Properties

Label 6027.2.a.y
Level $6027$
Weight $2$
Character orbit 6027.a
Self dual yes
Analytic conductor $48.126$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 6x^{5} + 16x^{4} + 14x^{3} - 20x^{2} - 10x + 4 \) 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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} - \beta_{5} q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} - \beta_{5} q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{8} + q^{9} + ( - \beta_{5} - \beta_{3} + 1) q^{10} + ( - \beta_{5} + \beta_{4} + 2) q^{11} + ( - \beta_{2} + \beta_1 - 2) q^{12} + ( - \beta_{6} + 1) q^{13} + \beta_{5} q^{15} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 3) q^{16} + (\beta_{6} + \beta_{4} - \beta_{3} - 2 \beta_1) q^{17} + ( - \beta_1 + 1) q^{18} + (\beta_{6} + \beta_{4} + 2 \beta_{2} - \beta_1 + 1) q^{19} + (\beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{20} + ( - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{22} + (\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{23} + (\beta_{3} - \beta_{2} + \beta_1 - 3) q^{24} + ( - \beta_{6} + \beta_{3} - 2 \beta_{2} - 1) q^{25} + ( - 2 \beta_{6} + \beta_{4} - \beta_{3} - \beta_1 + 3) q^{26} - q^{27} + (\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{29} + (\beta_{5} + \beta_{3} - 1) q^{30} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}) q^{31} + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 4) q^{32} + (\beta_{5} - \beta_{4} - 2) q^{33} + (\beta_{6} - \beta_{4} + 2 \beta_{2} + \beta_1 + 3) q^{34} + (\beta_{2} - \beta_1 + 2) q^{36} + (\beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2) q^{37} + (\beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_1) q^{38} + (\beta_{6} - 1) q^{39} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{40} - q^{41} + (\beta_{6} + \beta_{4} + \beta_{3}) q^{43} + ( - \beta_{6} + \beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_1 + 5) q^{44} - \beta_{5} q^{45} + (3 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 3) q^{46} + (\beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} - 2) q^{47} + (\beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 3) q^{48} + ( - 2 \beta_{6} + \beta_{5} + 2 \beta_{3} - \beta_{2} + 3 \beta_1) q^{50} + ( - \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_1) q^{51} + ( - 3 \beta_{6} + 2 \beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_1 + 7) q^{52} + (\beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 3) q^{53} + (\beta_1 - 1) q^{54} + ( - \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 3) q^{55} + ( - \beta_{6} - \beta_{4} - 2 \beta_{2} + \beta_1 - 1) q^{57} + (\beta_{6} - \beta_{5} - \beta_{4} + \beta_1 - 1) q^{58} + ( - \beta_{5} + \beta_{4} + \beta_{2} - 3 \beta_1 + 1) q^{59} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 3) q^{60} + (\beta_{6} + \beta_{5} - 2 \beta_{4} + 2 \beta_{2} - \beta_1 + 1) q^{61} + (\beta_{6} - \beta_{5} + \beta_{3} + 2) q^{62} + ( - 3 \beta_{6} - \beta_{4} - \beta_{3} + 1) q^{64} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1 + 4) q^{65} + (\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{66} + ( - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{67} + (\beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_1) q^{68} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{69} + (\beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_1 + 1) q^{71} + ( - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{72} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{73} + ( - \beta_{6} - \beta_{5} + \beta_{4} - 4 \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{74} + (\beta_{6} - \beta_{3} + 2 \beta_{2} + 1) q^{75} + (2 \beta_{6} - 2 \beta_{5} + \beta_{3} + \beta_{2} + 6) q^{76} + (2 \beta_{6} - \beta_{4} + \beta_{3} + \beta_1 - 3) q^{78} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{79} + ( - 3 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_1 + 7) q^{80} + q^{81} + (\beta_1 - 1) q^{82} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{83} + ( - \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2) q^{85} + (\beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 5) q^{86} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{87} + ( - \beta_{6} + 3 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 7) q^{88} + ( - \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 7) q^{89} + ( - \beta_{5} - \beta_{3} + 1) q^{90} + (5 \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{92} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{93} + (\beta_{6} + 2 \beta_{5} - 4 \beta_{4} + \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 7) q^{94} + (4 \beta_{5} - \beta_{3} + 3 \beta_{2}) q^{95} + (\beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 4) q^{96} + (2 \beta_{6} + \beta_{4} - 4 \beta_{3} - \beta_1 + 6) q^{97} + ( - \beta_{5} + \beta_{4} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 4 q^{2} - 7 q^{3} + 8 q^{4} - q^{5} - 4 q^{6} + 12 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 4 q^{2} - 7 q^{3} + 8 q^{4} - q^{5} - 4 q^{6} + 12 q^{8} + 7 q^{9} + 3 q^{10} + 11 q^{11} - 8 q^{12} + 7 q^{13} + q^{15} + 6 q^{16} - 11 q^{17} + 4 q^{18} - 4 q^{19} + 7 q^{20} + 6 q^{22} + 7 q^{23} - 12 q^{24} + 2 q^{25} + 13 q^{26} - 7 q^{27} + 4 q^{29} - 3 q^{30} + 7 q^{31} + 18 q^{32} - 11 q^{33} + 20 q^{34} + 8 q^{36} - 4 q^{38} - 7 q^{39} + 9 q^{40} - 7 q^{41} + q^{43} + 18 q^{44} - q^{45} - 17 q^{46} - 14 q^{47} - 6 q^{48} + 19 q^{50} + 11 q^{51} + 27 q^{52} + 23 q^{53} - 4 q^{54} + 30 q^{55} + 4 q^{57} - 3 q^{58} - 8 q^{59} - 7 q^{60} + 3 q^{61} + 16 q^{62} + 6 q^{64} + 15 q^{65} - 6 q^{66} + 3 q^{67} - 7 q^{69} + 7 q^{71} + 12 q^{72} + 11 q^{73} - 13 q^{74} - 2 q^{75} + 40 q^{76} - 13 q^{78} - q^{79} + 43 q^{80} + 7 q^{81} - 4 q^{82} - 10 q^{85} - 12 q^{86} - 4 q^{87} + 10 q^{88} - 32 q^{89} + 3 q^{90} - 19 q^{92} - 7 q^{93} - 21 q^{94} - 8 q^{95} - 18 q^{96} + 25 q^{97} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 6x^{5} + 16x^{4} + 14x^{3} - 20x^{2} - 10x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 4\nu^{4} + 12\nu^{3} + 4\nu^{2} - 8\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 6\nu^{4} + 16\nu^{3} + 12\nu^{2} - 16\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 5\nu^{5} - 2\nu^{4} - 18\nu^{3} + 14\nu^{2} + 14\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{5} + \beta_{4} + 2\beta_{3} + 8\beta_{2} + 10\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} - 3\beta_{5} + 4\beta_{4} + 9\beta_{3} + 21\beta_{2} + 33\beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{6} - 13\beta_{5} + 18\beta_{4} + 23\beta_{3} + 67\beta_{2} + 83\beta _1 + 115 \) 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
2.95889
2.35311
1.19009
0.281557
−0.762978
−1.32604
−1.69463
−1.95889 −1.00000 1.83724 0.513172 1.95889 0 0.318823 1.00000 −1.00525
1.2 −1.35311 −1.00000 −0.169102 −1.31916 1.35311 0 2.93503 1.00000 1.78497
1.3 −0.190092 −1.00000 −1.96387 −2.28332 0.190092 0 0.753499 1.00000 0.434041
1.4 0.718443 −1.00000 −1.48384 3.61951 −0.718443 0 −2.50294 1.00000 2.60041
1.5 1.76298 −1.00000 1.10809 −3.51322 −1.76298 0 −1.57241 1.00000 −6.19372
1.6 2.32604 −1.00000 3.41045 −0.0978008 −2.32604 0 3.28076 1.00000 −0.227488
1.7 2.69463 −1.00000 5.26102 2.08082 −2.69463 0 8.78724 1.00000 5.60704
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.y 7
7.b odd 2 1 861.2.a.m 7
21.c even 2 1 2583.2.a.u 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.a.m 7 7.b odd 2 1
2583.2.a.u 7 21.c even 2 1
6027.2.a.y 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{7} - 4T_{2}^{6} - 3T_{2}^{5} + 24T_{2}^{4} - 7T_{2}^{3} - 34T_{2}^{2} + 15T_{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{7} + T_{5}^{6} - 18T_{5}^{5} - 18T_{5}^{4} + 69T_{5}^{3} + 57T_{5}^{2} - 36T_{5} - 4 \) Copy content Toggle raw display
\( T_{13}^{7} - 7T_{13}^{6} - 14T_{13}^{5} + 114T_{13}^{4} - 83T_{13}^{3} - 223T_{13}^{2} + 294T_{13} - 72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 4 T^{6} - 3 T^{5} + 24 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T + 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} + T^{6} - 18 T^{5} - 18 T^{4} + \cdots - 4 \) Copy content Toggle raw display
$7$ \( T^{7} \) Copy content Toggle raw display
$11$ \( T^{7} - 11 T^{6} + 16 T^{5} + 176 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{7} - 7 T^{6} - 14 T^{5} + 114 T^{4} + \cdots - 72 \) Copy content Toggle raw display
$17$ \( T^{7} + 11 T^{6} - 6 T^{5} + \cdots + 2168 \) Copy content Toggle raw display
$19$ \( T^{7} + 4 T^{6} - 78 T^{5} + \cdots + 1440 \) Copy content Toggle raw display
$23$ \( T^{7} - 7 T^{6} - 70 T^{5} + \cdots + 30592 \) Copy content Toggle raw display
$29$ \( T^{7} - 4 T^{6} - 53 T^{5} + 92 T^{4} + \cdots + 10 \) Copy content Toggle raw display
$31$ \( T^{7} - 7 T^{6} - 40 T^{5} + 98 T^{4} + \cdots + 288 \) Copy content Toggle raw display
$37$ \( T^{7} - 141 T^{5} + 22 T^{4} + \cdots + 32738 \) Copy content Toggle raw display
$41$ \( (T + 1)^{7} \) Copy content Toggle raw display
$43$ \( T^{7} - T^{6} - 102 T^{5} + 332 T^{4} + \cdots - 7632 \) Copy content Toggle raw display
$47$ \( T^{7} + 14 T^{6} - 99 T^{5} + \cdots + 3632 \) Copy content Toggle raw display
$53$ \( T^{7} - 23 T^{6} + 32 T^{5} + \cdots + 102708 \) Copy content Toggle raw display
$59$ \( T^{7} + 8 T^{6} - 104 T^{5} + \cdots + 320 \) Copy content Toggle raw display
$61$ \( T^{7} - 3 T^{6} - 174 T^{5} + \cdots + 39128 \) Copy content Toggle raw display
$67$ \( T^{7} - 3 T^{6} - 274 T^{5} + \cdots + 227592 \) Copy content Toggle raw display
$71$ \( T^{7} - 7 T^{6} - 174 T^{5} + \cdots - 33696 \) Copy content Toggle raw display
$73$ \( T^{7} - 11 T^{6} - 78 T^{5} + \cdots - 1448 \) Copy content Toggle raw display
$79$ \( T^{7} + T^{6} - 222 T^{5} + \cdots - 89440 \) Copy content Toggle raw display
$83$ \( T^{7} - 388 T^{5} + 1296 T^{4} + \cdots - 79616 \) Copy content Toggle raw display
$89$ \( T^{7} + 32 T^{6} + 224 T^{5} + \cdots + 659520 \) Copy content Toggle raw display
$97$ \( T^{7} - 25 T^{6} - 14 T^{5} + \cdots + 783908 \) Copy content Toggle raw display
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