LMFDB - Embedded newform 588.4.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,4,Mod(1,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, 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("588.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	588.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:	$34.6931230834$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{193}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 48 $ x^2 - x - 48
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.44622$ of defining polynomial
Character	$\chi$	$=$	588.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +1.44622 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +1.44622 q^{5} +9.00000 q^{9} +46.1236 q^{11} +32.2311 q^{13} -4.33867 q^{15} -77.7849 q^{17} -12.6613 q^{19} -100.924 q^{23} -122.908 q^{25} -27.0000 q^{27} +213.908 q^{29} -42.0756 q^{31} -138.371 q^{33} +310.080 q^{37} -96.6933 q^{39} -44.0320 q^{41} +381.339 q^{43} +13.0160 q^{45} +358.064 q^{47} +233.355 q^{51} +184.984 q^{53} +66.7049 q^{55} +37.9840 q^{57} -454.650 q^{59} -11.8489 q^{61} +46.6133 q^{65} +590.359 q^{67} +302.773 q^{69} +494.366 q^{71} -975.650 q^{73} +368.725 q^{75} +299.334 q^{79} +81.0000 q^{81} +1406.07 q^{83} -112.494 q^{85} -641.725 q^{87} +695.259 q^{89} +126.227 q^{93} -18.3111 q^{95} -481.940 q^{97} +415.112 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 11 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 11 * q^5 + 18 * q^9	$ 2 q - 6 q^{3} - 11 q^{5} + 18 q^{9} - 5 q^{11} - 5 q^{13} + 33 q^{15} - 100 q^{17} - 67 q^{19} + 76 q^{23} - 93 q^{25} - 54 q^{27} + 275 q^{29} - 362 q^{31} + 15 q^{33} - 5 q^{37} + 15 q^{39} + 162 q^{41} + 721 q^{43} - 99 q^{45} + 216 q^{47} + 300 q^{51} + 495 q^{53} + 703 q^{55} + 201 q^{57} - 173 q^{59} + 532 q^{61} + 510 q^{65} + 111 q^{67} - 228 q^{69} + 1600 q^{71} - 1215 q^{73} + 279 q^{75} + 1460 q^{79} + 162 q^{81} + 1409 q^{83} + 164 q^{85} - 825 q^{87} + 1974 q^{89} + 1086 q^{93} + 658 q^{95} - 561 q^{97} - 45 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 11 * q^5 + 18 * q^9 - 5 * q^11 - 5 * q^13 + 33 * q^15 - 100 * q^17 - 67 * q^19 + 76 * q^23 - 93 * q^25 - 54 * q^27 + 275 * q^29 - 362 * q^31 + 15 * q^33 - 5 * q^37 + 15 * q^39 + 162 * q^41 + 721 * q^43 - 99 * q^45 + 216 * q^47 + 300 * q^51 + 495 * q^53 + 703 * q^55 + 201 * q^57 - 173 * q^59 + 532 * q^61 + 510 * q^65 + 111 * q^67 - 228 * q^69 + 1600 * q^71 - 1215 * q^73 + 279 * q^75 + 1460 * q^79 + 162 * q^81 + 1409 * q^83 + 164 * q^85 - 825 * q^87 + 1974 * q^89 + 1086 * q^93 + 658 * q^95 - 561 * q^97 - 45 * 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
$5$	1.44622	0.129354	0.0646770	−	0.997906i	$-0.479398\pi$
$5$	1.44622	0.129354	0.0646770	+	0.997906i	$0.479398\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	46.1236	1.26425	0.632126	−	0.774866i	$-0.282182\pi$
$11$	46.1236	1.26425	0.632126	+	0.774866i	$0.282182\pi$
$12$	0	0
$13$	32.2311	0.687639	0.343819	−	0.939036i	$-0.388279\pi$
$13$	32.2311	0.687639	0.343819	+	0.939036i	$0.388279\pi$
$14$	0	0
$15$	−4.33867	−0.0746826
$16$	0	0
$17$	−77.7849	−1.10974	−0.554871	−	0.831937i	$-0.687232\pi$
$17$	−77.7849	−1.10974	−0.554871	+	0.831937i	$0.687232\pi$
$18$	0	0
$19$	−12.6613	−0.152879	−0.0764397	−	0.997074i	$-0.524355\pi$
$19$	−12.6613	−0.152879	−0.0764397	+	0.997074i	$0.524355\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−100.924	−0.914965	−0.457483	−	0.889219i	$-0.651249\pi$
$23$	−100.924	−0.914965	−0.457483	+	0.889219i	$0.651249\pi$
$24$	0	0
$25$	−122.908	−0.983268
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	213.908	1.36972	0.684859	−	0.728676i	$-0.259864\pi$
$29$	213.908	1.36972	0.684859	+	0.728676i	$0.259864\pi$
$30$	0	0
$31$	−42.0756	−0.243774	−0.121887	−	0.992544i	$-0.538895\pi$
$31$	−42.0756	−0.243774	−0.121887	+	0.992544i	$0.538895\pi$
$32$	0	0
$33$	−138.371	−0.729916
$34$	0	0
$35$	0	0
$36$	0	0
$37$	310.080	1.37775	0.688876	−	0.724879i	$-0.258104\pi$
$37$	310.080	1.37775	0.688876	+	0.724879i	$0.258104\pi$
$38$	0	0
$39$	−96.6933	−0.397008
$40$	0	0
$41$	−44.0320	−0.167723	−0.0838615	−	0.996477i	$-0.526725\pi$
$41$	−44.0320	−0.167723	−0.0838615	+	0.996477i	$0.526725\pi$
$42$	0	0
$43$	381.339	1.35241	0.676205	−	0.736714i	$-0.263624\pi$
$43$	381.339	1.35241	0.676205	+	0.736714i	$0.263624\pi$
$44$	0	0
$45$	13.0160	0.0431180
$46$	0	0
$47$	358.064	1.11126	0.555628	−	0.831431i	$-0.312478\pi$
$47$	358.064	1.11126	0.555628	+	0.831431i	$0.312478\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	233.355	0.640710
$52$	0	0
$53$	184.984	0.479425	0.239712	−	0.970844i	$-0.422947\pi$
$53$	184.984	0.479425	0.239712	+	0.970844i	$0.422947\pi$
$54$	0	0
$55$	66.7049	0.163536
$56$	0	0
$57$	37.9840	0.0882650
$58$	0	0
$59$	−454.650	−1.00323	−0.501613	−	0.865092i	$-0.667260\pi$
$59$	−454.650	−1.00323	−0.501613	+	0.865092i	$0.667260\pi$
$60$	0	0
$61$	−11.8489	−0.0248704	−0.0124352	−	0.999923i	$-0.503958\pi$
$61$	−11.8489	−0.0248704	−0.0124352	+	0.999923i	$0.503958\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	46.6133	0.0889488
$66$	0	0
$67$	590.359	1.07648	0.538238	−	0.842793i	$-0.319090\pi$
$67$	590.359	1.07648	0.538238	+	0.842793i	$0.319090\pi$
$68$	0	0
$69$	302.773	0.528255
$70$	0	0
$71$	494.366	0.826345	0.413172	−	0.910653i	$-0.364421\pi$
$71$	494.366	0.826345	0.413172	+	0.910653i	$0.364421\pi$
$72$	0	0
$73$	−975.650	−1.56426	−0.782131	−	0.623114i	$-0.785867\pi$
$73$	−975.650	−1.56426	−0.782131	+	0.623114i	$0.785867\pi$
$74$	0	0
$75$	368.725	0.567690
$76$	0	0
$77$	0	0
$78$	0	0
$79$	299.334	0.426300	0.213150	−	0.977019i	$-0.431628\pi$
$79$	299.334	0.426300	0.213150	+	0.977019i	$0.431628\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1406.07	1.85947	0.929735	−	0.368229i	$-0.120036\pi$
$83$	1406.07	1.85947	0.929735	+	0.368229i	$0.120036\pi$
$84$	0	0
$85$	−112.494	−0.143550
$86$	0	0
$87$	−641.725	−0.790807
$88$	0	0
$89$	695.259	0.828059	0.414030	−	0.910263i	$-0.364121\pi$
$89$	695.259	0.828059	0.414030	+	0.910263i	$0.364121\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	126.227	0.140743
$94$	0	0
$95$	−18.3111	−0.0197756
$96$	0	0
$97$	−481.940	−0.504470	−0.252235	−	0.967666i	$-0.581166\pi$
$97$	−481.940	−0.504470	−0.252235	+	0.967666i	$0.581166\pi$
$98$	0	0
$99$	415.112	0.421417
$100$	0	0
$101$	1184.10	1.16656	0.583281	−	0.812270i	$-0.301769\pi$
$101$	1184.10	1.16656	0.583281	+	0.812270i	$0.301769\pi$
$102$	0	0
$103$	1283.53	1.22786	0.613932	−	0.789359i	$-0.289587\pi$
$103$	1283.53	1.22786	0.613932	+	0.789359i	$0.289587\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1613.37	1.45767	0.728833	−	0.684692i	$-0.240063\pi$
$107$	1613.37	1.45767	0.728833	+	0.684692i	$0.240063\pi$
$108$	0	0
$109$	−153.833	−0.135179	−0.0675895	−	0.997713i	$-0.521531\pi$
$109$	−153.833	−0.135179	−0.0675895	+	0.997713i	$0.521531\pi$
$110$	0	0
$111$	−930.240	−0.795446
$112$	0	0
$113$	1581.08	1.31625	0.658123	−	0.752910i	$-0.271350\pi$
$113$	1581.08	1.31625	0.658123	+	0.752910i	$0.271350\pi$
$114$	0	0
$115$	−145.959	−0.118354
$116$	0	0
$117$	290.080	0.229213
$118$	0	0
$119$	0	0
$120$	0	0
$121$	796.382	0.598334
$122$	0	0
$123$	132.096	0.0968349
$124$	0	0
$125$	−358.531	−0.256544
$126$	0	0
$127$	1916.30	1.33893	0.669465	−	0.742844i	$-0.266523\pi$
$127$	1916.30	1.33893	0.669465	+	0.742844i	$0.266523\pi$
$128$	0	0
$129$	−1144.02	−0.780814
$130$	0	0
$131$	−2500.48	−1.66770	−0.833849	−	0.551993i	$-0.813867\pi$
$131$	−2500.48	−1.66770	−0.833849	+	0.551993i	$0.813867\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−39.0480	−0.0248942
$136$	0	0
$137$	290.892	0.181406	0.0907030	−	0.995878i	$-0.471089\pi$
$137$	290.892	0.181406	0.0907030	+	0.995878i	$0.471089\pi$
$138$	0	0
$139$	1348.77	0.823028	0.411514	−	0.911403i	$-0.365000\pi$
$139$	1348.77	0.823028	0.411514	+	0.911403i	$0.365000\pi$
$140$	0	0
$141$	−1074.19	−0.641584
$142$	0	0
$143$	1486.61	0.869349
$144$	0	0
$145$	309.359	0.177178
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2848.81	−1.56633	−0.783165	−	0.621814i	$-0.786396\pi$
$149$	−2848.81	−1.56633	−0.783165	+	0.621814i	$0.786396\pi$
$150$	0	0
$151$	−1489.31	−0.802639	−0.401320	−	0.915938i	$-0.631448\pi$
$151$	−1489.31	−0.802639	−0.401320	+	0.915938i	$0.631448\pi$
$152$	0	0
$153$	−700.064	−0.369914
$154$	0	0
$155$	−60.8506	−0.0315331
$156$	0	0
$157$	−3643.38	−1.85206	−0.926030	−	0.377449i	$-0.876801\pi$
$157$	−3643.38	−1.85206	−0.926030	+	0.377449i	$0.876801\pi$
$158$	0	0
$159$	−554.952	−0.276796
$160$	0	0
$161$	0	0
$162$	0	0
$163$	769.163	0.369604	0.184802	−	0.982776i	$-0.440836\pi$
$163$	769.163	0.369604	0.184802	+	0.982776i	$0.440836\pi$
$164$	0	0
$165$	−200.115	−0.0944176
$166$	0	0
$167$	2399.78	1.11198	0.555991	−	0.831188i	$-0.312339\pi$
$167$	2399.78	1.11198	0.555991	+	0.831188i	$0.312339\pi$
$168$	0	0
$169$	−1158.16	−0.527153
$170$	0	0
$171$	−113.952	−0.0509598
$172$	0	0
$173$	3336.65	1.46636	0.733181	−	0.680033i	$-0.238035\pi$
$173$	3336.65	1.46636	0.733181	+	0.680033i	$0.238035\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1363.95	0.579213
$178$	0	0
$179$	2461.77	1.02794	0.513970	−	0.857808i	$-0.328174\pi$
$179$	2461.77	1.02794	0.513970	+	0.857808i	$0.328174\pi$
$180$	0	0
$181$	−1316.74	−0.540732	−0.270366	−	0.962758i	$-0.587145\pi$
$181$	−1316.74	−0.540732	−0.270366	+	0.962758i	$0.587145\pi$
$182$	0	0
$183$	35.5466	0.0143589
$184$	0	0
$185$	448.445	0.178218
$186$	0	0
$187$	−3587.72	−1.40299
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3267.97	−1.23802	−0.619010	−	0.785383i	$-0.712466\pi$
$191$	−3267.97	−1.23802	−0.619010	+	0.785383i	$0.712466\pi$
$192$	0	0
$193$	233.672	0.0871506	0.0435753	−	0.999050i	$-0.486125\pi$
$193$	233.672	0.0871506	0.0435753	+	0.999050i	$0.486125\pi$
$194$	0	0
$195$	−139.840	−0.0513546
$196$	0	0
$197$	−31.8632	−0.0115236	−0.00576182	−	0.999983i	$-0.501834\pi$
$197$	−31.8632	−0.0115236	−0.00576182	+	0.999983i	$0.501834\pi$
$198$	0	0
$199$	1478.90	0.526817	0.263408	−	0.964684i	$-0.415153\pi$
$199$	1478.90	0.526817	0.263408	+	0.964684i	$0.415153\pi$
$200$	0	0
$201$	−1771.08	−0.621503
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−63.6800	−0.0216956
$206$	0	0
$207$	−908.320	−0.304988
$208$	0	0
$209$	−583.986	−0.193278
$210$	0	0
$211$	−4498.67	−1.46778	−0.733889	−	0.679269i	$-0.762297\pi$
$211$	−4498.67	−1.46778	−0.733889	+	0.679269i	$0.762297\pi$
$212$	0	0
$213$	−1483.10	−0.477090
$214$	0	0
$215$	551.500	0.174940
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2926.95	0.903127
$220$	0	0
$221$	−2507.09	−0.763101
$222$	0	0
$223$	−5382.75	−1.61639	−0.808196	−	0.588913i	$-0.799556\pi$
$223$	−5382.75	−1.61639	−0.808196	+	0.588913i	$0.799556\pi$
$224$	0	0
$225$	−1106.18	−0.327756
$226$	0	0
$227$	5425.57	1.58638	0.793188	−	0.608977i	$-0.208420\pi$
$227$	5425.57	1.58638	0.793188	+	0.608977i	$0.208420\pi$
$228$	0	0
$229$	−1989.75	−0.574175	−0.287088	−	0.957904i	$-0.592687\pi$
$229$	−1989.75	−0.574175	−0.287088	+	0.957904i	$0.592687\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6297.66	−1.77070	−0.885351	−	0.464924i	$-0.846082\pi$
$233$	−6297.66	−1.77070	−0.885351	+	0.464924i	$0.846082\pi$
$234$	0	0
$235$	517.840	0.143745
$236$	0	0
$237$	−898.003	−0.246125
$238$	0	0
$239$	3395.77	0.919054	0.459527	−	0.888164i	$-0.348019\pi$
$239$	3395.77	0.919054	0.459527	+	0.888164i	$0.348019\pi$
$240$	0	0
$241$	6373.94	1.70366	0.851829	−	0.523820i	$-0.175494\pi$
$241$	6373.94	1.70366	0.851829	+	0.523820i	$0.175494\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−408.089	−0.105126
$248$	0	0
$249$	−4218.21	−1.07357
$250$	0	0
$251$	2650.91	0.666630	0.333315	−	0.942815i	$-0.391833\pi$
$251$	2650.91	0.666630	0.333315	+	0.942815i	$0.391833\pi$
$252$	0	0
$253$	−4654.99	−1.15675
$254$	0	0
$255$	337.483	0.0828784
$256$	0	0
$257$	1872.07	0.454383	0.227192	−	0.973850i	$-0.427046\pi$
$257$	1872.07	0.454383	0.227192	+	0.973850i	$0.427046\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1925.18	0.456572
$262$	0	0
$263$	6126.10	1.43632	0.718158	−	0.695880i	$-0.244985\pi$
$263$	6126.10	1.43632	0.718158	+	0.695880i	$0.244985\pi$
$264$	0	0
$265$	267.528	0.0620155
$266$	0	0
$267$	−2085.78	−0.478080
$268$	0	0
$269$	8504.12	1.92753	0.963764	−	0.266755i	$-0.0859515\pi$
$269$	8504.12	1.92753	0.963764	+	0.266755i	$0.0859515\pi$
$270$	0	0
$271$	−2121.54	−0.475552	−0.237776	−	0.971320i	$-0.576418\pi$
$271$	−2121.54	−0.475552	−0.237776	+	0.971320i	$0.576418\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5668.97	−1.24310
$276$	0	0
$277$	−8021.37	−1.73992	−0.869959	−	0.493125i	$-0.835855\pi$
$277$	−8021.37	−1.73992	−0.869959	+	0.493125i	$0.835855\pi$
$278$	0	0
$279$	−378.680	−0.0812580
$280$	0	0
$281$	−8244.17	−1.75020	−0.875100	−	0.483943i	$-0.839204\pi$
$281$	−8244.17	−1.75020	−0.875100	+	0.483943i	$0.839204\pi$
$282$	0	0
$283$	−6101.28	−1.28157	−0.640784	−	0.767722i	$-0.721390\pi$
$283$	−6101.28	−1.28157	−0.640784	+	0.767722i	$0.721390\pi$
$284$	0	0
$285$	54.9333	0.0114174
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1137.49	0.231526
$290$	0	0
$291$	1445.82	0.291256
$292$	0	0
$293$	1965.98	0.391993	0.195996	−	0.980605i	$-0.437206\pi$
$293$	1965.98	0.391993	0.195996	+	0.980605i	$0.437206\pi$
$294$	0	0
$295$	−657.524	−0.129771
$296$	0	0
$297$	−1245.34	−0.243305
$298$	0	0
$299$	−3252.91	−0.629165
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−3552.31	−0.673515
$304$	0	0
$305$	−17.1361	−0.00321708
$306$	0	0
$307$	−997.810	−0.185498	−0.0927492	−	0.995690i	$-0.529565\pi$
$307$	−997.810	−0.185498	−0.0927492	+	0.995690i	$0.529565\pi$
$308$	0	0
$309$	−3850.59	−0.708908
$310$	0	0
$311$	6901.43	1.25834	0.629171	−	0.777267i	$-0.283395\pi$
$311$	6901.43	1.25834	0.629171	+	0.777267i	$0.283395\pi$
$312$	0	0
$313$	6341.71	1.14522	0.572612	−	0.819827i	$-0.305930\pi$
$313$	6341.71	1.14522	0.572612	+	0.819827i	$0.305930\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1224.97	0.217039	0.108519	−	0.994094i	$-0.465389\pi$
$317$	1224.97	0.217039	0.108519	+	0.994094i	$0.465389\pi$
$318$	0	0
$319$	9866.22	1.73167
$320$	0	0
$321$	−4840.10	−0.841583
$322$	0	0
$323$	984.860	0.169657
$324$	0	0
$325$	−3961.48	−0.676133
$326$	0	0
$327$	461.499	0.0780457
$328$	0	0
$329$	0	0
$330$	0	0
$331$	3736.30	0.620440	0.310220	−	0.950665i	$-0.399597\pi$
$331$	3736.30	0.620440	0.310220	+	0.950665i	$0.399597\pi$
$332$	0	0
$333$	2790.72	0.459251
$334$	0	0
$335$	853.790	0.139246
$336$	0	0
$337$	−3928.18	−0.634960	−0.317480	−	0.948265i	$-0.602837\pi$
$337$	−3928.18	−0.634960	−0.317480	+	0.948265i	$0.602837\pi$
$338$	0	0
$339$	−4743.25	−0.759936
$340$	0	0
$341$	−1940.67	−0.308192
$342$	0	0
$343$	0	0
$344$	0	0
$345$	437.877	0.0683320
$346$	0	0
$347$	542.005	0.0838512	0.0419256	−	0.999121i	$-0.486651\pi$
$347$	542.005	0.0838512	0.0419256	+	0.999121i	$0.486651\pi$
$348$	0	0
$349$	−2331.24	−0.357561	−0.178780	−	0.983889i	$-0.557215\pi$
$349$	−2331.24	−0.357561	−0.178780	+	0.983889i	$0.557215\pi$
$350$	0	0
$351$	−870.240	−0.132336
$352$	0	0
$353$	2560.30	0.386037	0.193018	−	0.981195i	$-0.438172\pi$
$353$	2560.30	0.386037	0.193018	+	0.981195i	$0.438172\pi$
$354$	0	0
$355$	714.963	0.106891
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4568.53	−0.671638	−0.335819	−	0.941927i	$-0.609013\pi$
$359$	−4568.53	−0.671638	−0.335819	+	0.941927i	$0.609013\pi$
$360$	0	0
$361$	−6698.69	−0.976628
$362$	0	0
$363$	−2389.15	−0.345448
$364$	0	0
$365$	−1411.01	−0.202344
$366$	0	0
$367$	6381.83	0.907707	0.453854	−	0.891076i	$-0.350049\pi$
$367$	6381.83	0.907707	0.453854	+	0.891076i	$0.350049\pi$
$368$	0	0
$369$	−396.288	−0.0559077
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6372.01	−0.884532	−0.442266	−	0.896884i	$-0.645825\pi$
$373$	−6372.01	−0.884532	−0.442266	+	0.896884i	$0.645825\pi$
$374$	0	0
$375$	1075.59	0.148116
$376$	0	0
$377$	6894.51	0.941870
$378$	0	0
$379$	1494.59	0.202564	0.101282	−	0.994858i	$-0.467706\pi$
$379$	1494.59	0.202564	0.101282	+	0.994858i	$0.467706\pi$
$380$	0	0
$381$	−5748.90	−0.773031
$382$	0	0
$383$	−8820.26	−1.17675	−0.588374	−	0.808589i	$-0.700232\pi$
$383$	−8820.26	−1.17675	−0.588374	+	0.808589i	$0.700232\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3432.05	0.450803
$388$	0	0
$389$	−9878.47	−1.28755	−0.643777	−	0.765213i	$-0.722634\pi$
$389$	−9878.47	−1.28755	−0.643777	+	0.765213i	$0.722634\pi$
$390$	0	0
$391$	7850.40	1.01537
$392$	0	0
$393$	7501.45	0.962845
$394$	0	0
$395$	432.904	0.0551437
$396$	0	0
$397$	−1941.80	−0.245481	−0.122740	−	0.992439i	$-0.539168\pi$
$397$	−1941.80	−0.245481	−0.122740	+	0.992439i	$0.539168\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7584.96	0.944576	0.472288	−	0.881444i	$-0.343428\pi$
$401$	7584.96	0.944576	0.472288	+	0.881444i	$0.343428\pi$
$402$	0	0
$403$	−1356.14	−0.167628
$404$	0	0
$405$	117.144	0.0143727
$406$	0	0
$407$	14302.0	1.74183
$408$	0	0
$409$	−8707.80	−1.05275	−0.526373	−	0.850254i	$-0.676448\pi$
$409$	−8707.80	−1.05275	−0.526373	+	0.850254i	$0.676448\pi$
$410$	0	0
$411$	−872.677	−0.104735
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2033.49	0.240530
$416$	0	0
$417$	−4046.30	−0.475175
$418$	0	0
$419$	−6647.96	−0.775117	−0.387558	−	0.921845i	$-0.626681\pi$
$419$	−6647.96	−0.775117	−0.387558	+	0.921845i	$0.626681\pi$
$420$	0	0
$421$	11670.6	1.35105	0.675524	−	0.737338i	$-0.263917\pi$
$421$	11670.6	1.35105	0.675524	+	0.737338i	$0.263917\pi$
$422$	0	0
$423$	3222.58	0.370418
$424$	0	0
$425$	9560.42	1.09117
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4459.84	−0.501919
$430$	0	0
$431$	−1565.64	−0.174975	−0.0874876	−	0.996166i	$-0.527884\pi$
$431$	−1565.64	−0.174975	−0.0874876	+	0.996166i	$0.527884\pi$
$432$	0	0
$433$	−15446.4	−1.71434	−0.857168	−	0.515037i	$-0.827778\pi$
$433$	−15446.4	−1.71434	−0.857168	+	0.515037i	$0.827778\pi$
$434$	0	0
$435$	−928.077	−0.102294
$436$	0	0
$437$	1277.84	0.139879
$438$	0	0
$439$	−10697.4	−1.16300	−0.581502	−	0.813545i	$-0.697535\pi$
$439$	−10697.4	−1.16300	−0.581502	+	0.813545i	$0.697535\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1053.96	−0.113037	−0.0565183	−	0.998402i	$-0.518000\pi$
$443$	−1053.96	−0.113037	−0.0565183	+	0.998402i	$0.518000\pi$
$444$	0	0
$445$	1005.50	0.107113
$446$	0	0
$447$	8546.42	0.904321
$448$	0	0
$449$	1139.33	0.119752	0.0598759	−	0.998206i	$-0.480930\pi$
$449$	1139.33	0.119752	0.0598759	+	0.998206i	$0.480930\pi$
$450$	0	0
$451$	−2030.91	−0.212044
$452$	0	0
$453$	4467.94	0.463404
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5704.16	0.583872	0.291936	−	0.956438i	$-0.405701\pi$
$457$	5704.16	0.583872	0.291936	+	0.956438i	$0.405701\pi$
$458$	0	0
$459$	2100.19	0.213570
$460$	0	0
$461$	6476.39	0.654307	0.327154	−	0.944971i	$-0.393911\pi$
$461$	6476.39	0.654307	0.327154	+	0.944971i	$0.393911\pi$
$462$	0	0
$463$	−232.366	−0.0233239	−0.0116619	−	0.999932i	$-0.503712\pi$
$463$	−232.366	−0.0233239	−0.0116619	+	0.999932i	$0.503712\pi$
$464$	0	0
$465$	182.552	0.0182057
$466$	0	0
$467$	1037.17	0.102772	0.0513858	−	0.998679i	$-0.483636\pi$
$467$	1037.17	0.102772	0.0513858	+	0.998679i	$0.483636\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	10930.1	1.06929
$472$	0	0
$473$	17588.7	1.70979
$474$	0	0
$475$	1556.18	0.150321
$476$	0	0
$477$	1664.86	0.159808
$478$	0	0
$479$	−10284.6	−0.981039	−0.490519	−	0.871430i	$-0.663193\pi$
$479$	−10284.6	−0.981039	−0.490519	+	0.871430i	$0.663193\pi$
$480$	0	0
$481$	9994.22	0.947396
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−696.993	−0.0652553
$486$	0	0
$487$	10845.8	1.00918	0.504591	−	0.863359i	$-0.331643\pi$
$487$	10845.8	1.00918	0.504591	+	0.863359i	$0.331643\pi$
$488$	0	0
$489$	−2307.49	−0.213391
$490$	0	0
$491$	8442.11	0.775941	0.387971	−	0.921672i	$-0.373176\pi$
$491$	8442.11	0.775941	0.387971	+	0.921672i	$0.373176\pi$
$492$	0	0
$493$	−16638.8	−1.52003
$494$	0	0
$495$	600.344	0.0545120
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9810.85	0.880148	0.440074	−	0.897961i	$-0.354952\pi$
$499$	9810.85	0.880148	0.440074	+	0.897961i	$0.354952\pi$
$500$	0	0
$501$	−7199.35	−0.642003
$502$	0	0
$503$	6433.96	0.570330	0.285165	−	0.958478i	$-0.407952\pi$
$503$	6433.96	0.570330	0.285165	+	0.958478i	$0.407952\pi$
$504$	0	0
$505$	1712.48	0.150900
$506$	0	0
$507$	3474.47	0.304352
$508$	0	0
$509$	20560.6	1.79044	0.895220	−	0.445624i	$-0.147018\pi$
$509$	20560.6	1.79044	0.895220	+	0.445624i	$0.147018\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	341.856	0.0294217
$514$	0	0
$515$	1856.27	0.158829
$516$	0	0
$517$	16515.2	1.40491
$518$	0	0
$519$	−10009.9	−0.846605
$520$	0	0
$521$	−19126.1	−1.60831	−0.804156	−	0.594418i	$-0.797383\pi$
$521$	−19126.1	−1.60831	−0.804156	+	0.594418i	$0.797383\pi$
$522$	0	0
$523$	3044.66	0.254558	0.127279	−	0.991867i	$-0.459376\pi$
$523$	3044.66	0.254558	0.127279	+	0.991867i	$0.459376\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3272.84	0.270526
$528$	0	0
$529$	−1981.26	−0.162839
$530$	0	0
$531$	−4091.85	−0.334409
$532$	0	0
$533$	−1419.20	−0.115333
$534$	0	0
$535$	2333.29	0.188555
$536$	0	0
$537$	−7385.31	−0.593482
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5895.09	−0.468484	−0.234242	−	0.972178i	$-0.575261\pi$
$541$	−5895.09	−0.468484	−0.234242	+	0.972178i	$0.575261\pi$
$542$	0	0
$543$	3950.22	0.312192
$544$	0	0
$545$	−222.476	−0.0174860
$546$	0	0
$547$	11151.1	0.871641	0.435821	−	0.900034i	$-0.356458\pi$
$547$	11151.1	0.871641	0.435821	+	0.900034i	$0.356458\pi$
$548$	0	0
$549$	−106.640	−0.00829013
$550$	0	0
$551$	−2708.37	−0.209402
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1345.33	−0.102894
$556$	0	0
$557$	1744.35	0.132694	0.0663468	−	0.997797i	$-0.478866\pi$
$557$	1744.35	0.132694	0.0663468	+	0.997797i	$0.478866\pi$
$558$	0	0
$559$	12291.0	0.929969
$560$	0	0
$561$	10763.1	0.810019
$562$	0	0
$563$	−11524.4	−0.862691	−0.431345	−	0.902187i	$-0.641961\pi$
$563$	−11524.4	−0.862691	−0.431345	+	0.902187i	$0.641961\pi$
$564$	0	0
$565$	2286.60	0.170262
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2822.76	0.207973	0.103986	−	0.994579i	$-0.466840\pi$
$569$	2822.76	0.207973	0.103986	+	0.994579i	$0.466840\pi$
$570$	0	0
$571$	−5167.64	−0.378737	−0.189369	−	0.981906i	$-0.560644\pi$
$571$	−5167.64	−0.378737	−0.189369	+	0.981906i	$0.560644\pi$
$572$	0	0
$573$	9803.90	0.714771
$574$	0	0
$575$	12404.5	0.899656
$576$	0	0
$577$	14715.2	1.06170	0.530851	−	0.847465i	$-0.321872\pi$
$577$	14715.2	1.06170	0.530851	+	0.847465i	$0.321872\pi$
$578$	0	0
$579$	−701.016	−0.0503164
$580$	0	0
$581$	0	0
$582$	0	0
$583$	8532.12	0.606114
$584$	0	0
$585$	419.520	0.0296496
$586$	0	0
$587$	−9981.64	−0.701851	−0.350925	−	0.936403i	$-0.614133\pi$
$587$	−9981.64	−0.701851	−0.350925	+	0.936403i	$0.614133\pi$
$588$	0	0
$589$	532.733	0.0372680
$590$	0	0
$591$	95.5895	0.00665317
$592$	0	0
$593$	1675.21	0.116008	0.0580039	−	0.998316i	$-0.481526\pi$
$593$	1675.21	0.116008	0.0580039	+	0.998316i	$0.481526\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4436.70	−0.304158
$598$	0	0
$599$	1039.91	0.0709341	0.0354671	−	0.999371i	$-0.488708\pi$
$599$	1039.91	0.0709341	0.0354671	+	0.999371i	$0.488708\pi$
$600$	0	0
$601$	−4472.61	−0.303563	−0.151782	−	0.988414i	$-0.548501\pi$
$601$	−4472.61	−0.303563	−0.151782	+	0.988414i	$0.548501\pi$
$602$	0	0
$603$	5313.23	0.358825
$604$	0	0
$605$	1151.75	0.0773969
$606$	0	0
$607$	17790.8	1.18963	0.594814	−	0.803863i	$-0.297226\pi$
$607$	17790.8	1.18963	0.594814	+	0.803863i	$0.297226\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11540.8	0.764142
$612$	0	0
$613$	−4326.74	−0.285082	−0.142541	−	0.989789i	$-0.545527\pi$
$613$	−4326.74	−0.285082	−0.142541	+	0.989789i	$0.545527\pi$
$614$	0	0
$615$	191.040	0.0125260
$616$	0	0
$617$	−18866.2	−1.23100	−0.615498	−	0.788139i	$-0.711045\pi$
$617$	−18866.2	−1.23100	−0.615498	+	0.788139i	$0.711045\pi$
$618$	0	0
$619$	10179.4	0.660975	0.330488	−	0.943810i	$-0.392787\pi$
$619$	10179.4	0.660975	0.330488	+	0.943810i	$0.392787\pi$
$620$	0	0
$621$	2724.96	0.176085
$622$	0	0
$623$	0	0
$624$	0	0
$625$	14845.0	0.950083
$626$	0	0
$627$	1751.96	0.111589
$628$	0	0
$629$	−24119.5	−1.52895
$630$	0	0
$631$	−1661.72	−0.104837	−0.0524184	−	0.998625i	$-0.516693\pi$
$631$	−1661.72	−0.104837	−0.0524184	+	0.998625i	$0.516693\pi$
$632$	0	0
$633$	13496.0	0.847423
$634$	0	0
$635$	2771.39	0.173196
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4449.30	0.275448
$640$	0	0
$641$	−2277.64	−0.140346	−0.0701728	−	0.997535i	$-0.522355\pi$
$641$	−2277.64	−0.140346	−0.0701728	+	0.997535i	$0.522355\pi$
$642$	0	0
$643$	−1217.38	−0.0746638	−0.0373319	−	0.999303i	$-0.511886\pi$
$643$	−1217.38	−0.0746638	−0.0373319	+	0.999303i	$0.511886\pi$
$644$	0	0
$645$	−1654.50	−0.101001
$646$	0	0
$647$	−10731.8	−0.652103	−0.326052	−	0.945352i	$-0.605718\pi$
$647$	−10731.8	−0.652103	−0.326052	+	0.945352i	$0.605718\pi$
$648$	0	0
$649$	−20970.1	−1.26833
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15228.1	0.912588	0.456294	−	0.889829i	$-0.349177\pi$
$653$	15228.1	0.912588	0.456294	+	0.889829i	$0.349177\pi$
$654$	0	0
$655$	−3616.26	−0.215723
$656$	0	0
$657$	−8780.85	−0.521421
$658$	0	0
$659$	10590.8	0.626038	0.313019	−	0.949747i	$-0.398660\pi$
$659$	10590.8	0.626038	0.313019	+	0.949747i	$0.398660\pi$
$660$	0	0
$661$	3868.30	0.227624	0.113812	−	0.993502i	$-0.463694\pi$
$661$	3868.30	0.227624	0.113812	+	0.993502i	$0.463694\pi$
$662$	0	0
$663$	7521.28	0.440577
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−21588.6	−1.25324
$668$	0	0
$669$	16148.2	0.933225
$670$	0	0
$671$	−546.512	−0.0314424
$672$	0	0
$673$	11028.7	0.631687	0.315843	−	0.948811i	$-0.397713\pi$
$673$	11028.7	0.631687	0.315843	+	0.948811i	$0.397713\pi$
$674$	0	0
$675$	3318.53	0.189230
$676$	0	0
$677$	−18792.5	−1.06684	−0.533422	−	0.845849i	$-0.679094\pi$
$677$	−18792.5	−1.06684	−0.533422	+	0.845849i	$0.679094\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−16276.7	−0.915895
$682$	0	0
$683$	−20243.8	−1.13412	−0.567062	−	0.823675i	$-0.691920\pi$
$683$	−20243.8	−1.13412	−0.567062	+	0.823675i	$0.691920\pi$
$684$	0	0
$685$	420.695	0.0234656
$686$	0	0
$687$	5969.24	0.331500
$688$	0	0
$689$	5962.24	0.329671
$690$	0	0
$691$	19202.9	1.05718	0.528591	−	0.848877i	$-0.322721\pi$
$691$	19202.9	1.05718	0.528591	+	0.848877i	$0.322721\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1950.62	0.106462
$696$	0	0
$697$	3425.02	0.186129
$698$	0	0
$699$	18893.0	1.02231
$700$	0	0
$701$	−22156.9	−1.19380	−0.596900	−	0.802316i	$-0.703601\pi$
$701$	−22156.9	−1.19380	−0.596900	+	0.802316i	$0.703601\pi$
$702$	0	0
$703$	−3926.03	−0.210630
$704$	0	0
$705$	−1553.52	−0.0829914
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−27209.1	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$709$	−27209.1	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$710$	0	0
$711$	2694.01	0.142100
$712$	0	0
$713$	4246.45	0.223045
$714$	0	0
$715$	2149.97	0.112454
$716$	0	0
$717$	−10187.3	−0.530616
$718$	0	0
$719$	−21552.3	−1.11789	−0.558947	−	0.829203i	$-0.688794\pi$
$719$	−21552.3	−1.11789	−0.558947	+	0.829203i	$0.688794\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−19121.8	−0.983607
$724$	0	0
$725$	−26291.2	−1.34680
$726$	0	0
$727$	20599.1	1.05086	0.525431	−	0.850836i	$-0.323904\pi$
$727$	20599.1	1.05086	0.525431	+	0.850836i	$0.323904\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−29662.4	−1.50082
$732$	0	0
$733$	−19761.2	−0.995764	−0.497882	−	0.867245i	$-0.665889\pi$
$733$	−19761.2	−0.995764	−0.497882	+	0.867245i	$0.665889\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	27229.5	1.36094
$738$	0	0
$739$	33408.9	1.66301	0.831506	−	0.555516i	$-0.187479\pi$
$739$	33408.9	1.66301	0.831506	+	0.555516i	$0.187479\pi$
$740$	0	0
$741$	1224.27	0.0606944
$742$	0	0
$743$	−36225.4	−1.78867	−0.894335	−	0.447398i	$-0.852351\pi$
$743$	−36225.4	−1.78867	−0.894335	+	0.447398i	$0.852351\pi$
$744$	0	0
$745$	−4120.00	−0.202611
$746$	0	0
$747$	12654.6	0.619824
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−27320.4	−1.32748	−0.663740	−	0.747964i	$-0.731032\pi$
$751$	−27320.4	−1.32748	−0.663740	+	0.747964i	$0.731032\pi$
$752$	0	0
$753$	−7952.74	−0.384879
$754$	0	0
$755$	−2153.88	−0.103825
$756$	0	0
$757$	−9918.43	−0.476211	−0.238105	−	0.971239i	$-0.576526\pi$
$757$	−9918.43	−0.476211	−0.238105	+	0.971239i	$0.576526\pi$
$758$	0	0
$759$	13965.0	0.667848
$760$	0	0
$761$	5635.04	0.268423	0.134211	−	0.990953i	$-0.457150\pi$
$761$	5635.04	0.268423	0.134211	+	0.990953i	$0.457150\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1012.45	−0.0478498
$766$	0	0
$767$	−14653.9	−0.689857
$768$	0	0
$769$	−12089.3	−0.566907	−0.283453	−	0.958986i	$-0.591480\pi$
$769$	−12089.3	−0.566907	−0.283453	+	0.958986i	$0.591480\pi$
$770$	0	0
$771$	−5616.21	−0.262338
$772$	0	0
$773$	14171.4	0.659390	0.329695	−	0.944087i	$-0.393054\pi$
$773$	14171.4	0.659390	0.329695	+	0.944087i	$0.393054\pi$
$774$	0	0
$775$	5171.44	0.239695
$776$	0	0
$777$	0	0
$778$	0	0
$779$	557.504	0.0256414
$780$	0	0
$781$	22801.9	1.04471
$782$	0	0
$783$	−5775.53	−0.263602
$784$	0	0
$785$	−5269.14	−0.239571
$786$	0	0
$787$	25707.6	1.16439	0.582197	−	0.813047i	$-0.302193\pi$
$787$	25707.6	1.16439	0.582197	+	0.813047i	$0.302193\pi$
$788$	0	0
$789$	−18378.3	−0.829258
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−381.903	−0.0171018
$794$	0	0
$795$	−802.584	−0.0358047
$796$	0	0
$797$	6194.68	0.275316	0.137658	−	0.990480i	$-0.456043\pi$
$797$	6194.68	0.275316	0.137658	+	0.990480i	$0.456043\pi$
$798$	0	0
$799$	−27852.0	−1.23321
$800$	0	0
$801$	6257.33	0.276020
$802$	0	0
$803$	−45000.4	−1.97762
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−25512.4	−1.11286
$808$	0	0
$809$	30815.1	1.33919	0.669593	−	0.742729i	$-0.266469\pi$
$809$	30815.1	1.33919	0.669593	+	0.742729i	$0.266469\pi$
$810$	0	0
$811$	−43024.1	−1.86286	−0.931431	−	0.363917i	$-0.881439\pi$
$811$	−43024.1	−1.86286	−0.931431	+	0.363917i	$0.881439\pi$
$812$	0	0
$813$	6364.63	0.274560
$814$	0	0
$815$	1112.38	0.0478098
$816$	0	0
$817$	−4828.26	−0.206756
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−11152.2	−0.474075	−0.237038	−	0.971500i	$-0.576176\pi$
$821$	−11152.2	−0.474075	−0.237038	+	0.971500i	$0.576176\pi$
$822$	0	0
$823$	−44912.8	−1.90226	−0.951130	−	0.308790i	$-0.900076\pi$
$823$	−44912.8	−1.90226	−0.951130	+	0.308790i	$0.900076\pi$
$824$	0	0
$825$	17006.9	0.717703
$826$	0	0
$827$	−3213.42	−0.135117	−0.0675584	−	0.997715i	$-0.521521\pi$
$827$	−3213.42	−0.135117	−0.0675584	+	0.997715i	$0.521521\pi$
$828$	0	0
$829$	8794.91	0.368468	0.184234	−	0.982882i	$-0.441020\pi$
$829$	8794.91	0.368468	0.184234	+	0.982882i	$0.441020\pi$
$830$	0	0
$831$	24064.1	1.00454
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3470.62	0.143839
$836$	0	0
$837$	1136.04	0.0469143
$838$	0	0
$839$	−45817.0	−1.88532	−0.942658	−	0.333761i	$-0.891682\pi$
$839$	−45817.0	−1.88532	−0.942658	+	0.333761i	$0.891682\pi$
$840$	0	0
$841$	21367.8	0.876125
$842$	0	0
$843$	24732.5	1.01048
$844$	0	0
$845$	−1674.95	−0.0681894
$846$	0	0
$847$	0	0
$848$	0	0
$849$	18303.8	0.739913
$850$	0	0
$851$	−31294.6	−1.26060
$852$	0	0
$853$	16373.6	0.657234	0.328617	−	0.944463i	$-0.393417\pi$
$853$	16373.6	0.657234	0.328617	+	0.944463i	$0.393417\pi$
$854$	0	0
$855$	−164.800	−0.00659186
$856$	0	0
$857$	4430.20	0.176584	0.0882922	−	0.996095i	$-0.471859\pi$
$857$	4430.20	0.176584	0.0882922	+	0.996095i	$0.471859\pi$
$858$	0	0
$859$	−4944.39	−0.196392	−0.0981958	−	0.995167i	$-0.531307\pi$
$859$	−4944.39	−0.196392	−0.0981958	+	0.995167i	$0.531307\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	13178.6	0.519821	0.259911	−	0.965633i	$-0.416307\pi$
$863$	13178.6	0.519821	0.259911	+	0.965633i	$0.416307\pi$
$864$	0	0
$865$	4825.53	0.189680
$866$	0	0
$867$	−3412.47	−0.133672
$868$	0	0
$869$	13806.4	0.538951
$870$	0	0
$871$	19027.9	0.740226
$872$	0	0
$873$	−4337.46	−0.168157
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12985.9	−0.500003	−0.250001	−	0.968245i	$-0.580431\pi$
$877$	−12985.9	−0.500003	−0.250001	+	0.968245i	$0.580431\pi$
$878$	0	0
$879$	−5897.94	−0.226317
$880$	0	0
$881$	−36877.4	−1.41025	−0.705126	−	0.709082i	$-0.749110\pi$
$881$	−36877.4	−1.41025	−0.705126	+	0.709082i	$0.749110\pi$
$882$	0	0
$883$	−24874.5	−0.948012	−0.474006	−	0.880522i	$-0.657192\pi$
$883$	−24874.5	−0.948012	−0.474006	+	0.880522i	$0.657192\pi$
$884$	0	0
$885$	1972.57	0.0749235
$886$	0	0
$887$	−4410.64	−0.166961	−0.0834806	−	0.996509i	$-0.526604\pi$
$887$	−4410.64	−0.166961	−0.0834806	+	0.996509i	$0.526604\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3736.01	0.140472
$892$	0	0
$893$	−4533.57	−0.169888
$894$	0	0
$895$	3560.27	0.132968
$896$	0	0
$897$	9758.72	0.363249
$898$	0	0
$899$	−9000.32	−0.333901
$900$	0	0
$901$	−14389.0	−0.532037
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1904.30	−0.0699459
$906$	0	0
$907$	−7273.99	−0.266294	−0.133147	−	0.991096i	$-0.542508\pi$
$907$	−7273.99	−0.266294	−0.133147	+	0.991096i	$0.542508\pi$
$908$	0	0
$909$	10656.9	0.388854
$910$	0	0
$911$	−49491.9	−1.79993	−0.899967	−	0.435957i	$-0.856410\pi$
$911$	−49491.9	−1.79993	−0.899967	+	0.435957i	$0.856410\pi$
$912$	0	0
$913$	64852.9	2.35084
$914$	0	0
$915$	51.4083	0.00185738
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−5015.90	−0.180043	−0.0900213	−	0.995940i	$-0.528694\pi$
$919$	−5015.90	−0.180043	−0.0900213	+	0.995940i	$0.528694\pi$
$920$	0	0
$921$	2993.43	0.107098
$922$	0	0
$923$	15934.0	0.568227
$924$	0	0
$925$	−38111.4	−1.35470
$926$	0	0
$927$	11551.8	0.409288
$928$	0	0
$929$	52158.2	1.84204	0.921021	−	0.389514i	$-0.127357\pi$
$929$	52158.2	1.84204	0.921021	+	0.389514i	$0.127357\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−20704.3	−0.726504
$934$	0	0
$935$	−5188.63	−0.181483
$936$	0	0
$937$	14821.9	0.516766	0.258383	−	0.966042i	$-0.416810\pi$
$937$	14821.9	0.516766	0.258383	+	0.966042i	$0.416810\pi$
$938$	0	0
$939$	−19025.1	−0.661195
$940$	0	0
$941$	−26492.3	−0.917772	−0.458886	−	0.888495i	$-0.651751\pi$
$941$	−26492.3	−0.917772	−0.458886	+	0.888495i	$0.651751\pi$
$942$	0	0
$943$	4443.90	0.153461
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6410.25	0.219963	0.109982	−	0.993934i	$-0.464921\pi$
$947$	6410.25	0.219963	0.109982	+	0.993934i	$0.464921\pi$
$948$	0	0
$949$	−31446.3	−1.07565
$950$	0	0
$951$	−3674.92	−0.125307
$952$	0	0
$953$	−25108.5	−0.853458	−0.426729	−	0.904380i	$-0.640334\pi$
$953$	−25108.5	−0.853458	−0.426729	+	0.904380i	$0.640334\pi$
$954$	0	0
$955$	−4726.21	−0.160143
$956$	0	0
$957$	−29598.7	−0.999779
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28020.6	−0.940574
$962$	0	0
$963$	14520.3	0.485888
$964$	0	0
$965$	337.941	0.0112733
$966$	0	0
$967$	32928.1	1.09503	0.547516	−	0.836795i	$-0.315574\pi$
$967$	32928.1	1.09503	0.547516	+	0.836795i	$0.315574\pi$
$968$	0	0
$969$	−2954.58	−0.0979513
$970$	0	0
$971$	31543.2	1.04250	0.521251	−	0.853403i	$-0.325465\pi$
$971$	31543.2	1.04250	0.521251	+	0.853403i	$0.325465\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	11884.4	0.390365
$976$	0	0
$977$	53881.8	1.76441	0.882206	−	0.470864i	$-0.156058\pi$
$977$	53881.8	1.76441	0.882206	+	0.470864i	$0.156058\pi$
$978$	0	0
$979$	32067.8	1.04688
$980$	0	0
$981$	−1384.50	−0.0450597
$982$	0	0
$983$	−35688.6	−1.15798	−0.578988	−	0.815336i	$-0.696552\pi$
$983$	−35688.6	−1.15798	−0.578988	+	0.815336i	$0.696552\pi$
$984$	0	0
$985$	−46.0812	−0.00149063
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−38486.4	−1.23741
$990$	0	0
$991$	14839.7	0.475681	0.237840	−	0.971304i	$-0.423561\pi$
$991$	14839.7	0.475681	0.237840	+	0.971304i	$0.423561\pi$
$992$	0	0
$993$	−11208.9	−0.358211
$994$	0	0
$995$	2138.82	0.0681459
$996$	0	0
$997$	−17372.1	−0.551834	−0.275917	−	0.961181i	$-0.588982\pi$
$997$	−17372.1	−0.551834	−0.275917	+	0.961181i	$0.588982\pi$
$998$	0	0
$999$	−8372.16	−0.265149

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.4.a.f.1.2		2
3.2	odd	2		1764.4.a.y.1.1		2
4.3	odd	2		2352.4.a.bx.1.2		2
7.2	even	3		588.4.i.j.361.1		4
7.3	odd	6		84.4.i.a.37.2	yes	4
7.4	even	3		588.4.i.j.373.1		4
7.5	odd	6		84.4.i.a.25.2	✓	4
7.6	odd	2		588.4.a.i.1.1		2
21.2	odd	6		1764.4.k.q.361.2		4
21.5	even	6		252.4.k.f.109.1		4
21.11	odd	6		1764.4.k.q.1549.2		4
21.17	even	6		252.4.k.f.37.1		4
21.20	even	2		1764.4.a.o.1.2		2
28.3	even	6		336.4.q.i.289.2		4
28.19	even	6		336.4.q.i.193.2		4
28.27	even	2		2352.4.a.bt.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.i.a.25.2	✓	4	7.5	odd	6
84.4.i.a.37.2	yes	4	7.3	odd	6
252.4.k.f.37.1		4	21.17	even	6
252.4.k.f.109.1		4	21.5	even	6
336.4.q.i.193.2		4	28.19	even	6
336.4.q.i.289.2		4	28.3	even	6
588.4.a.f.1.2		2	1.1	even	1	trivial
588.4.a.i.1.1		2	7.6	odd	2
588.4.i.j.361.1		4	7.2	even	3
588.4.i.j.373.1		4	7.4	even	3
1764.4.a.o.1.2		2	21.20	even	2
1764.4.a.y.1.1		2	3.2	odd	2
1764.4.k.q.361.2		4	21.2	odd	6
1764.4.k.q.1549.2		4	21.11	odd	6
2352.4.a.bt.1.1		2	28.27	even	2
2352.4.a.bx.1.2		2	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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	588.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +1.44622 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +1.44622 q^{5} +9.00000 q^{9} +46.1236 q^{11} +32.2311 q^{13} -4.33867 q^{15} -77.7849 q^{17} -12.6613 q^{19} -100.924 q^{23} -122.908 q^{25} -27.0000 q^{27} +213.908 q^{29} -42.0756 q^{31} -138.371 q^{33} +310.080 q^{37} -96.6933 q^{39} -44.0320 q^{41} +381.339 q^{43} +13.0160 q^{45} +358.064 q^{47} +233.355 q^{51} +184.984 q^{53} +66.7049 q^{55} +37.9840 q^{57} -454.650 q^{59} -11.8489 q^{61} +46.6133 q^{65} +590.359 q^{67} +302.773 q^{69} +494.366 q^{71} -975.650 q^{73} +368.725 q^{75} +299.334 q^{79} +81.0000 q^{81} +1406.07 q^{83} -112.494 q^{85} -641.725 q^{87} +695.259 q^{89} +126.227 q^{93} -18.3111 q^{95} -481.940 q^{97} +415.112 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 11 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 11 * q^5 + 18 * q^9	\( 2 q - 6 q^{3} - 11 q^{5} + 18 q^{9} - 5 q^{11} - 5 q^{13} + 33 q^{15} - 100 q^{17} - 67 q^{19} + 76 q^{23} - 93 q^{25} - 54 q^{27} + 275 q^{29} - 362 q^{31} + 15 q^{33} - 5 q^{37} + 15 q^{39} + 162 q^{41} + 721 q^{43} - 99 q^{45} + 216 q^{47} + 300 q^{51} + 495 q^{53} + 703 q^{55} + 201 q^{57} - 173 q^{59} + 532 q^{61} + 510 q^{65} + 111 q^{67} - 228 q^{69} + 1600 q^{71} - 1215 q^{73} + 279 q^{75} + 1460 q^{79} + 162 q^{81} + 1409 q^{83} + 164 q^{85} - 825 q^{87} + 1974 q^{89} + 1086 q^{93} + 658 q^{95} - 561 q^{97} - 45 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 11 * q^5 + 18 * q^9 - 5 * q^11 - 5 * q^13 + 33 * q^15 - 100 * q^17 - 67 * q^19 + 76 * q^23 - 93 * q^25 - 54 * q^27 + 275 * q^29 - 362 * q^31 + 15 * q^33 - 5 * q^37 + 15 * q^39 + 162 * q^41 + 721 * q^43 - 99 * q^45 + 216 * q^47 + 300 * q^51 + 495 * q^53 + 703 * q^55 + 201 * q^57 - 173 * q^59 + 532 * q^61 + 510 * q^65 + 111 * q^67 - 228 * q^69 + 1600 * q^71 - 1215 * q^73 + 279 * q^75 + 1460 * q^79 + 162 * q^81 + 1409 * q^83 + 164 * q^85 - 825 * q^87 + 1974 * q^89 + 1086 * q^93 + 658 * q^95 - 561 * q^97 - 45 * q^99

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