Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8670,2,Mod(1,8670)] 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("8670.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8670, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8670.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,1,1,1,1,4,1,1,1,4,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(69.2302985525\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 510)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8670.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} +4.00000 q^{21} +4.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} -2.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +4.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} +4.00000 q^{42} -12.0000 q^{43} +4.00000 q^{44} +1.00000 q^{45} +4.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} +4.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} -2.00000 q^{61} -4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} +4.00000 q^{69} +4.00000 q^{70} +4.00000 q^{71} +1.00000 q^{72} +14.0000 q^{73} +6.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +16.0000 q^{77} -2.00000 q^{78} +12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -4.00000 q^{83} +4.00000 q^{84} -12.0000 q^{86} -2.00000 q^{87} +4.00000 q^{88} +10.0000 q^{89} +1.00000 q^{90} -8.00000 q^{91} +4.00000 q^{92} -4.00000 q^{93} +8.00000 q^{94} -4.00000 q^{95} +1.00000 q^{96} -18.0000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.00000 1.06904
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 0.223607
\(21\) 4.00000 0.872872
\(22\) 4.00000 0.852803
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) −2.00000 −0.320256
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 4.00000 0.617213
\(43\) −12.0000 −1.82998 −0.914991 − 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.00000 0.149071
\(46\) 4.00000 0.589768
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.00000 0.539360
\(56\) 4.00000 0.534522
\(57\) −4.00000 −0.529813
\(58\) −2.00000 −0.262613
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 1.00000 0.129099
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −4.00000 −0.508001
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 4.00000 0.492366
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 4.00000 0.478091
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 1.00000 0.117851
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 16.0000 1.82337
\(78\) −2.00000 −0.226455
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) −2.00000 −0.214423
\(88\) 4.00000 0.426401
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 1.00000 0.105409
\(91\) −8.00000 −0.838628
\(92\) 4.00000 0.417029
\(93\) −4.00000 −0.414781
\(94\) 8.00000 0.825137
\(95\) −4.00000 −0.410391
\(96\) 1.00000 0.102062
\(97\) −18.0000 −1.82762 −0.913812 − 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 9.00000 0.909137
\(99\) 4.00000 0.402015
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 8670.2.a.bb.1.1 1
17.16 even 2 510.2.a.c.1.1 ✓ 1
51.50 odd 2 1530.2.a.d.1.1 1
68.67 odd 2 4080.2.a.x.1.1 1
85.33 odd 4 2550.2.d.b.2449.1 2
85.67 odd 4 2550.2.d.b.2449.2 2
85.84 even 2 2550.2.a.n.1.1 1
255.254 odd 2 7650.2.a.cn.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.c.1.1 ✓ 1 17.16 even 2
1530.2.a.d.1.1 1 51.50 odd 2
2550.2.a.n.1.1 1 85.84 even 2
2550.2.d.b.2449.1 2 85.33 odd 4
2550.2.d.b.2449.2 2 85.67 odd 4
4080.2.a.x.1.1 1 68.67 odd 2
7650.2.a.cn.1.1 1 255.254 odd 2
8670.2.a.bb.1.1 1 1.1 even 1 trivial