Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(89,522)] 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("522.89"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([14, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.r (of order \(28\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72] 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: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(6\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label 251.3
Character \(\chi\) \(=\) 522.251
Dual form 522.2.r.b.287.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.846724 + 0.532032i) q^{2} +(0.433884 - 0.900969i) q^{4} +(0.641917 - 2.81242i) q^{5} +(-3.82892 + 1.84391i) q^{7} +(0.111964 + 0.993712i) q^{8} +(0.952772 + 2.72287i) q^{10} +(0.298225 - 2.64682i) q^{11} +(-0.572763 + 0.456763i) q^{13} +(2.26102 - 3.59839i) q^{14} +(-0.623490 - 0.781831i) q^{16} +(-0.273539 - 0.273539i) q^{17} +(-6.83780 + 2.39265i) q^{19} +(-2.25539 - 1.79861i) q^{20} +(1.15568 + 2.39979i) q^{22} +(-4.49427 + 1.02579i) q^{23} +(-2.99282 - 1.44127i) q^{25} +(0.241960 - 0.691481i) q^{26} +4.24978i q^{28} +(-4.00418 - 3.60091i) q^{29} +(2.00102 + 3.18461i) q^{31} +(0.943883 + 0.330279i) q^{32} +(0.377144 + 0.0860807i) q^{34} +(2.72801 + 11.9522i) q^{35} +(1.77576 - 0.200080i) q^{37} +(4.51676 - 5.66384i) q^{38} +(2.86661 + 0.322989i) q^{40} +(-4.20728 + 4.20728i) q^{41} +(-7.93803 - 4.98779i) q^{43} +(-2.25531 - 1.41710i) q^{44} +(3.25965 - 3.25965i) q^{46} +(-6.34581 - 0.715001i) q^{47} +(6.89619 - 8.64755i) q^{49} +(3.30089 - 0.371921i) q^{50} +(0.163017 + 0.714224i) q^{52} +(0.672029 + 0.153386i) q^{53} +(-7.25254 - 2.53777i) q^{55} +(-2.26102 - 3.59839i) q^{56} +(5.30624 + 0.918623i) q^{58} -15.1636i q^{59} +(4.51708 - 12.9091i) q^{61} +(-3.38863 - 1.63188i) q^{62} +(-0.974928 + 0.222521i) q^{64} +(0.916945 + 1.90406i) q^{65} +(4.79121 + 3.82087i) q^{67} +(-0.365135 + 0.127766i) q^{68} +(-8.66881 - 8.66881i) q^{70} +(-2.08475 - 2.61420i) q^{71} +(-3.08849 + 4.91530i) q^{73} +(-1.39713 + 1.11417i) q^{74} +(-0.811107 + 7.19878i) q^{76} +(3.73862 + 10.6844i) q^{77} +(0.790904 + 7.01946i) q^{79} +(-2.59907 + 1.25165i) q^{80} +(1.32400 - 5.80082i) q^{82} +(-0.0649045 + 0.134776i) q^{83} +(-0.944898 + 0.593719i) q^{85} +9.37499 q^{86} +2.66357 q^{88} +(8.51977 - 5.35333i) q^{89} +(1.35083 - 2.80503i) q^{91} +(-1.02579 + 4.49427i) q^{92} +(5.75356 - 2.77077i) q^{94} +(2.33984 + 20.7667i) q^{95} +(6.15540 + 17.5911i) q^{97} +(-1.23840 + 10.9911i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 8 q^{7} + 4 q^{10} + 12 q^{16} - 40 q^{19} - 20 q^{25} - 28 q^{31} - 12 q^{37} - 4 q^{40} - 72 q^{43} + 16 q^{46} + 52 q^{49} + 4 q^{52} + 140 q^{55} + 44 q^{58} + 24 q^{61} + 140 q^{67} - 20 q^{70}+ \cdots + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/522\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(407\)
\(\chi(n)\) \(e\left(\frac{9}{28}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.846724 + 0.532032i −0.598724 + 0.376203i
\(3\) 0 0
\(4\) 0.433884 − 0.900969i 0.216942 − 0.450484i
\(5\) 0.641917 − 2.81242i 0.287074 − 1.25775i −0.601445 − 0.798914i \(-0.705408\pi\)
0.888519 − 0.458840i \(-0.151735\pi\)
\(6\) 0 0
\(7\) −3.82892 + 1.84391i −1.44720 + 0.696933i −0.982106 − 0.188331i \(-0.939692\pi\)
−0.465090 + 0.885264i \(0.653978\pi\)
\(8\) 0.111964 + 0.993712i 0.0395854 + 0.351330i
\(9\) 0 0
\(10\) 0.952772 + 2.72287i 0.301293 + 0.861046i
\(11\) 0.298225 − 2.64682i 0.0899181 − 0.798046i −0.865120 − 0.501564i \(-0.832758\pi\)
0.955039 − 0.296481i \(-0.0958133\pi\)
\(12\) 0 0
\(13\) −0.572763 + 0.456763i −0.158856 + 0.126683i −0.699692 − 0.714445i \(-0.746679\pi\)
0.540836 + 0.841128i \(0.318108\pi\)
\(14\) 2.26102 − 3.59839i 0.604283 − 0.961711i
\(15\) 0 0
\(16\) −0.623490 − 0.781831i −0.155872 − 0.195458i
\(17\) −0.273539 − 0.273539i −0.0663430 − 0.0663430i 0.673157 − 0.739500i \(-0.264938\pi\)
−0.739500 + 0.673157i \(0.764938\pi\)
\(18\) 0 0
\(19\) −6.83780 + 2.39265i −1.56870 + 0.548912i −0.968084 − 0.250627i \(-0.919363\pi\)
−0.600615 + 0.799538i \(0.705078\pi\)
\(20\) −2.25539 − 1.79861i −0.504320 − 0.402182i
\(21\) 0 0
\(22\) 1.15568 + 2.39979i 0.246391 + 0.511637i
\(23\) −4.49427 + 1.02579i −0.937119 + 0.213891i −0.663696 − 0.748002i \(-0.731013\pi\)
−0.273423 + 0.961894i \(0.588156\pi\)
\(24\) 0 0
\(25\) −2.99282 − 1.44127i −0.598564 − 0.288253i
\(26\) 0.241960 − 0.691481i 0.0474522 − 0.135611i
\(27\) 0 0
\(28\) 4.24978i 0.803133i
\(29\) −4.00418 − 3.60091i −0.743558 − 0.668672i
\(30\) 0 0
\(31\) 2.00102 + 3.18461i 0.359394 + 0.571973i 0.976662 − 0.214780i \(-0.0689035\pi\)
−0.617268 + 0.786753i \(0.711761\pi\)
\(32\) 0.943883 + 0.330279i 0.166857 + 0.0583856i
\(33\) 0 0
\(34\) 0.377144 + 0.0860807i 0.0646797 + 0.0147627i
\(35\) 2.72801 + 11.9522i 0.461117 + 2.02029i
\(36\) 0 0
\(37\) 1.77576 − 0.200080i 0.291933 − 0.0328930i 0.0352166 − 0.999380i \(-0.488788\pi\)
0.256717 + 0.966487i \(0.417359\pi\)
\(38\) 4.51676 − 5.66384i 0.732716 − 0.918797i
\(39\) 0 0
\(40\) 2.86661 + 0.322989i 0.453251 + 0.0510691i
\(41\) −4.20728 + 4.20728i −0.657067 + 0.657067i −0.954685 − 0.297618i \(-0.903808\pi\)
0.297618 + 0.954685i \(0.403808\pi\)
\(42\) 0 0
\(43\) −7.93803 − 4.98779i −1.21054 − 0.760631i −0.233223 − 0.972423i \(-0.574927\pi\)
−0.977315 + 0.211792i \(0.932070\pi\)
\(44\) −2.25531 − 1.41710i −0.340000 − 0.213636i
\(45\) 0 0
\(46\) 3.25965 − 3.25965i 0.480609 − 0.480609i
\(47\) −6.34581 − 0.715001i −0.925632 − 0.104294i −0.363736 − 0.931502i \(-0.618499\pi\)
−0.561895 + 0.827208i \(0.689928\pi\)
\(48\) 0 0
\(49\) 6.89619 − 8.64755i 0.985170 − 1.23536i
\(50\) 3.30089 − 0.371921i 0.466817 − 0.0525976i
\(51\) 0 0
\(52\) 0.163017 + 0.714224i 0.0226064 + 0.0990451i
\(53\) 0.672029 + 0.153386i 0.0923103 + 0.0210692i 0.268427 − 0.963300i \(-0.413496\pi\)
−0.176116 + 0.984369i \(0.556353\pi\)
\(54\) 0 0
\(55\) −7.25254 − 2.53777i −0.977932 − 0.342193i
\(56\) −2.26102 − 3.59839i −0.302141 − 0.480855i
\(57\) 0 0
\(58\) 5.30624 + 0.918623i 0.696743 + 0.120621i
\(59\) − 15.1636i − 1.97413i −0.160320 − 0.987065i \(-0.551253\pi\)
0.160320 − 0.987065i \(-0.448747\pi\)
\(60\) 0 0
\(61\) 4.51708 − 12.9091i 0.578353 − 1.65284i −0.166835 − 0.985985i \(-0.553355\pi\)
0.745189 − 0.666854i \(-0.232359\pi\)
\(62\) −3.38863 − 1.63188i −0.430356 − 0.207249i
\(63\) 0 0
\(64\) −0.974928 + 0.222521i −0.121866 + 0.0278151i
\(65\) 0.916945 + 1.90406i 0.113733 + 0.236169i
\(66\) 0 0
\(67\) 4.79121 + 3.82087i 0.585340 + 0.466793i 0.870827 − 0.491589i \(-0.163584\pi\)
−0.285487 + 0.958383i \(0.592155\pi\)
\(68\) −0.365135 + 0.127766i −0.0442791 + 0.0154939i
\(69\) 0 0
\(70\) −8.66881 − 8.66881i −1.03612 − 1.03612i
\(71\) −2.08475 − 2.61420i −0.247415 − 0.310248i 0.642580 − 0.766218i \(-0.277864\pi\)
−0.889995 + 0.455970i \(0.849292\pi\)
\(72\) 0 0
\(73\) −3.08849 + 4.91530i −0.361480 + 0.575292i −0.977108 − 0.212745i \(-0.931760\pi\)
0.615628 + 0.788037i \(0.288903\pi\)
\(74\) −1.39713 + 1.11417i −0.162413 + 0.129520i
\(75\) 0 0
\(76\) −0.811107 + 7.19878i −0.0930404 + 0.825756i
\(77\) 3.73862 + 10.6844i 0.426055 + 1.21759i
\(78\) 0 0
\(79\) 0.790904 + 7.01946i 0.0889836 + 0.789751i 0.956378 + 0.292131i \(0.0943642\pi\)
−0.867395 + 0.497621i \(0.834207\pi\)
\(80\) −2.59907 + 1.25165i −0.290585 + 0.139938i
\(81\) 0 0
\(82\) 1.32400 − 5.80082i 0.146211 − 0.640593i
\(83\) −0.0649045 + 0.134776i −0.00712420 + 0.0147935i −0.904501 − 0.426472i \(-0.859756\pi\)
0.897376 + 0.441266i \(0.145470\pi\)
\(84\) 0 0
\(85\) −0.944898 + 0.593719i −0.102489 + 0.0643978i
\(86\) 9.37499 1.01093
\(87\) 0 0
\(88\) 2.66357 0.283937
\(89\) 8.51977 − 5.35333i 0.903094 − 0.567451i 0.00148264 − 0.999999i \(-0.499528\pi\)
0.901611 + 0.432547i \(0.142385\pi\)
\(90\) 0 0
\(91\) 1.35083 − 2.80503i 0.141606 − 0.294047i
\(92\) −1.02579 + 4.49427i −0.106946 + 0.468560i
\(93\) 0 0
\(94\) 5.75356 − 2.77077i 0.593434 − 0.285783i
\(95\) 2.33984 + 20.7667i 0.240063 + 2.13062i
\(96\) 0 0
\(97\) 6.15540 + 17.5911i 0.624986 + 1.78611i 0.618059 + 0.786132i \(0.287919\pi\)
0.00692767 + 0.999976i \(0.497795\pi\)
\(98\) −1.23840 + 10.9911i −0.125097 + 1.11027i
\(99\) 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 522.2.r.b.251.3 ✓ 72
3.2 odd 2 inner 522.2.r.b.251.4 yes 72
29.26 odd 28 inner 522.2.r.b.287.4 yes 72
87.26 even 28 inner 522.2.r.b.287.3 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
522.2.r.b.251.3 ✓ 72 1.1 even 1 trivial
522.2.r.b.251.4 yes 72 3.2 odd 2 inner
522.2.r.b.287.3 yes 72 87.26 even 28 inner
522.2.r.b.287.4 yes 72 29.26 odd 28 inner