Properties

Label 8085.2.a.bn
Level 8085
Weight 2
Character orbit 8085.a
Self dual yes
Analytic conductor 64.559
Analytic rank 1
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8085.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.5590500342\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.13448.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1155)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + q^{5} -\beta_{1} q^{6} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + q^{5} -\beta_{1} q^{6} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + q^{9} + \beta_{1} q^{10} - q^{11} + ( -2 - \beta_{2} ) q^{12} + ( -2 - \beta_{1} - \beta_{3} ) q^{13} - q^{15} + ( 6 + \beta_{2} ) q^{16} + ( 1 + \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( -3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{19} + ( 2 + \beta_{2} ) q^{20} -\beta_{1} q^{22} + ( -1 - \beta_{2} - 2 \beta_{3} ) q^{23} + ( -2 \beta_{1} - \beta_{3} ) q^{24} + q^{25} + ( -6 - 2 \beta_{1} - 2 \beta_{2} ) q^{26} - q^{27} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{29} -\beta_{1} q^{30} + ( -6 - \beta_{1} + \beta_{3} ) q^{31} + ( 4 \beta_{1} - \beta_{3} ) q^{32} + q^{33} + ( 3 \beta_{1} + \beta_{3} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( 2 - 3 \beta_{1} + \beta_{3} ) q^{37} + ( -4 - 5 \beta_{1} - \beta_{3} ) q^{38} + ( 2 + \beta_{1} + \beta_{3} ) q^{39} + ( 2 \beta_{1} + \beta_{3} ) q^{40} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{41} + ( 1 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + ( -2 - \beta_{2} ) q^{44} + q^{45} + ( -4 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{46} + ( -2 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{47} + ( -6 - \beta_{2} ) q^{48} + \beta_{1} q^{50} + ( -1 - \beta_{2} ) q^{51} + ( -4 - 8 \beta_{1} - 2 \beta_{2} ) q^{52} + ( 3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{53} -\beta_{1} q^{54} - q^{55} + ( 3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{57} + ( -6 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{58} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{59} + ( -2 - \beta_{2} ) q^{60} + ( -1 + 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{61} + ( -2 - 6 \beta_{1} ) q^{62} + ( 2 + \beta_{2} ) q^{64} + ( -2 - \beta_{1} - \beta_{3} ) q^{65} + \beta_{1} q^{66} + ( -2 - 4 \beta_{1} + 2 \beta_{3} ) q^{67} + ( 12 + 2 \beta_{2} ) q^{68} + ( 1 + \beta_{2} + 2 \beta_{3} ) q^{69} + ( -\beta_{1} - \beta_{3} ) q^{71} + ( 2 \beta_{1} + \beta_{3} ) q^{72} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{73} + ( -10 + 2 \beta_{1} - 2 \beta_{2} ) q^{74} - q^{75} + ( -16 - 4 \beta_{2} - 4 \beta_{3} ) q^{76} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{78} + ( \beta_{1} + \beta_{3} ) q^{79} + ( 6 + \beta_{2} ) q^{80} + q^{81} + 6 q^{82} + ( -1 + \beta_{2} + 4 \beta_{3} ) q^{83} + ( 1 + \beta_{2} ) q^{85} + ( -10 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{86} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{87} + ( -2 \beta_{1} - \beta_{3} ) q^{88} + ( -5 + 5 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{89} + \beta_{1} q^{90} + ( -12 - 8 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{92} + ( 6 + \beta_{1} - \beta_{3} ) q^{93} + ( 10 - 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{94} + ( -3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{95} + ( -4 \beta_{1} + \beta_{3} ) q^{96} + ( 3 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 6q^{4} + 4q^{5} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 6q^{4} + 4q^{5} + 4q^{9} - 4q^{11} - 6q^{12} - 8q^{13} - 4q^{15} + 22q^{16} + 2q^{17} - 10q^{19} + 6q^{20} - 2q^{23} + 4q^{25} - 20q^{26} - 4q^{27} - 2q^{29} - 24q^{31} + 4q^{33} + 6q^{36} + 8q^{37} - 16q^{38} + 8q^{39} + 6q^{43} - 6q^{44} + 4q^{45} - 12q^{46} - 4q^{47} - 22q^{48} - 2q^{51} - 12q^{52} + 14q^{53} - 4q^{55} + 10q^{57} - 20q^{58} + 2q^{59} - 6q^{60} - 6q^{61} - 8q^{62} + 6q^{64} - 8q^{65} - 8q^{67} + 44q^{68} + 2q^{69} - 4q^{73} - 36q^{74} - 4q^{75} - 56q^{76} + 20q^{78} + 22q^{80} + 4q^{81} + 24q^{82} - 6q^{83} + 2q^{85} - 36q^{86} + 2q^{87} - 18q^{89} - 44q^{92} + 24q^{93} + 36q^{94} - 10q^{95} + 6q^{97} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 7 x^{2} + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 6 \beta_{1}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.58874
−0.546295
0.546295
2.58874
−2.58874 −1.00000 4.70156 1.00000 2.58874 0 −6.99364 1.00000 −2.58874
1.2 −0.546295 −1.00000 −1.70156 1.00000 0.546295 0 2.02214 1.00000 −0.546295
1.3 0.546295 −1.00000 −1.70156 1.00000 −0.546295 0 −2.02214 1.00000 0.546295
1.4 2.58874 −1.00000 4.70156 1.00000 −2.58874 0 6.99364 1.00000 2.58874
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.bn 4
7.b odd 2 1 1155.2.a.u 4
21.c even 2 1 3465.2.a.bl 4
35.c odd 2 1 5775.2.a.bz 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.u 4 7.b odd 2 1
3465.2.a.bl 4 21.c even 2 1
5775.2.a.bz 4 35.c odd 2 1
8085.2.a.bn 4 1.a even 1 1 trivial

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):

\( T_{2}^{4} - 7 T_{2}^{2} + 2 \)
\( T_{13}^{4} + 8 T_{13}^{3} - 2 T_{13}^{2} - 72 T_{13} + 40 \)
\( T_{17}^{2} - T_{17} - 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} - 2 T^{4} + 4 T^{6} + 16 T^{8} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( ( 1 - T )^{4} \)
$7$ 1
$11$ \( ( 1 + T )^{4} \)
$13$ \( 1 + 8 T + 50 T^{2} + 240 T^{3} + 1002 T^{4} + 3120 T^{5} + 8450 T^{6} + 17576 T^{7} + 28561 T^{8} \)
$17$ \( ( 1 - T + 24 T^{2} - 17 T^{3} + 289 T^{4} )^{2} \)
$19$ \( 1 + 10 T + 37 T^{2} - 78 T^{3} - 916 T^{4} - 1482 T^{5} + 13357 T^{6} + 68590 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 2 T + 21 T^{2} - 98 T^{3} - 108 T^{4} - 2254 T^{5} + 11109 T^{6} + 24334 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 2 T + 71 T^{2} + 210 T^{3} + 2432 T^{4} + 6090 T^{5} + 59711 T^{6} + 48778 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 24 T + 326 T^{2} + 2928 T^{3} + 19090 T^{4} + 90768 T^{5} + 313286 T^{6} + 714984 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 8 T + 114 T^{2} - 688 T^{3} + 6282 T^{4} - 25456 T^{5} + 156066 T^{6} - 405224 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 38 T^{2} + 402 T^{4} + 63878 T^{6} + 2825761 T^{8} \)
$43$ \( 1 - 6 T + 107 T^{2} - 470 T^{3} + 5520 T^{4} - 20210 T^{5} + 197843 T^{6} - 477042 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 4 T + 54 T^{2} + 452 T^{3} + 786 T^{4} + 21244 T^{5} + 119286 T^{6} + 415292 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 14 T + 161 T^{2} - 1198 T^{3} + 10012 T^{4} - 63494 T^{5} + 452249 T^{6} - 2084278 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 2 T + 203 T^{2} - 238 T^{3} + 16912 T^{4} - 14042 T^{5} + 706643 T^{6} - 410758 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 6 T + 121 T^{2} + 866 T^{3} + 9380 T^{4} + 52826 T^{5} + 450241 T^{6} + 1361886 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 8 T + 176 T^{2} + 1176 T^{3} + 17358 T^{4} + 78792 T^{5} + 790064 T^{6} + 2406104 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 258 T^{2} + 26682 T^{4} + 1300578 T^{6} + 25411681 T^{8} \)
$73$ \( 1 + 4 T + 112 T^{2} - 148 T^{3} + 4318 T^{4} - 10804 T^{5} + 596848 T^{6} + 1556068 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 290 T^{2} + 33466 T^{4} + 1809890 T^{6} + 38950081 T^{8} \)
$83$ \( 1 + 6 T + 117 T^{2} + 1478 T^{3} + 12284 T^{4} + 122674 T^{5} + 806013 T^{6} + 3430722 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 18 T + 255 T^{2} + 2594 T^{3} + 29096 T^{4} + 230866 T^{5} + 2019855 T^{6} + 12689442 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 6 T + 111 T^{2} + 834 T^{3} - 984 T^{4} + 80898 T^{5} + 1044399 T^{6} - 5476038 T^{7} + 88529281 T^{8} \)
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