Properties

Label 786.2.e.a
Level $786$
Weight $2$
Character orbit 786.e
Analytic conductor $6.276$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [786,2,Mod(61,786)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(786, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("786.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 786 = 2 \cdot 3 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 786.e (of order \(5\), 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: \(6.27624159887\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} - \zeta_{10}^{3} q^{3} - \zeta_{10}^{3} q^{4} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{5} - \zeta_{10}^{2} q^{6} - \zeta_{10}^{2} q^{8} - \zeta_{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} - \zeta_{10}^{3} q^{3} - \zeta_{10}^{3} q^{4} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{5} - \zeta_{10}^{2} q^{6} - \zeta_{10}^{2} q^{8} - \zeta_{10} q^{9} + 2 \zeta_{10}^{3} q^{10} + ( - 4 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 4) q^{11} - \zeta_{10} q^{12} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{13} + 2 \zeta_{10}^{2} q^{15} - \zeta_{10} q^{16} + (\zeta_{10}^{3} - 6 \zeta_{10}^{2} + \zeta_{10}) q^{17} - q^{18} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 2) q^{19} + 2 \zeta_{10}^{2} q^{20} + ( - 4 \zeta_{10}^{3} + \zeta_{10} - 1) q^{22} + (2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 2 \zeta_{10}) q^{23} - q^{24} + \zeta_{10}^{3} q^{25} + ( - 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{26} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{27} + ( - 6 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 6) q^{29} + 2 \zeta_{10} q^{30} + ( - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 6) q^{31} - q^{32} + (\zeta_{10}^{3} - 5 \zeta_{10}^{2} + \zeta_{10}) q^{33} + (\zeta_{10}^{2} - 6 \zeta_{10} + 1) q^{34} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{36} + (2 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 2) q^{37} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 2) q^{38} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 4) q^{39} + 2 \zeta_{10} q^{40} + ( - 2 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 2) q^{41} + ( - 7 \zeta_{10}^{2} + 8 \zeta_{10} - 7) q^{43} + (\zeta_{10}^{3} - 5 \zeta_{10}^{2} + \zeta_{10}) q^{44} + 2 q^{45} + (2 \zeta_{10}^{2} - 4 \zeta_{10} + 2) q^{46} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2}) q^{47} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{48} + 7 \zeta_{10}^{3} q^{49} + \zeta_{10}^{2} q^{50} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 5) q^{51} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 4) q^{52} + ( - 2 \zeta_{10}^{3} + 2) q^{53} + \zeta_{10}^{3} q^{54} + (8 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{55} + (2 \zeta_{10}^{3} + \zeta_{10} - 1) q^{57} + ( - 6 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{58} + (9 \zeta_{10}^{3} - \zeta_{10} + 1) q^{59} + 2 q^{60} + (4 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 4 \zeta_{10}) q^{61} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2}) q^{62} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{64} + (4 \zeta_{10}^{2} + 4 \zeta_{10} + 4) q^{65} + (\zeta_{10}^{2} - 5 \zeta_{10} + 1) q^{66} + ( - 6 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 6) q^{67} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 5) q^{68} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2) q^{69} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 8) q^{71} + \zeta_{10}^{3} q^{72} + (10 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{73} + (2 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{74} + \zeta_{10} q^{75} + (2 \zeta_{10}^{3} + \zeta_{10} - 1) q^{76} + (4 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 4) q^{78} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2}) q^{79} + 2 q^{80} + \zeta_{10}^{2} q^{81} + ( - 2 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{82} + ( - 9 \zeta_{10}^{3} + 14 \zeta_{10}^{2} - 14 \zeta_{10} + 9) q^{83} + ( - 2 \zeta_{10}^{2} + 12 \zeta_{10} - 2) q^{85} + (7 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 1) q^{86} + ( - 4 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 4 \zeta_{10}) q^{87} + (\zeta_{10}^{2} - 5 \zeta_{10} + 1) q^{88} + ( - 3 \zeta_{10}^{2} - 2 \zeta_{10} - 3) q^{89} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{90} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2) q^{92} + (6 \zeta_{10}^{2} - 6 \zeta_{10}) q^{93} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{94} + ( - 4 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 4) q^{95} + \zeta_{10}^{3} q^{96} + (7 \zeta_{10}^{2} + 8 \zeta_{10} + 7) q^{97} + 7 \zeta_{10}^{2} q^{98} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} + q^{8} - q^{9} + 2 q^{10} + 2 q^{11} - q^{12} - 2 q^{13} - 2 q^{15} - q^{16} + 8 q^{17} - 4 q^{18} - 6 q^{19} - 2 q^{20} - 7 q^{22} + 8 q^{23} - 4 q^{24} + q^{25} - 8 q^{26} - q^{27} + 14 q^{29} + 2 q^{30} - 12 q^{31} - 4 q^{32} + 7 q^{33} - 3 q^{34} - q^{36} + 6 q^{37} - 4 q^{38} - 12 q^{39} + 2 q^{40} + 8 q^{41} - 13 q^{43} + 7 q^{44} + 8 q^{45} + 2 q^{46} - 4 q^{47} - q^{48} + 7 q^{49} - q^{50} - 22 q^{51} - 12 q^{52} + 6 q^{53} + q^{54} + 14 q^{55} - q^{57} + 6 q^{58} + 12 q^{59} + 8 q^{60} + 10 q^{61} + 12 q^{62} - q^{64} + 16 q^{65} - 2 q^{66} + 8 q^{67} - 22 q^{68} - 12 q^{69} + 40 q^{71} + q^{72} - 2 q^{73} + 14 q^{74} + q^{75} - q^{76} - 8 q^{78} - 16 q^{79} + 8 q^{80} - q^{81} + 7 q^{82} - q^{83} + 6 q^{85} + 18 q^{86} - 6 q^{87} - 2 q^{88} - 11 q^{89} + 2 q^{90} - 12 q^{92} - 12 q^{93} + 4 q^{94} + 8 q^{95} + q^{96} + 29 q^{97} - 7 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/786\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(263\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i −0.809017 0.587785i −0.809017 0.587785i 0.618034 1.90211i 0.809017 0.587785i 0 0.809017 0.587785i 0.309017 + 0.951057i 1.61803 + 1.17557i
451.1 −0.309017 0.951057i −0.809017 + 0.587785i −0.809017 + 0.587785i 0.618034 + 1.90211i 0.809017 + 0.587785i 0 0.809017 + 0.587785i 0.309017 0.951057i 1.61803 1.17557i
577.1 0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 + 0.951057i −1.61803 1.17557i −0.309017 + 0.951057i 0 −0.309017 + 0.951057i −0.809017 + 0.587785i −0.618034 1.90211i
613.1 0.809017 0.587785i 0.309017 0.951057i 0.309017 0.951057i −1.61803 + 1.17557i −0.309017 0.951057i 0 −0.309017 0.951057i −0.809017 0.587785i −0.618034 + 1.90211i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
131.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 786.2.e.a 4
131.c even 5 1 inner 786.2.e.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
786.2.e.a 4 1.a even 1 1 trivial
786.2.e.a 4 131.c even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2T_{5}^{3} + 4T_{5}^{2} + 8T_{5} + 16 \) acting on \(S_{2}^{\mathrm{new}}(786, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + 24 T^{2} - 133 T + 361 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} + 24 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + 34 T^{2} - 87 T + 841 \) Copy content Toggle raw display
$19$ \( (T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} + 24 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$29$ \( T^{4} - 14 T^{3} + 76 T^{2} - 24 T + 16 \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + 144 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} + 76 T^{2} - 56 T + 16 \) Copy content Toggle raw display
$41$ \( T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121 \) Copy content Toggle raw display
$43$ \( T^{4} + 13 T^{3} + 204 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$47$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + 16 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + 94 T^{2} + \cdots + 7921 \) Copy content Toggle raw display
$61$ \( T^{4} - 10 T^{3} + 60 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$67$ \( T^{4} - 8 T^{3} + 34 T^{2} - 87 T + 841 \) Copy content Toggle raw display
$71$ \( (T^{2} - 20 T + 80)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} + 124 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T - 64)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + T^{3} + 226 T^{2} + \cdots + 10201 \) Copy content Toggle raw display
$89$ \( T^{4} + 11 T^{3} + 46 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$97$ \( T^{4} - 29 T^{3} + 316 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
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