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Properties

Label 6036.2.a.f
Level $6036$
Weight $2$
Character orbit 6036.a
Self dual yes
Analytic conductor $48.198$
Analytic rank $1$
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] = [6036,2,Mod(1,6036)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6036, 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("6036.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6036.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: \(48.1977026600\)
Analytic rank: \(1\)
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: \( x^{14} - 6 x^{13} - 12 x^{12} + 112 x^{11} + 7 x^{10} - 710 x^{9} + 281 x^{8} + 1850 x^{7} - 830 x^{6} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 12 x^{12} + 112 x^{11} + 7 x^{10} - 710 x^{9} + 281 x^{8} + 1850 x^{7} - 830 x^{6} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7952245 \nu^{13} + 3028708645 \nu^{12} - 18771968543 \nu^{11} - 30186210676 \nu^{10} + \cdots - 90550432649 ) / 27475685729 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 322100129 \nu^{13} - 2221309782 \nu^{12} - 1385989722 \nu^{11} + 33379320665 \nu^{10} + \cdots - 60056755878 ) / 27475685729 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2007881369 \nu^{13} - 11372279696 \nu^{12} - 27653630613 \nu^{11} + 212995039129 \nu^{10} + \cdots - 56412665837 ) / 27475685729 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3152210972 \nu^{13} - 20759780039 \nu^{12} - 24374867588 \nu^{11} + 357762732425 \nu^{10} + \cdots + 3219219179 ) / 27475685729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4165887700 \nu^{13} + 24780738655 \nu^{12} + 51258093215 \nu^{11} - 465960412783 \nu^{10} + \cdots - 119337497143 ) / 27475685729 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4212643376 \nu^{13} + 23573883354 \nu^{12} + 58572637137 \nu^{11} - 436114908343 \nu^{10} + \cdots + 176609367247 ) / 27475685729 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4905000419 \nu^{13} + 31618495960 \nu^{12} + 44138491000 \nu^{11} - 564543134100 \nu^{10} + \cdots + 2484730916 ) / 27475685729 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5439301880 \nu^{13} + 31641095191 \nu^{12} + 72188909248 \nu^{11} - 605726146495 \nu^{10} + \cdots + 82980122956 ) / 27475685729 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5590186212 \nu^{13} - 31970020417 \nu^{12} - 78495590618 \nu^{11} + 621759608457 \nu^{10} + \cdots - 146248145768 ) / 27475685729 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6957537558 \nu^{13} - 41934570665 \nu^{12} - 81776887223 \nu^{11} + 778020971724 \nu^{10} + \cdots + 29960854643 ) / 27475685729 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 9079109911 \nu^{13} - 55197598148 \nu^{12} - 103938633805 \nu^{11} + 1019970626770 \nu^{10} + \cdots - 77123318663 ) / 27475685729 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 21130993205 \nu^{13} + 130464228246 \nu^{12} + 228630777256 \nu^{11} - 2390891047375 \nu^{10} + \cdots + 70165343107 ) / 27475685729 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} - \beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{13} - \beta_{11} - 2 \beta_{10} - \beta_{9} + 4 \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 19 \beta_{13} - 13 \beta_{12} - 7 \beta_{11} - 17 \beta_{10} - 11 \beta_{9} + 21 \beta_{8} + \cdots + 39 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 61 \beta_{13} - 21 \beta_{12} - 34 \beta_{11} - 48 \beta_{10} - 17 \beta_{9} + 87 \beta_{8} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 351 \beta_{13} - 201 \beta_{12} - 176 \beta_{11} - 283 \beta_{10} - 125 \beta_{9} + 384 \beta_{8} + \cdots + 509 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1313 \beta_{13} - 577 \beta_{12} - 758 \beta_{11} - 956 \beta_{10} - 256 \beta_{9} + 1607 \beta_{8} + \cdots + 1568 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 6332 \beta_{13} - 3363 \beta_{12} - 3506 \beta_{11} - 4769 \beta_{10} - 1530 \beta_{9} + 6821 \beta_{8} + \cdots + 7649 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 25174 \beta_{13} - 11936 \beta_{12} - 14774 \beta_{11} - 17822 \beta_{10} - 3989 \beta_{9} + \cdots + 27544 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 112550 \beta_{13} - 57882 \beta_{12} - 64910 \beta_{11} - 81496 \beta_{10} - 20608 \beta_{9} + \cdots + 123472 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 459495 \beta_{13} - 224498 \beta_{12} - 272049 \beta_{11} - 321593 \beta_{10} - 64717 \beta_{9} + \cdots + 479390 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1983492 \beta_{13} - 1005713 \beta_{12} - 1165533 \beta_{11} - 1405827 \beta_{10} - 303735 \beta_{9} + \cdots + 2068261 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 8195838 \beta_{13} - 4057544 \beta_{12} - 4876133 \beta_{11} - 5706295 \beta_{10} - 1080214 \beta_{9} + \cdots + 8330892 \) 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
4.17501
3.05416
2.98659
1.79209
1.63333
0.736579
0.176671
0.0590520
−0.246625
−0.362365
−1.38751
−1.41050
−2.28353
−2.92296
0 −1.00000 0 −4.17501 0 −2.39518 0 1.00000 0
1.2 0 −1.00000 0 −3.05416 0 −1.32476 0 1.00000 0
1.3 0 −1.00000 0 −2.98659 0 −4.28847 0 1.00000 0
1.4 0 −1.00000 0 −1.79209 0 1.22660 0 1.00000 0
1.5 0 −1.00000 0 −1.63333 0 0.404256 0 1.00000 0
1.6 0 −1.00000 0 −0.736579 0 1.91291 0 1.00000 0
1.7 0 −1.00000 0 −0.176671 0 4.02875 0 1.00000 0
1.8 0 −1.00000 0 −0.0590520 0 −1.12902 0 1.00000 0
1.9 0 −1.00000 0 0.246625 0 −4.31906 0 1.00000 0
1.10 0 −1.00000 0 0.362365 0 0.748260 0 1.00000 0
1.11 0 −1.00000 0 1.38751 0 −0.715830 0 1.00000 0
1.12 0 −1.00000 0 1.41050 0 1.12091 0 1.00000 0
1.13 0 −1.00000 0 2.28353 0 −2.96859 0 1.00000 0
1.14 0 −1.00000 0 2.92296 0 0.699214 0 1.00000 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
\(2\) \(-1\)
\(3\) \(1\)
\(503\) \(-1\)

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 6036.2.a.f ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6036.2.a.f ✓ 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6036))\):

\( T_{5}^{14} + 6 T_{5}^{13} - 12 T_{5}^{12} - 112 T_{5}^{11} + 7 T_{5}^{10} + 710 T_{5}^{9} + 281 T_{5}^{8} + \cdots + 1 \) Copy content Toggle raw display
\( T_{7}^{14} + 7 T_{7}^{13} - 15 T_{7}^{12} - 177 T_{7}^{11} - 62 T_{7}^{10} + 1248 T_{7}^{9} + 786 T_{7}^{8} + \cdots - 316 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T + 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 6 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{14} + 7 T^{13} + \cdots - 316 \) Copy content Toggle raw display
$11$ \( T^{14} - T^{13} + \cdots + 37 \) Copy content Toggle raw display
$13$ \( T^{14} + T^{13} + \cdots - 373 \) Copy content Toggle raw display
$17$ \( T^{14} + 6 T^{13} + \cdots + 740 \) Copy content Toggle raw display
$19$ \( T^{14} - T^{13} + \cdots - 12133 \) Copy content Toggle raw display
$23$ \( T^{14} - 10 T^{13} + \cdots - 14723633 \) Copy content Toggle raw display
$29$ \( T^{14} + 6 T^{13} + \cdots - 48365125 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 177733783 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 176384576 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 958233761 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 238857152 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 22735463201 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 734370907 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 4381982971 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 1785934579 \) Copy content Toggle raw display
$67$ \( T^{14} - 17 T^{13} + \cdots + 38988604 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 70984438352 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 55019690009 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 45100114732 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 218929559383 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 8556118324 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 31015238351 \) Copy content Toggle raw display
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