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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 443.12
Character \(\chi\) \(=\) 888.443
Dual form 888.2.c.d.443.9

$q$-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} +(1.55485 - 1.89274i) 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} +(-1.34839 + 3.19090i) 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} +(-0.391860 + 4.22451i) 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} +(0.547191 - 4.86832i) 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} +(-4.72397 - 3.88064i) 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} +(-1.16491 - 5.88583i) 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} +(-5.56441 - 4.57105i) 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} +(1.21279 + 6.82123i) 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} +(-3.54053 - 6.43930i) 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} +(7.96397 + 3.36538i) 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} +(-10.5794 - 8.69075i) 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} +(3.94299 + 7.51351i) q^{72} -6.56860 q^{73} +(-6.86457 - 5.18437i) q^{74} +(1.76293 - 1.19372i) q^{75} +(-3.94578 - 6.07843i) q^{76} -16.4323 q^{77} +(4.60027 - 5.59998i) 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} +(9.38084 + 3.96411i) 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} +(10.5437 + 0.978019i) 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} +(-4.38351 - 8.76270i) 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 \(-0.811536\pi\)
0.829783 − 0.558087i \(-0.188464\pi\)
\(6\) 1.55485 − 1.89274i 0.634763 − 0.772707i
\(7\) − 2.93987i − 1.11117i −0.831460 − 0.555584i \(-0.812495\pi\)
0.831460 − 0.555584i \(-0.187505\pi\)
\(8\) −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 \(-0.319002\pi\)
\(12\) −1.34839 + 3.19090i −0.389248 + 0.921133i
\(13\) 2.95867 0.820587 0.410293 − 0.911954i \(-0.365426\pi\)
0.410293 + 0.911954i \(0.365426\pi\)
\(14\) 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 \(0.793622\pi\)
\(18\) −0.391860 + 4.22451i −0.0923622 + 0.995725i
\(19\) − 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 \(0.115921\pi\)
−0.934418 + 0.356179i \(0.884079\pi\)
\(24\) 0.547191 − 4.86832i 0.111695 − 0.993743i
\(25\) −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 \(0.0607572\pi\)
−0.981839 + 0.189717i \(0.939243\pi\)
\(30\) −4.72397 − 3.88064i −0.862475 − 0.708506i
\(31\) −8.34338 −1.49852 −0.749258 − 0.662279i \(-0.769589\pi\)
−0.749258 + 0.662279i \(0.769589\pi\)
\(32\) −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
\(36\) −1.16491 − 5.88583i −0.194152 − 0.980972i
\(37\) 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 \(0.954791\pi\)
\(42\) −5.56441 − 4.57105i −0.858607 − 0.705328i
\(43\) 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 \(-0.748756\pi\)
−0.704337 + 0.709865i \(0.748756\pi\)
\(48\) 1.21279 + 6.82123i 0.175052 + 0.984559i
\(49\) −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 \(-0.292116\pi\)
0.607641 + 0.794212i \(0.292116\pi\)
\(54\) −3.54053 − 6.43930i −0.481805 − 0.876278i
\(55\) −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 \(0.574926\pi\)
\(60\) 7.96397 + 3.36538i 1.02814 + 0.434468i
\(61\) 8.04325 1.02983 0.514916 − 0.857240i \(-0.327823\pi\)
0.514916 + 0.857240i \(0.327823\pi\)
\(62\) 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
\(66\) −10.5794 − 8.69075i −1.30223 − 1.06976i
\(67\) 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 \(-0.407009\pi\)
0.288001 + 0.957630i \(0.407009\pi\)
\(72\) 3.94299 + 7.51351i 0.464685 + 0.885476i
\(73\) −6.56860 −0.768796 −0.384398 − 0.923167i \(-0.625591\pi\)
−0.384398 + 0.923167i \(0.625591\pi\)
\(74\) −6.86457 − 5.18437i −0.797990 − 0.602670i
\(75\) 1.76293 − 1.19372i 0.203566 − 0.137839i
\(76\) −3.94578 − 6.07843i −0.452612 − 0.697244i
\(77\) −16.4323 −1.87264
\(78\) 4.60027 − 5.59998i 0.520878 − 0.634073i
\(79\) −1.48811 −0.167425 −0.0837126 − 0.996490i \(-0.526678\pi\)
−0.0837126 + 0.996490i \(0.526678\pi\)
\(80\) −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 \(-0.353639\pi\)
\(84\) 9.38084 + 3.96411i 1.02353 + 0.432520i
\(85\) 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 \(0.420977\pi\)
\(90\) 10.5437 + 0.978019i 1.11140 + 0.103092i
\(91\) − 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
\(96\) −4.38351 − 8.76270i −0.447390 − 0.894339i
\(97\) − 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.443.12 yes 96
3.2 odd 2 inner 888.2.c.d.443.85 yes 96
8.3 odd 2 inner 888.2.c.d.443.10 yes 96
24.11 even 2 inner 888.2.c.d.443.87 yes 96
37.36 even 2 inner 888.2.c.d.443.86 yes 96
111.110 odd 2 inner 888.2.c.d.443.11 yes 96
296.147 odd 2 inner 888.2.c.d.443.88 yes 96
888.443 even 2 inner 888.2.c.d.443.9 ✓ 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.c.d.443.9 ✓ 96 888.443 even 2 inner
888.2.c.d.443.10 yes 96 8.3 odd 2 inner
888.2.c.d.443.11 yes 96 111.110 odd 2 inner
888.2.c.d.443.12 yes 96 1.1 even 1 trivial
888.2.c.d.443.85 yes 96 3.2 odd 2 inner
888.2.c.d.443.86 yes 96 37.36 even 2 inner
888.2.c.d.443.87 yes 96 24.11 even 2 inner
888.2.c.d.443.88 yes 96 296.147 odd 2 inner