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 877.55
Character \(\chi\) \(=\) 888.877
Dual form 888.2.bh.a.565.55

$q$-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+(0.830663 - 1.14455i) q^{2} +(-0.866025 - 0.500000i) q^{3} +(-0.619998 - 1.90147i) q^{4} +(-2.32201 - 1.34061i) q^{5} +(-1.29165 + 0.575879i) q^{6} +(-0.950490 + 1.64630i) q^{7} +(-2.69134 - 0.869864i) q^{8} +(0.500000 + 0.866025i) q^{9} +(-3.46321 + 1.54406i) q^{10} +2.68734i q^{11} +(-0.413803 + 1.95672i) q^{12} +(0.596656 + 0.344479i) q^{13} +(1.09474 + 2.45540i) q^{14} +(1.34061 + 2.32201i) q^{15} +(-3.23121 + 2.35782i) q^{16} +(3.33284 + 5.77266i) q^{17} +(1.40654 + 0.147099i) q^{18} +(-2.71109 - 1.56525i) q^{19} +(-1.10950 + 5.24642i) q^{20} +(1.64630 - 0.950490i) q^{21} +(3.07580 + 2.23227i) q^{22} -5.58435 q^{23} +(1.89584 + 2.09900i) q^{24} +(1.09448 + 1.89570i) q^{25} +(0.889894 - 0.396757i) q^{26} -1.00000i q^{27} +(3.71969 + 0.786632i) q^{28} -9.64659i q^{29} +(3.77126 + 0.394407i) q^{30} +9.33663 q^{31} +(0.0146032 + 5.65684i) q^{32} +(1.34367 - 2.32730i) q^{33} +(9.37558 + 0.980519i) q^{34} +(4.41409 - 2.54848i) q^{35} +(1.33673 - 1.48767i) q^{36} +(1.31902 + 5.93803i) q^{37} +(-4.04351 + 1.80279i) q^{38} +(-0.344479 - 0.596656i) q^{39} +(5.08317 + 5.62788i) q^{40} +(-1.95058 + 3.37851i) q^{41} +(0.279633 - 2.67381i) q^{42} +8.09354i q^{43} +(5.10991 - 1.66614i) q^{44} -2.68122i q^{45} +(-4.63871 + 6.39158i) q^{46} -6.22565 q^{47} +(3.97722 - 0.426329i) q^{48} +(1.69314 + 2.93260i) q^{49} +(3.07887 + 0.321995i) q^{50} -6.66569i q^{51} +(0.285093 - 1.34810i) q^{52} +(-1.10121 + 0.635786i) q^{53} +(-1.14455 - 0.830663i) q^{54} +(3.60268 - 6.24003i) q^{55} +(3.99015 - 3.60396i) q^{56} +(1.56525 + 2.71109i) q^{57} +(-11.0410 - 8.01307i) q^{58} +(-9.29826 + 5.36836i) q^{59} +(3.58406 - 3.98878i) q^{60} +(-2.74019 - 1.58205i) q^{61} +(7.75560 - 10.6863i) q^{62} -1.90098 q^{63} +(6.48667 + 4.68221i) q^{64} +(-0.923626 - 1.59977i) q^{65} +(-1.54758 - 3.47111i) q^{66} +(3.41298 + 1.97049i) q^{67} +(8.91020 - 9.91635i) q^{68} +(4.83619 + 2.79217i) q^{69} +(0.749759 - 7.16908i) q^{70} +(-7.20941 + 12.4871i) q^{71} +(-0.592348 - 2.76571i) q^{72} -10.6925 q^{73} +(7.89204 + 3.42282i) q^{74} -2.18897i q^{75} +(-1.29541 + 6.12552i) q^{76} +(-4.42416 - 2.55429i) q^{77} +(-0.969049 - 0.101345i) q^{78} +(-1.61778 + 2.80208i) q^{79} +(10.6638 - 1.14308i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(2.24660 + 5.03894i) q^{82} +(-7.47378 + 4.31499i) q^{83} +(-2.82803 - 2.54109i) q^{84} -17.8722i q^{85} +(9.26348 + 6.72301i) q^{86} +(-4.82330 + 8.35419i) q^{87} +(2.33762 - 7.23256i) q^{88} +(0.292372 + 0.506402i) q^{89} +(-3.06880 - 2.22719i) q^{90} +(-1.13423 + 0.654848i) q^{91} +(3.46228 + 10.6185i) q^{92} +(-8.08576 - 4.66832i) q^{93} +(-5.17142 + 7.12558i) q^{94} +(4.19678 + 7.26904i) q^{95} +(2.81577 - 4.90626i) q^{96} +3.33204 q^{97} +(4.76294 + 0.498119i) q^{98} +(-2.32730 + 1.34367i) 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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.830663 − 1.14455i 0.587367 − 0.809320i
\(3\) −0.866025 − 0.500000i −0.500000 − 0.288675i
\(4\) −0.619998 − 1.90147i −0.309999 − 0.950737i
\(5\) −2.32201 − 1.34061i −1.03843 − 0.599540i −0.119045 − 0.992889i \(-0.537983\pi\)
−0.919389 + 0.393349i \(0.871316\pi\)
\(6\) −1.29165 + 0.575879i −0.527314 + 0.235102i
\(7\) −0.950490 + 1.64630i −0.359252 + 0.622242i −0.987836 − 0.155499i \(-0.950301\pi\)
0.628584 + 0.777741i \(0.283635\pi\)
\(8\) −2.69134 − 0.869864i −0.951534 − 0.307544i
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) −3.46321 + 1.54406i −1.09516 + 0.488275i
\(11\) 2.68734i 0.810264i 0.914258 + 0.405132i \(0.132774\pi\)
−0.914258 + 0.405132i \(0.867226\pi\)
\(12\) −0.413803 + 1.95672i −0.119455 + 0.564857i
\(13\) 0.596656 + 0.344479i 0.165482 + 0.0955414i 0.580454 − 0.814293i \(-0.302875\pi\)
−0.414971 + 0.909834i \(0.636208\pi\)
\(14\) 1.09474 + 2.45540i 0.292580 + 0.656234i
\(15\) 1.34061 + 2.32201i 0.346145 + 0.599540i
\(16\) −3.23121 + 2.35782i −0.807801 + 0.589455i
\(17\) 3.33284 + 5.77266i 0.808334 + 1.40007i 0.914017 + 0.405675i \(0.132964\pi\)
−0.105684 + 0.994400i \(0.533703\pi\)
\(18\) 1.40654 + 0.147099i 0.331525 + 0.0346717i
\(19\) −2.71109 − 1.56525i −0.621967 − 0.359093i 0.155667 − 0.987810i \(-0.450247\pi\)
−0.777634 + 0.628717i \(0.783581\pi\)
\(20\) −1.10950 + 5.24642i −0.248092 + 1.17313i
\(21\) 1.64630 − 0.950490i 0.359252 − 0.207414i
\(22\) 3.07580 + 2.23227i 0.655763 + 0.475922i
\(23\) −5.58435 −1.16442 −0.582209 − 0.813039i \(-0.697811\pi\)
−0.582209 + 0.813039i \(0.697811\pi\)
\(24\) 1.89584 + 2.09900i 0.386987 + 0.428456i
\(25\) 1.09448 + 1.89570i 0.218897 + 0.379140i
\(26\) 0.889894 − 0.396757i 0.174523 − 0.0778105i
\(27\) − 1.00000i − 0.192450i
\(28\) 3.71969 + 0.786632i 0.702956 + 0.148659i
\(29\) − 9.64659i − 1.79133i −0.444732 − 0.895663i \(-0.646701\pi\)
0.444732 − 0.895663i \(-0.353299\pi\)
\(30\) 3.77126 + 0.394407i 0.688534 + 0.0720084i
\(31\) 9.33663 1.67691 0.838454 − 0.544972i \(-0.183460\pi\)
0.838454 + 0.544972i \(0.183460\pi\)
\(32\) 0.0146032 + 5.65684i 0.00258151 + 0.999997i
\(33\) 1.34367 − 2.32730i 0.233903 − 0.405132i
\(34\) 9.37558 + 0.980519i 1.60790 + 0.168158i
\(35\) 4.41409 − 2.54848i 0.746118 − 0.430771i
\(36\) 1.33673 − 1.48767i 0.222788 − 0.247945i
\(37\) 1.31902 + 5.93803i 0.216845 + 0.976206i
\(38\) −4.04351 + 1.80279i −0.655944 + 0.292451i
\(39\) −0.344479 − 0.596656i −0.0551608 − 0.0955414i
\(40\) 5.08317 + 5.62788i 0.803720 + 0.889846i
\(41\) −1.95058 + 3.37851i −0.304630 + 0.527634i −0.977179 − 0.212418i \(-0.931866\pi\)
0.672549 + 0.740052i \(0.265199\pi\)
\(42\) 0.279633 − 2.67381i 0.0431483 − 0.412578i
\(43\) 8.09354i 1.23425i 0.786864 + 0.617127i \(0.211703\pi\)
−0.786864 + 0.617127i \(0.788297\pi\)
\(44\) 5.10991 − 1.66614i 0.770347 − 0.251181i
\(45\) − 2.68122i − 0.399693i
\(46\) −4.63871 + 6.39158i −0.683941 + 0.942387i
\(47\) −6.22565 −0.908105 −0.454052 − 0.890975i \(-0.650022\pi\)
−0.454052 + 0.890975i \(0.650022\pi\)
\(48\) 3.97722 − 0.426329i 0.574062 − 0.0615352i
\(49\) 1.69314 + 2.93260i 0.241877 + 0.418943i
\(50\) 3.07887 + 0.321995i 0.435418 + 0.0455370i
\(51\) − 6.66569i − 0.933383i
\(52\) 0.285093 − 1.34810i 0.0395353 − 0.186948i
\(53\) −1.10121 + 0.635786i −0.151263 + 0.0873319i −0.573721 − 0.819051i \(-0.694501\pi\)
0.422458 + 0.906383i \(0.361167\pi\)
\(54\) −1.14455 − 0.830663i −0.155754 − 0.113039i
\(55\) 3.60268 − 6.24003i 0.485785 − 0.841405i
\(56\) 3.99015 − 3.60396i 0.533207 − 0.481599i
\(57\) 1.56525 + 2.71109i 0.207322 + 0.359093i
\(58\) −11.0410 − 8.01307i −1.44976 − 1.05217i
\(59\) −9.29826 + 5.36836i −1.21053 + 0.698900i −0.962875 − 0.269948i \(-0.912994\pi\)
−0.247656 + 0.968848i \(0.579660\pi\)
\(60\) 3.58406 − 3.98878i 0.462700 − 0.514949i
\(61\) −2.74019 − 1.58205i −0.350845 − 0.202560i 0.314212 − 0.949353i \(-0.398260\pi\)
−0.665057 + 0.746792i \(0.731593\pi\)
\(62\) 7.75560 − 10.6863i 0.984962 − 1.35716i
\(63\) −1.90098 −0.239501
\(64\) 6.48667 + 4.68221i 0.810834 + 0.585276i
\(65\) −0.923626 − 1.59977i −0.114562 − 0.198427i
\(66\) −1.54758 − 3.47111i −0.190494 − 0.427264i
\(67\) 3.41298 + 1.97049i 0.416962 + 0.240733i 0.693777 − 0.720190i \(-0.255945\pi\)
−0.276815 + 0.960923i \(0.589279\pi\)
\(68\) 8.91020 − 9.91635i 1.08052 − 1.20253i
\(69\) 4.83619 + 2.79217i 0.582209 + 0.336138i
\(70\) 0.749759 − 7.16908i 0.0896133 − 0.856870i
\(71\) −7.20941 + 12.4871i −0.855600 + 1.48194i 0.0204872 + 0.999790i \(0.493478\pi\)
−0.876087 + 0.482153i \(0.839855\pi\)
\(72\) −0.592348 − 2.76571i −0.0698088 − 0.325941i
\(73\) −10.6925 −1.25146 −0.625730 − 0.780040i \(-0.715199\pi\)
−0.625730 + 0.780040i \(0.715199\pi\)
\(74\) 7.89204 + 3.42282i 0.917431 + 0.397894i
\(75\) − 2.18897i − 0.252760i
\(76\) −1.29541 + 6.12552i −0.148594 + 0.702645i
\(77\) −4.42416 − 2.55429i −0.504180 − 0.291088i
\(78\) −0.969049 − 0.101345i −0.109723 − 0.0114751i
\(79\) −1.61778 + 2.80208i −0.182015 + 0.315259i −0.942567 − 0.334018i \(-0.891595\pi\)
0.760552 + 0.649277i \(0.224929\pi\)
\(80\) 10.6638 − 1.14308i 1.19225 − 0.127801i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 2.24660 + 5.03894i 0.248095 + 0.556458i
\(83\) −7.47378 + 4.31499i −0.820354 + 0.473631i −0.850538 − 0.525913i \(-0.823724\pi\)
0.0301847 + 0.999544i \(0.490390\pi\)
\(84\) −2.82803 − 2.54109i −0.308564 − 0.277256i
\(85\) − 17.8722i − 1.93851i
\(86\) 9.26348 + 6.72301i 0.998907 + 0.724961i
\(87\) −4.82330 + 8.35419i −0.517112 + 0.895663i
\(88\) 2.33762 − 7.23256i 0.249191 − 0.770993i
\(89\) 0.292372 + 0.506402i 0.0309913 + 0.0536785i 0.881105 − 0.472921i \(-0.156800\pi\)
−0.850114 + 0.526599i \(0.823467\pi\)
\(90\) −3.06880 − 2.22719i −0.323480 − 0.234767i
\(91\) −1.13423 + 0.654848i −0.118900 + 0.0686468i
\(92\) 3.46228 + 10.6185i 0.360968 + 1.10705i
\(93\) −8.08576 − 4.66832i −0.838454 − 0.484082i
\(94\) −5.17142 + 7.12558i −0.533391 + 0.734948i
\(95\) 4.19678 + 7.26904i 0.430581 + 0.745788i
\(96\) 2.81577 − 4.90626i 0.287383 − 0.500744i
\(97\) 3.33204 0.338317 0.169158 − 0.985589i \(-0.445895\pi\)
0.169158 + 0.985589i \(0.445895\pi\)
\(98\) 4.76294 + 0.498119i 0.481129 + 0.0503176i
\(99\) −2.32730 + 1.34367i −0.233903 + 0.135044i
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.877.55 yes 152
8.5 even 2 inner 888.2.bh.a.877.6 yes 152
37.10 even 3 inner 888.2.bh.a.565.6 ✓ 152
296.269 even 6 inner 888.2.bh.a.565.55 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.6 ✓ 152 37.10 even 3 inner
888.2.bh.a.565.55 yes 152 296.269 even 6 inner
888.2.bh.a.877.6 yes 152 8.5 even 2 inner
888.2.bh.a.877.55 yes 152 1.1 even 1 trivial