Properties

Label 1520.1.bh.a
Level $1520$
Weight $1$
Character orbit 1520.bh
Analytic conductor $0.759$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,1,Mod(189,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.189");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1520.bh (of order \(4\), degree \(2\), 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: \(0.758578819202\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.972800.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{8}^{3} q^{2} - \zeta_{8} q^{3} - \zeta_{8}^{2} q^{4} - \zeta_{8}^{2} q^{5} - q^{6} - q^{7} - \zeta_{8} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8}^{3} q^{2} - \zeta_{8} q^{3} - \zeta_{8}^{2} q^{4} - \zeta_{8}^{2} q^{5} - q^{6} - q^{7} - \zeta_{8} q^{8} - \zeta_{8} q^{10} + \zeta_{8}^{3} q^{12} + \zeta_{8} q^{13} + \zeta_{8}^{3} q^{14} + \zeta_{8}^{3} q^{15} - q^{16} - \zeta_{8}^{2} q^{17} + \zeta_{8}^{3} q^{19} - q^{20} + \zeta_{8} q^{21} + q^{23} + \zeta_{8}^{2} q^{24} - q^{25} + q^{26} + \zeta_{8}^{3} q^{27} + \zeta_{8}^{2} q^{28} - \zeta_{8}^{3} q^{29} + \zeta_{8}^{2} q^{30} + \zeta_{8}^{3} q^{32} - \zeta_{8} q^{34} + \zeta_{8}^{2} q^{35} + \zeta_{8}^{2} q^{38} - \zeta_{8}^{2} q^{39} + \zeta_{8}^{3} q^{40} + q^{42} + (\zeta_{8}^{2} + 1) q^{43} - \zeta_{8}^{3} q^{46} - \zeta_{8}^{2} q^{47} + \zeta_{8} q^{48} + \zeta_{8}^{3} q^{50} + \zeta_{8}^{3} q^{51} - \zeta_{8}^{3} q^{52} + \zeta_{8}^{3} q^{53} + \zeta_{8}^{2} q^{54} + \zeta_{8} q^{56} + q^{57} - \zeta_{8}^{2} q^{58} + \zeta_{8} q^{59} + \zeta_{8} q^{60} + ( - \zeta_{8}^{2} - 1) q^{61} + \zeta_{8}^{2} q^{64} - \zeta_{8}^{3} q^{65} - \zeta_{8} q^{67} - q^{68} - \zeta_{8} q^{69} + \zeta_{8} q^{70} + q^{73} + \zeta_{8} q^{75} + \zeta_{8} q^{76} - \zeta_{8} q^{78} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{79} + \zeta_{8}^{2} q^{80} + q^{81} - \zeta_{8}^{3} q^{84} - q^{85} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{86} - q^{87} + (\zeta_{8}^{3} - \zeta_{8}) q^{89} - \zeta_{8} q^{91} - \zeta_{8}^{2} q^{92} - 2 \zeta_{8} q^{94} + \zeta_{8} q^{95} + q^{96} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{6} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{6} - 4 q^{7} - 4 q^{16} - 4 q^{20} + 4 q^{23} - 4 q^{25} + 4 q^{26} + 4 q^{42} + 4 q^{43} + 4 q^{57} - 4 q^{61} - 4 q^{68} + 4 q^{73} + 4 q^{81} - 4 q^{85} - 4 q^{87} + 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{8}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
189.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 1.00000i −1.00000 −1.00000 0.707107 + 0.707107i 0 0.707107 + 0.707107i
189.2 0.707107 0.707107i −0.707107 0.707107i 1.00000i 1.00000i −1.00000 −1.00000 −0.707107 0.707107i 0 −0.707107 0.707107i
949.1 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 1.00000i −1.00000 −1.00000 0.707107 0.707107i 0 0.707107 0.707107i
949.2 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 1.00000i −1.00000 −1.00000 −0.707107 + 0.707107i 0 −0.707107 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner
80.q even 4 1 inner
1520.bh odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.1.bh.a 4
5.b even 2 1 1520.1.bh.b yes 4
16.e even 4 1 1520.1.bh.b yes 4
19.b odd 2 1 inner 1520.1.bh.a 4
80.q even 4 1 inner 1520.1.bh.a 4
95.d odd 2 1 1520.1.bh.b yes 4
304.j odd 4 1 1520.1.bh.b yes 4
1520.bh odd 4 1 inner 1520.1.bh.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1520.1.bh.a 4 1.a even 1 1 trivial
1520.1.bh.a 4 19.b odd 2 1 inner
1520.1.bh.a 4 80.q even 4 1 inner
1520.1.bh.a 4 1520.bh odd 4 1 inner
1520.1.bh.b yes 4 5.b even 2 1
1520.1.bh.b yes 4 16.e even 4 1
1520.1.bh.b yes 4 95.d odd 2 1
1520.1.bh.b yes 4 304.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1520, [\chi])\):

\( T_{3}^{4} + 1 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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