↑

Properties

Label 8001.2.a.r
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{13} q^{5} - q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{13} q^{5} - q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{8} + (\beta_{13} - \beta_{11} + \cdots - \beta_{2}) q^{10}+ \cdots - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 34171 \nu^{15} - 45717 \nu^{14} - 952649 \nu^{13} + 996036 \nu^{12} + 10901947 \nu^{11} + \cdots + 10476522 ) / 3260350 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 95749 \nu^{15} + 241013 \nu^{14} + 1943846 \nu^{13} - 4685549 \nu^{12} - 14893598 \nu^{11} + \cdots + 5521822 ) / 1630175 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 138556 \nu^{15} - 479387 \nu^{14} - 2447289 \nu^{13} + 9459921 \nu^{12} + 14196617 \nu^{11} + \cdots - 10011933 ) / 1630175 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 312927 \nu^{15} - 1689159 \nu^{14} - 4050143 \nu^{13} + 33947762 \nu^{12} + 2433199 \nu^{11} + \cdots - 7559716 ) / 3260350 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 174111 \nu^{15} - 416897 \nu^{14} - 3848034 \nu^{13} + 8756876 \nu^{12} + 33610177 \nu^{11} + \cdots + 12678877 ) / 1630175 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 62553 \nu^{15} + 185357 \nu^{14} + 1244303 \nu^{13} - 3782409 \nu^{12} - 9231198 \nu^{11} + \cdots - 3490290 ) / 326035 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 322256 \nu^{15} - 1375567 \nu^{14} - 5035219 \nu^{13} + 27365426 \nu^{12} + 19868527 \nu^{11} + \cdots - 11457663 ) / 1630175 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 387602 \nu^{15} + 1093459 \nu^{14} + 7892793 \nu^{13} - 22401737 \nu^{12} - 60887074 \nu^{11} + \cdots + 2433991 ) / 1630175 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 397191 \nu^{15} - 1572742 \nu^{14} - 6632689 \nu^{13} + 31550366 \nu^{12} + 32948807 \nu^{11} + \cdots - 6777548 ) / 1630175 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 443728 \nu^{15} - 1561421 \nu^{14} - 8006447 \nu^{13} + 31434138 \nu^{12} + 49072551 \nu^{11} + \cdots + 3556906 ) / 1630175 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 509784 \nu^{15} - 2052553 \nu^{14} - 8419656 \nu^{13} + 41075579 \nu^{12} + 40710358 \nu^{11} + \cdots - 4486597 ) / 1630175 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 540201 \nu^{15} - 2114267 \nu^{14} - 9125559 \nu^{13} + 42440381 \nu^{12} + 47185962 \nu^{11} + \cdots - 15755608 ) / 1630175 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{11} - \beta_{10} + \beta_{8} + \beta_{6} + \beta_{3} + 8\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + \beta_{13} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} + 9\beta_{3} + 10\beta_{2} + 28\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} - 2 \beta_{13} + 10 \beta_{12} + 11 \beta_{11} - 10 \beta_{10} + 13 \beta_{8} + 12 \beta_{6} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 11 \beta_{15} + \beta_{14} + 8 \beta_{13} + 3 \beta_{11} + \beta_{10} + 10 \beta_{9} + 17 \beta_{8} + \cdots + 40 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3 \beta_{15} + 15 \beta_{14} - 28 \beta_{13} + 76 \beta_{12} + 94 \beta_{11} - 76 \beta_{10} + \cdots + 514 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 90 \beta_{15} + 13 \beta_{14} + 41 \beta_{13} + \beta_{12} + 56 \beta_{11} + 16 \beta_{10} + \cdots + 386 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 47 \beta_{15} + 152 \beta_{14} - 283 \beta_{13} + 524 \beta_{12} + 745 \beta_{11} - 524 \beta_{10} + \cdots + 3243 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 659 \beta_{15} + 124 \beta_{14} + 113 \beta_{13} + 11 \beta_{12} + 693 \beta_{11} + 168 \beta_{10} + \cdots + 3289 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 488 \beta_{15} + 1308 \beta_{14} - 2518 \beta_{13} + 3447 \beta_{12} + 5755 \beta_{11} - 3453 \beta_{10} + \cdots + 21084 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4564 \beta_{15} + 1055 \beta_{14} - 621 \beta_{13} + 40 \beta_{12} + 7185 \beta_{11} + 1480 \beta_{10} + \cdots + 26378 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 4226 \beta_{15} + 10308 \beta_{14} - 20979 \beta_{13} + 22018 \beta_{12} + 44116 \beta_{11} + \cdots + 140173 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 30550 \beta_{15} + 8435 \beta_{14} - 14647 \beta_{13} - 580 \beta_{12} + 67671 \beta_{11} + \cdots + 204740 \) 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
2.76723
2.62936
2.41798
1.84800
1.68977
1.55853
1.09349
0.639731
0.298113
0.0296587
−0.415886
−0.823041
−1.82937
−2.04658
−2.36717
−2.48981
−2.76723 0 5.65757 3.65468 0 −1.00000 −10.1213 0 −10.1133
1.2 −2.62936 0 4.91351 0.281520 0 −1.00000 −7.66067 0 −0.740216
1.3 −2.41798 0 3.84663 −2.86533 0 −1.00000 −4.46511 0 6.92831
1.4 −1.84800 0 1.41512 3.49192 0 −1.00000 1.08086 0 −6.45308
1.5 −1.68977 0 0.855315 −2.75353 0 −1.00000 1.93425 0 4.65282
1.6 −1.55853 0 0.429028 1.78044 0 −1.00000 2.44841 0 −2.77487
1.7 −1.09349 0 −0.804288 −1.42245 0 −1.00000 3.06645 0 1.55543
1.8 −0.639731 0 −1.59074 2.86363 0 −1.00000 2.29711 0 −1.83195
1.9 −0.298113 0 −1.91113 −2.54722 0 −1.00000 1.16596 0 0.759360
1.10 −0.0296587 0 −1.99912 −3.07938 0 −1.00000 0.118609 0 0.0913304
1.11 0.415886 0 −1.82704 0.233776 0 −1.00000 −1.59161 0 0.0972242
1.12 0.823041 0 −1.32260 3.74037 0 −1.00000 −2.73464 0 3.07848
1.13 1.82937 0 1.34661 1.45514 0 −1.00000 −1.19530 0 2.66200
1.14 2.04658 0 2.18847 2.05615 0 −1.00000 0.385720 0 4.20807
1.15 2.36717 0 3.60352 −4.44142 0 −1.00000 3.79580 0 −10.5136
1.16 2.48981 0 4.19916 −1.44828 0 −1.00000 5.47548 0 −3.60594
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

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

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.r 16
3.b odd 2 1 2667.2.a.o ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.o ✓ 16 3.b odd 2 1
8001.2.a.r 16 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{16} + 5 T_{2}^{15} - 13 T_{2}^{14} - 98 T_{2}^{13} + 9 T_{2}^{12} + 712 T_{2}^{11} + 565 T_{2}^{10} + \cdots + 4 \) Copy content Toggle raw display
\( T_{5}^{16} - T_{5}^{15} - 56 T_{5}^{14} + 56 T_{5}^{13} + 1249 T_{5}^{12} - 1185 T_{5}^{11} + \cdots - 27136 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 5 T^{15} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - T^{15} + \cdots - 27136 \) Copy content Toggle raw display
$7$ \( (T + 1)^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + 11 T^{15} + \cdots - 1179648 \) Copy content Toggle raw display
$13$ \( T^{16} - 18 T^{15} + \cdots - 61239104 \) Copy content Toggle raw display
$17$ \( T^{16} - 5 T^{15} + \cdots + 939096 \) Copy content Toggle raw display
$19$ \( T^{16} + 11 T^{15} + \cdots - 18205664 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 885006336 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 24336668672 \) Copy content Toggle raw display
$31$ \( T^{16} + 42 T^{15} + \cdots + 63774720 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 39421737428 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 16289945096 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 41489887232 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 14109523968 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots - 443170186752 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots - 219737088000 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 15419695152 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots - 50604450816 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 2604300724224 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 46641155536 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 39713124416832 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 18683904000 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 105203797590016 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 14259174215040 \) Copy content Toggle raw display
show more
show less