Properties

Label 8993.2.a.g
Level $8993$
Weight $2$
Character orbit 8993.a
Self dual yes
Analytic conductor $71.809$
Analytic rank $1$
Dimension $9$
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] = [8993,2,Mod(1,8993)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8993, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8993.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8993 = 17 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8993.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: \(71.8094665377\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 12x^{7} + 23x^{6} + 43x^{5} - 79x^{4} - 43x^{3} + 78x^{2} + 11x - 21 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 391)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{7} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{6} - 1) q^{5} + ( - \beta_{8} + \beta_{4} - \beta_{3} + \beta_{2}) q^{6} + ( - \beta_{7} - \beta_{6} - \beta_{3}) q^{7} + (\beta_{8} - \beta_{5} - \beta_{4} + 1) q^{8} + (2 \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{7} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{6} - 1) q^{5} + ( - \beta_{8} + \beta_{4} - \beta_{3} + \beta_{2}) q^{6} + ( - \beta_{7} - \beta_{6} - \beta_{3}) q^{7} + (\beta_{8} - \beta_{5} - \beta_{4} + 1) q^{8} + (2 \beta_{4} + 1) q^{9} + (\beta_{7} + 2 \beta_{6} + \beta_{2} - 2 \beta_1) q^{10} + ( - \beta_{6} - \beta_{5} + \beta_{4} - 1) q^{11} + (\beta_{8} + \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{2} - 1) q^{12} + ( - \beta_{8} - \beta_{6} - \beta_{4} - \beta_{2}) q^{13} + (\beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2}) q^{14} + ( - \beta_{8} - \beta_{6} - \beta_{5} + \beta_{2} - 1) q^{15} + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{16} + q^{17} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + \beta_1 - 2) q^{18} + ( - \beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2}) q^{19} + (2 \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 - 3) q^{20} + (\beta_{8} - 2 \beta_{7} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{21} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{22} + (\beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{24} + (\beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{25} + ( - \beta_{8} - 2 \beta_{6} + 2 \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{26} + (2 \beta_{8} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 2) q^{27} + ( - \beta_{8} - \beta_{7} - \beta_{6} + 3 \beta_{4} + \beta_1) q^{28} + ( - 2 \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 3) q^{29} + (\beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 1) q^{30} + (\beta_{8} - \beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{31} + ( - \beta_{8} + 2 \beta_{7} + 3 \beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{2}) q^{32} + (2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 1) q^{33} + \beta_1 q^{34} + (2 \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 - 1) q^{35} + ( - 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \cdots - 1) q^{36}+ \cdots + (2 \beta_{8} + 4 \beta_{7} - \beta_{6} - 5 \beta_{5} + \beta_{4} + 2 \beta_{2} - 6 \beta_1 + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 2 q^{2} + 2 q^{3} + 10 q^{4} - 7 q^{5} + 4 q^{6} - 3 q^{7} + 3 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 2 q^{2} + 2 q^{3} + 10 q^{4} - 7 q^{5} + 4 q^{6} - 3 q^{7} + 3 q^{8} + 17 q^{9} + 3 q^{10} - 11 q^{11} - 10 q^{12} - 9 q^{13} - q^{14} - 16 q^{15} + 8 q^{16} + 9 q^{17} - 18 q^{18} + 4 q^{19} - 23 q^{20} - 12 q^{21} - 10 q^{22} + 12 q^{24} + 2 q^{25} - 9 q^{26} + 8 q^{27} + 8 q^{28} + 24 q^{29} - 4 q^{30} - 6 q^{31} + 13 q^{32} + 10 q^{33} + 2 q^{34} - 18 q^{35} - 14 q^{36} - 15 q^{37} - 18 q^{38} - 4 q^{39} - 2 q^{40} + 14 q^{41} + 52 q^{42} + 2 q^{43} - 11 q^{44} - 11 q^{45} + 9 q^{47} - 32 q^{48} + 32 q^{50} + 2 q^{51} - 36 q^{52} - 8 q^{53} + 16 q^{54} - 24 q^{55} - 18 q^{56} - 4 q^{57} - 18 q^{58} - 7 q^{59} + 48 q^{60} - 17 q^{61} - 28 q^{62} - 11 q^{63} - 7 q^{64} - 14 q^{65} - 6 q^{67} + 10 q^{68} - 41 q^{70} + 20 q^{71} - 57 q^{72} - 2 q^{73} + 12 q^{74} - 40 q^{75} + 40 q^{76} + 16 q^{77} + 8 q^{78} - 7 q^{79} + 28 q^{80} + 37 q^{81} - 42 q^{82} - 6 q^{83} - 32 q^{84} - 7 q^{85} - 12 q^{86} - 76 q^{87} + 13 q^{88} - 34 q^{89} + 31 q^{90} + 29 q^{91} - 28 q^{93} - 68 q^{94} - 32 q^{95} - 12 q^{96} - 25 q^{97} - 11 q^{98} + 29 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 12x^{7} + 23x^{6} + 43x^{5} - 79x^{4} - 43x^{3} + 78x^{2} + 11x - 21 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 12\nu^{5} - 40\nu^{3} + 2\nu^{2} + 26\nu - 3 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} - 2\nu^{7} - 11\nu^{6} + 21\nu^{5} + 33\nu^{4} - 61\nu^{3} - 17\nu^{2} + 34\nu + 3 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} - \nu^{7} - 13\nu^{6} + 11\nu^{5} + 51\nu^{4} - 35\nu^{3} - 57\nu^{2} + 26\nu + 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{8} - \nu^{7} - 13\nu^{6} + 11\nu^{5} + 55\nu^{4} - 35\nu^{3} - 85\nu^{2} + 26\nu + 40 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{8} + \nu^{7} + 14\nu^{6} - 10\nu^{5} - 66\nu^{4} + 28\nu^{3} + 114\nu^{2} - 19\nu - 47 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2\nu^{8} - 3\nu^{7} - 24\nu^{6} + 32\nu^{5} + 84\nu^{4} - 92\nu^{3} - 74\nu^{2} + 44\nu + 15 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{5} - \beta_{4} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - \beta_{5} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{8} + 2\beta_{7} + 3\beta_{6} - 7\beta_{5} - 8\beta_{4} + \beta_{3} + 2\beta_{2} + 20\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} + 12\beta_{6} - 11\beta_{5} + \beta_{4} - \beta_{3} + 46\beta_{2} + \beta _1 + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 44\beta_{8} + 24\beta_{7} + 36\beta_{6} - 44\beta_{5} - 56\beta_{4} + 8\beta_{3} + 26\beta_{2} + 106\beta _1 + 59 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{8} + 28\beta_{7} + 108\beta_{6} - 90\beta_{5} + 10\beta_{4} - 16\beta_{3} + 302\beta_{2} + 13\beta _1 + 488 \) Copy content Toggle raw display

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
−2.41113
−2.13419
−0.982337
−0.655886
0.694427
0.928716
1.60303
2.34766
2.60971
−2.41113 −2.97582 3.81355 −1.50981 7.17508 2.86216 −4.37269 5.85548 3.64036
1.2 −2.13419 1.24506 2.55475 −2.19346 −2.65719 −2.50345 −1.18395 −1.44983 4.68126
1.3 −0.982337 3.41233 −1.03501 −1.79121 −3.35206 −2.92427 2.98141 8.64402 1.75957
1.4 −0.655886 −0.842334 −1.56981 0.0377941 0.552475 2.13226 2.34139 −2.29047 −0.0247886
1.5 0.694427 −2.80315 −1.51777 3.60744 −1.94659 −2.92544 −2.44284 4.85767 2.50511
1.6 0.928716 2.28175 −1.13749 −0.267749 2.11910 −2.64533 −2.91383 2.20640 −0.248663
1.7 1.60303 2.57788 0.569692 −2.37174 4.13241 4.07654 −2.29282 3.64546 −3.80196
1.8 2.34766 −1.19956 3.51149 −3.97920 −2.81616 0.624291 3.54845 −1.56105 −9.34178
1.9 2.60971 0.303841 4.81061 1.46794 0.792939 −1.69676 7.33489 −2.90768 3.83089
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)
\(23\) \(-1\)

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 8993.2.a.g 9
23.b odd 2 1 391.2.a.d 9
69.c even 2 1 3519.2.a.r 9
92.b even 2 1 6256.2.a.bc 9
115.c odd 2 1 9775.2.a.n 9
391.c odd 2 1 6647.2.a.e 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
391.2.a.d 9 23.b odd 2 1
3519.2.a.r 9 69.c even 2 1
6256.2.a.bc 9 92.b even 2 1
6647.2.a.e 9 391.c odd 2 1
8993.2.a.g 9 1.a even 1 1 trivial
9775.2.a.n 9 115.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8993))\):

\( T_{2}^{9} - 2T_{2}^{8} - 12T_{2}^{7} + 23T_{2}^{6} + 43T_{2}^{5} - 79T_{2}^{4} - 43T_{2}^{3} + 78T_{2}^{2} + 11T_{2} - 21 \) Copy content Toggle raw display
\( T_{5}^{9} + 7T_{5}^{8} + T_{5}^{7} - 92T_{5}^{6} - 216T_{5}^{5} - 15T_{5}^{4} + 421T_{5}^{3} + 391T_{5}^{2} + 64T_{5} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 2 T^{8} - 12 T^{7} + 23 T^{6} + \cdots - 21 \) Copy content Toggle raw display
$3$ \( T^{9} - 2 T^{8} - 20 T^{7} + 36 T^{6} + \cdots - 64 \) Copy content Toggle raw display
$5$ \( T^{9} + 7 T^{8} + T^{7} - 92 T^{6} + \cdots - 3 \) Copy content Toggle raw display
$7$ \( T^{9} + 3 T^{8} - 27 T^{7} + \cdots + 1493 \) Copy content Toggle raw display
$11$ \( T^{9} + 11 T^{8} + 11 T^{7} + \cdots - 3723 \) Copy content Toggle raw display
$13$ \( T^{9} + 9 T^{8} - 35 T^{7} + \cdots + 5161 \) Copy content Toggle raw display
$17$ \( (T - 1)^{9} \) Copy content Toggle raw display
$19$ \( T^{9} - 4 T^{8} - 48 T^{7} + \cdots - 7232 \) Copy content Toggle raw display
$23$ \( T^{9} \) Copy content Toggle raw display
$29$ \( T^{9} - 24 T^{8} + 58 T^{7} + \cdots + 265536 \) Copy content Toggle raw display
$31$ \( T^{9} + 6 T^{8} - 72 T^{7} + \cdots + 4672 \) Copy content Toggle raw display
$37$ \( T^{9} + 15 T^{8} - 39 T^{7} + \cdots + 42731 \) Copy content Toggle raw display
$41$ \( T^{9} - 14 T^{8} - 166 T^{7} + \cdots + 2297664 \) Copy content Toggle raw display
$43$ \( T^{9} - 2 T^{8} - 178 T^{7} + \cdots + 6976 \) Copy content Toggle raw display
$47$ \( T^{9} - 9 T^{8} - 153 T^{7} + \cdots + 123201 \) Copy content Toggle raw display
$53$ \( T^{9} + 8 T^{8} - 212 T^{7} + \cdots + 8671296 \) Copy content Toggle raw display
$59$ \( T^{9} + 7 T^{8} - 177 T^{7} + \cdots - 1114527 \) Copy content Toggle raw display
$61$ \( T^{9} + 17 T^{8} - 5 T^{7} - 808 T^{6} + \cdots + 133 \) Copy content Toggle raw display
$67$ \( T^{9} + 6 T^{8} - 288 T^{7} + \cdots - 60736 \) Copy content Toggle raw display
$71$ \( T^{9} - 20 T^{8} - 174 T^{7} + \cdots - 13311168 \) Copy content Toggle raw display
$73$ \( T^{9} + 2 T^{8} - 314 T^{7} + \cdots - 962496 \) Copy content Toggle raw display
$79$ \( T^{9} + 7 T^{8} - 393 T^{7} + \cdots - 10331077 \) Copy content Toggle raw display
$83$ \( T^{9} + 6 T^{8} - 590 T^{7} + \cdots + 998968512 \) Copy content Toggle raw display
$89$ \( T^{9} + 34 T^{8} + \cdots - 207748032 \) Copy content Toggle raw display
$97$ \( T^{9} + 25 T^{8} - 135 T^{7} + \cdots + 87671729 \) Copy content Toggle raw display
show more
show less