Properties

Label 8993.2.a.c
Level $8993$
Weight $2$
Character orbit 8993.a
Self dual yes
Analytic conductor $71.809$
Analytic rank $1$
Dimension $3$
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] = [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: \(3\)
Coefficient field: 3.3.257.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 2 q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} + 1) q^{5} + 2 \beta_1 q^{6} - \beta_1 q^{7} - \beta_{2} q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 2 q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} + 1) q^{5} + 2 \beta_1 q^{6} - \beta_1 q^{7} - \beta_{2} q^{8} + q^{9} + ( - \beta_{2} - 2 \beta_1) q^{10} + (\beta_{2} - \beta_1) q^{11} + ( - 2 \beta_{2} - 2) q^{12} + (2 \beta_{2} + \beta_1) q^{13} + (\beta_{2} + 3) q^{14} + ( - 2 \beta_{2} - 2) q^{15} + ( - \beta_{2} + \beta_1 - 2) q^{16} + q^{17} - \beta_1 q^{18} + ( - 2 \beta_{2} + 2 \beta_1) q^{19} + (\beta_{2} + \beta_1 + 4) q^{20} + 2 \beta_1 q^{21} + ( - \beta_1 + 3) q^{22} + 2 \beta_{2} q^{24} + (\beta_{2} + \beta_1 - 1) q^{25} + ( - 3 \beta_{2} - 2 \beta_1 - 3) q^{26} + 4 q^{27} + ( - \beta_{2} - 2 \beta_1) q^{28} + (2 \beta_{2} - 4) q^{29} + (2 \beta_{2} + 4 \beta_1) q^{30} + ( - 2 \beta_1 - 2) q^{31} + (2 \beta_{2} + 3 \beta_1 - 3) q^{32} + ( - 2 \beta_{2} + 2 \beta_1) q^{33} - \beta_1 q^{34} + ( - \beta_{2} - 2 \beta_1) q^{35} + (\beta_{2} + 1) q^{36} + (3 \beta_{2} - \beta_1 + 2) q^{37} + (2 \beta_1 - 6) q^{38} + ( - 4 \beta_{2} - 2 \beta_1) q^{39} + ( - \beta_1 - 3) q^{40} - 8 q^{41} + ( - 2 \beta_{2} - 6) q^{42} + (4 \beta_1 - 2) q^{43} + ( - \beta_{2} - \beta_1 + 3) q^{44} + (\beta_{2} + 1) q^{45} + ( - \beta_{2} - 3 \beta_1 + 4) q^{47} + (2 \beta_{2} - 2 \beta_1 + 4) q^{48} + (\beta_{2} - 4) q^{49} + ( - 2 \beta_{2} - 3) q^{50} - 2 q^{51} + (\beta_{2} + 4 \beta_1 + 6) q^{52} + ( - 4 \beta_{2} + 2 \beta_1 + 2) q^{53} - 4 \beta_1 q^{54} + ( - \beta_{2} - \beta_1 + 3) q^{55} + (\beta_{2} + \beta_1) q^{56} + (4 \beta_{2} - 4 \beta_1) q^{57} + ( - 2 \beta_{2} + 2 \beta_1) q^{58} + ( - 3 \beta_{2} + 6 \beta_1 + 1) q^{59} + ( - 2 \beta_{2} - 2 \beta_1 - 8) q^{60} + (2 \beta_{2} - \beta_1 - 4) q^{61} + (2 \beta_{2} + 2 \beta_1 + 6) q^{62} - \beta_1 q^{63} + ( - 3 \beta_{2} - \beta_1 - 5) q^{64} + (\beta_{2} + 4 \beta_1 + 6) q^{65} + (2 \beta_1 - 6) q^{66} + ( - 4 \beta_{2} - 2 \beta_1 + 2) q^{67} + (\beta_{2} + 1) q^{68} + (3 \beta_{2} + \beta_1 + 6) q^{70} + (2 \beta_{2} - 6) q^{71} - \beta_{2} q^{72} + ( - 4 \beta_{2} - 2 \beta_1 + 4) q^{73} + ( - 2 \beta_{2} - 5 \beta_1 + 3) q^{74} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{75} + (2 \beta_{2} + 2 \beta_1 - 6) q^{76} + ( - \beta_1 + 3) q^{77} + (6 \beta_{2} + 4 \beta_1 + 6) q^{78} + (\beta_{2} + 2 \beta_1 + 5) q^{79} + ( - \beta_{2} + \beta_1 - 5) q^{80} - 11 q^{81} + 8 \beta_1 q^{82} + (2 \beta_{2} - 6) q^{83} + (2 \beta_{2} + 4 \beta_1) q^{84} + (\beta_{2} + 1) q^{85} + ( - 4 \beta_{2} + 2 \beta_1 - 12) q^{86} + ( - 4 \beta_{2} + 8) q^{87} + (2 \beta_{2} - 3) q^{88} + (4 \beta_1 + 4) q^{89} + ( - \beta_{2} - 2 \beta_1) q^{90} + ( - 3 \beta_{2} - 2 \beta_1 - 3) q^{91} + (4 \beta_1 + 4) q^{93} + (4 \beta_{2} - 3 \beta_1 + 9) q^{94} + (2 \beta_{2} + 2 \beta_1 - 6) q^{95} + ( - 4 \beta_{2} - 6 \beta_1 + 6) q^{96} + ( - 7 \beta_{2} - 3 \beta_1 + 2) q^{97} + ( - \beta_{2} + 3 \beta_1) q^{98} + (\beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - 6 q^{3} + 3 q^{4} + 3 q^{5} + 2 q^{6} - q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - 6 q^{3} + 3 q^{4} + 3 q^{5} + 2 q^{6} - q^{7} + 3 q^{9} - 2 q^{10} - q^{11} - 6 q^{12} + q^{13} + 9 q^{14} - 6 q^{15} - 5 q^{16} + 3 q^{17} - q^{18} + 2 q^{19} + 13 q^{20} + 2 q^{21} + 8 q^{22} - 2 q^{25} - 11 q^{26} + 12 q^{27} - 2 q^{28} - 12 q^{29} + 4 q^{30} - 8 q^{31} - 6 q^{32} + 2 q^{33} - q^{34} - 2 q^{35} + 3 q^{36} + 5 q^{37} - 16 q^{38} - 2 q^{39} - 10 q^{40} - 24 q^{41} - 18 q^{42} - 2 q^{43} + 8 q^{44} + 3 q^{45} + 9 q^{47} + 10 q^{48} - 12 q^{49} - 9 q^{50} - 6 q^{51} + 22 q^{52} + 8 q^{53} - 4 q^{54} + 8 q^{55} + q^{56} - 4 q^{57} + 2 q^{58} + 9 q^{59} - 26 q^{60} - 13 q^{61} + 20 q^{62} - q^{63} - 16 q^{64} + 22 q^{65} - 16 q^{66} + 4 q^{67} + 3 q^{68} + 19 q^{70} - 18 q^{71} + 10 q^{73} + 4 q^{74} + 4 q^{75} - 16 q^{76} + 8 q^{77} + 22 q^{78} + 17 q^{79} - 14 q^{80} - 33 q^{81} + 8 q^{82} - 18 q^{83} + 4 q^{84} + 3 q^{85} - 34 q^{86} + 24 q^{87} - 9 q^{88} + 16 q^{89} - 2 q^{90} - 11 q^{91} + 16 q^{93} + 24 q^{94} - 16 q^{95} + 12 q^{96} + 3 q^{97} + 3 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
2.19869
0.713538
−1.91223
−2.19869 −2.00000 2.83424 2.83424 4.39738 −2.19869 −1.83424 1.00000 −6.23163
1.2 −0.713538 −2.00000 −1.49086 −1.49086 1.42708 −0.713538 2.49086 1.00000 1.06379
1.3 1.91223 −2.00000 1.65662 1.65662 −3.82446 1.91223 −0.656620 1.00000 3.16784
\(n\): e.g. 2-40 or 990-1000
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.c 3
23.b odd 2 1 391.2.a.b 3
69.c even 2 1 3519.2.a.l 3
92.b even 2 1 6256.2.a.s 3
115.c odd 2 1 9775.2.a.m 3
391.c odd 2 1 6647.2.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
391.2.a.b 3 23.b odd 2 1
3519.2.a.l 3 69.c even 2 1
6256.2.a.s 3 92.b even 2 1
6647.2.a.d 3 391.c odd 2 1
8993.2.a.c 3 1.a even 1 1 trivial
9775.2.a.m 3 115.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8993))\):

\( T_{2}^{3} + T_{2}^{2} - 4T_{2} - 3 \) Copy content Toggle raw display
\( T_{5}^{3} - 3T_{5}^{2} - 2T_{5} + 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 4T - 3 \) Copy content Toggle raw display
$3$ \( (T + 2)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 3 T^{2} - 2 T + 7 \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 4T - 3 \) Copy content Toggle raw display
$11$ \( T^{3} + T^{2} - 8T - 3 \) Copy content Toggle raw display
$13$ \( T^{3} - T^{2} - 26 T - 15 \) Copy content Toggle raw display
$17$ \( (T - 1)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 2 T^{2} - 32 T + 24 \) Copy content Toggle raw display
$23$ \( T^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 12 T^{2} + 28 T + 8 \) Copy content Toggle raw display
$31$ \( T^{3} + 8 T^{2} + 4 T - 40 \) Copy content Toggle raw display
$37$ \( T^{3} - 5 T^{2} - 38 T + 193 \) Copy content Toggle raw display
$41$ \( (T + 8)^{3} \) Copy content Toggle raw display
$43$ \( T^{3} + 2 T^{2} - 68 T + 56 \) Copy content Toggle raw display
$47$ \( T^{3} - 9 T^{2} - 20 T + 175 \) Copy content Toggle raw display
$53$ \( T^{3} - 8 T^{2} - 68 T - 56 \) Copy content Toggle raw display
$59$ \( T^{3} - 9 T^{2} - 156 T + 1379 \) Copy content Toggle raw display
$61$ \( T^{3} + 13 T^{2} + 34 T + 19 \) Copy content Toggle raw display
$67$ \( T^{3} - 4 T^{2} - 100 T + 328 \) Copy content Toggle raw display
$71$ \( T^{3} + 18 T^{2} + 88 T + 120 \) Copy content Toggle raw display
$73$ \( T^{3} - 10 T^{2} - 72 T + 504 \) Copy content Toggle raw display
$79$ \( T^{3} - 17 T^{2} + 72 T - 81 \) Copy content Toggle raw display
$83$ \( T^{3} + 18 T^{2} + 88 T + 120 \) Copy content Toggle raw display
$89$ \( T^{3} - 16 T^{2} + 16 T + 320 \) Copy content Toggle raw display
$97$ \( T^{3} - 3 T^{2} - 302 T + 947 \) Copy content Toggle raw display
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