Properties

Label 8085.2.a.bz
Level $8085$
Weight $2$
Character orbit 8085.a
Self dual yes
Analytic conductor $64.559$
Analytic rank $1$
Dimension $6$
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] = [8085,2,Mod(1,8085)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8085, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8085.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8085.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: \(64.5590500342\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.127775712.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 9x^{4} + 20x^{2} - 2x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1155)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - q^{5} - \beta_1 q^{6} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - q^{5} - \beta_1 q^{6} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{8} + q^{9} + \beta_1 q^{10} + q^{11} + (\beta_{2} + 1) q^{12} + ( - \beta_{4} + \beta_1 - 1) q^{13} - q^{15} + (\beta_{4} + \beta_1) q^{16} + (\beta_{5} - \beta_{3} + \beta_1) q^{17} - \beta_1 q^{18} + (\beta_{3} - 2) q^{19} + ( - \beta_{2} - 1) q^{20} - \beta_1 q^{22} + (\beta_{3} + \beta_1 - 1) q^{23} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{24} + q^{25} + (\beta_{5} + 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{26} + q^{27} + (\beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{29} + \beta_1 q^{30} + (\beta_{5} - \beta_{4} - \beta_{2} - 3) q^{31} + (\beta_{5} - \beta_{3} + \beta_1 - 2) q^{32} + q^{33} + (\beta_{4} - 2 \beta_{2} + \beta_1 - 2) q^{34} + (\beta_{2} + 1) q^{36} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{37} + ( - \beta_{4} + \beta_1) q^{38} + ( - \beta_{4} + \beta_1 - 1) q^{39} + (\beta_{5} + \beta_{3} + \beta_1) q^{40} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} + 1) q^{41} + (\beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 - 1) q^{43} + (\beta_{2} + 1) q^{44} - q^{45} + ( - \beta_{4} - \beta_{2} - 3) q^{46} + ( - \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 4) q^{47} + (\beta_{4} + \beta_1) q^{48} - \beta_1 q^{50} + (\beta_{5} - \beta_{3} + \beta_1) q^{51} + (2 \beta_{5} - \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{52} + ( - 2 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1) q^{53} - \beta_1 q^{54} - q^{55} + (\beta_{3} - 2) q^{57} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{58} + (2 \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{59} + ( - \beta_{2} - 1) q^{60} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2) q^{61} + (2 \beta_{5} + 4 \beta_{3} - 2 \beta_{2} + 6 \beta_1) q^{62} + ( - \beta_{4} - 2 \beta_{2} + \beta_1 - 2) q^{64} + (\beta_{4} - \beta_1 + 1) q^{65} - \beta_1 q^{66} + (\beta_{4} + \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 3) q^{67} + ( - \beta_{5} + \beta_{3} + 3 \beta_1 - 2) q^{68} + (\beta_{3} + \beta_1 - 1) q^{69} + (\beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_1 + 2) q^{71} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{72} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 1) q^{73} + ( - \beta_{5} + \beta_{4} - 5 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 3) q^{74} + q^{75} + (\beta_{5} + \beta_{3} - 2 \beta_{2} + \beta_1) q^{76} + (\beta_{5} + 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{78} + ( - \beta_{5} - 2 \beta_{2} - 3 \beta_1 - 2) q^{79} + ( - \beta_{4} - \beta_1) q^{80} + q^{81} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - \beta_1 - 1) q^{82} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_1 - 2) q^{83} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{85} + (\beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_1 + 6) q^{86} + (\beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{87} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{88} + (\beta_{5} + 5 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{89} + \beta_1 q^{90} + (2 \beta_{5} + 2 \beta_{3} - \beta_{2} + 4 \beta_1 + 1) q^{92} + (\beta_{5} - \beta_{4} - \beta_{2} - 3) q^{93} + ( - 2 \beta_{5} + 3 \beta_{4} - \beta_{2} + 2 \beta_1 - 1) q^{94} + ( - \beta_{3} + 2) q^{95} + (\beta_{5} - \beta_{3} + \beta_1 - 2) q^{96} + (\beta_{5} + \beta_{3} - \beta_{2} - 1) q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 6 q^{4} - 6 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + 6 q^{4} - 6 q^{5} + 6 q^{9} + 6 q^{11} + 6 q^{12} - 4 q^{13} - 6 q^{15} - 2 q^{16} + 2 q^{17} - 13 q^{19} - 6 q^{20} - 7 q^{23} + 6 q^{25} - 26 q^{26} + 6 q^{27} + 3 q^{29} - 15 q^{31} - 10 q^{32} + 6 q^{33} - 14 q^{34} + 6 q^{36} - 11 q^{37} + 2 q^{38} - 4 q^{39} + 3 q^{41} - 4 q^{43} + 6 q^{44} - 6 q^{45} - 16 q^{46} - 19 q^{47} - 2 q^{48} + 2 q^{51} - 16 q^{52} - q^{53} - 6 q^{55} - 13 q^{57} - 22 q^{59} - 6 q^{60} - 14 q^{61} - 2 q^{62} - 10 q^{64} + 4 q^{65} - 21 q^{67} - 14 q^{68} - 7 q^{69} + 8 q^{71} - 11 q^{73} + 20 q^{74} + 6 q^{75} - 26 q^{78} - 13 q^{79} + 2 q^{80} + 6 q^{81} - 6 q^{82} - 10 q^{83} - 2 q^{85} + 36 q^{86} + 3 q^{87} - 28 q^{89} + 6 q^{92} - 15 q^{93} - 14 q^{94} + 13 q^{95} - 10 q^{96} - 6 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 9x^{4} + 20x^{2} - 2x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 7\nu^{3} + 8\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 6\nu^{2} - \nu + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 9\nu^{3} - 18\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 6\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{5} + 9\beta_{3} + 27\beta _1 + 2 \) 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
2.45865
1.49028
0.816751
−0.638364
−1.79707
−2.33025
−2.45865 1.00000 4.04495 −1.00000 −2.45865 0 −5.02782 1.00000 2.45865
1.2 −1.49028 1.00000 0.220943 −1.00000 −1.49028 0 2.65130 1.00000 1.49028
1.3 −0.816751 1.00000 −1.33292 −1.00000 −0.816751 0 2.72216 1.00000 0.816751
1.4 0.638364 1.00000 −1.59249 −1.00000 0.638364 0 −2.29332 1.00000 −0.638364
1.5 1.79707 1.00000 1.22946 −1.00000 1.79707 0 −1.38471 1.00000 −1.79707
1.6 2.33025 1.00000 3.43006 −1.00000 2.33025 0 3.33238 1.00000 −2.33025
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)

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 8085.2.a.bz 6
7.b odd 2 1 8085.2.a.bx 6
7.c even 3 2 1155.2.q.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.q.h 12 7.c even 3 2
8085.2.a.bx 6 7.b odd 2 1
8085.2.a.bz 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):

\( T_{2}^{6} - 9T_{2}^{4} + 20T_{2}^{2} + 2T_{2} - 8 \) Copy content Toggle raw display
\( T_{13}^{6} + 4T_{13}^{5} - 42T_{13}^{4} - 160T_{13}^{3} + 37T_{13}^{2} + 20T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{6} - 2T_{17}^{5} - 53T_{17}^{4} - 8T_{17}^{3} + 600T_{17}^{2} + 288T_{17} - 1552 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 9 T^{4} + 20 T^{2} + 2 T - 8 \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T - 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 4 T^{5} - 42 T^{4} - 160 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$17$ \( T^{6} - 2 T^{5} - 53 T^{4} + \cdots - 1552 \) Copy content Toggle raw display
$19$ \( T^{6} + 13 T^{5} + 57 T^{4} + 89 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$23$ \( T^{6} + 7 T^{5} - 76 T^{3} - 103 T^{2} + \cdots + 2 \) Copy content Toggle raw display
$29$ \( T^{6} - 3 T^{5} - 90 T^{4} + \cdots + 5238 \) Copy content Toggle raw display
$31$ \( T^{6} + 15 T^{5} + 28 T^{4} + \cdots - 704 \) Copy content Toggle raw display
$37$ \( T^{6} + 11 T^{5} - 89 T^{4} + \cdots + 39686 \) Copy content Toggle raw display
$41$ \( T^{6} - 3 T^{5} - 77 T^{4} + 97 T^{3} + \cdots - 384 \) Copy content Toggle raw display
$43$ \( T^{6} + 4 T^{5} - 65 T^{4} - 404 T^{3} + \cdots + 293 \) Copy content Toggle raw display
$47$ \( T^{6} + 19 T^{5} + 14 T^{4} + \cdots + 12074 \) Copy content Toggle raw display
$53$ \( T^{6} + T^{5} - 123 T^{4} + 315 T^{3} + \cdots + 8128 \) Copy content Toggle raw display
$59$ \( T^{6} + 22 T^{5} + 49 T^{4} + \cdots + 171984 \) Copy content Toggle raw display
$61$ \( T^{6} + 14 T^{5} - 136 T^{4} + \cdots - 134144 \) Copy content Toggle raw display
$67$ \( T^{6} + 21 T^{5} + 68 T^{4} + \cdots + 40744 \) Copy content Toggle raw display
$71$ \( T^{6} - 8 T^{5} - 139 T^{4} + 1236 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$73$ \( T^{6} + 11 T^{5} - 210 T^{4} + \cdots - 108208 \) Copy content Toggle raw display
$79$ \( T^{6} + 13 T^{5} - 68 T^{4} + \cdots - 256 \) Copy content Toggle raw display
$83$ \( T^{6} + 10 T^{5} - 244 T^{4} + \cdots + 6912 \) Copy content Toggle raw display
$89$ \( T^{6} + 28 T^{5} + 84 T^{4} + \cdots - 48672 \) Copy content Toggle raw display
$97$ \( T^{6} + 6 T^{5} - 17 T^{4} - 146 T^{3} + \cdots + 556 \) Copy content Toggle raw display
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