Properties

Label 180.3.f.g
Level $180$
Weight $3$
Character orbit 180.f
Analytic conductor $4.905$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [180,3,Mod(19,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.f (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: \(4.90464475849\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 22x^{2} + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + (2 \beta_{2} - 2) q^{4} + ( - \beta_{2} - \beta_1) q^{5} - 2 \beta_1 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + (2 \beta_{2} - 2) q^{4} + ( - \beta_{2} - \beta_1) q^{5} - 2 \beta_1 q^{7} - 8 q^{8} + (\beta_{3} - \beta_{2} - \beta_1 + 3) q^{10} + 2 \beta_{3} q^{11} - 2 \beta_{3} q^{13} + (2 \beta_{3} - 2 \beta_1) q^{14} + ( - 8 \beta_{2} - 8) q^{16} + 6 \beta_{2} q^{17} + 12 \beta_{2} q^{19} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 6) q^{20} + (2 \beta_{3} + 6 \beta_1) q^{22} - 28 q^{23} + ( - 2 \beta_{3} + 19) q^{25} + ( - 2 \beta_{3} - 6 \beta_1) q^{26} + (4 \beta_{3} + 4 \beta_1) q^{28} - 2 \beta_1 q^{29} - 20 \beta_{2} q^{31} + ( - 16 \beta_{2} + 16) q^{32} + (6 \beta_{2} - 18) q^{34} + ( - 2 \beta_{3} + 44) q^{35} - 6 \beta_{3} q^{37} + (12 \beta_{2} - 36) q^{38} + (8 \beta_{2} + 8 \beta_1) q^{40} + 4 \beta_1 q^{41} - 8 \beta_1 q^{43} + ( - 4 \beta_{3} + 12 \beta_1) q^{44} + ( - 28 \beta_{2} - 28) q^{46} - 4 q^{47} + 39 q^{49} + ( - 2 \beta_{3} + 19 \beta_{2} - 6 \beta_1 + 19) q^{50} + (4 \beta_{3} - 12 \beta_1) q^{52} - 18 \beta_{2} q^{53} + (44 \beta_{2} - 6 \beta_1) q^{55} + 16 \beta_1 q^{56} + (2 \beta_{3} - 2 \beta_1) q^{58} - 2 \beta_{3} q^{59} - 58 q^{61} + ( - 20 \beta_{2} + 60) q^{62} + 64 q^{64} + ( - 44 \beta_{2} + 6 \beta_1) q^{65} + 4 \beta_1 q^{67} + ( - 12 \beta_{2} - 36) q^{68} + ( - 2 \beta_{3} + 44 \beta_{2} - 6 \beta_1 + 44) q^{70} - 12 \beta_{3} q^{71} + ( - 6 \beta_{3} - 18 \beta_1) q^{74} + ( - 24 \beta_{2} - 72) q^{76} + 88 \beta_{2} q^{77} - 4 \beta_{2} q^{79} + ( - 8 \beta_{3} + 8 \beta_{2} + 8 \beta_1 - 24) q^{80} + ( - 4 \beta_{3} + 4 \beta_1) q^{82} + 32 q^{83} + (6 \beta_{3} + 18) q^{85} + (8 \beta_{3} - 8 \beta_1) q^{86} - 16 \beta_{3} q^{88} + 16 \beta_1 q^{89} - 88 \beta_{2} q^{91} + ( - 56 \beta_{2} + 56) q^{92} + ( - 4 \beta_{2} - 4) q^{94} + (12 \beta_{3} + 36) q^{95} + 20 \beta_{3} q^{97} + (39 \beta_{2} + 39) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{4} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 8 q^{4} - 32 q^{8} + 12 q^{10} - 32 q^{16} + 24 q^{20} - 112 q^{23} + 76 q^{25} + 64 q^{32} - 72 q^{34} + 176 q^{35} - 144 q^{38} - 112 q^{46} - 16 q^{47} + 156 q^{49} + 76 q^{50} - 232 q^{61} + 240 q^{62} + 256 q^{64} - 144 q^{68} + 176 q^{70} - 288 q^{76} - 96 q^{80} + 128 q^{83} + 72 q^{85} + 224 q^{92} - 16 q^{94} + 144 q^{95} + 156 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 22x^{2} + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 22 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 11 ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 44\nu ) / 22 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 11\beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 22\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
−2.34521 + 4.06202i
2.34521 4.06202i
−2.34521 4.06202i
2.34521 + 4.06202i
1.00000 1.73205i 0 −2.00000 3.46410i −4.69042 + 1.73205i 0 −9.38083 −8.00000 0 −1.69042 + 9.85609i
19.2 1.00000 1.73205i 0 −2.00000 3.46410i 4.69042 + 1.73205i 0 9.38083 −8.00000 0 7.69042 6.39199i
19.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −4.69042 1.73205i 0 −9.38083 −8.00000 0 −1.69042 9.85609i
19.4 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 4.69042 1.73205i 0 9.38083 −8.00000 0 7.69042 + 6.39199i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.f.g yes 4
3.b odd 2 1 180.3.f.d 4
4.b odd 2 1 180.3.f.d 4
5.b even 2 1 180.3.f.d 4
5.c odd 4 2 900.3.c.s 8
12.b even 2 1 inner 180.3.f.g yes 4
15.d odd 2 1 inner 180.3.f.g yes 4
15.e even 4 2 900.3.c.s 8
20.d odd 2 1 inner 180.3.f.g yes 4
20.e even 4 2 900.3.c.s 8
60.h even 2 1 180.3.f.d 4
60.l odd 4 2 900.3.c.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.f.d 4 3.b odd 2 1
180.3.f.d 4 4.b odd 2 1
180.3.f.d 4 5.b even 2 1
180.3.f.d 4 60.h even 2 1
180.3.f.g yes 4 1.a even 1 1 trivial
180.3.f.g yes 4 12.b even 2 1 inner
180.3.f.g yes 4 15.d odd 2 1 inner
180.3.f.g yes 4 20.d odd 2 1 inner
900.3.c.s 8 5.c odd 4 2
900.3.c.s 8 15.e even 4 2
900.3.c.s 8 20.e even 4 2
900.3.c.s 8 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(180, [\chi])\):

\( T_{7}^{2} - 88 \) Copy content Toggle raw display
\( T_{13}^{2} + 264 \) Copy content Toggle raw display
\( T_{23} + 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 38T^{2} + 625 \) Copy content Toggle raw display
$7$ \( (T^{2} - 88)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 264)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 264)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 432)^{2} \) Copy content Toggle raw display
$23$ \( (T + 28)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 88)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1200)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2376)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 352)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 1408)^{2} \) Copy content Toggle raw display
$47$ \( (T + 4)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 972)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 264)^{2} \) Copy content Toggle raw display
$61$ \( (T + 58)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 352)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 9504)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$83$ \( (T - 32)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 5632)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 26400)^{2} \) Copy content Toggle raw display
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