Properties

Label 2883.1.o.c
Level $2883$
Weight $1$
Character orbit 2883.o
Analytic conductor $1.439$
Analytic rank $0$
Dimension $8$
Projective image $D_{5}$
CM discriminant -3
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2883,1,Mod(338,2883)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2883, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 16]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2883.338");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2883 = 3 \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2883.o (of order \(30\), degree \(8\), not 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: \(1.43880443142\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 93)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.8311689.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{30}^{2} q^{3} + \zeta_{30}^{6} q^{4} + ( - \zeta_{30}^{5} + \zeta_{30}^{2}) q^{7} + \zeta_{30}^{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{30}^{2} q^{3} + \zeta_{30}^{6} q^{4} + ( - \zeta_{30}^{5} + \zeta_{30}^{2}) q^{7} + \zeta_{30}^{4} q^{9} + \zeta_{30}^{8} q^{12} + (\zeta_{30}^{10} + \zeta_{30}^{4}) q^{13} + \zeta_{30}^{12} q^{16} + ( - \zeta_{30}^{11} - \zeta_{30}^{5}) q^{19} + ( - \zeta_{30}^{7} + \zeta_{30}^{4}) q^{21} - \zeta_{30}^{5} q^{25} + \zeta_{30}^{6} q^{27} + ( - \zeta_{30}^{11} + \zeta_{30}^{8}) q^{28} + \zeta_{30}^{10} q^{36} + (\zeta_{30}^{14} - \zeta_{30}^{11}) q^{37} + (\zeta_{30}^{12} + \zeta_{30}^{6}) q^{39} + (\zeta_{30}^{14} + \zeta_{30}^{2}) q^{43} + \zeta_{30}^{14} q^{48} + (\zeta_{30}^{10} - \zeta_{30}^{7} + \zeta_{30}^{4}) q^{49} + (\zeta_{30}^{10} - \zeta_{30}) q^{52} + ( - \zeta_{30}^{13} - \zeta_{30}^{7}) q^{57} + (\zeta_{30}^{12} - \zeta_{30}^{3}) q^{61} + ( - \zeta_{30}^{9} + \zeta_{30}^{6}) q^{63} - \zeta_{30}^{3} q^{64} + ( - \zeta_{30}^{13} - \zeta_{30}^{7}) q^{67} + (\zeta_{30}^{10} - \zeta_{30}^{7}) q^{73} - \zeta_{30}^{7} q^{75} + ( - \zeta_{30}^{11} + \zeta_{30}^{2}) q^{76} + ( - \zeta_{30}^{11} + \zeta_{30}^{2}) q^{79} + \zeta_{30}^{8} q^{81} + ( - \zeta_{30}^{13} + \zeta_{30}^{10}) q^{84} + (\zeta_{30}^{12} - \zeta_{30}^{9} + \zeta_{30}^{6} + 1) q^{91} + ( - \zeta_{30}^{9} - \zeta_{30}^{3}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} - 2 q^{4} - 3 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} - 2 q^{4} - 3 q^{7} + q^{9} + q^{12} - 3 q^{13} - 2 q^{16} - 3 q^{19} + 2 q^{21} - 4 q^{25} - 2 q^{27} + 2 q^{28} - 4 q^{36} + 2 q^{37} - 4 q^{39} + 2 q^{43} + q^{48} - 2 q^{49} - 3 q^{52} + 2 q^{57} - 4 q^{61} - 4 q^{63} - 2 q^{64} + 2 q^{67} - 3 q^{73} + q^{75} + 2 q^{76} + 2 q^{79} + q^{81} - 3 q^{84} + 2 q^{91} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2883\mathbb{Z}\right)^\times\).

\(n\) \(962\) \(964\)
\(\chi(n)\) \(-1\) \(\zeta_{30}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
338.1
0.669131 + 0.743145i
−0.978148 + 0.207912i
0.913545 0.406737i
0.913545 + 0.406737i
−0.104528 0.994522i
−0.978148 0.207912i
0.669131 0.743145i
−0.104528 + 0.994522i
0 −0.104528 + 0.994522i 0.309017 0.951057i 0 0 −0.604528 + 0.128496i 0 −0.978148 0.207912i 0
1196.1 0 0.913545 0.406737i 0.309017 0.951057i 0 0 0.413545 + 0.459289i 0 0.669131 0.743145i 0
1409.1 0 0.669131 0.743145i −0.809017 0.587785i 0 0 0.169131 1.60917i 0 −0.104528 0.994522i 0
1508.1 0 0.669131 + 0.743145i −0.809017 + 0.587785i 0 0 0.169131 + 1.60917i 0 −0.104528 + 0.994522i 0
1805.1 0 −0.978148 + 0.207912i −0.809017 + 0.587785i 0 0 −1.47815 0.658114i 0 0.913545 0.406737i 0
2654.1 0 0.913545 + 0.406737i 0.309017 + 0.951057i 0 0 0.413545 0.459289i 0 0.669131 + 0.743145i 0
2738.1 0 −0.104528 0.994522i 0.309017 + 0.951057i 0 0 −0.604528 0.128496i 0 −0.978148 + 0.207912i 0
2768.1 0 −0.978148 0.207912i −0.809017 0.587785i 0 0 −1.47815 + 0.658114i 0 0.913545 + 0.406737i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 338.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
31.c even 3 1 inner
31.d even 5 1 inner
31.g even 15 1 inner
93.h odd 6 1 inner
93.l odd 10 1 inner
93.o odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2883.1.o.c 8
3.b odd 2 1 CM 2883.1.o.c 8
31.b odd 2 1 2883.1.o.a 8
31.c even 3 1 2883.1.l.a 4
31.c even 3 1 inner 2883.1.o.c 8
31.d even 5 1 2883.1.h.a 4
31.d even 5 1 inner 2883.1.o.c 8
31.d even 5 2 2883.1.o.d 8
31.e odd 6 1 2883.1.l.c 4
31.e odd 6 1 2883.1.o.a 8
31.f odd 10 1 2883.1.h.b 4
31.f odd 10 1 2883.1.o.a 8
31.f odd 10 2 2883.1.o.b 8
31.g even 15 2 93.1.l.a 4
31.g even 15 1 2883.1.b.b 2
31.g even 15 1 2883.1.h.a 4
31.g even 15 1 2883.1.l.a 4
31.g even 15 1 inner 2883.1.o.c 8
31.g even 15 2 2883.1.o.d 8
31.h odd 30 1 2883.1.b.a 2
31.h odd 30 1 2883.1.h.b 4
31.h odd 30 2 2883.1.l.b 4
31.h odd 30 1 2883.1.l.c 4
31.h odd 30 1 2883.1.o.a 8
31.h odd 30 2 2883.1.o.b 8
93.c even 2 1 2883.1.o.a 8
93.g even 6 1 2883.1.l.c 4
93.g even 6 1 2883.1.o.a 8
93.h odd 6 1 2883.1.l.a 4
93.h odd 6 1 inner 2883.1.o.c 8
93.k even 10 1 2883.1.h.b 4
93.k even 10 1 2883.1.o.a 8
93.k even 10 2 2883.1.o.b 8
93.l odd 10 1 2883.1.h.a 4
93.l odd 10 1 inner 2883.1.o.c 8
93.l odd 10 2 2883.1.o.d 8
93.o odd 30 2 93.1.l.a 4
93.o odd 30 1 2883.1.b.b 2
93.o odd 30 1 2883.1.h.a 4
93.o odd 30 1 2883.1.l.a 4
93.o odd 30 1 inner 2883.1.o.c 8
93.o odd 30 2 2883.1.o.d 8
93.p even 30 1 2883.1.b.a 2
93.p even 30 1 2883.1.h.b 4
93.p even 30 2 2883.1.l.b 4
93.p even 30 1 2883.1.l.c 4
93.p even 30 1 2883.1.o.a 8
93.p even 30 2 2883.1.o.b 8
124.n odd 30 2 1488.1.br.a 4
155.u even 30 2 2325.1.ca.a 4
155.w odd 60 4 2325.1.bq.a 8
279.ba even 15 2 2511.1.bu.a 8
279.bb even 15 2 2511.1.bu.a 8
279.bd odd 30 2 2511.1.bu.a 8
279.bi odd 30 2 2511.1.bu.a 8
372.bd even 30 2 1488.1.br.a 4
465.bl odd 30 2 2325.1.ca.a 4
465.bt even 60 4 2325.1.bq.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.1.l.a 4 31.g even 15 2
93.1.l.a 4 93.o odd 30 2
1488.1.br.a 4 124.n odd 30 2
1488.1.br.a 4 372.bd even 30 2
2325.1.bq.a 8 155.w odd 60 4
2325.1.bq.a 8 465.bt even 60 4
2325.1.ca.a 4 155.u even 30 2
2325.1.ca.a 4 465.bl odd 30 2
2511.1.bu.a 8 279.ba even 15 2
2511.1.bu.a 8 279.bb even 15 2
2511.1.bu.a 8 279.bd odd 30 2
2511.1.bu.a 8 279.bi odd 30 2
2883.1.b.a 2 31.h odd 30 1
2883.1.b.a 2 93.p even 30 1
2883.1.b.b 2 31.g even 15 1
2883.1.b.b 2 93.o odd 30 1
2883.1.h.a 4 31.d even 5 1
2883.1.h.a 4 31.g even 15 1
2883.1.h.a 4 93.l odd 10 1
2883.1.h.a 4 93.o odd 30 1
2883.1.h.b 4 31.f odd 10 1
2883.1.h.b 4 31.h odd 30 1
2883.1.h.b 4 93.k even 10 1
2883.1.h.b 4 93.p even 30 1
2883.1.l.a 4 31.c even 3 1
2883.1.l.a 4 31.g even 15 1
2883.1.l.a 4 93.h odd 6 1
2883.1.l.a 4 93.o odd 30 1
2883.1.l.b 4 31.h odd 30 2
2883.1.l.b 4 93.p even 30 2
2883.1.l.c 4 31.e odd 6 1
2883.1.l.c 4 31.h odd 30 1
2883.1.l.c 4 93.g even 6 1
2883.1.l.c 4 93.p even 30 1
2883.1.o.a 8 31.b odd 2 1
2883.1.o.a 8 31.e odd 6 1
2883.1.o.a 8 31.f odd 10 1
2883.1.o.a 8 31.h odd 30 1
2883.1.o.a 8 93.c even 2 1
2883.1.o.a 8 93.g even 6 1
2883.1.o.a 8 93.k even 10 1
2883.1.o.a 8 93.p even 30 1
2883.1.o.b 8 31.f odd 10 2
2883.1.o.b 8 31.h odd 30 2
2883.1.o.b 8 93.k even 10 2
2883.1.o.b 8 93.p even 30 2
2883.1.o.c 8 1.a even 1 1 trivial
2883.1.o.c 8 3.b odd 2 1 CM
2883.1.o.c 8 31.c even 3 1 inner
2883.1.o.c 8 31.d even 5 1 inner
2883.1.o.c 8 31.g even 15 1 inner
2883.1.o.c 8 93.h odd 6 1 inner
2883.1.o.c 8 93.l odd 10 1 inner
2883.1.o.c 8 93.o odd 30 1 inner
2883.1.o.d 8 31.d even 5 2
2883.1.o.d 8 31.g even 15 2
2883.1.o.d 8 93.l odd 10 2
2883.1.o.d 8 93.o odd 30 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2883, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7}^{8} + 3T_{7}^{7} + 5T_{7}^{6} + 8T_{7}^{5} + 9T_{7}^{4} + 2T_{7}^{3} + 2T_{7} + 1 \) Copy content Toggle raw display
\( T_{13}^{8} + 3T_{13}^{7} + 5T_{13}^{6} + 8T_{13}^{5} + 9T_{13}^{4} + 2T_{13}^{3} + 2T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 3 T^{7} + 5 T^{6} + 8 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 3 T^{7} + 5 T^{6} + 8 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 3 T^{7} + 5 T^{6} + 8 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - T^{3} + 2 T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} - 2 T^{7} - 2 T^{5} + 9 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{2} + T - 1)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - T^{3} + 2 T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 3 T^{7} + 5 T^{6} + 8 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{8} - 2 T^{7} - 2 T^{5} + 9 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
show more
show less