Properties

Label 4400.2.a.bs
Level $4400$
Weight $2$
Character orbit 4400.a
Self dual yes
Analytic conductor $35.134$
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] = [4400,2,Mod(1,4400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4400.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.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: \(35.1341768894\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( - \beta + 3) q^{7} + \beta q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + ( - \beta + 3) q^{7} + \beta q^{9} + q^{11} - 5 q^{13} - 3 \beta q^{17} + q^{19} + (2 \beta - 3) q^{21} + ( - \beta - 5) q^{23} + ( - 2 \beta + 3) q^{27} + (3 \beta - 6) q^{29} + ( - 4 \beta - 1) q^{31} + \beta q^{33} + (2 \beta - 7) q^{37} - 5 \beta q^{39} + ( - 2 \beta - 1) q^{41} + (4 \beta - 2) q^{43} - 3 q^{47} + ( - 5 \beta + 5) q^{49} + ( - 3 \beta - 9) q^{51} + (\beta - 1) q^{53} + \beta q^{57} + (4 \beta + 5) q^{59} + ( - 3 \beta - 1) q^{61} + (2 \beta - 3) q^{63} - 4 q^{67} + ( - 6 \beta - 3) q^{69} + (2 \beta - 2) q^{71} + (3 \beta + 1) q^{73} + ( - \beta + 3) q^{77} + (3 \beta + 4) q^{79} + ( - 2 \beta - 6) q^{81} + ( - 5 \beta + 8) q^{83} + ( - 3 \beta + 9) q^{87} + ( - \beta + 4) q^{89} + (5 \beta - 15) q^{91} + ( - 5 \beta - 12) q^{93} + ( - \beta - 13) q^{97} + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 5 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 5 q^{7} + q^{9} + 2 q^{11} - 10 q^{13} - 3 q^{17} + 2 q^{19} - 4 q^{21} - 11 q^{23} + 4 q^{27} - 9 q^{29} - 6 q^{31} + q^{33} - 12 q^{37} - 5 q^{39} - 4 q^{41} - 6 q^{47} + 5 q^{49} - 21 q^{51} - q^{53} + q^{57} + 14 q^{59} - 5 q^{61} - 4 q^{63} - 8 q^{67} - 12 q^{69} - 2 q^{71} + 5 q^{73} + 5 q^{77} + 11 q^{79} - 14 q^{81} + 11 q^{83} + 15 q^{87} + 7 q^{89} - 25 q^{91} - 29 q^{93} - 27 q^{97} + q^{99}+O(q^{100}) \) 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
−1.30278
2.30278
0 −1.30278 0 0 0 4.30278 0 −1.30278 0
1.2 0 2.30278 0 0 0 0.697224 0 2.30278 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(11\) \( -1 \)

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 4400.2.a.bs 2
4.b odd 2 1 275.2.a.e 2
5.b even 2 1 4400.2.a.bh 2
5.c odd 4 2 4400.2.b.y 4
12.b even 2 1 2475.2.a.t 2
20.d odd 2 1 275.2.a.f yes 2
20.e even 4 2 275.2.b.c 4
44.c even 2 1 3025.2.a.n 2
60.h even 2 1 2475.2.a.o 2
60.l odd 4 2 2475.2.c.k 4
220.g even 2 1 3025.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.e 2 4.b odd 2 1
275.2.a.f yes 2 20.d odd 2 1
275.2.b.c 4 20.e even 4 2
2475.2.a.o 2 60.h even 2 1
2475.2.a.t 2 12.b even 2 1
2475.2.c.k 4 60.l odd 4 2
3025.2.a.h 2 220.g even 2 1
3025.2.a.n 2 44.c even 2 1
4400.2.a.bh 2 5.b even 2 1
4400.2.a.bs 2 1.a even 1 1 trivial
4400.2.b.y 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4400))\):

\( T_{3}^{2} - T_{3} - 3 \) Copy content Toggle raw display
\( T_{7}^{2} - 5T_{7} + 3 \) Copy content Toggle raw display
\( T_{13} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 5T + 3 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T + 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 27 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 11T + 27 \) Copy content Toggle raw display
$29$ \( T^{2} + 9T - 9 \) Copy content Toggle raw display
$31$ \( T^{2} + 6T - 43 \) Copy content Toggle raw display
$37$ \( T^{2} + 12T + 23 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 52 \) Copy content Toggle raw display
$47$ \( (T + 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$59$ \( T^{2} - 14T - 3 \) Copy content Toggle raw display
$61$ \( T^{2} + 5T - 23 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$73$ \( T^{2} - 5T - 23 \) Copy content Toggle raw display
$79$ \( T^{2} - 11T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} - 11T - 51 \) Copy content Toggle raw display
$89$ \( T^{2} - 7T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} + 27T + 179 \) Copy content Toggle raw display
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