LMFDB - Embedded newform 1980.2.y.c.1693.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1980,2,Mod(1297,1980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1980, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))

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

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

chi := DirichletCharacter("1980.1297");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1980.y (of order $4$, degree $2$, 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:	$15.8103796002$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 660)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1693.2
Character	$\chi$	$=$	1980.1693
Dual form			1980.2.y.c.1297.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.05777 + 0.874978i) q^{5} +(2.32148 + 2.32148i) q^{7} +O(q^{10})$	$q+(-2.05777 + 0.874978i) q^{5} +(2.32148 + 2.32148i) q^{7} +(-2.23741 + 2.44827i) q^{11} +(-3.15629 + 3.15629i) q^{13} +(-4.61362 - 4.61362i) q^{17} +0.600117 q^{19} +(0.436835 + 0.436835i) q^{23} +(3.46883 - 3.60100i) q^{25} +1.40139 q^{29} -7.67097 q^{31} +(-6.80832 - 2.74583i) q^{35} +(3.84287 - 3.84287i) q^{37} -9.19555i q^{41} +(8.66683 - 8.66683i) q^{43} +(-8.22477 + 8.22477i) q^{47} +3.77856i q^{49} +(-3.72320 - 3.72320i) q^{53} +(2.46188 - 6.99565i) q^{55} +7.10736i q^{59} -2.28178i q^{61} +(3.73323 - 9.25659i) q^{65} +(-1.71070 + 1.71070i) q^{67} +2.25804 q^{71} +(1.34887 - 1.34887i) q^{73} +(-10.8777 + 0.489514i) q^{77} -5.85526 q^{79} +(11.7602 - 11.7602i) q^{83} +(13.5306 + 5.45695i) q^{85} -5.41573i q^{89} -14.6545 q^{91} +(-1.23490 + 0.525089i) q^{95} +(-12.3177 + 12.3177i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 8 q^{5}+O(q^{10}) $ 24 * q - 8 * q^5	$ 24 q - 8 q^{5} - 8 q^{11} - 16 q^{23} + 24 q^{25} + 16 q^{31} - 40 q^{37} - 64 q^{47} - 24 q^{53} + 24 q^{55} + 48 q^{67} - 24 q^{77} - 48 q^{91} - 72 q^{97}+O(q^{100}) $ 24 * q - 8 * q^5 - 8 * q^11 - 16 * q^23 + 24 * q^25 + 16 * q^31 - 40 * q^37 - 64 * q^47 - 24 * q^53 + 24 * q^55 + 48 * q^67 - 24 * q^77 - 48 * q^91 - 72 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1980\mathbb{Z}\right)^\times$.

$n$	$397$	$541$	$991$	$1541$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	−2.05777	+	0.874978i	−0.920262	+	0.391302i
$5$	−2.05777	+	0.874978i	−0.920262	+	0.391302i
$6$		0			0
$7$	2.32148	+	2.32148i	0.877438	+	0.877438i	0.993269	−	0.115831i	$-0.0369531\pi$
$7$	2.32148	+	2.32148i	0.877438	+	0.877438i	−0.115831	+	0.993269i	$0.536953\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.23741	+	2.44827i	−0.674603	+	0.738181i
$11$	−2.23741	+	2.44827i	−0.674603	+	0.738181i
$12$		0			0
$13$	−3.15629	+	3.15629i	−0.875396	+	0.875396i	−0.993054	−	0.117658i	$-0.962461\pi$
$13$	−3.15629	+	3.15629i	−0.875396	+	0.875396i	0.117658	+	0.993054i	$0.462461\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.61362	−	4.61362i	−1.11897	−	1.11897i	−0.991893	−	0.127075i	$-0.959441\pi$
$17$	−4.61362	−	4.61362i	−1.11897	−	1.11897i	−0.127075	−	0.991893i	$-0.540559\pi$
$18$		0			0
$19$	0.600117			0.137676			0.0688382	−	0.997628i	$-0.478071\pi$
$19$	0.600117			0.137676			0.0688382	+	0.997628i	$0.478071\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.436835	+	0.436835i	0.0910865	+	0.0910865i	0.751182	−	0.660095i	$-0.229484\pi$
$23$	0.436835	+	0.436835i	0.0910865	+	0.0910865i	−0.660095	+	0.751182i	$0.729484\pi$
$24$		0			0
$25$	3.46883	−	3.60100i	0.693766	−	0.720201i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.40139			0.260231			0.130116	−	0.991499i	$-0.458465\pi$
$29$	1.40139			0.260231			0.130116	+	0.991499i	$0.458465\pi$
$30$		0			0
$31$	−7.67097			−1.37775			−0.688874	−	0.724881i	$-0.741895\pi$
$31$	−7.67097			−1.37775			−0.688874	+	0.724881i	$0.741895\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.80832	−	2.74583i	−1.15082	−	0.464130i
$36$		0			0
$37$	3.84287	−	3.84287i	0.631764	−	0.631764i	−0.316746	−	0.948510i	$-0.602590\pi$
$37$	3.84287	−	3.84287i	0.631764	−	0.631764i	0.948510	+	0.316746i	$0.102590\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	9.19555i		−	1.43610i	−0.695990	−	0.718051i	$-0.745034\pi$
$41$		−	9.19555i		−	1.43610i	0.695990	−	0.718051i	$-0.254966\pi$
$42$		0			0
$43$	8.66683	−	8.66683i	1.32168	−	1.32168i	0.409262	−	0.912417i	$-0.365786\pi$
$43$	8.66683	−	8.66683i	1.32168	−	1.32168i	0.912417	−	0.409262i	$-0.134214\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.22477	+	8.22477i	−1.19971	+	1.19971i	−0.225451	+	0.974255i	$0.572386\pi$
$47$	−8.22477	+	8.22477i	−1.19971	+	1.19971i	−0.974255	+	0.225451i	$0.927614\pi$
$48$		0			0
$49$			3.77856i			0.539794i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.72320	−	3.72320i	−0.511421	−	0.511421i	0.403541	−	0.914962i	$-0.367779\pi$
$53$	−3.72320	−	3.72320i	−0.511421	−	0.511421i	−0.914962	+	0.403541i	$0.867779\pi$
$54$		0			0
$55$	2.46188	−	6.99565i	0.331960	−	0.943293i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			7.10736i			0.925300i	0.886541	+	0.462650i	$0.153101\pi$
$59$			7.10736i			0.925300i	−0.886541	+	0.462650i	$0.846899\pi$
$60$		0			0
$61$		−	2.28178i		−	0.292152i	−0.989273	−	0.146076i	$-0.953336\pi$
$61$		−	2.28178i		−	0.292152i	0.989273	−	0.146076i	$-0.0466645\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.73323	−	9.25659i	0.463050	−	1.14814i
$66$		0			0
$67$	−1.71070	+	1.71070i	−0.208995	+	0.208995i	−0.803840	−	0.594845i	$-0.797213\pi$
$67$	−1.71070	+	1.71070i	−0.208995	+	0.208995i	0.594845	+	0.803840i	$0.297213\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.25804			0.267981			0.133990	−	0.990983i	$-0.457221\pi$
$71$	2.25804			0.267981			0.133990	+	0.990983i	$0.457221\pi$
$72$		0			0
$73$	1.34887	−	1.34887i	0.157873	−	0.157873i	−0.623750	−	0.781624i	$-0.714392\pi$
$73$	1.34887	−	1.34887i	0.157873	−	0.157873i	0.781624	+	0.623750i	$0.214392\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.8777	+	0.489514i	−1.23963	+	0.0557853i
$78$		0			0
$79$	−5.85526			−0.658768			−0.329384	−	0.944196i	$-0.606841\pi$
$79$	−5.85526			−0.658768			−0.329384	+	0.944196i	$0.606841\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.7602	−	11.7602i	1.29085	−	1.29085i	0.356595	−	0.934259i	$-0.383938\pi$
$83$	11.7602	−	11.7602i	1.29085	−	1.29085i	0.934259	−	0.356595i	$-0.116062\pi$
$84$		0			0
$85$	13.5306	+	5.45695i	1.46760	+	0.591890i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	5.41573i		−	0.574067i	−0.957921	−	0.287033i	$-0.907331\pi$
$89$		−	5.41573i		−	0.574067i	0.957921	−	0.287033i	$-0.0926690\pi$
$90$		0			0
$91$	−14.6545			−1.53621
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.23490	+	0.525089i	−0.126698	+	0.0538730i
$96$		0			0
$97$	−12.3177	+	12.3177i	−1.25067	+	1.25067i	−0.295252	+	0.955420i	$0.595404\pi$
$97$	−12.3177	+	12.3177i	−1.25067	+	1.25067i	−0.955420	+	0.295252i	$0.904596\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			13.8178i			1.37492i	0.726221	+	0.687461i	$0.241275\pi$
$101$			13.8178i			1.37492i	−0.726221	+	0.687461i	$0.758725\pi$
$102$		0			0
$103$	−3.60147	−	3.60147i	−0.354863	−	0.354863i	0.507052	−	0.861915i	$-0.330735\pi$
$103$	−3.60147	−	3.60147i	−0.354863	−	0.354863i	−0.861915	+	0.507052i	$0.830735\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.73323	+	3.73323i	0.360905	+	0.360905i	0.864146	−	0.503241i	$-0.167859\pi$
$107$	3.73323	+	3.73323i	0.360905	+	0.360905i	−0.503241	+	0.864146i	$0.667859\pi$
$108$		0			0
$109$	−19.6154			−1.87882			−0.939409	−	0.342798i	$-0.888626\pi$
$109$	−19.6154			−1.87882			−0.939409	+	0.342798i	$0.888626\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.0119	−	10.0119i	−0.941845	−	0.941845i	0.0565545	−	0.998400i	$-0.481989\pi$
$113$	−10.0119	−	10.0119i	−0.941845	−	0.941845i	−0.998400	+	0.0565545i	$0.981989\pi$
$114$		0			0
$115$	−1.28113	−	0.516685i	−0.119466	−	0.0481811i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	21.4209i		−	1.96365i
$120$		0			0
$121$	−0.988033	−	10.9555i	−0.0898212	−	0.995958i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.98725	+	10.4452i	−0.356630	+	0.934246i
$126$		0			0
$127$	−9.26695	−	9.26695i	−0.822309	−	0.822309i	0.164130	−	0.986439i	$-0.447518\pi$
$127$	−9.26695	−	9.26695i	−0.822309	−	0.822309i	−0.986439	+	0.164130i	$0.947518\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			3.79427i			0.331507i	0.986167	+	0.165753i	$0.0530056\pi$
$131$			3.79427i			0.331507i	−0.986167	+	0.165753i	$0.946994\pi$
$132$		0			0
$133$	1.39316	+	1.39316i	0.120802	+	0.120802i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.53010	−	1.53010i	0.130725	−	0.130725i	−0.638717	−	0.769442i	$-0.720534\pi$
$137$	1.53010	−	1.53010i	0.130725	−	0.130725i	0.769442	+	0.638717i	$0.220534\pi$
$138$		0			0
$139$	−14.3446			−1.21669			−0.608346	−	0.793672i	$-0.708167\pi$
$139$	−14.3446			−1.21669			−0.608346	+	0.793672i	$0.708167\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.665543	−	14.7893i	−0.0556555	−	1.23675i
$144$		0			0
$145$	−2.88373	+	1.22618i	−0.239481	+	0.101829i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.50365			−0.205107			−0.102554	−	0.994727i	$-0.532701\pi$
$149$	−2.50365			−0.205107			−0.102554	+	0.994727i	$0.532701\pi$
$150$		0			0
$151$		−	17.3472i		−	1.41170i	−0.708362	−	0.705849i	$-0.750566\pi$
$151$		−	17.3472i		−	1.41170i	0.708362	−	0.705849i	$-0.249434\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	15.7851	−	6.71193i	1.26789	−	0.539115i
$156$		0			0
$157$	−8.57371	+	8.57371i	−0.684257	+	0.684257i	−0.960956	−	0.276700i	$-0.910759\pi$
$157$	−8.57371	+	8.57371i	−0.684257	+	0.684257i	0.276700	+	0.960956i	$0.410759\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			2.02821i			0.159845i
$162$		0			0
$163$	6.32282	+	6.32282i	0.495241	+	0.495241i	0.909953	−	0.414712i	$-0.136118\pi$
$163$	6.32282	+	6.32282i	0.495241	+	0.495241i	−0.414712	+	0.909953i	$0.636118\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.23430	−	4.23430i	−0.327660	−	0.327660i	0.524036	−	0.851696i	$-0.324426\pi$
$167$	−4.23430	−	4.23430i	−0.327660	−	0.327660i	−0.851696	+	0.524036i	$0.824426\pi$
$168$		0			0
$169$		−	6.92428i		−	0.532637i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.22074	+	2.22074i	−0.168840	+	0.168840i	−0.786469	−	0.617629i	$-0.788093\pi$
$173$	−2.22074	+	2.22074i	−0.168840	+	0.168840i	0.617629	+	0.786469i	$0.288093\pi$
$174$		0			0
$175$	16.4125	−	0.306846i	1.24067	−	0.0231954i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			5.44971i			0.407331i	0.979041	+	0.203665i	$0.0652854\pi$
$179$			5.44971i			0.407331i	−0.979041	+	0.203665i	$0.934715\pi$
$180$		0			0
$181$	9.35615			0.695437			0.347718	−	0.937599i	$-0.386956\pi$
$181$	9.35615			0.695437			0.347718	+	0.937599i	$0.386956\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.54532	+	11.2702i	−0.334178	+	0.828599i
$186$		0			0
$187$	21.6179	−	0.972841i	1.58086	−	0.0711412i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.55894			−0.257516			−0.128758	−	0.991676i	$-0.541099\pi$
$191$	−3.55894			−0.257516			−0.128758	+	0.991676i	$0.541099\pi$
$192$		0			0
$193$	−6.03819	+	6.03819i	−0.434638	+	0.434638i	−0.890203	−	0.455565i	$-0.849437\pi$
$193$	−6.03819	+	6.03819i	−0.434638	+	0.434638i	0.455565	+	0.890203i	$0.349437\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	0.909737	+	0.909737i	0.0648161	+	0.0648161i	0.738772	−	0.673956i	$-0.235406\pi$
$197$	0.909737	+	0.909737i	0.0648161	+	0.0648161i	−0.673956	+	0.738772i	$0.735406\pi$
$198$		0			0
$199$			27.5780i			1.95495i	0.211049	+	0.977476i	$0.432312\pi$
$199$			27.5780i			1.95495i	−0.211049	+	0.977476i	$0.567688\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.25330	+	3.25330i	0.228337	+	0.228337i
$204$		0			0
$205$	8.04590	+	18.9223i	0.561950	+	1.32159i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.34271	+	1.46925i	−0.0928769	+	0.101630i
$210$		0			0
$211$			12.7166i			0.875447i	0.899110	+	0.437723i	$0.144215\pi$
$211$			12.7166i			0.875447i	−0.899110	+	0.437723i	$0.855785\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−10.2511	+	25.4176i	−0.699116	+	1.73347i
$216$		0			0
$217$	−17.8080	−	17.8080i	−1.20889	−	1.20889i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	29.1238			1.95908
$222$		0			0
$223$	−16.3544	−	16.3544i	−1.09517	−	1.09517i	−0.994967	−	0.100207i	$-0.968050\pi$
$223$	−16.3544	−	16.3544i	−1.09517	−	1.09517i	−0.100207	−	0.994967i	$-0.531950\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.33341	−	7.33341i	−0.486735	−	0.486735i	0.420539	−	0.907274i	$-0.361841\pi$
$227$	−7.33341	−	7.33341i	−0.486735	−	0.486735i	−0.907274	+	0.420539i	$0.861841\pi$
$228$		0			0
$229$			8.09535i			0.534955i	0.963564	+	0.267478i	$0.0861902\pi$
$229$			8.09535i			0.534955i	−0.963564	+	0.267478i	$0.913810\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	17.2241	−	17.2241i	1.12839	−	1.12839i	0.137950	−	0.990439i	$-0.455949\pi$
$233$	17.2241	−	17.2241i	1.12839	−	1.12839i	0.990439	−	0.137950i	$-0.0440515\pi$
$234$		0			0
$235$	9.72818	−	24.1212i	0.634597	−	1.57349i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.86607			0.444129			0.222064	−	0.975032i	$-0.428720\pi$
$239$	6.86607			0.444129			0.222064	+	0.975032i	$0.428720\pi$
$240$		0			0
$241$			6.35034i			0.409061i	0.978860	+	0.204531i	$0.0655668\pi$
$241$			6.35034i			0.409061i	−0.978860	+	0.204531i	$0.934433\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.30616	−	7.77541i	−0.211223	−	0.496752i
$246$		0			0
$247$	−1.89414	+	1.89414i	−0.120521	+	0.120521i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7.72752			−0.487757			−0.243878	−	0.969806i	$-0.578420\pi$
$251$	−7.72752			−0.487757			−0.243878	+	0.969806i	$0.578420\pi$
$252$		0			0
$253$	−2.04687	+	0.0921123i	−0.128686	+	0.00579105i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−12.9325	+	12.9325i	−0.806708	+	0.806708i	−0.984134	−	0.177426i	$-0.943223\pi$
$257$	−12.9325	+	12.9325i	−0.806708	+	0.806708i	0.177426	+	0.984134i	$0.443223\pi$
$258$		0			0
$259$	17.8423			1.10867
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.80627	−	5.80627i	0.358030	−	0.358030i	−0.505056	−	0.863086i	$-0.668528\pi$
$263$	5.80627	−	5.80627i	0.358030	−	0.358030i	0.863086	+	0.505056i	$0.168528\pi$
$264$		0			0
$265$	10.9192	+	4.40377i	0.670761	+	0.270521i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			26.0590i			1.58884i	0.607367	+	0.794421i	$0.292226\pi$
$269$			26.0590i			1.58884i	−0.607367	+	0.794421i	$0.707774\pi$
$270$		0			0
$271$			12.5992i			0.765349i	0.923883	+	0.382675i	$0.124997\pi$
$271$			12.5992i			0.765349i	−0.923883	+	0.382675i	$0.875003\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.05505	+	16.5495i	0.0636220	+	0.997974i
$276$		0			0
$277$	−1.96105	−	1.96105i	−0.117828	−	0.117828i	0.645734	−	0.763562i	$-0.276551\pi$
$277$	−1.96105	−	1.96105i	−0.117828	−	0.117828i	−0.763562	+	0.645734i	$0.776551\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			19.8559i			1.18450i	0.805753	+	0.592251i	$0.201761\pi$
$281$			19.8559i			1.18450i	−0.805753	+	0.592251i	$0.798239\pi$
$282$		0			0
$283$	7.47159	−	7.47159i	0.444140	−	0.444140i	−0.449261	−	0.893401i	$-0.648313\pi$
$283$	7.47159	−	7.47159i	0.444140	−	0.444140i	0.893401	+	0.449261i	$0.148313\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	21.3473	−	21.3473i	1.26009	−	1.26009i
$288$		0			0
$289$			25.5710i			1.50418i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−8.89798	+	8.89798i	−0.519826	+	0.519826i	−0.917519	−	0.397693i	$-0.869811\pi$
$293$	−8.89798	+	8.89798i	−0.519826	+	0.519826i	0.397693	+	0.917519i	$0.369811\pi$
$294$		0			0
$295$	−6.21878	−	14.6253i	−0.362072	−	0.851518i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.75756			−0.159474
$300$		0			0
$301$	40.2398			2.31938
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.99651	+	4.69538i	0.114320	+	0.268857i
$306$		0			0
$307$	13.3561	+	13.3561i	0.762275	+	0.762275i	0.976733	−	0.214458i	$-0.0687984\pi$
$307$	13.3561	+	13.3561i	0.762275	+	0.762275i	−0.214458	+	0.976733i	$0.568798\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	14.5350			0.824202			0.412101	−	0.911138i	$-0.364795\pi$
$311$	14.5350			0.824202			0.412101	+	0.911138i	$0.364795\pi$
$312$		0			0
$313$	−14.2330	−	14.2330i	−0.804499	−	0.804499i	0.179296	−	0.983795i	$-0.442618\pi$
$313$	−14.2330	−	14.2330i	−0.804499	−	0.804499i	−0.983795	+	0.179296i	$0.942618\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.7296	−	10.7296i	0.602634	−	0.602634i	−0.338376	−	0.941011i	$-0.609878\pi$
$317$	10.7296	−	10.7296i	0.602634	−	0.602634i	0.941011	+	0.338376i	$0.109878\pi$
$318$		0			0
$319$	−3.13547	+	3.43097i	−0.175553	+	0.192098i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.76871	−	2.76871i	−0.154055	−	0.154055i
$324$		0			0
$325$	0.417188	+	22.3144i	0.0231414	+	1.23778i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−38.1873			−2.10533
$330$		0			0
$331$	18.1262			0.996306			0.498153	−	0.867089i	$-0.334012\pi$
$331$	18.1262			0.996306			0.498153	+	0.867089i	$0.334012\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.02340	−	5.01704i	0.110550	−	0.274110i
$336$		0			0
$337$	−22.2353	−	22.2353i	−1.21123	−	1.21123i	−0.970621	−	0.240612i	$-0.922652\pi$
$337$	−22.2353	−	22.2353i	−1.21123	−	1.21123i	−0.240612	−	0.970621i	$-0.577348\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	17.1631	−	18.7806i	0.929433	−	1.01703i
$342$		0			0
$343$	7.47851	−	7.47851i	0.403802	−	0.403802i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.4387	−	12.4387i	−0.667744	−	0.667744i	0.289450	−	0.957193i	$-0.406528\pi$
$347$	−12.4387	−	12.4387i	−0.667744	−	0.667744i	−0.957193	+	0.289450i	$0.906528\pi$
$348$		0			0
$349$	9.30138			0.497891			0.248946	−	0.968517i	$-0.419916\pi$
$349$	9.30138			0.497891			0.248946	+	0.968517i	$0.419916\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5.79023	+	5.79023i	0.308183	+	0.308183i	0.844204	−	0.536022i	$-0.180073\pi$
$353$	5.79023	+	5.79023i	0.308183	+	0.308183i	−0.536022	+	0.844204i	$0.680073\pi$
$354$		0			0
$355$	−4.64653	+	1.97574i	−0.246612	+	0.104861i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	31.5247			1.66381			0.831905	−	0.554918i	$-0.187250\pi$
$359$	31.5247			1.66381			0.831905	+	0.554918i	$0.187250\pi$
$360$		0			0
$361$	−18.6399			−0.981045
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.59543	+	3.95590i	−0.0835088	+	0.207061i
$366$		0			0
$367$	16.2922	−	16.2922i	0.850447	−	0.850447i	−0.139741	−	0.990188i	$-0.544627\pi$
$367$	16.2922	−	16.2922i	0.850447	−	0.850447i	0.990188	+	0.139741i	$0.0446270\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	17.2867i		−	0.897480i
$372$		0			0
$373$	−1.46963	+	1.46963i	−0.0760943	+	0.0760943i	−0.744130	−	0.668035i	$-0.767136\pi$
$373$	−1.46963	+	1.46963i	−0.0760943	+	0.0760943i	0.668035	+	0.744130i	$0.267136\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.42318	+	4.42318i	−0.227805	+	0.227805i
$378$		0			0
$379$		−	20.8638i		−	1.07170i	−0.844312	−	0.535851i	$-0.819991\pi$
$379$		−	20.8638i		−	1.07170i	0.844312	−	0.535851i	$-0.180009\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−18.8995	−	18.8995i	−0.965717	−	0.965717i	0.0337146	−	0.999432i	$-0.489266\pi$
$383$	−18.8995	−	18.8995i	−0.965717	−	0.965717i	−0.999432	+	0.0337146i	$0.989266\pi$
$384$		0			0
$385$	21.9555	−	10.5251i	1.11896	−	0.536407i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			28.1471i			1.42711i	0.700597	+	0.713557i	$0.252917\pi$
$389$			28.1471i			1.42711i	−0.700597	+	0.713557i	$0.747083\pi$
$390$		0			0
$391$		−	4.03079i		−	0.203846i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.0488	−	5.12322i	0.606239	−	0.257777i
$396$		0			0
$397$	−7.04885	+	7.04885i	−0.353772	+	0.353772i	−0.861511	−	0.507739i	$-0.830481\pi$
$397$	−7.04885	+	7.04885i	−0.353772	+	0.353772i	0.507739	+	0.861511i	$0.330481\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−26.1684			−1.30679			−0.653394	−	0.757018i	$-0.726656\pi$
$401$	−26.1684			−1.30679			−0.653394	+	0.757018i	$0.726656\pi$
$402$		0			0
$403$	24.2118	−	24.2118i	1.20607	−	1.20607i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.810318	+	18.0064i	0.0401660	+	0.892546i
$408$		0			0
$409$	16.7193			0.826718			0.413359	−	0.910568i	$-0.364355\pi$
$409$	16.7193			0.826718			0.413359	+	0.910568i	$0.364355\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−16.4996	+	16.4996i	−0.811893	+	0.811893i
$414$		0			0
$415$	−13.9099	+	34.4898i	−0.682811	+	1.69304i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	12.1843i		−	0.595241i	−0.954684	−	0.297621i	$-0.903807\pi$
$419$		−	12.1843i		−	0.595241i	0.954684	−	0.297621i	$-0.0961931\pi$
$420$		0			0
$421$	−29.6783			−1.44643			−0.723215	−	0.690622i	$-0.757337\pi$
$421$	−29.6783			−1.44643			−0.723215	+	0.690622i	$0.757337\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−32.6175	+	0.609815i	−1.58218	+	0.0295804i
$426$		0			0
$427$	5.29712	−	5.29712i	0.256346	−	0.256346i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			28.1186i			1.35443i	0.735787	+	0.677213i	$0.236812\pi$
$431$			28.1186i			1.35443i	−0.735787	+	0.677213i	$0.763188\pi$
$432$		0			0
$433$	12.8525	+	12.8525i	0.617652	+	0.617652i	0.944929	−	0.327276i	$-0.106131\pi$
$433$	12.8525	+	12.8525i	0.617652	+	0.617652i	−0.327276	+	0.944929i	$0.606131\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.262152	+	0.262152i	0.0125405	+	0.0125405i
$438$		0			0
$439$	−21.9331			−1.04681			−0.523405	−	0.852084i	$-0.675339\pi$
$439$	−21.9331			−1.04681			−0.523405	+	0.852084i	$0.675339\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.2383	−	10.2383i	−0.486438	−	0.486438i	0.420742	−	0.907180i	$-0.361770\pi$
$443$	−10.2383	−	10.2383i	−0.486438	−	0.486438i	−0.907180	+	0.420742i	$0.861770\pi$
$444$		0			0
$445$	4.73865	+	11.1443i	0.224633	+	0.528292i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			17.1824i			0.810887i	0.914120	+	0.405444i	$0.132883\pi$
$449$			17.1824i			0.810887i	−0.914120	+	0.405444i	$0.867117\pi$
$450$		0			0
$451$	22.5132	+	20.5742i	1.06010	+	0.968800i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	30.1556	−	12.8224i	1.41372	−	0.601123i
$456$		0			0
$457$	15.2020	+	15.2020i	0.711121	+	0.711121i	0.966770	−	0.255649i	$-0.0822891\pi$
$457$	15.2020	+	15.2020i	0.711121	+	0.711121i	−0.255649	+	0.966770i	$0.582289\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			34.6990i			1.61609i	0.589118	+	0.808047i	$0.299475\pi$
$461$			34.6990i			1.61609i	−0.589118	+	0.808047i	$0.700525\pi$
$462$		0			0
$463$	−9.09179	−	9.09179i	−0.422531	−	0.422531i	0.463543	−	0.886074i	$-0.346578\pi$
$463$	−9.09179	−	9.09179i	−0.422531	−	0.422531i	−0.886074	+	0.463543i	$0.846578\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.17044	+	6.17044i	−0.285534	+	0.285534i	−0.835311	−	0.549777i	$-0.814713\pi$
$467$	−6.17044	+	6.17044i	−0.285534	+	0.285534i	0.549777	+	0.835311i	$0.314713\pi$
$468$		0			0
$469$	−7.94270			−0.366760
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.82751	+	40.6099i	0.0840290	+	1.86725i
$474$		0			0
$475$	2.08170	−	2.16102i	0.0955151	−	0.0991546i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	32.7249			1.49524			0.747620	−	0.664126i	$-0.231196\pi$
$479$	32.7249			1.49524			0.747620	+	0.664126i	$0.231196\pi$
$480$		0			0
$481$			24.2584i			1.10609i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.5692	−	36.1246i	0.661556	−	1.64034i
$486$		0			0
$487$	−15.6586	+	15.6586i	−0.709561	+	0.709561i	−0.966443	−	0.256882i	$-0.917305\pi$
$487$	−15.6586	+	15.6586i	−0.709561	+	0.709561i	0.256882	+	0.966443i	$0.417305\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			5.91600i			0.266985i	0.991050	+	0.133493i	$0.0426193\pi$
$491$			5.91600i			0.266985i	−0.991050	+	0.133493i	$0.957381\pi$
$492$		0			0
$493$	−6.46547	−	6.46547i	−0.291190	−	0.291190i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.24201	+	5.24201i	0.235136	+	0.235136i
$498$		0			0
$499$		−	4.66957i		−	0.209039i	−0.994523	−	0.104519i	$-0.966670\pi$
$499$		−	4.66957i		−	0.209039i	0.994523	−	0.104519i	$-0.0333304\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−17.5379	+	17.5379i	−0.781976	+	0.781976i	−0.980164	−	0.198188i	$-0.936494\pi$
$503$	−17.5379	+	17.5379i	−0.781976	+	0.781976i	0.198188	+	0.980164i	$0.436494\pi$
$504$		0			0
$505$	−12.0903	−	28.4338i	−0.538010	−	1.26529i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	19.3977i		−	0.859786i	−0.902880	−	0.429893i	$-0.858551\pi$
$509$		−	19.3977i		−	0.859786i	0.902880	−	0.429893i	$-0.141449\pi$
$510$		0			0
$511$	6.26276			0.277048
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.5622	+	4.25978i	0.465426	+	0.187708i
$516$		0			0
$517$	−1.73430	−	38.5386i	−0.0762742	−	1.69492i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.47145			−0.152087			−0.0760434	−	0.997105i	$-0.524229\pi$
$521$	−3.47145			−0.152087			−0.0760434	+	0.997105i	$0.524229\pi$
$522$		0			0
$523$	0.636153	−	0.636153i	0.0278170	−	0.0278170i	−0.693061	−	0.720879i	$-0.743739\pi$
$523$	0.636153	−	0.636153i	0.0278170	−	0.0278170i	0.720879	+	0.693061i	$0.243739\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	35.3910	+	35.3910i	1.54166	+	1.54166i
$528$		0			0
$529$		−	22.6183i		−	0.983407i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	29.0238	+	29.0238i	1.25716	+	1.25716i
$534$		0			0
$535$	−10.9486	−	4.41563i	−0.473350	−	0.190904i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−9.25093	−	8.45417i	−0.398466	−	0.364147i
$540$		0			0
$541$			6.74032i			0.289789i	0.989447	+	0.144894i	$0.0462843\pi$
$541$			6.74032i			0.289789i	−0.989447	+	0.144894i	$0.953716\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	40.3641	−	17.1631i	1.72901	−	0.735185i
$546$		0			0
$547$	10.2901	+	10.2901i	0.439972	+	0.439972i	0.892003	−	0.452030i	$-0.149300\pi$
$547$	10.2901	+	10.2901i	0.439972	+	0.439972i	−0.452030	+	0.892003i	$0.649300\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.840997			0.0358277
$552$		0			0
$553$	−13.5929	−	13.5929i	−0.578028	−	0.578028i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.23038	+	1.23038i	0.0521327	+	0.0521327i	0.732693	−	0.680560i	$-0.238263\pi$
$557$	1.23038	+	1.23038i	0.0521327	+	0.0521327i	−0.680560	+	0.732693i	$0.738263\pi$
$558$		0			0
$559$			54.7100i			2.31399i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9.38234	−	9.38234i	0.395418	−	0.395418i	−0.481195	−	0.876614i	$-0.659797\pi$
$563$	9.38234	−	9.38234i	0.395418	−	0.395418i	0.876614	+	0.481195i	$0.159797\pi$
$564$		0			0
$565$	29.3625	+	11.8420i	1.23529	+	0.498199i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−25.2820			−1.05988			−0.529938	−	0.848036i	$-0.677785\pi$
$569$	−25.2820			−1.05988			−0.529938	+	0.848036i	$0.677785\pi$
$570$		0			0
$571$			21.9231i			0.917455i	0.888577	+	0.458727i	$0.151695\pi$
$571$			21.9231i			0.917455i	−0.888577	+	0.458727i	$0.848305\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.08835	−	0.0577396i	0.128793	−	0.00240791i
$576$		0			0
$577$	−29.1377	+	29.1377i	−1.21302	+	1.21302i	−0.242992	+	0.970028i	$0.578129\pi$
$577$	−29.1377	+	29.1377i	−1.21302	+	1.21302i	−0.970028	+	0.242992i	$0.921871\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	54.6024			2.26529
$582$		0			0
$583$	17.4457	−	0.785083i	0.722527	−	0.0325148i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.93109	−	7.93109i	0.327351	−	0.327351i	−0.524227	−	0.851578i	$-0.675646\pi$
$587$	7.93109	−	7.93109i	0.327351	−	0.327351i	0.851578	+	0.524227i	$0.175646\pi$
$588$		0			0
$589$	−4.60348			−0.189683
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14.8447	−	14.8447i	0.609599	−	0.609599i	−0.333242	−	0.942841i	$-0.608143\pi$
$593$	14.8447	−	14.8447i	0.609599	−	0.609599i	0.942841	+	0.333242i	$0.108143\pi$
$594$		0			0
$595$	18.7428	+	44.0793i	0.768380	+	1.80707i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	17.0318i		−	0.695899i	−0.937513	−	0.347949i	$-0.886878\pi$
$599$		−	17.0318i		−	0.695899i	0.937513	−	0.347949i	$-0.113122\pi$
$600$		0			0
$601$			7.44422i			0.303656i	0.988407	+	0.151828i	$0.0485160\pi$
$601$			7.44422i			0.303656i	−0.988407	+	0.151828i	$0.951484\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.6190	+	21.6795i	0.472379	+	0.881395i
$606$		0			0
$607$	4.00814	+	4.00814i	0.162686	+	0.162686i	0.783755	−	0.621070i	$-0.213302\pi$
$607$	4.00814	+	4.00814i	0.162686	+	0.162686i	−0.621070	+	0.783755i	$0.713302\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	51.9194i		−	2.10044i
$612$		0			0
$613$	−4.90502	+	4.90502i	−0.198112	+	0.198112i	−0.799190	−	0.601078i	$-0.794738\pi$
$613$	−4.90502	+	4.90502i	−0.198112	+	0.198112i	0.601078	+	0.799190i	$0.294738\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	11.7436	−	11.7436i	0.472779	−	0.472779i	−0.430034	−	0.902813i	$-0.641498\pi$
$617$	11.7436	−	11.7436i	0.472779	−	0.472779i	0.902813	+	0.430034i	$0.141498\pi$
$618$		0			0
$619$			18.9601i			0.762070i	0.924561	+	0.381035i	$0.124432\pi$
$619$			18.9601i			0.762070i	−0.924561	+	0.381035i	$0.875568\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	12.5725	−	12.5725i	0.503708	−	0.503708i
$624$		0			0
$625$	−0.934469	−	24.9825i	−0.0373788	−	0.999301i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−35.4591			−1.41385
$630$		0			0
$631$	17.0398			0.678345			0.339173	−	0.940724i	$-0.389853\pi$
$631$	17.0398			0.678345			0.339173	+	0.940724i	$0.389853\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	27.1776	+	10.9609i	1.07851	+	0.434969i
$636$		0			0
$637$	−11.9262	−	11.9262i	−0.472534	−	0.472534i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.30146			−0.367386			−0.183693	−	0.982984i	$-0.558805\pi$
$641$	−9.30146			−0.367386			−0.183693	+	0.982984i	$0.558805\pi$
$642$		0			0
$643$	−24.5501	−	24.5501i	−0.968163	−	0.968163i	0.0313460	−	0.999509i	$-0.490021\pi$
$643$	−24.5501	−	24.5501i	−0.968163	−	0.968163i	−0.999509	+	0.0313460i	$0.990021\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	17.9876	−	17.9876i	0.707167	−	0.707167i	−0.258771	−	0.965939i	$-0.583318\pi$
$647$	17.9876	−	17.9876i	0.707167	−	0.707167i	0.965939	+	0.258771i	$0.0833176\pi$
$648$		0			0
$649$	−17.4007	−	15.9020i	−0.683038	−	0.624210i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−17.7279	−	17.7279i	−0.693746	−	0.693746i	0.269308	−	0.963054i	$-0.413205\pi$
$653$	−17.7279	−	17.7279i	−0.693746	−	0.693746i	−0.963054	+	0.269308i	$0.913205\pi$
$654$		0			0
$655$	−3.31990	−	7.80773i	−0.129719	−	0.305073i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−28.2994			−1.10239			−0.551194	−	0.834377i	$-0.685827\pi$
$659$	−28.2994			−1.10239			−0.551194	+	0.834377i	$0.685827\pi$
$660$		0			0
$661$	51.1753			1.99049			0.995245	−	0.0974080i	$-0.0310552\pi$
$661$	51.1753			1.99049			0.995245	+	0.0974080i	$0.0310552\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.08579	−	1.64782i	−0.158440	−	0.0638997i
$666$		0			0
$667$	0.612176	+	0.612176i	0.0237035	+	0.0237035i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.58642	+	5.10527i	0.215661	+	0.197087i
$672$		0			0
$673$	31.4918	−	31.4918i	1.21392	−	1.21392i	0.244195	−	0.969726i	$-0.421476\pi$
$673$	31.4918	−	31.4918i	1.21392	−	1.21392i	0.969726	−	0.244195i	$-0.0785236\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.01767	+	9.01767i	0.346577	+	0.346577i	0.858833	−	0.512256i	$-0.171190\pi$
$677$	9.01767	+	9.01767i	0.346577	+	0.346577i	−0.512256	+	0.858833i	$0.671190\pi$
$678$		0			0
$679$	−57.1906			−2.19477
$680$		0			0
$681$		0			0
$682$		0			0
$683$	25.6006	+	25.6006i	0.979581	+	0.979581i	0.999796	−	0.0202143i	$-0.00643486\pi$
$683$	25.6006	+	25.6006i	0.979581	+	0.979581i	−0.0202143	+	0.999796i	$0.506435\pi$
$684$		0			0
$685$	−1.80979	+	4.48739i	−0.0691484	+	0.171454i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	23.5030			0.895391
$690$		0			0
$691$	15.4669			0.588390			0.294195	−	0.955745i	$-0.404949\pi$
$691$	15.4669			0.588390			0.294195	+	0.955745i	$0.404949\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	29.5178	−	12.5512i	1.11968	−	0.476094i
$696$		0			0
$697$	−42.4248	+	42.4248i	−1.60695	+	1.60695i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			49.7833i			1.88029i	0.340775	+	0.940145i	$0.389311\pi$
$701$			49.7833i			1.88029i	−0.340775	+	0.940145i	$0.610689\pi$
$702$		0			0
$703$	2.30617	−	2.30617i	0.0869790	−	0.0869790i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−32.0778	+	32.0778i	−1.20641	+	1.20641i
$708$		0			0
$709$		−	41.7982i		−	1.56976i	−0.619645	−	0.784882i	$-0.712723\pi$
$709$		−	41.7982i		−	1.56976i	0.619645	−	0.784882i	$-0.287277\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.35095	−	3.35095i	−0.125494	−	0.125494i
$714$		0			0
$715$	14.3099	+	29.8507i	0.535159	+	1.11635i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0	0.000382835i		0	1.42773e-5i	−1.00000		7.13867e-6i	$-0.999998\pi$
$719$		0	0.000382835i		0	1.42773e-5i	1.00000		7.13867e-6i	$-2.27231e-6\pi$
$720$		0			0
$721$		−	16.7215i		−	0.622741i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.86117	−	5.04640i	0.180539	−	0.187419i
$726$		0			0
$727$	4.03870	−	4.03870i	0.149787	−	0.149787i	−0.628236	−	0.778023i	$-0.716223\pi$
$727$	4.03870	−	4.03870i	0.149787	−	0.149787i	0.778023	+	0.628236i	$0.216223\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−79.9710			−2.95783
$732$		0			0
$733$	8.45805	−	8.45805i	0.312405	−	0.312405i	−0.533436	−	0.845841i	$-0.679099\pi$
$733$	8.45805	−	8.45805i	0.312405	−	0.312405i	0.845841	+	0.533436i	$0.179099\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.360722	−	8.01576i	−0.0132874	−	0.295264i
$738$		0			0
$739$	−44.8229			−1.64884			−0.824418	−	0.565982i	$-0.808497\pi$
$739$	−44.8229			−1.64884			−0.824418	+	0.565982i	$0.808497\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.44418	+	7.44418i	−0.273100	+	0.273100i	−0.830347	−	0.557247i	$-0.811858\pi$
$743$	−7.44418	+	7.44418i	−0.273100	+	0.273100i	0.557247	+	0.830347i	$0.311858\pi$
$744$		0			0
$745$	5.15194	−	2.19064i	0.188752	−	0.0802589i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			17.3332i			0.633343i
$750$		0			0
$751$	−34.0581			−1.24280			−0.621398	−	0.783495i	$-0.713435\pi$
$751$	−34.0581			−1.24280			−0.621398	+	0.783495i	$0.713435\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15.1784	+	35.6966i	0.552400	+	1.29913i
$756$		0			0
$757$	−31.8456	+	31.8456i	−1.15745	+	1.15745i	−0.172423	+	0.985023i	$0.555160\pi$
$757$	−31.8456	+	31.8456i	−1.15745	+	1.15745i	−0.985023	+	0.172423i	$0.944840\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	15.1288i		−	0.548418i	−0.961670	−	0.274209i	$-0.911584\pi$
$761$		−	15.1288i		−	0.548418i	0.961670	−	0.274209i	$-0.0884161\pi$
$762$		0			0
$763$	−45.5369	−	45.5369i	−1.64855	−	1.64855i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−22.4329	−	22.4329i	−0.810004	−	0.810004i
$768$		0			0
$769$	25.2742			0.911411			0.455706	−	0.890131i	$-0.349387\pi$
$769$	25.2742			0.911411			0.455706	+	0.890131i	$0.349387\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−14.4697	−	14.4697i	−0.520438	−	0.520438i	0.397266	−	0.917704i	$-0.369959\pi$
$773$	−14.4697	−	14.4697i	−0.520438	−	0.520438i	−0.917704	+	0.397266i	$0.869959\pi$
$774$		0			0
$775$	−26.6093	+	27.6232i	−0.955834	+	0.992255i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	5.51840i		−	0.197717i
$780$		0			0
$781$	−5.05216	+	5.52830i	−0.180780	+	0.197818i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	10.1409	−	25.1445i	0.361945	−	0.897447i
$786$		0			0
$787$	−28.6583	−	28.6583i	−1.02156	−	1.02156i	−0.999762	−	0.0217962i	$-0.993062\pi$
$787$	−28.6583	−	28.6583i	−1.02156	−	1.02156i	−0.0217962	−	0.999762i	$-0.506938\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	46.4851i		−	1.65282i
$792$		0			0
$793$	7.20196	+	7.20196i	0.255749	+	0.255749i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.21628	−	6.21628i	0.220192	−	0.220192i	−0.588387	−	0.808579i	$-0.700237\pi$
$797$	6.21628	−	6.21628i	0.220192	−	0.220192i	0.808579	+	0.588387i	$0.200237\pi$
$798$		0			0
$799$	75.8920			2.68486
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.284426	+	6.32037i	0.0100372	+	0.223041i
$804$		0			0
$805$	−1.77464	−	4.17359i	−0.0625478	−	0.147100i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−36.8786			−1.29658			−0.648292	−	0.761392i	$-0.724516\pi$
$809$	−36.8786			−1.29658			−0.648292	+	0.761392i	$0.724516\pi$
$810$		0			0
$811$		−	4.94183i		−	0.173531i	−0.996229	−	0.0867656i	$-0.972347\pi$
$811$		−	4.94183i		−	0.173531i	0.996229	−	0.0867656i	$-0.0276531\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−18.5432	−	7.47857i	−0.649541	−	0.261963i
$816$		0			0
$817$	5.20111	−	5.20111i	0.181964	−	0.181964i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	22.1356i		−	0.772537i	−0.922386	−	0.386269i	$-0.873764\pi$
$821$		−	22.1356i		−	0.772537i	0.922386	−	0.386269i	$-0.126236\pi$
$822$		0			0
$823$	10.7373	+	10.7373i	0.374280	+	0.374280i	0.869033	−	0.494753i	$-0.164742\pi$
$823$	10.7373	+	10.7373i	0.374280	+	0.374280i	−0.494753	+	0.869033i	$0.664742\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.30961	+	2.30961i	0.0803129	+	0.0803129i	0.746122	−	0.665809i	$-0.231914\pi$
$827$	2.30961	+	2.30961i	0.0803129	+	0.0803129i	−0.665809	+	0.746122i	$0.731914\pi$
$828$		0			0
$829$			4.61750i			0.160372i	0.996780	+	0.0801862i	$0.0255515\pi$
$829$			4.61750i			0.160372i	−0.996780	+	0.0801862i	$0.974449\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	17.4329	−	17.4329i	0.604013	−	0.604013i
$834$		0			0
$835$	12.4181	+	5.00830i	0.429747	+	0.173319i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			27.3332i			0.943647i	0.881693	+	0.471824i	$0.156404\pi$
$839$			27.3332i			0.943647i	−0.881693	+	0.471824i	$0.843596\pi$
$840$		0			0
$841$	−27.0361			−0.932280
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.05859	+	14.2486i	0.208422	+	0.490166i
$846$		0			0
$847$	23.1394	−	27.7268i	0.795079	−	0.952704i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.35741			0.115090
$852$		0			0
$853$	27.8254	−	27.8254i	0.952723	−	0.952723i	−0.0462088	−	0.998932i	$-0.514714\pi$
$853$	27.8254	−	27.8254i	0.952723	−	0.952723i	0.998932	+	0.0462088i	$0.0147140\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−21.1653	−	21.1653i	−0.722994	−	0.722994i	0.246220	−	0.969214i	$-0.420811\pi$
$857$	−21.1653	−	21.1653i	−0.722994	−	0.722994i	−0.969214	+	0.246220i	$0.920811\pi$
$858$		0			0
$859$		−	25.1745i		−	0.858944i	−0.903080	−	0.429472i	$-0.858700\pi$
$859$		−	25.1745i		−	0.858944i	0.903080	−	0.429472i	$-0.141300\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−13.6570	−	13.6570i	−0.464891	−	0.464891i	0.435364	−	0.900255i	$-0.356620\pi$
$863$	−13.6570	−	13.6570i	−0.464891	−	0.464891i	−0.900255	+	0.435364i	$0.856620\pi$
$864$		0			0
$865$	2.62667	−	6.51287i	0.0893095	−	0.221444i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	13.1006	−	14.3352i	0.444407	−	0.486290i
$870$		0			0
$871$		−	10.7989i		−	0.365906i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−33.5046	+	14.9920i	−1.13266	+	0.506822i
$876$		0			0
$877$	21.0523	+	21.0523i	0.710885	+	0.710885i	0.966720	−	0.255835i	$-0.0823504\pi$
$877$	21.0523	+	21.0523i	0.710885	+	0.710885i	−0.255835	+	0.966720i	$0.582350\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−49.1627			−1.65633			−0.828167	−	0.560481i	$-0.810616\pi$
$881$	−49.1627			−1.65633			−0.828167	+	0.560481i	$0.810616\pi$
$882$		0			0
$883$	−16.0264	−	16.0264i	−0.539331	−	0.539331i	0.384002	−	0.923332i	$-0.374546\pi$
$883$	−16.0264	−	16.0264i	−0.539331	−	0.539331i	−0.923332	+	0.384002i	$0.874546\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−23.8705	−	23.8705i	−0.801492	−	0.801492i	0.181837	−	0.983329i	$-0.441796\pi$
$887$	−23.8705	−	23.8705i	−0.801492	−	0.801492i	−0.983329	+	0.181837i	$0.941796\pi$
$888$		0			0
$889$		−	43.0261i		−	1.44305i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.93582	+	4.93582i	−0.165171	+	0.165171i
$894$		0			0
$895$	−4.76838	−	11.2142i	−0.159389	−	0.374851i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−10.7500			−0.358533
$900$		0			0
$901$			34.3549i			1.14453i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−19.2528	+	8.18642i	−0.639984	+	0.272126i
$906$		0			0
$907$	−21.1178	+	21.1178i	−0.701206	+	0.701206i	−0.964669	−	0.263463i	$-0.915135\pi$
$907$	−21.1178	+	21.1178i	−0.701206	+	0.701206i	0.263463	+	0.964669i	$0.415135\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.22816			−0.0406908			−0.0203454	−	0.999793i	$-0.506477\pi$
$911$	−1.22816			−0.0406908			−0.0203454	+	0.999793i	$0.506477\pi$
$912$		0			0
$913$	2.47980	+	55.1047i	0.0820693	+	1.82370i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8.80833	+	8.80833i	−0.290877	+	0.290877i
$918$		0			0
$919$	−14.8532			−0.489963			−0.244982	−	0.969528i	$-0.578782\pi$
$919$	−14.8532			−0.489963			−0.244982	+	0.969528i	$0.578782\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−7.12703	+	7.12703i	−0.234589	+	0.234589i
$924$		0			0
$925$	−0.507939	−	27.1685i	−0.0167009	−	0.893293i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			33.9859i			1.11504i	0.830163	+	0.557520i	$0.188247\pi$
$929$			33.9859i			1.11504i	−0.830163	+	0.557520i	$0.811753\pi$
$930$		0			0
$931$			2.26758i			0.0743169i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−43.6335	+	20.9171i	−1.42697	+	0.684062i
$936$		0			0
$937$	−23.8274	−	23.8274i	−0.778406	−	0.778406i	0.201153	−	0.979560i	$-0.435531\pi$
$937$	−23.8274	−	23.8274i	−0.778406	−	0.778406i	−0.979560	+	0.201153i	$0.935531\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	12.3719i		−	0.403313i	−0.979456	−	0.201656i	$-0.935368\pi$
$941$		−	12.3719i		−	0.403313i	0.979456	−	0.201656i	$-0.0646324\pi$
$942$		0			0
$943$	4.01694	−	4.01694i	0.130810	−	0.130810i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.94411	+	4.94411i	−0.160662	+	0.160662i	−0.782860	−	0.622198i	$-0.786240\pi$
$947$	−4.94411	+	4.94411i	−0.160662	+	0.160662i	0.622198	+	0.782860i	$0.286240\pi$
$948$		0			0
$949$			8.51485i			0.276404i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	23.7717	−	23.7717i	0.770041	−	0.770041i	−0.208073	−	0.978113i	$-0.566719\pi$
$953$	23.7717	−	23.7717i	0.770041	−	0.770041i	0.978113	+	0.208073i	$0.0667191\pi$
$954$		0			0
$955$	7.32347	−	3.11399i	0.236982	−	0.100766i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.10419			0.229406
$960$		0			0
$961$	27.8438			0.898188
$962$		0			0
$963$		0			0
$964$		0			0
$965$	7.14191	−	17.7085i	0.229906	−	0.570056i
$966$		0			0
$967$	8.66496	+	8.66496i	0.278646	+	0.278646i	0.832568	−	0.553922i	$-0.186870\pi$
$967$	8.66496	+	8.66496i	0.278646	+	0.278646i	−0.553922	+	0.832568i	$0.686870\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.4796			−0.400491			−0.200245	−	0.979746i	$-0.564174\pi$
$971$	−12.4796			−0.400491			−0.200245	+	0.979746i	$0.564174\pi$
$972$		0			0
$973$	−33.3007	−	33.3007i	−1.06757	−	1.06757i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−37.4364	+	37.4364i	−1.19770	+	1.19770i	−0.222843	+	0.974854i	$0.571534\pi$
$977$	−37.4364	+	37.4364i	−1.19770	+	1.19770i	−0.974854	+	0.222843i	$0.928466\pi$
$978$		0			0
$979$	13.2592	+	12.1172i	0.423765	+	0.387267i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−7.20828	−	7.20828i	−0.229908	−	0.229908i	0.582746	−	0.812654i	$-0.301978\pi$
$983$	−7.20828	−	7.20828i	−0.229908	−	0.229908i	−0.812654	+	0.582746i	$0.801978\pi$
$984$		0			0
$985$	−2.66803	−	1.07603i	−0.0850104	−	0.0342851i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.57196			0.240774
$990$		0			0
$991$	40.7072			1.29311			0.646554	−	0.762868i	$-0.276209\pi$
$991$	40.7072			1.29311			0.646554	+	0.762868i	$0.276209\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−24.1301	−	56.7491i	−0.764976	−	1.79907i
$996$		0			0
$997$	−2.59413	−	2.59413i	−0.0821571	−	0.0821571i	0.664834	−	0.746991i	$-0.268502\pi$
$997$	−2.59413	−	2.59413i	−0.0821571	−	0.0821571i	−0.746991	+	0.664834i	$0.768502\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1980.2.y.c.1693.2		24
3.2	odd	2		660.2.x.a.373.6	yes	24
5.2	odd	4	inner	1980.2.y.c.1297.1		24
11.10	odd	2	inner	1980.2.y.c.1693.1		24
15.2	even	4		660.2.x.a.637.5	yes	24
15.8	even	4		3300.2.x.c.1957.8		24
15.14	odd	2		3300.2.x.c.1693.7		24
33.32	even	2		660.2.x.a.373.5	✓	24
55.32	even	4	inner	1980.2.y.c.1297.2		24
165.32	odd	4		660.2.x.a.637.6	yes	24
165.98	odd	4		3300.2.x.c.1957.7		24
165.164	even	2		3300.2.x.c.1693.8		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
660.2.x.a.373.5	✓	24	33.32	even	2
660.2.x.a.373.6	yes	24	3.2	odd	2
660.2.x.a.637.5	yes	24	15.2	even	4
660.2.x.a.637.6	yes	24	165.32	odd	4
1980.2.y.c.1297.1		24	5.2	odd	4	inner
1980.2.y.c.1297.2		24	55.32	even	4	inner
1980.2.y.c.1693.1		24	11.10	odd	2	inner
1980.2.y.c.1693.2		24	1.1	even	1	trivial
3300.2.x.c.1693.7		24	15.14	odd	2
3300.2.x.c.1693.8		24	165.164	even	2
3300.2.x.c.1957.7		24	165.98	odd	4
3300.2.x.c.1957.8		24	15.8	even	4

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:	\( N \)	\(=\)	\( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1980.y (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.05777 + 0.874978i) q^{5} +(2.32148 + 2.32148i) q^{7} +O(q^{10})\)	\(q+(-2.05777 + 0.874978i) q^{5} +(2.32148 + 2.32148i) q^{7} +(-2.23741 + 2.44827i) q^{11} +(-3.15629 + 3.15629i) q^{13} +(-4.61362 - 4.61362i) q^{17} +0.600117 q^{19} +(0.436835 + 0.436835i) q^{23} +(3.46883 - 3.60100i) q^{25} +1.40139 q^{29} -7.67097 q^{31} +(-6.80832 - 2.74583i) q^{35} +(3.84287 - 3.84287i) q^{37} -9.19555i q^{41} +(8.66683 - 8.66683i) q^{43} +(-8.22477 + 8.22477i) q^{47} +3.77856i q^{49} +(-3.72320 - 3.72320i) q^{53} +(2.46188 - 6.99565i) q^{55} +7.10736i q^{59} -2.28178i q^{61} +(3.73323 - 9.25659i) q^{65} +(-1.71070 + 1.71070i) q^{67} +2.25804 q^{71} +(1.34887 - 1.34887i) q^{73} +(-10.8777 + 0.489514i) q^{77} -5.85526 q^{79} +(11.7602 - 11.7602i) q^{83} +(13.5306 + 5.45695i) q^{85} -5.41573i q^{89} -14.6545 q^{91} +(-1.23490 + 0.525089i) q^{95} +(-12.3177 + 12.3177i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 8 q^{5}+O(q^{10}) \) 24 * q - 8 * q^5	\( 24 q - 8 q^{5} - 8 q^{11} - 16 q^{23} + 24 q^{25} + 16 q^{31} - 40 q^{37} - 64 q^{47} - 24 q^{53} + 24 q^{55} + 48 q^{67} - 24 q^{77} - 48 q^{91} - 72 q^{97}+O(q^{100}) \) 24 * q - 8 * q^5 - 8 * q^11 - 16 * q^23 + 24 * q^25 + 16 * q^31 - 40 * q^37 - 64 * q^47 - 24 * q^53 + 24 * q^55 + 48 * q^67 - 24 * q^77 - 48 * q^91 - 72 * q^97

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