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Properties

Label 8085.2.a.cm
Level $8085$
Weight $2$
Character orbit 8085.a
Self dual yes
Analytic conductor $64.559$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 16x^{8} + 85x^{6} - 168x^{4} - 8x^{3} + 100x^{2} + 24x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 16x^{8} + 85x^{6} - 168x^{4} - 8x^{3} + 100x^{2} + 24x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 3\nu^{8} - 7\nu^{7} + 25\nu^{6} - 2\nu^{5} - 34\nu^{4} + 42\nu^{3} - 34\nu^{2} - 14\nu + 18 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{9} - 3\nu^{8} - 11\nu^{7} + 33\nu^{6} + 38\nu^{5} - 106\nu^{4} - 58\nu^{3} + 110\nu^{2} + 34\nu - 14 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{9} - 5\nu^{8} - 33\nu^{7} + 47\nu^{6} + 106\nu^{5} - 110\nu^{4} - 118\nu^{3} + 66\nu^{2} + 38\nu - 2 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{9} + 5\nu^{8} + 33\nu^{7} - 47\nu^{6} - 106\nu^{5} + 110\nu^{4} + 126\nu^{3} - 66\nu^{2} - 78\nu + 2 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} - 2\nu^{7} - 11\nu^{6} + 20\nu^{5} + 34\nu^{4} - 52\nu^{3} - 30\nu^{2} + 32\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} - \nu^{8} - 15\nu^{7} + 11\nu^{6} + 78\nu^{5} - 34\nu^{4} - 166\nu^{3} + 22\nu^{2} + 118\nu + 22 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5\nu^{9} + 3\nu^{8} + 71\nu^{7} - 33\nu^{6} - 326\nu^{5} + 114\nu^{4} + 530\nu^{3} - 134\nu^{2} - 226\nu - 10 ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - 2\beta_{4} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} + \beta_{8} + 8\beta_{6} + 10\beta_{5} - \beta_{4} - \beta_{3} + 30\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} + 10 \beta_{8} - 10 \beta_{7} + 9 \beta_{6} + 12 \beta_{5} - 20 \beta_{4} - 2 \beta_{3} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{9} + 12\beta_{8} - 2\beta_{7} + 55\beta_{6} + 81\beta_{5} - 16\beta_{4} - 13\beta_{3} + 191\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15 \beta_{9} + 80 \beta_{8} - 78 \beta_{7} + 67 \beta_{6} + 112 \beta_{5} - 164 \beta_{4} - 28 \beta_{3} + \cdots + 524 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 106 \beta_{9} + 110 \beta_{8} - 32 \beta_{7} + 369 \beta_{6} + 615 \beta_{5} - 174 \beta_{4} - 123 \beta_{3} + \cdots - 42 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−2.51164
−2.50368
−1.61819
−0.751303
−0.357712
0.0656170
1.19735
1.53283
2.26161
2.68512
−2.51164 1.00000 4.30832 −1.00000 −2.51164 0 −5.79766 1.00000 2.51164
1.2 −2.50368 1.00000 4.26844 −1.00000 −2.50368 0 −5.67945 1.00000 2.50368
1.3 −1.61819 1.00000 0.618551 −1.00000 −1.61819 0 2.23545 1.00000 1.61819
1.4 −0.751303 1.00000 −1.43554 −1.00000 −0.751303 0 2.58113 1.00000 0.751303
1.5 −0.357712 1.00000 −1.87204 −1.00000 −0.357712 0 1.38508 1.00000 0.357712
1.6 0.0656170 1.00000 −1.99569 −1.00000 0.0656170 0 −0.262186 1.00000 −0.0656170
1.7 1.19735 1.00000 −0.566344 −1.00000 1.19735 0 −3.07282 1.00000 −1.19735
1.8 1.53283 1.00000 0.349565 −1.00000 1.53283 0 −2.52983 1.00000 −1.53283
1.9 2.26161 1.00000 3.11490 −1.00000 2.26161 0 2.52147 1.00000 −2.26161
1.10 2.68512 1.00000 5.20985 −1.00000 2.68512 0 8.61882 1.00000 −2.68512
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.cm yes 10
7.b odd 2 1 8085.2.a.cl ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8085.2.a.cl ✓ 10 7.b odd 2 1
8085.2.a.cm yes 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):

\( T_{2}^{10} - 16T_{2}^{8} + 85T_{2}^{6} - 168T_{2}^{4} - 8T_{2}^{3} + 100T_{2}^{2} + 24T_{2} - 2 \) Copy content Toggle raw display
\( T_{13}^{10} - 8 T_{13}^{9} - 28 T_{13}^{8} + 376 T_{13}^{7} - 382 T_{13}^{6} - 4412 T_{13}^{5} + \cdots - 15176 \) Copy content Toggle raw display
\( T_{17}^{10} + 2 T_{17}^{9} - 71 T_{17}^{8} - 24 T_{17}^{7} + 1267 T_{17}^{6} - 1074 T_{17}^{5} + \cdots - 784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 16 T^{8} + \cdots - 2 \) Copy content Toggle raw display
$3$ \( (T - 1)^{10} \) Copy content Toggle raw display
$5$ \( (T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T + 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - 8 T^{9} + \cdots - 15176 \) Copy content Toggle raw display
$17$ \( T^{10} + 2 T^{9} + \cdots - 784 \) Copy content Toggle raw display
$19$ \( T^{10} - 2 T^{9} + \cdots + 16784 \) Copy content Toggle raw display
$23$ \( T^{10} - 10 T^{9} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{10} - 22 T^{9} + \cdots + 30724 \) Copy content Toggle raw display
$31$ \( T^{10} - 4 T^{9} + \cdots - 2454016 \) Copy content Toggle raw display
$37$ \( T^{10} - 12 T^{9} + \cdots + 254944 \) Copy content Toggle raw display
$41$ \( T^{10} + 8 T^{9} + \cdots - 1043968 \) Copy content Toggle raw display
$43$ \( T^{10} - 22 T^{9} + \cdots - 13888 \) Copy content Toggle raw display
$47$ \( T^{10} + 8 T^{9} + \cdots + 3968 \) Copy content Toggle raw display
$53$ \( T^{10} + 2 T^{9} + \cdots + 2314588 \) Copy content Toggle raw display
$59$ \( T^{10} + 30 T^{9} + \cdots + 17814404 \) Copy content Toggle raw display
$61$ \( T^{10} - 22 T^{9} + \cdots + 41744 \) Copy content Toggle raw display
$67$ \( T^{10} - 24 T^{9} + \cdots + 451712 \) Copy content Toggle raw display
$71$ \( T^{10} - 20 T^{9} + \cdots + 56096 \) Copy content Toggle raw display
$73$ \( T^{10} - 24 T^{9} + \cdots + 2735584 \) Copy content Toggle raw display
$79$ \( T^{10} - 36 T^{9} + \cdots + 42221896 \) Copy content Toggle raw display
$83$ \( T^{10} + 14 T^{9} + \cdots + 115772 \) Copy content Toggle raw display
$89$ \( T^{10} + 10 T^{9} + \cdots - 8176 \) Copy content Toggle raw display
$97$ \( T^{10} - 22 T^{9} + \cdots + 3512176 \) Copy content Toggle raw display
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