LMFDB - Embedded newform 1380.2.n.a.689.14

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			689.14
Character	$\chi$	$=$	1380.689
Dual form			1380.2.n.a.689.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.888021 - 1.48708i) q^{3} +(1.29815 - 1.82066i) q^{5} -0.470527 q^{7} +(-1.42284 + 2.64112i) q^{9} +O(q^{10})$	\(q+(-0.888021 - 1.48708i) q^{3} +(1.29815 - 1.82066i) q^{5} -0.470527 q^{7} +(-1.42284 + 2.64112i) q^{9} +2.71789 q^{11} -2.59395i q^{13} +(-3.86026 - 0.313680i) q^{15} -0.294363i q^{17} -0.759425i q^{19} +(0.417838 + 0.699714i) q^{21} +(3.33187 - 3.44944i) q^{23} +(-1.62959 - 4.72699i) q^{25} +(5.19108 - 0.229499i) q^{27} -6.71717i q^{29} +2.75201 q^{31} +(-2.41354 - 4.04173i) q^{33} +(-0.610817 + 0.856670i) q^{35} -10.9455 q^{37} +(-3.85742 + 2.30348i) q^{39} -1.64338i q^{41} -3.08303 q^{43} +(2.96152 + 6.01908i) q^{45} +9.53862 q^{47} -6.77860 q^{49} +(-0.437743 + 0.261401i) q^{51} +2.10802i q^{53} +(3.52824 - 4.94835i) q^{55} +(-1.12933 + 0.674385i) q^{57} +5.16407i q^{59} +8.99054i q^{61} +(0.669483 - 1.24272i) q^{63} +(-4.72269 - 3.36734i) q^{65} -9.92409 q^{67} +(-8.08838 - 1.89159i) q^{69} -2.98391i q^{71} -8.50311i q^{73} +(-5.58232 + 6.62101i) q^{75} -1.27884 q^{77} -14.8177i q^{79} +(-4.95108 - 7.51577i) q^{81} +10.6851i q^{83} +(-0.535935 - 0.382129i) q^{85} +(-9.98899 + 5.96499i) q^{87} -8.27339 q^{89} +1.22052i q^{91} +(-2.44385 - 4.09248i) q^{93} +(-1.38265 - 0.985850i) q^{95} +0.506765 q^{97} +(-3.86711 + 7.17828i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.888021	−	1.48708i	−0.512699	−	0.858568i
$3$	−0.888021	−	1.48708i	−0.512699	−	0.858568i
$4$		0			0
$5$	1.29815	−	1.82066i	0.580552	−	0.814223i
$5$	1.29815	−	1.82066i	0.580552	−	0.814223i
$6$		0			0
$7$	−0.470527			−0.177843			−0.0889213	−	0.996039i	$-0.528342\pi$
$7$	−0.470527			−0.177843			−0.0889213	+	0.996039i	$0.528342\pi$
$8$		0			0
$9$	−1.42284	+	2.64112i	−0.474279	+	0.880375i
$10$		0			0
$11$	2.71789			0.819475			0.409737	−	0.912204i	$-0.365620\pi$
$11$	2.71789			0.819475			0.409737	+	0.912204i	$0.365620\pi$
$12$		0			0
$13$		−	2.59395i		−	0.719432i	−0.933062	−	0.359716i	$-0.882874\pi$
$13$		−	2.59395i		−	0.719432i	0.933062	−	0.359716i	$-0.117126\pi$
$14$		0			0
$15$	−3.86026	−	0.313680i	−0.996715	−	0.0809919i
$16$		0			0
$17$		−	0.294363i		−	0.0713936i	−0.999363	−	0.0356968i	$-0.988635\pi$
$17$		−	0.294363i		−	0.0713936i	0.999363	−	0.0356968i	$-0.0113651\pi$
$18$		0			0
$19$		−	0.759425i		−	0.174224i	−0.996199	−	0.0871120i	$-0.972236\pi$
$19$		−	0.759425i		−	0.174224i	0.996199	−	0.0871120i	$-0.0277638\pi$
$20$		0			0
$21$	0.417838	+	0.699714i	0.0911798	+	0.152690i
$22$		0			0
$23$	3.33187	−	3.44944i	0.694743	−	0.719258i
$23$	3.33187	−	3.44944i	0.694743	−	0.719258i
$24$		0			0
$25$	−1.62959	−	4.72699i	−0.325919	−	0.945398i
$26$		0			0
$27$	5.19108	−	0.229499i	0.999024	−	0.0441670i
$28$		0			0
$29$		−	6.71717i		−	1.24735i	−0.781685	−	0.623674i	$-0.785639\pi$
$29$		−	6.71717i		−	1.24735i	0.781685	−	0.623674i	$-0.214361\pi$
$30$		0			0
$31$	2.75201			0.494276			0.247138	−	0.968980i	$-0.420510\pi$
$31$	2.75201			0.494276			0.247138	+	0.968980i	$0.420510\pi$
$32$		0			0
$33$	−2.41354	−	4.04173i	−0.420144	−	0.703575i
$34$		0			0
$35$	−0.610817	+	0.856670i	−0.103247	+	0.144804i
$36$		0			0
$37$	−10.9455			−1.79942			−0.899711	−	0.436487i	$-0.856223\pi$
$37$	−10.9455			−1.79942			−0.899711	+	0.436487i	$0.856223\pi$
$38$		0			0
$39$	−3.85742	+	2.30348i	−0.617681	+	0.368852i
$40$		0			0
$41$		−	1.64338i		−	0.256653i	−0.991732	−	0.128327i	$-0.959039\pi$
$41$		−	1.64338i		−	0.256653i	0.991732	−	0.128327i	$-0.0409606\pi$
$42$		0			0
$43$	−3.08303			−0.470157			−0.235079	−	0.971976i	$-0.575535\pi$
$43$	−3.08303			−0.470157			−0.235079	+	0.971976i	$0.575535\pi$
$44$		0			0
$45$	2.96152	+	6.01908i	0.441478	+	0.897272i
$46$		0			0
$47$	9.53862			1.39135			0.695675	−	0.718357i	$-0.255105\pi$
$47$	9.53862			1.39135			0.695675	+	0.718357i	$0.255105\pi$
$48$		0			0
$49$	−6.77860			−0.968372
$50$		0			0
$51$	−0.437743	+	0.261401i	−0.0612963	+	0.0366035i
$52$		0			0
$53$			2.10802i			0.289558i	0.989464	+	0.144779i	$0.0462472\pi$
$53$			2.10802i			0.289558i	−0.989464	+	0.144779i	$0.953753\pi$
$54$		0			0
$55$	3.52824	−	4.94835i	0.475748	−	0.667235i
$56$		0			0
$57$	−1.12933	+	0.674385i	−0.149583	+	0.0893245i
$58$		0			0
$59$			5.16407i			0.672304i	0.941808	+	0.336152i	$0.109126\pi$
$59$			5.16407i			0.672304i	−0.941808	+	0.336152i	$0.890874\pi$
$60$		0			0
$61$			8.99054i			1.15112i	0.817759	+	0.575560i	$0.195216\pi$
$61$			8.99054i			1.15112i	−0.817759	+	0.575560i	$0.804784\pi$
$62$		0			0
$63$	0.669483	−	1.24272i	0.0843470	−	0.156568i
$64$		0			0
$65$	−4.72269	−	3.36734i	−0.585778	−	0.417668i
$66$		0			0
$67$	−9.92409			−1.21242			−0.606210	−	0.795304i	$-0.707311\pi$
$67$	−9.92409			−1.21242			−0.606210	+	0.795304i	$0.707311\pi$
$68$		0			0
$69$	−8.08838	−	1.89159i	−0.973726	−	0.227721i
$70$		0			0
$71$		−	2.98391i		−	0.354124i	−0.984200	−	0.177062i	$-0.943341\pi$
$71$		−	2.98391i		−	0.354124i	0.984200	−	0.177062i	$-0.0566594\pi$
$72$		0			0
$73$		−	8.50311i		−	0.995214i	−0.867403	−	0.497607i	$-0.834212\pi$
$73$		−	8.50311i		−	0.995214i	0.867403	−	0.497607i	$-0.165788\pi$
$74$		0			0
$75$	−5.58232	+	6.62101i	−0.644590	+	0.764528i
$76$		0			0
$77$	−1.27884			−0.145738
$78$		0			0
$79$		−	14.8177i		−	1.66712i	−0.552431	−	0.833559i	$-0.686300\pi$
$79$		−	14.8177i		−	1.66712i	0.552431	−	0.833559i	$-0.313700\pi$
$80$		0			0
$81$	−4.95108	−	7.51577i	−0.550119	−	0.835086i
$82$		0			0
$83$			10.6851i			1.17284i	0.810008	+	0.586419i	$0.199463\pi$
$83$			10.6851i			1.17284i	−0.810008	+	0.586419i	$0.800537\pi$
$84$		0			0
$85$	−0.535935	−	0.382129i	−0.0581303	−	0.0414477i
$86$		0			0
$87$	−9.98899	+	5.96499i	−1.07093	+	0.639514i
$88$		0			0
$89$	−8.27339			−0.876977			−0.438489	−	0.898737i	$-0.644486\pi$
$89$	−8.27339			−0.876977			−0.438489	+	0.898737i	$0.644486\pi$
$90$		0			0
$91$			1.22052i			0.127946i
$92$		0			0
$93$	−2.44385	−	4.09248i	−0.253415	−	0.424370i
$94$		0			0
$95$	−1.38265	−	0.985850i	−0.141857	−	0.101146i
$96$		0			0
$97$	0.506765			0.0514542			0.0257271	−	0.999669i	$-0.491810\pi$
$97$	0.506765			0.0514542			0.0257271	+	0.999669i	$0.491810\pi$
$98$		0			0
$99$	−3.86711	+	7.17828i	−0.388659	+	0.721445i
$100$		0			0
$101$		−	6.25035i		−	0.621933i	−0.950421	−	0.310966i	$-0.899347\pi$
$101$		−	6.25035i		−	0.621933i	0.950421	−	0.310966i	$-0.100653\pi$
$102$		0			0
$103$	1.74406			0.171847			0.0859237	−	0.996302i	$-0.472616\pi$
$103$	1.74406			0.171847			0.0859237	+	0.996302i	$0.472616\pi$
$104$		0			0
$105$	1.81636	+	0.147595i	0.177258	+	0.0144038i
$106$		0			0
$107$			6.93391i			0.670326i	0.942160	+	0.335163i	$0.108791\pi$
$107$			6.93391i			0.670326i	−0.942160	+	0.335163i	$0.891209\pi$
$108$		0			0
$109$			14.9349i			1.43050i	0.698868	+	0.715250i	$0.253687\pi$
$109$			14.9349i			1.43050i	−0.698868	+	0.715250i	$0.746313\pi$
$110$		0			0
$111$	9.71980	+	16.2768i	0.922562	+	1.54493i
$112$		0			0
$113$		−	6.96261i		−	0.654987i	−0.944853	−	0.327494i	$-0.893796\pi$
$113$		−	6.96261i		−	0.654987i	0.944853	−	0.327494i	$-0.106204\pi$
$114$		0			0
$115$	−1.95497	−	10.5441i	−0.182302	−	0.983243i
$116$		0			0
$117$	6.85094	+	3.69076i	0.633370	+	0.341211i
$118$		0			0
$119$			0.138506i			0.0126968i
$120$		0			0
$121$	−3.61308			−0.328461
$122$		0			0
$123$	−2.44385	+	1.45936i	−0.220354	+	0.131586i
$124$		0			0
$125$	−10.7217	−	3.16943i	−0.958977	−	0.283482i
$126$		0			0
$127$		−	5.91915i		−	0.525239i	−0.964899	−	0.262620i	$-0.915414\pi$
$127$		−	5.91915i		−	0.525239i	0.964899	−	0.262620i	$-0.0845865\pi$
$128$		0			0
$129$	2.73779	+	4.58472i	0.241049	+	0.403662i
$130$		0			0
$131$		−	10.5848i		−	0.924800i	−0.886671	−	0.462400i	$-0.846988\pi$
$131$		−	10.5848i		−	0.924800i	0.886671	−	0.462400i	$-0.153012\pi$
$132$		0			0
$133$			0.357330i			0.0309845i
$134$		0			0
$135$	6.32099	−	9.74911i	0.544024	−	0.839070i
$136$		0			0
$137$		−	6.04370i		−	0.516348i	−0.966098	−	0.258174i	$-0.916879\pi$
$137$		−	6.04370i		−	0.516348i	0.966098	−	0.258174i	$-0.0831207\pi$
$138$		0			0
$139$	14.4216			1.22322			0.611611	−	0.791159i	$-0.290522\pi$
$139$	14.4216			1.22322			0.611611	+	0.791159i	$0.290522\pi$
$140$		0			0
$141$	−8.47049	−	14.1847i	−0.713344	−	1.19457i
$142$		0			0
$143$		−	7.05006i		−	0.589556i
$144$		0			0
$145$	−12.2297	−	8.71992i	−1.01562	−	0.724150i
$146$		0			0
$147$	6.01955	+	10.0804i	0.496484	+	0.831413i
$148$		0			0
$149$	−1.83274			−0.150144			−0.0750721	−	0.997178i	$-0.523919\pi$
$149$	−1.83274			−0.150144			−0.0750721	+	0.997178i	$0.523919\pi$
$150$		0			0
$151$	18.6590			1.51845			0.759223	−	0.650831i	$-0.225579\pi$
$151$	18.6590			1.51845			0.759223	+	0.650831i	$0.225579\pi$
$152$		0			0
$153$	0.777450	+	0.418831i	0.0628531	+	0.0338605i
$154$		0			0
$155$	3.57254	−	5.01048i	0.286953	−	0.402451i
$156$		0			0
$157$	3.11927			0.248944			0.124472	−	0.992223i	$-0.460276\pi$
$157$	3.11927			0.248944			0.124472	+	0.992223i	$0.460276\pi$
$158$		0			0
$159$	3.13480	−	1.87196i	0.248605	−	0.148456i
$160$		0			0
$161$	−1.56774	+	1.62306i	−0.123555	+	0.127915i
$162$		0			0
$163$		−	1.25275i		−	0.0981226i	−0.998796	−	0.0490613i	$-0.984377\pi$
$163$		−	1.25275i		−	0.0981226i	0.998796	−	0.0490613i	$-0.0156230\pi$
$164$		0			0
$165$	−10.4918	−	0.852548i	−0.816782	−	0.0663708i
$166$		0			0
$167$	−20.4752			−1.58442			−0.792209	−	0.610250i	$-0.791069\pi$
$167$	−20.4752			−1.58442			−0.792209	+	0.610250i	$0.791069\pi$
$168$		0			0
$169$	6.27143			0.482418
$170$		0			0
$171$	2.00574	+	1.08054i	0.153382	+	0.0826307i
$172$		0			0
$173$	5.44210			0.413756			0.206878	−	0.978367i	$-0.433670\pi$
$173$	5.44210			0.413756			0.206878	+	0.978367i	$0.433670\pi$
$174$		0			0
$175$	0.766768	+	2.22418i	0.0579622	+	0.168132i
$176$		0			0
$177$	7.67940	−	4.58580i	0.577219	−	0.344690i
$178$		0			0
$179$			13.2700i			0.991845i	0.868367	+	0.495923i	$0.165170\pi$
$179$			13.2700i			0.991845i	−0.868367	+	0.495923i	$0.834830\pi$
$180$		0			0
$181$		−	18.7821i		−	1.39606i	−0.716069	−	0.698029i	$-0.754060\pi$
$181$		−	18.7821i		−	1.39606i	0.716069	−	0.698029i	$-0.245940\pi$
$182$		0			0
$183$	13.3697	−	7.98379i	0.988315	−	0.590179i
$184$		0			0
$185$	−14.2089	+	19.9279i	−1.04466	+	1.46513i
$186$		0			0
$187$		−	0.800047i		−	0.0585053i
$188$		0			0
$189$	−2.44255	+	0.107985i	−0.177669	+	0.00785478i
$190$		0			0
$191$	17.8404			1.29088			0.645441	−	0.763810i	$-0.276673\pi$
$191$	17.8404			1.29088			0.645441	+	0.763810i	$0.276673\pi$
$192$		0			0
$193$		−	6.93881i		−	0.499466i	−0.968315	−	0.249733i	$-0.919657\pi$
$193$		−	6.93881i		−	0.499466i	0.968315	−	0.249733i	$-0.0803429\pi$
$194$		0			0
$195$	−0.813670	+	10.0133i	−0.0582681	+	0.717068i
$196$		0			0
$197$	6.71610			0.478502			0.239251	−	0.970958i	$-0.423098\pi$
$197$	6.71610			0.478502			0.239251	+	0.970958i	$0.423098\pi$
$198$		0			0
$199$		−	18.1393i		−	1.28586i	−0.765924	−	0.642931i	$-0.777718\pi$
$199$		−	18.1393i		−	1.28586i	0.765924	−	0.642931i	$-0.222282\pi$
$200$		0			0
$201$	8.81281	+	14.7580i	0.621607	+	1.04095i
$202$		0			0
$203$			3.16061i			0.221831i
$204$		0			0
$205$	−2.99204	−	2.13336i	−0.208973	−	0.149001i
$206$		0			0
$207$	4.36970	+	13.7079i	0.303715	+	0.952763i
$208$		0			0
$209$		−	2.06403i		−	0.142772i
$210$		0			0
$211$	5.85377			0.402990			0.201495	−	0.979489i	$-0.435420\pi$
$211$	5.85377			0.402990			0.201495	+	0.979489i	$0.435420\pi$
$212$		0			0
$213$	−4.43732	+	2.64977i	−0.304040	+	0.181559i
$214$		0			0
$215$	−4.00224	+	5.61314i	−0.272951	+	0.382813i
$216$		0			0
$217$	−1.29490			−0.0879034
$218$		0			0
$219$	−12.6448	+	7.55095i	−0.854459	+	0.510246i
$220$		0			0
$221$	−0.763563			−0.0513628
$222$		0			0
$223$			21.4919i			1.43921i	0.694386	+	0.719603i	$0.255676\pi$
$223$			21.4919i			1.43921i	−0.694386	+	0.719603i	$0.744324\pi$
$224$		0			0
$225$	14.8032	+	2.42177i	0.986881	+	0.161452i
$226$		0			0
$227$		−	3.82471i		−	0.253855i	−0.991912	−	0.126928i	$-0.959488\pi$
$227$		−	3.82471i		−	0.253855i	0.991912	−	0.126928i	$-0.0405116\pi$
$228$		0			0
$229$		−	0.876625i		−	0.0579290i	−0.999580	−	0.0289645i	$-0.990779\pi$
$229$		−	0.876625i		−	0.0579290i	0.999580	−	0.0289645i	$-0.00922098\pi$
$230$		0			0
$231$	1.13564	+	1.90174i	0.0747195	+	0.125126i
$232$		0			0
$233$	5.23404			0.342894			0.171447	−	0.985193i	$-0.445156\pi$
$233$	5.23404			0.342894			0.171447	+	0.985193i	$0.445156\pi$
$234$		0			0
$235$	12.3826	−	17.3666i	0.807751	−	1.13287i
$236$		0			0
$237$	−22.0351	+	13.1584i	−1.43133	+	0.854730i
$238$		0			0
$239$			12.7332i			0.823643i	0.911265	+	0.411821i	$0.135107\pi$
$239$			12.7332i			0.823643i	−0.911265	+	0.411821i	$0.864893\pi$
$240$		0			0
$241$			0.343898i			0.0221524i	0.999939	+	0.0110762i	$0.00352574\pi$
$241$			0.343898i			0.0221524i	−0.999939	+	0.0110762i	$0.996474\pi$
$242$		0			0
$243$	−6.77992	+	14.0368i	−0.434932	+	0.900463i
$244$		0			0
$245$	−8.79967	+	12.3415i	−0.562190	+	0.788471i
$246$		0			0
$247$	−1.96991			−0.125342
$248$		0			0
$249$	15.8896	−	9.48856i	1.00696	−	0.601313i
$250$		0			0
$251$	17.3426			1.09466			0.547329	−	0.836918i	$-0.315645\pi$
$251$	17.3426			1.09466			0.547329	+	0.836918i	$0.315645\pi$
$252$		0			0
$253$	9.05565	−	9.37520i	0.569324	−	0.589414i
$254$		0			0
$255$	−0.0923360	+	1.13632i	−0.00578230	+	0.0711591i
$256$		0			0
$257$	19.0455			1.18803			0.594013	−	0.804456i	$-0.297543\pi$
$257$	19.0455			1.18803			0.594013	+	0.804456i	$0.297543\pi$
$258$		0			0
$259$	5.15014			0.320014
$260$		0			0
$261$	17.7409	+	9.55743i	1.09813	+	0.591590i
$262$		0			0
$263$			18.0915i			1.11557i	0.829985	+	0.557786i	$0.188349\pi$
$263$			18.0915i			1.11557i	−0.829985	+	0.557786i	$0.811651\pi$
$264$		0			0
$265$	3.83798	+	2.73653i	0.235765	+	0.168104i
$266$		0			0
$267$	7.34694	+	12.3032i	0.449626	+	0.752945i
$268$		0			0
$269$			7.39246i			0.450726i	0.974275	+	0.225363i	$0.0723568\pi$
$269$			7.39246i			0.450726i	−0.974275	+	0.225363i	$0.927643\pi$
$270$		0			0
$271$	27.1023			1.64635			0.823175	−	0.567788i	$-0.192201\pi$
$271$	27.1023			1.64635			0.823175	+	0.567788i	$0.192201\pi$
$272$		0			0
$273$	1.81502	−	1.08385i	0.109850	−	0.0655976i
$274$		0			0
$275$	−4.42905	−	12.8474i	−0.267082	−	0.774729i
$276$		0			0
$277$			17.8783i			1.07420i	0.843517	+	0.537102i	$0.180481\pi$
$277$			17.8783i			1.07420i	−0.843517	+	0.537102i	$0.819519\pi$
$278$		0			0
$279$	−3.91567	+	7.26841i	−0.234425	+	0.435149i
$280$		0			0
$281$	24.7140			1.47431			0.737156	−	0.675722i	$-0.236168\pi$
$281$	24.7140			1.47431			0.737156	+	0.675722i	$0.236168\pi$
$282$		0			0
$283$	−13.9367			−0.828452			−0.414226	−	0.910174i	$-0.635948\pi$
$283$	−13.9367			−0.828452			−0.414226	+	0.910174i	$0.635948\pi$
$284$		0			0
$285$	−0.238217	+	2.93158i	−0.0141107	+	0.173652i
$286$		0			0
$287$			0.773257i			0.0456439i
$288$		0			0
$289$	16.9134			0.994903
$290$		0			0
$291$	−0.450019	−	0.753603i	−0.0263806	−	0.0441770i
$292$		0			0
$293$			19.6494i			1.14793i	0.818881	+	0.573964i	$0.194595\pi$
$293$			19.6494i			1.14793i	−0.818881	+	0.573964i	$0.805405\pi$
$294$		0			0
$295$	9.40200	+	6.70376i	0.547406	+	0.390308i
$296$		0			0
$297$	14.1088	−	0.623752i	0.818675	−	0.0361937i
$298$		0			0
$299$	−8.94767	−	8.64270i	−0.517457	−	0.499820i
$300$		0			0
$301$	1.45065			0.0836140
$302$		0			0
$303$	−9.29479	+	5.55044i	−0.533972	+	0.318864i
$304$		0			0
$305$	16.3687	+	11.6711i	0.937269	+	0.668285i
$306$		0			0
$307$			0.855496i			0.0488258i	0.999702	+	0.0244129i	$0.00777163\pi$
$307$			0.855496i			0.0488258i	−0.999702	+	0.0244129i	$0.992228\pi$
$308$		0			0
$309$	−1.54876	−	2.59356i	−0.0881061	−	0.147543i
$310$		0			0
$311$		−	25.5437i		−	1.44845i	−0.689563	−	0.724225i	$-0.742198\pi$
$311$		−	25.5437i		−	1.44845i	0.689563	−	0.724225i	$-0.257802\pi$
$312$		0			0
$313$	−17.7271			−1.00200			−0.500998	−	0.865448i	$-0.667034\pi$
$313$	−17.7271			−1.00200			−0.500998	+	0.865448i	$0.667034\pi$
$314$		0			0
$315$	−1.39348	−	2.83214i	−0.0785136	−	0.159573i
$316$		0			0
$317$	−16.1134			−0.905019			−0.452510	−	0.891760i	$-0.649471\pi$
$317$	−16.1134			−0.905019			−0.452510	+	0.891760i	$0.649471\pi$
$318$		0			0
$319$		−	18.2565i		−	1.02217i
$320$		0			0
$321$	10.3113	−	6.15746i	0.575521	−	0.343676i
$322$		0			0
$323$	−0.223547			−0.0124385
$324$		0			0
$325$	−12.2616	+	4.22708i	−0.680149	+	0.234476i
$326$		0			0
$327$	22.2094	−	13.2625i	1.22818	−	0.733417i
$328$		0			0
$329$	−4.48818			−0.247441
$330$		0			0
$331$	6.40122			0.351843			0.175921	−	0.984404i	$-0.443710\pi$
$331$	6.40122			0.351843			0.175921	+	0.984404i	$0.443710\pi$
$332$		0			0
$333$	15.5736	−	28.9083i	0.853427	−	1.58417i
$334$		0			0
$335$	−12.8830	+	18.0684i	−0.703873	+	0.987181i
$336$		0			0
$337$	−17.5553			−0.956296			−0.478148	−	0.878279i	$-0.658692\pi$
$337$	−17.5553			−0.956296			−0.478148	+	0.878279i	$0.658692\pi$
$338$		0			0
$339$	−10.3540	+	6.18295i	−0.562351	+	0.335812i
$340$		0			0
$341$	7.47967			0.405047
$342$		0			0
$343$	6.48321			0.350060
$344$		0			0
$345$	−13.9439	+	12.2706i	−0.750714	+	0.660627i
$346$		0			0
$347$	3.89843			0.209279			0.104639	−	0.994510i	$-0.466631\pi$
$347$	3.89843			0.209279			0.104639	+	0.994510i	$0.466631\pi$
$348$		0			0
$349$	−23.1848			−1.24105			−0.620527	−	0.784185i	$-0.713081\pi$
$349$	−23.1848			−1.24105			−0.620527	+	0.784185i	$0.713081\pi$
$350$		0			0
$351$	−0.595307	−	13.4654i	−0.0317752	−	0.718730i
$352$		0			0
$353$	9.37228			0.498836			0.249418	−	0.968396i	$-0.419761\pi$
$353$	9.37228			0.498836			0.249418	+	0.968396i	$0.419761\pi$
$354$		0			0
$355$	−5.43267	−	3.87357i	−0.288336	−	0.205588i
$356$		0			0
$357$	0.205970	−	0.122996i	0.0109011	−	0.00650966i
$358$		0			0
$359$	−24.0966			−1.27177			−0.635884	−	0.771785i	$-0.719364\pi$
$359$	−24.0966			−1.27177			−0.635884	+	0.771785i	$0.719364\pi$
$360$		0			0
$361$	18.4233			0.969646
$362$		0			0
$363$	3.20849	+	5.37295i	0.168402	+	0.282007i
$364$		0			0
$365$	−15.4813	−	11.0384i	−0.810326	−	0.577774i
$366$		0			0
$367$	36.9693			1.92978			0.964891	−	0.262651i	$-0.0845969\pi$
$367$	36.9693			1.92978			0.964891	+	0.262651i	$0.0845969\pi$
$368$		0			0
$369$	4.34038	+	2.33826i	0.225951	+	0.121725i
$370$		0			0
$371$		−	0.991879i		−	0.0514958i
$372$		0			0
$373$	29.2888			1.51652			0.758260	−	0.651953i	$-0.226050\pi$
$373$	29.2888			1.51652			0.758260	+	0.651953i	$0.226050\pi$
$374$		0			0
$375$	4.80789	+	18.7586i	0.248278	+	0.968689i
$376$		0			0
$377$	−17.4240			−0.897381
$378$		0			0
$379$		−	13.9410i		−	0.716103i	−0.933702	−	0.358052i	$-0.883441\pi$
$379$		−	13.9410i		−	0.716103i	0.933702	−	0.358052i	$-0.116559\pi$
$380$		0			0
$381$	−8.80227	+	5.25633i	−0.450954	+	0.269290i
$382$		0			0
$383$			17.7556i			0.907268i	0.891188	+	0.453634i	$0.149873\pi$
$383$			17.7556i			0.907268i	−0.891188	+	0.453634i	$0.850127\pi$
$384$		0			0
$385$	−1.66013	+	2.32833i	−0.0846082	+	0.118663i
$386$		0			0
$387$	4.38664	−	8.14266i	0.222986	−	0.413915i
$388$		0			0
$389$	25.3747			1.28655			0.643275	−	0.765635i	$-0.277575\pi$
$389$	25.3747			1.28655			0.643275	+	0.765635i	$0.277575\pi$
$390$		0			0
$391$	−1.01539	−	0.980780i	−0.0513505	−	0.0496002i
$392$		0			0
$393$	−15.7405	+	9.39955i	−0.794004	+	0.474145i
$394$		0			0
$395$	−26.9779	−	19.2356i	−1.35741	−	0.967849i
$396$		0			0
$397$		−	12.8480i		−	0.644821i	−0.946600	−	0.322411i	$-0.895507\pi$
$397$		−	12.8480i		−	0.644821i	0.946600	−	0.322411i	$-0.104493\pi$
$398$		0			0
$399$	0.531380	−	0.317317i	0.0266023	−	0.0158857i
$400$		0			0
$401$	−2.59631			−0.129653			−0.0648267	−	0.997897i	$-0.520649\pi$
$401$	−2.59631			−0.129653			−0.0648267	+	0.997897i	$0.520649\pi$
$402$		0			0
$403$		−	7.13858i		−	0.355598i
$404$		0			0
$405$	−20.1109	−	0.742416i	−0.999319	−	0.0368909i
$406$		0			0
$407$	−29.7485			−1.47458
$408$		0			0
$409$	−3.62758			−0.179372			−0.0896860	−	0.995970i	$-0.528586\pi$
$409$	−3.62758			−0.179372			−0.0896860	+	0.995970i	$0.528586\pi$
$410$		0			0
$411$	−8.98748	+	5.36693i	−0.443320	+	0.264731i
$412$		0			0
$413$		−	2.42984i		−	0.119564i
$414$		0			0
$415$	19.4538	+	13.8709i	0.954951	+	0.680893i
$416$		0			0
$417$	−12.8067	−	21.4461i	−0.627145	−	1.05022i
$418$		0			0
$419$	−0.394895			−0.0192919			−0.00964595	−	0.999953i	$-0.503070\pi$
$419$	−0.394895			−0.0192919			−0.00964595	+	0.999953i	$0.503070\pi$
$420$		0			0
$421$			25.7426i			1.25462i	0.778771	+	0.627308i	$0.215843\pi$
$421$			25.7426i			1.25462i	−0.778771	+	0.627308i	$0.784157\pi$
$422$		0			0
$423$	−13.5719	+	25.1927i	−0.659888	+	1.22491i
$424$		0			0
$425$	−1.39145	+	0.479692i	−0.0674954	+	0.0232685i
$426$		0			0
$427$		−	4.23029i		−	0.204718i
$428$		0			0
$429$	−10.4840	+	6.26061i	−0.506174	+	0.302265i
$430$		0			0
$431$	33.6918			1.62288			0.811439	−	0.584438i	$-0.198685\pi$
$431$	33.6918			1.62288			0.811439	+	0.584438i	$0.198685\pi$
$432$		0			0
$433$	−4.89010			−0.235003			−0.117502	−	0.993073i	$-0.537489\pi$
$433$	−4.89010			−0.235003			−0.117502	+	0.993073i	$0.537489\pi$
$434$		0			0
$435$	−2.10704	+	25.9300i	−0.101025	+	1.24325i
$436$		0			0
$437$	−2.61959	−	2.53030i	−0.125312	−	0.121041i
$438$		0			0
$439$	−21.2938			−1.01630			−0.508150	−	0.861269i	$-0.669670\pi$
$439$	−21.2938			−1.01630			−0.508150	+	0.861269i	$0.669670\pi$
$440$		0			0
$441$	9.64484	−	17.9031i	0.459278	−	0.852530i
$442$		0			0
$443$	34.9103			1.65864			0.829320	−	0.558774i	$-0.188728\pi$
$443$	34.9103			1.65864			0.829320	+	0.558774i	$0.188728\pi$
$444$		0			0
$445$	−10.7401	+	15.0630i	−0.509131	+	0.714055i
$446$		0			0
$447$	1.62752	+	2.72544i	0.0769789	+	0.128909i
$448$		0			0
$449$			5.02761i			0.237268i	0.992938	+	0.118634i	$0.0378515\pi$
$449$			5.02761i			0.237268i	−0.992938	+	0.118634i	$0.962149\pi$
$450$		0			0
$451$		−	4.46653i		−	0.210321i
$452$		0			0
$453$	−16.5696	−	27.7474i	−0.778506	−	1.30369i
$454$		0			0
$455$	2.22216	+	1.58443i	0.104176	+	0.0742791i
$456$		0			0
$457$	26.7256			1.25017			0.625086	−	0.780556i	$-0.285064\pi$
$457$	26.7256			1.25017			0.625086	+	0.780556i	$0.285064\pi$
$458$		0			0
$459$	−0.0675560	−	1.52806i	−0.00315324	−	0.0713239i
$460$		0			0
$461$		−	41.7818i		−	1.94597i	−0.230866	−	0.972986i	$-0.574156\pi$
$461$		−	41.7818i		−	1.94597i	0.230866	−	0.972986i	$-0.425844\pi$
$462$		0			0
$463$			10.5946i			0.492375i	0.969222	+	0.246187i	$0.0791778\pi$
$463$			10.5946i			0.492375i	−0.969222	+	0.246187i	$0.920822\pi$
$464$		0			0
$465$	−10.6235	−	0.863253i	−0.492653	−	0.0400324i
$466$		0			0
$467$			1.92981i			0.0893009i	0.999003	+	0.0446504i	$0.0142174\pi$
$467$			1.92981i			0.0893009i	−0.999003	+	0.0446504i	$0.985783\pi$
$468$		0			0
$469$	4.66956			0.215620
$470$		0			0
$471$	−2.76997	−	4.63861i	−0.127634	−	0.213736i
$472$		0			0
$473$	−8.37933			−0.385282
$474$		0			0
$475$	−3.58979	+	1.23755i	−0.164711	+	0.0567828i
$476$		0			0
$477$	−5.56753	−	2.99936i	−0.254920	−	0.137331i
$478$		0			0
$479$	1.45475			0.0664692			0.0332346	−	0.999448i	$-0.489419\pi$
$479$	1.45475			0.0664692			0.0332346	+	0.999448i	$0.489419\pi$
$480$		0			0
$481$			28.3919i			1.29456i
$482$		0			0
$483$	3.80580	+	0.890045i	0.173170	+	0.0404985i
$484$		0			0
$485$	0.657860	−	0.922647i	0.0298719	−	0.0418952i
$486$		0			0
$487$			13.4249i			0.608341i	0.952618	+	0.304171i	$0.0983793\pi$
$487$			13.4249i			0.608341i	−0.952618	+	0.304171i	$0.901621\pi$
$488$		0			0
$489$	−1.86294	+	1.11247i	−0.0842450	+	0.0503074i
$490$		0			0
$491$			2.32212i			0.104796i	0.998626	+	0.0523980i	$0.0166864\pi$
$491$			2.32212i			0.104796i	−0.998626	+	0.0523980i	$0.983314\pi$
$492$		0			0
$493$	−1.97729			−0.0890526
$494$		0			0
$495$	8.04910	+	16.3592i	0.361780	+	0.735292i
$496$		0			0
$497$			1.40401i			0.0629784i
$498$		0			0
$499$	−7.57904			−0.339284			−0.169642	−	0.985506i	$-0.554261\pi$
$499$	−7.57904			−0.339284			−0.169642	+	0.985506i	$0.554261\pi$
$500$		0			0
$501$	18.1824	+	30.4483i	0.812330	+	1.36033i
$502$		0			0
$503$			33.5867i			1.49756i	0.662821	+	0.748778i	$0.269359\pi$
$503$			33.5867i			1.49756i	−0.662821	+	0.748778i	$0.730641\pi$
$504$		0			0
$505$	−11.3797	−	8.11391i	−0.506392	−	0.361064i
$506$		0			0
$507$	−5.56917	−	9.32615i	−0.247335	−	0.414189i
$508$		0			0
$509$		−	19.8690i		−	0.880677i	−0.897832	−	0.440338i	$-0.854858\pi$
$509$		−	19.8690i		−	0.880677i	0.897832	−	0.440338i	$-0.145142\pi$
$510$		0			0
$511$			4.00095i			0.176992i
$512$		0			0
$513$	−0.174287	−	3.94224i	−0.00769495	−	0.174054i
$514$		0			0
$515$	2.26406	−	3.17534i	0.0997664	−	0.139922i
$516$		0			0
$517$	25.9249			1.14018
$518$		0			0
$519$	−4.83271	−	8.09287i	−0.212132	−	0.355237i
$520$		0			0
$521$	−16.3058			−0.714368			−0.357184	−	0.934034i	$-0.616263\pi$
$521$	−16.3058			−0.714368			−0.357184	+	0.934034i	$0.616263\pi$
$522$		0			0
$523$	−41.3526			−1.80822			−0.904112	−	0.427295i	$-0.859466\pi$
$523$	−41.3526			−1.80822			−0.904112	+	0.427295i	$0.859466\pi$
$524$		0			0
$525$	2.62663	−	3.11537i	0.114636	−	0.135966i
$526$		0			0
$527$		−	0.810093i		−	0.0352882i
$528$		0			0
$529$	−0.797293	−	22.9862i	−0.0346649	−	0.999399i
$530$		0			0
$531$	−13.6389	−	7.34762i	−0.591880	−	0.318860i
$532$		0			0
$533$	−4.26285			−0.184645
$534$		0			0
$535$	12.6243	+	9.00128i	0.545795	+	0.389159i
$536$		0			0
$537$	19.7336	−	11.7840i	0.851567	−	0.508518i
$538$		0			0
$539$	−18.4235			−0.793556
$540$		0			0
$541$	−13.3949			−0.575893			−0.287947	−	0.957646i	$-0.592973\pi$
$541$	−13.3949			−0.575893			−0.287947	+	0.957646i	$0.592973\pi$
$542$		0			0
$543$	−27.9305	+	16.6789i	−1.19861	+	0.715758i
$544$		0			0
$545$	27.1913	+	19.3878i	1.16475	+	0.830480i
$546$		0			0
$547$		−	18.9768i		−	0.811387i	−0.914009	−	0.405694i	$-0.867030\pi$
$547$		−	18.9768i		−	0.811387i	0.914009	−	0.405694i	$-0.132970\pi$
$548$		0			0
$549$	−23.7451	−	12.7921i	−1.01342	−	0.545952i
$550$		0			0
$551$	−5.10118			−0.217318
$552$		0			0
$553$			6.97212i			0.296485i
$554$		0			0
$555$	42.2523	+	3.43337i	1.79351	+	0.145739i
$556$		0			0
$557$		−	40.4609i		−	1.71438i	−0.514998	−	0.857192i	$-0.672207\pi$
$557$		−	40.4609i		−	1.71438i	0.514998	−	0.857192i	$-0.327793\pi$
$558$		0			0
$559$			7.99721i			0.338246i
$560$		0			0
$561$	−1.18974	+	0.710459i	−0.0502307	+	0.0299956i
$562$		0			0
$563$			2.62726i			0.110726i	0.998466	+	0.0553629i	$0.0176316\pi$
$563$			2.62726i			0.110726i	−0.998466	+	0.0553629i	$0.982368\pi$
$564$		0			0
$565$	−12.6765	−	9.03854i	−0.533306	−	0.380254i
$566$		0			0
$567$	2.32962	+	3.53638i	0.0978347	+	0.148514i
$568$		0			0
$569$	45.1617			1.89327			0.946637	−	0.322300i	$-0.104456\pi$
$569$	45.1617			1.89327			0.946637	+	0.322300i	$0.104456\pi$
$570$		0			0
$571$			12.0554i			0.504504i	0.967662	+	0.252252i	$0.0811711\pi$
$571$			12.0554i			0.504504i	−0.967662	+	0.252252i	$0.918829\pi$
$572$		0			0
$573$	−15.8426	−	26.5301i	−0.661835	−	1.10831i
$574$		0			0
$575$	−21.7351	−	10.1285i	−0.906415	−	0.422389i
$576$		0			0
$577$		−	21.8658i		−	0.910287i	−0.890418	−	0.455144i	$-0.849588\pi$
$577$		−	21.8658i		−	0.910287i	0.890418	−	0.455144i	$-0.150412\pi$
$578$		0			0
$579$	−10.3186	+	6.16181i	−0.428826	+	0.256076i
$580$		0			0
$581$		−	5.02761i		−	0.208581i
$582$		0			0
$583$			5.72935i			0.237286i
$584$		0			0
$585$	15.6132	−	7.68204i	0.645526	−	0.317613i
$586$		0			0
$587$	16.5450			0.682886			0.341443	−	0.939902i	$-0.389084\pi$
$587$	16.5450			0.682886			0.341443	+	0.939902i	$0.389084\pi$
$588$		0			0
$589$		−	2.08995i		−	0.0861148i
$590$		0			0
$591$	−5.96404	−	9.98741i	−0.245328	−	0.410827i
$592$		0			0
$593$	17.6571			0.725089			0.362545	−	0.931966i	$-0.381908\pi$
$593$	17.6571			0.725089			0.362545	+	0.931966i	$0.381908\pi$
$594$		0			0
$595$	0.252172	+	0.179802i	0.0103381	+	0.00737117i
$596$		0			0
$597$	−26.9747	+	16.1081i	−1.10400	+	0.659261i
$598$		0			0
$599$			32.0528i			1.30964i	0.755784	+	0.654821i	$0.227256\pi$
$599$			32.0528i			1.30964i	−0.755784	+	0.654821i	$0.772744\pi$
$600$		0			0
$601$	−25.6282			−1.04540			−0.522698	−	0.852518i	$-0.675074\pi$
$601$	−25.6282			−1.04540			−0.522698	+	0.852518i	$0.675074\pi$
$602$		0			0
$603$	14.1204	−	26.2108i	0.575025	−	1.06738i
$604$		0			0
$605$	−4.69033	+	6.57818i	−0.190689	+	0.267441i
$606$		0			0
$607$		−	24.1937i		−	0.981992i	−0.871162	−	0.490996i	$-0.836633\pi$
$607$		−	24.1937i		−	0.981992i	0.871162	−	0.490996i	$-0.163367\pi$
$608$		0			0
$609$	4.70009	−	2.80669i	0.190457	−	0.113733i
$610$		0			0
$611$		−	24.7427i		−	1.00098i
$612$		0			0
$613$	−19.9078			−0.804067			−0.402033	−	0.915625i	$-0.631696\pi$
$613$	−19.9078			−0.804067			−0.402033	+	0.915625i	$0.631696\pi$
$614$		0			0
$615$	−0.515497	+	6.34388i	−0.0207868	+	0.255810i
$616$		0			0
$617$		−	39.9069i		−	1.60659i	−0.595581	−	0.803295i	$-0.703078\pi$
$617$		−	39.9069i		−	1.60659i	0.595581	−	0.803295i	$-0.296922\pi$
$618$		0			0
$619$			41.3469i			1.66187i	0.556369	+	0.830935i	$0.312194\pi$
$619$			41.3469i			1.66187i	−0.556369	+	0.830935i	$0.687806\pi$
$620$		0			0
$621$	16.5044	−	18.6710i	0.662297	−	0.749241i
$622$		0			0
$623$	3.89285			0.155964
$624$		0			0
$625$	−19.6889	+	15.4061i	−0.787554	+	0.616245i
$626$		0			0
$627$	−3.06939	+	1.83291i	−0.122580	+	0.0731992i
$628$		0			0
$629$			3.22194i			0.128467i
$630$		0			0
$631$		−	7.27753i		−	0.289714i	−0.989453	−	0.144857i	$-0.953728\pi$
$631$		−	7.27753i		−	0.289714i	0.989453	−	0.144857i	$-0.0462722\pi$
$632$		0			0
$633$	−5.19828	−	8.70505i	−0.206613	−	0.345995i
$634$		0			0
$635$	−10.7767	−	7.68396i	−0.427662	−	0.304929i
$636$		0			0
$637$			17.5833i			0.696678i
$638$		0			0
$639$	7.88087	+	4.24561i	0.311762	+	0.167954i
$640$		0			0
$641$	−15.9598			−0.630373			−0.315187	−	0.949030i	$-0.602067\pi$
$641$	−15.9598			−0.630373			−0.315187	+	0.949030i	$0.602067\pi$
$642$		0			0
$643$	10.7217			0.422822			0.211411	−	0.977397i	$-0.432194\pi$
$643$	10.7217			0.422822			0.211411	+	0.977397i	$0.432194\pi$
$644$		0			0
$645$	11.9013	+	0.967085i	0.468613	+	0.0380789i
$646$		0			0
$647$	−3.95520			−0.155495			−0.0777475	−	0.996973i	$-0.524773\pi$
$647$	−3.95520			−0.155495			−0.0777475	+	0.996973i	$0.524773\pi$
$648$		0			0
$649$			14.0354i			0.550936i
$650$		0			0
$651$	1.14990	+	1.92562i	0.0450680	+	0.0754711i
$652$		0			0
$653$	20.4228			0.799207			0.399603	−	0.916688i	$-0.369148\pi$
$653$	20.4228			0.799207			0.399603	+	0.916688i	$0.369148\pi$
$654$		0			0
$655$	−19.2713	−	13.7407i	−0.752994	−	0.536895i
$656$		0			0
$657$	22.4578	+	12.0985i	0.876161	+	0.472009i
$658$		0			0
$659$	−30.9648			−1.20622			−0.603109	−	0.797659i	$-0.706071\pi$
$659$	−30.9648			−1.20622			−0.603109	+	0.797659i	$0.706071\pi$
$660$		0			0
$661$			30.2091i			1.17500i	0.809225	+	0.587499i	$0.199887\pi$
$661$			30.2091i			1.17500i	−0.809225	+	0.587499i	$0.800113\pi$
$662$		0			0
$663$	0.678061	+	1.13548i	0.0263337	+	0.0440985i
$664$		0			0
$665$	0.650576	+	0.463870i	0.0252283	+	0.0179881i
$666$		0			0
$667$	−23.1705	−	22.3807i	−0.897165	−	0.866585i
$668$		0			0
$669$	31.9603	−	19.0853i	1.23566	−	0.737880i
$670$		0			0
$671$			24.4353i			0.943314i
$672$		0			0
$673$		−	34.6381i		−	1.33520i	−0.744520	−	0.667601i	$-0.767321\pi$
$673$		−	34.6381i		−	1.33520i	0.744520	−	0.667601i	$-0.232679\pi$
$674$		0			0
$675$	−9.54419	−	24.1642i	−0.367356	−	0.930080i
$676$		0			0
$677$			4.28688i			0.164758i	0.996601	+	0.0823791i	$0.0262518\pi$
$677$			4.28688i			0.164758i	−0.996601	+	0.0823791i	$0.973748\pi$
$678$		0			0
$679$	−0.238447			−0.00915076
$680$		0			0
$681$	−5.68767	+	3.39643i	−0.217952	+	0.130151i
$682$		0			0
$683$	35.3046			1.35089			0.675447	−	0.737409i	$-0.263951\pi$
$683$	35.3046			1.35089			0.675447	+	0.737409i	$0.263951\pi$
$684$		0			0
$685$	−11.0035	−	7.84565i	−0.420422	−	0.299767i
$686$		0			0
$687$	−1.30362	+	0.778462i	−0.0497360	+	0.0297002i
$688$		0			0
$689$	5.46808			0.208317
$690$		0			0
$691$	−37.9438			−1.44345			−0.721725	−	0.692180i	$-0.756651\pi$
$691$	−37.9438			−1.44345			−0.721725	+	0.692180i	$0.756651\pi$
$692$		0			0
$693$	1.81958	−	3.37758i	0.0691202	−	0.128304i
$694$		0			0
$695$	18.7214	−	26.2567i	0.710144	−	0.995975i
$696$		0			0
$697$	−0.483752			−0.0183234
$698$		0			0
$699$	−4.64794	−	7.78346i	−0.175801	−	0.294398i
$700$		0			0
$701$	11.0595			0.417712			0.208856	−	0.977946i	$-0.433026\pi$
$701$	11.0595			0.417712			0.208856	+	0.977946i	$0.433026\pi$
$702$		0			0
$703$			8.31225i			0.313502i
$704$		0			0
$705$	−36.8215	−	2.99208i	−1.38678	−	0.112688i
$706$		0			0
$707$			2.94096i			0.110606i
$708$		0			0
$709$			31.1952i			1.17156i	0.810470	+	0.585781i	$0.199212\pi$
$709$			31.1952i			1.17156i	−0.810470	+	0.585781i	$0.800788\pi$
$710$		0			0
$711$	39.1353	+	21.0831i	1.46769	+	0.790678i
$712$		0			0
$713$	9.16935	−	9.49291i	0.343395	−	0.355512i
$714$		0			0
$715$	−12.8358	−	9.15207i	−0.480030	−	0.342268i
$716$		0			0
$717$	18.9353	−	11.3074i	0.707153	−	0.422281i
$718$		0			0
$719$			28.6275i			1.06763i	0.845602	+	0.533813i	$0.179242\pi$
$719$			28.6275i			1.06763i	−0.845602	+	0.533813i	$0.820758\pi$
$720$		0			0
$721$	−0.820628			−0.0305618
$722$		0			0
$723$	0.511406	−	0.305389i	0.0190194	−	0.0113575i
$724$		0			0
$725$	−31.7520	+	10.9462i	−1.17924	+	0.406533i
$726$		0			0
$727$	−32.0328			−1.18803			−0.594015	−	0.804454i	$-0.702458\pi$
$727$	−32.0328			−1.18803			−0.594015	+	0.804454i	$0.702458\pi$
$728$		0			0
$729$	26.8947	−	2.38269i	0.996099	−	0.0882478i
$730$		0			0
$731$			0.907531i			0.0335662i
$732$		0			0
$733$	10.5996			0.391503			0.195752	−	0.980653i	$-0.437285\pi$
$733$	10.5996			0.391503			0.195752	+	0.980653i	$0.437285\pi$
$734$		0			0
$735$	26.1672	+	2.12631i	0.965191	+	0.0784303i
$736$		0			0
$737$	−26.9726			−0.993548
$738$		0			0
$739$	21.8330			0.803140			0.401570	−	0.915828i	$-0.368465\pi$
$739$	21.8330			0.803140			0.401570	+	0.915828i	$0.368465\pi$
$740$		0			0
$741$	1.74932	+	2.92942i	0.0642629	+	0.107615i
$742$		0			0
$743$		−	0.424532i		−	0.0155746i	−0.999970	−	0.00778728i	$-0.997521\pi$
$743$		−	0.424532i		−	0.0155746i	0.999970	−	0.00778728i	$-0.00247879\pi$
$744$		0			0
$745$	−2.37918	+	3.33680i	−0.0871666	+	0.122251i
$746$		0			0
$747$	−28.2206	−	15.2031i	−1.03254	−	0.556252i
$748$		0			0
$749$		−	3.26259i		−	0.119213i
$750$		0			0
$751$		−	36.3391i		−	1.32603i	−0.748605	−	0.663016i	$-0.769276\pi$
$751$		−	36.3391i		−	1.32603i	0.748605	−	0.663016i	$-0.230724\pi$
$752$		0			0
$753$	−15.4006	−	25.7900i	−0.561230	−	0.939838i
$754$		0			0
$755$	24.2222	−	33.9716i	0.881537	−	1.23635i
$756$		0			0
$757$	−3.12855			−0.113709			−0.0568545	−	0.998382i	$-0.518107\pi$
$757$	−3.12855			−0.113709			−0.0568545	+	0.998382i	$0.518107\pi$
$758$		0			0
$759$	−21.9833	−	5.14114i	−0.797944	−	0.186611i
$760$		0			0
$761$			39.0859i			1.41686i	0.705779	+	0.708432i	$0.250597\pi$
$761$			39.0859i			1.41686i	−0.705779	+	0.708432i	$0.749403\pi$
$762$		0			0
$763$		−	7.02726i		−	0.254404i
$764$		0			0
$765$	1.77180	−	0.871764i	0.0640595	−	0.0315187i
$766$		0			0
$767$	13.3953			0.483677
$768$		0			0
$769$			45.8619i			1.65382i	0.562334	+	0.826910i	$0.309904\pi$
$769$			45.8619i			1.65382i	−0.562334	+	0.826910i	$0.690096\pi$
$770$		0			0
$771$	−16.9128	−	28.3222i	−0.609100	−	1.02000i
$772$		0			0
$773$			36.9125i			1.32765i	0.747888	+	0.663825i	$0.231068\pi$
$773$			36.9125i			1.32765i	−0.747888	+	0.663825i	$0.768932\pi$
$774$		0			0
$775$	−4.48466	−	13.0087i	−0.161094	−	0.467288i
$776$		0			0
$777$	−4.57343	−	7.65868i	−0.164071	−	0.274754i
$778$		0			0
$779$	−1.24803			−0.0447152
$780$		0			0
$781$		−	8.10993i		−	0.290196i
$782$		0			0
$783$	−1.54158	−	34.8694i	−0.0550916	−	1.24613i
$784$		0			0
$785$	4.04929	−	5.67912i	0.144525	−	0.202696i
$786$		0			0
$787$	15.3328			0.546557			0.273278	−	0.961935i	$-0.411892\pi$
$787$	15.3328			0.546557			0.273278	+	0.961935i	$0.411892\pi$
$788$		0			0
$789$	26.9036	−	16.0657i	0.957795	−	0.571953i
$790$		0			0
$791$			3.27610i			0.116485i
$792$		0			0
$793$	23.3210			0.828152
$794$		0			0
$795$	0.661243	−	8.13749i	0.0234519	−	0.288607i
$796$		0			0
$797$			45.0345i			1.59520i	0.603185	+	0.797601i	$0.293898\pi$
$797$			45.0345i			1.59520i	−0.603185	+	0.797601i	$0.706102\pi$
$798$		0			0
$799$		−	2.80782i		−	0.0993335i
$800$		0			0
$801$	11.7717	−	21.8510i	0.415932	−	0.772069i
$802$		0			0
$803$		−	23.1105i		−	0.815553i
$804$		0			0
$805$	0.919869	+	4.96129i	0.0324211	+	0.174862i
$806$		0			0
$807$	10.9932	−	6.56466i	0.386979	−	0.231087i
$808$		0			0
$809$		−	24.1332i		−	0.848479i	−0.905550	−	0.424240i	$-0.860541\pi$
$809$		−	24.1332i		−	0.848479i	0.905550	−	0.424240i	$-0.139459\pi$
$810$		0			0
$811$	5.67375			0.199232			0.0996161	−	0.995026i	$-0.468239\pi$
$811$	5.67375			0.199232			0.0996161	+	0.995026i	$0.468239\pi$
$812$		0			0
$813$	−24.0674	−	40.3034i	−0.844082	−	1.41350i
$814$		0			0
$815$	−2.28082	−	1.62626i	−0.0798937	−	0.0569653i
$816$		0			0
$817$			2.34133i			0.0819127i
$818$		0			0
$819$	−3.22355	−	1.73660i	−0.112640	−	0.0606819i
$820$		0			0
$821$		−	10.4857i		−	0.365952i	−0.983117	−	0.182976i	$-0.941427\pi$
$821$		−	10.4857i		−	0.365952i	0.983117	−	0.182976i	$-0.0585731\pi$
$822$		0			0
$823$			4.66303i			0.162543i	0.996692	+	0.0812715i	$0.0258981\pi$
$823$			4.66303i			0.162543i	−0.996692	+	0.0812715i	$0.974102\pi$
$824$		0			0
$825$	−15.1721	+	17.9952i	−0.528225	+	0.626511i
$826$		0			0
$827$		−	49.9806i		−	1.73800i	−0.494816	−	0.868998i	$-0.664764\pi$
$827$		−	49.9806i		−	1.73800i	0.494816	−	0.868998i	$-0.335236\pi$
$828$		0			0
$829$	−6.39168			−0.221992			−0.110996	−	0.993821i	$-0.535404\pi$
$829$	−6.39168			−0.221992			−0.110996	+	0.993821i	$0.535404\pi$
$830$		0			0
$831$	26.5866	−	15.8763i	0.922277	−	0.550744i
$832$		0			0
$833$			1.99537i			0.0691356i
$834$		0			0
$835$	−26.5799	+	37.2783i	−0.919837	+	1.29007i
$836$		0			0
$837$	14.2859	−	0.631583i	0.493794	−	0.0218307i
$838$		0			0
$839$	−27.0331			−0.933285			−0.466642	−	0.884446i	$-0.654536\pi$
$839$	−27.0331			−0.933285			−0.466642	+	0.884446i	$0.654536\pi$
$840$		0			0
$841$	−16.1204			−0.555875
$842$		0			0
$843$	−21.9465	−	36.7518i	−0.755879	−	1.26580i
$844$		0			0
$845$	8.14129	−	11.4181i	0.280069	−	0.392796i
$846$		0			0
$847$	1.70005			0.0584145
$848$		0			0
$849$	12.3761	+	20.7251i	0.424747	+	0.711283i
$850$		0			0
$851$	−36.4688	+	37.7557i	−1.25014	+	1.29425i
$852$		0			0
$853$			46.4110i			1.58908i	0.607211	+	0.794541i	$0.292288\pi$
$853$			46.4110i			1.58908i	−0.607211	+	0.794541i	$0.707712\pi$
$854$		0			0
$855$	4.57104	−	2.24905i	0.156326	−	0.0769161i
$856$		0			0
$857$	−20.0436			−0.684676			−0.342338	−	0.939577i	$-0.611219\pi$
$857$	−20.0436			−0.684676			−0.342338	+	0.939577i	$0.611219\pi$
$858$		0			0
$859$	−49.2450			−1.68022			−0.840108	−	0.542419i	$-0.817509\pi$
$859$	−49.2450			−1.68022			−0.840108	+	0.542419i	$0.817509\pi$
$860$		0			0
$861$	1.14990	−	0.686668i	0.0391884	−	0.0234016i
$862$		0			0
$863$	−8.42611			−0.286828			−0.143414	−	0.989663i	$-0.545808\pi$
$863$	−8.42611			−0.286828			−0.143414	+	0.989663i	$0.545808\pi$
$864$		0			0
$865$	7.06469	−	9.90821i	0.240207	−	0.336889i
$866$		0			0
$867$	−15.0194	−	25.1516i	−0.510086	−	0.854192i
$868$		0			0
$869$		−	40.2728i		−	1.36616i
$870$		0			0
$871$			25.7426i			0.872254i
$872$		0			0
$873$	−0.721044	+	1.33843i	−0.0244036	+	0.0452990i
$874$		0			0
$875$	5.04485	+	1.49130i	0.170547	+	0.0504152i
$876$		0			0
$877$			27.9659i			0.944341i	0.881507	+	0.472170i	$0.156529\pi$
$877$			27.9659i			0.944341i	−0.881507	+	0.472170i	$0.843471\pi$
$878$		0			0
$879$	29.2202	−	17.4491i	0.985575	−	0.588542i
$880$		0			0
$881$	−50.2217			−1.69201			−0.846006	−	0.533174i	$-0.820999\pi$
$881$	−50.2217			−1.69201			−0.846006	+	0.533174i	$0.820999\pi$
$882$		0			0
$883$		−	33.4586i		−	1.12597i	−0.826467	−	0.562986i	$-0.809653\pi$
$883$		−	33.4586i		−	1.12597i	0.826467	−	0.562986i	$-0.190347\pi$
$884$		0			0
$885$	1.61987	−	19.9346i	0.0544512	−	0.670096i
$886$		0			0
$887$	45.9037			1.54129			0.770647	−	0.637262i	$-0.219933\pi$
$887$	45.9037			1.54129			0.770647	+	0.637262i	$0.219933\pi$
$888$		0			0
$889$			2.78512i			0.0934100i
$890$		0			0
$891$	−13.4565	−	20.4270i	−0.450809	−	0.684332i
$892$		0			0
$893$		−	7.24386i		−	0.242407i
$894$		0			0
$895$	24.1601	+	17.2265i	0.807583	+	0.575818i
$896$		0			0
$897$	−4.90669	+	20.9808i	−0.163830	+	0.700530i
$898$		0			0
$899$		−	18.4857i		−	0.616534i
$900$		0			0
$901$	0.620523			0.0206726
$902$		0			0
$903$	−1.28821	−	2.15724i	−0.0428689	−	0.0717883i
$904$		0			0
$905$	−34.1957	−	24.3820i	−1.13670	−	0.810485i
$906$		0			0
$907$	23.2790			0.772967			0.386483	−	0.922296i	$-0.373690\pi$
$907$	23.2790			0.772967			0.386483	+	0.922296i	$0.373690\pi$
$908$		0			0
$909$	16.5079	+	8.89322i	0.547534	+	0.294969i
$910$		0			0
$911$	−29.7049			−0.984167			−0.492083	−	0.870548i	$-0.663764\pi$
$911$	−29.7049			−0.984167			−0.492083	+	0.870548i	$0.663764\pi$
$912$		0			0
$913$			29.0408i			0.961110i
$914$		0			0
$915$	2.82015	−	34.7058i	0.0932314	−	1.14734i
$916$		0			0
$917$			4.98045i			0.164469i
$918$		0			0
$919$		−	18.3966i		−	0.606848i	−0.952856	−	0.303424i	$-0.901870\pi$
$919$		−	18.3966i		−	0.606848i	0.952856	−	0.303424i	$-0.0981298\pi$
$920$		0			0
$921$	1.27219	−	0.759699i	0.0419202	−	0.0250329i
$922$		0			0
$923$	−7.74010			−0.254768
$924$		0			0
$925$	17.8366	+	51.7390i	0.586465	+	1.70117i
$926$		0			0
$927$	−2.48151	+	4.60628i	−0.0815036	+	0.151290i
$928$		0			0
$929$		−	19.5644i		−	0.641888i	−0.947098	−	0.320944i	$-0.896000\pi$
$929$		−	19.5644i		−	0.641888i	0.947098	−	0.320944i	$-0.104000\pi$
$930$		0			0
$931$			5.14784i			0.168714i
$932$		0			0
$933$	−37.9856	+	22.6834i	−1.24359	+	0.742620i
$934$		0			0
$935$	−1.45661	−	1.03858i	−0.0476363	−	0.0339653i
$936$		0			0
$937$	9.72907			0.317835			0.158917	−	0.987292i	$-0.449200\pi$
$937$	9.72907			0.317835			0.158917	+	0.987292i	$0.449200\pi$
$938$		0			0
$939$	15.7421	+	26.3617i	0.513723	+	0.860282i
$940$		0			0
$941$	20.7025			0.674883			0.337441	−	0.941347i	$-0.390439\pi$
$941$	20.7025			0.674883			0.337441	+	0.941347i	$0.390439\pi$
$942$		0			0
$943$	−5.66875	−	5.47554i	−0.184600	−	0.178308i
$944$		0			0
$945$	−2.97420	+	4.58722i	−0.0967506	+	0.149222i
$946$		0			0
$947$	4.15842			0.135130			0.0675652	−	0.997715i	$-0.478477\pi$
$947$	4.15842			0.135130			0.0675652	+	0.997715i	$0.478477\pi$
$948$		0			0
$949$	−22.0566			−0.715989
$950$		0			0
$951$	14.3091	+	23.9620i	0.464003	+	0.777021i
$952$		0			0
$953$			13.6049i			0.440705i	0.975420	+	0.220352i	$0.0707208\pi$
$953$			13.6049i			0.440705i	−0.975420	+	0.220352i	$0.929279\pi$
$954$		0			0
$955$	23.1595	−	32.4812i	0.749425	−	1.05107i
$956$		0			0
$957$	−27.1490	+	16.2122i	−0.877602	+	0.524065i
$958$		0			0
$959$			2.84372i			0.0918286i
$960$		0			0
$961$	−23.4264			−0.755691
$962$		0			0
$963$	−18.3133	−	9.86581i	−0.590138	−	0.317921i
$964$		0			0
$965$	−12.6332	−	9.00764i	−0.406677	−	0.289966i
$966$		0			0
$967$			6.31539i			0.203089i	0.994831	+	0.101545i	$0.0323784\pi$
$967$			6.31539i			0.203089i	−0.994831	+	0.101545i	$0.967622\pi$
$968$		0			0
$969$	0.198514	+	0.332433i	0.00637720	+	0.0106793i
$970$		0			0
$971$	12.0773			0.387578			0.193789	−	0.981043i	$-0.437922\pi$
$971$	12.0773			0.387578			0.193789	+	0.981043i	$0.437922\pi$
$972$		0			0
$973$	−6.78574			−0.217541
$974$		0			0
$975$	17.1746	+	14.4802i	0.550026	+	0.463739i
$976$		0			0
$977$			15.8674i			0.507642i	0.967251	+	0.253821i	$0.0816874\pi$
$977$			15.8674i			0.507642i	−0.967251	+	0.253821i	$0.918313\pi$
$978$		0			0
$979$	−22.4861			−0.718660
$980$		0			0
$981$	−39.4448	−	21.2499i	−1.25938	−	0.678456i
$982$		0			0
$983$			27.6415i			0.881628i	0.897599	+	0.440814i	$0.145310\pi$
$983$			27.6415i			0.881628i	−0.897599	+	0.440814i	$0.854690\pi$
$984$		0			0
$985$	8.71853	−	12.2277i	0.277796	−	0.389608i
$986$		0			0
$987$	3.98560	+	6.67430i	0.126863	+	0.212445i
$988$		0			0
$989$	−10.2722	+	10.6347i	−0.326638	+	0.338165i
$990$		0			0
$991$	21.6111			0.686499			0.343249	−	0.939244i	$-0.388472\pi$
$991$	21.6111			0.686499			0.343249	+	0.939244i	$0.388472\pi$
$992$		0			0
$993$	−5.68442	−	9.51915i	−0.180390	−	0.302081i
$994$		0			0
$995$	−33.0255	−	23.5476i	−1.04698	−	0.746510i
$996$		0			0
$997$			49.7984i			1.57713i	0.614950	+	0.788566i	$0.289176\pi$
$997$			49.7984i			1.57713i	−0.614950	+	0.788566i	$0.710824\pi$
$998$		0			0
$999$	−56.8187	+	2.51197i	−1.79767	+	0.0794751i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.n.a.689.14	yes	48
3.2	odd	2	inner	1380.2.n.a.689.33	yes	48
5.4	even	2	inner	1380.2.n.a.689.36	yes	48
15.14	odd	2	inner	1380.2.n.a.689.15	yes	48
23.22	odd	2	inner	1380.2.n.a.689.13	✓	48
69.68	even	2	inner	1380.2.n.a.689.34	yes	48
115.114	odd	2	inner	1380.2.n.a.689.35	yes	48
345.344	even	2	inner	1380.2.n.a.689.16	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.n.a.689.13	✓	48	23.22	odd	2	inner
1380.2.n.a.689.14	yes	48	1.1	even	1	trivial
1380.2.n.a.689.15	yes	48	15.14	odd	2	inner
1380.2.n.a.689.16	yes	48	345.344	even	2	inner
1380.2.n.a.689.33	yes	48	3.2	odd	2	inner
1380.2.n.a.689.34	yes	48	69.68	even	2	inner
1380.2.n.a.689.35	yes	48	115.114	odd	2	inner
1380.2.n.a.689.36	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)

\(f(q)\)	\(=\)	\(q+(-0.888021 - 1.48708i) q^{3} +(1.29815 - 1.82066i) q^{5} -0.470527 q^{7} +(-1.42284 + 2.64112i) q^{9} +O(q^{10})\)	\(q+(-0.888021 - 1.48708i) q^{3} +(1.29815 - 1.82066i) q^{5} -0.470527 q^{7} +(-1.42284 + 2.64112i) q^{9} +2.71789 q^{11} -2.59395i q^{13} +(-3.86026 - 0.313680i) q^{15} -0.294363i q^{17} -0.759425i q^{19} +(0.417838 + 0.699714i) q^{21} +(3.33187 - 3.44944i) q^{23} +(-1.62959 - 4.72699i) q^{25} +(5.19108 - 0.229499i) q^{27} -6.71717i q^{29} +2.75201 q^{31} +(-2.41354 - 4.04173i) q^{33} +(-0.610817 + 0.856670i) q^{35} -10.9455 q^{37} +(-3.85742 + 2.30348i) q^{39} -1.64338i q^{41} -3.08303 q^{43} +(2.96152 + 6.01908i) q^{45} +9.53862 q^{47} -6.77860 q^{49} +(-0.437743 + 0.261401i) q^{51} +2.10802i q^{53} +(3.52824 - 4.94835i) q^{55} +(-1.12933 + 0.674385i) q^{57} +5.16407i q^{59} +8.99054i q^{61} +(0.669483 - 1.24272i) q^{63} +(-4.72269 - 3.36734i) q^{65} -9.92409 q^{67} +(-8.08838 - 1.89159i) q^{69} -2.98391i q^{71} -8.50311i q^{73} +(-5.58232 + 6.62101i) q^{75} -1.27884 q^{77} -14.8177i q^{79} +(-4.95108 - 7.51577i) q^{81} +10.6851i q^{83} +(-0.535935 - 0.382129i) q^{85} +(-9.98899 + 5.96499i) q^{87} -8.27339 q^{89} +1.22052i q^{91} +(-2.44385 - 4.09248i) q^{93} +(-1.38265 - 0.985850i) q^{95} +0.506765 q^{97} +(-3.86711 + 7.17828i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 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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