Properties

Label 507.2.b.g
Level $507$
Weight $2$
Character orbit 507.b
Analytic conductor $4.048$
Analytic rank $0$
Dimension $6$
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(337,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(2))
 
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("507.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), 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: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + \beta_{2} q^{4} + (\beta_{5} - \beta_{3}) q^{5} + \beta_1 q^{6} + ( - 3 \beta_{5} + \beta_{3}) q^{7} + (\beta_{3} + \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + \beta_{2} q^{4} + (\beta_{5} - \beta_{3}) q^{5} + \beta_1 q^{6} + ( - 3 \beta_{5} + \beta_{3}) q^{7} + (\beta_{3} + \beta_1) q^{8} + q^{9} + ( - 2 \beta_{4} + 1) q^{10} + ( - 2 \beta_{5} - 4 \beta_{3} + 3 \beta_1) q^{11} + \beta_{2} q^{12} + (4 \beta_{4} - 1) q^{14} + (\beta_{5} - \beta_{3}) q^{15} + (\beta_{4} + 3 \beta_{2} - 3) q^{16} + (\beta_{2} + 2) q^{17} + \beta_1 q^{18} + (3 \beta_{5} + \beta_{3} + 3 \beta_1) q^{19} - \beta_1 q^{20} + ( - 3 \beta_{5} + \beta_{3}) q^{21} + ( - 2 \beta_{4} + 3 \beta_{2} - 2) q^{22} + ( - 3 \beta_{4} - 5 \beta_{2} + 2) q^{23} + (\beta_{3} + \beta_1) q^{24} + (3 \beta_{4} + 2 \beta_{2}) q^{25} + q^{27} + ( - 2 \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{28} + ( - 3 \beta_{4} - 2 \beta_{2} - 1) q^{29} + ( - 2 \beta_{4} + 1) q^{30} + (5 \beta_{5} + 2 \beta_{3} - 5 \beta_1) q^{31} + (\beta_{5} + 4 \beta_{3} - 3 \beta_1) q^{32} + ( - 2 \beta_{5} - 4 \beta_{3} + 3 \beta_1) q^{33} + (\beta_{3} + \beta_1) q^{34} + ( - 5 \beta_{4} - 4 \beta_{2} + 9) q^{35} + \beta_{2} q^{36} + ( - 4 \beta_{5} + \beta_{3} - \beta_1) q^{37} + ( - 2 \beta_{4} + 3 \beta_{2} - 7) q^{38} + ( - 4 \beta_{4} - \beta_{2} + 4) q^{40} + \beta_1 q^{41} + (4 \beta_{4} - 1) q^{42} + (2 \beta_{4} - 2 \beta_{2} + 1) q^{43} + ( - 6 \beta_{5} - 3 \beta_{3} - \beta_1) q^{44} + (\beta_{5} - \beta_{3}) q^{45} + ( - 3 \beta_{5} - 2 \beta_{3} + 4 \beta_1) q^{46} + (7 \beta_{5} + 3 \beta_{3} - 9 \beta_1) q^{47} + (\beta_{4} + 3 \beta_{2} - 3) q^{48} + (7 \beta_{4} + 6 \beta_{2} - 10) q^{49} + (3 \beta_{5} - \beta_{3} + \beta_1) q^{50} + (\beta_{2} + 2) q^{51} + (3 \beta_{4} + 2 \beta_{2} - 6) q^{53} + \beta_1 q^{54} + (2 \beta_{2} - 5) q^{55} + (8 \beta_{4} + 3 \beta_{2} - 6) q^{56} + (3 \beta_{5} + \beta_{3} + 3 \beta_1) q^{57} + ( - 3 \beta_{5} + \beta_{3} - 2 \beta_1) q^{58} + (8 \beta_{5} + 6 \beta_{3} - 4 \beta_1) q^{59} - \beta_1 q^{60} + (2 \beta_{4} - 3 \beta_{2} - 4) q^{61} + ( - 3 \beta_{4} - 5 \beta_{2} + 8) q^{62} + ( - 3 \beta_{5} + \beta_{3}) q^{63} + (5 \beta_{4} + 3 \beta_{2} - 4) q^{64} + ( - 2 \beta_{4} + 3 \beta_{2} - 2) q^{66} + (4 \beta_{5} + 3 \beta_{3} - 4 \beta_1) q^{67} + (\beta_{4} + 3 \beta_{2} + 1) q^{68} + ( - 3 \beta_{4} - 5 \beta_{2} + 2) q^{69} + ( - 5 \beta_{5} + \beta_{3} + 8 \beta_1) q^{70} + ( - \beta_{5} - 2 \beta_{3} - 5 \beta_1) q^{71} + (\beta_{3} + \beta_1) q^{72} + ( - 7 \beta_{5} - \beta_{3} + 2 \beta_1) q^{73} + (5 \beta_{4} - \beta_{2} + 1) q^{74} + (3 \beta_{4} + 2 \beta_{2}) q^{75} + (4 \beta_{5} + 7 \beta_{3} - 6 \beta_1) q^{76} + ( - 2 \beta_{4} - 10 \beta_{2} + 9) q^{77} + (\beta_{4} + 5 \beta_{2} - 5) q^{79} + ( - 4 \beta_{5} + 3 \beta_{3} - \beta_1) q^{80} + q^{81} + (\beta_{2} - 2) q^{82} + ( - 6 \beta_{5} - 3 \beta_{3} - \beta_1) q^{83} + ( - 2 \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{84} + (2 \beta_{5} - 2 \beta_{3} - \beta_1) q^{85} + (2 \beta_{5} - 4 \beta_{3} + 5 \beta_1) q^{86} + ( - 3 \beta_{4} - 2 \beta_{2} - 1) q^{87} + ( - \beta_{4} + 5 \beta_{2} + 1) q^{88} + (2 \beta_{5} - \beta_{3} - 2 \beta_1) q^{89} + ( - 2 \beta_{4} + 1) q^{90} + ( - 5 \beta_{4} - 6 \beta_{2} - 2) q^{92} + (5 \beta_{5} + 2 \beta_{3} - 5 \beta_1) q^{93} + ( - 4 \beta_{4} - 9 \beta_{2} + 15) q^{94} + ( - 5 \beta_{4} + 2 \beta_{2}) q^{95} + (\beta_{5} + 4 \beta_{3} - 3 \beta_1) q^{96} + ( - 3 \beta_{5} - 10 \beta_{3} + 4 \beta_1) q^{97} + (7 \beta_{5} - \beta_{3} - 9 \beta_1) q^{98} + ( - 2 \beta_{5} - 4 \beta_{3} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 2 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + 2 q^{4} + 6 q^{9} + 2 q^{10} + 2 q^{12} + 2 q^{14} - 10 q^{16} + 14 q^{17} - 10 q^{22} - 4 q^{23} + 10 q^{25} + 6 q^{27} - 16 q^{29} + 2 q^{30} + 36 q^{35} + 2 q^{36} - 40 q^{38} + 14 q^{40} + 2 q^{42} + 6 q^{43} - 10 q^{48} - 34 q^{49} + 14 q^{51} - 26 q^{53} - 26 q^{55} - 14 q^{56} - 26 q^{61} + 32 q^{62} - 8 q^{64} - 10 q^{66} + 14 q^{68} - 4 q^{69} + 14 q^{74} + 10 q^{75} + 30 q^{77} - 18 q^{79} + 6 q^{81} - 10 q^{82} - 16 q^{87} + 14 q^{88} + 2 q^{90} - 34 q^{92} + 64 q^{94} - 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 5x^{4} + 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 3\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 4\nu^{3} + 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 3\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{3} + 9\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.80194i
1.24698i
0.445042i
0.445042i
1.24698i
1.80194i
1.80194i 1.00000 −1.24698 1.44504i 1.80194i 3.44504i 1.35690i 1.00000 −2.60388
337.2 1.24698i 1.00000 0.445042 2.80194i 1.24698i 4.80194i 3.04892i 1.00000 3.49396
337.3 0.445042i 1.00000 1.80194 0.246980i 0.445042i 1.75302i 1.69202i 1.00000 0.109916
337.4 0.445042i 1.00000 1.80194 0.246980i 0.445042i 1.75302i 1.69202i 1.00000 0.109916
337.5 1.24698i 1.00000 0.445042 2.80194i 1.24698i 4.80194i 3.04892i 1.00000 3.49396
337.6 1.80194i 1.00000 −1.24698 1.44504i 1.80194i 3.44504i 1.35690i 1.00000 −2.60388
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.b.g 6
3.b odd 2 1 1521.2.b.m 6
13.b even 2 1 inner 507.2.b.g 6
13.c even 3 2 507.2.j.h 12
13.d odd 4 1 507.2.a.j 3
13.d odd 4 1 507.2.a.k yes 3
13.e even 6 2 507.2.j.h 12
13.f odd 12 2 507.2.e.j 6
13.f odd 12 2 507.2.e.k 6
39.d odd 2 1 1521.2.b.m 6
39.f even 4 1 1521.2.a.p 3
39.f even 4 1 1521.2.a.q 3
52.f even 4 1 8112.2.a.by 3
52.f even 4 1 8112.2.a.cf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.j 3 13.d odd 4 1
507.2.a.k yes 3 13.d odd 4 1
507.2.b.g 6 1.a even 1 1 trivial
507.2.b.g 6 13.b even 2 1 inner
507.2.e.j 6 13.f odd 12 2
507.2.e.k 6 13.f odd 12 2
507.2.j.h 12 13.c even 3 2
507.2.j.h 12 13.e even 6 2
1521.2.a.p 3 39.f even 4 1
1521.2.a.q 3 39.f even 4 1
1521.2.b.m 6 3.b odd 2 1
1521.2.b.m 6 39.d odd 2 1
8112.2.a.by 3 52.f even 4 1
8112.2.a.cf 3 52.f even 4 1

Hecke kernels

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

\( T_{2}^{6} + 5T_{2}^{4} + 6T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} + 10T_{5}^{4} + 17T_{5}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 10 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} + 38 T^{4} + \cdots + 841 \) Copy content Toggle raw display
$11$ \( T^{6} + 61 T^{4} + \cdots + 1849 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} - 7 T^{2} + 14 T - 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 101 T^{4} + \cdots + 12769 \) Copy content Toggle raw display
$23$ \( (T^{3} + 2 T^{2} - 43 T + 83)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 8 T^{2} + 5 T - 43)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 110 T^{4} + \cdots + 38809 \) Copy content Toggle raw display
$37$ \( T^{6} + 70 T^{4} + \cdots + 8281 \) Copy content Toggle raw display
$41$ \( T^{6} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( (T^{3} - 3 T^{2} - 25 T - 29)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 321 T^{4} + \cdots + 829921 \) Copy content Toggle raw display
$53$ \( (T^{3} + 13 T^{2} + \cdots + 29)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 196 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$61$ \( (T^{3} + 13 T^{2} + \cdots - 223)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 69 T^{4} + \cdots + 9409 \) Copy content Toggle raw display
$71$ \( T^{6} + 194 T^{4} + \cdots + 212521 \) Copy content Toggle raw display
$73$ \( T^{6} + 122 T^{4} + \cdots + 27889 \) Copy content Toggle raw display
$79$ \( (T^{3} + 9 T^{2} + \cdots - 169)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 146 T^{4} + \cdots + 1849 \) Copy content Toggle raw display
$89$ \( T^{6} + 41 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{6} + 363 T^{4} + \cdots + 1413721 \) Copy content Toggle raw display
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