Properties

Label 280.6.a.e
Level $280$
Weight $6$
Character orbit 280.a
Self dual yes
Analytic conductor $44.907$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - 463 x - 1890\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 - \beta_{1} ) q^{3} -25 q^{5} + 49 q^{7} + ( 70 + 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 2 - \beta_{1} ) q^{3} -25 q^{5} + 49 q^{7} + ( 70 + 2 \beta_{1} + \beta_{2} ) q^{9} + ( -193 - 4 \beta_{1} - \beta_{2} ) q^{11} + ( 26 + 23 \beta_{1} - 2 \beta_{2} ) q^{13} + ( -50 + 25 \beta_{1} ) q^{15} + ( -416 - 13 \beta_{1} - 4 \beta_{2} ) q^{17} + ( 315 + 105 \beta_{1} + \beta_{2} ) q^{19} + ( 98 - 49 \beta_{1} ) q^{21} + ( 371 + 41 \beta_{1} + 17 \beta_{2} ) q^{23} + 625 q^{25} + ( -1000 + 47 \beta_{1} + 6 \beta_{2} ) q^{27} + ( 625 + 48 \beta_{1} + 7 \beta_{2} ) q^{29} + ( -2215 + 177 \beta_{1} - 25 \beta_{2} ) q^{31} + ( 886 + 327 \beta_{1} - 4 \beta_{2} ) q^{33} -1225 q^{35} + ( -1351 + 115 \beta_{1} + 5 \beta_{2} ) q^{37} + ( -6983 + 118 \beta_{1} - 39 \beta_{2} ) q^{39} + ( -3195 - 319 \beta_{1} + 17 \beta_{2} ) q^{41} + ( -4099 - \beta_{1} + 43 \beta_{2} ) q^{43} + ( -1750 - 50 \beta_{1} - 25 \beta_{2} ) q^{45} + ( -13688 + 93 \beta_{1} - 38 \beta_{2} ) q^{47} + 2401 q^{49} + ( 3329 + 940 \beta_{1} - 19 \beta_{2} ) q^{51} + ( -12516 - 1060 \beta_{1} + 62 \beta_{2} ) q^{53} + ( 4825 + 100 \beta_{1} + 25 \beta_{2} ) q^{55} + ( -31851 - 853 \beta_{1} - 97 \beta_{2} ) q^{57} + ( -12564 - 644 \beta_{1} - 28 \beta_{2} ) q^{59} + ( -2769 - 633 \beta_{1} + 79 \beta_{2} ) q^{61} + ( 3430 + 98 \beta_{1} + 49 \beta_{2} ) q^{63} + ( -650 - 575 \beta_{1} + 50 \beta_{2} ) q^{65} + ( 23136 + 996 \beta_{1} + 68 \beta_{2} ) q^{67} + ( -12539 - 2541 \beta_{1} + 95 \beta_{2} ) q^{69} + ( -11352 + 152 \beta_{1} - 40 \beta_{2} ) q^{71} + ( -15144 + 926 \beta_{1} - 118 \beta_{2} ) q^{73} + ( 1250 - 625 \beta_{1} ) q^{75} + ( -9457 - 196 \beta_{1} - 49 \beta_{2} ) q^{77} + ( 537 - 2168 \beta_{1} + 229 \beta_{2} ) q^{79} + ( -33749 - 382 \beta_{1} - 242 \beta_{2} ) q^{81} + ( -3414 + 2174 \beta_{1} - 114 \beta_{2} ) q^{83} + ( 10400 + 325 \beta_{1} + 100 \beta_{2} ) q^{85} + ( -13834 - 1643 \beta_{1} + 8 \beta_{2} ) q^{87} + ( -74331 - 1743 \beta_{1} - 183 \beta_{2} ) q^{89} + ( 1274 + 1127 \beta_{1} - 98 \beta_{2} ) q^{91} + ( -58223 + 4457 \beta_{1} - 377 \beta_{2} ) q^{93} + ( -7875 - 2625 \beta_{1} - 25 \beta_{2} ) q^{95} + ( -53080 - 2889 \beta_{1} + 236 \beta_{2} ) q^{97} + ( -52228 - 750 \beta_{1} - 116 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{3} - 75 q^{5} + 147 q^{7} + 209 q^{9} + O(q^{10}) \) \( 3 q + 6 q^{3} - 75 q^{5} + 147 q^{7} + 209 q^{9} - 578 q^{11} + 80 q^{13} - 150 q^{15} - 1244 q^{17} + 944 q^{19} + 294 q^{21} + 1096 q^{23} + 1875 q^{25} - 3006 q^{27} + 1868 q^{29} - 6620 q^{31} + 2662 q^{33} - 3675 q^{35} - 4058 q^{37} - 20910 q^{39} - 9602 q^{41} - 12340 q^{43} - 5225 q^{45} - 41026 q^{47} + 7203 q^{49} + 10006 q^{51} - 37610 q^{53} + 14450 q^{55} - 95456 q^{57} - 37664 q^{59} - 8386 q^{61} + 10241 q^{63} - 2000 q^{65} + 69340 q^{67} - 37712 q^{69} - 34016 q^{71} - 45314 q^{73} + 3750 q^{75} - 28322 q^{77} + 1382 q^{79} - 101005 q^{81} - 10128 q^{83} + 31100 q^{85} - 41510 q^{87} - 222810 q^{89} + 3920 q^{91} - 174292 q^{93} - 23600 q^{95} - 159476 q^{97} - 156568 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 463 x - 1890\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 6 \nu - 309 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 6 \beta_{1} + 309\)

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
23.3245
−4.24759
−19.0769
0 −21.3245 0 −25.0000 0 49.0000 0 211.733 0
1.2 0 6.24759 0 −25.0000 0 49.0000 0 −203.968 0
1.3 0 21.0769 0 −25.0000 0 49.0000 0 201.235 0
\(n\): e.g. 2-40 or 990-1000
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.e 3
4.b odd 2 1 560.6.a.s 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.6.a.e 3 1.a even 1 1 trivial
560.6.a.s 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 6 T_{3}^{2} - 451 T_{3} + 2808 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(280))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 2808 - 451 T - 6 T^{2} + T^{3} \)
$5$ \( ( 25 + T )^{3} \)
$7$ \( ( -49 + T )^{3} \)
$11$ \( -9760932 + 49393 T + 578 T^{2} + T^{3} \)
$13$ \( 128496494 - 453967 T - 80 T^{2} + T^{3} \)
$17$ \( -651843290 - 433999 T + 1244 T^{2} + T^{3} \)
$19$ \( -1721529888 - 4873604 T - 944 T^{2} + T^{3} \)
$23$ \( 31363349376 - 16093508 T - 1096 T^{2} + T^{3} \)
$29$ \( 3100258778 - 2593027 T - 1868 T^{2} + T^{3} \)
$31$ \( -16721033600 - 33207952 T + 6620 T^{2} + T^{3} \)
$37$ \( -15436371144 - 2052212 T + 4058 T^{2} + T^{3} \)
$41$ \( -330620399712 - 31400472 T + 9602 T^{2} + T^{3} \)
$43$ \( -1675236784 - 49275940 T + 12340 T^{2} + T^{3} \)
$47$ \( 1323866448632 + 479315957 T + 41026 T^{2} + T^{3} \)
$53$ \( -13966928161632 - 249207744 T + 37610 T^{2} + T^{3} \)
$59$ \( 363808567296 + 236362752 T + 37664 T^{2} + T^{3} \)
$61$ \( -4786876164960 - 494041688 T + 8386 T^{2} + T^{3} \)
$67$ \( -705690174528 + 885475696 T - 69340 T^{2} + T^{3} \)
$71$ \( 336586973184 + 289125120 T + 34016 T^{2} + T^{3} \)
$73$ \( -3081736975464 - 453450708 T + 45314 T^{2} + T^{3} \)
$79$ \( -116381435931432 - 4956254607 T - 1382 T^{2} + T^{3} \)
$83$ \( 41224716201984 - 2828960272 T + 10128 T^{2} + T^{3} \)
$89$ \( 175059765854496 + 13293219528 T + 222810 T^{2} + T^{3} \)
$97$ \( -426037755101066 + 1677515609 T + 159476 T^{2} + T^{3} \)
show more
show less