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Properties

Label 2205.1.bz.a
Level $2205$
Weight $1$
Character orbit 2205.bz
Analytic conductor $1.100$
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] = [2205,1,Mod(148,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 9, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.148");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2205.bz (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: \(1.10043835286\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.3472875.1

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{24}^{10} - \zeta_{24}^{4}) q^{2} + \zeta_{24}^{3} q^{3} + \zeta_{24}^{2} q^{4} - \zeta_{24}^{11} q^{5} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{6} + q^{8} + \zeta_{24}^{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{24}^{10} - \zeta_{24}^{4}) q^{2} + \zeta_{24}^{3} q^{3} + \zeta_{24}^{2} q^{4} - \zeta_{24}^{11} q^{5} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{6} + q^{8} + \zeta_{24}^{6} q^{9} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{10} + \zeta_{24}^{4} q^{11} + \zeta_{24}^{5} q^{12} - \zeta_{24}^{11} q^{13} + \zeta_{24}^{2} q^{15} - \zeta_{24}^{4} q^{16} - \zeta_{24}^{9} q^{17} + ( - \zeta_{24}^{10} - \zeta_{24}^{4}) q^{18} + \zeta_{24} q^{20} + ( - \zeta_{24}^{8} - \zeta_{24}^{2}) q^{22} + (\zeta_{24}^{8} - \zeta_{24}^{2}) q^{23} - \zeta_{24}^{10} q^{25} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{26} + \zeta_{24}^{9} q^{27} + ( - \zeta_{24}^{6} - 1) q^{30} + (\zeta_{24}^{8} + \zeta_{24}^{2}) q^{32} + \zeta_{24}^{7} q^{33} + (\zeta_{24}^{7} - \zeta_{24}) q^{34} + \zeta_{24}^{8} q^{36} + ( - \zeta_{24}^{6} + 1) q^{37} + \zeta_{24}^{2} q^{39} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{41} + \zeta_{24}^{6} q^{44} + \zeta_{24}^{5} q^{45} + ( - \zeta_{24}^{6} + 2) q^{46} - \zeta_{24} q^{47} - \zeta_{24}^{7} q^{48} + (\zeta_{24}^{8} - \zeta_{24}^{2}) q^{50} + q^{51} + \zeta_{24} q^{52} + ( - \zeta_{24}^{6} - 1) q^{53} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{54} + \zeta_{24}^{3} q^{55} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{59} + \zeta_{24}^{4} q^{60} - \zeta_{24}^{6} q^{64} - \zeta_{24}^{10} q^{65} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{66} + ( - \zeta_{24}^{8} - \zeta_{24}^{2}) q^{67} - \zeta_{24}^{11} q^{68} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{69} - q^{71} - \zeta_{24}^{3} q^{73} + (2 \zeta_{24}^{10} + \zeta_{24}^{4}) q^{74} + \zeta_{24} q^{75} + ( - \zeta_{24}^{6} - 1) q^{78} + \zeta_{24}^{10} q^{79} - \zeta_{24}^{3} q^{80} - q^{81} - \zeta_{24}^{9} q^{82} + \zeta_{24}^{7} q^{83} - \zeta_{24}^{8} q^{85} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{90} + (\zeta_{24}^{10} - \zeta_{24}^{4}) q^{92} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{94} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{96} + \zeta_{24} q^{97} + \zeta_{24}^{10} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 4 q^{11} - 4 q^{16} - 4 q^{18} + 4 q^{22} - 4 q^{23} - 8 q^{30} - 4 q^{32} - 4 q^{36} + 8 q^{37} + 16 q^{46} - 4 q^{50} + 8 q^{51} - 8 q^{53} + 4 q^{60} + 4 q^{67} - 8 q^{71} - 8 q^{78} - 8 q^{81} + 4 q^{85} - 4 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(1081\) \(1226\)
\(\chi(n)\) \(-\zeta_{24}^{6}\) \(1\) \(\zeta_{24}^{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1
−0.965926 − 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−1.36603 − 0.366025i −0.707107 − 0.707107i 0.866025 + 0.500000i −0.965926 + 0.258819i 0.707107 + 1.22474i 0 0 1.00000i 1.41421
148.2 −1.36603 − 0.366025i 0.707107 + 0.707107i 0.866025 + 0.500000i 0.965926 − 0.258819i −0.707107 − 1.22474i 0 0 1.00000i −1.41421
1177.1 −1.36603 + 0.366025i −0.707107 + 0.707107i 0.866025 − 0.500000i −0.965926 − 0.258819i 0.707107 − 1.22474i 0 0 − 1.00000i 1.41421
1177.2 −1.36603 + 0.366025i 0.707107 − 0.707107i 0.866025 − 0.500000i 0.965926 + 0.258819i −0.707107 + 1.22474i 0 0 − 1.00000i −1.41421
1618.1 0.366025 + 1.36603i −0.707107 − 0.707107i −0.866025 + 0.500000i 0.258819 − 0.965926i 0.707107 − 1.22474i 0 0 1.00000i 1.41421
1618.2 0.366025 + 1.36603i 0.707107 + 0.707107i −0.866025 + 0.500000i −0.258819 + 0.965926i −0.707107 + 1.22474i 0 0 1.00000i −1.41421
1912.1 0.366025 − 1.36603i −0.707107 + 0.707107i −0.866025 − 0.500000i 0.258819 + 0.965926i 0.707107 + 1.22474i 0 0 − 1.00000i 1.41421
1912.2 0.366025 − 1.36603i 0.707107 − 0.707107i −0.866025 − 0.500000i −0.258819 − 0.965926i −0.707107 − 1.22474i 0 0 − 1.00000i −1.41421
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 148.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
9.c even 3 1 inner
35.f even 4 1 inner
45.k odd 12 1 inner
63.l odd 6 1 inner
315.cb even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.1.bz.a ✓ 8
5.c odd 4 1 inner 2205.1.bz.a ✓ 8
7.b odd 2 1 inner 2205.1.bz.a ✓ 8
7.c even 3 1 2205.1.bu.a 8
7.c even 3 1 2205.1.ci.a 8
7.d odd 6 1 2205.1.bu.a 8
7.d odd 6 1 2205.1.ci.a 8
9.c even 3 1 inner 2205.1.bz.a ✓ 8
35.f even 4 1 inner 2205.1.bz.a ✓ 8
35.k even 12 1 2205.1.bu.a 8
35.k even 12 1 2205.1.ci.a 8
35.l odd 12 1 2205.1.bu.a 8
35.l odd 12 1 2205.1.ci.a 8
45.k odd 12 1 inner 2205.1.bz.a ✓ 8
63.g even 3 1 2205.1.bu.a 8
63.h even 3 1 2205.1.ci.a 8
63.k odd 6 1 2205.1.bu.a 8
63.l odd 6 1 inner 2205.1.bz.a ✓ 8
63.t odd 6 1 2205.1.ci.a 8
315.bs even 12 1 2205.1.ci.a 8
315.bt odd 12 1 2205.1.ci.a 8
315.cb even 12 1 inner 2205.1.bz.a ✓ 8
315.cg even 12 1 2205.1.bu.a 8
315.ch odd 12 1 2205.1.bu.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2205.1.bu.a 8 7.c even 3 1
2205.1.bu.a 8 7.d odd 6 1
2205.1.bu.a 8 35.k even 12 1
2205.1.bu.a 8 35.l odd 12 1
2205.1.bu.a 8 63.g even 3 1
2205.1.bu.a 8 63.k odd 6 1
2205.1.bu.a 8 315.cg even 12 1
2205.1.bu.a 8 315.ch odd 12 1
2205.1.bz.a ✓ 8 1.a even 1 1 trivial
2205.1.bz.a ✓ 8 5.c odd 4 1 inner
2205.1.bz.a ✓ 8 7.b odd 2 1 inner
2205.1.bz.a ✓ 8 9.c even 3 1 inner
2205.1.bz.a ✓ 8 35.f even 4 1 inner
2205.1.bz.a ✓ 8 45.k odd 12 1 inner
2205.1.bz.a ✓ 8 63.l odd 6 1 inner
2205.1.bz.a ✓ 8 315.cb even 12 1 inner
2205.1.ci.a 8 7.c even 3 1
2205.1.ci.a 8 7.d odd 6 1
2205.1.ci.a 8 35.k even 12 1
2205.1.ci.a 8 35.l odd 12 1
2205.1.ci.a 8 63.h even 3 1
2205.1.ci.a 8 63.t odd 6 1
2205.1.ci.a 8 315.bs even 12 1
2205.1.ci.a 8 315.bt odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2205, [\chi])\).

Hecke characteristic polynomials

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