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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(443,888)] 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("888.443"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0,-4,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(96\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 443.10
Character \(\chi\) \(=\) 888.443
Dual form 888.2.c.d.443.11

$q$-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+(-1.35601 - 0.401533i) q^{2} +(-1.43419 + 0.971128i) q^{3} +(1.67754 + 1.08897i) q^{4} +2.49584i q^{5} +(2.33472 - 0.740986i) q^{6} +2.93987i q^{7} +(-1.83751 - 2.15024i) q^{8} +(1.11382 - 2.78557i) q^{9} +(1.00216 - 3.38439i) q^{10} -5.58946i q^{11} +(-3.46345 + 0.0673184i) q^{12} -2.95867 q^{13} +(1.18046 - 3.98651i) q^{14} +(-2.42378 - 3.57952i) q^{15} +(1.62830 + 3.65358i) q^{16} -6.57288 q^{17} +(-2.62885 + 3.33003i) q^{18} -3.62342i q^{19} +(-2.71789 + 4.18688i) q^{20} +(-2.85499 - 4.21635i) q^{21} +(-2.24435 + 7.57939i) q^{22} -3.41635i q^{23} +(4.72351 + 1.29940i) q^{24} -1.22921 q^{25} +(4.01199 + 1.18800i) q^{26} +(1.10771 + 5.07671i) q^{27} +(-3.20143 + 4.93177i) q^{28} -2.04332i q^{29} +(1.84938 + 5.82710i) q^{30} +8.34338 q^{31} +(-0.740965 - 5.60812i) q^{32} +(5.42808 + 8.01637i) q^{33} +(8.91291 + 2.63923i) q^{34} -7.33745 q^{35} +(4.90188 - 3.46000i) q^{36} +(-3.61338 - 4.89321i) q^{37} +(-1.45492 + 4.91340i) q^{38} +(4.24330 - 2.87324i) q^{39} +(5.36666 - 4.58614i) q^{40} +1.81275i q^{41} +(2.17841 + 6.86380i) q^{42} +1.44109i q^{43} +(6.08674 - 9.37656i) q^{44} +(6.95233 + 2.77992i) q^{45} +(-1.37178 + 4.63261i) q^{46} +9.65739 q^{47} +(-5.88339 - 3.65865i) q^{48} -1.64286 q^{49} +(1.66683 + 0.493570i) q^{50} +(9.42678 - 6.38310i) q^{51} +(-4.96329 - 3.22189i) q^{52} -8.84739 q^{53} +(0.536399 - 7.32887i) q^{54} +13.9504 q^{55} +(6.32144 - 5.40206i) q^{56} +(3.51880 + 5.19668i) q^{57} +(-0.820459 + 2.77077i) q^{58} -3.58277 q^{59} +(-0.168016 - 8.64421i) q^{60} -8.04325 q^{61} +(-11.3137 - 3.35014i) q^{62} +(8.18923 + 3.27450i) q^{63} +(-1.24708 + 7.90220i) q^{64} -7.38436i q^{65} +(-4.14171 - 13.0499i) q^{66} +0.198413 q^{67} +(-11.0263 - 7.15765i) q^{68} +(3.31771 + 4.89970i) q^{69} +(9.94968 + 2.94623i) q^{70} -4.85348 q^{71} +(-8.03631 + 2.72354i) q^{72} -6.56860 q^{73} +(2.93500 + 8.08615i) q^{74} +(1.76293 - 1.19372i) q^{75} +(3.94578 - 6.07843i) q^{76} +16.4323 q^{77} +(-6.90767 + 2.19233i) q^{78} +1.48811 q^{79} +(-9.11875 + 4.06398i) q^{80} +(-6.51880 - 6.20526i) q^{81} +(0.727878 - 2.45811i) q^{82} -16.3284i q^{83} +(-0.197908 - 10.1821i) q^{84} -16.4048i q^{85} +(0.578646 - 1.95414i) q^{86} +(1.98432 + 2.93051i) q^{87} +(-12.0187 + 10.2707i) q^{88} +4.63614 q^{89} +(-8.31123 - 6.56120i) q^{90} -8.69811i q^{91} +(3.72029 - 5.73107i) q^{92} +(-11.9660 + 8.10249i) q^{93} +(-13.0955 - 3.87776i) q^{94} +9.04346 q^{95} +(6.50888 + 7.32355i) q^{96} -8.25890i q^{97} +(2.22774 + 0.659663i) q^{98} +(-15.5698 - 6.22567i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q - 4 q^{3} + 24 q^{4} - 4 q^{9} - 20 q^{10} - 18 q^{12} - 56 q^{16} + 72 q^{25} - 4 q^{27} - 64 q^{28} - 16 q^{33} - 8 q^{34} + 42 q^{36} + 44 q^{40} - 68 q^{46} - 94 q^{48} - 104 q^{49} - 28 q^{58}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.35601 − 0.401533i −0.958846 − 0.283927i
\(3\) −1.43419 + 0.971128i −0.828032 + 0.560681i
\(4\) 1.67754 + 1.08897i 0.838771 + 0.544484i
\(5\) 2.49584i 1.11617i 0.829783 + 0.558087i \(0.188464\pi\)
−0.829783 + 0.558087i \(0.811536\pi\)
\(6\) 2.33472 − 0.740986i 0.953147 − 0.302506i
\(7\) 2.93987i 1.11117i 0.831460 + 0.555584i \(0.187505\pi\)
−0.831460 + 0.555584i \(0.812495\pi\)
\(8\) −1.83751 − 2.15024i −0.649659 − 0.760225i
\(9\) 1.11382 − 2.78557i 0.371274 − 0.928523i
\(10\) 1.00216 − 3.38439i 0.316911 − 1.07024i
\(11\) − 5.58946i − 1.68529i −0.538472 − 0.842643i \(-0.680998\pi\)
0.538472 − 0.842643i \(-0.319002\pi\)
\(12\) −3.46345 + 0.0673184i −0.999811 + 0.0194331i
\(13\) −2.95867 −0.820587 −0.410293 − 0.911954i \(-0.634574\pi\)
−0.410293 + 0.911954i \(0.634574\pi\)
\(14\) 1.18046 − 3.98651i 0.315490 − 1.06544i
\(15\) −2.42378 − 3.57952i −0.625817 − 0.924227i
\(16\) 1.62830 + 3.65358i 0.407075 + 0.913395i
\(17\) −6.57288 −1.59416 −0.797079 − 0.603876i \(-0.793622\pi\)
−0.797079 + 0.603876i \(0.793622\pi\)
\(18\) −2.62885 + 3.33003i −0.619627 + 0.784896i
\(19\) − 3.62342i − 0.831268i −0.909532 − 0.415634i \(-0.863560\pi\)
0.909532 − 0.415634i \(-0.136440\pi\)
\(20\) −2.71789 + 4.18688i −0.607738 + 0.936214i
\(21\) −2.85499 − 4.21635i −0.623011 − 0.920083i
\(22\) −2.24435 + 7.57939i −0.478498 + 1.61593i
\(23\) − 3.41635i − 0.712358i −0.934418 − 0.356179i \(-0.884079\pi\)
0.934418 − 0.356179i \(-0.115921\pi\)
\(24\) 4.72351 + 1.29940i 0.964183 + 0.265240i
\(25\) −1.22921 −0.245843
\(26\) 4.01199 + 1.18800i 0.786816 + 0.232986i
\(27\) 1.10771 + 5.07671i 0.213178 + 0.977013i
\(28\) −3.20143 + 4.93177i −0.605013 + 0.932016i
\(29\) − 2.04332i − 0.379435i −0.981839 − 0.189717i \(-0.939243\pi\)
0.981839 − 0.189717i \(-0.0607572\pi\)
\(30\) 1.84938 + 5.82710i 0.337649 + 1.06388i
\(31\) 8.34338 1.49852 0.749258 − 0.662279i \(-0.230411\pi\)
0.749258 + 0.662279i \(0.230411\pi\)
\(32\) −0.740965 − 5.60812i −0.130985 − 0.991384i
\(33\) 5.42808 + 8.01637i 0.944908 + 1.39547i
\(34\) 8.91291 + 2.63923i 1.52855 + 0.452624i
\(35\) −7.33745 −1.24026
\(36\) 4.90188 − 3.46000i 0.816980 − 0.576666i
\(37\) −3.61338 − 4.89321i −0.594036 − 0.804439i
\(38\) −1.45492 + 4.91340i −0.236019 + 0.797058i
\(39\) 4.24330 − 2.87324i 0.679472 − 0.460087i
\(40\) 5.36666 − 4.58614i 0.848543 − 0.725132i
\(41\) 1.81275i 0.283104i 0.989931 + 0.141552i \(0.0452092\pi\)
−0.989931 + 0.141552i \(0.954791\pi\)
\(42\) 2.17841 + 6.86380i 0.336135 + 1.05911i
\(43\) 1.44109i 0.219765i 0.993945 + 0.109882i \(0.0350474\pi\)
−0.993945 + 0.109882i \(0.964953\pi\)
\(44\) 6.08674 − 9.37656i 0.917611 − 1.41357i
\(45\) 6.95233 + 2.77992i 1.03639 + 0.414406i
\(46\) −1.37178 + 4.63261i −0.202257 + 0.683041i
\(47\) 9.65739 1.40867 0.704337 − 0.709865i \(-0.251244\pi\)
0.704337 + 0.709865i \(0.251244\pi\)
\(48\) −5.88339 − 3.65865i −0.849194 − 0.528081i
\(49\) −1.64286 −0.234695
\(50\) 1.66683 + 0.493570i 0.235725 + 0.0698013i
\(51\) 9.42678 − 6.38310i 1.32001 − 0.893813i
\(52\) −4.96329 − 3.22189i −0.688285 − 0.446796i
\(53\) −8.84739 −1.21528 −0.607641 − 0.794212i \(-0.707884\pi\)
−0.607641 + 0.794212i \(0.707884\pi\)
\(54\) 0.536399 − 7.32887i 0.0729947 − 0.997332i
\(55\) 13.9504 1.88107
\(56\) 6.32144 − 5.40206i 0.844738 − 0.721881i
\(57\) 3.51880 + 5.19668i 0.466076 + 0.688317i
\(58\) −0.820459 + 2.77077i −0.107732 + 0.363819i
\(59\) −3.58277 −0.466437 −0.233218 − 0.972424i \(-0.574926\pi\)
−0.233218 + 0.972424i \(0.574926\pi\)
\(60\) −0.168016 − 8.64421i −0.0216908 − 1.11596i
\(61\) −8.04325 −1.02983 −0.514916 − 0.857240i \(-0.672177\pi\)
−0.514916 + 0.857240i \(0.672177\pi\)
\(62\) −11.3137 − 3.35014i −1.43685 − 0.425468i
\(63\) 8.18923 + 3.27450i 1.03175 + 0.412548i
\(64\) −1.24708 + 7.90220i −0.155886 + 0.987775i
\(65\) − 7.38436i − 0.915917i
\(66\) −4.14171 − 13.0499i −0.509810 − 1.60633i
\(67\) 0.198413 0.0242400 0.0121200 − 0.999927i \(-0.496142\pi\)
0.0121200 + 0.999927i \(0.496142\pi\)
\(68\) −11.0263 − 7.15765i −1.33713 − 0.867993i
\(69\) 3.31771 + 4.89970i 0.399405 + 0.589855i
\(70\) 9.94968 + 2.94623i 1.18921 + 0.352142i
\(71\) −4.85348 −0.576002 −0.288001 − 0.957630i \(-0.592991\pi\)
−0.288001 + 0.957630i \(0.592991\pi\)
\(72\) −8.03631 + 2.72354i −0.947089 + 0.320972i
\(73\) −6.56860 −0.768796 −0.384398 − 0.923167i \(-0.625591\pi\)
−0.384398 + 0.923167i \(0.625591\pi\)
\(74\) 2.93500 + 8.08615i 0.341187 + 0.939995i
\(75\) 1.76293 − 1.19372i 0.203566 − 0.137839i
\(76\) 3.94578 − 6.07843i 0.452612 − 0.697244i
\(77\) 16.4323 1.87264
\(78\) −6.90767 + 2.19233i −0.782140 + 0.248233i
\(79\) 1.48811 0.167425 0.0837126 − 0.996490i \(-0.473322\pi\)
0.0837126 + 0.996490i \(0.473322\pi\)
\(80\) −9.11875 + 4.06398i −1.01951 + 0.454366i
\(81\) −6.51880 − 6.20526i −0.724311 − 0.689473i
\(82\) 0.727878 − 2.45811i 0.0803806 − 0.271453i
\(83\) − 16.3284i − 1.79228i −0.443774 − 0.896139i \(-0.646361\pi\)
0.443774 − 0.896139i \(-0.353639\pi\)
\(84\) −0.197908 − 10.1821i −0.0215935 − 1.11096i
\(85\) − 16.4048i − 1.77936i
\(86\) 0.578646 − 1.95414i 0.0623970 − 0.210720i
\(87\) 1.98432 + 2.93051i 0.212742 + 0.314184i
\(88\) −12.0187 + 10.2707i −1.28120 + 1.09486i
\(89\) 4.63614 0.491430 0.245715 − 0.969342i \(-0.420977\pi\)
0.245715 + 0.969342i \(0.420977\pi\)
\(90\) −8.31123 − 6.56120i −0.876080 − 0.691611i
\(91\) − 8.69811i − 0.911810i
\(92\) 3.72029 − 5.73107i 0.387867 − 0.597505i
\(93\) −11.9660 + 8.10249i −1.24082 + 0.840189i
\(94\) −13.0955 − 3.87776i −1.35070 − 0.399960i
\(95\) 9.04346 0.927840
\(96\) 6.50888 + 7.32355i 0.664310 + 0.747457i
\(97\) − 8.25890i − 0.838564i −0.907856 − 0.419282i \(-0.862282\pi\)
0.907856 − 0.419282i \(-0.137718\pi\)
\(98\) 2.22774 + 0.659663i 0.225036 + 0.0666360i
\(99\) −15.5698 − 6.22567i −1.56483 − 0.625703i
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 888.2.c.d.443.10 yes 96
3.2 odd 2 inner 888.2.c.d.443.87 yes 96
8.3 odd 2 inner 888.2.c.d.443.12 yes 96
24.11 even 2 inner 888.2.c.d.443.85 yes 96
37.36 even 2 inner 888.2.c.d.443.88 yes 96
111.110 odd 2 inner 888.2.c.d.443.9 ✓ 96
296.147 odd 2 inner 888.2.c.d.443.86 yes 96
888.443 even 2 inner 888.2.c.d.443.11 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.c.d.443.9 ✓ 96 111.110 odd 2 inner
888.2.c.d.443.10 yes 96 1.1 even 1 trivial
888.2.c.d.443.11 yes 96 888.443 even 2 inner
888.2.c.d.443.12 yes 96 8.3 odd 2 inner
888.2.c.d.443.85 yes 96 24.11 even 2 inner
888.2.c.d.443.86 yes 96 296.147 odd 2 inner
888.2.c.d.443.87 yes 96 3.2 odd 2 inner
888.2.c.d.443.88 yes 96 37.36 even 2 inner