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Properties

Label 6024.2.a.r
Level $6024$
Weight $2$
Character orbit 6024.a
Self dual yes
Analytic conductor $48.102$
Analytic rank $0$
Dimension $20$
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] = [6024,2,Mod(1,6024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6024, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6024.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: \(48.1018821776\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 9 x^{19} - 31 x^{18} + 471 x^{17} - 82 x^{16} - 9476 x^{15} + 12881 x^{14} + 94079 x^{13} + \cdots + 24832 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{5} - \beta_{3} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_1 q^{5} - \beta_{3} q^{7} + q^{9} - \beta_{13} q^{11} + ( - \beta_{12} + 1) q^{13} + \beta_1 q^{15} + (\beta_{11} + 1) q^{17} - \beta_{17} q^{19} - \beta_{3} q^{21} + \beta_{4} q^{23} + (\beta_{19} + \beta_{15} + \beta_{13} + \cdots + 2) q^{25}+ \cdots - \beta_{13} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9} + 4 q^{11} + 21 q^{13} + 9 q^{15} + 10 q^{17} + 8 q^{19} + 9 q^{21} + 9 q^{23} + 43 q^{25} + 20 q^{27} + 18 q^{29} + 27 q^{31} + 4 q^{33} - 7 q^{35} + 33 q^{37} + 21 q^{39} + 14 q^{41} - 6 q^{43} + 9 q^{45} + 21 q^{47} + 47 q^{49} + 10 q^{51} + 23 q^{53} + 24 q^{55} + 8 q^{57} + 10 q^{59} + 28 q^{61} + 9 q^{63} + 2 q^{65} + 15 q^{67} + 9 q^{69} + 33 q^{71} + 50 q^{73} + 43 q^{75} + 20 q^{77} + 17 q^{79} + 20 q^{81} - 19 q^{83} + 41 q^{85} + 18 q^{87} + 21 q^{89} + 30 q^{91} + 27 q^{93} + 27 q^{95} + 47 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 9 x^{19} - 31 x^{18} + 471 x^{17} - 82 x^{16} - 9476 x^{15} + 12881 x^{14} + 94079 x^{13} + \cdots + 24832 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!07 \nu^{19} + \cdots - 40\!\cdots\!52 ) / 38\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 70\!\cdots\!27 \nu^{19} + \cdots - 29\!\cdots\!24 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18\!\cdots\!89 \nu^{19} + \cdots - 99\!\cdots\!44 ) / 38\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!53 \nu^{19} + \cdots - 20\!\cdots\!76 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\!\cdots\!21 \nu^{19} + \cdots + 81\!\cdots\!12 ) / 38\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\!\cdots\!65 \nu^{19} + \cdots - 14\!\cdots\!92 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 93\!\cdots\!42 \nu^{19} + \cdots - 19\!\cdots\!92 ) / 96\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 32\!\cdots\!45 \nu^{19} + \cdots - 40\!\cdots\!04 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 99\!\cdots\!39 \nu^{19} + \cdots - 31\!\cdots\!48 ) / 48\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!70 \nu^{19} + \cdots + 15\!\cdots\!52 ) / 68\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!05 \nu^{19} + \cdots + 10\!\cdots\!56 ) / 38\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 28\!\cdots\!13 \nu^{19} + \cdots - 13\!\cdots\!16 ) / 96\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 28\!\cdots\!89 \nu^{19} + \cdots + 47\!\cdots\!28 ) / 96\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 18\!\cdots\!69 \nu^{19} + \cdots - 16\!\cdots\!04 ) / 38\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 94\!\cdots\!53 \nu^{19} + \cdots + 11\!\cdots\!72 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 98\!\cdots\!37 \nu^{19} + \cdots + 86\!\cdots\!08 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 25\!\cdots\!59 \nu^{19} + \cdots + 37\!\cdots\!44 ) / 38\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 26\!\cdots\!85 \nu^{19} + \cdots + 19\!\cdots\!44 ) / 38\!\cdots\!84 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} + \beta_{15} + \beta_{13} - \beta_{8} + \beta_{6} + \beta_{4} + \beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{19} + \beta_{18} - 2 \beta_{17} - 2 \beta_{16} + \beta_{13} + \beta_{11} + \beta_{8} - \beta_{7} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 21 \beta_{19} - \beta_{18} - \beta_{17} - 4 \beta_{16} + 20 \beta_{15} + 4 \beta_{14} + 16 \beta_{13} + \cdots + 96 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 53 \beta_{19} + 18 \beta_{18} - 44 \beta_{17} - 45 \beta_{16} + 3 \beta_{15} + 2 \beta_{14} + 35 \beta_{13} + \cdots + 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 399 \beta_{19} - 30 \beta_{18} - 20 \beta_{17} - 107 \beta_{16} + 366 \beta_{15} + 100 \beta_{14} + \cdots + 1556 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1147 \beta_{19} + 272 \beta_{18} - 818 \beta_{17} - 874 \beta_{16} + 118 \beta_{15} + 107 \beta_{14} + \cdots + 1093 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 7433 \beta_{19} - 758 \beta_{18} - 250 \beta_{17} - 2307 \beta_{16} + 6564 \beta_{15} + 2107 \beta_{14} + \cdots + 26808 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 23242 \beta_{19} + 3879 \beta_{18} - 14550 \beta_{17} - 16454 \beta_{16} + 3076 \beta_{15} + \cdots + 24000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 138196 \beta_{19} - 17646 \beta_{18} - 1693 \beta_{17} - 46816 \beta_{16} + 117174 \beta_{15} + \cdots + 475381 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 457878 \beta_{19} + 52126 \beta_{18} - 255043 \beta_{17} - 307432 \beta_{16} + 69205 \beta_{15} + \cdots + 510755 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2576151 \beta_{19} - 389933 \beta_{18} + 20627 \beta_{17} - 926883 \beta_{16} + 2092657 \beta_{15} + \cdots + 8568397 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8899212 \beta_{19} + 628325 \beta_{18} - 4449474 \beta_{17} - 5739069 \beta_{16} + 1458467 \beta_{15} + \cdots + 10620443 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 48195391 \beta_{19} - 8321865 \beta_{18} + 1205585 \beta_{17} - 18114234 \beta_{16} + 37473372 \beta_{15} + \cdots + 156044666 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 171778708 \beta_{19} + 5758273 \beta_{18} - 77561636 \beta_{17} - 107254757 \beta_{16} + \cdots + 217260447 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 904774360 \beta_{19} - 173406944 \beta_{18} + 34663333 \beta_{17} - 351199008 \beta_{16} + \cdots + 2862083727 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 3303997969 \beta_{19} + 2308883 \beta_{18} - 1353331821 \beta_{17} - 2007632198 \beta_{16} + \cdots + 4393716710 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 17037005368 \beta_{19} - 3552823207 \beta_{18} + 819203440 \beta_{17} - 6772878174 \beta_{16} + \cdots + 52767726574 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 63433494478 \beta_{19} - 1833217772 \beta_{18} - 23659744477 \beta_{17} - 37638208977 \beta_{16} + \cdots + 88126279520 \) 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
−4.22324
−3.69479
−3.19440
−1.98459
−1.83693
−1.23330
−1.06426
−0.553985
−0.458931
−0.129395
0.273378
1.32210
2.20464
2.35321
2.60908
2.61062
3.18186
4.06601
4.35600
4.39693
0 1.00000 0 −4.22324 0 1.38831 0 1.00000 0
1.2 0 1.00000 0 −3.69479 0 −1.23254 0 1.00000 0
1.3 0 1.00000 0 −3.19440 0 5.21224 0 1.00000 0
1.4 0 1.00000 0 −1.98459 0 −4.60442 0 1.00000 0
1.5 0 1.00000 0 −1.83693 0 2.82191 0 1.00000 0
1.6 0 1.00000 0 −1.23330 0 3.99613 0 1.00000 0
1.7 0 1.00000 0 −1.06426 0 1.22980 0 1.00000 0
1.8 0 1.00000 0 −0.553985 0 −2.25153 0 1.00000 0
1.9 0 1.00000 0 −0.458931 0 −2.87812 0 1.00000 0
1.10 0 1.00000 0 −0.129395 0 2.39454 0 1.00000 0
1.11 0 1.00000 0 0.273378 0 −2.92800 0 1.00000 0
1.12 0 1.00000 0 1.32210 0 −1.93279 0 1.00000 0
1.13 0 1.00000 0 2.20464 0 −1.71268 0 1.00000 0
1.14 0 1.00000 0 2.35321 0 2.49191 0 1.00000 0
1.15 0 1.00000 0 2.60908 0 3.89020 0 1.00000 0
1.16 0 1.00000 0 2.61062 0 4.62670 0 1.00000 0
1.17 0 1.00000 0 3.18186 0 2.54652 0 1.00000 0
1.18 0 1.00000 0 4.06601 0 1.32234 0 1.00000 0
1.19 0 1.00000 0 4.35600 0 −4.92117 0 1.00000 0
1.20 0 1.00000 0 4.39693 0 −0.459336 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(251\) \(1\)

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 6024.2.a.r ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6024.2.a.r ✓ 20 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6024))\):

\( T_{5}^{20} - 9 T_{5}^{19} - 31 T_{5}^{18} + 471 T_{5}^{17} - 82 T_{5}^{16} - 9476 T_{5}^{15} + \cdots + 24832 \) Copy content Toggle raw display
\( T_{7}^{20} - 9 T_{7}^{19} - 53 T_{7}^{18} + 671 T_{7}^{17} + 525 T_{7}^{16} - 19487 T_{7}^{15} + \cdots - 29241856 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T - 1)^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 9 T^{19} + \cdots + 24832 \) Copy content Toggle raw display
$7$ \( T^{20} - 9 T^{19} + \cdots - 29241856 \) Copy content Toggle raw display
$11$ \( T^{20} - 4 T^{19} + \cdots + 91205632 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots - 124730368 \) Copy content Toggle raw display
$17$ \( T^{20} - 10 T^{19} + \cdots - 3701888 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 22382608384 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 886263296 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 3335817837184 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 1727292928 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 83694977024 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 1521065441024 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 27604557824 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 39882529439744 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 17895297130496 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 6004213302272 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 43\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 12250866408448 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 42\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 13\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 28\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 97\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 29\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 75\!\cdots\!96 \) Copy content Toggle raw display
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