LMFDB - Embedded newform 1521.4.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 25x^{2} + 24x + 78 $ x^4 - 2x^3 - 25x^2 + 24*x + 78
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.36176$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.36176 q^{2} -2.42208 q^{4} +6.42208 q^{5} +29.4938 q^{7} -24.6145 q^{8} +O(q^{10})$	$q+2.36176 q^{2} -2.42208 q^{4} +6.42208 q^{5} +29.4938 q^{7} -24.6145 q^{8} +15.1674 q^{10} -0.624715 q^{11} +69.6575 q^{14} -38.7569 q^{16} -87.7291 q^{17} -82.8018 q^{19} -15.5548 q^{20} -1.47543 q^{22} +74.7977 q^{23} -83.7569 q^{25} -71.4364 q^{28} -226.329 q^{29} -173.660 q^{31} +105.381 q^{32} -207.195 q^{34} +189.412 q^{35} -112.020 q^{37} -195.558 q^{38} -158.076 q^{40} -267.011 q^{41} +383.450 q^{43} +1.51311 q^{44} +176.654 q^{46} +337.380 q^{47} +526.887 q^{49} -197.814 q^{50} +146.354 q^{53} -4.01197 q^{55} -725.975 q^{56} -534.536 q^{58} -529.173 q^{59} +203.272 q^{61} -410.144 q^{62} +558.941 q^{64} -121.497 q^{67} +212.487 q^{68} +447.346 q^{70} -661.314 q^{71} -167.341 q^{73} -264.565 q^{74} +200.552 q^{76} -18.4253 q^{77} -101.399 q^{79} -248.900 q^{80} -630.617 q^{82} -506.985 q^{83} -563.403 q^{85} +905.617 q^{86} +15.3770 q^{88} +1402.33 q^{89} -181.166 q^{92} +796.810 q^{94} -531.760 q^{95} -1902.89 q^{97} +1244.38 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 22 q^{4} - 6 q^{5} + 14 q^{7} + 54 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 22 * q^4 - 6 * q^5 + 14 * q^7 + 54 * q^8	$ 4 q + 2 q^{2} + 22 q^{4} - 6 q^{5} + 14 q^{7} + 54 q^{8} - 62 q^{10} + 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} - 124 q^{19} - 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} + 144 q^{28} - 194 q^{29} - 26 q^{31} + 654 q^{32} - 1062 q^{34} - 88 q^{35} - 102 q^{37} - 332 q^{38} - 998 q^{40} - 1054 q^{41} + 450 q^{43} - 44 q^{44} + 172 q^{46} - 96 q^{47} + 1070 q^{49} + 996 q^{50} - 262 q^{53} + 204 q^{55} - 2164 q^{56} - 722 q^{58} + 308 q^{59} - 928 q^{61} - 2780 q^{62} + 1026 q^{64} + 1134 q^{67} - 1786 q^{68} + 2324 q^{70} + 1064 q^{71} - 952 q^{73} - 1158 q^{74} + 1708 q^{76} - 2508 q^{77} - 746 q^{79} - 2922 q^{80} + 1734 q^{82} - 404 q^{83} + 1394 q^{85} + 3168 q^{86} + 3060 q^{88} + 1620 q^{89} - 332 q^{92} - 772 q^{94} - 2204 q^{95} - 2166 q^{97} - 1906 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 22 * q^4 - 6 * q^5 + 14 * q^7 + 54 * q^8 - 62 * q^10 + 40 * q^11 - 40 * q^14 + 122 * q^16 - 98 * q^17 - 124 * q^19 - 466 * q^20 + 220 * q^22 - 104 * q^23 - 58 * q^25 + 144 * q^28 - 194 * q^29 - 26 * q^31 + 654 * q^32 - 1062 * q^34 - 88 * q^35 - 102 * q^37 - 332 * q^38 - 998 * q^40 - 1054 * q^41 + 450 * q^43 - 44 * q^44 + 172 * q^46 - 96 * q^47 + 1070 * q^49 + 996 * q^50 - 262 * q^53 + 204 * q^55 - 2164 * q^56 - 722 * q^58 + 308 * q^59 - 928 * q^61 - 2780 * q^62 + 1026 * q^64 + 1134 * q^67 - 1786 * q^68 + 2324 * q^70 + 1064 * q^71 - 952 * q^73 - 1158 * q^74 + 1708 * q^76 - 2508 * q^77 - 746 * q^79 - 2922 * q^80 + 1734 * q^82 - 404 * q^83 + 1394 * q^85 + 3168 * q^86 + 3060 * q^88 + 1620 * q^89 - 332 * q^92 - 772 * q^94 - 2204 * q^95 - 2166 * q^97 - 1906 * 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 $

$2$	2.36176	0.835009	0.417505	−	0.908675i	$-0.362905\pi$
$2$	2.36176	0.835009	0.417505	+	0.908675i	$0.362905\pi$
$3$	0	0
$3$	0	0
$4$	−2.42208	−0.302760
$5$	6.42208	0.574408	0.287204	−	0.957869i	$-0.407274\pi$
$5$	6.42208	0.574408	0.287204	+	0.957869i	$0.407274\pi$
$6$	0	0
$7$	29.4938	1.59252	0.796259	−	0.604956i	$-0.206809\pi$
$7$	29.4938	1.59252	0.796259	+	0.604956i	$0.206809\pi$
$8$	−24.6145	−1.08782
$9$	0	0
$10$	15.1674	0.479636
$11$	−0.624715	−0.0171235	−0.00856176	−	0.999963i	$-0.502725\pi$
$11$	−0.624715	−0.0171235	−0.00856176	+	0.999963i	$0.502725\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	69.6575	1.32977
$15$	0	0
$16$	−38.7569	−0.605577
$17$	−87.7291	−1.25161	−0.625807	−	0.779978i	$-0.715230\pi$
$17$	−87.7291	−1.25161	−0.625807	+	0.779978i	$0.715230\pi$
$18$	0	0
$19$	−82.8018	−0.999792	−0.499896	−	0.866085i	$-0.666628\pi$
$19$	−82.8018	−0.999792	−0.499896	+	0.866085i	$0.666628\pi$
$20$	−15.5548	−0.173908
$21$	0	0
$22$	−1.47543	−0.0142983
$23$	74.7977	0.678104	0.339052	−	0.940768i	$-0.389894\pi$
$23$	74.7977	0.678104	0.339052	+	0.940768i	$0.389894\pi$
$24$	0	0
$25$	−83.7569	−0.670055
$26$	0	0
$27$	0	0
$28$	−71.4364	−0.482150
$29$	−226.329	−1.44925	−0.724625	−	0.689143i	$-0.757987\pi$
$29$	−226.329	−1.44925	−0.724625	+	0.689143i	$0.757987\pi$
$30$	0	0
$31$	−173.660	−1.00614	−0.503070	−	0.864246i	$-0.667796\pi$
$31$	−173.660	−1.00614	−0.503070	+	0.864246i	$0.667796\pi$
$32$	105.381	0.582154
$33$	0	0
$34$	−207.195	−1.04511
$35$	189.412	0.914755
$36$	0	0
$37$	−112.020	−0.497730	−0.248865	−	0.968538i	$-0.580058\pi$
$37$	−112.020	−0.497730	−0.248865	+	0.968538i	$0.580058\pi$
$38$	−195.558	−0.834835
$39$	0	0
$40$	−158.076	−0.624850
$41$	−267.011	−1.01708	−0.508538	−	0.861040i	$-0.669814\pi$
$41$	−267.011	−1.01708	−0.508538	+	0.861040i	$0.669814\pi$
$42$	0	0
$43$	383.450	1.35990	0.679948	−	0.733260i	$-0.262002\pi$
$43$	383.450	1.35990	0.679948	+	0.733260i	$0.262002\pi$
$44$	1.51311	0.00518431
$45$	0	0
$46$	176.654	0.566223
$47$	337.380	1.04706	0.523530	−	0.852007i	$-0.324615\pi$
$47$	337.380	1.04706	0.523530	+	0.852007i	$0.324615\pi$
$48$	0	0
$49$	526.887	1.53611
$50$	−197.814	−0.559502
$51$	0	0
$52$	0	0
$53$	146.354	0.379308	0.189654	−	0.981851i	$-0.439263\pi$
$53$	146.354	0.379308	0.189654	+	0.981851i	$0.439263\pi$
$54$	0	0
$55$	−4.01197	−0.00983589
$56$	−725.975	−1.73237
$57$	0	0
$58$	−534.536	−1.21014
$59$	−529.173	−1.16767	−0.583834	−	0.811873i	$-0.698448\pi$
$59$	−529.173	−1.16767	−0.583834	+	0.811873i	$0.698448\pi$
$60$	0	0
$61$	203.272	0.426660	0.213330	−	0.976980i	$-0.431569\pi$
$61$	203.272	0.426660	0.213330	+	0.976980i	$0.431569\pi$
$62$	−410.144	−0.840135
$63$	0	0
$64$	558.941	1.09168
$65$	0	0
$66$	0	0
$67$	−121.497	−0.221540	−0.110770	−	0.993846i	$-0.535332\pi$
$67$	−121.497	−0.221540	−0.110770	+	0.993846i	$0.535332\pi$
$68$	212.487	0.378938
$69$	0	0
$70$	447.346	0.763829
$71$	−661.314	−1.10540	−0.552701	−	0.833380i	$-0.686403\pi$
$71$	−661.314	−1.10540	−0.552701	+	0.833380i	$0.686403\pi$
$72$	0	0
$73$	−167.341	−0.268299	−0.134150	−	0.990961i	$-0.542830\pi$
$73$	−167.341	−0.268299	−0.134150	+	0.990961i	$0.542830\pi$
$74$	−264.565	−0.415609
$75$	0	0
$76$	200.552	0.302697
$77$	−18.4253	−0.0272695
$78$	0	0
$79$	−101.399	−0.144408	−0.0722042	−	0.997390i	$-0.523003\pi$
$79$	−101.399	−0.144408	−0.0722042	+	0.997390i	$0.523003\pi$
$80$	−248.900	−0.347848
$81$	0	0
$82$	−630.617	−0.849268
$83$	−506.985	−0.670468	−0.335234	−	0.942135i	$-0.608815\pi$
$83$	−506.985	−0.670468	−0.335234	+	0.942135i	$0.608815\pi$
$84$	0	0
$85$	−563.403	−0.718937
$86$	905.617	1.13553
$87$	0	0
$88$	15.3770	0.0186272
$89$	1402.33	1.67019	0.835095	−	0.550106i	$-0.185413\pi$
$89$	1402.33	1.67019	0.835095	+	0.550106i	$0.185413\pi$
$90$	0	0
$91$	0	0
$92$	−181.166	−0.205303
$93$	0	0
$94$	796.810	0.874306
$95$	−531.760	−0.574289
$96$	0	0
$97$	−1902.89	−1.99185	−0.995924	−	0.0901969i	$-0.971250\pi$
$97$	−1902.89	−1.99185	−0.995924	+	0.0901969i	$0.971250\pi$
$98$	1244.38	1.28267
$99$	0	0
$100$	202.866	0.202866
$101$	−1833.09	−1.80594	−0.902968	−	0.429708i	$-0.858617\pi$
$101$	−1833.09	−1.80594	−0.902968	+	0.429708i	$0.858617\pi$
$102$	0	0
$103$	1446.99	1.38423	0.692115	−	0.721787i	$-0.256679\pi$
$103$	1446.99	1.38423	0.692115	+	0.721787i	$0.256679\pi$
$104$	0	0
$105$	0	0
$106$	345.654	0.316725
$107$	369.286	0.333647	0.166823	−	0.985987i	$-0.446649\pi$
$107$	369.286	0.333647	0.166823	+	0.985987i	$0.446649\pi$
$108$	0	0
$109$	815.694	0.716782	0.358391	−	0.933572i	$-0.383325\pi$
$109$	815.694	0.716782	0.358391	+	0.933572i	$0.383325\pi$
$110$	−9.47532	−0.00821306
$111$	0	0
$112$	−1143.09	−0.964392
$113$	−1790.56	−1.49064	−0.745319	−	0.666708i	$-0.767703\pi$
$113$	−1790.56	−1.49064	−0.745319	+	0.666708i	$0.767703\pi$
$114$	0	0
$115$	480.357	0.389509
$116$	548.187	0.438775
$117$	0	0
$118$	−1249.78	−0.975014
$119$	−2587.47	−1.99322
$120$	0	0
$121$	−1330.61	−0.999707
$122$	480.079	0.356265
$123$	0	0
$124$	420.619	0.304618
$125$	−1340.65	−0.959293
$126$	0	0
$127$	−45.2900	−0.0316444	−0.0158222	−	0.999875i	$-0.505037\pi$
$127$	−45.2900	−0.0316444	−0.0158222	+	0.999875i	$0.505037\pi$
$128$	477.036	0.329410
$129$	0	0
$130$	0	0
$131$	−1051.82	−0.701511	−0.350756	−	0.936467i	$-0.614075\pi$
$131$	−1051.82	−0.701511	−0.350756	+	0.936467i	$0.614075\pi$
$132$	0	0
$133$	−2442.14	−1.59219
$134$	−286.946	−0.184988
$135$	0	0
$136$	2159.41	1.36153
$137$	−1542.94	−0.962208	−0.481104	−	0.876664i	$-0.659764\pi$
$137$	−1542.94	−0.962208	−0.481104	+	0.876664i	$0.659764\pi$
$138$	0	0
$139$	37.8644	0.0231052	0.0115526	−	0.999933i	$-0.496323\pi$
$139$	37.8644	0.0231052	0.0115526	+	0.999933i	$0.496323\pi$
$140$	−458.770	−0.276951
$141$	0	0
$142$	−1561.87	−0.923020
$143$	0	0
$144$	0	0
$145$	−1453.50	−0.832461
$146$	−395.221	−0.224032
$147$	0	0
$148$	271.322	0.150693
$149$	−1822.40	−1.00199	−0.500995	−	0.865450i	$-0.667033\pi$
$149$	−1822.40	−1.00199	−0.500995	+	0.865450i	$0.667033\pi$
$150$	0	0
$151$	3239.36	1.74580	0.872900	−	0.487899i	$-0.162237\pi$
$151$	3239.36	1.74580	0.872900	+	0.487899i	$0.162237\pi$
$152$	2038.12	1.08759
$153$	0	0
$154$	−43.5161	−0.0227703
$155$	−1115.26	−0.577934
$156$	0	0
$157$	830.565	0.422206	0.211103	−	0.977464i	$-0.432295\pi$
$157$	830.565	0.422206	0.211103	+	0.977464i	$0.432295\pi$
$158$	−239.480	−0.120582
$159$	0	0
$160$	676.766	0.334394
$161$	2206.07	1.07989
$162$	0	0
$163$	2079.90	0.999451	0.499725	−	0.866184i	$-0.333434\pi$
$163$	2079.90	0.999451	0.499725	+	0.866184i	$0.333434\pi$
$164$	646.721	0.307930
$165$	0	0
$166$	−1197.38	−0.559847
$167$	85.9790	0.0398398	0.0199199	−	0.999802i	$-0.493659\pi$
$167$	85.9790	0.0398398	0.0199199	+	0.999802i	$0.493659\pi$
$168$	0	0
$169$	0	0
$170$	−1330.62	−0.600319
$171$	0	0
$172$	−928.745	−0.411722
$173$	−2706.47	−1.18942	−0.594708	−	0.803942i	$-0.702732\pi$
$173$	−2706.47	−1.18942	−0.594708	+	0.803942i	$0.702732\pi$
$174$	0	0
$175$	−2470.31	−1.06708
$176$	24.2120	0.0103696
$177$	0	0
$178$	3311.97	1.39462
$179$	4402.10	1.83815	0.919074	−	0.394085i	$-0.128938\pi$
$179$	4402.10	1.83815	0.919074	+	0.394085i	$0.128938\pi$
$180$	0	0
$181$	−1673.98	−0.687435	−0.343718	−	0.939073i	$-0.611686\pi$
$181$	−1673.98	−0.687435	−0.343718	+	0.939073i	$0.611686\pi$
$182$	0	0
$183$	0	0
$184$	−1841.11	−0.737653
$185$	−719.403	−0.285900
$186$	0	0
$187$	54.8057	0.0214320
$188$	−817.159	−0.317008
$189$	0	0
$190$	−1255.89	−0.479536
$191$	290.117	0.109907	0.0549533	−	0.998489i	$-0.482499\pi$
$191$	290.117	0.109907	0.0549533	+	0.998489i	$0.482499\pi$
$192$	0	0
$193$	−1039.50	−0.387693	−0.193847	−	0.981032i	$-0.562096\pi$
$193$	−1039.50	−0.387693	−0.193847	+	0.981032i	$0.562096\pi$
$194$	−4494.18	−1.66321
$195$	0	0
$196$	−1276.16	−0.465073
$197$	1418.70	0.513089	0.256544	−	0.966532i	$-0.417416\pi$
$197$	1418.70	0.513089	0.256544	+	0.966532i	$0.417416\pi$
$198$	0	0
$199$	−2388.39	−0.850795	−0.425398	−	0.905007i	$-0.639866\pi$
$199$	−2388.39	−0.850795	−0.425398	+	0.905007i	$0.639866\pi$
$200$	2061.63	0.728897
$201$	0	0
$202$	−4329.33	−1.50797
$203$	−6675.32	−2.30796
$204$	0	0
$205$	−1714.77	−0.584217
$206$	3417.44	1.15584
$207$	0	0
$208$	0	0
$209$	51.7276	0.0171200
$210$	0	0
$211$	4341.45	1.41648	0.708241	−	0.705971i	$-0.249489\pi$
$211$	4341.45	1.41648	0.708241	+	0.705971i	$0.249489\pi$
$212$	−354.481	−0.114839
$213$	0	0
$214$	872.165	0.278598
$215$	2462.54	0.781136
$216$	0	0
$217$	−5121.91	−1.60229
$218$	1926.47	0.598520
$219$	0	0
$220$	9.71730	0.00297791
$221$	0	0
$222$	0	0
$223$	−4615.37	−1.38596	−0.692978	−	0.720959i	$-0.743702\pi$
$223$	−4615.37	−1.38596	−0.692978	+	0.720959i	$0.743702\pi$
$224$	3108.09	0.927091
$225$	0	0
$226$	−4228.89	−1.24470
$227$	−2163.44	−0.632565	−0.316283	−	0.948665i	$-0.602435\pi$
$227$	−2163.44	−0.632565	−0.316283	+	0.948665i	$0.602435\pi$
$228$	0	0
$229$	−1859.48	−0.536584	−0.268292	−	0.963338i	$-0.586459\pi$
$229$	−1859.48	−0.536584	−0.268292	+	0.963338i	$0.586459\pi$
$230$	1134.49	0.325243
$231$	0	0
$232$	5570.97	1.57652
$233$	−2866.87	−0.806073	−0.403037	−	0.915184i	$-0.632045\pi$
$233$	−2866.87	−0.806073	−0.403037	+	0.915184i	$0.632045\pi$
$234$	0	0
$235$	2166.68	0.601440
$236$	1281.70	0.353523
$237$	0	0
$238$	−6110.99	−1.66435
$239$	1893.55	0.512485	0.256242	−	0.966613i	$-0.417516\pi$
$239$	1893.55	0.512485	0.256242	+	0.966613i	$0.417516\pi$
$240$	0	0
$241$	1813.58	0.484741	0.242371	−	0.970184i	$-0.422075\pi$
$241$	1813.58	0.484741	0.242371	+	0.970184i	$0.422075\pi$
$242$	−3142.58	−0.834764
$243$	0	0
$244$	−492.340	−0.129176
$245$	3383.71	0.882356
$246$	0	0
$247$	0	0
$248$	4274.56	1.09449
$249$	0	0
$250$	−3166.30	−0.801019
$251$	−4162.32	−1.04671	−0.523353	−	0.852116i	$-0.675319\pi$
$251$	−4162.32	−1.04671	−0.523353	+	0.852116i	$0.675319\pi$
$252$	0	0
$253$	−46.7273	−0.0116115
$254$	−106.964	−0.0264234
$255$	0	0
$256$	−3344.88	−0.816621
$257$	−5985.30	−1.45274	−0.726368	−	0.687306i	$-0.758793\pi$
$257$	−5985.30	−1.45274	−0.726368	+	0.687306i	$0.758793\pi$
$258$	0	0
$259$	−3303.91	−0.792645
$260$	0	0
$261$	0	0
$262$	−2484.15	−0.585768
$263$	−574.618	−0.134724	−0.0673621	−	0.997729i	$-0.521458\pi$
$263$	−574.618	−0.134724	−0.0673621	+	0.997729i	$0.521458\pi$
$264$	0	0
$265$	939.899	0.217877
$266$	−5767.77	−1.32949
$267$	0	0
$268$	294.275	0.0670734
$269$	−6348.61	−1.43896	−0.719482	−	0.694511i	$-0.755621\pi$
$269$	−6348.61	−1.43896	−0.719482	+	0.694511i	$0.755621\pi$
$270$	0	0
$271$	3278.38	0.734861	0.367431	−	0.930051i	$-0.380238\pi$
$271$	3278.38	0.734861	0.367431	+	0.930051i	$0.380238\pi$
$272$	3400.11	0.757948
$273$	0	0
$274$	−3644.06	−0.803452
$275$	52.3242	0.0114737
$276$	0	0
$277$	3952.17	0.857267	0.428633	−	0.903478i	$-0.358995\pi$
$277$	3952.17	0.857267	0.428633	+	0.903478i	$0.358995\pi$
$278$	89.4267	0.0192930
$279$	0	0
$280$	−4662.27	−0.995085
$281$	−411.389	−0.0873360	−0.0436680	−	0.999046i	$-0.513904\pi$
$281$	−411.389	−0.0873360	−0.0436680	+	0.999046i	$0.513904\pi$
$282$	0	0
$283$	−5872.78	−1.23357	−0.616785	−	0.787131i	$-0.711565\pi$
$283$	−5872.78	−1.23357	−0.616785	+	0.787131i	$0.711565\pi$
$284$	1601.75	0.334671
$285$	0	0
$286$	0	0
$287$	−7875.18	−1.61971
$288$	0	0
$289$	2783.39	0.566537
$290$	−3432.83	−0.695113
$291$	0	0
$292$	405.314	0.0812301
$293$	−500.957	−0.0998847	−0.0499423	−	0.998752i	$-0.515904\pi$
$293$	−500.957	−0.0998847	−0.0499423	+	0.998752i	$0.515904\pi$
$294$	0	0
$295$	−3398.39	−0.670718
$296$	2757.32	0.541439
$297$	0	0
$298$	−4304.07	−0.836671
$299$	0	0
$300$	0	0
$301$	11309.4	2.16566
$302$	7650.61	1.45776
$303$	0	0
$304$	3209.14	0.605451
$305$	1305.43	0.245077
$306$	0	0
$307$	5975.57	1.11089	0.555446	−	0.831553i	$-0.312548\pi$
$307$	5975.57	1.11089	0.555446	+	0.831553i	$0.312548\pi$
$308$	44.6274	0.00825611
$309$	0	0
$310$	−2633.98	−0.482581
$311$	−44.4925	−0.00811234	−0.00405617	−	0.999992i	$-0.501291\pi$
$311$	−44.4925	−0.00811234	−0.00405617	+	0.999992i	$0.501291\pi$
$312$	0	0
$313$	9957.78	1.79823	0.899117	−	0.437709i	$-0.144210\pi$
$313$	9957.78	1.79823	0.899117	+	0.437709i	$0.144210\pi$
$314$	1961.60	0.352546
$315$	0	0
$316$	245.596	0.0437211
$317$	7752.29	1.37354	0.686770	−	0.726875i	$-0.259028\pi$
$317$	7752.29	1.37354	0.686770	+	0.726875i	$0.259028\pi$
$318$	0	0
$319$	141.391	0.0248163
$320$	3589.56	0.627070
$321$	0	0
$322$	5210.22	0.901721
$323$	7264.13	1.25135
$324$	0	0
$325$	0	0
$326$	4912.23	0.834550
$327$	0	0
$328$	6572.34	1.10639
$329$	9950.62	1.66746
$330$	0	0
$331$	1338.85	0.222326	0.111163	−	0.993802i	$-0.464542\pi$
$331$	1338.85	0.222326	0.111163	+	0.993802i	$0.464542\pi$
$332$	1227.96	0.202991
$333$	0	0
$334$	203.062	0.0332666
$335$	−780.261	−0.127254
$336$	0	0
$337$	3788.95	0.612454	0.306227	−	0.951958i	$-0.400933\pi$
$337$	3788.95	0.612454	0.306227	+	0.951958i	$0.400933\pi$
$338$	0	0
$339$	0	0
$340$	1364.61	0.217665
$341$	108.488	0.0172286
$342$	0	0
$343$	5423.53	0.853770
$344$	−9438.42	−1.47932
$345$	0	0
$346$	−6392.03	−0.993173
$347$	−7795.15	−1.20595	−0.602977	−	0.797759i	$-0.706019\pi$
$347$	−7795.15	−1.20595	−0.602977	+	0.797759i	$0.706019\pi$
$348$	0	0
$349$	−134.533	−0.0206344	−0.0103172	−	0.999947i	$-0.503284\pi$
$349$	−134.533	−0.0206344	−0.0103172	+	0.999947i	$0.503284\pi$
$350$	−5834.29	−0.891018
$351$	0	0
$352$	−65.8332	−0.00996853
$353$	−2973.71	−0.448370	−0.224185	−	0.974547i	$-0.571972\pi$
$353$	−2973.71	−0.448370	−0.224185	+	0.974547i	$0.571972\pi$
$354$	0	0
$355$	−4247.01	−0.634952
$356$	−3396.56	−0.505666
$357$	0	0
$358$	10396.7	1.53487
$359$	8671.60	1.27485	0.637423	−	0.770514i	$-0.280001\pi$
$359$	8671.60	1.27485	0.637423	+	0.770514i	$0.280001\pi$
$360$	0	0
$361$	−2.85364	−0.000416043 0
$362$	−3953.54	−0.574015
$363$	0	0
$364$	0	0
$365$	−1074.68	−0.154113
$366$	0	0
$367$	4514.32	0.642086	0.321043	−	0.947065i	$-0.395967\pi$
$367$	4514.32	0.642086	0.321043	+	0.947065i	$0.395967\pi$
$368$	−2898.93	−0.410644
$369$	0	0
$370$	−1699.06	−0.238729
$371$	4316.55	0.604054
$372$	0	0
$373$	6570.60	0.912099	0.456049	−	0.889955i	$-0.349264\pi$
$373$	6570.60	0.912099	0.456049	+	0.889955i	$0.349264\pi$
$374$	129.438	0.0178959
$375$	0	0
$376$	−8304.42	−1.13901
$377$	0	0
$378$	0	0
$379$	5490.38	0.744121	0.372060	−	0.928209i	$-0.378651\pi$
$379$	5490.38	0.744121	0.372060	+	0.928209i	$0.378651\pi$
$380$	1287.96	0.173871
$381$	0	0
$382$	685.188	0.0917730
$383$	10187.3	1.35912	0.679562	−	0.733618i	$-0.262170\pi$
$383$	10187.3	1.35912	0.679562	+	0.733618i	$0.262170\pi$
$384$	0	0
$385$	−118.328	−0.0156638
$386$	−2455.05	−0.323728
$387$	0	0
$388$	4608.95	0.603051
$389$	−4883.97	−0.636573	−0.318287	−	0.947995i	$-0.603107\pi$
$389$	−4883.97	−0.636573	−0.318287	+	0.947995i	$0.603107\pi$
$390$	0	0
$391$	−6561.94	−0.848725
$392$	−12969.0	−1.67101
$393$	0	0
$394$	3350.64	0.428434
$395$	−651.192	−0.0829494
$396$	0	0
$397$	2115.04	0.267382	0.133691	−	0.991023i	$-0.457317\pi$
$397$	2115.04	0.267382	0.133691	+	0.991023i	$0.457317\pi$
$398$	−5640.80	−0.710422
$399$	0	0
$400$	3246.16	0.405770
$401$	−674.254	−0.0839667	−0.0419834	−	0.999118i	$-0.513368\pi$
$401$	−674.254	−0.0839667	−0.0419834	+	0.999118i	$0.513368\pi$
$402$	0	0
$403$	0	0
$404$	4439.89	0.546765
$405$	0	0
$406$	−15765.5	−1.92717
$407$	69.9808	0.00852290
$408$	0	0
$409$	−4673.33	−0.564991	−0.282496	−	0.959269i	$-0.591162\pi$
$409$	−4673.33	−0.564991	−0.282496	+	0.959269i	$0.591162\pi$
$410$	−4049.87	−0.487826
$411$	0	0
$412$	−3504.71	−0.419089
$413$	−15607.3	−1.85953
$414$	0	0
$415$	−3255.90	−0.385122
$416$	0	0
$417$	0	0
$418$	122.168	0.0142953
$419$	256.853	0.0299477	0.0149739	−	0.999888i	$-0.495233\pi$
$419$	256.853	0.0299477	0.0149739	+	0.999888i	$0.495233\pi$
$420$	0	0
$421$	8746.82	1.01257	0.506287	−	0.862365i	$-0.331018\pi$
$421$	8746.82	1.01257	0.506287	+	0.862365i	$0.331018\pi$
$422$	10253.5	1.18278
$423$	0	0
$424$	−3602.43	−0.412617
$425$	7347.92	0.838650
$426$	0	0
$427$	5995.26	0.679464
$428$	−894.439	−0.101015
$429$	0	0
$430$	5815.95	0.652255
$431$	8762.72	0.979316	0.489658	−	0.871914i	$-0.337122\pi$
$431$	8762.72	0.979316	0.489658	+	0.871914i	$0.337122\pi$
$432$	0	0
$433$	6425.50	0.713140	0.356570	−	0.934269i	$-0.383946\pi$
$433$	6425.50	0.713140	0.356570	+	0.934269i	$0.383946\pi$
$434$	−12096.7	−1.33793
$435$	0	0
$436$	−1975.67	−0.217013
$437$	−6193.39	−0.677963
$438$	0	0
$439$	6820.28	0.741490	0.370745	−	0.928735i	$-0.379102\pi$
$439$	6820.28	0.741490	0.370745	+	0.928735i	$0.379102\pi$
$440$	98.7525	0.0106996
$441$	0	0
$442$	0	0
$443$	−5062.48	−0.542948	−0.271474	−	0.962446i	$-0.587511\pi$
$443$	−5062.48	−0.542948	−0.271474	+	0.962446i	$0.587511\pi$
$444$	0	0
$445$	9005.88	0.959370
$446$	−10900.4	−1.15729
$447$	0	0
$448$	16485.3	1.73852
$449$	−6590.82	−0.692740	−0.346370	−	0.938098i	$-0.612586\pi$
$449$	−6590.82	−0.692740	−0.346370	+	0.938098i	$0.612586\pi$
$450$	0	0
$451$	166.806	0.0174159
$452$	4336.89	0.451305
$453$	0	0
$454$	−5109.52	−0.528198
$455$	0	0
$456$	0	0
$457$	2254.75	0.230793	0.115397	−	0.993319i	$-0.463186\pi$
$457$	2254.75	0.230793	0.115397	+	0.993319i	$0.463186\pi$
$458$	−4391.65	−0.448053
$459$	0	0
$460$	−1163.46	−0.117928
$461$	7358.79	0.743456	0.371728	−	0.928342i	$-0.378765\pi$
$461$	7358.79	0.743456	0.371728	+	0.928342i	$0.378765\pi$
$462$	0	0
$463$	−11598.3	−1.16419	−0.582096	−	0.813120i	$-0.697767\pi$
$463$	−11598.3	−1.16419	−0.582096	+	0.813120i	$0.697767\pi$
$464$	8771.82	0.877633
$465$	0	0
$466$	−6770.87	−0.673079
$467$	302.150	0.0299397	0.0149698	−	0.999888i	$-0.495235\pi$
$467$	302.150	0.0299397	0.0149698	+	0.999888i	$0.495235\pi$
$468$	0	0
$469$	−3583.41	−0.352807
$470$	5117.18	0.502208
$471$	0	0
$472$	13025.3	1.27021
$473$	−239.547	−0.0232862
$474$	0	0
$475$	6935.23	0.669916
$476$	6267.05	0.603466
$477$	0	0
$478$	4472.12	0.427929
$479$	−3146.15	−0.300107	−0.150053	−	0.988678i	$-0.547945\pi$
$479$	−3146.15	−0.300107	−0.150053	+	0.988678i	$0.547945\pi$
$480$	0	0
$481$	0	0
$482$	4283.24	0.404764
$483$	0	0
$484$	3222.84	0.302671
$485$	−12220.5	−1.14413
$486$	0	0
$487$	−3068.16	−0.285486	−0.142743	−	0.989760i	$-0.545592\pi$
$487$	−3068.16	−0.285486	−0.142743	+	0.989760i	$0.545592\pi$
$488$	−5003.43	−0.464128
$489$	0	0
$490$	7991.51	0.736775
$491$	100.558	0.00924262	0.00462131	−	0.999989i	$-0.498529\pi$
$491$	100.558	0.00924262	0.00462131	+	0.999989i	$0.498529\pi$
$492$	0	0
$493$	19855.7	1.81390
$494$	0	0
$495$	0	0
$496$	6730.54	0.609295
$497$	−19504.7	−1.76037
$498$	0	0
$499$	−3616.55	−0.324447	−0.162223	−	0.986754i	$-0.551867\pi$
$499$	−3616.55	−0.324447	−0.162223	+	0.986754i	$0.551867\pi$
$500$	3247.17	0.290435
$501$	0	0
$502$	−9830.40	−0.874009
$503$	5372.64	0.476251	0.238125	−	0.971234i	$-0.423467\pi$
$503$	5372.64	0.476251	0.238125	+	0.971234i	$0.423467\pi$
$504$	0	0
$505$	−11772.3	−1.03734
$506$	−110.359	−0.00969574
$507$	0	0
$508$	109.696	0.00958065
$509$	−11314.7	−0.985298	−0.492649	−	0.870228i	$-0.663971\pi$
$509$	−11314.7	−0.985298	−0.492649	+	0.870228i	$0.663971\pi$
$510$	0	0
$511$	−4935.54	−0.427271
$512$	−11716.1	−1.01130
$513$	0	0
$514$	−14135.9	−1.21305
$515$	9292.65	0.795113
$516$	0	0
$517$	−210.766	−0.0179294
$518$	−7803.05	−0.661865
$519$	0	0
$520$	0	0
$521$	18470.9	1.55321	0.776606	−	0.629986i	$-0.216939\pi$
$521$	18470.9	1.55321	0.776606	+	0.629986i	$0.216939\pi$
$522$	0	0
$523$	10891.0	0.910576	0.455288	−	0.890344i	$-0.349536\pi$
$523$	10891.0	0.910576	0.455288	+	0.890344i	$0.349536\pi$
$524$	2547.59	0.212389
$525$	0	0
$526$	−1357.11	−0.112496
$527$	15235.1	1.25930
$528$	0	0
$529$	−6572.30	−0.540174
$530$	2219.82	0.181930
$531$	0	0
$532$	5915.06	0.482050
$533$	0	0
$534$	0	0
$535$	2371.58	0.191649
$536$	2990.58	0.240995
$537$	0	0
$538$	−14993.9	−1.20155
$539$	−329.154	−0.0263037
$540$	0	0
$541$	13416.3	1.06620	0.533099	−	0.846053i	$-0.321027\pi$
$541$	13416.3	1.06620	0.533099	+	0.846053i	$0.321027\pi$
$542$	7742.75	0.613616
$543$	0	0
$544$	−9244.99	−0.728632
$545$	5238.45	0.411726
$546$	0	0
$547$	−17849.0	−1.39519	−0.697593	−	0.716495i	$-0.745745\pi$
$547$	−17849.0	−1.39519	−0.697593	+	0.716495i	$0.745745\pi$
$548$	3737.13	0.291318
$549$	0	0
$550$	123.577	0.00958065
$551$	18740.5	1.44895
$552$	0	0
$553$	−2990.64	−0.229973
$554$	9334.09	0.715826
$555$	0	0
$556$	−91.7105	−0.00699531
$557$	−20131.4	−1.53141	−0.765703	−	0.643195i	$-0.777609\pi$
$557$	−20131.4	−1.53141	−0.765703	+	0.643195i	$0.777609\pi$
$558$	0	0
$559$	0	0
$560$	−7341.02	−0.553955
$561$	0	0
$562$	−971.603	−0.0729263
$563$	22345.8	1.67276	0.836380	−	0.548150i	$-0.184668\pi$
$563$	22345.8	1.67276	0.836380	+	0.548150i	$0.184668\pi$
$564$	0	0
$565$	−11499.1	−0.856235
$566$	−13870.1	−1.03004
$567$	0	0
$568$	16277.9	1.20247
$569$	8455.72	0.622992	0.311496	−	0.950248i	$-0.399170\pi$
$569$	8455.72	0.622992	0.311496	+	0.950248i	$0.399170\pi$
$570$	0	0
$571$	12813.0	0.939069	0.469534	−	0.882914i	$-0.344422\pi$
$571$	12813.0	0.939069	0.469534	+	0.882914i	$0.344422\pi$
$572$	0	0
$573$	0	0
$574$	−18599.3	−1.35247
$575$	−6264.83	−0.454367
$576$	0	0
$577$	−1971.59	−0.142251	−0.0711253	−	0.997467i	$-0.522659\pi$
$577$	−1971.59	−0.142251	−0.0711253	+	0.997467i	$0.522659\pi$
$578$	6573.72	0.473063
$579$	0	0
$580$	3520.50	0.252036
$581$	−14952.9	−1.06773
$582$	0	0
$583$	−91.4298	−0.00649508
$584$	4119.02	0.291860
$585$	0	0
$586$	−1183.14	−0.0834046
$587$	8585.69	0.603696	0.301848	−	0.953356i	$-0.402397\pi$
$587$	8585.69	0.603696	0.301848	+	0.953356i	$0.402397\pi$
$588$	0	0
$589$	14379.4	1.00593
$590$	−8026.19	−0.560056
$591$	0	0
$592$	4341.56	0.301414
$593$	−1746.73	−0.120961	−0.0604803	−	0.998169i	$-0.519263\pi$
$593$	−1746.73	−0.120961	−0.0604803	+	0.998169i	$0.519263\pi$
$594$	0	0
$595$	−16616.9	−1.14492
$596$	4413.99	0.303362
$597$	0	0
$598$	0	0
$599$	−27531.1	−1.87794	−0.938972	−	0.343994i	$-0.888220\pi$
$599$	−27531.1	−1.87794	−0.938972	+	0.343994i	$0.888220\pi$
$600$	0	0
$601$	−17539.1	−1.19041	−0.595203	−	0.803575i	$-0.702928\pi$
$601$	−17539.1	−1.19041	−0.595203	+	0.803575i	$0.702928\pi$
$602$	26710.1	1.80835
$603$	0	0
$604$	−7845.99	−0.528558
$605$	−8545.28	−0.574240
$606$	0	0
$607$	−22691.1	−1.51730	−0.758651	−	0.651497i	$-0.774141\pi$
$607$	−22691.1	−1.51730	−0.758651	+	0.651497i	$0.774141\pi$
$608$	−8725.75	−0.582033
$609$	0	0
$610$	3083.11	0.204642
$611$	0	0
$612$	0	0
$613$	−7215.30	−0.475405	−0.237702	−	0.971338i	$-0.576394\pi$
$613$	−7215.30	−0.475405	−0.237702	+	0.971338i	$0.576394\pi$
$614$	14112.9	0.927605
$615$	0	0
$616$	453.528	0.0296642
$617$	−16870.4	−1.10077	−0.550387	−	0.834909i	$-0.685520\pi$
$617$	−16870.4	−1.10077	−0.550387	+	0.834909i	$0.685520\pi$
$618$	0	0
$619$	−2244.53	−0.145743	−0.0728717	−	0.997341i	$-0.523216\pi$
$619$	−2244.53	−0.145743	−0.0728717	+	0.997341i	$0.523216\pi$
$620$	2701.25	0.174975
$621$	0	0
$622$	−105.081	−0.00677388
$623$	41360.1	2.65981
$624$	0	0
$625$	1859.84	0.119030
$626$	23517.9	1.50154
$627$	0	0
$628$	−2011.69	−0.127827
$629$	9827.44	0.622966
$630$	0	0
$631$	3669.22	0.231488	0.115744	−	0.993279i	$-0.463075\pi$
$631$	3669.22	0.231488	0.115744	+	0.993279i	$0.463075\pi$
$632$	2495.88	0.157090
$633$	0	0
$634$	18309.1	1.14692
$635$	−290.856	−0.0181768
$636$	0	0
$637$	0	0
$638$	333.933	0.0207218
$639$	0	0
$640$	3063.56	0.189215
$641$	−14678.6	−0.904477	−0.452239	−	0.891897i	$-0.649374\pi$
$641$	−14678.6	−0.904477	−0.452239	+	0.891897i	$0.649374\pi$
$642$	0	0
$643$	−5519.72	−0.338533	−0.169266	−	0.985570i	$-0.554140\pi$
$643$	−5519.72	−0.338533	−0.169266	+	0.985570i	$0.554140\pi$
$644$	−5343.28	−0.326948
$645$	0	0
$646$	17156.2	1.04489
$647$	11326.9	0.688261	0.344131	−	0.938922i	$-0.388174\pi$
$647$	11326.9	0.688261	0.344131	+	0.938922i	$0.388174\pi$
$648$	0	0
$649$	330.582	0.0199946
$650$	0	0
$651$	0	0
$652$	−5037.68	−0.302593
$653$	−3902.89	−0.233893	−0.116946	−	0.993138i	$-0.537311\pi$
$653$	−3902.89	−0.233893	−0.116946	+	0.993138i	$0.537311\pi$
$654$	0	0
$655$	−6754.87	−0.402954
$656$	10348.5	0.615918
$657$	0	0
$658$	23501.0	1.39235
$659$	−8022.47	−0.474220	−0.237110	−	0.971483i	$-0.576200\pi$
$659$	−8022.47	−0.474220	−0.237110	+	0.971483i	$0.576200\pi$
$660$	0	0
$661$	5168.76	0.304147	0.152074	−	0.988369i	$-0.451405\pi$
$661$	5168.76	0.304147	0.152074	+	0.988369i	$0.451405\pi$
$662$	3162.05	0.185645
$663$	0	0
$664$	12479.2	0.729346
$665$	−15683.6	−0.914565
$666$	0	0
$667$	−16928.9	−0.982743
$668$	−208.248	−0.0120619
$669$	0	0
$670$	−1842.79	−0.106259
$671$	−126.987	−0.00730593
$672$	0	0
$673$	−6654.10	−0.381124	−0.190562	−	0.981675i	$-0.561031\pi$
$673$	−6654.10	−0.381124	−0.190562	+	0.981675i	$0.561031\pi$
$674$	8948.59	0.511405
$675$	0	0
$676$	0	0
$677$	20649.4	1.17226	0.586130	−	0.810217i	$-0.300651\pi$
$677$	20649.4	1.17226	0.586130	+	0.810217i	$0.300651\pi$
$678$	0	0
$679$	−56123.6	−3.17205
$680$	13867.9	0.782071
$681$	0	0
$682$	256.223	0.0143861
$683$	28475.6	1.59530	0.797649	−	0.603122i	$-0.206077\pi$
$683$	28475.6	1.59530	0.797649	+	0.603122i	$0.206077\pi$
$684$	0	0
$685$	−9908.90	−0.552700
$686$	12809.1	0.712905
$687$	0	0
$688$	−14861.3	−0.823522
$689$	0	0
$690$	0	0
$691$	−10610.8	−0.584160	−0.292080	−	0.956394i	$-0.594347\pi$
$691$	−10610.8	−0.584160	−0.292080	+	0.956394i	$0.594347\pi$
$692$	6555.27	0.360107
$693$	0	0
$694$	−18410.3	−1.00698
$695$	243.168	0.0132718
$696$	0	0
$697$	23424.6	1.27299
$698$	−317.736	−0.0172299
$699$	0	0
$700$	5983.29	0.323067
$701$	13518.9	0.728390	0.364195	−	0.931323i	$-0.381344\pi$
$701$	13518.9	0.728390	0.364195	+	0.931323i	$0.381344\pi$
$702$	0	0
$703$	9275.49	0.497627
$704$	−349.179	−0.0186934
$705$	0	0
$706$	−7023.19	−0.374393
$707$	−54064.9	−2.87599
$708$	0	0
$709$	−14845.4	−0.786361	−0.393180	−	0.919461i	$-0.628625\pi$
$709$	−14845.4	−0.786361	−0.393180	+	0.919461i	$0.628625\pi$
$710$	−10030.4	−0.530190
$711$	0	0
$712$	−34517.7	−1.81686
$713$	−12989.4	−0.682267
$714$	0	0
$715$	0	0
$716$	−10662.2	−0.556517
$717$	0	0
$718$	20480.3	1.06451
$719$	35900.1	1.86210	0.931049	−	0.364893i	$-0.118894\pi$
$719$	35900.1	1.86210	0.931049	+	0.364893i	$0.118894\pi$
$720$	0	0
$721$	42677.2	2.20441
$722$	−6.73962	−0.000347400 0
$723$	0	0
$724$	4054.50	0.208128
$725$	18956.6	0.971078
$726$	0	0
$727$	12951.4	0.660715	0.330357	−	0.943856i	$-0.392831\pi$
$727$	12951.4	0.660715	0.330357	+	0.943856i	$0.392831\pi$
$728$	0	0
$729$	0	0
$730$	−2538.14	−0.128686
$731$	−33639.7	−1.70206
$732$	0	0
$733$	1105.36	0.0556989	0.0278494	−	0.999612i	$-0.491134\pi$
$733$	1105.36	0.0556989	0.0278494	+	0.999612i	$0.491134\pi$
$734$	10661.8	0.536148
$735$	0	0
$736$	7882.27	0.394761
$737$	75.9009	0.00379355
$738$	0	0
$739$	−13638.1	−0.678869	−0.339434	−	0.940630i	$-0.610236\pi$
$739$	−13638.1	−0.678869	−0.339434	+	0.940630i	$0.610236\pi$
$740$	1742.45	0.0865591
$741$	0	0
$742$	10194.7	0.504391
$743$	10154.9	0.501408	0.250704	−	0.968064i	$-0.419338\pi$
$743$	10154.9	0.501408	0.250704	+	0.968064i	$0.419338\pi$
$744$	0	0
$745$	−11703.6	−0.575552
$746$	15518.2	0.761611
$747$	0	0
$748$	−132.744	−0.00648876
$749$	10891.7	0.531338
$750$	0	0
$751$	26579.5	1.29148	0.645738	−	0.763559i	$-0.276550\pi$
$751$	26579.5	1.29148	0.645738	+	0.763559i	$0.276550\pi$
$752$	−13075.8	−0.634076
$753$	0	0
$754$	0	0
$755$	20803.5	1.00280
$756$	0	0
$757$	−13677.7	−0.656705	−0.328352	−	0.944555i	$-0.606493\pi$
$757$	−13677.7	−0.656705	−0.328352	+	0.944555i	$0.606493\pi$
$758$	12967.0	0.621348
$759$	0	0
$760$	13089.0	0.624720
$761$	17996.7	0.857268	0.428634	−	0.903478i	$-0.358995\pi$
$761$	17996.7	0.857268	0.428634	+	0.903478i	$0.358995\pi$
$762$	0	0
$763$	24057.9	1.14149
$764$	−702.687	−0.0332753
$765$	0	0
$766$	24059.9	1.13488
$767$	0	0
$768$	0	0
$769$	−2995.04	−0.140447	−0.0702236	−	0.997531i	$-0.522371\pi$
$769$	−2995.04	−0.140447	−0.0702236	+	0.997531i	$0.522371\pi$
$770$	−279.464	−0.0130794
$771$	0	0
$772$	2517.75	0.117378
$773$	−26055.4	−1.21235	−0.606177	−	0.795330i	$-0.707298\pi$
$773$	−26055.4	−1.21235	−0.606177	+	0.795330i	$0.707298\pi$
$774$	0	0
$775$	14545.3	0.674169
$776$	46838.6	2.16676
$777$	0	0
$778$	−11534.8	−0.531545
$779$	22109.0	1.01686
$780$	0	0
$781$	413.133	0.0189284
$782$	−15497.7	−0.708693
$783$	0	0
$784$	−20420.5	−0.930235
$785$	5333.95	0.242518
$786$	0	0
$787$	22992.0	1.04139	0.520697	−	0.853741i	$-0.325672\pi$
$787$	22992.0	1.04139	0.520697	+	0.853741i	$0.325672\pi$
$788$	−3436.21	−0.155343
$789$	0	0
$790$	−1537.96	−0.0692635
$791$	−52810.6	−2.37387
$792$	0	0
$793$	0	0
$794$	4995.22	0.223267
$795$	0	0
$796$	5784.86	0.257587
$797$	25826.3	1.14782	0.573910	−	0.818918i	$-0.305426\pi$
$797$	25826.3	1.14782	0.573910	+	0.818918i	$0.305426\pi$
$798$	0	0
$799$	−29598.0	−1.31052
$800$	−8826.40	−0.390075
$801$	0	0
$802$	−1592.43	−0.0701130
$803$	104.541	0.00459423
$804$	0	0
$805$	14167.6	0.620299
$806$	0	0
$807$	0	0
$808$	45120.6	1.96453
$809$	28495.8	1.23839	0.619195	−	0.785237i	$-0.287459\pi$
$809$	28495.8	1.23839	0.619195	+	0.785237i	$0.287459\pi$
$810$	0	0
$811$	−6992.41	−0.302758	−0.151379	−	0.988476i	$-0.548371\pi$
$811$	−6992.41	−0.302758	−0.151379	+	0.988476i	$0.548371\pi$
$812$	16168.1	0.698757
$813$	0	0
$814$	165.278	0.00711670
$815$	13357.3	0.574093
$816$	0	0
$817$	−31750.4	−1.35961
$818$	−11037.3	−0.471773
$819$	0	0
$820$	4153.29	0.176877
$821$	31512.7	1.33959	0.669793	−	0.742547i	$-0.266383\pi$
$821$	31512.7	1.33959	0.669793	+	0.742547i	$0.266383\pi$
$822$	0	0
$823$	39159.6	1.65859	0.829293	−	0.558814i	$-0.188743\pi$
$823$	39159.6	1.65859	0.829293	+	0.558814i	$0.188743\pi$
$824$	−35616.8	−1.50579
$825$	0	0
$826$	−36860.8	−1.55273
$827$	−36557.6	−1.53716	−0.768581	−	0.639752i	$-0.779037\pi$
$827$	−36557.6	−1.53716	−0.768581	+	0.639752i	$0.779037\pi$
$828$	0	0
$829$	14675.2	0.614825	0.307413	−	0.951576i	$-0.400537\pi$
$829$	14675.2	0.614825	0.307413	+	0.951576i	$0.400537\pi$
$830$	−7689.66	−0.321581
$831$	0	0
$832$	0	0
$833$	−46223.3	−1.92262
$834$	0	0
$835$	552.164	0.0228843
$836$	−125.288	−0.00518323
$837$	0	0
$838$	606.626	0.0250066
$839$	−8515.56	−0.350405	−0.175202	−	0.984532i	$-0.556058\pi$
$839$	−8515.56	−0.350405	−0.175202	+	0.984532i	$0.556058\pi$
$840$	0	0
$841$	26835.9	1.10033
$842$	20657.9	0.845509
$843$	0	0
$844$	−10515.3	−0.428854
$845$	0	0
$846$	0	0
$847$	−39244.8	−1.59205
$848$	−5672.24	−0.229700
$849$	0	0
$850$	17354.0	0.700281
$851$	−8378.86	−0.337513
$852$	0	0
$853$	−9645.93	−0.387187	−0.193593	−	0.981082i	$-0.562014\pi$
$853$	−9645.93	−0.387187	−0.193593	+	0.981082i	$0.562014\pi$
$854$	14159.4	0.567359
$855$	0	0
$856$	−9089.77	−0.362946
$857$	−36139.6	−1.44050	−0.720248	−	0.693717i	$-0.755972\pi$
$857$	−36139.6	−1.44050	−0.720248	+	0.693717i	$0.755972\pi$
$858$	0	0
$859$	−7108.04	−0.282332	−0.141166	−	0.989986i	$-0.545085\pi$
$859$	−7108.04	−0.282332	−0.141166	+	0.989986i	$0.545085\pi$
$860$	−5964.47	−0.236496
$861$	0	0
$862$	20695.5	0.817738
$863$	16225.8	0.640016	0.320008	−	0.947415i	$-0.396314\pi$
$863$	16225.8	0.640016	0.320008	+	0.947415i	$0.396314\pi$
$864$	0	0
$865$	−17381.1	−0.683210
$866$	15175.5	0.595478
$867$	0	0
$868$	12405.7	0.485110
$869$	63.3455	0.00247278
$870$	0	0
$871$	0	0
$872$	−20077.9	−0.779727
$873$	0	0
$874$	−14627.3	−0.566106
$875$	−39541.0	−1.52769
$876$	0	0
$877$	30983.0	1.19295	0.596477	−	0.802630i	$-0.296567\pi$
$877$	30983.0	1.19295	0.596477	+	0.802630i	$0.296567\pi$
$878$	16107.9	0.619151
$879$	0	0
$880$	155.492	0.00595639
$881$	7670.76	0.293342	0.146671	−	0.989185i	$-0.453144\pi$
$881$	7670.76	0.293342	0.146671	+	0.989185i	$0.453144\pi$
$882$	0	0
$883$	−34340.6	−1.30878	−0.654390	−	0.756157i	$-0.727075\pi$
$883$	−34340.6	−1.30878	−0.654390	+	0.756157i	$0.727075\pi$
$884$	0	0
$885$	0	0
$886$	−11956.4	−0.453366
$887$	19208.3	0.727118	0.363559	−	0.931571i	$-0.381562\pi$
$887$	19208.3	0.727118	0.363559	+	0.931571i	$0.381562\pi$
$888$	0	0
$889$	−1335.78	−0.0503943
$890$	21269.8	0.801083
$891$	0	0
$892$	11178.8	0.419611
$893$	−27935.7	−1.04684
$894$	0	0
$895$	28270.6	1.05585
$896$	14069.6	0.524591
$897$	0	0
$898$	−15566.0	−0.578444
$899$	39304.4	1.45815
$900$	0	0
$901$	−12839.5	−0.474747
$902$	393.956	0.0145425
$903$	0	0
$904$	44073.8	1.62154
$905$	−10750.4	−0.394868
$906$	0	0
$907$	−46481.0	−1.70163	−0.850813	−	0.525468i	$-0.823890\pi$
$907$	−46481.0	−1.70163	−0.850813	+	0.525468i	$0.823890\pi$
$908$	5240.01	0.191515
$909$	0	0
$910$	0	0
$911$	34109.3	1.24050	0.620248	−	0.784406i	$-0.287032\pi$
$911$	34109.3	1.24050	0.620248	+	0.784406i	$0.287032\pi$
$912$	0	0
$913$	316.721	0.0114808
$914$	5325.18	0.192715
$915$	0	0
$916$	4503.80	0.162456
$917$	−31022.2	−1.11717
$918$	0	0
$919$	−37533.6	−1.34725	−0.673623	−	0.739075i	$-0.735263\pi$
$919$	−37533.6	−1.34725	−0.673623	+	0.739075i	$0.735263\pi$
$920$	−11823.7	−0.423714
$921$	0	0
$922$	17379.7	0.620793
$923$	0	0
$924$	0	0
$925$	9382.48	0.333507
$926$	−27392.5	−0.972111
$927$	0	0
$928$	−23850.8	−0.843687
$929$	4966.75	0.175408	0.0877038	−	0.996147i	$-0.472047\pi$
$929$	4966.75	0.175408	0.0877038	+	0.996147i	$0.472047\pi$
$930$	0	0
$931$	−43627.2	−1.53579
$932$	6943.79	0.244047
$933$	0	0
$934$	713.606	0.0249999
$935$	351.966	0.0123107
$936$	0	0
$937$	−5096.90	−0.177704	−0.0888519	−	0.996045i	$-0.528320\pi$
$937$	−5096.90	−0.177704	−0.0888519	+	0.996045i	$0.528320\pi$
$938$	−8463.15	−0.294597
$939$	0	0
$940$	−5247.86	−0.182092
$941$	54774.8	1.89756	0.948781	−	0.315933i	$-0.102318\pi$
$941$	54774.8	1.89756	0.948781	+	0.315933i	$0.102318\pi$
$942$	0	0
$943$	−19971.8	−0.689684
$944$	20509.1	0.707113
$945$	0	0
$946$	−565.753	−0.0194442
$947$	−13768.5	−0.472456	−0.236228	−	0.971698i	$-0.575911\pi$
$947$	−13768.5	−0.472456	−0.236228	+	0.971698i	$0.575911\pi$
$948$	0	0
$949$	0	0
$950$	16379.4	0.559386
$951$	0	0
$952$	63689.2	2.16825
$953$	−35866.6	−1.21913	−0.609566	−	0.792735i	$-0.708656\pi$
$953$	−35866.6	−1.21913	−0.609566	+	0.792735i	$0.708656\pi$
$954$	0	0
$955$	1863.16	0.0631312
$956$	−4586.33	−0.155160
$957$	0	0
$958$	−7430.46	−0.250592
$959$	−45507.3	−1.53233
$960$	0	0
$961$	366.899	0.0123158
$962$	0	0
$963$	0	0
$964$	−4392.62	−0.146760
$965$	−6675.75	−0.222694
$966$	0	0
$967$	24476.5	0.813972	0.406986	−	0.913434i	$-0.366580\pi$
$967$	24476.5	0.813972	0.406986	+	0.913434i	$0.366580\pi$
$968$	32752.3	1.08750
$969$	0	0
$970$	−28861.9	−0.955362
$971$	8976.07	0.296659	0.148329	−	0.988938i	$-0.452610\pi$
$971$	8976.07	0.296659	0.148329	+	0.988938i	$0.452610\pi$
$972$	0	0
$973$	1116.77	0.0367954
$974$	−7246.26	−0.238383
$975$	0	0
$976$	−7878.18	−0.258376
$977$	−42002.8	−1.37542	−0.687711	−	0.725984i	$-0.741385\pi$
$977$	−42002.8	−1.37542	−0.687711	+	0.725984i	$0.741385\pi$
$978$	0	0
$979$	−876.058	−0.0285995
$980$	−8195.60	−0.267142
$981$	0	0
$982$	237.495	0.00771767
$983$	43240.7	1.40301	0.701507	−	0.712662i	$-0.252511\pi$
$983$	43240.7	1.40301	0.701507	+	0.712662i	$0.252511\pi$
$984$	0	0
$985$	9111.03	0.294722
$986$	46894.3	1.51462
$987$	0	0
$988$	0	0
$989$	28681.2	0.922152
$990$	0	0
$991$	37916.5	1.21540	0.607698	−	0.794168i	$-0.292093\pi$
$991$	37916.5	1.21540	0.607698	+	0.794168i	$0.292093\pi$
$992$	−18300.5	−0.585728
$993$	0	0
$994$	−46065.4	−1.46993
$995$	−15338.4	−0.488704
$996$	0	0
$997$	6634.86	0.210760	0.105380	−	0.994432i	$-0.466394\pi$
$997$	6634.86	0.210760	0.105380	+	0.994432i	$0.466394\pi$
$998$	−8541.43	−0.270916
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bb.1.3		4
3.2	odd	2		507.4.a.i.1.2		4
13.4	even	6		117.4.g.e.55.3		8
13.10	even	6		117.4.g.e.100.3		8
13.12	even	2		1521.4.a.v.1.2		4
39.5	even	4		507.4.b.h.337.6		8
39.8	even	4		507.4.b.h.337.3		8
39.17	odd	6		39.4.e.c.16.2	✓	8
39.23	odd	6		39.4.e.c.22.2	yes	8
39.38	odd	2		507.4.a.m.1.3		4
156.23	even	6		624.4.q.i.529.3		8
156.95	even	6		624.4.q.i.289.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.c.16.2	✓	8	39.17	odd	6
39.4.e.c.22.2	yes	8	39.23	odd	6
117.4.g.e.55.3		8	13.4	even	6
117.4.g.e.100.3		8	13.10	even	6
507.4.a.i.1.2		4	3.2	odd	2
507.4.a.m.1.3		4	39.38	odd	2
507.4.b.h.337.3		8	39.8	even	4
507.4.b.h.337.6		8	39.5	even	4
624.4.q.i.289.3		8	156.95	even	6
624.4.q.i.529.3		8	156.23	even	6
1521.4.a.v.1.2		4	13.12	even	2
1521.4.a.bb.1.3		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+2.36176 q^{2} -2.42208 q^{4} +6.42208 q^{5} +29.4938 q^{7} -24.6145 q^{8} +O(q^{10})\)	\(q+2.36176 q^{2} -2.42208 q^{4} +6.42208 q^{5} +29.4938 q^{7} -24.6145 q^{8} +15.1674 q^{10} -0.624715 q^{11} +69.6575 q^{14} -38.7569 q^{16} -87.7291 q^{17} -82.8018 q^{19} -15.5548 q^{20} -1.47543 q^{22} +74.7977 q^{23} -83.7569 q^{25} -71.4364 q^{28} -226.329 q^{29} -173.660 q^{31} +105.381 q^{32} -207.195 q^{34} +189.412 q^{35} -112.020 q^{37} -195.558 q^{38} -158.076 q^{40} -267.011 q^{41} +383.450 q^{43} +1.51311 q^{44} +176.654 q^{46} +337.380 q^{47} +526.887 q^{49} -197.814 q^{50} +146.354 q^{53} -4.01197 q^{55} -725.975 q^{56} -534.536 q^{58} -529.173 q^{59} +203.272 q^{61} -410.144 q^{62} +558.941 q^{64} -121.497 q^{67} +212.487 q^{68} +447.346 q^{70} -661.314 q^{71} -167.341 q^{73} -264.565 q^{74} +200.552 q^{76} -18.4253 q^{77} -101.399 q^{79} -248.900 q^{80} -630.617 q^{82} -506.985 q^{83} -563.403 q^{85} +905.617 q^{86} +15.3770 q^{88} +1402.33 q^{89} -181.166 q^{92} +796.810 q^{94} -531.760 q^{95} -1902.89 q^{97} +1244.38 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 22 q^{4} - 6 q^{5} + 14 q^{7} + 54 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 22 * q^4 - 6 * q^5 + 14 * q^7 + 54 * q^8	\( 4 q + 2 q^{2} + 22 q^{4} - 6 q^{5} + 14 q^{7} + 54 q^{8} - 62 q^{10} + 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} - 124 q^{19} - 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} + 144 q^{28} - 194 q^{29} - 26 q^{31} + 654 q^{32} - 1062 q^{34} - 88 q^{35} - 102 q^{37} - 332 q^{38} - 998 q^{40} - 1054 q^{41} + 450 q^{43} - 44 q^{44} + 172 q^{46} - 96 q^{47} + 1070 q^{49} + 996 q^{50} - 262 q^{53} + 204 q^{55} - 2164 q^{56} - 722 q^{58} + 308 q^{59} - 928 q^{61} - 2780 q^{62} + 1026 q^{64} + 1134 q^{67} - 1786 q^{68} + 2324 q^{70} + 1064 q^{71} - 952 q^{73} - 1158 q^{74} + 1708 q^{76} - 2508 q^{77} - 746 q^{79} - 2922 q^{80} + 1734 q^{82} - 404 q^{83} + 1394 q^{85} + 3168 q^{86} + 3060 q^{88} + 1620 q^{89} - 332 q^{92} - 772 q^{94} - 2204 q^{95} - 2166 q^{97} - 1906 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 22 * q^4 - 6 * q^5 + 14 * q^7 + 54 * q^8 - 62 * q^10 + 40 * q^11 - 40 * q^14 + 122 * q^16 - 98 * q^17 - 124 * q^19 - 466 * q^20 + 220 * q^22 - 104 * q^23 - 58 * q^25 + 144 * q^28 - 194 * q^29 - 26 * q^31 + 654 * q^32 - 1062 * q^34 - 88 * q^35 - 102 * q^37 - 332 * q^38 - 998 * q^40 - 1054 * q^41 + 450 * q^43 - 44 * q^44 + 172 * q^46 - 96 * q^47 + 1070 * q^49 + 996 * q^50 - 262 * q^53 + 204 * q^55 - 2164 * q^56 - 722 * q^58 + 308 * q^59 - 928 * q^61 - 2780 * q^62 + 1026 * q^64 + 1134 * q^67 - 1786 * q^68 + 2324 * q^70 + 1064 * q^71 - 952 * q^73 - 1158 * q^74 + 1708 * q^76 - 2508 * q^77 - 746 * q^79 - 2922 * q^80 + 1734 * q^82 - 404 * q^83 + 1394 * q^85 + 3168 * q^86 + 3060 * q^88 + 1620 * q^89 - 332 * q^92 - 772 * q^94 - 2204 * q^95 - 2166 * q^97 - 1906 * q^98

Introduction

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