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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 47.9
Character \(\chi\) \(=\) 160.47
Dual form 160.4.o.a.143.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.56085 - 1.56085i) q^{3} +(-10.5634 - 3.66270i) q^{5} +(18.5221 - 18.5221i) q^{7} +22.1275i q^{9} -13.0709 q^{11} +(-51.6789 - 51.6789i) q^{13} +(-22.2047 + 10.7709i) q^{15} +(-57.6873 - 57.6873i) q^{17} -28.4226i q^{19} -57.8204i q^{21} +(-102.782 - 102.782i) q^{23} +(98.1693 + 77.3808i) q^{25} +(76.6805 + 76.6805i) q^{27} -9.08265 q^{29} -115.940i q^{31} +(-20.4017 + 20.4017i) q^{33} +(-263.497 + 127.815i) q^{35} +(-19.9147 + 19.9147i) q^{37} -161.325 q^{39} -96.6271 q^{41} +(285.182 - 285.182i) q^{43} +(81.0464 - 233.741i) q^{45} +(-5.83951 + 5.83951i) q^{47} -343.139i q^{49} -180.082 q^{51} +(291.191 + 291.191i) q^{53} +(138.073 + 47.8749i) q^{55} +(-44.3632 - 44.3632i) q^{57} +184.669i q^{59} +270.307i q^{61} +(409.849 + 409.849i) q^{63} +(356.619 + 735.187i) q^{65} +(151.797 + 151.797i) q^{67} -320.854 q^{69} +742.577i q^{71} +(40.1228 - 40.1228i) q^{73} +(274.007 - 32.4477i) q^{75} +(-242.102 + 242.102i) q^{77} +1106.08 q^{79} -358.070 q^{81} +(807.395 - 807.395i) q^{83} +(398.081 + 820.663i) q^{85} +(-14.1766 + 14.1766i) q^{87} -1122.56i q^{89} -1914.41 q^{91} +(-180.965 - 180.965i) q^{93} +(-104.103 + 300.238i) q^{95} +(424.428 + 424.428i) q^{97} -289.228i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 4 q^{3} + 8 q^{11} + 48 q^{17} + 40 q^{25} - 104 q^{27} - 112 q^{33} + 460 q^{35} - 8 q^{41} + 868 q^{43} - 1480 q^{51} + 104 q^{57} + 520 q^{65} + 1852 q^{67} - 744 q^{73} - 3300 q^{75} - 1240 q^{81}+ \cdots - 584 q^{97}+O(q^{100}) \) 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
\(3\) 1.56085 − 1.56085i 0.300385 − 0.300385i −0.540779 − 0.841164i \(-0.681871\pi\)
0.841164 + 0.540779i \(0.181871\pi\)
\(4\) 0 0
\(5\) −10.5634 − 3.66270i −0.944816 − 0.327601i
\(6\) 0 0
\(7\) 18.5221 − 18.5221i 1.00010 − 1.00010i 0.000101443 − 1.00000i \(-0.499968\pi\)
1.00000 0.000101443i \(-3.22903e-5\pi\)
\(8\) 0 0
\(9\) 22.1275i 0.819538i
\(10\) 0 0
\(11\) −13.0709 −0.358276 −0.179138 − 0.983824i \(-0.557331\pi\)
−0.179138 + 0.983824i \(0.557331\pi\)
\(12\) 0 0
\(13\) −51.6789 − 51.6789i −1.10255 − 1.10255i −0.994102 − 0.108447i \(-0.965412\pi\)
−0.108447 − 0.994102i \(-0.534588\pi\)
\(14\) 0 0
\(15\) −22.2047 + 10.7709i −0.382215 + 0.185402i
\(16\) 0 0
\(17\) −57.6873 − 57.6873i −0.823013 − 0.823013i 0.163526 − 0.986539i \(-0.447713\pi\)
−0.986539 + 0.163526i \(0.947713\pi\)
\(18\) 0 0
\(19\) − 28.4226i − 0.343188i −0.985168 − 0.171594i \(-0.945108\pi\)
0.985168 − 0.171594i \(-0.0548918\pi\)
\(20\) 0 0
\(21\) − 57.8204i − 0.600831i
\(22\) 0 0
\(23\) −102.782 − 102.782i −0.931806 − 0.931806i 0.0660125 − 0.997819i \(-0.478972\pi\)
−0.997819 + 0.0660125i \(0.978972\pi\)
\(24\) 0 0
\(25\) 98.1693 + 77.3808i 0.785355 + 0.619046i
\(26\) 0 0
\(27\) 76.6805 + 76.6805i 0.546562 + 0.546562i
\(28\) 0 0
\(29\) −9.08265 −0.0581588 −0.0290794 − 0.999577i \(-0.509258\pi\)
−0.0290794 + 0.999577i \(0.509258\pi\)
\(30\) 0 0
\(31\) − 115.940i − 0.671726i −0.941911 − 0.335863i \(-0.890972\pi\)
0.941911 − 0.335863i \(-0.109028\pi\)
\(32\) 0 0
\(33\) −20.4017 + 20.4017i −0.107621 + 0.107621i
\(34\) 0 0
\(35\) −263.497 + 127.815i −1.27255 + 0.617277i
\(36\) 0 0
\(37\) −19.9147 + 19.9147i −0.0884853 + 0.0884853i −0.749964 − 0.661479i \(-0.769929\pi\)
0.661479 + 0.749964i \(0.269929\pi\)
\(38\) 0 0
\(39\) −161.325 −0.662378
\(40\) 0 0
\(41\) −96.6271 −0.368064 −0.184032 − 0.982920i \(-0.558915\pi\)
−0.184032 + 0.982920i \(0.558915\pi\)
\(42\) 0 0
\(43\) 285.182 − 285.182i 1.01139 − 1.01139i 0.0114565 − 0.999934i \(-0.496353\pi\)
0.999934 − 0.0114565i \(-0.00364679\pi\)
\(44\) 0 0
\(45\) 81.0464 − 233.741i 0.268482 − 0.774312i
\(46\) 0 0
\(47\) −5.83951 + 5.83951i −0.0181230 + 0.0181230i −0.716110 − 0.697987i \(-0.754079\pi\)
0.697987 + 0.716110i \(0.254079\pi\)
\(48\) 0 0
\(49\) − 343.139i − 1.00041i
\(50\) 0 0
\(51\) −180.082 −0.494442
\(52\) 0 0
\(53\) 291.191 + 291.191i 0.754683 + 0.754683i 0.975349 − 0.220666i \(-0.0708232\pi\)
−0.220666 + 0.975349i \(0.570823\pi\)
\(54\) 0 0
\(55\) 138.073 + 47.8749i 0.338505 + 0.117372i
\(56\) 0 0
\(57\) −44.3632 − 44.3632i −0.103089 − 0.103089i
\(58\) 0 0
\(59\) 184.669i 0.407489i 0.979024 + 0.203744i \(0.0653111\pi\)
−0.979024 + 0.203744i \(0.934689\pi\)
\(60\) 0 0
\(61\) 270.307i 0.567365i 0.958918 + 0.283682i \(0.0915561\pi\)
−0.958918 + 0.283682i \(0.908444\pi\)
\(62\) 0 0
\(63\) 409.849 + 409.849i 0.819621 + 0.819621i
\(64\) 0 0
\(65\) 356.619 + 735.187i 0.680509 + 1.40290i
\(66\) 0 0
\(67\) 151.797 + 151.797i 0.276790 + 0.276790i 0.831826 − 0.555036i \(-0.187296\pi\)
−0.555036 + 0.831826i \(0.687296\pi\)
\(68\) 0 0
\(69\) −320.854 −0.559801
\(70\) 0 0
\(71\) 742.577i 1.24123i 0.784114 + 0.620617i \(0.213118\pi\)
−0.784114 + 0.620617i \(0.786882\pi\)
\(72\) 0 0
\(73\) 40.1228 − 40.1228i 0.0643289 − 0.0643289i −0.674210 − 0.738539i \(-0.735516\pi\)
0.738539 + 0.674210i \(0.235516\pi\)
\(74\) 0 0
\(75\) 274.007 − 32.4477i 0.421861 − 0.0499565i
\(76\) 0 0
\(77\) −242.102 + 242.102i −0.358313 + 0.358313i
\(78\) 0 0
\(79\) 1106.08 1.57524 0.787618 − 0.616164i \(-0.211314\pi\)
0.787618 + 0.616164i \(0.211314\pi\)
\(80\) 0 0
\(81\) −358.070 −0.491180
\(82\) 0 0
\(83\) 807.395 − 807.395i 1.06775 − 1.06775i 0.0702169 − 0.997532i \(-0.477631\pi\)
0.997532 − 0.0702169i \(-0.0223691\pi\)
\(84\) 0 0
\(85\) 398.081 + 820.663i 0.507976 + 1.04722i
\(86\) 0 0
\(87\) −14.1766 + 14.1766i −0.0174700 + 0.0174700i
\(88\) 0 0
\(89\) − 1122.56i − 1.33698i −0.743722 − 0.668489i \(-0.766941\pi\)
0.743722 − 0.668489i \(-0.233059\pi\)
\(90\) 0 0
\(91\) −1914.41 −2.20532
\(92\) 0 0
\(93\) −180.965 − 180.965i −0.201776 − 0.201776i
\(94\) 0 0
\(95\) −104.103 + 300.238i −0.112429 + 0.324250i
\(96\) 0 0
\(97\) 424.428 + 424.428i 0.444270 + 0.444270i 0.893444 − 0.449175i \(-0.148282\pi\)
−0.449175 + 0.893444i \(0.648282\pi\)
\(98\) 0 0
\(99\) − 289.228i − 0.293621i
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.47.9 32
4.3 odd 2 40.4.k.a.27.6 yes 32
5.3 odd 4 inner 160.4.o.a.143.10 32
8.3 odd 2 inner 160.4.o.a.47.10 32
8.5 even 2 40.4.k.a.27.4 yes 32
20.3 even 4 40.4.k.a.3.4 ✓ 32
20.7 even 4 200.4.k.j.43.13 32
20.19 odd 2 200.4.k.j.107.11 32
40.3 even 4 inner 160.4.o.a.143.9 32
40.13 odd 4 40.4.k.a.3.6 yes 32
40.29 even 2 200.4.k.j.107.13 32
40.37 odd 4 200.4.k.j.43.11 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.k.a.3.4 ✓ 32 20.3 even 4
40.4.k.a.3.6 yes 32 40.13 odd 4
40.4.k.a.27.4 yes 32 8.5 even 2
40.4.k.a.27.6 yes 32 4.3 odd 2
160.4.o.a.47.9 32 1.1 even 1 trivial
160.4.o.a.47.10 32 8.3 odd 2 inner
160.4.o.a.143.9 32 40.3 even 4 inner
160.4.o.a.143.10 32 5.3 odd 4 inner
200.4.k.j.43.11 32 40.37 odd 4
200.4.k.j.43.13 32 20.7 even 4
200.4.k.j.107.11 32 20.19 odd 2
200.4.k.j.107.13 32 40.29 even 2