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Properties

Label 1424.1.bt.a
Level $1424$
Weight $1$
Character orbit 1424.bt
Analytic conductor $0.711$
Analytic rank $0$
Dimension $20$
Projective image $D_{44}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1424,1,Mod(47,1424)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1424, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([22, 0, 27]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1424.47");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1424 = 2^{4} \cdot 89 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1424.bt (of order \(44\), degree \(20\), 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: \(0.710668577989\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{44})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{44}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{44} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{44}^{8} + \zeta_{44}^{2}) q^{5} + \zeta_{44}^{19} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{44}^{8} + \zeta_{44}^{2}) q^{5} + \zeta_{44}^{19} q^{9} + ( - \zeta_{44}^{12} + \zeta_{44}^{7}) q^{13} + (\zeta_{44}^{3} - \zeta_{44}) q^{17} + (\zeta_{44}^{16} + \cdots + \zeta_{44}^{4}) q^{25}+ \cdots + ( - \zeta_{44}^{9} + \zeta_{44}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{13} - 2 q^{25} - 2 q^{29} - 2 q^{37} + 2 q^{41} + 2 q^{61} - 4 q^{73} + 2 q^{81} + 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1424\mathbb{Z}\right)^\times\).

\(n\) \(357\) \(1247\) \(1249\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{44}^{19}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−0.281733 + 0.959493i
−0.540641 + 0.841254i
0.755750 + 0.654861i
−0.281733 − 0.959493i
0.989821 + 0.142315i
0.909632 + 0.415415i
−0.909632 + 0.415415i
0.540641 + 0.841254i
−0.540641 − 0.841254i
0.909632 − 0.415415i
−0.909632 − 0.415415i
−0.989821 − 0.142315i
0.281733 + 0.959493i
−0.755750 − 0.654861i
0.540641 − 0.841254i
0.281733 − 0.959493i
−0.989821 + 0.142315i
0.755750 − 0.654861i
−0.755750 + 0.654861i
0.989821 − 0.142315i
0 0 0 −1.49611 + 0.215109i 0 0 0 −0.755750 − 0.654861i 0
79.1 0 0 0 −0.557730 − 1.89945i 0 0 0 −0.989821 + 0.142315i 0
287.1 0 0 0 0.983568 + 0.449181i 0 0 0 0.540641 + 0.841254i 0
303.1 0 0 0 −1.49611 − 0.215109i 0 0 0 −0.755750 + 0.654861i 0
335.1 0 0 0 1.37491 + 1.19136i 0 0 0 −0.909632 + 0.415415i 0
351.1 0 0 0 −0.304632 + 0.474017i 0 0 0 −0.281733 + 0.959493i 0
463.1 0 0 0 −0.304632 − 0.474017i 0 0 0 0.281733 + 0.959493i 0
543.1 0 0 0 −0.557730 + 1.89945i 0 0 0 0.989821 + 0.142315i 0
703.1 0 0 0 −0.557730 + 1.89945i 0 0 0 −0.989821 − 0.142315i 0
783.1 0 0 0 −0.304632 − 0.474017i 0 0 0 −0.281733 − 0.959493i 0
895.1 0 0 0 −0.304632 + 0.474017i 0 0 0 0.281733 − 0.959493i 0
911.1 0 0 0 1.37491 + 1.19136i 0 0 0 0.909632 − 0.415415i 0
943.1 0 0 0 −1.49611 − 0.215109i 0 0 0 0.755750 − 0.654861i 0
959.1 0 0 0 0.983568 + 0.449181i 0 0 0 −0.540641 − 0.841254i 0
1167.1 0 0 0 −0.557730 − 1.89945i 0 0 0 0.989821 − 0.142315i 0
1199.1 0 0 0 −1.49611 + 0.215109i 0 0 0 0.755750 + 0.654861i 0
1263.1 0 0 0 1.37491 − 1.19136i 0 0 0 0.909632 + 0.415415i 0
1295.1 0 0 0 0.983568 − 0.449181i 0 0 0 0.540641 − 0.841254i 0
1375.1 0 0 0 0.983568 − 0.449181i 0 0 0 −0.540641 + 0.841254i 0
1407.1 0 0 0 1.37491 − 1.19136i 0 0 0 −0.909632 − 0.415415i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
89.g even 44 1 inner
356.n odd 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1424.1.bt.a ✓ 20
4.b odd 2 1 CM 1424.1.bt.a ✓ 20
89.g even 44 1 inner 1424.1.bt.a ✓ 20
356.n odd 44 1 inner 1424.1.bt.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1424.1.bt.a ✓ 20 1.a even 1 1 trivial
1424.1.bt.a ✓ 20 4.b odd 2 1 CM
1424.1.bt.a ✓ 20 89.g even 44 1 inner
1424.1.bt.a ✓ 20 356.n odd 44 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1424, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} + 22 T^{5} + \cdots + 11)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} - 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{20} - 4 T^{18} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{20} \) Copy content Toggle raw display
$23$ \( T^{20} \) Copy content Toggle raw display
$29$ \( T^{20} + 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{20} \) Copy content Toggle raw display
$37$ \( T^{20} + 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{20} - 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{20} \) Copy content Toggle raw display
$47$ \( T^{20} \) Copy content Toggle raw display
$53$ \( T^{20} - 4 T^{18} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{20} \) Copy content Toggle raw display
$61$ \( T^{20} - 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{20} \) Copy content Toggle raw display
$71$ \( T^{20} \) Copy content Toggle raw display
$73$ \( (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} \) Copy content Toggle raw display
$83$ \( T^{20} \) Copy content Toggle raw display
$89$ \( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + 55 T^{14} + \cdots + 121 \) Copy content Toggle raw display
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