Properties

Label 8085.2.a.cp
Level $8085$
Weight $2$
Character orbit 8085.a
Self dual yes
Analytic conductor $64.559$
Analytic rank $0$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} - 16 x^{12} + 76 x^{11} + 78 x^{10} - 532 x^{9} - 56 x^{8} + 1684 x^{7} - 471 x^{6} - 2352 x^{5} + 950 x^{4} + 1184 x^{3} - 340 x^{2} - 152 x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} + (\beta_{3} + \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} + (\beta_{3} + \beta_1 + 1) q^{8} + q^{9} + \beta_1 q^{10} + q^{11} + (\beta_{2} + 1) q^{12} + (\beta_{7} + 1) q^{13} + q^{15} + (\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{16} + ( - \beta_{13} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_1 - 1) q^{17} + \beta_1 q^{18} + ( - \beta_{8} - \beta_{3}) q^{19} + (\beta_{2} + 1) q^{20} + \beta_1 q^{22} + ( - \beta_{10} - \beta_{7} - \beta_{5} + \beta_{3} - \beta_{2} + 2) q^{23} + (\beta_{3} + \beta_1 + 1) q^{24} + q^{25} + (\beta_{13} + \beta_{12} + \beta_{10} + \beta_{7} + \beta_{6} + \beta_{2} + 1) q^{26} + q^{27} + (\beta_{10} + \beta_{9} + \beta_{5} - \beta_{3} + \beta_{2} + \beta_1) q^{29} + \beta_1 q^{30} + ( - \beta_{13} + \beta_{10} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{31} + ( - \beta_{13} - \beta_{11} + \beta_{10} + \beta_{8} + 2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{32}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 4 q^{2} + 14 q^{3} + 20 q^{4} + 14 q^{5} + 4 q^{6} + 12 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 4 q^{2} + 14 q^{3} + 20 q^{4} + 14 q^{5} + 4 q^{6} + 12 q^{8} + 14 q^{9} + 4 q^{10} + 14 q^{11} + 20 q^{12} + 8 q^{13} + 14 q^{15} + 32 q^{16} - 6 q^{17} + 4 q^{18} + 10 q^{19} + 20 q^{20} + 4 q^{22} + 22 q^{23} + 12 q^{24} + 14 q^{25} + 12 q^{26} + 14 q^{27} + 10 q^{29} + 4 q^{30} + 20 q^{31} + 28 q^{32} + 14 q^{33} + 16 q^{34} + 20 q^{36} + 20 q^{37} - 16 q^{38} + 8 q^{39} + 12 q^{40} + 22 q^{43} + 20 q^{44} + 14 q^{45} + 4 q^{46} - 8 q^{47} + 32 q^{48} + 4 q^{50} - 6 q^{51} + 4 q^{52} + 34 q^{53} + 4 q^{54} + 14 q^{55} + 10 q^{57} + 20 q^{58} + 22 q^{59} + 20 q^{60} + 6 q^{61} - 28 q^{62} + 32 q^{64} + 8 q^{65} + 4 q^{66} + 32 q^{67} - 56 q^{68} + 22 q^{69} + 4 q^{71} + 12 q^{72} + 8 q^{73} + 36 q^{74} + 14 q^{75} + 24 q^{76} + 12 q^{78} + 12 q^{79} + 32 q^{80} + 14 q^{81} - 28 q^{82} - 26 q^{83} - 6 q^{85} + 40 q^{86} + 10 q^{87} + 12 q^{88} + 10 q^{89} + 4 q^{90} + 8 q^{92} + 20 q^{93} + 28 q^{94} + 10 q^{95} + 28 q^{96} + 14 q^{97} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} - 16 x^{12} + 76 x^{11} + 78 x^{10} - 532 x^{9} - 56 x^{8} + 1684 x^{7} - 471 x^{6} - 2352 x^{5} + 950 x^{4} + 1184 x^{3} - 340 x^{2} - 152 x + 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 129 \nu^{13} + 689 \nu^{12} + 3291 \nu^{11} - 14955 \nu^{10} - 34071 \nu^{9} + 121695 \nu^{8} + 177181 \nu^{7} - 454605 \nu^{6} - 458856 \nu^{5} + 761688 \nu^{4} + \cdots + 59786 ) / 31712 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 173 \nu^{13} + 1227 \nu^{12} - 5151 \nu^{11} - 24009 \nu^{10} + 53067 \nu^{9} + 169957 \nu^{8} - 237369 \nu^{7} - 519615 \nu^{6} + 458280 \nu^{5} + 623848 \nu^{4} + \cdots + 1806 ) / 31712 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 151 \nu^{13} + 269 \nu^{12} - 4221 \nu^{11} - 4527 \nu^{10} + 43569 \nu^{9} + 24131 \nu^{8} - 207275 \nu^{7} - 32505 \nu^{6} + 458568 \nu^{5} - 53064 \nu^{4} - 429214 \nu^{3} + \cdots + 66146 ) / 15856 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2883 \nu^{13} - 29595 \nu^{12} - 11601 \nu^{11} + 587001 \nu^{10} - 397147 \nu^{9} - 4328757 \nu^{8} + 3852425 \nu^{7} + 14568783 \nu^{6} - 12543080 \nu^{5} + \cdots - 1262030 ) / 221984 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1857 \nu^{13} + 11209 \nu^{12} + 21563 \nu^{11} - 219523 \nu^{10} + 21353 \nu^{9} + 1568023 \nu^{8} - 1124691 \nu^{7} - 4875077 \nu^{6} + 4814984 \nu^{5} + \cdots - 181062 ) / 110992 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1697 \nu^{13} + 3041 \nu^{12} + 34259 \nu^{11} - 52739 \nu^{10} - 254799 \nu^{9} + 304951 \nu^{8} + 822229 \nu^{7} - 574741 \nu^{6} - 896712 \nu^{5} - 315680 \nu^{4} + \cdots - 126462 ) / 55496 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1153 \nu^{13} + 2225 \nu^{12} + 23515 \nu^{11} - 42219 \nu^{10} - 182103 \nu^{9} + 296383 \nu^{8} + 671245 \nu^{7} - 951789 \nu^{6} - 1203656 \nu^{5} + 1387416 \nu^{4} + \cdots + 90890 ) / 15856 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11143 \nu^{13} - 28111 \nu^{12} - 209421 \nu^{11} + 512101 \nu^{10} + 1433409 \nu^{9} - 3383617 \nu^{8} - 4278187 \nu^{7} + 9858595 \nu^{6} + 4824984 \nu^{5} + \cdots - 475846 ) / 110992 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 4557 \nu^{13} - 13277 \nu^{12} - 83807 \nu^{11} + 247311 \nu^{10} + 561963 \nu^{9} - 1674963 \nu^{8} - 1682169 \nu^{7} + 4987273 \nu^{6} + 2191176 \nu^{5} + \cdots - 95682 ) / 31712 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 38817 \nu^{13} + 122145 \nu^{12} + 696027 \nu^{11} - 2285755 \nu^{10} - 4392119 \nu^{9} + 15580175 \nu^{8} + 11140989 \nu^{7} - 46884765 \nu^{6} + \cdots + 136906 ) / 221984 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{13} - \beta_{11} + \beta_{10} + \beta_{8} + 2\beta_{5} - \beta_{4} + 9\beta_{3} + 2\beta_{2} + 29\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{8} + \beta_{7} + 10 \beta_{6} - 11 \beta_{5} + 12 \beta_{4} + 12 \beta_{3} + 48 \beta_{2} + 98 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12 \beta_{13} - 12 \beta_{11} + 13 \beta_{10} - 2 \beta_{9} + 14 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 24 \beta_{5} - 12 \beta_{4} + 71 \beta_{3} + 26 \beta_{2} + 179 \beta _1 + 73 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 14 \beta_{13} - 13 \beta_{12} - 16 \beta_{11} - 14 \beta_{10} - 16 \beta_{9} + 28 \beta_{8} + 14 \beta_{7} + 81 \beta_{6} - 95 \beta_{5} + 110 \beta_{4} + 111 \beta_{3} + 333 \beta_{2} + \beta _1 + 640 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 106 \beta_{13} + 3 \beta_{12} - 109 \beta_{11} + 123 \beta_{10} - 31 \beta_{9} + 140 \beta_{8} + 32 \beta_{7} + 35 \beta_{6} + 211 \beta_{5} - 102 \beta_{4} + 543 \beta_{3} + 252 \beta_{2} + 1150 \beta _1 + 575 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 137 \beta_{13} - 119 \beta_{12} - 175 \beta_{11} - 138 \beta_{10} - 174 \beta_{9} + 278 \beta_{8} + 144 \beta_{7} + 621 \beta_{6} - 749 \beta_{5} + 907 \beta_{4} + 936 \beta_{3} + 2340 \beta_{2} + 23 \beta _1 + 4340 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 836 \beta_{13} + 60 \beta_{12} - 899 \beta_{11} + 1029 \beta_{10} - 333 \beta_{9} + 1226 \beta_{8} + 357 \beta_{7} + 412 \beta_{6} + 1657 \beta_{5} - 738 \beta_{4} + 4105 \beta_{3} + 2192 \beta_{2} + 7607 \beta _1 + 4474 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1158 \beta_{13} - 939 \beta_{12} - 1638 \beta_{11} - 1181 \beta_{10} - 1619 \beta_{9} + 2420 \beta_{8} + 1319 \beta_{7} + 4682 \beta_{6} - 5637 \beta_{5} + 7123 \beta_{4} + 7551 \beta_{3} + 16623 \beta_{2} + \cdots + 30166 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 6242 \beta_{13} + 779 \beta_{12} - 7102 \beta_{11} + 8071 \beta_{10} - 3100 \beta_{9} + 10036 \beta_{8} + 3437 \beta_{7} + 4105 \beta_{6} + 12358 \beta_{5} - 4776 \beta_{4} + 30851 \beta_{3} + \cdots + 34608 \) 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
−2.64188
−2.24953
−1.96781
−1.47936
−0.709211
−0.347588
0.0816533
0.537407
1.30829
1.53024
2.17507
2.32113
2.68384
2.75776
−2.64188 1.00000 4.97951 1.00000 −2.64188 0 −7.87151 1.00000 −2.64188
1.2 −2.24953 1.00000 3.06040 1.00000 −2.24953 0 −2.38541 1.00000 −2.24953
1.3 −1.96781 1.00000 1.87227 1.00000 −1.96781 0 0.251340 1.00000 −1.96781
1.4 −1.47936 1.00000 0.188515 1.00000 −1.47936 0 2.67984 1.00000 −1.47936
1.5 −0.709211 1.00000 −1.49702 1.00000 −0.709211 0 2.48013 1.00000 −0.709211
1.6 −0.347588 1.00000 −1.87918 1.00000 −0.347588 0 1.34836 1.00000 −0.347588
1.7 0.0816533 1.00000 −1.99333 1.00000 0.0816533 0 −0.326069 1.00000 0.0816533
1.8 0.537407 1.00000 −1.71119 1.00000 0.537407 0 −1.99442 1.00000 0.537407
1.9 1.30829 1.00000 −0.288382 1.00000 1.30829 0 −2.99386 1.00000 1.30829
1.10 1.53024 1.00000 0.341649 1.00000 1.53024 0 −2.53768 1.00000 1.53024
1.11 2.17507 1.00000 2.73091 1.00000 2.17507 0 1.58978 1.00000 2.17507
1.12 2.32113 1.00000 3.38763 1.00000 2.32113 0 3.22086 1.00000 2.32113
1.13 2.68384 1.00000 5.20298 1.00000 2.68384 0 8.59627 1.00000 2.68384
1.14 2.75776 1.00000 5.60524 1.00000 2.75776 0 9.94239 1.00000 2.75776
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.cp yes 14
7.b odd 2 1 8085.2.a.co 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8085.2.a.co 14 7.b odd 2 1
8085.2.a.cp yes 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):

\( T_{2}^{14} - 4 T_{2}^{13} - 16 T_{2}^{12} + 76 T_{2}^{11} + 78 T_{2}^{10} - 532 T_{2}^{9} - 56 T_{2}^{8} + 1684 T_{2}^{7} - 471 T_{2}^{6} - 2352 T_{2}^{5} + 950 T_{2}^{4} + 1184 T_{2}^{3} - 340 T_{2}^{2} - 152 T_{2} + 14 \) Copy content Toggle raw display
\( T_{13}^{14} - 8 T_{13}^{13} - 68 T_{13}^{12} + 640 T_{13}^{11} + 1170 T_{13}^{10} - 17492 T_{13}^{9} + 6982 T_{13}^{8} + 188032 T_{13}^{7} - 325072 T_{13}^{6} - 502352 T_{13}^{5} + 1961816 T_{13}^{4} - 2130880 T_{13}^{3} + \cdots - 20608 \) Copy content Toggle raw display
\( T_{17}^{14} + 6 T_{17}^{13} - 151 T_{17}^{12} - 976 T_{17}^{11} + 8139 T_{17}^{10} + 58026 T_{17}^{9} - 182889 T_{17}^{8} - 1536112 T_{17}^{7} + 1427512 T_{17}^{6} + 17553792 T_{17}^{5} - 699024 T_{17}^{4} + \cdots + 2766848 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 4 T^{13} - 16 T^{12} + 76 T^{11} + \cdots + 14 \) Copy content Toggle raw display
$3$ \( (T - 1)^{14} \) Copy content Toggle raw display
$5$ \( (T - 1)^{14} \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( (T - 1)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} - 8 T^{13} - 68 T^{12} + \cdots - 20608 \) Copy content Toggle raw display
$17$ \( T^{14} + 6 T^{13} - 151 T^{12} + \cdots + 2766848 \) Copy content Toggle raw display
$19$ \( T^{14} - 10 T^{13} - 117 T^{12} + \cdots - 7683008 \) Copy content Toggle raw display
$23$ \( T^{14} - 22 T^{13} + 101 T^{12} + \cdots - 23302144 \) Copy content Toggle raw display
$29$ \( T^{14} - 10 T^{13} - 129 T^{12} + \cdots + 53114944 \) Copy content Toggle raw display
$31$ \( T^{14} - 20 T^{13} - 16 T^{12} + \cdots + 6909952 \) Copy content Toggle raw display
$37$ \( T^{14} - 20 T^{13} + \cdots + 600366592 \) Copy content Toggle raw display
$41$ \( T^{14} - 418 T^{12} + \cdots - 115702988800 \) Copy content Toggle raw display
$43$ \( T^{14} - 22 T^{13} + \cdots + 634482688 \) Copy content Toggle raw display
$47$ \( T^{14} + 8 T^{13} - 252 T^{12} + \cdots + 808204288 \) Copy content Toggle raw display
$53$ \( T^{14} - 34 T^{13} + \cdots - 636433552 \) Copy content Toggle raw display
$59$ \( T^{14} - 22 T^{13} + \cdots - 187579264 \) Copy content Toggle raw display
$61$ \( T^{14} - 6 T^{13} - 453 T^{12} + \cdots - 24005056 \) Copy content Toggle raw display
$67$ \( T^{14} - 32 T^{13} + \cdots + 2834403328 \) Copy content Toggle raw display
$71$ \( T^{14} - 4 T^{13} + \cdots + 39094765568 \) Copy content Toggle raw display
$73$ \( T^{14} - 8 T^{13} - 514 T^{12} + \cdots + 121036288 \) Copy content Toggle raw display
$79$ \( T^{14} - 12 T^{13} + \cdots - 949213385600 \) Copy content Toggle raw display
$83$ \( T^{14} + 26 T^{13} + \cdots + 286643607488 \) Copy content Toggle raw display
$89$ \( T^{14} - 10 T^{13} + \cdots + 99797668096 \) Copy content Toggle raw display
$97$ \( T^{14} - 14 T^{13} + \cdots + 2826434309888 \) Copy content Toggle raw display
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