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Properties

Label 888.2.r.e.43.11
Level $888$
Weight $2$
Character 888.43
Analytic conductor $7.091$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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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(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 43.11
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.11

$q$-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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −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 −0.747644 − 0.664100i \(-0.768815\pi\)
0.664100 + 0.747644i \(0.268815\pi\)
\(6\) −1.18943 + 0.765018i −0.485583 + 0.312317i
\(7\) − 1.09998i − 0.415754i −0.978155 − 0.207877i \(-0.933345\pi\)
0.978155 − 0.207877i \(-0.0666553\pi\)
\(8\) 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 \(0.184114\pi\)
−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 −0.854564 − 0.519347i \(-0.826175\pi\)
0.519347 + 0.854564i \(0.326175\pi\)
\(14\) −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 0.898368 − 0.439245i \(-0.144754\pi\)
−0.439245 + 0.898368i \(0.644754\pi\)
\(18\) 0.765018 + 1.18943i 0.180317 + 0.280352i
\(19\) 1.92753 + 1.92753i 0.442206 + 0.442206i 0.892753 − 0.450547i \(-0.148771\pi\)
−0.450547 + 0.892753i \(0.648771\pi\)
\(20\) −0.185011 − 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 −0.940343 − 0.340228i \(-0.889496\pi\)
0.340228 + 0.940343i \(0.389496\pi\)
\(24\) −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 −0.995840 − 0.0911212i \(-0.970955\pi\)
−0.0911212 − 0.995840i \(-0.529045\pi\)
\(30\) 0.0792839 − 0.365108i 0.0144752 − 0.0666592i
\(31\) 5.38406 + 5.38406i 0.967007 + 0.967007i 0.999473 − 0.0324662i \(-0.0103361\pi\)
−0.0324662 + 0.999473i \(0.510336\pi\)
\(32\) −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 \(-0.767579\pi\)
0.745060 − 0.666997i \(-0.232421\pi\)
\(42\) 0.841505 + 1.30835i 0.129847 + 0.201883i
\(43\) 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 \(-0.942767\pi\)
0.983879 − 0.178836i \(-0.0572331\pi\)
\(48\) −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 \(0.169371\pi\)
−0.861747 + 0.507339i \(0.830629\pi\)
\(54\) 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 0.924522 − 0.381129i \(-0.124465\pi\)
−0.381129 + 0.924522i \(0.624465\pi\)
\(60\) −0.494924 + 0.185011i −0.0638944 + 0.0238848i
\(61\) 3.57366 + 3.57366i 0.457561 + 0.457561i 0.897854 − 0.440293i \(-0.145126\pi\)
−0.440293 + 0.897854i \(0.645126\pi\)
\(62\) 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 \(0.219795\pi\)
−0.770924 + 0.636927i \(0.780205\pi\)
\(68\) −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 \(0.0646771\pi\)
−0.979428 + 0.201794i \(0.935323\pi\)
\(72\) −2.79919 + 0.405610i −0.329888 + 0.0478016i
\(73\) − 12.5188i − 1.46521i −0.680654 − 0.732605i \(-0.738304\pi\)
0.680654 − 0.732605i \(-0.261696\pi\)
\(74\) 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 0.0568366 − 0.998383i \(-0.481899\pi\)
0.998383 − 0.0568366i \(-0.0181014\pi\)
\(80\) 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 −0.510655 − 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
\(92\) −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 −0.971799 − 0.235809i \(-0.924226\pi\)
−0.235809 − 0.971799i \(-0.575774\pi\)
\(98\) −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.43.11 ✓ 72
8.3 odd 2 inner 888.2.r.e.43.30 yes 72
37.31 odd 4 inner 888.2.r.e.475.30 yes 72
296.179 even 4 inner 888.2.r.e.475.11 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.11 ✓ 72 1.1 even 1 trivial
888.2.r.e.43.30 yes 72 8.3 odd 2 inner
888.2.r.e.475.11 yes 72 296.179 even 4 inner
888.2.r.e.475.30 yes 72 37.31 odd 4 inner