Properties

Label 6039.2.a.d
Level $6039$
Weight $2$
Character orbit 6039.a
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $11$
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] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, 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("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.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: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 4x^{10} - 6x^{9} + 37x^{8} - 2x^{7} - 109x^{6} + 55x^{5} + 115x^{4} - 76x^{3} - 29x^{2} + 14x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2013)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{5} + 1) q^{4} + ( - \beta_{10} + 1) q^{5} + ( - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{10} - \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{5} + 1) q^{4} + ( - \beta_{10} + 1) q^{5} + ( - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{10} - \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} + 2) q^{8} + (\beta_{8} - \beta_{7} - \beta_{6} + 3 \beta_1 - 1) q^{10} + q^{11} + ( - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{13} + (\beta_{10} + \beta_{9} + \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 + 2) q^{14} + (\beta_{10} - \beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_1 + 2) q^{16} + (\beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_1 + 2) q^{17} + (\beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{19} + ( - \beta_{9} + \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 5) q^{20} + \beta_1 q^{22} + (\beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2) q^{23} + ( - 3 \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3}) q^{25} + ( - 2 \beta_{10} - 2 \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \cdots - 1) q^{26}+ \cdots + ( - 5 \beta_{10} - 3 \beta_{9} - 3 \beta_{8} + 6 \beta_{7} + \beta_{6} - 4 \beta_{5} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8} + 6 q^{10} + 11 q^{11} - 3 q^{13} + 9 q^{14} + 4 q^{16} + 7 q^{17} - 8 q^{19} + 25 q^{20} + 4 q^{22} + 15 q^{23} + 4 q^{25} + 2 q^{26} + 13 q^{28} + 8 q^{29} - 17 q^{31} + 27 q^{32} - 18 q^{34} + 2 q^{35} - 10 q^{37} + 30 q^{38} + 10 q^{40} + 25 q^{41} - 7 q^{43} + 6 q^{44} + 32 q^{46} + 30 q^{47} - 2 q^{49} - 11 q^{50} - 7 q^{52} + 18 q^{53} + 13 q^{55} + 20 q^{56} - 13 q^{58} + 43 q^{59} + 11 q^{61} - 7 q^{62} + 25 q^{64} + 27 q^{65} - 30 q^{67} - 10 q^{68} - 4 q^{70} + 7 q^{71} + 6 q^{73} + 44 q^{74} - 19 q^{76} - 5 q^{77} + 17 q^{79} + 22 q^{80} + 8 q^{82} + 34 q^{83} + 10 q^{85} - 2 q^{86} + 9 q^{88} + 41 q^{89} - 39 q^{91} + 32 q^{92} + 55 q^{94} + 9 q^{95} - 41 q^{97} + 29 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 4x^{10} - 6x^{9} + 37x^{8} - 2x^{7} - 109x^{6} + 55x^{5} + 115x^{4} - 76x^{3} - 29x^{2} + 14x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} - 2\nu^{9} - 10\nu^{8} + 17\nu^{7} + 32\nu^{6} - 45\nu^{5} - 35\nu^{4} + 54\nu^{3} + 14\nu^{2} - 37\nu - 6 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5 \nu^{10} - 16 \nu^{9} - 41 \nu^{8} + 145 \nu^{7} + 97 \nu^{6} - 408 \nu^{5} - 55 \nu^{4} + 378 \nu^{3} - 38 \nu^{2} - 35 \nu + 6 ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{10} - 16 \nu^{9} - 41 \nu^{8} + 145 \nu^{7} + 97 \nu^{6} - 408 \nu^{5} - 64 \nu^{4} + 387 \nu^{3} + 16 \nu^{2} - 62 \nu - 30 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4 \nu^{10} + 11 \nu^{9} + 40 \nu^{8} - 107 \nu^{7} - 137 \nu^{6} + 339 \nu^{5} + 188 \nu^{4} - 396 \nu^{3} - 83 \nu^{2} + 109 \nu + 6 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4 \nu^{10} - 11 \nu^{9} - 40 \nu^{8} + 107 \nu^{7} + 137 \nu^{6} - 339 \nu^{5} - 188 \nu^{4} + 396 \nu^{3} + 92 \nu^{2} - 109 \nu - 33 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7 \nu^{10} - 20 \nu^{9} - 61 \nu^{8} + 179 \nu^{7} + 152 \nu^{6} - 489 \nu^{5} - 53 \nu^{4} + 432 \nu^{3} - 136 \nu^{2} - 19 \nu + 21 ) / 9 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2 \nu^{10} + 5 \nu^{9} + 20 \nu^{8} - 44 \nu^{7} - 73 \nu^{6} + 119 \nu^{5} + 125 \nu^{4} - 114 \nu^{3} - 94 \nu^{2} + 28 \nu + 15 ) / 3 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7 \nu^{10} - 14 \nu^{9} - 79 \nu^{8} + 128 \nu^{7} + 314 \nu^{6} - 360 \nu^{5} - 506 \nu^{4} + 342 \nu^{3} + 296 \nu^{2} - 43 \nu - 42 ) / 9 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 16 \nu^{10} - 47 \nu^{9} - 142 \nu^{8} + 431 \nu^{7} + 386 \nu^{6} - 1236 \nu^{5} - 305 \nu^{4} + 1224 \nu^{3} - 46 \nu^{2} - 226 \nu - 6 ) / 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{7} + 7\beta_{6} + 8\beta_{5} - 2\beta_{4} + \beta_{3} + \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{10} + \beta_{9} + \beta_{8} - 9\beta_{7} + 9\beta_{6} + 18\beta_{5} - 8\beta_{4} + 2\beta_{3} + 21\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{10} + \beta_{9} + \beta_{8} - 12 \beta_{7} + 45 \beta_{6} + 56 \beta_{5} - 18 \beta_{4} + 11 \beta_{3} + 2 \beta_{2} + 15 \beta _1 + 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 53 \beta_{10} + 10 \beta_{9} + 11 \beta_{8} - 65 \beta_{7} + 72 \beta_{6} + 134 \beta_{5} - 56 \beta_{4} + 25 \beta_{3} + 6 \beta_{2} + 124 \beta _1 + 154 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 80 \beta_{10} + 14 \beta_{9} + 16 \beta_{8} - 107 \beta_{7} + 292 \beta_{6} + 386 \beta_{5} - 134 \beta_{4} + 96 \beta_{3} + 33 \beta_{2} + 148 \beta _1 + 613 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 339 \beta_{10} + 80 \beta_{9} + 93 \beta_{8} - 448 \beta_{7} + 551 \beta_{6} + 960 \beta_{5} - 386 \beta_{4} + 236 \beta_{3} + 96 \beta_{2} + 781 \beta _1 + 1180 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 598 \beta_{10} + 143 \beta_{9} + 172 \beta_{8} - 863 \beta_{7} + 1940 \beta_{6} + 2678 \beta_{5} - 960 \beta_{4} + 780 \beta_{3} + 365 \beta_{2} + 1255 \beta _1 + 4071 \) 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
−2.14662
−1.57504
−1.35090
−0.423080
−0.186189
0.536504
1.14470
1.36137
1.53948
2.39607
2.70371
−2.14662 0 2.60797 4.24506 0 0.408872 −1.30509 0 −9.11254
1.2 −1.57504 0 0.480756 −0.664958 0 −4.15209 2.39287 0 1.04734
1.3 −1.35090 0 −0.175068 0.638467 0 −0.931245 2.93830 0 −0.862505
1.4 −0.423080 0 −1.82100 1.36483 0 0.865577 1.61659 0 −0.577432
1.5 −0.186189 0 −1.96533 −1.94515 0 0.0328573 0.738300 0 0.362164
1.6 0.536504 0 −1.71216 3.00877 0 0.211107 −1.99159 0 1.61422
1.7 1.14470 0 −0.689651 −1.33318 0 −3.52170 −3.07886 0 −1.52610
1.8 1.36137 0 −0.146675 3.87776 0 −4.18708 −2.92242 0 5.27907
1.9 1.53948 0 0.369988 0.258725 0 4.37144 −2.50937 0 0.398300
1.10 2.39607 0 3.74115 0.714370 0 −0.642303 4.17191 0 1.71168
1.11 2.70371 0 5.31003 2.83529 0 2.54457 8.94935 0 7.66580
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)
\(61\) \(-1\)

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 6039.2.a.d 11
3.b odd 2 1 2013.2.a.a 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.a 11 3.b odd 2 1
6039.2.a.d 11 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{11} - 4 T_{2}^{10} - 6 T_{2}^{9} + 37 T_{2}^{8} - 2 T_{2}^{7} - 109 T_{2}^{6} + 55 T_{2}^{5} + 115 T_{2}^{4} - 76 T_{2}^{3} - 29 T_{2}^{2} + 14 T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} - 4 T^{10} - 6 T^{9} + 37 T^{8} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( T^{11} \) Copy content Toggle raw display
$5$ \( T^{11} - 13 T^{10} + 55 T^{9} - 42 T^{8} + \cdots + 39 \) Copy content Toggle raw display
$7$ \( T^{11} + 5 T^{10} - 25 T^{9} - 137 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T - 1)^{11} \) Copy content Toggle raw display
$13$ \( T^{11} + 3 T^{10} - 65 T^{9} + \cdots - 113383 \) Copy content Toggle raw display
$17$ \( T^{11} - 7 T^{10} - 55 T^{9} + \cdots - 21027 \) Copy content Toggle raw display
$19$ \( T^{11} + 8 T^{10} - 75 T^{9} + \cdots + 10163 \) Copy content Toggle raw display
$23$ \( T^{11} - 15 T^{10} - 28 T^{9} + \cdots + 198891 \) Copy content Toggle raw display
$29$ \( T^{11} - 8 T^{10} - 138 T^{9} + \cdots + 529149 \) Copy content Toggle raw display
$31$ \( T^{11} + 17 T^{10} - 31 T^{9} + \cdots - 2875241 \) Copy content Toggle raw display
$37$ \( T^{11} + 10 T^{10} - 200 T^{9} + \cdots - 7025731 \) Copy content Toggle raw display
$41$ \( T^{11} - 25 T^{10} + \cdots + 793812771 \) Copy content Toggle raw display
$43$ \( T^{11} + 7 T^{10} - 216 T^{9} + \cdots - 2670529 \) Copy content Toggle raw display
$47$ \( T^{11} - 30 T^{10} + \cdots - 149717619 \) Copy content Toggle raw display
$53$ \( T^{11} - 18 T^{10} - 122 T^{9} + \cdots - 4627299 \) Copy content Toggle raw display
$59$ \( T^{11} - 43 T^{10} + 589 T^{9} + \cdots + 91481937 \) Copy content Toggle raw display
$61$ \( (T - 1)^{11} \) Copy content Toggle raw display
$67$ \( T^{11} + 30 T^{10} + 95 T^{9} + \cdots + 1573457 \) Copy content Toggle raw display
$71$ \( T^{11} - 7 T^{10} + \cdots + 28070149767 \) Copy content Toggle raw display
$73$ \( T^{11} - 6 T^{10} - 489 T^{9} + \cdots + 126285977 \) Copy content Toggle raw display
$79$ \( T^{11} - 17 T^{10} - 359 T^{9} + \cdots - 15718043 \) Copy content Toggle raw display
$83$ \( T^{11} - 34 T^{10} + \cdots - 18319432683 \) Copy content Toggle raw display
$89$ \( T^{11} - 41 T^{10} + \cdots + 78347255019 \) Copy content Toggle raw display
$97$ \( T^{11} + 41 T^{10} + \cdots - 7174782977 \) Copy content Toggle raw display
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