Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [720,2,Mod(109,720)] 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("720.109"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.bm (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-4,8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.74922894553\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 469.6
Root \(1.32661 - 0.490008i\) of defining polynomial
Character \(\chi\) \(=\) 720.469
Dual form 720.2.bm.f.109.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.490008 - 1.32661i) q^{2} +(-1.51978 - 1.30010i) q^{4} +(0.569800 + 2.16225i) q^{5} -1.09033 q^{7} +(-2.46943 + 1.37910i) q^{8} +(3.14767 + 0.303617i) q^{10} +(2.33225 + 2.33225i) q^{11} +(1.80775 + 1.80775i) q^{13} +(-0.534268 + 1.44644i) q^{14} +(0.619491 + 3.95174i) q^{16} +4.93886i q^{17} +(2.03957 - 2.03957i) q^{19} +(1.94516 - 4.02695i) q^{20} +(4.23680 - 1.95116i) q^{22} -1.45791 q^{23} +(-4.35066 + 2.46410i) q^{25} +(3.28398 - 1.51236i) q^{26} +(1.65706 + 1.41753i) q^{28} +(0.707323 - 0.707323i) q^{29} +10.1286 q^{31} +(5.54597 + 1.11456i) q^{32} +(6.55193 + 2.42008i) q^{34} +(-0.621268 - 2.35756i) q^{35} +(4.35066 - 4.35066i) q^{37} +(-1.70631 - 3.70512i) q^{38} +(-4.38905 - 4.55371i) q^{40} +10.2753i q^{41} +(-2.22457 + 2.22457i) q^{43} +(-0.512364 - 6.57666i) q^{44} +(-0.714386 + 1.93407i) q^{46} -2.09458i q^{47} -5.81119 q^{49} +(1.13705 + 6.97905i) q^{50} +(-0.397138 - 5.09763i) q^{52} +(-0.215297 + 0.215297i) q^{53} +(-3.71399 + 6.37182i) q^{55} +(2.69248 - 1.50367i) q^{56} +(-0.591747 - 1.28493i) q^{58} +(1.16082 + 1.16082i) q^{59} +(-3.46410 + 3.46410i) q^{61} +(4.96309 - 13.4367i) q^{62} +(4.19615 - 6.81119i) q^{64} +(-2.87875 + 4.93886i) q^{65} +(-5.04189 - 5.04189i) q^{67} +(6.42100 - 7.50600i) q^{68} +(-3.43198 - 0.331042i) q^{70} -6.40078i q^{71} +5.24343 q^{73} +(-3.63977 - 7.90348i) q^{74} +(-5.75135 + 0.448067i) q^{76} +(-2.54291 - 2.54291i) q^{77} +2.61504 q^{79} +(-8.19166 + 3.59120i) q^{80} +(13.6313 + 5.03497i) q^{82} +(-5.67856 - 5.67856i) q^{83} +(-10.6790 + 2.81416i) q^{85} +(1.86108 + 4.04120i) q^{86} +(-8.97572 - 2.54291i) q^{88} -6.87875i q^{89} +(-1.97103 - 1.97103i) q^{91} +(2.21570 + 1.89542i) q^{92} +(-2.77868 - 1.02636i) q^{94} +(5.57221 + 3.24791i) q^{95} +3.77310i q^{97} +(-2.84753 + 7.70918i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} + 8 q^{5} - 12 q^{10} - 8 q^{11} + 4 q^{14} + 16 q^{16} - 8 q^{19} + 4 q^{20} - 32 q^{26} + 16 q^{29} + 16 q^{31} + 48 q^{34} + 24 q^{35} + 24 q^{40} + 8 q^{44} - 28 q^{46} + 16 q^{49} - 24 q^{50}+ \cdots - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) −1.51978 − 1.30010i −0.759892 − 0.650049i
\(5\) 0.569800 + 2.16225i 0.254822 + 0.966988i
\(6\) 0 0
\(7\) −1.09033 −0.412104 −0.206052 − 0.978541i \(-0.566062\pi\)
−0.206052 + 0.978541i \(0.566062\pi\)
\(8\) −2.46943 + 1.37910i −0.873075 + 0.487586i
\(9\) 0 0
\(10\) 3.14767 + 0.303617i 0.995380 + 0.0960123i
\(11\) 2.33225 + 2.33225i 0.703199 + 0.703199i 0.965096 − 0.261897i \(-0.0843481\pi\)
−0.261897 + 0.965096i \(0.584348\pi\)
\(12\) 0 0
\(13\) 1.80775 + 1.80775i 0.501379 + 0.501379i 0.911866 − 0.410487i \(-0.134641\pi\)
−0.410487 + 0.911866i \(0.634641\pi\)
\(14\) −0.534268 + 1.44644i −0.142789 + 0.386576i
\(15\) 0 0
\(16\) 0.619491 + 3.95174i 0.154873 + 0.987934i
\(17\) 4.93886i 1.19785i 0.800806 + 0.598924i \(0.204405\pi\)
−0.800806 + 0.598924i \(0.795595\pi\)
\(18\) 0 0
\(19\) 2.03957 − 2.03957i 0.467909 − 0.467909i −0.433327 − 0.901237i \(-0.642661\pi\)
0.901237 + 0.433327i \(0.142661\pi\)
\(20\) 1.94516 − 4.02695i 0.434952 − 0.900454i
\(21\) 0 0
\(22\) 4.23680 − 1.95116i 0.903289 − 0.415989i
\(23\) −1.45791 −0.303995 −0.151997 − 0.988381i \(-0.548570\pi\)
−0.151997 + 0.988381i \(0.548570\pi\)
\(24\) 0 0
\(25\) −4.35066 + 2.46410i −0.870131 + 0.492820i
\(26\) 3.28398 − 1.51236i 0.644042 − 0.296599i
\(27\) 0 0
\(28\) 1.65706 + 1.41753i 0.313155 + 0.267888i
\(29\) 0.707323 − 0.707323i 0.131347 − 0.131347i −0.638377 − 0.769724i \(-0.720394\pi\)
0.769724 + 0.638377i \(0.220394\pi\)
\(30\) 0 0
\(31\) 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
\(33\) 0 0
\(34\) 6.55193 + 2.42008i 1.12365 + 0.415040i
\(35\) −0.621268 − 2.35756i −0.105013 − 0.398500i
\(36\) 0 0
\(37\) 4.35066 − 4.35066i 0.715243 − 0.715243i −0.252384 − 0.967627i \(-0.581215\pi\)
0.967627 + 0.252384i \(0.0812146\pi\)
\(38\) −1.70631 − 3.70512i −0.276800 − 0.601049i
\(39\) 0 0
\(40\) −4.38905 − 4.55371i −0.693969 − 0.720005i
\(41\) 10.2753i 1.60473i 0.596833 + 0.802365i \(0.296425\pi\)
−0.596833 + 0.802365i \(0.703575\pi\)
\(42\) 0 0
\(43\) −2.22457 + 2.22457i −0.339244 + 0.339244i −0.856083 − 0.516839i \(-0.827109\pi\)
0.516839 + 0.856083i \(0.327109\pi\)
\(44\) −0.512364 − 6.57666i −0.0772418 − 0.991469i
\(45\) 0 0
\(46\) −0.714386 + 1.93407i −0.105330 + 0.285163i
\(47\) − 2.09458i − 0.305525i −0.988263 − 0.152763i \(-0.951183\pi\)
0.988263 − 0.152763i \(-0.0488170\pi\)
\(48\) 0 0
\(49\) −5.81119 −0.830170
\(50\) 1.13705 + 6.97905i 0.160802 + 0.986987i
\(51\) 0 0
\(52\) −0.397138 − 5.09763i −0.0550732 − 0.706915i
\(53\) −0.215297 + 0.215297i −0.0295733 + 0.0295733i −0.721739 − 0.692166i \(-0.756657\pi\)
0.692166 + 0.721739i \(0.256657\pi\)
\(54\) 0 0
\(55\) −3.71399 + 6.37182i −0.500794 + 0.859175i
\(56\) 2.69248 − 1.50367i 0.359798 − 0.200936i
\(57\) 0 0
\(58\) −0.591747 − 1.28493i −0.0777002 − 0.168720i
\(59\) 1.16082 + 1.16082i 0.151126 + 0.151126i 0.778621 − 0.627495i \(-0.215920\pi\)
−0.627495 + 0.778621i \(0.715920\pi\)
\(60\) 0 0
\(61\) −3.46410 + 3.46410i −0.443533 + 0.443533i −0.893197 − 0.449665i \(-0.851543\pi\)
0.449665 + 0.893197i \(0.351543\pi\)
\(62\) 4.96309 − 13.4367i 0.630313 − 1.70646i
\(63\) 0 0
\(64\) 4.19615 − 6.81119i 0.524519 − 0.851399i
\(65\) −2.87875 + 4.93886i −0.357065 + 0.612590i
\(66\) 0 0
\(67\) −5.04189 − 5.04189i −0.615965 − 0.615965i 0.328529 − 0.944494i \(-0.393447\pi\)
−0.944494 + 0.328529i \(0.893447\pi\)
\(68\) 6.42100 − 7.50600i 0.778660 − 0.910236i
\(69\) 0 0
\(70\) −3.43198 − 0.331042i −0.410201 − 0.0395671i
\(71\) − 6.40078i − 0.759633i −0.925062 − 0.379817i \(-0.875987\pi\)
0.925062 − 0.379817i \(-0.124013\pi\)
\(72\) 0 0
\(73\) 5.24343 0.613696 0.306848 − 0.951758i \(-0.400726\pi\)
0.306848 + 0.951758i \(0.400726\pi\)
\(74\) −3.63977 − 7.90348i −0.423114 − 0.918760i
\(75\) 0 0
\(76\) −5.75135 + 0.448067i −0.659725 + 0.0513968i
\(77\) −2.54291 − 2.54291i −0.289791 − 0.289791i
\(78\) 0 0
\(79\) 2.61504 0.294215 0.147107 − 0.989121i \(-0.453004\pi\)
0.147107 + 0.989121i \(0.453004\pi\)
\(80\) −8.19166 + 3.59120i −0.915856 + 0.401508i
\(81\) 0 0
\(82\) 13.6313 + 5.03497i 1.50532 + 0.556020i
\(83\) −5.67856 − 5.67856i −0.623303 − 0.623303i 0.323071 − 0.946375i \(-0.395285\pi\)
−0.946375 + 0.323071i \(0.895285\pi\)
\(84\) 0 0
\(85\) −10.6790 + 2.81416i −1.15831 + 0.305239i
\(86\) 1.86108 + 4.04120i 0.200686 + 0.435774i
\(87\) 0 0
\(88\) −8.97572 − 2.54291i −0.956815 − 0.271075i
\(89\) − 6.87875i − 0.729146i −0.931175 − 0.364573i \(-0.881215\pi\)
0.931175 − 0.364573i \(-0.118785\pi\)
\(90\) 0 0
\(91\) −1.97103 − 1.97103i −0.206620 − 0.206620i
\(92\) 2.21570 + 1.89542i 0.231003 + 0.197611i
\(93\) 0 0
\(94\) −2.77868 − 1.02636i −0.286599 − 0.105861i
\(95\) 5.57221 + 3.24791i 0.571696 + 0.333229i
\(96\) 0 0
\(97\) 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
\(99\) 0 0
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 720.2.bm.f.469.6 16
3.2 odd 2 80.2.q.c.69.3 yes 16
5.4 even 2 inner 720.2.bm.f.469.3 16
12.11 even 2 320.2.q.c.49.8 16
15.2 even 4 400.2.l.i.101.2 16
15.8 even 4 400.2.l.i.101.7 16
15.14 odd 2 80.2.q.c.69.6 yes 16
16.13 even 4 inner 720.2.bm.f.109.3 16
24.5 odd 2 640.2.q.e.609.8 16
24.11 even 2 640.2.q.f.609.1 16
48.5 odd 4 640.2.q.e.289.1 16
48.11 even 4 640.2.q.f.289.8 16
48.29 odd 4 80.2.q.c.29.6 yes 16
48.35 even 4 320.2.q.c.209.1 16
60.23 odd 4 1600.2.l.h.1201.1 16
60.47 odd 4 1600.2.l.h.1201.8 16
60.59 even 2 320.2.q.c.49.1 16
80.29 even 4 inner 720.2.bm.f.109.6 16
120.29 odd 2 640.2.q.e.609.1 16
120.59 even 2 640.2.q.f.609.8 16
240.29 odd 4 80.2.q.c.29.3 ✓ 16
240.59 even 4 640.2.q.f.289.1 16
240.77 even 4 400.2.l.i.301.2 16
240.83 odd 4 1600.2.l.h.401.1 16
240.149 odd 4 640.2.q.e.289.8 16
240.173 even 4 400.2.l.i.301.7 16
240.179 even 4 320.2.q.c.209.8 16
240.227 odd 4 1600.2.l.h.401.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.q.c.29.3 ✓ 16 240.29 odd 4
80.2.q.c.29.6 yes 16 48.29 odd 4
80.2.q.c.69.3 yes 16 3.2 odd 2
80.2.q.c.69.6 yes 16 15.14 odd 2
320.2.q.c.49.1 16 60.59 even 2
320.2.q.c.49.8 16 12.11 even 2
320.2.q.c.209.1 16 48.35 even 4
320.2.q.c.209.8 16 240.179 even 4
400.2.l.i.101.2 16 15.2 even 4
400.2.l.i.101.7 16 15.8 even 4
400.2.l.i.301.2 16 240.77 even 4
400.2.l.i.301.7 16 240.173 even 4
640.2.q.e.289.1 16 48.5 odd 4
640.2.q.e.289.8 16 240.149 odd 4
640.2.q.e.609.1 16 120.29 odd 2
640.2.q.e.609.8 16 24.5 odd 2
640.2.q.f.289.1 16 240.59 even 4
640.2.q.f.289.8 16 48.11 even 4
640.2.q.f.609.1 16 24.11 even 2
640.2.q.f.609.8 16 120.59 even 2
720.2.bm.f.109.3 16 16.13 even 4 inner
720.2.bm.f.109.6 16 80.29 even 4 inner
720.2.bm.f.469.3 16 5.4 even 2 inner
720.2.bm.f.469.6 16 1.1 even 1 trivial
1600.2.l.h.401.1 16 240.83 odd 4
1600.2.l.h.401.8 16 240.227 odd 4
1600.2.l.h.1201.1 16 60.23 odd 4
1600.2.l.h.1201.8 16 60.47 odd 4