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Properties

Label 8001.2.a.o
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} + \cdots - 8 \) 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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{10} - 1) q^{5} + q^{7} + (\beta_{7} - \beta_{6} - \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{10} - 1) q^{5} + q^{7} + (\beta_{7} - \beta_{6} - \beta_1 - 1) q^{8} + ( - \beta_{11} + \beta_1) q^{10} + \beta_{8} q^{11} + ( - \beta_{12} - \beta_{7} + \beta_{4} + 2) q^{13} - \beta_1 q^{14} + ( - \beta_{12} - \beta_{9} - \beta_{7} + \cdots + 1) q^{16}+ \cdots - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} + \cdots - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{12} - 2 \nu^{11} - 18 \nu^{10} + 29 \nu^{9} + 125 \nu^{8} - 144 \nu^{7} - 411 \nu^{6} + 285 \nu^{5} + \cdots + 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{12} - 2 \nu^{11} - 15 \nu^{10} + 26 \nu^{9} + 84 \nu^{8} - 114 \nu^{7} - 224 \nu^{6} + 198 \nu^{5} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{12} + 2 \nu^{11} - 22 \nu^{10} - 27 \nu^{9} + 169 \nu^{8} + 120 \nu^{7} - 571 \nu^{6} - 203 \nu^{5} + \cdots + 48 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{12} - 8 \nu^{11} - 40 \nu^{10} + 105 \nu^{9} + 187 \nu^{8} - 470 \nu^{7} - 387 \nu^{6} + 863 \nu^{5} + \cdots + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{12} - 8 \nu^{11} - 40 \nu^{10} + 105 \nu^{9} + 187 \nu^{8} - 470 \nu^{7} - 387 \nu^{6} + \cdots + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{12} + 4 \nu^{11} + 8 \nu^{10} - 51 \nu^{9} + 9 \nu^{8} + 220 \nu^{7} - 193 \nu^{6} - 387 \nu^{5} + \cdots + 28 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2 \nu^{12} + 5 \nu^{11} + 27 \nu^{10} - 65 \nu^{9} - 128 \nu^{8} + 285 \nu^{7} + 269 \nu^{6} + \cdots - 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{12} - 8 \nu^{11} - 44 \nu^{10} + 109 \nu^{9} + 243 \nu^{8} - 510 \nu^{7} - 655 \nu^{6} + \cdots + 48 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2 \nu^{12} - 7 \nu^{11} - 25 \nu^{10} + 93 \nu^{9} + 108 \nu^{8} - 419 \nu^{7} - 213 \nu^{6} + 758 \nu^{5} + \cdots + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3 \nu^{12} - 7 \nu^{11} - 42 \nu^{10} + 91 \nu^{9} + 212 \nu^{8} - 399 \nu^{7} - 493 \nu^{6} + 696 \nu^{5} + \cdots + 18 ) / 2 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} - \beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} + 6\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{9} - 8 \beta_{7} + 9 \beta_{6} + \beta_{5} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{12} + 2 \beta_{11} - 4 \beta_{10} - 11 \beta_{9} - \beta_{8} - 10 \beta_{7} + 11 \beta_{6} + \cdots + 89 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 16 \beta_{12} + 14 \beta_{11} - 27 \beta_{10} - 15 \beta_{9} - 2 \beta_{8} - 58 \beta_{7} + 68 \beta_{6} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 109 \beta_{12} + 31 \beta_{11} - 59 \beta_{10} - 95 \beta_{9} - 14 \beta_{8} - 84 \beta_{7} + \cdots + 571 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 174 \beta_{12} + 139 \beta_{11} - 262 \beta_{10} - 156 \beta_{9} - 31 \beta_{8} - 414 \beta_{7} + \cdots + 694 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 894 \beta_{12} + 327 \beta_{11} - 607 \beta_{10} - 757 \beta_{9} - 140 \beta_{8} - 673 \beta_{7} + \cdots + 3811 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1613 \beta_{12} + 1210 \beta_{11} - 2248 \beta_{10} - 1404 \beta_{9} - 328 \beta_{8} - 2952 \beta_{7} + \cdots + 5431 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6983 \beta_{12} + 2953 \beta_{11} - 5411 \beta_{10} - 5816 \beta_{9} - 1226 \beta_{8} - 5269 \beta_{7} + \cdots + 26023 \) 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
2.70878
2.29288
2.28328
1.45503
1.27342
1.18273
0.307326
−0.305711
−0.423652
−0.910949
−1.58856
−1.82297
−2.45160
−2.70878 0 5.33747 −2.87857 0 1.00000 −9.04047 0 7.79741
1.2 −2.29288 0 3.25728 0.132682 0 1.00000 −2.88278 0 −0.304223
1.3 −2.28328 0 3.21337 −2.58645 0 1.00000 −2.77047 0 5.90560
1.4 −1.45503 0 0.117115 0.420019 0 1.00000 2.73966 0 −0.611141
1.5 −1.27342 0 −0.378389 2.08575 0 1.00000 3.02870 0 −2.65604
1.6 −1.18273 0 −0.601156 −4.39766 0 1.00000 3.07646 0 5.20123
1.7 −0.307326 0 −1.90555 0.988649 0 1.00000 1.20028 0 −0.303838
1.8 0.305711 0 −1.90654 0.276458 0 1.00000 −1.19427 0 0.0845162
1.9 0.423652 0 −1.82052 −2.66196 0 1.00000 −1.61857 0 −1.12774
1.10 0.910949 0 −1.17017 2.06751 0 1.00000 −2.88787 0 1.88339
1.11 1.58856 0 0.523519 −3.57671 0 1.00000 −2.34548 0 −5.68181
1.12 1.82297 0 1.32323 −0.630622 0 1.00000 −1.23374 0 −1.14961
1.13 2.45160 0 4.01034 −1.23909 0 1.00000 4.92856 0 −3.03775
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.o 13
3.b odd 2 1 2667.2.a.l ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.l ✓ 13 3.b odd 2 1
8001.2.a.o 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{13} + 4 T_{2}^{12} - 10 T_{2}^{11} - 53 T_{2}^{10} + 19 T_{2}^{9} + 242 T_{2}^{8} + 61 T_{2}^{7} + \cdots + 8 \) Copy content Toggle raw display
\( T_{5}^{13} + 12 T_{5}^{12} + 39 T_{5}^{11} - 50 T_{5}^{10} - 457 T_{5}^{9} - 340 T_{5}^{8} + 1492 T_{5}^{7} + \cdots + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + 4 T^{12} + \cdots + 8 \) Copy content Toggle raw display
$3$ \( T^{13} \) Copy content Toggle raw display
$5$ \( T^{13} + 12 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T - 1)^{13} \) Copy content Toggle raw display
$11$ \( T^{13} + 3 T^{12} + \cdots - 284672 \) Copy content Toggle raw display
$13$ \( T^{13} - 21 T^{12} + \cdots - 22336 \) Copy content Toggle raw display
$17$ \( T^{13} + 17 T^{12} + \cdots - 27656 \) Copy content Toggle raw display
$19$ \( T^{13} - 5 T^{12} + \cdots - 16768 \) Copy content Toggle raw display
$23$ \( T^{13} + 4 T^{12} + \cdots - 975616 \) Copy content Toggle raw display
$29$ \( T^{13} + 21 T^{12} + \cdots - 92930644 \) Copy content Toggle raw display
$31$ \( T^{13} + 7 T^{12} + \cdots + 15003392 \) Copy content Toggle raw display
$37$ \( T^{13} - 7 T^{12} + \cdots + 1844308 \) Copy content Toggle raw display
$41$ \( T^{13} + 21 T^{12} + \cdots - 184216 \) Copy content Toggle raw display
$43$ \( T^{13} + 9 T^{12} + \cdots - 28880896 \) Copy content Toggle raw display
$47$ \( T^{13} + 23 T^{12} + \cdots + 9569152 \) Copy content Toggle raw display
$53$ \( T^{13} + 31 T^{12} + \cdots - 405812 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots - 43041792704 \) Copy content Toggle raw display
$61$ \( T^{13} - 29 T^{12} + \cdots - 20477872 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 49278934912 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots + 15605244416 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 169328048 \) Copy content Toggle raw display
$79$ \( T^{13} + 28 T^{12} + \cdots - 18990976 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots - 28884464384 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 180656384 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 416205928 \) Copy content Toggle raw display
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