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 disc. -8
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 2008 = 2^{3} \cdot 251 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2008.bd (of order \(250\) and degree \(100\))

Newform invariants

Self dual: No
Analytic conductor: \(1.00212254537\)
Analytic rank: \(0\)
Dimension: \(100\)
Coefficient field: \(\Q(\zeta_{250})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{125}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{125} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

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.

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 1979.1
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes
251.g Even 1 yes
2008.bd Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(2008, [\chi])\).