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Properties

Label 507.2.k.h
Level $507$
Weight $2$
Character orbit 507.k
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(80,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.80");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.k (of order \(12\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (\beta_{3} + 2 \beta_{2}) q^{3} + ( - \beta_{4} + 2 \beta_{2} + 1) q^{4} + (2 \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + ( - 3 \beta_{4} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (\beta_{3} + 2 \beta_{2}) q^{3} + ( - \beta_{4} + 2 \beta_{2} + 1) q^{4} + (2 \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + (3 \beta_{6} + 3 \beta_{5} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{4} - 12 q^{9} - 36 q^{10} - 4 q^{16} + 28 q^{22} + 12 q^{30} - 36 q^{36} - 72 q^{40} - 24 q^{48} + 32 q^{55} + 120 q^{66} + 60 q^{75} - 36 q^{81} + 12 q^{82} + 84 q^{88} - 20 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 427\nu^{2} + 335\nu - 181 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 29\nu^{6} - 89\nu^{5} + 261\nu^{4} - 373\nu^{3} + 498\nu^{2} - 294\nu + 152 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8\nu^{7} - 28\nu^{6} + 114\nu^{5} - 215\nu^{4} + 378\nu^{3} - 366\nu^{2} + 266\nu - 97 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -21\nu^{7} + 55\nu^{6} - 216\nu^{5} + 273\nu^{4} - 428\nu^{3} + 156\nu^{2} - 97\nu - 46 ) / 37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -24\nu^{7} + 84\nu^{6} - 305\nu^{5} + 534\nu^{4} - 801\nu^{3} + 617\nu^{2} - 317\nu - 5 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -24\nu^{7} + 84\nu^{6} - 305\nu^{5} + 571\nu^{4} - 875\nu^{3} + 876\nu^{2} - 539\nu + 217 ) / 37 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - 4\beta_{6} + 3\beta_{5} - 3\beta_{4} + 7\beta_{3} + 4\beta_{2} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} - 3\beta_{4} + 4\beta_{3} + 8\beta_{2} - 4\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{7} + 13\beta_{6} - 14\beta_{5} + 15\beta_{4} - 26\beta_{3} - \beta_{2} - 11\beta _1 + 41 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5\beta_{7} + 16\beta_{6} - 4\beta_{5} + 30\beta_{4} - 31\beta_{3} - 40\beta_{2} + 11\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -46\beta_{7} - 25\beta_{6} + 63\beta_{5} - 14\beta_{4} + 71\beta_{3} - 83\beta_{2} + 77\beta _1 - 167 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
0.500000 − 0.564882i
0.500000 + 1.56488i
0.500000 − 2.19293i
0.500000 + 1.19293i
0.500000 + 2.19293i
0.500000 − 1.19293i
0.500000 + 0.564882i
0.500000 − 1.56488i
−0.389774 + 1.45466i −0.866025 − 1.50000i −0.232051 − 0.133975i 2.90931 + 2.90931i 2.51954 − 0.675108i 0 −1.84443 + 1.84443i −1.50000 + 2.59808i −5.36603 + 3.09808i
80.2 0.389774 − 1.45466i −0.866025 − 1.50000i −0.232051 − 0.133975i −2.90931 − 2.90931i −2.51954 + 0.675108i 0 1.84443 − 1.84443i −1.50000 + 2.59808i −5.36603 + 3.09808i
89.1 −2.31259 − 0.619657i 0.866025 + 1.50000i 3.23205 + 1.86603i 1.23931 − 1.23931i −1.07328 − 4.00552i 0 −2.93225 − 2.93225i −1.50000 + 2.59808i −3.63397 + 2.09808i
89.2 2.31259 + 0.619657i 0.866025 + 1.50000i 3.23205 + 1.86603i −1.23931 + 1.23931i 1.07328 + 4.00552i 0 2.93225 + 2.93225i −1.50000 + 2.59808i −3.63397 + 2.09808i
188.1 −2.31259 + 0.619657i 0.866025 − 1.50000i 3.23205 − 1.86603i 1.23931 + 1.23931i −1.07328 + 4.00552i 0 −2.93225 + 2.93225i −1.50000 − 2.59808i −3.63397 − 2.09808i
188.2 2.31259 − 0.619657i 0.866025 − 1.50000i 3.23205 − 1.86603i −1.23931 − 1.23931i 1.07328 − 4.00552i 0 2.93225 − 2.93225i −1.50000 − 2.59808i −3.63397 − 2.09808i
488.1 −0.389774 − 1.45466i −0.866025 + 1.50000i −0.232051 + 0.133975i 2.90931 − 2.90931i 2.51954 + 0.675108i 0 −1.84443 − 1.84443i −1.50000 − 2.59808i −5.36603 − 3.09808i
488.2 0.389774 + 1.45466i −0.866025 + 1.50000i −0.232051 + 0.133975i −2.90931 + 2.90931i −2.51954 − 0.675108i 0 1.84443 + 1.84443i −1.50000 − 2.59808i −5.36603 − 3.09808i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.2
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.f odd 12 2 inner
39.k even 12 2 inner

Twists

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

Hecke kernels

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

\( T_{2}^{8} - 6T_{2}^{6} - T_{2}^{4} + 78T_{2}^{2} + 169 \) 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

$p$ $F_p(T)$
$2$ \( T^{8} - 6 T^{6} + \cdots + 169 \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 296T^{4} + 2704 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 72 T^{6} + \cdots + 2704 \) 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
$41$ \( T^{8} + 72 T^{6} + \cdots + 39589264 \) Copy content Toggle raw display
$43$ \( (T^{4} - 16 T^{2} + 256)^{2} \) 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
$59$ \( T^{8} + 408 T^{6} + \cdots + 2704 \) Copy content Toggle raw display
$61$ \( (T^{4} + 192 T^{2} + 36864)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + 408 T^{6} + \cdots + 39589264 \) 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
$89$ \( T^{8} - 552 T^{6} + \cdots + 39589264 \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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