LMFDB - Embedded newform 490.6.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [490,6,Mod(1,490)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(490, 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("490.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 490 = 2 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	490.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:	$78.5880717084$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 438x - 1536 $ x^3 - x^2 - 438*x - 1536
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.64808$ of defining polynomial
Character	$\chi$	$=$	490.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} -4.64808 q^{3} +16.0000 q^{4} +25.0000 q^{5} -18.5923 q^{6} +64.0000 q^{8} -221.395 q^{9} +O(q^{10})$	\(q+4.00000 q^{2} -4.64808 q^{3} +16.0000 q^{4} +25.0000 q^{5} -18.5923 q^{6} +64.0000 q^{8} -221.395 q^{9} +100.000 q^{10} +394.589 q^{11} -74.3693 q^{12} +837.521 q^{13} -116.202 q^{15} +256.000 q^{16} -1328.25 q^{17} -885.581 q^{18} -674.397 q^{19} +400.000 q^{20} +1578.35 q^{22} +3135.55 q^{23} -297.477 q^{24} +625.000 q^{25} +3350.08 q^{26} +2158.55 q^{27} -8053.53 q^{29} -464.808 q^{30} +9768.93 q^{31} +1024.00 q^{32} -1834.08 q^{33} -5313.01 q^{34} -3542.33 q^{36} -8847.77 q^{37} -2697.59 q^{38} -3892.86 q^{39} +1600.00 q^{40} +34.8719 q^{41} +5707.72 q^{43} +6313.42 q^{44} -5534.88 q^{45} +12542.2 q^{46} +28075.8 q^{47} -1189.91 q^{48} +2500.00 q^{50} +6173.82 q^{51} +13400.3 q^{52} +17211.3 q^{53} +8634.19 q^{54} +9864.72 q^{55} +3134.65 q^{57} -32214.1 q^{58} +27955.0 q^{59} -1859.23 q^{60} +628.995 q^{61} +39075.7 q^{62} +4096.00 q^{64} +20938.0 q^{65} -7336.32 q^{66} -47091.4 q^{67} -21252.0 q^{68} -14574.3 q^{69} -4491.20 q^{71} -14169.3 q^{72} +60781.0 q^{73} -35391.1 q^{74} -2905.05 q^{75} -10790.3 q^{76} -15571.5 q^{78} -17617.6 q^{79} +6400.00 q^{80} +43766.0 q^{81} +139.487 q^{82} -26491.6 q^{83} -33206.3 q^{85} +22830.9 q^{86} +37433.5 q^{87} +25253.7 q^{88} -60919.6 q^{89} -22139.5 q^{90} +50168.8 q^{92} -45406.7 q^{93} +112303. q^{94} -16859.9 q^{95} -4759.63 q^{96} +27843.7 q^{97} -87360.1 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 12 q^{2} - 2 q^{3} + 48 q^{4} + 75 q^{5} - 8 q^{6} + 192 q^{8} + 149 q^{9}+O(q^{10}) $ 3 * q + 12 * q^2 - 2 * q^3 + 48 * q^4 + 75 * q^5 - 8 * q^6 + 192 * q^8 + 149 * q^9	\( 3 q + 12 q^{2} - 2 q^{3} + 48 q^{4} + 75 q^{5} - 8 q^{6} + 192 q^{8} + 149 q^{9} + 300 q^{10} - 144 q^{11} - 32 q^{12} + 288 q^{13} - 50 q^{15} + 768 q^{16} + 1182 q^{17} + 596 q^{18} + 1134 q^{19} + 1200 q^{20} - 576 q^{22} + 1872 q^{23} - 128 q^{24} + 1875 q^{25} + 1152 q^{26} + 4264 q^{27} - 5724 q^{29} - 200 q^{30} + 2184 q^{31} + 3072 q^{32} + 14258 q^{33} + 4728 q^{34} + 2384 q^{36} + 10698 q^{37} + 4536 q^{38} + 3956 q^{39} + 4800 q^{40} + 6204 q^{41} - 10728 q^{43} - 2304 q^{44} + 3725 q^{45} + 7488 q^{46} + 23556 q^{47} - 512 q^{48} + 7500 q^{50} - 12220 q^{51} + 4608 q^{52} - 13218 q^{53} + 17056 q^{54} - 3600 q^{55} + 7740 q^{57} - 22896 q^{58} + 83226 q^{59} - 800 q^{60} + 24330 q^{61} + 8736 q^{62} + 12288 q^{64} + 7200 q^{65} + 57032 q^{66} - 37836 q^{67} + 18912 q^{68} + 3812 q^{69} - 996 q^{71} + 9536 q^{72} + 167256 q^{73} + 42792 q^{74} - 1250 q^{75} + 18144 q^{76} + 15824 q^{78} - 8796 q^{79} + 19200 q^{80} - 89761 q^{81} + 24816 q^{82} - 2418 q^{83} + 29550 q^{85} - 42912 q^{86} + 296652 q^{87} - 9216 q^{88} - 14292 q^{89} + 14900 q^{90} + 29952 q^{92} - 94712 q^{93} + 94224 q^{94} + 28350 q^{95} - 2048 q^{96} + 103710 q^{97} - 142604 q^{99}+O(q^{100}) \) 3 * q + 12 * q^2 - 2 * q^3 + 48 * q^4 + 75 * q^5 - 8 * q^6 + 192 * q^8 + 149 * q^9 + 300 * q^10 - 144 * q^11 - 32 * q^12 + 288 * q^13 - 50 * q^15 + 768 * q^16 + 1182 * q^17 + 596 * q^18 + 1134 * q^19 + 1200 * q^20 - 576 * q^22 + 1872 * q^23 - 128 * q^24 + 1875 * q^25 + 1152 * q^26 + 4264 * q^27 - 5724 * q^29 - 200 * q^30 + 2184 * q^31 + 3072 * q^32 + 14258 * q^33 + 4728 * q^34 + 2384 * q^36 + 10698 * q^37 + 4536 * q^38 + 3956 * q^39 + 4800 * q^40 + 6204 * q^41 - 10728 * q^43 - 2304 * q^44 + 3725 * q^45 + 7488 * q^46 + 23556 * q^47 - 512 * q^48 + 7500 * q^50 - 12220 * q^51 + 4608 * q^52 - 13218 * q^53 + 17056 * q^54 - 3600 * q^55 + 7740 * q^57 - 22896 * q^58 + 83226 * q^59 - 800 * q^60 + 24330 * q^61 + 8736 * q^62 + 12288 * q^64 + 7200 * q^65 + 57032 * q^66 - 37836 * q^67 + 18912 * q^68 + 3812 * q^69 - 996 * q^71 + 9536 * q^72 + 167256 * q^73 + 42792 * q^74 - 1250 * q^75 + 18144 * q^76 + 15824 * q^78 - 8796 * q^79 + 19200 * q^80 - 89761 * q^81 + 24816 * q^82 - 2418 * q^83 + 29550 * q^85 - 42912 * q^86 + 296652 * q^87 - 9216 * q^88 - 14292 * q^89 + 14900 * q^90 + 29952 * q^92 - 94712 * q^93 + 94224 * q^94 + 28350 * q^95 - 2048 * q^96 + 103710 * q^97 - 142604 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−4.64808	−0.298174	−0.149087	−	0.988824i	$-0.547634\pi$
$3$	−4.64808	−0.298174	−0.149087	+	0.988824i	$0.547634\pi$
$4$	16.0000	0.500000
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	−18.5923	−0.210841
$7$	0	0
$7$	0	0
$8$	64.0000	0.353553
$9$	−221.395	−0.911092
$10$	100.000	0.316228
$11$	394.589	0.983248	0.491624	−	0.870808i	$-0.336404\pi$
$11$	394.589	0.983248	0.491624	+	0.870808i	$0.336404\pi$
$12$	−74.3693	−0.149087
$13$	837.521	1.37448	0.687238	−	0.726432i	$-0.258823\pi$
$13$	837.521	1.37448	0.687238	+	0.726432i	$0.258823\pi$
$14$	0	0
$15$	−116.202	−0.133348
$16$	256.000	0.250000
$17$	−1328.25	−1.11470	−0.557350	−	0.830278i	$-0.688182\pi$
$17$	−1328.25	−1.11470	−0.557350	+	0.830278i	$0.688182\pi$
$18$	−885.581	−0.644239
$19$	−674.397	−0.428580	−0.214290	−	0.976770i	$-0.568744\pi$
$19$	−674.397	−0.428580	−0.214290	+	0.976770i	$0.568744\pi$
$20$	400.000	0.223607
$21$	0	0
$22$	1578.35	0.695261
$23$	3135.55	1.23593	0.617965	−	0.786205i	$-0.287957\pi$
$23$	3135.55	1.23593	0.617965	+	0.786205i	$0.287957\pi$
$24$	−297.477	−0.105421
$25$	625.000	0.200000
$26$	3350.08	0.971901
$27$	2158.55	0.569839
$28$	0	0
$29$	−8053.53	−1.77824	−0.889122	−	0.457670i	$-0.848684\pi$
$29$	−8053.53	−1.77824	−0.889122	+	0.457670i	$0.848684\pi$
$30$	−464.808	−0.0942910
$31$	9768.93	1.82576	0.912878	−	0.408233i	$-0.133855\pi$
$31$	9768.93	1.82576	0.912878	+	0.408233i	$0.133855\pi$
$32$	1024.00	0.176777
$33$	−1834.08	−0.293179
$34$	−5313.01	−0.788212
$35$	0	0
$36$	−3542.33	−0.455546
$37$	−8847.77	−1.06250	−0.531250	−	0.847215i	$-0.678278\pi$
$37$	−8847.77	−1.06250	−0.531250	+	0.847215i	$0.678278\pi$
$38$	−2697.59	−0.303052
$39$	−3892.86	−0.409834
$40$	1600.00	0.158114
$41$	34.8719	0.00323978	0.00161989	−	0.999999i	$-0.499484\pi$
$41$	34.8719	0.00323978	0.00161989	+	0.999999i	$0.499484\pi$
$42$	0	0
$43$	5707.72	0.470752	0.235376	−	0.971904i	$-0.424368\pi$
$43$	5707.72	0.470752	0.235376	+	0.971904i	$0.424368\pi$
$44$	6313.42	0.491624
$45$	−5534.88	−0.407453
$46$	12542.2	0.873935
$47$	28075.8	1.85390	0.926952	−	0.375180i	$-0.122419\pi$
$47$	28075.8	1.85390	0.926952	+	0.375180i	$0.122419\pi$
$48$	−1189.91	−0.0745436
$49$	0	0
$50$	2500.00	0.141421
$51$	6173.82	0.332375
$52$	13400.3	0.687238
$53$	17211.3	0.841638	0.420819	−	0.907145i	$-0.361743\pi$
$53$	17211.3	0.841638	0.420819	+	0.907145i	$0.361743\pi$
$54$	8634.19	0.402937
$55$	9864.72	0.439722
$56$	0	0
$57$	3134.65	0.127791
$58$	−32214.1	−1.25741
$59$	27955.0	1.04551	0.522757	−	0.852482i	$-0.324903\pi$
$59$	27955.0	1.04551	0.522757	+	0.852482i	$0.324903\pi$
$60$	−1859.23	−0.0666738
$61$	628.995	0.0216433	0.0108216	−	0.999941i	$-0.496555\pi$
$61$	628.995	0.0216433	0.0108216	+	0.999941i	$0.496555\pi$
$62$	39075.7	1.29100
$63$	0	0
$64$	4096.00	0.125000
$65$	20938.0	0.614684
$66$	−7336.32	−0.207309
$67$	−47091.4	−1.28161	−0.640803	−	0.767706i	$-0.721398\pi$
$67$	−47091.4	−1.28161	−0.640803	+	0.767706i	$0.721398\pi$
$68$	−21252.0	−0.557350
$69$	−14574.3	−0.368523
$70$	0	0
$71$	−4491.20	−0.105735	−0.0528673	−	0.998602i	$-0.516836\pi$
$71$	−4491.20	−0.105735	−0.0528673	+	0.998602i	$0.516836\pi$
$72$	−14169.3	−0.322120
$73$	60781.0	1.33494	0.667468	−	0.744638i	$-0.267378\pi$
$73$	60781.0	1.33494	0.667468	+	0.744638i	$0.267378\pi$
$74$	−35391.1	−0.751302
$75$	−2905.05	−0.0596349
$76$	−10790.3	−0.214290
$77$	0	0
$78$	−15571.5	−0.289796
$79$	−17617.6	−0.317598	−0.158799	−	0.987311i	$-0.550762\pi$
$79$	−17617.6	−0.317598	−0.158799	+	0.987311i	$0.550762\pi$
$80$	6400.00	0.111803
$81$	43766.0	0.741181
$82$	139.487	0.00229087
$83$	−26491.6	−0.422097	−0.211049	−	0.977476i	$-0.567688\pi$
$83$	−26491.6	−0.422097	−0.211049	+	0.977476i	$0.567688\pi$
$84$	0	0
$85$	−33206.3	−0.498509
$86$	22830.9	0.332872
$87$	37433.5	0.530227
$88$	25253.7	0.347631
$89$	−60919.6	−0.815233	−0.407617	−	0.913153i	$-0.633640\pi$
$89$	−60919.6	−0.815233	−0.407617	+	0.913153i	$0.633640\pi$
$90$	−22139.5	−0.288113
$91$	0	0
$92$	50168.8	0.617965
$93$	−45406.7	−0.544394
$94$	112303.	1.31091
$95$	−16859.9	−0.191667
$96$	−4759.63	−0.0527103
$97$	27843.7	0.300468	0.150234	−	0.988650i	$-0.451997\pi$
$97$	27843.7	0.300468	0.150234	+	0.988650i	$0.451997\pi$
$98$	0	0
$99$	−87360.1	−0.895829
$100$	10000.0	0.100000
$101$	192626.	1.87893	0.939467	−	0.342640i	$-0.111321\pi$
$101$	192626.	1.87893	0.939467	+	0.342640i	$0.111321\pi$
$102$	24695.3	0.235025
$103$	127671.	1.18576	0.592882	−	0.805290i	$-0.297990\pi$
$103$	127671.	1.18576	0.592882	+	0.805290i	$0.297990\pi$
$104$	53601.3	0.485951
$105$	0	0
$106$	68845.4	0.595128
$107$	86802.4	0.732947	0.366473	−	0.930429i	$-0.380565\pi$
$107$	86802.4	0.732947	0.366473	+	0.930429i	$0.380565\pi$
$108$	34536.7	0.284919
$109$	34447.9	0.277713	0.138857	−	0.990313i	$-0.455657\pi$
$109$	34447.9	0.277713	0.138857	+	0.990313i	$0.455657\pi$
$110$	39458.9	0.310930
$111$	41125.1	0.316811
$112$	0	0
$113$	221654.	1.63297	0.816486	−	0.577366i	$-0.195919\pi$
$113$	221654.	1.63297	0.816486	+	0.577366i	$0.195919\pi$
$114$	12538.6	0.0903622
$115$	78388.7	0.552725
$116$	−128857.	−0.889122
$117$	−185423.	−1.25227
$118$	111820.	0.739290
$119$	0	0
$120$	−7436.93	−0.0471455
$121$	−5350.74	−0.0332239
$122$	2515.98	0.0153041
$123$	−162.087	−0.000966020 0
$124$	156303.	0.912878
$125$	15625.0	0.0894427
$126$	0	0
$127$	−67269.5	−0.370091	−0.185046	−	0.982730i	$-0.559243\pi$
$127$	−67269.5	−0.370091	−0.185046	+	0.982730i	$0.559243\pi$
$128$	16384.0	0.0883883
$129$	−26529.9	−0.140366
$130$	83752.1	0.434647
$131$	−202386.	−1.03039	−0.515195	−	0.857073i	$-0.672280\pi$
$131$	−202386.	−1.03039	−0.515195	+	0.857073i	$0.672280\pi$
$132$	−29345.3	−0.146590
$133$	0	0
$134$	−188365.	−0.906232
$135$	53963.7	0.254840
$136$	−85008.1	−0.394106
$137$	101041.	0.459933	0.229967	−	0.973199i	$-0.426138\pi$
$137$	101041.	0.459933	0.229967	+	0.973199i	$0.426138\pi$
$138$	−58297.1	−0.260585
$139$	357608.	1.56989	0.784946	−	0.619564i	$-0.212691\pi$
$139$	357608.	1.56989	0.784946	+	0.619564i	$0.212691\pi$
$140$	0	0
$141$	−130498.	−0.552787
$142$	−17964.8	−0.0747656
$143$	330476.	1.35145
$144$	−56677.2	−0.227773
$145$	−201338.	−0.795255
$146$	243124.	0.943942
$147$	0	0
$148$	−141564.	−0.531250
$149$	321387.	1.18594	0.592969	−	0.805225i	$-0.297956\pi$
$149$	321387.	1.18594	0.592969	+	0.805225i	$0.297956\pi$
$150$	−11620.2	−0.0421682
$151$	−169217.	−0.603950	−0.301975	−	0.953316i	$-0.597646\pi$
$151$	−169217.	−0.603950	−0.301975	+	0.953316i	$0.597646\pi$
$152$	−43161.4	−0.151526
$153$	294069.	1.01559
$154$	0	0
$155$	244223.	0.816503
$156$	−62285.8	−0.204917
$157$	357834.	1.15860	0.579299	−	0.815115i	$-0.303326\pi$
$157$	357834.	1.15860	0.579299	+	0.815115i	$0.303326\pi$
$158$	−70470.3	−0.224576
$159$	−79999.7	−0.250955
$160$	25600.0	0.0790569
$161$	0	0
$162$	175064.	0.524094
$163$	−144826.	−0.426950	−0.213475	−	0.976949i	$-0.568478\pi$
$163$	−144826.	−0.426950	−0.213475	+	0.976949i	$0.568478\pi$
$164$	557.950	0.00161989
$165$	−45852.0	−0.131114
$166$	−105966.	−0.298468
$167$	181648.	0.504011	0.252006	−	0.967726i	$-0.418910\pi$
$167$	181648.	0.504011	0.252006	+	0.967726i	$0.418910\pi$
$168$	0	0
$169$	330148.	0.889184
$170$	−132825.	−0.352499
$171$	149308.	0.390475
$172$	91323.6	0.235376
$173$	−555255.	−1.41051	−0.705257	−	0.708952i	$-0.749168\pi$
$173$	−555255.	−1.41051	−0.705257	+	0.708952i	$0.749168\pi$
$174$	149734.	0.374927
$175$	0	0
$176$	101015.	0.245812
$177$	−129937.	−0.311746
$178$	−243678.	−0.576457
$179$	697804.	1.62780	0.813900	−	0.581005i	$-0.197340\pi$
$179$	697804.	1.62780	0.813900	+	0.581005i	$0.197340\pi$
$180$	−88558.1	−0.203726
$181$	71536.7	0.162305	0.0811526	−	0.996702i	$-0.474140\pi$
$181$	71536.7	0.162305	0.0811526	+	0.996702i	$0.474140\pi$
$182$	0	0
$183$	−2923.62	−0.00645347
$184$	200675.	0.436967
$185$	−221194.	−0.475165
$186$	−181627.	−0.384944
$187$	−524113.	−1.09603
$188$	449213.	0.926952
$189$	0	0
$190$	−67439.7	−0.135529
$191$	−589929.	−1.17008	−0.585041	−	0.811004i	$-0.698922\pi$
$191$	−589929.	−1.17008	−0.585041	+	0.811004i	$0.698922\pi$
$192$	−19038.5	−0.0372718
$193$	4131.03	0.00798298	0.00399149	−	0.999992i	$-0.498729\pi$
$193$	4131.03	0.00798298	0.00399149	+	0.999992i	$0.498729\pi$
$194$	111375.	0.212463
$195$	−97321.6	−0.183283
$196$	0	0
$197$	−869836.	−1.59688	−0.798439	−	0.602076i	$-0.794340\pi$
$197$	−869836.	−1.59688	−0.798439	+	0.602076i	$0.794340\pi$
$198$	−349440.	−0.633447
$199$	870534.	1.55831	0.779154	−	0.626833i	$-0.215649\pi$
$199$	870534.	1.55831	0.779154	+	0.626833i	$0.215649\pi$
$200$	40000.0	0.0707107
$201$	218884.	0.382142
$202$	770504.	1.32861
$203$	0	0
$204$	98781.1	0.166188
$205$	871.797	0.00144887
$206$	510683.	0.838461
$207$	−694196.	−1.12605
$208$	214405.	0.343619
$209$	−266109.	−0.421400
$210$	0	0
$211$	132411.	0.204747	0.102374	−	0.994746i	$-0.467356\pi$
$211$	132411.	0.204747	0.102374	+	0.994746i	$0.467356\pi$
$212$	275382.	0.420819
$213$	20875.5	0.0315273
$214$	347210.	0.518272
$215$	142693.	0.210526
$216$	138147.	0.201468
$217$	0	0
$218$	137792.	0.196373
$219$	−282515.	−0.398044
$220$	157835.	0.219861
$221$	−1.11244e6	−1.53213
$222$	164500.	0.224019
$223$	−560626.	−0.754938	−0.377469	−	0.926022i	$-0.623206\pi$
$223$	−560626.	−0.754938	−0.377469	+	0.926022i	$0.623206\pi$
$224$	0	0
$225$	−138372.	−0.182218
$226$	886614.	1.15468
$227$	−623657.	−0.803306	−0.401653	−	0.915792i	$-0.631564\pi$
$227$	−623657.	−0.803306	−0.401653	+	0.915792i	$0.631564\pi$
$228$	50154.4	0.0638957
$229$	−948401.	−1.19510	−0.597549	−	0.801833i	$-0.703859\pi$
$229$	−948401.	−1.19510	−0.597549	+	0.801833i	$0.703859\pi$
$230$	313555.	0.390836
$231$	0	0
$232$	−515426.	−0.628704
$233$	63389.4	0.0764939	0.0382469	−	0.999268i	$-0.487823\pi$
$233$	63389.4	0.0764939	0.0382469	+	0.999268i	$0.487823\pi$
$234$	−741693.	−0.885491
$235$	701895.	0.829091
$236$	447281.	0.522757
$237$	81887.9	0.0946997
$238$	0	0
$239$	−668798.	−0.757357	−0.378678	−	0.925528i	$-0.623621\pi$
$239$	−668798.	−0.757357	−0.378678	+	0.925528i	$0.623621\pi$
$240$	−29747.7	−0.0333369
$241$	−192786.	−0.213813	−0.106906	−	0.994269i	$-0.534094\pi$
$241$	−192786.	−0.213813	−0.106906	+	0.994269i	$0.534094\pi$
$242$	−21403.0	−0.0234928
$243$	−727955.	−0.790840
$244$	10063.9	0.0108216
$245$	0	0
$246$	−648.349	−0.000683079 0
$247$	−564821.	−0.589072
$248$	625211.	0.645502
$249$	123135.	0.125859
$250$	62500.0	0.0632456
$251$	−917135.	−0.918859	−0.459430	−	0.888214i	$-0.651946\pi$
$251$	−917135.	−0.918859	−0.459430	+	0.888214i	$0.651946\pi$
$252$	0	0
$253$	1.23725e6	1.21523
$254$	−269078.	−0.261694
$255$	154346.	0.148643
$256$	65536.0	0.0625000
$257$	−275088.	−0.259800	−0.129900	−	0.991527i	$-0.541466\pi$
$257$	−275088.	−0.259800	−0.129900	+	0.991527i	$0.541466\pi$
$258$	−106120.	−0.0992538
$259$	0	0
$260$	335008.	0.307342
$261$	1.78301e6	1.62014
$262$	−809542.	−0.728595
$263$	178152.	0.158818	0.0794092	−	0.996842i	$-0.474697\pi$
$263$	178152.	0.158818	0.0794092	+	0.996842i	$0.474697\pi$
$264$	−117381.	−0.103655
$265$	430284.	0.376392
$266$	0	0
$267$	283159.	0.243082
$268$	−753462.	−0.640803
$269$	1.18713e6	1.00027	0.500133	−	0.865948i	$-0.333284\pi$
$269$	1.18713e6	1.00027	0.500133	+	0.865948i	$0.333284\pi$
$270$	215855.	0.180199
$271$	−2.16649e6	−1.79198	−0.895990	−	0.444073i	$-0.853533\pi$
$271$	−2.16649e6	−1.79198	−0.895990	+	0.444073i	$0.853533\pi$
$272$	−340033.	−0.278675
$273$	0	0
$274$	404163.	0.325222
$275$	246618.	0.196650
$276$	−233189.	−0.184261
$277$	838481.	0.656590	0.328295	−	0.944575i	$-0.393526\pi$
$277$	838481.	0.656590	0.328295	+	0.944575i	$0.393526\pi$
$278$	1.43043e6	1.11008
$279$	−2.16279e6	−1.66343
$280$	0	0
$281$	−965395.	−0.729356	−0.364678	−	0.931134i	$-0.618821\pi$
$281$	−965395.	−0.729356	−0.364678	+	0.931134i	$0.618821\pi$
$282$	−521994.	−0.390879
$283$	−66704.0	−0.0495092	−0.0247546	−	0.999694i	$-0.507880\pi$
$283$	−66704.0	−0.0495092	−0.0247546	+	0.999694i	$0.507880\pi$
$284$	−71859.2	−0.0528673
$285$	78366.2	0.0571501
$286$	1.32190e6	0.955620
$287$	0	0
$288$	−226709.	−0.161060
$289$	344396.	0.242557
$290$	−805353.	−0.562330
$291$	−129420.	−0.0895919
$292$	972495.	0.667468
$293$	−1.94134e6	−1.32109	−0.660545	−	0.750787i	$-0.729675\pi$
$293$	−1.94134e6	−1.32109	−0.660545	+	0.750787i	$0.729675\pi$
$294$	0	0
$295$	698876.	0.467568
$296$	−566257.	−0.375651
$297$	851738.	0.560293
$298$	1.28555e6	0.838585
$299$	2.62609e6	1.69876
$300$	−46480.8	−0.0298174
$301$	0	0
$302$	−676867.	−0.427057
$303$	−895341.	−0.560250
$304$	−172646.	−0.107145
$305$	15724.9	0.00967916
$306$	1.17628e6	0.718134
$307$	−644714.	−0.390410	−0.195205	−	0.980762i	$-0.562537\pi$
$307$	−644714.	−0.390410	−0.195205	+	0.980762i	$0.562537\pi$
$308$	0	0
$309$	−593423.	−0.353564
$310$	976893.	0.577355
$311$	−636135.	−0.372948	−0.186474	−	0.982460i	$-0.559706\pi$
$311$	−636135.	−0.372948	−0.186474	+	0.982460i	$0.559706\pi$
$312$	−249143.	−0.144898
$313$	2.24505e6	1.29529	0.647643	−	0.761944i	$-0.275755\pi$
$313$	2.24505e6	1.29529	0.647643	+	0.761944i	$0.275755\pi$
$314$	1.43134e6	0.819253
$315$	0	0
$316$	−281881.	−0.158799
$317$	−1.15545e6	−0.645806	−0.322903	−	0.946432i	$-0.604659\pi$
$317$	−1.15545e6	−0.645806	−0.322903	+	0.946432i	$0.604659\pi$
$318$	−319999.	−0.177452
$319$	−3.17783e6	−1.74845
$320$	102400.	0.0559017
$321$	−403465.	−0.218546
$322$	0	0
$323$	895769.	0.477738
$324$	700256.	0.370590
$325$	523450.	0.274895
$326$	−579303.	−0.301899
$327$	−160116.	−0.0828069
$328$	2231.80	0.00114544
$329$	0	0
$330$	−183408.	−0.0927115
$331$	2.17600e6	1.09166	0.545832	−	0.837895i	$-0.316214\pi$
$331$	2.17600e6	1.09166	0.545832	+	0.837895i	$0.316214\pi$
$332$	−423865.	−0.211049
$333$	1.95885e6	0.968036
$334$	726593.	0.356390
$335$	−1.17728e6	−0.573151
$336$	0	0
$337$	−362513.	−0.173880	−0.0869399	−	0.996214i	$-0.527709\pi$
$337$	−362513.	−0.173880	−0.0869399	+	0.996214i	$0.527709\pi$
$338$	1.32059e6	0.628748
$339$	−1.03026e6	−0.486910
$340$	−531301.	−0.249255
$341$	3.85471e6	1.79517
$342$	597233.	0.276108
$343$	0	0
$344$	365294.	0.166436
$345$	−364357.	−0.164808
$346$	−2.22102e6	−0.997384
$347$	−213411.	−0.0951464	−0.0475732	−	0.998868i	$-0.515149\pi$
$347$	−213411.	−0.0951464	−0.0475732	+	0.998868i	$0.515149\pi$
$348$	598935.	0.265113
$349$	−1.49494e6	−0.656990	−0.328495	−	0.944506i	$-0.606541\pi$
$349$	−1.49494e6	−0.656990	−0.328495	+	0.944506i	$0.606541\pi$
$350$	0	0
$351$	1.80783e6	0.783230
$352$	404059.	0.173815
$353$	−1.04707e6	−0.447240	−0.223620	−	0.974676i	$-0.571787\pi$
$353$	−1.04707e6	−0.447240	−0.223620	+	0.974676i	$0.571787\pi$
$354$	−519749.	−0.220438
$355$	−112280.	−0.0472859
$356$	−974713.	−0.407617
$357$	0	0
$358$	2.79122e6	1.15103
$359$	−151663.	−0.0621076	−0.0310538	−	0.999518i	$-0.509886\pi$
$359$	−151663.	−0.0621076	−0.0310538	+	0.999518i	$0.509886\pi$
$360$	−354233.	−0.144056
$361$	−2.02129e6	−0.816320
$362$	286147.	0.114767
$363$	24870.7	0.00990652
$364$	0	0
$365$	1.51952e6	0.597002
$366$	−11694.5	−0.00456329
$367$	−2.35117e6	−0.911212	−0.455606	−	0.890181i	$-0.650577\pi$
$367$	−2.35117e6	−0.911212	−0.455606	+	0.890181i	$0.650577\pi$
$368$	802701.	0.308983
$369$	−7720.47	−0.00295174
$370$	−884777.	−0.335992
$371$	0	0
$372$	−726508.	−0.272197
$373$	−3.80222e6	−1.41503	−0.707515	−	0.706698i	$-0.750184\pi$
$373$	−3.80222e6	−1.41503	−0.707515	+	0.706698i	$0.750184\pi$
$374$	−2.09645e6	−0.775008
$375$	−72626.2	−0.0266695
$376$	1.79685e6	0.655454
$377$	−6.74500e6	−2.44415
$378$	0	0
$379$	−4.03663e6	−1.44351	−0.721757	−	0.692146i	$-0.756665\pi$
$379$	−4.03663e6	−1.44351	−0.721757	+	0.692146i	$0.756665\pi$
$380$	−269759.	−0.0958333
$381$	312674.	0.110352
$382$	−2.35972e6	−0.827373
$383$	−2.18801e6	−0.762171	−0.381085	−	0.924540i	$-0.624450\pi$
$383$	−2.18801e6	−0.762171	−0.381085	+	0.924540i	$0.624450\pi$
$384$	−76154.1	−0.0263551
$385$	0	0
$386$	16524.1	0.00564482
$387$	−1.26366e6	−0.428898
$388$	445500.	0.150234
$389$	2.93226e6	0.982491	0.491246	−	0.871021i	$-0.336542\pi$
$389$	2.93226e6	0.982491	0.491246	+	0.871021i	$0.336542\pi$
$390$	−389286.	−0.129601
$391$	−4.16480e6	−1.37769
$392$	0	0
$393$	940704.	0.307236
$394$	−3.47934e6	−1.12916
$395$	−440439.	−0.142034
$396$	−1.39776e6	−0.447915
$397$	−3.82235e6	−1.21718	−0.608590	−	0.793485i	$-0.708264\pi$
$397$	−3.82235e6	−1.21718	−0.608590	+	0.793485i	$0.708264\pi$
$398$	3.48214e6	1.10189
$399$	0	0
$400$	160000.	0.0500000
$401$	4.86541e6	1.51098	0.755490	−	0.655161i	$-0.227399\pi$
$401$	4.86541e6	1.51098	0.755490	+	0.655161i	$0.227399\pi$
$402$	875538.	0.270215
$403$	8.18168e6	2.50946
$404$	3.08202e6	0.939467
$405$	1.09415e6	0.331466
$406$	0	0
$407$	−3.49123e6	−1.04470
$408$	395125.	0.117512
$409$	−2.97218e6	−0.878552	−0.439276	−	0.898352i	$-0.644765\pi$
$409$	−2.97218e6	−0.878552	−0.439276	+	0.898352i	$0.644765\pi$
$410$	3487.19	0.00102451
$411$	−469645.	−0.137140
$412$	2.04273e6	0.592882
$413$	0	0
$414$	−2.77678e6	−0.796235
$415$	−662289.	−0.188768
$416$	857621.	0.242975
$417$	−1.66219e6	−0.468102
$418$	−1.06444e6	−0.297975
$419$	1.35527e6	0.377128	0.188564	−	0.982061i	$-0.439617\pi$
$419$	1.35527e6	0.377128	0.188564	+	0.982061i	$0.439617\pi$
$420$	0	0
$421$	4.43802e6	1.22035	0.610175	−	0.792267i	$-0.291099\pi$
$421$	4.43802e6	1.22035	0.610175	+	0.792267i	$0.291099\pi$
$422$	529644.	0.144778
$423$	−6.21585e6	−1.68908
$424$	1.10153e6	0.297564
$425$	−830157.	−0.222940
$426$	83501.9	0.0222932
$427$	0	0
$428$	1.38884e6	0.366473
$429$	−1.53608e6	−0.402968
$430$	570772.	0.148865
$431$	−6.55501e6	−1.69973	−0.849865	−	0.527001i	$-0.823317\pi$
$431$	−6.55501e6	−1.69973	−0.849865	+	0.527001i	$0.823317\pi$
$432$	552588.	0.142460
$433$	−3.76486e6	−0.965003	−0.482502	−	0.875895i	$-0.660272\pi$
$433$	−3.76486e6	−0.965003	−0.482502	+	0.875895i	$0.660272\pi$
$434$	0	0
$435$	935836.	0.237125
$436$	551166.	0.138857
$437$	−2.11460e6	−0.529695
$438$	−1.13006e6	−0.281459
$439$	−1.55115e6	−0.384144	−0.192072	−	0.981381i	$-0.561521\pi$
$439$	−1.55115e6	−0.384144	−0.192072	+	0.981381i	$0.561521\pi$
$440$	631342.	0.155465
$441$	0	0
$442$	−4.44975e6	−1.08338
$443$	3.22557e6	0.780904	0.390452	−	0.920623i	$-0.372319\pi$
$443$	3.22557e6	0.780904	0.390452	+	0.920623i	$0.372319\pi$
$444$	658002.	0.158405
$445$	−1.52299e6	−0.364583
$446$	−2.24251e6	−0.533822
$447$	−1.49383e6	−0.353617
$448$	0	0
$449$	1.28331e6	0.300410	0.150205	−	0.988655i	$-0.452007\pi$
$449$	1.28331e6	0.300410	0.150205	+	0.988655i	$0.452007\pi$
$450$	−553488.	−0.128848
$451$	13760.0	0.00318551
$452$	3.54646e6	0.816486
$453$	786533.	0.180083
$454$	−2.49463e6	−0.568023
$455$	0	0
$456$	200618.	0.0451811
$457$	3.54457e6	0.793912	0.396956	−	0.917838i	$-0.370066\pi$
$457$	3.54457e6	0.793912	0.396956	+	0.917838i	$0.370066\pi$
$458$	−3.79360e6	−0.845062
$459$	−2.86709e6	−0.635200
$460$	1.25422e6	0.276362
$461$	3.28279e6	0.719434	0.359717	−	0.933061i	$-0.382873\pi$
$461$	3.28279e6	0.719434	0.359717	+	0.933061i	$0.382873\pi$
$462$	0	0
$463$	−5.90114e6	−1.27933	−0.639666	−	0.768653i	$-0.720927\pi$
$463$	−5.90114e6	−1.27933	−0.639666	+	0.768653i	$0.720927\pi$
$464$	−2.06170e6	−0.444561
$465$	−1.13517e6	−0.243460
$466$	253558.	0.0540893
$467$	8.89304e6	1.88694	0.943469	−	0.331460i	$-0.107541\pi$
$467$	8.89304e6	1.88694	0.943469	+	0.331460i	$0.107541\pi$
$468$	−2.96677e6	−0.626137
$469$	0	0
$470$	2.80758e6	0.586256
$471$	−1.66324e6	−0.345465
$472$	1.78912e6	0.369645
$473$	2.25220e6	0.462865
$474$	327551.	0.0669628
$475$	−421498.	−0.0857159
$476$	0	0
$477$	−3.81051e6	−0.766809
$478$	−2.67519e6	−0.535532
$479$	2.33411e6	0.464817	0.232408	−	0.972618i	$-0.425339\pi$
$479$	2.33411e6	0.464817	0.232408	+	0.972618i	$0.425339\pi$
$480$	−118991.	−0.0235728
$481$	−7.41019e6	−1.46038
$482$	−771145.	−0.151188
$483$	0	0
$484$	−85611.9	−0.0166120
$485$	696094.	0.134373
$486$	−2.91182e6	−0.559208
$487$	7.91210e6	1.51171	0.755857	−	0.654737i	$-0.227221\pi$
$487$	7.91210e6	1.51171	0.755857	+	0.654737i	$0.227221\pi$
$488$	40255.7	0.00765205
$489$	673162.	0.127306
$490$	0	0
$491$	1.40334e6	0.262699	0.131349	−	0.991336i	$-0.458069\pi$
$491$	1.40334e6	0.262699	0.131349	+	0.991336i	$0.458069\pi$
$492$	−2593.40	−0.000483010 0
$493$	1.06971e7	1.98221
$494$	−2.25928e6	−0.416537
$495$	−2.18400e6	−0.400627
$496$	2.50085e6	0.456439
$497$	0	0
$498$	492540.	0.0889954
$499$	538241.	0.0967666	0.0483833	−	0.998829i	$-0.484593\pi$
$499$	538241.	0.0967666	0.0483833	+	0.998829i	$0.484593\pi$
$500$	250000.	0.0447214
$501$	−844316.	−0.150283
$502$	−3.66854e6	−0.649732
$503$	−383280.	−0.0675455	−0.0337728	−	0.999430i	$-0.510752\pi$
$503$	−383280.	−0.0675455	−0.0337728	+	0.999430i	$0.510752\pi$
$504$	0	0
$505$	4.81565e6	0.840285
$506$	4.94901e6	0.859295
$507$	−1.53455e6	−0.265132
$508$	−1.07631e6	−0.185046
$509$	−1.16377e7	−1.99100	−0.995500	−	0.0947589i	$-0.969792\pi$
$509$	−1.16377e7	−1.99100	−0.995500	+	0.0947589i	$0.969792\pi$
$510$	617382.	0.105106
$511$	0	0
$512$	262144.	0.0441942
$513$	−1.45572e6	−0.244221
$514$	−1.10035e6	−0.183706
$515$	3.19177e6	0.530290
$516$	−424479.	−0.0701830
$517$	1.10784e7	1.82285
$518$	0	0
$519$	2.58087e6	0.420579
$520$	1.34003e6	0.217324
$521$	−5.77834e6	−0.932628	−0.466314	−	0.884619i	$-0.654418\pi$
$521$	−5.77834e6	−0.932628	−0.466314	+	0.884619i	$0.654418\pi$
$522$	7.13206e6	1.14561
$523$	1.69185e6	0.270463	0.135232	−	0.990814i	$-0.456822\pi$
$523$	1.69185e6	0.270463	0.135232	+	0.990814i	$0.456822\pi$
$524$	−3.23817e6	−0.515195
$525$	0	0
$526$	712608.	0.112302
$527$	−1.29756e7	−2.03517
$528$	−469524.	−0.0732948
$529$	3.39533e6	0.527524
$530$	1.72113e6	0.266149
$531$	−6.18912e6	−0.952560
$532$	0	0
$533$	29205.9	0.00445300
$534$	1.13264e6	0.171885
$535$	2.17006e6	0.327784
$536$	−3.01385e6	−0.453116
$537$	−3.24345e6	−0.485368
$538$	4.74850e6	0.707295
$539$	0	0
$540$	863419.	0.127420
$541$	1.14959e7	1.68868	0.844342	−	0.535805i	$-0.179992\pi$
$541$	1.14959e7	1.68868	0.844342	+	0.535805i	$0.179992\pi$
$542$	−8.66596e6	−1.26712
$543$	−332508.	−0.0483952
$544$	−1.36013e6	−0.197053
$545$	861197.	0.124197
$546$	0	0
$547$	−8.75977e6	−1.25177	−0.625885	−	0.779915i	$-0.715262\pi$
$547$	−8.75977e6	−1.25177	−0.625885	+	0.779915i	$0.715262\pi$
$548$	1.61665e6	0.229967
$549$	−139257.	−0.0197190
$550$	986472.	0.139052
$551$	5.43128e6	0.762119
$552$	−932754.	−0.130293
$553$	0	0
$554$	3.35393e6	0.464279
$555$	1.02813e6	0.141682
$556$	5.72172e6	0.784946
$557$	−1.18871e7	−1.62345	−0.811725	−	0.584039i	$-0.801471\pi$
$557$	−1.18871e7	−1.62345	−0.811725	+	0.584039i	$0.801471\pi$
$558$	−8.65118e6	−1.17622
$559$	4.78034e6	0.647037
$560$	0	0
$561$	2.43612e6	0.326807
$562$	−3.86158e6	−0.515732
$563$	−4.25802e6	−0.566157	−0.283079	−	0.959097i	$-0.591356\pi$
$563$	−4.25802e6	−0.566157	−0.283079	+	0.959097i	$0.591356\pi$
$564$	−2.08798e6	−0.276393
$565$	5.54134e6	0.730287
$566$	−266816.	−0.0350083
$567$	0	0
$568$	−287437.	−0.0373828
$569$	−1.50508e7	−1.94885	−0.974425	−	0.224712i	$-0.927856\pi$
$569$	−1.50508e7	−1.94885	−0.974425	+	0.224712i	$0.927856\pi$
$570$	313465.	0.0404112
$571$	−1.18959e7	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$571$	−1.18959e7	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$572$	5.28762e6	0.675725
$573$	2.74204e6	0.348889
$574$	0	0
$575$	1.95972e6	0.247186
$576$	−906835.	−0.113886
$577$	1.25552e7	1.56994	0.784969	−	0.619535i	$-0.212679\pi$
$577$	1.25552e7	1.56994	0.784969	+	0.619535i	$0.212679\pi$
$578$	1.37759e6	0.171514
$579$	−19201.4	−0.00238032
$580$	−3.22141e6	−0.397627
$581$	0	0
$582$	−517680.	−0.0633510
$583$	6.79140e6	0.827538
$584$	3.88998e6	0.471971
$585$	−4.63558e6	−0.560034
$586$	−7.76536e6	−0.934151
$587$	1.49646e7	1.79254	0.896271	−	0.443506i	$-0.146266\pi$
$587$	1.49646e7	1.79254	0.896271	+	0.443506i	$0.146266\pi$
$588$	0	0
$589$	−6.58813e6	−0.782482
$590$	2.79550e6	0.330621
$591$	4.04307e6	0.476148
$592$	−2.26503e6	−0.265625
$593$	−1.11149e6	−0.129799	−0.0648993	−	0.997892i	$-0.520673\pi$
$593$	−1.11149e6	−0.129799	−0.0648993	+	0.997892i	$0.520673\pi$
$594$	3.40695e6	0.396187
$595$	0	0
$596$	5.14219e6	0.592969
$597$	−4.04631e6	−0.464647
$598$	1.05043e7	1.20120
$599$	1.35953e7	1.54819	0.774093	−	0.633072i	$-0.218206\pi$
$599$	1.35953e7	1.54819	0.774093	+	0.633072i	$0.218206\pi$
$600$	−185923.	−0.0210841
$601$	1.45048e7	1.63804	0.819021	−	0.573763i	$-0.194517\pi$
$601$	1.45048e7	1.63804	0.819021	+	0.573763i	$0.194517\pi$
$602$	0	0
$603$	1.04258e7	1.16766
$604$	−2.70747e6	−0.301975
$605$	−133769.	−0.0148582
$606$	−3.58136e6	−0.396157
$607$	−587517.	−0.0647215	−0.0323608	−	0.999476i	$-0.510303\pi$
$607$	−587517.	−0.0647215	−0.0323608	+	0.999476i	$0.510303\pi$
$608$	−690582.	−0.0757629
$609$	0	0
$610$	62899.5	0.00684420
$611$	2.35140e7	2.54815
$612$	4.70510e6	0.507797
$613$	−5.86854e6	−0.630782	−0.315391	−	0.948962i	$-0.602136\pi$
$613$	−5.86854e6	−0.630782	−0.315391	+	0.948962i	$0.602136\pi$
$614$	−2.57885e6	−0.276062
$615$	−4052.18	−0.000432017 0
$616$	0	0
$617$	−1.65388e7	−1.74901	−0.874505	−	0.485017i	$-0.838813\pi$
$617$	−1.65388e7	−1.74901	−0.874505	+	0.485017i	$0.838813\pi$
$618$	−2.37369e6	−0.250008
$619$	302199.	0.0317005	0.0158503	−	0.999874i	$-0.494954\pi$
$619$	302199.	0.0317005	0.0158503	+	0.999874i	$0.494954\pi$
$620$	3.90757e6	0.408251
$621$	6.76823e6	0.704281
$622$	−2.54454e6	−0.263714
$623$	0	0
$624$	−996573.	−0.102458
$625$	390625.	0.0400000
$626$	8.98021e6	0.915906
$627$	1.23690e6	0.125651
$628$	5.72535e6	0.579299
$629$	1.17521e7	1.18437
$630$	0	0
$631$	1.79953e7	1.79923	0.899613	−	0.436688i	$-0.143849\pi$
$631$	1.79953e7	1.79923	0.899613	+	0.436688i	$0.143849\pi$
$632$	−1.12752e6	−0.112288
$633$	−615457.	−0.0610504
$634$	−4.62179e6	−0.456654
$635$	−1.68174e6	−0.165510
$636$	−1.28000e6	−0.125477
$637$	0	0
$638$	−1.27113e7	−1.23634
$639$	994331.	0.0963339
$640$	409600.	0.0395285
$641$	6.87969e6	0.661339	0.330669	−	0.943747i	$-0.392725\pi$
$641$	6.87969e6	0.661339	0.330669	+	0.943747i	$0.392725\pi$
$642$	−1.61386e6	−0.154535
$643$	5.79235e6	0.552493	0.276247	−	0.961087i	$-0.410909\pi$
$643$	5.79235e6	0.552493	0.276247	+	0.961087i	$0.410909\pi$
$644$	0	0
$645$	−663249.	−0.0627736
$646$	3.58308e6	0.337812
$647$	9.51698e6	0.893796	0.446898	−	0.894585i	$-0.352529\pi$
$647$	9.51698e6	0.893796	0.446898	+	0.894585i	$0.352529\pi$
$648$	2.80102e6	0.262047
$649$	1.10307e7	1.02800
$650$	2.09380e6	0.194380
$651$	0	0
$652$	−2.31721e6	−0.213475
$653$	−8.53081e6	−0.782902	−0.391451	−	0.920199i	$-0.628027\pi$
$653$	−8.53081e6	−0.782902	−0.391451	+	0.920199i	$0.628027\pi$
$654$	−640466.	−0.0585533
$655$	−5.05964e6	−0.460804
$656$	8927.20	0.000809945 0
$657$	−1.34566e7	−1.21625
$658$	0	0
$659$	−4.29154e6	−0.384946	−0.192473	−	0.981302i	$-0.561651\pi$
$659$	−4.29154e6	−0.384946	−0.192473	+	0.981302i	$0.561651\pi$
$660$	−733632.	−0.0655569
$661$	1.34304e7	1.19559	0.597797	−	0.801647i	$-0.296043\pi$
$661$	1.34304e7	1.19559	0.597797	+	0.801647i	$0.296043\pi$
$662$	8.70400e6	0.771923
$663$	5.17070e6	0.456842
$664$	−1.69546e6	−0.149234
$665$	0	0
$666$	7.83542e6	0.684505
$667$	−2.52522e7	−2.19779
$668$	2.90637e6	0.252006
$669$	2.60584e6	0.225103
$670$	−4.70914e6	−0.405279
$671$	248194.	0.0212807
$672$	0	0
$673$	6.49021e6	0.552359	0.276179	−	0.961106i	$-0.410932\pi$
$673$	6.49021e6	0.552359	0.276179	+	0.961106i	$0.410932\pi$
$674$	−1.45005e6	−0.122952
$675$	1.34909e6	0.113968
$676$	5.28236e6	0.444592
$677$	6.08808e6	0.510515	0.255258	−	0.966873i	$-0.417840\pi$
$677$	6.08808e6	0.510515	0.255258	+	0.966873i	$0.417840\pi$
$678$	−4.12105e6	−0.344298
$679$	0	0
$680$	−2.12520e6	−0.176250
$681$	2.89881e6	0.239525
$682$	1.54188e7	1.26938
$683$	626787.	0.0514125	0.0257062	−	0.999670i	$-0.491817\pi$
$683$	626787.	0.0514125	0.0257062	+	0.999670i	$0.491817\pi$
$684$	2.38893e6	0.195238
$685$	2.52602e6	0.205688
$686$	0	0
$687$	4.40824e6	0.356348
$688$	1.46118e6	0.117688
$689$	1.44149e7	1.15681
$690$	−1.45743e6	−0.116537
$691$	−9.60387e6	−0.765158	−0.382579	−	0.923923i	$-0.624964\pi$
$691$	−9.60387e6	−0.765158	−0.382579	+	0.923923i	$0.624964\pi$
$692$	−8.88408e6	−0.705257
$693$	0	0
$694$	−853643.	−0.0672787
$695$	8.94019e6	0.702077
$696$	2.39574e6	0.187464
$697$	−46318.6	−0.00361138
$698$	−5.97974e6	−0.464562
$699$	−294639.	−0.0228085
$700$	0	0
$701$	2.24366e7	1.72449	0.862247	−	0.506488i	$-0.169056\pi$
$701$	2.24366e7	1.72449	0.862247	+	0.506488i	$0.169056\pi$
$702$	7.23131e6	0.553827
$703$	5.96690e6	0.455366
$704$	1.61624e6	0.122906
$705$	−3.26246e6	−0.247214
$706$	−4.18829e6	−0.316246
$707$	0	0
$708$	−2.07900e6	−0.155873
$709$	2.08674e7	1.55902	0.779511	−	0.626388i	$-0.215467\pi$
$709$	2.08674e7	1.55902	0.779511	+	0.626388i	$0.215467\pi$
$710$	−449120.	−0.0334362
$711$	3.90045e6	0.289361
$712$	−3.89885e6	−0.288228
$713$	3.06310e7	2.25651
$714$	0	0
$715$	8.26190e6	0.604387
$716$	1.11649e7	0.813900
$717$	3.10863e6	0.225824
$718$	−606653.	−0.0439167
$719$	1.03499e7	0.746643	0.373322	−	0.927702i	$-0.378219\pi$
$719$	1.03499e7	0.746643	0.373322	+	0.927702i	$0.378219\pi$
$720$	−1.41693e6	−0.101863
$721$	0	0
$722$	−8.08515e6	−0.577225
$723$	896085.	0.0637534
$724$	1.14459e6	0.0811526
$725$	−5.03346e6	−0.355649
$726$	99482.7	0.00700497
$727$	−1.70257e7	−1.19473	−0.597364	−	0.801970i	$-0.703785\pi$
$727$	−1.70257e7	−1.19473	−0.597364	+	0.801970i	$0.703785\pi$
$728$	0	0
$729$	−7.25154e6	−0.505372
$730$	6.07810e6	0.422144
$731$	−7.58129e6	−0.524747
$732$	−46777.9	−0.00322673
$733$	−1.70108e7	−1.16940	−0.584701	−	0.811249i	$-0.698788\pi$
$733$	−1.70108e7	−1.16940	−0.584701	+	0.811249i	$0.698788\pi$
$734$	−9.40469e6	−0.644324
$735$	0	0
$736$	3.21080e6	0.218484
$737$	−1.85817e7	−1.26014
$738$	−30881.9	−0.00208719
$739$	−5.59388e6	−0.376792	−0.188396	−	0.982093i	$-0.560329\pi$
$739$	−5.59388e6	−0.376792	−0.188396	+	0.982093i	$0.560329\pi$
$740$	−3.53911e6	−0.237582
$741$	2.62533e6	0.175646
$742$	0	0
$743$	−2.91460e7	−1.93690	−0.968450	−	0.249208i	$-0.919830\pi$
$743$	−2.91460e7	−1.93690	−0.968450	+	0.249208i	$0.919830\pi$
$744$	−2.90603e6	−0.192472
$745$	8.03467e6	0.530368
$746$	−1.52089e7	−1.00058
$747$	5.86511e6	0.384569
$748$	−8.38581e6	−0.548013
$749$	0	0
$750$	−290505.	−0.0188582
$751$	−1.08792e7	−0.703881	−0.351940	−	0.936022i	$-0.614478\pi$
$751$	−1.08792e7	−0.703881	−0.351940	+	0.936022i	$0.614478\pi$
$752$	7.18740e6	0.463476
$753$	4.26292e6	0.273980
$754$	−2.69800e7	−1.72828
$755$	−4.23042e6	−0.270095
$756$	0	0
$757$	−2.53390e7	−1.60713	−0.803563	−	0.595220i	$-0.797065\pi$
$757$	−2.53390e7	−1.60713	−0.803563	+	0.595220i	$0.797065\pi$
$758$	−1.61465e7	−1.02072
$759$	−5.75085e6	−0.362349
$760$	−1.07903e6	−0.0677644
$761$	1.62613e7	1.01788	0.508938	−	0.860803i	$-0.330038\pi$
$761$	1.62613e7	1.01788	0.508938	+	0.860803i	$0.330038\pi$
$762$	1.25070e6	0.0780305
$763$	0	0
$764$	−9.43886e6	−0.585041
$765$	7.35172e6	0.454188
$766$	−8.75204e6	−0.538936
$767$	2.34129e7	1.43703
$768$	−304617.	−0.0186359
$769$	2.32964e7	1.42060	0.710302	−	0.703897i	$-0.248559\pi$
$769$	2.32964e7	1.42060	0.710302	+	0.703897i	$0.248559\pi$
$770$	0	0
$771$	1.27863e6	0.0774656
$772$	66096.5	0.00399149
$773$	−1.50117e7	−0.903607	−0.451804	−	0.892117i	$-0.649219\pi$
$773$	−1.50117e7	−0.903607	−0.451804	+	0.892117i	$0.649219\pi$
$774$	−5.05465e6	−0.303277
$775$	6.10558e6	0.365151
$776$	1.78200e6	0.106231
$777$	0	0
$778$	1.17290e7	0.694726
$779$	−23517.5	−0.00138850
$780$	−1.55715e6	−0.0916416
$781$	−1.77218e6	−0.103963
$782$	−1.66592e7	−0.974176
$783$	−1.73839e7	−1.01331
$784$	0	0
$785$	8.94586e6	0.518141
$786$	3.76282e6	0.217248
$787$	−1.98401e7	−1.14184	−0.570922	−	0.821004i	$-0.693414\pi$
$787$	−1.98401e7	−1.14184	−0.570922	+	0.821004i	$0.693414\pi$
$788$	−1.39174e7	−0.798439
$789$	−828064.	−0.0473556
$790$	−1.76176e6	−0.100433
$791$	0	0
$792$	−5.59105e6	−0.316723
$793$	526796.	0.0297481
$794$	−1.52894e7	−0.860676
$795$	−1.99999e6	−0.112230
$796$	1.39285e7	0.779154
$797$	3.01598e7	1.68183	0.840917	−	0.541164i	$-0.182016\pi$
$797$	3.01598e7	1.68183	0.840917	+	0.541164i	$0.182016\pi$
$798$	0	0
$799$	−3.72917e7	−2.06655
$800$	640000.	0.0353553
$801$	1.34873e7	0.742752
$802$	1.94616e7	1.06842
$803$	2.39835e7	1.31257
$804$	3.50215e6	0.191071
$805$	0	0
$806$	3.27267e7	1.77445
$807$	−5.51785e6	−0.298254
$808$	1.23281e7	0.664303
$809$	1.23997e7	0.666101	0.333050	−	0.942909i	$-0.391922\pi$
$809$	1.23997e7	0.666101	0.333050	+	0.942909i	$0.391922\pi$
$810$	4.37660e6	0.234382
$811$	1.46016e7	0.779559	0.389779	−	0.920908i	$-0.372551\pi$
$811$	1.46016e7	0.779559	0.389779	+	0.920908i	$0.372551\pi$
$812$	0	0
$813$	1.00700e7	0.534323
$814$	−1.39649e7	−0.738716
$815$	−3.62065e6	−0.190938
$816$	1.58050e6	0.0830938
$817$	−3.84927e6	−0.201754
$818$	−1.18887e7	−0.621230
$819$	0	0
$820$	13948.7	0.000724437 0
$821$	−9.00439e6	−0.466226	−0.233113	−	0.972450i	$-0.574891\pi$
$821$	−9.00439e6	−0.466226	−0.233113	+	0.972450i	$0.574891\pi$
$822$	−1.87858e6	−0.0969728
$823$	−1.25209e7	−0.644371	−0.322186	−	0.946677i	$-0.604418\pi$
$823$	−1.25209e7	−0.644371	−0.322186	+	0.946677i	$0.604418\pi$
$824$	8.17092e6	0.419231
$825$	−1.14630e6	−0.0586359
$826$	0	0
$827$	1.39981e7	0.711712	0.355856	−	0.934541i	$-0.384189\pi$
$827$	1.39981e7	0.711712	0.355856	+	0.934541i	$0.384189\pi$
$828$	−1.11071e7	−0.563023
$829$	3.07176e7	1.55239	0.776195	−	0.630493i	$-0.217147\pi$
$829$	3.07176e7	1.55239	0.776195	+	0.630493i	$0.217147\pi$
$830$	−2.64916e6	−0.133479
$831$	−3.89733e6	−0.195778
$832$	3.43048e6	0.171809
$833$	0	0
$834$	−6.64875e6	−0.330998
$835$	4.54121e6	0.225401
$836$	−4.25775e6	−0.210700
$837$	2.10867e7	1.04039
$838$	5.42106e6	0.266670
$839$	−1.22269e7	−0.599667	−0.299833	−	0.953992i	$-0.596931\pi$
$839$	−1.22269e7	−0.599667	−0.299833	+	0.953992i	$0.596931\pi$
$840$	0	0
$841$	4.43482e7	2.16215
$842$	1.77521e7	0.862918
$843$	4.48723e6	0.217475
$844$	2.11858e6	0.102374
$845$	8.25370e6	0.397655
$846$	−2.48634e7	−1.19436
$847$	0	0
$848$	4.40611e6	0.210409
$849$	310045.	0.0147624
$850$	−3.32063e6	−0.157642
$851$	−2.77426e7	−1.31318
$852$	334007.	0.0157637
$853$	−4.50757e6	−0.212114	−0.106057	−	0.994360i	$-0.533823\pi$
$853$	−4.50757e6	−0.212114	−0.106057	+	0.994360i	$0.533823\pi$
$854$	0	0
$855$	3.73271e6	0.174626
$856$	5.55536e6	0.259136
$857$	−8.11790e6	−0.377565	−0.188782	−	0.982019i	$-0.560454\pi$
$857$	−8.11790e6	−0.377565	−0.188782	+	0.982019i	$0.560454\pi$
$858$	−6.14432e6	−0.284941
$859$	−6.47162e6	−0.299247	−0.149623	−	0.988743i	$-0.547806\pi$
$859$	−6.47162e6	−0.299247	−0.149623	+	0.988743i	$0.547806\pi$
$860$	2.28309e6	0.105263
$861$	0	0
$862$	−2.62200e7	−1.20189
$863$	2.36975e7	1.08312	0.541558	−	0.840664i	$-0.317835\pi$
$863$	2.36975e7	1.08312	0.541558	+	0.840664i	$0.317835\pi$
$864$	2.21035e6	0.100734
$865$	−1.38814e7	−0.630801
$866$	−1.50594e7	−0.682360
$867$	−1.60078e6	−0.0723243
$868$	0	0
$869$	−6.95169e6	−0.312278
$870$	3.74335e6	0.167672
$871$	−3.94400e7	−1.76154
$872$	2.20466e6	0.0981864
$873$	−6.16448e6	−0.273754
$874$	−8.45842e6	−0.374551
$875$	0	0
$876$	−4.52024e6	−0.199022
$877$	−3.07810e7	−1.35140	−0.675701	−	0.737176i	$-0.736159\pi$
$877$	−3.07810e7	−1.35140	−0.675701	+	0.737176i	$0.736159\pi$
$878$	−6.20461e6	−0.271630
$879$	9.02350e6	0.393915
$880$	2.52537e6	0.109930
$881$	7.91877e6	0.343730	0.171865	−	0.985120i	$-0.445021\pi$
$881$	7.91877e6	0.343730	0.171865	+	0.985120i	$0.445021\pi$
$882$	0	0
$883$	−1.50199e7	−0.648282	−0.324141	−	0.946009i	$-0.605075\pi$
$883$	−1.50199e7	−0.648282	−0.324141	+	0.946009i	$0.605075\pi$
$884$	−1.77990e7	−0.766064
$885$	−3.24843e6	−0.139417
$886$	1.29023e7	0.552183
$887$	5.38207e6	0.229689	0.114844	−	0.993383i	$-0.463363\pi$
$887$	5.38207e6	0.229689	0.114844	+	0.993383i	$0.463363\pi$
$888$	2.63201e6	0.112009
$889$	0	0
$890$	−6.09196e6	−0.257799
$891$	1.72696e7	0.728764
$892$	−8.97002e6	−0.377469
$893$	−1.89342e7	−0.794545
$894$	−5.97532e6	−0.250045
$895$	1.74451e7	0.727974
$896$	0	0
$897$	−1.22063e7	−0.506526
$898$	5.13322e6	0.212422
$899$	−7.86744e7	−3.24664
$900$	−2.21395e6	−0.0911092
$901$	−2.28610e7	−0.938174
$902$	55040.2	0.00225249
$903$	0	0
$904$	1.41858e7	0.577342
$905$	1.78842e6	0.0725851
$906$	3.14613e6	0.127338
$907$	−2.11264e7	−0.852723	−0.426361	−	0.904553i	$-0.640205\pi$
$907$	−2.11264e7	−0.852723	−0.426361	+	0.904553i	$0.640205\pi$
$908$	−9.97851e6	−0.401653
$909$	−4.26465e7	−1.71188
$910$	0	0
$911$	4.55219e7	1.81729	0.908645	−	0.417570i	$-0.137118\pi$
$911$	4.55219e7	1.81729	0.908645	+	0.417570i	$0.137118\pi$
$912$	802470.	0.0319479
$913$	−1.04533e7	−0.415026
$914$	1.41783e7	0.561381
$915$	−73090.5	−0.00288608
$916$	−1.51744e7	−0.597549
$917$	0	0
$918$	−1.14684e7	−0.449154
$919$	5.89538e6	0.230262	0.115131	−	0.993350i	$-0.463271\pi$
$919$	5.89538e6	0.230262	0.115131	+	0.993350i	$0.463271\pi$
$920$	5.01688e6	0.195418
$921$	2.99668e6	0.116410
$922$	1.31312e7	0.508717
$923$	−3.76147e6	−0.145330
$924$	0	0
$925$	−5.52985e6	−0.212500
$926$	−2.36046e7	−0.904624
$927$	−2.82657e7	−1.08034
$928$	−8.24682e6	−0.314352
$929$	1.82133e7	0.692389	0.346195	−	0.938163i	$-0.387474\pi$
$929$	1.82133e7	0.692389	0.346195	+	0.938163i	$0.387474\pi$
$930$	−4.54067e6	−0.172152
$931$	0	0
$932$	1.01423e6	0.0382469
$933$	2.95681e6	0.111204
$934$	3.55721e7	1.33427
$935$	−1.31028e7	−0.490158
$936$	−1.18671e7	−0.442746
$937$	4.77248e6	0.177580	0.0887902	−	0.996050i	$-0.471700\pi$
$937$	4.77248e6	0.177580	0.0887902	+	0.996050i	$0.471700\pi$
$938$	0	0
$939$	−1.04352e7	−0.386221
$940$	1.12303e7	0.414546
$941$	−2.89165e7	−1.06456	−0.532281	−	0.846567i	$-0.678665\pi$
$941$	−2.89165e7	−1.06456	−0.532281	+	0.846567i	$0.678665\pi$
$942$	−6.65297e6	−0.244280
$943$	109342.	0.00400414
$944$	7.15649e6	0.261379
$945$	0	0
$946$	9.00881e6	0.327295
$947$	−1.75917e7	−0.637431	−0.318716	−	0.947850i	$-0.603251\pi$
$947$	−1.75917e7	−0.637431	−0.318716	+	0.947850i	$0.603251\pi$
$948$	1.31021e6	0.0473499
$949$	5.09053e7	1.83484
$950$	−1.68599e6	−0.0606103
$951$	5.37061e6	0.192563
$952$	0	0
$953$	−4.46816e7	−1.59366	−0.796831	−	0.604202i	$-0.793492\pi$
$953$	−4.46816e7	−1.59366	−0.796831	+	0.604202i	$0.793492\pi$
$954$	−1.52420e7	−0.542216
$955$	−1.47482e7	−0.523277
$956$	−1.07008e7	−0.378678
$957$	1.47708e7	0.521344
$958$	9.33642e6	0.328675
$959$	0	0
$960$	−475963.	−0.0166685
$961$	6.68028e7	2.33338
$962$	−2.96407e7	−1.03265
$963$	−1.92177e7	−0.667782
$964$	−3.08458e6	−0.106906
$965$	103276.	0.00357010
$966$	0	0
$967$	1.61536e6	0.0555524	0.0277762	−	0.999614i	$-0.491157\pi$
$967$	1.61536e6	0.0555524	0.0277762	+	0.999614i	$0.491157\pi$
$968$	−342448.	−0.0117464
$969$	−4.16361e6	−0.142449
$970$	2.78437e6	0.0950163
$971$	3.07957e7	1.04819	0.524097	−	0.851659i	$-0.324403\pi$
$971$	3.07957e7	1.04819	0.524097	+	0.851659i	$0.324403\pi$
$972$	−1.16473e7	−0.395420
$973$	0	0
$974$	3.16484e7	1.06894
$975$	−2.43304e6	−0.0819667
$976$	161023.	0.00541082
$977$	389307.	0.0130484	0.00652418	−	0.999979i	$-0.497923\pi$
$977$	389307.	0.0130484	0.00652418	+	0.999979i	$0.497923\pi$
$978$	2.69265e6	0.0900186
$979$	−2.40382e7	−0.801576
$980$	0	0
$981$	−7.62660e6	−0.253022
$982$	5.61335e6	0.185756
$983$	−3.83464e7	−1.26573	−0.632865	−	0.774262i	$-0.718121\pi$
$983$	−3.83464e7	−1.26573	−0.632865	+	0.774262i	$0.718121\pi$
$984$	−10373.6	−0.000341540 0
$985$	−2.17459e7	−0.714145
$986$	4.27885e7	1.40163
$987$	0	0
$988$	−9.03714e6	−0.294536
$989$	1.78968e7	0.581816
$990$	−8.73601e6	−0.283286
$991$	3.62302e7	1.17189	0.585944	−	0.810351i	$-0.300724\pi$
$991$	3.62302e7	1.17189	0.585944	+	0.810351i	$0.300724\pi$
$992$	1.00034e7	0.322751
$993$	−1.01142e7	−0.325506
$994$	0	0
$995$	2.17634e7	0.696896
$996$	1.97016e6	0.0629293
$997$	−5.92901e6	−0.188905	−0.0944527	−	0.995529i	$-0.530110\pi$
$997$	−5.92901e6	−0.188905	−0.0944527	+	0.995529i	$0.530110\pi$
$998$	2.15296e6	0.0684243
$999$	−1.90983e7	−0.605454

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.6.a.x.1.2	✓	3
7.6	odd	2		490.6.a.y.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.6.a.x.1.2	✓	3	1.1	even	1	trivial
490.6.a.y.1.2	yes	3	7.6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	490.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} -4.64808 q^{3} +16.0000 q^{4} +25.0000 q^{5} -18.5923 q^{6} +64.0000 q^{8} -221.395 q^{9} +O(q^{10})\)	\(q+4.00000 q^{2} -4.64808 q^{3} +16.0000 q^{4} +25.0000 q^{5} -18.5923 q^{6} +64.0000 q^{8} -221.395 q^{9} +100.000 q^{10} +394.589 q^{11} -74.3693 q^{12} +837.521 q^{13} -116.202 q^{15} +256.000 q^{16} -1328.25 q^{17} -885.581 q^{18} -674.397 q^{19} +400.000 q^{20} +1578.35 q^{22} +3135.55 q^{23} -297.477 q^{24} +625.000 q^{25} +3350.08 q^{26} +2158.55 q^{27} -8053.53 q^{29} -464.808 q^{30} +9768.93 q^{31} +1024.00 q^{32} -1834.08 q^{33} -5313.01 q^{34} -3542.33 q^{36} -8847.77 q^{37} -2697.59 q^{38} -3892.86 q^{39} +1600.00 q^{40} +34.8719 q^{41} +5707.72 q^{43} +6313.42 q^{44} -5534.88 q^{45} +12542.2 q^{46} +28075.8 q^{47} -1189.91 q^{48} +2500.00 q^{50} +6173.82 q^{51} +13400.3 q^{52} +17211.3 q^{53} +8634.19 q^{54} +9864.72 q^{55} +3134.65 q^{57} -32214.1 q^{58} +27955.0 q^{59} -1859.23 q^{60} +628.995 q^{61} +39075.7 q^{62} +4096.00 q^{64} +20938.0 q^{65} -7336.32 q^{66} -47091.4 q^{67} -21252.0 q^{68} -14574.3 q^{69} -4491.20 q^{71} -14169.3 q^{72} +60781.0 q^{73} -35391.1 q^{74} -2905.05 q^{75} -10790.3 q^{76} -15571.5 q^{78} -17617.6 q^{79} +6400.00 q^{80} +43766.0 q^{81} +139.487 q^{82} -26491.6 q^{83} -33206.3 q^{85} +22830.9 q^{86} +37433.5 q^{87} +25253.7 q^{88} -60919.6 q^{89} -22139.5 q^{90} +50168.8 q^{92} -45406.7 q^{93} +112303. q^{94} -16859.9 q^{95} -4759.63 q^{96} +27843.7 q^{97} -87360.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 12 q^{2} - 2 q^{3} + 48 q^{4} + 75 q^{5} - 8 q^{6} + 192 q^{8} + 149 q^{9}+O(q^{10}) \) 3 * q + 12 * q^2 - 2 * q^3 + 48 * q^4 + 75 * q^5 - 8 * q^6 + 192 * q^8 + 149 * q^9	\( 3 q + 12 q^{2} - 2 q^{3} + 48 q^{4} + 75 q^{5} - 8 q^{6} + 192 q^{8} + 149 q^{9} + 300 q^{10} - 144 q^{11} - 32 q^{12} + 288 q^{13} - 50 q^{15} + 768 q^{16} + 1182 q^{17} + 596 q^{18} + 1134 q^{19} + 1200 q^{20} - 576 q^{22} + 1872 q^{23} - 128 q^{24} + 1875 q^{25} + 1152 q^{26} + 4264 q^{27} - 5724 q^{29} - 200 q^{30} + 2184 q^{31} + 3072 q^{32} + 14258 q^{33} + 4728 q^{34} + 2384 q^{36} + 10698 q^{37} + 4536 q^{38} + 3956 q^{39} + 4800 q^{40} + 6204 q^{41} - 10728 q^{43} - 2304 q^{44} + 3725 q^{45} + 7488 q^{46} + 23556 q^{47} - 512 q^{48} + 7500 q^{50} - 12220 q^{51} + 4608 q^{52} - 13218 q^{53} + 17056 q^{54} - 3600 q^{55} + 7740 q^{57} - 22896 q^{58} + 83226 q^{59} - 800 q^{60} + 24330 q^{61} + 8736 q^{62} + 12288 q^{64} + 7200 q^{65} + 57032 q^{66} - 37836 q^{67} + 18912 q^{68} + 3812 q^{69} - 996 q^{71} + 9536 q^{72} + 167256 q^{73} + 42792 q^{74} - 1250 q^{75} + 18144 q^{76} + 15824 q^{78} - 8796 q^{79} + 19200 q^{80} - 89761 q^{81} + 24816 q^{82} - 2418 q^{83} + 29550 q^{85} - 42912 q^{86} + 296652 q^{87} - 9216 q^{88} - 14292 q^{89} + 14900 q^{90} + 29952 q^{92} - 94712 q^{93} + 94224 q^{94} + 28350 q^{95} - 2048 q^{96} + 103710 q^{97} - 142604 q^{99}+O(q^{100}) \) 3 * q + 12 * q^2 - 2 * q^3 + 48 * q^4 + 75 * q^5 - 8 * q^6 + 192 * q^8 + 149 * q^9 + 300 * q^10 - 144 * q^11 - 32 * q^12 + 288 * q^13 - 50 * q^15 + 768 * q^16 + 1182 * q^17 + 596 * q^18 + 1134 * q^19 + 1200 * q^20 - 576 * q^22 + 1872 * q^23 - 128 * q^24 + 1875 * q^25 + 1152 * q^26 + 4264 * q^27 - 5724 * q^29 - 200 * q^30 + 2184 * q^31 + 3072 * q^32 + 14258 * q^33 + 4728 * q^34 + 2384 * q^36 + 10698 * q^37 + 4536 * q^38 + 3956 * q^39 + 4800 * q^40 + 6204 * q^41 - 10728 * q^43 - 2304 * q^44 + 3725 * q^45 + 7488 * q^46 + 23556 * q^47 - 512 * q^48 + 7500 * q^50 - 12220 * q^51 + 4608 * q^52 - 13218 * q^53 + 17056 * q^54 - 3600 * q^55 + 7740 * q^57 - 22896 * q^58 + 83226 * q^59 - 800 * q^60 + 24330 * q^61 + 8736 * q^62 + 12288 * q^64 + 7200 * q^65 + 57032 * q^66 - 37836 * q^67 + 18912 * q^68 + 3812 * q^69 - 996 * q^71 + 9536 * q^72 + 167256 * q^73 + 42792 * q^74 - 1250 * q^75 + 18144 * q^76 + 15824 * q^78 - 8796 * q^79 + 19200 * q^80 - 89761 * q^81 + 24816 * q^82 - 2418 * q^83 + 29550 * q^85 - 42912 * q^86 + 296652 * q^87 - 9216 * q^88 - 14292 * q^89 + 14900 * q^90 + 29952 * q^92 - 94712 * q^93 + 94224 * q^94 + 28350 * q^95 - 2048 * q^96 + 103710 * q^97 - 142604 * q^99

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