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Properties

Label 80.2.s.b.27.6
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 27.6
Root \(1.41303 - 0.0578659i\) of defining polynomial
Character \(\chi\) \(=\) 80.27
Dual form 80.2.s.b.3.6

$q$-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+(0.567819 + 1.29521i) q^{2} +1.96251 q^{3} +(-1.35516 + 1.47090i) q^{4} +(-1.42182 - 1.72581i) q^{5} +(1.11435 + 2.54187i) q^{6} +(-1.60205 - 1.60205i) q^{7} +(-2.67461 - 0.920026i) q^{8} +0.851447 q^{9} +(1.42795 - 2.82151i) q^{10} +(0.754587 - 0.754587i) q^{11} +(-2.65952 + 2.88665i) q^{12} +5.94580i q^{13} +(1.16532 - 2.98467i) q^{14} +(-2.79034 - 3.38692i) q^{15} +(-0.327065 - 3.98661i) q^{16} +(1.95574 + 1.95574i) q^{17} +(0.483468 + 1.10281i) q^{18} +(0.780680 - 0.780680i) q^{19} +(4.46529 + 0.247399i) q^{20} +(-3.14404 - 3.14404i) q^{21} +(1.40582 + 0.548884i) q^{22} +(4.93121 - 4.93121i) q^{23} +(-5.24896 - 1.80556i) q^{24} +(-0.956833 + 4.90759i) q^{25} +(-7.70109 + 3.37614i) q^{26} -4.21656 q^{27} +(4.52748 - 0.185408i) q^{28} +(-1.44802 - 1.44802i) q^{29} +(2.80238 - 5.53725i) q^{30} -3.60859i q^{31} +(4.97780 - 2.68729i) q^{32} +(1.48089 - 1.48089i) q^{33} +(-1.42260 + 3.64361i) q^{34} +(-0.486998 + 5.04266i) q^{35} +(-1.15385 + 1.25239i) q^{36} +10.2364i q^{37} +(1.45443 + 0.567864i) q^{38} +11.6687i q^{39} +(2.21504 + 5.92398i) q^{40} -6.93334i q^{41} +(2.28696 - 5.85745i) q^{42} +9.91344i q^{43} +(0.0873298 + 2.13251i) q^{44} +(-1.21061 - 1.46944i) q^{45} +(9.18700 + 3.58694i) q^{46} +(0.104270 - 0.104270i) q^{47} +(-0.641868 - 7.82376i) q^{48} -1.86688i q^{49} +(-6.89970 + 1.54732i) q^{50} +(3.83816 + 3.83816i) q^{51} +(-8.74565 - 8.05753i) q^{52} -4.03213 q^{53} +(-2.39424 - 5.46135i) q^{54} +(-2.37516 - 0.229383i) q^{55} +(2.81093 + 5.75878i) q^{56} +(1.53209 - 1.53209i) q^{57} +(1.05328 - 2.69771i) q^{58} +(-3.46736 - 3.46736i) q^{59} +(8.76317 + 0.485523i) q^{60} +(0.680578 - 0.680578i) q^{61} +(4.67390 - 2.04902i) q^{62} +(-1.36406 - 1.36406i) q^{63} +(6.30711 + 4.92142i) q^{64} +(10.2613 - 8.45388i) q^{65} +(2.75894 + 1.07719i) q^{66} -9.04721i q^{67} +(-5.52703 + 0.226341i) q^{68} +(9.67754 - 9.67754i) q^{69} +(-6.80785 + 2.23255i) q^{70} -3.64007 q^{71} +(-2.27729 - 0.783353i) q^{72} +(-2.94030 - 2.94030i) q^{73} +(-13.2583 + 5.81242i) q^{74} +(-1.87779 + 9.63120i) q^{75} +(0.0903496 + 2.20625i) q^{76} -2.41777 q^{77} +(-15.1135 + 6.62570i) q^{78} +10.7140 q^{79} +(-6.41509 + 6.23270i) q^{80} -10.8294 q^{81} +(8.98016 - 3.93688i) q^{82} -4.23845 q^{83} +(8.88523 - 0.363865i) q^{84} +(0.594515 - 6.15595i) q^{85} +(-12.8400 + 5.62904i) q^{86} +(-2.84176 - 2.84176i) q^{87} +(-2.71247 + 1.32399i) q^{88} +0.0426256 q^{89} +(1.21583 - 2.40237i) q^{90} +(9.52546 - 9.52546i) q^{91} +(0.570698 + 13.9359i) q^{92} -7.08189i q^{93} +(0.194258 + 0.0758455i) q^{94} +(-2.45730 - 0.237315i) q^{95} +(9.76898 - 5.27383i) q^{96} +(-1.91173 - 1.91173i) q^{97} +(2.41802 - 1.06005i) q^{98} +(0.642491 - 0.642491i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.567819 + 1.29521i 0.401509 + 0.915855i
\(3\) 1.96251 1.13306 0.566528 − 0.824043i \(-0.308286\pi\)
0.566528 + 0.824043i \(0.308286\pi\)
\(4\) −1.35516 + 1.47090i −0.677582 + 0.735448i
\(5\) −1.42182 − 1.72581i −0.635859 − 0.771805i
\(6\) 1.11435 + 2.54187i 0.454932 + 1.03772i
\(7\) −1.60205 − 1.60205i −0.605517 − 0.605517i 0.336254 − 0.941771i \(-0.390840\pi\)
−0.941771 + 0.336254i \(0.890840\pi\)
\(8\) −2.67461 − 0.920026i −0.945618 − 0.325278i
\(9\) 0.851447 0.283816
\(10\) 1.42795 − 2.82151i 0.451559 − 0.892241i
\(11\) 0.754587 − 0.754587i 0.227517 − 0.227517i −0.584138 − 0.811654i \(-0.698567\pi\)
0.811654 + 0.584138i \(0.198567\pi\)
\(12\) −2.65952 + 2.88665i −0.767738 + 0.833303i
\(13\) 5.94580i 1.64907i 0.565812 + 0.824534i \(0.308563\pi\)
−0.565812 + 0.824534i \(0.691437\pi\)
\(14\) 1.16532 − 2.98467i 0.311446 − 0.797687i
\(15\) −2.79034 − 3.38692i −0.720464 − 0.874498i
\(16\) −0.327065 − 3.98661i −0.0817662 − 0.996652i
\(17\) 1.95574 + 1.95574i 0.474336 + 0.474336i 0.903315 − 0.428978i \(-0.141126\pi\)
−0.428978 + 0.903315i \(0.641126\pi\)
\(18\) 0.483468 + 1.10281i 0.113954 + 0.259934i
\(19\) 0.780680 − 0.780680i 0.179100 − 0.179100i −0.611863 − 0.790964i \(-0.709580\pi\)
0.790964 + 0.611863i \(0.209580\pi\)
\(20\) 4.46529 + 0.247399i 0.998469 + 0.0553201i
\(21\) −3.14404 − 3.14404i −0.686085 − 0.686085i
\(22\) 1.40582 + 0.548884i 0.299722 + 0.117022i
\(23\) 4.93121 − 4.93121i 1.02823 − 1.02823i 0.0286378 − 0.999590i \(-0.490883\pi\)
0.999590 − 0.0286378i \(-0.00911693\pi\)
\(24\) −5.24896 − 1.80556i −1.07144 − 0.368558i
\(25\) −0.956833 + 4.90759i −0.191367 + 0.981519i
\(26\) −7.70109 + 3.37614i −1.51031 + 0.662115i
\(27\) −4.21656 −0.811477
\(28\) 4.52748 − 0.185408i 0.855614 − 0.0350388i
\(29\) −1.44802 − 1.44802i −0.268891 − 0.268891i 0.559762 − 0.828653i \(-0.310892\pi\)
−0.828653 + 0.559762i \(0.810892\pi\)
\(30\) 2.80238 − 5.53725i 0.511642 − 1.01096i
\(31\) − 3.60859i − 0.648121i −0.946036 − 0.324061i \(-0.894952\pi\)
0.946036 − 0.324061i \(-0.105048\pi\)
\(32\) 4.97780 − 2.68729i 0.879959 − 0.475050i
\(33\) 1.48089 − 1.48089i 0.257789 − 0.257789i
\(34\) −1.42260 + 3.64361i −0.243973 + 0.624874i
\(35\) −0.486998 + 5.04266i −0.0823177 + 0.852365i
\(36\) −1.15385 + 1.25239i −0.192308 + 0.208732i
\(37\) 10.2364i 1.68285i 0.540371 + 0.841427i \(0.318284\pi\)
−0.540371 + 0.841427i \(0.681716\pi\)
\(38\) 1.45443 + 0.567864i 0.235940 + 0.0921197i
\(39\) 11.6687i 1.86849i
\(40\) 2.21504 + 5.92398i 0.350229 + 0.936664i
\(41\) − 6.93334i − 1.08281i −0.840763 − 0.541403i \(-0.817893\pi\)
0.840763 − 0.541403i \(-0.182107\pi\)
\(42\) 2.28696 − 5.85745i 0.352885 − 0.903823i
\(43\) 9.91344i 1.51179i 0.654695 + 0.755893i \(0.272797\pi\)
−0.654695 + 0.755893i \(0.727203\pi\)
\(44\) 0.0873298 + 2.13251i 0.0131655 + 0.321488i
\(45\) −1.21061 − 1.46944i −0.180467 − 0.219050i
\(46\) 9.18700 + 3.58694i 1.35455 + 0.528865i
\(47\) 0.104270 − 0.104270i 0.0152093 − 0.0152093i −0.699461 − 0.714671i \(-0.746577\pi\)
0.714671 + 0.699461i \(0.246577\pi\)
\(48\) −0.641868 − 7.82376i −0.0926457 − 1.12926i
\(49\) − 1.86688i − 0.266698i
\(50\) −6.89970 + 1.54732i −0.975764 + 0.218824i
\(51\) 3.83816 + 3.83816i 0.537450 + 0.537450i
\(52\) −8.74565 − 8.05753i −1.21280 − 1.11738i
\(53\) −4.03213 −0.553856 −0.276928 − 0.960891i \(-0.589316\pi\)
−0.276928 + 0.960891i \(0.589316\pi\)
\(54\) −2.39424 − 5.46135i −0.325815 − 0.743195i
\(55\) −2.37516 − 0.229383i −0.320267 − 0.0309300i
\(56\) 2.81093 + 5.75878i 0.375627 + 0.769550i
\(57\) 1.53209 − 1.53209i 0.202931 − 0.202931i
\(58\) 1.05328 − 2.69771i 0.138303 − 0.354227i
\(59\) −3.46736 − 3.46736i −0.451412 − 0.451412i 0.444411 − 0.895823i \(-0.353413\pi\)
−0.895823 + 0.444411i \(0.853413\pi\)
\(60\) 8.76317 + 0.485523i 1.13132 + 0.0626807i
\(61\) 0.680578 − 0.680578i 0.0871391 − 0.0871391i −0.662194 − 0.749333i \(-0.730374\pi\)
0.749333 + 0.662194i \(0.230374\pi\)
\(62\) 4.67390 − 2.04902i 0.593585 − 0.260226i
\(63\) −1.36406 − 1.36406i −0.171855 − 0.171855i
\(64\) 6.30711 + 4.92142i 0.788388 + 0.615178i
\(65\) 10.2613 − 8.45388i 1.27276 − 1.04857i
\(66\) 2.75894 + 1.07719i 0.339602 + 0.132593i
\(67\) − 9.04721i − 1.10529i −0.833416 − 0.552646i \(-0.813618\pi\)
0.833416 − 0.552646i \(-0.186382\pi\)
\(68\) −5.52703 + 0.226341i −0.670251 + 0.0274479i
\(69\) 9.67754 − 9.67754i 1.16504 − 1.16504i
\(70\) −6.80785 + 2.23255i −0.813694 + 0.266841i
\(71\) −3.64007 −0.431997 −0.215998 − 0.976394i \(-0.569301\pi\)
−0.215998 + 0.976394i \(0.569301\pi\)
\(72\) −2.27729 − 0.783353i −0.268381 − 0.0923191i
\(73\) −2.94030 − 2.94030i −0.344136 − 0.344136i 0.513784 − 0.857920i \(-0.328243\pi\)
−0.857920 + 0.513784i \(0.828243\pi\)
\(74\) −13.2583 + 5.81242i −1.54125 + 0.675681i
\(75\) −1.87779 + 9.63120i −0.216829 + 1.11212i
\(76\) 0.0903496 + 2.20625i 0.0103638 + 0.253074i
\(77\) −2.41777 −0.275530
\(78\) −15.1135 + 6.62570i −1.71126 + 0.750213i
\(79\) 10.7140 1.20542 0.602711 − 0.797960i \(-0.294087\pi\)
0.602711 + 0.797960i \(0.294087\pi\)
\(80\) −6.41509 + 6.23270i −0.717229 + 0.696838i
\(81\) −10.8294 −1.20326
\(82\) 8.98016 − 3.93688i 0.991693 − 0.434756i
\(83\) −4.23845 −0.465230 −0.232615 − 0.972569i \(-0.574728\pi\)
−0.232615 + 0.972569i \(0.574728\pi\)
\(84\) 8.88523 − 0.363865i 0.969458 − 0.0397009i
\(85\) 0.594515 − 6.15595i 0.0644842 − 0.667707i
\(86\) −12.8400 + 5.62904i −1.38458 + 0.606995i
\(87\) −2.84176 − 2.84176i −0.304668 − 0.304668i
\(88\) −2.71247 + 1.32399i −0.289150 + 0.141138i
\(89\) 0.0426256 0.00451831 0.00225915 − 0.999997i \(-0.499281\pi\)
0.00225915 + 0.999997i \(0.499281\pi\)
\(90\) 1.21583 − 2.40237i 0.128160 − 0.253232i
\(91\) 9.52546 − 9.52546i 0.998539 − 0.998539i
\(92\) 0.570698 + 13.9359i 0.0594993 + 1.45292i
\(93\) − 7.08189i − 0.734358i
\(94\) 0.194258 + 0.0758455i 0.0200362 + 0.00782287i
\(95\) −2.45730 − 0.237315i −0.252113 − 0.0243480i
\(96\) 9.76898 − 5.27383i 0.997042 − 0.538258i
\(97\) −1.91173 − 1.91173i −0.194106 − 0.194106i 0.603362 − 0.797468i \(-0.293828\pi\)
−0.797468 + 0.603362i \(0.793828\pi\)
\(98\) 2.41802 − 1.06005i 0.244257 − 0.107081i
\(99\) 0.642491 − 0.642491i 0.0645728 − 0.0645728i
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.27.6 yes 18
3.2 odd 2 720.2.z.g.667.4 18
4.3 odd 2 320.2.s.b.207.2 18
5.2 odd 4 400.2.j.d.43.1 18
5.3 odd 4 80.2.j.b.43.9 ✓ 18
5.4 even 2 400.2.s.d.107.4 18
8.3 odd 2 640.2.s.c.287.8 18
8.5 even 2 640.2.s.d.287.2 18
15.8 even 4 720.2.bd.g.523.1 18
16.3 odd 4 80.2.j.b.67.9 yes 18
16.5 even 4 640.2.j.c.607.2 18
16.11 odd 4 640.2.j.d.607.8 18
16.13 even 4 320.2.j.b.47.8 18
20.3 even 4 320.2.j.b.143.2 18
20.7 even 4 1600.2.j.d.143.8 18
20.19 odd 2 1600.2.s.d.207.8 18
40.3 even 4 640.2.j.c.543.8 18
40.13 odd 4 640.2.j.d.543.2 18
48.35 even 4 720.2.bd.g.307.1 18
80.3 even 4 inner 80.2.s.b.3.6 yes 18
80.13 odd 4 320.2.s.b.303.2 18
80.19 odd 4 400.2.j.d.307.1 18
80.29 even 4 1600.2.j.d.1007.2 18
80.43 even 4 640.2.s.d.223.2 18
80.53 odd 4 640.2.s.c.223.8 18
80.67 even 4 400.2.s.d.243.4 18
80.77 odd 4 1600.2.s.d.943.8 18
240.83 odd 4 720.2.z.g.163.4 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.9 ✓ 18 5.3 odd 4
80.2.j.b.67.9 yes 18 16.3 odd 4
80.2.s.b.3.6 yes 18 80.3 even 4 inner
80.2.s.b.27.6 yes 18 1.1 even 1 trivial
320.2.j.b.47.8 18 16.13 even 4
320.2.j.b.143.2 18 20.3 even 4
320.2.s.b.207.2 18 4.3 odd 2
320.2.s.b.303.2 18 80.13 odd 4
400.2.j.d.43.1 18 5.2 odd 4
400.2.j.d.307.1 18 80.19 odd 4
400.2.s.d.107.4 18 5.4 even 2
400.2.s.d.243.4 18 80.67 even 4
640.2.j.c.543.8 18 40.3 even 4
640.2.j.c.607.2 18 16.5 even 4
640.2.j.d.543.2 18 40.13 odd 4
640.2.j.d.607.8 18 16.11 odd 4
640.2.s.c.223.8 18 80.53 odd 4
640.2.s.c.287.8 18 8.3 odd 2
640.2.s.d.223.2 18 80.43 even 4
640.2.s.d.287.2 18 8.5 even 2
720.2.z.g.163.4 18 240.83 odd 4
720.2.z.g.667.4 18 3.2 odd 2
720.2.bd.g.307.1 18 48.35 even 4
720.2.bd.g.523.1 18 15.8 even 4
1600.2.j.d.143.8 18 20.7 even 4
1600.2.j.d.1007.2 18 80.29 even 4
1600.2.s.d.207.8 18 20.19 odd 2
1600.2.s.d.943.8 18 80.77 odd 4