Properties

Label 4410.2.a.ca
Level $4410$
Weight $2$
Character orbit 4410.a
Self dual yes
Analytic conductor $35.214$
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] = [4410,2,Mod(1,4410)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4410, 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("4410.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.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: \(35.2140272914\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 630)
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} + (\beta + 2) q^{11} + ( - \beta + 2) q^{13} + q^{16} + ( - 2 \beta + 2) q^{17} + (2 \beta - 1) q^{19} + q^{20} + (\beta + 2) q^{22} + 3 q^{23} + q^{25} + ( - \beta + 2) q^{26} - 2 q^{31} + q^{32} + ( - 2 \beta + 2) q^{34} + ( - 3 \beta + 2) q^{37} + (2 \beta - 1) q^{38} + q^{40} + (\beta - 4) q^{41} + (2 \beta + 6) q^{43} + (\beta + 2) q^{44} + 3 q^{46} + (4 \beta - 1) q^{47} + q^{50} + ( - \beta + 2) q^{52} + ( - 2 \beta + 5) q^{53} + (\beta + 2) q^{55} + (2 \beta + 4) q^{59} + ( - 2 \beta - 6) q^{61} - 2 q^{62} + q^{64} + ( - \beta + 2) q^{65} + ( - 2 \beta + 10) q^{67} + ( - 2 \beta + 2) q^{68} + (2 \beta - 8) q^{71} - 2 q^{73} + ( - 3 \beta + 2) q^{74} + (2 \beta - 1) q^{76} + (2 \beta + 6) q^{79} + q^{80} + (\beta - 4) q^{82} + ( - 2 \beta - 10) q^{83} + ( - 2 \beta + 2) q^{85} + (2 \beta + 6) q^{86} + (\beta + 2) q^{88} + ( - 4 \beta + 4) q^{89} + 3 q^{92} + (4 \beta - 1) q^{94} + (2 \beta - 1) q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} + 4 q^{11} + 4 q^{13} + 2 q^{16} + 4 q^{17} - 2 q^{19} + 2 q^{20} + 4 q^{22} + 6 q^{23} + 2 q^{25} + 4 q^{26} - 4 q^{31} + 2 q^{32} + 4 q^{34} + 4 q^{37} - 2 q^{38} + 2 q^{40} - 8 q^{41} + 12 q^{43} + 4 q^{44} + 6 q^{46} - 2 q^{47} + 2 q^{50} + 4 q^{52} + 10 q^{53} + 4 q^{55} + 8 q^{59} - 12 q^{61} - 4 q^{62} + 2 q^{64} + 4 q^{65} + 20 q^{67} + 4 q^{68} - 16 q^{71} - 4 q^{73} + 4 q^{74} - 2 q^{76} + 12 q^{79} + 2 q^{80} - 8 q^{82} - 20 q^{83} + 4 q^{85} + 12 q^{86} + 4 q^{88} + 8 q^{89} + 6 q^{92} - 2 q^{94} - 2 q^{95} + 8 q^{97}+O(q^{100}) \) 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
−2.64575
2.64575
1.00000 0 1.00000 1.00000 0 0 1.00000 0 1.00000
1.2 1.00000 0 1.00000 1.00000 0 0 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 4410.2.a.ca 2
3.b odd 2 1 4410.2.a.bo 2
7.b odd 2 1 4410.2.a.bv 2
7.d odd 6 2 630.2.k.i 4
21.c even 2 1 4410.2.a.bq 2
21.g even 6 2 630.2.k.j yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.k.i 4 7.d odd 6 2
630.2.k.j yes 4 21.g even 6 2
4410.2.a.bo 2 3.b odd 2 1
4410.2.a.bq 2 21.c even 2 1
4410.2.a.bv 2 7.b odd 2 1
4410.2.a.ca 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4410))\):

\( T_{11}^{2} - 4T_{11} - 3 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} - 3 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} - 24 \) Copy content Toggle raw display
\( T_{19}^{2} + 2T_{19} - 27 \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display
\( T_{31} + 2 \) 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
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 3 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 3 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 24 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 27 \) Copy content Toggle raw display
$23$ \( (T - 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 59 \) Copy content Toggle raw display
$41$ \( T^{2} + 8T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 12T + 8 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 111 \) Copy content Toggle raw display
$53$ \( T^{2} - 10T - 3 \) Copy content Toggle raw display
$59$ \( T^{2} - 8T - 12 \) Copy content Toggle raw display
$61$ \( T^{2} + 12T + 8 \) Copy content Toggle raw display
$67$ \( T^{2} - 20T + 72 \) Copy content Toggle raw display
$71$ \( T^{2} + 16T + 36 \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 12T + 8 \) Copy content Toggle raw display
$83$ \( T^{2} + 20T + 72 \) Copy content Toggle raw display
$89$ \( T^{2} - 8T - 96 \) Copy content Toggle raw display
$97$ \( (T - 4)^{2} \) Copy content Toggle raw display
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