LMFDB - Embedded newform 161.6.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [161,6,Mod(1,161)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(161, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("161.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 161 = 7 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	161.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:	$25.8217949899$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 187 x^{8} + 322 x^{7} + 11471 x^{6} - 16782 x^{5} - 253209 x^{4} + 251398 x^{3} + \cdots - 6912 $ x^10 - 2x^9 - 187x^8 + 322x^7 + 11471x^6 - 16782x^5 - 253209x^4 + 251398x^3 + 1470116x^2 + 974056*x - 6912
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-5.68327$ of defining polynomial
Character	$\chi$	$=$	161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-6.68327 q^{2} +5.69706 q^{3} +12.6661 q^{4} -77.3301 q^{5} -38.0750 q^{6} +49.0000 q^{7} +129.213 q^{8} -210.543 q^{9} +516.818 q^{10} +359.837 q^{11} +72.1598 q^{12} +811.735 q^{13} -327.480 q^{14} -440.555 q^{15} -1268.89 q^{16} +260.323 q^{17} +1407.12 q^{18} -1023.19 q^{19} -979.474 q^{20} +279.156 q^{21} -2404.89 q^{22} +529.000 q^{23} +736.137 q^{24} +2854.95 q^{25} -5425.05 q^{26} -2583.87 q^{27} +620.641 q^{28} +7814.84 q^{29} +2944.35 q^{30} +249.001 q^{31} +4345.48 q^{32} +2050.02 q^{33} -1739.81 q^{34} -3789.18 q^{35} -2666.77 q^{36} -9566.88 q^{37} +6838.23 q^{38} +4624.50 q^{39} -9992.09 q^{40} -2811.95 q^{41} -1865.68 q^{42} -7852.72 q^{43} +4557.75 q^{44} +16281.4 q^{45} -3535.45 q^{46} -3119.88 q^{47} -7228.92 q^{48} +2401.00 q^{49} -19080.4 q^{50} +1483.08 q^{51} +10281.5 q^{52} -38401.2 q^{53} +17268.7 q^{54} -27826.3 q^{55} +6331.46 q^{56} -5829.15 q^{57} -52228.7 q^{58} +16911.6 q^{59} -5580.13 q^{60} -41916.5 q^{61} -1664.14 q^{62} -10316.6 q^{63} +11562.3 q^{64} -62771.6 q^{65} -13700.8 q^{66} -59003.5 q^{67} +3297.29 q^{68} +3013.75 q^{69} +25324.1 q^{70} -17855.0 q^{71} -27205.1 q^{72} -61628.3 q^{73} +63938.1 q^{74} +16264.8 q^{75} -12959.8 q^{76} +17632.0 q^{77} -30906.8 q^{78} +18843.8 q^{79} +98123.1 q^{80} +36441.6 q^{81} +18793.0 q^{82} +95284.6 q^{83} +3535.83 q^{84} -20130.8 q^{85} +52481.9 q^{86} +44521.6 q^{87} +46495.8 q^{88} -80436.2 q^{89} -108813. q^{90} +39775.0 q^{91} +6700.39 q^{92} +1418.57 q^{93} +20851.0 q^{94} +79123.1 q^{95} +24756.5 q^{96} +55807.5 q^{97} -16046.5 q^{98} -75761.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{2} - 2 q^{3} + 64 q^{4} - 92 q^{5} - 162 q^{6} + 490 q^{7} - 462 q^{8} + 140 q^{9} - 566 q^{10} - 750 q^{11} + 2348 q^{12} + 124 q^{13} - 392 q^{14} - 1306 q^{15} - 524 q^{16} - 2500 q^{17}+ \cdots - 460326 q^{99}+O(q^{100}) $ 10 * q - 8 * q^2 - 2 * q^3 + 64 * q^4 - 92 * q^5 - 162 * q^6 + 490 * q^7 - 462 * q^8 + 140 * q^9 - 566 * q^10 - 750 * q^11 + 2348 * q^12 + 124 * q^13 - 392 * q^14 - 1306 * q^15 - 524 * q^16 - 2500 * q^17 - 3646 * q^18 - 3888 * q^19 - 538 * q^20 - 98 * q^21 + 2454 * q^22 + 5290 * q^23 - 8208 * q^24 - 5050 * q^25 + 202 * q^26 - 2906 * q^27 + 3136 * q^28 - 4190 * q^29 - 3200 * q^30 - 28126 * q^31 - 5582 * q^32 - 27780 * q^33 - 10088 * q^34 - 4508 * q^35 - 9720 * q^36 - 34096 * q^37 + 6394 * q^38 - 23128 * q^39 - 26846 * q^40 + 13612 * q^41 - 7938 * q^42 - 36330 * q^43 - 20998 * q^44 - 27538 * q^45 - 4232 * q^46 - 29486 * q^47 - 51088 * q^48 + 24010 * q^49 + 5632 * q^50 - 21224 * q^51 - 87640 * q^52 - 67608 * q^53 - 66042 * q^54 - 19776 * q^55 - 22638 * q^56 - 135614 * q^57 - 197486 * q^58 + 4004 * q^59 - 98584 * q^60 + 13134 * q^61 - 9854 * q^62 + 6860 * q^63 - 107420 * q^64 - 66994 * q^65 + 80436 * q^66 - 95740 * q^67 - 57992 * q^68 - 1058 * q^69 - 27734 * q^70 - 66740 * q^71 - 52746 * q^72 + 46044 * q^73 + 99746 * q^74 + 68958 * q^75 - 155098 * q^76 - 36750 * q^77 + 175854 * q^78 - 199082 * q^79 + 237246 * q^80 + 149810 * q^81 - 52286 * q^82 + 24150 * q^83 + 115052 * q^84 - 50060 * q^85 + 219598 * q^86 + 160446 * q^87 + 149210 * q^88 - 98804 * q^89 + 204470 * q^90 + 6076 * q^91 + 33856 * q^92 + 43906 * q^93 + 214806 * q^94 + 228204 * q^95 + 284944 * q^96 - 63914 * q^97 - 19208 * q^98 - 460326 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−6.68327	−1.18145	−0.590723	−	0.806874i	$-0.701158\pi$
$2$	−6.68327	−1.18145	−0.590723	+	0.806874i	$0.701158\pi$
$3$	5.69706	0.365467	0.182733	−	0.983163i	$-0.441505\pi$
$3$	5.69706	0.365467	0.182733	+	0.983163i	$0.441505\pi$
$4$	12.6661	0.395817
$5$	−77.3301	−1.38332	−0.691662	−	0.722222i	$-0.743121\pi$
$5$	−77.3301	−1.38332	−0.691662	+	0.722222i	$0.743121\pi$
$6$	−38.0750	−0.431780
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	129.213	0.713810
$9$	−210.543	−0.866434
$10$	516.818	1.63432
$11$	359.837	0.896653	0.448326	−	0.893870i	$-0.352020\pi$
$11$	359.837	0.896653	0.448326	+	0.893870i	$0.352020\pi$
$12$	72.1598	0.144658
$13$	811.735	1.33216	0.666079	−	0.745881i	$-0.267971\pi$
$13$	811.735	1.33216	0.666079	+	0.745881i	$0.267971\pi$
$14$	−327.480	−0.446545
$15$	−440.555	−0.505559
$16$	−1268.89	−1.23915
$17$	260.323	0.218469	0.109235	−	0.994016i	$-0.465160\pi$
$17$	260.323	0.218469	0.109235	+	0.994016i	$0.465160\pi$
$18$	1407.12	1.02365
$19$	−1023.19	−0.650235	−0.325118	−	0.945674i	$-0.605404\pi$
$19$	−1023.19	−0.650235	−0.325118	+	0.945674i	$0.605404\pi$
$20$	−979.474	−0.547543
$21$	279.156	0.138133
$22$	−2404.89	−1.05935
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	736.137	0.260874
$25$	2854.95	0.913583
$26$	−5425.05	−1.57387
$27$	−2583.87	−0.682120
$28$	620.641	0.149605
$29$	7814.84	1.72554	0.862770	−	0.505596i	$-0.168728\pi$
$29$	7814.84	1.72554	0.862770	+	0.505596i	$0.168728\pi$
$30$	2944.35	0.597291
$31$	249.001	0.0465368	0.0232684	−	0.999729i	$-0.492593\pi$
$31$	249.001	0.0465368	0.0232684	+	0.999729i	$0.492593\pi$
$32$	4345.48	0.750175
$33$	2050.02	0.327697
$34$	−1739.81	−0.258110
$35$	−3789.18	−0.522847
$36$	−2666.77	−0.342949
$37$	−9566.88	−1.14886	−0.574429	−	0.818555i	$-0.694776\pi$
$37$	−9566.88	−1.14886	−0.574429	+	0.818555i	$0.694776\pi$
$38$	6838.23	0.768218
$39$	4624.50	0.486860
$40$	−9992.09	−0.987430
$41$	−2811.95	−0.261245	−0.130622	−	0.991432i	$-0.541698\pi$
$41$	−2811.95	−0.261245	−0.130622	+	0.991432i	$0.541698\pi$
$42$	−1865.68	−0.163197
$43$	−7852.72	−0.647663	−0.323831	−	0.946115i	$-0.604971\pi$
$43$	−7852.72	−0.647663	−0.323831	+	0.946115i	$0.604971\pi$
$44$	4557.75	0.354910
$45$	16281.4	1.19856
$46$	−3535.45	−0.246349
$47$	−3119.88	−0.206013	−0.103006	−	0.994681i	$-0.532846\pi$
$47$	−3119.88	−0.206013	−0.103006	+	0.994681i	$0.532846\pi$
$48$	−7228.92	−0.452867
$49$	2401.00	0.142857
$50$	−19080.4	−1.07935
$51$	1483.08	0.0798433
$52$	10281.5	0.527291
$53$	−38401.2	−1.87782	−0.938912	−	0.344157i	$-0.888165\pi$
$53$	−38401.2	−1.87782	−0.938912	+	0.344157i	$0.888165\pi$
$54$	17268.7	0.805888
$55$	−27826.3	−1.24036
$56$	6331.46	0.269795
$57$	−5829.15	−0.237639
$58$	−52228.7	−2.03863
$59$	16911.6	0.632492	0.316246	−	0.948677i	$-0.397577\pi$
$59$	16911.6	0.632492	0.316246	+	0.948677i	$0.397577\pi$
$60$	−5580.13	−0.200109
$61$	−41916.5	−1.44232	−0.721158	−	0.692771i	$-0.756390\pi$
$61$	−41916.5	−1.44232	−0.721158	+	0.692771i	$0.756390\pi$
$62$	−1664.14	−0.0549808
$63$	−10316.6	−0.327481
$64$	11562.3	0.352854
$65$	−62771.6	−1.84281
$66$	−13700.8	−0.387156
$67$	−59003.5	−1.60580	−0.802899	−	0.596115i	$-0.796710\pi$
$67$	−59003.5	−1.60580	−0.802899	+	0.596115i	$0.796710\pi$
$68$	3297.29	0.0864739
$69$	3013.75	0.0762051
$70$	25324.1	0.617716
$71$	−17855.0	−0.420353	−0.210177	−	0.977663i	$-0.567404\pi$
$71$	−17855.0	−0.420353	−0.210177	+	0.977663i	$0.567404\pi$
$72$	−27205.1	−0.618470
$73$	−61628.3	−1.35355	−0.676773	−	0.736191i	$-0.736622\pi$
$73$	−61628.3	−1.35355	−0.676773	+	0.736191i	$0.736622\pi$
$74$	63938.1	1.35731
$75$	16264.8	0.333884
$76$	−12959.8	−0.257374
$77$	17632.0	0.338903
$78$	−30906.8	−0.575199
$79$	18843.8	0.339704	0.169852	−	0.985470i	$-0.445671\pi$
$79$	18843.8	0.339704	0.169852	+	0.985470i	$0.445671\pi$
$80$	98123.1	1.71414
$81$	36441.6	0.617142
$82$	18793.0	0.308647
$83$	95284.6	1.51820	0.759098	−	0.650977i	$-0.225640\pi$
$83$	95284.6	1.51820	0.759098	+	0.650977i	$0.225640\pi$
$84$	3535.83	0.0546756
$85$	−20130.8	−0.302214
$86$	52481.9	0.765179
$87$	44521.6	0.630628
$88$	46495.8	0.640040
$89$	−80436.2	−1.07641	−0.538203	−	0.842815i	$-0.680897\pi$
$89$	−80436.2	−1.07641	−0.538203	+	0.842815i	$0.680897\pi$
$90$	−108813.	−1.41603
$91$	39775.0	0.503508
$92$	6700.39	0.0825335
$93$	1418.57	0.0170077
$94$	20851.0	0.243393
$95$	79123.1	0.899485
$96$	24756.5	0.274164
$97$	55807.5	0.602231	0.301116	−	0.953588i	$-0.402641\pi$
$97$	55807.5	0.602231	0.301116	+	0.953588i	$0.402641\pi$
$98$	−16046.5	−0.168778
$99$	−75761.4	−0.776891
$100$	36161.2	0.361612
$101$	1770.69	0.0172718	0.00863592	−	0.999963i	$-0.497251\pi$
$101$	1770.69	0.0172718	0.00863592	+	0.999963i	$0.497251\pi$
$102$	−9911.82	−0.0943307
$103$	34333.3	0.318876	0.159438	−	0.987208i	$-0.449032\pi$
$103$	34333.3	0.318876	0.159438	+	0.987208i	$0.449032\pi$
$104$	104887.	0.950908
$105$	−21587.2	−0.191083
$106$	256646.	2.21855
$107$	126238.	1.06593	0.532966	−	0.846137i	$-0.321077\pi$
$107$	126238.	1.06593	0.532966	+	0.846137i	$0.321077\pi$
$108$	−32727.6	−0.269994
$109$	30610.6	0.246777	0.123389	−	0.992358i	$-0.460624\pi$
$109$	30610.6	0.246777	0.123389	+	0.992358i	$0.460624\pi$
$110$	185970.	1.46542
$111$	−54503.1	−0.419869
$112$	−62175.4	−0.468353
$113$	−258837.	−1.90691	−0.953457	−	0.301530i	$-0.902503\pi$
$113$	−258837.	−1.90691	−0.953457	+	0.301530i	$0.902503\pi$
$114$	38957.8	0.280758
$115$	−40907.6	−0.288443
$116$	98983.9	0.682998
$117$	−170905.	−1.15423
$118$	−113025.	−0.747256
$119$	12755.8	0.0825737
$120$	−56925.6	−0.360873
$121$	−31568.2	−0.196014
$122$	280139.	1.70402
$123$	−16019.8	−0.0954762
$124$	3153.88	0.0184201
$125$	20883.2	0.119542
$126$	68948.9	0.386902
$127$	47117.9	0.259225	0.129612	−	0.991565i	$-0.458627\pi$
$127$	47117.9	0.259225	0.129612	+	0.991565i	$0.458627\pi$
$128$	−216329.	−1.16705
$129$	−44737.4	−0.236699
$130$	419519.	2.17718
$131$	−86926.9	−0.442564	−0.221282	−	0.975210i	$-0.571024\pi$
$131$	−86926.9	−0.442564	−0.221282	+	0.975210i	$0.571024\pi$
$132$	25965.8	0.129708
$133$	−50136.1	−0.245766
$134$	394337.	1.89717
$135$	199811.	0.943592
$136$	33637.3	0.155946
$137$	44143.4	0.200939	0.100470	−	0.994940i	$-0.467966\pi$
$137$	44143.4	0.200939	0.100470	+	0.994940i	$0.467966\pi$
$138$	−20141.7	−0.0900323
$139$	−283427.	−1.24424	−0.622119	−	0.782922i	$-0.713728\pi$
$139$	−283427.	−1.24424	−0.622119	+	0.782922i	$0.713728\pi$
$140$	−47994.2	−0.206952
$141$	−17774.2	−0.0752907
$142$	119330.	0.496625
$143$	292092.	1.19448
$144$	267156.	1.07364
$145$	−604323.	−2.38698
$146$	411879.	1.59914
$147$	13678.6	0.0522095
$148$	−121175.	−0.454737
$149$	267917.	0.988631	0.494315	−	0.869283i	$-0.335419\pi$
$149$	267917.	0.988631	0.494315	+	0.869283i	$0.335419\pi$
$150$	−108702.	−0.394467
$151$	−353403.	−1.26133	−0.630663	−	0.776057i	$-0.717217\pi$
$151$	−353403.	−1.26133	−0.630663	+	0.776057i	$0.717217\pi$
$152$	−132209.	−0.464145
$153$	−54809.4	−0.189289
$154$	−117840.	−0.400396
$155$	−19255.3	−0.0643755
$156$	58574.6	0.192707
$157$	−235308.	−0.761881	−0.380940	−	0.924600i	$-0.624400\pi$
$157$	−235308.	−0.761881	−0.380940	+	0.924600i	$0.624400\pi$
$158$	−125938.	−0.401342
$159$	−218774.	−0.686282
$160$	−336036.	−1.03773
$161$	25921.0	0.0788110
$162$	−243549.	−0.729121
$163$	392817.	1.15803	0.579016	−	0.815316i	$-0.303437\pi$
$163$	392817.	1.15803	0.579016	+	0.815316i	$0.303437\pi$
$164$	−35616.5	−0.103405
$165$	−158528.	−0.453311
$166$	−636813.	−1.79367
$167$	213017.	0.591049	0.295524	−	0.955335i	$-0.404506\pi$
$167$	213017.	0.591049	0.295524	+	0.955335i	$0.404506\pi$
$168$	36070.7	0.0986011
$169$	287620.	0.774646
$170$	134540.	0.357050
$171$	215425.	0.563386
$172$	−99463.7	−0.256356
$173$	−234406.	−0.595461	−0.297730	−	0.954650i	$-0.596230\pi$
$173$	−234406.	−0.595461	−0.297730	+	0.954650i	$0.596230\pi$
$174$	−297550.	−0.745053
$175$	139892.	0.345302
$176$	−456592.	−1.11108
$177$	96346.6	0.231155
$178$	537577.	1.27172
$179$	−426482.	−0.994873	−0.497436	−	0.867500i	$-0.665725\pi$
$179$	−426482.	−0.994873	−0.497436	+	0.867500i	$0.665725\pi$
$180$	206222.	0.474410
$181$	−377682.	−0.856899	−0.428450	−	0.903566i	$-0.640940\pi$
$181$	−377682.	−0.856899	−0.428450	+	0.903566i	$0.640940\pi$
$182$	−265827.	−0.594869
$183$	−238801.	−0.527118
$184$	68353.9	0.148840
$185$	739808.	1.58924
$186$	−9480.72	−0.0200937
$187$	93674.0	0.195891
$188$	−39516.9	−0.0815433
$189$	−126609.	−0.257817
$190$	−528801.	−1.06269
$191$	576023.	1.14250	0.571250	−	0.820776i	$-0.306459\pi$
$191$	576023.	1.14250	0.571250	+	0.820776i	$0.306459\pi$
$192$	65871.3	0.128956
$193$	42519.4	0.0821662	0.0410831	−	0.999156i	$-0.486919\pi$
$193$	42519.4	0.0821662	0.0410831	+	0.999156i	$0.486919\pi$
$194$	−372977.	−0.711504
$195$	−357614.	−0.673484
$196$	30411.4	0.0565453
$197$	−406471.	−0.746215	−0.373107	−	0.927788i	$-0.621708\pi$
$197$	−406471.	−0.746215	−0.373107	+	0.927788i	$0.621708\pi$
$198$	506334.	0.917855
$199$	447282.	0.800660	0.400330	−	0.916371i	$-0.368895\pi$
$199$	447282.	0.800660	0.400330	+	0.916371i	$0.368895\pi$
$200$	368898.	0.652125
$201$	−336147.	−0.586866
$202$	−11834.0	−0.0204058
$203$	382927.	0.652193
$204$	18784.9	0.0316033
$205$	217448.	0.361386
$206$	−229459.	−0.376735
$207$	−111377.	−0.180664
$208$	−1.03000e6	−1.65074
$209$	−368180.	−0.583035
$210$	144273.	0.225755
$211$	−608033.	−0.940202	−0.470101	−	0.882613i	$-0.655783\pi$
$211$	−608033.	−0.940202	−0.470101	+	0.882613i	$0.655783\pi$
$212$	−486395.	−0.743275
$213$	−101721.	−0.153625
$214$	−843681.	−1.25934
$215$	607252.	0.895927
$216$	−333870.	−0.486904
$217$	12201.0	0.0175893
$218$	−204579.	−0.291554
$219$	−351100.	−0.494676
$220$	−352451.	−0.490956
$221$	211314.	0.291036
$222$	364259.	0.496053
$223$	335834.	0.452233	0.226117	−	0.974100i	$-0.427397\pi$
$223$	335834.	0.452233	0.226117	+	0.974100i	$0.427397\pi$
$224$	212928.	0.283539
$225$	−601091.	−0.791560
$226$	1.72988e6	2.25292
$227$	−779261.	−1.00373	−0.501867	−	0.864945i	$-0.667353\pi$
$227$	−779261.	−1.00373	−0.501867	+	0.864945i	$0.667353\pi$
$228$	−73832.9	−0.0940617
$229$	646862.	0.815123	0.407561	−	0.913178i	$-0.366379\pi$
$229$	646862.	0.815123	0.407561	+	0.913178i	$0.366379\pi$
$230$	273397.	0.340780
$231$	100451.	0.123858
$232$	1.00978e6	1.23171
$233$	1.13590e6	1.37073	0.685364	−	0.728201i	$-0.259643\pi$
$233$	1.13590e6	1.37073	0.685364	+	0.728201i	$0.259643\pi$
$234$	1.14221e6	1.36366
$235$	241261.	0.284982
$236$	214205.	0.250351
$237$	107354.	0.124150
$238$	−85250.8	−0.0975565
$239$	−450126.	−0.509729	−0.254865	−	0.966977i	$-0.582031\pi$
$239$	−450126.	−0.509729	−0.254865	+	0.966977i	$0.582031\pi$
$240$	559013.	0.626461
$241$	−954805.	−1.05894	−0.529471	−	0.848328i	$-0.677610\pi$
$241$	−954805.	−1.05894	−0.529471	+	0.848328i	$0.677610\pi$
$242$	210979.	0.231580
$243$	835490.	0.907664
$244$	−530920.	−0.570893
$245$	−185670.	−0.197618
$246$	107065.	0.112800
$247$	−830555.	−0.866216
$248$	32174.3	0.0332185
$249$	542843.	0.554850
$250$	−139568.	−0.141233
$251$	−1.40972e6	−1.41237	−0.706187	−	0.708025i	$-0.749586\pi$
$251$	−1.40972e6	−1.41237	−0.706187	+	0.708025i	$0.749586\pi$
$252$	−130672.	−0.129623
$253$	190354.	0.186965
$254$	−314902.	−0.306260
$255$	−114687.	−0.110449
$256$	1.07579e6	1.02596
$257$	1.24666e6	1.17738	0.588688	−	0.808360i	$-0.299645\pi$
$257$	1.24666e6	1.17738	0.588688	+	0.808360i	$0.299645\pi$
$258$	298993.	0.279648
$259$	−468777.	−0.434227
$260$	−795073.	−0.729414
$261$	−1.64536e6	−1.49507
$262$	580956.	0.522866
$263$	−1.83332e6	−1.63437	−0.817183	−	0.576379i	$-0.804465\pi$
$263$	−1.83332e6	−1.63437	−0.817183	+	0.576379i	$0.804465\pi$
$264$	264890.	0.233913
$265$	2.96957e6	2.59764
$266$	335073.	0.290359
$267$	−458250.	−0.393391
$268$	−747347.	−0.635602
$269$	1.39805e6	1.17799	0.588994	−	0.808138i	$-0.299524\pi$
$269$	1.39805e6	1.17799	0.588994	+	0.808138i	$0.299524\pi$
$270$	−1.33539e6	−1.11480
$271$	−1.80572e6	−1.49358	−0.746789	−	0.665061i	$-0.768405\pi$
$271$	−1.80572e6	−1.49358	−0.746789	+	0.665061i	$0.768405\pi$
$272$	−330320.	−0.270716
$273$	226601.	0.184016
$274$	−295022.	−0.237399
$275$	1.02732e6	0.819167
$276$	38172.5	0.0301633
$277$	−1.91234e6	−1.49750	−0.748748	−	0.662855i	$-0.769344\pi$
$277$	−1.91234e6	−1.49750	−0.748748	+	0.662855i	$0.769344\pi$
$278$	1.89422e6	1.47000
$279$	−52425.5	−0.0403211
$280$	−489613.	−0.373214
$281$	−1.60891e6	−1.21553	−0.607766	−	0.794116i	$-0.707934\pi$
$281$	−1.60891e6	−1.21553	−0.607766	+	0.794116i	$0.707934\pi$
$282$	118790.	0.0889520
$283$	−300506.	−0.223042	−0.111521	−	0.993762i	$-0.535572\pi$
$283$	−300506.	−0.223042	−0.111521	+	0.993762i	$0.535572\pi$
$284$	−226154.	−0.166383
$285$	450769.	0.328732
$286$	−1.95213e6	−1.41122
$287$	−137785.	−0.0987412
$288$	−914912.	−0.649977
$289$	−1.35209e6	−0.952271
$290$	4.03885e6	2.82009
$291$	317939.	0.220095
$292$	−780593.	−0.535757
$293$	−2.17093e6	−1.47733	−0.738663	−	0.674075i	$-0.764542\pi$
$293$	−2.17093e6	−1.47733	−0.738663	+	0.674075i	$0.764542\pi$
$294$	−91418.1	−0.0616828
$295$	−1.30778e6	−0.874941
$296$	−1.23617e6	−0.820066
$297$	−929771.	−0.611624
$298$	−1.79056e6	−1.16801
$299$	429408.	0.277774
$300$	206012.	0.132157
$301$	−384783.	−0.244794
$302$	2.36189e6	1.49019
$303$	10087.7	0.00631228
$304$	1.29831e6	0.805736
$305$	3.24141e6	1.99519
$306$	366306.	0.223635
$307$	−727722.	−0.440676	−0.220338	−	0.975424i	$-0.570716\pi$
$307$	−727722.	−0.440676	−0.220338	+	0.975424i	$0.570716\pi$
$308$	223330.	0.134144
$309$	195599.	0.116539
$310$	128688.	0.0760562
$311$	2.99933e6	1.75842	0.879212	−	0.476431i	$-0.158070\pi$
$311$	2.99933e6	1.75842	0.879212	+	0.476431i	$0.158070\pi$
$312$	597548.	0.347525
$313$	41859.7	0.0241510	0.0120755	−	0.999927i	$-0.496156\pi$
$313$	41859.7	0.0241510	0.0120755	+	0.999927i	$0.496156\pi$
$314$	1.57263e6	0.900122
$315$	797786.	0.453013
$316$	238678.	0.134461
$317$	−1.76976e6	−0.989159	−0.494579	−	0.869133i	$-0.664678\pi$
$317$	−1.76976e6	−0.989159	−0.494579	+	0.869133i	$0.664678\pi$
$318$	1.46213e6	0.810806
$319$	2.81207e6	1.54721
$320$	−894116.	−0.488111
$321$	719184.	0.389563
$322$	−173237.	−0.0931111
$323$	−266359.	−0.142057
$324$	461575.	0.244275
$325$	2.31746e6	1.21704
$326$	−2.62530e6	−1.36815
$327$	174390.	0.0901889
$328$	−363341.	−0.186479
$329$	−152874.	−0.0778654
$330$	1.05949e6	0.535562
$331$	1.69623e6	0.850970	0.425485	−	0.904965i	$-0.360103\pi$
$331$	1.69623e6	0.850970	0.425485	+	0.904965i	$0.360103\pi$
$332$	1.20689e6	0.600927
$333$	2.01424e6	0.995409
$334$	−1.42365e6	−0.698293
$335$	4.56275e6	2.22134
$336$	−354217.	−0.171167
$337$	−1.78727e6	−0.857264	−0.428632	−	0.903479i	$-0.641004\pi$
$337$	−1.78727e6	−0.857264	−0.428632	+	0.903479i	$0.641004\pi$
$338$	−1.92225e6	−0.915203
$339$	−1.47461e6	−0.696913
$340$	−254980.	−0.119621
$341$	89599.8	0.0417274
$342$	−1.43974e6	−0.665611
$343$	117649.	0.0539949
$344$	−1.01468e6	−0.462308
$345$	−233053.	−0.105416
$346$	1.56660e6	0.703506
$347$	−87985.7	−0.0392273	−0.0196137	−	0.999808i	$-0.506244\pi$
$347$	−87985.7	−0.0392273	−0.0196137	+	0.999808i	$0.506244\pi$
$348$	563917.	0.249613
$349$	−1.95914e6	−0.860997	−0.430499	−	0.902591i	$-0.641662\pi$
$349$	−1.95914e6	−0.860997	−0.430499	+	0.902591i	$0.641662\pi$
$350$	−934939.	−0.407956
$351$	−2.09741e6	−0.908691
$352$	1.56366e6	0.672646
$353$	−1.39184e6	−0.594499	−0.297250	−	0.954800i	$-0.596069\pi$
$353$	−1.39184e6	−0.594499	−0.297250	+	0.954800i	$0.596069\pi$
$354$	−643911.	−0.273097
$355$	1.38073e6	0.581485
$356$	−1.01882e6	−0.426060
$357$	72670.8	0.0301779
$358$	2.85029e6	1.17539
$359$	2.72511e6	1.11596	0.557979	−	0.829855i	$-0.311577\pi$
$359$	2.72511e6	1.11596	0.557979	+	0.829855i	$0.311577\pi$
$360$	2.10377e6	0.855543
$361$	−1.42919e6	−0.577194
$362$	2.52415e6	1.01238
$363$	−179846.	−0.0716365
$364$	503796.	0.199297
$365$	4.76573e6	1.87239
$366$	1.59597e6	0.622763
$367$	−4.33344e6	−1.67945	−0.839726	−	0.543011i	$-0.817284\pi$
$367$	−4.33344e6	−1.67945	−0.839726	+	0.543011i	$0.817284\pi$
$368$	−671240.	−0.258380
$369$	592037.	0.226351
$370$	−4.94434e6	−1.87760
$371$	−1.88166e6	−0.709751
$372$	17967.9	0.00673192
$373$	2.45591e6	0.913990	0.456995	−	0.889469i	$-0.348926\pi$
$373$	2.45591e6	0.913990	0.456995	+	0.889469i	$0.348926\pi$
$374$	−626049.	−0.231435
$375$	118973.	0.0436887
$376$	−403131.	−0.147054
$377$	6.34358e6	2.29869
$378$	846165.	0.304597
$379$	−2.84350e6	−1.01684	−0.508422	−	0.861108i	$-0.669771\pi$
$379$	−2.84350e6	−1.01684	−0.508422	+	0.861108i	$0.669771\pi$
$380$	1.00218e6	0.356032
$381$	268433.	0.0947380
$382$	−3.84972e6	−1.34980
$383$	5.15142e6	1.79444	0.897222	−	0.441581i	$-0.145582\pi$
$383$	5.15142e6	1.79444	0.897222	+	0.441581i	$0.145582\pi$
$384$	−1.23244e6	−0.426519
$385$	−1.36349e6	−0.468812
$386$	−284168.	−0.0970751
$387$	1.65334e6	0.561157
$388$	706866.	0.238373
$389$	−1.29287e6	−0.433191	−0.216595	−	0.976261i	$-0.569495\pi$
$389$	−1.29287e6	−0.433191	−0.216595	+	0.976261i	$0.569495\pi$
$390$	2.39003e6	0.795686
$391$	137711.	0.0455540
$392$	310242.	0.101973
$393$	−495228.	−0.161742
$394$	2.71656e6	0.881613
$395$	−1.45719e6	−0.469920
$396$	−959604.	−0.307506
$397$	2.10419e6	0.670051	0.335026	−	0.942209i	$-0.391255\pi$
$397$	2.10419e6	0.670051	0.335026	+	0.942209i	$0.391255\pi$
$398$	−2.98931e6	−0.945938
$399$	−285629.	−0.0898192
$400$	−3.62260e6	−1.13206
$401$	−2.32723e6	−0.722735	−0.361367	−	0.932424i	$-0.617690\pi$
$401$	−2.32723e6	−0.722735	−0.361367	+	0.932424i	$0.617690\pi$
$402$	2.24656e6	0.693351
$403$	202123.	0.0619944
$404$	22427.8	0.00683649
$405$	−2.81803e6	−0.853707
$406$	−2.55921e6	−0.770531
$407$	−3.44252e6	−1.03013
$408$	191634.	0.0569930
$409$	1.89123e6	0.559032	0.279516	−	0.960141i	$-0.409826\pi$
$409$	1.89123e6	0.559032	0.279516	+	0.960141i	$0.409826\pi$
$410$	−1.45327e6	−0.426958
$411$	251488.	0.0734365
$412$	434870.	0.126217
$413$	828670.	0.239060
$414$	744366.	0.213445
$415$	−7.36837e6	−2.10016
$416$	3.52738e6	0.999352
$417$	−1.61470e6	−0.454728
$418$	2.46065e6	0.688825
$419$	4.15812e6	1.15708	0.578538	−	0.815655i	$-0.303623\pi$
$419$	4.15812e6	1.15708	0.578538	+	0.815655i	$0.303623\pi$
$420$	−273426.	−0.0756340
$421$	−4.26633e6	−1.17314	−0.586569	−	0.809899i	$-0.699522\pi$
$421$	−4.26633e6	−1.17314	−0.586569	+	0.809899i	$0.699522\pi$
$422$	4.06365e6	1.11080
$423$	656871.	0.178496
$424$	−4.96195e6	−1.34041
$425$	743210.	0.199590
$426$	679831.	0.181500
$427$	−2.05391e6	−0.545144
$428$	1.59894e6	0.421914
$429$	1.66407e6	0.436544
$430$	−4.05843e6	−1.05849
$431$	−5.57140e6	−1.44468	−0.722339	−	0.691539i	$-0.756933\pi$
$431$	−5.57140e6	−1.44468	−0.722339	+	0.691539i	$0.756933\pi$
$432$	3.27863e6	0.845246
$433$	−365124.	−0.0935881	−0.0467940	−	0.998905i	$-0.514900\pi$
$433$	−365124.	−0.0935881	−0.0467940	+	0.998905i	$0.514900\pi$
$434$	−81542.9	−0.0207808
$435$	−3.44286e6	−0.872362
$436$	387718.	0.0976786
$437$	−541265.	−0.135583
$438$	2.34650e6	0.584434
$439$	878974.	0.217678	0.108839	−	0.994059i	$-0.465287\pi$
$439$	878974.	0.217678	0.108839	+	0.994059i	$0.465287\pi$
$440$	−3.59553e6	−0.885382
$441$	−505515.	−0.123776
$442$	−1.41227e6	−0.343844
$443$	−5.35277e6	−1.29589	−0.647947	−	0.761686i	$-0.724372\pi$
$443$	−5.35277e6	−1.29589	−0.647947	+	0.761686i	$0.724372\pi$
$444$	−690344.	−0.166191
$445$	6.22014e6	1.48902
$446$	−2.24447e6	−0.534290
$447$	1.52634e6	0.361312
$448$	566554.	0.133366
$449$	6.41194e6	1.50098	0.750488	−	0.660884i	$-0.229819\pi$
$449$	6.41194e6	1.50098	0.750488	+	0.660884i	$0.229819\pi$
$450$	4.01725e6	0.935186
$451$	−1.01184e6	−0.234246
$452$	−3.27847e6	−0.754789
$453$	−2.01336e6	−0.460973
$454$	5.20802e6	1.18586
$455$	−3.07581e6	−0.696515
$456$	−753205.	−0.169629
$457$	7.40166e6	1.65782	0.828912	−	0.559378i	$-0.188960\pi$
$457$	7.40166e6	1.65782	0.828912	+	0.559378i	$0.188960\pi$
$458$	−4.32316e6	−0.963024
$459$	−672641.	−0.149022
$460$	−518142.	−0.114171
$461$	1.62888e6	0.356975	0.178487	−	0.983942i	$-0.442880\pi$
$461$	1.62888e6	0.356975	0.178487	+	0.983942i	$0.442880\pi$
$462$	−671340.	−0.146331
$463$	−1.42073e6	−0.308006	−0.154003	−	0.988070i	$-0.549217\pi$
$463$	−1.42073e6	−0.308006	−0.154003	+	0.988070i	$0.549217\pi$
$464$	−9.91614e6	−2.13820
$465$	−109699.	−0.0235271
$466$	−7.59154e6	−1.61944
$467$	6.70860e6	1.42344	0.711721	−	0.702462i	$-0.247916\pi$
$467$	6.70860e6	1.42344	0.711721	+	0.702462i	$0.247916\pi$
$468$	−2.16471e6	−0.456863
$469$	−2.89117e6	−0.606935
$470$	−1.61241e6	−0.336691
$471$	−1.34056e6	−0.278442
$472$	2.18521e6	0.451480
$473$	−2.82570e6	−0.580729
$474$	−717478.	−0.146677
$475$	−2.92114e6	−0.594044
$476$	161567.	0.0326841
$477$	8.08512e6	1.62701
$478$	3.00832e6	0.602218
$479$	3.84819e6	0.766332	0.383166	−	0.923679i	$-0.374834\pi$
$479$	3.84819e6	0.766332	0.383166	+	0.923679i	$0.374834\pi$
$480$	−1.91442e6	−0.379257
$481$	−7.76577e6	−1.53046
$482$	6.38123e6	1.25108
$483$	147674.	0.0288028
$484$	−399847.	−0.0775855
$485$	−4.31560e6	−0.833080
$486$	−5.58381e6	−1.07236
$487$	−7.85905e6	−1.50158	−0.750789	−	0.660542i	$-0.770326\pi$
$487$	−7.85905e6	−1.50158	−0.750789	+	0.660542i	$0.770326\pi$
$488$	−5.41618e6	−1.02954
$489$	2.23790e6	0.423222
$490$	1.24088e6	0.233475
$491$	1.34364e6	0.251523	0.125762	−	0.992060i	$-0.459863\pi$
$491$	1.34364e6	0.251523	0.125762	+	0.992060i	$0.459863\pi$
$492$	−202909.	−0.0377911
$493$	2.03439e6	0.376978
$494$	5.55083e6	1.02339
$495$	5.85864e6	1.07469
$496$	−315954.	−0.0576659
$497$	−874896.	−0.158879
$498$	−3.62797e6	−0.655526
$499$	−3.77600e6	−0.678860	−0.339430	−	0.940631i	$-0.610234\pi$
$499$	−3.77600e6	−0.678860	−0.339430	+	0.940631i	$0.610234\pi$
$500$	264509.	0.0473168
$501$	1.21357e6	0.216009
$502$	9.42157e6	1.66865
$503$	−216483.	−0.0381509	−0.0190754	−	0.999818i	$-0.506072\pi$
$503$	−216483.	−0.0381509	−0.0190754	+	0.999818i	$0.506072\pi$
$504$	−1.33305e6	−0.233760
$505$	−136928.	−0.0238925
$506$	−1.27219e6	−0.220889
$507$	1.63859e6	0.283107
$508$	596801.	0.102605
$509$	−1.00134e7	−1.71311	−0.856556	−	0.516053i	$-0.827401\pi$
$509$	−1.00134e7	−1.71311	−0.856556	+	0.516053i	$0.827401\pi$
$510$	766482.	0.130490
$511$	−3.01979e6	−0.511593
$512$	−267286.	−0.0450611
$513$	2.64377e6	0.443538
$514$	−8.33177e6	−1.39101
$515$	−2.65499e6	−0.441109
$516$	−566651.	−0.0936896
$517$	−1.12265e6	−0.184722
$518$	3.13297e6	0.513016
$519$	−1.33543e6	−0.217621
$520$	−8.11093e6	−1.31541
$521$	4.45441e6	0.718946	0.359473	−	0.933156i	$-0.382957\pi$
$521$	4.45441e6	0.718946	0.359473	+	0.933156i	$0.382957\pi$
$522$	1.09964e7	1.76634
$523$	2.47637e6	0.395878	0.197939	−	0.980214i	$-0.436575\pi$
$523$	2.47637e6	0.395878	0.197939	+	0.980214i	$0.436575\pi$
$524$	−1.10103e6	−0.175174
$525$	796976.	0.126196
$526$	1.22526e7	1.93092
$527$	64820.8	0.0101669
$528$	−2.60123e6	−0.406064
$529$	279841.	0.0434783
$530$	−1.98464e7	−3.06897
$531$	−3.56063e6	−0.548013
$532$	−635031.	−0.0972783
$533$	−2.28255e6	−0.348019
$534$	3.06261e6	0.464770
$535$	−9.76198e6	−1.47453
$536$	−7.62405e6	−1.14624
$537$	−2.42969e6	−0.363593
$538$	−9.34352e6	−1.39173
$539$	863969.	0.128093
$540$	2.53083e6	0.373490
$541$	1.13650e7	1.66946	0.834729	−	0.550660i	$-0.185624\pi$
$541$	1.13650e7	1.66946	0.834729	+	0.550660i	$0.185624\pi$
$542$	1.20681e7	1.76458
$543$	−2.15168e6	−0.313168
$544$	1.13123e6	0.163890
$545$	−2.36712e6	−0.341373
$546$	−1.51443e6	−0.217405
$547$	1.03518e7	1.47928	0.739639	−	0.673004i	$-0.234996\pi$
$547$	1.03518e7	1.47928	0.739639	+	0.673004i	$0.234996\pi$
$548$	559126.	0.0795351
$549$	8.82524e6	1.24967
$550$	−6.86584e6	−0.967803
$551$	−7.99603e6	−1.12201
$552$	389417.	0.0543960
$553$	923346.	0.128396
$554$	1.27807e7	1.76921
$555$	4.21473e6	0.580815
$556$	−3.58992e6	−0.492491
$557$	−7.76504e6	−1.06049	−0.530244	−	0.847845i	$-0.677900\pi$
$557$	−7.76504e6	−1.06049	−0.530244	+	0.847845i	$0.677900\pi$
$558$	350374.	0.0476372
$559$	−6.37433e6	−0.862789
$560$	4.80803e6	0.647884
$561$	533667.	0.0715918
$562$	1.07528e7	1.43609
$563$	1.41201e7	1.87745	0.938723	−	0.344674i	$-0.112010\pi$
$563$	1.41201e7	1.87745	0.938723	+	0.344674i	$0.112010\pi$
$564$	−225130.	−0.0298014
$565$	2.00159e7	2.63788
$566$	2.00836e6	0.263513
$567$	1.78564e6	0.233258
$568$	−2.30711e6	−0.300053
$569$	−8.53936e6	−1.10572	−0.552859	−	0.833275i	$-0.686463\pi$
$569$	−8.53936e6	−1.10572	−0.552859	+	0.833275i	$0.686463\pi$
$570$	−3.01261e6	−0.388379
$571$	542125.	0.0695839	0.0347920	−	0.999395i	$-0.488923\pi$
$571$	542125.	0.0695839	0.0347920	+	0.999395i	$0.488923\pi$
$572$	3.69968e6	0.472797
$573$	3.28164e6	0.417546
$574$	920857.	0.116657
$575$	1.51027e6	0.190495
$576$	−2.43437e6	−0.305725
$577$	1.42394e7	1.78054	0.890270	−	0.455433i	$-0.150516\pi$
$577$	1.42394e7	1.78054	0.890270	+	0.455433i	$0.150516\pi$
$578$	9.03638e6	1.12506
$579$	242235.	0.0300290
$580$	−7.65443e6	−0.944807
$581$	4.66895e6	0.573824
$582$	−2.12487e6	−0.260031
$583$	−1.38182e7	−1.68376
$584$	−7.96321e6	−0.966176
$585$	1.32161e7	1.59667
$586$	1.45089e7	1.74538
$587$	8.45283e6	1.01253	0.506264	−	0.862379i	$-0.331026\pi$
$587$	8.45283e6	1.01253	0.506264	+	0.862379i	$0.331026\pi$
$588$	173256.	0.0206654
$589$	−254774.	−0.0302599
$590$	8.74024e6	1.03370
$591$	−2.31569e6	−0.272717
$592$	1.21393e7	1.42360
$593$	−6.78761e6	−0.792647	−0.396324	−	0.918111i	$-0.629714\pi$
$593$	−6.78761e6	−0.792647	−0.396324	+	0.918111i	$0.629714\pi$
$594$	6.21391e6	0.722602
$595$	−986411.	−0.114226
$596$	3.39347e6	0.391317
$597$	2.54819e6	0.292615
$598$	−2.86985e6	−0.328175
$599$	9.34030e6	1.06364	0.531819	−	0.846858i	$-0.321509\pi$
$599$	9.34030e6	1.06364	0.531819	+	0.846858i	$0.321509\pi$
$600$	2.10163e6	0.238330
$601$	1.55790e7	1.75935	0.879677	−	0.475572i	$-0.157759\pi$
$601$	1.55790e7	1.75935	0.879677	+	0.475572i	$0.157759\pi$
$602$	2.57161e6	0.289211
$603$	1.24228e7	1.39132
$604$	−4.47625e6	−0.499254
$605$	2.44117e6	0.271150
$606$	−67419.0	−0.00745763
$607$	1.14766e7	1.26428	0.632138	−	0.774856i	$-0.282178\pi$
$607$	1.14766e7	1.26428	0.632138	+	0.774856i	$0.282178\pi$
$608$	−4.44623e6	−0.487790
$609$	2.18156e6	0.238355
$610$	−2.16632e7	−2.35721
$611$	−2.53252e6	−0.274441
$612$	−694223.	−0.0749239
$613$	4.57897e6	0.492172	0.246086	−	0.969248i	$-0.420856\pi$
$613$	4.57897e6	0.492172	0.246086	+	0.969248i	$0.420856\pi$
$614$	4.86356e6	0.520635
$615$	1.23882e6	0.132074
$616$	2.27829e6	0.241912
$617$	5.39646e6	0.570685	0.285342	−	0.958426i	$-0.407893\pi$
$617$	5.39646e6	0.570685	0.285342	+	0.958426i	$0.407893\pi$
$618$	−1.30724e6	−0.137684
$619$	9.45484e6	0.991808	0.495904	−	0.868377i	$-0.334837\pi$
$619$	9.45484e6	0.991808	0.495904	+	0.868377i	$0.334837\pi$
$620$	−243890.	−0.0254809
$621$	−1.36686e6	−0.142232
$622$	−2.00454e7	−2.07748
$623$	−3.94137e6	−0.406843
$624$	−5.86797e6	−0.603290
$625$	−1.05366e7	−1.07895
$626$	−279760.	−0.0285331
$627$	−2.09755e6	−0.213080
$628$	−2.98044e6	−0.301565
$629$	−2.49048e6	−0.250990
$630$	−5.33182e6	−0.535210
$631$	−1.16868e7	−1.16848	−0.584242	−	0.811580i	$-0.698608\pi$
$631$	−1.16868e7	−1.16848	−0.584242	+	0.811580i	$0.698608\pi$
$632$	2.43487e6	0.242484
$633$	−3.46400e6	−0.343613
$634$	1.18278e7	1.16864
$635$	−3.64363e6	−0.358591
$636$	−2.77102e6	−0.271642
$637$	1.94898e6	0.190308
$638$	−1.87938e7	−1.82795
$639$	3.75926e6	0.364208
$640$	1.67288e7	1.61441
$641$	−3.61240e6	−0.347257	−0.173629	−	0.984811i	$-0.555549\pi$
$641$	−3.61240e6	−0.347257	−0.173629	+	0.984811i	$0.555549\pi$
$642$	−4.80650e6	−0.460248
$643$	−1.02519e6	−0.0977858	−0.0488929	−	0.998804i	$-0.515569\pi$
$643$	−1.02519e6	−0.0977858	−0.0488929	+	0.998804i	$0.515569\pi$
$644$	328319.	0.0311947
$645$	3.45955e6	0.327432
$646$	1.78015e6	0.167832
$647$	−1.14753e7	−1.07772	−0.538859	−	0.842396i	$-0.681144\pi$
$647$	−1.14753e7	−1.07772	−0.538859	+	0.842396i	$0.681144\pi$
$648$	4.70875e6	0.440522
$649$	6.08543e6	0.567126
$650$	−1.54882e7	−1.43787
$651$	69510.1	0.00642829
$652$	4.97547e6	0.458369
$653$	−5.21459e6	−0.478561	−0.239280	−	0.970951i	$-0.576911\pi$
$653$	−5.21459e6	−0.478561	−0.239280	+	0.970951i	$0.576911\pi$
$654$	−1.16550e6	−0.106553
$655$	6.72207e6	0.612209
$656$	3.56804e6	0.323720
$657$	1.29754e7	1.17276
$658$	1.02170e6	0.0919939
$659$	−9.04891e6	−0.811676	−0.405838	−	0.913945i	$-0.633020\pi$
$659$	−9.04891e6	−0.811676	−0.405838	+	0.913945i	$0.633020\pi$
$660$	−2.00794e6	−0.179428
$661$	835159.	0.0743474	0.0371737	−	0.999309i	$-0.488165\pi$
$661$	835159.	0.0743474	0.0371737	+	0.999309i	$0.488165\pi$
$662$	−1.13364e7	−1.00538
$663$	1.20387e6	0.106364
$664$	1.23121e7	1.08370
$665$	3.87703e6	0.339974
$666$	−1.34617e7	−1.17602
$667$	4.13405e6	0.359800
$668$	2.69811e6	0.233947
$669$	1.91327e6	0.165276
$670$	−3.04941e7	−2.62439
$671$	−1.50831e7	−1.29326
$672$	1.21307e6	0.103624
$673$	2.11530e6	0.180026	0.0900129	−	0.995941i	$-0.471309\pi$
$673$	2.11530e6	0.180026	0.0900129	+	0.995941i	$0.471309\pi$
$674$	1.19448e7	1.01281
$675$	−7.37680e6	−0.623173
$676$	3.64304e6	0.306618
$677$	−6.61179e6	−0.554431	−0.277215	−	0.960808i	$-0.589412\pi$
$677$	−6.61179e6	−0.554431	−0.277215	+	0.960808i	$0.589412\pi$
$678$	9.85524e6	0.823366
$679$	2.73457e6	0.227622
$680$	−2.60117e6	−0.215723
$681$	−4.43950e6	−0.366831
$682$	−598820.	−0.0492987
$683$	5.06301e6	0.415295	0.207648	−	0.978204i	$-0.433419\pi$
$683$	5.06301e6	0.415295	0.207648	+	0.978204i	$0.433419\pi$
$684$	2.72860e6	0.222998
$685$	−3.41361e6	−0.277964
$686$	−786280.	−0.0637921
$687$	3.68521e6	0.297900
$688$	9.96420e6	0.802549
$689$	−3.11716e7	−2.50156
$690$	1.55756e6	0.124544
$691$	1.33295e7	1.06198	0.530991	−	0.847377i	$-0.321820\pi$
$691$	1.33295e7	1.06198	0.530991	+	0.847377i	$0.321820\pi$
$692$	−2.96902e6	−0.235694
$693$	−3.71231e6	−0.293637
$694$	588033.	0.0463450
$695$	2.19174e7	1.72118
$696$	5.75279e6	0.450148
$697$	−732015.	−0.0570740
$698$	1.30935e7	1.01722
$699$	6.47130e6	0.500955
$700$	1.77190e6	0.136676
$701$	2.62291e6	0.201599	0.100800	−	0.994907i	$-0.467860\pi$
$701$	2.62291e6	0.201599	0.100800	+	0.994907i	$0.467860\pi$
$702$	1.40176e7	1.07357
$703$	9.78869e6	0.747027
$704$	4.16055e6	0.316388
$705$	1.37448e6	0.104151
$706$	9.30202e6	0.702369
$707$	86763.7	0.00652814
$708$	1.22034e6	0.0914950
$709$	100816.	0.00753210	0.00376605	−	0.999993i	$-0.498801\pi$
$709$	100816.	0.00753210	0.00376605	+	0.999993i	$0.498801\pi$
$710$	−9.22780e6	−0.686993
$711$	−3.96744e6	−0.294331
$712$	−1.03934e7	−0.768350
$713$	131722.	0.00970360
$714$	−485679.	−0.0356536
$715$	−2.25875e7	−1.65236
$716$	−5.40188e6	−0.393788
$717$	−2.56440e6	−0.186289
$718$	−1.82127e7	−1.31845
$719$	−2.34693e7	−1.69308	−0.846539	−	0.532326i	$-0.821318\pi$
$719$	−2.34693e7	−1.69308	−0.846539	+	0.532326i	$0.821318\pi$
$720$	−2.06592e7	−1.48519
$721$	1.68233e6	0.120524
$722$	9.55167e6	0.681924
$723$	−5.43959e6	−0.387008
$724$	−4.78377e6	−0.339175
$725$	2.23110e7	1.57642
$726$	1.20196e6	0.0846347
$727$	−2.10697e7	−1.47850	−0.739252	−	0.673429i	$-0.764821\pi$
$727$	−2.10697e7	−1.47850	−0.739252	+	0.673429i	$0.764821\pi$
$728$	5.13947e6	0.359410
$729$	−4.09548e6	−0.285421
$730$	−3.18507e7	−2.21213
$731$	−2.04425e6	−0.141495
$732$	−3.02469e6	−0.208642
$733$	−5.45237e6	−0.374822	−0.187411	−	0.982282i	$-0.560010\pi$
$733$	−5.45237e6	−0.374822	−0.187411	+	0.982282i	$0.560010\pi$
$734$	2.89615e7	1.98418
$735$	−1.05777e6	−0.0722227
$736$	2.29876e6	0.156422
$737$	−2.12317e7	−1.43984
$738$	−3.95674e6	−0.267422
$739$	1.31042e7	0.882674	0.441337	−	0.897342i	$-0.354504\pi$
$739$	1.31042e7	0.882674	0.441337	+	0.897342i	$0.354504\pi$
$740$	9.37051e6	0.629048
$741$	−4.73173e6	−0.316573
$742$	1.25756e7	0.838533
$743$	−8.32658e6	−0.553343	−0.276672	−	0.960965i	$-0.589231\pi$
$743$	−8.32658e6	−0.553343	−0.276672	+	0.960965i	$0.589231\pi$
$744$	183299.	0.0121402
$745$	−2.07180e7	−1.36760
$746$	−1.64136e7	−1.07983
$747$	−2.00616e7	−1.31542
$748$	1.18649e6	0.0775371
$749$	6.18565e6	0.402885
$750$	−795127.	−0.0516159
$751$	−2.41029e7	−1.55944	−0.779721	−	0.626127i	$-0.784639\pi$
$751$	−2.41029e7	−1.55944	−0.779721	+	0.626127i	$0.784639\pi$
$752$	3.95877e6	0.255280
$753$	−8.03129e6	−0.516176
$754$	−4.23959e7	−2.71578
$755$	2.73287e7	1.74482
$756$	−1.60365e6	−0.102048
$757$	−6.38709e6	−0.405101	−0.202550	−	0.979272i	$-0.564923\pi$
$757$	−6.38709e6	−0.405101	−0.202550	+	0.979272i	$0.564923\pi$
$758$	1.90039e7	1.20135
$759$	1.08446e6	0.0683295
$760$	1.02238e7	0.642062
$761$	−1.67115e7	−1.04605	−0.523025	−	0.852317i	$-0.675197\pi$
$761$	−1.67115e7	−1.04605	−0.523025	+	0.852317i	$0.675197\pi$
$762$	−1.79401e6	−0.111928
$763$	1.49992e6	0.0932730
$764$	7.29599e6	0.452221
$765$	4.23842e6	0.261848
$766$	−3.44283e7	−2.12004
$767$	1.37278e7	0.842580
$768$	6.12887e6	0.374953
$769$	1.66237e7	1.01371	0.506854	−	0.862032i	$-0.330808\pi$
$769$	1.66237e7	1.01371	0.506854	+	0.862032i	$0.330808\pi$
$770$	9.11255e6	0.553877
$771$	7.10230e6	0.430292
$772$	538556.	0.0325228
$773$	2.72819e7	1.64220	0.821101	−	0.570783i	$-0.193360\pi$
$773$	2.72819e7	1.64220	0.821101	+	0.570783i	$0.193360\pi$
$774$	−1.10497e7	−0.662977
$775$	710885.	0.0425153
$776$	7.21108e6	0.429879
$777$	−2.67065e6	−0.158696
$778$	8.64057e6	0.511792
$779$	2.87714e6	0.169870
$780$	−4.52958e6	−0.266576
$781$	−6.42490e6	−0.376911
$782$	−920361.	−0.0538197
$783$	−2.01925e7	−1.17702
$784$	−3.04659e6	−0.177021
$785$	1.81964e7	1.05393
$786$	3.30975e6	0.191090
$787$	1.07446e7	0.618377	0.309189	−	0.951001i	$-0.399943\pi$
$787$	1.07446e7	0.618377	0.309189	+	0.951001i	$0.399943\pi$
$788$	−5.14842e6	−0.295364
$789$	−1.04445e7	−0.597306
$790$	9.73882e6	0.555186
$791$	−1.26830e7	−0.720746
$792$	−9.78939e6	−0.554552
$793$	−3.40251e7	−1.92139
$794$	−1.40629e7	−0.791630
$795$	1.69178e7	0.949350
$796$	5.66533e6	0.316915
$797$	2.15172e7	1.19989	0.599944	−	0.800042i	$-0.295190\pi$
$797$	2.15172e7	1.19989	0.599944	+	0.800042i	$0.295190\pi$
$798$	1.90893e6	0.106117
$799$	−812178.	−0.0450075
$800$	1.24061e7	0.685347
$801$	1.69353e7	0.932635
$802$	1.55535e7	0.853873
$803$	−2.21762e7	−1.21366
$804$	−4.25768e6	−0.232291
$805$	−2.00447e6	−0.109021
$806$	−1.35084e6	−0.0732431
$807$	7.96475e6	0.430515
$808$	228797.	0.0123288
$809$	3.13181e6	0.168238	0.0841189	−	0.996456i	$-0.473192\pi$
$809$	3.13181e6	0.168238	0.0841189	+	0.996456i	$0.473192\pi$
$810$	1.88337e7	1.00861
$811$	1.07790e7	0.575473	0.287736	−	0.957710i	$-0.407097\pi$
$811$	1.07790e7	0.575473	0.287736	+	0.957710i	$0.407097\pi$
$812$	4.85021e6	0.258149
$813$	−1.02873e7	−0.545853
$814$	2.30073e7	1.21704
$815$	−3.03766e7	−1.60193
$816$	−1.88186e6	−0.0989375
$817$	8.03479e6	0.421133
$818$	−1.26396e7	−0.660467
$819$	−8.37437e6	−0.436257
$820$	2.75423e6	0.143043
$821$	−3.05138e7	−1.57993	−0.789967	−	0.613149i	$-0.789902\pi$
$821$	−3.05138e7	−1.57993	−0.789967	+	0.613149i	$0.789902\pi$
$822$	−1.68076e6	−0.0867614
$823$	−2.50014e6	−0.128666	−0.0643332	−	0.997928i	$-0.520492\pi$
$823$	−2.50014e6	−0.128666	−0.0643332	+	0.997928i	$0.520492\pi$
$824$	4.43632e6	0.227617
$825$	5.85269e6	0.299378
$826$	−5.53823e6	−0.282436
$827$	4.34060e6	0.220692	0.110346	−	0.993893i	$-0.464804\pi$
$827$	4.34060e6	0.220692	0.110346	+	0.993893i	$0.464804\pi$
$828$	−1.41072e6	−0.0715099
$829$	1.50388e7	0.760024	0.380012	−	0.924981i	$-0.375920\pi$
$829$	1.50388e7	0.760024	0.380012	+	0.924981i	$0.375920\pi$
$830$	4.92449e7	2.48122
$831$	−1.08947e7	−0.547285
$832$	9.38554e6	0.470057
$833$	625036.	0.0312099
$834$	1.07915e7	0.537237
$835$	−1.64726e7	−0.817612
$836$	−4.66342e6	−0.230775
$837$	−643385.	−0.0317437
$838$	−2.77899e7	−1.36702
$839$	−2.35956e7	−1.15725	−0.578623	−	0.815595i	$-0.696410\pi$
$839$	−2.35956e7	−1.15725	−0.578623	+	0.815595i	$0.696410\pi$
$840$	−2.78935e6	−0.136397
$841$	4.05606e7	1.97749
$842$	2.85130e7	1.38600
$843$	−9.16607e6	−0.444237
$844$	−7.70143e6	−0.372148
$845$	−2.22417e7	−1.07159
$846$	−4.39005e6	−0.210884
$847$	−1.54684e6	−0.0740862
$848$	4.87267e7	2.32690
$849$	−1.71200e6	−0.0815145
$850$	−4.96707e6	−0.235805
$851$	−5.06088e6	−0.239553
$852$	−1.28842e6	−0.0608075
$853$	1.06675e6	0.0501986	0.0250993	−	0.999685i	$-0.492010\pi$
$853$	1.06675e6	0.0501986	0.0250993	+	0.999685i	$0.492010\pi$
$854$	1.37268e7	0.644059
$855$	−1.66588e7	−0.779345
$856$	1.63116e7	0.760873
$857$	−1.51836e7	−0.706191	−0.353096	−	0.935587i	$-0.614871\pi$
$857$	−1.51836e7	−0.706191	−0.353096	+	0.935587i	$0.614871\pi$
$858$	−1.11214e7	−0.515754
$859$	3.17872e7	1.46984	0.734919	−	0.678155i	$-0.237220\pi$
$859$	3.17872e7	1.46984	0.734919	+	0.678155i	$0.237220\pi$
$860$	7.69154e6	0.354623
$861$	−784972.	−0.0360866
$862$	3.72352e7	1.70681
$863$	1.02922e7	0.470415	0.235207	−	0.971945i	$-0.424423\pi$
$863$	1.02922e7	0.470415	0.235207	+	0.971945i	$0.424423\pi$
$864$	−1.12281e7	−0.511709
$865$	1.81266e7	0.823715
$866$	2.44022e6	0.110569
$867$	−7.70293e6	−0.348023
$868$	154540.	0.00696213
$869$	6.78070e6	0.304596
$870$	2.30096e7	1.03065
$871$	−4.78952e7	−2.13918
$872$	3.95530e6	0.176152
$873$	−1.17499e7	−0.521794
$874$	3.61742e6	0.160185
$875$	1.02327e6	0.0451827
$876$	−4.44709e6	−0.195801
$877$	1.28214e7	0.562907	0.281453	−	0.959575i	$-0.409183\pi$
$877$	1.28214e7	0.562907	0.281453	+	0.959575i	$0.409183\pi$
$878$	−5.87442e6	−0.257175
$879$	−1.23679e7	−0.539913
$880$	3.53083e7	1.53699
$881$	1.74956e7	0.759432	0.379716	−	0.925103i	$-0.376022\pi$
$881$	1.74956e7	0.759432	0.379716	+	0.925103i	$0.376022\pi$
$882$	3.37849e6	0.146235
$883$	1.81531e6	0.0783518	0.0391759	−	0.999232i	$-0.487527\pi$
$883$	1.81531e6	0.0783518	0.0391759	+	0.999232i	$0.487527\pi$
$884$	2.67653e6	0.115197
$885$	−7.45049e6	−0.319762
$886$	3.57740e7	1.53103
$887$	−2.84627e7	−1.21470	−0.607348	−	0.794436i	$-0.707767\pi$
$887$	−2.84627e7	−1.21470	−0.607348	+	0.794436i	$0.707767\pi$
$888$	−7.04254e6	−0.299707
$889$	2.30877e6	0.0979777
$890$	−4.15709e7	−1.75920
$891$	1.31130e7	0.553362
$892$	4.25372e6	0.179002
$893$	3.19222e6	0.133957
$894$	−1.02009e7	−0.426871
$895$	3.29799e7	1.37623
$896$	−1.06001e7	−0.441105
$897$	2.44636e6	0.101517
$898$	−4.28528e7	−1.77332
$899$	1.94590e6	0.0803012
$900$	−7.61350e6	−0.313313
$901$	−9.99672e6	−0.410247
$902$	6.76242e6	0.276749
$903$	−2.19213e6	−0.0894639
$904$	−3.34453e7	−1.36117
$905$	2.92062e7	1.18537
$906$	1.34558e7	0.544615
$907$	3.85851e7	1.55740	0.778701	−	0.627395i	$-0.215879\pi$
$907$	3.85851e7	1.55740	0.778701	+	0.627395i	$0.215879\pi$
$908$	−9.87023e6	−0.397295
$909$	−372807.	−0.0149649
$910$	2.05565e7	0.822896
$911$	2.66980e6	0.106582	0.0532909	−	0.998579i	$-0.483029\pi$
$911$	2.66980e6	0.106582	0.0532909	+	0.998579i	$0.483029\pi$
$912$	7.39653e6	0.294470
$913$	3.42870e7	1.36129
$914$	−4.94673e7	−1.95863
$915$	1.84665e7	0.729175
$916$	8.19325e6	0.322639
$917$	−4.25942e6	−0.167273
$918$	4.49544e6	0.176062
$919$	1.69742e7	0.662981	0.331491	−	0.943459i	$-0.392448\pi$
$919$	1.69742e7	0.662981	0.331491	+	0.943459i	$0.392448\pi$
$920$	−5.28582e6	−0.205893
$921$	−4.14588e6	−0.161052
$922$	−1.08863e7	−0.421747
$923$	−1.44935e7	−0.559977
$924$	1.27232e6	0.0490250
$925$	−2.73129e7	−1.04958
$926$	9.49513e6	0.363893
$927$	−7.22864e6	−0.276285
$928$	3.39592e7	1.29446
$929$	5.79682e6	0.220369	0.110184	−	0.993911i	$-0.464856\pi$
$929$	5.79682e6	0.220369	0.110184	+	0.993911i	$0.464856\pi$
$930$	733145.	0.0277960
$931$	−2.45667e6	−0.0928907
$932$	1.43875e7	0.542557
$933$	1.70874e7	0.642645
$934$	−4.48354e7	−1.68172
$935$	−7.24382e6	−0.270981
$936$	−2.20833e7	−0.823899
$937$	2.25276e7	0.838235	0.419117	−	0.907932i	$-0.362340\pi$
$937$	2.25276e7	0.838235	0.419117	+	0.907932i	$0.362340\pi$
$938$	1.93225e7	0.717061
$939$	238477.	0.00882639
$940$	3.05585e6	0.112801
$941$	−3.68545e7	−1.35680	−0.678400	−	0.734693i	$-0.737326\pi$
$941$	−3.68545e7	−1.35680	−0.678400	+	0.734693i	$0.737326\pi$
$942$	8.95935e6	0.328965
$943$	−1.48752e6	−0.0544733
$944$	−2.14589e7	−0.783750
$945$	9.79072e6	0.356644
$946$	1.88849e7	0.686100
$947$	−4.55162e7	−1.64927	−0.824633	−	0.565668i	$-0.808618\pi$
$947$	−4.55162e7	−1.64927	−0.824633	+	0.565668i	$0.808618\pi$
$948$	1.35976e6	0.0491409
$949$	−5.00259e7	−1.80314
$950$	1.95228e7	0.701832
$951$	−1.00824e7	−0.361505
$952$	1.64823e6	0.0589420
$953$	4.99430e7	1.78132	0.890660	−	0.454669i	$-0.150243\pi$
$953$	4.99430e7	1.78132	0.890660	+	0.454669i	$0.150243\pi$
$954$	−5.40351e7	−1.92223
$955$	−4.45439e7	−1.58045
$956$	−5.70136e6	−0.201759
$957$	1.60205e7	0.565454
$958$	−2.57185e7	−0.905381
$959$	2.16303e6	0.0759478
$960$	−5.09383e6	−0.178388
$961$	−2.85671e7	−0.997834
$962$	5.19008e7	1.80816
$963$	−2.65785e7	−0.923560
$964$	−1.20937e7	−0.419147
$965$	−3.28803e6	−0.113662
$966$	−986943.	−0.0340290
$967$	−1.45315e7	−0.499740	−0.249870	−	0.968279i	$-0.580388\pi$
$967$	−1.45315e7	−0.499740	−0.249870	+	0.968279i	$0.580388\pi$
$968$	−4.07904e6	−0.139917
$969$	−1.51746e6	−0.0519169
$970$	2.88423e7	0.984240
$971$	−2.60662e7	−0.887217	−0.443609	−	0.896221i	$-0.646302\pi$
$971$	−2.60662e7	−0.887217	−0.443609	+	0.896221i	$0.646302\pi$
$972$	1.05824e7	0.359269
$973$	−1.38879e7	−0.470278
$974$	5.25242e7	1.77403
$975$	1.32027e7	0.444787
$976$	5.31872e7	1.78724
$977$	3.10232e6	0.103980	0.0519900	−	0.998648i	$-0.483444\pi$
$977$	3.10232e6	0.103980	0.0519900	+	0.998648i	$0.483444\pi$
$978$	−1.49565e7	−0.500015
$979$	−2.89439e7	−0.965163
$980$	−2.35172e6	−0.0782204
$981$	−6.44485e6	−0.213816
$982$	−8.97990e6	−0.297162
$983$	−1.27189e7	−0.419823	−0.209912	−	0.977720i	$-0.567318\pi$
$983$	−1.27189e7	−0.419823	−0.209912	+	0.977720i	$0.567318\pi$
$984$	−2.06998e6	−0.0681519
$985$	3.14324e7	1.03226
$986$	−1.35964e7	−0.445379
$987$	−870934.	−0.0284572
$988$	−1.05199e7	−0.342863
$989$	−4.15409e6	−0.135047
$990$	−3.91549e7	−1.26969
$991$	1.54629e7	0.500159	0.250080	−	0.968225i	$-0.419543\pi$
$991$	1.54629e7	0.500159	0.250080	+	0.968225i	$0.419543\pi$
$992$	1.08203e6	0.0349108
$993$	9.66352e6	0.311001
$994$	5.84717e6	0.187707
$995$	−3.45883e7	−1.10757
$996$	6.87572e6	0.219619
$997$	2.99514e7	0.954287	0.477143	−	0.878825i	$-0.341672\pi$
$997$	2.99514e7	0.954287	0.477143	+	0.878825i	$0.341672\pi$
$998$	2.52360e7	0.802037
$999$	2.47195e7	0.783658

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.6.a.a.1.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.6.a.a.1.3	✓	10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 161 = 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	161.a (trivial)

\(f(q)\)	\(=\)	\(q-6.68327 q^{2} +5.69706 q^{3} +12.6661 q^{4} -77.3301 q^{5} -38.0750 q^{6} +49.0000 q^{7} +129.213 q^{8} -210.543 q^{9} +516.818 q^{10} +359.837 q^{11} +72.1598 q^{12} +811.735 q^{13} -327.480 q^{14} -440.555 q^{15} -1268.89 q^{16} +260.323 q^{17} +1407.12 q^{18} -1023.19 q^{19} -979.474 q^{20} +279.156 q^{21} -2404.89 q^{22} +529.000 q^{23} +736.137 q^{24} +2854.95 q^{25} -5425.05 q^{26} -2583.87 q^{27} +620.641 q^{28} +7814.84 q^{29} +2944.35 q^{30} +249.001 q^{31} +4345.48 q^{32} +2050.02 q^{33} -1739.81 q^{34} -3789.18 q^{35} -2666.77 q^{36} -9566.88 q^{37} +6838.23 q^{38} +4624.50 q^{39} -9992.09 q^{40} -2811.95 q^{41} -1865.68 q^{42} -7852.72 q^{43} +4557.75 q^{44} +16281.4 q^{45} -3535.45 q^{46} -3119.88 q^{47} -7228.92 q^{48} +2401.00 q^{49} -19080.4 q^{50} +1483.08 q^{51} +10281.5 q^{52} -38401.2 q^{53} +17268.7 q^{54} -27826.3 q^{55} +6331.46 q^{56} -5829.15 q^{57} -52228.7 q^{58} +16911.6 q^{59} -5580.13 q^{60} -41916.5 q^{61} -1664.14 q^{62} -10316.6 q^{63} +11562.3 q^{64} -62771.6 q^{65} -13700.8 q^{66} -59003.5 q^{67} +3297.29 q^{68} +3013.75 q^{69} +25324.1 q^{70} -17855.0 q^{71} -27205.1 q^{72} -61628.3 q^{73} +63938.1 q^{74} +16264.8 q^{75} -12959.8 q^{76} +17632.0 q^{77} -30906.8 q^{78} +18843.8 q^{79} +98123.1 q^{80} +36441.6 q^{81} +18793.0 q^{82} +95284.6 q^{83} +3535.83 q^{84} -20130.8 q^{85} +52481.9 q^{86} +44521.6 q^{87} +46495.8 q^{88} -80436.2 q^{89} -108813. q^{90} +39775.0 q^{91} +6700.39 q^{92} +1418.57 q^{93} +20851.0 q^{94} +79123.1 q^{95} +24756.5 q^{96} +55807.5 q^{97} -16046.5 q^{98} -75761.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} - 2 q^{3} + 64 q^{4} - 92 q^{5} - 162 q^{6} + 490 q^{7} - 462 q^{8} + 140 q^{9} - 566 q^{10} - 750 q^{11} + 2348 q^{12} + 124 q^{13} - 392 q^{14} - 1306 q^{15} - 524 q^{16} - 2500 q^{17}+ \cdots - 460326 q^{99}+O(q^{100}) \) 10 * q - 8 * q^2 - 2 * q^3 + 64 * q^4 - 92 * q^5 - 162 * q^6 + 490 * q^7 - 462 * q^8 + 140 * q^9 - 566 * q^10 - 750 * q^11 + 2348 * q^12 + 124 * q^13 - 392 * q^14 - 1306 * q^15 - 524 * q^16 - 2500 * q^17 - 3646 * q^18 - 3888 * q^19 - 538 * q^20 - 98 * q^21 + 2454 * q^22 + 5290 * q^23 - 8208 * q^24 - 5050 * q^25 + 202 * q^26 - 2906 * q^27 + 3136 * q^28 - 4190 * q^29 - 3200 * q^30 - 28126 * q^31 - 5582 * q^32 - 27780 * q^33 - 10088 * q^34 - 4508 * q^35 - 9720 * q^36 - 34096 * q^37 + 6394 * q^38 - 23128 * q^39 - 26846 * q^40 + 13612 * q^41 - 7938 * q^42 - 36330 * q^43 - 20998 * q^44 - 27538 * q^45 - 4232 * q^46 - 29486 * q^47 - 51088 * q^48 + 24010 * q^49 + 5632 * q^50 - 21224 * q^51 - 87640 * q^52 - 67608 * q^53 - 66042 * q^54 - 19776 * q^55 - 22638 * q^56 - 135614 * q^57 - 197486 * q^58 + 4004 * q^59 - 98584 * q^60 + 13134 * q^61 - 9854 * q^62 + 6860 * q^63 - 107420 * q^64 - 66994 * q^65 + 80436 * q^66 - 95740 * q^67 - 57992 * q^68 - 1058 * q^69 - 27734 * q^70 - 66740 * q^71 - 52746 * q^72 + 46044 * q^73 + 99746 * q^74 + 68958 * q^75 - 155098 * q^76 - 36750 * q^77 + 175854 * q^78 - 199082 * q^79 + 237246 * q^80 + 149810 * q^81 - 52286 * q^82 + 24150 * q^83 + 115052 * q^84 - 50060 * q^85 + 219598 * q^86 + 160446 * q^87 + 149210 * q^88 - 98804 * q^89 + 204470 * q^90 + 6076 * q^91 + 33856 * q^92 + 43906 * q^93 + 214806 * q^94 + 228204 * q^95 + 284944 * q^96 - 63914 * q^97 - 19208 * q^98 - 460326 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more