LMFDB - Embedded newform 1849.4.a.j.1.31

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,4,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1849.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:	$109.094531601$
Analytic rank:	$1$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.31
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.26350 q^{2} +9.40007 q^{3} -6.40356 q^{4} +0.325119 q^{5} +11.8770 q^{6} +8.80478 q^{7} -18.1989 q^{8} +61.3613 q^{9} +O(q^{10})$	\(q+1.26350 q^{2} +9.40007 q^{3} -6.40356 q^{4} +0.325119 q^{5} +11.8770 q^{6} +8.80478 q^{7} -18.1989 q^{8} +61.3613 q^{9} +0.410789 q^{10} -23.6733 q^{11} -60.1939 q^{12} -23.2530 q^{13} +11.1249 q^{14} +3.05614 q^{15} +28.2341 q^{16} -125.680 q^{17} +77.5301 q^{18} +16.8288 q^{19} -2.08192 q^{20} +82.7656 q^{21} -29.9113 q^{22} +127.751 q^{23} -171.071 q^{24} -124.894 q^{25} -29.3802 q^{26} +322.998 q^{27} -56.3820 q^{28} -220.686 q^{29} +3.86145 q^{30} -303.073 q^{31} +181.265 q^{32} -222.531 q^{33} -158.797 q^{34} +2.86261 q^{35} -392.931 q^{36} +245.352 q^{37} +21.2632 q^{38} -218.580 q^{39} -5.91683 q^{40} +185.397 q^{41} +104.574 q^{42} +151.594 q^{44} +19.9497 q^{45} +161.414 q^{46} -1.86186 q^{47} +265.403 q^{48} -265.476 q^{49} -157.804 q^{50} -1181.40 q^{51} +148.902 q^{52} +261.722 q^{53} +408.109 q^{54} -7.69666 q^{55} -160.238 q^{56} +158.192 q^{57} -278.837 q^{58} -417.774 q^{59} -19.5702 q^{60} -699.788 q^{61} -382.934 q^{62} +540.273 q^{63} +3.15625 q^{64} -7.56001 q^{65} -281.168 q^{66} -279.849 q^{67} +804.800 q^{68} +1200.87 q^{69} +3.61691 q^{70} -675.761 q^{71} -1116.71 q^{72} -1029.66 q^{73} +310.002 q^{74} -1174.01 q^{75} -107.764 q^{76} -208.439 q^{77} -276.176 q^{78} -732.866 q^{79} +9.17946 q^{80} +1379.45 q^{81} +234.250 q^{82} +1094.07 q^{83} -529.994 q^{84} -40.8610 q^{85} -2074.46 q^{87} +430.829 q^{88} -200.368 q^{89} +25.2065 q^{90} -204.738 q^{91} -818.061 q^{92} -2848.91 q^{93} -2.35247 q^{94} +5.47136 q^{95} +1703.91 q^{96} +1442.40 q^{97} -335.429 q^{98} -1452.63 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) $ 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} + 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} + 386 q^{18} - 12 q^{19} - 108 q^{20} - 408 q^{21} - 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} - 1493 q^{26} - 10 q^{27} + 242 q^{28} - 208 q^{29} + 48 q^{30} - 932 q^{31} + 1124 q^{32} - 254 q^{33} - 765 q^{34} - 1452 q^{35} + 747 q^{36} + 90 q^{37} - 1213 q^{38} + 1610 q^{39} - 1693 q^{40} - 1354 q^{41} + 16 q^{42} - 2704 q^{44} - 4508 q^{45} - 233 q^{46} - 3484 q^{47} + 376 q^{48} + 1324 q^{49} + 408 q^{50} - 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} + 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} - 1172 q^{61} + 1546 q^{62} + 3686 q^{63} + 606 q^{64} - 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} - 136 q^{69} + 1310 q^{70} - 162 q^{71} + 5814 q^{72} - 746 q^{73} - 4332 q^{74} - 236 q^{75} - 1338 q^{76} - 2024 q^{77} - 2782 q^{78} - 2656 q^{79} + 5713 q^{80} - 86 q^{81} + 4168 q^{82} - 3514 q^{83} - 4269 q^{84} + 7558 q^{85} - 10278 q^{87} - 11692 q^{88} - 2640 q^{89} - 8286 q^{90} + 5946 q^{91} - 4271 q^{92} + 2 q^{93} - 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} + 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 + 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 + 386 * q^18 - 12 * q^19 - 108 * q^20 - 408 * q^21 - 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 - 1493 * q^26 - 10 * q^27 + 242 * q^28 - 208 * q^29 + 48 * q^30 - 932 * q^31 + 1124 * q^32 - 254 * q^33 - 765 * q^34 - 1452 * q^35 + 747 * q^36 + 90 * q^37 - 1213 * q^38 + 1610 * q^39 - 1693 * q^40 - 1354 * q^41 + 16 * q^42 - 2704 * q^44 - 4508 * q^45 - 233 * q^46 - 3484 * q^47 + 376 * q^48 + 1324 * q^49 + 408 * q^50 - 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 + 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 - 1172 * q^61 + 1546 * q^62 + 3686 * q^63 + 606 * q^64 - 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 - 136 * q^69 + 1310 * q^70 - 162 * q^71 + 5814 * q^72 - 746 * q^73 - 4332 * q^74 - 236 * q^75 - 1338 * q^76 - 2024 * q^77 - 2782 * q^78 - 2656 * q^79 + 5713 * q^80 - 86 * q^81 + 4168 * q^82 - 3514 * q^83 - 4269 * q^84 + 7558 * q^85 - 10278 * q^87 - 11692 * q^88 - 2640 * q^89 - 8286 * q^90 + 5946 * q^91 - 4271 * q^92 + 2 * q^93 - 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 + 2826 * q^98 - 8174 * 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 $

$2$	1.26350	0.446715	0.223358	−	0.974737i	$-0.428298\pi$
$2$	1.26350	0.446715	0.223358	+	0.974737i	$0.428298\pi$
$3$	9.40007	1.80904	0.904522	−	0.426427i	$-0.140228\pi$
$3$	9.40007	1.80904	0.904522	+	0.426427i	$0.140228\pi$
$4$	−6.40356	−0.800445
$5$	0.325119	0.0290796	0.0145398	−	0.999894i	$-0.495372\pi$
$5$	0.325119	0.0290796	0.0145398	+	0.999894i	$0.495372\pi$
$6$	11.8770	0.808128
$7$	8.80478	0.475414	0.237707	−	0.971337i	$-0.423604\pi$
$7$	8.80478	0.475414	0.237707	+	0.971337i	$0.423604\pi$
$8$	−18.1989	−0.804287
$9$	61.3613	2.27264
$10$	0.410789	0.0129903
$11$	−23.6733	−0.648889	−0.324444	−	0.945905i	$-0.605177\pi$
$11$	−23.6733	−0.648889	−0.324444	+	0.945905i	$0.605177\pi$
$12$	−60.1939	−1.44804
$13$	−23.2530	−0.496095	−0.248047	−	0.968748i	$-0.579789\pi$
$13$	−23.2530	−0.496095	−0.248047	+	0.968748i	$0.579789\pi$
$14$	11.1249	0.212375
$15$	3.05614	0.0526062
$16$	28.2341	0.441158
$17$	−125.680	−1.79305	−0.896526	−	0.442991i	$-0.853917\pi$
$17$	−125.680	−1.79305	−0.896526	+	0.442991i	$0.853917\pi$
$18$	77.5301	1.01522
$19$	16.8288	0.203199	0.101600	−	0.994825i	$-0.467604\pi$
$19$	16.8288	0.203199	0.101600	+	0.994825i	$0.467604\pi$
$20$	−2.08192	−0.0232766
$21$	82.7656	0.860044
$22$	−29.9113	−0.289869
$23$	127.751	1.15817	0.579085	−	0.815267i	$-0.303410\pi$
$23$	127.751	1.15817	0.579085	+	0.815267i	$0.303410\pi$
$24$	−171.071	−1.45499
$25$	−124.894	−0.999154
$26$	−29.3802	−0.221613
$27$	322.998	2.30226
$28$	−56.3820	−0.380543
$29$	−220.686	−1.41311	−0.706556	−	0.707657i	$-0.749752\pi$
$29$	−220.686	−1.41311	−0.706556	+	0.707657i	$0.749752\pi$
$30$	3.86145	0.0235000
$31$	−303.073	−1.75592	−0.877961	−	0.478733i	$-0.841096\pi$
$31$	−303.073	−1.75592	−0.877961	+	0.478733i	$0.841096\pi$
$32$	181.265	1.00136
$33$	−222.531	−1.17387
$34$	−158.797	−0.800984
$35$	2.86261	0.0138248
$36$	−392.931	−1.81912
$37$	245.352	1.09015	0.545075	−	0.838387i	$-0.316501\pi$
$37$	245.352	1.09015	0.545075	+	0.838387i	$0.316501\pi$
$38$	21.2632	0.0907722
$39$	−218.580	−0.897457
$40$	−5.91683	−0.0233883
$41$	185.397	0.706200	0.353100	−	0.935586i	$-0.385127\pi$
$41$	185.397	0.706200	0.353100	+	0.935586i	$0.385127\pi$
$42$	104.574	0.384195
$43$	0	0
$43$	0	0
$44$	151.594	0.519400
$45$	19.9497	0.0660874
$46$	161.414	0.517372
$47$	−1.86186	−0.00577831	−0.00288915	−	0.999996i	$-0.500920\pi$
$47$	−1.86186	−0.00577831	−0.00288915	+	0.999996i	$0.500920\pi$
$48$	265.403	0.798074
$49$	−265.476	−0.773982
$50$	−157.804	−0.446338
$51$	−1181.40	−3.24371
$52$	148.902	0.397097
$53$	261.722	0.678307	0.339153	−	0.940731i	$-0.389859\pi$
$53$	261.722	0.678307	0.339153	+	0.940731i	$0.389859\pi$
$54$	408.109	1.02845
$55$	−7.69666	−0.0188694
$56$	−160.238	−0.382369
$57$	158.192	0.367596
$58$	−278.837	−0.631259
$59$	−417.774	−0.921857	−0.460929	−	0.887437i	$-0.652484\pi$
$59$	−417.774	−0.921857	−0.460929	+	0.887437i	$0.652484\pi$
$60$	−19.5702	−0.0421084
$61$	−699.788	−1.46883	−0.734416	−	0.678700i	$-0.762544\pi$
$61$	−699.788	−1.46883	−0.734416	+	0.678700i	$0.762544\pi$
$62$	−382.934	−0.784397
$63$	540.273	1.08044
$64$	3.15625	0.00616454
$65$	−7.56001	−0.0144262
$66$	−281.168	−0.524385
$67$	−279.849	−0.510284	−0.255142	−	0.966904i	$-0.582122\pi$
$67$	−279.849	−0.510284	−0.255142	+	0.966904i	$0.582122\pi$
$68$	804.800	1.43524
$69$	1200.87	2.09518
$70$	3.61691	0.00617576
$71$	−675.761	−1.12955	−0.564776	−	0.825245i	$-0.691037\pi$
$71$	−675.761	−1.12955	−0.564776	+	0.825245i	$0.691037\pi$
$72$	−1116.71	−1.82785
$73$	−1029.66	−1.65086	−0.825432	−	0.564502i	$-0.809068\pi$
$73$	−1029.66	−1.65086	−0.825432	+	0.564502i	$0.809068\pi$
$74$	310.002	0.486987
$75$	−1174.01	−1.80751
$76$	−107.764	−0.162650
$77$	−208.439	−0.308491
$78$	−276.176	−0.400908
$79$	−732.866	−1.04372	−0.521860	−	0.853031i	$-0.674762\pi$
$79$	−732.866	−1.04372	−0.521860	+	0.853031i	$0.674762\pi$
$80$	9.17946	0.0128287
$81$	1379.45	1.89225
$82$	234.250	0.315471
$83$	1094.07	1.44686	0.723431	−	0.690397i	$-0.242564\pi$
$83$	1094.07	1.44686	0.723431	+	0.690397i	$0.242564\pi$
$84$	−529.994	−0.688418
$85$	−40.8610	−0.0521412
$86$	0	0
$87$	−2074.46	−2.55638
$88$	430.829	0.521893
$89$	−200.368	−0.238639	−0.119320	−	0.992856i	$-0.538071\pi$
$89$	−200.368	−0.238639	−0.119320	+	0.992856i	$0.538071\pi$
$90$	25.2065	0.0295222
$91$	−204.738	−0.235850
$92$	−818.061	−0.927052
$93$	−2848.91	−3.17654
$94$	−2.35247	−0.00258126
$95$	5.47136	0.00590895
$96$	1703.91	1.81150
$97$	1442.40	1.50983	0.754916	−	0.655822i	$-0.227678\pi$
$97$	1442.40	1.50983	0.754916	+	0.655822i	$0.227678\pi$
$98$	−335.429	−0.345750
$99$	−1452.63	−1.47469
$100$	799.768	0.799768
$101$	909.111	0.895643	0.447822	−	0.894123i	$-0.352200\pi$
$101$	909.111	0.895643	0.447822	+	0.894123i	$0.352200\pi$
$102$	−1492.70	−1.44902
$103$	1263.28	1.20849	0.604245	−	0.796798i	$-0.293475\pi$
$103$	1263.28	1.20849	0.604245	+	0.796798i	$0.293475\pi$
$104$	423.180	0.399002
$105$	26.9087	0.0250097
$106$	330.686	0.303010
$107$	333.716	0.301509	0.150755	−	0.988571i	$-0.451830\pi$
$107$	333.716	0.301509	0.150755	+	0.988571i	$0.451830\pi$
$108$	−2068.34	−1.84283
$109$	115.351	0.101364	0.0506818	−	0.998715i	$-0.483861\pi$
$109$	115.351	0.101364	0.0506818	+	0.998715i	$0.483861\pi$
$110$	−9.72475	−0.00842926
$111$	2306.32	1.97213
$112$	248.595	0.209733
$113$	−719.984	−0.599384	−0.299692	−	0.954036i	$-0.596884\pi$
$113$	−719.984	−0.599384	−0.299692	+	0.954036i	$0.596884\pi$
$114$	199.875	0.164211
$115$	41.5343	0.0336791
$116$	1413.17	1.13112
$117$	−1426.83	−1.12744
$118$	−527.859	−0.411808
$119$	−1106.59	−0.852441
$120$	−55.6186	−0.0423105
$121$	−770.573	−0.578943
$122$	−884.184	−0.656150
$123$	1742.75	1.27755
$124$	1940.75	1.40552
$125$	−81.2455	−0.0581345
$126$	682.636	0.482651
$127$	1823.69	1.27422	0.637111	−	0.770772i	$-0.280129\pi$
$127$	1823.69	1.27422	0.637111	+	0.770772i	$0.280129\pi$
$128$	−1446.13	−0.998605
$129$	0	0
$130$	−9.55209	−0.00644441
$131$	827.828	0.552120	0.276060	−	0.961140i	$-0.410971\pi$
$131$	827.828	0.552120	0.276060	+	0.961140i	$0.410971\pi$
$132$	1424.99	0.939617
$133$	148.174	0.0966037
$134$	−353.590	−0.227952
$135$	105.013	0.0669487
$136$	2287.24	1.44213
$137$	−1732.64	−1.08051	−0.540254	−	0.841502i	$-0.681672\pi$
$137$	−1732.64	−1.08051	−0.540254	+	0.841502i	$0.681672\pi$
$138$	1517.30	0.935949
$139$	−2409.59	−1.47035	−0.735177	−	0.677875i	$-0.762901\pi$
$139$	−2409.59	−1.47035	−0.735177	+	0.677875i	$0.762901\pi$
$140$	−18.3309	−0.0110660
$141$	−17.5016	−0.0104532
$142$	−853.826	−0.504588
$143$	550.477	0.321910
$144$	1732.48	1.00259
$145$	−71.7492	−0.0410927
$146$	−1300.98	−0.737466
$147$	−2495.49	−1.40017
$148$	−1571.13	−0.872606
$149$	1235.52	0.679316	0.339658	−	0.940549i	$-0.389689\pi$
$149$	1235.52	0.679316	0.339658	+	0.940549i	$0.389689\pi$
$150$	−1483.37	−0.807444
$151$	76.5960	0.0412801	0.0206400	−	0.999787i	$-0.493430\pi$
$151$	76.5960	0.0412801	0.0206400	+	0.999787i	$0.493430\pi$
$152$	−306.266	−0.163430
$153$	−7711.88	−4.07496
$154$	−263.363	−0.137808
$155$	−98.5350	−0.0510614
$156$	1399.69	0.718365
$157$	1307.88	0.664840	0.332420	−	0.943131i	$-0.392135\pi$
$157$	1307.88	0.664840	0.332420	+	0.943131i	$0.392135\pi$
$158$	−925.978	−0.466246
$159$	2460.20	1.22709
$160$	58.9329	0.0291191
$161$	1124.82	0.550610
$162$	1742.94	0.845297
$163$	−2230.12	−1.07164	−0.535818	−	0.844334i	$-0.679997\pi$
$163$	−2230.12	−1.07164	−0.535818	+	0.844334i	$0.679997\pi$
$164$	−1187.20	−0.565275
$165$	−72.3491	−0.0341356
$166$	1382.36	0.646335
$167$	−1109.52	−0.514115	−0.257058	−	0.966396i	$-0.582753\pi$
$167$	−1109.52	−0.514115	−0.257058	+	0.966396i	$0.582753\pi$
$168$	−1506.24	−0.691722
$169$	−1656.30	−0.753890
$170$	−51.6280	−0.0232923
$171$	1032.63	0.461799
$172$	0	0
$173$	226.055	0.0993447	0.0496723	−	0.998766i	$-0.484182\pi$
$173$	226.055	0.0993447	0.0496723	+	0.998766i	$0.484182\pi$
$174$	−2621.08	−1.14198
$175$	−1099.67	−0.475012
$176$	−668.395	−0.286262
$177$	−3927.11	−1.66768
$178$	−253.165	−0.106604
$179$	−1174.15	−0.490280	−0.245140	−	0.969488i	$-0.578834\pi$
$179$	−1174.15	−0.490280	−0.245140	+	0.969488i	$0.578834\pi$
$180$	−127.749	−0.0528993
$181$	−948.581	−0.389544	−0.194772	−	0.980849i	$-0.562397\pi$
$181$	−948.581	−0.389544	−0.194772	+	0.980849i	$0.562397\pi$
$182$	−258.687	−0.105358
$183$	−6578.06	−2.65718
$184$	−2324.93	−0.931501
$185$	79.7686	0.0317011
$186$	−3599.60	−1.41901
$187$	2975.26	1.16349
$188$	11.9225	0.00462522
$189$	2843.93	1.09453
$190$	6.91308	0.00263962
$191$	241.015	0.0913047	0.0456524	−	0.998957i	$-0.485463\pi$
$191$	241.015	0.0913047	0.0456524	+	0.998957i	$0.485463\pi$
$192$	29.6689	0.0111519
$193$	−754.139	−0.281265	−0.140632	−	0.990062i	$-0.544914\pi$
$193$	−754.139	−0.281265	−0.140632	+	0.990062i	$0.544914\pi$
$194$	1822.48	0.674465
$195$	−71.0646	−0.0260977
$196$	1699.99	0.619530
$197$	−4583.62	−1.65771	−0.828856	−	0.559462i	$-0.811008\pi$
$197$	−4583.62	−1.65771	−0.828856	+	0.559462i	$0.811008\pi$
$198$	−1835.40	−0.658767
$199$	2564.26	0.913445	0.456723	−	0.889609i	$-0.349023\pi$
$199$	2564.26	0.913445	0.456723	+	0.889609i	$0.349023\pi$
$200$	2272.94	0.803607
$201$	−2630.60	−0.923125
$202$	1148.66	0.400098
$203$	−1943.09	−0.671813
$204$	7565.17	2.59641
$205$	60.2763	0.0205360
$206$	1596.15	0.539851
$207$	7838.96	2.63210
$208$	−656.528	−0.218856
$209$	−398.393	−0.131854
$210$	33.9992	0.0111722
$211$	1951.52	0.636722	0.318361	−	0.947970i	$-0.396868\pi$
$211$	1951.52	0.636722	0.318361	+	0.947970i	$0.396868\pi$
$212$	−1675.95	−0.542947
$213$	−6352.20	−2.04341
$214$	421.650	0.134689
$215$	0	0
$216$	−5878.22	−1.85168
$217$	−2668.49	−0.834789
$218$	145.746	0.0452807
$219$	−9678.91	−2.98648
$220$	49.2860	0.0151039
$221$	2922.44	0.889523
$222$	2914.04	0.880981
$223$	2379.09	0.714421	0.357211	−	0.934024i	$-0.383728\pi$
$223$	2379.09	0.714421	0.357211	+	0.934024i	$0.383728\pi$
$224$	1596.00	0.476060
$225$	−7663.67	−2.27072
$226$	−909.702	−0.267754
$227$	665.520	0.194591	0.0972955	−	0.995256i	$-0.468981\pi$
$227$	665.520	0.194591	0.0972955	+	0.995256i	$0.468981\pi$
$228$	−1012.99	−0.294241
$229$	3250.55	0.938002	0.469001	−	0.883198i	$-0.344614\pi$
$229$	3250.55	0.938002	0.469001	+	0.883198i	$0.344614\pi$
$230$	52.4787	0.0150450
$231$	−1959.34	−0.558073
$232$	4016.24	1.13655
$233$	−5961.66	−1.67623	−0.838115	−	0.545494i	$-0.816342\pi$
$233$	−5961.66	−1.67623	−0.838115	+	0.545494i	$0.816342\pi$
$234$	−1802.81	−0.503647
$235$	−0.605327	−0.000168031 0
$236$	2675.24	0.737896
$237$	−6888.99	−1.88814
$238$	−1398.17	−0.380799
$239$	1651.94	0.447092	0.223546	−	0.974693i	$-0.428237\pi$
$239$	1651.94	0.447092	0.223546	+	0.974693i	$0.428237\pi$
$240$	86.2875	0.0232076
$241$	−524.998	−0.140324	−0.0701621	−	0.997536i	$-0.522352\pi$
$241$	−524.998	−0.140324	−0.0701621	+	0.997536i	$0.522352\pi$
$242$	−973.621	−0.258623
$243$	4245.97	1.12090
$244$	4481.14	1.17572
$245$	−86.3113	−0.0225071
$246$	2201.97	0.570700
$247$	−391.320	−0.100806
$248$	5515.61	1.41226
$249$	10284.3	2.61744
$250$	−102.654	−0.0259696
$251$	5790.68	1.45619	0.728096	−	0.685475i	$-0.240405\pi$
$251$	5790.68	1.45619	0.728096	+	0.685475i	$0.240405\pi$
$252$	−3459.67	−0.864836
$253$	−3024.29	−0.751524
$254$	2304.24	0.569215
$255$	−384.096	−0.0943257
$256$	−1852.44	−0.452257
$257$	−4622.57	−1.12198	−0.560989	−	0.827823i	$-0.689579\pi$
$257$	−4622.57	−1.12198	−0.560989	+	0.827823i	$0.689579\pi$
$258$	0	0
$259$	2160.27	0.518273
$260$	48.4110	0.0115474
$261$	−13541.5	−3.21150
$262$	1045.96	0.246640
$263$	−468.704	−0.109892	−0.0549458	−	0.998489i	$-0.517499\pi$
$263$	−468.704	−0.109892	−0.0549458	+	0.998489i	$0.517499\pi$
$264$	4049.82	0.944127
$265$	85.0908	0.0197249
$266$	187.218	0.0431544
$267$	−1883.47	−0.431709
$268$	1792.03	0.408454
$269$	−1162.74	−0.263546	−0.131773	−	0.991280i	$-0.542067\pi$
$269$	−1162.74	−0.263546	−0.131773	+	0.991280i	$0.542067\pi$
$270$	132.684	0.0299070
$271$	−5794.03	−1.29875	−0.649377	−	0.760467i	$-0.724970\pi$
$271$	−5794.03	−1.29875	−0.649377	+	0.760467i	$0.724970\pi$
$272$	−3548.46	−0.791019
$273$	−1924.55	−0.426663
$274$	−2189.20	−0.482680
$275$	2956.66	0.648340
$276$	−7689.83	−1.67708
$277$	1078.12	0.233855	0.116928	−	0.993140i	$-0.462695\pi$
$277$	1078.12	0.233855	0.116928	+	0.993140i	$0.462695\pi$
$278$	−3044.53	−0.656830
$279$	−18597.0	−3.99058
$280$	−52.0964	−0.0111191
$281$	4395.46	0.933136	0.466568	−	0.884485i	$-0.345490\pi$
$281$	4395.46	0.933136	0.466568	+	0.884485i	$0.345490\pi$
$282$	−22.1133	−0.00466961
$283$	1861.43	0.390990	0.195495	−	0.980705i	$-0.437369\pi$
$283$	1861.43	0.390990	0.195495	+	0.980705i	$0.437369\pi$
$284$	4327.28	0.904144
$285$	51.4312	0.0106895
$286$	695.528	0.143802
$287$	1632.38	0.335737
$288$	11122.7	2.27573
$289$	10882.5	2.21504
$290$	−90.6552	−0.0183568
$291$	13558.7	2.73135
$292$	6593.52	1.32143
$293$	−480.757	−0.0958572	−0.0479286	−	0.998851i	$-0.515262\pi$
$293$	−480.757	−0.0958572	−0.0479286	+	0.998851i	$0.515262\pi$
$294$	−3153.06	−0.625476
$295$	−135.827	−0.0268072
$296$	−4465.14	−0.876794
$297$	−7646.44	−1.49391
$298$	1561.09	0.303461
$299$	−2970.60	−0.574562
$300$	7517.88	1.44682
$301$	0	0
$302$	96.7792	0.0184404
$303$	8545.71	1.62026
$304$	475.145	0.0896430
$305$	−227.515	−0.0427130
$306$	−9743.98	−1.82035
$307$	6933.15	1.28891	0.644456	−	0.764641i	$-0.277084\pi$
$307$	6933.15	1.28891	0.644456	+	0.764641i	$0.277084\pi$
$308$	1334.75	0.246930
$309$	11874.9	2.18621
$310$	−124.499	−0.0228099
$311$	−2059.92	−0.375586	−0.187793	−	0.982209i	$-0.560133\pi$
$311$	−2059.92	−0.375586	−0.187793	+	0.982209i	$0.560133\pi$
$312$	3977.92	0.721813
$313$	5359.79	0.967901	0.483950	−	0.875095i	$-0.339201\pi$
$313$	5359.79	0.967901	0.483950	+	0.875095i	$0.339201\pi$
$314$	1652.50	0.296994
$315$	175.653	0.0314188
$316$	4692.96	0.835441
$317$	−7019.47	−1.24370	−0.621849	−	0.783137i	$-0.713618\pi$
$317$	−7019.47	−1.24370	−0.621849	+	0.783137i	$0.713618\pi$
$318$	3108.47	0.548158
$319$	5224.36	0.916953
$320$	1.02616	0.000179262 0
$321$	3136.95	0.545444
$322$	1421.21	0.245966
$323$	−2115.04	−0.364347
$324$	−8833.39	−1.51464
$325$	2904.17	0.495675
$326$	−2817.76	−0.478716
$327$	1084.31	0.183371
$328$	−3374.04	−0.567988
$329$	−16.3933	−0.00274709
$330$	−91.4133	−0.0152489
$331$	5551.69	0.921898	0.460949	−	0.887427i	$-0.347509\pi$
$331$	5551.69	0.921898	0.460949	+	0.887427i	$0.347509\pi$
$332$	−7005.93	−1.15813
$333$	15055.1	2.47752
$334$	−1401.88	−0.229663
$335$	−90.9844	−0.0148388
$336$	2336.81	0.379415
$337$	1460.34	0.236053	0.118026	−	0.993010i	$-0.462343\pi$
$337$	1460.34	0.236053	0.118026	+	0.993010i	$0.462343\pi$
$338$	−2092.73	−0.336774
$339$	−6767.90	−1.08431
$340$	261.656	0.0417362
$341$	7174.75	1.13940
$342$	1304.74	0.206293
$343$	−5357.50	−0.843375
$344$	0	0
$345$	390.425	0.0609269
$346$	285.621	0.0443788
$347$	−7150.17	−1.10617	−0.553085	−	0.833125i	$-0.686550\pi$
$347$	−7150.17	−1.10617	−0.553085	+	0.833125i	$0.686550\pi$
$348$	13283.9	2.04624
$349$	−5169.61	−0.792902	−0.396451	−	0.918056i	$-0.629758\pi$
$349$	−5169.61	−0.792902	−0.396451	+	0.918056i	$0.629758\pi$
$350$	−1389.43	−0.212195
$351$	−7510.68	−1.14214
$352$	−4291.15	−0.649771
$353$	1936.50	0.291981	0.145991	−	0.989286i	$-0.453363\pi$
$353$	1936.50	0.291981	0.145991	+	0.989286i	$0.453363\pi$
$354$	−4961.91	−0.744978
$355$	−219.703	−0.0328469
$356$	1283.07	0.191018
$357$	−10402.0	−1.54210
$358$	−1483.54	−0.219016
$359$	1557.07	0.228911	0.114456	−	0.993428i	$-0.463488\pi$
$359$	1557.07	0.228911	0.114456	+	0.993428i	$0.463488\pi$
$360$	−363.064	−0.0531532
$361$	−6575.79	−0.958710
$362$	−1198.53	−0.174015
$363$	−7243.44	−1.04733
$364$	1311.05	0.188785
$365$	−334.764	−0.0480064
$366$	−8311.39	−1.18700
$367$	5890.68	0.837851	0.418925	−	0.908021i	$-0.362407\pi$
$367$	5890.68	0.837851	0.418925	+	0.908021i	$0.362407\pi$
$368$	3606.93	0.510936
$369$	11376.2	1.60494
$370$	100.788	0.0141614
$371$	2304.40	0.322476
$372$	18243.2	2.54265
$373$	−1361.91	−0.189053	−0.0945265	−	0.995522i	$-0.530134\pi$
$373$	−1361.91	−0.189053	−0.0945265	+	0.995522i	$0.530134\pi$
$374$	3759.25	0.519750
$375$	−763.713	−0.105168
$376$	33.8839	0.00464741
$377$	5131.61	0.701038
$378$	3593.31	0.488942
$379$	−2673.20	−0.362304	−0.181152	−	0.983455i	$-0.557983\pi$
$379$	−2673.20	−0.362304	−0.181152	+	0.983455i	$0.557983\pi$
$380$	−35.0362	−0.00472979
$381$	17142.8	2.30512
$382$	304.522	0.0407872
$383$	3017.72	0.402607	0.201303	−	0.979529i	$-0.435482\pi$
$383$	3017.72	0.402607	0.201303	+	0.979529i	$0.435482\pi$
$384$	−13593.8	−1.80652
$385$	−67.7674	−0.00897078
$386$	−952.857	−0.125645
$387$	0	0
$388$	−9236.51	−1.20854
$389$	−15220.5	−1.98384	−0.991918	−	0.126879i	$-0.959504\pi$
$389$	−15220.5	−1.98384	−0.991918	+	0.126879i	$0.959504\pi$
$390$	−89.7903	−0.0116582
$391$	−16055.7	−2.07666
$392$	4831.38	0.622503
$393$	7781.64	0.998809
$394$	−5791.41	−0.740526
$395$	−238.269	−0.0303509
$396$	9301.98	1.18041
$397$	−9599.48	−1.21356	−0.606781	−	0.794869i	$-0.707540\pi$
$397$	−9599.48	−1.21356	−0.606781	+	0.794869i	$0.707540\pi$
$398$	3239.95	0.408050
$399$	1392.84	0.174760
$400$	−3526.28	−0.440785
$401$	15516.8	1.93235	0.966177	−	0.257879i	$-0.0830236\pi$
$401$	15516.8	1.93235	0.966177	+	0.257879i	$0.0830236\pi$
$402$	−3323.77	−0.412374
$403$	7047.37	0.871103
$404$	−5821.55	−0.716913
$405$	448.486	0.0550258
$406$	−2455.10	−0.300109
$407$	−5808.29	−0.707387
$408$	21500.2	2.60887
$409$	13082.4	1.58162	0.790812	−	0.612059i	$-0.209658\pi$
$409$	13082.4	1.58162	0.790812	+	0.612059i	$0.209658\pi$
$410$	76.1593	0.00917375
$411$	−16287.0	−1.95469
$412$	−8089.48	−0.967330
$413$	−3678.41	−0.438263
$414$	9904.54	1.17580
$415$	355.703	0.0420741
$416$	−4214.97	−0.496769
$417$	−22650.4	−2.65993
$418$	−503.371	−0.0589011
$419$	−4000.38	−0.466424	−0.233212	−	0.972426i	$-0.574924\pi$
$419$	−4000.38	−0.466424	−0.233212	+	0.972426i	$0.574924\pi$
$420$	−172.312	−0.0200189
$421$	−14443.5	−1.67204	−0.836022	−	0.548695i	$-0.815125\pi$
$421$	−14443.5	−1.67204	−0.836022	+	0.548695i	$0.815125\pi$
$422$	2465.75	0.284434
$423$	−114.246	−0.0131320
$424$	−4763.06	−0.545553
$425$	15696.7	1.79154
$426$	−8026.02	−0.912822
$427$	−6161.49	−0.698303
$428$	−2136.97	−0.241342
$429$	5174.52	0.582350
$430$	0	0
$431$	8704.07	0.972762	0.486381	−	0.873747i	$-0.338317\pi$
$431$	8704.07	0.972762	0.486381	+	0.873747i	$0.338317\pi$
$432$	9119.56	1.01566
$433$	13149.7	1.45943	0.729717	−	0.683749i	$-0.239652\pi$
$433$	13149.7	1.45943	0.729717	+	0.683749i	$0.239652\pi$
$434$	−3371.65	−0.372913
$435$	−674.447	−0.0743385
$436$	−738.658	−0.0811360
$437$	2149.89	0.235339
$438$	−12229.3	−1.33411
$439$	11391.2	1.23844	0.619218	−	0.785219i	$-0.287450\pi$
$439$	11391.2	1.23844	0.619218	+	0.785219i	$0.287450\pi$
$440$	140.071	0.0151764
$441$	−16289.9	−1.75898
$442$	3692.51	0.397364
$443$	−4305.83	−0.461797	−0.230898	−	0.972978i	$-0.574166\pi$
$443$	−4305.83	−0.461797	−0.230898	+	0.972978i	$0.574166\pi$
$444$	−14768.7	−1.57858
$445$	−65.1434	−0.00693953
$446$	3005.99	0.319143
$447$	11614.0	1.22891
$448$	27.7901	0.00293071
$449$	−3738.80	−0.392973	−0.196487	−	0.980506i	$-0.562953\pi$
$449$	−3738.80	−0.392973	−0.196487	+	0.980506i	$0.562953\pi$
$450$	−9683.06	−1.01436
$451$	−4388.98	−0.458246
$452$	4610.47	0.479774
$453$	720.007	0.0746775
$454$	840.886	0.0869268
$455$	−66.5643	−0.00685842
$456$	−2878.92	−0.295653
$457$	8908.27	0.911841	0.455920	−	0.890021i	$-0.349310\pi$
$457$	8908.27	0.911841	0.455920	+	0.890021i	$0.349310\pi$
$458$	4107.08	0.419020
$459$	−40594.4	−4.12807
$460$	−265.968	−0.0269583
$461$	2694.69	0.272244	0.136122	−	0.990692i	$-0.456536\pi$
$461$	2694.69	0.272244	0.136122	+	0.990692i	$0.456536\pi$
$462$	−2475.63	−0.249300
$463$	−6709.80	−0.673500	−0.336750	−	0.941594i	$-0.609328\pi$
$463$	−6709.80	−0.673500	−0.336750	+	0.941594i	$0.609328\pi$
$464$	−6230.86	−0.623406
$465$	−926.236	−0.0923724
$466$	−7532.57	−0.748798
$467$	16767.9	1.66151	0.830754	−	0.556639i	$-0.187909\pi$
$467$	16767.9	1.66151	0.830754	+	0.556639i	$0.187909\pi$
$468$	9136.83	0.902457
$469$	−2464.01	−0.242596
$470$	−0.764832	−7.50619e−5 0
$471$	12294.1	1.20272
$472$	7603.04	0.741437
$473$	0	0
$474$	−8704.26	−0.843460
$475$	−2101.82	−0.203027
$476$	7086.09	0.682333
$477$	16059.6	1.54155
$478$	2087.23	0.199723
$479$	−13204.9	−1.25959	−0.629797	−	0.776760i	$-0.716862\pi$
$479$	−13204.9	−1.25959	−0.629797	+	0.776760i	$0.716862\pi$
$480$	553.973	0.0526777
$481$	−5705.17	−0.540818
$482$	−663.336	−0.0626849
$483$	10573.4	0.996077
$484$	4934.41	0.463412
$485$	468.953	0.0439052
$486$	5364.79	0.500724
$487$	−2291.19	−0.213191	−0.106595	−	0.994302i	$-0.533995\pi$
$487$	−2291.19	−0.213191	−0.106595	+	0.994302i	$0.533995\pi$
$488$	12735.4	1.18136
$489$	−20963.3	−1.93864
$490$	−109.055	−0.0100543
$491$	8817.80	0.810472	0.405236	−	0.914212i	$-0.367189\pi$
$491$	8817.80	0.810472	0.405236	+	0.914212i	$0.367189\pi$
$492$	−11159.8	−1.02261
$493$	27735.8	2.53378
$494$	−494.434	−0.0450316
$495$	−472.277	−0.0428834
$496$	−8557.00	−0.774639
$497$	−5949.93	−0.537004
$498$	12994.2	1.16925
$499$	16011.1	1.43638	0.718192	−	0.695845i	$-0.244970\pi$
$499$	16011.1	1.43638	0.718192	+	0.695845i	$0.244970\pi$
$500$	520.261	0.0465335
$501$	−10429.6	−0.930057
$502$	7316.53	0.650504
$503$	−55.8134	−0.00494751	−0.00247375	−	0.999997i	$-0.500787\pi$
$503$	−55.8134	−0.00494751	−0.00247375	+	0.999997i	$0.500787\pi$
$504$	−9832.38	−0.868987
$505$	295.570	0.0260449
$506$	−3821.20	−0.335717
$507$	−15569.3	−1.36382
$508$	−11678.1	−1.01995
$509$	9212.25	0.802212	0.401106	−	0.916032i	$-0.368626\pi$
$509$	9212.25	0.802212	0.401106	+	0.916032i	$0.368626\pi$
$510$	−485.307	−0.0421367
$511$	−9065.97	−0.784843
$512$	9228.51	0.796575
$513$	5435.66	0.467817
$514$	−5840.63	−0.501205
$515$	410.716	0.0351424
$516$	0	0
$517$	44.0765	0.00374948
$518$	2729.50	0.231520
$519$	2124.93	0.179719
$520$	137.584	0.0116028
$521$	14038.9	1.18053	0.590266	−	0.807209i	$-0.299023\pi$
$521$	14038.9	1.18053	0.590266	+	0.807209i	$0.299023\pi$
$522$	−17109.8	−1.43462
$523$	−2476.00	−0.207013	−0.103507	−	0.994629i	$-0.533006\pi$
$523$	−2476.00	−0.207013	−0.103507	+	0.994629i	$0.533006\pi$
$524$	−5301.05	−0.441942
$525$	−10336.9	−0.859317
$526$	−592.208	−0.0490903
$527$	38090.3	3.14846
$528$	−6282.96	−0.517861
$529$	4153.30	0.341358
$530$	107.512	0.00881140
$531$	−25635.1	−2.09505
$532$	−948.840	−0.0773260
$533$	−4311.05	−0.350342
$534$	−2379.77	−0.192851
$535$	108.497	0.00876776
$536$	5092.95	0.410414
$537$	−11037.1	−0.886938
$538$	−1469.13	−0.117730
$539$	6284.70	0.502228
$540$	−672.457	−0.0535888
$541$	5185.20	0.412069	0.206034	−	0.978545i	$-0.433944\pi$
$541$	5185.20	0.412069	0.206034	+	0.978545i	$0.433944\pi$
$542$	−7320.77	−0.580173
$543$	−8916.73	−0.704702
$544$	−22781.4	−1.79549
$545$	37.5029	0.00294761
$546$	−2431.67	−0.190597
$547$	−7098.82	−0.554888	−0.277444	−	0.960742i	$-0.589487\pi$
$547$	−7098.82	−0.554888	−0.277444	+	0.960742i	$0.589487\pi$
$548$	11095.1	0.864888
$549$	−42939.9	−3.33812
$550$	3735.75	0.289624
$551$	−3713.87	−0.287143
$552$	−21854.5	−1.68513
$553$	−6452.73	−0.496199
$554$	1362.21	0.104467
$555$	749.830	0.0573487
$556$	15430.0	1.17694
$557$	−1358.66	−0.103354	−0.0516772	−	0.998664i	$-0.516457\pi$
$557$	−1358.66	−0.103354	−0.0516772	+	0.998664i	$0.516457\pi$
$558$	−23497.3	−1.78265
$559$	0	0
$560$	80.8232	0.00609893
$561$	27967.7	2.10481
$562$	5553.67	0.416846
$563$	−13603.7	−1.01834	−0.509172	−	0.860665i	$-0.670048\pi$
$563$	−13603.7	−1.01834	−0.509172	+	0.860665i	$0.670048\pi$
$564$	112.073	0.00836722
$565$	−234.081	−0.0174298
$566$	2351.91	0.174661
$567$	12145.8	0.899601
$568$	12298.1	0.908483
$569$	3316.81	0.244373	0.122186	−	0.992507i	$-0.461009\pi$
$569$	3316.81	0.244373	0.122186	+	0.992507i	$0.461009\pi$
$570$	64.9834	0.00477518
$571$	7745.82	0.567693	0.283846	−	0.958870i	$-0.408389\pi$
$571$	7745.82	0.567693	0.283846	+	0.958870i	$0.408389\pi$
$572$	−3525.01	−0.257672
$573$	2265.55	0.165174
$574$	2062.52	0.149979
$575$	−15955.4	−1.15719
$576$	193.671	0.0140098
$577$	−24031.9	−1.73390	−0.866952	−	0.498391i	$-0.833924\pi$
$577$	−24031.9	−1.73390	−0.866952	+	0.498391i	$0.833924\pi$
$578$	13750.0	0.989491
$579$	−7088.96	−0.508821
$580$	459.450	0.0328925
$581$	9633.03	0.687858
$582$	17131.4	1.22014
$583$	−6195.83	−0.440146
$584$	18738.8	1.32777
$585$	−463.892	−0.0327856
$586$	−607.438	−0.0428209
$587$	−2119.09	−0.149002	−0.0745011	−	0.997221i	$-0.523736\pi$
$587$	−2119.09	−0.149002	−0.0745011	+	0.997221i	$0.523736\pi$
$588$	15980.0	1.12076
$589$	−5100.35	−0.356802
$590$	−171.617	−0.0119752
$591$	−43086.3	−2.99887
$592$	6927.29	0.480929
$593$	6253.12	0.433027	0.216513	−	0.976280i	$-0.430531\pi$
$593$	6253.12	0.433027	0.216513	+	0.976280i	$0.430531\pi$
$594$	−9661.29	−0.667353
$595$	−359.773	−0.0247886
$596$	−7911.76	−0.543756
$597$	24104.2	1.65246
$598$	−3753.35	−0.256666
$599$	10339.6	0.705284	0.352642	−	0.935758i	$-0.385283\pi$
$599$	10339.6	0.705284	0.352642	+	0.935758i	$0.385283\pi$
$600$	21365.8	1.45376
$601$	5782.91	0.392495	0.196248	−	0.980554i	$-0.437124\pi$
$601$	5782.91	0.392495	0.196248	+	0.980554i	$0.437124\pi$
$602$	0	0
$603$	−17171.9	−1.15969
$604$	−490.487	−0.0330424
$605$	−250.528	−0.0168354
$606$	10797.5	0.723794
$607$	20822.9	1.39238	0.696191	−	0.717856i	$-0.254877\pi$
$607$	20822.9	1.39238	0.696191	+	0.717856i	$0.254877\pi$
$608$	3050.47	0.203475
$609$	−18265.2	−1.21534
$610$	−287.465	−0.0190806
$611$	43.2939	0.00286659
$612$	49383.5	3.26178
$613$	11051.2	0.728148	0.364074	−	0.931370i	$-0.381386\pi$
$613$	11051.2	0.728148	0.364074	+	0.931370i	$0.381386\pi$
$614$	8760.05	0.575777
$615$	566.601	0.0371505
$616$	3793.36	0.248115
$617$	1118.45	0.0729775	0.0364888	−	0.999334i	$-0.488383\pi$
$617$	1118.45	0.0729775	0.0364888	+	0.999334i	$0.488383\pi$
$618$	15004.0	0.976615
$619$	−15760.9	−1.02340	−0.511700	−	0.859164i	$-0.670984\pi$
$619$	−15760.9	−1.02340	−0.511700	+	0.859164i	$0.670984\pi$
$620$	630.975	0.0408719
$621$	41263.3	2.66641
$622$	−2602.71	−0.167780
$623$	−1764.19	−0.113452
$624$	−6171.41	−0.395920
$625$	15585.4	0.997464
$626$	6772.10	0.432376
$627$	−3744.92	−0.238529
$628$	−8375.06	−0.532168
$629$	−30835.8	−1.95470
$630$	221.938	0.0140353
$631$	17527.6	1.10581	0.552904	−	0.833245i	$-0.313520\pi$
$631$	17527.6	1.10581	0.552904	+	0.833245i	$0.313520\pi$
$632$	13337.4	0.839451
$633$	18344.4	1.15186
$634$	−8869.11	−0.555579
$635$	592.917	0.0370539
$636$	−15754.1	−0.982215
$637$	6173.11	0.383968
$638$	6600.99	0.409617
$639$	−41465.6	−2.56706
$640$	−470.167	−0.0290390
$641$	9258.01	0.570467	0.285233	−	0.958458i	$-0.407929\pi$
$641$	9258.01	0.570467	0.285233	+	0.958458i	$0.407929\pi$
$642$	3963.54	0.243658
$643$	3332.29	0.204374	0.102187	−	0.994765i	$-0.467416\pi$
$643$	3332.29	0.204374	0.102187	+	0.994765i	$0.467416\pi$
$644$	−7202.85	−0.440733
$645$	0	0
$646$	−2672.36	−0.162759
$647$	−4208.18	−0.255704	−0.127852	−	0.991793i	$-0.540808\pi$
$647$	−4208.18	−0.255704	−0.127852	+	0.991793i	$0.540808\pi$
$648$	−25104.5	−1.52191
$649$	9890.11	0.598183
$650$	3669.43	0.221426
$651$	−25084.0	−1.51017
$652$	14280.7	0.857786
$653$	−25071.4	−1.50248	−0.751241	−	0.660028i	$-0.770544\pi$
$653$	−25071.4	−1.50248	−0.751241	+	0.660028i	$0.770544\pi$
$654$	1370.03	0.0819147
$655$	269.143	0.0160554
$656$	5234.53	0.311546
$657$	−63181.5	−3.75182
$658$	−20.7130	−0.00122717
$659$	−16578.9	−0.980001	−0.490000	−	0.871722i	$-0.663003\pi$
$659$	−16578.9	−0.980001	−0.490000	+	0.871722i	$0.663003\pi$
$660$	463.292	0.0273237
$661$	−13221.0	−0.777966	−0.388983	−	0.921245i	$-0.627174\pi$
$661$	−13221.0	−0.777966	−0.388983	+	0.921245i	$0.627174\pi$
$662$	7014.57	0.411826
$663$	27471.1	1.60919
$664$	−19910.9	−1.16369
$665$	48.1742	0.00280919
$666$	19022.1	1.10675
$667$	−28192.8	−1.63662
$668$	7104.88	0.411521
$669$	22363.6	1.29242
$670$	−114.959	−0.00662873
$671$	16566.3	0.953108
$672$	15002.5	0.861213
$673$	−9297.40	−0.532524	−0.266262	−	0.963901i	$-0.585789\pi$
$673$	−9297.40	−0.532524	−0.266262	+	0.963901i	$0.585789\pi$
$674$	1845.14	0.105449
$675$	−40340.6	−2.30031
$676$	10606.2	0.603448
$677$	−5546.69	−0.314884	−0.157442	−	0.987528i	$-0.550325\pi$
$677$	−5546.69	−0.314884	−0.157442	+	0.987528i	$0.550325\pi$
$678$	−8551.26	−0.484379
$679$	12700.0	0.717795
$680$	743.627	0.0419365
$681$	6255.94	0.352023
$682$	9065.32	0.508987
$683$	−13290.9	−0.744603	−0.372301	−	0.928112i	$-0.621431\pi$
$683$	−13290.9	−0.744603	−0.372301	+	0.928112i	$0.621431\pi$
$684$	−6612.54	−0.369644
$685$	−563.316	−0.0314207
$686$	−6769.21	−0.376749
$687$	30555.4	1.69689
$688$	0	0
$689$	−6085.82	−0.336504
$690$	493.303	0.0272170
$691$	16442.2	0.905199	0.452599	−	0.891714i	$-0.350497\pi$
$691$	16442.2	0.905199	0.452599	+	0.891714i	$0.350497\pi$
$692$	−1447.56	−0.0795200
$693$	−12790.1	−0.701088
$694$	−9034.25	−0.494144
$695$	−783.406	−0.0427573
$696$	37752.9	2.05607
$697$	−23300.8	−1.26625
$698$	−6531.81	−0.354201
$699$	−56040.0	−3.03237
$700$	7041.79	0.380221
$701$	25096.3	1.35217	0.676087	−	0.736822i	$-0.263675\pi$
$701$	25096.3	1.35217	0.676087	+	0.736822i	$0.263675\pi$
$702$	−9489.76	−0.510211
$703$	4128.97	0.221518
$704$	−74.7189	−0.00400010
$705$	−5.69012	−0.000303975 0
$706$	2446.77	0.130433
$707$	8004.53	0.425801
$708$	25147.5	1.33489
$709$	15705.5	0.831920	0.415960	−	0.909383i	$-0.363446\pi$
$709$	15705.5	0.831920	0.415960	+	0.909383i	$0.363446\pi$
$710$	−277.595	−0.0146732
$711$	−44969.6	−2.37200
$712$	3646.47	0.191935
$713$	−38717.9	−2.03366
$714$	−13142.9	−0.688882
$715$	178.971	0.00936101
$716$	7518.74	0.392442
$717$	15528.3	0.808809
$718$	1967.36	0.102258
$719$	−37095.9	−1.92412	−0.962061	−	0.272836i	$-0.912038\pi$
$719$	−37095.9	−1.92412	−0.962061	+	0.272836i	$0.912038\pi$
$720$	563.263	0.0291550
$721$	11122.9	0.574533
$722$	−8308.53	−0.428271
$723$	−4935.02	−0.253852
$724$	6074.30	0.311809
$725$	27562.4	1.41192
$726$	−9152.10	−0.467860
$727$	29835.3	1.52205	0.761024	−	0.648724i	$-0.224697\pi$
$727$	29835.3	1.52205	0.761024	+	0.648724i	$0.224697\pi$
$728$	3726.01	0.189691
$729$	2667.26	0.135511
$730$	−422.975	−0.0214452
$731$	0	0
$732$	42123.0	2.12693
$733$	−33242.9	−1.67511	−0.837553	−	0.546356i	$-0.816015\pi$
$733$	−33242.9	−1.67511	−0.837553	+	0.546356i	$0.816015\pi$
$734$	7442.89	0.374281
$735$	−811.332	−0.0407163
$736$	23156.8	1.15974
$737$	6624.96	0.331117
$738$	14373.9	0.716951
$739$	−19988.7	−0.994990	−0.497495	−	0.867467i	$-0.665747\pi$
$739$	−19988.7	−0.994990	−0.497495	+	0.867467i	$0.665747\pi$
$740$	−510.803	−0.0253750
$741$	−3678.43	−0.182363
$742$	2911.62	0.144055
$743$	19510.4	0.963346	0.481673	−	0.876351i	$-0.340029\pi$
$743$	19510.4	0.963346	0.481673	+	0.876351i	$0.340029\pi$
$744$	51847.1	2.55485
$745$	401.693	0.0197542
$746$	−1720.77	−0.0844529
$747$	67133.3	3.28819
$748$	−19052.3	−0.931311
$749$	2938.29	0.143342
$750$	−964.953	−0.0469801
$751$	−3423.45	−0.166343	−0.0831714	−	0.996535i	$-0.526505\pi$
$751$	−3423.45	−0.166343	−0.0831714	+	0.996535i	$0.526505\pi$
$752$	−52.5680	−0.00254915
$753$	54432.7	2.63432
$754$	6483.80	0.313164
$755$	24.9028	0.00120041
$756$	−18211.3	−0.876108
$757$	−27084.2	−1.30039	−0.650193	−	0.759769i	$-0.725312\pi$
$757$	−27084.2	−1.30039	−0.650193	+	0.759769i	$0.725312\pi$
$758$	−3377.60	−0.161847
$759$	−28428.5	−1.35954
$760$	−99.5730	−0.00475249
$761$	14059.4	0.669714	0.334857	−	0.942269i	$-0.391312\pi$
$761$	14059.4	0.669714	0.334857	+	0.942269i	$0.391312\pi$
$762$	21660.0	1.02973
$763$	1015.64	0.0481896
$764$	−1543.35	−0.0730844
$765$	−2507.28	−0.118498
$766$	3812.90	0.179851
$767$	9714.51	0.457328
$768$	−17413.1	−0.818152
$769$	−37181.7	−1.74357	−0.871786	−	0.489886i	$-0.837038\pi$
$769$	−37181.7	−1.74357	−0.871786	+	0.489886i	$0.837038\pi$
$770$	−85.6243	−0.00400738
$771$	−43452.5	−2.02971
$772$	4829.18	0.225137
$773$	−4717.31	−0.219495	−0.109748	−	0.993959i	$-0.535004\pi$
$773$	−4717.31	−0.219495	−0.109748	+	0.993959i	$0.535004\pi$
$774$	0	0
$775$	37852.1	1.75444
$776$	−26250.2	−1.21434
$777$	20306.7	0.937578
$778$	−19231.2	−0.886210
$779$	3120.01	0.143499
$780$	455.067	0.0208897
$781$	15997.5	0.732953
$782$	−20286.5	−0.927676
$783$	−71281.0	−3.25335
$784$	−7495.47	−0.341448
$785$	425.216	0.0193333
$786$	9832.12	0.446183
$787$	−28567.6	−1.29393	−0.646966	−	0.762519i	$-0.723963\pi$
$787$	−28567.6	−1.29393	−0.646966	+	0.762519i	$0.723963\pi$
$788$	29351.5	1.32691
$789$	−4405.85	−0.198799
$790$	−301.054	−0.0135582
$791$	−6339.31	−0.284956
$792$	26436.2	1.18607
$793$	16272.2	0.728679
$794$	−12129.0	−0.542117
$795$	799.860	0.0356831
$796$	−16420.4	−0.731163
$797$	−25388.9	−1.12838	−0.564191	−	0.825644i	$-0.690812\pi$
$797$	−25388.9	−1.12838	−0.564191	+	0.825644i	$0.690812\pi$
$798$	1759.86	0.0780681
$799$	233.999	0.0103608
$800$	−22639.0	−1.00051
$801$	−12294.8	−0.542341
$802$	19605.6	0.863213
$803$	24375.6	1.07123
$804$	16845.2	0.738911
$805$	365.701	0.0160115
$806$	8904.37	0.389135
$807$	−10929.9	−0.476765
$808$	−16544.9	−0.720354
$809$	−6922.63	−0.300849	−0.150424	−	0.988622i	$-0.548064\pi$
$809$	−6922.63	−0.300849	−0.150424	+	0.988622i	$0.548064\pi$
$810$	566.663	0.0245809
$811$	5155.99	0.223245	0.111622	−	0.993751i	$-0.464395\pi$
$811$	5155.99	0.223245	0.111622	+	0.993751i	$0.464395\pi$
$812$	12442.7	0.537750
$813$	−54464.3	−2.34950
$814$	−7338.79	−0.316001
$815$	−725.056	−0.0311627
$816$	−33355.8	−1.43099
$817$	0	0
$818$	16529.7	0.706536
$819$	−12563.0	−0.536002
$820$	−385.983	−0.0164379
$821$	−28808.7	−1.22464	−0.612320	−	0.790610i	$-0.709764\pi$
$821$	−28808.7	−1.22464	−0.612320	+	0.790610i	$0.709764\pi$
$822$	−20578.6	−0.873189
$823$	14065.1	0.595720	0.297860	−	0.954610i	$-0.403727\pi$
$823$	14065.1	0.595720	0.297860	+	0.954610i	$0.403727\pi$
$824$	−22990.3	−0.971973
$825$	27792.8	1.17288
$826$	−4647.68	−0.195779
$827$	−20288.1	−0.853068	−0.426534	−	0.904471i	$-0.640266\pi$
$827$	−20288.1	−0.853068	−0.426534	+	0.904471i	$0.640266\pi$
$828$	−50197.2	−2.10685
$829$	2531.77	0.106070	0.0530349	−	0.998593i	$-0.483111\pi$
$829$	2531.77	0.106070	0.0530349	+	0.998593i	$0.483111\pi$
$830$	449.431	0.0187952
$831$	10134.4	0.423055
$832$	−73.3923	−0.00305820
$833$	33365.0	1.38779
$834$	−28618.8	−1.18823
$835$	−360.726	−0.0149502
$836$	2551.14	0.105542
$837$	−97892.1	−4.04259
$838$	−5054.49	−0.208359
$839$	46374.3	1.90824	0.954122	−	0.299417i	$-0.0967921\pi$
$839$	46374.3	1.90824	0.954122	+	0.299417i	$0.0967921\pi$
$840$	−489.710	−0.0201150
$841$	24313.1	0.996888
$842$	−18249.3	−0.746928
$843$	41317.6	1.68808
$844$	−12496.7	−0.509661
$845$	−538.494	−0.0219228
$846$	−144.350	−0.00586627
$847$	−6784.73	−0.275238
$848$	7389.48	0.299240
$849$	17497.5	0.707318
$850$	19832.8	0.800307
$851$	31343.9	1.26258
$852$	40676.7	1.63564
$853$	3436.56	0.137943	0.0689716	−	0.997619i	$-0.478028\pi$
$853$	3436.56	0.137943	0.0689716	+	0.997619i	$0.478028\pi$
$854$	−7785.05	−0.311943
$855$	335.730	0.0134289
$856$	−6073.27	−0.242500
$857$	−29311.5	−1.16833	−0.584167	−	0.811633i	$-0.698579\pi$
$857$	−29311.5	−1.16833	−0.584167	+	0.811633i	$0.698579\pi$
$858$	6538.01	0.260145
$859$	31911.1	1.26751	0.633756	−	0.773533i	$-0.281512\pi$
$859$	31911.1	1.26751	0.633756	+	0.773533i	$0.281512\pi$
$860$	0	0
$861$	15344.5	0.607364
$862$	10997.6	0.434548
$863$	16307.5	0.643239	0.321619	−	0.946869i	$-0.395773\pi$
$863$	16307.5	0.643239	0.321619	+	0.946869i	$0.395773\pi$
$864$	58548.3	2.30539
$865$	73.4948	0.00288890
$866$	16614.7	0.651952
$867$	102296.	4.00710
$868$	17087.9	0.668203
$869$	17349.4	0.677259
$870$	−852.165	−0.0332082
$871$	6507.34	0.253149
$872$	−2099.27	−0.0815254
$873$	88507.5	3.43130
$874$	2716.39	0.105130
$875$	−715.349	−0.0276380
$876$	61979.5	2.39052
$877$	34825.8	1.34092	0.670458	−	0.741948i	$-0.266098\pi$
$877$	34825.8	1.34092	0.670458	+	0.741948i	$0.266098\pi$
$878$	14392.8	0.553229
$879$	−4519.15	−0.173410
$880$	−217.308	−0.00832439
$881$	3505.74	0.134065	0.0670326	−	0.997751i	$-0.478647\pi$
$881$	3505.74	0.134065	0.0670326	+	0.997751i	$0.478647\pi$
$882$	−20582.4	−0.785764
$883$	23731.6	0.904451	0.452226	−	0.891904i	$-0.350630\pi$
$883$	23731.6	0.904451	0.452226	+	0.891904i	$0.350630\pi$
$884$	−18714.0	−0.712015
$885$	−1276.78	−0.0484954
$886$	−5440.42	−0.206292
$887$	15367.8	0.581736	0.290868	−	0.956763i	$-0.406056\pi$
$887$	15367.8	0.581736	0.290868	+	0.956763i	$0.406056\pi$
$888$	−41972.6	−1.58616
$889$	16057.2	0.605783
$890$	−82.3088	−0.00310000
$891$	−32656.2	−1.22786
$892$	−15234.7	−0.571855
$893$	−31.3328	−0.00117415
$894$	14674.3	0.548974
$895$	−381.739	−0.0142571
$896$	−12732.9	−0.474751
$897$	−27923.8	−1.03941
$898$	−4723.99	−0.175547
$899$	66883.9	2.48132
$900$	49074.8	1.81758
$901$	−32893.2	−1.21624
$902$	−5545.48	−0.204705
$903$	0	0
$904$	13102.9	0.482077
$905$	−308.402	−0.0113278
$906$	909.731	0.0333596
$907$	36989.5	1.35415	0.677076	−	0.735913i	$-0.263247\pi$
$907$	36989.5	1.35415	0.677076	+	0.735913i	$0.263247\pi$
$908$	−4261.70	−0.155759
$909$	55784.2	2.03547
$910$	−84.1041	−0.00306376
$911$	52269.2	1.90094	0.950469	−	0.310819i	$-0.100603\pi$
$911$	52269.2	1.90094	0.950469	+	0.310819i	$0.100603\pi$
$912$	4466.40	0.162168
$913$	−25900.2	−0.938852
$914$	11255.6	0.407333
$915$	−2138.65	−0.0772697
$916$	−20815.1	−0.750819
$917$	7288.85	0.262485
$918$	−51291.1	−1.84407
$919$	17187.9	0.616948	0.308474	−	0.951233i	$-0.400182\pi$
$919$	17187.9	0.616948	0.308474	+	0.951233i	$0.400182\pi$
$920$	−755.880	−0.0270876
$921$	65172.1	2.33170
$922$	3404.75	0.121615
$923$	15713.5	0.560364
$924$	12546.7	0.446707
$925$	−30643.0	−1.08923
$926$	−8477.84	−0.300863
$927$	77516.3	2.74646
$928$	−40002.6	−1.41503
$929$	18064.1	0.637960	0.318980	−	0.947761i	$-0.396660\pi$
$929$	18064.1	0.637960	0.318980	+	0.947761i	$0.396660\pi$
$930$	−1170.30	−0.0412642
$931$	−4467.63	−0.157273
$932$	38175.9	1.34173
$933$	−19363.4	−0.679451
$934$	21186.2	0.742222
$935$	967.317	0.0338338
$936$	25966.9	0.906788
$937$	12776.9	0.445467	0.222734	−	0.974879i	$-0.428502\pi$
$937$	12776.9	0.445467	0.222734	+	0.974879i	$0.428502\pi$
$938$	−3113.28	−0.108371
$939$	50382.3	1.75098
$940$	3.87625	0.000134499 0
$941$	12576.8	0.435698	0.217849	−	0.975982i	$-0.430096\pi$
$941$	12576.8	0.435698	0.217849	+	0.975982i	$0.430096\pi$
$942$	15533.6	0.537276
$943$	23684.7	0.817900
$944$	−11795.5	−0.406685
$945$	924.617	0.0318283
$946$	0	0
$947$	9064.67	0.311048	0.155524	−	0.987832i	$-0.450293\pi$
$947$	9064.67	0.311048	0.155524	+	0.987832i	$0.450293\pi$
$948$	44114.1	1.51135
$949$	23942.8	0.818984
$950$	−2655.65	−0.0906955
$951$	−65983.4	−2.24990
$952$	20138.7	0.685607
$953$	−24942.4	−0.847811	−0.423905	−	0.905707i	$-0.639341\pi$
$953$	−24942.4	−0.847811	−0.423905	+	0.905707i	$0.639341\pi$
$954$	20291.3	0.688632
$955$	78.3585	0.00265510
$956$	−10578.3	−0.357873
$957$	49109.3	1.65881
$958$	−16684.4	−0.562680
$959$	−15255.5	−0.513689
$960$	9.64595	0.000324293 0
$961$	62062.4	2.08326
$962$	−7208.49	−0.241592
$963$	20477.2	0.685222
$964$	3361.86	0.112322
$965$	−245.185	−0.00817906
$966$	13359.5	0.444963
$967$	−8748.78	−0.290943	−0.145471	−	0.989362i	$-0.546470\pi$
$967$	−8748.78	−0.290943	−0.145471	+	0.989362i	$0.546470\pi$
$968$	14023.6	0.465636
$969$	−19881.5	−0.659119
$970$	592.523	0.0196132
$971$	14600.8	0.482557	0.241279	−	0.970456i	$-0.422433\pi$
$971$	14600.8	0.482557	0.241279	+	0.970456i	$0.422433\pi$
$972$	−27189.3	−0.897220
$973$	−21216.0	−0.699026
$974$	−2894.93	−0.0952356
$975$	27299.4	0.896698
$976$	−19757.9	−0.647987
$977$	22071.1	0.722741	0.361371	−	0.932422i	$-0.382309\pi$
$977$	22071.1	0.722741	0.361371	+	0.932422i	$0.382309\pi$
$978$	−26487.2	−0.866019
$979$	4743.37	0.154851
$980$	552.700	0.0180157
$981$	7078.09	0.230363
$982$	11141.3	0.362050
$983$	−36744.9	−1.19225	−0.596125	−	0.802892i	$-0.703294\pi$
$983$	−36744.9	−1.19225	−0.596125	+	0.802892i	$0.703294\pi$
$984$	−31716.2	−1.02751
$985$	−1490.22	−0.0482056
$986$	35044.2	1.13188
$987$	−154.098	−0.00496960
$988$	2505.84	0.0806897
$989$	0	0
$990$	−596.723	−0.0191567
$991$	3840.07	0.123092	0.0615458	−	0.998104i	$-0.480397\pi$
$991$	3840.07	0.123092	0.0615458	+	0.998104i	$0.480397\pi$
$992$	−54936.7	−1.75831
$993$	52186.2	1.66775
$994$	−7517.75	−0.239888
$995$	833.691	0.0265626
$996$	−65856.2	−2.09511
$997$	−11059.7	−0.351318	−0.175659	−	0.984451i	$-0.556206\pi$
$997$	−11059.7	−0.351318	−0.175659	+	0.984451i	$0.556206\pi$
$998$	20230.1	0.641655
$999$	79248.1	2.50981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.j.1.31	yes	50
43.42	odd	2		1849.4.a.i.1.20	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.20	✓	50	43.42	odd	2
1849.4.a.j.1.31	yes	50	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+1.26350 q^{2} +9.40007 q^{3} -6.40356 q^{4} +0.325119 q^{5} +11.8770 q^{6} +8.80478 q^{7} -18.1989 q^{8} +61.3613 q^{9} +O(q^{10})\)	\(q+1.26350 q^{2} +9.40007 q^{3} -6.40356 q^{4} +0.325119 q^{5} +11.8770 q^{6} +8.80478 q^{7} -18.1989 q^{8} +61.3613 q^{9} +0.410789 q^{10} -23.6733 q^{11} -60.1939 q^{12} -23.2530 q^{13} +11.1249 q^{14} +3.05614 q^{15} +28.2341 q^{16} -125.680 q^{17} +77.5301 q^{18} +16.8288 q^{19} -2.08192 q^{20} +82.7656 q^{21} -29.9113 q^{22} +127.751 q^{23} -171.071 q^{24} -124.894 q^{25} -29.3802 q^{26} +322.998 q^{27} -56.3820 q^{28} -220.686 q^{29} +3.86145 q^{30} -303.073 q^{31} +181.265 q^{32} -222.531 q^{33} -158.797 q^{34} +2.86261 q^{35} -392.931 q^{36} +245.352 q^{37} +21.2632 q^{38} -218.580 q^{39} -5.91683 q^{40} +185.397 q^{41} +104.574 q^{42} +151.594 q^{44} +19.9497 q^{45} +161.414 q^{46} -1.86186 q^{47} +265.403 q^{48} -265.476 q^{49} -157.804 q^{50} -1181.40 q^{51} +148.902 q^{52} +261.722 q^{53} +408.109 q^{54} -7.69666 q^{55} -160.238 q^{56} +158.192 q^{57} -278.837 q^{58} -417.774 q^{59} -19.5702 q^{60} -699.788 q^{61} -382.934 q^{62} +540.273 q^{63} +3.15625 q^{64} -7.56001 q^{65} -281.168 q^{66} -279.849 q^{67} +804.800 q^{68} +1200.87 q^{69} +3.61691 q^{70} -675.761 q^{71} -1116.71 q^{72} -1029.66 q^{73} +310.002 q^{74} -1174.01 q^{75} -107.764 q^{76} -208.439 q^{77} -276.176 q^{78} -732.866 q^{79} +9.17946 q^{80} +1379.45 q^{81} +234.250 q^{82} +1094.07 q^{83} -529.994 q^{84} -40.8610 q^{85} -2074.46 q^{87} +430.829 q^{88} -200.368 q^{89} +25.2065 q^{90} -204.738 q^{91} -818.061 q^{92} -2848.91 q^{93} -2.35247 q^{94} +5.47136 q^{95} +1703.91 q^{96} +1442.40 q^{97} -335.429 q^{98} -1452.63 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} + 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} + 386 q^{18} - 12 q^{19} - 108 q^{20} - 408 q^{21} - 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} - 1493 q^{26} - 10 q^{27} + 242 q^{28} - 208 q^{29} + 48 q^{30} - 932 q^{31} + 1124 q^{32} - 254 q^{33} - 765 q^{34} - 1452 q^{35} + 747 q^{36} + 90 q^{37} - 1213 q^{38} + 1610 q^{39} - 1693 q^{40} - 1354 q^{41} + 16 q^{42} - 2704 q^{44} - 4508 q^{45} - 233 q^{46} - 3484 q^{47} + 376 q^{48} + 1324 q^{49} + 408 q^{50} - 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} + 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} - 1172 q^{61} + 1546 q^{62} + 3686 q^{63} + 606 q^{64} - 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} - 136 q^{69} + 1310 q^{70} - 162 q^{71} + 5814 q^{72} - 746 q^{73} - 4332 q^{74} - 236 q^{75} - 1338 q^{76} - 2024 q^{77} - 2782 q^{78} - 2656 q^{79} + 5713 q^{80} - 86 q^{81} + 4168 q^{82} - 3514 q^{83} - 4269 q^{84} + 7558 q^{85} - 10278 q^{87} - 11692 q^{88} - 2640 q^{89} - 8286 q^{90} + 5946 q^{91} - 4271 q^{92} + 2 q^{93} - 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} + 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 + 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 + 386 * q^18 - 12 * q^19 - 108 * q^20 - 408 * q^21 - 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 - 1493 * q^26 - 10 * q^27 + 242 * q^28 - 208 * q^29 + 48 * q^30 - 932 * q^31 + 1124 * q^32 - 254 * q^33 - 765 * q^34 - 1452 * q^35 + 747 * q^36 + 90 * q^37 - 1213 * q^38 + 1610 * q^39 - 1693 * q^40 - 1354 * q^41 + 16 * q^42 - 2704 * q^44 - 4508 * q^45 - 233 * q^46 - 3484 * q^47 + 376 * q^48 + 1324 * q^49 + 408 * q^50 - 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 + 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 - 1172 * q^61 + 1546 * q^62 + 3686 * q^63 + 606 * q^64 - 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 - 136 * q^69 + 1310 * q^70 - 162 * q^71 + 5814 * q^72 - 746 * q^73 - 4332 * q^74 - 236 * q^75 - 1338 * q^76 - 2024 * q^77 - 2782 * q^78 - 2656 * q^79 + 5713 * q^80 - 86 * q^81 + 4168 * q^82 - 3514 * q^83 - 4269 * q^84 + 7558 * q^85 - 10278 * q^87 - 11692 * q^88 - 2640 * q^89 - 8286 * q^90 + 5946 * q^91 - 4271 * q^92 + 2 * q^93 - 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 + 2826 * q^98 - 8174 * q^99

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