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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 565.5
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.5

$q$-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+(-1.40730 + 0.139705i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(1.96096 - 0.393214i) 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} +(-1.50164 + 1.32102i) q^{12} +(1.47455 - 0.851334i) q^{13} +(1.35616 + 1.88944i) q^{14} +(-1.75436 + 3.03864i) q^{15} +(3.69077 - 1.54216i) q^{16} +(2.13202 - 3.69276i) q^{17} +(-0.582660 + 1.28861i) q^{18} +(3.79717 - 2.19230i) q^{19} +(5.26883 - 4.63508i) q^{20} +(1.42423 + 0.822280i) q^{21} +(-0.295394 - 2.97560i) q^{22} +1.62477 q^{23} +(1.92870 - 2.06885i) q^{24} +(3.65557 - 6.33163i) q^{25} +(-1.95620 + 1.40408i) q^{26} +1.00000i q^{27} +(-2.17249 - 2.46953i) q^{28} +10.2769i q^{29} +(2.04439 - 4.52136i) q^{30} -10.6128 q^{31} +(-4.97855 + 2.68589i) q^{32} +(-1.05720 - 1.83113i) q^{33} +(-2.48448 + 5.49467i) 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} +(-6.76726 + 7.25901i) q^{40} +(-3.17421 - 5.49790i) q^{41} +(-2.11919 - 0.958219i) q^{42} -2.08617i q^{43} +(0.831414 + 4.14628i) q^{44} -3.50872i q^{45} +(-2.28653 + 0.226989i) q^{46} +0.00681765 q^{47} +(-2.42522 + 3.18093i) q^{48} +(2.14771 - 3.71995i) q^{49} +(-4.25991 + 9.42118i) q^{50} +4.26404i q^{51} +(2.55679 - 2.24925i) q^{52} +(-6.76763 - 3.90729i) q^{53} +(-0.139705 - 1.40730i) q^{54} +(3.70943 + 6.42493i) q^{55} +(3.40234 + 3.17186i) q^{56} +(-2.19230 + 3.79717i) q^{57} +(-1.43573 - 14.4626i) q^{58} +(3.97296 + 2.29379i) q^{59} +(-2.24541 + 6.64851i) q^{60} +(12.6497 - 7.30331i) q^{61} +(14.9353 - 1.48266i) 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} +(7.43565 + 3.36212i) q^{70} +(5.41041 + 9.37110i) q^{71} +(-0.635877 + 2.75602i) q^{72} +3.95517 q^{73} +(-1.46481 + 8.47669i) q^{74} +7.31114i q^{75} +(6.58408 - 5.79212i) q^{76} +(3.01140 - 1.73863i) q^{77} +(0.992076 - 2.19407i) 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} +(0.291449 + 2.93585i) q^{86} +(-5.13844 - 8.90003i) q^{87} +(-1.74930 - 5.71889i) q^{88} +(-6.52807 + 11.3069i) q^{89} +(0.490187 + 4.93781i) q^{90} +(-2.42499 - 1.40007i) q^{91} +(3.18612 - 0.638882i) q^{92} +(9.19092 - 5.30638i) q^{93} +(-0.00959446 + 0.000952463i) q^{94} +(7.69217 - 13.3232i) q^{95} +(2.96861 - 4.81533i) q^{96} -5.02686 q^{97} +(-2.50277 + 5.53511i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40730 + 0.139705i −0.995109 + 0.0987866i
\(3\) −0.866025 + 0.500000i −0.500000 + 0.288675i
\(4\) 1.96096 − 0.393214i 0.980482 − 0.196607i
\(5\) 3.03864 − 1.75436i 1.35892 − 0.784574i 0.369444 − 0.929253i \(-0.379548\pi\)
0.989479 + 0.144679i \(0.0462148\pi\)
\(6\) 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 \(0.103266\pi\)
−0.947836 + 0.318759i \(0.896734\pi\)
\(12\) −1.50164 + 1.32102i −0.433486 + 0.381344i
\(13\) 1.47455 − 0.851334i 0.408968 − 0.236118i −0.281378 − 0.959597i \(-0.590792\pi\)
0.690346 + 0.723479i \(0.257458\pi\)
\(14\) 1.35616 + 1.88944i 0.362450 + 0.504973i
\(15\) −1.75436 + 3.03864i −0.452974 + 0.784574i
\(16\) 3.69077 − 1.54216i 0.922692 − 0.385539i
\(17\) 2.13202 − 3.69276i 0.517090 − 0.895627i −0.482713 − 0.875779i \(-0.660348\pi\)
0.999803 − 0.0198480i \(-0.00631824\pi\)
\(18\) −0.582660 + 1.28861i −0.137334 + 0.303728i
\(19\) 3.79717 − 2.19230i 0.871132 − 0.502948i 0.00340751 − 0.999994i \(-0.498915\pi\)
0.867724 + 0.497046i \(0.165582\pi\)
\(20\) 5.26883 − 4.63508i 1.17815 − 1.03643i
\(21\) 1.42423 + 0.822280i 0.310793 + 0.179436i
\(22\) −0.295394 − 2.97560i −0.0629782 − 0.634399i
\(23\) 1.62477 0.338788 0.169394 − 0.985548i \(-0.445819\pi\)
0.169394 + 0.985548i \(0.445819\pi\)
\(24\) 1.92870 − 2.06885i 0.393694 − 0.422302i
\(25\) 3.65557 − 6.33163i 0.731114 − 1.26633i
\(26\) −1.95620 + 1.40408i −0.383642 + 0.275363i
\(27\) 1.00000i 0.192450i
\(28\) −2.17249 − 2.46953i −0.410562 − 0.466698i
\(29\) 10.2769i 1.90837i 0.299222 + 0.954183i \(0.403273\pi\)
−0.299222 + 0.954183i \(0.596727\pi\)
\(30\) 2.04439 − 4.52136i 0.373253 − 0.825485i
\(31\) −10.6128 −1.90611 −0.953054 − 0.302800i \(-0.902079\pi\)
−0.953054 + 0.302800i \(0.902079\pi\)
\(32\) −4.97855 + 2.68589i −0.880092 + 0.474803i
\(33\) −1.05720 − 1.83113i −0.184036 − 0.318759i
\(34\) −2.48448 + 5.49467i −0.426085 + 0.942328i
\(35\) −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
\(40\) −6.76726 + 7.25901i −1.07000 + 1.14775i
\(41\) −3.17421 − 5.49790i −0.495729 − 0.858628i 0.504259 − 0.863552i \(-0.331766\pi\)
−0.999988 + 0.00492484i \(0.998432\pi\)
\(42\) −2.11919 − 0.958219i −0.326998 − 0.147856i
\(43\) − 2.08617i − 0.318137i −0.987268 − 0.159069i \(-0.949151\pi\)
0.987268 − 0.159069i \(-0.0508491\pi\)
\(44\) 0.831414 + 4.14628i 0.125340 + 0.625075i
\(45\) − 3.50872i − 0.523050i
\(46\) −2.28653 + 0.226989i −0.337131 + 0.0334677i
\(47\) 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
\(50\) −4.25991 + 9.42118i −0.602442 + 1.33236i
\(51\) 4.26404i 0.597085i
\(52\) 2.55679 − 2.24925i 0.354563 − 0.311915i
\(53\) −6.76763 − 3.90729i −0.929605 − 0.536708i −0.0429186 − 0.999079i \(-0.513666\pi\)
−0.886687 + 0.462371i \(0.846999\pi\)
\(54\) −0.139705 − 1.40730i −0.0190115 − 0.191509i
\(55\) 3.70943 + 6.42493i 0.500180 + 0.866337i
\(56\) 3.40234 + 3.17186i 0.454657 + 0.423857i
\(57\) −2.19230 + 3.79717i −0.290377 + 0.502948i
\(58\) −1.43573 − 14.4626i −0.188521 − 1.89903i
\(59\) 3.97296 + 2.29379i 0.517235 + 0.298626i 0.735803 − 0.677196i \(-0.236805\pi\)
−0.218567 + 0.975822i \(0.570138\pi\)
\(60\) −2.24541 + 6.64851i −0.289881 + 0.858319i
\(61\) 12.6497 − 7.30331i 1.61963 − 0.935093i 0.632614 − 0.774468i \(-0.281982\pi\)
0.987015 − 0.160626i \(-0.0513512\pi\)
\(62\) 14.9353 − 1.48266i 1.89678 − 0.188298i
\(63\) −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 −0.0381166 − 0.999273i \(-0.512136\pi\)
0.846338 + 0.532647i \(0.178802\pi\)
\(68\) 2.72877 − 8.07972i 0.330912 − 0.979810i
\(69\) −1.40709 + 0.812385i −0.169394 + 0.0977997i
\(70\) 7.43565 + 3.36212i 0.888731 + 0.401851i
\(71\) 5.41041 + 9.37110i 0.642098 + 1.11215i 0.984964 + 0.172761i \(0.0552689\pi\)
−0.342866 + 0.939384i \(0.611398\pi\)
\(72\) −0.635877 + 2.75602i −0.0749389 + 0.324800i
\(73\) 3.95517 0.462917 0.231459 − 0.972845i \(-0.425650\pi\)
0.231459 + 0.972845i \(0.425650\pi\)
\(74\) −1.46481 + 8.47669i −0.170281 + 0.985396i
\(75\) 7.31114i 0.844218i
\(76\) 6.58408 − 5.79212i 0.755246 − 0.664402i
\(77\) 3.01140 − 1.73863i 0.343181 − 0.198136i
\(78\) 0.992076 − 2.19407i 0.112331 − 0.248430i
\(79\) −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 0.579164 − 0.815211i \(-0.303379\pi\)
−0.416412 + 0.909176i \(0.636712\pi\)
\(84\) 3.11620 + 1.05244i 0.340005 + 0.114830i
\(85\) − 14.9613i − 1.62278i
\(86\) 0.291449 + 2.93585i 0.0314277 + 0.316581i
\(87\) −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 \(0.923408\pi\)
\(90\) 0.490187 + 4.93781i 0.0516703 + 0.520491i
\(91\) −2.42499 − 1.40007i −0.254208 − 0.146767i
\(92\) 3.18612 − 0.638882i 0.332176 − 0.0666080i
\(93\) 9.19092 − 5.30638i 0.953054 − 0.550246i
\(94\) −0.00959446 0.000952463i −0.000989593 9.82390e-5i
\(95\) 7.69217 − 13.3232i 0.789200 − 1.36694i
\(96\) 2.96861 − 4.81533i 0.302982 − 0.491462i
\(97\) −5.02686 −0.510400 −0.255200 − 0.966888i \(-0.582141\pi\)
−0.255200 + 0.966888i \(0.582141\pi\)
\(98\) −2.50277 + 5.53511i −0.252818 + 0.559131i
\(99\) 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.565.5 ✓ 152
8.5 even 2 inner 888.2.bh.a.565.50 yes 152
37.26 even 3 inner 888.2.bh.a.877.50 yes 152
296.285 even 6 inner 888.2.bh.a.877.5 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.5 ✓ 152 1.1 even 1 trivial
888.2.bh.a.565.50 yes 152 8.5 even 2 inner
888.2.bh.a.877.5 yes 152 296.285 even 6 inner
888.2.bh.a.877.50 yes 152 37.26 even 3 inner