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Properties

Label 888.2.c.d.443.8
Level $888$
Weight $2$
Character 888.443
Analytic conductor $7.091$
Analytic rank $0$
Dimension $96$
Inner twists $8$

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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(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 443.8
Character \(\chi\) \(=\) 888.443
Dual form 888.2.c.d.443.5

$q$-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+(-1.39209 + 0.249187i) q^{2} +(1.03985 + 1.38518i) q^{3} +(1.87581 - 0.693779i) q^{4} +0.556466i q^{5} +(-1.79273 - 1.66917i) 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} +(2.91157 + 1.87690i) 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} +(0.447924 - 4.21893i) 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} +(-4.52086 - 1.88729i) 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} +(0.928836 - 0.997593i) 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} +(0.427752 + 5.98473i) 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} +(-6.12291 + 6.57616i) 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} +(6.76371 + 1.50073i) 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} +(6.30973 - 3.76660i) 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} +(-1.04443 + 1.62019i) 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} +(-7.01894 + 7.53852i) 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} +(-2.08678 - 8.22468i) 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} +(3.96147 + 3.68843i) 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} +(6.88494 - 10.6803i) 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} +(2.34769 + 0.249255i) 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} +(-9.78964 - 0.403723i) 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 \(0.960290\pi\)
\(6\) −1.79273 − 1.66917i −0.731878 − 0.681435i
\(7\) − 3.66824i − 1.38647i −0.720714 − 0.693233i \(-0.756186\pi\)
0.720714 − 0.693233i \(-0.243814\pi\)
\(8\) −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 \(-0.218562\pi\)
\(12\) 2.91157 + 1.87690i 0.840498 + 0.541815i
\(13\) −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 \(-0.375707\pi\)
0.380631 + 0.924727i \(0.375707\pi\)
\(18\) 0.447924 − 4.21893i 0.105577 − 0.994411i
\(19\) 1.33708i 0.306748i 0.988168 + 0.153374i \(0.0490139\pi\)
−0.988168 + 0.153374i \(0.950986\pi\)
\(20\) 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 \(-0.216404\pi\)
\(24\) −4.52086 − 1.88729i −0.922816 − 0.385241i
\(25\) 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 \(-0.294973\pi\)
\(30\) 0.928836 − 0.997593i 0.169581 − 0.182135i
\(31\) 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
\(36\) 0.427752 + 5.98473i 0.0712920 + 0.997455i
\(37\) 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 \(0.864299\pi\)
\(42\) −6.12291 + 6.57616i −0.944786 + 1.01472i
\(43\) − 7.58400i − 1.15655i −0.815842 − 0.578275i \(-0.803726\pi\)
0.815842 − 0.578275i \(-0.196274\pi\)
\(44\) −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 \(-0.371894\pi\)
0.391682 + 0.920101i \(0.371894\pi\)
\(48\) 6.76371 + 1.50073i 0.976258 + 0.216612i
\(49\) −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 \(-0.869793\pi\)
−0.917496 + 0.397744i \(0.869793\pi\)
\(54\) 6.30973 − 3.76660i 0.858646 − 0.512569i
\(55\) 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 \(-0.606836\pi\)
−0.329369 + 0.944201i \(0.606836\pi\)
\(60\) −1.04443 + 1.62019i −0.134836 + 0.209166i
\(61\) 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
\(66\) −7.01894 + 7.53852i −0.863972 + 0.927928i
\(67\) 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 \(-0.259486\pi\)
0.685724 + 0.727861i \(0.259486\pi\)
\(72\) −2.08678 − 8.22468i −0.245930 − 0.969288i
\(73\) 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
\(78\) 3.96147 + 3.68843i 0.448548 + 0.417633i
\(79\) 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 \(0.870996\pi\)
\(84\) 6.88494 − 10.6803i 0.751208 − 1.16532i
\(85\) 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 \(-0.559099\pi\)
−0.184600 + 0.982814i \(0.559099\pi\)
\(90\) 2.34769 + 0.249255i 0.247468 + 0.0262737i
\(91\) 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
\(96\) −9.78964 − 0.403723i −0.999151 − 0.0412048i
\(97\) 7.96066i 0.808283i 0.914697 + 0.404141i \(0.132430\pi\)
−0.914697 + 0.404141i \(0.867570\pi\)
\(98\) 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.443.8 yes 96
3.2 odd 2 inner 888.2.c.d.443.89 yes 96
8.3 odd 2 inner 888.2.c.d.443.6 yes 96
24.11 even 2 inner 888.2.c.d.443.91 yes 96
37.36 even 2 inner 888.2.c.d.443.90 yes 96
111.110 odd 2 inner 888.2.c.d.443.7 yes 96
296.147 odd 2 inner 888.2.c.d.443.92 yes 96
888.443 even 2 inner 888.2.c.d.443.5 ✓ 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.c.d.443.5 ✓ 96 888.443 even 2 inner
888.2.c.d.443.6 yes 96 8.3 odd 2 inner
888.2.c.d.443.7 yes 96 111.110 odd 2 inner
888.2.c.d.443.8 yes 96 1.1 even 1 trivial
888.2.c.d.443.89 yes 96 3.2 odd 2 inner
888.2.c.d.443.90 yes 96 37.36 even 2 inner
888.2.c.d.443.91 yes 96 24.11 even 2 inner
888.2.c.d.443.92 yes 96 296.147 odd 2 inner