Properties

Label 6039.2.a.i
Level $6039$
Weight $2$
Character orbit 6039.a
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47 \) 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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{8} q^{5} + (\beta_{11} + 1) q^{7} + (\beta_{7} - \beta_{6} + \beta_{3} - \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{8} q^{5} + (\beta_{11} + 1) q^{7} + (\beta_{7} - \beta_{6} + \beta_{3} - \beta_{2} - \beta_1) q^{8} + (\beta_{11} + 2 \beta_{10} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{10} - q^{11} + (\beta_{12} + \beta_{11} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{3} - \beta_1 + 1) q^{13} + ( - \beta_{12} - 2 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} - \beta_1 + 1) q^{14} + ( - \beta_{12} + \beta_{10} - \beta_{9} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2}) q^{16} + (\beta_{12} + \beta_{11} - \beta_{9} - \beta_{7} - 2) q^{17} + (\beta_{12} - \beta_{11} - 2 \beta_{10} + \beta_{9} + 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{2} + 2) q^{19} + (\beta_{12} + \beta_{8} + 2 \beta_{7} - \beta_{6} - \beta_{2} + 1) q^{20} + \beta_1 q^{22} + (\beta_{12} + \beta_{11} + \beta_{8} - \beta_{7} + \beta_{4} - 1) q^{23} + (2 \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{7} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + \cdots + 3) q^{25}+ \cdots + (3 \beta_{12} - \beta_{11} - 4 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} + 2 \beta_{7} + 5 \beta_{6} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 38 \nu^{12} + 47 \nu^{11} + 646 \nu^{10} - 577 \nu^{9} - 3970 \nu^{8} + 1615 \nu^{7} + 11430 \nu^{6} + 2278 \nu^{5} - 18122 \nu^{4} - 10328 \nu^{3} + 14993 \nu^{2} + 9272 \nu - 4830 ) / 1261 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 80 \nu^{12} - 192 \nu^{11} - 1217 \nu^{10} + 2967 \nu^{9} + 5684 \nu^{8} - 15191 \nu^{7} - 6221 \nu^{6} + 29789 \nu^{5} - 11934 \nu^{4} - 24751 \nu^{3} + 21328 \nu^{2} + 12785 \nu - 1964 ) / 1261 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{12} + 275 \nu^{11} - 352 \nu^{10} - 5029 \nu^{9} + 5943 \nu^{8} + 33414 \nu^{7} - 31352 \nu^{6} - 99199 \nu^{5} + 60892 \nu^{4} + 131903 \nu^{3} - 38492 \nu^{2} + \cdots + 5338 ) / 1261 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 74 \nu^{12} - 448 \nu^{11} - 738 \nu^{10} + 7586 \nu^{9} - 695 \nu^{8} - 44875 \nu^{7} + 24220 \nu^{6} + 111281 \nu^{5} - 66105 \nu^{4} - 114506 \nu^{3} + 52816 \nu^{2} + \cdots - 9293 ) / 1261 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 112 \nu^{12} - 495 \nu^{11} - 1384 \nu^{10} + 8163 \nu^{9} + 3275 \nu^{8} - 46490 \nu^{7} + 12790 \nu^{6} + 109003 \nu^{5} - 47983 \nu^{4} - 105439 \nu^{3} + 39084 \nu^{2} + \cdots - 8246 ) / 1261 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 128 \nu^{12} - 42 \nu^{11} - 2670 \nu^{10} + 699 \nu^{9} + 20498 \nu^{8} - 4257 \nu^{7} - 69946 \nu^{6} + 11200 \nu^{5} + 100321 \nu^{4} - 10180 \nu^{3} - 46002 \nu^{2} + \cdots + 1171 ) / 1261 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 162 \nu^{12} - 524 \nu^{11} - 2507 \nu^{10} + 8823 \nu^{9} + 12632 \nu^{8} - 52242 \nu^{7} - 22121 \nu^{6} + 132371 \nu^{5} + 6149 \nu^{4} - 147391 \nu^{3} + 16859 \nu^{2} + \cdots - 11183 ) / 1261 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 191 \nu^{12} + 448 \nu^{11} + 3260 \nu^{10} - 7625 \nu^{9} - 19377 \nu^{8} + 45850 \nu^{7} + 47579 \nu^{6} - 118535 \nu^{5} - 45357 \nu^{4} + 134188 \nu^{3} + 6321 \nu^{2} + \cdots + 8136 ) / 1261 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 187 \nu^{12} - 883 \nu^{11} - 2451 \nu^{10} + 15172 \nu^{9} + 7541 \nu^{8} - 92220 \nu^{7} + 11676 \nu^{6} + 240723 \nu^{5} - 65676 \nu^{4} - 270378 \nu^{3} + 62706 \nu^{2} + \cdots - 17754 ) / 1261 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 344 \nu^{12} + 953 \nu^{11} + 5640 \nu^{10} - 16038 \nu^{9} - 31573 \nu^{8} + 94362 \nu^{7} + 70611 \nu^{6} - 233199 \nu^{5} - 59488 \nu^{4} + 240692 \nu^{3} + 6554 \nu^{2} + \cdots + 8700 ) / 1261 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} - \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} + \beta_{10} - \beta_{9} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + 9\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{12} - \beta_{11} - \beta_{8} - 10 \beta_{7} + 9 \beta_{6} - \beta_{5} + \beta_{4} - 9 \beta_{3} + 11 \beta_{2} + 29 \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{12} - 2 \beta_{11} + 11 \beta_{10} - 8 \beta_{9} + \beta_{8} - 11 \beta_{7} + 3 \beta_{6} - 11 \beta_{5} + 10 \beta_{4} - 10 \beta_{3} + 69 \beta_{2} - \beta _1 + 79 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13 \beta_{12} - 16 \beta_{11} - \beta_{10} - 12 \beta_{8} - 79 \beta_{7} + 72 \beta_{6} - 13 \beta_{5} + 10 \beta_{4} - 67 \beta_{3} + 94 \beta_{2} + 181 \beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 81 \beta_{12} - 33 \beta_{11} + 92 \beta_{10} - 51 \beta_{9} + 13 \beta_{8} - 92 \beta_{7} + 44 \beta_{6} - 97 \beta_{5} + 76 \beta_{4} - 83 \beta_{3} + 506 \beta_{2} - 5 \beta _1 + 488 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 120 \beta_{12} - 176 \beta_{11} - 22 \beta_{10} + \beta_{9} - 105 \beta_{8} - 580 \beta_{7} + 560 \beta_{6} - 128 \beta_{5} + 71 \beta_{4} - 478 \beta_{3} + 739 \beta_{2} + 1179 \beta _1 - 29 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 606 \beta_{12} - 375 \beta_{11} + 688 \beta_{10} - 303 \beta_{9} + 127 \beta_{8} - 697 \beta_{7} + 471 \beta_{6} - 793 \beta_{5} + 523 \beta_{4} - 655 \beta_{3} + 3644 \beta_{2} + 34 \beta _1 + 3165 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 967 \beta_{12} - 1670 \beta_{11} - 303 \beta_{10} + 23 \beta_{9} - 815 \beta_{8} - 4127 \beta_{7} + 4302 \beta_{6} - 1137 \beta_{5} + 427 \beta_{4} - 3386 \beta_{3} + 5608 \beta_{2} + 7898 \beta _1 + 124 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4357 \beta_{12} - 3662 \beta_{11} + 4833 \beta_{10} - 1739 \beta_{9} + 1118 \beta_{8} - 5052 \beta_{7} + 4458 \beta_{6} - 6254 \beta_{5} + 3417 \beta_{4} - 5073 \beta_{3} + 26039 \beta_{2} + 982 \beta _1 + 21158 \) 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
2.73913
2.53773
2.14727
1.61181
1.50067
0.822526
0.171582
−0.805107
−0.948254
−1.33092
−1.46794
−2.37960
−2.59890
−2.73913 0 5.50285 −0.604616 0 −1.91298 −9.59477 0 1.65612
1.2 −2.53773 0 4.44007 −0.994065 0 4.88646 −6.19224 0 2.52267
1.3 −2.14727 0 2.61077 3.62125 0 2.48742 −1.31149 0 −7.77580
1.4 −1.61181 0 0.597924 −4.27930 0 1.98799 2.25988 0 6.89741
1.5 −1.50067 0 0.252024 0.569898 0 −3.97554 2.62314 0 −0.855232
1.6 −0.822526 0 −1.32345 −2.11599 0 0.404149 2.73363 0 1.74046
1.7 −0.171582 0 −1.97056 0.133072 0 0.615329 0.681275 0 −0.0228327
1.8 0.805107 0 −1.35180 −1.06503 0 −0.203035 −2.69856 0 −0.857467
1.9 0.948254 0 −1.10081 2.25122 0 5.24025 −2.94036 0 2.13473
1.10 1.33092 0 −0.228660 2.36819 0 −2.92169 −2.96616 0 3.15187
1.11 1.46794 0 0.154851 −3.84216 0 2.46223 −2.70857 0 −5.64006
1.12 2.37960 0 3.66252 −2.55185 0 1.67712 3.95613 0 −6.07239
1.13 2.59890 0 4.75428 3.50938 0 0.252288 7.15810 0 9.12053
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.i 13
3.b odd 2 1 2013.2.a.e 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.e 13 3.b odd 2 1
6039.2.a.i 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} + 2 T_{2}^{12} - 19 T_{2}^{11} - 35 T_{2}^{10} + 136 T_{2}^{9} + 220 T_{2}^{8} - 469 T_{2}^{7} - 610 T_{2}^{6} + 841 T_{2}^{5} + 760 T_{2}^{4} - 742 T_{2}^{3} - 366 T_{2}^{2} + 236 T_{2} + 47 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + 2 T^{12} - 19 T^{11} - 35 T^{10} + \cdots + 47 \) Copy content Toggle raw display
$3$ \( T^{13} \) Copy content Toggle raw display
$5$ \( T^{13} + 3 T^{12} - 37 T^{11} - 104 T^{10} + \cdots + 292 \) Copy content Toggle raw display
$7$ \( T^{13} - 11 T^{12} + 11 T^{11} + \cdots - 148 \) Copy content Toggle raw display
$11$ \( (T + 1)^{13} \) Copy content Toggle raw display
$13$ \( T^{13} - 13 T^{12} - 3 T^{11} + 657 T^{10} + \cdots + 388 \) Copy content Toggle raw display
$17$ \( T^{13} + 17 T^{12} + 51 T^{11} + \cdots - 4744 \) Copy content Toggle raw display
$19$ \( T^{13} - 14 T^{12} - 43 T^{11} + \cdots - 10215742 \) Copy content Toggle raw display
$23$ \( T^{13} + 7 T^{12} - 126 T^{11} + \cdots + 1840672 \) Copy content Toggle raw display
$29$ \( T^{13} - 6 T^{12} - 130 T^{11} + \cdots - 11772848 \) Copy content Toggle raw display
$31$ \( T^{13} - 27 T^{12} + 197 T^{11} + \cdots - 1170458 \) Copy content Toggle raw display
$37$ \( T^{13} - 10 T^{12} - 194 T^{11} + \cdots + 76337468 \) Copy content Toggle raw display
$41$ \( T^{13} + 3 T^{12} - 207 T^{11} + \cdots + 98100918 \) Copy content Toggle raw display
$43$ \( T^{13} - 29 T^{12} + \cdots - 195974572 \) Copy content Toggle raw display
$47$ \( T^{13} + 8 T^{12} - 229 T^{11} + \cdots + 26421804 \) Copy content Toggle raw display
$53$ \( T^{13} - 24 T^{12} + \cdots - 93111825698 \) Copy content Toggle raw display
$59$ \( T^{13} + 13 T^{12} - 219 T^{11} + \cdots - 7051312 \) Copy content Toggle raw display
$61$ \( (T + 1)^{13} \) Copy content Toggle raw display
$67$ \( T^{13} - 44 T^{12} + \cdots - 6958847006 \) Copy content Toggle raw display
$71$ \( T^{13} + 3 T^{12} - 438 T^{11} + \cdots + 48814344 \) Copy content Toggle raw display
$73$ \( T^{13} - 48 T^{12} + \cdots + 28332933844 \) Copy content Toggle raw display
$79$ \( T^{13} + 17 T^{12} + \cdots + 388386798444 \) Copy content Toggle raw display
$83$ \( T^{13} + 50 T^{12} + \cdots - 58130319032 \) Copy content Toggle raw display
$89$ \( T^{13} - 15 T^{12} + \cdots + 120747458 \) Copy content Toggle raw display
$97$ \( T^{13} - 27 T^{12} + \cdots - 87087118744 \) Copy content Toggle raw display
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