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 517.11
Character \(\chi\) \(=\) 888.517
Dual form 888.2.o.a.517.12

$q$-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} +(-7.64018 - 3.95318i) 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 \(-0.864618\pi\)
−0.910908 + 0.412609i \(0.864618\pi\)
\(6\) −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 \(-0.611993\pi\)
−0.344623 + 0.938741i \(0.611993\pi\)
\(14\) 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 \(-0.870976\pi\)
0.918968 − 0.394333i \(-0.129024\pi\)
\(18\) 1.28459 + 0.591458i 0.302781 + 0.139408i
\(19\) −3.69970 −0.848770 −0.424385 − 0.905482i \(-0.639510\pi\)
−0.424385 + 0.905482i \(0.639510\pi\)
\(20\) −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 \(-0.964580\pi\)
0.993815 − 0.111045i \(-0.0354197\pi\)
\(24\) −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 \(-0.424154\pi\)
0.236029 + 0.971746i \(0.424154\pi\)
\(30\) 2.40942 − 5.23305i 0.439899 − 0.955420i
\(31\) 10.6374i 1.91053i 0.295751 + 0.955265i \(0.404430\pi\)
−0.295751 + 0.955265i \(0.595570\pi\)
\(32\) 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 \(-0.328388\pi\)
0.513393 + 0.858153i \(0.328388\pi\)
\(44\) 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 \(-0.559037\pi\)
−0.184408 + 0.982850i \(0.559037\pi\)
\(60\) −6.19026 + 5.29727i −0.799159 + 0.683874i
\(61\) 0.792322 0.101446 0.0507232 − 0.998713i \(-0.483847\pi\)
0.0507232 + 0.998713i \(0.483847\pi\)
\(62\) 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 \(0.324212\pi\)
\(74\) −7.64018 − 3.95318i −0.888153 − 0.459548i
\(75\) − 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 \(-0.875355\pi\)
0.924305 − 0.381654i \(-0.124645\pi\)
\(80\) 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 \(0.222572\pi\)
−0.765337 + 0.643629i \(0.777428\pi\)
\(90\) −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 \(-0.949949\pi\)
0.987663 − 0.156594i \(-0.0500514\pi\)
\(98\) −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.517.11 ✓ 76
4.3 odd 2 3552.2.o.a.2737.26 76
8.3 odd 2 3552.2.o.a.2737.3 76
8.5 even 2 inner 888.2.o.a.517.65 yes 76
37.36 even 2 inner 888.2.o.a.517.66 yes 76
148.147 odd 2 3552.2.o.a.2737.4 76
296.147 odd 2 3552.2.o.a.2737.25 76
296.221 even 2 inner 888.2.o.a.517.12 yes 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.11 ✓ 76 1.1 even 1 trivial
888.2.o.a.517.12 yes 76 296.221 even 2 inner
888.2.o.a.517.65 yes 76 8.5 even 2 inner
888.2.o.a.517.66 yes 76 37.36 even 2 inner
3552.2.o.a.2737.3 76 8.3 odd 2
3552.2.o.a.2737.4 76 148.147 odd 2
3552.2.o.a.2737.25 76 296.147 odd 2
3552.2.o.a.2737.26 76 4.3 odd 2