Properties

Label 80.10.a.k
Level $80$
Weight $10$
Character orbit 80.a
Self dual yes
Analytic conductor $41.203$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 80.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(41.2028668931\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.7117.1
Defining polynomial: \(x^{3} - x^{2} - 19 x - 22\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{3}\cdot 5 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -28 - \beta_{1} ) q^{3} + 625 q^{5} + ( 1840 - 17 \beta_{1} + \beta_{2} ) q^{7} + ( 15693 + 76 \beta_{1} + 4 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -28 - \beta_{1} ) q^{3} + 625 q^{5} + ( 1840 - 17 \beta_{1} + \beta_{2} ) q^{7} + ( 15693 + 76 \beta_{1} + 4 \beta_{2} ) q^{9} + ( -1852 - 292 \beta_{1} - \beta_{2} ) q^{11} + ( -27698 - 28 \beta_{1} - 16 \beta_{2} ) q^{13} + ( -17500 - 625 \beta_{1} ) q^{15} + ( 122354 - 1596 \beta_{1} ) q^{17} + ( -496372 + 378 \beta_{1} - 81 \beta_{2} ) q^{19} + ( 524576 - 5248 \beta_{1} + 60 \beta_{2} ) q^{21} + ( 166640 + 1417 \beta_{1} + 319 \beta_{2} ) q^{23} + 390625 q^{25} + ( -2565144 - 16554 \beta_{1} - 336 \beta_{2} ) q^{27} + ( 1744894 - 6128 \beta_{1} - 380 \beta_{2} ) q^{29} + ( -4236304 - 6022 \beta_{1} - 238 \beta_{2} ) q^{31} + ( 10164688 + 20092 \beta_{1} + 1176 \beta_{2} ) q^{33} + ( 1150000 - 10625 \beta_{1} + 625 \beta_{2} ) q^{35} + ( 7241478 + 50680 \beta_{1} - 1544 \beta_{2} ) q^{37} + ( 1935608 + 96626 \beta_{1} + 240 \beta_{2} ) q^{39} + ( 9146826 + 3188 \beta_{1} - 556 \beta_{2} ) q^{41} + ( 7739420 + 162615 \beta_{1} - 1134 \beta_{2} ) q^{43} + ( 9808125 + 47500 \beta_{1} + 2500 \beta_{2} ) q^{45} + ( 9567176 - 115901 \beta_{1} - 2081 \beta_{2} ) q^{47} + ( 9031641 - 218524 \beta_{1} + 1292 \beta_{2} ) q^{49} + ( 51782920 - 45746 \beta_{1} + 6384 \beta_{2} ) q^{51} + ( -15209994 - 175444 \beta_{1} - 5104 \beta_{2} ) q^{53} + ( -1157500 - 182500 \beta_{1} - 625 \beta_{2} ) q^{55} + ( 1792048 + 820372 \beta_{1} - 864 \beta_{2} ) q^{57} + ( 89573956 - 183246 \beta_{1} - 5775 \beta_{2} ) q^{59} + ( 51990046 - 362224 \beta_{1} + 12800 \beta_{2} ) q^{61} + ( 129915888 - 191501 \beta_{1} + 829 \beta_{2} ) q^{63} + ( -17311250 - 17500 \beta_{1} - 10000 \beta_{2} ) q^{65} + ( 175638868 - 277043 \beta_{1} + 4270 \beta_{2} ) q^{67} + ( -57500576 - 1582112 \beta_{1} - 8220 \beta_{2} ) q^{69} + ( 79964808 + 772366 \beta_{1} - 12320 \beta_{2} ) q^{71} + ( 66120810 + 460180 \beta_{1} - 4832 \beta_{2} ) q^{73} + ( -10937500 - 390625 \beta_{1} ) q^{75} + ( 128606336 - 1549988 \beta_{1} + 25888 \beta_{2} ) q^{77} + ( 137946576 - 298676 \beta_{1} + 29392 \beta_{2} ) q^{79} + ( 339595929 + 3283092 \beta_{1} - 9828 \beta_{2} ) q^{81} + ( -123983276 + 2169743 \beta_{1} - 21868 \beta_{2} ) q^{83} + ( 76471250 - 997500 \beta_{1} ) q^{85} + ( 167670584 + 154370 \beta_{1} + 27552 \beta_{2} ) q^{87} + ( 251642202 + 2366280 \beta_{1} + 20184 \beta_{2} ) q^{89} + ( -614129888 + 1749570 \beta_{1} + 7470 \beta_{2} ) q^{91} + ( 329777920 + 5530672 \beta_{1} + 25992 \beta_{2} ) q^{93} + ( -310232500 + 236250 \beta_{1} - 50625 \beta_{2} ) q^{95} + ( -301150334 - 1338708 \beta_{1} - 18648 \beta_{2} ) q^{97} + ( -957255180 - 10349092 \beta_{1} - 70093 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 84q^{3} + 1875q^{5} + 5520q^{7} + 47079q^{9} + O(q^{10}) \) \( 3q - 84q^{3} + 1875q^{5} + 5520q^{7} + 47079q^{9} - 5556q^{11} - 83094q^{13} - 52500q^{15} + 367062q^{17} - 1489116q^{19} + 1573728q^{21} + 499920q^{23} + 1171875q^{25} - 7695432q^{27} + 5234682q^{29} - 12708912q^{31} + 30494064q^{33} + 3450000q^{35} + 21724434q^{37} + 5806824q^{39} + 27440478q^{41} + 23218260q^{43} + 29424375q^{45} + 28701528q^{47} + 27094923q^{49} + 155348760q^{51} - 45629982q^{53} - 3472500q^{55} + 5376144q^{57} + 268721868q^{59} + 155970138q^{61} + 389747664q^{63} - 51933750q^{65} + 526916604q^{67} - 172501728q^{69} + 239894424q^{71} + 198362430q^{73} - 32812500q^{75} + 385819008q^{77} + 413839728q^{79} + 1018787787q^{81} - 371949828q^{83} + 229413750q^{85} + 503011752q^{87} + 754926606q^{89} - 1842389664q^{91} + 989333760q^{93} - 930697500q^{95} - 903451002q^{97} - 2871765540q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 19 x - 22\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -12 \nu^{2} + 84 \nu + 128 \)
\(\beta_{2}\)\(=\)\( 1272 \nu^{2} - 3144 \nu - 15488 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 106 \beta_{1} + 1920\)\()/5760\)
\(\nu^{2}\)\(=\)\((\)\(7 \beta_{2} + 262 \beta_{1} + 74880\)\()/5760\)

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
5.33466
−1.41013
−2.92453
0 −262.608 0 625.000 0 1790.86 0 49280.0 0
1.2 0 −13.6876 0 625.000 0 −6441.91 0 −19495.7 0
1.3 0 192.296 0 625.000 0 10171.1 0 17294.6 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)

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 80.10.a.k 3
4.b odd 2 1 40.10.a.d 3
5.b even 2 1 400.10.a.x 3
5.c odd 4 2 400.10.c.r 6
8.b even 2 1 320.10.a.v 3
8.d odd 2 1 320.10.a.u 3
12.b even 2 1 360.10.a.j 3
20.d odd 2 1 200.10.a.f 3
20.e even 4 2 200.10.c.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.10.a.d 3 4.b odd 2 1
80.10.a.k 3 1.a even 1 1 trivial
200.10.a.f 3 20.d odd 2 1
200.10.c.f 6 20.e even 4 2
320.10.a.u 3 8.d odd 2 1
320.10.a.v 3 8.b even 2 1
360.10.a.j 3 12.b even 2 1
400.10.a.x 3 5.b even 2 1
400.10.c.r 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 84 T_{3}^{2} - 49536 T_{3} - 691200 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(80))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 84 T + 9513 T^{2} + 2615544 T^{3} + 187244379 T^{4} + 32543321076 T^{5} + 7625597484987 T^{6} \)
$5$ \( ( 1 - 625 T )^{3} \)
$7$ \( 1 - 5520 T + 62218149 T^{2} - 328164567136 T^{3} + 2510726733013443 T^{4} - 8988843060465678480 T^{5} + \)\(65\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 + 5556 T + 2594856177 T^{2} + 72877014191416 T^{3} + 6118535131034237307 T^{4} + \)\(30\!\cdots\!36\)\( T^{5} + \)\(13\!\cdots\!71\)\( T^{6} \)
$13$ \( 1 + 83094 T + 20077033299 T^{2} + 754871996338436 T^{3} + \)\(21\!\cdots\!27\)\( T^{4} + \)\(93\!\cdots\!26\)\( T^{5} + \)\(11\!\cdots\!17\)\( T^{6} \)
$17$ \( 1 - 367062 T + 268505189631 T^{2} - 69711192496917428 T^{3} + \)\(31\!\cdots\!07\)\( T^{4} - \)\(51\!\cdots\!58\)\( T^{5} + \)\(16\!\cdots\!73\)\( T^{6} \)
$19$ \( 1 + 1489116 T + 1342580105481 T^{2} + 830560424045529832 T^{3} + \)\(43\!\cdots\!99\)\( T^{4} + \)\(15\!\cdots\!56\)\( T^{5} + \)\(33\!\cdots\!39\)\( T^{6} \)
$23$ \( 1 - 499920 T - 191246545323 T^{2} + 4150866877865471776 T^{3} - \)\(34\!\cdots\!49\)\( T^{4} - \)\(16\!\cdots\!80\)\( T^{5} + \)\(58\!\cdots\!47\)\( T^{6} \)
$29$ \( 1 - 5234682 T + 42736985914563 T^{2} - \)\(14\!\cdots\!08\)\( T^{3} + \)\(61\!\cdots\!47\)\( T^{4} - \)\(11\!\cdots\!02\)\( T^{5} + \)\(30\!\cdots\!09\)\( T^{6} \)
$31$ \( 1 + 12708912 T + 128130963574173 T^{2} + \)\(72\!\cdots\!04\)\( T^{3} + \)\(33\!\cdots\!83\)\( T^{4} + \)\(88\!\cdots\!92\)\( T^{5} + \)\(18\!\cdots\!11\)\( T^{6} \)
$37$ \( 1 - 21724434 T + 286542561812475 T^{2} - \)\(28\!\cdots\!32\)\( T^{3} + \)\(37\!\cdots\!75\)\( T^{4} - \)\(36\!\cdots\!86\)\( T^{5} + \)\(21\!\cdots\!33\)\( T^{6} \)
$41$ \( 1 - 27440478 T + 1215792148252263 T^{2} - \)\(18\!\cdots\!72\)\( T^{3} + \)\(39\!\cdots\!43\)\( T^{4} - \)\(29\!\cdots\!38\)\( T^{5} + \)\(35\!\cdots\!81\)\( T^{6} \)
$43$ \( 1 - 23218260 T + 251755624778721 T^{2} - \)\(25\!\cdots\!68\)\( T^{3} + \)\(12\!\cdots\!03\)\( T^{4} - \)\(58\!\cdots\!40\)\( T^{5} + \)\(12\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 - 28701528 T + 2689800471894429 T^{2} - \)\(45\!\cdots\!28\)\( T^{3} + \)\(30\!\cdots\!43\)\( T^{4} - \)\(35\!\cdots\!92\)\( T^{5} + \)\(14\!\cdots\!63\)\( T^{6} \)
$53$ \( 1 + 45629982 T + 7541283402476907 T^{2} + \)\(30\!\cdots\!96\)\( T^{3} + \)\(24\!\cdots\!31\)\( T^{4} + \)\(49\!\cdots\!98\)\( T^{5} + \)\(35\!\cdots\!37\)\( T^{6} \)
$59$ \( 1 - 268721868 T + 46457702853914817 T^{2} - \)\(50\!\cdots\!40\)\( T^{3} + \)\(40\!\cdots\!63\)\( T^{4} - \)\(20\!\cdots\!28\)\( T^{5} + \)\(65\!\cdots\!19\)\( T^{6} \)
$61$ \( 1 - 155970138 T + 27601780143557283 T^{2} - \)\(34\!\cdots\!16\)\( T^{3} + \)\(32\!\cdots\!03\)\( T^{4} - \)\(21\!\cdots\!78\)\( T^{5} + \)\(15\!\cdots\!21\)\( T^{6} \)
$67$ \( 1 - 526916604 T + 169230786613422441 T^{2} - \)\(33\!\cdots\!48\)\( T^{3} + \)\(46\!\cdots\!27\)\( T^{4} - \)\(39\!\cdots\!36\)\( T^{5} + \)\(20\!\cdots\!23\)\( T^{6} \)
$71$ \( 1 - 239894424 T + 117827037562982037 T^{2} - \)\(16\!\cdots\!88\)\( T^{3} + \)\(54\!\cdots\!47\)\( T^{4} - \)\(50\!\cdots\!64\)\( T^{5} + \)\(96\!\cdots\!91\)\( T^{6} \)
$73$ \( 1 - 198362430 T + 177547240540777287 T^{2} - \)\(22\!\cdots\!56\)\( T^{3} + \)\(10\!\cdots\!31\)\( T^{4} - \)\(68\!\cdots\!70\)\( T^{5} + \)\(20\!\cdots\!97\)\( T^{6} \)
$79$ \( 1 - 413839728 T + 365148310414214253 T^{2} - \)\(92\!\cdots\!64\)\( T^{3} + \)\(43\!\cdots\!07\)\( T^{4} - \)\(59\!\cdots\!08\)\( T^{5} + \)\(17\!\cdots\!59\)\( T^{6} \)
$83$ \( 1 + 371949828 T + 338245334081122329 T^{2} + \)\(14\!\cdots\!04\)\( T^{3} + \)\(63\!\cdots\!87\)\( T^{4} + \)\(12\!\cdots\!52\)\( T^{5} + \)\(65\!\cdots\!27\)\( T^{6} \)
$89$ \( 1 - 754926606 T + 926540806511526711 T^{2} - \)\(52\!\cdots\!12\)\( T^{3} + \)\(32\!\cdots\!99\)\( T^{4} - \)\(92\!\cdots\!86\)\( T^{5} + \)\(43\!\cdots\!29\)\( T^{6} \)
$97$ \( 1 + 903451002 T + 2439888940908067119 T^{2} + \)\(13\!\cdots\!72\)\( T^{3} + \)\(18\!\cdots\!23\)\( T^{4} + \)\(52\!\cdots\!78\)\( T^{5} + \)\(43\!\cdots\!13\)\( T^{6} \)
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