Properties

Label 507.2.f.e
Level $507$
Weight $2$
Character orbit 507.f
Analytic conductor $4.048$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{2} + \beta_1) q^{2} + (\beta_{5} + \beta_{4}) q^{3} + (\beta_{5} + 3 \beta_{3} + \beta_{2} - 1) q^{4} + (\beta_{4} + \beta_{2} + \beta_1) q^{5} + ( - \beta_{6} + \beta_{5} + 2 \beta_{3} - 2) q^{6} + ( - \beta_{3} - 1) q^{7} + (2 \beta_{7} + \beta_{5} + \beta_{4} + \cdots - 2) q^{8}+ \cdots + ( - \beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{2} + \beta_1) q^{2} + (\beta_{5} + \beta_{4}) q^{3} + (\beta_{5} + 3 \beta_{3} + \beta_{2} - 1) q^{4} + (\beta_{4} + \beta_{2} + \beta_1) q^{5} + ( - \beta_{6} + \beta_{5} + 2 \beta_{3} - 2) q^{6} + ( - \beta_{3} - 1) q^{7} + (2 \beta_{7} + \beta_{5} + \beta_{4} + \cdots - 2) q^{8}+ \cdots + (2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 16 q^{6} - 8 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 16 q^{6} - 8 q^{7} - 8 q^{9} - 16 q^{15} + 8 q^{16} - 4 q^{18} - 8 q^{19} - 4 q^{21} - 8 q^{22} - 12 q^{24} + 4 q^{27} + 16 q^{28} - 8 q^{31} - 4 q^{33} + 36 q^{34} + 4 q^{37} - 24 q^{40} + 32 q^{42} - 4 q^{45} + 28 q^{48} + 8 q^{54} - 8 q^{55} - 16 q^{57} + 68 q^{58} - 44 q^{60} - 56 q^{61} + 8 q^{63} - 40 q^{66} - 8 q^{67} + 32 q^{70} + 36 q^{72} + 28 q^{73} + 8 q^{76} + 16 q^{79} - 8 q^{81} - 4 q^{84} + 36 q^{85} + 68 q^{87} + 20 q^{93} - 128 q^{94} - 16 q^{96} - 32 q^{97} - 40 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{7} - \nu^{6} - 25\nu^{5} - 46\nu^{4} - 5\nu^{3} - 132\nu^{2} + 28\nu - 55 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -6\nu^{7} + 21\nu^{6} - 67\nu^{5} + 115\nu^{4} - 117\nu^{3} + 71\nu^{2} + 41\nu - 29 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 7\nu^{6} + 47\nu^{5} - 100\nu^{4} + 261\nu^{3} - 295\nu^{2} + 344\nu - 126 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 36\nu^{6} - 136\nu^{5} + 361\nu^{4} - 634\nu^{3} + 793\nu^{2} - 601\nu + 278 ) / 37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -18\nu^{7} + 63\nu^{6} - 238\nu^{5} + 419\nu^{4} - 684\nu^{3} + 546\nu^{2} - 395\nu + 61 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -18\nu^{7} + 63\nu^{6} - 238\nu^{5} + 456\nu^{4} - 758\nu^{3} + 805\nu^{2} - 617\nu + 283 ) / 37 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + 2\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} - \beta_{5} - 5\beta_{4} + 6\beta_{3} - \beta_{2} + 3\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{6} - 3\beta_{5} - 12\beta_{4} + 13\beta_{3} - 3\beta_{2} - 8\beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{7} - \beta_{6} - 2\beta_{5} + 11\beta_{4} - 17\beta_{3} - 2\beta_{2} - 25\beta _1 + 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5\beta_{7} + 12\beta_{6} + 2\beta_{5} + 32\beta_{4} - 42\beta_{3} + 7\beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -46\beta_{7} + 38\beta_{6} + 17\beta_{5} + 6\beta_{4} - 3\beta_{3} + 3\beta_{2} + 140\beta _1 - 96 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(-\beta_{3}\)

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 −0.366025 1.69293i 3.73205i −1.69293 1.69293i −2.24637 + 3.48568i −1.00000 1.00000i 2.93225 2.93225i −2.73205 + 1.23931i 5.73205i
239.2 −1.06488 1.06488i 1.36603 1.06488i 0.267949i −1.06488 1.06488i −2.58863 0.320682i −1.00000 1.00000i −1.84443 + 1.84443i 0.732051 2.90931i 2.26795i
239.3 1.06488 + 1.06488i 1.36603 + 1.06488i 0.267949i 1.06488 + 1.06488i 0.320682 + 2.58863i −1.00000 1.00000i 1.84443 1.84443i 0.732051 + 2.90931i 2.26795i
239.4 1.69293 + 1.69293i −0.366025 + 1.69293i 3.73205i 1.69293 + 1.69293i −3.48568 + 2.24637i −1.00000 1.00000i −2.93225 + 2.93225i −2.73205 1.23931i 5.73205i
437.1 −1.69293 + 1.69293i −0.366025 + 1.69293i 3.73205i −1.69293 + 1.69293i −2.24637 3.48568i −1.00000 + 1.00000i 2.93225 + 2.93225i −2.73205 1.23931i 5.73205i
437.2 −1.06488 + 1.06488i 1.36603 + 1.06488i 0.267949i −1.06488 + 1.06488i −2.58863 + 0.320682i −1.00000 + 1.00000i −1.84443 1.84443i 0.732051 + 2.90931i 2.26795i
437.3 1.06488 1.06488i 1.36603 1.06488i 0.267949i 1.06488 1.06488i 0.320682 2.58863i −1.00000 + 1.00000i 1.84443 + 1.84443i 0.732051 2.90931i 2.26795i
437.4 1.69293 1.69293i −0.366025 1.69293i 3.73205i 1.69293 1.69293i −3.48568 2.24637i −1.00000 + 1.00000i −2.93225 2.93225i −2.73205 + 1.23931i 5.73205i
\(n\): 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
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.f.e 8
3.b odd 2 1 inner 507.2.f.e 8
13.b even 2 1 507.2.f.f 8
13.c even 3 1 507.2.k.d 8
13.c even 3 1 507.2.k.f 8
13.d odd 4 1 inner 507.2.f.e 8
13.d odd 4 1 507.2.f.f 8
13.e even 6 1 39.2.k.b 8
13.e even 6 1 507.2.k.e 8
13.f odd 12 1 39.2.k.b 8
13.f odd 12 1 507.2.k.d 8
13.f odd 12 1 507.2.k.e 8
13.f odd 12 1 507.2.k.f 8
39.d odd 2 1 507.2.f.f 8
39.f even 4 1 inner 507.2.f.e 8
39.f even 4 1 507.2.f.f 8
39.h odd 6 1 39.2.k.b 8
39.h odd 6 1 507.2.k.e 8
39.i odd 6 1 507.2.k.d 8
39.i odd 6 1 507.2.k.f 8
39.k even 12 1 39.2.k.b 8
39.k even 12 1 507.2.k.d 8
39.k even 12 1 507.2.k.e 8
39.k even 12 1 507.2.k.f 8
52.i odd 6 1 624.2.cn.c 8
52.l even 12 1 624.2.cn.c 8
65.l even 6 1 975.2.bo.d 8
65.o even 12 1 975.2.bp.e 8
65.r odd 12 1 975.2.bp.e 8
65.r odd 12 1 975.2.bp.f 8
65.s odd 12 1 975.2.bo.d 8
65.t even 12 1 975.2.bp.f 8
156.r even 6 1 624.2.cn.c 8
156.v odd 12 1 624.2.cn.c 8
195.y odd 6 1 975.2.bo.d 8
195.bc odd 12 1 975.2.bp.f 8
195.bf even 12 1 975.2.bp.e 8
195.bf even 12 1 975.2.bp.f 8
195.bh even 12 1 975.2.bo.d 8
195.bn odd 12 1 975.2.bp.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.b 8 13.e even 6 1
39.2.k.b 8 13.f odd 12 1
39.2.k.b 8 39.h odd 6 1
39.2.k.b 8 39.k even 12 1
507.2.f.e 8 1.a even 1 1 trivial
507.2.f.e 8 3.b odd 2 1 inner
507.2.f.e 8 13.d odd 4 1 inner
507.2.f.e 8 39.f even 4 1 inner
507.2.f.f 8 13.b even 2 1
507.2.f.f 8 13.d odd 4 1
507.2.f.f 8 39.d odd 2 1
507.2.f.f 8 39.f even 4 1
507.2.k.d 8 13.c even 3 1
507.2.k.d 8 13.f odd 12 1
507.2.k.d 8 39.i odd 6 1
507.2.k.d 8 39.k even 12 1
507.2.k.e 8 13.e even 6 1
507.2.k.e 8 13.f odd 12 1
507.2.k.e 8 39.h odd 6 1
507.2.k.e 8 39.k even 12 1
507.2.k.f 8 13.c even 3 1
507.2.k.f 8 13.f odd 12 1
507.2.k.f 8 39.i odd 6 1
507.2.k.f 8 39.k even 12 1
624.2.cn.c 8 52.i odd 6 1
624.2.cn.c 8 52.l even 12 1
624.2.cn.c 8 156.r even 6 1
624.2.cn.c 8 156.v odd 12 1
975.2.bo.d 8 65.l even 6 1
975.2.bo.d 8 65.s odd 12 1
975.2.bo.d 8 195.y odd 6 1
975.2.bo.d 8 195.bh even 12 1
975.2.bp.e 8 65.o even 12 1
975.2.bp.e 8 65.r odd 12 1
975.2.bp.e 8 195.bf even 12 1
975.2.bp.e 8 195.bn odd 12 1
975.2.bp.f 8 65.r odd 12 1
975.2.bp.f 8 65.t even 12 1
975.2.bp.f 8 195.bc odd 12 1
975.2.bp.f 8 195.bf even 12 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 38T^{4} + 169 \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 38T^{4} + 169 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 296T^{4} + 2704 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 30 T^{2} + 117)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{3} + 8 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} + 82 T^{2} + 1573)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} + \cdots + 484)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2 T^{3} + \cdots + 1369)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 998T^{4} + 169 \) Copy content Toggle raw display
$43$ \( (T^{4} + 72 T^{2} + 324)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 9728 T^{4} + 11075584 \) Copy content Toggle raw display
$53$ \( (T^{4} + 22 T^{2} + 13)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 608 T^{4} + 43264 \) Copy content Toggle raw display
$61$ \( (T + 7)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 4 T^{3} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 608 T^{4} + 43264 \) Copy content Toggle raw display
$73$ \( (T^{4} - 14 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$79$ \( (T - 2)^{8} \) Copy content Toggle raw display
$83$ \( T^{8} + 296T^{4} + 2704 \) Copy content Toggle raw display
$89$ \( T^{8} + 17768 T^{4} + 77228944 \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} + \cdots + 484)^{2} \) Copy content Toggle raw display
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