Properties

Label 36.3.d.c
Level $36$
Weight $3$
Character orbit 36.d
Analytic conductor $0.981$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [36,3,Mod(19,36)]
 
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([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 36.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(0.980928951697\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (2 \beta - 2) q^{4} + 2 q^{5} - 4 \beta q^{7} - 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + (2 \beta - 2) q^{4} + 2 q^{5} - 4 \beta q^{7} - 8 q^{8} + (2 \beta + 2) q^{10} - 4 \beta q^{11} + 2 q^{13} + ( - 4 \beta + 12) q^{14} + ( - 8 \beta - 8) q^{16} - 10 q^{17} + 12 \beta q^{19} + (4 \beta - 4) q^{20} + ( - 4 \beta + 12) q^{22} + 16 \beta q^{23} - 21 q^{25} + (2 \beta + 2) q^{26} + (8 \beta + 24) q^{28} + 26 q^{29} + 4 \beta q^{31} + ( - 16 \beta + 16) q^{32} + ( - 10 \beta - 10) q^{34} - 8 \beta q^{35} + 26 q^{37} + (12 \beta - 36) q^{38} - 16 q^{40} - 58 q^{41} - 28 \beta q^{43} + (8 \beta + 24) q^{44} + (16 \beta - 48) q^{46} - 40 \beta q^{47} + q^{49} + ( - 21 \beta - 21) q^{50} + (4 \beta - 4) q^{52} + 74 q^{53} - 8 \beta q^{55} + 32 \beta q^{56} + (26 \beta + 26) q^{58} + 52 \beta q^{59} + 26 q^{61} + (4 \beta - 12) q^{62} + 64 q^{64} + 4 q^{65} + 4 \beta q^{67} + ( - 20 \beta + 20) q^{68} + ( - 8 \beta + 24) q^{70} - 46 q^{73} + (26 \beta + 26) q^{74} + ( - 24 \beta - 72) q^{76} - 48 q^{77} + 68 \beta q^{79} + ( - 16 \beta - 16) q^{80} + ( - 58 \beta - 58) q^{82} - 28 \beta q^{83} - 20 q^{85} + ( - 28 \beta + 84) q^{86} + 32 \beta q^{88} - 82 q^{89} - 8 \beta q^{91} + ( - 32 \beta - 96) q^{92} + ( - 40 \beta + 120) q^{94} + 24 \beta q^{95} + 2 q^{97} + (\beta + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{4} + 4 q^{5} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{4} + 4 q^{5} - 16 q^{8} + 4 q^{10} + 4 q^{13} + 24 q^{14} - 16 q^{16} - 20 q^{17} - 8 q^{20} + 24 q^{22} - 42 q^{25} + 4 q^{26} + 48 q^{28} + 52 q^{29} + 32 q^{32} - 20 q^{34} + 52 q^{37} - 72 q^{38} - 32 q^{40} - 116 q^{41} + 48 q^{44} - 96 q^{46} + 2 q^{49} - 42 q^{50} - 8 q^{52} + 148 q^{53} + 52 q^{58} + 52 q^{61} - 24 q^{62} + 128 q^{64} + 8 q^{65} + 40 q^{68} + 48 q^{70} - 92 q^{73} + 52 q^{74} - 144 q^{76} - 96 q^{77} - 32 q^{80} - 116 q^{82} - 40 q^{85} + 168 q^{86} - 164 q^{89} - 192 q^{92} + 240 q^{94} + 4 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 1.73205i 0 −2.00000 3.46410i 2.00000 0 6.92820i −8.00000 0 2.00000 3.46410i
19.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.00000 0 6.92820i −8.00000 0 2.00000 + 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.3.d.c 2
3.b odd 2 1 12.3.d.a 2
4.b odd 2 1 inner 36.3.d.c 2
5.b even 2 1 900.3.c.e 2
5.c odd 4 2 900.3.f.c 4
8.b even 2 1 576.3.g.e 2
8.d odd 2 1 576.3.g.e 2
9.c even 3 1 324.3.f.a 2
9.c even 3 1 324.3.f.g 2
9.d odd 6 1 324.3.f.d 2
9.d odd 6 1 324.3.f.j 2
12.b even 2 1 12.3.d.a 2
15.d odd 2 1 300.3.c.b 2
15.e even 4 2 300.3.f.a 4
16.e even 4 2 2304.3.b.l 4
16.f odd 4 2 2304.3.b.l 4
20.d odd 2 1 900.3.c.e 2
20.e even 4 2 900.3.f.c 4
21.c even 2 1 588.3.g.b 2
24.f even 2 1 192.3.g.b 2
24.h odd 2 1 192.3.g.b 2
36.f odd 6 1 324.3.f.a 2
36.f odd 6 1 324.3.f.g 2
36.h even 6 1 324.3.f.d 2
36.h even 6 1 324.3.f.j 2
48.i odd 4 2 768.3.b.c 4
48.k even 4 2 768.3.b.c 4
60.h even 2 1 300.3.c.b 2
60.l odd 4 2 300.3.f.a 4
84.h odd 2 1 588.3.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.d.a 2 3.b odd 2 1
12.3.d.a 2 12.b even 2 1
36.3.d.c 2 1.a even 1 1 trivial
36.3.d.c 2 4.b odd 2 1 inner
192.3.g.b 2 24.f even 2 1
192.3.g.b 2 24.h odd 2 1
300.3.c.b 2 15.d odd 2 1
300.3.c.b 2 60.h even 2 1
300.3.f.a 4 15.e even 4 2
300.3.f.a 4 60.l odd 4 2
324.3.f.a 2 9.c even 3 1
324.3.f.a 2 36.f odd 6 1
324.3.f.d 2 9.d odd 6 1
324.3.f.d 2 36.h even 6 1
324.3.f.g 2 9.c even 3 1
324.3.f.g 2 36.f odd 6 1
324.3.f.j 2 9.d odd 6 1
324.3.f.j 2 36.h even 6 1
576.3.g.e 2 8.b even 2 1
576.3.g.e 2 8.d odd 2 1
588.3.g.b 2 21.c even 2 1
588.3.g.b 2 84.h odd 2 1
768.3.b.c 4 48.i odd 4 2
768.3.b.c 4 48.k even 4 2
900.3.c.e 2 5.b even 2 1
900.3.c.e 2 20.d odd 2 1
900.3.f.c 4 5.c odd 4 2
900.3.f.c 4 20.e even 4 2
2304.3.b.l 4 16.e even 4 2
2304.3.b.l 4 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 2 \) acting on \(S_{3}^{\mathrm{new}}(36, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 48 \) Copy content Toggle raw display
$11$ \( T^{2} + 48 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( (T + 10)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 432 \) Copy content Toggle raw display
$23$ \( T^{2} + 768 \) Copy content Toggle raw display
$29$ \( (T - 26)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 48 \) Copy content Toggle raw display
$37$ \( (T - 26)^{2} \) Copy content Toggle raw display
$41$ \( (T + 58)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2352 \) Copy content Toggle raw display
$47$ \( T^{2} + 4800 \) Copy content Toggle raw display
$53$ \( (T - 74)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 8112 \) Copy content Toggle raw display
$61$ \( (T - 26)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 48 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 46)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 13872 \) Copy content Toggle raw display
$83$ \( T^{2} + 2352 \) Copy content Toggle raw display
$89$ \( (T + 82)^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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