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Properties

Label 1260.1.eb.b
Level $1260$
Weight $1$
Character orbit 1260.eb
Analytic conductor $0.629$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1260,1,Mod(223,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 8, 9, 6]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.223");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1260.eb (of order \(12\), degree \(4\), 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: \(0.628821915918\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.1134000.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{24}^{10} q^{2} + \zeta_{24}^{3} q^{3} - \zeta_{24}^{8} q^{4} - \zeta_{24}^{5} q^{5} - \zeta_{24} q^{6} + \zeta_{24} q^{7} + \zeta_{24}^{6} q^{8} + \zeta_{24}^{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{10} q^{2} + \zeta_{24}^{3} q^{3} - \zeta_{24}^{8} q^{4} - \zeta_{24}^{5} q^{5} - \zeta_{24} q^{6} + \zeta_{24} q^{7} + \zeta_{24}^{6} q^{8} + \zeta_{24}^{6} q^{9} + \zeta_{24}^{3} q^{10} - \zeta_{24}^{10} q^{11} - \zeta_{24}^{11} q^{12} + \zeta_{24}^{5} q^{13} + \zeta_{24}^{11} q^{14} - \zeta_{24}^{8} q^{15} - \zeta_{24}^{4} q^{16} - \zeta_{24}^{3} q^{17} - \zeta_{24}^{4} q^{18} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{19} - \zeta_{24} q^{20} + \zeta_{24}^{4} q^{21} + \zeta_{24}^{8} q^{22} + \zeta_{24}^{9} q^{24} + \zeta_{24}^{10} q^{25} - \zeta_{24}^{3} q^{26} + \zeta_{24}^{9} q^{27} - \zeta_{24}^{9} q^{28} + \zeta_{24}^{6} q^{30} + \zeta_{24}^{2} q^{32} + \zeta_{24} q^{33} + \zeta_{24} q^{34} - \zeta_{24}^{6} q^{35} + \zeta_{24}^{2} q^{36} + (\zeta_{24}^{7} + \zeta_{24}) q^{38} + \zeta_{24}^{8} q^{39} - \zeta_{24}^{11} q^{40} - \zeta_{24}^{2} q^{42} + ( - \zeta_{24}^{10} + \zeta_{24}^{4}) q^{43} - \zeta_{24}^{6} q^{44} - \zeta_{24}^{11} q^{45} - \zeta_{24} q^{47} - \zeta_{24}^{7} q^{48} + \zeta_{24}^{2} q^{49} - \zeta_{24}^{8} q^{50} - \zeta_{24}^{6} q^{51} + \zeta_{24} q^{52} + (\zeta_{24}^{6} + 1) q^{53} - \zeta_{24}^{7} q^{54} - \zeta_{24}^{3} q^{55} + \zeta_{24}^{7} q^{56} + ( - \zeta_{24}^{6} + 1) q^{57} - \zeta_{24}^{4} q^{60} + (\zeta_{24}^{7} - \zeta_{24}) q^{61} + \zeta_{24}^{7} q^{63} - q^{64} - \zeta_{24}^{10} q^{65} + \zeta_{24}^{11} q^{66} + \zeta_{24}^{11} q^{68} + \zeta_{24}^{4} q^{70} + \zeta_{24}^{6} q^{71} - q^{72} + \zeta_{24}^{9} q^{73} - \zeta_{24} q^{75} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{76} - \zeta_{24}^{11} q^{77} - \zeta_{24}^{6} q^{78} - \zeta_{24}^{4} q^{79} + \zeta_{24}^{9} q^{80} - q^{81} - \zeta_{24}^{7} q^{83} + q^{84} + \zeta_{24}^{8} q^{85} + (\zeta_{24}^{8} - \zeta_{24}^{2}) q^{86} + \zeta_{24}^{4} q^{88} + \zeta_{24}^{9} q^{90} + \zeta_{24}^{6} q^{91} - \zeta_{24}^{11} q^{94} + (\zeta_{24}^{8} - \zeta_{24}^{2}) q^{95} + \zeta_{24}^{5} q^{96} - \zeta_{24}^{7} q^{97} - q^{98} + \zeta_{24}^{4} q^{99} +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 \( 8 q + 4 q^{4} + 4 q^{15} - 4 q^{16} - 4 q^{18} + 4 q^{21} - 4 q^{22} - 4 q^{39} + 4 q^{43} + 4 q^{50} + 8 q^{53} + 8 q^{57} - 4 q^{60} - 8 q^{64} + 4 q^{70} - 8 q^{72} - 4 q^{79} - 8 q^{81} + 8 q^{84} - 4 q^{85} - 4 q^{86} + 4 q^{88} - 4 q^{95} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(\zeta_{24}^{8}\) \(-1\) \(-\zeta_{24}^{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.965926 − 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.965926 − 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
0.866025 + 0.500000i −0.707107 − 0.707107i 0.500000 + 0.866025i −0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i 1.00000i 1.00000i −0.707107 − 0.707107i
223.2 0.866025 + 0.500000i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i 1.00000i 1.00000i 0.707107 + 0.707107i
643.1 −0.866025 + 0.500000i −0.707107 − 0.707107i 0.500000 − 0.866025i 0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i 1.00000i 1.00000i −0.707107 − 0.707107i
643.2 −0.866025 + 0.500000i 0.707107 + 0.707107i 0.500000 − 0.866025i −0.258819 − 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i 1.00000i 1.00000i 0.707107 + 0.707107i
727.1 −0.866025 − 0.500000i −0.707107 + 0.707107i 0.500000 + 0.866025i 0.258819 − 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i − 1.00000i − 1.00000i −0.707107 + 0.707107i
727.2 −0.866025 − 0.500000i 0.707107 − 0.707107i 0.500000 + 0.866025i −0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i − 1.00000i − 1.00000i 0.707107 − 0.707107i
1147.1 0.866025 − 0.500000i −0.707107 + 0.707107i 0.500000 − 0.866025i −0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i − 1.00000i − 1.00000i −0.707107 + 0.707107i
1147.2 0.866025 − 0.500000i 0.707107 − 0.707107i 0.500000 − 0.866025i 0.965926 − 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i − 1.00000i − 1.00000i 0.707107 − 0.707107i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1147.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.c even 3 1 inner
20.e even 4 1 inner
63.l odd 6 1 inner
140.j odd 4 1 inner
180.x even 12 1 inner
1260.eb odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.1.eb.b yes 8
3.b odd 2 1 3780.1.ee.a 8
4.b odd 2 1 1260.1.eb.a ✓ 8
5.c odd 4 1 1260.1.eb.a ✓ 8
7.b odd 2 1 inner 1260.1.eb.b yes 8
9.c even 3 1 inner 1260.1.eb.b yes 8
9.d odd 6 1 3780.1.ee.a 8
12.b even 2 1 3780.1.ee.b 8
15.e even 4 1 3780.1.ee.b 8
20.e even 4 1 inner 1260.1.eb.b yes 8
21.c even 2 1 3780.1.ee.a 8
28.d even 2 1 1260.1.eb.a ✓ 8
35.f even 4 1 1260.1.eb.a ✓ 8
36.f odd 6 1 1260.1.eb.a ✓ 8
36.h even 6 1 3780.1.ee.b 8
45.k odd 12 1 1260.1.eb.a ✓ 8
45.l even 12 1 3780.1.ee.b 8
60.l odd 4 1 3780.1.ee.a 8
63.l odd 6 1 inner 1260.1.eb.b yes 8
63.o even 6 1 3780.1.ee.a 8
84.h odd 2 1 3780.1.ee.b 8
105.k odd 4 1 3780.1.ee.b 8
140.j odd 4 1 inner 1260.1.eb.b yes 8
180.v odd 12 1 3780.1.ee.a 8
180.x even 12 1 inner 1260.1.eb.b yes 8
252.s odd 6 1 3780.1.ee.b 8
252.bi even 6 1 1260.1.eb.a ✓ 8
315.cb even 12 1 1260.1.eb.a ✓ 8
315.cf odd 12 1 3780.1.ee.b 8
420.w even 4 1 3780.1.ee.a 8
1260.do even 12 1 3780.1.ee.a 8
1260.eb odd 12 1 inner 1260.1.eb.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.eb.a ✓ 8 4.b odd 2 1
1260.1.eb.a ✓ 8 5.c odd 4 1
1260.1.eb.a ✓ 8 28.d even 2 1
1260.1.eb.a ✓ 8 35.f even 4 1
1260.1.eb.a ✓ 8 36.f odd 6 1
1260.1.eb.a ✓ 8 45.k odd 12 1
1260.1.eb.a ✓ 8 252.bi even 6 1
1260.1.eb.a ✓ 8 315.cb even 12 1
1260.1.eb.b yes 8 1.a even 1 1 trivial
1260.1.eb.b yes 8 7.b odd 2 1 inner
1260.1.eb.b yes 8 9.c even 3 1 inner
1260.1.eb.b yes 8 20.e even 4 1 inner
1260.1.eb.b yes 8 63.l odd 6 1 inner
1260.1.eb.b yes 8 140.j odd 4 1 inner
1260.1.eb.b yes 8 180.x even 12 1 inner
1260.1.eb.b yes 8 1260.eb odd 12 1 inner
3780.1.ee.a 8 3.b odd 2 1
3780.1.ee.a 8 9.d odd 6 1
3780.1.ee.a 8 21.c even 2 1
3780.1.ee.a 8 60.l odd 4 1
3780.1.ee.a 8 63.o even 6 1
3780.1.ee.a 8 180.v odd 12 1
3780.1.ee.a 8 420.w even 4 1
3780.1.ee.a 8 1260.do even 12 1
3780.1.ee.b 8 12.b even 2 1
3780.1.ee.b 8 15.e even 4 1
3780.1.ee.b 8 36.h even 6 1
3780.1.ee.b 8 45.l even 12 1
3780.1.ee.b 8 84.h odd 2 1
3780.1.ee.b 8 105.k odd 4 1
3780.1.ee.b 8 252.s odd 6 1
3780.1.ee.b 8 315.cf odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{43}^{4} - 2T_{43}^{3} + 2T_{43}^{2} - 4T_{43} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1260, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$7$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$17$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$53$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
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