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Properties

Label 2583.1.ba.d
Level $2583$
Weight $1$
Character orbit 2583.ba
Analytic conductor $1.289$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -287
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2583,1,Mod(286,2583)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2583, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2583.286");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2583 = 3^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2583.ba (of order \(6\), 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: \(1.28908492763\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{21}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{21} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{42}^{8} + \zeta_{42}^{6}) q^{2} + \zeta_{42}^{9} q^{3} + (\zeta_{42}^{16} + \cdots + \zeta_{42}^{12}) q^{4}+ \cdots + \zeta_{42}^{18} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{42}^{8} + \zeta_{42}^{6}) q^{2} + \zeta_{42}^{9} q^{3} + (\zeta_{42}^{16} + \cdots + \zeta_{42}^{12}) q^{4}+ \cdots + (\zeta_{42}^{20} - \zeta_{42}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + 2 q^{3} - 7 q^{4} + q^{6} + 6 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} + 2 q^{3} - 7 q^{4} + q^{6} + 6 q^{7} - 2 q^{8} - 2 q^{9} - 14 q^{12} - 2 q^{13} + q^{14} - 8 q^{16} - 2 q^{17} - q^{18} + 4 q^{19} + q^{21} - q^{23} + 2 q^{24} - 6 q^{25} + 10 q^{26} + 2 q^{27} - 14 q^{28} - q^{34} + 2 q^{37} + 2 q^{38} + 2 q^{39} + 6 q^{41} - q^{42} - q^{43} - 2 q^{46} - 2 q^{47} + 8 q^{48} - 6 q^{49} - q^{50} + 2 q^{51} + 8 q^{54} - q^{56} - 4 q^{57} - q^{63} + 12 q^{64} - 13 q^{69} + 12 q^{72} + 8 q^{74} - q^{75} - 3 q^{78} - 2 q^{81} - 2 q^{82} - 7 q^{84} + q^{86} - 2 q^{89} - 4 q^{91} - 14 q^{92} - 5 q^{94} + 7 q^{96} + q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2583\mathbb{Z}\right)^\times\).

\(n\) \(1072\) \(2215\) \(2297\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{42}^{14}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
286.1
0.955573 + 0.294755i
0.826239 − 0.563320i
0.365341 + 0.930874i
0.0747301 − 0.997204i
−0.733052 − 0.680173i
−0.988831 + 0.149042i
0.955573 − 0.294755i
0.826239 + 0.563320i
0.365341 − 0.930874i
0.0747301 + 0.997204i
−0.733052 + 0.680173i
−0.988831 − 0.149042i
−0.955573 + 1.65510i 0.900969 − 0.433884i −1.32624 − 2.29711i 0 −0.142820 + 1.90580i 0.500000 − 0.866025i 3.15813 0.623490 − 0.781831i 0
286.2 −0.826239 + 1.43109i −0.623490 − 0.781831i −0.865341 − 1.49881i 0 1.63402 − 0.246289i 0.500000 − 0.866025i 1.20744 −0.222521 + 0.974928i 0
286.3 −0.365341 + 0.632789i 0.222521 + 0.974928i 0.233052 + 0.403658i 0 −0.698220 − 0.215372i 0.500000 − 0.866025i −1.07126 −0.900969 + 0.433884i 0
286.4 −0.0747301 + 0.129436i −0.623490 + 0.781831i 0.488831 + 0.846680i 0 −0.0546039 − 0.139129i 0.500000 − 0.866025i −0.295582 −0.222521 − 0.974928i 0
286.5 0.733052 − 1.26968i 0.900969 + 0.433884i −0.574730 − 0.995462i 0 1.21135 − 0.825886i 0.500000 − 0.866025i −0.219124 0.623490 + 0.781831i 0
286.6 0.988831 − 1.71271i 0.222521 − 0.974928i −1.45557 − 2.52113i 0 −1.44973 − 1.34515i 0.500000 − 0.866025i −3.77960 −0.900969 − 0.433884i 0
1147.1 −0.955573 − 1.65510i 0.900969 + 0.433884i −1.32624 + 2.29711i 0 −0.142820 − 1.90580i 0.500000 + 0.866025i 3.15813 0.623490 + 0.781831i 0
1147.2 −0.826239 − 1.43109i −0.623490 + 0.781831i −0.865341 + 1.49881i 0 1.63402 + 0.246289i 0.500000 + 0.866025i 1.20744 −0.222521 − 0.974928i 0
1147.3 −0.365341 − 0.632789i 0.222521 − 0.974928i 0.233052 − 0.403658i 0 −0.698220 + 0.215372i 0.500000 + 0.866025i −1.07126 −0.900969 − 0.433884i 0
1147.4 −0.0747301 − 0.129436i −0.623490 − 0.781831i 0.488831 − 0.846680i 0 −0.0546039 + 0.139129i 0.500000 + 0.866025i −0.295582 −0.222521 + 0.974928i 0
1147.5 0.733052 + 1.26968i 0.900969 − 0.433884i −0.574730 + 0.995462i 0 1.21135 + 0.825886i 0.500000 + 0.866025i −0.219124 0.623490 − 0.781831i 0
1147.6 0.988831 + 1.71271i 0.222521 + 0.974928i −1.45557 + 2.52113i 0 −1.44973 + 1.34515i 0.500000 + 0.866025i −3.77960 −0.900969 + 0.433884i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 286.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
287.d odd 2 1 CM by \(\Q(\sqrt{-287}) \)
9.c even 3 1 inner
2583.ba odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2583.1.ba.d yes 12
7.b odd 2 1 2583.1.ba.c ✓ 12
9.c even 3 1 inner 2583.1.ba.d yes 12
41.b even 2 1 2583.1.ba.c ✓ 12
63.l odd 6 1 2583.1.ba.c ✓ 12
287.d odd 2 1 CM 2583.1.ba.d yes 12
369.i even 6 1 2583.1.ba.c ✓ 12
2583.ba odd 6 1 inner 2583.1.ba.d yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2583.1.ba.c ✓ 12 7.b odd 2 1
2583.1.ba.c ✓ 12 41.b even 2 1
2583.1.ba.c ✓ 12 63.l odd 6 1
2583.1.ba.c ✓ 12 369.i even 6 1
2583.1.ba.d yes 12 1.a even 1 1 trivial
2583.1.ba.d yes 12 9.c even 3 1 inner
2583.1.ba.d yes 12 287.d odd 2 1 CM
2583.1.ba.d yes 12 2583.ba odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2583, [\chi])\):

\( T_{2}^{12} + T_{2}^{11} + 7 T_{2}^{10} + 6 T_{2}^{9} + 34 T_{2}^{8} + 28 T_{2}^{7} + 78 T_{2}^{6} + \cdots + 1 \) Copy content Toggle raw display
\( T_{13}^{6} + T_{13}^{5} + 3T_{13}^{4} + 5T_{13}^{2} + 2T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} + T^{5} + 3 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + T^{5} - 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - T^{2} - 2 T + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( (T^{6} - T^{5} - 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$43$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( (T^{6} + T^{5} + 3 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( (T^{6} + T^{5} - 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
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