LMFDB - Embedded newform 588.6.a.n.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [588,6,Mod(1,588)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("588.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	588.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-36,0,0,0,0,0,324,0,462] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$94.3056860500$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 699x^{2} - 686x + 59664 $ x^4 - 2x^3 - 699x^2 - 686*x + 59664
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2}\cdot 7 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-10.9924$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} +46.1154 q^{5} +81.0000 q^{9} -631.165 q^{11} -1079.22 q^{13} -415.038 q^{15} +161.156 q^{17} +1176.86 q^{19} +2163.46 q^{23} -998.372 q^{25} -729.000 q^{27} -4492.01 q^{29} +318.260 q^{31} +5680.48 q^{33} +15186.8 q^{37} +9713.02 q^{39} +20587.2 q^{41} -455.118 q^{43} +3735.35 q^{45} -20762.8 q^{47} -1450.40 q^{51} -19300.1 q^{53} -29106.4 q^{55} -10591.7 q^{57} +6368.31 q^{59} -49145.1 q^{61} -49768.9 q^{65} +34054.0 q^{67} -19471.1 q^{69} +62962.4 q^{71} -8867.76 q^{73} +8985.34 q^{75} +34413.2 q^{79} +6561.00 q^{81} -7041.42 q^{83} +7431.75 q^{85} +40428.1 q^{87} -20242.8 q^{89} -2864.34 q^{93} +54271.1 q^{95} +54066.6 q^{97} -51124.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{3} + 324 q^{9} + 462 q^{11} - 602 q^{13} - 228 q^{17} - 358 q^{19} + 2148 q^{23} + 5454 q^{25} - 2916 q^{27} - 5532 q^{29} - 830 q^{31} - 4158 q^{33} + 3914 q^{37} + 5418 q^{39} - 8316 q^{41}+ \cdots + 37422 q^{99}+O(q^{100}) $ 4 * q - 36 * q^3 + 324 * q^9 + 462 * q^11 - 602 * q^13 - 228 * q^17 - 358 * q^19 + 2148 * q^23 + 5454 * q^25 - 2916 * q^27 - 5532 * q^29 - 830 * q^31 - 4158 * q^33 + 3914 * q^37 + 5418 * q^39 - 8316 * q^41 - 14518 * q^43 - 41700 * q^47 + 2052 * q^51 - 22164 * q^53 + 3892 * q^55 + 3222 * q^57 - 32886 * q^59 - 83732 * q^61 + 93192 * q^65 + 80034 * q^67 - 19332 * q^69 + 44772 * q^71 + 22470 * q^73 - 49086 * q^75 + 75286 * q^79 + 26244 * q^81 - 17418 * q^83 + 139252 * q^85 + 49788 * q^87 - 28944 * q^89 + 7470 * q^93 + 144120 * q^95 - 216678 * q^97 + 37422 * q^99

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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	46.1154	0.824937	0.412469	−	0.910972i	$-0.364667\pi$
$5$	46.1154	0.824937	0.412469	+	0.910972i	$0.364667\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−631.165	−1.57276	−0.786378	−	0.617746i	$-0.788046\pi$
$11$	−631.165	−1.57276	−0.786378	+	0.617746i	$0.788046\pi$
$12$	0	0
$13$	−1079.22	−1.77114	−0.885571	−	0.464503i	$-0.846233\pi$
$13$	−1079.22	−1.77114	−0.885571	+	0.464503i	$0.846233\pi$
$14$	0	0
$15$	−415.038	−0.476278
$16$	0	0
$17$	161.156	0.135246	0.0676228	−	0.997711i	$-0.478459\pi$
$17$	161.156	0.135246	0.0676228	+	0.997711i	$0.478459\pi$
$18$	0	0
$19$	1176.86	0.747892	0.373946	−	0.927450i	$-0.378004\pi$
$19$	1176.86	0.747892	0.373946	+	0.927450i	$0.378004\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2163.46	0.852765	0.426382	−	0.904543i	$-0.359788\pi$
$23$	2163.46	0.852765	0.426382	+	0.904543i	$0.359788\pi$
$24$	0	0
$25$	−998.372	−0.319479
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−4492.01	−0.991850	−0.495925	−	0.868365i	$-0.665171\pi$
$29$	−4492.01	−0.991850	−0.495925	+	0.868365i	$0.665171\pi$
$30$	0	0
$31$	318.260	0.0594809	0.0297405	−	0.999558i	$-0.490532\pi$
$31$	318.260	0.0594809	0.0297405	+	0.999558i	$0.490532\pi$
$32$	0	0
$33$	5680.48	0.908031
$34$	0	0
$35$	0	0
$36$	0	0
$37$	15186.8	1.82374	0.911869	−	0.410482i	$-0.134639\pi$
$37$	15186.8	1.82374	0.911869	+	0.410482i	$0.134639\pi$
$38$	0	0
$39$	9713.02	1.02257
$40$	0	0
$41$	20587.2	1.91266	0.956330	−	0.292289i	$-0.0944169\pi$
$41$	20587.2	1.91266	0.956330	+	0.292289i	$0.0944169\pi$
$42$	0	0
$43$	−455.118	−0.0375364	−0.0187682	−	0.999824i	$-0.505974\pi$
$43$	−455.118	−0.0375364	−0.0187682	+	0.999824i	$0.505974\pi$
$44$	0	0
$45$	3735.35	0.274979
$46$	0	0
$47$	−20762.8	−1.37101	−0.685504	−	0.728068i	$-0.740418\pi$
$47$	−20762.8	−1.37101	−0.685504	+	0.728068i	$0.740418\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1450.40	−0.0780841
$52$	0	0
$53$	−19300.1	−0.943776	−0.471888	−	0.881658i	$-0.656427\pi$
$53$	−19300.1	−0.943776	−0.471888	+	0.881658i	$0.656427\pi$
$54$	0	0
$55$	−29106.4	−1.29742
$56$	0	0
$57$	−10591.7	−0.431796
$58$	0	0
$59$	6368.31	0.238174	0.119087	−	0.992884i	$-0.462003\pi$
$59$	6368.31	0.238174	0.119087	+	0.992884i	$0.462003\pi$
$60$	0	0
$61$	−49145.1	−1.69105	−0.845524	−	0.533938i	$-0.820712\pi$
$61$	−49145.1	−1.69105	−0.845524	+	0.533938i	$0.820712\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−49768.9	−1.46108
$66$	0	0
$67$	34054.0	0.926790	0.463395	−	0.886152i	$-0.346631\pi$
$67$	34054.0	0.926790	0.463395	+	0.886152i	$0.346631\pi$
$68$	0	0
$69$	−19471.1	−0.492344
$70$	0	0
$71$	62962.4	1.48230	0.741149	−	0.671341i	$-0.234281\pi$
$71$	62962.4	1.48230	0.741149	+	0.671341i	$0.234281\pi$
$72$	0	0
$73$	−8867.76	−0.194763	−0.0973815	−	0.995247i	$-0.531047\pi$
$73$	−8867.76	−0.194763	−0.0973815	+	0.995247i	$0.531047\pi$
$74$	0	0
$75$	8985.34	0.184451
$76$	0	0
$77$	0	0
$78$	0	0
$79$	34413.2	0.620380	0.310190	−	0.950675i	$-0.399607\pi$
$79$	34413.2	0.620380	0.310190	+	0.950675i	$0.399607\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−7041.42	−0.112193	−0.0560964	−	0.998425i	$-0.517865\pi$
$83$	−7041.42	−0.112193	−0.0560964	+	0.998425i	$0.517865\pi$
$84$	0	0
$85$	7431.75	0.111569
$86$	0	0
$87$	40428.1	0.572645
$88$	0	0
$89$	−20242.8	−0.270891	−0.135445	−	0.990785i	$-0.543247\pi$
$89$	−20242.8	−0.270891	−0.135445	+	0.990785i	$0.543247\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2864.34	−0.0343413
$94$	0	0
$95$	54271.1	0.616964
$96$	0	0
$97$	54066.6	0.583444	0.291722	−	0.956503i	$-0.405772\pi$
$97$	54066.6	0.583444	0.291722	+	0.956503i	$0.405772\pi$
$98$	0	0
$99$	−51124.4	−0.524252
$100$	0	0
$101$	−91486.3	−0.892386	−0.446193	−	0.894937i	$-0.647220\pi$
$101$	−91486.3	−0.892386	−0.446193	+	0.894937i	$0.647220\pi$
$102$	0	0
$103$	75381.4	0.700118	0.350059	−	0.936728i	$-0.386162\pi$
$103$	75381.4	0.700118	0.350059	+	0.936728i	$0.386162\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	152133.	1.28459	0.642293	−	0.766459i	$-0.277983\pi$
$107$	152133.	1.28459	0.642293	+	0.766459i	$0.277983\pi$
$108$	0	0
$109$	76671.9	0.618116	0.309058	−	0.951043i	$-0.399986\pi$
$109$	76671.9	0.618116	0.309058	+	0.951043i	$0.399986\pi$
$110$	0	0
$111$	−136681.	−1.05294
$112$	0	0
$113$	228515.	1.68352	0.841760	−	0.539852i	$-0.181520\pi$
$113$	228515.	1.68352	0.841760	+	0.539852i	$0.181520\pi$
$114$	0	0
$115$	99768.8	0.703477
$116$	0	0
$117$	−87417.2	−0.590381
$118$	0	0
$119$	0	0
$120$	0	0
$121$	237318.	1.47356
$122$	0	0
$123$	−185285.	−1.10427
$124$	0	0
$125$	−190151.	−1.08849
$126$	0	0
$127$	122111.	0.671809	0.335905	−	0.941896i	$-0.390958\pi$
$127$	122111.	0.671809	0.335905	+	0.941896i	$0.390958\pi$
$128$	0	0
$129$	4096.06	0.0216717
$130$	0	0
$131$	75795.3	0.385890	0.192945	−	0.981210i	$-0.438196\pi$
$131$	75795.3	0.385890	0.192945	+	0.981210i	$0.438196\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−33618.1	−0.158759
$136$	0	0
$137$	−241611.	−1.09981	−0.549903	−	0.835229i	$-0.685335\pi$
$137$	−241611.	−1.09981	−0.549903	+	0.835229i	$0.685335\pi$
$138$	0	0
$139$	125657.	0.551634	0.275817	−	0.961210i	$-0.411052\pi$
$139$	125657.	0.551634	0.275817	+	0.961210i	$0.411052\pi$
$140$	0	0
$141$	186865.	0.791552
$142$	0	0
$143$	681169.	2.78557
$144$	0	0
$145$	−207151.	−0.818214
$146$	0	0
$147$	0	0
$148$	0	0
$149$	36046.6	0.133014	0.0665071	−	0.997786i	$-0.478814\pi$
$149$	36046.6	0.133014	0.0665071	+	0.997786i	$0.478814\pi$
$150$	0	0
$151$	150668.	0.537748	0.268874	−	0.963175i	$-0.413349\pi$
$151$	150668.	0.537748	0.268874	+	0.963175i	$0.413349\pi$
$152$	0	0
$153$	13053.6	0.0450819
$154$	0	0
$155$	14676.7	0.0490680
$156$	0	0
$157$	426414.	1.38065	0.690323	−	0.723501i	$-0.257468\pi$
$157$	426414.	1.38065	0.690323	+	0.723501i	$0.257468\pi$
$158$	0	0
$159$	173701.	0.544889
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−192510.	−0.567524	−0.283762	−	0.958895i	$-0.591583\pi$
$163$	−192510.	−0.567524	−0.283762	+	0.958895i	$0.591583\pi$
$164$	0	0
$165$	261958.	0.749068
$166$	0	0
$167$	164987.	0.457782	0.228891	−	0.973452i	$-0.426490\pi$
$167$	164987.	0.457782	0.228891	+	0.973452i	$0.426490\pi$
$168$	0	0
$169$	793433.	2.13695
$170$	0	0
$171$	95325.3	0.249297
$172$	0	0
$173$	−329498.	−0.837022	−0.418511	−	0.908212i	$-0.637448\pi$
$173$	−329498.	−0.837022	−0.418511	+	0.908212i	$0.637448\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−57314.7	−0.137510
$178$	0	0
$179$	368348.	0.859262	0.429631	−	0.903005i	$-0.358644\pi$
$179$	368348.	0.859262	0.429631	+	0.903005i	$0.358644\pi$
$180$	0	0
$181$	−79607.3	−0.180616	−0.0903080	−	0.995914i	$-0.528785\pi$
$181$	−79607.3	−0.180616	−0.0903080	+	0.995914i	$0.528785\pi$
$182$	0	0
$183$	442306.	0.976327
$184$	0	0
$185$	700346.	1.50447
$186$	0	0
$187$	−101716.	−0.212708
$188$	0	0
$189$	0	0
$190$	0	0
$191$	565914.	1.12245	0.561225	−	0.827663i	$-0.310330\pi$
$191$	565914.	1.12245	0.561225	+	0.827663i	$0.310330\pi$
$192$	0	0
$193$	−38665.8	−0.0747194	−0.0373597	−	0.999302i	$-0.511895\pi$
$193$	−38665.8	−0.0747194	−0.0373597	+	0.999302i	$0.511895\pi$
$194$	0	0
$195$	447920.	0.843556
$196$	0	0
$197$	−334957.	−0.614927	−0.307463	−	0.951560i	$-0.599480\pi$
$197$	−334957.	−0.614927	−0.307463	+	0.951560i	$0.599480\pi$
$198$	0	0
$199$	−600245.	−1.07447	−0.537237	−	0.843431i	$-0.680532\pi$
$199$	−600245.	−1.07447	−0.537237	+	0.843431i	$0.680532\pi$
$200$	0	0
$201$	−306486.	−0.535083
$202$	0	0
$203$	0	0
$204$	0	0
$205$	949387.	1.57782
$206$	0	0
$207$	175240.	0.284255
$208$	0	0
$209$	−742790.	−1.17625
$210$	0	0
$211$	1.06504e6	1.64687	0.823433	−	0.567414i	$-0.192056\pi$
$211$	1.06504e6	1.64687	0.823433	+	0.567414i	$0.192056\pi$
$212$	0	0
$213$	−566662.	−0.855805
$214$	0	0
$215$	−20987.9	−0.0309652
$216$	0	0
$217$	0	0
$218$	0	0
$219$	79809.8	0.112446
$220$	0	0
$221$	−173923.	−0.239539
$222$	0	0
$223$	−1.37380e6	−1.84995	−0.924976	−	0.380027i	$-0.875915\pi$
$223$	−1.37380e6	−1.84995	−0.924976	+	0.380027i	$0.875915\pi$
$224$	0	0
$225$	−80868.1	−0.106493
$226$	0	0
$227$	324880.	0.418464	0.209232	−	0.977866i	$-0.432904\pi$
$227$	324880.	0.418464	0.209232	+	0.977866i	$0.432904\pi$
$228$	0	0
$229$	−822392.	−1.03631	−0.518155	−	0.855286i	$-0.673381\pi$
$229$	−822392.	−1.03631	−0.518155	+	0.855286i	$0.673381\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.13814e6	1.37343	0.686713	−	0.726928i	$-0.259053\pi$
$233$	1.13814e6	1.37343	0.686713	+	0.726928i	$0.259053\pi$
$234$	0	0
$235$	−957482.	−1.13100
$236$	0	0
$237$	−309719.	−0.358176
$238$	0	0
$239$	483125.	0.547097	0.273549	−	0.961858i	$-0.411803\pi$
$239$	483125.	0.547097	0.273549	+	0.961858i	$0.411803\pi$
$240$	0	0
$241$	1.01481e6	1.12549	0.562747	−	0.826629i	$-0.309745\pi$
$241$	1.01481e6	1.12549	0.562747	+	0.826629i	$0.309745\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.27009e6	−1.32462
$248$	0	0
$249$	63372.7	0.0647745
$250$	0	0
$251$	415812.	0.416593	0.208297	−	0.978066i	$-0.433208\pi$
$251$	415812.	0.416593	0.208297	+	0.978066i	$0.433208\pi$
$252$	0	0
$253$	−1.36550e6	−1.34119
$254$	0	0
$255$	−66885.8	−0.0644144
$256$	0	0
$257$	1.01200e6	0.955753	0.477876	−	0.878427i	$-0.341407\pi$
$257$	1.01200e6	0.955753	0.477876	+	0.878427i	$0.341407\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−363853.	−0.330617
$262$	0	0
$263$	2.05769e6	1.83439	0.917195	−	0.398439i	$-0.130448\pi$
$263$	2.05769e6	1.83439	0.917195	+	0.398439i	$0.130448\pi$
$264$	0	0
$265$	−890030.	−0.778556
$266$	0	0
$267$	182185.	0.156399
$268$	0	0
$269$	−807996.	−0.680814	−0.340407	−	0.940278i	$-0.610565\pi$
$269$	−807996.	−0.680814	−0.340407	+	0.940278i	$0.610565\pi$
$270$	0	0
$271$	196023.	0.162137	0.0810687	−	0.996709i	$-0.474167\pi$
$271$	196023.	0.162137	0.0810687	+	0.996709i	$0.474167\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	630137.	0.502462
$276$	0	0
$277$	302584.	0.236945	0.118472	−	0.992957i	$-0.462200\pi$
$277$	302584.	0.236945	0.118472	+	0.992957i	$0.462200\pi$
$278$	0	0
$279$	25779.0	0.0198270
$280$	0	0
$281$	646014.	0.488063	0.244032	−	0.969767i	$-0.421530\pi$
$281$	646014.	0.488063	0.244032	+	0.969767i	$0.421530\pi$
$282$	0	0
$283$	−1.10750e6	−0.822008	−0.411004	−	0.911634i	$-0.634822\pi$
$283$	−1.10750e6	−0.822008	−0.411004	+	0.911634i	$0.634822\pi$
$284$	0	0
$285$	−488440.	−0.356204
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.39389e6	−0.981709
$290$	0	0
$291$	−486599.	−0.336852
$292$	0	0
$293$	1.89396e6	1.28885	0.644425	−	0.764668i	$-0.277097\pi$
$293$	1.89396e6	1.28885	0.644425	+	0.764668i	$0.277097\pi$
$294$	0	0
$295$	293677.	0.196478
$296$	0	0
$297$	460119.	0.302677
$298$	0	0
$299$	−2.33486e6	−1.51037
$300$	0	0
$301$	0	0
$302$	0	0
$303$	823377.	0.515219
$304$	0	0
$305$	−2.26635e6	−1.39501
$306$	0	0
$307$	1.97803e6	1.19781	0.598905	−	0.800820i	$-0.295603\pi$
$307$	1.97803e6	1.19781	0.598905	+	0.800820i	$0.295603\pi$
$308$	0	0
$309$	−678432.	−0.404213
$310$	0	0
$311$	2.58787e6	1.51719	0.758596	−	0.651561i	$-0.225885\pi$
$311$	2.58787e6	1.51719	0.758596	+	0.651561i	$0.225885\pi$
$312$	0	0
$313$	−141357.	−0.0815561	−0.0407781	−	0.999168i	$-0.512984\pi$
$313$	−141357.	−0.0815561	−0.0407781	+	0.999168i	$0.512984\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.17285e6	0.655530	0.327765	−	0.944759i	$-0.393705\pi$
$317$	1.17285e6	0.655530	0.327765	+	0.944759i	$0.393705\pi$
$318$	0	0
$319$	2.83520e6	1.55994
$320$	0	0
$321$	−1.36919e6	−0.741656
$322$	0	0
$323$	189657.	0.101149
$324$	0	0
$325$	1.07747e6	0.565843
$326$	0	0
$327$	−690047.	−0.356870
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.30661e6	0.655505	0.327752	−	0.944764i	$-0.393709\pi$
$331$	1.30661e6	0.655505	0.327752	+	0.944764i	$0.393709\pi$
$332$	0	0
$333$	1.23013e6	0.607913
$334$	0	0
$335$	1.57041e6	0.764544
$336$	0	0
$337$	−265059.	−0.127136	−0.0635679	−	0.997978i	$-0.520248\pi$
$337$	−265059.	−0.127136	−0.0635679	+	0.997978i	$0.520248\pi$
$338$	0	0
$339$	−2.05663e6	−0.971981
$340$	0	0
$341$	−200874.	−0.0935489
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−897919.	−0.406153
$346$	0	0
$347$	−3.05583e6	−1.36240	−0.681200	−	0.732097i	$-0.738542\pi$
$347$	−3.05583e6	−1.36240	−0.681200	+	0.732097i	$0.738542\pi$
$348$	0	0
$349$	−1.47164e6	−0.646753	−0.323377	−	0.946270i	$-0.604818\pi$
$349$	−1.47164e6	−0.646753	−0.323377	+	0.946270i	$0.604818\pi$
$350$	0	0
$351$	786755.	0.340857
$352$	0	0
$353$	−385183.	−0.164525	−0.0822623	−	0.996611i	$-0.526215\pi$
$353$	−385183.	−0.164525	−0.0822623	+	0.996611i	$0.526215\pi$
$354$	0	0
$355$	2.90354e6	1.22280
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.64188e6	−0.672364	−0.336182	−	0.941797i	$-0.609136\pi$
$359$	−1.64188e6	−0.672364	−0.336182	+	0.941797i	$0.609136\pi$
$360$	0	0
$361$	−1.09111e6	−0.440657
$362$	0	0
$363$	−2.13586e6	−0.850760
$364$	0	0
$365$	−408940.	−0.160667
$366$	0	0
$367$	−1.12323e6	−0.435315	−0.217658	−	0.976025i	$-0.569842\pi$
$367$	−1.12323e6	−0.435315	−0.217658	+	0.976025i	$0.569842\pi$
$368$	0	0
$369$	1.66756e6	0.637553
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−4.22105e6	−1.57090	−0.785450	−	0.618926i	$-0.787568\pi$
$373$	−4.22105e6	−1.57090	−0.785450	+	0.618926i	$0.787568\pi$
$374$	0	0
$375$	1.71136e6	0.628438
$376$	0	0
$377$	4.84789e6	1.75671
$378$	0	0
$379$	−4.49923e6	−1.60894	−0.804471	−	0.593993i	$-0.797551\pi$
$379$	−4.49923e6	−1.60894	−0.804471	+	0.593993i	$0.797551\pi$
$380$	0	0
$381$	−1.09900e6	−0.387869
$382$	0	0
$383$	4.43114e6	1.54354	0.771771	−	0.635900i	$-0.219371\pi$
$383$	4.43114e6	1.54354	0.771771	+	0.635900i	$0.219371\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−36864.6	−0.0125121
$388$	0	0
$389$	−5.44223e6	−1.82349	−0.911744	−	0.410759i	$-0.865264\pi$
$389$	−5.44223e6	−1.82349	−0.911744	+	0.410759i	$0.865264\pi$
$390$	0	0
$391$	348654.	0.115333
$392$	0	0
$393$	−682158.	−0.222794
$394$	0	0
$395$	1.58698e6	0.511774
$396$	0	0
$397$	2.68794e6	0.855939	0.427969	−	0.903793i	$-0.359229\pi$
$397$	2.68794e6	0.855939	0.427969	+	0.903793i	$0.359229\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	443294.	0.137667	0.0688337	−	0.997628i	$-0.478072\pi$
$401$	443294.	0.137667	0.0688337	+	0.997628i	$0.478072\pi$
$402$	0	0
$403$	−343474.	−0.105349
$404$	0	0
$405$	302563.	0.0916597
$406$	0	0
$407$	−9.58539e6	−2.86829
$408$	0	0
$409$	−4.89578e6	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$409$	−4.89578e6	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$410$	0	0
$411$	2.17450e6	0.634973
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−324718.	−0.0925520
$416$	0	0
$417$	−1.13092e6	−0.318486
$418$	0	0
$419$	−2.23186e6	−0.621058	−0.310529	−	0.950564i	$-0.600506\pi$
$419$	−2.23186e6	−0.621058	−0.310529	+	0.950564i	$0.600506\pi$
$420$	0	0
$421$	5.48208e6	1.50744	0.753721	−	0.657195i	$-0.228257\pi$
$421$	5.48208e6	1.50744	0.753721	+	0.657195i	$0.228257\pi$
$422$	0	0
$423$	−1.68178e6	−0.457003
$424$	0	0
$425$	−160893.	−0.0432081
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−6.13052e6	−1.60825
$430$	0	0
$431$	4.62366e6	1.19893	0.599463	−	0.800402i	$-0.295381\pi$
$431$	4.62366e6	1.19893	0.599463	+	0.800402i	$0.295381\pi$
$432$	0	0
$433$	−3.08314e6	−0.790267	−0.395134	−	0.918624i	$-0.629302\pi$
$433$	−3.08314e6	−0.790267	−0.395134	+	0.918624i	$0.629302\pi$
$434$	0	0
$435$	1.86436e6	0.472396
$436$	0	0
$437$	2.54608e6	0.637776
$438$	0	0
$439$	434470.	0.107597	0.0537983	−	0.998552i	$-0.482867\pi$
$439$	434470.	0.107597	0.0537983	+	0.998552i	$0.482867\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.72063e6	0.416562	0.208281	−	0.978069i	$-0.433213\pi$
$443$	1.72063e6	0.416562	0.208281	+	0.978069i	$0.433213\pi$
$444$	0	0
$445$	−933502.	−0.223468
$446$	0	0
$447$	−324419.	−0.0767958
$448$	0	0
$449$	−4.06508e6	−0.951598	−0.475799	−	0.879554i	$-0.657841\pi$
$449$	−4.06508e6	−0.951598	−0.475799	+	0.879554i	$0.657841\pi$
$450$	0	0
$451$	−1.29939e7	−3.00815
$452$	0	0
$453$	−1.35601e6	−0.310469
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.26184e6	0.506607	0.253303	−	0.967387i	$-0.418483\pi$
$457$	2.26184e6	0.506607	0.253303	+	0.967387i	$0.418483\pi$
$458$	0	0
$459$	−117482.	−0.0260280
$460$	0	0
$461$	7.80980e6	1.71154	0.855771	−	0.517355i	$-0.173083\pi$
$461$	7.80980e6	1.71154	0.855771	+	0.517355i	$0.173083\pi$
$462$	0	0
$463$	−525518.	−0.113929	−0.0569647	−	0.998376i	$-0.518142\pi$
$463$	−525518.	−0.113929	−0.0569647	+	0.998376i	$0.518142\pi$
$464$	0	0
$465$	−132090.	−0.0283294
$466$	0	0
$467$	5.59376e6	1.18689	0.593447	−	0.804873i	$-0.297767\pi$
$467$	5.59376e6	1.18689	0.593447	+	0.804873i	$0.297767\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.83773e6	−0.797116
$472$	0	0
$473$	287255.	0.0590356
$474$	0	0
$475$	−1.17494e6	−0.238936
$476$	0	0
$477$	−1.56330e6	−0.314592
$478$	0	0
$479$	1.81246e6	0.360936	0.180468	−	0.983581i	$-0.442239\pi$
$479$	1.81246e6	0.360936	0.180468	+	0.983581i	$0.442239\pi$
$480$	0	0
$481$	−1.63900e7	−3.23010
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.49330e6	0.481305
$486$	0	0
$487$	1.44068e6	0.275261	0.137631	−	0.990484i	$-0.456051\pi$
$487$	1.44068e6	0.275261	0.137631	+	0.990484i	$0.456051\pi$
$488$	0	0
$489$	1.73259e6	0.327660
$490$	0	0
$491$	5.25026e6	0.982827	0.491413	−	0.870926i	$-0.336480\pi$
$491$	5.25026e6	0.982827	0.491413	+	0.870926i	$0.336480\pi$
$492$	0	0
$493$	−723913.	−0.134143
$494$	0	0
$495$	−2.35762e6	−0.432475
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−8.14160e6	−1.46372	−0.731861	−	0.681454i	$-0.761348\pi$
$499$	−8.14160e6	−1.46372	−0.731861	+	0.681454i	$0.761348\pi$
$500$	0	0
$501$	−1.48488e6	−0.264301
$502$	0	0
$503$	−7.98029e6	−1.40637	−0.703183	−	0.711009i	$-0.748239\pi$
$503$	−7.98029e6	−1.40637	−0.703183	+	0.711009i	$0.748239\pi$
$504$	0	0
$505$	−4.21893e6	−0.736162
$506$	0	0
$507$	−7.14090e6	−1.23377
$508$	0	0
$509$	8.14283e6	1.39309	0.696547	−	0.717511i	$-0.254719\pi$
$509$	8.14283e6	1.39309	0.696547	+	0.717511i	$0.254719\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−857928.	−0.143932
$514$	0	0
$515$	3.47624e6	0.577553
$516$	0	0
$517$	1.31047e7	2.15626
$518$	0	0
$519$	2.96548e6	0.483255
$520$	0	0
$521$	5.11967e6	0.826319	0.413160	−	0.910659i	$-0.364425\pi$
$521$	5.11967e6	0.826319	0.413160	+	0.910659i	$0.364425\pi$
$522$	0	0
$523$	4.73473e6	0.756904	0.378452	−	0.925621i	$-0.376456\pi$
$523$	4.73473e6	0.756904	0.378452	+	0.925621i	$0.376456\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	51289.3	0.00804453
$528$	0	0
$529$	−1.75579e6	−0.272792
$530$	0	0
$531$	515833.	0.0793912
$532$	0	0
$533$	−2.22182e7	−3.38759
$534$	0	0
$535$	7.01566e6	1.05970
$536$	0	0
$537$	−3.31513e6	−0.496095
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8.20453e6	−1.20520	−0.602602	−	0.798042i	$-0.705869\pi$
$541$	−8.20453e6	−1.20520	−0.602602	+	0.798042i	$0.705869\pi$
$542$	0	0
$543$	716465.	0.104279
$544$	0	0
$545$	3.53575e6	0.509907
$546$	0	0
$547$	−1.57733e6	−0.225400	−0.112700	−	0.993629i	$-0.535950\pi$
$547$	−1.57733e6	−0.225400	−0.112700	+	0.993629i	$0.535950\pi$
$548$	0	0
$549$	−3.98076e6	−0.563683
$550$	0	0
$551$	−5.28645e6	−0.741797
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−6.30311e6	−0.868605
$556$	0	0
$557$	−2.10411e6	−0.287363	−0.143681	−	0.989624i	$-0.545894\pi$
$557$	−2.10411e6	−0.287363	−0.143681	+	0.989624i	$0.545894\pi$
$558$	0	0
$559$	491175.	0.0664824
$560$	0	0
$561$	915442.	0.122807
$562$	0	0
$563$	7.69167e6	1.02270	0.511351	−	0.859372i	$-0.329145\pi$
$563$	7.69167e6	1.02270	0.511351	+	0.859372i	$0.329145\pi$
$564$	0	0
$565$	1.05381e7	1.38880
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.30139e6	0.945421	0.472710	−	0.881218i	$-0.343276\pi$
$569$	7.30139e6	0.945421	0.472710	+	0.881218i	$0.343276\pi$
$570$	0	0
$571$	3.41934e6	0.438887	0.219443	−	0.975625i	$-0.429576\pi$
$571$	3.41934e6	0.438887	0.219443	+	0.975625i	$0.429576\pi$
$572$	0	0
$573$	−5.09323e6	−0.648047
$574$	0	0
$575$	−2.15994e6	−0.272440
$576$	0	0
$577$	−1.09289e7	−1.36659	−0.683295	−	0.730142i	$-0.739454\pi$
$577$	−1.09289e7	−1.36659	−0.683295	+	0.730142i	$0.739454\pi$
$578$	0	0
$579$	347992.	0.0431393
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.21815e7	1.48433
$584$	0	0
$585$	−4.03128e6	−0.487027
$586$	0	0
$587$	2.25268e6	0.269839	0.134919	−	0.990857i	$-0.456922\pi$
$587$	2.25268e6	0.269839	0.134919	+	0.990857i	$0.456922\pi$
$588$	0	0
$589$	374546.	0.0444853
$590$	0	0
$591$	3.01461e6	0.355028
$592$	0	0
$593$	−5.68240e6	−0.663583	−0.331792	−	0.943353i	$-0.607653\pi$
$593$	−5.68240e6	−0.663583	−0.331792	+	0.943353i	$0.607653\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.40221e6	0.620348
$598$	0	0
$599$	−1.28026e7	−1.45792	−0.728958	−	0.684559i	$-0.759995\pi$
$599$	−1.28026e7	−1.45792	−0.728958	+	0.684559i	$0.759995\pi$
$600$	0	0
$601$	−9.25870e6	−1.04560	−0.522798	−	0.852457i	$-0.675112\pi$
$601$	−9.25870e6	−1.04560	−0.522798	+	0.852457i	$0.675112\pi$
$602$	0	0
$603$	2.75838e6	0.308930
$604$	0	0
$605$	1.09440e7	1.21559
$606$	0	0
$607$	−401320.	−0.0442098	−0.0221049	−	0.999756i	$-0.507037\pi$
$607$	−401320.	−0.0442098	−0.0221049	+	0.999756i	$0.507037\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.24077e7	2.42825
$612$	0	0
$613$	6.82396e6	0.733475	0.366738	−	0.930324i	$-0.380475\pi$
$613$	6.82396e6	0.733475	0.366738	+	0.930324i	$0.380475\pi$
$614$	0	0
$615$	−8.54448e6	−0.910957
$616$	0	0
$617$	−336274.	−0.0355615	−0.0177808	−	0.999842i	$-0.505660\pi$
$617$	−336274.	−0.0355615	−0.0177808	+	0.999842i	$0.505660\pi$
$618$	0	0
$619$	−9.61151e6	−1.00824	−0.504121	−	0.863633i	$-0.668183\pi$
$619$	−9.61151e6	−1.00824	−0.504121	+	0.863633i	$0.668183\pi$
$620$	0	0
$621$	−1.57716e6	−0.164115
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−5.64897e6	−0.578454
$626$	0	0
$627$	6.68511e6	0.679109
$628$	0	0
$629$	2.44744e6	0.246652
$630$	0	0
$631$	1.28813e7	1.28791	0.643957	−	0.765062i	$-0.277292\pi$
$631$	1.28813e7	1.28791	0.643957	+	0.765062i	$0.277292\pi$
$632$	0	0
$633$	−9.58532e6	−0.950818
$634$	0	0
$635$	5.63120e6	0.554200
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.09996e6	0.494099
$640$	0	0
$641$	−1.49496e7	−1.43709	−0.718545	−	0.695480i	$-0.755192\pi$
$641$	−1.49496e7	−1.43709	−0.718545	+	0.695480i	$0.755192\pi$
$642$	0	0
$643$	−8.03892e6	−0.766780	−0.383390	−	0.923587i	$-0.625243\pi$
$643$	−8.03892e6	−0.766780	−0.383390	+	0.923587i	$0.625243\pi$
$644$	0	0
$645$	188892.	0.0178778
$646$	0	0
$647$	−6.68775e6	−0.628086	−0.314043	−	0.949409i	$-0.601684\pi$
$647$	−6.68775e6	−0.628086	−0.314043	+	0.949409i	$0.601684\pi$
$648$	0	0
$649$	−4.01945e6	−0.374589
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.04443e6	−0.0958505	−0.0479253	−	0.998851i	$-0.515261\pi$
$653$	−1.04443e6	−0.0958505	−0.0479253	+	0.998851i	$0.515261\pi$
$654$	0	0
$655$	3.49533e6	0.318335
$656$	0	0
$657$	−718288.	−0.0649210
$658$	0	0
$659$	2.10237e7	1.88580	0.942902	−	0.333071i	$-0.108085\pi$
$659$	2.10237e7	1.88580	0.942902	+	0.333071i	$0.108085\pi$
$660$	0	0
$661$	9.46680e6	0.842752	0.421376	−	0.906886i	$-0.361547\pi$
$661$	9.46680e6	0.842752	0.421376	+	0.906886i	$0.361547\pi$
$662$	0	0
$663$	1.56531e6	0.138298
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−9.71829e6	−0.845815
$668$	0	0
$669$	1.23642e7	1.06807
$670$	0	0
$671$	3.10187e7	2.65960
$672$	0	0
$673$	−1.95188e7	−1.66117	−0.830587	−	0.556889i	$-0.811995\pi$
$673$	−1.95188e7	−1.66117	−0.830587	+	0.556889i	$0.811995\pi$
$674$	0	0
$675$	727813.	0.0614837
$676$	0	0
$677$	−1.70544e7	−1.43010	−0.715049	−	0.699075i	$-0.753596\pi$
$677$	−1.70544e7	−1.43010	−0.715049	+	0.699075i	$0.753596\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−2.92392e6	−0.241600
$682$	0	0
$683$	1.09475e7	0.897972	0.448986	−	0.893539i	$-0.351785\pi$
$683$	1.09475e7	0.897972	0.448986	+	0.893539i	$0.351785\pi$
$684$	0	0
$685$	−1.11420e7	−0.907270
$686$	0	0
$687$	7.40153e6	0.598314
$688$	0	0
$689$	2.08291e7	1.67156
$690$	0	0
$691$	1.04538e7	0.832874	0.416437	−	0.909165i	$-0.363279\pi$
$691$	1.04538e7	0.832874	0.416437	+	0.909165i	$0.363279\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.79474e6	0.455063
$696$	0	0
$697$	3.31774e6	0.258679
$698$	0	0
$699$	−1.02433e7	−0.792948
$700$	0	0
$701$	8.71564e6	0.669891	0.334946	−	0.942237i	$-0.391282\pi$
$701$	8.71564e6	0.669891	0.334946	+	0.942237i	$0.391282\pi$
$702$	0	0
$703$	1.78727e7	1.36396
$704$	0	0
$705$	8.61734e6	0.652981
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−938439.	−0.0701117	−0.0350559	−	0.999385i	$-0.511161\pi$
$709$	−938439.	−0.0701117	−0.0350559	+	0.999385i	$0.511161\pi$
$710$	0	0
$711$	2.78747e6	0.206793
$712$	0	0
$713$	688542.	0.0507232
$714$	0	0
$715$	3.14124e7	2.29792
$716$	0	0
$717$	−4.34812e6	−0.315867
$718$	0	0
$719$	6.59073e6	0.475457	0.237729	−	0.971332i	$-0.423597\pi$
$719$	6.59073e6	0.475457	0.237729	+	0.971332i	$0.423597\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−9.13331e6	−0.649804
$724$	0	0
$725$	4.48470e6	0.316875
$726$	0	0
$727$	−2.32586e7	−1.63210	−0.816052	−	0.577979i	$-0.803842\pi$
$727$	−2.32586e7	−1.63210	−0.816052	+	0.577979i	$0.803842\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−73344.8	−0.00507664
$732$	0	0
$733$	−2.02854e7	−1.39452	−0.697259	−	0.716819i	$-0.745597\pi$
$733$	−2.02854e7	−1.39452	−0.697259	+	0.716819i	$0.745597\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.14937e7	−1.45761
$738$	0	0
$739$	593427.	0.0399720	0.0199860	−	0.999800i	$-0.493638\pi$
$739$	593427.	0.0399720	0.0199860	+	0.999800i	$0.493638\pi$
$740$	0	0
$741$	1.14308e7	0.764772
$742$	0	0
$743$	7.70228e6	0.511855	0.255928	−	0.966696i	$-0.417619\pi$
$743$	7.70228e6	0.511855	0.255928	+	0.966696i	$0.417619\pi$
$744$	0	0
$745$	1.66230e6	0.109728
$746$	0	0
$747$	−570355.	−0.0373976
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2.68333e7	−1.73610	−0.868048	−	0.496481i	$-0.834625\pi$
$751$	−2.68333e7	−1.73610	−0.868048	+	0.496481i	$0.834625\pi$
$752$	0	0
$753$	−3.74230e6	−0.240520
$754$	0	0
$755$	6.94811e6	0.443608
$756$	0	0
$757$	2.34943e7	1.49013	0.745063	−	0.666994i	$-0.232419\pi$
$757$	2.34943e7	1.49013	0.745063	+	0.666994i	$0.232419\pi$
$758$	0	0
$759$	1.22895e7	0.774337
$760$	0	0
$761$	−1.26192e7	−0.789895	−0.394947	−	0.918704i	$-0.629237\pi$
$761$	−1.26192e7	−0.789895	−0.394947	+	0.918704i	$0.629237\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	601972.	0.0371897
$766$	0	0
$767$	−6.87283e6	−0.421840
$768$	0	0
$769$	2.73674e6	0.166885	0.0834425	−	0.996513i	$-0.473408\pi$
$769$	2.73674e6	0.166885	0.0834425	+	0.996513i	$0.473408\pi$
$770$	0	0
$771$	−9.10796e6	−0.551804
$772$	0	0
$773$	2.15505e7	1.29720	0.648602	−	0.761128i	$-0.275354\pi$
$773$	2.15505e7	1.29720	0.648602	+	0.761128i	$0.275354\pi$
$774$	0	0
$775$	−317742.	−0.0190029
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.42282e7	1.43046
$780$	0	0
$781$	−3.97397e7	−2.33129
$782$	0	0
$783$	3.27468e6	0.190882
$784$	0	0
$785$	1.96642e7	1.13895
$786$	0	0
$787$	2.89891e6	0.166839	0.0834195	−	0.996515i	$-0.473416\pi$
$787$	2.89891e6	0.166839	0.0834195	+	0.996515i	$0.473416\pi$
$788$	0	0
$789$	−1.85192e7	−1.05909
$790$	0	0
$791$	0	0
$792$	0	0
$793$	5.30386e7	2.99509
$794$	0	0
$795$	8.01027e6	0.449499
$796$	0	0
$797$	−1.57797e7	−0.879937	−0.439969	−	0.898013i	$-0.645010\pi$
$797$	−1.57797e7	−0.879937	−0.439969	+	0.898013i	$0.645010\pi$
$798$	0	0
$799$	−3.34603e6	−0.185423
$800$	0	0
$801$	−1.63966e6	−0.0902970
$802$	0	0
$803$	5.59702e6	0.306315
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.27197e6	0.393068
$808$	0	0
$809$	7.95560e6	0.427367	0.213684	−	0.976903i	$-0.431454\pi$
$809$	7.95560e6	0.427367	0.213684	+	0.976903i	$0.431454\pi$
$810$	0	0
$811$	−2.48725e7	−1.32791	−0.663953	−	0.747774i	$-0.731123\pi$
$811$	−2.48725e7	−1.32791	−0.663953	+	0.747774i	$0.731123\pi$
$812$	0	0
$813$	−1.76420e6	−0.0936101
$814$	0	0
$815$	−8.87767e6	−0.468171
$816$	0	0
$817$	−535608.	−0.0280732
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.19168e7	1.13480	0.567401	−	0.823442i	$-0.307949\pi$
$821$	2.19168e7	1.13480	0.567401	+	0.823442i	$0.307949\pi$
$822$	0	0
$823$	1.95072e7	1.00391	0.501955	−	0.864894i	$-0.332614\pi$
$823$	1.95072e7	1.00391	0.501955	+	0.864894i	$0.332614\pi$
$824$	0	0
$825$	−5.67123e6	−0.290097
$826$	0	0
$827$	9.90134e6	0.503420	0.251710	−	0.967803i	$-0.419007\pi$
$827$	9.90134e6	0.503420	0.251710	+	0.967803i	$0.419007\pi$
$828$	0	0
$829$	2.39789e7	1.21183	0.605916	−	0.795529i	$-0.292807\pi$
$829$	2.39789e7	1.21183	0.605916	+	0.795529i	$0.292807\pi$
$830$	0	0
$831$	−2.72326e6	−0.136800
$832$	0	0
$833$	0	0
$834$	0	0
$835$	7.60844e6	0.377641
$836$	0	0
$837$	−232011.	−0.0114471
$838$	0	0
$839$	−1.26483e6	−0.0620334	−0.0310167	−	0.999519i	$-0.509875\pi$
$839$	−1.26483e6	−0.0620334	−0.0310167	+	0.999519i	$0.509875\pi$
$840$	0	0
$841$	−332952.	−0.0162327
$842$	0	0
$843$	−5.81413e6	−0.281784
$844$	0	0
$845$	3.65895e7	1.76285
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9.96746e6	0.474586
$850$	0	0
$851$	3.28561e7	1.55522
$852$	0	0
$853$	2.75502e7	1.29644	0.648219	−	0.761454i	$-0.275514\pi$
$853$	2.75502e7	1.29644	0.648219	+	0.761454i	$0.275514\pi$
$854$	0	0
$855$	4.39596e6	0.205655
$856$	0	0
$857$	3.59074e7	1.67006	0.835030	−	0.550204i	$-0.185450\pi$
$857$	3.59074e7	1.67006	0.835030	+	0.550204i	$0.185450\pi$
$858$	0	0
$859$	−2.34341e7	−1.08359	−0.541795	−	0.840511i	$-0.682255\pi$
$859$	−2.34341e7	−1.08359	−0.541795	+	0.840511i	$0.682255\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.35409e7	1.53302	0.766511	−	0.642231i	$-0.221991\pi$
$863$	3.35409e7	1.53302	0.766511	+	0.642231i	$0.221991\pi$
$864$	0	0
$865$	−1.51949e7	−0.690491
$866$	0	0
$867$	1.25450e7	0.566790
$868$	0	0
$869$	−2.17204e7	−0.975705
$870$	0	0
$871$	−3.67520e7	−1.64148
$872$	0	0
$873$	4.37939e6	0.194481
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.54557e7	−0.678561	−0.339281	−	0.940685i	$-0.610184\pi$
$877$	−1.54557e7	−0.678561	−0.339281	+	0.940685i	$0.610184\pi$
$878$	0	0
$879$	−1.70457e7	−0.744118
$880$	0	0
$881$	1.02425e7	0.444599	0.222299	−	0.974978i	$-0.428644\pi$
$881$	1.02425e7	0.444599	0.222299	+	0.974978i	$0.428644\pi$
$882$	0	0
$883$	−6.85044e6	−0.295676	−0.147838	−	0.989012i	$-0.547231\pi$
$883$	−6.85044e6	−0.295676	−0.147838	+	0.989012i	$0.547231\pi$
$884$	0	0
$885$	−2.64309e6	−0.113437
$886$	0	0
$887$	6.08759e6	0.259799	0.129899	−	0.991527i	$-0.458535\pi$
$887$	6.08759e6	0.259799	0.129899	+	0.991527i	$0.458535\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.14107e6	−0.174751
$892$	0	0
$893$	−2.44348e7	−1.02537
$894$	0	0
$895$	1.69865e7	0.708837
$896$	0	0
$897$	2.10137e7	0.872011
$898$	0	0
$899$	−1.42963e6	−0.0589962
$900$	0	0
$901$	−3.11031e6	−0.127642
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.67112e6	−0.148997
$906$	0	0
$907$	4.62504e7	1.86680	0.933398	−	0.358843i	$-0.116829\pi$
$907$	4.62504e7	1.86680	0.933398	+	0.358843i	$0.116829\pi$
$908$	0	0
$909$	−7.41039e6	−0.297462
$910$	0	0
$911$	−4.09674e7	−1.63547	−0.817734	−	0.575596i	$-0.804770\pi$
$911$	−4.09674e7	−1.63547	−0.817734	+	0.575596i	$0.804770\pi$
$912$	0	0
$913$	4.44430e6	0.176452
$914$	0	0
$915$	2.03971e7	0.805408
$916$	0	0
$917$	0	0
$918$	0	0
$919$	3.05857e7	1.19462	0.597309	−	0.802011i	$-0.296237\pi$
$919$	3.05857e7	1.19462	0.597309	+	0.802011i	$0.296237\pi$
$920$	0	0
$921$	−1.78023e7	−0.691556
$922$	0	0
$923$	−6.79506e7	−2.62536
$924$	0	0
$925$	−1.51621e7	−0.582646
$926$	0	0
$927$	6.10589e6	0.233373
$928$	0	0
$929$	−3.16851e7	−1.20453	−0.602263	−	0.798298i	$-0.705734\pi$
$929$	−3.16851e7	−1.20453	−0.602263	+	0.798298i	$0.705734\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−2.32908e7	−0.875952
$934$	0	0
$935$	−4.69066e6	−0.175471
$936$	0	0
$937$	2.43042e7	0.904342	0.452171	−	0.891931i	$-0.350650\pi$
$937$	2.43042e7	0.904342	0.452171	+	0.891931i	$0.350650\pi$
$938$	0	0
$939$	1.27221e6	0.0470865
$940$	0	0
$941$	−2.48459e7	−0.914704	−0.457352	−	0.889286i	$-0.651202\pi$
$941$	−2.48459e7	−0.914704	−0.457352	+	0.889286i	$0.651202\pi$
$942$	0	0
$943$	4.45396e7	1.63105
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.91617e7	−1.78136	−0.890681	−	0.454629i	$-0.849772\pi$
$947$	−4.91617e7	−1.78136	−0.890681	+	0.454629i	$0.849772\pi$
$948$	0	0
$949$	9.57030e6	0.344953
$950$	0	0
$951$	−1.05556e7	−0.378471
$952$	0	0
$953$	1.84741e7	0.658918	0.329459	−	0.944170i	$-0.393134\pi$
$953$	1.84741e7	0.658918	0.329459	+	0.944170i	$0.393134\pi$
$954$	0	0
$955$	2.60973e7	0.925951
$956$	0	0
$957$	−2.55168e7	−0.900631
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.85279e7	−0.996462
$962$	0	0
$963$	1.23227e7	0.428195
$964$	0	0
$965$	−1.78309e6	−0.0616388
$966$	0	0
$967$	3.78820e7	1.30277	0.651384	−	0.758748i	$-0.274189\pi$
$967$	3.78820e7	1.30277	0.651384	+	0.758748i	$0.274189\pi$
$968$	0	0
$969$	−1.70691e6	−0.0583985
$970$	0	0
$971$	−3.72621e7	−1.26829	−0.634146	−	0.773213i	$-0.718648\pi$
$971$	−3.72621e7	−1.26829	−0.634146	+	0.773213i	$0.718648\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−9.69721e6	−0.326689
$976$	0	0
$977$	−3.70590e6	−0.124210	−0.0621052	−	0.998070i	$-0.519781\pi$
$977$	−3.70590e6	−0.124210	−0.0621052	+	0.998070i	$0.519781\pi$
$978$	0	0
$979$	1.27765e7	0.426045
$980$	0	0
$981$	6.21043e6	0.206039
$982$	0	0
$983$	8.28765e6	0.273557	0.136778	−	0.990602i	$-0.456325\pi$
$983$	8.28765e6	0.273557	0.136778	+	0.990602i	$0.456325\pi$
$984$	0	0
$985$	−1.54467e7	−0.507276
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−984630.	−0.0320097
$990$	0	0
$991$	1.62498e7	0.525612	0.262806	−	0.964849i	$-0.415352\pi$
$991$	1.62498e7	0.525612	0.262806	+	0.964849i	$0.415352\pi$
$992$	0	0
$993$	−1.17595e7	−0.378456
$994$	0	0
$995$	−2.76805e7	−0.886374
$996$	0	0
$997$	−4.35858e7	−1.38870	−0.694348	−	0.719640i	$-0.744307\pi$
$997$	−4.35858e7	−1.38870	−0.694348	+	0.719640i	$0.744307\pi$
$998$	0	0
$999$	−1.10712e7	−0.350978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.6.a.n.1.3		4
7.2	even	3		84.6.i.c.25.2	✓	8
7.3	odd	6		588.6.i.o.373.3		8
7.4	even	3		84.6.i.c.37.2	yes	8
7.5	odd	6		588.6.i.o.361.3		8
7.6	odd	2		588.6.a.p.1.2		4
21.2	odd	6		252.6.k.f.109.3		8
21.11	odd	6		252.6.k.f.37.3		8
28.11	odd	6		336.6.q.i.289.2		8
28.23	odd	6		336.6.q.i.193.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.i.c.25.2	✓	8	7.2	even	3
84.6.i.c.37.2	yes	8	7.4	even	3
252.6.k.f.37.3		8	21.11	odd	6
252.6.k.f.109.3		8	21.2	odd	6
336.6.q.i.193.2		8	28.23	odd	6
336.6.q.i.289.2		8	28.11	odd	6
588.6.a.n.1.3		4	1.1	even	1	trivial
588.6.a.p.1.2		4	7.6	odd	2
588.6.i.o.361.3		8	7.5	odd	6
588.6.i.o.373.3		8	7.3	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	588.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +46.1154 q^{5} +81.0000 q^{9} -631.165 q^{11} -1079.22 q^{13} -415.038 q^{15} +161.156 q^{17} +1176.86 q^{19} +2163.46 q^{23} -998.372 q^{25} -729.000 q^{27} -4492.01 q^{29} +318.260 q^{31} +5680.48 q^{33} +15186.8 q^{37} +9713.02 q^{39} +20587.2 q^{41} -455.118 q^{43} +3735.35 q^{45} -20762.8 q^{47} -1450.40 q^{51} -19300.1 q^{53} -29106.4 q^{55} -10591.7 q^{57} +6368.31 q^{59} -49145.1 q^{61} -49768.9 q^{65} +34054.0 q^{67} -19471.1 q^{69} +62962.4 q^{71} -8867.76 q^{73} +8985.34 q^{75} +34413.2 q^{79} +6561.00 q^{81} -7041.42 q^{83} +7431.75 q^{85} +40428.1 q^{87} -20242.8 q^{89} -2864.34 q^{93} +54271.1 q^{95} +54066.6 q^{97} -51124.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{3} + 324 q^{9} + 462 q^{11} - 602 q^{13} - 228 q^{17} - 358 q^{19} + 2148 q^{23} + 5454 q^{25} - 2916 q^{27} - 5532 q^{29} - 830 q^{31} - 4158 q^{33} + 3914 q^{37} + 5418 q^{39} - 8316 q^{41}+ \cdots + 37422 q^{99}+O(q^{100}) \) 4 * q - 36 * q^3 + 324 * q^9 + 462 * q^11 - 602 * q^13 - 228 * q^17 - 358 * q^19 + 2148 * q^23 + 5454 * q^25 - 2916 * q^27 - 5532 * q^29 - 830 * q^31 - 4158 * q^33 + 3914 * q^37 + 5418 * q^39 - 8316 * q^41 - 14518 * q^43 - 41700 * q^47 + 2052 * q^51 - 22164 * q^53 + 3892 * q^55 + 3222 * q^57 - 32886 * q^59 - 83732 * q^61 + 93192 * q^65 + 80034 * q^67 - 19332 * q^69 + 44772 * q^71 + 22470 * q^73 - 49086 * q^75 + 75286 * q^79 + 26244 * q^81 - 17418 * q^83 + 139252 * q^85 + 49788 * q^87 - 28944 * q^89 + 7470 * q^93 + 144120 * q^95 - 216678 * q^97 + 37422 * q^99

Introduction

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