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Properties

Label 6042.2.a.bf
Level $6042$
Weight $2$
Character orbit 6042.a
Self dual yes
Analytic conductor $48.246$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6042,2,Mod(1,6042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, 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("6042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.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: \(48.2456129013\)
Analytic rank: \(0\)
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: \( x^{12} - 5 x^{11} - 27 x^{10} + 134 x^{9} + 294 x^{8} - 1313 x^{7} - 1685 x^{6} + 5910 x^{5} + \cdots + 4048 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + (\beta_1 - 1) q^{5} - q^{6} - \beta_{7} q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} + (\beta_1 - 1) q^{5} - q^{6} - \beta_{7} q^{7} - q^{8} + q^{9} + ( - \beta_1 + 1) q^{10} + \beta_{2} q^{11} + q^{12} + ( - \beta_{7} - \beta_{5}) q^{13} + \beta_{7} q^{14} + (\beta_1 - 1) q^{15} + q^{16} + (\beta_{11} + \beta_{8} + \beta_{7} + \cdots + 1) q^{17}+ \cdots + \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 7 q^{5} - 12 q^{6} + 5 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 7 q^{5} - 12 q^{6} + 5 q^{7} - 12 q^{8} + 12 q^{9} + 7 q^{10} - 5 q^{11} + 12 q^{12} + 10 q^{13} - 5 q^{14} - 7 q^{15} + 12 q^{16} + 6 q^{17} - 12 q^{18} + 12 q^{19} - 7 q^{20} + 5 q^{21} + 5 q^{22} - 12 q^{24} + 21 q^{25} - 10 q^{26} + 12 q^{27} + 5 q^{28} + 7 q^{30} + 2 q^{31} - 12 q^{32} - 5 q^{33} - 6 q^{34} - 17 q^{35} + 12 q^{36} + 16 q^{37} - 12 q^{38} + 10 q^{39} + 7 q^{40} + 7 q^{41} - 5 q^{42} + 24 q^{43} - 5 q^{44} - 7 q^{45} + 4 q^{47} + 12 q^{48} + 29 q^{49} - 21 q^{50} + 6 q^{51} + 10 q^{52} + 12 q^{53} - 12 q^{54} - 2 q^{55} - 5 q^{56} + 12 q^{57} + 2 q^{59} - 7 q^{60} + 29 q^{61} - 2 q^{62} + 5 q^{63} + 12 q^{64} - 17 q^{65} + 5 q^{66} + 36 q^{67} + 6 q^{68} + 17 q^{70} - 3 q^{71} - 12 q^{72} + 7 q^{73} - 16 q^{74} + 21 q^{75} + 12 q^{76} - 35 q^{77} - 10 q^{78} + 40 q^{79} - 7 q^{80} + 12 q^{81} - 7 q^{82} + 24 q^{83} + 5 q^{84} - 22 q^{85} - 24 q^{86} + 5 q^{88} - 45 q^{89} + 7 q^{90} + 71 q^{91} + 2 q^{93} - 4 q^{94} - 7 q^{95} - 12 q^{96} + q^{97} - 29 q^{98} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} - 27 x^{10} + 134 x^{9} + 294 x^{8} - 1313 x^{7} - 1685 x^{6} + 5910 x^{5} + \cdots + 4048 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 33394 \nu^{11} - 28307 \nu^{10} + 3132030 \nu^{9} - 8852987 \nu^{8} - 42987049 \nu^{7} + \cdots - 434774552 ) / 29065268 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 240415 \nu^{11} + 881782 \nu^{10} + 9164831 \nu^{9} - 32921225 \nu^{8} - 111040173 \nu^{7} + \cdots - 1413490944 ) / 116261072 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 240415 \nu^{11} - 881782 \nu^{10} - 9164831 \nu^{9} + 32921225 \nu^{8} + 111040173 \nu^{7} + \cdots + 715924512 ) / 116261072 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 189989 \nu^{11} - 1107305 \nu^{10} - 4357903 \nu^{9} + 30837332 \nu^{8} + 28149920 \nu^{7} + \cdots + 975911880 ) / 58130536 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 706435 \nu^{11} + 1519886 \nu^{10} + 27452799 \nu^{9} - 41360357 \nu^{8} - 388277645 \nu^{7} + \cdots + 1249035152 ) / 116261072 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 719715 \nu^{11} + 3372970 \nu^{10} + 19188627 \nu^{9} - 87318673 \nu^{8} - 187114117 \nu^{7} + \cdots - 413025456 ) / 116261072 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 933169 \nu^{11} + 5051586 \nu^{10} + 21846301 \nu^{9} - 130167559 \nu^{8} - 171505199 \nu^{7} + \cdots - 579260304 ) / 116261072 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 486498 \nu^{11} + 2895903 \nu^{10} + 10801602 \nu^{9} - 78900597 \nu^{8} - 65415103 \nu^{7} + \cdots - 1147126304 ) / 58130536 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1217761 \nu^{11} + 8309250 \nu^{10} + 15914015 \nu^{9} - 180804043 \nu^{8} + 3535843 \nu^{7} + \cdots - 1441942312 ) / 58130536 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7036535 \nu^{11} + 49323472 \nu^{10} + 87625363 \nu^{9} - 1101810507 \nu^{8} + \cdots - 13604819008 ) / 116261072 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + 3\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} + 3\beta_{3} + 10\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} + \beta_{8} + 7\beta_{7} - 3\beta_{6} + 6\beta_{5} + 20\beta_{4} + 21\beta_{3} + \beta_{2} + 22\beta _1 + 69 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} - 3 \beta_{10} - 8 \beta_{9} - 16 \beta_{8} + 61 \beta_{7} - 21 \beta_{6} + 54 \beta_{5} + \cdots + 173 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7 \beta_{11} - 14 \beta_{10} - 65 \beta_{9} + 16 \beta_{8} + 195 \beta_{7} - 83 \beta_{6} + 190 \beta_{5} + \cdots + 1060 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 65 \beta_{11} - 120 \beta_{10} - 269 \beta_{9} - 177 \beta_{8} + 1154 \beta_{7} - 432 \beta_{6} + \cdots + 3579 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 259 \beta_{11} - 553 \beta_{10} - 1565 \beta_{9} + 301 \beta_{8} + 4335 \beta_{7} - 1881 \beta_{6} + \cdots + 18406 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1600 \beta_{11} - 3295 \beta_{10} - 6760 \beta_{9} - 1248 \beta_{8} + 22111 \beta_{7} - 9011 \beta_{6} + \cdots + 70877 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6855 \beta_{11} - 15257 \beta_{10} - 34617 \beta_{9} + 7268 \beta_{8} + 90252 \beta_{7} - 40470 \beta_{6} + \cdots + 337587 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 36164 \beta_{11} - 79401 \beta_{10} - 153606 \beta_{9} + 7995 \beta_{8} + 430534 \beta_{7} + \cdots + 1387136 \) 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
−3.22349
−2.24876
−2.19598
−1.72799
−1.60709
−0.392682
1.26637
1.70127
2.22012
2.70802
4.00527
4.49493
−1.00000 1.00000 1.00000 −4.22349 −1.00000 3.89357 −1.00000 1.00000 4.22349
1.2 −1.00000 1.00000 1.00000 −3.24876 −1.00000 1.74785 −1.00000 1.00000 3.24876
1.3 −1.00000 1.00000 1.00000 −3.19598 −1.00000 2.75709 −1.00000 1.00000 3.19598
1.4 −1.00000 1.00000 1.00000 −2.72799 −1.00000 −0.326815 −1.00000 1.00000 2.72799
1.5 −1.00000 1.00000 1.00000 −2.60709 −1.00000 −4.26018 −1.00000 1.00000 2.60709
1.6 −1.00000 1.00000 1.00000 −1.39268 −1.00000 −2.74135 −1.00000 1.00000 1.39268
1.7 −1.00000 1.00000 1.00000 0.266375 −1.00000 1.01292 −1.00000 1.00000 −0.266375
1.8 −1.00000 1.00000 1.00000 0.701272 −1.00000 5.18729 −1.00000 1.00000 −0.701272
1.9 −1.00000 1.00000 1.00000 1.22012 −1.00000 −3.11298 −1.00000 1.00000 −1.22012
1.10 −1.00000 1.00000 1.00000 1.70802 −1.00000 3.64339 −1.00000 1.00000 −1.70802
1.11 −1.00000 1.00000 1.00000 3.00527 −1.00000 −3.21582 −1.00000 1.00000 −3.00527
1.12 −1.00000 1.00000 1.00000 3.49493 −1.00000 0.415035 −1.00000 1.00000 −3.49493
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(19\) \(-1\)
\(53\) \(-1\)

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 6042.2.a.bf ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.bf ✓ 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{12} + 7 T_{5}^{11} - 16 T_{5}^{10} - 191 T_{5}^{9} - 45 T_{5}^{8} + 1765 T_{5}^{7} + 1556 T_{5}^{6} + \cdots + 1776 \) Copy content Toggle raw display
\( T_{7}^{12} - 5 T_{7}^{11} - 44 T_{7}^{10} + 224 T_{7}^{9} + 658 T_{7}^{8} - 3596 T_{7}^{7} - 3494 T_{7}^{6} + \cdots - 5696 \) Copy content Toggle raw display
\( T_{11}^{12} + 5 T_{11}^{11} - 84 T_{11}^{10} - 329 T_{11}^{9} + 2978 T_{11}^{8} + 7230 T_{11}^{7} + \cdots - 1767424 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 7 T^{11} + \cdots + 1776 \) Copy content Toggle raw display
$7$ \( T^{12} - 5 T^{11} + \cdots - 5696 \) Copy content Toggle raw display
$11$ \( T^{12} + 5 T^{11} + \cdots - 1767424 \) Copy content Toggle raw display
$13$ \( T^{12} - 10 T^{11} + \cdots + 1152 \) Copy content Toggle raw display
$17$ \( T^{12} - 6 T^{11} + \cdots - 302496 \) Copy content Toggle raw display
$19$ \( (T - 1)^{12} \) Copy content Toggle raw display
$23$ \( T^{12} - 151 T^{10} + \cdots - 29104128 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots - 100789184 \) Copy content Toggle raw display
$31$ \( T^{12} - 2 T^{11} + \cdots - 70151424 \) Copy content Toggle raw display
$37$ \( T^{12} - 16 T^{11} + \cdots + 5254592 \) Copy content Toggle raw display
$41$ \( T^{12} - 7 T^{11} + \cdots + 26157056 \) Copy content Toggle raw display
$43$ \( T^{12} - 24 T^{11} + \cdots - 7168 \) Copy content Toggle raw display
$47$ \( T^{12} - 4 T^{11} + \cdots - 91152 \) Copy content Toggle raw display
$53$ \( (T - 1)^{12} \) Copy content Toggle raw display
$59$ \( T^{12} - 2 T^{11} + \cdots + 25016304 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 4908411744 \) Copy content Toggle raw display
$67$ \( T^{12} - 36 T^{11} + \cdots + 50937856 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 6161796096 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 38000489472 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 104194048 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 170989056 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 1368038528 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 271347847168 \) Copy content Toggle raw display
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