↑

Properties

Label 3520.1.cg.a
Level $3520$
Weight $1$
Character orbit 3520.cg
Analytic conductor $1.757$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -40
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3520,1,Mod(1249,3520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3520, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 5, 9]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3520.1249");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3520.cg (of order \(10\), degree \(4\), 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: \(1.75670884447\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.241453843558400000.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{20}^{8} q^{5} + (\zeta_{20}^{5} + \zeta_{20}^{3}) q^{7} + \zeta_{20}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{20}^{8} q^{5} + (\zeta_{20}^{5} + \zeta_{20}^{3}) q^{7} + \zeta_{20}^{2} q^{9} - \zeta_{20}^{9} q^{11} + (\zeta_{20}^{4} - 1) q^{13} + (\zeta_{20}^{9} + \zeta_{20}^{3}) q^{19} + ( - \zeta_{20}^{9} - \zeta_{20}) q^{23} - \zeta_{20}^{6} q^{25} + ( - \zeta_{20}^{3} - \zeta_{20}) q^{35} + ( - \zeta_{20}^{6} - \zeta_{20}^{2}) q^{37} + ( - \zeta_{20}^{2} - 1) q^{41} - q^{45} + ( - \zeta_{20}^{9} + \zeta_{20}^{3}) q^{47} + (\zeta_{20}^{8} + \zeta_{20}^{6} - 1) q^{49} + (\zeta_{20}^{8} - \zeta_{20}^{6}) q^{53} + \zeta_{20}^{7} q^{55} + ( - \zeta_{20}^{7} + \zeta_{20}) q^{59} + (\zeta_{20}^{7} + \zeta_{20}^{5}) q^{63} + ( - \zeta_{20}^{8} - \zeta_{20}^{2}) q^{65} + (\zeta_{20}^{4} + \zeta_{20}^{2}) q^{77} + \zeta_{20}^{4} q^{81} + (\zeta_{20}^{6} - \zeta_{20}^{4}) q^{89} + (\zeta_{20}^{9} + \zeta_{20}^{7} - \zeta_{20}^{5} - \zeta_{20}^{3}) q^{91} + ( - \zeta_{20}^{7} - \zeta_{20}) q^{95} + \zeta_{20} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{5} + 2 q^{9} - 10 q^{13} - 2 q^{25} - 4 q^{37} - 10 q^{41} - 8 q^{45} - 8 q^{49} - 4 q^{53} - 2 q^{81} + 4 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(1541\) \(2751\) \(2817\)
\(\chi(n)\) \(-\zeta_{20}^{4}\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
0.587785 − 0.809017i
−0.587785 + 0.809017i
−0.951057 − 0.309017i
0.951057 + 0.309017i
−0.951057 + 0.309017i
0.951057 − 0.309017i
0.587785 + 0.809017i
−0.587785 − 0.809017i
0 0 0 0.309017 − 0.951057i 0 −0.951057 + 0.690983i 0 −0.309017 − 0.951057i 0
1249.2 0 0 0 0.309017 − 0.951057i 0 0.951057 − 0.690983i 0 −0.309017 − 0.951057i 0
1569.1 0 0 0 −0.809017 + 0.587785i 0 −0.587785 − 1.80902i 0 0.809017 + 0.587785i 0
1569.2 0 0 0 −0.809017 + 0.587785i 0 0.587785 + 1.80902i 0 0.809017 + 0.587785i 0
1889.1 0 0 0 −0.809017 − 0.587785i 0 −0.587785 + 1.80902i 0 0.809017 − 0.587785i 0
1889.2 0 0 0 −0.809017 − 0.587785i 0 0.587785 − 1.80902i 0 0.809017 − 0.587785i 0
3489.1 0 0 0 0.309017 + 0.951057i 0 −0.951057 − 0.690983i 0 −0.309017 + 0.951057i 0
3489.2 0 0 0 0.309017 + 0.951057i 0 0.951057 + 0.690983i 0 −0.309017 + 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
4.b odd 2 1 inner
11.d odd 10 1 inner
40.f even 2 1 inner
44.g even 10 1 inner
440.ba odd 10 1 inner
440.bm even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3520.1.cg.a ✓ 8
4.b odd 2 1 inner 3520.1.cg.a ✓ 8
5.b even 2 1 3520.1.cg.b yes 8
8.b even 2 1 3520.1.cg.b yes 8
8.d odd 2 1 3520.1.cg.b yes 8
11.d odd 10 1 inner 3520.1.cg.a ✓ 8
20.d odd 2 1 3520.1.cg.b yes 8
40.e odd 2 1 CM 3520.1.cg.a ✓ 8
40.f even 2 1 inner 3520.1.cg.a ✓ 8
44.g even 10 1 inner 3520.1.cg.a ✓ 8
55.h odd 10 1 3520.1.cg.b yes 8
88.k even 10 1 3520.1.cg.b yes 8
88.p odd 10 1 3520.1.cg.b yes 8
220.o even 10 1 3520.1.cg.b yes 8
440.ba odd 10 1 inner 3520.1.cg.a ✓ 8
440.bm even 10 1 inner 3520.1.cg.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3520.1.cg.a ✓ 8 1.a even 1 1 trivial
3520.1.cg.a ✓ 8 4.b odd 2 1 inner
3520.1.cg.a ✓ 8 11.d odd 10 1 inner
3520.1.cg.a ✓ 8 40.e odd 2 1 CM
3520.1.cg.a ✓ 8 40.f even 2 1 inner
3520.1.cg.a ✓ 8 44.g even 10 1 inner
3520.1.cg.a ✓ 8 440.ba odd 10 1 inner
3520.1.cg.a ✓ 8 440.bm even 10 1 inner
3520.1.cg.b yes 8 5.b even 2 1
3520.1.cg.b yes 8 8.b even 2 1
3520.1.cg.b yes 8 8.d odd 2 1
3520.1.cg.b yes 8 20.d odd 2 1
3520.1.cg.b yes 8 55.h odd 10 1
3520.1.cg.b yes 8 88.k even 10 1
3520.1.cg.b yes 8 88.p odd 10 1
3520.1.cg.b yes 8 220.o even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{4} + 5T_{13}^{3} + 10T_{13}^{2} + 10T_{13} + 5 \) acting on \(S_{1}^{\mathrm{new}}(3520, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 5 T^{6} + 10 T^{4} + 25 \) Copy content Toggle raw display
$11$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T^{4} + 5 T^{3} + 10 T^{2} + 10 T + 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 10 T^{4} + 25 T^{2} + 25 \) Copy content Toggle raw display
$23$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 5 T^{3} + 10 T^{2} + 10 T + 5)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \) Copy content Toggle raw display
$53$ \( (T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \) Copy content Toggle raw display
$61$ \( T^{8} \) 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
$89$ \( (T^{2} - T - 1)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
show more
show less