Properties

Label 180.4.k.d
Level $180$
Weight $4$
Character orbit 180.k
Analytic conductor $10.620$
Analytic rank $0$
Dimension $8$
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] = [180,4,Mod(127,180)]
 
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([2, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.127");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 180.k (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: \(10.6203438010\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 60)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{2}) q^{2} + (2 \beta_{6} + 6 \beta_{3}) q^{4} + (\beta_{7} - 9 \beta_{3} - 2 \beta_{2} + 3) q^{5} + (5 \beta_{6} + 6 \beta_{4} - 5 \beta_1) q^{7} + (4 \beta_{7} + 20 \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{2}) q^{2} + (2 \beta_{6} + 6 \beta_{3}) q^{4} + (\beta_{7} - 9 \beta_{3} - 2 \beta_{2} + 3) q^{5} + (5 \beta_{6} + 6 \beta_{4} - 5 \beta_1) q^{7} + (4 \beta_{7} + 20 \beta_{5}) q^{8} + ( - 9 \beta_{7} - 2 \beta_{6} - 9 \beta_{5} + 3 \beta_{4} - 14 \beta_{3} + 3 \beta_{2} + \cdots - 7) q^{10}+ \cdots + ( - 43 \beta_{7} - 60 \beta_{6} - 43 \beta_{5} - 420 \beta_{3} + \cdots + 420) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{5} - 56 q^{10} + 320 q^{13} - 64 q^{16} - 240 q^{17} + 432 q^{20} - 40 q^{22} - 352 q^{25} + 336 q^{26} + 560 q^{28} - 640 q^{37} + 240 q^{38} + 448 q^{40} - 1104 q^{41} - 304 q^{46} - 2352 q^{50} + 1920 q^{52} + 1200 q^{53} + 960 q^{56} - 1960 q^{58} - 272 q^{61} + 1200 q^{62} - 2592 q^{65} + 1440 q^{68} - 712 q^{70} + 440 q^{73} + 2464 q^{76} + 3120 q^{77} - 192 q^{80} - 1680 q^{82} - 2320 q^{85} - 3168 q^{86} + 800 q^{88} + 3360 q^{92} - 40 q^{97} + 3360 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{4} + 1 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 11\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 5\nu^{2} ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 7\nu^{3} ) / 24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 3\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 13\nu^{3} ) / 24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 3\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 5\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{6} + 9\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7\beta_{7} - 13\beta_{5} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(\beta_{3}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−1.28897 0.581861i
−0.581861 1.28897i
0.581861 + 1.28897i
1.28897 + 0.581861i
−1.28897 + 0.581861i
−0.581861 + 1.28897i
0.581861 1.28897i
1.28897 0.581861i
−2.57794 1.16372i 0 5.29150 + 6.00000i 8.61249 7.12917i 0 8.98612 8.98612i −6.65882 21.6255i 0 −30.4988 + 8.35600i
127.2 −1.16372 2.57794i 0 −5.29150 + 6.00000i 8.61249 7.12917i 0 −8.98612 + 8.98612i 21.6255 + 6.65882i 0 −28.4011 13.9061i
127.3 1.16372 + 2.57794i 0 −5.29150 + 6.00000i −2.61249 10.8708i 0 −17.4714 + 17.4714i −21.6255 6.65882i 0 24.9841 19.3854i
127.4 2.57794 + 1.16372i 0 5.29150 + 6.00000i −2.61249 10.8708i 0 17.4714 17.4714i 6.65882 + 21.6255i 0 5.91580 31.0645i
163.1 −2.57794 + 1.16372i 0 5.29150 6.00000i 8.61249 + 7.12917i 0 8.98612 + 8.98612i −6.65882 + 21.6255i 0 −30.4988 8.35600i
163.2 −1.16372 + 2.57794i 0 −5.29150 6.00000i 8.61249 + 7.12917i 0 −8.98612 8.98612i 21.6255 6.65882i 0 −28.4011 + 13.9061i
163.3 1.16372 2.57794i 0 −5.29150 6.00000i −2.61249 + 10.8708i 0 −17.4714 17.4714i −21.6255 + 6.65882i 0 24.9841 + 19.3854i
163.4 2.57794 1.16372i 0 5.29150 6.00000i −2.61249 + 10.8708i 0 17.4714 + 17.4714i 6.65882 21.6255i 0 5.91580 + 31.0645i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.4.k.d 8
3.b odd 2 1 60.4.j.a 8
4.b odd 2 1 inner 180.4.k.d 8
5.c odd 4 1 inner 180.4.k.d 8
12.b even 2 1 60.4.j.a 8
15.e even 4 1 60.4.j.a 8
20.e even 4 1 inner 180.4.k.d 8
60.l odd 4 1 60.4.j.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.j.a 8 3.b odd 2 1
60.4.j.a 8 12.b even 2 1
60.4.j.a 8 15.e even 4 1
60.4.j.a 8 60.l odd 4 1
180.4.k.d 8 1.a even 1 1 trivial
180.4.k.d 8 4.b odd 2 1 inner
180.4.k.d 8 5.c odd 4 1 inner
180.4.k.d 8 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(180, [\chi])\):

\( T_{7}^{8} + 398792T_{7}^{4} + 9721171216 \) Copy content Toggle raw display
\( T_{13}^{4} - 160T_{13}^{3} + 12800T_{13}^{2} - 471680T_{13} + 8690704 \) Copy content Toggle raw display
\( T_{17}^{4} + 120T_{17}^{3} + 7200T_{17}^{2} + 132000T_{17} + 1210000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 16T^{4} + 4096 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 12 T^{3} + 160 T^{2} + \cdots + 15625)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 398792 T^{4} + \cdots + 9721171216 \) Copy content Toggle raw display
$11$ \( (T^{4} + 2116 T^{2} + 917764)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 160 T^{3} + 12800 T^{2} + \cdots + 8690704)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 120 T^{3} + 7200 T^{2} + \cdots + 1210000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 10376 T^{2} + 2521744)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 540023072 T^{4} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{4} + 46468 T^{2} + \cdots + 122456356)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 92016 T^{2} + \cdots + 1935296064)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 320 T^{3} + 51200 T^{2} + \cdots + 313006864)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 276 T + 6444)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + 51820944512 T^{4} + \cdots + 58\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{8} + 93187220000 T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} - 600 T^{3} + \cdots + 40968998464)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 423396 T^{2} + \cdots + 2342753604)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 68 T - 313844)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} + 659635774592 T^{4} + \cdots + 53\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{4} + 460000 T^{2} + \cdots + 32256160000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 220 T^{3} + 24200 T^{2} + \cdots + 101566084)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 449936 T^{2} + \cdots + 44365839424)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 415186437152 T^{4} + \cdots + 42\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{4} + 547912 T^{2} + \cdots + 36529936)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 258502084)^{2} \) Copy content Toggle raw display
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