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Properties

Label 6027.2.a.bf
Level $6027$
Weight $2$
Character orbit 6027.a
Self dual yes
Analytic conductor $48.126$
Analytic rank $1$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 15 x^{10} + 30 x^{9} + 74 x^{8} - 149 x^{7} - 140 x^{6} + 278 x^{5} + 126 x^{4} + \cdots + 18 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{4} - 1) q^{5} + \beta_1 q^{6} + (\beta_{11} - \beta_{8} + \cdots + \beta_{4}) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{4} - 1) q^{5} + \beta_1 q^{6} + (\beta_{11} - \beta_{8} + \cdots + \beta_{4}) q^{8}+ \cdots + (\beta_{10} + \beta_{6} - \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 12 q^{3} + 10 q^{4} - 12 q^{5} + 2 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} - 12 q^{3} + 10 q^{4} - 12 q^{5} + 2 q^{6} + 12 q^{9} - 11 q^{10} + 10 q^{11} - 10 q^{12} - 15 q^{13} + 12 q^{15} + 14 q^{16} - 8 q^{17} - 2 q^{18} - 2 q^{19} - 16 q^{20} - 7 q^{22} + 5 q^{23} + 20 q^{25} - 12 q^{27} + 20 q^{29} + 11 q^{30} - 10 q^{31} + 3 q^{32} - 10 q^{33} + 23 q^{34} + 10 q^{36} - 17 q^{37} - 6 q^{38} + 15 q^{39} - 39 q^{40} + 12 q^{41} + 12 q^{43} + 20 q^{44} - 12 q^{45} - 36 q^{46} - 34 q^{47} - 14 q^{48} + 59 q^{50} + 8 q^{51} - 26 q^{52} + 6 q^{53} + 2 q^{54} + q^{55} + 2 q^{57} - 11 q^{58} - 27 q^{59} + 16 q^{60} - 22 q^{61} + 45 q^{62} + 26 q^{64} + 7 q^{66} - 26 q^{67} - 33 q^{68} - 5 q^{69} + 50 q^{71} - 21 q^{73} - 35 q^{74} - 20 q^{75} + 24 q^{76} - 10 q^{79} - 22 q^{80} + 12 q^{81} - 2 q^{82} - 8 q^{83} + 8 q^{85} - 17 q^{86} - 20 q^{87} - 46 q^{88} - 11 q^{89} - 11 q^{90} + 63 q^{92} + 10 q^{93} - 10 q^{94} + 35 q^{95} - 3 q^{96} - 32 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 15 x^{10} + 30 x^{9} + 74 x^{8} - 149 x^{7} - 140 x^{6} + 278 x^{5} + 126 x^{4} + \cdots + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{6} - 2\nu^{5} - 8\nu^{4} + 14\nu^{3} + 12\nu^{2} - 13\nu - 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{9} - 2\nu^{8} - 13\nu^{7} + 27\nu^{6} + 47\nu^{5} - 109\nu^{4} - 30\nu^{3} + 115\nu^{2} - 9\nu - 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{11} - 2 \nu^{10} - 14 \nu^{9} + 28 \nu^{8} + 60 \nu^{7} - 121 \nu^{6} - 80 \nu^{5} + 158 \nu^{4} + \cdots + 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{10} + 2\nu^{9} + 14\nu^{8} - 28\nu^{7} - 60\nu^{6} + 121\nu^{5} + 81\nu^{4} - 158\nu^{3} - 53\nu^{2} + 60\nu + 23 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{11} - 2 \nu^{10} - 14 \nu^{9} + 28 \nu^{8} + 60 \nu^{7} - 122 \nu^{6} - 79 \nu^{5} + 167 \nu^{4} + \cdots + 11 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{11} - \nu^{10} - 16 \nu^{9} + 15 \nu^{8} + 87 \nu^{7} - 75 \nu^{6} - 188 \nu^{5} + 137 \nu^{4} + \cdots + 9 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( \nu^{11} - 4 \nu^{10} - 11 \nu^{9} + 58 \nu^{8} + 18 \nu^{7} - 269 \nu^{6} + 104 \nu^{5} + 435 \nu^{4} + \cdots + 62 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( - 2 \nu^{11} + 4 \nu^{10} + 29 \nu^{9} - 59 \nu^{8} - 132 \nu^{7} + 283 \nu^{6} + 194 \nu^{5} + \cdots - 54 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( 3 \nu^{11} - 5 \nu^{10} - 45 \nu^{9} + 73 \nu^{8} + 220 \nu^{7} - 345 \nu^{6} - 394 \nu^{5} + 571 \nu^{4} + \cdots + 48 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} + 7\beta_{2} - \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{11} + \beta_{10} + 9 \beta_{8} + 7 \beta_{7} + \beta_{6} + 10 \beta_{5} - 9 \beta_{4} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{11} + 10 \beta_{10} + 12 \beta_{8} + 10 \beta_{6} + 14 \beta_{5} - 12 \beta_{4} - \beta_{3} + \cdots + 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 56 \beta_{11} + 15 \beta_{10} + \beta_{9} + 71 \beta_{8} + 43 \beta_{7} + 13 \beta_{6} + 83 \beta_{5} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 29 \beta_{11} + 82 \beta_{10} + \beta_{9} + 112 \beta_{8} + \beta_{7} + 81 \beta_{6} + 137 \beta_{5} + \cdots + 549 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 386 \beta_{11} + 151 \beta_{10} + 15 \beta_{9} + 539 \beta_{8} + 262 \beta_{7} + 123 \beta_{6} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 300 \beta_{11} + 632 \beta_{10} + 16 \beta_{9} + 950 \beta_{8} + 23 \beta_{7} + 617 \beta_{6} + \cdots + 3540 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2669 \beta_{11} + 1314 \beta_{10} + 154 \beta_{9} + 4019 \beta_{8} + 1621 \beta_{7} + 1040 \beta_{6} + \cdots + 774 \) 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
2.71780
2.11101
1.94261
1.22157
1.14271
0.837060
−0.420502
−0.467496
−1.05133
−1.09852
−2.44902
−2.48589
−2.71780 −1.00000 5.38643 −0.137890 2.71780 0 −9.20364 1.00000 0.374758
1.2 −2.11101 −1.00000 2.45637 2.44925 2.11101 0 −0.963413 1.00000 −5.17041
1.3 −1.94261 −1.00000 1.77372 −2.73840 1.94261 0 0.439573 1.00000 5.31964
1.4 −1.22157 −1.00000 −0.507765 −0.145411 1.22157 0 3.06341 1.00000 0.177630
1.5 −1.14271 −1.00000 −0.694209 −1.38626 1.14271 0 3.07871 1.00000 1.58409
1.6 −0.837060 −1.00000 −1.29933 −2.51468 0.837060 0 2.76174 1.00000 2.10494
1.7 0.420502 −1.00000 −1.82318 1.49913 −0.420502 0 −1.60765 1.00000 0.630387
1.8 0.467496 −1.00000 −1.78145 −2.48991 −0.467496 0 −1.76781 1.00000 −1.16402
1.9 1.05133 −1.00000 −0.894704 −4.26226 −1.05133 0 −3.04329 1.00000 −4.48105
1.10 1.09852 −1.00000 −0.793257 3.51189 −1.09852 0 −3.06844 1.00000 3.85788
1.11 2.44902 −1.00000 3.99771 −4.01828 −2.44902 0 4.89242 1.00000 −9.84085
1.12 2.48589 −1.00000 4.17966 −1.76717 −2.48589 0 5.41840 1.00000 −4.39300
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.bf 12
7.b odd 2 1 6027.2.a.bg 12
7.d odd 6 2 861.2.i.e ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.i.e ✓ 24 7.d odd 6 2
6027.2.a.bf 12 1.a even 1 1 trivial
6027.2.a.bg 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{12} + 2 T_{2}^{11} - 15 T_{2}^{10} - 30 T_{2}^{9} + 74 T_{2}^{8} + 149 T_{2}^{7} - 140 T_{2}^{6} + \cdots + 18 \) Copy content Toggle raw display
\( T_{5}^{12} + 12 T_{5}^{11} + 32 T_{5}^{10} - 145 T_{5}^{9} - 885 T_{5}^{8} - 699 T_{5}^{7} + 4208 T_{5}^{6} + \cdots - 186 \) Copy content Toggle raw display
\( T_{13}^{12} + 15 T_{13}^{11} + 24 T_{13}^{10} - 543 T_{13}^{9} - 1938 T_{13}^{8} + 6310 T_{13}^{7} + \cdots - 18481 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{11} + \cdots + 18 \) Copy content Toggle raw display
$3$ \( (T + 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 12 T^{11} + \cdots - 186 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 10 T^{11} + \cdots + 8922 \) Copy content Toggle raw display
$13$ \( T^{12} + 15 T^{11} + \cdots - 18481 \) Copy content Toggle raw display
$17$ \( T^{12} + 8 T^{11} + \cdots - 136368 \) Copy content Toggle raw display
$19$ \( T^{12} + 2 T^{11} + \cdots + 437917 \) Copy content Toggle raw display
$23$ \( T^{12} - 5 T^{11} + \cdots + 4248 \) Copy content Toggle raw display
$29$ \( T^{12} - 20 T^{11} + \cdots - 25668 \) Copy content Toggle raw display
$31$ \( T^{12} + 10 T^{11} + \cdots - 23267457 \) Copy content Toggle raw display
$37$ \( T^{12} + 17 T^{11} + \cdots - 54407 \) Copy content Toggle raw display
$41$ \( (T - 1)^{12} \) Copy content Toggle raw display
$43$ \( T^{12} - 12 T^{11} + \cdots + 8 \) Copy content Toggle raw display
$47$ \( T^{12} + 34 T^{11} + \cdots + 55242 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 11922809124 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 10812386196 \) Copy content Toggle raw display
$61$ \( T^{12} + 22 T^{11} + \cdots + 33014146 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 166129819199 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 9819752802 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 587324459849 \) Copy content Toggle raw display
$79$ \( T^{12} + 10 T^{11} + \cdots + 47599213 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 4772473782 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 134049432 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 19801106226 \) Copy content Toggle raw display
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