Properties

Label 2888.1.bs.a
Level $2888$
Weight $1$
Character orbit 2888.bs
Analytic conductor $1.441$
Analytic rank $0$
Dimension $108$
Projective image $D_{171}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2888,1,Mod(35,2888)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2888, base_ring=CyclotomicField(342))
 
chi = DirichletCharacter(H, H._module([171, 171, 40]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2888.35");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2888 = 2^{3} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2888.bs (of order \(342\), degree \(108\), 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: \(1.44129975648\)
Analytic rank: \(0\)
Dimension: \(108\)
Coefficient field: \(\Q(\zeta_{171})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{108} - x^{105} + x^{99} - x^{96} + x^{90} - x^{87} + x^{81} - x^{78} + x^{72} - x^{69} + x^{63} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{171}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{171} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{342}^{70} q^{2} + (\zeta_{342}^{156} + \zeta_{342}^{152}) q^{3} + \zeta_{342}^{140} q^{4} + ( - \zeta_{342}^{55} - \zeta_{342}^{51}) q^{6} - \zeta_{342}^{39} q^{8} + ( - \zeta_{342}^{141} + \cdots - \zeta_{342}^{133}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{342}^{70} q^{2} + (\zeta_{342}^{156} + \zeta_{342}^{152}) q^{3} + \zeta_{342}^{140} q^{4} + ( - \zeta_{342}^{55} - \zeta_{342}^{51}) q^{6} - \zeta_{342}^{39} q^{8} + ( - \zeta_{342}^{141} + \cdots - \zeta_{342}^{133}) q^{9} + \cdots + ( - \zeta_{342}^{163} + \cdots - \zeta_{342}^{27}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q + 3 q^{3} + 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 108 q + 3 q^{3} + 3 q^{6} + 3 q^{8} + 3 q^{9} - 6 q^{18} - 6 q^{22} + 3 q^{24} - 60 q^{27} + 3 q^{33} + 3 q^{36} - 54 q^{38} + 3 q^{41} - 6 q^{44} - 6 q^{48} + 3 q^{49} + 3 q^{50} - 3 q^{51} + 3 q^{54} + 3 q^{59} + 3 q^{64} + 3 q^{66} + 3 q^{67} - 3 q^{68} - 6 q^{72} - 6 q^{73} - 3 q^{81} + 3 q^{82} + 3 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2888\mathbb{Z}\right)^\times\).

\(n\) \(1445\) \(2167\) \(2529\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{342}^{140}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
−0.227635 + 0.973746i
−0.467849 0.883809i
−0.703852 0.710347i
−0.367783 0.929912i
−0.951623 + 0.307269i
−0.951623 0.307269i
−0.995784 0.0917303i
−0.155527 + 0.987832i
0.280931 + 0.959728i
−0.861396 0.507934i
0.418451 0.908239i
0.384804 0.922998i
−0.621436 + 0.783465i
−0.467849 + 0.883809i
0.209708 0.977764i
−0.991742 + 0.128249i
0.384804 + 0.922998i
0.888069 + 0.459710i
0.315998 + 0.948760i
−0.729471 + 0.684011i
0.933251 + 0.359225i −1.23821 1.29643i 0.741914 + 0.670495i 0 −0.689849 1.65469i 0 0.451533 + 0.892254i −0.101647 + 2.21154i 0
43.1 0.888069 + 0.459710i 1.02557 + 0.461135i 0.577333 + 0.816509i 0 0.698786 + 0.880984i 0 0.137354 + 0.990522i 0.175492 + 0.197808i 0
123.1 0.315998 0.948760i 0.0119378 0.0139644i −0.800291 0.599612i 0 −0.00947655 0.0157389i 0 −0.821778 + 0.569808i 0.155475 + 0.987498i 0
131.1 −0.333374 + 0.942795i 0.173809 1.44855i −0.777724 0.628606i 0 1.30774 + 0.646776i 0 0.851919 0.523673i −1.09648 0.266973i 0
139.1 −0.991742 0.128249i −0.912138 + 1.34164i 0.967104 + 0.254380i 0 1.07667 1.21358i 0 −0.926494 0.376309i −0.600220 1.51761i 0
187.1 −0.991742 + 0.128249i −0.912138 1.34164i 0.967104 0.254380i 0 1.07667 + 1.21358i 0 −0.926494 + 0.376309i −0.600220 + 1.51761i 0
195.1 0.989219 + 0.146447i −0.0180625 + 1.96626i 0.957107 + 0.289735i 0 −0.305820 + 1.94242i 0 0.904357 + 0.426776i −2.86602 0.0526601i 0
251.1 0.0642573 + 0.997933i 0.890431 1.68210i −0.991742 + 0.128249i 0 1.73584 + 0.780503i 0 −0.191711 0.981451i −1.47437 2.16862i 0
275.1 −0.467849 + 0.883809i 1.67040 + 0.216011i −0.562235 0.826977i 0 −0.972408 + 1.37525i 0 0.993931 0.110008i 1.77648 + 0.467271i 0
283.1 0.919425 0.393266i 0.903398 + 0.347734i 0.690683 0.723158i 0 0.967359 0.0355604i 0 0.350638 0.936511i −0.0467056 0.0422095i 0
291.1 −0.367783 + 0.929912i −1.13140 0.639431i −0.729471 0.684011i 0 1.01073 0.816933i 0 0.904357 0.426776i 0.355376 + 0.590216i 0
339.1 0.811171 0.584809i −0.589055 1.27853i 0.315998 0.948760i 0 −1.22552 0.692623i 0 −0.298515 0.954405i −0.637859 + 0.746143i 0
347.1 0.983171 0.182687i −0.418587 + 0.179043i 0.933251 0.359225i 0 −0.378834 + 0.252500i 0 0.851919 0.523673i −0.547524 + 0.573268i 0
403.1 0.888069 0.459710i 1.02557 0.461135i 0.577333 0.816509i 0 0.698786 0.880984i 0 0.137354 0.990522i 0.175492 0.197808i 0
427.1 0.606938 0.794749i 0.793599 + 1.64241i −0.263253 0.964727i 0 1.78697 + 0.366128i 0 −0.926494 0.376309i −1.44627 + 1.82336i 0
435.1 −0.912045 0.410091i 1.11668 1.57930i 0.663651 + 0.748042i 0 −1.66612 + 0.982450i 0 −0.298515 0.954405i −0.913832 2.58435i 0
443.1 0.811171 + 0.584809i −0.589055 + 1.27853i 0.315998 + 0.948760i 0 −1.22552 + 0.692623i 0 −0.298515 + 0.954405i −0.637859 0.746143i 0
491.1 −0.435066 + 0.900399i −0.303969 1.11394i −0.621436 0.783465i 0 1.13523 + 0.210942i 0 0.975796 0.218681i −0.287062 + 0.169270i 0
499.1 0.870582 0.492024i 1.16758 + 1.09482i 0.515825 0.856694i 0 1.55515 + 0.378650i 0 0.0275543 0.999620i 0.100361 + 1.55863i 0
555.1 −0.777724 0.628606i −0.124867 0.0304028i 0.209708 + 0.977764i 0 0.0780004 + 0.102137i 0 0.451533 0.892254i −0.873402 0.452118i 0
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
361.k even 171 1 inner
2888.bs odd 342 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2888.1.bs.a 108
8.d odd 2 1 CM 2888.1.bs.a 108
361.k even 171 1 inner 2888.1.bs.a 108
2888.bs odd 342 1 inner 2888.1.bs.a 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2888.1.bs.a 108 1.a even 1 1 trivial
2888.1.bs.a 108 8.d odd 2 1 CM
2888.1.bs.a 108 361.k even 171 1 inner
2888.1.bs.a 108 2888.bs odd 342 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2888, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{108} - T^{105} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{108} - 3 T^{107} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{108} \) Copy content Toggle raw display
$7$ \( T^{108} \) Copy content Toggle raw display
$11$ \( T^{108} - 3 T^{106} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{108} \) Copy content Toggle raw display
$17$ \( T^{108} + T^{105} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{108} - T^{105} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{108} \) Copy content Toggle raw display
$29$ \( T^{108} \) Copy content Toggle raw display
$31$ \( T^{108} \) Copy content Toggle raw display
$37$ \( T^{108} \) Copy content Toggle raw display
$41$ \( T^{108} - 3 T^{107} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{108} + T^{105} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{108} \) Copy content Toggle raw display
$53$ \( T^{108} \) Copy content Toggle raw display
$59$ \( T^{108} - 3 T^{107} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{108} \) Copy content Toggle raw display
$67$ \( T^{108} - 3 T^{107} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{108} \) Copy content Toggle raw display
$73$ \( T^{108} + 6 T^{107} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{108} \) Copy content Toggle raw display
$83$ \( T^{108} - 3 T^{106} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{108} + T^{105} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{108} - 3 T^{107} + \cdots + 1 \) Copy content Toggle raw display
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