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Properties

Label 888.2.f.b.445.1
Level $888$
Weight $2$
Character 888.445
Analytic conductor $7.091$
Analytic rank $0$
Dimension $44$
Inner twists $2$

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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(445,888)] 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("888.445"); 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([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 445.1
Character \(\chi\) \(=\) 888.445
Dual form 888.2.f.b.445.2

$q$-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+(-1.39660 - 0.222517i) q^{2} -1.00000i q^{3} +(1.90097 + 0.621534i) q^{4} +4.40147i q^{5} +(-0.222517 + 1.39660i) q^{6} -1.69571 q^{7} +(-2.51659 - 1.29103i) q^{8} -1.00000 q^{9} +(0.979403 - 6.14709i) q^{10} +2.08645i q^{11} +(0.621534 - 1.90097i) q^{12} +0.687546i q^{13} +(2.36822 + 0.377324i) q^{14} +4.40147 q^{15} +(3.22739 + 2.36304i) q^{16} +3.68856 q^{17} +(1.39660 + 0.222517i) q^{18} -2.86790i q^{19} +(-2.73566 + 8.36708i) q^{20} +1.69571i q^{21} +(0.464271 - 2.91393i) q^{22} -3.21526 q^{23} +(-1.29103 + 2.51659i) q^{24} -14.3730 q^{25} +(0.152991 - 0.960226i) q^{26} +1.00000i q^{27} +(-3.22349 - 1.05394i) q^{28} +6.84958i q^{29} +(-6.14709 - 0.979403i) q^{30} -2.02754 q^{31} +(-3.98155 - 4.01836i) q^{32} +2.08645 q^{33} +(-5.15143 - 0.820768i) q^{34} -7.46360i q^{35} +(-1.90097 - 0.621534i) q^{36} -1.00000i q^{37} +(-0.638156 + 4.00530i) q^{38} +0.687546 q^{39} +(5.68244 - 11.0767i) q^{40} -6.69225 q^{41} +(0.377324 - 2.36822i) q^{42} -7.41574i q^{43} +(-1.29680 + 3.96628i) q^{44} -4.40147i q^{45} +(4.49042 + 0.715450i) q^{46} +1.07133 q^{47} +(2.36304 - 3.22739i) q^{48} -4.12458 q^{49} +(20.0732 + 3.19823i) q^{50} -3.68856i q^{51} +(-0.427333 + 1.30701i) q^{52} -2.99631i q^{53} +(0.222517 - 1.39660i) q^{54} -9.18345 q^{55} +(4.26740 + 2.18921i) q^{56} -2.86790 q^{57} +(1.52415 - 9.56611i) q^{58} -6.10312i q^{59} +(8.36708 + 2.73566i) q^{60} +13.1839i q^{61} +(2.83166 + 0.451162i) q^{62} +1.69571 q^{63} +(4.66647 + 6.49800i) q^{64} -3.02621 q^{65} +(-2.91393 - 0.464271i) q^{66} -6.43114i q^{67} +(7.01185 + 2.29257i) q^{68} +3.21526i q^{69} +(-1.66078 + 10.4237i) q^{70} -7.80505 q^{71} +(2.51659 + 1.29103i) q^{72} -13.8509 q^{73} +(-0.222517 + 1.39660i) q^{74} +14.3730i q^{75} +(1.78250 - 5.45179i) q^{76} -3.53801i q^{77} +(-0.960226 - 0.152991i) q^{78} -7.93106 q^{79} +(-10.4008 + 14.2053i) q^{80} +1.00000 q^{81} +(9.34638 + 1.48914i) q^{82} +14.2158i q^{83} +(-1.05394 + 3.22349i) q^{84} +16.2351i q^{85} +(-1.65013 + 10.3568i) q^{86} +6.84958 q^{87} +(2.69367 - 5.25074i) q^{88} -1.81697 q^{89} +(-0.979403 + 6.14709i) q^{90} -1.16588i q^{91} +(-6.11211 - 1.99839i) q^{92} +2.02754i q^{93} +(-1.49621 - 0.238388i) q^{94} +12.6230 q^{95} +(-4.01836 + 3.98155i) q^{96} -3.61017 q^{97} +(5.76038 + 0.917790i) q^{98} -2.08645i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q + 2 q^{4} + 2 q^{6} - 6 q^{8} - 44 q^{9} - 4 q^{10} - 4 q^{12} - 2 q^{14} - 4 q^{15} + 10 q^{16} + 20 q^{17} + 8 q^{20} + 4 q^{24} - 76 q^{25} + 8 q^{26} - 10 q^{28} - 4 q^{30} - 4 q^{31} + 10 q^{32}+ \cdots + 106 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39660 − 0.222517i −0.987544 − 0.157343i
\(3\) − 1.00000i − 0.577350i
\(4\) 1.90097 + 0.621534i 0.950486 + 0.310767i
\(5\) 4.40147i 1.96840i 0.177066 + 0.984199i \(0.443339\pi\)
−0.177066 + 0.984199i \(0.556661\pi\)
\(6\) −0.222517 + 1.39660i −0.0908423 + 0.570159i
\(7\) −1.69571 −0.640917 −0.320458 − 0.947263i \(-0.603837\pi\)
−0.320458 + 0.947263i \(0.603837\pi\)
\(8\) −2.51659 − 1.29103i −0.889750 − 0.456449i
\(9\) −1.00000 −0.333333
\(10\) 0.979403 − 6.14709i 0.309714 − 1.94388i
\(11\) 2.08645i 0.629088i 0.949243 + 0.314544i \(0.101852\pi\)
−0.949243 + 0.314544i \(0.898148\pi\)
\(12\) 0.621534 − 1.90097i 0.179421 − 0.548763i
\(13\) 0.687546i 0.190691i 0.995444 + 0.0953455i \(0.0303956\pi\)
−0.995444 + 0.0953455i \(0.969604\pi\)
\(14\) 2.36822 + 0.377324i 0.632933 + 0.100844i
\(15\) 4.40147 1.13646
\(16\) 3.22739 + 2.36304i 0.806848 + 0.590760i
\(17\) 3.68856 0.894607 0.447303 − 0.894382i \(-0.352384\pi\)
0.447303 + 0.894382i \(0.352384\pi\)
\(18\) 1.39660 + 0.222517i 0.329181 + 0.0524478i
\(19\) − 2.86790i − 0.657940i −0.944340 − 0.328970i \(-0.893298\pi\)
0.944340 − 0.328970i \(-0.106702\pi\)
\(20\) −2.73566 + 8.36708i −0.611713 + 1.87094i
\(21\) 1.69571i 0.370033i
\(22\) 0.464271 − 2.91393i 0.0989829 − 0.621252i
\(23\) −3.21526 −0.670427 −0.335214 − 0.942142i \(-0.608809\pi\)
−0.335214 + 0.942142i \(0.608809\pi\)
\(24\) −1.29103 + 2.51659i −0.263531 + 0.513697i
\(25\) −14.3730 −2.87459
\(26\) 0.152991 − 0.960226i 0.0300040 − 0.188316i
\(27\) 1.00000i 0.192450i
\(28\) −3.22349 − 1.05394i −0.609182 − 0.199176i
\(29\) 6.84958i 1.27193i 0.771716 + 0.635967i \(0.219399\pi\)
−0.771716 + 0.635967i \(0.780601\pi\)
\(30\) −6.14709 − 0.979403i −1.12230 − 0.178814i
\(31\) −2.02754 −0.364157 −0.182078 − 0.983284i \(-0.558282\pi\)
−0.182078 + 0.983284i \(0.558282\pi\)
\(32\) −3.98155 − 4.01836i −0.703845 − 0.710353i
\(33\) 2.08645 0.363204
\(34\) −5.15143 − 0.820768i −0.883464 − 0.140761i
\(35\) − 7.46360i − 1.26158i
\(36\) −1.90097 − 0.621534i −0.316829 − 0.103589i
\(37\) − 1.00000i − 0.164399i
\(38\) −0.638156 + 4.00530i −0.103523 + 0.649745i
\(39\) 0.687546 0.110095
\(40\) 5.68244 − 11.0767i 0.898473 − 1.75138i
\(41\) −6.69225 −1.04515 −0.522577 − 0.852592i \(-0.675029\pi\)
−0.522577 + 0.852592i \(0.675029\pi\)
\(42\) 0.377324 − 2.36822i 0.0582223 − 0.365424i
\(43\) − 7.41574i − 1.13089i −0.824786 − 0.565445i \(-0.808705\pi\)
0.824786 − 0.565445i \(-0.191295\pi\)
\(44\) −1.29680 + 3.96628i −0.195500 + 0.597940i
\(45\) − 4.40147i − 0.656133i
\(46\) 4.49042 + 0.715450i 0.662077 + 0.105487i
\(47\) 1.07133 0.156269 0.0781345 − 0.996943i \(-0.475104\pi\)
0.0781345 + 0.996943i \(0.475104\pi\)
\(48\) 2.36304 − 3.22739i 0.341075 − 0.465834i
\(49\) −4.12458 −0.589226
\(50\) 20.0732 + 3.19823i 2.83878 + 0.452298i
\(51\) − 3.68856i − 0.516502i
\(52\) −0.427333 + 1.30701i −0.0592605 + 0.181249i
\(53\) − 2.99631i − 0.411575i −0.978597 − 0.205788i \(-0.934024\pi\)
0.978597 − 0.205788i \(-0.0659756\pi\)
\(54\) 0.222517 − 1.39660i 0.0302808 − 0.190053i
\(55\) −9.18345 −1.23830
\(56\) 4.26740 + 2.18921i 0.570255 + 0.292546i
\(57\) −2.86790 −0.379862
\(58\) 1.52415 − 9.56611i 0.200131 − 1.25609i
\(59\) − 6.10312i − 0.794559i −0.917698 − 0.397279i \(-0.869954\pi\)
0.917698 − 0.397279i \(-0.130046\pi\)
\(60\) 8.36708 + 2.73566i 1.08018 + 0.353173i
\(61\) 13.1839i 1.68802i 0.536326 + 0.844011i \(0.319812\pi\)
−0.536326 + 0.844011i \(0.680188\pi\)
\(62\) 2.83166 + 0.451162i 0.359621 + 0.0572977i
\(63\) 1.69571 0.213639
\(64\) 4.66647 + 6.49800i 0.583309 + 0.812250i
\(65\) −3.02621 −0.375356
\(66\) −2.91393 − 0.464271i −0.358680 − 0.0571478i
\(67\) − 6.43114i − 0.785689i −0.919605 − 0.392844i \(-0.871491\pi\)
0.919605 − 0.392844i \(-0.128509\pi\)
\(68\) 7.01185 + 2.29257i 0.850311 + 0.278014i
\(69\) 3.21526i 0.387071i
\(70\) −1.66078 + 10.4237i −0.198501 + 1.24586i
\(71\) −7.80505 −0.926289 −0.463144 − 0.886283i \(-0.653279\pi\)
−0.463144 + 0.886283i \(0.653279\pi\)
\(72\) 2.51659 + 1.29103i 0.296583 + 0.152150i
\(73\) −13.8509 −1.62113 −0.810563 − 0.585652i \(-0.800839\pi\)
−0.810563 + 0.585652i \(0.800839\pi\)
\(74\) −0.222517 + 1.39660i −0.0258671 + 0.162351i
\(75\) 14.3730i 1.65965i
\(76\) 1.78250 − 5.45179i 0.204466 − 0.625363i
\(77\) − 3.53801i − 0.403193i
\(78\) −0.960226 − 0.152991i −0.108724 − 0.0173228i
\(79\) −7.93106 −0.892313 −0.446157 − 0.894955i \(-0.647208\pi\)
−0.446157 + 0.894955i \(0.647208\pi\)
\(80\) −10.4008 + 14.2053i −1.16285 + 1.58820i
\(81\) 1.00000 0.111111
\(82\) 9.34638 + 1.48914i 1.03213 + 0.164448i
\(83\) 14.2158i 1.56038i 0.625542 + 0.780191i \(0.284878\pi\)
−0.625542 + 0.780191i \(0.715122\pi\)
\(84\) −1.05394 + 3.22349i −0.114994 + 0.351712i
\(85\) 16.2351i 1.76094i
\(86\) −1.65013 + 10.3568i −0.177938 + 1.11680i
\(87\) 6.84958 0.734352
\(88\) 2.69367 − 5.25074i 0.287147 − 0.559731i
\(89\) −1.81697 −0.192599 −0.0962993 − 0.995352i \(-0.530701\pi\)
−0.0962993 + 0.995352i \(0.530701\pi\)
\(90\) −0.979403 + 6.14709i −0.103238 + 0.647960i
\(91\) − 1.16588i − 0.122217i
\(92\) −6.11211 − 1.99839i −0.637232 − 0.208347i
\(93\) 2.02754i 0.210246i
\(94\) −1.49621 − 0.238388i −0.154322 − 0.0245879i
\(95\) 12.6230 1.29509
\(96\) −4.01836 + 3.98155i −0.410123 + 0.406365i
\(97\) −3.61017 −0.366557 −0.183278 − 0.983061i \(-0.558671\pi\)
−0.183278 + 0.983061i \(0.558671\pi\)
\(98\) 5.76038 + 0.917790i 0.581886 + 0.0927108i
\(99\) − 2.08645i − 0.209696i
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.f.b.445.1 ✓ 44
4.3 odd 2 3552.2.f.b.1777.44 44
8.3 odd 2 3552.2.f.b.1777.1 44
8.5 even 2 inner 888.2.f.b.445.2 yes 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.f.b.445.1 ✓ 44 1.1 even 1 trivial
888.2.f.b.445.2 yes 44 8.5 even 2 inner
3552.2.f.b.1777.1 44 8.3 odd 2
3552.2.f.b.1777.44 44 4.3 odd 2