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 517.73
Character \(\chi\) \(=\) 888.517
Dual form 888.2.o.a.517.74

$q$-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+(1.39994 - 0.200399i) q^{2} -1.00000i q^{3} +(1.91968 - 0.561094i) q^{4} -2.44038 q^{5} +(-0.200399 - 1.39994i) q^{6} +4.61327 q^{7} +(2.57500 - 1.17020i) q^{8} -1.00000 q^{9} +(-3.41639 + 0.489048i) q^{10} -5.06341i q^{11} +(-0.561094 - 1.91968i) q^{12} -0.712581 q^{13} +(6.45831 - 0.924493i) q^{14} +2.44038i q^{15} +(3.37035 - 2.15424i) q^{16} +7.27568i q^{17} +(-1.39994 + 0.200399i) q^{18} -0.0331828 q^{19} +(-4.68474 + 1.36928i) q^{20} -4.61327i q^{21} +(-1.01470 - 7.08848i) q^{22} -4.39595i q^{23} +(-1.17020 - 2.57500i) q^{24} +0.955437 q^{25} +(-0.997572 + 0.142800i) q^{26} +1.00000i q^{27} +(8.85600 - 2.58847i) q^{28} -0.282192 q^{29} +(0.489048 + 3.41639i) q^{30} -7.20229i q^{31} +(4.28659 - 3.69123i) q^{32} -5.06341 q^{33} +(1.45804 + 10.1855i) q^{34} -11.2581 q^{35} +(-1.91968 + 0.561094i) q^{36} +(6.04784 + 0.650853i) q^{37} +(-0.0464541 + 0.00664980i) q^{38} +0.712581i q^{39} +(-6.28397 + 2.85573i) q^{40} -10.2375 q^{41} +(-0.924493 - 6.45831i) q^{42} +5.99710 q^{43} +(-2.84104 - 9.72012i) q^{44} +2.44038 q^{45} +(-0.880944 - 6.15408i) q^{46} +8.51773 q^{47} +(-2.15424 - 3.37035i) q^{48} +14.2822 q^{49} +(1.33756 - 0.191468i) q^{50} +7.27568 q^{51} +(-1.36793 + 0.399825i) q^{52} +3.34248i q^{53} +(0.200399 + 1.39994i) q^{54} +12.3566i q^{55} +(11.8792 - 5.39845i) q^{56} +0.0331828i q^{57} +(-0.395053 + 0.0565510i) q^{58} -2.99492 q^{59} +(1.36928 + 4.68474i) q^{60} -10.1290 q^{61} +(-1.44333 - 10.0828i) q^{62} -4.61327 q^{63} +(5.26126 - 6.02654i) q^{64} +1.73897 q^{65} +(-7.08848 + 1.01470i) q^{66} +12.4307i q^{67} +(4.08234 + 13.9670i) q^{68} -4.39595 q^{69} +(-15.7607 + 2.25611i) q^{70} -12.9471 q^{71} +(-2.57500 + 1.17020i) q^{72} +5.32731 q^{73} +(8.59706 - 0.300823i) q^{74} -0.955437i q^{75} +(-0.0637005 + 0.0186187i) q^{76} -23.3588i q^{77} +(0.142800 + 0.997572i) q^{78} +1.15117i q^{79} +(-8.22492 + 5.25716i) q^{80} +1.00000 q^{81} +(-14.3319 + 2.05159i) q^{82} +3.97545i q^{83} +(-2.58847 - 8.85600i) q^{84} -17.7554i q^{85} +(8.39560 - 1.20181i) q^{86} +0.282192i q^{87} +(-5.92520 - 13.0383i) q^{88} +4.16072i q^{89} +(3.41639 - 0.489048i) q^{90} -3.28733 q^{91} +(-2.46654 - 8.43883i) q^{92} -7.20229 q^{93} +(11.9243 - 1.70694i) q^{94} +0.0809786 q^{95} +(-3.69123 - 4.28659i) q^{96} +13.9820i q^{97} +(19.9943 - 2.86214i) q^{98} +5.06341i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.39994 − 0.200399i 0.989909 − 0.141703i
\(3\) − 1.00000i − 0.577350i
\(4\) 1.91968 − 0.561094i 0.959840 − 0.280547i
\(5\) −2.44038 −1.09137 −0.545685 − 0.837991i \(-0.683730\pi\)
−0.545685 + 0.837991i \(0.683730\pi\)
\(6\) −0.200399 − 1.39994i −0.0818124 − 0.571524i
\(7\) 4.61327 1.74365 0.871826 − 0.489817i \(-0.162936\pi\)
0.871826 + 0.489817i \(0.162936\pi\)
\(8\) 2.57500 − 1.17020i 0.910400 − 0.413728i
\(9\) −1.00000 −0.333333
\(10\) −3.41639 + 0.489048i −1.08036 + 0.154651i
\(11\) − 5.06341i − 1.52667i −0.646000 − 0.763337i \(-0.723560\pi\)
0.646000 − 0.763337i \(-0.276440\pi\)
\(12\) −0.561094 − 1.91968i −0.161974 − 0.554164i
\(13\) −0.712581 −0.197634 −0.0988172 − 0.995106i \(-0.531506\pi\)
−0.0988172 + 0.995106i \(0.531506\pi\)
\(14\) 6.45831 − 0.924493i 1.72606 − 0.247081i
\(15\) 2.44038i 0.630102i
\(16\) 3.37035 − 2.15424i 0.842587 − 0.538560i
\(17\) 7.27568i 1.76461i 0.470678 + 0.882305i \(0.344009\pi\)
−0.470678 + 0.882305i \(0.655991\pi\)
\(18\) −1.39994 + 0.200399i −0.329970 + 0.0472344i
\(19\) −0.0331828 −0.00761267 −0.00380633 − 0.999993i \(-0.501212\pi\)
−0.00380633 + 0.999993i \(0.501212\pi\)
\(20\) −4.68474 + 1.36928i −1.04754 + 0.306180i
\(21\) − 4.61327i − 1.00670i
\(22\) −1.01470 − 7.08848i −0.216335 − 1.51127i
\(23\) − 4.39595i − 0.916620i −0.888792 − 0.458310i \(-0.848455\pi\)
0.888792 − 0.458310i \(-0.151545\pi\)
\(24\) −1.17020 − 2.57500i −0.238866 − 0.525620i
\(25\) 0.955437 0.191087
\(26\) −0.997572 + 0.142800i −0.195640 + 0.0280054i
\(27\) 1.00000i 0.192450i
\(28\) 8.85600 − 2.58847i 1.67363 − 0.489176i
\(29\) −0.282192 −0.0524018 −0.0262009 − 0.999657i \(-0.508341\pi\)
−0.0262009 + 0.999657i \(0.508341\pi\)
\(30\) 0.489048 + 3.41639i 0.0892876 + 0.623744i
\(31\) − 7.20229i − 1.29357i −0.762673 − 0.646784i \(-0.776113\pi\)
0.762673 − 0.646784i \(-0.223887\pi\)
\(32\) 4.28659 − 3.69123i 0.757769 − 0.652523i
\(33\) −5.06341 −0.881426
\(34\) 1.45804 + 10.1855i 0.250051 + 1.74680i
\(35\) −11.2581 −1.90297
\(36\) −1.91968 + 0.561094i −0.319947 + 0.0935156i
\(37\) 6.04784 + 0.650853i 0.994259 + 0.107000i
\(38\) −0.0464541 + 0.00664980i −0.00753585 + 0.00107874i
\(39\) 0.712581i 0.114104i
\(40\) −6.28397 + 2.85573i −0.993583 + 0.451531i
\(41\) −10.2375 −1.59883 −0.799416 − 0.600778i \(-0.794858\pi\)
−0.799416 + 0.600778i \(0.794858\pi\)
\(42\) −0.924493 − 6.45831i −0.142652 − 0.996539i
\(43\) 5.99710 0.914549 0.457274 − 0.889326i \(-0.348826\pi\)
0.457274 + 0.889326i \(0.348826\pi\)
\(44\) −2.84104 − 9.72012i −0.428304 − 1.46536i
\(45\) 2.44038 0.363790
\(46\) −0.880944 − 6.15408i −0.129888 − 0.907370i
\(47\) 8.51773 1.24244 0.621219 − 0.783637i \(-0.286638\pi\)
0.621219 + 0.783637i \(0.286638\pi\)
\(48\) −2.15424 − 3.37035i −0.310938 − 0.486468i
\(49\) 14.2822 2.04032
\(50\) 1.33756 − 0.191468i 0.189159 − 0.0270777i
\(51\) 7.27568 1.01880
\(52\) −1.36793 + 0.399825i −0.189697 + 0.0554457i
\(53\) 3.34248i 0.459125i 0.973294 + 0.229562i \(0.0737295\pi\)
−0.973294 + 0.229562i \(0.926271\pi\)
\(54\) 0.200399 + 1.39994i 0.0272708 + 0.190508i
\(55\) 12.3566i 1.66617i
\(56\) 11.8792 − 5.39845i 1.58742 − 0.721398i
\(57\) 0.0331828i 0.00439518i
\(58\) −0.395053 + 0.0565510i −0.0518730 + 0.00742550i
\(59\) −2.99492 −0.389906 −0.194953 − 0.980813i \(-0.562455\pi\)
−0.194953 + 0.980813i \(0.562455\pi\)
\(60\) 1.36928 + 4.68474i 0.176773 + 0.604798i
\(61\) −10.1290 −1.29689 −0.648445 − 0.761262i \(-0.724580\pi\)
−0.648445 + 0.761262i \(0.724580\pi\)
\(62\) −1.44333 − 10.0828i −0.183303 − 1.28052i
\(63\) −4.61327 −0.581217
\(64\) 5.26126 − 6.02654i 0.657658 − 0.753317i
\(65\) 1.73897 0.215692
\(66\) −7.08848 + 1.01470i −0.872531 + 0.124901i
\(67\) 12.4307i 1.51865i 0.650710 + 0.759326i \(0.274471\pi\)
−0.650710 + 0.759326i \(0.725529\pi\)
\(68\) 4.08234 + 13.9670i 0.495056 + 1.69374i
\(69\) −4.39595 −0.529211
\(70\) −15.7607 + 2.25611i −1.88377 + 0.269657i
\(71\) −12.9471 −1.53654 −0.768268 − 0.640128i \(-0.778881\pi\)
−0.768268 + 0.640128i \(0.778881\pi\)
\(72\) −2.57500 + 1.17020i −0.303467 + 0.137909i
\(73\) 5.32731 0.623514 0.311757 − 0.950162i \(-0.399083\pi\)
0.311757 + 0.950162i \(0.399083\pi\)
\(74\) 8.59706 − 0.300823i 0.999388 − 0.0349699i
\(75\) − 0.955437i − 0.110324i
\(76\) −0.0637005 + 0.0186187i −0.00730694 + 0.00213571i
\(77\) − 23.3588i − 2.66199i
\(78\) 0.142800 + 0.997572i 0.0161689 + 0.112953i
\(79\) 1.15117i 0.129517i 0.997901 + 0.0647584i \(0.0206277\pi\)
−0.997901 + 0.0647584i \(0.979372\pi\)
\(80\) −8.22492 + 5.25716i −0.919574 + 0.587768i
\(81\) 1.00000 0.111111
\(82\) −14.3319 + 2.05159i −1.58270 + 0.226560i
\(83\) 3.97545i 0.436363i 0.975908 + 0.218181i \(0.0700125\pi\)
−0.975908 + 0.218181i \(0.929988\pi\)
\(84\) −2.58847 − 8.85600i −0.282426 − 0.966269i
\(85\) − 17.7554i − 1.92584i
\(86\) 8.39560 − 1.20181i 0.905320 − 0.129595i
\(87\) 0.282192i 0.0302542i
\(88\) −5.92520 − 13.0383i −0.631628 − 1.38988i
\(89\) 4.16072i 0.441036i 0.975383 + 0.220518i \(0.0707748\pi\)
−0.975383 + 0.220518i \(0.929225\pi\)
\(90\) 3.41639 − 0.489048i 0.360119 − 0.0515502i
\(91\) −3.28733 −0.344605
\(92\) −2.46654 − 8.43883i −0.257155 − 0.879809i
\(93\) −7.20229 −0.746842
\(94\) 11.9243 − 1.70694i 1.22990 − 0.176058i
\(95\) 0.0809786 0.00830823
\(96\) −3.69123 − 4.28659i −0.376734 − 0.437498i
\(97\) 13.9820i 1.41966i 0.704373 + 0.709830i \(0.251228\pi\)
−0.704373 + 0.709830i \(0.748772\pi\)
\(98\) 19.9943 − 2.86214i 2.01973 − 0.289120i
\(99\) 5.06341i 0.508891i
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.517.73 yes 76
4.3 odd 2 3552.2.o.a.2737.76 76
8.3 odd 2 3552.2.o.a.2737.59 76
8.5 even 2 inner 888.2.o.a.517.3 ✓ 76
37.36 even 2 inner 888.2.o.a.517.4 yes 76
148.147 odd 2 3552.2.o.a.2737.60 76
296.147 odd 2 3552.2.o.a.2737.75 76
296.221 even 2 inner 888.2.o.a.517.74 yes 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.3 ✓ 76 8.5 even 2 inner
888.2.o.a.517.4 yes 76 37.36 even 2 inner
888.2.o.a.517.73 yes 76 1.1 even 1 trivial
888.2.o.a.517.74 yes 76 296.221 even 2 inner
3552.2.o.a.2737.59 76 8.3 odd 2
3552.2.o.a.2737.60 76 148.147 odd 2
3552.2.o.a.2737.75 76 296.147 odd 2
3552.2.o.a.2737.76 76 4.3 odd 2