Properties

Label 507.2.k.e
Level $507$
Weight $2$
Character orbit 507.k
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(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: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{5} + \beta_{2}) q^{2} + (\beta_{4} - \beta_{2}) q^{3} + (\beta_{6} + 2 \beta_{4} + 1) q^{4} + ( - \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{5}+ \cdots + (\beta_{7} - 2 \beta_{6} + \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{5} + \beta_{2}) q^{2} + (\beta_{4} - \beta_{2}) q^{3} + (\beta_{6} + 2 \beta_{4} + 1) q^{4} + ( - \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{5}+ \cdots + (4 \beta_{6} + 3 \beta_{5} - 7 \beta_{4} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 12 q^{4} - 14 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + 12 q^{4} - 14 q^{6} - 4 q^{7} + 4 q^{9} - 12 q^{10} - 2 q^{15} - 4 q^{16} + 4 q^{18} + 8 q^{19} + 4 q^{21} + 4 q^{22} - 30 q^{24} + 4 q^{27} - 4 q^{28} - 18 q^{30} + 8 q^{31} - 20 q^{33} - 36 q^{34} + 36 q^{36} + 32 q^{37} - 24 q^{40} - 16 q^{42} - 36 q^{43} + 16 q^{45} - 14 q^{48} - 38 q^{54} + 4 q^{55} + 16 q^{57} + 40 q^{58} + 44 q^{60} + 28 q^{61} + 16 q^{63} - 40 q^{66} + 32 q^{67} - 32 q^{70} + 24 q^{72} - 28 q^{73} - 12 q^{75} + 16 q^{76} + 16 q^{79} + 4 q^{81} + 24 q^{82} - 8 q^{84} + 24 q^{85} - 34 q^{87} + 12 q^{88} + 16 q^{93} + 64 q^{94} + 16 q^{96} - 52 q^{97} + 40 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 15\nu^{6} + 32\nu^{5} - 172\nu^{4} + 221\nu^{3} - 426\nu^{2} + 235\nu - 159 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 390\nu^{2} + 298\nu - 70 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 427\nu^{2} + 335\nu - 181 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 29\nu^{6} + 89\nu^{5} - 261\nu^{4} + 373\nu^{3} - 498\nu^{2} + 294\nu - 152 ) / 37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 28\nu^{6} - 114\nu^{5} + 215\nu^{4} - 378\nu^{3} + 366\nu^{2} - 266\nu + 97 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\nu^{7} - 41\nu^{6} + 159\nu^{5} - 184\nu^{4} + 276\nu^{3} - 84\nu^{2} + 38\nu + 39 ) / 37 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{3} - 2\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 3\beta_{6} + 6\beta_{4} - 2\beta_{3} - 2\beta_{2} - 6\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{7} - 3\beta_{6} + 7\beta_{5} + 6\beta_{4} - 12\beta_{3} - 5\beta_{2} + \beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{7} - 25\beta_{6} + 3\beta_{5} - 24\beta_{4} - 5\beta_{3} + 7\beta_{2} + 27\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{7} - 16\beta_{6} - 42\beta_{5} - 54\beta_{4} + 51\beta_{3} + 42\beta_{2} + 26\beta _1 - 122 \) Copy content Toggle raw display

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(-\beta_{4}\)

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 0.239203 1.71545i −0.232051 0.133975i −1.06488 1.06488i 2.40216 + 1.01660i −1.36603 + 0.366025i −1.84443 + 1.84443i −2.88556 0.820682i 1.96410 1.13397i
80.2 0.389774 1.45466i −1.60523 0.650571i −0.232051 0.133975i 1.06488 + 1.06488i −1.57203 + 2.08148i −1.36603 + 0.366025i 1.84443 1.84443i 2.15351 + 2.08863i 1.96410 1.13397i
89.1 −2.31259 0.619657i 1.64914 0.529480i 3.23205 + 1.86603i 1.69293 1.69293i −4.14187 + 0.202571i 0.366025 + 1.36603i −2.93225 2.93225i 2.43930 1.74637i −4.96410 + 2.86603i
89.2 2.31259 + 0.619657i −1.28311 + 1.16345i 3.23205 + 1.86603i −1.69293 + 1.69293i −3.68825 + 1.89551i 0.366025 + 1.36603i 2.93225 + 2.93225i 0.292748 2.98568i −4.96410 + 2.86603i
188.1 −2.31259 + 0.619657i 1.64914 + 0.529480i 3.23205 1.86603i 1.69293 + 1.69293i −4.14187 0.202571i 0.366025 1.36603i −2.93225 + 2.93225i 2.43930 + 1.74637i −4.96410 2.86603i
188.2 2.31259 0.619657i −1.28311 1.16345i 3.23205 1.86603i −1.69293 1.69293i −3.68825 1.89551i 0.366025 1.36603i 2.93225 2.93225i 0.292748 + 2.98568i −4.96410 2.86603i
488.1 −0.389774 1.45466i 0.239203 + 1.71545i −0.232051 + 0.133975i −1.06488 + 1.06488i 2.40216 1.01660i −1.36603 0.366025i −1.84443 1.84443i −2.88556 + 0.820682i 1.96410 + 1.13397i
488.2 0.389774 + 1.45466i −1.60523 + 0.650571i −0.232051 + 0.133975i 1.06488 1.06488i −1.57203 2.08148i −1.36603 0.366025i 1.84443 + 1.84443i 2.15351 2.08863i 1.96410 + 1.13397i
\(n\): 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
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.k.e 8
3.b odd 2 1 inner 507.2.k.e 8
13.b even 2 1 507.2.k.f 8
13.c even 3 1 39.2.k.b 8
13.c even 3 1 507.2.f.f 8
13.d odd 4 1 39.2.k.b 8
13.d odd 4 1 507.2.k.d 8
13.e even 6 1 507.2.f.e 8
13.e even 6 1 507.2.k.d 8
13.f odd 12 1 507.2.f.e 8
13.f odd 12 1 507.2.f.f 8
13.f odd 12 1 inner 507.2.k.e 8
13.f odd 12 1 507.2.k.f 8
39.d odd 2 1 507.2.k.f 8
39.f even 4 1 39.2.k.b 8
39.f even 4 1 507.2.k.d 8
39.h odd 6 1 507.2.f.e 8
39.h odd 6 1 507.2.k.d 8
39.i odd 6 1 39.2.k.b 8
39.i odd 6 1 507.2.f.f 8
39.k even 12 1 507.2.f.e 8
39.k even 12 1 507.2.f.f 8
39.k even 12 1 inner 507.2.k.e 8
39.k even 12 1 507.2.k.f 8
52.f even 4 1 624.2.cn.c 8
52.j odd 6 1 624.2.cn.c 8
65.f even 4 1 975.2.bp.e 8
65.g odd 4 1 975.2.bo.d 8
65.k even 4 1 975.2.bp.f 8
65.n even 6 1 975.2.bo.d 8
65.q odd 12 1 975.2.bp.e 8
65.q odd 12 1 975.2.bp.f 8
156.l odd 4 1 624.2.cn.c 8
156.p even 6 1 624.2.cn.c 8
195.j odd 4 1 975.2.bp.f 8
195.n even 4 1 975.2.bo.d 8
195.u odd 4 1 975.2.bp.e 8
195.x odd 6 1 975.2.bo.d 8
195.bl even 12 1 975.2.bp.e 8
195.bl even 12 1 975.2.bp.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.b 8 13.c even 3 1
39.2.k.b 8 13.d odd 4 1
39.2.k.b 8 39.f even 4 1
39.2.k.b 8 39.i odd 6 1
507.2.f.e 8 13.e even 6 1
507.2.f.e 8 13.f odd 12 1
507.2.f.e 8 39.h odd 6 1
507.2.f.e 8 39.k even 12 1
507.2.f.f 8 13.c even 3 1
507.2.f.f 8 13.f odd 12 1
507.2.f.f 8 39.i odd 6 1
507.2.f.f 8 39.k even 12 1
507.2.k.d 8 13.d odd 4 1
507.2.k.d 8 13.e even 6 1
507.2.k.d 8 39.f even 4 1
507.2.k.d 8 39.h odd 6 1
507.2.k.e 8 1.a even 1 1 trivial
507.2.k.e 8 3.b odd 2 1 inner
507.2.k.e 8 13.f odd 12 1 inner
507.2.k.e 8 39.k even 12 1 inner
507.2.k.f 8 13.b even 2 1
507.2.k.f 8 13.f odd 12 1
507.2.k.f 8 39.d odd 2 1
507.2.k.f 8 39.k even 12 1
624.2.cn.c 8 52.f even 4 1
624.2.cn.c 8 52.j odd 6 1
624.2.cn.c 8 156.l odd 4 1
624.2.cn.c 8 156.p even 6 1
975.2.bo.d 8 65.g odd 4 1
975.2.bo.d 8 65.n even 6 1
975.2.bo.d 8 195.n even 4 1
975.2.bo.d 8 195.x odd 6 1
975.2.bp.e 8 65.f even 4 1
975.2.bp.e 8 65.q odd 12 1
975.2.bp.e 8 195.u odd 4 1
975.2.bp.e 8 195.bl even 12 1
975.2.bp.f 8 65.k even 4 1
975.2.bp.f 8 65.q odd 12 1
975.2.bp.f 8 195.j odd 4 1
975.2.bp.f 8 195.bl even 12 1

Hecke kernels

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

\( T_{2}^{8} - 6T_{2}^{6} - T_{2}^{4} + 78T_{2}^{2} + 169 \) Copy content Toggle raw display
\( T_{5}^{8} + 38T_{5}^{4} + 169 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 2T_{7}^{2} + 4T_{7} + 4 \) 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
$3$ \( T^{8} + 2 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 38T^{4} + 169 \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 24 T^{6} + \cdots + 2704 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 30 T^{6} + \cdots + 13689 \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} + 20 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 82 T^{6} + \cdots + 2474329 \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} + \cdots + 484)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 16 T^{3} + \cdots + 1369)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 54 T^{6} + \cdots + 169 \) Copy content Toggle raw display
$43$ \( (T^{4} + 18 T^{3} + \cdots + 324)^{2} \) 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
$59$ \( T^{8} - 24 T^{6} + \cdots + 43264 \) Copy content Toggle raw display
$61$ \( (T^{2} - 7 T + 49)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 16 T^{3} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 24 T^{6} + \cdots + 43264 \) 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
$89$ \( T^{8} + 24 T^{6} + \cdots + 77228944 \) Copy content Toggle raw display
$97$ \( (T^{4} + 26 T^{3} + \cdots + 484)^{2} \) Copy content Toggle raw display
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