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Properties

Label 6027.2.a.bh
Level $6027$
Weight $2$
Character orbit 6027.a
Self dual yes
Analytic conductor $48.126$
Analytic rank $0$
Dimension $13$
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: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 11 x^{11} + 56 x^{10} + 26 x^{9} - 263 x^{8} + 50 x^{7} + 478 x^{6} - 174 x^{5} + \cdots - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 861)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{8} + 1) q^{5} + \beta_1 q^{6} + ( - \beta_{9} - \beta_{8} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{8} + 1) q^{5} + \beta_1 q^{6} + ( - \beta_{9} - \beta_{8} - \beta_1) q^{8} + q^{9} + ( - \beta_{12} - \beta_{10} - \beta_{9} + \cdots + 1) q^{10}+ \cdots + ( - \beta_{12} - \beta_{10} + \beta_{6} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 4 q^{2} - 13 q^{3} + 12 q^{4} + 8 q^{5} + 4 q^{6} - 12 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 4 q^{2} - 13 q^{3} + 12 q^{4} + 8 q^{5} + 4 q^{6} - 12 q^{8} + 13 q^{9} + q^{10} - 10 q^{11} - 12 q^{12} + 16 q^{13} - 8 q^{15} + 26 q^{16} + 12 q^{17} - 4 q^{18} + 11 q^{19} + 6 q^{20} + q^{22} - 15 q^{23} + 12 q^{24} + 15 q^{25} + 18 q^{26} - 13 q^{27} - 8 q^{29} - q^{30} + 9 q^{31} - 23 q^{32} + 10 q^{33} - 7 q^{34} + 12 q^{36} - 2 q^{37} + 20 q^{38} - 16 q^{39} + 49 q^{40} - 13 q^{41} - 7 q^{43} - 22 q^{44} + 8 q^{45} - 4 q^{46} + 26 q^{47} - 26 q^{48} - 15 q^{50} - 12 q^{51} + 24 q^{52} + 4 q^{53} + 4 q^{54} + q^{55} - 11 q^{57} + 39 q^{58} - 3 q^{59} - 6 q^{60} + 28 q^{61} + 7 q^{62} + 2 q^{64} - 20 q^{65} - q^{66} + 7 q^{67} + 55 q^{68} + 15 q^{69} - 40 q^{71} - 12 q^{72} - 2 q^{73} + q^{74} - 15 q^{75} - 26 q^{76} - 18 q^{78} + 13 q^{79} + 22 q^{80} + 13 q^{81} + 4 q^{82} + 14 q^{83} + 48 q^{85} - 49 q^{86} + 8 q^{87} + 20 q^{88} + 35 q^{89} + q^{90} - 105 q^{92} - 9 q^{93} - 2 q^{94} + 7 q^{95} + 23 q^{96} + 64 q^{97} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 4 x^{12} - 11 x^{11} + 56 x^{10} + 26 x^{9} - 263 x^{8} + 50 x^{7} + 478 x^{6} - 174 x^{5} + \cdots - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 105 \nu^{12} + 402 \nu^{11} + 1471 \nu^{10} - 6122 \nu^{9} - 7424 \nu^{8} + 32643 \nu^{7} + \cdots - 4612 ) / 2162 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 944 \nu^{12} - 3120 \nu^{11} - 12772 \nu^{10} + 44868 \nu^{9} + 58087 \nu^{8} - 219999 \nu^{7} + \cdots + 11577 ) / 1081 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2617 \nu^{12} - 8228 \nu^{11} - 37363 \nu^{10} + 120936 \nu^{9} + 187114 \nu^{8} - 614931 \nu^{7} + \cdots + 44488 ) / 2162 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1537 \nu^{12} - 5483 \nu^{11} - 19453 \nu^{10} + 78197 \nu^{9} + 75852 \nu^{8} - 378472 \nu^{7} + \cdots + 9559 ) / 1081 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1680 \nu^{12} - 6432 \nu^{11} - 19212 \nu^{10} + 89304 \nu^{9} + 55005 \nu^{8} - 412026 \nu^{7} + \cdots + 7851 ) / 1081 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4041 \nu^{12} + 14730 \nu^{11} + 49447 \nu^{10} - 207380 \nu^{9} - 177062 \nu^{8} + 981221 \nu^{7} + \cdots - 31592 ) / 2162 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4041 \nu^{12} - 14730 \nu^{11} - 49447 \nu^{10} + 207380 \nu^{9} + 177062 \nu^{8} - 981221 \nu^{7} + \cdots + 31592 ) / 2162 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2764 \nu^{12} + 10088 \nu^{11} + 33585 \nu^{10} - 141614 \nu^{9} - 117946 \nu^{8} + 666901 \nu^{7} + \cdots - 23055 ) / 1081 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5977 \nu^{12} + 21092 \nu^{11} + 75787 \nu^{10} - 299544 \nu^{9} - 297124 \nu^{8} + 1438983 \nu^{7} + \cdots - 47218 ) / 2162 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 6745 \nu^{12} - 23044 \nu^{11} - 88523 \nu^{10} + 329744 \nu^{9} + 376628 \nu^{8} - 1604297 \nu^{7} + \cdots + 78110 ) / 2162 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{12} + 2 \beta_{11} + 9 \beta_{9} + 9 \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{12} + 11 \beta_{11} + 12 \beta_{9} + 12 \beta_{8} + 9 \beta_{7} + 2 \beta_{6} + 8 \beta_{5} + \cdots + 82 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 25 \beta_{12} + 26 \beta_{11} + 66 \beta_{9} + 70 \beta_{8} + 3 \beta_{7} + 14 \beta_{6} - 13 \beta_{5} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 17 \beta_{12} + 98 \beta_{11} - 4 \beta_{10} + 104 \beta_{9} + 112 \beta_{8} + 67 \beta_{7} + 32 \beta_{6} + \cdots + 471 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 229 \beta_{12} + 251 \beta_{11} - 8 \beta_{10} + 454 \beta_{9} + 529 \beta_{8} + 49 \beta_{7} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 196 \beta_{12} + 811 \beta_{11} - 75 \beta_{10} + 797 \beta_{9} + 963 \beta_{8} + 488 \beta_{7} + \cdots + 2806 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1867 \beta_{12} + 2161 \beta_{11} - 166 \beta_{10} + 3042 \beta_{9} + 3979 \beta_{8} + 550 \beta_{7} + \cdots + 202 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1927 \beta_{12} + 6467 \beta_{11} - 937 \beta_{10} + 5748 \beta_{9} + 7966 \beta_{8} + 3595 \beta_{7} + \cdots + 17210 \) Copy content Toggle raw display

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

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
2.73393
2.58904
2.20098
1.84587
1.21702
0.467051
0.441481
−0.196565
−0.558375
−0.646272
−1.46649
−2.15509
−2.47258
−2.73393 −1.00000 5.47440 −2.29019 2.73393 0 −9.49877 1.00000 6.26122
1.2 −2.58904 −1.00000 4.70310 3.45213 2.58904 0 −6.99843 1.00000 −8.93769
1.3 −2.20098 −1.00000 2.84432 −2.77621 2.20098 0 −1.85834 1.00000 6.11039
1.4 −1.84587 −1.00000 1.40722 1.41311 1.84587 0 1.09419 1.00000 −2.60841
1.5 −1.21702 −1.00000 −0.518850 3.45145 1.21702 0 3.06550 1.00000 −4.20050
1.6 −0.467051 −1.00000 −1.78186 −0.377698 0.467051 0 1.76632 1.00000 0.176404
1.7 −0.441481 −1.00000 −1.80509 3.83895 0.441481 0 1.67988 1.00000 −1.69482
1.8 0.196565 −1.00000 −1.96136 1.00666 −0.196565 0 −0.778664 1.00000 0.197874
1.9 0.558375 −1.00000 −1.68822 0.297196 −0.558375 0 −2.05941 1.00000 0.165947
1.10 0.646272 −1.00000 −1.58233 −2.20733 −0.646272 0 −2.31516 1.00000 −1.42653
1.11 1.46649 −1.00000 0.150578 −1.27945 −1.46649 0 −2.71215 1.00000 −1.87630
1.12 2.15509 −1.00000 2.64442 −0.784696 −2.15509 0 1.38879 1.00000 −1.69109
1.13 2.47258 −1.00000 4.11367 4.25608 −2.47258 0 5.22624 1.00000 10.5235
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.bh 13
7.b odd 2 1 6027.2.a.bi 13
7.d odd 6 2 861.2.i.f ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.i.f ✓ 26 7.d odd 6 2
6027.2.a.bh 13 1.a even 1 1 trivial
6027.2.a.bi 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{13} + 4 T_{2}^{12} - 11 T_{2}^{11} - 56 T_{2}^{10} + 26 T_{2}^{9} + 263 T_{2}^{8} + 50 T_{2}^{7} + \cdots + 4 \) Copy content Toggle raw display
\( T_{5}^{13} - 8 T_{5}^{12} - 8 T_{5}^{11} + 177 T_{5}^{10} - 85 T_{5}^{9} - 1489 T_{5}^{8} + 980 T_{5}^{7} + \cdots - 438 \) Copy content Toggle raw display
\( T_{13}^{13} - 16 T_{13}^{12} + 29 T_{13}^{11} + 615 T_{13}^{10} - 2063 T_{13}^{9} - 9174 T_{13}^{8} + \cdots - 2025 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + 4 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T + 1)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} - 8 T^{12} + \cdots - 438 \) Copy content Toggle raw display
$7$ \( T^{13} \) Copy content Toggle raw display
$11$ \( T^{13} + 10 T^{12} + \cdots - 16326 \) Copy content Toggle raw display
$13$ \( T^{13} - 16 T^{12} + \cdots - 2025 \) Copy content Toggle raw display
$17$ \( T^{13} - 12 T^{12} + \cdots + 657072 \) Copy content Toggle raw display
$19$ \( T^{13} - 11 T^{12} + \cdots + 5007501 \) Copy content Toggle raw display
$23$ \( T^{13} + 15 T^{12} + \cdots - 2755350 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots - 188753976 \) Copy content Toggle raw display
$31$ \( T^{13} - 9 T^{12} + \cdots + 2755215 \) Copy content Toggle raw display
$37$ \( T^{13} + 2 T^{12} + \cdots - 4739247 \) Copy content Toggle raw display
$41$ \( (T + 1)^{13} \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 2098432440 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 6292135854 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 172955736 \) Copy content Toggle raw display
$59$ \( T^{13} + 3 T^{12} + \cdots - 62082528 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 14237334360 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 595976103 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 7925020020 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots - 1188570609 \) Copy content Toggle raw display
$79$ \( T^{13} - 13 T^{12} + \cdots - 56835 \) Copy content Toggle raw display
$83$ \( T^{13} - 14 T^{12} + \cdots - 953334 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 69854349618 \) Copy content Toggle raw display
$97$ \( T^{13} - 64 T^{12} + \cdots + 84516900 \) Copy content Toggle raw display
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