Properties

Label 6021.2.a.l
Level $6021$
Weight $2$
Character orbit 6021.a
Self dual yes
Analytic conductor $48.078$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6021,2,Mod(1,6021)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6021, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6021.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6021 = 3^{3} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6021.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: \(48.0779270570\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 20x^{8} + 139x^{6} - 384x^{4} + 331x^{2} - 63 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{8} + \beta_{6} + \beta_{3} + 2) q^{4} + ( - \beta_{5} + \beta_{4}) q^{5} + \beta_{6} q^{7} + (\beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{8} + \beta_{6} + \beta_{3} + 2) q^{4} + ( - \beta_{5} + \beta_{4}) q^{5} + \beta_{6} q^{7} + (\beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{8} + (2 \beta_{9} - \beta_{8} - \beta_{3} - 2) q^{10} + ( - 2 \beta_{7} + \beta_{2} - \beta_1) q^{11} + (\beta_{9} - \beta_{8}) q^{13} + (\beta_{4} + 2 \beta_{2}) q^{14} + (2 \beta_{8} + 3 \beta_{6} - \beta_{3} + 3) q^{16} + (\beta_{5} + \beta_{2}) q^{17} + (\beta_{8} + \beta_{3} + 3) q^{19} + ( - 2 \beta_{7} - \beta_{5} + 3 \beta_{2} - \beta_1) q^{20} + (4 \beta_{9} - 3 \beta_{8} - \beta_{6} - 5) q^{22} + (2 \beta_{7} + \beta_{5} + \beta_{4} - 2 \beta_{2} - \beta_1) q^{23} + ( - \beta_{9} - 2 \beta_{6} + 3) q^{25} + ( - 2 \beta_{7} - 2 \beta_{5} + \beta_{2} + \beta_1) q^{26} + (\beta_{9} + \beta_{8} + 3 \beta_{6} - 2 \beta_{3} + 2) q^{28} + ( - \beta_{7} - \beta_{4} - \beta_{2} + 3 \beta_1) q^{29} + (\beta_{8} - \beta_{6} + 2 \beta_{3} - 1) q^{31} + (3 \beta_{7} + 4 \beta_{4} + 5 \beta_{2}) q^{32} + ( - \beta_{9} + 2 \beta_{8} + 3 \beta_{6} + 3) q^{34} + ( - \beta_{7} + 2 \beta_{5} - \beta_{4}) q^{35} + ( - \beta_{9} + \beta_{6} + 2 \beta_{3} + 3) q^{37} + (\beta_{5} - \beta_{2} + 4 \beta_1) q^{38} + (\beta_{9} - 3 \beta_{8} + 2 \beta_{6} - \beta_{3} - 1) q^{40} + (\beta_{4} + \beta_{2} - 3 \beta_1) q^{41} + (\beta_{9} - \beta_{8} - 2 \beta_{6} - \beta_{3}) q^{43} + ( - 3 \beta_{7} - 7 \beta_{5} + \beta_1) q^{44} + ( - 4 \beta_{9} + 4 \beta_{8} - \beta_{6} - 2) q^{46} + (2 \beta_{7} + 2 \beta_{5} - 4 \beta_{2} - \beta_1) q^{47} + (\beta_{8} + \beta_{6} - \beta_{3} - 4) q^{49} + (\beta_{7} + \beta_{5} - 3 \beta_{4} - 5 \beta_{2} + 2 \beta_1) q^{50} + (4 \beta_{9} - 3 \beta_{8} - \beta_{6} - 1) q^{52} + (\beta_{5} + \beta_{4} - 2 \beta_{2} - \beta_1) q^{53} + ( - 4 \beta_{9} + \beta_{8} + 2 \beta_{6} + 7 \beta_{3} + 9) q^{55} + (2 \beta_{7} + 5 \beta_{4} + 5 \beta_{2} + \beta_1) q^{56} + (\beta_{9} + \beta_{8} - \beta_{6} + 5 \beta_{3} + 10) q^{58} + ( - \beta_{7} - 2 \beta_{5} + \beta_{4} - 3 \beta_{2} + 3 \beta_1) q^{59} + (2 \beta_{8} + \beta_{3} + 4) q^{61} + ( - \beta_{7} + \beta_{5} - 2 \beta_{4} - 4 \beta_{2} + \beta_1) q^{62} + ( - 2 \beta_{9} + 3 \beta_{8} + 11 \beta_{6} - 6 \beta_{3} + 2) q^{64} + ( - \beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{2} + 3 \beta_1) q^{65} + ( - \beta_{9} - \beta_{8} - 3 \beta_{6} + 3) q^{67} + (3 \beta_{7} + \beta_{5} + 4 \beta_{4} + 3 \beta_{2} + 2 \beta_1) q^{68} + ( - \beta_{9} + 2 \beta_{8} + 3 \beta_{3} + 3) q^{70} + ( - 3 \beta_{5} + \beta_{4} + \beta_{2} + 4 \beta_1) q^{71} + ( - 3 \beta_{9} - \beta_{8} - 2 \beta_{6} + 5) q^{73} + ( - \beta_{7} + \beta_{5} - 2 \beta_{4} - \beta_{2} + 4 \beta_1) q^{74} + ( - \beta_{9} + 4 \beta_{8} + 3 \beta_{6} + 4 \beta_{3} + 11) q^{76} + (\beta_{7} - 2 \beta_{5} - 5 \beta_{2} + 3 \beta_1) q^{77} + (\beta_{9} + 2 \beta_{8} + \beta_{6} - \beta_{3} - 3) q^{79} + (\beta_{7} - 2 \beta_{5} + \beta_{4} + \beta_1) q^{80} + (\beta_{9} - 2 \beta_{8} - 4 \beta_{3} - 11) q^{82} + (2 \beta_{7} + 2 \beta_{2} - 2 \beta_1) q^{83} + (2 \beta_{6} + \beta_{3} - 4) q^{85} + ( - \beta_{7} - 2 \beta_{5} - \beta_{4} - 2 \beta_{2}) q^{86} + (5 \beta_{9} - 10 \beta_{8} - 7 \beta_{6} - 3 \beta_{3} - 3) q^{88} + ( - \beta_{7} - 4 \beta_{4} + \beta_{2} - \beta_1) q^{89} + ( - \beta_{6} + \beta_{3} + 2) q^{91} + (4 \beta_{7} + 6 \beta_{5} - 3 \beta_{4} - 2 \beta_{2} - 4 \beta_1) q^{92} + ( - 6 \beta_{9} + 5 \beta_{8} - 5 \beta_{6} + 3 \beta_{3} - 2) q^{94} + ( - \beta_{7} - 4 \beta_{5} + 2 \beta_{4} + 3 \beta_{2} - \beta_1) q^{95} + ( - 3 \beta_{9} + 2 \beta_{6} - 1) q^{97} + (2 \beta_{7} + \beta_{5} + 3 \beta_{4} + 3 \beta_{2} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 20 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 20 q^{4} + 2 q^{7} - 10 q^{10} + 2 q^{13} + 44 q^{16} + 28 q^{19} - 42 q^{22} + 22 q^{25} + 40 q^{28} - 18 q^{31} + 36 q^{34} + 20 q^{37} - 4 q^{40} + 2 q^{43} - 30 q^{46} - 32 q^{49} - 2 q^{52} + 52 q^{55} + 84 q^{58} + 40 q^{61} + 64 q^{64} + 18 q^{67} + 18 q^{70} + 32 q^{73} + 104 q^{76} - 16 q^{79} - 94 q^{82} - 40 q^{85} - 32 q^{88} + 14 q^{91} - 56 q^{94} - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 20x^{8} + 139x^{6} - 384x^{4} + 331x^{2} - 63 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} - 14\nu^{7} + 52\nu^{5} - 30\nu^{3} - 2\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{8} - 29\nu^{6} + 116\nu^{4} - 95\nu^{2} + 2 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8\nu^{9} - 121\nu^{7} + 539\nu^{5} - 675\nu^{3} + 173\nu ) / 15 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{9} + 191\nu^{7} - 799\nu^{5} + 840\nu^{3} - 238\nu ) / 15 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6\nu^{8} - 87\nu^{6} + 353\nu^{4} - 325\nu^{2} + 51 ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -19\nu^{9} + 278\nu^{7} - 1147\nu^{5} + 1110\nu^{3} - 154\nu ) / 15 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -8\nu^{8} + 116\nu^{6} - 469\nu^{4} + 425\nu^{2} - 73 ) / 5 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 3\nu^{8} - 44\nu^{6} + 183\nu^{4} - 185\nu^{2} + 38 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{6} + \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{8} + 9\beta_{6} + 5\beta_{3} + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{7} + 8\beta_{5} + 12\beta_{4} + 13\beta_{2} + 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{9} + 59\beta_{8} + 77\beta_{6} + 20\beta_{3} + 144 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 41\beta_{7} + 61\beta_{5} + 114\beta_{4} + 132\beta_{2} + 162\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -29\beta_{9} + 439\beta_{8} + 642\beta_{6} + 50\beta_{3} + 943 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 418\beta_{7} + 468\beta_{5} + 1002\beta_{4} + 1205\beta_{2} + 964\beta_1 \) 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
−2.81643
−2.32314
−2.31657
−1.02020
−0.513295
0.513295
1.02020
2.31657
2.32314
2.81643
−2.81643 0 5.93226 −0.124866 0 3.22829 −11.0749 0 0.351677
1.2 −2.32314 0 3.39697 3.53749 0 −1.40218 −3.24535 0 −8.21807
1.3 −2.31657 0 3.36649 −1.78959 0 0.694189 −3.16557 0 4.14570
1.4 −1.02020 0 −0.959195 2.96046 0 0.816945 3.01897 0 −3.02025
1.5 −0.513295 0 −1.73653 −3.39171 0 −2.33725 1.91794 0 1.74095
1.6 0.513295 0 −1.73653 3.39171 0 −2.33725 −1.91794 0 1.74095
1.7 1.02020 0 −0.959195 −2.96046 0 0.816945 −3.01897 0 −3.02025
1.8 2.31657 0 3.36649 1.78959 0 0.694189 3.16557 0 4.14570
1.9 2.32314 0 3.39697 −3.53749 0 −1.40218 3.24535 0 −8.21807
1.10 2.81643 0 5.93226 0.124866 0 3.22829 11.0749 0 0.351677
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(223\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6021.2.a.l 10
3.b odd 2 1 inner 6021.2.a.l 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6021.2.a.l 10 1.a even 1 1 trivial
6021.2.a.l 10 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):

\( T_{2}^{10} - 20T_{2}^{8} + 139T_{2}^{6} - 384T_{2}^{4} + 331T_{2}^{2} - 63 \) Copy content Toggle raw display
\( T_{5}^{10} - 36T_{5}^{8} + 460T_{5}^{6} - 2404T_{5}^{4} + 4078T_{5}^{2} - 63 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 20 T^{8} + 139 T^{6} + \cdots - 63 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 36 T^{8} + 460 T^{6} + \cdots - 63 \) Copy content Toggle raw display
$7$ \( (T^{5} - T^{4} - 9 T^{3} + 3 T^{2} + 11 T - 6)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} - 140 T^{8} + \cdots - 10832752 \) Copy content Toggle raw display
$13$ \( (T^{5} - T^{4} - 17 T^{3} - 5 T^{2} + 17 T + 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} - 48 T^{8} + 548 T^{6} + \cdots - 112 \) Copy content Toggle raw display
$19$ \( (T^{5} - 14 T^{4} + 66 T^{3} - 109 T^{2} + \cdots + 98)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} - 186 T^{8} + 11802 T^{6} + \cdots - 828352 \) Copy content Toggle raw display
$29$ \( T^{10} - 143 T^{8} + 5185 T^{6} + \cdots - 2268 \) Copy content Toggle raw display
$31$ \( (T^{5} + 9 T^{4} - 6 T^{3} - 198 T^{2} + \cdots + 400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 10 T^{4} - 14 T^{3} + 428 T^{2} + \cdots + 1086)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 146 T^{8} + 6644 T^{6} + \cdots - 1153852 \) Copy content Toggle raw display
$43$ \( (T^{5} - T^{4} - 44 T^{3} - 22 T^{2} + 91 T + 24)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 424 T^{8} + \cdots - 1133123103 \) Copy content Toggle raw display
$53$ \( T^{10} - 218 T^{8} + 5806 T^{6} + \cdots - 16128 \) Copy content Toggle raw display
$59$ \( T^{10} - 363 T^{8} + \cdots - 300684412 \) Copy content Toggle raw display
$61$ \( (T^{5} - 20 T^{4} + 119 T^{3} - 60 T^{2} + \cdots + 2842)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} - 9 T^{4} - 90 T^{3} + 1117 T^{2} + \cdots - 4371)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} - 396 T^{8} + \cdots - 129670912 \) Copy content Toggle raw display
$73$ \( (T^{5} - 16 T^{4} - 95 T^{3} + 1704 T^{2} + \cdots - 32221)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + 8 T^{4} - 83 T^{3} - 532 T^{2} + \cdots + 3084)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} - 284 T^{8} + 25184 T^{6} + \cdots - 7340032 \) Copy content Toggle raw display
$89$ \( T^{10} - 551 T^{8} + \cdots - 202912192 \) Copy content Toggle raw display
$97$ \( (T^{5} + 9 T^{4} - 82 T^{3} - 566 T^{2} + \cdots - 818)^{2} \) Copy content Toggle raw display
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