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(29,80)] 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("80.29"); 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, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 80.q (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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: 16.0.534694406811304329216.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2x^{14} - 2x^{12} + 4x^{10} + 4x^{8} + 16x^{6} - 32x^{4} - 128x^{2} + 256 \) 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 29.6
Root \(1.32661 + 0.490008i\) of defining polynomial
Character \(\chi\) \(=\) 80.29
Dual form 80.2.q.c.69.6

$q$-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+(0.490008 + 1.32661i) q^{2} +(1.99154 - 1.99154i) q^{3} +(-1.51978 + 1.30010i) q^{4} +(-2.16225 + 0.569800i) q^{5} +(3.61786 + 1.66612i) q^{6} +1.09033 q^{7} +(-2.46943 - 1.37910i) q^{8} -4.93244i q^{9} +(-1.81542 - 2.58926i) q^{10} +(-2.33225 + 2.33225i) q^{11} +(-0.437515 + 5.61590i) q^{12} +(-1.80775 + 1.80775i) q^{13} +(0.534268 + 1.44644i) q^{14} +(-3.17142 + 5.44098i) q^{15} +(0.619491 - 3.95174i) q^{16} -4.93886i q^{17} +(6.54342 - 2.41694i) q^{18} +(2.03957 + 2.03957i) q^{19} +(2.54536 - 3.67711i) q^{20} +(2.17142 - 2.17142i) q^{21} +(-4.23680 - 1.95116i) q^{22} -1.45791 q^{23} +(-7.66449 + 2.17142i) q^{24} +(4.35066 - 2.46410i) q^{25} +(-3.28398 - 1.51236i) q^{26} +(-3.84853 - 3.84853i) q^{27} +(-1.65706 + 1.41753i) q^{28} +(-0.707323 - 0.707323i) q^{29} +(-8.77208 - 1.54112i) q^{30} +10.1286 q^{31} +(5.54597 - 1.11456i) q^{32} +9.28951i q^{33} +(6.55193 - 2.42008i) q^{34} +(-2.35756 + 0.621268i) q^{35} +(6.41266 + 7.49625i) q^{36} +(-4.35066 - 4.35066i) q^{37} +(-1.70631 + 3.70512i) q^{38} +7.20039i q^{39} +(6.12534 + 1.57488i) q^{40} +10.2753i q^{41} +(3.94465 + 1.81662i) q^{42} +(2.22457 + 2.22457i) q^{43} +(0.512364 - 6.57666i) q^{44} +(2.81051 + 10.6652i) q^{45} +(-0.714386 - 1.93407i) q^{46} +2.09458i q^{47} +(-6.63629 - 9.10377i) q^{48} -5.81119 q^{49} +(5.40076 + 4.56419i) q^{50} +(-9.83592 - 9.83592i) q^{51} +(0.397138 - 5.09763i) q^{52} +(-0.215297 - 0.215297i) q^{53} +(3.21969 - 6.99131i) q^{54} +(3.71399 - 6.37182i) q^{55} +(-2.69248 - 1.50367i) q^{56} +8.12376 q^{57} +(0.591747 - 1.28493i) q^{58} +(-1.16082 + 1.16082i) q^{59} +(-2.25393 - 12.3923i) q^{60} +(-3.46410 - 3.46410i) q^{61} +(4.96309 + 13.4367i) q^{62} -5.37797i q^{63} +(4.19615 + 6.81119i) q^{64} +(2.87875 - 4.93886i) q^{65} +(-12.3236 + 4.55193i) q^{66} +(5.04189 - 5.04189i) q^{67} +(6.42100 + 7.50600i) q^{68} +(-2.90348 + 2.90348i) q^{69} +(-1.97940 - 2.82313i) q^{70} -6.40078i q^{71} +(-6.80234 + 12.1803i) q^{72} -5.24343 q^{73} +(3.63977 - 7.90348i) q^{74} +(3.75714 - 13.5718i) q^{75} +(-5.75135 - 0.448067i) q^{76} +(-2.54291 + 2.54291i) q^{77} +(-9.55211 + 3.52825i) q^{78} +2.61504 q^{79} +(0.912206 + 8.89763i) q^{80} -0.531659 q^{81} +(-13.6313 + 5.03497i) q^{82} +(-5.67856 + 5.67856i) q^{83} +(-0.477033 + 6.12316i) q^{84} +(2.81416 + 10.6790i) q^{85} +(-1.86108 + 4.04120i) q^{86} -2.81732 q^{87} +(8.97572 - 2.54291i) q^{88} -6.87875i q^{89} +(-12.7714 + 8.95446i) q^{90} +(-1.97103 + 1.97103i) q^{91} +(2.21570 - 1.89542i) q^{92} +(20.1715 - 20.1715i) q^{93} +(-2.77868 + 1.02636i) q^{94} +(-5.57221 - 3.24791i) q^{95} +(8.82531 - 13.2647i) q^{96} +3.77310i q^{97} +(-2.84753 - 7.70918i) q^{98} +(11.5037 + 11.5037i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} - 8 q^{5} - 4 q^{6} - 12 q^{10} + 8 q^{11} - 4 q^{14} + 16 q^{16} - 8 q^{19} - 4 q^{20} - 16 q^{21} - 32 q^{24} + 32 q^{26} - 16 q^{29} - 36 q^{30} + 16 q^{31} + 48 q^{34} - 24 q^{35} + 60 q^{36}+ \cdots + 88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.490008 + 1.32661i 0.346488 + 0.938054i
\(3\) 1.99154 − 1.99154i 1.14981 − 1.14981i 0.163226 − 0.986589i \(-0.447810\pi\)
0.986589 − 0.163226i \(-0.0521899\pi\)
\(4\) −1.51978 + 1.30010i −0.759892 + 0.650049i
\(5\) −2.16225 + 0.569800i −0.966988 + 0.254822i
\(6\) 3.61786 + 1.66612i 1.47699 + 0.680192i
\(7\) 1.09033 0.412104 0.206052 − 0.978541i \(-0.433938\pi\)
0.206052 + 0.978541i \(0.433938\pi\)
\(8\) −2.46943 − 1.37910i −0.873075 − 0.487586i
\(9\) − 4.93244i − 1.64415i
\(10\) −1.81542 − 2.58926i −0.574087 − 0.818794i
\(11\) −2.33225 + 2.33225i −0.703199 + 0.703199i −0.965096 − 0.261897i \(-0.915652\pi\)
0.261897 + 0.965096i \(0.415652\pi\)
\(12\) −0.437515 + 5.61590i −0.126300 + 1.62117i
\(13\) −1.80775 + 1.80775i −0.501379 + 0.501379i −0.911866 − 0.410487i \(-0.865359\pi\)
0.410487 + 0.911866i \(0.365359\pi\)
\(14\) 0.534268 + 1.44644i 0.142789 + 0.386576i
\(15\) −3.17142 + 5.44098i −0.818858 + 1.40486i
\(16\) 0.619491 − 3.95174i 0.154873 − 0.987934i
\(17\) − 4.93886i − 1.19785i −0.800806 − 0.598924i \(-0.795595\pi\)
0.800806 − 0.598924i \(-0.204405\pi\)
\(18\) 6.54342 − 2.41694i 1.54230 − 0.569677i
\(19\) 2.03957 + 2.03957i 0.467909 + 0.467909i 0.901237 − 0.433327i \(-0.142661\pi\)
−0.433327 + 0.901237i \(0.642661\pi\)
\(20\) 2.54536 − 3.67711i 0.569160 − 0.822227i
\(21\) 2.17142 − 2.17142i 0.473844 − 0.473844i
\(22\) −4.23680 − 1.95116i −0.903289 − 0.415989i
\(23\) −1.45791 −0.303995 −0.151997 − 0.988381i \(-0.548570\pi\)
−0.151997 + 0.988381i \(0.548570\pi\)
\(24\) −7.66449 + 2.17142i −1.56451 + 0.443240i
\(25\) 4.35066 − 2.46410i 0.870131 − 0.492820i
\(26\) −3.28398 − 1.51236i −0.644042 − 0.296599i
\(27\) −3.84853 − 3.84853i −0.740650 − 0.740650i
\(28\) −1.65706 + 1.41753i −0.313155 + 0.267888i
\(29\) −0.707323 − 0.707323i −0.131347 − 0.131347i 0.638377 − 0.769724i \(-0.279606\pi\)
−0.769724 + 0.638377i \(0.779606\pi\)
\(30\) −8.77208 − 1.54112i −1.60156 − 0.281368i
\(31\) 10.1286 1.81915 0.909575 − 0.415541i \(-0.136408\pi\)
0.909575 + 0.415541i \(0.136408\pi\)
\(32\) 5.54597 − 1.11456i 0.980398 − 0.197028i
\(33\) 9.28951i 1.61710i
\(34\) 6.55193 − 2.42008i 1.12365 − 0.415040i
\(35\) −2.35756 + 0.621268i −0.398500 + 0.105013i
\(36\) 6.41266 + 7.49625i 1.06878 + 1.24938i
\(37\) −4.35066 − 4.35066i −0.715243 − 0.715243i 0.252384 − 0.967627i \(-0.418785\pi\)
−0.967627 + 0.252384i \(0.918785\pi\)
\(38\) −1.70631 + 3.70512i −0.276800 + 0.601049i
\(39\) 7.20039i 1.15299i
\(40\) 6.12534 + 1.57488i 0.968501 + 0.249011i
\(41\) 10.2753i 1.60473i 0.596833 + 0.802365i \(0.296425\pi\)
−0.596833 + 0.802365i \(0.703575\pi\)
\(42\) 3.94465 + 1.81662i 0.608672 + 0.280310i
\(43\) 2.22457 + 2.22457i 0.339244 + 0.339244i 0.856083 − 0.516839i \(-0.172891\pi\)
−0.516839 + 0.856083i \(0.672891\pi\)
\(44\) 0.512364 − 6.57666i 0.0772418 − 0.991469i
\(45\) 2.81051 + 10.6652i 0.418966 + 1.58987i
\(46\) −0.714386 − 1.93407i −0.105330 − 0.285163i
\(47\) 2.09458i 0.305525i 0.988263 + 0.152763i \(0.0488170\pi\)
−0.988263 + 0.152763i \(0.951183\pi\)
\(48\) −6.63629 − 9.10377i −0.957866 − 1.31402i
\(49\) −5.81119 −0.830170
\(50\) 5.40076 + 4.56419i 0.763782 + 0.645474i
\(51\) −9.83592 − 9.83592i −1.37730 − 1.37730i
\(52\) 0.397138 − 5.09763i 0.0550732 − 0.706915i
\(53\) −0.215297 − 0.215297i −0.0295733 − 0.0295733i 0.692166 − 0.721739i \(-0.256657\pi\)
−0.721739 + 0.692166i \(0.756657\pi\)
\(54\) 3.21969 − 6.99131i 0.438144 − 0.951396i
\(55\) 3.71399 − 6.37182i 0.500794 − 0.859175i
\(56\) −2.69248 − 1.50367i −0.359798 − 0.200936i
\(57\) 8.12376 1.07602
\(58\) 0.591747 − 1.28493i 0.0777002 − 0.168720i
\(59\) −1.16082 + 1.16082i −0.151126 + 0.151126i −0.778621 − 0.627495i \(-0.784080\pi\)
0.627495 + 0.778621i \(0.284080\pi\)
\(60\) −2.25393 − 12.3923i −0.290981 − 1.59984i
\(61\) −3.46410 − 3.46410i −0.443533 − 0.443533i 0.449665 − 0.893197i \(-0.351543\pi\)
−0.893197 + 0.449665i \(0.851543\pi\)
\(62\) 4.96309 + 13.4367i 0.630313 + 1.70646i
\(63\) − 5.37797i − 0.677560i
\(64\) 4.19615 + 6.81119i 0.524519 + 0.851399i
\(65\) 2.87875 − 4.93886i 0.357065 − 0.612590i
\(66\) −12.3236 + 4.55193i −1.51692 + 0.560304i
\(67\) 5.04189 − 5.04189i 0.615965 − 0.615965i −0.328529 − 0.944494i \(-0.606553\pi\)
0.944494 + 0.328529i \(0.106553\pi\)
\(68\) 6.42100 + 7.50600i 0.778660 + 0.910236i
\(69\) −2.90348 + 2.90348i −0.349537 + 0.349537i
\(70\) −1.97940 − 2.82313i −0.236584 − 0.337429i
\(71\) − 6.40078i − 0.759633i −0.925062 − 0.379817i \(-0.875987\pi\)
0.925062 − 0.379817i \(-0.124013\pi\)
\(72\) −6.80234 + 12.1803i −0.801664 + 1.43546i
\(73\) −5.24343 −0.613696 −0.306848 − 0.951758i \(-0.599274\pi\)
−0.306848 + 0.951758i \(0.599274\pi\)
\(74\) 3.63977 − 7.90348i 0.423114 − 0.918760i
\(75\) 3.75714 − 13.5718i 0.433837 − 1.56714i
\(76\) −5.75135 − 0.448067i −0.659725 − 0.0513968i
\(77\) −2.54291 + 2.54291i −0.289791 + 0.289791i
\(78\) −9.55211 + 3.52825i −1.08156 + 0.399495i
\(79\) 2.61504 0.294215 0.147107 − 0.989121i \(-0.453004\pi\)
0.147107 + 0.989121i \(0.453004\pi\)
\(80\) 0.912206 + 8.89763i 0.101988 + 0.994786i
\(81\) −0.531659 −0.0590733
\(82\) −13.6313 + 5.03497i −1.50532 + 0.556020i
\(83\) −5.67856 + 5.67856i −0.623303 + 0.623303i −0.946375 − 0.323071i \(-0.895285\pi\)
0.323071 + 0.946375i \(0.395285\pi\)
\(84\) −0.477033 + 6.12316i −0.0520486 + 0.668092i
\(85\) 2.81416 + 10.6790i 0.305239 + 1.15831i
\(86\) −1.86108 + 4.04120i −0.200686 + 0.435774i
\(87\) −2.81732 −0.302048
\(88\) 8.97572 − 2.54291i 0.956815 − 0.271075i
\(89\) − 6.87875i − 0.729146i −0.931175 − 0.364573i \(-0.881215\pi\)
0.931175 − 0.364573i \(-0.118785\pi\)
\(90\) −12.7714 + 8.95446i −1.34622 + 0.943883i
\(91\) −1.97103 + 1.97103i −0.206620 + 0.206620i
\(92\) 2.21570 − 1.89542i 0.231003 − 0.197611i
\(93\) 20.1715 − 20.1715i 2.09168 − 2.09168i
\(94\) −2.77868 + 1.02636i −0.286599 + 0.105861i
\(95\) −5.57221 − 3.24791i −0.571696 − 0.333229i
\(96\) 8.82531 − 13.2647i 0.900730 − 1.35382i
\(97\) 3.77310i 0.383100i 0.981483 + 0.191550i \(0.0613515\pi\)
−0.981483 + 0.191550i \(0.938649\pi\)
\(98\) −2.84753 − 7.70918i −0.287644 − 0.778745i
\(99\) 11.5037 + 11.5037i 1.15616 + 1.15616i
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.q.c.29.6 yes 16
3.2 odd 2 720.2.bm.f.109.3 16
4.3 odd 2 320.2.q.c.209.1 16
5.2 odd 4 400.2.l.i.301.2 16
5.3 odd 4 400.2.l.i.301.7 16
5.4 even 2 inner 80.2.q.c.29.3 ✓ 16
8.3 odd 2 640.2.q.f.289.8 16
8.5 even 2 640.2.q.e.289.1 16
15.14 odd 2 720.2.bm.f.109.6 16
16.3 odd 4 640.2.q.f.609.1 16
16.5 even 4 inner 80.2.q.c.69.3 yes 16
16.11 odd 4 320.2.q.c.49.8 16
16.13 even 4 640.2.q.e.609.8 16
20.3 even 4 1600.2.l.h.401.1 16
20.7 even 4 1600.2.l.h.401.8 16
20.19 odd 2 320.2.q.c.209.8 16
40.19 odd 2 640.2.q.f.289.1 16
40.29 even 2 640.2.q.e.289.8 16
48.5 odd 4 720.2.bm.f.469.6 16
80.19 odd 4 640.2.q.f.609.8 16
80.27 even 4 1600.2.l.h.1201.8 16
80.29 even 4 640.2.q.e.609.1 16
80.37 odd 4 400.2.l.i.101.2 16
80.43 even 4 1600.2.l.h.1201.1 16
80.53 odd 4 400.2.l.i.101.7 16
80.59 odd 4 320.2.q.c.49.1 16
80.69 even 4 inner 80.2.q.c.69.6 yes 16
240.149 odd 4 720.2.bm.f.469.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.q.c.29.3 ✓ 16 5.4 even 2 inner
80.2.q.c.29.6 yes 16 1.1 even 1 trivial
80.2.q.c.69.3 yes 16 16.5 even 4 inner
80.2.q.c.69.6 yes 16 80.69 even 4 inner
320.2.q.c.49.1 16 80.59 odd 4
320.2.q.c.49.8 16 16.11 odd 4
320.2.q.c.209.1 16 4.3 odd 2
320.2.q.c.209.8 16 20.19 odd 2
400.2.l.i.101.2 16 80.37 odd 4
400.2.l.i.101.7 16 80.53 odd 4
400.2.l.i.301.2 16 5.2 odd 4
400.2.l.i.301.7 16 5.3 odd 4
640.2.q.e.289.1 16 8.5 even 2
640.2.q.e.289.8 16 40.29 even 2
640.2.q.e.609.1 16 80.29 even 4
640.2.q.e.609.8 16 16.13 even 4
640.2.q.f.289.1 16 40.19 odd 2
640.2.q.f.289.8 16 8.3 odd 2
640.2.q.f.609.1 16 16.3 odd 4
640.2.q.f.609.8 16 80.19 odd 4
720.2.bm.f.109.3 16 3.2 odd 2
720.2.bm.f.109.6 16 15.14 odd 2
720.2.bm.f.469.3 16 240.149 odd 4
720.2.bm.f.469.6 16 48.5 odd 4
1600.2.l.h.401.1 16 20.3 even 4
1600.2.l.h.401.8 16 20.7 even 4
1600.2.l.h.1201.1 16 80.43 even 4
1600.2.l.h.1201.8 16 80.27 even 4