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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [160,4,Mod(47,160)] 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("160.47"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(160, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 160.o (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.44030560092\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 47.5
Character \(\chi\) \(=\) 160.47
Dual form 160.4.o.a.143.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.49003 + 3.49003i) q^{3} +(-4.99441 + 10.0028i) q^{5} +(4.97302 - 4.97302i) q^{7} +2.63942i q^{9} -29.8229 q^{11} +(13.5734 + 13.5734i) q^{13} +(-17.4794 - 52.3406i) q^{15} +(-61.7929 - 61.7929i) q^{17} -131.323i q^{19} +34.7119i q^{21} +(1.13009 + 1.13009i) q^{23} +(-75.1118 - 99.9161i) q^{25} +(-103.442 - 103.442i) q^{27} -179.304 q^{29} +276.096i q^{31} +(104.083 - 104.083i) q^{33} +(24.9068 + 74.5813i) q^{35} +(278.964 - 278.964i) q^{37} -94.7429 q^{39} -225.140 q^{41} +(-171.874 + 171.874i) q^{43} +(-26.4016 - 13.1824i) q^{45} +(86.1627 - 86.1627i) q^{47} +293.538i q^{49} +431.318 q^{51} +(-108.361 - 108.361i) q^{53} +(148.948 - 298.312i) q^{55} +(458.322 + 458.322i) q^{57} +157.272i q^{59} +791.867i q^{61} +(13.1259 + 13.1259i) q^{63} +(-203.563 + 67.9807i) q^{65} +(3.80296 + 3.80296i) q^{67} -7.88811 q^{69} -58.6097i q^{71} +(452.731 - 452.731i) q^{73} +(610.852 + 86.5677i) q^{75} +(-148.310 + 148.310i) q^{77} -821.590 q^{79} +650.769 q^{81} +(-512.512 + 512.512i) q^{83} +(926.721 - 309.483i) q^{85} +(625.775 - 625.775i) q^{87} +500.262i q^{89} +135.001 q^{91} +(-963.581 - 963.581i) q^{93} +(1313.60 + 655.882i) q^{95} +(-60.7068 - 60.7068i) q^{97} -78.7153i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 4 q^{3} + 8 q^{11} + 48 q^{17} + 40 q^{25} - 104 q^{27} - 112 q^{33} + 460 q^{35} - 8 q^{41} + 868 q^{43} - 1480 q^{51} + 104 q^{57} + 520 q^{65} + 1852 q^{67} - 744 q^{73} - 3300 q^{75} - 1240 q^{81}+ \cdots - 584 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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
\(3\) −3.49003 + 3.49003i −0.671656 + 0.671656i −0.958098 − 0.286442i \(-0.907528\pi\)
0.286442 + 0.958098i \(0.407528\pi\)
\(4\) 0 0
\(5\) −4.99441 + 10.0028i −0.446713 + 0.894677i
\(6\) 0 0
\(7\) 4.97302 − 4.97302i 0.268518 − 0.268518i −0.559985 − 0.828503i \(-0.689193\pi\)
0.828503 + 0.559985i \(0.189193\pi\)
\(8\) 0 0
\(9\) 2.63942i 0.0977564i
\(10\) 0 0
\(11\) −29.8229 −0.817449 −0.408725 − 0.912658i \(-0.634026\pi\)
−0.408725 + 0.912658i \(0.634026\pi\)
\(12\) 0 0
\(13\) 13.5734 + 13.5734i 0.289583 + 0.289583i 0.836915 − 0.547332i \(-0.184357\pi\)
−0.547332 + 0.836915i \(0.684357\pi\)
\(14\) 0 0
\(15\) −17.4794 − 52.3406i −0.300877 − 0.900953i
\(16\) 0 0
\(17\) −61.7929 − 61.7929i −0.881588 − 0.881588i 0.112108 − 0.993696i \(-0.464240\pi\)
−0.993696 + 0.112108i \(0.964240\pi\)
\(18\) 0 0
\(19\) − 131.323i − 1.58566i −0.609440 − 0.792832i \(-0.708606\pi\)
0.609440 − 0.792832i \(-0.291394\pi\)
\(20\) 0 0
\(21\) 34.7119i 0.360703i
\(22\) 0 0
\(23\) 1.13009 + 1.13009i 0.0102452 + 0.0102452i 0.712211 − 0.701966i \(-0.247694\pi\)
−0.701966 + 0.712211i \(0.747694\pi\)
\(24\) 0 0
\(25\) −75.1118 − 99.9161i −0.600894 − 0.799329i
\(26\) 0 0
\(27\) −103.442 − 103.442i −0.737315 − 0.737315i
\(28\) 0 0
\(29\) −179.304 −1.14813 −0.574067 − 0.818809i \(-0.694635\pi\)
−0.574067 + 0.818809i \(0.694635\pi\)
\(30\) 0 0
\(31\) 276.096i 1.59962i 0.600253 + 0.799810i \(0.295067\pi\)
−0.600253 + 0.799810i \(0.704933\pi\)
\(32\) 0 0
\(33\) 104.083 − 104.083i 0.549045 − 0.549045i
\(34\) 0 0
\(35\) 24.9068 + 74.5813i 0.120286 + 0.360187i
\(36\) 0 0
\(37\) 278.964 − 278.964i 1.23950 − 1.23950i 0.279289 − 0.960207i \(-0.409901\pi\)
0.960207 − 0.279289i \(-0.0900988\pi\)
\(38\) 0 0
\(39\) −94.7429 −0.389000
\(40\) 0 0
\(41\) −225.140 −0.857585 −0.428793 − 0.903403i \(-0.641061\pi\)
−0.428793 + 0.903403i \(0.641061\pi\)
\(42\) 0 0
\(43\) −171.874 + 171.874i −0.609546 + 0.609546i −0.942827 − 0.333281i \(-0.891844\pi\)
0.333281 + 0.942827i \(0.391844\pi\)
\(44\) 0 0
\(45\) −26.4016 − 13.1824i −0.0874604 − 0.0436691i
\(46\) 0 0
\(47\) 86.1627 − 86.1627i 0.267407 − 0.267407i −0.560648 − 0.828055i \(-0.689448\pi\)
0.828055 + 0.560648i \(0.189448\pi\)
\(48\) 0 0
\(49\) 293.538i 0.855797i
\(50\) 0 0
\(51\) 431.318 1.18425
\(52\) 0 0
\(53\) −108.361 − 108.361i −0.280841 − 0.280841i 0.552603 − 0.833444i \(-0.313634\pi\)
−0.833444 + 0.552603i \(0.813634\pi\)
\(54\) 0 0
\(55\) 148.948 − 298.312i 0.365166 − 0.731353i
\(56\) 0 0
\(57\) 458.322 + 458.322i 1.06502 + 1.06502i
\(58\) 0 0
\(59\) 157.272i 0.347034i 0.984831 + 0.173517i \(0.0555132\pi\)
−0.984831 + 0.173517i \(0.944487\pi\)
\(60\) 0 0
\(61\) 791.867i 1.66210i 0.556197 + 0.831051i \(0.312260\pi\)
−0.556197 + 0.831051i \(0.687740\pi\)
\(62\) 0 0
\(63\) 13.1259 + 13.1259i 0.0262493 + 0.0262493i
\(64\) 0 0
\(65\) −203.563 + 67.9807i −0.388444 + 0.129723i
\(66\) 0 0
\(67\) 3.80296 + 3.80296i 0.00693440 + 0.00693440i 0.710565 − 0.703631i \(-0.248439\pi\)
−0.703631 + 0.710565i \(0.748439\pi\)
\(68\) 0 0
\(69\) −7.88811 −0.0137626
\(70\) 0 0
\(71\) − 58.6097i − 0.0979676i −0.998800 − 0.0489838i \(-0.984402\pi\)
0.998800 − 0.0489838i \(-0.0155983\pi\)
\(72\) 0 0
\(73\) 452.731 − 452.731i 0.725865 − 0.725865i −0.243928 − 0.969793i \(-0.578436\pi\)
0.969793 + 0.243928i \(0.0784361\pi\)
\(74\) 0 0
\(75\) 610.852 + 86.5677i 0.940468 + 0.133280i
\(76\) 0 0
\(77\) −148.310 + 148.310i −0.219500 + 0.219500i
\(78\) 0 0
\(79\) −821.590 −1.17008 −0.585039 − 0.811005i \(-0.698921\pi\)
−0.585039 + 0.811005i \(0.698921\pi\)
\(80\) 0 0
\(81\) 650.769 0.892687
\(82\) 0 0
\(83\) −512.512 + 512.512i −0.677777 + 0.677777i −0.959497 − 0.281720i \(-0.909095\pi\)
0.281720 + 0.959497i \(0.409095\pi\)
\(84\) 0 0
\(85\) 926.721 − 309.483i 1.18255 − 0.394919i
\(86\) 0 0
\(87\) 625.775 − 625.775i 0.771151 − 0.771151i
\(88\) 0 0
\(89\) 500.262i 0.595817i 0.954594 + 0.297908i \(0.0962890\pi\)
−0.954594 + 0.297908i \(0.903711\pi\)
\(90\) 0 0
\(91\) 135.001 0.155516
\(92\) 0 0
\(93\) −963.581 − 963.581i −1.07439 − 1.07439i
\(94\) 0 0
\(95\) 1313.60 + 655.882i 1.41866 + 0.708337i
\(96\) 0 0
\(97\) −60.7068 − 60.7068i −0.0635448 − 0.0635448i 0.674620 − 0.738165i \(-0.264307\pi\)
−0.738165 + 0.674620i \(0.764307\pi\)
\(98\) 0 0
\(99\) − 78.7153i − 0.0799109i
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 160.4.o.a.47.5 32
4.3 odd 2 40.4.k.a.27.13 yes 32
5.3 odd 4 inner 160.4.o.a.143.6 32
8.3 odd 2 inner 160.4.o.a.47.6 32
8.5 even 2 40.4.k.a.27.5 yes 32
20.3 even 4 40.4.k.a.3.5 ✓ 32
20.7 even 4 200.4.k.j.43.12 32
20.19 odd 2 200.4.k.j.107.4 32
40.3 even 4 inner 160.4.o.a.143.5 32
40.13 odd 4 40.4.k.a.3.13 yes 32
40.29 even 2 200.4.k.j.107.12 32
40.37 odd 4 200.4.k.j.43.4 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.k.a.3.5 ✓ 32 20.3 even 4
40.4.k.a.3.13 yes 32 40.13 odd 4
40.4.k.a.27.5 yes 32 8.5 even 2
40.4.k.a.27.13 yes 32 4.3 odd 2
160.4.o.a.47.5 32 1.1 even 1 trivial
160.4.o.a.47.6 32 8.3 odd 2 inner
160.4.o.a.143.5 32 40.3 even 4 inner
160.4.o.a.143.6 32 5.3 odd 4 inner
200.4.k.j.43.4 32 40.37 odd 4
200.4.k.j.43.12 32 20.7 even 4
200.4.k.j.107.4 32 20.19 odd 2
200.4.k.j.107.12 32 40.29 even 2