↑

Properties

Label 560.4.a.u.1.2
Level $560$
Weight $4$
Character 560.1
Self dual yes
Analytic conductor $33.041$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [560,4,Mod(1,560)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("560.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 560.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-2,0,15,0,-21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(33.0410696032\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.14360.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 17x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.48565\) of defining polynomial
Character \(\chi\) \(=\) 560.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.850238 q^{3} +5.00000 q^{5} -7.00000 q^{7} -26.2771 q^{9} +6.90764 q^{11} -22.1364 q^{13} -4.25119 q^{15} +88.3030 q^{17} -36.9560 q^{19} +5.95167 q^{21} +95.5283 q^{23} +25.0000 q^{25} +45.2982 q^{27} +269.029 q^{29} -197.114 q^{31} -5.87314 q^{33} -35.0000 q^{35} +2.14546 q^{37} +18.8212 q^{39} +174.127 q^{41} +17.0345 q^{43} -131.385 q^{45} +528.029 q^{47} +49.0000 q^{49} -75.0786 q^{51} -641.114 q^{53} +34.5382 q^{55} +31.4214 q^{57} +642.975 q^{59} +142.967 q^{61} +183.940 q^{63} -110.682 q^{65} -478.797 q^{67} -81.2218 q^{69} -105.550 q^{71} +986.512 q^{73} -21.2560 q^{75} -48.3534 q^{77} +1099.86 q^{79} +670.967 q^{81} +1236.62 q^{83} +441.515 q^{85} -228.739 q^{87} -711.698 q^{89} +154.955 q^{91} +167.594 q^{93} -184.780 q^{95} -636.553 q^{97} -181.513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 15 q^{5} - 21 q^{7} + 81 q^{9} + 74 q^{11} + 44 q^{13} - 10 q^{15} - 52 q^{17} - 168 q^{19} + 14 q^{21} + 124 q^{23} + 75 q^{25} - 170 q^{27} + 332 q^{29} - 320 q^{31} - 106 q^{33} - 105 q^{35}+ \cdots + 3488 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.850238 −0.163628 −0.0818142 − 0.996648i \(-0.526071\pi\)
−0.0818142 + 0.996648i \(0.526071\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) −26.2771 −0.973226
\(10\) 0 0
\(11\) 6.90764 0.189339 0.0946696 − 0.995509i \(-0.469821\pi\)
0.0946696 + 0.995509i \(0.469821\pi\)
\(12\) 0 0
\(13\) −22.1364 −0.472272 −0.236136 − 0.971720i \(-0.575881\pi\)
−0.236136 + 0.971720i \(0.575881\pi\)
\(14\) 0 0
\(15\) −4.25119 −0.0731769
\(16\) 0 0
\(17\) 88.3030 1.25980 0.629901 − 0.776676i \(-0.283096\pi\)
0.629901 + 0.776676i \(0.283096\pi\)
\(18\) 0 0
\(19\) −36.9560 −0.446225 −0.223113 − 0.974793i \(-0.571622\pi\)
−0.223113 + 0.974793i \(0.571622\pi\)
\(20\) 0 0
\(21\) 5.95167 0.0618457
\(22\) 0 0
\(23\) 95.5283 0.866045 0.433022 − 0.901383i \(-0.357447\pi\)
0.433022 + 0.901383i \(0.357447\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 45.2982 0.322876
\(28\) 0 0
\(29\) 269.029 1.72267 0.861336 − 0.508035i \(-0.169628\pi\)
0.861336 + 0.508035i \(0.169628\pi\)
\(30\) 0 0
\(31\) −197.114 −1.14202 −0.571012 − 0.820942i \(-0.693449\pi\)
−0.571012 + 0.820942i \(0.693449\pi\)
\(32\) 0 0
\(33\) −5.87314 −0.0309813
\(34\) 0 0
\(35\) −35.0000 −0.169031
\(36\) 0 0
\(37\) 2.14546 0.00953276 0.00476638 − 0.999989i \(-0.498483\pi\)
0.00476638 + 0.999989i \(0.498483\pi\)
\(38\) 0 0
\(39\) 18.8212 0.0772771
\(40\) 0 0
\(41\) 174.127 0.663271 0.331636 − 0.943408i \(-0.392400\pi\)
0.331636 + 0.943408i \(0.392400\pi\)
\(42\) 0 0
\(43\) 17.0345 0.0604125 0.0302062 − 0.999544i \(-0.490384\pi\)
0.0302062 + 0.999544i \(0.490384\pi\)
\(44\) 0 0
\(45\) −131.385 −0.435240
\(46\) 0 0
\(47\) 528.029 1.63874 0.819371 − 0.573264i \(-0.194323\pi\)
0.819371 + 0.573264i \(0.194323\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −75.0786 −0.206139
\(52\) 0 0
\(53\) −641.114 −1.66158 −0.830790 − 0.556586i \(-0.812111\pi\)
−0.830790 + 0.556586i \(0.812111\pi\)
\(54\) 0 0
\(55\) 34.5382 0.0846750
\(56\) 0 0
\(57\) 31.4214 0.0730151
\(58\) 0 0
\(59\) 642.975 1.41878 0.709391 − 0.704815i \(-0.248970\pi\)
0.709391 + 0.704815i \(0.248970\pi\)
\(60\) 0 0
\(61\) 142.967 0.300083 0.150042 − 0.988680i \(-0.452059\pi\)
0.150042 + 0.988680i \(0.452059\pi\)
\(62\) 0 0
\(63\) 183.940 0.367845
\(64\) 0 0
\(65\) −110.682 −0.211206
\(66\) 0 0
\(67\) −478.797 −0.873050 −0.436525 − 0.899692i \(-0.643791\pi\)
−0.436525 + 0.899692i \(0.643791\pi\)
\(68\) 0 0
\(69\) −81.2218 −0.141710
\(70\) 0 0
\(71\) −105.550 −0.176430 −0.0882150 − 0.996101i \(-0.528116\pi\)
−0.0882150 + 0.996101i \(0.528116\pi\)
\(72\) 0 0
\(73\) 986.512 1.58168 0.790839 − 0.612024i \(-0.209644\pi\)
0.790839 + 0.612024i \(0.209644\pi\)
\(74\) 0 0
\(75\) −21.2560 −0.0327257
\(76\) 0 0
\(77\) −48.3534 −0.0715635
\(78\) 0 0
\(79\) 1099.86 1.56638 0.783190 − 0.621783i \(-0.213591\pi\)
0.783190 + 0.621783i \(0.213591\pi\)
\(80\) 0 0
\(81\) 670.967 0.920394
\(82\) 0 0
\(83\) 1236.62 1.63538 0.817691 − 0.575657i \(-0.195254\pi\)
0.817691 + 0.575657i \(0.195254\pi\)
\(84\) 0 0
\(85\) 441.515 0.563400
\(86\) 0 0
\(87\) −228.739 −0.281878
\(88\) 0 0
\(89\) −711.698 −0.847638 −0.423819 − 0.905747i \(-0.639311\pi\)
−0.423819 + 0.905747i \(0.639311\pi\)
\(90\) 0 0
\(91\) 154.955 0.178502
\(92\) 0 0
\(93\) 167.594 0.186867
\(94\) 0 0
\(95\) −184.780 −0.199558
\(96\) 0 0
\(97\) −636.553 −0.666311 −0.333156 − 0.942872i \(-0.608113\pi\)
−0.333156 + 0.942872i \(0.608113\pi\)
\(98\) 0 0
\(99\) −181.513 −0.184270
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 560.4.a.u.1.2 3
4.3 odd 2 35.4.a.c.1.3 ✓ 3
8.3 odd 2 2240.4.a.bt.1.2 3
8.5 even 2 2240.4.a.bv.1.2 3
12.11 even 2 315.4.a.p.1.1 3
20.3 even 4 175.4.b.e.99.2 6
20.7 even 4 175.4.b.e.99.5 6
20.19 odd 2 175.4.a.f.1.1 3
28.3 even 6 245.4.e.n.226.1 6
28.11 odd 6 245.4.e.m.226.1 6
28.19 even 6 245.4.e.n.116.1 6
28.23 odd 6 245.4.e.m.116.1 6
28.27 even 2 245.4.a.l.1.3 3
60.59 even 2 1575.4.a.ba.1.3 3
84.83 odd 2 2205.4.a.bm.1.1 3
140.139 even 2 1225.4.a.y.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.c.1.3 ✓ 3 4.3 odd 2
175.4.a.f.1.1 3 20.19 odd 2
175.4.b.e.99.2 6 20.3 even 4
175.4.b.e.99.5 6 20.7 even 4
245.4.a.l.1.3 3 28.27 even 2
245.4.e.m.116.1 6 28.23 odd 6
245.4.e.m.226.1 6 28.11 odd 6
245.4.e.n.116.1 6 28.19 even 6
245.4.e.n.226.1 6 28.3 even 6
315.4.a.p.1.1 3 12.11 even 2
560.4.a.u.1.2 3 1.1 even 1 trivial
1225.4.a.y.1.1 3 140.139 even 2
1575.4.a.ba.1.3 3 60.59 even 2
2205.4.a.bm.1.1 3 84.83 odd 2
2240.4.a.bt.1.2 3 8.3 odd 2
2240.4.a.bv.1.2 3 8.5 even 2