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Properties

Label 2001.2.a.n
Level $2001$
Weight $2$
Character orbit 2001.a
Self dual yes
Analytic conductor $15.978$
Analytic rank $0$
Dimension $16$
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] = [2001,2,Mod(1,2001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, 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("2001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.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: \(15.9780654445\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + \cdots - 192 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} - \beta_{8} q^{5} - \beta_1 q^{6} + ( - \beta_{10} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} - \beta_{8} q^{5} - \beta_1 q^{6} + ( - \beta_{10} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9} + ( - \beta_{15} - \beta_{13} - \beta_1) q^{10} + (\beta_{13} + 1) q^{11} + (\beta_{2} + 2) q^{12} + ( - \beta_{7} - \beta_{2} + 1) q^{13} + (\beta_{10} + \beta_{6} - \beta_1 + 1) q^{14} - \beta_{8} q^{15} + ( - \beta_{14} + \beta_{12} + \beta_{8} + \cdots + 3) q^{16}+ \cdots + (\beta_{13} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9} + 11 q^{10} + 8 q^{11} + 25 q^{12} + 19 q^{13} + 16 q^{14} + 3 q^{15} + 31 q^{16} - 4 q^{17} - q^{18} + 19 q^{19} + 16 q^{20} + 13 q^{21} + 6 q^{22} - 16 q^{23} + 23 q^{25} - 15 q^{26} + 16 q^{27} + 18 q^{28} - 16 q^{29} + 11 q^{30} + 24 q^{31} - 21 q^{32} + 8 q^{33} - 9 q^{34} - 13 q^{35} + 25 q^{36} + 26 q^{37} + 19 q^{39} - 22 q^{40} - 15 q^{41} + 16 q^{42} + 33 q^{43} + 6 q^{44} + 3 q^{45} + q^{46} + 13 q^{47} + 31 q^{48} + 41 q^{49} + 13 q^{50} - 4 q^{51} - 26 q^{52} + 5 q^{53} - q^{54} + 9 q^{55} + 40 q^{56} + 19 q^{57} + q^{58} + 2 q^{59} + 16 q^{60} + 29 q^{61} - 32 q^{62} + 13 q^{63} + 28 q^{64} + 18 q^{65} + 6 q^{66} + 32 q^{67} - 26 q^{68} - 16 q^{69} + 18 q^{70} + 29 q^{71} + 19 q^{73} - 16 q^{74} + 23 q^{75} + 64 q^{76} - 21 q^{77} - 15 q^{78} + 56 q^{79} + 16 q^{81} + 14 q^{82} + 5 q^{83} + 18 q^{84} + 16 q^{85} - 20 q^{86} - 16 q^{87} + q^{88} + 7 q^{89} + 11 q^{90} - 6 q^{91} - 25 q^{92} + 24 q^{93} - 11 q^{94} + 39 q^{95} - 21 q^{96} + 35 q^{97} - 109 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + \cdots - 192 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 35 \nu^{15} - 86 \nu^{14} + 850 \nu^{13} + 2133 \nu^{12} - 7813 \nu^{11} - 20506 \nu^{10} + \cdots + 4864 ) / 3904 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5 \nu^{15} + 40 \nu^{14} + 226 \nu^{13} - 985 \nu^{12} - 3713 \nu^{11} + 9288 \nu^{10} + \cdots - 7392 ) / 1952 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19 \nu^{15} - 30 \nu^{14} - 566 \nu^{13} + 815 \nu^{12} + 6765 \nu^{11} - 8674 \nu^{10} - 41247 \nu^{9} + \cdots + 7008 ) / 976 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 141 \nu^{15} + 30 \nu^{14} + 3982 \nu^{13} - 693 \nu^{12} - 45683 \nu^{11} + 6722 \nu^{10} + \cdots - 36288 ) / 3904 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 147 \nu^{15} - 78 \nu^{14} - 4302 \nu^{13} + 2119 \nu^{12} + 51261 \nu^{11} - 23626 \nu^{10} + \cdots + 113088 ) / 3904 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 207 \nu^{15} - 314 \nu^{14} - 5794 \nu^{13} + 7839 \nu^{12} + 65561 \nu^{11} - 77254 \nu^{10} + \cdots + 108096 ) / 3904 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 71 \nu^{15} + 19 \nu^{14} + 1977 \nu^{13} - 506 \nu^{12} - 22127 \nu^{11} + 5583 \nu^{10} + \cdots - 21616 ) / 976 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 163 \nu^{15} - 84 \nu^{14} - 4830 \nu^{13} + 2343 \nu^{12} + 58043 \nu^{11} - 26532 \nu^{10} + \cdots + 116832 ) / 1952 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 411 \nu^{15} - 238 \nu^{14} - 11550 \nu^{13} + 6303 \nu^{12} + 130773 \nu^{11} - 67610 \nu^{10} + \cdots + 236352 ) / 3904 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 551 \nu^{15} - 382 \nu^{14} - 15438 \nu^{13} + 9971 \nu^{12} + 174713 \nu^{11} - 105146 \nu^{10} + \cdots + 298880 ) / 3904 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 279 \nu^{15} - 158 \nu^{14} - 7926 \nu^{13} + 4211 \nu^{12} + 91017 \nu^{11} - 45618 \nu^{10} + \cdots + 164960 ) / 1952 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 155 \nu^{15} + 142 \nu^{14} + 4322 \nu^{13} - 3695 \nu^{12} - 48613 \nu^{11} + 38438 \nu^{10} + \cdots - 81776 ) / 976 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} + \beta_{12} + \beta_{8} + 7\beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{14} + \beta_{11} + \beta_{9} + \beta_{5} - \beta_{4} + 9\beta_{3} + 30\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{15} - 11 \beta_{14} + 2 \beta_{13} + 10 \beta_{12} + 10 \beta_{8} + \beta_{6} + \beta_{3} + \cdots + 146 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12 \beta_{14} + \beta_{12} + 13 \beta_{11} + 2 \beta_{10} + 12 \beta_{9} - 3 \beta_{8} - 2 \beta_{7} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{15} - 93 \beta_{14} + 30 \beta_{13} + 80 \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{9} + \cdots + 972 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{15} - 107 \beta_{14} - 2 \beta_{13} + 16 \beta_{12} + 124 \beta_{11} + 34 \beta_{10} + 112 \beta_{9} + \cdots + 296 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 174 \beta_{15} - 721 \beta_{14} + 310 \beta_{13} + 602 \beta_{12} - 19 \beta_{11} + 40 \beta_{10} + \cdots + 6647 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 20 \beta_{15} - 859 \beta_{14} - 38 \beta_{13} + 177 \beta_{12} + 1050 \beta_{11} + 400 \beta_{10} + \cdots + 2797 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1614 \beta_{15} - 5389 \beta_{14} + 2758 \beta_{13} + 4436 \beta_{12} - 229 \beta_{11} + 532 \beta_{10} + \cdots + 46238 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 265 \beta_{15} - 6590 \beta_{14} - 456 \beta_{13} + 1693 \beta_{12} + 8376 \beta_{11} + 4048 \beta_{10} + \cdots + 24859 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 13794 \beta_{15} - 39596 \beta_{14} + 22738 \beta_{13} + 32438 \beta_{12} - 2242 \beta_{11} + \cdots + 325551 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2965 \beta_{15} - 49525 \beta_{14} - 4410 \beta_{13} + 15062 \beta_{12} + 64576 \beta_{11} + \cdots + 212580 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
2.77164
2.63319
2.18082
2.03156
1.78406
1.40306
0.563182
0.386616
0.300211
−0.510814
−1.44866
−1.53756
−2.00482
−2.39038
−2.50224
−2.65986
−2.77164 1.00000 5.68196 2.29773 −2.77164 −5.27354 −10.2051 1.00000 −6.36848
1.2 −2.63319 1.00000 4.93371 0.107081 −2.63319 4.37350 −7.72501 1.00000 −0.281966
1.3 −2.18082 1.00000 2.75597 −3.64804 −2.18082 2.94825 −1.64864 1.00000 7.95571
1.4 −2.03156 1.00000 2.12722 −2.52723 −2.03156 −2.32870 −0.258453 1.00000 5.13421
1.5 −1.78406 1.00000 1.18288 3.42296 −1.78406 3.39829 1.45779 1.00000 −6.10677
1.6 −1.40306 1.00000 −0.0314276 1.13298 −1.40306 −1.90949 2.85021 1.00000 −1.58963
1.7 −0.563182 1.00000 −1.68283 −1.59433 −0.563182 4.95339 2.07410 1.00000 0.897897
1.8 −0.386616 1.00000 −1.85053 0.287925 −0.386616 −2.57107 1.48868 1.00000 −0.111317
1.9 −0.300211 1.00000 −1.90987 −3.45593 −0.300211 −1.15345 1.17379 1.00000 1.03751
1.10 0.510814 1.00000 −1.73907 2.52081 0.510814 1.21289 −1.90997 1.00000 1.28766
1.11 1.44866 1.00000 0.0986233 2.08765 1.44866 4.31885 −2.75445 1.00000 3.02430
1.12 1.53756 1.00000 0.364091 −2.97145 1.53756 1.52213 −2.51531 1.00000 −4.56878
1.13 2.00482 1.00000 2.01932 3.94433 2.00482 −2.43681 0.0387322 1.00000 7.90768
1.14 2.39038 1.00000 3.71391 3.15865 2.39038 2.42988 4.09689 1.00000 7.55038
1.15 2.50224 1.00000 4.26120 0.499377 2.50224 0.377265 5.65806 1.00000 1.24956
1.16 2.65986 1.00000 5.07484 −2.26251 2.65986 3.13860 8.17864 1.00000 −6.01796
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(1\)

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 2001.2.a.n ✓ 16
3.b odd 2 1 6003.2.a.r 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.n ✓ 16 1.a even 1 1 trivial
6003.2.a.r 16 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2001))\):

\( T_{2}^{16} + T_{2}^{15} - 28 T_{2}^{14} - 27 T_{2}^{13} + 316 T_{2}^{12} + 295 T_{2}^{11} - 1835 T_{2}^{10} + \cdots - 192 \) Copy content Toggle raw display
\( T_{5}^{16} - 3 T_{5}^{15} - 47 T_{5}^{14} + 142 T_{5}^{13} + 862 T_{5}^{12} - 2674 T_{5}^{11} + \cdots + 3072 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + T^{15} + \cdots - 192 \) Copy content Toggle raw display
$3$ \( (T - 1)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 3 T^{15} + \cdots + 3072 \) Copy content Toggle raw display
$7$ \( T^{16} - 13 T^{15} + \cdots + 843776 \) Copy content Toggle raw display
$11$ \( T^{16} - 8 T^{15} + \cdots + 6912 \) Copy content Toggle raw display
$13$ \( T^{16} - 19 T^{15} + \cdots - 79048 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots - 1542733824 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots - 474324992 \) Copy content Toggle raw display
$23$ \( (T + 1)^{16} \) Copy content Toggle raw display
$29$ \( (T + 1)^{16} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots - 664365824 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 692734264 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 15815731488 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 5860667264 \) Copy content Toggle raw display
$47$ \( T^{16} - 13 T^{15} + \cdots - 27131904 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots - 537119225856 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 962614616064 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots - 262090689152 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 37036651159552 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 412266061824 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 3421018126336 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 180646877312 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 9511719825408 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 1186015739904 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 1415545856 \) Copy content Toggle raw display
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