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Properties

Label 507.2.j.d
Level $507$
Weight $2$
Character orbit 507.j
Analytic conductor $4.048$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(316,507)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.316");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.j (of order \(6\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i 0.500000 − 0.866025i −0.500000 − 0.866025i − 1.00000i −0.866025 + 0.500000i −1.73205 + 1.00000i 3.00000i −0.500000 − 0.866025i −0.500000 + 0.866025i
316.2 0.866025 + 0.500000i 0.500000 − 0.866025i −0.500000 − 0.866025i 1.00000i 0.866025 − 0.500000i 1.73205 − 1.00000i − 3.00000i −0.500000 − 0.866025i −0.500000 + 0.866025i
361.1 −0.866025 + 0.500000i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000i −0.866025 − 0.500000i −1.73205 − 1.00000i − 3.00000i −0.500000 + 0.866025i −0.500000 − 0.866025i
361.2 0.866025 − 0.500000i 0.500000 + 0.866025i −0.500000 + 0.866025i − 1.00000i 0.866025 + 0.500000i 1.73205 + 1.00000i 3.00000i −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.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

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

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$43$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$53$ \( (T + 9)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$73$ \( (T^{2} + 121)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 196 T^{2} + 38416 \) Copy content Toggle raw display
$97$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
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