Properties

Label 180.3.l.a
Level $180$
Weight $3$
Character orbit 180.l
Analytic conductor $4.905$
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] = [180,3,Mod(37,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 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("180.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.l (of order \(4\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 4 i + 3) q^{5} + ( - 7 i - 7) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 4 i + 3) q^{5} + ( - 7 i - 7) q^{7} - 10 q^{11} + ( - 9 i + 9) q^{13} + ( - i - 1) q^{17} - 8 i q^{19} + ( - 23 i + 23) q^{23} + ( - 24 i - 7) q^{25} + 8 i q^{29} - 14 q^{31} + (7 i - 49) q^{35} + (33 i + 33) q^{37} + 14 q^{41} + (15 i - 15) q^{43} + (39 i + 39) q^{47} + 49 i q^{49} + ( - 7 i + 7) q^{53} + (40 i - 30) q^{55} + 56 i q^{59} + 42 q^{61} + ( - 63 i - 9) q^{65} + ( - 7 i - 7) q^{67} - 98 q^{71} + ( - 49 i + 49) q^{73} + (70 i + 70) q^{77} + 96 i q^{79} + ( - 63 i + 63) q^{83} + (i - 7) q^{85} + 112 i q^{89} - 126 q^{91} + ( - 24 i - 32) q^{95} + (33 i + 33) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} - 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{5} - 14 q^{7} - 20 q^{11} + 18 q^{13} - 2 q^{17} + 46 q^{23} - 14 q^{25} - 28 q^{31} - 98 q^{35} + 66 q^{37} + 28 q^{41} - 30 q^{43} + 78 q^{47} + 14 q^{53} - 60 q^{55} + 84 q^{61} - 18 q^{65} - 14 q^{67} - 196 q^{71} + 98 q^{73} + 140 q^{77} + 126 q^{83} - 14 q^{85} - 252 q^{91} - 64 q^{95} + 66 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
1.00000i
1.00000i
0 0 0 3.00000 4.00000i 0 −7.00000 7.00000i 0 0 0
73.1 0 0 0 3.00000 + 4.00000i 0 −7.00000 + 7.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.l.a 2
3.b odd 2 1 20.3.f.a 2
4.b odd 2 1 720.3.bh.e 2
5.b even 2 1 900.3.l.a 2
5.c odd 4 1 inner 180.3.l.a 2
5.c odd 4 1 900.3.l.a 2
12.b even 2 1 80.3.p.a 2
15.d odd 2 1 100.3.f.a 2
15.e even 4 1 20.3.f.a 2
15.e even 4 1 100.3.f.a 2
20.e even 4 1 720.3.bh.e 2
21.c even 2 1 980.3.l.a 2
24.f even 2 1 320.3.p.g 2
24.h odd 2 1 320.3.p.c 2
60.h even 2 1 400.3.p.d 2
60.l odd 4 1 80.3.p.a 2
60.l odd 4 1 400.3.p.d 2
105.k odd 4 1 980.3.l.a 2
120.q odd 4 1 320.3.p.g 2
120.w even 4 1 320.3.p.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.f.a 2 3.b odd 2 1
20.3.f.a 2 15.e even 4 1
80.3.p.a 2 12.b even 2 1
80.3.p.a 2 60.l odd 4 1
100.3.f.a 2 15.d odd 2 1
100.3.f.a 2 15.e even 4 1
180.3.l.a 2 1.a even 1 1 trivial
180.3.l.a 2 5.c odd 4 1 inner
320.3.p.c 2 24.h odd 2 1
320.3.p.c 2 120.w even 4 1
320.3.p.g 2 24.f even 2 1
320.3.p.g 2 120.q odd 4 1
400.3.p.d 2 60.h even 2 1
400.3.p.d 2 60.l odd 4 1
720.3.bh.e 2 4.b odd 2 1
720.3.bh.e 2 20.e even 4 1
900.3.l.a 2 5.b even 2 1
900.3.l.a 2 5.c odd 4 1
980.3.l.a 2 21.c even 2 1
980.3.l.a 2 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 14T_{7} + 98 \) acting on \(S_{3}^{\mathrm{new}}(180, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$11$ \( (T + 10)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{2} - 46T + 1058 \) Copy content Toggle raw display
$29$ \( T^{2} + 64 \) Copy content Toggle raw display
$31$ \( (T + 14)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 66T + 2178 \) Copy content Toggle raw display
$41$ \( (T - 14)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 30T + 450 \) Copy content Toggle raw display
$47$ \( T^{2} - 78T + 3042 \) Copy content Toggle raw display
$53$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$59$ \( T^{2} + 3136 \) Copy content Toggle raw display
$61$ \( (T - 42)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$71$ \( (T + 98)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 98T + 4802 \) Copy content Toggle raw display
$79$ \( T^{2} + 9216 \) Copy content Toggle raw display
$83$ \( T^{2} - 126T + 7938 \) Copy content Toggle raw display
$89$ \( T^{2} + 12544 \) Copy content Toggle raw display
$97$ \( T^{2} - 66T + 2178 \) Copy content Toggle raw display
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