Properties

Label 513.2.g.b
Level $513$
Weight $2$
Character orbit 513.g
Analytic conductor $4.096$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [513,2,Mod(64,513)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(513, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("513.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.g (of order \(3\), degree \(2\), not 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: \(4.09632562369\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + q^{5} + (3 \zeta_{6} - 3) q^{7} - 3 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + q^{5} + (3 \zeta_{6} - 3) q^{7} - 3 q^{8} - \zeta_{6} q^{10} + ( - 3 \zeta_{6} + 3) q^{11} + ( - 6 \zeta_{6} + 6) q^{13} + 3 q^{14} + \zeta_{6} q^{16} + ( - 3 \zeta_{6} + 3) q^{17} + ( - 2 \zeta_{6} - 3) q^{19} + ( - \zeta_{6} + 1) q^{20} - 3 q^{22} + ( - 8 \zeta_{6} + 8) q^{23} - 4 q^{25} - 6 q^{26} + 3 \zeta_{6} q^{28} + 5 q^{29} + 7 \zeta_{6} q^{31} + (5 \zeta_{6} - 5) q^{32} - 3 q^{34} + (3 \zeta_{6} - 3) q^{35} + 2 q^{37} + (5 \zeta_{6} - 2) q^{38} - 3 q^{40} + q^{41} - 8 \zeta_{6} q^{43} - 3 \zeta_{6} q^{44} - 8 q^{46} - 9 q^{47} - 2 \zeta_{6} q^{49} + 4 \zeta_{6} q^{50} - 6 \zeta_{6} q^{52} + 3 \zeta_{6} q^{53} + ( - 3 \zeta_{6} + 3) q^{55} + ( - 9 \zeta_{6} + 9) q^{56} - 5 \zeta_{6} q^{58} - 3 q^{59} + 7 q^{61} + ( - 7 \zeta_{6} + 7) q^{62} + 7 q^{64} + ( - 6 \zeta_{6} + 6) q^{65} + ( - 4 \zeta_{6} + 4) q^{67} - 3 \zeta_{6} q^{68} + 3 q^{70} + (15 \zeta_{6} - 15) q^{71} + ( - 5 \zeta_{6} + 5) q^{73} - 2 \zeta_{6} q^{74} + (3 \zeta_{6} - 5) q^{76} + 9 \zeta_{6} q^{77} + 12 \zeta_{6} q^{79} + \zeta_{6} q^{80} - \zeta_{6} q^{82} + (\zeta_{6} - 1) q^{83} + ( - 3 \zeta_{6} + 3) q^{85} + (8 \zeta_{6} - 8) q^{86} + (9 \zeta_{6} - 9) q^{88} - \zeta_{6} q^{89} + 18 \zeta_{6} q^{91} - 8 \zeta_{6} q^{92} + 9 \zeta_{6} q^{94} + ( - 2 \zeta_{6} - 3) q^{95} + 2 \zeta_{6} q^{97} + (2 \zeta_{6} - 2) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{4} + 2 q^{5} - 3 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{4} + 2 q^{5} - 3 q^{7} - 6 q^{8} - q^{10} + 3 q^{11} + 6 q^{13} + 6 q^{14} + q^{16} + 3 q^{17} - 8 q^{19} + q^{20} - 6 q^{22} + 8 q^{23} - 8 q^{25} - 12 q^{26} + 3 q^{28} + 10 q^{29} + 7 q^{31} - 5 q^{32} - 6 q^{34} - 3 q^{35} + 4 q^{37} + q^{38} - 6 q^{40} + 2 q^{41} - 8 q^{43} - 3 q^{44} - 16 q^{46} - 18 q^{47} - 2 q^{49} + 4 q^{50} - 6 q^{52} + 3 q^{53} + 3 q^{55} + 9 q^{56} - 5 q^{58} - 6 q^{59} + 14 q^{61} + 7 q^{62} + 14 q^{64} + 6 q^{65} + 4 q^{67} - 3 q^{68} + 6 q^{70} - 15 q^{71} + 5 q^{73} - 2 q^{74} - 7 q^{76} + 9 q^{77} + 12 q^{79} + q^{80} - q^{82} - q^{83} + 3 q^{85} - 8 q^{86} - 9 q^{88} - q^{89} + 18 q^{91} - 8 q^{92} + 9 q^{94} - 8 q^{95} + 2 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/513\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(-\zeta_{6}\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 0 0.500000 + 0.866025i 1.00000 0 −1.50000 2.59808i −3.00000 0 −0.500000 + 0.866025i
505.1 −0.500000 0.866025i 0 0.500000 0.866025i 1.00000 0 −1.50000 + 2.59808i −3.00000 0 −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 513.2.g.b 2
3.b odd 2 1 171.2.g.b 2
9.c even 3 1 513.2.h.a 2
9.d odd 6 1 171.2.h.b yes 2
19.c even 3 1 513.2.h.a 2
57.h odd 6 1 171.2.h.b yes 2
171.g even 3 1 inner 513.2.g.b 2
171.n odd 6 1 171.2.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.g.b 2 3.b odd 2 1
171.2.g.b 2 171.n odd 6 1
171.2.h.b yes 2 9.d odd 6 1
171.2.h.b yes 2 57.h odd 6 1
513.2.g.b 2 1.a even 1 1 trivial
513.2.g.b 2 171.g even 3 1 inner
513.2.h.a 2 9.c even 3 1
513.2.h.a 2 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(513, [\chi])\):

\( T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$29$ \( (T - 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( (T - 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$47$ \( (T + 9)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$59$ \( (T + 3)^{2} \) Copy content Toggle raw display
$61$ \( (T - 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$73$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$79$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$83$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$89$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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