Properties

Label 1875.2.a.a
Level $1875$
Weight $2$
Character orbit 1875.a
Self dual yes
Analytic conductor $14.972$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1875,2,Mod(1,1875)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1875.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.a (trivial)

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: yes
Analytic conductor: \(14.9719503790\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.61803
−0.618034
−1.00000 −1.00000 −1.00000 0 1.00000 −4.47214 3.00000 1.00000 0
1.2 −1.00000 −1.00000 −1.00000 0 1.00000 4.47214 3.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1875.2.a.a 2
3.b odd 2 1 5625.2.a.h 2
5.b even 2 1 1875.2.a.d 2
5.c odd 4 2 1875.2.b.b 4
15.d odd 2 1 5625.2.a.a 2
25.d even 5 2 375.2.g.a 4
25.e even 10 2 75.2.g.a 4
25.f odd 20 4 375.2.i.a 8
75.h odd 10 2 225.2.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.a 4 25.e even 10 2
225.2.h.a 4 75.h odd 10 2
375.2.g.a 4 25.d even 5 2
375.2.i.a 8 25.f odd 20 4
1875.2.a.a 2 1.a even 1 1 trivial
1875.2.a.d 2 5.b even 2 1
1875.2.b.b 4 5.c odd 4 2
5625.2.a.a 2 15.d odd 2 1
5625.2.a.h 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 20 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$17$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 20 \) Copy content Toggle raw display
$29$ \( T^{2} - 11T + 29 \) Copy content Toggle raw display
$31$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$37$ \( T^{2} + 5T - 25 \) Copy content Toggle raw display
$41$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$73$ \( T^{2} - 5T - 25 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
$89$ \( T^{2} - 13T + 41 \) Copy content Toggle raw display
$97$ \( T^{2} + 11T + 19 \) Copy content Toggle raw display
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