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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 251.2
Character \(\chi\) \(=\) 522.251
Dual form 522.2.r.b.287.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.846724 + 0.532032i) q^{2} +(0.433884 - 0.900969i) q^{4} +(0.0961712 - 0.421354i) q^{5} +(0.802475 - 0.386452i) q^{7} +(0.111964 + 0.993712i) q^{8} +(0.142743 + 0.407937i) q^{10} +(-0.661325 + 5.86942i) q^{11} +(-0.249434 + 0.198917i) q^{13} +(-0.473870 + 0.754160i) q^{14} +(-0.623490 - 0.781831i) q^{16} +(-1.53614 - 1.53614i) q^{17} +(7.52763 - 2.63403i) q^{19} +(-0.337899 - 0.269466i) q^{20} +(-2.56276 - 5.32163i) q^{22} +(2.90702 - 0.663509i) q^{23} +(4.33655 + 2.08837i) q^{25} +(0.105372 - 0.301135i) q^{26} -0.890680i q^{28} +(5.06335 + 1.83370i) q^{29} +(2.77228 + 4.41206i) q^{31} +(0.943883 + 0.330279i) q^{32} +(2.11796 + 0.483410i) q^{34} +(-0.0856578 - 0.375291i) q^{35} +(-3.52226 + 0.396863i) q^{37} +(-4.97244 + 6.23524i) q^{38} +(0.429472 + 0.0483899i) q^{40} +(-4.29076 + 4.29076i) q^{41} +(-2.14344 - 1.34681i) q^{43} +(5.00123 + 3.14248i) q^{44} +(-2.10844 + 2.10844i) q^{46} +(5.31362 + 0.598701i) q^{47} +(-3.86981 + 4.85258i) q^{49} +(-4.78295 + 0.538909i) q^{50} +(0.0709926 + 0.311039i) q^{52} +(8.85323 + 2.02069i) q^{53} +(2.40950 + 0.843121i) q^{55} +(0.473870 + 0.754160i) q^{56} +(-5.26285 + 1.14123i) q^{58} -9.94204i q^{59} +(0.0148600 - 0.0424675i) q^{61} +(-4.69472 - 2.26086i) q^{62} +(-0.974928 + 0.222521i) q^{64} +(0.0598260 + 0.124230i) q^{65} +(4.09809 + 3.26812i) q^{67} +(-2.05051 + 0.717506i) q^{68} +(0.272196 + 0.272196i) q^{70} +(-5.68125 - 7.12406i) q^{71} +(-2.43545 + 3.87599i) q^{73} +(2.77124 - 2.20999i) q^{74} +(0.892935 - 7.92502i) q^{76} +(1.73755 + 4.96563i) q^{77} +(1.16061 + 10.3007i) q^{79} +(-0.389389 + 0.187520i) q^{80} +(1.35027 - 5.91591i) q^{82} +(3.22506 - 6.69690i) q^{83} +(-0.794988 + 0.499524i) q^{85} +2.53145 q^{86} -5.90656 q^{88} +(-7.70400 + 4.84074i) q^{89} +(-0.123293 + 0.256020i) q^{91} +(0.663509 - 2.90702i) q^{92} +(-4.81770 + 2.32008i) q^{94} +(-0.385917 - 3.42511i) q^{95} +(0.399749 + 1.14242i) q^{97} +(0.694929 - 6.16766i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 8 q^{7} + 4 q^{10} + 12 q^{16} - 40 q^{19} - 20 q^{25} - 28 q^{31} - 12 q^{37} - 4 q^{40} - 72 q^{43} + 16 q^{46} + 52 q^{49} + 4 q^{52} + 140 q^{55} + 44 q^{58} + 24 q^{61} + 140 q^{67} - 20 q^{70}+ \cdots + 28 q^{97}+O(q^{100}) \) 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
\(5\) 0.0961712 − 0.421354i 0.0430091 − 0.188435i −0.948860 − 0.315697i \(-0.897762\pi\)
0.991869 + 0.127262i \(0.0406188\pi\)
\(6\) 0 0
\(7\) 0.802475 − 0.386452i 0.303307 − 0.146065i −0.276040 − 0.961146i \(-0.589022\pi\)
0.579347 + 0.815081i \(0.303308\pi\)
\(8\) 0.111964 + 0.993712i 0.0395854 + 0.351330i
\(9\) 0 0
\(10\) 0.142743 + 0.407937i 0.0451394 + 0.129001i
\(11\) −0.661325 + 5.86942i −0.199397 + 1.76970i 0.352526 + 0.935802i \(0.385323\pi\)
−0.551923 + 0.833895i \(0.686106\pi\)
\(12\) 0 0
\(13\) −0.249434 + 0.198917i −0.0691805 + 0.0551696i −0.657470 − 0.753481i \(-0.728373\pi\)
0.588289 + 0.808651i \(0.299802\pi\)
\(14\) −0.473870 + 0.754160i −0.126647 + 0.201558i
\(15\) 0 0
\(16\) −0.623490 − 0.781831i −0.155872 − 0.195458i
\(17\) −1.53614 − 1.53614i −0.372568 − 0.372568i 0.495844 − 0.868412i \(-0.334859\pi\)
−0.868412 + 0.495844i \(0.834859\pi\)
\(18\) 0 0
\(19\) 7.52763 − 2.63403i 1.72696 − 0.604288i 0.730087 − 0.683354i \(-0.239480\pi\)
0.996869 + 0.0790663i \(0.0251939\pi\)
\(20\) −0.337899 − 0.269466i −0.0755566 − 0.0602544i
\(21\) 0 0
\(22\) −2.56276 − 5.32163i −0.546382 − 1.13457i
\(23\) 2.90702 − 0.663509i 0.606157 − 0.138351i 0.0915829 − 0.995797i \(-0.470807\pi\)
0.514574 + 0.857446i \(0.327950\pi\)
\(24\) 0 0
\(25\) 4.33655 + 2.08837i 0.867311 + 0.417675i
\(26\) 0.105372 − 0.301135i 0.0206651 − 0.0590574i
\(27\) 0 0
\(28\) − 0.890680i − 0.168323i
\(29\) 5.06335 + 1.83370i 0.940241 + 0.340510i
\(30\) 0 0
\(31\) 2.77228 + 4.41206i 0.497917 + 0.792430i 0.997260 − 0.0739741i \(-0.0235682\pi\)
−0.499343 + 0.866404i \(0.666425\pi\)
\(32\) 0.943883 + 0.330279i 0.166857 + 0.0583856i
\(33\) 0 0
\(34\) 2.11796 + 0.483410i 0.363227 + 0.0829041i
\(35\) −0.0856578 − 0.375291i −0.0144788 − 0.0634358i
\(36\) 0 0
\(37\) −3.52226 + 0.396863i −0.579056 + 0.0652439i −0.396635 − 0.917977i \(-0.629822\pi\)
−0.182421 + 0.983220i \(0.558393\pi\)
\(38\) −4.97244 + 6.23524i −0.806636 + 1.01149i
\(39\) 0 0
\(40\) 0.429472 + 0.0483899i 0.0679055 + 0.00765111i
\(41\) −4.29076 + 4.29076i −0.670103 + 0.670103i −0.957740 − 0.287636i \(-0.907131\pi\)
0.287636 + 0.957740i \(0.407131\pi\)
\(42\) 0 0
\(43\) −2.14344 − 1.34681i −0.326871 − 0.205387i 0.358591 − 0.933495i \(-0.383257\pi\)
−0.685463 + 0.728108i \(0.740400\pi\)
\(44\) 5.00123 + 3.14248i 0.753963 + 0.473747i
\(45\) 0 0
\(46\) −2.10844 + 2.10844i −0.310872 + 0.310872i
\(47\) 5.31362 + 0.598701i 0.775071 + 0.0873296i 0.490635 − 0.871365i \(-0.336765\pi\)
0.284437 + 0.958695i \(0.408194\pi\)
\(48\) 0 0
\(49\) −3.86981 + 4.85258i −0.552830 + 0.693226i
\(50\) −4.78295 + 0.538909i −0.676411 + 0.0762132i
\(51\) 0 0
\(52\) 0.0709926 + 0.311039i 0.00984491 + 0.0431334i
\(53\) 8.85323 + 2.02069i 1.21609 + 0.277564i 0.781983 − 0.623300i \(-0.214208\pi\)
0.434102 + 0.900863i \(0.357066\pi\)
\(54\) 0 0
\(55\) 2.40950 + 0.843121i 0.324897 + 0.113686i
\(56\) 0.473870 + 0.754160i 0.0633236 + 0.100779i
\(57\) 0 0
\(58\) −5.26285 + 1.14123i −0.691046 + 0.149850i
\(59\) − 9.94204i − 1.29434i −0.762344 − 0.647172i \(-0.775952\pi\)
0.762344 − 0.647172i \(-0.224048\pi\)
\(60\) 0 0
\(61\) 0.0148600 − 0.0424675i 0.00190263 − 0.00543741i −0.942928 − 0.332996i \(-0.891940\pi\)
0.944831 + 0.327559i \(0.106226\pi\)
\(62\) −4.69472 − 2.26086i −0.596230 − 0.287129i
\(63\) 0 0
\(64\) −0.974928 + 0.222521i −0.121866 + 0.0278151i
\(65\) 0.0598260 + 0.124230i 0.00742050 + 0.0154088i
\(66\) 0 0
\(67\) 4.09809 + 3.26812i 0.500662 + 0.399264i 0.840997 − 0.541040i \(-0.181969\pi\)
−0.340335 + 0.940304i \(0.610541\pi\)
\(68\) −2.05051 + 0.717506i −0.248661 + 0.0870104i
\(69\) 0 0
\(70\) 0.272196 + 0.272196i 0.0325336 + 0.0325336i
\(71\) −5.68125 − 7.12406i −0.674240 − 0.845471i 0.320569 − 0.947225i \(-0.396126\pi\)
−0.994810 + 0.101754i \(0.967554\pi\)
\(72\) 0 0
\(73\) −2.43545 + 3.87599i −0.285047 + 0.453650i −0.958301 − 0.285762i \(-0.907753\pi\)
0.673253 + 0.739412i \(0.264896\pi\)
\(74\) 2.77124 − 2.20999i 0.322150 − 0.256906i
\(75\) 0 0
\(76\) 0.892935 − 7.92502i 0.102427 − 0.909062i
\(77\) 1.73755 + 4.96563i 0.198012 + 0.565887i
\(78\) 0 0
\(79\) 1.16061 + 10.3007i 0.130578 + 1.15892i 0.872564 + 0.488500i \(0.162456\pi\)
−0.741986 + 0.670416i \(0.766116\pi\)
\(80\) −0.389389 + 0.187520i −0.0435351 + 0.0209654i
\(81\) 0 0
\(82\) 1.35027 − 5.91591i 0.149112 − 0.653303i
\(83\) 3.22506 − 6.69690i 0.353996 − 0.735080i −0.645595 − 0.763680i \(-0.723391\pi\)
0.999591 + 0.0285997i \(0.00910482\pi\)
\(84\) 0 0
\(85\) −0.794988 + 0.499524i −0.0862286 + 0.0541810i
\(86\) 2.53145 0.272973
\(87\) 0 0
\(88\) −5.90656 −0.629642
\(89\) −7.70400 + 4.84074i −0.816622 + 0.513118i −0.874397 − 0.485212i \(-0.838742\pi\)
0.0577745 + 0.998330i \(0.481600\pi\)
\(90\) 0 0
\(91\) −0.123293 + 0.256020i −0.0129246 + 0.0268382i
\(92\) 0.663509 − 2.90702i 0.0691756 − 0.303078i
\(93\) 0 0
\(94\) −4.81770 + 2.32008i −0.496908 + 0.239298i
\(95\) −0.385917 − 3.42511i −0.0395943 − 0.351409i
\(96\) 0 0
\(97\) 0.399749 + 1.14242i 0.0405884 + 0.115995i 0.962394 − 0.271657i \(-0.0875718\pi\)
−0.921806 + 0.387652i \(0.873286\pi\)
\(98\) 0.694929 − 6.16766i 0.0701984 − 0.623028i
\(99\) 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.251.2 ✓ 72
3.2 odd 2 inner 522.2.r.b.251.5 yes 72
29.26 odd 28 inner 522.2.r.b.287.5 yes 72
87.26 even 28 inner 522.2.r.b.287.2 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
522.2.r.b.251.2 ✓ 72 1.1 even 1 trivial
522.2.r.b.251.5 yes 72 3.2 odd 2 inner
522.2.r.b.287.2 yes 72 87.26 even 28 inner
522.2.r.b.287.5 yes 72 29.26 odd 28 inner