LMFDB - Embedded newform 189.4.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,4,Mod(1,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, 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("189.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	189.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:	$11.1513609911$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	189.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.26795 q^{2} -6.39230 q^{4} +3.92820 q^{5} +7.00000 q^{7} +18.2487 q^{8} -4.98076 q^{10} -51.7128 q^{11} +67.5692 q^{13} -8.87564 q^{14} +28.0000 q^{16} -63.4974 q^{17} -71.9230 q^{19} -25.1103 q^{20} +65.5692 q^{22} -147.779 q^{23} -109.569 q^{25} -85.6743 q^{26} -44.7461 q^{28} -117.646 q^{29} +54.7077 q^{31} -181.492 q^{32} +80.5115 q^{34} +27.4974 q^{35} +9.70766 q^{37} +91.1948 q^{38} +71.6846 q^{40} +236.338 q^{41} -489.785 q^{43} +330.564 q^{44} +187.377 q^{46} -613.841 q^{47} +49.0000 q^{49} +138.928 q^{50} -431.923 q^{52} -316.543 q^{53} -203.138 q^{55} +127.741 q^{56} +149.169 q^{58} -2.43594 q^{59} +482.831 q^{61} -69.3665 q^{62} +6.12297 q^{64} +265.426 q^{65} +646.123 q^{67} +405.895 q^{68} -34.8653 q^{70} +459.846 q^{71} +137.615 q^{73} -12.3088 q^{74} +459.754 q^{76} -361.990 q^{77} -816.307 q^{79} +109.990 q^{80} -299.665 q^{82} +1009.25 q^{83} -249.431 q^{85} +621.022 q^{86} -943.692 q^{88} +255.682 q^{89} +472.985 q^{91} +944.651 q^{92} +778.319 q^{94} -282.528 q^{95} -62.9076 q^{97} -62.1295 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{2} + 8 q^{4} - 6 q^{5} + 14 q^{7} - 12 q^{8} + 42 q^{10} - 48 q^{11} + 52 q^{13} - 42 q^{14} + 56 q^{16} - 30 q^{17} + 64 q^{19} - 168 q^{20} + 48 q^{22} - 60 q^{23} - 136 q^{25} - 12 q^{26}+ \cdots - 294 q^{98}+O(q^{100}) $ 2 * q - 6 * q^2 + 8 * q^4 - 6 * q^5 + 14 * q^7 - 12 * q^8 + 42 * q^10 - 48 * q^11 + 52 * q^13 - 42 * q^14 + 56 * q^16 - 30 * q^17 + 64 * q^19 - 168 * q^20 + 48 * q^22 - 60 * q^23 - 136 * q^25 - 12 * q^26 + 56 * q^28 - 360 * q^29 - 140 * q^31 - 72 * q^32 - 78 * q^34 - 42 * q^35 - 230 * q^37 - 552 * q^38 + 372 * q^40 - 234 * q^41 - 938 * q^43 + 384 * q^44 - 228 * q^46 - 618 * q^47 + 98 * q^49 + 264 * q^50 - 656 * q^52 + 420 * q^53 - 240 * q^55 - 84 * q^56 + 1296 * q^58 - 282 * q^59 - 32 * q^61 + 852 * q^62 - 736 * q^64 + 420 * q^65 + 544 * q^67 + 888 * q^68 + 294 * q^70 + 504 * q^71 - 764 * q^73 + 1122 * q^74 + 2416 * q^76 - 336 * q^77 + 238 * q^79 - 168 * q^80 + 1926 * q^82 + 522 * q^83 - 582 * q^85 + 2742 * q^86 - 1056 * q^88 - 708 * q^89 + 364 * q^91 + 2208 * q^92 + 798 * q^94 - 1632 * q^95 + 664 * q^97 - 294 * q^98

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 $

$2$	−1.26795	−0.448288	−0.224144	−	0.974556i	$-0.571959\pi$
$2$	−1.26795	−0.448288	−0.224144	+	0.974556i	$0.571959\pi$
$3$	0	0
$3$	0	0
$4$	−6.39230	−0.799038
$5$	3.92820	0.351349	0.175675	−	0.984448i	$-0.443789\pi$
$5$	3.92820	0.351349	0.175675	+	0.984448i	$0.443789\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	18.2487	0.806487
$9$	0	0
$10$	−4.98076	−0.157506
$11$	−51.7128	−1.41745	−0.708727	−	0.705483i	$-0.750730\pi$
$11$	−51.7128	−1.41745	−0.708727	+	0.705483i	$0.750730\pi$
$12$	0	0
$13$	67.5692	1.44156	0.720782	−	0.693162i	$-0.243783\pi$
$13$	67.5692	1.44156	0.720782	+	0.693162i	$0.243783\pi$
$14$	−8.87564	−0.169437
$15$	0	0
$16$	28.0000	0.437500
$17$	−63.4974	−0.905905	−0.452953	−	0.891535i	$-0.649629\pi$
$17$	−63.4974	−0.905905	−0.452953	+	0.891535i	$0.649629\pi$
$18$	0	0
$19$	−71.9230	−0.868436	−0.434218	−	0.900808i	$-0.642975\pi$
$19$	−71.9230	−0.868436	−0.434218	+	0.900808i	$0.642975\pi$
$20$	−25.1103	−0.280741
$21$	0	0
$22$	65.5692	0.635427
$23$	−147.779	−1.33975	−0.669873	−	0.742476i	$-0.733651\pi$
$23$	−147.779	−1.33975	−0.669873	+	0.742476i	$0.733651\pi$
$24$	0	0
$25$	−109.569	−0.876554
$26$	−85.6743	−0.646235
$27$	0	0
$28$	−44.7461	−0.302008
$29$	−117.646	−0.753322	−0.376661	−	0.926351i	$-0.622928\pi$
$29$	−117.646	−0.753322	−0.376661	+	0.926351i	$0.622928\pi$
$30$	0	0
$31$	54.7077	0.316961	0.158480	−	0.987362i	$-0.449341\pi$
$31$	54.7077	0.316961	0.158480	+	0.987362i	$0.449341\pi$
$32$	−181.492	−1.00261
$33$	0	0
$34$	80.5115	0.406106
$35$	27.4974	0.132798
$36$	0	0
$37$	9.70766	0.0431332	0.0215666	−	0.999767i	$-0.493135\pi$
$37$	9.70766	0.0431332	0.0215666	+	0.999767i	$0.493135\pi$
$38$	91.1948	0.389309
$39$	0	0
$40$	71.6846	0.283358
$41$	236.338	0.900240	0.450120	−	0.892968i	$-0.351381\pi$
$41$	236.338	0.900240	0.450120	+	0.892968i	$0.351381\pi$
$42$	0	0
$43$	−489.785	−1.73701	−0.868505	−	0.495680i	$-0.834919\pi$
$43$	−489.785	−1.73701	−0.868505	+	0.495680i	$0.834919\pi$
$44$	330.564	1.13260
$45$	0	0
$46$	187.377	0.600591
$47$	−613.841	−1.90506	−0.952531	−	0.304442i	$-0.901530\pi$
$47$	−613.841	−1.90506	−0.952531	+	0.304442i	$0.901530\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	138.928	0.392948
$51$	0	0
$52$	−431.923	−1.15186
$53$	−316.543	−0.820388	−0.410194	−	0.911998i	$-0.634539\pi$
$53$	−316.543	−0.820388	−0.410194	+	0.911998i	$0.634539\pi$
$54$	0	0
$55$	−203.138	−0.498021
$56$	127.741	0.304823
$57$	0	0
$58$	149.169	0.337705
$59$	−2.43594	−0.00537511	−0.00268756	−	0.999996i	$-0.500855\pi$
$59$	−2.43594	−0.00537511	−0.00268756	+	0.999996i	$0.500855\pi$
$60$	0	0
$61$	482.831	1.01344	0.506722	−	0.862109i	$-0.330857\pi$
$61$	482.831	1.01344	0.506722	+	0.862109i	$0.330857\pi$
$62$	−69.3665	−0.142090
$63$	0	0
$64$	6.12297	0.0119589
$65$	265.426	0.506492
$66$	0	0
$67$	646.123	1.17816	0.589078	−	0.808076i	$-0.299491\pi$
$67$	646.123	1.17816	0.589078	+	0.808076i	$0.299491\pi$
$68$	405.895	0.723853
$69$	0	0
$70$	−34.8653	−0.0595315
$71$	459.846	0.768644	0.384322	−	0.923199i	$-0.374435\pi$
$71$	459.846	0.768644	0.384322	+	0.923199i	$0.374435\pi$
$72$	0	0
$73$	137.615	0.220639	0.110319	−	0.993896i	$-0.464813\pi$
$73$	137.615	0.220639	0.110319	+	0.993896i	$0.464813\pi$
$74$	−12.3088	−0.0193361
$75$	0	0
$76$	459.754	0.693913
$77$	−361.990	−0.535747
$78$	0	0
$79$	−816.307	−1.16255	−0.581277	−	0.813706i	$-0.697447\pi$
$79$	−816.307	−1.16255	−0.581277	+	0.813706i	$0.697447\pi$
$80$	109.990	0.153715
$81$	0	0
$82$	−299.665	−0.403567
$83$	1009.25	1.33469	0.667344	−	0.744749i	$-0.267431\pi$
$83$	1009.25	1.33469	0.667344	+	0.744749i	$0.267431\pi$
$84$	0	0
$85$	−249.431	−0.318289
$86$	621.022	0.778681
$87$	0	0
$88$	−943.692	−1.14316
$89$	255.682	0.304519	0.152260	−	0.988341i	$-0.451345\pi$
$89$	255.682	0.304519	0.152260	+	0.988341i	$0.451345\pi$
$90$	0	0
$91$	472.985	0.544860
$92$	944.651	1.07051
$93$	0	0
$94$	778.319	0.854016
$95$	−282.528	−0.305124
$96$	0	0
$97$	−62.9076	−0.0658484	−0.0329242	−	0.999458i	$-0.510482\pi$
$97$	−62.9076	−0.0658484	−0.0329242	+	0.999458i	$0.510482\pi$
$98$	−62.1295	−0.0640411
$99$	0	0
$100$	700.400	0.700400
$101$	−823.128	−0.810934	−0.405467	−	0.914110i	$-0.632891\pi$
$101$	−823.128	−0.810934	−0.405467	+	0.914110i	$0.632891\pi$
$102$	0	0
$103$	983.646	0.940986	0.470493	−	0.882404i	$-0.344076\pi$
$103$	983.646	0.940986	0.470493	+	0.882404i	$0.344076\pi$
$104$	1233.05	1.16260
$105$	0	0
$106$	401.361	0.367770
$107$	1176.47	1.06293	0.531464	−	0.847081i	$-0.321642\pi$
$107$	1176.47	1.06293	0.531464	+	0.847081i	$0.321642\pi$
$108$	0	0
$109$	−1171.83	−1.02973	−0.514867	−	0.857270i	$-0.672159\pi$
$109$	−1171.83	−1.02973	−0.514867	+	0.857270i	$0.672159\pi$
$110$	257.569	0.223257
$111$	0	0
$112$	196.000	0.165359
$113$	−426.231	−0.354836	−0.177418	−	0.984136i	$-0.556774\pi$
$113$	−426.231	−0.354836	−0.177418	+	0.984136i	$0.556774\pi$
$114$	0	0
$115$	−580.508	−0.470718
$116$	752.030	0.601933
$117$	0	0
$118$	3.08864	0.00240960
$119$	−444.482	−0.342400
$120$	0	0
$121$	1343.22	1.00918
$122$	−612.205	−0.454315
$123$	0	0
$124$	−349.708	−0.253264
$125$	−921.436	−0.659326
$126$	0	0
$127$	−1832.03	−1.28005	−0.640025	−	0.768354i	$-0.721076\pi$
$127$	−1832.03	−1.28005	−0.640025	+	0.768354i	$0.721076\pi$
$128$	1444.17	0.997252
$129$	0	0
$130$	−336.546	−0.227054
$131$	−2801.24	−1.86829	−0.934143	−	0.356898i	$-0.883834\pi$
$131$	−2801.24	−1.86829	−0.934143	+	0.356898i	$0.883834\pi$
$132$	0	0
$133$	−503.461	−0.328238
$134$	−819.251	−0.528153
$135$	0	0
$136$	−1158.75	−0.730600
$137$	−898.113	−0.560080	−0.280040	−	0.959988i	$-0.590348\pi$
$137$	−898.113	−0.560080	−0.280040	+	0.959988i	$0.590348\pi$
$138$	0	0
$139$	1141.11	0.696313	0.348156	−	0.937436i	$-0.386808\pi$
$139$	1141.11	0.696313	0.348156	+	0.937436i	$0.386808\pi$
$140$	−175.772	−0.106110
$141$	0	0
$142$	−583.061	−0.344573
$143$	−3494.19	−2.04335
$144$	0	0
$145$	−462.138	−0.264679
$146$	−174.489	−0.0989098
$147$	0	0
$148$	−62.0543	−0.0344651
$149$	3603.59	1.98133	0.990663	−	0.136335i	$-0.0435323\pi$
$149$	3603.59	1.98133	0.990663	+	0.136335i	$0.0435323\pi$
$150$	0	0
$151$	145.447	0.0783859	0.0391930	−	0.999232i	$-0.487521\pi$
$151$	145.447	0.0783859	0.0391930	+	0.999232i	$0.487521\pi$
$152$	−1312.50	−0.700382
$153$	0	0
$154$	458.985	0.240169
$155$	214.903	0.111364
$156$	0	0
$157$	−1180.17	−0.599922	−0.299961	−	0.953951i	$-0.596974\pi$
$157$	−1180.17	−0.599922	−0.299961	+	0.953951i	$0.596974\pi$
$158$	1035.04	0.521159
$159$	0	0
$160$	−712.939	−0.352267
$161$	−1034.46	−0.506376
$162$	0	0
$163$	2116.31	1.01694	0.508472	−	0.861078i	$-0.330210\pi$
$163$	2116.31	1.01694	0.508472	+	0.861078i	$0.330210\pi$
$164$	−1510.75	−0.719326
$165$	0	0
$166$	−1279.67	−0.598324
$167$	−3247.22	−1.50465	−0.752326	−	0.658790i	$-0.771068\pi$
$167$	−3247.22	−1.50465	−0.752326	+	0.658790i	$0.771068\pi$
$168$	0	0
$169$	2368.60	1.07811
$170$	316.266	0.142685
$171$	0	0
$172$	3130.85	1.38794
$173$	−990.195	−0.435163	−0.217581	−	0.976042i	$-0.569817\pi$
$173$	−990.195	−0.435163	−0.217581	+	0.976042i	$0.569817\pi$
$174$	0	0
$175$	−766.985	−0.331306
$176$	−1447.96	−0.620136
$177$	0	0
$178$	−324.192	−0.136512
$179$	−743.267	−0.310360	−0.155180	−	0.987886i	$-0.549596\pi$
$179$	−743.267	−0.310360	−0.155180	+	0.987886i	$0.549596\pi$
$180$	0	0
$181$	−2422.14	−0.994675	−0.497337	−	0.867557i	$-0.665689\pi$
$181$	−2422.14	−0.994675	−0.497337	+	0.867557i	$0.665689\pi$
$182$	−599.720	−0.244254
$183$	0	0
$184$	−2696.78	−1.08049
$185$	38.1337	0.0151548
$186$	0	0
$187$	3283.63	1.28408
$188$	3923.86	1.52222
$189$	0	0
$190$	358.232	0.136783
$191$	361.897	0.137099	0.0685496	−	0.997648i	$-0.478163\pi$
$191$	361.897	0.137099	0.0685496	+	0.997648i	$0.478163\pi$
$192$	0	0
$193$	1128.39	0.420844	0.210422	−	0.977611i	$-0.432516\pi$
$193$	1128.39	0.420844	0.210422	+	0.977611i	$0.432516\pi$
$194$	79.7636	0.0295190
$195$	0	0
$196$	−313.223	−0.114148
$197$	1972.54	0.713389	0.356694	−	0.934221i	$-0.383904\pi$
$197$	1972.54	0.713389	0.356694	+	0.934221i	$0.383904\pi$
$198$	0	0
$199$	1100.51	0.392024	0.196012	−	0.980601i	$-0.437201\pi$
$199$	1100.51	0.392024	0.196012	+	0.980601i	$0.437201\pi$
$200$	−1999.50	−0.706929
$201$	0	0
$202$	1043.68	0.363532
$203$	−823.523	−0.284729
$204$	0	0
$205$	928.385	0.316299
$206$	−1247.21	−0.421832
$207$	0	0
$208$	1891.94	0.630684
$209$	3719.34	1.23097
$210$	0	0
$211$	−3617.08	−1.18014	−0.590071	−	0.807352i	$-0.700900\pi$
$211$	−3617.08	−1.18014	−0.590071	+	0.807352i	$0.700900\pi$
$212$	2023.44	0.655522
$213$	0	0
$214$	−1491.70	−0.476498
$215$	−1923.97	−0.610297
$216$	0	0
$217$	382.954	0.119800
$218$	1485.82	0.461617
$219$	0	0
$220$	1298.52	0.397938
$221$	−4290.47	−1.30592
$222$	0	0
$223$	4224.15	1.26848	0.634238	−	0.773138i	$-0.281314\pi$
$223$	4224.15	1.26848	0.634238	+	0.773138i	$0.281314\pi$
$224$	−1270.45	−0.378952
$225$	0	0
$226$	540.439	0.159068
$227$	−6255.65	−1.82908	−0.914542	−	0.404491i	$-0.867449\pi$
$227$	−6255.65	−1.82908	−0.914542	+	0.404491i	$0.867449\pi$
$228$	0	0
$229$	−5237.89	−1.51148	−0.755742	−	0.654870i	$-0.772723\pi$
$229$	−5237.89	−1.51148	−0.755742	+	0.654870i	$0.772723\pi$
$230$	736.054	0.211017
$231$	0	0
$232$	−2146.89	−0.607544
$233$	5433.71	1.52779	0.763893	−	0.645342i	$-0.223285\pi$
$233$	5433.71	1.52779	0.763893	+	0.645342i	$0.223285\pi$
$234$	0	0
$235$	−2411.29	−0.669342
$236$	15.5712	0.00429492
$237$	0	0
$238$	563.581	0.153494
$239$	2927.15	0.792225	0.396112	−	0.918202i	$-0.370359\pi$
$239$	2927.15	0.792225	0.396112	+	0.918202i	$0.370359\pi$
$240$	0	0
$241$	3922.03	1.04830	0.524150	−	0.851626i	$-0.324383\pi$
$241$	3922.03	1.04830	0.524150	+	0.851626i	$0.324383\pi$
$242$	−1703.13	−0.452402
$243$	0	0
$244$	−3086.40	−0.809781
$245$	192.482	0.0501927
$246$	0	0
$247$	−4859.78	−1.25191
$248$	998.344	0.255625
$249$	0	0
$250$	1168.33	0.295568
$251$	3827.00	0.962383	0.481191	−	0.876616i	$-0.340204\pi$
$251$	3827.00	0.962383	0.481191	+	0.876616i	$0.340204\pi$
$252$	0	0
$253$	7642.09	1.89903
$254$	2322.92	0.573831
$255$	0	0
$256$	−1880.12	−0.459015
$257$	1718.54	0.417120	0.208560	−	0.978010i	$-0.433122\pi$
$257$	1718.54	0.417120	0.208560	+	0.978010i	$0.433122\pi$
$258$	0	0
$259$	67.9536	0.0163028
$260$	−1696.68	−0.404707
$261$	0	0
$262$	3551.83	0.837530
$263$	4181.27	0.980336	0.490168	−	0.871628i	$-0.336935\pi$
$263$	4181.27	0.980336	0.490168	+	0.871628i	$0.336935\pi$
$264$	0	0
$265$	−1243.45	−0.288243
$266$	638.363	0.147145
$267$	0	0
$268$	−4130.22	−0.941392
$269$	6158.35	1.39584	0.697920	−	0.716175i	$-0.254109\pi$
$269$	6158.35	1.39584	0.697920	+	0.716175i	$0.254109\pi$
$270$	0	0
$271$	−4754.98	−1.06585	−0.532924	−	0.846163i	$-0.678907\pi$
$271$	−4754.98	−1.06585	−0.532924	+	0.846163i	$0.678907\pi$
$272$	−1777.93	−0.396333
$273$	0	0
$274$	1138.76	0.251077
$275$	5666.13	1.24248
$276$	0	0
$277$	4991.83	1.08278	0.541390	−	0.840772i	$-0.317898\pi$
$277$	4991.83	1.08278	0.541390	+	0.840772i	$0.317898\pi$
$278$	−1446.87	−0.312148
$279$	0	0
$280$	501.793	0.107099
$281$	6068.29	1.28827	0.644135	−	0.764912i	$-0.277217\pi$
$281$	6068.29	1.28827	0.644135	+	0.764912i	$0.277217\pi$
$282$	0	0
$283$	−1854.58	−0.389553	−0.194777	−	0.980848i	$-0.562398\pi$
$283$	−1854.58	−0.389553	−0.194777	+	0.980848i	$0.562398\pi$
$284$	−2939.48	−0.614175
$285$	0	0
$286$	4430.46	0.916009
$287$	1654.37	0.340259
$288$	0	0
$289$	−881.077	−0.179336
$290$	585.968	0.118652
$291$	0	0
$292$	−879.679	−0.176299
$293$	5938.68	1.18410	0.592050	−	0.805901i	$-0.298319\pi$
$293$	5938.68	1.18410	0.592050	+	0.805901i	$0.298319\pi$
$294$	0	0
$295$	−9.56885	−0.00188854
$296$	177.152	0.0347864
$297$	0	0
$298$	−4569.17	−0.888204
$299$	−9985.34	−1.93133
$300$	0	0
$301$	−3428.49	−0.656528
$302$	−184.419	−0.0351395
$303$	0	0
$304$	−2013.85	−0.379941
$305$	1896.66	0.356073
$306$	0	0
$307$	−8870.08	−1.64900	−0.824498	−	0.565864i	$-0.808543\pi$
$307$	−8870.08	−1.64900	−0.824498	+	0.565864i	$0.808543\pi$
$308$	2313.95	0.428083
$309$	0	0
$310$	−272.486	−0.0499231
$311$	2343.32	0.427259	0.213629	−	0.976915i	$-0.431472\pi$
$311$	2343.32	0.427259	0.213629	+	0.976915i	$0.431472\pi$
$312$	0	0
$313$	0.262523	4.74080e−5 0	2.37040e−5	−	1.00000i	$-0.499992\pi$
$313$	0.262523	4.74080e−5 0	2.37040e−5		1.00000i	$0.499992\pi$
$314$	1496.39	0.268938
$315$	0	0
$316$	5218.09	0.928925
$317$	3051.05	0.540581	0.270290	−	0.962779i	$-0.412880\pi$
$317$	3051.05	0.540581	0.270290	+	0.962779i	$0.412880\pi$
$318$	0	0
$319$	6083.81	1.06780
$320$	24.0523	0.00420176
$321$	0	0
$322$	1311.64	0.227002
$323$	4566.93	0.786720
$324$	0	0
$325$	−7403.51	−1.26361
$326$	−2683.37	−0.455884
$327$	0	0
$328$	4312.87	0.726032
$329$	−4296.89	−0.720046
$330$	0	0
$331$	10952.6	1.81876	0.909381	−	0.415965i	$-0.136556\pi$
$331$	10952.6	1.81876	0.909381	+	0.415965i	$0.136556\pi$
$332$	−6451.41	−1.06647
$333$	0	0
$334$	4117.30	0.674517
$335$	2538.10	0.413944
$336$	0	0
$337$	−3551.61	−0.574091	−0.287046	−	0.957917i	$-0.592673\pi$
$337$	−3551.61	−0.574091	−0.287046	+	0.957917i	$0.592673\pi$
$338$	−3003.26	−0.483302
$339$	0	0
$340$	1594.44	0.254325
$341$	−2829.09	−0.449278
$342$	0	0
$343$	343.000	0.0539949
$344$	−8937.94	−1.40088
$345$	0	0
$346$	1255.52	0.195078
$347$	−7089.53	−1.09679	−0.548394	−	0.836220i	$-0.684761\pi$
$347$	−7089.53	−1.09679	−0.548394	+	0.836220i	$0.684761\pi$
$348$	0	0
$349$	−6493.29	−0.995925	−0.497963	−	0.867198i	$-0.665918\pi$
$349$	−6493.29	−0.995925	−0.497963	+	0.867198i	$0.665918\pi$
$350$	972.497	0.148520
$351$	0	0
$352$	9385.48	1.42116
$353$	5106.99	0.770022	0.385011	−	0.922912i	$-0.374198\pi$
$353$	5106.99	0.770022	0.385011	+	0.922912i	$0.374198\pi$
$354$	0	0
$355$	1806.37	0.270062
$356$	−1634.40	−0.243323
$357$	0	0
$358$	942.425	0.139130
$359$	−9598.78	−1.41115	−0.705577	−	0.708633i	$-0.749312\pi$
$359$	−9598.78	−1.41115	−0.705577	+	0.708633i	$0.749312\pi$
$360$	0	0
$361$	−1686.08	−0.245819
$362$	3071.15	0.445900
$363$	0	0
$364$	−3023.46	−0.435364
$365$	540.581	0.0775213
$366$	0	0
$367$	1267.23	0.180242	0.0901212	−	0.995931i	$-0.471275\pi$
$367$	1267.23	0.180242	0.0901212	+	0.995931i	$0.471275\pi$
$368$	−4137.82	−0.586139
$369$	0	0
$370$	−48.3515	−0.00679372
$371$	−2215.80	−0.310078
$372$	0	0
$373$	−9312.57	−1.29273	−0.646363	−	0.763030i	$-0.723711\pi$
$373$	−9312.57	−1.29273	−0.646363	+	0.763030i	$0.723711\pi$
$374$	−4163.48	−0.575637
$375$	0	0
$376$	−11201.8	−1.53641
$377$	−7949.26	−1.08596
$378$	0	0
$379$	1366.43	0.185195	0.0925973	−	0.995704i	$-0.470483\pi$
$379$	1366.43	0.185195	0.0925973	+	0.995704i	$0.470483\pi$
$380$	1806.01	0.243806
$381$	0	0
$382$	−458.867	−0.0614599
$383$	5096.46	0.679940	0.339970	−	0.940436i	$-0.389583\pi$
$383$	5096.46	0.679940	0.339970	+	0.940436i	$0.389583\pi$
$384$	0	0
$385$	−1421.97	−0.188234
$386$	−1430.74	−0.188659
$387$	0	0
$388$	402.124	0.0526154
$389$	8662.07	1.12901	0.564504	−	0.825430i	$-0.309067\pi$
$389$	8662.07	1.12901	0.564504	+	0.825430i	$0.309067\pi$
$390$	0	0
$391$	9383.61	1.21368
$392$	894.187	0.115212
$393$	0	0
$394$	−2501.08	−0.319803
$395$	−3206.62	−0.408462
$396$	0	0
$397$	15350.7	1.94063	0.970314	−	0.241850i	$-0.0777543\pi$
$397$	15350.7	1.94063	0.970314	+	0.241850i	$0.0777543\pi$
$398$	−1395.39	−0.175740
$399$	0	0
$400$	−3067.94	−0.383492
$401$	−5740.02	−0.714821	−0.357410	−	0.933947i	$-0.616340\pi$
$401$	−5740.02	−0.714821	−0.357410	+	0.933947i	$0.616340\pi$
$402$	0	0
$403$	3696.55	0.456919
$404$	5261.69	0.647967
$405$	0	0
$406$	1044.19	0.127641
$407$	−502.010	−0.0611394
$408$	0	0
$409$	8785.54	1.06214	0.531072	−	0.847327i	$-0.321789\pi$
$409$	8785.54	1.06214	0.531072	+	0.847327i	$0.321789\pi$
$410$	−1177.15	−0.141793
$411$	0	0
$412$	−6287.77	−0.751884
$413$	−17.0515	−0.00203160
$414$	0	0
$415$	3964.52	0.468942
$416$	−12263.3	−1.44533
$417$	0	0
$418$	−4715.94	−0.551828
$419$	−4404.08	−0.513492	−0.256746	−	0.966479i	$-0.582650\pi$
$419$	−4404.08	−0.513492	−0.256746	+	0.966479i	$0.582650\pi$
$420$	0	0
$421$	4027.26	0.466215	0.233108	−	0.972451i	$-0.425111\pi$
$421$	4027.26	0.466215	0.233108	+	0.972451i	$0.425111\pi$
$422$	4586.27	0.529043
$423$	0	0
$424$	−5776.51	−0.661632
$425$	6957.36	0.794075
$426$	0	0
$427$	3379.81	0.383046
$428$	−7520.33	−0.849320
$429$	0	0
$430$	2439.50	0.273589
$431$	−4733.90	−0.529058	−0.264529	−	0.964378i	$-0.585216\pi$
$431$	−4733.90	−0.529058	−0.264529	+	0.964378i	$0.585216\pi$
$432$	0	0
$433$	−15456.7	−1.71548	−0.857738	−	0.514087i	$-0.828131\pi$
$433$	−15456.7	−1.71548	−0.857738	+	0.514087i	$0.828131\pi$
$434$	−485.566	−0.0537048
$435$	0	0
$436$	7490.70	0.822797
$437$	10628.7	1.16348
$438$	0	0
$439$	−9779.64	−1.06323	−0.531614	−	0.846987i	$-0.678414\pi$
$439$	−9779.64	−1.06323	−0.531614	+	0.846987i	$0.678414\pi$
$440$	−3707.01	−0.401648
$441$	0	0
$442$	5440.10	0.585428
$443$	−14671.0	−1.57346	−0.786729	−	0.617298i	$-0.788227\pi$
$443$	−14671.0	−1.57346	−0.786729	+	0.617298i	$0.788227\pi$
$444$	0	0
$445$	1004.37	0.106993
$446$	−5356.01	−0.568642
$447$	0	0
$448$	42.8608	0.00452005
$449$	503.293	0.0528995	0.0264497	−	0.999650i	$-0.491580\pi$
$449$	503.293	0.0528995	0.0264497	+	0.999650i	$0.491580\pi$
$450$	0	0
$451$	−12221.7	−1.27605
$452$	2724.60	0.283527
$453$	0	0
$454$	7931.85	0.819956
$455$	1857.98	0.191436
$456$	0	0
$457$	−18037.2	−1.84627	−0.923136	−	0.384475i	$-0.874383\pi$
$457$	−18037.2	−1.84627	−0.923136	+	0.384475i	$0.874383\pi$
$458$	6641.38	0.677579
$459$	0	0
$460$	3710.78	0.376122
$461$	−10432.6	−1.05400	−0.527001	−	0.849864i	$-0.676684\pi$
$461$	−10432.6	−1.05400	−0.527001	+	0.849864i	$0.676684\pi$
$462$	0	0
$463$	206.922	0.0207700	0.0103850	−	0.999946i	$-0.496694\pi$
$463$	206.922	0.0207700	0.0103850	+	0.999946i	$0.496694\pi$
$464$	−3294.09	−0.329578
$465$	0	0
$466$	−6889.67	−0.684888
$467$	1808.79	0.179231	0.0896153	−	0.995976i	$-0.471436\pi$
$467$	1808.79	0.179231	0.0896153	+	0.995976i	$0.471436\pi$
$468$	0	0
$469$	4522.86	0.445301
$470$	3057.40	0.300058
$471$	0	0
$472$	−44.4527	−0.00433496
$473$	25328.1	2.46213
$474$	0	0
$475$	7880.55	0.761231
$476$	2841.26	0.273591
$477$	0	0
$478$	−3711.48	−0.355145
$479$	3269.26	0.311850	0.155925	−	0.987769i	$-0.450164\pi$
$479$	3269.26	0.311850	0.155925	+	0.987769i	$0.450164\pi$
$480$	0	0
$481$	655.939	0.0621793
$482$	−4972.94	−0.469940
$483$	0	0
$484$	−8586.24	−0.806371
$485$	−247.114	−0.0231358
$486$	0	0
$487$	−15589.4	−1.45056	−0.725282	−	0.688451i	$-0.758291\pi$
$487$	−15589.4	−1.45056	−0.725282	+	0.688451i	$0.758291\pi$
$488$	8811.04	0.817330
$489$	0	0
$490$	−244.057	−0.0225008
$491$	−12032.1	−1.10591	−0.552954	−	0.833212i	$-0.686499\pi$
$491$	−12032.1	−1.10591	−0.552954	+	0.833212i	$0.686499\pi$
$492$	0	0
$493$	7470.23	0.682438
$494$	6161.96	0.561214
$495$	0	0
$496$	1531.81	0.138670
$497$	3218.92	0.290520
$498$	0	0
$499$	−9958.64	−0.893407	−0.446704	−	0.894682i	$-0.647402\pi$
$499$	−9958.64	−0.893407	−0.446704	+	0.894682i	$0.647402\pi$
$500$	5890.10	0.526826
$501$	0	0
$502$	−4852.44	−0.431424
$503$	−7368.14	−0.653139	−0.326570	−	0.945173i	$-0.605893\pi$
$503$	−7368.14	−0.653139	−0.326570	+	0.945173i	$0.605893\pi$
$504$	0	0
$505$	−3233.41	−0.284921
$506$	−9689.78	−0.851311
$507$	0	0
$508$	11710.9	1.02281
$509$	−7725.18	−0.672716	−0.336358	−	0.941734i	$-0.609195\pi$
$509$	−7725.18	−0.672716	−0.336358	+	0.941734i	$0.609195\pi$
$510$	0	0
$511$	963.307	0.0833937
$512$	−9169.49	−0.791481
$513$	0	0
$514$	−2179.03	−0.186990
$515$	3863.96	0.330615
$516$	0	0
$517$	31743.4	2.70034
$518$	−86.1617	−0.00730836
$519$	0	0
$520$	4843.68	0.408479
$521$	3904.26	0.328308	0.164154	−	0.986435i	$-0.447511\pi$
$521$	3904.26	0.328308	0.164154	+	0.986435i	$0.447511\pi$
$522$	0	0
$523$	17029.3	1.42379	0.711893	−	0.702288i	$-0.247838\pi$
$523$	17029.3	1.42379	0.711893	+	0.702288i	$0.247838\pi$
$524$	17906.4	1.49283
$525$	0	0
$526$	−5301.64	−0.439472
$527$	−3473.80	−0.287136
$528$	0	0
$529$	9671.77	0.794918
$530$	1576.63	0.129216
$531$	0	0
$532$	3218.28	0.262275
$533$	15969.2	1.29775
$534$	0	0
$535$	4621.40	0.373459
$536$	11790.9	0.950168
$537$	0	0
$538$	−7808.47	−0.625738
$539$	−2533.93	−0.202494
$540$	0	0
$541$	15599.2	1.23967	0.619834	−	0.784733i	$-0.287200\pi$
$541$	15599.2	1.23967	0.619834	+	0.784733i	$0.287200\pi$
$542$	6029.08	0.477806
$543$	0	0
$544$	11524.3	0.908272
$545$	−4603.19	−0.361796
$546$	0	0
$547$	−22666.1	−1.77172	−0.885860	−	0.463953i	$-0.846431\pi$
$547$	−22666.1	−1.77172	−0.885860	+	0.463953i	$0.846431\pi$
$548$	5741.01	0.447525
$549$	0	0
$550$	−7184.37	−0.556986
$551$	8461.47	0.654212
$552$	0	0
$553$	−5714.15	−0.439404
$554$	−6329.39	−0.485397
$555$	0	0
$556$	−7294.31	−0.556380
$557$	15151.1	1.15255	0.576276	−	0.817255i	$-0.304505\pi$
$557$	15151.1	1.15255	0.576276	+	0.817255i	$0.304505\pi$
$558$	0	0
$559$	−33094.4	−2.50401
$560$	769.928	0.0580989
$561$	0	0
$562$	−7694.29	−0.577516
$563$	13957.9	1.04486	0.522429	−	0.852683i	$-0.325026\pi$
$563$	13957.9	1.04486	0.522429	+	0.852683i	$0.325026\pi$
$564$	0	0
$565$	−1674.32	−0.124671
$566$	2351.52	0.174632
$567$	0	0
$568$	8391.60	0.619901
$569$	−3463.93	−0.255212	−0.127606	−	0.991825i	$-0.540729\pi$
$569$	−3463.93	−0.255212	−0.127606	+	0.991825i	$0.540729\pi$
$570$	0	0
$571$	22177.2	1.62537	0.812687	−	0.582701i	$-0.198004\pi$
$571$	22177.2	1.62537	0.812687	+	0.582701i	$0.198004\pi$
$572$	22336.0	1.63272
$573$	0	0
$574$	−2097.66	−0.152534
$575$	16192.1	1.17436
$576$	0	0
$577$	−5083.18	−0.366752	−0.183376	−	0.983043i	$-0.558703\pi$
$577$	−5083.18	−0.366752	−0.183376	+	0.983043i	$0.558703\pi$
$578$	1117.16	0.0803941
$579$	0	0
$580$	2954.13	0.211489
$581$	7064.72	0.504465
$582$	0	0
$583$	16369.4	1.16286
$584$	2511.30	0.177942
$585$	0	0
$586$	−7529.94	−0.530817
$587$	−26283.4	−1.84809	−0.924047	−	0.382278i	$-0.875140\pi$
$587$	−26283.4	−1.84809	−0.924047	+	0.382278i	$0.875140\pi$
$588$	0	0
$589$	−3934.74	−0.275260
$590$	12.1328	0.000846610 0
$591$	0	0
$592$	271.814	0.0188708
$593$	23901.8	1.65519	0.827596	−	0.561324i	$-0.189708\pi$
$593$	23901.8	1.65519	0.827596	+	0.561324i	$0.189708\pi$
$594$	0	0
$595$	−1746.02	−0.120302
$596$	−23035.2	−1.58315
$597$	0	0
$598$	12660.9	0.865791
$599$	−15150.3	−1.03343	−0.516715	−	0.856158i	$-0.672845\pi$
$599$	−15150.3	−1.03343	−0.516715	+	0.856158i	$0.672845\pi$
$600$	0	0
$601$	11860.1	0.804962	0.402481	−	0.915428i	$-0.368148\pi$
$601$	11860.1	0.804962	0.402481	+	0.915428i	$0.368148\pi$
$602$	4347.15	0.294314
$603$	0	0
$604$	−929.739	−0.0626334
$605$	5276.42	0.354574
$606$	0	0
$607$	−7847.27	−0.524730	−0.262365	−	0.964969i	$-0.584502\pi$
$607$	−7847.27	−0.524730	−0.262365	+	0.964969i	$0.584502\pi$
$608$	13053.5	0.870705
$609$	0	0
$610$	−2404.86	−0.159623
$611$	−41476.8	−2.74627
$612$	0	0
$613$	2068.12	0.136265	0.0681327	−	0.997676i	$-0.478296\pi$
$613$	2068.12	0.136265	0.0681327	+	0.997676i	$0.478296\pi$
$614$	11246.8	0.739225
$615$	0	0
$616$	−6605.85	−0.432073
$617$	25241.6	1.64698	0.823492	−	0.567328i	$-0.192023\pi$
$617$	25241.6	1.64698	0.823492	+	0.567328i	$0.192023\pi$
$618$	0	0
$619$	−16776.6	−1.08935	−0.544677	−	0.838646i	$-0.683348\pi$
$619$	−16776.6	−1.08935	−0.544677	+	0.838646i	$0.683348\pi$
$620$	−1373.72	−0.0889840
$621$	0	0
$622$	−2971.21	−0.191535
$623$	1789.77	0.115098
$624$	0	0
$625$	10076.6	0.644900
$626$	−0.332866	−2.12524e−5 0
$627$	0	0
$628$	7544.00	0.479360
$629$	−616.411	−0.0390746
$630$	0	0
$631$	22011.8	1.38871	0.694354	−	0.719633i	$-0.255690\pi$
$631$	22011.8	1.38871	0.694354	+	0.719633i	$0.255690\pi$
$632$	−14896.6	−0.937584
$633$	0	0
$634$	−3868.58	−0.242336
$635$	−7196.59	−0.449745
$636$	0	0
$637$	3310.89	0.205938
$638$	−7713.97	−0.478682
$639$	0	0
$640$	5673.01	0.350384
$641$	−21485.3	−1.32390	−0.661950	−	0.749548i	$-0.730271\pi$
$641$	−21485.3	−1.32390	−0.661950	+	0.749548i	$0.730271\pi$
$642$	0	0
$643$	14305.3	0.877365	0.438682	−	0.898642i	$-0.355445\pi$
$643$	14305.3	0.877365	0.438682	+	0.898642i	$0.355445\pi$
$644$	6612.56	0.404614
$645$	0	0
$646$	−5790.63	−0.352677
$647$	13848.7	0.841498	0.420749	−	0.907177i	$-0.361767\pi$
$647$	13848.7	0.841498	0.420749	+	0.907177i	$0.361767\pi$
$648$	0	0
$649$	125.969	0.00761898
$650$	9387.27	0.566460
$651$	0	0
$652$	−13528.1	−0.812578
$653$	−24324.9	−1.45774	−0.728872	−	0.684649i	$-0.759955\pi$
$653$	−24324.9	−1.45774	−0.728872	+	0.684649i	$0.759955\pi$
$654$	0	0
$655$	−11003.8	−0.656421
$656$	6617.47	0.393855
$657$	0	0
$658$	5448.23	0.322788
$659$	−5232.82	−0.309320	−0.154660	−	0.987968i	$-0.549428\pi$
$659$	−5232.82	−0.309320	−0.154660	+	0.987968i	$0.549428\pi$
$660$	0	0
$661$	−33511.1	−1.97191	−0.985954	−	0.167016i	$-0.946587\pi$
$661$	−33511.1	−1.97191	−0.985954	+	0.167016i	$0.946587\pi$
$662$	−13887.4	−0.815328
$663$	0	0
$664$	18417.4	1.07641
$665$	−1977.70	−0.115326
$666$	0	0
$667$	17385.7	1.00926
$668$	20757.2	1.20228
$669$	0	0
$670$	−3218.18	−0.185566
$671$	−24968.5	−1.43651
$672$	0	0
$673$	−17861.8	−1.02306	−0.511530	−	0.859265i	$-0.670921\pi$
$673$	−17861.8	−1.02306	−0.511530	+	0.859265i	$0.670921\pi$
$674$	4503.27	0.257358
$675$	0	0
$676$	−15140.8	−0.861448
$677$	−28992.9	−1.64592	−0.822959	−	0.568101i	$-0.807678\pi$
$677$	−28992.9	−1.64592	−0.822959	+	0.568101i	$0.807678\pi$
$678$	0	0
$679$	−440.353	−0.0248884
$680$	−4551.79	−0.256696
$681$	0	0
$682$	3587.14	0.201406
$683$	2381.89	0.133441	0.0667206	−	0.997772i	$-0.478746\pi$
$683$	2381.89	0.133441	0.0667206	+	0.997772i	$0.478746\pi$
$684$	0	0
$685$	−3527.97	−0.196784
$686$	−434.907	−0.0242053
$687$	0	0
$688$	−13714.0	−0.759942
$689$	−21388.6	−1.18264
$690$	0	0
$691$	8734.63	0.480870	0.240435	−	0.970665i	$-0.422710\pi$
$691$	8734.63	0.480870	0.240435	+	0.970665i	$0.422710\pi$
$692$	6329.63	0.347712
$693$	0	0
$694$	8989.16	0.491677
$695$	4482.50	0.244649
$696$	0	0
$697$	−15006.9	−0.815532
$698$	8233.16	0.446461
$699$	0	0
$700$	4902.80	0.264726
$701$	−24618.5	−1.32643	−0.663215	−	0.748429i	$-0.730809\pi$
$701$	−24618.5	−1.32643	−0.663215	+	0.748429i	$0.730809\pi$
$702$	0	0
$703$	−698.204	−0.0374584
$704$	−316.636	−0.0169512
$705$	0	0
$706$	−6475.41	−0.345192
$707$	−5761.90	−0.306504
$708$	0	0
$709$	−460.784	−0.0244078	−0.0122039	−	0.999926i	$-0.503885\pi$
$709$	−460.784	−0.0244078	−0.0122039	+	0.999926i	$0.503885\pi$
$710$	−2290.38	−0.121066
$711$	0	0
$712$	4665.86	0.245591
$713$	−8084.67	−0.424647
$714$	0	0
$715$	−13725.9	−0.717930
$716$	4751.19	0.247989
$717$	0	0
$718$	12170.8	0.632603
$719$	−31274.8	−1.62219	−0.811094	−	0.584915i	$-0.801128\pi$
$719$	−31274.8	−1.62219	−0.811094	+	0.584915i	$0.801128\pi$
$720$	0	0
$721$	6885.52	0.355659
$722$	2137.86	0.110198
$723$	0	0
$724$	15483.0	0.794783
$725$	12890.4	0.660327
$726$	0	0
$727$	1523.12	0.0777021	0.0388510	−	0.999245i	$-0.487630\pi$
$727$	1523.12	0.0777021	0.0388510	+	0.999245i	$0.487630\pi$
$728$	8631.36	0.439422
$729$	0	0
$730$	−685.429	−0.0347519
$731$	31100.1	1.57357
$732$	0	0
$733$	−28363.5	−1.42923	−0.714617	−	0.699516i	$-0.753399\pi$
$733$	−28363.5	−1.42923	−0.714617	+	0.699516i	$0.753399\pi$
$734$	−1606.78	−0.0808004
$735$	0	0
$736$	26820.8	1.34325
$737$	−33412.8	−1.66998
$738$	0	0
$739$	4971.75	0.247481	0.123741	−	0.992315i	$-0.460511\pi$
$739$	4971.75	0.247481	0.123741	+	0.992315i	$0.460511\pi$
$740$	−243.762	−0.0121093
$741$	0	0
$742$	2809.53	0.139004
$743$	8803.96	0.434705	0.217353	−	0.976093i	$-0.430258\pi$
$743$	8803.96	0.434705	0.217353	+	0.976093i	$0.430258\pi$
$744$	0	0
$745$	14155.6	0.696137
$746$	11807.9	0.579513
$747$	0	0
$748$	−20990.0	−1.02603
$749$	8235.27	0.401749
$750$	0	0
$751$	−30759.0	−1.49456	−0.747278	−	0.664511i	$-0.768640\pi$
$751$	−30759.0	−1.49456	−0.747278	+	0.664511i	$0.768640\pi$
$752$	−17187.5	−0.833465
$753$	0	0
$754$	10079.3	0.486823
$755$	571.344	0.0275408
$756$	0	0
$757$	1104.42	0.0530261	0.0265131	−	0.999648i	$-0.491560\pi$
$757$	1104.42	0.0530261	0.0265131	+	0.999648i	$0.491560\pi$
$758$	−1732.56	−0.0830205
$759$	0	0
$760$	−5155.78	−0.246079
$761$	3722.02	0.177297	0.0886485	−	0.996063i	$-0.471745\pi$
$761$	3722.02	0.177297	0.0886485	+	0.996063i	$0.471745\pi$
$762$	0	0
$763$	−8202.81	−0.389203
$764$	−2313.36	−0.109547
$765$	0	0
$766$	−6462.06	−0.304809
$767$	−164.594	−0.00774857
$768$	0	0
$769$	12779.6	0.599279	0.299640	−	0.954052i	$-0.403134\pi$
$769$	12779.6	0.599279	0.299640	+	0.954052i	$0.403134\pi$
$770$	1802.98	0.0843832
$771$	0	0
$772$	−7212.98	−0.336271
$773$	24178.0	1.12500	0.562498	−	0.826798i	$-0.309840\pi$
$773$	24178.0	1.12500	0.562498	+	0.826798i	$0.309840\pi$
$774$	0	0
$775$	−5994.28	−0.277833
$776$	−1147.98	−0.0531059
$777$	0	0
$778$	−10983.1	−0.506121
$779$	−16998.2	−0.781801
$780$	0	0
$781$	−23779.9	−1.08952
$782$	−11897.9	−0.544079
$783$	0	0
$784$	1372.00	0.0625000
$785$	−4635.94	−0.210782
$786$	0	0
$787$	34814.0	1.57686	0.788428	−	0.615126i	$-0.210895\pi$
$787$	34814.0	1.57686	0.788428	+	0.615126i	$0.210895\pi$
$788$	−12609.1	−0.570025
$789$	0	0
$790$	4065.83	0.183109
$791$	−2983.62	−0.134115
$792$	0	0
$793$	32624.5	1.46095
$794$	−19463.9	−0.869959
$795$	0	0
$796$	−7034.78	−0.313243
$797$	1412.03	0.0627563	0.0313781	−	0.999508i	$-0.490010\pi$
$797$	1412.03	0.0627563	0.0313781	+	0.999508i	$0.490010\pi$
$798$	0	0
$799$	38977.3	1.72581
$800$	19886.0	0.878844
$801$	0	0
$802$	7278.06	0.320445
$803$	−7116.47	−0.312746
$804$	0	0
$805$	−4063.55	−0.177915
$806$	−4687.04	−0.204831
$807$	0	0
$808$	−15021.0	−0.654007
$809$	−7209.04	−0.313296	−0.156648	−	0.987655i	$-0.550069\pi$
$809$	−7209.04	−0.313296	−0.156648	+	0.987655i	$0.550069\pi$
$810$	0	0
$811$	−35027.8	−1.51664	−0.758318	−	0.651885i	$-0.773979\pi$
$811$	−35027.8	−1.51664	−0.758318	+	0.651885i	$0.773979\pi$
$812$	5264.21	0.227509
$813$	0	0
$814$	636.524	0.0274080
$815$	8313.29	0.357303
$816$	0	0
$817$	35226.8	1.50848
$818$	−11139.6	−0.476146
$819$	0	0
$820$	−5934.52	−0.252735
$821$	2387.69	0.101499	0.0507496	−	0.998711i	$-0.483839\pi$
$821$	2387.69	0.101499	0.0507496	+	0.998711i	$0.483839\pi$
$822$	0	0
$823$	11009.1	0.466287	0.233144	−	0.972442i	$-0.425099\pi$
$823$	11009.1	0.466287	0.233144	+	0.972442i	$0.425099\pi$
$824$	17950.3	0.758893
$825$	0	0
$826$	21.6205	0.000910742 0
$827$	−156.810	−0.00659348	−0.00329674	−	0.999995i	$-0.501049\pi$
$827$	−156.810	−0.00659348	−0.00329674	+	0.999995i	$0.501049\pi$
$828$	0	0
$829$	10244.8	0.429210	0.214605	−	0.976701i	$-0.431154\pi$
$829$	10244.8	0.429210	0.214605	+	0.976701i	$0.431154\pi$
$830$	−5026.81	−0.210221
$831$	0	0
$832$	413.725	0.0172396
$833$	−3111.37	−0.129415
$834$	0	0
$835$	−12755.7	−0.528659
$836$	−23775.2	−0.983590
$837$	0	0
$838$	5584.15	0.230192
$839$	−6146.56	−0.252924	−0.126462	−	0.991971i	$-0.540362\pi$
$839$	−6146.56	−0.252924	−0.126462	+	0.991971i	$0.540362\pi$
$840$	0	0
$841$	−10548.4	−0.432506
$842$	−5106.36	−0.208999
$843$	0	0
$844$	23121.5	0.942978
$845$	9304.34	0.378792
$846$	0	0
$847$	9402.51	0.381433
$848$	−8863.22	−0.358920
$849$	0	0
$850$	−8821.58	−0.355974
$851$	−1434.59	−0.0577875
$852$	0	0
$853$	−18316.2	−0.735212	−0.367606	−	0.929982i	$-0.619822\pi$
$853$	−18316.2	−0.735212	−0.367606	+	0.929982i	$0.619822\pi$
$854$	−4285.43	−0.171715
$855$	0	0
$856$	21469.0	0.857237
$857$	−657.584	−0.0262108	−0.0131054	−	0.999914i	$-0.504172\pi$
$857$	−657.584	−0.0262108	−0.0131054	+	0.999914i	$0.504172\pi$
$858$	0	0
$859$	−16467.4	−0.654085	−0.327042	−	0.945010i	$-0.606052\pi$
$859$	−16467.4	−0.654085	−0.327042	+	0.945010i	$0.606052\pi$
$860$	12298.6	0.487651
$861$	0	0
$862$	6002.34	0.237170
$863$	24470.4	0.965217	0.482609	−	0.875836i	$-0.339689\pi$
$863$	24470.4	0.965217	0.482609	+	0.875836i	$0.339689\pi$
$864$	0	0
$865$	−3889.69	−0.152894
$866$	19598.3	0.769027
$867$	0	0
$868$	−2447.96	−0.0957247
$869$	42213.6	1.64787
$870$	0	0
$871$	43658.0	1.69839
$872$	−21384.4	−0.830467
$873$	0	0
$874$	−13476.7	−0.521575
$875$	−6450.05	−0.249202
$876$	0	0
$877$	−23651.8	−0.910678	−0.455339	−	0.890318i	$-0.650482\pi$
$877$	−23651.8	−0.910678	−0.455339	+	0.890318i	$0.650482\pi$
$878$	12400.1	0.476632
$879$	0	0
$880$	−5687.88	−0.217884
$881$	−2325.38	−0.0889264	−0.0444632	−	0.999011i	$-0.514158\pi$
$881$	−2325.38	−0.0889264	−0.0444632	+	0.999011i	$0.514158\pi$
$882$	0	0
$883$	2193.02	0.0835798	0.0417899	−	0.999126i	$-0.486694\pi$
$883$	2193.02	0.0835798	0.0417899	+	0.999126i	$0.486694\pi$
$884$	27426.0	1.04348
$885$	0	0
$886$	18602.1	0.705362
$887$	38.3431	0.00145145	0.000725725	−	1.00000i	$-0.499769\pi$
$887$	38.3431	0.00145145	0.000725725		1.00000i	$0.499769\pi$
$888$	0	0
$889$	−12824.2	−0.483814
$890$	−1273.49	−0.0479635
$891$	0	0
$892$	−27002.1	−1.01356
$893$	44149.3	1.65442
$894$	0	0
$895$	−2919.70	−0.109045
$896$	10109.2	0.376926
$897$	0	0
$898$	−638.150	−0.0237142
$899$	−6436.15	−0.238774
$900$	0	0
$901$	20099.7	0.743194
$902$	15496.5	0.572037
$903$	0	0
$904$	−7778.16	−0.286170
$905$	−9514.65	−0.349478
$906$	0	0
$907$	18032.2	0.660142	0.330071	−	0.943956i	$-0.392927\pi$
$907$	18032.2	0.660142	0.330071	+	0.943956i	$0.392927\pi$
$908$	39988.0	1.46151
$909$	0	0
$910$	−2355.82	−0.0858184
$911$	12877.7	0.468338	0.234169	−	0.972196i	$-0.424763\pi$
$911$	12877.7	0.468338	0.234169	+	0.972196i	$0.424763\pi$
$912$	0	0
$913$	−52190.9	−1.89186
$914$	22870.3	0.827661
$915$	0	0
$916$	33482.2	1.20773
$917$	−19608.7	−0.706146
$918$	0	0
$919$	−16267.1	−0.583899	−0.291950	−	0.956434i	$-0.594304\pi$
$919$	−16267.1	−0.583899	−0.291950	+	0.956434i	$0.594304\pi$
$920$	−10593.5	−0.379628
$921$	0	0
$922$	13228.0	0.472497
$923$	31071.4	1.10805
$924$	0	0
$925$	−1063.66	−0.0378086
$926$	−262.367	−0.00931092
$927$	0	0
$928$	21351.9	0.755290
$929$	14942.6	0.527718	0.263859	−	0.964561i	$-0.415005\pi$
$929$	14942.6	0.527718	0.263859	+	0.964561i	$0.415005\pi$
$930$	0	0
$931$	−3524.23	−0.124062
$932$	−34733.9	−1.22076
$933$	0	0
$934$	−2293.45	−0.0803469
$935$	12898.8	0.451160
$936$	0	0
$937$	16005.5	0.558032	0.279016	−	0.960287i	$-0.409992\pi$
$937$	16005.5	0.558032	0.279016	+	0.960287i	$0.409992\pi$
$938$	−5734.76	−0.199623
$939$	0	0
$940$	15413.7	0.534830
$941$	−35515.1	−1.23035	−0.615175	−	0.788390i	$-0.710915\pi$
$941$	−35515.1	−1.23035	−0.615175	+	0.788390i	$0.710915\pi$
$942$	0	0
$943$	−34926.0	−1.20609
$944$	−68.2062	−0.00235161
$945$	0	0
$946$	−32114.8	−1.10374
$947$	15151.4	0.519911	0.259955	−	0.965621i	$-0.416292\pi$
$947$	15151.4	0.519911	0.259955	+	0.965621i	$0.416292\pi$
$948$	0	0
$949$	9298.55	0.318065
$950$	−9992.14	−0.341250
$951$	0	0
$952$	−8111.22	−0.276141
$953$	38421.2	1.30596	0.652982	−	0.757374i	$-0.273518\pi$
$953$	38421.2	1.30596	0.652982	+	0.757374i	$0.273518\pi$
$954$	0	0
$955$	1421.60	0.0481697
$956$	−18711.3	−0.633018
$957$	0	0
$958$	−4145.25	−0.139799
$959$	−6286.79	−0.211690
$960$	0	0
$961$	−26798.1	−0.899536
$962$	−831.697	−0.0278742
$963$	0	0
$964$	−25070.8	−0.837631
$965$	4432.53	0.147863
$966$	0	0
$967$	32575.3	1.08330	0.541650	−	0.840604i	$-0.317800\pi$
$967$	32575.3	1.08330	0.541650	+	0.840604i	$0.317800\pi$
$968$	24511.9	0.813888
$969$	0	0
$970$	313.328	0.0103715
$971$	−4374.24	−0.144569	−0.0722843	−	0.997384i	$-0.523029\pi$
$971$	−4374.24	−0.144569	−0.0722843	+	0.997384i	$0.523029\pi$
$972$	0	0
$973$	7987.75	0.263181
$974$	19766.6	0.650270
$975$	0	0
$976$	13519.3	0.443382
$977$	37493.4	1.22776	0.613880	−	0.789399i	$-0.289608\pi$
$977$	37493.4	1.22776	0.613880	+	0.789399i	$0.289608\pi$
$978$	0	0
$979$	−13222.0	−0.431642
$980$	−1230.40	−0.0401059
$981$	0	0
$982$	15256.1	0.495765
$983$	38197.3	1.23938	0.619688	−	0.784848i	$-0.287259\pi$
$983$	38197.3	1.23938	0.619688	+	0.784848i	$0.287259\pi$
$984$	0	0
$985$	7748.53	0.250649
$986$	−9471.87	−0.305929
$987$	0	0
$988$	31065.2	1.00032
$989$	72380.1	2.32715
$990$	0	0
$991$	−39675.4	−1.27178	−0.635889	−	0.771781i	$-0.719366\pi$
$991$	−39675.4	−1.27178	−0.635889	+	0.771781i	$0.719366\pi$
$992$	−9929.02	−0.317789
$993$	0	0
$994$	−4081.43	−0.130237
$995$	4323.02	0.137737
$996$	0	0
$997$	−21444.3	−0.681193	−0.340596	−	0.940210i	$-0.610629\pi$
$997$	−21444.3	−0.681193	−0.340596	+	0.940210i	$0.610629\pi$
$998$	12627.1	0.400503
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.4.a.e.1.2	✓	2
3.2	odd	2		189.4.a.i.1.1	yes	2
7.6	odd	2		1323.4.a.o.1.2		2
21.20	even	2		1323.4.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.a.e.1.2	✓	2	1.1	even	1	trivial
189.4.a.i.1.1	yes	2	3.2	odd	2
1323.4.a.o.1.2		2	7.6	odd	2
1323.4.a.x.1.1		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	189.a (trivial)

\(f(q)\)	\(=\)	\(q-1.26795 q^{2} -6.39230 q^{4} +3.92820 q^{5} +7.00000 q^{7} +18.2487 q^{8} -4.98076 q^{10} -51.7128 q^{11} +67.5692 q^{13} -8.87564 q^{14} +28.0000 q^{16} -63.4974 q^{17} -71.9230 q^{19} -25.1103 q^{20} +65.5692 q^{22} -147.779 q^{23} -109.569 q^{25} -85.6743 q^{26} -44.7461 q^{28} -117.646 q^{29} +54.7077 q^{31} -181.492 q^{32} +80.5115 q^{34} +27.4974 q^{35} +9.70766 q^{37} +91.1948 q^{38} +71.6846 q^{40} +236.338 q^{41} -489.785 q^{43} +330.564 q^{44} +187.377 q^{46} -613.841 q^{47} +49.0000 q^{49} +138.928 q^{50} -431.923 q^{52} -316.543 q^{53} -203.138 q^{55} +127.741 q^{56} +149.169 q^{58} -2.43594 q^{59} +482.831 q^{61} -69.3665 q^{62} +6.12297 q^{64} +265.426 q^{65} +646.123 q^{67} +405.895 q^{68} -34.8653 q^{70} +459.846 q^{71} +137.615 q^{73} -12.3088 q^{74} +459.754 q^{76} -361.990 q^{77} -816.307 q^{79} +109.990 q^{80} -299.665 q^{82} +1009.25 q^{83} -249.431 q^{85} +621.022 q^{86} -943.692 q^{88} +255.682 q^{89} +472.985 q^{91} +944.651 q^{92} +778.319 q^{94} -282.528 q^{95} -62.9076 q^{97} -62.1295 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{2} + 8 q^{4} - 6 q^{5} + 14 q^{7} - 12 q^{8} + 42 q^{10} - 48 q^{11} + 52 q^{13} - 42 q^{14} + 56 q^{16} - 30 q^{17} + 64 q^{19} - 168 q^{20} + 48 q^{22} - 60 q^{23} - 136 q^{25} - 12 q^{26}+ \cdots - 294 q^{98}+O(q^{100}) \) 2 * q - 6 * q^2 + 8 * q^4 - 6 * q^5 + 14 * q^7 - 12 * q^8 + 42 * q^10 - 48 * q^11 + 52 * q^13 - 42 * q^14 + 56 * q^16 - 30 * q^17 + 64 * q^19 - 168 * q^20 + 48 * q^22 - 60 * q^23 - 136 * q^25 - 12 * q^26 + 56 * q^28 - 360 * q^29 - 140 * q^31 - 72 * q^32 - 78 * q^34 - 42 * q^35 - 230 * q^37 - 552 * q^38 + 372 * q^40 - 234 * q^41 - 938 * q^43 + 384 * q^44 - 228 * q^46 - 618 * q^47 + 98 * q^49 + 264 * q^50 - 656 * q^52 + 420 * q^53 - 240 * q^55 - 84 * q^56 + 1296 * q^58 - 282 * q^59 - 32 * q^61 + 852 * q^62 - 736 * q^64 + 420 * q^65 + 544 * q^67 + 888 * q^68 + 294 * q^70 + 504 * q^71 - 764 * q^73 + 1122 * q^74 + 2416 * q^76 - 336 * q^77 + 238 * q^79 - 168 * q^80 + 1926 * q^82 + 522 * q^83 - 582 * q^85 + 2742 * q^86 - 1056 * q^88 - 708 * q^89 + 364 * q^91 + 2208 * q^92 + 798 * q^94 - 1632 * q^95 + 664 * q^97 - 294 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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