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Properties

Label 80.2.s.b.27.4
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 27.4
Root \(1.41323 + 0.0526497i\) of defining polynomial
Character \(\chi\) \(=\) 80.27
Dual form 80.2.s.b.3.4

$q$-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 \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −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 0.458111 − 0.888895i \(-0.348526\pi\)
−0.888895 + 0.458111i \(0.848526\pi\)
\(8\) 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 −0.964436 − 0.264316i \(-0.914854\pi\)
0.264316 + 0.964436i \(0.414854\pi\)
\(12\) −1.87794 + 1.74304i −0.542114 + 0.503172i
\(13\) − 1.36502i − 0.378589i −0.981920 − 0.189294i \(-0.939380\pi\)
0.981920 − 0.189294i \(-0.0606201\pi\)
\(14\) −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 0.943845 + 0.330389i \(0.107180\pi\)
0.330389 + 0.943845i \(0.392820\pi\)
\(18\) 0.702192 + 1.78873i 0.165508 + 0.421608i
\(19\) −3.69752 + 3.69752i −0.848269 + 0.848269i −0.989917 − 0.141648i \(-0.954760\pi\)
0.141648 + 0.989917i \(0.454760\pi\)
\(20\) −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 −0.795688 − 0.605707i \(-0.792890\pi\)
0.605707 + 0.795688i \(0.292890\pi\)
\(24\) 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 0.451524 − 0.892259i \(-0.350881\pi\)
−0.892259 + 0.451524i \(0.850881\pi\)
\(30\) −2.79098 − 2.93641i −0.509560 − 0.536114i
\(31\) 0.242577i 0.0435681i 0.999763 + 0.0217841i \(0.00693463\pi\)
−0.999763 + 0.0217841i \(0.993065\pi\)
\(32\) −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 \(-0.911338\pi\)
0.961458 − 0.274953i \(-0.0886621\pi\)
\(38\) 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 \(0.0668442\pi\)
−0.978031 + 0.208457i \(0.933156\pi\)
\(42\) −1.16757 + 2.67669i −0.180160 + 0.413023i
\(43\) − 9.04874i − 1.37992i −0.723847 − 0.689960i \(-0.757628\pi\)
0.723847 − 0.689960i \(-0.242372\pi\)
\(44\) 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 0.162435 − 0.986719i \(-0.448065\pi\)
0.986719 − 0.162435i \(-0.0519348\pi\)
\(48\) 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 0.978112 − 0.208079i \(-0.0667210\pi\)
−0.208079 + 0.978112i \(0.566721\pi\)
\(60\) −2.42322 + 5.19155i −0.312836 + 0.670226i
\(61\) −6.67404 + 6.67404i −0.854523 + 0.854523i −0.990686 − 0.136163i \(-0.956523\pi\)
0.136163 + 0.990686i \(0.456523\pi\)
\(62\) 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 \(-0.910416\pi\)
0.960658 − 0.277736i \(-0.0895839\pi\)
\(68\) −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 0.614313 − 0.789063i \(-0.289433\pi\)
−0.789063 + 0.614313i \(0.789433\pi\)
\(74\) −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 0.619164 − 0.785262i \(-0.287472\pi\)
−0.785262 + 0.619164i \(0.787472\pi\)
\(98\) −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.27.4 yes 18
3.2 odd 2 720.2.z.g.667.6 18
4.3 odd 2 320.2.s.b.207.3 18
5.2 odd 4 400.2.j.d.43.8 18
5.3 odd 4 80.2.j.b.43.2 ✓ 18
5.4 even 2 400.2.s.d.107.6 18
8.3 odd 2 640.2.s.c.287.7 18
8.5 even 2 640.2.s.d.287.3 18
15.8 even 4 720.2.bd.g.523.8 18
16.3 odd 4 80.2.j.b.67.2 yes 18
16.5 even 4 640.2.j.c.607.3 18
16.11 odd 4 640.2.j.d.607.7 18
16.13 even 4 320.2.j.b.47.7 18
20.3 even 4 320.2.j.b.143.3 18
20.7 even 4 1600.2.j.d.143.7 18
20.19 odd 2 1600.2.s.d.207.7 18
40.3 even 4 640.2.j.c.543.7 18
40.13 odd 4 640.2.j.d.543.3 18
48.35 even 4 720.2.bd.g.307.8 18
80.3 even 4 inner 80.2.s.b.3.4 yes 18
80.13 odd 4 320.2.s.b.303.3 18
80.19 odd 4 400.2.j.d.307.8 18
80.29 even 4 1600.2.j.d.1007.3 18
80.43 even 4 640.2.s.d.223.3 18
80.53 odd 4 640.2.s.c.223.7 18
80.67 even 4 400.2.s.d.243.6 18
80.77 odd 4 1600.2.s.d.943.7 18
240.83 odd 4 720.2.z.g.163.6 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.2 ✓ 18 5.3 odd 4
80.2.j.b.67.2 yes 18 16.3 odd 4
80.2.s.b.3.4 yes 18 80.3 even 4 inner
80.2.s.b.27.4 yes 18 1.1 even 1 trivial
320.2.j.b.47.7 18 16.13 even 4
320.2.j.b.143.3 18 20.3 even 4
320.2.s.b.207.3 18 4.3 odd 2
320.2.s.b.303.3 18 80.13 odd 4
400.2.j.d.43.8 18 5.2 odd 4
400.2.j.d.307.8 18 80.19 odd 4
400.2.s.d.107.6 18 5.4 even 2
400.2.s.d.243.6 18 80.67 even 4
640.2.j.c.543.7 18 40.3 even 4
640.2.j.c.607.3 18 16.5 even 4
640.2.j.d.543.3 18 40.13 odd 4
640.2.j.d.607.7 18 16.11 odd 4
640.2.s.c.223.7 18 80.53 odd 4
640.2.s.c.287.7 18 8.3 odd 2
640.2.s.d.223.3 18 80.43 even 4
640.2.s.d.287.3 18 8.5 even 2
720.2.z.g.163.6 18 240.83 odd 4
720.2.z.g.667.6 18 3.2 odd 2
720.2.bd.g.307.8 18 48.35 even 4
720.2.bd.g.523.8 18 15.8 even 4
1600.2.j.d.143.7 18 20.7 even 4
1600.2.j.d.1007.3 18 80.29 even 4
1600.2.s.d.207.7 18 20.19 odd 2
1600.2.s.d.943.7 18 80.77 odd 4