Properties

Label 280.6.a.i
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: \(x^{5} - 2 x^{4} - 791 x^{3} + 280 x^{2} + 24832 x + 39040\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{3} -25 q^{5} -49 q^{7} + ( 74 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{3} -25 q^{5} -49 q^{7} + ( 74 + \beta_{3} ) q^{9} + ( -54 + 3 \beta_{1} - \beta_{2} - \beta_{4} ) q^{11} + ( -152 + 18 \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{13} + ( 25 - 25 \beta_{1} ) q^{15} + ( -225 + 59 \beta_{1} + 4 \beta_{2} ) q^{17} + ( -517 + 34 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} ) q^{19} + ( 49 - 49 \beta_{1} ) q^{21} + ( -241 + 86 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} + 10 \beta_{4} ) q^{23} + 625 q^{25} + ( 26 + 240 \beta_{1} + 9 \beta_{2} - \beta_{3} + \beta_{4} ) q^{27} + ( 695 - 8 \beta_{2} - 11 \beta_{4} ) q^{29} + ( 1477 + 262 \beta_{1} - 15 \beta_{2} + 3 \beta_{3} + 10 \beta_{4} ) q^{31} + ( 982 - 18 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} - \beta_{4} ) q^{33} + 1225 q^{35} + ( 2674 - 117 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} + 15 \beta_{4} ) q^{37} + ( 5493 - 164 \beta_{1} + 6 \beta_{2} + 16 \beta_{3} - 7 \beta_{4} ) q^{39} + ( 4683 + 226 \beta_{1} + 31 \beta_{2} + 15 \beta_{3} - 22 \beta_{4} ) q^{41} + ( 544 + 595 \beta_{1} + 2 \beta_{2} - 12 \beta_{3} - 45 \beta_{4} ) q^{43} + ( -1850 - 25 \beta_{3} ) q^{45} + ( 1860 - 116 \beta_{1} + 33 \beta_{2} + 22 \beta_{4} ) q^{47} + 2401 q^{49} + ( 19385 - 214 \beta_{1} + 83 \beta_{3} + 12 \beta_{4} ) q^{51} + ( 523 - 167 \beta_{1} + 19 \beta_{2} - 63 \beta_{3} + 23 \beta_{4} ) q^{53} + ( 1350 - 75 \beta_{1} + 25 \beta_{2} + 25 \beta_{4} ) q^{55} + ( 12989 - 1644 \beta_{1} - 45 \beta_{2} + 55 \beta_{3} + 24 \beta_{4} ) q^{57} + ( -3070 + 86 \beta_{1} - 102 \beta_{2} - 60 \beta_{3} + 56 \beta_{4} ) q^{59} + ( 1159 - 448 \beta_{1} - 49 \beta_{2} - 75 \beta_{3} + 70 \beta_{4} ) q^{61} + ( -3626 - 49 \beta_{3} ) q^{63} + ( 3800 - 450 \beta_{1} + 25 \beta_{2} - 50 \beta_{4} ) q^{65} + ( 1169 + 903 \beta_{1} - 15 \beta_{2} - 51 \beta_{3} - 123 \beta_{4} ) q^{67} + ( 25853 - 1508 \beta_{1} + 3 \beta_{2} + 67 \beta_{3} - 44 \beta_{4} ) q^{69} + ( 6311 + 477 \beta_{1} + 27 \beta_{2} + 21 \beta_{3} - 87 \beta_{4} ) q^{71} + ( 38163 - 839 \beta_{1} + 93 \beta_{2} + 39 \beta_{3} - 95 \beta_{4} ) q^{73} + ( -625 + 625 \beta_{1} ) q^{75} + ( 2646 - 147 \beta_{1} + 49 \beta_{2} + 49 \beta_{4} ) q^{77} + ( 9260 - 2271 \beta_{1} + 171 \beta_{2} + 33 \beta_{3} - 30 \beta_{4} ) q^{79} + ( 59027 - 272 \beta_{1} - 6 \beta_{2} + 54 \beta_{3} + 24 \beta_{4} ) q^{81} + ( 11521 + 669 \beta_{1} - 167 \beta_{2} + 21 \beta_{3} + 199 \beta_{4} ) q^{83} + ( 5625 - 1475 \beta_{1} - 100 \beta_{2} ) q^{85} + ( -528 + 1022 \beta_{1} - 33 \beta_{2} - 70 \beta_{3} - 2 \beta_{4} ) q^{87} + ( 47579 - 3178 \beta_{1} - 17 \beta_{2} + 33 \beta_{3} + 132 \beta_{4} ) q^{89} + ( 7448 - 882 \beta_{1} + 49 \beta_{2} - 98 \beta_{4} ) q^{91} + ( 77861 + 2936 \beta_{1} + 57 \beta_{2} + 189 \beta_{3} - 62 \beta_{4} ) q^{93} + ( 12925 - 850 \beta_{1} - 125 \beta_{2} + 75 \beta_{3} + 150 \beta_{4} ) q^{95} + ( 48859 + 1551 \beta_{1} - 196 \beta_{2} + 108 \beta_{3} - 76 \beta_{4} ) q^{97} + ( 6889 - 1753 \beta_{1} + 195 \beta_{2} - 33 \beta_{3} + 231 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{3} - 125 q^{5} - 245 q^{7} + 372 q^{9} + O(q^{10}) \) \( 5 q - 3 q^{3} - 125 q^{5} - 245 q^{7} + 372 q^{9} - 263 q^{11} - 729 q^{13} + 75 q^{15} - 1003 q^{17} - 2506 q^{19} + 147 q^{21} - 1066 q^{23} + 3125 q^{25} + 615 q^{27} + 3489 q^{29} + 7880 q^{31} + 4863 q^{33} + 6125 q^{35} + 13118 q^{37} + 27189 q^{39} + 23972 q^{41} + 3978 q^{43} - 9300 q^{45} + 9057 q^{47} + 12005 q^{49} + 96639 q^{51} + 2128 q^{53} + 6575 q^{55} + 61674 q^{57} - 15512 q^{59} + 4560 q^{61} - 18228 q^{63} + 18225 q^{65} + 7780 q^{67} + 126474 q^{69} + 32752 q^{71} + 189498 q^{73} - 1875 q^{75} + 12887 q^{77} + 42055 q^{79} + 294645 q^{81} + 58420 q^{83} + 25075 q^{85} - 765 q^{87} + 231324 q^{89} + 35721 q^{91} + 395736 q^{93} + 62650 q^{95} + 247569 q^{97} + 30606 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 791 x^{3} + 280 x^{2} + 24832 x + 39040\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} + 28 \nu^{3} + 729 \nu^{2} - 19234 \nu - 12926 \)\()/222\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 2 \nu - 316 \)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{4} - 10 \nu^{3} - 2335 \nu^{2} + 4052 \nu + 49360 \)\()/74\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 2 \beta_{1} + 316\)
\(\nu^{3}\)\(=\)\(\beta_{4} + 2 \beta_{3} + 9 \beta_{2} + 729 \beta_{1} + 489\)
\(\nu^{4}\)\(=\)\(28 \beta_{4} + 785 \beta_{3} + 30 \beta_{2} + 2636 \beta_{1} + 231130\)

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
−26.7452
−4.39132
−1.78912
6.54694
28.3787
0 −27.7452 0 −25.0000 0 −49.0000 0 526.795 0
1.2 0 −5.39132 0 −25.0000 0 −49.0000 0 −213.934 0
1.3 0 −2.78912 0 −25.0000 0 −49.0000 0 −235.221 0
1.4 0 5.54694 0 −25.0000 0 −49.0000 0 −212.231 0
1.5 0 27.3787 0 −25.0000 0 −49.0000 0 506.591 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.i 5
4.b odd 2 1 560.6.a.z 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.6.a.i 5 1.a even 1 1 trivial
560.6.a.z 5 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} + 3 T_{3}^{4} - 789 T_{3}^{3} - 2095 T_{3}^{2} + 23016 T_{3} + 63360 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(280))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( 63360 + 23016 T - 2095 T^{2} - 789 T^{3} + 3 T^{4} + T^{5} \)
$5$ \( ( 25 + T )^{5} \)
$7$ \( ( 49 + T )^{5} \)
$11$ \( 144603094960 - 6765547024 T - 156276923 T^{2} - 465077 T^{3} + 263 T^{4} + T^{5} \)
$13$ \( 16843416666300 + 71030006352 T - 337552377 T^{2} - 601965 T^{3} + 729 T^{4} + T^{5} \)
$17$ \( -650694463696732 + 7135935288320 T - 2889097555 T^{2} - 5544197 T^{3} + 1003 T^{4} + T^{5} \)
$19$ \( 166563647234048 - 8422687257472 T - 22247271880 T^{2} - 8124308 T^{3} + 2506 T^{4} + T^{5} \)
$23$ \( -39109431666688000 + 105868493090432 T - 22901567560 T^{2} - 24570260 T^{3} + 1066 T^{4} + T^{5} \)
$29$ \( 8173830646541988 + 57453935362992 T + 53474470161 T^{2} - 38771613 T^{3} - 3489 T^{4} + T^{5} \)
$31$ \( -4054992663402422272 + 456696901271552 T + 522867356288 T^{2} - 68360192 T^{3} - 7880 T^{4} + T^{5} \)
$37$ \( -604935778749836896 - 1292225469830320 T + 631736139536 T^{2} - 29544344 T^{3} - 13118 T^{4} + T^{5} \)
$41$ \( -36527847479911805440 + 5829203940220928 T + 2687881258688 T^{2} - 65088476 T^{3} - 23972 T^{4} + T^{5} \)
$43$ \( -\)\(73\!\cdots\!40\)\( + 99596388679947072 T + 3334556966376 T^{2} - 645112260 T^{3} - 3978 T^{4} + T^{5} \)
$47$ \( 3043740583883797056 - 8616051590003568 T + 4500008008173 T^{2} - 358556805 T^{3} - 9057 T^{4} + T^{5} \)
$53$ \( -\)\(12\!\cdots\!00\)\( + 595941744488599232 T + 2026707625744 T^{2} - 1596800492 T^{3} - 2128 T^{4} + T^{5} \)
$59$ \( \)\(37\!\cdots\!00\)\( + 1752861584729329664 T - 54462342984704 T^{2} - 2794686848 T^{3} + 15512 T^{4} + T^{5} \)
$61$ \( \)\(44\!\cdots\!04\)\( + 1041404248474937280 T - 1740344935344 T^{2} - 2915865756 T^{3} - 4560 T^{4} + T^{5} \)
$67$ \( -\)\(45\!\cdots\!80\)\( + 2869135268358993152 T - 12878787471488 T^{2} - 4106766176 T^{3} - 7780 T^{4} + T^{5} \)
$71$ \( \)\(14\!\cdots\!20\)\( - 587570105150599168 T + 51956333314048 T^{2} - 1052097152 T^{3} - 32752 T^{4} + T^{5} \)
$73$ \( \)\(10\!\cdots\!00\)\( - 648838738520092080 T - 204318382934736 T^{2} + 11395038888 T^{3} - 189498 T^{4} + T^{5} \)
$79$ \( -\)\(84\!\cdots\!44\)\( + 9774186196609757576 T + 323870192349499 T^{2} - 8389624637 T^{3} - 42055 T^{4} + T^{5} \)
$83$ \( \)\(76\!\cdots\!56\)\( - 11279124407682953728 T + 509025476875840 T^{2} - 6442415024 T^{3} - 58420 T^{4} + T^{5} \)
$89$ \( -\)\(13\!\cdots\!20\)\( - 7614820945226278656 T + 393150855540768 T^{2} + 10253181924 T^{3} - 231324 T^{4} + T^{5} \)
$97$ \( -\)\(30\!\cdots\!08\)\( - 44999444284380112752 T + 1459897974159537 T^{2} + 8032414851 T^{3} - 247569 T^{4} + T^{5} \)
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