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 251.4
Character \(\chi\) \(=\) 522.251
Dual form 522.2.r.b.287.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.846724 - 0.532032i) q^{2} +(0.433884 - 0.900969i) q^{4} +(-0.641917 + 2.81242i) q^{5} +(-3.82892 + 1.84391i) q^{7} +(-0.111964 - 0.993712i) q^{8} +(0.952772 + 2.72287i) q^{10} +(-0.298225 + 2.64682i) q^{11} +(-0.572763 + 0.456763i) q^{13} +(-2.26102 + 3.59839i) q^{14} +(-0.623490 - 0.781831i) q^{16} +(0.273539 + 0.273539i) q^{17} +(-6.83780 + 2.39265i) q^{19} +(2.25539 + 1.79861i) q^{20} +(1.15568 + 2.39979i) q^{22} +(4.49427 - 1.02579i) q^{23} +(-2.99282 - 1.44127i) q^{25} +(-0.241960 + 0.691481i) q^{26} +4.24978i q^{28} +(4.00418 + 3.60091i) q^{29} +(2.00102 + 3.18461i) q^{31} +(-0.943883 - 0.330279i) q^{32} +(0.377144 + 0.0860807i) q^{34} +(-2.72801 - 11.9522i) q^{35} +(1.77576 - 0.200080i) q^{37} +(-4.51676 + 5.66384i) q^{38} +(2.86661 + 0.322989i) q^{40} +(4.20728 - 4.20728i) q^{41} +(-7.93803 - 4.98779i) q^{43} +(2.25531 + 1.41710i) q^{44} +(3.25965 - 3.25965i) q^{46} +(6.34581 + 0.715001i) q^{47} +(6.89619 - 8.64755i) q^{49} +(-3.30089 + 0.371921i) q^{50} +(0.163017 + 0.714224i) q^{52} +(-0.672029 - 0.153386i) q^{53} +(-7.25254 - 2.53777i) q^{55} +(2.26102 + 3.59839i) q^{56} +(5.30624 + 0.918623i) q^{58} +15.1636i q^{59} +(4.51708 - 12.9091i) q^{61} +(3.38863 + 1.63188i) q^{62} +(-0.974928 + 0.222521i) q^{64} +(-0.916945 - 1.90406i) q^{65} +(4.79121 + 3.82087i) q^{67} +(0.365135 - 0.127766i) q^{68} +(-8.66881 - 8.66881i) q^{70} +(2.08475 + 2.61420i) q^{71} +(-3.08849 + 4.91530i) q^{73} +(1.39713 - 1.11417i) q^{74} +(-0.811107 + 7.19878i) q^{76} +(-3.73862 - 10.6844i) q^{77} +(0.790904 + 7.01946i) q^{79} +(2.59907 - 1.25165i) q^{80} +(1.32400 - 5.80082i) q^{82} +(0.0649045 - 0.134776i) q^{83} +(-0.944898 + 0.593719i) q^{85} -9.37499 q^{86} +2.66357 q^{88} +(-8.51977 + 5.35333i) q^{89} +(1.35083 - 2.80503i) q^{91} +(1.02579 - 4.49427i) q^{92} +(5.75356 - 2.77077i) q^{94} +(-2.33984 - 20.7667i) q^{95} +(6.15540 + 17.5911i) q^{97} +(1.23840 - 10.9911i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 8 q^{7} + 4 q^{10} + 12 q^{16} - 40 q^{19} - 20 q^{25} - 28 q^{31} - 12 q^{37} - 4 q^{40} - 72 q^{43} + 16 q^{46} + 52 q^{49} + 4 q^{52} + 140 q^{55} + 44 q^{58} + 24 q^{61} + 140 q^{67} - 20 q^{70}+ \cdots + 28 q^{97}+O(q^{100}) \) 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 \(0.294592\pi\)
−0.888519 + 0.458840i \(0.848265\pi\)
\(6\) 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 \(0.167242\pi\)
−0.955039 + 0.296481i \(0.904187\pi\)
\(12\) 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 0.739500 − 0.673157i \(-0.235062\pi\)
−0.673157 + 0.739500i \(0.735062\pi\)
\(18\) 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 0.273423 − 0.961894i \(-0.411844\pi\)
0.663696 + 0.748002i \(0.268987\pi\)
\(24\) 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 −0.297618 − 0.954685i \(-0.596192\pi\)
0.954685 + 0.297618i \(0.0961921\pi\)
\(42\) 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 0.561895 − 0.827208i \(-0.310072\pi\)
0.363736 + 0.931502i \(0.381501\pi\)
\(48\) 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 0.176116 − 0.984369i \(-0.443647\pi\)
−0.268427 + 0.963300i \(0.586504\pi\)
\(54\) 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 \(0.448747\pi\)
−0.160320 + 0.987065i \(0.551253\pi\)
\(60\) 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 0.889995 − 0.455970i \(-0.150708\pi\)
−0.642580 + 0.766218i \(0.722136\pi\)
\(72\) 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 −0.897376 − 0.441266i \(-0.854530\pi\)
0.904501 + 0.426472i \(0.140244\pi\)
\(84\) 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 −0.901611 − 0.432547i \(-0.857615\pi\)
−0.00148264 + 0.999999i \(0.500472\pi\)
\(90\) 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.251.4 yes 72
3.2 odd 2 inner 522.2.r.b.251.3 ✓ 72
29.26 odd 28 inner 522.2.r.b.287.3 yes 72
87.26 even 28 inner 522.2.r.b.287.4 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
522.2.r.b.251.3 ✓ 72 3.2 odd 2 inner
522.2.r.b.251.4 yes 72 1.1 even 1 trivial
522.2.r.b.287.3 yes 72 29.26 odd 28 inner
522.2.r.b.287.4 yes 72 87.26 even 28 inner