↑

Properties

Label 1875.2.b.b
Level $1875$
Weight $2$
Character orbit 1875.b
Analytic conductor $14.972$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1875,2,Mod(1249,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1875.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

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

Display 100 coefficients 10 coefficients

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

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) 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

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).

\(n\) \(626\) \(1252\)
\(\chi(n)\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
1.61803i
− 0.618034i
0.618034i
− 1.61803i
− 1.00000i 1.00000i 1.00000 0 1.00000 − 4.47214i − 3.00000i −1.00000 0
1249.2 − 1.00000i 1.00000i 1.00000 0 1.00000 4.47214i − 3.00000i −1.00000 0
1249.3 1.00000i − 1.00000i 1.00000 0 1.00000 − 4.47214i 3.00000i −1.00000 0
1249.4 1.00000i − 1.00000i 1.00000 0 1.00000 4.47214i 3.00000i −1.00000 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 1875.2.b.b 4
5.b even 2 1 inner 1875.2.b.b 4
5.c odd 4 1 1875.2.a.a 2
5.c odd 4 1 1875.2.a.d 2
15.e even 4 1 5625.2.a.a 2
15.e even 4 1 5625.2.a.h 2
25.d even 5 2 375.2.i.a 8
25.e even 10 2 375.2.i.a 8
25.f odd 20 2 75.2.g.a ✓ 4
25.f odd 20 2 375.2.g.a 4
75.l even 20 2 225.2.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.a ✓ 4 25.f odd 20 2
225.2.h.a 4 75.l even 20 2
375.2.g.a 4 25.f odd 20 2
375.2.i.a 8 25.d even 5 2
375.2.i.a 8 25.e even 10 2
1875.2.a.a 2 5.c odd 4 1
1875.2.a.d 2 5.c odd 4 1
1875.2.b.b 4 1.a even 1 1 trivial
1875.2.b.b 4 5.b even 2 1 inner
5625.2.a.a 2 15.e even 4 1
5625.2.a.h 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1875, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 43T^{2} + 361 \) Copy content Toggle raw display
$17$ \( T^{4} + 23T^{2} + 121 \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 11 T + 29)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T + 20)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 75T^{2} + 625 \) Copy content Toggle raw display
$41$ \( (T^{2} + 5 T + 5)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 92T^{2} + 1936 \) Copy content Toggle raw display
$47$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$53$ \( T^{4} + 15T^{2} + 25 \) Copy content Toggle raw display
$59$ \( (T - 4)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - T - 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 75T^{2} + 625 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 168T^{2} + 1936 \) Copy content Toggle raw display
$89$ \( (T^{2} + 13 T + 41)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 83T^{2} + 361 \) Copy content Toggle raw display
show more
show less