# Properties

 Label 3675.1.u.f Level $3675$ Weight $1$ Character orbit 3675.u Analytic conductor $1.834$ Analytic rank $0$ Dimension $8$ Projective image $D_{4}$ CM discriminant -15 Inner twists $16$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,1,Mod(851,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 4]))

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

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

chi := DirichletCharacter("3675.851");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3675 = 3 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3675.u (of order $$6$$, degree $$2$$, 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: $$1.83406392143$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 735) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.15435.1 Artin image: $C_{12}\times D_8$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{96} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{24}^{7} - \zeta_{24}) q^{2} - \zeta_{24}^{2} q^{3} - \zeta_{24}^{8} q^{4} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{6} - \zeta_{24}^{3} q^{8} + \zeta_{24}^{4} q^{9} +O(q^{10})$$ q + (z^7 - z) * q^2 - z^2 * q^3 - z^8 * q^4 + (-z^9 + z^3) * q^6 - z^3 * q^8 + z^4 * q^9 $$q + (\zeta_{24}^{7} - \zeta_{24}) q^{2} - \zeta_{24}^{2} q^{3} - \zeta_{24}^{8} q^{4} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{6} - \zeta_{24}^{3} q^{8} + \zeta_{24}^{4} q^{9} + \zeta_{24}^{10} q^{12} + \zeta_{24}^{4} q^{16} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{18} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{19} + (\zeta_{24}^{7} - \zeta_{24}) q^{23} - \zeta_{24}^{6} q^{27} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{31} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{32} + q^{36} + \zeta_{24}^{2} q^{38} + ( - 2 \zeta_{24}^{8} - \zeta_{24}^{2}) q^{46} - \zeta_{24}^{6} q^{48} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{53} + (\zeta_{24}^{7} + \zeta_{24}) q^{54} + (\zeta_{24}^{9} + \zeta_{24}^{3}) q^{57} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{61} - \zeta_{24}^{6} q^{62} - q^{64} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{69} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{76} + \zeta_{24}^{8} q^{81} + (\zeta_{24}^{9} + \zeta_{24}^{3}) q^{92} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{93} + (\zeta_{24}^{7} + \zeta_{24}) q^{96} +O(q^{100})$$ q + (z^7 - z) * q^2 - z^2 * q^3 - z^8 * q^4 + (-z^9 + z^3) * q^6 - z^3 * q^8 + z^4 * q^9 + z^10 * q^12 + z^4 * q^16 + (z^11 - z^5) * q^18 + (-z^7 - z) * q^19 + (z^7 - z) * q^23 - z^6 * q^27 + (z^11 + z^5) * q^31 + (z^11 - z^5) * q^32 + q^36 + z^2 * q^38 + (-2*z^8 - z^2) * q^46 - z^6 * q^48 + (-z^11 + z^5) * q^53 + (z^7 + z) * q^54 + (z^9 + z^3) * q^57 + (-z^7 - z) * q^61 - z^6 * q^62 - q^64 + (-z^9 + z^3) * q^69 + (z^9 - z^3) * q^76 + z^8 * q^81 + (z^9 + z^3) * q^92 + (-z^7 + z) * q^93 + (z^7 + z) * q^96 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4} + 4 q^{9}+O(q^{10})$$ 8 * q + 4 * q^4 + 4 * q^9 $$8 q + 4 q^{4} + 4 q^{9} + 4 q^{16} + 8 q^{36} + 8 q^{46} + 8 q^{64} - 4 q^{81}+O(q^{100})$$ 8 * q + 4 * q^4 + 4 * q^9 + 4 * q^16 + 8 * q^36 + 8 * q^46 + 8 * q^64 - 4 * q^81

## Character values

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

 $$n$$ $$1177$$ $$1226$$ $$2551$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{24}^{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
851.1
 0.965926 + 0.258819i 0.258819 − 0.965926i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.965926 + 0.258819i −0.258819 − 0.965926i
−1.22474 + 0.707107i −0.866025 0.500000i 0.500000 0.866025i 0 1.41421 0 0 0.500000 + 0.866025i 0
851.2 −1.22474 + 0.707107i 0.866025 + 0.500000i 0.500000 0.866025i 0 −1.41421 0 0 0.500000 + 0.866025i 0
851.3 1.22474 0.707107i −0.866025 0.500000i 0.500000 0.866025i 0 −1.41421 0 0 0.500000 + 0.866025i 0
851.4 1.22474 0.707107i 0.866025 + 0.500000i 0.500000 0.866025i 0 1.41421 0 0 0.500000 + 0.866025i 0
1451.1 −1.22474 0.707107i −0.866025 + 0.500000i 0.500000 + 0.866025i 0 1.41421 0 0 0.500000 0.866025i 0
1451.2 −1.22474 0.707107i 0.866025 0.500000i 0.500000 + 0.866025i 0 −1.41421 0 0 0.500000 0.866025i 0
1451.3 1.22474 + 0.707107i −0.866025 + 0.500000i 0.500000 + 0.866025i 0 −1.41421 0 0 0.500000 0.866025i 0
1451.4 1.22474 + 0.707107i 0.866025 0.500000i 0.500000 + 0.866025i 0 1.41421 0 0 0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 851.4 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
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
35.j even 6 1 inner
105.g even 2 1 inner
105.o odd 6 1 inner
105.p even 6 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{4} - 2T_{2}^{2} + 4$$ T2^4 - 2*T2^2 + 4 $$T_{13}$$ T13 $$T_{37}$$ T37

## Hecke characteristic polynomials

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