 4
35.f even 4 1 735.1.f.c 2
35.f even 4 1 735.1.f.d yes 2
35.i odd 6 2 3675.1.u.f 8
35.j even 6 2 3675.1.u.f 8
35.k even 12 2 735.1.o.c 4
35.k even 12 2 735.1.o.d 4
35.l odd 12 2 735.1.o.c 4
35.l odd 12 2 735.1.o.d 4
105.g even 2 1 inner 3675.1.c.f 4
105.k odd 4 1 735.1.f.c 2
105.k odd 4 1 735.1.f.d yes 2
105.o odd 6 2 3675.1.u.f 8
105.p even 6 2 3675.1.u.f 8
105.w odd 12 2 735.1.o.c 4
105.w odd 12 2 735.1.o.d 4
105.x even 12 2 735.1.o.c 4
105.x even 12 2 735.1.o.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.1.f.c 2 5.c odd 4 1
735.1.f.c 2 15.e even 4 1
735.1.f.c 2 35.f even 4 1
735.1.f.c 2 105.k odd 4 1
735.1.f.d yes 2 5.c odd 4 1
735.1.f.d yes 2 15.e even 4 1
735.1.f.d yes 2 35.f even 4 1
735.1.f.d yes 2 105.k odd 4 1
735.1.o.c 4 35.k even 12 2
735.1.o.c 4 35.l odd 12 2
735.1.o.c 4 105.w odd 12 2
735.1.o.c 4 105.x even 12 2
735.1.o.d 4 35.k even 12 2
735.1.o.d 4 35.l odd 12 2
735.1.o.d 4 105.w odd 12 2
735.1.o.d 4 105.x even 12 2
3675.1.c.f 4 1.a even 1 1 trivial
3675.1.c.f 4 3.b odd 2 1 inner
3675.1.c.f 4 5.b even 2 1 inner
3675.1.c.f 4 7.b odd 2 1 inner
3675.1.c.f 4 15.d odd 2 1 CM
3675.1.c.f 4 21.c even 2 1 inner
3675.1.c.f 4 35.c odd 2 1 inner
3675.1.c.f 4 105.g even 2 1 inner
3675.1.u.f 8 7.c even 3 2
3675.1.u.f 8 7.d odd 6 2
3675.1.u.f 8 21.g even 6 2
3675.1.u.f 8 21.h odd 6 2
3675.1.u.f 8 35.i odd 6 2
3675.1.u.f 8 35.j even 6 2
3675.1.u.f 8 105.o odd 6 2
3675.1.u.f 8 105.p even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3675, [\chi])$$:

 $$T_{2}^{2} + 2$$ T2^2 + 2 $$T_{13}$$ T13 $$T_{19}^{2} - 2$$ T19^2 - 2 $$T_{37}$$ T37

## Hecke characteristic polynomials

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