Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(517,888)] 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("888.517"); 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([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 517.66
Character \(\chi\) \(=\) 888.517
Dual form 888.2.o.a.517.65

$q$-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+(1.28459 + 0.591458i) q^{2} -1.00000i q^{3} +(1.30036 + 1.51956i) q^{4} +4.07371 q^{5} +(0.591458 - 1.28459i) q^{6} -2.80969 q^{7} +(0.771669 + 2.72113i) q^{8} -1.00000 q^{9} +(5.23305 + 2.40942i) q^{10} -2.23997i q^{11} +(1.51956 - 1.30036i) q^{12} +2.48511 q^{13} +(-3.60930 - 1.66181i) q^{14} -4.07371i q^{15} +(-0.618151 + 3.95195i) q^{16} +3.25175i q^{17} +(-1.28459 - 0.591458i) q^{18} +3.69970 q^{19} +(5.29727 + 6.19026i) q^{20} +2.80969i q^{21} +(1.32485 - 2.87745i) q^{22} +1.06510i q^{23} +(2.72113 - 0.771669i) q^{24} +11.5951 q^{25} +(3.19235 + 1.46984i) q^{26} +1.00000i q^{27} +(-3.65359 - 4.26950i) q^{28} -2.54211 q^{29} +(2.40942 - 5.23305i) q^{30} -10.6374i q^{31} +(-3.13148 + 4.71103i) q^{32} -2.23997 q^{33} +(-1.92327 + 4.17718i) q^{34} -11.4458 q^{35} +(-1.30036 - 1.51956i) q^{36} +(-6.07633 + 0.279692i) q^{37} +(4.75261 + 2.18822i) q^{38} -2.48511i q^{39} +(3.14355 + 11.0851i) q^{40} -3.79086 q^{41} +(-1.66181 + 3.60930i) q^{42} -6.73309 q^{43} +(3.40378 - 2.91276i) q^{44} -4.07371 q^{45} +(-0.629964 + 1.36823i) q^{46} +4.97048 q^{47} +(3.95195 + 0.618151i) q^{48} +0.894338 q^{49} +(14.8950 + 6.85800i) q^{50} +3.25175 q^{51} +(3.23152 + 3.77628i) q^{52} +7.06006i q^{53} +(-0.591458 + 1.28459i) q^{54} -9.12498i q^{55} +(-2.16815 - 7.64551i) q^{56} -3.69970i q^{57} +(-3.26558 - 1.50355i) q^{58} +2.83293 q^{59} +(6.19026 - 5.29727i) q^{60} -0.792322 q^{61} +(6.29156 - 13.6647i) q^{62} +2.80969 q^{63} +(-6.80905 + 4.19962i) q^{64} +10.1236 q^{65} +(-2.87745 - 1.32485i) q^{66} -2.28484i q^{67} +(-4.94125 + 4.22843i) q^{68} +1.06510 q^{69} +(-14.7032 - 6.76973i) q^{70} -13.9183 q^{71} +(-0.771669 - 2.72113i) q^{72} +8.96449 q^{73} +(-7.97103 - 3.23460i) q^{74} -11.5951i q^{75} +(4.81093 + 5.62193i) q^{76} +6.29361i q^{77} +(1.46984 - 3.19235i) q^{78} +6.78443i q^{79} +(-2.51817 + 16.0991i) q^{80} +1.00000 q^{81} +(-4.86970 - 2.24213i) q^{82} -6.40228i q^{83} +(-4.26950 + 3.65359i) q^{84} +13.2467i q^{85} +(-8.64928 - 3.98234i) q^{86} +2.54211i q^{87} +(6.09524 - 1.72851i) q^{88} -12.1440i q^{89} +(-5.23305 - 2.40942i) q^{90} -6.98238 q^{91} +(-1.61849 + 1.38501i) q^{92} -10.6374 q^{93} +(6.38504 + 2.93983i) q^{94} +15.0715 q^{95} +(4.71103 + 3.13148i) q^{96} +3.08454i q^{97} +(1.14886 + 0.528963i) q^{98} +2.23997i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q + 4 q^{4} + 8 q^{7} - 76 q^{9} + 4 q^{16} + 84 q^{25} + 12 q^{28} + 8 q^{30} - 12 q^{34} - 4 q^{36} + 8 q^{38} - 56 q^{40} - 8 q^{41} - 24 q^{44} - 44 q^{46} - 8 q^{48} + 60 q^{49} - 40 q^{58} + 32 q^{62}+ \cdots - 56 q^{86}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.28459 + 0.591458i 0.908344 + 0.418224i
\(3\) − 1.00000i − 0.577350i
\(4\) 1.30036 + 1.51956i 0.650178 + 0.759782i
\(5\) 4.07371 1.82182 0.910908 − 0.412609i \(-0.135382\pi\)
0.910908 + 0.412609i \(0.135382\pi\)
\(6\) 0.591458 − 1.28459i 0.241462 − 0.524433i
\(7\) −2.80969 −1.06196 −0.530981 − 0.847384i \(-0.678176\pi\)
−0.530981 + 0.847384i \(0.678176\pi\)
\(8\) 0.771669 + 2.72113i 0.272826 + 0.962063i
\(9\) −1.00000 −0.333333
\(10\) 5.23305 + 2.40942i 1.65484 + 0.761927i
\(11\) − 2.23997i − 0.675376i −0.941258 − 0.337688i \(-0.890355\pi\)
0.941258 − 0.337688i \(-0.109645\pi\)
\(12\) 1.51956 − 1.30036i 0.438660 − 0.375380i
\(13\) 2.48511 0.689245 0.344623 − 0.938741i \(-0.388007\pi\)
0.344623 + 0.938741i \(0.388007\pi\)
\(14\) −3.60930 − 1.66181i −0.964627 − 0.444138i
\(15\) − 4.07371i − 1.05183i
\(16\) −0.618151 + 3.95195i −0.154538 + 0.987987i
\(17\) 3.25175i 0.788666i 0.918968 + 0.394333i \(0.129024\pi\)
−0.918968 + 0.394333i \(0.870976\pi\)
\(18\) −1.28459 − 0.591458i −0.302781 − 0.139408i
\(19\) 3.69970 0.848770 0.424385 − 0.905482i \(-0.360490\pi\)
0.424385 + 0.905482i \(0.360490\pi\)
\(20\) 5.29727 + 6.19026i 1.18450 + 1.38418i
\(21\) 2.80969i 0.613124i
\(22\) 1.32485 − 2.87745i 0.282458 − 0.613474i
\(23\) 1.06510i 0.222090i 0.993815 + 0.111045i \(0.0354197\pi\)
−0.993815 + 0.111045i \(0.964580\pi\)
\(24\) 2.72113 − 0.771669i 0.555448 − 0.157516i
\(25\) 11.5951 2.31902
\(26\) 3.19235 + 1.46984i 0.626072 + 0.288259i
\(27\) 1.00000i 0.192450i
\(28\) −3.65359 − 4.26950i −0.690464 − 0.806860i
\(29\) −2.54211 −0.472058 −0.236029 − 0.971746i \(-0.575846\pi\)
−0.236029 + 0.971746i \(0.575846\pi\)
\(30\) 2.40942 − 5.23305i 0.439899 − 0.955420i
\(31\) − 10.6374i − 1.91053i −0.295751 − 0.955265i \(-0.595570\pi\)
0.295751 − 0.955265i \(-0.404430\pi\)
\(32\) −3.13148 + 4.71103i −0.553573 + 0.832801i
\(33\) −2.23997 −0.389929
\(34\) −1.92327 + 4.17718i −0.329839 + 0.716380i
\(35\) −11.4458 −1.93470
\(36\) −1.30036 − 1.51956i −0.216726 − 0.253261i
\(37\) −6.07633 + 0.279692i −0.998942 + 0.0459811i
\(38\) 4.75261 + 2.18822i 0.770975 + 0.354976i
\(39\) − 2.48511i − 0.397936i
\(40\) 3.14355 + 11.0851i 0.497039 + 1.75270i
\(41\) −3.79086 −0.592032 −0.296016 − 0.955183i \(-0.595658\pi\)
−0.296016 + 0.955183i \(0.595658\pi\)
\(42\) −1.66181 + 3.60930i −0.256423 + 0.556927i
\(43\) −6.73309 −1.02679 −0.513393 − 0.858153i \(-0.671612\pi\)
−0.513393 + 0.858153i \(0.671612\pi\)
\(44\) 3.40378 − 2.91276i 0.513139 − 0.439115i
\(45\) −4.07371 −0.607272
\(46\) −0.629964 + 1.36823i −0.0928832 + 0.201734i
\(47\) 4.97048 0.725018 0.362509 − 0.931980i \(-0.381920\pi\)
0.362509 + 0.931980i \(0.381920\pi\)
\(48\) 3.95195 + 0.618151i 0.570414 + 0.0892225i
\(49\) 0.894338 0.127763
\(50\) 14.8950 + 6.85800i 2.10646 + 0.969868i
\(51\) 3.25175 0.455337
\(52\) 3.23152 + 3.77628i 0.448132 + 0.523676i
\(53\) 7.06006i 0.969773i 0.874577 + 0.484887i \(0.161139\pi\)
−0.874577 + 0.484887i \(0.838861\pi\)
\(54\) −0.591458 + 1.28459i −0.0804872 + 0.174811i
\(55\) − 9.12498i − 1.23041i
\(56\) −2.16815 − 7.64551i −0.289731 − 1.02167i
\(57\) − 3.69970i − 0.490037i
\(58\) −3.26558 − 1.50355i −0.428791 − 0.197426i
\(59\) 2.83293 0.368816 0.184408 − 0.982850i \(-0.440963\pi\)
0.184408 + 0.982850i \(0.440963\pi\)
\(60\) 6.19026 − 5.29727i 0.799159 − 0.683874i
\(61\) −0.792322 −0.101446 −0.0507232 − 0.998713i \(-0.516153\pi\)
−0.0507232 + 0.998713i \(0.516153\pi\)
\(62\) 6.29156 − 13.6647i 0.799029 − 1.73542i
\(63\) 2.80969 0.353987
\(64\) −6.80905 + 4.19962i −0.851132 + 0.524952i
\(65\) 10.1236 1.25568
\(66\) −2.87745 − 1.32485i −0.354189 − 0.163077i
\(67\) − 2.28484i − 0.279137i −0.990212 − 0.139569i \(-0.955428\pi\)
0.990212 − 0.139569i \(-0.0445715\pi\)
\(68\) −4.94125 + 4.22843i −0.599214 + 0.512773i
\(69\) 1.06510 0.128224
\(70\) −14.7032 − 6.76973i −1.75737 − 0.809137i
\(71\) −13.9183 −1.65180 −0.825900 − 0.563817i \(-0.809332\pi\)
−0.825900 + 0.563817i \(0.809332\pi\)
\(72\) −0.771669 − 2.72113i −0.0909420 − 0.320688i
\(73\) 8.96449 1.04921 0.524607 − 0.851344i \(-0.324212\pi\)
0.524607 + 0.851344i \(0.324212\pi\)
\(74\) −7.97103 − 3.23460i −0.926614 − 0.376015i
\(75\) − 11.5951i − 1.33888i
\(76\) 4.81093 + 5.62193i 0.551851 + 0.644880i
\(77\) 6.29361i 0.717224i
\(78\) 1.46984 − 3.19235i 0.166426 − 0.361463i
\(79\) 6.78443i 0.763308i 0.924305 + 0.381654i \(0.124645\pi\)
−0.924305 + 0.381654i \(0.875355\pi\)
\(80\) −2.51817 + 16.0991i −0.281540 + 1.79993i
\(81\) 1.00000 0.111111
\(82\) −4.86970 − 2.24213i −0.537769 − 0.247602i
\(83\) − 6.40228i − 0.702741i −0.936236 − 0.351371i \(-0.885716\pi\)
0.936236 − 0.351371i \(-0.114284\pi\)
\(84\) −4.26950 + 3.65359i −0.465841 + 0.398639i
\(85\) 13.2467i 1.43680i
\(86\) −8.64928 − 3.98234i −0.932676 − 0.429427i
\(87\) 2.54211i 0.272543i
\(88\) 6.09524 − 1.72851i 0.649755 − 0.184260i
\(89\) − 12.1440i − 1.28726i −0.765337 − 0.643629i \(-0.777428\pi\)
0.765337 − 0.643629i \(-0.222572\pi\)
\(90\) −5.23305 − 2.40942i −0.551612 − 0.253976i
\(91\) −6.98238 −0.731952
\(92\) −1.61849 + 1.38501i −0.168740 + 0.144398i
\(93\) −10.6374 −1.10305
\(94\) 6.38504 + 2.93983i 0.658566 + 0.303220i
\(95\) 15.0715 1.54630
\(96\) 4.71103 + 3.13148i 0.480818 + 0.319606i
\(97\) 3.08454i 0.313188i 0.987663 + 0.156594i \(0.0500514\pi\)
−0.987663 + 0.156594i \(0.949949\pi\)
\(98\) 1.14886 + 0.528963i 0.116052 + 0.0534333i
\(99\) 2.23997i 0.225125i
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.o.a.517.66 yes 76
4.3 odd 2 3552.2.o.a.2737.4 76
8.3 odd 2 3552.2.o.a.2737.25 76
8.5 even 2 inner 888.2.o.a.517.12 yes 76
37.36 even 2 inner 888.2.o.a.517.11 ✓ 76
148.147 odd 2 3552.2.o.a.2737.26 76
296.147 odd 2 3552.2.o.a.2737.3 76
296.221 even 2 inner 888.2.o.a.517.65 yes 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.11 ✓ 76 37.36 even 2 inner
888.2.o.a.517.12 yes 76 8.5 even 2 inner
888.2.o.a.517.65 yes 76 296.221 even 2 inner
888.2.o.a.517.66 yes 76 1.1 even 1 trivial
3552.2.o.a.2737.3 76 296.147 odd 2
3552.2.o.a.2737.4 76 4.3 odd 2
3552.2.o.a.2737.25 76 8.3 odd 2
3552.2.o.a.2737.26 76 148.147 odd 2