Properties

Label 560.6.a.x
Level $560$
Weight $6$
Character orbit 560.a
Self dual yes
Analytic conductor $89.815$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.8149390953\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 2 x^{3} - 232 x^{2} + 60 x + 5808\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 3 + \beta_{1} ) q^{3} + 25 q^{5} -49 q^{7} + ( -49 + 5 \beta_{1} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 3 + \beta_{1} ) q^{3} + 25 q^{5} -49 q^{7} + ( -49 + 5 \beta_{1} - \beta_{3} ) q^{9} + ( 150 + 5 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{11} + ( -243 + 17 \beta_{1} + 9 \beta_{2} - 7 \beta_{3} ) q^{13} + ( 75 + 25 \beta_{1} ) q^{15} + ( -353 + 105 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{17} + ( 293 - 34 \beta_{1} - 7 \beta_{2} - 24 \beta_{3} ) q^{19} + ( -147 - 49 \beta_{1} ) q^{21} + ( 359 + 26 \beta_{1} + 35 \beta_{2} + 36 \beta_{3} ) q^{23} + 625 q^{25} + ( 19 - 257 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{27} + ( 980 + 63 \beta_{1} - 67 \beta_{2} + 58 \beta_{3} ) q^{29} + ( 1217 - 84 \beta_{1} - 99 \beta_{2} + 18 \beta_{3} ) q^{31} + ( 953 + 273 \beta_{1} + 38 \beta_{2} + 32 \beta_{3} ) q^{33} -1225 q^{35} + ( 1951 + 60 \beta_{1} + 101 \beta_{2} - 86 \beta_{3} ) q^{37} + ( 1594 - 61 \beta_{1} + 111 \beta_{2} + 28 \beta_{3} ) q^{39} + ( -2513 + 394 \beta_{1} - 135 \beta_{2} - 32 \beta_{3} ) q^{41} + ( 5069 + 526 \beta_{1} - 23 \beta_{2} - 96 \beta_{3} ) q^{43} + ( -1225 + 125 \beta_{1} - 25 \beta_{3} ) q^{45} + ( 91 + 975 \beta_{1} - 137 \beta_{2} + 161 \beta_{3} ) q^{47} + 2401 q^{49} + ( 18382 - 37 \beta_{1} - 58 \beta_{2} - 65 \beta_{3} ) q^{51} + ( -7302 - 122 \beta_{1} + 118 \beta_{2} + 216 \beta_{3} ) q^{53} + ( 3750 + 125 \beta_{1} + 100 \beta_{2} - 125 \beta_{3} ) q^{55} + ( -5655 + 846 \beta_{1} - 263 \beta_{2} + 264 \beta_{3} ) q^{57} + ( 25516 + 132 \beta_{1} + 258 \beta_{2} - 162 \beta_{3} ) q^{59} + ( -6095 + 106 \beta_{1} - 47 \beta_{2} - 234 \beta_{3} ) q^{61} + ( 2401 - 245 \beta_{1} + 49 \beta_{3} ) q^{63} + ( -6075 + 425 \beta_{1} + 225 \beta_{2} - 175 \beta_{3} ) q^{65} + ( 19952 + 1824 \beta_{1} + 218 \beta_{2} + 214 \beta_{3} ) q^{67} + ( 4587 - 594 \beta_{1} + 811 \beta_{2} - 420 \beta_{3} ) q^{69} + ( 25296 + 1336 \beta_{1} + 172 \beta_{2} + 132 \beta_{3} ) q^{71} + ( -19344 - 2508 \beta_{1} - 114 \beta_{2} - 384 \beta_{3} ) q^{73} + ( 1875 + 625 \beta_{1} ) q^{75} + ( -7350 - 245 \beta_{1} - 196 \beta_{2} + 245 \beta_{3} ) q^{77} + ( 30040 - 2283 \beta_{1} - 643 \beta_{2} - 38 \beta_{3} ) q^{79} + ( -35053 - 1792 \beta_{1} - 78 \beta_{2} + 476 \beta_{3} ) q^{81} + ( 37606 + 5460 \beta_{1} + 204 \beta_{2} - 222 \beta_{3} ) q^{83} + ( -8825 + 2625 \beta_{1} - 50 \beta_{2} - 100 \beta_{3} ) q^{85} + ( 20891 - 143 \beta_{1} - 791 \beta_{2} - 451 \beta_{3} ) q^{87} + ( 12899 + 4394 \beta_{1} - 465 \beta_{2} + 218 \beta_{3} ) q^{89} + ( 11907 - 833 \beta_{1} - 441 \beta_{2} + 343 \beta_{3} ) q^{91} + ( -4617 + 896 \beta_{1} - 1575 \beta_{2} + 120 \beta_{3} ) q^{93} + ( 7325 - 850 \beta_{1} - 175 \beta_{2} - 600 \beta_{3} ) q^{95} + ( -64633 - 2195 \beta_{1} + 270 \beta_{2} - 152 \beta_{3} ) q^{97} + ( 15290 - 630 \beta_{1} - 134 \beta_{2} + 578 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 13 q^{3} + 100 q^{5} - 196 q^{7} - 193 q^{9} + O(q^{10}) \) \( 4 q + 13 q^{3} + 100 q^{5} - 196 q^{7} - 193 q^{9} + 595 q^{11} - 969 q^{13} + 325 q^{15} - 1315 q^{17} + 1090 q^{19} - 637 q^{21} + 1534 q^{23} + 2500 q^{25} - 173 q^{27} + 4099 q^{29} + 4820 q^{31} + 4149 q^{33} - 4900 q^{35} + 7692 q^{37} + 6371 q^{39} - 9722 q^{41} + 20610 q^{43} - 4825 q^{45} + 1661 q^{47} + 9604 q^{49} + 73361 q^{51} - 28898 q^{53} + 14875 q^{55} - 21246 q^{57} + 101872 q^{59} - 24742 q^{61} + 9457 q^{63} - 24225 q^{65} + 82060 q^{67} + 16914 q^{69} + 102784 q^{71} - 80652 q^{73} + 8125 q^{75} - 29155 q^{77} + 117801 q^{79} - 141052 q^{81} + 155440 q^{83} - 32875 q^{85} + 82519 q^{87} + 56426 q^{89} + 47481 q^{91} - 17332 q^{93} + 27250 q^{95} - 261031 q^{97} + 61686 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 232 x^{2} + 60 x + 5808\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3 \nu^{3} + 10 \nu^{2} + 496 \nu - 468 \)\()/140\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 20 \nu^{2} - 352 \nu - 2469 \)\()/35\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 50 \nu^{2} + 352 \nu - 5686 \)\()/70\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2} - 2 \beta_{1} + 4\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} + \beta_{2} + 233\)\()/2\)
\(\nu^{3}\)\(=\)\(24 \beta_{3} - 19 \beta_{2} - 88 \beta_{1} + 315\)

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
−5.33412
15.2911
−13.3078
5.35079
0 −13.9563 0 25.0000 0 −49.0000 0 −48.2216 0
1.2 0 −6.08191 0 25.0000 0 −49.0000 0 −206.010 0
1.3 0 15.6617 0 25.0000 0 −49.0000 0 2.28996 0
1.4 0 17.3765 0 25.0000 0 −49.0000 0 58.9420 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 560.6.a.x 4
4.b odd 2 1 280.6.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.6.a.g 4 4.b odd 2 1
560.6.a.x 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 13 T_{3}^{3} - 305 T_{3}^{2} + 2649 T_{3} + 23100 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(560))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 23100 + 2649 T - 305 T^{2} - 13 T^{3} + T^{4} \)
$5$ \( ( -25 + T )^{4} \)
$7$ \( ( 49 + T )^{4} \)
$11$ \( 1284964060 + 102574535 T - 164881 T^{2} - 595 T^{3} + T^{4} \)
$13$ \( 12745535770 - 542830965 T - 630407 T^{2} + 969 T^{3} + T^{4} \)
$17$ \( 3501744505322 - 3154510191 T - 3705119 T^{2} + 1315 T^{3} + T^{4} \)
$19$ \( 9842142584032 + 3078957224 T - 7690444 T^{2} - 1090 T^{3} + T^{4} \)
$23$ \( 287121438493120 + 27821718488 T - 33705508 T^{2} - 1534 T^{3} + T^{4} \)
$29$ \( 784038032585906 + 115910770719 T - 52251287 T^{2} - 4099 T^{3} + T^{4} \)
$31$ \( -404999125572608 + 394217664448 T - 82762896 T^{2} - 4820 T^{3} + T^{4} \)
$37$ \( 2865972224519856 + 419704433328 T - 101875568 T^{2} - 7692 T^{3} + T^{4} \)
$41$ \( -13073850297889920 - 3898031802528 T - 260369624 T^{2} + 9722 T^{3} + T^{4} \)
$43$ \( -7373470654575520 + 2239577081384 T - 60187372 T^{2} - 20610 T^{3} + T^{4} \)
$47$ \( -5382392451807952 + 5836578288225 T - 717592013 T^{2} - 1661 T^{3} + T^{4} \)
$53$ \( 70949286560194560 - 11931163246080 T - 499403104 T^{2} + 28898 T^{3} + T^{4} \)
$59$ \( 108086380388235520 - 33500021857024 T + 3217283808 T^{2} - 101872 T^{3} + T^{4} \)
$61$ \( 44171240723833984 - 3677368349792 T - 450261160 T^{2} + 24742 T^{3} + T^{4} \)
$67$ \( -958163383738316800 + 75368302888704 T + 105852448 T^{2} - 82060 T^{3} + T^{4} \)
$71$ \( -376555989609676800 - 3596150046720 T + 2720281088 T^{2} - 102784 T^{3} + T^{4} \)
$73$ \( -4579104301939314000 - 254069477394480 T - 1851686064 T^{2} + 80652 T^{3} + T^{4} \)
$79$ \( -11874091357017913464 + 450886298354901 T - 692506397 T^{2} - 117801 T^{3} + T^{4} \)
$83$ \( 16246363578526673152 + 622177884531968 T - 2171809056 T^{2} - 155440 T^{3} + T^{4} \)
$89$ \( 1884246731367194880 + 171111836999040 T - 8588628992 T^{2} - 56426 T^{3} + T^{4} \)
$97$ \( 7448509412577504306 + 772292429955189 T + 22872619633 T^{2} + 261031 T^{3} + T^{4} \)
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