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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 49.3
Character \(\chi\) \(=\) 888.49
Dual form 888.2.bo.b.145.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.766044 + 0.642788i) q^{3} +(0.534466 - 0.194530i) q^{5} +(-3.76120 + 1.36896i) q^{7} +(0.173648 + 0.984808i) q^{9} +(1.68235 + 2.91392i) q^{11} +(-0.109761 + 0.622487i) q^{13} +(0.534466 + 0.194530i) q^{15} +(-0.529320 - 3.00192i) q^{17} +(1.56809 + 1.31578i) q^{19} +(-3.76120 - 1.36896i) q^{21} +(-4.17133 + 7.22495i) q^{23} +(-3.58241 + 3.00600i) q^{25} +(-0.500000 + 0.866025i) q^{27} +(0.606754 + 1.05093i) q^{29} -1.49928 q^{31} +(-0.584275 + 3.31359i) q^{33} +(-1.74393 + 1.46333i) q^{35} +(-2.99421 + 5.29478i) q^{37} +(-0.484209 + 0.406300i) q^{39} +(0.218572 - 1.23958i) q^{41} +4.84076 q^{43} +(0.284383 + 0.492566i) q^{45} +(-4.22858 + 7.32412i) q^{47} +(6.91023 - 5.79837i) q^{49} +(1.52412 - 2.63985i) q^{51} +(4.49383 + 1.63562i) q^{53} +(1.46600 + 1.23012i) q^{55} +(0.355457 + 2.01590i) q^{57} +(-5.76083 - 2.09677i) q^{59} +(0.932073 - 5.28605i) q^{61} +(-2.00129 - 3.46634i) q^{63} +(0.0624286 + 0.354050i) q^{65} +(15.0446 - 5.47579i) q^{67} +(-7.83953 + 2.85336i) q^{69} +(-5.47680 - 4.59558i) q^{71} +2.55239 q^{73} -4.67650 q^{75} +(-10.3167 - 8.65675i) q^{77} +(3.61095 - 1.31428i) q^{79} +(-0.939693 + 0.342020i) q^{81} +(2.60380 + 14.7669i) q^{83} +(-0.866866 - 1.50146i) q^{85} +(-0.210724 + 1.19507i) q^{87} +(-10.3942 - 3.78319i) q^{89} +(-0.439329 - 2.49156i) q^{91} +(-1.14852 - 0.963720i) q^{93} +(1.09405 + 0.398201i) q^{95} +(1.48433 - 2.57093i) q^{97} +(-2.57751 + 2.16279i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 3 q^{5} - 3 q^{7} - 6 q^{11} - 12 q^{13} - 3 q^{15} + 15 q^{17} - 3 q^{19} - 3 q^{21} - 6 q^{23} - 9 q^{25} - 12 q^{27} - 12 q^{29} + 6 q^{31} + 3 q^{33} - 3 q^{35} - 9 q^{37} + 15 q^{39} + 9 q^{41}+ \cdots - 6 q^{99}+O(q^{100}) \) 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
\(5\) 0.534466 − 0.194530i 0.239020 − 0.0869963i −0.219733 − 0.975560i \(-0.570518\pi\)
0.458753 + 0.888564i \(0.348296\pi\)
\(6\) 0 0
\(7\) −3.76120 + 1.36896i −1.42160 + 0.517420i −0.934511 − 0.355933i \(-0.884163\pi\)
−0.487088 + 0.873353i \(0.661941\pi\)
\(8\) 0 0
\(9\) 0.173648 + 0.984808i 0.0578827 + 0.328269i
\(10\) 0 0
\(11\) 1.68235 + 2.91392i 0.507249 + 0.878580i 0.999965 + 0.00839025i \(0.00267073\pi\)
−0.492716 + 0.870190i \(0.663996\pi\)
\(12\) 0 0
\(13\) −0.109761 + 0.622487i −0.0304423 + 0.172647i −0.996238 − 0.0866563i \(-0.972382\pi\)
0.965796 + 0.259303i \(0.0834929\pi\)
\(14\) 0 0
\(15\) 0.534466 + 0.194530i 0.137998 + 0.0502273i
\(16\) 0 0
\(17\) −0.529320 − 3.00192i −0.128379 − 0.728073i −0.979243 − 0.202688i \(-0.935032\pi\)
0.850864 − 0.525385i \(-0.176079\pi\)
\(18\) 0 0
\(19\) 1.56809 + 1.31578i 0.359744 + 0.301861i 0.804689 − 0.593697i \(-0.202332\pi\)
−0.444944 + 0.895558i \(0.646777\pi\)
\(20\) 0 0
\(21\) −3.76120 − 1.36896i −0.820761 − 0.298732i
\(22\) 0 0
\(23\) −4.17133 + 7.22495i −0.869782 + 1.50651i −0.00756193 + 0.999971i \(0.502407\pi\)
−0.862220 + 0.506535i \(0.830926\pi\)
\(24\) 0 0
\(25\) −3.58241 + 3.00600i −0.716482 + 0.601200i
\(26\) 0 0
\(27\) −0.500000 + 0.866025i −0.0962250 + 0.166667i
\(28\) 0 0
\(29\) 0.606754 + 1.05093i 0.112671 + 0.195153i 0.916847 − 0.399240i \(-0.130726\pi\)
−0.804175 + 0.594392i \(0.797393\pi\)
\(30\) 0 0
\(31\) −1.49928 −0.269279 −0.134639 − 0.990895i \(-0.542988\pi\)
−0.134639 + 0.990895i \(0.542988\pi\)
\(32\) 0 0
\(33\) −0.584275 + 3.31359i −0.101709 + 0.576822i
\(34\) 0 0
\(35\) −1.74393 + 1.46333i −0.294778 + 0.247348i
\(36\) 0 0
\(37\) −2.99421 + 5.29478i −0.492245 + 0.870457i
\(38\) 0 0
\(39\) −0.484209 + 0.406300i −0.0775355 + 0.0650600i
\(40\) 0 0
\(41\) 0.218572 − 1.23958i 0.0341352 − 0.193590i −0.962972 − 0.269603i \(-0.913108\pi\)
0.997107 + 0.0760122i \(0.0242188\pi\)
\(42\) 0 0
\(43\) 4.84076 0.738209 0.369104 − 0.929388i \(-0.379664\pi\)
0.369104 + 0.929388i \(0.379664\pi\)
\(44\) 0 0
\(45\) 0.284383 + 0.492566i 0.0423934 + 0.0734275i
\(46\) 0 0
\(47\) −4.22858 + 7.32412i −0.616802 + 1.06833i 0.373263 + 0.927726i \(0.378239\pi\)
−0.990065 + 0.140608i \(0.955094\pi\)
\(48\) 0 0
\(49\) 6.91023 − 5.79837i 0.987176 − 0.828339i
\(50\) 0 0
\(51\) 1.52412 − 2.63985i 0.213419 − 0.369652i
\(52\) 0 0
\(53\) 4.49383 + 1.63562i 0.617275 + 0.224670i 0.631684 − 0.775226i \(-0.282364\pi\)
−0.0144082 + 0.999896i \(0.504586\pi\)
\(54\) 0 0
\(55\) 1.46600 + 1.23012i 0.197676 + 0.165870i
\(56\) 0 0
\(57\) 0.355457 + 2.01590i 0.0470814 + 0.267012i
\(58\) 0 0
\(59\) −5.76083 − 2.09677i −0.749997 − 0.272976i −0.0613922 − 0.998114i \(-0.519554\pi\)
−0.688604 + 0.725137i \(0.741776\pi\)
\(60\) 0 0
\(61\) 0.932073 − 5.28605i 0.119340 − 0.676809i −0.865170 − 0.501479i \(-0.832790\pi\)
0.984510 − 0.175330i \(-0.0560993\pi\)
\(62\) 0 0
\(63\) −2.00129 − 3.46634i −0.252139 − 0.436718i
\(64\) 0 0
\(65\) 0.0624286 + 0.354050i 0.00774331 + 0.0439145i
\(66\) 0 0
\(67\) 15.0446 − 5.47579i 1.83799 − 0.668974i 0.847607 − 0.530625i \(-0.178043\pi\)
0.990383 − 0.138349i \(-0.0441797\pi\)
\(68\) 0 0
\(69\) −7.83953 + 2.85336i −0.943768 + 0.343504i
\(70\) 0 0
\(71\) −5.47680 − 4.59558i −0.649976 − 0.545395i 0.257088 − 0.966388i \(-0.417237\pi\)
−0.907064 + 0.420993i \(0.861682\pi\)
\(72\) 0 0
\(73\) 2.55239 0.298735 0.149367 − 0.988782i \(-0.452276\pi\)
0.149367 + 0.988782i \(0.452276\pi\)
\(74\) 0 0
\(75\) −4.67650 −0.539996
\(76\) 0 0
\(77\) −10.3167 − 8.65675i −1.17570 − 0.986529i
\(78\) 0 0
\(79\) 3.61095 − 1.31428i 0.406263 − 0.147868i −0.130802 − 0.991409i \(-0.541755\pi\)
0.537065 + 0.843541i \(0.319533\pi\)
\(80\) 0 0
\(81\) −0.939693 + 0.342020i −0.104410 + 0.0380022i
\(82\) 0 0
\(83\) 2.60380 + 14.7669i 0.285804 + 1.62088i 0.702399 + 0.711783i \(0.252112\pi\)
−0.416595 + 0.909092i \(0.636777\pi\)
\(84\) 0 0
\(85\) −0.866866 − 1.50146i −0.0940249 − 0.162856i
\(86\) 0 0
\(87\) −0.210724 + 1.19507i −0.0225919 + 0.128125i
\(88\) 0 0
\(89\) −10.3942 − 3.78319i −1.10179 − 0.401018i −0.273812 − 0.961783i \(-0.588285\pi\)
−0.827975 + 0.560765i \(0.810507\pi\)
\(90\) 0 0
\(91\) −0.439329 − 2.49156i −0.0460542 − 0.261186i
\(92\) 0 0
\(93\) −1.14852 − 0.963720i −0.119096 − 0.0999331i
\(94\) 0 0
\(95\) 1.09405 + 0.398201i 0.112247 + 0.0408546i
\(96\) 0 0
\(97\) 1.48433 − 2.57093i 0.150711 − 0.261039i −0.780778 − 0.624808i \(-0.785177\pi\)
0.931489 + 0.363770i \(0.118510\pi\)
\(98\) 0 0
\(99\) −2.57751 + 2.16279i −0.259050 + 0.217369i
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.49.3 ✓ 24
37.34 even 9 inner 888.2.bo.b.145.3 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bo.b.49.3 ✓ 24 1.1 even 1 trivial
888.2.bo.b.145.3 yes 24 37.34 even 9 inner