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 475.20
Character \(\chi\) \(=\) 888.475
Dual form 888.2.r.e.43.20

$q$-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 \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

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

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 0.470157 − 0.882583i \(-0.344197\pi\)
−0.882583 + 0.470157i \(0.844197\pi\)
\(6\) 1.41194 + 0.0802186i 0.576421 + 0.0327491i
\(7\) 1.23837i 0.468059i 0.972230 + 0.234029i \(0.0751912\pi\)
−0.972230 + 0.234029i \(0.924809\pi\)
\(8\) −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 \(-0.945817\pi\)
0.985548 − 0.169399i \(-0.0541826\pi\)
\(12\) 0.226527 − 1.98713i 0.0653927 − 0.573635i
\(13\) 1.45319 + 1.45319i 0.403042 + 0.403042i 0.879304 − 0.476261i \(-0.158008\pi\)
−0.476261 + 0.879304i \(0.658008\pi\)
\(14\) 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 −0.292338 − 0.956315i \(-0.594433\pi\)
0.956315 + 0.292338i \(0.0944333\pi\)
\(18\) −0.0802186 + 1.41194i −0.0189077 + 0.332797i
\(19\) 4.43525 − 4.43525i 1.01752 − 1.01752i 0.0176731 − 0.999844i \(-0.494374\pi\)
0.999844 − 0.0176731i \(-0.00562582\pi\)
\(20\) 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 −0.994983 − 0.100040i \(-0.968103\pi\)
−0.100040 − 0.994983i \(-0.531897\pi\)
\(24\) −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 −0.472774 − 0.881184i \(-0.656747\pi\)
0.881184 + 0.472774i \(0.156747\pi\)
\(30\) −1.22813 − 1.37608i −0.224224 − 0.251237i
\(31\) −4.60962 + 4.60962i −0.827912 + 0.827912i −0.987228 − 0.159316i \(-0.949071\pi\)
0.159316 + 0.987228i \(0.449071\pi\)
\(32\) 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 \(-0.928604\pi\)
0.974951 − 0.222421i \(-0.0713958\pi\)
\(42\) −0.0993400 + 1.74850i −0.0153285 + 0.269799i
\(43\) 1.25673 − 1.25673i 0.191650 − 0.191650i −0.604759 − 0.796409i \(-0.706730\pi\)
0.796409 + 0.604759i \(0.206730\pi\)
\(44\) −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
\(50\) −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 \(-0.810298\pi\)
0.827607 − 0.561308i \(-0.189702\pi\)
\(54\) −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 −0.668319 − 0.743875i \(-0.732986\pi\)
0.743875 + 0.668319i \(0.232986\pi\)
\(60\) −2.04146 + 1.62365i −0.263551 + 0.209612i
\(61\) −2.21909 + 2.21909i −0.284126 + 0.284126i −0.834752 − 0.550626i \(-0.814389\pi\)
0.550626 + 0.834752i \(0.314389\pi\)
\(62\) 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 \(0.199898\pi\)
−0.809206 + 0.587525i \(0.800102\pi\)
\(68\) −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 \(-0.826320\pi\)
0.854800 − 0.518958i \(-0.173680\pi\)
\(72\) 0.479246 − 2.78753i 0.0564797 − 0.328514i
\(73\) − 12.1926i − 1.42704i −0.700636 − 0.713518i \(-0.747100\pi\)
0.700636 − 0.713518i \(-0.252900\pi\)
\(74\) −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 0.418421 − 0.908253i \(-0.362584\pi\)
−0.908253 + 0.418421i \(0.862584\pi\)
\(80\) −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 0.319096 − 0.947722i \(-0.396621\pi\)
−0.947722 + 0.319096i \(0.896621\pi\)
\(90\) 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 −0.939208 − 0.343348i \(-0.888439\pi\)
0.343348 + 0.939208i \(0.388439\pi\)
\(98\) 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.475.20 yes 72
8.3 odd 2 inner 888.2.r.e.475.2 yes 72
37.6 odd 4 inner 888.2.r.e.43.2 ✓ 72
296.43 even 4 inner 888.2.r.e.43.20 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.2 ✓ 72 37.6 odd 4 inner
888.2.r.e.43.20 yes 72 296.43 even 4 inner
888.2.r.e.475.2 yes 72 8.3 odd 2 inner
888.2.r.e.475.20 yes 72 1.1 even 1 trivial