Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [90,5,Mod(7,90)] 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("90.7"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([8, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 90.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,-48] 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: \(9.30329667755\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 13.1
Character \(\chi\) \(=\) 90.13
Dual form 90.5.k.a.7.1

$q$-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+(-2.73205 - 0.732051i) q^{2} +(-8.99776 - 0.200617i) q^{3} +(6.92820 + 4.00000i) q^{4} +(-5.51437 - 24.3843i) q^{5} +(24.4355 + 7.13492i) q^{6} +(72.5536 + 19.4407i) q^{7} +(-16.0000 - 16.0000i) q^{8} +(80.9195 + 3.61021i) q^{9} +(-2.78496 + 70.6558i) q^{10} +(-49.4867 - 85.7134i) q^{11} +(-61.5359 - 37.3810i) q^{12} +(-242.471 + 64.9698i) q^{13} +(-183.989 - 106.226i) q^{14} +(44.7251 + 220.510i) q^{15} +(32.0000 + 55.4256i) q^{16} +(60.1943 - 60.1943i) q^{17} +(-218.433 - 69.1005i) q^{18} -392.553i q^{19} +(59.3323 - 190.997i) q^{20} +(-648.920 - 189.478i) q^{21} +(72.4535 + 270.400i) q^{22} +(-817.438 + 219.032i) q^{23} +(140.754 + 147.174i) q^{24} +(-564.183 + 268.928i) q^{25} +710.003 q^{26} +(-727.370 - 48.7177i) q^{27} +(424.904 + 424.904i) q^{28} +(-175.946 + 101.583i) q^{29} +(39.2332 - 635.186i) q^{30} +(-39.1711 + 67.8463i) q^{31} +(-46.8513 - 174.851i) q^{32} +(428.074 + 781.157i) q^{33} +(-208.519 + 120.389i) q^{34} +(73.9587 - 1876.37i) q^{35} +(546.186 + 348.690i) q^{36} +(-1656.33 + 1656.33i) q^{37} +(-287.369 + 1072.48i) q^{38} +(2194.73 - 535.939i) q^{39} +(-301.918 + 478.378i) q^{40} +(1206.44 - 2089.62i) q^{41} +(1634.18 + 992.707i) q^{42} +(175.226 - 653.954i) q^{43} -791.787i q^{44} +(-358.188 - 1993.07i) q^{45} +2393.62 q^{46} +(-446.287 - 119.582i) q^{47} +(-276.809 - 505.126i) q^{48} +(2806.76 + 1620.49i) q^{49} +(1738.25 - 321.713i) q^{50} +(-553.690 + 529.538i) q^{51} +(-1939.77 - 519.759i) q^{52} +(-296.250 - 296.250i) q^{53} +(1951.55 + 665.571i) q^{54} +(-1817.17 + 1679.35i) q^{55} +(-849.807 - 1471.91i) q^{56} +(-78.7529 + 3532.10i) q^{57} +(555.057 - 148.727i) q^{58} +(-3902.69 - 2253.22i) q^{59} +(-572.175 + 1706.64i) q^{60} +(534.192 + 925.248i) q^{61} +(156.684 - 156.684i) q^{62} +(5800.82 + 1835.07i) q^{63} +512.000i q^{64} +(2921.31 + 5554.20i) q^{65} +(-597.673 - 2447.53i) q^{66} +(-1842.03 - 6874.54i) q^{67} +(657.815 - 176.261i) q^{68} +(7399.06 - 1806.80i) q^{69} +(-1575.66 + 5072.20i) q^{70} -8524.10 q^{71} +(-1236.95 - 1352.48i) q^{72} +(2993.00 + 2993.00i) q^{73} +(5737.71 - 3312.67i) q^{74} +(5130.34 - 2306.56i) q^{75} +(1570.21 - 2719.69i) q^{76} +(-1924.11 - 7180.88i) q^{77} +(-6388.44 - 142.439i) q^{78} +(-2798.55 + 1615.75i) q^{79} +(1175.05 - 1085.93i) q^{80} +(6534.93 + 584.273i) q^{81} +(-4825.77 + 4825.77i) q^{82} +(-870.223 + 3247.72i) q^{83} +(-3737.94 - 3908.43i) q^{84} +(-1799.73 - 1135.86i) q^{85} +(-957.455 + 1658.36i) q^{86} +(1603.50 - 878.718i) q^{87} +(-579.628 + 2163.20i) q^{88} +6328.76i q^{89} +(-480.440 + 5707.38i) q^{90} -18855.2 q^{91} +(-6539.50 - 1752.25i) q^{92} +(366.063 - 602.606i) q^{93} +(1131.74 + 653.410i) q^{94} +(-9572.12 + 2164.69i) q^{95} +(386.478 + 1582.67i) q^{96} +(-5218.11 - 1398.19i) q^{97} +(-6481.95 - 6481.95i) q^{98} +(-3694.99 - 7114.55i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 48 q^{2} - 2 q^{3} + 16 q^{6} - 24 q^{7} - 768 q^{8} + 24 q^{10} + 156 q^{11} - 16 q^{12} - 506 q^{15} + 1536 q^{16} + 900 q^{17} + 424 q^{18} - 48 q^{20} - 1292 q^{21} + 312 q^{22} - 1284 q^{23}+ \cdots - 63840 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.73205 − 0.732051i −0.683013 − 0.183013i
\(3\) −8.99776 − 0.200617i −0.999752 − 0.0222908i
\(4\) 6.92820 + 4.00000i 0.433013 + 0.250000i
\(5\) −5.51437 − 24.3843i −0.220575 − 0.975370i
\(6\) 24.4355 + 7.13492i 0.678763 + 0.198192i
\(7\) 72.5536 + 19.4407i 1.48069 + 0.396749i 0.906581 − 0.422032i \(-0.138683\pi\)
0.574106 + 0.818781i \(0.305350\pi\)
\(8\) −16.0000 − 16.0000i −0.250000 − 0.250000i
\(9\) 80.9195 + 3.61021i 0.999006 + 0.0445705i
\(10\) −2.78496 + 70.6558i −0.0278496 + 0.706558i
\(11\) −49.4867 − 85.7134i −0.408981 − 0.708376i 0.585795 − 0.810459i \(-0.300782\pi\)
−0.994776 + 0.102084i \(0.967449\pi\)
\(12\) −61.5359 − 37.3810i −0.427332 − 0.259590i
\(13\) −242.471 + 64.9698i −1.43474 + 0.384437i −0.890688 − 0.454616i \(-0.849777\pi\)
−0.544050 + 0.839053i \(0.683110\pi\)
\(14\) −183.989 − 106.226i −0.938718 − 0.541969i
\(15\) 44.7251 + 220.510i 0.198778 + 0.980044i
\(16\) 32.0000 + 55.4256i 0.125000 + 0.216506i
\(17\) 60.1943 − 60.1943i 0.208285 − 0.208285i −0.595253 − 0.803538i \(-0.702948\pi\)
0.803538 + 0.595253i \(0.202948\pi\)
\(18\) −218.433 − 69.1005i −0.674177 − 0.213273i
\(19\) − 392.553i − 1.08741i −0.839278 − 0.543703i \(-0.817022\pi\)
0.839278 − 0.543703i \(-0.182978\pi\)
\(20\) 59.3323 − 190.997i 0.148331 − 0.477491i
\(21\) −648.920 − 189.478i −1.47147 − 0.429656i
\(22\) 72.4535 + 270.400i 0.149697 + 0.558678i
\(23\) −817.438 + 219.032i −1.54525 + 0.414049i −0.927958 − 0.372684i \(-0.878437\pi\)
−0.617293 + 0.786733i \(0.711771\pi\)
\(24\) 140.754 + 147.174i 0.244365 + 0.255511i
\(25\) −564.183 + 268.928i −0.902693 + 0.430284i
\(26\) 710.003 1.05030
\(27\) −727.370 − 48.7177i −0.997765 − 0.0668281i
\(28\) 424.904 + 424.904i 0.541969 + 0.541969i
\(29\) −175.946 + 101.583i −0.209211 + 0.120788i −0.600944 − 0.799291i \(-0.705209\pi\)
0.391734 + 0.920079i \(0.371875\pi\)
\(30\) 39.2332 − 635.186i 0.0435924 − 0.705762i
\(31\) −39.1711 + 67.8463i −0.0407607 + 0.0705996i −0.885686 − 0.464285i \(-0.846311\pi\)
0.844925 + 0.534884i \(0.179645\pi\)
\(32\) −46.8513 − 174.851i −0.0457532 − 0.170753i
\(33\) 428.074 + 781.157i 0.393089 + 0.717316i
\(34\) −208.519 + 120.389i −0.180380 + 0.104142i
\(35\) 73.9587 − 1876.37i 0.0603745 − 1.53173i
\(36\) 546.186 + 348.690i 0.421440 + 0.269051i
\(37\) −1656.33 + 1656.33i −1.20989 + 1.20989i −0.238823 + 0.971063i \(0.576762\pi\)
−0.971063 + 0.238823i \(0.923238\pi\)
\(38\) −287.369 + 1072.48i −0.199009 + 0.742711i
\(39\) 2194.73 − 535.939i 1.44295 − 0.352360i
\(40\) −301.918 + 478.378i −0.188699 + 0.298986i
\(41\) 1206.44 − 2089.62i 0.717693 − 1.24308i −0.244219 − 0.969720i \(-0.578531\pi\)
0.961912 − 0.273361i \(-0.0881352\pi\)
\(42\) 1634.18 + 992.707i 0.926404 + 0.562759i
\(43\) 175.226 − 653.954i 0.0947682 − 0.353680i −0.902216 − 0.431284i \(-0.858061\pi\)
0.996984 + 0.0776047i \(0.0247272\pi\)
\(44\) − 791.787i − 0.408981i
\(45\) −358.188 − 1993.07i −0.176883 − 0.984232i
\(46\) 2393.62 1.13120
\(47\) −446.287 − 119.582i −0.202031 − 0.0541341i 0.156384 − 0.987696i \(-0.450016\pi\)
−0.358415 + 0.933562i \(0.616683\pi\)
\(48\) −276.809 − 505.126i −0.120143 − 0.219239i
\(49\) 2806.76 + 1620.49i 1.16900 + 0.674921i
\(50\) 1738.25 − 321.713i 0.695299 − 0.128685i
\(51\) −553.690 + 529.538i −0.212876 + 0.203590i
\(52\) −1939.77 − 519.759i −0.717369 − 0.192218i
\(53\) −296.250 − 296.250i −0.105465 − 0.105465i 0.652405 − 0.757870i \(-0.273760\pi\)
−0.757870 + 0.652405i \(0.773760\pi\)
\(54\) 1951.55 + 665.571i 0.669255 + 0.228248i
\(55\) −1817.17 + 1679.35i −0.600717 + 0.555158i
\(56\) −849.807 − 1471.91i −0.270984 − 0.469359i
\(57\) −78.7529 + 3532.10i −0.0242391 + 1.08713i
\(58\) 555.057 − 148.727i 0.164999 − 0.0442114i
\(59\) −3902.69 − 2253.22i −1.12114 − 0.647291i −0.179449 − 0.983767i \(-0.557431\pi\)
−0.941692 + 0.336476i \(0.890765\pi\)
\(60\) −572.175 + 1706.64i −0.158938 + 0.474066i
\(61\) 534.192 + 925.248i 0.143562 + 0.248656i 0.928835 − 0.370493i \(-0.120811\pi\)
−0.785274 + 0.619149i \(0.787478\pi\)
\(62\) 156.684 − 156.684i 0.0407607 − 0.0407607i
\(63\) 5800.82 + 1835.07i 1.46153 + 0.462350i
\(64\) 512.000i 0.125000i
\(65\) 2921.31 + 5554.20i 0.691435 + 1.31460i
\(66\) −597.673 − 2447.53i −0.137207 − 0.561876i
\(67\) −1842.03 − 6874.54i −0.410343 − 1.53142i −0.793985 − 0.607938i \(-0.791997\pi\)
0.383642 − 0.923482i \(-0.374670\pi\)
\(68\) 657.815 − 176.261i 0.142261 − 0.0381187i
\(69\) 7399.06 − 1806.80i 1.55410 − 0.379501i
\(70\) −1575.66 + 5072.20i −0.321563 + 1.03514i
\(71\) −8524.10 −1.69095 −0.845477 − 0.534012i \(-0.820684\pi\)
−0.845477 + 0.534012i \(0.820684\pi\)
\(72\) −1236.95 − 1352.48i −0.238609 − 0.260894i
\(73\) 2993.00 + 2993.00i 0.561643 + 0.561643i 0.929774 − 0.368131i \(-0.120002\pi\)
−0.368131 + 0.929774i \(0.620002\pi\)
\(74\) 5737.71 − 3312.67i 1.04779 − 0.604943i
\(75\) 5130.34 − 2306.56i 0.912060 − 0.410056i
\(76\) 1570.21 − 2719.69i 0.271851 − 0.470860i
\(77\) −1924.11 − 7180.88i −0.324525 − 1.21114i
\(78\) −6388.44 − 142.439i −1.05004 − 0.0234120i
\(79\) −2798.55 + 1615.75i −0.448414 + 0.258892i −0.707160 − 0.707053i \(-0.750024\pi\)
0.258746 + 0.965945i \(0.416691\pi\)
\(80\) 1175.05 − 1085.93i 0.183602 − 0.169677i
\(81\) 6534.93 + 584.273i 0.996027 + 0.0890524i
\(82\) −4825.77 + 4825.77i −0.717693 + 0.717693i
\(83\) −870.223 + 3247.72i −0.126321 + 0.471435i −0.999883 − 0.0152766i \(-0.995137\pi\)
0.873563 + 0.486712i \(0.161804\pi\)
\(84\) −3737.94 − 3908.43i −0.529753 − 0.553915i
\(85\) −1799.73 − 1135.86i −0.249097 − 0.157212i
\(86\) −957.455 + 1658.36i −0.129456 + 0.224224i
\(87\) 1603.50 − 878.718i 0.211851 − 0.116094i
\(88\) −579.628 + 2163.20i −0.0748487 + 0.279339i
\(89\) 6328.76i 0.798985i 0.916737 + 0.399492i \(0.130814\pi\)
−0.916737 + 0.399492i \(0.869186\pi\)
\(90\) −480.440 + 5707.38i −0.0593136 + 0.704615i
\(91\) −18855.2 −2.27692
\(92\) −6539.50 − 1752.25i −0.772626 − 0.207024i
\(93\) 366.063 − 602.606i 0.0423243 − 0.0696735i
\(94\) 1131.74 + 653.410i 0.128083 + 0.0739486i
\(95\) −9572.12 + 2164.69i −1.06062 + 0.239854i
\(96\) 386.478 + 1582.67i 0.0419356 + 0.171731i
\(97\) −5218.11 − 1398.19i −0.554587 − 0.148601i −0.0293688 − 0.999569i \(-0.509350\pi\)
−0.525218 + 0.850967i \(0.676016\pi\)
\(98\) −6481.95 − 6481.95i −0.674921 − 0.674921i
\(99\) −3694.99 − 7114.55i −0.377002 − 0.725900i
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 90.5.k.a.13.1 yes 48
3.2 odd 2 270.5.l.b.253.8 48
5.2 odd 4 inner 90.5.k.a.67.6 yes 48
9.2 odd 6 270.5.l.b.73.1 48
9.7 even 3 inner 90.5.k.a.43.6 yes 48
15.2 even 4 270.5.l.b.37.1 48
45.2 even 12 270.5.l.b.127.8 48
45.7 odd 12 inner 90.5.k.a.7.1 ✓ 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.5.k.a.7.1 ✓ 48 45.7 odd 12 inner
90.5.k.a.13.1 yes 48 1.1 even 1 trivial
90.5.k.a.43.6 yes 48 9.7 even 3 inner
90.5.k.a.67.6 yes 48 5.2 odd 4 inner
270.5.l.b.37.1 48 15.2 even 4
270.5.l.b.73.1 48 9.2 odd 6
270.5.l.b.127.8 48 45.2 even 12
270.5.l.b.253.8 48 3.2 odd 2