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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(49,888)] 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("888.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 0, 14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.bo (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 49.4
Character \(\chi\) \(=\) 888.49
Dual form 888.2.bo.b.145.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.766044 + 0.642788i) q^{3} +(1.14695 - 0.417455i) q^{5} +(2.87145 - 1.04512i) q^{7} +(0.173648 + 0.984808i) q^{9} +(1.56671 + 2.71362i) q^{11} +(0.745857 - 4.22997i) q^{13} +(1.14695 + 0.417455i) q^{15} +(-0.134148 - 0.760794i) q^{17} +(-1.67441 - 1.40500i) q^{19} +(2.87145 + 1.04512i) q^{21} +(1.58894 - 2.75212i) q^{23} +(-2.68900 + 2.25634i) q^{25} +(-0.500000 + 0.866025i) q^{27} +(3.24143 + 5.61432i) q^{29} +0.647976 q^{31} +(-0.544113 + 3.08582i) q^{33} +(2.85711 - 2.39740i) q^{35} +(-3.71610 - 4.81566i) q^{37} +(3.29033 - 2.76091i) q^{39} +(-0.332472 + 1.88554i) q^{41} +2.44274 q^{43} +(0.610278 + 1.05703i) q^{45} +(3.35534 - 5.81163i) q^{47} +(1.79065 - 1.50253i) q^{49} +(0.386265 - 0.669031i) q^{51} +(-3.64231 - 1.32569i) q^{53} +(2.92975 + 2.45835i) q^{55} +(-0.379558 - 2.15258i) q^{57} +(9.24842 + 3.36615i) q^{59} +(-1.87004 + 10.6055i) q^{61} +(1.52787 + 2.64635i) q^{63} +(-0.910361 - 5.16291i) q^{65} +(-3.95921 + 1.44104i) q^{67} +(2.98622 - 1.08690i) q^{69} +(5.01902 + 4.21146i) q^{71} -10.1473 q^{73} -3.51024 q^{75} +(7.33481 + 6.15463i) q^{77} +(8.65783 - 3.15119i) q^{79} +(-0.939693 + 0.342020i) q^{81} +(1.16863 + 6.62762i) q^{83} +(-0.471458 - 0.816590i) q^{85} +(-1.12574 + 6.38437i) q^{87} +(-0.424793 - 0.154612i) q^{89} +(-2.27914 - 12.9257i) q^{91} +(0.496379 + 0.416511i) q^{93} +(-2.50698 - 0.912467i) q^{95} +(0.760878 - 1.31788i) q^{97} +(-2.40034 + 2.01412i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 3 q^{5} - 3 q^{7} - 6 q^{11} - 12 q^{13} - 3 q^{15} + 15 q^{17} - 3 q^{19} - 3 q^{21} - 6 q^{23} - 9 q^{25} - 12 q^{27} - 12 q^{29} + 6 q^{31} + 3 q^{33} - 3 q^{35} - 9 q^{37} + 15 q^{39} + 9 q^{41}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{7}{9}\right)\) \(1\) \(1\)

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 0
\(3\) 0.766044 + 0.642788i 0.442276 + 0.371114i
\(4\) 0 0
\(5\) 1.14695 − 0.417455i 0.512931 − 0.186691i −0.0725701 − 0.997363i \(-0.523120\pi\)
0.585501 + 0.810672i \(0.300898\pi\)
\(6\) 0 0
\(7\) 2.87145 − 1.04512i 1.08531 − 0.395020i 0.263427 − 0.964679i \(-0.415147\pi\)
0.821881 + 0.569660i \(0.192925\pi\)
\(8\) 0 0
\(9\) 0.173648 + 0.984808i 0.0578827 + 0.328269i
\(10\) 0 0
\(11\) 1.56671 + 2.71362i 0.472381 + 0.818188i 0.999500 − 0.0316033i \(-0.0100613\pi\)
−0.527119 + 0.849791i \(0.676728\pi\)
\(12\) 0 0
\(13\) 0.745857 − 4.22997i 0.206864 − 1.17318i −0.687617 − 0.726074i \(-0.741343\pi\)
0.894480 − 0.447108i \(-0.147546\pi\)
\(14\) 0 0
\(15\) 1.14695 + 0.417455i 0.296141 + 0.107786i
\(16\) 0 0
\(17\) −0.134148 − 0.760794i −0.0325358 − 0.184520i 0.964209 − 0.265144i \(-0.0854196\pi\)
−0.996745 + 0.0806247i \(0.974308\pi\)
\(18\) 0 0
\(19\) −1.67441 − 1.40500i −0.384136 − 0.322328i 0.430188 − 0.902739i \(-0.358447\pi\)
−0.814323 + 0.580411i \(0.802892\pi\)
\(20\) 0 0
\(21\) 2.87145 + 1.04512i 0.626602 + 0.228065i
\(22\) 0 0
\(23\) 1.58894 − 2.75212i 0.331316 − 0.573857i −0.651454 − 0.758688i \(-0.725841\pi\)
0.982770 + 0.184832i \(0.0591740\pi\)
\(24\) 0 0
\(25\) −2.68900 + 2.25634i −0.537800 + 0.451268i
\(26\) 0 0
\(27\) −0.500000 + 0.866025i −0.0962250 + 0.166667i
\(28\) 0 0
\(29\) 3.24143 + 5.61432i 0.601918 + 1.04255i 0.992530 + 0.121997i \(0.0389300\pi\)
−0.390612 + 0.920555i \(0.627737\pi\)
\(30\) 0 0
\(31\) 0.647976 0.116380 0.0581900 − 0.998306i \(-0.481467\pi\)
0.0581900 + 0.998306i \(0.481467\pi\)
\(32\) 0 0
\(33\) −0.544113 + 3.08582i −0.0947179 + 0.537172i
\(34\) 0 0
\(35\) 2.85711 − 2.39740i 0.482941 − 0.405235i
\(36\) 0 0
\(37\) −3.71610 − 4.81566i −0.610923 − 0.791690i
\(38\) 0 0
\(39\) 3.29033 − 2.76091i 0.526874 − 0.442100i
\(40\) 0 0
\(41\) −0.332472 + 1.88554i −0.0519233 + 0.294472i −0.999701 − 0.0244606i \(-0.992213\pi\)
0.947777 + 0.318932i \(0.103324\pi\)
\(42\) 0 0
\(43\) 2.44274 0.372514 0.186257 − 0.982501i \(-0.440364\pi\)
0.186257 + 0.982501i \(0.440364\pi\)
\(44\) 0 0
\(45\) 0.610278 + 1.05703i 0.0909749 + 0.157573i
\(46\) 0 0
\(47\) 3.35534 − 5.81163i 0.489427 − 0.847713i −0.510499 − 0.859879i \(-0.670539\pi\)
0.999926 + 0.0121656i \(0.00387251\pi\)
\(48\) 0 0
\(49\) 1.79065 − 1.50253i 0.255807 − 0.214647i
\(50\) 0 0
\(51\) 0.386265 − 0.669031i 0.0540879 − 0.0936831i
\(52\) 0 0
\(53\) −3.64231 − 1.32569i −0.500310 − 0.182098i 0.0795236 − 0.996833i \(-0.474660\pi\)
−0.579833 + 0.814735i \(0.696882\pi\)
\(54\) 0 0
\(55\) 2.92975 + 2.45835i 0.395047 + 0.331484i
\(56\) 0 0
\(57\) −0.379558 − 2.15258i −0.0502736 − 0.285116i
\(58\) 0 0
\(59\) 9.24842 + 3.36615i 1.20404 + 0.438235i 0.864633 − 0.502404i \(-0.167551\pi\)
0.339408 + 0.940639i \(0.389773\pi\)
\(60\) 0 0
\(61\) −1.87004 + 10.6055i −0.239434 + 1.35790i 0.593638 + 0.804732i \(0.297691\pi\)
−0.833072 + 0.553165i \(0.813420\pi\)
\(62\) 0 0
\(63\) 1.52787 + 2.64635i 0.192493 + 0.333408i
\(64\) 0 0
\(65\) −0.910361 − 5.16291i −0.112916 − 0.640380i
\(66\) 0 0
\(67\) −3.95921 + 1.44104i −0.483695 + 0.176050i −0.572346 − 0.820012i \(-0.693967\pi\)
0.0886513 + 0.996063i \(0.471744\pi\)
\(68\) 0 0
\(69\) 2.98622 − 1.08690i 0.359499 − 0.130847i
\(70\) 0 0
\(71\) 5.01902 + 4.21146i 0.595649 + 0.499809i 0.890044 − 0.455875i \(-0.150674\pi\)
−0.294395 + 0.955684i \(0.595118\pi\)
\(72\) 0 0
\(73\) −10.1473 −1.18765 −0.593825 − 0.804594i \(-0.702383\pi\)
−0.593825 + 0.804594i \(0.702383\pi\)
\(74\) 0 0
\(75\) −3.51024 −0.405328
\(76\) 0 0
\(77\) 7.33481 + 6.15463i 0.835879 + 0.701385i
\(78\) 0 0
\(79\) 8.65783 − 3.15119i 0.974083 − 0.354537i 0.194546 − 0.980893i \(-0.437677\pi\)
0.779537 + 0.626356i \(0.215455\pi\)
\(80\) 0 0
\(81\) −0.939693 + 0.342020i −0.104410 + 0.0380022i
\(82\) 0 0
\(83\) 1.16863 + 6.62762i 0.128274 + 0.727476i 0.979310 + 0.202368i \(0.0648638\pi\)
−0.851036 + 0.525108i \(0.824025\pi\)
\(84\) 0 0
\(85\) −0.471458 − 0.816590i −0.0511368 − 0.0885716i
\(86\) 0 0
\(87\) −1.12574 + 6.38437i −0.120692 + 0.684476i
\(88\) 0 0
\(89\) −0.424793 − 0.154612i −0.0450280 − 0.0163888i 0.319408 − 0.947617i \(-0.396516\pi\)
−0.364436 + 0.931228i \(0.618738\pi\)
\(90\) 0 0
\(91\) −2.27914 − 12.9257i −0.238919 − 1.35498i
\(92\) 0 0
\(93\) 0.496379 + 0.416511i 0.0514721 + 0.0431902i
\(94\) 0 0
\(95\) −2.50698 − 0.912467i −0.257211 − 0.0936171i
\(96\) 0 0
\(97\) 0.760878 − 1.31788i 0.0772554 − 0.133810i −0.824809 − 0.565411i \(-0.808718\pi\)
0.902065 + 0.431601i \(0.142051\pi\)
\(98\) 0 0
\(99\) −2.40034 + 2.01412i −0.241243 + 0.202427i
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.bo.b.49.4 ✓ 24
37.34 even 9 inner 888.2.bo.b.145.4 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bo.b.49.4 ✓ 24 1.1 even 1 trivial
888.2.bo.b.145.4 yes 24 37.34 even 9 inner