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Properties

Label 888.2.r.e.43.20
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 43.20
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.20

$q$-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+(0.0802186 + 1.41194i) q^{2} -1.00000i q^{3} +(-1.98713 + 0.226527i) q^{4} +(-0.922211 + 0.922211i) q^{5} +(1.41194 - 0.0802186i) q^{6} -1.23837i q^{7} +(-0.479246 - 2.78753i) q^{8} -1.00000 q^{9} +(-1.37608 - 1.22813i) q^{10} +1.12366i q^{11} +(0.226527 + 1.98713i) q^{12} +(1.45319 - 1.45319i) q^{13} +(1.74850 - 0.0993400i) q^{14} +(0.922211 + 0.922211i) q^{15} +(3.89737 - 0.900277i) q^{16} +(2.73765 + 2.73765i) q^{17} +(-0.0802186 - 1.41194i) q^{18} +(4.43525 + 4.43525i) q^{19} +(1.62365 - 2.04146i) q^{20} -1.23837 q^{21} +(-1.58654 + 0.0901387i) q^{22} +(-5.25155 + 5.25155i) q^{23} +(-2.78753 + 0.479246i) q^{24} +3.29905i q^{25} +(2.16839 + 1.93524i) q^{26} +1.00000i q^{27} +(0.280524 + 2.46080i) q^{28} +(2.19936 + 2.19936i) q^{29} +(-1.22813 + 1.37608i) q^{30} +(-4.60962 - 4.60962i) q^{31} +(1.58378 + 5.43062i) q^{32} +1.12366 q^{33} +(-3.64577 + 4.08499i) q^{34} +(1.14204 + 1.14204i) q^{35} +(1.98713 - 0.226527i) q^{36} +(5.15996 + 3.22100i) q^{37} +(-5.90651 + 6.61809i) q^{38} +(-1.45319 - 1.45319i) q^{39} +(3.01266 + 2.12872i) q^{40} +2.84837i q^{41} +(-0.0993400 - 1.74850i) q^{42} +(1.25673 + 1.25673i) q^{43} +(-0.254540 - 2.23287i) q^{44} +(0.922211 - 0.922211i) q^{45} +(-7.83612 - 6.99358i) q^{46} -2.71087i q^{47} +(-0.900277 - 3.89737i) q^{48} +5.46645 q^{49} +(-4.65805 + 0.264645i) q^{50} +(2.73765 - 2.73765i) q^{51} +(-2.55849 + 3.21686i) q^{52} +8.17277i q^{53} +(-1.41194 + 0.0802186i) q^{54} +(-1.03626 - 1.03626i) q^{55} +(-3.45198 + 0.593483i) q^{56} +(4.43525 - 4.43525i) q^{57} +(-2.92892 + 3.28178i) q^{58} +(0.580360 + 0.580360i) q^{59} +(-2.04146 - 1.62365i) q^{60} +(-2.21909 - 2.21909i) q^{61} +(6.13871 - 6.87827i) q^{62} +1.23837i q^{63} +(-7.54065 + 2.67183i) q^{64} +2.68030i q^{65} +(0.0901387 + 1.58654i) q^{66} -9.61820i q^{67} +(-6.06021 - 4.81991i) q^{68} +(5.25155 + 5.25155i) q^{69} +(-1.52087 + 1.70409i) q^{70} +8.74563i q^{71} +(0.479246 + 2.78753i) q^{72} +12.1926i q^{73} +(-4.13393 + 7.54391i) q^{74} +3.29905 q^{75} +(-9.81813 - 7.80872i) q^{76} +1.39151 q^{77} +(1.93524 - 2.16839i) q^{78} +(-4.35373 + 4.35373i) q^{79} +(-2.76395 + 4.42444i) q^{80} +1.00000 q^{81} +(-4.02172 + 0.228492i) q^{82} +11.8731 q^{83} +(2.46080 - 0.280524i) q^{84} -5.04937 q^{85} +(-1.67362 + 1.87524i) q^{86} +(2.19936 - 2.19936i) q^{87} +(3.13225 - 0.538512i) q^{88} +(-5.93045 + 5.93045i) q^{89} +(1.37608 + 1.22813i) q^{90} +(-1.79958 - 1.79958i) q^{91} +(9.24589 - 11.6251i) q^{92} +(-4.60962 + 4.60962i) q^{93} +(3.82757 - 0.217462i) q^{94} -8.18048 q^{95} +(5.43062 - 1.58378i) q^{96} +(-5.86854 - 5.86854i) q^{97} +(0.438511 + 7.71828i) q^{98} -1.12366i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0802186 + 1.41194i 0.0567231 + 0.998390i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.98713 + 0.226527i −0.993565 + 0.113264i
\(5\) −0.922211 + 0.922211i −0.412425 + 0.412425i −0.882583 − 0.470157i \(-0.844197\pi\)
0.470157 + 0.882583i \(0.344197\pi\)
\(6\) 1.41194 − 0.0802186i 0.576421 − 0.0327491i
\(7\) − 1.23837i − 0.468059i −0.972230 − 0.234029i \(-0.924809\pi\)
0.972230 − 0.234029i \(-0.0751912\pi\)
\(8\) −0.479246 − 2.78753i −0.169439 − 0.985541i
\(9\) −1.00000 −0.333333
\(10\) −1.37608 − 1.22813i −0.435155 − 0.388367i
\(11\) 1.12366i 0.338797i 0.985548 + 0.169399i \(0.0541826\pi\)
−0.985548 + 0.169399i \(0.945817\pi\)
\(12\) 0.226527 + 1.98713i 0.0653927 + 0.573635i
\(13\) 1.45319 − 1.45319i 0.403042 − 0.403042i −0.476261 − 0.879304i \(-0.658008\pi\)
0.879304 + 0.476261i \(0.158008\pi\)
\(14\) 1.74850 − 0.0993400i 0.467305 − 0.0265497i
\(15\) 0.922211 + 0.922211i 0.238114 + 0.238114i
\(16\) 3.89737 − 0.900277i 0.974343 − 0.225069i
\(17\) 2.73765 + 2.73765i 0.663977 + 0.663977i 0.956315 − 0.292338i \(-0.0944333\pi\)
−0.292338 + 0.956315i \(0.594433\pi\)
\(18\) −0.0802186 − 1.41194i −0.0189077 − 0.332797i
\(19\) 4.43525 + 4.43525i 1.01752 + 1.01752i 0.999844 + 0.0176731i \(0.00562582\pi\)
0.0176731 + 0.999844i \(0.494374\pi\)
\(20\) 1.62365 − 2.04146i 0.363059 − 0.456484i
\(21\) −1.23837 −0.270234
\(22\) −1.58654 + 0.0901387i −0.338252 + 0.0192176i
\(23\) −5.25155 + 5.25155i −1.09502 + 1.09502i −0.100040 + 0.994983i \(0.531897\pi\)
−0.994983 + 0.100040i \(0.968103\pi\)
\(24\) −2.78753 + 0.479246i −0.569002 + 0.0978258i
\(25\) 3.29905i 0.659811i
\(26\) 2.16839 + 1.93524i 0.425255 + 0.379532i
\(27\) 1.00000i 0.192450i
\(28\) 0.280524 + 2.46080i 0.0530140 + 0.465047i
\(29\) 2.19936 + 2.19936i 0.408410 + 0.408410i 0.881184 − 0.472774i \(-0.156747\pi\)
−0.472774 + 0.881184i \(0.656747\pi\)
\(30\) −1.22813 + 1.37608i −0.224224 + 0.251237i
\(31\) −4.60962 − 4.60962i −0.827912 − 0.827912i 0.159316 − 0.987228i \(-0.449071\pi\)
−0.987228 + 0.159316i \(0.949071\pi\)
\(32\) 1.58378 + 5.43062i 0.279975 + 0.960007i
\(33\) 1.12366 0.195605
\(34\) −3.64577 + 4.08499i −0.625245 + 0.700570i
\(35\) 1.14204 + 1.14204i 0.193039 + 0.193039i
\(36\) 1.98713 − 0.226527i 0.331188 − 0.0377545i
\(37\) 5.15996 + 3.22100i 0.848292 + 0.529529i
\(38\) −5.90651 + 6.61809i −0.958162 + 1.07360i
\(39\) −1.45319 − 1.45319i −0.232697 − 0.232697i
\(40\) 3.01266 + 2.12872i 0.476343 + 0.336581i
\(41\) 2.84837i 0.444841i 0.974951 + 0.222421i \(0.0713958\pi\)
−0.974951 + 0.222421i \(0.928604\pi\)
\(42\) −0.0993400 − 1.74850i −0.0153285 − 0.269799i
\(43\) 1.25673 + 1.25673i 0.191650 + 0.191650i 0.796409 − 0.604759i \(-0.206730\pi\)
−0.604759 + 0.796409i \(0.706730\pi\)
\(44\) −0.254540 − 2.23287i −0.0383734 − 0.336617i
\(45\) 0.922211 − 0.922211i 0.137475 − 0.137475i
\(46\) −7.83612 − 6.99358i −1.15537 − 1.03115i
\(47\) − 2.71087i − 0.395421i −0.980260 − 0.197710i \(-0.936649\pi\)
0.980260 − 0.197710i \(-0.0633506\pi\)
\(48\) −0.900277 − 3.89737i −0.129944 − 0.562537i
\(49\) 5.46645 0.780921
\(50\) −4.65805 + 0.264645i −0.658748 + 0.0374265i
\(51\) 2.73765 − 2.73765i 0.383347 − 0.383347i
\(52\) −2.55849 + 3.21686i −0.354799 + 0.446099i
\(53\) 8.17277i 1.12262i 0.827607 + 0.561308i \(0.189702\pi\)
−0.827607 + 0.561308i \(0.810298\pi\)
\(54\) −1.41194 + 0.0802186i −0.192140 + 0.0109164i
\(55\) −1.03626 − 1.03626i −0.139729 − 0.139729i
\(56\) −3.45198 + 0.593483i −0.461291 + 0.0793075i
\(57\) 4.43525 − 4.43525i 0.587464 − 0.587464i
\(58\) −2.92892 + 3.28178i −0.384586 + 0.430919i
\(59\) 0.580360 + 0.580360i 0.0755565 + 0.0755565i 0.743875 − 0.668319i \(-0.232986\pi\)
−0.668319 + 0.743875i \(0.732986\pi\)
\(60\) −2.04146 − 1.62365i −0.263551 − 0.209612i
\(61\) −2.21909 − 2.21909i −0.284126 − 0.284126i 0.550626 − 0.834752i \(-0.314389\pi\)
−0.834752 + 0.550626i \(0.814389\pi\)
\(62\) 6.13871 − 6.87827i 0.779617 − 0.873541i
\(63\) 1.23837i 0.156020i
\(64\) −7.54065 + 2.67183i −0.942581 + 0.333978i
\(65\) 2.68030i 0.332450i
\(66\) 0.0901387 + 1.58654i 0.0110953 + 0.195290i
\(67\) − 9.61820i − 1.17505i −0.809206 − 0.587525i \(-0.800102\pi\)
0.809206 − 0.587525i \(-0.199898\pi\)
\(68\) −6.06021 − 4.81991i −0.734908 − 0.584500i
\(69\) 5.25155 + 5.25155i 0.632212 + 0.632212i
\(70\) −1.52087 + 1.70409i −0.181779 + 0.203678i
\(71\) 8.74563i 1.03792i 0.854800 + 0.518958i \(0.173680\pi\)
−0.854800 + 0.518958i \(0.826320\pi\)
\(72\) 0.479246 + 2.78753i 0.0564797 + 0.328514i
\(73\) 12.1926i 1.42704i 0.700636 + 0.713518i \(0.252900\pi\)
−0.700636 + 0.713518i \(0.747100\pi\)
\(74\) −4.13393 + 7.54391i −0.480559 + 0.876962i
\(75\) 3.29905 0.380942
\(76\) −9.81813 − 7.80872i −1.12622 − 0.895722i
\(77\) 1.39151 0.158577
\(78\) 1.93524 − 2.16839i 0.219123 − 0.245521i
\(79\) −4.35373 + 4.35373i −0.489832 + 0.489832i −0.908253 − 0.418421i \(-0.862584\pi\)
0.418421 + 0.908253i \(0.362584\pi\)
\(80\) −2.76395 + 4.42444i −0.309019 + 0.494668i
\(81\) 1.00000 0.111111
\(82\) −4.02172 + 0.228492i −0.444125 + 0.0252328i
\(83\) 11.8731 1.30324 0.651621 − 0.758545i \(-0.274089\pi\)
0.651621 + 0.758545i \(0.274089\pi\)
\(84\) 2.46080 − 0.280524i 0.268495 − 0.0306076i
\(85\) −5.04937 −0.547682
\(86\) −1.67362 + 1.87524i −0.180471 + 0.202213i
\(87\) 2.19936 − 2.19936i 0.235796 − 0.235796i
\(88\) 3.13225 − 0.538512i 0.333899 − 0.0574056i
\(89\) −5.93045 + 5.93045i −0.628626 + 0.628626i −0.947722 − 0.319096i \(-0.896621\pi\)
0.319096 + 0.947722i \(0.396621\pi\)
\(90\) 1.37608 + 1.22813i 0.145052 + 0.129456i
\(91\) −1.79958 − 1.79958i −0.188647 − 0.188647i
\(92\) 9.24589 − 11.6251i 0.963951 − 1.21200i
\(93\) −4.60962 + 4.60962i −0.477995 + 0.477995i
\(94\) 3.82757 − 0.217462i 0.394784 − 0.0224295i
\(95\) −8.18048 −0.839299
\(96\) 5.43062 − 1.58378i 0.554261 − 0.161643i
\(97\) −5.86854 − 5.86854i −0.595860 − 0.595860i 0.343348 − 0.939208i \(-0.388439\pi\)
−0.939208 + 0.343348i \(0.888439\pi\)
\(98\) 0.438511 + 7.71828i 0.0442963 + 0.779664i
\(99\) − 1.12366i − 0.112932i
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.43.20 yes 72
8.3 odd 2 inner 888.2.r.e.43.2 ✓ 72
37.31 odd 4 inner 888.2.r.e.475.2 yes 72
296.179 even 4 inner 888.2.r.e.475.20 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.2 ✓ 72 8.3 odd 2 inner
888.2.r.e.43.20 yes 72 1.1 even 1 trivial
888.2.r.e.475.2 yes 72 37.31 odd 4 inner
888.2.r.e.475.20 yes 72 296.179 even 4 inner