↑

Properties

Label 3675.1.p.b
Level $3675$
Weight $1$
Character orbit 3675.p
Analytic conductor $1.834$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3675,1,Mod(2174,3675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3675, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 4])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3675.2174"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3675.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content 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: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content 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: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.77175.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{24}^{4} q^{2} - \zeta_{24}^{5} q^{3} + \zeta_{24}^{9} q^{6} - q^{8} + \zeta_{24}^{10} q^{9} + \zeta_{24}^{2} q^{11} + (\zeta_{24}^{9} + \zeta_{24}^{3}) q^{13} + \zeta_{24}^{4} q^{16} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{17} + \cdots - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 8 q^{8} + 4 q^{16} - 4 q^{23} + 4 q^{39} - 4 q^{46} + 4 q^{51} + 8 q^{64} - 8 q^{78} - 4 q^{79} + 4 q^{81} - 8 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2174.1
0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
−0.258819 + 0.965926i
0.258819 + 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
−0.258819 − 0.965926i
−0.500000 − 0.866025i −0.965926 + 0.258819i 0 0 0.707107 + 0.707107i 0 −1.00000 0.866025 − 0.500000i 0
2174.2 −0.500000 − 0.866025i −0.258819 − 0.965926i 0 0 −0.707107 + 0.707107i 0 −1.00000 −0.866025 + 0.500000i 0
2174.3 −0.500000 − 0.866025i 0.258819 + 0.965926i 0 0 0.707107 − 0.707107i 0 −1.00000 −0.866025 + 0.500000i 0
2174.4 −0.500000 − 0.866025i 0.965926 − 0.258819i 0 0 −0.707107 − 0.707107i 0 −1.00000 0.866025 − 0.500000i 0
2774.1 −0.500000 + 0.866025i −0.965926 − 0.258819i 0 0 0.707107 − 0.707107i 0 −1.00000 0.866025 + 0.500000i 0
2774.2 −0.500000 + 0.866025i −0.258819 + 0.965926i 0 0 −0.707107 − 0.707107i 0 −1.00000 −0.866025 − 0.500000i 0
2774.3 −0.500000 + 0.866025i 0.258819 − 0.965926i 0 0 0.707107 + 0.707107i 0 −1.00000 −0.866025 − 0.500000i 0
2774.4 −0.500000 + 0.866025i 0.965926 + 0.258819i 0 0 −0.707107 + 0.707107i 0 −1.00000 0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2174.4
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3675, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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