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Properties

Label 6016.2.a.r
Level $6016$
Weight $2$
Character orbit 6016.a
Self dual yes
Analytic conductor $48.038$
Analytic rank $0$
Dimension $14$
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] = [6016,2,Mod(1,6016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6016, 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("6016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6016 = 2^{7} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6016.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: \(48.0380018560\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 30 x^{12} + 56 x^{11} + 331 x^{10} - 562 x^{9} - 1630 x^{8} + 2458 x^{7} + \cdots + 128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{6} q^{5} + \beta_{12} q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{6} q^{5} + \beta_{12} q^{7} + (\beta_{2} + 2) q^{9} + (\beta_{5} - 1) q^{11} + \beta_{9} q^{13} + ( - \beta_{10} - \beta_{9} + \beta_{5} + \cdots + 2) q^{15}+ \cdots + ( - 2 \beta_{13} - 3 \beta_{12} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{3} + 6 q^{5} + 2 q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{3} + 6 q^{5} + 2 q^{7} + 22 q^{9} - 14 q^{11} + 4 q^{13} + 6 q^{15} + 8 q^{17} - 8 q^{19} + 6 q^{21} + 18 q^{23} + 22 q^{25} - 8 q^{27} + 22 q^{29} + 4 q^{31} + 2 q^{33} - 26 q^{35} + 6 q^{37} + 20 q^{39} + 16 q^{41} - 12 q^{43} + 30 q^{45} + 14 q^{47} + 34 q^{49} - 18 q^{51} + 20 q^{53} + 2 q^{55} - 4 q^{57} - 32 q^{59} + 12 q^{61} + 40 q^{65} - 16 q^{67} + 46 q^{69} + 16 q^{71} - 12 q^{73} + 16 q^{75} + 10 q^{77} + 16 q^{79} + 74 q^{81} - 14 q^{83} + 12 q^{85} + 14 q^{87} - 28 q^{91} - 16 q^{93} + 52 q^{95} + 28 q^{97} - 50 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 30 x^{12} + 56 x^{11} + 331 x^{10} - 562 x^{9} - 1630 x^{8} + 2458 x^{7} + \cdots + 128 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{13} - 2 \nu^{12} - 30 \nu^{11} + 56 \nu^{10} + 331 \nu^{9} - 562 \nu^{8} - 1630 \nu^{7} + \cdots - 480 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 484993 \nu^{13} + 730282 \nu^{12} + 14955638 \nu^{11} - 19876816 \nu^{10} - 171836347 \nu^{9} + \cdots + 125207648 ) / 4518368 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 289541 \nu^{13} - 260388 \nu^{12} - 9039874 \nu^{11} + 6300544 \nu^{10} + 104858487 \nu^{9} + \cdots + 9662480 ) / 2259184 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 78274 \nu^{13} + 126585 \nu^{12} + 2398951 \nu^{11} - 3472995 \nu^{10} - 27321652 \nu^{9} + \cdots + 24122896 ) / 564796 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 777001 \nu^{13} - 1214150 \nu^{12} - 23823670 \nu^{11} + 32998120 \nu^{10} + 271394307 \nu^{9} + \cdots - 209711264 ) / 4518368 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 656282 \nu^{13} + 1127343 \nu^{12} + 20060086 \nu^{11} - 31133002 \nu^{10} - 227638294 \nu^{9} + \cdots + 244559088 ) / 2259184 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1455971 \nu^{13} + 2703562 \nu^{12} + 44317498 \nu^{11} - 75509272 \nu^{10} - 500716337 \nu^{9} + \cdots + 596942400 ) / 4518368 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2421541 \nu^{13} - 3831170 \nu^{12} - 74306062 \nu^{11} + 104590656 \nu^{10} + 846983271 \nu^{9} + \cdots - 704003136 ) / 4518368 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3701455 \nu^{13} - 5681996 \nu^{12} - 113745806 \nu^{11} + 154386324 \nu^{10} + 1298611333 \nu^{9} + \cdots - 966338624 ) / 4518368 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2565158 \nu^{13} - 3704225 \nu^{12} - 78967818 \nu^{11} + 99735398 \nu^{10} + 903129002 \nu^{9} + \cdots - 615133760 ) / 2259184 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3577454 \nu^{13} + 5343897 \nu^{12} + 110025850 \nu^{11} - 144594126 \nu^{10} + \cdots + 932056000 ) / 2259184 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{4} - \beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{9} + 2\beta_{8} + 3\beta_{6} - \beta_{4} + \beta_{3} + 11\beta_{2} + 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} - 11 \beta_{6} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{13} + \beta_{11} + 10 \beta_{10} - 17 \beta_{9} + 33 \beta_{8} - \beta_{7} + 51 \beta_{6} + \cdots + 370 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 41 \beta_{13} + 41 \beta_{12} + 4 \beta_{11} + 38 \beta_{10} + 28 \beta_{9} - 40 \beta_{8} + 18 \beta_{7} + \cdots - 64 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 37 \beta_{13} + 4 \beta_{12} + 32 \beta_{11} + 62 \beta_{10} - 219 \beta_{9} + 417 \beta_{8} + \cdots + 3505 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 604 \beta_{13} + 608 \beta_{12} + 99 \beta_{11} + 511 \beta_{10} + 486 \beta_{9} - 593 \beta_{8} + \cdots - 1208 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 496 \beta_{13} + 112 \beta_{12} + 589 \beta_{11} + 124 \beta_{10} - 2569 \beta_{9} + 4831 \beta_{8} + \cdots + 34317 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7833 \beta_{13} + 7945 \beta_{12} + 1642 \beta_{11} + 6026 \beta_{10} + 6952 \beta_{9} - 7834 \beta_{8} + \cdots - 18676 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 5941 \beta_{13} + 2004 \beta_{12} + 8680 \beta_{11} - 3955 \beta_{10} - 29012 \beta_{9} + \cdots + 344150 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 95286 \beta_{13} + 97290 \beta_{12} + 22913 \beta_{11} + 66875 \beta_{10} + 90057 \beta_{9} + \cdots - 260556 \) 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
3.19506
3.07939
2.93973
2.25884
0.738787
0.665022
0.491078
0.424329
−0.516354
−0.890986
−1.84392
−2.48560
−2.72539
−3.32999
0 −3.19506 0 1.68931 0 2.62839 0 7.20843 0
1.2 0 −3.07939 0 2.92934 0 −3.46404 0 6.48265 0
1.3 0 −2.93973 0 −3.39956 0 1.84135 0 5.64200 0
1.4 0 −2.25884 0 −1.99023 0 −2.05528 0 2.10235 0
1.5 0 −0.738787 0 4.01447 0 3.86183 0 −2.45419 0
1.6 0 −0.665022 0 1.90873 0 −4.05540 0 −2.55775 0
1.7 0 −0.491078 0 −0.776229 0 4.57937 0 −2.75884 0
1.8 0 −0.424329 0 1.21269 0 −4.06208 0 −2.81994 0
1.9 0 0.516354 0 −1.00514 0 0.337483 0 −2.73338 0
1.10 0 0.890986 0 −3.51901 0 −0.769608 0 −2.20614 0
1.11 0 1.84392 0 1.93931 0 −0.0400407 0 0.400051 0
1.12 0 2.48560 0 1.36605 0 3.24812 0 3.17820 0
1.13 0 2.72539 0 −2.56607 0 3.53774 0 4.42776 0
1.14 0 3.32999 0 4.19635 0 −3.58783 0 8.08881 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(47\) \(-1\)

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 6016.2.a.r yes 14
4.b odd 2 1 6016.2.a.t yes 14
8.b even 2 1 6016.2.a.s yes 14
8.d odd 2 1 6016.2.a.q ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6016.2.a.q ✓ 14 8.d odd 2 1
6016.2.a.r yes 14 1.a even 1 1 trivial
6016.2.a.s yes 14 8.b even 2 1
6016.2.a.t yes 14 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6016))\):

\( T_{3}^{14} + 2 T_{3}^{13} - 30 T_{3}^{12} - 56 T_{3}^{11} + 331 T_{3}^{10} + 562 T_{3}^{9} - 1630 T_{3}^{8} + \cdots + 128 \) Copy content Toggle raw display
\( T_{5}^{14} - 6 T_{5}^{13} - 28 T_{5}^{12} + 214 T_{5}^{11} + 182 T_{5}^{10} - 2780 T_{5}^{9} + \cdots + 24368 \) Copy content Toggle raw display
\( T_{13}^{14} - 4 T_{13}^{13} - 86 T_{13}^{12} + 402 T_{13}^{11} + 2508 T_{13}^{10} - 14124 T_{13}^{9} + \cdots + 1016944 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 2 T^{13} + \cdots + 128 \) Copy content Toggle raw display
$5$ \( T^{14} - 6 T^{13} + \cdots + 24368 \) Copy content Toggle raw display
$7$ \( T^{14} - 2 T^{13} + \cdots - 4304 \) Copy content Toggle raw display
$11$ \( T^{14} + 14 T^{13} + \cdots - 372656 \) Copy content Toggle raw display
$13$ \( T^{14} - 4 T^{13} + \cdots + 1016944 \) Copy content Toggle raw display
$17$ \( T^{14} - 8 T^{13} + \cdots + 12130688 \) Copy content Toggle raw display
$19$ \( T^{14} + 8 T^{13} + \cdots + 2087024 \) Copy content Toggle raw display
$23$ \( T^{14} - 18 T^{13} + \cdots - 9706496 \) Copy content Toggle raw display
$29$ \( T^{14} - 22 T^{13} + \cdots + 34106608 \) Copy content Toggle raw display
$31$ \( T^{14} - 4 T^{13} + \cdots - 10248192 \) Copy content Toggle raw display
$37$ \( T^{14} - 6 T^{13} + \cdots + 76015296 \) Copy content Toggle raw display
$41$ \( T^{14} - 16 T^{13} + \cdots - 33266432 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 2232468272 \) Copy content Toggle raw display
$47$ \( (T - 1)^{14} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 135056055488 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 18604160512 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 242176541504 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 884204528 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 826203767888 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 18918942464 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 1540489216 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 667693316096 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 68676302336 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 6010929088256 \) Copy content Toggle raw display
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