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Properties

Label 8001.2.a.p
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5 x^{13} - 9 x^{12} + 76 x^{11} - 12 x^{10} - 414 x^{9} + 331 x^{8} + 959 x^{7} - 1067 x^{6} + \cdots + 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{3} q^{5} - q^{7} + ( - \beta_{11} + \beta_{10} - \beta_{9} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{3} q^{5} - q^{7} + ( - \beta_{11} + \beta_{10} - \beta_{9} + \cdots + 1) q^{8}+ \cdots + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8} + 4 q^{10} + 3 q^{11} - 13 q^{13} - 5 q^{14} + 13 q^{16} + 5 q^{17} + 21 q^{19} + 3 q^{20} - 3 q^{22} + 10 q^{23} + 4 q^{25} + 6 q^{26} - 15 q^{28} + 15 q^{29} + 33 q^{31} + 29 q^{32} + 28 q^{34} - 4 q^{35} - 29 q^{37} + 15 q^{38} + 3 q^{40} + q^{41} - 25 q^{43} + 26 q^{44} - 4 q^{46} + 9 q^{47} + 14 q^{49} + 28 q^{50} - 13 q^{52} + 35 q^{53} + 14 q^{55} - 12 q^{56} - 23 q^{58} - 10 q^{59} + q^{61} + 43 q^{62} - 2 q^{64} + 24 q^{65} - 38 q^{67} + 2 q^{68} - 4 q^{70} + 10 q^{71} + 8 q^{73} + 25 q^{74} + 26 q^{76} - 3 q^{77} + 26 q^{79} + 48 q^{80} + 6 q^{82} + 30 q^{83} - 32 q^{85} + 50 q^{86} - 29 q^{88} - 4 q^{89} + 13 q^{91} + 32 q^{92} - 7 q^{94} + 32 q^{95} + 15 q^{97} + 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 5 x^{13} - 9 x^{12} + 76 x^{11} - 12 x^{10} - 414 x^{9} + 331 x^{8} + 959 x^{7} - 1067 x^{6} + \cdots + 44 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2 \nu^{13} + 34 \nu^{12} + 21 \nu^{11} - 541 \nu^{10} - 60 \nu^{9} + 3055 \nu^{8} - 58 \nu^{7} + \cdots + 1454 ) / 274 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12 \nu^{13} + 67 \nu^{12} + 126 \nu^{11} - 917 \nu^{10} - 223 \nu^{9} + 3945 \nu^{8} - 1581 \nu^{7} + \cdots - 2510 ) / 274 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20 \nu^{13} - 66 \nu^{12} - 347 \nu^{11} + 1163 \nu^{10} + 2244 \nu^{9} - 7671 \nu^{8} - 6544 \nu^{7} + \cdots - 2210 ) / 274 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13 \nu^{13} - 84 \nu^{12} - 68 \nu^{11} + 1256 \nu^{10} - 843 \nu^{9} - 6637 \nu^{8} + 7501 \nu^{7} + \cdots - 135 ) / 137 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 67 \nu^{13} + 180 \nu^{12} + 909 \nu^{11} - 2711 \nu^{10} - 3791 \nu^{9} + 14731 \nu^{8} + \cdots + 896 ) / 274 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13 \nu^{13} - 53 \nu^{12} + 479 \nu^{11} + 662 \nu^{10} - 5322 \nu^{9} - 2268 \nu^{8} + 25379 \nu^{7} + \cdots + 2190 ) / 274 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 69 \nu^{13} + 214 \nu^{12} + 930 \nu^{11} - 3252 \nu^{10} - 3851 \nu^{9} + 17786 \nu^{8} + \cdots + 1254 ) / 274 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27 \nu^{13} - 48 \nu^{12} - 489 \nu^{11} + 796 \nu^{10} + 3413 \nu^{9} - 5006 \nu^{8} - 11410 \nu^{7} + \cdots - 2230 ) / 137 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 40 \nu^{13} - 132 \nu^{12} - 557 \nu^{11} + 2052 \nu^{10} + 2570 \nu^{9} - 11643 \nu^{8} - 3909 \nu^{7} + \cdots - 2091 ) / 137 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 51 \nu^{13} - 319 \nu^{12} - 193 \nu^{11} + 4548 \nu^{10} - 4498 \nu^{9} - 21938 \nu^{8} + 36551 \nu^{7} + \cdots + 4160 ) / 274 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 120 \nu^{13} - 259 \nu^{12} - 1945 \nu^{11} + 3964 \nu^{10} + 11683 \nu^{9} - 21914 \nu^{8} + \cdots - 2026 ) / 274 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} - \beta_{9} + \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{10} - 2\beta_{9} + 2\beta_{7} + \beta_{6} + 2\beta_{3} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - \beta_{12} - 8 \beta_{11} + 8 \beta_{10} - 9 \beta_{9} + 2 \beta_{8} + 10 \beta_{7} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} - 11 \beta_{11} + 12 \beta_{10} - 21 \beta_{9} + 2 \beta_{8} + 23 \beta_{7} + 11 \beta_{6} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11 \beta_{13} - 10 \beta_{12} - 57 \beta_{11} + 59 \beta_{10} - 69 \beta_{9} + 23 \beta_{8} + 82 \beta_{7} + \cdots + 73 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15 \beta_{13} - 95 \beta_{11} + 109 \beta_{10} - 171 \beta_{9} + 29 \beta_{8} + 202 \beta_{7} + \cdots + 496 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 92 \beta_{13} - 74 \beta_{12} - 397 \beta_{11} + 429 \beta_{10} - 504 \beta_{9} + 196 \beta_{8} + \cdots + 568 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 154 \beta_{13} - 3 \beta_{12} - 753 \beta_{11} + 892 \beta_{10} - 1285 \beta_{9} + 295 \beta_{8} + \cdots + 3048 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 703 \beta_{13} - 494 \beta_{12} - 2755 \beta_{11} + 3097 \beta_{10} - 3614 \beta_{9} + 1508 \beta_{8} + \cdots + 4304 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1357 \beta_{13} - 62 \beta_{12} - 5725 \beta_{11} + 6926 \beta_{10} - 9353 \beta_{9} + 2601 \beta_{8} + \cdots + 19350 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5181 \beta_{13} - 3167 \beta_{12} - 19157 \beta_{11} + 22239 \beta_{10} - 25724 \beta_{9} + 11112 \beta_{8} + \cdots + 32026 \) 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
−2.33388
−2.16813
−1.87889
−1.00753
−0.579209
−0.477041
0.369092
0.789779
1.09407
1.75826
1.86702
2.23233
2.66571
2.66839
−2.33388 0 3.44698 0.335557 0 −1.00000 −3.37706 0 −0.783148
1.2 −2.16813 0 2.70077 1.98599 0 −1.00000 −1.51936 0 −4.30589
1.3 −1.87889 0 1.53022 −2.15857 0 −1.00000 0.882670 0 4.05571
1.4 −1.00753 0 −0.984892 −2.84275 0 −1.00000 3.00736 0 2.86415
1.5 −0.579209 0 −1.66452 1.12937 0 −1.00000 2.12252 0 −0.654144
1.6 −0.477041 0 −1.77243 1.98652 0 −1.00000 1.79961 0 −0.947652
1.7 0.369092 0 −1.86377 3.55869 0 −1.00000 −1.42609 0 1.31348
1.8 0.789779 0 −1.37625 0.366860 0 −1.00000 −2.66649 0 0.289738
1.9 1.09407 0 −0.803001 −2.53877 0 −1.00000 −3.06669 0 −2.77760
1.10 1.75826 0 1.09149 −1.44353 0 −1.00000 −1.59740 0 −2.53810
1.11 1.86702 0 1.48577 4.06976 0 −1.00000 −0.960078 0 7.59833
1.12 2.23233 0 2.98331 −2.48067 0 −1.00000 2.19508 0 −5.53768
1.13 2.66571 0 5.10602 −0.697799 0 −1.00000 8.27977 0 −1.86013
1.14 2.66839 0 5.12030 2.72934 0 −1.00000 8.32617 0 7.28293
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.p 14
3.b odd 2 1 2667.2.a.m ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.m ✓ 14 3.b odd 2 1
8001.2.a.p 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{14} - 5 T_{2}^{13} - 9 T_{2}^{12} + 76 T_{2}^{11} - 12 T_{2}^{10} - 414 T_{2}^{9} + 331 T_{2}^{8} + \cdots + 44 \) Copy content Toggle raw display
\( T_{5}^{14} - 4 T_{5}^{13} - 29 T_{5}^{12} + 110 T_{5}^{11} + 335 T_{5}^{10} - 1158 T_{5}^{9} + \cdots + 844 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 5 T^{13} + \cdots + 44 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} - 4 T^{13} + \cdots + 844 \) Copy content Toggle raw display
$7$ \( (T + 1)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} - 3 T^{13} + \cdots - 256 \) Copy content Toggle raw display
$13$ \( T^{14} + 13 T^{13} + \cdots - 512 \) Copy content Toggle raw display
$17$ \( T^{14} - 5 T^{13} + \cdots + 1964992 \) Copy content Toggle raw display
$19$ \( T^{14} - 21 T^{13} + \cdots + 11651104 \) Copy content Toggle raw display
$23$ \( T^{14} - 10 T^{13} + \cdots - 14984800 \) Copy content Toggle raw display
$29$ \( T^{14} - 15 T^{13} + \cdots + 11323600 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 614457824 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 299125948 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 3726826112 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 3855391552 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 637784464 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 19702384400 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 49597767424 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 181770499328 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 2652304192 \) Copy content Toggle raw display
$71$ \( T^{14} - 10 T^{13} + \cdots - 54609728 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 4865757632 \) Copy content Toggle raw display
$79$ \( T^{14} - 26 T^{13} + \cdots + 37260800 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 84701372800 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 208920272 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 1491113902028 \) Copy content Toggle raw display
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