Properties

Label 500.1.p.a
Level $500$
Weight $1$
Character orbit 500.p
Analytic conductor $0.250$
Analytic rank $0$
Dimension $20$
Projective image $D_{25}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 500.p (of order \(50\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.249532506317\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{50})\)
Defining polynomial: \(x^{20} - x^{15} + x^{10} - x^{5} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\zeta_{50}^{13} q^{2} -\zeta_{50} q^{4} -\zeta_{50}^{9} q^{5} + \zeta_{50}^{14} q^{8} -\zeta_{50}^{7} q^{9} +O(q^{10})\) \( q -\zeta_{50}^{13} q^{2} -\zeta_{50} q^{4} -\zeta_{50}^{9} q^{5} + \zeta_{50}^{14} q^{8} -\zeta_{50}^{7} q^{9} + \zeta_{50}^{22} q^{10} + ( -\zeta_{50}^{3} - \zeta_{50}^{11} ) q^{13} + \zeta_{50}^{2} q^{16} + ( \zeta_{50}^{6} - \zeta_{50}^{17} ) q^{17} + \zeta_{50}^{20} q^{18} + \zeta_{50}^{10} q^{20} + \zeta_{50}^{18} q^{25} + ( \zeta_{50}^{16} + \zeta_{50}^{24} ) q^{26} + ( \zeta_{50}^{4} + \zeta_{50}^{8} ) q^{29} -\zeta_{50}^{15} q^{32} + ( -\zeta_{50}^{5} - \zeta_{50}^{19} ) q^{34} + \zeta_{50}^{8} q^{36} + ( -\zeta_{50}^{5} + \zeta_{50}^{24} ) q^{37} -\zeta_{50}^{23} q^{40} + ( -\zeta_{50}^{21} - \zeta_{50}^{23} ) q^{41} + \zeta_{50}^{16} q^{45} + \zeta_{50}^{10} q^{49} + \zeta_{50}^{6} q^{50} + ( \zeta_{50}^{4} + \zeta_{50}^{12} ) q^{52} + ( \zeta_{50}^{2} - \zeta_{50}^{15} ) q^{53} + ( -\zeta_{50}^{17} - \zeta_{50}^{21} ) q^{58} + ( \zeta_{50}^{12} - \zeta_{50}^{19} ) q^{61} -\zeta_{50}^{3} q^{64} + ( \zeta_{50}^{12} + \zeta_{50}^{20} ) q^{65} + ( -\zeta_{50}^{7} + \zeta_{50}^{18} ) q^{68} -\zeta_{50}^{21} q^{72} + ( -\zeta_{50}^{19} + \zeta_{50}^{22} ) q^{73} + ( \zeta_{50}^{12} + \zeta_{50}^{18} ) q^{74} -\zeta_{50}^{11} q^{80} + \zeta_{50}^{14} q^{81} + ( -\zeta_{50}^{9} - \zeta_{50}^{11} ) q^{82} + ( -\zeta_{50} - \zeta_{50}^{15} ) q^{85} + ( \zeta_{50}^{20} - \zeta_{50}^{21} ) q^{89} + \zeta_{50}^{4} q^{90} + ( \zeta_{50}^{14} - \zeta_{50}^{23} ) q^{97} -\zeta_{50}^{23} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + O(q^{10}) \) \( 20q - 5q^{18} - 5q^{20} - 5q^{32} - 5q^{34} - 5q^{37} - 5q^{49} - 5q^{53} - 5q^{65} - 5q^{85} - 5q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/500\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(377\)
\(\chi(n)\) \(-1\) \(-\zeta_{50}\)

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
−0.0627905 + 0.998027i
0.992115 0.125333i
0.187381 + 0.982287i
−0.0627905 0.998027i
0.637424 0.770513i
−0.876307 0.481754i
0.425779 0.904827i
−0.535827 + 0.844328i
−0.968583 + 0.248690i
0.425779 + 0.904827i
−0.876307 + 0.481754i
0.929776 0.368125i
0.929776 + 0.368125i
0.187381 0.982287i
0.992115 + 0.125333i
−0.968583 0.248690i
−0.535827 0.844328i
−0.728969 + 0.684547i
−0.728969 0.684547i
0.637424 + 0.770513i
0.728969 0.684547i 0 0.0627905 0.998027i 0.535827 0.844328i 0 0 −0.637424 0.770513i −0.425779 + 0.904827i −0.187381 0.982287i
31.1 0.0627905 + 0.998027i 0 −0.992115 + 0.125333i −0.425779 + 0.904827i 0 0 −0.187381 0.982287i −0.637424 + 0.770513i −0.929776 0.368125i
71.1 −0.637424 + 0.770513i 0 −0.187381 0.982287i −0.992115 + 0.125333i 0 0 0.876307 + 0.481754i 0.968583 + 0.248690i 0.535827 0.844328i
91.1 0.728969 + 0.684547i 0 0.0627905 + 0.998027i 0.535827 + 0.844328i 0 0 −0.637424 + 0.770513i −0.425779 0.904827i −0.187381 + 0.982287i
111.1 −0.425779 0.904827i 0 −0.637424 + 0.770513i 0.0627905 + 0.998027i 0 0 0.968583 + 0.248690i −0.992115 0.125333i 0.876307 0.481754i
131.1 0.968583 + 0.248690i 0 0.876307 + 0.481754i −0.187381 0.982287i 0 0 0.728969 + 0.684547i −0.929776 0.368125i 0.0627905 0.998027i
171.1 0.535827 + 0.844328i 0 −0.425779 + 0.904827i 0.728969 0.684547i 0 0 −0.992115 + 0.125333i 0.0627905 + 0.998027i 0.968583 + 0.248690i
191.1 0.876307 0.481754i 0 0.535827 0.844328i −0.929776 0.368125i 0 0 0.0627905 0.998027i 0.728969 0.684547i −0.992115 + 0.125333i
211.1 −0.992115 + 0.125333i 0 0.968583 0.248690i −0.637424 0.770513i 0 0 −0.929776 + 0.368125i −0.187381 0.982287i 0.728969 + 0.684547i
231.1 0.535827 0.844328i 0 −0.425779 0.904827i 0.728969 + 0.684547i 0 0 −0.992115 0.125333i 0.0627905 0.998027i 0.968583 0.248690i
271.1 0.968583 0.248690i 0 0.876307 0.481754i −0.187381 + 0.982287i 0 0 0.728969 0.684547i −0.929776 + 0.368125i 0.0627905 + 0.998027i
291.1 −0.187381 0.982287i 0 −0.929776 + 0.368125i 0.968583 0.248690i 0 0 0.535827 + 0.844328i 0.876307 + 0.481754i −0.425779 0.904827i
311.1 −0.187381 + 0.982287i 0 −0.929776 0.368125i 0.968583 + 0.248690i 0 0 0.535827 0.844328i 0.876307 0.481754i −0.425779 + 0.904827i
331.1 −0.637424 0.770513i 0 −0.187381 + 0.982287i −0.992115 0.125333i 0 0 0.876307 0.481754i 0.968583 0.248690i 0.535827 + 0.844328i
371.1 0.0627905 0.998027i 0 −0.992115 0.125333i −0.425779 0.904827i 0 0 −0.187381 + 0.982287i −0.637424 0.770513i −0.929776 + 0.368125i
391.1 −0.992115 0.125333i 0 0.968583 + 0.248690i −0.637424 + 0.770513i 0 0 −0.929776 0.368125i −0.187381 + 0.982287i 0.728969 0.684547i
411.1 0.876307 + 0.481754i 0 0.535827 + 0.844328i −0.929776 + 0.368125i 0 0 0.0627905 + 0.998027i 0.728969 + 0.684547i −0.992115 0.125333i
431.1 −0.929776 + 0.368125i 0 0.728969 0.684547i 0.876307 0.481754i 0 0 −0.425779 + 0.904827i 0.535827 + 0.844328i −0.637424 + 0.770513i
471.1 −0.929776 0.368125i 0 0.728969 + 0.684547i 0.876307 + 0.481754i 0 0 −0.425779 0.904827i 0.535827 0.844328i −0.637424 0.770513i
491.1 −0.425779 + 0.904827i 0 −0.637424 0.770513i 0.0627905 0.998027i 0 0 0.968583 0.248690i −0.992115 + 0.125333i 0.876307 + 0.481754i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 491.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
125.g even 25 1 inner
500.p odd 50 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.1.p.a 20
4.b odd 2 1 CM 500.1.p.a 20
5.b even 2 1 2500.1.p.a 20
5.c odd 4 2 2500.1.n.a 40
20.d odd 2 1 2500.1.p.a 20
20.e even 4 2 2500.1.n.a 40
125.g even 25 1 inner 500.1.p.a 20
125.h even 50 1 2500.1.p.a 20
125.i odd 100 2 2500.1.n.a 40
500.n odd 50 1 2500.1.p.a 20
500.p odd 50 1 inner 500.1.p.a 20
500.r even 100 2 2500.1.n.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.1.p.a 20 1.a even 1 1 trivial
500.1.p.a 20 4.b odd 2 1 CM
500.1.p.a 20 125.g even 25 1 inner
500.1.p.a 20 500.p odd 50 1 inner
2500.1.n.a 40 5.c odd 4 2
2500.1.n.a 40 20.e even 4 2
2500.1.n.a 40 125.i odd 100 2
2500.1.n.a 40 500.r even 100 2
2500.1.p.a 20 5.b even 2 1
2500.1.p.a 20 20.d odd 2 1
2500.1.p.a 20 125.h even 50 1
2500.1.p.a 20 500.n odd 50 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(500, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
$3$ \( T^{20} \)
$5$ \( 1 + T^{5} + T^{10} + T^{15} + T^{20} \)
$7$ \( T^{20} \)
$11$ \( T^{20} \)
$13$ \( 1 - 15 T + 85 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 360 T^{6} - 640 T^{7} + 800 T^{8} - 100 T^{9} + 379 T^{10} - 505 T^{11} - 5 T^{12} - 75 T^{13} + 150 T^{14} + 2 T^{15} + 5 T^{16} - 20 T^{17} + T^{20} \)
$17$ \( 1 + 10 T + 85 T^{2} + 200 T^{3} - 25 T^{4} - 122 T^{5} + 385 T^{6} - 615 T^{7} + 675 T^{8} - 225 T^{9} - 246 T^{10} + 145 T^{11} + 20 T^{12} - 75 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20} \)
$19$ \( T^{20} \)
$23$ \( T^{20} \)
$29$ \( 1 + 10 T + 110 T^{2} + 575 T^{3} + 1850 T^{4} + 3628 T^{5} + 4160 T^{6} + 2685 T^{7} + 1050 T^{8} + 525 T^{9} + 379 T^{10} + 145 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20} \)
$31$ \( T^{20} \)
$37$ \( 1 - 10 T + 70 T^{2} - 45 T^{3} - 340 T^{4} - 252 T^{5} + 665 T^{6} + 1520 T^{7} + 1655 T^{8} + 1235 T^{9} + 629 T^{10} + 205 T^{11} + 165 T^{12} + 185 T^{13} + 170 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20} \)
$41$ \( 1 + 10 T + 110 T^{2} + 575 T^{3} + 1850 T^{4} + 3628 T^{5} + 4160 T^{6} + 2685 T^{7} + 1050 T^{8} + 525 T^{9} + 379 T^{10} + 145 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20} \)
$43$ \( T^{20} \)
$47$ \( T^{20} \)
$53$ \( 1 - 10 T + 70 T^{2} - 45 T^{3} - 340 T^{4} - 252 T^{5} + 665 T^{6} + 1520 T^{7} + 1655 T^{8} + 1235 T^{9} + 629 T^{10} + 205 T^{11} + 165 T^{12} + 185 T^{13} + 170 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20} \)
$59$ \( T^{20} \)
$61$ \( 1 - 15 T + 85 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 360 T^{6} - 640 T^{7} + 800 T^{8} - 100 T^{9} + 379 T^{10} - 505 T^{11} - 5 T^{12} - 75 T^{13} + 150 T^{14} + 2 T^{15} + 5 T^{16} - 20 T^{17} + T^{20} \)
$67$ \( T^{20} \)
$71$ \( T^{20} \)
$73$ \( 1 + 10 T + 110 T^{2} + 575 T^{3} + 1850 T^{4} + 3628 T^{5} + 4160 T^{6} + 2685 T^{7} + 1050 T^{8} + 525 T^{9} + 379 T^{10} + 145 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20} \)
$79$ \( T^{20} \)
$83$ \( T^{20} \)
$89$ \( 1 - 10 T + 70 T^{2} - 45 T^{3} - 340 T^{4} - 252 T^{5} + 665 T^{6} + 1520 T^{7} + 1655 T^{8} + 1235 T^{9} + 629 T^{10} + 205 T^{11} + 165 T^{12} + 185 T^{13} + 170 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20} \)
$97$ \( 1 + 10 T + 85 T^{2} + 200 T^{3} - 25 T^{4} - 122 T^{5} + 385 T^{6} - 615 T^{7} + 675 T^{8} - 225 T^{9} - 246 T^{10} + 145 T^{11} + 20 T^{12} - 75 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20} \)
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