Properties

Label 36.10.a.b
Level $36$
Weight $10$
Character orbit 36.a
Self dual yes
Analytic conductor $18.541$
Analytic rank $1$
Dimension $1$
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] = [36,10,Mod(1,36)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(36, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 36.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: \(18.5412901019\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4)
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)
 
\(f(q)\) \(=\) \( q + 666 q^{5} - 6328 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 666 q^{5} - 6328 q^{7} + 30420 q^{11} - 32338 q^{13} - 590994 q^{17} + 34676 q^{19} - 1048536 q^{23} - 1509569 q^{25} - 4409406 q^{29} - 7401184 q^{31} - 4214448 q^{35} + 10234502 q^{37} - 18352746 q^{41} - 252340 q^{43} + 49517136 q^{47} - 310023 q^{49} + 66396906 q^{53} + 20259720 q^{55} + 61523748 q^{59} + 35638622 q^{61} - 21537108 q^{65} + 181742372 q^{67} - 90904968 q^{71} - 262978678 q^{73} - 192497760 q^{77} - 116502832 q^{79} + 9563724 q^{83} - 393602004 q^{85} - 611826714 q^{89} + 204634864 q^{91} + 23094216 q^{95} - 259312798 q^{97}+O(q^{100}) \) 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
0
0 0 0 666.000 0 −6328.00 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

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 36.10.a.b 1
3.b odd 2 1 4.10.a.a 1
4.b odd 2 1 144.10.a.j 1
9.c even 3 2 324.10.e.b 2
9.d odd 6 2 324.10.e.e 2
12.b even 2 1 16.10.a.a 1
15.d odd 2 1 100.10.a.a 1
15.e even 4 2 100.10.c.a 2
21.c even 2 1 196.10.a.a 1
21.g even 6 2 196.10.e.b 2
21.h odd 6 2 196.10.e.a 2
24.f even 2 1 64.10.a.i 1
24.h odd 2 1 64.10.a.a 1
48.i odd 4 2 256.10.b.j 2
48.k even 4 2 256.10.b.b 2
60.h even 2 1 400.10.a.k 1
60.l odd 4 2 400.10.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.10.a.a 1 3.b odd 2 1
16.10.a.a 1 12.b even 2 1
36.10.a.b 1 1.a even 1 1 trivial
64.10.a.a 1 24.h odd 2 1
64.10.a.i 1 24.f even 2 1
100.10.a.a 1 15.d odd 2 1
100.10.c.a 2 15.e even 4 2
144.10.a.j 1 4.b odd 2 1
196.10.a.a 1 21.c even 2 1
196.10.e.a 2 21.h odd 6 2
196.10.e.b 2 21.g even 6 2
256.10.b.b 2 48.k even 4 2
256.10.b.j 2 48.i odd 4 2
324.10.e.b 2 9.c even 3 2
324.10.e.e 2 9.d odd 6 2
400.10.a.k 1 60.h even 2 1
400.10.c.a 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 666 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(36))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 666 \) Copy content Toggle raw display
$7$ \( T + 6328 \) Copy content Toggle raw display
$11$ \( T - 30420 \) Copy content Toggle raw display
$13$ \( T + 32338 \) Copy content Toggle raw display
$17$ \( T + 590994 \) Copy content Toggle raw display
$19$ \( T - 34676 \) Copy content Toggle raw display
$23$ \( T + 1048536 \) Copy content Toggle raw display
$29$ \( T + 4409406 \) Copy content Toggle raw display
$31$ \( T + 7401184 \) Copy content Toggle raw display
$37$ \( T - 10234502 \) Copy content Toggle raw display
$41$ \( T + 18352746 \) Copy content Toggle raw display
$43$ \( T + 252340 \) Copy content Toggle raw display
$47$ \( T - 49517136 \) Copy content Toggle raw display
$53$ \( T - 66396906 \) Copy content Toggle raw display
$59$ \( T - 61523748 \) Copy content Toggle raw display
$61$ \( T - 35638622 \) Copy content Toggle raw display
$67$ \( T - 181742372 \) Copy content Toggle raw display
$71$ \( T + 90904968 \) Copy content Toggle raw display
$73$ \( T + 262978678 \) Copy content Toggle raw display
$79$ \( T + 116502832 \) Copy content Toggle raw display
$83$ \( T - 9563724 \) Copy content Toggle raw display
$89$ \( T + 611826714 \) Copy content Toggle raw display
$97$ \( T + 259312798 \) Copy content Toggle raw display
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