Properties

Label 100.1.j.a
Level 100
Weight 1
Character orbit 100.j
Analytic conductor 0.050
Analytic rank 0
Dimension 4
Projective image \(D_{5}\)
CM disc. -4
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 100.j (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0499065012633\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.6250000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\zeta_{10}^{3} q^{2} \) \( -\zeta_{10} q^{4} \) \( + \zeta_{10}^{4} q^{5} \) \( + \zeta_{10}^{4} q^{8} \) \( + \zeta_{10}^{2} q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{10}^{3} q^{2} \) \( -\zeta_{10} q^{4} \) \( + \zeta_{10}^{4} q^{5} \) \( + \zeta_{10}^{4} q^{8} \) \( + \zeta_{10}^{2} q^{9} \) \( + \zeta_{10}^{2} q^{10} \) \( + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{13} \) \( + \zeta_{10}^{2} q^{16} \) \( + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{17} \) \(+ q^{18}\) \(+ q^{20}\) \( -\zeta_{10}^{3} q^{25} \) \( + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{26} \) \( + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{29} \) \(+ q^{32}\) \( + ( 1 + \zeta_{10}^{4} ) q^{34} \) \( -\zeta_{10}^{3} q^{36} \) \( + ( 1 + \zeta_{10}^{4} ) q^{37} \) \( -\zeta_{10}^{3} q^{40} \) \( + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{41} \) \( -\zeta_{10} q^{45} \) \(+ q^{49}\) \( -\zeta_{10} q^{50} \) \( + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{52} \) \( + ( 1 + \zeta_{10}^{2} ) q^{53} \) \( + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{58} \) \( + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{61} \) \( -\zeta_{10}^{3} q^{64} \) \( + ( 1 + \zeta_{10}^{2} ) q^{65} \) \( + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{68} \) \( -\zeta_{10} q^{72} \) \( + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{73} \) \( + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{74} \) \( -\zeta_{10} q^{80} \) \( + \zeta_{10}^{4} q^{81} \) \( + ( -\zeta_{10} + \zeta_{10}^{4} ) q^{82} \) \( + ( 1 - \zeta_{10} ) q^{85} \) \( + ( 1 - \zeta_{10} ) q^{89} \) \( + \zeta_{10}^{4} q^{90} \) \( + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{97} \) \( -\zeta_{10}^{3} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{8} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{8} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut -\mathstrut q^{10} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut +\mathstrut 4q^{18} \) \(\mathstrut +\mathstrut 4q^{20} \) \(\mathstrut -\mathstrut q^{25} \) \(\mathstrut -\mathstrut 2q^{26} \) \(\mathstrut -\mathstrut 2q^{29} \) \(\mathstrut +\mathstrut 4q^{32} \) \(\mathstrut +\mathstrut 3q^{34} \) \(\mathstrut -\mathstrut q^{36} \) \(\mathstrut +\mathstrut 3q^{37} \) \(\mathstrut -\mathstrut q^{40} \) \(\mathstrut -\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut q^{45} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut q^{50} \) \(\mathstrut -\mathstrut 2q^{52} \) \(\mathstrut +\mathstrut 3q^{53} \) \(\mathstrut -\mathstrut 2q^{58} \) \(\mathstrut -\mathstrut 2q^{61} \) \(\mathstrut -\mathstrut q^{64} \) \(\mathstrut +\mathstrut 3q^{65} \) \(\mathstrut -\mathstrut 2q^{68} \) \(\mathstrut -\mathstrut q^{72} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut 2q^{74} \) \(\mathstrut -\mathstrut q^{80} \) \(\mathstrut -\mathstrut q^{81} \) \(\mathstrut -\mathstrut 2q^{82} \) \(\mathstrut +\mathstrut 3q^{85} \) \(\mathstrut +\mathstrut 3q^{89} \) \(\mathstrut -\mathstrut q^{90} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut -\mathstrut q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(-\zeta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.309017 + 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 0.951057i
−0.809017 + 0.587785i 0 0.309017 0.951057i 0.309017 + 0.951057i 0 0 0.309017 + 0.951057i −0.809017 0.587785i −0.809017 0.587785i
31.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −0.809017 0.587785i 0 0 −0.809017 0.587785i 0.309017 0.951057i 0.309017 0.951057i
71.1 0.309017 0.951057i 0 −0.809017 0.587785i −0.809017 + 0.587785i 0 0 −0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 + 0.951057i
91.1 −0.809017 0.587785i 0 0.309017 + 0.951057i 0.309017 0.951057i 0 0 0.309017 0.951057i −0.809017 + 0.587785i −0.809017 + 0.587785i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes
25.d Even 1 yes
100.j Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(100, [\chi])\).