Properties

Label 6020.2.a.k
Level $6020$
Weight $2$
Character orbit 6020.a
Self dual yes
Analytic conductor $48.070$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6020,2,Mod(1,6020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6020, 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("6020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6020.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: \(48.0699420168\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 27 x^{11} - 2 x^{10} + 268 x^{9} + 37 x^{8} - 1201 x^{7} - 189 x^{6} + 2384 x^{5} + 231 x^{4} - 1729 x^{3} + 20 x^{2} + 105 x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + q^{5} - q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + q^{5} - q^{7} + (\beta_{2} + 1) q^{9} + \beta_{8} q^{11} + (\beta_{10} + 1) q^{13} + \beta_1 q^{15} + (\beta_{11} + 1) q^{17} + \beta_{9} q^{19} - \beta_1 q^{21} + ( - \beta_{6} - \beta_{5} + \beta_{3} + \beta_1) q^{23} + q^{25} + ( - \beta_{11} - \beta_{10} + \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{27} + (\beta_{12} - \beta_{9} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{29} + ( - \beta_{12} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{31} + ( - \beta_{12} - \beta_{11} + \beta_{8} + \beta_{6} + \beta_{5} + \beta_{3} - \beta_1 + 1) q^{33} - q^{35} + (\beta_{12} + \beta_{11} + \beta_{5} - \beta_{3} + 1) q^{37} + (\beta_{11} + 2 \beta_{4} - 2 \beta_{2} - 1) q^{39} + (\beta_{7} + \beta_{5} - \beta_{4} + 1) q^{41} + q^{43} + (\beta_{2} + 1) q^{45} + (\beta_{11} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3}) q^{47} + q^{49} + ( - \beta_{11} - \beta_{9} + \beta_{6} + 3 \beta_1 + 1) q^{51} + (\beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} + \beta_1 + 2) q^{53} + \beta_{8} q^{55} + ( - \beta_{11} + \beta_{7} + \beta_{5} + \beta_{2} - \beta_1 + 1) q^{57} + ( - \beta_{12} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{59} + (\beta_{10} - \beta_{9} + \beta_{6} + \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{61} + ( - \beta_{2} - 1) q^{63} + (\beta_{10} + 1) q^{65} + (\beta_{10} + \beta_{9} - \beta_{6} + 2 \beta_{4} + \beta_{3} - \beta_1 + 1) q^{67} + ( - \beta_{12} - \beta_{7} + \beta_{6} + 2 \beta_{4} - \beta_{2} - \beta_1 + 2) q^{69} + (\beta_{12} - \beta_{11} - \beta_{9} - \beta_{5} + \beta_{2} + 2 \beta_1 + 1) q^{71} + ( - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{4} + \beta_{2} + 3) q^{73} + \beta_1 q^{75} - \beta_{8} q^{77} + (\beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{9} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{79} + (\beta_{12} - \beta_{10} + \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1) q^{81} + ( - \beta_{12} + \beta_{9} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{83} + (\beta_{11} + 1) q^{85} + (\beta_{12} - \beta_{10} - \beta_{9} - \beta_{7} + 2 \beta_{6} + 3 \beta_1 + 1) q^{87} + ( - \beta_{10} - \beta_{9} + \beta_{6} - 2 \beta_{4} - 3 \beta_{3} + \beta_1 + 1) q^{89} + ( - \beta_{10} - 1) q^{91} + ( - \beta_{12} + \beta_{11} - \beta_{8} - 2 \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} + \cdots + 2) q^{93}+ \cdots + ( - 2 \beta_{10} + \beta_{7} + \beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9} + 11 q^{13} + 16 q^{17} + 3 q^{23} + 13 q^{25} + 6 q^{27} + 10 q^{29} - q^{31} + 14 q^{33} - 13 q^{35} + 16 q^{37} - 14 q^{39} + 23 q^{41} + 13 q^{43} + 15 q^{45} + 2 q^{47} + 13 q^{49} + 4 q^{51} + 20 q^{53} + 22 q^{57} + 2 q^{59} + 5 q^{61} - 15 q^{63} + 11 q^{65} + 19 q^{67} + 16 q^{69} + 4 q^{71} + 34 q^{73} - 15 q^{79} + 17 q^{81} + 27 q^{83} + 16 q^{85} - 5 q^{87} + 3 q^{89} - 11 q^{91} + 35 q^{93} + 45 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 27 x^{11} - 2 x^{10} + 268 x^{9} + 37 x^{8} - 1201 x^{7} - 189 x^{6} + 2384 x^{5} + 231 x^{4} - 1729 x^{3} + 20 x^{2} + 105 x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1537654 \nu^{12} - 33962 \nu^{11} + 42386523 \nu^{10} + 2329850 \nu^{9} - 430739109 \nu^{8} - 29216373 \nu^{7} + 1977473357 \nu^{6} + \cdots - 61265760 ) / 42614911 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3379072 \nu^{12} + 657965 \nu^{11} + 90038475 \nu^{10} - 6078641 \nu^{9} - 883869052 \nu^{8} - 53662673 \nu^{7} + 3942799622 \nu^{6} + \cdots - 407264550 ) / 85229822 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3634961 \nu^{12} - 1032220 \nu^{11} - 98400044 \nu^{10} + 19594321 \nu^{9} + 981840509 \nu^{8} - 115849143 \nu^{7} - 4449167492 \nu^{6} + \cdots + 532793576 ) / 85229822 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4068317 \nu^{12} + 2428772 \nu^{11} - 108151396 \nu^{10} - 71287003 \nu^{9} + 1045858413 \nu^{8} + 738896929 \nu^{7} - 4477485048 \nu^{6} + \cdots - 56489344 ) / 85229822 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4383099 \nu^{12} + 2682991 \nu^{11} - 114991907 \nu^{10} - 79451502 \nu^{9} + 1081452331 \nu^{8} + 830625394 \nu^{7} - 4359230160 \nu^{6} + \cdots - 56900248 ) / 85229822 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2205646 \nu^{12} + 200934 \nu^{11} + 58451359 \nu^{10} + 796160 \nu^{9} - 567105876 \nu^{8} - 68621002 \nu^{7} + 2478773524 \nu^{6} + \cdots - 238977264 ) / 42614911 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2264279 \nu^{12} + 1297400 \nu^{11} + 60374111 \nu^{10} - 27283586 \nu^{9} - 591766125 \nu^{8} + 184521935 \nu^{7} + 2625107725 \nu^{6} + \cdots - 115635328 ) / 42614911 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5099437 \nu^{12} - 1334884 \nu^{11} - 133195254 \nu^{10} + 20110413 \nu^{9} + 1264513359 \nu^{8} - 52366611 \nu^{7} - 5328390224 \nu^{6} + \cdots - 20700834 ) / 85229822 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5423260 \nu^{12} + 3173615 \nu^{11} - 149767663 \nu^{10} - 89978475 \nu^{9} + 1526692324 \nu^{8} + 903358959 \nu^{7} - 7046841160 \nu^{6} + \cdots + 238318094 ) / 85229822 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5608450 \nu^{12} + 759047 \nu^{11} + 152122251 \nu^{10} - 3476691 \nu^{9} - 1520659172 \nu^{8} - 135571279 \nu^{7} + 6888121624 \nu^{6} + \cdots - 259624920 ) / 85229822 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} + \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - \beta_{10} + \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} + 10\beta_{2} + \beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{12} - 13 \beta_{11} - 12 \beta_{10} + 2 \beta_{8} - \beta_{7} + 15 \beta_{6} + 2 \beta_{5} - 11 \beta_{4} - 11 \beta_{3} + 11 \beta_{2} + 55 \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16 \beta_{12} + \beta_{11} - 14 \beta_{10} - 3 \beta_{9} + 5 \beta_{8} + 15 \beta_{7} - 14 \beta_{6} + 13 \beta_{5} - 32 \beta_{4} + 96 \beta_{2} + 14 \beta _1 + 213 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 16 \beta_{12} - 140 \beta_{11} - 126 \beta_{10} - 5 \beta_{9} + 35 \beta_{8} - 17 \beta_{7} + 177 \beta_{6} + 34 \beta_{5} - 105 \beta_{4} - 104 \beta_{3} + 102 \beta_{2} + 466 \beta _1 + 145 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 200 \beta_{12} + 25 \beta_{11} - 154 \beta_{10} - 56 \beta_{9} + 87 \beta_{8} + 172 \beta_{7} - 161 \beta_{6} + 135 \beta_{5} - 391 \beta_{4} + 3 \beta_{3} + 921 \beta_{2} + 148 \beta _1 + 1822 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 178 \beta_{12} - 1427 \beta_{11} - 1278 \beta_{10} - 114 \beta_{9} + 444 \beta_{8} - 217 \beta_{7} + 1923 \beta_{6} + 422 \beta_{5} - 978 \beta_{4} - 972 \beta_{3} + 904 \beta_{2} + 4153 \beta _1 + 1355 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2291 \beta_{12} + 416 \beta_{11} - 1549 \beta_{10} - 758 \beta_{9} + 1089 \beta_{8} + 1809 \beta_{7} - 1752 \beta_{6} + 1308 \beta_{5} - 4316 \beta_{4} + 90 \beta_{3} + 8866 \beta_{2} + 1400 \beta _1 + 16329 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1730 \beta_{12} - 14222 \beta_{11} - 12771 \beta_{10} - 1743 \beta_{9} + 4994 \beta_{8} - 2510 \beta_{7} + 20128 \beta_{6} + 4647 \beta_{5} - 9084 \beta_{4} - 9175 \beta_{3} + 7846 \beta_{2} + \cdots + 12073 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 25137 \beta_{12} + 5835 \beta_{11} - 14912 \beta_{10} - 9105 \beta_{9} + 12060 \beta_{8} + 18385 \beta_{7} - 18684 \beta_{6} + 12335 \beta_{5} - 45331 \beta_{4} + 1676 \beta_{3} + 85618 \beta_{2} + \cdots + 150610 \) Copy content Toggle raw display

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
−3.18036
−2.54322
−2.12558
−1.90637
−1.30831
−0.265767
0.0387611
0.247295
1.13823
1.47357
2.41335
2.91397
3.10444
0 −3.18036 0 1.00000 0 −1.00000 0 7.11471 0
1.2 0 −2.54322 0 1.00000 0 −1.00000 0 3.46796 0
1.3 0 −2.12558 0 1.00000 0 −1.00000 0 1.51810 0
1.4 0 −1.90637 0 1.00000 0 −1.00000 0 0.634257 0
1.5 0 −1.30831 0 1.00000 0 −1.00000 0 −1.28831 0
1.6 0 −0.265767 0 1.00000 0 −1.00000 0 −2.92937 0
1.7 0 0.0387611 0 1.00000 0 −1.00000 0 −2.99850 0
1.8 0 0.247295 0 1.00000 0 −1.00000 0 −2.93885 0
1.9 0 1.13823 0 1.00000 0 −1.00000 0 −1.70443 0
1.10 0 1.47357 0 1.00000 0 −1.00000 0 −0.828594 0
1.11 0 2.41335 0 1.00000 0 −1.00000 0 2.82427 0
1.12 0 2.91397 0 1.00000 0 −1.00000 0 5.49125 0
1.13 0 3.10444 0 1.00000 0 −1.00000 0 6.63752 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)
\(43\) \(-1\)

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 6020.2.a.k 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6020.2.a.k 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6020))\):

\( T_{3}^{13} - 27 T_{3}^{11} - 2 T_{3}^{10} + 268 T_{3}^{9} + 37 T_{3}^{8} - 1201 T_{3}^{7} - 189 T_{3}^{6} + 2384 T_{3}^{5} + 231 T_{3}^{4} - 1729 T_{3}^{3} + 20 T_{3}^{2} + 105 T_{3} - 4 \) Copy content Toggle raw display
\( T_{11}^{13} - 90 T_{11}^{11} + 50 T_{11}^{10} + 2953 T_{11}^{9} - 3472 T_{11}^{8} - 43924 T_{11}^{7} + 83075 T_{11}^{6} + 276481 T_{11}^{5} - 818440 T_{11}^{4} - 203300 T_{11}^{3} + 2734016 T_{11}^{2} - 3233168 T_{11} + 1212800 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} \) Copy content Toggle raw display
$3$ \( T^{13} - 27 T^{11} - 2 T^{10} + 268 T^{9} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{13} \) Copy content Toggle raw display
$7$ \( (T + 1)^{13} \) Copy content Toggle raw display
$11$ \( T^{13} - 90 T^{11} + 50 T^{10} + \cdots + 1212800 \) Copy content Toggle raw display
$13$ \( T^{13} - 11 T^{12} - 45 T^{11} + \cdots - 36992 \) Copy content Toggle raw display
$17$ \( T^{13} - 16 T^{12} + 11 T^{11} + \cdots - 16360 \) Copy content Toggle raw display
$19$ \( T^{13} - 130 T^{11} + 5 T^{10} + \cdots - 2596864 \) Copy content Toggle raw display
$23$ \( T^{13} - 3 T^{12} - 148 T^{11} + \cdots - 2598608 \) Copy content Toggle raw display
$29$ \( T^{13} - 10 T^{12} - 139 T^{11} + \cdots - 13319136 \) Copy content Toggle raw display
$31$ \( T^{13} + T^{12} - 315 T^{11} + \cdots + 33628624 \) Copy content Toggle raw display
$37$ \( T^{13} - 16 T^{12} + \cdots + 10344616960 \) Copy content Toggle raw display
$41$ \( T^{13} - 23 T^{12} + 43 T^{11} + \cdots - 93736256 \) Copy content Toggle raw display
$43$ \( (T - 1)^{13} \) Copy content Toggle raw display
$47$ \( T^{13} - 2 T^{12} + \cdots - 1966768064 \) Copy content Toggle raw display
$53$ \( T^{13} - 20 T^{12} + \cdots - 533806784 \) Copy content Toggle raw display
$59$ \( T^{13} - 2 T^{12} + \cdots + 8018407640 \) Copy content Toggle raw display
$61$ \( T^{13} - 5 T^{12} + \cdots + 13507683320 \) Copy content Toggle raw display
$67$ \( T^{13} - 19 T^{12} + \cdots + 32115774744 \) Copy content Toggle raw display
$71$ \( T^{13} - 4 T^{12} - 375 T^{11} + \cdots - 21390896 \) Copy content Toggle raw display
$73$ \( T^{13} - 34 T^{12} + 172 T^{11} + \cdots - 9479744 \) Copy content Toggle raw display
$79$ \( T^{13} + 15 T^{12} + \cdots - 442238670208 \) Copy content Toggle raw display
$83$ \( T^{13} - 27 T^{12} + \cdots + 154375792640 \) Copy content Toggle raw display
$89$ \( T^{13} - 3 T^{12} - 632 T^{11} + \cdots + 682505452 \) Copy content Toggle raw display
$97$ \( T^{13} - 45 T^{12} + \cdots + 7093265644 \) Copy content Toggle raw display
show more
show less