LMFDB - Embedded newform 2366.4.a.s.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2366,4,Mod(1,2366)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2366, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2366.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 2366 = 2 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2366.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-10,1,20,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$139.598519074$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 95x^{3} + 122x^{2} + 2074x - 2668 $ x^5 - x^4 - 95x^3 + 122x^2 + 2074*x - 2668
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.70204$ of defining polynomial
Character	$\chi$	$=$	2366.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -5.70204 q^{3} +4.00000 q^{4} +9.31361 q^{5} +11.4041 q^{6} +7.00000 q^{7} -8.00000 q^{8} +5.51321 q^{9} -18.6272 q^{10} -43.9477 q^{11} -22.8081 q^{12} -14.0000 q^{14} -53.1065 q^{15} +16.0000 q^{16} -66.0485 q^{17} -11.0264 q^{18} +41.0983 q^{19} +37.2544 q^{20} -39.9143 q^{21} +87.8953 q^{22} +79.3462 q^{23} +45.6163 q^{24} -38.2567 q^{25} +122.518 q^{27} +28.0000 q^{28} +94.2019 q^{29} +106.213 q^{30} +53.4525 q^{31} -32.0000 q^{32} +250.591 q^{33} +132.097 q^{34} +65.1952 q^{35} +22.0528 q^{36} +56.6658 q^{37} -82.1965 q^{38} -74.5089 q^{40} -93.8897 q^{41} +79.8285 q^{42} +11.9989 q^{43} -175.791 q^{44} +51.3479 q^{45} -158.692 q^{46} +155.931 q^{47} -91.2326 q^{48} +49.0000 q^{49} +76.5134 q^{50} +376.611 q^{51} +431.367 q^{53} -245.037 q^{54} -409.311 q^{55} -56.0000 q^{56} -234.344 q^{57} -188.404 q^{58} -516.538 q^{59} -212.426 q^{60} +126.857 q^{61} -106.905 q^{62} +38.5925 q^{63} +64.0000 q^{64} -501.182 q^{66} -779.421 q^{67} -264.194 q^{68} -452.435 q^{69} -130.390 q^{70} -338.952 q^{71} -44.1057 q^{72} -73.6584 q^{73} -113.332 q^{74} +218.141 q^{75} +164.393 q^{76} -307.634 q^{77} +584.714 q^{79} +149.018 q^{80} -847.461 q^{81} +187.779 q^{82} +633.536 q^{83} -159.657 q^{84} -615.150 q^{85} -23.9979 q^{86} -537.142 q^{87} +351.581 q^{88} +1603.47 q^{89} -102.696 q^{90} +317.385 q^{92} -304.788 q^{93} -311.862 q^{94} +382.773 q^{95} +182.465 q^{96} -1587.64 q^{97} -98.0000 q^{98} -242.293 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{2} + q^{3} + 20 q^{4} + 6 q^{5} - 2 q^{6} + 35 q^{7} - 40 q^{8} + 56 q^{9} - 12 q^{10} - 9 q^{11} + 4 q^{12} - 70 q^{14} + 137 q^{15} + 80 q^{16} - 89 q^{17} - 112 q^{18} - 49 q^{19} + 24 q^{20}+ \cdots + 81 q^{99}+O(q^{100}) $ 5 * q - 10 * q^2 + q^3 + 20 * q^4 + 6 * q^5 - 2 * q^6 + 35 * q^7 - 40 * q^8 + 56 * q^9 - 12 * q^10 - 9 * q^11 + 4 * q^12 - 70 * q^14 + 137 * q^15 + 80 * q^16 - 89 * q^17 - 112 * q^18 - 49 * q^19 + 24 * q^20 + 7 * q^21 + 18 * q^22 - 60 * q^23 - 8 * q^24 - 59 * q^25 - 134 * q^27 + 140 * q^28 - 196 * q^29 - 274 * q^30 + 154 * q^31 - 160 * q^32 - 433 * q^33 + 178 * q^34 + 42 * q^35 + 224 * q^36 + 61 * q^37 + 98 * q^38 - 48 * q^40 - 237 * q^41 - 14 * q^42 - 459 * q^43 - 36 * q^44 + 384 * q^45 + 120 * q^46 - 314 * q^47 + 16 * q^48 + 245 * q^49 + 118 * q^50 - 1069 * q^51 - 143 * q^53 + 268 * q^54 - 1899 * q^55 - 280 * q^56 + 211 * q^57 + 392 * q^58 + 178 * q^59 + 548 * q^60 - 811 * q^61 - 308 * q^62 + 392 * q^63 + 320 * q^64 + 866 * q^66 - 1078 * q^67 - 356 * q^68 - 654 * q^69 - 84 * q^70 + 1670 * q^71 - 448 * q^72 + 249 * q^73 - 122 * q^74 + 771 * q^75 - 196 * q^76 - 63 * q^77 + 4 * q^79 + 96 * q^80 - 2179 * q^81 + 474 * q^82 + 1986 * q^83 + 28 * q^84 - 2910 * q^85 + 918 * q^86 - 1427 * q^87 + 72 * q^88 + 461 * q^89 - 768 * q^90 - 240 * q^92 - 1055 * q^93 + 628 * q^94 - 543 * q^95 - 32 * q^96 + 2415 * q^97 - 490 * q^98 + 81 * q^99

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−5.70204	−1.09736	−0.548679	−	0.836033i	$-0.684869\pi$
$3$	−5.70204	−1.09736	−0.548679	+	0.836033i	$0.684869\pi$
$4$	4.00000	0.500000
$5$	9.31361	0.833034	0.416517	−	0.909128i	$-0.363251\pi$
$5$	9.31361	0.833034	0.416517	+	0.909128i	$0.363251\pi$
$6$	11.4041	0.775949
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	5.51321	0.204193
$10$	−18.6272	−0.589044
$11$	−43.9477	−1.20461	−0.602305	−	0.798266i	$-0.705751\pi$
$11$	−43.9477	−1.20461	−0.602305	+	0.798266i	$0.705751\pi$
$12$	−22.8081	−0.548679
$13$	0	0
$13$	0	0
$14$	−14.0000	−0.267261
$15$	−53.1065	−0.914136
$16$	16.0000	0.250000
$17$	−66.0485	−0.942301	−0.471150	−	0.882053i	$-0.656161\pi$
$17$	−66.0485	−0.942301	−0.471150	+	0.882053i	$0.656161\pi$
$18$	−11.0264	−0.144386
$19$	41.0983	0.496242	0.248121	−	0.968729i	$-0.420187\pi$
$19$	41.0983	0.496242	0.248121	+	0.968729i	$0.420187\pi$
$20$	37.2544	0.416517
$21$	−39.9143	−0.414762
$22$	87.8953	0.851788
$23$	79.3462	0.719340	0.359670	−	0.933079i	$-0.382889\pi$
$23$	79.3462	0.719340	0.359670	+	0.933079i	$0.382889\pi$
$24$	45.6163	0.387974
$25$	−38.2567	−0.306054
$26$	0	0
$27$	122.518	0.873285
$28$	28.0000	0.188982
$29$	94.2019	0.603202	0.301601	−	0.953434i	$-0.402479\pi$
$29$	94.2019	0.603202	0.301601	+	0.953434i	$0.402479\pi$
$30$	106.213	0.646392
$31$	53.4525	0.309689	0.154844	−	0.987939i	$-0.450512\pi$
$31$	53.4525	0.309689	0.154844	+	0.987939i	$0.450512\pi$
$32$	−32.0000	−0.176777
$33$	250.591	1.32189
$34$	132.097	0.666307
$35$	65.1952	0.314857
$36$	22.0528	0.102097
$37$	56.6658	0.251778	0.125889	−	0.992044i	$-0.459822\pi$
$37$	56.6658	0.251778	0.125889	+	0.992044i	$0.459822\pi$
$38$	−82.1965	−0.350896
$39$	0	0
$40$	−74.5089	−0.294522
$41$	−93.8897	−0.357637	−0.178818	−	0.983882i	$-0.557227\pi$
$41$	−93.8897	−0.357637	−0.178818	+	0.983882i	$0.557227\pi$
$42$	79.8285	0.293281
$43$	11.9989	0.0425540	0.0212770	−	0.999774i	$-0.493227\pi$
$43$	11.9989	0.0425540	0.0212770	+	0.999774i	$0.493227\pi$
$44$	−175.791	−0.602305
$45$	51.3479	0.170100
$46$	−158.692	−0.508650
$47$	155.931	0.483933	0.241966	−	0.970285i	$-0.422208\pi$
$47$	155.931	0.483933	0.241966	+	0.970285i	$0.422208\pi$
$48$	−91.2326	−0.274339
$49$	49.0000	0.142857
$50$	76.5134	0.216413
$51$	376.611	1.03404
$52$	0	0
$53$	431.367	1.11798	0.558989	−	0.829175i	$-0.311189\pi$
$53$	431.367	1.11798	0.558989	+	0.829175i	$0.311189\pi$
$54$	−245.037	−0.617505
$55$	−409.311	−1.00348
$56$	−56.0000	−0.133631
$57$	−234.344	−0.544554
$58$	−188.404	−0.426528
$59$	−516.538	−1.13979	−0.569895	−	0.821718i	$-0.693016\pi$
$59$	−516.538	−1.13979	−0.569895	+	0.821718i	$0.693016\pi$
$60$	−212.426	−0.457068
$61$	126.857	0.266269	0.133134	−	0.991098i	$-0.457496\pi$
$61$	126.857	0.266269	0.133134	+	0.991098i	$0.457496\pi$
$62$	−106.905	−0.218983
$63$	38.5925	0.0771777
$64$	64.0000	0.125000
$65$	0	0
$66$	−501.182	−0.934716
$67$	−779.421	−1.42121	−0.710607	−	0.703589i	$-0.751580\pi$
$67$	−779.421	−1.42121	−0.710607	+	0.703589i	$0.751580\pi$
$68$	−264.194	−0.471150
$69$	−452.435	−0.789373
$70$	−130.390	−0.222638
$71$	−338.952	−0.566566	−0.283283	−	0.959036i	$-0.591424\pi$
$71$	−338.952	−0.566566	−0.283283	+	0.959036i	$0.591424\pi$
$72$	−44.1057	−0.0721931
$73$	−73.6584	−0.118097	−0.0590484	−	0.998255i	$-0.518807\pi$
$73$	−73.6584	−0.118097	−0.0590484	+	0.998255i	$0.518807\pi$
$74$	−113.332	−0.178034
$75$	218.141	0.335850
$76$	164.393	0.248121
$77$	−307.634	−0.455300
$78$	0	0
$79$	584.714	0.832727	0.416364	−	0.909198i	$-0.363304\pi$
$79$	584.714	0.832727	0.416364	+	0.909198i	$0.363304\pi$
$80$	149.018	0.208259
$81$	−847.461	−1.16250
$82$	187.779	0.252887
$83$	633.536	0.837827	0.418913	−	0.908026i	$-0.362411\pi$
$83$	633.536	0.837827	0.418913	+	0.908026i	$0.362411\pi$
$84$	−159.657	−0.207381
$85$	−615.150	−0.784969
$86$	−23.9979	−0.0300902
$87$	−537.142	−0.661928
$88$	351.581	0.425894
$89$	1603.47	1.90975	0.954875	−	0.297008i	$-0.0959887\pi$
$89$	1603.47	1.90975	0.954875	+	0.297008i	$0.0959887\pi$
$90$	−102.696	−0.120279
$91$	0	0
$92$	317.385	0.359670
$93$	−304.788	−0.339839
$94$	−311.862	−0.342192
$95$	382.773	0.413386
$96$	182.465	0.193987
$97$	−1587.64	−1.66186	−0.830930	−	0.556377i	$-0.812191\pi$
$97$	−1587.64	−1.66186	−0.830930	+	0.556377i	$0.812191\pi$
$98$	−98.0000	−0.101015
$99$	−242.293	−0.245973
$100$	−153.027	−0.153027
$101$	−203.646	−0.200629	−0.100315	−	0.994956i	$-0.531985\pi$
$101$	−203.646	−0.200629	−0.100315	+	0.994956i	$0.531985\pi$
$102$	−753.222	−0.731177
$103$	−820.107	−0.784540	−0.392270	−	0.919850i	$-0.628310\pi$
$103$	−820.107	−0.784540	−0.392270	+	0.919850i	$0.628310\pi$
$104$	0	0
$105$	−371.746	−0.345511
$106$	−862.734	−0.790530
$107$	−1117.96	−1.01007	−0.505034	−	0.863099i	$-0.668520\pi$
$107$	−1117.96	−1.01007	−0.505034	+	0.863099i	$0.668520\pi$
$108$	490.074	0.436642
$109$	1488.63	1.30812	0.654061	−	0.756442i	$-0.273064\pi$
$109$	1488.63	1.30812	0.654061	+	0.756442i	$0.273064\pi$
$110$	818.622	0.709569
$111$	−323.110	−0.276291
$112$	112.000	0.0944911
$113$	−780.189	−0.649504	−0.324752	−	0.945799i	$-0.605281\pi$
$113$	−780.189	−0.649504	−0.324752	+	0.945799i	$0.605281\pi$
$114$	468.688	0.385058
$115$	738.999	0.599235
$116$	376.808	0.301601
$117$	0	0
$118$	1033.08	0.805953
$119$	−462.339	−0.356156
$120$	424.852	0.323196
$121$	600.396	0.451087
$122$	−253.714	−0.188280
$123$	535.362	0.392455
$124$	213.810	0.154844
$125$	−1520.51	−1.08799
$126$	−77.1850	−0.0545729
$127$	1838.12	1.28430	0.642152	−	0.766577i	$-0.278042\pi$
$127$	1838.12	1.28430	0.642152	+	0.766577i	$0.278042\pi$
$128$	−128.000	−0.0883883
$129$	−68.4184	−0.0466969
$130$	0	0
$131$	347.583	0.231820	0.115910	−	0.993260i	$-0.463022\pi$
$131$	347.583	0.231820	0.115910	+	0.993260i	$0.463022\pi$
$132$	1002.36	0.660944
$133$	287.688	0.187562
$134$	1558.84	1.00495
$135$	1141.09	0.727476
$136$	528.388	0.333154
$137$	2560.83	1.59698	0.798491	−	0.602007i	$-0.205632\pi$
$137$	2560.83	1.59698	0.798491	+	0.602007i	$0.205632\pi$
$138$	904.870	0.558171
$139$	1048.09	0.639553	0.319776	−	0.947493i	$-0.396392\pi$
$139$	1048.09	0.639553	0.319776	+	0.947493i	$0.396392\pi$
$140$	260.781	0.157429
$141$	−889.123	−0.531047
$142$	677.903	0.400622
$143$	0	0
$144$	88.2114	0.0510483
$145$	877.359	0.502488
$146$	147.317	0.0835070
$147$	−279.400	−0.156765
$148$	226.663	0.125889
$149$	3581.03	1.96892	0.984462	−	0.175597i	$-0.0561856\pi$
$149$	3581.03	1.96892	0.984462	+	0.175597i	$0.0561856\pi$
$150$	−436.282	−0.237482
$151$	−1444.52	−0.778497	−0.389249	−	0.921133i	$-0.627265\pi$
$151$	−1444.52	−0.778497	−0.389249	+	0.921133i	$0.627265\pi$
$152$	−328.786	−0.175448
$153$	−364.139	−0.192411
$154$	615.267	0.321946
$155$	497.836	0.257981
$156$	0	0
$157$	−2441.16	−1.24093	−0.620465	−	0.784234i	$-0.713056\pi$
$157$	−2441.16	−1.24093	−0.620465	+	0.784234i	$0.713056\pi$
$158$	−1169.43	−0.588827
$159$	−2459.67	−1.22682
$160$	−298.035	−0.147261
$161$	555.424	0.271885
$162$	1694.92	0.822010
$163$	−2667.17	−1.28165	−0.640826	−	0.767686i	$-0.721408\pi$
$163$	−2667.17	−1.28165	−0.640826	+	0.767686i	$0.721408\pi$
$164$	−375.559	−0.178818
$165$	2333.91	1.10118
$166$	−1267.07	−0.592433
$167$	3824.25	1.77203	0.886015	−	0.463656i	$-0.153463\pi$
$167$	3824.25	1.77203	0.886015	+	0.463656i	$0.153463\pi$
$168$	319.314	0.146641
$169$	0	0
$170$	1230.30	0.555057
$171$	226.583	0.101329
$172$	47.9958	0.0212770
$173$	1707.92	0.750584	0.375292	−	0.926907i	$-0.377542\pi$
$173$	1707.92	0.750584	0.375292	+	0.926907i	$0.377542\pi$
$174$	1074.28	0.468054
$175$	−267.797	−0.115677
$176$	−703.162	−0.301153
$177$	2945.32	1.25076
$178$	−3206.95	−1.35040
$179$	−2151.89	−0.898547	−0.449273	−	0.893394i	$-0.648317\pi$
$179$	−2151.89	−0.898547	−0.449273	+	0.893394i	$0.648317\pi$
$180$	205.392	0.0850499
$181$	−4083.35	−1.67687	−0.838435	−	0.545002i	$-0.816529\pi$
$181$	−4083.35	−1.67687	−0.838435	+	0.545002i	$0.816529\pi$
$182$	0	0
$183$	−723.344	−0.292192
$184$	−634.770	−0.254325
$185$	527.763	0.209740
$186$	609.576	0.240303
$187$	2902.68	1.13511
$188$	623.723	0.241966
$189$	857.629	0.330071
$190$	−765.546	−0.292308
$191$	1928.79	0.730693	0.365346	−	0.930872i	$-0.380951\pi$
$191$	1928.79	0.730693	0.365346	+	0.930872i	$0.380951\pi$
$192$	−364.930	−0.137170
$193$	−210.728	−0.0785934	−0.0392967	−	0.999228i	$-0.512512\pi$
$193$	−210.728	−0.0785934	−0.0392967	+	0.999228i	$0.512512\pi$
$194$	3175.28	1.17511
$195$	0	0
$196$	196.000	0.0714286
$197$	−1095.68	−0.396266	−0.198133	−	0.980175i	$-0.563488\pi$
$197$	−1095.68	−0.396266	−0.198133	+	0.980175i	$0.563488\pi$
$198$	484.586	0.173929
$199$	−3255.01	−1.15950	−0.579752	−	0.814793i	$-0.696851\pi$
$199$	−3255.01	−1.15950	−0.579752	+	0.814793i	$0.696851\pi$
$200$	306.054	0.108206
$201$	4444.28	1.55958
$202$	407.292	0.141866
$203$	659.413	0.227989
$204$	1506.44	0.517020
$205$	−874.452	−0.297924
$206$	1640.21	0.554753
$207$	437.453	0.146884
$208$	0	0
$209$	−1806.17	−0.597778
$210$	743.491	0.244313
$211$	1087.31	0.354757	0.177379	−	0.984143i	$-0.443238\pi$
$211$	1087.31	0.354757	0.177379	+	0.984143i	$0.443238\pi$
$212$	1725.47	0.558989
$213$	1932.71	0.621725
$214$	2235.92	0.714226
$215$	111.753	0.0354489
$216$	−980.147	−0.308753
$217$	374.168	0.117051
$218$	−2977.27	−0.924982
$219$	420.003	0.129594
$220$	−1637.24	−0.501741
$221$	0	0
$222$	646.221	0.195367
$223$	6403.37	1.92288	0.961438	−	0.275020i	$-0.0886845\pi$
$223$	6403.37	1.92288	0.961438	+	0.275020i	$0.0886845\pi$
$224$	−224.000	−0.0668153
$225$	−210.917	−0.0624941
$226$	1560.38	0.459269
$227$	−2744.12	−0.802350	−0.401175	−	0.916001i	$-0.631398\pi$
$227$	−2744.12	−0.802350	−0.401175	+	0.916001i	$0.631398\pi$
$228$	−937.375	−0.272277
$229$	993.609	0.286723	0.143361	−	0.989670i	$-0.454209\pi$
$229$	993.609	0.286723	0.143361	+	0.989670i	$0.454209\pi$
$230$	−1478.00	−0.423723
$231$	1754.14	0.499627
$232$	−753.615	−0.213264
$233$	−5503.91	−1.54752	−0.773761	−	0.633477i	$-0.781627\pi$
$233$	−5503.91	−1.54752	−0.773761	+	0.633477i	$0.781627\pi$
$234$	0	0
$235$	1452.28	0.403133
$236$	−2066.15	−0.569895
$237$	−3334.06	−0.913799
$238$	924.679	0.251840
$239$	5926.80	1.60407	0.802035	−	0.597277i	$-0.203751\pi$
$239$	5926.80	1.60407	0.802035	+	0.597277i	$0.203751\pi$
$240$	−849.704	−0.228534
$241$	−29.2500	−0.00781808	−0.00390904	−	0.999992i	$-0.501244\pi$
$241$	−29.2500	−0.00781808	−0.00390904	+	0.999992i	$0.501244\pi$
$242$	−1200.79	−0.318966
$243$	1524.26	0.402391
$244$	507.429	0.133134
$245$	456.367	0.119005
$246$	−1070.72	−0.277508
$247$	0	0
$248$	−427.620	−0.109492
$249$	−3612.44	−0.919395
$250$	3041.02	0.769323
$251$	−2681.08	−0.674216	−0.337108	−	0.941466i	$-0.609449\pi$
$251$	−2681.08	−0.674216	−0.337108	+	0.941466i	$0.609449\pi$
$252$	154.370	0.0385889
$253$	−3487.08	−0.866525
$254$	−3676.24	−0.908141
$255$	3507.61	0.861391
$256$	256.000	0.0625000
$257$	−1407.90	−0.341721	−0.170860	−	0.985295i	$-0.554655\pi$
$257$	−1407.90	−0.341721	−0.170860	+	0.985295i	$0.554655\pi$
$258$	136.837	0.0330197
$259$	396.660	0.0951632
$260$	0	0
$261$	519.355	0.123170
$262$	−695.165	−0.163922
$263$	3355.50	0.786726	0.393363	−	0.919383i	$-0.371312\pi$
$263$	3355.50	0.786726	0.393363	+	0.919383i	$0.371312\pi$
$264$	−2004.73	−0.467358
$265$	4017.58	0.931314
$266$	−575.376	−0.132626
$267$	−9143.06	−2.09568
$268$	−3117.68	−0.710607
$269$	−1873.33	−0.424606	−0.212303	−	0.977204i	$-0.568096\pi$
$269$	−1873.33	−0.424606	−0.212303	+	0.977204i	$0.568096\pi$
$270$	−2282.18	−0.514403
$271$	−3797.76	−0.851283	−0.425642	−	0.904892i	$-0.639952\pi$
$271$	−3797.76	−0.851283	−0.425642	+	0.904892i	$0.639952\pi$
$272$	−1056.78	−0.235575
$273$	0	0
$274$	−5121.66	−1.12924
$275$	1681.29	0.368676
$276$	−1809.74	−0.394687
$277$	−5523.72	−1.19815	−0.599076	−	0.800692i	$-0.704465\pi$
$277$	−5523.72	−1.19815	−0.599076	+	0.800692i	$0.704465\pi$
$278$	−2096.18	−0.452232
$279$	294.695	0.0632363
$280$	−521.562	−0.111319
$281$	2613.32	0.554796	0.277398	−	0.960755i	$-0.410528\pi$
$281$	2613.32	0.554796	0.277398	+	0.960755i	$0.410528\pi$
$282$	1778.25	0.375507
$283$	4399.28	0.924063	0.462032	−	0.886863i	$-0.347121\pi$
$283$	4399.28	0.924063	0.462032	+	0.886863i	$0.347121\pi$
$284$	−1355.81	−0.283283
$285$	−2182.59	−0.453632
$286$	0	0
$287$	−657.228	−0.135174
$288$	−176.423	−0.0360966
$289$	−550.597	−0.112069
$290$	−1754.72	−0.355312
$291$	9052.78	1.82365
$292$	−294.633	−0.0590484
$293$	−2316.91	−0.461963	−0.230981	−	0.972958i	$-0.574194\pi$
$293$	−2316.91	−0.461963	−0.230981	+	0.972958i	$0.574194\pi$
$294$	558.800	0.110850
$295$	−4810.84	−0.949484
$296$	−453.326	−0.0890171
$297$	−5384.40	−1.05197
$298$	−7162.07	−1.39224
$299$	0	0
$300$	872.565	0.167925
$301$	83.9926	0.0160839
$302$	2889.03	0.550481
$303$	1161.20	0.220162
$304$	657.572	0.124060
$305$	1181.50	0.221811
$306$	728.279	0.136055
$307$	999.339	0.185783	0.0928913	−	0.995676i	$-0.470389\pi$
$307$	999.339	0.185783	0.0928913	+	0.995676i	$0.470389\pi$
$308$	−1230.53	−0.227650
$309$	4676.28	0.860920
$310$	−995.671	−0.182420
$311$	−8702.74	−1.58677	−0.793387	−	0.608717i	$-0.791684\pi$
$311$	−8702.74	−1.58677	−0.793387	+	0.608717i	$0.791684\pi$
$312$	0	0
$313$	−6409.06	−1.15739	−0.578693	−	0.815546i	$-0.696437\pi$
$313$	−6409.06	−1.15739	−0.578693	+	0.815546i	$0.696437\pi$
$314$	4882.33	0.877470
$315$	359.435	0.0642917
$316$	2338.86	0.416364
$317$	−9233.74	−1.63602	−0.818010	−	0.575204i	$-0.804923\pi$
$317$	−9233.74	−1.63602	−0.818010	+	0.575204i	$0.804923\pi$
$318$	4919.34	0.867494
$319$	−4139.95	−0.726623
$320$	596.071	0.104129
$321$	6374.65	1.10841
$322$	−1110.85	−0.192252
$323$	−2714.48	−0.467609
$324$	−3389.84	−0.581249
$325$	0	0
$326$	5334.35	0.906265
$327$	−8488.25	−1.43548
$328$	751.117	0.126444
$329$	1091.52	0.182909
$330$	−4667.81	−0.778651
$331$	8022.96	1.33227	0.666135	−	0.745831i	$-0.267947\pi$
$331$	8022.96	1.33227	0.666135	+	0.745831i	$0.267947\pi$
$332$	2534.14	0.418913
$333$	312.410	0.0514114
$334$	−7648.49	−1.25301
$335$	−7259.22	−1.18392
$336$	−638.628	−0.103691
$337$	−8565.08	−1.38448	−0.692240	−	0.721667i	$-0.743376\pi$
$337$	−8565.08	−1.38448	−0.692240	+	0.721667i	$0.743376\pi$
$338$	0	0
$339$	4448.66	0.712738
$340$	−2460.60	−0.392484
$341$	−2349.11	−0.373054
$342$	−453.167	−0.0716505
$343$	343.000	0.0539949
$344$	−95.9915	−0.0150451
$345$	−4213.80	−0.657575
$346$	−3415.85	−0.530743
$347$	5310.82	0.821613	0.410807	−	0.911722i	$-0.365247\pi$
$347$	5310.82	0.821613	0.410807	+	0.911722i	$0.365247\pi$
$348$	−2148.57	−0.330964
$349$	−4736.28	−0.726439	−0.363220	−	0.931704i	$-0.618322\pi$
$349$	−4736.28	−0.726439	−0.363220	+	0.931704i	$0.618322\pi$
$350$	535.594	0.0817963
$351$	0	0
$352$	1406.32	0.212947
$353$	−8380.32	−1.26357	−0.631784	−	0.775145i	$-0.717677\pi$
$353$	−8380.32	−1.26357	−0.631784	+	0.775145i	$0.717677\pi$
$354$	−5890.64	−0.884418
$355$	−3156.86	−0.471969
$356$	6413.89	0.954875
$357$	2636.28	0.390831
$358$	4303.78	0.635369
$359$	774.846	0.113913	0.0569566	−	0.998377i	$-0.481860\pi$
$359$	774.846	0.113913	0.0569566	+	0.998377i	$0.481860\pi$
$360$	−410.783	−0.0601394
$361$	−5169.93	−0.753744
$362$	8166.71	1.18573
$363$	−3423.48	−0.495003
$364$	0	0
$365$	−686.025	−0.0983786
$366$	1446.69	0.206611
$367$	1660.76	0.236216	0.118108	−	0.993001i	$-0.462317\pi$
$367$	1660.76	0.236216	0.118108	+	0.993001i	$0.462317\pi$
$368$	1269.54	0.179835
$369$	−517.634	−0.0730269
$370$	−1055.53	−0.148309
$371$	3019.57	0.422556
$372$	−1219.15	−0.169920
$373$	7199.78	0.999438	0.499719	−	0.866188i	$-0.333437\pi$
$373$	7199.78	0.999438	0.499719	+	0.866188i	$0.333437\pi$
$374$	−5805.35	−0.802641
$375$	8670.00	1.19391
$376$	−1247.45	−0.171096
$377$	0	0
$378$	−1715.26	−0.233395
$379$	−6386.48	−0.865571	−0.432785	−	0.901497i	$-0.642469\pi$
$379$	−6386.48	−0.865571	−0.432785	+	0.901497i	$0.642469\pi$
$380$	1531.09	0.206693
$381$	−10481.0	−1.40934
$382$	−3857.58	−0.516678
$383$	13384.3	1.78565	0.892825	−	0.450403i	$-0.148720\pi$
$383$	13384.3	1.78565	0.892825	+	0.450403i	$0.148720\pi$
$384$	729.861	0.0969936
$385$	−2865.18	−0.379281
$386$	421.456	0.0555739
$387$	66.1527	0.00868923
$388$	−6350.56	−0.830930
$389$	−158.624	−0.0206749	−0.0103375	−	0.999947i	$-0.503291\pi$
$389$	−158.624	−0.0206749	−0.0103375	+	0.999947i	$0.503291\pi$
$390$	0	0
$391$	−5240.70	−0.677835
$392$	−392.000	−0.0505076
$393$	−1981.93	−0.254390
$394$	2191.37	0.280202
$395$	5445.79	0.693690
$396$	−969.171	−0.122987
$397$	−14688.0	−1.85685	−0.928423	−	0.371524i	$-0.878835\pi$
$397$	−14688.0	−1.85685	−0.928423	+	0.371524i	$0.878835\pi$
$398$	6510.01	0.819893
$399$	−1640.41	−0.205822
$400$	−612.108	−0.0765134
$401$	−8191.71	−1.02014	−0.510068	−	0.860134i	$-0.670380\pi$
$401$	−8191.71	−1.02014	−0.510068	+	0.860134i	$0.670380\pi$
$402$	−8888.57	−1.10279
$403$	0	0
$404$	−814.584	−0.100315
$405$	−7892.92	−0.968401
$406$	−1318.83	−0.161212
$407$	−2490.33	−0.303295
$408$	−3012.89	−0.365589
$409$	−4642.23	−0.561231	−0.280615	−	0.959820i	$-0.590539\pi$
$409$	−4642.23	−0.561231	−0.280615	+	0.959820i	$0.590539\pi$
$410$	1748.90	0.210664
$411$	−14601.9	−1.75246
$412$	−3280.43	−0.392270
$413$	−3615.77	−0.430800
$414$	−874.905	−0.103863
$415$	5900.50	0.697938
$416$	0	0
$417$	−5976.25	−0.701818
$418$	3612.35	0.422693
$419$	−5304.72	−0.618503	−0.309251	−	0.950980i	$-0.600078\pi$
$419$	−5304.72	−0.618503	−0.309251	+	0.950980i	$0.600078\pi$
$420$	−1486.98	−0.172756
$421$	−10880.3	−1.25956	−0.629780	−	0.776774i	$-0.716855\pi$
$421$	−10880.3	−1.25956	−0.629780	+	0.776774i	$0.716855\pi$
$422$	−2174.63	−0.250851
$423$	859.679	0.0988157
$424$	−3450.94	−0.395265
$425$	2526.80	0.288395
$426$	−3865.43	−0.439626
$427$	888.000	0.100640
$428$	−4471.84	−0.505034
$429$	0	0
$430$	−223.507	−0.0250662
$431$	−7973.88	−0.891156	−0.445578	−	0.895243i	$-0.647002\pi$
$431$	−7973.88	−0.891156	−0.445578	+	0.895243i	$0.647002\pi$
$432$	1960.29	0.218321
$433$	−1988.78	−0.220727	−0.110363	−	0.993891i	$-0.535201\pi$
$433$	−1988.78	−0.220727	−0.110363	+	0.993891i	$0.535201\pi$
$434$	−748.335	−0.0827678
$435$	−5002.73	−0.551409
$436$	5954.54	0.654061
$437$	3260.99	0.356967
$438$	−840.005	−0.0916370
$439$	1254.30	0.136365	0.0681825	−	0.997673i	$-0.478280\pi$
$439$	1254.30	0.136365	0.0681825	+	0.997673i	$0.478280\pi$
$440$	3274.49	0.354784
$441$	270.147	0.0291704
$442$	0	0
$443$	−1918.96	−0.205807	−0.102904	−	0.994691i	$-0.532813\pi$
$443$	−1918.96	−0.205807	−0.102904	+	0.994691i	$0.532813\pi$
$444$	−1292.44	−0.138145
$445$	14934.1	1.59089
$446$	−12806.7	−1.35968
$447$	−20419.2	−2.16061
$448$	448.000	0.0472456
$449$	−9778.85	−1.02782	−0.513911	−	0.857843i	$-0.671804\pi$
$449$	−9778.85	−1.02782	−0.513911	+	0.857843i	$0.671804\pi$
$450$	421.835	0.0441900
$451$	4126.23	0.430813
$452$	−3120.76	−0.324752
$453$	8236.69	0.854290
$454$	5488.23	0.567347
$455$	0	0
$456$	1874.75	0.192529
$457$	10690.9	1.09431	0.547153	−	0.837032i	$-0.315711\pi$
$457$	10690.9	1.09431	0.547153	+	0.837032i	$0.315711\pi$
$458$	−1987.22	−0.202744
$459$	−8092.16	−0.822897
$460$	2956.00	0.299618
$461$	−5804.85	−0.586462	−0.293231	−	0.956042i	$-0.594730\pi$
$461$	−5804.85	−0.586462	−0.293231	+	0.956042i	$0.594730\pi$
$462$	−3508.28	−0.353289
$463$	11738.5	1.17826	0.589128	−	0.808040i	$-0.299472\pi$
$463$	11738.5	1.17826	0.589128	+	0.808040i	$0.299472\pi$
$464$	1507.23	0.150800
$465$	−2838.68	−0.283098
$466$	11007.8	1.09426
$467$	2584.12	0.256057	0.128029	−	0.991770i	$-0.459135\pi$
$467$	2584.12	0.256057	0.128029	+	0.991770i	$0.459135\pi$
$468$	0	0
$469$	−5455.94	−0.537169
$470$	−2904.56	−0.285058
$471$	13919.6	1.36174
$472$	4132.31	0.402976
$473$	−527.325	−0.0512610
$474$	6668.12	0.646154
$475$	−1572.29	−0.151877
$476$	−1849.36	−0.178078
$477$	2378.22	0.228283
$478$	−11853.6	−1.13425
$479$	3410.64	0.325336	0.162668	−	0.986681i	$-0.447990\pi$
$479$	3410.64	0.325336	0.162668	+	0.986681i	$0.447990\pi$
$480$	1699.41	0.161598
$481$	0	0
$482$	58.5000	0.00552822
$483$	−3167.04	−0.298355
$484$	2401.59	0.225543
$485$	−14786.7	−1.38439
$486$	−3048.51	−0.284534
$487$	4959.17	0.461440	0.230720	−	0.973020i	$-0.425892\pi$
$487$	4959.17	0.461440	0.230720	+	0.973020i	$0.425892\pi$
$488$	−1014.86	−0.0941402
$489$	15208.3	1.40643
$490$	−912.733	−0.0841492
$491$	−18694.8	−1.71830	−0.859148	−	0.511727i	$-0.829006\pi$
$491$	−18694.8	−1.71830	−0.859148	+	0.511727i	$0.829006\pi$
$492$	2141.45	0.196228
$493$	−6221.89	−0.568397
$494$	0	0
$495$	−2256.62	−0.204904
$496$	855.240	0.0774222
$497$	−2372.66	−0.214142
$498$	7224.89	0.650110
$499$	16434.4	1.47436	0.737179	−	0.675697i	$-0.236157\pi$
$499$	16434.4	1.47436	0.737179	+	0.675697i	$0.236157\pi$
$500$	−6082.04	−0.543994
$501$	−21806.0	−1.94455
$502$	5362.16	0.476743
$503$	15590.7	1.38202	0.691009	−	0.722846i	$-0.257167\pi$
$503$	15590.7	1.38202	0.691009	+	0.722846i	$0.257167\pi$
$504$	−308.740	−0.0272864
$505$	−1896.68	−0.167131
$506$	6974.16	0.612726
$507$	0	0
$508$	7352.48	0.642152
$509$	5617.81	0.489205	0.244602	−	0.969624i	$-0.421343\pi$
$509$	5617.81	0.489205	0.244602	+	0.969624i	$0.421343\pi$
$510$	−7015.21	−0.609096
$511$	−515.609	−0.0446364
$512$	−512.000	−0.0441942
$513$	5035.30	0.433360
$514$	2815.79	0.241633
$515$	−7638.16	−0.653548
$516$	−273.674	−0.0233485
$517$	−6852.79	−0.582951
$518$	−793.321	−0.0672906
$519$	−9738.64	−0.823659
$520$	0	0
$521$	−15034.4	−1.26424	−0.632119	−	0.774872i	$-0.717814\pi$
$521$	−15034.4	−1.26424	−0.632119	+	0.774872i	$0.717814\pi$
$522$	−1038.71	−0.0870941
$523$	−15255.0	−1.27544	−0.637719	−	0.770269i	$-0.720122\pi$
$523$	−15255.0	−1.27544	−0.637719	+	0.770269i	$0.720122\pi$
$524$	1390.33	0.115910
$525$	1526.99	0.126940
$526$	−6711.00	−0.556299
$527$	−3530.46	−0.291820
$528$	4009.46	0.330472
$529$	−5871.18	−0.482549
$530$	−8035.17	−0.658539
$531$	−2847.79	−0.232737
$532$	1150.75	0.0937808
$533$	0	0
$534$	18286.1	1.48187
$535$	−10412.2	−0.841422
$536$	6235.36	0.502475
$537$	12270.2	0.986027
$538$	3746.66	0.300242
$539$	−2153.44	−0.172087
$540$	4564.35	0.363738
$541$	−6788.16	−0.539456	−0.269728	−	0.962937i	$-0.586934\pi$
$541$	−6788.16	−0.539456	−0.269728	+	0.962937i	$0.586934\pi$
$542$	7595.53	0.601948
$543$	23283.4	1.84013
$544$	2113.55	0.166577
$545$	13864.6	1.08971
$546$	0	0
$547$	5410.68	0.422932	0.211466	−	0.977385i	$-0.432176\pi$
$547$	5410.68	0.422932	0.211466	+	0.977385i	$0.432176\pi$
$548$	10243.3	0.798491
$549$	699.390	0.0543702
$550$	−3362.59	−0.260693
$551$	3871.53	0.299334
$552$	3619.48	0.279086
$553$	4093.00	0.314741
$554$	11047.4	0.847222
$555$	−3009.32	−0.230160
$556$	4192.36	0.319776
$557$	11170.2	0.849725	0.424863	−	0.905258i	$-0.360322\pi$
$557$	11170.2	0.849725	0.424863	+	0.905258i	$0.360322\pi$
$558$	−589.390	−0.0447148
$559$	0	0
$560$	1043.12	0.0787143
$561$	−16551.2	−1.24562
$562$	−5226.64	−0.392300
$563$	11716.9	0.877102	0.438551	−	0.898706i	$-0.355492\pi$
$563$	11716.9	0.877102	0.438551	+	0.898706i	$0.355492\pi$
$564$	−3556.49	−0.265524
$565$	−7266.37	−0.541059
$566$	−8798.55	−0.653411
$567$	−5932.23	−0.439383
$568$	2711.61	0.200311
$569$	−6040.44	−0.445041	−0.222521	−	0.974928i	$-0.571428\pi$
$569$	−6040.44	−0.445041	−0.222521	+	0.974928i	$0.571428\pi$
$570$	4365.17	0.320767
$571$	−9726.47	−0.712855	−0.356427	−	0.934323i	$-0.616005\pi$
$571$	−9726.47	−0.712855	−0.356427	+	0.934323i	$0.616005\pi$
$572$	0	0
$573$	−10998.0	−0.801831
$574$	1314.46	0.0955824
$575$	−3035.53	−0.220157
$576$	352.846	0.0255241
$577$	10838.8	0.782016	0.391008	−	0.920387i	$-0.372126\pi$
$577$	10838.8	0.782016	0.391008	+	0.920387i	$0.372126\pi$
$578$	1101.19	0.0792450
$579$	1201.58	0.0862450
$580$	3509.44	0.251244
$581$	4434.75	0.316669
$582$	−18105.6	−1.28952
$583$	−18957.6	−1.34673
$584$	589.267	0.0417535
$585$	0	0
$586$	4633.81	0.326657
$587$	17711.2	1.24535	0.622673	−	0.782482i	$-0.286047\pi$
$587$	17711.2	1.24535	0.622673	+	0.782482i	$0.286047\pi$
$588$	−1117.60	−0.0783827
$589$	2196.81	0.153680
$590$	9621.67	0.671386
$591$	6247.63	0.434845
$592$	906.652	0.0629446
$593$	−20065.3	−1.38952	−0.694759	−	0.719243i	$-0.744489\pi$
$593$	−20065.3	−1.38952	−0.694759	+	0.719243i	$0.744489\pi$
$594$	10768.8	0.743854
$595$	−4306.05	−0.296690
$596$	14324.1	0.984462
$597$	18560.2	1.27239
$598$	0	0
$599$	−18078.5	−1.23317	−0.616585	−	0.787289i	$-0.711484\pi$
$599$	−18078.5	−1.23317	−0.616585	+	0.787289i	$0.711484\pi$
$600$	−1745.13	−0.118741
$601$	−23058.3	−1.56500	−0.782501	−	0.622649i	$-0.786057\pi$
$601$	−23058.3	−1.56500	−0.782501	+	0.622649i	$0.786057\pi$
$602$	−167.985	−0.0113730
$603$	−4297.11	−0.290202
$604$	−5778.07	−0.389249
$605$	5591.86	0.375771
$606$	−2322.39	−0.155678
$607$	−17731.8	−1.18568	−0.592842	−	0.805319i	$-0.701994\pi$
$607$	−17731.8	−1.18568	−0.592842	+	0.805319i	$0.701994\pi$
$608$	−1315.14	−0.0877239
$609$	−3760.00	−0.250185
$610$	−2363.00	−0.156844
$611$	0	0
$612$	−1456.56	−0.0962056
$613$	9772.69	0.643908	0.321954	−	0.946755i	$-0.395660\pi$
$613$	9772.69	0.643908	0.321954	+	0.946755i	$0.395660\pi$
$614$	−1998.68	−0.131368
$615$	4986.15	0.326929
$616$	2461.07	0.160973
$617$	−28475.5	−1.85799	−0.928994	−	0.370094i	$-0.879326\pi$
$617$	−28475.5	−1.85799	−0.928994	+	0.370094i	$0.879326\pi$
$618$	−9352.56	−0.608763
$619$	−13533.3	−0.878752	−0.439376	−	0.898303i	$-0.644800\pi$
$619$	−13533.3	−0.878752	−0.439376	+	0.898303i	$0.644800\pi$
$620$	1991.34	0.128991
$621$	9721.37	0.628189
$622$	17405.5	1.12202
$623$	11224.3	0.721818
$624$	0	0
$625$	−9379.33	−0.600277
$626$	12818.1	0.818395
$627$	10298.9	0.655976
$628$	−9764.65	−0.620465
$629$	−3742.69	−0.237251
$630$	−718.870	−0.0454611
$631$	−17830.8	−1.12493	−0.562465	−	0.826821i	$-0.690147\pi$
$631$	−17830.8	−1.12493	−0.562465	+	0.826821i	$0.690147\pi$
$632$	−4677.71	−0.294414
$633$	−6199.90	−0.389295
$634$	18467.5	1.15684
$635$	17119.5	1.06987
$636$	−9838.68	−0.613411
$637$	0	0
$638$	8279.90	0.513800
$639$	−1868.71	−0.115689
$640$	−1192.14	−0.0736305
$641$	9469.08	0.583473	0.291736	−	0.956499i	$-0.405767\pi$
$641$	9469.08	0.583473	0.291736	+	0.956499i	$0.405767\pi$
$642$	−12749.3	−0.783762
$643$	24907.3	1.52760	0.763802	−	0.645451i	$-0.223331\pi$
$643$	24907.3	1.52760	0.763802	+	0.645451i	$0.223331\pi$
$644$	2221.69	0.135943
$645$	−637.222	−0.0389001
$646$	5428.96	0.330649
$647$	15062.5	0.915249	0.457625	−	0.889145i	$-0.348700\pi$
$647$	15062.5	0.915249	0.457625	+	0.889145i	$0.348700\pi$
$648$	6779.69	0.411005
$649$	22700.7	1.37300
$650$	0	0
$651$	−2133.52	−0.128447
$652$	−10668.7	−0.640826
$653$	−2322.39	−0.139176	−0.0695880	−	0.997576i	$-0.522168\pi$
$653$	−2322.39	−0.139176	−0.0695880	+	0.997576i	$0.522168\pi$
$654$	16976.5	1.01504
$655$	3237.25	0.193114
$656$	−1502.23	−0.0894092
$657$	−406.094	−0.0241145
$658$	−2183.03	−0.129336
$659$	−20890.9	−1.23489	−0.617447	−	0.786613i	$-0.711833\pi$
$659$	−20890.9	−1.23489	−0.617447	+	0.786613i	$0.711833\pi$
$660$	9335.63	0.550589
$661$	−10200.5	−0.600235	−0.300117	−	0.953902i	$-0.597026\pi$
$661$	−10200.5	−0.600235	−0.300117	+	0.953902i	$0.597026\pi$
$662$	−16045.9	−0.942057
$663$	0	0
$664$	−5068.29	−0.296216
$665$	2679.41	0.156245
$666$	−624.821	−0.0363533
$667$	7474.56	0.433907
$668$	15297.0	0.886015
$669$	−36512.3	−2.11008
$670$	14518.4	0.837158
$671$	−5575.07	−0.320750
$672$	1277.26	0.0733203
$673$	−23440.2	−1.34258	−0.671289	−	0.741196i	$-0.734259\pi$
$673$	−23440.2	−1.34258	−0.671289	+	0.741196i	$0.734259\pi$
$674$	17130.2	0.978975
$675$	−4687.15	−0.267272
$676$	0	0
$677$	−25944.0	−1.47283	−0.736416	−	0.676528i	$-0.763484\pi$
$677$	−25944.0	−1.47283	−0.736416	+	0.676528i	$0.763484\pi$
$678$	−8897.33	−0.503982
$679$	−11113.5	−0.628124
$680$	4921.20	0.277528
$681$	15647.1	0.880465
$682$	4698.22	0.263789
$683$	12371.2	0.693074	0.346537	−	0.938036i	$-0.387358\pi$
$683$	12371.2	0.693074	0.346537	+	0.938036i	$0.387358\pi$
$684$	906.334	0.0506645
$685$	23850.6	1.33034
$686$	−686.000	−0.0381802
$687$	−5665.59	−0.314637
$688$	191.983	0.0106385
$689$	0	0
$690$	8427.60	0.464976
$691$	25322.2	1.39407	0.697035	−	0.717037i	$-0.254502\pi$
$691$	25322.2	1.39407	0.697035	+	0.717037i	$0.254502\pi$
$692$	6831.69	0.375292
$693$	−1696.05	−0.0929691
$694$	−10621.6	−0.580968
$695$	9761.50	0.532769
$696$	4297.14	0.234027
$697$	6201.27	0.337001
$698$	9472.56	0.513670
$699$	31383.5	1.69819
$700$	−1071.19	−0.0578387
$701$	13634.6	0.734625	0.367312	−	0.930098i	$-0.380278\pi$
$701$	13634.6	0.734625	0.367312	+	0.930098i	$0.380278\pi$
$702$	0	0
$703$	2328.87	0.124943
$704$	−2812.65	−0.150576
$705$	−8280.94	−0.442381
$706$	16760.6	0.893477
$707$	−1425.52	−0.0758307
$708$	11781.3	0.625378
$709$	−6855.71	−0.363147	−0.181574	−	0.983377i	$-0.558119\pi$
$709$	−6855.71	−0.363147	−0.181574	+	0.983377i	$0.558119\pi$
$710$	6313.73	0.333732
$711$	3223.65	0.170037
$712$	−12827.8	−0.675199
$713$	4241.25	0.222772
$714$	−5272.55	−0.276359
$715$	0	0
$716$	−8607.56	−0.449273
$717$	−33794.8	−1.76024
$718$	−1549.69	−0.0805487
$719$	−13282.5	−0.688948	−0.344474	−	0.938796i	$-0.611943\pi$
$719$	−13282.5	−0.688948	−0.344474	+	0.938796i	$0.611943\pi$
$720$	821.566	0.0425250
$721$	−5740.75	−0.296528
$722$	10339.9	0.532978
$723$	166.785	0.00857923
$724$	−16333.4	−0.838435
$725$	−3603.86	−0.184612
$726$	6846.96	0.350020
$727$	21170.4	1.08001	0.540003	−	0.841663i	$-0.318423\pi$
$727$	21170.4	1.08001	0.540003	+	0.841663i	$0.318423\pi$
$728$	0	0
$729$	14190.1	0.720931
$730$	1372.05	0.0695642
$731$	−792.512	−0.0400986
$732$	−2893.38	−0.146096
$733$	−8482.75	−0.427445	−0.213723	−	0.976894i	$-0.568559\pi$
$733$	−8482.75	−0.427445	−0.213723	+	0.976894i	$0.568559\pi$
$734$	−3321.53	−0.167030
$735$	−2602.22	−0.130591
$736$	−2539.08	−0.127163
$737$	34253.7	1.71201
$738$	1035.27	0.0516378
$739$	−19695.8	−0.980409	−0.490205	−	0.871607i	$-0.663078\pi$
$739$	−19695.8	−0.980409	−0.490205	+	0.871607i	$0.663078\pi$
$740$	2111.05	0.104870
$741$	0	0
$742$	−6039.14	−0.298792
$743$	−18213.0	−0.899286	−0.449643	−	0.893208i	$-0.648449\pi$
$743$	−18213.0	−0.899286	−0.449643	+	0.893208i	$0.648449\pi$
$744$	2438.30	0.120151
$745$	33352.3	1.64018
$746$	−14399.6	−0.706709
$747$	3492.82	0.171078
$748$	11610.7	0.567553
$749$	−7825.73	−0.381770
$750$	−17340.0	−0.844223
$751$	−24649.3	−1.19769	−0.598845	−	0.800865i	$-0.704373\pi$
$751$	−24649.3	−1.19769	−0.598845	+	0.800865i	$0.704373\pi$
$752$	2494.89	0.120983
$753$	15287.6	0.739856
$754$	0	0
$755$	−13453.7	−0.648515
$756$	3430.52	0.165035
$757$	27328.2	1.31210	0.656050	−	0.754717i	$-0.272226\pi$
$757$	27328.2	1.31210	0.656050	+	0.754717i	$0.272226\pi$
$758$	12773.0	0.612051
$759$	19883.5	0.950888
$760$	−3062.19	−0.146154
$761$	−15777.9	−0.751574	−0.375787	−	0.926706i	$-0.622628\pi$
$761$	−15777.9	−0.751574	−0.375787	+	0.926706i	$0.622628\pi$
$762$	20962.0	0.996555
$763$	10420.4	0.494424
$764$	7715.16	0.365346
$765$	−3391.45	−0.160285
$766$	−26768.5	−1.26265
$767$	0	0
$768$	−1459.72	−0.0685848
$769$	−27331.1	−1.28165	−0.640823	−	0.767688i	$-0.721407\pi$
$769$	−27331.1	−1.28165	−0.640823	+	0.767688i	$0.721407\pi$
$770$	5730.36	0.268192
$771$	8027.88	0.374990
$772$	−842.911	−0.0392967
$773$	−25696.7	−1.19566	−0.597831	−	0.801622i	$-0.703971\pi$
$773$	−25696.7	−1.19566	−0.597831	+	0.801622i	$0.703971\pi$
$774$	−132.305	−0.00614421
$775$	−2044.92	−0.0947814
$776$	12701.1	0.587556
$777$	−2261.77	−0.104428
$778$	317.248	0.0146194
$779$	−3858.70	−0.177474
$780$	0	0
$781$	14896.1	0.682491
$782$	10481.4	0.479302
$783$	11541.5	0.526767
$784$	784.000	0.0357143
$785$	−22736.0	−1.03374
$786$	3963.86	0.179881
$787$	9837.00	0.445554	0.222777	−	0.974869i	$-0.428488\pi$
$787$	9837.00	0.445554	0.222777	+	0.974869i	$0.428488\pi$
$788$	−4382.74	−0.198133
$789$	−19133.2	−0.863319
$790$	−10891.6	−0.490513
$791$	−5461.32	−0.245490
$792$	1938.34	0.0869646
$793$	0	0
$794$	29375.9	1.31299
$795$	−22908.4	−1.02198
$796$	−13020.0	−0.579752
$797$	−17598.7	−0.782153	−0.391077	−	0.920358i	$-0.627897\pi$
$797$	−17598.7	−0.782153	−0.391077	+	0.920358i	$0.627897\pi$
$798$	3280.81	0.145538
$799$	−10299.0	−0.456010
$800$	1224.22	0.0541032
$801$	8840.29	0.389958
$802$	16383.4	0.721345
$803$	3237.11	0.142261
$804$	17777.1	0.779790
$805$	5173.00	0.226490
$806$	0	0
$807$	10681.8	0.465944
$808$	1629.17	0.0709331
$809$	39275.5	1.70686	0.853431	−	0.521206i	$-0.174518\pi$
$809$	39275.5	1.70686	0.853431	+	0.521206i	$0.174518\pi$
$810$	15785.8	0.684763
$811$	29158.8	1.26252	0.631259	−	0.775572i	$-0.282538\pi$
$811$	29158.8	1.26252	0.631259	+	0.775572i	$0.282538\pi$
$812$	2637.65	0.113994
$813$	21655.0	0.934162
$814$	4980.66	0.214462
$815$	−24841.0	−1.06766
$816$	6025.77	0.258510
$817$	493.136	0.0211171
$818$	9284.46	0.396850
$819$	0	0
$820$	−3497.81	−0.148962
$821$	33894.2	1.44082	0.720412	−	0.693546i	$-0.243953\pi$
$821$	33894.2	1.44082	0.720412	+	0.693546i	$0.243953\pi$
$822$	29203.9	1.23918
$823$	24281.6	1.02844	0.514218	−	0.857660i	$-0.328082\pi$
$823$	24281.6	1.02844	0.514218	+	0.857660i	$0.328082\pi$
$824$	6560.86	0.277377
$825$	−9586.79	−0.404569
$826$	7231.54	0.304622
$827$	−44432.6	−1.86828	−0.934142	−	0.356901i	$-0.883833\pi$
$827$	−44432.6	−1.86828	−0.934142	+	0.356901i	$0.883833\pi$
$828$	1749.81	0.0734422
$829$	27052.3	1.13337	0.566687	−	0.823933i	$-0.308225\pi$
$829$	27052.3	1.13337	0.566687	+	0.823933i	$0.308225\pi$
$830$	−11801.0	−0.493517
$831$	31496.5	1.31480
$832$	0	0
$833$	−3236.38	−0.134614
$834$	11952.5	0.496260
$835$	35617.5	1.47616
$836$	−7224.69	−0.298889
$837$	6548.92	0.270446
$838$	10609.4	0.437347
$839$	−19634.2	−0.807922	−0.403961	−	0.914776i	$-0.632367\pi$
$839$	−19634.2	−0.807922	−0.403961	+	0.914776i	$0.632367\pi$
$840$	2973.97	0.122157
$841$	−15515.0	−0.636148
$842$	21760.7	0.890643
$843$	−14901.2	−0.608809
$844$	4349.26	0.177379
$845$	0	0
$846$	−1719.36	−0.0698733
$847$	4202.78	0.170495
$848$	6901.87	0.279495
$849$	−25084.8	−1.01403
$850$	−5053.60	−0.203926
$851$	4496.21	0.181114
$852$	7730.86	0.310862
$853$	11550.2	0.463624	0.231812	−	0.972761i	$-0.425535\pi$
$853$	11550.2	0.463624	0.231812	+	0.972761i	$0.425535\pi$
$854$	−1776.00	−0.0711633
$855$	2110.31	0.0844106
$856$	8943.69	0.357113
$857$	34565.4	1.37775	0.688875	−	0.724880i	$-0.258105\pi$
$857$	34565.4	1.37775	0.688875	+	0.724880i	$0.258105\pi$
$858$	0	0
$859$	8682.32	0.344862	0.172431	−	0.985022i	$-0.444838\pi$
$859$	8682.32	0.344862	0.172431	+	0.985022i	$0.444838\pi$
$860$	447.014	0.0177245
$861$	3747.54	0.148334
$862$	15947.8	0.630143
$863$	5319.60	0.209828	0.104914	−	0.994481i	$-0.466543\pi$
$863$	5319.60	0.209828	0.104914	+	0.994481i	$0.466543\pi$
$864$	−3920.59	−0.154376
$865$	15906.9	0.625262
$866$	3977.56	0.156077
$867$	3139.52	0.122980
$868$	1496.67	0.0585257
$869$	−25696.8	−1.00311
$870$	10005.5	0.389905
$871$	0	0
$872$	−11909.1	−0.462491
$873$	−8753.00	−0.339340
$874$	−6521.98	−0.252414
$875$	−10643.6	−0.411221
$876$	1680.01	0.0647971
$877$	−12921.2	−0.497512	−0.248756	−	0.968566i	$-0.580022\pi$
$877$	−12921.2	−0.497512	−0.248756	+	0.968566i	$0.580022\pi$
$878$	−2508.59	−0.0964246
$879$	13211.1	0.506938
$880$	−6548.98	−0.250871
$881$	12996.9	0.497022	0.248511	−	0.968629i	$-0.420059\pi$
$881$	12996.9	0.497022	0.248511	+	0.968629i	$0.420059\pi$
$882$	−540.295	−0.0206266
$883$	−45885.0	−1.74876	−0.874379	−	0.485244i	$-0.838731\pi$
$883$	−45885.0	−1.74876	−0.874379	+	0.485244i	$0.838731\pi$
$884$	0	0
$885$	27431.6	1.04192
$886$	3837.93	0.145528
$887$	−40508.1	−1.53340	−0.766702	−	0.642004i	$-0.778103\pi$
$887$	−40508.1	−1.53340	−0.766702	+	0.642004i	$0.778103\pi$
$888$	2584.88	0.0976835
$889$	12866.8	0.485422
$890$	−29868.2	−1.12493
$891$	37243.9	1.40036
$892$	25613.5	0.961438
$893$	6408.49	0.240148
$894$	40838.4	1.52778
$895$	−20041.9	−0.748520
$896$	−896.000	−0.0334077
$897$	0	0
$898$	19557.7	0.726780
$899$	5035.33	0.186805
$900$	−843.670	−0.0312470
$901$	−28491.1	−1.05347
$902$	−8252.46	−0.304631
$903$	−478.929	−0.0176498
$904$	6241.51	0.229634
$905$	−38030.8	−1.39689
$906$	−16473.4	−0.604074
$907$	−25270.2	−0.925120	−0.462560	−	0.886588i	$-0.653069\pi$
$907$	−25270.2	−0.925120	−0.462560	+	0.886588i	$0.653069\pi$
$908$	−10976.5	−0.401175
$909$	−1122.74	−0.0409671
$910$	0	0
$911$	10985.8	0.399534	0.199767	−	0.979843i	$-0.435981\pi$
$911$	10985.8	0.399534	0.199767	+	0.979843i	$0.435981\pi$
$912$	−3749.50	−0.136139
$913$	−27842.4	−1.00925
$914$	−21381.8	−0.773792
$915$	−6736.94	−0.243406
$916$	3974.43	0.143361
$917$	2433.08	0.0876198
$918$	16184.3	0.581876
$919$	−34899.9	−1.25271	−0.626356	−	0.779537i	$-0.715454\pi$
$919$	−34899.9	−1.25271	−0.626356	+	0.779537i	$0.715454\pi$
$920$	−5912.00	−0.211862
$921$	−5698.27	−0.203870
$922$	11609.7	0.414691
$923$	0	0
$924$	7016.55	0.249813
$925$	−2167.85	−0.0770577
$926$	−23476.9	−0.833152
$927$	−4521.43	−0.160198
$928$	−3014.46	−0.106632
$929$	20551.8	0.725814	0.362907	−	0.931825i	$-0.381784\pi$
$929$	20551.8	0.725814	0.362907	+	0.931825i	$0.381784\pi$
$930$	5677.35	0.200180
$931$	2013.82	0.0708917
$932$	−22015.6	−0.773761
$933$	49623.3	1.74126
$934$	−5168.24	−0.181060
$935$	27034.4	0.945582
$936$	0	0
$937$	36992.4	1.28974	0.644871	−	0.764291i	$-0.276911\pi$
$937$	36992.4	1.28974	0.644871	+	0.764291i	$0.276911\pi$
$938$	10911.9	0.379836
$939$	36544.7	1.27006
$940$	5809.11	0.201566
$941$	−17550.3	−0.607995	−0.303997	−	0.952673i	$-0.598321\pi$
$941$	−17550.3	−0.607995	−0.303997	+	0.952673i	$0.598321\pi$
$942$	−27839.2	−0.962898
$943$	−7449.79	−0.257263
$944$	−8264.61	−0.284947
$945$	7987.62	0.274960
$946$	1054.65	0.0362470
$947$	−39173.8	−1.34422	−0.672110	−	0.740451i	$-0.734612\pi$
$947$	−39173.8	−1.34422	−0.672110	+	0.740451i	$0.734612\pi$
$948$	−13336.2	−0.456900
$949$	0	0
$950$	3144.57	0.107393
$951$	52651.1	1.79530
$952$	3698.72	0.125920
$953$	7508.84	0.255231	0.127615	−	0.991824i	$-0.459268\pi$
$953$	7508.84	0.255231	0.127615	+	0.991824i	$0.459268\pi$
$954$	−4756.44	−0.161421
$955$	17964.0	0.608692
$956$	23707.2	0.802035
$957$	23606.2	0.797365
$958$	−6821.28	−0.230047
$959$	17925.8	0.603602
$960$	−3398.82	−0.114267
$961$	−26933.8	−0.904093
$962$	0	0
$963$	−6163.55	−0.206249
$964$	−117.000	−0.00390904
$965$	−1962.64	−0.0654710
$966$	6334.09	0.210969
$967$	−43530.7	−1.44763	−0.723813	−	0.689996i	$-0.757612\pi$
$967$	−43530.7	−1.44763	−0.723813	+	0.689996i	$0.757612\pi$
$968$	−4803.17	−0.159483
$969$	15478.1	0.513134
$970$	29573.3	0.978909
$971$	3890.59	0.128584	0.0642920	−	0.997931i	$-0.479521\pi$
$971$	3890.59	0.128584	0.0642920	+	0.997931i	$0.479521\pi$
$972$	6097.03	0.201196
$973$	7336.63	0.241728
$974$	−9918.33	−0.326287
$975$	0	0
$976$	2029.71	0.0665672
$977$	25191.0	0.824905	0.412453	−	0.910979i	$-0.364672\pi$
$977$	25191.0	0.824905	0.412453	+	0.910979i	$0.364672\pi$
$978$	−30416.7	−0.994496
$979$	−70468.9	−2.30051
$980$	1825.47	0.0595025
$981$	8207.16	0.267109
$982$	37389.6	1.21502
$983$	3340.06	0.108374	0.0541869	−	0.998531i	$-0.482743\pi$
$983$	3340.06	0.108374	0.0541869	+	0.998531i	$0.482743\pi$
$984$	−4282.90	−0.138754
$985$	−10204.8	−0.330103
$986$	12443.8	0.401918
$987$	−6223.86	−0.200717
$988$	0	0
$989$	952.070	0.0306108
$990$	4513.24	0.144889
$991$	−33181.1	−1.06360	−0.531802	−	0.846868i	$-0.678485\pi$
$991$	−33181.1	−1.06360	−0.531802	+	0.846868i	$0.678485\pi$
$992$	−1710.48	−0.0547458
$993$	−45747.2	−1.46198
$994$	4745.32	0.151421
$995$	−30315.9	−0.965907
$996$	−14449.8	−0.459698
$997$	−25239.9	−0.801760	−0.400880	−	0.916131i	$-0.631296\pi$
$997$	−25239.9	−0.801760	−0.400880	+	0.916131i	$0.631296\pi$
$998$	−32868.8	−1.04253
$999$	6942.60	0.219874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.4.a.s.1.2		5
13.3	even	3		182.4.g.b.113.4	yes	10
13.9	even	3		182.4.g.b.29.4	✓	10
13.12	even	2		2366.4.a.t.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.4.g.b.29.4	✓	10	13.9	even	3
182.4.g.b.113.4	yes	10	13.3	even	3
2366.4.a.s.1.2		5	1.1	even	1	trivial
2366.4.a.t.1.2		5	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2366.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -5.70204 q^{3} +4.00000 q^{4} +9.31361 q^{5} +11.4041 q^{6} +7.00000 q^{7} -8.00000 q^{8} +5.51321 q^{9} -18.6272 q^{10} -43.9477 q^{11} -22.8081 q^{12} -14.0000 q^{14} -53.1065 q^{15} +16.0000 q^{16} -66.0485 q^{17} -11.0264 q^{18} +41.0983 q^{19} +37.2544 q^{20} -39.9143 q^{21} +87.8953 q^{22} +79.3462 q^{23} +45.6163 q^{24} -38.2567 q^{25} +122.518 q^{27} +28.0000 q^{28} +94.2019 q^{29} +106.213 q^{30} +53.4525 q^{31} -32.0000 q^{32} +250.591 q^{33} +132.097 q^{34} +65.1952 q^{35} +22.0528 q^{36} +56.6658 q^{37} -82.1965 q^{38} -74.5089 q^{40} -93.8897 q^{41} +79.8285 q^{42} +11.9989 q^{43} -175.791 q^{44} +51.3479 q^{45} -158.692 q^{46} +155.931 q^{47} -91.2326 q^{48} +49.0000 q^{49} +76.5134 q^{50} +376.611 q^{51} +431.367 q^{53} -245.037 q^{54} -409.311 q^{55} -56.0000 q^{56} -234.344 q^{57} -188.404 q^{58} -516.538 q^{59} -212.426 q^{60} +126.857 q^{61} -106.905 q^{62} +38.5925 q^{63} +64.0000 q^{64} -501.182 q^{66} -779.421 q^{67} -264.194 q^{68} -452.435 q^{69} -130.390 q^{70} -338.952 q^{71} -44.1057 q^{72} -73.6584 q^{73} -113.332 q^{74} +218.141 q^{75} +164.393 q^{76} -307.634 q^{77} +584.714 q^{79} +149.018 q^{80} -847.461 q^{81} +187.779 q^{82} +633.536 q^{83} -159.657 q^{84} -615.150 q^{85} -23.9979 q^{86} -537.142 q^{87} +351.581 q^{88} +1603.47 q^{89} -102.696 q^{90} +317.385 q^{92} -304.788 q^{93} -311.862 q^{94} +382.773 q^{95} +182.465 q^{96} -1587.64 q^{97} -98.0000 q^{98} -242.293 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{2} + q^{3} + 20 q^{4} + 6 q^{5} - 2 q^{6} + 35 q^{7} - 40 q^{8} + 56 q^{9} - 12 q^{10} - 9 q^{11} + 4 q^{12} - 70 q^{14} + 137 q^{15} + 80 q^{16} - 89 q^{17} - 112 q^{18} - 49 q^{19} + 24 q^{20}+ \cdots + 81 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 + q^3 + 20 * q^4 + 6 * q^5 - 2 * q^6 + 35 * q^7 - 40 * q^8 + 56 * q^9 - 12 * q^10 - 9 * q^11 + 4 * q^12 - 70 * q^14 + 137 * q^15 + 80 * q^16 - 89 * q^17 - 112 * q^18 - 49 * q^19 + 24 * q^20 + 7 * q^21 + 18 * q^22 - 60 * q^23 - 8 * q^24 - 59 * q^25 - 134 * q^27 + 140 * q^28 - 196 * q^29 - 274 * q^30 + 154 * q^31 - 160 * q^32 - 433 * q^33 + 178 * q^34 + 42 * q^35 + 224 * q^36 + 61 * q^37 + 98 * q^38 - 48 * q^40 - 237 * q^41 - 14 * q^42 - 459 * q^43 - 36 * q^44 + 384 * q^45 + 120 * q^46 - 314 * q^47 + 16 * q^48 + 245 * q^49 + 118 * q^50 - 1069 * q^51 - 143 * q^53 + 268 * q^54 - 1899 * q^55 - 280 * q^56 + 211 * q^57 + 392 * q^58 + 178 * q^59 + 548 * q^60 - 811 * q^61 - 308 * q^62 + 392 * q^63 + 320 * q^64 + 866 * q^66 - 1078 * q^67 - 356 * q^68 - 654 * q^69 - 84 * q^70 + 1670 * q^71 - 448 * q^72 + 249 * q^73 - 122 * q^74 + 771 * q^75 - 196 * q^76 - 63 * q^77 + 4 * q^79 + 96 * q^80 - 2179 * q^81 + 474 * q^82 + 1986 * q^83 + 28 * q^84 - 2910 * q^85 + 918 * q^86 - 1427 * q^87 + 72 * q^88 + 461 * q^89 - 768 * q^90 - 240 * q^92 - 1055 * q^93 + 628 * q^94 - 543 * q^95 - 32 * q^96 + 2415 * q^97 - 490 * q^98 + 81 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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