LMFDB - Embedded newform 4536.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4536,2,Mod(1,4536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4536, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4536.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4536 = 2^{3} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4536.a (trivial)

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:	yes
Analytic conductor:	$36.2201423569$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.45729.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} + 12x - 3 $ x^4 - x^3 - 11x^2 + 12x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 504)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.38095$ of defining polynomial
Character	$\chi$	$=$	4536.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.38095 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-4.38095 q^{5} -1.00000 q^{7} +5.38095 q^{11} -2.54348 q^{13} +2.58058 q^{17} -6.72479 q^{19} +0.800370 q^{23} +14.1927 q^{25} -3.74311 q^{29} +3.39370 q^{31} +4.38095 q^{35} +4.38095 q^{37} +6.39370 q^{41} -0.763270 q^{43} +8.26827 q^{47} +1.00000 q^{49} +4.94877 q^{53} -23.5736 q^{55} -5.57502 q^{59} -8.28659 q^{61} +11.1428 q^{65} -1.89288 q^{67} -1.34385 q^{71} -8.65060 q^{73} -5.38095 q^{77} -13.2810 q^{79} -3.72479 q^{83} -11.3054 q^{85} +6.99862 q^{89} +2.54348 q^{91} +29.4610 q^{95} +2.96152 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q - 3 * q^5 - 4 * q^7	$ 4 q - 3 q^{5} - 4 q^{7} + 7 q^{11} - 3 q^{13} - 3 q^{17} - 4 q^{19} + 2 q^{23} + 5 q^{25} - 9 q^{29} - 3 q^{31} + 3 q^{35} + 3 q^{37} + 9 q^{41} - 8 q^{43} + 3 q^{47} + 4 q^{49} - 6 q^{53} - 28 q^{55} + 10 q^{59} - 20 q^{61} + q^{65} - 11 q^{67} + 3 q^{71} - 24 q^{73} - 7 q^{77} - 21 q^{79} + 8 q^{83} - 9 q^{85} - 6 q^{89} + 3 q^{91} + 36 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q - 3 * q^5 - 4 * q^7 + 7 * q^11 - 3 * q^13 - 3 * q^17 - 4 * q^19 + 2 * q^23 + 5 * q^25 - 9 * q^29 - 3 * q^31 + 3 * q^35 + 3 * q^37 + 9 * q^41 - 8 * q^43 + 3 * q^47 + 4 * q^49 - 6 * q^53 - 28 * q^55 + 10 * q^59 - 20 * q^61 + q^65 - 11 * q^67 + 3 * q^71 - 24 * q^73 - 7 * q^77 - 21 * q^79 + 8 * q^83 - 9 * q^85 - 6 * q^89 + 3 * q^91 + 36 * q^95 - 16 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−4.38095	−1.95922	−0.979610	−	0.200911i	$-0.935610\pi$
$5$	−4.38095	−1.95922	−0.979610	+	0.200911i	$0.935610\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.38095	1.62242	0.811208	−	0.584757i	$-0.198810\pi$
$11$	5.38095	1.62242	0.811208	+	0.584757i	$0.198810\pi$
$12$	0	0
$13$	−2.54348	−0.705434	−0.352717	−	0.935730i	$-0.614742\pi$
$13$	−2.54348	−0.705434	−0.352717	+	0.935730i	$0.614742\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.58058	0.625882	0.312941	−	0.949773i	$-0.398686\pi$
$17$	2.58058	0.625882	0.312941	+	0.949773i	$0.398686\pi$
$18$	0	0
$19$	−6.72479	−1.54277	−0.771387	−	0.636366i	$-0.780437\pi$
$19$	−6.72479	−1.54277	−0.771387	+	0.636366i	$0.780437\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.800370	0.166889	0.0834443	−	0.996512i	$-0.473408\pi$
$23$	0.800370	0.166889	0.0834443	+	0.996512i	$0.473408\pi$
$24$	0	0
$25$	14.1927	2.83854
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.74311	−0.695078	−0.347539	−	0.937666i	$-0.612983\pi$
$29$	−3.74311	−0.695078	−0.347539	+	0.937666i	$0.612983\pi$
$30$	0	0
$31$	3.39370	0.609527	0.304764	−	0.952428i	$-0.401423\pi$
$31$	3.39370	0.609527	0.304764	+	0.952428i	$0.401423\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.38095	0.740515
$36$	0	0
$37$	4.38095	0.720223	0.360112	−	0.932909i	$-0.382739\pi$
$37$	4.38095	0.720223	0.360112	+	0.932909i	$0.382739\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.39370	0.998529	0.499264	−	0.866450i	$-0.333604\pi$
$41$	6.39370	0.998529	0.499264	+	0.866450i	$0.333604\pi$
$42$	0	0
$43$	−0.763270	−0.116398	−0.0581988	−	0.998305i	$-0.518536\pi$
$43$	−0.763270	−0.116398	−0.0581988	+	0.998305i	$0.518536\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.26827	1.20605	0.603026	−	0.797722i	$-0.293962\pi$
$47$	8.26827	1.20605	0.603026	+	0.797722i	$0.293962\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.94877	0.679766	0.339883	−	0.940468i	$-0.389613\pi$
$53$	4.94877	0.679766	0.339883	+	0.940468i	$0.389613\pi$
$54$	0	0
$55$	−23.5736	−3.17867
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.57502	−0.725806	−0.362903	−	0.931827i	$-0.618214\pi$
$59$	−5.57502	−0.725806	−0.362903	+	0.931827i	$0.618214\pi$
$60$	0	0
$61$	−8.28659	−1.06099	−0.530494	−	0.847689i	$-0.677994\pi$
$61$	−8.28659	−1.06099	−0.530494	+	0.847689i	$0.677994\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	11.1428	1.38210
$66$	0	0
$67$	−1.89288	−0.231252	−0.115626	−	0.993293i	$-0.536887\pi$
$67$	−1.89288	−0.231252	−0.115626	+	0.993293i	$0.536887\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.34385	−0.159485	−0.0797427	−	0.996815i	$-0.525410\pi$
$71$	−1.34385	−0.159485	−0.0797427	+	0.996815i	$0.525410\pi$
$72$	0	0
$73$	−8.65060	−1.01248	−0.506238	−	0.862394i	$-0.668964\pi$
$73$	−8.65060	−1.01248	−0.506238	+	0.862394i	$0.668964\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.38095	−0.613216
$78$	0	0
$79$	−13.2810	−1.49423	−0.747116	−	0.664693i	$-0.768562\pi$
$79$	−13.2810	−1.49423	−0.747116	+	0.664693i	$0.768562\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.72479	−0.408849	−0.204425	−	0.978882i	$-0.565532\pi$
$83$	−3.72479	−0.408849	−0.204425	+	0.978882i	$0.565532\pi$
$84$	0	0
$85$	−11.3054	−1.22624
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.99862	0.741853	0.370926	−	0.928662i	$-0.379040\pi$
$89$	6.99862	0.741853	0.370926	+	0.928662i	$0.379040\pi$
$90$	0	0
$91$	2.54348	0.266629
$92$	0	0
$93$	0	0
$94$	0	0
$95$	29.4610	3.02263
$96$	0	0
$97$	2.96152	0.300697	0.150349	−	0.988633i	$-0.451960\pi$
$97$	2.96152	0.300697	0.150349	+	0.988633i	$0.451960\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.1299	1.70449	0.852243	−	0.523146i	$-0.175242\pi$
$101$	17.1299	1.70449	0.852243	+	0.523146i	$0.175242\pi$
$102$	0	0
$103$	−12.0357	−1.18592	−0.592958	−	0.805234i	$-0.702040\pi$
$103$	−12.0357	−1.18592	−0.592958	+	0.805234i	$0.702040\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.92580	0.669542	0.334771	−	0.942299i	$-0.391341\pi$
$107$	6.92580	0.669542	0.334771	+	0.942299i	$0.391341\pi$
$108$	0	0
$109$	−4.41942	−0.423304	−0.211652	−	0.977345i	$-0.567884\pi$
$109$	−4.41942	−0.423304	−0.211652	+	0.977345i	$0.567884\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.8361	−1.20752	−0.603759	−	0.797167i	$-0.706331\pi$
$113$	−12.8361	−1.20752	−0.603759	+	0.797167i	$0.706331\pi$
$114$	0	0
$115$	−3.50638	−0.326971
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.58058	−0.236561
$120$	0	0
$121$	17.9546	1.63224
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−40.2727	−3.60210
$126$	0	0
$127$	−2.82471	−0.250653	−0.125326	−	0.992116i	$-0.539998\pi$
$127$	−2.82471	−0.250653	−0.125326	+	0.992116i	$0.539998\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−17.1185	−1.49565	−0.747825	−	0.663896i	$-0.768902\pi$
$131$	−17.1185	−1.49565	−0.747825	+	0.663896i	$0.768902\pi$
$132$	0	0
$133$	6.72479	0.583114
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.2129	1.21429	0.607143	−	0.794592i	$-0.292315\pi$
$137$	14.2129	1.21429	0.607143	+	0.794592i	$0.292315\pi$
$138$	0	0
$139$	−3.20101	−0.271506	−0.135753	−	0.990743i	$-0.543345\pi$
$139$	−3.20101	−0.271506	−0.135753	+	0.990743i	$0.543345\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−13.6863	−1.14451
$144$	0	0
$145$	16.3984	1.36181
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.0983	−1.40075	−0.700375	−	0.713775i	$-0.746984\pi$
$149$	−17.0983	−1.40075	−0.700375	+	0.713775i	$0.746984\pi$
$150$	0	0
$151$	0.105742	0.00860514	0.00430257	−	0.999991i	$-0.498630\pi$
$151$	0.105742	0.00860514	0.00430257	+	0.999991i	$0.498630\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−14.8676	−1.19420
$156$	0	0
$157$	−21.0047	−1.67635	−0.838177	−	0.545398i	$-0.816378\pi$
$157$	−21.0047	−1.67635	−0.838177	+	0.545398i	$0.816378\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.800370	−0.0630780
$162$	0	0
$163$	2.58915	0.202798	0.101399	−	0.994846i	$-0.467668\pi$
$163$	2.58915	0.202798	0.101399	+	0.994846i	$0.467668\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.61768	−0.202562	−0.101281	−	0.994858i	$-0.532294\pi$
$167$	−2.61768	−0.202562	−0.101281	+	0.994858i	$0.532294\pi$
$168$	0	0
$169$	−6.53072	−0.502363
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.4551	−1.09900	−0.549502	−	0.835492i	$-0.685183\pi$
$173$	−14.4551	−1.09900	−0.549502	+	0.835492i	$0.685183\pi$
$174$	0	0
$175$	−14.1927	−1.07287
$176$	0	0
$177$	0	0
$178$	0	0
$179$	23.7179	1.77276	0.886378	−	0.462962i	$-0.153213\pi$
$179$	23.7179	1.77276	0.886378	+	0.462962i	$0.153213\pi$
$180$	0	0
$181$	−6.60630	−0.491042	−0.245521	−	0.969391i	$-0.578959\pi$
$181$	−6.60630	−0.491042	−0.245521	+	0.969391i	$0.578959\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−19.1927	−1.41108
$186$	0	0
$187$	13.8859	1.01544
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.16671	0.301493	0.150746	−	0.988572i	$-0.451832\pi$
$191$	4.16671	0.301493	0.150746	+	0.988572i	$0.451832\pi$
$192$	0	0
$193$	−16.1885	−1.16527	−0.582637	−	0.812732i	$-0.697979\pi$
$193$	−16.1885	−1.16527	−0.582637	+	0.812732i	$0.697979\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.33109	−0.308577	−0.154289	−	0.988026i	$-0.549309\pi$
$197$	−4.33109	−0.308577	−0.154289	+	0.988026i	$0.549309\pi$
$198$	0	0
$199$	13.7849	0.977183	0.488591	−	0.872513i	$-0.337511\pi$
$199$	13.7849	0.977183	0.488591	+	0.872513i	$0.337511\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.74311	0.262715
$204$	0	0
$205$	−28.0105	−1.95634
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−36.1858	−2.50302
$210$	0	0
$211$	12.3668	0.851367	0.425683	−	0.904872i	$-0.360034\pi$
$211$	12.3668	0.851367	0.425683	+	0.904872i	$0.360034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.34385	0.228049
$216$	0	0
$217$	−3.39370	−0.230380
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.56364	−0.441518
$222$	0	0
$223$	−14.8064	−0.991510	−0.495755	−	0.868462i	$-0.665109\pi$
$223$	−14.8064	−0.991510	−0.495755	+	0.868462i	$0.665109\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.7976	−0.915780	−0.457890	−	0.889009i	$-0.651395\pi$
$227$	−13.7976	−0.915780	−0.457890	+	0.889009i	$0.651395\pi$
$228$	0	0
$229$	0.942738	0.0622979	0.0311489	−	0.999515i	$-0.490083\pi$
$229$	0.942738	0.0622979	0.0311489	+	0.999515i	$0.490083\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.6608	−1.09149	−0.545743	−	0.837953i	$-0.683752\pi$
$233$	−16.6608	−1.09149	−0.545743	+	0.837953i	$0.683752\pi$
$234$	0	0
$235$	−36.2229	−2.36292
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.35681	−0.0877648	−0.0438824	−	0.999037i	$-0.513973\pi$
$239$	−1.35681	−0.0877648	−0.0438824	+	0.999037i	$0.513973\pi$
$240$	0	0
$241$	−9.21657	−0.593691	−0.296846	−	0.954925i	$-0.595935\pi$
$241$	−9.21657	−0.593691	−0.296846	+	0.954925i	$0.595935\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.38095	−0.279888
$246$	0	0
$247$	17.1044	1.08832
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.53189	0.159812	0.0799058	−	0.996802i	$-0.474538\pi$
$251$	2.53189	0.159812	0.0799058	+	0.996802i	$0.474538\pi$
$252$	0	0
$253$	4.30675	0.270763
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.16809	0.384755	0.192377	−	0.981321i	$-0.438380\pi$
$257$	6.16809	0.384755	0.192377	+	0.981321i	$0.438380\pi$
$258$	0	0
$259$	−4.38095	−0.272219
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.9815	−0.677147	−0.338574	−	0.940940i	$-0.609944\pi$
$263$	−10.9815	−0.677147	−0.338574	+	0.940940i	$0.609944\pi$
$264$	0	0
$265$	−21.6803	−1.33181
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.73035	−0.288415	−0.144207	−	0.989547i	$-0.546063\pi$
$269$	−4.73035	−0.288415	−0.144207	+	0.989547i	$0.546063\pi$
$270$	0	0
$271$	−13.5379	−0.822370	−0.411185	−	0.911552i	$-0.634885\pi$
$271$	−13.5379	−0.822370	−0.411185	+	0.911552i	$0.634885\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	76.3702	4.60529
$276$	0	0
$277$	−29.3228	−1.76184	−0.880918	−	0.473270i	$-0.843074\pi$
$277$	−29.3228	−1.76184	−0.880918	+	0.473270i	$0.843074\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−14.6267	−0.872557	−0.436279	−	0.899812i	$-0.643704\pi$
$281$	−14.6267	−0.872557	−0.436279	+	0.899812i	$0.643704\pi$
$282$	0	0
$283$	15.2912	0.908970	0.454485	−	0.890754i	$-0.349823\pi$
$283$	15.2912	0.908970	0.454485	+	0.890754i	$0.349823\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.39370	−0.377408
$288$	0	0
$289$	−10.3406	−0.608272
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.88039	0.109854	0.0549268	−	0.998490i	$-0.482507\pi$
$293$	1.88039	0.109854	0.0549268	+	0.998490i	$0.482507\pi$
$294$	0	0
$295$	24.4239	1.42201
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.03572	−0.117729
$300$	0	0
$301$	0.763270	0.0439942
$302$	0	0
$303$	0	0
$304$	0	0
$305$	36.3031	2.07871
$306$	0	0
$307$	11.8451	0.676038	0.338019	−	0.941139i	$-0.390243\pi$
$307$	11.8451	0.676038	0.338019	+	0.941139i	$0.390243\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−35.0845	−1.98946	−0.994729	−	0.102540i	$-0.967303\pi$
$311$	−35.0845	−1.98946	−0.994729	+	0.102540i	$0.967303\pi$
$312$	0	0
$313$	−0.861812	−0.0487125	−0.0243563	−	0.999703i	$-0.507754\pi$
$313$	−0.861812	−0.0487125	−0.0243563	+	0.999703i	$0.507754\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.62763	−0.484576	−0.242288	−	0.970204i	$-0.577898\pi$
$317$	−8.62763	−0.484576	−0.242288	+	0.970204i	$0.577898\pi$
$318$	0	0
$319$	−20.1415	−1.12771
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−17.3539	−0.965594
$324$	0	0
$325$	−36.0988	−2.00240
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.26827	−0.455845
$330$	0	0
$331$	−17.5370	−0.963922	−0.481961	−	0.876193i	$-0.660075\pi$
$331$	−17.5370	−0.963922	−0.481961	+	0.876193i	$0.660075\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.29262	0.453074
$336$	0	0
$337$	33.5723	1.82880	0.914399	−	0.404814i	$-0.132664\pi$
$337$	33.5723	1.82880	0.914399	+	0.404814i	$0.132664\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	18.2613	0.988907
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.61233	0.193920	0.0969599	−	0.995288i	$-0.469088\pi$
$347$	3.61233	0.193920	0.0969599	+	0.995288i	$0.469088\pi$
$348$	0	0
$349$	−9.33782	−0.499842	−0.249921	−	0.968266i	$-0.580405\pi$
$349$	−9.33782	−0.499842	−0.249921	+	0.968266i	$0.580405\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−19.3539	−1.03010	−0.515051	−	0.857160i	$-0.672227\pi$
$353$	−19.3539	−1.03010	−0.515051	+	0.857160i	$0.672227\pi$
$354$	0	0
$355$	5.88733	0.312467
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.1716	−1.64518	−0.822588	−	0.568638i	$-0.807471\pi$
$359$	−31.1716	−1.64518	−0.822588	+	0.568638i	$0.807471\pi$
$360$	0	0
$361$	26.2229	1.38015
$362$	0	0
$363$	0	0
$364$	0	0
$365$	37.8978	1.98366
$366$	0	0
$367$	5.50823	0.287527	0.143764	−	0.989612i	$-0.454080\pi$
$367$	5.50823	0.287527	0.143764	+	0.989612i	$0.454080\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.94877	−0.256927
$372$	0	0
$373$	−11.5337	−0.597194	−0.298597	−	0.954379i	$-0.596519\pi$
$373$	−11.5337	−0.597194	−0.298597	+	0.954379i	$0.596519\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.52051	0.490331
$378$	0	0
$379$	35.4614	1.82153	0.910766	−	0.412923i	$-0.135492\pi$
$379$	35.4614	1.82153	0.910766	+	0.412923i	$0.135492\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	29.2801	1.49614	0.748072	−	0.663617i	$-0.230980\pi$
$383$	29.2801	1.49614	0.748072	+	0.663617i	$0.230980\pi$
$384$	0	0
$385$	23.5736	1.20142
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.90728	0.350213	0.175107	−	0.984549i	$-0.443973\pi$
$389$	6.90728	0.350213	0.175107	+	0.984549i	$0.443973\pi$
$390$	0	0
$391$	2.06542	0.104453
$392$	0	0
$393$	0	0
$394$	0	0
$395$	58.1835	2.92753
$396$	0	0
$397$	24.6291	1.23610	0.618048	−	0.786140i	$-0.287924\pi$
$397$	24.6291	1.23610	0.618048	+	0.786140i	$0.287924\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	33.8039	1.68809	0.844043	−	0.536275i	$-0.180169\pi$
$401$	33.8039	1.68809	0.844043	+	0.536275i	$0.180169\pi$
$402$	0	0
$403$	−8.63181	−0.429981
$404$	0	0
$405$	0	0
$406$	0	0
$407$	23.5736	1.16850
$408$	0	0
$409$	−24.3026	−1.20169	−0.600844	−	0.799367i	$-0.705169\pi$
$409$	−24.3026	−1.20169	−0.600844	+	0.799367i	$0.705169\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.57502	0.274329
$414$	0	0
$415$	16.3181	0.801025
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−19.0374	−0.930036	−0.465018	−	0.885301i	$-0.653952\pi$
$419$	−19.0374	−0.930036	−0.465018	+	0.885301i	$0.653952\pi$
$420$	0	0
$421$	−11.7179	−0.571094	−0.285547	−	0.958365i	$-0.592175\pi$
$421$	−11.7179	−0.571094	−0.285547	+	0.958365i	$0.592175\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	36.6253	1.77659
$426$	0	0
$427$	8.28659	0.401016
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.89426	0.476590	0.238295	−	0.971193i	$-0.423412\pi$
$431$	9.89426	0.476590	0.238295	+	0.971193i	$0.423412\pi$
$432$	0	0
$433$	38.0986	1.83090	0.915451	−	0.402429i	$-0.131834\pi$
$433$	38.0986	1.83090	0.915451	+	0.402429i	$0.131834\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.38232	−0.257471
$438$	0	0
$439$	19.9949	0.954305	0.477153	−	0.878820i	$-0.341669\pi$
$439$	19.9949	0.954305	0.477153	+	0.878820i	$0.341669\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.31648	0.395128	0.197564	−	0.980290i	$-0.436697\pi$
$443$	8.31648	0.395128	0.197564	+	0.980290i	$0.436697\pi$
$444$	0	0
$445$	−30.6606	−1.45345
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−22.5036	−1.06201	−0.531006	−	0.847368i	$-0.678186\pi$
$449$	−22.5036	−1.06201	−0.531006	+	0.847368i	$0.678186\pi$
$450$	0	0
$451$	34.4042	1.62003
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−11.1428	−0.522384
$456$	0	0
$457$	15.8229	0.740162	0.370081	−	0.928999i	$-0.379330\pi$
$457$	15.8229	0.740162	0.370081	+	0.928999i	$0.379330\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.7587	0.733954	0.366977	−	0.930230i	$-0.380393\pi$
$461$	15.7587	0.733954	0.366977	+	0.930230i	$0.380393\pi$
$462$	0	0
$463$	−17.1482	−0.796944	−0.398472	−	0.917180i	$-0.630459\pi$
$463$	−17.1482	−0.796944	−0.398472	+	0.917180i	$0.630459\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.41270	0.111646	0.0558231	−	0.998441i	$-0.482222\pi$
$467$	2.41270	0.111646	0.0558231	+	0.998441i	$0.482222\pi$
$468$	0	0
$469$	1.89288	0.0874052
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.10712	−0.188845
$474$	0	0
$475$	−95.4430	−4.37922
$476$	0	0
$477$	0	0
$478$	0	0
$479$	26.3569	1.20428	0.602138	−	0.798392i	$-0.294316\pi$
$479$	26.3569	1.20428	0.602138	+	0.798392i	$0.294316\pi$
$480$	0	0
$481$	−11.1428	−0.508070
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12.9743	−0.589132
$486$	0	0
$487$	16.1141	0.730200	0.365100	−	0.930968i	$-0.381035\pi$
$487$	16.1141	0.730200	0.365100	+	0.930968i	$0.381035\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7.32533	−0.330587	−0.165294	−	0.986244i	$-0.552857\pi$
$491$	−7.32533	−0.330587	−0.165294	+	0.986244i	$0.552857\pi$
$492$	0	0
$493$	−9.65938	−0.435037
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.34385	0.0602798
$498$	0	0
$499$	6.03175	0.270018	0.135009	−	0.990844i	$-0.456894\pi$
$499$	6.03175	0.270018	0.135009	+	0.990844i	$0.456894\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.4106	0.731714	0.365857	−	0.930671i	$-0.380776\pi$
$503$	16.4106	0.731714	0.365857	+	0.930671i	$0.380776\pi$
$504$	0	0
$505$	−75.0451	−3.33946
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11.4139	−0.505911	−0.252955	−	0.967478i	$-0.581403\pi$
$509$	−11.4139	−0.505911	−0.252955	+	0.967478i	$0.581403\pi$
$510$	0	0
$511$	8.65060	0.382680
$512$	0	0
$513$	0	0
$514$	0	0
$515$	52.7279	2.32347
$516$	0	0
$517$	44.4911	1.95672
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−41.1600	−1.80325	−0.901627	−	0.432514i	$-0.857627\pi$
$521$	−41.1600	−1.80325	−0.901627	+	0.432514i	$0.857627\pi$
$522$	0	0
$523$	−29.2303	−1.27815	−0.639075	−	0.769144i	$-0.720683\pi$
$523$	−29.2303	−1.27815	−0.639075	+	0.769144i	$0.720683\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.75771	0.381492
$528$	0	0
$529$	−22.3594	−0.972148
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−16.2622	−0.704396
$534$	0	0
$535$	−30.3416	−1.31178
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.38095	0.231774
$540$	0	0
$541$	−6.44101	−0.276921	−0.138460	−	0.990368i	$-0.544215\pi$
$541$	−6.44101	−0.276921	−0.138460	+	0.990368i	$0.544215\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	19.3613	0.829345
$546$	0	0
$547$	−1.37936	−0.0589774	−0.0294887	−	0.999565i	$-0.509388\pi$
$547$	−1.37936	−0.0589774	−0.0294887	+	0.999565i	$0.509388\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	25.1716	1.07235
$552$	0	0
$553$	13.2810	0.564767
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.5692	−1.08340	−0.541701	−	0.840571i	$-0.682219\pi$
$557$	−25.5692	−1.08340	−0.541701	+	0.840571i	$0.682219\pi$
$558$	0	0
$559$	1.94136	0.0821108
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15.0666	0.634981	0.317490	−	0.948261i	$-0.397160\pi$
$563$	15.0666	0.634981	0.317490	+	0.948261i	$0.397160\pi$
$564$	0	0
$565$	56.2342	2.36579
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14.3388	−0.601112	−0.300556	−	0.953764i	$-0.597172\pi$
$569$	−14.3388	−0.601112	−0.300556	+	0.953764i	$0.597172\pi$
$570$	0	0
$571$	−2.58819	−0.108312	−0.0541562	−	0.998532i	$-0.517247\pi$
$571$	−2.58819	−0.108312	−0.0541562	+	0.998532i	$0.517247\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	11.3594	0.473720
$576$	0	0
$577$	17.5050	0.728743	0.364371	−	0.931254i	$-0.381284\pi$
$577$	17.5050	0.728743	0.364371	+	0.931254i	$0.381284\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.72479	0.154531
$582$	0	0
$583$	26.6291	1.10286
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.36819	−0.345392	−0.172696	−	0.984975i	$-0.555248\pi$
$587$	−8.36819	−0.345392	−0.172696	+	0.984975i	$0.555248\pi$
$588$	0	0
$589$	−22.8220	−0.940362
$590$	0	0
$591$	0	0
$592$	0	0
$593$	38.5284	1.58217	0.791087	−	0.611704i	$-0.209516\pi$
$593$	38.5284	1.58217	0.791087	+	0.611704i	$0.209516\pi$
$594$	0	0
$595$	11.3054	0.463475
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−13.1584	−0.537638	−0.268819	−	0.963191i	$-0.586633\pi$
$599$	−13.1584	−0.537638	−0.268819	+	0.963191i	$0.586633\pi$
$600$	0	0
$601$	−10.0227	−0.408835	−0.204417	−	0.978884i	$-0.565530\pi$
$601$	−10.0227	−0.408835	−0.204417	+	0.978884i	$0.565530\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−78.6581	−3.19791
$606$	0	0
$607$	−36.2760	−1.47240	−0.736199	−	0.676766i	$-0.763381\pi$
$607$	−36.2760	−1.47240	−0.736199	+	0.676766i	$0.763381\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21.0302	−0.850789
$612$	0	0
$613$	−24.2523	−0.979541	−0.489770	−	0.871851i	$-0.662919\pi$
$613$	−24.2523	−0.979541	−0.489770	+	0.871851i	$0.662919\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.4185	0.540209	0.270105	−	0.962831i	$-0.412942\pi$
$617$	13.4185	0.540209	0.270105	+	0.962831i	$0.412942\pi$
$618$	0	0
$619$	36.6036	1.47122	0.735612	−	0.677404i	$-0.236895\pi$
$619$	36.6036	1.47122	0.735612	+	0.677404i	$0.236895\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.99862	−0.280394
$624$	0	0
$625$	105.469	4.21877
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.3054	0.450775
$630$	0	0
$631$	26.9365	1.07232	0.536162	−	0.844115i	$-0.319874\pi$
$631$	26.9365	1.07232	0.536162	+	0.844115i	$0.319874\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.3749	0.491084
$636$	0	0
$637$	−2.54348	−0.100776
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−20.2780	−0.800933	−0.400467	−	0.916311i	$-0.631152\pi$
$641$	−20.2780	−0.800933	−0.400467	+	0.916311i	$0.631152\pi$
$642$	0	0
$643$	12.8846	0.508118	0.254059	−	0.967189i	$-0.418234\pi$
$643$	12.8846	0.508118	0.254059	+	0.967189i	$0.418234\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	41.4633	1.63009	0.815045	−	0.579397i	$-0.196712\pi$
$647$	41.4633	1.63009	0.815045	+	0.579397i	$0.196712\pi$
$648$	0	0
$649$	−29.9989	−1.17756
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18.7607	−0.734164	−0.367082	−	0.930189i	$-0.619643\pi$
$653$	−18.7607	−0.734164	−0.367082	+	0.930189i	$0.619643\pi$
$654$	0	0
$655$	74.9952	2.93031
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−8.67865	−0.338072	−0.169036	−	0.985610i	$-0.554065\pi$
$659$	−8.67865	−0.338072	−0.169036	+	0.985610i	$0.554065\pi$
$660$	0	0
$661$	−14.3905	−0.559725	−0.279862	−	0.960040i	$-0.590289\pi$
$661$	−14.3905	−0.559725	−0.279862	+	0.960040i	$0.590289\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−29.4610	−1.14245
$666$	0	0
$667$	−2.99587	−0.116001
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−44.5897	−1.72137
$672$	0	0
$673$	−28.2118	−1.08748	−0.543742	−	0.839253i	$-0.682993\pi$
$673$	−28.2118	−1.08748	−0.543742	+	0.839253i	$0.682993\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.733104	0.0281755	0.0140877	−	0.999901i	$-0.495516\pi$
$677$	0.733104	0.0281755	0.0140877	+	0.999901i	$0.495516\pi$
$678$	0	0
$679$	−2.96152	−0.113653
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−16.4171	−0.628185	−0.314092	−	0.949392i	$-0.601700\pi$
$683$	−16.4171	−0.628185	−0.314092	+	0.949392i	$0.601700\pi$
$684$	0	0
$685$	−62.2658	−2.37905
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12.5871	−0.479530
$690$	0	0
$691$	−12.2001	−0.464114	−0.232057	−	0.972702i	$-0.574546\pi$
$691$	−12.2001	−0.464114	−0.232057	+	0.972702i	$0.574546\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.0234	0.531939
$696$	0	0
$697$	16.4994	0.624961
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−0.337121	−0.0127329	−0.00636644	−	0.999980i	$-0.502027\pi$
$701$	−0.337121	−0.0127329	−0.00636644	+	0.999980i	$0.502027\pi$
$702$	0	0
$703$	−29.4610	−1.11114
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.1299	−0.644235
$708$	0	0
$709$	20.0631	0.753485	0.376743	−	0.926318i	$-0.377044\pi$
$709$	20.0631	0.753485	0.376743	+	0.926318i	$0.377044\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.71622	0.101723
$714$	0	0
$715$	59.9590	2.24234
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10.7220	−0.399863	−0.199931	−	0.979810i	$-0.564072\pi$
$719$	−10.7220	−0.399863	−0.199931	+	0.979810i	$0.564072\pi$
$720$	0	0
$721$	12.0357	0.448234
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−53.1248	−1.97301
$726$	0	0
$727$	−10.4551	−0.387760	−0.193880	−	0.981025i	$-0.562107\pi$
$727$	−10.4551	−0.387760	−0.193880	+	0.981025i	$0.562107\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.96968	−0.0728512
$732$	0	0
$733$	42.3302	1.56350	0.781751	−	0.623591i	$-0.214327\pi$
$733$	42.3302	1.56350	0.781751	+	0.623591i	$0.214327\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.1855	−0.375188
$738$	0	0
$739$	−12.6281	−0.464532	−0.232266	−	0.972652i	$-0.574614\pi$
$739$	−12.6281	−0.464532	−0.232266	+	0.972652i	$0.574614\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	38.7264	1.42074	0.710368	−	0.703831i	$-0.248529\pi$
$743$	38.7264	1.42074	0.710368	+	0.703831i	$0.248529\pi$
$744$	0	0
$745$	74.9069	2.74438
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6.92580	−0.253063
$750$	0	0
$751$	21.2810	0.776556	0.388278	−	0.921542i	$-0.373070\pi$
$751$	21.2810	0.776556	0.388278	+	0.921542i	$0.373070\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−0.463249	−0.0168594
$756$	0	0
$757$	22.8632	0.830977	0.415488	−	0.909598i	$-0.363611\pi$
$757$	22.8632	0.830977	0.415488	+	0.909598i	$0.363611\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12.0970	−0.438514	−0.219257	−	0.975667i	$-0.570363\pi$
$761$	−12.0970	−0.438514	−0.219257	+	0.975667i	$0.570363\pi$
$762$	0	0
$763$	4.41942	0.159994
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.1799	0.512008
$768$	0	0
$769$	−40.1841	−1.44908	−0.724538	−	0.689235i	$-0.757947\pi$
$769$	−40.1841	−1.44908	−0.724538	+	0.689235i	$0.757947\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−34.1442	−1.22808	−0.614041	−	0.789274i	$-0.710457\pi$
$773$	−34.1442	−1.22808	−0.614041	+	0.789274i	$0.710457\pi$
$774$	0	0
$775$	48.1658	1.73017
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−42.9963	−1.54050
$780$	0	0
$781$	−7.23117	−0.258752
$782$	0	0
$783$	0	0
$784$	0	0
$785$	92.0203	3.28434
$786$	0	0
$787$	−20.0555	−0.714900	−0.357450	−	0.933932i	$-0.616354\pi$
$787$	−20.0555	−0.714900	−0.357450	+	0.933932i	$0.616354\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.8361	0.456399
$792$	0	0
$793$	21.0767	0.748457
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.1405	0.465461	0.232730	−	0.972541i	$-0.425234\pi$
$797$	13.1405	0.465461	0.232730	+	0.972541i	$0.425234\pi$
$798$	0	0
$799$	21.3369	0.754846
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−46.5484	−1.64266
$804$	0	0
$805$	3.50638	0.123584
$806$	0	0
$807$	0	0
$808$	0	0
$809$	54.7893	1.92629	0.963145	−	0.268984i	$-0.0866878\pi$
$809$	54.7893	1.92629	0.963145	+	0.268984i	$0.0866878\pi$
$810$	0	0
$811$	−44.1157	−1.54911	−0.774557	−	0.632505i	$-0.782027\pi$
$811$	−44.1157	−1.54911	−0.774557	+	0.632505i	$0.782027\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11.3429	−0.397326
$816$	0	0
$817$	5.13284	0.179575
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−54.7645	−1.91129	−0.955647	−	0.294515i	$-0.904842\pi$
$821$	−54.7645	−1.91129	−0.955647	+	0.294515i	$0.904842\pi$
$822$	0	0
$823$	50.5630	1.76252	0.881258	−	0.472636i	$-0.156697\pi$
$823$	50.5630	1.76252	0.881258	+	0.472636i	$0.156697\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.38142	0.256677	0.128339	−	0.991730i	$-0.459036\pi$
$827$	7.38142	0.256677	0.128339	+	0.991730i	$0.459036\pi$
$828$	0	0
$829$	−33.0566	−1.14810	−0.574052	−	0.818819i	$-0.694629\pi$
$829$	−33.0566	−1.14810	−0.574052	+	0.818819i	$0.694629\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.58058	0.0894117
$834$	0	0
$835$	11.4679	0.396863
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30.4327	1.05065	0.525326	−	0.850901i	$-0.323943\pi$
$839$	30.4327	1.05065	0.525326	+	0.850901i	$0.323943\pi$
$840$	0	0
$841$	−14.9891	−0.516867
$842$	0	0
$843$	0	0
$844$	0	0
$845$	28.6107	0.984240
$846$	0	0
$847$	−17.9546	−0.616927
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.50638	0.120197
$852$	0	0
$853$	13.4549	0.460688	0.230344	−	0.973109i	$-0.426015\pi$
$853$	13.4549	0.460688	0.230344	+	0.973109i	$0.426015\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.2838	0.556245	0.278123	−	0.960546i	$-0.410288\pi$
$857$	16.2838	0.556245	0.278123	+	0.960546i	$0.410288\pi$
$858$	0	0
$859$	−33.9706	−1.15906	−0.579531	−	0.814950i	$-0.696764\pi$
$859$	−33.9706	−1.15906	−0.579531	+	0.814950i	$0.696764\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−20.8721	−0.710494	−0.355247	−	0.934772i	$-0.615603\pi$
$863$	−20.8721	−0.710494	−0.355247	+	0.934772i	$0.615603\pi$
$864$	0	0
$865$	63.3272	2.15319
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−71.4645	−2.42427
$870$	0	0
$871$	4.81450	0.163133
$872$	0	0
$873$	0	0
$874$	0	0
$875$	40.2727	1.36147
$876$	0	0
$877$	−0.486689	−0.0164343	−0.00821716	−	0.999966i	$-0.502616\pi$
$877$	−0.486689	−0.0164343	−0.00821716	+	0.999966i	$0.502616\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−45.6511	−1.53803	−0.769013	−	0.639234i	$-0.779252\pi$
$881$	−45.6511	−1.53803	−0.769013	+	0.639234i	$0.779252\pi$
$882$	0	0
$883$	27.2622	0.917448	0.458724	−	0.888579i	$-0.348307\pi$
$883$	27.2622	0.917448	0.458724	+	0.888579i	$0.348307\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32.2244	−1.08199	−0.540996	−	0.841025i	$-0.681952\pi$
$887$	−32.2244	−1.08199	−0.540996	+	0.841025i	$0.681952\pi$
$888$	0	0
$889$	2.82471	0.0947378
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−55.6024	−1.86066
$894$	0	0
$895$	−103.907	−3.47322
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.7030	−0.423669
$900$	0	0
$901$	12.7707	0.425453
$902$	0	0
$903$	0	0
$904$	0	0
$905$	28.9418	0.962059
$906$	0	0
$907$	−15.0631	−0.500162	−0.250081	−	0.968225i	$-0.580457\pi$
$907$	−15.0631	−0.500162	−0.250081	+	0.968225i	$0.580457\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−8.25689	−0.273563	−0.136782	−	0.990601i	$-0.543676\pi$
$911$	−8.25689	−0.273563	−0.136782	+	0.990601i	$0.543676\pi$
$912$	0	0
$913$	−20.0429	−0.663324
$914$	0	0
$915$	0	0
$916$	0	0
$917$	17.1185	0.565303
$918$	0	0
$919$	−10.2256	−0.337312	−0.168656	−	0.985675i	$-0.553943\pi$
$919$	−10.2256	−0.337312	−0.168656	+	0.985675i	$0.553943\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.41805	0.112506
$924$	0	0
$925$	62.1775	2.04438
$926$	0	0
$927$	0	0
$928$	0	0
$929$	21.2840	0.698307	0.349153	−	0.937066i	$-0.386469\pi$
$929$	21.2840	0.698307	0.349153	+	0.937066i	$0.386469\pi$
$930$	0	0
$931$	−6.72479	−0.220396
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−60.8336	−1.98947
$936$	0	0
$937$	−33.7879	−1.10380	−0.551901	−	0.833910i	$-0.686097\pi$
$937$	−33.7879	−1.10380	−0.551901	+	0.833910i	$0.686097\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	24.5044	0.798819	0.399410	−	0.916773i	$-0.369215\pi$
$941$	24.5044	0.798819	0.399410	+	0.916773i	$0.369215\pi$
$942$	0	0
$943$	5.11733	0.166643
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.73310	0.251292	0.125646	−	0.992075i	$-0.459900\pi$
$947$	7.73310	0.251292	0.125646	+	0.992075i	$0.459900\pi$
$948$	0	0
$949$	22.0026	0.714235
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−6.66525	−0.215909	−0.107954	−	0.994156i	$-0.534430\pi$
$953$	−6.66525	−0.215909	−0.107954	+	0.994156i	$0.534430\pi$
$954$	0	0
$955$	−18.2541	−0.590690
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−14.2129	−0.458957
$960$	0	0
$961$	−19.4828	−0.628477
$962$	0	0
$963$	0	0
$964$	0	0
$965$	70.9210	2.28303
$966$	0	0
$967$	50.6717	1.62949	0.814746	−	0.579818i	$-0.196876\pi$
$967$	50.6717	1.62949	0.814746	+	0.579818i	$0.196876\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.04780	0.0336255	0.0168128	−	0.999859i	$-0.494648\pi$
$971$	1.04780	0.0336255	0.0168128	+	0.999859i	$0.494648\pi$
$972$	0	0
$973$	3.20101	0.102620
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−40.9723	−1.31082	−0.655409	−	0.755274i	$-0.727504\pi$
$977$	−40.9723	−1.31082	−0.655409	+	0.755274i	$0.727504\pi$
$978$	0	0
$979$	37.6592	1.20359
$980$	0	0
$981$	0	0
$982$	0	0
$983$	53.5081	1.70664	0.853321	−	0.521386i	$-0.174585\pi$
$983$	53.5081	1.70664	0.853321	+	0.521386i	$0.174585\pi$
$984$	0	0
$985$	18.9743	0.604571
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.610899	−0.0194255
$990$	0	0
$991$	−32.9532	−1.04679	−0.523397	−	0.852089i	$-0.675335\pi$
$991$	−32.9532	−1.04679	−0.523397	+	0.852089i	$0.675335\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−60.3907	−1.91452
$996$	0	0
$997$	−5.54930	−0.175748	−0.0878741	−	0.996132i	$-0.528007\pi$
$997$	−5.54930	−0.175748	−0.0878741	+	0.996132i	$0.528007\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4536.2.a.y.1.1		4
3.2	odd	2		4536.2.a.z.1.4		4
4.3	odd	2		9072.2.a.cg.1.1		4
9.2	odd	6		504.2.r.e.337.2	yes	8
9.4	even	3		1512.2.r.e.505.4		8
9.5	odd	6		504.2.r.e.169.2	✓	8
9.7	even	3		1512.2.r.e.1009.4		8
12.11	even	2		9072.2.a.cj.1.4		4
36.7	odd	6		3024.2.r.m.1009.4		8
36.11	even	6		1008.2.r.l.337.3		8
36.23	even	6		1008.2.r.l.673.3		8
36.31	odd	6		3024.2.r.m.2017.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.r.e.169.2	✓	8	9.5	odd	6
504.2.r.e.337.2	yes	8	9.2	odd	6
1008.2.r.l.337.3		8	36.11	even	6
1008.2.r.l.673.3		8	36.23	even	6
1512.2.r.e.505.4		8	9.4	even	3
1512.2.r.e.1009.4		8	9.7	even	3
3024.2.r.m.1009.4		8	36.7	odd	6
3024.2.r.m.2017.4		8	36.31	odd	6
4536.2.a.y.1.1		4	1.1	even	1	trivial
4536.2.a.z.1.4		4	3.2	odd	2
9072.2.a.cg.1.1		4	4.3	odd	2
9072.2.a.cj.1.4		4	12.11	even	2

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:	\( N \)	\(=\)	\( 4536 = 2^{3} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4536.a (trivial)

\(f(q)\)	\(=\)	\(q-4.38095 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-4.38095 q^{5} -1.00000 q^{7} +5.38095 q^{11} -2.54348 q^{13} +2.58058 q^{17} -6.72479 q^{19} +0.800370 q^{23} +14.1927 q^{25} -3.74311 q^{29} +3.39370 q^{31} +4.38095 q^{35} +4.38095 q^{37} +6.39370 q^{41} -0.763270 q^{43} +8.26827 q^{47} +1.00000 q^{49} +4.94877 q^{53} -23.5736 q^{55} -5.57502 q^{59} -8.28659 q^{61} +11.1428 q^{65} -1.89288 q^{67} -1.34385 q^{71} -8.65060 q^{73} -5.38095 q^{77} -13.2810 q^{79} -3.72479 q^{83} -11.3054 q^{85} +6.99862 q^{89} +2.54348 q^{91} +29.4610 q^{95} +2.96152 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q - 3 * q^5 - 4 * q^7	\( 4 q - 3 q^{5} - 4 q^{7} + 7 q^{11} - 3 q^{13} - 3 q^{17} - 4 q^{19} + 2 q^{23} + 5 q^{25} - 9 q^{29} - 3 q^{31} + 3 q^{35} + 3 q^{37} + 9 q^{41} - 8 q^{43} + 3 q^{47} + 4 q^{49} - 6 q^{53} - 28 q^{55} + 10 q^{59} - 20 q^{61} + q^{65} - 11 q^{67} + 3 q^{71} - 24 q^{73} - 7 q^{77} - 21 q^{79} + 8 q^{83} - 9 q^{85} - 6 q^{89} + 3 q^{91} + 36 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q - 3 * q^5 - 4 * q^7 + 7 * q^11 - 3 * q^13 - 3 * q^17 - 4 * q^19 + 2 * q^23 + 5 * q^25 - 9 * q^29 - 3 * q^31 + 3 * q^35 + 3 * q^37 + 9 * q^41 - 8 * q^43 + 3 * q^47 + 4 * q^49 - 6 * q^53 - 28 * q^55 + 10 * q^59 - 20 * q^61 + q^65 - 11 * q^67 + 3 * q^71 - 24 * q^73 - 7 * q^77 - 21 * q^79 + 8 * q^83 - 9 * q^85 - 6 * q^89 + 3 * q^91 + 36 * q^95 - 16 * q^97

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