LMFDB - Embedded newform 45.3.d.a.44.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,3,Mod(44,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("45.44");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	45.d (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:	$1.22616118962$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			44.3
Root			$1.16372i$ of defining polynomial
Character	$\chi$	$=$	45.44
Dual form			45.3.d.a.44.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{2} +3.00000 q^{4} +(-2.64575 - 4.24264i) q^{5} +11.2250i q^{7} -2.64575 q^{8} +O(q^{10})$	\(q+2.64575 q^{2} +3.00000 q^{4} +(-2.64575 - 4.24264i) q^{5} +11.2250i q^{7} -2.64575 q^{8} +(-7.00000 - 11.2250i) q^{10} -4.24264i q^{11} -11.2250i q^{13} +29.6985i q^{14} -19.0000 q^{16} +10.5830 q^{17} +20.0000 q^{19} +(-7.93725 - 12.7279i) q^{20} -11.2250i q^{22} -5.29150 q^{23} +(-11.0000 + 22.4499i) q^{25} -29.6985i q^{26} +33.6749i q^{28} +8.48528i q^{29} +26.0000 q^{31} -39.6863 q^{32} +28.0000 q^{34} +(47.6235 - 29.6985i) q^{35} -33.6749i q^{37} +52.9150 q^{38} +(7.00000 + 11.2250i) q^{40} +55.1543i q^{41} -22.4499i q^{43} -12.7279i q^{44} -14.0000 q^{46} -21.1660 q^{47} -77.0000 q^{49} +(-29.1033 + 59.3970i) q^{50} -33.6749i q^{52} -84.6640 q^{53} +(-18.0000 + 11.2250i) q^{55} -29.6985i q^{56} +22.4499i q^{58} -46.6690i q^{59} -22.0000 q^{61} +68.7895 q^{62} -29.0000 q^{64} +(-47.6235 + 29.6985i) q^{65} +89.7998i q^{67} +31.7490 q^{68} +(126.000 - 78.5748i) q^{70} +50.9117i q^{71} -67.3498i q^{73} -89.0955i q^{74} +60.0000 q^{76} +47.6235 q^{77} +14.0000 q^{79} +(50.2693 + 80.6102i) q^{80} +145.925i q^{82} +74.0810 q^{83} +(-28.0000 - 44.8999i) q^{85} -59.3970i q^{86} +11.2250i q^{88} -89.0955i q^{89} +126.000 q^{91} -15.8745 q^{92} -56.0000 q^{94} +(-52.9150 - 84.8528i) q^{95} -22.4499i q^{97} -203.723 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{4}+O(q^{10}) $ 4 * q + 12 * q^4	$ 4 q + 12 q^{4} - 28 q^{10} - 76 q^{16} + 80 q^{19} - 44 q^{25} + 104 q^{31} + 112 q^{34} + 28 q^{40} - 56 q^{46} - 308 q^{49} - 72 q^{55} - 88 q^{61} - 116 q^{64} + 504 q^{70} + 240 q^{76} + 56 q^{79} - 112 q^{85} + 504 q^{91} - 224 q^{94}+O(q^{100}) $ 4 * q + 12 * q^4 - 28 * q^10 - 76 * q^16 + 80 * q^19 - 44 * q^25 + 104 * q^31 + 112 * q^34 + 28 * q^40 - 56 * q^46 - 308 * q^49 - 72 * q^55 - 88 * q^61 - 116 * q^64 + 504 * q^70 + 240 * q^76 + 56 * q^79 - 112 * q^85 + 504 * q^91 - 224 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$-1$	$-1$

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 $

$2$	2.64575			1.32288			0.661438	−	0.750000i	$-0.269947\pi$
$2$	2.64575			1.32288			0.661438	+	0.750000i	$0.269947\pi$
$3$		0			0
$3$		0			0
$4$	3.00000			0.750000
$5$	−2.64575	−	4.24264i	−0.529150	−	0.848528i
$5$	−2.64575	−	4.24264i	−0.529150	−	0.848528i
$6$		0			0
$7$			11.2250i			1.60357i	0.597614	+	0.801784i	$0.296115\pi$
$7$			11.2250i			1.60357i	−0.597614	+	0.801784i	$0.703885\pi$
$8$	−2.64575			−0.330719
$9$		0			0
$10$	−7.00000	−	11.2250i	−0.700000	−	1.12250i
$11$		−	4.24264i		−	0.385695i	−0.981229	−	0.192847i	$-0.938228\pi$
$11$		−	4.24264i		−	0.385695i	0.981229	−	0.192847i	$-0.0617722\pi$
$12$		0			0
$13$		−	11.2250i		−	0.863459i	−0.902003	−	0.431730i	$-0.857903\pi$
$13$		−	11.2250i		−	0.863459i	0.902003	−	0.431730i	$-0.142097\pi$
$14$			29.6985i			2.12132i
$15$		0			0
$16$	−19.0000			−1.18750
$17$	10.5830			0.622530			0.311265	−	0.950323i	$-0.399247\pi$
$17$	10.5830			0.622530			0.311265	+	0.950323i	$0.399247\pi$
$18$		0			0
$19$	20.0000			1.05263			0.526316	−	0.850289i	$-0.323573\pi$
$19$	20.0000			1.05263			0.526316	+	0.850289i	$0.323573\pi$
$20$	−7.93725	−	12.7279i	−0.396863	−	0.636396i
$21$		0			0
$22$		−	11.2250i		−	0.510226i
$23$	−5.29150			−0.230065			−0.115033	−	0.993362i	$-0.536697\pi$
$23$	−5.29150			−0.230065			−0.115033	+	0.993362i	$0.536697\pi$
$24$		0			0
$25$	−11.0000	+	22.4499i	−0.440000	+	0.897998i
$26$		−	29.6985i		−	1.14225i
$27$		0			0
$28$			33.6749i			1.20268i
$29$			8.48528i			0.292596i	0.989241	+	0.146298i	$0.0467358\pi$
$29$			8.48528i			0.292596i	−0.989241	+	0.146298i	$0.953264\pi$
$30$		0			0
$31$	26.0000			0.838710			0.419355	−	0.907822i	$-0.362256\pi$
$31$	26.0000			0.838710			0.419355	+	0.907822i	$0.362256\pi$
$32$	−39.6863			−1.24020
$33$		0			0
$34$	28.0000			0.823529
$35$	47.6235	−	29.6985i	1.36067	−	0.848528i
$36$		0			0
$37$		−	33.6749i		−	0.910133i	−0.890457	−	0.455066i	$-0.849616\pi$
$37$		−	33.6749i		−	0.910133i	0.890457	−	0.455066i	$-0.150384\pi$
$38$	52.9150			1.39250
$39$		0			0
$40$	7.00000	+	11.2250i	0.175000	+	0.280624i
$41$			55.1543i			1.34523i	0.739994	+	0.672614i	$0.234828\pi$
$41$			55.1543i			1.34523i	−0.739994	+	0.672614i	$0.765172\pi$
$42$		0			0
$43$		−	22.4499i		−	0.522092i	−0.965326	−	0.261046i	$-0.915933\pi$
$43$		−	22.4499i		−	0.522092i	0.965326	−	0.261046i	$-0.0840674\pi$
$44$		−	12.7279i		−	0.289271i
$45$		0			0
$46$	−14.0000			−0.304348
$47$	−21.1660			−0.450341			−0.225170	−	0.974319i	$-0.572294\pi$
$47$	−21.1660			−0.450341			−0.225170	+	0.974319i	$0.572294\pi$
$48$		0			0
$49$	−77.0000			−1.57143
$50$	−29.1033	+	59.3970i	−0.582065	+	1.18794i
$51$		0			0
$52$		−	33.6749i		−	0.647595i
$53$	−84.6640			−1.59743			−0.798717	−	0.601706i	$-0.794488\pi$
$53$	−84.6640			−1.59743			−0.798717	+	0.601706i	$0.794488\pi$
$54$		0			0
$55$	−18.0000	+	11.2250i	−0.327273	+	0.204090i
$56$		−	29.6985i		−	0.530330i
$57$		0			0
$58$			22.4499i			0.387068i
$59$		−	46.6690i		−	0.791001i	−0.918466	−	0.395500i	$-0.870571\pi$
$59$		−	46.6690i		−	0.791001i	0.918466	−	0.395500i	$-0.129429\pi$
$60$		0			0
$61$	−22.0000			−0.360656			−0.180328	−	0.983607i	$-0.557716\pi$
$61$	−22.0000			−0.360656			−0.180328	+	0.983607i	$0.557716\pi$
$62$	68.7895			1.10951
$63$		0			0
$64$	−29.0000			−0.453125
$65$	−47.6235	+	29.6985i	−0.732670	+	0.456900i
$66$		0			0
$67$			89.7998i			1.34030i	0.742228	+	0.670148i	$0.233769\pi$
$67$			89.7998i			1.34030i	−0.742228	+	0.670148i	$0.766231\pi$
$68$	31.7490			0.466897
$69$		0			0
$70$	126.000	−	78.5748i	1.80000	−	1.12250i
$71$			50.9117i			0.717066i	0.933517	+	0.358533i	$0.116723\pi$
$71$			50.9117i			0.717066i	−0.933517	+	0.358533i	$0.883277\pi$
$72$		0			0
$73$		−	67.3498i		−	0.922600i	−0.887244	−	0.461300i	$-0.847383\pi$
$73$		−	67.3498i		−	0.922600i	0.887244	−	0.461300i	$-0.152617\pi$
$74$		−	89.0955i		−	1.20399i
$75$		0			0
$76$	60.0000			0.789474
$77$	47.6235			0.618487
$78$		0			0
$79$	14.0000			0.177215			0.0886076	−	0.996067i	$-0.471758\pi$
$79$	14.0000			0.177215			0.0886076	+	0.996067i	$0.471758\pi$
$80$	50.2693	+	80.6102i	0.628366	+	1.00763i
$81$		0			0
$82$			145.925i			1.77957i
$83$	74.0810			0.892543			0.446271	−	0.894898i	$-0.352752\pi$
$83$	74.0810			0.892543			0.446271	+	0.894898i	$0.352752\pi$
$84$		0			0
$85$	−28.0000	−	44.8999i	−0.329412	−	0.528234i
$86$		−	59.3970i		−	0.690662i
$87$		0			0
$88$			11.2250i			0.127557i
$89$		−	89.0955i		−	1.00107i	−0.865716	−	0.500536i	$-0.833136\pi$
$89$		−	89.0955i		−	1.00107i	0.865716	−	0.500536i	$-0.166864\pi$
$90$		0			0
$91$	126.000			1.38462
$92$	−15.8745			−0.172549
$93$		0			0
$94$	−56.0000			−0.595745
$95$	−52.9150	−	84.8528i	−0.557000	−	0.893188i
$96$		0			0
$97$		−	22.4499i		−	0.231443i	−0.993282	−	0.115721i	$-0.963082\pi$
$97$		−	22.4499i		−	0.231443i	0.993282	−	0.115721i	$-0.0369180\pi$
$98$	−203.723			−2.07880
$99$		0			0
$100$	−33.0000	+	67.3498i	−0.330000	+	0.673498i
$101$			135.765i			1.34420i	0.740459	+	0.672101i	$0.234608\pi$
$101$			135.765i			1.34420i	−0.740459	+	0.672101i	$0.765392\pi$
$102$		0			0
$103$			56.1249i			0.544902i	0.962170	+	0.272451i	$0.0878342\pi$
$103$			56.1249i			0.544902i	−0.962170	+	0.272451i	$0.912166\pi$
$104$			29.6985i			0.285562i
$105$		0			0
$106$	−224.000			−2.11321
$107$	10.5830			0.0989066			0.0494533	−	0.998776i	$-0.484252\pi$
$107$	10.5830			0.0989066			0.0494533	+	0.998776i	$0.484252\pi$
$108$		0			0
$109$	−70.0000			−0.642202			−0.321101	−	0.947045i	$-0.604053\pi$
$109$	−70.0000			−0.642202			−0.321101	+	0.947045i	$0.604053\pi$
$110$	−47.6235	+	29.6985i	−0.432941	+	0.269986i
$111$		0			0
$112$		−	213.274i		−	1.90424i
$113$	137.579			1.21751			0.608757	−	0.793357i	$-0.291668\pi$
$113$	137.579			1.21751			0.608757	+	0.793357i	$0.291668\pi$
$114$		0			0
$115$	14.0000	+	22.4499i	0.121739	+	0.195217i
$116$			25.4558i			0.219447i
$117$		0			0
$118$		−	123.475i		−	1.04640i
$119$			118.794i			0.998268i
$120$		0			0
$121$	103.000			0.851240
$122$	−58.2065			−0.477103
$123$		0			0
$124$	78.0000			0.629032
$125$	124.350	−	12.7279i	0.994802	−	0.101823i
$126$		0			0
$127$		−	168.375i		−	1.32578i	−0.748715	−	0.662892i	$-0.769329\pi$
$127$		−	168.375i		−	1.32578i	0.748715	−	0.662892i	$-0.230671\pi$
$128$	82.0183			0.640768
$129$		0			0
$130$	−126.000	+	78.5748i	−0.969231	+	0.604422i
$131$		−	148.492i		−	1.13353i	−0.823880	−	0.566765i	$-0.808195\pi$
$131$		−	148.492i		−	1.13353i	0.823880	−	0.566765i	$-0.191805\pi$
$132$		0			0
$133$			224.499i			1.68797i
$134$			237.588i			1.77304i
$135$		0			0
$136$	−28.0000			−0.205882
$137$	−211.660			−1.54496			−0.772482	−	0.635036i	$-0.780985\pi$
$137$	−211.660			−1.54496			−0.772482	+	0.635036i	$0.780985\pi$
$138$		0			0
$139$	206.000			1.48201			0.741007	−	0.671497i	$-0.234348\pi$
$139$	206.000			1.48201			0.741007	+	0.671497i	$0.234348\pi$
$140$	142.871	−	89.0955i	1.02050	−	0.636396i
$141$		0			0
$142$			134.700i			0.948589i
$143$	−47.6235			−0.333032
$144$		0			0
$145$	36.0000	−	22.4499i	0.248276	−	0.154827i
$146$		−	178.191i		−	1.22049i
$147$		0			0
$148$		−	101.025i		−	0.682600i
$149$		−	135.765i		−	0.911171i	−0.890192	−	0.455586i	$-0.849430\pi$
$149$		−	135.765i		−	0.911171i	0.890192	−	0.455586i	$-0.150570\pi$
$150$		0			0
$151$	−202.000			−1.33775			−0.668874	−	0.743376i	$-0.733224\pi$
$151$	−202.000			−1.33775			−0.668874	+	0.743376i	$0.733224\pi$
$152$	−52.9150			−0.348125
$153$		0			0
$154$	126.000			0.818182
$155$	−68.7895	−	110.309i	−0.443803	−	0.711669i
$156$		0			0
$157$			56.1249i			0.357483i	0.983896	+	0.178742i	$0.0572026\pi$
$157$			56.1249i			0.357483i	−0.983896	+	0.178742i	$0.942797\pi$
$158$	37.0405			0.234434
$159$		0			0
$160$	105.000	+	168.375i	0.656250	+	1.05234i
$161$		−	59.3970i		−	0.368925i
$162$		0			0
$163$			202.049i			1.23957i	0.784773	+	0.619784i	$0.212780\pi$
$163$			202.049i			1.23957i	−0.784773	+	0.619784i	$0.787220\pi$
$164$			165.463i			1.00892i
$165$		0			0
$166$	196.000			1.18072
$167$	185.203			1.10900			0.554499	−	0.832185i	$-0.312910\pi$
$167$	185.203			1.10900			0.554499	+	0.832185i	$0.312910\pi$
$168$		0			0
$169$	43.0000			0.254438
$170$	−74.0810	−	118.794i	−0.435771	−	0.698788i
$171$		0			0
$172$		−	67.3498i		−	0.391569i
$173$	−21.1660			−0.122347			−0.0611734	−	0.998127i	$-0.519484\pi$
$173$	−21.1660			−0.122347			−0.0611734	+	0.998127i	$0.519484\pi$
$174$		0			0
$175$	−252.000	−	123.475i	−1.44000	−	0.705570i
$176$			80.6102i			0.458012i
$177$		0			0
$178$		−	235.724i		−	1.32429i
$179$			241.831i			1.35101i	0.737356	+	0.675504i	$0.236074\pi$
$179$			241.831i			1.35101i	−0.737356	+	0.675504i	$0.763926\pi$
$180$		0			0
$181$	74.0000			0.408840			0.204420	−	0.978883i	$-0.434469\pi$
$181$	74.0000			0.408840			0.204420	+	0.978883i	$0.434469\pi$
$182$	333.365			1.83167
$183$		0			0
$184$	14.0000			0.0760870
$185$	−142.871	+	89.0955i	−0.772273	+	0.481597i
$186$		0			0
$187$		−	44.8999i		−	0.240106i
$188$	−63.4980			−0.337755
$189$		0			0
$190$	−140.000	−	224.499i	−0.736842	−	1.18158i
$191$			161.220i			0.844086i	0.906576	+	0.422043i	$0.138687\pi$
$191$			161.220i			0.844086i	−0.906576	+	0.422043i	$0.861313\pi$
$192$		0			0
$193$		−	179.600i		−	0.930568i	−0.885162	−	0.465284i	$-0.845952\pi$
$193$		−	179.600i		−	0.930568i	0.885162	−	0.465284i	$-0.154048\pi$
$194$		−	59.3970i		−	0.306170i
$195$		0			0
$196$	−231.000			−1.17857
$197$	−37.0405			−0.188023			−0.0940115	−	0.995571i	$-0.529969\pi$
$197$	−37.0405			−0.188023			−0.0940115	+	0.995571i	$0.529969\pi$
$198$		0			0
$199$	−250.000			−1.25628			−0.628141	−	0.778100i	$-0.716184\pi$
$199$	−250.000			−1.25628			−0.628141	+	0.778100i	$0.716184\pi$
$200$	29.1033	−	59.3970i	0.145516	−	0.296985i
$201$		0			0
$202$			359.199i			1.77821i
$203$	−95.2470			−0.469197
$204$		0			0
$205$	234.000	−	145.925i	1.14146	−	0.711828i
$206$			148.492i			0.720837i
$207$		0			0
$208$			213.274i			1.02536i
$209$		−	84.8528i		−	0.405994i
$210$		0			0
$211$	−154.000			−0.729858			−0.364929	−	0.931035i	$-0.618907\pi$
$211$	−154.000			−0.729858			−0.364929	+	0.931035i	$0.618907\pi$
$212$	−253.992			−1.19808
$213$		0			0
$214$	28.0000			0.130841
$215$	−95.2470	+	59.3970i	−0.443010	+	0.276265i
$216$		0			0
$217$			291.849i			1.34493i
$218$	−185.203			−0.849553
$219$		0			0
$220$	−54.0000	+	33.6749i	−0.245455	+	0.153068i
$221$		−	118.794i		−	0.537529i
$222$		0			0
$223$		−	392.874i		−	1.76177i	−0.473333	−	0.880883i	$-0.656949\pi$
$223$		−	392.874i		−	1.76177i	0.473333	−	0.880883i	$-0.343051\pi$
$224$		−	445.477i		−	1.98874i
$225$		0			0
$226$	364.000			1.61062
$227$	−21.1660			−0.0932423			−0.0466212	−	0.998913i	$-0.514845\pi$
$227$	−21.1660			−0.0932423			−0.0466212	+	0.998913i	$0.514845\pi$
$228$		0			0
$229$	−118.000			−0.515284			−0.257642	−	0.966240i	$-0.582945\pi$
$229$	−118.000			−0.515284			−0.257642	+	0.966240i	$0.582945\pi$
$230$	37.0405	+	59.3970i	0.161046	+	0.258248i
$231$		0			0
$232$		−	22.4499i		−	0.0967670i
$233$	391.571			1.68056			0.840282	−	0.542150i	$-0.182390\pi$
$233$	391.571			1.68056			0.840282	+	0.542150i	$0.182390\pi$
$234$		0			0
$235$	56.0000	+	89.7998i	0.238298	+	0.382127i
$236$		−	140.007i		−	0.593251i
$237$		0			0
$238$			314.299i			1.32058i
$239$			347.897i			1.45563i	0.685771	+	0.727817i	$0.259465\pi$
$239$			347.897i			1.45563i	−0.685771	+	0.727817i	$0.740535\pi$
$240$		0			0
$241$	−40.0000			−0.165975			−0.0829876	−	0.996551i	$-0.526446\pi$
$241$	−40.0000			−0.165975			−0.0829876	+	0.996551i	$0.526446\pi$
$242$	272.512			1.12608
$243$		0			0
$244$	−66.0000			−0.270492
$245$	203.723	+	326.683i	0.831522	+	1.33340i
$246$		0			0
$247$		−	224.499i		−	0.908905i
$248$	−68.7895			−0.277377
$249$		0			0
$250$	329.000	−	33.6749i	1.31600	−	0.134700i
$251$		−	241.831i		−	0.963468i	−0.876317	−	0.481734i	$-0.840007\pi$
$251$		−	241.831i		−	0.963468i	0.876317	−	0.481734i	$-0.159993\pi$
$252$		0			0
$253$			22.4499i			0.0887350i
$254$		−	445.477i		−	1.75385i
$255$		0			0
$256$	333.000			1.30078
$257$	232.826			0.905938			0.452969	−	0.891526i	$-0.350365\pi$
$257$	232.826			0.905938			0.452969	+	0.891526i	$0.350365\pi$
$258$		0			0
$259$	378.000			1.45946
$260$	−142.871	+	89.0955i	−0.549502	+	0.342675i
$261$		0			0
$262$		−	392.874i		−	1.49952i
$263$	−164.037			−0.623713			−0.311857	−	0.950129i	$-0.600951\pi$
$263$	−164.037			−0.623713			−0.311857	+	0.950129i	$0.600951\pi$
$264$		0			0
$265$	224.000	+	359.199i	0.845283	+	1.35547i
$266$			593.970i			2.23297i
$267$		0			0
$268$			269.399i			1.00522i
$269$		−	534.573i		−	1.98726i	−0.112695	−	0.993630i	$-0.535948\pi$
$269$		−	534.573i		−	1.98726i	0.112695	−	0.993630i	$-0.464052\pi$
$270$		0			0
$271$	−286.000			−1.05535			−0.527675	−	0.849446i	$-0.676936\pi$
$271$	−286.000			−1.05535			−0.527675	+	0.849446i	$0.676936\pi$
$272$	−201.077			−0.739254
$273$		0			0
$274$	−560.000			−2.04380
$275$	95.2470	+	46.6690i	0.346353	+	0.169706i
$276$		0			0
$277$		−	190.825i		−	0.688897i	−0.938805	−	0.344449i	$-0.888066\pi$
$277$		−	190.825i		−	0.688897i	0.938805	−	0.344449i	$-0.111934\pi$
$278$	545.025			1.96052
$279$		0			0
$280$	−126.000	+	78.5748i	−0.450000	+	0.280624i
$281$		−	80.6102i		−	0.286869i	−0.989660	−	0.143434i	$-0.954185\pi$
$281$		−	80.6102i		−	0.286869i	0.989660	−	0.143434i	$-0.0458146\pi$
$282$		0			0
$283$			89.7998i			0.317314i	0.987334	+	0.158657i	$0.0507164\pi$
$283$			89.7998i			0.317314i	−0.987334	+	0.158657i	$0.949284\pi$
$284$			152.735i			0.537800i
$285$		0			0
$286$	−126.000			−0.440559
$287$	−619.106			−2.15716
$288$		0			0
$289$	−177.000			−0.612457
$290$	95.2470	−	59.3970i	0.328438	−	0.204817i
$291$		0			0
$292$		−	202.049i		−	0.691950i
$293$	−576.774			−1.96851			−0.984256	−	0.176751i	$-0.943441\pi$
$293$	−576.774			−1.96851			−0.984256	+	0.176751i	$0.943441\pi$
$294$		0			0
$295$	−198.000	+	123.475i	−0.671186	+	0.418558i
$296$			89.0955i			0.300998i
$297$		0			0
$298$		−	359.199i		−	1.20537i
$299$			59.3970i			0.198652i
$300$		0			0
$301$	252.000			0.837209
$302$	−534.442			−1.76967
$303$		0			0
$304$	−380.000			−1.25000
$305$	58.2065	+	93.3381i	0.190841	+	0.306027i
$306$		0			0
$307$			269.399i			0.877522i	0.898604	+	0.438761i	$0.144583\pi$
$307$			269.399i			0.877522i	−0.898604	+	0.438761i	$0.855417\pi$
$308$	142.871			0.463865
$309$		0			0
$310$	−182.000	−	291.849i	−0.587097	−	0.941449i
$311$		−	59.3970i		−	0.190987i	−0.995430	−	0.0954935i	$-0.969557\pi$
$311$		−	59.3970i		−	0.190987i	0.995430	−	0.0954935i	$-0.0304429\pi$
$312$		0			0
$313$			179.600i			0.573800i	0.957960	+	0.286900i	$0.0926248\pi$
$313$			179.600i			0.573800i	−0.957960	+	0.286900i	$0.907375\pi$
$314$			148.492i			0.472906i
$315$		0			0
$316$	42.0000			0.132911
$317$	312.199			0.984854			0.492427	−	0.870354i	$-0.336110\pi$
$317$	312.199			0.984854			0.492427	+	0.870354i	$0.336110\pi$
$318$		0			0
$319$	36.0000			0.112853
$320$	76.7268	+	123.037i	0.239771	+	0.384489i
$321$		0			0
$322$		−	157.150i		−	0.488042i
$323$	211.660			0.655294
$324$		0			0
$325$	252.000	+	123.475i	0.775385	+	0.379922i
$326$			534.573i			1.63979i
$327$		0			0
$328$		−	145.925i		−	0.444892i
$329$		−	237.588i		−	0.722152i
$330$		0			0
$331$	−112.000			−0.338369			−0.169184	−	0.985584i	$-0.554113\pi$
$331$	−112.000			−0.338369			−0.169184	+	0.985584i	$0.554113\pi$
$332$	222.243			0.669407
$333$		0			0
$334$	490.000			1.46707
$335$	380.988	−	237.588i	1.13728	−	0.709218i
$336$		0			0
$337$		−	112.250i		−	0.333085i	−0.986034	−	0.166543i	$-0.946740\pi$
$337$		−	112.250i		−	0.333085i	0.986034	−	0.166543i	$-0.0532603\pi$
$338$	113.767			0.336590
$339$		0			0
$340$	−84.0000	−	134.700i	−0.247059	−	0.396175i
$341$		−	110.309i		−	0.323486i
$342$		0			0
$343$		−	314.299i		−	0.916324i
$344$			59.3970i			0.172666i
$345$		0			0
$346$	−56.0000			−0.161850
$347$	518.567			1.49443			0.747215	−	0.664582i	$-0.231390\pi$
$347$	518.567			1.49443			0.747215	+	0.664582i	$0.231390\pi$
$348$		0			0
$349$	122.000			0.349570			0.174785	−	0.984607i	$-0.444077\pi$
$349$	122.000			0.349570			0.174785	+	0.984607i	$0.444077\pi$
$350$	−666.729	−	326.683i	−1.90494	−	0.933381i
$351$		0			0
$352$			168.375i			0.478337i
$353$	−402.154			−1.13925			−0.569624	−	0.821906i	$-0.692911\pi$
$353$	−402.154			−1.13925			−0.569624	+	0.821906i	$0.692911\pi$
$354$		0			0
$355$	216.000	−	134.700i	0.608451	−	0.379436i
$356$		−	267.286i		−	0.750804i
$357$		0			0
$358$			639.823i			1.78722i
$359$			636.396i			1.77269i	0.463024	+	0.886346i	$0.346764\pi$
$359$			636.396i			1.77269i	−0.463024	+	0.886346i	$0.653236\pi$
$360$		0			0
$361$	39.0000			0.108033
$362$	195.786			0.540844
$363$		0			0
$364$	378.000			1.03846
$365$	−285.741	+	178.191i	−0.782852	+	0.488194i
$366$		0			0
$367$			684.723i			1.86573i	0.360225	+	0.932866i	$0.382700\pi$
$367$			684.723i			1.86573i	−0.360225	+	0.932866i	$0.617300\pi$
$368$	100.539			0.273203
$369$		0			0
$370$	−378.000	+	235.724i	−1.02162	+	0.637093i
$371$		−	950.352i		−	2.56159i
$372$		0			0
$373$		−	145.925i		−	0.391219i	−0.980682	−	0.195609i	$-0.937332\pi$
$373$		−	145.925i		−	0.391219i	0.980682	−	0.195609i	$-0.0626685\pi$
$374$		−	118.794i		−	0.317631i
$375$		0			0
$376$	56.0000			0.148936
$377$	95.2470			0.252645
$378$		0			0
$379$	362.000			0.955145			0.477573	−	0.878592i	$-0.341517\pi$
$379$	362.000			0.955145			0.477573	+	0.878592i	$0.341517\pi$
$380$	−158.745	−	254.558i	−0.417750	−	0.669891i
$381$		0			0
$382$			426.549i			1.11662i
$383$	42.3320			0.110527			0.0552637	−	0.998472i	$-0.482400\pi$
$383$	42.3320			0.110527			0.0552637	+	0.998472i	$0.482400\pi$
$384$		0			0
$385$	−126.000	−	202.049i	−0.327273	−	0.524804i
$386$		−	475.176i		−	1.23103i
$387$		0			0
$388$		−	67.3498i		−	0.173582i
$389$			263.044i			0.676205i	0.941109	+	0.338102i	$0.109785\pi$
$389$			263.044i			0.676205i	−0.941109	+	0.338102i	$0.890215\pi$
$390$		0			0
$391$	−56.0000			−0.143223
$392$	203.723			0.519701
$393$		0			0
$394$	−98.0000			−0.248731
$395$	−37.0405	−	59.3970i	−0.0937735	−	0.150372i
$396$		0			0
$397$			33.6749i			0.0848235i	0.999100	+	0.0424117i	$0.0135041\pi$
$397$			33.6749i			0.0848235i	−0.999100	+	0.0424117i	$0.986496\pi$
$398$	−661.438			−1.66190
$399$		0			0
$400$	209.000	−	426.549i	0.522500	−	1.06637i
$401$			462.448i			1.15324i	0.817014	+	0.576618i	$0.195628\pi$
$401$			462.448i			1.15324i	−0.817014	+	0.576618i	$0.804372\pi$
$402$		0			0
$403$		−	291.849i		−	0.724192i
$404$			407.294i			1.00815i
$405$		0			0
$406$	−252.000			−0.620690
$407$	−142.871			−0.351033
$408$		0			0
$409$	−82.0000			−0.200489			−0.100244	−	0.994963i	$-0.531962\pi$
$409$	−82.0000			−0.200489			−0.100244	+	0.994963i	$0.531962\pi$
$410$	619.106	−	386.080i	1.51001	−	0.941659i
$411$		0			0
$412$			168.375i			0.408676i
$413$	523.859			1.26842
$414$		0			0
$415$	−196.000	−	314.299i	−0.472289	−	0.757348i
$416$			445.477i			1.07086i
$417$		0			0
$418$		−	224.499i		−	0.537080i
$419$			207.889i			0.496156i	0.968740	+	0.248078i	$0.0797989\pi$
$419$			207.889i			0.496156i	−0.968740	+	0.248078i	$0.920201\pi$
$420$		0			0
$421$	−490.000			−1.16390			−0.581948	−	0.813226i	$-0.697709\pi$
$421$	−490.000			−1.16390			−0.581948	+	0.813226i	$0.697709\pi$
$422$	−407.446			−0.965511
$423$		0			0
$424$	224.000			0.528302
$425$	−116.413	+	237.588i	−0.273913	+	0.559030i
$426$		0			0
$427$		−	246.949i		−	0.578336i
$428$	31.7490			0.0741799
$429$		0			0
$430$	−252.000	+	157.150i	−0.586047	+	0.365464i
$431$			687.308i			1.59468i	0.603529	+	0.797341i	$0.293761\pi$
$431$			687.308i			1.59468i	−0.603529	+	0.797341i	$0.706239\pi$
$432$		0			0
$433$		−	202.049i		−	0.466627i	−0.972402	−	0.233314i	$-0.925043\pi$
$433$		−	202.049i		−	0.466627i	0.972402	−	0.233314i	$-0.0749568\pi$
$434$			772.161i			1.77917i
$435$		0			0
$436$	−210.000			−0.481651
$437$	−105.830			−0.242174
$438$		0			0
$439$	302.000			0.687927			0.343964	−	0.938983i	$-0.388230\pi$
$439$	302.000			0.687927			0.343964	+	0.938983i	$0.388230\pi$
$440$	47.6235	−	29.6985i	0.108235	−	0.0674966i
$441$		0			0
$442$		−	314.299i		−	0.711084i
$443$	264.575			0.597235			0.298618	−	0.954373i	$-0.403475\pi$
$443$	264.575			0.597235			0.298618	+	0.954373i	$0.403475\pi$
$444$		0			0
$445$	−378.000	+	235.724i	−0.849438	+	0.529718i
$446$		−	1039.45i		−	2.33060i
$447$		0			0
$448$		−	325.524i		−	0.726617i
$449$		−	216.375i		−	0.481904i	−0.970537	−	0.240952i	$-0.922540\pi$
$449$		−	216.375i		−	0.481904i	0.970537	−	0.240952i	$-0.0774596\pi$
$450$		0			0
$451$	234.000			0.518847
$452$	412.737			0.913135
$453$		0			0
$454$	−56.0000			−0.123348
$455$	−333.365	−	534.573i	−0.732670	−	1.17489i
$456$		0			0
$457$		−	561.249i		−	1.22812i	−0.789261	−	0.614058i	$-0.789536\pi$
$457$		−	561.249i		−	1.22812i	0.789261	−	0.614058i	$-0.210464\pi$
$458$	−312.199			−0.681656
$459$		0			0
$460$	42.0000	+	67.3498i	0.0913043	+	0.146413i
$461$			237.588i			0.515375i	0.966228	+	0.257688i	$0.0829605\pi$
$461$			237.588i			0.515375i	−0.966228	+	0.257688i	$0.917039\pi$
$462$		0			0
$463$			729.623i			1.57586i	0.615765	+	0.787930i	$0.288847\pi$
$463$			729.623i			1.57586i	−0.615765	+	0.787930i	$0.711153\pi$
$464$		−	161.220i		−	0.347458i
$465$		0			0
$466$	1036.00			2.22318
$467$	−465.652			−0.997114			−0.498557	−	0.866857i	$-0.666137\pi$
$467$	−465.652			−0.997114			−0.498557	+	0.866857i	$0.666137\pi$
$468$		0			0
$469$	−1008.00			−2.14925
$470$	148.162	+	237.588i	0.315238	+	0.505506i
$471$		0			0
$472$			123.475i			0.261599i
$473$	−95.2470			−0.201368
$474$		0			0
$475$	−220.000	+	448.999i	−0.463158	+	0.945261i
$476$			356.382i			0.748701i
$477$		0			0
$478$			920.448i			1.92562i
$479$		−	475.176i		−	0.992016i	−0.868318	−	0.496008i	$-0.834799\pi$
$479$		−	475.176i		−	0.992016i	0.868318	−	0.496008i	$-0.165201\pi$
$480$		0			0
$481$	−378.000			−0.785863
$482$	−105.830			−0.219564
$483$		0			0
$484$	309.000			0.638430
$485$	−95.2470	+	59.3970i	−0.196386	+	0.122468i
$486$		0			0
$487$		−	505.124i		−	1.03722i	−0.855012	−	0.518608i	$-0.826451\pi$
$487$		−	505.124i		−	1.03722i	0.855012	−	0.518608i	$-0.173549\pi$
$488$	58.2065			0.119276
$489$		0			0
$490$	539.000	+	864.323i	1.10000	+	1.76392i
$491$		−	275.772i		−	0.561653i	−0.959759	−	0.280827i	$-0.909391\pi$
$491$		−	275.772i		−	0.561653i	0.959759	−	0.280827i	$-0.0906086\pi$
$492$		0			0
$493$			89.7998i			0.182150i
$494$		−	593.970i		−	1.20237i
$495$		0			0
$496$	−494.000			−0.995968
$497$	−571.482			−1.14986
$498$		0			0
$499$	368.000			0.737475			0.368737	−	0.929534i	$-0.379790\pi$
$499$	368.000			0.737475			0.368737	+	0.929534i	$0.379790\pi$
$500$	373.051	−	38.1838i	0.746102	−	0.0763675i
$501$		0			0
$502$		−	639.823i		−	1.27455i
$503$	−275.158			−0.547034			−0.273517	−	0.961867i	$-0.588187\pi$
$503$	−275.158			−0.547034			−0.273517	+	0.961867i	$0.588187\pi$
$504$		0			0
$505$	576.000	−	359.199i	1.14059	−	0.711285i
$506$			59.3970i			0.117385i
$507$		0			0
$508$		−	505.124i		−	0.994338i
$509$			118.794i			0.233387i	0.993168	+	0.116693i	$0.0372295\pi$
$509$			118.794i			0.233387i	−0.993168	+	0.116693i	$0.962770\pi$
$510$		0			0
$511$	756.000			1.47945
$512$	552.962			1.08000
$513$		0			0
$514$	616.000			1.19844
$515$	238.118	−	148.492i	0.462364	−	0.288335i
$516$		0			0
$517$			89.7998i			0.173694i
$518$	1000.09			1.93068
$519$		0			0
$520$	126.000	−	78.5748i	0.242308	−	0.151105i
$521$		−	89.0955i		−	0.171009i	−0.996338	−	0.0855043i	$-0.972750\pi$
$521$		−	89.0955i		−	0.171009i	0.996338	−	0.0855043i	$-0.0272501\pi$
$522$		0			0
$523$			875.548i			1.67409i	0.547136	+	0.837044i	$0.315718\pi$
$523$			875.548i			1.67409i	−0.547136	+	0.837044i	$0.684282\pi$
$524$		−	445.477i		−	0.850147i
$525$		0			0
$526$	−434.000			−0.825095
$527$	275.158			0.522122
$528$		0			0
$529$	−501.000			−0.947070
$530$	592.648	+	950.352i	1.11820	+	1.79312i
$531$		0			0
$532$			673.498i			1.26597i
$533$	619.106			1.16155
$534$		0			0
$535$	−28.0000	−	44.8999i	−0.0523364	−	0.0839250i
$536$		−	237.588i		−	0.443261i
$537$		0			0
$538$		−	1414.35i		−	2.62890i
$539$			326.683i			0.606092i
$540$		0			0
$541$	434.000			0.802218			0.401109	−	0.916030i	$-0.368625\pi$
$541$	434.000			0.802218			0.401109	+	0.916030i	$0.368625\pi$
$542$	−756.685			−1.39610
$543$		0			0
$544$	−420.000			−0.772059
$545$	185.203	+	296.985i	0.339821	+	0.544926i
$546$		0			0
$547$			112.250i			0.205210i	0.994722	+	0.102605i	$0.0327177\pi$
$547$			112.250i			0.205210i	−0.994722	+	0.102605i	$0.967282\pi$
$548$	−634.980			−1.15872
$549$		0			0
$550$	252.000	+	123.475i	0.458182	+	0.224499i
$551$			169.706i			0.307996i
$552$		0			0
$553$			157.150i			0.284177i
$554$		−	504.874i		−	0.911325i
$555$		0			0
$556$	618.000			1.11151
$557$	−465.652			−0.836000			−0.418000	−	0.908447i	$-0.637269\pi$
$557$	−465.652			−0.836000			−0.418000	+	0.908447i	$0.637269\pi$
$558$		0			0
$559$	−252.000			−0.450805
$560$	−904.847	+	564.271i	−1.61580	+	1.00763i
$561$		0			0
$562$		−	213.274i		−	0.379492i
$563$	−52.9150			−0.0939876			−0.0469938	−	0.998895i	$-0.514964\pi$
$563$	−52.9150			−0.0939876			−0.0469938	+	0.998895i	$0.514964\pi$
$564$		0			0
$565$	−364.000	−	583.699i	−0.644248	−	1.03309i
$566$			237.588i			0.419767i
$567$		0			0
$568$		−	134.700i		−	0.237147i
$569$		−	640.639i		−	1.12590i	−0.826490	−	0.562951i	$-0.809666\pi$
$569$		−	640.639i		−	1.12590i	0.826490	−	0.562951i	$-0.190334\pi$
$570$		0			0
$571$	−568.000			−0.994746			−0.497373	−	0.867537i	$-0.665702\pi$
$571$	−568.000			−0.994746			−0.497373	+	0.867537i	$0.665702\pi$
$572$	−142.871			−0.249774
$573$		0			0
$574$	−1638.00			−2.85366
$575$	58.2065	−	118.794i	0.101229	−	0.206598i
$576$		0			0
$577$			67.3498i			0.116724i	0.998295	+	0.0583621i	$0.0185878\pi$
$577$			67.3498i			0.116724i	−0.998295	+	0.0583621i	$0.981412\pi$
$578$	−468.298			−0.810204
$579$		0			0
$580$	108.000	−	67.3498i	0.186207	−	0.116120i
$581$			831.558i			1.43125i
$582$		0			0
$583$			359.199i			0.616122i
$584$			178.191i			0.305121i
$585$		0			0
$586$	−1526.00			−2.60410
$587$	1026.55			1.74881			0.874405	−	0.485197i	$-0.161252\pi$
$587$	1026.55			1.74881			0.874405	+	0.485197i	$0.161252\pi$
$588$		0			0
$589$	520.000			0.882852
$590$	−523.859	+	326.683i	−0.887896	+	0.553701i
$591$		0			0
$592$			639.823i			1.08078i
$593$	−656.146			−1.10649			−0.553243	−	0.833020i	$-0.686610\pi$
$593$	−656.146			−1.10649			−0.553243	+	0.833020i	$0.686610\pi$
$594$		0			0
$595$	504.000	−	314.299i	0.847059	−	0.528234i
$596$		−	407.294i		−	0.683378i
$597$		0			0
$598$			157.150i			0.262792i
$599$		−	924.896i		−	1.54407i	−0.635582	−	0.772033i	$-0.719240\pi$
$599$		−	924.896i		−	1.54407i	0.635582	−	0.772033i	$-0.280760\pi$
$600$		0			0
$601$	788.000			1.31115			0.655574	−	0.755131i	$-0.272427\pi$
$601$	788.000			1.31115			0.655574	+	0.755131i	$0.272427\pi$
$602$	666.729			1.10752
$603$		0			0
$604$	−606.000			−1.00331
$605$	−272.512	−	436.992i	−0.450434	−	0.722301i
$606$		0			0
$607$		−	11.2250i		−	0.0184925i	−0.999957	−	0.00924627i	$-0.997057\pi$
$607$		−	11.2250i		−	0.0184925i	0.999957	−	0.00924627i	$-0.00294322\pi$
$608$	−793.725			−1.30547
$609$		0			0
$610$	154.000	+	246.949i	0.252459	+	0.404835i
$611$			237.588i			0.388851i
$612$		0			0
$613$		−	572.474i		−	0.933888i	−0.884287	−	0.466944i	$-0.845355\pi$
$613$		−	572.474i		−	0.933888i	0.884287	−	0.466944i	$-0.154645\pi$
$614$			712.764i			1.16085i
$615$		0			0
$616$	−126.000			−0.204545
$617$	423.320			0.686094			0.343047	−	0.939318i	$-0.388541\pi$
$617$	423.320			0.686094			0.343047	+	0.939318i	$0.388541\pi$
$618$		0			0
$619$	194.000			0.313409			0.156704	−	0.987646i	$-0.449913\pi$
$619$	194.000			0.313409			0.156704	+	0.987646i	$0.449913\pi$
$620$	−206.369	−	330.926i	−0.332853	−	0.533752i
$621$		0			0
$622$		−	157.150i		−	0.252652i
$623$	1000.09			1.60529
$624$		0			0
$625$	−383.000	−	493.899i	−0.612800	−	0.790238i
$626$			475.176i			0.759067i
$627$		0			0
$628$			168.375i			0.268112i
$629$		−	356.382i		−	0.566585i
$630$		0			0
$631$	1190.00			1.88590			0.942948	−	0.332941i	$-0.108041\pi$
$631$	1190.00			1.88590			0.942948	+	0.332941i	$0.108041\pi$
$632$	−37.0405			−0.0586084
$633$		0			0
$634$	826.000			1.30284
$635$	−714.353	+	445.477i	−1.12497	+	0.701539i
$636$		0			0
$637$			864.323i			1.35686i
$638$	95.2470			0.149290
$639$		0			0
$640$	−217.000	−	347.974i	−0.339062	−	0.543710i
$641$			708.521i			1.10534i	0.833401	+	0.552668i	$0.186390\pi$
$641$			708.521i			1.10534i	−0.833401	+	0.552668i	$0.813610\pi$
$642$		0			0
$643$		−	651.048i		−	1.01252i	−0.862382	−	0.506258i	$-0.831028\pi$
$643$		−	651.048i		−	1.01252i	0.862382	−	0.506258i	$-0.168972\pi$
$644$		−	178.191i		−	0.276694i
$645$		0			0
$646$	560.000			0.866873
$647$	1058.30			1.63570			0.817852	−	0.575429i	$-0.195165\pi$
$647$	1058.30			1.63570			0.817852	+	0.575429i	$0.195165\pi$
$648$		0			0
$649$	−198.000			−0.305085
$650$	666.729	+	326.683i	1.02574	+	0.502590i
$651$		0			0
$652$			606.148i			0.929676i
$653$	470.944			0.721200			0.360600	−	0.932721i	$-0.382572\pi$
$653$	470.944			0.721200			0.360600	+	0.932721i	$0.382572\pi$
$654$		0			0
$655$	−630.000	+	392.874i	−0.961832	+	0.599808i
$656$		−	1047.93i		−	1.59746i
$657$		0			0
$658$		−	628.598i		−	0.955317i
$659$		−	80.6102i		−	0.122322i	−0.998128	−	0.0611610i	$-0.980520\pi$
$659$		−	80.6102i		−	0.122322i	0.998128	−	0.0611610i	$-0.0194803\pi$
$660$		0			0
$661$	170.000			0.257186			0.128593	−	0.991697i	$-0.458954\pi$
$661$	170.000			0.257186			0.128593	+	0.991697i	$0.458954\pi$
$662$	−296.324			−0.447620
$663$		0			0
$664$	−196.000			−0.295181
$665$	952.470	−	593.970i	1.43229	−	0.893188i
$666$		0			0
$667$		−	44.8999i		−	0.0673162i
$668$	555.608			0.831748
$669$		0			0
$670$	1008.00	−	628.598i	1.50448	−	0.938207i
$671$			93.3381i			0.139103i
$672$		0			0
$673$			1055.15i			1.56783i	0.620870	+	0.783913i	$0.286779\pi$
$673$			1055.15i			1.56783i	−0.620870	+	0.783913i	$0.713221\pi$
$674$		−	296.985i		−	0.440630i
$675$		0			0
$676$	129.000			0.190828
$677$	−830.766			−1.22713			−0.613564	−	0.789645i	$-0.710265\pi$
$677$	−830.766			−1.22713			−0.613564	+	0.789645i	$0.710265\pi$
$678$		0			0
$679$	252.000			0.371134
$680$	74.0810	+	118.794i	0.108943	+	0.174697i
$681$		0			0
$682$		−	291.849i		−	0.427931i
$683$	−1037.13			−1.51850			−0.759249	−	0.650800i	$-0.774434\pi$
$683$	−1037.13			−1.51850			−0.759249	+	0.650800i	$0.774434\pi$
$684$		0			0
$685$	560.000	+	897.998i	0.817518	+	1.31095i
$686$		−	831.558i		−	1.21218i
$687$		0			0
$688$			426.549i			0.619984i
$689$			950.352i			1.37932i
$690$		0			0
$691$	−652.000			−0.943560			−0.471780	−	0.881716i	$-0.656388\pi$
$691$	−652.000			−0.943560			−0.471780	+	0.881716i	$0.656388\pi$
$692$	−63.4980			−0.0917602
$693$		0			0
$694$	1372.00			1.97695
$695$	−545.025	−	873.984i	−0.784208	−	1.25753i
$696$		0			0
$697$			583.699i			0.837444i
$698$	322.782			0.462438
$699$		0			0
$700$	−756.000	−	370.424i	−1.08000	−	0.529177i
$701$		−	763.675i		−	1.08941i	−0.838628	−	0.544704i	$-0.816642\pi$
$701$		−	763.675i		−	1.08941i	0.838628	−	0.544704i	$-0.183358\pi$
$702$		0			0
$703$		−	673.498i		−	0.958035i
$704$			123.037i			0.174768i
$705$		0			0
$706$	−1064.00			−1.50708
$707$	−1523.95			−2.15552
$708$		0			0
$709$	158.000			0.222849			0.111425	−	0.993773i	$-0.464459\pi$
$709$	158.000			0.222849			0.111425	+	0.993773i	$0.464459\pi$
$710$	571.482	−	356.382i	0.804905	−	0.501946i
$711$		0			0
$712$			235.724i			0.331074i
$713$	−137.579			−0.192958
$714$		0			0
$715$	126.000	+	202.049i	0.176224	+	0.282587i
$716$			725.492i			1.01326i
$717$		0			0
$718$			1683.75i			2.34505i
$719$			127.279i			0.177023i	0.996075	+	0.0885113i	$0.0282109\pi$
$719$			127.279i			0.177023i	−0.996075	+	0.0885113i	$0.971789\pi$
$720$		0			0
$721$	−630.000			−0.873786
$722$	103.184			0.142915
$723$		0			0
$724$	222.000			0.306630
$725$	−190.494	−	93.3381i	−0.262750	−	0.128742i
$726$		0			0
$727$		−	662.273i		−	0.910967i	−0.890244	−	0.455484i	$-0.849466\pi$
$727$		−	662.273i		−	0.910967i	0.890244	−	0.455484i	$-0.150534\pi$
$728$	−333.365			−0.457918
$729$		0			0
$730$	−756.000	+	471.449i	−1.03562	+	0.645820i
$731$		−	237.588i		−	0.325018i
$732$		0			0
$733$		−	684.723i		−	0.934138i	−0.884221	−	0.467069i	$-0.845310\pi$
$733$		−	684.723i		−	0.934138i	0.884221	−	0.467069i	$-0.154690\pi$
$734$			1811.61i			2.46813i
$735$		0			0
$736$	210.000			0.285326
$737$	380.988			0.516945
$738$		0			0
$739$	−1240.00			−1.67794			−0.838972	−	0.544175i	$-0.816843\pi$
$739$	−1240.00			−1.67794			−0.838972	+	0.544175i	$0.816843\pi$
$740$	−428.612	+	267.286i	−0.579205	+	0.361198i
$741$		0			0
$742$		−	2514.39i		−	3.38867i
$743$	42.3320			0.0569745			0.0284872	−	0.999594i	$-0.490931\pi$
$743$	42.3320			0.0569745			0.0284872	+	0.999594i	$0.490931\pi$
$744$		0			0
$745$	−576.000	+	359.199i	−0.773154	+	0.482146i
$746$		−	386.080i		−	0.517534i
$747$		0			0
$748$		−	134.700i		−	0.180080i
$749$			118.794i			0.158603i
$750$		0			0
$751$	−154.000			−0.205060			−0.102530	−	0.994730i	$-0.532694\pi$
$751$	−154.000			−0.205060			−0.102530	+	0.994730i	$0.532694\pi$
$752$	402.154			0.534780
$753$		0			0
$754$	252.000			0.334218
$755$	534.442	+	857.013i	0.707870	+	1.13512i
$756$		0			0
$757$		−	1313.32i		−	1.73490i	−0.497522	−	0.867452i	$-0.665756\pi$
$757$		−	1313.32i		−	1.73490i	0.497522	−	0.867452i	$-0.334244\pi$
$758$	957.762			1.26354
$759$		0			0
$760$	140.000	+	224.499i	0.184211	+	0.295394i
$761$		−	504.874i		−	0.663435i	−0.943379	−	0.331718i	$-0.892372\pi$
$761$		−	504.874i		−	0.663435i	0.943379	−	0.331718i	$-0.107628\pi$
$762$		0			0
$763$		−	785.748i		−	1.02981i
$764$			483.661i			0.633064i
$765$		0			0
$766$	112.000			0.146214
$767$	−523.859			−0.682997
$768$		0			0
$769$	368.000			0.478544			0.239272	−	0.970953i	$-0.423091\pi$
$769$	368.000			0.478544			0.239272	+	0.970953i	$0.423091\pi$
$770$	−333.365	−	534.573i	−0.432941	−	0.694250i
$771$		0			0
$772$		−	538.799i		−	0.697926i
$773$	153.454			0.198517			0.0992585	−	0.995062i	$-0.468353\pi$
$773$	153.454			0.198517			0.0992585	+	0.995062i	$0.468353\pi$
$774$		0			0
$775$	−286.000	+	583.699i	−0.369032	+	0.753159i
$776$			59.3970i			0.0765425i
$777$		0			0
$778$			695.948i			0.894535i
$779$			1103.09i			1.41603i
$780$		0			0
$781$	216.000			0.276569
$782$	−148.162			−0.189466
$783$		0			0
$784$	1463.00			1.86607
$785$	238.118	−	148.492i	0.303335	−	0.189162i
$786$		0			0
$787$			426.549i			0.541994i	0.962580	+	0.270997i	$0.0873533\pi$
$787$			426.549i			0.541994i	−0.962580	+	0.270997i	$0.912647\pi$
$788$	−111.122

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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	45.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.64575 q^{2} +3.00000 q^{4} +(-2.64575 - 4.24264i) q^{5} +11.2250i q^{7} -2.64575 q^{8} +O(q^{10})\)	\(q+2.64575 q^{2} +3.00000 q^{4} +(-2.64575 - 4.24264i) q^{5} +11.2250i q^{7} -2.64575 q^{8} +(-7.00000 - 11.2250i) q^{10} -4.24264i q^{11} -11.2250i q^{13} +29.6985i q^{14} -19.0000 q^{16} +10.5830 q^{17} +20.0000 q^{19} +(-7.93725 - 12.7279i) q^{20} -11.2250i q^{22} -5.29150 q^{23} +(-11.0000 + 22.4499i) q^{25} -29.6985i q^{26} +33.6749i q^{28} +8.48528i q^{29} +26.0000 q^{31} -39.6863 q^{32} +28.0000 q^{34} +(47.6235 - 29.6985i) q^{35} -33.6749i q^{37} +52.9150 q^{38} +(7.00000 + 11.2250i) q^{40} +55.1543i q^{41} -22.4499i q^{43} -12.7279i q^{44} -14.0000 q^{46} -21.1660 q^{47} -77.0000 q^{49} +(-29.1033 + 59.3970i) q^{50} -33.6749i q^{52} -84.6640 q^{53} +(-18.0000 + 11.2250i) q^{55} -29.6985i q^{56} +22.4499i q^{58} -46.6690i q^{59} -22.0000 q^{61} +68.7895 q^{62} -29.0000 q^{64} +(-47.6235 + 29.6985i) q^{65} +89.7998i q^{67} +31.7490 q^{68} +(126.000 - 78.5748i) q^{70} +50.9117i q^{71} -67.3498i q^{73} -89.0955i q^{74} +60.0000 q^{76} +47.6235 q^{77} +14.0000 q^{79} +(50.2693 + 80.6102i) q^{80} +145.925i q^{82} +74.0810 q^{83} +(-28.0000 - 44.8999i) q^{85} -59.3970i q^{86} +11.2250i q^{88} -89.0955i q^{89} +126.000 q^{91} -15.8745 q^{92} -56.0000 q^{94} +(-52.9150 - 84.8528i) q^{95} -22.4499i q^{97} -203.723 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{4}+O(q^{10}) \) 4 * q + 12 * q^4	\( 4 q + 12 q^{4} - 28 q^{10} - 76 q^{16} + 80 q^{19} - 44 q^{25} + 104 q^{31} + 112 q^{34} + 28 q^{40} - 56 q^{46} - 308 q^{49} - 72 q^{55} - 88 q^{61} - 116 q^{64} + 504 q^{70} + 240 q^{76} + 56 q^{79} - 112 q^{85} + 504 q^{91} - 224 q^{94}+O(q^{100}) \) 4 * q + 12 * q^4 - 28 * q^10 - 76 * q^16 + 80 * q^19 - 44 * q^25 + 104 * q^31 + 112 * q^34 + 28 * q^40 - 56 * q^46 - 308 * q^49 - 72 * q^55 - 88 * q^61 - 116 * q^64 + 504 * q^70 + 240 * q^76 + 56 * q^79 - 112 * q^85 + 504 * q^91 - 224 * q^94

Introduction

L-functions

Modular forms

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