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Properties

Label 888.2.r.e.43.19
Level $888$
Weight $2$
Character 888.43
Analytic conductor $7.091$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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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(43,888)] 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("888.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,2,0,0,0,-2,0,2,-72,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.19
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.19

$q$-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+(0.0571571 - 1.41306i) q^{2} -1.00000i q^{3} +(-1.99347 - 0.161533i) q^{4} +(-0.322645 + 0.322645i) q^{5} +(-1.41306 - 0.0571571i) q^{6} +0.857064i q^{7} +(-0.342196 + 2.80765i) q^{8} -1.00000 q^{9} +(0.437474 + 0.474357i) q^{10} +1.26841i q^{11} +(-0.161533 + 1.99347i) q^{12} +(-2.24823 + 2.24823i) q^{13} +(1.21108 + 0.0489873i) q^{14} +(0.322645 + 0.322645i) q^{15} +(3.94781 + 0.644020i) q^{16} +(-1.69473 - 1.69473i) q^{17} +(-0.0571571 + 1.41306i) q^{18} +(1.36739 + 1.36739i) q^{19} +(0.695299 - 0.591064i) q^{20} +0.857064 q^{21} +(1.79234 + 0.0724989i) q^{22} +(-5.38631 + 5.38631i) q^{23} +(2.80765 + 0.342196i) q^{24} +4.79180i q^{25} +(3.04837 + 3.30538i) q^{26} +1.00000i q^{27} +(0.138444 - 1.70853i) q^{28} +(4.56858 + 4.56858i) q^{29} +(0.474357 - 0.437474i) q^{30} +(3.15323 + 3.15323i) q^{31} +(1.13568 - 5.54168i) q^{32} +1.26841 q^{33} +(-2.49161 + 2.29788i) q^{34} +(-0.276527 - 0.276527i) q^{35} +(1.99347 + 0.161533i) q^{36} +(1.50659 - 5.89323i) q^{37} +(2.01036 - 1.85405i) q^{38} +(2.24823 + 2.24823i) q^{39} +(-0.795466 - 1.01628i) q^{40} +7.84929i q^{41} +(0.0489873 - 1.21108i) q^{42} +(-3.36109 - 3.36109i) q^{43} +(0.204890 - 2.52854i) q^{44} +(0.322645 - 0.322645i) q^{45} +(7.30330 + 7.91903i) q^{46} +1.36069i q^{47} +(0.644020 - 3.94781i) q^{48} +6.26544 q^{49} +(6.77109 + 0.273885i) q^{50} +(-1.69473 + 1.69473i) q^{51} +(4.84493 - 4.11860i) q^{52} -9.41268i q^{53} +(1.41306 + 0.0571571i) q^{54} +(-0.409247 - 0.409247i) q^{55} +(-2.40634 - 0.293284i) q^{56} +(1.36739 - 1.36739i) q^{57} +(6.71679 - 6.19454i) q^{58} +(-9.64918 - 9.64918i) q^{59} +(-0.591064 - 0.695299i) q^{60} +(2.02989 + 2.02989i) q^{61} +(4.63593 - 4.27547i) q^{62} -0.857064i q^{63} +(-7.76580 - 1.92153i) q^{64} -1.45076i q^{65} +(0.0724989 - 1.79234i) q^{66} +13.4603i q^{67} +(3.10463 + 3.65213i) q^{68} +(5.38631 + 5.38631i) q^{69} +(-0.406554 + 0.374943i) q^{70} -8.52826i q^{71} +(0.342196 - 2.80765i) q^{72} +8.93020i q^{73} +(-8.24137 - 2.46574i) q^{74} +4.79180 q^{75} +(-2.50498 - 2.94673i) q^{76} -1.08711 q^{77} +(3.30538 - 3.04837i) q^{78} +(-10.1752 + 10.1752i) q^{79} +(-1.48153 + 1.06595i) q^{80} +1.00000 q^{81} +(11.0915 + 0.448643i) q^{82} -3.47336 q^{83} +(-1.70853 - 0.138444i) q^{84} +1.09359 q^{85} +(-4.94152 + 4.55730i) q^{86} +(4.56858 - 4.56858i) q^{87} +(-3.56126 - 0.434046i) q^{88} +(1.69985 - 1.69985i) q^{89} +(-0.437474 - 0.474357i) q^{90} +(-1.92688 - 1.92688i) q^{91} +(11.6075 - 9.86736i) q^{92} +(3.15323 - 3.15323i) q^{93} +(1.92274 + 0.0777733i) q^{94} -0.882365 q^{95} +(-5.54168 - 1.13568i) q^{96} +(6.72129 + 6.72129i) q^{97} +(0.358114 - 8.85343i) q^{98} -1.26841i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 2 q^{2} - 2 q^{6} + 2 q^{8} - 72 q^{9} + 8 q^{10} - 8 q^{12} + 20 q^{14} + 12 q^{16} + 12 q^{17} - 2 q^{18} + 20 q^{19} - 22 q^{20} - 16 q^{22} + 2 q^{24} - 32 q^{26} - 8 q^{32} - 16 q^{33} + 8 q^{34}+ \cdots - 110 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0571571 − 1.41306i 0.0404162 − 0.999183i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.99347 − 0.161533i −0.996733 − 0.0807663i
\(5\) −0.322645 + 0.322645i −0.144291 + 0.144291i −0.775562 − 0.631271i \(-0.782534\pi\)
0.631271 + 0.775562i \(0.282534\pi\)
\(6\) −1.41306 − 0.0571571i −0.576879 − 0.0233343i
\(7\) 0.857064i 0.323940i 0.986796 + 0.161970i \(0.0517847\pi\)
−0.986796 + 0.161970i \(0.948215\pi\)
\(8\) −0.342196 + 2.80765i −0.120984 + 0.992654i
\(9\) −1.00000 −0.333333
\(10\) 0.437474 + 0.474357i 0.138341 + 0.150005i
\(11\) 1.26841i 0.382441i 0.981547 + 0.191221i \(0.0612446\pi\)
−0.981547 + 0.191221i \(0.938755\pi\)
\(12\) −0.161533 + 1.99347i −0.0466304 + 0.575464i
\(13\) −2.24823 + 2.24823i −0.623546 + 0.623546i −0.946436 − 0.322890i \(-0.895346\pi\)
0.322890 + 0.946436i \(0.395346\pi\)
\(14\) 1.21108 + 0.0489873i 0.323675 + 0.0130924i
\(15\) 0.322645 + 0.322645i 0.0833065 + 0.0833065i
\(16\) 3.94781 + 0.644020i 0.986954 + 0.161005i
\(17\) −1.69473 − 1.69473i −0.411032 − 0.411032i 0.471066 − 0.882098i \(-0.343869\pi\)
−0.882098 + 0.471066i \(0.843869\pi\)
\(18\) −0.0571571 + 1.41306i −0.0134721 + 0.333061i
\(19\) 1.36739 + 1.36739i 0.313702 + 0.313702i 0.846342 − 0.532640i \(-0.178800\pi\)
−0.532640 + 0.846342i \(0.678800\pi\)
\(20\) 0.695299 − 0.591064i 0.155474 − 0.132166i
\(21\) 0.857064 0.187027
\(22\) 1.79234 + 0.0724989i 0.382129 + 0.0154568i
\(23\) −5.38631 + 5.38631i −1.12312 + 1.12312i −0.131854 + 0.991269i \(0.542093\pi\)
−0.991269 + 0.131854i \(0.957907\pi\)
\(24\) 2.80765 + 0.342196i 0.573109 + 0.0698504i
\(25\) 4.79180i 0.958360i
\(26\) 3.04837 + 3.30538i 0.597835 + 0.648238i
\(27\) 1.00000i 0.192450i
\(28\) 0.138444 − 1.70853i 0.0261634 − 0.322881i
\(29\) 4.56858 + 4.56858i 0.848364 + 0.848364i 0.989929 − 0.141565i \(-0.0452135\pi\)
−0.141565 + 0.989929i \(0.545214\pi\)
\(30\) 0.474357 − 0.437474i 0.0866054 − 0.0798715i
\(31\) 3.15323 + 3.15323i 0.566337 + 0.566337i 0.931100 − 0.364763i \(-0.118850\pi\)
−0.364763 + 0.931100i \(0.618850\pi\)
\(32\) 1.13568 − 5.54168i 0.200762 − 0.979640i
\(33\) 1.26841 0.220803
\(34\) −2.49161 + 2.29788i −0.427308 + 0.394083i
\(35\) −0.276527 − 0.276527i −0.0467416 − 0.0467416i
\(36\) 1.99347 + 0.161533i 0.332244 + 0.0269221i
\(37\) 1.50659 − 5.89323i 0.247681 − 0.968842i
\(38\) 2.01036 − 1.85405i 0.326124 − 0.300767i
\(39\) 2.24823 + 2.24823i 0.360005 + 0.360005i
\(40\) −0.795466 − 1.01628i −0.125774 − 0.160688i
\(41\) 7.84929i 1.22585i 0.790140 + 0.612926i \(0.210008\pi\)
−0.790140 + 0.612926i \(0.789992\pi\)
\(42\) 0.0489873 − 1.21108i 0.00755890 − 0.186874i
\(43\) −3.36109 − 3.36109i −0.512561 − 0.512561i 0.402749 − 0.915310i \(-0.368055\pi\)
−0.915310 + 0.402749i \(0.868055\pi\)
\(44\) 0.204890 − 2.52854i 0.0308884 − 0.381192i
\(45\) 0.322645 − 0.322645i 0.0480970 − 0.0480970i
\(46\) 7.30330 + 7.91903i 1.07681 + 1.16760i
\(47\) 1.36069i 0.198478i 0.995064 + 0.0992388i \(0.0316408\pi\)
−0.995064 + 0.0992388i \(0.968359\pi\)
\(48\) 0.644020 − 3.94781i 0.0929562 − 0.569818i
\(49\) 6.26544 0.895063
\(50\) 6.77109 + 0.273885i 0.957577 + 0.0387333i
\(51\) −1.69473 + 1.69473i −0.237309 + 0.237309i
\(52\) 4.84493 − 4.11860i 0.671871 − 0.571148i
\(53\) − 9.41268i − 1.29293i −0.762943 − 0.646466i \(-0.776246\pi\)
0.762943 − 0.646466i \(-0.223754\pi\)
\(54\) 1.41306 + 0.0571571i 0.192293 + 0.00777810i
\(55\) −0.409247 − 0.409247i −0.0551829 − 0.0551829i
\(56\) −2.40634 − 0.293284i −0.321560 − 0.0391917i
\(57\) 1.36739 − 1.36739i 0.181116 − 0.181116i
\(58\) 6.71679 − 6.19454i 0.881958 − 0.813383i
\(59\) −9.64918 − 9.64918i −1.25622 − 1.25622i −0.952885 − 0.303331i \(-0.901901\pi\)
−0.303331 − 0.952885i \(-0.598099\pi\)
\(60\) −0.591064 − 0.695299i −0.0763060 − 0.0897627i
\(61\) 2.02989 + 2.02989i 0.259901 + 0.259901i 0.825014 − 0.565112i \(-0.191167\pi\)
−0.565112 + 0.825014i \(0.691167\pi\)
\(62\) 4.63593 − 4.27547i 0.588764 − 0.542985i
\(63\) − 0.857064i − 0.107980i
\(64\) −7.76580 − 1.92153i −0.970726 − 0.240191i
\(65\) − 1.45076i − 0.179944i
\(66\) 0.0724989 − 1.79234i 0.00892400 − 0.220622i
\(67\) 13.4603i 1.64443i 0.569176 + 0.822216i \(0.307262\pi\)
−0.569176 + 0.822216i \(0.692738\pi\)
\(68\) 3.10463 + 3.65213i 0.376491 + 0.442886i
\(69\) 5.38631 + 5.38631i 0.648435 + 0.648435i
\(70\) −0.406554 + 0.374943i −0.0485925 + 0.0448143i
\(71\) − 8.52826i − 1.01212i −0.862499 − 0.506059i \(-0.831102\pi\)
0.862499 − 0.506059i \(-0.168898\pi\)
\(72\) 0.342196 − 2.80765i 0.0403282 − 0.330885i
\(73\) 8.93020i 1.04520i 0.852578 + 0.522601i \(0.175038\pi\)
−0.852578 + 0.522601i \(0.824962\pi\)
\(74\) −8.24137 − 2.46574i −0.958040 − 0.286636i
\(75\) 4.79180 0.553310
\(76\) −2.50498 − 2.94673i −0.287340 − 0.338013i
\(77\) −1.08711 −0.123888
\(78\) 3.30538 − 3.04837i 0.374260 − 0.345160i
\(79\) −10.1752 + 10.1752i −1.14480 + 1.14480i −0.157245 + 0.987560i \(0.550261\pi\)
−0.987560 + 0.157245i \(0.949739\pi\)
\(80\) −1.48153 + 1.06595i −0.165640 + 0.119177i
\(81\) 1.00000 0.111111
\(82\) 11.0915 + 0.448643i 1.22485 + 0.0495443i
\(83\) −3.47336 −0.381250 −0.190625 − 0.981663i \(-0.561052\pi\)
−0.190625 + 0.981663i \(0.561052\pi\)
\(84\) −1.70853 − 0.138444i −0.186416 − 0.0151055i
\(85\) 1.09359 0.118616
\(86\) −4.94152 + 4.55730i −0.532858 + 0.491426i
\(87\) 4.56858 − 4.56858i 0.489803 − 0.489803i
\(88\) −3.56126 − 0.434046i −0.379632 − 0.0462695i
\(89\) 1.69985 − 1.69985i 0.180184 − 0.180184i −0.611252 − 0.791436i \(-0.709334\pi\)
0.791436 + 0.611252i \(0.209334\pi\)
\(90\) −0.437474 − 0.474357i −0.0461138 − 0.0500016i
\(91\) −1.92688 − 1.92688i −0.201991 − 0.201991i
\(92\) 11.6075 − 9.86736i 1.21016 − 1.02874i
\(93\) 3.15323 − 3.15323i 0.326975 − 0.326975i
\(94\) 1.92274 + 0.0777733i 0.198315 + 0.00802171i
\(95\) −0.882365 −0.0905287
\(96\) −5.54168 − 1.13568i −0.565595 − 0.115910i
\(97\) 6.72129 + 6.72129i 0.682444 + 0.682444i 0.960550 − 0.278106i \(-0.0897067\pi\)
−0.278106 + 0.960550i \(0.589707\pi\)
\(98\) 0.358114 − 8.85343i 0.0361750 − 0.894332i
\(99\) − 1.26841i − 0.127480i
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.r.e.43.19 ✓ 72
8.3 odd 2 inner 888.2.r.e.43.36 yes 72
37.31 odd 4 inner 888.2.r.e.475.36 yes 72
296.179 even 4 inner 888.2.r.e.475.19 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.19 ✓ 72 1.1 even 1 trivial
888.2.r.e.43.36 yes 72 8.3 odd 2 inner
888.2.r.e.475.19 yes 72 296.179 even 4 inner
888.2.r.e.475.36 yes 72 37.31 odd 4 inner