Properties

Label 2200.2.a.o
Level $2200$
Weight $2$
Character orbit 2200.a
Self dual yes
Analytic conductor $17.567$
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] = [2200,2,Mod(1,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, 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("2200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2200.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: \(17.5670884447\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 88)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + 2 \beta q^{7} + (\beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + 2 \beta q^{7} + (\beta + 1) q^{9} - q^{11} + ( - 2 \beta + 2) q^{13} - 2 q^{17} - 4 q^{19} + ( - 2 \beta - 8) q^{21} + ( - \beta - 4) q^{23} + (\beta - 4) q^{27} + (2 \beta - 2) q^{29} + (\beta - 4) q^{31} + \beta q^{33} + ( - \beta + 6) q^{37} + 8 q^{39} + (2 \beta + 2) q^{41} + ( - 2 \beta + 4) q^{43} - 8 q^{47} + (4 \beta + 9) q^{49} + 2 \beta q^{51} + (4 \beta - 6) q^{53} + 4 \beta q^{57} - 5 \beta q^{59} + ( - 2 \beta - 2) q^{61} + (4 \beta + 8) q^{63} + (\beta - 8) q^{67} + (5 \beta + 4) q^{69} + (3 \beta - 4) q^{71} + (2 \beta - 2) q^{73} - 2 \beta q^{77} + (2 \beta - 8) q^{79} - 7 q^{81} + ( - 2 \beta - 4) q^{83} - 8 q^{87} + ( - 3 \beta - 2) q^{89} - 16 q^{91} + (3 \beta - 4) q^{93} + (\beta - 14) q^{97} + ( - \beta - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 4 q^{17} - 8 q^{19} - 18 q^{21} - 9 q^{23} - 7 q^{27} - 2 q^{29} - 7 q^{31} + q^{33} + 11 q^{37} + 16 q^{39} + 6 q^{41} + 6 q^{43} - 16 q^{47} + 22 q^{49} + 2 q^{51} - 8 q^{53} + 4 q^{57} - 5 q^{59} - 6 q^{61} + 20 q^{63} - 15 q^{67} + 13 q^{69} - 5 q^{71} - 2 q^{73} - 2 q^{77} - 14 q^{79} - 14 q^{81} - 10 q^{83} - 16 q^{87} - 7 q^{89} - 32 q^{91} - 5 q^{93} - 27 q^{97} - 3 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
0 −2.56155 0 0 0 5.12311 0 3.56155 0
1.2 0 1.56155 0 0 0 −3.12311 0 −0.561553 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 2200.2.a.o 2
4.b odd 2 1 4400.2.a.bp 2
5.b even 2 1 88.2.a.b 2
5.c odd 4 2 2200.2.b.g 4
15.d odd 2 1 792.2.a.h 2
20.d odd 2 1 176.2.a.d 2
20.e even 4 2 4400.2.b.v 4
35.c odd 2 1 4312.2.a.n 2
40.e odd 2 1 704.2.a.p 2
40.f even 2 1 704.2.a.m 2
55.d odd 2 1 968.2.a.j 2
55.h odd 10 4 968.2.i.q 8
55.j even 10 4 968.2.i.r 8
60.h even 2 1 1584.2.a.t 2
80.k odd 4 2 2816.2.c.p 4
80.q even 4 2 2816.2.c.w 4
120.i odd 2 1 6336.2.a.cu 2
120.m even 2 1 6336.2.a.cx 2
140.c even 2 1 8624.2.a.cb 2
165.d even 2 1 8712.2.a.bb 2
220.g even 2 1 1936.2.a.r 2
440.c even 2 1 7744.2.a.cl 2
440.o odd 2 1 7744.2.a.by 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.b 2 5.b even 2 1
176.2.a.d 2 20.d odd 2 1
704.2.a.m 2 40.f even 2 1
704.2.a.p 2 40.e odd 2 1
792.2.a.h 2 15.d odd 2 1
968.2.a.j 2 55.d odd 2 1
968.2.i.q 8 55.h odd 10 4
968.2.i.r 8 55.j even 10 4
1584.2.a.t 2 60.h even 2 1
1936.2.a.r 2 220.g even 2 1
2200.2.a.o 2 1.a even 1 1 trivial
2200.2.b.g 4 5.c odd 4 2
2816.2.c.p 4 80.k odd 4 2
2816.2.c.w 4 80.q even 4 2
4312.2.a.n 2 35.c odd 2 1
4400.2.a.bp 2 4.b odd 2 1
4400.2.b.v 4 20.e even 4 2
6336.2.a.cu 2 120.i odd 2 1
6336.2.a.cx 2 120.m even 2 1
7744.2.a.by 2 440.o odd 2 1
7744.2.a.cl 2 440.c even 2 1
8624.2.a.cb 2 140.c even 2 1
8712.2.a.bb 2 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2200))\):

\( T_{3}^{2} + T_{3} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 16 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$31$ \( T^{2} + 7T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 26 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 52 \) Copy content Toggle raw display
$59$ \( T^{2} + 5T - 100 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$67$ \( T^{2} + 15T + 52 \) Copy content Toggle raw display
$71$ \( T^{2} + 5T - 32 \) Copy content Toggle raw display
$73$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 32 \) Copy content Toggle raw display
$83$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} + 7T - 26 \) Copy content Toggle raw display
$97$ \( T^{2} + 27T + 178 \) Copy content Toggle raw display
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