LMFDB - Embedded newform 1536.4.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1536,4,Mod(1,1536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1536, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1536.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1536 = 2^{9} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1536.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:	$90.6269337688$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 16x^{2} + 49 $ x^4 - 16*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.03151$ of defining polynomial
Character	$\chi$	$=$	1536.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -2.26874 q^{5} -20.9692 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -2.26874 q^{5} -20.9692 q^{7} +9.00000 q^{9} +1.51472 q^{11} -3.20848 q^{13} +6.80622 q^{15} -10.2010 q^{17} +31.1127 q^{19} +62.9075 q^{21} +197.959 q^{23} -119.853 q^{25} -27.0000 q^{27} -152.973 q^{29} +188.722 q^{31} -4.54416 q^{33} +47.5736 q^{35} +274.868 q^{37} +9.62545 q^{39} -95.5736 q^{41} +263.563 q^{43} -20.4187 q^{45} -116.284 q^{47} +96.7056 q^{49} +30.6030 q^{51} -87.2464 q^{53} -3.43650 q^{55} -93.3381 q^{57} +323.029 q^{59} +409.435 q^{61} -188.722 q^{63} +7.27922 q^{65} -264.833 q^{67} -593.876 q^{69} -308.604 q^{71} -331.529 q^{73} +359.558 q^{75} -31.7624 q^{77} +1.58468 q^{79} +81.0000 q^{81} +25.8680 q^{83} +23.1435 q^{85} +458.919 q^{87} -842.548 q^{89} +67.2792 q^{91} -566.167 q^{93} -70.5867 q^{95} -519.088 q^{97} +13.6325 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 36 * q^9	$ 4 q - 12 q^{3} + 36 q^{9} + 40 q^{11} - 120 q^{17} - 140 q^{25} - 108 q^{27} - 120 q^{33} + 360 q^{35} - 552 q^{41} + 432 q^{43} - 292 q^{49} + 360 q^{51} + 1360 q^{59} - 480 q^{65} + 864 q^{67} - 240 q^{73} + 420 q^{75} + 324 q^{81} + 952 q^{83} - 1560 q^{89} - 240 q^{91} - 2280 q^{97} + 360 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 + 36 * q^9 + 40 * q^11 - 120 * q^17 - 140 * q^25 - 108 * q^27 - 120 * q^33 + 360 * q^35 - 552 * q^41 + 432 * q^43 - 292 * q^49 + 360 * q^51 + 1360 * q^59 - 480 * q^65 + 864 * q^67 - 240 * q^73 + 420 * q^75 + 324 * q^81 + 952 * q^83 - 1560 * q^89 - 240 * q^91 - 2280 * q^97 + 360 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−2.26874	−0.202922	−0.101461	−	0.994839i	$-0.532352\pi$
$5$	−2.26874	−0.202922	−0.101461	+	0.994839i	$0.532352\pi$
$6$	0	0
$7$	−20.9692	−1.13223	−0.566114	−	0.824327i	$-0.691554\pi$
$7$	−20.9692	−1.13223	−0.566114	+	0.824327i	$0.691554\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	1.51472	0.0415186	0.0207593	−	0.999785i	$-0.493392\pi$
$11$	1.51472	0.0415186	0.0207593	+	0.999785i	$0.493392\pi$
$12$	0	0
$13$	−3.20848	−0.0684518	−0.0342259	−	0.999414i	$-0.510897\pi$
$13$	−3.20848	−0.0684518	−0.0342259	+	0.999414i	$0.510897\pi$
$14$	0	0
$15$	6.80622	0.117157
$16$	0	0
$17$	−10.2010	−0.145536	−0.0727679	−	0.997349i	$-0.523183\pi$
$17$	−10.2010	−0.145536	−0.0727679	+	0.997349i	$0.523183\pi$
$18$	0	0
$19$	31.1127	0.375671	0.187835	−	0.982201i	$-0.439853\pi$
$19$	31.1127	0.375671	0.187835	+	0.982201i	$0.439853\pi$
$20$	0	0
$21$	62.9075	0.653692
$22$	0	0
$23$	197.959	1.79466	0.897331	−	0.441358i	$-0.145503\pi$
$23$	197.959	1.79466	0.897331	+	0.441358i	$0.145503\pi$
$24$	0	0
$25$	−119.853	−0.958823
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−152.973	−0.979530	−0.489765	−	0.871854i	$-0.662918\pi$
$29$	−152.973	−0.979530	−0.489765	+	0.871854i	$0.662918\pi$
$30$	0	0
$31$	188.722	1.09340	0.546702	−	0.837327i	$-0.315883\pi$
$31$	188.722	1.09340	0.546702	+	0.837327i	$0.315883\pi$
$32$	0	0
$33$	−4.54416	−0.0239708
$34$	0	0
$35$	47.5736	0.229754
$36$	0	0
$37$	274.868	1.22130	0.610649	−	0.791902i	$-0.290909\pi$
$37$	274.868	1.22130	0.610649	+	0.791902i	$0.290909\pi$
$38$	0	0
$39$	9.62545	0.0395207
$40$	0	0
$41$	−95.5736	−0.364051	−0.182025	−	0.983294i	$-0.558265\pi$
$41$	−95.5736	−0.364051	−0.182025	+	0.983294i	$0.558265\pi$
$42$	0	0
$43$	263.563	0.934722	0.467361	−	0.884067i	$-0.345205\pi$
$43$	263.563	0.934722	0.467361	+	0.884067i	$0.345205\pi$
$44$	0	0
$45$	−20.4187	−0.0676408
$46$	0	0
$47$	−116.284	−0.360888	−0.180444	−	0.983585i	$-0.557754\pi$
$47$	−116.284	−0.360888	−0.180444	+	0.983585i	$0.557754\pi$
$48$	0	0
$49$	96.7056	0.281941
$50$	0	0
$51$	30.6030	0.0840251
$52$	0	0
$53$	−87.2464	−0.226117	−0.113059	−	0.993588i	$-0.536065\pi$
$53$	−87.2464	−0.226117	−0.113059	+	0.993588i	$0.536065\pi$
$54$	0	0
$55$	−3.43650	−0.00842506
$56$	0	0
$57$	−93.3381	−0.216894
$58$	0	0
$59$	323.029	0.712794	0.356397	−	0.934335i	$-0.384005\pi$
$59$	323.029	0.712794	0.356397	+	0.934335i	$0.384005\pi$
$60$	0	0
$61$	409.435	0.859390	0.429695	−	0.902974i	$-0.358621\pi$
$61$	409.435	0.859390	0.429695	+	0.902974i	$0.358621\pi$
$62$	0	0
$63$	−188.722	−0.377409
$64$	0	0
$65$	7.27922	0.0138904
$66$	0	0
$67$	−264.833	−0.482902	−0.241451	−	0.970413i	$-0.577623\pi$
$67$	−264.833	−0.482902	−0.241451	+	0.970413i	$0.577623\pi$
$68$	0	0
$69$	−593.876	−1.03615
$70$	0	0
$71$	−308.604	−0.515839	−0.257920	−	0.966166i	$-0.583037\pi$
$71$	−308.604	−0.515839	−0.257920	+	0.966166i	$0.583037\pi$
$72$	0	0
$73$	−331.529	−0.531542	−0.265771	−	0.964036i	$-0.585626\pi$
$73$	−331.529	−0.531542	−0.265771	+	0.964036i	$0.585626\pi$
$74$	0	0
$75$	359.558	0.553576
$76$	0	0
$77$	−31.7624	−0.0470086
$78$	0	0
$79$	1.58468	0.00225684	0.00112842	−	0.999999i	$-0.499641\pi$
$79$	1.58468	0.00225684	0.00112842	+	0.999999i	$0.499641\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	25.8680	0.0342094	0.0171047	−	0.999854i	$-0.494555\pi$
$83$	25.8680	0.0342094	0.0171047	+	0.999854i	$0.494555\pi$
$84$	0	0
$85$	23.1435	0.0295325
$86$	0	0
$87$	458.919	0.565532
$88$	0	0
$89$	−842.548	−1.00348	−0.501741	−	0.865018i	$-0.667307\pi$
$89$	−842.548	−1.00348	−0.501741	+	0.865018i	$0.667307\pi$
$90$	0	0
$91$	67.2792	0.0775031
$92$	0	0
$93$	−566.167	−0.631278
$94$	0	0
$95$	−70.5867	−0.0762320
$96$	0	0
$97$	−519.088	−0.543355	−0.271677	−	0.962388i	$-0.587578\pi$
$97$	−519.088	−0.543355	−0.271677	+	0.962388i	$0.587578\pi$
$98$	0	0
$99$	13.6325	0.0138395
$100$	0	0
$101$	431.561	0.425167	0.212584	−	0.977143i	$-0.431812\pi$
$101$	431.561	0.425167	0.212584	+	0.977143i	$0.431812\pi$
$102$	0	0
$103$	−1317.62	−1.26048	−0.630238	−	0.776402i	$-0.717043\pi$
$103$	−1317.62	−1.26048	−0.630238	+	0.776402i	$0.717043\pi$
$104$	0	0
$105$	−142.721	−0.132649
$106$	0	0
$107$	1593.65	1.43985	0.719923	−	0.694054i	$-0.244177\pi$
$107$	1593.65	1.43985	0.719923	+	0.694054i	$0.244177\pi$
$108$	0	0
$109$	1163.92	1.02279	0.511393	−	0.859347i	$-0.329130\pi$
$109$	1163.92	1.02279	0.511393	+	0.859347i	$0.329130\pi$
$110$	0	0
$111$	−824.603	−0.705116
$112$	0	0
$113$	−286.325	−0.238364	−0.119182	−	0.992872i	$-0.538027\pi$
$113$	−286.325	−0.238364	−0.119182	+	0.992872i	$0.538027\pi$
$114$	0	0
$115$	−449.117	−0.364177
$116$	0	0
$117$	−28.8764	−0.0228173
$118$	0	0
$119$	213.907	0.164780
$120$	0	0
$121$	−1328.71	−0.998276
$122$	0	0
$123$	286.721	0.210185
$124$	0	0
$125$	555.508	0.397489
$126$	0	0
$127$	219.117	0.153098	0.0765491	−	0.997066i	$-0.475610\pi$
$127$	219.117	0.153098	0.0765491	+	0.997066i	$0.475610\pi$
$128$	0	0
$129$	−790.690	−0.539662
$130$	0	0
$131$	−1463.70	−0.976217	−0.488108	−	0.872783i	$-0.662313\pi$
$131$	−1463.70	−0.976217	−0.488108	+	0.872783i	$0.662313\pi$
$132$	0	0
$133$	−652.407	−0.425345
$134$	0	0
$135$	61.2560	0.0390524
$136$	0	0
$137$	−957.868	−0.597344	−0.298672	−	0.954356i	$-0.596544\pi$
$137$	−957.868	−0.597344	−0.298672	+	0.954356i	$0.596544\pi$
$138$	0	0
$139$	1631.21	0.995379	0.497689	−	0.867355i	$-0.334182\pi$
$139$	1631.21	0.995379	0.497689	+	0.867355i	$0.334182\pi$
$140$	0	0
$141$	348.852	0.208359
$142$	0	0
$143$	−4.85995	−0.00284202
$144$	0	0
$145$	347.056	0.198769
$146$	0	0
$147$	−290.117	−0.162778
$148$	0	0
$149$	−2722.62	−1.49695	−0.748476	−	0.663162i	$-0.769214\pi$
$149$	−2722.62	−1.49695	−0.748476	+	0.663162i	$0.769214\pi$
$150$	0	0
$151$	−1556.20	−0.838690	−0.419345	−	0.907827i	$-0.637740\pi$
$151$	−1556.20	−0.838690	−0.419345	+	0.907827i	$0.637740\pi$
$152$	0	0
$153$	−91.8091	−0.0485119
$154$	0	0
$155$	−428.162	−0.221876
$156$	0	0
$157$	−1337.39	−0.679845	−0.339923	−	0.940453i	$-0.610401\pi$
$157$	−1337.39	−0.679845	−0.339923	+	0.940453i	$0.610401\pi$
$158$	0	0
$159$	261.739	0.130549
$160$	0	0
$161$	−4151.03	−2.03197
$162$	0	0
$163$	−2234.93	−1.07395	−0.536974	−	0.843599i	$-0.680433\pi$
$163$	−2234.93	−1.07395	−0.536974	+	0.843599i	$0.680433\pi$
$164$	0	0
$165$	10.3095	0.00486421
$166$	0	0
$167$	2661.95	1.23346	0.616731	−	0.787174i	$-0.288457\pi$
$167$	2661.95	1.23346	0.616731	+	0.787174i	$0.288457\pi$
$168$	0	0
$169$	−2186.71	−0.995314
$170$	0	0
$171$	280.014	0.125224
$172$	0	0
$173$	−3433.94	−1.50912	−0.754559	−	0.656232i	$-0.772149\pi$
$173$	−3433.94	−1.50912	−0.754559	+	0.656232i	$0.772149\pi$
$174$	0	0
$175$	2513.21	1.08561
$176$	0	0
$177$	−969.088	−0.411532
$178$	0	0
$179$	−861.263	−0.359630	−0.179815	−	0.983700i	$-0.557550\pi$
$179$	−861.263	−0.359630	−0.179815	+	0.983700i	$0.557550\pi$
$180$	0	0
$181$	4452.81	1.82859	0.914294	−	0.405051i	$-0.132746\pi$
$181$	4452.81	1.82859	0.914294	+	0.405051i	$0.132746\pi$
$182$	0	0
$183$	−1228.31	−0.496169
$184$	0	0
$185$	−623.604	−0.247828
$186$	0	0
$187$	−15.4517	−0.00604245
$188$	0	0
$189$	566.167	0.217897
$190$	0	0
$191$	−20.8079	−0.00788277	−0.00394138	−	0.999992i	$-0.501255\pi$
$191$	−20.8079	−0.00788277	−0.00394138	+	0.999992i	$0.501255\pi$
$192$	0	0
$193$	−3823.76	−1.42612	−0.713058	−	0.701105i	$-0.752690\pi$
$193$	−3823.76	−1.42612	−0.713058	+	0.701105i	$0.752690\pi$
$194$	0	0
$195$	−21.8377	−0.00801963
$196$	0	0
$197$	3929.22	1.42104	0.710522	−	0.703675i	$-0.248459\pi$
$197$	3929.22	1.42104	0.710522	+	0.703675i	$0.248459\pi$
$198$	0	0
$199$	4711.35	1.67829	0.839143	−	0.543910i	$-0.183057\pi$
$199$	4711.35	1.67829	0.839143	+	0.543910i	$0.183057\pi$
$200$	0	0
$201$	794.498	0.278804
$202$	0	0
$203$	3207.72	1.10905
$204$	0	0
$205$	216.832	0.0738741
$206$	0	0
$207$	1781.63	0.598221
$208$	0	0
$209$	47.1270	0.0155973
$210$	0	0
$211$	−4873.25	−1.58999	−0.794996	−	0.606614i	$-0.792527\pi$
$211$	−4873.25	−1.58999	−0.794996	+	0.606614i	$0.792527\pi$
$212$	0	0
$213$	925.812	0.297820
$214$	0	0
$215$	−597.957	−0.189676
$216$	0	0
$217$	−3957.35	−1.23798
$218$	0	0
$219$	994.587	0.306886
$220$	0	0
$221$	32.7298	0.00996219
$222$	0	0
$223$	−2177.17	−0.653786	−0.326893	−	0.945061i	$-0.606002\pi$
$223$	−2177.17	−0.653786	−0.326893	+	0.945061i	$0.606002\pi$
$224$	0	0
$225$	−1078.68	−0.319608
$226$	0	0
$227$	−3934.74	−1.15048	−0.575238	−	0.817986i	$-0.695091\pi$
$227$	−3934.74	−1.15048	−0.575238	+	0.817986i	$0.695091\pi$
$228$	0	0
$229$	−4057.08	−1.17074	−0.585370	−	0.810767i	$-0.699051\pi$
$229$	−4057.08	−1.17074	−0.585370	+	0.810767i	$0.699051\pi$
$230$	0	0
$231$	95.2871	0.0271404
$232$	0	0
$233$	1508.56	0.424160	0.212080	−	0.977252i	$-0.431976\pi$
$233$	1508.56	0.424160	0.212080	+	0.977252i	$0.431976\pi$
$234$	0	0
$235$	263.818	0.0732323
$236$	0	0
$237$	−4.75404	−0.00130299
$238$	0	0
$239$	1135.85	0.307414	0.153707	−	0.988116i	$-0.450879\pi$
$239$	1135.85	0.307414	0.153707	+	0.988116i	$0.450879\pi$
$240$	0	0
$241$	288.671	0.0771574	0.0385787	−	0.999256i	$-0.487717\pi$
$241$	288.671	0.0771574	0.0385787	+	0.999256i	$0.487717\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−219.400	−0.0572121
$246$	0	0
$247$	−99.8246	−0.0257153
$248$	0	0
$249$	−77.6039	−0.0197508
$250$	0	0
$251$	4547.16	1.14348	0.571741	−	0.820434i	$-0.306268\pi$
$251$	4547.16	1.14348	0.571741	+	0.820434i	$0.306268\pi$
$252$	0	0
$253$	299.852	0.0745119
$254$	0	0
$255$	−69.4304	−0.0170506
$256$	0	0
$257$	−6856.90	−1.66429	−0.832144	−	0.554560i	$-0.812887\pi$
$257$	−6856.90	−1.66429	−0.832144	+	0.554560i	$0.812887\pi$
$258$	0	0
$259$	−5763.75	−1.38279
$260$	0	0
$261$	−1376.76	−0.326510
$262$	0	0
$263$	−5473.62	−1.28334	−0.641669	−	0.766981i	$-0.721758\pi$
$263$	−5473.62	−1.28334	−0.641669	+	0.766981i	$0.721758\pi$
$264$	0	0
$265$	197.939	0.0458842
$266$	0	0
$267$	2527.65	0.579361
$268$	0	0
$269$	5249.36	1.18981	0.594906	−	0.803795i	$-0.297189\pi$
$269$	5249.36	1.18981	0.594906	+	0.803795i	$0.297189\pi$
$270$	0	0
$271$	−6829.69	−1.53090	−0.765450	−	0.643495i	$-0.777484\pi$
$271$	−6829.69	−1.53090	−0.765450	+	0.643495i	$0.777484\pi$
$272$	0	0
$273$	−201.838	−0.0447464
$274$	0	0
$275$	−181.543	−0.0398090
$276$	0	0
$277$	−6282.87	−1.36282	−0.681410	−	0.731902i	$-0.738633\pi$
$277$	−6282.87	−1.36282	−0.681410	+	0.731902i	$0.738633\pi$
$278$	0	0
$279$	1698.50	0.364468
$280$	0	0
$281$	−5470.24	−1.16131	−0.580654	−	0.814151i	$-0.697203\pi$
$281$	−5470.24	−1.16131	−0.580654	+	0.814151i	$0.697203\pi$
$282$	0	0
$283$	−2263.84	−0.475516	−0.237758	−	0.971324i	$-0.576413\pi$
$283$	−2263.84	−0.475516	−0.237758	+	0.971324i	$0.576413\pi$
$284$	0	0
$285$	211.760	0.0440126
$286$	0	0
$287$	2004.10	0.412189
$288$	0	0
$289$	−4808.94	−0.978819
$290$	0	0
$291$	1557.26	0.313706
$292$	0	0
$293$	2760.69	0.550448	0.275224	−	0.961380i	$-0.411248\pi$
$293$	2760.69	0.550448	0.275224	+	0.961380i	$0.411248\pi$
$294$	0	0
$295$	−732.870	−0.144642
$296$	0	0
$297$	−40.8974	−0.00799026
$298$	0	0
$299$	−635.147	−0.122848
$300$	0	0
$301$	−5526.70	−1.05832
$302$	0	0
$303$	−1294.68	−0.245470
$304$	0	0
$305$	−928.903	−0.174390
$306$	0	0
$307$	−6617.77	−1.23028	−0.615140	−	0.788418i	$-0.710901\pi$
$307$	−6617.77	−1.23028	−0.615140	+	0.788418i	$0.710901\pi$
$308$	0	0
$309$	3952.86	0.727736
$310$	0	0
$311$	−896.109	−0.163388	−0.0816940	−	0.996657i	$-0.526033\pi$
$311$	−896.109	−0.163388	−0.0816940	+	0.996657i	$0.526033\pi$
$312$	0	0
$313$	−6257.93	−1.13009	−0.565047	−	0.825059i	$-0.691142\pi$
$313$	−6257.93	−1.13009	−0.565047	+	0.825059i	$0.691142\pi$
$314$	0	0
$315$	428.162	0.0765848
$316$	0	0
$317$	5996.04	1.06237	0.531185	−	0.847256i	$-0.321747\pi$
$317$	5996.04	1.06237	0.531185	+	0.847256i	$0.321747\pi$
$318$	0	0
$319$	−231.711	−0.0406688
$320$	0	0
$321$	−4780.94	−0.831295
$322$	0	0
$323$	−317.381	−0.0546735
$324$	0	0
$325$	384.546	0.0656331
$326$	0	0
$327$	−3491.77	−0.590506
$328$	0	0
$329$	2438.38	0.408608
$330$	0	0
$331$	−585.068	−0.0971548	−0.0485774	−	0.998819i	$-0.515469\pi$
$331$	−585.068	−0.0971548	−0.0485774	+	0.998819i	$0.515469\pi$
$332$	0	0
$333$	2473.81	0.407099
$334$	0	0
$335$	600.837	0.0979917
$336$	0	0
$337$	10603.5	1.71398	0.856989	−	0.515335i	$-0.172333\pi$
$337$	10603.5	1.71398	0.856989	+	0.515335i	$0.172333\pi$
$338$	0	0
$339$	858.974	0.137620
$340$	0	0
$341$	285.861	0.0453967
$342$	0	0
$343$	5164.59	0.813007
$344$	0	0
$345$	1347.35	0.210258
$346$	0	0
$347$	6576.57	1.01743	0.508716	−	0.860935i	$-0.330120\pi$
$347$	6576.57	1.01743	0.508716	+	0.860935i	$0.330120\pi$
$348$	0	0
$349$	−1752.56	−0.268804	−0.134402	−	0.990927i	$-0.542911\pi$
$349$	−1752.56	−0.268804	−0.134402	+	0.990927i	$0.542911\pi$
$350$	0	0
$351$	86.6291	0.0131736
$352$	0	0
$353$	−9605.58	−1.44831	−0.724155	−	0.689638i	$-0.757770\pi$
$353$	−9605.58	−1.44831	−0.724155	+	0.689638i	$0.757770\pi$
$354$	0	0
$355$	700.143	0.104675
$356$	0	0
$357$	−641.720	−0.0951356
$358$	0	0
$359$	689.753	0.101403	0.0507016	−	0.998714i	$-0.483854\pi$
$359$	689.753	0.101403	0.0507016	+	0.998714i	$0.483854\pi$
$360$	0	0
$361$	−5891.00	−0.858872
$362$	0	0
$363$	3986.12	0.576355
$364$	0	0
$365$	752.153	0.107862
$366$	0	0
$367$	1361.33	0.193626	0.0968130	−	0.995303i	$-0.469135\pi$
$367$	1361.33	0.193626	0.0968130	+	0.995303i	$0.469135\pi$
$368$	0	0
$369$	−860.162	−0.121350
$370$	0	0
$371$	1829.48	0.256016
$372$	0	0
$373$	−2967.35	−0.411913	−0.205957	−	0.978561i	$-0.566031\pi$
$373$	−2967.35	−0.411913	−0.205957	+	0.978561i	$0.566031\pi$
$374$	0	0
$375$	−1666.52	−0.229490
$376$	0	0
$377$	490.812	0.0670506
$378$	0	0
$379$	537.530	0.0728524	0.0364262	−	0.999336i	$-0.488403\pi$
$379$	537.530	0.0728524	0.0364262	+	0.999336i	$0.488403\pi$
$380$	0	0
$381$	−657.350	−0.0883912
$382$	0	0
$383$	−4851.60	−0.647272	−0.323636	−	0.946182i	$-0.604905\pi$
$383$	−4851.60	−0.647272	−0.323636	+	0.946182i	$0.604905\pi$
$384$	0	0
$385$	72.0606	0.00953909
$386$	0	0
$387$	2372.07	0.311574
$388$	0	0
$389$	−11567.8	−1.50774	−0.753868	−	0.657026i	$-0.771814\pi$
$389$	−11567.8	−1.50774	−0.753868	+	0.657026i	$0.771814\pi$
$390$	0	0
$391$	−2019.38	−0.261188
$392$	0	0
$393$	4391.11	0.563619
$394$	0	0
$395$	−3.59523	−0.000457964 0
$396$	0	0
$397$	2444.46	0.309028	0.154514	−	0.987991i	$-0.450619\pi$
$397$	2444.46	0.309028	0.154514	+	0.987991i	$0.450619\pi$
$398$	0	0
$399$	1957.22	0.245573
$400$	0	0
$401$	9707.07	1.20885	0.604424	−	0.796663i	$-0.293403\pi$
$401$	9707.07	1.20885	0.604424	+	0.796663i	$0.293403\pi$
$402$	0	0
$403$	−605.513	−0.0748455
$404$	0	0
$405$	−183.768	−0.0225469
$406$	0	0
$407$	416.347	0.0507066
$408$	0	0
$409$	−13592.7	−1.64331	−0.821655	−	0.569985i	$-0.806949\pi$
$409$	−13592.7	−1.64331	−0.821655	+	0.569985i	$0.806949\pi$
$410$	0	0
$411$	2873.60	0.344877
$412$	0	0
$413$	−6773.66	−0.807046
$414$	0	0
$415$	−58.6877	−0.00694185
$416$	0	0
$417$	−4893.64	−0.574682
$418$	0	0
$419$	1914.14	0.223178	0.111589	−	0.993754i	$-0.464406\pi$
$419$	1914.14	0.223178	0.111589	+	0.993754i	$0.464406\pi$
$420$	0	0
$421$	9614.08	1.11297	0.556486	−	0.830857i	$-0.312149\pi$
$421$	9614.08	1.11297	0.556486	+	0.830857i	$0.312149\pi$
$422$	0	0
$423$	−1046.56	−0.120296
$424$	0	0
$425$	1222.62	0.139543
$426$	0	0
$427$	−8585.51	−0.973026
$428$	0	0
$429$	14.5799	0.00164084
$430$	0	0
$431$	−6515.13	−0.728127	−0.364064	−	0.931374i	$-0.618611\pi$
$431$	−6515.13	−0.728127	−0.364064	+	0.931374i	$0.618611\pi$
$432$	0	0
$433$	14836.2	1.64661	0.823306	−	0.567598i	$-0.192127\pi$
$433$	14836.2	1.64661	0.823306	+	0.567598i	$0.192127\pi$
$434$	0	0
$435$	−1041.17	−0.114759
$436$	0	0
$437$	6159.03	0.674202
$438$	0	0
$439$	757.009	0.0823008	0.0411504	−	0.999153i	$-0.486898\pi$
$439$	757.009	0.0823008	0.0411504	+	0.999153i	$0.486898\pi$
$440$	0	0
$441$	870.351	0.0939802
$442$	0	0
$443$	8729.10	0.936189	0.468095	−	0.883678i	$-0.344941\pi$
$443$	8729.10	0.936189	0.468095	+	0.883678i	$0.344941\pi$
$444$	0	0
$445$	1911.52	0.203629
$446$	0	0
$447$	8167.86	0.864265
$448$	0	0
$449$	−6574.28	−0.691001	−0.345500	−	0.938419i	$-0.612291\pi$
$449$	−6574.28	−0.691001	−0.345500	+	0.938419i	$0.612291\pi$
$450$	0	0
$451$	−144.767	−0.0151149
$452$	0	0
$453$	4668.61	0.484218
$454$	0	0
$455$	−152.639	−0.0157271
$456$	0	0
$457$	−31.0173	−0.00317490	−0.00158745	−	0.999999i	$-0.500505\pi$
$457$	−31.0173	−0.00317490	−0.00158745	+	0.999999i	$0.500505\pi$
$458$	0	0
$459$	275.427	0.0280084
$460$	0	0
$461$	−14702.5	−1.48539	−0.742694	−	0.669631i	$-0.766452\pi$
$461$	−14702.5	−1.48539	−0.742694	+	0.669631i	$0.766452\pi$
$462$	0	0
$463$	9162.02	0.919644	0.459822	−	0.888011i	$-0.347913\pi$
$463$	9162.02	0.919644	0.459822	+	0.888011i	$0.347913\pi$
$464$	0	0
$465$	1284.49	0.128100
$466$	0	0
$467$	15539.7	1.53981	0.769903	−	0.638161i	$-0.220305\pi$
$467$	15539.7	1.53981	0.769903	+	0.638161i	$0.220305\pi$
$468$	0	0
$469$	5553.32	0.546756
$470$	0	0
$471$	4012.18	0.392509
$472$	0	0
$473$	399.225	0.0388084
$474$	0	0
$475$	−3728.94	−0.360201
$476$	0	0
$477$	−785.217	−0.0753724
$478$	0	0
$479$	15249.7	1.45464	0.727322	−	0.686296i	$-0.240765\pi$
$479$	15249.7	1.45464	0.727322	+	0.686296i	$0.240765\pi$
$480$	0	0
$481$	−881.909	−0.0836000
$482$	0	0
$483$	12453.1	1.17316
$484$	0	0
$485$	1177.68	0.110259
$486$	0	0
$487$	−2321.14	−0.215977	−0.107988	−	0.994152i	$-0.534441\pi$
$487$	−2321.14	−0.215977	−0.107988	+	0.994152i	$0.534441\pi$
$488$	0	0
$489$	6704.80	0.620044
$490$	0	0
$491$	−2296.72	−0.211099	−0.105550	−	0.994414i	$-0.533660\pi$
$491$	−2296.72	−0.211099	−0.105550	+	0.994414i	$0.533660\pi$
$492$	0	0
$493$	1560.48	0.142557
$494$	0	0
$495$	−30.9285	−0.00280835
$496$	0	0
$497$	6471.17	0.584047
$498$	0	0
$499$	−16770.7	−1.50453	−0.752264	−	0.658862i	$-0.771038\pi$
$499$	−16770.7	−1.50453	−0.752264	+	0.658862i	$0.771038\pi$
$500$	0	0
$501$	−7985.86	−0.712140
$502$	0	0
$503$	13487.8	1.19561	0.597805	−	0.801641i	$-0.296040\pi$
$503$	13487.8	1.19561	0.597805	+	0.801641i	$0.296040\pi$
$504$	0	0
$505$	−979.100	−0.0862760
$506$	0	0
$507$	6560.12	0.574645
$508$	0	0
$509$	−4223.36	−0.367774	−0.183887	−	0.982947i	$-0.558868\pi$
$509$	−4223.36	−0.367774	−0.183887	+	0.982947i	$0.558868\pi$
$510$	0	0
$511$	6951.88	0.601826
$512$	0	0
$513$	−840.043	−0.0722979
$514$	0	0
$515$	2989.34	0.255779
$516$	0	0
$517$	−176.137	−0.0149836
$518$	0	0
$519$	10301.8	0.871290
$520$	0	0
$521$	13328.7	1.12080	0.560402	−	0.828221i	$-0.310647\pi$
$521$	13328.7	1.12080	0.560402	+	0.828221i	$0.310647\pi$
$522$	0	0
$523$	−11372.5	−0.950833	−0.475416	−	0.879761i	$-0.657703\pi$
$523$	−11372.5	−0.950833	−0.475416	+	0.879761i	$0.657703\pi$
$524$	0	0
$525$	−7539.64	−0.626775
$526$	0	0
$527$	−1925.16	−0.159130
$528$	0	0
$529$	27020.6	2.22081
$530$	0	0
$531$	2907.26	0.237598
$532$	0	0
$533$	306.646	0.0249199
$534$	0	0
$535$	−3615.57	−0.292177
$536$	0	0
$537$	2583.79	0.207633
$538$	0	0
$539$	146.482	0.0117058
$540$	0	0
$541$	405.445	0.0322208	0.0161104	−	0.999870i	$-0.494872\pi$
$541$	405.445	0.0322208	0.0161104	+	0.999870i	$0.494872\pi$
$542$	0	0
$543$	−13358.4	−1.05574
$544$	0	0
$545$	−2640.64	−0.207546
$546$	0	0
$547$	−1506.93	−0.117791	−0.0588956	−	0.998264i	$-0.518758\pi$
$547$	−1506.93	−0.117791	−0.0588956	+	0.998264i	$0.518758\pi$
$548$	0	0
$549$	3684.92	0.286463
$550$	0	0
$551$	−4759.40	−0.367981
$552$	0	0
$553$	−33.2294	−0.00255526
$554$	0	0
$555$	1870.81	0.143084
$556$	0	0
$557$	25711.9	1.95592	0.977960	−	0.208791i	$-0.0669528\pi$
$557$	25711.9	1.95592	0.977960	+	0.208791i	$0.0669528\pi$
$558$	0	0
$559$	−845.639	−0.0639834
$560$	0	0
$561$	46.3550	0.00348861
$562$	0	0
$563$	15804.0	1.18305	0.591525	−	0.806286i	$-0.298526\pi$
$563$	15804.0	1.18305	0.591525	+	0.806286i	$0.298526\pi$
$564$	0	0
$565$	649.597	0.0483694
$566$	0	0
$567$	−1698.50	−0.125803
$568$	0	0
$569$	12466.7	0.918505	0.459253	−	0.888306i	$-0.348117\pi$
$569$	12466.7	0.918505	0.459253	+	0.888306i	$0.348117\pi$
$570$	0	0
$571$	−15570.2	−1.14114	−0.570572	−	0.821248i	$-0.693278\pi$
$571$	−15570.2	−1.14114	−0.570572	+	0.821248i	$0.693278\pi$
$572$	0	0
$573$	62.4238	0.00455112
$574$	0	0
$575$	−23725.9	−1.72076
$576$	0	0
$577$	6957.65	0.501994	0.250997	−	0.967988i	$-0.419242\pi$
$577$	6957.65	0.501994	0.250997	+	0.967988i	$0.419242\pi$
$578$	0	0
$579$	11471.3	0.823368
$580$	0	0
$581$	−542.429	−0.0387328
$582$	0	0
$583$	−132.154	−0.00938807
$584$	0	0
$585$	65.5130	0.00463013
$586$	0	0
$587$	−295.840	−0.0208017	−0.0104009	−	0.999946i	$-0.503311\pi$
$587$	−295.840	−0.0208017	−0.0104009	+	0.999946i	$0.503311\pi$
$588$	0	0
$589$	5871.66	0.410760
$590$	0	0
$591$	−11787.7	−0.820440
$592$	0	0
$593$	14779.0	1.02344	0.511720	−	0.859152i	$-0.329009\pi$
$593$	14779.0	1.02344	0.511720	+	0.859152i	$0.329009\pi$
$594$	0	0
$595$	−485.299	−0.0334375
$596$	0	0
$597$	−14134.1	−0.968959
$598$	0	0
$599$	−21096.5	−1.43903	−0.719514	−	0.694478i	$-0.755635\pi$
$599$	−21096.5	−1.43903	−0.719514	+	0.694478i	$0.755635\pi$
$600$	0	0
$601$	−9409.10	−0.638611	−0.319305	−	0.947652i	$-0.603450\pi$
$601$	−9409.10	−0.638611	−0.319305	+	0.947652i	$0.603450\pi$
$602$	0	0
$603$	−2383.49	−0.160967
$604$	0	0
$605$	3014.49	0.202573
$606$	0	0
$607$	−6091.78	−0.407344	−0.203672	−	0.979039i	$-0.565288\pi$
$607$	−6091.78	−0.407344	−0.203672	+	0.979039i	$0.565288\pi$
$608$	0	0
$609$	−9623.15	−0.640311
$610$	0	0
$611$	373.095	0.0247035
$612$	0	0
$613$	−7884.11	−0.519472	−0.259736	−	0.965680i	$-0.583636\pi$
$613$	−7884.11	−0.519472	−0.259736	+	0.965680i	$0.583636\pi$
$614$	0	0
$615$	−650.495	−0.0426512
$616$	0	0
$617$	14573.9	0.950928	0.475464	−	0.879735i	$-0.342280\pi$
$617$	14573.9	0.950928	0.475464	+	0.879735i	$0.342280\pi$
$618$	0	0
$619$	−26984.9	−1.75220	−0.876101	−	0.482128i	$-0.839864\pi$
$619$	−26984.9	−1.75220	−0.876101	+	0.482128i	$0.839864\pi$
$620$	0	0
$621$	−5344.88	−0.345383
$622$	0	0
$623$	17667.5	1.13617
$624$	0	0
$625$	13721.3	0.878163
$626$	0	0
$627$	−141.381	−0.00900512
$628$	0	0
$629$	−2803.93	−0.177742
$630$	0	0
$631$	−28881.6	−1.82212	−0.911061	−	0.412272i	$-0.864735\pi$
$631$	−28881.6	−1.82212	−0.911061	+	0.412272i	$0.864735\pi$
$632$	0	0
$633$	14619.8	0.917983
$634$	0	0
$635$	−497.119	−0.0310670
$636$	0	0
$637$	−310.279	−0.0192993
$638$	0	0
$639$	−2777.44	−0.171946
$640$	0	0
$641$	−16448.9	−1.01356	−0.506781	−	0.862075i	$-0.669165\pi$
$641$	−16448.9	−1.01356	−0.506781	+	0.862075i	$0.669165\pi$
$642$	0	0
$643$	8947.26	0.548749	0.274375	−	0.961623i	$-0.411529\pi$
$643$	8947.26	0.548749	0.274375	+	0.961623i	$0.411529\pi$
$644$	0	0
$645$	1793.87	0.109510
$646$	0	0
$647$	18260.7	1.10959	0.554794	−	0.831988i	$-0.312797\pi$
$647$	18260.7	1.10959	0.554794	+	0.831988i	$0.312797\pi$
$648$	0	0
$649$	489.299	0.0295942
$650$	0	0
$651$	11872.1	0.714750
$652$	0	0
$653$	7847.64	0.470294	0.235147	−	0.971960i	$-0.424443\pi$
$653$	7847.64	0.470294	0.235147	+	0.971960i	$0.424443\pi$
$654$	0	0
$655$	3320.77	0.198096
$656$	0	0
$657$	−2983.76	−0.177181
$658$	0	0
$659$	31707.2	1.87426	0.937131	−	0.348978i	$-0.113471\pi$
$659$	31707.2	1.87426	0.937131	+	0.348978i	$0.113471\pi$
$660$	0	0
$661$	14271.9	0.839805	0.419903	−	0.907569i	$-0.362064\pi$
$661$	14271.9	0.839805	0.419903	+	0.907569i	$0.362064\pi$
$662$	0	0
$663$	−98.1893	−0.00575167
$664$	0	0
$665$	1480.14	0.0863120
$666$	0	0
$667$	−30282.3	−1.75793
$668$	0	0
$669$	6531.52	0.377463
$670$	0	0
$671$	620.179	0.0356807
$672$	0	0
$673$	8792.89	0.503627	0.251813	−	0.967776i	$-0.418973\pi$
$673$	8792.89	0.503627	0.251813	+	0.967776i	$0.418973\pi$
$674$	0	0
$675$	3236.03	0.184525
$676$	0	0
$677$	−24521.0	−1.39205	−0.696026	−	0.718017i	$-0.745050\pi$
$677$	−24521.0	−1.39205	−0.696026	+	0.718017i	$0.745050\pi$
$678$	0	0
$679$	10884.8	0.615202
$680$	0	0
$681$	11804.2	0.664228
$682$	0	0
$683$	−23406.0	−1.31128	−0.655641	−	0.755072i	$-0.727602\pi$
$683$	−23406.0	−1.31128	−0.655641	+	0.755072i	$0.727602\pi$
$684$	0	0
$685$	2173.15	0.121215
$686$	0	0
$687$	12171.2	0.675927
$688$	0	0
$689$	279.929	0.0154781
$690$	0	0
$691$	−13121.2	−0.722367	−0.361183	−	0.932495i	$-0.617627\pi$
$691$	−13121.2	−0.722367	−0.361183	+	0.932495i	$0.617627\pi$
$692$	0	0
$693$	−285.861	−0.0156695
$694$	0	0
$695$	−3700.80	−0.201985
$696$	0	0
$697$	974.947	0.0529824
$698$	0	0
$699$	−4525.69	−0.244889
$700$	0	0
$701$	−16603.5	−0.894585	−0.447293	−	0.894388i	$-0.647612\pi$
$701$	−16603.5	−0.894585	−0.447293	+	0.894388i	$0.647612\pi$
$702$	0	0
$703$	8551.88	0.458805
$704$	0	0
$705$	−791.455	−0.0422807
$706$	0	0
$707$	−9049.47	−0.481386
$708$	0	0
$709$	16453.8	0.871557	0.435778	−	0.900054i	$-0.356473\pi$
$709$	16453.8	0.871557	0.435778	+	0.900054i	$0.356473\pi$
$710$	0	0
$711$	14.2621	0.000752281 0
$712$	0	0
$713$	37359.2	1.96229
$714$	0	0
$715$	11.0260	0.000576710 0
$716$	0	0
$717$	−3407.55	−0.177486
$718$	0	0
$719$	−17050.6	−0.884393	−0.442197	−	0.896918i	$-0.645801\pi$
$719$	−17050.6	−0.884393	−0.442197	+	0.896918i	$0.645801\pi$
$720$	0	0
$721$	27629.4	1.42715
$722$	0	0
$723$	−866.013	−0.0445468
$724$	0	0
$725$	18334.3	0.939196
$726$	0	0
$727$	20024.2	1.02153	0.510767	−	0.859719i	$-0.329361\pi$
$727$	20024.2	1.02153	0.510767	+	0.859719i	$0.329361\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−2688.61	−0.136036
$732$	0	0
$733$	34181.9	1.72243	0.861213	−	0.508243i	$-0.169705\pi$
$733$	34181.9	1.72243	0.861213	+	0.508243i	$0.169705\pi$
$734$	0	0
$735$	658.200	0.0330314
$736$	0	0
$737$	−401.147	−0.0200494
$738$	0	0
$739$	−951.634	−0.0473700	−0.0236850	−	0.999719i	$-0.507540\pi$
$739$	−951.634	−0.0473700	−0.0236850	+	0.999719i	$0.507540\pi$
$740$	0	0
$741$	299.474	0.0148468
$742$	0	0
$743$	16905.7	0.834737	0.417368	−	0.908737i	$-0.362953\pi$
$743$	16905.7	0.834737	0.417368	+	0.908737i	$0.362953\pi$
$744$	0	0
$745$	6176.92	0.303765
$746$	0	0
$747$	232.812	0.0114031
$748$	0	0
$749$	−33417.4	−1.63023
$750$	0	0
$751$	−23154.0	−1.12503	−0.562517	−	0.826786i	$-0.690167\pi$
$751$	−23154.0	−1.12503	−0.562517	+	0.826786i	$0.690167\pi$
$752$	0	0
$753$	−13641.5	−0.660190
$754$	0	0
$755$	3530.63	0.170189
$756$	0	0
$757$	−29839.7	−1.43268	−0.716342	−	0.697749i	$-0.754185\pi$
$757$	−29839.7	−1.43268	−0.716342	+	0.697749i	$0.754185\pi$
$758$	0	0
$759$	−899.555	−0.0430195
$760$	0	0
$761$	15872.3	0.756074	0.378037	−	0.925791i	$-0.376599\pi$
$761$	15872.3	0.756074	0.378037	+	0.925791i	$0.376599\pi$
$762$	0	0
$763$	−24406.5	−1.15803
$764$	0	0
$765$	208.291	0.00984416
$766$	0	0
$767$	−1036.43	−0.0487920
$768$	0	0
$769$	27878.8	1.30733	0.653664	−	0.756785i	$-0.273231\pi$
$769$	27878.8	1.30733	0.653664	+	0.756785i	$0.273231\pi$
$770$	0	0
$771$	20570.7	0.960877
$772$	0	0
$773$	9894.81	0.460403	0.230201	−	0.973143i	$-0.426061\pi$
$773$	9894.81	0.460403	0.230201	+	0.973143i	$0.426061\pi$
$774$	0	0
$775$	−22618.9	−1.04838
$776$	0	0
$777$	17291.2	0.798352
$778$	0	0
$779$	−2973.55	−0.136763
$780$	0	0
$781$	−467.448	−0.0214169
$782$	0	0
$783$	4130.27	0.188511
$784$	0	0
$785$	3034.20	0.137956
$786$	0	0
$787$	−5394.90	−0.244355	−0.122177	−	0.992508i	$-0.538988\pi$
$787$	−5394.90	−0.244355	−0.122177	+	0.992508i	$0.538988\pi$
$788$	0	0
$789$	16420.9	0.740936
$790$	0	0
$791$	6003.99	0.269883
$792$	0	0
$793$	−1313.67	−0.0588268
$794$	0	0
$795$	−593.818	−0.0264913
$796$	0	0
$797$	4227.39	0.187882	0.0939410	−	0.995578i	$-0.470053\pi$
$797$	4227.39	0.187882	0.0939410	+	0.995578i	$0.470053\pi$
$798$	0	0
$799$	1186.21	0.0525222
$800$	0	0
$801$	−7582.94	−0.334494
$802$	0	0
$803$	−502.173	−0.0220689
$804$	0	0
$805$	9417.60	0.412332
$806$	0	0
$807$	−15748.1	−0.686938
$808$	0	0
$809$	−3420.55	−0.148653	−0.0743264	−	0.997234i	$-0.523681\pi$
$809$	−3420.55	−0.148653	−0.0743264	+	0.997234i	$0.523681\pi$
$810$	0	0
$811$	−14917.4	−0.645894	−0.322947	−	0.946417i	$-0.604674\pi$
$811$	−14917.4	−0.645894	−0.322947	+	0.946417i	$0.604674\pi$
$812$	0	0
$813$	20489.1	0.883866
$814$	0	0
$815$	5070.49	0.217928
$816$	0	0
$817$	8200.17	0.351148
$818$	0	0
$819$	605.513	0.0258344
$820$	0	0
$821$	−448.581	−0.0190689	−0.00953447	−	0.999955i	$-0.503035\pi$
$821$	−448.581	−0.0190689	−0.00953447	+	0.999955i	$0.503035\pi$
$822$	0	0
$823$	20199.4	0.855536	0.427768	−	0.903889i	$-0.359300\pi$
$823$	20199.4	0.855536	0.427768	+	0.903889i	$0.359300\pi$
$824$	0	0
$825$	544.630	0.0229837
$826$	0	0
$827$	12961.0	0.544979	0.272490	−	0.962159i	$-0.412153\pi$
$827$	12961.0	0.544979	0.272490	+	0.962159i	$0.412153\pi$
$828$	0	0
$829$	−31314.2	−1.31193	−0.655964	−	0.754792i	$-0.727738\pi$
$829$	−31314.2	−1.31193	−0.655964	+	0.754792i	$0.727738\pi$
$830$	0	0
$831$	18848.6	0.786824
$832$	0	0
$833$	−986.495	−0.0410324
$834$	0	0
$835$	−6039.29	−0.250297
$836$	0	0
$837$	−5095.51	−0.210426
$838$	0	0
$839$	−33718.0	−1.38746	−0.693728	−	0.720237i	$-0.744033\pi$
$839$	−33718.0	−1.38746	−0.693728	+	0.720237i	$0.744033\pi$
$840$	0	0
$841$	−988.242	−0.0405200
$842$	0	0
$843$	16410.7	0.670481
$844$	0	0
$845$	4961.07	0.201972
$846$	0	0
$847$	27861.8	1.13028
$848$	0	0
$849$	6791.51	0.274540
$850$	0	0
$851$	54412.5	2.19182
$852$	0	0
$853$	−13279.6	−0.533040	−0.266520	−	0.963829i	$-0.585874\pi$
$853$	−13279.6	−0.533040	−0.266520	+	0.963829i	$0.585874\pi$
$854$	0	0
$855$	−635.280	−0.0254107
$856$	0	0
$857$	20285.5	0.808564	0.404282	−	0.914634i	$-0.367522\pi$
$857$	20285.5	0.808564	0.404282	+	0.914634i	$0.367522\pi$
$858$	0	0
$859$	−42382.1	−1.68342	−0.841711	−	0.539929i	$-0.818451\pi$
$859$	−42382.1	−1.68342	−0.841711	+	0.539929i	$0.818451\pi$
$860$	0	0
$861$	−6012.29	−0.237977
$862$	0	0
$863$	19943.5	0.786657	0.393328	−	0.919398i	$-0.371324\pi$
$863$	19943.5	0.786657	0.393328	+	0.919398i	$0.371324\pi$
$864$	0	0
$865$	7790.71	0.306234
$866$	0	0
$867$	14426.8	0.565122
$868$	0	0
$869$	2.40035	9.37010e−5 0
$870$	0	0
$871$	849.711	0.0330555
$872$	0	0
$873$	−4671.79	−0.181118
$874$	0	0
$875$	−11648.5	−0.450048
$876$	0	0
$877$	−25607.7	−0.985988	−0.492994	−	0.870033i	$-0.664097\pi$
$877$	−25607.7	−0.985988	−0.492994	+	0.870033i	$0.664097\pi$
$878$	0	0
$879$	−8282.07	−0.317801
$880$	0	0
$881$	−4881.96	−0.186694	−0.0933470	−	0.995634i	$-0.529757\pi$
$881$	−4881.96	−0.186694	−0.0933470	+	0.995634i	$0.529757\pi$
$882$	0	0
$883$	−34096.9	−1.29949	−0.649747	−	0.760151i	$-0.725125\pi$
$883$	−34096.9	−1.29949	−0.649747	+	0.760151i	$0.725125\pi$
$884$	0	0
$885$	2198.61	0.0835090
$886$	0	0
$887$	42658.9	1.61482	0.807410	−	0.589991i	$-0.200869\pi$
$887$	42658.9	1.61482	0.807410	+	0.589991i	$0.200869\pi$
$888$	0	0
$889$	−4594.69	−0.173342
$890$	0	0
$891$	122.692	0.00461318
$892$	0	0
$893$	−3617.91	−0.135575
$894$	0	0
$895$	1953.98	0.0729770
$896$	0	0
$897$	1905.44	0.0709262
$898$	0	0
$899$	−28869.4	−1.07102
$900$	0	0
$901$	890.001	0.0329081
$902$	0	0
$903$	16580.1	0.611021
$904$	0	0
$905$	−10102.3	−0.371062
$906$	0	0
$907$	14321.8	0.524310	0.262155	−	0.965026i	$-0.415567\pi$
$907$	14321.8	0.524310	0.262155	+	0.965026i	$0.415567\pi$
$908$	0	0
$909$	3884.05	0.141722
$910$	0	0
$911$	15159.6	0.551329	0.275664	−	0.961254i	$-0.411102\pi$
$911$	15159.6	0.551329	0.275664	+	0.961254i	$0.411102\pi$
$912$	0	0
$913$	39.1827	0.00142033
$914$	0	0
$915$	2786.71	0.100684
$916$	0	0
$917$	30692.6	1.10530
$918$	0	0
$919$	21391.5	0.767833	0.383917	−	0.923368i	$-0.374575\pi$
$919$	21391.5	0.767833	0.383917	+	0.923368i	$0.374575\pi$
$920$	0	0
$921$	19853.3	0.710302
$922$	0	0
$923$	990.152	0.0353101
$924$	0	0
$925$	−32943.7	−1.17101
$926$	0	0
$927$	−11858.6	−0.420159
$928$	0	0
$929$	16845.7	0.594931	0.297465	−	0.954733i	$-0.403859\pi$
$929$	16845.7	0.594931	0.297465	+	0.954733i	$0.403859\pi$
$930$	0	0
$931$	3008.77	0.105917
$932$	0	0
$933$	2688.33	0.0943321
$934$	0	0
$935$	35.0558	0.00122615
$936$	0	0
$937$	−25445.1	−0.887147	−0.443573	−	0.896238i	$-0.646289\pi$
$937$	−25445.1	−0.887147	−0.443573	+	0.896238i	$0.646289\pi$
$938$	0	0
$939$	18773.8	0.652460
$940$	0	0
$941$	16942.8	0.586948	0.293474	−	0.955967i	$-0.405189\pi$
$941$	16942.8	0.586948	0.293474	+	0.955967i	$0.405189\pi$
$942$	0	0
$943$	−18919.6	−0.653348
$944$	0	0
$945$	−1284.49	−0.0442163
$946$	0	0
$947$	21984.0	0.754367	0.377184	−	0.926139i	$-0.376893\pi$
$947$	21984.0	0.754367	0.377184	+	0.926139i	$0.376893\pi$
$948$	0	0
$949$	1063.71	0.0363850
$950$	0	0
$951$	−17988.1	−0.613360
$952$	0	0
$953$	17078.9	0.580523	0.290262	−	0.956947i	$-0.406258\pi$
$953$	17078.9	0.580523	0.290262	+	0.956947i	$0.406258\pi$
$954$	0	0
$955$	47.2078	0.00159959
$956$	0	0
$957$	695.133	0.0234801
$958$	0	0
$959$	20085.7	0.676330
$960$	0	0
$961$	5825.16	0.195534
$962$	0	0
$963$	14342.8	0.479949
$964$	0	0
$965$	8675.12	0.289391
$966$	0	0
$967$	21385.4	0.711178	0.355589	−	0.934642i	$-0.384280\pi$
$967$	21385.4	0.711178	0.355589	+	0.934642i	$0.384280\pi$
$968$	0	0
$969$	952.143	0.0315658
$970$	0	0
$971$	32629.9	1.07842	0.539209	−	0.842172i	$-0.318723\pi$
$971$	32629.9	1.07842	0.539209	+	0.842172i	$0.318723\pi$
$972$	0	0
$973$	−34205.2	−1.12700
$974$	0	0
$975$	−1153.64	−0.0378933
$976$	0	0
$977$	−13577.8	−0.444618	−0.222309	−	0.974976i	$-0.571359\pi$
$977$	−13577.8	−0.444618	−0.222309	+	0.974976i	$0.571359\pi$
$978$	0	0
$979$	−1276.22	−0.0416632
$980$	0	0
$981$	10475.3	0.340929
$982$	0	0
$983$	−50233.2	−1.62990	−0.814949	−	0.579533i	$-0.803235\pi$
$983$	−50233.2	−1.62990	−0.814949	+	0.579533i	$0.803235\pi$
$984$	0	0
$985$	−8914.39	−0.288361
$986$	0	0
$987$	−7315.13	−0.235910
$988$	0	0
$989$	52174.7	1.67751
$990$	0	0
$991$	−38733.4	−1.24158	−0.620791	−	0.783976i	$-0.713188\pi$
$991$	−38733.4	−1.24158	−0.620791	+	0.783976i	$0.713188\pi$
$992$	0	0
$993$	1755.20	0.0560924
$994$	0	0
$995$	−10688.8	−0.340562
$996$	0	0
$997$	−42613.4	−1.35364	−0.676820	−	0.736148i	$-0.736643\pi$
$997$	−42613.4	−1.35364	−0.676820	+	0.736148i	$0.736643\pi$
$998$	0	0
$999$	−7421.43	−0.235039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.4.a.i.1.2	✓	4
4.3	odd	2		1536.4.a.l.1.2	yes	4
8.3	odd	2	inner	1536.4.a.i.1.3	yes	4
8.5	even	2		1536.4.a.l.1.3	yes	4
16.3	odd	4		1536.4.d.f.769.7		8
16.5	even	4		1536.4.d.f.769.6		8
16.11	odd	4		1536.4.d.f.769.2		8
16.13	even	4		1536.4.d.f.769.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.4.a.i.1.2	✓	4	1.1	even	1	trivial
1536.4.a.i.1.3	yes	4	8.3	odd	2	inner
1536.4.a.l.1.2	yes	4	4.3	odd	2
1536.4.a.l.1.3	yes	4	8.5	even	2
1536.4.d.f.769.2		8	16.11	odd	4
1536.4.d.f.769.3		8	16.13	even	4
1536.4.d.f.769.6		8	16.5	even	4
1536.4.d.f.769.7		8	16.3	odd	4

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:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1536.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -2.26874 q^{5} -20.9692 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -2.26874 q^{5} -20.9692 q^{7} +9.00000 q^{9} +1.51472 q^{11} -3.20848 q^{13} +6.80622 q^{15} -10.2010 q^{17} +31.1127 q^{19} +62.9075 q^{21} +197.959 q^{23} -119.853 q^{25} -27.0000 q^{27} -152.973 q^{29} +188.722 q^{31} -4.54416 q^{33} +47.5736 q^{35} +274.868 q^{37} +9.62545 q^{39} -95.5736 q^{41} +263.563 q^{43} -20.4187 q^{45} -116.284 q^{47} +96.7056 q^{49} +30.6030 q^{51} -87.2464 q^{53} -3.43650 q^{55} -93.3381 q^{57} +323.029 q^{59} +409.435 q^{61} -188.722 q^{63} +7.27922 q^{65} -264.833 q^{67} -593.876 q^{69} -308.604 q^{71} -331.529 q^{73} +359.558 q^{75} -31.7624 q^{77} +1.58468 q^{79} +81.0000 q^{81} +25.8680 q^{83} +23.1435 q^{85} +458.919 q^{87} -842.548 q^{89} +67.2792 q^{91} -566.167 q^{93} -70.5867 q^{95} -519.088 q^{97} +13.6325 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 36 * q^9	\( 4 q - 12 q^{3} + 36 q^{9} + 40 q^{11} - 120 q^{17} - 140 q^{25} - 108 q^{27} - 120 q^{33} + 360 q^{35} - 552 q^{41} + 432 q^{43} - 292 q^{49} + 360 q^{51} + 1360 q^{59} - 480 q^{65} + 864 q^{67} - 240 q^{73} + 420 q^{75} + 324 q^{81} + 952 q^{83} - 1560 q^{89} - 240 q^{91} - 2280 q^{97} + 360 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 + 36 * q^9 + 40 * q^11 - 120 * q^17 - 140 * q^25 - 108 * q^27 - 120 * q^33 + 360 * q^35 - 552 * q^41 + 432 * q^43 - 292 * q^49 + 360 * q^51 + 1360 * q^59 - 480 * q^65 + 864 * q^67 - 240 * q^73 + 420 * q^75 + 324 * q^81 + 952 * q^83 - 1560 * q^89 - 240 * q^91 - 2280 * q^97 + 360 * q^99

Introduction

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