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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 877.6
Character \(\chi\) \(=\) 888.877
Dual form 888.2.bh.a.565.6

$q$-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+(-1.40654 + 0.147099i) q^{2} +(0.866025 + 0.500000i) q^{3} +(1.95672 - 0.413803i) 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} +(1.90147 + 0.619998i) q^{12} +(-0.596656 - 0.344479i) q^{13} +(1.09474 - 2.45540i) q^{14} +(1.34061 + 2.32201i) q^{15} +(3.65753 - 1.61940i) q^{16} +(3.33284 + 5.77266i) q^{17} +(-0.830663 - 1.14455i) q^{18} +(2.71109 + 1.56525i) q^{19} +(5.09828 + 1.66235i) q^{20} +(-1.64630 + 0.950490i) q^{21} +(0.395306 + 3.77986i) q^{22} -5.58435 q^{23} +(-2.76571 - 0.592348i) q^{24} +(1.09448 + 1.89570i) q^{25} +(0.889894 + 0.396757i) q^{26} +1.00000i q^{27} +(-1.17860 + 3.61466i) q^{28} +9.64659i q^{29} +(-2.22719 - 3.06880i) q^{30} +9.33663 q^{31} +(-4.90626 + 2.81577i) q^{32} +(1.34367 - 2.32730i) q^{33} +(-5.53694 - 7.62923i) 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} +(-7.41548 - 1.58822i) q^{40} +(-1.95058 + 3.37851i) q^{41} +(2.17577 - 1.57907i) q^{42} -8.09354i q^{43} +(-1.11203 - 5.25838i) q^{44} +2.68122i q^{45} +(7.85462 - 0.821454i) q^{46} -6.22565 q^{47} +(3.97722 + 0.426329i) q^{48} +(1.69314 + 2.93260i) q^{49} +(-1.81829 - 2.50538i) q^{50} +6.66569i q^{51} +(-1.31004 - 0.427153i) q^{52} +(1.10121 - 0.635786i) q^{53} +(-0.147099 - 1.40654i) q^{54} +(3.60268 - 6.24003i) q^{55} +(1.12604 - 5.25755i) q^{56} +(1.56525 + 2.71109i) q^{57} +(-1.41901 - 13.5683i) q^{58} +(9.29826 - 5.36836i) q^{59} +(3.58406 + 3.98878i) q^{60} +(2.74019 + 1.58205i) q^{61} +(-13.1324 + 1.37341i) 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} +(5.83373 - 4.23385i) q^{70} +(-7.20941 + 12.4871i) q^{71} +(-2.09900 - 1.89584i) q^{72} -10.6925 q^{73} +(2.72874 + 8.15806i) q^{74} +2.18897i q^{75} +(5.95256 + 1.94090i) q^{76} +(4.42416 + 2.55429i) q^{77} +(0.572292 + 0.788549i) 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} +(1.19056 + 11.3839i) q^{86} +(-4.82330 + 8.35419i) q^{87} +(2.33762 + 7.23256i) q^{88} +(0.292372 + 0.506402i) q^{89} +(-0.394407 - 3.77126i) q^{90} +(1.13423 - 0.654848i) q^{91} +(-10.9270 + 2.31082i) q^{92} +(8.08576 + 4.66832i) q^{93} +(8.75665 - 0.915790i) q^{94} +(4.19678 + 7.26904i) q^{95} +(-5.65684 - 0.0146032i) q^{96} +3.33204 q^{97} +(-2.81285 - 3.87577i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40654 + 0.147099i −0.994576 + 0.104015i
\(3\) 0.866025 + 0.500000i 0.500000 + 0.288675i
\(4\) 1.95672 − 0.413803i 0.978362 − 0.206902i
\(5\) 2.32201 + 1.34061i 1.03843 + 0.599540i 0.919389 − 0.393349i \(-0.128684\pi\)
0.119045 + 0.992889i \(0.462017\pi\)
\(6\) −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 \(-0.867226\pi\)
0.914258 − 0.405132i \(-0.132774\pi\)
\(12\) 1.90147 + 0.619998i 0.548908 + 0.178978i
\(13\) −0.596656 − 0.344479i −0.165482 − 0.0955414i 0.414971 − 0.909834i \(-0.363792\pi\)
−0.580454 + 0.814293i \(0.697125\pi\)
\(14\) 1.09474 − 2.45540i 0.292580 − 0.656234i
\(15\) 1.34061 + 2.32201i 0.346145 + 0.599540i
\(16\) 3.65753 − 1.61940i 0.914384 − 0.404849i
\(17\) 3.33284 + 5.77266i 0.808334 + 1.40007i 0.914017 + 0.405675i \(0.132964\pi\)
−0.105684 + 0.994400i \(0.533703\pi\)
\(18\) −0.830663 − 1.14455i −0.195789 − 0.269773i
\(19\) 2.71109 + 1.56525i 0.621967 + 0.359093i 0.777634 − 0.628717i \(-0.216419\pi\)
−0.155667 + 0.987810i \(0.549753\pi\)
\(20\) 5.09828 + 1.66235i 1.14001 + 0.371714i
\(21\) −1.64630 + 0.950490i −0.359252 + 0.207414i
\(22\) 0.395306 + 3.77986i 0.0842795 + 0.805868i
\(23\) −5.58435 −1.16442 −0.582209 − 0.813039i \(-0.697811\pi\)
−0.582209 + 0.813039i \(0.697811\pi\)
\(24\) −2.76571 − 0.592348i −0.564547 − 0.120912i
\(25\) 1.09448 + 1.89570i 0.218897 + 0.379140i
\(26\) 0.889894 + 0.396757i 0.174523 + 0.0778105i
\(27\) 1.00000i 0.192450i
\(28\) −1.17860 + 3.61466i −0.222735 + 0.683107i
\(29\) 9.64659i 1.79133i 0.444732 + 0.895663i \(0.353299\pi\)
−0.444732 + 0.895663i \(0.646701\pi\)
\(30\) −2.22719 − 3.06880i −0.406628 − 0.560284i
\(31\) 9.33663 1.67691 0.838454 − 0.544972i \(-0.183460\pi\)
0.838454 + 0.544972i \(0.183460\pi\)
\(32\) −4.90626 + 2.81577i −0.867313 + 0.497763i
\(33\) 1.34367 − 2.32730i 0.233903 − 0.405132i
\(34\) −5.53694 − 7.62923i −0.949578 − 1.30840i
\(35\) −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
\(40\) −7.41548 − 1.58822i −1.17249 − 0.251119i
\(41\) −1.95058 + 3.37851i −0.304630 + 0.527634i −0.977179 − 0.212418i \(-0.931866\pi\)
0.672549 + 0.740052i \(0.265199\pi\)
\(42\) 2.17577 − 1.57907i 0.335729 − 0.243656i
\(43\) − 8.09354i − 1.23425i −0.786864 − 0.617127i \(-0.788297\pi\)
0.786864 − 0.617127i \(-0.211703\pi\)
\(44\) −1.11203 − 5.25838i −0.167645 − 0.792731i
\(45\) 2.68122i 0.399693i
\(46\) 7.85462 − 0.821454i 1.15810 − 0.121117i
\(47\) −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
\(50\) −1.81829 − 2.50538i −0.257145 − 0.354315i
\(51\) 6.66569i 0.933383i
\(52\) −1.31004 − 0.427153i −0.181669 − 0.0592354i
\(53\) 1.10121 − 0.635786i 0.151263 − 0.0873319i −0.422458 − 0.906383i \(-0.638833\pi\)
0.573721 + 0.819051i \(0.305499\pi\)
\(54\) −0.147099 − 1.40654i −0.0200177 − 0.191406i
\(55\) 3.60268 − 6.24003i 0.485785 − 0.841405i
\(56\) 1.12604 − 5.25755i 0.150474 − 0.702570i
\(57\) 1.56525 + 2.71109i 0.207322 + 0.359093i
\(58\) −1.41901 − 13.5683i −0.186325 − 1.78161i
\(59\) 9.29826 − 5.36836i 1.21053 − 0.698900i 0.247656 − 0.968848i \(-0.420340\pi\)
0.962875 + 0.269948i \(0.0870064\pi\)
\(60\) 3.58406 + 3.98878i 0.462700 + 0.514949i
\(61\) 2.74019 + 1.58205i 0.350845 + 0.202560i 0.665057 − 0.746792i \(-0.268407\pi\)
−0.314212 + 0.949353i \(0.601740\pi\)
\(62\) −13.1324 + 1.37341i −1.66781 + 0.174424i
\(63\) −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 0.276815 − 0.960923i \(-0.410721\pi\)
−0.693777 + 0.720190i \(0.744055\pi\)
\(68\) 8.91020 + 9.91635i 1.08052 + 1.20253i
\(69\) −4.83619 − 2.79217i −0.582209 − 0.336138i
\(70\) 5.83373 − 4.23385i 0.697264 − 0.506042i
\(71\) −7.20941 + 12.4871i −0.855600 + 1.48194i 0.0204872 + 0.999790i \(0.493478\pi\)
−0.876087 + 0.482153i \(0.839855\pi\)
\(72\) −2.09900 − 1.89584i −0.247369 − 0.223427i
\(73\) −10.6925 −1.25146 −0.625730 − 0.780040i \(-0.715199\pi\)
−0.625730 + 0.780040i \(0.715199\pi\)
\(74\) 2.72874 + 8.15806i 0.317209 + 0.948356i
\(75\) 2.18897i 0.252760i
\(76\) 5.95256 + 1.94090i 0.682805 + 0.222637i
\(77\) 4.42416 + 2.55429i 0.504180 + 0.291088i
\(78\) 0.572292 + 0.788549i 0.0647993 + 0.0892856i
\(79\) −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 −0.0301847 − 0.999544i \(-0.509610\pi\)
0.850538 + 0.525913i \(0.176276\pi\)
\(84\) −2.82803 + 2.54109i −0.308564 + 0.277256i
\(85\) 17.8722i 1.93851i
\(86\) 1.19056 + 11.3839i 0.128381 + 1.22756i
\(87\) −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 \(0.823467\pi\)
\(90\) −0.394407 − 3.77126i −0.0415741 − 0.397525i
\(91\) 1.13423 − 0.654848i 0.118900 − 0.0686468i
\(92\) −10.9270 + 2.31082i −1.13922 + 0.240920i
\(93\) 8.08576 + 4.66832i 0.838454 + 0.484082i
\(94\) 8.75665 − 0.915790i 0.903179 − 0.0944565i
\(95\) 4.19678 + 7.26904i 0.430581 + 0.745788i
\(96\) −5.65684 − 0.0146032i −0.577348 − 0.00149043i
\(97\) 3.33204 0.338317 0.169158 − 0.985589i \(-0.445895\pi\)
0.169158 + 0.985589i \(0.445895\pi\)
\(98\) −2.81285 − 3.87577i −0.284141 − 0.391511i
\(99\) 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.877.6 yes 152
8.5 even 2 inner 888.2.bh.a.877.55 yes 152
37.10 even 3 inner 888.2.bh.a.565.55 yes 152
296.269 even 6 inner 888.2.bh.a.565.6 ✓ 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.6 ✓ 152 296.269 even 6 inner
888.2.bh.a.565.55 yes 152 37.10 even 3 inner
888.2.bh.a.877.6 yes 152 1.1 even 1 trivial
888.2.bh.a.877.55 yes 152 8.5 even 2 inner