Properties

Label 8085.2.a.bv
Level $8085$
Weight $2$
Character orbit 8085.a
Self dual yes
Analytic conductor $64.559$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.352076.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 8x^{3} + 3x^{2} + 8x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} + q^{5} + \beta_1 q^{6} + ( - \beta_{4} - 2 \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} + q^{5} + \beta_1 q^{6} + ( - \beta_{4} - 2 \beta_1) q^{8} + q^{9} - \beta_1 q^{10} + q^{11} + ( - \beta_{2} - 2) q^{12} + ( - \beta_{4} - \beta_1 - 2) q^{13} - q^{15} + (\beta_{3} + \beta_{2} + 3) q^{16} + (\beta_{3} - \beta_{2}) q^{17} - \beta_1 q^{18} + (\beta_{2} - 1) q^{19} + (\beta_{2} + 2) q^{20} - \beta_1 q^{22} + ( - \beta_{2} - 2 \beta_1 - 1) q^{23} + (\beta_{4} + 2 \beta_1) q^{24} + q^{25} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 + 3) q^{26} - q^{27} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 2) q^{29} + \beta_1 q^{30} + (\beta_{4} - \beta_1 - 2) q^{31} + ( - \beta_{4} + \beta_{3} - 2 \beta_1 + 1) q^{32} - q^{33} + ( - \beta_{4} + \beta_{3} + \beta_1 + 1) q^{34} + (\beta_{2} + 2) q^{36} + (\beta_{4} - \beta_{3} + \beta_1 + 1) q^{37} + ( - \beta_{4} - \beta_1) q^{38} + (\beta_{4} + \beta_1 + 2) q^{39} + ( - \beta_{4} - 2 \beta_1) q^{40} + ( - \beta_{4} - \beta_{3} - 3 \beta_1 - 1) q^{41} + ( - \beta_{4} - 3 \beta_{2} - \beta_1 - 1) q^{43} + (\beta_{2} + 2) q^{44} + q^{45} + (\beta_{4} + 2 \beta_{2} + 3 \beta_1 + 8) q^{46} + (\beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{47} + ( - \beta_{3} - \beta_{2} - 3) q^{48} - \beta_1 q^{50} + ( - \beta_{3} + \beta_{2}) q^{51} + ( - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 6 \beta_1 - 3) q^{52} + ( - \beta_{3} - \beta_{2} - 2) q^{53} + \beta_1 q^{54} + q^{55} + ( - \beta_{2} + 1) q^{57} + ( - 3 \beta_{4} - 2 \beta_{2} - 5 \beta_1 - 2) q^{58} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1) q^{59} + ( - \beta_{2} - 2) q^{60} + (2 \beta_{4} - \beta_{2} - 3) q^{61} + ( - \beta_{3} + 2 \beta_1 + 5) q^{62} + ( - 2 \beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{64} + ( - \beta_{4} - \beta_1 - 2) q^{65} + \beta_1 q^{66} + (\beta_{3} - 2 \beta_1 + 3) q^{67} + ( - 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1 - 4) q^{68} + (\beta_{2} + 2 \beta_1 + 1) q^{69} + ( - \beta_{4} - \beta_{3} + 3 \beta_1 - 1) q^{71} + ( - \beta_{4} - 2 \beta_1) q^{72} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 1) q^{73} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 4) q^{74} - q^{75} + (\beta_{3} + 5) q^{76} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 3) q^{78} + ( - \beta_{4} - \beta_1 + 8) q^{79} + (\beta_{3} + \beta_{2} + 3) q^{80} + q^{81} + (2 \beta_{4} + 4 \beta_{2} + 2 \beta_1 + 10) q^{82} + (2 \beta_{4} - \beta_{2} + 2 \beta_1 + 5) q^{83} + (\beta_{3} - \beta_{2}) q^{85} + (3 \beta_{4} + \beta_{3} + 2 \beta_{2} + 7 \beta_1 + 3) q^{86} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{87} + ( - \beta_{4} - 2 \beta_1) q^{88} + ( - \beta_{4} + 3 \beta_{2} + 3 \beta_1 + 3) q^{89} - \beta_1 q^{90} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 8 \beta_1 - 9) q^{92} + ( - \beta_{4} + \beta_1 + 2) q^{93} + ( - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 4) q^{94} + (\beta_{2} - 1) q^{95} + (\beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{96} + ( - \beta_{4} - 5 \beta_{2} - \beta_1 - 1) q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} - 5 q^{3} + 9 q^{4} + 5 q^{5} - q^{6} + 3 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{2} - 5 q^{3} + 9 q^{4} + 5 q^{5} - q^{6} + 3 q^{8} + 5 q^{9} + q^{10} + 5 q^{11} - 9 q^{12} - 8 q^{13} - 5 q^{15} + 13 q^{16} + q^{18} - 6 q^{19} + 9 q^{20} + q^{22} - 2 q^{23} - 3 q^{24} + 5 q^{25} + 10 q^{26} - 5 q^{27} + 6 q^{29} - q^{30} - 10 q^{31} + 7 q^{32} - 5 q^{33} + 4 q^{34} + 9 q^{36} + 4 q^{37} + 2 q^{38} + 8 q^{39} + 3 q^{40} + 9 q^{44} + 5 q^{45} + 34 q^{46} + 2 q^{47} - 13 q^{48} + q^{50} - 6 q^{52} - 8 q^{53} - q^{54} + 5 q^{55} + 6 q^{57} + 4 q^{59} - 9 q^{60} - 16 q^{61} + 24 q^{62} + 13 q^{64} - 8 q^{65} - q^{66} + 16 q^{67} - 18 q^{68} + 2 q^{69} - 6 q^{71} + 3 q^{72} - 2 q^{73} - 18 q^{74} - 5 q^{75} + 24 q^{76} - 10 q^{78} + 42 q^{79} + 13 q^{80} + 5 q^{81} + 42 q^{82} + 22 q^{83} + 2 q^{86} - 6 q^{87} + 3 q^{88} + 10 q^{89} + q^{90} - 32 q^{92} + 10 q^{93} - 12 q^{94} - 6 q^{95} - 7 q^{96} + 2 q^{97} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 8x^{3} + 3x^{2} + 8x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} - \nu^{3} - 8\nu^{2} + 3\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - 9\nu^{2} - 5\nu + 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} + \nu^{3} - 10\nu^{2} - 9\nu + 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{4} - \nu^{3} - 15\nu^{2} - 4\nu + 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 2\beta_{2} + 2\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{4} + \beta_{3} - 4\beta_{2} - 2\beta _1 + 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{4} + 9\beta_{3} - 16\beta_{2} + 6\beta _1 + 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{4} + 7\beta_{3} - 21\beta_{2} - 4\beta _1 + 45 ) / 2 \) 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
0.567739
0.844040
3.06028
−2.27097
−1.20109
−2.52275 −1.00000 4.36426 1.00000 2.52275 0 −5.96443 1.00000 −2.52275
1.2 −1.36955 −1.00000 −0.124319 1.00000 1.36955 0 2.90937 1.00000 −1.36955
1.3 0.346466 −1.00000 −1.87996 1.00000 −0.346466 0 −1.34427 1.00000 0.346466
1.4 1.88068 −1.00000 1.53695 1.00000 −1.88068 0 −0.870842 1.00000 1.88068
1.5 2.66516 −1.00000 5.10307 1.00000 −2.66516 0 8.27017 1.00000 2.66516
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.bv 5
7.b odd 2 1 1155.2.a.w 5
21.c even 2 1 3465.2.a.bm 5
35.c odd 2 1 5775.2.a.cg 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.w 5 7.b odd 2 1
3465.2.a.bm 5 21.c even 2 1
5775.2.a.cg 5 35.c odd 2 1
8085.2.a.bv 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):

\( T_{2}^{5} - T_{2}^{4} - 9T_{2}^{3} + 7T_{2}^{2} + 16T_{2} - 6 \) Copy content Toggle raw display
\( T_{13}^{5} + 8T_{13}^{4} - 10T_{13}^{3} - 164T_{13}^{2} - 40T_{13} + 784 \) Copy content Toggle raw display
\( T_{17}^{5} - 63T_{17}^{3} - 70T_{17}^{2} + 380T_{17} + 504 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - T^{4} - 9 T^{3} + 7 T^{2} + 16 T - 6 \) Copy content Toggle raw display
$3$ \( (T + 1)^{5} \) Copy content Toggle raw display
$5$ \( (T - 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( (T - 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} + 8 T^{4} - 10 T^{3} - 164 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$17$ \( T^{5} - 63 T^{3} - 70 T^{2} + \cdots + 504 \) Copy content Toggle raw display
$19$ \( T^{5} + 6 T^{4} - 3 T^{3} - 44 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{5} + 2 T^{4} - 47 T^{3} + 76 T^{2} + \cdots - 192 \) Copy content Toggle raw display
$29$ \( T^{5} - 6 T^{4} - 87 T^{3} + \cdots - 4872 \) Copy content Toggle raw display
$31$ \( T^{5} + 10 T^{4} - 10 T^{3} + \cdots + 128 \) Copy content Toggle raw display
$37$ \( T^{5} - 4 T^{4} - 114 T^{3} + \cdots - 784 \) Copy content Toggle raw display
$41$ \( T^{5} - 122 T^{3} + 260 T^{2} + \cdots - 10128 \) Copy content Toggle raw display
$43$ \( T^{5} - 167 T^{3} + 12 T^{2} + \cdots + 11072 \) Copy content Toggle raw display
$47$ \( T^{5} - 2 T^{4} - 138 T^{3} + \cdots + 9216 \) Copy content Toggle raw display
$53$ \( T^{5} + 8 T^{4} - 71 T^{3} + \cdots + 3576 \) Copy content Toggle raw display
$59$ \( T^{5} - 4 T^{4} - 131 T^{3} + \cdots - 16704 \) Copy content Toggle raw display
$61$ \( T^{5} + 16 T^{4} - 59 T^{3} + \cdots + 17144 \) Copy content Toggle raw display
$67$ \( T^{5} - 16 T^{4} - 12 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( T^{5} + 6 T^{4} - 194 T^{3} + \cdots + 24576 \) Copy content Toggle raw display
$73$ \( T^{5} + 2 T^{4} - 244 T^{3} + \cdots + 67264 \) Copy content Toggle raw display
$79$ \( T^{5} - 42 T^{4} + 670 T^{3} + \cdots - 25216 \) Copy content Toggle raw display
$83$ \( T^{5} - 22 T^{4} + 17 T^{3} + \cdots - 17088 \) Copy content Toggle raw display
$89$ \( T^{5} - 10 T^{4} - 243 T^{3} + \cdots - 78696 \) Copy content Toggle raw display
$97$ \( T^{5} - 2 T^{4} - 427 T^{3} + \cdots + 59192 \) Copy content Toggle raw display
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