Properties

Label 1287.1.ba.c
Level $1287$
Weight $1$
Character orbit 1287.ba
Analytic conductor $0.642$
Analytic rank $0$
Dimension $8$
Projective image $D_{15}$
CM discriminant -143
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1287,1,Mod(142,1287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1287, 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("1287.142");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1287.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: \(0.642296671259\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{30}^{6} + \zeta_{30}^{4}) q^{2} - \zeta_{30}^{11} q^{3} + (\zeta_{30}^{12} + \zeta_{30}^{10} + \zeta_{30}^{8}) q^{4} + (\zeta_{30}^{2} + 1) q^{6} + (\zeta_{30}^{8} + \zeta_{30}^{2}) q^{7} + (\zeta_{30}^{14} + \zeta_{30}^{12} - \zeta_{30}^{3} + \zeta_{30}) q^{8} - \zeta_{30}^{7} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{30}^{6} + \zeta_{30}^{4}) q^{2} - \zeta_{30}^{11} q^{3} + (\zeta_{30}^{12} + \zeta_{30}^{10} + \zeta_{30}^{8}) q^{4} + (\zeta_{30}^{2} + 1) q^{6} + (\zeta_{30}^{8} + \zeta_{30}^{2}) q^{7} + (\zeta_{30}^{14} + \zeta_{30}^{12} - \zeta_{30}^{3} + \zeta_{30}) q^{8} - \zeta_{30}^{7} q^{9} - \zeta_{30}^{5} q^{11} + (\zeta_{30}^{8} + \zeta_{30}^{6} + \zeta_{30}^{4}) q^{12} + \zeta_{30}^{10} q^{13} + (\zeta_{30}^{14} + \zeta_{30}^{12} + \zeta_{30}^{8} + \zeta_{30}^{6}) q^{14} + ( - \zeta_{30}^{9} + \zeta_{30}^{7} + \zeta_{30}^{5} + \zeta_{30}^{3} - \zeta_{30}) q^{16} + ( - \zeta_{30}^{13} - \zeta_{30}^{11}) q^{18} + ( - \zeta_{30}^{11} + \zeta_{30}^{4}) q^{19} + ( - \zeta_{30}^{13} + \zeta_{30}^{4}) q^{21} + ( - \zeta_{30}^{11} - \zeta_{30}^{9}) q^{22} + ( - \zeta_{30}^{3} + \zeta_{30}^{2}) q^{23} + (\zeta_{30}^{14} + \zeta_{30}^{12} + \zeta_{30}^{10} + \zeta_{30}^{8}) q^{24} - \zeta_{30}^{5} q^{25} + (\zeta_{30}^{14} - \zeta_{30}) q^{26} - \zeta_{30}^{3} q^{27} + (\zeta_{30}^{14} + \zeta_{30}^{12} + \zeta_{30}^{10} - \zeta_{30}^{5} - \zeta_{30}^{3} - \zeta_{30}) q^{28} + ( - \zeta_{30}^{13} - \zeta_{30}^{11} + \zeta_{30}^{9} + \zeta_{30}^{7} - \zeta_{30}^{5} + 1) q^{32} - \zeta_{30} q^{33} + (\zeta_{30}^{4} + \zeta_{30}^{2} + 1) q^{36} + (\zeta_{30}^{10} + \zeta_{30}^{8} + \zeta_{30}^{2} + 1) q^{38} + \zeta_{30}^{6} q^{39} + ( - \zeta_{30}^{3} + \zeta_{30}^{2}) q^{41} + (\zeta_{30}^{10} + \zeta_{30}^{8} + \zeta_{30}^{4} + \zeta_{30}^{2}) q^{42} + ( - \zeta_{30}^{13} + \zeta_{30}^{2} + 1) q^{44} + ( - \zeta_{30}^{9} + \zeta_{30}^{8} - \zeta_{30}^{7} + \zeta_{30}^{6}) q^{46} + (\zeta_{30}^{14} + \zeta_{30}^{12} - \zeta_{30}^{5} - \zeta_{30}^{3} - \zeta_{30}) q^{48} + (\zeta_{30}^{10} + \zeta_{30}^{4} - \zeta_{30}) q^{49} + ( - \zeta_{30}^{11} - \zeta_{30}^{9}) q^{50} + ( - \zeta_{30}^{7} - \zeta_{30}^{5} - \zeta_{30}^{3}) q^{52} + ( - \zeta_{30}^{9} + \zeta_{30}^{6}) q^{53} + ( - \zeta_{30}^{9} - \zeta_{30}^{7}) q^{54} + (\zeta_{30}^{14} - \zeta_{30}^{11} - \zeta_{30}^{9} - \zeta_{30}^{7} - \zeta_{30}^{5} - \zeta_{30}^{3} - \zeta_{30}) q^{56} + ( - \zeta_{30}^{7} + 1) q^{57} + ( - \zeta_{30}^{9} + 1) q^{63} + ( - \zeta_{30}^{13} - \zeta_{30}^{11} - \zeta_{30}^{9} + \zeta_{30}^{6} - \zeta_{30}^{4} - \zeta_{30}^{2} - 1) q^{64} + ( - \zeta_{30}^{7} - \zeta_{30}^{5}) q^{66} + (\zeta_{30}^{14} - \zeta_{30}^{13}) q^{69} + (\zeta_{30}^{10} + \zeta_{30}^{8} + \zeta_{30}^{6} + \zeta_{30}^{4}) q^{72} + ( - \zeta_{30}^{9} + \zeta_{30}^{6}) q^{73} - \zeta_{30} q^{75} + (\zeta_{30}^{14} + \zeta_{30}^{12} + \zeta_{30}^{8} + \zeta_{30}^{6} + \zeta_{30}^{4} - \zeta_{30}) q^{76} + ( - \zeta_{30}^{13} - \zeta_{30}^{7}) q^{77} + (\zeta_{30}^{12} + \zeta_{30}^{10}) q^{78} + \zeta_{30}^{14} q^{81} + ( - \zeta_{30}^{9} + \zeta_{30}^{8} - \zeta_{30}^{7} + \zeta_{30}^{6}) q^{82} + ( - \zeta_{30}^{7} - \zeta_{30}^{3}) q^{83} + (\zeta_{30}^{14} + \zeta_{30}^{12} + \zeta_{30}^{10} + \zeta_{30}^{8} + \zeta_{30}^{6} - \zeta_{30}) q^{84} + (\zeta_{30}^{8} + \zeta_{30}^{6} + \zeta_{30}^{4} + \zeta_{30}^{2}) q^{88} + (\zeta_{30}^{12} - \zeta_{30}^{3}) q^{91} + (\zeta_{30}^{14} - \zeta_{30}^{13} + \zeta_{30}^{12} - \zeta_{30}^{11} + \zeta_{30}^{10} + 1) q^{92} + ( - \zeta_{30}^{11} - \zeta_{30}^{9} - \zeta_{30}^{7} - \zeta_{30}^{5} - \zeta_{30}^{3} - \zeta_{30}) q^{96} + (\zeta_{30}^{14} + \zeta_{30}^{10} + \zeta_{30}^{8} - \zeta_{30}^{7} - \zeta_{30}^{5} - \zeta_{30}) q^{98} + \zeta_{30}^{12} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + q^{3} - 5 q^{4} + 9 q^{6} + 2 q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} + q^{3} - 5 q^{4} + 9 q^{6} + 2 q^{7} - 2 q^{8} + q^{9} - 4 q^{11} - 4 q^{13} - 2 q^{14} - 6 q^{16} + 2 q^{18} + 2 q^{19} + 2 q^{21} - q^{22} - q^{23} - 4 q^{24} - 4 q^{25} + 2 q^{26} - 2 q^{27} - 10 q^{28} + 5 q^{32} + q^{33} + 10 q^{36} + 6 q^{38} - 2 q^{39} - q^{41} - q^{42} + 10 q^{44} - 2 q^{46} - 6 q^{48} - 2 q^{49} - q^{50} - 5 q^{52} - 4 q^{53} - q^{54} - 8 q^{56} + 9 q^{57} + 6 q^{63} + 8 q^{64} - 3 q^{66} + 2 q^{69} - 4 q^{72} - 4 q^{73} + q^{75} + 2 q^{77} - 6 q^{78} + q^{81} - 2 q^{82} - q^{83} - 5 q^{84} + q^{88} - 4 q^{91} + 5 q^{92} - 5 q^{96} - 4 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1287\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(937\) \(1145\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{30}^{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
142.1
0.913545 + 0.406737i
0.669131 0.743145i
−0.104528 + 0.994522i
−0.978148 + 0.207912i
0.913545 0.406737i
0.669131 + 0.743145i
−0.104528 0.994522i
−0.978148 0.207912i
−0.913545 + 1.58231i −0.104528 0.994522i −1.16913 2.02499i 0 1.66913 + 0.743145i −0.309017 + 0.535233i 2.44512 −0.978148 + 0.207912i 0
142.2 −0.669131 + 1.15897i −0.978148 0.207912i −0.395472 0.684977i 0 0.895472 0.994522i 0.809017 1.40126i −0.279773 0.913545 + 0.406737i 0
142.3 0.104528 0.181049i 0.913545 0.406737i 0.478148 + 0.828176i 0 0.0218524 0.207912i −0.309017 + 0.535233i 0.408977 0.669131 0.743145i 0
142.4 0.978148 1.69420i 0.669131 + 0.743145i −1.41355 2.44833i 0 1.91355 0.406737i 0.809017 1.40126i −3.57433 −0.104528 + 0.994522i 0
571.1 −0.913545 1.58231i −0.104528 + 0.994522i −1.16913 + 2.02499i 0 1.66913 0.743145i −0.309017 0.535233i 2.44512 −0.978148 0.207912i 0
571.2 −0.669131 1.15897i −0.978148 + 0.207912i −0.395472 + 0.684977i 0 0.895472 + 0.994522i 0.809017 + 1.40126i −0.279773 0.913545 0.406737i 0
571.3 0.104528 + 0.181049i 0.913545 + 0.406737i 0.478148 0.828176i 0 0.0218524 + 0.207912i −0.309017 0.535233i 0.408977 0.669131 + 0.743145i 0
571.4 0.978148 + 1.69420i 0.669131 0.743145i −1.41355 + 2.44833i 0 1.91355 + 0.406737i 0.809017 + 1.40126i −3.57433 −0.104528 0.994522i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 142.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
143.d odd 2 1 CM by \(\Q(\sqrt{-143}) \)
9.c even 3 1 inner
1287.ba odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1287.1.ba.c 8
3.b odd 2 1 3861.1.ba.d 8
9.c even 3 1 inner 1287.1.ba.c 8
9.d odd 6 1 3861.1.ba.d 8
11.b odd 2 1 1287.1.ba.d yes 8
13.b even 2 1 1287.1.ba.d yes 8
33.d even 2 1 3861.1.ba.c 8
39.d odd 2 1 3861.1.ba.c 8
99.g even 6 1 3861.1.ba.c 8
99.h odd 6 1 1287.1.ba.d yes 8
117.n odd 6 1 3861.1.ba.c 8
117.t even 6 1 1287.1.ba.d yes 8
143.d odd 2 1 CM 1287.1.ba.c 8
429.e even 2 1 3861.1.ba.d 8
1287.ba odd 6 1 inner 1287.1.ba.c 8
1287.bh even 6 1 3861.1.ba.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1287.1.ba.c 8 1.a even 1 1 trivial
1287.1.ba.c 8 9.c even 3 1 inner
1287.1.ba.c 8 143.d odd 2 1 CM
1287.1.ba.c 8 1287.ba odd 6 1 inner
1287.1.ba.d yes 8 11.b odd 2 1
1287.1.ba.d yes 8 13.b even 2 1
1287.1.ba.d yes 8 99.h odd 6 1
1287.1.ba.d yes 8 117.t even 6 1
3861.1.ba.c 8 33.d even 2 1
3861.1.ba.c 8 39.d odd 2 1
3861.1.ba.c 8 99.g even 6 1
3861.1.ba.c 8 117.n odd 6 1
3861.1.ba.d 8 3.b odd 2 1
3861.1.ba.d 8 9.d odd 6 1
3861.1.ba.d 8 429.e even 2 1
3861.1.ba.d 8 1287.bh even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + T_{2}^{7} + 5T_{2}^{6} + 4T_{2}^{5} + 19T_{2}^{4} + 14T_{2}^{3} + 20T_{2}^{2} - 4T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1287, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} + 5 T^{6} + 4 T^{5} + 19 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} + 2 T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{3} - 4 T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + T^{7} + 5 T^{6} + 4 T^{5} + 19 T^{4} + \cdots + 1 \) 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
$41$ \( T^{8} + T^{7} + 5 T^{6} + 4 T^{5} + 19 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{2} + T - 1)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} + T - 1)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} + T^{7} + 5 T^{6} + 4 T^{5} + 19 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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