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Properties

Label 80.2.s.b.27.9
Level $80$
Weight $2$
Character 80.27
Analytic conductor $0.639$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,2,Mod(3,80)] 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("80.3"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 80.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.638803216170\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 27.9
Root \(-1.08900 - 0.902261i\) of defining polynomial
Character \(\chi\) \(=\) 80.27
Dual form 80.2.s.b.3.9

$q$-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+(1.41267 - 0.0660953i) q^{2} -0.496487 q^{3} +(1.99126 - 0.186742i) q^{4} +(-2.00635 - 0.987189i) q^{5} +(-0.701372 + 0.0328155i) q^{6} +(1.55426 + 1.55426i) q^{7} +(2.80065 - 0.395417i) q^{8} -2.75350 q^{9} +(-2.89956 - 1.26196i) q^{10} +(-4.19607 + 4.19607i) q^{11} +(-0.988637 + 0.0927148i) q^{12} -5.09530i q^{13} +(2.29838 + 2.09292i) q^{14} +(0.996130 + 0.490127i) q^{15} +(3.93026 - 0.743703i) q^{16} +(0.213542 + 0.213542i) q^{17} +(-3.88978 + 0.181993i) q^{18} +(0.844754 - 0.844754i) q^{19} +(-4.17953 - 1.59108i) q^{20} +(-0.771668 - 0.771668i) q^{21} +(-5.65031 + 6.20499i) q^{22} +(1.70744 - 1.70744i) q^{23} +(-1.39049 + 0.196320i) q^{24} +(3.05092 + 3.96130i) q^{25} +(-0.336775 - 7.19797i) q^{26} +2.85654 q^{27} +(3.38518 + 2.80469i) q^{28} +(2.24750 + 2.24750i) q^{29} +(1.43960 + 0.626547i) q^{30} +0.818209i q^{31} +(5.50299 - 1.31038i) q^{32} +(2.08329 - 2.08329i) q^{33} +(0.315778 + 0.287550i) q^{34} +(-1.58404 - 4.65273i) q^{35} +(-5.48294 + 0.514193i) q^{36} -5.12639i q^{37} +(1.13752 - 1.24919i) q^{38} +2.52975i q^{39} +(-6.00945 - 1.97142i) q^{40} +3.34727i q^{41} +(-1.14111 - 1.03911i) q^{42} +4.49131i q^{43} +(-7.57189 + 9.13905i) q^{44} +(5.52450 + 2.71822i) q^{45} +(2.29920 - 2.52490i) q^{46} +(4.29355 - 4.29355i) q^{47} +(-1.95132 + 0.369239i) q^{48} -2.16858i q^{49} +(4.57176 + 5.39435i) q^{50} +(-0.106021 - 0.106021i) q^{51} +(-0.951504 - 10.1461i) q^{52} -1.00653 q^{53} +(4.03534 - 0.188804i) q^{54} +(12.5611 - 4.27649i) q^{55} +(4.96751 + 3.73835i) q^{56} +(-0.419410 + 0.419410i) q^{57} +(3.32352 + 3.02642i) q^{58} +(-7.65005 - 7.65005i) q^{59} +(2.07508 + 0.789952i) q^{60} +(-1.90291 + 1.90291i) q^{61} +(0.0540798 + 1.15586i) q^{62} +(-4.27964 - 4.27964i) q^{63} +(7.68729 - 2.21485i) q^{64} +(-5.03002 + 10.2230i) q^{65} +(2.80531 - 3.08070i) q^{66} +11.0221i q^{67} +(0.465096 + 0.385341i) q^{68} +(-0.847724 + 0.847724i) q^{69} +(-2.54525 - 6.46807i) q^{70} -10.5331 q^{71} +(-7.71159 + 1.08878i) q^{72} +(2.70854 + 2.70854i) q^{73} +(-0.338831 - 7.24189i) q^{74} +(-1.51474 - 1.96674i) q^{75} +(1.52438 - 1.83988i) q^{76} -13.0435 q^{77} +(0.167205 + 3.57370i) q^{78} -8.32010 q^{79} +(-8.61966 - 2.38777i) q^{80} +6.84226 q^{81} +(0.221239 + 4.72858i) q^{82} -9.17237 q^{83} +(-1.68070 - 1.39249i) q^{84} +(-0.217635 - 0.639248i) q^{85} +(0.296855 + 6.34474i) q^{86} +(-1.11585 - 1.11585i) q^{87} +(-10.0925 + 13.4109i) q^{88} -4.25101 q^{89} +(7.98394 + 3.47481i) q^{90} +(7.91940 - 7.91940i) q^{91} +(3.08112 - 3.71882i) q^{92} -0.406230i q^{93} +(5.78157 - 6.34914i) q^{94} +(-2.52881 + 0.860944i) q^{95} +(-2.73217 + 0.650586i) q^{96} +(-7.16000 - 7.16000i) q^{97} +(-0.143333 - 3.06348i) q^{98} +(11.5539 - 11.5539i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 4 q^{4} + 2 q^{5} - 8 q^{6} + 2 q^{7} - 12 q^{8} + 10 q^{9} - 2 q^{11} - 12 q^{14} - 20 q^{15} - 6 q^{17} - 24 q^{18} - 2 q^{19} - 12 q^{20} - 16 q^{21} + 12 q^{22} - 2 q^{23} - 4 q^{24} - 6 q^{25}+ \cdots + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41267 − 0.0660953i 0.998907 − 0.0467365i
\(3\) −0.496487 −0.286647 −0.143324 − 0.989676i \(-0.545779\pi\)
−0.143324 + 0.989676i \(0.545779\pi\)
\(4\) 1.99126 − 0.186742i 0.995631 − 0.0933708i
\(5\) −2.00635 − 0.987189i −0.897269 − 0.441484i
\(6\) −0.701372 + 0.0328155i −0.286334 + 0.0133969i
\(7\) 1.55426 + 1.55426i 0.587453 + 0.587453i 0.936941 − 0.349488i \(-0.113644\pi\)
−0.349488 + 0.936941i \(0.613644\pi\)
\(8\) 2.80065 − 0.395417i 0.990180 − 0.139801i
\(9\) −2.75350 −0.917833
\(10\) −2.89956 − 1.26196i −0.916922 − 0.399067i
\(11\) −4.19607 + 4.19607i −1.26516 + 1.26516i −0.316604 + 0.948558i \(0.602543\pi\)
−0.948558 + 0.316604i \(0.897457\pi\)
\(12\) −0.988637 + 0.0927148i −0.285395 + 0.0267645i
\(13\) − 5.09530i − 1.41318i −0.707622 − 0.706591i \(-0.750232\pi\)
0.707622 − 0.706591i \(-0.249768\pi\)
\(14\) 2.29838 + 2.09292i 0.614267 + 0.559356i
\(15\) 0.996130 + 0.490127i 0.257200 + 0.126550i
\(16\) 3.93026 − 0.743703i 0.982564 − 0.185926i
\(17\) 0.213542 + 0.213542i 0.0517916 + 0.0517916i 0.732528 − 0.680737i \(-0.238340\pi\)
−0.680737 + 0.732528i \(0.738340\pi\)
\(18\) −3.88978 + 0.181993i −0.916830 + 0.0428963i
\(19\) 0.844754 − 0.844754i 0.193800 − 0.193800i −0.603536 − 0.797336i \(-0.706242\pi\)
0.797336 + 0.603536i \(0.206242\pi\)
\(20\) −4.17953 − 1.59108i −0.934571 − 0.355777i
\(21\) −0.771668 − 0.771668i −0.168392 − 0.168392i
\(22\) −5.65031 + 6.20499i −1.20465 + 1.32291i
\(23\) 1.70744 − 1.70744i 0.356027 − 0.356027i −0.506319 − 0.862346i \(-0.668994\pi\)
0.862346 + 0.506319i \(0.168994\pi\)
\(24\) −1.39049 + 0.196320i −0.283832 + 0.0400736i
\(25\) 3.05092 + 3.96130i 0.610183 + 0.792260i
\(26\) −0.336775 − 7.19797i −0.0660471 − 1.41164i
\(27\) 2.85654 0.549741
\(28\) 3.38518 + 2.80469i 0.639738 + 0.530036i
\(29\) 2.24750 + 2.24750i 0.417350 + 0.417350i 0.884289 − 0.466939i \(-0.154643\pi\)
−0.466939 + 0.884289i \(0.654643\pi\)
\(30\) 1.43960 + 0.626547i 0.262833 + 0.114391i
\(31\) 0.818209i 0.146955i 0.997297 + 0.0734773i \(0.0234097\pi\)
−0.997297 + 0.0734773i \(0.976590\pi\)
\(32\) 5.50299 − 1.31038i 0.972801 − 0.231644i
\(33\) 2.08329 − 2.08329i 0.362655 − 0.362655i
\(34\) 0.315778 + 0.287550i 0.0541556 + 0.0493144i
\(35\) −1.58404 − 4.65273i −0.267752 − 0.786455i
\(36\) −5.48294 + 0.514193i −0.913824 + 0.0856988i
\(37\) − 5.12639i − 0.842774i −0.906881 − 0.421387i \(-0.861543\pi\)
0.906881 − 0.421387i \(-0.138457\pi\)
\(38\) 1.13752 − 1.24919i 0.184531 − 0.202646i
\(39\) 2.52975i 0.405084i
\(40\) −6.00945 − 1.97142i −0.950177 − 0.311710i
\(41\) 3.34727i 0.522756i 0.965237 + 0.261378i \(0.0841769\pi\)
−0.965237 + 0.261378i \(0.915823\pi\)
\(42\) −1.14111 − 1.03911i −0.176078 − 0.160338i
\(43\) 4.49131i 0.684919i 0.939533 + 0.342460i \(0.111260\pi\)
−0.939533 + 0.342460i \(0.888740\pi\)
\(44\) −7.57189 + 9.13905i −1.14151 + 1.37776i
\(45\) 5.52450 + 2.71822i 0.823543 + 0.405209i
\(46\) 2.29920 − 2.52490i 0.338998 − 0.372277i
\(47\) 4.29355 − 4.29355i 0.626278 − 0.626278i −0.320851 − 0.947130i \(-0.603969\pi\)
0.947130 + 0.320851i \(0.103969\pi\)
\(48\) −1.95132 + 0.369239i −0.281649 + 0.0532951i
\(49\) − 2.16858i − 0.309797i
\(50\) 4.57176 + 5.39435i 0.646544 + 0.762877i
\(51\) −0.106021 − 0.106021i −0.0148459 − 0.0148459i
\(52\) −0.951504 − 10.1461i −0.131950 − 1.40701i
\(53\) −1.00653 −0.138258 −0.0691291 − 0.997608i \(-0.522022\pi\)
−0.0691291 + 0.997608i \(0.522022\pi\)
\(54\) 4.03534 − 0.188804i 0.549141 − 0.0256930i
\(55\) 12.5611 − 4.27649i 1.69374 − 0.576642i
\(56\) 4.96751 + 3.73835i 0.663811 + 0.499558i
\(57\) −0.419410 + 0.419410i −0.0555521 + 0.0555521i
\(58\) 3.32352 + 3.02642i 0.436399 + 0.397388i
\(59\) −7.65005 − 7.65005i −0.995952 − 0.995952i 0.00404030 − 0.999992i \(-0.498714\pi\)
−0.999992 + 0.00404030i \(0.998714\pi\)
\(60\) 2.07508 + 0.789952i 0.267892 + 0.101982i
\(61\) −1.90291 + 1.90291i −0.243643 + 0.243643i −0.818355 − 0.574712i \(-0.805114\pi\)
0.574712 + 0.818355i \(0.305114\pi\)
\(62\) 0.0540798 + 1.15586i 0.00686814 + 0.146794i
\(63\) −4.27964 − 4.27964i −0.539184 − 0.539184i
\(64\) 7.68729 − 2.21485i 0.960911 − 0.276856i
\(65\) −5.03002 + 10.2230i −0.623897 + 1.26800i
\(66\) 2.80531 − 3.08070i 0.345310 − 0.379208i
\(67\) 11.0221i 1.34656i 0.739387 + 0.673280i \(0.235115\pi\)
−0.739387 + 0.673280i \(0.764885\pi\)
\(68\) 0.465096 + 0.385341i 0.0564012 + 0.0467295i
\(69\) −0.847724 + 0.847724i −0.102054 + 0.102054i
\(70\) −2.54525 − 6.46807i −0.304216 − 0.773082i
\(71\) −10.5331 −1.25005 −0.625027 − 0.780604i \(-0.714912\pi\)
−0.625027 + 0.780604i \(0.714912\pi\)
\(72\) −7.71159 + 1.08878i −0.908820 + 0.128314i
\(73\) 2.70854 + 2.70854i 0.317010 + 0.317010i 0.847618 − 0.530607i \(-0.178036\pi\)
−0.530607 + 0.847618i \(0.678036\pi\)
\(74\) −0.338831 − 7.24189i −0.0393883 − 0.841853i
\(75\) −1.51474 − 1.96674i −0.174907 − 0.227099i
\(76\) 1.52438 − 1.83988i 0.174858 − 0.211048i
\(77\) −13.0435 −1.48645
\(78\) 0.167205 + 3.57370i 0.0189322 + 0.404642i
\(79\) −8.32010 −0.936085 −0.468042 − 0.883706i \(-0.655041\pi\)
−0.468042 + 0.883706i \(0.655041\pi\)
\(80\) −8.61966 − 2.38777i −0.963707 − 0.266961i
\(81\) 6.84226 0.760252
\(82\) 0.221239 + 4.72858i 0.0244317 + 0.522185i
\(83\) −9.17237 −1.00680 −0.503399 − 0.864054i \(-0.667917\pi\)
−0.503399 + 0.864054i \(0.667917\pi\)
\(84\) −1.68070 − 1.39249i −0.183379 − 0.151933i
\(85\) −0.217635 − 0.639248i −0.0236058 − 0.0693362i
\(86\) 0.296855 + 6.34474i 0.0320107 + 0.684171i
\(87\) −1.11585 − 1.11585i −0.119632 − 0.119632i
\(88\) −10.0925 + 13.4109i −1.07587 + 1.42961i
\(89\) −4.25101 −0.450606 −0.225303 − 0.974289i \(-0.572337\pi\)
−0.225303 + 0.974289i \(0.572337\pi\)
\(90\) 7.98394 + 3.47481i 0.841582 + 0.366277i
\(91\) 7.91940 − 7.91940i 0.830178 − 0.830178i
\(92\) 3.08112 − 3.71882i 0.321229 − 0.387714i
\(93\) − 0.406230i − 0.0421241i
\(94\) 5.78157 − 6.34914i 0.596324 − 0.654864i
\(95\) −2.52881 + 0.860944i −0.259450 + 0.0883310i
\(96\) −2.73217 + 0.650586i −0.278850 + 0.0664001i
\(97\) −7.16000 − 7.16000i −0.726987 − 0.726987i 0.243031 − 0.970019i \(-0.421858\pi\)
−0.970019 + 0.243031i \(0.921858\pi\)
\(98\) −0.143333 − 3.06348i −0.0144788 − 0.309458i
\(99\) 11.5539 − 11.5539i 1.16121 − 1.16121i
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 80.2.s.b.27.9 yes 18
3.2 odd 2 720.2.z.g.667.1 18
4.3 odd 2 320.2.s.b.207.6 18
5.2 odd 4 400.2.j.d.43.4 18
5.3 odd 4 80.2.j.b.43.6 ✓ 18
5.4 even 2 400.2.s.d.107.1 18
8.3 odd 2 640.2.s.c.287.4 18
8.5 even 2 640.2.s.d.287.6 18
15.8 even 4 720.2.bd.g.523.4 18
16.3 odd 4 80.2.j.b.67.6 yes 18
16.5 even 4 640.2.j.c.607.6 18
16.11 odd 4 640.2.j.d.607.4 18
16.13 even 4 320.2.j.b.47.4 18
20.3 even 4 320.2.j.b.143.6 18
20.7 even 4 1600.2.j.d.143.4 18
20.19 odd 2 1600.2.s.d.207.4 18
40.3 even 4 640.2.j.c.543.4 18
40.13 odd 4 640.2.j.d.543.6 18
48.35 even 4 720.2.bd.g.307.4 18
80.3 even 4 inner 80.2.s.b.3.9 yes 18
80.13 odd 4 320.2.s.b.303.6 18
80.19 odd 4 400.2.j.d.307.4 18
80.29 even 4 1600.2.j.d.1007.6 18
80.43 even 4 640.2.s.d.223.6 18
80.53 odd 4 640.2.s.c.223.4 18
80.67 even 4 400.2.s.d.243.1 18
80.77 odd 4 1600.2.s.d.943.4 18
240.83 odd 4 720.2.z.g.163.1 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.6 ✓ 18 5.3 odd 4
80.2.j.b.67.6 yes 18 16.3 odd 4
80.2.s.b.3.9 yes 18 80.3 even 4 inner
80.2.s.b.27.9 yes 18 1.1 even 1 trivial
320.2.j.b.47.4 18 16.13 even 4
320.2.j.b.143.6 18 20.3 even 4
320.2.s.b.207.6 18 4.3 odd 2
320.2.s.b.303.6 18 80.13 odd 4
400.2.j.d.43.4 18 5.2 odd 4
400.2.j.d.307.4 18 80.19 odd 4
400.2.s.d.107.1 18 5.4 even 2
400.2.s.d.243.1 18 80.67 even 4
640.2.j.c.543.4 18 40.3 even 4
640.2.j.c.607.6 18 16.5 even 4
640.2.j.d.543.6 18 40.13 odd 4
640.2.j.d.607.4 18 16.11 odd 4
640.2.s.c.223.4 18 80.53 odd 4
640.2.s.c.287.4 18 8.3 odd 2
640.2.s.d.223.6 18 80.43 even 4
640.2.s.d.287.6 18 8.5 even 2
720.2.z.g.163.1 18 240.83 odd 4
720.2.z.g.667.1 18 3.2 odd 2
720.2.bd.g.307.4 18 48.35 even 4
720.2.bd.g.523.4 18 15.8 even 4
1600.2.j.d.143.4 18 20.7 even 4
1600.2.j.d.1007.6 18 80.29 even 4
1600.2.s.d.207.4 18 20.19 odd 2
1600.2.s.d.943.4 18 80.77 odd 4