LMFDB - Embedded newform 900.3.c.s.451.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(451,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("900.451");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.c (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:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.77720518656.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 121x^{4} + 14641 $ x^8 - 121*x^4 + 14641
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			451.6
Root			$3.20361 + 0.858406i$ of defining polynomial
Character	$\chi$	$=$	900.451
Dual form			900.3.c.s.451.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +9.38083i q^{7} -8.00000i q^{8} +O(q^{10})$	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +9.38083i q^{7} -8.00000i q^{8} -16.2481i q^{11} -16.2481 q^{13} +(9.38083 + 16.2481i) q^{14} +(-8.00000 - 13.8564i) q^{16} +10.3923 q^{17} -20.7846i q^{19} +(-16.2481 - 28.1425i) q^{22} -28.0000i q^{23} +(-28.1425 + 16.2481i) q^{26} +(32.4962 + 18.7617i) q^{28} +9.38083 q^{29} -34.6410i q^{31} +(-27.7128 - 16.0000i) q^{32} +(18.0000 - 10.3923i) q^{34} +48.7442 q^{37} +(-20.7846 - 36.0000i) q^{38} +18.7617 q^{41} -37.5233i q^{43} +(-56.2850 - 32.4962i) q^{44} +(-28.0000 - 48.4974i) q^{46} +4.00000i q^{47} -39.0000 q^{49} +(-32.4962 + 56.2850i) q^{52} +31.1769 q^{53} +75.0467 q^{56} +(16.2481 - 9.38083i) q^{58} -16.2481i q^{59} -58.0000 q^{61} +(-34.6410 - 60.0000i) q^{62} -64.0000 q^{64} -18.7617i q^{67} +(20.7846 - 36.0000i) q^{68} +97.4885i q^{71} +(84.4275 - 48.7442i) q^{74} +(-72.0000 - 41.5692i) q^{76} +152.420 q^{77} +6.92820i q^{79} +(32.4962 - 18.7617i) q^{82} +32.0000i q^{83} +(-37.5233 - 64.9923i) q^{86} -129.985 q^{88} -75.0467 q^{89} -152.420i q^{91} +(-96.9948 - 56.0000i) q^{92} +(4.00000 + 6.92820i) q^{94} -162.481 q^{97} +(-67.5500 + 39.0000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4}+O(q^{10}) $ 8 * q + 16 * q^4	$ 8 q + 16 q^{4} - 64 q^{16} + 144 q^{34} - 224 q^{46} - 312 q^{49} - 464 q^{61} - 512 q^{64} - 576 q^{76} + 32 q^{94}+O(q^{100}) $ 8 * q + 16 * q^4 - 64 * q^16 + 144 * q^34 - 224 * q^46 - 312 * q^49 - 464 * q^61 - 512 * q^64 - 576 * q^76 + 32 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	1.73205	−	1.00000i	0.866025	−	0.500000i
$2$	1.73205	−	1.00000i	0.866025	−	0.500000i
$3$		0			0
$3$		0			0
$4$	2.00000	−	3.46410i	0.500000	−	0.866025i
$5$		0			0
$5$		0			0
$6$		0			0
$7$			9.38083i			1.34012i	0.742307	+	0.670059i	$0.233731\pi$
$7$			9.38083i			1.34012i	−0.742307	+	0.670059i	$0.766269\pi$
$8$		−	8.00000i		−	1.00000i
$9$		0			0
$10$		0			0
$11$		−	16.2481i		−	1.47710i	−0.674200	−	0.738549i	$-0.735511\pi$
$11$		−	16.2481i		−	1.47710i	0.674200	−	0.738549i	$-0.264489\pi$
$12$		0			0
$13$	−16.2481			−1.24985			−0.624926	−	0.780684i	$-0.714871\pi$
$13$	−16.2481			−1.24985			−0.624926	+	0.780684i	$0.714871\pi$
$14$	9.38083	+	16.2481i	0.670059	+	1.16058i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$17$	10.3923			0.611312			0.305656	−	0.952142i	$-0.401124\pi$
$17$	10.3923			0.611312			0.305656	+	0.952142i	$0.401124\pi$
$18$		0			0
$19$		−	20.7846i		−	1.09393i	−0.837157	−	0.546963i	$-0.815784\pi$
$19$		−	20.7846i		−	1.09393i	0.837157	−	0.546963i	$-0.184216\pi$
$20$		0			0
$21$		0			0
$22$	−16.2481	−	28.1425i	−0.738549	−	1.27920i
$23$		−	28.0000i		−	1.21739i	−0.793404	−	0.608696i	$-0.791693\pi$
$23$		−	28.0000i		−	1.21739i	0.793404	−	0.608696i	$-0.208307\pi$
$24$		0			0
$25$		0			0
$26$	−28.1425	+	16.2481i	−1.08240	+	0.624926i
$27$		0			0
$28$	32.4962	+	18.7617i	1.16058	+	0.670059i
$29$	9.38083			0.323477			0.161738	−	0.986834i	$-0.448290\pi$
$29$	9.38083			0.323477			0.161738	+	0.986834i	$0.448290\pi$
$30$		0			0
$31$		−	34.6410i		−	1.11745i	−0.829352	−	0.558726i	$-0.811290\pi$
$31$		−	34.6410i		−	1.11745i	0.829352	−	0.558726i	$-0.188710\pi$
$32$	−27.7128	−	16.0000i	−0.866025	−	0.500000i
$33$		0			0
$34$	18.0000	−	10.3923i	0.529412	−	0.305656i
$35$		0			0
$36$		0			0
$37$	48.7442			1.31741			0.658706	−	0.752401i	$-0.271104\pi$
$37$	48.7442			1.31741			0.658706	+	0.752401i	$0.271104\pi$
$38$	−20.7846	−	36.0000i	−0.546963	−	0.947368i
$39$		0			0
$40$		0			0
$41$	18.7617			0.457602			0.228801	−	0.973473i	$-0.426520\pi$
$41$	18.7617			0.457602			0.228801	+	0.973473i	$0.426520\pi$
$42$		0			0
$43$		−	37.5233i		−	0.872635i	−0.899793	−	0.436318i	$-0.856282\pi$
$43$		−	37.5233i		−	0.872635i	0.899793	−	0.436318i	$-0.143718\pi$
$44$	−56.2850	−	32.4962i	−1.27920	−	0.738549i
$45$		0			0
$46$	−28.0000	−	48.4974i	−0.608696	−	1.05429i
$47$			4.00000i			0.0851064i	0.999094	+	0.0425532i	$0.0135492\pi$
$47$			4.00000i			0.0851064i	−0.999094	+	0.0425532i	$0.986451\pi$
$48$		0			0
$49$	−39.0000			−0.795918
$50$		0			0
$51$		0			0
$52$	−32.4962	+	56.2850i	−0.624926	+	1.08240i
$53$	31.1769			0.588244			0.294122	−	0.955768i	$-0.404973\pi$
$53$	31.1769			0.588244			0.294122	+	0.955768i	$0.404973\pi$
$54$		0			0
$55$		0			0
$56$	75.0467			1.34012
$57$		0			0
$58$	16.2481	−	9.38083i	0.280139	−	0.161738i
$59$		−	16.2481i		−	0.275391i	−0.990475	−	0.137696i	$-0.956030\pi$
$59$		−	16.2481i		−	0.275391i	0.990475	−	0.137696i	$-0.0439696\pi$
$60$		0			0
$61$	−58.0000			−0.950820			−0.475410	−	0.879764i	$-0.657700\pi$
$61$	−58.0000			−0.950820			−0.475410	+	0.879764i	$0.657700\pi$
$62$	−34.6410	−	60.0000i	−0.558726	−	0.967742i
$63$		0			0
$64$	−64.0000			−1.00000
$65$		0			0
$66$		0			0
$67$		−	18.7617i		−	0.280025i	−0.990150	−	0.140012i	$-0.955286\pi$
$67$		−	18.7617i		−	0.280025i	0.990150	−	0.140012i	$-0.0447142\pi$
$68$	20.7846	−	36.0000i	0.305656	−	0.529412i
$69$		0			0
$70$		0			0
$71$			97.4885i			1.37308i	0.727093	+	0.686538i	$0.240871\pi$
$71$			97.4885i			1.37308i	−0.727093	+	0.686538i	$0.759129\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$	84.4275	−	48.7442i	1.14091	−	0.658706i
$75$		0			0
$76$	−72.0000	−	41.5692i	−0.947368	−	0.546963i
$77$	152.420			1.97949
$78$		0			0
$79$			6.92820i			0.0876988i	0.999038	+	0.0438494i	$0.0139622\pi$
$79$			6.92820i			0.0876988i	−0.999038	+	0.0438494i	$0.986038\pi$
$80$		0			0
$81$		0			0
$82$	32.4962	−	18.7617i	0.396295	−	0.228801i
$83$			32.0000i			0.385542i	0.981244	+	0.192771i	$0.0617475\pi$
$83$			32.0000i			0.385542i	−0.981244	+	0.192771i	$0.938253\pi$
$84$		0			0
$85$		0			0
$86$	−37.5233	−	64.9923i	−0.436318	−	0.755725i
$87$		0			0
$88$	−129.985			−1.47710
$89$	−75.0467			−0.843221			−0.421610	−	0.906777i	$-0.638535\pi$
$89$	−75.0467			−0.843221			−0.421610	+	0.906777i	$0.638535\pi$
$90$		0			0
$91$		−	152.420i		−	1.67495i
$92$	−96.9948	−	56.0000i	−1.05429	−	0.608696i
$93$		0			0
$94$	4.00000	+	6.92820i	0.0425532	+	0.0737043i
$95$		0			0
$96$		0			0
$97$	−162.481			−1.67506			−0.837530	−	0.546392i	$-0.816001\pi$
$97$	−162.481			−1.67506			−0.837530	+	0.546392i	$0.816001\pi$
$98$	−67.5500	+	39.0000i	−0.689286	+	0.397959i
$99$		0			0
$100$		0			0
$101$	−121.951			−1.20743			−0.603717	−	0.797199i	$-0.706314\pi$
$101$	−121.951			−1.20743			−0.603717	+	0.797199i	$0.706314\pi$
$102$		0			0
$103$			159.474i			1.54829i	0.633007	+	0.774146i	$0.281821\pi$
$103$			159.474i			1.54829i	−0.633007	+	0.774146i	$0.718179\pi$
$104$			129.985i			1.24985i
$105$		0			0
$106$	54.0000	−	31.1769i	0.509434	−	0.294122i
$107$			184.000i			1.71963i	0.510609	+	0.859813i	$0.329420\pi$
$107$			184.000i			1.71963i	−0.510609	+	0.859813i	$0.670580\pi$
$108$		0			0
$109$	22.0000			0.201835			0.100917	−	0.994895i	$-0.467822\pi$
$109$	22.0000			0.201835			0.100917	+	0.994895i	$0.467822\pi$
$110$		0			0
$111$		0			0
$112$	129.985	−	75.0467i	1.16058	−	0.670059i
$113$	86.6025			0.766394			0.383197	−	0.923667i	$-0.374823\pi$
$113$	86.6025			0.766394			0.383197	+	0.923667i	$0.374823\pi$
$114$		0			0
$115$		0			0
$116$	18.7617	−	32.4962i	0.161738	−	0.280139i
$117$		0			0
$118$	−16.2481	−	28.1425i	−0.137696	−	0.238496i
$119$			97.4885i			0.819231i
$120$		0			0
$121$	−143.000			−1.18182
$122$	−100.459	+	58.0000i	−0.823434	+	0.475410i
$123$		0			0
$124$	−120.000	−	69.2820i	−0.967742	−	0.558726i
$125$		0			0
$126$		0			0
$127$			9.38083i			0.0738648i	0.999318	+	0.0369324i	$0.0117586\pi$
$127$			9.38083i			0.0738648i	−0.999318	+	0.0369324i	$0.988241\pi$
$128$	−110.851	+	64.0000i	−0.866025	+	0.500000i
$129$		0			0
$130$		0			0
$131$		−	178.729i		−	1.36434i	−0.731193	−	0.682171i	$-0.761036\pi$
$131$		−	178.729i		−	1.36434i	0.731193	−	0.682171i	$-0.238964\pi$
$132$		0			0
$133$	194.977			1.46599
$134$	−18.7617	−	32.4962i	−0.140012	−	0.242509i
$135$		0			0
$136$		−	83.1384i		−	0.611312i
$137$	211.310			1.54241			0.771205	−	0.636587i	$-0.219654\pi$
$137$	211.310			1.54241			0.771205	+	0.636587i	$0.219654\pi$
$138$		0			0
$139$		−	90.0666i		−	0.647961i	−0.946064	−	0.323981i	$-0.894979\pi$
$139$		−	90.0666i		−	0.647961i	0.946064	−	0.323981i	$-0.105021\pi$
$140$		0			0
$141$		0			0
$142$	97.4885	+	168.855i	0.686538	+	1.18912i
$143$			264.000i			1.84615i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	97.4885	−	168.855i	0.658706	−	1.14091i
$149$	234.521			1.57397			0.786983	−	0.616975i	$-0.211642\pi$
$149$	234.521			1.57397			0.786983	+	0.616975i	$0.211642\pi$
$150$		0			0
$151$		−	48.4974i		−	0.321175i	−0.987022	−	0.160587i	$-0.948661\pi$
$151$		−	48.4974i		−	0.321175i	0.987022	−	0.160587i	$-0.0513389\pi$
$152$	−166.277			−1.09393
$153$		0			0
$154$	264.000	−	152.420i	1.71429	−	0.989743i
$155$		0			0
$156$		0			0
$157$	16.2481			0.103491			0.0517455	−	0.998660i	$-0.483522\pi$
$157$	16.2481			0.103491			0.0517455	+	0.998660i	$0.483522\pi$
$158$	6.92820	+	12.0000i	0.0438494	+	0.0759494i
$159$		0			0
$160$		0			0
$161$	262.663			1.63145
$162$		0			0
$163$		−	206.378i		−	1.26612i	−0.774101	−	0.633062i	$-0.781798\pi$
$163$		−	206.378i		−	1.26612i	0.774101	−	0.633062i	$-0.218202\pi$
$164$	37.5233	−	64.9923i	0.228801	−	0.396295i
$165$		0			0
$166$	32.0000	+	55.4256i	0.192771	+	0.333889i
$167$			244.000i			1.46108i	0.682871	+	0.730539i	$0.260731\pi$
$167$			244.000i			1.46108i	−0.682871	+	0.730539i	$0.739269\pi$
$168$		0			0
$169$	95.0000			0.562130
$170$		0			0
$171$		0			0
$172$	−129.985	−	75.0467i	−0.755725	−	0.436318i
$173$	38.1051			0.220261			0.110130	−	0.993917i	$-0.464873\pi$
$173$	38.1051			0.220261			0.110130	+	0.993917i	$0.464873\pi$
$174$		0			0
$175$		0			0
$176$	−225.140	+	129.985i	−1.27920	+	0.738549i
$177$		0			0
$178$	−129.985	+	75.0467i	−0.730251	+	0.421610i
$179$		−	146.233i		−	0.816942i	−0.912771	−	0.408471i	$-0.866062\pi$
$179$		−	146.233i		−	0.816942i	0.912771	−	0.408471i	$-0.133938\pi$
$180$		0			0
$181$	230.000			1.27072			0.635359	−	0.772217i	$-0.280852\pi$
$181$	230.000			1.27072			0.635359	+	0.772217i	$0.280852\pi$
$182$	−152.420	−	264.000i	−0.837475	−	1.45055i
$183$		0			0
$184$	−224.000			−1.21739
$185$		0			0
$186$		0			0
$187$		−	168.855i		−	0.902968i
$188$	13.8564	+	8.00000i	0.0737043	+	0.0425532i
$189$		0			0
$190$		0			0
$191$			32.4962i			0.170137i	0.996375	+	0.0850685i	$0.0271109\pi$
$191$			32.4962i			0.170137i	−0.996375	+	0.0850685i	$0.972889\pi$
$192$		0			0
$193$	−162.481			−0.841869			−0.420935	−	0.907091i	$-0.638298\pi$
$193$	−162.481			−0.841869			−0.420935	+	0.907091i	$0.638298\pi$
$194$	−281.425	+	162.481i	−1.45064	+	0.837530i
$195$		0			0
$196$	−78.0000	+	135.100i	−0.397959	+	0.689286i
$197$	342.946			1.74084			0.870421	−	0.492307i	$-0.163846\pi$
$197$	342.946			1.74084			0.870421	+	0.492307i	$0.163846\pi$
$198$		0			0
$199$		−	20.7846i		−	0.104445i	−0.998635	−	0.0522226i	$-0.983369\pi$
$199$		−	20.7846i		−	0.104445i	0.998635	−	0.0522226i	$-0.0166305\pi$
$200$		0			0
$201$		0			0
$202$	−211.225	+	121.951i	−1.04567	+	0.603717i
$203$			88.0000i			0.433498i
$204$		0			0
$205$		0			0
$206$	159.474	+	276.217i	0.774146	+	1.34086i
$207$		0			0
$208$	129.985	+	225.140i	0.624926	+	1.08240i
$209$	−337.710			−1.61584
$210$		0			0
$211$			297.913i			1.41191i	0.708257	+	0.705954i	$0.249482\pi$
$211$			297.913i			1.41191i	−0.708257	+	0.705954i	$0.750518\pi$
$212$	62.3538	−	108.000i	0.294122	−	0.509434i
$213$		0			0
$214$	184.000	+	318.697i	0.859813	+	1.48924i
$215$		0			0
$216$		0			0
$217$	324.962			1.49752
$218$	38.1051	−	22.0000i	0.174794	−	0.100917i
$219$		0			0
$220$		0			0
$221$	−168.855			−0.764050
$222$		0			0
$223$			159.474i			0.715131i	0.933888	+	0.357565i	$0.116393\pi$
$223$			159.474i			0.715131i	−0.933888	+	0.357565i	$0.883607\pi$
$224$	150.093	−	259.969i	0.670059	−	1.16058i
$225$		0			0
$226$	150.000	−	86.6025i	0.663717	−	0.383197i
$227$		−	128.000i		−	0.563877i	−0.959433	−	0.281938i	$-0.909023\pi$
$227$		−	128.000i		−	0.563877i	0.959433	−	0.281938i	$-0.0909774\pi$
$228$		0			0
$229$	−134.000			−0.585153			−0.292576	−	0.956242i	$-0.594513\pi$
$229$	−134.000			−0.585153			−0.292576	+	0.956242i	$0.594513\pi$
$230$		0			0
$231$		0			0
$232$		−	75.0467i		−	0.323477i
$233$	301.377			1.29346			0.646731	−	0.762718i	$-0.276135\pi$
$233$	301.377			1.29346			0.646731	+	0.762718i	$0.276135\pi$
$234$		0			0
$235$		0			0
$236$	−56.2850	−	32.4962i	−0.238496	−	0.137696i
$237$		0			0
$238$	97.4885	+	168.855i	0.409615	+	0.709475i
$239$		−	162.481i		−	0.679836i	−0.940455	−	0.339918i	$-0.889601\pi$
$239$		−	162.481i		−	0.679836i	0.940455	−	0.339918i	$-0.110399\pi$
$240$		0			0
$241$	50.0000			0.207469			0.103734	−	0.994605i	$-0.466921\pi$
$241$	50.0000			0.207469			0.103734	+	0.994605i	$0.466921\pi$
$242$	−247.683	+	143.000i	−1.02348	+	0.590909i
$243$		0			0
$244$	−116.000	+	200.918i	−0.475410	+	0.823434i
$245$		0			0
$246$		0			0
$247$			337.710i			1.36725i
$248$	−277.128			−1.11745
$249$		0			0
$250$		0			0
$251$			146.233i			0.582600i	0.956632	+	0.291300i	$0.0940878\pi$
$251$			146.233i			0.582600i	−0.956632	+	0.291300i	$0.905912\pi$
$252$		0			0
$253$	−454.946			−1.79821
$254$	9.38083	+	16.2481i	0.0369324	+	0.0639688i
$255$		0			0
$256$	−128.000	+	221.703i	−0.500000	+	0.866025i
$257$	−24.2487			−0.0943530			−0.0471765	−	0.998887i	$-0.515022\pi$
$257$	−24.2487			−0.0943530			−0.0471765	+	0.998887i	$0.515022\pi$
$258$		0			0
$259$			457.261i			1.76549i
$260$		0			0
$261$		0			0
$262$	−178.729	−	309.567i	−0.682171	−	1.18156i
$263$		−	340.000i		−	1.29278i	−0.763009	−	0.646388i	$-0.776279\pi$
$263$		−	340.000i		−	1.29278i	0.763009	−	0.646388i	$-0.223721\pi$
$264$		0			0
$265$		0			0
$266$	337.710	−	194.977i	1.26959	−	0.732996i
$267$		0			0
$268$	−64.9923	−	37.5233i	−0.242509	−	0.140012i
$269$	178.236			0.662587			0.331293	−	0.943528i	$-0.392515\pi$
$269$	178.236			0.662587			0.331293	+	0.943528i	$0.392515\pi$
$270$		0			0
$271$			394.908i			1.45722i	0.684927	+	0.728612i	$0.259834\pi$
$271$			394.908i			1.45722i	−0.684927	+	0.728612i	$0.740166\pi$
$272$	−83.1384	−	144.000i	−0.305656	−	0.529412i
$273$		0			0
$274$	366.000	−	211.310i	1.33577	−	0.771205i
$275$		0			0
$276$		0			0
$277$	−113.737			−0.410601			−0.205301	−	0.978699i	$-0.565817\pi$
$277$	−113.737			−0.410601			−0.205301	+	0.978699i	$0.565817\pi$
$278$	−90.0666	−	156.000i	−0.323981	−	0.561151i
$279$		0			0
$280$		0			0
$281$	75.0467			0.267070			0.133535	−	0.991044i	$-0.457367\pi$
$281$	75.0467			0.267070			0.133535	+	0.991044i	$0.457367\pi$
$282$		0			0
$283$		−	318.948i		−	1.12703i	−0.826107	−	0.563513i	$-0.809450\pi$
$283$		−	318.948i		−	1.12703i	0.826107	−	0.563513i	$-0.190550\pi$
$284$	337.710	+	194.977i	1.18912	+	0.686538i
$285$		0			0
$286$	264.000	+	457.261i	0.923077	+	1.59882i
$287$			176.000i			0.613240i
$288$		0			0
$289$	−181.000			−0.626298
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−349.874			−1.19411			−0.597055	−	0.802200i	$-0.703663\pi$
$293$	−349.874			−1.19411			−0.597055	+	0.802200i	$0.703663\pi$
$294$		0			0
$295$		0			0
$296$		−	389.954i		−	1.31741i
$297$		0			0
$298$	406.202	−	234.521i	1.36309	−	0.786983i
$299$			454.946i			1.52156i
$300$		0			0
$301$	352.000			1.16944
$302$	−48.4974	−	84.0000i	−0.160587	−	0.278146i
$303$		0			0
$304$	−288.000	+	166.277i	−0.947368	+	0.546963i
$305$		0			0
$306$		0			0
$307$			262.663i			0.855581i	0.903878	+	0.427790i	$0.140708\pi$
$307$			262.663i			0.855581i	−0.903878	+	0.427790i	$0.859292\pi$
$308$	304.841	−	528.000i	0.989743	−	1.71429i
$309$		0			0
$310$		0			0
$311$		−	32.4962i		−	0.104489i	−0.998634	−	0.0522446i	$-0.983362\pi$
$311$		−	32.4962i		−	0.104489i	0.998634	−	0.0522446i	$-0.0166376\pi$
$312$		0			0
$313$	357.458			1.14204			0.571019	−	0.820937i	$-0.306548\pi$
$313$	357.458			1.14204			0.571019	+	0.820937i	$0.306548\pi$
$314$	28.1425	−	16.2481i	0.0896258	−	0.0517455i
$315$		0			0
$316$	24.0000	+	13.8564i	0.0759494	+	0.0438494i
$317$	45.0333			0.142061			0.0710305	−	0.997474i	$-0.477371\pi$
$317$	45.0333			0.142061			0.0710305	+	0.997474i	$0.477371\pi$
$318$		0			0
$319$		−	152.420i		−	0.477807i
$320$		0			0
$321$		0			0
$322$	454.946	−	262.663i	1.41288	−	0.815724i
$323$		−	216.000i		−	0.668731i
$324$		0			0
$325$		0			0
$326$	−206.378	−	357.458i	−0.633062	−	1.09650i
$327$		0			0
$328$		−	150.093i		−	0.457602i
$329$	−37.5233			−0.114053
$330$		0			0
$331$			325.626i			0.983763i	0.870662	+	0.491881i	$0.163691\pi$
$331$			325.626i			0.983763i	−0.870662	+	0.491881i	$0.836309\pi$
$332$	110.851	+	64.0000i	0.333889	+	0.192771i
$333$		0			0
$334$	244.000	+	422.620i	0.730539	+	1.26533i
$335$		0			0
$336$		0			0
$337$	64.9923			0.192856			0.0964278	−	0.995340i	$-0.469258\pi$
$337$	64.9923			0.192856			0.0964278	+	0.995340i	$0.469258\pi$
$338$	164.545	−	95.0000i	0.486819	−	0.281065i
$339$		0			0
$340$		0			0
$341$	−562.850			−1.65059
$342$		0			0
$343$			93.8083i			0.273494i
$344$	−300.187			−0.872635
$345$		0			0
$346$	66.0000	−	38.1051i	0.190751	−	0.110130i
$347$		−	104.000i		−	0.299712i	−0.988708	−	0.149856i	$-0.952119\pi$
$347$		−	104.000i		−	0.299712i	0.988708	−	0.149856i	$-0.0478810\pi$
$348$		0			0
$349$	598.000			1.71347			0.856734	−	0.515759i	$-0.172490\pi$
$349$	598.000			1.71347			0.856734	+	0.515759i	$0.172490\pi$
$350$		0			0
$351$		0			0
$352$	−259.969	+	450.280i	−0.738549	+	1.27920i
$353$	79.6743			0.225706			0.112853	−	0.993612i	$-0.464001\pi$
$353$	79.6743			0.225706			0.112853	+	0.993612i	$0.464001\pi$
$354$		0			0
$355$		0			0
$356$	−150.093	+	259.969i	−0.421610	+	0.730251i
$357$		0			0
$358$	−146.233	−	253.282i	−0.408471	−	0.707493i
$359$			194.977i			0.543111i	0.962423	+	0.271556i	$0.0875381\pi$
$359$			194.977i			0.543111i	−0.962423	+	0.271556i	$0.912462\pi$
$360$		0			0
$361$	−71.0000			−0.196676
$362$	398.372	−	230.000i	1.10047	−	0.635359i
$363$		0			0
$364$	−528.000	−	304.841i	−1.45055	−	0.837475i
$365$		0			0
$366$		0			0
$367$		−	553.469i		−	1.50809i	−0.656823	−	0.754045i	$-0.728100\pi$
$367$		−	553.469i		−	1.50809i	0.656823	−	0.754045i	$-0.271900\pi$
$368$	−387.979	+	224.000i	−1.05429	+	0.608696i
$369$		0			0
$370$		0			0
$371$			292.465i			0.788316i
$372$		0			0
$373$	276.217			0.740529			0.370264	−	0.928926i	$-0.379267\pi$
$373$	276.217			0.740529			0.370264	+	0.928926i	$0.379267\pi$
$374$	−168.855	−	292.465i	−0.451484	−	0.781993i
$375$		0			0
$376$	32.0000			0.0851064
$377$	−152.420			−0.404298
$378$		0			0
$379$		−	103.923i		−	0.274203i	−0.990557	−	0.137102i	$-0.956221\pi$
$379$		−	103.923i		−	0.274203i	0.990557	−	0.137102i	$-0.0437787\pi$
$380$		0			0
$381$		0			0
$382$	32.4962	+	56.2850i	0.0850685	+	0.147343i
$383$		−	220.000i		−	0.574413i	−0.957869	−	0.287206i	$-0.907273\pi$
$383$		−	220.000i		−	0.574413i	0.957869	−	0.287206i	$-0.0927265\pi$
$384$		0			0
$385$		0			0
$386$	−281.425	+	162.481i	−0.729080	+	0.420935i
$387$		0			0
$388$	−324.962	+	562.850i	−0.837530	+	1.45064i
$389$	347.091			0.892264			0.446132	−	0.894967i	$-0.352801\pi$
$389$	347.091			0.892264			0.446132	+	0.894967i	$0.352801\pi$
$390$		0			0
$391$		−	290.985i		−	0.744206i
$392$			312.000i			0.795918i
$393$		0			0
$394$	594.000	−	342.946i	1.50761	−	0.870421i
$395$		0			0
$396$		0			0
$397$	−341.210			−0.859470			−0.429735	−	0.902955i	$-0.641393\pi$
$397$	−341.210			−0.859470			−0.429735	+	0.902955i	$0.641393\pi$
$398$	−20.7846	−	36.0000i	−0.0522226	−	0.0904523i
$399$		0			0
$400$		0			0
$401$	581.612			1.45040			0.725201	−	0.688537i	$-0.241747\pi$
$401$	581.612			1.45040			0.725201	+	0.688537i	$0.241747\pi$
$402$		0			0
$403$			562.850i			1.39665i
$404$	−243.902	+	422.450i	−0.603717	+	1.04567i
$405$		0			0
$406$	88.0000	+	152.420i	0.216749	+	0.375420i
$407$		−	792.000i		−	1.94595i
$408$		0			0
$409$	−122.000			−0.298289			−0.149144	−	0.988815i	$-0.547652\pi$
$409$	−122.000			−0.298289			−0.149144	+	0.988815i	$0.547652\pi$
$410$		0			0
$411$		0			0
$412$	552.435	+	318.948i	1.34086	+	0.774146i
$413$	152.420			0.369057
$414$		0			0
$415$		0			0
$416$	450.280	+	259.969i	1.08240	+	0.624926i
$417$		0			0
$418$	−584.931	+	337.710i	−1.39936	+	0.807919i
$419$		−	503.690i		−	1.20213i	−0.799202	−	0.601063i	$-0.794744\pi$
$419$		−	503.690i		−	1.20213i	0.799202	−	0.601063i	$-0.205256\pi$
$420$		0			0
$421$	74.0000			0.175772			0.0878860	−	0.996131i	$-0.471989\pi$
$421$	74.0000			0.175772			0.0878860	+	0.996131i	$0.471989\pi$
$422$	297.913	+	516.000i	0.705954	+	1.22275i
$423$		0			0
$424$		−	249.415i		−	0.588244i
$425$		0			0
$426$		0			0
$427$		−	544.088i		−	1.27421i
$428$	637.395	+	368.000i	1.48924	+	0.859813i
$429$		0			0
$430$		0			0
$431$		−	779.908i		−	1.80953i	−0.425911	−	0.904765i	$-0.640046\pi$
$431$		−	779.908i		−	1.80953i	0.425911	−	0.904765i	$-0.359954\pi$
$432$		0			0
$433$	97.4885			0.225147			0.112573	−	0.993643i	$-0.464091\pi$
$433$	97.4885			0.225147			0.112573	+	0.993643i	$0.464091\pi$
$434$	562.850	−	324.962i	1.29689	−	0.748759i
$435$		0			0
$436$	44.0000	−	76.2102i	0.100917	−	0.174794i
$437$	−581.969			−1.33174
$438$		0			0
$439$			339.482i			0.773307i	0.922225	+	0.386654i	$0.126369\pi$
$439$			339.482i			0.773307i	−0.922225	+	0.386654i	$0.873631\pi$
$440$		0			0
$441$		0			0
$442$	−292.465	+	168.855i	−0.661686	+	0.382025i
$443$			176.000i			0.397291i	0.980071	+	0.198646i	$0.0636543\pi$
$443$			176.000i			0.397291i	−0.980071	+	0.198646i	$0.936346\pi$
$444$		0			0
$445$		0			0
$446$	159.474	+	276.217i	0.357565	+	0.619321i
$447$		0			0
$448$		−	600.373i		−	1.34012i
$449$	−469.042			−1.04464			−0.522318	−	0.852751i	$-0.674933\pi$
$449$	−469.042			−1.04464			−0.522318	+	0.852751i	$0.674933\pi$
$450$		0			0
$451$		−	304.841i		−	0.675922i
$452$	173.205	−	300.000i	0.383197	−	0.663717i
$453$		0			0
$454$	−128.000	−	221.703i	−0.281938	−	0.488332i
$455$		0			0
$456$		0			0
$457$	−357.458			−0.782183			−0.391092	−	0.920352i	$-0.627902\pi$
$457$	−357.458			−0.782183			−0.391092	+	0.920352i	$0.627902\pi$
$458$	−232.095	+	134.000i	−0.506757	+	0.292576i
$459$		0			0
$460$		0			0
$461$	497.184			1.07849			0.539245	−	0.842149i	$-0.318710\pi$
$461$	497.184			1.07849			0.539245	+	0.842149i	$0.318710\pi$
$462$		0			0
$463$		−	347.091i		−	0.749656i	−0.927094	−	0.374828i	$-0.877702\pi$
$463$		−	347.091i		−	0.749656i	0.927094	−	0.374828i	$-0.122298\pi$
$464$	−75.0467	−	129.985i	−0.161738	−	0.280139i
$465$		0			0
$466$	522.000	−	301.377i	1.12017	−	0.646731i
$467$			520.000i			1.11349i	0.830683	+	0.556745i	$0.187950\pi$
$467$			520.000i			1.11349i	−0.830683	+	0.556745i	$0.812050\pi$
$468$		0			0
$469$	176.000			0.375267
$470$		0			0
$471$		0			0
$472$	−129.985			−0.275391
$473$	−609.682			−1.28897
$474$		0			0
$475$		0			0
$476$	337.710	+	194.977i	0.709475	+	0.409615i
$477$		0			0
$478$	−162.481	−	281.425i	−0.339918	−	0.588755i
$479$		−	812.404i		−	1.69604i	−0.529963	−	0.848021i	$-0.677794\pi$
$479$		−	812.404i		−	1.69604i	0.529963	−	0.848021i	$-0.322206\pi$
$480$		0			0
$481$	−792.000			−1.64657
$482$	86.6025	−	50.0000i	0.179673	−	0.103734i
$483$		0			0
$484$	−286.000	+	495.367i	−0.590909	+	1.02348i
$485$		0			0
$486$		0			0
$487$			290.806i			0.597137i	0.954388	+	0.298569i	$0.0965092\pi$
$487$			290.806i			0.597137i	−0.954388	+	0.298569i	$0.903491\pi$
$488$			464.000i			0.950820i
$489$		0			0
$490$		0			0
$491$			406.202i			0.827295i	0.910437	+	0.413648i	$0.135745\pi$
$491$			406.202i			0.827295i	−0.910437	+	0.413648i	$0.864255\pi$
$492$		0			0
$493$	97.4885			0.197745
$494$	337.710	+	584.931i	0.683623	+	1.18407i
$495$		0			0
$496$	−480.000	+	277.128i	−0.967742	+	0.558726i
$497$	−914.523			−1.84009
$498$		0			0
$499$		−	838.313i		−	1.67999i	−0.542598	−	0.839993i	$-0.682559\pi$
$499$		−	838.313i		−	1.67999i	0.542598	−	0.839993i	$-0.317441\pi$
$500$		0			0
$501$		0			0
$502$	146.233	+	253.282i	0.291300	+	0.504547i
$503$			932.000i			1.85288i	0.376439	+	0.926441i	$0.377148\pi$
$503$			932.000i			1.85288i	−0.376439	+	0.926441i	$0.622852\pi$
$504$		0			0
$505$		0			0
$506$	−787.990	+	454.946i	−1.55729	+	0.899103i
$507$		0			0
$508$	32.4962	+	18.7617i	0.0639688	+	0.0369324i
$509$	−440.899			−0.866206			−0.433103	−	0.901344i	$-0.642581\pi$
$509$	−440.899			−0.866206			−0.433103	+	0.901344i	$0.642581\pi$
$510$		0			0
$511$		0			0
$512$			512.000i			1.00000i
$513$		0			0
$514$	−42.0000	+	24.2487i	−0.0817121	+	0.0471765i
$515$		0			0
$516$		0			0
$517$	64.9923			0.125710
$518$	457.261	+	792.000i	0.882744	+	1.52896i
$519$		0			0
$520$		0			0
$521$	−431.518			−0.828250			−0.414125	−	0.910220i	$-0.635912\pi$
$521$	−431.518			−0.828250			−0.414125	+	0.910220i	$0.635912\pi$
$522$		0			0
$523$			243.902i			0.466351i	0.972435	+	0.233176i	$0.0749117\pi$
$523$			243.902i			0.466351i	−0.972435	+	0.233176i	$0.925088\pi$
$524$	−619.135	−	357.458i	−1.18156	−	0.682171i
$525$		0			0
$526$	−340.000	−	588.897i	−0.646388	−	1.11958i
$527$		−	360.000i		−	0.683112i
$528$		0			0
$529$	−255.000			−0.482042
$530$		0			0
$531$		0			0
$532$	389.954	−	675.420i	0.732996	−	1.26959i
$533$	−304.841			−0.571934
$534$		0			0
$535$		0			0
$536$	−150.093			−0.280025
$537$		0			0
$538$	308.713	−	178.236i	0.573817	−	0.331293i
$539$			633.675i			1.17565i
$540$		0			0
$541$	374.000			0.691312			0.345656	−	0.938361i	$-0.387656\pi$
$541$	374.000			0.691312			0.345656	+	0.938361i	$0.387656\pi$
$542$	394.908	+	684.000i	0.728612	+	1.26199i
$543$		0			0
$544$	−288.000	−	166.277i	−0.529412	−	0.305656i
$545$		0			0
$546$		0			0
$547$			938.083i			1.71496i	0.514517	+	0.857480i	$0.327971\pi$
$547$			938.083i			1.71496i	−0.514517	+	0.857480i	$0.672029\pi$
$548$	422.620	−	732.000i	0.771205	−	1.33577i
$549$		0			0
$550$		0			0
$551$		−	194.977i		−	0.353860i
$552$		0			0
$553$	−64.9923			−0.117527
$554$	−196.997	+	113.737i	−0.355591	+	0.205301i
$555$		0			0
$556$	−312.000	−	180.133i	−0.561151	−	0.323981i
$557$	−426.084			−0.764963			−0.382482	−	0.923963i	$-0.624930\pi$
$557$	−426.084			−0.764963			−0.382482	+	0.923963i	$0.624930\pi$
$558$		0			0
$559$			609.682i			1.09067i
$560$		0			0
$561$		0			0
$562$	129.985	−	75.0467i	0.231289	−	0.133535i
$563$		−	136.000i		−	0.241563i	−0.992679	−	0.120782i	$-0.961460\pi$
$563$		−	136.000i		−	0.241563i	0.992679	−	0.120782i	$-0.0385400\pi$
$564$		0			0
$565$		0			0
$566$	−318.948	−	552.435i	−0.563513	−	0.976033i
$567$		0			0
$568$	779.908			1.37308
$569$	881.798			1.54973			0.774867	−	0.632125i	$-0.217817\pi$
$569$	881.798			1.54973			0.774867	+	0.632125i	$0.217817\pi$
$570$		0			0
$571$		−	949.164i		−	1.66228i	−0.556061	−	0.831142i	$-0.687688\pi$
$571$		−	949.164i		−	1.66228i	0.556061	−	0.831142i	$-0.312312\pi$
$572$	914.523	+	528.000i	1.59882	+	0.923077i
$573$		0			0
$574$	176.000	+	304.841i	0.306620	+	0.531082i
$575$		0			0
$576$		0			0
$577$	389.954			0.675830			0.337915	−	0.941177i	$-0.390278\pi$
$577$	389.954			0.675830			0.337915	+	0.941177i	$0.390278\pi$
$578$	−313.501	+	181.000i	−0.542390	+	0.313149i
$579$		0			0
$580$		0			0
$581$	−300.187			−0.516672
$582$		0			0
$583$		−	506.565i		−	0.868893i
$584$		0			0
$585$		0			0
$586$	−606.000	+	349.874i	−1.03413	+	0.597055i
$587$		−	680.000i		−	1.15843i	−0.815174	−	0.579216i	$-0.803359\pi$
$587$		−	680.000i		−	1.15843i	0.815174	−	0.579216i	$-0.196641\pi$
$588$		0			0
$589$	−720.000			−1.22241
$590$		0			0
$591$		0			0
$592$	−389.954	−	675.420i	−0.658706	−	1.14091i
$593$	114.315			0.192775			0.0963873	−	0.995344i	$-0.469271\pi$
$593$	114.315			0.192775			0.0963873	+	0.995344i	$0.469271\pi$
$594$		0			0
$595$		0			0
$596$	469.042	−	812.404i	0.786983	−	1.36309i
$597$		0			0
$598$	454.946	+	787.990i	0.760780	+	1.31771i
$599$			1104.87i			1.84452i	0.386567	+	0.922261i	$0.373661\pi$
$599$			1104.87i			1.84452i	−0.386567	+	0.922261i	$0.626339\pi$
$600$		0			0
$601$	−682.000			−1.13478			−0.567388	−	0.823451i	$-0.692046\pi$
$601$	−682.000			−1.13478			−0.567388	+	0.823451i	$0.692046\pi$
$602$	609.682	−	352.000i	1.01276	−	0.584718i
$603$		0			0
$604$	−168.000	−	96.9948i	−0.278146	−	0.160587i
$605$		0			0
$606$		0			0
$607$		−	272.044i		−	0.448178i	−0.974569	−	0.224089i	$-0.928059\pi$
$607$		−	272.044i		−	0.448178i	0.974569	−	0.224089i	$-0.0719407\pi$
$608$	−332.554	+	576.000i	−0.546963	+	0.947368i
$609$		0			0
$610$		0			0
$611$		−	64.9923i		−	0.106370i
$612$		0			0
$613$	−341.210			−0.556623			−0.278311	−	0.960491i	$-0.589775\pi$
$613$	−341.210			−0.556623			−0.278311	+	0.960491i	$0.589775\pi$
$614$	262.663	+	454.946i	0.427790	+	0.740955i
$615$		0			0
$616$		−	1219.36i		−	1.97949i
$617$	−336.018			−0.544599			−0.272300	−	0.962212i	$-0.587784\pi$
$617$	−336.018			−0.544599			−0.272300	+	0.962212i	$0.587784\pi$
$618$		0			0
$619$			381.051i			0.615592i	0.951452	+	0.307796i	$0.0995914\pi$
$619$			381.051i			0.615592i	−0.951452	+	0.307796i	$0.900409\pi$
$620$		0			0
$621$		0			0
$622$	−32.4962	−	56.2850i	−0.0522446	−	0.0904903i
$623$		−	704.000i		−	1.13002i
$624$		0			0
$625$		0			0
$626$	619.135	−	357.458i	0.989033	−	0.571019i
$627$		0			0
$628$	32.4962	−	56.2850i	0.0517455	−	0.0896258i
$629$	506.565			0.805350
$630$		0			0
$631$			145.492i			0.230574i	0.993332	+	0.115287i	$0.0367788\pi$
$631$			145.492i			0.230574i	−0.993332	+	0.115287i	$0.963221\pi$
$632$	55.4256			0.0876988
$633$		0			0
$634$	78.0000	−	45.0333i	0.123028	−	0.0710305i
$635$		0			0
$636$		0			0
$637$	633.675			0.994780
$638$	−152.420	−	264.000i	−0.238904	−	0.413793i
$639$		0			0
$640$		0			0
$641$	−431.518			−0.673195			−0.336598	−	0.941649i	$-0.609276\pi$
$641$	−431.518			−0.673195			−0.336598	+	0.941649i	$0.609276\pi$
$642$		0			0
$643$			356.472i			0.554388i	0.960814	+	0.277194i	$0.0894045\pi$
$643$			356.472i			0.554388i	−0.960814	+	0.277194i	$0.910595\pi$
$644$	525.327	−	909.892i	0.815724	−	1.41288i
$645$		0			0
$646$	−216.000	−	374.123i	−0.334365	−	0.579138i
$647$			364.000i			0.562597i	0.959620	+	0.281298i	$0.0907650\pi$
$647$			364.000i			0.562597i	−0.959620	+	0.281298i	$0.909235\pi$
$648$		0			0
$649$	−264.000			−0.406780
$650$		0			0
$651$		0			0
$652$	−714.915	−	412.757i	−1.09650	−	0.633062i
$653$	−162.813			−0.249330			−0.124665	−	0.992199i	$-0.539786\pi$
$653$	−162.813			−0.249330			−0.124665	+	0.992199i	$0.539786\pi$
$654$		0			0
$655$		0			0
$656$	−150.093	−	259.969i	−0.228801	−	0.396295i
$657$		0			0
$658$	−64.9923	+	37.5233i	−0.0987725	+	0.0570263i
$659$		−	178.729i		−	0.271212i	−0.990763	−	0.135606i	$-0.956702\pi$
$659$		−	178.729i		−	0.271212i	0.990763	−	0.135606i	$-0.0432982\pi$
$660$		0			0
$661$	662.000			1.00151			0.500756	−	0.865588i	$-0.333055\pi$
$661$	662.000			1.00151			0.500756	+	0.865588i	$0.333055\pi$
$662$	325.626	+	564.000i	0.491881	+	0.851964i
$663$		0			0
$664$	256.000			0.385542
$665$		0			0
$666$		0			0
$667$		−	262.663i		−	0.393798i
$668$	845.241	+	488.000i	1.26533	+	0.730539i
$669$		0			0
$670$		0			0
$671$			942.388i			1.40445i
$672$		0			0
$673$	−1202.36			−1.78656			−0.893282	−	0.449497i	$-0.851603\pi$
$673$	−1202.36			−1.78656			−0.893282	+	0.449497i	$0.851603\pi$
$674$	112.570	−	64.9923i	0.167018	−	0.0964278i
$675$		0			0
$676$	190.000	−	329.090i	0.281065	−	0.486819i
$677$	−495.367			−0.731708			−0.365854	−	0.930672i	$-0.619223\pi$
$677$	−495.367			−0.731708			−0.365854	+	0.930672i	$0.619223\pi$
$678$		0			0
$679$		−	1524.20i		−	2.24478i
$680$		0			0
$681$		0			0
$682$	−974.885	+	562.850i	−1.42945	+	0.825293i
$683$		−	88.0000i		−	0.128843i	−0.997923	−	0.0644217i	$-0.979480\pi$
$683$		−	88.0000i		−	0.128843i	0.997923	−	0.0644217i	$-0.0205203\pi$
$684$		0			0
$685$		0			0
$686$	93.8083	+	162.481i	0.136747	+	0.236852i
$687$		0			0
$688$	−519.938	+	300.187i	−0.755725	+	0.436318i
$689$	−506.565			−0.735218
$690$		0			0
$691$		−	339.482i		−	0.491291i	−0.969360	−	0.245645i	$-0.921000\pi$
$691$		−	339.482i		−	0.491291i	0.969360	−	0.245645i	$-0.0789999\pi$
$692$	76.2102	−	132.000i	0.110130	−	0.190751i
$693$		0			0
$694$	−104.000	−	180.133i	−0.149856	−	0.259558i
$695$		0			0
$696$		0			0
$697$	194.977			0.279737
$698$	1035.77	−	598.000i	1.48391	−	0.856734i
$699$		0			0
$700$		0			0
$701$	834.894			1.19100			0.595502	−	0.803354i	$-0.296953\pi$
$701$	834.894			1.19100			0.595502	+	0.803354i	$0.296953\pi$
$702$		0			0
$703$		−	1013.13i		−	1.44115i
$704$			1039.88i			1.47710i
$705$		0			0
$706$	138.000	−	79.6743i	0.195467	−	0.112853i
$707$		−	1144.00i		−	1.61810i
$708$		0			0
$709$	490.000			0.691114			0.345557	−	0.938398i	$-0.387690\pi$
$709$	490.000			0.691114			0.345557	+	0.938398i	$0.387690\pi$
$710$		0			0
$711$		0			0
$712$			600.373i			0.843221i
$713$	−969.948			−1.36038
$714$		0			0
$715$		0			0
$716$	−506.565	−	292.465i	−0.707493	−	0.408471i
$717$		0			0
$718$	194.977	+	337.710i	0.271556	+	0.470348i
$719$		−	584.931i		−	0.813534i	−0.913532	−	0.406767i	$-0.866656\pi$
$719$		−	584.931i		−	0.813534i	0.913532	−	0.406767i	$-0.133344\pi$
$720$		0			0
$721$	−1496.00			−2.07490
$722$	−122.976	+	71.0000i	−0.170326	+	0.0983380i
$723$		0			0
$724$	460.000	−	796.743i	0.635359	−	1.10047i
$725$		0			0
$726$		0			0
$727$			178.236i			0.245166i	0.992458	+	0.122583i	$0.0391178\pi$
$727$			178.236i			0.245166i	−0.992458	+	0.122583i	$0.960882\pi$
$728$	−1219.36			−1.67495
$729$		0			0
$730$		0			0
$731$		−	389.954i		−	0.533453i
$732$		0			0
$733$	81.2404			0.110833			0.0554164	−	0.998463i	$-0.482351\pi$
$733$	81.2404			0.110833			0.0554164	+	0.998463i	$0.482351\pi$
$734$	−553.469	−	958.637i	−0.754045	−	1.30604i
$735$		0			0
$736$	−448.000	+	775.959i	−0.608696	+	1.05429i
$737$	−304.841			−0.413624
$738$		0			0
$739$		−	311.769i		−	0.421880i	−0.977499	−	0.210940i	$-0.932348\pi$
$739$		−	311.769i		−	0.421880i	0.977499	−	0.210940i	$-0.0676524\pi$
$740$		0			0
$741$		0			0
$742$	292.465	+	506.565i	0.394158	+	0.682702i
$743$		−	676.000i		−	0.909825i	−0.890536	−	0.454913i	$-0.849671\pi$
$743$		−	676.000i		−	0.909825i	0.890536	−	0.454913i	$-0.150329\pi$
$744$		0			0
$745$		0			0
$746$	478.422	−	276.217i	0.641317	−	0.370264i
$747$		0			0
$748$	−584.931	−	337.710i	−0.781993	−	0.451484i
$749$	−1726.07			−2.30450
$750$		0			0
$751$		−	1281.72i		−	1.70668i	−0.521354	−	0.853341i	$-0.674573\pi$
$751$		−	1281.72i		−	1.70668i	0.521354	−	0.853341i	$-0.325427\pi$
$752$	55.4256	−	32.0000i	0.0737043	−	0.0425532i
$753$		0			0
$754$	−264.000	+	152.420i	−0.350133	+	0.202149i
$755$		0			0
$756$		0			0
$757$	341.210			0.450739			0.225370	−	0.974273i	$-0.427641\pi$
$757$	341.210			0.450739			0.225370	+	0.974273i	$0.427641\pi$
$758$	−103.923	−	180.000i	−0.137102	−	0.237467i
$759$		0			0
$760$		0			0
$761$	−994.368			−1.30666			−0.653330	−	0.757073i	$-0.726629\pi$
$761$	−994.368			−1.30666			−0.653330	+	0.757073i	$0.726629\pi$
$762$		0			0
$763$			206.378i			0.270483i
$764$	112.570	+	64.9923i	0.147343	+	0.0850685i
$765$		0			0
$766$	−220.000	−	381.051i	−0.287206	−	0.497456i
$767$			264.000i			0.344198i
$768$		0			0
$769$	−314.000			−0.408322			−0.204161	−	0.978937i	$-0.565447\pi$
$769$	−314.000			−0.408322			−0.204161	+	0.978937i	$0.565447\pi$
$770$		0			0
$771$		0			0
$772$	−324.962	+	562.850i	−0.420935	+	0.729080i
$773$	904.131			1.16964			0.584819	−	0.811164i	$-0.301165\pi$
$773$	904.131			1.16964			0.584819	+	0.811164i	$0.301165\pi$
$774$		0			0
$775$		0			0
$776$			1299.85i			1.67506i
$777$		0			0
$778$	601.179	−	347.091i	0.772723	−	0.446132i
$779$		−	389.954i		−	0.500583i
$780$		0			0
$781$	1584.00			2.02817
$782$	−290.985	−	504.000i	−0.372103	−	0.644501i
$783$		0			0
$784$	312.000	+	540.400i	0.397959	+	0.689286i
$785$		0			0
$786$		0			0
$787$			431.518i			0.548308i	0.961686	+	0.274154i	$0.0883978\pi$
$787$			431.518i			0.548308i	−0.961686	+	0.274154i	$0.911602\pi$
$788$	685.892	−	1188.00i	0.870421	−	1.50761i
$789$		0			0
$790$		0			0
$791$			812.404i			1.02706i
$792$		0			0
$793$	942.388			1.18838
$794$	−590.992	+	341.210i	−0.744323	+	0.429735i
$795$		0			0
$796$	−72.0000	−	41.5692i	−0.0904523	−	0.0522226i
$797$	−1292.11			−1.62122			−0.810608	−	0.585589i	$-0.800863\pi$
$797$	−1292.11			−1.62122			−0.810608	+	0.585589i	$0.800863\pi$
$798$		0			0
$799$			41.5692i			0.0520266i
$800$		0			0
$801$		0			0
$802$	1007.38	−	581.612i	1.25609	−	0.725201i
$803$		0			0
$804$		0			0
$805$		0			0
$806$	562.850	+	974.885i	0.698325	+	1.20953i
$807$		0			0
$808$			975.606i			1.20743i
$809$	1219.51			1.50743			0.753713	−	0.657203i	$-0.228261\pi$
$809$	1219.51			1.50743			0.753713	+	0.657203i	$0.228261\pi$
$810$		0			0
$811$			1018.45i			1.25579i	0.778298	+	0.627895i	$0.216083\pi$
$811$			1018.45i			1.25579i	−0.778298	+	0.627895i	$0.783917\pi$
$812$	304.841	+	176.000i	0.375420	+	0.216749i
$813$		0			0
$814$	−792.000	−	1371.78i	−0.972973	−	1.68524i
$815$		0			0
$816$		0			0
$817$	−779.908			−0.954599
$818$	−211.310	+	122.000i	−0.258325	+	0.149144i
$819$		0			0
$820$		0			0
$821$	497.184			0.605584			0.302792	−	0.953057i	$-0.402081\pi$
$821$	497.184			0.605584			0.302792	+	0.953057i	$0.402081\pi$
$822$		0			0
$823$			440.899i			0.535722i	0.963458	+	0.267861i	$0.0863168\pi$
$823$			440.899i			0.535722i	−0.963458	+	0.267861i	$0.913683\pi$
$824$	1275.79			1.54829
$825$		0			0
$826$	264.000	−	152.420i	0.319613	−	0.184528i
$827$		−	752.000i		−	0.909311i	−0.890667	−	0.454655i	$-0.849762\pi$
$827$		−	752.000i		−	0.909311i	0.890667	−	0.454655i	$-0.150238\pi$
$828$		0			0
$829$	1030.00			1.24246			0.621230	−	0.783628i	$-0.286633\pi$
$829$	1030.00			1.24246			0.621230	+	0.783628i	$0.286633\pi$
$830$		0			0
$831$		0			0
$832$	1039.88			1.24985
$833$	−405.300			−0.486554
$834$		0			0
$835$		0			0
$836$	−675.420	+	1169.86i	−0.807919	+	1.39936i
$837$		0			0
$838$	−503.690	−	872.417i	−0.601063	−	1.04107i
$839$		−	519.938i		−	0.619712i	−0.950783	−	0.309856i	$-0.899719\pi$
$839$		−	519.938i		−	0.619712i	0.950783	−	0.309856i	$-0.100281\pi$
$840$		0			0
$841$	−753.000			−0.895363
$842$	128.172	−	74.0000i	0.152223	−	0.0878860i
$843$		0			0
$844$	1032.00	+	595.825i	1.22275	+	0.705954i
$845$		0			0
$846$		0			0
$847$		−	1341.46i		−	1.58378i
$848$	−249.415	−	432.000i	−0.294122	−	0.509434i
$849$		0			0
$850$		0			0
$851$		−	1364.84i		−	1.60381i
$852$		0			0
$853$	698.667			0.819071			0.409535	−	0.912294i	$-0.365691\pi$
$853$	698.667			0.819071			0.409535	+	0.912294i	$0.365691\pi$
$854$	−544.088	−	942.388i	−0.637106	−	1.10350i
$855$		0			0
$856$	1472.00			1.71963
$857$	897.202			1.04691			0.523455	−	0.852053i	$-0.324643\pi$
$857$	897.202			1.04691			0.523455	+	0.852053i	$0.324643\pi$
$858$		0			0
$859$			533.472i			0.621038i	0.950567	+	0.310519i	$0.100503\pi$
$859$			533.472i			0.621038i	−0.950567	+	0.310519i	$0.899497\pi$
$860$		0			0
$861$		0			0
$862$	−779.908	−	1350.84i	−0.904765	−	1.56710i
$863$		−	220.000i		−	0.254925i	−0.991843	−	0.127462i	$-0.959317\pi$
$863$		−	220.000i		−	0.254925i	0.991843	−	0.127462i	$-0.0406832\pi$
$864$		0			0
$865$		0			0
$866$	168.855	−	97.4885i	0.194983	−	0.112573i
$867$		0			0
$868$	649.923	−	1125.70i	0.748759	−	1.29689i
$869$	112.570			0.129540
$870$		0			0
$871$			304.841i			0.349990i
$872$		−	176.000i		−	0.201835i
$873$		0			0
$874$	−1008.00	+	581.969i	−1.15332	+	0.665869i
$875$		0			0
$876$		0			0
$877$	−763.660			−0.870764			−0.435382	−	0.900246i	$-0.643387\pi$
$877$	−763.660			−0.870764			−0.435382	+	0.900246i	$0.643387\pi$
$878$	339.482	+	588.000i	0.386654	+	0.669704i
$879$		0			0
$880$		0			0
$881$	1088.18			1.23516			0.617580	−	0.786508i	$-0.288113\pi$
$881$	1088.18			1.23516			0.617580	+	0.786508i	$0.288113\pi$
$882$		0			0
$883$			1257.03i			1.42359i	0.702386	+	0.711796i	$0.252118\pi$
$883$			1257.03i			1.42359i	−0.702386	+	0.711796i	$0.747882\pi$
$884$	−337.710	+	584.931i	−0.382025	+	0.661686i
$885$		0			0
$886$	176.000	+	304.841i	0.198646	+	0.344064i
$887$		−	140.000i		−	0.157835i	−0.996881	−	0.0789177i	$-0.974854\pi$
$887$		−	140.000i		−	0.157835i	0.996881	−	0.0789177i	$-0.0251464\pi$
$888$		0			0
$889$	−88.0000			−0.0989876
$890$		0			0
$891$		0			0
$892$	552.435	+	318.948i	0.619321	+	0.357565i
$893$	83.1384			0.0931002
$894$		0			0
$895$		0			0
$896$	−600.373	−	1039.88i	−0.670059	−	1.16058i
$897$		0			0
$898$	−812.404	+	469.042i	−0.904681	+	0.522318i
$899$		−	324.962i		−	0.361470i
$900$		0			0
$901$	324.000			0.359600
$902$	−304.841	−	528.000i	−0.337961	−	0.585366i
$903$		0			0
$904$		−	692.820i		−	0.766394i
$905$		0			0
$906$		0			0
$907$		−	75.0467i		−	0.0827416i	−0.999144	−	0.0413708i	$-0.986828\pi$
$907$		−	75.0467i		−	0.0827416i	0.999144	−	0.0413708i	$-0.0131725\pi$
$908$	−443.405	−	256.000i	−0.488332	−	0.281938i
$909$		0			0
$910$		0			0
$911$		−	649.923i		−	0.713417i	−0.934216	−	0.356709i	$-0.883899\pi$
$911$		−	649.923i		−	0.713417i	0.934216	−	0.356709i	$-0.116101\pi$
$912$		0			0
$913$	519.938			0.569484
$914$	−619.135	+	357.458i	−0.677390	+	0.391092i
$915$		0			0
$916$	−268.000	+	464.190i	−0.292576	+	0.506757i
$917$	1676.63			1.82838
$918$		0			0
$919$			1101.58i			1.19868i	0.800496	+	0.599339i	$0.204570\pi$
$919$			1101.58i			1.19868i	−0.800496	+	0.599339i	$0.795430\pi$
$920$		0			0
$921$		0			0
$922$	861.148	−	497.184i	0.934000	−	0.539245i
$923$		−	1584.00i		−	1.71614i
$924$		0			0
$925$		0			0
$926$	−347.091	−	601.179i	−0.374828	−	0.649221i
$927$		0			0
$928$	−259.969	−	150.093i	−0.280139	−	0.161738i
$929$	1219.51			1.31271			0.656355	−	0.754452i	$-0.272097\pi$
$929$	1219.51			1.31271			0.656355	+	0.754452i	$0.272097\pi$
$930$		0			0
$931$			810.600i			0.870676i
$932$	602.754	−	1044.00i	0.646731	−	1.12017i
$933$		0			0
$934$	520.000	+	900.666i	0.556745	+	0.964311i
$935$		0			0
$936$		0			0
$937$	−1754.79			−1.87278			−0.936389	−	0.350965i	$-0.885854\pi$
$937$	−1754.79			−1.87278			−0.936389	+	0.350965i	$0.885854\pi$
$938$	304.841	−	176.000i	0.324990	−	0.187633i
$939$		0			0
$940$		0			0
$941$	1285.17			1.36575			0.682877	−	0.730534i	$-0.260729\pi$
$941$	1285.17			1.36575			0.682877	+	0.730534i	$0.260729\pi$
$942$		0			0
$943$		−	525.327i		−	0.557080i
$944$	−225.140	+	129.985i	−0.238496	+	0.137696i
$945$		0			0
$946$	−1056.00	+	609.682i	−1.11628	+	0.644484i
$947$			1480.00i			1.56283i	0.624012	+	0.781415i	$0.285502\pi$
$947$			1480.00i			1.56283i	−0.624012	+	0.781415i	$0.714498\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$	779.908			0.819231
$953$	1486.10			1.55939			0.779695	−	0.626159i	$-0.215374\pi$
$953$	1486.10			1.55939			0.779695	+	0.626159i	$0.215374\pi$
$954$		0			0
$955$		0			0
$956$	−562.850	−	324.962i	−0.588755	−	0.339918i
$957$		0			0
$958$	−812.404	−	1407.12i	−0.848021	−	1.46881i
$959$			1982.27i			2.06701i
$960$		0			0
$961$	−239.000			−0.248699
$962$	−1371.78	+	792.000i	−1.42597	+	0.823285i
$963$		0			0
$964$	100.000	−	173.205i	0.103734	−	0.179673i
$965$		0			0
$966$		0			0
$967$			1247.65i			1.29023i	0.764086	+	0.645114i	$0.223190\pi$
$967$			1247.65i			1.29023i	−0.764086	+	0.645114i	$0.776810\pi$
$968$			1144.00i			1.18182i
$969$		0			0
$970$		0			0
$971$		−	438.698i		−	0.451800i	−0.974150	−	0.225900i	$-0.927468\pi$
$971$		−	438.698i		−	0.451800i	0.974150	−	0.225900i	$-0.0725323\pi$
$972$		0			0
$973$	844.900			0.868345
$974$	290.806	+	503.690i	0.298569	+	0.517136i
$975$		0			0
$976$	464.000	+	803.672i	0.475410	+	0.823434i
$977$	869.490			0.889959			0.444979	−	0.895541i	$-0.353211\pi$
$977$	869.490			0.889959			0.444979	+	0.895541i	$0.353211\pi$
$978$		0			0
$979$			1219.36i			1.24552i
$980$		0			0
$981$		0			0
$982$	406.202	+	703.562i	0.413648	+	0.716459i
$983$			860.000i			0.874873i	0.899249	+	0.437436i	$0.144113\pi$
$983$			860.000i			0.874873i	−0.899249	+	0.437436i	$0.855887\pi$
$984$		0			0
$985$		0			0
$986$	168.855	−	97.4885i	0.171253	−	0.0988727i
$987$		0			0
$988$	1169.86	+	675.420i	1.18407	+	0.683623i
$989$	−1050.65			−1.06234
$990$		0			0
$991$			436.477i			0.440441i	0.975450	+	0.220220i	$0.0706777\pi$
$991$			436.477i			0.440441i	−0.975450	+	0.220220i	$0.929322\pi$
$992$	−554.256	+	960.000i	−0.558726	+	0.967742i
$993$		0			0
$994$	−1584.00	+	914.523i	−1.59356	+	0.920043i
$995$		0			0
$996$		0			0
$997$	1056.12			1.05930			0.529651	−	0.848215i	$-0.322323\pi$
$997$	1056.12			1.05930			0.529651	+	0.848215i	$0.322323\pi$
$998$	−838.313	−	1452.00i	−0.839993	−	1.45491i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.c.s.451.6		8
3.2	odd	2	inner	900.3.c.s.451.4		8
4.3	odd	2	inner	900.3.c.s.451.7		8
5.2	odd	4		180.3.f.g.19.3	yes	4
5.3	odd	4		180.3.f.d.19.1	✓	4
5.4	even	2	inner	900.3.c.s.451.3		8
12.11	even	2	inner	900.3.c.s.451.1		8
15.2	even	4		180.3.f.d.19.2	yes	4
15.8	even	4		180.3.f.g.19.4	yes	4
15.14	odd	2	inner	900.3.c.s.451.5		8
20.3	even	4		180.3.f.g.19.1	yes	4
20.7	even	4		180.3.f.d.19.3	yes	4
20.19	odd	2	inner	900.3.c.s.451.2		8
60.23	odd	4		180.3.f.d.19.4	yes	4
60.47	odd	4		180.3.f.g.19.2	yes	4
60.59	even	2	inner	900.3.c.s.451.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.f.d.19.1	✓	4	5.3	odd	4
180.3.f.d.19.2	yes	4	15.2	even	4
180.3.f.d.19.3	yes	4	20.7	even	4
180.3.f.d.19.4	yes	4	60.23	odd	4
180.3.f.g.19.1	yes	4	20.3	even	4
180.3.f.g.19.2	yes	4	60.47	odd	4
180.3.f.g.19.3	yes	4	5.2	odd	4
180.3.f.g.19.4	yes	4	15.8	even	4
900.3.c.s.451.1		8	12.11	even	2	inner
900.3.c.s.451.2		8	20.19	odd	2	inner
900.3.c.s.451.3		8	5.4	even	2	inner
900.3.c.s.451.4		8	3.2	odd	2	inner
900.3.c.s.451.5		8	15.14	odd	2	inner
900.3.c.s.451.6		8	1.1	even	1	trivial
900.3.c.s.451.7		8	4.3	odd	2	inner
900.3.c.s.451.8		8	60.59	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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +9.38083i q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+(1.73205 - 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +9.38083i q^{7} -8.00000i q^{8} -16.2481i q^{11} -16.2481 q^{13} +(9.38083 + 16.2481i) q^{14} +(-8.00000 - 13.8564i) q^{16} +10.3923 q^{17} -20.7846i q^{19} +(-16.2481 - 28.1425i) q^{22} -28.0000i q^{23} +(-28.1425 + 16.2481i) q^{26} +(32.4962 + 18.7617i) q^{28} +9.38083 q^{29} -34.6410i q^{31} +(-27.7128 - 16.0000i) q^{32} +(18.0000 - 10.3923i) q^{34} +48.7442 q^{37} +(-20.7846 - 36.0000i) q^{38} +18.7617 q^{41} -37.5233i q^{43} +(-56.2850 - 32.4962i) q^{44} +(-28.0000 - 48.4974i) q^{46} +4.00000i q^{47} -39.0000 q^{49} +(-32.4962 + 56.2850i) q^{52} +31.1769 q^{53} +75.0467 q^{56} +(16.2481 - 9.38083i) q^{58} -16.2481i q^{59} -58.0000 q^{61} +(-34.6410 - 60.0000i) q^{62} -64.0000 q^{64} -18.7617i q^{67} +(20.7846 - 36.0000i) q^{68} +97.4885i q^{71} +(84.4275 - 48.7442i) q^{74} +(-72.0000 - 41.5692i) q^{76} +152.420 q^{77} +6.92820i q^{79} +(32.4962 - 18.7617i) q^{82} +32.0000i q^{83} +(-37.5233 - 64.9923i) q^{86} -129.985 q^{88} -75.0467 q^{89} -152.420i q^{91} +(-96.9948 - 56.0000i) q^{92} +(4.00000 + 6.92820i) q^{94} -162.481 q^{97} +(-67.5500 + 39.0000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4}+O(q^{10}) \) 8 * q + 16 * q^4	\( 8 q + 16 q^{4} - 64 q^{16} + 144 q^{34} - 224 q^{46} - 312 q^{49} - 464 q^{61} - 512 q^{64} - 576 q^{76} + 32 q^{94}+O(q^{100}) \) 8 * q + 16 * q^4 - 64 * q^16 + 144 * q^34 - 224 * q^46 - 312 * q^49 - 464 * q^61 - 512 * q^64 - 576 * q^76 + 32 * q^94

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