LMFDB - Embedded newform 1020.2.g.d.409.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1020,2,Mod(409,1020)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1020, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1020.409"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1020.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,0,0,0,0,-10,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$8.14474100617$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} + 2x^{8} + 14x^{7} + 42x^{6} + 2x^{5} + 10x^{4} + 54x^{3} + 121x^{2} + 44x + 8 $ x^10 - 2x^9 + 2x^8 + 14x^7 + 42x^6 + 2x^5 + 10x^4 + 54x^3 + 121x^2 + 44*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			409.4
Root			$-0.841453 - 0.841453i$ of defining polynomial
Character	$\chi$	$=$	1020.409
Dual form			1020.2.g.d.409.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(1.85384 + 1.25031i) q^{5} +1.12377i q^{7} -1.00000 q^{9} -0.781901 q^{11} -6.90690i q^{13} +(1.25031 - 1.85384i) q^{15} -1.00000i q^{17} +4.99020 q^{19} +1.12377 q^{21} -2.69860i q^{23} +(1.87347 + 4.63574i) q^{25} +1.00000i q^{27} +4.48959 q^{29} +5.36581 q^{31} +0.781901i q^{33} +(-1.40506 + 2.08330i) q^{35} -1.87745i q^{37} -6.90690 q^{39} +12.1973 q^{41} +4.10366i q^{43} +(-1.85384 - 1.25031i) q^{45} +12.1029i q^{47} +5.73713 q^{49} -1.00000 q^{51} -7.12377i q^{53} +(-1.44952 - 0.977616i) q^{55} -4.99020i q^{57} -8.61213 q^{59} -13.0106 q^{61} -1.12377i q^{63} +(8.63574 - 12.8043i) q^{65} -8.61213i q^{67} -2.69860 q^{69} +13.8138 q^{71} -7.29037i q^{73} +(4.63574 - 1.87347i) q^{75} -0.878680i q^{77} -12.7316 q^{79} +1.00000 q^{81} +13.0601i q^{83} +(1.25031 - 1.85384i) q^{85} -4.48959i q^{87} -10.7630 q^{89} +7.76179 q^{91} -5.36581i q^{93} +(9.25105 + 6.23928i) q^{95} -18.2279i q^{97} +0.781901 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{9} + 12 q^{11} + 4 q^{15} - 24 q^{19} - 4 q^{21} + 12 q^{25} - 12 q^{29} + 12 q^{31} + 4 q^{35} + 28 q^{41} - 30 q^{49} - 10 q^{51} + 26 q^{55} - 48 q^{59} - 28 q^{61} + 48 q^{65} - 12 q^{69}+ \cdots - 12 q^{99}+O(q^{100}) $ 10 * q - 10 * q^9 + 12 * q^11 + 4 * q^15 - 24 * q^19 - 4 * q^21 + 12 * q^25 - 12 * q^29 + 12 * q^31 + 4 * q^35 + 28 * q^41 - 30 * q^49 - 10 * q^51 + 26 * q^55 - 48 * q^59 - 28 * q^61 + 48 * q^65 - 12 * q^69 + 8 * q^75 - 44 * q^79 + 10 * q^81 + 4 * q^85 - 36 * q^89 + 40 * q^91 + 44 * q^95 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1020\mathbb{Z}\right)^\times$.

$n$	$241$	$341$	$511$	$817$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	1.85384	+	1.25031i	0.829064	+	0.559154i
$5$	1.85384	+	1.25031i	0.829064	+	0.559154i
$6$		0			0
$7$			1.12377i			0.424746i	0.977189	+	0.212373i	$0.0681192\pi$
$7$			1.12377i			0.424746i	−0.977189	+	0.212373i	$0.931881\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−0.781901			−0.235752			−0.117876	−	0.993028i	$-0.537609\pi$
$11$	−0.781901			−0.235752			−0.117876	+	0.993028i	$0.537609\pi$
$12$		0			0
$13$		−	6.90690i		−	1.91563i	−0.287386	−	0.957815i	$-0.592786\pi$
$13$		−	6.90690i		−	1.91563i	0.287386	−	0.957815i	$-0.407214\pi$
$14$		0			0
$15$	1.25031	−	1.85384i	0.322828	−	0.478660i
$16$		0			0
$17$		−	1.00000i		−	0.242536i
$17$		−	1.00000i		−	0.242536i
$18$		0			0
$19$	4.99020			1.14483			0.572415	−	0.819964i	$-0.306007\pi$
$19$	4.99020			1.14483			0.572415	+	0.819964i	$0.306007\pi$
$20$		0			0
$21$	1.12377			0.245227
$22$		0			0
$23$		−	2.69860i		−	0.562697i	−0.959606	−	0.281349i	$-0.909218\pi$
$23$		−	2.69860i		−	0.562697i	0.959606	−	0.281349i	$-0.0907818\pi$
$24$		0			0
$25$	1.87347	+	4.63574i	0.374693	+	0.927149i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	4.48959			0.833695			0.416848	−	0.908976i	$-0.363135\pi$
$29$	4.48959			0.833695			0.416848	+	0.908976i	$0.363135\pi$
$30$		0			0
$31$	5.36581			0.963729			0.481864	−	0.876246i	$-0.339960\pi$
$31$	5.36581			0.963729			0.481864	+	0.876246i	$0.339960\pi$
$32$		0			0
$33$			0.781901i			0.136112i
$34$		0			0
$35$	−1.40506	+	2.08330i	−0.237499	+	0.352142i
$36$		0			0
$37$		−	1.87745i		−	0.308651i	−0.988020	−	0.154326i	$-0.950679\pi$
$37$		−	1.87745i		−	0.308651i	0.988020	−	0.154326i	$-0.0493205\pi$
$38$		0			0
$39$	−6.90690			−1.10599
$40$		0			0
$41$	12.1973			1.90489			0.952447	−	0.304704i	$-0.0985577\pi$
$41$	12.1973			1.90489			0.952447	+	0.304704i	$0.0985577\pi$
$42$		0			0
$43$			4.10366i			0.625803i	0.949786	+	0.312901i	$0.101301\pi$
$43$			4.10366i			0.625803i	−0.949786	+	0.312901i	$0.898699\pi$
$44$		0			0
$45$	−1.85384	−	1.25031i	−0.276355	−	0.186385i
$46$		0			0
$47$			12.1029i			1.76540i	0.469940	+	0.882698i	$0.344275\pi$
$47$			12.1029i			1.76540i	−0.469940	+	0.882698i	$0.655725\pi$
$48$		0			0
$49$	5.73713			0.819591
$50$		0			0
$51$	−1.00000			−0.140028
$52$		0			0
$53$		−	7.12377i		−	0.978526i	−0.872136	−	0.489263i	$-0.837266\pi$
$53$		−	7.12377i		−	0.978526i	0.872136	−	0.489263i	$-0.162734\pi$
$54$		0			0
$55$	−1.44952	−	0.977616i	−0.195453	−	0.131822i
$56$		0			0
$57$		−	4.99020i		−	0.660968i
$58$		0			0
$59$	−8.61213			−1.12120			−0.560602	−	0.828085i	$-0.689430\pi$
$59$	−8.61213			−1.12120			−0.560602	+	0.828085i	$0.689430\pi$
$60$		0			0
$61$	−13.0106			−1.66583			−0.832916	−	0.553399i	$-0.813330\pi$
$61$	−13.0106			−1.66583			−0.832916	+	0.553399i	$0.813330\pi$
$62$		0			0
$63$		−	1.12377i		−	0.141582i
$64$		0			0
$65$	8.63574	−	12.8043i	1.07113	−	1.58818i
$66$		0			0
$67$		−	8.61213i		−	1.05214i	−0.850441	−	0.526070i	$-0.823665\pi$
$67$		−	8.61213i		−	1.05214i	0.850441	−	0.526070i	$-0.176335\pi$
$68$		0			0
$69$	−2.69860			−0.324874
$70$		0			0
$71$	13.8138			1.63940			0.819698	−	0.572795i	$-0.194141\pi$
$71$	13.8138			1.63940			0.819698	+	0.572795i	$0.194141\pi$
$72$		0			0
$73$		−	7.29037i		−	0.853273i	−0.904423	−	0.426637i	$-0.859698\pi$
$73$		−	7.29037i		−	0.853273i	0.904423	−	0.426637i	$-0.140302\pi$
$74$		0			0
$75$	4.63574	−	1.87347i	0.535290	−	0.216329i
$76$		0			0
$77$		−	0.878680i		−	0.100135i
$78$		0			0
$79$	−12.7316			−1.43242			−0.716210	−	0.697885i	$-0.754125\pi$
$79$	−12.7316			−1.43242			−0.716210	+	0.697885i	$0.754125\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			13.0601i			1.43353i	0.697312	+	0.716767i	$0.254379\pi$
$83$			13.0601i			1.43353i	−0.697312	+	0.716767i	$0.745621\pi$
$84$		0			0
$85$	1.25031	−	1.85384i	0.135615	−	0.201077i
$86$		0			0
$87$		−	4.48959i		−	0.481334i
$88$		0			0
$89$	−10.7630			−1.14088			−0.570439	−	0.821340i	$-0.693227\pi$
$89$	−10.7630			−1.14088			−0.570439	+	0.821340i	$0.693227\pi$
$90$		0			0
$91$	7.76179			0.813657
$92$		0			0
$93$		−	5.36581i		−	0.556409i
$94$		0			0
$95$	9.25105	+	6.23928i	0.949137	+	0.640137i
$96$		0			0
$97$		−	18.2279i		−	1.85077i	−0.379031	−	0.925384i	$-0.623743\pi$
$97$		−	18.2279i		−	1.85077i	0.379031	−	0.925384i	$-0.376257\pi$
$98$		0			0
$99$	0.781901			0.0785840
$100$		0			0
$101$	5.24877			0.522272			0.261136	−	0.965302i	$-0.415903\pi$
$101$	5.24877			0.522272			0.261136	+	0.965302i	$0.415903\pi$
$102$		0			0
$103$		−	4.10366i		−	0.404346i	−0.979350	−	0.202173i	$-0.935200\pi$
$103$		−	4.10366i		−	0.404346i	0.979350	−	0.202173i	$-0.0648003\pi$
$104$		0			0
$105$	2.08330	+	1.40506i	0.203309	+	0.137120i
$106$		0			0
$107$			2.25307i			0.217812i	0.994052	+	0.108906i	$0.0347348\pi$
$107$			2.25307i			0.217812i	−0.994052	+	0.108906i	$0.965265\pi$
$108$		0			0
$109$	11.9804			1.14751			0.573757	−	0.819026i	$-0.305485\pi$
$109$	11.9804			1.14751			0.573757	+	0.819026i	$0.305485\pi$
$110$		0			0
$111$	−1.87745			−0.178200
$112$		0			0
$113$			8.28057i			0.778971i	0.921033	+	0.389485i	$0.127347\pi$
$113$			8.28057i			0.778971i	−0.921033	+	0.389485i	$0.872653\pi$
$114$		0			0
$115$	3.37408	−	5.00278i	0.314635	−	0.466512i
$116$		0			0
$117$			6.90690i			0.638543i
$118$		0			0
$119$	1.12377			0.103016
$120$		0			0
$121$	−10.3886			−0.944421
$122$		0			0
$123$		−	12.1973i		−	1.09979i
$124$		0			0
$125$	−2.32299	+	10.9363i	−0.207774	+	0.978177i
$126$		0			0
$127$			13.6385i			1.21022i	0.796140	+	0.605112i	$0.206872\pi$
$127$			13.6385i			1.21022i	−0.796140	+	0.605112i	$0.793128\pi$
$128$		0			0
$129$	4.10366			0.361307
$130$		0			0
$131$	−22.5007			−1.96589			−0.982946	−	0.183895i	$-0.941129\pi$
$131$	−22.5007			−1.96589			−0.982946	+	0.183895i	$0.941129\pi$
$132$		0			0
$133$			5.60785i			0.486263i
$134$		0			0
$135$	−1.25031	+	1.85384i	−0.107609	+	0.159553i
$136$		0			0
$137$			9.29282i			0.793940i	0.917832	+	0.396970i	$0.129938\pi$
$137$			9.29282i			0.793940i	−0.917832	+	0.396970i	$0.870062\pi$
$138$		0			0
$139$	−14.2500			−1.20867			−0.604335	−	0.796731i	$-0.706561\pi$
$139$	−14.2500			−1.20867			−0.604335	+	0.796731i	$0.706561\pi$
$140$		0			0
$141$	12.1029			1.01925
$142$		0			0
$143$			5.40051i			0.451614i
$144$		0			0
$145$	8.32299	+	5.61336i	0.691187	+	0.466164i
$146$		0			0
$147$		−	5.73713i		−	0.473191i
$148$		0			0
$149$	7.36827			0.603632			0.301816	−	0.953366i	$-0.402407\pi$
$149$	7.36827			0.603632			0.301816	+	0.953366i	$0.402407\pi$
$150$		0			0
$151$	−6.34116			−0.516036			−0.258018	−	0.966140i	$-0.583069\pi$
$151$	−6.34116			−0.516036			−0.258018	+	0.966140i	$0.583069\pi$
$152$		0			0
$153$			1.00000i			0.0808452i
$154$		0			0
$155$	9.94738	+	6.70891i	0.798992	+	0.538873i
$156$		0			0
$157$		−	9.95523i		−	0.794514i	−0.917707	−	0.397257i	$-0.869962\pi$
$157$		−	9.95523i		−	0.794514i	0.917707	−	0.397257i	$-0.130038\pi$
$158$		0			0
$159$	−7.12377			−0.564952
$160$		0			0
$161$	3.03262			0.239004
$162$		0			0
$163$		−	13.4192i		−	1.05107i	−0.850771	−	0.525537i	$-0.823865\pi$
$163$		−	13.4192i		−	1.05107i	0.850771	−	0.525537i	$-0.176135\pi$
$164$		0			0
$165$	−0.977616	+	1.44952i	−0.0761073	+	0.112845i
$166$		0			0
$167$			6.35218i			0.491547i	0.969327	+	0.245773i	$0.0790419\pi$
$167$			6.35218i			0.491547i	−0.969327	+	0.245773i	$0.920958\pi$
$168$		0			0
$169$	−34.7053			−2.66964
$170$		0			0
$171$	−4.99020			−0.381610
$172$		0			0
$173$		−	1.30385i		−	0.0991300i	−0.998771	−	0.0495650i	$-0.984217\pi$
$173$		−	1.30385i		−	0.0991300i	0.998771	−	0.0495650i	$-0.0157835\pi$
$174$		0			0
$175$	−5.20953	+	2.10535i	−0.393803	+	0.159150i
$176$		0			0
$177$			8.61213i			0.647328i
$178$		0			0
$179$	−6.60280			−0.493516			−0.246758	−	0.969077i	$-0.579365\pi$
$179$	−6.60280			−0.493516			−0.246758	+	0.969077i	$0.579365\pi$
$180$		0			0
$181$	4.95412			0.368237			0.184118	−	0.982904i	$-0.441057\pi$
$181$	4.95412			0.368237			0.184118	+	0.982904i	$0.441057\pi$
$182$		0			0
$183$			13.0106i			0.961769i
$184$		0			0
$185$	2.34739	−	3.48050i	0.172584	−	0.255892i
$186$		0			0
$187$			0.781901i			0.0571783i
$188$		0			0
$189$	−1.12377			−0.0817425
$190$		0			0
$191$	19.0987			1.38193			0.690966	−	0.722888i	$-0.257186\pi$
$191$	19.0987			1.38193			0.690966	+	0.722888i	$0.257186\pi$
$192$		0			0
$193$			16.3984i			1.18038i	0.807263	+	0.590192i	$0.200948\pi$
$193$			16.3984i			1.18038i	−0.807263	+	0.590192i	$0.799052\pi$
$194$		0			0
$195$	−12.8043	−	8.63574i	−0.916936	−	0.618418i
$196$		0			0
$197$			18.2335i			1.29908i	0.760327	+	0.649540i	$0.225039\pi$
$197$			18.2335i			1.29908i	−0.760327	+	0.649540i	$0.774961\pi$
$198$		0			0
$199$	−25.7329			−1.82415			−0.912077	−	0.410019i	$-0.865522\pi$
$199$	−25.7329			−1.82415			−0.912077	+	0.410019i	$0.865522\pi$
$200$		0			0
$201$	−8.61213			−0.607453
$202$		0			0
$203$			5.04528i			0.354109i
$204$		0			0
$205$	22.6118	+	15.2503i	1.57928	+	1.06513i
$206$		0			0
$207$			2.69860i			0.187566i
$208$		0			0
$209$	−3.90184			−0.269896
$210$		0			0
$211$	3.18988			0.219600			0.109800	−	0.993954i	$-0.464979\pi$
$211$	3.18988			0.219600			0.109800	+	0.993954i	$0.464979\pi$
$212$		0			0
$213$		−	13.8138i		−	0.946506i
$214$		0			0
$215$	−5.13084	+	7.60755i	−0.349920	+	0.518830i
$216$		0			0
$217$			6.02996i			0.409340i
$218$		0			0
$219$	−7.29037			−0.492638
$220$		0			0
$221$	−6.90690			−0.464608
$222$		0			0
$223$			18.6466i			1.24867i	0.781156	+	0.624336i	$0.214630\pi$
$223$			18.6466i			1.24867i	−0.781156	+	0.624336i	$0.785370\pi$
$224$		0			0
$225$	−1.87347	−	4.63574i	−0.124898	−	0.309050i
$226$		0			0
$227$		−	5.27935i		−	0.350402i	−0.984533	−	0.175201i	$-0.943942\pi$
$227$		−	5.27935i		−	0.350402i	0.984533	−	0.175201i	$-0.0560576\pi$
$228$		0			0
$229$	−16.9871			−1.12254			−0.561271	−	0.827632i	$-0.689687\pi$
$229$	−16.9871			−1.12254			−0.561271	+	0.827632i	$0.689687\pi$
$230$		0			0
$231$	−0.878680			−0.0578129
$232$		0			0
$233$		−	8.52567i		−	0.558535i	−0.960213	−	0.279267i	$-0.909908\pi$
$233$		−	8.52567i		−	0.558535i	0.960213	−	0.279267i	$-0.0900916\pi$
$234$		0			0
$235$	−15.1324	+	22.4370i	−0.987129	+	1.46363i
$236$		0			0
$237$			12.7316i			0.827008i
$238$		0			0
$239$	6.95412			0.449825			0.224912	−	0.974379i	$-0.427790\pi$
$239$	6.95412			0.449825			0.224912	+	0.974379i	$0.427790\pi$
$240$		0			0
$241$	7.25811			0.467536			0.233768	−	0.972292i	$-0.424894\pi$
$241$	7.25811			0.467536			0.233768	+	0.972292i	$0.424894\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	10.6357	+	7.17318i	0.679493	+	0.458277i
$246$		0			0
$247$		−	34.4668i		−	2.19307i
$248$		0			0
$249$	13.0601			0.827652
$250$		0			0
$251$	8.63419			0.544985			0.272493	−	0.962158i	$-0.412152\pi$
$251$	8.63419			0.544985			0.272493	+	0.962158i	$0.412152\pi$
$252$		0			0
$253$			2.11004i			0.132657i
$254$		0			0
$255$	−1.85384	−	1.25031i	−0.116092	−	0.0782972i
$256$		0			0
$257$		−	0.542976i		−	0.0338699i	−0.999857	−	0.0169350i	$-0.994609\pi$
$257$		−	0.542976i		−	0.0338699i	0.999857	−	0.0169350i	$-0.00539082\pi$
$258$		0			0
$259$	2.10983			0.131099
$260$		0			0
$261$	−4.48959			−0.277898
$262$		0			0
$263$			2.74985i			0.169563i	0.996400	+	0.0847815i	$0.0270192\pi$
$263$			2.74985i			0.169563i	−0.996400	+	0.0847815i	$0.972981\pi$
$264$		0			0
$265$	8.90690	−	13.2064i	0.547147	−	0.811260i
$266$		0			0
$267$			10.7630i			0.658686i
$268$		0			0
$269$	−11.8531			−0.722694			−0.361347	−	0.932431i	$-0.617683\pi$
$269$	−11.8531			−0.722694			−0.361347	+	0.932431i	$0.617683\pi$
$270$		0			0
$271$	4.52267			0.274732			0.137366	−	0.990520i	$-0.456136\pi$
$271$	4.52267			0.274732			0.137366	+	0.990520i	$0.456136\pi$
$272$		0			0
$273$		−	7.76179i		−	0.469765i
$274$		0			0
$275$	−1.46487	−	3.62469i	−0.0883347	−	0.218577i
$276$		0			0
$277$		−	12.7316i		−	0.764969i	−0.923962	−	0.382485i	$-0.875068\pi$
$277$		−	12.7316i		−	0.764969i	0.923962	−	0.382485i	$-0.124932\pi$
$278$		0			0
$279$	−5.36581			−0.321243
$280$		0			0
$281$	−5.08217			−0.303177			−0.151589	−	0.988444i	$-0.548439\pi$
$281$	−5.08217			−0.303177			−0.151589	+	0.988444i	$0.548439\pi$
$282$		0			0
$283$			10.2759i			0.610838i	0.952218	+	0.305419i	$0.0987965\pi$
$283$			10.2759i			0.610838i	−0.952218	+	0.305419i	$0.901203\pi$
$284$		0			0
$285$	6.23928	−	9.25105i	0.369583	−	0.547985i
$286$		0			0
$287$			13.7070i			0.809097i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	−18.2279			−1.06854
$292$		0			0
$293$		−	15.7290i		−	0.918899i	−0.888204	−	0.459450i	$-0.848047\pi$
$293$		−	15.7290i		−	0.918899i	0.888204	−	0.459450i	$-0.151953\pi$
$294$		0			0
$295$	−15.9655	−	10.7678i	−0.929550	−	0.626926i
$296$		0			0
$297$		−	0.781901i		−	0.0453705i
$298$		0			0
$299$	−18.6390			−1.07792
$300$		0			0
$301$	−4.61159			−0.265807
$302$		0			0
$303$		−	5.24877i		−	0.301534i
$304$		0			0
$305$	−24.1195	−	16.2672i	−1.38108	−	0.931457i
$306$		0			0
$307$		−	10.5871i		−	0.604237i	−0.953270	−	0.302118i	$-0.902306\pi$
$307$		−	10.5871i		−	0.604237i	0.953270	−	0.302118i	$-0.0976938\pi$
$308$		0			0
$309$	−4.10366			−0.233449
$310$		0			0
$311$	−5.65741			−0.320802			−0.160401	−	0.987052i	$-0.551279\pi$
$311$	−5.65741			−0.320802			−0.160401	+	0.987052i	$0.551279\pi$
$312$		0			0
$313$			3.80752i			0.215214i	0.994194	+	0.107607i	$0.0343188\pi$
$313$			3.80752i			0.215214i	−0.994194	+	0.107607i	$0.965681\pi$
$314$		0			0
$315$	1.40506	−	2.08330i	0.0791662	−	0.117381i
$316$		0			0
$317$			22.5226i			1.26500i	0.774562	+	0.632498i	$0.217970\pi$
$317$			22.5226i			1.26500i	−0.774562	+	0.632498i	$0.782030\pi$
$318$		0			0
$319$	−3.51041			−0.196545
$320$		0			0
$321$	2.25307			0.125754
$322$		0			0
$323$		−	4.99020i		−	0.277662i
$324$		0			0
$325$	32.0186	−	12.9398i	1.77607	−	0.717774i
$326$		0			0
$327$		−	11.9804i		−	0.662518i
$328$		0			0
$329$	−13.6010			−0.749846
$330$		0			0
$331$	19.6337			1.07917			0.539584	−	0.841932i	$-0.318582\pi$
$331$	19.6337			1.07917			0.539584	+	0.841932i	$0.318582\pi$
$332$		0			0
$333$			1.87745i			0.102884i
$334$		0			0
$335$	10.7678	−	15.9655i	0.588308	−	0.872291i
$336$		0			0
$337$			18.5014i			1.00783i	0.863752	+	0.503917i	$0.168108\pi$
$337$			18.5014i			1.00783i	−0.863752	+	0.503917i	$0.831892\pi$
$338$		0			0
$339$	8.28057			0.449739
$340$		0			0
$341$	−4.19554			−0.227201
$342$		0			0
$343$			14.3137i			0.772865i
$344$		0			0
$345$	−5.00278	−	3.37408i	−0.269341	−	0.181654i
$346$		0			0
$347$		−	3.69431i		−	0.198321i	−0.995071	−	0.0991604i	$-0.968384\pi$
$347$		−	3.69431i		−	0.198321i	0.995071	−	0.0991604i	$-0.0316157\pi$
$348$		0			0
$349$	11.3234			0.606128			0.303064	−	0.952970i	$-0.401990\pi$
$349$	11.3234			0.606128			0.303064	+	0.952970i	$0.401990\pi$
$350$		0			0
$351$	6.90690			0.368663
$352$		0			0
$353$			5.04282i			0.268402i	0.990954	+	0.134201i	$0.0428469\pi$
$353$			5.04282i			0.268402i	−0.990954	+	0.134201i	$0.957153\pi$
$354$		0			0
$355$	25.6086	+	17.2715i	1.35916	+	0.916675i
$356$		0			0
$357$		−	1.12377i		−	0.0594764i
$358$		0			0
$359$	−12.5901			−0.664479			−0.332240	−	0.943195i	$-0.607804\pi$
$359$	−12.5901			−0.664479			−0.332240	+	0.943195i	$0.607804\pi$
$360$		0			0
$361$	5.90210			0.310637
$362$		0			0
$363$			10.3886i			0.545262i
$364$		0			0
$365$	9.11520	−	13.5152i	0.477111	−	0.707418i
$366$		0			0
$367$		−	13.5231i		−	0.705899i	−0.935642	−	0.352949i	$-0.885179\pi$
$367$		−	13.5231i		−	0.705899i	0.935642	−	0.352949i	$-0.114821\pi$
$368$		0			0
$369$	−12.1973			−0.634965
$370$		0			0
$371$	8.00551			0.415625
$372$		0			0
$373$			14.2475i			0.737710i	0.929487	+	0.368855i	$0.120250\pi$
$373$			14.2475i			0.737710i	−0.929487	+	0.368855i	$0.879750\pi$
$374$		0			0
$375$	10.9363	+	2.32299i	0.564751	+	0.119959i
$376$		0			0
$377$		−	31.0091i		−	1.59705i
$378$		0			0
$379$	−29.2881			−1.50443			−0.752214	−	0.658919i	$-0.771014\pi$
$379$	−29.2881			−1.50443			−0.752214	+	0.658919i	$0.771014\pi$
$380$		0			0
$381$	13.6385			0.698723
$382$		0			0
$383$			35.8201i			1.83032i	0.403090	+	0.915160i	$0.367936\pi$
$383$			35.8201i			1.83032i	−0.403090	+	0.915160i	$0.632064\pi$
$384$		0			0
$385$	1.09862	−	1.62893i	0.0559908	−	0.0830182i
$386$		0			0
$387$		−	4.10366i		−	0.208601i
$388$		0			0
$389$	6.83463			0.346529			0.173265	−	0.984875i	$-0.444568\pi$
$389$	6.83463			0.346529			0.173265	+	0.984875i	$0.444568\pi$
$390$		0			0
$391$	−2.69860			−0.136474
$392$		0			0
$393$			22.5007i			1.13501i
$394$		0			0
$395$	−23.6024	−	15.9184i	−1.18757	−	0.800944i
$396$		0			0
$397$			1.66675i			0.0836518i	0.999125	+	0.0418259i	$0.0133175\pi$
$397$			1.66675i			0.0836518i	−0.999125	+	0.0418259i	$0.986683\pi$
$398$		0			0
$399$	5.60785			0.280744
$400$		0			0
$401$	−15.4681			−0.772440			−0.386220	−	0.922407i	$-0.626219\pi$
$401$	−15.4681			−0.772440			−0.386220	+	0.922407i	$0.626219\pi$
$402$		0			0
$403$		−	37.0611i		−	1.84615i
$404$		0			0
$405$	1.85384	+	1.25031i	0.0921182	+	0.0621282i
$406$		0			0
$407$			1.46798i			0.0727652i
$408$		0			0
$409$	10.7451			0.531311			0.265656	−	0.964068i	$-0.414412\pi$
$409$	10.7451			0.531311			0.265656	+	0.964068i	$0.414412\pi$
$410$		0			0
$411$	9.29282			0.458381
$412$		0			0
$413$		−	9.67809i		−	0.476227i
$414$		0			0
$415$	−16.3292	+	24.2114i	−0.801567	+	1.18849i
$416$		0			0
$417$			14.2500i			0.697825i
$418$		0			0
$419$	12.0041			0.586437			0.293219	−	0.956045i	$-0.405274\pi$
$419$	12.0041			0.586437			0.293219	+	0.956045i	$0.405274\pi$
$420$		0			0
$421$	34.8222			1.69713			0.848564	−	0.529092i	$-0.177467\pi$
$421$	34.8222			1.69713			0.848564	+	0.529092i	$0.177467\pi$
$422$		0			0
$423$		−	12.1029i		−	0.588465i
$424$		0			0
$425$	4.63574	−	1.87347i	0.224867	−	0.0908765i
$426$		0			0
$427$		−	14.6209i		−	0.707556i
$428$		0			0
$429$	5.40051			0.260739
$430$		0			0
$431$	2.43069			0.117082			0.0585411	−	0.998285i	$-0.481355\pi$
$431$	2.43069			0.117082			0.0585411	+	0.998285i	$0.481355\pi$
$432$		0			0
$433$			26.7271i			1.28442i	0.766528	+	0.642211i	$0.221983\pi$
$433$			26.7271i			1.28442i	−0.766528	+	0.642211i	$0.778017\pi$
$434$		0			0
$435$	5.61336	−	8.32299i	0.269140	−	0.399057i
$436$		0			0
$437$		−	13.4666i		−	0.644193i
$438$		0			0
$439$	12.2029			0.582412			0.291206	−	0.956660i	$-0.405943\pi$
$439$	12.2029			0.582412			0.291206	+	0.956660i	$0.405943\pi$
$440$		0			0
$441$	−5.73713			−0.273197
$442$		0			0
$443$			34.6225i			1.64496i	0.568791	+	0.822482i	$0.307411\pi$
$443$			34.6225i			1.64496i	−0.568791	+	0.822482i	$0.692589\pi$
$444$		0			0
$445$	−19.9529	−	13.4571i	−0.945860	−	0.637926i
$446$		0			0
$447$		−	7.36827i		−	0.348507i
$448$		0			0
$449$	2.34666			0.110746			0.0553729	−	0.998466i	$-0.482365\pi$
$449$	2.34666			0.110746			0.0553729	+	0.998466i	$0.482365\pi$
$450$		0			0
$451$	−9.53706			−0.449083
$452$		0			0
$453$			6.34116i			0.297933i
$454$		0			0
$455$	14.3891	+	9.70462i	0.674573	+	0.454960i
$456$		0			0
$457$		−	4.07616i		−	0.190675i	−0.995445	−	0.0953373i	$-0.969607\pi$
$457$		−	4.07616i		−	0.190675i	0.995445	−	0.0953373i	$-0.0303930\pi$
$458$		0			0
$459$	1.00000			0.0466760
$460$		0			0
$461$	−2.38909			−0.111271			−0.0556356	−	0.998451i	$-0.517719\pi$
$461$	−2.38909			−0.111271			−0.0556356	+	0.998451i	$0.517719\pi$
$462$		0			0
$463$		−	32.6445i		−	1.51712i	−0.651603	−	0.758560i	$-0.725903\pi$
$463$		−	32.6445i		−	1.51712i	0.651603	−	0.758560i	$-0.274097\pi$
$464$		0			0
$465$	6.70891	−	9.94738i	0.311118	−	0.461298i
$466$		0			0
$467$			5.93390i			0.274588i	0.990530	+	0.137294i	$0.0438405\pi$
$467$			5.93390i			0.274588i	−0.990530	+	0.137294i	$0.956159\pi$
$468$		0			0
$469$	9.67809			0.446892
$470$		0			0
$471$	−9.95523			−0.458713
$472$		0			0
$473$		−	3.20866i		−	0.147534i
$474$		0			0
$475$	9.34897	+	23.1333i	0.428960	+	1.06143i
$476$		0			0
$477$			7.12377i			0.326175i
$478$		0			0
$479$	5.98199			0.273324			0.136662	−	0.990618i	$-0.456363\pi$
$479$	5.98199			0.273324			0.136662	+	0.990618i	$0.456363\pi$
$480$		0			0
$481$	−12.9674			−0.591262
$482$		0			0
$483$		−	3.03262i		−	0.137989i
$484$		0			0
$485$	22.7905	−	33.7917i	1.03486	−	1.53440i
$486$		0			0
$487$		−	12.6033i		−	0.571110i	−0.958362	−	0.285555i	$-0.907822\pi$
$487$		−	12.6033i		−	0.571110i	0.958362	−	0.285555i	$-0.0921780\pi$
$488$		0			0
$489$	−13.4192			−0.606837
$490$		0			0
$491$	−16.1398			−0.728381			−0.364190	−	0.931324i	$-0.618654\pi$
$491$	−16.1398			−0.728381			−0.364190	+	0.931324i	$0.618654\pi$
$492$		0			0
$493$		−	4.48959i		−	0.202201i
$494$		0			0
$495$	1.44952	+	0.977616i	0.0651512	+	0.0439406i
$496$		0			0
$497$			15.5236i			0.696328i
$498$		0			0
$499$	−9.02160			−0.403862			−0.201931	−	0.979400i	$-0.564722\pi$
$499$	−9.02160			−0.403862			−0.201931	+	0.979400i	$0.564722\pi$
$500$		0			0
$501$	6.35218			0.283795
$502$		0			0
$503$			0.646591i			0.0288301i	0.999896	+	0.0144150i	$0.00458861\pi$
$503$			0.646591i			0.0288301i	−0.999896	+	0.0144150i	$0.995411\pi$
$504$		0			0
$505$	9.73040	+	6.56258i	0.432997	+	0.292031i
$506$		0			0
$507$			34.7053i			1.54132i
$508$		0			0
$509$	22.4014			0.992926			0.496463	−	0.868058i	$-0.334632\pi$
$509$	22.4014			0.992926			0.496463	+	0.868058i	$0.334632\pi$
$510$		0			0
$511$	8.19273			0.362425
$512$		0			0
$513$			4.99020i			0.220323i
$514$		0			0
$515$	5.13084	−	7.60755i	0.226092	−	0.335229i
$516$		0			0
$517$		−	9.46331i		−	0.416196i
$518$		0			0
$519$	−1.30385			−0.0572327
$520$		0			0
$521$	11.8531			0.519292			0.259646	−	0.965704i	$-0.416394\pi$
$521$	11.8531			0.519292			0.259646	+	0.965704i	$0.416394\pi$
$522$		0			0
$523$			35.2253i			1.54029i	0.637866	+	0.770147i	$0.279817\pi$
$523$			35.2253i			1.54029i	−0.637866	+	0.770147i	$0.720183\pi$
$524$		0			0
$525$	2.10535	+	5.20953i	0.0918851	+	0.227362i
$526$		0			0
$527$		−	5.36581i		−	0.233739i
$528$		0			0
$529$	15.7175			0.683372
$530$		0			0
$531$	8.61213			0.373735
$532$		0			0
$533$		−	84.2454i		−	3.64907i
$534$		0			0
$535$	−2.81702	+	4.17683i	−0.121791	+	0.180580i
$536$		0			0
$537$			6.60280i			0.284932i
$538$		0			0
$539$	−4.48587			−0.193220
$540$		0			0
$541$	0.765471			0.0329101			0.0164551	−	0.999865i	$-0.494762\pi$
$541$	0.765471			0.0329101			0.0164551	+	0.999865i	$0.494762\pi$
$542$		0			0
$543$		−	4.95412i		−	0.212602i
$544$		0			0
$545$	22.2098	+	14.9792i	0.951362	+	0.641637i
$546$		0			0
$547$			3.36521i			0.143886i	0.997409	+	0.0719431i	$0.0229200\pi$
$547$			3.36521i			0.143886i	−0.997409	+	0.0719431i	$0.977080\pi$
$548$		0			0
$549$	13.0106			0.555277
$550$		0			0
$551$	22.4039			0.954440
$552$		0			0
$553$		−	14.3075i		−	0.608415i
$554$		0			0
$555$	−3.48050	−	2.34739i	−0.147739	−	0.0996412i
$556$		0			0
$557$		−	9.62270i		−	0.407727i	−0.978999	−	0.203863i	$-0.934650\pi$
$557$		−	9.62270i		−	0.407727i	0.978999	−	0.203863i	$-0.0653498\pi$
$558$		0			0
$559$	28.3436			1.19881
$560$		0			0
$561$	0.781901			0.0330119
$562$		0			0
$563$		−	38.9023i		−	1.63953i	−0.572697	−	0.819767i	$-0.694103\pi$
$563$		−	38.9023i		−	1.63953i	0.572697	−	0.819767i	$-0.305897\pi$
$564$		0			0
$565$	−10.3533	+	15.3509i	−0.435565	+	0.645816i
$566$		0			0
$567$			1.12377i			0.0471940i
$568$		0			0
$569$	−15.9914			−0.670395			−0.335197	−	0.942148i	$-0.608803\pi$
$569$	−15.9914			−0.670395			−0.335197	+	0.942148i	$0.608803\pi$
$570$		0			0
$571$	5.08906			0.212971			0.106485	−	0.994314i	$-0.466040\pi$
$571$	5.08906			0.212971			0.106485	+	0.994314i	$0.466040\pi$
$572$		0			0
$573$		−	19.0987i		−	0.797858i
$574$		0			0
$575$	12.5100	−	5.05574i	0.521704	−	0.210839i
$576$		0			0
$577$			14.0056i			0.583059i	0.956562	+	0.291530i	$0.0941642\pi$
$577$			14.0056i			0.583059i	−0.956562	+	0.291530i	$0.905836\pi$
$578$		0			0
$579$	16.3984			0.681496
$580$		0			0
$581$	−14.6766			−0.608889
$582$		0			0
$583$			5.57009i			0.230689i
$584$		0			0
$585$	−8.63574	+	12.8043i	−0.357044	+	0.529393i
$586$		0			0
$587$		−	29.7769i		−	1.22902i	−0.788908	−	0.614512i	$-0.789353\pi$
$587$		−	29.7769i		−	1.22902i	0.788908	−	0.614512i	$-0.210647\pi$
$588$		0			0
$589$	26.7765			1.10331
$590$		0			0
$591$	18.2335			0.750025
$592$		0			0
$593$			32.2587i			1.32471i	0.749191	+	0.662354i	$0.230442\pi$
$593$			32.2587i			1.32471i	−0.749191	+	0.662354i	$0.769558\pi$
$594$		0			0
$595$	2.08330	+	1.40506i	0.0854069	+	0.0576019i
$596$		0			0
$597$			25.7329i			1.05318i
$598$		0			0
$599$	−7.07116			−0.288920			−0.144460	−	0.989511i	$-0.546144\pi$
$599$	−7.07116			−0.288920			−0.144460	+	0.989511i	$0.546144\pi$
$600$		0			0
$601$	−30.9910			−1.26415			−0.632074	−	0.774908i	$-0.717796\pi$
$601$	−30.9910			−1.26415			−0.632074	+	0.774908i	$0.717796\pi$
$602$		0			0
$603$			8.61213i			0.350713i
$604$		0			0
$605$	−19.2589	−	12.9890i	−0.782985	−	0.528077i
$606$		0			0
$607$		−	19.5806i		−	0.794752i	−0.917656	−	0.397376i	$-0.869921\pi$
$607$		−	19.5806i		−	0.794752i	0.917656	−	0.397376i	$-0.130079\pi$
$608$		0			0
$609$	5.04528			0.204445
$610$		0			0
$611$	83.5939			3.38185
$612$		0			0
$613$			36.2578i			1.46444i	0.681069	+	0.732219i	$0.261515\pi$
$613$			36.2578i			1.46444i	−0.681069	+	0.732219i	$0.738485\pi$
$614$		0			0
$615$	15.2503	−	22.6118i	0.614953	−	0.911797i
$616$		0			0
$617$		−	1.63153i		−	0.0656828i	−0.999461	−	0.0328414i	$-0.989544\pi$
$617$		−	1.63153i		−	0.0656828i	0.999461	−	0.0328414i	$-0.0104556\pi$
$618$		0			0
$619$	4.39843			0.176788			0.0883939	−	0.996086i	$-0.471827\pi$
$619$	4.39843			0.176788			0.0883939	+	0.996086i	$0.471827\pi$
$620$		0			0
$621$	2.69860			0.108291
$622$		0			0
$623$		−	12.0952i		−	0.484584i
$624$		0			0
$625$	−17.9802	+	17.3698i	−0.719210	+	0.694793i
$626$		0			0
$627$			3.90184i			0.155825i
$628$		0			0
$629$	−1.87745			−0.0748590
$630$		0			0
$631$	32.3381			1.28736			0.643679	−	0.765295i	$-0.277407\pi$
$631$	32.3381			1.28736			0.643679	+	0.765295i	$0.277407\pi$
$632$		0			0
$633$		−	3.18988i		−	0.126786i
$634$		0			0
$635$	−17.0523	+	25.2837i	−0.676702	+	1.00335i
$636$		0			0
$637$		−	39.6258i		−	1.57003i
$638$		0			0
$639$	−13.8138			−0.546466
$640$		0			0
$641$	−45.6200			−1.80188			−0.900941	−	0.433942i	$-0.857122\pi$
$641$	−45.6200			−1.80188			−0.900941	+	0.433942i	$0.857122\pi$
$642$		0			0
$643$		−	23.3446i		−	0.920621i	−0.887758	−	0.460310i	$-0.847738\pi$
$643$		−	23.3446i		−	0.920621i	0.887758	−	0.460310i	$-0.152262\pi$
$644$		0			0
$645$	7.60755	+	5.13084i	0.299547	+	0.202027i
$646$		0			0
$647$			28.6985i			1.12826i	0.825687	+	0.564128i	$0.190788\pi$
$647$			28.6985i			1.12826i	−0.825687	+	0.564128i	$0.809212\pi$
$648$		0			0
$649$	6.73384			0.264326
$650$		0			0
$651$	6.02996			0.236333
$652$		0			0
$653$		−	28.6270i		−	1.12026i	−0.828404	−	0.560130i	$-0.810751\pi$
$653$		−	28.6270i		−	1.12026i	0.828404	−	0.560130i	$-0.189249\pi$
$654$		0			0
$655$	−41.7127	−	28.1327i	−1.62985	−	1.09924i
$656$		0			0
$657$			7.29037i			0.284424i
$658$		0			0
$659$	4.91783			0.191571			0.0957856	−	0.995402i	$-0.469464\pi$
$659$	4.91783			0.191571			0.0957856	+	0.995402i	$0.469464\pi$
$660$		0			0
$661$	−17.6855			−0.687886			−0.343943	−	0.938991i	$-0.611763\pi$
$661$	−17.6855			−0.687886			−0.343943	+	0.938991i	$0.611763\pi$
$662$		0			0
$663$			6.90690i			0.268242i
$664$		0			0
$665$	−7.01154	+	10.3961i	−0.271896	+	0.403143i
$666$		0			0
$667$		−	12.1156i		−	0.469118i
$668$		0			0
$669$	18.6466			0.720921
$670$		0			0
$671$	10.1730			0.392723
$672$		0			0
$673$		−	3.66435i		−	0.141250i	−0.997503	−	0.0706252i	$-0.977501\pi$
$673$		−	3.66435i		−	0.141250i	0.997503	−	0.0706252i	$-0.0224994\pi$
$674$		0			0
$675$	−4.63574	+	1.87347i	−0.178430	+	0.0721098i
$676$		0			0
$677$		−	50.8292i		−	1.95353i	−0.214325	−	0.976763i	$-0.568755\pi$
$677$		−	50.8292i		−	1.95353i	0.214325	−	0.976763i	$-0.431245\pi$
$678$		0			0
$679$	20.4841			0.786107
$680$		0			0
$681$	−5.27935			−0.202305
$682$		0			0
$683$		−	44.8133i		−	1.71473i	−0.514707	−	0.857366i	$-0.672099\pi$
$683$		−	44.8133i		−	1.71473i	0.514707	−	0.857366i	$-0.327901\pi$
$684$		0			0
$685$	−11.6189	+	17.2274i	−0.443935	+	0.658227i
$686$		0			0
$687$			16.9871i			0.648100i
$688$		0			0
$689$	−49.2032			−1.87449
$690$		0			0
$691$	17.9779			0.683913			0.341957	−	0.939716i	$-0.388910\pi$
$691$	17.9779			0.683913			0.341957	+	0.939716i	$0.388910\pi$
$692$		0			0
$693$			0.878680i			0.0333783i
$694$		0			0
$695$	−26.4173	−	17.8169i	−1.00206	−	0.675832i
$696$		0			0
$697$		−	12.1973i		−	0.462005i
$698$		0			0
$699$	−8.52567			−0.322470
$700$		0			0
$701$	−32.1980			−1.21610			−0.608051	−	0.793898i	$-0.708048\pi$
$701$	−32.1980			−1.21610			−0.608051	+	0.793898i	$0.708048\pi$
$702$		0			0
$703$		−	9.36887i		−	0.353354i
$704$		0			0
$705$	22.4370	+	15.1324i	0.845025	+	0.569919i
$706$		0			0
$707$			5.89843i			0.221833i
$708$		0			0
$709$	40.8651			1.53472			0.767361	−	0.641215i	$-0.221569\pi$
$709$	40.8651			1.53472			0.767361	+	0.641215i	$0.221569\pi$
$710$		0			0
$711$	12.7316			0.477473
$712$		0			0
$713$		−	14.4802i		−	0.542288i
$714$		0			0
$715$	−6.75230	+	10.0117i	−0.252522	+	0.374416i
$716$		0			0
$717$		−	6.95412i		−	0.259706i
$718$		0			0
$719$	−12.4487			−0.464258			−0.232129	−	0.972685i	$-0.574569\pi$
$719$	−12.4487			−0.464258			−0.232129	+	0.972685i	$0.574569\pi$
$720$		0			0
$721$	4.61159			0.171744
$722$		0			0
$723$		−	7.25811i		−	0.269932i
$724$		0			0
$725$	8.41109	+	20.8126i	0.312380	+	0.772960i
$726$		0			0
$727$		−	13.5687i		−	0.503236i	−0.967827	−	0.251618i	$-0.919037\pi$
$727$		−	13.5687i		−	0.503236i	0.967827	−	0.251618i	$-0.0809626\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	4.10366			0.151779
$732$		0			0
$733$		−	19.1364i		−	0.706820i	−0.935468	−	0.353410i	$-0.885022\pi$
$733$		−	19.1364i		−	0.706820i	0.935468	−	0.353410i	$-0.114978\pi$
$734$		0			0
$735$	7.17318	−	10.6357i	0.264587	−	0.392305i
$736$		0			0
$737$			6.73384i			0.248044i
$738$		0			0
$739$	15.2592			0.561319			0.280659	−	0.959807i	$-0.409447\pi$
$739$	15.2592			0.561319			0.280659	+	0.959807i	$0.409447\pi$
$740$		0			0
$741$	−34.4668			−1.26617
$742$		0			0
$743$		−	5.68967i		−	0.208734i	−0.994539	−	0.104367i	$-0.966718\pi$
$743$		−	5.68967i		−	0.208734i	0.994539	−	0.104367i	$-0.0332816\pi$
$744$		0			0
$745$	13.6596	+	9.21259i	0.500449	+	0.337523i
$746$		0			0
$747$		−	13.0601i		−	0.477845i
$748$		0			0
$749$	−2.53194			−0.0925149
$750$		0			0
$751$	−44.9416			−1.63994			−0.819972	−	0.572404i	$-0.806011\pi$
$751$	−44.9416			−1.63994			−0.819972	+	0.572404i	$0.806011\pi$
$752$		0			0
$753$		−	8.63419i		−	0.314647i
$754$		0			0
$755$	−11.7555	−	7.92839i	−0.427827	−	0.288544i
$756$		0			0
$757$		−	3.21402i		−	0.116816i	−0.998293	−	0.0584079i	$-0.981398\pi$
$757$		−	3.21402i		−	0.116816i	0.998293	−	0.0584079i	$-0.0186024\pi$
$758$		0			0
$759$	2.11004			0.0765896
$760$		0			0
$761$	−49.6110			−1.79840			−0.899199	−	0.437540i	$-0.855850\pi$
$761$	−49.6110			−1.79840			−0.899199	+	0.437540i	$0.855850\pi$
$762$		0			0
$763$			13.4633i			0.487403i
$764$		0			0
$765$	−1.25031	+	1.85384i	−0.0452049	+	0.0670258i
$766$		0			0
$767$			59.4832i			2.14781i
$768$		0			0
$769$	−45.2790			−1.63280			−0.816401	−	0.577485i	$-0.804034\pi$
$769$	−45.2790			−1.63280			−0.816401	+	0.577485i	$0.804034\pi$
$770$		0			0
$771$	−0.542976			−0.0195548
$772$		0			0
$773$			34.7968i			1.25155i	0.780002	+	0.625777i	$0.215218\pi$
$773$			34.7968i			1.25155i	−0.780002	+	0.625777i	$0.784782\pi$
$774$		0			0
$775$	10.0527	+	24.8745i	0.361103	+	0.893520i
$776$		0			0
$777$		−	2.10983i		−	0.0756898i
$778$		0			0
$779$	60.8668			2.18078
$780$		0			0
$781$	−10.8010			−0.386491
$782$		0			0
$783$			4.48959i			0.160445i
$784$		0			0
$785$	12.4471	−	18.4554i	0.444256	−	0.658703i
$786$		0			0
$787$		−	2.99106i		−	0.106620i	−0.998578	−	0.0533098i	$-0.983023\pi$
$787$		−	2.99106i		−	0.106620i	0.998578	−	0.0533098i	$-0.0169771\pi$
$788$		0			0
$789$	2.74985			0.0978972
$790$		0			0
$791$	−9.30549			−0.330865
$792$		0			0
$793$			89.8627i			3.19112i
$794$		0			0
$795$	−13.2064	−	8.90690i	−0.468381	−	0.315895i
$796$		0			0
$797$		−	15.5293i		−	0.550077i	−0.961433	−	0.275039i	$-0.911309\pi$
$797$		−	15.5293i		−	0.550077i	0.961433	−	0.275039i	$-0.0886906\pi$
$798$		0			0
$799$	12.1029			0.428171
$800$		0			0
$801$	10.7630			0.380293
$802$		0			0
$803$			5.70035i			0.201161i
$804$		0			0
$805$	5.62200	+	3.79170i	0.198149	+	0.133640i
$806$		0			0
$807$			11.8531i			0.417248i
$808$		0			0
$809$	−5.31416			−0.186836			−0.0934180	−	0.995627i	$-0.529779\pi$
$809$	−5.31416			−0.186836			−0.0934180	+	0.995627i	$0.529779\pi$
$810$		0			0
$811$	38.6831			1.35835			0.679174	−	0.733978i	$-0.262338\pi$
$811$	38.6831			1.35835			0.679174	+	0.733978i	$0.262338\pi$
$812$		0			0
$813$		−	4.52267i		−	0.158617i
$814$		0			0
$815$	16.7781	−	24.8771i	0.587712	−	0.871407i
$816$		0			0
$817$			20.4781i			0.716438i
$818$		0			0
$819$	−7.76179			−0.271219
$820$		0			0
$821$	−36.8426			−1.28582			−0.642909	−	0.765943i	$-0.722273\pi$
$821$	−36.8426			−1.28582			−0.642909	+	0.765943i	$0.722273\pi$
$822$		0			0
$823$			7.36521i			0.256735i	0.991727	+	0.128368i	$0.0409737\pi$
$823$			7.36521i			0.256735i	−0.991727	+	0.128368i	$0.959026\pi$
$824$		0			0
$825$	−3.62469	+	1.46487i	−0.126196	+	0.0510001i
$826$		0			0
$827$			32.4229i			1.12745i	0.825961	+	0.563727i	$0.190633\pi$
$827$			32.4229i			1.12745i	−0.825961	+	0.563727i	$0.809367\pi$
$828$		0			0
$829$	30.8905			1.07287			0.536435	−	0.843942i	$-0.319771\pi$
$829$	30.8905			1.07287			0.536435	+	0.843942i	$0.319771\pi$
$830$		0			0
$831$	−12.7316			−0.441655
$832$		0			0
$833$		−	5.73713i		−	0.198780i
$834$		0			0
$835$	−7.94218	+	11.7759i	−0.274850	+	0.407523i
$836$		0			0
$837$			5.36581i			0.185470i
$838$		0			0
$839$	−16.5933			−0.572863			−0.286431	−	0.958101i	$-0.592469\pi$
$839$	−16.5933			−0.572863			−0.286431	+	0.958101i	$0.592469\pi$
$840$		0			0
$841$	−8.84361			−0.304952
$842$		0			0
$843$			5.08217i			0.175039i
$844$		0			0
$845$	−64.3381	−	43.3922i	−2.21330	−	1.49274i
$846$		0			0
$847$		−	11.6745i		−	0.401139i
$848$		0			0
$849$	10.2759			0.352667
$850$		0			0
$851$	−5.06650			−0.173677
$852$		0			0
$853$			49.8460i			1.70669i	0.521344	+	0.853347i	$0.325431\pi$
$853$			49.8460i			1.70669i	−0.521344	+	0.853347i	$0.674569\pi$
$854$		0			0
$855$	−9.25105	−	6.23928i	−0.316379	−	0.213379i
$856$		0			0
$857$			44.1973i			1.50975i	0.655867	+	0.754876i	$0.272303\pi$
$857$			44.1973i			1.50975i	−0.655867	+	0.754876i	$0.727697\pi$
$858$		0			0
$859$	−11.0703			−0.377715			−0.188857	−	0.982005i	$-0.560478\pi$
$859$	−11.0703			−0.377715			−0.188857	+	0.982005i	$0.560478\pi$
$860$		0			0
$861$	13.7070			0.467132
$862$		0			0
$863$			1.97166i			0.0671162i	0.999437	+	0.0335581i	$0.0106839\pi$
$863$			1.97166i			0.0671162i	−0.999437	+	0.0335581i	$0.989316\pi$
$864$		0			0
$865$	1.63021	−	2.41713i	0.0554289	−	0.0821850i
$866$		0			0
$867$			1.00000i			0.0339618i
$868$		0			0
$869$	9.95487			0.337696
$870$		0			0
$871$	−59.4832			−2.01551
$872$		0			0
$873$			18.2279i			0.616923i
$874$		0			0
$875$	−12.2900	−	2.61051i	−0.415477	−	0.0882514i
$876$		0			0
$877$		−	18.6005i		−	0.628094i	−0.949407	−	0.314047i	$-0.898315\pi$
$877$		−	18.6005i		−	0.628094i	0.949407	−	0.314047i	$-0.101685\pi$
$878$		0			0
$879$	−15.7290			−0.530527
$880$		0			0
$881$	−26.9871			−0.909220			−0.454610	−	0.890691i	$-0.650221\pi$
$881$	−26.9871			−0.909220			−0.454610	+	0.890691i	$0.650221\pi$
$882$		0			0
$883$			32.5684i			1.09601i	0.836474	+	0.548007i	$0.184613\pi$
$883$			32.5684i			1.09601i	−0.836474	+	0.548007i	$0.815387\pi$
$884$		0			0
$885$	−10.7678	+	15.9655i	−0.361956	+	0.536676i
$886$		0			0
$887$			15.9614i			0.535932i	0.963428	+	0.267966i	$0.0863515\pi$
$887$			15.9614i			0.535932i	−0.963428	+	0.267966i	$0.913649\pi$
$888$		0			0
$889$	−15.3266			−0.514038
$890$		0			0
$891$	−0.781901			−0.0261947
$892$		0			0
$893$			60.3961i			2.02108i
$894$		0			0
$895$	−12.2405	−	8.25552i	−0.409156	−	0.275952i
$896$		0			0
$897$			18.6390i			0.622337i
$898$		0			0
$899$	24.0903			0.803456
$900$		0			0
$901$	−7.12377			−0.237327
$902$		0			0
$903$			4.61159i			0.153464i
$904$		0			0
$905$	9.18416	+	6.19417i	0.305292	+	0.205901i
$906$		0			0
$907$		−	46.1926i		−	1.53380i	−0.641767	−	0.766900i	$-0.721798\pi$
$907$		−	46.1926i		−	1.53380i	0.641767	−	0.766900i	$-0.278202\pi$
$908$		0			0
$909$	−5.24877			−0.174091
$910$		0			0
$911$	3.03435			0.100533			0.0502663	−	0.998736i	$-0.483993\pi$
$911$	3.03435			0.100533			0.0502663	+	0.998736i	$0.483993\pi$
$912$		0			0
$913$		−	10.2117i		−	0.337959i
$914$		0			0
$915$	−16.2672	+	24.1195i	−0.537777	+	0.797367i
$916$		0			0
$917$		−	25.2856i		−	0.835005i
$918$		0			0
$919$	32.7231			1.07943			0.539717	−	0.841847i	$-0.318531\pi$
$919$	32.7231			1.07943			0.539717	+	0.841847i	$0.318531\pi$
$920$		0			0
$921$	−10.5871			−0.348856
$922$		0			0
$923$		−	95.4106i		−	3.14048i
$924$		0			0
$925$	8.70339	−	3.51735i	0.286166	−	0.115650i
$926$		0			0
$927$			4.10366i			0.134782i
$928$		0			0
$929$	29.6401			0.972460			0.486230	−	0.873831i	$-0.338372\pi$
$929$	29.6401			0.972460			0.486230	+	0.873831i	$0.338372\pi$
$930$		0			0
$931$	28.6294			0.938292
$932$		0			0
$933$			5.65741i			0.185215i
$934$		0			0
$935$	−0.977616	+	1.44952i	−0.0319715	+	0.0474044i
$936$		0			0
$937$		−	13.5349i		−	0.442165i	−0.975255	−	0.221082i	$-0.929041\pi$
$937$		−	13.5349i		−	0.442165i	0.975255	−	0.221082i	$-0.0709590\pi$
$938$		0			0
$939$	3.80752			0.124254
$940$		0			0
$941$	55.0553			1.79475			0.897376	−	0.441267i	$-0.145471\pi$
$941$	55.0553			1.79475			0.897376	+	0.441267i	$0.145471\pi$
$942$		0			0
$943$		−	32.9156i		−	1.07188i
$944$		0			0
$945$	−2.08330	−	1.40506i	−0.0677697	−	0.0457066i
$946$		0			0
$947$		−	43.7344i		−	1.42118i	−0.703608	−	0.710588i	$-0.748429\pi$
$947$		−	43.7344i		−	1.42118i	0.703608	−	0.710588i	$-0.251571\pi$
$948$		0			0
$949$	−50.3539			−1.63456
$950$		0			0
$951$	22.5226			0.730345
$952$		0			0
$953$		−	21.4795i		−	0.695790i	−0.937533	−	0.347895i	$-0.886897\pi$
$953$		−	21.4795i		−	0.695790i	0.937533	−	0.347895i	$-0.113103\pi$
$954$		0			0
$955$	35.4059	+	23.8792i	1.14571	+	0.772712i
$956$		0			0
$957$			3.51041i			0.113476i
$958$		0			0
$959$	−10.4430			−0.337223
$960$		0			0
$961$	−2.20804			−0.0712273
$962$		0			0
$963$		−	2.25307i		−	0.0726041i
$964$		0			0
$965$	−20.5031	+	30.4001i	−0.660017	+	0.978614i
$966$		0			0
$967$			3.21334i			0.103334i	0.998664	+	0.0516671i	$0.0164535\pi$
$967$			3.21334i			0.103334i	−0.998664	+	0.0516671i	$0.983547\pi$
$968$		0			0
$969$	−4.99020			−0.160308
$970$		0			0
$971$	−7.02628			−0.225484			−0.112742	−	0.993624i	$-0.535963\pi$
$971$	−7.02628			−0.225484			−0.112742	+	0.993624i	$0.535963\pi$
$972$		0			0
$973$		−	16.0138i		−	0.513378i
$974$		0			0
$975$	−12.9398	−	32.0186i	−0.414407	−	1.02542i
$976$		0			0
$977$			37.8209i			1.21000i	0.796226	+	0.604999i	$0.206827\pi$
$977$			37.8209i			1.21000i	−0.796226	+	0.604999i	$0.793173\pi$
$978$		0			0
$979$	8.41562			0.268964
$980$		0			0
$981$	−11.9804			−0.382505
$982$		0			0
$983$			28.1150i			0.896729i	0.893851	+	0.448365i	$0.147993\pi$
$983$			28.1150i			0.896729i	−0.893851	+	0.448365i	$0.852007\pi$
$984$		0			0
$985$	−22.7974	+	33.8020i	−0.726386	+	1.07702i
$986$		0			0
$987$			13.6010i			0.432924i
$988$		0			0
$989$	11.0742			0.352138
$990$		0			0
$991$	3.96666			0.126005			0.0630026	−	0.998013i	$-0.479932\pi$
$991$	3.96666			0.126005			0.0630026	+	0.998013i	$0.479932\pi$
$992$		0			0
$993$		−	19.6337i		−	0.623057i
$994$		0			0
$995$	−47.7047	−	32.1740i	−1.51234	−	1.01998i
$996$		0			0
$997$			22.4622i			0.711384i	0.934603	+	0.355692i	$0.115755\pi$
$997$			22.4622i			0.711384i	−0.934603	+	0.355692i	$0.884245\pi$
$998$		0			0
$999$	1.87745			0.0594000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1020.2.g.d.409.4	✓	10
3.2	odd	2		3060.2.g.g.2449.3		10
4.3	odd	2		4080.2.m.r.2449.9		10
5.2	odd	4		5100.2.a.bc.1.3		5
5.3	odd	4		5100.2.a.bd.1.3		5
5.4	even	2	inner	1020.2.g.d.409.9	yes	10
15.14	odd	2		3060.2.g.g.2449.4		10
20.19	odd	2		4080.2.m.r.2449.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1020.2.g.d.409.4	✓	10	1.1	even	1	trivial
1020.2.g.d.409.9	yes	10	5.4	even	2	inner
3060.2.g.g.2449.3		10	3.2	odd	2
3060.2.g.g.2449.4		10	15.14	odd	2
4080.2.m.r.2449.4		10	20.19	odd	2
4080.2.m.r.2449.9		10	4.3	odd	2
5100.2.a.bc.1.3		5	5.2	odd	4
5100.2.a.bd.1.3		5	5.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1020.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(1.85384 + 1.25031i) q^{5} +1.12377i q^{7} -1.00000 q^{9} -0.781901 q^{11} -6.90690i q^{13} +(1.25031 - 1.85384i) q^{15} -1.00000i q^{17} +4.99020 q^{19} +1.12377 q^{21} -2.69860i q^{23} +(1.87347 + 4.63574i) q^{25} +1.00000i q^{27} +4.48959 q^{29} +5.36581 q^{31} +0.781901i q^{33} +(-1.40506 + 2.08330i) q^{35} -1.87745i q^{37} -6.90690 q^{39} +12.1973 q^{41} +4.10366i q^{43} +(-1.85384 - 1.25031i) q^{45} +12.1029i q^{47} +5.73713 q^{49} -1.00000 q^{51} -7.12377i q^{53} +(-1.44952 - 0.977616i) q^{55} -4.99020i q^{57} -8.61213 q^{59} -13.0106 q^{61} -1.12377i q^{63} +(8.63574 - 12.8043i) q^{65} -8.61213i q^{67} -2.69860 q^{69} +13.8138 q^{71} -7.29037i q^{73} +(4.63574 - 1.87347i) q^{75} -0.878680i q^{77} -12.7316 q^{79} +1.00000 q^{81} +13.0601i q^{83} +(1.25031 - 1.85384i) q^{85} -4.48959i q^{87} -10.7630 q^{89} +7.76179 q^{91} -5.36581i q^{93} +(9.25105 + 6.23928i) q^{95} -18.2279i q^{97} +0.781901 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{9} + 12 q^{11} + 4 q^{15} - 24 q^{19} - 4 q^{21} + 12 q^{25} - 12 q^{29} + 12 q^{31} + 4 q^{35} + 28 q^{41} - 30 q^{49} - 10 q^{51} + 26 q^{55} - 48 q^{59} - 28 q^{61} + 48 q^{65} - 12 q^{69}+ \cdots - 12 q^{99}+O(q^{100}) \) 10 * q - 10 * q^9 + 12 * q^11 + 4 * q^15 - 24 * q^19 - 4 * q^21 + 12 * q^25 - 12 * q^29 + 12 * q^31 + 4 * q^35 + 28 * q^41 - 30 * q^49 - 10 * q^51 + 26 * q^55 - 48 * q^59 - 28 * q^61 + 48 * q^65 - 12 * q^69 + 8 * q^75 - 44 * q^79 + 10 * q^81 + 4 * q^85 - 36 * q^89 + 40 * q^91 + 44 * q^95 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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