Properties

Label 9200.2.a.bo
Level $9200$
Weight $2$
Character orbit 9200.a
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9200,2,Mod(1,9200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9200, 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("9200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.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: \(73.4623698596\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 230)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + (3 \beta - 3) q^{7} + (\beta - 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + (3 \beta - 3) q^{7} + (\beta - 2) q^{9} + (\beta + 4) q^{11} + (\beta + 1) q^{13} + (3 \beta - 4) q^{17} + ( - 3 \beta + 5) q^{19} - 3 q^{21} - q^{23} + (4 \beta - 1) q^{27} - 6 \beta q^{29} + ( - 3 \beta + 7) q^{31} + ( - 5 \beta - 1) q^{33} - 6 \beta q^{37} + ( - 2 \beta - 1) q^{39} + ( - \beta - 4) q^{41} + ( - 2 \beta - 8) q^{43} + ( - 6 \beta + 8) q^{47} + ( - 9 \beta + 11) q^{49} + (\beta - 3) q^{51} - 2 q^{53} + ( - 2 \beta + 3) q^{57} + 6 q^{59} + (3 \beta - 2) q^{61} + ( - 6 \beta + 9) q^{63} + ( - 2 \beta - 2) q^{67} + \beta q^{69} + (\beta - 2) q^{71} + ( - 4 \beta - 10) q^{73} + (12 \beta - 9) q^{77} + (6 \beta - 2) q^{79} + ( - 6 \beta + 2) q^{81} + ( - 6 \beta + 2) q^{83} + (6 \beta + 6) q^{87} + (6 \beta - 6) q^{89} + 3 \beta q^{91} + ( - 4 \beta + 3) q^{93} + (13 \beta - 8) q^{97} + (3 \beta - 7) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 3 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 3 q^{7} - 3 q^{9} + 9 q^{11} + 3 q^{13} - 5 q^{17} + 7 q^{19} - 6 q^{21} - 2 q^{23} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 7 q^{33} - 6 q^{37} - 4 q^{39} - 9 q^{41} - 18 q^{43} + 10 q^{47} + 13 q^{49} - 5 q^{51} - 4 q^{53} + 4 q^{57} + 12 q^{59} - q^{61} + 12 q^{63} - 6 q^{67} + q^{69} - 3 q^{71} - 24 q^{73} - 6 q^{77} + 2 q^{79} - 2 q^{81} - 2 q^{83} + 18 q^{87} - 6 q^{89} + 3 q^{91} + 2 q^{93} - 3 q^{97} - 11 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
0 −1.61803 0 0 0 1.85410 0 −0.381966 0
1.2 0 0.618034 0 0 0 −4.85410 0 −2.61803 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

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 9200.2.a.bo 2
4.b odd 2 1 1150.2.a.l 2
5.b even 2 1 9200.2.a.by 2
5.c odd 4 2 1840.2.e.c 4
20.d odd 2 1 1150.2.a.n 2
20.e even 4 2 230.2.b.a 4
60.l odd 4 2 2070.2.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.b.a 4 20.e even 4 2
1150.2.a.l 2 4.b odd 2 1
1150.2.a.n 2 20.d odd 2 1
1840.2.e.c 4 5.c odd 4 2
2070.2.d.c 4 60.l odd 4 2
9200.2.a.bo 2 1.a even 1 1 trivial
9200.2.a.by 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9200))\):

\( T_{3}^{2} + T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 3T_{7} - 9 \) Copy content Toggle raw display
\( T_{11}^{2} - 9T_{11} + 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$11$ \( T^{2} - 9T + 19 \) Copy content Toggle raw display
$13$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 5T - 5 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 1 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 36 \) Copy content Toggle raw display
$31$ \( T^{2} - 11T + 19 \) Copy content Toggle raw display
$37$ \( T^{2} + 6T - 36 \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$43$ \( T^{2} + 18T + 76 \) Copy content Toggle raw display
$47$ \( T^{2} - 10T - 20 \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$73$ \( T^{2} + 24T + 124 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$83$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 3T - 209 \) Copy content Toggle raw display
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