Properties

Label 9450.2.a.ei
Level $9450$
Weight $2$
Character orbit 9450.a
Self dual yes
Analytic conductor $75.459$
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] = [9450,2,Mod(1,9450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9450, 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("9450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9450.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: \(75.4586299101\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1890)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + q^{7} - q^{8} + \beta q^{11} + ( - 2 \beta - 2) q^{13} - q^{14} + q^{16} - 4 q^{17} + (\beta + 4) q^{19} - \beta q^{22} + ( - 2 \beta + 1) q^{23} + (2 \beta + 2) q^{26} + q^{28} + (2 \beta - 2) q^{29} + (\beta - 2) q^{31} - q^{32} + 4 q^{34} + (3 \beta - 2) q^{37} + ( - \beta - 4) q^{38} + ( - 4 \beta + 3) q^{41} + ( - 2 \beta + 8) q^{43} + \beta q^{44} + (2 \beta - 1) q^{46} + ( - 2 \beta - 8) q^{47} + q^{49} + ( - 2 \beta - 2) q^{52} + 8 \beta q^{53} - q^{56} + ( - 2 \beta + 2) q^{58} + (4 \beta - 2) q^{59} - 8 \beta q^{61} + ( - \beta + 2) q^{62} + q^{64} + (6 \beta + 2) q^{67} - 4 q^{68} + ( - \beta + 6) q^{71} + ( - 4 \beta - 10) q^{73} + ( - 3 \beta + 2) q^{74} + (\beta + 4) q^{76} + \beta q^{77} + (4 \beta + 2) q^{79} + (4 \beta - 3) q^{82} + ( - 2 \beta - 6) q^{83} + (2 \beta - 8) q^{86} - \beta q^{88} + q^{89} + ( - 2 \beta - 2) q^{91} + ( - 2 \beta + 1) q^{92} + (2 \beta + 8) q^{94} + 4 \beta q^{97} - q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 4 q^{13} - 2 q^{14} + 2 q^{16} - 8 q^{17} + 8 q^{19} + 2 q^{23} + 4 q^{26} + 2 q^{28} - 4 q^{29} - 4 q^{31} - 2 q^{32} + 8 q^{34} - 4 q^{37} - 8 q^{38} + 6 q^{41} + 16 q^{43} - 2 q^{46} - 16 q^{47} + 2 q^{49} - 4 q^{52} - 2 q^{56} + 4 q^{58} - 4 q^{59} + 4 q^{62} + 2 q^{64} + 4 q^{67} - 8 q^{68} + 12 q^{71} - 20 q^{73} + 4 q^{74} + 8 q^{76} + 4 q^{79} - 6 q^{82} - 12 q^{83} - 16 q^{86} + 2 q^{89} - 4 q^{91} + 2 q^{92} + 16 q^{94} - 2 q^{98}+O(q^{100}) \) 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
−1.73205
1.73205
−1.00000 0 1.00000 0 0 1.00000 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 1.00000 −1.00000 0 0
\(n\): 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 9450.2.a.ei 2
3.b odd 2 1 9450.2.a.eu 2
5.b even 2 1 9450.2.a.er 2
5.c odd 4 2 1890.2.g.q yes 4
15.d odd 2 1 9450.2.a.eb 2
15.e even 4 2 1890.2.g.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1890.2.g.n 4 15.e even 4 2
1890.2.g.q yes 4 5.c odd 4 2
9450.2.a.eb 2 15.d odd 2 1
9450.2.a.ei 2 1.a even 1 1 trivial
9450.2.a.er 2 5.b even 2 1
9450.2.a.eu 2 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9450))\):

\( T_{11}^{2} - 3 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 8 \) Copy content Toggle raw display
\( T_{17} + 4 \) Copy content Toggle raw display
\( T_{19}^{2} - 8T_{19} + 13 \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 3 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 13 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 11 \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 1 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 23 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 39 \) Copy content Toggle raw display
$43$ \( T^{2} - 16T + 52 \) Copy content Toggle raw display
$47$ \( T^{2} + 16T + 52 \) Copy content Toggle raw display
$53$ \( T^{2} - 192 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$61$ \( T^{2} - 192 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 104 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 33 \) Copy content Toggle raw display
$73$ \( T^{2} + 20T + 52 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$89$ \( (T - 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 48 \) Copy content Toggle raw display
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