Properties

Label 625.2.b.a
Level $625$
Weight $2$
Character orbit 625.b
Analytic conductor $4.991$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
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.

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

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 \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
624.1
1.61803i
0.618034i
0.618034i
1.61803i
1.61803i 1.00000i −0.618034 0 −1.61803 0.618034i 2.23607i 2.00000 0
624.2 0.618034i 1.00000i 1.61803 0 0.618034 1.61803i 2.23607i 2.00000 0
624.3 0.618034i 1.00000i 1.61803 0 0.618034 1.61803i 2.23607i 2.00000 0
624.4 1.61803i 1.00000i −0.618034 0 −1.61803 0.618034i 2.23607i 2.00000 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 625.2.b.a 4
5.b even 2 1 inner 625.2.b.a 4
5.c odd 4 1 625.2.a.b 2
5.c odd 4 1 625.2.a.c 2
15.e even 4 1 5625.2.a.d 2
15.e even 4 1 5625.2.a.f 2
20.e even 4 1 10000.2.a.c 2
20.e even 4 1 10000.2.a.l 2
25.d even 5 2 125.2.e.a 8
25.d even 5 2 625.2.e.c 8
25.e even 10 2 125.2.e.a 8
25.e even 10 2 625.2.e.c 8
25.f odd 20 2 25.2.d.a 4
25.f odd 20 2 125.2.d.a 4
25.f odd 20 2 625.2.d.b 4
25.f odd 20 2 625.2.d.h 4
75.l even 20 2 225.2.h.b 4
100.l even 20 2 400.2.u.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 25.f odd 20 2
125.2.d.a 4 25.f odd 20 2
125.2.e.a 8 25.d even 5 2
125.2.e.a 8 25.e even 10 2
225.2.h.b 4 75.l even 20 2
400.2.u.b 4 100.l even 20 2
625.2.a.b 2 5.c odd 4 1
625.2.a.c 2 5.c odd 4 1
625.2.b.a 4 1.a even 1 1 trivial
625.2.b.a 4 5.b even 2 1 inner
625.2.d.b 4 25.f odd 20 2
625.2.d.h 4 25.f odd 20 2
625.2.e.c 8 25.d even 5 2
625.2.e.c 8 25.e even 10 2
5625.2.a.d 2 15.e even 4 1
5625.2.a.f 2 15.e even 4 1
10000.2.a.c 2 20.e even 4 1
10000.2.a.l 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\):

\( T_{2}^{4} + 3 T_{2}^{2} + 1 \)
\( T_{3}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} + T^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 1 + 3 T^{2} + T^{4} \)
$11$ \( ( 4 + 6 T + T^{2} )^{2} \)
$13$ \( 81 + 27 T^{2} + T^{4} \)
$17$ \( 16 + 28 T^{2} + T^{4} \)
$19$ \( ( -5 - 5 T + T^{2} )^{2} \)
$23$ \( 961 + 82 T^{2} + T^{4} \)
$29$ \( ( 5 - 5 T + T^{2} )^{2} \)
$31$ \( ( 3 + T )^{4} \)
$37$ \( 1 + 18 T^{2} + T^{4} \)
$41$ \( ( 4 + 6 T + T^{2} )^{2} \)
$43$ \( 81 + 27 T^{2} + T^{4} \)
$47$ \( 1 + 3 T^{2} + T^{4} \)
$53$ \( 361 + 42 T^{2} + T^{4} \)
$59$ \( ( 45 - 15 T + T^{2} )^{2} \)
$61$ \( ( -41 - 4 T + T^{2} )^{2} \)
$67$ \( 1936 + 108 T^{2} + T^{4} \)
$71$ \( ( 29 + 11 T + T^{2} )^{2} \)
$73$ \( ( 81 + T^{2} )^{2} \)
$79$ \( ( -25 - 5 T + T^{2} )^{2} \)
$83$ \( 121 + 42 T^{2} + T^{4} \)
$89$ \( ( -80 + T^{2} )^{2} \)
$97$ \( 121 + 23 T^{2} + T^{4} \)
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