Properties

Label 280.6.a.d
Level $280$
Weight $6$
Character orbit 280.a
Self dual yes
Analytic conductor $44.907$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{37}) \)
Defining polynomial: \(x^{2} - x - 9\)
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 \(\beta = 2\sqrt{37}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 13 - \beta ) q^{3} + 25 q^{5} -49 q^{7} + ( 74 - 26 \beta ) q^{9} +O(q^{10})\) \( q + ( 13 - \beta ) q^{3} + 25 q^{5} -49 q^{7} + ( 74 - 26 \beta ) q^{9} + ( -7 - 22 \beta ) q^{11} + ( 157 + 5 \beta ) q^{13} + ( 325 - 25 \beta ) q^{15} + ( -73 + 21 \beta ) q^{17} + ( 1424 + 71 \beta ) q^{19} + ( -637 + 49 \beta ) q^{21} + ( 1996 - 71 \beta ) q^{23} + 625 q^{25} + ( 1651 - 169 \beta ) q^{27} + ( 3003 - 96 \beta ) q^{29} + ( 610 + 3 \beta ) q^{31} + ( 3165 - 279 \beta ) q^{33} -1225 q^{35} + ( 3616 - 825 \beta ) q^{37} + ( 1301 - 92 \beta ) q^{39} + ( 3258 + 611 \beta ) q^{41} + ( -1168 - 509 \beta ) q^{43} + ( 1850 - 650 \beta ) q^{45} + ( 6465 + 1521 \beta ) q^{47} + 2401 q^{49} + ( -4057 + 346 \beta ) q^{51} + ( 1608 - 1232 \beta ) q^{53} + ( -175 - 550 \beta ) q^{55} + ( 8004 - 501 \beta ) q^{57} + ( 3364 + 3036 \beta ) q^{59} + ( -378 + 2157 \beta ) q^{61} + ( -3626 + 1274 \beta ) q^{63} + ( 3925 + 125 \beta ) q^{65} + ( 5872 + 2608 \beta ) q^{67} + ( 36456 - 2919 \beta ) q^{69} + ( -952 - 952 \beta ) q^{71} + ( 11306 + 2554 \beta ) q^{73} + ( 8125 - 625 \beta ) q^{75} + ( 343 + 1078 \beta ) q^{77} + ( 31243 - 3218 \beta ) q^{79} + ( 28493 + 2470 \beta ) q^{81} + ( 7600 - 462 \beta ) q^{83} + ( -1825 + 525 \beta ) q^{85} + ( 53247 - 4251 \beta ) q^{87} + ( 21422 - 3005 \beta ) q^{89} + ( -7693 - 245 \beta ) q^{91} + ( 7486 - 571 \beta ) q^{93} + ( 35600 + 1775 \beta ) q^{95} + ( 41411 + 2365 \beta ) q^{97} + ( 84138 - 1446 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 26 q^{3} + 50 q^{5} - 98 q^{7} + 148 q^{9} + O(q^{10}) \) \( 2 q + 26 q^{3} + 50 q^{5} - 98 q^{7} + 148 q^{9} - 14 q^{11} + 314 q^{13} + 650 q^{15} - 146 q^{17} + 2848 q^{19} - 1274 q^{21} + 3992 q^{23} + 1250 q^{25} + 3302 q^{27} + 6006 q^{29} + 1220 q^{31} + 6330 q^{33} - 2450 q^{35} + 7232 q^{37} + 2602 q^{39} + 6516 q^{41} - 2336 q^{43} + 3700 q^{45} + 12930 q^{47} + 4802 q^{49} - 8114 q^{51} + 3216 q^{53} - 350 q^{55} + 16008 q^{57} + 6728 q^{59} - 756 q^{61} - 7252 q^{63} + 7850 q^{65} + 11744 q^{67} + 72912 q^{69} - 1904 q^{71} + 22612 q^{73} + 16250 q^{75} + 686 q^{77} + 62486 q^{79} + 56986 q^{81} + 15200 q^{83} - 3650 q^{85} + 106494 q^{87} + 42844 q^{89} - 15386 q^{91} + 14972 q^{93} + 71200 q^{95} + 82822 q^{97} + 168276 q^{99} + O(q^{100}) \)

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
3.54138
−2.54138
0 0.834475 0 25.0000 0 −49.0000 0 −242.304 0
1.2 0 25.1655 0 25.0000 0 −49.0000 0 390.304 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.d 2
4.b odd 2 1 560.6.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.6.a.d 2 1.a even 1 1 trivial
560.6.a.j 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 26 T_{3} + 21 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(280))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 21 - 26 T + T^{2} \)
$5$ \( ( -25 + T )^{2} \)
$7$ \( ( 49 + T )^{2} \)
$11$ \( -71583 + 14 T + T^{2} \)
$13$ \( 20949 - 314 T + T^{2} \)
$17$ \( -59939 + 146 T + T^{2} \)
$19$ \( 1281708 - 2848 T + T^{2} \)
$23$ \( 3237948 - 3992 T + T^{2} \)
$29$ \( 7654041 - 6006 T + T^{2} \)
$31$ \( 370768 - 1220 T + T^{2} \)
$37$ \( -87657044 - 7232 T + T^{2} \)
$41$ \( -44636944 - 6516 T + T^{2} \)
$43$ \( -36979764 + 2336 T + T^{2} \)
$47$ \( -300593043 - 12930 T + T^{2} \)
$53$ \( -222052288 - 3216 T + T^{2} \)
$59$ \( -1352843312 - 6728 T + T^{2} \)
$61$ \( -688449168 + 756 T + T^{2} \)
$67$ \( -972165888 - 11744 T + T^{2} \)
$71$ \( -133226688 + 1904 T + T^{2} \)
$73$ \( -837565932 - 22612 T + T^{2} \)
$79$ \( -556492503 - 62486 T + T^{2} \)
$83$ \( 26170288 - 15200 T + T^{2} \)
$89$ \( -877541616 - 42844 T + T^{2} \)
$97$ \( 887073621 - 82822 T + T^{2} \)
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