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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(443,888)] 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("888.443"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0,-4,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(96\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 443.92
Character \(\chi\) \(=\) 888.443
Dual form 888.2.c.d.443.89

$q$-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+(1.39209 + 0.249187i) q^{2} +(1.03985 + 1.38518i) q^{3} +(1.87581 + 0.693779i) q^{4} +0.556466i q^{5} +(1.10239 + 2.18740i) q^{6} +3.66824i q^{7} +(2.43841 + 1.43323i) q^{8} +(-0.837425 + 2.88075i) q^{9} +(-0.138664 + 0.774649i) q^{10} -4.20505i q^{11} +(0.989556 + 3.31976i) q^{12} -2.20974 q^{13} +(-0.914077 + 5.10651i) q^{14} +(-0.770804 + 0.578641i) q^{15} +(3.03734 + 2.60280i) q^{16} -3.13876 q^{17} +(-1.88361 + 3.80158i) q^{18} -1.33708i q^{19} +(-0.386065 + 1.04383i) q^{20} +(-5.08116 + 3.81442i) q^{21} +(1.04784 - 5.85380i) q^{22} -6.03009i q^{23} +(0.550309 + 4.86797i) q^{24} +4.69035 q^{25} +(-3.07615 - 0.550638i) q^{26} +(-4.86114 + 1.83557i) q^{27} +(-2.54495 + 6.88093i) q^{28} -8.61231i q^{29} +(-1.21722 + 0.613445i) q^{30} +2.20682 q^{31} +(3.57966 + 4.38019i) q^{32} +(5.82474 - 4.37262i) q^{33} +(-4.36943 - 0.782138i) q^{34} -2.04125 q^{35} +(-3.56946 + 4.82276i) q^{36} +(6.08247 - 0.0599166i) q^{37} +(0.333183 - 1.86133i) q^{38} +(-2.29780 - 3.06088i) q^{39} +(-0.797543 + 1.35689i) q^{40} +5.29566i q^{41} +(-8.02392 + 4.04385i) q^{42} +7.58400i q^{43} +(2.91738 - 7.88789i) q^{44} +(-1.60304 - 0.465999i) q^{45} +(1.50262 - 8.39440i) q^{46} -5.37047 q^{47} +(-0.446956 + 6.91377i) q^{48} -6.45600 q^{49} +(6.52937 + 1.16877i) q^{50} +(-3.26384 - 4.34774i) q^{51} +(-4.14506 - 1.53307i) q^{52} +13.3589 q^{53} +(-7.22453 + 1.34393i) q^{54} +2.33997 q^{55} +(-5.25743 + 8.94469i) q^{56} +(1.85209 - 1.39036i) q^{57} +(2.14607 - 11.9891i) q^{58} +5.05987 q^{59} +(-1.84733 + 0.550655i) q^{60} +1.07963 q^{61} +(3.07208 + 0.549909i) q^{62} +(-10.5673 - 3.07188i) q^{63} +(3.89171 + 6.98960i) q^{64} -1.22965i q^{65} +(9.19814 - 4.63562i) q^{66} +0.856672 q^{67} +(-5.88773 - 2.17761i) q^{68} +(8.35273 - 6.27038i) q^{69} +(-2.84160 - 0.508653i) q^{70} -11.5560 q^{71} +(-6.17076 + 5.82424i) q^{72} +0.501387 q^{73} +(8.48225 + 1.43226i) q^{74} +(4.87725 + 6.49695i) q^{75} +(0.927639 - 2.50811i) q^{76} +15.4252 q^{77} +(-2.43601 - 4.83360i) q^{78} +11.0324 q^{79} +(-1.44837 + 1.69018i) q^{80} +(-7.59744 - 4.82482i) q^{81} +(-1.31961 + 7.37202i) q^{82} +7.18404i q^{83} +(-12.1777 + 3.62993i) q^{84} -1.74662i q^{85} +(-1.88983 + 10.5576i) q^{86} +(11.9296 - 8.95551i) q^{87} +(6.02680 - 10.2537i) q^{88} +3.48303 q^{89} +(-2.11545 - 1.04817i) q^{90} -8.10587i q^{91} +(4.18355 - 11.3113i) q^{92} +(2.29476 + 3.05683i) q^{93} +(-7.47616 - 1.33825i) q^{94} +0.744041 q^{95} +(-2.34502 + 9.51319i) q^{96} -7.96066i q^{97} +(-8.98731 - 1.60875i) q^{98} +(12.1137 + 3.52142i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q - 4 q^{3} + 24 q^{4} - 4 q^{9} - 20 q^{10} - 18 q^{12} - 56 q^{16} + 72 q^{25} - 4 q^{27} - 64 q^{28} - 16 q^{33} - 8 q^{34} + 42 q^{36} + 44 q^{40} - 68 q^{46} - 94 q^{48} - 104 q^{49} - 28 q^{58}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.39209 + 0.249187i 0.984354 + 0.176202i
\(3\) 1.03985 + 1.38518i 0.600358 + 0.799732i
\(4\) 1.87581 + 0.693779i 0.937906 + 0.346890i
\(5\) 0.556466i 0.248859i 0.992228 + 0.124430i \(0.0397101\pi\)
−0.992228 + 0.124430i \(0.960290\pi\)
\(6\) 1.10239 + 2.18740i 0.450050 + 0.893003i
\(7\) 3.66824i 1.38647i 0.720714 + 0.693233i \(0.243814\pi\)
−0.720714 + 0.693233i \(0.756186\pi\)
\(8\) 2.43841 + 1.43323i 0.862109 + 0.506723i
\(9\) −0.837425 + 2.88075i −0.279142 + 0.960250i
\(10\) −0.138664 + 0.774649i −0.0438494 + 0.244966i
\(11\) − 4.20505i − 1.26787i −0.773386 − 0.633936i \(-0.781438\pi\)
0.773386 − 0.633936i \(-0.218562\pi\)
\(12\) 0.989556 + 3.31976i 0.285660 + 0.958331i
\(13\) −2.20974 −0.612872 −0.306436 − 0.951891i \(-0.599137\pi\)
−0.306436 + 0.951891i \(0.599137\pi\)
\(14\) −0.914077 + 5.10651i −0.244297 + 1.36477i
\(15\) −0.770804 + 0.578641i −0.199021 + 0.149405i
\(16\) 3.03734 + 2.60280i 0.759335 + 0.650700i
\(17\) −3.13876 −0.761262 −0.380631 − 0.924727i \(-0.624293\pi\)
−0.380631 + 0.924727i \(0.624293\pi\)
\(18\) −1.88361 + 3.80158i −0.443972 + 0.896041i
\(19\) − 1.33708i − 0.306748i −0.988168 − 0.153374i \(-0.950986\pi\)
0.988168 − 0.153374i \(-0.0490139\pi\)
\(20\) −0.386065 + 1.04383i −0.0863267 + 0.233407i
\(21\) −5.08116 + 3.81442i −1.10880 + 0.832375i
\(22\) 1.04784 − 5.85380i 0.223401 − 1.24803i
\(23\) − 6.03009i − 1.25736i −0.777664 − 0.628680i \(-0.783596\pi\)
0.777664 − 0.628680i \(-0.216404\pi\)
\(24\) 0.550309 + 4.86797i 0.112331 + 0.993671i
\(25\) 4.69035 0.938069
\(26\) −3.07615 − 0.550638i −0.603283 − 0.107989i
\(27\) −4.86114 + 1.83557i −0.935527 + 0.353255i
\(28\) −2.54495 + 6.88093i −0.480950 + 1.30037i
\(29\) − 8.61231i − 1.59927i −0.600489 − 0.799633i \(-0.705027\pi\)
0.600489 − 0.799633i \(-0.294973\pi\)
\(30\) −1.21722 + 0.613445i −0.222232 + 0.111999i
\(31\) 2.20682 0.396356 0.198178 − 0.980166i \(-0.436498\pi\)
0.198178 + 0.980166i \(0.436498\pi\)
\(32\) 3.57966 + 4.38019i 0.632800 + 0.774315i
\(33\) 5.82474 − 4.37262i 1.01396 − 0.761176i
\(34\) −4.36943 − 0.782138i −0.749352 − 0.134136i
\(35\) −2.04125 −0.345035
\(36\) −3.56946 + 4.82276i −0.594909 + 0.803793i
\(37\) 6.08247 − 0.0599166i 0.999951 − 0.00985023i
\(38\) 0.333183 − 1.86133i 0.0540494 − 0.301948i
\(39\) −2.29780 − 3.06088i −0.367942 − 0.490133i
\(40\) −0.797543 + 1.35689i −0.126103 + 0.214544i
\(41\) 5.29566i 0.827043i 0.910494 + 0.413521i \(0.135701\pi\)
−0.910494 + 0.413521i \(0.864299\pi\)
\(42\) −8.02392 + 4.04385i −1.23812 + 0.623979i
\(43\) 7.58400i 1.15655i 0.815842 + 0.578275i \(0.196274\pi\)
−0.815842 + 0.578275i \(0.803726\pi\)
\(44\) 2.91738 − 7.88789i 0.439811 − 1.18914i
\(45\) −1.60304 − 0.465999i −0.238967 − 0.0694670i
\(46\) 1.50262 − 8.39440i 0.221549 − 1.23769i
\(47\) −5.37047 −0.783363 −0.391682 − 0.920101i \(-0.628106\pi\)
−0.391682 + 0.920101i \(0.628106\pi\)
\(48\) −0.446956 + 6.91377i −0.0645126 + 0.997917i
\(49\) −6.45600 −0.922286
\(50\) 6.52937 + 1.16877i 0.923392 + 0.165289i
\(51\) −3.26384 − 4.34774i −0.457029 − 0.608806i
\(52\) −4.14506 − 1.53307i −0.574817 − 0.212599i
\(53\) 13.3589 1.83499 0.917496 − 0.397744i \(-0.130207\pi\)
0.917496 + 0.397744i \(0.130207\pi\)
\(54\) −7.22453 + 1.34393i −0.983134 + 0.182886i
\(55\) 2.33997 0.315522
\(56\) −5.25743 + 8.94469i −0.702553 + 1.19528i
\(57\) 1.85209 − 1.39036i 0.245316 − 0.184158i
\(58\) 2.14607 − 11.9891i 0.281793 − 1.57424i
\(59\) 5.05987 0.658739 0.329369 − 0.944201i \(-0.393164\pi\)
0.329369 + 0.944201i \(0.393164\pi\)
\(60\) −1.84733 + 0.550655i −0.238490 + 0.0710892i
\(61\) 1.07963 0.138233 0.0691165 − 0.997609i \(-0.477982\pi\)
0.0691165 + 0.997609i \(0.477982\pi\)
\(62\) 3.07208 + 0.549909i 0.390154 + 0.0698385i
\(63\) −10.5673 − 3.07188i −1.33135 − 0.387020i
\(64\) 3.89171 + 6.98960i 0.486464 + 0.873701i
\(65\) − 1.22965i − 0.152519i
\(66\) 9.19814 − 4.63562i 1.13221 − 0.570606i
\(67\) 0.856672 0.104659 0.0523296 − 0.998630i \(-0.483335\pi\)
0.0523296 + 0.998630i \(0.483335\pi\)
\(68\) −5.88773 − 2.17761i −0.713992 − 0.264074i
\(69\) 8.35273 − 6.27038i 1.00555 − 0.754865i
\(70\) −2.84160 − 0.508653i −0.339636 − 0.0607957i
\(71\) −11.5560 −1.37145 −0.685724 − 0.727861i \(-0.740514\pi\)
−0.685724 + 0.727861i \(0.740514\pi\)
\(72\) −6.17076 + 5.82424i −0.727231 + 0.686393i
\(73\) 0.501387 0.0586829 0.0293415 − 0.999569i \(-0.490659\pi\)
0.0293415 + 0.999569i \(0.490659\pi\)
\(74\) 8.48225 + 1.43226i 0.986042 + 0.166497i
\(75\) 4.87725 + 6.49695i 0.563177 + 0.750204i
\(76\) 0.927639 − 2.50811i 0.106408 − 0.287700i
\(77\) 15.4252 1.75786
\(78\) −2.43601 − 4.83360i −0.275823 − 0.547297i
\(79\) 11.0324 1.24124 0.620622 − 0.784110i \(-0.286880\pi\)
0.620622 + 0.784110i \(0.286880\pi\)
\(80\) −1.44837 + 1.69018i −0.161933 + 0.188968i
\(81\) −7.59744 − 4.82482i −0.844160 − 0.536092i
\(82\) −1.31961 + 7.37202i −0.145726 + 0.814103i
\(83\) 7.18404i 0.788551i 0.918992 + 0.394276i \(0.129004\pi\)
−0.918992 + 0.394276i \(0.870996\pi\)
\(84\) −12.1777 + 3.62993i −1.32869 + 0.396058i
\(85\) − 1.74662i − 0.189447i
\(86\) −1.88983 + 10.5576i −0.203786 + 1.13845i
\(87\) 11.9296 − 8.95551i 1.27898 − 0.960131i
\(88\) 6.02680 − 10.2537i 0.642459 − 1.09304i
\(89\) 3.48303 0.369201 0.184600 − 0.982814i \(-0.440901\pi\)
0.184600 + 0.982814i \(0.440901\pi\)
\(90\) −2.11545 − 1.04817i −0.222988 − 0.110487i
\(91\) − 8.10587i − 0.849726i
\(92\) 4.18355 − 11.3113i 0.436165 − 1.17929i
\(93\) 2.29476 + 3.05683i 0.237955 + 0.316978i
\(94\) −7.47616 − 1.33825i −0.771107 − 0.138030i
\(95\) 0.744041 0.0763370
\(96\) −2.34502 + 9.51319i −0.239338 + 0.970936i
\(97\) − 7.96066i − 0.808283i −0.914697 − 0.404141i \(-0.867570\pi\)
0.914697 − 0.404141i \(-0.132430\pi\)
\(98\) −8.98731 − 1.60875i −0.907856 − 0.162508i
\(99\) 12.1137 + 3.52142i 1.21747 + 0.353916i
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.c.d.443.92 yes 96
3.2 odd 2 inner 888.2.c.d.443.5 ✓ 96
8.3 odd 2 inner 888.2.c.d.443.90 yes 96
24.11 even 2 inner 888.2.c.d.443.7 yes 96
37.36 even 2 inner 888.2.c.d.443.6 yes 96
111.110 odd 2 inner 888.2.c.d.443.91 yes 96
296.147 odd 2 inner 888.2.c.d.443.8 yes 96
888.443 even 2 inner 888.2.c.d.443.89 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.c.d.443.5 ✓ 96 3.2 odd 2 inner
888.2.c.d.443.6 yes 96 37.36 even 2 inner
888.2.c.d.443.7 yes 96 24.11 even 2 inner
888.2.c.d.443.8 yes 96 296.147 odd 2 inner
888.2.c.d.443.89 yes 96 888.443 even 2 inner
888.2.c.d.443.90 yes 96 8.3 odd 2 inner
888.2.c.d.443.91 yes 96 111.110 odd 2 inner
888.2.c.d.443.92 yes 96 1.1 even 1 trivial