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Properties

Label 6027.2.a.bc
Level $6027$
Weight $2$
Character orbit 6027.a
Self dual yes
Analytic conductor $48.126$
Analytic rank $0$
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: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.7457527933.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 6x^{6} + 23x^{5} - 4x^{4} - 27x^{3} + 8x^{2} + 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 6x^{6} + 23x^{5} - 4x^{4} - 27x^{3} + 8x^{2} + 8x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{7} - 2\nu^{6} - 7\nu^{5} + 14\nu^{4} + 3\nu^{3} - 10\nu^{2} + \nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{6} + 2\nu^{5} + 8\nu^{4} - 15\nu^{3} - 10\nu^{2} + 16\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} - 2\nu^{5} - 8\nu^{4} + 15\nu^{3} + 11\nu^{2} - 16\nu - 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{7} + 2\nu^{6} + 8\nu^{5} - 15\nu^{4} - 11\nu^{3} + 16\nu^{2} + 7\nu - 1 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\nu^{6} - 4\nu^{5} - 15\nu^{4} + 29\nu^{3} + 14\nu^{2} - 26\nu - 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -3\nu^{7} + 6\nu^{6} + 23\nu^{5} - 44\nu^{4} - 24\nu^{3} + 42\nu^{2} + 8\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{5} + \beta_{2} + 5\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} - 2\beta_{5} + 6\beta_{4} + 8\beta_{3} + \beta_{2} - \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{7} + \beta_{6} - 17\beta_{5} + 2\beta_{3} + 10\beta_{2} + 31\beta _1 - 17 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{7} + 10\beta_{6} - 20\beta_{5} + 38\beta_{4} + 57\beta_{3} + 13\beta_{2} - 5\beta _1 + 97 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 68\beta_{7} + 13\beta_{6} - 125\beta_{5} + 2\beta_{4} + 26\beta_{3} + 80\beta_{2} + 205\beta _1 - 113 \) 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
2.70360
1.35554
1.21768
−0.978012
−0.169079
1.70821
−2.52358
−0.314356
−2.23115 1.00000 2.97801 −0.893036 −2.23115 0 −2.18209 1.00000 1.99249
1.2 −2.04183 1.00000 2.16908 3.68950 −2.04183 0 −0.345232 1.00000 −7.53333
1.3 −1.51387 1.00000 0.291794 −3.80505 −1.51387 0 2.58600 1.00000 5.76034
1.4 −1.13860 1.00000 −0.703600 2.27123 −1.13860 0 3.07831 1.00000 −2.58601
1.5 1.62618 1.00000 0.644462 1.67363 1.62618 0 −2.20435 1.00000 2.72162
1.6 1.66803 1.00000 0.782321 4.32429 1.66803 0 −2.03112 1.00000 7.21304
1.7 2.07710 1.00000 2.31436 −2.30039 2.07710 0 0.652949 1.00000 −4.77814
1.8 2.55413 1.00000 4.52358 2.03983 2.55413 0 6.44554 1.00000 5.20999
\(n\): 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.bc 8
7.b odd 2 1 6027.2.a.bb 8
7.d odd 6 2 861.2.i.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.i.d ✓ 16 7.d odd 6 2
6027.2.a.bb 8 7.b odd 2 1
6027.2.a.bc 8 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{8} - T_{2}^{7} - 14T_{2}^{6} + 11T_{2}^{5} + 70T_{2}^{4} - 39T_{2}^{3} - 148T_{2}^{2} + 44T_{2} + 113 \) Copy content Toggle raw display
\( T_{5}^{8} - 7T_{5}^{7} - 8T_{5}^{6} + 141T_{5}^{5} - 169T_{5}^{4} - 589T_{5}^{3} + 1153T_{5}^{2} + 206T_{5} - 967 \) Copy content Toggle raw display
\( T_{13}^{4} - 5T_{13}^{3} - 10T_{13}^{2} + 44T_{13} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + \cdots + 113 \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 7 T^{7} + \cdots - 967 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 11 T^{7} + \cdots + 263 \) Copy content Toggle raw display
$13$ \( (T^{4} - 5 T^{3} - 10 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 3 T^{7} + \cdots - 433 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + \cdots + 4291 \) Copy content Toggle raw display
$23$ \( T^{8} - 14 T^{7} + \cdots - 13693 \) Copy content Toggle raw display
$29$ \( T^{8} - 2 T^{7} + \cdots + 78769 \) Copy content Toggle raw display
$31$ \( T^{8} - 16 T^{7} + \cdots - 1933 \) Copy content Toggle raw display
$37$ \( T^{8} + 20 T^{7} + \cdots - 22981 \) Copy content Toggle raw display
$41$ \( (T - 1)^{8} \) Copy content Toggle raw display
$43$ \( T^{8} - 7 T^{7} + \cdots + 316351 \) Copy content Toggle raw display
$47$ \( T^{8} - 14 T^{7} + \cdots - 1591 \) Copy content Toggle raw display
$53$ \( T^{8} - 7 T^{7} + \cdots - 150589 \) Copy content Toggle raw display
$59$ \( T^{8} - 22 T^{7} + \cdots - 460973 \) Copy content Toggle raw display
$61$ \( T^{8} - 253 T^{6} + \cdots + 77443 \) Copy content Toggle raw display
$67$ \( T^{8} - 12 T^{7} + \cdots - 5340679 \) Copy content Toggle raw display
$71$ \( T^{8} + 5 T^{7} + \cdots + 13689 \) Copy content Toggle raw display
$73$ \( T^{8} - 2 T^{7} + \cdots - 144257 \) Copy content Toggle raw display
$79$ \( T^{8} + 15 T^{7} + \cdots - 685351 \) Copy content Toggle raw display
$83$ \( T^{8} - 15 T^{7} + \cdots - 63343 \) Copy content Toggle raw display
$89$ \( T^{8} - 29 T^{7} + \cdots + 1006021 \) Copy content Toggle raw display
$97$ \( T^{8} - 19 T^{7} + \cdots - 250733 \) Copy content Toggle raw display
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