LMFDB - Embedded newform 585.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,2,Mod(1,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("585.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 585 = 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	585.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:	$4.67124851824$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	585.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{5} -0.438447 q^{7} -6.56155 q^{8} +O(q^{10})$	\(q-2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{5} -0.438447 q^{7} -6.56155 q^{8} +2.56155 q^{10} +1.56155 q^{11} -1.00000 q^{13} +1.12311 q^{14} +7.68466 q^{16} -1.56155 q^{17} -5.12311 q^{19} -4.56155 q^{20} -4.00000 q^{22} -2.43845 q^{23} +1.00000 q^{25} +2.56155 q^{26} -2.00000 q^{28} +7.12311 q^{29} +6.00000 q^{31} -6.56155 q^{32} +4.00000 q^{34} +0.438447 q^{35} -10.6847 q^{37} +13.1231 q^{38} +6.56155 q^{40} -3.56155 q^{41} +3.12311 q^{43} +7.12311 q^{44} +6.24621 q^{46} -11.1231 q^{47} -6.80776 q^{49} -2.56155 q^{50} -4.56155 q^{52} +4.68466 q^{53} -1.56155 q^{55} +2.87689 q^{56} -18.2462 q^{58} -12.0000 q^{59} -6.68466 q^{61} -15.3693 q^{62} +1.43845 q^{64} +1.00000 q^{65} -11.3693 q^{67} -7.12311 q^{68} -1.12311 q^{70} -10.4384 q^{71} -6.00000 q^{73} +27.3693 q^{74} -23.3693 q^{76} -0.684658 q^{77} +4.68466 q^{79} -7.68466 q^{80} +9.12311 q^{82} +16.4924 q^{83} +1.56155 q^{85} -8.00000 q^{86} -10.2462 q^{88} -10.6847 q^{89} +0.438447 q^{91} -11.1231 q^{92} +28.4924 q^{94} +5.12311 q^{95} +16.9309 q^{97} +17.4384 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 5 q^{4} - 2 q^{5} - 5 q^{7} - 9 q^{8}+O(q^{10}) $ 2 * q - q^2 + 5 * q^4 - 2 * q^5 - 5 * q^7 - 9 * q^8	$ 2 q - q^{2} + 5 q^{4} - 2 q^{5} - 5 q^{7} - 9 q^{8} + q^{10} - q^{11} - 2 q^{13} - 6 q^{14} + 3 q^{16} + q^{17} - 2 q^{19} - 5 q^{20} - 8 q^{22} - 9 q^{23} + 2 q^{25} + q^{26} - 4 q^{28} + 6 q^{29} + 12 q^{31} - 9 q^{32} + 8 q^{34} + 5 q^{35} - 9 q^{37} + 18 q^{38} + 9 q^{40} - 3 q^{41} - 2 q^{43} + 6 q^{44} - 4 q^{46} - 14 q^{47} + 7 q^{49} - q^{50} - 5 q^{52} - 3 q^{53} + q^{55} + 14 q^{56} - 20 q^{58} - 24 q^{59} - q^{61} - 6 q^{62} + 7 q^{64} + 2 q^{65} + 2 q^{67} - 6 q^{68} + 6 q^{70} - 25 q^{71} - 12 q^{73} + 30 q^{74} - 22 q^{76} + 11 q^{77} - 3 q^{79} - 3 q^{80} + 10 q^{82} - q^{85} - 16 q^{86} - 4 q^{88} - 9 q^{89} + 5 q^{91} - 14 q^{92} + 24 q^{94} + 2 q^{95} + 5 q^{97} + 39 q^{98}+O(q^{100}) $ 2 * q - q^2 + 5 * q^4 - 2 * q^5 - 5 * q^7 - 9 * q^8 + q^10 - q^11 - 2 * q^13 - 6 * q^14 + 3 * q^16 + q^17 - 2 * q^19 - 5 * q^20 - 8 * q^22 - 9 * q^23 + 2 * q^25 + q^26 - 4 * q^28 + 6 * q^29 + 12 * q^31 - 9 * q^32 + 8 * q^34 + 5 * q^35 - 9 * q^37 + 18 * q^38 + 9 * q^40 - 3 * q^41 - 2 * q^43 + 6 * q^44 - 4 * q^46 - 14 * q^47 + 7 * q^49 - q^50 - 5 * q^52 - 3 * q^53 + q^55 + 14 * q^56 - 20 * q^58 - 24 * q^59 - q^61 - 6 * q^62 + 7 * q^64 + 2 * q^65 + 2 * q^67 - 6 * q^68 + 6 * q^70 - 25 * q^71 - 12 * q^73 + 30 * q^74 - 22 * q^76 + 11 * q^77 - 3 * q^79 - 3 * q^80 + 10 * q^82 - q^85 - 16 * q^86 - 4 * q^88 - 9 * q^89 + 5 * q^91 - 14 * q^92 + 24 * q^94 + 2 * q^95 + 5 * q^97 + 39 * q^98

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 $

$2$	−2.56155	−1.81129	−0.905646	−	0.424035i	$-0.860613\pi$
$2$	−2.56155	−1.81129	−0.905646	+	0.424035i	$0.860613\pi$
$3$	0	0
$3$	0	0
$4$	4.56155	2.28078
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.438447	−0.165717	−0.0828587	−	0.996561i	$-0.526405\pi$
$7$	−0.438447	−0.165717	−0.0828587	+	0.996561i	$0.526405\pi$
$8$	−6.56155	−2.31986
$9$	0	0
$10$	2.56155	0.810034
$11$	1.56155	0.470826	0.235413	−	0.971895i	$-0.424356\pi$
$11$	1.56155	0.470826	0.235413	+	0.971895i	$0.424356\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.12311	0.300163
$15$	0	0
$16$	7.68466	1.92116
$17$	−1.56155	−0.378732	−0.189366	−	0.981907i	$-0.560643\pi$
$17$	−1.56155	−0.378732	−0.189366	+	0.981907i	$0.560643\pi$
$18$	0	0
$19$	−5.12311	−1.17532	−0.587661	−	0.809108i	$-0.699951\pi$
$19$	−5.12311	−1.17532	−0.587661	+	0.809108i	$0.699951\pi$
$20$	−4.56155	−1.01999
$21$	0	0
$22$	−4.00000	−0.852803
$23$	−2.43845	−0.508451	−0.254226	−	0.967145i	$-0.581821\pi$
$23$	−2.43845	−0.508451	−0.254226	+	0.967145i	$0.581821\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.56155	0.502362
$27$	0	0
$28$	−2.00000	−0.377964
$29$	7.12311	1.32273	0.661364	−	0.750065i	$-0.269978\pi$
$29$	7.12311	1.32273	0.661364	+	0.750065i	$0.269978\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−6.56155	−1.15993
$33$	0	0
$34$	4.00000	0.685994
$35$	0.438447	0.0741111
$36$	0	0
$37$	−10.6847	−1.75655	−0.878274	−	0.478159i	$-0.841304\pi$
$37$	−10.6847	−1.75655	−0.878274	+	0.478159i	$0.841304\pi$
$38$	13.1231	2.12885
$39$	0	0
$40$	6.56155	1.03747
$41$	−3.56155	−0.556221	−0.278111	−	0.960549i	$-0.589708\pi$
$41$	−3.56155	−0.556221	−0.278111	+	0.960549i	$0.589708\pi$
$42$	0	0
$43$	3.12311	0.476269	0.238135	−	0.971232i	$-0.423464\pi$
$43$	3.12311	0.476269	0.238135	+	0.971232i	$0.423464\pi$
$44$	7.12311	1.07385
$45$	0	0
$46$	6.24621	0.920954
$47$	−11.1231	−1.62247	−0.811236	−	0.584719i	$-0.801205\pi$
$47$	−11.1231	−1.62247	−0.811236	+	0.584719i	$0.801205\pi$
$48$	0	0
$49$	−6.80776	−0.972538
$50$	−2.56155	−0.362258
$51$	0	0
$52$	−4.56155	−0.632574
$53$	4.68466	0.643487	0.321744	−	0.946827i	$-0.395731\pi$
$53$	4.68466	0.643487	0.321744	+	0.946827i	$0.395731\pi$
$54$	0	0
$55$	−1.56155	−0.210560
$56$	2.87689	0.384441
$57$	0	0
$58$	−18.2462	−2.39584
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−6.68466	−0.855883	−0.427941	−	0.903806i	$-0.640761\pi$
$61$	−6.68466	−0.855883	−0.427941	+	0.903806i	$0.640761\pi$
$62$	−15.3693	−1.95191
$63$	0	0
$64$	1.43845	0.179806
$65$	1.00000	0.124035
$66$	0	0
$67$	−11.3693	−1.38898	−0.694492	−	0.719501i	$-0.744371\pi$
$67$	−11.3693	−1.38898	−0.694492	+	0.719501i	$0.744371\pi$
$68$	−7.12311	−0.863803
$69$	0	0
$70$	−1.12311	−0.134237
$71$	−10.4384	−1.23882	−0.619408	−	0.785069i	$-0.712627\pi$
$71$	−10.4384	−1.23882	−0.619408	+	0.785069i	$0.712627\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	27.3693	3.18162
$75$	0	0
$76$	−23.3693	−2.68064
$77$	−0.684658	−0.0780241
$78$	0	0
$79$	4.68466	0.527065	0.263533	−	0.964650i	$-0.415112\pi$
$79$	4.68466	0.527065	0.263533	+	0.964650i	$0.415112\pi$
$80$	−7.68466	−0.859171
$81$	0	0
$82$	9.12311	1.00748
$83$	16.4924	1.81028	0.905139	−	0.425115i	$-0.139766\pi$
$83$	16.4924	1.81028	0.905139	+	0.425115i	$0.139766\pi$
$84$	0	0
$85$	1.56155	0.169374
$86$	−8.00000	−0.862662
$87$	0	0
$88$	−10.2462	−1.09225
$89$	−10.6847	−1.13257	−0.566286	−	0.824209i	$-0.691620\pi$
$89$	−10.6847	−1.13257	−0.566286	+	0.824209i	$0.691620\pi$
$90$	0	0
$91$	0.438447	0.0459618
$92$	−11.1231	−1.15966
$93$	0	0
$94$	28.4924	2.93877
$95$	5.12311	0.525620
$96$	0	0
$97$	16.9309	1.71907	0.859535	−	0.511077i	$-0.170753\pi$
$97$	16.9309	1.71907	0.859535	+	0.511077i	$0.170753\pi$
$98$	17.4384	1.76155
$99$	0	0
$100$	4.56155	0.456155
$101$	−10.2462	−1.01954	−0.509768	−	0.860312i	$-0.670269\pi$
$101$	−10.2462	−1.01954	−0.509768	+	0.860312i	$0.670269\pi$
$102$	0	0
$103$	−15.1231	−1.49012	−0.745062	−	0.666995i	$-0.767580\pi$
$103$	−15.1231	−1.49012	−0.745062	+	0.666995i	$0.767580\pi$
$104$	6.56155	0.643413
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−10.9309	−1.05673	−0.528364	−	0.849018i	$-0.677194\pi$
$107$	−10.9309	−1.05673	−0.528364	+	0.849018i	$0.677194\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	−3.36932	−0.318371
$113$	4.87689	0.458780	0.229390	−	0.973335i	$-0.426327\pi$
$113$	4.87689	0.458780	0.229390	+	0.973335i	$0.426327\pi$
$114$	0	0
$115$	2.43845	0.227386
$116$	32.4924	3.01685
$117$	0	0
$118$	30.7386	2.82972
$119$	0.684658	0.0627625
$120$	0	0
$121$	−8.56155	−0.778323
$122$	17.1231	1.55025
$123$	0	0
$124$	27.3693	2.45784
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.75379	0.155624	0.0778118	−	0.996968i	$-0.475207\pi$
$127$	1.75379	0.155624	0.0778118	+	0.996968i	$0.475207\pi$
$128$	9.43845	0.834249
$129$	0	0
$130$	−2.56155	−0.224663
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	2.24621	0.194771
$134$	29.1231	2.51585
$135$	0	0
$136$	10.2462	0.878605
$137$	−1.12311	−0.0959534	−0.0479767	−	0.998848i	$-0.515277\pi$
$137$	−1.12311	−0.0959534	−0.0479767	+	0.998848i	$0.515277\pi$
$138$	0	0
$139$	3.31534	0.281204	0.140602	−	0.990066i	$-0.455096\pi$
$139$	3.31534	0.281204	0.140602	+	0.990066i	$0.455096\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	26.7386	2.24386
$143$	−1.56155	−0.130584
$144$	0	0
$145$	−7.12311	−0.591542
$146$	15.3693	1.27197
$147$	0	0
$148$	−48.7386	−4.00629
$149$	17.8078	1.45887	0.729434	−	0.684051i	$-0.239783\pi$
$149$	17.8078	1.45887	0.729434	+	0.684051i	$0.239783\pi$
$150$	0	0
$151$	11.3693	0.925222	0.462611	−	0.886561i	$-0.346913\pi$
$151$	11.3693	0.925222	0.462611	+	0.886561i	$0.346913\pi$
$152$	33.6155	2.72658
$153$	0	0
$154$	1.75379	0.141324
$155$	−6.00000	−0.481932
$156$	0	0
$157$	3.36932	0.268901	0.134450	−	0.990920i	$-0.457073\pi$
$157$	3.36932	0.268901	0.134450	+	0.990920i	$0.457073\pi$
$158$	−12.0000	−0.954669
$159$	0	0
$160$	6.56155	0.518736
$161$	1.06913	0.0842593
$162$	0	0
$163$	16.0540	1.25744	0.628722	−	0.777630i	$-0.283578\pi$
$163$	16.0540	1.25744	0.628722	+	0.777630i	$0.283578\pi$
$164$	−16.2462	−1.26862
$165$	0	0
$166$	−42.2462	−3.27894
$167$	−4.87689	−0.377385	−0.188693	−	0.982036i	$-0.560425\pi$
$167$	−4.87689	−0.377385	−0.188693	+	0.982036i	$0.560425\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−4.00000	−0.306786
$171$	0	0
$172$	14.2462	1.08626
$173$	−12.8769	−0.979012	−0.489506	−	0.872000i	$-0.662823\pi$
$173$	−12.8769	−0.979012	−0.489506	+	0.872000i	$0.662823\pi$
$174$	0	0
$175$	−0.438447	−0.0331435
$176$	12.0000	0.904534
$177$	0	0
$178$	27.3693	2.05142
$179$	4.87689	0.364516	0.182258	−	0.983251i	$-0.441659\pi$
$179$	4.87689	0.364516	0.182258	+	0.983251i	$0.441659\pi$
$180$	0	0
$181$	13.3153	0.989722	0.494861	−	0.868972i	$-0.335219\pi$
$181$	13.3153	0.989722	0.494861	+	0.868972i	$0.335219\pi$
$182$	−1.12311	−0.0832501
$183$	0	0
$184$	16.0000	1.17954
$185$	10.6847	0.785552
$186$	0	0
$187$	−2.43845	−0.178317
$188$	−50.7386	−3.70050
$189$	0	0
$190$	−13.1231	−0.952050
$191$	−19.6155	−1.41933	−0.709665	−	0.704539i	$-0.751154\pi$
$191$	−19.6155	−1.41933	−0.709665	+	0.704539i	$0.751154\pi$
$192$	0	0
$193$	19.5616	1.40807	0.704036	−	0.710165i	$-0.251379\pi$
$193$	19.5616	1.40807	0.704036	+	0.710165i	$0.251379\pi$
$194$	−43.3693	−3.11374
$195$	0	0
$196$	−31.0540	−2.21814
$197$	3.36932	0.240054	0.120027	−	0.992771i	$-0.461702\pi$
$197$	3.36932	0.240054	0.120027	+	0.992771i	$0.461702\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	−6.56155	−0.463972
$201$	0	0
$202$	26.2462	1.84668
$203$	−3.12311	−0.219199
$204$	0	0
$205$	3.56155	0.248750
$206$	38.7386	2.69905
$207$	0	0
$208$	−7.68466	−0.532835
$209$	−8.00000	−0.553372
$210$	0	0
$211$	6.24621	0.430007	0.215003	−	0.976613i	$-0.431024\pi$
$211$	6.24621	0.430007	0.215003	+	0.976613i	$0.431024\pi$
$212$	21.3693	1.46765
$213$	0	0
$214$	28.0000	1.91404
$215$	−3.12311	−0.212994
$216$	0	0
$217$	−2.63068	−0.178582
$218$	5.12311	0.346980
$219$	0	0
$220$	−7.12311	−0.480240
$221$	1.56155	0.105041
$222$	0	0
$223$	−15.3693	−1.02921	−0.514603	−	0.857429i	$-0.672061\pi$
$223$	−15.3693	−1.02921	−0.514603	+	0.857429i	$0.672061\pi$
$224$	2.87689	0.192221
$225$	0	0
$226$	−12.4924	−0.830984
$227$	−5.75379	−0.381892	−0.190946	−	0.981601i	$-0.561156\pi$
$227$	−5.75379	−0.381892	−0.190946	+	0.981601i	$0.561156\pi$
$228$	0	0
$229$	17.1231	1.13153	0.565763	−	0.824568i	$-0.308582\pi$
$229$	17.1231	1.13153	0.565763	+	0.824568i	$0.308582\pi$
$230$	−6.24621	−0.411863
$231$	0	0
$232$	−46.7386	−3.06854
$233$	27.8078	1.82175	0.910874	−	0.412685i	$-0.135409\pi$
$233$	27.8078	1.82175	0.910874	+	0.412685i	$0.135409\pi$
$234$	0	0
$235$	11.1231	0.725591
$236$	−54.7386	−3.56318
$237$	0	0
$238$	−1.75379	−0.113681
$239$	−22.9309	−1.48327	−0.741637	−	0.670801i	$-0.765950\pi$
$239$	−22.9309	−1.48327	−0.741637	+	0.670801i	$0.765950\pi$
$240$	0	0
$241$	24.7386	1.59356	0.796778	−	0.604272i	$-0.206536\pi$
$241$	24.7386	1.59356	0.796778	+	0.604272i	$0.206536\pi$
$242$	21.9309	1.40977
$243$	0	0
$244$	−30.4924	−1.95208
$245$	6.80776	0.434932
$246$	0	0
$247$	5.12311	0.325975
$248$	−39.3693	−2.49995
$249$	0	0
$250$	2.56155	0.162007
$251$	−26.2462	−1.65665	−0.828323	−	0.560251i	$-0.810705\pi$
$251$	−26.2462	−1.65665	−0.828323	+	0.560251i	$0.810705\pi$
$252$	0	0
$253$	−3.80776	−0.239392
$254$	−4.49242	−0.281880
$255$	0	0
$256$	−27.0540	−1.69087
$257$	12.8769	0.803239	0.401619	−	0.915807i	$-0.368448\pi$
$257$	12.8769	0.803239	0.401619	+	0.915807i	$0.368448\pi$
$258$	0	0
$259$	4.68466	0.291091
$260$	4.56155	0.282895
$261$	0	0
$262$	0	0
$263$	2.24621	0.138507	0.0692537	−	0.997599i	$-0.477938\pi$
$263$	2.24621	0.138507	0.0692537	+	0.997599i	$0.477938\pi$
$264$	0	0
$265$	−4.68466	−0.287776
$266$	−5.75379	−0.352787
$267$	0	0
$268$	−51.8617	−3.16796
$269$	0.876894	0.0534652	0.0267326	−	0.999643i	$-0.491490\pi$
$269$	0.876894	0.0534652	0.0267326	+	0.999643i	$0.491490\pi$
$270$	0	0
$271$	−19.3693	−1.17660	−0.588301	−	0.808642i	$-0.700203\pi$
$271$	−19.3693	−1.17660	−0.588301	+	0.808642i	$0.700203\pi$
$272$	−12.0000	−0.727607
$273$	0	0
$274$	2.87689	0.173800
$275$	1.56155	0.0941652
$276$	0	0
$277$	−12.2462	−0.735804	−0.367902	−	0.929865i	$-0.619924\pi$
$277$	−12.2462	−0.735804	−0.367902	+	0.929865i	$0.619924\pi$
$278$	−8.49242	−0.509342
$279$	0	0
$280$	−2.87689	−0.171927
$281$	−4.24621	−0.253308	−0.126654	−	0.991947i	$-0.540424\pi$
$281$	−4.24621	−0.253308	−0.126654	+	0.991947i	$0.540424\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−47.6155	−2.82546
$285$	0	0
$286$	4.00000	0.236525
$287$	1.56155	0.0921755
$288$	0	0
$289$	−14.5616	−0.856562
$290$	18.2462	1.07145
$291$	0	0
$292$	−27.3693	−1.60167
$293$	20.2462	1.18280	0.591398	−	0.806380i	$-0.298576\pi$
$293$	20.2462	1.18280	0.591398	+	0.806380i	$0.298576\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	70.1080	4.07494
$297$	0	0
$298$	−45.6155	−2.64244
$299$	2.43845	0.141019
$300$	0	0
$301$	−1.36932	−0.0789261
$302$	−29.1231	−1.67585
$303$	0	0
$304$	−39.3693	−2.25799
$305$	6.68466	0.382762
$306$	0	0
$307$	7.56155	0.431561	0.215780	−	0.976442i	$-0.430770\pi$
$307$	7.56155	0.431561	0.215780	+	0.976442i	$0.430770\pi$
$308$	−3.12311	−0.177955
$309$	0	0
$310$	15.3693	0.872919
$311$	−2.63068	−0.149172	−0.0745862	−	0.997215i	$-0.523764\pi$
$311$	−2.63068	−0.149172	−0.0745862	+	0.997215i	$0.523764\pi$
$312$	0	0
$313$	29.1231	1.64614	0.823068	−	0.567943i	$-0.192261\pi$
$313$	29.1231	1.64614	0.823068	+	0.567943i	$0.192261\pi$
$314$	−8.63068	−0.487058
$315$	0	0
$316$	21.3693	1.20212
$317$	−1.50758	−0.0846740	−0.0423370	−	0.999103i	$-0.513480\pi$
$317$	−1.50758	−0.0846740	−0.0423370	+	0.999103i	$0.513480\pi$
$318$	0	0
$319$	11.1231	0.622774
$320$	−1.43845	−0.0804116
$321$	0	0
$322$	−2.73863	−0.152618
$323$	8.00000	0.445132
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−41.1231	−2.27760
$327$	0	0
$328$	23.3693	1.29035
$329$	4.87689	0.268872
$330$	0	0
$331$	29.1231	1.60075	0.800375	−	0.599499i	$-0.204634\pi$
$331$	29.1231	1.60075	0.800375	+	0.599499i	$0.204634\pi$
$332$	75.2311	4.12884
$333$	0	0
$334$	12.4924	0.683555
$335$	11.3693	0.621172
$336$	0	0
$337$	−30.4924	−1.66103	−0.830514	−	0.556998i	$-0.811953\pi$
$337$	−30.4924	−1.66103	−0.830514	+	0.556998i	$0.811953\pi$
$338$	−2.56155	−0.139330
$339$	0	0
$340$	7.12311	0.386305
$341$	9.36932	0.507377
$342$	0	0
$343$	6.05398	0.326884
$344$	−20.4924	−1.10488
$345$	0	0
$346$	32.9848	1.77328
$347$	26.0540	1.39865	0.699325	−	0.714804i	$-0.253484\pi$
$347$	26.0540	1.39865	0.699325	+	0.714804i	$0.253484\pi$
$348$	0	0
$349$	−23.3693	−1.25093	−0.625465	−	0.780252i	$-0.715091\pi$
$349$	−23.3693	−1.25093	−0.625465	+	0.780252i	$0.715091\pi$
$350$	1.12311	0.0600325
$351$	0	0
$352$	−10.2462	−0.546125
$353$	22.4924	1.19715	0.598575	−	0.801066i	$-0.295734\pi$
$353$	22.4924	1.19715	0.598575	+	0.801066i	$0.295734\pi$
$354$	0	0
$355$	10.4384	0.554015
$356$	−48.7386	−2.58314
$357$	0	0
$358$	−12.4924	−0.660245
$359$	14.2462	0.751886	0.375943	−	0.926643i	$-0.377319\pi$
$359$	14.2462	0.751886	0.375943	+	0.926643i	$0.377319\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	−34.1080	−1.79267
$363$	0	0
$364$	2.00000	0.104828
$365$	6.00000	0.314054
$366$	0	0
$367$	−1.75379	−0.0915470	−0.0457735	−	0.998952i	$-0.514575\pi$
$367$	−1.75379	−0.0915470	−0.0457735	+	0.998952i	$0.514575\pi$
$368$	−18.7386	−0.976819
$369$	0	0
$370$	−27.3693	−1.42286
$371$	−2.05398	−0.106637
$372$	0	0
$373$	12.2462	0.634085	0.317042	−	0.948411i	$-0.397310\pi$
$373$	12.2462	0.634085	0.317042	+	0.948411i	$0.397310\pi$
$374$	6.24621	0.322984
$375$	0	0
$376$	72.9848	3.76391
$377$	−7.12311	−0.366859
$378$	0	0
$379$	−30.4924	−1.56629	−0.783145	−	0.621839i	$-0.786386\pi$
$379$	−30.4924	−1.56629	−0.783145	+	0.621839i	$0.786386\pi$
$380$	23.3693	1.19882
$381$	0	0
$382$	50.2462	2.57082
$383$	−3.50758	−0.179229	−0.0896144	−	0.995977i	$-0.528563\pi$
$383$	−3.50758	−0.179229	−0.0896144	+	0.995977i	$0.528563\pi$
$384$	0	0
$385$	0.684658	0.0348934
$386$	−50.1080	−2.55043
$387$	0	0
$388$	77.2311	3.92081
$389$	−37.8617	−1.91967	−0.959833	−	0.280571i	$-0.909476\pi$
$389$	−37.8617	−1.91967	−0.959833	+	0.280571i	$0.909476\pi$
$390$	0	0
$391$	3.80776	0.192567
$392$	44.6695	2.25615
$393$	0	0
$394$	−8.63068	−0.434808
$395$	−4.68466	−0.235711
$396$	0	0
$397$	−4.43845	−0.222759	−0.111380	−	0.993778i	$-0.535527\pi$
$397$	−4.43845	−0.222759	−0.111380	+	0.993778i	$0.535527\pi$
$398$	20.4924	1.02719
$399$	0	0
$400$	7.68466	0.384233
$401$	−3.75379	−0.187455	−0.0937276	−	0.995598i	$-0.529878\pi$
$401$	−3.75379	−0.187455	−0.0937276	+	0.995598i	$0.529878\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	−46.7386	−2.32533
$405$	0	0
$406$	8.00000	0.397033
$407$	−16.6847	−0.827028
$408$	0	0
$409$	−6.87689	−0.340041	−0.170020	−	0.985441i	$-0.554383\pi$
$409$	−6.87689	−0.340041	−0.170020	+	0.985441i	$0.554383\pi$
$410$	−9.12311	−0.450558
$411$	0	0
$412$	−68.9848	−3.39864
$413$	5.26137	0.258895
$414$	0	0
$415$	−16.4924	−0.809581
$416$	6.56155	0.321707
$417$	0	0
$418$	20.4924	1.00232
$419$	−7.61553	−0.372043	−0.186021	−	0.982546i	$-0.559559\pi$
$419$	−7.61553	−0.372043	−0.186021	+	0.982546i	$0.559559\pi$
$420$	0	0
$421$	39.3693	1.91874	0.959372	−	0.282146i	$-0.0910462\pi$
$421$	39.3693	1.91874	0.959372	+	0.282146i	$0.0910462\pi$
$422$	−16.0000	−0.778868
$423$	0	0
$424$	−30.7386	−1.49280
$425$	−1.56155	−0.0757464
$426$	0	0
$427$	2.93087	0.141835
$428$	−49.8617	−2.41016
$429$	0	0
$430$	8.00000	0.385794
$431$	3.50758	0.168954	0.0844770	−	0.996425i	$-0.473078\pi$
$431$	3.50758	0.168954	0.0844770	+	0.996425i	$0.473078\pi$
$432$	0	0
$433$	−9.61553	−0.462093	−0.231046	−	0.972943i	$-0.574215\pi$
$433$	−9.61553	−0.462093	−0.231046	+	0.972943i	$0.574215\pi$
$434$	6.73863	0.323465
$435$	0	0
$436$	−9.12311	−0.436918
$437$	12.4924	0.597594
$438$	0	0
$439$	22.0540	1.05258	0.526289	−	0.850306i	$-0.323583\pi$
$439$	22.0540	1.05258	0.526289	+	0.850306i	$0.323583\pi$
$440$	10.2462	0.488469
$441$	0	0
$442$	−4.00000	−0.190261
$443$	7.80776	0.370958	0.185479	−	0.982648i	$-0.440616\pi$
$443$	7.80776	0.370958	0.185479	+	0.982648i	$0.440616\pi$
$444$	0	0
$445$	10.6847	0.506501
$446$	39.3693	1.86419
$447$	0	0
$448$	−0.630683	−0.0297970
$449$	35.1771	1.66011	0.830055	−	0.557682i	$-0.188309\pi$
$449$	35.1771	1.66011	0.830055	+	0.557682i	$0.188309\pi$
$450$	0	0
$451$	−5.56155	−0.261883
$452$	22.2462	1.04637
$453$	0	0
$454$	14.7386	0.691718
$455$	−0.438447	−0.0205547
$456$	0	0
$457$	−13.3153	−0.622865	−0.311433	−	0.950268i	$-0.600809\pi$
$457$	−13.3153	−0.622865	−0.311433	+	0.950268i	$0.600809\pi$
$458$	−43.8617	−2.04952
$459$	0	0
$460$	11.1231	0.518617
$461$	−8.05398	−0.375111	−0.187556	−	0.982254i	$-0.560056\pi$
$461$	−8.05398	−0.375111	−0.187556	+	0.982254i	$0.560056\pi$
$462$	0	0
$463$	−28.9309	−1.34453	−0.672266	−	0.740310i	$-0.734679\pi$
$463$	−28.9309	−1.34453	−0.672266	+	0.740310i	$0.734679\pi$
$464$	54.7386	2.54118
$465$	0	0
$466$	−71.2311	−3.29971
$467$	−6.93087	−0.320722	−0.160361	−	0.987058i	$-0.551266\pi$
$467$	−6.93087	−0.320722	−0.160361	+	0.987058i	$0.551266\pi$
$468$	0	0
$469$	4.98485	0.230179
$470$	−28.4924	−1.31426
$471$	0	0
$472$	78.7386	3.62424
$473$	4.87689	0.224240
$474$	0	0
$475$	−5.12311	−0.235064
$476$	3.12311	0.143147
$477$	0	0
$478$	58.7386	2.68664
$479$	−26.0540	−1.19044	−0.595218	−	0.803564i	$-0.702934\pi$
$479$	−26.0540	−1.19044	−0.595218	+	0.803564i	$0.702934\pi$
$480$	0	0
$481$	10.6847	0.487178
$482$	−63.3693	−2.88639
$483$	0	0
$484$	−39.0540	−1.77518
$485$	−16.9309	−0.768791
$486$	0	0
$487$	−32.0540	−1.45250	−0.726252	−	0.687428i	$-0.758740\pi$
$487$	−32.0540	−1.45250	−0.726252	+	0.687428i	$0.758740\pi$
$488$	43.8617	1.98553
$489$	0	0
$490$	−17.4384	−0.787789
$491$	27.6155	1.24627	0.623136	−	0.782114i	$-0.285858\pi$
$491$	27.6155	1.24627	0.623136	+	0.782114i	$0.285858\pi$
$492$	0	0
$493$	−11.1231	−0.500959
$494$	−13.1231	−0.590436
$495$	0	0
$496$	46.1080	2.07031
$497$	4.57671	0.205293
$498$	0	0
$499$	34.9848	1.56614	0.783068	−	0.621936i	$-0.213653\pi$
$499$	34.9848	1.56614	0.783068	+	0.621936i	$0.213653\pi$
$500$	−4.56155	−0.203999
$501$	0	0
$502$	67.2311	3.00067
$503$	−13.7538	−0.613251	−0.306626	−	0.951830i	$-0.599200\pi$
$503$	−13.7538	−0.613251	−0.306626	+	0.951830i	$0.599200\pi$
$504$	0	0
$505$	10.2462	0.455950
$506$	9.75379	0.433609
$507$	0	0
$508$	8.00000	0.354943
$509$	23.5616	1.04435	0.522174	−	0.852839i	$-0.325121\pi$
$509$	23.5616	1.04435	0.522174	+	0.852839i	$0.325121\pi$
$510$	0	0
$511$	2.63068	0.116375
$512$	50.4233	2.22842
$513$	0	0
$514$	−32.9848	−1.45490
$515$	15.1231	0.666404
$516$	0	0
$517$	−17.3693	−0.763902
$518$	−12.0000	−0.527250
$519$	0	0
$520$	−6.56155	−0.287743
$521$	25.8617	1.13302	0.566512	−	0.824054i	$-0.308293\pi$
$521$	25.8617	1.13302	0.566512	+	0.824054i	$0.308293\pi$
$522$	0	0
$523$	−30.7386	−1.34411	−0.672053	−	0.740503i	$-0.734587\pi$
$523$	−30.7386	−1.34411	−0.672053	+	0.740503i	$0.734587\pi$
$524$	0	0
$525$	0	0
$526$	−5.75379	−0.250877
$527$	−9.36932	−0.408134
$528$	0	0
$529$	−17.0540	−0.741477
$530$	12.0000	0.521247
$531$	0	0
$532$	10.2462	0.444230
$533$	3.56155	0.154268
$534$	0	0
$535$	10.9309	0.472583
$536$	74.6004	3.22225
$537$	0	0
$538$	−2.24621	−0.0968410
$539$	−10.6307	−0.457896
$540$	0	0
$541$	18.8769	0.811581	0.405791	−	0.913966i	$-0.366996\pi$
$541$	18.8769	0.811581	0.405791	+	0.913966i	$0.366996\pi$
$542$	49.6155	2.13117
$543$	0	0
$544$	10.2462	0.439303
$545$	2.00000	0.0856706
$546$	0	0
$547$	−5.36932	−0.229575	−0.114788	−	0.993390i	$-0.536619\pi$
$547$	−5.36932	−0.229575	−0.114788	+	0.993390i	$0.536619\pi$
$548$	−5.12311	−0.218848
$549$	0	0
$550$	−4.00000	−0.170561
$551$	−36.4924	−1.55463
$552$	0	0
$553$	−2.05398	−0.0873439
$554$	31.3693	1.33275
$555$	0	0
$556$	15.1231	0.641363
$557$	−6.49242	−0.275093	−0.137546	−	0.990495i	$-0.543922\pi$
$557$	−6.49242	−0.275093	−0.137546	+	0.990495i	$0.543922\pi$
$558$	0	0
$559$	−3.12311	−0.132093
$560$	3.36932	0.142380
$561$	0	0
$562$	10.8769	0.458814
$563$	19.3153	0.814045	0.407022	−	0.913418i	$-0.366567\pi$
$563$	19.3153	0.814045	0.407022	+	0.913418i	$0.366567\pi$
$564$	0	0
$565$	−4.87689	−0.205172
$566$	10.2462	0.430680
$567$	0	0
$568$	68.4924	2.87388
$569$	32.8769	1.37827	0.689136	−	0.724632i	$-0.257990\pi$
$569$	32.8769	1.37827	0.689136	+	0.724632i	$0.257990\pi$
$570$	0	0
$571$	−22.0540	−0.922930	−0.461465	−	0.887158i	$-0.652676\pi$
$571$	−22.0540	−0.922930	−0.461465	+	0.887158i	$0.652676\pi$
$572$	−7.12311	−0.297832
$573$	0	0
$574$	−4.00000	−0.166957
$575$	−2.43845	−0.101690
$576$	0	0
$577$	24.4384	1.01739	0.508693	−	0.860948i	$-0.330129\pi$
$577$	24.4384	1.01739	0.508693	+	0.860948i	$0.330129\pi$
$578$	37.3002	1.55148
$579$	0	0
$580$	−32.4924	−1.34917
$581$	−7.23106	−0.299995
$582$	0	0
$583$	7.31534	0.302970
$584$	39.3693	1.62911
$585$	0	0
$586$	−51.8617	−2.14239
$587$	32.4924	1.34111	0.670553	−	0.741862i	$-0.266057\pi$
$587$	32.4924	1.34111	0.670553	+	0.741862i	$0.266057\pi$
$588$	0	0
$589$	−30.7386	−1.26656
$590$	−30.7386	−1.26549
$591$	0	0
$592$	−82.1080	−3.37462
$593$	−24.2462	−0.995673	−0.497836	−	0.867271i	$-0.665872\pi$
$593$	−24.2462	−0.995673	−0.497836	+	0.867271i	$0.665872\pi$
$594$	0	0
$595$	−0.684658	−0.0280683
$596$	81.2311	3.32735
$597$	0	0
$598$	−6.24621	−0.255427
$599$	9.36932	0.382820	0.191410	−	0.981510i	$-0.438694\pi$
$599$	9.36932	0.382820	0.191410	+	0.981510i	$0.438694\pi$
$600$	0	0
$601$	−28.5464	−1.16443	−0.582216	−	0.813034i	$-0.697814\pi$
$601$	−28.5464	−1.16443	−0.582216	+	0.813034i	$0.697814\pi$
$602$	3.50758	0.142958
$603$	0	0
$604$	51.8617	2.11022
$605$	8.56155	0.348077
$606$	0	0
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	33.6155	1.36329
$609$	0	0
$610$	−17.1231	−0.693294
$611$	11.1231	0.449993
$612$	0	0
$613$	−32.5464	−1.31454	−0.657268	−	0.753657i	$-0.728288\pi$
$613$	−32.5464	−1.31454	−0.657268	+	0.753657i	$0.728288\pi$
$614$	−19.3693	−0.781682
$615$	0	0
$616$	4.49242	0.181005
$617$	−0.738634	−0.0297363	−0.0148681	−	0.999889i	$-0.504733\pi$
$617$	−0.738634	−0.0297363	−0.0148681	+	0.999889i	$0.504733\pi$
$618$	0	0
$619$	−8.63068	−0.346896	−0.173448	−	0.984843i	$-0.555491\pi$
$619$	−8.63068	−0.346896	−0.173448	+	0.984843i	$0.555491\pi$
$620$	−27.3693	−1.09918
$621$	0	0
$622$	6.73863	0.270195
$623$	4.68466	0.187687
$624$	0	0
$625$	1.00000	0.0400000
$626$	−74.6004	−2.98163
$627$	0	0
$628$	15.3693	0.613303
$629$	16.6847	0.665261
$630$	0	0
$631$	−32.7386	−1.30330	−0.651652	−	0.758518i	$-0.725924\pi$
$631$	−32.7386	−1.30330	−0.651652	+	0.758518i	$0.725924\pi$
$632$	−30.7386	−1.22272
$633$	0	0
$634$	3.86174	0.153369
$635$	−1.75379	−0.0695970
$636$	0	0
$637$	6.80776	0.269733
$638$	−28.4924	−1.12803
$639$	0	0
$640$	−9.43845	−0.373087
$641$	32.9848	1.30282	0.651412	−	0.758725i	$-0.274177\pi$
$641$	32.9848	1.30282	0.651412	+	0.758725i	$0.274177\pi$
$642$	0	0
$643$	−10.6847	−0.421362	−0.210681	−	0.977555i	$-0.567568\pi$
$643$	−10.6847	−0.421362	−0.210681	+	0.977555i	$0.567568\pi$
$644$	4.87689	0.192177
$645$	0	0
$646$	−20.4924	−0.806264
$647$	−31.8078	−1.25049	−0.625246	−	0.780428i	$-0.715001\pi$
$647$	−31.8078	−1.25049	−0.625246	+	0.780428i	$0.715001\pi$
$648$	0	0
$649$	−18.7386	−0.735556
$650$	2.56155	0.100472
$651$	0	0
$652$	73.2311	2.86795
$653$	33.3693	1.30584	0.652921	−	0.757426i	$-0.273543\pi$
$653$	33.3693	1.30584	0.652921	+	0.757426i	$0.273543\pi$
$654$	0	0
$655$	0	0
$656$	−27.3693	−1.06859
$657$	0	0
$658$	−12.4924	−0.487005
$659$	−0.876894	−0.0341590	−0.0170795	−	0.999854i	$-0.505437\pi$
$659$	−0.876894	−0.0341590	−0.0170795	+	0.999854i	$0.505437\pi$
$660$	0	0
$661$	6.49242	0.252526	0.126263	−	0.991997i	$-0.459702\pi$
$661$	6.49242	0.252526	0.126263	+	0.991997i	$0.459702\pi$
$662$	−74.6004	−2.89943
$663$	0	0
$664$	−108.216	−4.19959
$665$	−2.24621	−0.0871043
$666$	0	0
$667$	−17.3693	−0.672543
$668$	−22.2462	−0.860732
$669$	0	0
$670$	−29.1231	−1.12512
$671$	−10.4384	−0.402972
$672$	0	0
$673$	16.7386	0.645227	0.322613	−	0.946531i	$-0.395439\pi$
$673$	16.7386	0.645227	0.322613	+	0.946531i	$0.395439\pi$
$674$	78.1080	3.00861
$675$	0	0
$676$	4.56155	0.175444
$677$	14.4384	0.554915	0.277457	−	0.960738i	$-0.410508\pi$
$677$	14.4384	0.554915	0.277457	+	0.960738i	$0.410508\pi$
$678$	0	0
$679$	−7.42329	−0.284880
$680$	−10.2462	−0.392924
$681$	0	0
$682$	−24.0000	−0.919007
$683$	−32.4924	−1.24329	−0.621644	−	0.783300i	$-0.713535\pi$
$683$	−32.4924	−1.24329	−0.621644	+	0.783300i	$0.713535\pi$
$684$	0	0
$685$	1.12311	0.0429117
$686$	−15.5076	−0.592082
$687$	0	0
$688$	24.0000	0.914991
$689$	−4.68466	−0.178471
$690$	0	0
$691$	−21.6155	−0.822293	−0.411147	−	0.911569i	$-0.634872\pi$
$691$	−21.6155	−0.822293	−0.411147	+	0.911569i	$0.634872\pi$
$692$	−58.7386	−2.23291
$693$	0	0
$694$	−66.7386	−2.53336
$695$	−3.31534	−0.125758
$696$	0	0
$697$	5.56155	0.210659
$698$	59.8617	2.26580
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	48.9848	1.85013	0.925066	−	0.379806i	$-0.124009\pi$
$701$	48.9848	1.85013	0.925066	+	0.379806i	$0.124009\pi$
$702$	0	0
$703$	54.7386	2.06451
$704$	2.24621	0.0846573
$705$	0	0
$706$	−57.6155	−2.16839
$707$	4.49242	0.168955
$708$	0	0
$709$	9.12311	0.342625	0.171313	−	0.985217i	$-0.445199\pi$
$709$	9.12311	0.342625	0.171313	+	0.985217i	$0.445199\pi$
$710$	−26.7386	−1.00348
$711$	0	0
$712$	70.1080	2.62741
$713$	−14.6307	−0.547923
$714$	0	0
$715$	1.56155	0.0583988
$716$	22.2462	0.831380
$717$	0	0
$718$	−36.4924	−1.36189
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	6.63068	0.246940
$722$	−18.5616	−0.690789
$723$	0	0
$724$	60.7386	2.25733
$725$	7.12311	0.264546
$726$	0	0
$727$	12.8769	0.477578	0.238789	−	0.971072i	$-0.423250\pi$
$727$	12.8769	0.477578	0.238789	+	0.971072i	$0.423250\pi$
$728$	−2.87689	−0.106625
$729$	0	0
$730$	−15.3693	−0.568844
$731$	−4.87689	−0.180378
$732$	0	0
$733$	−16.4384	−0.607168	−0.303584	−	0.952805i	$-0.598183\pi$
$733$	−16.4384	−0.607168	−0.303584	+	0.952805i	$0.598183\pi$
$734$	4.49242	0.165818
$735$	0	0
$736$	16.0000	0.589768
$737$	−17.7538	−0.653969
$738$	0	0
$739$	48.7386	1.79288	0.896440	−	0.443166i	$-0.146145\pi$
$739$	48.7386	1.79288	0.896440	+	0.443166i	$0.146145\pi$
$740$	48.7386	1.79167
$741$	0	0
$742$	5.26137	0.193151
$743$	−18.7386	−0.687454	−0.343727	−	0.939070i	$-0.611689\pi$
$743$	−18.7386	−0.687454	−0.343727	+	0.939070i	$0.611689\pi$
$744$	0	0
$745$	−17.8078	−0.652426
$746$	−31.3693	−1.14851
$747$	0	0
$748$	−11.1231	−0.406701
$749$	4.79261	0.175118
$750$	0	0
$751$	14.0540	0.512837	0.256418	−	0.966566i	$-0.417458\pi$
$751$	14.0540	0.512837	0.256418	+	0.966566i	$0.417458\pi$
$752$	−85.4773	−3.11704
$753$	0	0
$754$	18.2462	0.664488
$755$	−11.3693	−0.413772
$756$	0	0
$757$	−14.4924	−0.526736	−0.263368	−	0.964695i	$-0.584833\pi$
$757$	−14.4924	−0.526736	−0.263368	+	0.964695i	$0.584833\pi$
$758$	78.1080	2.83701
$759$	0	0
$760$	−33.6155	−1.21936
$761$	−45.2311	−1.63962	−0.819812	−	0.572632i	$-0.805922\pi$
$761$	−45.2311	−1.63962	−0.819812	+	0.572632i	$0.805922\pi$
$762$	0	0
$763$	0.876894	0.0317457
$764$	−89.4773	−3.23717
$765$	0	0
$766$	8.98485	0.324636
$767$	12.0000	0.433295
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	−1.75379	−0.0632022
$771$	0	0
$772$	89.2311	3.21150
$773$	9.12311	0.328135	0.164068	−	0.986449i	$-0.447538\pi$
$773$	9.12311	0.328135	0.164068	+	0.986449i	$0.447538\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	−111.093	−3.98800
$777$	0	0
$778$	96.9848	3.47708
$779$	18.2462	0.653738
$780$	0	0
$781$	−16.3002	−0.583267
$782$	−9.75379	−0.348795
$783$	0	0
$784$	−52.3153	−1.86841
$785$	−3.36932	−0.120256
$786$	0	0
$787$	11.3693	0.405272	0.202636	−	0.979254i	$-0.435049\pi$
$787$	11.3693	0.405272	0.202636	+	0.979254i	$0.435049\pi$
$788$	15.3693	0.547509
$789$	0	0
$790$	12.0000	0.426941
$791$	−2.13826	−0.0760278
$792$	0	0
$793$	6.68466	0.237379
$794$	11.3693	0.403482
$795$	0	0
$796$	−36.4924	−1.29344
$797$	−4.68466	−0.165939	−0.0829696	−	0.996552i	$-0.526440\pi$
$797$	−4.68466	−0.165939	−0.0829696	+	0.996552i	$0.526440\pi$
$798$	0	0
$799$	17.3693	0.614482
$800$	−6.56155	−0.231986
$801$	0	0
$802$	9.61553	0.339536
$803$	−9.36932	−0.330636
$804$	0	0
$805$	−1.06913	−0.0376819
$806$	15.3693	0.541361
$807$	0	0
$808$	67.2311	2.36518
$809$	−7.50758	−0.263952	−0.131976	−	0.991253i	$-0.542132\pi$
$809$	−7.50758	−0.263952	−0.131976	+	0.991253i	$0.542132\pi$
$810$	0	0
$811$	4.24621	0.149105	0.0745523	−	0.997217i	$-0.476247\pi$
$811$	4.24621	0.149105	0.0745523	+	0.997217i	$0.476247\pi$
$812$	−14.2462	−0.499944
$813$	0	0
$814$	42.7386	1.49799
$815$	−16.0540	−0.562346
$816$	0	0
$817$	−16.0000	−0.559769
$818$	17.6155	0.615912
$819$	0	0
$820$	16.2462	0.567342
$821$	−48.5464	−1.69428	−0.847140	−	0.531369i	$-0.821678\pi$
$821$	−48.5464	−1.69428	−0.847140	+	0.531369i	$0.821678\pi$
$822$	0	0
$823$	−29.7538	−1.03715	−0.518576	−	0.855032i	$-0.673538\pi$
$823$	−29.7538	−1.03715	−0.518576	+	0.855032i	$0.673538\pi$
$824$	99.2311	3.45688
$825$	0	0
$826$	−13.4773	−0.468934
$827$	−7.12311	−0.247695	−0.123847	−	0.992301i	$-0.539523\pi$
$827$	−7.12311	−0.247695	−0.123847	+	0.992301i	$0.539523\pi$
$828$	0	0
$829$	0.738634	0.0256538	0.0128269	−	0.999918i	$-0.495917\pi$
$829$	0.738634	0.0256538	0.0128269	+	0.999918i	$0.495917\pi$
$830$	42.2462	1.46639
$831$	0	0
$832$	−1.43845	−0.0498692
$833$	10.6307	0.368331
$834$	0	0
$835$	4.87689	0.168772
$836$	−36.4924	−1.26212
$837$	0	0
$838$	19.5076	0.673878
$839$	44.7926	1.54641	0.773206	−	0.634155i	$-0.218652\pi$
$839$	44.7926	1.54641	0.773206	+	0.634155i	$0.218652\pi$
$840$	0	0
$841$	21.7386	0.749608
$842$	−100.847	−3.47540
$843$	0	0
$844$	28.4924	0.980750
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	3.75379	0.128982
$848$	36.0000	1.23625
$849$	0	0
$850$	4.00000	0.137199
$851$	26.0540	0.893119
$852$	0	0
$853$	41.4233	1.41831	0.709153	−	0.705054i	$-0.249077\pi$
$853$	41.4233	1.41831	0.709153	+	0.705054i	$0.249077\pi$
$854$	−7.50758	−0.256904
$855$	0	0
$856$	71.7235	2.45146
$857$	−2.43845	−0.0832958	−0.0416479	−	0.999132i	$-0.513261\pi$
$857$	−2.43845	−0.0832958	−0.0416479	+	0.999132i	$0.513261\pi$
$858$	0	0
$859$	3.80776	0.129919	0.0649596	−	0.997888i	$-0.479308\pi$
$859$	3.80776	0.129919	0.0649596	+	0.997888i	$0.479308\pi$
$860$	−14.2462	−0.485792
$861$	0	0
$862$	−8.98485	−0.306025
$863$	9.36932	0.318935	0.159468	−	0.987203i	$-0.449022\pi$
$863$	9.36932	0.318935	0.159468	+	0.987203i	$0.449022\pi$
$864$	0	0
$865$	12.8769	0.437828
$866$	24.6307	0.836985
$867$	0	0
$868$	−12.0000	−0.407307
$869$	7.31534	0.248156
$870$	0	0
$871$	11.3693	0.385235
$872$	13.1231	0.444404
$873$	0	0
$874$	−32.0000	−1.08242
$875$	0.438447	0.0148222
$876$	0	0
$877$	46.9848	1.58657	0.793283	−	0.608853i	$-0.208370\pi$
$877$	46.9848	1.58657	0.793283	+	0.608853i	$0.208370\pi$
$878$	−56.4924	−1.90653
$879$	0	0
$880$	−12.0000	−0.404520
$881$	21.3693	0.719951	0.359975	−	0.932962i	$-0.382785\pi$
$881$	21.3693	0.719951	0.359975	+	0.932962i	$0.382785\pi$
$882$	0	0
$883$	−56.1080	−1.88818	−0.944091	−	0.329684i	$-0.893058\pi$
$883$	−56.1080	−1.88818	−0.944091	+	0.329684i	$0.893058\pi$
$884$	7.12311	0.239576
$885$	0	0
$886$	−20.0000	−0.671913
$887$	−5.56155	−0.186739	−0.0933693	−	0.995632i	$-0.529764\pi$
$887$	−5.56155	−0.186739	−0.0933693	+	0.995632i	$0.529764\pi$
$888$	0	0
$889$	−0.768944	−0.0257895
$890$	−27.3693	−0.917422
$891$	0	0
$892$	−70.1080	−2.34739
$893$	56.9848	1.90693
$894$	0	0
$895$	−4.87689	−0.163017
$896$	−4.13826	−0.138250
$897$	0	0
$898$	−90.1080	−3.00694
$899$	42.7386	1.42541
$900$	0	0
$901$	−7.31534	−0.243709
$902$	14.2462	0.474347
$903$	0	0
$904$	−32.0000	−1.06430
$905$	−13.3153	−0.442617
$906$	0	0
$907$	−2.63068	−0.0873504	−0.0436752	−	0.999046i	$-0.513907\pi$
$907$	−2.63068	−0.0873504	−0.0436752	+	0.999046i	$0.513907\pi$
$908$	−26.2462	−0.871011
$909$	0	0
$910$	1.12311	0.0372306
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	25.7538	0.852326
$914$	34.1080	1.12819
$915$	0	0
$916$	78.1080	2.58076
$917$	0	0
$918$	0	0
$919$	1.94602	0.0641934	0.0320967	−	0.999485i	$-0.489782\pi$
$919$	1.94602	0.0641934	0.0320967	+	0.999485i	$0.489782\pi$
$920$	−16.0000	−0.527504
$921$	0	0
$922$	20.6307	0.679435
$923$	10.4384	0.343586
$924$	0	0
$925$	−10.6847	−0.351309
$926$	74.1080	2.43534
$927$	0	0
$928$	−46.7386	−1.53427
$929$	−39.6695	−1.30151	−0.650757	−	0.759286i	$-0.725548\pi$
$929$	−39.6695	−1.30151	−0.650757	+	0.759286i	$0.725548\pi$
$930$	0	0
$931$	34.8769	1.14304
$932$	126.847	4.15500
$933$	0	0
$934$	17.7538	0.580922
$935$	2.43845	0.0797458
$936$	0	0
$937$	−27.3693	−0.894117	−0.447058	−	0.894505i	$-0.647528\pi$
$937$	−27.3693	−0.894117	−0.447058	+	0.894505i	$0.647528\pi$
$938$	−12.7689	−0.416921
$939$	0	0
$940$	50.7386	1.65491
$941$	41.8078	1.36289	0.681447	−	0.731867i	$-0.261351\pi$
$941$	41.8078	1.36289	0.681447	+	0.731867i	$0.261351\pi$
$942$	0	0
$943$	8.68466	0.282811
$944$	−92.2159	−3.00137
$945$	0	0
$946$	−12.4924	−0.406164
$947$	−60.6004	−1.96925	−0.984624	−	0.174688i	$-0.944108\pi$
$947$	−60.6004	−1.96925	−0.984624	+	0.174688i	$0.944108\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	13.1231	0.425770
$951$	0	0
$952$	−4.49242	−0.145600
$953$	−29.5616	−0.957593	−0.478796	−	0.877926i	$-0.658927\pi$
$953$	−29.5616	−0.957593	−0.478796	+	0.877926i	$0.658927\pi$
$954$	0	0
$955$	19.6155	0.634744
$956$	−104.600	−3.38302
$957$	0	0
$958$	66.7386	2.15623
$959$	0.492423	0.0159012
$960$	0	0
$961$	5.00000	0.161290
$962$	−27.3693	−0.882422
$963$	0	0
$964$	112.847	3.63454
$965$	−19.5616	−0.629709
$966$	0	0
$967$	−7.36932	−0.236981	−0.118491	−	0.992955i	$-0.537806\pi$
$967$	−7.36932	−0.236981	−0.118491	+	0.992955i	$0.537806\pi$
$968$	56.1771	1.80560
$969$	0	0
$970$	43.3693	1.39250
$971$	24.4924	0.785999	0.393000	−	0.919539i	$-0.371437\pi$
$971$	24.4924	0.785999	0.393000	+	0.919539i	$0.371437\pi$
$972$	0	0
$973$	−1.45360	−0.0466003
$974$	82.1080	2.63091
$975$	0	0
$976$	−51.3693	−1.64429
$977$	43.4773	1.39096	0.695481	−	0.718545i	$-0.255192\pi$
$977$	43.4773	1.39096	0.695481	+	0.718545i	$0.255192\pi$
$978$	0	0
$979$	−16.6847	−0.533244
$980$	31.0540	0.991983
$981$	0	0
$982$	−70.7386	−2.25736
$983$	−42.7386	−1.36315	−0.681575	−	0.731748i	$-0.738705\pi$
$983$	−42.7386	−1.36315	−0.681575	+	0.731748i	$0.738705\pi$
$984$	0	0
$985$	−3.36932	−0.107355
$986$	28.4924	0.907384
$987$	0	0
$988$	23.3693	0.743477
$989$	−7.61553	−0.242160
$990$	0	0
$991$	10.0540	0.319375	0.159688	−	0.987168i	$-0.448951\pi$
$991$	10.0540	0.319375	0.159688	+	0.987168i	$0.448951\pi$
$992$	−39.3693	−1.24998
$993$	0	0
$994$	−11.7235	−0.371846
$995$	8.00000	0.253617
$996$	0	0
$997$	28.2462	0.894566	0.447283	−	0.894392i	$-0.352392\pi$
$997$	28.2462	0.894566	0.447283	+	0.894392i	$0.352392\pi$
$998$	−89.6155	−2.83673
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.2.a.j.1.1	✓	2
3.2	odd	2		585.2.a.l.1.2	yes	2
4.3	odd	2		9360.2.a.cl.1.1		2
5.2	odd	4		2925.2.c.o.2224.1		4
5.3	odd	4		2925.2.c.o.2224.4		4
5.4	even	2		2925.2.a.bc.1.2		2
12.11	even	2		9360.2.a.cw.1.1		2
13.12	even	2		7605.2.a.bi.1.2		2
15.2	even	4		2925.2.c.p.2224.4		4
15.8	even	4		2925.2.c.p.2224.1		4
15.14	odd	2		2925.2.a.x.1.1		2
39.38	odd	2		7605.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.a.j.1.1	✓	2	1.1	even	1	trivial
585.2.a.l.1.2	yes	2	3.2	odd	2
2925.2.a.x.1.1		2	15.14	odd	2
2925.2.a.bc.1.2		2	5.4	even	2
2925.2.c.o.2224.1		4	5.2	odd	4
2925.2.c.o.2224.4		4	5.3	odd	4
2925.2.c.p.2224.1		4	15.8	even	4
2925.2.c.p.2224.4		4	15.2	even	4
7605.2.a.bd.1.1		2	39.38	odd	2
7605.2.a.bi.1.2		2	13.12	even	2
9360.2.a.cl.1.1		2	4.3	odd	2
9360.2.a.cw.1.1		2	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:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	585.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{5} -0.438447 q^{7} -6.56155 q^{8} +O(q^{10})\)	\(q-2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{5} -0.438447 q^{7} -6.56155 q^{8} +2.56155 q^{10} +1.56155 q^{11} -1.00000 q^{13} +1.12311 q^{14} +7.68466 q^{16} -1.56155 q^{17} -5.12311 q^{19} -4.56155 q^{20} -4.00000 q^{22} -2.43845 q^{23} +1.00000 q^{25} +2.56155 q^{26} -2.00000 q^{28} +7.12311 q^{29} +6.00000 q^{31} -6.56155 q^{32} +4.00000 q^{34} +0.438447 q^{35} -10.6847 q^{37} +13.1231 q^{38} +6.56155 q^{40} -3.56155 q^{41} +3.12311 q^{43} +7.12311 q^{44} +6.24621 q^{46} -11.1231 q^{47} -6.80776 q^{49} -2.56155 q^{50} -4.56155 q^{52} +4.68466 q^{53} -1.56155 q^{55} +2.87689 q^{56} -18.2462 q^{58} -12.0000 q^{59} -6.68466 q^{61} -15.3693 q^{62} +1.43845 q^{64} +1.00000 q^{65} -11.3693 q^{67} -7.12311 q^{68} -1.12311 q^{70} -10.4384 q^{71} -6.00000 q^{73} +27.3693 q^{74} -23.3693 q^{76} -0.684658 q^{77} +4.68466 q^{79} -7.68466 q^{80} +9.12311 q^{82} +16.4924 q^{83} +1.56155 q^{85} -8.00000 q^{86} -10.2462 q^{88} -10.6847 q^{89} +0.438447 q^{91} -11.1231 q^{92} +28.4924 q^{94} +5.12311 q^{95} +16.9309 q^{97} +17.4384 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 5 q^{4} - 2 q^{5} - 5 q^{7} - 9 q^{8}+O(q^{10}) \) 2 * q - q^2 + 5 * q^4 - 2 * q^5 - 5 * q^7 - 9 * q^8	\( 2 q - q^{2} + 5 q^{4} - 2 q^{5} - 5 q^{7} - 9 q^{8} + q^{10} - q^{11} - 2 q^{13} - 6 q^{14} + 3 q^{16} + q^{17} - 2 q^{19} - 5 q^{20} - 8 q^{22} - 9 q^{23} + 2 q^{25} + q^{26} - 4 q^{28} + 6 q^{29} + 12 q^{31} - 9 q^{32} + 8 q^{34} + 5 q^{35} - 9 q^{37} + 18 q^{38} + 9 q^{40} - 3 q^{41} - 2 q^{43} + 6 q^{44} - 4 q^{46} - 14 q^{47} + 7 q^{49} - q^{50} - 5 q^{52} - 3 q^{53} + q^{55} + 14 q^{56} - 20 q^{58} - 24 q^{59} - q^{61} - 6 q^{62} + 7 q^{64} + 2 q^{65} + 2 q^{67} - 6 q^{68} + 6 q^{70} - 25 q^{71} - 12 q^{73} + 30 q^{74} - 22 q^{76} + 11 q^{77} - 3 q^{79} - 3 q^{80} + 10 q^{82} - q^{85} - 16 q^{86} - 4 q^{88} - 9 q^{89} + 5 q^{91} - 14 q^{92} + 24 q^{94} + 2 q^{95} + 5 q^{97} + 39 q^{98}+O(q^{100}) \) 2 * q - q^2 + 5 * q^4 - 2 * q^5 - 5 * q^7 - 9 * q^8 + q^10 - q^11 - 2 * q^13 - 6 * q^14 + 3 * q^16 + q^17 - 2 * q^19 - 5 * q^20 - 8 * q^22 - 9 * q^23 + 2 * q^25 + q^26 - 4 * q^28 + 6 * q^29 + 12 * q^31 - 9 * q^32 + 8 * q^34 + 5 * q^35 - 9 * q^37 + 18 * q^38 + 9 * q^40 - 3 * q^41 - 2 * q^43 + 6 * q^44 - 4 * q^46 - 14 * q^47 + 7 * q^49 - q^50 - 5 * q^52 - 3 * q^53 + q^55 + 14 * q^56 - 20 * q^58 - 24 * q^59 - q^61 - 6 * q^62 + 7 * q^64 + 2 * q^65 + 2 * q^67 - 6 * q^68 + 6 * q^70 - 25 * q^71 - 12 * q^73 + 30 * q^74 - 22 * q^76 + 11 * q^77 - 3 * q^79 - 3 * q^80 + 10 * q^82 - q^85 - 16 * q^86 - 4 * q^88 - 9 * q^89 + 5 * q^91 - 14 * q^92 + 24 * q^94 + 2 * q^95 + 5 * q^97 + 39 * q^98

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