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(21,80)] 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("80.21"); 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([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 80.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.638803216170\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 21.4
Root \(1.32070 + 0.505727i\) of defining polynomial
Character \(\chi\) \(=\) 80.21
Dual form 80.2.l.a.61.4

$q$-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+(-0.257150 + 1.39064i) q^{2} +(-1.66366 + 1.66366i) q^{3} +(-1.86775 - 0.715205i) q^{4} +(-0.707107 - 0.707107i) q^{5} +(-1.88574 - 2.74137i) q^{6} +2.89402i q^{7} +(1.47488 - 2.41345i) q^{8} -2.53555i q^{9} +(1.16516 - 0.801497i) q^{10} +(1.84462 + 1.84462i) q^{11} +(4.29717 - 1.91744i) q^{12} +(-3.08011 + 3.08011i) q^{13} +(-4.02454 - 0.744198i) q^{14} +2.35278 q^{15} +(2.97696 + 2.67165i) q^{16} +7.29875 q^{17} +(3.52604 + 0.652018i) q^{18} +(-1.23593 + 1.23593i) q^{19} +(0.814970 + 1.82642i) q^{20} +(-4.81468 - 4.81468i) q^{21} +(-3.03955 + 2.09086i) q^{22} -4.60490i q^{23} +(1.56145 + 6.46887i) q^{24} +1.00000i q^{25} +(-3.49126 - 5.07536i) q^{26} +(-0.772683 - 0.772683i) q^{27} +(2.06982 - 5.40530i) q^{28} +(4.24680 - 4.24680i) q^{29} +(-0.605017 + 3.27186i) q^{30} +2.06299 q^{31} +(-4.48082 + 3.45286i) q^{32} -6.13767 q^{33} +(-1.87688 + 10.1499i) q^{34} +(2.04638 - 2.04638i) q^{35} +(-1.81344 + 4.73577i) q^{36} +(-1.17899 - 1.17899i) q^{37} +(-1.40091 - 2.03655i) q^{38} -10.2485i q^{39} +(-2.74946 + 0.663664i) q^{40} +4.61484i q^{41} +(7.93357 - 5.45738i) q^{42} +(3.03019 + 3.03019i) q^{43} +(-2.12601 - 4.76458i) q^{44} +(-1.79291 + 1.79291i) q^{45} +(6.40375 + 1.18415i) q^{46} -11.7111 q^{47} +(-9.39739 + 0.507943i) q^{48} -1.37537 q^{49} +(-1.39064 - 0.257150i) q^{50} +(-12.1427 + 12.1427i) q^{51} +(7.95577 - 3.54995i) q^{52} +(2.73048 + 2.73048i) q^{53} +(1.27322 - 0.875827i) q^{54} -2.60869i q^{55} +(6.98457 + 4.26835i) q^{56} -4.11235i q^{57} +(4.81369 + 6.99782i) q^{58} +(3.11306 + 3.11306i) q^{59} +(-4.39439 - 1.68272i) q^{60} +(2.34962 - 2.34962i) q^{61} +(-0.530498 + 2.86887i) q^{62} +7.33795 q^{63} +(-3.64944 - 7.11910i) q^{64} +4.35593 q^{65} +(1.57830 - 8.53528i) q^{66} +(8.24311 - 8.24311i) q^{67} +(-13.6322 - 5.22011i) q^{68} +(7.66101 + 7.66101i) q^{69} +(2.31955 + 3.37201i) q^{70} -3.25937i q^{71} +(-6.11942 - 3.73965i) q^{72} +12.6877i q^{73} +(1.94272 - 1.33637i) q^{74} +(-1.66366 - 1.66366i) q^{75} +(3.19235 - 1.42446i) q^{76} +(-5.33839 + 5.33839i) q^{77} +(14.2520 + 2.63541i) q^{78} -0.113885 q^{79} +(-0.215891 - 3.99417i) q^{80} +10.1776 q^{81} +(-6.41758 - 1.18671i) q^{82} +(9.76813 - 9.76813i) q^{83} +(5.54912 + 12.4361i) q^{84} +(-5.16100 - 5.16100i) q^{85} +(-4.99310 + 3.43468i) q^{86} +14.1305i q^{87} +(7.17251 - 1.73129i) q^{88} -3.74593i q^{89} +(-2.03224 - 2.95433i) q^{90} +(-8.91390 - 8.91390i) q^{91} +(-3.29345 + 8.60080i) q^{92} +(-3.43212 + 3.43212i) q^{93} +(3.01150 - 16.2858i) q^{94} +1.74787 q^{95} +(1.71017 - 13.1990i) q^{96} -13.9853 q^{97} +(0.353676 - 1.91264i) q^{98} +(4.67714 - 4.67714i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} - 12 q^{6} + 4 q^{10} - 8 q^{11} - 12 q^{12} + 4 q^{14} - 8 q^{15} + 16 q^{16} - 8 q^{19} + 8 q^{20} - 20 q^{22} + 8 q^{24} - 16 q^{26} + 24 q^{27} - 4 q^{28} - 16 q^{29} + 16 q^{34} - 4 q^{36}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.257150 + 1.39064i −0.181833 + 0.983329i
\(3\) −1.66366 + 1.66366i −0.960517 + 0.960517i −0.999250 − 0.0387330i \(-0.987668\pi\)
0.0387330 + 0.999250i \(0.487668\pi\)
\(4\) −1.86775 − 0.715205i −0.933874 − 0.357603i
\(5\) −0.707107 − 0.707107i −0.316228 − 0.316228i
\(6\) −1.88574 − 2.74137i −0.769851 − 1.11916i
\(7\) 2.89402i 1.09384i 0.837186 + 0.546919i \(0.184199\pi\)
−0.837186 + 0.546919i \(0.815801\pi\)
\(8\) 1.47488 − 2.41345i 0.521450 − 0.853282i
\(9\) − 2.53555i − 0.845184i
\(10\) 1.16516 − 0.801497i 0.368457 − 0.253456i
\(11\) 1.84462 + 1.84462i 0.556175 + 0.556175i 0.928216 − 0.372041i \(-0.121342\pi\)
−0.372041 + 0.928216i \(0.621342\pi\)
\(12\) 4.29717 − 1.91744i 1.24048 − 0.553518i
\(13\) −3.08011 + 3.08011i −0.854268 + 0.854268i −0.990656 − 0.136388i \(-0.956451\pi\)
0.136388 + 0.990656i \(0.456451\pi\)
\(14\) −4.02454 − 0.744198i −1.07560 − 0.198895i
\(15\) 2.35278 0.607484
\(16\) 2.97696 + 2.67165i 0.744241 + 0.667912i
\(17\) 7.29875 1.77021 0.885104 − 0.465393i \(-0.154087\pi\)
0.885104 + 0.465393i \(0.154087\pi\)
\(18\) 3.52604 + 0.652018i 0.831095 + 0.153682i
\(19\) −1.23593 + 1.23593i −0.283542 + 0.283542i −0.834520 − 0.550978i \(-0.814255\pi\)
0.550978 + 0.834520i \(0.314255\pi\)
\(20\) 0.814970 + 1.82642i 0.182233 + 0.408401i
\(21\) −4.81468 − 4.81468i −1.05065 − 1.05065i
\(22\) −3.03955 + 2.09086i −0.648034 + 0.445773i
\(23\) − 4.60490i − 0.960189i −0.877217 − 0.480094i \(-0.840602\pi\)
0.877217 − 0.480094i \(-0.159398\pi\)
\(24\) 1.56145 + 6.46887i 0.318730 + 1.32045i
\(25\) 1.00000i 0.200000i
\(26\) −3.49126 − 5.07536i −0.684693 − 0.995360i
\(27\) −0.772683 − 0.772683i −0.148703 − 0.148703i
\(28\) 2.06982 − 5.40530i 0.391159 − 1.02151i
\(29\) 4.24680 − 4.24680i 0.788611 − 0.788611i −0.192656 − 0.981266i \(-0.561710\pi\)
0.981266 + 0.192656i \(0.0617101\pi\)
\(30\) −0.605017 + 3.27186i −0.110460 + 0.597357i
\(31\) 2.06299 0.370524 0.185262 − 0.982689i \(-0.440687\pi\)
0.185262 + 0.982689i \(0.440687\pi\)
\(32\) −4.48082 + 3.45286i −0.792104 + 0.610386i
\(33\) −6.13767 −1.06843
\(34\) −1.87688 + 10.1499i −0.321882 + 1.74070i
\(35\) 2.04638 − 2.04638i 0.345902 − 0.345902i
\(36\) −1.81344 + 4.73577i −0.302240 + 0.789296i
\(37\) −1.17899 − 1.17899i −0.193825 − 0.193825i 0.603522 − 0.797346i \(-0.293764\pi\)
−0.797346 + 0.603522i \(0.793764\pi\)
\(38\) −1.40091 − 2.03655i −0.227258 − 0.330373i
\(39\) − 10.2485i − 1.64108i
\(40\) −2.74946 + 0.663664i −0.434728 + 0.104934i
\(41\) 4.61484i 0.720717i 0.932814 + 0.360359i \(0.117346\pi\)
−0.932814 + 0.360359i \(0.882654\pi\)
\(42\) 7.93357 − 5.45738i 1.22418 − 0.842092i
\(43\) 3.03019 + 3.03019i 0.462099 + 0.462099i 0.899343 − 0.437244i \(-0.144045\pi\)
−0.437244 + 0.899343i \(0.644045\pi\)
\(44\) −2.12601 − 4.76458i −0.320508 − 0.718287i
\(45\) −1.79291 + 1.79291i −0.267271 + 0.267271i
\(46\) 6.40375 + 1.18415i 0.944182 + 0.174594i
\(47\) −11.7111 −1.70823 −0.854117 − 0.520081i \(-0.825902\pi\)
−0.854117 + 0.520081i \(0.825902\pi\)
\(48\) −9.39739 + 0.507943i −1.35640 + 0.0733152i
\(49\) −1.37537 −0.196481
\(50\) −1.39064 − 0.257150i −0.196666 − 0.0363665i
\(51\) −12.1427 + 12.1427i −1.70031 + 1.70031i
\(52\) 7.95577 − 3.54995i 1.10327 − 0.492290i
\(53\) 2.73048 + 2.73048i 0.375061 + 0.375061i 0.869316 − 0.494256i \(-0.164559\pi\)
−0.494256 + 0.869316i \(0.664559\pi\)
\(54\) 1.27322 − 0.875827i 0.173263 − 0.119185i
\(55\) − 2.60869i − 0.351756i
\(56\) 6.98457 + 4.26835i 0.933352 + 0.570382i
\(57\) − 4.11235i − 0.544694i
\(58\) 4.81369 + 6.99782i 0.632069 + 0.918859i
\(59\) 3.11306 + 3.11306i 0.405285 + 0.405285i 0.880091 − 0.474805i \(-0.157482\pi\)
−0.474805 + 0.880091i \(0.657482\pi\)
\(60\) −4.39439 − 1.68272i −0.567313 − 0.217238i
\(61\) 2.34962 − 2.34962i 0.300838 − 0.300838i −0.540503 − 0.841342i \(-0.681766\pi\)
0.841342 + 0.540503i \(0.181766\pi\)
\(62\) −0.530498 + 2.86887i −0.0673733 + 0.364347i
\(63\) 7.33795 0.924495
\(64\) −3.64944 − 7.11910i −0.456180 − 0.889888i
\(65\) 4.35593 0.540286
\(66\) 1.57830 − 8.53528i 0.194276 − 1.05062i
\(67\) 8.24311 − 8.24311i 1.00706 − 1.00706i 0.00708173 − 0.999975i \(-0.497746\pi\)
0.999975 − 0.00708173i \(-0.00225420\pi\)
\(68\) −13.6322 − 5.22011i −1.65315 − 0.633031i
\(69\) 7.66101 + 7.66101i 0.922277 + 0.922277i
\(70\) 2.31955 + 3.37201i 0.277239 + 0.403032i
\(71\) − 3.25937i − 0.386816i −0.981118 − 0.193408i \(-0.938046\pi\)
0.981118 − 0.193408i \(-0.0619541\pi\)
\(72\) −6.11942 − 3.73965i −0.721180 − 0.440721i
\(73\) 12.6877i 1.48499i 0.669853 + 0.742494i \(0.266357\pi\)
−0.669853 + 0.742494i \(0.733643\pi\)
\(74\) 1.94272 − 1.33637i 0.225837 − 0.155350i
\(75\) −1.66366 − 1.66366i −0.192103 − 0.192103i
\(76\) 3.19235 − 1.42446i 0.366188 − 0.163397i
\(77\) −5.33839 + 5.33839i −0.608365 + 0.608365i
\(78\) 14.2520 + 2.63541i 1.61372 + 0.298401i
\(79\) −0.113885 −0.0128130 −0.00640652 − 0.999979i \(-0.502039\pi\)
−0.00640652 + 0.999979i \(0.502039\pi\)
\(80\) −0.215891 − 3.99417i −0.0241373 − 0.446562i
\(81\) 10.1776 1.13085
\(82\) −6.41758 − 1.18671i −0.708703 − 0.131050i
\(83\) 9.76813 − 9.76813i 1.07219 − 1.07219i 0.0750089 − 0.997183i \(-0.476101\pi\)
0.997183 − 0.0750089i \(-0.0238985\pi\)
\(84\) 5.54912 + 12.4361i 0.605459 + 1.35689i
\(85\) −5.16100 − 5.16100i −0.559789 − 0.559789i
\(86\) −4.99310 + 3.43468i −0.538420 + 0.370371i
\(87\) 14.1305i 1.51495i
\(88\) 7.17251 − 1.73129i 0.764592 − 0.184557i
\(89\) − 3.74593i − 0.397068i −0.980094 − 0.198534i \(-0.936382\pi\)
0.980094 − 0.198534i \(-0.0636180\pi\)
\(90\) −2.03224 − 2.95433i −0.214217 − 0.311414i
\(91\) −8.91390 − 8.91390i −0.934430 − 0.934430i
\(92\) −3.29345 + 8.60080i −0.343366 + 0.896695i
\(93\) −3.43212 + 3.43212i −0.355894 + 0.355894i
\(94\) 3.01150 − 16.2858i 0.310613 − 1.67976i
\(95\) 1.74787 0.179328
\(96\) 1.71017 − 13.1990i 0.174544 − 1.34711i
\(97\) −13.9853 −1.41999 −0.709995 − 0.704206i \(-0.751303\pi\)
−0.709995 + 0.704206i \(0.751303\pi\)
\(98\) 0.353676 − 1.91264i 0.0357267 − 0.193206i
\(99\) 4.67714 − 4.67714i 0.470071 − 0.470071i
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.l.a.21.4 ✓ 16
3.2 odd 2 720.2.t.c.181.5 16
4.3 odd 2 320.2.l.a.241.7 16
5.2 odd 4 400.2.q.g.149.2 16
5.3 odd 4 400.2.q.h.149.7 16
5.4 even 2 400.2.l.h.101.5 16
8.3 odd 2 640.2.l.a.481.2 16
8.5 even 2 640.2.l.b.481.7 16
12.11 even 2 2880.2.t.c.2161.6 16
16.3 odd 4 320.2.l.a.81.7 16
16.5 even 4 640.2.l.b.161.7 16
16.11 odd 4 640.2.l.a.161.2 16
16.13 even 4 inner 80.2.l.a.61.4 yes 16
20.3 even 4 1600.2.q.g.49.7 16
20.7 even 4 1600.2.q.h.49.2 16
20.19 odd 2 1600.2.l.i.1201.2 16
32.3 odd 8 5120.2.a.t.1.8 8
32.13 even 8 5120.2.a.s.1.8 8
32.19 odd 8 5120.2.a.u.1.1 8
32.29 even 8 5120.2.a.v.1.1 8
48.29 odd 4 720.2.t.c.541.5 16
48.35 even 4 2880.2.t.c.721.7 16
80.3 even 4 1600.2.q.h.849.2 16
80.13 odd 4 400.2.q.g.349.2 16
80.19 odd 4 1600.2.l.i.401.2 16
80.29 even 4 400.2.l.h.301.5 16
80.67 even 4 1600.2.q.g.849.7 16
80.77 odd 4 400.2.q.h.349.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.4 ✓ 16 1.1 even 1 trivial
80.2.l.a.61.4 yes 16 16.13 even 4 inner
320.2.l.a.81.7 16 16.3 odd 4
320.2.l.a.241.7 16 4.3 odd 2
400.2.l.h.101.5 16 5.4 even 2
400.2.l.h.301.5 16 80.29 even 4
400.2.q.g.149.2 16 5.2 odd 4
400.2.q.g.349.2 16 80.13 odd 4
400.2.q.h.149.7 16 5.3 odd 4
400.2.q.h.349.7 16 80.77 odd 4
640.2.l.a.161.2 16 16.11 odd 4
640.2.l.a.481.2 16 8.3 odd 2
640.2.l.b.161.7 16 16.5 even 4
640.2.l.b.481.7 16 8.5 even 2
720.2.t.c.181.5 16 3.2 odd 2
720.2.t.c.541.5 16 48.29 odd 4
1600.2.l.i.401.2 16 80.19 odd 4
1600.2.l.i.1201.2 16 20.19 odd 2
1600.2.q.g.49.7 16 20.3 even 4
1600.2.q.g.849.7 16 80.67 even 4
1600.2.q.h.49.2 16 20.7 even 4
1600.2.q.h.849.2 16 80.3 even 4
2880.2.t.c.721.7 16 48.35 even 4
2880.2.t.c.2161.6 16 12.11 even 2
5120.2.a.s.1.8 8 32.13 even 8
5120.2.a.t.1.8 8 32.3 odd 8
5120.2.a.u.1.1 8 32.19 odd 8
5120.2.a.v.1.1 8 32.29 even 8