Properties

Label 2624.1.dr.b
Level $2624$
Weight $1$
Character orbit 2624.dr
Analytic conductor $1.310$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
CM discriminant -8
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2624,1,Mod(97,2624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2624, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([0, 20, 37]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2624.97");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2624 = 2^{6} \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2624.dr (of order \(40\), degree \(16\), 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: \(1.30954659315\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{40}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{40} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{40}^{14} + \zeta_{40}) q^{3} + ( - \zeta_{40}^{15} - \zeta_{40}^{8} + \zeta_{40}^{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{40}^{14} + \zeta_{40}) q^{3} + ( - \zeta_{40}^{15} - \zeta_{40}^{8} + \zeta_{40}^{2}) q^{9} + ( - \zeta_{40}^{15} + \zeta_{40}^{4}) q^{11} + (\zeta_{40}^{17} - \zeta_{40}^{12}) q^{17} + (\zeta_{40}^{17} - 1) q^{19} + \zeta_{40}^{6} q^{25} + ( - \zeta_{40}^{16} + \zeta_{40}^{9} + \zeta_{40}^{3} - \zeta_{40}^{2}) q^{27} + ( - \zeta_{40}^{18} - \zeta_{40}^{16} - \zeta_{40}^{9} + \zeta_{40}^{5}) q^{33} + \zeta_{40}^{13} q^{41} + (\zeta_{40}^{13} - \zeta_{40}^{5}) q^{43} - \zeta_{40}^{7} q^{49} + (\zeta_{40}^{18} - \zeta_{40}^{13} + \zeta_{40}^{11} - \zeta_{40}^{6}) q^{51} + (\zeta_{40}^{18} + \zeta_{40}^{14} + \zeta_{40}^{11} - \zeta_{40}) q^{57} + (\zeta_{40}^{19} + \zeta_{40}^{9}) q^{59} + ( - \zeta_{40}^{11} - \zeta_{40}^{10}) q^{67} + (\zeta_{40}^{9} + \zeta_{40}) q^{73} + (\zeta_{40}^{7} + 1) q^{75} + ( - \zeta_{40}^{17} + \zeta_{40}^{16} + \zeta_{40}^{10} + \zeta_{40}^{4} + \zeta_{40}^{3}) q^{81} + ( - \zeta_{40}^{12} - \zeta_{40}^{8}) q^{83} + (\zeta_{40}^{4} + \zeta_{40}^{3}) q^{89} + ( - \zeta_{40}^{19} - \zeta_{40}^{2}) q^{97} + ( - \zeta_{40}^{19} - \zeta_{40}^{17} - \zeta_{40}^{12} - \zeta_{40}^{10} + \zeta_{40}^{6} - \zeta_{40}^{3}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{9} + 4 q^{11} - 4 q^{17} - 16 q^{19} + 4 q^{27} + 4 q^{33} + 16 q^{75} + 4 q^{89} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2624\mathbb{Z}\right)^\times\).

\(n\) \(129\) \(575\) \(1477\)
\(\chi(n)\) \(-\zeta_{40}^{13}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−0.156434 0.987688i
−0.891007 0.453990i
0.156434 0.987688i
−0.987688 + 0.156434i
0.891007 0.453990i
0.987688 + 0.156434i
−0.453990 + 0.891007i
0.453990 0.891007i
−0.987688 0.156434i
−0.891007 + 0.453990i
0.987688 0.156434i
−0.156434 + 0.987688i
0.891007 + 0.453990i
0.156434 + 0.987688i
0.453990 + 0.891007i
−0.453990 0.891007i
0 −0.744220 1.79671i 0 0 0 0 0 −1.96718 + 1.96718i 0
417.1 0 −1.84206 0.763007i 0 0 0 0 0 2.10391 + 2.10391i 0
481.1 0 −0.431351 0.178671i 0 0 0 0 0 −0.552967 0.552967i 0
545.1 0 −0.399903 + 0.965451i 0 0 0 0 0 −0.0650673 0.0650673i 0
609.1 0 −0.0600500 0.144974i 0 0 0 0 0 0.689695 0.689695i 0
673.1 0 1.57547 0.652583i 0 0 0 0 0 1.34915 1.34915i 0
801.1 0 0.497066 + 1.20002i 0 0 0 0 0 −0.485875 + 0.485875i 0
1249.1 0 1.40505 0.581990i 0 0 0 0 0 0.928339 0.928339i 0
1377.1 0 −0.399903 0.965451i 0 0 0 0 0 −0.0650673 + 0.0650673i 0
1441.1 0 −1.84206 + 0.763007i 0 0 0 0 0 2.10391 2.10391i 0
1505.1 0 1.57547 + 0.652583i 0 0 0 0 0 1.34915 + 1.34915i 0
1569.1 0 −0.744220 + 1.79671i 0 0 0 0 0 −1.96718 1.96718i 0
1633.1 0 −0.0600500 + 0.144974i 0 0 0 0 0 0.689695 + 0.689695i 0
1953.1 0 −0.431351 + 0.178671i 0 0 0 0 0 −0.552967 + 0.552967i 0
2145.1 0 1.40505 + 0.581990i 0 0 0 0 0 0.928339 + 0.928339i 0
2529.1 0 0.497066 1.20002i 0 0 0 0 0 −0.485875 0.485875i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
164.o even 40 1 inner
328.bf odd 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2624.1.dr.b yes 16
4.b odd 2 1 2624.1.dr.a 16
8.b even 2 1 2624.1.dr.a 16
8.d odd 2 1 CM 2624.1.dr.b yes 16
41.h odd 40 1 2624.1.dr.a 16
164.o even 40 1 inner 2624.1.dr.b yes 16
328.bd even 40 1 2624.1.dr.a 16
328.bf odd 40 1 inner 2624.1.dr.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2624.1.dr.a 16 4.b odd 2 1
2624.1.dr.a 16 8.b even 2 1
2624.1.dr.a 16 41.h odd 40 1
2624.1.dr.a 16 328.bd even 40 1
2624.1.dr.b yes 16 1.a even 1 1 trivial
2624.1.dr.b yes 16 8.d odd 2 1 CM
2624.1.dr.b yes 16 164.o even 40 1 inner
2624.1.dr.b yes 16 328.bf odd 40 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 2 T_{3}^{14} - 4 T_{3}^{13} + 2 T_{3}^{12} + 16 T_{3}^{11} + 16 T_{3}^{10} - 52 T_{3}^{9} + 75 T_{3}^{8} - 72 T_{3}^{7} + 28 T_{3}^{6} - 52 T_{3}^{5} + 82 T_{3}^{4} + 136 T_{3}^{3} + 58 T_{3}^{2} + 8 T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2624, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 2 T^{14} - 4 T^{13} + 2 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} - 4 T^{15} + 10 T^{14} - 20 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} + 4 T^{15} + 10 T^{14} + 20 T^{13} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{16} + 16 T^{15} + 120 T^{14} + 560 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} - T^{12} + T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$43$ \( T^{16} + 5 T^{12} + 150 T^{8} + \cdots + 625 \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} + 8 T^{14} - 4 T^{13} + 27 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{8} + 7 T^{4} + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( (T^{4} + 5 T^{2} + 5)^{4} \) Copy content Toggle raw display
$89$ \( T^{16} - 4 T^{15} + 10 T^{14} - 20 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{16} - 2 T^{14} + 16 T^{13} + 2 T^{12} + \cdots + 1 \) Copy content Toggle raw display
show more
show less