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Properties

Label 180.2.h.b
Level $180$
Weight $2$
Character orbit 180.h
Analytic conductor $1.437$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [180,2,Mod(179,180)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.179");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 180.h (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: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_{3} q^{4} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{5} + (\beta_{6} + \beta_1) q^{7} + ( - 2 \beta_{5} + \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + \beta_{3} q^{4} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{5} + (\beta_{6} + \beta_1) q^{7} + ( - 2 \beta_{5} + \beta_{4}) q^{8} + (\beta_{3} + \beta_1 + 2) q^{10} + ( - 2 \beta_{7} - \beta_{2}) q^{11} + (\beta_{6} - \beta_1) q^{13} + (\beta_{7} + 3 \beta_{2}) q^{14} + ( - \beta_{3} - 4) q^{16} + (\beta_{7} - 2 \beta_{5} + \cdots + \beta_{2}) q^{20}+ \cdots - 3 \beta_{4} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{4} + 12 q^{10} - 28 q^{16} + 8 q^{25} - 36 q^{40} + 40 q^{46} + 24 q^{49} - 80 q^{61} + 44 q^{64} + 40 q^{70} - 80 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 2\nu^{5} + 23\nu^{3} - 42\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} - 25\nu^{3} - 78\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 4\nu^{2} - 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} - 17\nu^{2} + 18 ) / 24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} - 5\nu^{2} - 6 ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 8\nu^{5} + 19\nu^{3} + 6\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 8\nu^{5} - 19\nu^{3} + 30\nu ) / 18 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 2\beta_{4} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 5\beta_{6} + 4\beta_{2} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{5} - 8\beta_{4} + 3\beta_{3} + 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 15\beta_{7} + 23\beta_{6} + 28\beta_{2} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{5} - 38\beta_{4} + 18\beta_{3} + 50 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 57\beta_{7} + 119\beta_{6} + 148\beta_{2} - 22\beta_1 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1
0.578737 − 0.965926i
−0.578737 + 0.965926i
−0.578737 − 0.965926i
0.578737 + 0.965926i
−2.15988 + 0.258819i
2.15988 − 0.258819i
2.15988 + 0.258819i
−2.15988 − 0.258819i
−0.866025 − 1.11803i 0 −0.500000 + 1.93649i −1.73205 − 1.41421i 0 −3.16228 2.59808 − 1.11803i 0 −0.0811388 + 3.16124i
179.2 −0.866025 − 1.11803i 0 −0.500000 + 1.93649i −1.73205 + 1.41421i 0 3.16228 2.59808 − 1.11803i 0 3.08114 + 0.711747i
179.3 −0.866025 + 1.11803i 0 −0.500000 − 1.93649i −1.73205 − 1.41421i 0 3.16228 2.59808 + 1.11803i 0 3.08114 − 0.711747i
179.4 −0.866025 + 1.11803i 0 −0.500000 − 1.93649i −1.73205 + 1.41421i 0 −3.16228 2.59808 + 1.11803i 0 −0.0811388 − 3.16124i
179.5 0.866025 − 1.11803i 0 −0.500000 − 1.93649i 1.73205 − 1.41421i 0 −3.16228 −2.59808 − 1.11803i 0 −0.0811388 − 3.16124i
179.6 0.866025 − 1.11803i 0 −0.500000 − 1.93649i 1.73205 + 1.41421i 0 3.16228 −2.59808 − 1.11803i 0 3.08114 − 0.711747i
179.7 0.866025 + 1.11803i 0 −0.500000 + 1.93649i 1.73205 − 1.41421i 0 3.16228 −2.59808 + 1.11803i 0 3.08114 + 0.711747i
179.8 0.866025 + 1.11803i 0 −0.500000 + 1.93649i 1.73205 + 1.41421i 0 −3.16228 −2.59808 + 1.11803i 0 −0.0811388 + 3.16124i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.8
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
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.2.h.b ✓ 8
3.b odd 2 1 inner 180.2.h.b ✓ 8
4.b odd 2 1 inner 180.2.h.b ✓ 8
5.b even 2 1 inner 180.2.h.b ✓ 8
5.c odd 4 2 900.2.e.f 8
8.b even 2 1 2880.2.o.c 8
8.d odd 2 1 2880.2.o.c 8
12.b even 2 1 inner 180.2.h.b ✓ 8
15.d odd 2 1 inner 180.2.h.b ✓ 8
15.e even 4 2 900.2.e.f 8
20.d odd 2 1 inner 180.2.h.b ✓ 8
20.e even 4 2 900.2.e.f 8
24.f even 2 1 2880.2.o.c 8
24.h odd 2 1 2880.2.o.c 8
40.e odd 2 1 2880.2.o.c 8
40.f even 2 1 2880.2.o.c 8
60.h even 2 1 inner 180.2.h.b ✓ 8
60.l odd 4 2 900.2.e.f 8
120.i odd 2 1 2880.2.o.c 8
120.m even 2 1 2880.2.o.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.h.b ✓ 8 1.a even 1 1 trivial
180.2.h.b ✓ 8 3.b odd 2 1 inner
180.2.h.b ✓ 8 4.b odd 2 1 inner
180.2.h.b ✓ 8 5.b even 2 1 inner
180.2.h.b ✓ 8 12.b even 2 1 inner
180.2.h.b ✓ 8 15.d odd 2 1 inner
180.2.h.b ✓ 8 20.d odd 2 1 inner
180.2.h.b ✓ 8 60.h even 2 1 inner
900.2.e.f 8 5.c odd 4 2
900.2.e.f 8 15.e even 4 2
900.2.e.f 8 20.e even 4 2
900.2.e.f 8 60.l odd 4 2
2880.2.o.c 8 8.b even 2 1
2880.2.o.c 8 8.d odd 2 1
2880.2.o.c 8 24.f even 2 1
2880.2.o.c 8 24.h odd 2 1
2880.2.o.c 8 40.e odd 2 1
2880.2.o.c 8 40.f even 2 1
2880.2.o.c 8 120.i odd 2 1
2880.2.o.c 8 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 10 \) acting on \(S_{2}^{\mathrm{new}}(180, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 10)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 30)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 54)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 40)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 80)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{2} - 30)^{4} \) Copy content Toggle raw display
$61$ \( (T + 10)^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} - 160)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} + 216)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 80)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 98)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
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