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Properties

Label 8085.2.a.cn
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: \( x^{10} - 16x^{8} + 84x^{6} - 2x^{5} - 169x^{4} + 8x^{3} + 128x^{2} - 4x - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + q^{5} - \beta_1 q^{6} + ( - \beta_{6} + \beta_{5} + \cdots - \beta_1) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + q^{5} - \beta_1 q^{6} + ( - \beta_{6} + \beta_{5} + \cdots - \beta_1) q^{8}+ \cdots + q^{99}+O(q^{100}) \) 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 \( 10 q + 10 q^{3} + 12 q^{4} + 10 q^{5} + 10 q^{9} + 10 q^{11} + 12 q^{12} + 4 q^{13} + 10 q^{15} + 24 q^{16} + 12 q^{17} + 9 q^{19} + 12 q^{20} - q^{23} + 10 q^{25} + 6 q^{26} + 10 q^{27} + 3 q^{29} + 9 q^{31} - 10 q^{32} + 10 q^{33} + 6 q^{34} + 12 q^{36} + 3 q^{37} + 42 q^{38} + 4 q^{39} + 3 q^{41} + 12 q^{44} + 10 q^{45} - 22 q^{46} + 21 q^{47} + 24 q^{48} + 12 q^{51} + 24 q^{52} - 5 q^{53} + 10 q^{55} + 9 q^{57} - 26 q^{58} + 16 q^{59} + 12 q^{60} + 20 q^{61} - 18 q^{62} + 70 q^{64} + 4 q^{65} - q^{67} + 30 q^{68} - q^{69} + 22 q^{71} - q^{73} - 34 q^{74} + 10 q^{75} - 28 q^{76} + 6 q^{78} + 19 q^{79} + 24 q^{80} + 10 q^{81} + 32 q^{82} + 20 q^{83} + 12 q^{85} + 3 q^{87} + 18 q^{89} - 8 q^{92} + 9 q^{93} - 6 q^{94} + 9 q^{95} - 10 q^{96} + 6 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 16x^{8} + 84x^{6} - 2x^{5} - 169x^{4} + 8x^{3} + 128x^{2} - 4x - 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 16\nu^{7} + 84\nu^{5} - 2\nu^{4} - 165\nu^{3} + 8\nu^{2} + 96\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} + 4\nu^{8} + 16\nu^{7} - 60\nu^{6} - 84\nu^{5} + 282\nu^{4} + 157\nu^{3} - 444\nu^{2} - 72\nu + 172 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{9} - 76\nu^{7} + 360\nu^{5} - 10\nu^{4} - 569\nu^{3} + 36\nu^{2} + 228\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{9} - 46\nu^{7} + 222\nu^{5} - 6\nu^{4} - 365\nu^{3} + 20\nu^{2} + 152\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{9} + 46\nu^{7} - 222\nu^{5} + 8\nu^{4} + 365\nu^{3} - 36\nu^{2} - 152\nu + 18 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{9} - 4\nu^{8} - 46\nu^{7} + 62\nu^{6} + 222\nu^{5} - 310\nu^{4} - 359\nu^{3} + 532\nu^{2} + 136\nu - 222 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7\nu^{9} - 4\nu^{8} - 108\nu^{7} + 62\nu^{6} + 526\nu^{5} - 318\nu^{4} - 871\nu^{3} + 562\nu^{2} + 356\nu - 230 ) / 2 \) 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
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} + 8\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{9} + \beta_{8} + 10\beta_{6} - 9\beta_{5} - 7\beta_{3} + 10\beta_{2} + 31\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{8} + 12\beta_{7} + 12\beta_{6} - \beta_{5} + 2\beta_{4} + \beta_{3} + 59\beta_{2} + \beta _1 + 104 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -15\beta_{9} + 15\beta_{8} + 86\beta_{6} - 70\beta_{5} - 46\beta_{3} + 87\beta_{2} + 208\beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15\beta_{8} + 110\beta_{7} + 112\beta_{6} - 17\beta_{5} + 31\beta_{4} + 14\beta_{3} + 436\beta_{2} + 19\beta _1 + 725 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 156 \beta_{9} + 156 \beta_{8} + 2 \beta_{7} + 703 \beta_{6} - 529 \beta_{5} - 309 \beta_{3} + \cdots + 340 \) 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
2.78919
2.20260
1.39293
0.896329
0.761016
−0.743153
−0.786121
−1.50738
−2.30699
−2.69843
−2.78919 1.00000 5.77960 1.00000 −2.78919 0 −10.5420 1.00000 −2.78919
1.2 −2.20260 1.00000 2.85145 1.00000 −2.20260 0 −1.87541 1.00000 −2.20260
1.3 −1.39293 1.00000 −0.0597583 1.00000 −1.39293 0 2.86909 1.00000 −1.39293
1.4 −0.896329 1.00000 −1.19659 1.00000 −0.896329 0 2.86520 1.00000 −0.896329
1.5 −0.761016 1.00000 −1.42085 1.00000 −0.761016 0 2.60333 1.00000 −0.761016
1.6 0.743153 1.00000 −1.44772 1.00000 0.743153 0 −2.56219 1.00000 0.743153
1.7 0.786121 1.00000 −1.38201 1.00000 0.786121 0 −2.65867 1.00000 0.786121
1.8 1.50738 1.00000 0.272195 1.00000 1.50738 0 −2.60446 1.00000 1.50738
1.9 2.30699 1.00000 3.32219 1.00000 2.30699 0 3.05028 1.00000 2.30699
1.10 2.69843 1.00000 5.28150 1.00000 2.69843 0 8.85488 1.00000 2.69843
\(n\): 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.cn 10
7.b odd 2 1 8085.2.a.ck 10
7.c even 3 2 1155.2.q.l ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.q.l ✓ 20 7.c even 3 2
8085.2.a.ck 10 7.b odd 2 1
8085.2.a.cn 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):

\( T_{2}^{10} - 16T_{2}^{8} + 84T_{2}^{6} + 2T_{2}^{5} - 169T_{2}^{4} - 8T_{2}^{3} + 128T_{2}^{2} + 4T_{2} - 32 \) Copy content Toggle raw display
\( T_{13}^{10} - 4 T_{13}^{9} - 86 T_{13}^{8} + 396 T_{13}^{7} + 2016 T_{13}^{6} - 11328 T_{13}^{5} + \cdots - 1788 \) Copy content Toggle raw display
\( T_{17}^{10} - 12 T_{17}^{9} - 17 T_{17}^{8} + 610 T_{17}^{7} - 908 T_{17}^{6} - 9192 T_{17}^{5} + \cdots + 77824 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 16 T^{8} + \cdots - 32 \) 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
$13$ \( T^{10} - 4 T^{9} + \cdots - 1788 \) Copy content Toggle raw display
$17$ \( T^{10} - 12 T^{9} + \cdots + 77824 \) Copy content Toggle raw display
$19$ \( T^{10} - 9 T^{9} + \cdots + 2786176 \) Copy content Toggle raw display
$23$ \( T^{10} + T^{9} + \cdots + 72544 \) Copy content Toggle raw display
$29$ \( T^{10} - 3 T^{9} + \cdots - 34016 \) Copy content Toggle raw display
$31$ \( T^{10} - 9 T^{9} + \cdots + 6980096 \) Copy content Toggle raw display
$37$ \( T^{10} - 3 T^{9} + \cdots - 3465664 \) Copy content Toggle raw display
$41$ \( T^{10} - 3 T^{9} + \cdots - 297984 \) Copy content Toggle raw display
$43$ \( T^{10} - 305 T^{8} + \cdots - 2408797 \) Copy content Toggle raw display
$47$ \( T^{10} - 21 T^{9} + \cdots - 9597664 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 202857728 \) Copy content Toggle raw display
$59$ \( T^{10} - 16 T^{9} + \cdots - 5422208 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 3365900288 \) Copy content Toggle raw display
$67$ \( T^{10} + T^{9} + \cdots - 1093376 \) Copy content Toggle raw display
$71$ \( T^{10} - 22 T^{9} + \cdots + 57359712 \) Copy content Toggle raw display
$73$ \( T^{10} + T^{9} + \cdots + 5037856 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 8696262656 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 189485312 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 2285291328 \) Copy content Toggle raw display
$97$ \( T^{10} - 6 T^{9} + \cdots - 51240576 \) Copy content Toggle raw display
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