Properties

Label 2008.1.bd.a
Level $2008$
Weight $1$
Character orbit 2008.bd
Analytic conductor $1.002$
Analytic rank $0$
Dimension $100$
Projective image $D_{125}$
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] = [2008,1,Mod(3,2008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2008, base_ring=CyclotomicField(250))
 
chi = DirichletCharacter(H, H._module([125, 125, 16]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2008.3");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2008 = 2^{3} \cdot 251 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2008.bd (of order \(250\), degree \(100\), 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: \(1.00212254537\)
Analytic rank: \(0\)
Dimension: \(100\)
Coefficient field: \(\Q(\zeta_{250})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{100} - x^{75} + x^{50} - x^{25} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{125}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{125} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{250}^{35} q^{2} + ( - \zeta_{250}^{53} + \zeta_{250}^{14}) q^{3} + \zeta_{250}^{70} q^{4} + (\zeta_{250}^{88} - \zeta_{250}^{49}) q^{6} - \zeta_{250}^{105} q^{8} + (\zeta_{250}^{106} - \zeta_{250}^{67} + \zeta_{250}^{28}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{250}^{35} q^{2} + ( - \zeta_{250}^{53} + \zeta_{250}^{14}) q^{3} + \zeta_{250}^{70} q^{4} + (\zeta_{250}^{88} - \zeta_{250}^{49}) q^{6} - \zeta_{250}^{105} q^{8} + (\zeta_{250}^{106} - \zeta_{250}^{67} + \zeta_{250}^{28}) q^{9} + ( - \zeta_{250}^{117} + \zeta_{250}^{40}) q^{11} + ( - \zeta_{250}^{123} + \zeta_{250}^{84}) q^{12} - \zeta_{250}^{15} q^{16} + (\zeta_{250}^{94} + \zeta_{250}^{44}) q^{17} + (\zeta_{250}^{102} - \zeta_{250}^{63} + \zeta_{250}^{16}) q^{18} + ( - \zeta_{250}^{43} + \zeta_{250}^{38}) q^{19} + ( - \zeta_{250}^{75} - \zeta_{250}^{27}) q^{22} + ( - \zeta_{250}^{119} - \zeta_{250}^{33}) q^{24} + \zeta_{250}^{60} q^{25} + (\zeta_{250}^{120} - \zeta_{250}^{81} + \zeta_{250}^{42} + \zeta_{250}^{34}) q^{27} + \zeta_{250}^{50} q^{32} + ( - \zeta_{250}^{93} + \zeta_{250}^{54} - \zeta_{250}^{45} + \zeta_{250}^{6}) q^{33} + ( - \zeta_{250}^{79} + \zeta_{250}^{4}) q^{34} + (\zeta_{250}^{98} - \zeta_{250}^{51} + \zeta_{250}^{12}) q^{36} + (\zeta_{250}^{78} - \zeta_{250}^{73}) q^{38} + ( - \zeta_{250}^{37} - \zeta_{250}^{11}) q^{41} + (\zeta_{250}^{76} - \zeta_{250}^{13}) q^{43} + (\zeta_{250}^{110} + \zeta_{250}^{62}) q^{44} + (\zeta_{250}^{68} - \zeta_{250}^{29}) q^{48} - \zeta_{250}^{101} q^{49} - \zeta_{250}^{95} q^{50} + (\zeta_{250}^{108} - \zeta_{250}^{97} + \zeta_{250}^{58} + \zeta_{250}^{22}) q^{51} + (\zeta_{250}^{116} - \zeta_{250}^{77} - \zeta_{250}^{69} + \zeta_{250}^{30}) q^{54} + (\zeta_{250}^{96} - \zeta_{250}^{91} - \zeta_{250}^{57} + \zeta_{250}^{52}) q^{57} + ( - \zeta_{250}^{65} - \zeta_{250}^{47}) q^{59} - \zeta_{250}^{85} q^{64} + ( - \zeta_{250}^{89} + \zeta_{250}^{80} - \zeta_{250}^{41} - \zeta_{250}^{3}) q^{66} + ( - \zeta_{250}^{121} + \zeta_{250}^{2}) q^{67} + (\zeta_{250}^{114} - \zeta_{250}^{39}) q^{68} + (\zeta_{250}^{86} - \zeta_{250}^{47} + \zeta_{250}^{8}) q^{72} + ( - \zeta_{250}^{89} - \zeta_{250}^{69}) q^{73} + ( - \zeta_{250}^{113} + \zeta_{250}^{74}) q^{75} + ( - \zeta_{250}^{113} + \zeta_{250}^{108}) q^{76} + ( - \zeta_{250}^{95} - \zeta_{250}^{87} + \zeta_{250}^{56} - \zeta_{250}^{48} + \zeta_{250}^{9}) q^{81} + (\zeta_{250}^{72} + \zeta_{250}^{46}) q^{82} + (\zeta_{250}^{118} + \zeta_{250}^{10}) q^{83} + ( - \zeta_{250}^{111} + \zeta_{250}^{48}) q^{86} + ( - \zeta_{250}^{97} + \zeta_{250}^{20}) q^{88} + ( - \zeta_{250}^{71} - \zeta_{250}^{17}) q^{89} + ( - \zeta_{250}^{103} + \zeta_{250}^{64}) q^{96} + ( - \zeta_{250}^{77} - \zeta_{250}) q^{97} - \zeta_{250}^{11} q^{98} + ( - \zeta_{250}^{107} + \zeta_{250}^{98} + \zeta_{250}^{68} - \zeta_{250}^{59} - \zeta_{250}^{21} + \zeta_{250}^{20}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 100 q+O(q^{10}) \) Copy content Toggle raw display \( 100 q - 25 q^{22} - 25 q^{32}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2008\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(503\) \(1005\)
\(\chi(n)\) \(\zeta_{250}^{12}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.470704 + 0.882291i
−0.994951 0.100362i
0.711536 0.702650i
0.675333 0.737513i
0.837528 + 0.546394i
0.947098 0.320944i
−0.979855 + 0.199710i
−0.998737 + 0.0502443i
0.999684 + 0.0251301i
0.137790 0.990461i
0.285019 + 0.958522i
−0.162637 + 0.986686i
−0.979855 0.199710i
−0.577573 0.816339i
0.984564 0.175023i
−0.793990 + 0.607930i
−0.448383 + 0.893841i
0.962028 0.272952i
0.778462 0.627691i
0.711536 + 0.702650i
−0.992115 0.125333i −1.58347 0.119618i 0.968583 + 0.248690i 0 1.55599 + 0.317136i 0 −0.929776 0.368125i 1.50441 + 0.228596i 0
27.1 −0.929776 0.368125i 0.740210 + 0.170347i 0.728969 + 0.684547i 0 −0.625521 0.430874i 0 −0.425779 0.904827i −0.380513 0.184931i 0
35.1 0.535827 + 0.844328i 0.811554 + 0.559018i −0.425779 + 0.904827i 0 −0.0371420 + 0.984754i 0 −0.992115 + 0.125333i −0.0102928 0.0269825i 0
67.1 0.728969 0.684547i −0.422111 + 0.791209i 0.0627905 0.998027i 0 0.233914 + 0.865722i 0 −0.637424 0.770513i 0.109042 + 0.162639i 0
75.1 −0.187381 0.982287i −0.948035 + 1.67428i −0.929776 + 0.368125i 0 1.82227 + 0.617513i 0 0.535827 + 0.844328i −1.39001 2.31703i 0
83.1 −0.425779 0.904827i −0.175480 0.00882805i −0.637424 + 0.770513i 0 0.0667281 + 0.162538i 0 0.968583 + 0.248690i −0.964236 0.0972634i 0
115.1 0.728969 0.684547i −1.27992 + 0.622047i 0.0627905 0.998027i 0 −0.507200 + 1.32962i 0 −0.637424 0.770513i 0.633388 0.806049i 0
131.1 −0.187381 0.982287i −0.125694 1.10664i −0.929776 + 0.368125i 0 −1.06348 + 0.330830i 0 0.535827 + 0.844328i −0.234317 + 0.0539241i 0
147.1 −0.637424 0.770513i 0.702235 0.626989i −0.187381 + 0.982287i 0 −0.930725 0.141424i 0 0.876307 0.481754i −0.0128376 + 0.113025i 0
155.1 −0.992115 0.125333i −0.507512 0.430706i 0.968583 + 0.248690i 0 0.449528 + 0.490918i 0 −0.929776 0.368125i −0.0905766 0.549510i 0
179.1 −0.637424 0.770513i 0.238081 + 0.138789i −0.187381 + 0.982287i 0 −0.0448196 0.271911i 0 0.876307 0.481754i −0.455307 0.804098i 0
195.1 0.535827 + 0.844328i 1.35024 0.0339424i −0.425779 + 0.904827i 0 0.752153 + 1.12186i 0 −0.992115 + 0.125333i 0.823256 0.0414163i 0
227.1 0.728969 + 0.684547i −1.27992 0.622047i 0.0627905 + 0.998027i 0 −0.507200 1.32962i 0 −0.637424 + 0.770513i 0.633388 + 0.806049i 0
299.1 −0.425779 + 0.904827i 1.63239 + 1.06495i −0.637424 0.770513i 0 −1.65863 + 1.02359i 0 0.968583 0.248690i 1.12766 + 2.56159i 0
339.1 −0.992115 0.125333i 0.216489 0.527330i 0.968583 + 0.248690i 0 −0.280874 + 0.496038i 0 −0.929776 0.368125i 0.480326 + 0.474328i 0
363.1 −0.637424 + 0.770513i −1.95919 0.197625i −0.187381 0.982287i 0 1.40111 1.38361i 0 0.876307 + 0.481754i 2.81950 + 0.574659i 0
395.1 0.535827 0.844328i −1.44523 1.10656i −0.425779 0.904827i 0 −1.70869 + 0.627323i 0 −0.992115 0.125333i 0.603371 + 2.23309i 0
403.1 0.968583 0.248690i −0.253214 + 1.53620i 0.876307 0.481754i 0 0.136778 + 1.55090i 0 0.728969 0.684547i −1.34868 0.457028i 0
443.1 −0.187381 0.982287i −0.834522 0.911359i −0.929776 + 0.368125i 0 −0.738843 + 0.990512i 0 0.535827 + 0.844328i −0.0462978 + 0.524964i 0
459.1 0.535827 0.844328i 0.811554 0.559018i −0.425779 0.904827i 0 −0.0371420 0.984754i 0 −0.992115 0.125333i −0.0102928 + 0.0269825i 0
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
251.g even 125 1 inner
2008.bd odd 250 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2008.1.bd.a 100
8.d odd 2 1 CM 2008.1.bd.a 100
251.g even 125 1 inner 2008.1.bd.a 100
2008.bd odd 250 1 inner 2008.1.bd.a 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2008.1.bd.a 100 1.a even 1 1 trivial
2008.1.bd.a 100 8.d odd 2 1 CM
2008.1.bd.a 100 251.g even 125 1 inner
2008.1.bd.a 100 2008.bd odd 250 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2008, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{20} + T^{15} + T^{10} + T^{5} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{100} + 25 T^{87} - 2600 T^{86} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{100} \) Copy content Toggle raw display
$7$ \( T^{100} \) Copy content Toggle raw display
$11$ \( T^{100} + 5 T^{95} - 100 T^{94} + 15 T^{90} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{100} \) Copy content Toggle raw display
$17$ \( T^{100} - 271443 T^{75} + 28143753124 T^{50} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{100} + 50 T^{95} + 1800 T^{90} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{100} \) Copy content Toggle raw display
$29$ \( T^{100} \) Copy content Toggle raw display
$31$ \( T^{100} \) Copy content Toggle raw display
$37$ \( T^{100} \) Copy content Toggle raw display
$41$ \( T^{100} - 100 T^{91} + 750 T^{89} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{100} + 25 T^{87} - 2600 T^{86} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{100} \) Copy content Toggle raw display
$53$ \( T^{100} \) Copy content Toggle raw display
$59$ \( T^{100} + 5 T^{95} + 25 T^{94} + 15 T^{90} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{100} \) Copy content Toggle raw display
$67$ \( T^{100} - 100 T^{97} + 4750 T^{94} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{100} \) Copy content Toggle raw display
$73$ \( T^{100} + 25 T^{95} - 775 T^{90} - 12925 T^{85} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{100} \) Copy content Toggle raw display
$83$ \( T^{100} + 5 T^{95} + 25 T^{94} + 15 T^{90} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{100} + 25 T^{87} + 650 T^{86} + 5005 T^{85} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{100} + 25 T^{87} + 650 T^{86} + 5005 T^{85} + \cdots + 1 \) Copy content Toggle raw display
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