LMFDB - Embedded newform 1521.4.a.bi.1.8

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 $ x^9 - 56x^7 - 27x^6 + 945x^5 + 763x^4 - 4139x^3 - 2478x^2 + 63*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-4.14324$ 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)

$f(q)$	$=$	$q+4.69820 q^{2} +14.0731 q^{4} +4.47249 q^{5} -27.2096 q^{7} +28.5326 q^{8} +O(q^{10})$	$q+4.69820 q^{2} +14.0731 q^{4} +4.47249 q^{5} -27.2096 q^{7} +28.5326 q^{8} +21.0127 q^{10} -5.99207 q^{11} -127.836 q^{14} +21.4672 q^{16} -105.037 q^{17} +156.462 q^{19} +62.9418 q^{20} -28.1519 q^{22} +175.423 q^{23} -104.997 q^{25} -382.923 q^{28} -204.886 q^{29} -31.9570 q^{31} -127.404 q^{32} -493.483 q^{34} -121.695 q^{35} -344.140 q^{37} +735.088 q^{38} +127.612 q^{40} +46.5921 q^{41} -173.286 q^{43} -84.3269 q^{44} +824.175 q^{46} -265.613 q^{47} +397.361 q^{49} -493.296 q^{50} -172.912 q^{53} -26.7995 q^{55} -776.360 q^{56} -962.596 q^{58} -137.566 q^{59} -58.9384 q^{61} -150.140 q^{62} -770.306 q^{64} -211.668 q^{67} -1478.19 q^{68} -571.746 q^{70} -436.317 q^{71} -1159.11 q^{73} -1616.84 q^{74} +2201.90 q^{76} +163.042 q^{77} -1017.51 q^{79} +96.0118 q^{80} +218.899 q^{82} -150.251 q^{83} -469.775 q^{85} -814.131 q^{86} -170.969 q^{88} +565.984 q^{89} +2468.75 q^{92} -1247.90 q^{94} +699.773 q^{95} +286.741 q^{97} +1866.88 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8}+O(q^{10}) $ 9 * q + 6 * q^2 + 44 * q^4 + 33 * q^5 - 83 * q^7 + 87 * q^8	$ 9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8} - 54 q^{10} + 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} - 352 q^{19} + 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} - 940 q^{28} + 97 q^{29} - 717 q^{31} + 707 q^{32} - 632 q^{34} + 418 q^{35} - 1108 q^{37} + 660 q^{38} - 1506 q^{40} + 334 q^{41} + 242 q^{43} - 307 q^{44} - 979 q^{46} - 184 q^{47} - 38 q^{49} - 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} - 1161 q^{58} + 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} - 2308 q^{67} - 2785 q^{68} + 1420 q^{70} + 96 q^{71} - 2505 q^{73} + 1191 q^{74} - 2409 q^{76} + 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 1517 q^{82} + 1539 q^{83} - 4296 q^{85} - 3763 q^{86} - 3716 q^{88} - 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} - 1445 q^{97} + 1457 q^{98}+O(q^{100}) $ 9 * q + 6 * q^2 + 44 * q^4 + 33 * q^5 - 83 * q^7 + 87 * q^8 - 54 * q^10 + 85 * q^11 - 158 * q^14 + 216 * q^16 - 178 * q^17 - 352 * q^19 + 402 * q^20 - 630 * q^22 - 150 * q^23 - 20 * q^25 - 940 * q^28 + 97 * q^29 - 717 * q^31 + 707 * q^32 - 632 * q^34 + 418 * q^35 - 1108 * q^37 + 660 * q^38 - 1506 * q^40 + 334 * q^41 + 242 * q^43 - 307 * q^44 - 979 * q^46 - 184 * q^47 - 38 * q^49 - 2031 * q^50 + 151 * q^53 + 2064 * q^55 - 2276 * q^56 - 1161 * q^58 + 537 * q^59 - 1340 * q^61 - 347 * q^62 + 893 * q^64 - 2308 * q^67 - 2785 * q^68 + 1420 * q^70 + 96 * q^71 - 2505 * q^73 + 1191 * q^74 - 2409 * q^76 + 2142 * q^77 - 1591 * q^79 - 2671 * q^80 + 1517 * q^82 + 1539 * q^83 - 4296 * q^85 - 3763 * q^86 - 3716 * q^88 - 592 * q^89 - 515 * q^92 - 692 * q^94 - 4158 * q^95 - 1445 * q^97 + 1457 * 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 $

$2$	4.69820	1.66106	0.830532	−	0.556970i	$-0.188036\pi$
$2$	4.69820	1.66106	0.830532	+	0.556970i	$0.188036\pi$
$3$	0	0
$3$	0	0
$4$	14.0731	1.75914
$5$	4.47249	0.400032	0.200016	−	0.979793i	$-0.435901\pi$
$5$	4.47249	0.400032	0.200016	+	0.979793i	$0.435901\pi$
$6$	0	0
$7$	−27.2096	−1.46918	−0.734589	−	0.678512i	$-0.762625\pi$
$7$	−27.2096	−1.46918	−0.734589	+	0.678512i	$0.762625\pi$
$8$	28.5326	1.26098
$9$	0	0
$10$	21.0127	0.664479
$11$	−5.99207	−0.164243	−0.0821216	−	0.996622i	$-0.526170\pi$
$11$	−5.99207	−0.164243	−0.0821216	+	0.996622i	$0.526170\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−127.836	−2.44040
$15$	0	0
$16$	21.4672	0.335425
$17$	−105.037	−1.49854	−0.749268	−	0.662267i	$-0.769594\pi$
$17$	−105.037	−1.49854	−0.749268	+	0.662267i	$0.769594\pi$
$18$	0	0
$19$	156.462	1.88920	0.944599	−	0.328227i	$-0.106451\pi$
$19$	156.462	1.88920	0.944599	+	0.328227i	$0.106451\pi$
$20$	62.9418	0.703711
$21$	0	0
$22$	−28.1519	−0.272819
$23$	175.423	1.59036	0.795181	−	0.606372i	$-0.207376\pi$
$23$	175.423	1.59036	0.795181	+	0.606372i	$0.207376\pi$
$24$	0	0
$25$	−104.997	−0.839975
$26$	0	0
$27$	0	0
$28$	−382.923	−2.58449
$29$	−204.886	−1.31195	−0.655973	−	0.754785i	$-0.727741\pi$
$29$	−204.886	−1.31195	−0.655973	+	0.754785i	$0.727741\pi$
$30$	0	0
$31$	−31.9570	−0.185150	−0.0925748	−	0.995706i	$-0.529510\pi$
$31$	−31.9570	−0.185150	−0.0925748	+	0.995706i	$0.529510\pi$
$32$	−127.404	−0.703813
$33$	0	0
$34$	−493.483	−2.48916
$35$	−121.695	−0.587718
$36$	0	0
$37$	−344.140	−1.52909	−0.764545	−	0.644571i	$-0.777036\pi$
$37$	−344.140	−1.52909	−0.764545	+	0.644571i	$0.777036\pi$
$38$	735.088	3.13808
$39$	0	0
$40$	127.612	0.504430
$41$	46.5921	0.177475	0.0887373	−	0.996055i	$-0.471717\pi$
$41$	46.5921	0.177475	0.0887373	+	0.996055i	$0.471717\pi$
$42$	0	0
$43$	−173.286	−0.614554	−0.307277	−	0.951620i	$-0.599418\pi$
$43$	−173.286	−0.614554	−0.307277	+	0.951620i	$0.599418\pi$
$44$	−84.3269	−0.288926
$45$	0	0
$46$	824.175	2.64169
$47$	−265.613	−0.824334	−0.412167	−	0.911108i	$-0.635228\pi$
$47$	−265.613	−0.824334	−0.412167	+	0.911108i	$0.635228\pi$
$48$	0	0
$49$	397.361	1.15849
$50$	−493.296	−1.39525
$51$	0	0
$52$	0	0
$53$	−172.912	−0.448137	−0.224068	−	0.974573i	$-0.571934\pi$
$53$	−172.912	−0.448137	−0.224068	+	0.974573i	$0.571934\pi$
$54$	0	0
$55$	−26.7995	−0.0657025
$56$	−776.360	−1.85260
$57$	0	0
$58$	−962.596	−2.17923
$59$	−137.566	−0.303551	−0.151775	−	0.988415i	$-0.548499\pi$
$59$	−137.566	−0.303551	−0.151775	+	0.988415i	$0.548499\pi$
$60$	0	0
$61$	−58.9384	−0.123710	−0.0618548	−	0.998085i	$-0.519702\pi$
$61$	−58.9384	−0.123710	−0.0618548	+	0.998085i	$0.519702\pi$
$62$	−150.140	−0.307546
$63$	0	0
$64$	−770.306	−1.50450
$65$	0	0
$66$	0	0
$67$	−211.668	−0.385961	−0.192980	−	0.981203i	$-0.561815\pi$
$67$	−211.668	−0.385961	−0.192980	+	0.981203i	$0.561815\pi$
$68$	−1478.19	−2.63613
$69$	0	0
$70$	−571.746	−0.976238
$71$	−436.317	−0.729314	−0.364657	−	0.931142i	$-0.618814\pi$
$71$	−436.317	−0.729314	−0.364657	+	0.931142i	$0.618814\pi$
$72$	0	0
$73$	−1159.11	−1.85840	−0.929199	−	0.369579i	$-0.879502\pi$
$73$	−1159.11	−1.85840	−0.929199	+	0.369579i	$0.879502\pi$
$74$	−1616.84	−2.53992
$75$	0	0
$76$	2201.90	3.32336
$77$	163.042	0.241303
$78$	0	0
$79$	−1017.51	−1.44910	−0.724548	−	0.689224i	$-0.757952\pi$
$79$	−1017.51	−1.44910	−0.724548	+	0.689224i	$0.757952\pi$
$80$	96.0118	0.134181
$81$	0	0
$82$	218.899	0.294797
$83$	−150.251	−0.198701	−0.0993504	−	0.995053i	$-0.531676\pi$
$83$	−150.251	−0.198701	−0.0993504	+	0.995053i	$0.531676\pi$
$84$	0	0
$85$	−469.775	−0.599462
$86$	−814.131	−1.02081
$87$	0	0
$88$	−170.969	−0.207107
$89$	565.984	0.674092	0.337046	−	0.941488i	$-0.390572\pi$
$89$	565.984	0.674092	0.337046	+	0.941488i	$0.390572\pi$
$90$	0	0
$91$	0	0
$92$	2468.75	2.79766
$93$	0	0
$94$	−1247.90	−1.36927
$95$	699.773	0.755739
$96$	0	0
$97$	286.741	0.300145	0.150073	−	0.988675i	$-0.452049\pi$
$97$	286.741	0.300145	0.150073	+	0.988675i	$0.452049\pi$
$98$	1866.88	1.92432
$99$	0	0
$100$	−1477.63	−1.47763
$101$	1218.57	1.20052	0.600258	−	0.799807i	$-0.295065\pi$
$101$	1218.57	1.20052	0.600258	+	0.799807i	$0.295065\pi$
$102$	0	0
$103$	74.2485	0.0710283	0.0355142	−	0.999369i	$-0.488693\pi$
$103$	74.2485	0.0710283	0.0355142	+	0.999369i	$0.488693\pi$
$104$	0	0
$105$	0	0
$106$	−812.374	−0.744384
$107$	1253.46	1.13249	0.566246	−	0.824237i	$-0.308395\pi$
$107$	1253.46	1.13249	0.566246	+	0.824237i	$0.308395\pi$
$108$	0	0
$109$	−722.643	−0.635015	−0.317507	−	0.948256i	$-0.602846\pi$
$109$	−722.643	−0.635015	−0.317507	+	0.948256i	$0.602846\pi$
$110$	−125.909	−0.109136
$111$	0	0
$112$	−584.113	−0.492799
$113$	855.913	0.712544	0.356272	−	0.934382i	$-0.384048\pi$
$113$	855.913	0.712544	0.356272	+	0.934382i	$0.384048\pi$
$114$	0	0
$115$	784.580	0.636195
$116$	−2883.38	−2.30789
$117$	0	0
$118$	−646.310	−0.504218
$119$	2858.00	2.20162
$120$	0	0
$121$	−1295.10	−0.973024
$122$	−276.905	−0.205490
$123$	0	0
$124$	−449.733	−0.325703
$125$	−1028.66	−0.736048
$126$	0	0
$127$	726.104	0.507333	0.253667	−	0.967292i	$-0.418363\pi$
$127$	726.104	0.507333	0.253667	+	0.967292i	$0.418363\pi$
$128$	−2599.82	−1.79526
$129$	0	0
$130$	0	0
$131$	1456.91	0.971685	0.485843	−	0.874046i	$-0.338513\pi$
$131$	1456.91	0.971685	0.485843	+	0.874046i	$0.338513\pi$
$132$	0	0
$133$	−4257.25	−2.77557
$134$	−994.460	−0.641106
$135$	0	0
$136$	−2996.97	−1.88962
$137$	1806.80	1.12675	0.563377	−	0.826200i	$-0.309502\pi$
$137$	1806.80	1.12675	0.563377	+	0.826200i	$0.309502\pi$
$138$	0	0
$139$	1229.28	0.750117	0.375059	−	0.927001i	$-0.377623\pi$
$139$	1229.28	0.750117	0.375059	+	0.927001i	$0.377623\pi$
$140$	−1712.62	−1.03388
$141$	0	0
$142$	−2049.90	−1.21144
$143$	0	0
$144$	0	0
$145$	−916.352	−0.524820
$146$	−5445.71	−3.08692
$147$	0	0
$148$	−4843.12	−2.68988
$149$	446.127	0.245290	0.122645	−	0.992451i	$-0.460862\pi$
$149$	446.127	0.245290	0.122645	+	0.992451i	$0.460862\pi$
$150$	0	0
$151$	−207.208	−0.111671	−0.0558355	−	0.998440i	$-0.517782\pi$
$151$	−207.208	−0.111671	−0.0558355	+	0.998440i	$0.517782\pi$
$152$	4464.26	2.38223
$153$	0	0
$154$	766.002	0.400819
$155$	−142.927	−0.0740658
$156$	0	0
$157$	1096.60	0.557442	0.278721	−	0.960372i	$-0.410089\pi$
$157$	1096.60	0.557442	0.278721	+	0.960372i	$0.410089\pi$
$158$	−4780.46	−2.40704
$159$	0	0
$160$	−569.812	−0.281547
$161$	−4773.20	−2.33653
$162$	0	0
$163$	3154.25	1.51571	0.757853	−	0.652425i	$-0.226248\pi$
$163$	3154.25	1.51571	0.757853	+	0.652425i	$0.226248\pi$
$164$	655.694	0.312202
$165$	0	0
$166$	−705.908	−0.330055
$167$	3679.65	1.70503	0.852514	−	0.522705i	$-0.175077\pi$
$167$	3679.65	1.70503	0.852514	+	0.522705i	$0.175077\pi$
$168$	0	0
$169$	0	0
$170$	−2207.10	−0.995745
$171$	0	0
$172$	−2438.66	−1.08108
$173$	3666.99	1.61154	0.805768	−	0.592231i	$-0.201753\pi$
$173$	3666.99	1.61154	0.805768	+	0.592231i	$0.201753\pi$
$174$	0	0
$175$	2856.92	1.23407
$176$	−128.633	−0.0550913
$177$	0	0
$178$	2659.11	1.11971
$179$	−4173.50	−1.74269	−0.871347	−	0.490666i	$-0.836753\pi$
$179$	−4173.50	−1.74269	−0.871347	+	0.490666i	$0.836753\pi$
$180$	0	0
$181$	−499.197	−0.205000	−0.102500	−	0.994733i	$-0.532684\pi$
$181$	−499.197	−0.205000	−0.102500	+	0.994733i	$0.532684\pi$
$182$	0	0
$183$	0	0
$184$	5005.29	2.00541
$185$	−1539.16	−0.611685
$186$	0	0
$187$	629.386	0.246124
$188$	−3738.00	−1.45012
$189$	0	0
$190$	3287.68	1.25533
$191$	3086.68	1.16934	0.584671	−	0.811271i	$-0.301224\pi$
$191$	3086.68	1.16934	0.584671	+	0.811271i	$0.301224\pi$
$192$	0	0
$193$	−1644.14	−0.613201	−0.306601	−	0.951838i	$-0.599192\pi$
$193$	−1644.14	−0.613201	−0.306601	+	0.951838i	$0.599192\pi$
$194$	1347.17	0.498561
$195$	0	0
$196$	5592.10	2.03794
$197$	−1371.21	−0.495911	−0.247956	−	0.968771i	$-0.579759\pi$
$197$	−1371.21	−0.495911	−0.247956	+	0.968771i	$0.579759\pi$
$198$	0	0
$199$	−4627.33	−1.64836	−0.824178	−	0.566332i	$-0.808362\pi$
$199$	−4627.33	−1.64836	−0.824178	+	0.566332i	$0.808362\pi$
$200$	−2995.83	−1.05919
$201$	0	0
$202$	5725.08	1.99413
$203$	5574.87	1.92748
$204$	0	0
$205$	208.383	0.0709955
$206$	348.834	0.117983
$207$	0	0
$208$	0	0
$209$	−937.528	−0.310288
$210$	0	0
$211$	5088.31	1.66016	0.830080	−	0.557644i	$-0.188294\pi$
$211$	5088.31	1.66016	0.830080	+	0.557644i	$0.188294\pi$
$212$	−2433.40	−0.788334
$213$	0	0
$214$	5889.01	1.88114
$215$	−775.019	−0.245841
$216$	0	0
$217$	869.535	0.272018
$218$	−3395.12	−1.05480
$219$	0	0
$220$	−377.151	−0.115580
$221$	0	0
$222$	0	0
$223$	−4744.82	−1.42483	−0.712415	−	0.701759i	$-0.752398\pi$
$223$	−4744.82	−1.42483	−0.712415	+	0.701759i	$0.752398\pi$
$224$	3466.60	1.03403
$225$	0	0
$226$	4021.25	1.18358
$227$	1145.52	0.334937	0.167469	−	0.985877i	$-0.446441\pi$
$227$	1145.52	0.334937	0.167469	+	0.985877i	$0.446441\pi$
$228$	0	0
$229$	−1348.47	−0.389123	−0.194561	−	0.980890i	$-0.562328\pi$
$229$	−1348.47	−0.389123	−0.194561	+	0.980890i	$0.562328\pi$
$230$	3686.12	1.05676
$231$	0	0
$232$	−5845.94	−1.65433
$233$	952.002	0.267673	0.133836	−	0.991003i	$-0.457270\pi$
$233$	952.002	0.267673	0.133836	+	0.991003i	$0.457270\pi$
$234$	0	0
$235$	−1187.95	−0.329760
$236$	−1935.97	−0.533987
$237$	0	0
$238$	13427.5	3.65703
$239$	−3069.12	−0.830647	−0.415323	−	0.909674i	$-0.636332\pi$
$239$	−3069.12	−0.830647	−0.415323	+	0.909674i	$0.636332\pi$
$240$	0	0
$241$	−2508.17	−0.670396	−0.335198	−	0.942148i	$-0.608803\pi$
$241$	−2508.17	−0.670396	−0.335198	+	0.942148i	$0.608803\pi$
$242$	−6084.62	−1.61626
$243$	0	0
$244$	−829.446	−0.217622
$245$	1777.19	0.463432
$246$	0	0
$247$	0	0
$248$	−911.815	−0.233469
$249$	0	0
$250$	−4832.85	−1.22262
$251$	−3405.91	−0.856490	−0.428245	−	0.903663i	$-0.640868\pi$
$251$	−3405.91	−0.856490	−0.428245	+	0.903663i	$0.640868\pi$
$252$	0	0
$253$	−1051.15	−0.261206
$254$	3411.38	0.842714
$255$	0	0
$256$	−6052.04	−1.47755
$257$	4733.95	1.14901	0.574506	−	0.818501i	$-0.305194\pi$
$257$	4733.95	1.14901	0.574506	+	0.818501i	$0.305194\pi$
$258$	0	0
$259$	9363.91	2.24651
$260$	0	0
$261$	0	0
$262$	6844.85	1.61403
$263$	1866.11	0.437526	0.218763	−	0.975778i	$-0.429798\pi$
$263$	1866.11	0.437526	0.218763	+	0.975778i	$0.429798\pi$
$264$	0	0
$265$	−773.346	−0.179269
$266$	−20001.4	−4.61040
$267$	0	0
$268$	−2978.83	−0.678958
$269$	−1580.19	−0.358164	−0.179082	−	0.983834i	$-0.557313\pi$
$269$	−1580.19	−0.358164	−0.179082	+	0.983834i	$0.557313\pi$
$270$	0	0
$271$	−4922.09	−1.10330	−0.551652	−	0.834074i	$-0.686002\pi$
$271$	−4922.09	−1.10330	−0.551652	+	0.834074i	$0.686002\pi$
$272$	−2254.84	−0.502646
$273$	0	0
$274$	8488.71	1.87161
$275$	629.148	0.137960
$276$	0	0
$277$	2687.08	0.582856	0.291428	−	0.956593i	$-0.405870\pi$
$277$	2687.08	0.582856	0.291428	+	0.956593i	$0.405870\pi$
$278$	5775.41	1.24599
$279$	0	0
$280$	−3472.26	−0.741098
$281$	883.753	0.187617	0.0938083	−	0.995590i	$-0.470096\pi$
$281$	883.753	0.187617	0.0938083	+	0.995590i	$0.470096\pi$
$282$	0	0
$283$	−469.776	−0.0986759	−0.0493379	−	0.998782i	$-0.515711\pi$
$283$	−469.776	−0.0986759	−0.0493379	+	0.998782i	$0.515711\pi$
$284$	−6140.33	−1.28296
$285$	0	0
$286$	0	0
$287$	−1267.75	−0.260742
$288$	0	0
$289$	6119.68	1.24561
$290$	−4305.20	−0.871760
$291$	0	0
$292$	−16312.2	−3.26918
$293$	−3403.76	−0.678668	−0.339334	−	0.940666i	$-0.610202\pi$
$293$	−3403.76	−0.678668	−0.339334	+	0.940666i	$0.610202\pi$
$294$	0	0
$295$	−615.261	−0.121430
$296$	−9819.22	−1.92814
$297$	0	0
$298$	2095.99	0.407442
$299$	0	0
$300$	0	0
$301$	4715.03	0.902889
$302$	−973.504	−0.185493
$303$	0	0
$304$	3358.79	0.633684
$305$	−263.602	−0.0494878
$306$	0	0
$307$	888.862	0.165244	0.0826222	−	0.996581i	$-0.473671\pi$
$307$	888.862	0.165244	0.0826222	+	0.996581i	$0.473671\pi$
$308$	2294.50	0.424484
$309$	0	0
$310$	−671.501	−0.123028
$311$	1218.36	0.222145	0.111072	−	0.993812i	$-0.464571\pi$
$311$	1218.36	0.222145	0.111072	+	0.993812i	$0.464571\pi$
$312$	0	0
$313$	−4870.41	−0.879526	−0.439763	−	0.898114i	$-0.644938\pi$
$313$	−4870.41	−0.879526	−0.439763	+	0.898114i	$0.644938\pi$
$314$	5152.06	0.925948
$315$	0	0
$316$	−14319.5	−2.54916
$317$	4340.93	0.769120	0.384560	−	0.923100i	$-0.374353\pi$
$317$	4340.93	0.769120	0.384560	+	0.923100i	$0.374353\pi$
$318$	0	0
$319$	1227.69	0.215478
$320$	−3445.19	−0.601849
$321$	0	0
$322$	−22425.4	−3.88112
$323$	−16434.2	−2.83103
$324$	0	0
$325$	0	0
$326$	14819.3	2.51769
$327$	0	0
$328$	1329.39	0.223791
$329$	7227.23	1.21109
$330$	0	0
$331$	−907.289	−0.150662	−0.0753310	−	0.997159i	$-0.524001\pi$
$331$	−907.289	−0.150662	−0.0753310	+	0.997159i	$0.524001\pi$
$332$	−2114.49	−0.349542
$333$	0	0
$334$	17287.7	2.83216
$335$	−946.684	−0.154397
$336$	0	0
$337$	8660.88	1.39997	0.699983	−	0.714160i	$-0.253191\pi$
$337$	8660.88	1.39997	0.699983	+	0.714160i	$0.253191\pi$
$338$	0	0
$339$	0	0
$340$	−6611.19	−1.05454
$341$	191.488	0.0304096
$342$	0	0
$343$	−1479.14	−0.232846
$344$	−4944.29	−0.774937
$345$	0	0
$346$	17228.2	2.67687
$347$	347.605	0.0537763	0.0268882	−	0.999638i	$-0.491440\pi$
$347$	347.605	0.0537763	0.0268882	+	0.999638i	$0.491440\pi$
$348$	0	0
$349$	−10970.2	−1.68258	−0.841292	−	0.540581i	$-0.818205\pi$
$349$	−10970.2	−1.68258	−0.841292	+	0.540581i	$0.818205\pi$
$350$	13422.4	2.04988
$351$	0	0
$352$	763.411	0.115596
$353$	−10384.8	−1.56580	−0.782901	−	0.622146i	$-0.786261\pi$
$353$	−10384.8	−1.56580	−0.782901	+	0.622146i	$0.786261\pi$
$354$	0	0
$355$	−1951.42	−0.291749
$356$	7965.15	1.18582
$357$	0	0
$358$	−19608.0	−2.89473
$359$	−8665.80	−1.27399	−0.636996	−	0.770867i	$-0.719823\pi$
$359$	−8665.80	−1.27399	−0.636996	+	0.770867i	$0.719823\pi$
$360$	0	0
$361$	17621.2	2.56907
$362$	−2345.33	−0.340518
$363$	0	0
$364$	0	0
$365$	−5184.09	−0.743419
$366$	0	0
$367$	1234.22	0.175547	0.0877734	−	0.996140i	$-0.472025\pi$
$367$	1234.22	0.175547	0.0877734	+	0.996140i	$0.472025\pi$
$368$	3765.85	0.533447
$369$	0	0
$370$	−7231.31	−1.01605
$371$	4704.85	0.658393
$372$	0	0
$373$	−427.483	−0.0593410	−0.0296705	−	0.999560i	$-0.509446\pi$
$373$	−427.483	−0.0593410	−0.0296705	+	0.999560i	$0.509446\pi$
$374$	2956.98	0.408829
$375$	0	0
$376$	−7578.64	−1.03946
$377$	0	0
$378$	0	0
$379$	−124.241	−0.0168386	−0.00841929	−	0.999965i	$-0.502680\pi$
$379$	−124.241	−0.0168386	−0.00841929	+	0.999965i	$0.502680\pi$
$380$	9847.98	1.32945
$381$	0	0
$382$	14501.8	1.94235
$383$	9341.21	1.24625	0.623125	−	0.782122i	$-0.285863\pi$
$383$	9341.21	1.24625	0.623125	+	0.782122i	$0.285863\pi$
$384$	0	0
$385$	729.202	0.0965288
$386$	−7724.51	−1.01857
$387$	0	0
$388$	4035.33	0.527997
$389$	−11368.9	−1.48182	−0.740908	−	0.671607i	$-0.765604\pi$
$389$	−11368.9	−1.48182	−0.740908	+	0.671607i	$0.765604\pi$
$390$	0	0
$391$	−18425.9	−2.38321
$392$	11337.7	1.46082
$393$	0	0
$394$	−6442.21	−0.823741
$395$	−4550.80	−0.579685
$396$	0	0
$397$	12077.8	1.52687	0.763436	−	0.645883i	$-0.223511\pi$
$397$	12077.8	1.52687	0.763436	+	0.645883i	$0.223511\pi$
$398$	−21740.1	−2.73802
$399$	0	0
$400$	−2253.99	−0.281748
$401$	−4856.74	−0.604823	−0.302411	−	0.953178i	$-0.597792\pi$
$401$	−4856.74	−0.604823	−0.302411	+	0.953178i	$0.597792\pi$
$402$	0	0
$403$	0	0
$404$	17149.0	2.11187
$405$	0	0
$406$	26191.8	3.20167
$407$	2062.11	0.251143
$408$	0	0
$409$	−2981.80	−0.360490	−0.180245	−	0.983622i	$-0.557689\pi$
$409$	−2981.80	−0.360490	−0.180245	+	0.983622i	$0.557689\pi$
$410$	979.023	0.117928
$411$	0	0
$412$	1044.91	0.124949
$413$	3743.10	0.445971
$414$	0	0
$415$	−671.995	−0.0794866
$416$	0	0
$417$	0	0
$418$	−4404.70	−0.515409
$419$	−7774.01	−0.906408	−0.453204	−	0.891407i	$-0.649719\pi$
$419$	−7774.01	−0.906408	−0.453204	+	0.891407i	$0.649719\pi$
$420$	0	0
$421$	−3959.71	−0.458396	−0.229198	−	0.973380i	$-0.573610\pi$
$421$	−3959.71	−0.458396	−0.229198	+	0.973380i	$0.573610\pi$
$422$	23905.9	2.75763
$423$	0	0
$424$	−4933.62	−0.565089
$425$	11028.5	1.25873
$426$	0	0
$427$	1603.69	0.181752
$428$	17640.1	1.99221
$429$	0	0
$430$	−3641.19	−0.408358
$431$	4375.12	0.488961	0.244480	−	0.969654i	$-0.421383\pi$
$431$	4375.12	0.488961	0.244480	+	0.969654i	$0.421383\pi$
$432$	0	0
$433$	8992.74	0.998068	0.499034	−	0.866582i	$-0.333688\pi$
$433$	8992.74	0.998068	0.499034	+	0.866582i	$0.333688\pi$
$434$	4085.25	0.451839
$435$	0	0
$436$	−10169.8	−1.11708
$437$	27447.0	3.00451
$438$	0	0
$439$	−195.465	−0.0212506	−0.0106253	−	0.999944i	$-0.503382\pi$
$439$	−195.465	−0.0212506	−0.0106253	+	0.999944i	$0.503382\pi$
$440$	−764.659	−0.0828493
$441$	0	0
$442$	0	0
$443$	−7369.97	−0.790424	−0.395212	−	0.918590i	$-0.629329\pi$
$443$	−7369.97	−0.790424	−0.395212	+	0.918590i	$0.629329\pi$
$444$	0	0
$445$	2531.36	0.269658
$446$	−22292.1	−2.36673
$447$	0	0
$448$	20959.7	2.21038
$449$	−164.281	−0.0172670	−0.00863351	−	0.999963i	$-0.502748\pi$
$449$	−164.281	−0.0172670	−0.00863351	+	0.999963i	$0.502748\pi$
$450$	0	0
$451$	−279.183	−0.0291490
$452$	12045.3	1.25346
$453$	0	0
$454$	5381.88	0.556352
$455$	0	0
$456$	0	0
$457$	−11687.0	−1.19627	−0.598133	−	0.801397i	$-0.704091\pi$
$457$	−11687.0	−1.19627	−0.598133	+	0.801397i	$0.704091\pi$
$458$	−6335.36	−0.646358
$459$	0	0
$460$	11041.5	1.11915
$461$	1057.53	0.106842	0.0534211	−	0.998572i	$-0.482987\pi$
$461$	1057.53	0.106842	0.0534211	+	0.998572i	$0.482987\pi$
$462$	0	0
$463$	−8554.74	−0.858688	−0.429344	−	0.903141i	$-0.641255\pi$
$463$	−8554.74	−0.858688	−0.429344	+	0.903141i	$0.641255\pi$
$464$	−4398.33	−0.440059
$465$	0	0
$466$	4472.70	0.444622
$467$	7705.26	0.763506	0.381753	−	0.924264i	$-0.375321\pi$
$467$	7705.26	0.763506	0.381753	+	0.924264i	$0.375321\pi$
$468$	0	0
$469$	5759.40	0.567046
$470$	−5581.24	−0.547752
$471$	0	0
$472$	−3925.10	−0.382770
$473$	1038.34	0.100936
$474$	0	0
$475$	−16428.0	−1.58688
$476$	40220.9	3.87295
$477$	0	0
$478$	−14419.3	−1.37976
$479$	−4508.59	−0.430069	−0.215034	−	0.976606i	$-0.568986\pi$
$479$	−4508.59	−0.430069	−0.215034	+	0.976606i	$0.568986\pi$
$480$	0	0
$481$	0	0
$482$	−11783.9	−1.11357
$483$	0	0
$484$	−18226.0	−1.71168
$485$	1282.45	0.120068
$486$	0	0
$487$	−6725.73	−0.625815	−0.312908	−	0.949784i	$-0.601303\pi$
$487$	−6725.73	−0.625815	−0.312908	+	0.949784i	$0.601303\pi$
$488$	−1681.67	−0.155995
$489$	0	0
$490$	8349.61	0.769790
$491$	−11517.5	−1.05861	−0.529303	−	0.848433i	$-0.677546\pi$
$491$	−11517.5	−1.05861	−0.529303	+	0.848433i	$0.677546\pi$
$492$	0	0
$493$	21520.5	1.96600
$494$	0	0
$495$	0	0
$496$	−686.026	−0.0621038
$497$	11872.0	1.07149
$498$	0	0
$499$	−19907.9	−1.78598	−0.892988	−	0.450080i	$-0.851395\pi$
$499$	−19907.9	−1.78598	−0.892988	+	0.450080i	$0.851395\pi$
$500$	−14476.4	−1.29481
$501$	0	0
$502$	−16001.6	−1.42269
$503$	5735.48	0.508415	0.254207	−	0.967150i	$-0.418185\pi$
$503$	5735.48	0.508415	0.254207	+	0.967150i	$0.418185\pi$
$504$	0	0
$505$	5450.04	0.480244
$506$	−4938.51	−0.433881
$507$	0	0
$508$	10218.5	0.892469
$509$	9253.84	0.805834	0.402917	−	0.915237i	$-0.367996\pi$
$509$	9253.84	0.805834	0.402917	+	0.915237i	$0.367996\pi$
$510$	0	0
$511$	31538.8	2.73032
$512$	−7635.12	−0.659039
$513$	0	0
$514$	22241.1	1.90858
$515$	332.076	0.0284136
$516$	0	0
$517$	1591.57	0.135391
$518$	43993.5	3.73159
$519$	0	0
$520$	0	0
$521$	−3887.42	−0.326892	−0.163446	−	0.986552i	$-0.552261\pi$
$521$	−3887.42	−0.326892	−0.163446	+	0.986552i	$0.552261\pi$
$522$	0	0
$523$	4782.27	0.399836	0.199918	−	0.979813i	$-0.435932\pi$
$523$	4782.27	0.399836	0.199918	+	0.979813i	$0.435932\pi$
$524$	20503.2	1.70933
$525$	0	0
$526$	8767.37	0.726759
$527$	3356.65	0.277453
$528$	0	0
$529$	18606.4	1.52925
$530$	−3633.34	−0.297777
$531$	0	0
$532$	−59912.7	−4.88261
$533$	0	0
$534$	0	0
$535$	5606.09	0.453033
$536$	−6039.44	−0.486687
$537$	0	0
$538$	−7424.06	−0.594933
$539$	−2381.01	−0.190274
$540$	0	0
$541$	14872.6	1.18192	0.590962	−	0.806699i	$-0.298748\pi$
$541$	14872.6	1.18192	0.590962	+	0.806699i	$0.298748\pi$
$542$	−23125.0	−1.83266
$543$	0	0
$544$	13382.0	1.05469
$545$	−3232.01	−0.254026
$546$	0	0
$547$	16965.3	1.32611	0.663055	−	0.748570i	$-0.269259\pi$
$547$	16965.3	1.32611	0.663055	+	0.748570i	$0.269259\pi$
$548$	25427.3	1.98212
$549$	0	0
$550$	2955.86	0.229161
$551$	−32056.8	−2.47852
$552$	0	0
$553$	27686.0	2.12898
$554$	12624.5	0.968162
$555$	0	0
$556$	17299.8	1.31956
$557$	1934.05	0.147125	0.0735623	−	0.997291i	$-0.476563\pi$
$557$	1934.05	0.147125	0.0735623	+	0.997291i	$0.476563\pi$
$558$	0	0
$559$	0	0
$560$	−2612.44	−0.197135
$561$	0	0
$562$	4152.05	0.311643
$563$	−6592.13	−0.493473	−0.246736	−	0.969083i	$-0.579358\pi$
$563$	−6592.13	−0.493473	−0.246736	+	0.969083i	$0.579358\pi$
$564$	0	0
$565$	3828.06	0.285040
$566$	−2207.10	−0.163907
$567$	0	0
$568$	−12449.3	−0.919646
$569$	14477.7	1.06667	0.533336	−	0.845903i	$-0.320938\pi$
$569$	14477.7	1.06667	0.533336	+	0.845903i	$0.320938\pi$
$570$	0	0
$571$	−4998.21	−0.366320	−0.183160	−	0.983083i	$-0.558633\pi$
$571$	−4998.21	−0.366320	−0.183160	+	0.983083i	$0.558633\pi$
$572$	0	0
$573$	0	0
$574$	−5956.15	−0.433109
$575$	−18418.9	−1.33586
$576$	0	0
$577$	−9291.73	−0.670398	−0.335199	−	0.942147i	$-0.608804\pi$
$577$	−9291.73	−0.670398	−0.335199	+	0.942147i	$0.608804\pi$
$578$	28751.5	2.06904
$579$	0	0
$580$	−12895.9	−0.923230
$581$	4088.26	0.291927
$582$	0	0
$583$	1036.10	0.0736034
$584$	−33072.3	−2.34339
$585$	0	0
$586$	−15991.5	−1.12731
$587$	−5602.64	−0.393945	−0.196972	−	0.980409i	$-0.563111\pi$
$587$	−5602.64	−0.393945	−0.196972	+	0.980409i	$0.563111\pi$
$588$	0	0
$589$	−5000.04	−0.349784
$590$	−2890.62	−0.201703
$591$	0	0
$592$	−7387.73	−0.512895
$593$	−10885.8	−0.753839	−0.376919	−	0.926246i	$-0.623017\pi$
$593$	−10885.8	−0.753839	−0.376919	+	0.926246i	$0.623017\pi$
$594$	0	0
$595$	12782.4	0.880717
$596$	6278.38	0.431498
$597$	0	0
$598$	0	0
$599$	−20403.2	−1.39174	−0.695872	−	0.718166i	$-0.744982\pi$
$599$	−20403.2	−1.39174	−0.695872	+	0.718166i	$0.744982\pi$
$600$	0	0
$601$	−6312.19	−0.428419	−0.214209	−	0.976788i	$-0.568717\pi$
$601$	−6312.19	−0.428419	−0.214209	+	0.976788i	$0.568717\pi$
$602$	22152.2	1.49976
$603$	0	0
$604$	−2916.05	−0.196445
$605$	−5792.30	−0.389241
$606$	0	0
$607$	−21848.6	−1.46097	−0.730484	−	0.682930i	$-0.760705\pi$
$607$	−21848.6	−1.46097	−0.730484	+	0.682930i	$0.760705\pi$
$608$	−19933.8	−1.32964
$609$	0	0
$610$	−1238.45	−0.0822025
$611$	0	0
$612$	0	0
$613$	1335.14	0.0879704	0.0439852	−	0.999032i	$-0.485995\pi$
$613$	1335.14	0.0879704	0.0439852	+	0.999032i	$0.485995\pi$
$614$	4176.05	0.274482
$615$	0	0
$616$	4652.00	0.304277
$617$	18908.3	1.23374	0.616871	−	0.787064i	$-0.288400\pi$
$617$	18908.3	1.23374	0.616871	+	0.787064i	$0.288400\pi$
$618$	0	0
$619$	−6722.49	−0.436510	−0.218255	−	0.975892i	$-0.570036\pi$
$619$	−6722.49	−0.436510	−0.218255	+	0.975892i	$0.570036\pi$
$620$	−2011.43	−0.130292
$621$	0	0
$622$	5724.11	0.368997
$623$	−15400.2	−0.990362
$624$	0	0
$625$	8523.93	0.545532
$626$	−22882.2	−1.46095
$627$	0	0
$628$	15432.6	0.980617
$629$	36147.3	2.29140
$630$	0	0
$631$	8098.44	0.510925	0.255463	−	0.966819i	$-0.417772\pi$
$631$	8098.44	0.510925	0.255463	+	0.966819i	$0.417772\pi$
$632$	−29032.2	−1.82727
$633$	0	0
$634$	20394.6	1.27756
$635$	3247.50	0.202950
$636$	0	0
$637$	0	0
$638$	5767.94	0.357923
$639$	0	0
$640$	−11627.7	−0.718163
$641$	10955.9	0.675090	0.337545	−	0.941309i	$-0.390403\pi$
$641$	10955.9	0.675090	0.337545	+	0.941309i	$0.390403\pi$
$642$	0	0
$643$	28125.0	1.72495	0.862473	−	0.506104i	$-0.168915\pi$
$643$	28125.0	1.72495	0.862473	+	0.506104i	$0.168915\pi$
$644$	−67173.7	−4.11027
$645$	0	0
$646$	−77211.1	−4.70253
$647$	29001.4	1.76223	0.881115	−	0.472901i	$-0.156793\pi$
$647$	29001.4	1.76223	0.881115	+	0.472901i	$0.156793\pi$
$648$	0	0
$649$	824.302	0.0498562
$650$	0	0
$651$	0	0
$652$	44390.1	2.66634
$653$	−19506.3	−1.16898	−0.584488	−	0.811402i	$-0.698705\pi$
$653$	−19506.3	−1.16898	−0.584488	+	0.811402i	$0.698705\pi$
$654$	0	0
$655$	6516.01	0.388705
$656$	1000.20	0.0595294
$657$	0	0
$658$	33955.0	2.01171
$659$	−5985.86	−0.353833	−0.176917	−	0.984226i	$-0.556612\pi$
$659$	−5985.86	−0.353833	−0.176917	+	0.984226i	$0.556612\pi$
$660$	0	0
$661$	280.836	0.0165254	0.00826268	−	0.999966i	$-0.497370\pi$
$661$	280.836	0.0165254	0.00826268	+	0.999966i	$0.497370\pi$
$662$	−4262.63	−0.250259
$663$	0	0
$664$	−4287.04	−0.250557
$665$	−19040.5	−1.11032
$666$	0	0
$667$	−35941.8	−2.08647
$668$	51784.0	2.99938
$669$	0	0
$670$	−4447.71	−0.256463
$671$	353.163	0.0203185
$672$	0	0
$673$	−14868.9	−0.851640	−0.425820	−	0.904808i	$-0.640014\pi$
$673$	−14868.9	−0.851640	−0.425820	+	0.904808i	$0.640014\pi$
$674$	40690.6	2.32543
$675$	0	0
$676$	0	0
$677$	20769.5	1.17908	0.589540	−	0.807739i	$-0.299309\pi$
$677$	20769.5	1.17908	0.589540	+	0.807739i	$0.299309\pi$
$678$	0	0
$679$	−7802.09	−0.440967
$680$	−13403.9	−0.755907
$681$	0	0
$682$	899.650	0.0505123
$683$	14980.8	0.839275	0.419638	−	0.907692i	$-0.362157\pi$
$683$	14980.8	0.839275	0.419638	+	0.907692i	$0.362157\pi$
$684$	0	0
$685$	8080.90	0.450738
$686$	−6949.30	−0.386772
$687$	0	0
$688$	−3719.96	−0.206137
$689$	0	0
$690$	0	0
$691$	−9472.45	−0.521489	−0.260745	−	0.965408i	$-0.583968\pi$
$691$	−9472.45	−0.521489	−0.260745	+	0.965408i	$0.583968\pi$
$692$	51605.8	2.83491
$693$	0	0
$694$	1633.12	0.0893260
$695$	5497.95	0.300071
$696$	0	0
$697$	−4893.87	−0.265952
$698$	−51540.3	−2.79488
$699$	0	0
$700$	40205.7	2.17090
$701$	−1035.34	−0.0557834	−0.0278917	−	0.999611i	$-0.508879\pi$
$701$	−1035.34	−0.0557834	−0.0278917	+	0.999611i	$0.508879\pi$
$702$	0	0
$703$	−53844.8	−2.88875
$704$	4615.72	0.247105
$705$	0	0
$706$	−48790.0	−2.60090
$707$	−33156.7	−1.76377
$708$	0	0
$709$	−16800.4	−0.889917	−0.444958	−	0.895551i	$-0.646781\pi$
$709$	−16800.4	−0.889917	−0.444958	+	0.895551i	$0.646781\pi$
$710$	−9168.18	−0.484614
$711$	0	0
$712$	16149.0	0.850013
$713$	−5606.00	−0.294455
$714$	0	0
$715$	0	0
$716$	−58734.1	−3.06564
$717$	0	0
$718$	−40713.7	−2.11618
$719$	−14873.1	−0.771449	−0.385724	−	0.922614i	$-0.626048\pi$
$719$	−14873.1	−0.771449	−0.385724	+	0.922614i	$0.626048\pi$
$720$	0	0
$721$	−2020.27	−0.104353
$722$	82788.1	4.26739
$723$	0	0
$724$	−7025.24	−0.360623
$725$	21512.4	1.10200
$726$	0	0
$727$	−2318.92	−0.118300	−0.0591500	−	0.998249i	$-0.518839\pi$
$727$	−2318.92	−0.118300	−0.0591500	+	0.998249i	$0.518839\pi$
$728$	0	0
$729$	0	0
$730$	−24355.9	−1.23487
$731$	18201.3	0.920931
$732$	0	0
$733$	3350.95	0.168854	0.0844270	−	0.996430i	$-0.473094\pi$
$733$	3350.95	0.168854	0.0844270	+	0.996430i	$0.473094\pi$
$734$	5798.61	0.291595
$735$	0	0
$736$	−22349.6	−1.11932
$737$	1268.33	0.0633915
$738$	0	0
$739$	29348.1	1.46088	0.730438	−	0.682979i	$-0.239316\pi$
$739$	29348.1	1.46088	0.730438	+	0.682979i	$0.239316\pi$
$740$	−21660.8	−1.07604
$741$	0	0
$742$	22104.3	1.09363
$743$	23622.2	1.16637	0.583187	−	0.812338i	$-0.301805\pi$
$743$	23622.2	1.16637	0.583187	+	0.812338i	$0.301805\pi$
$744$	0	0
$745$	1995.30	0.0981236
$746$	−2008.40	−0.0985693
$747$	0	0
$748$	8857.41	0.432966
$749$	−34106.1	−1.66383
$750$	0	0
$751$	29751.4	1.44560	0.722798	−	0.691059i	$-0.242856\pi$
$751$	29751.4	1.44560	0.722798	+	0.691059i	$0.242856\pi$
$752$	−5701.97	−0.276502
$753$	0	0
$754$	0	0
$755$	−926.735	−0.0446720
$756$	0	0
$757$	−36907.2	−1.77201	−0.886006	−	0.463673i	$-0.846531\pi$
$757$	−36907.2	−1.77201	−0.886006	+	0.463673i	$0.846531\pi$
$758$	−583.709	−0.0279700
$759$	0	0
$760$	19966.4	0.952968
$761$	−20208.2	−0.962613	−0.481306	−	0.876552i	$-0.659838\pi$
$761$	−20208.2	−0.962613	−0.481306	+	0.876552i	$0.659838\pi$
$762$	0	0
$763$	19662.8	0.932950
$764$	43439.1	2.05703
$765$	0	0
$766$	43886.9	2.07010
$767$	0	0
$768$	0	0
$769$	−24751.0	−1.16066	−0.580329	−	0.814382i	$-0.697076\pi$
$769$	−24751.0	−1.16066	−0.580329	+	0.814382i	$0.697076\pi$
$770$	3425.94	0.160341
$771$	0	0
$772$	−23138.2	−1.07871
$773$	−30673.4	−1.42722	−0.713612	−	0.700541i	$-0.752942\pi$
$773$	−30673.4	−1.42722	−0.713612	+	0.700541i	$0.752942\pi$
$774$	0	0
$775$	3355.38	0.155521
$776$	8181.46	0.378476
$777$	0	0
$778$	−53413.4	−2.46139
$779$	7289.87	0.335285
$780$	0	0
$781$	2614.44	0.119785
$782$	−86568.5	−3.95867
$783$	0	0
$784$	8530.23	0.388585
$785$	4904.55	0.222995
$786$	0	0
$787$	−29009.9	−1.31397	−0.656983	−	0.753905i	$-0.728168\pi$
$787$	−29009.9	−1.31397	−0.656983	+	0.753905i	$0.728168\pi$
$788$	−19297.1	−0.872375
$789$	0	0
$790$	−21380.6	−0.962894
$791$	−23289.0	−1.04685
$792$	0	0
$793$	0	0
$794$	56744.0	2.53623
$795$	0	0
$796$	−65120.8	−2.89968
$797$	−6778.24	−0.301252	−0.150626	−	0.988591i	$-0.548129\pi$
$797$	−6778.24	−0.301252	−0.150626	+	0.988591i	$0.548129\pi$
$798$	0	0
$799$	27899.1	1.23529
$800$	13377.0	0.591185
$801$	0	0
$802$	−22817.9	−1.00465
$803$	6945.44	0.305229
$804$	0	0
$805$	−21348.1	−0.934685
$806$	0	0
$807$	0	0
$808$	34768.9	1.51382
$809$	34862.9	1.51510	0.757549	−	0.652778i	$-0.226397\pi$
$809$	34862.9	1.51510	0.757549	+	0.652778i	$0.226397\pi$
$810$	0	0
$811$	−22665.4	−0.981370	−0.490685	−	0.871337i	$-0.663253\pi$
$811$	−22665.4	−0.981370	−0.490685	+	0.871337i	$0.663253\pi$
$812$	78455.6	3.39070
$813$	0	0
$814$	9688.21	0.417164
$815$	14107.4	0.606331
$816$	0	0
$817$	−27112.6	−1.16101
$818$	−14009.1	−0.598797
$819$	0	0
$820$	2932.59	0.124891
$821$	20748.7	0.882016	0.441008	−	0.897503i	$-0.354621\pi$
$821$	20748.7	0.882016	0.441008	+	0.897503i	$0.354621\pi$
$822$	0	0
$823$	6141.41	0.260117	0.130058	−	0.991506i	$-0.458484\pi$
$823$	6141.41	0.260117	0.130058	+	0.991506i	$0.458484\pi$
$824$	2118.50	0.0895650
$825$	0	0
$826$	17585.8	0.740786
$827$	−28383.5	−1.19346	−0.596730	−	0.802442i	$-0.703533\pi$
$827$	−28383.5	−1.19346	−0.596730	+	0.802442i	$0.703533\pi$
$828$	0	0
$829$	−908.734	−0.0380720	−0.0190360	−	0.999819i	$-0.506060\pi$
$829$	−908.734	−0.0380720	−0.0190360	+	0.999819i	$0.506060\pi$
$830$	−3157.17	−0.132032
$831$	0	0
$832$	0	0
$833$	−41737.4	−1.73603
$834$	0	0
$835$	16457.2	0.682065
$836$	−13193.9	−0.545839
$837$	0	0
$838$	−36523.8	−1.50560
$839$	−27820.3	−1.14477	−0.572385	−	0.819985i	$-0.693982\pi$
$839$	−27820.3	−1.14477	−0.572385	+	0.819985i	$0.693982\pi$
$840$	0	0
$841$	17589.3	0.721200
$842$	−18603.5	−0.761425
$843$	0	0
$844$	71608.3	2.92045
$845$	0	0
$846$	0	0
$847$	35239.0	1.42955
$848$	−3711.93	−0.150316
$849$	0	0
$850$	51814.1	2.09084
$851$	−60370.3	−2.43181
$852$	0	0
$853$	−5802.11	−0.232896	−0.116448	−	0.993197i	$-0.537151\pi$
$853$	−5802.11	−0.232896	−0.116448	+	0.993197i	$0.537151\pi$
$854$	7534.45	0.301901
$855$	0	0
$856$	35764.5	1.42804
$857$	43311.1	1.72635	0.863173	−	0.504909i	$-0.168474\pi$
$857$	43311.1	1.72635	0.863173	+	0.504909i	$0.168474\pi$
$858$	0	0
$859$	−16698.2	−0.663254	−0.331627	−	0.943411i	$-0.607598\pi$
$859$	−16698.2	−0.663254	−0.331627	+	0.943411i	$0.607598\pi$
$860$	−10906.9	−0.432468
$861$	0	0
$862$	20555.2	0.812196
$863$	19429.2	0.766369	0.383185	−	0.923672i	$-0.374827\pi$
$863$	19429.2	0.766369	0.383185	+	0.923672i	$0.374827\pi$
$864$	0	0
$865$	16400.6	0.644666
$866$	42249.7	1.65786
$867$	0	0
$868$	12237.0	0.478517
$869$	6096.98	0.238004
$870$	0	0
$871$	0	0
$872$	−20618.9	−0.800738
$873$	0	0
$874$	128952.	4.99068
$875$	27989.4	1.08139
$876$	0	0
$877$	−11619.7	−0.447400	−0.223700	−	0.974658i	$-0.571814\pi$
$877$	−11619.7	−0.447400	−0.223700	+	0.974658i	$0.571814\pi$
$878$	−918.332	−0.0352986
$879$	0	0
$880$	−575.309	−0.0220383
$881$	51102.0	1.95422	0.977112	−	0.212726i	$-0.0682341\pi$
$881$	51102.0	1.95422	0.977112	+	0.212726i	$0.0682341\pi$
$882$	0	0
$883$	−37838.5	−1.44209	−0.721046	−	0.692888i	$-0.756338\pi$
$883$	−37838.5	−1.44209	−0.721046	+	0.692888i	$0.756338\pi$
$884$	0	0
$885$	0	0
$886$	−34625.6	−1.31295
$887$	9626.92	0.364420	0.182210	−	0.983260i	$-0.441675\pi$
$887$	9626.92	0.364420	0.182210	+	0.983260i	$0.441675\pi$
$888$	0	0
$889$	−19757.0	−0.745364
$890$	11892.8	0.447920
$891$	0	0
$892$	−66774.3	−2.50647
$893$	−41558.3	−1.55733
$894$	0	0
$895$	−18666.0	−0.697133
$896$	70740.0	2.63757
$897$	0	0
$898$	−771.825	−0.0286816
$899$	6547.54	0.242906
$900$	0	0
$901$	18162.0	0.671549
$902$	−1311.66	−0.0484184
$903$	0	0
$904$	24421.4	0.898500
$905$	−2232.65	−0.0820065
$906$	0	0
$907$	17066.0	0.624772	0.312386	−	0.949955i	$-0.398872\pi$
$907$	17066.0	0.624772	0.312386	+	0.949955i	$0.398872\pi$
$908$	16121.0	0.589200
$909$	0	0
$910$	0	0
$911$	37423.9	1.36104	0.680521	−	0.732729i	$-0.261754\pi$
$911$	37423.9	1.36104	0.680521	+	0.732729i	$0.261754\pi$
$912$	0	0
$913$	900.312	0.0326353
$914$	−54907.7	−1.98708
$915$	0	0
$916$	−18977.1	−0.684520
$917$	−39641.9	−1.42758
$918$	0	0
$919$	−2783.88	−0.0999256	−0.0499628	−	0.998751i	$-0.515910\pi$
$919$	−2783.88	−0.0999256	−0.0499628	+	0.998751i	$0.515910\pi$
$920$	22386.1	0.802227
$921$	0	0
$922$	4968.50	0.177472
$923$	0	0
$924$	0	0
$925$	36133.6	1.28440
$926$	−40191.9	−1.42634
$927$	0	0
$928$	26103.3	0.923363
$929$	26585.2	0.938894	0.469447	−	0.882961i	$-0.344453\pi$
$929$	26585.2	0.938894	0.469447	+	0.882961i	$0.344453\pi$
$930$	0	0
$931$	62171.8	2.18861
$932$	13397.6	0.470873
$933$	0	0
$934$	36200.9	1.26823
$935$	2814.92	0.0984576
$936$	0	0
$937$	34474.0	1.20194	0.600970	−	0.799272i	$-0.294781\pi$
$937$	34474.0	1.20194	0.600970	+	0.799272i	$0.294781\pi$
$938$	27058.8	0.941900
$939$	0	0
$940$	−16718.2	−0.580092
$941$	−41994.3	−1.45481	−0.727404	−	0.686209i	$-0.759274\pi$
$941$	−41994.3	−1.45481	−0.727404	+	0.686209i	$0.759274\pi$
$942$	0	0
$943$	8173.34	0.282249
$944$	−2953.15	−0.101819
$945$	0	0
$946$	4878.32	0.167662
$947$	−49352.0	−1.69348	−0.846739	−	0.532008i	$-0.821438\pi$
$947$	−49352.0	−1.69348	−0.846739	+	0.532008i	$0.821438\pi$
$948$	0	0
$949$	0	0
$950$	−77181.9	−2.63591
$951$	0	0
$952$	81546.2	2.77618
$953$	51144.5	1.73844	0.869220	−	0.494425i	$-0.164621\pi$
$953$	51144.5	1.73844	0.869220	+	0.494425i	$0.164621\pi$
$954$	0	0
$955$	13805.2	0.467774
$956$	−43192.0	−1.46122
$957$	0	0
$958$	−21182.3	−0.714372
$959$	−49162.3	−1.65540
$960$	0	0
$961$	−28769.8	−0.965720
$962$	0	0
$963$	0	0
$964$	−35297.7	−1.17932
$965$	−7353.41	−0.245300
$966$	0	0
$967$	24895.1	0.827892	0.413946	−	0.910301i	$-0.364150\pi$
$967$	24895.1	0.827892	0.413946	+	0.910301i	$0.364150\pi$
$968$	−36952.4	−1.22696
$969$	0	0
$970$	6025.19	0.199440
$971$	−42942.9	−1.41926	−0.709630	−	0.704574i	$-0.751138\pi$
$971$	−42942.9	−1.41926	−0.709630	+	0.704574i	$0.751138\pi$
$972$	0	0
$973$	−33448.2	−1.10206
$974$	−31598.8	−1.03952
$975$	0	0
$976$	−1265.24	−0.0414953
$977$	42555.7	1.39353	0.696764	−	0.717301i	$-0.254623\pi$
$977$	42555.7	1.39353	0.696764	+	0.717301i	$0.254623\pi$
$978$	0	0
$979$	−3391.41	−0.110715
$980$	25010.6	0.815240
$981$	0	0
$982$	−54111.3	−1.75841
$983$	−4345.38	−0.140993	−0.0704965	−	0.997512i	$-0.522458\pi$
$983$	−4345.38	−0.140993	−0.0704965	+	0.997512i	$0.522458\pi$
$984$	0	0
$985$	−6132.71	−0.198380
$986$	101108.	3.26565
$987$	0	0
$988$	0	0
$989$	−30398.4	−0.977363
$990$	0	0
$991$	7934.89	0.254349	0.127175	−	0.991880i	$-0.459409\pi$
$991$	7934.89	0.254349	0.127175	+	0.991880i	$0.459409\pi$
$992$	4071.43	0.130311
$993$	0	0
$994$	55777.0	1.77982
$995$	−20695.7	−0.659395
$996$	0	0
$997$	2725.62	0.0865810	0.0432905	−	0.999063i	$-0.486216\pi$
$997$	2725.62	0.0865810	0.0432905	+	0.999063i	$0.486216\pi$
$998$	−93531.5	−2.96662
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bi.1.8		9
3.2	odd	2		507.4.a.o.1.2	✓	9
13.12	even	2		1521.4.a.bf.1.2		9
39.5	even	4		507.4.b.k.337.16		18
39.8	even	4		507.4.b.k.337.3		18
39.38	odd	2		507.4.a.p.1.8	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.2	✓	9	3.2	odd	2
507.4.a.p.1.8	yes	9	39.38	odd	2
507.4.b.k.337.3		18	39.8	even	4
507.4.b.k.337.16		18	39.5	even	4
1521.4.a.bf.1.2		9	13.12	even	2
1521.4.a.bi.1.8		9	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)

\(f(q)\)	\(=\)	\(q+4.69820 q^{2} +14.0731 q^{4} +4.47249 q^{5} -27.2096 q^{7} +28.5326 q^{8} +O(q^{10})\)	\(q+4.69820 q^{2} +14.0731 q^{4} +4.47249 q^{5} -27.2096 q^{7} +28.5326 q^{8} +21.0127 q^{10} -5.99207 q^{11} -127.836 q^{14} +21.4672 q^{16} -105.037 q^{17} +156.462 q^{19} +62.9418 q^{20} -28.1519 q^{22} +175.423 q^{23} -104.997 q^{25} -382.923 q^{28} -204.886 q^{29} -31.9570 q^{31} -127.404 q^{32} -493.483 q^{34} -121.695 q^{35} -344.140 q^{37} +735.088 q^{38} +127.612 q^{40} +46.5921 q^{41} -173.286 q^{43} -84.3269 q^{44} +824.175 q^{46} -265.613 q^{47} +397.361 q^{49} -493.296 q^{50} -172.912 q^{53} -26.7995 q^{55} -776.360 q^{56} -962.596 q^{58} -137.566 q^{59} -58.9384 q^{61} -150.140 q^{62} -770.306 q^{64} -211.668 q^{67} -1478.19 q^{68} -571.746 q^{70} -436.317 q^{71} -1159.11 q^{73} -1616.84 q^{74} +2201.90 q^{76} +163.042 q^{77} -1017.51 q^{79} +96.0118 q^{80} +218.899 q^{82} -150.251 q^{83} -469.775 q^{85} -814.131 q^{86} -170.969 q^{88} +565.984 q^{89} +2468.75 q^{92} -1247.90 q^{94} +699.773 q^{95} +286.741 q^{97} +1866.88 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8}+O(q^{10}) \) 9 * q + 6 * q^2 + 44 * q^4 + 33 * q^5 - 83 * q^7 + 87 * q^8	\( 9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8} - 54 q^{10} + 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} - 352 q^{19} + 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} - 940 q^{28} + 97 q^{29} - 717 q^{31} + 707 q^{32} - 632 q^{34} + 418 q^{35} - 1108 q^{37} + 660 q^{38} - 1506 q^{40} + 334 q^{41} + 242 q^{43} - 307 q^{44} - 979 q^{46} - 184 q^{47} - 38 q^{49} - 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} - 1161 q^{58} + 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} - 2308 q^{67} - 2785 q^{68} + 1420 q^{70} + 96 q^{71} - 2505 q^{73} + 1191 q^{74} - 2409 q^{76} + 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 1517 q^{82} + 1539 q^{83} - 4296 q^{85} - 3763 q^{86} - 3716 q^{88} - 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} - 1445 q^{97} + 1457 q^{98}+O(q^{100}) \) 9 * q + 6 * q^2 + 44 * q^4 + 33 * q^5 - 83 * q^7 + 87 * q^8 - 54 * q^10 + 85 * q^11 - 158 * q^14 + 216 * q^16 - 178 * q^17 - 352 * q^19 + 402 * q^20 - 630 * q^22 - 150 * q^23 - 20 * q^25 - 940 * q^28 + 97 * q^29 - 717 * q^31 + 707 * q^32 - 632 * q^34 + 418 * q^35 - 1108 * q^37 + 660 * q^38 - 1506 * q^40 + 334 * q^41 + 242 * q^43 - 307 * q^44 - 979 * q^46 - 184 * q^47 - 38 * q^49 - 2031 * q^50 + 151 * q^53 + 2064 * q^55 - 2276 * q^56 - 1161 * q^58 + 537 * q^59 - 1340 * q^61 - 347 * q^62 + 893 * q^64 - 2308 * q^67 - 2785 * q^68 + 1420 * q^70 + 96 * q^71 - 2505 * q^73 + 1191 * q^74 - 2409 * q^76 + 2142 * q^77 - 1591 * q^79 - 2671 * q^80 + 1517 * q^82 + 1539 * q^83 - 4296 * q^85 - 3763 * q^86 - 3716 * q^88 - 592 * q^89 - 515 * q^92 - 692 * q^94 - 4158 * q^95 - 1445 * q^97 + 1457 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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