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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 475.11
Character \(\chi\) \(=\) 888.475
Dual form 888.2.r.e.43.11

$q$-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+(-0.765018 + 1.18943i) q^{2} +1.00000i q^{3} +(-0.829494 - 1.81987i) q^{4} +(-0.186808 - 0.186808i) q^{5} +(-1.18943 - 0.765018i) q^{6} +1.09998i q^{7} +(2.79919 + 0.405610i) 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} +(-1.30835 - 0.841505i) q^{14} +(0.186808 - 0.186808i) q^{15} +(-2.62388 + 3.01915i) q^{16} +(1.89301 - 1.89301i) q^{17} +(0.765018 - 1.18943i) q^{18} +(1.92753 - 1.92753i) q^{19} +(-0.185011 + 0.494924i) q^{20} -1.09998 q^{21} +(4.31331 + 2.77424i) q^{22} +(-2.87805 - 2.87805i) q^{23} +(-0.405610 + 2.79919i) 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} +(-1.58375 - 5.43063i) 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} +(0.841505 - 1.30835i) 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} +(5.86414 + 3.77170i) q^{50} +(1.89301 + 1.89301i) q^{51} +(-1.19701 + 3.20213i) q^{52} -7.38697i q^{53} +(1.18943 + 0.765018i) q^{54} +(-0.677435 + 0.677435i) q^{55} +(-0.446163 + 3.07906i) q^{56} +(1.92753 + 1.92753i) q^{57} +(-2.48429 - 11.4403i) q^{58} +(4.17388 - 4.17388i) q^{59} +(-0.494924 - 0.185011i) 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} +(-2.77424 + 4.31331i) q^{66} -10.4269i q^{67} +(-5.01529 - 1.87480i) q^{68} +(2.87805 - 2.87805i) q^{69} +(0.0872108 + 0.401611i) q^{70} -3.40069i q^{71} +(-2.79919 - 0.405610i) q^{72} +12.5188i q^{73} +(2.55066 + 8.21548i) q^{74} +4.93021 q^{75} +(-5.10673 - 1.90899i) q^{76} +3.98893 q^{77} +(0.512963 + 2.36223i) q^{78} +(9.37900 + 9.37900i) q^{79} +(1.05416 - 0.0738396i) q^{80} +1.00000 q^{81} +(-10.1598 - 6.53458i) 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} +(1.47089 - 10.1509i) q^{88} +(3.29369 + 3.29369i) q^{89} +(-0.365108 + 0.0792839i) q^{90} +(1.32948 - 1.32948i) q^{91} +(-2.85036 + 7.62501i) q^{92} +(5.38406 + 5.38406i) q^{93} +(-2.91657 - 1.87588i) q^{94} -0.720157 q^{95} +(5.43063 - 1.58375i) q^{96} +(-11.8936 + 11.8936i) q^{97} +(-4.42949 + 6.88686i) 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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.765018 + 1.18943i −0.540950 + 0.841055i
\(3\) 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 \(-0.268815\pi\)
−0.747644 + 0.664100i \(0.768815\pi\)
\(6\) −1.18943 − 0.765018i −0.485583 − 0.312317i
\(7\) 1.09998i 0.415754i 0.978155 + 0.207877i \(0.0666553\pi\)
−0.978155 + 0.207877i \(0.933345\pi\)
\(8\) 2.79919 + 0.405610i 0.989664 + 0.143405i
\(9\) −1.00000 −0.333333
\(10\) 0.365108 − 0.0792839i 0.115457 − 0.0250718i
\(11\) − 3.62637i − 1.09339i −0.837332 − 0.546695i \(-0.815886\pi\)
0.837332 − 0.546695i \(-0.184114\pi\)
\(12\) 1.81987 − 0.829494i 0.525352 − 0.239454i
\(13\) −1.20864 − 1.20864i −0.335217 − 0.335217i 0.519347 − 0.854564i \(-0.326175\pi\)
−0.854564 + 0.519347i \(0.826175\pi\)
\(14\) −1.30835 − 0.841505i −0.349672 − 0.224902i
\(15\) 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 −0.439245 − 0.898368i \(-0.644754\pi\)
0.898368 + 0.439245i \(0.144754\pi\)
\(18\) 0.765018 − 1.18943i 0.180317 − 0.280352i
\(19\) 1.92753 − 1.92753i 0.442206 − 0.442206i −0.450547 − 0.892753i \(-0.648771\pi\)
0.892753 + 0.450547i \(0.148771\pi\)
\(20\) −0.185011 + 0.494924i −0.0413698 + 0.110668i
\(21\) −1.09998 −0.240035
\(22\) 4.31331 + 2.77424i 0.919601 + 0.591469i
\(23\) −2.87805 − 2.87805i −0.600115 − 0.600115i 0.340228 − 0.940343i \(-0.389496\pi\)
−0.940343 + 0.340228i \(0.889496\pi\)
\(24\) −0.405610 + 2.79919i −0.0827948 + 0.571383i
\(25\) − 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 −0.0911212 + 0.995840i \(0.529045\pi\)
−0.995840 + 0.0911212i \(0.970955\pi\)
\(30\) 0.0792839 + 0.365108i 0.0144752 + 0.0666592i
\(31\) 5.38406 − 5.38406i 0.967007 − 0.967007i −0.0324662 − 0.999473i \(-0.510336\pi\)
0.999473 + 0.0324662i \(0.0103361\pi\)
\(32\) −1.58375 − 5.43063i −0.279971 − 0.960008i
\(33\) 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 \(0.232421\pi\)
−0.745060 + 0.666997i \(0.767579\pi\)
\(42\) 0.841505 − 1.30835i 0.129847 − 0.201883i
\(43\) 4.19856 − 4.19856i 0.640275 − 0.640275i −0.310348 − 0.950623i \(-0.600446\pi\)
0.950623 + 0.310348i \(0.100446\pi\)
\(44\) −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
\(50\) 5.86414 + 3.77170i 0.829315 + 0.533399i
\(51\) 1.89301 + 1.89301i 0.265075 + 0.265075i
\(52\) −1.19701 + 3.20213i −0.165996 + 0.444056i
\(53\) − 7.38697i − 1.01468i −0.861747 − 0.507339i \(-0.830629\pi\)
0.861747 − 0.507339i \(-0.169371\pi\)
\(54\) 1.18943 + 0.765018i 0.161861 + 0.104106i
\(55\) −0.677435 + 0.677435i −0.0913454 + 0.0913454i
\(56\) −0.446163 + 3.07906i −0.0596211 + 0.411456i
\(57\) 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 −0.381129 − 0.924522i \(-0.624465\pi\)
0.924522 + 0.381129i \(0.124465\pi\)
\(60\) −0.494924 − 0.185011i −0.0638944 − 0.0238848i
\(61\) 3.57366 − 3.57366i 0.457561 − 0.457561i −0.440293 − 0.897854i \(-0.645126\pi\)
0.897854 + 0.440293i \(0.145126\pi\)
\(62\) 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
\(66\) −2.77424 + 4.31331i −0.341485 + 0.530932i
\(67\) − 10.4269i − 1.27385i −0.770924 − 0.636927i \(-0.780205\pi\)
0.770924 − 0.636927i \(-0.219795\pi\)
\(68\) −5.01529 − 1.87480i −0.608193 − 0.227353i
\(69\) 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 \(-0.0646771\pi\)
\(72\) −2.79919 − 0.405610i −0.329888 − 0.0478016i
\(73\) 12.5188i 1.46521i 0.680654 + 0.732605i \(0.261696\pi\)
−0.680654 + 0.732605i \(0.738304\pi\)
\(74\) 2.55066 + 8.21548i 0.296508 + 0.955030i
\(75\) 4.93021 0.569291
\(76\) −5.10673 − 1.90899i −0.585783 − 0.218976i
\(77\) 3.98893 0.454581
\(78\) 0.512963 + 2.36223i 0.0580816 + 0.267470i
\(79\) 9.37900 + 9.37900i 1.05522 + 1.05522i 0.998383 + 0.0568366i \(0.0181014\pi\)
0.0568366 + 0.998383i \(0.481899\pi\)
\(80\) 1.05416 − 0.0738396i 0.117859 − 0.00825552i
\(81\) 1.00000 0.111111
\(82\) −10.1598 − 6.53458i −1.12196 − 0.721623i
\(83\) −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
\(88\) 1.47089 − 10.1509i 0.156797 − 1.08209i
\(89\) 3.29369 + 3.29369i 0.349130 + 0.349130i 0.859786 − 0.510655i \(-0.170597\pi\)
−0.510655 + 0.859786i \(0.670597\pi\)
\(90\) −0.365108 + 0.0792839i −0.0384857 + 0.00835726i
\(91\) 1.32948 − 1.32948i 0.139368 − 0.139368i
\(92\) −2.85036 + 7.62501i −0.297171 + 0.794963i
\(93\) 5.38406 + 5.38406i 0.558302 + 0.558302i
\(94\) −2.91657 − 1.87588i −0.300821 − 0.193482i
\(95\) −0.720157 −0.0738866
\(96\) 5.43063 − 1.58375i 0.554261 − 0.161641i
\(97\) −11.8936 + 11.8936i −1.20761 + 1.20761i −0.235809 + 0.971799i \(0.575774\pi\)
−0.971799 + 0.235809i \(0.924226\pi\)
\(98\) −4.42949 + 6.88686i −0.447446 + 0.695678i
\(99\) 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.475.11 yes 72
8.3 odd 2 inner 888.2.r.e.475.30 yes 72
37.6 odd 4 inner 888.2.r.e.43.30 yes 72
296.43 even 4 inner 888.2.r.e.43.11 ✓ 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.11 ✓ 72 296.43 even 4 inner
888.2.r.e.43.30 yes 72 37.6 odd 4 inner
888.2.r.e.475.11 yes 72 1.1 even 1 trivial
888.2.r.e.475.30 yes 72 8.3 odd 2 inner