Properties

Label 507.2.e.c
Level $507$
Weight $2$
Character orbit 507.e
Analytic conductor $4.048$
Analytic rank $0$
Dimension $2$
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] = [507,2,Mod(22,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.22");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.e (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: \(4.04841538248\)
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 39)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{4} + q^{5} - \zeta_{6} q^{6} + 2 \zeta_{6} q^{7} + 3 q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{4} + q^{5} - \zeta_{6} q^{6} + 2 \zeta_{6} q^{7} + 3 q^{8} - \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{10} + (2 \zeta_{6} - 2) q^{11} + q^{12} + 2 q^{14} + ( - \zeta_{6} + 1) q^{15} + ( - \zeta_{6} + 1) q^{16} + 7 \zeta_{6} q^{17} - q^{18} - 6 \zeta_{6} q^{19} + \zeta_{6} q^{20} + 2 q^{21} + 2 \zeta_{6} q^{22} + ( - 6 \zeta_{6} + 6) q^{23} + ( - 3 \zeta_{6} + 3) q^{24} - 4 q^{25} - q^{27} + (2 \zeta_{6} - 2) q^{28} + ( - \zeta_{6} + 1) q^{29} - \zeta_{6} q^{30} - 4 q^{31} + 5 \zeta_{6} q^{32} + 2 \zeta_{6} q^{33} + 7 q^{34} + 2 \zeta_{6} q^{35} + ( - \zeta_{6} + 1) q^{36} + ( - \zeta_{6} + 1) q^{37} - 6 q^{38} + 3 q^{40} + ( - 9 \zeta_{6} + 9) q^{41} + ( - 2 \zeta_{6} + 2) q^{42} - 6 \zeta_{6} q^{43} - 2 q^{44} - \zeta_{6} q^{45} - 6 \zeta_{6} q^{46} - 6 q^{47} - \zeta_{6} q^{48} + ( - 3 \zeta_{6} + 3) q^{49} + (4 \zeta_{6} - 4) q^{50} + 7 q^{51} - 9 q^{53} + (\zeta_{6} - 1) q^{54} + (2 \zeta_{6} - 2) q^{55} + 6 \zeta_{6} q^{56} - 6 q^{57} - \zeta_{6} q^{58} + q^{60} - \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} + ( - 2 \zeta_{6} + 2) q^{63} + 7 q^{64} + 2 q^{66} + (2 \zeta_{6} - 2) q^{67} + (7 \zeta_{6} - 7) q^{68} - 6 \zeta_{6} q^{69} + 2 q^{70} + 6 \zeta_{6} q^{71} - 3 \zeta_{6} q^{72} - 11 q^{73} - \zeta_{6} q^{74} + (4 \zeta_{6} - 4) q^{75} + ( - 6 \zeta_{6} + 6) q^{76} - 4 q^{77} - 4 q^{79} + ( - \zeta_{6} + 1) q^{80} + (\zeta_{6} - 1) q^{81} - 9 \zeta_{6} q^{82} + 14 q^{83} + 2 \zeta_{6} q^{84} + 7 \zeta_{6} q^{85} - 6 q^{86} - \zeta_{6} q^{87} + (6 \zeta_{6} - 6) q^{88} + (14 \zeta_{6} - 14) q^{89} - q^{90} + 6 q^{92} + (4 \zeta_{6} - 4) q^{93} + (6 \zeta_{6} - 6) q^{94} - 6 \zeta_{6} q^{95} + 5 q^{96} - 2 \zeta_{6} q^{97} - 3 \zeta_{6} q^{98} + 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} + 2 q^{7} + 6 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} + 2 q^{7} + 6 q^{8} - q^{9} + q^{10} - 2 q^{11} + 2 q^{12} + 4 q^{14} + q^{15} + q^{16} + 7 q^{17} - 2 q^{18} - 6 q^{19} + q^{20} + 4 q^{21} + 2 q^{22} + 6 q^{23} + 3 q^{24} - 8 q^{25} - 2 q^{27} - 2 q^{28} + q^{29} - q^{30} - 8 q^{31} + 5 q^{32} + 2 q^{33} + 14 q^{34} + 2 q^{35} + q^{36} + q^{37} - 12 q^{38} + 6 q^{40} + 9 q^{41} + 2 q^{42} - 6 q^{43} - 4 q^{44} - q^{45} - 6 q^{46} - 12 q^{47} - q^{48} + 3 q^{49} - 4 q^{50} + 14 q^{51} - 18 q^{53} - q^{54} - 2 q^{55} + 6 q^{56} - 12 q^{57} - q^{58} + 2 q^{60} - q^{61} - 4 q^{62} + 2 q^{63} + 14 q^{64} + 4 q^{66} - 2 q^{67} - 7 q^{68} - 6 q^{69} + 4 q^{70} + 6 q^{71} - 3 q^{72} - 22 q^{73} - q^{74} - 4 q^{75} + 6 q^{76} - 8 q^{77} - 8 q^{79} + q^{80} - q^{81} - 9 q^{82} + 28 q^{83} + 2 q^{84} + 7 q^{85} - 12 q^{86} - q^{87} - 6 q^{88} - 14 q^{89} - 2 q^{90} + 12 q^{92} - 4 q^{93} - 6 q^{94} - 6 q^{95} + 10 q^{96} - 2 q^{97} - 3 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 1.00000 + 1.73205i 3.00000 −0.500000 0.866025i 0.500000 0.866025i
484.1 0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 1.00000 1.73205i 3.00000 −0.500000 + 0.866025i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.e.c 2
13.b even 2 1 39.2.e.a 2
13.c even 3 1 507.2.a.b 1
13.c even 3 1 inner 507.2.e.c 2
13.d odd 4 2 507.2.j.d 4
13.e even 6 1 39.2.e.a 2
13.e even 6 1 507.2.a.c 1
13.f odd 12 2 507.2.b.b 2
13.f odd 12 2 507.2.j.d 4
39.d odd 2 1 117.2.g.b 2
39.h odd 6 1 117.2.g.b 2
39.h odd 6 1 1521.2.a.a 1
39.i odd 6 1 1521.2.a.d 1
39.k even 12 2 1521.2.b.c 2
52.b odd 2 1 624.2.q.c 2
52.i odd 6 1 624.2.q.c 2
52.i odd 6 1 8112.2.a.w 1
52.j odd 6 1 8112.2.a.bc 1
65.d even 2 1 975.2.i.f 2
65.h odd 4 2 975.2.bb.d 4
65.l even 6 1 975.2.i.f 2
65.r odd 12 2 975.2.bb.d 4
156.h even 2 1 1872.2.t.j 2
156.r even 6 1 1872.2.t.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 13.b even 2 1
39.2.e.a 2 13.e even 6 1
117.2.g.b 2 39.d odd 2 1
117.2.g.b 2 39.h odd 6 1
507.2.a.b 1 13.c even 3 1
507.2.a.c 1 13.e even 6 1
507.2.b.b 2 13.f odd 12 2
507.2.e.c 2 1.a even 1 1 trivial
507.2.e.c 2 13.c even 3 1 inner
507.2.j.d 4 13.d odd 4 2
507.2.j.d 4 13.f odd 12 2
624.2.q.c 2 52.b odd 2 1
624.2.q.c 2 52.i odd 6 1
975.2.i.f 2 65.d even 2 1
975.2.i.f 2 65.l even 6 1
975.2.bb.d 4 65.h odd 4 2
975.2.bb.d 4 65.r odd 12 2
1521.2.a.a 1 39.h odd 6 1
1521.2.a.d 1 39.i odd 6 1
1521.2.b.c 2 39.k even 12 2
1872.2.t.j 2 156.h even 2 1
1872.2.t.j 2 156.r even 6 1
8112.2.a.w 1 52.i odd 6 1
8112.2.a.bc 1 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) 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
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( (T + 11)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T - 14)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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