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 discriminant -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 \(10\), degree \(4\), minimal)

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: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.6250000.1

$q$-expansion

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
25.d even 5 1 inner
100.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.1.j.a 4
3.b odd 2 1 900.1.x.a 4
4.b odd 2 1 CM 100.1.j.a 4
5.b even 2 1 500.1.j.a 4
5.c odd 4 2 500.1.h.a 8
8.b even 2 1 1600.1.bh.a 4
8.d odd 2 1 1600.1.bh.a 4
12.b even 2 1 900.1.x.a 4
20.d odd 2 1 500.1.j.a 4
20.e even 4 2 500.1.h.a 8
25.d even 5 1 inner 100.1.j.a 4
25.d even 5 1 2500.1.b.b 2
25.d even 5 2 2500.1.j.a 4
25.e even 10 1 500.1.j.a 4
25.e even 10 1 2500.1.b.a 2
25.e even 10 2 2500.1.j.b 4
25.f odd 20 2 500.1.h.a 8
25.f odd 20 2 2500.1.d.a 4
25.f odd 20 4 2500.1.h.e 8
75.j odd 10 1 900.1.x.a 4
100.h odd 10 1 500.1.j.a 4
100.h odd 10 1 2500.1.b.a 2
100.h odd 10 2 2500.1.j.b 4
100.j odd 10 1 inner 100.1.j.a 4
100.j odd 10 1 2500.1.b.b 2
100.j odd 10 2 2500.1.j.a 4
100.l even 20 2 500.1.h.a 8
100.l even 20 2 2500.1.d.a 4
100.l even 20 4 2500.1.h.e 8
200.n odd 10 1 1600.1.bh.a 4
200.t even 10 1 1600.1.bh.a 4
300.n even 10 1 900.1.x.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.1.j.a 4 1.a even 1 1 trivial
100.1.j.a 4 4.b odd 2 1 CM
100.1.j.a 4 25.d even 5 1 inner
100.1.j.a 4 100.j odd 10 1 inner
500.1.h.a 8 5.c odd 4 2
500.1.h.a 8 20.e even 4 2
500.1.h.a 8 25.f odd 20 2
500.1.h.a 8 100.l even 20 2
500.1.j.a 4 5.b even 2 1
500.1.j.a 4 20.d odd 2 1
500.1.j.a 4 25.e even 10 1
500.1.j.a 4 100.h odd 10 1
900.1.x.a 4 3.b odd 2 1
900.1.x.a 4 12.b even 2 1
900.1.x.a 4 75.j odd 10 1
900.1.x.a 4 300.n even 10 1
1600.1.bh.a 4 8.b even 2 1
1600.1.bh.a 4 8.d odd 2 1
1600.1.bh.a 4 200.n odd 10 1
1600.1.bh.a 4 200.t even 10 1
2500.1.b.a 2 25.e even 10 1
2500.1.b.a 2 100.h odd 10 1
2500.1.b.b 2 25.d even 5 1
2500.1.b.b 2 100.j odd 10 1
2500.1.d.a 4 25.f odd 20 2
2500.1.d.a 4 100.l even 20 2
2500.1.h.e 8 25.f odd 20 4
2500.1.h.e 8 100.l even 20 4
2500.1.j.a 4 25.d even 5 2
2500.1.j.a 4 100.j odd 10 2
2500.1.j.b 4 25.e even 10 2
2500.1.j.b 4 100.h odd 10 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(100, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$3$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$5$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$7$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$11$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$13$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$17$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$19$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$23$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$29$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$31$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$37$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$41$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$43$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$47$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$53$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$59$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$61$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$67$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$71$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$73$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$79$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$83$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$89$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$97$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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