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Properties

Label 6024.2.a.q
Level $6024$
Weight $2$
Character orbit 6024.a
Self dual yes
Analytic conductor $48.102$
Analytic rank $0$
Dimension $18$
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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} - 57 x^{16} + 51 x^{15} + 1328 x^{14} - 1116 x^{13} - 16275 x^{12} + 13699 x^{11} + \cdots + 44032 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{5} - \beta_{14} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_1 q^{5} - \beta_{14} q^{7} + q^{9} - \beta_{15} q^{11} - \beta_{3} q^{13} + \beta_1 q^{15} - \beta_{4} q^{17} + (\beta_{10} + 1) q^{19} - \beta_{14} q^{21} - \beta_{5} q^{23} + (\beta_{2} + 1) q^{25} + q^{27} + (\beta_{16} + \beta_{15} - \beta_{13} + \cdots + 2) q^{29}+ \cdots - \beta_{15} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9} + 8 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} + 21 q^{19} + 7 q^{21} - 2 q^{23} + 25 q^{25} + 18 q^{27} + 5 q^{29} + 23 q^{31} + 8 q^{33} + 17 q^{35} + 15 q^{37} + 2 q^{39} + 20 q^{41} + 37 q^{43} + q^{45} - q^{47} + 33 q^{49} + 2 q^{51} + 2 q^{53} + 26 q^{55} + 21 q^{57} + 24 q^{59} + 20 q^{61} + 7 q^{63} + 26 q^{65} + 49 q^{67} - 2 q^{69} + 15 q^{71} + 15 q^{73} + 25 q^{75} + 6 q^{77} + 23 q^{79} + 18 q^{81} + 38 q^{83} + 21 q^{85} + 5 q^{87} + 31 q^{89} + 48 q^{91} + 23 q^{93} + 13 q^{95} + 25 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} - 57 x^{16} + 51 x^{15} + 1328 x^{14} - 1116 x^{13} - 16275 x^{12} + 13699 x^{11} + \cdots + 44032 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14\!\cdots\!69 \nu^{17} + \cdots + 10\!\cdots\!88 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 46\!\cdots\!68 \nu^{17} + \cdots + 74\!\cdots\!64 ) / 10\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 88\!\cdots\!67 \nu^{17} + \cdots + 83\!\cdots\!16 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 64\!\cdots\!01 \nu^{17} + \cdots - 27\!\cdots\!08 ) / 27\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!79 \nu^{17} + \cdots + 72\!\cdots\!12 ) / 27\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26\!\cdots\!91 \nu^{17} + \cdots - 48\!\cdots\!88 ) / 54\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 27\!\cdots\!51 \nu^{17} + \cdots - 23\!\cdots\!72 ) / 54\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 58\!\cdots\!31 \nu^{17} + \cdots - 36\!\cdots\!48 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 42\!\cdots\!29 \nu^{17} + \cdots + 26\!\cdots\!60 ) / 54\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 51\!\cdots\!69 \nu^{17} + \cdots + 17\!\cdots\!72 ) / 54\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 11\!\cdots\!99 \nu^{17} + \cdots - 10\!\cdots\!72 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 73\!\cdots\!76 \nu^{17} + \cdots - 38\!\cdots\!48 ) / 54\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 79\!\cdots\!17 \nu^{17} + \cdots + 49\!\cdots\!56 ) / 54\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 35\!\cdots\!54 \nu^{17} + \cdots - 24\!\cdots\!92 ) / 18\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 11\!\cdots\!43 \nu^{17} + \cdots + 74\!\cdots\!64 ) / 54\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} - \beta_{16} - \beta_{15} + 2 \beta_{14} + 2 \beta_{13} + \beta_{11} - 3 \beta_{10} + \cdots + 11 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{17} - \beta_{16} - 2 \beta_{15} + 2 \beta_{14} + \beta_{12} - \beta_{10} + 2 \beta_{9} + \cdots + 58 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 19 \beta_{17} - 16 \beta_{16} - 18 \beta_{15} + 35 \beta_{14} + 33 \beta_{13} + 3 \beta_{12} + 18 \beta_{11} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37 \beta_{17} - 39 \beta_{16} - 65 \beta_{15} + 43 \beta_{14} + 8 \beta_{13} + 32 \beta_{12} + \cdots + 657 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 283 \beta_{17} - 290 \beta_{16} - 339 \beta_{15} + 521 \beta_{14} + 498 \beta_{13} + 95 \beta_{12} + \cdots + 517 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 572 \beta_{17} - 1026 \beta_{16} - 1547 \beta_{15} + 756 \beta_{14} + 365 \beta_{13} + 680 \beta_{12} + \cdots + 8037 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3938 \beta_{17} - 5554 \beta_{16} - 6489 \beta_{15} + 7527 \beta_{14} + 7625 \beta_{13} + 2160 \beta_{12} + \cdots + 9964 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 8515 \beta_{17} - 22545 \beta_{16} - 31986 \beta_{15} + 12702 \beta_{14} + 10037 \beta_{13} + \cdots + 103555 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 53730 \beta_{17} - 106099 \beta_{16} - 123267 \beta_{15} + 109097 \beta_{14} + 119779 \beta_{13} + \cdots + 173562 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 125913 \beta_{17} - 450720 \beta_{16} - 613797 \beta_{15} + 211525 \beta_{14} + 224722 \beta_{13} + \cdots + 1392193 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 733565 \beta_{17} - 1986331 \beta_{16} - 2303444 \beta_{15} + 1604488 \beta_{14} + 1921903 \beta_{13} + \cdots + 2899559 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1869123 \beta_{17} - 8530036 \beta_{16} - 11281062 \beta_{15} + 3528593 \beta_{14} + 4532679 \beta_{13} + \cdots + 19431168 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 10129166 \beta_{17} - 36393606 \beta_{16} - 42279036 \beta_{15} + 24036289 \beta_{14} + 31311162 \beta_{13} + \cdots + 47620335 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 27986180 \beta_{17} - 156013338 \beta_{16} - 201909588 \beta_{15} + 59111791 \beta_{14} + \cdots + 280429505 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 142315607 \beta_{17} - 654644465 \beta_{16} - 763417207 \beta_{15} + 366958297 \beta_{14} + \cdots + 777857687 \) 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
−3.61197
−3.36568
−3.09022
−3.06473
−2.48653
−2.19329
−1.35939
−0.493430
0.301624
0.693705
0.750756
1.51521
1.75663
1.86531
2.64131
3.41379
3.61923
4.10767
0 1.00000 0 −3.61197 0 3.30181 0 1.00000 0
1.2 0 1.00000 0 −3.36568 0 −2.47862 0 1.00000 0
1.3 0 1.00000 0 −3.09022 0 4.20154 0 1.00000 0
1.4 0 1.00000 0 −3.06473 0 −3.46092 0 1.00000 0
1.5 0 1.00000 0 −2.48653 0 0.647451 0 1.00000 0
1.6 0 1.00000 0 −2.19329 0 −1.02399 0 1.00000 0
1.7 0 1.00000 0 −1.35939 0 −0.0125376 0 1.00000 0
1.8 0 1.00000 0 −0.493430 0 −4.29258 0 1.00000 0
1.9 0 1.00000 0 0.301624 0 4.56963 0 1.00000 0
1.10 0 1.00000 0 0.693705 0 −3.39358 0 1.00000 0
1.11 0 1.00000 0 0.750756 0 2.83375 0 1.00000 0
1.12 0 1.00000 0 1.51521 0 0.0590047 0 1.00000 0
1.13 0 1.00000 0 1.75663 0 3.11001 0 1.00000 0
1.14 0 1.00000 0 1.86531 0 −3.57984 0 1.00000 0
1.15 0 1.00000 0 2.64131 0 4.83855 0 1.00000 0
1.16 0 1.00000 0 3.41379 0 −0.715150 0 1.00000 0
1.17 0 1.00000 0 3.61923 0 0.0319200 0 1.00000 0
1.18 0 1.00000 0 4.10767 0 2.36355 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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.q ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6024.2.a.q ✓ 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6024))\):

\( T_{5}^{18} - T_{5}^{17} - 57 T_{5}^{16} + 51 T_{5}^{15} + 1328 T_{5}^{14} - 1116 T_{5}^{13} + \cdots + 44032 \) Copy content Toggle raw display
\( T_{7}^{18} - 7 T_{7}^{17} - 55 T_{7}^{16} + 445 T_{7}^{15} + 1087 T_{7}^{14} - 11249 T_{7}^{13} + \cdots + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( (T - 1)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} - T^{17} + \cdots + 44032 \) Copy content Toggle raw display
$7$ \( T^{18} - 7 T^{17} + \cdots + 32 \) Copy content Toggle raw display
$11$ \( T^{18} - 8 T^{17} + \cdots + 3625216 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots - 338589184 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 104358832 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 102145792 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 496962496 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 547284832 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 731530368 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots - 94616100864 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 14532359324224 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 389161934080 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 1309285122048 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 95065791488 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 151590231712 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 375938359765888 \) Copy content Toggle raw display
$67$ \( T^{18} - 49 T^{17} + \cdots - 414784 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 1931688804352 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 16552701882 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 14\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 69465118769152 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 15\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 6852946432 \) Copy content Toggle raw display
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