LMFDB - Embedded newform 1380.2.n.a.689.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(689,1380)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1380.689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1380.n (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$11.0193554789$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			689.18
Character	$\chi$	$=$	1380.689
Dual form			1380.2.n.a.689.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.779801 - 1.54658i) q^{3} +(0.268100 + 2.21994i) q^{5} -2.49920 q^{7} +(-1.78382 + 2.41205i) q^{9} +O(q^{10})$	\(q+(-0.779801 - 1.54658i) q^{3} +(0.268100 + 2.21994i) q^{5} -2.49920 q^{7} +(-1.78382 + 2.41205i) q^{9} +3.17850 q^{11} +5.31475i q^{13} +(3.22425 - 2.14575i) q^{15} -2.70251i q^{17} -8.13115i q^{19} +(1.94888 + 3.86522i) q^{21} +(-4.12360 - 2.44865i) q^{23} +(-4.85624 + 1.19033i) q^{25} +(5.12145 + 0.877900i) q^{27} -2.82516i q^{29} -6.28802 q^{31} +(-2.47860 - 4.91580i) q^{33} +(-0.670036 - 5.54808i) q^{35} -3.82520 q^{37} +(8.21968 - 4.14445i) q^{39} +3.17048i q^{41} -6.93673 q^{43} +(-5.83284 - 3.31330i) q^{45} -8.10788 q^{47} -0.753980 q^{49} +(-4.17965 + 2.10742i) q^{51} -3.79254i q^{53} +(0.852155 + 7.05607i) q^{55} +(-12.5755 + 6.34069i) q^{57} -11.2202i q^{59} +0.443335i q^{61} +(4.45813 - 6.02821i) q^{63} +(-11.7984 + 1.42488i) q^{65} +6.68981 q^{67} +(-0.571450 + 8.28694i) q^{69} +14.4086i q^{71} -2.80940i q^{73} +(5.62785 + 6.58235i) q^{75} -7.94372 q^{77} +6.08039i q^{79} +(-2.63597 - 8.60533i) q^{81} +8.29322i q^{83} +(5.99941 - 0.724543i) q^{85} +(-4.36934 + 2.20306i) q^{87} +9.80790 q^{89} -13.2826i q^{91} +(4.90340 + 9.72492i) q^{93} +(18.0507 - 2.17996i) q^{95} -16.6155 q^{97} +(-5.66987 + 7.66670i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 16 q^{31} + 24 q^{39} + 112 q^{49} + 8 q^{55} - 16 q^{69} + 20 q^{75} - 8 q^{81} + 32 q^{85}+O(q^{100}) $ 48 * q + 16 * q^31 + 24 * q^39 + 112 * q^49 + 8 * q^55 - 16 * q^69 + 20 * q^75 - 8 * q^81 + 32 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$691$	$1201$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

Coefficient data

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

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
$3$	−0.779801	−	1.54658i	−0.450219	−	0.892918i
$3$	−0.779801	−	1.54658i	−0.450219	−	0.892918i
$4$		0			0
$5$	0.268100	+	2.21994i	0.119898	+	0.992786i
$5$	0.268100	+	2.21994i	0.119898	+	0.992786i
$6$		0			0
$7$	−2.49920			−0.944610			−0.472305	−	0.881435i	$-0.656578\pi$
$7$	−2.49920			−0.944610			−0.472305	+	0.881435i	$0.656578\pi$
$8$		0			0
$9$	−1.78382	+	2.41205i	−0.594607	+	0.804017i
$10$		0			0
$11$	3.17850			0.958353			0.479177	−	0.877718i	$-0.340935\pi$
$11$	3.17850			0.958353			0.479177	+	0.877718i	$0.340935\pi$
$12$		0			0
$13$			5.31475i			1.47405i	0.675868	+	0.737023i	$0.263769\pi$
$13$			5.31475i			1.47405i	−0.675868	+	0.737023i	$0.736231\pi$
$14$		0			0
$15$	3.22425	−	2.14575i	0.832497	−	0.554030i
$16$		0			0
$17$		−	2.70251i		−	0.655456i	−0.944772	−	0.327728i	$-0.893717\pi$
$17$		−	2.70251i		−	0.655456i	0.944772	−	0.327728i	$-0.106283\pi$
$18$		0			0
$19$		−	8.13115i		−	1.86541i	−0.360634	−	0.932707i	$-0.617440\pi$
$19$		−	8.13115i		−	1.86541i	0.360634	−	0.932707i	$-0.382560\pi$
$20$		0			0
$21$	1.94888	+	3.86522i	0.425281	+	0.843460i
$22$		0			0
$23$	−4.12360	−	2.44865i	−0.859830	−	0.510580i
$23$	−4.12360	−	2.44865i	−0.859830	−	0.510580i
$24$		0			0
$25$	−4.85624	+	1.19033i	−0.971249	+	0.238066i
$26$		0			0
$27$	5.12145	+	0.877900i	0.985624	+	0.168952i
$28$		0			0
$29$		−	2.82516i		−	0.524619i	−0.964984	−	0.262310i	$-0.915516\pi$
$29$		−	2.82516i		−	0.524619i	0.964984	−	0.262310i	$-0.0844842\pi$
$30$		0			0
$31$	−6.28802			−1.12936			−0.564680	−	0.825310i	$-0.691001\pi$
$31$	−6.28802			−1.12936			−0.564680	+	0.825310i	$0.691001\pi$
$32$		0			0
$33$	−2.47860	−	4.91580i	−0.431468	−	0.855731i
$34$		0			0
$35$	−0.670036	−	5.54808i	−0.113257	−	0.937796i
$36$		0			0
$37$	−3.82520			−0.628860			−0.314430	−	0.949281i	$-0.601813\pi$
$37$	−3.82520			−0.628860			−0.314430	+	0.949281i	$0.601813\pi$
$38$		0			0
$39$	8.21968	−	4.14445i	1.31620	−	0.663643i
$40$		0			0
$41$			3.17048i			0.495146i	0.968869	+	0.247573i	$0.0796330\pi$
$41$			3.17048i			0.495146i	−0.968869	+	0.247573i	$0.920367\pi$
$42$		0			0
$43$	−6.93673			−1.05784			−0.528920	−	0.848671i	$-0.677403\pi$
$43$	−6.93673			−1.05784			−0.528920	+	0.848671i	$0.677403\pi$
$44$		0			0
$45$	−5.83284	−	3.31330i	−0.869509	−	0.493917i
$46$		0			0
$47$	−8.10788			−1.18266			−0.591328	−	0.806431i	$-0.701396\pi$
$47$	−8.10788			−1.18266			−0.591328	+	0.806431i	$0.701396\pi$
$48$		0			0
$49$	−0.753980			−0.107711
$50$		0			0
$51$	−4.17965	+	2.10742i	−0.585268	+	0.295098i
$52$		0			0
$53$		−	3.79254i		−	0.520946i	−0.965481	−	0.260473i	$-0.916122\pi$
$53$		−	3.79254i		−	0.520946i	0.965481	−	0.260473i	$-0.0838784\pi$
$54$		0			0
$55$	0.852155	+	7.05607i	0.114905	+	0.951440i
$56$		0			0
$57$	−12.5755	+	6.34069i	−1.66566	+	0.839844i
$58$		0			0
$59$		−	11.2202i		−	1.46075i	−0.683049	−	0.730373i	$-0.739346\pi$
$59$		−	11.2202i		−	1.46075i	0.683049	−	0.730373i	$-0.260654\pi$
$60$		0			0
$61$			0.443335i			0.0567632i	0.999597	+	0.0283816i	$0.00903536\pi$
$61$			0.443335i			0.0567632i	−0.999597	+	0.0283816i	$0.990965\pi$
$62$		0			0
$63$	4.45813	−	6.02821i	0.561671	−	0.759483i
$64$		0			0
$65$	−11.7984	+	1.42488i	−1.46341	+	0.176735i
$66$		0			0
$67$	6.68981			0.817290			0.408645	−	0.912693i	$-0.366001\pi$
$67$	6.68981			0.817290			0.408645	+	0.912693i	$0.366001\pi$
$68$		0			0
$69$	−0.571450	+	8.28694i	−0.0687945	+	0.997631i
$70$		0			0
$71$			14.4086i			1.70998i	0.518641	+	0.854992i	$0.326438\pi$
$71$			14.4086i			1.70998i	−0.518641	+	0.854992i	$0.673562\pi$
$72$		0			0
$73$		−	2.80940i		−	0.328816i	−0.986392	−	0.164408i	$-0.947429\pi$
$73$		−	2.80940i		−	0.328816i	0.986392	−	0.164408i	$-0.0525713\pi$
$74$		0			0
$75$	5.62785	+	6.58235i	0.649848	+	0.760064i
$76$		0			0
$77$	−7.94372			−0.905271
$78$		0			0
$79$			6.08039i			0.684097i	0.939682	+	0.342049i	$0.111121\pi$
$79$			6.08039i			0.684097i	−0.939682	+	0.342049i	$0.888879\pi$
$80$		0			0
$81$	−2.63597	−	8.60533i	−0.292886	−	0.956147i
$82$		0			0
$83$			8.29322i			0.910299i	0.890415	+	0.455150i	$0.150414\pi$
$83$			8.29322i			0.910299i	−0.890415	+	0.455150i	$0.849586\pi$
$84$		0			0
$85$	5.99941	−	0.724543i	0.650727	−	0.0785877i
$86$		0			0
$87$	−4.36934	+	2.20306i	−0.468442	+	0.236193i
$88$		0			0
$89$	9.80790			1.03964			0.519818	−	0.854277i	$-0.326000\pi$
$89$	9.80790			1.03964			0.519818	+	0.854277i	$0.326000\pi$
$90$		0			0
$91$		−	13.2826i		−	1.39240i
$92$		0			0
$93$	4.90340	+	9.72492i	0.508459	+	1.00843i
$94$		0			0
$95$	18.0507	−	2.17996i	1.85196	−	0.223659i
$96$		0			0
$97$	−16.6155			−1.68705			−0.843523	−	0.537093i	$-0.819523\pi$
$97$	−16.6155			−1.68705			−0.843523	+	0.537093i	$0.819523\pi$
$98$		0			0
$99$	−5.66987	+	7.66670i	−0.569843	+	0.770532i
$100$		0			0
$101$			11.4594i			1.14025i	0.821558	+	0.570125i	$0.193105\pi$
$101$			11.4594i			1.14025i	−0.821558	+	0.570125i	$0.806895\pi$
$102$		0			0
$103$	−1.97953			−0.195049			−0.0975245	−	0.995233i	$-0.531092\pi$
$103$	−1.97953			−0.195049			−0.0975245	+	0.995233i	$0.531092\pi$
$104$		0			0
$105$	−8.05805	+	5.36266i	−0.786385	+	0.523342i
$106$		0			0
$107$		−	3.29812i		−	0.318841i	−0.987211	−	0.159421i	$-0.949037\pi$
$107$		−	3.29812i		−	0.318841i	0.987211	−	0.159421i	$-0.0509626\pi$
$108$		0			0
$109$		−	10.6960i		−	1.02449i	−0.858840	−	0.512243i	$-0.828815\pi$
$109$		−	10.6960i		−	1.02449i	0.858840	−	0.512243i	$-0.171185\pi$
$110$		0			0
$111$	2.98290	+	5.91598i	0.283124	+	0.561520i
$112$		0			0
$113$		−	18.1325i		−	1.70576i	−0.522108	−	0.852879i	$-0.674854\pi$
$113$		−	18.1325i		−	1.70576i	0.522108	−	0.852879i	$-0.325146\pi$
$114$		0			0
$115$	4.33032	−	9.81062i	0.403805	−	0.914845i
$116$		0			0
$117$	−12.8194	−	9.48055i	−1.18516	−	0.876477i
$118$		0			0
$119$			6.75413i			0.619150i
$120$		0			0
$121$	−0.897145			−0.0815586
$122$		0			0
$123$	4.90340	−	2.47235i	0.442125	−	0.222924i
$124$		0			0
$125$	−3.94442	−	10.4614i	−0.352799	−	0.935699i
$126$		0			0
$127$		−	18.3592i		−	1.62911i	−0.580083	−	0.814557i	$-0.696980\pi$
$127$		−	18.3592i		−	1.62911i	0.580083	−	0.814557i	$-0.303020\pi$
$128$		0			0
$129$	5.40927	+	10.7282i	0.476260	+	0.944566i
$130$		0			0
$131$		−	0.358033i		−	0.0312815i	−0.999878	−	0.0156408i	$-0.995021\pi$
$131$		−	0.358033i		−	0.0312815i	0.999878	−	0.0156408i	$-0.00497881\pi$
$132$		0			0
$133$			20.3214i			1.76209i
$134$		0			0
$135$	−0.575822	+	11.6047i	−0.0495588	+	0.998771i
$136$		0			0
$137$			5.52990i			0.472452i	0.971698	+	0.236226i	$0.0759105\pi$
$137$			5.52990i			0.472452i	−0.971698	+	0.236226i	$0.924090\pi$
$138$		0			0
$139$	−2.68656			−0.227871			−0.113936	−	0.993488i	$-0.536346\pi$
$139$	−2.68656			−0.227871			−0.113936	+	0.993488i	$0.536346\pi$
$140$		0			0
$141$	6.32254	+	12.5395i	0.532454	+	1.05602i
$142$		0			0
$143$			16.8929i			1.41266i
$144$		0			0
$145$	6.27168	−	0.757425i	0.520835	−	0.0629007i
$146$		0			0
$147$	0.587955	+	1.16609i	0.0484937	+	0.0961775i
$148$		0			0
$149$	−14.8994			−1.22060			−0.610302	−	0.792169i	$-0.708952\pi$
$149$	−14.8994			−1.22060			−0.610302	+	0.792169i	$0.708952\pi$
$150$		0			0
$151$	−14.8285			−1.20673			−0.603365	−	0.797465i	$-0.706174\pi$
$151$	−14.8285			−1.20673			−0.603365	+	0.797465i	$0.706174\pi$
$152$		0			0
$153$	6.51860	+	4.82080i	0.526997	+	0.389738i
$154$		0			0
$155$	−1.68582	−	13.9590i	−0.135408	−	1.12121i
$156$		0			0
$157$	−12.1780			−0.971907			−0.485953	−	0.873985i	$-0.661528\pi$
$157$	−12.1780			−0.971907			−0.485953	+	0.873985i	$0.661528\pi$
$158$		0			0
$159$	−5.86547	+	2.95743i	−0.465162	+	0.234539i
$160$		0			0
$161$	10.3057	+	6.11969i	0.812205	+	0.482299i
$162$		0			0
$163$		−	7.09350i		−	0.555606i	−0.960638	−	0.277803i	$-0.910394\pi$
$163$		−	7.09350i		−	0.555606i	0.960638	−	0.277803i	$-0.0896062\pi$
$164$		0			0
$165$	10.2483	−	6.82026i	0.797826	−	0.530956i
$166$		0			0
$167$	−2.35456			−0.182201			−0.0911006	−	0.995842i	$-0.529038\pi$
$167$	−2.35456			−0.182201			−0.0911006	+	0.995842i	$0.529038\pi$
$168$		0			0
$169$	−15.2465			−1.17281
$170$		0			0
$171$	19.6128	+	14.5045i	1.49982	+	1.10919i
$172$		0			0
$173$	−9.58728			−0.728907			−0.364454	−	0.931222i	$-0.618744\pi$
$173$	−9.58728			−0.728907			−0.364454	+	0.931222i	$0.618744\pi$
$174$		0			0
$175$	12.1367	−	2.97488i	0.917452	−	0.224880i
$176$		0			0
$177$	−17.3529	+	8.74953i	−1.30433	+	0.657655i
$178$		0			0
$179$			4.33683i			0.324150i	0.986778	+	0.162075i	$0.0518187\pi$
$179$			4.33683i			0.324150i	−0.986778	+	0.162075i	$0.948181\pi$
$180$		0			0
$181$			12.0682i			0.897022i	0.893777	+	0.448511i	$0.148046\pi$
$181$			12.0682i			0.897022i	−0.893777	+	0.448511i	$0.851954\pi$
$182$		0			0
$183$	0.685653	−	0.345713i	0.0506849	−	0.0255559i
$184$		0			0
$185$	−1.02554	−	8.49171i	−0.0753989	−	0.624323i
$186$		0			0
$187$		−	8.58993i		−	0.628158i
$188$		0			0
$189$	−12.7996	−	2.19405i	−0.931031	−	0.159594i
$190$		0			0
$191$	16.4312			1.18892			0.594460	−	0.804125i	$-0.297366\pi$
$191$	16.4312			1.18892			0.594460	+	0.804125i	$0.297366\pi$
$192$		0			0
$193$			7.22962i			0.520399i	0.965555	+	0.260200i	$0.0837884\pi$
$193$			7.22962i			0.520399i	−0.965555	+	0.260200i	$0.916212\pi$
$194$		0			0
$195$	11.4041	+	17.1361i	0.816665	+	1.22714i
$196$		0			0
$197$	21.3898			1.52396			0.761980	−	0.647601i	$-0.224228\pi$
$197$	21.3898			1.52396			0.761980	+	0.647601i	$0.224228\pi$
$198$		0			0
$199$		−	13.5953i		−	0.963748i	−0.876241	−	0.481874i	$-0.839956\pi$
$199$		−	13.5953i		−	0.963748i	0.876241	−	0.481874i	$-0.160044\pi$
$200$		0			0
$201$	−5.21672	−	10.3463i	−0.367959	−	0.729773i
$202$		0			0
$203$			7.06065i			0.495561i
$204$		0			0
$205$	−7.03827	+	0.850005i	−0.491574	+	0.0593670i
$206$		0			0
$207$	13.2620	−	5.57838i	0.921776	−	0.387724i
$208$		0			0
$209$		−	25.8449i		−	1.78773i
$210$		0			0
$211$	−24.6245			−1.69522			−0.847611	−	0.530618i	$-0.821960\pi$
$211$	−24.6245			−1.69522			−0.847611	+	0.530618i	$0.821960\pi$
$212$		0			0
$213$	22.2840	−	11.2358i	1.52688	−	0.769866i
$214$		0			0
$215$	−1.85974	−	15.3991i	−0.126833	−	1.05021i
$216$		0			0
$217$	15.7150			1.06681
$218$		0			0
$219$	−4.34496	+	2.19078i	−0.293606	+	0.148039i
$220$		0			0
$221$	14.3632			0.966171
$222$		0			0
$223$		−	20.3111i		−	1.36013i	−0.733151	−	0.680066i	$-0.761951\pi$
$223$		−	20.3111i		−	1.36013i	0.733151	−	0.680066i	$-0.238049\pi$
$224$		0			0
$225$	5.79153	−	13.8368i	0.386102	−	0.922456i
$226$		0			0
$227$			12.5641i			0.833910i	0.908927	+	0.416955i	$0.136903\pi$
$227$			12.5641i			0.833910i	−0.908927	+	0.416955i	$0.863097\pi$
$228$		0			0
$229$		−	3.51559i		−	0.232317i	−0.993231	−	0.116158i	$-0.962942\pi$
$229$		−	3.51559i		−	0.232317i	0.993231	−	0.116158i	$-0.0370580\pi$
$230$		0			0
$231$	6.19452	+	12.2856i	0.407570	+	0.808333i
$232$		0			0
$233$	14.9281			0.977972			0.488986	−	0.872292i	$-0.337367\pi$
$233$	14.9281			0.977972			0.488986	+	0.872292i	$0.337367\pi$
$234$		0			0
$235$	−2.17372	−	17.9990i	−0.141798	−	1.17412i
$236$		0			0
$237$	9.40381	−	4.74150i	0.610843	−	0.307993i
$238$		0			0
$239$		−	2.02203i		−	0.130794i	−0.997859	−	0.0653971i	$-0.979169\pi$
$239$		−	2.02203i		−	0.130794i	0.997859	−	0.0653971i	$-0.0208314\pi$
$240$		0			0
$241$		−	16.7982i		−	1.08207i	−0.841000	−	0.541034i	$-0.818033\pi$
$241$		−	16.7982i		−	1.08207i	0.841000	−	0.541034i	$-0.181967\pi$
$242$		0			0
$243$	−11.2533	+	10.7872i	−0.721899	+	0.691999i
$244$		0			0
$245$	−0.202142	−	1.67379i	−0.0129144	−	0.106934i
$246$		0			0
$247$	43.2150			2.74971
$248$		0			0
$249$	12.8261	−	6.46707i	0.812823	−	0.409834i
$250$		0			0
$251$	−20.1796			−1.27372			−0.636862	−	0.770978i	$-0.719768\pi$
$251$	−20.1796			−1.27372			−0.636862	+	0.770978i	$0.719768\pi$
$252$		0			0
$253$	−13.1069	−	7.78305i	−0.824021	−	0.489316i
$254$		0			0
$255$	−5.79891	−	8.71357i	−0.363142	−	0.545665i
$256$		0			0
$257$	25.5299			1.59251			0.796254	−	0.604962i	$-0.206812\pi$
$257$	25.5299			1.59251			0.796254	+	0.604962i	$0.206812\pi$
$258$		0			0
$259$	9.55996			0.594027
$260$		0			0
$261$	6.81443	+	5.03958i	0.421803	+	0.311942i
$262$		0			0
$263$			23.9949i			1.47959i	0.672832	+	0.739795i	$0.265078\pi$
$263$			23.9949i			1.47959i	−0.672832	+	0.739795i	$0.734922\pi$
$264$		0			0
$265$	8.41920	−	1.01678i	0.517188	−	0.0624603i
$266$		0			0
$267$	−7.64822	−	15.1687i	−0.468063	−	0.928310i
$268$		0			0
$269$			15.9382i			0.971766i	0.874024	+	0.485883i	$0.161502\pi$
$269$			15.9382i			0.971766i	−0.874024	+	0.485883i	$0.838498\pi$
$270$		0			0
$271$	−13.9813			−0.849302			−0.424651	−	0.905357i	$-0.639603\pi$
$271$	−13.9813			−0.849302			−0.424651	+	0.905357i	$0.639603\pi$
$272$		0			0
$273$	−20.5427	+	10.3578i	−1.24330	+	0.626884i
$274$		0			0
$275$	−15.4356	+	3.78346i	−0.930800	+	0.228151i
$276$		0			0
$277$			10.5924i			0.636439i	0.948017	+	0.318219i	$0.103085\pi$
$277$			10.5924i			0.636439i	−0.948017	+	0.318219i	$0.896915\pi$
$278$		0			0
$279$	11.2167	−	15.1670i	0.671525	−	0.908025i
$280$		0			0
$281$	5.21896			0.311337			0.155668	−	0.987809i	$-0.450247\pi$
$281$	5.21896			0.311337			0.155668	+	0.987809i	$0.450247\pi$
$282$		0			0
$283$	28.8748			1.71643			0.858214	−	0.513291i	$-0.171574\pi$
$283$	28.8748			1.71643			0.858214	+	0.513291i	$0.171574\pi$
$284$		0			0
$285$	−17.4474	−	26.2168i	−1.03350	−	1.55295i
$286$		0			0
$287$		−	7.92368i		−	0.467720i
$288$		0			0
$289$	9.69642			0.570378
$290$		0			0
$291$	12.9568	+	25.6972i	0.759540	+	1.50639i
$292$		0			0
$293$		−	1.16268i		−	0.0679248i	−0.999423	−	0.0339624i	$-0.989187\pi$
$293$		−	1.16268i		−	0.0679248i	0.999423	−	0.0339624i	$-0.0108126\pi$
$294$		0			0
$295$	24.9081	−	3.00813i	1.45021	−	0.175140i
$296$		0			0
$297$	16.2785	+	2.79040i	0.944576	+	0.161916i
$298$		0			0
$299$	13.0140	−	21.9159i	0.752618	−	1.26743i
$300$		0			0
$301$	17.3363			0.999247
$302$		0			0
$303$	17.7228	−	8.93603i	1.01815	−	0.513362i
$304$		0			0
$305$	−0.984176	+	0.118858i	−0.0563538	+	0.00680579i
$306$		0			0
$307$			16.6278i			0.948998i	0.880256	+	0.474499i	$0.157371\pi$
$307$			16.6278i			0.948998i	−0.880256	+	0.474499i	$0.842629\pi$
$308$		0			0
$309$	1.54364	+	3.06150i	0.0878147	+	0.174163i
$310$		0			0
$311$			23.5661i			1.33631i	0.744021	+	0.668156i	$0.232916\pi$
$311$			23.5661i			1.33631i	−0.744021	+	0.668156i	$0.767084\pi$
$312$		0			0
$313$	19.4502			1.09939			0.549694	−	0.835366i	$-0.314744\pi$
$313$	19.4502			1.09939			0.549694	+	0.835366i	$0.314744\pi$
$314$		0			0
$315$	14.5775	+	8.28061i	0.821347	+	0.466559i
$316$		0			0
$317$	−18.7869			−1.05518			−0.527588	−	0.849500i	$-0.676904\pi$
$317$	−18.7869			−1.05518			−0.527588	+	0.849500i	$0.676904\pi$
$318$		0			0
$319$		−	8.97977i		−	0.502771i
$320$		0			0
$321$	−5.10081	+	2.57188i	−0.284699	+	0.143548i
$322$		0			0
$323$	−21.9746			−1.22270
$324$		0			0
$325$	−6.32630	−	25.8097i	−0.350920	−	1.43167i
$326$		0			0
$327$	−16.5421	+	8.34072i	−0.914783	+	0.461243i
$328$		0			0
$329$	20.2632			1.11715
$330$		0			0
$331$	35.1440			1.93169			0.965843	−	0.259126i	$-0.0834345\pi$
$331$	35.1440			1.93169			0.965843	+	0.259126i	$0.0834345\pi$
$332$		0			0
$333$	6.82347	−	9.22659i	0.373924	−	0.505614i
$334$		0			0
$335$	1.79354	+	14.8510i	0.0979914	+	0.811394i
$336$		0			0
$337$	−1.06020			−0.0577527			−0.0288763	−	0.999583i	$-0.509193\pi$
$337$	−1.06020			−0.0577527			−0.0288763	+	0.999583i	$0.509193\pi$
$338$		0			0
$339$	−28.0433	+	14.1397i	−1.52310	+	0.767964i
$340$		0			0
$341$	−19.9864			−1.08233
$342$		0			0
$343$	19.3788			1.04636
$344$		0			0
$345$	−18.5497	+	0.953144i	−0.998682	+	0.0513155i
$346$		0			0
$347$	27.8541			1.49529			0.747643	−	0.664100i	$-0.231185\pi$
$347$	27.8541			1.49529			0.747643	+	0.664100i	$0.231185\pi$
$348$		0			0
$349$	5.55010			0.297090			0.148545	−	0.988906i	$-0.452541\pi$
$349$	5.55010			0.297090			0.148545	+	0.988906i	$0.452541\pi$
$350$		0			0
$351$	−4.66581	+	27.2192i	−0.249043	+	1.45285i
$352$		0			0
$353$	−23.4788			−1.24965			−0.624825	−	0.780765i	$-0.714830\pi$
$353$	−23.4788			−1.24965			−0.624825	+	0.780765i	$0.714830\pi$
$354$		0			0
$355$	−31.9861	+	3.86294i	−1.69765	+	0.205023i
$356$		0			0
$357$	10.4458	−	5.26688i	0.552851	−	0.278753i
$358$		0			0
$359$	12.0484			0.635892			0.317946	−	0.948109i	$-0.397007\pi$
$359$	12.0484			0.635892			0.317946	+	0.948109i	$0.397007\pi$
$360$		0			0
$361$	−47.1157			−2.47977
$362$		0			0
$363$	0.699595	+	1.38751i	0.0367192	+	0.0728252i
$364$		0			0
$365$	6.23670	−	0.753200i	0.326444	−	0.0394243i
$366$		0			0
$367$	2.87742			0.150200			0.0750999	−	0.997176i	$-0.476072\pi$
$367$	2.87742			0.150200			0.0750999	+	0.997176i	$0.476072\pi$
$368$		0			0
$369$	−7.64736	−	5.65557i	−0.398106	−	0.294417i
$370$		0			0
$371$			9.47833i			0.492091i
$372$		0			0
$373$	5.78585			0.299580			0.149790	−	0.988718i	$-0.452140\pi$
$373$	5.78585			0.299580			0.149790	+	0.988718i	$0.452140\pi$
$374$		0			0
$375$	−13.1036	+	14.2582i	−0.676666	+	0.736290i
$376$		0			0
$377$	15.0150			0.773312
$378$		0			0
$379$			9.59598i			0.492912i	0.969154	+	0.246456i	$0.0792661\pi$
$379$			9.59598i			0.492912i	−0.969154	+	0.246456i	$0.920734\pi$
$380$		0			0
$381$	−28.3940	+	14.3165i	−1.45467	+	0.733458i
$382$		0			0
$383$		−	13.8929i		−	0.709892i	−0.934887	−	0.354946i	$-0.884499\pi$
$383$		−	13.8929i		−	0.709892i	0.934887	−	0.354946i	$-0.115501\pi$
$384$		0			0
$385$	−2.12971	−	17.6346i	−0.108540	−	0.898740i
$386$		0			0
$387$	12.3739	−	16.7317i	0.628999	−	0.850522i
$388$		0			0
$389$	−13.2864			−0.673646			−0.336823	−	0.941568i	$-0.609352\pi$
$389$	−13.2864			−0.673646			−0.336823	+	0.941568i	$0.609352\pi$
$390$		0			0
$391$	−6.61752	+	11.1441i	−0.334662	+	0.563581i
$392$		0			0
$393$	−0.553727	+	0.279195i	−0.0279318	+	0.0140835i
$394$		0			0
$395$	−13.4981	+	1.63015i	−0.679162	+	0.0820218i
$396$		0			0
$397$		−	0.894529i		−	0.0448951i	−0.999748	−	0.0224476i	$-0.992854\pi$
$397$		−	0.894529i		−	0.0448951i	0.999748	−	0.0224476i	$-0.00714588\pi$
$398$		0			0
$399$	31.4287	−	15.8467i	1.57340	−	0.793326i
$400$		0			0
$401$	−0.536200			−0.0267765			−0.0133883	−	0.999910i	$-0.504262\pi$
$401$	−0.536200			−0.0267765			−0.0133883	+	0.999910i	$0.504262\pi$
$402$		0			0
$403$		−	33.4192i		−	1.66473i
$404$		0			0
$405$	18.3966	−	8.15879i	0.914133	−	0.405413i
$406$		0			0
$407$	−12.1584			−0.602670
$408$		0			0
$409$	13.0581			0.645680			0.322840	−	0.946454i	$-0.395362\pi$
$409$	13.0581			0.645680			0.322840	+	0.946454i	$0.395362\pi$
$410$		0			0
$411$	8.55244	−	4.31223i	0.421861	−	0.212706i
$412$		0			0
$413$			28.0416i			1.37984i
$414$		0			0
$415$	−18.4104	+	2.22341i	−0.903733	+	0.109143i
$416$		0			0
$417$	2.09498	+	4.15498i	0.102592	+	0.203470i
$418$		0			0
$419$	−32.4686			−1.58620			−0.793098	−	0.609094i	$-0.791533\pi$
$419$	−32.4686			−1.58620			−0.793098	+	0.609094i	$0.791533\pi$
$420$		0			0
$421$			35.5546i			1.73283i	0.499327	+	0.866413i	$0.333581\pi$
$421$			35.5546i			1.73283i	−0.499327	+	0.866413i	$0.666419\pi$
$422$		0			0
$423$	14.4630	−	19.5566i	0.703215	−	0.950875i
$424$		0			0
$425$	3.21688	+	13.1241i	0.156042	+	0.636611i
$426$		0			0
$427$		−	1.10798i		−	0.0536191i
$428$		0			0
$429$	26.1262	−	13.1731i	1.26139	−	0.636004i
$430$		0			0
$431$	16.7467			0.806660			0.403330	−	0.915055i	$-0.367853\pi$
$431$	16.7467			0.806660			0.403330	+	0.915055i	$0.367853\pi$
$432$		0			0
$433$	−4.98017			−0.239332			−0.119666	−	0.992814i	$-0.538182\pi$
$433$	−4.98017			−0.239332			−0.119666	+	0.992814i	$0.538182\pi$
$434$		0			0
$435$	−6.06208	−	9.10901i	−0.290655	−	0.436744i
$436$		0			0
$437$	−19.9104	+	33.5296i	−0.952443	+	1.60394i
$438$		0			0
$439$	5.39758			0.257612			0.128806	−	0.991670i	$-0.458886\pi$
$439$	5.39758			0.257612			0.128806	+	0.991670i	$0.458886\pi$
$440$		0			0
$441$	1.34496	−	1.81864i	0.0640459	−	0.0866018i
$442$		0			0
$443$	1.46631			0.0696664			0.0348332	−	0.999393i	$-0.488910\pi$
$443$	1.46631			0.0696664			0.0348332	+	0.999393i	$0.488910\pi$
$444$		0			0
$445$	2.62950	+	21.7729i	0.124650	+	1.03214i
$446$		0			0
$447$	11.6185	+	23.0431i	0.549538	+	1.08990i
$448$		0			0
$449$			20.7265i			0.978142i	0.872244	+	0.489071i	$0.162664\pi$
$449$			20.7265i			0.978142i	−0.872244	+	0.489071i	$0.837336\pi$
$450$		0			0
$451$			10.0774i			0.474525i
$452$		0			0
$453$	11.5633	+	22.9335i	0.543292	+	1.07751i
$454$		0			0
$455$	29.4866	−	3.56107i	1.38235	−	0.166946i
$456$		0			0
$457$	−9.31011			−0.435509			−0.217754	−	0.976004i	$-0.569873\pi$
$457$	−9.31011			−0.435509			−0.217754	+	0.976004i	$0.569873\pi$
$458$		0			0
$459$	2.37254	−	13.8408i	0.110740	−	0.646033i
$460$		0			0
$461$			2.79612i			0.130228i	0.997878	+	0.0651142i	$0.0207412\pi$
$461$			2.79612i			0.130228i	−0.997878	+	0.0651142i	$0.979259\pi$
$462$		0			0
$463$			22.8431i			1.06161i	0.847495	+	0.530803i	$0.178110\pi$
$463$			22.8431i			1.06161i	−0.847495	+	0.530803i	$0.821890\pi$
$464$		0			0
$465$	−20.2741	+	13.4925i	−0.940189	+	0.625700i
$466$		0			0
$467$		−	20.0898i		−	0.929647i	−0.885403	−	0.464824i	$-0.846118\pi$
$467$		−	20.0898i		−	0.929647i	0.885403	−	0.464824i	$-0.153882\pi$
$468$		0			0
$469$	−16.7192			−0.772021
$470$		0			0
$471$	9.49639	+	18.8342i	0.437570	+	0.867834i
$472$		0			0
$473$	−22.0484			−1.01379
$474$		0			0
$475$	9.67875	+	39.4869i	0.444092	+	1.81178i
$476$		0			0
$477$	9.14780	+	6.76521i	0.418849	+	0.309758i
$478$		0			0
$479$	−41.0548			−1.87584			−0.937921	−	0.346848i	$-0.887252\pi$
$479$	−41.0548			−1.87584			−0.937921	+	0.346848i	$0.887252\pi$
$480$		0			0
$481$		−	20.3300i		−	0.926968i
$482$		0			0
$483$	1.42817	−	20.7108i	0.0649840	−	0.942372i
$484$		0			0
$485$	−4.45461	−	36.8853i	−0.202273	−	1.67488i
$486$		0			0
$487$		−	12.9628i		−	0.587401i	−0.955898	−	0.293700i	$-0.905113\pi$
$487$		−	12.9628i		−	0.587401i	0.955898	−	0.293700i	$-0.0948868\pi$
$488$		0			0
$489$	−10.9707	+	5.53152i	−0.496111	+	0.250144i
$490$		0			0
$491$			26.0378i			1.17507i	0.809199	+	0.587535i	$0.199902\pi$
$491$			26.0378i			1.17507i	−0.809199	+	0.587535i	$0.800098\pi$
$492$		0			0
$493$	−7.63503			−0.343865
$494$		0			0
$495$	−18.5397	−	10.5313i	−0.833297	−	0.473347i
$496$		0			0
$497$		−	36.0100i		−	1.61527i
$498$		0			0
$499$	−10.0017			−0.447737			−0.223868	−	0.974619i	$-0.571869\pi$
$499$	−10.0017			−0.447737			−0.223868	+	0.974619i	$0.571869\pi$
$500$		0			0
$501$	1.83609	+	3.64152i	0.0820304	+	0.162691i
$502$		0			0
$503$			25.1067i			1.11945i	0.828678	+	0.559726i	$0.189093\pi$
$503$			25.1067i			1.11945i	−0.828678	+	0.559726i	$0.810907\pi$
$504$		0			0
$505$	−25.4391	+	3.07226i	−1.13202	+	0.136714i
$506$		0			0
$507$	11.8893	+	23.5800i	0.528021	+	1.04722i
$508$		0			0
$509$			8.88230i			0.393701i	0.980433	+	0.196851i	$0.0630714\pi$
$509$			8.88230i			0.393701i	−0.980433	+	0.196851i	$0.936929\pi$
$510$		0			0
$511$			7.02127i			0.310603i
$512$		0			0
$513$	7.13834	−	41.6433i	0.315165	−	1.83860i
$514$		0			0
$515$	−0.530712	−	4.39444i	−0.0233860	−	0.193642i
$516$		0			0
$517$	−25.7709			−1.13340
$518$		0			0
$519$	7.47617	+	14.8275i	0.328167	+	0.650855i
$520$		0			0
$521$	35.6427			1.56154			0.780768	−	0.624821i	$-0.214828\pi$
$521$	35.6427			1.56154			0.780768	+	0.624821i	$0.214828\pi$
$522$		0			0
$523$	−24.4731			−1.07013			−0.535066	−	0.844810i	$-0.679713\pi$
$523$	−24.4731			−1.07013			−0.535066	+	0.844810i	$0.679713\pi$
$524$		0			0
$525$	−14.0651	−	16.4506i	−0.613853	−	0.717965i
$526$		0			0
$527$			16.9934i			0.740246i
$528$		0			0
$529$	11.0082	+	20.1946i	0.478617	+	0.878024i
$530$		0			0
$531$	27.0637	+	20.0148i	1.17446	+	0.868569i
$532$		0			0
$533$	−16.8503			−0.729868
$534$		0			0
$535$	7.32162	−	0.884225i	0.316541	−	0.0382284i
$536$		0			0
$537$	6.70726	−	3.38187i	0.289440	−	0.145938i
$538$		0			0
$539$	−2.39653			−0.103226
$540$		0			0
$541$	11.6885			0.502526			0.251263	−	0.967919i	$-0.419154\pi$
$541$	11.6885			0.502526			0.251263	+	0.967919i	$0.419154\pi$
$542$		0			0
$543$	18.6644	−	9.41080i	0.800968	−	0.403856i
$544$		0			0
$545$	23.7443	−	2.86758i	1.01710	−	0.122834i
$546$		0			0
$547$			25.1601i			1.07577i	0.843019	+	0.537883i	$0.180776\pi$
$547$			25.1601i			1.07577i	−0.843019	+	0.537883i	$0.819224\pi$
$548$		0			0
$549$	−1.06935	−	0.790830i	−0.0456386	−	0.0337518i
$550$		0			0
$551$	−22.9718			−0.978632
$552$		0			0
$553$		−	15.1961i		−	0.646205i
$554$		0			0
$555$	−12.3334	+	8.20792i	−0.523524	+	0.348407i
$556$		0			0
$557$		−	35.0941i		−	1.48699i	−0.668744	−	0.743493i	$-0.733168\pi$
$557$		−	35.0941i		−	1.48699i	0.668744	−	0.743493i	$-0.266832\pi$
$558$		0			0
$559$		−	36.8669i		−	1.55931i
$560$		0			0
$561$	−13.2850	+	6.69844i	−0.560894	+	0.282808i
$562$		0			0
$563$		−	31.2531i		−	1.31716i	−0.752510	−	0.658581i	$-0.771157\pi$
$563$		−	31.2531i		−	1.31716i	0.752510	−	0.658581i	$-0.228843\pi$
$564$		0			0
$565$	40.2529	−	4.86131i	1.69345	−	0.204517i
$566$		0			0
$567$	6.58784	+	21.5065i	0.276663	+	0.903187i
$568$		0			0
$569$	−27.3858			−1.14807			−0.574035	−	0.818830i	$-0.694623\pi$
$569$	−27.3858			−1.14807			−0.574035	+	0.818830i	$0.694623\pi$
$570$		0			0
$571$		−	30.5119i		−	1.27688i	−0.769670	−	0.638442i	$-0.779579\pi$
$571$		−	30.5119i		−	1.27688i	0.769670	−	0.638442i	$-0.220421\pi$
$572$		0			0
$573$	−12.8131	−	25.4122i	−0.535274	−	1.06161i
$574$		0			0
$575$	22.9399	+	6.98282i	0.956661	+	0.291204i
$576$		0			0
$577$			11.6438i			0.484739i	0.970184	+	0.242370i	$0.0779247\pi$
$577$			11.6438i			0.484739i	−0.970184	+	0.242370i	$0.922075\pi$
$578$		0			0
$579$	11.1812	−	5.63767i	0.464674	−	0.234293i
$580$		0			0
$581$		−	20.7265i		−	0.859878i
$582$		0			0
$583$		−	12.0546i		−	0.499250i
$584$		0			0
$585$	17.6093	−	31.0001i	0.728056	−	1.28170i
$586$		0			0
$587$	16.2461			0.670547			0.335274	−	0.942121i	$-0.391171\pi$
$587$	16.2461			0.670547			0.335274	+	0.942121i	$0.391171\pi$
$588$		0			0
$589$			51.1288i			2.10673i
$590$		0			0
$591$	−16.6798	−	33.0810i	−0.686115	−	1.36077i
$592$		0			0
$593$	−18.6545			−0.766048			−0.383024	−	0.923738i	$-0.625117\pi$
$593$	−18.6545			−0.766048			−0.383024	+	0.923738i	$0.625117\pi$
$594$		0			0
$595$	−14.9937	+	1.81078i	−0.614684	+	0.0742348i
$596$		0			0
$597$	−21.0263	+	10.6017i	−0.860548	+	0.433897i
$598$		0			0
$599$		−	24.4728i		−	0.999933i	−0.866045	−	0.499967i	$-0.833346\pi$
$599$		−	24.4728i		−	0.999933i	0.866045	−	0.499967i	$-0.166654\pi$
$600$		0			0
$601$	5.74294			0.234259			0.117130	−	0.993117i	$-0.462631\pi$
$601$	5.74294			0.234259			0.117130	+	0.993117i	$0.462631\pi$
$602$		0			0
$603$	−11.9334	+	16.1362i	−0.485966	+	0.657115i
$604$		0			0
$605$	−0.240524	−	1.99161i	−0.00977871	−	0.0809703i
$606$		0			0
$607$			22.1013i			0.897064i	0.893767	+	0.448532i	$0.148053\pi$
$607$			22.1013i			0.897064i	−0.893767	+	0.448532i	$0.851947\pi$
$608$		0			0
$609$	10.9199	−	5.50591i	0.442495	−	0.223111i
$610$		0			0
$611$		−	43.0913i		−	1.74329i
$612$		0			0
$613$	23.2545			0.939241			0.469620	−	0.882868i	$-0.344391\pi$
$613$	23.2545			0.939241			0.469620	+	0.882868i	$0.344391\pi$
$614$		0			0
$615$	6.80305	+	10.2224i	0.274326	+	0.412207i
$616$		0			0
$617$		−	12.6444i		−	0.509045i	−0.967067	−	0.254522i	$-0.918082\pi$
$617$		−	12.6444i		−	0.509045i	0.967067	−	0.254522i	$-0.0819183\pi$
$618$		0			0
$619$		−	25.8745i		−	1.03998i	−0.854172	−	0.519991i	$-0.825935\pi$
$619$		−	25.8745i		−	1.03998i	0.854172	−	0.519991i	$-0.174065\pi$
$620$		0			0
$621$	−18.9692	−	16.1608i	−0.761206	−	0.648510i
$622$		0			0
$623$	−24.5120			−0.982051
$624$		0			0
$625$	22.1662	−	11.5611i	0.886649	−	0.462443i
$626$		0			0
$627$	−39.9712	+	20.1539i	−1.59629	+	0.804868i
$628$		0			0
$629$			10.3377i			0.412190i
$630$		0			0
$631$		−	34.6299i		−	1.37859i	−0.724478	−	0.689297i	$-0.757919\pi$
$631$		−	34.6299i		−	1.37859i	0.724478	−	0.689297i	$-0.242081\pi$
$632$		0			0
$633$	19.2022	+	38.0838i	0.763220	+	1.51369i
$634$		0			0
$635$	40.7563	−	4.92210i	1.61736	−	0.195327i
$636$		0			0
$637$		−	4.00721i		−	0.158772i
$638$		0			0
$639$	−34.7542	−	25.7023i	−1.37486	−	1.01677i
$640$		0			0
$641$	45.0695			1.78014			0.890069	−	0.455826i	$-0.150656\pi$
$641$	45.0695			1.78014			0.890069	+	0.455826i	$0.150656\pi$
$642$		0			0
$643$	−44.0417			−1.73684			−0.868418	−	0.495833i	$-0.834863\pi$
$643$	−44.0417			−1.73684			−0.868418	+	0.495833i	$0.834863\pi$
$644$		0			0
$645$	−22.3657	+	14.8845i	−0.880649	+	0.586075i
$646$		0			0
$647$	−6.98438			−0.274584			−0.137292	−	0.990531i	$-0.543840\pi$
$647$	−6.98438			−0.274584			−0.137292	+	0.990531i	$0.543840\pi$
$648$		0			0
$649$		−	35.6634i		−	1.39991i
$650$		0			0
$651$	−12.2546	−	24.3046i	−0.480296	−	0.952571i
$652$		0			0
$653$	−27.2824			−1.06764			−0.533822	−	0.845597i	$-0.679245\pi$
$653$	−27.2824			−1.06764			−0.533822	+	0.845597i	$0.679245\pi$
$654$		0			0
$655$	0.794812	−	0.0959887i	0.0310559	−	0.00375059i
$656$		0			0
$657$	6.77642	+	5.01147i	0.264373	+	0.195516i
$658$		0			0
$659$	27.6987			1.07899			0.539495	−	0.841989i	$-0.318615\pi$
$659$	27.6987			1.07899			0.539495	+	0.841989i	$0.318615\pi$
$660$		0			0
$661$			25.4773i			0.990951i	0.868622	+	0.495475i	$0.165006\pi$
$661$			25.4773i			0.990951i	−0.868622	+	0.495475i	$0.834994\pi$
$662$		0			0
$663$	−11.2004	−	22.2138i	−0.434988	−	0.862712i
$664$		0			0
$665$	−45.1123	+	5.44817i	−1.74938	+	0.211271i
$666$		0			0
$667$	−6.91784	+	11.6498i	−0.267860	+	0.451083i
$668$		0			0
$669$	−31.4128	+	15.8386i	−1.21449	+	0.612357i
$670$		0			0
$671$			1.40914i			0.0543993i
$672$		0			0
$673$		−	11.9655i		−	0.461238i	−0.973044	−	0.230619i	$-0.925925\pi$
$673$		−	11.9655i		−	0.461238i	0.973044	−	0.230619i	$-0.0740751\pi$
$674$		0			0
$675$	−25.9160	+	1.83292i	−0.997508	+	0.0705492i
$676$		0			0
$677$		−	0.225684i		−	0.00867374i	−0.999991	−	0.00433687i	$-0.998620\pi$
$677$		−	0.225684i		−	0.00867374i	0.999991	−	0.00433687i	$-0.00138047\pi$
$678$		0			0
$679$	41.5255			1.59360
$680$		0			0
$681$	19.4314	−	9.79752i	0.744614	−	0.375442i
$682$		0			0
$683$	−11.6950			−0.447495			−0.223747	−	0.974647i	$-0.571829\pi$
$683$	−11.6950			−0.447495			−0.223747	+	0.974647i	$0.571829\pi$
$684$		0			0
$685$	−12.2760	+	1.48257i	−0.469043	+	0.0566459i
$686$		0			0
$687$	−5.43714	+	2.74146i	−0.207440	+	0.104593i
$688$		0			0
$689$	20.1564			0.767897
$690$		0			0
$691$	−2.71396			−0.103244			−0.0516220	−	0.998667i	$-0.516439\pi$
$691$	−2.71396			−0.103244			−0.0516220	+	0.998667i	$0.516439\pi$
$692$		0			0
$693$	14.1702	−	19.1606i	0.538280	−	0.727853i
$694$		0			0
$695$	−0.720266	−	5.96400i	−0.0273213	−	0.226227i
$696$		0			0
$697$	8.56827			0.324546
$698$		0			0
$699$	−11.6409	−	23.0875i	−0.440301	−	0.873249i
$700$		0			0
$701$	−1.06128			−0.0400838			−0.0200419	−	0.999799i	$-0.506380\pi$
$701$	−1.06128			−0.0400838			−0.0200419	+	0.999799i	$0.506380\pi$
$702$		0			0
$703$			31.1033i			1.17308i
$704$		0			0
$705$	−26.1418	+	17.3975i	−0.984557	+	0.655226i
$706$		0			0
$707$		−	28.6393i		−	1.07709i
$708$		0			0
$709$			21.4258i			0.804661i	0.915494	+	0.402331i	$0.131800\pi$
$709$			21.4258i			0.804661i	−0.915494	+	0.402331i	$0.868200\pi$
$710$		0			0
$711$	−14.6662	−	10.8463i	−0.550026	−	0.406769i
$712$		0			0
$713$	25.9293	+	15.3972i	0.971059	+	0.576629i
$714$		0			0
$715$	−37.5012	+	4.52899i	−1.40247	+	0.169375i
$716$		0			0
$717$	−3.12723	+	1.57678i	−0.116788	+	0.0588859i
$718$		0			0
$719$		−	24.8860i		−	0.928092i	−0.885811	−	0.464046i	$-0.846397\pi$
$719$		−	24.8860i		−	0.928092i	0.885811	−	0.464046i	$-0.153603\pi$
$720$		0			0
$721$	4.94725			0.184245
$722$		0			0
$723$	−25.9798	+	13.0993i	−0.966199	+	0.487167i
$724$		0			0
$725$	3.36287	+	13.7197i	0.124894	+	0.509536i
$726$		0			0
$727$	11.2531			0.417354			0.208677	−	0.977985i	$-0.433084\pi$
$727$	11.2531			0.417354			0.208677	+	0.977985i	$0.433084\pi$
$728$		0			0
$729$	25.4586	+	8.99225i	0.942911	+	0.333046i
$730$		0			0
$731$			18.7466i			0.693368i
$732$		0			0
$733$	−41.2939			−1.52522			−0.762612	−	0.646856i	$-0.776083\pi$
$733$	−41.2939			−1.52522			−0.762612	+	0.646856i	$0.776083\pi$
$734$		0			0
$735$	−2.43102	+	1.61785i	−0.0896695	+	0.0596754i
$736$		0			0
$737$	21.2635			0.783253
$738$		0			0
$739$	−48.5017			−1.78417			−0.892083	−	0.451872i	$-0.850756\pi$
$739$	−48.5017			−1.78417			−0.892083	+	0.451872i	$0.850756\pi$
$740$		0			0
$741$	−33.6991	−	66.8355i	−1.23797	−	2.45526i
$742$		0			0
$743$			2.34205i			0.0859214i	0.999077	+	0.0429607i	$0.0136790\pi$
$743$			2.34205i			0.0859214i	−0.999077	+	0.0429607i	$0.986321\pi$
$744$		0			0
$745$	−3.99452	−	33.0757i	−0.146348	−	1.21180i
$746$		0			0
$747$	−20.0037	−	14.7936i	−0.731896	−	0.541270i
$748$		0			0
$749$			8.24267i			0.301181i
$750$		0			0
$751$			15.1689i			0.553521i	0.960939	+	0.276761i	$0.0892609\pi$
$751$			15.1689i			0.553521i	−0.960939	+	0.276761i	$0.910739\pi$
$752$		0			0
$753$	15.7361	+	31.2093i	0.573454	+	1.13733i
$754$		0			0
$755$	−3.97553	−	32.9184i	−0.144684	−	1.19802i
$756$		0			0
$757$	30.9066			1.12332			0.561660	−	0.827368i	$-0.310163\pi$
$757$	30.9066			1.12332			0.561660	+	0.827368i	$0.310163\pi$
$758$		0			0
$759$	−1.81635	+	26.3400i	−0.0659295	+	0.956083i
$760$		0			0
$761$			44.8828i			1.62700i	0.581566	+	0.813499i	$0.302440\pi$
$761$			44.8828i			1.62700i	−0.581566	+	0.813499i	$0.697560\pi$
$762$		0			0
$763$			26.7314i			0.967740i
$764$		0			0
$765$	−8.95423	+	15.7633i	−0.323741	+	0.569925i
$766$		0			0
$767$	59.6325			2.15320
$768$		0			0
$769$			43.9985i			1.58663i	0.608814	+	0.793313i	$0.291646\pi$
$769$			43.9985i			1.58663i	−0.608814	+	0.793313i	$0.708354\pi$
$770$		0			0
$771$	−19.9082	−	39.4840i	−0.716977	−	1.42198i
$772$		0			0
$773$			15.2898i			0.549936i	0.961453	+	0.274968i	$0.0886673\pi$
$773$			15.2898i			0.549936i	−0.961453	+	0.274968i	$0.911333\pi$
$774$		0			0
$775$	30.5361	−	7.48481i	1.09689	−	0.268862i
$776$		0			0
$777$	−7.45487	−	14.7853i	−0.267442	−	0.530418i
$778$		0			0
$779$	25.7797			0.923653
$780$		0			0
$781$			45.7977i			1.63877i
$782$		0			0
$783$	2.48021	−	14.4689i	0.0886354	−	0.517077i
$784$		0			0
$785$	−3.26491	−	27.0343i	−0.116530	−	0.964896i
$786$		0			0
$787$	−43.2054			−1.54011			−0.770054	−	0.637979i	$-0.779771\pi$
$787$	−43.2054			−1.54011			−0.770054	+	0.637979i	$0.779771\pi$
$788$		0			0
$789$	37.1101	−	18.7113i	1.32115	−	0.666139i
$790$		0			0
$791$			45.3167i			1.61128i
$792$		0			0
$793$	−2.35621			−0.0836716
$794$		0			0
$795$	−8.13784	−	12.2281i	−0.288619	−	0.433686i
$796$		0			0
$797$		−	38.4093i		−	1.36053i	−0.732968	−	0.680263i	$-0.761866\pi$
$797$		−	38.4093i		−	1.36053i	0.732968	−	0.680263i	$-0.238134\pi$
$798$		0			0
$799$			21.9116i			0.775178i
$800$		0			0
$801$	−17.4955	+	23.6572i	−0.618174	+	0.835885i
$802$		0			0
$803$		−	8.92968i		−	0.315122i
$804$		0			0
$805$	−10.8224	+	24.5187i	−0.381438	+	0.864172i
$806$		0			0
$807$	24.6496	−	12.4286i	0.867708	−	0.437507i
$808$		0			0
$809$			4.40234i			0.154778i	0.997001	+	0.0773891i	$0.0246584\pi$
$809$			4.40234i			0.154778i	−0.997001	+	0.0773891i	$0.975342\pi$
$810$		0			0
$811$	−7.52615			−0.264279			−0.132139	−	0.991231i	$-0.542185\pi$
$811$	−7.52615			−0.264279			−0.132139	+	0.991231i	$0.542185\pi$
$812$		0			0
$813$	10.9026	+	21.6232i	0.382372	+	0.758358i
$814$		0			0
$815$	15.7471	−	1.90177i	0.551598	−	0.0666160i
$816$		0			0
$817$			56.4036i			1.97331i
$818$		0			0
$819$	32.0384	+	23.6938i	1.11951	+	0.827929i
$820$		0			0
$821$		−	41.7227i		−	1.45613i	−0.685508	−	0.728065i	$-0.740420\pi$
$821$		−	41.7227i		−	1.45613i	0.685508	−	0.728065i	$-0.259580\pi$
$822$		0			0
$823$		−	20.8585i		−	0.727083i	−0.931578	−	0.363541i	$-0.881568\pi$
$823$		−	20.8585i		−	0.727083i	0.931578	−	0.363541i	$-0.118432\pi$
$824$		0			0
$825$	17.8881	+	20.9220i	0.622784	+	0.728410i
$826$		0			0
$827$			15.2222i			0.529329i	0.964341	+	0.264665i	$0.0852613\pi$
$827$			15.2222i			0.529329i	−0.964341	+	0.264665i	$0.914739\pi$
$828$		0			0
$829$	2.34887			0.0815798			0.0407899	−	0.999168i	$-0.487013\pi$
$829$	2.34887			0.0815798			0.0407899	+	0.999168i	$0.487013\pi$
$830$		0			0
$831$	16.3821	−	8.26001i	0.568288	−	0.286536i
$832$		0			0
$833$			2.03764i			0.0706001i
$834$		0			0
$835$	−0.631257	−	5.22698i	−0.0218455	−	0.180887i
$836$		0			0
$837$	−32.2038	−	5.52025i	−1.11313	−	0.190808i
$838$		0			0
$839$	−30.6875			−1.05945			−0.529725	−	0.848169i	$-0.677705\pi$
$839$	−30.6875			−1.05945			−0.529725	+	0.848169i	$0.677705\pi$
$840$		0			0
$841$	21.0185			0.724775
$842$		0			0
$843$	−4.06975	−	8.07153i	−0.140170	−	0.277998i
$844$		0			0
$845$	−4.08759	−	33.8463i	−0.140617	−	1.16435i
$846$		0			0
$847$	2.24215			0.0770411
$848$		0			0
$849$	−22.5166	−	44.6572i	−0.772768	−	1.53263i
$850$		0			0
$851$	15.7736	+	9.36660i	0.540713	+	0.321083i
$852$		0			0
$853$			30.4110i			1.04125i	0.853784	+	0.520627i	$0.174302\pi$
$853$			30.4110i			1.04125i	−0.853784	+	0.520627i	$0.825698\pi$
$854$		0			0
$855$	−26.9409	+	47.4277i	−0.921361	+	1.62199i
$856$		0			0
$857$	−4.89538			−0.167223			−0.0836115	−	0.996498i	$-0.526645\pi$
$857$	−4.89538			−0.167223			−0.0836115	+	0.996498i	$0.526645\pi$
$858$		0			0
$859$	−29.0860			−0.992400			−0.496200	−	0.868208i	$-0.665272\pi$
$859$	−29.0860			−0.992400			−0.496200	+	0.868208i	$0.665272\pi$
$860$		0			0
$861$	−12.2546	+	6.17890i	−0.417636	+	0.210576i
$862$		0			0
$863$	0.680690			0.0231709			0.0115855	−	0.999933i	$-0.496312\pi$
$863$	0.680690			0.0231709			0.0115855	+	0.999933i	$0.496312\pi$
$864$		0			0
$865$	−2.57035	−	21.2832i	−0.0873944	−	0.723649i
$866$		0			0
$867$	−7.56128	−	14.9963i	−0.256795	−	0.509301i
$868$		0			0
$869$			19.3265i			0.655607i
$870$		0			0
$871$			35.5546i			1.20472i
$872$		0			0
$873$	29.6390	−	40.0774i	1.00313	−	1.35641i
$874$		0			0
$875$	9.85790	+	26.1453i	0.333258	+	0.883871i
$876$		0			0
$877$		−	31.8136i		−	1.07427i	−0.843496	−	0.537135i	$-0.819507\pi$
$877$		−	31.8136i		−	1.07427i	0.843496	−	0.537135i	$-0.180493\pi$
$878$		0			0
$879$	−1.79819	+	0.906663i	−0.0606513	+	0.0305810i
$880$		0			0
$881$	−20.6165			−0.694586			−0.347293	−	0.937757i	$-0.612899\pi$
$881$	−20.6165			−0.694586			−0.347293	+	0.937757i	$0.612899\pi$
$882$		0			0
$883$			32.9502i			1.10886i	0.832230	+	0.554431i	$0.187064\pi$
$883$			32.9502i			1.10886i	−0.832230	+	0.554431i	$0.812936\pi$
$884$		0			0
$885$	−24.0757	−	36.1767i	−0.809296	−	1.21607i
$886$		0			0
$887$	−8.94098			−0.300209			−0.150104	−	0.988670i	$-0.547961\pi$
$887$	−8.94098			−0.300209			−0.150104	+	0.988670i	$0.547961\pi$
$888$		0			0
$889$			45.8834i			1.53888i
$890$		0			0
$891$	−8.37844	−	27.3520i	−0.280688	−	0.916327i
$892$		0			0
$893$			65.9264i			2.20614i
$894$		0			0
$895$	−9.62750	+	1.16270i	−0.321812	+	0.0388649i
$896$		0			0
$897$	−44.0430	−	3.03711i	−1.47055	−	0.101406i
$898$		0			0
$899$			17.7647i			0.592484i
$900$		0			0
$901$	−10.2494			−0.341457
$902$		0			0
$903$	−13.5189	−	26.8120i	−0.449880	−	0.892246i
$904$		0			0
$905$	−26.7906	+	3.23548i	−0.890551	+	0.107551i
$906$		0			0
$907$	−30.9025			−1.02610			−0.513050	−	0.858359i	$-0.671485\pi$
$907$	−30.9025			−1.02610			−0.513050	+	0.858359i	$0.671485\pi$
$908$		0			0
$909$	−27.6406	−	20.4415i	−0.916780	−	0.678000i
$910$		0			0
$911$	56.7517			1.88027			0.940133	−	0.340807i	$-0.110700\pi$
$911$	56.7517			1.88027			0.940133	+	0.340807i	$0.110700\pi$
$912$		0			0
$913$			26.3600i			0.872389i
$914$		0			0
$915$	0.951286	+	1.42942i	0.0314485	+	0.0472552i
$916$		0			0
$917$			0.894798i			0.0295488i
$918$		0			0
$919$		−	25.8160i		−	0.851593i	−0.904819	−	0.425796i	$-0.859994\pi$
$919$		−	25.8160i		−	0.851593i	0.904819	−	0.425796i	$-0.140006\pi$
$920$		0			0
$921$	25.7162	−	12.9664i	0.847378	−	0.427256i
$922$		0			0
$923$	−76.5779			−2.52059
$924$		0			0
$925$	18.5761	−	4.55325i	0.610779	−	0.149710i
$926$		0			0
$927$	3.53113	−	4.77473i	0.115977	−	0.156823i
$928$		0			0
$929$			9.60757i			0.315214i	0.987502	+	0.157607i	$0.0503779\pi$
$929$			9.60757i			0.315214i	−0.987502	+	0.157607i	$0.949622\pi$
$930$		0			0
$931$			6.13073i			0.200927i
$932$		0			0
$933$	36.4469	−	18.3769i	1.19322	−	0.601632i
$934$		0			0
$935$	19.0691	−	2.30296i	0.623627	−	0.0753148i
$936$		0			0
$937$	−14.9430			−0.488165			−0.244083	−	0.969754i	$-0.578487\pi$
$937$	−14.9430			−0.488165			−0.244083	+	0.969754i	$0.578487\pi$
$938$		0			0
$939$	−15.1673	−	30.0812i	−0.494965	−	0.981664i
$940$		0			0
$941$	−34.0066			−1.10858			−0.554291	−	0.832323i	$-0.687011\pi$
$941$	−34.0066			−1.10858			−0.554291	+	0.832323i	$0.687011\pi$
$942$		0			0
$943$	7.76341	−	13.0738i	0.252812	−	0.425742i
$944$		0			0
$945$	1.43910	−	29.0024i	0.0468138	−	0.943450i
$946$		0			0
$947$	5.50235			0.178802			0.0894012	−	0.995996i	$-0.471505\pi$
$947$	5.50235			0.178802			0.0894012	+	0.995996i	$0.471505\pi$
$948$		0			0
$949$	14.9313			0.484689
$950$		0			0
$951$	14.6500	+	29.0554i	0.475060	+	0.942187i
$952$		0			0
$953$		−	19.3830i		−	0.627878i	−0.949443	−	0.313939i	$-0.898351\pi$
$953$		−	19.3830i		−	0.627878i	0.949443	−	0.313939i	$-0.101649\pi$
$954$		0			0
$955$	4.40520	+	36.4762i	0.142549	+	1.18034i
$956$		0			0
$957$	−13.8879	+	7.00244i	−0.448933	+	0.226357i
$958$		0			0
$959$		−	13.8204i		−	0.446283i
$960$		0			0
$961$	8.53914			0.275456
$962$		0			0
$963$	7.95523	+	5.88325i	0.256354	+	0.189585i
$964$		0			0
$965$	−16.0493	+	1.93826i	−0.516645	+	0.0623948i
$966$		0			0
$967$		−	32.6937i		−	1.05136i	−0.850682	−	0.525680i	$-0.823811\pi$
$967$		−	32.6937i		−	1.05136i	0.850682	−	0.525680i	$-0.176189\pi$
$968$		0			0
$969$	17.1358	+	33.9854i	0.550481	+	1.09177i
$970$		0			0
$971$	−59.7120			−1.91625			−0.958124	−	0.286353i	$-0.907557\pi$
$971$	−59.7120			−1.91625			−0.958124	+	0.286353i	$0.907557\pi$
$972$		0			0
$973$	6.71426			0.215249
$974$		0			0
$975$	−34.9835	+	29.9106i	−1.12037	+	0.957905i
$976$		0			0
$977$			41.3823i			1.32394i	0.749532	+	0.661968i	$0.230278\pi$
$977$			41.3823i			1.32394i	−0.749532	+	0.661968i	$0.769722\pi$
$978$		0			0
$979$	31.1744			0.996338
$980$		0			0
$981$	25.7992	+	19.0797i	0.823704	+	0.609166i
$982$		0			0
$983$		−	31.2013i		−	0.995166i	−0.867416	−	0.497583i	$-0.834221\pi$
$983$		−	31.2013i		−	0.995166i	0.867416	−	0.497583i	$-0.165779\pi$
$984$		0			0
$985$	5.73460	+	47.4840i	0.182719	+	1.51297i
$986$		0			0
$987$	−15.8013	−	31.3387i	−0.502961	−	0.997523i
$988$		0			0
$989$	28.6043	+	16.9856i	0.909564	+	0.540112i
$990$		0			0
$991$	33.0687			1.05046			0.525230	−	0.850960i	$-0.323979\pi$
$991$	33.0687			1.05046			0.525230	+	0.850960i	$0.323979\pi$
$992$		0			0
$993$	−27.4053	−	54.3530i	−0.869681	−	1.72484i
$994$		0			0
$995$	30.1808	−	3.64491i	0.956795	−	0.115551i
$996$		0			0
$997$		−	35.4909i		−	1.12401i	−0.827134	−	0.562005i	$-0.810030\pi$
$997$		−	35.4909i		−	1.12401i	0.827134	−	0.562005i	$-0.189970\pi$
$998$		0			0
$999$	−19.5906	−	3.35815i	−0.619819	−	0.106247i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.n.a.689.18	yes	48
3.2	odd	2	inner	1380.2.n.a.689.29	yes	48
5.4	even	2	inner	1380.2.n.a.689.32	yes	48
15.14	odd	2	inner	1380.2.n.a.689.19	yes	48
23.22	odd	2	inner	1380.2.n.a.689.17	✓	48
69.68	even	2	inner	1380.2.n.a.689.30	yes	48
115.114	odd	2	inner	1380.2.n.a.689.31	yes	48
345.344	even	2	inner	1380.2.n.a.689.20	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.n.a.689.17	✓	48	23.22	odd	2	inner
1380.2.n.a.689.18	yes	48	1.1	even	1	trivial
1380.2.n.a.689.19	yes	48	15.14	odd	2	inner
1380.2.n.a.689.20	yes	48	345.344	even	2	inner
1380.2.n.a.689.29	yes	48	3.2	odd	2	inner
1380.2.n.a.689.30	yes	48	69.68	even	2	inner
1380.2.n.a.689.31	yes	48	115.114	odd	2	inner
1380.2.n.a.689.32	yes	48	5.4	even	2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.779801 - 1.54658i) q^{3} +(0.268100 + 2.21994i) q^{5} -2.49920 q^{7} +(-1.78382 + 2.41205i) q^{9} +O(q^{10})\)	\(q+(-0.779801 - 1.54658i) q^{3} +(0.268100 + 2.21994i) q^{5} -2.49920 q^{7} +(-1.78382 + 2.41205i) q^{9} +3.17850 q^{11} +5.31475i q^{13} +(3.22425 - 2.14575i) q^{15} -2.70251i q^{17} -8.13115i q^{19} +(1.94888 + 3.86522i) q^{21} +(-4.12360 - 2.44865i) q^{23} +(-4.85624 + 1.19033i) q^{25} +(5.12145 + 0.877900i) q^{27} -2.82516i q^{29} -6.28802 q^{31} +(-2.47860 - 4.91580i) q^{33} +(-0.670036 - 5.54808i) q^{35} -3.82520 q^{37} +(8.21968 - 4.14445i) q^{39} +3.17048i q^{41} -6.93673 q^{43} +(-5.83284 - 3.31330i) q^{45} -8.10788 q^{47} -0.753980 q^{49} +(-4.17965 + 2.10742i) q^{51} -3.79254i q^{53} +(0.852155 + 7.05607i) q^{55} +(-12.5755 + 6.34069i) q^{57} -11.2202i q^{59} +0.443335i q^{61} +(4.45813 - 6.02821i) q^{63} +(-11.7984 + 1.42488i) q^{65} +6.68981 q^{67} +(-0.571450 + 8.28694i) q^{69} +14.4086i q^{71} -2.80940i q^{73} +(5.62785 + 6.58235i) q^{75} -7.94372 q^{77} +6.08039i q^{79} +(-2.63597 - 8.60533i) q^{81} +8.29322i q^{83} +(5.99941 - 0.724543i) q^{85} +(-4.36934 + 2.20306i) q^{87} +9.80790 q^{89} -13.2826i q^{91} +(4.90340 + 9.72492i) q^{93} +(18.0507 - 2.17996i) q^{95} -16.6155 q^{97} +(-5.66987 + 7.66670i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 16 q^{31} + 24 q^{39} + 112 q^{49} + 8 q^{55} - 16 q^{69} + 20 q^{75} - 8 q^{81} + 32 q^{85}+O(q^{100}) \) 48 * q + 16 * q^31 + 24 * q^39 + 112 * q^49 + 8 * q^55 - 16 * q^69 + 20 * q^75 - 8 * q^81 + 32 * q^85

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