Properties

Label 6020.2.a.h
Level $6020$
Weight $2$
Character orbit 6020.a
Self dual yes
Analytic conductor $48.070$
Analytic rank $0$
Dimension $12$
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] = [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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 24 x^{10} + 66 x^{9} + 212 x^{8} - 497 x^{7} - 878 x^{6} + 1557 x^{5} + 1645 x^{4} - 1812 x^{3} - 929 x^{2} + 457 x + 76 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} + q^{7} + (\beta_{2} + \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - q^{5} + q^{7} + (\beta_{2} + \beta_1 + 2) q^{9} + (\beta_{11} + \beta_{4}) q^{11} + (\beta_{9} + 1) q^{13} - \beta_1 q^{15} + \beta_{7} q^{17} + ( - \beta_{8} - \beta_{5} - \beta_{4} + \beta_{2} + 1) q^{19} + \beta_1 q^{21} + ( - \beta_{11} - \beta_{8} + \beta_{6} - \beta_{3} + \beta_1 + 1) q^{23} + q^{25} + (\beta_{11} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + 2 \beta_1 + 2) q^{27} + (\beta_{8} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2}) q^{29} + (2 \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{2} + 3) q^{31} + ( - 2 \beta_{11} + \beta_{10} - \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots - 1) q^{33}+ \cdots + (2 \beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{3} - \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - 12 q^{5} + 12 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{3} - 12 q^{5} + 12 q^{7} + 21 q^{9} - 3 q^{11} + 8 q^{13} - 3 q^{15} - 3 q^{17} + 8 q^{19} + 3 q^{21} + 7 q^{23} + 12 q^{25} + 27 q^{27} - 3 q^{29} + 31 q^{31} - 5 q^{33} - 12 q^{35} + 14 q^{37} - 3 q^{39} + 3 q^{41} + 12 q^{43} - 21 q^{45} + 5 q^{47} + 12 q^{49} - 3 q^{51} + 16 q^{53} + 3 q^{55} - 34 q^{57} + 12 q^{59} + 23 q^{61} + 21 q^{63} - 8 q^{65} + 21 q^{67} + 20 q^{69} - 20 q^{71} + 10 q^{73} + 3 q^{75} - 3 q^{77} - 16 q^{79} + 52 q^{81} + 19 q^{83} + 3 q^{85} + 2 q^{87} + 15 q^{89} + 8 q^{91} + q^{93} - 8 q^{95} + 24 q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 24 x^{10} + 66 x^{9} + 212 x^{8} - 497 x^{7} - 878 x^{6} + 1557 x^{5} + 1645 x^{4} - 1812 x^{3} - 929 x^{2} + 457 x + 76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 33061 \nu^{11} - 481513 \nu^{10} + 569600 \nu^{9} + 9040238 \nu^{8} - 18051968 \nu^{7} - 52004719 \nu^{6} + 109729402 \nu^{5} + 102747306 \nu^{4} + \cdots - 1427891 ) / 2578211 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 81229 \nu^{11} - 14624 \nu^{10} - 2437857 \nu^{9} + 258931 \nu^{8} + 25930708 \nu^{7} - 1841294 \nu^{6} - 116440090 \nu^{5} + 11932951 \nu^{4} + \cdots + 7298934 ) / 5156422 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 95353 \nu^{11} + 155839 \nu^{10} + 2274685 \nu^{9} - 2508394 \nu^{8} - 19115141 \nu^{7} + 9380980 \nu^{6} + 67518612 \nu^{5} + 2239757 \nu^{4} + \cdots + 1492320 ) / 5156422 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 121675 \nu^{11} + 301821 \nu^{10} + 2908123 \nu^{9} - 6114618 \nu^{8} - 25749985 \nu^{7} + 41479690 \nu^{6} + 107366762 \nu^{5} - 120883707 \nu^{4} + \cdots - 36735176 ) / 5156422 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 157737 \nu^{11} + 1207445 \nu^{10} + 1327623 \nu^{9} - 23803900 \nu^{8} + 10956267 \nu^{7} + 149923588 \nu^{6} - 99623092 \nu^{5} + \cdots - 17498044 ) / 5156422 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 186382 \nu^{11} - 34793 \nu^{10} - 4846120 \nu^{9} - 872107 \nu^{8} + 42710317 \nu^{7} + 22661196 \nu^{6} - 140501734 \nu^{5} - 114015604 \nu^{4} + \cdots - 3920694 ) / 5156422 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 269509 \nu^{11} + 775558 \nu^{10} + 5572449 \nu^{9} - 14565847 \nu^{8} - 38348796 \nu^{7} + 84759424 \nu^{6} + 105467274 \nu^{5} - 184965197 \nu^{4} + \cdots - 23262838 ) / 5156422 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 279017 \nu^{11} + 528174 \nu^{10} + 5969339 \nu^{9} - 7797885 \nu^{8} - 43053494 \nu^{7} + 21146954 \nu^{6} + 123788924 \nu^{5} + 49144999 \nu^{4} + \cdots + 43467746 ) / 5156422 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 289549 \nu^{11} + 813826 \nu^{10} + 5866649 \nu^{9} - 14826643 \nu^{8} - 38750340 \nu^{7} + 80084310 \nu^{6} + 98873448 \nu^{5} - 142543719 \nu^{4} + \cdots - 3245206 ) / 5156422 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{4} + 8\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{4} + \beta_{3} + 11 \beta_{2} + 11 \beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17 \beta_{11} + \beta_{9} + 18 \beta_{8} - 17 \beta_{7} + 12 \beta_{6} + 15 \beta_{5} + 26 \beta_{4} - \beta_{3} + 3 \beta_{2} + 75 \beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17 \beta_{11} + 15 \beta_{10} + 34 \beta_{9} + 51 \beta_{8} - 22 \beta_{7} - 28 \beta_{6} + 3 \beta_{5} + 17 \beta_{4} + 12 \beta_{3} + 118 \beta_{2} + 120 \beta _1 + 374 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 216 \beta_{11} + 29 \beta_{9} + 243 \beta_{8} - 223 \beta_{7} + 122 \beta_{6} + 190 \beta_{5} + 296 \beta_{4} - 18 \beta_{3} + 59 \beta_{2} + 761 \beta _1 + 375 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 232 \beta_{11} + 178 \beta_{10} + 444 \beta_{9} + 683 \beta_{8} - 338 \beta_{7} - 322 \beta_{6} + 103 \beta_{5} + 233 \beta_{4} + 115 \beta_{3} + 1272 \beta_{2} + 1345 \beta _1 + 3791 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2495 \beta_{11} + 2 \beta_{10} + 517 \beta_{9} + 2988 \beta_{8} - 2683 \beta_{7} + 1185 \beta_{6} + 2310 \beta_{5} + 3240 \beta_{4} - 250 \beta_{3} + 887 \beta_{2} + 8062 \beta _1 + 4537 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2933 \beta_{11} + 1940 \beta_{10} + 5373 \beta_{9} + 8479 \beta_{8} - 4582 \beta_{7} - 3543 \beta_{6} + 2108 \beta_{5} + 2947 \beta_{4} + 999 \beta_{3} + 13781 \beta_{2} + 15383 \beta _1 + 40066 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 27742 \beta_{11} + 35 \beta_{10} + 7721 \beta_{9} + 35390 \beta_{8} - 31222 \beta_{7} + 11287 \beta_{6} + 27715 \beta_{5} + 34946 \beta_{4} - 3252 \beta_{3} + 12119 \beta_{2} + 87457 \beta _1 + 54673 \) 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
−3.18664
−2.11792
−1.97578
−1.82342
−0.697505
−0.138868
0.472855
1.11690
2.34127
2.37164
3.21868
3.41879
0 −3.18664 0 −1.00000 0 1.00000 0 7.15470 0
1.2 0 −2.11792 0 −1.00000 0 1.00000 0 1.48558 0
1.3 0 −1.97578 0 −1.00000 0 1.00000 0 0.903721 0
1.4 0 −1.82342 0 −1.00000 0 1.00000 0 0.324862 0
1.5 0 −0.697505 0 −1.00000 0 1.00000 0 −2.51349 0
1.6 0 −0.138868 0 −1.00000 0 1.00000 0 −2.98072 0
1.7 0 0.472855 0 −1.00000 0 1.00000 0 −2.77641 0
1.8 0 1.11690 0 −1.00000 0 1.00000 0 −1.75253 0
1.9 0 2.34127 0 −1.00000 0 1.00000 0 2.48156 0
1.10 0 2.37164 0 −1.00000 0 1.00000 0 2.62469 0
1.11 0 3.21868 0 −1.00000 0 1.00000 0 7.35990 0
1.12 0 3.41879 0 −1.00000 0 1.00000 0 8.68813 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6020.2.a.h 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6020))\):

\( T_{3}^{12} - 3 T_{3}^{11} - 24 T_{3}^{10} + 66 T_{3}^{9} + 212 T_{3}^{8} - 497 T_{3}^{7} - 878 T_{3}^{6} + 1557 T_{3}^{5} + 1645 T_{3}^{4} - 1812 T_{3}^{3} - 929 T_{3}^{2} + 457 T_{3} + 76 \) Copy content Toggle raw display
\( T_{11}^{12} + 3 T_{11}^{11} - 89 T_{11}^{10} - 237 T_{11}^{9} + 2946 T_{11}^{8} + 6414 T_{11}^{7} - 46166 T_{11}^{6} - 70947 T_{11}^{5} + 350924 T_{11}^{4} + 278572 T_{11}^{3} - 1086096 T_{11}^{2} - 92880 T_{11} + 500352 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 3 T^{11} - 24 T^{10} + 66 T^{9} + \cdots + 76 \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 3 T^{11} - 89 T^{10} + \cdots + 500352 \) Copy content Toggle raw display
$13$ \( T^{12} - 8 T^{11} - 45 T^{10} + \cdots - 16916 \) Copy content Toggle raw display
$17$ \( T^{12} + 3 T^{11} - 112 T^{10} + \cdots - 70284 \) Copy content Toggle raw display
$19$ \( T^{12} - 8 T^{11} - 132 T^{10} + \cdots - 746240 \) Copy content Toggle raw display
$23$ \( T^{12} - 7 T^{11} - 130 T^{10} + \cdots + 511608 \) Copy content Toggle raw display
$29$ \( T^{12} + 3 T^{11} - 238 T^{10} + \cdots + 32032992 \) Copy content Toggle raw display
$31$ \( T^{12} - 31 T^{11} + 233 T^{10} + \cdots - 38237618 \) Copy content Toggle raw display
$37$ \( T^{12} - 14 T^{11} - 206 T^{10} + \cdots + 3771848 \) Copy content Toggle raw display
$41$ \( T^{12} - 3 T^{11} - 301 T^{10} + \cdots + 102094368 \) Copy content Toggle raw display
$43$ \( (T - 1)^{12} \) Copy content Toggle raw display
$47$ \( T^{12} - 5 T^{11} - 314 T^{10} + \cdots - 234455040 \) Copy content Toggle raw display
$53$ \( T^{12} - 16 T^{11} + \cdots - 6724442016 \) Copy content Toggle raw display
$59$ \( T^{12} - 12 T^{11} - 159 T^{10} + \cdots - 1007490 \) Copy content Toggle raw display
$61$ \( T^{12} - 23 T^{11} + \cdots - 1140278990 \) Copy content Toggle raw display
$67$ \( T^{12} - 21 T^{11} - 50 T^{10} + \cdots + 9082972 \) Copy content Toggle raw display
$71$ \( T^{12} + 20 T^{11} + \cdots + 1191385434 \) Copy content Toggle raw display
$73$ \( T^{12} - 10 T^{11} - 332 T^{10} + \cdots - 2690720 \) Copy content Toggle raw display
$79$ \( T^{12} + 16 T^{11} + \cdots - 265066624 \) Copy content Toggle raw display
$83$ \( T^{12} - 19 T^{11} + \cdots - 113951808 \) Copy content Toggle raw display
$89$ \( T^{12} - 15 T^{11} - 232 T^{10} + \cdots + 5953830 \) Copy content Toggle raw display
$97$ \( T^{12} - 24 T^{11} + \cdots - 554895008 \) Copy content Toggle raw display
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