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Properties

Label 360.2.w.c
Level $360$
Weight $2$
Character orbit 360.w
Analytic conductor $2.875$
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] = [360,2,Mod(163,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.163");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.w (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: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) 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 40)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + \beta_1 q^{4} + ( - \beta_{7} - \beta_{5} + \beta_{2} + 1) q^{5} + (\beta_{6} - \beta_{5} - \beta_{3} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{7} - \beta_{6} + \cdots + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + \beta_1 q^{4} + ( - \beta_{7} - \beta_{5} + \beta_{2} + 1) q^{5} + (\beta_{6} - \beta_{5} - \beta_{3} + \cdots + 1) q^{7}+ \cdots + (\beta_{7} + 2 \beta_{6} + \cdots - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 4 q^{8} - 10 q^{10} + 8 q^{11} + 8 q^{16} + 8 q^{17} + 12 q^{22} - 20 q^{26} - 20 q^{28} + 32 q^{32} - 20 q^{35} + 4 q^{38} - 20 q^{40} + 8 q^{41} + 28 q^{43} - 40 q^{46} + 10 q^{50} + 20 q^{52} - 40 q^{56} + 20 q^{58} - 40 q^{62} - 28 q^{67} + 4 q^{68} + 20 q^{70} + 16 q^{73} - 8 q^{76} - 28 q^{82} + 44 q^{83} + 24 q^{86} + 16 q^{88} - 40 q^{91} - 20 q^{92} + 16 q^{97} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{20} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{20}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{20}^{5} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{20}^{7} + \zeta_{20}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{20}^{7} + \zeta_{20}^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{3} + \zeta_{20}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{20}^{6} + \zeta_{20}^{4} + \zeta_{20}^{3} - \zeta_{20}^{2} + 1 \) Copy content Toggle raw display

Display \(\zeta_{20}^j\) in terms of \(\beta_i\)

\(\zeta_{20}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{4}\)\(=\) \( ( \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{5}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{20}^{6}\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{20}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(\beta_{3}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.587785 + 0.809017i
−0.951057 − 0.309017i
−0.587785 + 0.809017i
0.951057 − 0.309017i
0.587785 − 0.809017i
−0.951057 + 0.309017i
−0.587785 − 0.809017i
0.951057 + 0.309017i
−1.26007 − 0.642040i 0 1.17557 + 1.61803i 1.90211 − 1.17557i 0 −1.17557 − 1.17557i −0.442463 − 2.79360i 0 −3.15156 + 0.260074i
163.2 0.221232 − 1.39680i 0 −1.90211 − 0.618034i 1.17557 + 1.90211i 0 1.90211 + 1.90211i −1.28408 + 2.52015i 0 2.91695 − 1.22123i
163.3 0.642040 + 1.26007i 0 −1.17557 + 1.61803i −1.90211 + 1.17557i 0 1.17557 + 1.17557i −2.79360 − 0.442463i 0 −2.70254 − 1.64204i
163.4 1.39680 − 0.221232i 0 1.90211 − 0.618034i −1.17557 − 1.90211i 0 −1.90211 − 1.90211i 2.52015 − 1.28408i 0 −2.06285 − 2.39680i
307.1 −1.26007 + 0.642040i 0 1.17557 − 1.61803i 1.90211 + 1.17557i 0 −1.17557 + 1.17557i −0.442463 + 2.79360i 0 −3.15156 − 0.260074i
307.2 0.221232 + 1.39680i 0 −1.90211 + 0.618034i 1.17557 − 1.90211i 0 1.90211 − 1.90211i −1.28408 − 2.52015i 0 2.91695 + 1.22123i
307.3 0.642040 − 1.26007i 0 −1.17557 − 1.61803i −1.90211 − 1.17557i 0 1.17557 − 1.17557i −2.79360 + 0.442463i 0 −2.70254 + 1.64204i
307.4 1.39680 + 0.221232i 0 1.90211 + 0.618034i −1.17557 + 1.90211i 0 −1.90211 + 1.90211i 2.52015 + 1.28408i 0 −2.06285 + 2.39680i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.w.c 8
3.b odd 2 1 40.2.k.a ✓ 8
4.b odd 2 1 1440.2.bi.c 8
5.c odd 4 1 inner 360.2.w.c 8
8.b even 2 1 1440.2.bi.c 8
8.d odd 2 1 inner 360.2.w.c 8
12.b even 2 1 160.2.o.a 8
15.d odd 2 1 200.2.k.h 8
15.e even 4 1 40.2.k.a ✓ 8
15.e even 4 1 200.2.k.h 8
20.e even 4 1 1440.2.bi.c 8
24.f even 2 1 40.2.k.a ✓ 8
24.h odd 2 1 160.2.o.a 8
40.i odd 4 1 1440.2.bi.c 8
40.k even 4 1 inner 360.2.w.c 8
48.i odd 4 1 1280.2.n.m 8
48.i odd 4 1 1280.2.n.q 8
48.k even 4 1 1280.2.n.m 8
48.k even 4 1 1280.2.n.q 8
60.h even 2 1 800.2.o.g 8
60.l odd 4 1 160.2.o.a 8
60.l odd 4 1 800.2.o.g 8
120.i odd 2 1 800.2.o.g 8
120.m even 2 1 200.2.k.h 8
120.q odd 4 1 40.2.k.a ✓ 8
120.q odd 4 1 200.2.k.h 8
120.w even 4 1 160.2.o.a 8
120.w even 4 1 800.2.o.g 8
240.z odd 4 1 1280.2.n.m 8
240.bb even 4 1 1280.2.n.q 8
240.bd odd 4 1 1280.2.n.q 8
240.bf even 4 1 1280.2.n.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.k.a ✓ 8 3.b odd 2 1
40.2.k.a ✓ 8 15.e even 4 1
40.2.k.a ✓ 8 24.f even 2 1
40.2.k.a ✓ 8 120.q odd 4 1
160.2.o.a 8 12.b even 2 1
160.2.o.a 8 24.h odd 2 1
160.2.o.a 8 60.l odd 4 1
160.2.o.a 8 120.w even 4 1
200.2.k.h 8 15.d odd 2 1
200.2.k.h 8 15.e even 4 1
200.2.k.h 8 120.m even 2 1
200.2.k.h 8 120.q odd 4 1
360.2.w.c 8 1.a even 1 1 trivial
360.2.w.c 8 5.c odd 4 1 inner
360.2.w.c 8 8.d odd 2 1 inner
360.2.w.c 8 40.k even 4 1 inner
800.2.o.g 8 60.h even 2 1
800.2.o.g 8 60.l odd 4 1
800.2.o.g 8 120.i odd 2 1
800.2.o.g 8 120.w even 4 1
1280.2.n.m 8 48.i odd 4 1
1280.2.n.m 8 48.k even 4 1
1280.2.n.m 8 240.z odd 4 1
1280.2.n.m 8 240.bf even 4 1
1280.2.n.q 8 48.i odd 4 1
1280.2.n.q 8 48.k even 4 1
1280.2.n.q 8 240.bb even 4 1
1280.2.n.q 8 240.bd odd 4 1
1440.2.bi.c 8 4.b odd 2 1
1440.2.bi.c 8 8.b even 2 1
1440.2.bi.c 8 20.e even 4 1
1440.2.bi.c 8 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 60T_{7}^{4} + 400 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 30T^{4} + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 60T^{4} + 400 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 4)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 360T^{4} + 400 \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 1500 T^{4} + 250000 \) Copy content Toggle raw display
$29$ \( (T^{4} - 40 T^{2} + 80)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 100 T^{2} + 2000)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 360T^{4} + 400 \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T - 44)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 14 T^{3} + \cdots + 484)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 12060 T^{4} + 5856400 \) Copy content Toggle raw display
$53$ \( T^{8} + 360T^{4} + 400 \) Copy content Toggle raw display
$59$ \( (T^{4} + 72 T^{2} + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 100 T^{2} + 80)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 14 T^{3} + \cdots + 484)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 180 T^{2} + 6480)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 8 T^{3} + \cdots + 6724)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 160 T^{2} + 1280)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 22 T^{3} + \cdots + 3364)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 48 T^{2} + 256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
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