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Properties

Label 2001.2.a.o
Level $2001$
Weight $2$
Character orbit 2001.a
Self dual yes
Analytic conductor $15.978$
Analytic rank $0$
Dimension $20$
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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + \cdots - 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + \beta_{6} q^{5} + \beta_1 q^{6} - \beta_{14} q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + \beta_{6} q^{5} + \beta_1 q^{6} - \beta_{14} q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + q^{9} - \beta_{12} q^{10} - \beta_{8} q^{11} + (\beta_{2} + 2) q^{12} + (\beta_{18} + 1) q^{13} + ( - \beta_{19} - \beta_{18} - \beta_{14} + \cdots - 1) q^{14}+ \cdots - \beta_{8} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} + 20 q^{3} + 30 q^{4} - q^{5} + 2 q^{6} + 9 q^{7} + 6 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{2} + 20 q^{3} + 30 q^{4} - q^{5} + 2 q^{6} + 9 q^{7} + 6 q^{8} + 20 q^{9} + 7 q^{10} + 30 q^{12} + 21 q^{13} - q^{14} - q^{15} + 58 q^{16} - 4 q^{17} + 2 q^{18} + 7 q^{19} - 20 q^{20} + 9 q^{21} + 7 q^{22} + 20 q^{23} + 6 q^{24} + 47 q^{25} + 8 q^{26} + 20 q^{27} + 11 q^{28} + 20 q^{29} + 7 q^{30} + 28 q^{31} + 14 q^{32} + 16 q^{34} + 9 q^{35} + 30 q^{36} + 14 q^{37} - 20 q^{38} + 21 q^{39} + 34 q^{40} + 7 q^{41} - q^{42} + 3 q^{43} - q^{44} - q^{45} + 2 q^{46} + 3 q^{47} + 58 q^{48} + 35 q^{49} - 24 q^{50} - 4 q^{51} + 73 q^{52} - 19 q^{53} + 2 q^{54} + 29 q^{55} - 30 q^{56} + 7 q^{57} + 2 q^{58} + 20 q^{59} - 20 q^{60} + 15 q^{61} + 12 q^{62} + 9 q^{63} + 82 q^{64} - 28 q^{65} + 7 q^{66} + 20 q^{67} - 23 q^{68} + 20 q^{69} - 24 q^{70} + 63 q^{71} + 6 q^{72} + 19 q^{73} + 16 q^{74} + 47 q^{75} - 44 q^{76} - 7 q^{77} + 8 q^{78} + 32 q^{79} - 56 q^{80} + 20 q^{81} - 20 q^{82} - 21 q^{83} + 11 q^{84} + 4 q^{85} - 6 q^{86} + 20 q^{87} + 55 q^{88} - 13 q^{89} + 7 q^{90} + 70 q^{91} + 30 q^{92} + 28 q^{93} - 12 q^{94} + 9 q^{95} + 14 q^{96} - 9 q^{97} + 31 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + \cdots - 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 7\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9977 \nu^{19} - 284189 \nu^{18} - 706 \nu^{17} + 9004494 \nu^{16} + 5676465 \nu^{15} + \cdots + 67052416 ) / 3404224 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 59655 \nu^{19} + 83835 \nu^{18} - 1977318 \nu^{17} - 2493558 \nu^{16} + 26985321 \nu^{15} + \cdots + 42998144 ) / 6808448 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 32190 \nu^{19} - 101041 \nu^{18} - 1039401 \nu^{17} + 3106330 \nu^{16} + 13951816 \nu^{15} + \cdots + 21780992 ) / 3404224 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 113117 \nu^{19} - 286133 \nu^{18} + 3436418 \nu^{17} + 9067866 \nu^{16} - 41870219 \nu^{15} + \cdots + 65443584 ) / 6808448 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 132421 \nu^{19} + 61925 \nu^{18} - 4039074 \nu^{17} - 1758962 \nu^{16} + 49733347 \nu^{15} + \cdots - 8160896 ) / 6808448 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 161261 \nu^{19} + 371971 \nu^{18} + 5495374 \nu^{17} - 11857542 \nu^{16} - 78622707 \nu^{15} + \cdots - 221220224 ) / 6808448 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 88710 \nu^{19} + 348943 \nu^{18} + 3305005 \nu^{17} - 10913794 \nu^{16} - 52040672 \nu^{15} + \cdots - 107471168 ) / 3404224 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 203145 \nu^{19} + 8703 \nu^{18} + 6311478 \nu^{17} + 38394 \nu^{16} - 80239695 \nu^{15} + \cdots - 15271680 ) / 6808448 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 294337 \nu^{19} - 270767 \nu^{18} - 9616398 \nu^{17} + 8494062 \nu^{16} + 130255487 \nu^{15} + \cdots + 110542464 ) / 6808448 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 148537 \nu^{19} + 338681 \nu^{18} + 5041136 \nu^{17} - 10572042 \nu^{16} - 71767335 \nu^{15} + \cdots - 97520832 ) / 3404224 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 380439 \nu^{19} + 254997 \nu^{18} + 12633678 \nu^{17} - 8079138 \nu^{16} - 174716497 \nu^{15} + \cdots - 178908672 ) / 6808448 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 202867 \nu^{19} + 156362 \nu^{18} + 6774905 \nu^{17} - 4911368 \nu^{16} - 94271295 \nu^{15} + \cdots - 69421504 ) / 3404224 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 423491 \nu^{19} + 589261 \nu^{18} + 14300798 \nu^{17} - 18316426 \nu^{16} - 202063709 \nu^{15} + \cdots - 197504512 ) / 6808448 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 595707 \nu^{19} + 195465 \nu^{18} + 19683950 \nu^{17} - 6229090 \nu^{16} - 270259309 \nu^{15} + \cdots - 174423552 ) / 6808448 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 975995 \nu^{19} - 593997 \nu^{18} - 31897570 \nu^{17} + 18413026 \nu^{16} + 432558821 \nu^{15} + \cdots + 286605824 ) / 6808448 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 7\beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} + \beta_{18} - \beta_{17} + \beta_{16} + \beta_{15} + \beta_{14} + \beta_{12} + \cdots + 37 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{19} + 2 \beta_{18} - \beta_{17} - \beta_{15} + 2 \beta_{14} - \beta_{12} + \beta_{10} + \cdots + 149 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15 \beta_{19} + 17 \beta_{18} - 13 \beta_{17} + 12 \beta_{16} + 12 \beta_{15} + 13 \beta_{14} - 2 \beta_{13} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15 \beta_{19} + 29 \beta_{18} - 16 \beta_{17} + 5 \beta_{16} - 16 \beta_{15} + 31 \beta_{14} + \cdots + 1019 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 163 \beta_{19} + 195 \beta_{18} - 125 \beta_{17} + 109 \beta_{16} + 107 \beta_{15} + 128 \beta_{14} + \cdots - 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 161 \beta_{19} + 298 \beta_{18} - 177 \beta_{17} + 101 \beta_{16} - 176 \beta_{15} + 341 \beta_{14} + \cdots + 7189 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1554 \beta_{19} + 1914 \beta_{18} - 1080 \beta_{17} + 900 \beta_{16} + 862 \beta_{15} + 1144 \beta_{14} + \cdots - 361 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1517 \beta_{19} + 2679 \beta_{18} - 1689 \beta_{17} + 1337 \beta_{16} - 1661 \beta_{15} + 3273 \beta_{14} + \cdots + 51776 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 13817 \beta_{19} + 17346 \beta_{18} - 8891 \beta_{17} + 7126 \beta_{16} + 6667 \beta_{15} + 9750 \beta_{14} + \cdots - 3564 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 13384 \beta_{19} + 22530 \beta_{18} - 14956 \beta_{17} + 14755 \beta_{16} - 14479 \beta_{15} + \cdots + 378519 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 117785 \beta_{19} + 150054 \beta_{18} - 71434 \beta_{17} + 55290 \beta_{16} + 50746 \beta_{15} + \cdots - 32922 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 113673 \beta_{19} + 182464 \beta_{18} - 126943 \beta_{17} + 147497 \beta_{16} - 120512 \beta_{15} + \cdots + 2799099 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 977165 \beta_{19} + 1260560 \beta_{18} - 566566 \beta_{17} + 424748 \beta_{16} + 384398 \beta_{15} + \cdots - 292518 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 942530 \beta_{19} + 1444290 \beta_{18} - 1050191 \beta_{17} + 1387506 \beta_{16} - 974521 \beta_{15} + \cdots + 20887382 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 7960315 \beta_{19} + 10386058 \beta_{18} - 4462429 \beta_{17} + 3248175 \beta_{16} + 2912926 \beta_{15} + \cdots - 2533741 \) 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
−2.80084
−2.63931
−2.40422
−1.97506
−1.94926
−1.41510
−1.19744
−0.852674
−0.302778
0.194668
0.458217
0.603983
0.762638
1.20801
1.74461
2.23669
2.44527
2.45061
2.62822
2.80377
−2.80084 1.00000 5.84471 −3.48103 −2.80084 −1.41384 −10.7684 1.00000 9.74980
1.2 −2.63931 1.00000 4.96597 −1.13791 −2.63931 1.99715 −7.82811 1.00000 3.00330
1.3 −2.40422 1.00000 3.78028 2.94625 −2.40422 2.89217 −4.28018 1.00000 −7.08343
1.4 −1.97506 1.00000 1.90086 −3.88400 −1.97506 2.64382 0.195801 1.00000 7.67114
1.5 −1.94926 1.00000 1.79960 3.69296 −1.94926 −2.45473 0.390627 1.00000 −7.19852
1.6 −1.41510 1.00000 0.00250280 0.0661489 −1.41510 1.68271 2.82665 1.00000 −0.0936071
1.7 −1.19744 1.00000 −0.566132 −2.06209 −1.19744 −5.04192 3.07279 1.00000 2.46923
1.8 −0.852674 1.00000 −1.27295 −0.278169 −0.852674 2.70488 2.79076 1.00000 0.237188
1.9 −0.302778 1.00000 −1.90833 3.34986 −0.302778 0.428154 1.18336 1.00000 −1.01426
1.10 0.194668 1.00000 −1.96210 4.10517 0.194668 4.22277 −0.771295 1.00000 0.799144
1.11 0.458217 1.00000 −1.79004 −0.388274 0.458217 −4.22376 −1.73666 1.00000 −0.177914
1.12 0.603983 1.00000 −1.63520 −2.98438 0.603983 4.14035 −2.19560 1.00000 −1.80252
1.13 0.762638 1.00000 −1.41838 −4.10409 0.762638 −2.37592 −2.60699 1.00000 −3.12993
1.14 1.20801 1.00000 −0.540713 2.72981 1.20801 −1.05258 −3.06921 1.00000 3.29764
1.15 1.74461 1.00000 1.04367 0.892571 1.74461 3.89655 −1.66842 1.00000 1.55719
1.16 2.23669 1.00000 3.00280 1.57347 2.23669 −1.14044 2.24296 1.00000 3.51938
1.17 2.44527 1.00000 3.97935 −1.74520 2.44527 5.10283 4.84005 1.00000 −4.26748
1.18 2.45061 1.00000 4.00547 1.21123 2.45061 0.971463 4.91461 1.00000 2.96824
1.19 2.62822 1.00000 4.90753 −4.03060 2.62822 −1.30880 7.64164 1.00000 −10.5933
1.20 2.80377 1.00000 5.86110 2.52828 2.80377 −2.67086 10.8256 1.00000 7.08870
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
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.o ✓ 20
3.b odd 2 1 6003.2.a.s 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.o ✓ 20 1.a even 1 1 trivial
6003.2.a.s 20 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2001))\):

\( T_{2}^{20} - 2 T_{2}^{19} - 33 T_{2}^{18} + 64 T_{2}^{17} + 453 T_{2}^{16} - 846 T_{2}^{15} - 3353 T_{2}^{14} + \cdots - 256 \) Copy content Toggle raw display
\( T_{5}^{20} + T_{5}^{19} - 73 T_{5}^{18} - 56 T_{5}^{17} + 2230 T_{5}^{16} + 1202 T_{5}^{15} + \cdots + 34304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 2 T^{19} + \cdots - 256 \) Copy content Toggle raw display
$3$ \( (T - 1)^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + T^{19} + \cdots + 34304 \) Copy content Toggle raw display
$7$ \( T^{20} - 9 T^{19} + \cdots - 7405568 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 465124864 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots - 1403714144 \) Copy content Toggle raw display
$17$ \( T^{20} + 4 T^{19} + \cdots + 2695168 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 207486976 \) Copy content Toggle raw display
$23$ \( (T - 1)^{20} \) Copy content Toggle raw display
$29$ \( (T - 1)^{20} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 1964490752 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 223976620672 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 626363992576 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 382110913107968 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 41\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 541517103104 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 14812315648 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 66044926662656 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 31\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 38\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 115332488794624 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 20\!\cdots\!64 \) Copy content Toggle raw display
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