Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9280,2,Mod(1,9280)] 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("9280.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9280.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-4,0,-2,0,4,0,2,0,-4,0,4,0,4,0,0,0,-4,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(74.1011730757\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9280.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} -1.00000 q^{5} -0.828427 q^{7} +1.00000 q^{9} -4.82843 q^{11} +2.00000 q^{13} +2.00000 q^{15} -2.82843 q^{17} +0.828427 q^{19} +1.65685 q^{21} +8.82843 q^{23} +1.00000 q^{25} +4.00000 q^{27} -1.00000 q^{29} +10.4853 q^{31} +9.65685 q^{33} +0.828427 q^{35} -8.48528 q^{37} -4.00000 q^{39} -6.00000 q^{41} -6.00000 q^{43} -1.00000 q^{45} +0.343146 q^{47} -6.31371 q^{49} +5.65685 q^{51} -7.65685 q^{53} +4.82843 q^{55} -1.65685 q^{57} -7.65685 q^{61} -0.828427 q^{63} -2.00000 q^{65} -10.4853 q^{67} -17.6569 q^{69} -7.31371 q^{71} -8.48528 q^{73} -2.00000 q^{75} +4.00000 q^{77} -14.4853 q^{79} -11.0000 q^{81} +12.8284 q^{83} +2.82843 q^{85} +2.00000 q^{87} +3.65685 q^{89} -1.65685 q^{91} -20.9706 q^{93} -0.828427 q^{95} +4.48528 q^{97} -4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{13} + 4 q^{15} - 4 q^{19} - 8 q^{21} + 12 q^{23} + 2 q^{25} + 8 q^{27} - 2 q^{29} + 4 q^{31} + 8 q^{33} - 4 q^{35} - 8 q^{39} - 12 q^{41} - 12 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) \) 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
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.828427 −0.313116 −0.156558 − 0.987669i \(-0.550040\pi\)
−0.156558 + 0.987669i \(0.550040\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.82843 −1.45583 −0.727913 − 0.685670i \(-0.759509\pi\)
−0.727913 + 0.685670i \(0.759509\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) −2.82843 −0.685994 −0.342997 − 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 0 0
\(19\) 0.828427 0.190054 0.0950271 − 0.995475i \(-0.469706\pi\)
0.0950271 + 0.995475i \(0.469706\pi\)
\(20\) 0 0
\(21\) 1.65685 0.361555
\(22\) 0 0
\(23\) 8.82843 1.84085 0.920427 − 0.390914i \(-0.127841\pi\)
0.920427 + 0.390914i \(0.127841\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 10.4853 1.88321 0.941606 − 0.336717i \(-0.109316\pi\)
0.941606 + 0.336717i \(0.109316\pi\)
\(32\) 0 0
\(33\) 9.65685 1.68104
\(34\) 0 0
\(35\) 0.828427 0.140030
\(36\) 0 0
\(37\) −8.48528 −1.39497 −0.697486 − 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0.343146 0.0500530 0.0250265 − 0.999687i \(-0.492033\pi\)
0.0250265 + 0.999687i \(0.492033\pi\)
\(48\) 0 0
\(49\) −6.31371 −0.901958
\(50\) 0 0
\(51\) 5.65685 0.792118
\(52\) 0 0
\(53\) −7.65685 −1.05175 −0.525875 − 0.850562i \(-0.676262\pi\)
−0.525875 + 0.850562i \(0.676262\pi\)
\(54\) 0 0
\(55\) 4.82843 0.651065
\(56\) 0 0
\(57\) −1.65685 −0.219456
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −7.65685 −0.980360 −0.490180 − 0.871621i \(-0.663069\pi\)
−0.490180 + 0.871621i \(0.663069\pi\)
\(62\) 0 0
\(63\) −0.828427 −0.104372
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −10.4853 −1.28098 −0.640490 − 0.767966i \(-0.721269\pi\)
−0.640490 + 0.767966i \(0.721269\pi\)
\(68\) 0 0
\(69\) −17.6569 −2.12564
\(70\) 0 0
\(71\) −7.31371 −0.867978 −0.433989 − 0.900918i \(-0.642894\pi\)
−0.433989 + 0.900918i \(0.642894\pi\)
\(72\) 0 0
\(73\) −8.48528 −0.993127 −0.496564 − 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) 0 0
\(75\) −2.00000 −0.230940
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −14.4853 −1.62972 −0.814861 − 0.579657i \(-0.803187\pi\)
−0.814861 + 0.579657i \(0.803187\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 12.8284 1.40810 0.704051 − 0.710149i \(-0.251372\pi\)
0.704051 + 0.710149i \(0.251372\pi\)
\(84\) 0 0
\(85\) 2.82843 0.306786
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 3.65685 0.387626 0.193813 − 0.981039i \(-0.437915\pi\)
0.193813 + 0.981039i \(0.437915\pi\)
\(90\) 0 0
\(91\) −1.65685 −0.173686
\(92\) 0 0
\(93\) −20.9706 −2.17455
\(94\) 0 0
\(95\) −0.828427 −0.0849948
\(96\) 0 0
\(97\) 4.48528 0.455411 0.227706 − 0.973730i \(-0.426878\pi\)
0.227706 + 0.973730i \(0.426878\pi\)
\(98\) 0 0
\(99\) −4.82843 −0.485275
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 9280.2.a.w.1.1 2
4.3 odd 2 9280.2.a.be.1.2 2
8.3 odd 2 145.2.a.b.1.1 ✓ 2
8.5 even 2 2320.2.a.k.1.1 2
24.11 even 2 1305.2.a.n.1.2 2
40.3 even 4 725.2.b.c.349.4 4
40.19 odd 2 725.2.a.c.1.2 2
40.27 even 4 725.2.b.c.349.1 4
56.27 even 2 7105.2.a.e.1.1 2
120.59 even 2 6525.2.a.p.1.1 2
232.115 odd 2 4205.2.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.1 ✓ 2 8.3 odd 2
725.2.a.c.1.2 2 40.19 odd 2
725.2.b.c.349.1 4 40.27 even 4
725.2.b.c.349.4 4 40.3 even 4
1305.2.a.n.1.2 2 24.11 even 2
2320.2.a.k.1.1 2 8.5 even 2
4205.2.a.d.1.2 2 232.115 odd 2
6525.2.a.p.1.1 2 120.59 even 2
7105.2.a.e.1.1 2 56.27 even 2
9280.2.a.w.1.1 2 1.1 even 1 trivial
9280.2.a.be.1.2 2 4.3 odd 2