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(565,888)] 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("888.565"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.bh (of order \(6\), degree \(2\), 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: \(152\)
Relative dimension: \(76\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 565.50
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.50

$q$-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+(0.582660 + 1.28861i) q^{2} +(0.866025 - 0.500000i) q^{3} +(-1.32102 + 1.50164i) q^{4} +(-3.03864 + 1.75436i) q^{5} +(1.14890 + 0.824636i) q^{6} +(-0.822280 - 1.42423i) q^{7} +(-2.70472 - 0.827325i) q^{8} +(0.500000 - 0.866025i) q^{9} +(-4.03118 - 2.89342i) q^{10} -2.11441i q^{11} +(-0.393214 + 1.96096i) q^{12} +(-1.47455 + 0.851334i) q^{13} +(1.35616 - 1.88944i) q^{14} +(-1.75436 + 3.03864i) q^{15} +(-0.509836 - 3.96738i) q^{16} +(2.13202 - 3.69276i) q^{17} +(1.40730 + 0.139705i) q^{18} +(-3.79717 + 2.19230i) q^{19} +(1.37968 - 6.88048i) q^{20} +(-1.42423 - 0.822280i) q^{21} +(2.72464 - 1.23198i) q^{22} +1.62477 q^{23} +(-2.75602 + 0.635877i) q^{24} +(3.65557 - 6.33163i) q^{25} +(-1.95620 - 1.40408i) q^{26} -1.00000i q^{27} +(3.22492 + 0.646663i) q^{28} -10.2769i q^{29} +(-4.93781 - 0.490187i) q^{30} -10.6128 q^{31} +(4.81533 - 2.96861i) q^{32} +(-1.05720 - 1.83113i) q^{33} +(6.00076 + 0.595709i) q^{34} +(4.99723 + 2.88515i) q^{35} +(0.639949 + 1.89485i) q^{36} +(-1.62283 + 5.86229i) q^{37} +(-5.03747 - 3.61570i) q^{38} +(-0.851334 + 1.47455i) q^{39} +(9.67012 - 2.23112i) q^{40} +(-3.17421 - 5.49790i) q^{41} +(0.229754 - 2.31438i) q^{42} +2.08617i q^{43} +(3.17508 + 2.79316i) q^{44} +3.50872i q^{45} +(0.946688 + 2.09369i) q^{46} +0.00681765 q^{47} +(-2.42522 - 3.18093i) q^{48} +(2.14771 - 3.71995i) q^{49} +(10.2889 + 1.02141i) q^{50} -4.26404i q^{51} +(0.669512 - 3.33887i) q^{52} +(6.76763 + 3.90729i) q^{53} +(1.28861 - 0.582660i) q^{54} +(3.70943 + 6.42493i) q^{55} +(1.04574 + 4.53244i) q^{56} +(-2.19230 + 3.79717i) q^{57} +(13.2428 - 5.98792i) q^{58} +(-3.97296 - 2.29379i) q^{59} +(-2.24541 - 6.64851i) q^{60} +(-12.6497 + 7.30331i) q^{61} +(-6.18363 - 13.6757i) q^{62} -1.64456 q^{63} +(6.63107 + 4.47537i) q^{64} +(2.98710 - 5.17380i) q^{65} +(1.74362 - 2.42925i) q^{66} +(-6.61558 + 3.81950i) q^{67} +(2.72877 + 8.07972i) q^{68} +(1.40709 - 0.812385i) q^{69} +(-0.806143 + 8.12053i) q^{70} +(5.41041 + 9.37110i) q^{71} +(-2.06885 + 1.92870i) q^{72} +3.95517 q^{73} +(-8.49974 + 1.32453i) q^{74} -7.31114i q^{75} +(1.72408 - 8.59805i) q^{76} +(-3.01140 + 1.73863i) q^{77} +(-2.39616 - 0.237872i) q^{78} +(-7.31380 - 12.6679i) q^{79} +(8.50942 + 11.1610i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(5.23514 - 7.29372i) q^{82} +(-1.48274 - 0.856062i) q^{83} +(3.11620 - 1.05244i) q^{84} +14.9613i q^{85} +(-2.68825 + 1.21552i) q^{86} +(-5.13844 - 8.90003i) q^{87} +(-1.74930 + 5.71889i) q^{88} +(-6.52807 + 11.3069i) q^{89} +(-4.52136 + 2.04439i) q^{90} +(2.42499 + 1.40007i) q^{91} +(-2.14635 + 2.43982i) q^{92} +(-9.19092 + 5.30638i) q^{93} +(0.00397237 + 0.00878527i) q^{94} +(7.69217 - 13.3232i) q^{95} +(2.68589 - 4.97855i) q^{96} -5.02686 q^{97} +(6.04493 + 0.600094i) q^{98} +(-1.83113 - 1.05720i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q - 2 q^{2} + 2 q^{4} + 8 q^{7} - 8 q^{8} + 76 q^{9} - 4 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 6 q^{22} - 16 q^{23} + 80 q^{25} + 6 q^{28} + 8 q^{30} + 8 q^{32} + 2 q^{34} + 4 q^{36} + 76 q^{38}+ \cdots + 52 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.582660 + 1.28861i 0.412003 + 0.911183i
\(3\) 0.866025 − 0.500000i 0.500000 − 0.288675i
\(4\) −1.32102 + 1.50164i −0.660508 + 0.750819i
\(5\) −3.03864 + 1.75436i −1.35892 + 0.784574i −0.989479 − 0.144679i \(-0.953785\pi\)
−0.369444 + 0.929253i \(0.620452\pi\)
\(6\) 1.14890 + 0.824636i 0.469037 + 0.336656i
\(7\) −0.822280 − 1.42423i −0.310793 − 0.538309i 0.667742 − 0.744393i \(-0.267261\pi\)
−0.978534 + 0.206085i \(0.933928\pi\)
\(8\) −2.70472 − 0.827325i −0.956264 − 0.292504i
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) −4.03118 − 2.89342i −1.27477 − 0.914980i
\(11\) − 2.11441i − 0.637518i −0.947836 − 0.318759i \(-0.896734\pi\)
0.947836 − 0.318759i \(-0.103266\pi\)
\(12\) −0.393214 + 1.96096i −0.113511 + 0.566082i
\(13\) −1.47455 + 0.851334i −0.408968 + 0.236118i −0.690346 − 0.723479i \(-0.742542\pi\)
0.281378 + 0.959597i \(0.409208\pi\)
\(14\) 1.35616 − 1.88944i 0.362450 − 0.504973i
\(15\) −1.75436 + 3.03864i −0.452974 + 0.784574i
\(16\) −0.509836 − 3.96738i −0.127459 − 0.991844i
\(17\) 2.13202 − 3.69276i 0.517090 − 0.895627i −0.482713 − 0.875779i \(-0.660348\pi\)
0.999803 − 0.0198480i \(-0.00631824\pi\)
\(18\) 1.40730 + 0.139705i 0.331703 + 0.0329289i
\(19\) −3.79717 + 2.19230i −0.871132 + 0.502948i −0.867724 − 0.497046i \(-0.834418\pi\)
−0.00340751 + 0.999994i \(0.501085\pi\)
\(20\) 1.37968 − 6.88048i 0.308505 − 1.53852i
\(21\) −1.42423 − 0.822280i −0.310793 − 0.179436i
\(22\) 2.72464 − 1.23198i 0.580895 − 0.262659i
\(23\) 1.62477 0.338788 0.169394 − 0.985548i \(-0.445819\pi\)
0.169394 + 0.985548i \(0.445819\pi\)
\(24\) −2.75602 + 0.635877i −0.562571 + 0.129798i
\(25\) 3.65557 − 6.33163i 0.731114 − 1.26633i
\(26\) −1.95620 − 1.40408i −0.383642 − 0.275363i
\(27\) − 1.00000i − 0.192450i
\(28\) 3.22492 + 0.646663i 0.609453 + 0.122208i
\(29\) − 10.2769i − 1.90837i −0.299222 − 0.954183i \(-0.596727\pi\)
0.299222 − 0.954183i \(-0.403273\pi\)
\(30\) −4.93781 − 0.490187i −0.901517 − 0.0894956i
\(31\) −10.6128 −1.90611 −0.953054 − 0.302800i \(-0.902079\pi\)
−0.953054 + 0.302800i \(0.902079\pi\)
\(32\) 4.81533 − 2.96861i 0.851237 − 0.524781i
\(33\) −1.05720 − 1.83113i −0.184036 − 0.318759i
\(34\) 6.00076 + 0.595709i 1.02912 + 0.102163i
\(35\) 4.99723 + 2.88515i 0.844686 + 0.487680i
\(36\) 0.639949 + 1.89485i 0.106658 + 0.315809i
\(37\) −1.62283 + 5.86229i −0.266792 + 0.963754i
\(38\) −5.03747 − 3.61570i −0.817186 − 0.586544i
\(39\) −0.851334 + 1.47455i −0.136323 + 0.236118i
\(40\) 9.67012 − 2.23112i 1.52898 − 0.352771i
\(41\) −3.17421 − 5.49790i −0.495729 − 0.858628i 0.504259 − 0.863552i \(-0.331766\pi\)
−0.999988 + 0.00492484i \(0.998432\pi\)
\(42\) 0.229754 − 2.31438i 0.0354518 − 0.357117i
\(43\) 2.08617i 0.318137i 0.987268 + 0.159069i \(0.0508491\pi\)
−0.987268 + 0.159069i \(0.949151\pi\)
\(44\) 3.17508 + 2.79316i 0.478661 + 0.421085i
\(45\) 3.50872i 0.523050i
\(46\) 0.946688 + 2.09369i 0.139582 + 0.308698i
\(47\) 0.00681765 0.000994457 0.000497228 − 1.00000i \(-0.499842\pi\)
0.000497228 1.00000i \(0.499842\pi\)
\(48\) −2.42522 − 3.18093i −0.350050 − 0.459128i
\(49\) 2.14771 − 3.71995i 0.306816 − 0.531421i
\(50\) 10.2889 + 1.02141i 1.45508 + 0.144449i
\(51\) − 4.26404i − 0.597085i
\(52\) 0.669512 − 3.33887i 0.0928447 − 0.463018i
\(53\) 6.76763 + 3.90729i 0.929605 + 0.536708i 0.886687 − 0.462371i \(-0.153001\pi\)
0.0429186 + 0.999079i \(0.486334\pi\)
\(54\) 1.28861 − 0.582660i 0.175357 − 0.0792899i
\(55\) 3.70943 + 6.42493i 0.500180 + 0.866337i
\(56\) 1.04574 + 4.53244i 0.139743 + 0.605673i
\(57\) −2.19230 + 3.79717i −0.290377 + 0.502948i
\(58\) 13.2428 − 5.98792i 1.73887 − 0.786252i
\(59\) −3.97296 − 2.29379i −0.517235 − 0.298626i 0.218567 − 0.975822i \(-0.429862\pi\)
−0.735803 + 0.677196i \(0.763195\pi\)
\(60\) −2.24541 − 6.64851i −0.289881 − 0.858319i
\(61\) −12.6497 + 7.30331i −1.61963 + 0.935093i −0.632614 + 0.774468i \(0.718018\pi\)
−0.987015 + 0.160626i \(0.948649\pi\)
\(62\) −6.18363 − 13.6757i −0.785322 − 1.73681i
\(63\) −1.64456 −0.207195
\(64\) 6.63107 + 4.47537i 0.828883 + 0.559422i
\(65\) 2.98710 − 5.17380i 0.370504 − 0.641731i
\(66\) 1.74362 − 2.42925i 0.214624 − 0.299019i
\(67\) −6.61558 + 3.81950i −0.808221 + 0.466627i −0.846338 − 0.532647i \(-0.821198\pi\)
0.0381166 + 0.999273i \(0.487864\pi\)
\(68\) 2.72877 + 8.07972i 0.330912 + 0.979810i
\(69\) 1.40709 − 0.812385i 0.169394 − 0.0977997i
\(70\) −0.806143 + 8.12053i −0.0963525 + 0.970589i
\(71\) 5.41041 + 9.37110i 0.642098 + 1.11215i 0.984964 + 0.172761i \(0.0552689\pi\)
−0.342866 + 0.939384i \(0.611398\pi\)
\(72\) −2.06885 + 1.92870i −0.243816 + 0.227299i
\(73\) 3.95517 0.462917 0.231459 − 0.972845i \(-0.425650\pi\)
0.231459 + 0.972845i \(0.425650\pi\)
\(74\) −8.49974 + 1.32453i −0.988075 + 0.153973i
\(75\) − 7.31114i − 0.844218i
\(76\) 1.72408 − 8.59805i 0.197766 − 0.986264i
\(77\) −3.01140 + 1.73863i −0.343181 + 0.198136i
\(78\) −2.39616 − 0.237872i −0.271312 − 0.0269337i
\(79\) −7.31380 − 12.6679i −0.822866 − 1.42525i −0.903539 − 0.428506i \(-0.859040\pi\)
0.0806725 − 0.996741i \(-0.474293\pi\)
\(80\) 8.50942 + 11.1610i 0.951382 + 1.24784i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 5.23514 − 7.29372i 0.578125 − 0.805456i
\(83\) −1.48274 − 0.856062i −0.162752 − 0.0939650i 0.416412 − 0.909176i \(-0.363288\pi\)
−0.579164 + 0.815211i \(0.696621\pi\)
\(84\) 3.11620 − 1.05244i 0.340005 − 0.114830i
\(85\) 14.9613i 1.62278i
\(86\) −2.68825 + 1.21552i −0.289881 + 0.131073i
\(87\) −5.13844 − 8.90003i −0.550898 − 0.954183i
\(88\) −1.74930 + 5.71889i −0.186476 + 0.609636i
\(89\) −6.52807 + 11.3069i −0.691974 + 1.19853i 0.279217 + 0.960228i \(0.409925\pi\)
−0.971190 + 0.238305i \(0.923408\pi\)
\(90\) −4.52136 + 2.04439i −0.476594 + 0.215498i
\(91\) 2.42499 + 1.40007i 0.254208 + 0.146767i
\(92\) −2.14635 + 2.43982i −0.223772 + 0.254369i
\(93\) −9.19092 + 5.30638i −0.953054 + 0.550246i
\(94\) 0.00397237 + 0.00878527i 0.000409719 + 0.000906132i
\(95\) 7.69217 − 13.3232i 0.789200 − 1.36694i
\(96\) 2.68589 − 4.97855i 0.274128 − 0.508121i
\(97\) −5.02686 −0.510400 −0.255200 − 0.966888i \(-0.582141\pi\)
−0.255200 + 0.966888i \(0.582141\pi\)
\(98\) 6.04493 + 0.600094i 0.610630 + 0.0606186i
\(99\) −1.83113 − 1.05720i −0.184036 − 0.106253i
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.bh.a.565.50 yes 152
8.5 even 2 inner 888.2.bh.a.565.5 ✓ 152
37.26 even 3 inner 888.2.bh.a.877.5 yes 152
296.285 even 6 inner 888.2.bh.a.877.50 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.5 ✓ 152 8.5 even 2 inner
888.2.bh.a.565.50 yes 152 1.1 even 1 trivial
888.2.bh.a.877.5 yes 152 37.26 even 3 inner
888.2.bh.a.877.50 yes 152 296.285 even 6 inner