Properties

Label 507.2.f.d
Level $507$
Weight $2$
Character orbit 507.f
Analytic conductor $4.048$
Analytic rank $0$
Dimension $8$
CM discriminant -39
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(239,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.f (of order \(4\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.56070144.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{4} q^{3} + ( - \beta_{5} - 2 \beta_{2}) q^{4} + \beta_{6} q^{5} + ( - \beta_{6} + \beta_1) q^{6} + (\beta_{7} + \beta_{3}) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{4} q^{3} + ( - \beta_{5} - 2 \beta_{2}) q^{4} + \beta_{6} q^{5} + ( - \beta_{6} + \beta_1) q^{6} + (\beta_{7} + \beta_{3}) q^{8} + 3 q^{9} + ( - 3 \beta_{5} + \beta_{2}) q^{10} + (\beta_{7} + 2 \beta_{3}) q^{11} + (2 \beta_{5} + 3 \beta_{2}) q^{12} + (\beta_{6} + 2 \beta_1) q^{15} + (2 \beta_{4} + 1) q^{16} - 3 \beta_1 q^{18} + (\beta_{7} + 2 \beta_{3}) q^{20} + (5 \beta_{4} - 7) q^{22} - 3 \beta_{3} q^{24} + (4 \beta_{5} - 5 \beta_{2}) q^{25} + 3 \beta_{4} q^{27} + ( - \beta_{5} + 9 \beta_{2}) q^{30} - \beta_1 q^{32} + ( - \beta_{7} - 4 \beta_{3}) q^{33} + ( - 3 \beta_{5} - 6 \beta_{2}) q^{36} + ( - \beta_{4} - 9) q^{40} + ( - \beta_{6} - 4 \beta_1) q^{41} - 4 \beta_{2} q^{43} + ( - 3 \beta_{6} + 8 \beta_1) q^{44} + 3 \beta_{6} q^{45} + ( - 3 \beta_{7} - 2 \beta_{3}) q^{47} + (\beta_{4} + 6) q^{48} + 7 \beta_{2} q^{49} + ( - 4 \beta_{7} + \beta_{3}) q^{50} + ( - 3 \beta_{6} + 3 \beta_1) q^{54} + (2 \beta_{4} - 8) q^{55} + ( - 3 \beta_{7} + 2 \beta_{3}) q^{59} + ( - \beta_{7} - 4 \beta_{3}) q^{60} - 8 \beta_{4} q^{61} + ( - 5 \beta_{5} - 2 \beta_{2}) q^{64} + ( - 7 \beta_{4} + 15) q^{66} + (\beta_{6} - 6 \beta_1) q^{71} + (3 \beta_{7} + 3 \beta_{3}) q^{72} + (5 \beta_{5} - 12 \beta_{2}) q^{75} + 6 \beta_{4} q^{79} + (3 \beta_{6} + 4 \beta_1) q^{80} + 9 q^{81} + ( - \beta_{5} - 17 \beta_{2}) q^{82} + ( - \beta_{6} - 6 \beta_1) q^{83} + 4 \beta_{3} q^{86} + (7 \beta_{5} + 15 \beta_{2}) q^{88} + (3 \beta_{7} - 4 \beta_{3}) q^{89} + ( - 9 \beta_{5} + 3 \beta_{2}) q^{90} + ( - 11 \beta_{4} + 5) q^{94} + ( - \beta_{6} + \beta_1) q^{96} - 7 \beta_{3} q^{98} + (3 \beta_{7} + 6 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} + 8 q^{16} - 56 q^{22} - 72 q^{40} + 48 q^{48} - 64 q^{55} + 120 q^{66} + 72 q^{81} + 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{7} + 8\nu^{6} - 22\nu^{5} + 146\nu^{4} - 256\nu^{3} + 390\nu^{2} - 335\nu + 107 ) / 37 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6\nu^{7} - 21\nu^{6} + 67\nu^{5} - 115\nu^{4} + 117\nu^{3} - 71\nu^{2} - 41\nu + 29 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 29\nu^{6} - 89\nu^{5} + 261\nu^{4} - 373\nu^{3} + 461\nu^{2} - 220\nu + 41 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} - 3\nu^{5} + 11\nu^{4} - 17\nu^{3} + 25\nu^{2} - 17\nu + 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\nu^{7} - 56\nu^{6} + 228\nu^{5} - 430\nu^{4} + 756\nu^{3} - 732\nu^{2} + 532\nu - 157 ) / 37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -42\nu^{7} + 147\nu^{6} - 543\nu^{5} + 953\nu^{4} - 1485\nu^{3} + 1163\nu^{2} - 749\nu + 56 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -42\nu^{7} + 147\nu^{6} - 543\nu^{5} + 1027\nu^{4} - 1633\nu^{3} + 1681\nu^{2} - 1193\nu + 500 ) / 37 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} - 2\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} - 3\beta_{5} + 3\beta_{4} - 8\beta_{3} - 11\beta_{2} + 2\beta _1 - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{6} - 3\beta_{5} - 4\beta_{4} - 2\beta_{3} - 12\beta_{2} + 10\beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{7} - \beta_{6} + 15\beta_{5} - 25\beta_{4} + 36\beta_{3} + 27\beta_{2} + 14\beta _1 + 67 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5\beta_{7} + 12\beta_{6} + 30\beta_{5} + 9\beta_{4} + 25\beta_{3} + 71\beta_{2} - 39\beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -23\beta_{7} + 19\beta_{6} - 7\beta_{5} + 77\beta_{4} - 67\beta_{3} + 6\beta_{2} - 73\beta _1 - 160 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
0.500000 2.19293i
0.500000 1.56488i
0.500000 + 0.564882i
0.500000 + 1.19293i
0.500000 + 2.19293i
0.500000 + 1.56488i
0.500000 0.564882i
0.500000 1.19293i
−1.69293 1.69293i −1.73205 3.73205i −1.23931 1.23931i 2.93225 + 2.93225i 0 2.93225 2.93225i 3.00000 4.19615i
239.2 −1.06488 1.06488i 1.73205 0.267949i 2.90931 + 2.90931i −1.84443 1.84443i 0 −1.84443 + 1.84443i 3.00000 6.19615i
239.3 1.06488 + 1.06488i 1.73205 0.267949i −2.90931 2.90931i 1.84443 + 1.84443i 0 1.84443 1.84443i 3.00000 6.19615i
239.4 1.69293 + 1.69293i −1.73205 3.73205i 1.23931 + 1.23931i −2.93225 2.93225i 0 −2.93225 + 2.93225i 3.00000 4.19615i
437.1 −1.69293 + 1.69293i −1.73205 3.73205i −1.23931 + 1.23931i 2.93225 2.93225i 0 2.93225 + 2.93225i 3.00000 4.19615i
437.2 −1.06488 + 1.06488i 1.73205 0.267949i 2.90931 2.90931i −1.84443 + 1.84443i 0 −1.84443 1.84443i 3.00000 6.19615i
437.3 1.06488 1.06488i 1.73205 0.267949i −2.90931 + 2.90931i 1.84443 1.84443i 0 1.84443 + 1.84443i 3.00000 6.19615i
437.4 1.69293 1.69293i −1.73205 3.73205i 1.23931 1.23931i −2.93225 + 2.93225i 0 −2.93225 2.93225i 3.00000 4.19615i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
13.b even 2 1 inner
13.d odd 4 2 inner
39.f even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.f.d 8
3.b odd 2 1 inner 507.2.f.d 8
13.b even 2 1 inner 507.2.f.d 8
13.c even 3 1 507.2.k.g 8
13.c even 3 1 507.2.k.h 8
13.d odd 4 2 inner 507.2.f.d 8
13.e even 6 1 507.2.k.g 8
13.e even 6 1 507.2.k.h 8
13.f odd 12 2 507.2.k.g 8
13.f odd 12 2 507.2.k.h 8
39.d odd 2 1 CM 507.2.f.d 8
39.f even 4 2 inner 507.2.f.d 8
39.h odd 6 1 507.2.k.g 8
39.h odd 6 1 507.2.k.h 8
39.i odd 6 1 507.2.k.g 8
39.i odd 6 1 507.2.k.h 8
39.k even 12 2 507.2.k.g 8
39.k even 12 2 507.2.k.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.f.d 8 1.a even 1 1 trivial
507.2.f.d 8 3.b odd 2 1 inner
507.2.f.d 8 13.b even 2 1 inner
507.2.f.d 8 13.d odd 4 2 inner
507.2.f.d 8 39.d odd 2 1 CM
507.2.f.d 8 39.f even 4 2 inner
507.2.k.g 8 13.c even 3 1
507.2.k.g 8 13.e even 6 1
507.2.k.g 8 13.f odd 12 2
507.2.k.g 8 39.h odd 6 1
507.2.k.g 8 39.i odd 6 1
507.2.k.g 8 39.k even 12 2
507.2.k.h 8 13.c even 3 1
507.2.k.h 8 13.e even 6 1
507.2.k.h 8 13.f odd 12 2
507.2.k.h 8 39.h odd 6 1
507.2.k.h 8 39.i odd 6 1
507.2.k.h 8 39.k even 12 2

Hecke kernels

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

\( T_{2}^{8} + 38T_{2}^{4} + 169 \) Copy content Toggle raw display
\( T_{5}^{8} + 296T_{5}^{4} + 2704 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 38T^{4} + 169 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 296T^{4} + 2704 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 1832 T^{4} + 2704 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} + 14312 T^{4} + 39589264 \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 17768 T^{4} + 77228944 \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} + 55592 T^{4} + 2704 \) Copy content Toggle raw display
$61$ \( (T^{2} - 192)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + 68072 T^{4} + 39589264 \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 55208 T^{4} + 756690064 \) Copy content Toggle raw display
$89$ \( T^{8} + 114152 T^{4} + 39589264 \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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