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Properties

Label 8001.2.a.q
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $15$
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: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{8} q^{5} - q^{7} + (\beta_{14} + \beta_{12} + \cdots - \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{8} q^{5} - q^{7} + (\beta_{14} + \beta_{12} + \cdots - \beta_{4}) q^{8}+ \cdots + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5995 \nu^{14} - 50387 \nu^{13} - 70689 \nu^{12} + 1136059 \nu^{11} - 104425 \nu^{10} + \cdots - 1488901 ) / 624662 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13615 \nu^{14} - 201176 \nu^{13} - 414259 \nu^{12} + 4188080 \nu^{11} + 4802080 \nu^{10} + \cdots - 2677270 ) / 312331 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 46461 \nu^{14} + 346281 \nu^{13} - 856573 \nu^{12} - 7272379 \nu^{11} + 5275825 \nu^{10} + \cdots + 3303757 ) / 624662 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 61025 \nu^{14} + 251381 \nu^{13} - 1185327 \nu^{12} - 5203625 \nu^{11} + 8156911 \nu^{10} + \cdots + 1161803 ) / 624662 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 89367 \nu^{14} + 251991 \nu^{13} - 1775425 \nu^{12} - 5236867 \nu^{11} + 12760035 \nu^{10} + \cdots - 424393 ) / 624662 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 124199 \nu^{14} - 36911 \nu^{13} - 2688475 \nu^{12} + 848163 \nu^{11} + 22201561 \nu^{10} + \cdots - 4275743 ) / 624662 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 71314 \nu^{14} + 21453 \nu^{13} + 1521763 \nu^{12} - 459426 \nu^{11} - 12246648 \nu^{10} + \cdots + 295618 ) / 312331 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 124624 \nu^{14} + 196305 \nu^{13} - 2558030 \nu^{12} - 4051663 \nu^{11} + 19403758 \nu^{10} + \cdots + 2534792 ) / 312331 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 165788 \nu^{14} + 202150 \nu^{13} - 3394240 \nu^{12} - 4150019 \nu^{11} + 25626527 \nu^{10} + \cdots + 2727176 ) / 312331 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 457793 \nu^{14} - 140549 \nu^{13} - 9783541 \nu^{12} + 3108413 \nu^{11} + 79045069 \nu^{10} + \cdots - 6116335 ) / 624662 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 243430 \nu^{14} + 83019 \nu^{13} - 5093930 \nu^{12} - 1622009 \nu^{11} + 39878673 \nu^{10} + \cdots - 283714 ) / 312331 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 317108 \nu^{14} - 235139 \nu^{13} + 6591938 \nu^{12} + 4776260 \nu^{11} - 51045558 \nu^{10} + \cdots + 95614 ) / 312331 \) 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
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{12} - \beta_{9} + \beta_{6} - \beta_{4} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + 8\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{14} + 9 \beta_{12} + \beta_{11} - 10 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 8 \beta_{6} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{14} + 9 \beta_{13} - 11 \beta_{12} - 10 \beta_{11} + 9 \beta_{10} - 13 \beta_{9} + 2 \beta_{8} + \cdots + 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 79 \beta_{14} - \beta_{13} + 68 \beta_{12} + 13 \beta_{11} - 2 \beta_{10} - 81 \beta_{9} + 26 \beta_{8} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 8 \beta_{14} + 66 \beta_{13} - 91 \beta_{12} - 77 \beta_{11} + 67 \beta_{10} - 128 \beta_{9} + \cdots + 601 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 580 \beta_{14} - 13 \beta_{13} + 486 \beta_{12} + 117 \beta_{11} - 25 \beta_{10} - 616 \beta_{9} + \cdots - 208 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 24 \beta_{14} + 461 \beta_{13} - 673 \beta_{12} - 545 \beta_{11} + 473 \beta_{10} - 1123 \beta_{9} + \cdots + 3848 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4133 \beta_{14} - 121 \beta_{13} + 3390 \beta_{12} + 919 \beta_{11} - 212 \beta_{10} - 4567 \beta_{9} + \cdots - 1566 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 235 \beta_{14} + 3186 \beta_{13} - 4698 \beta_{12} - 3722 \beta_{11} + 3269 \beta_{10} - 9269 \beta_{9} + \cdots + 25023 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 29066 \beta_{14} - 987 \beta_{13} + 23369 \beta_{12} + 6765 \beta_{11} - 1512 \beta_{10} - 33474 \beta_{9} + \cdots - 10818 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5213 \beta_{14} + 22030 \beta_{13} - 31683 \beta_{12} - 24988 \beta_{11} + 22382 \beta_{10} + \cdots + 164749 \) 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
−2.59170
−2.51564
−2.07483
−1.64397
−0.955258
−0.735585
−0.601235
−0.146355
0.371530
1.15799
1.22206
1.32572
2.16969
2.30266
2.71493
−2.59170 0 4.71692 −3.45466 0 −1.00000 −7.04143 0 8.95345
1.2 −2.51564 0 4.32847 1.19101 0 −1.00000 −5.85760 0 −2.99615
1.3 −2.07483 0 2.30492 −1.11193 0 −1.00000 −0.632654 0 2.30706
1.4 −1.64397 0 0.702628 −1.50234 0 −1.00000 2.13284 0 2.46980
1.5 −0.955258 0 −1.08748 2.56419 0 −1.00000 2.94934 0 −2.44946
1.6 −0.735585 0 −1.45891 −0.430466 0 −1.00000 2.54433 0 0.316644
1.7 −0.601235 0 −1.63852 −4.40463 0 −1.00000 2.18760 0 2.64822
1.8 −0.146355 0 −1.97858 2.93285 0 −1.00000 0.582285 0 −0.429238
1.9 0.371530 0 −1.86197 −0.962757 0 −1.00000 −1.43484 0 −0.357693
1.10 1.15799 0 −0.659052 −3.10556 0 −1.00000 −3.07916 0 −3.59621
1.11 1.22206 0 −0.506578 −1.90746 0 −1.00000 −3.06318 0 −2.33103
1.12 1.32572 0 −0.242472 1.07970 0 −1.00000 −2.97289 0 1.43138
1.13 2.16969 0 2.70756 2.09371 0 −1.00000 1.53520 0 4.54272
1.14 2.30266 0 3.30224 1.35662 0 −1.00000 2.99860 0 3.12384
1.15 2.71493 0 5.37083 −1.33828 0 −1.00000 9.15156 0 −3.63333
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.q 15
3.b odd 2 1 889.2.a.b ✓ 15
21.c even 2 1 6223.2.a.j 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
889.2.a.b ✓ 15 3.b odd 2 1
6223.2.a.j 15 21.c even 2 1
8001.2.a.q 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{15} - 22 T_{2}^{13} + 186 T_{2}^{11} - 763 T_{2}^{9} - 7 T_{2}^{8} + 1588 T_{2}^{7} + 64 T_{2}^{6} + \cdots - 13 \) Copy content Toggle raw display
\( T_{5}^{15} + 7 T_{5}^{14} - 13 T_{5}^{13} - 167 T_{5}^{12} - 42 T_{5}^{11} + 1477 T_{5}^{10} + \cdots + 2294 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 22 T^{13} + \cdots - 13 \) Copy content Toggle raw display
$3$ \( T^{15} \) Copy content Toggle raw display
$5$ \( T^{15} + 7 T^{14} + \cdots + 2294 \) Copy content Toggle raw display
$7$ \( (T + 1)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} + 14 T^{14} + \cdots + 302112 \) Copy content Toggle raw display
$13$ \( T^{15} - 6 T^{14} + \cdots - 296 \) Copy content Toggle raw display
$17$ \( T^{15} + 10 T^{14} + \cdots - 13032 \) Copy content Toggle raw display
$19$ \( T^{15} - 13 T^{14} + \cdots - 2261282 \) Copy content Toggle raw display
$23$ \( T^{15} + 15 T^{14} + \cdots - 159348 \) Copy content Toggle raw display
$29$ \( T^{15} + 16 T^{14} + \cdots + 24350536 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots - 508362642 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 40940005196 \) Copy content Toggle raw display
$41$ \( T^{15} + 19 T^{14} + \cdots - 14583848 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 2881799908 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 127047598350 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 33403676712 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 925256454912 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 26580006408 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 38013704592 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 23410866384 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 1903703128 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 1041479699184 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 328479194135568 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 217040265526 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 1094692262 \) Copy content Toggle raw display
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