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Properties

Label 8001.2.a.w
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
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] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, 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("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.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: \(63.8883066572\)
Analytic rank: \(1\)
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: \( x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} + \cdots + 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 889)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} + \cdots + 31 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 80574 \nu^{19} + 88265 \nu^{18} + 4448116 \nu^{17} - 11956345 \nu^{16} - 63248616 \nu^{15} + \cdots + 123446515 ) / 4334246 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 112041 \nu^{19} - 1197675 \nu^{18} + 2264887 \nu^{17} + 17783216 \nu^{16} - 73262421 \nu^{15} + \cdots + 2196193 ) / 2167123 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16028 \nu^{19} - 143864 \nu^{18} + 122280 \nu^{17} + 2470224 \nu^{16} - 6859382 \nu^{15} + \cdots + 1383690 ) / 309589 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 248879 \nu^{19} + 3978132 \nu^{18} - 13988931 \nu^{17} - 48139158 \nu^{16} + 338768723 \nu^{15} + \cdots - 44644594 ) / 4334246 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 130444 \nu^{19} - 1062848 \nu^{18} + 203009 \nu^{17} + 19282840 \nu^{16} - 38105845 \nu^{15} + \cdots + 4552939 ) / 2167123 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 146869 \nu^{19} + 1305396 \nu^{18} - 1062848 \nu^{17} - 22121079 \nu^{16} + 59524946 \nu^{15} + \cdots + 9434934 ) / 2167123 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 452378 \nu^{19} - 3636757 \nu^{18} + 479452 \nu^{17} + 64942855 \nu^{16} - 120306036 \nu^{15} + \cdots + 51386023 ) / 4334246 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 239299 \nu^{19} - 839918 \nu^{18} - 6727183 \nu^{17} + 25057160 \nu^{16} + 75577807 \nu^{15} + \cdots + 33626793 ) / 2167123 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 513169 \nu^{19} - 3359961 \nu^{18} - 4253023 \nu^{17} + 67902593 \nu^{16} - 49716311 \nu^{15} + \cdots + 14104789 ) / 4334246 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 600491 \nu^{19} - 5014091 \nu^{18} + 1433657 \nu^{17} + 92851141 \nu^{16} - 196388497 \nu^{15} + \cdots + 34267467 ) / 4334246 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 410993 \nu^{19} + 2213625 \nu^{18} + 6917810 \nu^{17} - 52563575 \nu^{16} - 29025299 \nu^{15} + \cdots + 31062379 ) / 2167123 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 413388 \nu^{19} + 3462562 \nu^{18} - 1511903 \nu^{17} - 60346403 \nu^{16} + 134466805 \nu^{15} + \cdots - 3502341 ) / 2167123 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 1008515 \nu^{19} + 6530918 \nu^{18} + 8850087 \nu^{17} - 132965184 \nu^{16} + 86306181 \nu^{15} + \cdots - 10476190 ) / 4334246 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 579954 \nu^{19} + 3614942 \nu^{18} + 6341910 \nu^{17} - 77052369 \nu^{16} + 25536710 \nu^{15} + \cdots + 42358963 ) / 2167123 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 170912 \nu^{19} - 1102231 \nu^{18} - 1613038 \nu^{17} + 22982443 \nu^{16} - 12939568 \nu^{15} + \cdots - 10261733 ) / 619178 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 1492257 \nu^{19} - 9628117 \nu^{18} - 13332635 \nu^{17} + 194372481 \nu^{16} - 107507193 \nu^{15} + \cdots - 78687283 ) / 4334246 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{18} + \beta_{15} + \beta_{14} + \beta_{10} + \beta_{3} + 8\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} + 2 \beta_{18} + \beta_{17} + 2 \beta_{16} + \beta_{15} - \beta_{11} + \beta_{10} - \beta_{9} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{18} + \beta_{17} + \beta_{16} + 11 \beta_{15} + 9 \beta_{14} - \beta_{13} + \beta_{12} + \cdots + 74 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11 \beta_{19} + 25 \beta_{18} + 12 \beta_{17} + 23 \beta_{16} + 14 \beta_{15} - \beta_{14} - \beta_{13} + \cdots + 69 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3 \beta_{19} + 107 \beta_{18} + 16 \beta_{17} + 17 \beta_{16} + 95 \beta_{15} + 61 \beta_{14} + \cdots + 427 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 94 \beta_{19} + 233 \beta_{18} + 112 \beta_{17} + 200 \beta_{16} + 143 \beta_{15} - 18 \beta_{14} + \cdots + 508 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 55 \beta_{19} + 861 \beta_{18} + 181 \beta_{17} + 195 \beta_{16} + 758 \beta_{15} + 368 \beta_{14} + \cdots + 2627 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 749 \beta_{19} + 1951 \beta_{18} + 964 \beta_{17} + 1587 \beta_{16} + 1289 \beta_{15} - 215 \beta_{14} + \cdots + 3700 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 682 \beta_{19} + 6626 \beta_{18} + 1783 \beta_{17} + 1903 \beta_{16} + 5847 \beta_{15} + 2054 \beta_{14} + \cdots + 16903 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5849 \beta_{19} + 15528 \beta_{18} + 8007 \beta_{17} + 12128 \beta_{16} + 10902 \beta_{15} + \cdots + 26897 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 7165 \beta_{19} + 49927 \beta_{18} + 16332 \beta_{17} + 17066 \beta_{16} + 44367 \beta_{15} + \cdots + 112151 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 45504 \beta_{19} + 120296 \beta_{18} + 65244 \beta_{17} + 91153 \beta_{16} + 88859 \beta_{15} + \cdots + 195625 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 68741 \beta_{19} + 372420 \beta_{18} + 143070 \beta_{17} + 145540 \beta_{16} + 333767 \beta_{15} + \cdots + 760028 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 354430 \beta_{19} + 917916 \beta_{18} + 525239 \beta_{17} + 680149 \beta_{16} + 707465 \beta_{15} + \cdots + 1424146 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 623089 \beta_{19} + 2765411 \beta_{18} + 1215932 \beta_{17} + 1201403 \beta_{16} + 2498983 \beta_{15} + \cdots + 5228188 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 2765954 \beta_{19} + 6943503 \beta_{18} + 4191957 \beta_{17} + 5061055 \beta_{16} + 5543561 \beta_{15} + \cdots + 10376694 \) 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
2.75954
2.70334
2.58654
2.24329
2.11976
2.02020
1.78791
1.72132
0.831296
0.697877
0.170762
0.102309
−0.343534
−0.723268
−0.731838
−1.60447
−1.86440
−1.86718
−2.12365
−2.48579
−2.75954 0 5.61505 −0.308409 0 1.00000 −9.97588 0 0.851066
1.2 −2.70334 0 5.30805 3.37987 0 1.00000 −8.94278 0 −9.13693
1.3 −2.58654 0 4.69021 −1.18795 0 1.00000 −6.95835 0 3.07267
1.4 −2.24329 0 3.03233 1.67114 0 1.00000 −2.31581 0 −3.74884
1.5 −2.11976 0 2.49337 −3.20666 0 1.00000 −1.04583 0 6.79735
1.6 −2.02020 0 2.08119 3.61502 0 1.00000 −0.164016 0 −7.30304
1.7 −1.78791 0 1.19664 −2.61034 0 1.00000 1.43634 0 4.66707
1.8 −1.72132 0 0.962958 −2.01516 0 1.00000 1.78509 0 3.46875
1.9 −0.831296 0 −1.30895 1.71517 0 1.00000 2.75071 0 −1.42582
1.10 −0.697877 0 −1.51297 0.682552 0 1.00000 2.45162 0 −0.476337
1.11 −0.170762 0 −1.97084 −2.09248 0 1.00000 0.678071 0 0.357317
1.12 −0.102309 0 −1.98953 −2.62692 0 1.00000 0.408164 0 0.268756
1.13 0.343534 0 −1.88198 4.21457 0 1.00000 −1.33360 0 1.44785
1.14 0.723268 0 −1.47688 1.71672 0 1.00000 −2.51472 0 1.24165
1.15 0.731838 0 −1.46441 −3.93264 0 1.00000 −2.53539 0 −2.87806
1.16 1.60447 0 0.574339 −0.248535 0 1.00000 −2.28744 0 −0.398768
1.17 1.86440 0 1.47599 1.88221 0 1.00000 −0.976959 0 3.50920
1.18 1.86718 0 1.48637 −2.52504 0 1.00000 −0.959047 0 −4.71472
1.19 2.12365 0 2.50989 2.22935 0 1.00000 1.08284 0 4.73436
1.20 2.48579 0 4.17918 −3.35246 0 1.00000 5.41698 0 −8.33354
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(1\)

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 8001.2.a.w 20
3.b odd 2 1 889.2.a.d ✓ 20
21.c even 2 1 6223.2.a.l 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
889.2.a.d ✓ 20 3.b odd 2 1
6223.2.a.l 20 21.c even 2 1
8001.2.a.w 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{20} + 8 T_{2}^{19} - 152 T_{2}^{17} - 274 T_{2}^{16} + 1061 T_{2}^{15} + 3125 T_{2}^{14} + \cdots + 31 \) Copy content Toggle raw display
\( T_{5}^{20} + 3 T_{5}^{19} - 59 T_{5}^{18} - 183 T_{5}^{17} + 1412 T_{5}^{16} + 4519 T_{5}^{15} + \cdots - 203968 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 8 T^{19} + \cdots + 31 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + 3 T^{19} + \cdots - 203968 \) Copy content Toggle raw display
$7$ \( (T - 1)^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + 26 T^{19} + \cdots - 11392 \) Copy content Toggle raw display
$13$ \( T^{20} + 4 T^{19} + \cdots - 11509712 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 7648759472 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 3061190104 \) Copy content Toggle raw display
$23$ \( T^{20} + 31 T^{19} + \cdots - 37130752 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 847113116416 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 767665605056 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 1123613199576 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 9083203177744 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 2352002970368 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 7160930336 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 4131688997632 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 11947401797632 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 1381335112304 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 17\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 67\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots - 11\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 443538923776 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 952366014608384 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 1207053990976 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 52934605786048 \) Copy content Toggle raw display
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