LMFDB - Embedded newform 1521.4.a.bi.1.2

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			1.2
Root			$2.37150$ 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)

$f(q)$	$=$	$q-3.17344 q^{2} +2.07074 q^{4} +6.74147 q^{5} +14.1726 q^{7} +18.8162 q^{8} +O(q^{10})$	$q-3.17344 q^{2} +2.07074 q^{4} +6.74147 q^{5} +14.1726 q^{7} +18.8162 q^{8} -21.3937 q^{10} +62.4956 q^{11} -44.9761 q^{14} -76.2779 q^{16} +58.6172 q^{17} -64.1652 q^{19} +13.9598 q^{20} -198.326 q^{22} -10.9221 q^{23} -79.5526 q^{25} +29.3478 q^{28} -216.316 q^{29} +38.6271 q^{31} +91.5342 q^{32} -186.018 q^{34} +95.5445 q^{35} -423.770 q^{37} +203.625 q^{38} +126.849 q^{40} -366.126 q^{41} -128.297 q^{43} +129.412 q^{44} +34.6605 q^{46} -93.1169 q^{47} -142.136 q^{49} +252.455 q^{50} -131.909 q^{53} +421.313 q^{55} +266.675 q^{56} +686.467 q^{58} +386.729 q^{59} -621.077 q^{61} -122.581 q^{62} +319.745 q^{64} -865.273 q^{67} +121.381 q^{68} -303.205 q^{70} -607.506 q^{71} -980.958 q^{73} +1344.81 q^{74} -132.869 q^{76} +885.728 q^{77} +1331.91 q^{79} -514.226 q^{80} +1161.88 q^{82} +907.633 q^{83} +395.166 q^{85} +407.142 q^{86} +1175.93 q^{88} -1033.67 q^{89} -22.6167 q^{92} +295.501 q^{94} -432.568 q^{95} -1046.17 q^{97} +451.061 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.17344	−1.12198	−0.560991	−	0.827822i	$-0.689580\pi$
$2$	−3.17344	−1.12198	−0.560991	+	0.827822i	$0.689580\pi$
$3$	0	0
$3$	0	0
$4$	2.07074	0.258842
$5$	6.74147	0.602975	0.301488	−	0.953470i	$-0.402517\pi$
$5$	6.74147	0.602975	0.301488	+	0.953470i	$0.402517\pi$
$6$	0	0
$7$	14.1726	0.765251	0.382625	−	0.923904i	$-0.375020\pi$
$7$	14.1726	0.765251	0.382625	+	0.923904i	$0.375020\pi$
$8$	18.8162	0.831565
$9$	0	0
$10$	−21.3937	−0.676527
$11$	62.4956	1.71301	0.856507	−	0.516136i	$-0.172630\pi$
$11$	62.4956	1.71301	0.856507	+	0.516136i	$0.172630\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−44.9761	−0.858597
$15$	0	0
$16$	−76.2779	−1.19184
$17$	58.6172	0.836280	0.418140	−	0.908383i	$-0.362682\pi$
$17$	58.6172	0.836280	0.418140	+	0.908383i	$0.362682\pi$
$18$	0	0
$19$	−64.1652	−0.774764	−0.387382	−	0.921919i	$-0.626620\pi$
$19$	−64.1652	−0.774764	−0.387382	+	0.921919i	$0.626620\pi$
$20$	13.9598	0.156075
$21$	0	0
$22$	−198.326	−1.92197
$23$	−10.9221	−0.0990177	−0.0495088	−	0.998774i	$-0.515766\pi$
$23$	−10.9221	−0.0990177	−0.0495088	+	0.998774i	$0.515766\pi$
$24$	0	0
$25$	−79.5526	−0.636421
$26$	0	0
$27$	0	0
$28$	29.3478	0.198079
$29$	−216.316	−1.38514	−0.692568	−	0.721353i	$-0.743521\pi$
$29$	−216.316	−1.38514	−0.692568	+	0.721353i	$0.743521\pi$
$30$	0	0
$31$	38.6271	0.223795	0.111897	−	0.993720i	$-0.464307\pi$
$31$	38.6271	0.223795	0.111897	+	0.993720i	$0.464307\pi$
$32$	91.5342	0.505660
$33$	0	0
$34$	−186.018	−0.938290
$35$	95.5445	0.461427
$36$	0	0
$37$	−423.770	−1.88290	−0.941452	−	0.337147i	$-0.890538\pi$
$37$	−423.770	−1.88290	−0.941452	+	0.337147i	$0.890538\pi$
$38$	203.625	0.869270
$39$	0	0
$40$	126.849	0.501414
$41$	−366.126	−1.39461	−0.697307	−	0.716772i	$-0.745619\pi$
$41$	−366.126	−1.39461	−0.697307	+	0.716772i	$0.745619\pi$
$42$	0	0
$43$	−128.297	−0.455001	−0.227501	−	0.973778i	$-0.573055\pi$
$43$	−128.297	−0.455001	−0.227501	+	0.973778i	$0.573055\pi$
$44$	129.412	0.443400
$45$	0	0
$46$	34.6605	0.111096
$47$	−93.1169	−0.288989	−0.144495	−	0.989506i	$-0.546156\pi$
$47$	−93.1169	−0.288989	−0.144495	+	0.989506i	$0.546156\pi$
$48$	0	0
$49$	−142.136	−0.414391
$50$	252.455	0.714052
$51$	0	0
$52$	0	0
$53$	−131.909	−0.341869	−0.170934	−	0.985282i	$-0.554679\pi$
$53$	−131.909	−0.341869	−0.170934	+	0.985282i	$0.554679\pi$
$54$	0	0
$55$	421.313	1.03290
$56$	266.675	0.636356
$57$	0	0
$58$	686.467	1.55410
$59$	386.729	0.853353	0.426677	−	0.904404i	$-0.359684\pi$
$59$	386.729	0.853353	0.426677	+	0.904404i	$0.359684\pi$
$60$	0	0
$61$	−621.077	−1.30362	−0.651810	−	0.758382i	$-0.725990\pi$
$61$	−621.077	−1.30362	−0.651810	+	0.758382i	$0.725990\pi$
$62$	−122.581	−0.251094
$63$	0	0
$64$	319.745	0.624502
$65$	0	0
$66$	0	0
$67$	−865.273	−1.57776	−0.788880	−	0.614547i	$-0.789339\pi$
$67$	−865.273	−1.57776	−0.788880	+	0.614547i	$0.789339\pi$
$68$	121.381	0.216464
$69$	0	0
$70$	−303.205	−0.517713
$71$	−607.506	−1.01546	−0.507730	−	0.861516i	$-0.669515\pi$
$71$	−607.506	−1.01546	−0.507730	+	0.861516i	$0.669515\pi$
$72$	0	0
$73$	−980.958	−1.57277	−0.786387	−	0.617735i	$-0.788051\pi$
$73$	−980.958	−1.57277	−0.786387	+	0.617735i	$0.788051\pi$
$74$	1344.81	2.11258
$75$	0	0
$76$	−132.869	−0.200541
$77$	885.728	1.31088
$78$	0	0
$79$	1331.91	1.89685	0.948425	−	0.317003i	$-0.102676\pi$
$79$	1331.91	1.89685	0.948425	+	0.317003i	$0.102676\pi$
$80$	−514.226	−0.718652
$81$	0	0
$82$	1161.88	1.56473
$83$	907.633	1.20031	0.600155	−	0.799884i	$-0.295106\pi$
$83$	907.633	1.20031	0.600155	+	0.799884i	$0.295106\pi$
$84$	0	0
$85$	395.166	0.504256
$86$	407.142	0.510503
$87$	0	0
$88$	1175.93	1.42448
$89$	−1033.67	−1.23110	−0.615552	−	0.788096i	$-0.711067\pi$
$89$	−1033.67	−1.23110	−0.615552	+	0.788096i	$0.711067\pi$
$90$	0	0
$91$	0	0
$92$	−22.6167	−0.0256299
$93$	0	0
$94$	295.501	0.324240
$95$	−432.568	−0.467163
$96$	0	0
$97$	−1046.17	−1.09508	−0.547538	−	0.836781i	$-0.684435\pi$
$97$	−1046.17	−1.09508	−0.547538	+	0.836781i	$0.684435\pi$
$98$	451.061	0.464939
$99$	0	0
$100$	−164.732	−0.164732
$101$	−1416.64	−1.39566	−0.697828	−	0.716265i	$-0.745850\pi$
$101$	−1416.64	−1.39566	−0.697828	+	0.716265i	$0.745850\pi$
$102$	0	0
$103$	387.629	0.370818	0.185409	−	0.982661i	$-0.440639\pi$
$103$	387.629	0.370818	0.185409	+	0.982661i	$0.440639\pi$
$104$	0	0
$105$	0	0
$106$	418.605	0.383570
$107$	−86.4526	−0.0781092	−0.0390546	−	0.999237i	$-0.512435\pi$
$107$	−86.4526	−0.0781092	−0.0390546	+	0.999237i	$0.512435\pi$
$108$	0	0
$109$	940.072	0.826079	0.413039	−	0.910713i	$-0.364467\pi$
$109$	940.072	0.826079	0.413039	+	0.910713i	$0.364467\pi$
$110$	−1337.01	−1.15890
$111$	0	0
$112$	−1081.06	−0.912059
$113$	−960.499	−0.799612	−0.399806	−	0.916600i	$-0.630922\pi$
$113$	−960.499	−0.799612	−0.399806	+	0.916600i	$0.630922\pi$
$114$	0	0
$115$	−73.6307	−0.0597052
$116$	−447.934	−0.358531
$117$	0	0
$118$	−1227.26	−0.957447
$119$	830.760	0.639964
$120$	0	0
$121$	2574.71	1.93441
$122$	1970.95	1.46264
$123$	0	0
$124$	79.9866	0.0579275
$125$	−1378.99	−0.986721
$126$	0	0
$127$	2022.18	1.41291	0.706456	−	0.707757i	$-0.250293\pi$
$127$	2022.18	1.41291	0.706456	+	0.707757i	$0.250293\pi$
$128$	−1746.97	−1.20634
$129$	0	0
$130$	0	0
$131$	−1857.90	−1.23912	−0.619561	−	0.784948i	$-0.712690\pi$
$131$	−1857.90	−1.23912	−0.619561	+	0.784948i	$0.712690\pi$
$132$	0	0
$133$	−909.390	−0.592888
$134$	2745.90	1.77022
$135$	0	0
$136$	1102.95	0.695421
$137$	1894.12	1.18121	0.590604	−	0.806961i	$-0.298890\pi$
$137$	1894.12	1.18121	0.590604	+	0.806961i	$0.298890\pi$
$138$	0	0
$139$	−1226.08	−0.748165	−0.374082	−	0.927395i	$-0.622042\pi$
$139$	−1226.08	−0.748165	−0.374082	+	0.927395i	$0.622042\pi$
$140$	197.847	0.119437
$141$	0	0
$142$	1927.89	1.13933
$143$	0	0
$144$	0	0
$145$	−1458.29	−0.835203
$146$	3113.01	1.76462
$147$	0	0
$148$	−877.517	−0.487375
$149$	3195.65	1.75703	0.878517	−	0.477711i	$-0.158533\pi$
$149$	3195.65	1.75703	0.878517	+	0.477711i	$0.158533\pi$
$150$	0	0
$151$	−508.232	−0.273903	−0.136951	−	0.990578i	$-0.543730\pi$
$151$	−508.232	−0.273903	−0.136951	+	0.990578i	$0.543730\pi$
$152$	−1207.34	−0.644267
$153$	0	0
$154$	−2810.81	−1.47079
$155$	260.404	0.134943
$156$	0	0
$157$	−1243.08	−0.631900	−0.315950	−	0.948776i	$-0.602323\pi$
$157$	−1243.08	−0.631900	−0.315950	+	0.948776i	$0.602323\pi$
$158$	−4226.73	−2.12823
$159$	0	0
$160$	617.075	0.304901
$161$	−154.794	−0.0757734
$162$	0	0
$163$	33.9996	0.0163378	0.00816888	−	0.999967i	$-0.497400\pi$
$163$	33.9996	0.0163378	0.00816888	+	0.999967i	$0.497400\pi$
$164$	−758.149	−0.360985
$165$	0	0
$166$	−2880.32	−1.34673
$167$	−2210.67	−1.02435	−0.512176	−	0.858880i	$-0.671161\pi$
$167$	−2210.67	−1.02435	−0.512176	+	0.858880i	$0.671161\pi$
$168$	0	0
$169$	0	0
$170$	−1254.04	−0.565766
$171$	0	0
$172$	−265.668	−0.117773
$173$	661.307	0.290626	0.145313	−	0.989386i	$-0.453581\pi$
$173$	661.307	0.290626	0.145313	+	0.989386i	$0.453581\pi$
$174$	0	0
$175$	−1127.47	−0.487021
$176$	−4767.04	−2.04164
$177$	0	0
$178$	3280.28	1.38128
$179$	2325.05	0.970850	0.485425	−	0.874278i	$-0.338665\pi$
$179$	2325.05	0.970850	0.485425	+	0.874278i	$0.338665\pi$
$180$	0	0
$181$	2122.20	0.871503	0.435752	−	0.900067i	$-0.356483\pi$
$181$	2122.20	0.871503	0.435752	+	0.900067i	$0.356483\pi$
$182$	0	0
$183$	0	0
$184$	−205.511	−0.0823397
$185$	−2856.84	−1.13534
$186$	0	0
$187$	3663.32	1.43256
$188$	−192.820	−0.0748025
$189$	0	0
$190$	1372.73	0.524149
$191$	2484.37	0.941166	0.470583	−	0.882356i	$-0.344044\pi$
$191$	2484.37	0.941166	0.470583	+	0.882356i	$0.344044\pi$
$192$	0	0
$193$	266.771	0.0994955	0.0497478	−	0.998762i	$-0.484158\pi$
$193$	266.771	0.0994955	0.0497478	+	0.998762i	$0.484158\pi$
$194$	3319.95	1.22865
$195$	0	0
$196$	−294.327	−0.107262
$197$	−1231.03	−0.445216	−0.222608	−	0.974908i	$-0.571457\pi$
$197$	−1231.03	−0.445216	−0.222608	+	0.974908i	$0.571457\pi$
$198$	0	0
$199$	−3246.14	−1.15635	−0.578173	−	0.815914i	$-0.696234\pi$
$199$	−3246.14	−1.15635	−0.578173	+	0.815914i	$0.696234\pi$
$200$	−1496.88	−0.529225
$201$	0	0
$202$	4495.63	1.56590
$203$	−3065.77	−1.05998
$204$	0	0
$205$	−2468.23	−0.840919
$206$	−1230.12	−0.416051
$207$	0	0
$208$	0	0
$209$	−4010.05	−1.32718
$210$	0	0
$211$	330.708	0.107900	0.0539500	−	0.998544i	$-0.482819\pi$
$211$	330.708	0.107900	0.0539500	+	0.998544i	$0.482819\pi$
$212$	−273.148	−0.0884900
$213$	0	0
$214$	274.352	0.0876371
$215$	−864.908	−0.274355
$216$	0	0
$217$	547.449	0.171259
$218$	−2983.26	−0.926845
$219$	0	0
$220$	872.427	0.267359
$221$	0	0
$222$	0	0
$223$	−5785.86	−1.73744	−0.868722	−	0.495300i	$-0.835058\pi$
$223$	−5785.86	−1.73744	−0.868722	+	0.495300i	$0.835058\pi$
$224$	1297.28	0.386957
$225$	0	0
$226$	3048.09	0.897149
$227$	2945.35	0.861189	0.430595	−	0.902545i	$-0.358304\pi$
$227$	2945.35	0.861189	0.430595	+	0.902545i	$0.358304\pi$
$228$	0	0
$229$	3541.26	1.02189	0.510945	−	0.859613i	$-0.329296\pi$
$229$	3541.26	1.02189	0.510945	+	0.859613i	$0.329296\pi$
$230$	233.663	0.0669882
$231$	0	0
$232$	−4070.25	−1.15183
$233$	−2340.76	−0.658148	−0.329074	−	0.944304i	$-0.606737\pi$
$233$	−2340.76	−0.658148	−0.329074	+	0.944304i	$0.606737\pi$
$234$	0	0
$235$	−627.745	−0.174253
$236$	800.814	0.220884
$237$	0	0
$238$	−2636.37	−0.718027
$239$	−1515.70	−0.410218	−0.205109	−	0.978739i	$-0.565755\pi$
$239$	−1515.70	−0.410218	−0.205109	+	0.978739i	$0.565755\pi$
$240$	0	0
$241$	−2392.47	−0.639472	−0.319736	−	0.947507i	$-0.603594\pi$
$241$	−2392.47	−0.639472	−0.319736	+	0.947507i	$0.603594\pi$
$242$	−8170.68	−2.17038
$243$	0	0
$244$	−1286.09	−0.337432
$245$	−958.207	−0.249868
$246$	0	0
$247$	0	0
$248$	726.815	0.186100
$249$	0	0
$250$	4376.13	1.10708
$251$	2198.78	0.552931	0.276465	−	0.961024i	$-0.410837\pi$
$251$	2198.78	0.552931	0.276465	+	0.961024i	$0.410837\pi$
$252$	0	0
$253$	−682.581	−0.169619
$254$	−6417.29	−1.58526
$255$	0	0
$256$	2985.94	0.728988
$257$	6194.26	1.50345	0.751727	−	0.659475i	$-0.229221\pi$
$257$	6194.26	1.50345	0.751727	+	0.659475i	$0.229221\pi$
$258$	0	0
$259$	−6005.95	−1.44089
$260$	0	0
$261$	0	0
$262$	5895.92	1.39027
$263$	4181.74	0.980445	0.490222	−	0.871597i	$-0.336916\pi$
$263$	4181.74	0.980445	0.490222	+	0.871597i	$0.336916\pi$
$264$	0	0
$265$	−889.258	−0.206139
$266$	2885.90	0.665210
$267$	0	0
$268$	−1791.75	−0.408391
$269$	−2767.69	−0.627320	−0.313660	−	0.949535i	$-0.601555\pi$
$269$	−2767.69	−0.627320	−0.313660	+	0.949535i	$0.601555\pi$
$270$	0	0
$271$	7191.36	1.61197	0.805986	−	0.591935i	$-0.201636\pi$
$271$	7191.36	1.61197	0.805986	+	0.591935i	$0.201636\pi$
$272$	−4471.20	−0.996714
$273$	0	0
$274$	−6010.88	−1.32529
$275$	−4971.69	−1.09020
$276$	0	0
$277$	1317.27	0.285729	0.142864	−	0.989742i	$-0.454369\pi$
$277$	1317.27	0.285729	0.142864	+	0.989742i	$0.454369\pi$
$278$	3890.90	0.839427
$279$	0	0
$280$	1797.78	0.383707
$281$	−3948.92	−0.838338	−0.419169	−	0.907908i	$-0.637679\pi$
$281$	−3948.92	−0.838338	−0.419169	+	0.907908i	$0.637679\pi$
$282$	0	0
$283$	−4981.52	−1.04636	−0.523181	−	0.852221i	$-0.675255\pi$
$283$	−4981.52	−1.04636	−0.523181	+	0.852221i	$0.675255\pi$
$284$	−1257.99	−0.262844
$285$	0	0
$286$	0	0
$287$	−5188.97	−1.06723
$288$	0	0
$289$	−1477.03	−0.300636
$290$	4627.80	0.937082
$291$	0	0
$292$	−2031.31	−0.407100
$293$	−3203.02	−0.638644	−0.319322	−	0.947646i	$-0.603455\pi$
$293$	−3203.02	−0.638644	−0.319322	+	0.947646i	$0.603455\pi$
$294$	0	0
$295$	2607.12	0.514551
$296$	−7973.74	−1.56576
$297$	0	0
$298$	−10141.2	−1.97136
$299$	0	0
$300$	0	0
$301$	−1818.30	−0.348190
$302$	1612.84	0.307314
$303$	0	0
$304$	4894.39	0.923396
$305$	−4186.97	−0.786051
$306$	0	0
$307$	−4795.67	−0.891542	−0.445771	−	0.895147i	$-0.647070\pi$
$307$	−4795.67	−0.891542	−0.445771	+	0.895147i	$0.647070\pi$
$308$	1834.11	0.339312
$309$	0	0
$310$	−826.376	−0.151403
$311$	−630.213	−0.114907	−0.0574535	−	0.998348i	$-0.518298\pi$
$311$	−630.213	−0.114907	−0.0574535	+	0.998348i	$0.518298\pi$
$312$	0	0
$313$	−9314.73	−1.68211	−0.841054	−	0.540952i	$-0.818064\pi$
$313$	−9314.73	−1.68211	−0.841054	+	0.540952i	$0.818064\pi$
$314$	3944.83	0.708980
$315$	0	0
$316$	2758.02	0.490984
$317$	576.333	0.102114	0.0510569	−	0.998696i	$-0.483741\pi$
$317$	576.333	0.102114	0.0510569	+	0.998696i	$0.483741\pi$
$318$	0	0
$319$	−13518.8	−2.37275
$320$	2155.55	0.376559
$321$	0	0
$322$	491.231	0.0850163
$323$	−3761.18	−0.647919
$324$	0	0
$325$	0	0
$326$	−107.896	−0.0183307
$327$	0	0
$328$	−6889.08	−1.15971
$329$	−1319.71	−0.221149
$330$	0	0
$331$	−1575.95	−0.261699	−0.130849	−	0.991402i	$-0.541770\pi$
$331$	−1575.95	−0.261699	−0.130849	+	0.991402i	$0.541770\pi$
$332$	1879.47	0.310691
$333$	0	0
$334$	7015.44	1.14930
$335$	−5833.22	−0.951351
$336$	0	0
$337$	9289.32	1.50155	0.750774	−	0.660559i	$-0.229681\pi$
$337$	9289.32	1.50155	0.750774	+	0.660559i	$0.229681\pi$
$338$	0	0
$339$	0	0
$340$	818.284	0.130523
$341$	2414.03	0.383363
$342$	0	0
$343$	−6875.66	−1.08236
$344$	−2414.05	−0.378363
$345$	0	0
$346$	−2098.62	−0.326077
$347$	7701.82	1.19151	0.595757	−	0.803164i	$-0.296852\pi$
$347$	7701.82	1.19151	0.595757	+	0.803164i	$0.296852\pi$
$348$	0	0
$349$	4972.89	0.762730	0.381365	−	0.924425i	$-0.375454\pi$
$349$	4972.89	0.762730	0.381365	+	0.924425i	$0.375454\pi$
$350$	3577.96	0.546429
$351$	0	0
$352$	5720.49	0.866202
$353$	1575.34	0.237526	0.118763	−	0.992923i	$-0.462107\pi$
$353$	1575.34	0.237526	0.118763	+	0.992923i	$0.462107\pi$
$354$	0	0
$355$	−4095.49	−0.612298
$356$	−2140.45	−0.318661
$357$	0	0
$358$	−7378.41	−1.08928
$359$	7567.42	1.11252	0.556258	−	0.831010i	$-0.312237\pi$
$359$	7567.42	1.11252	0.556258	+	0.831010i	$0.312237\pi$
$360$	0	0
$361$	−2741.83	−0.399741
$362$	−6734.69	−0.977810
$363$	0	0
$364$	0	0
$365$	−6613.10	−0.948344
$366$	0	0
$367$	4368.25	0.621310	0.310655	−	0.950523i	$-0.399452\pi$
$367$	4368.25	0.621310	0.310655	+	0.950523i	$0.399452\pi$
$368$	833.112	0.118014
$369$	0	0
$370$	9066.01	1.27384
$371$	−1869.49	−0.261615
$372$	0	0
$373$	−801.944	−0.111322	−0.0556610	−	0.998450i	$-0.517727\pi$
$373$	−801.944	−0.111322	−0.0556610	+	0.998450i	$0.517727\pi$
$374$	−11625.3	−1.60730
$375$	0	0
$376$	−1752.10	−0.240313
$377$	0	0
$378$	0	0
$379$	−68.0819	−0.00922727	−0.00461363	−	0.999989i	$-0.501469\pi$
$379$	−68.0819	−0.00922727	−0.00461363	+	0.999989i	$0.501469\pi$
$380$	−895.734	−0.120922
$381$	0	0
$382$	−7884.01	−1.05597
$383$	1549.01	0.206659	0.103330	−	0.994647i	$-0.467050\pi$
$383$	1549.01	0.206659	0.103330	+	0.994647i	$0.467050\pi$
$384$	0	0
$385$	5971.11	0.790431
$386$	−846.584	−0.111632
$387$	0	0
$388$	−2166.34	−0.283451
$389$	−7300.51	−0.951544	−0.475772	−	0.879569i	$-0.657831\pi$
$389$	−7300.51	−0.951544	−0.475772	+	0.879569i	$0.657831\pi$
$390$	0	0
$391$	−640.220	−0.0828065
$392$	−2674.46	−0.344594
$393$	0	0
$394$	3906.61	0.499524
$395$	8979.00	1.14375
$396$	0	0
$397$	−6096.27	−0.770688	−0.385344	−	0.922773i	$-0.625917\pi$
$397$	−6096.27	−0.770688	−0.385344	+	0.922773i	$0.625917\pi$
$398$	10301.4	1.29740
$399$	0	0
$400$	6068.11	0.758513
$401$	7592.37	0.945498	0.472749	−	0.881197i	$-0.343262\pi$
$401$	7592.37	0.945498	0.472749	+	0.881197i	$0.343262\pi$
$402$	0	0
$403$	0	0
$404$	−2933.49	−0.361254
$405$	0	0
$406$	9729.05	1.18927
$407$	−26483.8	−3.22544
$408$	0	0
$409$	7233.86	0.874551	0.437275	−	0.899328i	$-0.355943\pi$
$409$	7233.86	0.874551	0.437275	+	0.899328i	$0.355943\pi$
$410$	7832.77	0.943495
$411$	0	0
$412$	802.678	0.0959832
$413$	5480.97	0.653029
$414$	0	0
$415$	6118.78	0.723757
$416$	0	0
$417$	0	0
$418$	12725.6	1.48907
$419$	5312.55	0.619416	0.309708	−	0.950832i	$-0.399769\pi$
$419$	5312.55	0.619416	0.309708	+	0.950832i	$0.399769\pi$
$420$	0	0
$421$	−15028.1	−1.73973	−0.869865	−	0.493290i	$-0.835794\pi$
$421$	−15028.1	−1.73973	−0.869865	+	0.493290i	$0.835794\pi$
$422$	−1049.48	−0.121062
$423$	0	0
$424$	−2482.02	−0.284286
$425$	−4663.15	−0.532226
$426$	0	0
$427$	−8802.31	−0.997596
$428$	−179.020	−0.0202179
$429$	0	0
$430$	2744.73	0.307821
$431$	−7154.66	−0.799600	−0.399800	−	0.916602i	$-0.630920\pi$
$431$	−7154.66	−0.799600	−0.399800	+	0.916602i	$0.630920\pi$
$432$	0	0
$433$	−9542.58	−1.05909	−0.529546	−	0.848281i	$-0.677638\pi$
$433$	−9542.58	−1.05909	−0.529546	+	0.848281i	$0.677638\pi$
$434$	−1737.30	−0.192150
$435$	0	0
$436$	1946.64	0.213824
$437$	700.816	0.0767153
$438$	0	0
$439$	7070.70	0.768715	0.384358	−	0.923184i	$-0.374423\pi$
$439$	7070.70	0.768715	0.384358	+	0.923184i	$0.374423\pi$
$440$	7927.49	0.858928
$441$	0	0
$442$	0	0
$443$	−2092.58	−0.224428	−0.112214	−	0.993684i	$-0.535794\pi$
$443$	−2092.58	−0.224428	−0.112214	+	0.993684i	$0.535794\pi$
$444$	0	0
$445$	−6968.42	−0.742326
$446$	18361.1	1.94938
$447$	0	0
$448$	4531.63	0.477901
$449$	5842.05	0.614038	0.307019	−	0.951703i	$-0.400668\pi$
$449$	5842.05	0.614038	0.307019	+	0.951703i	$0.400668\pi$
$450$	0	0
$451$	−22881.3	−2.38899
$452$	−1988.94	−0.206973
$453$	0	0
$454$	−9346.91	−0.966238
$455$	0	0
$456$	0	0
$457$	−5954.40	−0.609486	−0.304743	−	0.952435i	$-0.598571\pi$
$457$	−5954.40	−0.609486	−0.304743	+	0.952435i	$0.598571\pi$
$458$	−11238.0	−1.14654
$459$	0	0
$460$	−152.470	−0.0154542
$461$	−1865.94	−0.188515	−0.0942576	−	0.995548i	$-0.530048\pi$
$461$	−1865.94	−0.188515	−0.0942576	+	0.995548i	$0.530048\pi$
$462$	0	0
$463$	6700.05	0.672522	0.336261	−	0.941769i	$-0.390838\pi$
$463$	6700.05	0.672522	0.336261	+	0.941769i	$0.390838\pi$
$464$	16500.2	1.65086
$465$	0	0
$466$	7428.28	0.738430
$467$	16585.8	1.64347	0.821734	−	0.569871i	$-0.193007\pi$
$467$	16585.8	1.64347	0.821734	+	0.569871i	$0.193007\pi$
$468$	0	0
$469$	−12263.2	−1.20738
$470$	1992.11	0.195509
$471$	0	0
$472$	7276.77	0.709619
$473$	−8017.98	−0.779423
$474$	0	0
$475$	5104.51	0.493075
$476$	1720.29	0.165649
$477$	0	0
$478$	4809.97	0.460257
$479$	−6166.88	−0.588250	−0.294125	−	0.955767i	$-0.595028\pi$
$479$	−6166.88	−0.588250	−0.294125	+	0.955767i	$0.595028\pi$
$480$	0	0
$481$	0	0
$482$	7592.38	0.717476
$483$	0	0
$484$	5331.53	0.500708
$485$	−7052.71	−0.660303
$486$	0	0
$487$	−5718.51	−0.532095	−0.266047	−	0.963960i	$-0.585718\pi$
$487$	−5718.51	−0.532095	−0.266047	+	0.963960i	$0.585718\pi$
$488$	−11686.3	−1.08405
$489$	0	0
$490$	3040.82	0.280347
$491$	21060.9	1.93578	0.967888	−	0.251383i	$-0.0808854\pi$
$491$	21060.9	1.93578	0.967888	+	0.251383i	$0.0808854\pi$
$492$	0	0
$493$	−12679.8	−1.15836
$494$	0	0
$495$	0	0
$496$	−2946.40	−0.266728
$497$	−8609.97	−0.777082
$498$	0	0
$499$	7863.87	0.705481	0.352741	−	0.935721i	$-0.385250\pi$
$499$	7863.87	0.705481	0.352741	+	0.935721i	$0.385250\pi$
$500$	−2855.51	−0.255405
$501$	0	0
$502$	−6977.70	−0.620378
$503$	6504.06	0.576544	0.288272	−	0.957549i	$-0.406919\pi$
$503$	6504.06	0.576544	0.288272	+	0.957549i	$0.406919\pi$
$504$	0	0
$505$	−9550.26	−0.841546
$506$	2166.13	0.190309
$507$	0	0
$508$	4187.41	0.365721
$509$	14799.3	1.28873	0.644367	−	0.764717i	$-0.277121\pi$
$509$	14799.3	1.28873	0.644367	+	0.764717i	$0.277121\pi$
$510$	0	0
$511$	−13902.8	−1.20357
$512$	4500.03	0.388428
$513$	0	0
$514$	−19657.1	−1.68685
$515$	2613.19	0.223594
$516$	0	0
$517$	−5819.40	−0.495042
$518$	19059.5	1.61666
$519$	0	0
$520$	0	0
$521$	6633.65	0.557822	0.278911	−	0.960317i	$-0.410026\pi$
$521$	6633.65	0.557822	0.278911	+	0.960317i	$0.410026\pi$
$522$	0	0
$523$	4527.41	0.378527	0.189264	−	0.981926i	$-0.439390\pi$
$523$	4527.41	0.378527	0.189264	+	0.981926i	$0.439390\pi$
$524$	−3847.21	−0.320737
$525$	0	0
$526$	−13270.5	−1.10004
$527$	2264.21	0.187155
$528$	0	0
$529$	−12047.7	−0.990195
$530$	2822.01	0.231284
$531$	0	0
$532$	−1883.11	−0.153464
$533$	0	0
$534$	0	0
$535$	−582.818	−0.0470980
$536$	−16281.1	−1.31201
$537$	0	0
$538$	8783.11	0.703841
$539$	−8882.89	−0.709858
$540$	0	0
$541$	−8685.42	−0.690232	−0.345116	−	0.938560i	$-0.612160\pi$
$541$	−8685.42	−0.690232	−0.345116	+	0.938560i	$0.612160\pi$
$542$	−22821.4	−1.80860
$543$	0	0
$544$	5365.48	0.422873
$545$	6337.47	0.498105
$546$	0	0
$547$	24656.6	1.92731	0.963657	−	0.267142i	$-0.0860793\pi$
$547$	24656.6	1.92731	0.963657	+	0.267142i	$0.0860793\pi$
$548$	3922.22	0.305746
$549$	0	0
$550$	15777.4	1.22318
$551$	13880.0	1.07315
$552$	0	0
$553$	18876.6	1.45157
$554$	−4180.27	−0.320582
$555$	0	0
$556$	−2538.89	−0.193656
$557$	9215.90	0.701059	0.350530	−	0.936552i	$-0.386002\pi$
$557$	9215.90	0.701059	0.350530	+	0.936552i	$0.386002\pi$
$558$	0	0
$559$	0	0
$560$	−7287.93	−0.549949
$561$	0	0
$562$	12531.7	0.940600
$563$	19686.7	1.47370	0.736850	−	0.676056i	$-0.236312\pi$
$563$	19686.7	1.47370	0.736850	+	0.676056i	$0.236312\pi$
$564$	0	0
$565$	−6475.17	−0.482146
$566$	15808.6	1.17400
$567$	0	0
$568$	−11430.9	−0.844422
$569$	3559.36	0.262243	0.131121	−	0.991366i	$-0.458142\pi$
$569$	3559.36	0.262243	0.131121	+	0.991366i	$0.458142\pi$
$570$	0	0
$571$	−710.968	−0.0521070	−0.0260535	−	0.999661i	$-0.508294\pi$
$571$	−710.968	−0.0521070	−0.0260535	+	0.999661i	$0.508294\pi$
$572$	0	0
$573$	0	0
$574$	16466.9	1.19741
$575$	868.878	0.0630169
$576$	0	0
$577$	8041.67	0.580206	0.290103	−	0.956995i	$-0.406310\pi$
$577$	8041.67	0.580206	0.290103	+	0.956995i	$0.406310\pi$
$578$	4687.26	0.337308
$579$	0	0
$580$	−3019.73	−0.216186
$581$	12863.6	0.918538
$582$	0	0
$583$	−8243.72	−0.585626
$584$	−18457.9	−1.30786
$585$	0	0
$586$	10164.6	0.716546
$587$	−14641.3	−1.02949	−0.514744	−	0.857344i	$-0.672113\pi$
$587$	−14641.3	−1.02949	−0.514744	+	0.857344i	$0.672113\pi$
$588$	0	0
$589$	−2478.52	−0.173388
$590$	−8273.56	−0.577317
$591$	0	0
$592$	32324.3	2.24413
$593$	5735.76	0.397200	0.198600	−	0.980081i	$-0.436361\pi$
$593$	5735.76	0.397200	0.198600	+	0.980081i	$0.436361\pi$
$594$	0	0
$595$	5600.55	0.385882
$596$	6617.35	0.454794
$597$	0	0
$598$	0	0
$599$	16109.3	1.09884	0.549422	−	0.835545i	$-0.314848\pi$
$599$	16109.3	1.09884	0.549422	+	0.835545i	$0.314848\pi$
$600$	0	0
$601$	−21005.4	−1.42567	−0.712837	−	0.701330i	$-0.752590\pi$
$601$	−21005.4	−1.42567	−0.712837	+	0.701330i	$0.752590\pi$
$602$	5770.28	0.390663
$603$	0	0
$604$	−1052.41	−0.0708975
$605$	17357.3	1.16640
$606$	0	0
$607$	7478.16	0.500048	0.250024	−	0.968240i	$-0.419561\pi$
$607$	7478.16	0.500048	0.250024	+	0.968240i	$0.419561\pi$
$608$	−5873.31	−0.391767
$609$	0	0
$610$	13287.1	0.881934
$611$	0	0
$612$	0	0
$613$	−16435.5	−1.08291	−0.541455	−	0.840730i	$-0.682126\pi$
$613$	−16435.5	−1.08291	−0.541455	+	0.840730i	$0.682126\pi$
$614$	15218.8	1.00029
$615$	0	0
$616$	16666.0	1.09009
$617$	1290.89	0.0842289	0.0421145	−	0.999113i	$-0.486591\pi$
$617$	1290.89	0.0842289	0.0421145	+	0.999113i	$0.486591\pi$
$618$	0	0
$619$	26719.0	1.73494	0.867470	−	0.497490i	$-0.165745\pi$
$619$	26719.0	1.73494	0.867470	+	0.497490i	$0.165745\pi$
$620$	539.227	0.0349289
$621$	0	0
$622$	1999.94	0.128924
$623$	−14649.8	−0.942103
$624$	0	0
$625$	647.683	0.0414517
$626$	29559.8	1.88729
$627$	0	0
$628$	−2574.08	−0.163562
$629$	−24840.2	−1.57463
$630$	0	0
$631$	10697.4	0.674893	0.337447	−	0.941345i	$-0.390437\pi$
$631$	10697.4	0.674893	0.337447	+	0.941345i	$0.390437\pi$
$632$	25061.4	1.57735
$633$	0	0
$634$	−1828.96	−0.114570
$635$	13632.5	0.851951
$636$	0	0
$637$	0	0
$638$	42901.2	2.66219
$639$	0	0
$640$	−11777.1	−0.727393
$641$	−23572.8	−1.45253	−0.726264	−	0.687416i	$-0.758745\pi$
$641$	−23572.8	−1.45253	−0.726264	+	0.687416i	$0.758745\pi$
$642$	0	0
$643$	14000.3	0.858661	0.429331	−	0.903147i	$-0.358750\pi$
$643$	14000.3	0.858661	0.429331	+	0.903147i	$0.358750\pi$
$644$	−320.538	−0.0196133
$645$	0	0
$646$	11935.9	0.726953
$647$	−614.196	−0.0373207	−0.0186604	−	0.999826i	$-0.505940\pi$
$647$	−614.196	−0.0373207	−0.0186604	+	0.999826i	$0.505940\pi$
$648$	0	0
$649$	24168.9	1.46181
$650$	0	0
$651$	0	0
$652$	70.4042	0.00422890
$653$	5333.42	0.319622	0.159811	−	0.987148i	$-0.448912\pi$
$653$	5333.42	0.319622	0.159811	+	0.987148i	$0.448912\pi$
$654$	0	0
$655$	−12524.9	−0.747161
$656$	27927.3	1.66216
$657$	0	0
$658$	4188.03	0.248125
$659$	−21396.8	−1.26480	−0.632398	−	0.774644i	$-0.717929\pi$
$659$	−21396.8	−1.26480	−0.632398	+	0.774644i	$0.717929\pi$
$660$	0	0
$661$	16107.9	0.947841	0.473921	−	0.880568i	$-0.342838\pi$
$661$	16107.9	0.947841	0.473921	+	0.880568i	$0.342838\pi$
$662$	5001.20	0.293621
$663$	0	0
$664$	17078.2	0.998136
$665$	−6130.63	−0.357497
$666$	0	0
$667$	2362.62	0.137153
$668$	−4577.72	−0.265145
$669$	0	0
$670$	18511.4	1.06740
$671$	−38814.6	−2.23312
$672$	0	0
$673$	20329.9	1.16443	0.582213	−	0.813036i	$-0.302187\pi$
$673$	20329.9	1.16443	0.582213	+	0.813036i	$0.302187\pi$
$674$	−29479.1	−1.68471
$675$	0	0
$676$	0	0
$677$	24883.5	1.41263	0.706314	−	0.707899i	$-0.250357\pi$
$677$	24883.5	1.41263	0.706314	+	0.707899i	$0.250357\pi$
$678$	0	0
$679$	−14827.0	−0.838007
$680$	7435.51	0.419322
$681$	0	0
$682$	−7660.78	−0.430127
$683$	−258.953	−0.0145074	−0.00725369	−	0.999974i	$-0.502309\pi$
$683$	−258.953	−0.0145074	−0.00725369	+	0.999974i	$0.502309\pi$
$684$	0	0
$685$	12769.1	0.712240
$686$	21819.5	1.21439
$687$	0	0
$688$	9786.20	0.542290
$689$	0	0
$690$	0	0
$691$	658.193	0.0362357	0.0181178	−	0.999836i	$-0.494233\pi$
$691$	658.193	0.0362357	0.0181178	+	0.999836i	$0.494233\pi$
$692$	1369.39	0.0752262
$693$	0	0
$694$	−24441.3	−1.33686
$695$	−8265.60	−0.451125
$696$	0	0
$697$	−21461.2	−1.16629
$698$	−15781.2	−0.855768
$699$	0	0
$700$	−2334.69	−0.126062
$701$	8222.16	0.443005	0.221503	−	0.975160i	$-0.428904\pi$
$701$	8222.16	0.443005	0.221503	+	0.975160i	$0.428904\pi$
$702$	0	0
$703$	27191.3	1.45881
$704$	19982.7	1.06978
$705$	0	0
$706$	−4999.25	−0.266500
$707$	−20077.6	−1.06803
$708$	0	0
$709$	−6817.51	−0.361124	−0.180562	−	0.983564i	$-0.557792\pi$
$709$	−6817.51	−0.361124	−0.180562	+	0.983564i	$0.557792\pi$
$710$	12996.8	0.686987
$711$	0	0
$712$	−19449.6	−1.02374
$713$	−421.888	−0.0221596
$714$	0	0
$715$	0	0
$716$	4814.56	0.251297
$717$	0	0
$718$	−24014.8	−1.24822
$719$	−23385.7	−1.21299	−0.606496	−	0.795087i	$-0.707425\pi$
$719$	−23385.7	−1.21299	−0.606496	+	0.795087i	$0.707425\pi$
$720$	0	0
$721$	5493.73	0.283769
$722$	8701.03	0.448502
$723$	0	0
$724$	4394.52	0.225582
$725$	17208.5	0.881529
$726$	0	0
$727$	20488.6	1.04523	0.522613	−	0.852570i	$-0.324957\pi$
$727$	20488.6	1.04523	0.522613	+	0.852570i	$0.324957\pi$
$728$	0	0
$729$	0	0
$730$	20986.3	1.06402
$731$	−7520.38	−0.380508
$732$	0	0
$733$	16993.4	0.856299	0.428149	−	0.903708i	$-0.359166\pi$
$733$	16993.4	0.856299	0.428149	+	0.903708i	$0.359166\pi$
$734$	−13862.4	−0.697099
$735$	0	0
$736$	−999.742	−0.0500693
$737$	−54075.8	−2.70272
$738$	0	0
$739$	14014.3	0.697600	0.348800	−	0.937197i	$-0.386589\pi$
$739$	14014.3	0.697600	0.348800	+	0.937197i	$0.386589\pi$
$740$	−5915.75	−0.293875
$741$	0	0
$742$	5932.73	0.293528
$743$	−141.560	−0.00698971	−0.00349485	−	0.999994i	$-0.501112\pi$
$743$	−141.560	−0.00698971	−0.00349485	+	0.999994i	$0.501112\pi$
$744$	0	0
$745$	21543.4	1.05945
$746$	2544.92	0.124901
$747$	0	0
$748$	7585.76	0.370806
$749$	−1225.26	−0.0597731
$750$	0	0
$751$	20734.3	1.00746	0.503732	−	0.863860i	$-0.331960\pi$
$751$	20734.3	1.00746	0.503732	+	0.863860i	$0.331960\pi$
$752$	7102.76	0.344430
$753$	0	0
$754$	0	0
$755$	−3426.23	−0.165157
$756$	0	0
$757$	17449.8	0.837812	0.418906	−	0.908030i	$-0.362414\pi$
$757$	17449.8	0.837812	0.418906	+	0.908030i	$0.362414\pi$
$758$	216.054	0.0103528
$759$	0	0
$760$	−8139.27	−0.388477
$761$	−424.121	−0.0202028	−0.0101014	−	0.999949i	$-0.503215\pi$
$761$	−424.121	−0.0202028	−0.0101014	+	0.999949i	$0.503215\pi$
$762$	0	0
$763$	13323.3	0.632157
$764$	5144.48	0.243613
$765$	0	0
$766$	−4915.68	−0.231868
$767$	0	0
$768$	0	0
$769$	−38060.7	−1.78479	−0.892396	−	0.451252i	$-0.850977\pi$
$769$	−38060.7	−1.78479	−0.892396	+	0.451252i	$0.850977\pi$
$770$	−18949.0	−0.886849
$771$	0	0
$772$	552.413	0.0257536
$773$	−16683.3	−0.776268	−0.388134	−	0.921603i	$-0.626880\pi$
$773$	−16683.3	−0.776268	−0.388134	+	0.921603i	$0.626880\pi$
$774$	0	0
$775$	−3072.89	−0.142428
$776$	−19684.9	−0.910627
$777$	0	0
$778$	23167.7	1.06761
$779$	23492.5	1.08050
$780$	0	0
$781$	−37966.5	−1.73950
$782$	2031.70	0.0929073
$783$	0	0
$784$	10841.9	0.493889
$785$	−8380.17	−0.381020
$786$	0	0
$787$	39105.2	1.77122	0.885609	−	0.464432i	$-0.153741\pi$
$787$	39105.2	1.77122	0.885609	+	0.464432i	$0.153741\pi$
$788$	−2549.14	−0.115240
$789$	0	0
$790$	−28494.3	−1.28327
$791$	−13612.8	−0.611903
$792$	0	0
$793$	0	0
$794$	19346.2	0.864697
$795$	0	0
$796$	−6721.90	−0.299311
$797$	4346.93	0.193195	0.0965974	−	0.995324i	$-0.469204\pi$
$797$	4346.93	0.193195	0.0965974	+	0.995324i	$0.469204\pi$
$798$	0	0
$799$	−5458.25	−0.241676
$800$	−7281.78	−0.321812
$801$	0	0
$802$	−24093.9	−1.06083
$803$	−61305.6	−2.69418
$804$	0	0
$805$	−1043.54	−0.0456895
$806$	0	0
$807$	0	0
$808$	−26655.8	−1.16058
$809$	23030.2	1.00086	0.500432	−	0.865776i	$-0.333174\pi$
$809$	23030.2	1.00086	0.500432	+	0.865776i	$0.333174\pi$
$810$	0	0
$811$	−7898.11	−0.341973	−0.170987	−	0.985273i	$-0.554695\pi$
$811$	−7898.11	−0.341973	−0.170987	+	0.985273i	$0.554695\pi$
$812$	−6348.41	−0.274366
$813$	0	0
$814$	84044.8	3.61888
$815$	229.207	0.00985127
$816$	0	0
$817$	8232.18	0.352518
$818$	−22956.2	−0.981230
$819$	0	0
$820$	−5111.04	−0.217665
$821$	−3939.61	−0.167471	−0.0837353	−	0.996488i	$-0.526685\pi$
$821$	−3939.61	−0.167471	−0.0837353	+	0.996488i	$0.526685\pi$
$822$	0	0
$823$	−17599.8	−0.745430	−0.372715	−	0.927946i	$-0.621573\pi$
$823$	−17599.8	−0.745430	−0.372715	+	0.927946i	$0.621573\pi$
$824$	7293.70	0.308359
$825$	0	0
$826$	−17393.6	−0.732687
$827$	12510.6	0.526042	0.263021	−	0.964790i	$-0.415281\pi$
$827$	12510.6	0.526042	0.263021	+	0.964790i	$0.415281\pi$
$828$	0	0
$829$	28630.8	1.19950	0.599752	−	0.800186i	$-0.295266\pi$
$829$	28630.8	1.19950	0.599752	+	0.800186i	$0.295266\pi$
$830$	−19417.6	−0.812042
$831$	0	0
$832$	0	0
$833$	−8331.62	−0.346547
$834$	0	0
$835$	−14903.2	−0.617660
$836$	−8303.75	−0.343530
$837$	0	0
$838$	−16859.1	−0.694973
$839$	21250.3	0.874426	0.437213	−	0.899358i	$-0.355966\pi$
$839$	21250.3	0.874426	0.437213	+	0.899358i	$0.355966\pi$
$840$	0	0
$841$	22403.7	0.918600
$842$	47690.9	1.95194
$843$	0	0
$844$	684.810	0.0279291
$845$	0	0
$846$	0	0
$847$	36490.4	1.48031
$848$	10061.7	0.407454
$849$	0	0
$850$	14798.2	0.597147
$851$	4628.45	0.186441
$852$	0	0
$853$	−34721.1	−1.39370	−0.696852	−	0.717215i	$-0.745416\pi$
$853$	−34721.1	−1.39370	−0.696852	+	0.717215i	$0.745416\pi$
$854$	27933.6	1.11928
$855$	0	0
$856$	−1626.71	−0.0649529
$857$	−4898.06	−0.195233	−0.0976163	−	0.995224i	$-0.531122\pi$
$857$	−4898.06	−0.195233	−0.0976163	+	0.995224i	$0.531122\pi$
$858$	0	0
$859$	−12564.2	−0.499051	−0.249526	−	0.968368i	$-0.580275\pi$
$859$	−12564.2	−0.499051	−0.249526	+	0.968368i	$0.580275\pi$
$860$	−1791.00	−0.0710145
$861$	0	0
$862$	22704.9	0.897137
$863$	−22826.6	−0.900380	−0.450190	−	0.892933i	$-0.648644\pi$
$863$	−22826.6	−0.900380	−0.450190	+	0.892933i	$0.648644\pi$
$864$	0	0
$865$	4458.18	0.175240
$866$	30282.8	1.18828
$867$	0	0
$868$	1133.62	0.0443291
$869$	83238.3	3.24933
$870$	0	0
$871$	0	0
$872$	17688.6	0.686939
$873$	0	0
$874$	−2224.00	−0.0860731
$875$	−19543.9	−0.755089
$876$	0	0
$877$	−25212.7	−0.970776	−0.485388	−	0.874299i	$-0.661322\pi$
$877$	−25212.7	−0.970776	−0.485388	+	0.874299i	$0.661322\pi$
$878$	−22438.5	−0.862484
$879$	0	0
$880$	−32136.9	−1.23106
$881$	18026.2	0.689352	0.344676	−	0.938722i	$-0.387989\pi$
$881$	18026.2	0.689352	0.344676	+	0.938722i	$0.387989\pi$
$882$	0	0
$883$	−18833.1	−0.717764	−0.358882	−	0.933383i	$-0.616842\pi$
$883$	−18833.1	−0.717764	−0.358882	+	0.933383i	$0.616842\pi$
$884$	0	0
$885$	0	0
$886$	6640.68	0.251803
$887$	−38451.4	−1.45555	−0.727775	−	0.685816i	$-0.759445\pi$
$887$	−38451.4	−1.45555	−0.727775	+	0.685816i	$0.759445\pi$
$888$	0	0
$889$	28659.7	1.08123
$890$	22113.9	0.832876
$891$	0	0
$892$	−11981.0	−0.449723
$893$	5974.86	0.223898
$894$	0	0
$895$	15674.2	0.585399
$896$	−24759.1	−0.923152
$897$	0	0
$898$	−18539.4	−0.688940
$899$	−8355.68	−0.309986
$900$	0	0
$901$	−7732.11	−0.285898
$902$	72612.3	2.68041
$903$	0	0
$904$	−18072.9	−0.664929
$905$	14306.8	0.525495
$906$	0	0
$907$	−5531.31	−0.202496	−0.101248	−	0.994861i	$-0.532284\pi$
$907$	−5531.31	−0.202496	−0.101248	+	0.994861i	$0.532284\pi$
$908$	6099.05	0.222912
$909$	0	0
$910$	0	0
$911$	−15695.2	−0.570806	−0.285403	−	0.958408i	$-0.592128\pi$
$911$	−15695.2	−0.570806	−0.285403	+	0.958408i	$0.592128\pi$
$912$	0	0
$913$	56723.1	2.05615
$914$	18895.9	0.683832
$915$	0	0
$916$	7333.01	0.264508
$917$	−26331.3	−0.948240
$918$	0	0
$919$	−51503.5	−1.84869	−0.924344	−	0.381561i	$-0.875386\pi$
$919$	−51503.5	−1.84869	−0.924344	+	0.381561i	$0.875386\pi$
$920$	−1385.45	−0.0496488
$921$	0	0
$922$	5921.45	0.211510
$923$	0	0
$924$	0	0
$925$	33712.0	1.19832
$926$	−21262.2	−0.754557
$927$	0	0
$928$	−19800.3	−0.700407
$929$	−42927.0	−1.51603	−0.758014	−	0.652238i	$-0.773830\pi$
$929$	−42927.0	−1.51603	−0.758014	+	0.652238i	$0.773830\pi$
$930$	0	0
$931$	9120.20	0.321055
$932$	−4847.10	−0.170356
$933$	0	0
$934$	−52634.1	−1.84394
$935$	24696.2	0.863797
$936$	0	0
$937$	44206.6	1.54127	0.770633	−	0.637280i	$-0.219940\pi$
$937$	44206.6	1.54127	0.770633	+	0.637280i	$0.219940\pi$
$938$	38916.6	1.35466
$939$	0	0
$940$	−1299.89	−0.0451041
$941$	−44067.7	−1.52664	−0.763319	−	0.646022i	$-0.776432\pi$
$941$	−44067.7	−1.52664	−0.763319	+	0.646022i	$0.776432\pi$
$942$	0	0
$943$	3998.85	0.138092
$944$	−29498.9	−1.01706
$945$	0	0
$946$	25444.6	0.874498
$947$	−44402.5	−1.52364	−0.761820	−	0.647789i	$-0.775694\pi$
$947$	−44402.5	−1.52364	−0.761820	+	0.647789i	$0.775694\pi$
$948$	0	0
$949$	0	0
$950$	−16198.9	−0.553221
$951$	0	0
$952$	15631.7	0.532172
$953$	10361.7	0.352202	0.176101	−	0.984372i	$-0.443651\pi$
$953$	10361.7	0.352202	0.176101	+	0.984372i	$0.443651\pi$
$954$	0	0
$955$	16748.3	0.567500
$956$	−3138.60	−0.106182
$957$	0	0
$958$	19570.2	0.660006
$959$	26844.7	0.903920
$960$	0	0
$961$	−28298.9	−0.949916
$962$	0	0
$963$	0	0
$964$	−4954.18	−0.165522
$965$	1798.43	0.0599933
$966$	0	0
$967$	−8432.54	−0.280426	−0.140213	−	0.990121i	$-0.544779\pi$
$967$	−8432.54	−0.280426	−0.140213	+	0.990121i	$0.544779\pi$
$968$	48446.1	1.60859
$969$	0	0
$970$	22381.4	0.740848
$971$	36917.1	1.22011	0.610055	−	0.792359i	$-0.291147\pi$
$971$	36917.1	1.22011	0.610055	+	0.792359i	$0.291147\pi$
$972$	0	0
$973$	−17376.8	−0.572534
$974$	18147.4	0.597001
$975$	0	0
$976$	47374.5	1.55371
$977$	−38306.9	−1.25440	−0.627199	−	0.778859i	$-0.715799\pi$
$977$	−38306.9	−1.25440	−0.627199	+	0.778859i	$0.715799\pi$
$978$	0	0
$979$	−64599.6	−2.10890
$980$	−1984.19	−0.0646763
$981$	0	0
$982$	−66835.6	−2.17190
$983$	−18810.9	−0.610350	−0.305175	−	0.952296i	$-0.598715\pi$
$983$	−18810.9	−0.610350	−0.305175	+	0.952296i	$0.598715\pi$
$984$	0	0
$985$	−8298.97	−0.268454
$986$	40238.8	1.29966
$987$	0	0
$988$	0	0
$989$	1401.26	0.0450532
$990$	0	0
$991$	−6492.09	−0.208101	−0.104051	−	0.994572i	$-0.533180\pi$
$991$	−6492.09	−0.208101	−0.104051	+	0.994572i	$0.533180\pi$
$992$	3535.71	0.113164
$993$	0	0
$994$	27323.2	0.871872
$995$	−21883.8	−0.697249
$996$	0	0
$997$	48552.1	1.54229	0.771143	−	0.636662i	$-0.219685\pi$
$997$	48552.1	1.54229	0.771143	+	0.636662i	$0.219685\pi$
$998$	−24955.5	−0.791537
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bi.1.2		9
3.2	odd	2		507.4.a.o.1.8	✓	9
13.12	even	2		1521.4.a.bf.1.8		9
39.5	even	4		507.4.b.k.337.5		18
39.8	even	4		507.4.b.k.337.14		18
39.38	odd	2		507.4.a.p.1.2	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.8	✓	9	3.2	odd	2
507.4.a.p.1.2	yes	9	39.38	odd	2
507.4.b.k.337.5		18	39.5	even	4
507.4.b.k.337.14		18	39.8	even	4
1521.4.a.bf.1.8		9	13.12	even	2
1521.4.a.bi.1.2		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)

\(f(q)\)	\(=\)	\(q-3.17344 q^{2} +2.07074 q^{4} +6.74147 q^{5} +14.1726 q^{7} +18.8162 q^{8} +O(q^{10})\)	\(q-3.17344 q^{2} +2.07074 q^{4} +6.74147 q^{5} +14.1726 q^{7} +18.8162 q^{8} -21.3937 q^{10} +62.4956 q^{11} -44.9761 q^{14} -76.2779 q^{16} +58.6172 q^{17} -64.1652 q^{19} +13.9598 q^{20} -198.326 q^{22} -10.9221 q^{23} -79.5526 q^{25} +29.3478 q^{28} -216.316 q^{29} +38.6271 q^{31} +91.5342 q^{32} -186.018 q^{34} +95.5445 q^{35} -423.770 q^{37} +203.625 q^{38} +126.849 q^{40} -366.126 q^{41} -128.297 q^{43} +129.412 q^{44} +34.6605 q^{46} -93.1169 q^{47} -142.136 q^{49} +252.455 q^{50} -131.909 q^{53} +421.313 q^{55} +266.675 q^{56} +686.467 q^{58} +386.729 q^{59} -621.077 q^{61} -122.581 q^{62} +319.745 q^{64} -865.273 q^{67} +121.381 q^{68} -303.205 q^{70} -607.506 q^{71} -980.958 q^{73} +1344.81 q^{74} -132.869 q^{76} +885.728 q^{77} +1331.91 q^{79} -514.226 q^{80} +1161.88 q^{82} +907.633 q^{83} +395.166 q^{85} +407.142 q^{86} +1175.93 q^{88} -1033.67 q^{89} -22.6167 q^{92} +295.501 q^{94} -432.568 q^{95} -1046.17 q^{97} +451.061 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more