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 517.3
Character \(\chi\) \(=\) 888.517
Dual form 888.2.o.a.517.4

$q$-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} +(8.33620 + 2.12314i) 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 \(-0.316270\pi\)
0.545685 + 0.837991i \(0.316270\pi\)
\(6\) 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 \(0.276440\pi\)
−0.646000 + 0.763337i \(0.723560\pi\)
\(12\) −0.561094 + 1.91968i −0.161974 + 0.554164i
\(13\) 0.712581 0.197634 0.0988172 − 0.995106i \(-0.468494\pi\)
0.0988172 + 0.995106i \(0.468494\pi\)
\(14\) −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 \(-0.498788\pi\)
0.00380633 + 0.999993i \(0.498788\pi\)
\(20\) 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 \(-0.491659\pi\)
0.0262009 + 0.999657i \(0.491659\pi\)
\(30\) 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 \(-0.651174\pi\)
−0.457274 + 0.889326i \(0.651174\pi\)
\(44\) −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 \(-0.926271\pi\)
0.973294 − 0.229562i \(-0.0737295\pi\)
\(54\) −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 \(-0.437545\pi\)
0.194953 + 0.980813i \(0.437545\pi\)
\(60\) −1.36928 + 4.68474i −0.176773 + 0.604798i
\(61\) 10.1290 1.29689 0.648445 − 0.761262i \(-0.275420\pi\)
0.648445 + 0.761262i \(0.275420\pi\)
\(62\) −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 \(-0.725529\pi\)
0.650710 − 0.759326i \(-0.274471\pi\)
\(68\) −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 \(0.399083\pi\)
\(74\) 8.33620 + 2.12314i 0.969064 + 0.246810i
\(75\) 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 \(-0.929988\pi\)
0.975908 − 0.218181i \(-0.0700125\pi\)
\(84\) −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.517.3 ✓ 76
4.3 odd 2 3552.2.o.a.2737.59 76
8.3 odd 2 3552.2.o.a.2737.76 76
8.5 even 2 inner 888.2.o.a.517.73 yes 76
37.36 even 2 inner 888.2.o.a.517.74 yes 76
148.147 odd 2 3552.2.o.a.2737.75 76
296.147 odd 2 3552.2.o.a.2737.60 76
296.221 even 2 inner 888.2.o.a.517.4 yes 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.3 ✓ 76 1.1 even 1 trivial
888.2.o.a.517.4 yes 76 296.221 even 2 inner
888.2.o.a.517.73 yes 76 8.5 even 2 inner
888.2.o.a.517.74 yes 76 37.36 even 2 inner
3552.2.o.a.2737.59 76 4.3 odd 2
3552.2.o.a.2737.60 76 296.147 odd 2
3552.2.o.a.2737.75 76 148.147 odd 2
3552.2.o.a.2737.76 76 8.3 odd 2