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Properties

Label 80.2.s.b.27.8
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 27.8
Root \(0.482716 - 1.32928i\) of defining polynomial
Character \(\chi\) \(=\) 80.27
Dual form 80.2.s.b.3.8

$q$-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+(1.19301 + 0.759419i) q^{2} -1.39319 q^{3} +(0.846564 + 1.81200i) q^{4} +(2.17104 + 0.535339i) q^{5} +(-1.66209 - 1.05801i) q^{6} +(-2.13436 - 2.13436i) q^{7} +(-0.366101 + 2.80463i) q^{8} -1.05903 q^{9} +(2.18353 + 2.28740i) q^{10} +(2.17074 - 2.17074i) q^{11} +(-1.17942 - 2.52445i) q^{12} -1.54663i q^{13} +(-0.925449 - 4.16720i) q^{14} +(-3.02466 - 0.745827i) q^{15} +(-2.56666 + 3.06794i) q^{16} +(-3.86386 - 3.86386i) q^{17} +(-1.26344 - 0.804250i) q^{18} +(0.0136865 - 0.0136865i) q^{19} +(0.867892 + 4.38711i) q^{20} +(2.97357 + 2.97357i) q^{21} +(4.23822 - 0.941219i) q^{22} +(-3.15240 + 3.15240i) q^{23} +(0.510047 - 3.90738i) q^{24} +(4.42682 + 2.32449i) q^{25} +(1.17454 - 1.84515i) q^{26} +5.65499 q^{27} +(2.06058 - 5.67434i) q^{28} +(3.33787 + 3.33787i) q^{29} +(-3.04207 - 3.18677i) q^{30} +8.92639i q^{31} +(-5.39191 + 1.71093i) q^{32} +(-3.02424 + 3.02424i) q^{33} +(-1.67535 - 7.54394i) q^{34} +(-3.49118 - 5.77640i) q^{35} +(-0.896540 - 1.91896i) q^{36} +7.24737i q^{37} +(0.0267220 - 0.00593441i) q^{38} +2.15475i q^{39} +(-2.29625 + 5.89298i) q^{40} -10.3771i q^{41} +(1.28932 + 5.80569i) q^{42} +2.02975i q^{43} +(5.77103 + 2.09570i) q^{44} +(-2.29920 - 0.566942i) q^{45} +(-6.15484 + 1.36686i) q^{46} +(3.34313 - 3.34313i) q^{47} +(3.57583 - 4.27421i) q^{48} +2.11103i q^{49} +(3.51600 + 6.13496i) q^{50} +(5.38308 + 5.38308i) q^{51} +(2.80249 - 1.30932i) q^{52} -7.30702 q^{53} +(6.74648 + 4.29451i) q^{54} +(5.87483 - 3.55067i) q^{55} +(6.76751 - 5.20472i) q^{56} +(-0.0190679 + 0.0190679i) q^{57} +(1.44728 + 6.51696i) q^{58} +(-3.52732 - 3.52732i) q^{59} +(-1.20914 - 6.11206i) q^{60} +(1.41629 - 1.41629i) q^{61} +(-6.77887 + 10.6493i) q^{62} +(2.26036 + 2.26036i) q^{63} +(-7.73194 - 2.05356i) q^{64} +(0.827973 - 3.35780i) q^{65} +(-5.90462 + 1.31129i) q^{66} +0.748197i q^{67} +(3.73030 - 10.2723i) q^{68} +(4.39187 - 4.39187i) q^{69} +(0.221682 - 9.54260i) q^{70} -0.269603 q^{71} +(0.387713 - 2.97020i) q^{72} +(0.811870 + 0.811870i) q^{73} +(-5.50380 + 8.64622i) q^{74} +(-6.16739 - 3.23844i) q^{75} +(0.0363865 + 0.0132134i) q^{76} -9.26628 q^{77} +(-1.63636 + 2.57064i) q^{78} -2.80567 q^{79} +(-7.21470 + 5.28659i) q^{80} -4.70135 q^{81} +(7.88056 - 12.3800i) q^{82} +12.8279 q^{83} +(-2.87077 + 7.90541i) q^{84} +(-6.32012 - 10.4571i) q^{85} +(-1.54143 + 2.42152i) q^{86} +(-4.65027 - 4.65027i) q^{87} +(5.29341 + 6.88283i) q^{88} -13.3732 q^{89} +(-2.31243 - 2.42243i) q^{90} +(-3.30108 + 3.30108i) q^{91} +(-8.38083 - 3.04342i) q^{92} -12.4361i q^{93} +(6.52724 - 1.44956i) q^{94} +(0.0370409 - 0.0223871i) q^{95} +(7.51194 - 2.38364i) q^{96} +(6.33466 + 6.33466i) q^{97} +(-1.60315 + 2.51848i) q^{98} +(-2.29888 + 2.29888i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.19301 + 0.759419i 0.843588 + 0.536991i
\(3\) −1.39319 −0.804356 −0.402178 − 0.915561i \(-0.631747\pi\)
−0.402178 + 0.915561i \(0.631747\pi\)
\(4\) 0.846564 + 1.81200i 0.423282 + 0.905998i
\(5\) 2.17104 + 0.535339i 0.970918 + 0.239411i
\(6\) −1.66209 − 1.05801i −0.678546 − 0.431932i
\(7\) −2.13436 − 2.13436i −0.806714 − 0.806714i 0.177421 − 0.984135i \(-0.443225\pi\)
−0.984135 + 0.177421i \(0.943225\pi\)
\(8\) −0.366101 + 2.80463i −0.129436 + 0.991588i
\(9\) −1.05903 −0.353011
\(10\) 2.18353 + 2.28740i 0.690494 + 0.723338i
\(11\) 2.17074 − 2.17074i 0.654501 − 0.654501i −0.299572 − 0.954074i \(-0.596844\pi\)
0.954074 + 0.299572i \(0.0968440\pi\)
\(12\) −1.17942 − 2.52445i −0.340470 − 0.728745i
\(13\) − 1.54663i − 0.428958i −0.976729 − 0.214479i \(-0.931195\pi\)
0.976729 − 0.214479i \(-0.0688054\pi\)
\(14\) −0.925449 − 4.16720i −0.247337 − 1.11373i
\(15\) −3.02466 − 0.745827i −0.780964 − 0.192572i
\(16\) −2.56666 + 3.06794i −0.641664 + 0.766986i
\(17\) −3.86386 − 3.86386i −0.937125 − 0.937125i 0.0610123 − 0.998137i \(-0.480567\pi\)
−0.998137 + 0.0610123i \(0.980567\pi\)
\(18\) −1.26344 − 0.804250i −0.297796 − 0.189564i
\(19\) 0.0136865 − 0.0136865i 0.00313991 − 0.00313991i −0.705535 − 0.708675i \(-0.749293\pi\)
0.708675 + 0.705535i \(0.249293\pi\)
\(20\) 0.867892 + 4.38711i 0.194067 + 0.980988i
\(21\) 2.97357 + 2.97357i 0.648886 + 0.648886i
\(22\) 4.23822 − 0.941219i 0.903591 − 0.200669i
\(23\) −3.15240 + 3.15240i −0.657320 + 0.657320i −0.954745 − 0.297425i \(-0.903872\pi\)
0.297425 + 0.954745i \(0.403872\pi\)
\(24\) 0.510047 − 3.90738i 0.104113 − 0.797590i
\(25\) 4.42682 + 2.32449i 0.885365 + 0.464897i
\(26\) 1.17454 − 1.84515i 0.230347 − 0.361864i
\(27\) 5.65499 1.08830
\(28\) 2.06058 − 5.67434i 0.389413 − 1.07235i
\(29\) 3.33787 + 3.33787i 0.619826 + 0.619826i 0.945487 − 0.325660i \(-0.105587\pi\)
−0.325660 + 0.945487i \(0.605587\pi\)
\(30\) −3.04207 − 3.18677i −0.555403 − 0.581822i
\(31\) 8.92639i 1.60323i 0.597843 + 0.801613i \(0.296025\pi\)
−0.597843 + 0.801613i \(0.703975\pi\)
\(32\) −5.39191 + 1.71093i −0.953164 + 0.302452i
\(33\) −3.02424 + 3.02424i −0.526452 + 0.526452i
\(34\) −1.67535 − 7.54394i −0.287320 − 1.29377i
\(35\) −3.49118 − 5.77640i −0.590117 − 0.976390i
\(36\) −0.896540 − 1.91896i −0.149423 − 0.319827i
\(37\) 7.24737i 1.19146i 0.803184 + 0.595730i \(0.203137\pi\)
−0.803184 + 0.595730i \(0.796863\pi\)
\(38\) 0.0267220 − 0.00593441i 0.00433489 − 0.000962688i
\(39\) 2.15475i 0.345035i
\(40\) −2.29625 + 5.89298i −0.363069 + 0.931762i
\(41\) − 10.3771i − 1.62063i −0.585996 − 0.810314i \(-0.699296\pi\)
0.585996 − 0.810314i \(-0.300704\pi\)
\(42\) 1.28932 + 5.80569i 0.198947 + 0.895838i
\(43\) 2.02975i 0.309534i 0.987951 + 0.154767i \(0.0494627\pi\)
−0.987951 + 0.154767i \(0.950537\pi\)
\(44\) 5.77103 + 2.09570i 0.870015 + 0.315938i
\(45\) −2.29920 − 0.566942i −0.342745 − 0.0845147i
\(46\) −6.15484 + 1.36686i −0.907482 + 0.201533i
\(47\) 3.34313 − 3.34313i 0.487646 − 0.487646i −0.419917 − 0.907563i \(-0.637941\pi\)
0.907563 + 0.419917i \(0.137941\pi\)
\(48\) 3.57583 − 4.27421i 0.516127 − 0.616930i
\(49\) 2.11103i 0.301575i
\(50\) 3.51600 + 6.13496i 0.497238 + 0.867614i
\(51\) 5.38308 + 5.38308i 0.753782 + 0.753782i
\(52\) 2.80249 − 1.30932i 0.388635 − 0.181570i
\(53\) −7.30702 −1.00370 −0.501848 − 0.864956i \(-0.667346\pi\)
−0.501848 + 0.864956i \(0.667346\pi\)
\(54\) 6.74648 + 4.29451i 0.918080 + 0.584408i
\(55\) 5.87483 − 3.55067i 0.792162 − 0.478772i
\(56\) 6.76751 − 5.20472i 0.904346 − 0.695510i
\(57\) −0.0190679 + 0.0190679i −0.00252560 + 0.00252560i
\(58\) 1.44728 + 6.51696i 0.190037 + 0.855719i
\(59\) −3.52732 − 3.52732i −0.459218 − 0.459218i 0.439181 − 0.898399i \(-0.355269\pi\)
−0.898399 + 0.439181i \(0.855269\pi\)
\(60\) −1.20914 − 6.11206i −0.156099 − 0.789064i
\(61\) 1.41629 − 1.41629i 0.181338 − 0.181338i −0.610601 − 0.791939i \(-0.709072\pi\)
0.791939 + 0.610601i \(0.209072\pi\)
\(62\) −6.77887 + 10.6493i −0.860918 + 1.35246i
\(63\) 2.26036 + 2.26036i 0.284779 + 0.284779i
\(64\) −7.73194 − 2.05356i −0.966492 − 0.256695i
\(65\) 0.827973 − 3.35780i 0.102697 − 0.416484i
\(66\) −5.90462 + 1.31129i −0.726809 + 0.161409i
\(67\) 0.748197i 0.0914068i 0.998955 + 0.0457034i \(0.0145529\pi\)
−0.998955 + 0.0457034i \(0.985447\pi\)
\(68\) 3.73030 − 10.2723i 0.452365 − 1.24570i
\(69\) 4.39187 − 4.39187i 0.528719 − 0.528719i
\(70\) 0.221682 − 9.54260i 0.0264961 − 1.14056i
\(71\) −0.269603 −0.0319960 −0.0159980 − 0.999872i \(-0.505093\pi\)
−0.0159980 + 0.999872i \(0.505093\pi\)
\(72\) 0.387713 − 2.97020i 0.0456925 − 0.350041i
\(73\) 0.811870 + 0.811870i 0.0950222 + 0.0950222i 0.753020 − 0.657998i \(-0.228596\pi\)
−0.657998 + 0.753020i \(0.728596\pi\)
\(74\) −5.50380 + 8.64622i −0.639803 + 1.00510i
\(75\) −6.16739 − 3.23844i −0.712149 − 0.373943i
\(76\) 0.0363865 + 0.0132134i 0.00417381 + 0.00151568i
\(77\) −9.26628 −1.05599
\(78\) −1.63636 + 2.57064i −0.185281 + 0.291068i
\(79\) −2.80567 −0.315662 −0.157831 − 0.987466i \(-0.550450\pi\)
−0.157831 + 0.987466i \(0.550450\pi\)
\(80\) −7.21470 + 5.28659i −0.806628 + 0.591059i
\(81\) −4.70135 −0.522372
\(82\) 7.88056 − 12.3800i 0.870262 − 1.36714i
\(83\) 12.8279 1.40804 0.704022 − 0.710178i \(-0.251386\pi\)
0.704022 + 0.710178i \(0.251386\pi\)
\(84\) −2.87077 + 7.90541i −0.313227 + 0.862551i
\(85\) −6.32012 − 10.4571i −0.685514 − 1.13423i
\(86\) −1.54143 + 2.42152i −0.166217 + 0.261119i
\(87\) −4.65027 − 4.65027i −0.498561 − 0.498561i
\(88\) 5.29341 + 6.88283i 0.564279 + 0.733712i
\(89\) −13.3732 −1.41755 −0.708777 − 0.705432i \(-0.750753\pi\)
−0.708777 + 0.705432i \(0.750753\pi\)
\(90\) −2.31243 − 2.42243i −0.243752 − 0.255346i
\(91\) −3.30108 + 3.30108i −0.346047 + 0.346047i
\(92\) −8.38083 − 3.04342i −0.873762 − 0.317299i
\(93\) − 12.4361i − 1.28957i
\(94\) 6.52724 − 1.44956i 0.673233 − 0.149511i
\(95\) 0.0370409 − 0.0223871i 0.00380032 − 0.00229686i
\(96\) 7.51194 − 2.38364i 0.766684 − 0.243279i
\(97\) 6.33466 + 6.33466i 0.643187 + 0.643187i 0.951338 − 0.308151i \(-0.0997101\pi\)
−0.308151 + 0.951338i \(0.599710\pi\)
\(98\) −1.60315 + 2.51848i −0.161943 + 0.254405i
\(99\) −2.29888 + 2.29888i −0.231046 + 0.231046i
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.27.8 yes 18
3.2 odd 2 720.2.z.g.667.2 18
4.3 odd 2 320.2.s.b.207.7 18
5.2 odd 4 400.2.j.d.43.2 18
5.3 odd 4 80.2.j.b.43.8 ✓ 18
5.4 even 2 400.2.s.d.107.2 18
8.3 odd 2 640.2.s.c.287.3 18
8.5 even 2 640.2.s.d.287.7 18
15.8 even 4 720.2.bd.g.523.2 18
16.3 odd 4 80.2.j.b.67.8 yes 18
16.5 even 4 640.2.j.c.607.7 18
16.11 odd 4 640.2.j.d.607.3 18
16.13 even 4 320.2.j.b.47.3 18
20.3 even 4 320.2.j.b.143.7 18
20.7 even 4 1600.2.j.d.143.3 18
20.19 odd 2 1600.2.s.d.207.3 18
40.3 even 4 640.2.j.c.543.3 18
40.13 odd 4 640.2.j.d.543.7 18
48.35 even 4 720.2.bd.g.307.2 18
80.3 even 4 inner 80.2.s.b.3.8 yes 18
80.13 odd 4 320.2.s.b.303.7 18
80.19 odd 4 400.2.j.d.307.2 18
80.29 even 4 1600.2.j.d.1007.7 18
80.43 even 4 640.2.s.d.223.7 18
80.53 odd 4 640.2.s.c.223.3 18
80.67 even 4 400.2.s.d.243.2 18
80.77 odd 4 1600.2.s.d.943.3 18
240.83 odd 4 720.2.z.g.163.2 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.8 ✓ 18 5.3 odd 4
80.2.j.b.67.8 yes 18 16.3 odd 4
80.2.s.b.3.8 yes 18 80.3 even 4 inner
80.2.s.b.27.8 yes 18 1.1 even 1 trivial
320.2.j.b.47.3 18 16.13 even 4
320.2.j.b.143.7 18 20.3 even 4
320.2.s.b.207.7 18 4.3 odd 2
320.2.s.b.303.7 18 80.13 odd 4
400.2.j.d.43.2 18 5.2 odd 4
400.2.j.d.307.2 18 80.19 odd 4
400.2.s.d.107.2 18 5.4 even 2
400.2.s.d.243.2 18 80.67 even 4
640.2.j.c.543.3 18 40.3 even 4
640.2.j.c.607.7 18 16.5 even 4
640.2.j.d.543.7 18 40.13 odd 4
640.2.j.d.607.3 18 16.11 odd 4
640.2.s.c.223.3 18 80.53 odd 4
640.2.s.c.287.3 18 8.3 odd 2
640.2.s.d.223.7 18 80.43 even 4
640.2.s.d.287.7 18 8.5 even 2
720.2.z.g.163.2 18 240.83 odd 4
720.2.z.g.667.2 18 3.2 odd 2
720.2.bd.g.307.2 18 48.35 even 4
720.2.bd.g.523.2 18 15.8 even 4
1600.2.j.d.143.3 18 20.7 even 4
1600.2.j.d.1007.7 18 80.29 even 4
1600.2.s.d.207.3 18 20.19 odd 2
1600.2.s.d.943.3 18 80.77 odd 4