Properties

Label 125.2.b.a
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: 4.0.4400.1
Defining polynomial: \( x^{4} + 7x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) 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
1.54336i
2.14896i
2.14896i
1.54336i
2.49721i 1.54336i −4.23607 0 −3.85410 0.953850i 5.58394i 0.618034 0
124.2 1.32813i 2.14896i 0.236068 0 2.85410 3.47709i 2.96979i −1.61803 0
124.3 1.32813i 2.14896i 0.236068 0 2.85410 3.47709i 2.96979i −1.61803 0
124.4 2.49721i 1.54336i −4.23607 0 −3.85410 0.953850i 5.58394i 0.618034 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.a 4
3.b odd 2 1 1125.2.b.a 4
4.b odd 2 1 2000.2.c.c 4
5.b even 2 1 inner 125.2.b.a 4
5.c odd 4 2 125.2.a.c 4
15.d odd 2 1 1125.2.b.a 4
15.e even 4 2 1125.2.a.k 4
20.d odd 2 1 2000.2.c.c 4
20.e even 4 2 2000.2.a.o 4
25.d even 5 2 625.2.e.b 8
25.d even 5 2 625.2.e.h 8
25.e even 10 2 625.2.e.b 8
25.e even 10 2 625.2.e.h 8
25.f odd 20 4 625.2.d.k 8
25.f odd 20 4 625.2.d.l 8
35.f even 4 2 6125.2.a.o 4
40.i odd 4 2 8000.2.a.bj 4
40.k even 4 2 8000.2.a.bk 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.2.a.c 4 5.c odd 4 2
125.2.b.a 4 1.a even 1 1 trivial
125.2.b.a 4 5.b even 2 1 inner
625.2.d.k 8 25.f odd 20 4
625.2.d.l 8 25.f odd 20 4
625.2.e.b 8 25.d even 5 2
625.2.e.b 8 25.e even 10 2
625.2.e.h 8 25.d even 5 2
625.2.e.h 8 25.e even 10 2
1125.2.a.k 4 15.e even 4 2
1125.2.b.a 4 3.b odd 2 1
1125.2.b.a 4 15.d odd 2 1
2000.2.a.o 4 20.e even 4 2
2000.2.c.c 4 4.b odd 2 1
2000.2.c.c 4 20.d odd 2 1
6125.2.a.o 4 35.f even 4 2
8000.2.a.bj 4 40.i odd 4 2
8000.2.a.bk 4 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 8T_{2}^{2} + 11 \) acting on \(S_{2}^{\mathrm{new}}(125, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 8T^{2} + 11 \) Copy content Toggle raw display
$3$ \( T^{4} + 7T^{2} + 11 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 13T^{2} + 11 \) Copy content Toggle raw display
$11$ \( (T - 2)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 32T^{2} + 176 \) Copy content Toggle raw display
$17$ \( T^{4} + 28T^{2} + 176 \) Copy content Toggle raw display
$19$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 17T^{2} + 11 \) Copy content Toggle raw display
$29$ \( (T^{2} - 5 T - 5)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 68T^{2} + 176 \) Copy content Toggle raw display
$41$ \( (T^{2} + T - 31)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 107T^{2} + 1331 \) Copy content Toggle raw display
$47$ \( T^{4} + 43T^{2} + 11 \) Copy content Toggle raw display
$53$ \( T^{4} + 112T^{2} + 2816 \) Copy content Toggle raw display
$59$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + T - 31)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 28T^{2} + 176 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T - 116)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 352 T^{2} + 21296 \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 77T^{2} + 1331 \) Copy content Toggle raw display
$89$ \( (T^{2} - 15 T + 55)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 128T^{2} + 176 \) Copy content Toggle raw display
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