Properties

Label 280.6.a.j
Level $280$
Weight $6$
Character orbit 280.a
Self dual yes
Analytic conductor $44.907$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.9074695476\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 1151 x^{3} - 5642 x^{2} + 193596 x + 1258056\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - \beta_{1} ) q^{3} + 25 q^{5} + 49 q^{7} + ( 226 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 3 - \beta_{1} ) q^{3} + 25 q^{5} + 49 q^{7} + ( 226 + \beta_{1} + \beta_{2} ) q^{9} + ( 56 - 8 \beta_{1} - \beta_{4} ) q^{11} + ( 182 - 6 \beta_{1} - \beta_{3} + \beta_{4} ) q^{13} + ( 75 - 25 \beta_{1} ) q^{15} + ( 299 + 13 \beta_{1} + 2 \beta_{3} ) q^{17} + ( -83 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{19} + ( 147 - 49 \beta_{1} ) q^{21} + ( -11 - 13 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{23} + 625 q^{25} + ( -674 - 294 \beta_{1} + \beta_{2} - 10 \beta_{3} - 5 \beta_{4} ) q^{27} + ( -405 - 133 \beta_{1} - 6 \beta_{2} - 9 \beta_{3} + 10 \beta_{4} ) q^{29} + ( 329 - 53 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} + 3 \beta_{4} ) q^{31} + ( 3824 - 28 \beta_{1} + 23 \beta_{2} + 6 \beta_{3} - 15 \beta_{4} ) q^{33} + 1225 q^{35} + ( -2068 - 28 \beta_{1} - 18 \beta_{2} + 7 \beta_{3} - 18 \beta_{4} ) q^{37} + ( 3135 - 203 \beta_{1} + 12 \beta_{2} - 5 \beta_{3} + 14 \beta_{4} ) q^{39} + ( 1291 + 271 \beta_{1} - 15 \beta_{2} + 7 \beta_{3} + \beta_{4} ) q^{41} + ( 5650 - 56 \beta_{1} + 30 \beta_{2} + 13 \beta_{3} + 4 \beta_{4} ) q^{43} + ( 5650 + 25 \beta_{1} + 25 \beta_{2} ) q^{45} + ( 4184 - 238 \beta_{1} + 18 \beta_{2} - 9 \beta_{3} + \beta_{4} ) q^{47} + 2401 q^{49} + ( -4693 - 253 \beta_{1} - 55 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{51} + ( 8767 + 829 \beta_{1} - 21 \beta_{2} - 34 \beta_{3} + 45 \beta_{4} ) q^{53} + ( 1400 - 200 \beta_{1} - 25 \beta_{4} ) q^{55} + ( 2759 + 1077 \beta_{1} - 37 \beta_{2} + 23 \beta_{3} + 31 \beta_{4} ) q^{57} + ( 9140 - 396 \beta_{1} + 30 \beta_{2} + 10 \beta_{3} - 28 \beta_{4} ) q^{59} + ( 10083 + 539 \beta_{1} - 9 \beta_{2} - 9 \beta_{3} - 35 \beta_{4} ) q^{61} + ( 11074 + 49 \beta_{1} + 49 \beta_{2} ) q^{63} + ( 4550 - 150 \beta_{1} - 25 \beta_{3} + 25 \beta_{4} ) q^{65} + ( 8149 + 781 \beta_{1} - 33 \beta_{2} + 38 \beta_{3} - 33 \beta_{4} ) q^{67} + ( 7045 + 1135 \beta_{1} - 71 \beta_{2} + 21 \beta_{3} + 33 \beta_{4} ) q^{69} + ( -561 + 255 \beta_{1} - 105 \beta_{2} - 72 \beta_{3} + 81 \beta_{4} ) q^{71} + ( 27057 + 1403 \beta_{1} + 51 \beta_{2} - 26 \beta_{3} - 51 \beta_{4} ) q^{73} + ( 1875 - 625 \beta_{1} ) q^{75} + ( 2744 - 392 \beta_{1} - 49 \beta_{4} ) q^{77} + ( 2654 - 624 \beta_{1} - 93 \beta_{2} + 53 \beta_{3} - 107 \beta_{4} ) q^{79} + ( 76067 + 790 \beta_{1} + 338 \beta_{2} + 30 \beta_{3} - 90 \beta_{4} ) q^{81} + ( 7017 + 2085 \beta_{1} - 9 \beta_{2} + 32 \beta_{3} - 41 \beta_{4} ) q^{83} + ( 7475 + 325 \beta_{1} + 50 \beta_{3} ) q^{85} + ( 59428 + 2378 \beta_{1} + 160 \beta_{2} + 9 \beta_{3} + 171 \beta_{4} ) q^{87} + ( 42723 - 301 \beta_{1} + 99 \beta_{2} + 89 \beta_{3} + 41 \beta_{4} ) q^{89} + ( 8918 - 294 \beta_{1} - 49 \beta_{3} + 49 \beta_{4} ) q^{91} + ( 26903 + 1061 \beta_{1} - 103 \beta_{2} + 7 \beta_{3} + 65 \beta_{4} ) q^{93} + ( -2075 - 125 \beta_{1} - 75 \beta_{2} + 25 \beta_{3} + 25 \beta_{4} ) q^{95} + ( 2187 - 819 \beta_{1} - 96 \beta_{2} - 174 \beta_{3} + 84 \beta_{4} ) q^{97} + ( 7805 - 8595 \beta_{1} + 173 \beta_{2} - 146 \beta_{3} - 91 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 15 q^{3} + 125 q^{5} + 245 q^{7} + 1132 q^{9} + O(q^{10}) \) \( 5 q + 15 q^{3} + 125 q^{5} + 245 q^{7} + 1132 q^{9} + 281 q^{11} + 909 q^{13} + 375 q^{15} + 1495 q^{17} - 422 q^{19} + 735 q^{21} - 62 q^{23} + 3125 q^{25} - 3363 q^{27} - 2047 q^{29} + 1636 q^{31} + 19181 q^{33} + 6125 q^{35} - 10358 q^{37} + 15685 q^{39} + 6424 q^{41} + 28306 q^{43} + 28300 q^{45} + 20955 q^{47} + 12005 q^{49} - 23577 q^{51} + 43748 q^{53} + 7025 q^{55} + 13690 q^{57} + 45788 q^{59} + 50432 q^{61} + 55468 q^{63} + 22725 q^{65} + 40712 q^{67} + 35050 q^{69} - 3096 q^{71} + 135438 q^{73} + 9375 q^{75} + 13769 q^{77} + 13191 q^{79} + 381101 q^{81} + 35108 q^{83} + 37375 q^{85} + 297289 q^{87} + 213772 q^{89} + 44541 q^{91} + 134244 q^{93} - 10550 q^{95} + 10659 q^{97} + 39462 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 1151 x^{3} - 5642 x^{2} + 193596 x + 1258056\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 7 \nu - 460 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} + 6 \nu^{3} - 1157 \nu^{2} - 7838 \nu + 137328 \)\()/210\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{4} + 15 \nu^{3} + 989 \nu^{2} - 8122 \nu - 131091 \)\()/105\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 7 \beta_{1} + 460\)
\(\nu^{3}\)\(=\)\(5 \beta_{4} + 10 \beta_{3} + 8 \beta_{2} + 816 \beta_{1} + 3383\)
\(\nu^{4}\)\(=\)\(-30 \beta_{4} + 150 \beta_{3} + 1109 \beta_{2} + 11041 \beta_{1} + 374594\)

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
33.3496
14.5231
−7.03843
−13.5022
−27.3320
0 −30.3496 0 25.0000 0 49.0000 0 678.095 0
1.2 0 −11.5231 0 25.0000 0 49.0000 0 −110.219 0
1.3 0 10.0384 0 25.0000 0 49.0000 0 −142.230 0
1.4 0 16.5022 0 25.0000 0 49.0000 0 29.3221 0
1.5 0 30.3320 0 25.0000 0 49.0000 0 677.031 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 280.6.a.j 5
4.b odd 2 1 560.6.a.y 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.6.a.j 5 1.a even 1 1 trivial
560.6.a.y 5 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 15 T_{3}^{4} - 1061 T_{3}^{3} + 15731 T_{3}^{2} + 129072 T_{3} - 1757232 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(280))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( -1757232 + 129072 T + 15731 T^{2} - 1061 T^{3} - 15 T^{4} + T^{5} \)
$5$ \( ( -25 + T )^{5} \)
$7$ \( ( -49 + T )^{5} \)
$11$ \( -20384754585232 + 63367401216 T + 185591429 T^{2} - 583653 T^{3} - 281 T^{4} + T^{5} \)
$13$ \( -9709872570412 - 71810997360 T + 491497477 T^{2} - 419845 T^{3} - 909 T^{4} + T^{5} \)
$17$ \( -871869567890580 + 787735162456 T + 2448896695 T^{2} - 1731109 T^{3} - 1495 T^{4} + T^{5} \)
$19$ \( 1154217750713984 + 5160112134784 T - 943633880 T^{2} - 4963636 T^{3} + 422 T^{4} + T^{5} \)
$23$ \( 23919527367683584 + 32402819214336 T - 4517414552 T^{2} - 11601492 T^{3} + 62 T^{4} + T^{5} \)
$29$ \( -443670632506543804 + 1272525503092896 T - 181872689743 T^{2} - 89950125 T^{3} + 2047 T^{4} + T^{5} \)
$31$ \( 239402324126760960 + 279372488455168 T + 14837225152 T^{2} - 33911056 T^{3} - 1636 T^{4} + T^{5} \)
$37$ \( 72098634143924435552 + 4533156181472656 T - 2671439032688 T^{2} - 232067560 T^{3} + 10358 T^{4} + T^{5} \)
$41$ \( 12058950673153968896 - 9279417704685632 T + 2330766159664 T^{2} - 184163452 T^{3} - 6424 T^{4} + T^{5} \)
$43$ \( 91574596835805337472 - 56286628439160896 T + 7757251433416 T^{2} - 92561668 T^{3} - 28306 T^{4} + T^{5} \)
$47$ \( 623374189880908032 - 1273871822967680 T + 677231573991 T^{2} - 57381133 T^{3} - 20955 T^{4} + T^{5} \)
$53$ \( -\)\(20\!\cdots\!48\)\( + 385735325494789120 T + 60488885604800 T^{2} - 1252775020 T^{3} - 43748 T^{4} + T^{5} \)
$59$ \( -\)\(20\!\cdots\!44\)\( - 14253941241301760 T + 9160887185536 T^{2} + 18456480 T^{3} - 45788 T^{4} + T^{5} \)
$61$ \( \)\(15\!\cdots\!80\)\( - 451389774038655552 T + 33984856977008 T^{2} - 192434652 T^{3} - 50432 T^{4} + T^{5} \)
$67$ \( -\)\(83\!\cdots\!84\)\( + 271542412408689664 T + 46339122911360 T^{2} - 1407144016 T^{3} - 40712 T^{4} + T^{5} \)
$71$ \( \)\(34\!\cdots\!00\)\( + 13829221610861690880 T - 106168050487296 T^{2} - 7340613120 T^{3} + 3096 T^{4} + T^{5} \)
$73$ \( -\)\(23\!\cdots\!32\)\( - 10186029667190855536 T + 362130153328944 T^{2} + 1539203032 T^{3} - 135438 T^{4} + T^{5} \)
$79$ \( \)\(78\!\cdots\!16\)\( + 7860251218620928344 T - 180890250103973 T^{2} - 9317209837 T^{3} - 13191 T^{4} + T^{5} \)
$83$ \( -\)\(16\!\cdots\!04\)\( + 6422068805775660288 T + 137017712423808 T^{2} - 5454516576 T^{3} - 35108 T^{4} + T^{5} \)
$89$ \( \)\(18\!\cdots\!44\)\( - 88280980812895312640 T + 1008285101356000 T^{2} + 8222175332 T^{3} - 213772 T^{4} + T^{5} \)
$97$ \( -\)\(96\!\cdots\!88\)\( + 68194465007819650632 T + 45896316774507 T^{2} - 18250367373 T^{3} - 10659 T^{4} + T^{5} \)
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