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Properties

Label 360.1.u.a
Level $360$
Weight $1$
Character orbit 360.u
Analytic conductor $0.180$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,1,Mod(37,360)]
 
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([0, 2, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.37");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 360.u (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: \(0.179663404548\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.9000.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} + \zeta_{8} q^{5} + ( - \zeta_{8}^{2} - 1) q^{7} - \zeta_{8} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} + \zeta_{8} q^{5} + ( - \zeta_{8}^{2} - 1) q^{7} - \zeta_{8} q^{8} + q^{10} + (\zeta_{8}^{3} + \zeta_{8}) q^{11} + (\zeta_{8}^{3} - \zeta_{8}) q^{14} - q^{16} - \zeta_{8}^{3} q^{20} + (\zeta_{8}^{2} + 1) q^{22} + \zeta_{8}^{2} q^{25} + (\zeta_{8}^{2} - 1) q^{28} + (\zeta_{8}^{3} - \zeta_{8}) q^{29} + \zeta_{8}^{3} q^{32} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{35} - \zeta_{8}^{2} q^{40} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{44} + \zeta_{8}^{2} q^{49} + \zeta_{8} q^{50} + (\zeta_{8}^{2} - 1) q^{55} + (\zeta_{8}^{3} + \zeta_{8}) q^{56} + (\zeta_{8}^{2} - 1) q^{58} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{59} + \zeta_{8}^{2} q^{64} + ( - \zeta_{8}^{2} - 1) q^{70} + ( - \zeta_{8}^{2} + 1) q^{73} + ( - 2 \zeta_{8}^{3} + \zeta_{8}) q^{77} - \zeta_{8} q^{80} - \zeta_{8} q^{83} + ( - \zeta_{8}^{2} + 1) q^{88} + ( - \zeta_{8}^{2} - 1) q^{97} + \zeta_{8} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{7} + 4 q^{10} - 4 q^{16} + 4 q^{22} - 4 q^{28} - 4 q^{55} - 4 q^{58} - 4 q^{70} + 4 q^{73} + 4 q^{88} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(\zeta_{8}^{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−0.707107 − 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 − 0.707107i
−0.707107 + 0.707107i 0 − 1.00000i −0.707107 − 0.707107i 0 −1.00000 − 1.00000i 0.707107 + 0.707107i 0 1.00000
37.2 0.707107 − 0.707107i 0 − 1.00000i 0.707107 + 0.707107i 0 −1.00000 − 1.00000i −0.707107 − 0.707107i 0 1.00000
253.1 −0.707107 − 0.707107i 0 1.00000i −0.707107 + 0.707107i 0 −1.00000 + 1.00000i 0.707107 − 0.707107i 0 1.00000
253.2 0.707107 + 0.707107i 0 1.00000i 0.707107 − 0.707107i 0 −1.00000 + 1.00000i −0.707107 + 0.707107i 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
5.c odd 4 1 inner
8.b even 2 1 inner
15.e even 4 1 inner
40.i odd 4 1 inner
120.w even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.1.u.a ✓ 4
3.b odd 2 1 inner 360.1.u.a ✓ 4
4.b odd 2 1 1440.1.y.a 4
5.b even 2 1 1800.1.u.b 4
5.c odd 4 1 inner 360.1.u.a ✓ 4
5.c odd 4 1 1800.1.u.b 4
8.b even 2 1 inner 360.1.u.a ✓ 4
8.d odd 2 1 1440.1.y.a 4
9.c even 3 2 3240.1.bv.c 8
9.d odd 6 2 3240.1.bv.c 8
12.b even 2 1 1440.1.y.a 4
15.d odd 2 1 1800.1.u.b 4
15.e even 4 1 inner 360.1.u.a ✓ 4
15.e even 4 1 1800.1.u.b 4
20.e even 4 1 1440.1.y.a 4
24.f even 2 1 1440.1.y.a 4
24.h odd 2 1 CM 360.1.u.a ✓ 4
40.f even 2 1 1800.1.u.b 4
40.i odd 4 1 inner 360.1.u.a ✓ 4
40.i odd 4 1 1800.1.u.b 4
40.k even 4 1 1440.1.y.a 4
45.k odd 12 2 3240.1.bv.c 8
45.l even 12 2 3240.1.bv.c 8
60.l odd 4 1 1440.1.y.a 4
72.j odd 6 2 3240.1.bv.c 8
72.n even 6 2 3240.1.bv.c 8
120.i odd 2 1 1800.1.u.b 4
120.q odd 4 1 1440.1.y.a 4
120.w even 4 1 inner 360.1.u.a ✓ 4
120.w even 4 1 1800.1.u.b 4
360.br even 12 2 3240.1.bv.c 8
360.bu odd 12 2 3240.1.bv.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.1.u.a ✓ 4 1.a even 1 1 trivial
360.1.u.a ✓ 4 3.b odd 2 1 inner
360.1.u.a ✓ 4 5.c odd 4 1 inner
360.1.u.a ✓ 4 8.b even 2 1 inner
360.1.u.a ✓ 4 15.e even 4 1 inner
360.1.u.a ✓ 4 24.h odd 2 1 CM
360.1.u.a ✓ 4 40.i odd 4 1 inner
360.1.u.a ✓ 4 120.w even 4 1 inner
1440.1.y.a 4 4.b odd 2 1
1440.1.y.a 4 8.d odd 2 1
1440.1.y.a 4 12.b even 2 1
1440.1.y.a 4 20.e even 4 1
1440.1.y.a 4 24.f even 2 1
1440.1.y.a 4 40.k even 4 1
1440.1.y.a 4 60.l odd 4 1
1440.1.y.a 4 120.q odd 4 1
1800.1.u.b 4 5.b even 2 1
1800.1.u.b 4 5.c odd 4 1
1800.1.u.b 4 15.d odd 2 1
1800.1.u.b 4 15.e even 4 1
1800.1.u.b 4 40.f even 2 1
1800.1.u.b 4 40.i odd 4 1
1800.1.u.b 4 120.i odd 2 1
1800.1.u.b 4 120.w even 4 1
3240.1.bv.c 8 9.c even 3 2
3240.1.bv.c 8 9.d odd 6 2
3240.1.bv.c 8 45.k odd 12 2
3240.1.bv.c 8 45.l even 12 2
3240.1.bv.c 8 72.j odd 6 2
3240.1.bv.c 8 72.n even 6 2
3240.1.bv.c 8 360.br even 12 2
3240.1.bv.c 8 360.bu odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(360, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 16 \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
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