Properties

Label 6012.2.a.j
Level $6012$
Weight $2$
Character orbit 6012.a
Self dual yes
Analytic conductor $48.006$
Analytic rank $1$
Dimension $10$
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] = [6012,2,Mod(1,6012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6012, 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("6012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6012.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: \(48.0060616952\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 26x^{8} + 82x^{7} + 211x^{6} - 340x^{5} - 593x^{4} + 192x^{3} + 423x^{2} + 126x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{5} + \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{5} + \beta_1 q^{7} + (\beta_{7} + \beta_{6} + \beta_{3} - \beta_{2}) q^{11} - \beta_{9} q^{13} + (\beta_{9} - \beta_{7} - \beta_{5} + \beta_{3}) q^{17} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_1) q^{19} + (\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 3) q^{23} + ( - \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{2} + 2) q^{25} + ( - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{2} - 1) q^{29} + ( - \beta_{9} - \beta_{8} + \beta_{6} + \beta_{2} - \beta_1) q^{31} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{35} + (\beta_{9} - \beta_{7} - \beta_{4} - \beta_1) q^{37} + ( - \beta_{9} - \beta_{8} - 2 \beta_{5} - 2 \beta_{4} + \beta_{2} - \beta_1 + 3) q^{41} + (2 \beta_{8} - \beta_{6} + \beta_{5} - \beta_{2} - \beta_1 + 2) q^{43} + ( - \beta_{9} + \beta_{7} - \beta_{3} + \beta_{2} - \beta_1 - 5) q^{47} + ( - 2 \beta_{8} + \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_1 + 1) q^{49} + (\beta_{9} + 2 \beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} - 3) q^{53} + (2 \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - \beta_{2} + \cdots - 1) q^{55}+ \cdots + ( - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \cdots - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6 q^{5} + 4 q^{7} - 8 q^{11} - 2 q^{13} - 6 q^{17} - 20 q^{23} + 24 q^{25} - 8 q^{29} - 4 q^{31} - 4 q^{37} + 14 q^{41} + 20 q^{43} - 48 q^{47} - 2 q^{49} - 22 q^{53} - 6 q^{55} - 2 q^{59} - 8 q^{61} - 28 q^{65} - 6 q^{67} - 20 q^{71} + 20 q^{73} - 24 q^{77} - 4 q^{79} - 46 q^{83} - 18 q^{85} + 8 q^{89} + 28 q^{91} - 36 q^{95} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} - 26x^{8} + 82x^{7} + 211x^{6} - 340x^{5} - 593x^{4} + 192x^{3} + 423x^{2} + 126x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1438 \nu^{9} + 6681 \nu^{8} + 32441 \nu^{7} - 136919 \nu^{6} - 197274 \nu^{5} + 584470 \nu^{4} + 319314 \nu^{3} - 469017 \nu^{2} + 48114 \nu + 66348 ) / 44982 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1499 \nu^{9} - 6987 \nu^{8} - 34945 \nu^{7} + 148276 \nu^{6} + 234612 \nu^{5} - 710756 \nu^{4} - 561669 \nu^{3} + 837054 \nu^{2} + 435294 \nu - 151056 ) / 44982 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2840 \nu^{9} + 9945 \nu^{8} + 81184 \nu^{7} - 204046 \nu^{6} - 752934 \nu^{5} + 832712 \nu^{4} + 2390865 \nu^{3} - 463230 \nu^{2} - 1880118 \nu - 239175 ) / 44982 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1167 \nu^{9} + 5185 \nu^{8} + 27571 \nu^{7} - 105787 \nu^{6} - 188846 \nu^{5} + 438577 \nu^{4} + 436038 \nu^{3} - 257619 \nu^{2} - 276453 \nu - 92196 ) / 14994 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 426 \nu^{9} - 1700 \nu^{8} - 11178 \nu^{7} + 35355 \nu^{6} + 91532 \nu^{5} - 154395 \nu^{4} - 254713 \nu^{3} + 119881 \nu^{2} + 168813 \nu + 29004 ) / 4998 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1849 \nu^{9} - 8415 \nu^{8} - 43331 \nu^{7} + 174848 \nu^{6} + 291942 \nu^{5} - 775366 \nu^{4} - 667089 \nu^{3} + 651792 \nu^{2} + 460620 \nu + 60750 ) / 14994 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2419 \nu^{9} - 9622 \nu^{8} - 63438 \nu^{7} + 198654 \nu^{6} + 521801 \nu^{5} - 846911 \nu^{4} - 1490655 \nu^{3} + 595053 \nu^{2} + 1098198 \nu + 204129 ) / 14994 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14219 \nu^{9} - 61455 \nu^{8} - 349141 \nu^{7} + 1274200 \nu^{6} + 2574117 \nu^{5} - 5573576 \nu^{4} - 6555282 \nu^{3} + 4419333 \nu^{2} + 4422843 \nu + 679113 ) / 44982 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{8} + \beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} + \beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{9} - 3\beta_{8} - 6\beta_{7} + \beta_{6} - 2\beta_{5} + 2\beta_{3} + \beta_{2} + 18\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5 \beta_{9} - 43 \beta_{8} - 11 \beta_{7} + 25 \beta_{6} - 26 \beta_{5} - 34 \beta_{4} + 21 \beta_{3} + 9 \beta_{2} + 38 \beta _1 + 126 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 65 \beta_{9} - 93 \beta_{8} - 146 \beta_{7} + 46 \beta_{6} - 79 \beta_{5} - 18 \beta_{4} + 77 \beta_{3} + 41 \beta_{2} + 354 \beta _1 + 226 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 138 \beta_{9} - 866 \beta_{8} - 381 \beta_{7} + 571 \beta_{6} - 625 \beta_{5} - 584 \beta_{4} + 520 \beta_{3} + 264 \beta_{2} + 1030 \beta _1 + 2345 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1227 \beta_{9} - 2433 \beta_{8} - 3207 \beta_{7} + 1585 \beta_{6} - 2366 \beta_{5} - 684 \beta_{4} + 2249 \beta_{3} + 1165 \beta_{2} + 7317 \beta _1 + 6043 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3143 \beta_{9} - 17755 \beta_{8} - 10508 \beta_{7} + 13075 \beta_{6} - 15044 \beta_{5} - 10450 \beta_{4} + 13002 \beta_{3} + 6621 \beta_{2} + 25514 \beta _1 + 46794 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 22925 \beta_{9} - 59655 \beta_{8} - 70667 \beta_{7} + 45883 \beta_{6} - 63610 \beta_{5} - 19074 \beta_{4} + 59723 \beta_{3} + 30131 \beta_{2} + 156123 \beta _1 + 149656 \) 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
1.07621
4.79792
−2.20804
−3.81111
−1.19255
2.09707
−0.617293
4.25727
−0.108196
−0.291281
0 0 0 −3.98604 0 1.07621 0 0 0
1.2 0 0 0 −3.67303 0 4.79792 0 0 0
1.3 0 0 0 −3.55061 0 −2.20804 0 0 0
1.4 0 0 0 −2.67502 0 −3.81111 0 0 0
1.5 0 0 0 −0.553287 0 −1.19255 0 0 0
1.6 0 0 0 −0.399927 0 2.09707 0 0 0
1.7 0 0 0 0.860385 0 −0.617293 0 0 0
1.8 0 0 0 1.39178 0 4.25727 0 0 0
1.9 0 0 0 3.17375 0 −0.108196 0 0 0
1.10 0 0 0 3.41200 0 −0.291281 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(167\) \(-1\)

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 6012.2.a.j 10
3.b odd 2 1 6012.2.a.k yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6012.2.a.j 10 1.a even 1 1 trivial
6012.2.a.k yes 10 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 6 T_{5}^{9} - 19 T_{5}^{8} - 152 T_{5}^{7} + 55 T_{5}^{6} + 1182 T_{5}^{5} + 404 T_{5}^{4} - 2728 T_{5}^{3} - 587 T_{5}^{2} + 1196 T_{5} + 399 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 6 T^{9} - 19 T^{8} - 152 T^{7} + \cdots + 399 \) Copy content Toggle raw display
$7$ \( T^{10} - 4 T^{9} - 26 T^{8} + 82 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{10} + 8 T^{9} - 37 T^{8} + \cdots - 50841 \) Copy content Toggle raw display
$13$ \( T^{10} + 2 T^{9} - 58 T^{8} + \cdots + 9289 \) Copy content Toggle raw display
$17$ \( T^{10} + 6 T^{9} - 97 T^{8} + \cdots - 655837 \) Copy content Toggle raw display
$19$ \( T^{10} - 118 T^{8} + 4201 T^{6} + \cdots - 62927 \) Copy content Toggle raw display
$23$ \( T^{10} + 20 T^{9} + 97 T^{8} + \cdots - 2401 \) Copy content Toggle raw display
$29$ \( T^{10} + 8 T^{9} - 99 T^{8} + \cdots + 199927 \) Copy content Toggle raw display
$31$ \( T^{10} + 4 T^{9} - 168 T^{8} + \cdots - 496399 \) Copy content Toggle raw display
$37$ \( T^{10} + 4 T^{9} - 167 T^{8} + \cdots - 7450623 \) Copy content Toggle raw display
$41$ \( T^{10} - 14 T^{9} - 120 T^{8} + \cdots + 1233827 \) Copy content Toggle raw display
$43$ \( T^{10} - 20 T^{9} - 6 T^{8} + \cdots + 1501993 \) Copy content Toggle raw display
$47$ \( T^{10} + 48 T^{9} + 913 T^{8} + \cdots - 720657 \) Copy content Toggle raw display
$53$ \( T^{10} + 22 T^{9} + 21 T^{8} + \cdots - 3701117 \) Copy content Toggle raw display
$59$ \( T^{10} + 2 T^{9} - 275 T^{8} + \cdots + 304311 \) Copy content Toggle raw display
$61$ \( T^{10} + 8 T^{9} - 354 T^{8} + \cdots - 64024443 \) Copy content Toggle raw display
$67$ \( T^{10} + 6 T^{9} - 314 T^{8} + \cdots + 92973717 \) Copy content Toggle raw display
$71$ \( T^{10} + 20 T^{9} + \cdots + 609118083 \) Copy content Toggle raw display
$73$ \( T^{10} - 20 T^{9} - 78 T^{8} + \cdots + 9538237 \) Copy content Toggle raw display
$79$ \( T^{10} + 4 T^{9} - 106 T^{8} + \cdots + 189 \) Copy content Toggle raw display
$83$ \( T^{10} + 46 T^{9} + 797 T^{8} + \cdots + 3298771 \) Copy content Toggle raw display
$89$ \( T^{10} - 8 T^{9} - 462 T^{8} + \cdots + 11998287 \) Copy content Toggle raw display
$97$ \( T^{10} + 34 T^{9} - 51 T^{8} + \cdots + 95931549 \) Copy content Toggle raw display
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