LMFDB - Embedded newform 82.6.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [82,6,Mod(1,82)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(82, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("82.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 82 = 2 \cdot 41 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	82.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$13.1514732247$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 730x^{3} - 4674x^{2} + 68790x + 487116 $ x^5 - 730x^3 - 4674x^2 + 68790*x + 487116
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$10.3879$ of defining polynomial
Character	$\chi$	$=$	82.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} -12.3879 q^{3} +16.0000 q^{4} -48.5965 q^{5} +49.5517 q^{6} +247.165 q^{7} -64.0000 q^{8} -89.5391 q^{9} +194.386 q^{10} +578.385 q^{11} -198.207 q^{12} -730.193 q^{13} -988.662 q^{14} +602.010 q^{15} +256.000 q^{16} -661.308 q^{17} +358.157 q^{18} +1187.50 q^{19} -777.544 q^{20} -3061.87 q^{21} -2313.54 q^{22} -3639.95 q^{23} +792.828 q^{24} -763.381 q^{25} +2920.77 q^{26} +4119.47 q^{27} +3954.65 q^{28} -8181.04 q^{29} -2408.04 q^{30} -6778.86 q^{31} -1024.00 q^{32} -7165.00 q^{33} +2645.23 q^{34} -12011.4 q^{35} -1432.63 q^{36} +3832.01 q^{37} -4749.98 q^{38} +9045.58 q^{39} +3110.18 q^{40} -1681.00 q^{41} +12247.5 q^{42} +2523.46 q^{43} +9254.16 q^{44} +4351.29 q^{45} +14559.8 q^{46} -650.878 q^{47} -3171.31 q^{48} +44283.8 q^{49} +3053.52 q^{50} +8192.24 q^{51} -11683.1 q^{52} -31106.7 q^{53} -16477.9 q^{54} -28107.5 q^{55} -15818.6 q^{56} -14710.6 q^{57} +32724.2 q^{58} -7172.42 q^{59} +9632.16 q^{60} +9005.64 q^{61} +27115.5 q^{62} -22131.0 q^{63} +4096.00 q^{64} +35484.8 q^{65} +28660.0 q^{66} +34036.5 q^{67} -10580.9 q^{68} +45091.5 q^{69} +48045.5 q^{70} -21998.3 q^{71} +5730.50 q^{72} -73110.7 q^{73} -15328.0 q^{74} +9456.71 q^{75} +18999.9 q^{76} +142957. q^{77} -36182.3 q^{78} -59345.8 q^{79} -12440.7 q^{80} -29273.7 q^{81} +6724.00 q^{82} -17996.6 q^{83} -48989.9 q^{84} +32137.2 q^{85} -10093.9 q^{86} +101346. q^{87} -37016.7 q^{88} +26533.5 q^{89} -17405.2 q^{90} -180478. q^{91} -58239.3 q^{92} +83976.1 q^{93} +2603.51 q^{94} -57708.1 q^{95} +12685.2 q^{96} -122783. q^{97} -177135. q^{98} -51788.1 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 20 q^{2} - 10 q^{3} + 80 q^{4} - 38 q^{5} + 40 q^{6} - 38 q^{7} - 320 q^{8} + 265 q^{9} + 152 q^{10} + 416 q^{11} - 160 q^{12} + 268 q^{13} + 152 q^{14} + 22 q^{15} + 1280 q^{16} - 2198 q^{17} - 1060 q^{18}+ \cdots - 139442 q^{99}+O(q^{100}) $ 5 * q - 20 * q^2 - 10 * q^3 + 80 * q^4 - 38 * q^5 + 40 * q^6 - 38 * q^7 - 320 * q^8 + 265 * q^9 + 152 * q^10 + 416 * q^11 - 160 * q^12 + 268 * q^13 + 152 * q^14 + 22 * q^15 + 1280 * q^16 - 2198 * q^17 - 1060 * q^18 - 2250 * q^19 - 608 * q^20 - 6344 * q^21 - 1664 * q^22 - 12008 * q^23 + 640 * q^24 - 9861 * q^25 - 1072 * q^26 - 17962 * q^27 - 608 * q^28 - 11684 * q^29 - 88 * q^30 - 6780 * q^31 - 5120 * q^32 - 15876 * q^33 + 8792 * q^34 - 23158 * q^35 + 4240 * q^36 - 5822 * q^37 + 9000 * q^38 - 14440 * q^39 + 2432 * q^40 - 8405 * q^41 + 25376 * q^42 + 13248 * q^43 + 6656 * q^44 + 5058 * q^45 + 48032 * q^46 - 2000 * q^47 - 2560 * q^48 + 51789 * q^49 + 39444 * q^50 + 36628 * q^51 + 4288 * q^52 - 32036 * q^53 + 71848 * q^54 + 28670 * q^55 + 2432 * q^56 + 5476 * q^57 + 46736 * q^58 - 41376 * q^59 + 352 * q^60 + 45314 * q^61 + 27120 * q^62 + 25568 * q^63 + 20480 * q^64 + 20012 * q^65 + 63504 * q^66 + 47244 * q^67 - 35168 * q^68 + 77140 * q^69 + 92632 * q^70 - 87348 * q^71 - 16960 * q^72 + 44550 * q^73 + 23288 * q^74 + 23370 * q^75 - 36000 * q^76 - 19040 * q^77 + 57760 * q^78 - 6864 * q^79 - 9728 * q^80 + 154261 * q^81 + 33620 * q^82 - 119888 * q^83 - 101504 * q^84 - 22700 * q^85 - 52992 * q^86 - 242980 * q^87 - 26624 * q^88 - 136626 * q^89 - 20232 * q^90 - 30964 * q^91 - 192128 * q^92 - 180684 * q^93 + 8000 * q^94 - 153770 * q^95 + 10240 * q^96 + 7086 * q^97 - 207156 * q^98 - 139442 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	−12.3879	−0.794686	−0.397343	−	0.917670i	$-0.630068\pi$
$3$	−12.3879	−0.794686	−0.397343	+	0.917670i	$0.630068\pi$
$4$	16.0000	0.500000
$5$	−48.5965	−0.869321	−0.434660	−	0.900595i	$-0.643132\pi$
$5$	−48.5965	−0.869321	−0.434660	+	0.900595i	$0.643132\pi$
$6$	49.5517	0.561928
$7$	247.165	1.90653	0.953263	−	0.302142i	$-0.0977017\pi$
$7$	247.165	1.90653	0.953263	+	0.302142i	$0.0977017\pi$
$8$	−64.0000	−0.353553
$9$	−89.5391	−0.368474
$10$	194.386	0.614702
$11$	578.385	1.44124	0.720619	−	0.693332i	$-0.243858\pi$
$11$	578.385	1.44124	0.720619	+	0.693332i	$0.243858\pi$
$12$	−198.207	−0.397343
$13$	−730.193	−1.19834	−0.599169	−	0.800623i	$-0.704502\pi$
$13$	−730.193	−1.19834	−0.599169	+	0.800623i	$0.704502\pi$
$14$	−988.662	−1.34812
$15$	602.010	0.690837
$16$	256.000	0.250000
$17$	−661.308	−0.554985	−0.277493	−	0.960728i	$-0.589503\pi$
$17$	−661.308	−0.554985	−0.277493	+	0.960728i	$0.589503\pi$
$18$	358.157	0.260550
$19$	1187.50	0.754654	0.377327	−	0.926080i	$-0.376843\pi$
$19$	1187.50	0.754654	0.377327	+	0.926080i	$0.376843\pi$
$20$	−777.544	−0.434660
$21$	−3061.87	−1.51509
$22$	−2313.54	−1.01911
$23$	−3639.95	−1.43475	−0.717375	−	0.696687i	$-0.754657\pi$
$23$	−3639.95	−1.43475	−0.717375	+	0.696687i	$0.754657\pi$
$24$	792.828	0.280964
$25$	−763.381	−0.244282
$26$	2920.77	0.847352
$27$	4119.47	1.08751
$28$	3954.65	0.953263
$29$	−8181.04	−1.80640	−0.903199	−	0.429222i	$-0.858788\pi$
$29$	−8181.04	−1.80640	−0.903199	+	0.429222i	$0.858788\pi$
$30$	−2408.04	−0.488496
$31$	−6778.86	−1.26693	−0.633465	−	0.773771i	$-0.718368\pi$
$31$	−6778.86	−1.26693	−0.633465	+	0.773771i	$0.718368\pi$
$32$	−1024.00	−0.176777
$33$	−7165.00	−1.14533
$34$	2645.23	0.392434
$35$	−12011.4	−1.65738
$36$	−1432.63	−0.184237
$37$	3832.01	0.460174	0.230087	−	0.973170i	$-0.426099\pi$
$37$	3832.01	0.460174	0.230087	+	0.973170i	$0.426099\pi$
$38$	−4749.98	−0.533621
$39$	9045.58	0.952302
$40$	3110.18	0.307351
$41$	−1681.00	−0.156174
$41$	−1681.00	−0.156174
$42$	12247.5	1.07133
$43$	2523.46	0.208126	0.104063	−	0.994571i	$-0.466816\pi$
$43$	2523.46	0.208126	0.104063	+	0.994571i	$0.466816\pi$
$44$	9254.16	0.720619
$45$	4351.29	0.320322
$46$	14559.8	1.01452
$47$	−650.878	−0.0429788	−0.0214894	−	0.999769i	$-0.506841\pi$
$47$	−650.878	−0.0429788	−0.0214894	+	0.999769i	$0.506841\pi$
$48$	−3171.31	−0.198672
$49$	44283.8	2.63484
$50$	3053.52	0.172733
$51$	8192.24	0.441039
$52$	−11683.1	−0.599169
$53$	−31106.7	−1.52112	−0.760562	−	0.649265i	$-0.775076\pi$
$53$	−31106.7	−1.52112	−0.760562	+	0.649265i	$0.775076\pi$
$54$	−16477.9	−0.768984
$55$	−28107.5	−1.25290
$56$	−15818.6	−0.674059
$57$	−14710.6	−0.599713
$58$	32724.2	1.27732
$59$	−7172.42	−0.268248	−0.134124	−	0.990965i	$-0.542822\pi$
$59$	−7172.42	−0.268248	−0.134124	+	0.990965i	$0.542822\pi$
$60$	9632.16	0.345419
$61$	9005.64	0.309878	0.154939	−	0.987924i	$-0.450482\pi$
$61$	9005.64	0.309878	0.154939	+	0.987924i	$0.450482\pi$
$62$	27115.5	0.895855
$63$	−22131.0	−0.702505
$64$	4096.00	0.125000
$65$	35484.8	1.04174
$66$	28660.0	0.809872
$67$	34036.5	0.926314	0.463157	−	0.886276i	$-0.346717\pi$
$67$	34036.5	0.926314	0.463157	+	0.886276i	$0.346717\pi$
$68$	−10580.9	−0.277493
$69$	45091.5	1.14018
$70$	48045.5	1.17195
$71$	−21998.3	−0.517898	−0.258949	−	0.965891i	$-0.583376\pi$
$71$	−21998.3	−0.517898	−0.258949	+	0.965891i	$0.583376\pi$
$72$	5730.50	0.130275
$73$	−73110.7	−1.60573	−0.802867	−	0.596158i	$-0.796693\pi$
$73$	−73110.7	−1.60573	−0.802867	+	0.596158i	$0.796693\pi$
$74$	−15328.0	−0.325392
$75$	9456.71	0.194127
$76$	18999.9	0.377327
$77$	142957.	2.74776
$78$	−36182.3	−0.673379
$79$	−59345.8	−1.06985	−0.534925	−	0.844900i	$-0.679660\pi$
$79$	−59345.8	−1.06985	−0.534925	+	0.844900i	$0.679660\pi$
$80$	−12440.7	−0.217330
$81$	−29273.7	−0.495753
$82$	6724.00	0.110432
$83$	−17996.6	−0.286744	−0.143372	−	0.989669i	$-0.545795\pi$
$83$	−17996.6	−0.286744	−0.143372	+	0.989669i	$0.545795\pi$
$84$	−48989.9	−0.757545
$85$	32137.2	0.482460
$86$	−10093.9	−0.147167
$87$	101346.	1.43552
$88$	−37016.7	−0.509554
$89$	26533.5	0.355075	0.177538	−	0.984114i	$-0.443187\pi$
$89$	26533.5	0.355075	0.177538	+	0.984114i	$0.443187\pi$
$90$	−17405.2	−0.226502
$91$	−180478.	−2.28466
$92$	−58239.3	−0.717375
$93$	83976.1	1.00681
$94$	2603.51	0.0303906
$95$	−57708.1	−0.656036
$96$	12685.2	0.140482
$97$	−122783.	−1.32498	−0.662491	−	0.749070i	$-0.730501\pi$
$97$	−122783.	−1.32498	−0.662491	+	0.749070i	$0.730501\pi$
$98$	−177135.	−1.86311
$99$	−51788.1	−0.531058
$100$	−12214.1	−0.122141
$101$	11474.1	0.111922	0.0559609	−	0.998433i	$-0.482178\pi$
$101$	11474.1	0.111922	0.0559609	+	0.998433i	$0.482178\pi$
$102$	−32768.9	−0.311862
$103$	68759.1	0.638613	0.319306	−	0.947652i	$-0.396550\pi$
$103$	68759.1	0.638613	0.319306	+	0.947652i	$0.396550\pi$
$104$	46732.3	0.423676
$105$	148796.	1.31710
$106$	124427.	1.07560
$107$	−128149.	−1.08207	−0.541037	−	0.840999i	$-0.681968\pi$
$107$	−128149.	−1.08207	−0.541037	+	0.840999i	$0.681968\pi$
$108$	65911.6	0.543754
$109$	−194280.	−1.56625	−0.783126	−	0.621863i	$-0.786376\pi$
$109$	−194280.	−1.56625	−0.783126	+	0.621863i	$0.786376\pi$
$110$	112430.	0.885932
$111$	−47470.6	−0.365694
$112$	63274.4	0.476631
$113$	−261458.	−1.92622	−0.963110	−	0.269109i	$-0.913271\pi$
$113$	−261458.	−1.92622	−0.963110	+	0.269109i	$0.913271\pi$
$114$	58842.4	0.424061
$115$	176889.	1.24726
$116$	−130897.	−0.903199
$117$	65380.8	0.441556
$118$	28689.7	0.189680
$119$	−163452.	−1.05809
$120$	−38528.6	−0.244248
$121$	173478.	1.07716
$122$	−36022.6	−0.219117
$123$	20824.1	0.124109
$124$	−108462.	−0.633465
$125$	188962.	1.08168
$126$	88523.9	0.496746
$127$	100571.	0.553302	0.276651	−	0.960970i	$-0.410775\pi$
$127$	100571.	0.553302	0.276651	+	0.960970i	$0.410775\pi$
$128$	−16384.0	−0.0883883
$129$	−31260.5	−0.165395
$130$	−141939.	−0.736621
$131$	363152.	1.84889	0.924444	−	0.381318i	$-0.124530\pi$
$131$	363152.	1.84889	0.924444	+	0.381318i	$0.124530\pi$
$132$	−114640.	−0.572666
$133$	293508.	1.43877
$134$	−136146.	−0.655003
$135$	−200192.	−0.945392
$136$	42323.7	0.196217
$137$	−40857.5	−0.185982	−0.0929910	−	0.995667i	$-0.529643\pi$
$137$	−40857.5	−0.185982	−0.0929910	+	0.995667i	$0.529643\pi$
$138$	−180366.	−0.806226
$139$	383567.	1.68385	0.841926	−	0.539593i	$-0.181422\pi$
$139$	383567.	1.68385	0.841926	+	0.539593i	$0.181422\pi$
$140$	−192182.	−0.828691
$141$	8063.03	0.0341547
$142$	87993.3	0.366209
$143$	−422333.	−1.72709
$144$	−22922.0	−0.0921184
$145$	397570.	1.57034
$146$	292443.	1.13543
$147$	−548584.	−2.09387
$148$	61312.1	0.230087
$149$	−113425.	−0.418546	−0.209273	−	0.977857i	$-0.567110\pi$
$149$	−113425.	−0.418546	−0.209273	+	0.977857i	$0.567110\pi$
$150$	−37826.8	−0.137269
$151$	305548.	1.09053	0.545264	−	0.838264i	$-0.316429\pi$
$151$	305548.	1.09053	0.545264	+	0.838264i	$0.316429\pi$
$152$	−75999.7	−0.266810
$153$	59212.9	0.204497
$154$	−571827.	−1.94296
$155$	329429.	1.10137
$156$	144729.	0.476151
$157$	−393383.	−1.27370	−0.636850	−	0.770988i	$-0.719763\pi$
$157$	−393383.	−1.27370	−0.636850	+	0.770988i	$0.719763\pi$
$158$	237383.	0.756498
$159$	385348.	1.20882
$160$	49762.8	0.153676
$161$	−899671.	−2.73539
$162$	117095.	0.350551
$163$	143096.	0.421851	0.210925	−	0.977502i	$-0.432352\pi$
$163$	143096.	0.421851	0.210925	+	0.977502i	$0.432352\pi$
$164$	−26896.0	−0.0780869
$165$	348194.	0.995660
$166$	71986.3	0.202759
$167$	114692.	0.318231	0.159116	−	0.987260i	$-0.449136\pi$
$167$	114692.	0.318231	0.159116	+	0.987260i	$0.449136\pi$
$168$	195960.	0.535665
$169$	161888.	0.436012
$170$	−128549.	−0.341151
$171$	−106327.	−0.278070
$172$	40375.4	0.104063
$173$	296638.	0.753548	0.376774	−	0.926305i	$-0.377033\pi$
$173$	296638.	0.753548	0.376774	+	0.926305i	$0.377033\pi$
$174$	−405385.	−1.01507
$175$	−188681.	−0.465730
$176$	148067.	0.360309
$177$	88851.5	0.213173
$178$	−106134.	−0.251076
$179$	486960.	1.13595	0.567977	−	0.823044i	$-0.307726\pi$
$179$	486960.	1.13595	0.567977	+	0.823044i	$0.307726\pi$
$180$	69620.6	0.160161
$181$	709246.	1.60916	0.804582	−	0.593842i	$-0.202390\pi$
$181$	709246.	1.60916	0.804582	+	0.593842i	$0.202390\pi$
$182$	721914.	1.61550
$183$	−111561.	−0.246255
$184$	232957.	0.507261
$185$	−186222.	−0.400039
$186$	−335904.	−0.711924
$187$	−382491.	−0.799865
$188$	−10414.0	−0.0214894
$189$	1.01819e6	2.07336
$190$	230832.	0.463888
$191$	−581986.	−1.15433	−0.577164	−	0.816628i	$-0.695841\pi$
$191$	−581986.	−1.15433	−0.577164	+	0.816628i	$0.695841\pi$
$192$	−50741.0	−0.0993358
$193$	−4733.90	−0.00914799	−0.00457399	−	0.999990i	$-0.501456\pi$
$193$	−4733.90	−0.00914799	−0.00457399	+	0.999990i	$0.501456\pi$
$194$	491133.	0.936904
$195$	−439583.	−0.827856
$196$	708540.	1.31742
$197$	−227022.	−0.416775	−0.208388	−	0.978046i	$-0.566822\pi$
$197$	−227022.	−0.416775	−0.208388	+	0.978046i	$0.566822\pi$
$198$	207152.	0.375515
$199$	−372393.	−0.666605	−0.333302	−	0.942820i	$-0.608163\pi$
$199$	−372393.	−0.666605	−0.333302	+	0.942820i	$0.608163\pi$
$200$	48856.4	0.0863667
$201$	−421642.	−0.736129
$202$	−45896.3	−0.0791406
$203$	−2.02207e6	−3.44394
$204$	131076.	0.220519
$205$	81690.7	0.135765
$206$	−275037.	−0.451567
$207$	325918.	0.528668
$208$	−186929.	−0.299584
$209$	686830.	1.08764
$210$	−595184.	−0.931329
$211$	443085.	0.685142	0.342571	−	0.939492i	$-0.388702\pi$
$211$	443085.	0.685142	0.342571	+	0.939492i	$0.388702\pi$
$212$	−497708.	−0.760562
$213$	272514.	0.411566
$214$	512597.	0.765141
$215$	−122631.	−0.180928
$216$	−263646.	−0.384492
$217$	−1.67550e6	−2.41544
$218$	777120.	1.10751
$219$	905690.	1.27606
$220$	−449720.	−0.626448
$221$	482882.	0.665059
$222$	189883.	0.258585
$223$	−445098.	−0.599367	−0.299684	−	0.954039i	$-0.596881\pi$
$223$	−445098.	−0.599367	−0.299684	+	0.954039i	$0.596881\pi$
$224$	−253097.	−0.337029
$225$	68352.4	0.0900115
$226$	1.04583e6	1.36204
$227$	−194492.	−0.250517	−0.125258	−	0.992124i	$-0.539976\pi$
$227$	−194492.	−0.250517	−0.125258	+	0.992124i	$0.539976\pi$
$228$	−235370.	−0.299857
$229$	27003.2	0.0340272	0.0170136	−	0.999855i	$-0.494584\pi$
$229$	27003.2	0.0340272	0.0170136	+	0.999855i	$0.494584\pi$
$230$	−707556.	−0.881945
$231$	−1.77094e6	−2.18360
$232$	523587.	0.638658
$233$	−423773.	−0.511380	−0.255690	−	0.966759i	$-0.582303\pi$
$233$	−423773.	−0.511380	−0.255690	+	0.966759i	$0.582303\pi$
$234$	−261523.	−0.312227
$235$	31630.4	0.0373624
$236$	−114759.	−0.134124
$237$	735172.	0.850194
$238$	653810.	0.748185
$239$	97400.3	0.110297	0.0551487	−	0.998478i	$-0.482437\pi$
$239$	97400.3	0.110297	0.0551487	+	0.998478i	$0.482437\pi$
$240$	154115.	0.172709
$241$	481557.	0.534078	0.267039	−	0.963686i	$-0.413955\pi$
$241$	481557.	0.534078	0.267039	+	0.963686i	$0.413955\pi$
$242$	−693914.	−0.761670
$243$	−638391.	−0.693539
$244$	144090.	0.154939
$245$	−2.15204e6	−2.29052
$246$	−83296.5	−0.0877584
$247$	−867100.	−0.904330
$248$	433847.	0.447927
$249$	222941.	0.227872
$250$	−755847.	−0.764863
$251$	718165.	0.719515	0.359758	−	0.933046i	$-0.382859\pi$
$251$	718165.	0.719515	0.359758	+	0.933046i	$0.382859\pi$
$252$	−354096.	−0.351252
$253$	−2.10530e6	−2.06782
$254$	−402283.	−0.391243
$255$	−398114.	−0.383404
$256$	65536.0	0.0625000
$257$	24515.5	0.0231530	0.0115765	−	0.999933i	$-0.496315\pi$
$257$	24515.5	0.0231530	0.0115765	+	0.999933i	$0.496315\pi$
$258$	125042.	0.116952
$259$	947140.	0.877333
$260$	567757.	0.520870
$261$	732523.	0.665610
$262$	−1.45261e6	−1.30736
$263$	−862558.	−0.768952	−0.384476	−	0.923135i	$-0.625618\pi$
$263$	−862558.	−0.768952	−0.384476	+	0.923135i	$0.625618\pi$
$264$	458560.	0.404936
$265$	1.51168e6	1.32234
$266$	−1.17403e6	−1.01736
$267$	−328696.	−0.282173
$268$	544584.	0.463157
$269$	169348.	0.142692	0.0713459	−	0.997452i	$-0.477271\pi$
$269$	169348.	0.142692	0.0713459	+	0.997452i	$0.477271\pi$
$270$	800768.	0.668493
$271$	558866.	0.462258	0.231129	−	0.972923i	$-0.425758\pi$
$271$	558866.	0.462258	0.231129	+	0.972923i	$0.425758\pi$
$272$	−169295.	−0.138746
$273$	2.23575e6	1.81559
$274$	163430.	0.131509
$275$	−441528.	−0.352068
$276$	721464.	0.570088
$277$	510621.	0.399852	0.199926	−	0.979811i	$-0.435930\pi$
$277$	510621.	0.399852	0.199926	+	0.979811i	$0.435930\pi$
$278$	−1.53427e6	−1.19066
$279$	606974.	0.466831
$280$	768728.	0.585973
$281$	1.48924e6	1.12512	0.562560	−	0.826757i	$-0.309817\pi$
$281$	1.48924e6	1.12512	0.562560	+	0.826757i	$0.309817\pi$
$282$	−32252.1	−0.0241510
$283$	210948.	0.156570	0.0782852	−	0.996931i	$-0.475056\pi$
$283$	210948.	0.156570	0.0782852	+	0.996931i	$0.475056\pi$
$284$	−351973.	−0.258949
$285$	714884.	0.521343
$286$	1.68933e6	1.22124
$287$	−415485.	−0.297749
$288$	91688.1	0.0651376
$289$	−982529.	−0.691992
$290$	−1.59028e6	−1.11040
$291$	1.52103e6	1.05294
$292$	−1.16977e6	−0.802867
$293$	1.99745e6	1.35927	0.679636	−	0.733549i	$-0.262138\pi$
$293$	1.99745e6	1.35927	0.679636	+	0.733549i	$0.262138\pi$
$294$	2.19434e6	1.48059
$295$	348554.	0.233193
$296$	−245248.	−0.162696
$297$	2.38264e6	1.56736
$298$	453700.	0.295956
$299$	2.65787e6	1.71931
$300$	151307.	0.0970637
$301$	623713.	0.396797
$302$	−1.22219e6	−0.771120
$303$	−142140.	−0.0889427
$304$	303999.	0.188663
$305$	−437643.	−0.269383
$306$	−236852.	−0.144602
$307$	−3.05903e6	−1.85241	−0.926206	−	0.377019i	$-0.876949\pi$
$307$	−3.05903e6	−1.85241	−0.926206	+	0.377019i	$0.876949\pi$
$308$	2.28731e6	1.37388
$309$	−851784.	−0.507497
$310$	−1.31772e6	−0.778785
$311$	1.79184e6	1.05051	0.525253	−	0.850946i	$-0.323971\pi$
$311$	1.79184e6	1.05051	0.525253	+	0.850946i	$0.323971\pi$
$312$	−578917.	−0.336690
$313$	−607904.	−0.350731	−0.175365	−	0.984503i	$-0.556111\pi$
$313$	−607904.	−0.350731	−0.175365	+	0.984503i	$0.556111\pi$
$314$	1.57353e6	0.900642
$315$	1.07549e6	0.610702
$316$	−949533.	−0.534925
$317$	−1.23260e6	−0.688931	−0.344465	−	0.938799i	$-0.611940\pi$
$317$	−1.23260e6	−0.688931	−0.344465	+	0.938799i	$0.611940\pi$
$318$	−1.54139e6	−0.854762
$319$	−4.73179e6	−2.60345
$320$	−199051.	−0.108665
$321$	1.58750e6	0.859909
$322$	3.59868e6	1.93421
$323$	−785300.	−0.418822
$324$	−468380.	−0.247877
$325$	557415.	0.292732
$326$	−572384.	−0.298293
$327$	2.40673e6	1.24468
$328$	107584.	0.0552158
$329$	−160874.	−0.0819403
$330$	−1.39277e6	−0.704038
$331$	−3.89464e6	−1.95388	−0.976938	−	0.213523i	$-0.931506\pi$
$331$	−3.89464e6	−1.95388	−0.976938	+	0.213523i	$0.931506\pi$
$332$	−287945.	−0.143372
$333$	−343115.	−0.169562
$334$	−458769.	−0.225024
$335$	−1.65406e6	−0.805263
$336$	−783838.	−0.378772
$337$	2.60213e6	1.24811	0.624057	−	0.781379i	$-0.285483\pi$
$337$	2.60213e6	1.24811	0.624057	+	0.781379i	$0.285483\pi$
$338$	−647553.	−0.308307
$339$	3.23892e6	1.53074
$340$	514196.	0.241230
$341$	−3.92079e6	−1.82595
$342$	425309.	0.196625
$343$	6.79131e6	3.11687
$344$	−161502.	−0.0735836
$345$	−2.19129e6	−0.991179
$346$	−1.18655e6	−0.532839
$347$	1.45707e6	0.649618	0.324809	−	0.945780i	$-0.394700\pi$
$347$	1.45707e6	0.649618	0.324809	+	0.945780i	$0.394700\pi$
$348$	1.62154e6	0.717760
$349$	1.21466e6	0.533815	0.266908	−	0.963722i	$-0.413998\pi$
$349$	1.21466e6	0.533815	0.266908	+	0.963722i	$0.413998\pi$
$350$	754725.	0.329321
$351$	−3.00801e6	−1.30320
$352$	−592266.	−0.254777
$353$	217642.	0.0929621	0.0464811	−	0.998919i	$-0.485199\pi$
$353$	217642.	0.0929621	0.0464811	+	0.998919i	$0.485199\pi$
$354$	−355406.	−0.150736
$355$	1.06904e6	0.450219
$356$	424537.	0.177538
$357$	2.02484e6	0.840852
$358$	−1.94784e6	−0.803241
$359$	2.70007e6	1.10570	0.552852	−	0.833280i	$-0.313540\pi$
$359$	2.70007e6	1.10570	0.552852	+	0.833280i	$0.313540\pi$
$360$	−278482.	−0.113251
$361$	−1.06595e6	−0.430497
$362$	−2.83698e6	−1.13785
$363$	−2.14904e6	−0.856008
$364$	−2.88765e6	−1.14233
$365$	3.55292e6	1.39590
$366$	446245.	0.174129
$367$	−811222.	−0.314394	−0.157197	−	0.987567i	$-0.550246\pi$
$367$	−811222.	−0.314394	−0.157197	+	0.987567i	$0.550246\pi$
$368$	−931828.	−0.358688
$369$	150515.	0.0575459
$370$	744888.	0.282870
$371$	−7.68851e6	−2.90006
$372$	1.34362e6	0.503406
$373$	−1.79672e6	−0.668666	−0.334333	−	0.942455i	$-0.608511\pi$
$373$	−1.79672e6	−0.668666	−0.334333	+	0.942455i	$0.608511\pi$
$374$	1.52996e6	0.565590
$375$	−2.34084e6	−0.859596
$376$	41656.2	0.0151953
$377$	5.97373e6	2.16467
$378$	−4.07277e6	−1.46609
$379$	1.95509e6	0.699147	0.349573	−	0.936909i	$-0.386327\pi$
$379$	1.95509e6	0.699147	0.349573	+	0.936909i	$0.386327\pi$
$380$	−923330.	−0.328018
$381$	−1.24586e6	−0.439701
$382$	2.32795e6	0.816233
$383$	−558745.	−0.194633	−0.0973166	−	0.995253i	$-0.531026\pi$
$383$	−558745.	−0.194633	−0.0973166	+	0.995253i	$0.531026\pi$
$384$	202964.	0.0702410
$385$	−6.94720e6	−2.38868
$386$	18935.6	0.00646860
$387$	−225949.	−0.0766889
$388$	−1.96453e6	−0.662491
$389$	−1.74964e6	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$389$	−1.74964e6	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$390$	1.75833e6	0.585382
$391$	2.40713e6	0.796265
$392$	−2.83416e6	−0.931557
$393$	−4.49871e6	−1.46929
$394$	908087.	0.294705
$395$	2.88400e6	0.930042
$396$	−828610.	−0.265529
$397$	−3.62583e6	−1.15460	−0.577299	−	0.816533i	$-0.695893\pi$
$397$	−3.62583e6	−1.15460	−0.577299	+	0.816533i	$0.695893\pi$
$398$	1.48957e6	0.471361
$399$	−3.63595e6	−1.14337
$400$	−195425.	−0.0610705
$401$	3.57153e6	1.10916	0.554578	−	0.832132i	$-0.312880\pi$
$401$	3.57153e6	1.10916	0.554578	+	0.832132i	$0.312880\pi$
$402$	1.68657e6	0.520522
$403$	4.94988e6	1.51821
$404$	183585.	0.0559609
$405$	1.42260e6	0.430969
$406$	8.08828e6	2.43524
$407$	2.21638e6	0.663220
$408$	−524303.	−0.155931
$409$	−2.77747e6	−0.820996	−0.410498	−	0.911861i	$-0.634645\pi$
$409$	−2.77747e6	−0.820996	−0.410498	+	0.911861i	$0.634645\pi$
$410$	−326763.	−0.0960004
$411$	506140.	0.147797
$412$	1.10015e6	0.319306
$413$	−1.77277e6	−0.511421
$414$	−1.30367e6	−0.373825
$415$	874571.	0.249273
$416$	747717.	0.211838
$417$	−4.75160e6	−1.33813
$418$	−2.74732e6	−0.769074
$419$	2.70148e6	0.751738	0.375869	−	0.926673i	$-0.377344\pi$
$419$	2.70148e6	0.751738	0.375869	+	0.926673i	$0.377344\pi$
$420$	2.38074e6	0.658549
$421$	−382826.	−0.105268	−0.0526340	−	0.998614i	$-0.516762\pi$
$421$	−382826.	−0.105268	−0.0526340	+	0.998614i	$0.516762\pi$
$422$	−1.77234e6	−0.484469
$423$	58279.0	0.0158366
$424$	1.99083e6	0.537799
$425$	504830.	0.135573
$426$	−1.09006e6	−0.291021
$427$	2.22588e6	0.590790
$428$	−2.05039e6	−0.541037
$429$	5.23183e6	1.37249
$430$	490526.	0.127935
$431$	−1.17244e6	−0.304016	−0.152008	−	0.988379i	$-0.548574\pi$
$431$	−1.17244e6	−0.304016	−0.152008	+	0.988379i	$0.548574\pi$
$432$	1.05458e6	0.271877
$433$	−6.83222e6	−1.75123	−0.875613	−	0.483014i	$-0.839542\pi$
$433$	−6.83222e6	−1.75123	−0.875613	+	0.483014i	$0.839542\pi$
$434$	6.70200e6	1.70797
$435$	−4.92507e6	−1.24793
$436$	−3.10848e6	−0.783126
$437$	−4.32243e6	−1.08274
$438$	−3.62276e6	−0.902307
$439$	5.49132e6	1.35993	0.679963	−	0.733246i	$-0.261996\pi$
$439$	5.49132e6	1.35993	0.679963	+	0.733246i	$0.261996\pi$
$440$	1.79888e6	0.442966
$441$	−3.96513e6	−0.970870
$442$	−1.93153e6	−0.470268
$443$	3.85772e6	0.933945	0.466972	−	0.884272i	$-0.345345\pi$
$443$	3.85772e6	0.933945	0.466972	+	0.884272i	$0.345345\pi$
$444$	−759530.	−0.182847
$445$	−1.28944e6	−0.308674
$446$	1.78039e6	0.423817
$447$	1.40510e6	0.332612
$448$	1.01239e6	0.238316
$449$	−3.75507e6	−0.879027	−0.439514	−	0.898236i	$-0.644849\pi$
$449$	−3.75507e6	−0.879027	−0.439514	+	0.898236i	$0.644849\pi$
$450$	−273410.	−0.0636477
$451$	−972265.	−0.225083
$452$	−4.18333e6	−0.963110
$453$	−3.78511e6	−0.866628
$454$	777968.	0.177142
$455$	8.77062e6	1.98610
$456$	941479.	0.212031
$457$	2.92297e6	0.654687	0.327344	−	0.944905i	$-0.393847\pi$
$457$	2.92297e6	0.654687	0.327344	+	0.944905i	$0.393847\pi$
$458$	−108013.	−0.0240608
$459$	−2.72424e6	−0.603550
$460$	2.83022e6	0.623629
$461$	−2.53925e6	−0.556485	−0.278242	−	0.960511i	$-0.589752\pi$
$461$	−2.53925e6	−0.556485	−0.278242	+	0.960511i	$0.589752\pi$
$462$	7.08376e6	1.54404
$463$	−2.02151e6	−0.438251	−0.219126	−	0.975697i	$-0.570320\pi$
$463$	−2.02151e6	−0.438251	−0.219126	+	0.975697i	$0.570320\pi$
$464$	−2.09435e6	−0.451600
$465$	−4.08094e6	−0.875242
$466$	1.69509e6	0.361600
$467$	−2.05096e6	−0.435175	−0.217588	−	0.976041i	$-0.569819\pi$
$467$	−2.05096e6	−0.435175	−0.217588	+	0.976041i	$0.569819\pi$
$468$	1.04609e6	0.220778
$469$	8.41265e6	1.76604
$470$	−126522.	−0.0264192
$471$	4.87321e6	1.01219
$472$	459035.	0.0948398
$473$	1.45953e6	0.299959
$474$	−2.94069e6	−0.601178
$475$	−906511.	−0.184348
$476$	−2.61524e6	−0.529047
$477$	2.78527e6	0.560494
$478$	−389601.	−0.0779921
$479$	99964.4	0.0199070	0.00995352	−	0.999950i	$-0.496832\pi$
$479$	99964.4	0.0199070	0.00995352	+	0.999950i	$0.496832\pi$
$480$	−616458.	−0.122124
$481$	−2.79810e6	−0.551443
$482$	−1.92623e6	−0.377650
$483$	1.11451e7	2.17378
$484$	2.77565e6	0.538582
$485$	5.96684e6	1.15183
$486$	2.55356e6	0.490406
$487$	−5.14751e6	−0.983500	−0.491750	−	0.870736i	$-0.663643\pi$
$487$	−5.14751e6	−0.983500	−0.491750	+	0.870736i	$0.663643\pi$
$488$	−576361.	−0.109558
$489$	−1.77266e6	−0.335239
$490$	8.60814e6	1.61964
$491$	−2.90230e6	−0.543298	−0.271649	−	0.962396i	$-0.587569\pi$
$491$	−2.90230e6	−0.543298	−0.271649	+	0.962396i	$0.587569\pi$
$492$	333186.	0.0620546
$493$	5.41018e6	1.00252
$494$	3.46840e6	0.639458
$495$	2.51672e6	0.461660
$496$	−1.73539e6	−0.316733
$497$	−5.43723e6	−0.987385
$498$	−891762.	−0.161130
$499$	−1.91686e6	−0.344619	−0.172310	−	0.985043i	$-0.555123\pi$
$499$	−1.91686e6	−0.344619	−0.172310	+	0.985043i	$0.555123\pi$
$500$	3.02339e6	0.540840
$501$	−1.42080e6	−0.252894
$502$	−2.87266e6	−0.508774
$503$	3.75083e6	0.661009	0.330504	−	0.943804i	$-0.392781\pi$
$503$	3.75083e6	0.661009	0.330504	+	0.943804i	$0.392781\pi$
$504$	1.41638e6	0.248373
$505$	−557600.	−0.0972959
$506$	8.42118e6	1.46217
$507$	−2.00546e6	−0.346493
$508$	1.60913e6	0.276651
$509$	−803464.	−0.137459	−0.0687293	−	0.997635i	$-0.521894\pi$
$509$	−803464.	−0.137459	−0.0687293	+	0.997635i	$0.521894\pi$
$510$	1.59246e6	0.271108
$511$	−1.80704e7	−3.06137
$512$	−262144.	−0.0441942
$513$	4.89185e6	0.820692
$514$	−98062.0	−0.0163717
$515$	−3.34145e6	−0.555159
$516$	−500168.	−0.0826973
$517$	−376458.	−0.0619427
$518$	−3.78856e6	−0.620368
$519$	−3.67473e6	−0.598834
$520$	−2.27103e6	−0.368310
$521$	−1.17315e7	−1.89348	−0.946738	−	0.322005i	$-0.895643\pi$
$521$	−1.17315e7	−1.89348	−0.946738	+	0.322005i	$0.895643\pi$
$522$	−2.93009e6	−0.470658
$523$	805211.	0.128723	0.0643614	−	0.997927i	$-0.479499\pi$
$523$	805211.	0.128723	0.0643614	+	0.997927i	$0.479499\pi$
$524$	5.81044e6	0.924444
$525$	2.33737e6	0.370109
$526$	3.45023e6	0.543731
$527$	4.48292e6	0.703127
$528$	−1.83424e6	−0.286333
$529$	6.81292e6	1.05851
$530$	−6.04671e6	−0.935039
$531$	642212.	0.0988422
$532$	4.69612e6	0.719384
$533$	1.22745e6	0.187149
$534$	1.31478e6	0.199527
$535$	6.22760e6	0.940668
$536$	−2.17834e6	−0.327501
$537$	−6.03243e6	−0.902727
$538$	−677392.	−0.100898
$539$	2.56131e7	3.79743
$540$	−3.20307e6	−0.472696
$541$	4.98587e6	0.732399	0.366199	−	0.930536i	$-0.380659\pi$
$541$	4.98587e6	0.732399	0.366199	+	0.930536i	$0.380659\pi$
$542$	−2.23546e6	−0.326866
$543$	−8.78609e6	−1.27878
$544$	677179.	0.0981084
$545$	9.44132e6	1.36158
$546$	−8.94302e6	−1.28381
$547$	−1.69661e6	−0.242445	−0.121223	−	0.992625i	$-0.538681\pi$
$547$	−1.69661e6	−0.242445	−0.121223	+	0.992625i	$0.538681\pi$
$548$	−653720.	−0.0929910
$549$	−806358.	−0.114182
$550$	1.76611e6	0.248950
$551$	−9.71494e6	−1.36321
$552$	−2.88586e6	−0.403113
$553$	−1.46682e7	−2.03970
$554$	−2.04248e6	−0.282738
$555$	2.30691e6	0.317905
$556$	6.13707e6	0.841926
$557$	−1.01622e7	−1.38787	−0.693936	−	0.720036i	$-0.744125\pi$
$557$	−1.01622e7	−1.38787	−0.693936	+	0.720036i	$0.744125\pi$
$558$	−2.42789e6	−0.330099
$559$	−1.84261e6	−0.249405
$560$	−3.07491e6	−0.414346
$561$	4.73827e6	0.635642
$562$	−5.95696e6	−0.795580
$563$	7.08368e6	0.941863	0.470931	−	0.882170i	$-0.343918\pi$
$563$	7.08368e6	0.941863	0.470931	+	0.882170i	$0.343918\pi$
$564$	129008.	0.0170773
$565$	1.27059e7	1.67450
$566$	−843792.	−0.110712
$567$	−7.23546e6	−0.945166
$568$	1.40789e6	0.183104
$569$	−1.31059e6	−0.169702	−0.0848509	−	0.996394i	$-0.527041\pi$
$569$	−1.31059e6	−0.169702	−0.0848509	+	0.996394i	$0.527041\pi$
$570$	−2.85954e6	−0.368645
$571$	1.00475e7	1.28964	0.644819	−	0.764336i	$-0.276933\pi$
$571$	1.00475e7	1.28964	0.644819	+	0.764336i	$0.276933\pi$
$572$	−6.75732e6	−0.863544
$573$	7.20961e6	0.917329
$574$	1.66194e6	0.210541
$575$	2.77867e6	0.350483
$576$	−366752.	−0.0460592
$577$	4.30086e6	0.537794	0.268897	−	0.963169i	$-0.413341\pi$
$577$	4.30086e6	0.537794	0.268897	+	0.963169i	$0.413341\pi$
$578$	3.93012e6	0.489312
$579$	58643.2	0.00726978
$580$	6.36112e6	0.785170
$581$	−4.44813e6	−0.546686
$582$	−6.08412e6	−0.744544
$583$	−1.79917e7	−2.19230
$584$	4.67908e6	0.567713
$585$	−3.17728e6	−0.383854
$586$	−7.98979e6	−0.961151
$587$	2.17261e6	0.260247	0.130124	−	0.991498i	$-0.458463\pi$
$587$	2.17261e6	0.260247	0.130124	+	0.991498i	$0.458463\pi$
$588$	−8.77735e6	−1.04694
$589$	−8.04987e6	−0.956094
$590$	−1.39422e6	−0.164892
$591$	2.81233e6	0.331206
$592$	980994.	0.115043
$593$	1.26237e7	1.47418	0.737089	−	0.675795i	$-0.236200\pi$
$593$	1.26237e7	1.47418	0.737089	+	0.675795i	$0.236200\pi$
$594$	−9.53057e6	−1.10829
$595$	7.94321e6	0.919822
$596$	−1.81480e6	−0.209273
$597$	4.61318e6	0.529742
$598$	−1.06315e7	−1.21574
$599$	9.29500e6	1.05848	0.529240	−	0.848472i	$-0.322477\pi$
$599$	9.29500e6	1.05848	0.529240	+	0.848472i	$0.322477\pi$
$600$	−605229.	−0.0686344
$601$	1.39937e6	0.158033	0.0790164	−	0.996873i	$-0.474822\pi$
$601$	1.39937e6	0.158033	0.0790164	+	0.996873i	$0.474822\pi$
$602$	−2.49485e6	−0.280578
$603$	−3.04760e6	−0.341322
$604$	4.88877e6	0.545264
$605$	−8.43044e6	−0.936401
$606$	568561.	0.0628920
$607$	−4.76397e6	−0.524804	−0.262402	−	0.964959i	$-0.584515\pi$
$607$	−4.76397e6	−0.524804	−0.262402	+	0.964959i	$0.584515\pi$
$608$	−1.21599e6	−0.133405
$609$	2.50493e7	2.73686
$610$	1.75057e6	0.190483
$611$	475266.	0.0515032
$612$	947407.	0.102249
$613$	−8.18845e6	−0.880137	−0.440069	−	0.897964i	$-0.645046\pi$
$613$	−8.18845e6	−0.880137	−0.440069	+	0.897964i	$0.645046\pi$
$614$	1.22361e7	1.30985
$615$	−1.01198e6	−0.107891
$616$	−9.14924e6	−0.971478
$617$	1.10755e7	1.17125	0.585624	−	0.810583i	$-0.300850\pi$
$617$	1.10755e7	1.17125	0.585624	+	0.810583i	$0.300850\pi$
$618$	3.40713e6	0.358854
$619$	−1.57167e7	−1.64867	−0.824335	−	0.566102i	$-0.808451\pi$
$619$	−1.57167e7	−1.64867	−0.824335	+	0.566102i	$0.808451\pi$
$620$	5.27086e6	0.550684
$621$	−1.49947e7	−1.56030
$622$	−7.16736e6	−0.742820
$623$	6.55818e6	0.676960
$624$	2.31567e6	0.238076
$625$	−6.79731e6	−0.696045
$626$	2.43162e6	0.248004
$627$	−8.50840e6	−0.864329
$628$	−6.29414e6	−0.636850
$629$	−2.53414e6	−0.255390
$630$	−4.30195e6	−0.431831
$631$	−1.16843e7	−1.16823	−0.584115	−	0.811671i	$-0.698558\pi$
$631$	−1.16843e7	−1.16823	−0.584115	+	0.811671i	$0.698558\pi$
$632$	3.79813e6	0.378249
$633$	−5.48891e6	−0.544473
$634$	4.93042e6	0.487147
$635$	−4.88738e6	−0.480997
$636$	6.16557e6	0.604408
$637$	−3.23357e7	−3.15743
$638$	1.89272e7	1.84092
$639$	1.96971e6	0.190832
$640$	796205.	0.0768378
$641$	4.68810e6	0.450663	0.225331	−	0.974282i	$-0.427654\pi$
$641$	4.68810e6	0.450663	0.225331	+	0.974282i	$0.427654\pi$
$642$	−6.35002e6	−0.608047
$643$	1.02346e7	0.976214	0.488107	−	0.872784i	$-0.337688\pi$
$643$	1.02346e7	0.976214	0.488107	+	0.872784i	$0.337688\pi$
$644$	−1.43947e7	−1.36769
$645$	1.51915e6	0.143781
$646$	3.14120e6	0.296152
$647$	9.69864e6	0.910857	0.455428	−	0.890272i	$-0.349486\pi$
$647$	9.69864e6	0.910857	0.455428	+	0.890272i	$0.349486\pi$
$648$	1.87352e6	0.175275
$649$	−4.14842e6	−0.386608
$650$	−2.22966e6	−0.206993
$651$	2.07560e7	1.91951
$652$	2.28954e6	0.210925
$653$	−2.07808e6	−0.190712	−0.0953562	−	0.995443i	$-0.530399\pi$
$653$	−2.07808e6	−0.190712	−0.0953562	+	0.995443i	$0.530399\pi$
$654$	−9.62691e6	−0.880121
$655$	−1.76479e7	−1.60728
$656$	−430336.	−0.0390434
$657$	6.54627e6	0.591671
$658$	643498.	0.0579405
$659$	1.83169e7	1.64300	0.821501	−	0.570207i	$-0.193137\pi$
$659$	1.83169e7	1.64300	0.821501	+	0.570207i	$0.193137\pi$
$660$	5.57110e6	0.497830
$661$	1.46371e7	1.30302	0.651512	−	0.758638i	$-0.274135\pi$
$661$	1.46371e7	1.30302	0.651512	+	0.758638i	$0.274135\pi$
$662$	1.55785e7	1.38160
$663$	−5.98191e6	−0.528513
$664$	1.15178e6	0.101379
$665$	−1.42634e7	−1.25075
$666$	1.37246e6	0.119898
$667$	2.97786e7	2.59173
$668$	1.83508e6	0.159116
$669$	5.51384e6	0.476309
$670$	6.61622e6	0.569407
$671$	5.20873e6	0.446607
$672$	3.13535e6	0.267833
$673$	−1.79082e6	−0.152411	−0.0762053	−	0.997092i	$-0.524280\pi$
$673$	−1.79082e6	−0.152411	−0.0762053	+	0.997092i	$0.524280\pi$
$674$	−1.04085e7	−0.882550
$675$	−3.14473e6	−0.265658
$676$	2.59021e6	0.218006
$677$	−1.92866e7	−1.61727	−0.808637	−	0.588307i	$-0.799794\pi$
$677$	−1.92866e7	−1.61727	−0.808637	+	0.588307i	$0.799794\pi$
$678$	−1.29557e7	−1.08240
$679$	−3.03478e7	−2.52611
$680$	−2.05678e6	−0.170575
$681$	2.40935e6	0.199082
$682$	1.56832e7	1.29114
$683$	−1.77228e6	−0.145372	−0.0726862	−	0.997355i	$-0.523157\pi$
$683$	−1.77228e6	−0.145372	−0.0726862	+	0.997355i	$0.523157\pi$
$684$	−1.70124e6	−0.139035
$685$	1.98553e6	0.161678
$686$	−2.71652e7	−2.20396
$687$	−334513.	−0.0270409
$688$	646007.	0.0520314
$689$	2.27139e7	1.82282
$690$	8.76516e6	0.700869
$691$	−3.22531e6	−0.256966	−0.128483	−	0.991712i	$-0.541011\pi$
$691$	−3.22531e6	−0.256966	−0.128483	+	0.991712i	$0.541011\pi$
$692$	4.74620e6	0.376774
$693$	−1.28002e7	−1.01248
$694$	−5.82829e6	−0.459349
$695$	−1.86400e7	−1.46381
$696$	−6.48615e6	−0.507533
$697$	1.11166e6	0.0866741
$698$	−4.85864e6	−0.377465
$699$	5.24968e6	0.406387
$700$	−3.01890e6	−0.232865
$701$	−9.00705e6	−0.692289	−0.346145	−	0.938181i	$-0.612509\pi$
$701$	−9.00705e6	−0.692289	−0.346145	+	0.938181i	$0.612509\pi$
$702$	1.20320e7	0.921502
$703$	4.55049e6	0.347272
$704$	2.36907e6	0.180155
$705$	−391835.	−0.0296914
$706$	−870568.	−0.0657341
$707$	2.83600e6	0.213382
$708$	1.42162e6	0.106586
$709$	7.30590e6	0.545831	0.272916	−	0.962038i	$-0.412012\pi$
$709$	7.30590e6	0.545831	0.272916	+	0.962038i	$0.412012\pi$
$710$	−4.27617e6	−0.318353
$711$	5.31377e6	0.394211
$712$	−1.69815e6	−0.125538
$713$	2.46748e7	1.81773
$714$	−8.09935e6	−0.594572
$715$	2.05239e7	1.50139
$716$	7.79136e6	0.567977
$717$	−1.20659e6	−0.0876519
$718$	−1.08003e7	−0.781850
$719$	−2.01264e7	−1.45193	−0.725964	−	0.687733i	$-0.758606\pi$
$719$	−2.01264e7	−1.45193	−0.725964	+	0.687733i	$0.758606\pi$
$720$	1.11393e6	0.0800805
$721$	1.69949e7	1.21753
$722$	4.26382e6	0.304408
$723$	−5.96549e6	−0.424425
$724$	1.13479e7	0.804582
$725$	6.24525e6	0.441270
$726$	8.59615e6	0.605289
$727$	2.23615e7	1.56915	0.784574	−	0.620035i	$-0.212882\pi$
$727$	2.23615e7	1.56915	0.784574	+	0.620035i	$0.212882\pi$
$728$	1.15506e7	0.807750
$729$	1.50219e7	1.04690
$730$	−1.42117e7	−0.987049
$731$	−1.66879e6	−0.115507
$732$	−1.78498e6	−0.123128
$733$	2.07541e7	1.42674	0.713369	−	0.700789i	$-0.247168\pi$
$733$	2.07541e7	1.42674	0.713369	+	0.700789i	$0.247168\pi$
$734$	3.24489e6	0.222310
$735$	2.66593e7	1.82025
$736$	3.72731e6	0.253630
$737$	1.96862e7	1.33504
$738$	−602061.	−0.0406911
$739$	1.47054e7	0.990524	0.495262	−	0.868744i	$-0.335072\pi$
$739$	1.47054e7	0.990524	0.495262	+	0.868744i	$0.335072\pi$
$740$	−2.97955e6	−0.200019
$741$	1.07416e7	0.718658
$742$	3.07540e7	2.05065
$743$	2.37184e7	1.57621	0.788105	−	0.615541i	$-0.211062\pi$
$743$	2.37184e7	1.57621	0.788105	+	0.615541i	$0.211062\pi$
$744$	−5.37447e6	−0.355962
$745$	5.51205e6	0.363850
$746$	7.18690e6	0.472818
$747$	1.61140e6	0.105658
$748$	−6.11985e6	−0.399933
$749$	−3.16741e7	−2.06300
$750$	9.36338e6	0.607826
$751$	−1.62485e7	−1.05127	−0.525635	−	0.850710i	$-0.676172\pi$
$751$	−1.62485e7	−1.05127	−0.525635	+	0.850710i	$0.676172\pi$
$752$	−166625.	−0.0107447
$753$	−8.89658e6	−0.571789
$754$	−2.38949e7	−1.53066
$755$	−1.48486e7	−0.948019
$756$	1.62911e7	1.03668
$757$	−8.02827e6	−0.509193	−0.254596	−	0.967047i	$-0.581943\pi$
$757$	−8.02827e6	−0.509193	−0.254596	+	0.967047i	$0.581943\pi$
$758$	−7.82035e6	−0.494371
$759$	2.60803e7	1.64326
$760$	3.69332e6	0.231944
$761$	−160140.	−0.0100239	−0.00501195	−	0.999987i	$-0.501595\pi$
$761$	−160140.	−0.0100239	−0.00501195	+	0.999987i	$0.501595\pi$
$762$	4.98345e6	0.310916
$763$	−4.80193e7	−2.98610
$764$	−9.31178e6	−0.577164
$765$	−2.87754e6	−0.177774
$766$	2.23498e6	0.137626
$767$	5.23725e6	0.321451
$768$	−811856.	−0.0496679
$769$	−2.65975e7	−1.62191	−0.810953	−	0.585112i	$-0.801051\pi$
$769$	−2.65975e7	−1.62191	−0.810953	+	0.585112i	$0.801051\pi$
$770$	2.77888e7	1.68905
$771$	−303696.	−0.0183994
$772$	−75742.3	−0.00457399
$773$	1.66849e6	0.100433	0.0502164	−	0.998738i	$-0.484009\pi$
$773$	1.66849e6	0.100433	0.0502164	+	0.998738i	$0.484009\pi$
$774$	903795.	0.0542272
$775$	5.17485e6	0.309488
$776$	7.85813e6	0.468452
$777$	−1.17331e7	−0.697205
$778$	6.99854e6	0.414533
$779$	−1.99618e6	−0.117857
$780$	−7.03333e6	−0.413928
$781$	−1.27235e7	−0.746413
$782$	−9.62852e6	−0.563044
$783$	−3.37016e7	−1.96447
$784$	1.13366e7	0.658710
$785$	1.91171e7	1.10725
$786$	1.79948e7	1.03894
$787$	−1.57735e7	−0.907805	−0.453902	−	0.891051i	$-0.649968\pi$
$787$	−1.57735e7	−0.907805	−0.453902	+	0.891051i	$0.649968\pi$
$788$	−3.63235e6	−0.208388
$789$	1.06853e7	0.611075
$790$	−1.15360e7	−0.657639
$791$	−6.46234e7	−3.67239
$792$	3.31444e6	0.187757
$793$	−6.57585e6	−0.371338
$794$	1.45033e7	0.816425
$795$	−1.87266e7	−1.05085
$796$	−5.95828e6	−0.333302
$797$	1.51975e7	0.847473	0.423736	−	0.905786i	$-0.360718\pi$
$797$	1.51975e7	0.847473	0.423736	+	0.905786i	$0.360718\pi$
$798$	1.45438e7	0.808484
$799$	430430.	0.0238526
$800$	781702.	0.0431833
$801$	−2.37579e6	−0.130836
$802$	−1.42861e7	−0.784292
$803$	−4.22861e7	−2.31424
$804$	−6.74627e6	−0.368064
$805$	4.37209e7	2.37793
$806$	−1.97995e7	−1.07354
$807$	−2.09787e6	−0.113395
$808$	−734341.	−0.0395703
$809$	−2.97276e7	−1.59694	−0.798469	−	0.602036i	$-0.794356\pi$
$809$	−2.97276e7	−1.59694	−0.798469	+	0.602036i	$0.794356\pi$
$810$	−5.69040e6	−0.304741
$811$	−6.48896e6	−0.346436	−0.173218	−	0.984884i	$-0.555417\pi$
$811$	−6.48896e6	−0.346436	−0.173218	+	0.984884i	$0.555417\pi$
$812$	−3.23531e7	−1.72197
$813$	−6.92319e6	−0.367350
$814$	−8.86550e6	−0.468967
$815$	−6.95397e6	−0.366723
$816$	2.09721e6	0.110260
$817$	2.99660e6	0.157063
$818$	1.11099e7	0.580532
$819$	1.61599e7	0.841838
$820$	1.30705e6	0.0678825
$821$	2.17528e6	0.112631	0.0563155	−	0.998413i	$-0.482065\pi$
$821$	2.17528e6	0.112631	0.0563155	+	0.998413i	$0.482065\pi$
$822$	−2.02456e6	−0.104508
$823$	−3.19547e7	−1.64450	−0.822252	−	0.569124i	$-0.807283\pi$
$823$	−3.19547e7	−1.64450	−0.822252	+	0.569124i	$0.807283\pi$
$824$	−4.40059e6	−0.225784
$825$	5.46962e6	0.279784
$826$	7.09110e6	0.361629
$827$	2.95696e7	1.50342	0.751712	−	0.659492i	$-0.229229\pi$
$827$	2.95696e7	1.50342	0.751712	+	0.659492i	$0.229229\pi$
$828$	5.21469e6	0.264334
$829$	−5.97210e6	−0.301815	−0.150907	−	0.988548i	$-0.548220\pi$
$829$	−5.97210e6	−0.301815	−0.150907	+	0.988548i	$0.548220\pi$
$830$	−3.49828e6	−0.176262
$831$	−6.32553e6	−0.317757
$832$	−2.99087e6	−0.149792
$833$	−2.92852e7	−1.46230
$834$	1.90064e7	0.946204
$835$	−5.57364e6	−0.276645
$836$	1.09893e7	0.543818
$837$	−2.79253e7	−1.37780
$838$	−1.08059e7	−0.531559
$839$	9.79030e6	0.480166	0.240083	−	0.970752i	$-0.422825\pi$
$839$	9.79030e6	0.480166	0.240083	+	0.970752i	$0.422825\pi$
$840$	−9.52295e6	−0.465665
$841$	4.64183e7	2.26307
$842$	1.53130e6	0.0744357
$843$	−1.84486e7	−0.894117
$844$	7.08936e6	0.342571
$845$	−7.86720e6	−0.379034
$846$	−233116.	−0.0111982
$847$	4.28779e7	2.05364
$848$	−7.96332e6	−0.380281
$849$	−2.61321e6	−0.124424
$850$	−2.01932e6	−0.0958644
$851$	−1.39483e7	−0.660235
$852$	4.36022e6	0.205783
$853$	−6.81098e6	−0.320507	−0.160253	−	0.987076i	$-0.551231\pi$
$853$	−6.81098e6	−0.320507	−0.160253	+	0.987076i	$0.551231\pi$
$854$	−8.90354e6	−0.417751
$855$	5.16713e6	0.241732
$856$	8.20155e6	0.382571
$857$	1.55658e7	0.723970	0.361985	−	0.932184i	$-0.382099\pi$
$857$	1.55658e7	0.723970	0.361985	+	0.932184i	$0.382099\pi$
$858$	−2.09273e7	−0.970499
$859$	3.09084e7	1.42920	0.714601	−	0.699533i	$-0.246608\pi$
$859$	3.09084e7	1.42920	0.714601	+	0.699533i	$0.246608\pi$
$860$	−1.96210e6	−0.0904640
$861$	5.14700e6	0.236617
$862$	4.68975e6	0.214972
$863$	4.79573e6	0.219194	0.109597	−	0.993976i	$-0.465044\pi$
$863$	4.79573e6	0.219194	0.109597	+	0.993976i	$0.465044\pi$
$864$	−4.21834e6	−0.192246
$865$	−1.44155e7	−0.655075
$866$	2.73289e7	1.23830
$867$	1.21715e7	0.549916
$868$	−2.68080e7	−1.20772
$869$	−3.43248e7	−1.54191
$870$	1.97003e7	0.882418
$871$	−2.48532e7	−1.11004
$872$	1.24339e7	0.553754
$873$	1.09939e7	0.488221
$874$	1.72897e7	0.765613
$875$	4.67048e7	2.06225
$876$	1.44910e7	0.638028
$877$	−1.53698e6	−0.0674789	−0.0337395	−	0.999431i	$-0.510742\pi$
$877$	−1.53698e6	−0.0674789	−0.0337395	+	0.999431i	$0.510742\pi$
$878$	−2.19653e7	−0.961613
$879$	−2.47443e7	−1.08020
$880$	−7.19552e6	−0.313224
$881$	2.29260e7	0.995150	0.497575	−	0.867421i	$-0.334224\pi$
$881$	2.29260e7	0.995150	0.497575	+	0.867421i	$0.334224\pi$
$882$	1.58605e7	0.686509
$883$	1.14096e7	0.492459	0.246229	−	0.969212i	$-0.420808\pi$
$883$	1.14096e7	0.492459	0.246229	+	0.969212i	$0.420808\pi$
$884$	7.72611e6	0.332530
$885$	−4.31787e6	−0.185315
$886$	−1.54309e7	−0.660399
$887$	−2.52073e7	−1.07577	−0.537883	−	0.843020i	$-0.680776\pi$
$887$	−2.52073e7	−1.07577	−0.537883	+	0.843020i	$0.680776\pi$
$888$	3.03812e6	0.129292
$889$	2.48576e7	1.05488
$890$	5.15775e6	0.218266
$891$	−1.69315e7	−0.714498
$892$	−7.12156e6	−0.299684
$893$	−772914.	−0.0324342
$894$	−5.62040e6	−0.235193
$895$	−2.36645e7	−0.987508
$896$	−4.04956e6	−0.168515
$897$	−3.29255e7	−1.36632
$898$	1.50203e7	0.621566
$899$	5.54582e7	2.28858
$900$	1.09364e6	0.0450057
$901$	2.05711e7	0.844201
$902$	3.88906e6	0.159158
$903$	−7.72651e6	−0.315329
$904$	1.67333e7	0.681021
$905$	−3.44668e7	−1.39888
$906$	1.51404e7	0.612798
$907$	−4.31698e7	−1.74246	−0.871229	−	0.490877i	$-0.836677\pi$
$907$	−4.31698e7	−1.74246	−0.871229	+	0.490877i	$0.836677\pi$
$908$	−3.11187e6	−0.125258
$909$	−1.02738e6	−0.0412402
$910$	−3.50825e7	−1.40439
$911$	−4.82799e7	−1.92739	−0.963697	−	0.266997i	$-0.913969\pi$
$911$	−4.82799e7	−1.92739	−0.963697	+	0.266997i	$0.913969\pi$
$912$	−3.76592e6	−0.149928
$913$	−1.04090e7	−0.413267
$914$	−1.16919e7	−0.462934
$915$	5.42149e6	0.214075
$916$	432051.	0.0170136
$917$	8.97587e7	3.52495
$918$	1.08970e7	0.426774
$919$	−1.43630e6	−0.0560992	−0.0280496	−	0.999607i	$-0.508930\pi$
$919$	−1.43630e6	−0.0560992	−0.0280496	+	0.999607i	$0.508930\pi$
$920$	−1.13209e7	−0.440972
$921$	3.78950e7	1.47209
$922$	1.01570e7	0.393494
$923$	1.60630e7	0.620616
$924$	−2.83350e7	−1.09180
$925$	−2.92528e6	−0.112412
$926$	8.08604e6	0.309891
$927$	−6.15663e6	−0.235312
$928$	8.37738e6	0.319329
$929$	2.02185e7	0.768617	0.384308	−	0.923205i	$-0.374440\pi$
$929$	2.02185e7	0.768617	0.384308	+	0.923205i	$0.374440\pi$
$930$	1.63238e7	0.618890
$931$	5.25868e7	1.98839
$932$	−6.78037e6	−0.255690
$933$	−2.21972e7	−0.834823
$934$	8.20383e6	0.307715
$935$	1.85877e7	0.695339
$936$	−4.18437e6	−0.156114
$937$	−5.07141e7	−1.88703	−0.943517	−	0.331325i	$-0.892504\pi$
$937$	−5.07141e7	−1.88703	−0.943517	+	0.331325i	$0.892504\pi$
$938$	−3.36506e7	−1.24878
$939$	7.53067e6	0.278721
$940$	506086.	0.0186812
$941$	−2.49099e7	−0.917059	−0.458530	−	0.888679i	$-0.651624\pi$
$941$	−2.49099e7	−0.917059	−0.458530	+	0.888679i	$0.651624\pi$
$942$	−1.94928e7	−0.715727
$943$	6.11876e6	0.224070
$944$	−1.83614e6	−0.0670619
$945$	−4.94805e7	−1.80242
$946$	−5.83813e6	−0.212103
$947$	4.67107e7	1.69255	0.846274	−	0.532748i	$-0.178841\pi$
$947$	4.67107e7	1.69255	0.846274	+	0.532748i	$0.178841\pi$
$948$	1.17628e7	0.425097
$949$	5.33849e7	1.92421
$950$	3.62604e6	0.130354
$951$	1.52694e7	0.547484
$952$	1.04610e7	0.374092
$953$	2.65207e7	0.945916	0.472958	−	0.881085i	$-0.343186\pi$
$953$	2.65207e7	0.945916	0.472958	+	0.881085i	$0.343186\pi$
$954$	−1.11411e7	−0.396329
$955$	2.82825e7	1.00348
$956$	1.55840e6	0.0551487
$957$	5.86171e7	2.06892
$958$	−399858.	−0.0140764
$959$	−1.00986e7	−0.354579
$960$	2.46583e6	0.0863546
$961$	1.73239e7	0.605112
$962$	1.11924e7	0.389929
$963$	1.14744e7	0.398716
$964$	7.70491e6	0.267039
$965$	230051.	0.00795253
$966$	−4.45803e7	−1.53709
$967$	−3.83145e7	−1.31764	−0.658820	−	0.752301i	$-0.728944\pi$
$967$	−3.83145e7	−1.31764	−0.658820	+	0.752301i	$0.728944\pi$
$968$	−1.11026e7	−0.380835
$969$	9.72824e6	0.332832
$970$	−2.38673e7	−0.814469
$971$	−2.31694e7	−0.788619	−0.394309	−	0.918978i	$-0.629016\pi$
$971$	−2.31694e7	−0.788619	−0.394309	+	0.918978i	$0.629016\pi$
$972$	−1.02143e7	−0.346769
$973$	9.48044e7	3.21031
$974$	2.05900e7	0.695440
$975$	−6.90522e6	−0.232630
$976$	2.30544e6	0.0774694
$977$	−2.96660e7	−0.994312	−0.497156	−	0.867661i	$-0.665622\pi$
$977$	−2.96660e7	−0.994312	−0.497156	+	0.867661i	$0.665622\pi$
$978$	7.09066e6	0.237050
$979$	1.53466e7	0.511747
$980$	−3.44326e7	−1.14526
$981$	1.73957e7	0.577123
$982$	1.16092e7	0.384170
$983$	−3.02838e6	−0.0999602	−0.0499801	−	0.998750i	$-0.515916\pi$
$983$	−3.02838e6	−0.0999602	−0.0499801	+	0.998750i	$0.515916\pi$
$984$	−1.33274e6	−0.0438792
$985$	1.10325e7	0.362311
$986$	−2.16407e7	−0.708892
$987$	1.99290e6	0.0651168
$988$	−1.38736e7	−0.452165
$989$	−9.18529e6	−0.298609
$990$	−1.00669e7	−0.326443
$991$	−1.17709e7	−0.380738	−0.190369	−	0.981713i	$-0.560969\pi$
$991$	−1.17709e7	−0.380738	−0.190369	+	0.981713i	$0.560969\pi$
$992$	6.94156e6	0.223964
$993$	4.82465e7	1.55272
$994$	2.17489e7	0.698187
$995$	1.80970e7	0.579493
$996$	3.56705e6	0.113936
$997$	−1.39658e7	−0.444969	−0.222484	−	0.974936i	$-0.571417\pi$
$997$	−1.39658e7	−0.444969	−0.222484	+	0.974936i	$0.571417\pi$
$998$	7.66745e6	0.243683
$999$	1.57858e7	0.500442

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	82.6.a.c.1.2	✓	5
3.2	odd	2		738.6.a.k.1.5		5
4.3	odd	2		656.6.a.c.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
82.6.a.c.1.2	✓	5	1.1	even	1	trivial
656.6.a.c.1.4		5	4.3	odd	2
738.6.a.k.1.5		5	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 82 = 2 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	82.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} -12.3879 q^{3} +16.0000 q^{4} -48.5965 q^{5} +49.5517 q^{6} +247.165 q^{7} -64.0000 q^{8} -89.5391 q^{9} +194.386 q^{10} +578.385 q^{11} -198.207 q^{12} -730.193 q^{13} -988.662 q^{14} +602.010 q^{15} +256.000 q^{16} -661.308 q^{17} +358.157 q^{18} +1187.50 q^{19} -777.544 q^{20} -3061.87 q^{21} -2313.54 q^{22} -3639.95 q^{23} +792.828 q^{24} -763.381 q^{25} +2920.77 q^{26} +4119.47 q^{27} +3954.65 q^{28} -8181.04 q^{29} -2408.04 q^{30} -6778.86 q^{31} -1024.00 q^{32} -7165.00 q^{33} +2645.23 q^{34} -12011.4 q^{35} -1432.63 q^{36} +3832.01 q^{37} -4749.98 q^{38} +9045.58 q^{39} +3110.18 q^{40} -1681.00 q^{41} +12247.5 q^{42} +2523.46 q^{43} +9254.16 q^{44} +4351.29 q^{45} +14559.8 q^{46} -650.878 q^{47} -3171.31 q^{48} +44283.8 q^{49} +3053.52 q^{50} +8192.24 q^{51} -11683.1 q^{52} -31106.7 q^{53} -16477.9 q^{54} -28107.5 q^{55} -15818.6 q^{56} -14710.6 q^{57} +32724.2 q^{58} -7172.42 q^{59} +9632.16 q^{60} +9005.64 q^{61} +27115.5 q^{62} -22131.0 q^{63} +4096.00 q^{64} +35484.8 q^{65} +28660.0 q^{66} +34036.5 q^{67} -10580.9 q^{68} +45091.5 q^{69} +48045.5 q^{70} -21998.3 q^{71} +5730.50 q^{72} -73110.7 q^{73} -15328.0 q^{74} +9456.71 q^{75} +18999.9 q^{76} +142957. q^{77} -36182.3 q^{78} -59345.8 q^{79} -12440.7 q^{80} -29273.7 q^{81} +6724.00 q^{82} -17996.6 q^{83} -48989.9 q^{84} +32137.2 q^{85} -10093.9 q^{86} +101346. q^{87} -37016.7 q^{88} +26533.5 q^{89} -17405.2 q^{90} -180478. q^{91} -58239.3 q^{92} +83976.1 q^{93} +2603.51 q^{94} -57708.1 q^{95} +12685.2 q^{96} -122783. q^{97} -177135. q^{98} -51788.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 20 q^{2} - 10 q^{3} + 80 q^{4} - 38 q^{5} + 40 q^{6} - 38 q^{7} - 320 q^{8} + 265 q^{9} + 152 q^{10} + 416 q^{11} - 160 q^{12} + 268 q^{13} + 152 q^{14} + 22 q^{15} + 1280 q^{16} - 2198 q^{17} - 1060 q^{18}+ \cdots - 139442 q^{99}+O(q^{100}) \) 5 * q - 20 * q^2 - 10 * q^3 + 80 * q^4 - 38 * q^5 + 40 * q^6 - 38 * q^7 - 320 * q^8 + 265 * q^9 + 152 * q^10 + 416 * q^11 - 160 * q^12 + 268 * q^13 + 152 * q^14 + 22 * q^15 + 1280 * q^16 - 2198 * q^17 - 1060 * q^18 - 2250 * q^19 - 608 * q^20 - 6344 * q^21 - 1664 * q^22 - 12008 * q^23 + 640 * q^24 - 9861 * q^25 - 1072 * q^26 - 17962 * q^27 - 608 * q^28 - 11684 * q^29 - 88 * q^30 - 6780 * q^31 - 5120 * q^32 - 15876 * q^33 + 8792 * q^34 - 23158 * q^35 + 4240 * q^36 - 5822 * q^37 + 9000 * q^38 - 14440 * q^39 + 2432 * q^40 - 8405 * q^41 + 25376 * q^42 + 13248 * q^43 + 6656 * q^44 + 5058 * q^45 + 48032 * q^46 - 2000 * q^47 - 2560 * q^48 + 51789 * q^49 + 39444 * q^50 + 36628 * q^51 + 4288 * q^52 - 32036 * q^53 + 71848 * q^54 + 28670 * q^55 + 2432 * q^56 + 5476 * q^57 + 46736 * q^58 - 41376 * q^59 + 352 * q^60 + 45314 * q^61 + 27120 * q^62 + 25568 * q^63 + 20480 * q^64 + 20012 * q^65 + 63504 * q^66 + 47244 * q^67 - 35168 * q^68 + 77140 * q^69 + 92632 * q^70 - 87348 * q^71 - 16960 * q^72 + 44550 * q^73 + 23288 * q^74 + 23370 * q^75 - 36000 * q^76 - 19040 * q^77 + 57760 * q^78 - 6864 * q^79 - 9728 * q^80 + 154261 * q^81 + 33620 * q^82 - 119888 * q^83 - 101504 * q^84 - 22700 * q^85 - 52992 * q^86 - 242980 * q^87 - 26624 * q^88 - 136626 * q^89 - 20232 * q^90 - 30964 * q^91 - 192128 * q^92 - 180684 * q^93 + 8000 * q^94 - 153770 * q^95 + 10240 * q^96 + 7086 * q^97 - 207156 * q^98 - 139442 * q^99

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