Properties

Label 125.2.b.b
Level $125$
Weight $2$
Character orbit 125.b
Analytic conductor $0.998$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 125.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.998130025266\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (2 \beta_{3} + \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} - 1) q^{6} - 3 \beta_{3} q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + ( - 3 \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (2 \beta_{3} + \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} - 1) q^{6} - 3 \beta_{3} q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + ( - 3 \beta_{2} - 2) q^{9} - 3 q^{11} + (3 \beta_{3} + 2 \beta_1) q^{12} - 3 \beta_1 q^{13} + 3 \beta_{2} q^{14} + 3 \beta_{2} q^{16} + (\beta_{3} - 2 \beta_1) q^{17} + ( - 3 \beta_{3} + \beta_1) q^{18} + ( - \beta_{2} + 2) q^{19} + (3 \beta_{2} + 6) q^{21} - 3 \beta_1 q^{22} + ( - 2 \beta_{3} - 2 \beta_1) q^{23} + ( - 3 \beta_{2} - 4) q^{24} + ( - 3 \beta_{2} + 3) q^{26} + ( - \beta_{3} - 2 \beta_1) q^{27} + ( - 3 \beta_{3} - 3 \beta_1) q^{28} + (6 \beta_{2} + 3) q^{29} + ( - 5 \beta_{2} - 3) q^{31} + (5 \beta_{3} + \beta_1) q^{32} + ( - 6 \beta_{3} - 3 \beta_1) q^{33} + ( - 3 \beta_{2} + 2) q^{34} + ( - 2 \beta_{2} - 5) q^{36} + ( - 6 \beta_{3} - 6 \beta_1) q^{37} + ( - \beta_{3} + 3 \beta_1) q^{38} + (3 \beta_{2} + 3) q^{39} - 3 q^{41} + (3 \beta_{3} + 3 \beta_1) q^{42} + 9 \beta_{3} q^{43} + ( - 3 \beta_{2} - 3) q^{44} + 2 q^{46} + (3 \beta_{3} + 7 \beta_1) q^{47} + (3 \beta_{3} + 3 \beta_1) q^{48} - 2 q^{49} + \beta_{2} q^{51} - 3 \beta_{3} q^{52} + ( - 3 \beta_{3} + \beta_1) q^{53} + ( - \beta_{2} + 2) q^{54} + (6 \beta_{2} + 3) q^{56} + (3 \beta_{3} + \beta_1) q^{57} + (6 \beta_{3} - 3 \beta_1) q^{58} + ( - 3 \beta_{2} - 9) q^{59} + (5 \beta_{2} + 2) q^{61} + ( - 5 \beta_{3} + 2 \beta_1) q^{62} + (6 \beta_{3} + 9 \beta_1) q^{63} + (2 \beta_{2} - 1) q^{64} + (3 \beta_{2} + 3) q^{66} + ( - 9 \beta_{3} + 3 \beta_1) q^{67} + ( - \beta_{3} + \beta_1) q^{68} + (4 \beta_{2} + 6) q^{69} - 3 q^{71} + ( - 8 \beta_{3} - \beta_1) q^{72} + (3 \beta_{3} + 3 \beta_1) q^{73} + 6 q^{74} + (2 \beta_{2} + 1) q^{76} + 9 \beta_{3} q^{77} + 3 \beta_{3} q^{78} + ( - 4 \beta_{2} - 7) q^{79} + ( - 6 \beta_{2} - 2) q^{81} - 3 \beta_1 q^{82} + (6 \beta_{3} + 4 \beta_1) q^{83} + (6 \beta_{2} + 9) q^{84} - 9 \beta_{2} q^{86} + (12 \beta_{3} + 9 \beta_1) q^{87} + ( - 3 \beta_{3} - 6 \beta_1) q^{88} + ( - 12 \beta_{2} - 6) q^{89} - 9 \beta_{2} q^{91} + ( - 4 \beta_{3} - 2 \beta_1) q^{92} + ( - 11 \beta_{3} - 8 \beta_1) q^{93} + (4 \beta_{2} - 7) q^{94} + ( - 6 \beta_{2} - 11) q^{96} + (3 \beta_{3} - 3 \beta_1) q^{97} - 2 \beta_1 q^{98} + (9 \beta_{2} + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 12 q^{11} - 6 q^{14} - 6 q^{16} + 10 q^{19} + 18 q^{21} - 10 q^{24} + 18 q^{26} - 2 q^{31} + 14 q^{34} - 16 q^{36} + 6 q^{39} - 12 q^{41} - 6 q^{44} + 8 q^{46} - 8 q^{49} - 2 q^{51} + 10 q^{54} - 30 q^{59} - 2 q^{61} - 8 q^{64} + 6 q^{66} + 16 q^{69} - 12 q^{71} + 24 q^{74} - 20 q^{79} + 4 q^{81} + 24 q^{84} + 18 q^{86} + 18 q^{91} - 36 q^{94} - 32 q^{96} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/125\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
1.61803i
0.618034i
0.618034i
1.61803i
1.61803i 0.381966i −0.618034 0 0.618034 3.00000i 2.23607i 2.85410 0
124.2 0.618034i 2.61803i 1.61803 0 −1.61803 3.00000i 2.23607i −3.85410 0
124.3 0.618034i 2.61803i 1.61803 0 −1.61803 3.00000i 2.23607i −3.85410 0
124.4 1.61803i 0.381966i −0.618034 0 0.618034 3.00000i 2.23607i 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 125.2.b.b 4
3.b odd 2 1 1125.2.b.f 4
4.b odd 2 1 2000.2.c.e 4
5.b even 2 1 inner 125.2.b.b 4
5.c odd 4 1 125.2.a.a 2
5.c odd 4 1 125.2.a.b yes 2
15.d odd 2 1 1125.2.b.f 4
15.e even 4 1 1125.2.a.c 2
15.e even 4 1 1125.2.a.d 2
20.d odd 2 1 2000.2.c.e 4
20.e even 4 1 2000.2.a.a 2
20.e even 4 1 2000.2.a.l 2
25.d even 5 2 625.2.e.d 8
25.d even 5 2 625.2.e.g 8
25.e even 10 2 625.2.e.d 8
25.e even 10 2 625.2.e.g 8
25.f odd 20 2 625.2.d.a 4
25.f odd 20 2 625.2.d.d 4
25.f odd 20 2 625.2.d.g 4
25.f odd 20 2 625.2.d.j 4
35.f even 4 1 6125.2.a.d 2
35.f even 4 1 6125.2.a.g 2
40.i odd 4 1 8000.2.a.d 2
40.i odd 4 1 8000.2.a.v 2
40.k even 4 1 8000.2.a.c 2
40.k even 4 1 8000.2.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.2.a.a 2 5.c odd 4 1
125.2.a.b yes 2 5.c odd 4 1
125.2.b.b 4 1.a even 1 1 trivial
125.2.b.b 4 5.b even 2 1 inner
625.2.d.a 4 25.f odd 20 2
625.2.d.d 4 25.f odd 20 2
625.2.d.g 4 25.f odd 20 2
625.2.d.j 4 25.f odd 20 2
625.2.e.d 8 25.d even 5 2
625.2.e.d 8 25.e even 10 2
625.2.e.g 8 25.d even 5 2
625.2.e.g 8 25.e even 10 2
1125.2.a.c 2 15.e even 4 1
1125.2.a.d 2 15.e even 4 1
1125.2.b.f 4 3.b odd 2 1
1125.2.b.f 4 15.d odd 2 1
2000.2.a.a 2 20.e even 4 1
2000.2.a.l 2 20.e even 4 1
2000.2.c.e 4 4.b odd 2 1
2000.2.c.e 4 20.d odd 2 1
6125.2.a.d 2 35.f even 4 1
6125.2.a.g 2 35.f even 4 1
8000.2.a.c 2 40.k even 4 1
8000.2.a.d 2 40.i odd 4 1
8000.2.a.u 2 40.k even 4 1
8000.2.a.v 2 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(125, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 7T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$11$ \( (T + 3)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 27T^{2} + 81 \) Copy content Toggle raw display
$17$ \( T^{4} + 18T^{2} + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} - 5 T + 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + T - 31)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 108T^{2} + 1296 \) Copy content Toggle raw display
$41$ \( (T + 3)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 123T^{2} + 3721 \) Copy content Toggle raw display
$53$ \( T^{4} + 27T^{2} + 121 \) Copy content Toggle raw display
$59$ \( (T^{2} + 15 T + 45)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + T - 31)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 243T^{2} + 9801 \) Copy content Toggle raw display
$71$ \( (T + 3)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 27T^{2} + 81 \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T + 5)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T^{2} - 180)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 63T^{2} + 81 \) Copy content Toggle raw display
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