LMFDB - Embedded newform 1404.2.k.a.1153.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1404,2,Mod(1153,1404)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1404.1153"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1404, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1404 = 2^{2} \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1404.k (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.2109964438$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 468)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1153.11
Character	$\chi$	$=$	1404.1153
Dual form			1404.2.k.a.1225.11

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.913993 + 1.58308i) q^{5} +3.54787 q^{7} +(0.394633 + 0.683524i) q^{11} +(3.60317 - 0.131005i) q^{13} +(1.48507 + 2.57222i) q^{17} +(1.51934 + 2.63157i) q^{19} -5.81581 q^{23} +(0.829233 - 1.43627i) q^{25} +(-4.09470 - 7.09223i) q^{29} +(-0.129332 - 0.224010i) q^{31} +(3.24273 + 5.61657i) q^{35} +(-1.90793 + 3.30463i) q^{37} -0.0912193 q^{41} +8.61643 q^{43} +(-1.16369 + 2.01557i) q^{47} +5.58740 q^{49} +6.58958 q^{53} +(-0.721383 + 1.24947i) q^{55} +(-0.605518 + 1.04879i) q^{59} -3.76705 q^{61} +(3.50066 + 5.58438i) q^{65} -8.47022 q^{67} +(6.51214 + 11.2794i) q^{71} -7.67537 q^{73} +(1.40011 + 2.42506i) q^{77} +(5.76726 - 9.98919i) q^{79} +(-7.82491 + 13.5531i) q^{83} +(-2.71469 + 4.70199i) q^{85} +(2.17897 - 3.77409i) q^{89} +(12.7836 - 0.464789i) q^{91} +(-2.77733 + 4.81047i) q^{95} +0.113077 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 4 q^{7} + 4 q^{11} + q^{13} + 8 q^{17} - q^{19} - 8 q^{23} - 14 q^{25} + 13 q^{29} + 2 q^{31} - 3 q^{35} - q^{37} + 8 q^{41} - 4 q^{43} - 11 q^{47} + 24 q^{49} - 52 q^{53} + 8 q^{59} + 14 q^{61}+ \cdots + 50 q^{97}+O(q^{100}) $ 28 * q - 4 * q^7 + 4 * q^11 + q^13 + 8 * q^17 - q^19 - 8 * q^23 - 14 * q^25 + 13 * q^29 + 2 * q^31 - 3 * q^35 - q^37 + 8 * q^41 - 4 * q^43 - 11 * q^47 + 24 * q^49 - 52 * q^53 + 8 * q^59 + 14 * q^61 - 38 * q^65 + 14 * q^67 + 12 * q^71 + 14 * q^73 + 28 * q^77 + 5 * q^79 - 9 * q^83 + 18 * q^85 + 11 * q^89 - q^91 - 28 * q^95 + 50 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1404\mathbb{Z}\right)^\times$.

$n$	$677$	$703$	$1081$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.913993	+	1.58308i	0.408750	+	0.707976i	0.994750	−	0.102335i	$-0.0326314\pi$
$5$	0.913993	+	1.58308i	0.408750	+	0.707976i	−0.586000	+	0.810311i	$0.699298\pi$
$6$		0			0
$7$	3.54787			1.34097			0.670485	−	0.741923i	$-0.266086\pi$
$7$	3.54787			1.34097			0.670485	+	0.741923i	$0.266086\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.394633	+	0.683524i	0.118986	+	0.206090i	0.919366	−	0.393403i	$-0.128702\pi$
$11$	0.394633	+	0.683524i	0.118986	+	0.206090i	−0.800380	+	0.599493i	$0.795369\pi$
$12$		0			0
$13$	3.60317	−	0.131005i	0.999340	−	0.0363342i
$13$	3.60317	−	0.131005i	0.999340	−	0.0363342i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.48507	+	2.57222i	0.360183	+	0.623855i	0.987991	−	0.154513i	$-0.0493809\pi$
$17$	1.48507	+	2.57222i	0.360183	+	0.623855i	−0.627808	+	0.778369i	$0.716048\pi$
$18$		0			0
$19$	1.51934	+	2.63157i	0.348560	+	0.603724i	0.985994	−	0.166781i	$-0.0533374\pi$
$19$	1.51934	+	2.63157i	0.348560	+	0.603724i	−0.637434	+	0.770505i	$0.720004\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.81581			−1.21268			−0.606340	−	0.795206i	$-0.707363\pi$
$23$	−5.81581			−1.21268			−0.606340	+	0.795206i	$0.707363\pi$
$24$		0			0
$25$	0.829233	−	1.43627i	0.165847	−	0.287255i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.09470	−	7.09223i	−0.760367	−	1.31699i	−0.942662	−	0.333750i	$-0.891686\pi$
$29$	−4.09470	−	7.09223i	−0.760367	−	1.31699i	0.182295	−	0.983244i	$-0.441648\pi$
$30$		0			0
$31$	−0.129332	−	0.224010i	−0.0232288	−	0.0402334i	0.854177	−	0.519982i	$-0.174061\pi$
$31$	−0.129332	−	0.224010i	−0.0232288	−	0.0402334i	−0.877406	+	0.479748i	$0.840728\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.24273	+	5.61657i	0.548122	+	0.949374i
$36$		0			0
$37$	−1.90793	+	3.30463i	−0.313661	+	0.543277i	−0.979152	−	0.203129i	$-0.934889\pi$
$37$	−1.90793	+	3.30463i	−0.313661	+	0.543277i	0.665491	+	0.746406i	$0.268222\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.0912193			−0.0142461			−0.00712303	−	0.999975i	$-0.502267\pi$
$41$	−0.0912193			−0.0142461			−0.00712303	+	0.999975i	$0.502267\pi$
$42$		0			0
$43$	8.61643			1.31399			0.656997	−	0.753893i	$-0.271826\pi$
$43$	8.61643			1.31399			0.656997	+	0.753893i	$0.271826\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.16369	+	2.01557i	−0.169741	+	0.294001i	−0.938329	−	0.345744i	$-0.887627\pi$
$47$	−1.16369	+	2.01557i	−0.169741	+	0.294001i	0.768587	+	0.639745i	$0.220960\pi$
$48$		0			0
$49$	5.58740			0.798200
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.58958			0.905149			0.452574	−	0.891727i	$-0.350506\pi$
$53$	6.58958			0.905149			0.452574	+	0.891727i	$0.350506\pi$
$54$		0			0
$55$	−0.721383	+	1.24947i	−0.0972713	+	0.168479i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.605518	+	1.04879i	−0.0788318	+	0.136541i	−0.902746	−	0.430174i	$-0.858452\pi$
$59$	−0.605518	+	1.04879i	−0.0788318	+	0.136541i	0.823914	+	0.566714i	$0.191786\pi$
$60$		0			0
$61$	−3.76705			−0.482321			−0.241161	−	0.970485i	$-0.577528\pi$
$61$	−3.76705			−0.482321			−0.241161	+	0.970485i	$0.577528\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.50066	+	5.58438i	0.434204	+	0.692657i
$66$		0			0
$67$	−8.47022			−1.03480			−0.517401	−	0.855743i	$-0.673100\pi$
$67$	−8.47022			−1.03480			−0.517401	+	0.855743i	$0.673100\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.51214	+	11.2794i	0.772849	+	1.33861i	0.935995	+	0.352012i	$0.114502\pi$
$71$	6.51214	+	11.2794i	0.772849	+	1.33861i	−0.163146	+	0.986602i	$0.552164\pi$
$72$		0			0
$73$	−7.67537			−0.898334			−0.449167	−	0.893448i	$-0.648279\pi$
$73$	−7.67537			−0.898334			−0.449167	+	0.893448i	$0.648279\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.40011	+	2.42506i	0.159557	+	0.276361i
$78$		0			0
$79$	5.76726	−	9.98919i	0.648868	−	1.12387i	−0.334526	−	0.942387i	$-0.608576\pi$
$79$	5.76726	−	9.98919i	0.648868	−	1.12387i	0.983394	−	0.181485i	$-0.0580905\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.82491	+	13.5531i	−0.858896	+	1.48765i	0.0140878	+	0.999901i	$0.495516\pi$
$83$	−7.82491	+	13.5531i	−0.858896	+	1.48765i	−0.872983	+	0.487750i	$0.837818\pi$
$84$		0			0
$85$	−2.71469	+	4.70199i	−0.294450	+	0.510002i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.17897	−	3.77409i	0.230970	−	0.400053i	−0.727124	−	0.686507i	$-0.759143\pi$
$89$	2.17897	−	3.77409i	0.230970	−	0.400053i	0.958094	+	0.286454i	$0.0924766\pi$
$90$		0			0
$91$	12.7836	−	0.464789i	1.34008	−	0.0487231i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.77733	+	4.81047i	−0.284948	+	0.493544i
$96$		0			0
$97$	0.113077			0.0114813			0.00574063	−	0.999984i	$-0.498173\pi$
$97$	0.113077			0.0114813			0.00574063	+	0.999984i	$0.498173\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.07159	+	7.05221i	0.405139	+	0.701721i	0.994338	−	0.106267i	$-0.0338899\pi$
$101$	4.07159	+	7.05221i	0.405139	+	0.701721i	−0.589199	+	0.807988i	$0.700557\pi$
$102$		0			0
$103$	−1.51452	−	2.62322i	−0.149230	−	0.258474i	0.781713	−	0.623638i	$-0.214346\pi$
$103$	−1.51452	−	2.62322i	−0.149230	−	0.258474i	−0.930943	+	0.365164i	$0.881013\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.94326	−	17.2222i	0.961252	−	1.66494i	0.241886	−	0.970305i	$-0.422234\pi$
$107$	9.94326	−	17.2222i	0.961252	−	1.66494i	0.719365	−	0.694632i	$-0.244433\pi$
$108$		0			0
$109$	18.7499			1.79591			0.897957	−	0.440082i	$-0.145051\pi$
$109$	18.7499			1.79591			0.897957	+	0.440082i	$0.145051\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.25463	+	2.17308i	−0.118025	+	0.204426i	−0.918985	−	0.394292i	$-0.870990\pi$
$113$	−1.25463	+	2.17308i	−0.118025	+	0.204426i	0.800960	+	0.598718i	$0.204323\pi$
$114$		0			0
$115$	−5.31561	−	9.20690i	−0.495683	−	0.858548i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.26885	+	9.12591i	0.482995	+	0.836571i
$120$		0			0
$121$	5.18853	−	8.98680i	0.471685	−	0.816982i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.1716			1.08866
$126$		0			0
$127$	−10.9444	+	18.9563i	−0.971160	+	1.68210i	−0.279096	+	0.960263i	$0.590035\pi$
$127$	−10.9444	+	18.9563i	−0.971160	+	1.68210i	−0.692065	+	0.721836i	$0.743299\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.14623	−	14.1097i	−0.711740	−	1.23277i	−0.964204	−	0.265163i	$-0.914574\pi$
$131$	−8.14623	−	14.1097i	−0.711740	−	1.23277i	0.252464	−	0.967606i	$-0.418759\pi$
$132$		0			0
$133$	5.39042	+	9.33647i	0.467408	+	0.809575i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	15.1730			1.29631			0.648157	−	0.761507i	$-0.275540\pi$
$137$	15.1730			1.29631			0.648157	+	0.761507i	$0.275540\pi$
$138$		0			0
$139$	−9.03063	+	15.6415i	−0.765968	+	1.32670i	0.173765	+	0.984787i	$0.444407\pi$
$139$	−9.03063	+	15.6415i	−0.765968	+	1.32670i	−0.939733	+	0.341908i	$0.888927\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.51147	+	2.41115i	0.126396	+	0.201631i
$144$		0			0
$145$	7.48506	−	12.9645i	0.621600	−	1.07664i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.00434	−	8.66778i	0.409972	−	0.710092i	−0.584914	−	0.811095i	$-0.698872\pi$
$149$	5.00434	−	8.66778i	0.409972	−	0.710092i	0.994886	+	0.101003i	$0.0322052\pi$
$150$		0			0
$151$	−9.52036	+	16.4898i	−0.774756	+	1.34192i	0.160175	+	0.987089i	$0.448794\pi$
$151$	−9.52036	+	16.4898i	−0.774756	+	1.34192i	−0.934931	+	0.354829i	$0.884539\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.236418	−	0.409488i	0.0189895	−	0.0328908i
$156$		0			0
$157$	−5.01398	−	8.68447i	−0.400159	−	0.693096i	0.593585	−	0.804771i	$-0.297712\pi$
$157$	−5.01398	−	8.68447i	−0.400159	−	0.693096i	−0.993745	+	0.111675i	$0.964379\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−20.6337			−1.62617
$162$		0			0
$163$	−2.74172	−	4.74879i	−0.214748	−	0.371954i	0.738447	−	0.674312i	$-0.235560\pi$
$163$	−2.74172	−	4.74879i	−0.214748	−	0.371954i	−0.953195	+	0.302358i	$0.902226\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−15.1142			−1.16957			−0.584784	−	0.811189i	$-0.698821\pi$
$167$	−15.1142			−1.16957			−0.584784	+	0.811189i	$0.698821\pi$
$168$		0			0
$169$	12.9657	−	0.944067i	0.997360	−	0.0726205i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.88965			0.523811			0.261905	−	0.965094i	$-0.415649\pi$
$173$	6.88965			0.523811			0.261905	+	0.965094i	$0.415649\pi$
$174$		0			0
$175$	2.94201	−	5.09572i	0.222395	−	0.385200i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.90439	−	6.76259i	0.291827	−	0.505460i	−0.682414	−	0.730966i	$-0.739070\pi$
$179$	3.90439	−	6.76259i	0.291827	−	0.505460i	0.974242	+	0.225505i	$0.0724034\pi$
$180$		0			0
$181$	−9.35311			−0.695211			−0.347606	−	0.937641i	$-0.613005\pi$
$181$	−9.35311			−0.695211			−0.347606	+	0.937641i	$0.613005\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.97533			−0.512836
$186$		0			0
$187$	−1.17212	+	2.03017i	−0.0857137	+	0.148460i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.3660			−1.40127			−0.700636	−	0.713519i	$-0.747100\pi$
$191$	−19.3660			−1.40127			−0.700636	+	0.713519i	$0.747100\pi$
$192$		0			0
$193$	5.57023			0.400954			0.200477	−	0.979698i	$-0.435751\pi$
$193$	5.57023			0.400954			0.200477	+	0.979698i	$0.435751\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	11.3330	−	19.6294i	0.807445	−	1.39854i	−0.107182	−	0.994239i	$-0.534183\pi$
$197$	11.3330	−	19.6294i	0.807445	−	1.39854i	0.914628	−	0.404297i	$-0.132484\pi$
$198$		0			0
$199$	−5.10112	−	8.83540i	−0.361609	−	0.626325i	0.626617	−	0.779327i	$-0.284439\pi$
$199$	−5.10112	−	8.83540i	−0.361609	−	0.626325i	−0.988226	+	0.153002i	$0.951106\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−14.5275	−	25.1623i	−1.01963	−	1.76605i
$204$		0			0
$205$	−0.0833738	−	0.144408i	−0.00582308	−	0.0100859i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.19916	+	2.07701i	−0.0829477	+	0.143670i
$210$		0			0
$211$	4.13229			0.284478			0.142239	−	0.989832i	$-0.454570\pi$
$211$	4.13229			0.284478			0.142239	+	0.989832i	$0.454570\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.87536	+	13.6405i	0.537095	+	0.930276i
$216$		0			0
$217$	−0.458855	−	0.794760i	−0.0311491	−	0.0539518i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.68795	+	9.07360i	0.382613	+	0.610357i
$222$		0			0
$223$	−0.997068	−	1.72697i	−0.0667686	−	0.115647i	0.830709	−	0.556708i	$-0.187936\pi$
$223$	−0.997068	−	1.72697i	−0.0667686	−	0.115647i	−0.897477	+	0.441061i	$0.854602\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−18.3994			−1.22121			−0.610607	−	0.791934i	$-0.709074\pi$
$227$	−18.3994			−1.22121			−0.610607	+	0.791934i	$0.709074\pi$
$228$		0			0
$229$	−10.8242	−	18.7481i	−0.715284	−	1.23891i	−0.962850	−	0.270038i	$-0.912964\pi$
$229$	−10.8242	−	18.7481i	−0.715284	−	1.23891i	0.247565	−	0.968871i	$-0.420370\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.5670			−0.692265			−0.346133	−	0.938186i	$-0.612505\pi$
$233$	−10.5670			−0.692265			−0.346133	+	0.938186i	$0.612505\pi$
$234$		0			0
$235$	−4.25441			−0.277527
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.75441	+	3.03873i	0.113484	+	0.196559i	0.917173	−	0.398490i	$-0.130466\pi$
$239$	1.75441	+	3.03873i	0.113484	+	0.196559i	−0.803689	+	0.595050i	$0.797132\pi$
$240$		0			0
$241$	−26.1264			−1.68295			−0.841475	−	0.540296i	$-0.818312\pi$
$241$	−26.1264			−1.68295			−0.841475	+	0.540296i	$0.818312\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.10684	+	8.84531i	0.326264	+	0.565106i
$246$		0			0
$247$	5.81918	+	9.28295i	0.370266	+	0.590660i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−8.00649	−	13.8677i	−0.505365	−	0.875319i	−0.999981	−	0.00620652i	$-0.998024\pi$
$251$	−8.00649	−	13.8677i	−0.505365	−	0.875319i	0.494615	−	0.869112i	$-0.335309\pi$
$252$		0			0
$253$	−2.29511	−	3.97524i	−0.144292	−	0.249921i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.38690			0.273647			0.136824	−	0.990595i	$-0.456311\pi$
$257$	4.38690			0.273647			0.136824	+	0.990595i	$0.456311\pi$
$258$		0			0
$259$	−6.76908	+	11.7244i	−0.420610	+	0.728518i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−11.3028	−	19.5770i	−0.696958	−	1.20717i	−0.969516	−	0.245027i	$-0.921203\pi$
$263$	−11.3028	−	19.5770i	−0.696958	−	1.20717i	0.272558	−	0.962139i	$-0.412130\pi$
$264$		0			0
$265$	6.02283	+	10.4319i	0.369980	+	0.640824i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.33389	+	12.7027i	0.447155	+	0.774496i	0.998200	−	0.0599804i	$-0.0191038\pi$
$269$	7.33389	+	12.7027i	0.447155	+	0.774496i	−0.551044	+	0.834476i	$0.685771\pi$
$270$		0			0
$271$	2.35377	−	4.07684i	0.142981	−	0.247651i	−0.785637	−	0.618688i	$-0.787664\pi$
$271$	2.35377	−	4.07684i	0.142981	−	0.247651i	0.928618	+	0.371037i	$0.120998\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.30897			0.0789339
$276$		0			0
$277$	−15.8732			−0.953727			−0.476863	−	0.878977i	$-0.658226\pi$
$277$	−15.8732			−0.953727			−0.476863	+	0.878977i	$0.658226\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.96870	+	8.60604i	−0.296408	+	0.513393i	−0.975311	−	0.220834i	$-0.929122\pi$
$281$	−4.96870	+	8.60604i	−0.296408	+	0.513393i	0.678904	+	0.734227i	$0.262455\pi$
$282$		0			0
$283$	−6.51285			−0.387149			−0.193574	−	0.981086i	$-0.562008\pi$
$283$	−6.51285			−0.387149			−0.193574	+	0.981086i	$0.562008\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.323634			−0.0191035
$288$		0			0
$289$	4.08912	−	7.08256i	0.240536	−	0.416621i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	15.7842	−	27.3390i	0.922120	−	1.59716i	0.125993	−	0.992031i	$-0.459788\pi$
$293$	15.7842	−	27.3390i	0.922120	−	1.59716i	0.796128	−	0.605128i	$-0.206878\pi$
$294$		0			0
$295$	−2.21376			−0.128890
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−20.9553	+	0.761900i	−1.21188	+	0.0440618i
$300$		0			0
$301$	30.5700			1.76203
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.44306	−	5.96355i	−0.197149	−	0.341472i
$306$		0			0
$307$	−9.62508			−0.549332			−0.274666	−	0.961540i	$-0.588567\pi$
$307$	−9.62508			−0.549332			−0.274666	+	0.961540i	$0.588567\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.09612	−	5.36263i	−0.175565	−	0.304087i	0.764792	−	0.644277i	$-0.222842\pi$
$311$	−3.09612	−	5.36263i	−0.175565	−	0.304087i	−0.940357	+	0.340190i	$0.889508\pi$
$312$		0			0
$313$	−8.25806	+	14.3034i	−0.466773	+	0.808475i	−0.999280	−	0.0379511i	$-0.987917\pi$
$313$	−8.25806	+	14.3034i	−0.466773	+	0.808475i	0.532506	+	0.846426i	$0.321250\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.2779	+	19.5340i	−0.633432	+	1.09714i	0.353413	+	0.935467i	$0.385021\pi$
$317$	−11.2779	+	19.5340i	−0.633432	+	1.09714i	−0.986845	+	0.161669i	$0.948312\pi$
$318$		0			0
$319$	3.23181	−	5.59765i	0.180946	−	0.313408i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.51265	+	7.81615i	−0.251091	+	0.434902i
$324$		0			0
$325$	2.79971	−	5.28377i	0.155300	−	0.293091i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.12862	+	7.15098i	−0.227618	+	0.394246i
$330$		0			0
$331$	13.9793			0.768374			0.384187	−	0.923255i	$-0.374482\pi$
$331$	13.9793			0.768374			0.384187	+	0.923255i	$0.374482\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.74172	−	13.4091i	−0.422976	−	0.732615i
$336$		0			0
$337$	0.207169	+	0.358827i	0.0112852	+	0.0195466i	0.871613	−	0.490195i	$-0.163074\pi$
$337$	0.207169	+	0.358827i	0.0112852	+	0.0195466i	−0.860328	+	0.509741i	$0.829741\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.102078	−	0.176804i	0.00552781	−	0.00957445i
$342$		0			0
$343$	−5.01173			−0.270608
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.06201	−	8.76765i	0.271743	−	0.470672i	−0.697565	−	0.716521i	$-0.745733\pi$
$347$	5.06201	−	8.76765i	0.271743	−	0.470672i	0.969308	+	0.245849i	$0.0790667\pi$
$348$		0			0
$349$	−7.41055	−	12.8355i	−0.396678	−	0.687066i	0.596636	−	0.802512i	$-0.296504\pi$
$349$	−7.41055	−	12.8355i	−0.396678	−	0.687066i	−0.993314	+	0.115446i	$0.963170\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	15.0414	+	26.0525i	0.800573	+	1.38663i	0.919239	+	0.393699i	$0.128805\pi$
$353$	15.0414	+	26.0525i	0.800573	+	1.38663i	−0.118666	+	0.992934i	$0.537862\pi$
$354$		0			0
$355$	−11.9041	+	20.6185i	−0.631804	+	1.09432i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.20722			−0.169271			−0.0846353	−	0.996412i	$-0.526973\pi$
$359$	−3.20722			−0.169271			−0.0846353	+	0.996412i	$0.526973\pi$
$360$		0			0
$361$	4.88323	−	8.45800i	0.257012	−	0.445158i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.01523	−	12.1507i	−0.367194	−	0.635999i
$366$		0			0
$367$	1.08009	+	1.87076i	0.0563800	+	0.0976530i	0.892838	−	0.450378i	$-0.148711\pi$
$367$	1.08009	+	1.87076i	0.0563800	+	0.0976530i	−0.836458	+	0.548031i	$0.815377\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	23.3790			1.21378
$372$		0			0
$373$	−6.98479	+	12.0980i	−0.361659	+	0.626411i	−0.988234	−	0.152950i	$-0.951123\pi$
$373$	−6.98479	+	12.0980i	−0.361659	+	0.626411i	0.626575	+	0.779361i	$0.284456\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−15.6830	−	25.0181i	−0.807717	−	1.28850i
$378$		0			0
$379$	−3.47398	+	6.01711i	−0.178446	+	0.309078i	−0.941349	−	0.337436i	$-0.890440\pi$
$379$	−3.47398	+	6.01711i	−0.178446	+	0.309078i	0.762902	+	0.646514i	$0.223774\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.5333	−	21.7082i	0.640420	−	1.10924i	−0.344919	−	0.938632i	$-0.612094\pi$
$383$	12.5333	−	21.7082i	0.640420	−	1.10924i	0.985339	−	0.170607i	$-0.0545729\pi$
$384$		0			0
$385$	−2.55938	+	4.43297i	−0.130438	+	0.225925i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	14.9765	−	25.9401i	0.759339	−	1.31521i	−0.183850	−	0.982954i	$-0.558856\pi$
$389$	14.9765	−	25.9401i	0.759339	−	1.31521i	0.943188	−	0.332259i	$-0.107811\pi$
$390$		0			0
$391$	−8.63690	−	14.9596i	−0.436787	−	0.756537i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	21.0850			1.06090
$396$		0			0
$397$	−14.4399	−	25.0106i	−0.724718	−	1.25525i	−0.959090	−	0.283101i	$-0.908637\pi$
$397$	−14.4399	−	25.0106i	−0.724718	−	1.25525i	0.234372	−	0.972147i	$-0.424696\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.64590			−0.481693			−0.240847	−	0.970563i	$-0.577425\pi$
$401$	−9.64590			−0.481693			−0.240847	+	0.970563i	$0.577425\pi$
$402$		0			0
$403$	−0.495353	−	0.790204i	−0.0246753	−	0.0393629i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.01172			−0.149286
$408$		0			0
$409$	−13.2875	+	23.0146i	−0.657024	+	1.13800i	0.324358	+	0.945934i	$0.394852\pi$
$409$	−13.2875	+	23.0146i	−0.657024	+	1.13800i	−0.981382	+	0.192065i	$0.938482\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.14830	+	3.72097i	−0.105711	+	0.183097i
$414$		0			0
$415$	−28.6077			−1.40429
$416$		0			0
$417$		0			0
$418$		0			0
$419$	30.4742			1.48876			0.744380	−	0.667756i	$-0.232745\pi$
$419$	30.4742			1.48876			0.744380	+	0.667756i	$0.232745\pi$
$420$		0			0
$421$	15.2202	−	26.3622i	0.741788	−	1.28481i	−0.209893	−	0.977724i	$-0.567311\pi$
$421$	15.2202	−	26.3622i	0.741788	−	1.28481i	0.951681	−	0.307090i	$-0.0993552\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.92589			0.238941
$426$		0			0
$427$	−13.3650			−0.646778
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.842335	+	1.45897i	−0.0405738	+	0.0702759i	−0.885599	−	0.464450i	$-0.846252\pi$
$431$	−0.842335	+	1.45897i	−0.0405738	+	0.0702759i	0.845025	+	0.534726i	$0.179585\pi$
$432$		0			0
$433$	10.5289	+	18.2367i	0.505989	+	0.876398i	0.999976	+	0.00692888i	$0.00220555\pi$
$433$	10.5289	+	18.2367i	0.505989	+	0.876398i	−0.493987	+	0.869469i	$0.664461\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.83618	−	15.3047i	−0.422692	−	0.732123i
$438$		0			0
$439$	−2.94942	−	5.10855i	−0.140768	−	0.243818i	0.787018	−	0.616930i	$-0.211624\pi$
$439$	−2.94942	−	5.10855i	−0.140768	−	0.243818i	−0.927786	+	0.373112i	$0.878291\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.8651	−	20.5510i	0.563728	−	0.976406i	−0.433438	−	0.901183i	$-0.642700\pi$
$443$	11.8651	−	20.5510i	0.563728	−	0.976406i	0.997167	−	0.0752229i	$-0.0239668\pi$
$444$		0			0
$445$	7.96626			0.377637
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.3374	−	21.3689i	−0.582236	−	1.00846i	−0.995214	−	0.0977216i	$-0.968845\pi$
$449$	−12.3374	−	21.3689i	−0.582236	−	1.00846i	0.412977	−	0.910741i	$-0.364489\pi$
$450$		0			0
$451$	−0.0359981	−	0.0623506i	−0.00169509	−	0.00293597i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	12.4199	+	19.8127i	0.582254	+	0.928832i
$456$		0			0
$457$	−7.24537	−	12.5493i	−0.338924	−	0.587033i	0.645307	−	0.763924i	$-0.276730\pi$
$457$	−7.24537	−	12.5493i	−0.338924	−	0.587033i	−0.984231	+	0.176890i	$0.943396\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−27.9740			−1.30288			−0.651439	−	0.758701i	$-0.725834\pi$
$461$	−27.9740			−1.30288			−0.651439	+	0.758701i	$0.725834\pi$
$462$		0			0
$463$	−12.9541	−	22.4371i	−0.602027	−	1.04274i	−0.992514	−	0.122132i	$-0.961027\pi$
$463$	−12.9541	−	22.4371i	−0.602027	−	1.04274i	0.390487	−	0.920608i	$-0.372307\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.61901			−0.398840			−0.199420	−	0.979914i	$-0.563906\pi$
$467$	−8.61901			−0.398840			−0.199420	+	0.979914i	$0.563906\pi$
$468$		0			0
$469$	−30.0513			−1.38764
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.40033	+	5.88954i	0.156347	+	0.270801i
$474$		0			0
$475$	5.03954			0.231230
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.15223	+	15.8521i	0.418176	+	0.724302i	0.995756	−	0.0920318i	$-0.0293362\pi$
$479$	9.15223	+	15.8521i	0.418176	+	0.724302i	−0.577580	+	0.816334i	$0.696003\pi$
$480$		0			0
$481$	−6.44166	+	12.1571i	−0.293715	+	0.554315i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.103352	+	0.179011i	0.00469297	+	0.00812846i
$486$		0			0
$487$	16.8413	+	29.1699i	0.763150	+	1.32182i	0.941219	+	0.337797i	$0.109682\pi$
$487$	16.8413	+	29.1699i	0.763150	+	1.32182i	−0.178069	+	0.984018i	$0.556985\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−30.8537			−1.39241			−0.696204	−	0.717844i	$-0.745129\pi$
$491$	−30.8537			−1.39241			−0.696204	+	0.717844i	$0.745129\pi$
$492$		0			0
$493$	12.1619	−	21.0650i	0.547743	−	0.948718i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	23.1042	+	40.0177i	1.03637	+	1.79504i
$498$		0			0
$499$	−6.47626	−	11.2172i	−0.289917	−	0.502151i	0.683873	−	0.729601i	$-0.260294\pi$
$499$	−6.47626	−	11.2172i	−0.289917	−	0.502151i	−0.973790	+	0.227450i	$0.926961\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.8940	+	22.3331i	0.574916	+	0.995784i	0.996051	+	0.0887855i	$0.0282986\pi$
$503$	12.8940	+	22.3331i	0.574916	+	0.995784i	−0.421135	+	0.906998i	$0.638368\pi$
$504$		0			0
$505$	−7.44282	+	12.8913i	−0.331201	+	0.573657i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	20.0195			0.887350			0.443675	−	0.896188i	$-0.353674\pi$
$509$	20.0195			0.887350			0.443675	+	0.896188i	$0.353674\pi$
$510$		0			0
$511$	−27.2312			−1.20464
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.76852	−	4.79522i	0.121996	−	0.211302i
$516$		0			0
$517$	−1.83692			−0.0807876
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−24.1247			−1.05692			−0.528461	−	0.848958i	$-0.677231\pi$
$521$	−24.1247			−1.05692			−0.528461	+	0.848958i	$0.677231\pi$
$522$		0			0
$523$	0.701127	−	1.21439i	0.0306581	−	0.0531015i	−0.850289	−	0.526316i	$-0.823573\pi$
$523$	0.701127	−	1.21439i	0.0306581	−	0.0531015i	0.880947	+	0.473214i	$0.156906\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.384136	−	0.665343i	0.0167332	−	0.0289828i
$528$		0			0
$529$	10.8236			0.470593
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.328679	+	0.0119502i	−0.0142367	+	0.000517620i
$534$		0			0
$535$	36.3523			1.57165
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.20497	+	3.81912i	0.0949748	+	0.164501i
$540$		0			0
$541$	16.6942			0.717741			0.358871	−	0.933387i	$-0.383162\pi$
$541$	16.6942			0.717741			0.358871	+	0.933387i	$0.383162\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	17.1373	+	29.6826i	0.734080	+	1.27146i
$546$		0			0
$547$	−9.02605	+	15.6336i	−0.385926	+	0.668443i	−0.991897	−	0.127043i	$-0.959451\pi$
$547$	−9.02605	+	15.6336i	−0.385926	+	0.668443i	0.605971	+	0.795487i	$0.292785\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.4425	−	21.5510i	0.530067	−	0.918103i
$552$		0			0
$553$	20.4615	−	35.4404i	0.870112	−	1.50708i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.46050	−	16.3861i	0.400854	−	0.694300i	−0.592975	−	0.805221i	$-0.702047\pi$
$557$	9.46050	−	16.3861i	0.400854	−	0.694300i	0.993829	+	0.110921i	$0.0353800\pi$
$558$		0			0
$559$	31.0465	−	1.12880i	1.31313	−	0.0477430i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.84531	+	11.8564i	−0.288496	+	0.499689i	−0.973451	−	0.228896i	$-0.926488\pi$
$563$	−6.84531	+	11.8564i	−0.288496	+	0.499689i	0.684955	+	0.728585i	$0.259822\pi$
$564$		0			0
$565$	−4.58688			−0.192972
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.12083	−	14.0657i	−0.340443	−	0.589665i	0.644072	−	0.764965i	$-0.277244\pi$
$569$	−8.12083	−	14.0657i	−0.340443	−	0.589665i	−0.984515	+	0.175300i	$0.943910\pi$
$570$		0			0
$571$	18.7269	+	32.4359i	0.783696	+	1.35740i	0.929775	+	0.368129i	$0.120001\pi$
$571$	18.7269	+	32.4359i	0.783696	+	1.35740i	−0.146079	+	0.989273i	$0.546665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.82266	+	8.35309i	−0.201119	+	0.348348i
$576$		0			0
$577$	24.9058			1.03684			0.518421	−	0.855126i	$-0.326520\pi$
$577$	24.9058			1.03684			0.518421	+	0.855126i	$0.326520\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−27.7618	+	48.0848i	−1.15175	+	1.99489i
$582$		0			0
$583$	2.60047	+	4.50414i	0.107700	+	0.186542i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.9379	+	24.1412i	0.575279	+	0.996413i	0.996011	+	0.0892281i	$0.0284400\pi$
$587$	13.9379	+	24.1412i	0.575279	+	0.996413i	−0.420732	+	0.907185i	$0.638227\pi$
$588$		0			0
$589$	0.392999	−	0.680694i	0.0161932	−	0.0280475i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	39.1962			1.60959			0.804797	−	0.593550i	$-0.202274\pi$
$593$	39.1962			1.60959			0.804797	+	0.593550i	$0.202274\pi$
$594$		0			0
$595$	−9.63138	+	16.6820i	−0.394848	+	0.683897i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5.21215	+	9.02771i	0.212963	+	0.368862i	0.952640	−	0.304099i	$-0.0983554\pi$
$599$	5.21215	+	9.02771i	0.212963	+	0.368862i	−0.739678	+	0.672961i	$0.765022\pi$
$600$		0			0
$601$	14.0755	+	24.3795i	0.574152	+	0.994460i	0.996133	+	0.0878557i	$0.0280014\pi$
$601$	14.0755	+	24.3795i	0.574152	+	0.994460i	−0.421981	+	0.906605i	$0.638665\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	18.9691			0.771205
$606$		0			0
$607$	−10.7141	+	18.5573i	−0.434871	+	0.753219i	−0.997285	−	0.0736371i	$-0.976539\pi$
$607$	−10.7141	+	18.5573i	−0.434871	+	0.753219i	0.562414	+	0.826856i	$0.309873\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.92892	+	7.41488i	−0.158947	+	0.299974i
$612$		0			0
$613$	17.1769	−	29.7512i	0.693766	−	1.20164i	−0.276828	−	0.960919i	$-0.589283\pi$
$613$	17.1769	−	29.7512i	0.693766	−	1.20164i	0.970595	−	0.240719i	$-0.0773833\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.18881	+	10.7193i	−0.249152	+	0.431544i	−0.963291	−	0.268460i	$-0.913485\pi$
$617$	−6.18881	+	10.7193i	−0.249152	+	0.431544i	0.714139	+	0.700004i	$0.246819\pi$
$618$		0			0
$619$	−7.10653	+	12.3089i	−0.285635	+	0.494735i	−0.972763	−	0.231802i	$-0.925538\pi$
$619$	−7.10653	+	12.3089i	−0.285635	+	0.494735i	0.687128	+	0.726537i	$0.258871\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7.73071	−	13.3900i	0.309724	−	0.536458i
$624$		0			0
$625$	6.97858	+	12.0873i	0.279143	+	0.483490i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−11.3336			−0.451902
$630$		0			0
$631$	−19.7301	−	34.1735i	−0.785441	−	1.36042i	−0.928735	−	0.370744i	$-0.879103\pi$
$631$	−19.7301	−	34.1735i	−0.785441	−	1.36042i	0.143294	−	0.989680i	$-0.454231\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−40.0125			−1.58785
$636$		0			0
$637$	20.1323	−	0.731977i	0.797673	−	0.0290020i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−38.5836			−1.52396			−0.761980	−	0.647600i	$-0.775773\pi$
$641$	−38.5836			−1.52396			−0.761980	+	0.647600i	$0.775773\pi$
$642$		0			0
$643$	16.7399	−	28.9944i	0.660158	−	1.14343i	−0.320416	−	0.947277i	$-0.603823\pi$
$643$	16.7399	−	28.9944i	0.660158	−	1.14343i	0.980574	−	0.196150i	$-0.0628438\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.34123	−	16.1795i	0.367242	−	0.636081i	−0.621891	−	0.783103i	$-0.713636\pi$
$647$	9.34123	−	16.1795i	0.367242	−	0.636081i	0.989133	+	0.147022i	$0.0469688\pi$
$648$		0			0
$649$	−0.955829			−0.0375196
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−21.8099			−0.853486			−0.426743	−	0.904373i	$-0.640339\pi$
$653$	−21.8099			−0.853486			−0.426743	+	0.904373i	$0.640339\pi$
$654$		0			0
$655$	14.8912	−	25.7923i	0.581848	−	1.00779i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−42.8462			−1.66905			−0.834526	−	0.550969i	$-0.814258\pi$
$659$	−42.8462			−1.66905			−0.834526	+	0.550969i	$0.814258\pi$
$660$		0			0
$661$	−24.4668			−0.951650			−0.475825	−	0.879540i	$-0.657850\pi$
$661$	−24.4668			−0.951650			−0.475825	+	0.879540i	$0.657850\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−9.85361	+	17.0669i	−0.382106	+	0.661828i
$666$		0			0
$667$	23.8140	+	41.2471i	0.922082	+	1.59709i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.48660	−	2.57487i	−0.0573896	−	0.0994017i
$672$		0			0
$673$	4.26726	+	7.39111i	0.164491	+	0.284907i	0.936474	−	0.350736i	$-0.114069\pi$
$673$	4.26726	+	7.39111i	0.164491	+	0.284907i	−0.771983	+	0.635643i	$0.780735\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	21.1789	−	36.6829i	0.813970	−	1.40984i	−0.0960958	−	0.995372i	$-0.530636\pi$
$677$	21.1789	−	36.6829i	0.813970	−	1.40984i	0.910065	−	0.414465i	$-0.136031\pi$
$678$		0			0
$679$	0.401184			0.0153960
$680$		0			0
$681$		0			0
$682$		0			0
$683$	16.5016	+	28.5816i	0.631415	+	1.09364i	0.987263	+	0.159099i	$0.0508589\pi$
$683$	16.5016	+	28.5816i	0.631415	+	1.09364i	−0.355847	+	0.934544i	$0.615808\pi$
$684$		0			0
$685$	13.8680	+	24.0201i	0.529869	+	0.917759i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	23.7434	−	0.863268i	0.904551	−	0.0328879i
$690$		0			0
$691$	5.50867	+	9.54129i	0.209560	+	0.362968i	0.951576	−	0.307414i	$-0.0994637\pi$
$691$	5.50867	+	9.54129i	0.209560	+	0.362968i	−0.742016	+	0.670382i	$0.766130\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−33.0157			−1.25236
$696$		0			0
$697$	−0.135467	−	0.234636i	−0.00513119	−	0.00888748i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.41066			0.128819			0.0644094	−	0.997924i	$-0.479484\pi$
$701$	3.41066			0.128819			0.0644094	+	0.997924i	$0.479484\pi$
$702$		0			0
$703$	−11.5951			−0.437319
$704$		0			0
$705$		0			0
$706$		0			0
$707$	14.4455	+	25.0203i	0.543279	+	0.940986i
$708$		0			0
$709$	28.7937			1.08137			0.540684	−	0.841226i	$-0.318165\pi$
$709$	28.7937			1.08137			0.540684	+	0.841226i	$0.318165\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.752172	+	1.30280i	0.0281691	+	0.0487903i
$714$		0			0
$715$	−2.43558	+	4.59657i	−0.0910855	+	0.171902i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	6.18739	+	10.7169i	0.230751	+	0.399672i	0.958029	−	0.286670i	$-0.0925484\pi$
$719$	6.18739	+	10.7169i	0.230751	+	0.399672i	−0.727278	+	0.686343i	$0.759215\pi$
$720$		0			0
$721$	−5.37332	−	9.30686i	−0.200113	−	0.346606i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−13.5818			−0.504417
$726$		0			0
$727$	0.0603226	−	0.104482i	0.00223724	−	0.00387502i	−0.864905	−	0.501936i	$-0.832621\pi$
$727$	0.0603226	−	0.104482i	0.00223724	−	0.00387502i	0.867142	+	0.498061i	$0.165955\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12.7960	+	22.1634i	0.473278	+	0.819742i
$732$		0			0
$733$	−6.33410	−	10.9710i	−0.233955	−	0.405223i	0.725013	−	0.688735i	$-0.241834\pi$
$733$	−6.33410	−	10.9710i	−0.233955	−	0.405223i	−0.958969	+	0.283512i	$0.908500\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.34263	−	5.78960i	−0.123127	−	0.213263i
$738$		0			0
$739$	−1.93983	+	3.35988i	−0.0713578	+	0.123595i	−0.899497	−	0.436928i	$-0.856067\pi$
$739$	−1.93983	+	3.35988i	−0.0713578	+	0.123595i	0.828139	+	0.560523i	$0.189400\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	44.0182			1.61487			0.807435	−	0.589956i	$-0.200855\pi$
$743$	44.0182			1.61487			0.807435	+	0.589956i	$0.200855\pi$
$744$		0			0
$745$	18.2957			0.670304
$746$		0			0
$747$		0			0
$748$		0			0
$749$	35.2774	−	61.1023i	1.28901	−	2.23263i
$750$		0			0
$751$	−29.5451			−1.07812			−0.539058	−	0.842269i	$-0.681219\pi$
$751$	−29.5451			−1.07812			−0.539058	+	0.842269i	$0.681219\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−34.8062			−1.26673
$756$		0			0
$757$	−3.72486	+	6.45165i	−0.135382	+	0.234489i	−0.925743	−	0.378152i	$-0.876560\pi$
$757$	−3.72486	+	6.45165i	−0.135382	+	0.234489i	0.790361	+	0.612641i	$0.209893\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.73158	+	8.19534i	−0.171520	+	0.297081i	−0.938951	−	0.344050i	$-0.888201\pi$
$761$	−4.73158	+	8.19534i	−0.171520	+	0.297081i	0.767432	+	0.641131i	$0.221534\pi$
$762$		0			0
$763$	66.5223			2.40827
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.04439	+	3.85829i	−0.0738186	+	0.139315i
$768$		0			0
$769$	32.9002			1.18641			0.593205	−	0.805051i	$-0.297862\pi$
$769$	32.9002			1.18641			0.593205	+	0.805051i	$0.297862\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	25.4419	+	44.0667i	0.915082	+	1.58497i	0.806781	+	0.590851i	$0.201208\pi$
$773$	25.4419	+	44.0667i	0.915082	+	1.58497i	0.108302	+	0.994118i	$0.465459\pi$
$774$		0			0
$775$	−0.428987			−0.0154097
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.138593	−	0.240050i	−0.00496561	−	0.00860068i
$780$		0			0
$781$	−5.13981	+	8.90241i	−0.183917	+	0.318553i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	9.16549	−	15.8751i	0.327130	−	0.566606i
$786$		0			0
$787$	10.6565	−	18.4575i	0.379862	−	0.657940i	−0.611180	−	0.791492i	$-0.709305\pi$
$787$	10.6565	−	18.4575i	0.379862	−	0.657940i	0.991042	+	0.133552i	$0.0426382\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4.45126	+	7.70980i	−0.158268	+	0.274129i
$792$		0			0
$793$	−13.5733	+	0.493502i	−0.482003	+	0.0175248i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−17.2569	+	29.8898i	−0.611270	+	1.05875i	0.379757	+	0.925086i	$0.376008\pi$
$797$	−17.2569	+	29.8898i	−0.611270	+	1.05875i	−0.991027	+	0.133664i	$0.957326\pi$
$798$		0			0
$799$	−6.91265			−0.244552
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.02895	−	5.24630i	−0.106889	−	0.185138i
$804$		0			0
$805$	−18.8591	−	32.6649i	−0.664696	−	1.15129i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−17.9703	+	31.1254i	−0.631801	+	1.09431i	0.355382	+	0.934721i	$0.384351\pi$
$809$	−17.9703	+	31.1254i	−0.631801	+	1.09431i	−0.987183	+	0.159591i	$0.948983\pi$
$810$		0			0
$811$	25.8673			0.908323			0.454161	−	0.890919i	$-0.349939\pi$
$811$	25.8673			0.908323			0.454161	+	0.890919i	$0.349939\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5.01182	−	8.68073i	0.175556	−	0.304073i
$816$		0			0
$817$	13.0913	+	22.6747i	0.458006	+	0.793289i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−6.61949	−	11.4653i	−0.231022	−	0.400142i	0.727087	−	0.686545i	$-0.240874\pi$
$821$	−6.61949	−	11.4653i	−0.231022	−	0.400142i	−0.958109	+	0.286404i	$0.907540\pi$
$822$		0			0
$823$	8.00285	−	13.8614i	0.278962	−	0.483176i	−0.692165	−	0.721739i	$-0.743343\pi$
$823$	8.00285	−	13.8614i	0.278962	−	0.483176i	0.971127	+	0.238563i	$0.0766763\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−32.7803			−1.13988			−0.569942	−	0.821685i	$-0.693034\pi$
$827$	−32.7803			−1.13988			−0.569942	+	0.821685i	$0.693034\pi$
$828$		0			0
$829$	−6.77951	+	11.7424i	−0.235462	+	0.407832i	−0.959407	−	0.282026i	$-0.908994\pi$
$829$	−6.77951	+	11.7424i	−0.235462	+	0.407832i	0.723945	+	0.689858i	$0.242327\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	8.29769	+	14.3720i	0.287498	+	0.497961i
$834$		0			0
$835$	−13.8142	−	23.9269i	−0.478061	−	0.828026i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−22.1091			−0.763290			−0.381645	−	0.924309i	$-0.624642\pi$
$839$	−22.1091			−0.763290			−0.381645	+	0.924309i	$0.624642\pi$
$840$		0			0
$841$	−19.0332	+	32.9664i	−0.656316	+	1.13677i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	13.3451	+	19.6629i	0.459084	+	0.676423i
$846$		0			0
$847$	18.4082	−	31.8840i	0.632515	−	1.09555i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	11.0961	−	19.2191i	0.380371	−	0.658821i
$852$		0			0
$853$	25.0344	−	43.3608i	0.857160	−	1.48464i	−0.0174667	−	0.999847i	$-0.505560\pi$
$853$	25.0344	−	43.3608i	0.857160	−	1.48464i	0.874627	−	0.484797i	$-0.161107\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	20.7269	−	35.9000i	0.708018	−	1.22632i	−0.257574	−	0.966259i	$-0.582923\pi$
$857$	20.7269	−	35.9000i	0.708018	−	1.22632i	0.965591	−	0.260064i	$-0.0837436\pi$
$858$		0			0
$859$	−12.4494	−	21.5630i	−0.424769	−	0.735721i	0.571630	−	0.820511i	$-0.306311\pi$
$859$	−12.4494	−	21.5630i	−0.424769	−	0.735721i	−0.996399	+	0.0847904i	$0.972978\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	15.8767			0.540448			0.270224	−	0.962797i	$-0.412902\pi$
$863$	15.8767			0.540448			0.270224	+	0.962797i	$0.412902\pi$
$864$		0			0
$865$	6.29710	+	10.9069i	0.214108	+	0.370845i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9.10380			0.308825
$870$		0			0
$871$	−30.5197	+	1.10964i	−1.03412	+	0.0375988i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	43.1832			1.45986
$876$		0			0
$877$	5.00766	−	8.67352i	0.169097	−	0.292884i	−0.769006	−	0.639242i	$-0.779248\pi$
$877$	5.00766	−	8.67352i	0.169097	−	0.292884i	0.938103	+	0.346358i	$0.112582\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−11.0336	+	19.1108i	−0.371732	+	0.643859i	−0.989832	−	0.142240i	$-0.954569\pi$
$881$	−11.0336	+	19.1108i	−0.371732	+	0.643859i	0.618100	+	0.786100i	$0.287903\pi$
$882$		0			0
$883$	46.2340			1.55590			0.777950	−	0.628326i	$-0.216260\pi$
$883$	46.2340			1.55590			0.777950	+	0.628326i	$0.216260\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.92642			0.0646828			0.0323414	−	0.999477i	$-0.489704\pi$
$887$	1.92642			0.0646828			0.0323414	+	0.999477i	$0.489704\pi$
$888$		0			0
$889$	−38.8294	+	67.2545i	−1.30230	+	2.25564i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.07214			−0.236660
$894$		0			0
$895$	14.2743			0.477138
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.05915	+	1.83451i	−0.0353248	+	0.0611843i
$900$		0			0
$901$	9.78601	+	16.9499i	0.326019	+	0.564682i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−8.54868	−	14.8067i	−0.284168	−	0.492193i
$906$		0			0
$907$	−1.54206	−	2.67092i	−0.0512032	−	0.0886865i	0.839288	−	0.543687i	$-0.182972\pi$
$907$	−1.54206	−	2.67092i	−0.0512032	−	0.0886865i	−0.890491	+	0.455001i	$0.849639\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5.56268	−	9.63484i	0.184300	−	0.319216i	−0.759041	−	0.651043i	$-0.774332\pi$
$911$	5.56268	−	9.63484i	0.184300	−	0.319216i	0.943340	+	0.331827i	$0.107665\pi$
$912$		0			0
$913$	−12.3519			−0.408787
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−28.9018	−	50.0594i	−0.954422	−	1.65311i
$918$		0			0
$919$	28.4673	+	49.3069i	0.939051	+	1.62648i	0.767247	+	0.641352i	$0.221626\pi$
$919$	28.4673	+	49.3069i	0.939051	+	1.62648i	0.171804	+	0.985131i	$0.445040\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	24.9420	+	39.7883i	0.820976	+	1.30965i
$924$		0			0
$925$	3.16423	+	5.48061i	0.104039	+	0.180201i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.34805			0.0770370			0.0385185	−	0.999258i	$-0.487736\pi$
$929$	2.34805			0.0770370			0.0385185	+	0.999258i	$0.487736\pi$
$930$		0			0
$931$	8.48914	+	14.7036i	0.278220	+	0.481892i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.28523			−0.140142
$936$		0			0
$937$	1.23201			0.0402482			0.0201241	−	0.999797i	$-0.493594\pi$
$937$	1.23201			0.0402482			0.0201241	+	0.999797i	$0.493594\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	10.2312	+	17.7210i	0.333529	+	0.577689i	0.983201	−	0.182525i	$-0.0584272\pi$
$941$	10.2312	+	17.7210i	0.333529	+	0.577689i	−0.649672	+	0.760214i	$0.725094\pi$
$942$		0			0
$943$	0.530514			0.0172759
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−18.0932	−	31.3384i	−0.587951	−	1.01836i	−0.994500	−	0.104733i	$-0.966601\pi$
$947$	−18.0932	−	31.3384i	−0.587951	−	1.01836i	0.406549	−	0.913629i	$-0.366732\pi$
$948$		0			0
$949$	−27.6557	+	1.00551i	−0.897741	+	0.0326403i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−10.3996	−	18.0126i	−0.336875	−	0.583484i	0.646969	−	0.762517i	$-0.276036\pi$
$953$	−10.3996	−	18.0126i	−0.336875	−	0.583484i	−0.983843	+	0.179033i	$0.942703\pi$
$954$		0			0
$955$	−17.7004	−	30.6579i	−0.572770	−	0.992067i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	53.8317			1.73832
$960$		0			0
$961$	15.4665	−	26.7888i	0.498921	−	0.864156i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.09116	+	8.81814i	0.163890	+	0.283866i
$966$		0			0
$967$	6.25537	+	10.8346i	0.201159	+	0.348418i	0.948902	−	0.315570i	$-0.102196\pi$
$967$	6.25537	+	10.8346i	0.201159	+	0.348418i	−0.747743	+	0.663988i	$0.768862\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.25114	+	3.89909i	0.0722425	+	0.125128i	0.899884	−	0.436130i	$-0.143651\pi$
$971$	2.25114	+	3.89909i	0.0722425	+	0.125128i	−0.827641	+	0.561257i	$0.810318\pi$
$972$		0			0
$973$	−32.0395	+	55.4941i	−1.02714	+	1.77906i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	47.5050			1.51982			0.759910	−	0.650029i	$-0.225243\pi$
$977$	47.5050			1.51982			0.759910	+	0.650029i	$0.225243\pi$
$978$		0			0
$979$	3.43957			0.109929
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.94982	+	12.0374i	−0.221665	+	0.383935i	−0.955314	−	0.295594i	$-0.904482\pi$
$983$	−6.94982	+	12.0374i	−0.221665	+	0.383935i	0.733649	+	0.679529i	$0.237816\pi$
$984$		0			0
$985$	41.4333			1.32017
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−50.1115			−1.59345
$990$		0			0
$991$	9.92979	−	17.1989i	0.315430	−	0.546341i	−0.664099	−	0.747645i	$-0.731185\pi$
$991$	9.92979	−	17.1989i	0.315430	−	0.546341i	0.979529	+	0.201304i	$0.0645178\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	9.32478	−	16.1510i	0.295615	−	0.512021i
$996$		0			0
$997$	−52.8484			−1.67372			−0.836862	−	0.547414i	$-0.815612\pi$
$997$	−52.8484			−1.67372			−0.836862	+	0.547414i	$0.815612\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1404.2.k.a.1153.11		28
3.2	odd	2		468.2.k.a.61.5	yes	28
9.4	even	3		1404.2.j.a.685.11		28
9.5	odd	6		468.2.j.a.373.6	yes	28
13.3	even	3		1404.2.j.a.289.11		28
39.29	odd	6		468.2.j.a.133.6	✓	28
117.68	odd	6		468.2.k.a.445.5	yes	28
117.94	even	3	inner	1404.2.k.a.1225.11		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.2.j.a.133.6	✓	28	39.29	odd	6
468.2.j.a.373.6	yes	28	9.5	odd	6
468.2.k.a.61.5	yes	28	3.2	odd	2
468.2.k.a.445.5	yes	28	117.68	odd	6
1404.2.j.a.289.11		28	13.3	even	3
1404.2.j.a.685.11		28	9.4	even	3
1404.2.k.a.1153.11		28	1.1	even	1	trivial
1404.2.k.a.1225.11		28	117.94	even	3	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1404 = 2^{2} \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1404.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.913993 + 1.58308i) q^{5} +3.54787 q^{7} +(0.394633 + 0.683524i) q^{11} +(3.60317 - 0.131005i) q^{13} +(1.48507 + 2.57222i) q^{17} +(1.51934 + 2.63157i) q^{19} -5.81581 q^{23} +(0.829233 - 1.43627i) q^{25} +(-4.09470 - 7.09223i) q^{29} +(-0.129332 - 0.224010i) q^{31} +(3.24273 + 5.61657i) q^{35} +(-1.90793 + 3.30463i) q^{37} -0.0912193 q^{41} +8.61643 q^{43} +(-1.16369 + 2.01557i) q^{47} +5.58740 q^{49} +6.58958 q^{53} +(-0.721383 + 1.24947i) q^{55} +(-0.605518 + 1.04879i) q^{59} -3.76705 q^{61} +(3.50066 + 5.58438i) q^{65} -8.47022 q^{67} +(6.51214 + 11.2794i) q^{71} -7.67537 q^{73} +(1.40011 + 2.42506i) q^{77} +(5.76726 - 9.98919i) q^{79} +(-7.82491 + 13.5531i) q^{83} +(-2.71469 + 4.70199i) q^{85} +(2.17897 - 3.77409i) q^{89} +(12.7836 - 0.464789i) q^{91} +(-2.77733 + 4.81047i) q^{95} +0.113077 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{7} + 4 q^{11} + q^{13} + 8 q^{17} - q^{19} - 8 q^{23} - 14 q^{25} + 13 q^{29} + 2 q^{31} - 3 q^{35} - q^{37} + 8 q^{41} - 4 q^{43} - 11 q^{47} + 24 q^{49} - 52 q^{53} + 8 q^{59} + 14 q^{61}+ \cdots + 50 q^{97}+O(q^{100}) \) 28 * q - 4 * q^7 + 4 * q^11 + q^13 + 8 * q^17 - q^19 - 8 * q^23 - 14 * q^25 + 13 * q^29 + 2 * q^31 - 3 * q^35 - q^37 + 8 * q^41 - 4 * q^43 - 11 * q^47 + 24 * q^49 - 52 * q^53 + 8 * q^59 + 14 * q^61 - 38 * q^65 + 14 * q^67 + 12 * q^71 + 14 * q^73 + 28 * q^77 + 5 * q^79 - 9 * q^83 + 18 * q^85 + 11 * q^89 - q^91 - 28 * q^95 + 50 * q^97

Introduction

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