Properties

Label 1449.1.bq.b
Level $1449$
Weight $1$
Character orbit 1449.bq
Analytic conductor $0.723$
Analytic rank $0$
Dimension $20$
Projective image $D_{22}$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1449,1,Mod(55,1449)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1449, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 11, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1449.55");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1449.bq (of order \(22\), degree \(10\), 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: \(0.723145203305\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
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: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{22}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{22} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{44}^{7} - \zeta_{44}^{5}) q^{2} + (\zeta_{44}^{14} + \zeta_{44}^{12} + \zeta_{44}^{10}) q^{4} + \zeta_{44}^{2} q^{7} + ( - \zeta_{44}^{21} - \zeta_{44}^{19} - \zeta_{44}^{17} - \zeta_{44}^{15}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{44}^{7} - \zeta_{44}^{5}) q^{2} + (\zeta_{44}^{14} + \zeta_{44}^{12} + \zeta_{44}^{10}) q^{4} + \zeta_{44}^{2} q^{7} + ( - \zeta_{44}^{21} - \zeta_{44}^{19} - \zeta_{44}^{17} - \zeta_{44}^{15}) q^{8} + (\zeta_{44}^{9} - \zeta_{44}) q^{11} + ( - \zeta_{44}^{9} - \zeta_{44}^{7}) q^{14} + (\zeta_{44}^{20} - \zeta_{44}^{6} + \zeta_{44}^{4} + \zeta_{44}^{2} + 1) q^{16} + ( - \zeta_{44}^{16} - \zeta_{44}^{14} + \zeta_{44}^{8} + \zeta_{44}^{6}) q^{22} + \zeta_{44}^{3} q^{23} - \zeta_{44}^{6} q^{25} + (\zeta_{44}^{16} + \zeta_{44}^{14} + \zeta_{44}^{12}) q^{28} + (\zeta_{44}^{15} + \zeta_{44}^{5}) q^{29} + (\zeta_{44}^{13} - \zeta_{44}^{11} - \zeta_{44}^{9} - \zeta_{44}^{7} - \zeta_{44}^{5} + \zeta_{44}^{3}) q^{32} + (\zeta_{44}^{10} + \zeta_{44}^{6}) q^{37} + (\zeta_{44}^{18} - \zeta_{44}^{12}) q^{43} + (\zeta_{44}^{21} + \zeta_{44}^{19} - \zeta_{44}^{15} - \zeta_{44}^{13} - \zeta_{44}^{11} - \zeta_{44}) q^{44} + ( - \zeta_{44}^{10} - \zeta_{44}^{8}) q^{46} + \zeta_{44}^{4} q^{49} + (\zeta_{44}^{13} + \zeta_{44}^{11}) q^{50} + ( - \zeta_{44}^{19} + \zeta_{44}^{7}) q^{53} + ( - \zeta_{44}^{21} - \zeta_{44}^{19} - \zeta_{44}^{17} + \zeta_{44}) q^{56} + ( - \zeta_{44}^{20} - \zeta_{44}^{12} - \zeta_{44}^{10} + 1) q^{58} + ( - \zeta_{44}^{20} - \zeta_{44}^{18} - \zeta_{44}^{16} - \zeta_{44}^{14} - \zeta_{44}^{12} + \zeta_{44}^{10} - \zeta_{44}^{8}) q^{64} + ( - \zeta_{44}^{18} + \zeta_{44}^{16}) q^{67} + (\zeta_{44}^{21} - \zeta_{44}^{13}) q^{71} + ( - \zeta_{44}^{17} - \zeta_{44}^{15} - \zeta_{44}^{13} - \zeta_{44}^{11}) q^{74} + (\zeta_{44}^{11} - \zeta_{44}^{3}) q^{77} + ( - \zeta_{44}^{14} + \zeta_{44}^{4}) q^{79} + (\zeta_{44}^{19} + \zeta_{44}^{17} + \zeta_{44}^{3} + \zeta_{44}) q^{86} + (\zeta_{44}^{20} + \zeta_{44}^{18} + \zeta_{44}^{16} + \zeta_{44}^{8} + \zeta_{44}^{6} + \zeta_{44}^{4} + \zeta_{44}^{2} - 1) q^{88} + (\zeta_{44}^{17} + \zeta_{44}^{15} + \zeta_{44}^{13}) q^{92} + ( - \zeta_{44}^{11} - \zeta_{44}^{9}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{4} + 2 q^{7} - 24 q^{16} - 2 q^{25} - 2 q^{28} + 4 q^{37} + 4 q^{43} - 2 q^{49} + 22 q^{58} + 2 q^{64} - 4 q^{67} - 4 q^{79} - 22 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1449\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(829\) \(1289\)
\(\chi(n)\) \(-\zeta_{44}^{6}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.540641 + 0.841254i
−0.540641 0.841254i
0.281733 + 0.959493i
−0.281733 0.959493i
0.281733 0.959493i
−0.281733 + 0.959493i
−0.909632 + 0.415415i
0.909632 0.415415i
0.989821 + 0.142315i
−0.989821 0.142315i
0.540641 0.841254i
−0.540641 + 0.841254i
−0.909632 0.415415i
0.909632 + 0.415415i
−0.755750 + 0.654861i
0.755750 0.654861i
0.989821 0.142315i
−0.989821 + 0.142315i
−0.755750 0.654861i
0.755750 + 0.654861i
−1.03748 + 0.304632i 0 0.142315 0.0914602i 0 0 −0.415415 + 0.909632i 0.588302 0.678936i 0 0
55.2 1.03748 0.304632i 0 0.142315 0.0914602i 0 0 −0.415415 + 0.909632i −0.588302 + 0.678936i 0 0
118.1 −0.0801894 0.557730i 0 0.654861 0.192284i 0 0 −0.841254 + 0.540641i −0.393828 0.862362i 0 0
118.2 0.0801894 + 0.557730i 0 0.654861 0.192284i 0 0 −0.841254 + 0.540641i 0.393828 + 0.862362i 0 0
307.1 −0.0801894 + 0.557730i 0 0.654861 + 0.192284i 0 0 −0.841254 0.540641i −0.393828 + 0.862362i 0 0
307.2 0.0801894 0.557730i 0 0.654861 + 0.192284i 0 0 −0.841254 0.540641i 0.393828 0.862362i 0 0
370.1 −1.53046 0.983568i 0 0.959493 + 2.10100i 0 0 0.654861 0.755750i 0.339098 2.35848i 0 0
370.2 1.53046 + 0.983568i 0 0.959493 + 2.10100i 0 0 0.654861 0.755750i −0.339098 + 2.35848i 0 0
496.1 −1.29639 1.49611i 0 −0.415415 + 2.88927i 0 0 0.959493 + 0.281733i 3.19584 2.05384i 0 0
496.2 1.29639 + 1.49611i 0 −0.415415 + 2.88927i 0 0 0.959493 + 0.281733i −3.19584 + 2.05384i 0 0
685.1 −1.03748 0.304632i 0 0.142315 + 0.0914602i 0 0 −0.415415 0.909632i 0.588302 + 0.678936i 0 0
685.2 1.03748 + 0.304632i 0 0.142315 + 0.0914602i 0 0 −0.415415 0.909632i −0.588302 0.678936i 0 0
748.1 −1.53046 + 0.983568i 0 0.959493 2.10100i 0 0 0.654861 + 0.755750i 0.339098 + 2.35848i 0 0
748.2 1.53046 0.983568i 0 0.959493 2.10100i 0 0 0.654861 + 0.755750i −0.339098 2.35848i 0 0
811.1 −0.627899 + 1.37491i 0 −0.841254 0.970858i 0 0 0.142315 0.989821i 0.412791 0.121206i 0 0
811.2 0.627899 1.37491i 0 −0.841254 0.970858i 0 0 0.142315 0.989821i −0.412791 + 0.121206i 0 0
1189.1 −1.29639 + 1.49611i 0 −0.415415 2.88927i 0 0 0.959493 0.281733i 3.19584 + 2.05384i 0 0
1189.2 1.29639 1.49611i 0 −0.415415 2.88927i 0 0 0.959493 0.281733i −3.19584 2.05384i 0 0
1315.1 −0.627899 1.37491i 0 −0.841254 + 0.970858i 0 0 0.142315 + 0.989821i 0.412791 + 0.121206i 0 0
1315.2 0.627899 + 1.37491i 0 −0.841254 + 0.970858i 0 0 0.142315 + 0.989821i −0.412791 0.121206i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
21.c even 2 1 inner
23.c even 11 1 inner
69.h odd 22 1 inner
161.l odd 22 1 inner
483.v even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1449.1.bq.b 20
3.b odd 2 1 inner 1449.1.bq.b 20
7.b odd 2 1 CM 1449.1.bq.b 20
21.c even 2 1 inner 1449.1.bq.b 20
23.c even 11 1 inner 1449.1.bq.b 20
69.h odd 22 1 inner 1449.1.bq.b 20
161.l odd 22 1 inner 1449.1.bq.b 20
483.v even 22 1 inner 1449.1.bq.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1449.1.bq.b 20 1.a even 1 1 trivial
1449.1.bq.b 20 3.b odd 2 1 inner
1449.1.bq.b 20 7.b odd 2 1 CM
1449.1.bq.b 20 21.c even 2 1 inner
1449.1.bq.b 20 23.c even 11 1 inner
1449.1.bq.b 20 69.h odd 22 1 inner
1449.1.bq.b 20 161.l odd 22 1 inner
1449.1.bq.b 20 483.v even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 22T_{2}^{16} + 165T_{2}^{12} + 99T_{2}^{10} + 484T_{2}^{8} - 968T_{2}^{6} + 484T_{2}^{4} + 605T_{2}^{2} + 121 \) acting on \(S_{1}^{\mathrm{new}}(1449, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 22 T^{16} + 165 T^{12} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + 55 T^{14} - 264 T^{10} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{20} \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( T^{20} \) Copy content Toggle raw display
$23$ \( T^{20} - T^{18} + T^{16} - T^{14} + T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{20} + 22 T^{12} + 462 T^{10} + \cdots + 121 \) Copy content Toggle raw display
$31$ \( T^{20} \) Copy content Toggle raw display
$37$ \( (T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} - 6 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} \) Copy content Toggle raw display
$43$ \( (T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + 5 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} \) Copy content Toggle raw display
$53$ \( T^{20} + 55 T^{14} - 264 T^{10} + \cdots + 121 \) Copy content Toggle raw display
$59$ \( T^{20} \) Copy content Toggle raw display
$61$ \( T^{20} \) Copy content Toggle raw display
$67$ \( (T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + 5 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + 55 T^{14} - 264 T^{10} + \cdots + 121 \) Copy content Toggle raw display
$73$ \( T^{20} \) Copy content Toggle raw display
$79$ \( (T^{10} + 2 T^{9} + 4 T^{8} - 3 T^{7} - 6 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} \) Copy content Toggle raw display
$89$ \( T^{20} \) Copy content Toggle raw display
$97$ \( T^{20} \) Copy content Toggle raw display
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