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Properties

Label 80.2.s.b.3.4
Level $80$
Weight $2$
Character 80.3
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 3.4
Root \(1.41323 - 0.0526497i\) of defining polynomial
Character \(\chi\) \(=\) 80.3
Dual form 80.2.s.b.27.4

$q$-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+(-0.516777 + 1.31641i) q^{2} +1.28110 q^{3} +(-1.46588 - 1.36058i) q^{4} +(2.07160 + 0.841703i) q^{5} +(-0.662041 + 1.68645i) q^{6} +(-1.13975 + 1.13975i) q^{7} +(2.54862 - 1.22659i) q^{8} -1.35879 q^{9} +(-2.17858 + 2.29211i) q^{10} +(-2.32204 - 2.32204i) q^{11} +(-1.87794 - 1.74304i) q^{12} +1.36502i q^{13} +(-0.911384 - 2.08938i) q^{14} +(2.65392 + 1.07830i) q^{15} +(0.297625 + 3.98891i) q^{16} +(5.25380 - 5.25380i) q^{17} +(0.702192 - 1.78873i) q^{18} +(-3.69752 - 3.69752i) q^{19} +(-1.89152 - 4.05243i) q^{20} +(-1.46013 + 1.46013i) q^{21} +(4.25673 - 1.85678i) q^{22} +(-0.911118 - 0.911118i) q^{23} +(3.26503 - 1.57138i) q^{24} +(3.58307 + 3.48735i) q^{25} +(-1.79693 - 0.705412i) q^{26} -5.58403 q^{27} +(3.22146 - 0.120015i) q^{28} +(-2.37343 + 2.37343i) q^{29} +(-2.79098 + 2.93641i) q^{30} -0.242577i q^{31} +(-5.40486 - 1.66958i) q^{32} +(-2.97475 - 2.97475i) q^{33} +(4.20112 + 9.63121i) q^{34} +(-3.32044 + 1.40178i) q^{35} +(1.99183 + 1.84875i) q^{36} +3.34494i q^{37} +(6.77825 - 2.95666i) q^{38} +1.74872i q^{39} +(6.31216 - 0.395820i) q^{40} -2.66956i q^{41} +(-1.16757 - 2.67669i) q^{42} +9.04874i q^{43} +(0.244509 + 6.56316i) q^{44} +(-2.81488 - 1.14370i) q^{45} +(1.67025 - 0.728562i) q^{46} +(7.87820 + 7.87820i) q^{47} +(0.381287 + 5.11018i) q^{48} +4.40194i q^{49} +(-6.44244 + 2.91462i) q^{50} +(6.73063 - 6.73063i) q^{51} +(1.85723 - 2.00096i) q^{52} -5.80113 q^{53} +(2.88570 - 7.35089i) q^{54} +(-2.85587 - 6.76480i) q^{55} +(-1.50679 + 4.30279i) q^{56} +(-4.73688 - 4.73688i) q^{57} +(-1.89788 - 4.35095i) q^{58} +(5.91474 - 5.91474i) q^{59} +(-2.42322 - 5.19155i) q^{60} +(-6.67404 - 6.67404i) q^{61} +(0.319332 + 0.125358i) q^{62} +(1.54868 - 1.54868i) q^{63} +(4.99096 - 6.25222i) q^{64} +(-1.14894 + 2.82778i) q^{65} +(5.45328 - 2.37872i) q^{66} +4.54673i q^{67} +(-14.8497 + 0.553222i) q^{68} +(-1.16723 - 1.16723i) q^{69} +(-0.129391 - 5.09547i) q^{70} +15.4389 q^{71} +(-3.46305 + 1.66668i) q^{72} +(-1.49307 + 1.49307i) q^{73} +(-4.40332 - 1.72859i) q^{74} +(4.59026 + 4.46763i) q^{75} +(0.389347 + 10.4509i) q^{76} +5.29308 q^{77} +(-2.30204 - 0.903701i) q^{78} -10.3024 q^{79} +(-2.74092 + 8.51395i) q^{80} -3.07731 q^{81} +(3.51424 + 1.37957i) q^{82} +3.26589 q^{83} +(4.12701 - 0.153751i) q^{84} +(15.3059 - 6.46165i) q^{85} +(-11.9119 - 4.67618i) q^{86} +(-3.04060 + 3.04060i) q^{87} +(-8.76618 - 3.06981i) q^{88} +9.77206 q^{89} +(2.96024 - 3.11450i) q^{90} +(-1.55578 - 1.55578i) q^{91} +(0.0959403 + 2.57524i) q^{92} -0.310765i q^{93} +(-14.4422 + 6.29969i) q^{94} +(-4.54758 - 10.7720i) q^{95} +(-6.92415 - 2.13889i) q^{96} +(-1.63587 + 1.63587i) q^{97} +(-5.79477 - 2.27482i) q^{98} +(3.15516 + 3.15516i) 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 \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.516777 + 1.31641i −0.365417 + 0.930844i
\(3\) 1.28110 0.739642 0.369821 − 0.929103i \(-0.379419\pi\)
0.369821 + 0.929103i \(0.379419\pi\)
\(4\) −1.46588 − 1.36058i −0.732941 − 0.680292i
\(5\) 2.07160 + 0.841703i 0.926449 + 0.376421i
\(6\) −0.662041 + 1.68645i −0.270277 + 0.688491i
\(7\) −1.13975 + 1.13975i −0.430785 + 0.430785i −0.888895 − 0.458111i \(-0.848526\pi\)
0.458111 + 0.888895i \(0.348526\pi\)
\(8\) 2.54862 − 1.22659i 0.901074 − 0.433664i
\(9\) −1.35879 −0.452930
\(10\) −2.17858 + 2.29211i −0.688929 + 0.724829i
\(11\) −2.32204 − 2.32204i −0.700120 − 0.700120i 0.264316 − 0.964436i \(-0.414854\pi\)
−0.964436 + 0.264316i \(0.914854\pi\)
\(12\) −1.87794 − 1.74304i −0.542114 − 0.503172i
\(13\) 1.36502i 0.378589i 0.981920 + 0.189294i \(0.0606201\pi\)
−0.981920 + 0.189294i \(0.939380\pi\)
\(14\) −0.911384 − 2.08938i −0.243578 − 0.558409i
\(15\) 2.65392 + 1.07830i 0.685240 + 0.278416i
\(16\) 0.297625 + 3.98891i 0.0744064 + 0.997228i
\(17\) 5.25380 − 5.25380i 1.27423 − 1.27423i 0.330389 − 0.943845i \(-0.392820\pi\)
0.943845 − 0.330389i \(-0.107180\pi\)
\(18\) 0.702192 − 1.78873i 0.165508 − 0.421608i
\(19\) −3.69752 − 3.69752i −0.848269 − 0.848269i 0.141648 − 0.989917i \(-0.454760\pi\)
−0.989917 + 0.141648i \(0.954760\pi\)
\(20\) −1.89152 − 4.05243i −0.422957 − 0.906150i
\(21\) −1.46013 + 1.46013i −0.318626 + 0.318626i
\(22\) 4.25673 − 1.85678i 0.907538 − 0.395867i
\(23\) −0.911118 − 0.911118i −0.189981 − 0.189981i 0.605707 − 0.795688i \(-0.292890\pi\)
−0.795688 + 0.605707i \(0.792890\pi\)
\(24\) 3.26503 − 1.57138i 0.666472 − 0.320756i
\(25\) 3.58307 + 3.48735i 0.716615 + 0.697469i
\(26\) −1.79693 − 0.705412i −0.352407 − 0.138343i
\(27\) −5.58403 −1.07465
\(28\) 3.22146 − 0.120015i 0.608799 − 0.0226807i
\(29\) −2.37343 + 2.37343i −0.440736 + 0.440736i −0.892259 − 0.451524i \(-0.850881\pi\)
0.451524 + 0.892259i \(0.350881\pi\)
\(30\) −2.79098 + 2.93641i −0.509560 + 0.536114i
\(31\) − 0.242577i − 0.0435681i −0.999763 − 0.0217841i \(-0.993065\pi\)
0.999763 − 0.0217841i \(-0.00693463\pi\)
\(32\) −5.40486 − 1.66958i −0.955453 − 0.295143i
\(33\) −2.97475 − 2.97475i −0.517838 − 0.517838i
\(34\) 4.20112 + 9.63121i 0.720487 + 1.65174i
\(35\) −3.32044 + 1.40178i −0.561256 + 0.236944i
\(36\) 1.99183 + 1.84875i 0.331971 + 0.308125i
\(37\) 3.34494i 0.549905i 0.961458 + 0.274953i \(0.0886621\pi\)
−0.961458 + 0.274953i \(0.911338\pi\)
\(38\) 6.77825 − 2.95666i 1.09958 − 0.479634i
\(39\) 1.74872i 0.280020i
\(40\) 6.31216 − 0.395820i 0.998040 − 0.0625846i
\(41\) − 2.66956i − 0.416915i −0.978031 − 0.208457i \(-0.933156\pi\)
0.978031 − 0.208457i \(-0.0668442\pi\)
\(42\) −1.16757 − 2.67669i −0.180160 − 0.413023i
\(43\) 9.04874i 1.37992i 0.723847 + 0.689960i \(0.242372\pi\)
−0.723847 + 0.689960i \(0.757628\pi\)
\(44\) 0.244509 + 6.56316i 0.0368611 + 0.989433i
\(45\) −2.81488 − 1.14370i −0.419617 − 0.170492i
\(46\) 1.67025 − 0.728562i 0.246265 − 0.107421i
\(47\) 7.87820 + 7.87820i 1.14915 + 1.14915i 0.986719 + 0.162435i \(0.0519348\pi\)
0.162435 + 0.986719i \(0.448065\pi\)
\(48\) 0.381287 + 5.11018i 0.0550340 + 0.737591i
\(49\) 4.40194i 0.628849i
\(50\) −6.44244 + 2.91462i −0.911098 + 0.412190i
\(51\) 6.73063 − 6.73063i 0.942476 − 0.942476i
\(52\) 1.85723 − 2.00096i 0.257551 − 0.277483i
\(53\) −5.80113 −0.796846 −0.398423 − 0.917202i \(-0.630442\pi\)
−0.398423 + 0.917202i \(0.630442\pi\)
\(54\) 2.88570 − 7.35089i 0.392694 − 1.00033i
\(55\) −2.85587 − 6.76480i −0.385086 − 0.912165i
\(56\) −1.50679 + 4.30279i −0.201353 + 0.574985i
\(57\) −4.73688 − 4.73688i −0.627415 − 0.627415i
\(58\) −1.89788 − 4.35095i −0.249204 − 0.571308i
\(59\) 5.91474 − 5.91474i 0.770033 − 0.770033i −0.208079 − 0.978112i \(-0.566721\pi\)
0.978112 + 0.208079i \(0.0667210\pi\)
\(60\) −2.42322 − 5.19155i −0.312836 − 0.670226i
\(61\) −6.67404 − 6.67404i −0.854523 − 0.854523i 0.136163 − 0.990686i \(-0.456523\pi\)
−0.990686 + 0.136163i \(0.956523\pi\)
\(62\) 0.319332 + 0.125358i 0.0405551 + 0.0159205i
\(63\) 1.54868 − 1.54868i 0.195116 − 0.195116i
\(64\) 4.99096 − 6.25222i 0.623870 − 0.781528i
\(65\) −1.14894 + 2.82778i −0.142509 + 0.350743i
\(66\) 5.45328 − 2.37872i 0.671253 − 0.292800i
\(67\) 4.54673i 0.555471i 0.960658 + 0.277736i \(0.0895839\pi\)
−0.960658 + 0.277736i \(0.910416\pi\)
\(68\) −14.8497 + 0.553222i −1.80079 + 0.0670881i
\(69\) −1.16723 − 1.16723i −0.140518 − 0.140518i
\(70\) −0.129391 − 5.09547i −0.0154652 − 0.609025i
\(71\) 15.4389 1.83226 0.916128 − 0.400885i \(-0.131297\pi\)
0.916128 + 0.400885i \(0.131297\pi\)
\(72\) −3.46305 + 1.66668i −0.408124 + 0.196420i
\(73\) −1.49307 + 1.49307i −0.174750 + 0.174750i −0.789063 − 0.614313i \(-0.789433\pi\)
0.614313 + 0.789063i \(0.289433\pi\)
\(74\) −4.40332 − 1.72859i −0.511876 − 0.200944i
\(75\) 4.59026 + 4.46763i 0.530038 + 0.515877i
\(76\) 0.389347 + 10.4509i 0.0446611 + 1.19880i
\(77\) 5.29308 0.603202
\(78\) −2.30204 − 0.903701i −0.260655 − 0.102324i
\(79\) −10.3024 −1.15911 −0.579556 − 0.814932i \(-0.696774\pi\)
−0.579556 + 0.814932i \(0.696774\pi\)
\(80\) −2.74092 + 8.51395i −0.306444 + 0.951889i
\(81\) −3.07731 −0.341924
\(82\) 3.51424 + 1.37957i 0.388083 + 0.152348i
\(83\) 3.26589 0.358478 0.179239 − 0.983806i \(-0.442636\pi\)
0.179239 + 0.983806i \(0.442636\pi\)
\(84\) 4.12701 − 0.153751i 0.450293 − 0.0167756i
\(85\) 15.3059 − 6.46165i 1.66016 − 0.700864i
\(86\) −11.9119 − 4.67618i −1.28449 − 0.504246i
\(87\) −3.04060 + 3.04060i −0.325986 + 0.325986i
\(88\) −8.76618 − 3.06981i −0.934477 − 0.327243i
\(89\) 9.77206 1.03584 0.517918 − 0.855430i \(-0.326707\pi\)
0.517918 + 0.855430i \(0.326707\pi\)
\(90\) 2.96024 − 3.11450i 0.312037 − 0.328297i
\(91\) −1.55578 − 1.55578i −0.163090 − 0.163090i
\(92\) 0.0959403 + 2.57524i 0.0100025 + 0.268488i
\(93\) − 0.310765i − 0.0322248i
\(94\) −14.4422 + 6.29969i −1.48960 + 0.649763i
\(95\) −4.54758 − 10.7720i −0.466571 − 1.10518i
\(96\) −6.92415 − 2.13889i −0.706693 − 0.218300i
\(97\) −1.63587 + 1.63587i −0.166097 + 0.166097i −0.785262 − 0.619164i \(-0.787472\pi\)
0.619164 + 0.785262i \(0.287472\pi\)
\(98\) −5.79477 − 2.27482i −0.585360 − 0.229792i
\(99\) 3.15516 + 3.15516i 0.317106 + 0.317106i
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.3.4 yes 18
3.2 odd 2 720.2.z.g.163.6 18
4.3 odd 2 320.2.s.b.303.3 18
5.2 odd 4 80.2.j.b.67.2 yes 18
5.3 odd 4 400.2.j.d.307.8 18
5.4 even 2 400.2.s.d.243.6 18
8.3 odd 2 640.2.s.c.223.7 18
8.5 even 2 640.2.s.d.223.3 18
15.2 even 4 720.2.bd.g.307.8 18
16.3 odd 4 640.2.j.d.543.3 18
16.5 even 4 320.2.j.b.143.3 18
16.11 odd 4 80.2.j.b.43.2 ✓ 18
16.13 even 4 640.2.j.c.543.7 18
20.3 even 4 1600.2.j.d.1007.3 18
20.7 even 4 320.2.j.b.47.7 18
20.19 odd 2 1600.2.s.d.943.7 18
40.27 even 4 640.2.j.c.607.3 18
40.37 odd 4 640.2.j.d.607.7 18
48.11 even 4 720.2.bd.g.523.8 18
80.27 even 4 inner 80.2.s.b.27.4 yes 18
80.37 odd 4 320.2.s.b.207.3 18
80.43 even 4 400.2.s.d.107.6 18
80.53 odd 4 1600.2.s.d.207.7 18
80.59 odd 4 400.2.j.d.43.8 18
80.67 even 4 640.2.s.d.287.3 18
80.69 even 4 1600.2.j.d.143.7 18
80.77 odd 4 640.2.s.c.287.7 18
240.107 odd 4 720.2.z.g.667.6 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.2 ✓ 18 16.11 odd 4
80.2.j.b.67.2 yes 18 5.2 odd 4
80.2.s.b.3.4 yes 18 1.1 even 1 trivial
80.2.s.b.27.4 yes 18 80.27 even 4 inner
320.2.j.b.47.7 18 20.7 even 4
320.2.j.b.143.3 18 16.5 even 4
320.2.s.b.207.3 18 80.37 odd 4
320.2.s.b.303.3 18 4.3 odd 2
400.2.j.d.43.8 18 80.59 odd 4
400.2.j.d.307.8 18 5.3 odd 4
400.2.s.d.107.6 18 80.43 even 4
400.2.s.d.243.6 18 5.4 even 2
640.2.j.c.543.7 18 16.13 even 4
640.2.j.c.607.3 18 40.27 even 4
640.2.j.d.543.3 18 16.3 odd 4
640.2.j.d.607.7 18 40.37 odd 4
640.2.s.c.223.7 18 8.3 odd 2
640.2.s.c.287.7 18 80.77 odd 4
640.2.s.d.223.3 18 8.5 even 2
640.2.s.d.287.3 18 80.67 even 4
720.2.z.g.163.6 18 3.2 odd 2
720.2.z.g.667.6 18 240.107 odd 4
720.2.bd.g.307.8 18 15.2 even 4
720.2.bd.g.523.8 18 48.11 even 4
1600.2.j.d.143.7 18 80.69 even 4
1600.2.j.d.1007.3 18 20.3 even 4
1600.2.s.d.207.7 18 80.53 odd 4
1600.2.s.d.943.7 18 20.19 odd 2