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Properties

Label 888.2.c.d.443.18
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 443.18
Character \(\chi\) \(=\) 888.443
Dual form 888.2.c.d.443.19

$q$-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} +(-8.59176 + 0.426285i) 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 \(-0.709725\pi\)
0.612225 − 0.790684i \(-0.290275\pi\)
\(8\) −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 \(0.247684\pi\)
−0.712234 + 0.701942i \(0.752316\pi\)
\(12\) −2.89601 − 1.90082i −0.836007 − 0.548719i
\(13\) −5.63372 −1.56251 −0.781256 − 0.624211i \(-0.785421\pi\)
−0.781256 + 0.624211i \(0.785421\pi\)
\(14\) −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 \(-0.434591\pi\)
0.204046 + 0.978961i \(0.434591\pi\)
\(18\) −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 \(-0.642326\pi\)
−0.432379 + 0.901692i \(0.642326\pi\)
\(32\) 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 \(0.315276\pi\)
−0.548297 + 0.836284i \(0.684724\pi\)
\(42\) 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 \(-0.923228\pi\)
−0.971055 + 0.238854i \(0.923228\pi\)
\(60\) −0.545848 + 0.831632i −0.0704686 + 0.107363i
\(61\) −10.0628 −1.28841 −0.644203 − 0.764854i \(-0.722811\pi\)
−0.644203 + 0.764854i \(0.722811\pi\)
\(62\) 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 \(0.517241\pi\)
\(74\) −8.59176 + 0.426285i −0.998771 + 0.0495546i
\(75\) −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 \(-0.552083\pi\)
−0.162893 + 0.986644i \(0.552083\pi\)
\(80\) 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 \(0.0134945\pi\)
−0.999101 + 0.0423816i \(0.986505\pi\)
\(84\) −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 \(-0.644991\pi\)
−0.439915 + 0.898040i \(0.644991\pi\)
\(90\) 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.443.18 yes 96
3.2 odd 2 inner 888.2.c.d.443.79 yes 96
8.3 odd 2 inner 888.2.c.d.443.20 yes 96
24.11 even 2 inner 888.2.c.d.443.77 yes 96
37.36 even 2 inner 888.2.c.d.443.80 yes 96
111.110 odd 2 inner 888.2.c.d.443.17 ✓ 96
296.147 odd 2 inner 888.2.c.d.443.78 yes 96
888.443 even 2 inner 888.2.c.d.443.19 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.c.d.443.17 ✓ 96 111.110 odd 2 inner
888.2.c.d.443.18 yes 96 1.1 even 1 trivial
888.2.c.d.443.19 yes 96 888.443 even 2 inner
888.2.c.d.443.20 yes 96 8.3 odd 2 inner
888.2.c.d.443.77 yes 96 24.11 even 2 inner
888.2.c.d.443.78 yes 96 296.147 odd 2 inner
888.2.c.d.443.79 yes 96 3.2 odd 2 inner
888.2.c.d.443.80 yes 96 37.36 even 2 inner