LMFDB - Embedded newform 1080.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,4,Mod(1,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1080.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1080.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:	$63.7220628062$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.4281.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 15 $ x^3 - x^2 - 12*x + 15
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.55784$ of defining polynomial
Character	$\chi$	$=$	1080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} -34.6634 q^{7} +O(q^{10})$	$q+5.00000 q^{5} -34.6634 q^{7} -23.2350 q^{11} +60.0918 q^{13} -19.6634 q^{17} -10.2350 q^{19} -79.7553 q^{23} +25.0000 q^{25} -110.847 q^{29} -42.5202 q^{31} -173.317 q^{35} +308.451 q^{37} +106.950 q^{41} -467.868 q^{43} -37.7982 q^{47} +858.554 q^{49} +568.399 q^{53} -116.175 q^{55} +666.893 q^{59} -862.166 q^{61} +300.459 q^{65} +547.085 q^{67} +761.113 q^{71} -216.562 q^{73} +805.407 q^{77} +258.600 q^{79} +903.680 q^{83} -98.3172 q^{85} +1265.95 q^{89} -2082.99 q^{91} -51.1752 q^{95} +617.305 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 15 q^{5} - 8 q^{7}+O(q^{10}) $ 3 * q + 15 * q^5 - 8 * q^7	$ 3 q + 15 q^{5} - 8 q^{7} - 10 q^{11} + 48 q^{13} + 37 q^{17} + 29 q^{19} - 11 q^{23} + 75 q^{25} + 28 q^{29} + 41 q^{31} - 40 q^{35} + 230 q^{37} + 370 q^{41} - 130 q^{43} - 56 q^{47} + 547 q^{49} + 805 q^{53} - 50 q^{55} + 576 q^{59} - 257 q^{61} + 240 q^{65} - 14 q^{67} + 1238 q^{71} - 398 q^{73} + 1296 q^{77} - 321 q^{79} + 687 q^{83} + 185 q^{85} + 2358 q^{89} - 1968 q^{91} + 145 q^{95} + 576 q^{97}+O(q^{100}) $ 3 * q + 15 * q^5 - 8 * q^7 - 10 * q^11 + 48 * q^13 + 37 * q^17 + 29 * q^19 - 11 * q^23 + 75 * q^25 + 28 * q^29 + 41 * q^31 - 40 * q^35 + 230 * q^37 + 370 * q^41 - 130 * q^43 - 56 * q^47 + 547 * q^49 + 805 * q^53 - 50 * q^55 + 576 * q^59 - 257 * q^61 + 240 * q^65 - 14 * q^67 + 1238 * q^71 - 398 * q^73 + 1296 * q^77 - 321 * q^79 + 687 * q^83 + 185 * q^85 + 2358 * q^89 - 1968 * q^91 + 145 * q^95 + 576 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−34.6634	−1.87165	−0.935825	−	0.352465i	$-0.885343\pi$
$7$	−34.6634	−1.87165	−0.935825	+	0.352465i	$0.885343\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−23.2350	−0.636875	−0.318438	−	0.947944i	$-0.603158\pi$
$11$	−23.2350	−0.636875	−0.318438	+	0.947944i	$0.603158\pi$
$12$	0	0
$13$	60.0918	1.28204	0.641018	−	0.767526i	$-0.278512\pi$
$13$	60.0918	1.28204	0.641018	+	0.767526i	$0.278512\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−19.6634	−0.280534	−0.140267	−	0.990114i	$-0.544796\pi$
$17$	−19.6634	−0.280534	−0.140267	+	0.990114i	$0.544796\pi$
$18$	0	0
$19$	−10.2350	−0.123583	−0.0617916	−	0.998089i	$-0.519681\pi$
$19$	−10.2350	−0.123583	−0.0617916	+	0.998089i	$0.519681\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−79.7553	−0.723049	−0.361524	−	0.932363i	$-0.617744\pi$
$23$	−79.7553	−0.723049	−0.361524	+	0.932363i	$0.617744\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−110.847	−0.709786	−0.354893	−	0.934907i	$-0.615483\pi$
$29$	−110.847	−0.709786	−0.354893	+	0.934907i	$0.615483\pi$
$30$	0	0
$31$	−42.5202	−0.246350	−0.123175	−	0.992385i	$-0.539308\pi$
$31$	−42.5202	−0.246350	−0.123175	+	0.992385i	$0.539308\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−173.317	−0.837027
$36$	0	0
$37$	308.451	1.37051	0.685257	−	0.728302i	$-0.259690\pi$
$37$	308.451	1.37051	0.685257	+	0.728302i	$0.259690\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	106.950	0.407384	0.203692	−	0.979035i	$-0.434706\pi$
$41$	106.950	0.407384	0.203692	+	0.979035i	$0.434706\pi$
$42$	0	0
$43$	−467.868	−1.65928	−0.829642	−	0.558295i	$-0.811456\pi$
$43$	−467.868	−1.65928	−0.829642	+	0.558295i	$0.811456\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−37.7982	−0.117307	−0.0586536	−	0.998278i	$-0.518681\pi$
$47$	−37.7982	−0.117307	−0.0586536	+	0.998278i	$0.518681\pi$
$48$	0	0
$49$	858.554	2.50307
$50$	0	0
$51$	0	0
$52$	0	0
$53$	568.399	1.47313	0.736563	−	0.676369i	$-0.236448\pi$
$53$	568.399	1.47313	0.736563	+	0.676369i	$0.236448\pi$
$54$	0	0
$55$	−116.175	−0.284819
$56$	0	0
$57$	0	0
$58$	0	0
$59$	666.893	1.47156	0.735780	−	0.677220i	$-0.236816\pi$
$59$	666.893	1.47156	0.735780	+	0.677220i	$0.236816\pi$
$60$	0	0
$61$	−862.166	−1.80966	−0.904828	−	0.425777i	$-0.860001\pi$
$61$	−862.166	−1.80966	−0.904828	+	0.425777i	$0.860001\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	300.459	0.573344
$66$	0	0
$67$	547.085	0.997568	0.498784	−	0.866726i	$-0.333780\pi$
$67$	547.085	0.997568	0.498784	+	0.866726i	$0.333780\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	761.113	1.27222	0.636109	−	0.771599i	$-0.280543\pi$
$71$	761.113	1.27222	0.636109	+	0.771599i	$0.280543\pi$
$72$	0	0
$73$	−216.562	−0.347214	−0.173607	−	0.984815i	$-0.555542\pi$
$73$	−216.562	−0.347214	−0.173607	+	0.984815i	$0.555542\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	805.407	1.19201
$78$	0	0
$79$	258.600	0.368288	0.184144	−	0.982899i	$-0.441049\pi$
$79$	258.600	0.368288	0.184144	+	0.982899i	$0.441049\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	903.680	1.19508	0.597541	−	0.801838i	$-0.296145\pi$
$83$	903.680	1.19508	0.597541	+	0.801838i	$0.296145\pi$
$84$	0	0
$85$	−98.3172	−0.125459
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1265.95	1.50776	0.753880	−	0.657012i	$-0.228180\pi$
$89$	1265.95	1.50776	0.753880	+	0.657012i	$0.228180\pi$
$90$	0	0
$91$	−2082.99	−2.39952
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−51.1752	−0.0552681
$96$	0	0
$97$	617.305	0.646163	0.323082	−	0.946371i	$-0.395281\pi$
$97$	617.305	0.646163	0.323082	+	0.946371i	$0.395281\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−831.717	−0.819395	−0.409698	−	0.912221i	$-0.634366\pi$
$101$	−831.717	−0.819395	−0.409698	+	0.912221i	$0.634366\pi$
$102$	0	0
$103$	904.254	0.865036	0.432518	−	0.901625i	$-0.357625\pi$
$103$	904.254	0.865036	0.432518	+	0.901625i	$0.357625\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−897.290	−0.810695	−0.405347	−	0.914163i	$-0.632849\pi$
$107$	−897.290	−0.810695	−0.405347	+	0.914163i	$0.632849\pi$
$108$	0	0
$109$	−258.932	−0.227534	−0.113767	−	0.993507i	$-0.536292\pi$
$109$	−258.932	−0.227534	−0.113767	+	0.993507i	$0.536292\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−329.028	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$113$	−329.028	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$114$	0	0
$115$	−398.776	−0.323357
$116$	0	0
$117$	0	0
$118$	0	0
$119$	681.602	0.525062
$120$	0	0
$121$	−791.133	−0.594390
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1602.36	1.11958	0.559789	−	0.828635i	$-0.310882\pi$
$127$	1602.36	1.11958	0.559789	+	0.828635i	$0.310882\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1373.59	0.916116	0.458058	−	0.888922i	$-0.348545\pi$
$131$	1373.59	0.916116	0.458058	+	0.888922i	$0.348545\pi$
$132$	0	0
$133$	354.782	0.231304
$134$	0	0
$135$	0	0
$136$	0	0
$137$	690.827	0.430813	0.215406	−	0.976524i	$-0.430892\pi$
$137$	690.827	0.430813	0.215406	+	0.976524i	$0.430892\pi$
$138$	0	0
$139$	−1072.90	−0.654693	−0.327346	−	0.944904i	$-0.606154\pi$
$139$	−1072.90	−0.654693	−0.327346	+	0.944904i	$0.606154\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1396.24	−0.816498
$144$	0	0
$145$	−554.236	−0.317426
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2828.55	−1.55520	−0.777598	−	0.628762i	$-0.783562\pi$
$149$	−2828.55	−1.55520	−0.777598	+	0.628762i	$0.783562\pi$
$150$	0	0
$151$	1138.61	0.613633	0.306817	−	0.951769i	$-0.400736\pi$
$151$	1138.61	0.613633	0.306817	+	0.951769i	$0.400736\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−212.601	−0.110171
$156$	0	0
$157$	2489.69	1.26560	0.632799	−	0.774316i	$-0.281906\pi$
$157$	2489.69	1.26560	0.632799	+	0.774316i	$0.281906\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2764.59	1.35329
$162$	0	0
$163$	−1174.61	−0.564431	−0.282216	−	0.959351i	$-0.591069\pi$
$163$	−1174.61	−0.564431	−0.282216	+	0.959351i	$0.591069\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2249.28	1.04224	0.521121	−	0.853483i	$-0.325514\pi$
$167$	2249.28	1.04224	0.521121	+	0.853483i	$0.325514\pi$
$168$	0	0
$169$	1414.03	0.643618
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3692.19	1.62261	0.811305	−	0.584623i	$-0.198757\pi$
$173$	3692.19	1.62261	0.811305	+	0.584623i	$0.198757\pi$
$174$	0	0
$175$	−866.586	−0.374330
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4240.02	1.77047	0.885234	−	0.465145i	$-0.153998\pi$
$179$	4240.02	1.77047	0.885234	+	0.465145i	$0.153998\pi$
$180$	0	0
$181$	−748.267	−0.307283	−0.153641	−	0.988127i	$-0.549100\pi$
$181$	−748.267	−0.307283	−0.153641	+	0.988127i	$0.549100\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1542.25	0.612912
$186$	0	0
$187$	456.881	0.178665
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1134.04	0.429612	0.214806	−	0.976657i	$-0.431088\pi$
$191$	1134.04	0.429612	0.214806	+	0.976657i	$0.431088\pi$
$192$	0	0
$193$	1134.32	0.423059	0.211529	−	0.977372i	$-0.432156\pi$
$193$	1134.32	0.423059	0.211529	+	0.977372i	$0.432156\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4392.63	1.58864	0.794320	−	0.607499i	$-0.207827\pi$
$197$	4392.63	1.58864	0.794320	+	0.607499i	$0.207827\pi$
$198$	0	0
$199$	−1922.75	−0.684926	−0.342463	−	0.939531i	$-0.611261\pi$
$199$	−1922.75	−0.684926	−0.342463	+	0.939531i	$0.611261\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3842.34	1.32847
$204$	0	0
$205$	534.749	0.182188
$206$	0	0
$207$	0	0
$208$	0	0
$209$	237.812	0.0787071
$210$	0	0
$211$	−4135.95	−1.34944	−0.674718	−	0.738076i	$-0.735735\pi$
$211$	−4135.95	−1.34944	−0.674718	+	0.738076i	$0.735735\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2339.34	−0.742055
$216$	0	0
$217$	1473.90	0.461081
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1181.61	−0.359655
$222$	0	0
$223$	3060.01	0.918893	0.459446	−	0.888205i	$-0.348048\pi$
$223$	3060.01	0.918893	0.459446	+	0.888205i	$0.348048\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2670.37	−0.780787	−0.390393	−	0.920648i	$-0.627661\pi$
$227$	−2670.37	−0.780787	−0.390393	+	0.920648i	$0.627661\pi$
$228$	0	0
$229$	−177.855	−0.0513231	−0.0256615	−	0.999671i	$-0.508169\pi$
$229$	−177.855	−0.0513231	−0.0256615	+	0.999671i	$0.508169\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3014.80	−0.847667	−0.423833	−	0.905740i	$-0.639316\pi$
$233$	−3014.80	−0.847667	−0.423833	+	0.905740i	$0.639316\pi$
$234$	0	0
$235$	−188.991	−0.0524613
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2997.51	0.811268	0.405634	−	0.914036i	$-0.367051\pi$
$239$	2997.51	0.811268	0.405634	+	0.914036i	$0.367051\pi$
$240$	0	0
$241$	4320.99	1.15494	0.577468	−	0.816413i	$-0.304041\pi$
$241$	4320.99	1.15494	0.577468	+	0.816413i	$0.304041\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4292.77	1.11941
$246$	0	0
$247$	−615.043	−0.158438
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4285.02	1.07756	0.538781	−	0.842446i	$-0.318885\pi$
$251$	4285.02	1.07756	0.538781	+	0.842446i	$0.318885\pi$
$252$	0	0
$253$	1853.12	0.460492
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7371.27	1.78913	0.894567	−	0.446935i	$-0.147484\pi$
$257$	7371.27	1.78913	0.894567	+	0.446935i	$0.147484\pi$
$258$	0	0
$259$	−10692.0	−2.56512
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5789.77	1.35746	0.678731	−	0.734387i	$-0.262530\pi$
$263$	5789.77	1.35746	0.678731	+	0.734387i	$0.262530\pi$
$264$	0	0
$265$	2842.00	0.658802
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4402.25	−0.997806	−0.498903	−	0.866658i	$-0.666264\pi$
$269$	−4402.25	−0.997806	−0.498903	+	0.866658i	$0.666264\pi$
$270$	0	0
$271$	−1181.60	−0.264859	−0.132430	−	0.991192i	$-0.542278\pi$
$271$	−1181.60	−0.264859	−0.132430	+	0.991192i	$0.542278\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−580.876	−0.127375
$276$	0	0
$277$	4912.20	1.06551	0.532754	−	0.846270i	$-0.321157\pi$
$277$	4912.20	1.06551	0.532754	+	0.846270i	$0.321157\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4588.62	−0.974143	−0.487072	−	0.873362i	$-0.661935\pi$
$281$	−4588.62	−0.974143	−0.487072	+	0.873362i	$0.661935\pi$
$282$	0	0
$283$	−5277.69	−1.10857	−0.554287	−	0.832326i	$-0.687009\pi$
$283$	−5277.69	−1.10857	−0.554287	+	0.832326i	$0.687009\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3707.25	−0.762481
$288$	0	0
$289$	−4526.35	−0.921300
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9083.85	−1.81121	−0.905604	−	0.424124i	$-0.860582\pi$
$293$	−9083.85	−1.81121	−0.905604	+	0.424124i	$0.860582\pi$
$294$	0	0
$295$	3334.46	0.658102
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4792.64	−0.926975
$300$	0	0
$301$	16217.9	3.10560
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4310.83	−0.809303
$306$	0	0
$307$	−10311.4	−1.91694	−0.958469	−	0.285197i	$-0.907941\pi$
$307$	−10311.4	−1.91694	−0.958469	+	0.285197i	$0.907941\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1200.72	−0.218929	−0.109464	−	0.993991i	$-0.534914\pi$
$311$	−1200.72	−0.218929	−0.109464	+	0.993991i	$0.534914\pi$
$312$	0	0
$313$	−6270.23	−1.13231	−0.566157	−	0.824298i	$-0.691570\pi$
$313$	−6270.23	−1.13231	−0.566157	+	0.824298i	$0.691570\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2348.12	0.416037	0.208018	−	0.978125i	$-0.433299\pi$
$317$	2348.12	0.416037	0.208018	+	0.978125i	$0.433299\pi$
$318$	0	0
$319$	2575.54	0.452045
$320$	0	0
$321$	0	0
$322$	0	0
$323$	201.256	0.0346693
$324$	0	0
$325$	1502.30	0.256407
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1310.22	0.219558
$330$	0	0
$331$	6583.03	1.09316	0.546580	−	0.837407i	$-0.315929\pi$
$331$	6583.03	1.09316	0.546580	+	0.837407i	$0.315929\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2735.43	0.446126
$336$	0	0
$337$	1513.37	0.244624	0.122312	−	0.992492i	$-0.460969\pi$
$337$	1513.37	0.244624	0.122312	+	0.992492i	$0.460969\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	987.959	0.156894
$342$	0	0
$343$	−17870.9	−2.81323
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10534.4	−1.62972	−0.814862	−	0.579656i	$-0.803187\pi$
$347$	−10534.4	−1.62972	−0.814862	+	0.579656i	$0.803187\pi$
$348$	0	0
$349$	7617.91	1.16842	0.584208	−	0.811604i	$-0.301405\pi$
$349$	7617.91	1.16842	0.584208	+	0.811604i	$0.301405\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4882.40	0.736159	0.368079	−	0.929794i	$-0.380015\pi$
$353$	4882.40	0.736159	0.368079	+	0.929794i	$0.380015\pi$
$354$	0	0
$355$	3805.56	0.568953
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3230.35	0.474906	0.237453	−	0.971399i	$-0.423687\pi$
$359$	3230.35	0.474906	0.237453	+	0.971399i	$0.423687\pi$
$360$	0	0
$361$	−6754.24	−0.984727
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1082.81	−0.155279
$366$	0	0
$367$	−10910.4	−1.55182	−0.775912	−	0.630841i	$-0.782710\pi$
$367$	−10910.4	−1.55182	−0.775912	+	0.630841i	$0.782710\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−19702.7	−2.75718
$372$	0	0
$373$	9337.73	1.29622	0.648109	−	0.761548i	$-0.275560\pi$
$373$	9337.73	1.29622	0.648109	+	0.761548i	$0.275560\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6661.01	−0.909971
$378$	0	0
$379$	−2566.39	−0.347828	−0.173914	−	0.984761i	$-0.555641\pi$
$379$	−2566.39	−0.347828	−0.173914	+	0.984761i	$0.555641\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6734.56	−0.898485	−0.449243	−	0.893410i	$-0.648306\pi$
$383$	−6734.56	−0.898485	−0.449243	+	0.893410i	$0.648306\pi$
$384$	0	0
$385$	4027.03	0.533082
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5015.93	0.653773	0.326886	−	0.945064i	$-0.394001\pi$
$389$	5015.93	0.653773	0.326886	+	0.945064i	$0.394001\pi$
$390$	0	0
$391$	1568.26	0.202840
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1293.00	0.164703
$396$	0	0
$397$	−484.045	−0.0611927	−0.0305964	−	0.999532i	$-0.509741\pi$
$397$	−484.045	−0.0611927	−0.0305964	+	0.999532i	$0.509741\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8235.48	1.02559	0.512793	−	0.858512i	$-0.328611\pi$
$401$	8235.48	1.02559	0.512793	+	0.858512i	$0.328611\pi$
$402$	0	0
$403$	−2555.12	−0.315830
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7166.87	−0.872846
$408$	0	0
$409$	−7276.15	−0.879664	−0.439832	−	0.898080i	$-0.644962\pi$
$409$	−7276.15	−0.879664	−0.439832	+	0.898080i	$0.644962\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−23116.8	−2.75425
$414$	0	0
$415$	4518.40	0.534457
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4852.57	−0.565784	−0.282892	−	0.959152i	$-0.591294\pi$
$419$	−4852.57	−0.565784	−0.282892	+	0.959152i	$0.591294\pi$
$420$	0	0
$421$	−11401.6	−1.31990	−0.659952	−	0.751308i	$-0.729424\pi$
$421$	−11401.6	−1.31990	−0.659952	+	0.751308i	$0.729424\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−491.586	−0.0561069
$426$	0	0
$427$	29885.6	3.38704
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13132.5	1.46768	0.733838	−	0.679324i	$-0.237727\pi$
$431$	13132.5	1.46768	0.733838	+	0.679324i	$0.237727\pi$
$432$	0	0
$433$	−2695.20	−0.299129	−0.149564	−	0.988752i	$-0.547787\pi$
$433$	−2695.20	−0.299129	−0.149564	+	0.988752i	$0.547787\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	816.299	0.0893567
$438$	0	0
$439$	−9673.34	−1.05167	−0.525835	−	0.850587i	$-0.676247\pi$
$439$	−9673.34	−1.05167	−0.525835	+	0.850587i	$0.676247\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14747.7	1.58168	0.790838	−	0.612026i	$-0.209645\pi$
$443$	14747.7	1.58168	0.790838	+	0.612026i	$0.209645\pi$
$444$	0	0
$445$	6329.76	0.674291
$446$	0	0
$447$	0	0
$448$	0	0
$449$	872.877	0.0917453	0.0458726	−	0.998947i	$-0.485393\pi$
$449$	872.877	0.0917453	0.0458726	+	0.998947i	$0.485393\pi$
$450$	0	0
$451$	−2484.98	−0.259453
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10414.9	−1.07310
$456$	0	0
$457$	7841.08	0.802604	0.401302	−	0.915946i	$-0.368558\pi$
$457$	7841.08	0.802604	0.401302	+	0.915946i	$0.368558\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4767.81	0.481690	0.240845	−	0.970564i	$-0.422576\pi$
$461$	4767.81	0.481690	0.240845	+	0.970564i	$0.422576\pi$
$462$	0	0
$463$	5183.54	0.520302	0.260151	−	0.965568i	$-0.416228\pi$
$463$	5183.54	0.520302	0.260151	+	0.965568i	$0.416228\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19168.9	1.89942	0.949712	−	0.313124i	$-0.101376\pi$
$467$	19168.9	1.89942	0.949712	+	0.313124i	$0.101376\pi$
$468$	0	0
$469$	−18963.9	−1.86710
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10870.9	1.05676
$474$	0	0
$475$	−255.876	−0.0247166
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−13518.8	−1.28954	−0.644772	−	0.764375i	$-0.723048\pi$
$479$	−13518.8	−1.28954	−0.644772	+	0.764375i	$0.723048\pi$
$480$	0	0
$481$	18535.4	1.75705
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3086.53	0.288973
$486$	0	0
$487$	−1435.08	−0.133531	−0.0667655	−	0.997769i	$-0.521268\pi$
$487$	−1435.08	−0.133531	−0.0667655	+	0.997769i	$0.521268\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−9967.64	−0.916158	−0.458079	−	0.888912i	$-0.651462\pi$
$491$	−9967.64	−0.916158	−0.458079	+	0.888912i	$0.651462\pi$
$492$	0	0
$493$	2179.64	0.199119
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−26382.8	−2.38115
$498$	0	0
$499$	9711.73	0.871256	0.435628	−	0.900127i	$-0.356526\pi$
$499$	9711.73	0.871256	0.435628	+	0.900127i	$0.356526\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7305.12	−0.647553	−0.323777	−	0.946134i	$-0.604953\pi$
$503$	−7305.12	−0.647553	−0.323777	+	0.946134i	$0.604953\pi$
$504$	0	0
$505$	−4158.58	−0.366445
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−145.255	−0.0126489	−0.00632447	−	0.999980i	$-0.502013\pi$
$509$	−145.255	−0.0126489	−0.00632447	+	0.999980i	$0.502013\pi$
$510$	0	0
$511$	7506.78	0.649864
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4521.27	0.386856
$516$	0	0
$517$	878.243	0.0747100
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−13168.5	−1.10734	−0.553668	−	0.832737i	$-0.686772\pi$
$521$	−13168.5	−1.10734	−0.553668	+	0.832737i	$0.686772\pi$
$522$	0	0
$523$	−547.489	−0.0457745	−0.0228872	−	0.999738i	$-0.507286\pi$
$523$	−547.489	−0.0457745	−0.0228872	+	0.999738i	$0.507286\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	836.094	0.0691097
$528$	0	0
$529$	−5806.10	−0.477200
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6426.81	0.522282
$534$	0	0
$535$	−4486.45	−0.362554
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−19948.5	−1.59415
$540$	0	0
$541$	9170.89	0.728812	0.364406	−	0.931240i	$-0.381272\pi$
$541$	9170.89	0.728812	0.364406	+	0.931240i	$0.381272\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1294.66	−0.101756
$546$	0	0
$547$	15008.1	1.17313	0.586563	−	0.809903i	$-0.300481\pi$
$547$	15008.1	1.17313	0.586563	+	0.809903i	$0.300481\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1134.53	0.0877176
$552$	0	0
$553$	−8963.96	−0.689306
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2304.03	−0.175269	−0.0876346	−	0.996153i	$-0.527931\pi$
$557$	−2304.03	−0.175269	−0.0876346	+	0.996153i	$0.527931\pi$
$558$	0	0
$559$	−28115.1	−2.12726
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4631.88	0.346733	0.173366	−	0.984857i	$-0.444536\pi$
$563$	4631.88	0.346733	0.173366	+	0.984857i	$0.444536\pi$
$564$	0	0
$565$	−1645.14	−0.122498
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4713.01	0.347240	0.173620	−	0.984813i	$-0.444454\pi$
$569$	4713.01	0.347240	0.173620	+	0.984813i	$0.444454\pi$
$570$	0	0
$571$	−24290.9	−1.78028	−0.890141	−	0.455685i	$-0.849394\pi$
$571$	−24290.9	−1.78028	−0.890141	+	0.455685i	$0.849394\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1993.88	−0.144610
$576$	0	0
$577$	19147.1	1.38146	0.690730	−	0.723113i	$-0.257289\pi$
$577$	19147.1	1.38146	0.690730	+	0.723113i	$0.257289\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−31324.7	−2.23677
$582$	0	0
$583$	−13206.8	−0.938197
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20112.1	−1.41417	−0.707083	−	0.707131i	$-0.749989\pi$
$587$	−20112.1	−1.41417	−0.707083	+	0.707131i	$0.749989\pi$
$588$	0	0
$589$	435.196	0.0304447
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16455.1	−1.13951	−0.569756	−	0.821814i	$-0.692962\pi$
$593$	−16455.1	−1.13951	−0.569756	+	0.821814i	$0.692962\pi$
$594$	0	0
$595$	3408.01	0.234815
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2444.35	0.166734	0.0833669	−	0.996519i	$-0.473433\pi$
$599$	2444.35	0.166734	0.0833669	+	0.996519i	$0.473433\pi$
$600$	0	0
$601$	−20881.1	−1.41724	−0.708619	−	0.705592i	$-0.750681\pi$
$601$	−20881.1	−1.41724	−0.708619	+	0.705592i	$0.750681\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3955.66	−0.265819
$606$	0	0
$607$	14041.6	0.938928	0.469464	−	0.882952i	$-0.344447\pi$
$607$	14041.6	0.938928	0.469464	+	0.882952i	$0.344447\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2271.36	−0.150392
$612$	0	0
$613$	−7303.82	−0.481237	−0.240619	−	0.970620i	$-0.577350\pi$
$613$	−7303.82	−0.481237	−0.240619	+	0.970620i	$0.577350\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22409.4	−1.46218	−0.731092	−	0.682279i	$-0.760989\pi$
$617$	−22409.4	−1.46218	−0.731092	+	0.682279i	$0.760989\pi$
$618$	0	0
$619$	−6459.28	−0.419419	−0.209710	−	0.977764i	$-0.567252\pi$
$619$	−6459.28	−0.419419	−0.209710	+	0.977764i	$0.567252\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−43882.2	−2.82200
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6065.20	−0.384476
$630$	0	0
$631$	−28348.6	−1.78850	−0.894248	−	0.447571i	$-0.852289\pi$
$631$	−28348.6	−1.78850	−0.894248	+	0.447571i	$0.852289\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8011.80	0.500691
$636$	0	0
$637$	51592.1	3.20903
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1235.34	0.0761202	0.0380601	−	0.999275i	$-0.487882\pi$
$641$	1235.34	0.0761202	0.0380601	+	0.999275i	$0.487882\pi$
$642$	0	0
$643$	7476.80	0.458563	0.229282	−	0.973360i	$-0.426362\pi$
$643$	7476.80	0.458563	0.229282	+	0.973360i	$0.426362\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24330.3	1.47840	0.739199	−	0.673487i	$-0.235204\pi$
$647$	24330.3	1.47840	0.739199	+	0.673487i	$0.235204\pi$
$648$	0	0
$649$	−15495.3	−0.937201
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10165.2	−0.609180	−0.304590	−	0.952484i	$-0.598519\pi$
$653$	−10165.2	−0.609180	−0.304590	+	0.952484i	$0.598519\pi$
$654$	0	0
$655$	6867.96	0.409700
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−17160.8	−1.01440	−0.507199	−	0.861829i	$-0.669319\pi$
$659$	−17160.8	−1.01440	−0.507199	+	0.861829i	$0.669319\pi$
$660$	0	0
$661$	−4982.25	−0.293173	−0.146586	−	0.989198i	$-0.546829\pi$
$661$	−4982.25	−0.293173	−0.146586	+	0.989198i	$0.546829\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1773.91	0.103442
$666$	0	0
$667$	8840.64	0.513210
$668$	0	0
$669$	0	0
$670$	0	0
$671$	20032.5	1.15253
$672$	0	0
$673$	26651.2	1.52649	0.763245	−	0.646109i	$-0.223605\pi$
$673$	26651.2	1.52649	0.763245	+	0.646109i	$0.223605\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24313.1	1.38025	0.690124	−	0.723691i	$-0.257556\pi$
$677$	24313.1	1.38025	0.690124	+	0.723691i	$0.257556\pi$
$678$	0	0
$679$	−21397.9	−1.20939
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26112.7	1.46292	0.731461	−	0.681883i	$-0.238839\pi$
$683$	26112.7	1.46292	0.731461	+	0.681883i	$0.238839\pi$
$684$	0	0
$685$	3454.14	0.192665
$686$	0	0
$687$	0	0
$688$	0	0
$689$	34156.2	1.88860
$690$	0	0
$691$	−4874.70	−0.268368	−0.134184	−	0.990956i	$-0.542841\pi$
$691$	−4874.70	−0.268368	−0.134184	+	0.990956i	$0.542841\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5364.51	−0.292788
$696$	0	0
$697$	−2103.00	−0.114285
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4679.39	0.252123	0.126062	−	0.992022i	$-0.459766\pi$
$701$	4679.39	0.252123	0.126062	+	0.992022i	$0.459766\pi$
$702$	0	0
$703$	−3157.01	−0.169372
$704$	0	0
$705$	0	0
$706$	0	0
$707$	28830.2	1.53362
$708$	0	0
$709$	16966.1	0.898696	0.449348	−	0.893357i	$-0.351656\pi$
$709$	16966.1	0.898696	0.449348	+	0.893357i	$0.351656\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3391.21	0.178123
$714$	0	0
$715$	−6981.18	−0.365149
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−19586.9	−1.01595	−0.507974	−	0.861372i	$-0.669605\pi$
$719$	−19586.9	−1.01595	−0.507974	+	0.861372i	$0.669605\pi$
$720$	0	0
$721$	−31344.5	−1.61905
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2771.18	−0.141957
$726$	0	0
$727$	32623.5	1.66429	0.832145	−	0.554558i	$-0.187113\pi$
$727$	32623.5	1.66429	0.832145	+	0.554558i	$0.187113\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9199.90	0.465486
$732$	0	0
$733$	−7468.57	−0.376341	−0.188170	−	0.982136i	$-0.560256\pi$
$733$	−7468.57	−0.376341	−0.188170	+	0.982136i	$0.560256\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12711.5	−0.635327
$738$	0	0
$739$	2585.13	0.128682	0.0643408	−	0.997928i	$-0.479506\pi$
$739$	2585.13	0.128682	0.0643408	+	0.997928i	$0.479506\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9989.29	0.493232	0.246616	−	0.969113i	$-0.420681\pi$
$743$	9989.29	0.493232	0.246616	+	0.969113i	$0.420681\pi$
$744$	0	0
$745$	−14142.8	−0.695505
$746$	0	0
$747$	0	0
$748$	0	0
$749$	31103.2	1.51734
$750$	0	0
$751$	−1340.53	−0.0651354	−0.0325677	−	0.999470i	$-0.510368\pi$
$751$	−1340.53	−0.0651354	−0.0325677	+	0.999470i	$0.510368\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5693.04	0.274425
$756$	0	0
$757$	20580.1	0.988105	0.494053	−	0.869432i	$-0.335515\pi$
$757$	20580.1	0.988105	0.494053	+	0.869432i	$0.335515\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6562.00	−0.312579	−0.156289	−	0.987711i	$-0.549953\pi$
$761$	−6562.00	−0.312579	−0.156289	+	0.987711i	$0.549953\pi$
$762$	0	0
$763$	8975.48	0.425864
$764$	0	0
$765$	0	0
$766$	0	0
$767$	40074.8	1.88659
$768$	0	0
$769$	−16766.2	−0.786221	−0.393110	−	0.919491i	$-0.628601\pi$
$769$	−16766.2	−0.786221	−0.393110	+	0.919491i	$0.628601\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	40296.9	1.87500	0.937502	−	0.347980i	$-0.113132\pi$
$773$	40296.9	1.87500	0.937502	+	0.347980i	$0.113132\pi$
$774$	0	0
$775$	−1063.01	−0.0492701
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1094.64	−0.0503459
$780$	0	0
$781$	−17684.5	−0.810244
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12448.4	0.565992
$786$	0	0
$787$	42549.6	1.92723	0.963615	−	0.267292i	$-0.0861289\pi$
$787$	42549.6	1.92723	0.963615	+	0.267292i	$0.0861289\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	11405.2	0.512673
$792$	0	0
$793$	−51809.1	−2.32005
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5850.28	−0.260010	−0.130005	−	0.991513i	$-0.541499\pi$
$797$	−5850.28	−0.260010	−0.130005	+	0.991513i	$0.541499\pi$
$798$	0	0
$799$	743.243	0.0329087
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5031.83	0.221132
$804$	0	0
$805$	13823.0	0.605212
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30659.8	1.33244	0.666218	−	0.745757i	$-0.267912\pi$
$809$	30659.8	1.33244	0.666218	+	0.745757i	$0.267912\pi$
$810$	0	0
$811$	27546.6	1.19272	0.596358	−	0.802718i	$-0.296614\pi$
$811$	27546.6	1.19272	0.596358	+	0.802718i	$0.296614\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5873.03	−0.252421
$816$	0	0
$817$	4788.65	0.205060
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42270.3	1.79688	0.898442	−	0.439092i	$-0.144700\pi$
$821$	42270.3	1.79688	0.898442	+	0.439092i	$0.144700\pi$
$822$	0	0
$823$	8766.81	0.371314	0.185657	−	0.982615i	$-0.440559\pi$
$823$	8766.81	0.371314	0.185657	+	0.982615i	$0.440559\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13187.2	−0.554489	−0.277244	−	0.960799i	$-0.589421\pi$
$827$	−13187.2	−0.554489	−0.277244	+	0.960799i	$0.589421\pi$
$828$	0	0
$829$	−7127.69	−0.298619	−0.149309	−	0.988791i	$-0.547705\pi$
$829$	−7127.69	−0.298619	−0.149309	+	0.988791i	$0.547705\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−16882.1	−0.702198
$834$	0	0
$835$	11246.4	0.466105
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−38084.2	−1.56712	−0.783558	−	0.621318i	$-0.786597\pi$
$839$	−38084.2	−1.56712	−0.783558	+	0.621318i	$0.786597\pi$
$840$	0	0
$841$	−12101.9	−0.496204
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7070.14	0.287835
$846$	0	0
$847$	27423.4	1.11249
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24600.6	−0.990948
$852$	0	0
$853$	−42235.2	−1.69532	−0.847659	−	0.530542i	$-0.821988\pi$
$853$	−42235.2	−1.69532	−0.847659	+	0.530542i	$0.821988\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15377.2	−0.612923	−0.306461	−	0.951883i	$-0.599145\pi$
$857$	−15377.2	−0.612923	−0.306461	+	0.951883i	$0.599145\pi$
$858$	0	0
$859$	19130.3	0.759858	0.379929	−	0.925016i	$-0.375949\pi$
$859$	19130.3	0.759858	0.379929	+	0.925016i	$0.375949\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6298.68	−0.248447	−0.124223	−	0.992254i	$-0.539644\pi$
$863$	−6298.68	−0.248447	−0.124223	+	0.992254i	$0.539644\pi$
$864$	0	0
$865$	18460.9	0.725654
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−6008.58	−0.234554
$870$	0	0
$871$	32875.3	1.27892
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4332.93	−0.167405
$876$	0	0
$877$	18035.4	0.694426	0.347213	−	0.937786i	$-0.387128\pi$
$877$	18035.4	0.694426	0.347213	+	0.937786i	$0.387128\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−18819.3	−0.719680	−0.359840	−	0.933014i	$-0.617169\pi$
$881$	−18819.3	−0.719680	−0.359840	+	0.933014i	$0.617169\pi$
$882$	0	0
$883$	−9977.04	−0.380243	−0.190121	−	0.981761i	$-0.560888\pi$
$883$	−9977.04	−0.380243	−0.190121	+	0.981761i	$0.560888\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42364.0	1.60366	0.801829	−	0.597554i	$-0.203861\pi$
$887$	42364.0	1.60366	0.801829	+	0.597554i	$0.203861\pi$
$888$	0	0
$889$	−55543.3	−2.09546
$890$	0	0
$891$	0	0
$892$	0	0
$893$	386.866	0.0144972
$894$	0	0
$895$	21200.1	0.791778
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4713.24	0.174856
$900$	0	0
$901$	−11176.7	−0.413262
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3741.33	−0.137421
$906$	0	0
$907$	−30111.4	−1.10235	−0.551176	−	0.834389i	$-0.685821\pi$
$907$	−30111.4	−1.10235	−0.551176	+	0.834389i	$0.685821\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	49694.3	1.80729	0.903647	−	0.428279i	$-0.140880\pi$
$911$	49694.3	1.80729	0.903647	+	0.428279i	$0.140880\pi$
$912$	0	0
$913$	−20997.1	−0.761118
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−47613.4	−1.71465
$918$	0	0
$919$	25625.8	0.919822	0.459911	−	0.887965i	$-0.347881\pi$
$919$	25625.8	0.919822	0.459911	+	0.887965i	$0.347881\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	45736.7	1.63103
$924$	0	0
$925$	7711.27	0.274103
$926$	0	0
$927$	0	0
$928$	0	0
$929$	26165.4	0.924067	0.462033	−	0.886862i	$-0.347120\pi$
$929$	26165.4	0.924067	0.462033	+	0.886862i	$0.347120\pi$
$930$	0	0
$931$	−8787.34	−0.309338
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2284.40	0.0799016
$936$	0	0
$937$	−45347.5	−1.58104	−0.790522	−	0.612434i	$-0.790191\pi$
$937$	−45347.5	−1.58104	−0.790522	+	0.612434i	$0.790191\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−28081.8	−0.972837	−0.486418	−	0.873726i	$-0.661697\pi$
$941$	−28081.8	−0.972837	−0.486418	+	0.873726i	$0.661697\pi$
$942$	0	0
$943$	−8529.82	−0.294559
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−12696.9	−0.435684	−0.217842	−	0.975984i	$-0.569902\pi$
$947$	−12696.9	−0.435684	−0.217842	+	0.975984i	$0.569902\pi$
$948$	0	0
$949$	−13013.6	−0.445142
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1592.18	0.0541194	0.0270597	−	0.999634i	$-0.491386\pi$
$953$	1592.18	0.0541194	0.0270597	+	0.999634i	$0.491386\pi$
$954$	0	0
$955$	5670.18	0.192129
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−23946.4	−0.806331
$960$	0	0
$961$	−27983.0	−0.939312
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5671.61	0.189198
$966$	0	0
$967$	47529.1	1.58059	0.790296	−	0.612725i	$-0.209927\pi$
$967$	47529.1	1.58059	0.790296	+	0.612725i	$0.209927\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−23045.3	−0.761647	−0.380824	−	0.924648i	$-0.624359\pi$
$971$	−23045.3	−0.761647	−0.380824	+	0.924648i	$0.624359\pi$
$972$	0	0
$973$	37190.5	1.22536
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45865.1	1.50190	0.750950	−	0.660359i	$-0.229596\pi$
$977$	45865.1	1.50190	0.750950	+	0.660359i	$0.229596\pi$
$978$	0	0
$979$	−29414.4	−0.960255
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−32496.5	−1.05440	−0.527202	−	0.849740i	$-0.676759\pi$
$983$	−32496.5	−1.05440	−0.527202	+	0.849740i	$0.676759\pi$
$984$	0	0
$985$	21963.2	0.710462
$986$	0	0
$987$	0	0
$988$	0	0
$989$	37315.0	1.19974
$990$	0	0
$991$	26439.3	0.847500	0.423750	−	0.905779i	$-0.360714\pi$
$991$	26439.3	0.847500	0.423750	+	0.905779i	$0.360714\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9613.77	−0.306308
$996$	0	0
$997$	−1920.47	−0.0610049	−0.0305025	−	0.999535i	$-0.509711\pi$
$997$	−1920.47	−0.0610049	−0.0305025	+	0.999535i	$0.509711\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.a.k.1.1	yes	3
3.2	odd	2		1080.4.a.e.1.1	✓	3
4.3	odd	2		2160.4.a.br.1.3		3
12.11	even	2		2160.4.a.bj.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.e.1.1	✓	3	3.2	odd	2
1080.4.a.k.1.1	yes	3	1.1	even	1	trivial
2160.4.a.bj.1.3		3	12.11	even	2
2160.4.a.br.1.3		3	4.3	odd	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:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} -34.6634 q^{7} +O(q^{10})\)	\(q+5.00000 q^{5} -34.6634 q^{7} -23.2350 q^{11} +60.0918 q^{13} -19.6634 q^{17} -10.2350 q^{19} -79.7553 q^{23} +25.0000 q^{25} -110.847 q^{29} -42.5202 q^{31} -173.317 q^{35} +308.451 q^{37} +106.950 q^{41} -467.868 q^{43} -37.7982 q^{47} +858.554 q^{49} +568.399 q^{53} -116.175 q^{55} +666.893 q^{59} -862.166 q^{61} +300.459 q^{65} +547.085 q^{67} +761.113 q^{71} -216.562 q^{73} +805.407 q^{77} +258.600 q^{79} +903.680 q^{83} -98.3172 q^{85} +1265.95 q^{89} -2082.99 q^{91} -51.1752 q^{95} +617.305 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 15 q^{5} - 8 q^{7}+O(q^{10}) \) 3 * q + 15 * q^5 - 8 * q^7	\( 3 q + 15 q^{5} - 8 q^{7} - 10 q^{11} + 48 q^{13} + 37 q^{17} + 29 q^{19} - 11 q^{23} + 75 q^{25} + 28 q^{29} + 41 q^{31} - 40 q^{35} + 230 q^{37} + 370 q^{41} - 130 q^{43} - 56 q^{47} + 547 q^{49} + 805 q^{53} - 50 q^{55} + 576 q^{59} - 257 q^{61} + 240 q^{65} - 14 q^{67} + 1238 q^{71} - 398 q^{73} + 1296 q^{77} - 321 q^{79} + 687 q^{83} + 185 q^{85} + 2358 q^{89} - 1968 q^{91} + 145 q^{95} + 576 q^{97}+O(q^{100}) \) 3 * q + 15 * q^5 - 8 * q^7 - 10 * q^11 + 48 * q^13 + 37 * q^17 + 29 * q^19 - 11 * q^23 + 75 * q^25 + 28 * q^29 + 41 * q^31 - 40 * q^35 + 230 * q^37 + 370 * q^41 - 130 * q^43 - 56 * q^47 + 547 * q^49 + 805 * q^53 - 50 * q^55 + 576 * q^59 - 257 * q^61 + 240 * q^65 - 14 * q^67 + 1238 * q^71 - 398 * q^73 + 1296 * q^77 - 321 * q^79 + 687 * q^83 + 185 * q^85 + 2358 * q^89 - 1968 * q^91 + 145 * q^95 + 576 * q^97

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