Properties

Label 1890.2.a.bb
Level $1890$
Weight $2$
Character orbit 1890.a
Self dual yes
Analytic conductor $15.092$
Analytic rank $0$
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] = [1890,2,Mod(1,1890)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1890, 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("1890.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1890.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: \(15.0917259820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{73})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} + \beta q^{11} + ( - \beta + 2) q^{13} + q^{14} + q^{16} - \beta q^{17} + 2 q^{19} + q^{20} + \beta q^{22} + \beta q^{23} + q^{25} + ( - \beta + 2) q^{26} + q^{28} + 3 q^{29} + (2 \beta - 1) q^{31} + q^{32} - \beta q^{34} + q^{35} + ( - \beta + 5) q^{37} + 2 q^{38} + q^{40} + ( - \beta + 3) q^{41} + (\beta + 2) q^{43} + \beta q^{44} + \beta q^{46} + ( - 2 \beta - 3) q^{47} + q^{49} + q^{50} + ( - \beta + 2) q^{52} - 6 q^{53} + \beta q^{55} + q^{56} + 3 q^{58} + (\beta - 3) q^{59} + ( - \beta - 1) q^{61} + (2 \beta - 1) q^{62} + q^{64} + ( - \beta + 2) q^{65} + ( - 2 \beta + 2) q^{67} - \beta q^{68} + q^{70} + ( - \beta + 3) q^{71} + ( - \beta + 11) q^{73} + ( - \beta + 5) q^{74} + 2 q^{76} + \beta q^{77} + (\beta + 2) q^{79} + q^{80} + ( - \beta + 3) q^{82} - \beta q^{85} + (\beta + 2) q^{86} + \beta q^{88} + (2 \beta - 6) q^{89} + ( - \beta + 2) q^{91} + \beta q^{92} + ( - 2 \beta - 3) q^{94} + 2 q^{95} + (2 \beta - 10) q^{97} + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8} + 2 q^{10} + q^{11} + 3 q^{13} + 2 q^{14} + 2 q^{16} - q^{17} + 4 q^{19} + 2 q^{20} + q^{22} + q^{23} + 2 q^{25} + 3 q^{26} + 2 q^{28} + 6 q^{29} + 2 q^{32} - q^{34} + 2 q^{35} + 9 q^{37} + 4 q^{38} + 2 q^{40} + 5 q^{41} + 5 q^{43} + q^{44} + q^{46} - 8 q^{47} + 2 q^{49} + 2 q^{50} + 3 q^{52} - 12 q^{53} + q^{55} + 2 q^{56} + 6 q^{58} - 5 q^{59} - 3 q^{61} + 2 q^{64} + 3 q^{65} + 2 q^{67} - q^{68} + 2 q^{70} + 5 q^{71} + 21 q^{73} + 9 q^{74} + 4 q^{76} + q^{77} + 5 q^{79} + 2 q^{80} + 5 q^{82} - q^{85} + 5 q^{86} + q^{88} - 10 q^{89} + 3 q^{91} + q^{92} - 8 q^{94} + 4 q^{95} - 18 q^{97} + 2 q^{98}+O(q^{100}) \) 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
−3.77200
4.77200
1.00000 0 1.00000 1.00000 0 1.00000 1.00000 0 1.00000
1.2 1.00000 0 1.00000 1.00000 0 1.00000 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 1890.2.a.bb yes 2
3.b odd 2 1 1890.2.a.z 2
5.b even 2 1 9450.2.a.ed 2
15.d odd 2 1 9450.2.a.eo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1890.2.a.z 2 3.b odd 2 1
1890.2.a.bb yes 2 1.a even 1 1 trivial
9450.2.a.ed 2 5.b even 2 1
9450.2.a.eo 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1890))\):

\( T_{11}^{2} - T_{11} - 18 \) Copy content Toggle raw display
\( T_{13}^{2} - 3T_{13} - 16 \) Copy content Toggle raw display
\( T_{17}^{2} + T_{17} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T - 18 \) Copy content Toggle raw display
$13$ \( T^{2} - 3T - 16 \) Copy content Toggle raw display
$17$ \( T^{2} + T - 18 \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - T - 18 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 73 \) Copy content Toggle raw display
$37$ \( T^{2} - 9T + 2 \) Copy content Toggle raw display
$41$ \( T^{2} - 5T - 12 \) Copy content Toggle raw display
$43$ \( T^{2} - 5T - 12 \) Copy content Toggle raw display
$47$ \( T^{2} + 8T - 57 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 5T - 12 \) Copy content Toggle raw display
$61$ \( T^{2} + 3T - 16 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T - 72 \) Copy content Toggle raw display
$71$ \( T^{2} - 5T - 12 \) Copy content Toggle raw display
$73$ \( T^{2} - 21T + 92 \) Copy content Toggle raw display
$79$ \( T^{2} - 5T - 12 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 10T - 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 18T + 8 \) Copy content Toggle raw display
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