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Properties

Label 888.2.c.d.443.20
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 443.20
Character \(\chi\) \(=\) 888.443
Dual form 888.2.c.d.443.17

$q$-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} +(1.55963 - 1.88879i) 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} +(-0.698741 + 3.39290i) 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} +(-1.24082 + 4.05714i) 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} +(-1.37282 - 4.70270i) 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} +(0.542394 + 0.447871i) 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} +(-1.13509 - 5.89165i) 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} +(7.90254 + 6.52536i) 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} +(4.82873 + 4.96823i) 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} +(-0.708964 - 7.31419i) 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} +(-0.974320 - 0.200654i) 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} +(8.79452 + 7.26190i) 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} +(5.31871 + 6.61146i) 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} +(8.78654 - 10.6409i) 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} +(-14.1956 - 2.92347i) 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} +(-1.16507 - 0.356320i) 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} +(-9.32341 - 3.01232i) 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 \(0.0204534\pi\)
−0.997936 + 0.0642119i \(0.979547\pi\)
\(6\) 1.55963 − 1.88879i 0.636718 − 0.771097i
\(7\) 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 \(0.247684\pi\)
−0.712234 + 0.701942i \(0.752316\pi\)
\(12\) −0.698741 + 3.39290i −0.201709 + 0.979445i
\(13\) 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 \(-0.434591\pi\)
0.204046 + 0.978961i \(0.434591\pi\)
\(18\) −1.24082 + 4.05714i −0.292464 + 0.956276i
\(19\) − 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 \(0.0962172\pi\)
−0.954662 + 0.297693i \(0.903783\pi\)
\(24\) −1.37282 − 4.70270i −0.280225 − 0.959934i
\(25\) 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 \(-0.866355\pi\)
0.913147 − 0.407630i \(-0.133645\pi\)
\(30\) 0.542394 + 0.447871i 0.0990272 + 0.0817697i
\(31\) 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
\(36\) −1.13509 − 5.89165i −0.189181 − 0.981942i
\(37\) −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 \(0.684724\pi\)
\(42\) 7.90254 + 6.52536i 1.21939 + 1.00688i
\(43\) 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 \(-0.261958\pi\)
0.680050 + 0.733165i \(0.261958\pi\)
\(48\) 4.82873 + 4.96823i 0.696967 + 0.717103i
\(49\) −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 \(-0.525602\pi\)
−0.0803449 + 0.996767i \(0.525602\pi\)
\(54\) −0.708964 − 7.31419i −0.0964777 − 0.995335i
\(55\) −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 \(0.923228\pi\)
\(60\) −0.974320 − 0.200654i −0.125784 − 0.0259043i
\(61\) 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
\(66\) 8.79452 + 7.26190i 1.08253 + 0.893878i
\(67\) −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 \(-0.429055\pi\)
0.221041 + 0.975265i \(0.429055\pi\)
\(72\) 5.31871 + 6.61146i 0.626816 + 0.779168i
\(73\) −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
\(78\) 8.78654 − 10.6409i 0.994879 − 1.20485i
\(79\) 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 \(0.0134945\pi\)
−0.999101 + 0.0423816i \(0.986505\pi\)
\(84\) −14.1956 − 2.92347i −1.54886 − 0.318977i
\(85\) 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 \(0.644991\pi\)
\(90\) −1.16507 − 0.356320i −0.122809 − 0.0375594i
\(91\) 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
\(96\) −9.32341 − 3.01232i −0.951566 − 0.307444i
\(97\) 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.443.20 yes 96
3.2 odd 2 inner 888.2.c.d.443.77 yes 96
8.3 odd 2 inner 888.2.c.d.443.18 yes 96
24.11 even 2 inner 888.2.c.d.443.79 yes 96
37.36 even 2 inner 888.2.c.d.443.78 yes 96
111.110 odd 2 inner 888.2.c.d.443.19 yes 96
296.147 odd 2 inner 888.2.c.d.443.80 yes 96
888.443 even 2 inner 888.2.c.d.443.17 ✓ 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.c.d.443.17 ✓ 96 888.443 even 2 inner
888.2.c.d.443.18 yes 96 8.3 odd 2 inner
888.2.c.d.443.19 yes 96 111.110 odd 2 inner
888.2.c.d.443.20 yes 96 1.1 even 1 trivial
888.2.c.d.443.77 yes 96 3.2 odd 2 inner
888.2.c.d.443.78 yes 96 37.36 even 2 inner
888.2.c.d.443.79 yes 96 24.11 even 2 inner
888.2.c.d.443.80 yes 96 296.147 odd 2 inner