LMFDB - Embedded newform 43.6.a.a.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [43,6,Mod(1,43)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(43, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("43.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	43.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:	$6.89650425196$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 $ x^8 - 4x^7 - 173x^6 + 462x^5 + 9118x^4 - 14192x^3 - 167688x^2 + 106368*x + 681984
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$10.9591$ of defining polynomial
Character	$\chi$	$=$	43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.95911 q^{2} -25.1057 q^{3} +48.2657 q^{4} -61.2927 q^{5} -224.924 q^{6} -166.001 q^{7} +145.726 q^{8} +387.294 q^{9} +O(q^{10})$	\(q+8.95911 q^{2} -25.1057 q^{3} +48.2657 q^{4} -61.2927 q^{5} -224.924 q^{6} -166.001 q^{7} +145.726 q^{8} +387.294 q^{9} -549.129 q^{10} +607.827 q^{11} -1211.74 q^{12} -1039.54 q^{13} -1487.22 q^{14} +1538.79 q^{15} -238.923 q^{16} +1439.76 q^{17} +3469.81 q^{18} -1332.62 q^{19} -2958.34 q^{20} +4167.57 q^{21} +5445.59 q^{22} -437.244 q^{23} -3658.56 q^{24} +631.800 q^{25} -9313.39 q^{26} -3622.60 q^{27} -8012.16 q^{28} -87.2656 q^{29} +13786.2 q^{30} +2654.45 q^{31} -6803.79 q^{32} -15259.9 q^{33} +12898.9 q^{34} +10174.7 q^{35} +18693.0 q^{36} -4671.70 q^{37} -11939.1 q^{38} +26098.4 q^{39} -8931.98 q^{40} -9012.91 q^{41} +37337.7 q^{42} -1849.00 q^{43} +29337.2 q^{44} -23738.3 q^{45} -3917.32 q^{46} -8623.49 q^{47} +5998.32 q^{48} +10749.4 q^{49} +5660.37 q^{50} -36146.0 q^{51} -50174.3 q^{52} -28358.3 q^{53} -32455.3 q^{54} -37255.4 q^{55} -24190.8 q^{56} +33456.3 q^{57} -781.822 q^{58} +48066.0 q^{59} +74271.0 q^{60} -39750.4 q^{61} +23781.5 q^{62} -64291.2 q^{63} -53310.4 q^{64} +63716.5 q^{65} -136715. q^{66} +19233.9 q^{67} +69490.8 q^{68} +10977.3 q^{69} +91156.0 q^{70} +17008.5 q^{71} +56439.0 q^{72} +22236.4 q^{73} -41854.3 q^{74} -15861.8 q^{75} -64319.9 q^{76} -100900. q^{77} +233819. q^{78} +36951.1 q^{79} +14644.3 q^{80} -3164.77 q^{81} -80747.7 q^{82} -117990. q^{83} +201151. q^{84} -88246.5 q^{85} -16565.4 q^{86} +2190.86 q^{87} +88576.5 q^{88} +41537.6 q^{89} -212674. q^{90} +172565. q^{91} -21103.9 q^{92} -66641.7 q^{93} -77258.8 q^{94} +81679.9 q^{95} +170814. q^{96} +32593.9 q^{97} +96304.8 q^{98} +235408. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{2} - 26 q^{3} + 122 q^{4} - 212 q^{5} - 69 q^{6} - 136 q^{7} - 666 q^{8} + 546 q^{9}+O(q^{10}) $ 8 * q - 12 * q^2 - 26 * q^3 + 122 * q^4 - 212 * q^5 - 69 * q^6 - 136 * q^7 - 666 * q^8 + 546 * q^9	\( 8 q - 12 q^{2} - 26 q^{3} + 122 q^{4} - 212 q^{5} - 69 q^{6} - 136 q^{7} - 666 q^{8} + 546 q^{9} - 617 q^{10} - 532 q^{11} - 4195 q^{12} - 2492 q^{13} - 4240 q^{14} - 1780 q^{15} + 1882 q^{16} - 2534 q^{17} - 3711 q^{18} - 1678 q^{19} - 2607 q^{20} - 2256 q^{21} + 11502 q^{22} - 2488 q^{23} + 19953 q^{24} + 4378 q^{25} + 4586 q^{26} - 8960 q^{27} + 18640 q^{28} - 4360 q^{29} + 25092 q^{30} + 5704 q^{31} - 18294 q^{32} - 12852 q^{33} + 30007 q^{34} + 5640 q^{35} + 67969 q^{36} - 3772 q^{37} - 6559 q^{38} + 11120 q^{39} + 14869 q^{40} - 10698 q^{41} + 78698 q^{42} - 14792 q^{43} - 356 q^{44} - 44912 q^{45} - 19389 q^{46} - 77864 q^{47} - 118727 q^{48} + 7188 q^{49} + 26877 q^{50} - 80246 q^{51} - 60736 q^{52} - 62352 q^{53} + 61026 q^{54} - 49552 q^{55} - 144528 q^{56} - 808 q^{57} + 52951 q^{58} - 26224 q^{59} + 101500 q^{60} - 82540 q^{61} - 9023 q^{62} - 61768 q^{63} + 153858 q^{64} - 5000 q^{65} - 48516 q^{66} + 27784 q^{67} + 40507 q^{68} - 93776 q^{69} + 185910 q^{70} - 9504 q^{71} - 186687 q^{72} + 14260 q^{73} - 15239 q^{74} + 167420 q^{75} + 1279 q^{76} - 218140 q^{77} + 264170 q^{78} + 160248 q^{79} - 1291 q^{80} + 161076 q^{81} - 47781 q^{82} - 77176 q^{83} + 16382 q^{84} + 141096 q^{85} + 22188 q^{86} + 268136 q^{87} + 129544 q^{88} - 265692 q^{89} + 48990 q^{90} + 401148 q^{91} + 190391 q^{92} - 123860 q^{93} + 248737 q^{94} + 135884 q^{95} + 950817 q^{96} + 144742 q^{97} - 292244 q^{98} + 239516 q^{99}+O(q^{100}) \) 8 * q - 12 * q^2 - 26 * q^3 + 122 * q^4 - 212 * q^5 - 69 * q^6 - 136 * q^7 - 666 * q^8 + 546 * q^9 - 617 * q^10 - 532 * q^11 - 4195 * q^12 - 2492 * q^13 - 4240 * q^14 - 1780 * q^15 + 1882 * q^16 - 2534 * q^17 - 3711 * q^18 - 1678 * q^19 - 2607 * q^20 - 2256 * q^21 + 11502 * q^22 - 2488 * q^23 + 19953 * q^24 + 4378 * q^25 + 4586 * q^26 - 8960 * q^27 + 18640 * q^28 - 4360 * q^29 + 25092 * q^30 + 5704 * q^31 - 18294 * q^32 - 12852 * q^33 + 30007 * q^34 + 5640 * q^35 + 67969 * q^36 - 3772 * q^37 - 6559 * q^38 + 11120 * q^39 + 14869 * q^40 - 10698 * q^41 + 78698 * q^42 - 14792 * q^43 - 356 * q^44 - 44912 * q^45 - 19389 * q^46 - 77864 * q^47 - 118727 * q^48 + 7188 * q^49 + 26877 * q^50 - 80246 * q^51 - 60736 * q^52 - 62352 * q^53 + 61026 * q^54 - 49552 * q^55 - 144528 * q^56 - 808 * q^57 + 52951 * q^58 - 26224 * q^59 + 101500 * q^60 - 82540 * q^61 - 9023 * q^62 - 61768 * q^63 + 153858 * q^64 - 5000 * q^65 - 48516 * q^66 + 27784 * q^67 + 40507 * q^68 - 93776 * q^69 + 185910 * q^70 - 9504 * q^71 - 186687 * q^72 + 14260 * q^73 - 15239 * q^74 + 167420 * q^75 + 1279 * q^76 - 218140 * q^77 + 264170 * q^78 + 160248 * q^79 - 1291 * q^80 + 161076 * q^81 - 47781 * q^82 - 77176 * q^83 + 16382 * q^84 + 141096 * q^85 + 22188 * q^86 + 268136 * q^87 + 129544 * q^88 - 265692 * q^89 + 48990 * q^90 + 401148 * q^91 + 190391 * q^92 - 123860 * q^93 + 248737 * q^94 + 135884 * q^95 + 950817 * q^96 + 144742 * q^97 - 292244 * q^98 + 239516 * 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 $

$2$	8.95911	1.58376	0.791881	−	0.610675i	$-0.209102\pi$
$2$	8.95911	1.58376	0.791881	+	0.610675i	$0.209102\pi$
$3$	−25.1057	−1.61053	−0.805264	−	0.592916i	$-0.797977\pi$
$3$	−25.1057	−1.61053	−0.805264	+	0.592916i	$0.797977\pi$
$4$	48.2657	1.50830
$5$	−61.2927	−1.09644	−0.548219	−	0.836335i	$-0.684694\pi$
$5$	−61.2927	−1.09644	−0.548219	+	0.836335i	$0.684694\pi$
$6$	−224.924	−2.55070
$7$	−166.001	−1.28046	−0.640230	−	0.768183i	$-0.721161\pi$
$7$	−166.001	−1.28046	−0.640230	+	0.768183i	$0.721161\pi$
$8$	145.726	0.805033
$9$	387.294	1.59380
$10$	−549.129	−1.73650
$11$	607.827	1.51460	0.757301	−	0.653066i	$-0.226518\pi$
$11$	607.827	1.51460	0.757301	+	0.653066i	$0.226518\pi$
$12$	−1211.74	−2.42917
$13$	−1039.54	−1.70602	−0.853011	−	0.521894i	$-0.825226\pi$
$13$	−1039.54	−1.70602	−0.853011	+	0.521894i	$0.825226\pi$
$14$	−1487.22	−2.02794
$15$	1538.79	1.76584
$16$	−238.923	−0.233323
$17$	1439.76	1.20828	0.604138	−	0.796880i	$-0.293517\pi$
$17$	1439.76	1.20828	0.604138	+	0.796880i	$0.293517\pi$
$18$	3469.81	2.52421
$19$	−1332.62	−0.846881	−0.423440	−	0.905924i	$-0.639178\pi$
$19$	−1332.62	−0.846881	−0.423440	+	0.905924i	$0.639178\pi$
$20$	−2958.34	−1.65376
$21$	4167.57	2.06222
$22$	5445.59	2.39877
$23$	−437.244	−0.172347	−0.0861737	−	0.996280i	$-0.527464\pi$
$23$	−437.244	−0.172347	−0.0861737	+	0.996280i	$0.527464\pi$
$24$	−3658.56	−1.29653
$25$	631.800	0.202176
$26$	−9313.39	−2.70193
$27$	−3622.60	−0.956336
$28$	−8012.16	−1.93132
$29$	−87.2656	−0.0192685	−0.00963425	−	0.999954i	$-0.503067\pi$
$29$	−87.2656	−0.0192685	−0.00963425	+	0.999954i	$0.503067\pi$
$30$	13786.2	2.79668
$31$	2654.45	0.496101	0.248050	−	0.968747i	$-0.420210\pi$
$31$	2654.45	0.496101	0.248050	+	0.968747i	$0.420210\pi$
$32$	−6803.79	−1.17456
$33$	−15259.9	−2.43931
$34$	12898.9	1.91362
$35$	10174.7	1.40394
$36$	18693.0	2.40394
$37$	−4671.70	−0.561010	−0.280505	−	0.959853i	$-0.590502\pi$
$37$	−4671.70	−0.561010	−0.280505	+	0.959853i	$0.590502\pi$
$38$	−11939.1	−1.34126
$39$	26098.4	2.74760
$40$	−8931.98	−0.882668
$41$	−9012.91	−0.837347	−0.418674	−	0.908137i	$-0.637505\pi$
$41$	−9012.91	−0.837347	−0.418674	+	0.908137i	$0.637505\pi$
$42$	37337.7	3.26606
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	29337.2	2.28448
$45$	−23738.3	−1.74751
$46$	−3917.32	−0.272957
$47$	−8623.49	−0.569427	−0.284714	−	0.958613i	$-0.591899\pi$
$47$	−8623.49	−0.569427	−0.284714	+	0.958613i	$0.591899\pi$
$48$	5998.32	0.375774
$49$	10749.4	0.639577
$50$	5660.37	0.320199
$51$	−36146.0	−1.94596
$52$	−50174.3	−2.57320
$53$	−28358.3	−1.38672	−0.693362	−	0.720590i	$-0.743871\pi$
$53$	−28358.3	−1.38672	−0.693362	+	0.720590i	$0.743871\pi$
$54$	−32455.3	−1.51461
$55$	−37255.4	−1.66067
$56$	−24190.8	−1.03081
$57$	33456.3	1.36393
$58$	−781.822	−0.0305167
$59$	48066.0	1.79766	0.898832	−	0.438294i	$-0.144417\pi$
$59$	48066.0	1.79766	0.898832	+	0.438294i	$0.144417\pi$
$60$	74271.0	2.66343
$61$	−39750.4	−1.36778	−0.683891	−	0.729585i	$-0.739713\pi$
$61$	−39750.4	−1.36778	−0.683891	+	0.729585i	$0.739713\pi$
$62$	23781.5	0.785706
$63$	−64291.2	−2.04080
$64$	−53310.4	−1.62690
$65$	63716.5	1.87055
$66$	−136715.	−3.86329
$67$	19233.9	0.523456	0.261728	−	0.965142i	$-0.415708\pi$
$67$	19233.9	0.523456	0.261728	+	0.965142i	$0.415708\pi$
$68$	69490.8	1.82245
$69$	10977.3	0.277570
$70$	91156.0	2.22351
$71$	17008.5	0.400424	0.200212	−	0.979753i	$-0.435837\pi$
$71$	17008.5	0.400424	0.200212	+	0.979753i	$0.435837\pi$
$72$	56439.0	1.28306
$73$	22236.4	0.488380	0.244190	−	0.969727i	$-0.421478\pi$
$73$	22236.4	0.488380	0.244190	+	0.969727i	$0.421478\pi$
$74$	−41854.3	−0.888506
$75$	−15861.8	−0.325610
$76$	−64319.9	−1.27735
$77$	−100900.	−1.93939
$78$	233819.	4.35154
$79$	36951.1	0.666131	0.333066	−	0.942904i	$-0.391917\pi$
$79$	36951.1	0.666131	0.333066	+	0.942904i	$0.391917\pi$
$80$	14644.3	0.255825
$81$	−3164.77	−0.0535957
$82$	−80747.7	−1.32616
$83$	−117990.	−1.87997	−0.939985	−	0.341216i	$-0.889161\pi$
$83$	−117990.	−1.87997	−0.939985	+	0.341216i	$0.889161\pi$
$84$	201151.	3.11045
$85$	−88246.5	−1.32480
$86$	−16565.4	−0.241522
$87$	2190.86	0.0310325
$88$	88576.5	1.21930
$89$	41537.6	0.555861	0.277931	−	0.960601i	$-0.410351\pi$
$89$	41537.6	0.555861	0.277931	+	0.960601i	$0.410351\pi$
$90$	−212674.	−2.76763
$91$	172565.	2.18449
$92$	−21103.9	−0.259952
$93$	−66641.7	−0.798985
$94$	−77258.8	−0.901838
$95$	81679.9	0.928552
$96$	170814.	1.89166
$97$	32593.9	0.351728	0.175864	−	0.984415i	$-0.443728\pi$
$97$	32593.9	0.351728	0.175864	+	0.984415i	$0.443728\pi$
$98$	96304.8	1.01294
$99$	235408.	2.41398
$100$	30494.3	0.304943
$101$	−7934.83	−0.0773988	−0.0386994	−	0.999251i	$-0.512321\pi$
$101$	−7934.83	−0.0773988	−0.0386994	+	0.999251i	$0.512321\pi$
$102$	−323836.	−3.08194
$103$	132558.	1.23116	0.615579	−	0.788075i	$-0.288922\pi$
$103$	132558.	1.23116	0.615579	+	0.788075i	$0.288922\pi$
$104$	−151489.	−1.37340
$105$	−255442.	−2.26109
$106$	−254065.	−2.19624
$107$	120642.	1.01868	0.509342	−	0.860564i	$-0.329889\pi$
$107$	120642.	1.01868	0.509342	+	0.860564i	$0.329889\pi$
$108$	−174847.	−1.44245
$109$	16769.1	0.135190	0.0675950	−	0.997713i	$-0.478467\pi$
$109$	16769.1	0.135190	0.0675950	+	0.997713i	$0.478467\pi$
$110$	−333775.	−2.63010
$111$	117286.	0.903522
$112$	39661.5	0.298761
$113$	−18201.6	−0.134096	−0.0670478	−	0.997750i	$-0.521358\pi$
$113$	−18201.6	−0.134096	−0.0670478	+	0.997750i	$0.521358\pi$
$114$	299739.	2.16013
$115$	26799.9	0.188968
$116$	−4211.94	−0.0290628
$117$	−402609.	−2.71906
$118$	430629.	2.84707
$119$	−239001.	−1.54715
$120$	224243.	1.42156
$121$	208403.	1.29402
$122$	−356128.	−2.16624
$123$	226275.	1.34857
$124$	128119.	0.748271
$125$	152815.	0.874764
$126$	−575993.	−3.23214
$127$	−155225.	−0.853987	−0.426994	−	0.904255i	$-0.640427\pi$
$127$	−155225.	−0.853987	−0.426994	+	0.904255i	$0.640427\pi$
$128$	−259892.	−1.40207
$129$	46420.4	0.245603
$130$	570843.	2.96250
$131$	206749.	1.05261	0.526303	−	0.850297i	$-0.323578\pi$
$131$	206749.	1.05261	0.526303	+	0.850297i	$0.323578\pi$
$132$	−736530.	−3.67922
$133$	221216.	1.08440
$134$	172319.	0.829030
$135$	222039.	1.04856
$136$	209810.	0.972702
$137$	−219188.	−0.997736	−0.498868	−	0.866678i	$-0.666251\pi$
$137$	−219188.	−0.997736	−0.498868	+	0.866678i	$0.666251\pi$
$138$	98347.0	0.439606
$139$	83047.4	0.364577	0.182288	−	0.983245i	$-0.441650\pi$
$139$	83047.4	0.364577	0.182288	+	0.983245i	$0.441650\pi$
$140$	491087.	2.11757
$141$	216498.	0.917079
$142$	152381.	0.634176
$143$	−631863.	−2.58394
$144$	−92533.5	−0.371871
$145$	5348.75	0.0211267
$146$	199219.	0.773478
$147$	−269870.	−1.03006
$148$	−225483.	−0.846173
$149$	−505431.	−1.86507	−0.932537	−	0.361074i	$-0.882410\pi$
$149$	−505431.	−1.86507	−0.932537	+	0.361074i	$0.882410\pi$
$150$	−142107.	−0.515690
$151$	−222212.	−0.793095	−0.396548	−	0.918014i	$-0.629792\pi$
$151$	−222212.	−0.793095	−0.396548	+	0.918014i	$0.629792\pi$
$152$	−194198.	−0.681767
$153$	557609.	1.92575
$154$	−903974.	−3.07153
$155$	−162698.	−0.543944
$156$	1.25966e6	4.14421
$157$	18138.1	0.0587275	0.0293638	−	0.999569i	$-0.490652\pi$
$157$	18138.1	0.0587275	0.0293638	+	0.999569i	$0.490652\pi$
$158$	331049.	1.05499
$159$	711953.	2.23336
$160$	417023.	1.28783
$161$	72583.1	0.220684
$162$	−28353.5	−0.0848828
$163$	−428372.	−1.26285	−0.631425	−	0.775437i	$-0.717530\pi$
$163$	−428372.	−1.26285	−0.631425	+	0.775437i	$0.717530\pi$
$164$	−435015.	−1.26297
$165$	935321.	2.67455
$166$	−1.05709e6	−2.97743
$167$	590110.	1.63735	0.818676	−	0.574256i	$-0.194709\pi$
$167$	590110.	1.63735	0.818676	+	0.574256i	$0.194709\pi$
$168$	607325.	1.66015
$169$	709358.	1.91051
$170$	−790611.	−2.09817
$171$	−516116.	−1.34976
$172$	−89243.3	−0.230014
$173$	−194001.	−0.492821	−0.246410	−	0.969166i	$-0.579251\pi$
$173$	−194001.	−0.492821	−0.246410	+	0.969166i	$0.579251\pi$
$174$	19628.2	0.0491481
$175$	−104880.	−0.258878
$176$	−145224.	−0.353392
$177$	−1.20673e6	−2.89519
$178$	372140.	0.880353
$179$	−434140.	−1.01274	−0.506369	−	0.862317i	$-0.669012\pi$
$179$	−434140.	−1.01274	−0.506369	+	0.862317i	$0.669012\pi$
$180$	−1.14575e6	−2.63577
$181$	−626829.	−1.42217	−0.711087	−	0.703104i	$-0.751797\pi$
$181$	−626829.	−1.42217	−0.711087	+	0.703104i	$0.751797\pi$
$182$	1.54603e6	3.45972
$183$	997959.	2.20285
$184$	−63718.1	−0.138745
$185$	286341.	0.615112
$186$	−597050.	−1.26540
$187$	875122.	1.83006
$188$	−416219.	−0.858870
$189$	601355.	1.22455
$190$	731780.	1.47061
$191$	−109059.	−0.216310	−0.108155	−	0.994134i	$-0.534494\pi$
$191$	−109059.	−0.216310	−0.108155	+	0.994134i	$0.534494\pi$
$192$	1.33839e6	2.62017
$193$	−31563.2	−0.0609941	−0.0304971	−	0.999535i	$-0.509709\pi$
$193$	−31563.2	−0.0609941	−0.0304971	+	0.999535i	$0.509709\pi$
$194$	292012.	0.557053
$195$	−1.59964e6	−3.01257
$196$	518826.	0.964676
$197$	−276843.	−0.508238	−0.254119	−	0.967173i	$-0.581786\pi$
$197$	−276843.	−0.508238	−0.254119	+	0.967173i	$0.581786\pi$
$198$	2.10905e6	3.82316
$199$	−777649.	−1.39204	−0.696019	−	0.718024i	$-0.745047\pi$
$199$	−777649.	−1.39204	−0.696019	+	0.718024i	$0.745047\pi$
$200$	92070.0	0.162758
$201$	−482879.	−0.843041
$202$	−71089.0	−0.122581
$203$	14486.2	0.0246725
$204$	−1.74461e6	−2.93511
$205$	552426.	0.918099
$206$	1.18761e6	1.94986
$207$	−169342.	−0.274688
$208$	248371.	0.398055
$209$	−810002.	−1.28269
$210$	−2.28853e6	−3.58103
$211$	−89486.6	−0.138373	−0.0691866	−	0.997604i	$-0.522040\pi$
$211$	−89486.6	−0.138373	−0.0691866	+	0.997604i	$0.522040\pi$
$212$	−1.36873e6	−2.09160
$213$	−427009.	−0.644894
$214$	1.08085e6	1.61335
$215$	113330.	0.167205
$216$	−527908.	−0.769882
$217$	−440641.	−0.635237
$218$	150237.	0.214109
$219$	−558260.	−0.786550
$220$	−1.79816e6	−2.50479
$221$	−1.49669e6	−2.06135
$222$	1.05078e6	1.43096
$223$	−952284.	−1.28234	−0.641172	−	0.767398i	$-0.721551\pi$
$223$	−952284.	−1.28234	−0.641172	+	0.767398i	$0.721551\pi$
$224$	1.12944e6	1.50398
$225$	244693.	0.322229
$226$	−163071.	−0.212376
$227$	−465809.	−0.599988	−0.299994	−	0.953941i	$-0.596985\pi$
$227$	−465809.	−0.599988	−0.299994	+	0.953941i	$0.596985\pi$
$228$	1.61479e6	2.05721
$229$	477440.	0.601631	0.300815	−	0.953682i	$-0.402741\pi$
$229$	477440.	0.601631	0.300815	+	0.953682i	$0.402741\pi$
$230$	240103.	0.299281
$231$	2.53316e6	3.12344
$232$	−12716.9	−0.0155118
$233$	743589.	0.897312	0.448656	−	0.893705i	$-0.351903\pi$
$233$	743589.	0.897312	0.448656	+	0.893705i	$0.351903\pi$
$234$	−3.60702e6	−4.30635
$235$	528557.	0.624342
$236$	2.31994e6	2.71142
$237$	−927682.	−1.07282
$238$	−2.14124e6	−2.45032
$239$	1.27115e6	1.43947	0.719734	−	0.694250i	$-0.244264\pi$
$239$	1.27115e6	1.43947	0.719734	+	0.694250i	$0.244264\pi$
$240$	−367654.	−0.412013
$241$	−875960.	−0.971497	−0.485749	−	0.874099i	$-0.661453\pi$
$241$	−875960.	−0.971497	−0.485749	+	0.874099i	$0.661453\pi$
$242$	1.86710e6	2.04942
$243$	959745.	1.04265
$244$	−1.91858e6	−2.06303
$245$	−658858.	−0.701256
$246$	2.02722e6	2.13582
$247$	1.38532e6	1.44480
$248$	386823.	0.399377
$249$	2.96222e6	3.02774
$250$	1.36909e6	1.38542
$251$	−1.19674e6	−1.19899	−0.599494	−	0.800380i	$-0.704631\pi$
$251$	−1.19674e6	−1.19899	−0.599494	+	0.800380i	$0.704631\pi$
$252$	−3.10306e6	−3.07815
$253$	−265769.	−0.261038
$254$	−1.39068e6	−1.35251
$255$	2.21549e6	2.13363
$256$	−622474.	−0.593638
$257$	−1.31630e6	−1.24315	−0.621573	−	0.783357i	$-0.713506\pi$
$257$	−1.31630e6	−1.24315	−0.621573	+	0.783357i	$0.713506\pi$
$258$	415885.	0.388977
$259$	775507.	0.718350
$260$	3.07532e6	2.82135
$261$	−33797.4	−0.0307102
$262$	1.85229e6	1.66708
$263$	1.74612e6	1.55663	0.778314	−	0.627876i	$-0.216075\pi$
$263$	1.74612e6	1.55663	0.778314	+	0.627876i	$0.216075\pi$
$264$	−2.22377e6	−1.96372
$265$	1.73816e6	1.52046
$266$	1.98190e6	1.71743
$267$	−1.04283e6	−0.895231
$268$	928338.	0.789531
$269$	−470733.	−0.396637	−0.198319	−	0.980138i	$-0.563548\pi$
$269$	−470733.	−0.396637	−0.198319	+	0.980138i	$0.563548\pi$
$270$	1.98927e6	1.66068
$271$	29459.5	0.0243670	0.0121835	−	0.999926i	$-0.496122\pi$
$271$	29459.5	0.0243670	0.0121835	+	0.999926i	$0.496122\pi$
$272$	−343991.	−0.281919
$273$	−4.33237e6	−3.51819
$274$	−1.96373e6	−1.58018
$275$	384025.	0.306216
$276$	529828.	0.418661
$277$	1.01172e6	0.792244	0.396122	−	0.918198i	$-0.370356\pi$
$277$	1.01172e6	0.792244	0.396122	+	0.918198i	$0.370356\pi$
$278$	744031.	0.577403
$279$	1.02805e6	0.790687
$280$	1.48272e6	1.13022
$281$	−74592.9	−0.0563549	−0.0281775	−	0.999603i	$-0.508970\pi$
$281$	−74592.9	−0.0563549	−0.0281775	+	0.999603i	$0.508970\pi$
$282$	1.93963e6	1.45244
$283$	−1.14789e6	−0.851993	−0.425996	−	0.904725i	$-0.640076\pi$
$283$	−1.14789e6	−0.851993	−0.425996	+	0.904725i	$0.640076\pi$
$284$	820927.	0.603961
$285$	−2.05063e6	−1.49546
$286$	−5.66093e6	−4.09235
$287$	1.49615e6	1.07219
$288$	−2.63507e6	−1.87202
$289$	653038.	0.459932
$290$	47920.0	0.0334597
$291$	−818290.	−0.566467
$292$	1.07326e6	0.736625
$293$	−1.30694e6	−0.889376	−0.444688	−	0.895686i	$-0.646685\pi$
$293$	−1.30694e6	−0.889376	−0.444688	+	0.895686i	$0.646685\pi$
$294$	−2.41780e6	−1.63137
$295$	−2.94610e6	−1.97103
$296$	−680790.	−0.451631
$297$	−2.20191e6	−1.44847
$298$	−4.52821e6	−2.95383
$299$	454535.	0.294028
$300$	−765580.	−0.491120
$301$	306936.	0.195268
$302$	−1.99082e6	−1.25607
$303$	199209.	0.124653
$304$	318394.	0.197597
$305$	2.43641e6	1.49969
$306$	4.99568e6	3.04994
$307$	−1.50999e6	−0.914380	−0.457190	−	0.889369i	$-0.651144\pi$
$307$	−1.50999e6	−0.914380	−0.457190	+	0.889369i	$0.651144\pi$
$308$	−4.87001e6	−2.92518
$309$	−3.32796e6	−1.98282
$310$	−1.45763e6	−0.861478
$311$	1.69979e6	0.996539	0.498269	−	0.867022i	$-0.333969\pi$
$311$	1.69979e6	0.996539	0.498269	+	0.867022i	$0.333969\pi$
$312$	3.80323e6	2.21190
$313$	2.85190e6	1.64541	0.822703	−	0.568471i	$-0.192465\pi$
$313$	2.85190e6	1.64541	0.822703	+	0.568471i	$0.192465\pi$
$314$	162501.	0.0930104
$315$	3.94059e6	2.23761
$316$	1.78347e6	1.00473
$317$	−158403.	−0.0885351	−0.0442676	−	0.999020i	$-0.514095\pi$
$317$	−158403.	−0.0885351	−0.0442676	+	0.999020i	$0.514095\pi$
$318$	6.37847e6	3.53711
$319$	−53042.4	−0.0291841
$320$	3.26754e6	1.78380
$321$	−3.02880e6	−1.64062
$322$	650280.	0.349511
$323$	−1.91865e6	−1.02327
$324$	−152750.	−0.0808386
$325$	−656784.	−0.344917
$326$	−3.83783e6	−2.00006
$327$	−421000.	−0.217727
$328$	−1.31342e6	−0.674092
$329$	1.43151e6	0.729129
$330$	8.37965e6	4.23585
$331$	1.38726e6	0.695968	0.347984	−	0.937501i	$-0.386866\pi$
$331$	1.38726e6	0.695968	0.347984	+	0.937501i	$0.386866\pi$
$332$	−5.69488e6	−2.83557
$333$	−1.80932e6	−0.894139
$334$	5.28687e6	2.59318
$335$	−1.17890e6	−0.573937
$336$	−995728.	−0.481163
$337$	602990.	0.289225	0.144612	−	0.989488i	$-0.453807\pi$
$337$	602990.	0.289225	0.144612	+	0.989488i	$0.453807\pi$
$338$	6.35522e6	3.02579
$339$	456964.	0.215965
$340$	−4.25928e6	−1.99820
$341$	1.61345e6	0.751395
$342$	−4.62394e6	−2.13770
$343$	1.00557e6	0.461507
$344$	−269448.	−0.122766
$345$	−672829.	−0.304339
$346$	−1.73808e6	−0.780511
$347$	1.38847e6	0.619030	0.309515	−	0.950895i	$-0.399833\pi$
$347$	1.38847e6	0.619030	0.309515	+	0.950895i	$0.399833\pi$
$348$	105743.	0.0468064
$349$	1.90607e6	0.837676	0.418838	−	0.908061i	$-0.362437\pi$
$349$	1.90607e6	0.837676	0.418838	+	0.908061i	$0.362437\pi$
$350$	−939628.	−0.410002
$351$	3.76585e6	1.63153
$352$	−4.13553e6	−1.77899
$353$	−2.53782e6	−1.08399	−0.541993	−	0.840383i	$-0.682330\pi$
$353$	−2.53782e6	−1.08399	−0.541993	+	0.840383i	$0.682330\pi$
$354$	−1.08112e7	−4.58529
$355$	−1.04250e6	−0.439040
$356$	2.00484e6	0.838408
$357$	6.00028e6	2.49173
$358$	−3.88951e6	−1.60394
$359$	−589213.	−0.241288	−0.120644	−	0.992696i	$-0.538496\pi$
$359$	−589213.	−0.241288	−0.120644	+	0.992696i	$0.538496\pi$
$360$	−3.45930e6	−1.40680
$361$	−700223.	−0.282793
$362$	−5.61583e6	−2.25239
$363$	−5.23209e6	−2.08405
$364$	8.32900e6	3.29488
$365$	−1.36293e6	−0.535478
$366$	8.94083e6	3.48879
$367$	1.67673e6	0.649826	0.324913	−	0.945744i	$-0.394665\pi$
$367$	1.67673e6	0.649826	0.324913	+	0.945744i	$0.394665\pi$
$368$	104468.	0.0402127
$369$	−3.49065e6	−1.33457
$370$	2.56536e6	0.974192
$371$	4.70750e6	1.77564
$372$	−3.21651e6	−1.20511
$373$	3.12665e6	1.16361	0.581804	−	0.813329i	$-0.302347\pi$
$373$	3.12665e6	1.16361	0.581804	+	0.813329i	$0.302347\pi$
$374$	7.84032e6	2.89838
$375$	−3.83652e6	−1.40883
$376$	−1.25667e6	−0.458408
$377$	90716.4	0.0328725
$378$	5.38761e6	1.93940
$379$	−1.98431e6	−0.709596	−0.354798	−	0.934943i	$-0.615450\pi$
$379$	−1.98431e6	−0.709596	−0.354798	+	0.934943i	$0.615450\pi$
$380$	3.94234e6	1.40054
$381$	3.89702e6	1.37537
$382$	−977069.	−0.342584
$383$	321813.	0.112100	0.0560502	−	0.998428i	$-0.482149\pi$
$383$	321813.	0.112100	0.0560502	+	0.998428i	$0.482149\pi$
$384$	6.52477e6	2.25807
$385$	6.18444e6	2.12642
$386$	−282778.	−0.0966002
$387$	−716107.	−0.243053
$388$	1.57317e6	0.530512
$389$	−1.34564e6	−0.450874	−0.225437	−	0.974258i	$-0.572381\pi$
$389$	−1.34564e6	−0.450874	−0.225437	+	0.974258i	$0.572381\pi$
$390$	−1.43314e7	−4.77119
$391$	−629525.	−0.208243
$392$	1.56647e6	0.514880
$393$	−5.19057e6	−1.69525
$394$	−2.48026e6	−0.804929
$395$	−2.26484e6	−0.730372
$396$	1.13621e7	3.64101
$397$	1.40299e6	0.446763	0.223382	−	0.974731i	$-0.428290\pi$
$397$	1.40299e6	0.446763	0.223382	+	0.974731i	$0.428290\pi$
$398$	−6.96704e6	−2.20466
$399$	−5.55378e6	−1.74645
$400$	−150952.	−0.0471724
$401$	3.44258e6	1.06911	0.534556	−	0.845133i	$-0.320479\pi$
$401$	3.44258e6	1.06911	0.534556	+	0.845133i	$0.320479\pi$
$402$	−4.32617e6	−1.33518
$403$	−2.75942e6	−0.846359
$404$	−382980.	−0.116741
$405$	193977.	0.0587643
$406$	129783.	0.0390754
$407$	−2.83958e6	−0.849706
$408$	−5.26743e6	−1.56656
$409$	6.44614e6	1.90542	0.952711	−	0.303877i	$-0.0982811\pi$
$409$	6.44614e6	1.90542	0.952711	+	0.303877i	$0.0982811\pi$
$410$	4.94925e6	1.45405
$411$	5.50286e6	1.60688
$412$	6.39802e6	1.85696
$413$	−7.97902e6	−2.30184
$414$	−1.51716e6	−0.435040
$415$	7.23194e6	2.06127
$416$	7.07283e6	2.00383
$417$	−2.08496e6	−0.587161
$418$	−7.25690e6	−2.03147
$419$	−1.77865e6	−0.494943	−0.247472	−	0.968895i	$-0.579600\pi$
$419$	−1.77865e6	−0.494943	−0.247472	+	0.968895i	$0.579600\pi$
$420$	−1.23291e7	−3.41042
$421$	−3.37157e6	−0.927100	−0.463550	−	0.886071i	$-0.653425\pi$
$421$	−3.37157e6	−0.927100	−0.463550	+	0.886071i	$0.653425\pi$
$422$	−801721.	−0.219150
$423$	−3.33983e6	−0.907555
$424$	−4.13255e6	−1.11636
$425$	909638.	0.244285
$426$	−3.82562e6	−1.02136
$427$	6.59861e6	1.75139
$428$	5.82288e6	1.53649
$429$	1.58633e7	4.16151
$430$	1.01534e6	0.264813
$431$	1.38644e6	0.359508	0.179754	−	0.983712i	$-0.442470\pi$
$431$	1.38644e6	0.359508	0.179754	+	0.983712i	$0.442470\pi$
$432$	865522.	0.223136
$433$	5.74536e6	1.47264	0.736322	−	0.676632i	$-0.236561\pi$
$433$	5.74536e6	1.47264	0.736322	+	0.676632i	$0.236561\pi$
$434$	−3.94776e6	−1.00606
$435$	−134284.	−0.0340252
$436$	809375.	0.203908
$437$	582681.	0.145958
$438$	−5.00152e6	−1.24571
$439$	−5.14484e6	−1.27412	−0.637060	−	0.770814i	$-0.719850\pi$
$439$	−5.14484e6	−1.27412	−0.637060	+	0.770814i	$0.719850\pi$
$440$	−5.42910e6	−1.33689
$441$	4.16317e6	1.01936
$442$	−1.34090e7	−3.26468
$443$	−5.65845e6	−1.36990	−0.684949	−	0.728591i	$-0.740176\pi$
$443$	−5.65845e6	−1.36990	−0.684949	+	0.728591i	$0.740176\pi$
$444$	5.66090e6	1.36279
$445$	−2.54595e6	−0.609468
$446$	−8.53162e6	−2.03093
$447$	1.26892e7	3.00376
$448$	8.84958e6	2.08318
$449$	−956361.	−0.223875	−0.111938	−	0.993715i	$-0.535706\pi$
$449$	−956361.	−0.223875	−0.111938	+	0.993715i	$0.535706\pi$
$450$	2.19223e6	0.510334
$451$	−5.47829e6	−1.26825
$452$	−878515.	−0.202257
$453$	5.57878e6	1.27730
$454$	−4.17323e6	−0.950239
$455$	−1.05770e7	−2.39516
$456$	4.87547e6	1.09800
$457$	−5.50698e6	−1.23345	−0.616727	−	0.787177i	$-0.711542\pi$
$457$	−5.50698e6	−1.23345	−0.616727	+	0.787177i	$0.711542\pi$
$458$	4.27744e6	0.952840
$459$	−5.21565e6	−1.15552
$460$	1.29352e6	0.285021
$461$	2.10202e6	0.460665	0.230333	−	0.973112i	$-0.426019\pi$
$461$	2.10202e6	0.460665	0.230333	+	0.973112i	$0.426019\pi$
$462$	2.26949e7	4.94678
$463$	2.36108e6	0.511868	0.255934	−	0.966694i	$-0.417617\pi$
$463$	2.36108e6	0.511868	0.255934	+	0.966694i	$0.417617\pi$
$464$	20849.8	0.00449579
$465$	4.08465e6	0.876037
$466$	6.66190e6	1.42113
$467$	−8.56485e6	−1.81730	−0.908652	−	0.417554i	$-0.862887\pi$
$467$	−8.56485e6	−1.81730	−0.908652	+	0.417554i	$0.862887\pi$
$468$	−1.94322e7	−4.10117
$469$	−3.19285e6	−0.670264
$470$	4.73540e6	0.988809
$471$	−455368.	−0.0945823
$472$	7.00450e6	1.44718
$473$	−1.12387e6	−0.230975
$474$	−8.31121e6	−1.69910
$475$	−841950.	−0.171219
$476$	−1.15356e7	−2.33357
$477$	−1.09830e7	−2.21016
$478$	1.13884e7	2.27977
$479$	3.09570e6	0.616481	0.308241	−	0.951308i	$-0.400260\pi$
$479$	3.09570e6	0.616481	0.308241	+	0.951308i	$0.400260\pi$
$480$	−1.04696e7	−2.07409
$481$	4.85643e6	0.957095
$482$	−7.84782e6	−1.53862
$483$	−1.82225e6	−0.355418
$484$	1.00587e7	1.95177
$485$	−1.99777e6	−0.385648
$486$	8.59846e6	1.65132
$487$	−4.69724e6	−0.897471	−0.448736	−	0.893665i	$-0.648125\pi$
$487$	−4.69724e6	−0.897471	−0.448736	+	0.893665i	$0.648125\pi$
$488$	−5.79268e6	−1.10111
$489$	1.07546e7	2.03386
$490$	−5.90279e6	−1.11062
$491$	4.40851e6	0.825255	0.412628	−	0.910900i	$-0.364611\pi$
$491$	4.40851e6	0.825255	0.412628	+	0.910900i	$0.364611\pi$
$492$	1.09213e7	2.03406
$493$	−125641.	−0.0232817
$494$	1.24112e7	2.28821
$495$	−1.44288e7	−2.64677
$496$	−634209.	−0.115752
$497$	−2.82343e6	−0.512726
$498$	2.65389e7	4.79523
$499$	3.52789e6	0.634255	0.317127	−	0.948383i	$-0.397282\pi$
$499$	3.52789e6	0.634255	0.317127	+	0.948383i	$0.397282\pi$
$500$	7.37573e6	1.31941
$501$	−1.48151e7	−2.63700
$502$	−1.07217e7	−1.89891
$503$	79054.9	0.0139319	0.00696593	−	0.999976i	$-0.497783\pi$
$503$	79054.9	0.0139319	0.00696593	+	0.999976i	$0.497783\pi$
$504$	−9.36894e6	−1.64291
$505$	486347.	0.0848629
$506$	−2.38105e6	−0.413422
$507$	−1.78089e7	−3.07693
$508$	−7.49203e6	−1.28807
$509$	155909.	0.0266733	0.0133366	−	0.999911i	$-0.495755\pi$
$509$	155909.	0.0266733	0.0133366	+	0.999911i	$0.495755\pi$
$510$	1.98488e7	3.37916
$511$	−3.69127e6	−0.625351
$512$	2.73974e6	0.461885
$513$	4.82754e6	0.809903
$514$	−1.17929e7	−1.96885
$515$	−8.12486e6	−1.34989
$516$	2.24051e6	0.370444
$517$	−5.24159e6	−0.862456
$518$	6.94786e6	1.13770
$519$	4.87052e6	0.793702
$520$	9.28518e6	1.50585
$521$	−4.24897e6	−0.685787	−0.342893	−	0.939374i	$-0.611407\pi$
$521$	−4.24897e6	−0.685787	−0.342893	+	0.939374i	$0.611407\pi$
$522$	−302795.	−0.0486377
$523$	1.30285e6	0.208276	0.104138	−	0.994563i	$-0.466792\pi$
$523$	1.30285e6	0.208276	0.104138	+	0.994563i	$0.466792\pi$
$524$	9.97890e6	1.58765
$525$	2.63307e6	0.416931
$526$	1.56437e7	2.46533
$527$	3.82176e6	0.599427
$528$	3.64594e6	0.569148
$529$	−6.24516e6	−0.970296
$530$	1.55723e7	2.40804
$531$	1.86157e7	2.86512
$532$	1.06772e7	1.63560
$533$	9.36932e6	1.42853
$534$	−9.34283e6	−1.41783
$535$	−7.39449e6	−1.11692
$536$	2.80289e6	0.421399
$537$	1.08994e7	1.63104
$538$	−4.21735e6	−0.628179
$539$	6.53376e6	0.968704
$540$	1.07169e7	1.58155
$541$	1.58581e6	0.232948	0.116474	−	0.993194i	$-0.462841\pi$
$541$	1.58581e6	0.232948	0.116474	+	0.993194i	$0.462841\pi$
$542$	263931.	0.0385915
$543$	1.57370e7	2.29045
$544$	−9.79579e6	−1.41919
$545$	−1.02783e6	−0.148227
$546$	−3.88142e7	−5.57197
$547$	−5.60569e6	−0.801052	−0.400526	−	0.916285i	$-0.631173\pi$
$547$	−5.60569e6	−0.801052	−0.400526	+	0.916285i	$0.631173\pi$
$548$	−1.05793e7	−1.50489
$549$	−1.53951e7	−2.17997
$550$	3.44053e6	0.484974
$551$	116292.	0.0163181
$552$	1.59968e6	0.223453
$553$	−6.13393e6	−0.852954
$554$	9.06408e6	1.25473
$555$	−7.18878e6	−0.990656
$556$	4.00834e6	0.549892
$557$	−44346.9	−0.00605655	−0.00302828	−	0.999995i	$-0.500964\pi$
$557$	−44346.9	−0.00605655	−0.00302828	+	0.999995i	$0.500964\pi$
$558$	9.21043e6	1.25226
$559$	1.92212e6	0.260166
$560$	−2.43096e6	−0.327573
$561$	−2.19705e7	−2.94736
$562$	−668286.	−0.0892528
$563$	4.30007e6	0.571748	0.285874	−	0.958267i	$-0.407716\pi$
$563$	4.30007e6	0.571748	0.285874	+	0.958267i	$0.407716\pi$
$564$	1.04495e7	1.38323
$565$	1.11563e6	0.147027
$566$	−1.02841e7	−1.34935
$567$	525355.	0.0686271
$568$	2.47859e6	0.322354
$569$	6.54339e6	0.847271	0.423635	−	0.905833i	$-0.360754\pi$
$569$	6.54339e6	0.847271	0.423635	+	0.905833i	$0.360754\pi$
$570$	−1.83718e7	−2.36845
$571$	8.65292e6	1.11064	0.555319	−	0.831637i	$-0.312596\pi$
$571$	8.65292e6	1.11064	0.555319	+	0.831637i	$0.312596\pi$
$572$	−3.04973e7	−3.89737
$573$	2.73799e6	0.348374
$574$	1.34042e7	1.69809
$575$	−276251.	−0.0348445
$576$	−2.06468e7	−2.59296
$577$	3.48407e6	0.435660	0.217830	−	0.975987i	$-0.430102\pi$
$577$	3.48407e6	0.435660	0.217830	+	0.975987i	$0.430102\pi$
$578$	5.85064e6	0.728423
$579$	792415.	0.0982328
$580$	258161.	0.0318655
$581$	1.95865e7	2.40723
$582$	−7.33116e6	−0.897150
$583$	−1.72369e7	−2.10033
$584$	3.24044e6	0.393162
$585$	2.46770e7	2.98128
$586$	−1.17090e7	−1.40856
$587$	6.56208e6	0.786043	0.393021	−	0.919529i	$-0.371430\pi$
$587$	6.56208e6	0.786043	0.393021	+	0.919529i	$0.371430\pi$
$588$	−1.30255e7	−1.55364
$589$	−3.53737e6	−0.420138
$590$	−2.63944e7	−3.12164
$591$	6.95032e6	0.818532
$592$	1.11618e6	0.130897
$593$	8.08785e6	0.944488	0.472244	−	0.881468i	$-0.343444\pi$
$593$	8.08785e6	0.944488	0.472244	+	0.881468i	$0.343444\pi$
$594$	−1.97272e7	−2.29403
$595$	1.46490e7	1.69635
$596$	−2.43950e7	−2.81310
$597$	1.95234e7	2.24192
$598$	4.07223e6	0.465671
$599$	−9.16029e6	−1.04314	−0.521569	−	0.853209i	$-0.674653\pi$
$599$	−9.16029e6	−1.04314	−0.521569	+	0.853209i	$0.674653\pi$
$600$	−2.31148e6	−0.262127
$601$	−1.25891e7	−1.42171	−0.710853	−	0.703341i	$-0.751691\pi$
$601$	−1.25891e7	−1.42171	−0.710853	+	0.703341i	$0.751691\pi$
$602$	2.74988e6	0.309259
$603$	7.44917e6	0.834285
$604$	−1.07252e7	−1.19623
$605$	−1.27736e7	−1.41881
$606$	1.78474e6	0.197421
$607$	−1.05315e7	−1.16017	−0.580083	−	0.814558i	$-0.696980\pi$
$607$	−1.05315e7	−1.16017	−0.580083	+	0.814558i	$0.696980\pi$
$608$	9.06686e6	0.994713
$609$	−363685.	−0.0397358
$610$	2.18281e7	2.37515
$611$	8.96450e6	0.971455
$612$	2.69134e7	2.90462
$613$	2.50815e6	0.269589	0.134794	−	0.990874i	$-0.456963\pi$
$613$	2.50815e6	0.269589	0.134794	+	0.990874i	$0.456963\pi$
$614$	−1.35281e7	−1.44816
$615$	−1.38690e7	−1.47863
$616$	−1.47038e7	−1.56127
$617$	1.04940e7	1.10975	0.554876	−	0.831933i	$-0.312766\pi$
$617$	1.04940e7	1.10975	0.554876	+	0.831933i	$0.312766\pi$
$618$	−2.98156e7	−3.14031
$619$	−7.31973e6	−0.767836	−0.383918	−	0.923367i	$-0.625426\pi$
$619$	−7.31973e6	−0.767836	−0.383918	+	0.923367i	$0.625426\pi$
$620$	−7.85276e6	−0.820433
$621$	1.58396e6	0.164822
$622$	1.52286e7	1.57828
$623$	−6.89529e6	−0.711758
$624$	−6.23552e6	−0.641078
$625$	−1.13408e7	−1.16130
$626$	2.55505e7	2.60593
$627$	2.03356e7	2.06580
$628$	875446.	0.0885789
$629$	−6.72610e6	−0.677855
$630$	3.53042e7	3.54384
$631$	−6.30990e6	−0.630883	−0.315442	−	0.948945i	$-0.602153\pi$
$631$	−6.30990e6	−0.630883	−0.315442	+	0.948945i	$0.602153\pi$
$632$	5.38476e6	0.536258
$633$	2.24662e6	0.222854
$634$	−1.41915e6	−0.140219
$635$	9.51414e6	0.936344
$636$	3.43629e7	3.36858
$637$	−1.11744e7	−1.09113
$638$	−475213.	−0.0462207
$639$	6.58729e6	0.638196
$640$	1.59295e7	1.53728
$641$	176304.	0.0169479	0.00847395	−	0.999964i	$-0.497303\pi$
$641$	176304.	0.0169479	0.00847395	+	0.999964i	$0.497303\pi$
$642$	−2.71354e7	−2.59835
$643$	−4.73829e6	−0.451954	−0.225977	−	0.974133i	$-0.572557\pi$
$643$	−4.73829e6	−0.451954	−0.225977	+	0.974133i	$0.572557\pi$
$644$	3.50327e6	0.332858
$645$	−2.84523e6	−0.269289
$646$	−1.71894e7	−1.62061
$647$	−1.27894e7	−1.20113	−0.600563	−	0.799577i	$-0.705057\pi$
$647$	−1.27894e7	−1.20113	−0.600563	+	0.799577i	$0.705057\pi$
$648$	−461191.	−0.0431463
$649$	2.92158e7	2.72274
$650$	−5.88420e6	−0.546266
$651$	1.10626e7	1.02307
$652$	−2.06757e7	−1.90476
$653$	2.31597e6	0.212545	0.106272	−	0.994337i	$-0.466108\pi$
$653$	2.31597e6	0.212545	0.106272	+	0.994337i	$0.466108\pi$
$654$	−3.77179e6	−0.344828
$655$	−1.26722e7	−1.15412
$656$	2.15339e6	0.195373
$657$	8.61203e6	0.778381
$658$	1.28250e7	1.15477
$659$	−4.32880e6	−0.388288	−0.194144	−	0.980973i	$-0.562193\pi$
$659$	−4.32880e6	−0.388288	−0.194144	+	0.980973i	$0.562193\pi$
$660$	4.51440e7	4.03404
$661$	1.34429e7	1.19671	0.598355	−	0.801231i	$-0.295821\pi$
$661$	1.34429e7	1.19671	0.598355	+	0.801231i	$0.295821\pi$
$662$	1.24287e7	1.10225
$663$	3.75754e7	3.31986
$664$	−1.71943e7	−1.51344
$665$	−1.35590e7	−1.18897
$666$	−1.62099e7	−1.41610
$667$	38156.4	0.00332088
$668$	2.84821e7	2.46962
$669$	2.39077e7	2.06525
$670$	−1.05619e7	−0.908980
$671$	−2.41614e7	−2.07164
$672$	−2.83552e7	−2.42220
$673$	−1.30893e7	−1.11398	−0.556991	−	0.830518i	$-0.688044\pi$
$673$	−1.30893e7	−1.11398	−0.556991	+	0.830518i	$0.688044\pi$
$674$	5.40225e6	0.458063
$675$	−2.28876e6	−0.193348
$676$	3.42377e7	2.88163
$677$	1.26746e6	0.106283	0.0531415	−	0.998587i	$-0.483077\pi$
$677$	1.26746e6	0.106283	0.0531415	+	0.998587i	$0.483077\pi$
$678$	4.09399e6	0.342037
$679$	−5.41062e6	−0.450373
$680$	−1.28599e7	−1.06651
$681$	1.16944e7	0.966298
$682$	1.44550e7	1.19003
$683$	−1.77529e7	−1.45619	−0.728093	−	0.685479i	$-0.759593\pi$
$683$	−1.77529e7	−1.45619	−0.728093	+	0.685479i	$0.759593\pi$
$684$	−2.49107e7	−2.03585
$685$	1.34346e7	1.09396
$686$	9.00905e6	0.730918
$687$	−1.19864e7	−0.968944
$688$	441769.	0.0355815
$689$	2.94797e7	2.36578
$690$	−6.02796e6	−0.482000
$691$	5.47133e6	0.435911	0.217955	−	0.975959i	$-0.430061\pi$
$691$	5.47133e6	0.435911	0.217955	+	0.975959i	$0.430061\pi$
$692$	−9.36360e6	−0.743323
$693$	−3.90780e7	−3.09100
$694$	1.24394e7	0.980396
$695$	−5.09020e6	−0.399736
$696$	319266.	0.0249822
$697$	−1.29764e7	−1.01175
$698$	1.70767e7	1.32668
$699$	−1.86683e7	−1.44515
$700$	−5.06209e6	−0.390467
$701$	−1.16242e7	−0.893443	−0.446721	−	0.894673i	$-0.647408\pi$
$701$	−1.16242e7	−0.893443	−0.446721	+	0.894673i	$0.647408\pi$
$702$	3.37387e7	2.58396
$703$	6.22560e6	0.475108
$704$	−3.24035e7	−2.46411
$705$	−1.32698e7	−1.00552
$706$	−2.27366e7	−1.71678
$707$	1.31719e6	0.0991060
$708$	−5.82437e7	−4.36682
$709$	−4.30447e6	−0.321591	−0.160795	−	0.986988i	$-0.551406\pi$
$709$	−4.30447e6	−0.321591	−0.160795	+	0.986988i	$0.551406\pi$
$710$	−9.33985e6	−0.695335
$711$	1.43110e7	1.06168
$712$	6.05313e6	0.447487
$713$	−1.16064e6	−0.0855017
$714$	5.37572e7	3.94631
$715$	3.87286e7	2.83313
$716$	−2.09541e7	−1.52752
$717$	−3.19130e7	−2.31830
$718$	−5.27883e6	−0.382143
$719$	1.35107e7	0.974664	0.487332	−	0.873217i	$-0.337970\pi$
$719$	1.35107e7	0.974664	0.487332	+	0.873217i	$0.337970\pi$
$720$	5.67163e6	0.407734
$721$	−2.20048e7	−1.57645
$722$	−6.27338e6	−0.447877
$723$	2.19915e7	1.56462
$724$	−3.02544e7	−2.14507
$725$	−55134.4	−0.00389563
$726$	−4.68749e7	−3.30064
$727$	2.62118e7	1.83934	0.919668	−	0.392697i	$-0.128458\pi$
$727$	2.62118e7	1.83934	0.919668	+	0.392697i	$0.128458\pi$
$728$	2.51474e7	1.75859
$729$	−2.33260e7	−1.62563
$730$	−1.22107e7	−0.848070
$731$	−2.66211e6	−0.184260
$732$	4.81672e7	3.32257
$733$	−2.19136e7	−1.50644	−0.753222	−	0.657766i	$-0.771501\pi$
$733$	−2.19136e7	−1.50644	−0.753222	+	0.657766i	$0.771501\pi$
$734$	1.50220e7	1.02917
$735$	1.65411e7	1.12939
$736$	2.97492e6	0.202433
$737$	1.16909e7	0.792827
$738$	−3.12731e7	−2.11364
$739$	−2.72080e6	−0.183267	−0.0916337	−	0.995793i	$-0.529209\pi$
$739$	−2.72080e6	−0.183267	−0.0916337	+	0.995793i	$0.529209\pi$
$740$	1.38205e7	0.927776
$741$	−3.47793e7	−2.32689
$742$	4.21751e7	2.81220
$743$	−5.79913e6	−0.385381	−0.192691	−	0.981260i	$-0.561721\pi$
$743$	−5.79913e6	−0.385381	−0.192691	+	0.981260i	$0.561721\pi$
$744$	−9.71146e6	−0.643209
$745$	3.09792e7	2.04494
$746$	2.80120e7	1.84288
$747$	−4.56969e7	−2.99630
$748$	4.22384e7	2.76028
$749$	−2.00267e7	−1.30438
$750$	−3.43718e7	−2.23126
$751$	−2.22708e6	−0.144091	−0.0720455	−	0.997401i	$-0.522953\pi$
$751$	−2.22708e6	−0.144091	−0.0720455	+	0.997401i	$0.522953\pi$
$752$	2.06035e6	0.132861
$753$	3.00449e7	1.93100
$754$	812739.	0.0520622
$755$	1.36200e7	0.869579
$756$	2.90248e7	1.84699
$757$	1.68706e7	1.07002	0.535009	−	0.844846i	$-0.320308\pi$
$757$	1.68706e7	1.07002	0.535009	+	0.844846i	$0.320308\pi$
$758$	−1.77776e7	−1.12383
$759$	6.67231e6	0.420409
$760$	1.19029e7	0.747515
$761$	−6.73683e6	−0.421691	−0.210845	−	0.977519i	$-0.567622\pi$
$761$	−6.73683e6	−0.421691	−0.210845	+	0.977519i	$0.567622\pi$
$762$	3.49138e7	2.17826
$763$	−2.78370e6	−0.173105
$764$	−5.26380e6	−0.326261
$765$	−3.41774e7	−2.11147
$766$	2.88316e6	0.177540
$767$	−4.99668e7	−3.06685
$768$	1.56276e7	0.956071
$769$	2.99965e7	1.82917	0.914587	−	0.404390i	$-0.132516\pi$
$769$	2.99965e7	1.82917	0.914587	+	0.404390i	$0.132516\pi$
$770$	5.54071e7	3.36774
$771$	3.30466e7	2.00212
$772$	−1.52342e6	−0.0919977
$773$	2.03964e7	1.22773	0.613867	−	0.789409i	$-0.289613\pi$
$773$	2.03964e7	1.22773	0.613867	+	0.789409i	$0.289613\pi$
$774$	−6.41568e6	−0.384938
$775$	1.67708e6	0.100300
$776$	4.74979e6	0.283152
$777$	−1.94696e7	−1.15692
$778$	−1.20558e7	−0.714078
$779$	1.20108e7	0.709133
$780$	−7.72080e7	−4.54387
$781$	1.03382e7	0.606482
$782$	−5.63999e6	−0.329808
$783$	316128.	0.0184272
$784$	−2.56827e6	−0.149228
$785$	−1.11173e6	−0.0643911
$786$	−4.65029e7	−2.68488
$787$	−9.96285e6	−0.573386	−0.286693	−	0.958023i	$-0.592556\pi$
$787$	−9.96285e6	−0.573386	−0.286693	+	0.958023i	$0.592556\pi$
$788$	−1.33620e7	−0.766578
$789$	−4.38375e7	−2.50699
$790$	−2.02909e7	−1.15674
$791$	3.02149e6	0.171704
$792$	3.43052e7	1.94333
$793$	4.13223e7	2.33346
$794$	1.25695e7	0.707567
$795$	−4.36375e7	−2.44874
$796$	−3.75338e7	−2.09961
$797$	−4.33007e6	−0.241462	−0.120731	−	0.992685i	$-0.538524\pi$
$797$	−4.33007e6	−0.241462	−0.120731	+	0.992685i	$0.538524\pi$
$798$	−4.97570e7	−2.76597
$799$	−1.24157e7	−0.688026
$800$	−4.29863e6	−0.237468
$801$	1.60873e7	0.885933
$802$	3.08425e7	1.69322
$803$	1.35159e7	0.739701
$804$	−2.33065e7	−1.27156
$805$	−4.44881e6	−0.241966
$806$	−2.47219e7	−1.34043
$807$	1.18181e7	0.638796
$808$	−1.15631e6	−0.0623085
$809$	4.65045e6	0.249818	0.124909	−	0.992168i	$-0.460136\pi$
$809$	4.65045e6	0.249818	0.124909	+	0.992168i	$0.460136\pi$
$810$	1.73787e6	0.0930687
$811$	−2.23521e7	−1.19334	−0.596672	−	0.802486i	$-0.703510\pi$
$811$	−2.23521e7	−1.19334	−0.596672	+	0.802486i	$0.703510\pi$
$812$	699186.	0.0372137
$813$	−739599.	−0.0392437
$814$	−2.54402e7	−1.34573
$815$	2.62561e7	1.38464
$816$	8.63611e6	0.454039
$817$	2.46401e6	0.129148
$818$	5.77517e7	3.01774
$819$	6.68336e7	3.48165
$820$	2.66633e7	1.38477
$821$	6.86450e6	0.355427	0.177714	−	0.984082i	$-0.443130\pi$
$821$	6.86450e6	0.355427	0.177714	+	0.984082i	$0.443130\pi$
$822$	4.93008e7	2.54492
$823$	−3.78159e7	−1.94614	−0.973072	−	0.230502i	$-0.925963\pi$
$823$	−3.78159e7	−1.94614	−0.973072	+	0.230502i	$0.925963\pi$
$824$	1.93173e7	0.991123
$825$	−9.64121e6	−0.493170
$826$	−7.14849e7	−3.64556
$827$	3.11812e7	1.58536	0.792681	−	0.609637i	$-0.208685\pi$
$827$	3.11812e7	1.58536	0.792681	+	0.609637i	$0.208685\pi$
$828$	−8.17342e6	−0.414313
$829$	−7.56161e6	−0.382145	−0.191072	−	0.981576i	$-0.561196\pi$
$829$	−7.56161e6	−0.382145	−0.191072	+	0.981576i	$0.561196\pi$
$830$	6.47918e7	3.26456
$831$	−2.53998e7	−1.27593
$832$	5.54185e7	2.77553
$833$	1.54765e7	0.772786
$834$	−1.86794e7	−0.929924
$835$	−3.61695e7	−1.79525
$836$	−3.90954e7	−1.93468
$837$	−9.61600e6	−0.474439
$838$	−1.59351e7	−0.783873
$839$	1.20663e7	0.591792	0.295896	−	0.955220i	$-0.404382\pi$
$839$	1.20663e7	0.591792	0.295896	+	0.955220i	$0.404382\pi$
$840$	−3.72246e7	−1.82025
$841$	−2.05035e7	−0.999629
$842$	−3.02062e7	−1.46831
$843$	1.87270e6	0.0907612
$844$	−4.31914e6	−0.208709
$845$	−4.34785e7	−2.09475
$846$	−2.99219e7	−1.43735
$847$	−3.45951e7	−1.65694
$848$	6.77544e6	0.323555
$849$	2.88186e7	1.37216
$850$	8.14955e6	0.386889
$851$	2.04267e6	0.0966886
$852$	−2.06099e7	−0.972696
$853$	2.73307e7	1.28611	0.643055	−	0.765820i	$-0.277667\pi$
$853$	2.73307e7	1.28611	0.643055	+	0.765820i	$0.277667\pi$
$854$	5.91177e7	2.77378
$855$	3.16341e7	1.47993
$856$	1.75808e7	0.820074
$857$	−3.10213e7	−1.44280	−0.721402	−	0.692516i	$-0.756502\pi$
$857$	−3.10213e7	−1.44280	−0.721402	+	0.692516i	$0.756502\pi$
$858$	1.42121e8	6.59085
$859$	2.98878e7	1.38201	0.691006	−	0.722849i	$-0.257168\pi$
$859$	2.98878e7	1.38201	0.691006	+	0.722849i	$0.257168\pi$
$860$	5.46997e6	0.252196
$861$	−3.75619e7	−1.72679
$862$	1.24213e7	0.569375
$863$	−2.37019e7	−1.08332	−0.541660	−	0.840598i	$-0.682204\pi$
$863$	−2.37019e7	−1.08332	−0.541660	+	0.840598i	$0.682204\pi$
$864$	2.46474e7	1.12328
$865$	1.18909e7	0.540347
$866$	5.14733e7	2.33232
$867$	−1.63949e7	−0.740734
$868$	−2.12679e7	−0.958131
$869$	2.24599e7	1.00892
$870$	−1.20306e6	−0.0538878
$871$	−1.99945e7	−0.893027
$872$	2.44371e6	0.108832
$873$	1.26234e7	0.560584
$874$	5.22030e6	0.231162
$875$	−2.53675e7	−1.12010
$876$	−2.69448e7	−1.18636
$877$	3.85645e7	1.69312	0.846562	−	0.532290i	$-0.178668\pi$
$877$	3.85645e7	1.69312	0.846562	+	0.532290i	$0.178668\pi$
$878$	−4.60932e7	−2.01790
$879$	3.28115e7	1.43237
$880$	8.90117e6	0.387472
$881$	−3.45951e7	−1.50167	−0.750836	−	0.660489i	$-0.770349\pi$
$881$	−3.45951e7	−1.50167	−0.750836	+	0.660489i	$0.770349\pi$
$882$	3.72983e7	1.61442
$883$	−8.71550e6	−0.376175	−0.188088	−	0.982152i	$-0.560229\pi$
$883$	−8.71550e6	−0.376175	−0.188088	+	0.982152i	$0.560229\pi$
$884$	−7.22388e7	−3.10914
$885$	7.39638e7	3.17439
$886$	−5.06947e7	−2.16959
$887$	4.58909e6	0.195847	0.0979237	−	0.995194i	$-0.468780\pi$
$887$	4.58909e6	0.195847	0.0979237	+	0.995194i	$0.468780\pi$
$888$	1.70917e7	0.727365
$889$	2.57675e7	1.09350
$890$	−2.28095e7	−0.965252
$891$	−1.92363e6	−0.0811761
$892$	−4.59627e7	−1.93416
$893$	1.14918e7	0.482237
$894$	1.13684e8	4.75723
$895$	2.66096e7	1.11040
$896$	4.31424e7	1.79529
$897$	−1.14114e7	−0.473541
$898$	−8.56815e6	−0.354565
$899$	−231642.	−0.00955912
$900$	1.18103e7	0.486019
$901$	−4.08290e7	−1.67555
$902$	−4.90807e7	−2.00860
$903$	−7.70583e6	−0.314485
$904$	−2.65246e6	−0.107951
$905$	3.84201e7	1.55933
$906$	4.99809e7	2.02294
$907$	1.10880e7	0.447544	0.223772	−	0.974642i	$-0.428163\pi$
$907$	1.10880e7	0.447544	0.223772	+	0.974642i	$0.428163\pi$
$908$	−2.24826e7	−0.904965
$909$	−3.07311e6	−0.123358
$910$	−9.47606e7	−3.79336
$911$	3.54367e7	1.41468	0.707339	−	0.706875i	$-0.249896\pi$
$911$	3.54367e7	1.41468	0.707339	+	0.706875i	$0.249896\pi$
$912$	−7.99348e6	−0.318236
$913$	−7.17177e7	−2.84740
$914$	−4.93377e7	−1.95350
$915$	−6.11677e7	−2.41529
$916$	2.30440e7	0.907442
$917$	−3.43206e7	−1.34782
$918$	−4.67276e7	−1.83007
$919$	3.32990e7	1.30060	0.650298	−	0.759679i	$-0.274644\pi$
$919$	3.32990e7	1.30060	0.650298	+	0.759679i	$0.274644\pi$
$920$	3.90546e6	0.152126
$921$	3.79092e7	1.47264
$922$	1.88323e7	0.729584
$923$	−1.76811e7	−0.683131
$924$	1.22265e8	4.71109
$925$	−2.95158e6	−0.113423
$926$	2.11532e7	0.810677
$927$	5.13390e7	1.96222
$928$	593736.	0.0226320
$929$	3.43816e7	1.30703	0.653517	−	0.756912i	$-0.273293\pi$
$929$	3.43816e7	1.30703	0.653517	+	0.756912i	$0.273293\pi$
$930$	3.65949e7	1.38743
$931$	−1.43248e7	−0.541645
$932$	3.58899e7	1.35342
$933$	−4.26743e7	−1.60495
$934$	−7.67335e7	−2.87818
$935$	−5.36386e7	−2.00654
$936$	−5.86708e7	−2.18893
$937$	3.81989e7	1.42135	0.710677	−	0.703518i	$-0.248389\pi$
$937$	3.81989e7	1.42135	0.710677	+	0.703518i	$0.248389\pi$
$938$	−2.86051e7	−1.06154
$939$	−7.15988e7	−2.64997
$940$	2.55112e7	0.941697
$941$	2.92661e7	1.07744	0.538718	−	0.842486i	$-0.318909\pi$
$941$	2.92661e7	1.07744	0.538718	+	0.842486i	$0.318909\pi$
$942$	−4.07969e6	−0.149796
$943$	3.94085e6	0.144315
$944$	−1.14841e7	−0.419437
$945$	−3.68587e7	−1.34264
$946$	−1.00689e7	−0.365809
$947$	4.49447e6	0.162856	0.0814280	−	0.996679i	$-0.474052\pi$
$947$	4.49447e6	0.162856	0.0814280	+	0.996679i	$0.474052\pi$
$948$	−4.47753e7	−1.61814
$949$	−2.31157e7	−0.833186
$950$	−7.54312e6	−0.271170
$951$	3.97681e6	0.142588
$952$	−3.48288e7	−1.24551
$953$	−7.96508e6	−0.284091	−0.142046	−	0.989860i	$-0.545368\pi$
$953$	−7.96508e6	−0.284091	−0.142046	+	0.989860i	$0.545368\pi$
$954$	−9.83978e7	−3.50038
$955$	6.68451e6	0.237171
$956$	6.13529e7	2.17115
$957$	1.33166e6	0.0470018
$958$	2.77347e7	0.976360
$959$	3.63855e7	1.27756
$960$	−8.20337e7	−2.87286
$961$	−2.15831e7	−0.753884
$962$	4.35094e7	1.51581
$963$	4.67240e7	1.62358
$964$	−4.22788e7	−1.46531
$965$	1.93460e6	0.0668763
$966$	−1.63257e7	−0.562897
$967$	3.91114e6	0.134505	0.0672523	−	0.997736i	$-0.478577\pi$
$967$	3.91114e6	0.134505	0.0672523	+	0.997736i	$0.478577\pi$
$968$	3.03698e7	1.04173
$969$	4.81689e7	1.64800
$970$	−1.78982e7	−0.610774
$971$	−2.98964e7	−1.01759	−0.508793	−	0.860889i	$-0.669908\pi$
$971$	−2.98964e7	−1.01759	−0.508793	+	0.860889i	$0.669908\pi$
$972$	4.63228e7	1.57264
$973$	−1.37860e7	−0.466826
$974$	−4.20831e7	−1.42138
$975$	1.64890e7	0.555498
$976$	9.49728e6	0.319135
$977$	−4.36187e7	−1.46196	−0.730981	−	0.682398i	$-0.760937\pi$
$977$	−4.36187e7	−1.46196	−0.730981	+	0.682398i	$0.760937\pi$
$978$	9.63513e7	3.22115
$979$	2.52477e7	0.841909
$980$	−3.18003e7	−1.05771
$981$	6.49459e6	0.215466
$982$	3.94964e7	1.30701
$983$	−1.95377e7	−0.644895	−0.322448	−	0.946587i	$-0.604506\pi$
$983$	−1.95377e7	−0.644895	−0.322448	+	0.946587i	$0.604506\pi$
$984$	3.29743e7	1.08564
$985$	1.69684e7	0.557252
$986$	−1.12563e6	−0.0368727
$987$	−3.59390e7	−1.17428
$988$	6.68633e7	2.17919
$989$	808465.	0.0262827
$990$	−1.29269e8	−4.19186
$991$	3.45462e7	1.11742	0.558709	−	0.829364i	$-0.311297\pi$
$991$	3.45462e7	1.11742	0.558709	+	0.829364i	$0.311297\pi$
$992$	−1.80603e7	−0.582701
$993$	−3.48282e7	−1.12088
$994$	−2.52954e7	−0.812037
$995$	4.76642e7	1.52628
$996$	1.42974e8	4.56676
$997$	3.34457e7	1.06562	0.532810	−	0.846235i	$-0.321136\pi$
$997$	3.34457e7	1.06562	0.532810	+	0.846235i	$0.321136\pi$
$998$	3.16068e7	1.00451
$999$	1.69237e7	0.536514

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.a.a.1.8	✓	8
3.2	odd	2		387.6.a.c.1.1		8
4.3	odd	2		688.6.a.e.1.7		8
5.4	even	2		1075.6.a.a.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.a.1.8	✓	8	1.1	even	1	trivial
387.6.a.c.1.1		8	3.2	odd	2
688.6.a.e.1.7		8	4.3	odd	2
1075.6.a.a.1.1		8	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	43.a (trivial)

\(f(q)\)	\(=\)	\(q+8.95911 q^{2} -25.1057 q^{3} +48.2657 q^{4} -61.2927 q^{5} -224.924 q^{6} -166.001 q^{7} +145.726 q^{8} +387.294 q^{9} +O(q^{10})\)	\(q+8.95911 q^{2} -25.1057 q^{3} +48.2657 q^{4} -61.2927 q^{5} -224.924 q^{6} -166.001 q^{7} +145.726 q^{8} +387.294 q^{9} -549.129 q^{10} +607.827 q^{11} -1211.74 q^{12} -1039.54 q^{13} -1487.22 q^{14} +1538.79 q^{15} -238.923 q^{16} +1439.76 q^{17} +3469.81 q^{18} -1332.62 q^{19} -2958.34 q^{20} +4167.57 q^{21} +5445.59 q^{22} -437.244 q^{23} -3658.56 q^{24} +631.800 q^{25} -9313.39 q^{26} -3622.60 q^{27} -8012.16 q^{28} -87.2656 q^{29} +13786.2 q^{30} +2654.45 q^{31} -6803.79 q^{32} -15259.9 q^{33} +12898.9 q^{34} +10174.7 q^{35} +18693.0 q^{36} -4671.70 q^{37} -11939.1 q^{38} +26098.4 q^{39} -8931.98 q^{40} -9012.91 q^{41} +37337.7 q^{42} -1849.00 q^{43} +29337.2 q^{44} -23738.3 q^{45} -3917.32 q^{46} -8623.49 q^{47} +5998.32 q^{48} +10749.4 q^{49} +5660.37 q^{50} -36146.0 q^{51} -50174.3 q^{52} -28358.3 q^{53} -32455.3 q^{54} -37255.4 q^{55} -24190.8 q^{56} +33456.3 q^{57} -781.822 q^{58} +48066.0 q^{59} +74271.0 q^{60} -39750.4 q^{61} +23781.5 q^{62} -64291.2 q^{63} -53310.4 q^{64} +63716.5 q^{65} -136715. q^{66} +19233.9 q^{67} +69490.8 q^{68} +10977.3 q^{69} +91156.0 q^{70} +17008.5 q^{71} +56439.0 q^{72} +22236.4 q^{73} -41854.3 q^{74} -15861.8 q^{75} -64319.9 q^{76} -100900. q^{77} +233819. q^{78} +36951.1 q^{79} +14644.3 q^{80} -3164.77 q^{81} -80747.7 q^{82} -117990. q^{83} +201151. q^{84} -88246.5 q^{85} -16565.4 q^{86} +2190.86 q^{87} +88576.5 q^{88} +41537.6 q^{89} -212674. q^{90} +172565. q^{91} -21103.9 q^{92} -66641.7 q^{93} -77258.8 q^{94} +81679.9 q^{95} +170814. q^{96} +32593.9 q^{97} +96304.8 q^{98} +235408. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{2} - 26 q^{3} + 122 q^{4} - 212 q^{5} - 69 q^{6} - 136 q^{7} - 666 q^{8} + 546 q^{9}+O(q^{10}) \) 8 * q - 12 * q^2 - 26 * q^3 + 122 * q^4 - 212 * q^5 - 69 * q^6 - 136 * q^7 - 666 * q^8 + 546 * q^9	\( 8 q - 12 q^{2} - 26 q^{3} + 122 q^{4} - 212 q^{5} - 69 q^{6} - 136 q^{7} - 666 q^{8} + 546 q^{9} - 617 q^{10} - 532 q^{11} - 4195 q^{12} - 2492 q^{13} - 4240 q^{14} - 1780 q^{15} + 1882 q^{16} - 2534 q^{17} - 3711 q^{18} - 1678 q^{19} - 2607 q^{20} - 2256 q^{21} + 11502 q^{22} - 2488 q^{23} + 19953 q^{24} + 4378 q^{25} + 4586 q^{26} - 8960 q^{27} + 18640 q^{28} - 4360 q^{29} + 25092 q^{30} + 5704 q^{31} - 18294 q^{32} - 12852 q^{33} + 30007 q^{34} + 5640 q^{35} + 67969 q^{36} - 3772 q^{37} - 6559 q^{38} + 11120 q^{39} + 14869 q^{40} - 10698 q^{41} + 78698 q^{42} - 14792 q^{43} - 356 q^{44} - 44912 q^{45} - 19389 q^{46} - 77864 q^{47} - 118727 q^{48} + 7188 q^{49} + 26877 q^{50} - 80246 q^{51} - 60736 q^{52} - 62352 q^{53} + 61026 q^{54} - 49552 q^{55} - 144528 q^{56} - 808 q^{57} + 52951 q^{58} - 26224 q^{59} + 101500 q^{60} - 82540 q^{61} - 9023 q^{62} - 61768 q^{63} + 153858 q^{64} - 5000 q^{65} - 48516 q^{66} + 27784 q^{67} + 40507 q^{68} - 93776 q^{69} + 185910 q^{70} - 9504 q^{71} - 186687 q^{72} + 14260 q^{73} - 15239 q^{74} + 167420 q^{75} + 1279 q^{76} - 218140 q^{77} + 264170 q^{78} + 160248 q^{79} - 1291 q^{80} + 161076 q^{81} - 47781 q^{82} - 77176 q^{83} + 16382 q^{84} + 141096 q^{85} + 22188 q^{86} + 268136 q^{87} + 129544 q^{88} - 265692 q^{89} + 48990 q^{90} + 401148 q^{91} + 190391 q^{92} - 123860 q^{93} + 248737 q^{94} + 135884 q^{95} + 950817 q^{96} + 144742 q^{97} - 292244 q^{98} + 239516 q^{99}+O(q^{100}) \) 8 * q - 12 * q^2 - 26 * q^3 + 122 * q^4 - 212 * q^5 - 69 * q^6 - 136 * q^7 - 666 * q^8 + 546 * q^9 - 617 * q^10 - 532 * q^11 - 4195 * q^12 - 2492 * q^13 - 4240 * q^14 - 1780 * q^15 + 1882 * q^16 - 2534 * q^17 - 3711 * q^18 - 1678 * q^19 - 2607 * q^20 - 2256 * q^21 + 11502 * q^22 - 2488 * q^23 + 19953 * q^24 + 4378 * q^25 + 4586 * q^26 - 8960 * q^27 + 18640 * q^28 - 4360 * q^29 + 25092 * q^30 + 5704 * q^31 - 18294 * q^32 - 12852 * q^33 + 30007 * q^34 + 5640 * q^35 + 67969 * q^36 - 3772 * q^37 - 6559 * q^38 + 11120 * q^39 + 14869 * q^40 - 10698 * q^41 + 78698 * q^42 - 14792 * q^43 - 356 * q^44 - 44912 * q^45 - 19389 * q^46 - 77864 * q^47 - 118727 * q^48 + 7188 * q^49 + 26877 * q^50 - 80246 * q^51 - 60736 * q^52 - 62352 * q^53 + 61026 * q^54 - 49552 * q^55 - 144528 * q^56 - 808 * q^57 + 52951 * q^58 - 26224 * q^59 + 101500 * q^60 - 82540 * q^61 - 9023 * q^62 - 61768 * q^63 + 153858 * q^64 - 5000 * q^65 - 48516 * q^66 + 27784 * q^67 + 40507 * q^68 - 93776 * q^69 + 185910 * q^70 - 9504 * q^71 - 186687 * q^72 + 14260 * q^73 - 15239 * q^74 + 167420 * q^75 + 1279 * q^76 - 218140 * q^77 + 264170 * q^78 + 160248 * q^79 - 1291 * q^80 + 161076 * q^81 - 47781 * q^82 - 77176 * q^83 + 16382 * q^84 + 141096 * q^85 + 22188 * q^86 + 268136 * q^87 + 129544 * q^88 - 265692 * q^89 + 48990 * q^90 + 401148 * q^91 + 190391 * q^92 - 123860 * q^93 + 248737 * q^94 + 135884 * q^95 + 950817 * q^96 + 144742 * q^97 - 292244 * q^98 + 239516 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more