Properties

Label 6027.2.a.z
Level $6027$
Weight $2$
Character orbit 6027.a
Self dual yes
Analytic conductor $48.126$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6027,2,Mod(1,6027)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6027.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6027.a (trivial)

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: yes
Analytic conductor: \(48.1258372982\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 8x^{6} + 14x^{5} + 18x^{4} - 24x^{3} - 10x^{2} + 10x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

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

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{5} q^{5} + \beta_1 q^{6} + ( - \beta_{3} - \beta_{2} - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{5} q^{5} + \beta_1 q^{6} + ( - \beta_{3} - \beta_{2} - 1) q^{8} + q^{9} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{10} + ( - \beta_{7} + \beta_{6} + \beta_1 - 1) q^{11} + ( - \beta_{2} - 1) q^{12} + (\beta_{6} - \beta_{2}) q^{13} + \beta_{5} q^{15} + (\beta_{4} + \beta_1) q^{16} + ( - \beta_{7} + 2 \beta_{5} - \beta_{4} - \beta_{3} + 1) q^{17} - \beta_1 q^{18} + (\beta_{7} + \beta_{5} + \beta_{2} + 2) q^{19} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{20} + ( - \beta_{6} - 2 \beta_{5} + \beta_{3} - \beta_{2} - 3) q^{22} + (\beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{23} + (\beta_{3} + \beta_{2} + 1) q^{24} + ( - \beta_{4} + \beta_1 - 1) q^{25} + ( - \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{26} - q^{27} + (2 \beta_{7} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{29} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3}) q^{30} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{31} + ( - \beta_{5} + 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{32} + (\beta_{7} - \beta_{6} - \beta_1 + 1) q^{33} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 4 \beta_1 + 2) q^{34} + (\beta_{2} + 1) q^{36} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{2} + 2 \beta_1 - 5) q^{37} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_{2} - 2 \beta_1 - 2) q^{38} + ( - \beta_{6} + \beta_{2}) q^{39} + (2 \beta_{6} + 3 \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 2) q^{40} + q^{41} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_1 - 1) q^{43} + (\beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_1 + 3) q^{44} - \beta_{5} q^{45} + (\beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 1) q^{46} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 3 \beta_1) q^{47} + ( - \beta_{4} - \beta_1) q^{48} + (\beta_{5} - \beta_{2} - 3) q^{50} + (\beta_{7} - 2 \beta_{5} + \beta_{4} + \beta_{3} - 1) q^{51} + ( - \beta_{7} + 2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 - 5) q^{52} + (3 \beta_{7} - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{53} + \beta_1 q^{54} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{55} + ( - \beta_{7} - \beta_{5} - \beta_{2} - 2) q^{57} + (\beta_{7} - \beta_{6} - 3 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1) q^{58} + (2 \beta_{7} - \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 5) q^{59} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{60} + (2 \beta_{7} - 3 \beta_{6} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{61} + (\beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 4) q^{62} + ( - \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} - 4 \beta_{2} - 6) q^{64} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 2) q^{65} + (\beta_{6} + 2 \beta_{5} - \beta_{3} + \beta_{2} + 3) q^{66} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} + 3 \beta_1 - 4) q^{67} + (4 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 8) q^{68} + ( - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{69} + ( - 2 \beta_{7} - \beta_{5} + 3 \beta_{4} - \beta_{3} - 2 \beta_1 - 1) q^{71} + ( - \beta_{3} - \beta_{2} - 1) q^{72} + (\beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{2} + 2 \beta_1 - 2) q^{73} + ( - 2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - \beta_{2} + \cdots - 1) q^{74}+ \cdots + ( - \beta_{7} + \beta_{6} + \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 8 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} + 4 q^{13} - 2 q^{15} + 8 q^{17} - 2 q^{18} + 6 q^{19} + 4 q^{20} - 14 q^{22} - 12 q^{23} + 6 q^{24} - 4 q^{25} + 4 q^{26} - 8 q^{27} - 4 q^{29} + 2 q^{30} - 10 q^{31} - 4 q^{32} + 2 q^{33} + 4 q^{34} + 4 q^{36} - 20 q^{37} - 18 q^{38} - 4 q^{39} + 12 q^{40} + 8 q^{41} - 8 q^{43} + 20 q^{44} + 2 q^{45} - 12 q^{46} + 24 q^{47} - 22 q^{50} - 8 q^{51} - 30 q^{52} - 36 q^{53} + 2 q^{54} + 4 q^{55} - 6 q^{57} + 14 q^{58} + 10 q^{59} - 4 q^{60} - 22 q^{61} + 30 q^{62} - 24 q^{64} + 8 q^{65} + 14 q^{66} - 14 q^{67} + 38 q^{68} + 12 q^{69} - 10 q^{71} - 6 q^{72} - 12 q^{73} - 2 q^{74} + 4 q^{75} + 32 q^{76} - 4 q^{78} + 16 q^{79} - 14 q^{80} + 8 q^{81} - 2 q^{82} + 24 q^{83} - 44 q^{85} + 36 q^{86} + 4 q^{87} - 34 q^{88} + 2 q^{89} - 2 q^{90} - 48 q^{92} + 10 q^{93} - 34 q^{94} - 24 q^{95} + 4 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 8x^{6} + 14x^{5} + 18x^{4} - 24x^{3} - 10x^{2} + 10x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 6\nu^{2} - \nu + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 6\nu^{3} - \nu^{2} + 5\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - \nu^{6} - 9\nu^{5} + 6\nu^{4} + 24\nu^{3} - 7\nu^{2} - 17\nu + 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 2\nu^{6} - 8\nu^{5} + 13\nu^{4} + 18\nu^{3} - 18\nu^{2} - 9\nu + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 6\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 6\beta_{3} + 7\beta_{2} + 19\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{5} + 7\beta_{4} + 32\beta_{2} + 10\beta _1 + 70 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{7} + 2\beta_{6} + 10\beta_{5} + \beta_{4} + 30\beta_{3} + 42\beta_{2} + 96\beta _1 + 62 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.39346
2.28873
1.17091
0.487949
0.117246
−0.896239
−1.41849
−2.14356
−2.39346 −1.00000 3.72866 −2.51820 2.39346 0 −4.13748 1.00000 6.02722
1.2 −2.28873 −1.00000 3.23826 2.92705 2.28873 0 −2.83405 1.00000 −6.69922
1.3 −1.17091 −1.00000 −0.628975 2.94759 1.17091 0 3.07829 1.00000 −3.45135
1.4 −0.487949 −1.00000 −1.76191 −1.53225 0.487949 0 1.83562 1.00000 0.747657
1.5 −0.117246 −1.00000 −1.98625 −0.562834 0.117246 0 0.467371 1.00000 0.0659898
1.6 0.896239 −1.00000 −1.19676 1.54330 −0.896239 0 −2.86506 1.00000 1.38317
1.7 1.41849 −1.00000 0.0121162 −2.27752 −1.41849 0 −2.81979 1.00000 −3.23064
1.8 2.14356 −1.00000 2.59485 1.47286 −2.14356 0 1.27510 1.00000 3.15717
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(41\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6027.2.a.z 8
7.b odd 2 1 6027.2.a.ba yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6027.2.a.z 8 1.a even 1 1 trivial
6027.2.a.ba yes 8 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{8} + 2T_{2}^{7} - 8T_{2}^{6} - 14T_{2}^{5} + 18T_{2}^{4} + 24T_{2}^{3} - 10T_{2}^{2} - 10T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{8} - 2T_{5}^{7} - 16T_{5}^{6} + 26T_{5}^{5} + 86T_{5}^{4} - 102T_{5}^{3} - 168T_{5}^{2} + 122T_{5} + 97 \) Copy content Toggle raw display
\( T_{13}^{8} - 4T_{13}^{7} - 30T_{13}^{6} + 92T_{13}^{5} + 330T_{13}^{4} - 556T_{13}^{3} - 1330T_{13}^{2} + 558T_{13} + 521 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} - 8 T^{6} - 14 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} - 16 T^{6} + 26 T^{5} + \cdots + 97 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 2 T^{7} - 40 T^{6} - 120 T^{5} + \cdots - 688 \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{7} - 30 T^{6} + 92 T^{5} + \cdots + 521 \) Copy content Toggle raw display
$17$ \( T^{8} - 8 T^{7} - 54 T^{6} + \cdots + 47564 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} - 24 T^{6} + 198 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} - 40 T^{6} + \cdots + 12079 \) Copy content Toggle raw display
$29$ \( T^{8} + 4 T^{7} - 144 T^{6} + \cdots - 115159 \) Copy content Toggle raw display
$31$ \( T^{8} + 10 T^{7} - 6 T^{6} - 254 T^{5} + \cdots - 332 \) Copy content Toggle raw display
$37$ \( T^{8} + 20 T^{7} - 12 T^{6} + \cdots - 374767 \) Copy content Toggle raw display
$41$ \( (T - 1)^{8} \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} - 84 T^{6} + \cdots + 1196 \) Copy content Toggle raw display
$47$ \( T^{8} - 24 T^{7} - 8 T^{6} + \cdots - 11096641 \) Copy content Toggle raw display
$53$ \( T^{8} + 36 T^{7} + 388 T^{6} + \cdots + 6709873 \) Copy content Toggle raw display
$59$ \( T^{8} - 10 T^{7} - 252 T^{6} + \cdots + 215924 \) Copy content Toggle raw display
$61$ \( T^{8} + 22 T^{7} + 6 T^{6} + \cdots - 358604 \) Copy content Toggle raw display
$67$ \( T^{8} + 14 T^{7} - 60 T^{6} + \cdots + 285011 \) Copy content Toggle raw display
$71$ \( T^{8} + 10 T^{7} - 300 T^{6} + \cdots + 14673892 \) Copy content Toggle raw display
$73$ \( T^{8} + 12 T^{7} - 162 T^{6} + \cdots - 9244 \) Copy content Toggle raw display
$79$ \( T^{8} - 16 T^{7} - 268 T^{6} + \cdots - 6250981 \) Copy content Toggle raw display
$83$ \( T^{8} - 24 T^{7} - 56 T^{6} + \cdots + 1120064 \) Copy content Toggle raw display
$89$ \( T^{8} - 2 T^{7} - 404 T^{6} + \cdots + 20410352 \) Copy content Toggle raw display
$97$ \( T^{8} - 16 T^{7} - 218 T^{6} + \cdots - 407903 \) Copy content Toggle raw display
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