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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 443.77
Character \(\chi\) \(=\) 888.443
Dual form 888.2.c.d.443.80

$q$-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+(1.24971 - 0.661983i) q^{2} +(-1.59972 - 0.663998i) q^{3} +(1.12356 - 1.65458i) q^{4} -0.287164i q^{5} +(-2.43874 + 0.229182i) q^{6} +4.18391i q^{7} +(0.308820 - 2.81152i) q^{8} +(2.11821 + 2.12442i) q^{9} +(-0.190098 - 0.358873i) q^{10} -4.65616i q^{11} +(-2.89601 + 1.90082i) q^{12} +5.63372 q^{13} +(2.76968 + 5.22867i) q^{14} +(-0.190677 + 0.459383i) q^{15} +(-1.47524 - 3.71802i) q^{16} -1.68260 q^{17} +(4.05349 + 1.25269i) q^{18} -5.89870i q^{19} +(-0.475135 - 0.322645i) q^{20} +(2.77811 - 6.69308i) q^{21} +(-3.08230 - 5.81885i) q^{22} -2.85537i q^{23} +(-2.36087 + 4.29259i) q^{24} +4.91754 q^{25} +(7.04052 - 3.72943i) q^{26} +(-1.97794 - 4.80497i) q^{27} +(6.92259 + 4.70086i) q^{28} +4.39031i q^{29} +(0.0658129 + 0.700321i) q^{30} +4.81477 q^{31} +(-4.30489 - 3.66987i) q^{32} +(-3.09168 + 7.44855i) q^{33} +(-2.10277 + 1.11386i) q^{34} +1.20147 q^{35} +(5.89495 - 1.11783i) q^{36} +(-5.22751 + 3.11017i) q^{37} +(-3.90484 - 7.37167i) q^{38} +(-9.01238 - 3.74078i) q^{39} +(-0.807368 - 0.0886821i) q^{40} -10.7097i q^{41} +(-0.958876 - 10.2035i) q^{42} +2.32524i q^{43} +(-7.70397 - 5.23146i) q^{44} +(0.610059 - 0.608275i) q^{45} +(-1.89021 - 3.56839i) q^{46} -9.32438 q^{47} +(-0.108784 + 6.92735i) q^{48} -10.5051 q^{49} +(6.14550 - 3.25533i) q^{50} +(2.69170 + 1.11725i) q^{51} +(6.32980 - 9.32141i) q^{52} +1.16984 q^{53} +(-5.65266 - 4.69547i) q^{54} -1.33708 q^{55} +(11.7631 + 1.29207i) q^{56} +(-3.91673 + 9.43627i) q^{57} +(2.90631 + 5.48662i) q^{58} +14.9176 q^{59} +(0.545848 + 0.831632i) q^{60} +10.0628 q^{61} +(6.01707 - 3.18730i) q^{62} +(-8.88839 + 8.86240i) q^{63} +(-7.80926 - 1.73651i) q^{64} -1.61780i q^{65} +(1.06711 + 11.3552i) q^{66} -9.33020 q^{67} +(-1.89050 + 2.78400i) q^{68} +(-1.89596 + 4.56780i) q^{69} +(1.50149 - 0.795352i) q^{70} -3.72505 q^{71} +(6.62700 - 5.29933i) q^{72} -0.925082 q^{73} +(-4.47400 + 7.34733i) q^{74} +(-7.86669 - 3.26524i) q^{75} +(-9.75985 - 6.62752i) q^{76} +19.4809 q^{77} +(-13.7392 + 1.29115i) q^{78} +2.89565 q^{79} +(-1.06768 + 0.423637i) q^{80} +(-0.0263502 + 8.99996i) q^{81} +(-7.08961 - 13.3840i) q^{82} -0.772230i q^{83} +(-7.95285 - 12.1166i) q^{84} +0.483184i q^{85} +(1.53927 + 2.90588i) q^{86} +(2.91516 - 7.02327i) q^{87} +(-13.0909 - 1.43792i) q^{88} +8.30030 q^{89} +(0.359729 - 1.16402i) q^{90} +23.5709i q^{91} +(-4.72443 - 3.20817i) q^{92} +(-7.70229 - 3.19700i) q^{93} +(-11.6528 + 6.17258i) q^{94} -1.69390 q^{95} +(4.44984 + 8.72920i) q^{96} +15.3552i q^{97} +(-13.1283 + 6.95418i) q^{98} +(9.89165 - 9.86273i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.24971 − 0.661983i 0.883679 − 0.468093i
\(3\) −1.59972 − 0.663998i −0.923599 − 0.383360i
\(4\) 1.12356 − 1.65458i 0.561778 − 0.827288i
\(5\) − 0.287164i − 0.128424i −0.997936 − 0.0642119i \(-0.979547\pi\)
0.997936 − 0.0642119i \(-0.0204534\pi\)
\(6\) −2.43874 + 0.229182i −0.995613 + 0.0935632i
\(7\) 4.18391i 1.58137i 0.612225 + 0.790684i \(0.290275\pi\)
−0.612225 + 0.790684i \(0.709725\pi\)
\(8\) 0.308820 − 2.81152i 0.109184 − 0.994022i
\(9\) 2.11821 + 2.12442i 0.706071 + 0.708141i
\(10\) −0.190098 − 0.358873i −0.0601143 − 0.113485i
\(11\) − 4.65616i − 1.40388i −0.712234 − 0.701942i \(-0.752316\pi\)
0.712234 − 0.701942i \(-0.247684\pi\)
\(12\) −2.89601 + 1.90082i −0.836007 + 0.548719i
\(13\) 5.63372 1.56251 0.781256 − 0.624211i \(-0.214579\pi\)
0.781256 + 0.624211i \(0.214579\pi\)
\(14\) 2.76968 + 5.22867i 0.740227 + 1.39742i
\(15\) −0.190677 + 0.459383i −0.0492325 + 0.118612i
\(16\) −1.47524 − 3.71802i −0.368810 − 0.929505i
\(17\) −1.68260 −0.408091 −0.204046 − 0.978961i \(-0.565409\pi\)
−0.204046 + 0.978961i \(0.565409\pi\)
\(18\) 4.05349 + 1.25269i 0.955416 + 0.295263i
\(19\) − 5.89870i − 1.35325i −0.736326 − 0.676627i \(-0.763441\pi\)
0.736326 − 0.676627i \(-0.236559\pi\)
\(20\) −0.475135 − 0.322645i −0.106243 − 0.0721457i
\(21\) 2.77811 − 6.69308i 0.606232 − 1.46055i
\(22\) −3.08230 − 5.81885i −0.657148 − 1.24058i
\(23\) − 2.85537i − 0.595386i −0.954662 − 0.297693i \(-0.903783\pi\)
0.954662 − 0.297693i \(-0.0962172\pi\)
\(24\) −2.36087 + 4.29259i −0.481910 + 0.876221i
\(25\) 4.91754 0.983507
\(26\) 7.04052 − 3.72943i 1.38076 − 0.731401i
\(27\) −1.97794 − 4.80497i −0.380654 − 0.924718i
\(28\) 6.92259 + 4.70086i 1.30825 + 0.888378i
\(29\) 4.39031i 0.815261i 0.913147 + 0.407630i \(0.133645\pi\)
−0.913147 + 0.407630i \(0.866355\pi\)
\(30\) 0.0658129 + 0.700321i 0.0120157 + 0.127860i
\(31\) 4.81477 0.864758 0.432379 − 0.901692i \(-0.357674\pi\)
0.432379 + 0.901692i \(0.357674\pi\)
\(32\) −4.30489 − 3.66987i −0.761004 − 0.648747i
\(33\) −3.09168 + 7.44855i −0.538193 + 1.29663i
\(34\) −2.10277 + 1.11386i −0.360622 + 0.191025i
\(35\) 1.20147 0.203085
\(36\) 5.89495 − 1.11783i 0.982492 − 0.186306i
\(37\) −5.22751 + 3.11017i −0.859397 + 0.511308i
\(38\) −3.90484 − 7.37167i −0.633449 − 1.19584i
\(39\) −9.01238 − 3.74078i −1.44314 − 0.599004i
\(40\) −0.807368 − 0.0886821i −0.127656 − 0.0140219i
\(41\) − 10.7097i − 1.67257i −0.548297 − 0.836284i \(-0.684724\pi\)
0.548297 − 0.836284i \(-0.315276\pi\)
\(42\) −0.958876 − 10.2035i −0.147958 − 1.57443i
\(43\) 2.32524i 0.354596i 0.984157 + 0.177298i \(0.0567356\pi\)
−0.984157 + 0.177298i \(0.943264\pi\)
\(44\) −7.70397 − 5.23146i −1.16142 − 0.788672i
\(45\) 0.610059 − 0.608275i 0.0909422 − 0.0906763i
\(46\) −1.89021 − 3.56839i −0.278696 − 0.526130i
\(47\) −9.32438 −1.36010 −0.680050 − 0.733165i \(-0.738042\pi\)
−0.680050 + 0.733165i \(0.738042\pi\)
\(48\) −0.108784 + 6.92735i −0.0157016 + 0.999877i
\(49\) −10.5051 −1.50072
\(50\) 6.14550 − 3.25533i 0.869105 − 0.460373i
\(51\) 2.69170 + 1.11725i 0.376913 + 0.156446i
\(52\) 6.32980 − 9.32141i 0.877786 − 1.29265i
\(53\) 1.16984 0.160690 0.0803449 − 0.996767i \(-0.474398\pi\)
0.0803449 + 0.996767i \(0.474398\pi\)
\(54\) −5.65266 − 4.69547i −0.769230 − 0.638973i
\(55\) −1.33708 −0.180292
\(56\) 11.7631 + 1.29207i 1.57191 + 0.172661i
\(57\) −3.91673 + 9.43627i −0.518783 + 1.24987i
\(58\) 2.90631 + 5.48662i 0.381618 + 0.720429i
\(59\) 14.9176 1.94211 0.971055 − 0.238854i \(-0.0767717\pi\)
0.971055 + 0.238854i \(0.0767717\pi\)
\(60\) 0.545848 + 0.831632i 0.0704686 + 0.107363i
\(61\) 10.0628 1.28841 0.644203 − 0.764854i \(-0.277189\pi\)
0.644203 + 0.764854i \(0.277189\pi\)
\(62\) 6.01707 − 3.18730i 0.764169 − 0.404787i
\(63\) −8.88839 + 8.86240i −1.11983 + 1.11656i
\(64\) −7.80926 − 1.73651i −0.976158 − 0.217063i
\(65\) − 1.61780i − 0.200664i
\(66\) 1.06711 + 11.3552i 0.131352 + 1.39773i
\(67\) −9.33020 −1.13987 −0.569933 − 0.821691i \(-0.693031\pi\)
−0.569933 + 0.821691i \(0.693031\pi\)
\(68\) −1.89050 + 2.78400i −0.229257 + 0.337609i
\(69\) −1.89596 + 4.56780i −0.228247 + 0.549898i
\(70\) 1.50149 − 0.795352i 0.179462 − 0.0950628i
\(71\) −3.72505 −0.442082 −0.221041 − 0.975265i \(-0.570945\pi\)
−0.221041 + 0.975265i \(0.570945\pi\)
\(72\) 6.62700 − 5.29933i 0.780999 − 0.624532i
\(73\) −0.925082 −0.108273 −0.0541363 − 0.998534i \(-0.517241\pi\)
−0.0541363 + 0.998534i \(0.517241\pi\)
\(74\) −4.47400 + 7.34733i −0.520092 + 0.854110i
\(75\) −7.86669 − 3.26524i −0.908367 − 0.377037i
\(76\) −9.75985 − 6.62752i −1.11953 − 0.760229i
\(77\) 19.4809 2.22006
\(78\) −13.7392 + 1.29115i −1.55566 + 0.146194i
\(79\) 2.89565 0.325786 0.162893 − 0.986644i \(-0.447917\pi\)
0.162893 + 0.986644i \(0.447917\pi\)
\(80\) −1.06768 + 0.423637i −0.119371 + 0.0473640i
\(81\) −0.0263502 + 8.99996i −0.00292780 + 0.999996i
\(82\) −7.08961 − 13.3840i −0.782917 − 1.47801i
\(83\) − 0.772230i − 0.0847633i −0.999101 − 0.0423816i \(-0.986505\pi\)
0.999101 − 0.0423816i \(-0.0134945\pi\)
\(84\) −7.95285 − 12.1166i −0.867727 − 1.32203i
\(85\) 0.483184i 0.0524087i
\(86\) 1.53927 + 2.90588i 0.165984 + 0.313349i
\(87\) 2.91516 − 7.02327i 0.312538 − 0.752974i
\(88\) −13.0909 − 1.43792i −1.39549 − 0.153282i
\(89\) 8.30030 0.879830 0.439915 − 0.898040i \(-0.355009\pi\)
0.439915 + 0.898040i \(0.355009\pi\)
\(90\) 0.359729 − 1.16402i 0.0379188 − 0.122698i
\(91\) 23.5709i 2.47091i
\(92\) −4.72443 − 3.20817i −0.492556 − 0.334475i
\(93\) −7.70229 − 3.19700i −0.798690 − 0.331513i
\(94\) −11.6528 + 6.17258i −1.20189 + 0.636653i
\(95\) −1.69390 −0.173790
\(96\) 4.44984 + 8.72920i 0.454160 + 0.890920i
\(97\) 15.3552i 1.55908i 0.626351 + 0.779541i \(0.284548\pi\)
−0.626351 + 0.779541i \(0.715452\pi\)
\(98\) −13.1283 + 6.95418i −1.32616 + 0.702478i
\(99\) 9.89165 − 9.86273i 0.994148 − 0.991242i
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.443.77 yes 96
3.2 odd 2 inner 888.2.c.d.443.20 yes 96
8.3 odd 2 inner 888.2.c.d.443.79 yes 96
24.11 even 2 inner 888.2.c.d.443.18 yes 96
37.36 even 2 inner 888.2.c.d.443.19 yes 96
111.110 odd 2 inner 888.2.c.d.443.78 yes 96
296.147 odd 2 inner 888.2.c.d.443.17 ✓ 96
888.443 even 2 inner 888.2.c.d.443.80 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.c.d.443.17 ✓ 96 296.147 odd 2 inner
888.2.c.d.443.18 yes 96 24.11 even 2 inner
888.2.c.d.443.19 yes 96 37.36 even 2 inner
888.2.c.d.443.20 yes 96 3.2 odd 2 inner
888.2.c.d.443.77 yes 96 1.1 even 1 trivial
888.2.c.d.443.78 yes 96 111.110 odd 2 inner
888.2.c.d.443.79 yes 96 8.3 odd 2 inner
888.2.c.d.443.80 yes 96 888.443 even 2 inner