Properties

Label 8619.2.a.r
Level $8619$
Weight $2$
Character orbit 8619.a
Self dual yes
Analytic conductor $68.823$
Analytic rank $0$
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] = [8619,2,Mod(1,8619)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8619, 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("8619.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8619.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: \(68.8230615021\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + q^{3} + (\beta_{2} + \beta_1) q^{4} + \beta_{2} q^{5} + ( - \beta_{2} - 1) q^{6} + (\beta_{2} + 2) q^{7} + (\beta_{2} - 2 \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + q^{3} + (\beta_{2} + \beta_1) q^{4} + \beta_{2} q^{5} + ( - \beta_{2} - 1) q^{6} + (\beta_{2} + 2) q^{7} + (\beta_{2} - 2 \beta_1) q^{8} + q^{9} + ( - \beta_1 - 1) q^{10} + (\beta_{2} - \beta_1 - 2) q^{11} + (\beta_{2} + \beta_1) q^{12} + ( - 2 \beta_{2} - \beta_1 - 3) q^{14} + \beta_{2} q^{15} + ( - \beta_1 + 1) q^{16} - q^{17} + ( - \beta_{2} - 1) q^{18} + (\beta_{2} + 5) q^{19} + (\beta_1 + 2) q^{20} + (\beta_{2} + 2) q^{21} + (3 \beta_{2} + 2) q^{22} + (\beta_{2} + \beta_1 + 5) q^{23} + (\beta_{2} - 2 \beta_1) q^{24} + ( - \beta_{2} + \beta_1 - 4) q^{25} + q^{27} + (2 \beta_{2} + 3 \beta_1 + 2) q^{28} + (7 \beta_{2} - 4 \beta_1 + 6) q^{29} + ( - \beta_1 - 1) q^{30} + ( - 2 \beta_{2} - 2 \beta_1 + 1) q^{31} + ( - 2 \beta_{2} + 5 \beta_1) q^{32} + (\beta_{2} - \beta_1 - 2) q^{33} + (\beta_{2} + 1) q^{34} + (\beta_{2} + \beta_1 + 1) q^{35} + (\beta_{2} + \beta_1) q^{36} + (4 \beta_1 + 3) q^{37} + ( - 5 \beta_{2} - \beta_1 - 6) q^{38} + ( - 3 \beta_{2} + \beta_1 - 1) q^{40} + (3 \beta_{2} + 3 \beta_1 + 1) q^{41} + ( - 2 \beta_{2} - \beta_1 - 3) q^{42} + (2 \beta_{2} - 5 \beta_1 - 4) q^{43} + ( - 4 \beta_{2} - \beta_1 - 1) q^{44} + \beta_{2} q^{45} + ( - 6 \beta_{2} - 2 \beta_1 - 7) q^{46} + (3 \beta_{2} + 3 \beta_1) q^{47} + ( - \beta_1 + 1) q^{48} + (3 \beta_{2} + \beta_1 - 2) q^{49} + (3 \beta_{2} + 4) q^{50} - q^{51} + ( - \beta_{2} - 3 \beta_1 + 5) q^{53} + ( - \beta_{2} - 1) q^{54} + ( - 4 \beta_{2} + \beta_1) q^{55} + ( - \beta_{2} - 3 \beta_1 - 1) q^{56} + (\beta_{2} + 5) q^{57} + ( - 2 \beta_{2} - 3 \beta_1 - 9) q^{58} + (6 \beta_{2} + 6) q^{59} + (\beta_1 + 2) q^{60} + (4 \beta_{2} - 5 \beta_1 - 3) q^{61} + (\beta_{2} + 4 \beta_1 + 3) q^{62} + (\beta_{2} + 2) q^{63} + ( - 5 \beta_{2} - \beta_1 - 5) q^{64} + (3 \beta_{2} + 2) q^{66} + ( - 3 \beta_{2} - 4 \beta_1 - 1) q^{67} + ( - \beta_{2} - \beta_1) q^{68} + (\beta_{2} + \beta_1 + 5) q^{69} + ( - 2 \beta_{2} - 2 \beta_1 - 3) q^{70} + ( - 5 \beta_{2} + \beta_1 + 5) q^{71} + (\beta_{2} - 2 \beta_1) q^{72} + (3 \beta_{2} - 3 \beta_1 + 5) q^{73} + ( - 7 \beta_{2} - 4 \beta_1 - 7) q^{74} + ( - \beta_{2} + \beta_1 - 4) q^{75} + (5 \beta_{2} + 6 \beta_1 + 2) q^{76} + ( - 2 \beta_{2} - \beta_1 - 4) q^{77} + (6 \beta_{2} + 2 \beta_1 - 7) q^{79} - q^{80} + q^{81} + ( - 4 \beta_{2} - 6 \beta_1 - 7) q^{82} + ( - 13 \beta_{2} + 5 \beta_1 - 8) q^{83} + (2 \beta_{2} + 3 \beta_1 + 2) q^{84} - \beta_{2} q^{85} + (9 \beta_{2} + 3 \beta_1 + 7) q^{86} + (7 \beta_{2} - 4 \beta_1 + 6) q^{87} + ( - 4 \beta_{2} + 5 \beta_1 + 2) q^{88} + ( - 5 \beta_{2} - 5 \beta_1) q^{89} + ( - \beta_1 - 1) q^{90} + (7 \beta_{2} + 6 \beta_1 + 5) q^{92} + ( - 2 \beta_{2} - 2 \beta_1 + 1) q^{93} + ( - 3 \beta_{2} - 6 \beta_1 - 6) q^{94} + (4 \beta_{2} + \beta_1 + 1) q^{95} + ( - 2 \beta_{2} + 5 \beta_1) q^{96} + (\beta_{2} - 2 \beta_1 + 11) q^{97} + (\beta_{2} - 4 \beta_1 - 2) q^{98} + (\beta_{2} - \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 3 q^{3} - q^{5} - 2 q^{6} + 5 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 3 q^{3} - q^{5} - 2 q^{6} + 5 q^{7} - 3 q^{8} + 3 q^{9} - 4 q^{10} - 8 q^{11} - 8 q^{14} - q^{15} + 2 q^{16} - 3 q^{17} - 2 q^{18} + 14 q^{19} + 7 q^{20} + 5 q^{21} + 3 q^{22} + 15 q^{23} - 3 q^{24} - 10 q^{25} + 3 q^{27} + 7 q^{28} + 7 q^{29} - 4 q^{30} + 3 q^{31} + 7 q^{32} - 8 q^{33} + 2 q^{34} + 3 q^{35} + 13 q^{37} - 14 q^{38} + q^{40} + 3 q^{41} - 8 q^{42} - 19 q^{43} - q^{45} - 17 q^{46} + 2 q^{48} - 8 q^{49} + 9 q^{50} - 3 q^{51} + 13 q^{53} - 2 q^{54} + 5 q^{55} - 5 q^{56} + 14 q^{57} - 28 q^{58} + 12 q^{59} + 7 q^{60} - 18 q^{61} + 12 q^{62} + 5 q^{63} - 11 q^{64} + 3 q^{66} - 4 q^{67} + 15 q^{69} - 9 q^{70} + 21 q^{71} - 3 q^{72} + 9 q^{73} - 18 q^{74} - 10 q^{75} + 7 q^{76} - 11 q^{77} - 25 q^{79} - 3 q^{80} + 3 q^{81} - 23 q^{82} - 6 q^{83} + 7 q^{84} + q^{85} + 15 q^{86} + 7 q^{87} + 15 q^{88} - 4 q^{90} + 14 q^{92} + 3 q^{93} - 21 q^{94} + 7 q^{96} + 30 q^{97} - 11 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.80194
−1.24698
0.445042
−2.24698 1.00000 3.04892 1.24698 −2.24698 3.24698 −2.35690 1.00000 −2.80194
1.2 −0.554958 1.00000 −1.69202 −0.445042 −0.554958 1.55496 2.04892 1.00000 0.246980
1.3 0.801938 1.00000 −1.35690 −1.80194 0.801938 0.198062 −2.69202 1.00000 −1.44504
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(1\)
\(17\) \(1\)

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 8619.2.a.r 3
13.b even 2 1 8619.2.a.t yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8619.2.a.r 3 1.a even 1 1 trivial
8619.2.a.t yes 3 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):

\( T_{2}^{3} + 2T_{2}^{2} - T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{3} + T_{5}^{2} - 2T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{3} - 5T_{7}^{2} + 6T_{7} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 2T^{2} - T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$7$ \( T^{3} - 5 T^{2} + 6 T - 1 \) Copy content Toggle raw display
$11$ \( T^{3} + 8 T^{2} + 19 T + 13 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( (T + 1)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 14 T^{2} + 63 T - 91 \) Copy content Toggle raw display
$23$ \( T^{3} - 15 T^{2} + 68 T - 97 \) Copy content Toggle raw display
$29$ \( T^{3} - 7 T^{2} - 70 T + 497 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} - 25 T + 83 \) Copy content Toggle raw display
$37$ \( T^{3} - 13 T^{2} + 19 T + 97 \) Copy content Toggle raw display
$41$ \( T^{3} - 3 T^{2} - 60 T - 127 \) Copy content Toggle raw display
$43$ \( T^{3} + 19 T^{2} + 76 T - 139 \) Copy content Toggle raw display
$47$ \( T^{3} - 63T - 189 \) Copy content Toggle raw display
$53$ \( T^{3} - 13 T^{2} + 26 T + 83 \) Copy content Toggle raw display
$59$ \( T^{3} - 12 T^{2} - 36 T + 216 \) Copy content Toggle raw display
$61$ \( T^{3} + 18 T^{2} + 59 T - 127 \) Copy content Toggle raw display
$67$ \( T^{3} + 4 T^{2} - 81 T + 167 \) Copy content Toggle raw display
$71$ \( T^{3} - 21 T^{2} + 98 T - 49 \) Copy content Toggle raw display
$73$ \( T^{3} - 9 T^{2} + 6 T + 43 \) Copy content Toggle raw display
$79$ \( T^{3} + 25 T^{2} + 87 T - 841 \) Copy content Toggle raw display
$83$ \( T^{3} + 6 T^{2} - 289 T - 2267 \) Copy content Toggle raw display
$89$ \( T^{3} - 175T + 875 \) Copy content Toggle raw display
$97$ \( T^{3} - 30 T^{2} + 293 T - 937 \) Copy content Toggle raw display
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