LMFDB - Embedded newform 588.6.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,6,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, 6, names="a")

//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);

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: f = N[0] # Warning: the index may be different

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			1.1
Root			$-22.4831$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} -92.8255 q^{5} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -92.8255 q^{5} +81.0000 q^{9} +140.762 q^{11} -1111.24 q^{13} +835.430 q^{15} +54.8869 q^{17} -1711.86 q^{19} -3287.91 q^{23} +5491.58 q^{25} -729.000 q^{27} -3790.72 q^{29} -4847.32 q^{31} -1266.86 q^{33} -11366.7 q^{37} +10001.2 q^{39} -10385.6 q^{41} +7137.16 q^{43} -7518.87 q^{45} -16415.1 q^{47} -493.983 q^{51} -20974.3 q^{53} -13066.3 q^{55} +15406.7 q^{57} -36211.9 q^{59} +4948.70 q^{61} +103152. q^{65} -22965.6 q^{67} +29591.2 q^{69} -26341.8 q^{71} +55387.4 q^{73} -49424.2 q^{75} -49956.6 q^{79} +6561.00 q^{81} +44858.9 q^{83} -5094.91 q^{85} +34116.5 q^{87} +127945. q^{89} +43625.9 q^{93} +158904. q^{95} -65685.9 q^{97} +11401.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{3} + 324 q^{9}+O(q^{10}) $ 4 * q - 36 * q^3 + 324 * q^9	$ 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}+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

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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−92.8255	−1.66051	−0.830257	−	0.557381i	$-0.811806\pi$
$5$	−92.8255	−1.66051	−0.830257	+	0.557381i	$0.811806\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	140.762	0.350756	0.175378	−	0.984501i	$-0.443885\pi$
$11$	140.762	0.350756	0.175378	+	0.984501i	$0.443885\pi$
$12$	0	0
$13$	−1111.24	−1.82369	−0.911844	−	0.410537i	$-0.865341\pi$
$13$	−1111.24	−1.82369	−0.911844	+	0.410537i	$0.865341\pi$
$14$	0	0
$15$	835.430	0.958698
$16$	0	0
$17$	54.8869	0.0460624	0.0230312	−	0.999735i	$-0.492668\pi$
$17$	54.8869	0.0460624	0.0230312	+	0.999735i	$0.492668\pi$
$18$	0	0
$19$	−1711.86	−1.08789	−0.543943	−	0.839122i	$-0.683069\pi$
$19$	−1711.86	−1.08789	−0.543943	+	0.839122i	$0.683069\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3287.91	−1.29599	−0.647993	−	0.761646i	$-0.724391\pi$
$23$	−3287.91	−1.29599	−0.647993	+	0.761646i	$0.724391\pi$
$24$	0	0
$25$	5491.58	1.75731
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−3790.72	−0.837003	−0.418501	−	0.908216i	$-0.637445\pi$
$29$	−3790.72	−0.837003	−0.418501	+	0.908216i	$0.637445\pi$
$30$	0	0
$31$	−4847.32	−0.905936	−0.452968	−	0.891527i	$-0.649635\pi$
$31$	−4847.32	−0.905936	−0.452968	+	0.891527i	$0.649635\pi$
$32$	0	0
$33$	−1266.86	−0.202509
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11366.7	−1.36499	−0.682496	−	0.730889i	$-0.739106\pi$
$37$	−11366.7	−1.36499	−0.682496	+	0.730889i	$0.739106\pi$
$38$	0	0
$39$	10001.2	1.05291
$40$	0	0
$41$	−10385.6	−0.964881	−0.482440	−	0.875929i	$-0.660249\pi$
$41$	−10385.6	−0.964881	−0.482440	+	0.875929i	$0.660249\pi$
$42$	0	0
$43$	7137.16	0.588646	0.294323	−	0.955706i	$-0.404906\pi$
$43$	7137.16	0.588646	0.294323	+	0.955706i	$0.404906\pi$
$44$	0	0
$45$	−7518.87	−0.553505
$46$	0	0
$47$	−16415.1	−1.08392	−0.541961	−	0.840404i	$-0.682318\pi$
$47$	−16415.1	−1.08392	−0.541961	+	0.840404i	$0.682318\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−493.983	−0.0265942
$52$	0	0
$53$	−20974.3	−1.02565	−0.512824	−	0.858494i	$-0.671401\pi$
$53$	−20974.3	−1.02565	−0.512824	+	0.858494i	$0.671401\pi$
$54$	0	0
$55$	−13066.3	−0.582435
$56$	0	0
$57$	15406.7	0.628092
$58$	0	0
$59$	−36211.9	−1.35432	−0.677161	−	0.735835i	$-0.736790\pi$
$59$	−36211.9	−1.35432	−0.677161	+	0.735835i	$0.736790\pi$
$60$	0	0
$61$	4948.70	0.170281	0.0851405	−	0.996369i	$-0.472866\pi$
$61$	4948.70	0.170281	0.0851405	+	0.996369i	$0.472866\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	103152.	3.02826
$66$	0	0
$67$	−22965.6	−0.625015	−0.312507	−	0.949915i	$-0.601169\pi$
$67$	−22965.6	−0.625015	−0.312507	+	0.949915i	$0.601169\pi$
$68$	0	0
$69$	29591.2	0.748238
$70$	0	0
$71$	−26341.8	−0.620154	−0.310077	−	0.950711i	$-0.600355\pi$
$71$	−26341.8	−0.620154	−0.310077	+	0.950711i	$0.600355\pi$
$72$	0	0
$73$	55387.4	1.21648	0.608238	−	0.793755i	$-0.291877\pi$
$73$	55387.4	1.21648	0.608238	+	0.793755i	$0.291877\pi$
$74$	0	0
$75$	−49424.2	−1.01458
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−49956.6	−0.900586	−0.450293	−	0.892881i	$-0.648680\pi$
$79$	−49956.6	−0.900586	−0.450293	+	0.892881i	$0.648680\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	44858.9	0.714749	0.357374	−	0.933961i	$-0.383672\pi$
$83$	44858.9	0.714749	0.357374	+	0.933961i	$0.383672\pi$
$84$	0	0
$85$	−5094.91	−0.0764873
$86$	0	0
$87$	34116.5	0.483244
$88$	0	0
$89$	127945.	1.71217	0.856087	−	0.516831i	$-0.172889\pi$
$89$	127945.	1.71217	0.856087	+	0.516831i	$0.172889\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	43625.9	0.523042
$94$	0	0
$95$	158904.	1.80645
$96$	0	0
$97$	−65685.9	−0.708831	−0.354415	−	0.935088i	$-0.615320\pi$
$97$	−65685.9	−0.708831	−0.354415	+	0.935088i	$0.615320\pi$
$98$	0	0
$99$	11401.8	0.116919
$100$	0	0
$101$	175783.	1.71464	0.857320	−	0.514784i	$-0.172128\pi$
$101$	175783.	1.71464	0.857320	+	0.514784i	$0.172128\pi$
$102$	0	0
$103$	−154109.	−1.43132	−0.715659	−	0.698450i	$-0.753873\pi$
$103$	−154109.	−1.43132	−0.715659	+	0.698450i	$0.753873\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	183362.	1.54828	0.774141	−	0.633013i	$-0.218182\pi$
$107$	183362.	1.54828	0.774141	+	0.633013i	$0.218182\pi$
$108$	0	0
$109$	−134645.	−1.08548	−0.542741	−	0.839900i	$-0.682613\pi$
$109$	−134645.	−1.08548	−0.542741	+	0.839900i	$0.682613\pi$
$110$	0	0
$111$	102300.	0.788079
$112$	0	0
$113$	176955.	1.30367	0.651833	−	0.758362i	$-0.274000\pi$
$113$	176955.	1.30367	0.651833	+	0.758362i	$0.274000\pi$
$114$	0	0
$115$	305202.	2.15200
$116$	0	0
$117$	−90010.7	−0.607896
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−141237.	−0.876970
$122$	0	0
$123$	93470.8	0.557074
$124$	0	0
$125$	−219679.	−1.25752
$126$	0	0
$127$	144432.	0.794608	0.397304	−	0.917687i	$-0.369946\pi$
$127$	144432.	0.794608	0.397304	+	0.917687i	$0.369946\pi$
$128$	0	0
$129$	−64234.4	−0.339855
$130$	0	0
$131$	−237508.	−1.20920	−0.604602	−	0.796527i	$-0.706668\pi$
$131$	−237508.	−1.20920	−0.604602	+	0.796527i	$0.706668\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	67669.8	0.319566
$136$	0	0
$137$	−96203.3	−0.437914	−0.218957	−	0.975735i	$-0.570265\pi$
$137$	−96203.3	−0.437914	−0.218957	+	0.975735i	$0.570265\pi$
$138$	0	0
$139$	−391373.	−1.71812	−0.859060	−	0.511874i	$-0.828951\pi$
$139$	−391373.	−1.71812	−0.859060	+	0.511874i	$0.828951\pi$
$140$	0	0
$141$	147736.	0.625802
$142$	0	0
$143$	−156421.	−0.639669
$144$	0	0
$145$	351876.	1.38985
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−80678.8	−0.297710	−0.148855	−	0.988859i	$-0.547559\pi$
$149$	−80678.8	−0.297710	−0.148855	+	0.988859i	$0.547559\pi$
$150$	0	0
$151$	95682.6	0.341500	0.170750	−	0.985314i	$-0.445381\pi$
$151$	95682.6	0.341500	0.170750	+	0.985314i	$0.445381\pi$
$152$	0	0
$153$	4445.84	0.0153541
$154$	0	0
$155$	449955.	1.50432
$156$	0	0
$157$	−195625.	−0.633397	−0.316699	−	0.948526i	$-0.602574\pi$
$157$	−195625.	−0.633397	−0.316699	+	0.948526i	$0.602574\pi$
$158$	0	0
$159$	188769.	0.592158
$160$	0	0
$161$	0	0
$162$	0	0
$163$	37645.7	0.110980	0.0554902	−	0.998459i	$-0.482328\pi$
$163$	37645.7	0.110980	0.0554902	+	0.998459i	$0.482328\pi$
$164$	0	0
$165$	117597.	0.336269
$166$	0	0
$167$	646778.	1.79458	0.897292	−	0.441437i	$-0.145531\pi$
$167$	646778.	1.79458	0.897292	+	0.441437i	$0.145531\pi$
$168$	0	0
$169$	863568.	2.32584
$170$	0	0
$171$	−138661.	−0.362629
$172$	0	0
$173$	−386908.	−0.982863	−0.491431	−	0.870916i	$-0.663526\pi$
$173$	−386908.	−0.982863	−0.491431	+	0.870916i	$0.663526\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	325907.	0.781918
$178$	0	0
$179$	27410.5	0.0639418	0.0319709	−	0.999489i	$-0.489822\pi$
$179$	27410.5	0.0639418	0.0319709	+	0.999489i	$0.489822\pi$
$180$	0	0
$181$	−196155.	−0.445044	−0.222522	−	0.974928i	$-0.571429\pi$
$181$	−196155.	−0.445044	−0.222522	+	0.974928i	$0.571429\pi$
$182$	0	0
$183$	−44538.3	−0.0983118
$184$	0	0
$185$	1.05512e6	2.26659
$186$	0	0
$187$	7726.02	0.0161567
$188$	0	0
$189$	0	0
$190$	0	0
$191$	266982.	0.529540	0.264770	−	0.964312i	$-0.414704\pi$
$191$	266982.	0.529540	0.264770	+	0.964312i	$0.414704\pi$
$192$	0	0
$193$	−101364.	−0.195880	−0.0979398	−	0.995192i	$-0.531225\pi$
$193$	−101364.	−0.195880	−0.0979398	+	0.995192i	$0.531225\pi$
$194$	0	0
$195$	−928366.	−1.74837
$196$	0	0
$197$	362366.	0.665245	0.332623	−	0.943060i	$-0.392066\pi$
$197$	362366.	0.665245	0.332623	+	0.943060i	$0.392066\pi$
$198$	0	0
$199$	225413.	0.403502	0.201751	−	0.979437i	$-0.435337\pi$
$199$	225413.	0.403502	0.201751	+	0.979437i	$0.435337\pi$
$200$	0	0
$201$	206690.	0.360852
$202$	0	0
$203$	0	0
$204$	0	0
$205$	964053.	1.60220
$206$	0	0
$207$	−266321.	−0.431995
$208$	0	0
$209$	−240965.	−0.381583
$210$	0	0
$211$	−327801.	−0.506878	−0.253439	−	0.967351i	$-0.581562\pi$
$211$	−327801.	−0.506878	−0.253439	+	0.967351i	$0.581562\pi$
$212$	0	0
$213$	237076.	0.358046
$214$	0	0
$215$	−662511.	−0.977455
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−498486.	−0.702333
$220$	0	0
$221$	−60992.7	−0.0840035
$222$	0	0
$223$	109690.	0.147708	0.0738538	−	0.997269i	$-0.476470\pi$
$223$	109690.	0.147708	0.0738538	+	0.997269i	$0.476470\pi$
$224$	0	0
$225$	444818.	0.585769
$226$	0	0
$227$	1.21747e6	1.56817	0.784083	−	0.620656i	$-0.213134\pi$
$227$	1.21747e6	1.56817	0.784083	+	0.620656i	$0.213134\pi$
$228$	0	0
$229$	−1.33603e6	−1.68355	−0.841776	−	0.539827i	$-0.818490\pi$
$229$	−1.33603e6	−1.68355	−0.841776	+	0.539827i	$0.818490\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	46788.4	0.0564610	0.0282305	−	0.999601i	$-0.491013\pi$
$233$	46788.4	0.0564610	0.0282305	+	0.999601i	$0.491013\pi$
$234$	0	0
$235$	1.52374e6	1.79987
$236$	0	0
$237$	449609.	0.519954
$238$	0	0
$239$	86350.2	0.0977841	0.0488921	−	0.998804i	$-0.484431\pi$
$239$	86350.2	0.0977841	0.0488921	+	0.998804i	$0.484431\pi$
$240$	0	0
$241$	825299.	0.915311	0.457655	−	0.889130i	$-0.348689\pi$
$241$	825299.	0.915311	0.457655	+	0.889130i	$0.348689\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.90229e6	1.98397
$248$	0	0
$249$	−403730.	−0.412660
$250$	0	0
$251$	−130188.	−0.130432	−0.0652162	−	0.997871i	$-0.520774\pi$
$251$	−130188.	−0.130432	−0.0652162	+	0.997871i	$0.520774\pi$
$252$	0	0
$253$	−462814.	−0.454574
$254$	0	0
$255$	45854.2	0.0441600
$256$	0	0
$257$	102790.	0.0970775	0.0485388	−	0.998821i	$-0.484544\pi$
$257$	102790.	0.0970775	0.0485388	+	0.998821i	$0.484544\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−307048.	−0.279001
$262$	0	0
$263$	−35500.7	−0.0316481	−0.0158241	−	0.999875i	$-0.505037\pi$
$263$	−35500.7	−0.0316481	−0.0158241	+	0.999875i	$0.505037\pi$
$264$	0	0
$265$	1.94695e6	1.70310
$266$	0	0
$267$	−1.15150e6	−0.988524
$268$	0	0
$269$	1.04634e6	0.881643	0.440821	−	0.897595i	$-0.354687\pi$
$269$	1.04634e6	0.881643	0.440821	+	0.897595i	$0.354687\pi$
$270$	0	0
$271$	177278.	0.146633	0.0733167	−	0.997309i	$-0.476642\pi$
$271$	177278.	0.146633	0.0733167	+	0.997309i	$0.476642\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	773008.	0.616385
$276$	0	0
$277$	145586.	0.114004	0.0570020	−	0.998374i	$-0.481846\pi$
$277$	145586.	0.114004	0.0570020	+	0.998374i	$0.481846\pi$
$278$	0	0
$279$	−392633.	−0.301979
$280$	0	0
$281$	−1.03073e6	−0.778714	−0.389357	−	0.921087i	$-0.627303\pi$
$281$	−1.03073e6	−0.778714	−0.389357	+	0.921087i	$0.627303\pi$
$282$	0	0
$283$	−1.11726e6	−0.829257	−0.414628	−	0.909991i	$-0.636088\pi$
$283$	−1.11726e6	−0.829257	−0.414628	+	0.909991i	$0.636088\pi$
$284$	0	0
$285$	−1.43014e6	−1.04296
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.41684e6	−0.997878
$290$	0	0
$291$	591173.	0.409244
$292$	0	0
$293$	−2.02032e6	−1.37484	−0.687418	−	0.726262i	$-0.741256\pi$
$293$	−2.02032e6	−1.37484	−0.687418	+	0.726262i	$0.741256\pi$
$294$	0	0
$295$	3.36139e6	2.24887
$296$	0	0
$297$	−102616.	−0.0675030
$298$	0	0
$299$	3.65367e6	2.36347
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.58204e6	−0.989948
$304$	0	0
$305$	−459366.	−0.282754
$306$	0	0
$307$	−535150.	−0.324063	−0.162031	−	0.986786i	$-0.551805\pi$
$307$	−535150.	−0.324063	−0.162031	+	0.986786i	$0.551805\pi$
$308$	0	0
$309$	1.38698e6	0.826372
$310$	0	0
$311$	−2.30019e6	−1.34854	−0.674268	−	0.738487i	$-0.735541\pi$
$311$	−2.30019e6	−1.34854	−0.674268	+	0.738487i	$0.735541\pi$
$312$	0	0
$313$	−2.75727e6	−1.59081	−0.795404	−	0.606079i	$-0.792742\pi$
$313$	−2.75727e6	−1.59081	−0.795404	+	0.606079i	$0.792742\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.54809e6	1.42418	0.712092	−	0.702086i	$-0.247748\pi$
$317$	2.54809e6	1.42418	0.712092	+	0.702086i	$0.247748\pi$
$318$	0	0
$319$	−533591.	−0.293584
$320$	0	0
$321$	−1.65026e6	−0.893901
$322$	0	0
$323$	−93958.7	−0.0501107
$324$	0	0
$325$	−6.10248e6	−3.20478
$326$	0	0
$327$	1.21180e6	0.626703
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−489335.	−0.245491	−0.122746	−	0.992438i	$-0.539170\pi$
$331$	−489335.	−0.245491	−0.122746	+	0.992438i	$0.539170\pi$
$332$	0	0
$333$	−920703.	−0.454998
$334$	0	0
$335$	2.13179e6	1.03785
$336$	0	0
$337$	−1.47140e6	−0.705757	−0.352878	−	0.935669i	$-0.614797\pi$
$337$	−1.47140e6	−0.705757	−0.352878	+	0.935669i	$0.614797\pi$
$338$	0	0
$339$	−1.59259e6	−0.752672
$340$	0	0
$341$	−682320.	−0.317762
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−2.74682e6	−1.24246
$346$	0	0
$347$	−2.41925e6	−1.07859	−0.539296	−	0.842116i	$-0.681310\pi$
$347$	−2.41925e6	−1.07859	−0.539296	+	0.842116i	$0.681310\pi$
$348$	0	0
$349$	−2.58571e6	−1.13636	−0.568180	−	0.822905i	$-0.692352\pi$
$349$	−2.58571e6	−1.13636	−0.568180	+	0.822905i	$0.692352\pi$
$350$	0	0
$351$	810096.	0.350969
$352$	0	0
$353$	1.00440e6	0.429010	0.214505	−	0.976723i	$-0.431186\pi$
$353$	1.00440e6	0.429010	0.214505	+	0.976723i	$0.431186\pi$
$354$	0	0
$355$	2.44519e6	1.02977
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.26765e6	−0.928625	−0.464313	−	0.885671i	$-0.653699\pi$
$359$	−2.26765e6	−0.928625	−0.464313	+	0.885671i	$0.653699\pi$
$360$	0	0
$361$	454359.	0.183498
$362$	0	0
$363$	1.27113e6	0.506319
$364$	0	0
$365$	−5.14136e6	−2.01998
$366$	0	0
$367$	4.78623e6	1.85493	0.927467	−	0.373905i	$-0.121981\pi$
$367$	4.78623e6	1.85493	0.927467	+	0.373905i	$0.121981\pi$
$368$	0	0
$369$	−841237.	−0.321627
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−4.62701e6	−1.72198	−0.860991	−	0.508619i	$-0.830156\pi$
$373$	−4.62701e6	−1.72198	−0.860991	+	0.508619i	$0.830156\pi$
$374$	0	0
$375$	1.97711e6	0.726028
$376$	0	0
$377$	4.21241e6	1.52643
$378$	0	0
$379$	−497760.	−0.178001	−0.0890004	−	0.996032i	$-0.528367\pi$
$379$	−497760.	−0.178001	−0.0890004	+	0.996032i	$0.528367\pi$
$380$	0	0
$381$	−1.29988e6	−0.458767
$382$	0	0
$383$	1.38749e6	0.483318	0.241659	−	0.970361i	$-0.422308\pi$
$383$	1.38749e6	0.483318	0.241659	+	0.970361i	$0.422308\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	578110.	0.196215
$388$	0	0
$389$	−586055.	−0.196365	−0.0981826	−	0.995168i	$-0.531303\pi$
$389$	−586055.	−0.196365	−0.0981826	+	0.995168i	$0.531303\pi$
$390$	0	0
$391$	−180463.	−0.0596962
$392$	0	0
$393$	2.13757e6	0.698135
$394$	0	0
$395$	4.63725e6	1.49544
$396$	0	0
$397$	3.05336e6	0.972302	0.486151	−	0.873875i	$-0.338400\pi$
$397$	3.05336e6	0.972302	0.486151	+	0.873875i	$0.338400\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.52316e6	1.71525	0.857624	−	0.514278i	$-0.171940\pi$
$401$	5.52316e6	1.71525	0.857624	+	0.514278i	$0.171940\pi$
$402$	0	0
$403$	5.38655e6	1.65214
$404$	0	0
$405$	−609028.	−0.184502
$406$	0	0
$407$	−1.60000e6	−0.478779
$408$	0	0
$409$	−3.34927e6	−0.990016	−0.495008	−	0.868888i	$-0.664835\pi$
$409$	−3.34927e6	−0.990016	−0.495008	+	0.868888i	$0.664835\pi$
$410$	0	0
$411$	865829.	0.252830
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.16405e6	−1.18685
$416$	0	0
$417$	3.52236e6	0.991957
$418$	0	0
$419$	2.97012e6	0.826493	0.413247	−	0.910619i	$-0.364395\pi$
$419$	2.97012e6	0.826493	0.413247	+	0.910619i	$0.364395\pi$
$420$	0	0
$421$	−5.41478e6	−1.48894	−0.744468	−	0.667658i	$-0.767297\pi$
$421$	−5.41478e6	−1.48894	−0.744468	+	0.667658i	$0.767297\pi$
$422$	0	0
$423$	−1.32962e6	−0.361307
$424$	0	0
$425$	301416.	0.0809458
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.40779e6	0.369313
$430$	0	0
$431$	−5.05950e6	−1.31194	−0.655970	−	0.754787i	$-0.727740\pi$
$431$	−5.05950e6	−1.31194	−0.655970	+	0.754787i	$0.727740\pi$
$432$	0	0
$433$	3.84174e6	0.984711	0.492355	−	0.870394i	$-0.336136\pi$
$433$	3.84174e6	0.984711	0.492355	+	0.870394i	$0.336136\pi$
$434$	0	0
$435$	−3.16688e6	−0.802433
$436$	0	0
$437$	5.62843e6	1.40989
$438$	0	0
$439$	−1.52572e6	−0.377844	−0.188922	−	0.981992i	$-0.560499\pi$
$439$	−1.52572e6	−0.377844	−0.188922	+	0.981992i	$0.560499\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.47014e6	−0.355916	−0.177958	−	0.984038i	$-0.556949\pi$
$443$	−1.47014e6	−0.355916	−0.177958	+	0.984038i	$0.556949\pi$
$444$	0	0
$445$	−1.18766e7	−2.84309
$446$	0	0
$447$	726110.	0.171883
$448$	0	0
$449$	−6.05071e6	−1.41641	−0.708207	−	0.706004i	$-0.750496\pi$
$449$	−6.05071e6	−1.41641	−0.708207	+	0.706004i	$0.750496\pi$
$450$	0	0
$451$	−1.46191e6	−0.338437
$452$	0	0
$453$	−861144.	−0.197165
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.75420e6	−1.28883	−0.644413	−	0.764678i	$-0.722898\pi$
$457$	−5.75420e6	−1.28883	−0.644413	+	0.764678i	$0.722898\pi$
$458$	0	0
$459$	−40012.6	−0.00886472
$460$	0	0
$461$	2.83684e6	0.621703	0.310851	−	0.950458i	$-0.399386\pi$
$461$	2.83684e6	0.621703	0.310851	+	0.950458i	$0.399386\pi$
$462$	0	0
$463$	5.19089e6	1.12535	0.562677	−	0.826677i	$-0.309771\pi$
$463$	5.19089e6	1.12535	0.562677	+	0.826677i	$0.309771\pi$
$464$	0	0
$465$	−4.04960e6	−0.868519
$466$	0	0
$467$	1.08904e6	0.231074	0.115537	−	0.993303i	$-0.463141\pi$
$467$	1.08904e6	0.231074	0.115537	+	0.993303i	$0.463141\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1.76063e6	0.365692
$472$	0	0
$473$	1.00464e6	0.206471
$474$	0	0
$475$	−9.40081e6	−1.91175
$476$	0	0
$477$	−1.69892e6	−0.341883
$478$	0	0
$479$	−5.69411e6	−1.13393	−0.566966	−	0.823741i	$-0.691883\pi$
$479$	−5.69411e6	−1.13393	−0.566966	+	0.823741i	$0.691883\pi$
$480$	0	0
$481$	1.26312e7	2.48932
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.09733e6	1.17702
$486$	0	0
$487$	−4.11425e6	−0.786083	−0.393042	−	0.919521i	$-0.628577\pi$
$487$	−4.11425e6	−0.786083	−0.393042	+	0.919521i	$0.628577\pi$
$488$	0	0
$489$	−338811.	−0.0640745
$490$	0	0
$491$	−609530.	−0.114101	−0.0570507	−	0.998371i	$-0.518170\pi$
$491$	−609530.	−0.114101	−0.0570507	+	0.998371i	$0.518170\pi$
$492$	0	0
$493$	−208061.	−0.0385544
$494$	0	0
$495$	−1.05837e6	−0.194145
$496$	0	0
$497$	0	0
$498$	0	0
$499$	995793.	0.179027	0.0895133	−	0.995986i	$-0.471469\pi$
$499$	995793.	0.179027	0.0895133	+	0.995986i	$0.471469\pi$
$500$	0	0
$501$	−5.82100e6	−1.03610
$502$	0	0
$503$	1.02730e7	1.81042	0.905209	−	0.424966i	$-0.139714\pi$
$503$	1.02730e7	1.81042	0.905209	+	0.424966i	$0.139714\pi$
$504$	0	0
$505$	−1.63171e7	−2.84718
$506$	0	0
$507$	−7.77211e6	−1.34282
$508$	0	0
$509$	−2.78427e6	−0.476339	−0.238170	−	0.971224i	$-0.576547\pi$
$509$	−2.78427e6	−0.476339	−0.238170	+	0.971224i	$0.576547\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.24794e6	0.209364
$514$	0	0
$515$	1.43053e7	2.37672
$516$	0	0
$517$	−2.31062e6	−0.380192
$518$	0	0
$519$	3.48218e6	0.567456
$520$	0	0
$521$	1.04536e6	0.168721	0.0843607	−	0.996435i	$-0.473115\pi$
$521$	1.04536e6	0.168721	0.0843607	+	0.996435i	$0.473115\pi$
$522$	0	0
$523$	−3.63156e6	−0.580550	−0.290275	−	0.956943i	$-0.593747\pi$
$523$	−3.63156e6	−0.580550	−0.290275	+	0.956943i	$0.593747\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−266055.	−0.0417296
$528$	0	0
$529$	4.37400e6	0.679579
$530$	0	0
$531$	−2.93317e6	−0.451440
$532$	0	0
$533$	1.15410e7	1.75964
$534$	0	0
$535$	−1.70207e7	−2.57094
$536$	0	0
$537$	−246695.	−0.0369168
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−9.44884e6	−1.38799	−0.693994	−	0.719981i	$-0.744150\pi$
$541$	−9.44884e6	−1.38799	−0.693994	+	0.719981i	$0.744150\pi$
$542$	0	0
$543$	1.76540e6	0.256946
$544$	0	0
$545$	1.24985e7	1.80246
$546$	0	0
$547$	2.69014e6	0.384421	0.192210	−	0.981354i	$-0.438434\pi$
$547$	2.69014e6	0.384421	0.192210	+	0.981354i	$0.438434\pi$
$548$	0	0
$549$	400845.	0.0567604
$550$	0	0
$551$	6.48918e6	0.910564
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−9.49609e6	−1.30862
$556$	0	0
$557$	−8.88981e6	−1.21410	−0.607050	−	0.794664i	$-0.707647\pi$
$557$	−8.88981e6	−1.21410	−0.607050	+	0.794664i	$0.707647\pi$
$558$	0	0
$559$	−7.93112e6	−1.07351
$560$	0	0
$561$	−69534.1	−0.00932805
$562$	0	0
$563$	1.25993e6	0.167523	0.0837615	−	0.996486i	$-0.473307\pi$
$563$	1.25993e6	0.167523	0.0837615	+	0.996486i	$0.473307\pi$
$564$	0	0
$565$	−1.64259e7	−2.16476
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.82038e6	−0.753651	−0.376826	−	0.926284i	$-0.622984\pi$
$569$	−5.82038e6	−0.753651	−0.376826	+	0.926284i	$0.622984\pi$
$570$	0	0
$571$	4.65961e6	0.598080	0.299040	−	0.954241i	$-0.403334\pi$
$571$	4.65961e6	0.598080	0.299040	+	0.954241i	$0.403334\pi$
$572$	0	0
$573$	−2.40284e6	−0.305730
$574$	0	0
$575$	−1.80558e7	−2.27744
$576$	0	0
$577$	−1.52478e6	−0.190664	−0.0953319	−	0.995446i	$-0.530391\pi$
$577$	−1.52478e6	−0.190664	−0.0953319	+	0.995446i	$0.530391\pi$
$578$	0	0
$579$	912273.	0.113091
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.95240e6	−0.359752
$584$	0	0
$585$	8.35529e6	1.00942
$586$	0	0
$587$	−1.50087e7	−1.79783	−0.898914	−	0.438124i	$-0.855643\pi$
$587$	−1.50087e7	−1.79783	−0.898914	+	0.438124i	$0.855643\pi$
$588$	0	0
$589$	8.29792e6	0.985556
$590$	0	0
$591$	−3.26129e6	−0.384079
$592$	0	0
$593$	−7.07946e6	−0.826729	−0.413364	−	0.910566i	$-0.635646\pi$
$593$	−7.07946e6	−0.826729	−0.413364	+	0.910566i	$0.635646\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.02871e6	−0.232962
$598$	0	0
$599$	−2.17521e6	−0.247705	−0.123853	−	0.992301i	$-0.539525\pi$
$599$	−2.17521e6	−0.247705	−0.123853	+	0.992301i	$0.539525\pi$
$600$	0	0
$601$	2.69785e6	0.304671	0.152336	−	0.988329i	$-0.451321\pi$
$601$	2.69785e6	0.304671	0.152336	+	0.988329i	$0.451321\pi$
$602$	0	0
$603$	−1.86021e6	−0.208338
$604$	0	0
$605$	1.31104e7	1.45622
$606$	0	0
$607$	4.98834e6	0.549521	0.274760	−	0.961513i	$-0.411402\pi$
$607$	4.98834e6	0.549521	0.274760	+	0.961513i	$0.411402\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.82411e7	1.97673
$612$	0	0
$613$	1.11364e6	0.119700	0.0598500	−	0.998207i	$-0.480938\pi$
$613$	1.11364e6	0.119700	0.0598500	+	0.998207i	$0.480938\pi$
$614$	0	0
$615$	−8.67647e6	−0.925029
$616$	0	0
$617$	−5.14757e6	−0.544364	−0.272182	−	0.962246i	$-0.587745\pi$
$617$	−5.14757e6	−0.544364	−0.272182	+	0.962246i	$0.587745\pi$
$618$	0	0
$619$	243176.	0.0255091	0.0127545	−	0.999919i	$-0.495940\pi$
$619$	243176.	0.0255091	0.0127545	+	0.999919i	$0.495940\pi$
$620$	0	0
$621$	2.39689e6	0.249413
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3.23066e6	0.330819
$626$	0	0
$627$	2.16869e6	0.220307
$628$	0	0
$629$	−623884.	−0.0628749
$630$	0	0
$631$	9.94255e6	0.994087	0.497044	−	0.867726i	$-0.334419\pi$
$631$	9.94255e6	0.994087	0.497044	+	0.867726i	$0.334419\pi$
$632$	0	0
$633$	2.95021e6	0.292646
$634$	0	0
$635$	−1.34069e7	−1.31946
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.13369e6	−0.206718
$640$	0	0
$641$	−1.15553e7	−1.11080	−0.555402	−	0.831582i	$-0.687436\pi$
$641$	−1.15553e7	−1.11080	−0.555402	+	0.831582i	$0.687436\pi$
$642$	0	0
$643$	1.35139e6	0.128900	0.0644500	−	0.997921i	$-0.479471\pi$
$643$	1.35139e6	0.128900	0.0644500	+	0.997921i	$0.479471\pi$
$644$	0	0
$645$	5.96260e6	0.564334
$646$	0	0
$647$	1.75536e7	1.64856	0.824279	−	0.566183i	$-0.191581\pi$
$647$	1.75536e7	1.64856	0.824279	+	0.566183i	$0.191581\pi$
$648$	0	0
$649$	−5.09728e6	−0.475036
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.99873e6	−0.734071	−0.367035	−	0.930207i	$-0.619627\pi$
$653$	−7.99873e6	−0.734071	−0.367035	+	0.930207i	$0.619627\pi$
$654$	0	0
$655$	2.20468e7	2.00790
$656$	0	0
$657$	4.48638e6	0.405492
$658$	0	0
$659$	229419.	0.0205786	0.0102893	−	0.999947i	$-0.496725\pi$
$659$	229419.	0.0205786	0.0102893	+	0.999947i	$0.496725\pi$
$660$	0	0
$661$	−3.28557e6	−0.292488	−0.146244	−	0.989249i	$-0.546718\pi$
$661$	−3.28557e6	−0.292488	−0.146244	+	0.989249i	$0.546718\pi$
$662$	0	0
$663$	548935.	0.0484994
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.24635e7	1.08474
$668$	0	0
$669$	−987206.	−0.0852791
$670$	0	0
$671$	696590.	0.0597271
$672$	0	0
$673$	8.22188e6	0.699734	0.349867	−	0.936799i	$-0.386227\pi$
$673$	8.22188e6	0.699734	0.349867	+	0.936799i	$0.386227\pi$
$674$	0	0
$675$	−4.00336e6	−0.338194
$676$	0	0
$677$	2.15405e7	1.80628	0.903138	−	0.429351i	$-0.141258\pi$
$677$	2.15405e7	1.80628	0.903138	+	0.429351i	$0.141258\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.09572e7	−0.905381
$682$	0	0
$683$	−1.61741e7	−1.32669	−0.663344	−	0.748314i	$-0.730863\pi$
$683$	−1.61741e7	−1.32669	−0.663344	+	0.748314i	$0.730863\pi$
$684$	0	0
$685$	8.93012e6	0.727162
$686$	0	0
$687$	1.20242e7	0.971999
$688$	0	0
$689$	2.33076e7	1.87046
$690$	0	0
$691$	1.06464e7	0.848215	0.424108	−	0.905612i	$-0.360588\pi$
$691$	1.06464e7	0.848215	0.424108	+	0.905612i	$0.360588\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.63294e7	2.85296
$696$	0	0
$697$	−570036.	−0.0444447
$698$	0	0
$699$	−421096.	−0.0325978
$700$	0	0
$701$	−4.55461e6	−0.350071	−0.175035	−	0.984562i	$-0.556004\pi$
$701$	−4.55461e6	−0.350071	−0.175035	+	0.984562i	$0.556004\pi$
$702$	0	0
$703$	1.94582e7	1.48496
$704$	0	0
$705$	−1.37136e7	−1.03915
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.31035e7	0.978974	0.489487	−	0.872011i	$-0.337184\pi$
$709$	1.31035e7	0.978974	0.489487	+	0.872011i	$0.337184\pi$
$710$	0	0
$711$	−4.04649e6	−0.300195
$712$	0	0
$713$	1.59375e7	1.17408
$714$	0	0
$715$	1.45199e7	1.06218
$716$	0	0
$717$	−777151.	−0.0564557
$718$	0	0
$719$	1.79188e7	1.29267	0.646335	−	0.763054i	$-0.276301\pi$
$719$	1.79188e7	1.29267	0.646335	+	0.763054i	$0.276301\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−7.42769e6	−0.528455
$724$	0	0
$725$	−2.08171e7	−1.47087
$726$	0	0
$727$	1.41466e7	0.992693	0.496347	−	0.868125i	$-0.334675\pi$
$727$	1.41466e7	0.992693	0.496347	+	0.868125i	$0.334675\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	391737.	0.0271145
$732$	0	0
$733$	−1.24448e7	−0.855517	−0.427758	−	0.903893i	$-0.640697\pi$
$733$	−1.24448e7	−0.855517	−0.427758	+	0.903893i	$0.640697\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.23269e6	−0.219228
$738$	0	0
$739$	−1.76664e6	−0.118998	−0.0594988	−	0.998228i	$-0.518950\pi$
$739$	−1.76664e6	−0.118998	−0.0594988	+	0.998228i	$0.518950\pi$
$740$	0	0
$741$	−1.71206e7	−1.14544
$742$	0	0
$743$	5.73250e6	0.380953	0.190477	−	0.981692i	$-0.438997\pi$
$743$	5.73250e6	0.380953	0.190477	+	0.981692i	$0.438997\pi$
$744$	0	0
$745$	7.48906e6	0.494352
$746$	0	0
$747$	3.63357e6	0.238250
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.63132e7	1.70245	0.851224	−	0.524803i	$-0.175861\pi$
$751$	2.63132e7	1.70245	0.851224	+	0.524803i	$0.175861\pi$
$752$	0	0
$753$	1.17169e6	0.0753052
$754$	0	0
$755$	−8.88179e6	−0.567066
$756$	0	0
$757$	1.63104e7	1.03449	0.517245	−	0.855838i	$-0.326958\pi$
$757$	1.63104e7	1.03449	0.517245	+	0.855838i	$0.326958\pi$
$758$	0	0
$759$	4.16532e6	0.262449
$760$	0	0
$761$	1.68416e7	1.05420	0.527098	−	0.849804i	$-0.323280\pi$
$761$	1.68416e7	1.05420	0.527098	+	0.849804i	$0.323280\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−412688.	−0.0254958
$766$	0	0
$767$	4.02402e7	2.46986
$768$	0	0
$769$	1.19890e7	0.731084	0.365542	−	0.930795i	$-0.380884\pi$
$769$	1.19890e7	0.731084	0.365542	+	0.930795i	$0.380884\pi$
$770$	0	0
$771$	−925112.	−0.0560477
$772$	0	0
$773$	1.69774e7	1.02193	0.510965	−	0.859601i	$-0.329288\pi$
$773$	1.69774e7	1.02193	0.510965	+	0.859601i	$0.329288\pi$
$774$	0	0
$775$	−2.66195e7	−1.59201
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.77787e7	1.04968
$780$	0	0
$781$	−3.70793e6	−0.217523
$782$	0	0
$783$	2.76344e6	0.161081
$784$	0	0
$785$	1.81590e7	1.05177
$786$	0	0
$787$	−3.19500e7	−1.83880	−0.919400	−	0.393323i	$-0.871325\pi$
$787$	−3.19500e7	−1.83880	−0.919400	+	0.393323i	$0.871325\pi$
$788$	0	0
$789$	319507.	0.0182720
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5.49921e6	−0.310540
$794$	0	0
$795$	−1.75226e7	−0.983287
$796$	0	0
$797$	−1.63006e7	−0.908988	−0.454494	−	0.890750i	$-0.650180\pi$
$797$	−1.63006e7	−0.908988	−0.454494	+	0.890750i	$0.650180\pi$
$798$	0	0
$799$	−900972.	−0.0499280
$800$	0	0
$801$	1.03635e7	0.570725
$802$	0	0
$803$	7.79646e6	0.426686
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.41707e6	−0.509017
$808$	0	0
$809$	−2.51398e7	−1.35049	−0.675244	−	0.737594i	$-0.735962\pi$
$809$	−2.51398e7	−1.35049	−0.675244	+	0.737594i	$0.735962\pi$
$810$	0	0
$811$	−2.27245e7	−1.21323	−0.606613	−	0.794997i	$-0.707472\pi$
$811$	−2.27245e7	−1.21323	−0.606613	+	0.794997i	$0.707472\pi$
$812$	0	0
$813$	−1.59551e6	−0.0846588
$814$	0	0
$815$	−3.49448e6	−0.184284
$816$	0	0
$817$	−1.22178e7	−0.640380
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.10488e7	0.572080	0.286040	−	0.958218i	$-0.407661\pi$
$821$	1.10488e7	0.572080	0.286040	+	0.958218i	$0.407661\pi$
$822$	0	0
$823$	−7.30479e6	−0.375931	−0.187966	−	0.982176i	$-0.560189\pi$
$823$	−7.30479e6	−0.375931	−0.187966	+	0.982176i	$0.560189\pi$
$824$	0	0
$825$	−6.95707e6	−0.355870
$826$	0	0
$827$	3.01208e7	1.53145	0.765724	−	0.643170i	$-0.222381\pi$
$827$	3.01208e7	1.53145	0.765724	+	0.643170i	$0.222381\pi$
$828$	0	0
$829$	1.35804e7	0.686318	0.343159	−	0.939277i	$-0.388503\pi$
$829$	1.35804e7	0.686318	0.343159	+	0.939277i	$0.388503\pi$
$830$	0	0
$831$	−1.31027e6	−0.0658203
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−6.00375e7	−2.97993
$836$	0	0
$837$	3.53370e6	0.174347
$838$	0	0
$839$	−3.78208e7	−1.85492	−0.927460	−	0.373922i	$-0.878013\pi$
$839$	−3.78208e7	−1.85492	−0.927460	+	0.373922i	$0.878013\pi$
$840$	0	0
$841$	−6.14158e6	−0.299426
$842$	0	0
$843$	9.27654e6	0.449591
$844$	0	0
$845$	−8.01611e7	−3.86209
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.00554e7	0.478772
$850$	0	0
$851$	3.73727e7	1.76901
$852$	0	0
$853$	−3.36325e7	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$853$	−3.36325e7	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$854$	0	0
$855$	1.28712e7	0.602150
$856$	0	0
$857$	−1.71796e7	−0.799028	−0.399514	−	0.916727i	$-0.630821\pi$
$857$	−1.71796e7	−0.799028	−0.399514	+	0.916727i	$0.630821\pi$
$858$	0	0
$859$	2.49154e7	1.15209	0.576043	−	0.817420i	$-0.304596\pi$
$859$	2.49154e7	1.15209	0.576043	+	0.817420i	$0.304596\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.70839e7	−1.69495	−0.847477	−	0.530832i	$-0.821880\pi$
$863$	−3.70839e7	−1.69495	−0.847477	+	0.530832i	$0.821880\pi$
$864$	0	0
$865$	3.59150e7	1.63206
$866$	0	0
$867$	1.27516e7	0.576125
$868$	0	0
$869$	−7.03201e6	−0.315886
$870$	0	0
$871$	2.55203e7	1.13983
$872$	0	0
$873$	−5.32055e6	−0.236277
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.35684e7	1.03474	0.517370	−	0.855762i	$-0.326911\pi$
$877$	2.35684e7	1.03474	0.517370	+	0.855762i	$0.326911\pi$
$878$	0	0
$879$	1.81829e7	0.793762
$880$	0	0
$881$	3.52124e7	1.52847	0.764234	−	0.644939i	$-0.223117\pi$
$881$	3.52124e7	1.52847	0.764234	+	0.644939i	$0.223117\pi$
$882$	0	0
$883$	4.08063e7	1.76127	0.880634	−	0.473796i	$-0.157117\pi$
$883$	4.08063e7	1.76127	0.880634	+	0.473796i	$0.157117\pi$
$884$	0	0
$885$	−3.02525e7	−1.29839
$886$	0	0
$887$	6.02857e6	0.257280	0.128640	−	0.991691i	$-0.458939\pi$
$887$	6.02857e6	0.257280	0.128640	+	0.991691i	$0.458939\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	923542.	0.0389729
$892$	0	0
$893$	2.81003e7	1.17918
$894$	0	0
$895$	−2.54440e6	−0.106176
$896$	0	0
$897$	−3.28830e7	−1.36455
$898$	0	0
$899$	1.83748e7	0.758271
$900$	0	0
$901$	−1.15122e6	−0.0472438
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.82082e7	0.739002
$906$	0	0
$907$	1.42799e7	0.576377	0.288188	−	0.957574i	$-0.406947\pi$
$907$	1.42799e7	0.576377	0.288188	+	0.957574i	$0.406947\pi$
$908$	0	0
$909$	1.42384e7	0.571546
$910$	0	0
$911$	2.11130e7	0.842855	0.421428	−	0.906862i	$-0.361529\pi$
$911$	2.11130e7	0.842855	0.421428	+	0.906862i	$0.361529\pi$
$912$	0	0
$913$	6.31445e6	0.250702
$914$	0	0
$915$	4.13429e6	0.163248
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.65736e7	0.647334	0.323667	−	0.946171i	$-0.395084\pi$
$919$	1.65736e7	0.647334	0.323667	+	0.946171i	$0.395084\pi$
$920$	0	0
$921$	4.81635e6	0.187098
$922$	0	0
$923$	2.92721e7	1.13097
$924$	0	0
$925$	−6.24212e7	−2.39871
$926$	0	0
$927$	−1.24829e7	−0.477106
$928$	0	0
$929$	1.63934e7	0.623203	0.311601	−	0.950213i	$-0.399135\pi$
$929$	1.63934e7	0.623203	0.311601	+	0.950213i	$0.399135\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.07017e7	0.778577
$934$	0	0
$935$	−717172.	−0.0268284
$936$	0	0
$937$	−2.68057e7	−0.997420	−0.498710	−	0.866769i	$-0.666193\pi$
$937$	−2.68057e7	−0.997420	−0.498710	+	0.866769i	$0.666193\pi$
$938$	0	0
$939$	2.48154e7	0.918454
$940$	0	0
$941$	−5.62875e6	−0.207223	−0.103611	−	0.994618i	$-0.533040\pi$
$941$	−5.62875e6	−0.207223	−0.103611	+	0.994618i	$0.533040\pi$
$942$	0	0
$943$	3.41470e7	1.25047
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.76782e7	−1.00291	−0.501456	−	0.865183i	$-0.667202\pi$
$947$	−2.76782e7	−1.00291	−0.501456	+	0.865183i	$0.667202\pi$
$948$	0	0
$949$	−6.15488e7	−2.21847
$950$	0	0
$951$	−2.29328e7	−0.822253
$952$	0	0
$953$	−2.40114e7	−0.856416	−0.428208	−	0.903680i	$-0.640855\pi$
$953$	−2.40114e7	−0.856416	−0.428208	+	0.903680i	$0.640855\pi$
$954$	0	0
$955$	−2.47828e7	−0.879309
$956$	0	0
$957$	4.80232e6	0.169501
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−5.13264e6	−0.179280
$962$	0	0
$963$	1.48523e7	0.516094
$964$	0	0
$965$	9.40914e6	0.325261
$966$	0	0
$967$	−1.43801e7	−0.494535	−0.247267	−	0.968947i	$-0.579533\pi$
$967$	−1.43801e7	−0.494535	−0.247267	+	0.968947i	$0.579533\pi$
$968$	0	0
$969$	845628.	0.0289314
$970$	0	0
$971$	−4.87782e7	−1.66027	−0.830133	−	0.557566i	$-0.811735\pi$
$971$	−4.87782e7	−1.66027	−0.830133	+	0.557566i	$0.811735\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	5.49223e7	1.85028
$976$	0	0
$977$	−4.70305e7	−1.57632	−0.788158	−	0.615472i	$-0.788965\pi$
$977$	−4.70305e7	−1.57632	−0.788158	+	0.615472i	$0.788965\pi$
$978$	0	0
$979$	1.80098e7	0.600555
$980$	0	0
$981$	−1.09062e7	−0.361827
$982$	0	0
$983$	7.44416e6	0.245715	0.122858	−	0.992424i	$-0.460794\pi$
$983$	7.44416e6	0.245715	0.122858	+	0.992424i	$0.460794\pi$
$984$	0	0
$985$	−3.36368e7	−1.10465
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.34663e7	−0.762877
$990$	0	0
$991$	−4.62294e7	−1.49532	−0.747659	−	0.664082i	$-0.768822\pi$
$991$	−4.62294e7	−1.49532	−0.747659	+	0.664082i	$0.768822\pi$
$992$	0	0
$993$	4.40401e6	0.141734
$994$	0	0
$995$	−2.09241e7	−0.670021
$996$	0	0
$997$	−6.18781e6	−0.197151	−0.0985755	−	0.995130i	$-0.531429\pi$
$997$	−6.18781e6	−0.197151	−0.0985755	+	0.995130i	$0.531429\pi$
$998$	0	0
$999$	8.28633e6	0.262693

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.6.a.n.1.1		4
7.2	even	3		84.6.i.c.25.4	✓	8
7.3	odd	6		588.6.i.o.373.1		8
7.4	even	3		84.6.i.c.37.4	yes	8
7.5	odd	6		588.6.i.o.361.1		8
7.6	odd	2		588.6.a.p.1.4		4
21.2	odd	6		252.6.k.f.109.1		8
21.11	odd	6		252.6.k.f.37.1		8
28.11	odd	6		336.6.q.i.289.4		8
28.23	odd	6		336.6.q.i.193.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.i.c.25.4	✓	8	7.2	even	3
84.6.i.c.37.4	yes	8	7.4	even	3
252.6.k.f.37.1		8	21.11	odd	6
252.6.k.f.109.1		8	21.2	odd	6
336.6.q.i.193.4		8	28.23	odd	6
336.6.q.i.289.4		8	28.11	odd	6
588.6.a.n.1.1		4	1.1	even	1	trivial
588.6.a.p.1.4		4	7.6	odd	2
588.6.i.o.361.1		8	7.5	odd	6
588.6.i.o.373.1		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -92.8255 q^{5} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -92.8255 q^{5} +81.0000 q^{9} +140.762 q^{11} -1111.24 q^{13} +835.430 q^{15} +54.8869 q^{17} -1711.86 q^{19} -3287.91 q^{23} +5491.58 q^{25} -729.000 q^{27} -3790.72 q^{29} -4847.32 q^{31} -1266.86 q^{33} -11366.7 q^{37} +10001.2 q^{39} -10385.6 q^{41} +7137.16 q^{43} -7518.87 q^{45} -16415.1 q^{47} -493.983 q^{51} -20974.3 q^{53} -13066.3 q^{55} +15406.7 q^{57} -36211.9 q^{59} +4948.70 q^{61} +103152. q^{65} -22965.6 q^{67} +29591.2 q^{69} -26341.8 q^{71} +55387.4 q^{73} -49424.2 q^{75} -49956.6 q^{79} +6561.00 q^{81} +44858.9 q^{83} -5094.91 q^{85} +34116.5 q^{87} +127945. q^{89} +43625.9 q^{93} +158904. q^{95} -65685.9 q^{97} +11401.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{3} + 324 q^{9}+O(q^{10}) \) 4 * q - 36 * q^3 + 324 * q^9	\( 4 q - 36 q^{3} + 324 q^{9} + 462 q^{11} - 602 q^{13} - 228 q^{17} - 358 q^{19} + 2148 q^{23} + 5454 q^{25} - 2916 q^{27} - 5532 q^{29} - 830 q^{31} - 4158 q^{33} + 3914 q^{37} + 5418 q^{39} - 8316 q^{41} - 14518 q^{43} - 41700 q^{47} + 2052 q^{51} - 22164 q^{53} + 3892 q^{55} + 3222 q^{57} - 32886 q^{59} - 83732 q^{61} + 93192 q^{65} + 80034 q^{67} - 19332 q^{69} + 44772 q^{71} + 22470 q^{73} - 49086 q^{75} + 75286 q^{79} + 26244 q^{81} - 17418 q^{83} + 139252 q^{85} + 49788 q^{87} - 28944 q^{89} + 7470 q^{93} + 144120 q^{95} - 216678 q^{97} + 37422 q^{99}+O(q^{100}) \) 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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