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Properties

Label 8001.2.a.s
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
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] = [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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 \) : 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
\(\beta_{4}\)\(=\) \( \nu^{4} - 7\nu^{2} - \nu + 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16172 \nu^{15} - 233047 \nu^{14} + 545740 \nu^{13} + 3569234 \nu^{12} - 13969268 \nu^{11} + \cdots - 1833530 ) / 739814 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 55506 \nu^{15} + 171907 \nu^{14} + 1127781 \nu^{13} - 3471014 \nu^{12} - 8968081 \nu^{11} + \cdots - 3035278 ) / 739814 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 79649 \nu^{15} + 179031 \nu^{14} + 1937788 \nu^{13} - 4128253 \nu^{12} - 18731714 \nu^{11} + \cdots - 1805692 ) / 739814 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 49011 \nu^{15} + 202993 \nu^{14} + 777233 \nu^{13} - 3856011 \nu^{12} - 3798508 \nu^{11} + \cdots + 482636 ) / 369907 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 123393 \nu^{15} - 271059 \nu^{14} - 3028354 \nu^{13} + 6241792 \nu^{12} + 29665237 \nu^{11} + \cdots + 3426510 ) / 739814 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 125049 \nu^{15} - 467479 \nu^{14} - 2288559 \nu^{13} + 9361224 \nu^{12} + 15315132 \nu^{11} + \cdots + 4758460 ) / 739814 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 129018 \nu^{15} - 383386 \nu^{14} - 2688667 \nu^{13} + 7911672 \nu^{12} + 21971299 \nu^{11} + \cdots + 725548 ) / 739814 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 137239 \nu^{15} + 515922 \nu^{14} + 2401726 \nu^{13} - 10047088 \nu^{12} - 14807929 \nu^{11} + \cdots - 4842758 ) / 739814 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 145145 \nu^{15} - 426142 \nu^{14} - 3134036 \nu^{13} + 9144531 \nu^{12} + 26680262 \nu^{11} + \cdots + 5228678 ) / 739814 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 158942 \nu^{15} - 398038 \nu^{14} - 3767459 \nu^{13} + 8967247 \nu^{12} + 35638598 \nu^{11} + \cdots + 7379728 ) / 739814 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 322032 \nu^{15} - 1082571 \nu^{14} - 6442221 \nu^{13} + 22367158 \nu^{12} + 49725923 \nu^{11} + \cdots + 7468534 ) / 739814 \) 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
\(\nu^{4}\)\(=\) \( \beta_{4} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{6} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{15} - \beta_{14} + \beta_{12} - \beta_{11} - 4 \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15 \beta_{15} - 14 \beta_{14} - 13 \beta_{13} - 14 \beta_{12} - 17 \beta_{10} + 13 \beta_{9} + 14 \beta_{8} + \cdots + 23 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 33 \beta_{15} - 19 \beta_{14} - 3 \beta_{13} + 10 \beta_{12} - 14 \beta_{11} - 61 \beta_{10} + \cdots + 520 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 162 \beta_{15} - 146 \beta_{14} - 125 \beta_{13} - 139 \beta_{12} + 2 \beta_{11} - 196 \beta_{10} + \cdots + 203 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 378 \beta_{15} - 239 \beta_{14} - 59 \beta_{13} + 62 \beta_{12} - 129 \beta_{11} - 657 \beta_{10} + \cdots + 3232 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1538 \beta_{15} - 1365 \beta_{14} - 1076 \beta_{13} - 1217 \beta_{12} + 47 \beta_{11} - 1930 \beta_{10} + \cdots + 1641 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3733 \beta_{15} - 2530 \beta_{14} - 752 \beta_{13} + 223 \beta_{12} - 982 \beta_{11} - 6181 \beta_{10} + \cdots + 20471 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 13671 \beta_{15} - 12084 \beta_{14} - 8781 \beta_{13} - 10051 \beta_{12} + 703 \beta_{11} - 17520 \beta_{10} + \cdots + 12767 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 34127 \beta_{15} - 24451 \beta_{14} - 7956 \beta_{13} - 756 \beta_{12} - 6564 \beta_{11} - 54356 \beta_{10} + \cdots + 131697 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 116930 \beta_{15} - 103619 \beta_{14} - 69592 \beta_{13} - 80533 \beta_{12} + 8580 \beta_{11} + \cdots + 97481 \) 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
2.80317
2.63870
2.45059
2.35655
1.50110
1.17466
0.825012
0.716430
0.284398
−0.0643619
−1.14773
−1.35902
−1.40936
−1.84433
−2.35145
−2.57436
−2.80317 0 5.85777 −1.45069 0 −1.00000 −10.8140 0 4.06653
1.2 −2.63870 0 4.96274 −0.422965 0 −1.00000 −7.81779 0 1.11608
1.3 −2.45059 0 4.00539 2.70231 0 −1.00000 −4.91439 0 −6.62224
1.4 −2.35655 0 3.55331 −2.88159 0 −1.00000 −3.66045 0 6.79060
1.5 −1.50110 0 0.253312 2.96917 0 −1.00000 2.62196 0 −4.45703
1.6 −1.17466 0 −0.620175 −3.75278 0 −1.00000 3.07781 0 4.40823
1.7 −0.825012 0 −1.31936 1.66882 0 −1.00000 2.73851 0 −1.37680
1.8 −0.716430 0 −1.48673 −0.617976 0 −1.00000 2.49800 0 0.442737
1.9 −0.284398 0 −1.91912 3.66581 0 −1.00000 1.11459 0 −1.04255
1.10 0.0643619 0 −1.99586 −3.34910 0 −1.00000 −0.257181 0 −0.215554
1.11 1.14773 0 −0.682718 −1.07917 0 −1.00000 −3.07903 0 −1.23860
1.12 1.35902 0 −0.153057 2.54404 0 −1.00000 −2.92605 0 3.45741
1.13 1.40936 0 −0.0136985 −2.06019 0 −1.00000 −2.83803 0 −2.90356
1.14 1.84433 0 1.40154 −2.16633 0 −1.00000 −1.10376 0 −3.99542
1.15 2.35145 0 3.52933 2.01529 0 −1.00000 3.59614 0 4.73886
1.16 2.57436 0 4.62732 −2.78465 0 −1.00000 6.76367 0 −7.16870
\(n\): 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.s 16
3.b odd 2 1 2667.2.a.n ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.n ✓ 16 3.b odd 2 1
8001.2.a.s 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{16} + 4 T_{2}^{15} - 18 T_{2}^{14} - 83 T_{2}^{13} + 112 T_{2}^{12} + 668 T_{2}^{11} - 235 T_{2}^{10} + \cdots - 20 \) Copy content Toggle raw display
\( T_{5}^{16} + 5 T_{5}^{15} - 36 T_{5}^{14} - 210 T_{5}^{13} + 435 T_{5}^{12} + 3467 T_{5}^{11} + \cdots + 46352 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 4 T^{15} + \cdots - 20 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 5 T^{15} + \cdots + 46352 \) Copy content Toggle raw display
$7$ \( (T + 1)^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + T^{15} + \cdots - 4096 \) Copy content Toggle raw display
$13$ \( T^{16} - 20 T^{15} + \cdots - 143360 \) Copy content Toggle raw display
$17$ \( T^{16} + 3 T^{15} + \cdots - 27287872 \) Copy content Toggle raw display
$19$ \( T^{16} - 13 T^{15} + \cdots - 59975840 \) Copy content Toggle raw display
$23$ \( T^{16} + 5 T^{15} + \cdots - 92323840 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots - 265579040 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots - 2220064768 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 2281961720 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 6198211456 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots - 270850329856 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 80836384768 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 3330453274336 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 2877741056 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 9236458496 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 5716113605888 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 4708249600 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots - 30744464128 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 1732480000 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 6896650240 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 550072064 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 24693460838972 \) Copy content Toggle raw display
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