Properties

Label 1900.2.a.f
Level $1900$
Weight $2$
Character orbit 1900.a
Self dual yes
Analytic conductor $15.172$
Analytic rank $1$
Dimension $3$
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] = [1900,2,Mod(1,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1900.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.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: \(15.1715763840\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + (\beta_1 - 1) q^{7} + (\beta_{2} - 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} + (\beta_1 - 1) q^{7} + (\beta_{2} - 2 \beta_1 + 1) q^{9} + ( - 2 \beta_{2} - \beta_1) q^{11} + (\beta_{2} - \beta_1) q^{13} + (\beta_1 - 1) q^{17} - q^{19} + (\beta_{2} - 2 \beta_1 + 4) q^{21} + ( - \beta_{2} - 2 \beta_1 - 2) q^{23} + ( - 2 \beta_{2} + \beta_1 - 4) q^{27} + ( - 2 \beta_{2} - \beta_1 - 2) q^{29} + (3 \beta_{2} - \beta_1 + 4) q^{31} + ( - \beta_{2} - \beta_1 - 3) q^{33} + (4 \beta_{2} - \beta_1 + 2) q^{37} + ( - \beta_{2} + 2 \beta_1 - 3) q^{39} + (\beta_{2} - 2 \beta_1 + 5) q^{41} + ( - 2 \beta_1 - 3) q^{43} + ( - 3 \beta_{2} - \beta_1 - 6) q^{47} + (\beta_{2} - 2 \beta_1 - 3) q^{49} + (\beta_{2} - 2 \beta_1 + 4) q^{51} + ( - 2 \beta_1 - 3) q^{53} + ( - \beta_1 + 1) q^{57} + (2 \beta_{2} + 6 \beta_1 - 4) q^{59} + ( - 3 \beta_{2} + \beta_1 + 2) q^{61} + ( - 2 \beta_{2} + 4 \beta_1 - 7) q^{63} + (4 \beta_{2} + 3 \beta_1 - 2) q^{67} + ( - 2 \beta_{2} - \beta_1 - 4) q^{69} + ( - 2 \beta_{2} + 4 \beta_1 - 3) q^{71} + (3 \beta_{2} - 3) q^{73} + ( - \beta_{2} - \beta_1 - 3) q^{77} + ( - 4 \beta_{2} + 3 \beta_1 - 7) q^{79} + ( - 2 \beta_{2} - \beta_1 + 4) q^{81} + (5 \beta_{2} - 6 \beta_1 + 1) q^{83} + ( - \beta_{2} - 3 \beta_1 - 1) q^{87} + (3 \beta_{2} - 6 \beta_1 + 2) q^{89} + ( - \beta_{2} + 2 \beta_1 - 3) q^{91} + ( - \beta_{2} + 8 \beta_1 - 7) q^{93} + ( - 3 \beta_{2} + 3 \beta_1) q^{97} + 5 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} - 2 q^{7} + q^{9} - q^{11} - q^{13} - 2 q^{17} - 3 q^{19} + 10 q^{21} - 8 q^{23} - 11 q^{27} - 7 q^{29} + 11 q^{31} - 10 q^{33} + 5 q^{37} - 7 q^{39} + 13 q^{41} - 11 q^{43} - 19 q^{47} - 11 q^{49} + 10 q^{51} - 11 q^{53} + 2 q^{57} - 6 q^{59} + 7 q^{61} - 17 q^{63} - 3 q^{67} - 13 q^{69} - 5 q^{71} - 9 q^{73} - 10 q^{77} - 18 q^{79} + 11 q^{81} - 3 q^{83} - 6 q^{87} - 7 q^{91} - 13 q^{93} + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
−1.91223
0.713538
2.19869
0 −2.91223 0 0 0 −2.91223 0 5.48108 0
1.2 0 −0.286462 0 0 0 −0.286462 0 −2.91794 0
1.3 0 1.19869 0 0 0 1.19869 0 −1.56314 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(19\) \(1\)

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 1900.2.a.f 3
4.b odd 2 1 7600.2.a.by 3
5.b even 2 1 1900.2.a.h yes 3
5.c odd 4 2 1900.2.c.g 6
20.d odd 2 1 7600.2.a.bj 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.a.f 3 1.a even 1 1 trivial
1900.2.a.h yes 3 5.b even 2 1
1900.2.c.g 6 5.c odd 4 2
7600.2.a.bj 3 20.d odd 2 1
7600.2.a.by 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1900))\):

\( T_{3}^{3} + 2T_{3}^{2} - 3T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 3T_{7} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$11$ \( T^{3} + T^{2} + \cdots + 15 \) Copy content Toggle raw display
$13$ \( T^{3} + T^{2} - 8T - 3 \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 8 T^{2} + \cdots - 9 \) Copy content Toggle raw display
$29$ \( T^{3} + 7 T^{2} + \cdots - 25 \) Copy content Toggle raw display
$31$ \( T^{3} - 11 T^{2} + \cdots + 241 \) Copy content Toggle raw display
$37$ \( T^{3} - 5 T^{2} + \cdots + 405 \) Copy content Toggle raw display
$41$ \( T^{3} - 13 T^{2} + \cdots - 25 \) Copy content Toggle raw display
$43$ \( T^{3} + 11 T^{2} + \cdots - 27 \) Copy content Toggle raw display
$47$ \( T^{3} + 19 T^{2} + \cdots - 63 \) Copy content Toggle raw display
$53$ \( T^{3} + 11 T^{2} + \cdots - 27 \) Copy content Toggle raw display
$59$ \( T^{3} + 6 T^{2} + \cdots - 856 \) Copy content Toggle raw display
$61$ \( T^{3} - 7 T^{2} + \cdots - 25 \) Copy content Toggle raw display
$67$ \( T^{3} + 3 T^{2} + \cdots - 599 \) Copy content Toggle raw display
$71$ \( T^{3} + 5 T^{2} + \cdots + 123 \) Copy content Toggle raw display
$73$ \( T^{3} + 9 T^{2} + \cdots - 27 \) Copy content Toggle raw display
$79$ \( T^{3} + 18 T^{2} + \cdots - 607 \) Copy content Toggle raw display
$83$ \( T^{3} + 3 T^{2} + \cdots - 749 \) Copy content Toggle raw display
$89$ \( T^{3} - 183T - 857 \) Copy content Toggle raw display
$97$ \( T^{3} - 3 T^{2} + \cdots + 81 \) Copy content Toggle raw display
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