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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 43.2
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.2

$q$-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+(-1.41194 - 0.0802186i) q^{2} -1.00000i q^{3} +(1.98713 + 0.226527i) q^{4} +(0.922211 - 0.922211i) q^{5} +(-0.0802186 + 1.41194i) q^{6} +1.23837i q^{7} +(-2.78753 - 0.479246i) 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} +(0.0993400 - 1.74850i) q^{14} +(-0.922211 - 0.922211i) q^{15} +(3.89737 + 0.900277i) q^{16} +(2.73765 + 2.73765i) q^{17} +(1.41194 + 0.0802186i) q^{18} +(4.43525 + 4.43525i) q^{19} +(2.04146 - 1.62365i) q^{20} +1.23837 q^{21} +(0.0901387 - 1.58654i) q^{22} +(5.25155 - 5.25155i) q^{23} +(-0.479246 + 2.78753i) 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} +(-5.43062 - 1.58378i) 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} +(-1.74850 - 0.0993400i) 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} +(0.264645 - 4.65805i) q^{50} +(2.73765 - 2.73765i) q^{51} +(-3.21686 + 2.55849i) q^{52} -8.17277i q^{53} +(0.0802186 - 1.41194i) q^{54} +(1.03626 + 1.03626i) q^{55} +(0.593483 - 3.45198i) q^{56} +(4.43525 - 4.43525i) q^{57} +(2.92892 + 3.28178i) q^{58} +(0.580360 + 0.580360i) q^{59} +(-1.62365 - 2.04146i) 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} +(-1.58654 - 0.0901387i) q^{66} -9.61820i q^{67} +(4.81991 + 6.06021i) q^{68} +(-5.25155 - 5.25155i) q^{69} +(-1.52087 - 1.70409i) q^{70} -8.74563i q^{71} +(2.78753 + 0.479246i) q^{72} +12.1926i q^{73} +(7.02715 + 4.96177i) q^{74} +3.29905 q^{75} +(7.80872 + 9.81813i) q^{76} -1.39151 q^{77} +(-1.93524 - 2.16839i) q^{78} +(4.35373 - 4.35373i) q^{79} +(4.42444 - 2.76395i) q^{80} +1.00000 q^{81} +(0.228492 - 4.02172i) 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} +(0.538512 - 3.13225i) q^{88} +(-5.93045 + 5.93045i) q^{89} +(1.37608 - 1.22813i) q^{90} +(-1.79958 - 1.79958i) q^{91} +(11.6251 - 9.24589i) q^{92} +(4.60962 - 4.60962i) q^{93} +(0.217462 - 3.82757i) q^{94} +8.18048 q^{95} +(-1.58378 + 5.43062i) q^{96} +(-5.86854 - 5.86854i) q^{97} +(-7.71828 - 0.438511i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41194 − 0.0802186i −0.998390 − 0.0567231i
\(3\) − 1.00000i − 0.577350i
\(4\) 1.98713 + 0.226527i 0.993565 + 0.113264i
\(5\) 0.922211 − 0.922211i 0.412425 − 0.412425i −0.470157 − 0.882583i \(-0.655803\pi\)
0.882583 + 0.470157i \(0.155803\pi\)
\(6\) −0.0802186 + 1.41194i −0.0327491 + 0.576421i
\(7\) 1.23837i 0.468059i 0.972230 + 0.234029i \(0.0751912\pi\)
−0.972230 + 0.234029i \(0.924809\pi\)
\(8\) −2.78753 − 0.479246i −0.985541 − 0.169439i
\(9\) −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 −0.879304 − 0.476261i \(-0.841992\pi\)
0.476261 + 0.879304i \(0.341992\pi\)
\(14\) 0.0993400 − 1.74850i 0.0265497 − 0.467305i
\(15\) −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 \(0.594433\pi\)
\(18\) 1.41194 + 0.0802186i 0.332797 + 0.0189077i
\(19\) 4.43525 + 4.43525i 1.01752 + 1.01752i 0.999844 + 0.0176731i \(0.00562582\pi\)
0.0176731 + 0.999844i \(0.494374\pi\)
\(20\) 2.04146 − 1.62365i 0.456484 − 0.363059i
\(21\) 1.23837 0.270234
\(22\) 0.0901387 − 1.58654i 0.0192176 − 0.338252i
\(23\) 5.25155 − 5.25155i 1.09502 − 1.09502i 0.100040 − 0.994983i \(-0.468103\pi\)
0.994983 − 0.100040i \(-0.0318970\pi\)
\(24\) −0.479246 + 2.78753i −0.0978258 + 0.569002i
\(25\) 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 0.472774 − 0.881184i \(-0.343253\pi\)
−0.881184 + 0.472774i \(0.843253\pi\)
\(30\) 1.22813 + 1.37608i 0.224224 + 0.251237i
\(31\) 4.60962 + 4.60962i 0.827912 + 0.827912i 0.987228 − 0.159316i \(-0.0509287\pi\)
−0.159316 + 0.987228i \(0.550929\pi\)
\(32\) −5.43062 − 1.58378i −0.960007 − 0.279975i
\(33\) 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 \(0.928604\pi\)
\(42\) −1.74850 − 0.0993400i −0.269799 − 0.0153285i
\(43\) 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 \(0.0633506\pi\)
−0.980260 + 0.197710i \(0.936649\pi\)
\(48\) 0.900277 − 3.89737i 0.129944 − 0.562537i
\(49\) 5.46645 0.780921
\(50\) 0.264645 − 4.65805i 0.0374265 − 0.658748i
\(51\) 2.73765 − 2.73765i 0.383347 − 0.383347i
\(52\) −3.21686 + 2.55849i −0.446099 + 0.354799i
\(53\) − 8.17277i − 1.12262i −0.827607 − 0.561308i \(-0.810298\pi\)
0.827607 − 0.561308i \(-0.189702\pi\)
\(54\) 0.0802186 − 1.41194i 0.0109164 − 0.192140i
\(55\) 1.03626 + 1.03626i 0.139729 + 0.139729i
\(56\) 0.593483 − 3.45198i 0.0793075 − 0.461291i
\(57\) 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 \(0.732986\pi\)
\(60\) −1.62365 − 2.04146i −0.209612 − 0.263551i
\(61\) 2.21909 + 2.21909i 0.284126 + 0.284126i 0.834752 − 0.550626i \(-0.185611\pi\)
−0.550626 + 0.834752i \(0.685611\pi\)
\(62\) −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
\(66\) −1.58654 − 0.0901387i −0.195290 − 0.0110953i
\(67\) − 9.61820i − 1.17505i −0.809206 − 0.587525i \(-0.800102\pi\)
0.809206 − 0.587525i \(-0.199898\pi\)
\(68\) 4.81991 + 6.06021i 0.584500 + 0.734908i
\(69\) −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 \(-0.826320\pi\)
0.854800 − 0.518958i \(-0.173680\pi\)
\(72\) 2.78753 + 0.479246i 0.328514 + 0.0564797i
\(73\) 12.1926i 1.42704i 0.700636 + 0.713518i \(0.252900\pi\)
−0.700636 + 0.713518i \(0.747100\pi\)
\(74\) 7.02715 + 4.96177i 0.816889 + 0.576795i
\(75\) 3.29905 0.380942
\(76\) 7.80872 + 9.81813i 0.895722 + 1.12622i
\(77\) −1.39151 −0.158577
\(78\) −1.93524 − 2.16839i −0.219123 − 0.245521i
\(79\) 4.35373 − 4.35373i 0.489832 − 0.489832i −0.418421 − 0.908253i \(-0.637416\pi\)
0.908253 + 0.418421i \(0.137416\pi\)
\(80\) 4.42444 − 2.76395i 0.494668 − 0.309019i
\(81\) 1.00000 0.111111
\(82\) 0.228492 − 4.02172i 0.0252328 − 0.444125i
\(83\) 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
\(88\) 0.538512 − 3.13225i 0.0574056 − 0.333899i
\(89\) −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
\(92\) 11.6251 − 9.24589i 1.21200 − 0.963951i
\(93\) 4.60962 − 4.60962i 0.477995 − 0.477995i
\(94\) 0.217462 − 3.82757i 0.0224295 − 0.394784i
\(95\) 8.18048 0.839299
\(96\) −1.58378 + 5.43062i −0.161643 + 0.554261i
\(97\) −5.86854 − 5.86854i −0.595860 − 0.595860i 0.343348 − 0.939208i \(-0.388439\pi\)
−0.939208 + 0.343348i \(0.888439\pi\)
\(98\) −7.71828 − 0.438511i −0.779664 − 0.0442963i
\(99\) − 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.43.2 ✓ 72
8.3 odd 2 inner 888.2.r.e.43.20 yes 72
37.31 odd 4 inner 888.2.r.e.475.20 yes 72
296.179 even 4 inner 888.2.r.e.475.2 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.2 ✓ 72 1.1 even 1 trivial
888.2.r.e.43.20 yes 72 8.3 odd 2 inner
888.2.r.e.475.2 yes 72 296.179 even 4 inner
888.2.r.e.475.20 yes 72 37.31 odd 4 inner