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 $16$

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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: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
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.

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

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:

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 \(S_{1}^{\mathrm{new}}(3675, [\chi])\):

\( T_{2}^{4} - 2T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{37} \) Copy content Toggle raw display

Hecke characteristic polynomials

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