Properties

Label 1250.2.b.b
Level $1250$
Weight $2$
Character orbit 1250.b
Analytic conductor $9.981$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1250 = 2 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1250.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.98130025266\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1250\mathbb{Z}\right)^\times\).

\(n\) \(627\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
0.618034i
1.61803i
1.61803i
0.618034i
1.00000i 0.381966i −1.00000 0 0.381966 3.00000i 1.00000i 2.85410 0
1249.2 1.00000i 2.61803i −1.00000 0 2.61803 3.00000i 1.00000i −3.85410 0
1249.3 1.00000i 2.61803i −1.00000 0 2.61803 3.00000i 1.00000i −3.85410 0
1249.4 1.00000i 0.381966i −1.00000 0 0.381966 3.00000i 1.00000i 2.85410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 7 T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1250, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 1 + 7 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 9 + T^{2} )^{2} \)
$11$ \( ( -1 - 4 T + T^{2} )^{2} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( 81 + 63 T^{2} + T^{4} \)
$19$ \( ( -5 + 5 T + T^{2} )^{2} \)
$23$ \( 121 + 42 T^{2} + T^{4} \)
$29$ \( ( 5 - 10 T + T^{2} )^{2} \)
$31$ \( ( -1 - 4 T + T^{2} )^{2} \)
$37$ \( 3721 + 123 T^{2} + T^{4} \)
$41$ \( ( -11 + 6 T + T^{2} )^{2} \)
$43$ \( 121 + 42 T^{2} + T^{4} \)
$47$ \( 5041 + 178 T^{2} + T^{4} \)
$53$ \( 256 + 112 T^{2} + T^{4} \)
$59$ \( ( -20 + T^{2} )^{2} \)
$61$ \( ( 19 + 11 T + T^{2} )^{2} \)
$67$ \( 3481 + 138 T^{2} + T^{4} \)
$71$ \( ( 3 + T )^{4} \)
$73$ \( 1936 + 92 T^{2} + T^{4} \)
$79$ \( ( 20 - 10 T + T^{2} )^{2} \)
$83$ \( 3721 + 202 T^{2} + T^{4} \)
$89$ \( ( -20 + T^{2} )^{2} \)
$97$ \( 10201 + 203 T^{2} + T^{4} \)
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