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Properties

Label 180.3.c.c
Level $180$
Weight $3$
Character orbit 180.c
Analytic conductor $4.905$
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] = [180,3,Mod(91,180)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.91");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.c (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: \(8\)
Coefficient field: 8.0.15012375625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{6} + 28x^{4} + 112x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} - 2) q^{4} + \beta_{3} q^{5} + 2 \beta_{6} q^{7} + (\beta_{5} + \beta_{2} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{4} - 2) q^{4} + \beta_{3} q^{5} + 2 \beta_{6} q^{7} + (\beta_{5} + \beta_{2} - 2 \beta_1) q^{8} + ( - \beta_{6} - \beta_{4} + 1) q^{10} + (\beta_{5} + 2 \beta_{2}) q^{11} + (\beta_{7} - \beta_{6} - 3 \beta_{4} + 1) q^{13} + (2 \beta_{5} + 4 \beta_{3} - 2 \beta_1) q^{14} + ( - \beta_{7} - \beta_{6} + 2 \beta_{4} - 1) q^{16} + (3 \beta_{5} + 4 \beta_{3} - 6 \beta_1) q^{17} + (\beta_{7} + 3 \beta_{6} + 5 \beta_{4} + 1) q^{19} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{20} - 2 \beta_{7} q^{22} + (\beta_{5} - 2 \beta_{3} + 6 \beta_1) q^{23} + 5 q^{25} + (8 \beta_{3} + 4 \beta_{2}) q^{26} + ( - 8 \beta_{6} - 2 \beta_{4} - 12) q^{28} + ( - 2 \beta_{5} - 2 \beta_{3} + 4 \beta_1) q^{29} + (\beta_{7} + 7 \beta_{6} + 5 \beta_{4} + 1) q^{31} + ( - \beta_{5} - 12 \beta_{3} + \cdots + 2 \beta_1) q^{32}+ \cdots + (32 \beta_{3} + 16 \beta_{2} - 27 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 14 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 14 q^{4} + 10 q^{10} + 16 q^{13} - 14 q^{16} - 4 q^{22} + 40 q^{25} - 92 q^{28} - 116 q^{34} + 272 q^{37} - 10 q^{40} - 184 q^{46} - 152 q^{49} + 232 q^{52} + 84 q^{58} + 16 q^{61} - 182 q^{64} + 180 q^{70} - 240 q^{73} + 384 q^{76} + 208 q^{82} + 160 q^{85} - 652 q^{88} - 168 q^{94} + 240 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{6} + 28x^{4} + 112x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 7\nu^{5} + 28\nu^{3} + 112\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 7\nu^{5} + 36\nu^{3} - 48\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + \nu^{5} - 4\nu^{3} - 48\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 7\nu^{4} + 28\nu^{2} + 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 3\nu^{5} - 16\nu^{3} - 16\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 3\nu^{4} - 16\nu^{2} - 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{6} - 13\nu^{4} + 4\nu^{2} - 80 ) / 16 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{4} - 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{5} + 2\beta_{3} + 4\beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{7} + 11\beta_{6} + 13\beta_{4} - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{5} + 42\beta_{3} - 12\beta_{2} + 10\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -7\beta_{7} - 49\beta_{6} - 55\beta_{4} - 103 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -49\beta_{5} - 126\beta_{3} - 28\beta_{2} - 94\beta_1 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
1.46040 − 1.36646i
1.46040 + 1.36646i
0.342371 − 1.97048i
0.342371 + 1.97048i
−0.342371 − 1.97048i
−0.342371 + 1.97048i
−1.46040 − 1.36646i
−1.46040 + 1.36646i
−1.46040 − 1.36646i 0 0.265564 + 3.99117i −2.23607 0 1.87135i 5.06596 − 6.19161i 0 3.26556 + 3.05550i
91.2 −1.46040 + 1.36646i 0 0.265564 − 3.99117i −2.23607 0 − 1.87135i 5.06596 + 6.19161i 0 3.26556 − 3.05550i
91.3 −0.342371 − 1.97048i 0 −3.76556 + 1.34927i 2.23607 0 11.5108i 3.94792 + 6.95801i 0 −0.765564 − 4.40612i
91.4 −0.342371 + 1.97048i 0 −3.76556 − 1.34927i 2.23607 0 − 11.5108i 3.94792 − 6.95801i 0 −0.765564 + 4.40612i
91.5 0.342371 − 1.97048i 0 −3.76556 − 1.34927i −2.23607 0 − 11.5108i −3.94792 + 6.95801i 0 −0.765564 + 4.40612i
91.6 0.342371 + 1.97048i 0 −3.76556 + 1.34927i −2.23607 0 11.5108i −3.94792 − 6.95801i 0 −0.765564 − 4.40612i
91.7 1.46040 − 1.36646i 0 0.265564 − 3.99117i 2.23607 0 − 1.87135i −5.06596 − 6.19161i 0 3.26556 − 3.05550i
91.8 1.46040 + 1.36646i 0 0.265564 + 3.99117i 2.23607 0 1.87135i −5.06596 + 6.19161i 0 3.26556 + 3.05550i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.c.c ✓ 8
3.b odd 2 1 inner 180.3.c.c ✓ 8
4.b odd 2 1 inner 180.3.c.c ✓ 8
5.b even 2 1 900.3.c.o 8
5.c odd 4 2 900.3.f.i 16
8.b even 2 1 2880.3.e.i 8
8.d odd 2 1 2880.3.e.i 8
12.b even 2 1 inner 180.3.c.c ✓ 8
15.d odd 2 1 900.3.c.o 8
15.e even 4 2 900.3.f.i 16
20.d odd 2 1 900.3.c.o 8
20.e even 4 2 900.3.f.i 16
24.f even 2 1 2880.3.e.i 8
24.h odd 2 1 2880.3.e.i 8
60.h even 2 1 900.3.c.o 8
60.l odd 4 2 900.3.f.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.c.c ✓ 8 1.a even 1 1 trivial
180.3.c.c ✓ 8 3.b odd 2 1 inner
180.3.c.c ✓ 8 4.b odd 2 1 inner
180.3.c.c ✓ 8 12.b even 2 1 inner
900.3.c.o 8 5.b even 2 1
900.3.c.o 8 15.d odd 2 1
900.3.c.o 8 20.d odd 2 1
900.3.c.o 8 60.h even 2 1
900.3.f.i 16 5.c odd 4 2
900.3.f.i 16 15.e even 4 2
900.3.f.i 16 20.e even 4 2
900.3.f.i 16 60.l odd 4 2
2880.3.e.i 8 8.b even 2 1
2880.3.e.i 8 8.d odd 2 1
2880.3.e.i 8 24.f even 2 1
2880.3.e.i 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 136T_{7}^{2} + 464 \) acting on \(S_{3}^{\mathrm{new}}(180, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 7 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 136 T^{2} + 464)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 328 T^{2} + 22736)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T - 256)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 1096 T^{2} + 150544)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 944 T^{2} + 118784)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 368 T^{2} + 29696)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 456 T^{2} + 35344)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1840 T^{2} + 742400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 68 T + 896)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 832)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 3776 T^{2} + 1900544)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 5552 T^{2} + 7602176)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 3624 T^{2} + 21904)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 16488 T^{2} + 51452496)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 1036)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 21664 T^{2} + 75732224)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 2848 T^{2} + 363776)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 60 T - 140)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 16816 T^{2} + 65598464)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 24608 T^{2} + 69852416)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 28704 T^{2} + 160985344)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 60 T - 8460)^{4} \) Copy content Toggle raw display
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