LMFDB - Embedded newform 531.6.a.b.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,6,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("531.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	531.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:	$85.1638083207$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5 x^{10} - 238 x^{9} + 1067 x^{8} + 20782 x^{7} - 79077 x^{6} - 813818 x^{5} + 2364885 x^{4} + \cdots - 14846072 $ x^11 - 5x^10 - 238x^9 + 1067x^8 + 20782x^7 - 79077x^6 - 813818x^5 + 2364885x^4 + 13849341x^3 - 23890558x^2 - 74443300x - 14846072
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-8.44473$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.44473 q^{2} +57.2030 q^{4} -13.7903 q^{5} +67.4858 q^{7} +238.036 q^{8} +O(q^{10})$	\(q+9.44473 q^{2} +57.2030 q^{4} -13.7903 q^{5} +67.4858 q^{7} +238.036 q^{8} -130.245 q^{10} +138.804 q^{11} -485.356 q^{13} +637.385 q^{14} +417.688 q^{16} +2224.39 q^{17} +1850.55 q^{19} -788.844 q^{20} +1310.97 q^{22} +1008.66 q^{23} -2934.83 q^{25} -4584.06 q^{26} +3860.39 q^{28} -1534.86 q^{29} +5924.08 q^{31} -3672.19 q^{32} +21008.8 q^{34} -930.646 q^{35} +1074.65 q^{37} +17477.9 q^{38} -3282.57 q^{40} +10573.5 q^{41} +15070.3 q^{43} +7940.01 q^{44} +9526.48 q^{46} +14991.6 q^{47} -12252.7 q^{49} -27718.7 q^{50} -27763.8 q^{52} +40821.3 q^{53} -1914.14 q^{55} +16064.0 q^{56} -14496.4 q^{58} -3481.00 q^{59} -34218.1 q^{61} +55951.4 q^{62} -48048.9 q^{64} +6693.19 q^{65} +18426.3 q^{67} +127242. q^{68} -8789.71 q^{70} -17658.3 q^{71} -25695.8 q^{73} +10149.8 q^{74} +105857. q^{76} +9367.30 q^{77} -13512.5 q^{79} -5760.02 q^{80} +99863.6 q^{82} +80373.4 q^{83} -30675.0 q^{85} +142335. q^{86} +33040.3 q^{88} -43704.2 q^{89} -32754.7 q^{91} +57698.1 q^{92} +141591. q^{94} -25519.5 q^{95} +67602.6 q^{97} -115723. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) $ 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8	$ 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8} - 399 q^{10} + 698 q^{11} - 1556 q^{13} + 1679 q^{14} - 2662 q^{16} + 4793 q^{17} - 3753 q^{19} + 11023 q^{20} - 9534 q^{22} + 7323 q^{23} + 7867 q^{25} + 4844 q^{26} + 3650 q^{28} + 15467 q^{29} - 5151 q^{31} + 15368 q^{32} + 8452 q^{34} + 23285 q^{35} + 8623 q^{37} - 15205 q^{38} + 41530 q^{40} + 6369 q^{41} - 20506 q^{43} + 55632 q^{44} - 45191 q^{46} + 47899 q^{47} - 10322 q^{49} + 102147 q^{50} - 292 q^{52} + 80048 q^{53} - 2114 q^{55} + 108126 q^{56} - 58294 q^{58} - 38291 q^{59} - 82527 q^{61} + 67438 q^{62} - 51411 q^{64} + 167646 q^{65} - 166976 q^{67} + 136533 q^{68} + 76140 q^{70} + 183560 q^{71} - 36809 q^{73} + 116686 q^{74} + 55580 q^{76} + 164885 q^{77} - 281518 q^{79} + 32683 q^{80} + 178815 q^{82} + 254691 q^{83} + 4763 q^{85} - 349324 q^{86} + 251285 q^{88} + 89687 q^{89} + 34897 q^{91} + 20240 q^{92} + 96548 q^{94} + 155113 q^{95} - 45828 q^{97} - 465864 q^{98}+O(q^{100}) $ 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8 - 399 * q^10 + 698 * q^11 - 1556 * q^13 + 1679 * q^14 - 2662 * q^16 + 4793 * q^17 - 3753 * q^19 + 11023 * q^20 - 9534 * q^22 + 7323 * q^23 + 7867 * q^25 + 4844 * q^26 + 3650 * q^28 + 15467 * q^29 - 5151 * q^31 + 15368 * q^32 + 8452 * q^34 + 23285 * q^35 + 8623 * q^37 - 15205 * q^38 + 41530 * q^40 + 6369 * q^41 - 20506 * q^43 + 55632 * q^44 - 45191 * q^46 + 47899 * q^47 - 10322 * q^49 + 102147 * q^50 - 292 * q^52 + 80048 * q^53 - 2114 * q^55 + 108126 * q^56 - 58294 * q^58 - 38291 * q^59 - 82527 * q^61 + 67438 * q^62 - 51411 * q^64 + 167646 * q^65 - 166976 * q^67 + 136533 * q^68 + 76140 * q^70 + 183560 * q^71 - 36809 * q^73 + 116686 * q^74 + 55580 * q^76 + 164885 * q^77 - 281518 * q^79 + 32683 * q^80 + 178815 * q^82 + 254691 * q^83 + 4763 * q^85 - 349324 * q^86 + 251285 * q^88 + 89687 * q^89 + 34897 * q^91 + 20240 * q^92 + 96548 * q^94 + 155113 * q^95 - 45828 * q^97 - 465864 * q^98

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 $

$2$	9.44473	1.66961	0.834804	−	0.550547i	$-0.185581\pi$
$2$	9.44473	1.66961	0.834804	+	0.550547i	$0.185581\pi$
$3$	0	0
$3$	0	0
$4$	57.2030	1.78759
$5$	−13.7903	−0.246688	−0.123344	−	0.992364i	$-0.539362\pi$
$5$	−13.7903	−0.246688	−0.123344	+	0.992364i	$0.539362\pi$
$6$	0	0
$7$	67.4858	0.520556	0.260278	−	0.965534i	$-0.416186\pi$
$7$	67.4858	0.520556	0.260278	+	0.965534i	$0.416186\pi$
$8$	238.036	1.31497
$9$	0	0
$10$	−130.245	−0.411872
$11$	138.804	0.345876	0.172938	−	0.984933i	$-0.444674\pi$
$11$	138.804	0.345876	0.172938	+	0.984933i	$0.444674\pi$
$12$	0	0
$13$	−485.356	−0.796530	−0.398265	−	0.917270i	$-0.630388\pi$
$13$	−485.356	−0.796530	−0.398265	+	0.917270i	$0.630388\pi$
$14$	637.385	0.869125
$15$	0	0
$16$	417.688	0.407898
$17$	2224.39	1.86676	0.933382	−	0.358884i	$-0.116843\pi$
$17$	2224.39	1.86676	0.933382	+	0.358884i	$0.116843\pi$
$18$	0	0
$19$	1850.55	1.17602	0.588012	−	0.808852i	$-0.299911\pi$
$19$	1850.55	1.17602	0.588012	+	0.808852i	$0.299911\pi$
$20$	−788.844	−0.440977
$21$	0	0
$22$	1310.97	0.577478
$23$	1008.66	0.397579	0.198789	−	0.980042i	$-0.436299\pi$
$23$	1008.66	0.397579	0.198789	+	0.980042i	$0.436299\pi$
$24$	0	0
$25$	−2934.83	−0.939145
$26$	−4584.06	−1.32989
$27$	0	0
$28$	3860.39	0.930542
$29$	−1534.86	−0.338902	−0.169451	−	0.985539i	$-0.554200\pi$
$29$	−1534.86	−0.338902	−0.169451	+	0.985539i	$0.554200\pi$
$30$	0	0
$31$	5924.08	1.10718	0.553588	−	0.832790i	$-0.313258\pi$
$31$	5924.08	1.10718	0.553588	+	0.832790i	$0.313258\pi$
$32$	−3672.19	−0.633943
$33$	0	0
$34$	21008.8	3.11677
$35$	−930.646	−0.128415
$36$	0	0
$37$	1074.65	0.129051	0.0645257	−	0.997916i	$-0.479447\pi$
$37$	1074.65	0.129051	0.0645257	+	0.997916i	$0.479447\pi$
$38$	17477.9	1.96350
$39$	0	0
$40$	−3282.57	−0.324388
$41$	10573.5	0.982331	0.491165	−	0.871066i	$-0.336571\pi$
$41$	10573.5	0.982331	0.491165	+	0.871066i	$0.336571\pi$
$42$	0	0
$43$	15070.3	1.24294	0.621470	−	0.783438i	$-0.286536\pi$
$43$	15070.3	1.24294	0.621470	+	0.783438i	$0.286536\pi$
$44$	7940.01	0.618286
$45$	0	0
$46$	9526.48	0.663801
$47$	14991.6	0.989924	0.494962	−	0.868915i	$-0.335182\pi$
$47$	14991.6	0.989924	0.494962	+	0.868915i	$0.335182\pi$
$48$	0	0
$49$	−12252.7	−0.729022
$50$	−27718.7	−1.56801
$51$	0	0
$52$	−27763.8	−1.42387
$53$	40821.3	1.99617	0.998084	−	0.0618654i	$-0.0197050\pi$
$53$	40821.3	1.99617	0.998084	+	0.0618654i	$0.0197050\pi$
$54$	0	0
$55$	−1914.14	−0.0853233
$56$	16064.0	0.684517
$57$	0	0
$58$	−14496.4	−0.565835
$59$	−3481.00	−0.130189
$59$	−3481.00	−0.130189
$60$	0	0
$61$	−34218.1	−1.17742	−0.588710	−	0.808345i	$-0.700364\pi$
$61$	−34218.1	−1.17742	−0.588710	+	0.808345i	$0.700364\pi$
$62$	55951.4	1.84855
$63$	0	0
$64$	−48048.9	−1.46634
$65$	6693.19	0.196494
$66$	0	0
$67$	18426.3	0.501478	0.250739	−	0.968055i	$-0.419326\pi$
$67$	18426.3	0.501478	0.250739	+	0.968055i	$0.419326\pi$
$68$	127242.	3.33702
$69$	0	0
$70$	−8789.71	−0.214402
$71$	−17658.3	−0.415723	−0.207861	−	0.978158i	$-0.566650\pi$
$71$	−17658.3	−0.415723	−0.207861	+	0.978158i	$0.566650\pi$
$72$	0	0
$73$	−25695.8	−0.564357	−0.282179	−	0.959362i	$-0.591057\pi$
$73$	−25695.8	−0.564357	−0.282179	+	0.959362i	$0.591057\pi$
$74$	10149.8	0.215465
$75$	0	0
$76$	105857.	2.10225
$77$	9367.30	0.180048
$78$	0	0
$79$	−13512.5	−0.243594	−0.121797	−	0.992555i	$-0.538866\pi$
$79$	−13512.5	−0.243594	−0.121797	+	0.992555i	$0.538866\pi$
$80$	−5760.02	−0.100623
$81$	0	0
$82$	99863.6	1.64011
$83$	80373.4	1.28061	0.640306	−	0.768120i	$-0.278808\pi$
$83$	80373.4	1.28061	0.640306	+	0.768120i	$0.278808\pi$
$84$	0	0
$85$	−30675.0	−0.460508
$86$	142335.	2.07522
$87$	0	0
$88$	33040.3	0.454818
$89$	−43704.2	−0.584855	−0.292427	−	0.956288i	$-0.594463\pi$
$89$	−43704.2	−0.584855	−0.292427	+	0.956288i	$0.594463\pi$
$90$	0	0
$91$	−32754.7	−0.414638
$92$	57698.1	0.710710
$93$	0	0
$94$	141591.	1.65279
$95$	−25519.5	−0.290110
$96$	0	0
$97$	67602.6	0.729515	0.364757	−	0.931103i	$-0.381152\pi$
$97$	67602.6	0.729515	0.364757	+	0.931103i	$0.381152\pi$
$98$	−115723.	−1.21718
$99$	0	0
$100$	−167881.	−1.67881
$101$	−21691.1	−0.211582	−0.105791	−	0.994388i	$-0.533737\pi$
$101$	−21691.1	−0.211582	−0.105791	+	0.994388i	$0.533737\pi$
$102$	0	0
$103$	35953.6	0.333926	0.166963	−	0.985963i	$-0.446604\pi$
$103$	35953.6	0.333926	0.166963	+	0.985963i	$0.446604\pi$
$104$	−115532.	−1.04742
$105$	0	0
$106$	385546.	3.33282
$107$	120880.	1.02069	0.510347	−	0.859969i	$-0.329517\pi$
$107$	120880.	1.02069	0.510347	+	0.859969i	$0.329517\pi$
$108$	0	0
$109$	−183993.	−1.48332	−0.741661	−	0.670775i	$-0.765962\pi$
$109$	−183993.	−1.48332	−0.741661	+	0.670775i	$0.765962\pi$
$110$	−18078.6	−0.142457
$111$	0	0
$112$	28188.0	0.212334
$113$	68654.1	0.505790	0.252895	−	0.967494i	$-0.418617\pi$
$113$	68654.1	0.505790	0.252895	+	0.967494i	$0.418617\pi$
$114$	0	0
$115$	−13909.6	−0.0980778
$116$	−87798.8	−0.605820
$117$	0	0
$118$	−32877.1	−0.217365
$119$	150115.	0.971755
$120$	0	0
$121$	−141784.	−0.880370
$122$	−323181.	−1.96583
$123$	0	0
$124$	338875.	1.97918
$125$	83566.6	0.478363
$126$	0	0
$127$	139438.	0.767136	0.383568	−	0.923513i	$-0.374695\pi$
$127$	139438.	0.767136	0.383568	+	0.923513i	$0.374695\pi$
$128$	−336299.	−1.81426
$129$	0	0
$130$	63215.4	0.328068
$131$	45314.3	0.230705	0.115353	−	0.993325i	$-0.463200\pi$
$131$	45314.3	0.230705	0.115353	+	0.993325i	$0.463200\pi$
$132$	0	0
$133$	124886.	0.612186
$134$	174032.	0.837272
$135$	0	0
$136$	529485.	2.45475
$137$	−72094.5	−0.328171	−0.164086	−	0.986446i	$-0.552467\pi$
$137$	−72094.5	−0.328171	−0.164086	+	0.986446i	$0.552467\pi$
$138$	0	0
$139$	−437395.	−1.92016	−0.960078	−	0.279732i	$-0.909754\pi$
$139$	−437395.	−1.92016	−0.960078	+	0.279732i	$0.909754\pi$
$140$	−53235.8	−0.229553
$141$	0	0
$142$	−166778.	−0.694094
$143$	−67369.4	−0.275501
$144$	0	0
$145$	21166.2	0.0836030
$146$	−242690.	−0.942256
$147$	0	0
$148$	61473.2	0.230692
$149$	−236655.	−0.873271	−0.436636	−	0.899638i	$-0.643830\pi$
$149$	−236655.	−0.873271	−0.436636	+	0.899638i	$0.643830\pi$
$150$	0	0
$151$	378542.	1.35105	0.675526	−	0.737336i	$-0.263917\pi$
$151$	378542.	1.35105	0.675526	+	0.737336i	$0.263917\pi$
$152$	440496.	1.54644
$153$	0	0
$154$	88471.7	0.300609
$155$	−81694.6	−0.273127
$156$	0	0
$157$	103612.	0.335475	0.167738	−	0.985832i	$-0.446354\pi$
$157$	103612.	0.335475	0.167738	+	0.985832i	$0.446354\pi$
$158$	−127622.	−0.406707
$159$	0	0
$160$	50640.5	0.156386
$161$	68069.9	0.206962
$162$	0	0
$163$	−318233.	−0.938160	−0.469080	−	0.883156i	$-0.655414\pi$
$163$	−318233.	−0.938160	−0.469080	+	0.883156i	$0.655414\pi$
$164$	604834.	1.75601
$165$	0	0
$166$	759106.	2.13812
$167$	24793.8	0.0687942	0.0343971	−	0.999408i	$-0.489049\pi$
$167$	24793.8	0.0687942	0.0343971	+	0.999408i	$0.489049\pi$
$168$	0	0
$169$	−135722.	−0.365539
$170$	−289717.	−0.768868
$171$	0	0
$172$	862065.	2.22187
$173$	313813.	0.797178	0.398589	−	0.917130i	$-0.369500\pi$
$173$	313813.	0.797178	0.398589	+	0.917130i	$0.369500\pi$
$174$	0	0
$175$	−198059.	−0.488877
$176$	57976.8	0.141082
$177$	0	0
$178$	−412774.	−0.976478
$179$	313044.	0.730253	0.365126	−	0.930958i	$-0.381026\pi$
$179$	313044.	0.730253	0.365126	+	0.930958i	$0.381026\pi$
$180$	0	0
$181$	−240755.	−0.546235	−0.273117	−	0.961981i	$-0.588055\pi$
$181$	−240755.	−0.546235	−0.273117	+	0.961981i	$0.588055\pi$
$182$	−309359.	−0.692284
$183$	0	0
$184$	240096.	0.522806
$185$	−14819.7	−0.0318354
$186$	0	0
$187$	308755.	0.645669
$188$	857562.	1.76958
$189$	0	0
$190$	−241025.	−0.484371
$191$	−804807.	−1.59628	−0.798139	−	0.602474i	$-0.794182\pi$
$191$	−804807.	−1.59628	−0.798139	+	0.602474i	$0.794182\pi$
$192$	0	0
$193$	−724642.	−1.40033	−0.700164	−	0.713982i	$-0.746890\pi$
$193$	−724642.	−1.40033	−0.700164	+	0.713982i	$0.746890\pi$
$194$	638489.	1.21800
$195$	0	0
$196$	−700889.	−1.30319
$197$	483578.	0.887771	0.443886	−	0.896083i	$-0.353600\pi$
$197$	483578.	0.887771	0.443886	+	0.896083i	$0.353600\pi$
$198$	0	0
$199$	356378.	0.637937	0.318968	−	0.947765i	$-0.396664\pi$
$199$	356378.	0.637937	0.318968	+	0.947765i	$0.396664\pi$
$200$	−698594.	−1.23495
$201$	0	0
$202$	−204866.	−0.353258
$203$	−103581.	−0.176418
$204$	0	0
$205$	−145811.	−0.242329
$206$	339573.	0.557525
$207$	0	0
$208$	−202727.	−0.324903
$209$	256863.	0.406758
$210$	0	0
$211$	64318.7	0.0994561	0.0497280	−	0.998763i	$-0.484165\pi$
$211$	64318.7	0.0994561	0.0497280	+	0.998763i	$0.484165\pi$
$212$	2.33510e6	3.56834
$213$	0	0
$214$	1.14168e6	1.70416
$215$	−207823.	−0.306618
$216$	0	0
$217$	399791.	0.576347
$218$	−1.73777e6	−2.47657
$219$	0	0
$220$	−109495.	−0.152523
$221$	−1.07962e6	−1.48693
$222$	0	0
$223$	359718.	0.484396	0.242198	−	0.970227i	$-0.422132\pi$
$223$	359718.	0.484396	0.242198	+	0.970227i	$0.422132\pi$
$224$	−247821.	−0.330003
$225$	0	0
$226$	648420.	0.844472
$227$	−628492.	−0.809533	−0.404767	−	0.914420i	$-0.632647\pi$
$227$	−628492.	−0.809533	−0.404767	+	0.914420i	$0.632647\pi$
$228$	0	0
$229$	−712809.	−0.898223	−0.449112	−	0.893476i	$-0.648259\pi$
$229$	−712809.	−0.898223	−0.449112	+	0.893476i	$0.648259\pi$
$230$	−131373.	−0.163751
$231$	0	0
$232$	−365352.	−0.445648
$233$	−451301.	−0.544598	−0.272299	−	0.962213i	$-0.587784\pi$
$233$	−451301.	−0.544598	−0.272299	+	0.962213i	$0.587784\pi$
$234$	0	0
$235$	−206737.	−0.244202
$236$	−199124.	−0.232725
$237$	0	0
$238$	1.41780e6	1.62245
$239$	−479799.	−0.543331	−0.271665	−	0.962392i	$-0.587574\pi$
$239$	−479799.	−0.543331	−0.271665	+	0.962392i	$0.587574\pi$
$240$	0	0
$241$	−165266.	−0.183291	−0.0916456	−	0.995792i	$-0.529213\pi$
$241$	−165266.	−0.183291	−0.0916456	+	0.995792i	$0.529213\pi$
$242$	−1.33912e6	−1.46987
$243$	0	0
$244$	−1.95738e6	−2.10475
$245$	168967.	0.179841
$246$	0	0
$247$	−898174.	−0.936738
$248$	1.41014e6	1.45591
$249$	0	0
$250$	789264.	0.798679
$251$	−1.49986e6	−1.50268	−0.751340	−	0.659915i	$-0.770592\pi$
$251$	−1.49986e6	−1.50268	−0.751340	+	0.659915i	$0.770592\pi$
$252$	0	0
$253$	140005.	0.137513
$254$	1.31696e6	1.28082
$255$	0	0
$256$	−1.63869e6	−1.56278
$257$	602030.	0.568572	0.284286	−	0.958739i	$-0.408243\pi$
$257$	602030.	0.568572	0.284286	+	0.958739i	$0.408243\pi$
$258$	0	0
$259$	72523.6	0.0671785
$260$	382870.	0.351252
$261$	0	0
$262$	427982.	0.385187
$263$	136455.	0.121647	0.0608233	−	0.998149i	$-0.480627\pi$
$263$	136455.	0.121647	0.0608233	+	0.998149i	$0.480627\pi$
$264$	0	0
$265$	−562936.	−0.492430
$266$	1.17951e6	1.02211
$267$	0	0
$268$	1.05404e6	0.896439
$269$	−1.68628e6	−1.42085	−0.710425	−	0.703773i	$-0.751497\pi$
$269$	−1.68628e6	−1.42085	−0.710425	+	0.703773i	$0.751497\pi$
$270$	0	0
$271$	−459713.	−0.380245	−0.190123	−	0.981760i	$-0.560889\pi$
$271$	−459713.	−0.380245	−0.190123	+	0.981760i	$0.560889\pi$
$272$	929103.	0.761450
$273$	0	0
$274$	−680913.	−0.547918
$275$	−407366.	−0.324828
$276$	0	0
$277$	−2.13458e6	−1.67152	−0.835761	−	0.549093i	$-0.814973\pi$
$277$	−2.13458e6	−1.67152	−0.835761	+	0.549093i	$0.814973\pi$
$278$	−4.13108e6	−3.20591
$279$	0	0
$280$	−221527.	−0.168862
$281$	−2.25092e6	−1.70057	−0.850284	−	0.526325i	$-0.823570\pi$
$281$	−2.25092e6	−1.70057	−0.850284	+	0.526325i	$0.823570\pi$
$282$	0	0
$283$	−5499.56	−0.00408189	−0.00204095	−	0.999998i	$-0.500650\pi$
$283$	−5499.56	−0.00408189	−0.00204095	+	0.999998i	$0.500650\pi$
$284$	−1.01011e6	−0.743143
$285$	0	0
$286$	−636286.	−0.459978
$287$	713559.	0.511358
$288$	0	0
$289$	3.52808e6	2.48481
$290$	199909.	0.139584
$291$	0	0
$292$	−1.46987e6	−1.00884
$293$	902523.	0.614171	0.307085	−	0.951682i	$-0.400646\pi$
$293$	902523.	0.614171	0.307085	+	0.951682i	$0.400646\pi$
$294$	0	0
$295$	48003.9	0.0321160
$296$	255805.	0.169699
$297$	0	0
$298$	−2.23514e6	−1.45802
$299$	−489557.	−0.316684
$300$	0	0
$301$	1.01703e6	0.647019
$302$	3.57523e6	2.25573
$303$	0	0
$304$	772951.	0.479698
$305$	471876.	0.290455
$306$	0	0
$307$	500034.	0.302798	0.151399	−	0.988473i	$-0.451622\pi$
$307$	500034.	0.302798	0.151399	+	0.988473i	$0.451622\pi$
$308$	535838.	0.321852
$309$	0	0
$310$	−771584.	−0.456015
$311$	1.44596e6	0.847728	0.423864	−	0.905726i	$-0.360673\pi$
$311$	1.44596e6	0.847728	0.423864	+	0.905726i	$0.360673\pi$
$312$	0	0
$313$	−1.78320e6	−1.02882	−0.514410	−	0.857544i	$-0.671989\pi$
$313$	−1.78320e6	−1.02882	−0.514410	+	0.857544i	$0.671989\pi$
$314$	978587.	0.560113
$315$	0	0
$316$	−772953.	−0.435447
$317$	1.43302e6	0.800948	0.400474	−	0.916308i	$-0.368846\pi$
$317$	1.43302e6	0.800948	0.400474	+	0.916308i	$0.368846\pi$
$318$	0	0
$319$	−213045.	−0.117218
$320$	662606.	0.361727
$321$	0	0
$322$	642902.	0.345545
$323$	4.11634e6	2.19536
$324$	0	0
$325$	1.42444e6	0.748058
$326$	−3.00563e6	−1.56636
$327$	0	0
$328$	2.51686e6	1.29174
$329$	1.01172e6	0.515311
$330$	0	0
$331$	2.77285e6	1.39109	0.695547	−	0.718481i	$-0.255162\pi$
$331$	2.77285e6	1.39109	0.695547	+	0.718481i	$0.255162\pi$
$332$	4.59760e6	2.28921
$333$	0	0
$334$	234171.	0.114859
$335$	−254104.	−0.123708
$336$	0	0
$337$	−1.39166e6	−0.667509	−0.333754	−	0.942660i	$-0.608316\pi$
$337$	−1.39166e6	−0.667509	−0.333754	+	0.942660i	$0.608316\pi$
$338$	−1.28186e6	−0.610308
$339$	0	0
$340$	−1.75470e6	−0.823201
$341$	822287.	0.382946
$342$	0	0
$343$	−1.96111e6	−0.900052
$344$	3.58726e6	1.63443
$345$	0	0
$346$	2.96388e6	1.33098
$347$	3.44484e6	1.53584	0.767918	−	0.640548i	$-0.221293\pi$
$347$	3.44484e6	1.53584	0.767918	+	0.640548i	$0.221293\pi$
$348$	0	0
$349$	−1.33302e6	−0.585833	−0.292917	−	0.956138i	$-0.594626\pi$
$349$	−1.33302e6	−0.585833	−0.292917	+	0.956138i	$0.594626\pi$
$350$	−1.87062e6	−0.816234
$351$	0	0
$352$	−509715.	−0.219266
$353$	−3.77713e6	−1.61334	−0.806669	−	0.591003i	$-0.798732\pi$
$353$	−3.77713e6	−1.61334	−0.806669	+	0.591003i	$0.798732\pi$
$354$	0	0
$355$	243513.	0.102554
$356$	−2.50001e6	−1.04548
$357$	0	0
$358$	2.95662e6	1.21924
$359$	−331917.	−0.135923	−0.0679616	−	0.997688i	$-0.521650\pi$
$359$	−331917.	−0.135923	−0.0679616	+	0.997688i	$0.521650\pi$
$360$	0	0
$361$	948422.	0.383031
$362$	−2.27387e6	−0.911998
$363$	0	0
$364$	−1.87366e6	−0.741205
$365$	354351.	0.139220
$366$	0	0
$367$	−1.30195e6	−0.504578	−0.252289	−	0.967652i	$-0.581183\pi$
$367$	−1.30195e6	−0.504578	−0.252289	+	0.967652i	$0.581183\pi$
$368$	421303.	0.162172
$369$	0	0
$370$	−139968.	−0.0531526
$371$	2.75486e6	1.03912
$372$	0	0
$373$	5.01394e6	1.86598	0.932990	−	0.359903i	$-0.117190\pi$
$373$	5.01394e6	1.86598	0.932990	+	0.359903i	$0.117190\pi$
$374$	2.91611e6	1.07801
$375$	0	0
$376$	3.56852e6	1.30172
$377$	744956.	0.269946
$378$	0	0
$379$	4.28983e6	1.53406	0.767030	−	0.641611i	$-0.221734\pi$
$379$	4.28983e6	1.53406	0.767030	+	0.641611i	$0.221734\pi$
$380$	−1.45979e6	−0.518599
$381$	0	0
$382$	−7.60119e6	−2.66516
$383$	−2.99051e6	−1.04171	−0.520856	−	0.853645i	$-0.674387\pi$
$383$	−2.99051e6	−1.04171	−0.520856	+	0.853645i	$0.674387\pi$
$384$	0	0
$385$	−129177.	−0.0444155
$386$	−6.84405e6	−2.33800
$387$	0	0
$388$	3.86707e6	1.30408
$389$	−5.18724e6	−1.73805	−0.869026	−	0.494767i	$-0.835254\pi$
$389$	−5.18724e6	−1.73805	−0.869026	+	0.494767i	$0.835254\pi$
$390$	0	0
$391$	2.24365e6	0.742186
$392$	−2.91657e6	−0.958645
$393$	0	0
$394$	4.56727e6	1.48223
$395$	186340.	0.0600916
$396$	0	0
$397$	2.45853e6	0.782887	0.391443	−	0.920202i	$-0.371976\pi$
$397$	2.45853e6	0.782887	0.391443	+	0.920202i	$0.371976\pi$
$398$	3.36589e6	1.06511
$399$	0	0
$400$	−1.22584e6	−0.383076
$401$	1.41766e6	0.440261	0.220130	−	0.975470i	$-0.429352\pi$
$401$	1.41766e6	0.440261	0.220130	+	0.975470i	$0.429352\pi$
$402$	0	0
$403$	−2.87529e6	−0.881900
$404$	−1.24079e6	−0.378222
$405$	0	0
$406$	−978300.	−0.294548
$407$	149166.	0.0446358
$408$	0	0
$409$	−2.35592e6	−0.696391	−0.348195	−	0.937422i	$-0.613205\pi$
$409$	−2.35592e6	−0.696391	−0.348195	+	0.937422i	$0.613205\pi$
$410$	−1.37714e6	−0.404594
$411$	0	0
$412$	2.05666e6	0.596924
$413$	−234918.	−0.0677706
$414$	0	0
$415$	−1.10837e6	−0.315911
$416$	1.78232e6	0.504955
$417$	0	0
$418$	2.42601e6	0.679127
$419$	−4.25487e6	−1.18400	−0.592000	−	0.805938i	$-0.701661\pi$
$419$	−4.25487e6	−1.18400	−0.592000	+	0.805938i	$0.701661\pi$
$420$	0	0
$421$	−1.91147e6	−0.525607	−0.262804	−	0.964849i	$-0.584647\pi$
$421$	−1.91147e6	−0.525607	−0.262804	+	0.964849i	$0.584647\pi$
$422$	607473.	0.166053
$423$	0	0
$424$	9.71693e6	2.62491
$425$	−6.52822e6	−1.75316
$426$	0	0
$427$	−2.30923e6	−0.612912
$428$	6.91471e6	1.82459
$429$	0	0
$430$	−1.96283e6	−0.511932
$431$	5.28060e6	1.36927	0.684637	−	0.728885i	$-0.259961\pi$
$431$	5.28060e6	1.36927	0.684637	+	0.728885i	$0.259961\pi$
$432$	0	0
$433$	−3.92135e6	−1.00512	−0.502558	−	0.864543i	$-0.667608\pi$
$433$	−3.92135e6	−1.00512	−0.502558	+	0.864543i	$0.667608\pi$
$434$	3.77592e6	0.962274
$435$	0	0
$436$	−1.05250e7	−2.65158
$437$	1.86656e6	0.467562
$438$	0	0
$439$	3.26904e6	0.809578	0.404789	−	0.914410i	$-0.367345\pi$
$439$	3.26904e6	0.809578	0.404789	+	0.914410i	$0.367345\pi$
$440$	−455634.	−0.112198
$441$	0	0
$442$	−1.01968e7	−2.48260
$443$	−6.03039e6	−1.45994	−0.729972	−	0.683477i	$-0.760467\pi$
$443$	−6.03039e6	−1.45994	−0.729972	+	0.683477i	$0.760467\pi$
$444$	0	0
$445$	602692.	0.144276
$446$	3.39745e6	0.808752
$447$	0	0
$448$	−3.24262e6	−0.763309
$449$	−5.58738e6	−1.30795	−0.653976	−	0.756515i	$-0.726900\pi$
$449$	−5.58738e6	−1.30795	−0.653976	+	0.756515i	$0.726900\pi$
$450$	0	0
$451$	1.46764e6	0.339765
$452$	3.92722e6	0.904148
$453$	0	0
$454$	−5.93594e6	−1.35160
$455$	451695.	0.102286
$456$	0	0
$457$	−2.59965e6	−0.582270	−0.291135	−	0.956682i	$-0.594033\pi$
$457$	−2.59965e6	−0.582270	−0.291135	+	0.956682i	$0.594033\pi$
$458$	−6.73229e6	−1.49968
$459$	0	0
$460$	−795672.	−0.175323
$461$	4.29896e6	0.942130	0.471065	−	0.882099i	$-0.343870\pi$
$461$	4.29896e6	0.942130	0.471065	+	0.882099i	$0.343870\pi$
$462$	0	0
$463$	−4.21015e6	−0.912735	−0.456367	−	0.889791i	$-0.650850\pi$
$463$	−4.21015e6	−0.912735	−0.456367	+	0.889791i	$0.650850\pi$
$464$	−641094.	−0.138238
$465$	0	0
$466$	−4.26242e6	−0.909266
$467$	−5.46044e6	−1.15860	−0.579302	−	0.815113i	$-0.696675\pi$
$467$	−5.46044e6	−1.15860	−0.579302	+	0.815113i	$0.696675\pi$
$468$	0	0
$469$	1.24352e6	0.261047
$470$	−1.95258e6	−0.407722
$471$	0	0
$472$	−828602.	−0.171195
$473$	2.09182e6	0.429903
$474$	0	0
$475$	−5.43104e6	−1.10446
$476$	8.58703e6	1.73710
$477$	0	0
$478$	−4.53157e6	−0.907150
$479$	692304.	0.137866	0.0689331	−	0.997621i	$-0.478040\pi$
$479$	692304.	0.137866	0.0689331	+	0.997621i	$0.478040\pi$
$480$	0	0
$481$	−521588.	−0.102793
$482$	−1.56090e6	−0.306025
$483$	0	0
$484$	−8.11050e6	−1.57374
$485$	−932257.	−0.179962
$486$	0	0
$487$	9.19622e6	1.75706	0.878530	−	0.477686i	$-0.158524\pi$
$487$	9.19622e6	1.75706	0.878530	+	0.477686i	$0.158524\pi$
$488$	−8.14513e6	−1.54828
$489$	0	0
$490$	1.59585e6	0.300263
$491$	−454736.	−0.0851247	−0.0425623	−	0.999094i	$-0.513552\pi$
$491$	−454736.	−0.0851247	−0.0425623	+	0.999094i	$0.513552\pi$
$492$	0	0
$493$	−3.41414e6	−0.632651
$494$	−8.48302e6	−1.56399
$495$	0	0
$496$	2.47442e6	0.451615
$497$	−1.19169e6	−0.216407
$498$	0	0
$499$	−4.56440e6	−0.820601	−0.410301	−	0.911950i	$-0.634576\pi$
$499$	−4.56440e6	−0.820601	−0.410301	+	0.911950i	$0.634576\pi$
$500$	4.78026e6	0.855119
$501$	0	0
$502$	−1.41658e7	−2.50889
$503$	1.87618e6	0.330639	0.165320	−	0.986240i	$-0.447134\pi$
$503$	1.87618e6	0.330639	0.165320	+	0.986240i	$0.447134\pi$
$504$	0	0
$505$	299125.	0.0521945
$506$	1.32231e6	0.229593
$507$	0	0
$508$	7.97629e6	1.37133
$509$	−2.91027e6	−0.497897	−0.248948	−	0.968517i	$-0.580085\pi$
$509$	−2.91027e6	−0.497897	−0.248948	+	0.968517i	$0.580085\pi$
$510$	0	0
$511$	−1.73410e6	−0.293779
$512$	−4.71542e6	−0.794960
$513$	0	0
$514$	5.68602e6	0.949293
$515$	−495810.	−0.0823753
$516$	0	0
$517$	2.08089e6	0.342391
$518$	684966.	0.112162
$519$	0	0
$520$	1.59322e6	0.258385
$521$	6.05299e6	0.976958	0.488479	−	0.872576i	$-0.337552\pi$
$521$	6.05299e6	0.976958	0.488479	+	0.872576i	$0.337552\pi$
$522$	0	0
$523$	4.76464e6	0.761685	0.380843	−	0.924640i	$-0.375634\pi$
$523$	4.76464e6	0.761685	0.380843	+	0.924640i	$0.375634\pi$
$524$	2.59212e6	0.412407
$525$	0	0
$526$	1.28878e6	0.203102
$527$	1.31775e7	2.06684
$528$	0	0
$529$	−5.41896e6	−0.841931
$530$	−5.31678e6	−0.822166
$531$	0	0
$532$	7.14383e6	1.09434
$533$	−5.13190e6	−0.782456
$534$	0	0
$535$	−1.66697e6	−0.251792
$536$	4.38613e6	0.659431
$537$	0	0
$538$	−1.59264e7	−2.37226
$539$	−1.70072e6	−0.252151
$540$	0	0
$541$	1.03136e7	1.51502	0.757511	−	0.652822i	$-0.226415\pi$
$541$	1.03136e7	1.51502	0.757511	+	0.652822i	$0.226415\pi$
$542$	−4.34187e6	−0.634861
$543$	0	0
$544$	−8.16840e6	−1.18342
$545$	2.53731e6	0.365917
$546$	0	0
$547$	9.32864e6	1.33306	0.666530	−	0.745478i	$-0.267779\pi$
$547$	9.32864e6	1.33306	0.666530	+	0.745478i	$0.267779\pi$
$548$	−4.12402e6	−0.586637
$549$	0	0
$550$	−3.84746e6	−0.542335
$551$	−2.84034e6	−0.398557
$552$	0	0
$553$	−911899.	−0.126804
$554$	−2.01605e7	−2.79079
$555$	0	0
$556$	−2.50203e7	−3.43246
$557$	−1.72698e6	−0.235857	−0.117929	−	0.993022i	$-0.537625\pi$
$557$	−1.72698e6	−0.235857	−0.117929	+	0.993022i	$0.537625\pi$
$558$	0	0
$559$	−7.31445e6	−0.990039
$560$	−388720.	−0.0523801
$561$	0	0
$562$	−2.12593e7	−2.83928
$563$	2.82325e6	0.375386	0.187693	−	0.982228i	$-0.439899\pi$
$563$	2.82325e6	0.375386	0.187693	+	0.982228i	$0.439899\pi$
$564$	0	0
$565$	−946758.	−0.124772
$566$	−51941.9	−0.00681517
$567$	0	0
$568$	−4.20331e6	−0.546664
$569$	−1.12737e7	−1.45978	−0.729888	−	0.683567i	$-0.760428\pi$
$569$	−1.12737e7	−1.45978	−0.729888	+	0.683567i	$0.760428\pi$
$570$	0	0
$571$	7.15556e6	0.918445	0.459223	−	0.888321i	$-0.348128\pi$
$571$	7.15556e6	0.918445	0.459223	+	0.888321i	$0.348128\pi$
$572$	−3.85373e6	−0.492483
$573$	0	0
$574$	6.73937e6	0.853768
$575$	−2.96023e6	−0.373384
$576$	0	0
$577$	6.00795e6	0.751254	0.375627	−	0.926771i	$-0.377427\pi$
$577$	6.00795e6	0.751254	0.375627	+	0.926771i	$0.377427\pi$
$578$	3.33217e7	4.14866
$579$	0	0
$580$	1.21077e6	0.149448
$581$	5.42406e6	0.666629
$582$	0	0
$583$	5.66616e6	0.690427
$584$	−6.11651e6	−0.742115
$585$	0	0
$586$	8.52409e6	1.02542
$587$	6.67034e6	0.799011	0.399505	−	0.916731i	$-0.369182\pi$
$587$	6.67034e6	0.799011	0.399505	+	0.916731i	$0.369182\pi$
$588$	0	0
$589$	1.09628e7	1.30207
$590$	453384.	0.0536211
$591$	0	0
$592$	448868.	0.0526399
$593$	−1.86332e6	−0.217596	−0.108798	−	0.994064i	$-0.534700\pi$
$593$	−1.86332e6	−0.217596	−0.108798	+	0.994064i	$0.534700\pi$
$594$	0	0
$595$	−2.07012e6	−0.239720
$596$	−1.35374e7	−1.56105
$597$	0	0
$598$	−4.62374e6	−0.528738
$599$	−1.74289e6	−0.198474	−0.0992369	−	0.995064i	$-0.531640\pi$
$599$	−1.74289e6	−0.198474	−0.0992369	+	0.995064i	$0.531640\pi$
$600$	0	0
$601$	4.23218e6	0.477945	0.238972	−	0.971026i	$-0.423189\pi$
$601$	4.23218e6	0.477945	0.238972	+	0.971026i	$0.423189\pi$
$602$	9.60557e6	1.08027
$603$	0	0
$604$	2.16538e7	2.41513
$605$	1.95524e6	0.217176
$606$	0	0
$607$	−1.13339e7	−1.24856	−0.624279	−	0.781201i	$-0.714607\pi$
$607$	−1.13339e7	−1.24856	−0.624279	+	0.781201i	$0.714607\pi$
$608$	−6.79556e6	−0.745532
$609$	0	0
$610$	4.45674e6	0.484946
$611$	−7.27624e6	−0.788505
$612$	0	0
$613$	1.16293e7	1.24997	0.624987	−	0.780635i	$-0.285104\pi$
$613$	1.16293e7	1.24997	0.624987	+	0.780635i	$0.285104\pi$
$614$	4.72269e6	0.505555
$615$	0	0
$616$	2.22975e6	0.236758
$617$	1.03462e7	1.09413	0.547066	−	0.837090i	$-0.315745\pi$
$617$	1.03462e7	1.09413	0.547066	+	0.837090i	$0.315745\pi$
$618$	0	0
$619$	−1.09852e7	−1.15234	−0.576169	−	0.817331i	$-0.695453\pi$
$619$	−1.09852e7	−1.15234	−0.576169	+	0.817331i	$0.695453\pi$
$620$	−4.67318e6	−0.488240
$621$	0	0
$622$	1.36568e7	1.41537
$623$	−2.94941e6	−0.304449
$624$	0	0
$625$	8.01894e6	0.821139
$626$	−1.68419e7	−1.71773
$627$	0	0
$628$	5.92692e6	0.599694
$629$	2.39045e6	0.240909
$630$	0	0
$631$	1.54733e7	1.54707	0.773535	−	0.633753i	$-0.218487\pi$
$631$	1.54733e7	1.54707	0.773535	+	0.633753i	$0.218487\pi$
$632$	−3.21645e6	−0.320320
$633$	0	0
$634$	1.35345e7	1.33727
$635$	−1.92289e6	−0.189243
$636$	0	0
$637$	5.94691e6	0.580688
$638$	−2.01216e6	−0.195709
$639$	0	0
$640$	4.63765e6	0.447556
$641$	−1.22578e7	−1.17833	−0.589166	−	0.808012i	$-0.700544\pi$
$641$	−1.22578e7	−1.17833	−0.589166	+	0.808012i	$0.700544\pi$
$642$	0	0
$643$	−1.32338e7	−1.26228	−0.631141	−	0.775668i	$-0.717413\pi$
$643$	−1.32338e7	−1.26228	−0.631141	+	0.775668i	$0.717413\pi$
$644$	3.89380e6	0.369964
$645$	0	0
$646$	3.88778e7	3.66539
$647$	1.49533e7	1.40436	0.702178	−	0.712002i	$-0.252211\pi$
$647$	1.49533e7	1.40436	0.702178	+	0.712002i	$0.252211\pi$
$648$	0	0
$649$	−483177.	−0.0450292
$650$	1.34534e7	1.24896
$651$	0	0
$652$	−1.82039e7	−1.67705
$653$	1.46185e7	1.34159	0.670794	−	0.741643i	$-0.265953\pi$
$653$	1.46185e7	1.34159	0.670794	+	0.741643i	$0.265953\pi$
$654$	0	0
$655$	−624896.	−0.0569121
$656$	4.41641e6	0.400691
$657$	0	0
$658$	9.55539e6	0.860367
$659$	7.89681e6	0.708334	0.354167	−	0.935182i	$-0.384764\pi$
$659$	7.89681e6	0.708334	0.354167	+	0.935182i	$0.384764\pi$
$660$	0	0
$661$	2.56132e6	0.228013	0.114007	−	0.993480i	$-0.463631\pi$
$661$	2.56132e6	0.228013	0.114007	+	0.993480i	$0.463631\pi$
$662$	2.61888e7	2.32258
$663$	0	0
$664$	1.91317e7	1.68397
$665$	−1.72220e6	−0.151019
$666$	0	0
$667$	−1.54815e6	−0.134740
$668$	1.41828e6	0.122976
$669$	0	0
$670$	−2.39994e6	−0.206545
$671$	−4.74961e6	−0.407241
$672$	0	0
$673$	1.96637e7	1.67351	0.836754	−	0.547578i	$-0.184450\pi$
$673$	1.96637e7	1.67351	0.836754	+	0.547578i	$0.184450\pi$
$674$	−1.31438e7	−1.11448
$675$	0	0
$676$	−7.76372e6	−0.653436
$677$	−2.65672e6	−0.222779	−0.111389	−	0.993777i	$-0.535530\pi$
$677$	−2.65672e6	−0.222779	−0.111389	+	0.993777i	$0.535530\pi$
$678$	0	0
$679$	4.56222e6	0.379753
$680$	−7.30174e6	−0.605556
$681$	0	0
$682$	7.76628e6	0.639370
$683$	360849.	0.0295988	0.0147994	−	0.999890i	$-0.495289\pi$
$683$	360849.	0.0295988	0.0147994	+	0.999890i	$0.495289\pi$
$684$	0	0
$685$	994201.	0.0809558
$686$	−1.85222e7	−1.50274
$687$	0	0
$688$	6.29467e6	0.506993
$689$	−1.98129e7	−1.59001
$690$	0	0
$691$	1.57193e7	1.25238	0.626191	−	0.779670i	$-0.284613\pi$
$691$	1.57193e7	1.25238	0.626191	+	0.779670i	$0.284613\pi$
$692$	1.79510e7	1.42503
$693$	0	0
$694$	3.25356e7	2.56425
$695$	6.03179e6	0.473679
$696$	0	0
$697$	2.35196e7	1.83378
$698$	−1.25901e7	−0.978113
$699$	0	0
$700$	−1.13296e7	−0.873914
$701$	2.32970e7	1.79062	0.895312	−	0.445440i	$-0.146953\pi$
$701$	2.32970e7	1.79062	0.895312	+	0.445440i	$0.146953\pi$
$702$	0	0
$703$	1.98869e6	0.151767
$704$	−6.66938e6	−0.507170
$705$	0	0
$706$	−3.56740e7	−2.69364
$707$	−1.46384e6	−0.110140
$708$	0	0
$709$	5.36409e6	0.400756	0.200378	−	0.979719i	$-0.435783\pi$
$709$	5.36409e6	0.400756	0.200378	+	0.979719i	$0.435783\pi$
$710$	2.29991e6	0.171224
$711$	0	0
$712$	−1.04032e7	−0.769069
$713$	5.97536e6	0.440190
$714$	0	0
$715$	929041.	0.0679626
$716$	1.79071e7	1.30540
$717$	0	0
$718$	−3.13487e6	−0.226939
$719$	−2.10973e7	−1.52196	−0.760981	−	0.648774i	$-0.775282\pi$
$719$	−2.10973e7	−1.52196	−0.760981	+	0.648774i	$0.775282\pi$
$720$	0	0
$721$	2.42636e6	0.173827
$722$	8.95759e6	0.639511
$723$	0	0
$724$	−1.37719e7	−0.976446
$725$	4.50456e6	0.318279
$726$	0	0
$727$	1.02219e7	0.717291	0.358646	−	0.933474i	$-0.383239\pi$
$727$	1.02219e7	0.717291	0.358646	+	0.933474i	$0.383239\pi$
$728$	−7.79678e6	−0.545239
$729$	0	0
$730$	3.34675e6	0.232443
$731$	3.35222e7	2.32028
$732$	0	0
$733$	1.18831e7	0.816900	0.408450	−	0.912781i	$-0.366069\pi$
$733$	1.18831e7	0.816900	0.408450	+	0.912781i	$0.366069\pi$
$734$	−1.22965e7	−0.842447
$735$	0	0
$736$	−3.70398e6	−0.252042
$737$	2.55765e6	0.173449
$738$	0	0
$739$	−7.32133e6	−0.493150	−0.246575	−	0.969124i	$-0.579305\pi$
$739$	−7.32133e6	−0.493150	−0.246575	+	0.969124i	$0.579305\pi$
$740$	−847731.	−0.0569087
$741$	0	0
$742$	2.60189e7	1.73492
$743$	−1.09019e7	−0.724484	−0.362242	−	0.932084i	$-0.617989\pi$
$743$	−1.09019e7	−0.724484	−0.362242	+	0.932084i	$0.617989\pi$
$744$	0	0
$745$	3.26353e6	0.215425
$746$	4.73553e7	3.11546
$747$	0	0
$748$	1.76617e7	1.15419
$749$	8.15769e6	0.531328
$750$	0	0
$751$	−3.73740e6	−0.241808	−0.120904	−	0.992664i	$-0.538579\pi$
$751$	−3.73740e6	−0.241808	−0.120904	+	0.992664i	$0.538579\pi$
$752$	6.26179e6	0.403788
$753$	0	0
$754$	7.03591e6	0.450704
$755$	−5.22019e6	−0.333288
$756$	0	0
$757$	−1.93400e7	−1.22664	−0.613319	−	0.789835i	$-0.710166\pi$
$757$	−1.93400e7	−1.22664	−0.613319	+	0.789835i	$0.710166\pi$
$758$	4.05163e7	2.56128
$759$	0	0
$760$	−6.07455e6	−0.381488
$761$	−1.32949e7	−0.832190	−0.416095	−	0.909321i	$-0.636602\pi$
$761$	−1.32949e7	−0.832190	−0.416095	+	0.909321i	$0.636602\pi$
$762$	0	0
$763$	−1.24169e7	−0.772152
$764$	−4.60374e7	−2.85350
$765$	0	0
$766$	−2.82445e7	−1.73925
$767$	1.68953e6	0.103699
$768$	0	0
$769$	−1.41450e7	−0.862555	−0.431277	−	0.902219i	$-0.641937\pi$
$769$	−1.41450e7	−0.862555	−0.431277	+	0.902219i	$0.641937\pi$
$770$	−1.22005e6	−0.0741566
$771$	0	0
$772$	−4.14517e7	−2.50322
$773$	−1.48281e7	−0.892557	−0.446278	−	0.894894i	$-0.647251\pi$
$773$	−1.48281e7	−0.892557	−0.446278	+	0.894894i	$0.647251\pi$
$774$	0	0
$775$	−1.73862e7	−1.03980
$776$	1.60918e7	0.959293
$777$	0	0
$778$	−4.89921e7	−2.90187
$779$	1.95667e7	1.15524
$780$	0	0
$781$	−2.45105e6	−0.143788
$782$	2.11907e7	1.23916
$783$	0	0
$784$	−5.11779e6	−0.297367
$785$	−1.42884e6	−0.0827576
$786$	0	0
$787$	−3.31567e7	−1.90825	−0.954123	−	0.299414i	$-0.903209\pi$
$787$	−3.31567e7	−1.90825	−0.954123	+	0.299414i	$0.903209\pi$
$788$	2.76621e7	1.58697
$789$	0	0
$790$	1.75993e6	0.100330
$791$	4.63318e6	0.263292
$792$	0	0
$793$	1.66080e7	0.937850
$794$	2.32202e7	1.30712
$795$	0	0
$796$	2.03859e7	1.14037
$797$	−2.23535e7	−1.24652	−0.623261	−	0.782014i	$-0.714192\pi$
$797$	−2.23535e7	−1.24652	−0.623261	+	0.782014i	$0.714192\pi$
$798$	0	0
$799$	3.33471e7	1.84796
$800$	1.07773e7	0.595365
$801$	0	0
$802$	1.33894e7	0.735063
$803$	−3.56667e6	−0.195198
$804$	0	0
$805$	−938701.	−0.0510549
$806$	−2.71564e7	−1.47243
$807$	0	0
$808$	−5.16325e6	−0.278224
$809$	1.54732e6	0.0831206	0.0415603	−	0.999136i	$-0.486767\pi$
$809$	1.54732e6	0.0831206	0.0415603	+	0.999136i	$0.486767\pi$
$810$	0	0
$811$	−2.84004e7	−1.51625	−0.758127	−	0.652107i	$-0.773885\pi$
$811$	−2.84004e7	−1.51625	−0.758127	+	0.652107i	$0.773885\pi$
$812$	−5.92517e6	−0.315363
$813$	0	0
$814$	1.40883e6	0.0745243
$815$	4.38852e6	0.231432
$816$	0	0
$817$	2.78882e7	1.46173
$818$	−2.22511e7	−1.16270
$819$	0	0
$820$	−8.34082e6	−0.433186
$821$	3.12187e7	1.61643	0.808216	−	0.588886i	$-0.200433\pi$
$821$	3.12187e7	1.61643	0.808216	+	0.588886i	$0.200433\pi$
$822$	0	0
$823$	4.30753e6	0.221681	0.110841	−	0.993838i	$-0.464646\pi$
$823$	4.30753e6	0.221681	0.110841	+	0.993838i	$0.464646\pi$
$824$	8.55825e6	0.439104
$825$	0	0
$826$	−2.21874e6	−0.113150
$827$	3.37874e7	1.71787	0.858937	−	0.512081i	$-0.171125\pi$
$827$	3.37874e7	1.71787	0.858937	+	0.512081i	$0.171125\pi$
$828$	0	0
$829$	2.32056e7	1.17275	0.586377	−	0.810038i	$-0.300554\pi$
$829$	2.32056e7	1.17275	0.586377	+	0.810038i	$0.300554\pi$
$830$	−1.04683e7	−0.527448
$831$	0	0
$832$	2.33208e7	1.16798
$833$	−2.72548e7	−1.36091
$834$	0	0
$835$	−341913.	−0.0169707
$836$	1.46934e7	0.727119
$837$	0	0
$838$	−4.01861e7	−1.97682
$839$	3.03381e7	1.48794	0.743968	−	0.668216i	$-0.232942\pi$
$839$	3.03381e7	1.48794	0.743968	+	0.668216i	$0.232942\pi$
$840$	0	0
$841$	−1.81553e7	−0.885145
$842$	−1.80533e7	−0.877559
$843$	0	0
$844$	3.67922e6	0.177787
$845$	1.87164e6	0.0901740
$846$	0	0
$847$	−9.56843e6	−0.458282
$848$	1.70506e7	0.814234
$849$	0	0
$850$	−6.16573e7	−2.92710
$851$	1.08395e6	0.0513081
$852$	0	0
$853$	−2.55496e7	−1.20230	−0.601149	−	0.799137i	$-0.705290\pi$
$853$	−2.55496e7	−1.20230	−0.601149	+	0.799137i	$0.705290\pi$
$854$	−2.18101e7	−1.02332
$855$	0	0
$856$	2.87738e7	1.34219
$857$	3.85037e7	1.79081	0.895407	−	0.445250i	$-0.146885\pi$
$857$	3.85037e7	1.79081	0.895407	+	0.445250i	$0.146885\pi$
$858$	0	0
$859$	2.49038e7	1.15155	0.575774	−	0.817609i	$-0.304701\pi$
$859$	2.49038e7	1.15155	0.575774	+	0.817609i	$0.304701\pi$
$860$	−1.18881e7	−0.548108
$861$	0	0
$862$	4.98739e7	2.28615
$863$	3.38580e7	1.54751	0.773756	−	0.633483i	$-0.218375\pi$
$863$	3.38580e7	1.54751	0.773756	+	0.633483i	$0.218375\pi$
$864$	0	0
$865$	−4.32756e6	−0.196654
$866$	−3.70361e7	−1.67815
$867$	0	0
$868$	2.28693e7	1.03027
$869$	−1.87558e6	−0.0842533
$870$	0	0
$871$	−8.94334e6	−0.399442
$872$	−4.37969e7	−1.95053
$873$	0	0
$874$	1.76292e7	0.780646
$875$	5.63956e6	0.249015
$876$	0	0
$877$	4.91457e6	0.215768	0.107884	−	0.994163i	$-0.465593\pi$
$877$	4.91457e6	0.215768	0.107884	+	0.994163i	$0.465593\pi$
$878$	3.08752e7	1.35168
$879$	0	0
$880$	−799514.	−0.0348032
$881$	−3.12482e7	−1.35639	−0.678196	−	0.734881i	$-0.737238\pi$
$881$	−3.12482e7	−1.35639	−0.678196	+	0.734881i	$0.737238\pi$
$882$	0	0
$883$	2.31563e7	0.999466	0.499733	−	0.866179i	$-0.333431\pi$
$883$	2.31563e7	0.999466	0.499733	+	0.866179i	$0.333431\pi$
$884$	−6.17577e7	−2.65804
$885$	0	0
$886$	−5.69555e7	−2.43754
$887$	1.87182e6	0.0798832	0.0399416	−	0.999202i	$-0.487283\pi$
$887$	1.87182e6	0.0798832	0.0399416	+	0.999202i	$0.487283\pi$
$888$	0	0
$889$	9.41010e6	0.399337
$890$	5.69226e6	0.240885
$891$	0	0
$892$	2.05770e7	0.865903
$893$	2.77426e7	1.16417
$894$	0	0
$895$	−4.31696e6	−0.180144
$896$	−2.26954e7	−0.944426
$897$	0	0
$898$	−5.27713e7	−2.18377
$899$	−9.09266e6	−0.375225
$900$	0	0
$901$	9.08027e7	3.72638
$902$	1.38615e7	0.567274
$903$	0	0
$904$	1.63421e7	0.665101
$905$	3.32008e6	0.134749
$906$	0	0
$907$	−1.61068e7	−0.650117	−0.325059	−	0.945694i	$-0.605384\pi$
$907$	−1.61068e7	−0.650117	−0.325059	+	0.945694i	$0.605384\pi$
$908$	−3.59516e7	−1.44712
$909$	0	0
$910$	4.26614e6	0.170778
$911$	2.80661e7	1.12043	0.560217	−	0.828346i	$-0.310718\pi$
$911$	2.80661e7	1.12043	0.560217	+	0.828346i	$0.310718\pi$
$912$	0	0
$913$	1.11562e7	0.442933
$914$	−2.45530e7	−0.972163
$915$	0	0
$916$	−4.07748e7	−1.60566
$917$	3.05807e6	0.120095
$918$	0	0
$919$	1.01464e7	0.396301	0.198150	−	0.980172i	$-0.436507\pi$
$919$	1.01464e7	0.396301	0.198150	+	0.980172i	$0.436507\pi$
$920$	−3.31099e6	−0.128970
$921$	0	0
$922$	4.06025e7	1.57299
$923$	8.57058e6	0.331136
$924$	0	0
$925$	−3.15391e6	−0.121198
$926$	−3.97637e7	−1.52391
$927$	0	0
$928$	5.63631e6	0.214845
$929$	1.17387e6	0.0446254	0.0223127	−	0.999751i	$-0.492897\pi$
$929$	1.17387e6	0.0446254	0.0223127	+	0.999751i	$0.492897\pi$
$930$	0	0
$931$	−2.26741e7	−0.857346
$932$	−2.58158e7	−0.973521
$933$	0	0
$934$	−5.15724e7	−1.93442
$935$	−4.25781e6	−0.159279
$936$	0	0
$937$	−1.37307e7	−0.510908	−0.255454	−	0.966821i	$-0.582225\pi$
$937$	−1.37307e7	−0.510908	−0.255454	+	0.966821i	$0.582225\pi$
$938$	1.17447e7	0.435847
$939$	0	0
$940$	−1.18260e7	−0.436534
$941$	−8.16579e6	−0.300624	−0.150312	−	0.988639i	$-0.548028\pi$
$941$	−8.16579e6	−0.300624	−0.150312	+	0.988639i	$0.548028\pi$
$942$	0	0
$943$	1.06650e7	0.390554
$944$	−1.45397e6	−0.0531038
$945$	0	0
$946$	1.97566e7	0.717770
$947$	1.84586e6	0.0668844	0.0334422	−	0.999441i	$-0.489353\pi$
$947$	1.84586e6	0.0668844	0.0334422	+	0.999441i	$0.489353\pi$
$948$	0	0
$949$	1.24716e7	0.449528
$950$	−5.12947e7	−1.84401
$951$	0	0
$952$	3.57327e7	1.27783
$953$	−2.06466e7	−0.736403	−0.368201	−	0.929746i	$-0.620026\pi$
$953$	−2.06466e7	−0.736403	−0.368201	+	0.929746i	$0.620026\pi$
$954$	0	0
$955$	1.10985e7	0.393782
$956$	−2.74459e7	−0.971255
$957$	0	0
$958$	6.53862e6	0.230183
$959$	−4.86535e6	−0.170831
$960$	0	0
$961$	6.46561e6	0.225840
$962$	−4.92626e6	−0.171625
$963$	0	0
$964$	−9.45373e6	−0.327650
$965$	9.99299e6	0.345444
$966$	0	0
$967$	259971.	0.00894044	0.00447022	−	0.999990i	$-0.498577\pi$
$967$	259971.	0.00894044	0.00447022	+	0.999990i	$0.498577\pi$
$968$	−3.37498e7	−1.15766
$969$	0	0
$970$	−8.80492e6	−0.300467
$971$	4.97402e7	1.69301	0.846505	−	0.532380i	$-0.178702\pi$
$971$	4.97402e7	1.69301	0.846505	+	0.532380i	$0.178702\pi$
$972$	0	0
$973$	−2.95179e7	−0.999548
$974$	8.68558e7	2.93360
$975$	0	0
$976$	−1.42925e7	−0.480267
$977$	3.30137e6	0.110652	0.0553259	−	0.998468i	$-0.482380\pi$
$977$	3.30137e6	0.110652	0.0553259	+	0.998468i	$0.482380\pi$
$978$	0	0
$979$	−6.06632e6	−0.202287
$980$	9.66544e6	0.321482
$981$	0	0
$982$	−4.29486e6	−0.142125
$983$	−1.02213e7	−0.337383	−0.168692	−	0.985669i	$-0.553954\pi$
$983$	−1.02213e7	−0.337383	−0.168692	+	0.985669i	$0.553954\pi$
$984$	0	0
$985$	−6.66866e6	−0.219002
$986$	−3.22457e7	−1.05628
$987$	0	0
$988$	−5.13783e7	−1.67451
$989$	1.52007e7	0.494167
$990$	0	0
$991$	−1.38660e7	−0.448506	−0.224253	−	0.974531i	$-0.571994\pi$
$991$	−1.38660e7	−0.448506	−0.224253	+	0.974531i	$0.571994\pi$
$992$	−2.17544e7	−0.701887
$993$	0	0
$994$	−1.12552e7	−0.361315
$995$	−4.91454e6	−0.157371
$996$	0	0
$997$	−2.38978e7	−0.761412	−0.380706	−	0.924696i	$-0.624319\pi$
$997$	−2.38978e7	−0.761412	−0.380706	+	0.924696i	$0.624319\pi$
$998$	−4.31095e7	−1.37008
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.b.1.10		11
3.2	odd	2		177.6.a.a.1.2	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.a.1.2	✓	11	3.2	odd	2
531.6.a.b.1.10		11	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q+9.44473 q^{2} +57.2030 q^{4} -13.7903 q^{5} +67.4858 q^{7} +238.036 q^{8} +O(q^{10})\)	\(q+9.44473 q^{2} +57.2030 q^{4} -13.7903 q^{5} +67.4858 q^{7} +238.036 q^{8} -130.245 q^{10} +138.804 q^{11} -485.356 q^{13} +637.385 q^{14} +417.688 q^{16} +2224.39 q^{17} +1850.55 q^{19} -788.844 q^{20} +1310.97 q^{22} +1008.66 q^{23} -2934.83 q^{25} -4584.06 q^{26} +3860.39 q^{28} -1534.86 q^{29} +5924.08 q^{31} -3672.19 q^{32} +21008.8 q^{34} -930.646 q^{35} +1074.65 q^{37} +17477.9 q^{38} -3282.57 q^{40} +10573.5 q^{41} +15070.3 q^{43} +7940.01 q^{44} +9526.48 q^{46} +14991.6 q^{47} -12252.7 q^{49} -27718.7 q^{50} -27763.8 q^{52} +40821.3 q^{53} -1914.14 q^{55} +16064.0 q^{56} -14496.4 q^{58} -3481.00 q^{59} -34218.1 q^{61} +55951.4 q^{62} -48048.9 q^{64} +6693.19 q^{65} +18426.3 q^{67} +127242. q^{68} -8789.71 q^{70} -17658.3 q^{71} -25695.8 q^{73} +10149.8 q^{74} +105857. q^{76} +9367.30 q^{77} -13512.5 q^{79} -5760.02 q^{80} +99863.6 q^{82} +80373.4 q^{83} -30675.0 q^{85} +142335. q^{86} +33040.3 q^{88} -43704.2 q^{89} -32754.7 q^{91} +57698.1 q^{92} +141591. q^{94} -25519.5 q^{95} +67602.6 q^{97} -115723. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) \) 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8} - 399 q^{10} + 698 q^{11} - 1556 q^{13} + 1679 q^{14} - 2662 q^{16} + 4793 q^{17} - 3753 q^{19} + 11023 q^{20} - 9534 q^{22} + 7323 q^{23} + 7867 q^{25} + 4844 q^{26} + 3650 q^{28} + 15467 q^{29} - 5151 q^{31} + 15368 q^{32} + 8452 q^{34} + 23285 q^{35} + 8623 q^{37} - 15205 q^{38} + 41530 q^{40} + 6369 q^{41} - 20506 q^{43} + 55632 q^{44} - 45191 q^{46} + 47899 q^{47} - 10322 q^{49} + 102147 q^{50} - 292 q^{52} + 80048 q^{53} - 2114 q^{55} + 108126 q^{56} - 58294 q^{58} - 38291 q^{59} - 82527 q^{61} + 67438 q^{62} - 51411 q^{64} + 167646 q^{65} - 166976 q^{67} + 136533 q^{68} + 76140 q^{70} + 183560 q^{71} - 36809 q^{73} + 116686 q^{74} + 55580 q^{76} + 164885 q^{77} - 281518 q^{79} + 32683 q^{80} + 178815 q^{82} + 254691 q^{83} + 4763 q^{85} - 349324 q^{86} + 251285 q^{88} + 89687 q^{89} + 34897 q^{91} + 20240 q^{92} + 96548 q^{94} + 155113 q^{95} - 45828 q^{97} - 465864 q^{98}+O(q^{100}) \) 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8 - 399 * q^10 + 698 * q^11 - 1556 * q^13 + 1679 * q^14 - 2662 * q^16 + 4793 * q^17 - 3753 * q^19 + 11023 * q^20 - 9534 * q^22 + 7323 * q^23 + 7867 * q^25 + 4844 * q^26 + 3650 * q^28 + 15467 * q^29 - 5151 * q^31 + 15368 * q^32 + 8452 * q^34 + 23285 * q^35 + 8623 * q^37 - 15205 * q^38 + 41530 * q^40 + 6369 * q^41 - 20506 * q^43 + 55632 * q^44 - 45191 * q^46 + 47899 * q^47 - 10322 * q^49 + 102147 * q^50 - 292 * q^52 + 80048 * q^53 - 2114 * q^55 + 108126 * q^56 - 58294 * q^58 - 38291 * q^59 - 82527 * q^61 + 67438 * q^62 - 51411 * q^64 + 167646 * q^65 - 166976 * q^67 + 136533 * q^68 + 76140 * q^70 + 183560 * q^71 - 36809 * q^73 + 116686 * q^74 + 55580 * q^76 + 164885 * q^77 - 281518 * q^79 + 32683 * q^80 + 178815 * q^82 + 254691 * q^83 + 4763 * q^85 - 349324 * q^86 + 251285 * q^88 + 89687 * q^89 + 34897 * q^91 + 20240 * q^92 + 96548 * q^94 + 155113 * q^95 - 45828 * q^97 - 465864 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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