LMFDB - Embedded newform 165.6.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [165,6,Mod(1,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("165.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 165 = 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	165.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:	$26.4633302691$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 209x^{5} + 137x^{4} + 12724x^{3} - 1040x^{2} - 218208x - 8784 $ x^7 - x^6 - 209x^5 + 137x^4 + 12724x^3 - 1040x^2 - 218208*x - 8784
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.96291$ of defining polynomial
Character	$\chi$	$=$	165.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.96291 q^{2} +9.00000 q^{3} +16.4822 q^{4} +25.0000 q^{5} -62.6662 q^{6} +244.038 q^{7} +108.049 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-6.96291 q^{2} +9.00000 q^{3} +16.4822 q^{4} +25.0000 q^{5} -62.6662 q^{6} +244.038 q^{7} +108.049 q^{8} +81.0000 q^{9} -174.073 q^{10} +121.000 q^{11} +148.340 q^{12} +742.713 q^{13} -1699.22 q^{14} +225.000 q^{15} -1279.77 q^{16} +170.372 q^{17} -563.996 q^{18} +388.599 q^{19} +412.054 q^{20} +2196.34 q^{21} -842.513 q^{22} +224.659 q^{23} +972.444 q^{24} +625.000 q^{25} -5171.45 q^{26} +729.000 q^{27} +4022.28 q^{28} -5577.78 q^{29} -1566.66 q^{30} -6171.57 q^{31} +5453.33 q^{32} +1089.00 q^{33} -1186.28 q^{34} +6100.95 q^{35} +1335.06 q^{36} +4514.45 q^{37} -2705.78 q^{38} +6684.41 q^{39} +2701.23 q^{40} -8779.82 q^{41} -15292.9 q^{42} +8437.24 q^{43} +1994.34 q^{44} +2025.00 q^{45} -1564.28 q^{46} +7343.40 q^{47} -11517.9 q^{48} +42747.5 q^{49} -4351.82 q^{50} +1533.34 q^{51} +12241.5 q^{52} -32882.2 q^{53} -5075.96 q^{54} +3025.00 q^{55} +26368.1 q^{56} +3497.39 q^{57} +38837.6 q^{58} +34178.2 q^{59} +3708.49 q^{60} +44832.7 q^{61} +42972.1 q^{62} +19767.1 q^{63} +2981.46 q^{64} +18567.8 q^{65} -7582.61 q^{66} +14320.3 q^{67} +2808.10 q^{68} +2021.94 q^{69} -42480.4 q^{70} +31300.9 q^{71} +8751.99 q^{72} +28110.8 q^{73} -31433.7 q^{74} +5625.00 q^{75} +6404.96 q^{76} +29528.6 q^{77} -46543.0 q^{78} -71426.8 q^{79} -31994.2 q^{80} +6561.00 q^{81} +61133.1 q^{82} -116407. q^{83} +36200.5 q^{84} +4259.29 q^{85} -58747.8 q^{86} -50200.1 q^{87} +13074.0 q^{88} -100275. q^{89} -14099.9 q^{90} +181250. q^{91} +3702.88 q^{92} -55544.1 q^{93} -51131.5 q^{94} +9714.98 q^{95} +49080.0 q^{96} +137036. q^{97} -297647. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 63 q^{3} + 195 q^{4} + 175 q^{5} + 9 q^{6} + 153 q^{8} + 567 q^{9}+O(q^{10}) $ 7 * q + q^2 + 63 * q^3 + 195 * q^4 + 175 * q^5 + 9 * q^6 + 153 * q^8 + 567 * q^9	\( 7 q + q^{2} + 63 q^{3} + 195 q^{4} + 175 q^{5} + 9 q^{6} + 153 q^{8} + 567 q^{9} + 25 q^{10} + 847 q^{11} + 1755 q^{12} + 1418 q^{13} + 2548 q^{14} + 1575 q^{15} + 3699 q^{16} + 630 q^{17} + 81 q^{18} + 2572 q^{19} + 4875 q^{20} + 121 q^{22} + 536 q^{23} + 1377 q^{24} + 4375 q^{25} - 7626 q^{26} + 5103 q^{27} - 11368 q^{28} - 1038 q^{29} + 225 q^{30} + 1872 q^{31} - 7523 q^{32} + 7623 q^{33} + 20790 q^{34} + 15795 q^{36} + 24298 q^{37} - 18952 q^{38} + 12762 q^{39} + 3825 q^{40} - 17658 q^{41} + 22932 q^{42} + 7244 q^{43} + 23595 q^{44} + 14175 q^{45} + 31016 q^{46} + 34560 q^{47} + 33291 q^{48} + 78735 q^{49} + 625 q^{50} + 5670 q^{51} + 110222 q^{52} - 10214 q^{53} + 729 q^{54} + 21175 q^{55} + 81124 q^{56} + 23148 q^{57} - 5718 q^{58} + 94676 q^{59} + 43875 q^{60} + 69538 q^{61} - 4208 q^{62} + 112339 q^{64} + 35450 q^{65} + 1089 q^{66} + 64908 q^{67} - 136010 q^{68} + 4824 q^{69} + 63700 q^{70} + 61816 q^{71} + 12393 q^{72} - 11890 q^{73} - 124050 q^{74} + 39375 q^{75} - 47216 q^{76} - 68634 q^{78} + 18928 q^{79} + 92475 q^{80} + 45927 q^{81} + 36398 q^{82} + 17492 q^{83} - 102312 q^{84} + 15750 q^{85} - 216688 q^{86} - 9342 q^{87} + 18513 q^{88} + 25302 q^{89} + 2025 q^{90} + 3392 q^{91} - 27408 q^{92} + 16848 q^{93} - 30800 q^{94} + 64300 q^{95} - 67707 q^{96} - 172546 q^{97} - 615271 q^{98} + 68607 q^{99}+O(q^{100}) \) 7 * q + q^2 + 63 * q^3 + 195 * q^4 + 175 * q^5 + 9 * q^6 + 153 * q^8 + 567 * q^9 + 25 * q^10 + 847 * q^11 + 1755 * q^12 + 1418 * q^13 + 2548 * q^14 + 1575 * q^15 + 3699 * q^16 + 630 * q^17 + 81 * q^18 + 2572 * q^19 + 4875 * q^20 + 121 * q^22 + 536 * q^23 + 1377 * q^24 + 4375 * q^25 - 7626 * q^26 + 5103 * q^27 - 11368 * q^28 - 1038 * q^29 + 225 * q^30 + 1872 * q^31 - 7523 * q^32 + 7623 * q^33 + 20790 * q^34 + 15795 * q^36 + 24298 * q^37 - 18952 * q^38 + 12762 * q^39 + 3825 * q^40 - 17658 * q^41 + 22932 * q^42 + 7244 * q^43 + 23595 * q^44 + 14175 * q^45 + 31016 * q^46 + 34560 * q^47 + 33291 * q^48 + 78735 * q^49 + 625 * q^50 + 5670 * q^51 + 110222 * q^52 - 10214 * q^53 + 729 * q^54 + 21175 * q^55 + 81124 * q^56 + 23148 * q^57 - 5718 * q^58 + 94676 * q^59 + 43875 * q^60 + 69538 * q^61 - 4208 * q^62 + 112339 * q^64 + 35450 * q^65 + 1089 * q^66 + 64908 * q^67 - 136010 * q^68 + 4824 * q^69 + 63700 * q^70 + 61816 * q^71 + 12393 * q^72 - 11890 * q^73 - 124050 * q^74 + 39375 * q^75 - 47216 * q^76 - 68634 * q^78 + 18928 * q^79 + 92475 * q^80 + 45927 * q^81 + 36398 * q^82 + 17492 * q^83 - 102312 * q^84 + 15750 * q^85 - 216688 * q^86 - 9342 * q^87 + 18513 * q^88 + 25302 * q^89 + 2025 * q^90 + 3392 * q^91 - 27408 * q^92 + 16848 * q^93 - 30800 * q^94 + 64300 * q^95 - 67707 * q^96 - 172546 * q^97 - 615271 * q^98 + 68607 * 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 $

$2$	−6.96291	−1.23088	−0.615440	−	0.788183i	$-0.711022\pi$
$2$	−6.96291	−1.23088	−0.615440	+	0.788183i	$0.711022\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.4822	0.515068
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	−62.6662	−0.710649
$7$	244.038	1.88240	0.941201	−	0.337848i	$-0.109699\pi$
$7$	244.038	1.88240	0.941201	+	0.337848i	$0.109699\pi$
$8$	108.049	0.596894
$9$	81.0000	0.333333
$10$	−174.073	−0.550467
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	148.340	0.297375
$13$	742.713	1.21888	0.609442	−	0.792831i	$-0.291393\pi$
$13$	742.713	1.21888	0.609442	+	0.792831i	$0.291393\pi$
$14$	−1699.22	−2.31701
$15$	225.000	0.258199
$16$	−1279.77	−1.24977
$17$	170.372	0.142980	0.0714900	−	0.997441i	$-0.477225\pi$
$17$	170.372	0.142980	0.0714900	+	0.997441i	$0.477225\pi$
$18$	−563.996	−0.410294
$19$	388.599	0.246955	0.123478	−	0.992347i	$-0.460595\pi$
$19$	388.599	0.246955	0.123478	+	0.992347i	$0.460595\pi$
$20$	412.054	0.230345
$21$	2196.34	1.08681
$22$	−842.513	−0.371125
$23$	224.659	0.0885534	0.0442767	−	0.999019i	$-0.485902\pi$
$23$	224.659	0.0885534	0.0442767	+	0.999019i	$0.485902\pi$
$24$	972.444	0.344617
$25$	625.000	0.200000
$26$	−5171.45	−1.50030
$27$	729.000	0.192450
$28$	4022.28	0.969565
$29$	−5577.78	−1.23159	−0.615796	−	0.787906i	$-0.711165\pi$
$29$	−5577.78	−1.23159	−0.615796	+	0.787906i	$0.711165\pi$
$30$	−1566.66	−0.317812
$31$	−6171.57	−1.15343	−0.576715	−	0.816945i	$-0.695666\pi$
$31$	−6171.57	−1.15343	−0.576715	+	0.816945i	$0.695666\pi$
$32$	5453.33	0.941428
$33$	1089.00	0.174078
$34$	−1186.28	−0.175991
$35$	6100.95	0.841836
$36$	1335.06	0.171689
$37$	4514.45	0.542126	0.271063	−	0.962562i	$-0.412625\pi$
$37$	4514.45	0.542126	0.271063	+	0.962562i	$0.412625\pi$
$38$	−2705.78	−0.303972
$39$	6684.41	0.703723
$40$	2701.23	0.266939
$41$	−8779.82	−0.815692	−0.407846	−	0.913051i	$-0.633720\pi$
$41$	−8779.82	−0.815692	−0.407846	+	0.913051i	$0.633720\pi$
$42$	−15292.9	−1.33773
$43$	8437.24	0.695872	0.347936	−	0.937518i	$-0.386883\pi$
$43$	8437.24	0.695872	0.347936	+	0.937518i	$0.386883\pi$
$44$	1994.34	0.155299
$45$	2025.00	0.149071
$46$	−1564.28	−0.108999
$47$	7343.40	0.484901	0.242450	−	0.970164i	$-0.422049\pi$
$47$	7343.40	0.484901	0.242450	+	0.970164i	$0.422049\pi$
$48$	−11517.9	−0.721557
$49$	42747.5	2.54344
$50$	−4351.82	−0.246176
$51$	1533.34	0.0825495
$52$	12241.5	0.627808
$53$	−32882.2	−1.60794	−0.803972	−	0.594666i	$-0.797284\pi$
$53$	−32882.2	−1.60794	−0.803972	+	0.594666i	$0.797284\pi$
$54$	−5075.96	−0.236883
$55$	3025.00	0.134840
$56$	26368.1	1.12359
$57$	3497.39	0.142580
$58$	38837.6	1.51594
$59$	34178.2	1.27826	0.639131	−	0.769098i	$-0.279294\pi$
$59$	34178.2	1.27826	0.639131	+	0.769098i	$0.279294\pi$
$60$	3708.49	0.132990
$61$	44832.7	1.54266	0.771331	−	0.636435i	$-0.219592\pi$
$61$	44832.7	1.54266	0.771331	+	0.636435i	$0.219592\pi$
$62$	42972.1	1.41974
$63$	19767.1	0.627467
$64$	2981.46	0.0909871
$65$	18567.8	0.545102
$66$	−7582.61	−0.214269
$67$	14320.3	0.389732	0.194866	−	0.980830i	$-0.437573\pi$
$67$	14320.3	0.389732	0.194866	+	0.980830i	$0.437573\pi$
$68$	2808.10	0.0736444
$69$	2021.94	0.0511263
$70$	−42480.4	−1.03620
$71$	31300.9	0.736904	0.368452	−	0.929647i	$-0.379888\pi$
$71$	31300.9	0.736904	0.368452	+	0.929647i	$0.379888\pi$
$72$	8751.99	0.198965
$73$	28110.8	0.617400	0.308700	−	0.951159i	$-0.400106\pi$
$73$	28110.8	0.617400	0.308700	+	0.951159i	$0.400106\pi$
$74$	−31433.7	−0.667293
$75$	5625.00	0.115470
$76$	6404.96	0.127199
$77$	29528.6	0.567565
$78$	−46543.0	−0.866199
$79$	−71426.8	−1.28764	−0.643819	−	0.765178i	$-0.722651\pi$
$79$	−71426.8	−1.28764	−0.643819	+	0.765178i	$0.722651\pi$
$80$	−31994.2	−0.558915
$81$	6561.00	0.111111
$82$	61133.1	1.00402
$83$	−116407.	−1.85474	−0.927368	−	0.374151i	$-0.877934\pi$
$83$	−116407.	−1.85474	−0.927368	+	0.374151i	$0.877934\pi$
$84$	36200.5	0.559778
$85$	4259.29	0.0639426
$86$	−58747.8	−0.856535
$87$	−50200.1	−0.711060
$88$	13074.0	0.179970
$89$	−100275.	−1.34189	−0.670947	−	0.741505i	$-0.734112\pi$
$89$	−100275.	−1.34189	−0.670947	+	0.741505i	$0.734112\pi$
$90$	−14099.9	−0.183489
$91$	181250.	2.29443
$92$	3702.88	0.0456110
$93$	−55544.1	−0.665934
$94$	−51131.5	−0.596855
$95$	9714.98	0.110442
$96$	49080.0	0.543534
$97$	137036.	1.47879	0.739393	−	0.673274i	$-0.235113\pi$
$97$	137036.	1.47879	0.739393	+	0.673274i	$0.235113\pi$
$98$	−297647.	−3.13067
$99$	9801.00	0.100504
$100$	10301.4	0.103014
$101$	51751.0	0.504795	0.252398	−	0.967624i	$-0.418781\pi$
$101$	51751.0	0.504795	0.252398	+	0.967624i	$0.418781\pi$
$102$	−10676.5	−0.101609
$103$	−102474.	−0.951747	−0.475874	−	0.879514i	$-0.657868\pi$
$103$	−102474.	−0.951747	−0.475874	+	0.879514i	$0.657868\pi$
$104$	80249.6	0.727544
$105$	54908.5	0.486034
$106$	228956.	1.97919
$107$	−216996.	−1.83228	−0.916140	−	0.400859i	$-0.868712\pi$
$107$	−216996.	−1.83228	−0.916140	+	0.400859i	$0.868712\pi$
$108$	12015.5	0.0991249
$109$	−20534.7	−0.165547	−0.0827736	−	0.996568i	$-0.526378\pi$
$109$	−20534.7	−0.165547	−0.0827736	+	0.996568i	$0.526378\pi$
$110$	−21062.8	−0.165972
$111$	40630.0	0.312997
$112$	−312312.	−2.35257
$113$	−834.386	−0.00614711	−0.00307356	−	0.999995i	$-0.500978\pi$
$113$	−834.386	−0.00614711	−0.00307356	+	0.999995i	$0.500978\pi$
$114$	−24352.1	−0.175499
$115$	5616.49	0.0396023
$116$	−91934.0	−0.634353
$117$	60159.7	0.406295
$118$	−237980.	−1.57339
$119$	41577.1	0.269146
$120$	24311.1	0.154117
$121$	14641.0	0.0909091
$122$	−312166.	−1.89883
$123$	−79018.4	−0.470940
$124$	−101721.	−0.594095
$125$	15625.0	0.0894427
$126$	−137636.	−0.772337
$127$	217555.	1.19691	0.598453	−	0.801158i	$-0.295783\pi$
$127$	217555.	1.19691	0.598453	+	0.801158i	$0.295783\pi$
$128$	−195266.	−1.05342
$129$	75935.1	0.401762
$130$	−129286.	−0.670955
$131$	12990.3	0.0661366	0.0330683	−	0.999453i	$-0.489472\pi$
$131$	12990.3	0.0661366	0.0330683	+	0.999453i	$0.489472\pi$
$132$	17949.1	0.0896618
$133$	94833.0	0.464869
$134$	−99711.3	−0.479714
$135$	18225.0	0.0860663
$136$	18408.5	0.0853438
$137$	244127.	1.11126	0.555628	−	0.831431i	$-0.312478\pi$
$137$	244127.	1.11126	0.555628	+	0.831431i	$0.312478\pi$
$138$	−14078.6	−0.0629304
$139$	−314017.	−1.37853	−0.689265	−	0.724510i	$-0.742066\pi$
$139$	−314017.	−1.37853	−0.689265	+	0.724510i	$0.742066\pi$
$140$	100557.	0.433603
$141$	66090.6	0.279957
$142$	−217945.	−0.907041
$143$	89868.2	0.367507
$144$	−103661.	−0.416591
$145$	−139445.	−0.550784
$146$	−195733.	−0.759946
$147$	384728.	1.46845
$148$	74407.9	0.279232
$149$	410917.	1.51631	0.758155	−	0.652074i	$-0.226101\pi$
$149$	410917.	1.51631	0.758155	+	0.652074i	$0.226101\pi$
$150$	−39166.4	−0.142130
$151$	95830.1	0.342026	0.171013	−	0.985269i	$-0.445296\pi$
$151$	95830.1	0.342026	0.171013	+	0.985269i	$0.445296\pi$
$152$	41987.9	0.147406
$153$	13800.1	0.0476600
$154$	−205605.	−0.698605
$155$	−154289.	−0.515830
$156$	110174.	0.362465
$157$	−141319.	−0.457564	−0.228782	−	0.973478i	$-0.573474\pi$
$157$	−141319.	−0.457564	−0.228782	+	0.973478i	$0.573474\pi$
$158$	497339.	1.58493
$159$	−295940.	−0.928347
$160$	136333.	0.421019
$161$	54825.4	0.166693
$162$	−45683.7	−0.136765
$163$	48028.4	0.141589	0.0707945	−	0.997491i	$-0.477447\pi$
$163$	48028.4	0.141589	0.0707945	+	0.997491i	$0.477447\pi$
$164$	−144711.	−0.420137
$165$	27225.0	0.0778499
$166$	810529.	2.28296
$167$	138657.	0.384725	0.192362	−	0.981324i	$-0.438385\pi$
$167$	138657.	0.384725	0.192362	+	0.981324i	$0.438385\pi$
$168$	237313.	0.648707
$169$	180329.	0.485679
$170$	−29657.1	−0.0787057
$171$	31476.5	0.0823184
$172$	139064.	0.358421
$173$	−602874.	−1.53148	−0.765740	−	0.643150i	$-0.777627\pi$
$173$	−602874.	−1.53148	−0.765740	+	0.643150i	$0.777627\pi$
$174$	349539.	0.875230
$175$	152524.	0.376480
$176$	−154852.	−0.376821
$177$	307604.	0.738004
$178$	698208.	1.65171
$179$	48954.2	0.114198	0.0570988	−	0.998369i	$-0.481815\pi$
$179$	48954.2	0.114198	0.0570988	+	0.998369i	$0.481815\pi$
$180$	33376.4	0.0767818
$181$	64205.1	0.145671	0.0728355	−	0.997344i	$-0.476795\pi$
$181$	64205.1	0.145671	0.0728355	+	0.997344i	$0.476795\pi$
$182$	−1.26203e6	−2.82417
$183$	403494.	0.890656
$184$	24274.3	0.0528569
$185$	112861.	0.242446
$186$	386749.	0.819685
$187$	20615.0	0.0431101
$188$	121035.	0.249757
$189$	177904.	0.362268
$190$	−67644.6	−0.135941
$191$	617012.	1.22380	0.611899	−	0.790936i	$-0.290406\pi$
$191$	617012.	1.22380	0.611899	+	0.790936i	$0.290406\pi$
$192$	26833.2	0.0525314
$193$	−63042.6	−0.121826	−0.0609131	−	0.998143i	$-0.519401\pi$
$193$	−63042.6	−0.121826	−0.0609131	+	0.998143i	$0.519401\pi$
$194$	−954170.	−1.82021
$195$	167110.	0.314715
$196$	704572.	1.31004
$197$	436107.	0.800621	0.400311	−	0.916380i	$-0.368902\pi$
$197$	436107.	0.800621	0.400311	+	0.916380i	$0.368902\pi$
$198$	−68243.5	−0.123708
$199$	−877391.	−1.57058	−0.785291	−	0.619127i	$-0.787487\pi$
$199$	−877391.	−1.57058	−0.785291	+	0.619127i	$0.787487\pi$
$200$	67530.8	0.119379
$201$	128883.	0.225012
$202$	−360338.	−0.621343
$203$	−1.36119e6	−2.31835
$204$	25272.9	0.0425186
$205$	−219496.	−0.364788
$206$	713519.	1.17149
$207$	18197.4	0.0295178
$208$	−950500.	−1.52333
$209$	47020.5	0.0744598
$210$	−382323.	−0.598250
$211$	−1.10605e6	−1.71028	−0.855142	−	0.518393i	$-0.826530\pi$
$211$	−1.10605e6	−1.71028	−0.855142	+	0.518393i	$0.826530\pi$
$212$	−541970.	−0.828201
$213$	281708.	0.425452
$214$	1.51092e6	2.25532
$215$	210931.	0.311203
$216$	78767.9	0.114872
$217$	−1.50610e6	−2.17122
$218$	142981.	0.203769
$219$	252997.	0.356456
$220$	49858.6	0.0694517
$221$	126537.	0.174276
$222$	−282903.	−0.385262
$223$	700787.	0.943678	0.471839	−	0.881685i	$-0.343590\pi$
$223$	700787.	0.943678	0.471839	+	0.881685i	$0.343590\pi$
$224$	1.33082e6	1.77215
$225$	50625.0	0.0666667
$226$	5809.76	0.00756636
$227$	830277.	1.06945	0.534723	−	0.845028i	$-0.320416\pi$
$227$	830277.	1.06945	0.534723	+	0.845028i	$0.320416\pi$
$228$	57644.7	0.0734382
$229$	1.20051e6	1.51278	0.756392	−	0.654118i	$-0.226960\pi$
$229$	1.20051e6	1.51278	0.756392	+	0.654118i	$0.226960\pi$
$230$	−39107.1	−0.0487457
$231$	265757.	0.327684
$232$	−602676.	−0.735129
$233$	−185040.	−0.223294	−0.111647	−	0.993748i	$-0.535613\pi$
$233$	−185040.	−0.223294	−0.111647	+	0.993748i	$0.535613\pi$
$234$	−418887.	−0.500101
$235$	183585.	0.216854
$236$	563332.	0.658391
$237$	−642842.	−0.743418
$238$	−289498.	−0.331286
$239$	94442.7	0.106948	0.0534741	−	0.998569i	$-0.482971\pi$
$239$	94442.7	0.106948	0.0534741	+	0.998569i	$0.482971\pi$
$240$	−287948.	−0.322690
$241$	−867299.	−0.961891	−0.480946	−	0.876750i	$-0.659707\pi$
$241$	−867299.	−0.961891	−0.480946	+	0.876750i	$0.659707\pi$
$242$	−101944.	−0.111898
$243$	59049.0	0.0641500
$244$	738941.	0.794575
$245$	1.06869e6	1.13746
$246$	550198.	0.579671
$247$	288618.	0.301010
$248$	−666834.	−0.688475
$249$	−1.04766e6	−1.07083
$250$	−108796.	−0.110093
$251$	1.37620e6	1.37878	0.689391	−	0.724389i	$-0.257878\pi$
$251$	1.37620e6	1.37878	0.689391	+	0.724389i	$0.257878\pi$
$252$	325804.	0.323188
$253$	27183.8	0.0266998
$254$	−1.51482e6	−1.47325
$255$	38333.6	0.0369173
$256$	1.26422e6	1.20565
$257$	240397.	0.227037	0.113518	−	0.993536i	$-0.463788\pi$
$257$	240397.	0.227037	0.113518	+	0.993536i	$0.463788\pi$
$258$	−528730.	−0.494521
$259$	1.10170e6	1.02050
$260$	306038.	0.280764
$261$	−451801.	−0.410531
$262$	−90450.6	−0.0814063
$263$	−565380.	−0.504024	−0.252012	−	0.967724i	$-0.581092\pi$
$263$	−565380.	−0.504024	−0.252012	+	0.967724i	$0.581092\pi$
$264$	117666.	0.103906
$265$	−822055.	−0.719095
$266$	−660314.	−0.572198
$267$	−902477.	−0.774743
$268$	236030.	0.200739
$269$	−653711.	−0.550815	−0.275407	−	0.961328i	$-0.588813\pi$
$269$	−653711.	−0.550815	−0.275407	+	0.961328i	$0.588813\pi$
$270$	−126899.	−0.105937
$271$	1.52990e6	1.26543	0.632717	−	0.774383i	$-0.281940\pi$
$271$	1.52990e6	1.26543	0.632717	+	0.774383i	$0.281940\pi$
$272$	−218036.	−0.178692
$273$	1.63125e6	1.32469
$274$	−1.69984e6	−1.36782
$275$	75625.0	0.0603023
$276$	33325.9	0.0263335
$277$	−636472.	−0.498402	−0.249201	−	0.968452i	$-0.580168\pi$
$277$	−636472.	−0.498402	−0.249201	+	0.968452i	$0.580168\pi$
$278$	2.18647e6	1.69681
$279$	−499897.	−0.384477
$280$	659203.	0.502486
$281$	−1.29656e6	−0.979550	−0.489775	−	0.871849i	$-0.662921\pi$
$281$	−1.29656e6	−0.979550	−0.489775	+	0.871849i	$0.662921\pi$
$282$	−460183.	−0.344594
$283$	−1.33255e6	−0.989048	−0.494524	−	0.869164i	$-0.664658\pi$
$283$	−1.33255e6	−0.989048	−0.494524	+	0.869164i	$0.664658\pi$
$284$	515907.	0.379555
$285$	87434.9	0.0637635
$286$	−625745.	−0.452358
$287$	−2.14261e6	−1.53546
$288$	441720.	0.313809
$289$	−1.39083e6	−0.979557
$290$	970941.	0.677950
$291$	1.23332e6	0.853777
$292$	463328.	0.318003
$293$	2.21486e6	1.50722	0.753610	−	0.657322i	$-0.228311\pi$
$293$	2.21486e6	1.50722	0.753610	+	0.657322i	$0.228311\pi$
$294$	−2.67883e6	−1.80749
$295$	854456.	0.571656
$296$	487783.	0.323592
$297$	88209.0	0.0580259
$298$	−2.86118e6	−1.86640
$299$	166857.	0.107936
$300$	92712.2	0.0594749
$301$	2.05901e6	1.30991
$302$	−667256.	−0.420994
$303$	465759.	0.291444
$304$	−497317.	−0.308638
$305$	1.12082e6	0.689899
$306$	−96088.9	−0.0586637
$307$	1.65823e6	1.00415	0.502074	−	0.864824i	$-0.332570\pi$
$307$	1.65823e6	1.00415	0.502074	+	0.864824i	$0.332570\pi$
$308$	486695.	0.292335
$309$	−922268.	−0.549492
$310$	1.07430e6	0.634925
$311$	−474414.	−0.278135	−0.139068	−	0.990283i	$-0.544411\pi$
$311$	−474414.	−0.278135	−0.139068	+	0.990283i	$0.544411\pi$
$312$	722246.	0.420048
$313$	2.36019e6	1.36171	0.680856	−	0.732417i	$-0.261608\pi$
$313$	2.36019e6	1.36171	0.680856	+	0.732417i	$0.261608\pi$
$314$	983993.	0.563207
$315$	494177.	0.280612
$316$	−1.17727e6	−0.663221
$317$	−2.39067e6	−1.33620	−0.668100	−	0.744071i	$-0.732892\pi$
$317$	−2.39067e6	−1.33620	−0.668100	+	0.744071i	$0.732892\pi$
$318$	2.06060e6	1.14269
$319$	−674912.	−0.371339
$320$	74536.6	0.0406906
$321$	−1.95296e6	−1.05787
$322$	−381745.	−0.205179
$323$	66206.3	0.0353096
$324$	108140.	0.0572298
$325$	464195.	0.243777
$326$	−334418.	−0.174279
$327$	−184812.	−0.0955787
$328$	−948653.	−0.486881
$329$	1.79207e6	0.912778
$330$	−189565.	−0.0958240
$331$	−2.36731e6	−1.18764	−0.593820	−	0.804598i	$-0.702381\pi$
$331$	−2.36731e6	−1.18764	−0.593820	+	0.804598i	$0.702381\pi$
$332$	−1.91863e6	−0.955315
$333$	365670.	0.180709
$334$	−965456.	−0.473551
$335$	358009.	0.174294
$336$	−2.81081e6	−1.35826
$337$	−537837.	−0.257974	−0.128987	−	0.991646i	$-0.541173\pi$
$337$	−537837.	−0.257974	−0.128987	+	0.991646i	$0.541173\pi$
$338$	−1.25562e6	−0.597813
$339$	−7509.48	−0.00354904
$340$	70202.4	0.0329348
$341$	−746760.	−0.347772
$342$	−219168.	−0.101324
$343$	6.33047e6	2.90537
$344$	911638.	0.415361
$345$	50548.4	0.0228644
$346$	4.19776e6	1.88507
$347$	1.22580e6	0.546508	0.273254	−	0.961942i	$-0.411900\pi$
$347$	1.22580e6	0.546508	0.273254	+	0.961942i	$0.411900\pi$
$348$	−827406.	−0.366244
$349$	−1.83313e6	−0.805617	−0.402809	−	0.915284i	$-0.631966\pi$
$349$	−1.83313e6	−0.805617	−0.402809	+	0.915284i	$0.631966\pi$
$350$	−1.06201e6	−0.463402
$351$	541438.	0.234574
$352$	659853.	0.283851
$353$	−2.25160e6	−0.961732	−0.480866	−	0.876794i	$-0.659678\pi$
$353$	−2.25160e6	−0.961732	−0.480866	+	0.876794i	$0.659678\pi$
$354$	−2.14182e6	−0.908396
$355$	782522.	0.329553
$356$	−1.65275e6	−0.691167
$357$	374194.	0.155391
$358$	−340864.	−0.140564
$359$	3.20169e6	1.31112	0.655561	−	0.755143i	$-0.272432\pi$
$359$	3.20169e6	1.31112	0.655561	+	0.755143i	$0.272432\pi$
$360$	218800.	0.0889797
$361$	−2.32509e6	−0.939013
$362$	−447054.	−0.179304
$363$	131769.	0.0524864
$364$	2.98740e6	1.18179
$365$	702771.	0.276110
$366$	−2.80950e6	−1.09629
$367$	−3.60298e6	−1.39636	−0.698178	−	0.715924i	$-0.746006\pi$
$367$	−3.60298e6	−1.39636	−0.698178	+	0.715924i	$0.746006\pi$
$368$	−287512.	−0.110672
$369$	−711165.	−0.271897
$370$	−785843.	−0.298422
$371$	−8.02450e6	−3.02680
$372$	−915488.	−0.343001
$373$	−1.60113e6	−0.595874	−0.297937	−	0.954586i	$-0.596299\pi$
$373$	−1.60113e6	−0.595874	−0.297937	+	0.954586i	$0.596299\pi$
$374$	−143540.	−0.0530634
$375$	140625.	0.0516398
$376$	793450.	0.289434
$377$	−4.14269e6	−1.50117
$378$	−1.23873e6	−0.445909
$379$	−474443.	−0.169663	−0.0848313	−	0.996395i	$-0.527035\pi$
$379$	−474443.	−0.169663	−0.0848313	+	0.996395i	$0.527035\pi$
$380$	160124.	0.0568850
$381$	1.95799e6	0.691033
$382$	−4.29620e6	−1.50635
$383$	−4.97526e6	−1.73308	−0.866540	−	0.499107i	$-0.833661\pi$
$383$	−4.97526e6	−1.73308	−0.866540	+	0.499107i	$0.833661\pi$
$384$	−1.75740e6	−0.608194
$385$	738215.	0.253823
$386$	438960.	0.149954
$387$	683416.	0.231957
$388$	2.25865e6	0.761675
$389$	−2.59830e6	−0.870592	−0.435296	−	0.900287i	$-0.643356\pi$
$389$	−2.59830e6	−0.870592	−0.435296	+	0.900287i	$0.643356\pi$
$390$	−1.16358e6	−0.387376
$391$	38275.6	0.0126614
$392$	4.61884e6	1.51816
$393$	116913.	0.0381840
$394$	−3.03657e6	−0.985469
$395$	−1.78567e6	−0.575849
$396$	161542.	0.0517663
$397$	4.33628e6	1.38083	0.690416	−	0.723413i	$-0.257428\pi$
$397$	4.33628e6	1.38083	0.690416	+	0.723413i	$0.257428\pi$
$398$	6.10920e6	1.93320
$399$	853497.	0.268392
$400$	−799855.	−0.249955
$401$	4.16715e6	1.29413	0.647066	−	0.762434i	$-0.275996\pi$
$401$	4.16715e6	1.29413	0.647066	+	0.762434i	$0.275996\pi$
$402$	−897402.	−0.276963
$403$	−4.58370e6	−1.40590
$404$	852968.	0.260004
$405$	164025.	0.0496904
$406$	9.47786e6	2.85361
$407$	546248.	0.163457
$408$	165677.	0.0492733
$409$	−6.35982e6	−1.87991	−0.939954	−	0.341302i	$-0.889132\pi$
$409$	−6.35982e6	−1.87991	−0.939954	+	0.341302i	$0.889132\pi$
$410$	1.52833e6	0.449011
$411$	2.19714e6	0.641584
$412$	−1.68900e6	−0.490214
$413$	8.34079e6	2.40620
$414$	−126707.	−0.0363329
$415$	−2.91016e6	−0.829463
$416$	4.05026e6	1.14749
$417$	−2.82615e6	−0.795894
$418$	−327400.	−0.0916511
$419$	3.32788e6	0.926046	0.463023	−	0.886346i	$-0.346765\pi$
$419$	3.32788e6	0.926046	0.463023	+	0.886346i	$0.346765\pi$
$420$	905012.	0.250341
$421$	−3.11231e6	−0.855809	−0.427905	−	0.903824i	$-0.640748\pi$
$421$	−3.11231e6	−0.855809	−0.427905	+	0.903824i	$0.640748\pi$
$422$	7.70133e6	2.10516
$423$	594816.	0.161634
$424$	−3.55290e6	−0.959772
$425$	106482.	0.0285960
$426$	−1.96151e6	−0.523680
$427$	1.09409e7	2.90391
$428$	−3.57656e6	−0.943749
$429$	808814.	0.212181
$430$	−1.46869e6	−0.383054
$431$	5.31827e6	1.37904	0.689521	−	0.724266i	$-0.257821\pi$
$431$	5.31827e6	1.37904	0.689521	+	0.724266i	$0.257821\pi$
$432$	−932951.	−0.240519
$433$	1.79781e6	0.460812	0.230406	−	0.973095i	$-0.425995\pi$
$433$	1.79781e6	0.460812	0.230406	+	0.973095i	$0.425995\pi$
$434$	1.04868e7	2.67251
$435$	−1.25500e6	−0.317996
$436$	−338456.	−0.0852680
$437$	87302.5	0.0218687
$438$	−1.76160e6	−0.438755
$439$	−5.18777e6	−1.28475	−0.642376	−	0.766390i	$-0.722051\pi$
$439$	−5.18777e6	−1.28475	−0.642376	+	0.766390i	$0.722051\pi$
$440$	326849.	0.0804851
$441$	3.46255e6	0.847812
$442$	−881068.	−0.214513
$443$	−3.13149e6	−0.758128	−0.379064	−	0.925371i	$-0.623754\pi$
$443$	−3.13149e6	−0.758128	−0.379064	+	0.925371i	$0.623754\pi$
$444$	669671.	0.161215
$445$	−2.50688e6	−0.600114
$446$	−4.87952e6	−1.16156
$447$	3.69825e6	0.875442
$448$	727590.	0.171274
$449$	3.63242e6	0.850315	0.425157	−	0.905119i	$-0.360219\pi$
$449$	3.63242e6	0.850315	0.425157	+	0.905119i	$0.360219\pi$
$450$	−352498.	−0.0820587
$451$	−1.06236e6	−0.245940
$452$	−13752.5	−0.00316618
$453$	862471.	0.197469
$454$	−5.78115e6	−1.31636
$455$	4.53125e6	1.02610
$456$	377891.	0.0851049
$457$	−1.11756e6	−0.250312	−0.125156	−	0.992137i	$-0.539943\pi$
$457$	−1.11756e6	−0.250312	−0.125156	+	0.992137i	$0.539943\pi$
$458$	−8.35905e6	−1.86206
$459$	124201.	0.0275165
$460$	92571.9	0.0203979
$461$	466934.	0.102330	0.0511650	−	0.998690i	$-0.483707\pi$
$461$	466934.	0.102330	0.0511650	+	0.998690i	$0.483707\pi$
$462$	−1.85045e6	−0.403340
$463$	−7.12727e6	−1.54515	−0.772575	−	0.634923i	$-0.781032\pi$
$463$	−7.12727e6	−1.54515	−0.772575	+	0.634923i	$0.781032\pi$
$464$	7.13827e6	1.53921
$465$	−1.38860e6	−0.297815
$466$	1.28842e6	0.274848
$467$	−1.73988e6	−0.369171	−0.184585	−	0.982816i	$-0.559094\pi$
$467$	−1.73988e6	−0.369171	−0.184585	+	0.982816i	$0.559094\pi$
$468$	991563.	0.209269
$469$	3.49471e6	0.733633
$470$	−1.27829e6	−0.266922
$471$	−1.27187e6	−0.264175
$472$	3.69293e6	0.762986
$473$	1.02091e6	0.209813
$474$	4.47605e6	0.915059
$475$	242875.	0.0493910
$476$	685282.	0.138628
$477$	−2.66346e6	−0.535982
$478$	−657597.	−0.131641
$479$	−8.38529e6	−1.66986	−0.834928	−	0.550359i	$-0.814491\pi$
$479$	−8.38529e6	−1.66986	−0.834928	+	0.550359i	$0.814491\pi$
$480$	1.22700e6	0.243076
$481$	3.35294e6	0.660789
$482$	6.03893e6	1.18397
$483$	493429.	0.0962402
$484$	241316.	0.0468244
$485$	3.42590e6	0.661333
$486$	−411153.	−0.0789611
$487$	6.24834e6	1.19383	0.596914	−	0.802305i	$-0.296393\pi$
$487$	6.24834e6	1.19383	0.596914	+	0.802305i	$0.296393\pi$
$488$	4.84414e6	0.920805
$489$	432256.	0.0817464
$490$	−7.44118e6	−1.40008
$491$	−6.31595e6	−1.18232	−0.591160	−	0.806555i	$-0.701330\pi$
$491$	−6.31595e6	−1.18232	−0.591160	+	0.806555i	$0.701330\pi$
$492$	−1.30239e6	−0.242566
$493$	−950296.	−0.176093
$494$	−2.00962e6	−0.370507
$495$	245025.	0.0449467
$496$	7.89818e6	1.44153
$497$	7.63860e6	1.38715
$498$	7.29476e6	1.31807
$499$	7.76425e6	1.39588	0.697940	−	0.716156i	$-0.254100\pi$
$499$	7.76425e6	1.39588	0.697940	+	0.716156i	$0.254100\pi$
$500$	257534.	0.0460691
$501$	1.24791e6	0.222121
$502$	−9.58233e6	−1.69712
$503$	−3.81274e6	−0.671920	−0.335960	−	0.941876i	$-0.609061\pi$
$503$	−3.81274e6	−0.671920	−0.335960	+	0.941876i	$0.609061\pi$
$504$	2.13582e6	0.374531
$505$	1.29377e6	0.225751
$506$	−189278.	−0.0328643
$507$	1.62296e6	0.280407
$508$	3.58578e6	0.616487
$509$	8.05659e6	1.37834	0.689170	−	0.724599i	$-0.257975\pi$
$509$	8.05659e6	1.37834	0.689170	+	0.724599i	$0.257975\pi$
$510$	−266914.	−0.0454407
$511$	6.86011e6	1.16219
$512$	−2.55410e6	−0.430590
$513$	283289.	0.0475265
$514$	−1.67386e6	−0.279455
$515$	−2.56186e6	−0.425634
$516$	1.25158e6	0.206935
$517$	888552.	0.146203
$518$	−7.67102e6	−1.25611
$519$	−5.42587e6	−0.884201
$520$	2.00624e6	0.325368
$521$	−3.60376e6	−0.581651	−0.290825	−	0.956776i	$-0.593930\pi$
$521$	−3.60376e6	−0.581651	−0.290825	+	0.956776i	$0.593930\pi$
$522$	3.14585e6	0.505314
$523$	−3.07370e6	−0.491369	−0.245685	−	0.969350i	$-0.579013\pi$
$523$	−3.07370e6	−0.491369	−0.245685	+	0.969350i	$0.579013\pi$
$524$	214109.	0.0340648
$525$	1.37271e6	0.217361
$526$	3.93669e6	0.620393
$527$	−1.05146e6	−0.164917
$528$	−1.39367e6	−0.217558
$529$	−6.38587e6	−0.992158
$530$	5.72390e6	0.885120
$531$	2.76844e6	0.426087
$532$	1.56305e6	0.239439
$533$	−6.52088e6	−0.994234
$534$	6.28387e6	0.953617
$535$	−5.42489e6	−0.819420
$536$	1.54730e6	0.232629
$537$	440587.	0.0659320
$538$	4.55174e6	0.677987
$539$	5.17245e6	0.766875
$540$	300388.	0.0443300
$541$	−1.20414e7	−1.76883	−0.884413	−	0.466705i	$-0.845441\pi$
$541$	−1.20414e7	−1.76883	−0.884413	+	0.466705i	$0.845441\pi$
$542$	−1.06525e7	−1.55760
$543$	577846.	0.0841032
$544$	929093.	0.134605
$545$	−513367.	−0.0740349
$546$	−1.13583e7	−1.63054
$547$	6.71046e6	0.958924	0.479462	−	0.877563i	$-0.340832\pi$
$547$	6.71046e6	0.958924	0.479462	+	0.877563i	$0.340832\pi$
$548$	4.02374e6	0.572373
$549$	3.63145e6	0.514220
$550$	−526570.	−0.0742249
$551$	−2.16752e6	−0.304148
$552$	218469.	0.0305170
$553$	−1.74309e7	−2.42385
$554$	4.43170e6	0.613474
$555$	1.01575e6	0.139976
$556$	−5.17568e6	−0.710036
$557$	−1.56882e6	−0.214257	−0.107128	−	0.994245i	$-0.534166\pi$
$557$	−1.56882e6	−0.214257	−0.107128	+	0.994245i	$0.534166\pi$
$558$	3.48074e6	0.473245
$559$	6.26644e6	0.848187
$560$	−7.80780e6	−1.05210
$561$	185535.	0.0248896
$562$	9.02783e6	1.20571
$563$	4.75691e6	0.632491	0.316245	−	0.948677i	$-0.397578\pi$
$563$	4.75691e6	0.632491	0.316245	+	0.948677i	$0.397578\pi$
$564$	1.08932e6	0.144197
$565$	−20859.7	−0.00274907
$566$	9.27843e6	1.21740
$567$	1.60113e6	0.209156
$568$	3.38204e6	0.439853
$569$	−9.45773e6	−1.22463	−0.612317	−	0.790612i	$-0.709762\pi$
$569$	−9.45773e6	−1.22463	−0.612317	+	0.790612i	$0.709762\pi$
$570$	−608801.	−0.0784853
$571$	1.07089e7	1.37453	0.687263	−	0.726408i	$-0.258812\pi$
$571$	1.07089e7	1.37453	0.687263	+	0.726408i	$0.258812\pi$
$572$	1.48122e6	0.189291
$573$	5.55311e6	0.706561
$574$	1.49188e7	1.88997
$575$	140412.	0.0177107
$576$	241499.	0.0303290
$577$	241609.	0.0302116	0.0151058	−	0.999886i	$-0.495191\pi$
$577$	241609.	0.0302116	0.0151058	+	0.999886i	$0.495191\pi$
$578$	9.68423e6	1.20572
$579$	−567384.	−0.0703364
$580$	−2.29835e6	−0.283691
$581$	−2.84076e7	−3.49136
$582$	−8.58753e6	−1.05090
$583$	−3.97875e6	−0.484814
$584$	3.03735e6	0.368522
$585$	1.50399e6	0.181701
$586$	−1.54219e7	−1.85521
$587$	−119449.	−0.0143083	−0.00715415	−	0.999974i	$-0.502277\pi$
$587$	−119449.	−0.0143083	−0.00715415	+	0.999974i	$0.502277\pi$
$588$	6.34115e6	0.756353
$589$	−2.39827e6	−0.284846
$590$	−5.94950e6	−0.703640
$591$	3.92496e6	0.462239
$592$	−5.77744e6	−0.677535
$593$	−3.75509e6	−0.438514	−0.219257	−	0.975667i	$-0.570363\pi$
$593$	−3.75509e6	−0.438514	−0.219257	+	0.975667i	$0.570363\pi$
$594$	−614192.	−0.0714230
$595$	1.03943e6	0.120366
$596$	6.77280e6	0.781003
$597$	−7.89652e6	−0.906776
$598$	−1.16181e6	−0.132857
$599$	−5.42814e6	−0.618136	−0.309068	−	0.951040i	$-0.600017\pi$
$599$	−5.42814e6	−0.618136	−0.309068	+	0.951040i	$0.600017\pi$
$600$	607777.	0.0689233
$601$	−1.16356e7	−1.31402	−0.657011	−	0.753881i	$-0.728179\pi$
$601$	−1.16356e7	−1.31402	−0.657011	+	0.753881i	$0.728179\pi$
$602$	−1.43367e7	−1.61234
$603$	1.15995e6	0.129911
$604$	1.57949e6	0.176167
$605$	366025.	0.0406558
$606$	−3.24304e6	−0.358732
$607$	−1.33029e6	−0.146547	−0.0732733	−	0.997312i	$-0.523345\pi$
$607$	−1.33029e6	−0.146547	−0.0732733	+	0.997312i	$0.523345\pi$
$608$	2.11916e6	0.232491
$609$	−1.22507e7	−1.33850
$610$	−7.80416e6	−0.849184
$611$	5.45404e6	0.591038
$612$	227456.	0.0245481
$613$	−7.98531e6	−0.858303	−0.429152	−	0.903232i	$-0.641187\pi$
$613$	−7.98531e6	−0.858303	−0.429152	+	0.903232i	$0.641187\pi$
$614$	−1.15461e7	−1.23599
$615$	−1.97546e6	−0.210611
$616$	3.19054e6	0.338776
$617$	523235.	0.0553329	0.0276665	−	0.999617i	$-0.491192\pi$
$617$	523235.	0.0553329	0.0276665	+	0.999617i	$0.491192\pi$
$618$	6.42167e6	0.676359
$619$	−1.58301e7	−1.66056	−0.830282	−	0.557343i	$-0.811821\pi$
$619$	−1.58301e7	−1.66056	−0.830282	+	0.557343i	$0.811821\pi$
$620$	−2.54302e6	−0.265687
$621$	163777.	0.0170421
$622$	3.30330e6	0.342352
$623$	−2.44710e7	−2.52598
$624$	−8.55450e6	−0.879494
$625$	390625.	0.0400000
$626$	−1.64338e7	−1.67611
$627$	423185.	0.0429894
$628$	−2.32925e6	−0.235676
$629$	769134.	0.0775132
$630$	−3.44091e6	−0.345400
$631$	1.62790e7	1.62762	0.813811	−	0.581130i	$-0.197389\pi$
$631$	1.62790e7	1.62762	0.813811	+	0.581130i	$0.197389\pi$
$632$	−7.71762e6	−0.768583
$633$	−9.95445e6	−0.987433
$634$	1.66460e7	1.64470
$635$	5.43887e6	0.535272
$636$	−4.87773e6	−0.478162
$637$	3.17491e7	3.10015
$638$	4.69935e6	0.457074
$639$	2.53537e6	0.245635
$640$	−4.88166e6	−0.471105
$641$	−6.77947e6	−0.651704	−0.325852	−	0.945421i	$-0.605651\pi$
$641$	−6.77947e6	−0.651704	−0.325852	+	0.945421i	$0.605651\pi$
$642$	1.35983e7	1.30211
$643$	−6.73415e6	−0.642326	−0.321163	−	0.947024i	$-0.604074\pi$
$643$	−6.73415e6	−0.642326	−0.321163	+	0.947024i	$0.604074\pi$
$644$	903642.	0.0858582
$645$	1.89838e6	0.179673
$646$	−460989.	−0.0434619
$647$	−2.88479e6	−0.270928	−0.135464	−	0.990782i	$-0.543252\pi$
$647$	−2.88479e6	−0.270928	−0.135464	+	0.990782i	$0.543252\pi$
$648$	708911.	0.0663215
$649$	4.13557e6	0.385410
$650$	−3.23215e6	−0.300060
$651$	−1.35549e7	−1.25355
$652$	791613.	0.0729279
$653$	−5.48231e6	−0.503130	−0.251565	−	0.967840i	$-0.580945\pi$
$653$	−5.48231e6	−0.503130	−0.251565	+	0.967840i	$0.580945\pi$
$654$	1.28683e6	0.117646
$655$	324758.	0.0295772
$656$	1.12361e7	1.01943
$657$	2.27698e6	0.205800
$658$	−1.24780e7	−1.12352
$659$	9.32337e6	0.836294	0.418147	−	0.908379i	$-0.362680\pi$
$659$	9.32337e6	0.836294	0.418147	+	0.908379i	$0.362680\pi$
$660$	448727.	0.0400980
$661$	2.90655e6	0.258747	0.129373	−	0.991596i	$-0.458703\pi$
$661$	2.90655e6	0.258747	0.129373	+	0.991596i	$0.458703\pi$
$662$	1.64834e7	1.46184
$663$	1.13883e6	0.100618
$664$	−1.25776e7	−1.10708
$665$	2.37082e6	0.207896
$666$	−2.54613e6	−0.222431
$667$	−1.25310e6	−0.109062
$668$	2.28537e6	0.198159
$669$	6.30709e6	0.544833
$670$	−2.49278e6	−0.214535
$671$	5.42476e6	0.465130
$672$	1.19774e7	1.02315
$673$	3.40865e6	0.290098	0.145049	−	0.989424i	$-0.453666\pi$
$673$	3.40865e6	0.290098	0.145049	+	0.989424i	$0.453666\pi$
$674$	3.74491e6	0.317535
$675$	455625.	0.0384900
$676$	2.97222e6	0.250158
$677$	−1.16473e7	−0.976685	−0.488343	−	0.872652i	$-0.662398\pi$
$677$	−1.16473e7	−0.976685	−0.488343	+	0.872652i	$0.662398\pi$
$678$	52287.8	0.00436844
$679$	3.34420e7	2.78367
$680$	460213.	0.0381669
$681$	7.47250e6	0.617444
$682$	5.19963e6	0.428066
$683$	8.31525e6	0.682061	0.341031	−	0.940052i	$-0.389224\pi$
$683$	8.31525e6	0.682061	0.341031	+	0.940052i	$0.389224\pi$
$684$	518802.	0.0423996
$685$	6.10318e6	0.496969
$686$	−4.40785e7	−3.57616
$687$	1.08046e7	0.873407
$688$	−1.07977e7	−0.869682
$689$	−2.44220e7	−1.95990
$690$	−351964.	−0.0281433
$691$	1.89691e6	0.151131	0.0755653	−	0.997141i	$-0.475924\pi$
$691$	1.89691e6	0.151131	0.0755653	+	0.997141i	$0.475924\pi$
$692$	−9.93668e6	−0.788817
$693$	2.39182e6	0.189188
$694$	−8.53516e6	−0.672687
$695$	−7.85042e6	−0.616497
$696$	−5.42408e6	−0.424427
$697$	−1.49583e6	−0.116628
$698$	1.27639e7	0.991619
$699$	−1.66536e6	−0.128919
$700$	2.51392e6	0.193913
$701$	−5.22980e6	−0.401967	−0.200983	−	0.979595i	$-0.564414\pi$
$701$	−5.22980e6	−0.401967	−0.200983	+	0.979595i	$0.564414\pi$
$702$	−3.76998e6	−0.288733
$703$	1.75431e6	0.133881
$704$	360757.	0.0274336
$705$	1.65227e6	0.125201
$706$	1.56777e7	1.18378
$707$	1.26292e7	0.950227
$708$	5.06998e6	0.380122
$709$	2.24825e7	1.67969	0.839843	−	0.542829i	$-0.182647\pi$
$709$	2.24825e7	1.67969	0.839843	+	0.542829i	$0.182647\pi$
$710$	−5.44863e6	−0.405641
$711$	−5.78557e6	−0.429213
$712$	−1.08347e7	−0.800969
$713$	−1.38650e6	−0.102140
$714$	−2.60548e6	−0.191268
$715$	2.24671e6	0.164354
$716$	806871.	0.0588195
$717$	849985.	0.0617466
$718$	−2.22931e7	−1.61383
$719$	7.94218e6	0.572951	0.286475	−	0.958088i	$-0.407516\pi$
$719$	7.94218e6	0.572951	0.286475	+	0.958088i	$0.407516\pi$
$720$	−2.59153e6	−0.186305
$721$	−2.50076e7	−1.79157
$722$	1.61894e7	1.15581
$723$	−7.80569e6	−0.555348
$724$	1.05824e6	0.0750304
$725$	−3.48612e6	−0.246318
$726$	−917496.	−0.0646045
$727$	7.82158e6	0.548856	0.274428	−	0.961608i	$-0.411511\pi$
$727$	7.82158e6	0.548856	0.274428	+	0.961608i	$0.411511\pi$
$728$	1.95839e7	1.36953
$729$	531441.	0.0370370
$730$	−4.89333e6	−0.339858
$731$	1.43747e6	0.0994957
$732$	6.65047e6	0.458748
$733$	−4.06221e6	−0.279256	−0.139628	−	0.990204i	$-0.544591\pi$
$733$	−4.06221e6	−0.279256	−0.139628	+	0.990204i	$0.544591\pi$
$734$	2.50872e7	1.71875
$735$	9.61819e6	0.656712
$736$	1.22514e6	0.0833666
$737$	1.73276e6	0.117509
$738$	4.95178e6	0.334673
$739$	2.72992e7	1.83882	0.919409	−	0.393302i	$-0.128667\pi$
$739$	2.72992e7	1.83882	0.919409	+	0.393302i	$0.128667\pi$
$740$	1.86020e6	0.124876
$741$	2.59756e6	0.173788
$742$	5.58739e7	3.72563
$743$	2.27700e7	1.51318	0.756592	−	0.653888i	$-0.226863\pi$
$743$	2.27700e7	1.51318	0.756592	+	0.653888i	$0.226863\pi$
$744$	−6.00151e6	−0.397491
$745$	1.02729e7	0.678115
$746$	1.11485e7	0.733450
$747$	−9.42893e6	−0.618245
$748$	339780.	0.0222046
$749$	−5.29552e7	−3.44909
$750$	−979160.	−0.0635624
$751$	2.45531e7	1.58857	0.794286	−	0.607544i	$-0.207845\pi$
$751$	2.45531e7	1.58857	0.794286	+	0.607544i	$0.207845\pi$
$752$	−9.39785e6	−0.606016
$753$	1.23858e7	0.796040
$754$	2.88452e7	1.84776
$755$	2.39575e6	0.152959
$756$	2.93224e6	0.186593
$757$	−2.81309e6	−0.178420	−0.0892101	−	0.996013i	$-0.528434\pi$
$757$	−2.81309e6	−0.178420	−0.0892101	+	0.996013i	$0.528434\pi$
$758$	3.30351e6	0.208834
$759$	244654.	0.0154152
$760$	1.04970e6	0.0659219
$761$	−2.45137e7	−1.53443	−0.767215	−	0.641390i	$-0.778358\pi$
$761$	−2.45137e7	−1.53443	−0.767215	+	0.641390i	$0.778358\pi$
$762$	−1.36333e7	−0.850580
$763$	−5.01124e6	−0.311626
$764$	1.01697e7	0.630340
$765$	345003.	0.0213142
$766$	3.46423e7	2.13322
$767$	2.53846e7	1.55805
$768$	1.13779e7	0.696083
$769$	2.21352e6	0.134980	0.0674898	−	0.997720i	$-0.478501\pi$
$769$	2.21352e6	0.134980	0.0674898	+	0.997720i	$0.478501\pi$
$770$	−5.14013e6	−0.312426
$771$	2.16357e6	0.131080
$772$	−1.03908e6	−0.0627488
$773$	−8.32847e6	−0.501322	−0.250661	−	0.968075i	$-0.580648\pi$
$773$	−8.32847e6	−0.501322	−0.250661	+	0.968075i	$0.580648\pi$
$774$	−4.75857e6	−0.285512
$775$	−3.85723e6	−0.230686
$776$	1.48066e7	0.882678
$777$	9.91527e6	0.589185
$778$	1.80917e7	1.07160
$779$	−3.41183e6	−0.201439
$780$	2.75434e6	0.162099
$781$	3.78741e6	0.222185
$782$	−266510.	−0.0155846
$783$	−4.06620e6	−0.237020
$784$	−5.47069e7	−3.17872
$785$	−3.53298e6	−0.204629
$786$	−814055.	−0.0469999
$787$	−3.42361e7	−1.97037	−0.985183	−	0.171506i	$-0.945137\pi$
$787$	−3.42361e7	−1.97037	−0.985183	+	0.171506i	$0.945137\pi$
$788$	7.18798e6	0.412374
$789$	−5.08842e6	−0.290998
$790$	1.24335e7	0.708802
$791$	−203622.	−0.0115713
$792$	1.05899e6	0.0599901
$793$	3.32978e7	1.88033
$794$	−3.01931e7	−1.69964
$795$	−7.39849e6	−0.415170
$796$	−1.44613e7	−0.808957
$797$	−6.63504e6	−0.369997	−0.184998	−	0.982739i	$-0.559228\pi$
$797$	−6.63504e6	−0.369997	−0.184998	+	0.982739i	$0.559228\pi$
$798$	−5.94283e6	−0.330359
$799$	1.25111e6	0.0693310
$800$	3.40833e6	0.188286
$801$	−8.12229e6	−0.447298
$802$	−2.90155e7	−1.59292
$803$	3.40141e6	0.186153
$804$	2.12427e6	0.115897
$805$	1.37064e6	0.0745474
$806$	3.19159e7	1.73049
$807$	−5.88340e6	−0.318013
$808$	5.59166e6	0.301309
$809$	−1.90116e7	−1.02129	−0.510643	−	0.859793i	$-0.670593\pi$
$809$	−1.90116e7	−1.02129	−0.510643	+	0.859793i	$0.670593\pi$
$810$	−1.14209e6	−0.0611630
$811$	−2.87368e6	−0.153422	−0.0767108	−	0.997053i	$-0.524442\pi$
$811$	−2.87368e6	−0.153422	−0.0767108	+	0.997053i	$0.524442\pi$
$812$	−2.24354e7	−1.19411
$813$	1.37691e7	0.730598
$814$	−3.80348e6	−0.201196
$815$	1.20071e6	0.0633205
$816$	−1.96232e6	−0.103168
$817$	3.27871e6	0.171849
$818$	4.42829e7	2.31394
$819$	1.46813e7	0.764810
$820$	−3.61776e6	−0.187891
$821$	−7.48901e6	−0.387763	−0.193881	−	0.981025i	$-0.562108\pi$
$821$	−7.48901e6	−0.387763	−0.193881	+	0.981025i	$0.562108\pi$
$822$	−1.52985e7	−0.789714
$823$	1.94316e7	1.00002	0.500010	−	0.866020i	$-0.333330\pi$
$823$	1.94316e7	1.00002	0.500010	+	0.866020i	$0.333330\pi$
$824$	−1.10723e7	−0.568092
$825$	680625.	0.0348155
$826$	−5.80762e7	−2.96175
$827$	2.60244e7	1.32318	0.661588	−	0.749868i	$-0.269883\pi$
$827$	2.60244e7	1.32318	0.661588	+	0.749868i	$0.269883\pi$
$828$	299933.	0.0152037
$829$	3.61948e6	0.182919	0.0914597	−	0.995809i	$-0.470847\pi$
$829$	3.61948e6	0.182919	0.0914597	+	0.995809i	$0.470847\pi$
$830$	2.02632e7	1.02097
$831$	−5.72825e6	−0.287753
$832$	2.21437e6	0.110903
$833$	7.28297e6	0.363660
$834$	1.96782e7	0.979651
$835$	3.46642e6	0.172054
$836$	775000.	0.0383518
$837$	−4.49908e6	−0.221978
$838$	−2.31717e7	−1.13985
$839$	2.47149e7	1.21214	0.606071	−	0.795410i	$-0.292745\pi$
$839$	2.47149e7	1.21214	0.606071	+	0.795410i	$0.292745\pi$
$840$	5.93283e6	0.290111
$841$	1.06005e7	0.516818
$842$	2.16707e7	1.05340
$843$	−1.16690e7	−0.565543
$844$	−1.82301e7	−0.880913
$845$	4.50823e6	0.217202
$846$	−4.14165e6	−0.198952
$847$	3.57296e6	0.171127
$848$	4.20816e7	2.00957
$849$	−1.19930e7	−0.571027
$850$	−741427.	−0.0351982
$851$	1.01421e6	0.0480071
$852$	4.64316e6	0.219136
$853$	−3.66361e7	−1.72400	−0.861999	−	0.506911i	$-0.830787\pi$
$853$	−3.66361e7	−1.72400	−0.861999	+	0.506911i	$0.830787\pi$
$854$	−7.61804e7	−3.57436
$855$	786914.	0.0368139
$856$	−2.34462e7	−1.09368
$857$	−6.08724e6	−0.283119	−0.141559	−	0.989930i	$-0.545212\pi$
$857$	−6.08724e6	−0.283119	−0.141559	+	0.989930i	$0.545212\pi$
$858$	−5.63170e6	−0.261169
$859$	−528725.	−0.0244482	−0.0122241	−	0.999925i	$-0.503891\pi$
$859$	−528725.	−0.0244482	−0.0122241	+	0.999925i	$0.503891\pi$
$860$	3.47660e6	0.160291
$861$	−1.92835e7	−0.886498
$862$	−3.70307e7	−1.69744
$863$	−3.67927e7	−1.68165	−0.840823	−	0.541311i	$-0.817928\pi$
$863$	−3.67927e7	−1.68165	−0.840823	+	0.541311i	$0.817928\pi$
$864$	3.97548e6	0.181178
$865$	−1.50719e7	−0.684899
$866$	−1.25180e7	−0.567205
$867$	−1.25175e7	−0.565547
$868$	−2.48238e7	−1.11833
$869$	−8.64265e6	−0.388237
$870$	8.73847e6	0.391415
$871$	1.06359e7	0.475039
$872$	−2.21876e6	−0.0988140
$873$	1.10999e7	0.492929
$874$	−607880.	−0.0269178
$875$	3.81309e6	0.168367
$876$	4.16995e6	0.183599
$877$	−3.03856e7	−1.33404	−0.667020	−	0.745040i	$-0.732430\pi$
$877$	−3.03856e7	−1.33404	−0.667020	+	0.745040i	$0.732430\pi$
$878$	3.61220e7	1.58138
$879$	1.99337e7	0.870194
$880$	−3.87130e6	−0.168519
$881$	−2.64506e7	−1.14814	−0.574071	−	0.818806i	$-0.694637\pi$
$881$	−2.64506e7	−1.14814	−0.574071	+	0.818806i	$0.694637\pi$
$882$	−2.41094e7	−1.04356
$883$	2.78180e7	1.20067	0.600335	−	0.799749i	$-0.295034\pi$
$883$	2.78180e7	1.20067	0.600335	+	0.799749i	$0.295034\pi$
$884$	2.08561e6	0.0897640
$885$	7.69010e6	0.330046
$886$	2.18043e7	0.933165
$887$	2.72290e7	1.16205	0.581023	−	0.813887i	$-0.302653\pi$
$887$	2.72290e7	1.16205	0.581023	+	0.813887i	$0.302653\pi$
$888$	4.39005e6	0.186826
$889$	5.30917e7	2.25306
$890$	1.74552e7	0.738668
$891$	793881.	0.0335013
$892$	1.15505e7	0.486059
$893$	2.85364e6	0.119749
$894$	−2.57506e7	−1.07757
$895$	1.22385e6	0.0510707
$896$	−4.76524e7	−1.98296
$897$	1.50172e6	0.0623171
$898$	−2.52922e7	−1.04664
$899$	3.44237e7	1.42056
$900$	834410.	0.0343379
$901$	−5.60219e6	−0.229904
$902$	7.39711e6	0.302723
$903$	1.85311e7	0.756277
$904$	−90154.8	−0.00366917
$905$	1.60513e6	0.0651460
$906$	−6.00531e6	−0.243061
$907$	−1.69344e7	−0.683519	−0.341760	−	0.939787i	$-0.611023\pi$
$907$	−1.69344e7	−0.683519	−0.341760	+	0.939787i	$0.611023\pi$
$908$	1.36848e7	0.550837
$909$	4.19183e6	0.168265
$910$	−3.15507e7	−1.26301
$911$	−2.51901e7	−1.00562	−0.502810	−	0.864397i	$-0.667700\pi$
$911$	−2.51901e7	−1.00562	−0.502810	+	0.864397i	$0.667700\pi$
$912$	−4.47585e6	−0.178192
$913$	−1.40852e7	−0.559224
$914$	7.78149e6	0.308104
$915$	1.00874e7	0.398313
$916$	1.97870e7	0.779187
$917$	3.17013e6	0.124496
$918$	−864800.	−0.0338695
$919$	−1.56686e7	−0.611985	−0.305993	−	0.952034i	$-0.598988\pi$
$919$	−1.56686e7	−0.611985	−0.305993	+	0.952034i	$0.598988\pi$
$920$	606857.	0.0236383
$921$	1.49240e7	0.579746
$922$	−3.25122e6	−0.125956
$923$	2.32476e7	0.898200
$924$	4.38026e6	0.168780
$925$	2.82153e6	0.108425
$926$	4.96266e7	1.90190
$927$	−8.30041e6	−0.317249
$928$	−3.04175e7	−1.15945
$929$	3.86877e7	1.47073	0.735367	−	0.677669i	$-0.237010\pi$
$929$	3.86877e7	1.47073	0.735367	+	0.677669i	$0.237010\pi$
$930$	9.66873e6	0.366574
$931$	1.66117e7	0.628114
$932$	−3.04987e6	−0.115011
$933$	−4.26972e6	−0.160582
$934$	1.21146e7	0.454405
$935$	515374.	0.0192794
$936$	6.50022e6	0.242515
$937$	2.93005e7	1.09025	0.545125	−	0.838355i	$-0.316482\pi$
$937$	2.93005e7	1.09025	0.545125	+	0.838355i	$0.316482\pi$
$938$	−2.43334e7	−0.903015
$939$	2.12417e7	0.786185
$940$	3.02588e6	0.111695
$941$	1.10429e7	0.406547	0.203274	−	0.979122i	$-0.434842\pi$
$941$	1.10429e7	0.406547	0.203274	+	0.979122i	$0.434842\pi$
$942$	8.85593e6	0.325168
$943$	−1.97247e6	−0.0722322
$944$	−4.37402e7	−1.59754
$945$	4.44759e6	0.162011
$946$	−7.10848e6	−0.258255
$947$	−3.48768e7	−1.26375	−0.631877	−	0.775069i	$-0.717715\pi$
$947$	−3.48768e7	−1.26375	−0.631877	+	0.775069i	$0.717715\pi$
$948$	−1.05954e7	−0.382911
$949$	2.08783e7	0.752539
$950$	−1.69111e6	−0.0607945
$951$	−2.15160e7	−0.771455
$952$	4.49238e6	0.160651
$953$	2.10660e7	0.751362	0.375681	−	0.926749i	$-0.377409\pi$
$953$	2.10660e7	0.751362	0.375681	+	0.926749i	$0.377409\pi$
$954$	1.85454e7	0.659730
$955$	1.54253e7	0.547300
$956$	1.55662e6	0.0550856
$957$	−6.07421e6	−0.214393
$958$	5.83860e7	2.05539
$959$	5.95763e7	2.09183
$960$	670829.	0.0234928
$961$	9.45914e6	0.330402
$962$	−2.33462e7	−0.813353
$963$	−1.75767e7	−0.610760
$964$	−1.42950e7	−0.495439
$965$	−1.57607e6	−0.0544824
$966$	−3.43570e6	−0.118460
$967$	−1.78033e7	−0.612257	−0.306129	−	0.951990i	$-0.599034\pi$
$967$	−1.78033e7	−0.612257	−0.306129	+	0.951990i	$0.599034\pi$
$968$	1.58195e6	0.0542631
$969$	595857.	0.0203860
$970$	−2.38542e7	−0.814022
$971$	3.15481e7	1.07380	0.536902	−	0.843645i	$-0.319595\pi$
$971$	3.15481e7	1.07380	0.536902	+	0.843645i	$0.319595\pi$
$972$	973256.	0.0330416
$973$	−7.66320e7	−2.59495
$974$	−4.35066e7	−1.46946
$975$	4.17776e6	0.140745
$976$	−5.73755e7	−1.92798
$977$	−1.98879e7	−0.666581	−0.333290	−	0.942824i	$-0.608159\pi$
$977$	−1.98879e7	−0.666581	−0.333290	+	0.942824i	$0.608159\pi$
$978$	−3.00976e6	−0.100620
$979$	−1.21333e7	−0.404597
$980$	1.76143e7	0.585869
$981$	−1.66331e6	−0.0551824
$982$	4.39774e7	1.45529
$983$	1.08853e7	0.359301	0.179650	−	0.983731i	$-0.442503\pi$
$983$	1.08853e7	0.359301	0.179650	+	0.983731i	$0.442503\pi$
$984$	−8.53788e6	−0.281101
$985$	1.09027e7	0.358049
$986$	6.61683e6	0.216749
$987$	1.61286e7	0.526992
$988$	4.75705e6	0.155040
$989$	1.89551e6	0.0616218
$990$	−1.70609e6	−0.0553240
$991$	5.02174e7	1.62431	0.812156	−	0.583440i	$-0.198294\pi$
$991$	5.02174e7	1.62431	0.812156	+	0.583440i	$0.198294\pi$
$992$	−3.36556e7	−1.08587
$993$	−2.13058e7	−0.685685
$994$	−5.31869e7	−1.70741
$995$	−2.19348e7	−0.702386
$996$	−1.72677e7	−0.551551
$997$	1.50268e6	0.0478770	0.0239385	−	0.999713i	$-0.492379\pi$
$997$	1.50268e6	0.0478770	0.0239385	+	0.999713i	$0.492379\pi$
$998$	−5.40618e7	−1.71816
$999$	3.29103e6	0.104332

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.6.a.h.1.2	✓	7
3.2	odd	2		495.6.a.n.1.6		7
5.4	even	2		825.6.a.n.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.h.1.2	✓	7	1.1	even	1	trivial
495.6.a.n.1.6		7	3.2	odd	2
825.6.a.n.1.6		7	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:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	165.a (trivial)

\(f(q)\)	\(=\)	\(q-6.96291 q^{2} +9.00000 q^{3} +16.4822 q^{4} +25.0000 q^{5} -62.6662 q^{6} +244.038 q^{7} +108.049 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-6.96291 q^{2} +9.00000 q^{3} +16.4822 q^{4} +25.0000 q^{5} -62.6662 q^{6} +244.038 q^{7} +108.049 q^{8} +81.0000 q^{9} -174.073 q^{10} +121.000 q^{11} +148.340 q^{12} +742.713 q^{13} -1699.22 q^{14} +225.000 q^{15} -1279.77 q^{16} +170.372 q^{17} -563.996 q^{18} +388.599 q^{19} +412.054 q^{20} +2196.34 q^{21} -842.513 q^{22} +224.659 q^{23} +972.444 q^{24} +625.000 q^{25} -5171.45 q^{26} +729.000 q^{27} +4022.28 q^{28} -5577.78 q^{29} -1566.66 q^{30} -6171.57 q^{31} +5453.33 q^{32} +1089.00 q^{33} -1186.28 q^{34} +6100.95 q^{35} +1335.06 q^{36} +4514.45 q^{37} -2705.78 q^{38} +6684.41 q^{39} +2701.23 q^{40} -8779.82 q^{41} -15292.9 q^{42} +8437.24 q^{43} +1994.34 q^{44} +2025.00 q^{45} -1564.28 q^{46} +7343.40 q^{47} -11517.9 q^{48} +42747.5 q^{49} -4351.82 q^{50} +1533.34 q^{51} +12241.5 q^{52} -32882.2 q^{53} -5075.96 q^{54} +3025.00 q^{55} +26368.1 q^{56} +3497.39 q^{57} +38837.6 q^{58} +34178.2 q^{59} +3708.49 q^{60} +44832.7 q^{61} +42972.1 q^{62} +19767.1 q^{63} +2981.46 q^{64} +18567.8 q^{65} -7582.61 q^{66} +14320.3 q^{67} +2808.10 q^{68} +2021.94 q^{69} -42480.4 q^{70} +31300.9 q^{71} +8751.99 q^{72} +28110.8 q^{73} -31433.7 q^{74} +5625.00 q^{75} +6404.96 q^{76} +29528.6 q^{77} -46543.0 q^{78} -71426.8 q^{79} -31994.2 q^{80} +6561.00 q^{81} +61133.1 q^{82} -116407. q^{83} +36200.5 q^{84} +4259.29 q^{85} -58747.8 q^{86} -50200.1 q^{87} +13074.0 q^{88} -100275. q^{89} -14099.9 q^{90} +181250. q^{91} +3702.88 q^{92} -55544.1 q^{93} -51131.5 q^{94} +9714.98 q^{95} +49080.0 q^{96} +137036. q^{97} -297647. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 63 q^{3} + 195 q^{4} + 175 q^{5} + 9 q^{6} + 153 q^{8} + 567 q^{9}+O(q^{10}) \) 7 * q + q^2 + 63 * q^3 + 195 * q^4 + 175 * q^5 + 9 * q^6 + 153 * q^8 + 567 * q^9	\( 7 q + q^{2} + 63 q^{3} + 195 q^{4} + 175 q^{5} + 9 q^{6} + 153 q^{8} + 567 q^{9} + 25 q^{10} + 847 q^{11} + 1755 q^{12} + 1418 q^{13} + 2548 q^{14} + 1575 q^{15} + 3699 q^{16} + 630 q^{17} + 81 q^{18} + 2572 q^{19} + 4875 q^{20} + 121 q^{22} + 536 q^{23} + 1377 q^{24} + 4375 q^{25} - 7626 q^{26} + 5103 q^{27} - 11368 q^{28} - 1038 q^{29} + 225 q^{30} + 1872 q^{31} - 7523 q^{32} + 7623 q^{33} + 20790 q^{34} + 15795 q^{36} + 24298 q^{37} - 18952 q^{38} + 12762 q^{39} + 3825 q^{40} - 17658 q^{41} + 22932 q^{42} + 7244 q^{43} + 23595 q^{44} + 14175 q^{45} + 31016 q^{46} + 34560 q^{47} + 33291 q^{48} + 78735 q^{49} + 625 q^{50} + 5670 q^{51} + 110222 q^{52} - 10214 q^{53} + 729 q^{54} + 21175 q^{55} + 81124 q^{56} + 23148 q^{57} - 5718 q^{58} + 94676 q^{59} + 43875 q^{60} + 69538 q^{61} - 4208 q^{62} + 112339 q^{64} + 35450 q^{65} + 1089 q^{66} + 64908 q^{67} - 136010 q^{68} + 4824 q^{69} + 63700 q^{70} + 61816 q^{71} + 12393 q^{72} - 11890 q^{73} - 124050 q^{74} + 39375 q^{75} - 47216 q^{76} - 68634 q^{78} + 18928 q^{79} + 92475 q^{80} + 45927 q^{81} + 36398 q^{82} + 17492 q^{83} - 102312 q^{84} + 15750 q^{85} - 216688 q^{86} - 9342 q^{87} + 18513 q^{88} + 25302 q^{89} + 2025 q^{90} + 3392 q^{91} - 27408 q^{92} + 16848 q^{93} - 30800 q^{94} + 64300 q^{95} - 67707 q^{96} - 172546 q^{97} - 615271 q^{98} + 68607 q^{99}+O(q^{100}) \) 7 * q + q^2 + 63 * q^3 + 195 * q^4 + 175 * q^5 + 9 * q^6 + 153 * q^8 + 567 * q^9 + 25 * q^10 + 847 * q^11 + 1755 * q^12 + 1418 * q^13 + 2548 * q^14 + 1575 * q^15 + 3699 * q^16 + 630 * q^17 + 81 * q^18 + 2572 * q^19 + 4875 * q^20 + 121 * q^22 + 536 * q^23 + 1377 * q^24 + 4375 * q^25 - 7626 * q^26 + 5103 * q^27 - 11368 * q^28 - 1038 * q^29 + 225 * q^30 + 1872 * q^31 - 7523 * q^32 + 7623 * q^33 + 20790 * q^34 + 15795 * q^36 + 24298 * q^37 - 18952 * q^38 + 12762 * q^39 + 3825 * q^40 - 17658 * q^41 + 22932 * q^42 + 7244 * q^43 + 23595 * q^44 + 14175 * q^45 + 31016 * q^46 + 34560 * q^47 + 33291 * q^48 + 78735 * q^49 + 625 * q^50 + 5670 * q^51 + 110222 * q^52 - 10214 * q^53 + 729 * q^54 + 21175 * q^55 + 81124 * q^56 + 23148 * q^57 - 5718 * q^58 + 94676 * q^59 + 43875 * q^60 + 69538 * q^61 - 4208 * q^62 + 112339 * q^64 + 35450 * q^65 + 1089 * q^66 + 64908 * q^67 - 136010 * q^68 + 4824 * q^69 + 63700 * q^70 + 61816 * q^71 + 12393 * q^72 - 11890 * q^73 - 124050 * q^74 + 39375 * q^75 - 47216 * q^76 - 68634 * q^78 + 18928 * q^79 + 92475 * q^80 + 45927 * q^81 + 36398 * q^82 + 17492 * q^83 - 102312 * q^84 + 15750 * q^85 - 216688 * q^86 - 9342 * q^87 + 18513 * q^88 + 25302 * q^89 + 2025 * q^90 + 3392 * q^91 - 27408 * q^92 + 16848 * q^93 - 30800 * q^94 + 64300 * q^95 - 67707 * q^96 - 172546 * q^97 - 615271 * q^98 + 68607 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more