Properties

Label 180.4.i.a
Level $180$
Weight $4$
Character orbit 180.i
Analytic conductor $10.620$
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] = [180,4,Mod(61,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.61");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 180.i (of order \(3\), 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: \(10.6203438010\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \zeta_{6} + 6) q^{3} + 5 \zeta_{6} q^{5} + ( - 7 \zeta_{6} + 7) q^{7} + ( - 27 \zeta_{6} + 27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \zeta_{6} + 6) q^{3} + 5 \zeta_{6} q^{5} + ( - 7 \zeta_{6} + 7) q^{7} + ( - 27 \zeta_{6} + 27) q^{9} + ( - 30 \zeta_{6} + 30) q^{11} + 22 \zeta_{6} q^{13} + (15 \zeta_{6} + 15) q^{15} + 48 q^{17} + 68 q^{19} + ( - 42 \zeta_{6} + 21) q^{21} - 111 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + ( - 162 \zeta_{6} + 81) q^{27} + (87 \zeta_{6} - 87) q^{29} - 20 \zeta_{6} q^{31} + ( - 180 \zeta_{6} + 90) q^{33} + 35 q^{35} + 200 q^{37} + (66 \zeta_{6} + 66) q^{39} - 69 \zeta_{6} q^{41} + ( - 232 \zeta_{6} + 232) q^{43} + 135 q^{45} + (243 \zeta_{6} - 243) q^{47} + 294 \zeta_{6} q^{49} + ( - 144 \zeta_{6} + 288) q^{51} - 498 q^{53} + 150 q^{55} + ( - 204 \zeta_{6} + 408) q^{57} - 66 \zeta_{6} q^{59} + (359 \zeta_{6} - 359) q^{61} - 189 \zeta_{6} q^{63} + (110 \zeta_{6} - 110) q^{65} + 1063 \zeta_{6} q^{67} + ( - 333 \zeta_{6} - 333) q^{69} - 618 q^{71} - 532 q^{73} + (150 \zeta_{6} - 75) q^{75} - 210 \zeta_{6} q^{77} + (410 \zeta_{6} - 410) q^{79} - 729 \zeta_{6} q^{81} + (693 \zeta_{6} - 693) q^{83} + 240 \zeta_{6} q^{85} + (522 \zeta_{6} - 261) q^{87} - 1599 q^{89} + 154 q^{91} + ( - 60 \zeta_{6} - 60) q^{93} + 340 \zeta_{6} q^{95} + (50 \zeta_{6} - 50) q^{97} - 810 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{3} + 5 q^{5} + 7 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 9 q^{3} + 5 q^{5} + 7 q^{7} + 27 q^{9} + 30 q^{11} + 22 q^{13} + 45 q^{15} + 96 q^{17} + 136 q^{19} - 111 q^{23} - 25 q^{25} - 87 q^{29} - 20 q^{31} + 70 q^{35} + 400 q^{37} + 198 q^{39} - 69 q^{41} + 232 q^{43} + 270 q^{45} - 243 q^{47} + 294 q^{49} + 432 q^{51} - 996 q^{53} + 300 q^{55} + 612 q^{57} - 66 q^{59} - 359 q^{61} - 189 q^{63} - 110 q^{65} + 1063 q^{67} - 999 q^{69} - 1236 q^{71} - 1064 q^{73} - 210 q^{77} - 410 q^{79} - 729 q^{81} - 693 q^{83} + 240 q^{85} - 3198 q^{89} + 308 q^{91} - 180 q^{93} + 340 q^{95} - 50 q^{97} - 810 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
0.500000 + 0.866025i
0.500000 0.866025i
0 4.50000 2.59808i 0 2.50000 + 4.33013i 0 3.50000 6.06218i 0 13.5000 23.3827i 0
121.1 0 4.50000 + 2.59808i 0 2.50000 4.33013i 0 3.50000 + 6.06218i 0 13.5000 + 23.3827i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.4.i.a 2
3.b odd 2 1 540.4.i.a 2
9.c even 3 1 inner 180.4.i.a 2
9.c even 3 1 1620.4.a.a 1
9.d odd 6 1 540.4.i.a 2
9.d odd 6 1 1620.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.4.i.a 2 1.a even 1 1 trivial
180.4.i.a 2 9.c even 3 1 inner
540.4.i.a 2 3.b odd 2 1
540.4.i.a 2 9.d odd 6 1
1620.4.a.a 1 9.c even 3 1
1620.4.a.b 1 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 7T_{7} + 49 \) acting on \(S_{4}^{\mathrm{new}}(180, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$5$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} - 30T + 900 \) Copy content Toggle raw display
$13$ \( T^{2} - 22T + 484 \) Copy content Toggle raw display
$17$ \( (T - 48)^{2} \) Copy content Toggle raw display
$19$ \( (T - 68)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 111T + 12321 \) Copy content Toggle raw display
$29$ \( T^{2} + 87T + 7569 \) Copy content Toggle raw display
$31$ \( T^{2} + 20T + 400 \) Copy content Toggle raw display
$37$ \( (T - 200)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 69T + 4761 \) Copy content Toggle raw display
$43$ \( T^{2} - 232T + 53824 \) Copy content Toggle raw display
$47$ \( T^{2} + 243T + 59049 \) Copy content Toggle raw display
$53$ \( (T + 498)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 66T + 4356 \) Copy content Toggle raw display
$61$ \( T^{2} + 359T + 128881 \) Copy content Toggle raw display
$67$ \( T^{2} - 1063 T + 1129969 \) Copy content Toggle raw display
$71$ \( (T + 618)^{2} \) Copy content Toggle raw display
$73$ \( (T + 532)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 410T + 168100 \) Copy content Toggle raw display
$83$ \( T^{2} + 693T + 480249 \) Copy content Toggle raw display
$89$ \( (T + 1599)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 50T + 2500 \) Copy content Toggle raw display
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