LMFDB - Embedded newform 648.4.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(1,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, 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("648.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.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:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 19x^{3} + 4x^{2} + 81x + 12 $ x^5 - x^4 - 19x^3 + 4x^2 + 81*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$4.16963$ of defining polynomial
Character	$\chi$	$=$	648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.3184 q^{5} +14.9787 q^{7} +O(q^{10})$	$q+18.3184 q^{5} +14.9787 q^{7} +50.8308 q^{11} -26.1267 q^{13} +93.7208 q^{17} +71.2297 q^{19} -146.415 q^{23} +210.563 q^{25} +38.4247 q^{29} +11.2466 q^{31} +274.385 q^{35} -426.542 q^{37} -124.082 q^{41} +309.751 q^{43} +16.3244 q^{47} -118.640 q^{49} -529.667 q^{53} +931.137 q^{55} -439.491 q^{59} -54.0038 q^{61} -478.599 q^{65} +445.613 q^{67} +3.34338 q^{71} +820.855 q^{73} +761.377 q^{77} +133.636 q^{79} +265.127 q^{83} +1716.81 q^{85} -1268.64 q^{89} -391.343 q^{91} +1304.81 q^{95} +1212.74 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} + 3 q^{7}+O(q^{10}) $ 5 * q - 5 * q^5 + 3 * q^7	$ 5 q - 5 q^{5} + 3 q^{7} + 25 q^{11} + 29 q^{13} - 28 q^{17} + 64 q^{19} - 89 q^{23} + 322 q^{25} - 129 q^{29} + 241 q^{31} + 243 q^{35} + 366 q^{37} - 171 q^{41} + 803 q^{43} + 477 q^{47} + 1072 q^{49} - 374 q^{53} + 1469 q^{55} + 607 q^{59} + 1349 q^{61} - 527 q^{65} + 1549 q^{67} + 812 q^{71} + 1928 q^{73} - 903 q^{77} + 1727 q^{79} + 1025 q^{83} + 2902 q^{85} - 2310 q^{89} + 2985 q^{91} + 2012 q^{95} + 2875 q^{97}+O(q^{100}) $ 5 * q - 5 * q^5 + 3 * q^7 + 25 * q^11 + 29 * q^13 - 28 * q^17 + 64 * q^19 - 89 * q^23 + 322 * q^25 - 129 * q^29 + 241 * q^31 + 243 * q^35 + 366 * q^37 - 171 * q^41 + 803 * q^43 + 477 * q^47 + 1072 * q^49 - 374 * q^53 + 1469 * q^55 + 607 * q^59 + 1349 * q^61 - 527 * q^65 + 1549 * q^67 + 812 * q^71 + 1928 * q^73 - 903 * q^77 + 1727 * q^79 + 1025 * q^83 + 2902 * q^85 - 2310 * q^89 + 2985 * q^91 + 2012 * q^95 + 2875 * 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
$5$	18.3184	1.63845	0.819223	−	0.573475i	$-0.194405\pi$
$5$	18.3184	1.63845	0.819223	+	0.573475i	$0.194405\pi$
$6$	0	0
$7$	14.9787	0.808772	0.404386	−	0.914588i	$-0.367485\pi$
$7$	14.9787	0.808772	0.404386	+	0.914588i	$0.367485\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	50.8308	1.39328	0.696639	−	0.717422i	$-0.254678\pi$
$11$	50.8308	1.39328	0.696639	+	0.717422i	$0.254678\pi$
$12$	0	0
$13$	−26.1267	−0.557404	−0.278702	−	0.960378i	$-0.589904\pi$
$13$	−26.1267	−0.557404	−0.278702	+	0.960378i	$0.589904\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	93.7208	1.33710	0.668548	−	0.743669i	$-0.266916\pi$
$17$	93.7208	1.33710	0.668548	+	0.743669i	$0.266916\pi$
$18$	0	0
$19$	71.2297	0.860064	0.430032	−	0.902814i	$-0.358502\pi$
$19$	71.2297	0.860064	0.430032	+	0.902814i	$0.358502\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−146.415	−1.32737	−0.663687	−	0.748010i	$-0.731009\pi$
$23$	−146.415	−1.32737	−0.663687	+	0.748010i	$0.731009\pi$
$24$	0	0
$25$	210.563	1.68450
$26$	0	0
$27$	0	0
$28$	0	0
$29$	38.4247	0.246044	0.123022	−	0.992404i	$-0.460741\pi$
$29$	38.4247	0.246044	0.123022	+	0.992404i	$0.460741\pi$
$30$	0	0
$31$	11.2466	0.0651598	0.0325799	−	0.999469i	$-0.489628\pi$
$31$	11.2466	0.0651598	0.0325799	+	0.999469i	$0.489628\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	274.385	1.32513
$36$	0	0
$37$	−426.542	−1.89522	−0.947608	−	0.319435i	$-0.896507\pi$
$37$	−426.542	−1.89522	−0.947608	+	0.319435i	$0.896507\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−124.082	−0.472642	−0.236321	−	0.971675i	$-0.575942\pi$
$41$	−124.082	−0.472642	−0.236321	+	0.971675i	$0.575942\pi$
$42$	0	0
$43$	309.751	1.09853	0.549263	−	0.835649i	$-0.314908\pi$
$43$	309.751	1.09853	0.549263	+	0.835649i	$0.314908\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	16.3244	0.0506629	0.0253314	−	0.999679i	$-0.491936\pi$
$47$	16.3244	0.0506629	0.0253314	+	0.999679i	$0.491936\pi$
$48$	0	0
$49$	−118.640	−0.345888
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−529.667	−1.37274	−0.686371	−	0.727251i	$-0.740798\pi$
$53$	−529.667	−1.37274	−0.686371	+	0.727251i	$0.740798\pi$
$54$	0	0
$55$	931.137	2.28281
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−439.491	−0.969777	−0.484888	−	0.874576i	$-0.661140\pi$
$59$	−439.491	−0.969777	−0.484888	+	0.874576i	$0.661140\pi$
$60$	0	0
$61$	−54.0038	−0.113352	−0.0566761	−	0.998393i	$-0.518050\pi$
$61$	−54.0038	−0.113352	−0.0566761	+	0.998393i	$0.518050\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−478.599	−0.913275
$66$	0	0
$67$	445.613	0.812541	0.406270	−	0.913753i	$-0.366829\pi$
$67$	445.613	0.812541	0.406270	+	0.913753i	$0.366829\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.34338	0.00558854	0.00279427	−	0.999996i	$-0.499111\pi$
$71$	3.34338	0.00558854	0.00279427	+	0.999996i	$0.499111\pi$
$72$	0	0
$73$	820.855	1.31608	0.658040	−	0.752983i	$-0.271386\pi$
$73$	820.855	1.31608	0.658040	+	0.752983i	$0.271386\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	761.377	1.12684
$78$	0	0
$79$	133.636	0.190319	0.0951595	−	0.995462i	$-0.469664\pi$
$79$	133.636	0.190319	0.0951595	+	0.995462i	$0.469664\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	265.127	0.350621	0.175310	−	0.984513i	$-0.443907\pi$
$83$	265.127	0.350621	0.175310	+	0.984513i	$0.443907\pi$
$84$	0	0
$85$	1716.81	2.19076
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1268.64	−1.51096	−0.755482	−	0.655170i	$-0.772597\pi$
$89$	−1268.64	−1.51096	−0.755482	+	0.655170i	$0.772597\pi$
$90$	0	0
$91$	−391.343	−0.450812
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1304.81	1.40917
$96$	0	0
$97$	1212.74	1.26944	0.634718	−	0.772744i	$-0.281116\pi$
$97$	1212.74	1.26944	0.634718	+	0.772744i	$0.281116\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−803.946	−0.792035	−0.396018	−	0.918243i	$-0.629608\pi$
$101$	−803.946	−0.792035	−0.396018	+	0.918243i	$0.629608\pi$
$102$	0	0
$103$	597.972	0.572038	0.286019	−	0.958224i	$-0.407668\pi$
$103$	597.972	0.572038	0.286019	+	0.958224i	$0.407668\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1413.64	1.27722	0.638608	−	0.769533i	$-0.279511\pi$
$107$	1413.64	1.27722	0.638608	+	0.769533i	$0.279511\pi$
$108$	0	0
$109$	387.654	0.340647	0.170323	−	0.985388i	$-0.445519\pi$
$109$	387.654	0.340647	0.170323	+	0.985388i	$0.445519\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−345.051	−0.287254	−0.143627	−	0.989632i	$-0.545877\pi$
$113$	−345.051	−0.287254	−0.143627	+	0.989632i	$0.545877\pi$
$114$	0	0
$115$	−2682.08	−2.17483
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1403.81	1.08141
$120$	0	0
$121$	1252.77	0.941222
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1567.37	1.12152
$126$	0	0
$127$	2022.91	1.41342	0.706711	−	0.707503i	$-0.250178\pi$
$127$	2022.91	1.41342	0.706711	+	0.707503i	$0.250178\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1630.76	1.08763	0.543817	−	0.839204i	$-0.316979\pi$
$131$	1630.76	1.08763	0.543817	+	0.839204i	$0.316979\pi$
$132$	0	0
$133$	1066.93	0.695596
$134$	0	0
$135$	0	0
$136$	0	0
$137$	968.037	0.603686	0.301843	−	0.953358i	$-0.402398\pi$
$137$	968.037	0.603686	0.301843	+	0.953358i	$0.402398\pi$
$138$	0	0
$139$	−1369.80	−0.835864	−0.417932	−	0.908478i	$-0.637245\pi$
$139$	−1369.80	−0.835864	−0.417932	+	0.908478i	$0.637245\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1328.04	−0.776618
$144$	0	0
$145$	703.878	0.403130
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−297.749	−0.163708	−0.0818541	−	0.996644i	$-0.526084\pi$
$149$	−297.749	−0.163708	−0.0818541	+	0.996644i	$0.526084\pi$
$150$	0	0
$151$	3295.21	1.77590	0.887948	−	0.459944i	$-0.152130\pi$
$151$	3295.21	1.77590	0.887948	+	0.459944i	$0.152130\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	206.020	0.106761
$156$	0	0
$157$	−1656.42	−0.842019	−0.421009	−	0.907056i	$-0.638324\pi$
$157$	−1656.42	−0.842019	−0.421009	+	0.907056i	$0.638324\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2193.10	−1.07354
$162$	0	0
$163$	−2941.91	−1.41367	−0.706835	−	0.707379i	$-0.749878\pi$
$163$	−2941.91	−1.41367	−0.706835	+	0.707379i	$0.749878\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1164.94	−0.539795	−0.269898	−	0.962889i	$-0.586990\pi$
$167$	−1164.94	−0.539795	−0.269898	+	0.962889i	$0.586990\pi$
$168$	0	0
$169$	−1514.39	−0.689301
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3717.21	−1.63361	−0.816805	−	0.576914i	$-0.804257\pi$
$173$	−3717.21	−1.63361	−0.816805	+	0.576914i	$0.804257\pi$
$174$	0	0
$175$	3153.95	1.36238
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−190.442	−0.0795213	−0.0397606	−	0.999209i	$-0.512660\pi$
$179$	−190.442	−0.0795213	−0.0397606	+	0.999209i	$0.512660\pi$
$180$	0	0
$181$	1647.85	0.676706	0.338353	−	0.941019i	$-0.390130\pi$
$181$	1647.85	0.676706	0.338353	+	0.941019i	$0.390130\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−7813.55	−3.10521
$186$	0	0
$187$	4763.90	1.86295
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1783.05	−0.675483	−0.337741	−	0.941239i	$-0.609663\pi$
$191$	−1783.05	−0.675483	−0.337741	+	0.941239i	$0.609663\pi$
$192$	0	0
$193$	−4814.27	−1.79554	−0.897768	−	0.440468i	$-0.854812\pi$
$193$	−4814.27	−1.79554	−0.897768	+	0.440468i	$0.854812\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1522.27	0.550545	0.275273	−	0.961366i	$-0.411232\pi$
$197$	1522.27	0.550545	0.275273	+	0.961366i	$0.411232\pi$
$198$	0	0
$199$	−2726.42	−0.971209	−0.485604	−	0.874179i	$-0.661400\pi$
$199$	−2726.42	−0.971209	−0.485604	+	0.874179i	$0.661400\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	575.550	0.198994
$204$	0	0
$205$	−2272.98	−0.774398
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3620.66	1.19831
$210$	0	0
$211$	−2369.00	−0.772933	−0.386467	−	0.922303i	$-0.626305\pi$
$211$	−2369.00	−0.772933	−0.386467	+	0.922303i	$0.626305\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5674.14	1.79988
$216$	0	0
$217$	168.459	0.0526994
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2448.62	−0.745302
$222$	0	0
$223$	1056.95	0.317393	0.158696	−	0.987327i	$-0.449271\pi$
$223$	1056.95	0.317393	0.158696	+	0.987327i	$0.449271\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4042.89	1.18210	0.591049	−	0.806636i	$-0.298714\pi$
$227$	4042.89	1.18210	0.591049	+	0.806636i	$0.298714\pi$
$228$	0	0
$229$	4985.96	1.43878	0.719391	−	0.694605i	$-0.244421\pi$
$229$	4985.96	1.43878	0.719391	+	0.694605i	$0.244421\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1619.10	0.455239	0.227620	−	0.973750i	$-0.426906\pi$
$233$	1619.10	0.455239	0.227620	+	0.973750i	$0.426906\pi$
$234$	0	0
$235$	299.036	0.0830084
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3428.43	0.927894	0.463947	−	0.885863i	$-0.346433\pi$
$239$	3428.43	0.927894	0.463947	+	0.885863i	$0.346433\pi$
$240$	0	0
$241$	−750.910	−0.200707	−0.100353	−	0.994952i	$-0.531997\pi$
$241$	−750.910	−0.200707	−0.100353	+	0.994952i	$0.531997\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2173.29	−0.566719
$246$	0	0
$247$	−1861.00	−0.479403
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2390.17	−0.601060	−0.300530	−	0.953772i	$-0.597164\pi$
$251$	−2390.17	−0.601060	−0.300530	+	0.953772i	$0.597164\pi$
$252$	0	0
$253$	−7442.38	−1.84940
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3741.91	0.908225	0.454112	−	0.890944i	$-0.349956\pi$
$257$	3741.91	0.908225	0.454112	+	0.890944i	$0.349956\pi$
$258$	0	0
$259$	−6389.02	−1.53280
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1570.27	−0.368164	−0.184082	−	0.982911i	$-0.558931\pi$
$263$	−1570.27	−0.368164	−0.184082	+	0.982911i	$0.558931\pi$
$264$	0	0
$265$	−9702.64	−2.24916
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6619.72	−1.50041	−0.750207	−	0.661203i	$-0.770046\pi$
$269$	−6619.72	−1.50041	−0.750207	+	0.661203i	$0.770046\pi$
$270$	0	0
$271$	−1678.72	−0.376292	−0.188146	−	0.982141i	$-0.560248\pi$
$271$	−1678.72	−0.376292	−0.188146	+	0.982141i	$0.560248\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10703.1	2.34698
$276$	0	0
$277$	−4289.16	−0.930364	−0.465182	−	0.885215i	$-0.654011\pi$
$277$	−4289.16	−0.930364	−0.465182	+	0.885215i	$0.654011\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	826.320	0.175424	0.0877120	−	0.996146i	$-0.472044\pi$
$281$	826.320	0.175424	0.0877120	+	0.996146i	$0.472044\pi$
$282$	0	0
$283$	−410.164	−0.0861545	−0.0430773	−	0.999072i	$-0.513716\pi$
$283$	−410.164	−0.0861545	−0.0430773	+	0.999072i	$0.513716\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1858.58	−0.382260
$288$	0	0
$289$	3870.59	0.787825
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2325.42	0.463659	0.231830	−	0.972756i	$-0.425529\pi$
$293$	2325.42	0.463659	0.231830	+	0.972756i	$0.425529\pi$
$294$	0	0
$295$	−8050.76	−1.58893
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3825.34	0.739883
$300$	0	0
$301$	4639.66	0.888457
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−989.263	−0.185721
$306$	0	0
$307$	5477.45	1.01829	0.509144	−	0.860681i	$-0.329962\pi$
$307$	5477.45	1.01829	0.509144	+	0.860681i	$0.329962\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6343.50	−1.15661	−0.578307	−	0.815819i	$-0.696286\pi$
$311$	−6343.50	−1.15661	−0.578307	+	0.815819i	$0.696286\pi$
$312$	0	0
$313$	−2618.52	−0.472868	−0.236434	−	0.971648i	$-0.575979\pi$
$313$	−2618.52	−0.472868	−0.236434	+	0.971648i	$0.575979\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8419.20	−1.49170	−0.745851	−	0.666113i	$-0.767957\pi$
$317$	−8419.20	−1.49170	−0.745851	+	0.666113i	$0.767957\pi$
$318$	0	0
$319$	1953.16	0.342808
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6675.71	1.14999
$324$	0	0
$325$	−5501.32	−0.938948
$326$	0	0
$327$	0	0
$328$	0	0
$329$	244.517	0.0409747
$330$	0	0
$331$	−3150.52	−0.523166	−0.261583	−	0.965181i	$-0.584245\pi$
$331$	−3150.52	−0.523166	−0.261583	+	0.965181i	$0.584245\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8162.90	1.33130
$336$	0	0
$337$	−1922.91	−0.310823	−0.155412	−	0.987850i	$-0.549670\pi$
$337$	−1922.91	−0.310823	−0.155412	+	0.987850i	$0.549670\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	571.674	0.0907856
$342$	0	0
$343$	−6914.74	−1.08852
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5260.14	−0.813772	−0.406886	−	0.913479i	$-0.633385\pi$
$347$	−5260.14	−0.813772	−0.406886	+	0.913479i	$0.633385\pi$
$348$	0	0
$349$	−1790.90	−0.274684	−0.137342	−	0.990524i	$-0.543856\pi$
$349$	−1790.90	−0.274684	−0.137342	+	0.990524i	$0.543856\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4962.31	−0.748207	−0.374104	−	0.927387i	$-0.622050\pi$
$353$	−4962.31	−0.748207	−0.374104	+	0.927387i	$0.622050\pi$
$354$	0	0
$355$	61.2453	0.00915651
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7852.28	−1.15439	−0.577197	−	0.816605i	$-0.695854\pi$
$359$	−7852.28	−1.15439	−0.577197	+	0.816605i	$0.695854\pi$
$360$	0	0
$361$	−1785.32	−0.260289
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15036.7	2.15632
$366$	0	0
$367$	−1662.10	−0.236405	−0.118203	−	0.992989i	$-0.537713\pi$
$367$	−1662.10	−0.236405	−0.118203	+	0.992989i	$0.537713\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7933.70	−1.11024
$372$	0	0
$373$	9219.45	1.27980	0.639899	−	0.768459i	$-0.278976\pi$
$373$	9219.45	1.27980	0.639899	+	0.768459i	$0.278976\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1003.91	−0.137146
$378$	0	0
$379$	2115.54	0.286723	0.143362	−	0.989670i	$-0.454209\pi$
$379$	2115.54	0.286723	0.143362	+	0.989670i	$0.454209\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7193.81	−0.959756	−0.479878	−	0.877335i	$-0.659319\pi$
$383$	−7193.81	−0.959756	−0.479878	+	0.877335i	$0.659319\pi$
$384$	0	0
$385$	13947.2	1.84627
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5546.54	0.722932	0.361466	−	0.932385i	$-0.382276\pi$
$389$	5546.54	0.722932	0.361466	+	0.932385i	$0.382276\pi$
$390$	0	0
$391$	−13722.1	−1.77483
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2447.99	0.311827
$396$	0	0
$397$	−4062.62	−0.513594	−0.256797	−	0.966465i	$-0.582667\pi$
$397$	−4062.62	−0.513594	−0.256797	+	0.966465i	$0.582667\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8470.18	−1.05481	−0.527407	−	0.849613i	$-0.676836\pi$
$401$	−8470.18	−1.05481	−0.527407	+	0.849613i	$0.676836\pi$
$402$	0	0
$403$	−293.837	−0.0363203
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21681.4	−2.64056
$408$	0	0
$409$	7367.48	0.890705	0.445353	−	0.895355i	$-0.353078\pi$
$409$	7367.48	0.890705	0.445353	+	0.895355i	$0.353078\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6582.98	−0.784328
$414$	0	0
$415$	4856.71	0.574473
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7695.46	−0.897250	−0.448625	−	0.893720i	$-0.648086\pi$
$419$	−7695.46	−0.897250	−0.448625	+	0.893720i	$0.648086\pi$
$420$	0	0
$421$	6095.95	0.705697	0.352848	−	0.935681i	$-0.385213\pi$
$421$	6095.95	0.705697	0.352848	+	0.935681i	$0.385213\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	19734.1	2.25234
$426$	0	0
$427$	−808.905	−0.0916760
$428$	0	0
$429$	0	0
$430$	0	0
$431$	119.401	0.0133442	0.00667212	−	0.999978i	$-0.497876\pi$
$431$	119.401	0.0133442	0.00667212	+	0.999978i	$0.497876\pi$
$432$	0	0
$433$	5425.09	0.602109	0.301054	−	0.953607i	$-0.402661\pi$
$433$	5425.09	0.602109	0.301054	+	0.953607i	$0.402661\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10429.1	−1.14163
$438$	0	0
$439$	−3952.21	−0.429679	−0.214839	−	0.976649i	$-0.568923\pi$
$439$	−3952.21	−0.429679	−0.214839	+	0.976649i	$0.568923\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14908.5	1.59893	0.799463	−	0.600715i	$-0.205117\pi$
$443$	14908.5	1.59893	0.799463	+	0.600715i	$0.205117\pi$
$444$	0	0
$445$	−23239.5	−2.47563
$446$	0	0
$447$	0	0
$448$	0	0
$449$	814.100	0.0855674	0.0427837	−	0.999084i	$-0.486377\pi$
$449$	814.100	0.0855674	0.0427837	+	0.999084i	$0.486377\pi$
$450$	0	0
$451$	−6307.18	−0.658522
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7168.77	−0.738631
$456$	0	0
$457$	−3985.40	−0.407941	−0.203971	−	0.978977i	$-0.565385\pi$
$457$	−3985.40	−0.407941	−0.203971	+	0.978977i	$0.565385\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5336.20	−0.539114	−0.269557	−	0.962984i	$-0.586877\pi$
$461$	−5336.20	−0.539114	−0.269557	+	0.962984i	$0.586877\pi$
$462$	0	0
$463$	−9627.66	−0.966383	−0.483192	−	0.875515i	$-0.660523\pi$
$463$	−9627.66	−0.966383	−0.483192	+	0.875515i	$0.660523\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15687.2	1.55443	0.777214	−	0.629236i	$-0.216632\pi$
$467$	15687.2	1.55443	0.777214	+	0.629236i	$0.216632\pi$
$468$	0	0
$469$	6674.68	0.657160
$470$	0	0
$471$	0	0
$472$	0	0
$473$	15744.9	1.53055
$474$	0	0
$475$	14998.3	1.44878
$476$	0	0
$477$	0	0
$478$	0	0
$479$	860.440	0.0820762	0.0410381	−	0.999158i	$-0.486933\pi$
$479$	860.440	0.0820762	0.0410381	+	0.999158i	$0.486933\pi$
$480$	0	0
$481$	11144.1	1.05640
$482$	0	0
$483$	0	0
$484$	0	0
$485$	22215.5	2.07990
$486$	0	0
$487$	11473.1	1.06755	0.533774	−	0.845627i	$-0.320773\pi$
$487$	11473.1	1.06755	0.533774	+	0.845627i	$0.320773\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9941.12	0.913720	0.456860	−	0.889539i	$-0.348974\pi$
$491$	9941.12	0.913720	0.456860	+	0.889539i	$0.348974\pi$
$492$	0	0
$493$	3601.19	0.328985
$494$	0	0
$495$	0	0
$496$	0	0
$497$	50.0794	0.00451985
$498$	0	0
$499$	15596.9	1.39922	0.699610	−	0.714525i	$-0.253357\pi$
$499$	15596.9	1.39922	0.699610	+	0.714525i	$0.253357\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6580.73	−0.583341	−0.291670	−	0.956519i	$-0.594211\pi$
$503$	−6580.73	−0.583341	−0.291670	+	0.956519i	$0.594211\pi$
$504$	0	0
$505$	−14727.0	−1.29771
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4074.73	−0.354831	−0.177416	−	0.984136i	$-0.556774\pi$
$509$	−4074.73	−0.354831	−0.177416	+	0.984136i	$0.556774\pi$
$510$	0	0
$511$	12295.3	1.06441
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10953.9	0.937254
$516$	0	0
$517$	829.781	0.0705875
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.09007	−0.000259844 0	−0.000129922	−	1.00000i	$-0.500041\pi$
$521$	−3.09007	−0.000259844 0	−0.000129922		1.00000i	$0.500041\pi$
$522$	0	0
$523$	985.268	0.0823762	0.0411881	−	0.999151i	$-0.486886\pi$
$523$	985.268	0.0823762	0.0411881	+	0.999151i	$0.486886\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1054.04	0.0871248
$528$	0	0
$529$	9270.33	0.761924
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3241.85	0.263452
$534$	0	0
$535$	25895.6	2.09265
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6030.55	−0.481919
$540$	0	0
$541$	3258.14	0.258925	0.129463	−	0.991584i	$-0.458675\pi$
$541$	3258.14	0.258925	0.129463	+	0.991584i	$0.458675\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7101.19	0.558131
$546$	0	0
$547$	25389.8	1.98462	0.992311	−	0.123766i	$-0.0394971\pi$
$547$	25389.8	1.98462	0.992311	+	0.123766i	$0.0394971\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2736.98	0.211614
$552$	0	0
$553$	2001.68	0.153925
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22995.5	1.74928	0.874640	−	0.484774i	$-0.161098\pi$
$557$	22995.5	1.74928	0.874640	+	0.484774i	$0.161098\pi$
$558$	0	0
$559$	−8092.79	−0.612323
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11018.8	−0.824845	−0.412423	−	0.910993i	$-0.635317\pi$
$563$	−11018.8	−0.824845	−0.412423	+	0.910993i	$0.635317\pi$
$564$	0	0
$565$	−6320.78	−0.470650
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13351.4	−0.983691	−0.491845	−	0.870683i	$-0.663677\pi$
$569$	−13351.4	−0.983691	−0.491845	+	0.870683i	$0.663677\pi$
$570$	0	0
$571$	15365.6	1.12615	0.563074	−	0.826407i	$-0.309619\pi$
$571$	15365.6	1.12615	0.563074	+	0.826407i	$0.309619\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−30829.6	−2.23597
$576$	0	0
$577$	23783.8	1.71600	0.857999	−	0.513652i	$-0.171708\pi$
$577$	23783.8	1.71600	0.857999	+	0.513652i	$0.171708\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3971.25	0.283572
$582$	0	0
$583$	−26923.4	−1.91261
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25896.0	−1.82086	−0.910429	−	0.413665i	$-0.864249\pi$
$587$	−25896.0	−1.82086	−0.910429	+	0.413665i	$0.864249\pi$
$588$	0	0
$589$	801.094	0.0560416
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2039.41	−0.141229	−0.0706144	−	0.997504i	$-0.522496\pi$
$593$	−2039.41	−0.141229	−0.0706144	+	0.997504i	$0.522496\pi$
$594$	0	0
$595$	25715.6	1.77182
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20.7499	0.00141539	0.000707695	−	1.00000i	$-0.499775\pi$
$599$	20.7499	0.00141539	0.000707695		1.00000i	$0.499775\pi$
$600$	0	0
$601$	−23169.0	−1.57252	−0.786261	−	0.617895i	$-0.787986\pi$
$601$	−23169.0	−1.57252	−0.786261	+	0.617895i	$0.787986\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	22948.7	1.54214
$606$	0	0
$607$	−18608.0	−1.24428	−0.622139	−	0.782907i	$-0.713736\pi$
$607$	−18608.0	−1.24428	−0.622139	+	0.782907i	$0.713736\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−426.502	−0.0282397
$612$	0	0
$613$	21010.0	1.38432	0.692159	−	0.721745i	$-0.256660\pi$
$613$	21010.0	1.38432	0.692159	+	0.721745i	$0.256660\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11839.1	−0.772488	−0.386244	−	0.922397i	$-0.626228\pi$
$617$	−11839.1	−0.772488	−0.386244	+	0.922397i	$0.626228\pi$
$618$	0	0
$619$	−3220.13	−0.209092	−0.104546	−	0.994520i	$-0.533339\pi$
$619$	−3220.13	−0.209092	−0.104546	+	0.994520i	$0.533339\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−19002.5	−1.22202
$624$	0	0
$625$	2391.39	0.153049
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−39975.8	−2.53409
$630$	0	0
$631$	6470.43	0.408215	0.204107	−	0.978948i	$-0.434571\pi$
$631$	6470.43	0.408215	0.204107	+	0.978948i	$0.434571\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	37056.5	2.31581
$636$	0	0
$637$	3099.67	0.192799
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6036.24	−0.371945	−0.185973	−	0.982555i	$-0.559544\pi$
$641$	−6036.24	−0.371945	−0.185973	+	0.982555i	$0.559544\pi$
$642$	0	0
$643$	18781.6	1.15190	0.575951	−	0.817484i	$-0.304632\pi$
$643$	18781.6	1.15190	0.575951	+	0.817484i	$0.304632\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15139.9	−0.919955	−0.459978	−	0.887931i	$-0.652143\pi$
$647$	−15139.9	−0.919955	−0.459978	+	0.887931i	$0.652143\pi$
$648$	0	0
$649$	−22339.7	−1.35117
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3105.54	−0.186109	−0.0930546	−	0.995661i	$-0.529663\pi$
$653$	−3105.54	−0.186109	−0.0930546	+	0.995661i	$0.529663\pi$
$654$	0	0
$655$	29872.8	1.78203
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−11042.3	−0.652730	−0.326365	−	0.945244i	$-0.605824\pi$
$659$	−11042.3	−0.652730	−0.326365	+	0.945244i	$0.605824\pi$
$660$	0	0
$661$	14161.7	0.833321	0.416660	−	0.909062i	$-0.363200\pi$
$661$	14161.7	0.833321	0.416660	+	0.909062i	$0.363200\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	19544.4	1.13970
$666$	0	0
$667$	−5625.95	−0.326593
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2745.06	−0.157931
$672$	0	0
$673$	−9540.37	−0.546440	−0.273220	−	0.961952i	$-0.588089\pi$
$673$	−9540.37	−0.546440	−0.273220	+	0.961952i	$0.588089\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18584.2	1.05502	0.527509	−	0.849549i	$-0.323126\pi$
$677$	18584.2	1.05502	0.527509	+	0.849549i	$0.323126\pi$
$678$	0	0
$679$	18165.2	1.02668
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−34780.6	−1.94853	−0.974263	−	0.225412i	$-0.927627\pi$
$683$	−34780.6	−1.94853	−0.974263	+	0.225412i	$0.927627\pi$
$684$	0	0
$685$	17732.9	0.989107
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13838.5	0.765172
$690$	0	0
$691$	−25974.2	−1.42996	−0.714981	−	0.699144i	$-0.753565\pi$
$691$	−25974.2	−1.42996	−0.714981	+	0.699144i	$0.753565\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−25092.6	−1.36952
$696$	0	0
$697$	−11629.0	−0.631968
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7381.81	0.397728	0.198864	−	0.980027i	$-0.436275\pi$
$701$	7381.81	0.397728	0.198864	+	0.980027i	$0.436275\pi$
$702$	0	0
$703$	−30382.4	−1.63001
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12042.0	−0.640576
$708$	0	0
$709$	17117.4	0.906708	0.453354	−	0.891330i	$-0.350227\pi$
$709$	17117.4	0.906708	0.453354	+	0.891330i	$0.350227\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1646.67	−0.0864914
$714$	0	0
$715$	−24327.6	−1.27245
$716$	0	0
$717$	0	0
$718$	0	0
$719$	15224.0	0.789649	0.394825	−	0.918757i	$-0.370805\pi$
$719$	15224.0	0.789649	0.394825	+	0.918757i	$0.370805\pi$
$720$	0	0
$721$	8956.82	0.462648
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8090.82	0.414463
$726$	0	0
$727$	2793.74	0.142523	0.0712615	−	0.997458i	$-0.477298\pi$
$727$	2793.74	0.142523	0.0712615	+	0.997458i	$0.477298\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	29030.1	1.46884
$732$	0	0
$733$	11539.7	0.581487	0.290743	−	0.956801i	$-0.406097\pi$
$733$	11539.7	0.581487	0.290743	+	0.956801i	$0.406097\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22650.8	1.13210
$738$	0	0
$739$	−33616.3	−1.67334	−0.836669	−	0.547708i	$-0.815500\pi$
$739$	−33616.3	−1.67334	−0.836669	+	0.547708i	$0.815500\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10879.1	−0.537168	−0.268584	−	0.963256i	$-0.586556\pi$
$743$	−10879.1	−0.537168	−0.268584	+	0.963256i	$0.586556\pi$
$744$	0	0
$745$	−5454.28	−0.268227
$746$	0	0
$747$	0	0
$748$	0	0
$749$	21174.5	1.03298
$750$	0	0
$751$	−25418.5	−1.23506	−0.617532	−	0.786546i	$-0.711867\pi$
$751$	−25418.5	−1.23506	−0.617532	+	0.786546i	$0.711867\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	60362.9	2.90971
$756$	0	0
$757$	21446.1	1.02968	0.514842	−	0.857285i	$-0.327851\pi$
$757$	21446.1	1.02968	0.514842	+	0.857285i	$0.327851\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6008.77	0.286226	0.143113	−	0.989706i	$-0.454289\pi$
$761$	6008.77	0.286226	0.143113	+	0.989706i	$0.454289\pi$
$762$	0	0
$763$	5806.53	0.275505
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11482.5	0.540557
$768$	0	0
$769$	39178.7	1.83722	0.918609	−	0.395167i	$-0.129313\pi$
$769$	39178.7	1.83722	0.918609	+	0.395167i	$0.129313\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	12755.2	0.593497	0.296749	−	0.954956i	$-0.404098\pi$
$773$	12755.2	0.593497	0.296749	+	0.954956i	$0.404098\pi$
$774$	0	0
$775$	2368.12	0.109762
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8838.32	−0.406503
$780$	0	0
$781$	169.947	0.00778638
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−30343.0	−1.37960
$786$	0	0
$787$	−37184.1	−1.68421	−0.842104	−	0.539315i	$-0.818683\pi$
$787$	−37184.1	−1.68421	−0.842104	+	0.539315i	$0.818683\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5168.41	−0.232323
$792$	0	0
$793$	1410.94	0.0631829
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9940.74	−0.441806	−0.220903	−	0.975296i	$-0.570900\pi$
$797$	−9940.74	−0.441806	−0.220903	+	0.975296i	$0.570900\pi$
$798$	0	0
$799$	1529.93	0.0677411
$800$	0	0
$801$	0	0
$802$	0	0
$803$	41724.7	1.83366
$804$	0	0
$805$	−40174.0	−1.75894
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5386.80	0.234104	0.117052	−	0.993126i	$-0.462656\pi$
$809$	5386.80	0.234104	0.117052	+	0.993126i	$0.462656\pi$
$810$	0	0
$811$	−1954.88	−0.0846426	−0.0423213	−	0.999104i	$-0.513475\pi$
$811$	−1954.88	−0.0846426	−0.0423213	+	0.999104i	$0.513475\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−53891.0	−2.31622
$816$	0	0
$817$	22063.5	0.944804
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29588.2	1.25778	0.628889	−	0.777495i	$-0.283510\pi$
$821$	29588.2	1.25778	0.628889	+	0.777495i	$0.283510\pi$
$822$	0	0
$823$	37208.1	1.57593	0.787966	−	0.615718i	$-0.211134\pi$
$823$	37208.1	1.57593	0.787966	+	0.615718i	$0.211134\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7623.10	0.320533	0.160267	−	0.987074i	$-0.448765\pi$
$827$	7623.10	0.320533	0.160267	+	0.987074i	$0.448765\pi$
$828$	0	0
$829$	−28888.0	−1.21028	−0.605140	−	0.796119i	$-0.706883\pi$
$829$	−28888.0	−1.21028	−0.605140	+	0.796119i	$0.706883\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−11119.0	−0.462486
$834$	0	0
$835$	−21339.8	−0.884425
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−46361.7	−1.90773	−0.953864	−	0.300238i	$-0.902934\pi$
$839$	−46361.7	−1.90773	−0.953864	+	0.300238i	$0.902934\pi$
$840$	0	0
$841$	−22912.5	−0.939462
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−27741.3	−1.12938
$846$	0	0
$847$	18764.8	0.761234
$848$	0	0
$849$	0	0
$850$	0	0
$851$	62452.0	2.51566
$852$	0	0
$853$	40597.5	1.62958	0.814790	−	0.579756i	$-0.196852\pi$
$853$	40597.5	1.62958	0.814790	+	0.579756i	$0.196852\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38115.0	−1.51924	−0.759618	−	0.650370i	$-0.774614\pi$
$857$	−38115.0	−1.51924	−0.759618	+	0.650370i	$0.774614\pi$
$858$	0	0
$859$	−28441.7	−1.12971	−0.564853	−	0.825192i	$-0.691067\pi$
$859$	−28441.7	−1.12971	−0.564853	+	0.825192i	$0.691067\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38669.0	−1.52527	−0.762635	−	0.646830i	$-0.776094\pi$
$863$	−38669.0	−1.52527	−0.762635	+	0.646830i	$0.776094\pi$
$864$	0	0
$865$	−68093.3	−2.67658
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6792.81	0.265167
$870$	0	0
$871$	−11642.4	−0.452913
$872$	0	0
$873$	0	0
$874$	0	0
$875$	23477.2	0.907055
$876$	0	0
$877$	−37858.2	−1.45767	−0.728837	−	0.684688i	$-0.759939\pi$
$877$	−37858.2	−1.45767	−0.728837	+	0.684688i	$0.759939\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23867.1	−0.912715	−0.456358	−	0.889796i	$-0.650846\pi$
$881$	−23867.1	−0.912715	−0.456358	+	0.889796i	$0.650846\pi$
$882$	0	0
$883$	36516.3	1.39170	0.695849	−	0.718188i	$-0.255028\pi$
$883$	36516.3	1.39170	0.695849	+	0.718188i	$0.255028\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36631.7	−1.38666	−0.693332	−	0.720618i	$-0.743858\pi$
$887$	−36631.7	−1.38666	−0.693332	+	0.720618i	$0.743858\pi$
$888$	0	0
$889$	30300.5	1.14313
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1162.78	0.0435733
$894$	0	0
$895$	−3488.59	−0.130291
$896$	0	0
$897$	0	0
$898$	0	0
$899$	432.148	0.0160322
$900$	0	0
$901$	−49640.8	−1.83549
$902$	0	0
$903$	0	0
$904$	0	0
$905$	30185.9	1.10875
$906$	0	0
$907$	−13583.2	−0.497268	−0.248634	−	0.968598i	$-0.579982\pi$
$907$	−13583.2	−0.497268	−0.248634	+	0.968598i	$0.579982\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	30841.0	1.12163	0.560817	−	0.827940i	$-0.310487\pi$
$911$	30841.0	1.12163	0.560817	+	0.827940i	$0.310487\pi$
$912$	0	0
$913$	13476.6	0.488512
$914$	0	0
$915$	0	0
$916$	0	0
$917$	24426.6	0.879647
$918$	0	0
$919$	46772.3	1.67886	0.839432	−	0.543464i	$-0.182888\pi$
$919$	46772.3	1.67886	0.839432	+	0.543464i	$0.182888\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−87.3515	−0.00311507
$924$	0	0
$925$	−89813.9	−3.19250
$926$	0	0
$927$	0	0
$928$	0	0
$929$	40663.7	1.43610	0.718048	−	0.695994i	$-0.245036\pi$
$929$	40663.7	1.43610	0.718048	+	0.695994i	$0.245036\pi$
$930$	0	0
$931$	−8450.68	−0.297486
$932$	0	0
$933$	0	0
$934$	0	0
$935$	87266.9	3.05233
$936$	0	0
$937$	19756.8	0.688821	0.344410	−	0.938819i	$-0.388079\pi$
$937$	19756.8	0.688821	0.344410	+	0.938819i	$0.388079\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−50904.4	−1.76348	−0.881740	−	0.471736i	$-0.843628\pi$
$941$	−50904.4	−1.76348	−0.881740	+	0.471736i	$0.843628\pi$
$942$	0	0
$943$	18167.4	0.627373
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−35702.0	−1.22509	−0.612545	−	0.790436i	$-0.709854\pi$
$947$	−35702.0	−1.22509	−0.612545	+	0.790436i	$0.709854\pi$
$948$	0	0
$949$	−21446.2	−0.733587
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−27676.9	−0.940760	−0.470380	−	0.882464i	$-0.655883\pi$
$953$	−27676.9	−0.940760	−0.470380	+	0.882464i	$0.655883\pi$
$954$	0	0
$955$	−32662.6	−1.10674
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14499.9	0.488244
$960$	0	0
$961$	−29664.5	−0.995754
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−88189.6	−2.94189
$966$	0	0
$967$	12951.4	0.430702	0.215351	−	0.976537i	$-0.430911\pi$
$967$	12951.4	0.430702	0.215351	+	0.976537i	$0.430911\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3350.65	0.110739	0.0553694	−	0.998466i	$-0.482366\pi$
$971$	3350.65	0.110739	0.0553694	+	0.998466i	$0.482366\pi$
$972$	0	0
$973$	−20517.8	−0.676023
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8786.22	0.287713	0.143857	−	0.989599i	$-0.454050\pi$
$977$	8786.22	0.287713	0.143857	+	0.989599i	$0.454050\pi$
$978$	0	0
$979$	−64486.0	−2.10519
$980$	0	0
$981$	0	0
$982$	0	0
$983$	18194.5	0.590350	0.295175	−	0.955443i	$-0.404622\pi$
$983$	18194.5	0.590350	0.295175	+	0.955443i	$0.404622\pi$
$984$	0	0
$985$	27885.6	0.902039
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−45352.2	−1.45816
$990$	0	0
$991$	−39390.2	−1.26264	−0.631318	−	0.775524i	$-0.717486\pi$
$991$	−39390.2	−1.26264	−0.631318	+	0.775524i	$0.717486\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−49943.5	−1.59127
$996$	0	0
$997$	15416.5	0.489714	0.244857	−	0.969559i	$-0.421259\pi$
$997$	15416.5	0.489714	0.244857	+	0.969559i	$0.421259\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.a.k.1.5		5
3.2	odd	2		648.4.a.l.1.1		5
4.3	odd	2		1296.4.a.bc.1.5		5
9.2	odd	6		72.4.i.b.49.4	yes	10
9.4	even	3		216.4.i.b.73.1		10
9.5	odd	6		72.4.i.b.25.4	✓	10
9.7	even	3		216.4.i.b.145.1		10
12.11	even	2		1296.4.a.bd.1.1		5
36.7	odd	6		432.4.i.f.145.1		10
36.11	even	6		144.4.i.f.49.2		10
36.23	even	6		144.4.i.f.97.2		10
36.31	odd	6		432.4.i.f.289.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.i.b.25.4	✓	10	9.5	odd	6
72.4.i.b.49.4	yes	10	9.2	odd	6
144.4.i.f.49.2		10	36.11	even	6
144.4.i.f.97.2		10	36.23	even	6
216.4.i.b.73.1		10	9.4	even	3
216.4.i.b.145.1		10	9.7	even	3
432.4.i.f.145.1		10	36.7	odd	6
432.4.i.f.289.1		10	36.31	odd	6
648.4.a.k.1.5		5	1.1	even	1	trivial
648.4.a.l.1.1		5	3.2	odd	2
1296.4.a.bc.1.5		5	4.3	odd	2
1296.4.a.bd.1.1		5	12.11	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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.a (trivial)

\(f(q)\)	\(=\)	\(q+18.3184 q^{5} +14.9787 q^{7} +O(q^{10})\)	\(q+18.3184 q^{5} +14.9787 q^{7} +50.8308 q^{11} -26.1267 q^{13} +93.7208 q^{17} +71.2297 q^{19} -146.415 q^{23} +210.563 q^{25} +38.4247 q^{29} +11.2466 q^{31} +274.385 q^{35} -426.542 q^{37} -124.082 q^{41} +309.751 q^{43} +16.3244 q^{47} -118.640 q^{49} -529.667 q^{53} +931.137 q^{55} -439.491 q^{59} -54.0038 q^{61} -478.599 q^{65} +445.613 q^{67} +3.34338 q^{71} +820.855 q^{73} +761.377 q^{77} +133.636 q^{79} +265.127 q^{83} +1716.81 q^{85} -1268.64 q^{89} -391.343 q^{91} +1304.81 q^{95} +1212.74 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} + 3 q^{7}+O(q^{10}) \) 5 * q - 5 * q^5 + 3 * q^7	\( 5 q - 5 q^{5} + 3 q^{7} + 25 q^{11} + 29 q^{13} - 28 q^{17} + 64 q^{19} - 89 q^{23} + 322 q^{25} - 129 q^{29} + 241 q^{31} + 243 q^{35} + 366 q^{37} - 171 q^{41} + 803 q^{43} + 477 q^{47} + 1072 q^{49} - 374 q^{53} + 1469 q^{55} + 607 q^{59} + 1349 q^{61} - 527 q^{65} + 1549 q^{67} + 812 q^{71} + 1928 q^{73} - 903 q^{77} + 1727 q^{79} + 1025 q^{83} + 2902 q^{85} - 2310 q^{89} + 2985 q^{91} + 2012 q^{95} + 2875 q^{97}+O(q^{100}) \) 5 * q - 5 * q^5 + 3 * q^7 + 25 * q^11 + 29 * q^13 - 28 * q^17 + 64 * q^19 - 89 * q^23 + 322 * q^25 - 129 * q^29 + 241 * q^31 + 243 * q^35 + 366 * q^37 - 171 * q^41 + 803 * q^43 + 477 * q^47 + 1072 * q^49 - 374 * q^53 + 1469 * q^55 + 607 * q^59 + 1349 * q^61 - 527 * q^65 + 1549 * q^67 + 812 * q^71 + 1928 * q^73 - 903 * q^77 + 1727 * q^79 + 1025 * q^83 + 2902 * q^85 - 2310 * q^89 + 2985 * q^91 + 2012 * q^95 + 2875 * q^97

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