Properties

Label 50.2.d.a
Level $50$
Weight $2$
Character orbit 50.d
Analytic conductor $0.399$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 50.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.399252010106\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} + ( -1 + \zeta_{10} + 2 \zeta_{10}^{3} ) q^{3} -\zeta_{10}^{3} q^{4} + ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{5} + ( \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} -3 q^{7} -\zeta_{10}^{2} q^{8} + ( -3 + \zeta_{10} - 3 \zeta_{10}^{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} + ( -1 + \zeta_{10} + 2 \zeta_{10}^{3} ) q^{3} -\zeta_{10}^{3} q^{4} + ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{5} + ( \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} -3 q^{7} -\zeta_{10}^{2} q^{8} + ( -3 + \zeta_{10} - 3 \zeta_{10}^{2} ) q^{9} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{10} + ( -1 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{11} + ( 1 + \zeta_{10} + \zeta_{10}^{2} ) q^{12} -\zeta_{10} q^{13} + ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{14} + ( 3 + \zeta_{10} + 3 \zeta_{10}^{2} ) q^{15} -\zeta_{10} q^{16} + ( 3 \zeta_{10} + 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{17} + ( -2 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{18} + ( -3 \zeta_{10} + 4 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{19} + ( -2 + \zeta_{10} - 2 \zeta_{10}^{2} ) q^{20} + ( 3 - 3 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{21} + ( 2 - 2 \zeta_{10} + \zeta_{10}^{3} ) q^{22} + ( 5 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{23} + ( 2 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{24} -5 \zeta_{10} q^{25} - q^{26} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{27} + 3 \zeta_{10}^{3} q^{28} + ( 4 - 4 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{29} + ( 4 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{30} + ( 2 \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{31} - q^{32} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{33} + ( 3 + 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{34} + ( -6 + 6 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{35} + ( -2 - \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{36} + ( -7 + 4 \zeta_{10} - 7 \zeta_{10}^{2} ) q^{37} + ( -3 + 4 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{38} + ( 2 - \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{39} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{40} + ( 4 + \zeta_{10} + 4 \zeta_{10}^{2} ) q^{41} + ( -3 \zeta_{10} - 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{42} + ( -3 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{43} + ( -2 \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{44} + ( -8 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{45} + ( 2 - 2 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{46} + ( 8 - 8 \zeta_{10} - \zeta_{10}^{3} ) q^{47} + ( 2 - \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{48} + 2 q^{49} -5 q^{50} + ( -15 - 9 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{51} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{52} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{53} + ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{54} + ( 4 \zeta_{10} - 7 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{55} + 3 \zeta_{10}^{2} q^{56} + ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{57} + ( -4 \zeta_{10} + 7 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{58} + ( -4 + 2 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{59} + ( 4 - \zeta_{10} + \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{60} + ( 4 - 7 \zeta_{10} + 7 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{61} + ( 2 + \zeta_{10} + 2 \zeta_{10}^{2} ) q^{62} + ( 9 - 3 \zeta_{10} + 9 \zeta_{10}^{2} ) q^{63} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64} + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{65} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{66} + ( 2 \zeta_{10} - 9 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{67} + ( 6 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{68} + ( 7 \zeta_{10} + 5 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{69} + ( 6 \zeta_{10} - 3 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{70} + 3 \zeta_{10}^{3} q^{71} + ( -3 + 3 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{72} + ( 2 + 4 \zeta_{10} - 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{73} + ( -3 - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{74} + ( 10 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{75} + ( 1 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{76} + ( 3 - 9 \zeta_{10} + 9 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{77} + ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{78} + ( 2 - 2 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{79} + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{80} + ( -6 \zeta_{10} + 4 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{81} + ( 5 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{82} + ( -4 \zeta_{10} - 7 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{83} + ( -3 - 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{84} + ( 12 + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{85} + ( -3 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{86} + ( 1 - 3 \zeta_{10} + \zeta_{10}^{2} ) q^{87} + ( -2 + 3 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{88} + ( 2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{89} + ( -1 + \zeta_{10} + 8 \zeta_{10}^{3} ) q^{90} + 3 \zeta_{10} q^{91} + ( -2 \zeta_{10} - 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{92} + ( -8 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{93} + ( -8 \zeta_{10} + 7 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{94} + ( -5 + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{95} + ( 1 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{96} + ( -9 + 9 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{97} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{98} + ( -4 + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{2} - q^{3} - q^{4} + 5q^{5} + q^{6} - 12q^{7} + q^{8} - 8q^{9} + O(q^{10}) \) \( 4q + q^{2} - q^{3} - q^{4} + 5q^{5} + q^{6} - 12q^{7} + q^{8} - 8q^{9} - 5q^{10} + 3q^{11} + 4q^{12} - q^{13} - 3q^{14} + 10q^{15} - q^{16} + 3q^{17} - 2q^{18} - 10q^{19} - 5q^{20} + 3q^{21} + 7q^{22} + 9q^{23} + 6q^{24} - 5q^{25} - 4q^{26} + 5q^{27} + 3q^{28} + 15q^{29} + 10q^{30} + 3q^{31} - 4q^{32} + 3q^{33} + 12q^{34} - 15q^{35} - 8q^{36} - 17q^{37} - 5q^{38} + 4q^{39} + 13q^{41} - 3q^{42} - 16q^{43} - 7q^{44} - 10q^{45} + q^{46} + 23q^{47} + 4q^{48} + 8q^{49} - 20q^{50} - 42q^{51} - q^{52} - 16q^{53} + 5q^{54} + 15q^{55} - 3q^{56} - 15q^{58} - 10q^{59} + 10q^{60} - 2q^{61} + 7q^{62} + 24q^{63} - q^{64} - 5q^{65} + 2q^{66} + 13q^{67} + 18q^{68} + 9q^{69} + 15q^{70} + 3q^{71} - 7q^{72} + 14q^{73} + 2q^{74} + 20q^{75} + 10q^{76} - 9q^{77} + q^{78} + 10q^{79} - 5q^{80} - 16q^{81} + 12q^{82} - q^{83} - 12q^{84} + 30q^{85} + q^{86} - 3q^{88} + 10q^{89} + 5q^{90} + 3q^{91} - q^{92} - 22q^{93} - 23q^{94} - 30q^{95} + q^{96} - 22q^{97} + 2q^{98} - 26q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-\zeta_{10}^{3}\)

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
0.809017 + 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 0.587785i
0.809017 0.587785i −0.809017 + 2.48990i 0.309017 0.951057i 0.690983 2.12663i 0.809017 + 2.48990i −3.00000 −0.309017 0.951057i −3.11803 2.26538i −0.690983 2.12663i
21.1 −0.309017 + 0.951057i 0.309017 + 0.224514i −0.809017 0.587785i 1.80902 + 1.31433i −0.309017 + 0.224514i −3.00000 0.809017 0.587785i −0.881966 2.71441i −1.80902 + 1.31433i
31.1 −0.309017 0.951057i 0.309017 0.224514i −0.809017 + 0.587785i 1.80902 1.31433i −0.309017 0.224514i −3.00000 0.809017 + 0.587785i −0.881966 + 2.71441i −1.80902 1.31433i
41.1 0.809017 + 0.587785i −0.809017 2.48990i 0.309017 + 0.951057i 0.690983 + 2.12663i 0.809017 2.48990i −3.00000 −0.309017 + 0.951057i −3.11803 + 2.26538i −0.690983 + 2.12663i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.2.d.a 4
3.b odd 2 1 450.2.h.a 4
4.b odd 2 1 400.2.u.c 4
5.b even 2 1 250.2.d.a 4
5.c odd 4 2 250.2.e.b 8
25.d even 5 1 inner 50.2.d.a 4
25.d even 5 1 1250.2.a.a 2
25.e even 10 1 250.2.d.a 4
25.e even 10 1 1250.2.a.d 2
25.f odd 20 2 250.2.e.b 8
25.f odd 20 2 1250.2.b.b 4
75.j odd 10 1 450.2.h.a 4
100.h odd 10 1 10000.2.a.a 2
100.j odd 10 1 400.2.u.c 4
100.j odd 10 1 10000.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.a 4 1.a even 1 1 trivial
50.2.d.a 4 25.d even 5 1 inner
250.2.d.a 4 5.b even 2 1
250.2.d.a 4 25.e even 10 1
250.2.e.b 8 5.c odd 4 2
250.2.e.b 8 25.f odd 20 2
400.2.u.c 4 4.b odd 2 1
400.2.u.c 4 100.j odd 10 1
450.2.h.a 4 3.b odd 2 1
450.2.h.a 4 75.j odd 10 1
1250.2.a.a 2 25.d even 5 1
1250.2.a.d 2 25.e even 10 1
1250.2.b.b 4 25.f odd 20 2
10000.2.a.a 2 100.h odd 10 1
10000.2.a.n 2 100.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + T_{3}^{3} + 6 T_{3}^{2} - 4 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(50, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$3$ \( 1 - 4 T + 6 T^{2} + T^{3} + T^{4} \)
$5$ \( 25 - 25 T + 15 T^{2} - 5 T^{3} + T^{4} \)
$7$ \( ( 3 + T )^{4} \)
$11$ \( 1 - 7 T + 19 T^{2} - 3 T^{3} + T^{4} \)
$13$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$17$ \( 81 + 108 T + 54 T^{2} - 3 T^{3} + T^{4} \)
$19$ \( 25 + 25 T + 40 T^{2} + 10 T^{3} + T^{4} \)
$23$ \( 121 + 11 T + 31 T^{2} - 9 T^{3} + T^{4} \)
$29$ \( 25 + 25 T + 85 T^{2} - 15 T^{3} + T^{4} \)
$31$ \( 1 - 7 T + 19 T^{2} - 3 T^{3} + T^{4} \)
$37$ \( 3721 + 1098 T + 184 T^{2} + 17 T^{3} + T^{4} \)
$41$ \( 121 - 77 T + 69 T^{2} - 13 T^{3} + T^{4} \)
$43$ \( ( 11 + 8 T + T^{2} )^{2} \)
$47$ \( 5041 - 1207 T + 249 T^{2} - 23 T^{3} + T^{4} \)
$53$ \( 256 - 64 T + 96 T^{2} + 16 T^{3} + T^{4} \)
$59$ \( 400 + 200 T + 60 T^{2} + 10 T^{3} + T^{4} \)
$61$ \( 361 - 247 T + 64 T^{2} + 2 T^{3} + T^{4} \)
$67$ \( 3481 - 177 T + 79 T^{2} - 13 T^{3} + T^{4} \)
$71$ \( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} \)
$73$ \( 1936 - 704 T + 136 T^{2} - 14 T^{3} + T^{4} \)
$79$ \( 400 + 40 T^{2} - 10 T^{3} + T^{4} \)
$83$ \( 3721 - 1159 T + 141 T^{2} + T^{3} + T^{4} \)
$89$ \( 400 - 200 T + 60 T^{2} - 10 T^{3} + T^{4} \)
$97$ \( 10201 + 2323 T + 304 T^{2} + 22 T^{3} + T^{4} \)
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