Properties

Label 250.2.e.b
Level $250$
Weight $2$
Character orbit 250.e
Analytic conductor $1.996$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 250 = 2 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 250.e (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/250\mathbb{Z}\right)^\times\).

\(n\) \(127\)
\(\chi(n)\) \(\zeta_{20}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.587785 + 0.809017i
0.587785 0.809017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.951057 0.309017i
0.951057 + 0.309017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.587785 + 0.809017i −2.48990 + 0.809017i −0.309017 0.951057i 0 0.809017 2.48990i 3.00000i 0.951057 + 0.309017i 3.11803 2.26538i 0
49.2 0.587785 0.809017i 2.48990 0.809017i −0.309017 0.951057i 0 0.809017 2.48990i 3.00000i −0.951057 0.309017i 3.11803 2.26538i 0
99.1 −0.951057 + 0.309017i 0.224514 + 0.309017i 0.809017 0.587785i 0 −0.309017 0.224514i 3.00000i −0.587785 + 0.809017i 0.881966 2.71441i 0
99.2 0.951057 0.309017i −0.224514 0.309017i 0.809017 0.587785i 0 −0.309017 0.224514i 3.00000i 0.587785 0.809017i 0.881966 2.71441i 0
149.1 −0.951057 0.309017i 0.224514 0.309017i 0.809017 + 0.587785i 0 −0.309017 + 0.224514i 3.00000i −0.587785 0.809017i 0.881966 + 2.71441i 0
149.2 0.951057 + 0.309017i −0.224514 + 0.309017i 0.809017 + 0.587785i 0 −0.309017 + 0.224514i 3.00000i 0.587785 + 0.809017i 0.881966 + 2.71441i 0
199.1 −0.587785 0.809017i −2.48990 0.809017i −0.309017 + 0.951057i 0 0.809017 + 2.48990i 3.00000i 0.951057 0.309017i 3.11803 + 2.26538i 0
199.2 0.587785 + 0.809017i 2.48990 + 0.809017i −0.309017 + 0.951057i 0 0.809017 + 2.48990i 3.00000i −0.951057 + 0.309017i 3.11803 + 2.26538i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 250.2.e.b 8
5.b even 2 1 inner 250.2.e.b 8
5.c odd 4 1 50.2.d.a 4
5.c odd 4 1 250.2.d.a 4
15.e even 4 1 450.2.h.a 4
20.e even 4 1 400.2.u.c 4
25.d even 5 1 inner 250.2.e.b 8
25.d even 5 1 1250.2.b.b 4
25.e even 10 1 inner 250.2.e.b 8
25.e even 10 1 1250.2.b.b 4
25.f odd 20 1 50.2.d.a 4
25.f odd 20 1 250.2.d.a 4
25.f odd 20 1 1250.2.a.a 2
25.f odd 20 1 1250.2.a.d 2
75.l even 20 1 450.2.h.a 4
100.l even 20 1 400.2.u.c 4
100.l even 20 1 10000.2.a.a 2
100.l even 20 1 10000.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.a 4 5.c odd 4 1
50.2.d.a 4 25.f odd 20 1
250.2.d.a 4 5.c odd 4 1
250.2.d.a 4 25.f odd 20 1
250.2.e.b 8 1.a even 1 1 trivial
250.2.e.b 8 5.b even 2 1 inner
250.2.e.b 8 25.d even 5 1 inner
250.2.e.b 8 25.e even 10 1 inner
400.2.u.c 4 20.e even 4 1
400.2.u.c 4 100.l even 20 1
450.2.h.a 4 15.e even 4 1
450.2.h.a 4 75.l even 20 1
1250.2.a.a 2 25.f odd 20 1
1250.2.a.d 2 25.f odd 20 1
1250.2.b.b 4 25.d even 5 1
1250.2.b.b 4 25.e even 10 1
10000.2.a.a 2 100.l even 20 1
10000.2.a.n 2 100.l even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 11 T_{3}^{6} + 46 T_{3}^{4} + 4 T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(250, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$3$ \( 1 + 4 T^{2} + 46 T^{4} - 11 T^{6} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 9 + T^{2} )^{4} \)
$11$ \( ( 1 - 7 T + 19 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$13$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$17$ \( 6561 + 2916 T^{2} + 3726 T^{4} - 99 T^{6} + T^{8} \)
$19$ \( ( 25 - 25 T + 40 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$23$ \( 14641 - 7381 T^{2} + 1401 T^{4} + 19 T^{6} + T^{8} \)
$29$ \( ( 25 - 25 T + 85 T^{2} + 15 T^{3} + T^{4} )^{2} \)
$31$ \( ( 1 - 7 T + 19 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$37$ \( 13845841 - 163724 T^{2} + 3966 T^{4} - 79 T^{6} + T^{8} \)
$41$ \( ( 121 - 77 T + 69 T^{2} - 13 T^{3} + T^{4} )^{2} \)
$43$ \( ( 121 + 42 T^{2} + T^{4} )^{2} \)
$47$ \( 25411681 - 1053569 T^{2} + 16561 T^{4} + 31 T^{6} + T^{8} \)
$53$ \( 65536 - 45056 T^{2} + 11776 T^{4} + 64 T^{6} + T^{8} \)
$59$ \( ( 400 - 200 T + 60 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$61$ \( ( 361 - 247 T + 64 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$67$ \( 12117361 - 518669 T^{2} + 8601 T^{4} + 11 T^{6} + T^{8} \)
$71$ \( ( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$73$ \( 3748096 - 30976 T^{2} + 2656 T^{4} - 76 T^{6} + T^{8} \)
$79$ \( ( 400 + 40 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$83$ \( 13845841 + 293959 T^{2} + 29641 T^{4} - 281 T^{6} + T^{8} \)
$89$ \( ( 400 + 200 T + 60 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$97$ \( 104060401 - 805879 T^{2} + 10606 T^{4} - 124 T^{6} + T^{8} \)
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