Properties

Label 2008.1.w.a
Level 2008
Weight 1
Character orbit 2008.w
Analytic conductor 1.002
Analytic rank 0
Dimension 20
Projective image \(D_{25}\)
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.w (of order \(50\) and degree \(20\))

Newform invariants

Self dual: No
Analytic conductor: \(1.00212254537\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{50})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{25}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{25} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\zeta_{50}^{5} q^{2} \) \( + ( \zeta_{50}^{2} + \zeta_{50}^{4} ) q^{3} \) \( + \zeta_{50}^{10} q^{4} \) \( + ( -\zeta_{50}^{7} - \zeta_{50}^{9} ) q^{6} \) \( -\zeta_{50}^{15} q^{8} \) \( + ( \zeta_{50}^{4} + \zeta_{50}^{6} + \zeta_{50}^{8} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{50}^{5} q^{2} \) \( + ( \zeta_{50}^{2} + \zeta_{50}^{4} ) q^{3} \) \( + \zeta_{50}^{10} q^{4} \) \( + ( -\zeta_{50}^{7} - \zeta_{50}^{9} ) q^{6} \) \( -\zeta_{50}^{15} q^{8} \) \( + ( \zeta_{50}^{4} + \zeta_{50}^{6} + \zeta_{50}^{8} ) q^{9} \) \( + ( \zeta_{50}^{6} + \zeta_{50}^{20} ) q^{11} \) \( + ( \zeta_{50}^{12} + \zeta_{50}^{14} ) q^{12} \) \( + \zeta_{50}^{20} q^{16} \) \( -2 \zeta_{50}^{17} q^{17} \) \( + ( -\zeta_{50}^{9} - \zeta_{50}^{11} - \zeta_{50}^{13} ) q^{18} \) \( + ( -\zeta_{50}^{9} + \zeta_{50}^{24} ) q^{19} \) \( + ( 1 - \zeta_{50}^{11} ) q^{22} \) \( + ( -\zeta_{50}^{17} - \zeta_{50}^{19} ) q^{24} \) \( -\zeta_{50}^{5} q^{25} \) \( + ( \zeta_{50}^{6} + \zeta_{50}^{8} + \zeta_{50}^{10} + \zeta_{50}^{12} ) q^{27} \) \(+ q^{32}\) \( + ( \zeta_{50}^{8} + \zeta_{50}^{10} + \zeta_{50}^{22} + \zeta_{50}^{24} ) q^{33} \) \( + 2 \zeta_{50}^{22} q^{34} \) \( + ( \zeta_{50}^{14} + \zeta_{50}^{16} + \zeta_{50}^{18} ) q^{36} \) \( + ( \zeta_{50}^{4} + \zeta_{50}^{14} ) q^{38} \) \( + ( \zeta_{50}^{16} - \zeta_{50}^{23} ) q^{41} \) \( + ( -\zeta_{50}^{9} + \zeta_{50}^{18} ) q^{43} \) \( + ( -\zeta_{50}^{5} + \zeta_{50}^{16} ) q^{44} \) \( + ( \zeta_{50}^{22} + \zeta_{50}^{24} ) q^{48} \) \( + \zeta_{50}^{18} q^{49} \) \( + \zeta_{50}^{10} q^{50} \) \( + ( -2 \zeta_{50}^{19} - 2 \zeta_{50}^{21} ) q^{51} \) \( + ( -\zeta_{50}^{11} - \zeta_{50}^{13} - \zeta_{50}^{15} - \zeta_{50}^{17} ) q^{54} \) \( + ( -\zeta_{50} - \zeta_{50}^{3} - \zeta_{50}^{11} - \zeta_{50}^{13} ) q^{57} \) \( + ( \zeta_{50}^{20} - \zeta_{50}^{21} ) q^{59} \) \( -\zeta_{50}^{5} q^{64} \) \( + ( \zeta_{50}^{2} + \zeta_{50}^{4} - \zeta_{50}^{13} - \zeta_{50}^{15} ) q^{66} \) \( + ( -\zeta_{50}^{3} - \zeta_{50}^{11} ) q^{67} \) \( + 2 \zeta_{50}^{2} q^{68} \) \( + ( -\zeta_{50}^{19} - \zeta_{50}^{21} - \zeta_{50}^{23} ) q^{72} \) \( + ( \zeta_{50}^{2} - \zeta_{50}^{17} ) q^{73} \) \( + ( -\zeta_{50}^{7} - \zeta_{50}^{9} ) q^{75} \) \( + ( -\zeta_{50}^{9} - \zeta_{50}^{19} ) q^{76} \) \( + ( \zeta_{50}^{8} + \zeta_{50}^{10} + \zeta_{50}^{12} + \zeta_{50}^{14} + \zeta_{50}^{16} ) q^{81} \) \( + ( -\zeta_{50}^{3} - \zeta_{50}^{21} ) q^{82} \) \( + ( -\zeta_{50}^{5} + \zeta_{50}^{24} ) q^{83} \) \( + ( \zeta_{50}^{14} - \zeta_{50}^{23} ) q^{86} \) \( + ( \zeta_{50}^{10} - \zeta_{50}^{21} ) q^{88} \) \( + ( -\zeta_{50}^{3} + \zeta_{50}^{6} ) q^{89} \) \( + ( \zeta_{50}^{2} + \zeta_{50}^{4} ) q^{96} \) \( + ( -\zeta_{50}^{11} + \zeta_{50}^{18} ) q^{97} \) \( -\zeta_{50}^{23} q^{98} \) \( + ( -\zeta_{50} - \zeta_{50}^{3} + \zeta_{50}^{10} + \zeta_{50}^{12} + \zeta_{50}^{14} + \zeta_{50}^{24} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(20q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut -\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 5q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(20q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut -\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 5q^{8} \) \(\mathstrut -\mathstrut 5q^{11} \) \(\mathstrut -\mathstrut 5q^{16} \) \(\mathstrut +\mathstrut 20q^{22} \) \(\mathstrut -\mathstrut 5q^{25} \) \(\mathstrut -\mathstrut 5q^{27} \) \(\mathstrut +\mathstrut 20q^{32} \) \(\mathstrut -\mathstrut 5q^{33} \) \(\mathstrut -\mathstrut 5q^{44} \) \(\mathstrut -\mathstrut 5q^{50} \) \(\mathstrut -\mathstrut 5q^{54} \) \(\mathstrut -\mathstrut 5q^{59} \) \(\mathstrut -\mathstrut 5q^{64} \) \(\mathstrut -\mathstrut 5q^{66} \) \(\mathstrut -\mathstrut 5q^{81} \) \(\mathstrut -\mathstrut 5q^{83} \) \(\mathstrut -\mathstrut 5q^{88} \) \(\mathstrut -\mathstrut 5q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(257\) \(503\) \(1005\)
\(\chi(n)\) \(\zeta_{50}^{16}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
−0.968583 + 0.248690i
−0.876307 + 0.481754i
−0.728969 + 0.684547i
−0.0627905 0.998027i
0.992115 + 0.125333i
0.637424 0.770513i
−0.968583 0.248690i
−0.876307 0.481754i
0.187381 + 0.982287i
−0.0627905 + 0.998027i
−0.535827 0.844328i
0.425779 0.904827i
0.425779 + 0.904827i
−0.728969 0.684547i
0.637424 + 0.770513i
0.929776 0.368125i
0.187381 0.982287i
0.929776 + 0.368125i
−0.535827 + 0.844328i
0.992115 0.125333i
0.309017 0.951057i 1.41213 1.32608i −0.809017 0.587785i 0 −0.824805 1.75280i 0 −0.809017 + 0.587785i 0.172838 2.74718i 0
91.1 −0.809017 0.587785i 0.110048 1.74915i 0.309017 + 0.951057i 0 −1.11716 + 1.35041i 0 0.309017 0.951057i −2.05532 0.259647i 0
123.1 −0.809017 + 0.587785i −0.929324 1.12336i 0.309017 0.951057i 0 1.41213 + 0.362574i 0 0.309017 + 0.951057i −0.210913 + 1.10564i 0
211.1 0.309017 + 0.951057i −0.0235315 0.123357i −0.809017 + 0.587785i 0 0.110048 0.0604991i 0 −0.809017 0.587785i 0.915113 0.362319i 0
243.1 −0.809017 0.587785i 1.84489 + 0.730444i 0.309017 + 0.951057i 0 −1.06320 1.67534i 0 0.309017 0.951057i 2.14110 + 2.01063i 0
267.1 0.309017 0.951057i −1.11716 0.614163i −0.809017 0.587785i 0 −0.929324 + 0.872693i 0 −0.809017 + 0.587785i 0.335019 + 0.527905i 0
315.1 0.309017 + 0.951057i 1.41213 + 1.32608i −0.809017 + 0.587785i 0 −0.824805 + 1.75280i 0 −0.809017 0.587785i 0.172838 + 2.74718i 0
331.1 −0.809017 + 0.587785i 0.110048 + 1.74915i 0.309017 0.951057i 0 −1.11716 1.35041i 0 0.309017 + 0.951057i −2.05532 + 0.259647i 0
507.1 −0.809017 0.587785i −0.200808 0.316423i 0.309017 + 0.951057i 0 −0.0235315 + 0.374023i 0 0.309017 0.951057i 0.365980 0.777747i 0
571.1 0.309017 0.951057i −0.0235315 + 0.123357i −0.809017 0.587785i 0 0.110048 + 0.0604991i 0 −0.809017 + 0.587785i 0.915113 + 0.362319i 0
627.1 0.309017 0.951057i −1.06320 + 0.134314i −0.809017 0.587785i 0 −0.200808 + 1.05267i 0 −0.809017 + 0.587785i 0.143778 0.0369159i 0
1067.1 −0.809017 0.587785i −0.824805 + 0.211774i 0.309017 + 0.951057i 0 0.791759 + 0.313480i 0 0.309017 0.951057i −0.240851 + 0.132409i 0
1259.1 −0.809017 + 0.587785i −0.824805 0.211774i 0.309017 0.951057i 0 0.791759 0.313480i 0 0.309017 + 0.951057i −0.240851 0.132409i 0
1355.1 −0.809017 0.587785i −0.929324 + 1.12336i 0.309017 + 0.951057i 0 1.41213 0.362574i 0 0.309017 0.951057i −0.210913 1.10564i 0
1459.1 0.309017 + 0.951057i −1.11716 + 0.614163i −0.809017 + 0.587785i 0 −0.929324 0.872693i 0 −0.809017 0.587785i 0.335019 0.527905i 0
1531.1 0.309017 + 0.951057i 0.791759 1.68257i −0.809017 + 0.587785i 0 1.84489 + 0.233064i 0 −0.809017 0.587785i −1.56675 1.89387i 0
1707.1 −0.809017 + 0.587785i −0.200808 + 0.316423i 0.309017 0.951057i 0 −0.0235315 0.374023i 0 0.309017 + 0.951057i 0.365980 + 0.777747i 0
1747.1 0.309017 0.951057i 0.791759 + 1.68257i −0.809017 0.587785i 0 1.84489 0.233064i 0 −0.809017 + 0.587785i −1.56675 + 1.89387i 0
1755.1 0.309017 + 0.951057i −1.06320 0.134314i −0.809017 + 0.587785i 0 −0.200808 1.05267i 0 −0.809017 0.587785i 0.143778 + 0.0369159i 0
1851.1 −0.809017 + 0.587785i 1.84489 0.730444i 0.309017 0.951057i 0 −1.06320 + 1.67534i 0 0.309017 + 0.951057i 2.14110 2.01063i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1851.1
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.e Even 1 yes
2008.w Odd 1 yes

Hecke kernels

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