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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(43,888)] 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("888.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,2,0,0,0,-2,0,2,-72,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.30
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.30

$q$-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+(1.18943 + 0.765018i) q^{2} -1.00000i q^{3} +(0.829494 + 1.81987i) q^{4} +(0.186808 - 0.186808i) q^{5} +(0.765018 - 1.18943i) q^{6} +1.09998i q^{7} +(-0.405610 + 2.79919i) q^{8} -1.00000 q^{9} +(0.365108 - 0.0792839i) q^{10} +3.62637i q^{11} +(1.81987 - 0.829494i) q^{12} +(1.20864 - 1.20864i) q^{13} +(-0.841505 + 1.30835i) q^{14} +(-0.186808 - 0.186808i) q^{15} +(-2.62388 + 3.01915i) q^{16} +(1.89301 + 1.89301i) q^{17} +(-1.18943 - 0.765018i) q^{18} +(1.92753 + 1.92753i) q^{19} +(0.494924 + 0.185011i) q^{20} +1.09998 q^{21} +(-2.77424 + 4.31331i) q^{22} +(2.87805 - 2.87805i) q^{23} +(2.79919 + 0.405610i) q^{24} +4.93021i q^{25} +(2.36223 - 0.512963i) q^{26} +1.00000i q^{27} +(-2.00183 + 0.912427i) q^{28} +(5.85346 + 5.85346i) q^{29} +(-0.0792839 - 0.365108i) q^{30} +(-5.38406 - 5.38406i) q^{31} +(-5.43063 + 1.58375i) q^{32} +3.62637 q^{33} +(0.803419 + 3.69980i) q^{34} +(0.205486 + 0.205486i) q^{35} +(-0.829494 - 1.81987i) q^{36} +(-3.91023 - 4.65941i) q^{37} +(0.818069 + 3.76726i) q^{38} +(-1.20864 - 1.20864i) q^{39} +(0.447141 + 0.598684i) q^{40} -8.54173i q^{41} +(1.30835 + 0.841505i) q^{42} +(4.19856 + 4.19856i) q^{43} +(-6.59953 + 3.00805i) q^{44} +(-0.186808 + 0.186808i) q^{45} +(5.62501 - 1.22148i) q^{46} +2.45207i q^{47} +(3.01915 + 2.62388i) q^{48} +5.79004 q^{49} +(-3.77170 + 5.86414i) q^{50} +(1.89301 - 1.89301i) q^{51} +(3.20213 + 1.19701i) q^{52} -7.38697i q^{53} +(-0.765018 + 1.18943i) q^{54} +(0.677435 + 0.677435i) q^{55} +(-3.07906 - 0.446163i) q^{56} +(1.92753 - 1.92753i) q^{57} +(2.48429 + 11.4403i) q^{58} +(4.17388 + 4.17388i) q^{59} +(0.185011 - 0.494924i) q^{60} +(-3.57366 - 3.57366i) q^{61} +(-2.28507 - 10.5229i) q^{62} -1.09998i q^{63} +(-7.67096 - 2.27076i) q^{64} -0.451568i q^{65} +(4.31331 + 2.77424i) q^{66} +10.4269i q^{67} +(-1.87480 + 5.01529i) q^{68} +(-2.87805 - 2.87805i) q^{69} +(0.0872108 + 0.401611i) q^{70} -3.40069i q^{71} +(0.405610 - 2.79919i) q^{72} -12.5188i q^{73} +(-1.08641 - 8.53345i) q^{74} +4.93021 q^{75} +(-1.90899 + 5.10673i) q^{76} -3.98893 q^{77} +(-0.512963 - 2.36223i) q^{78} +(-9.37900 + 9.37900i) q^{79} +(0.0738396 + 1.05416i) q^{80} +1.00000 q^{81} +(6.53458 - 10.1598i) q^{82} -6.91415 q^{83} +(0.912427 + 2.00183i) q^{84} +0.707261 q^{85} +(1.78193 + 8.20588i) q^{86} +(5.85346 - 5.85346i) q^{87} +(-10.1509 - 1.47089i) q^{88} +(3.29369 - 3.29369i) q^{89} +(-0.365108 + 0.0792839i) q^{90} +(1.32948 + 1.32948i) q^{91} +(7.62501 + 2.85036i) q^{92} +(-5.38406 + 5.38406i) q^{93} +(-1.87588 + 2.91657i) q^{94} +0.720157 q^{95} +(1.58375 + 5.43063i) q^{96} +(-11.8936 - 11.8936i) q^{97} +(6.88686 + 4.42949i) q^{98} -3.62637i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 2 q^{2} - 2 q^{6} + 2 q^{8} - 72 q^{9} + 8 q^{10} - 8 q^{12} + 20 q^{14} + 12 q^{16} + 12 q^{17} - 2 q^{18} + 20 q^{19} - 22 q^{20} - 16 q^{22} + 2 q^{24} - 32 q^{26} - 8 q^{32} - 16 q^{33} + 8 q^{34}+ \cdots - 110 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.18943 + 0.765018i 0.841055 + 0.540950i
\(3\) − 1.00000i − 0.577350i
\(4\) 0.829494 + 1.81987i 0.414747 + 0.909937i
\(5\) 0.186808 − 0.186808i 0.0835432 − 0.0835432i −0.664100 − 0.747644i \(-0.731185\pi\)
0.747644 + 0.664100i \(0.231185\pi\)
\(6\) 0.765018 − 1.18943i 0.312317 − 0.485583i
\(7\) 1.09998i 0.415754i 0.978155 + 0.207877i \(0.0666553\pi\)
−0.978155 + 0.207877i \(0.933345\pi\)
\(8\) −0.405610 + 2.79919i −0.143405 + 0.989664i
\(9\) −1.00000 −0.333333
\(10\) 0.365108 − 0.0792839i 0.115457 − 0.0250718i
\(11\) 3.62637i 1.09339i 0.837332 + 0.546695i \(0.184114\pi\)
−0.837332 + 0.546695i \(0.815886\pi\)
\(12\) 1.81987 − 0.829494i 0.525352 − 0.239454i
\(13\) 1.20864 − 1.20864i 0.335217 − 0.335217i −0.519347 − 0.854564i \(-0.673825\pi\)
0.854564 + 0.519347i \(0.173825\pi\)
\(14\) −0.841505 + 1.30835i −0.224902 + 0.349672i
\(15\) −0.186808 − 0.186808i −0.0482337 − 0.0482337i
\(16\) −2.62388 + 3.01915i −0.655970 + 0.754787i
\(17\) 1.89301 + 1.89301i 0.459123 + 0.459123i 0.898368 − 0.439245i \(-0.144754\pi\)
−0.439245 + 0.898368i \(0.644754\pi\)
\(18\) −1.18943 − 0.765018i −0.280352 − 0.180317i
\(19\) 1.92753 + 1.92753i 0.442206 + 0.442206i 0.892753 − 0.450547i \(-0.148771\pi\)
−0.450547 + 0.892753i \(0.648771\pi\)
\(20\) 0.494924 + 0.185011i 0.110668 + 0.0413698i
\(21\) 1.09998 0.240035
\(22\) −2.77424 + 4.31331i −0.591469 + 0.919601i
\(23\) 2.87805 − 2.87805i 0.600115 − 0.600115i −0.340228 − 0.940343i \(-0.610504\pi\)
0.940343 + 0.340228i \(0.110504\pi\)
\(24\) 2.79919 + 0.405610i 0.571383 + 0.0827948i
\(25\) 4.93021i 0.986041i
\(26\) 2.36223 − 0.512963i 0.463271 − 0.100600i
\(27\) 1.00000i 0.192450i
\(28\) −2.00183 + 0.912427i −0.378309 + 0.172433i
\(29\) 5.85346 + 5.85346i 1.08696 + 1.08696i 0.995840 + 0.0911212i \(0.0290451\pi\)
0.0911212 + 0.995840i \(0.470955\pi\)
\(30\) −0.0792839 − 0.365108i −0.0144752 − 0.0666592i
\(31\) −5.38406 − 5.38406i −0.967007 − 0.967007i 0.0324662 − 0.999473i \(-0.489664\pi\)
−0.999473 + 0.0324662i \(0.989664\pi\)
\(32\) −5.43063 + 1.58375i −0.960008 + 0.279971i
\(33\) 3.62637 0.631269
\(34\) 0.803419 + 3.69980i 0.137785 + 0.634510i
\(35\) 0.205486 + 0.205486i 0.0347334 + 0.0347334i
\(36\) −0.829494 − 1.81987i −0.138249 − 0.303312i
\(37\) −3.91023 − 4.65941i −0.642837 − 0.766003i
\(38\) 0.818069 + 3.76726i 0.132708 + 0.611130i
\(39\) −1.20864 − 1.20864i −0.193537 − 0.193537i
\(40\) 0.447141 + 0.598684i 0.0706992 + 0.0946602i
\(41\) − 8.54173i − 1.33399i −0.745060 − 0.666997i \(-0.767579\pi\)
0.745060 − 0.666997i \(-0.232421\pi\)
\(42\) 1.30835 + 0.841505i 0.201883 + 0.129847i
\(43\) 4.19856 + 4.19856i 0.640275 + 0.640275i 0.950623 − 0.310348i \(-0.100446\pi\)
−0.310348 + 0.950623i \(0.600446\pi\)
\(44\) −6.59953 + 3.00805i −0.994916 + 0.453480i
\(45\) −0.186808 + 0.186808i −0.0278477 + 0.0278477i
\(46\) 5.62501 − 1.22148i 0.829362 − 0.180098i
\(47\) 2.45207i 0.357672i 0.983879 + 0.178836i \(0.0572331\pi\)
−0.983879 + 0.178836i \(0.942767\pi\)
\(48\) 3.01915 + 2.62388i 0.435777 + 0.378724i
\(49\) 5.79004 0.827149
\(50\) −3.77170 + 5.86414i −0.533399 + 0.829315i
\(51\) 1.89301 − 1.89301i 0.265075 − 0.265075i
\(52\) 3.20213 + 1.19701i 0.444056 + 0.165996i
\(53\) − 7.38697i − 1.01468i −0.861747 − 0.507339i \(-0.830629\pi\)
0.861747 − 0.507339i \(-0.169371\pi\)
\(54\) −0.765018 + 1.18943i −0.104106 + 0.161861i
\(55\) 0.677435 + 0.677435i 0.0913454 + 0.0913454i
\(56\) −3.07906 − 0.446163i −0.411456 − 0.0596211i
\(57\) 1.92753 − 1.92753i 0.255308 − 0.255308i
\(58\) 2.48429 + 11.4403i 0.326203 + 1.50219i
\(59\) 4.17388 + 4.17388i 0.543393 + 0.543393i 0.924522 − 0.381129i \(-0.124465\pi\)
−0.381129 + 0.924522i \(0.624465\pi\)
\(60\) 0.185011 − 0.494924i 0.0238848 − 0.0638944i
\(61\) −3.57366 − 3.57366i −0.457561 − 0.457561i 0.440293 − 0.897854i \(-0.354874\pi\)
−0.897854 + 0.440293i \(0.854874\pi\)
\(62\) −2.28507 − 10.5229i −0.290204 − 1.33641i
\(63\) − 1.09998i − 0.138585i
\(64\) −7.67096 − 2.27076i −0.958870 − 0.283845i
\(65\) − 0.451568i − 0.0560102i
\(66\) 4.31331 + 2.77424i 0.530932 + 0.341485i
\(67\) 10.4269i 1.27385i 0.770924 + 0.636927i \(0.219795\pi\)
−0.770924 + 0.636927i \(0.780205\pi\)
\(68\) −1.87480 + 5.01529i −0.227353 + 0.608193i
\(69\) −2.87805 − 2.87805i −0.346477 − 0.346477i
\(70\) 0.0872108 + 0.401611i 0.0104237 + 0.0480017i
\(71\) − 3.40069i − 0.403588i −0.979428 − 0.201794i \(-0.935323\pi\)
0.979428 − 0.201794i \(-0.0646771\pi\)
\(72\) 0.405610 − 2.79919i 0.0478016 − 0.329888i
\(73\) − 12.5188i − 1.46521i −0.680654 − 0.732605i \(-0.738304\pi\)
0.680654 − 0.732605i \(-0.261696\pi\)
\(74\) −1.08641 − 8.53345i −0.126292 − 0.991993i
\(75\) 4.93021 0.569291
\(76\) −1.90899 + 5.10673i −0.218976 + 0.585783i
\(77\) −3.98893 −0.454581
\(78\) −0.512963 − 2.36223i −0.0580816 − 0.267470i
\(79\) −9.37900 + 9.37900i −1.05522 + 1.05522i −0.0568366 + 0.998383i \(0.518101\pi\)
−0.998383 + 0.0568366i \(0.981899\pi\)
\(80\) 0.0738396 + 1.05416i 0.00825552 + 0.117859i
\(81\) 1.00000 0.111111
\(82\) 6.53458 − 10.1598i 0.721623 − 1.12196i
\(83\) −6.91415 −0.758927 −0.379463 − 0.925207i \(-0.623891\pi\)
−0.379463 + 0.925207i \(0.623891\pi\)
\(84\) 0.912427 + 2.00183i 0.0995540 + 0.218417i
\(85\) 0.707261 0.0767132
\(86\) 1.78193 + 8.20588i 0.192150 + 0.884863i
\(87\) 5.85346 − 5.85346i 0.627557 − 0.627557i
\(88\) −10.1509 − 1.47089i −1.08209 − 0.156797i
\(89\) 3.29369 − 3.29369i 0.349130 − 0.349130i −0.510655 − 0.859786i \(-0.670597\pi\)
0.859786 + 0.510655i \(0.170597\pi\)
\(90\) −0.365108 + 0.0792839i −0.0384857 + 0.00835726i
\(91\) 1.32948 + 1.32948i 0.139368 + 0.139368i
\(92\) 7.62501 + 2.85036i 0.794963 + 0.297171i
\(93\) −5.38406 + 5.38406i −0.558302 + 0.558302i
\(94\) −1.87588 + 2.91657i −0.193482 + 0.300821i
\(95\) 0.720157 0.0738866
\(96\) 1.58375 + 5.43063i 0.161641 + 0.554261i
\(97\) −11.8936 − 11.8936i −1.20761 − 1.20761i −0.971799 − 0.235809i \(-0.924226\pi\)
−0.235809 − 0.971799i \(-0.575774\pi\)
\(98\) 6.88686 + 4.42949i 0.695678 + 0.447446i
\(99\) − 3.62637i − 0.364463i
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.r.e.43.30 yes 72
8.3 odd 2 inner 888.2.r.e.43.11 ✓ 72
37.31 odd 4 inner 888.2.r.e.475.11 yes 72
296.179 even 4 inner 888.2.r.e.475.30 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.11 ✓ 72 8.3 odd 2 inner
888.2.r.e.43.30 yes 72 1.1 even 1 trivial
888.2.r.e.475.11 yes 72 37.31 odd 4 inner
888.2.r.e.475.30 yes 72 296.179 even 4 inner