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Properties

Label 8001.2.a.u
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $18$
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: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 2667)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{12} + 1) q^{5} + q^{7} + (\beta_{16} + \beta_{14} - \beta_{10} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{12} + 1) q^{5} + q^{7} + (\beta_{16} + \beta_{14} - \beta_{10} + \cdots + 1) q^{8}+ \cdots + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9056232 \nu^{17} + 56952713 \nu^{16} + 63697503 \nu^{15} - 1083863790 \nu^{14} + \cdots - 228715354 ) / 365415070 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12516449 \nu^{17} - 35576866 \nu^{16} - 322120881 \nu^{15} + 878461675 \nu^{14} + \cdots - 1868332582 ) / 365415070 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 25066381 \nu^{17} + 165457944 \nu^{16} + 197605069 \nu^{15} - 3294331535 \nu^{14} + \cdots - 2904833932 ) / 730830140 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32625911 \nu^{17} + 257402314 \nu^{16} + 38720489 \nu^{15} - 4899416105 \nu^{14} + \cdots - 523550232 ) / 730830140 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 40185441 \nu^{17} + 349346684 \nu^{16} - 120164091 \nu^{15} - 6504500675 \nu^{14} + \cdots - 2527247372 ) / 730830140 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20878693 \nu^{17} + 77409897 \nu^{16} + 451307397 \nu^{15} - 1771179645 \nu^{14} + \cdots + 723097034 ) / 365415070 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8395741 \nu^{17} - 64482942 \nu^{16} - 11290113 \nu^{15} + 1199594231 \nu^{14} - 1945179675 \nu^{13} + \cdots - 76030588 ) / 146166028 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 50383667 \nu^{17} - 161191148 \nu^{16} - 1204101853 \nu^{15} + 3787887405 \nu^{14} + \cdots - 2009629216 ) / 730830140 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 80549203 \nu^{17} - 492648012 \nu^{16} - 821028837 \nu^{15} + 9925282045 \nu^{14} + \cdots + 3906261256 ) / 730830140 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 100575831 \nu^{17} - 597870654 \nu^{16} - 1070376159 \nu^{15} + 11961026425 \nu^{14} + \cdots + 4483050272 ) / 730830140 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 101911933 \nu^{17} + 591076942 \nu^{16} + 1177422127 \nu^{15} - 11988946255 \nu^{14} + \cdots + 1025152864 ) / 730830140 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10957014 \nu^{17} + 55541023 \nu^{16} + 166873740 \nu^{15} - 1164950524 \nu^{14} + \cdots + 267274566 ) / 73083014 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 33634461 \nu^{17} - 212119814 \nu^{16} - 304035654 \nu^{15} + 4222090590 \nu^{14} + \cdots + 1153552087 ) / 182707535 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 100069624 \nu^{17} - 532974031 \nu^{16} - 1376337581 \nu^{15} + 10970946660 \nu^{14} + \cdots - 346733612 ) / 365415070 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 212880147 \nu^{17} - 1116211818 \nu^{16} - 2947927153 \nu^{15} + 22837760465 \nu^{14} + \cdots + 5373162964 ) / 730830140 \) 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
\(\nu^{3}\)\(=\) \( \beta_{16} + \beta_{14} - \beta_{10} + \beta_{7} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} + \beta_{15} + \beta_{14} + \beta_{13} - \beta_{10} + \beta_{7} + \beta_{6} + 8\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{16} - \beta_{15} + 9 \beta_{14} - \beta_{13} - 9 \beta_{10} + 8 \beta_{7} + \beta_{6} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{16} + 11 \beta_{15} + 11 \beta_{14} + 10 \beta_{13} - 2 \beta_{11} - 13 \beta_{10} + 12 \beta_{7} + \cdots + 89 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 70 \beta_{16} - 11 \beta_{15} + 69 \beta_{14} - 12 \beta_{13} - \beta_{12} - \beta_{11} - 72 \beta_{10} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{17} + 110 \beta_{16} + 92 \beta_{15} + 96 \beta_{14} + 77 \beta_{13} + 2 \beta_{12} - 29 \beta_{11} + \cdots + 575 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3 \beta_{17} + 525 \beta_{16} - 88 \beta_{15} + 510 \beta_{14} - 106 \beta_{13} - 14 \beta_{12} + \cdots + 698 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17 \beta_{17} + 926 \beta_{16} + 703 \beta_{15} + 786 \beta_{14} + 543 \beta_{13} + 32 \beta_{12} + \cdots + 3893 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 56 \beta_{17} + 3904 \beta_{16} - 606 \beta_{15} + 3746 \beta_{14} - 833 \beta_{13} - 137 \beta_{12} + \cdots + 5515 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 201 \beta_{17} + 7513 \beta_{16} + 5177 \beta_{15} + 6290 \beta_{14} + 3679 \beta_{13} + 330 \beta_{12} + \cdots + 27107 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 688 \beta_{17} + 29038 \beta_{16} - 3737 \beta_{15} + 27583 \beta_{14} - 6182 \beta_{13} - 1180 \beta_{12} + \cdots + 43005 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2053 \beta_{17} + 59783 \beta_{16} + 37549 \beta_{15} + 49823 \beta_{14} + 24374 \beta_{13} + \cdots + 192283 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 7080 \beta_{17} + 216596 \beta_{16} - 20422 \beta_{15} + 204059 \beta_{14} - 44489 \beta_{13} + \cdots + 333140 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 19439 \beta_{17} + 470153 \beta_{16} + 271056 \beta_{15} + 392329 \beta_{14} + 159039 \beta_{13} + \cdots + 1382442 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 66426 \beta_{17} + 1620828 \beta_{16} - 91207 \beta_{15} + 1516971 \beta_{14} - 314852 \beta_{13} + \cdots + 2572050 \) 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
−2.55461
−2.24824
−1.90615
−1.70725
−1.33766
−0.665348
−0.432278
−0.375443
0.0366427
0.803961
0.981962
1.52505
1.60620
1.77085
2.40246
2.59597
2.73355
2.77035
−2.55461 0 4.52602 2.12871 0 1.00000 −6.45298 0 −5.43801
1.2 −2.24824 0 3.05458 3.31434 0 1.00000 −2.37095 0 −7.45143
1.3 −1.90615 0 1.63341 −1.97979 0 1.00000 0.698768 0 3.77379
1.4 −1.70725 0 0.914706 −1.08728 0 1.00000 1.85287 0 1.85625
1.5 −1.33766 0 −0.210663 0.660635 0 1.00000 2.95712 0 −0.883705
1.6 −0.665348 0 −1.55731 3.70415 0 1.00000 2.36685 0 −2.46455
1.7 −0.432278 0 −1.81314 4.10000 0 1.00000 1.64833 0 −1.77234
1.8 −0.375443 0 −1.85904 −2.49678 0 1.00000 1.44885 0 0.937397
1.9 0.0366427 0 −1.99866 −1.02187 0 1.00000 −0.146522 0 −0.0374440
1.10 0.803961 0 −1.35365 −0.343335 0 1.00000 −2.69620 0 −0.276028
1.11 0.981962 0 −1.03575 −2.59441 0 1.00000 −2.98099 0 −2.54761
1.12 1.52505 0 0.325779 1.71268 0 1.00000 −2.55327 0 2.61192
1.13 1.60620 0 0.579871 3.54808 0 1.00000 −2.28101 0 5.69892
1.14 1.77085 0 1.13590 −1.38376 0 1.00000 −1.53019 0 −2.45042
1.15 2.40246 0 3.77179 0.666645 0 1.00000 4.25666 0 1.60159
1.16 2.59597 0 4.73905 1.63234 0 1.00000 7.11048 0 4.23750
1.17 2.73355 0 5.47227 −4.25416 0 1.00000 9.49162 0 −11.6289
1.18 2.77035 0 5.67483 3.69380 0 1.00000 10.1806 0 10.2331
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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.u 18
3.b odd 2 1 2667.2.a.p ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.p ✓ 18 3.b odd 2 1
8001.2.a.u 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{18} - 6 T_{2}^{17} - 11 T_{2}^{16} + 123 T_{2}^{15} - 35 T_{2}^{14} - 982 T_{2}^{13} + 988 T_{2}^{12} + \cdots + 16 \) Copy content Toggle raw display
\( T_{5}^{18} - 10 T_{5}^{17} - 9 T_{5}^{16} + 374 T_{5}^{15} - 651 T_{5}^{14} - 4832 T_{5}^{13} + \cdots + 49792 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 6 T^{17} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} - 10 T^{17} + \cdots + 49792 \) Copy content Toggle raw display
$7$ \( (T - 1)^{18} \) Copy content Toggle raw display
$11$ \( T^{18} - 9 T^{17} + \cdots + 1280000 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots - 623244800 \) Copy content Toggle raw display
$17$ \( T^{18} - 17 T^{17} + \cdots + 85910912 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots - 2183223712 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 48663006208 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 2717396224 \) Copy content Toggle raw display
$31$ \( T^{18} - 5 T^{17} + \cdots + 5492768 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots - 131904255436 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 1896176000 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 5710191104 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 31187414908224 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 1454748563200 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 711880908800 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 167749132480000 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 13848455168 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 514861187238656 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 80005666705024 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 17570271507200 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 30\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 89133416579072 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 25023269459072 \) Copy content Toggle raw display
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