LMFDB - Embedded newform 1849.4.a.k.1.12

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:	$60$
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-3.61784 q^{2} +0.230904 q^{3} +5.08877 q^{4} +16.9044 q^{5} -0.835373 q^{6} -0.941277 q^{7} +10.5324 q^{8} -26.9467 q^{9} +O(q^{10})$	\(q-3.61784 q^{2} +0.230904 q^{3} +5.08877 q^{4} +16.9044 q^{5} -0.835373 q^{6} -0.941277 q^{7} +10.5324 q^{8} -26.9467 q^{9} -61.1575 q^{10} +6.40255 q^{11} +1.17502 q^{12} -58.7355 q^{13} +3.40539 q^{14} +3.90330 q^{15} -78.8146 q^{16} -93.0890 q^{17} +97.4888 q^{18} +34.5816 q^{19} +86.0227 q^{20} -0.217344 q^{21} -23.1634 q^{22} +121.644 q^{23} +2.43196 q^{24} +160.760 q^{25} +212.496 q^{26} -12.4565 q^{27} -4.78994 q^{28} +148.422 q^{29} -14.1215 q^{30} +293.940 q^{31} +200.880 q^{32} +1.47837 q^{33} +336.781 q^{34} -15.9117 q^{35} -137.125 q^{36} -33.0033 q^{37} -125.111 q^{38} -13.5622 q^{39} +178.044 q^{40} -261.072 q^{41} +0.786317 q^{42} +32.5811 q^{44} -455.518 q^{45} -440.089 q^{46} -22.3220 q^{47} -18.1986 q^{48} -342.114 q^{49} -581.603 q^{50} -21.4946 q^{51} -298.891 q^{52} +152.319 q^{53} +45.0656 q^{54} +108.231 q^{55} -9.91388 q^{56} +7.98503 q^{57} -536.969 q^{58} +246.765 q^{59} +19.8630 q^{60} -198.297 q^{61} -1063.43 q^{62} +25.3643 q^{63} -96.2338 q^{64} -992.890 q^{65} -5.34852 q^{66} +381.327 q^{67} -473.708 q^{68} +28.0881 q^{69} +57.5662 q^{70} +419.795 q^{71} -283.812 q^{72} -1023.39 q^{73} +119.401 q^{74} +37.1200 q^{75} +175.978 q^{76} -6.02657 q^{77} +49.0661 q^{78} -1167.19 q^{79} -1332.32 q^{80} +724.684 q^{81} +944.515 q^{82} +1035.81 q^{83} -1.10602 q^{84} -1573.62 q^{85} +34.2713 q^{87} +67.4340 q^{88} +1072.32 q^{89} +1647.99 q^{90} +55.2864 q^{91} +619.019 q^{92} +67.8719 q^{93} +80.7573 q^{94} +584.582 q^{95} +46.3839 q^{96} -381.609 q^{97} +1237.71 q^{98} -172.528 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 60 q - 15 q^{2} - 37 q^{3} + 213 q^{4} - 51 q^{5} + 22 q^{6} - 54 q^{7} - 204 q^{8} + 387 q^{9}+O(q^{10}) $ 60 * q - 15 * q^2 - 37 * q^3 + 213 * q^4 - 51 * q^5 + 22 * q^6 - 54 * q^7 - 204 * q^8 + 387 * q^9	\( 60 q - 15 q^{2} - 37 q^{3} + 213 q^{4} - 51 q^{5} + 22 q^{6} - 54 q^{7} - 204 q^{8} + 387 q^{9} + 27 q^{10} + 43 q^{11} - 432 q^{12} + 13 q^{13} + 96 q^{14} + 65 q^{15} + 717 q^{16} - 138 q^{17} - 625 q^{18} - 610 q^{19} - 345 q^{20} + 611 q^{21} - 118 q^{22} - 243 q^{23} + 258 q^{24} + 899 q^{25} - 1071 q^{26} - 1609 q^{27} - 46 q^{28} - 773 q^{29} - 375 q^{30} - 97 q^{31} - 1967 q^{32} - 500 q^{33} - 217 q^{34} + 247 q^{35} + 175 q^{36} - 228 q^{37} + 1253 q^{38} - 1493 q^{39} + 2220 q^{40} - 951 q^{41} - 2643 q^{42} - 1378 q^{44} - 1086 q^{45} + 565 q^{46} - 2 q^{47} - 2303 q^{48} + 1264 q^{49} - 3273 q^{50} - 3076 q^{51} - 2825 q^{52} - 39 q^{53} + 5201 q^{54} - 1306 q^{55} + 3683 q^{56} + 1342 q^{57} + 2588 q^{58} - 1065 q^{59} + 2803 q^{60} - 2999 q^{61} - 5569 q^{62} - 2377 q^{63} + 2082 q^{64} - 5578 q^{65} + 3338 q^{66} + 961 q^{67} - 3754 q^{68} - 1817 q^{69} - 2738 q^{70} - 8003 q^{71} - 1412 q^{72} + 1011 q^{73} - 1413 q^{74} - 7457 q^{75} - 5516 q^{76} - 4052 q^{77} + 1091 q^{78} - 4422 q^{79} - 1610 q^{80} + 2108 q^{81} - 4676 q^{82} - 297 q^{83} - 54 q^{84} - 4333 q^{85} + 1377 q^{87} - 3652 q^{88} - 2480 q^{89} - 1414 q^{90} - 4551 q^{91} - 3286 q^{92} - 4 q^{93} - 4609 q^{94} - 835 q^{95} + 5864 q^{96} + 3785 q^{97} - 753 q^{98} + 5072 q^{99}+O(q^{100}) \) 60 * q - 15 * q^2 - 37 * q^3 + 213 * q^4 - 51 * q^5 + 22 * q^6 - 54 * q^7 - 204 * q^8 + 387 * q^9 + 27 * q^10 + 43 * q^11 - 432 * q^12 + 13 * q^13 + 96 * q^14 + 65 * q^15 + 717 * q^16 - 138 * q^17 - 625 * q^18 - 610 * q^19 - 345 * q^20 + 611 * q^21 - 118 * q^22 - 243 * q^23 + 258 * q^24 + 899 * q^25 - 1071 * q^26 - 1609 * q^27 - 46 * q^28 - 773 * q^29 - 375 * q^30 - 97 * q^31 - 1967 * q^32 - 500 * q^33 - 217 * q^34 + 247 * q^35 + 175 * q^36 - 228 * q^37 + 1253 * q^38 - 1493 * q^39 + 2220 * q^40 - 951 * q^41 - 2643 * q^42 - 1378 * q^44 - 1086 * q^45 + 565 * q^46 - 2 * q^47 - 2303 * q^48 + 1264 * q^49 - 3273 * q^50 - 3076 * q^51 - 2825 * q^52 - 39 * q^53 + 5201 * q^54 - 1306 * q^55 + 3683 * q^56 + 1342 * q^57 + 2588 * q^58 - 1065 * q^59 + 2803 * q^60 - 2999 * q^61 - 5569 * q^62 - 2377 * q^63 + 2082 * q^64 - 5578 * q^65 + 3338 * q^66 + 961 * q^67 - 3754 * q^68 - 1817 * q^69 - 2738 * q^70 - 8003 * q^71 - 1412 * q^72 + 1011 * q^73 - 1413 * q^74 - 7457 * q^75 - 5516 * q^76 - 4052 * q^77 + 1091 * q^78 - 4422 * q^79 - 1610 * q^80 + 2108 * q^81 - 4676 * q^82 - 297 * q^83 - 54 * q^84 - 4333 * q^85 + 1377 * q^87 - 3652 * q^88 - 2480 * q^89 - 1414 * q^90 - 4551 * q^91 - 3286 * q^92 - 4 * q^93 - 4609 * q^94 - 835 * q^95 + 5864 * q^96 + 3785 * q^97 - 753 * q^98 + 5072 * 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 $

$2$	−3.61784	−1.27910	−0.639550	−	0.768750i	$-0.720879\pi$
$2$	−3.61784	−1.27910	−0.639550	+	0.768750i	$0.720879\pi$
$3$	0.230904	0.0444375	0.0222187	−	0.999753i	$-0.492927\pi$
$3$	0.230904	0.0444375	0.0222187	+	0.999753i	$0.492927\pi$
$4$	5.08877	0.636096
$5$	16.9044	1.51198	0.755989	−	0.654584i	$-0.227156\pi$
$5$	16.9044	1.51198	0.755989	+	0.654584i	$0.227156\pi$
$6$	−0.835373	−0.0568399
$7$	−0.941277	−0.0508242	−0.0254121	−	0.999677i	$-0.508090\pi$
$7$	−0.941277	−0.0508242	−0.0254121	+	0.999677i	$0.508090\pi$
$8$	10.5324	0.465469
$9$	−26.9467	−0.998025
$10$	−61.1575	−1.93397
$11$	6.40255	0.175495	0.0877473	−	0.996143i	$-0.472033\pi$
$11$	6.40255	0.175495	0.0877473	+	0.996143i	$0.472033\pi$
$12$	1.17502	0.0282665
$13$	−58.7355	−1.25310	−0.626550	−	0.779381i	$-0.715534\pi$
$13$	−58.7355	−1.25310	−0.626550	+	0.779381i	$0.715534\pi$
$14$	3.40539	0.0650092
$15$	3.90330	0.0671885
$16$	−78.8146	−1.23148
$17$	−93.0890	−1.32808	−0.664041	−	0.747696i	$-0.731160\pi$
$17$	−93.0890	−1.32808	−0.664041	+	0.747696i	$0.731160\pi$
$18$	97.4888	1.27657
$19$	34.5816	0.417556	0.208778	−	0.977963i	$-0.433051\pi$
$19$	34.5816	0.417556	0.208778	+	0.977963i	$0.433051\pi$
$20$	86.0227	0.961764
$21$	−0.217344	−0.00225850
$22$	−23.1634	−0.224475
$23$	121.644	1.10281	0.551403	−	0.834239i	$-0.314093\pi$
$23$	121.644	1.10281	0.551403	+	0.834239i	$0.314093\pi$
$24$	2.43196	0.0206843
$25$	160.760	1.28608
$26$	212.496	1.60284
$27$	−12.4565	−0.0887872
$28$	−4.78994	−0.0323291
$29$	148.422	0.950392	0.475196	−	0.879880i	$-0.342377\pi$
$29$	148.422	0.950392	0.475196	+	0.879880i	$0.342377\pi$
$30$	−14.1215	−0.0859408
$31$	293.940	1.70301	0.851504	−	0.524349i	$-0.175691\pi$
$31$	293.940	1.70301	0.851504	+	0.524349i	$0.175691\pi$
$32$	200.880	1.10971
$33$	1.47837	0.00779854
$34$	336.781	1.69875
$35$	−15.9117	−0.0768450
$36$	−137.125	−0.634840
$37$	−33.0033	−0.146641	−0.0733205	−	0.997308i	$-0.523360\pi$
$37$	−33.0033	−0.146641	−0.0733205	+	0.997308i	$0.523360\pi$
$38$	−125.111	−0.534096
$39$	−13.5622	−0.0556846
$40$	178.044	0.703780
$41$	−261.072	−0.994452	−0.497226	−	0.867621i	$-0.665648\pi$
$41$	−261.072	−0.994452	−0.497226	+	0.867621i	$0.665648\pi$
$42$	0.786317	0.00288884
$43$	0	0
$43$	0	0
$44$	32.5811	0.111631
$45$	−455.518	−1.50899
$46$	−440.089	−1.41060
$47$	−22.3220	−0.0692764	−0.0346382	−	0.999400i	$-0.511028\pi$
$47$	−22.3220	−0.0692764	−0.0346382	+	0.999400i	$0.511028\pi$
$48$	−18.1986	−0.0547237
$49$	−342.114	−0.997417
$50$	−581.603	−1.64502
$51$	−21.4946	−0.0590166
$52$	−298.891	−0.797092
$53$	152.319	0.394767	0.197384	−	0.980326i	$-0.436756\pi$
$53$	152.319	0.394767	0.197384	+	0.980326i	$0.436756\pi$
$54$	45.0656	0.113568
$55$	108.231	0.265344
$56$	−9.91388	−0.0236571
$57$	7.98503	0.0185551
$58$	−536.969	−1.21565
$59$	246.765	0.544510	0.272255	−	0.962225i	$-0.412231\pi$
$59$	246.765	0.544510	0.272255	+	0.962225i	$0.412231\pi$
$60$	19.8630	0.0427383
$61$	−198.297	−0.416218	−0.208109	−	0.978106i	$-0.566731\pi$
$61$	−198.297	−0.416218	−0.208109	+	0.978106i	$0.566731\pi$
$62$	−1063.43	−2.17832
$63$	25.3643	0.0507238
$64$	−96.2338	−0.187957
$65$	−992.890	−1.89466
$66$	−5.34852	−0.00997511
$67$	381.327	0.695322	0.347661	−	0.937620i	$-0.386976\pi$
$67$	381.327	0.695322	0.347661	+	0.937620i	$0.386976\pi$
$68$	−473.708	−0.844788
$69$	28.0881	0.0490059
$70$	57.5662	0.0982925
$71$	419.795	0.701697	0.350848	−	0.936432i	$-0.385893\pi$
$71$	419.795	0.701697	0.350848	+	0.936432i	$0.385893\pi$
$72$	−283.812	−0.464550
$73$	−1023.39	−1.64080	−0.820399	−	0.571791i	$-0.806249\pi$
$73$	−1023.39	−1.64080	−0.820399	+	0.571791i	$0.806249\pi$
$74$	119.401	0.187568
$75$	37.1200	0.0571501
$76$	175.978	0.265606
$77$	−6.02657	−0.00891937
$78$	49.0661	0.0712261
$79$	−1167.19	−1.66226	−0.831132	−	0.556075i	$-0.812307\pi$
$79$	−1167.19	−1.66226	−0.831132	+	0.556075i	$0.812307\pi$
$80$	−1332.32	−1.86197
$81$	724.684	0.994080
$82$	944.515	1.27200
$83$	1035.81	1.36981	0.684906	−	0.728631i	$-0.259843\pi$
$83$	1035.81	1.36981	0.684906	+	0.728631i	$0.259843\pi$
$84$	−1.10602	−0.00143662
$85$	−1573.62	−2.00803
$86$	0	0
$87$	34.2713	0.0422330
$88$	67.4340	0.0816874
$89$	1072.32	1.27715	0.638573	−	0.769562i	$-0.279525\pi$
$89$	1072.32	1.27715	0.638573	+	0.769562i	$0.279525\pi$
$90$	1647.99	1.93015
$91$	55.2864	0.0636878
$92$	619.019	0.701491
$93$	67.8719	0.0756773
$94$	80.7573	0.0886115
$95$	584.582	0.631336
$96$	46.3839	0.0493129
$97$	−381.609	−0.399449	−0.199724	−	0.979852i	$-0.564005\pi$
$97$	−381.609	−0.399449	−0.199724	+	0.979852i	$0.564005\pi$
$98$	1237.71	1.27580
$99$	−172.528	−0.175148
$100$	818.069	0.818069
$101$	−1144.59	−1.12764	−0.563818	−	0.825899i	$-0.690668\pi$
$101$	−1144.59	−1.12764	−0.563818	+	0.825899i	$0.690668\pi$
$102$	77.7640	0.0754881
$103$	−831.875	−0.795797	−0.397899	−	0.917429i	$-0.630260\pi$
$103$	−831.875	−0.795797	−0.397899	+	0.917429i	$0.630260\pi$
$104$	−618.624	−0.583279
$105$	−3.67408	−0.00341480
$106$	−551.067	−0.504947
$107$	1947.45	1.75951	0.879753	−	0.475432i	$-0.157708\pi$
$107$	1947.45	1.75951	0.879753	+	0.475432i	$0.157708\pi$
$108$	−63.3882	−0.0564772
$109$	−1643.27	−1.44401	−0.722004	−	0.691889i	$-0.756779\pi$
$109$	−1643.27	−1.44401	−0.722004	+	0.691889i	$0.756779\pi$
$110$	−391.564	−0.339402
$111$	−7.62060	−0.00651635
$112$	74.1863	0.0625888
$113$	−43.4910	−0.0362061	−0.0181031	−	0.999836i	$-0.505763\pi$
$113$	−43.4910	−0.0362061	−0.0181031	+	0.999836i	$0.505763\pi$
$114$	−28.8885	−0.0237339
$115$	2056.32	1.66742
$116$	755.288	0.604540
$117$	1582.73	1.25063
$118$	−892.756	−0.696482
$119$	87.6225	0.0674987
$120$	41.1110	0.0312742
$121$	−1290.01	−0.969202
$122$	717.406	0.532384
$123$	−60.2824	−0.0441909
$124$	1495.79	1.08328
$125$	604.499	0.432544
$126$	−91.7640	−0.0648808
$127$	387.288	0.270600	0.135300	−	0.990805i	$-0.456800\pi$
$127$	387.288	0.270600	0.135300	+	0.990805i	$0.456800\pi$
$128$	−1258.88	−0.869298
$129$	0	0
$130$	3592.12	2.42346
$131$	−2618.87	−1.74665	−0.873327	−	0.487134i	$-0.838042\pi$
$131$	−2618.87	−1.74665	−0.873327	+	0.487134i	$0.838042\pi$
$132$	7.52310	0.00496062
$133$	−32.5509	−0.0212219
$134$	−1379.58	−0.889386
$135$	−210.570	−0.134244
$136$	−980.448	−0.618182
$137$	280.707	0.175054	0.0875271	−	0.996162i	$-0.472104\pi$
$137$	280.707	0.175054	0.0875271	+	0.996162i	$0.472104\pi$
$138$	−101.618	−0.0626835
$139$	−3063.81	−1.86956	−0.934781	−	0.355225i	$-0.884404\pi$
$139$	−3063.81	−1.86956	−0.934781	+	0.355225i	$0.884404\pi$
$140$	−80.9712	−0.0488808
$141$	−5.15423	−0.00307847
$142$	−1518.75	−0.897540
$143$	−376.057	−0.219912
$144$	2123.79	1.22905
$145$	2509.00	1.43697
$146$	3702.45	2.09875
$147$	−78.9954	−0.0443227
$148$	−167.946	−0.0932777
$149$	−1160.90	−0.638286	−0.319143	−	0.947707i	$-0.603395\pi$
$149$	−1160.90	−0.638286	−0.319143	+	0.947707i	$0.603395\pi$
$150$	−134.294	−0.0731006
$151$	2604.42	1.40361	0.701804	−	0.712370i	$-0.252378\pi$
$151$	2604.42	1.40361	0.701804	+	0.712370i	$0.252378\pi$
$152$	364.226	0.194360
$153$	2508.44	1.32546
$154$	21.8032	0.0114088
$155$	4968.89	2.57491
$156$	−69.0152	−0.0354207
$157$	−1853.70	−0.942302	−0.471151	−	0.882053i	$-0.656161\pi$
$157$	−1853.70	−0.942302	−0.471151	+	0.882053i	$0.656161\pi$
$158$	4222.70	2.12620
$159$	35.1711	0.0175425
$160$	3395.76	1.67786
$161$	−114.501	−0.0560492
$162$	−2621.79	−1.27153
$163$	−2220.06	−1.06680	−0.533400	−	0.845863i	$-0.679086\pi$
$163$	−2220.06	−1.06680	−0.533400	+	0.845863i	$0.679086\pi$
$164$	−1328.53	−0.632567
$165$	24.9911	0.0117912
$166$	−3747.38	−1.75213
$167$	1579.48	0.731880	0.365940	−	0.930638i	$-0.380748\pi$
$167$	1579.48	0.731880	0.365940	+	0.930638i	$0.380748\pi$
$168$	−2.28915	−0.00105126
$169$	1252.86	0.570259
$170$	5693.09	2.56847
$171$	−931.860	−0.416732
$172$	0	0
$173$	3661.41	1.60908	0.804542	−	0.593895i	$-0.202411\pi$
$173$	3661.41	1.60908	0.804542	+	0.593895i	$0.202411\pi$
$174$	−123.988	−0.0540202
$175$	−151.319	−0.0653639
$176$	−504.614	−0.216118
$177$	56.9790	0.0241966
$178$	−3879.49	−1.63360
$179$	−703.133	−0.293601	−0.146801	−	0.989166i	$-0.546898\pi$
$179$	−703.133	−0.293601	−0.146801	+	0.989166i	$0.546898\pi$
$180$	−2318.03	−0.959864
$181$	120.992	0.0496864	0.0248432	−	0.999691i	$-0.492091\pi$
$181$	120.992	0.0496864	0.0248432	+	0.999691i	$0.492091\pi$
$182$	−200.017	−0.0814630
$183$	−45.7875	−0.0184957
$184$	1281.20	0.513323
$185$	−557.903	−0.221718
$186$	−245.550	−0.0967988
$187$	−596.007	−0.233071
$188$	−113.591	−0.0440665
$189$	11.7250	0.00451253
$190$	−2114.93	−0.807541
$191$	3169.60	1.20076	0.600378	−	0.799716i	$-0.295017\pi$
$191$	3169.60	1.20076	0.600378	+	0.799716i	$0.295017\pi$
$192$	−22.2207	−0.00835231
$193$	−2445.21	−0.911970	−0.455985	−	0.889988i	$-0.650713\pi$
$193$	−2445.21	−0.911970	−0.455985	+	0.889988i	$0.650713\pi$
$194$	1380.60	0.510935
$195$	−229.262	−0.0841939
$196$	−1740.94	−0.634453
$197$	−1997.94	−0.722575	−0.361288	−	0.932454i	$-0.617663\pi$
$197$	−1997.94	−0.722575	−0.361288	+	0.932454i	$0.617663\pi$
$198$	624.177	0.224032
$199$	−2130.24	−0.758836	−0.379418	−	0.925225i	$-0.623876\pi$
$199$	−2130.24	−0.758836	−0.379418	+	0.925225i	$0.623876\pi$
$200$	1693.18	0.598630
$201$	88.0499	0.0308983
$202$	4140.96	1.44236
$203$	−139.707	−0.0483029
$204$	−109.381	−0.0375402
$205$	−4413.27	−1.50359
$206$	3009.59	1.01790
$207$	−3277.91	−1.10063
$208$	4629.21	1.54316
$209$	221.411	0.0732789
$210$	13.2922	0.00436787
$211$	2947.47	0.961669	0.480834	−	0.876811i	$-0.340334\pi$
$211$	2947.47	0.961669	0.480834	+	0.876811i	$0.340334\pi$
$212$	775.118	0.251110
$213$	96.9322	0.0311816
$214$	−7045.56	−2.25058
$215$	0	0
$216$	−131.196	−0.0413277
$217$	−276.679	−0.0865539
$218$	5945.09	1.84703
$219$	−236.304	−0.0729129
$220$	550.765	0.168784
$221$	5467.63	1.66422
$222$	27.5701	0.00833506
$223$	−516.457	−0.155088	−0.0775438	−	0.996989i	$-0.524708\pi$
$223$	−516.457	−0.155088	−0.0775438	+	0.996989i	$0.524708\pi$
$224$	−189.083	−0.0564003
$225$	−4331.94	−1.28354
$226$	157.344	0.0463113
$227$	2995.20	0.875763	0.437881	−	0.899033i	$-0.355729\pi$
$227$	2995.20	0.875763	0.437881	+	0.899033i	$0.355729\pi$
$228$	40.6339	0.0118028
$229$	3167.43	0.914017	0.457008	−	0.889462i	$-0.348921\pi$
$229$	3167.43	0.914017	0.457008	+	0.889462i	$0.348921\pi$
$230$	−7439.45	−2.13280
$231$	−1.39156	−0.000396354 0
$232$	1563.24	0.442378
$233$	−4135.44	−1.16275	−0.581377	−	0.813634i	$-0.697486\pi$
$233$	−4135.44	−1.16275	−0.581377	+	0.813634i	$0.697486\pi$
$234$	−5726.05	−1.59967
$235$	−377.340	−0.104744
$236$	1255.73	0.346360
$237$	−269.508	−0.0738668
$238$	−317.004	−0.0863375
$239$	−3256.27	−0.881301	−0.440650	−	0.897679i	$-0.645252\pi$
$239$	−3256.27	−0.881301	−0.440650	+	0.897679i	$0.645252\pi$
$240$	−307.637	−0.0827411
$241$	−1865.13	−0.498522	−0.249261	−	0.968436i	$-0.580188\pi$
$241$	−1865.13	−0.498522	−0.249261	+	0.968436i	$0.580188\pi$
$242$	4667.04	1.23971
$243$	503.658	0.132962
$244$	−1009.09	−0.264755
$245$	−5783.24	−1.50807
$246$	218.092	0.0565246
$247$	−2031.17	−0.523239
$248$	3095.89	0.792698
$249$	239.171	0.0608710
$250$	−2186.98	−0.553267
$251$	−4696.45	−1.18102	−0.590512	−	0.807029i	$-0.701074\pi$
$251$	−4696.45	−1.18102	−0.590512	+	0.807029i	$0.701074\pi$
$252$	129.073	0.0322652
$253$	778.833	0.193537
$254$	−1401.15	−0.346125
$255$	−363.354	−0.0892318
$256$	5324.29	1.29988
$257$	−934.940	−0.226926	−0.113463	−	0.993542i	$-0.536194\pi$
$257$	−934.940	−0.226926	−0.113463	+	0.993542i	$0.536194\pi$
$258$	0	0
$259$	31.0653	0.00745291
$260$	−5052.59	−1.20519
$261$	−3999.49	−0.948515
$262$	9474.65	2.23415
$263$	−7137.75	−1.67351	−0.836754	−	0.547580i	$-0.815549\pi$
$263$	−7137.75	−1.67351	−0.836754	+	0.547580i	$0.815549\pi$
$264$	15.5708	0.00362998
$265$	2574.87	0.596880
$266$	117.764	0.0271450
$267$	247.603	0.0567531
$268$	1940.49	0.442291
$269$	−987.301	−0.223780	−0.111890	−	0.993721i	$-0.535690\pi$
$269$	−987.301	−0.223780	−0.111890	+	0.993721i	$0.535690\pi$
$270$	761.808	0.171712
$271$	4724.65	1.05905	0.529525	−	0.848295i	$-0.322370\pi$
$271$	4724.65	1.05905	0.529525	+	0.848295i	$0.322370\pi$
$272$	7336.77	1.63550
$273$	12.7658	0.00283012
$274$	−1015.55	−0.223912
$275$	1029.27	0.225700
$276$	142.934	0.0311725
$277$	−6311.26	−1.36898	−0.684489	−	0.729023i	$-0.739975\pi$
$277$	−6311.26	−1.36898	−0.684489	+	0.729023i	$0.739975\pi$
$278$	11084.4	2.39136
$279$	−7920.71	−1.69964
$280$	−167.588	−0.0357690
$281$	−5191.74	−1.10218	−0.551091	−	0.834445i	$-0.685788\pi$
$281$	−5191.74	−1.10218	−0.551091	+	0.834445i	$0.685788\pi$
$282$	18.6472	0.00393767
$283$	−7336.25	−1.54097	−0.770486	−	0.637457i	$-0.779986\pi$
$283$	−7336.25	−1.54097	−0.770486	+	0.637457i	$0.779986\pi$
$284$	2136.24	0.446346
$285$	134.982	0.0280550
$286$	1360.51	0.281290
$287$	245.741	0.0505422
$288$	−5413.04	−1.10752
$289$	3752.56	0.763802
$290$	−9077.15	−1.83803
$291$	−88.1150	−0.0177505
$292$	−5207.78	−1.04371
$293$	2392.51	0.477038	0.238519	−	0.971138i	$-0.423338\pi$
$293$	2392.51	0.477038	0.238519	+	0.971138i	$0.423338\pi$
$294$	285.793	0.0566931
$295$	4171.42	0.823287
$296$	−347.603	−0.0682569
$297$	−79.7533	−0.0155817
$298$	4199.95	0.816431
$299$	−7144.83	−1.38193
$300$	188.895	0.0363529
$301$	0	0
$302$	−9422.38	−1.79535
$303$	−264.291	−0.0501093
$304$	−2725.54	−0.514211
$305$	−3352.09	−0.629312
$306$	−9075.13	−1.69540
$307$	−975.647	−0.181378	−0.0906892	−	0.995879i	$-0.528907\pi$
$307$	−975.647	−0.181378	−0.0906892	+	0.995879i	$0.528907\pi$
$308$	−30.6678	−0.00567358
$309$	−192.083	−0.0353632
$310$	−17976.7	−3.29357
$311$	6761.42	1.23281	0.616407	−	0.787428i	$-0.288588\pi$
$311$	6761.42	1.23281	0.616407	+	0.787428i	$0.288588\pi$
$312$	−142.843	−0.0259195
$313$	−8731.46	−1.57678	−0.788388	−	0.615178i	$-0.789084\pi$
$313$	−8731.46	−1.57678	−0.788388	+	0.615178i	$0.789084\pi$
$314$	6706.39	1.20530
$315$	428.769	0.0766933
$316$	−5939.55	−1.05736
$317$	−7576.00	−1.34230	−0.671152	−	0.741320i	$-0.734200\pi$
$317$	−7576.00	−1.34230	−0.671152	+	0.741320i	$0.734200\pi$
$318$	−127.243	−0.0224386
$319$	950.282	0.166789
$320$	−1626.78	−0.284186
$321$	449.673	0.0781879
$322$	414.246	0.0716926
$323$	−3219.17	−0.554549
$324$	3687.75	0.632330
$325$	−9442.31	−1.61158
$326$	8031.82	1.36454
$327$	−379.438	−0.0641680
$328$	−2749.70	−0.462887
$329$	21.0111	0.00352092
$330$	−90.4137	−0.0150821
$331$	9823.11	1.63120	0.815600	−	0.578617i	$-0.196407\pi$
$331$	9823.11	1.63120	0.815600	+	0.578617i	$0.196407\pi$
$332$	5270.97	0.871332
$333$	889.331	0.146351
$334$	−5714.31	−0.936147
$335$	6446.12	1.05131
$336$	17.1299	0.00278129
$337$	1364.08	0.220494	0.110247	−	0.993904i	$-0.464836\pi$
$337$	1364.08	0.220494	0.110247	+	0.993904i	$0.464836\pi$
$338$	−4532.64	−0.729418
$339$	−10.0422	−0.00160891
$340$	−8007.77	−1.27730
$341$	1881.97	0.298869
$342$	3371.32	0.533041
$343$	644.882	0.101517
$344$	0	0
$345$	474.813	0.0740959
$346$	−13246.4	−2.05818
$347$	1151.20	0.178097	0.0890487	−	0.996027i	$-0.471617\pi$
$347$	1151.20	0.178097	0.0890487	+	0.996027i	$0.471617\pi$
$348$	174.399	0.0268642
$349$	−347.963	−0.0533697	−0.0266849	−	0.999644i	$-0.508495\pi$
$349$	−347.963	−0.0533697	−0.0266849	+	0.999644i	$0.508495\pi$
$350$	547.450	0.0836069
$351$	731.638	0.111259
$352$	1286.14	0.194749
$353$	−2867.42	−0.432344	−0.216172	−	0.976355i	$-0.569357\pi$
$353$	−2867.42	−0.432344	−0.216172	+	0.976355i	$0.569357\pi$
$354$	−206.141	−0.0309499
$355$	7096.39	1.06095
$356$	5456.80	0.812387
$357$	20.2324	0.00299947
$358$	2543.82	0.375546
$359$	−5106.72	−0.750758	−0.375379	−	0.926871i	$-0.622487\pi$
$359$	−5106.72	−0.750758	−0.375379	+	0.926871i	$0.622487\pi$
$360$	−4797.69	−0.702390
$361$	−5663.11	−0.825647
$362$	−437.729	−0.0635539
$363$	−297.868	−0.0430689
$364$	281.340	0.0405115
$365$	−17299.8	−2.48085
$366$	165.652	0.0236578
$367$	1522.22	0.216511	0.108255	−	0.994123i	$-0.465474\pi$
$367$	1522.22	0.216511	0.108255	+	0.994123i	$0.465474\pi$
$368$	−9587.33	−1.35808
$369$	7035.01	0.992488
$370$	2018.40	0.283599
$371$	−143.375	−0.0200637
$372$	345.384	0.0481380
$373$	1814.09	0.251823	0.125912	−	0.992041i	$-0.459814\pi$
$373$	1814.09	0.251823	0.125912	+	0.992041i	$0.459814\pi$
$374$	2156.26	0.298122
$375$	139.581	0.0192212
$376$	−235.103	−0.0322461
$377$	−8717.67	−1.19094
$378$	−42.4192	−0.00577198
$379$	1393.68	0.188888	0.0944440	−	0.995530i	$-0.469893\pi$
$379$	1393.68	0.188888	0.0944440	+	0.995530i	$0.469893\pi$
$380$	2974.81	0.401590
$381$	89.4263	0.0120248
$382$	−11467.1	−1.53589
$383$	−1455.78	−0.194222	−0.0971111	−	0.995274i	$-0.530960\pi$
$383$	−1455.78	−0.194222	−0.0971111	+	0.995274i	$0.530960\pi$
$384$	−290.680	−0.0386294
$385$	−101.876	−0.0134859
$386$	8846.38	1.16650
$387$	0	0
$388$	−1941.92	−0.254088
$389$	5582.90	0.727672	0.363836	−	0.931463i	$-0.381467\pi$
$389$	5582.90	0.727672	0.363836	+	0.931463i	$0.381467\pi$
$390$	829.434	0.107692
$391$	−11323.7	−1.46462
$392$	−3603.27	−0.464267
$393$	−604.707	−0.0776169
$394$	7228.23	0.924246
$395$	−19730.6	−2.51331
$396$	−877.953	−0.111411
$397$	−8442.90	−1.06735	−0.533674	−	0.845691i	$-0.679189\pi$
$397$	−8442.90	−1.06735	−0.533674	+	0.845691i	$0.679189\pi$
$398$	7706.85	0.970627
$399$	−7.51612	−0.000943049 0
$400$	−12670.2	−1.58378
$401$	3590.15	0.447091	0.223546	−	0.974693i	$-0.428237\pi$
$401$	3590.15	0.447091	0.223546	+	0.974693i	$0.428237\pi$
$402$	−318.551	−0.0395220
$403$	−17264.7	−2.13404
$404$	−5824.57	−0.717285
$405$	12250.4	1.50303
$406$	505.436	0.0617842
$407$	−211.306	−0.0257347
$408$	−226.389	−0.0274704
$409$	11944.2	1.44401	0.722007	−	0.691886i	$-0.243220\pi$
$409$	11944.2	1.44401	0.722007	+	0.691886i	$0.243220\pi$
$410$	15966.5	1.92324
$411$	64.8163	0.00777896
$412$	−4233.22	−0.506204
$413$	−232.274	−0.0276742
$414$	11858.9	1.40781
$415$	17509.7	2.07113
$416$	−11798.8	−1.39058
$417$	−707.446	−0.0830786
$418$	−801.028	−0.0937310
$419$	−13696.2	−1.59691	−0.798454	−	0.602056i	$-0.794349\pi$
$419$	−13696.2	−1.59691	−0.798454	+	0.602056i	$0.794349\pi$
$420$	−18.6966	−0.00217214
$421$	4138.73	0.479120	0.239560	−	0.970882i	$-0.422997\pi$
$421$	4138.73	0.479120	0.239560	+	0.970882i	$0.422997\pi$
$422$	−10663.5	−1.23007
$423$	601.503	0.0691396
$424$	1604.28	0.183752
$425$	−14965.0	−1.70802
$426$	−350.685	−0.0398844
$427$	186.652	0.0211539
$428$	9910.12	1.11921
$429$	−86.8330	−0.00977235
$430$	0	0
$431$	−509.897	−0.0569858	−0.0284929	−	0.999594i	$-0.509071\pi$
$431$	−509.897	−0.0569858	−0.0284929	+	0.999594i	$0.509071\pi$
$432$	981.753	0.109339
$433$	1117.48	0.124025	0.0620123	−	0.998075i	$-0.480248\pi$
$433$	1117.48	0.124025	0.0620123	+	0.998075i	$0.480248\pi$
$434$	1000.98	0.110711
$435$	579.337	0.0638554
$436$	−8362.23	−0.918528
$437$	4206.65	0.460484
$438$	854.909	0.0932629
$439$	3458.28	0.375979	0.187989	−	0.982171i	$-0.439803\pi$
$439$	3458.28	0.375979	0.187989	+	0.982171i	$0.439803\pi$
$440$	1139.93	0.123510
$441$	9218.84	0.995447
$442$	−19781.0	−2.12870
$443$	11988.4	1.28574	0.642872	−	0.765974i	$-0.277743\pi$
$443$	11988.4	1.28574	0.642872	+	0.765974i	$0.277743\pi$
$444$	−38.7795	−0.00414503
$445$	18127.0	1.93102
$446$	1868.46	0.198372
$447$	−268.056	−0.0283638
$448$	90.5826	0.00955274
$449$	−6927.73	−0.728151	−0.364076	−	0.931369i	$-0.618615\pi$
$449$	−6927.73	−0.728151	−0.364076	+	0.931369i	$0.618615\pi$
$450$	15672.3	1.64177
$451$	−1671.52	−0.174521
$452$	−221.316	−0.0230306
$453$	601.371	0.0623728
$454$	−10836.1	−1.12019
$455$	934.585	0.0962945
$456$	84.1012	0.00863684
$457$	−5021.62	−0.514008	−0.257004	−	0.966410i	$-0.582735\pi$
$457$	−5021.62	−0.514008	−0.257004	+	0.966410i	$0.582735\pi$
$458$	−11459.3	−1.16912
$459$	1159.56	0.117917
$460$	10464.2	1.06064
$461$	−6892.65	−0.696362	−0.348181	−	0.937427i	$-0.613201\pi$
$461$	−6892.65	−0.696362	−0.348181	+	0.937427i	$0.613201\pi$
$462$	5.03444	0.000506977 0
$463$	3873.13	0.388768	0.194384	−	0.980926i	$-0.437729\pi$
$463$	3873.13	0.388768	0.194384	+	0.980926i	$0.437729\pi$
$464$	−11697.9	−1.17039
$465$	1147.34	0.114422
$466$	14961.4	1.48728
$467$	14911.7	1.47758	0.738791	−	0.673935i	$-0.235397\pi$
$467$	14911.7	1.47758	0.738791	+	0.673935i	$0.235397\pi$
$468$	8054.13	0.795518
$469$	−358.935	−0.0353391
$470$	1365.16	0.133979
$471$	−428.026	−0.0418735
$472$	2599.02	0.253453
$473$	0	0
$474$	975.037	0.0944830
$475$	5559.33	0.537010
$476$	445.891	0.0429356
$477$	−4104.50	−0.393988
$478$	11780.7	1.12727
$479$	−4974.45	−0.474506	−0.237253	−	0.971448i	$-0.576247\pi$
$479$	−4974.45	−0.474506	−0.237253	+	0.971448i	$0.576247\pi$
$480$	784.093	0.0745600
$481$	1938.47	0.183756
$482$	6747.76	0.637660
$483$	−26.4387	−0.00249069
$484$	−6564.55	−0.616505
$485$	−6450.88	−0.603958
$486$	−1822.15	−0.170071
$487$	7246.07	0.674232	0.337116	−	0.941463i	$-0.390549\pi$
$487$	7246.07	0.674232	0.337116	+	0.941463i	$0.390549\pi$
$488$	−2088.53	−0.193737
$489$	−512.620	−0.0474059
$490$	20922.8	1.92898
$491$	−10684.3	−0.982030	−0.491015	−	0.871151i	$-0.663374\pi$
$491$	−10684.3	−0.982030	−0.491015	+	0.871151i	$0.663374\pi$
$492$	−306.763	−0.0281097
$493$	−13816.5	−1.26220
$494$	7348.44	0.669275
$495$	−2916.48	−0.264820
$496$	−23166.8	−2.09722
$497$	−395.143	−0.0356631
$498$	−865.284	−0.0778600
$499$	−20001.9	−1.79440	−0.897202	−	0.441621i	$-0.854404\pi$
$499$	−20001.9	−1.79440	−0.897202	+	0.441621i	$0.854404\pi$
$500$	3076.16	0.275140
$501$	364.708	0.0325229
$502$	16991.0	1.51065
$503$	−6642.47	−0.588813	−0.294407	−	0.955680i	$-0.595122\pi$
$503$	−6642.47	−0.588813	−0.294407	+	0.955680i	$0.595122\pi$
$504$	267.146	0.0236104
$505$	−19348.7	−1.70496
$506$	−2817.69	−0.247553
$507$	289.290	0.0253409
$508$	1970.82	0.172128
$509$	−6114.70	−0.532474	−0.266237	−	0.963908i	$-0.585780\pi$
$509$	−6114.70	−0.532474	−0.266237	+	0.963908i	$0.585780\pi$
$510$	1314.56	0.114136
$511$	963.290	0.0833922
$512$	−9191.41	−0.793373
$513$	−430.766	−0.0370736
$514$	3382.46	0.290261
$515$	−14062.4	−1.20323
$516$	0	0
$517$	−142.917	−0.0121576
$518$	−112.389	−0.00953301
$519$	845.433	0.0715036
$520$	−10457.5	−0.881906
$521$	−11357.7	−0.955068	−0.477534	−	0.878613i	$-0.658469\pi$
$521$	−11357.7	−0.955068	−0.477534	+	0.878613i	$0.658469\pi$
$522$	14469.5	1.21325
$523$	1146.92	0.0958915	0.0479458	−	0.998850i	$-0.484733\pi$
$523$	1146.92	0.0958915	0.0479458	+	0.998850i	$0.484733\pi$
$524$	−13326.8	−1.11104
$525$	−34.9402	−0.00290460
$526$	25823.2	2.14058
$527$	−27362.6	−2.26173
$528$	−116.517	−0.00960373
$529$	2630.29	0.216182
$530$	−9315.48	−0.763469
$531$	−6649.50	−0.543434
$532$	−165.644	−0.0134992
$533$	15334.2	1.24615
$534$	−895.789	−0.0725929
$535$	32920.5	2.66033
$536$	4016.28	0.323651
$537$	−162.356	−0.0130469
$538$	3571.90	0.286237
$539$	−2190.40	−0.175041
$540$	−1071.54	−0.0853923
$541$	5337.90	0.424204	0.212102	−	0.977248i	$-0.431969\pi$
$541$	5337.90	0.424204	0.212102	+	0.977248i	$0.431969\pi$
$542$	−17093.0	−1.35463
$543$	27.9374	0.00220794
$544$	−18699.7	−1.47379
$545$	−27778.6	−2.18331
$546$	−46.1847	−0.00362001
$547$	−16015.7	−1.25189	−0.625943	−	0.779869i	$-0.715286\pi$
$547$	−16015.7	−1.25189	−0.625943	+	0.779869i	$0.715286\pi$
$548$	1428.45	0.111351
$549$	5343.44	0.415396
$550$	−3723.74	−0.288693
$551$	5132.69	0.396842
$552$	295.834	0.0228108
$553$	1098.65	0.0844832
$554$	22833.1	1.75106
$555$	−128.822	−0.00985258
$556$	−15591.0	−1.18922
$557$	−7498.36	−0.570405	−0.285202	−	0.958467i	$-0.592061\pi$
$557$	−7498.36	−0.570405	−0.285202	+	0.958467i	$0.592061\pi$
$558$	28655.9	2.17401
$559$	0	0
$560$	1254.08	0.0946330
$561$	−137.620	−0.0103571
$562$	18782.9	1.40980
$563$	8522.34	0.637964	0.318982	−	0.947761i	$-0.396659\pi$
$563$	8522.34	0.637964	0.318982	+	0.947761i	$0.396659\pi$
$564$	−26.2287	−0.00195820
$565$	−735.191	−0.0547429
$566$	26541.4	1.97106
$567$	−682.128	−0.0505233
$568$	4421.43	0.326618
$569$	−8616.32	−0.634824	−0.317412	−	0.948288i	$-0.602814\pi$
$569$	−8616.32	−0.634824	−0.317412	+	0.948288i	$0.602814\pi$
$570$	−488.344	−0.0358851
$571$	3338.38	0.244670	0.122335	−	0.992489i	$-0.460962\pi$
$571$	3338.38	0.244670	0.122335	+	0.992489i	$0.460962\pi$
$572$	−1913.67	−0.139885
$573$	731.873	0.0533586
$574$	−889.050	−0.0646485
$575$	19555.5	1.41830
$576$	2593.18	0.187585
$577$	−21058.8	−1.51939	−0.759696	−	0.650278i	$-0.774652\pi$
$577$	−21058.8	−1.51939	−0.759696	+	0.650278i	$0.774652\pi$
$578$	−13576.2	−0.976979
$579$	−564.608	−0.0405256
$580$	12767.7	0.914052
$581$	−974.980	−0.0696196
$582$	318.786	0.0227046
$583$	975.232	0.0692796
$584$	−10778.7	−0.763742
$585$	26755.1	1.89092
$586$	−8655.73	−0.610179
$587$	22522.3	1.58364	0.791820	−	0.610755i	$-0.209134\pi$
$587$	22522.3	1.58364	0.791820	+	0.610755i	$0.209134\pi$
$588$	−401.989	−0.0281935
$589$	10164.9	0.711101
$590$	−15091.5	−1.05307
$591$	−461.332	−0.0321094
$592$	2601.14	0.180585
$593$	−13802.1	−0.955789	−0.477894	−	0.878417i	$-0.658600\pi$
$593$	−13802.1	−0.955789	−0.477894	+	0.878417i	$0.658600\pi$
$594$	288.535	0.0199305
$595$	1481.21	0.102057
$596$	−5907.55	−0.406011
$597$	−491.879	−0.0337207
$598$	25848.8	1.76762
$599$	−6333.09	−0.431992	−0.215996	−	0.976394i	$-0.569300\pi$
$599$	−6333.09	−0.431992	−0.215996	+	0.976394i	$0.569300\pi$
$600$	390.962	0.0266016
$601$	−21847.3	−1.48281	−0.741406	−	0.671057i	$-0.765841\pi$
$601$	−21847.3	−1.48281	−0.741406	+	0.671057i	$0.765841\pi$
$602$	0	0
$603$	−10275.5	−0.693949
$604$	13253.3	0.892830
$605$	−21806.8	−1.46541
$606$	956.162	0.0640948
$607$	−11089.7	−0.741545	−0.370772	−	0.928724i	$-0.620907\pi$
$607$	−11089.7	−0.741545	−0.370772	+	0.928724i	$0.620907\pi$
$608$	6946.74	0.463368
$609$	−32.2588	−0.00214646
$610$	12127.3	0.804953
$611$	1311.09	0.0868103
$612$	12764.9	0.843120
$613$	7636.22	0.503138	0.251569	−	0.967839i	$-0.419053\pi$
$613$	7636.22	0.503138	0.251569	+	0.967839i	$0.419053\pi$
$614$	3529.74	0.232001
$615$	−1019.04	−0.0668157
$616$	−63.4741	−0.00415169
$617$	23000.3	1.50074	0.750371	−	0.661017i	$-0.229875\pi$
$617$	23000.3	1.50074	0.750371	+	0.661017i	$0.229875\pi$
$618$	694.926	0.0452331
$619$	−17331.4	−1.12538	−0.562688	−	0.826669i	$-0.690233\pi$
$619$	−17331.4	−1.12538	−0.562688	+	0.826669i	$0.690233\pi$
$620$	25285.5	1.63789
$621$	−1515.26	−0.0979151
$622$	−24461.8	−1.57689
$623$	−1009.35	−0.0649099
$624$	1068.90	0.0685743
$625$	−9876.26	−0.632081
$626$	31589.0	2.01685
$627$	51.1245	0.00325633
$628$	−9433.05	−0.599395
$629$	3072.25	0.194751
$630$	−1551.22	−0.0980984
$631$	18008.9	1.13617	0.568084	−	0.822970i	$-0.307685\pi$
$631$	18008.9	1.13617	0.568084	+	0.822970i	$0.307685\pi$
$632$	−12293.3	−0.773733
$633$	680.582	0.0427341
$634$	27408.7	1.71694
$635$	6546.89	0.409142
$636$	178.978	0.0111587
$637$	20094.2	1.24986
$638$	−3437.97	−0.213339
$639$	−11312.1	−0.700311
$640$	−21280.6	−1.31436
$641$	−7479.43	−0.460873	−0.230437	−	0.973087i	$-0.574015\pi$
$641$	−7479.43	−0.460873	−0.230437	+	0.973087i	$0.574015\pi$
$642$	−1626.85	−0.100010
$643$	12.8576	0.000788576 0	0.000394288	−	1.00000i	$-0.499874\pi$
$643$	12.8576	0.000788576 0	0.000394288		1.00000i	$0.499874\pi$
$644$	−582.668	−0.0356527
$645$	0	0
$646$	11646.4	0.709323
$647$	6154.73	0.373983	0.186992	−	0.982361i	$-0.440126\pi$
$647$	6154.73	0.373983	0.186992	+	0.982361i	$0.440126\pi$
$648$	7632.64	0.462714
$649$	1579.93	0.0955585
$650$	34160.8	2.06138
$651$	−63.8863	−0.00384624
$652$	−11297.4	−0.678588
$653$	30577.9	1.83247	0.916237	−	0.400637i	$-0.131211\pi$
$653$	30577.9	1.83247	0.916237	+	0.400637i	$0.131211\pi$
$654$	1372.74	0.0820773
$655$	−44270.5	−2.64090
$656$	20576.2	1.22465
$657$	27576.9	1.63756
$658$	−76.0150	−0.00450360
$659$	−23474.2	−1.38759	−0.693796	−	0.720171i	$-0.744063\pi$
$659$	−23474.2	−1.38759	−0.693796	+	0.720171i	$0.744063\pi$
$660$	127.174	0.00750035
$661$	11599.3	0.682544	0.341272	−	0.939965i	$-0.389142\pi$
$661$	11599.3	0.682544	0.341272	+	0.939965i	$0.389142\pi$
$662$	−35538.4	−2.08647
$663$	1262.50	0.0739537
$664$	10909.5	0.637606
$665$	−550.254	−0.0320871
$666$	−3217.46	−0.187198
$667$	18054.7	1.04810
$668$	8037.62	0.465546
$669$	−119.252	−0.00689169
$670$	−23321.0	−1.34473
$671$	−1269.60	−0.0730440
$672$	−43.6601	−0.00250628
$673$	−20103.0	−1.15143	−0.575716	−	0.817649i	$-0.695277\pi$
$673$	−20103.0	−1.15143	−0.575716	+	0.817649i	$0.695277\pi$
$674$	−4935.04	−0.282033
$675$	−2002.50	−0.114187
$676$	6375.51	0.362739
$677$	20889.0	1.18586	0.592932	−	0.805252i	$-0.297970\pi$
$677$	20889.0	1.18586	0.592932	+	0.805252i	$0.297970\pi$
$678$	36.3312	0.00205795
$679$	359.200	0.0203017
$680$	−16573.9	−0.934677
$681$	691.602	0.0389167
$682$	−6808.66	−0.382283
$683$	10279.8	0.575909	0.287954	−	0.957644i	$-0.407025\pi$
$683$	10279.8	0.575909	0.287954	+	0.957644i	$0.407025\pi$
$684$	−4742.02	−0.265081
$685$	4745.19	0.264678
$686$	−2333.08	−0.129850
$687$	731.372	0.0406166
$688$	0	0
$689$	−8946.55	−0.494683
$690$	−1717.80	−0.0947760
$691$	−6373.31	−0.350871	−0.175436	−	0.984491i	$-0.556133\pi$
$691$	−6373.31	−0.350871	−0.175436	+	0.984491i	$0.556133\pi$
$692$	18632.1	1.02353
$693$	162.396	0.00890176
$694$	−4164.87	−0.227804
$695$	−51792.0	−2.82674
$696$	360.958	0.0196582
$697$	24302.9	1.32071
$698$	1258.88	0.0682652
$699$	−954.889	−0.0516698
$700$	−770.030	−0.0415777
$701$	6890.38	0.371250	0.185625	−	0.982621i	$-0.440569\pi$
$701$	6890.38	0.371250	0.185625	+	0.982621i	$0.440569\pi$
$702$	−2646.95	−0.142312
$703$	−1141.31	−0.0612308
$704$	−616.142	−0.0329854
$705$	−87.1292	−0.00465458
$706$	10373.9	0.553011
$707$	1077.38	0.0573112
$708$	289.953	0.0153914
$709$	15982.8	0.846608	0.423304	−	0.905988i	$-0.360870\pi$
$709$	15982.8	0.846608	0.423304	+	0.905988i	$0.360870\pi$
$710$	−25673.6	−1.35706
$711$	31451.8	1.65898
$712$	11294.1	0.594472
$713$	35756.1	1.87809
$714$	−73.1975	−0.00383662
$715$	−6357.03	−0.332503
$716$	−3578.08	−0.186759
$717$	−751.886	−0.0391628
$718$	18475.3	0.960295
$719$	30191.8	1.56601	0.783007	−	0.622012i	$-0.213685\pi$
$719$	30191.8	1.56601	0.783007	+	0.622012i	$0.213685\pi$
$720$	35901.5	1.85829
$721$	783.025	0.0404457
$722$	20488.2	1.05608
$723$	−430.667	−0.0221531
$724$	615.699	0.0316053
$725$	23860.4	1.22228
$726$	1077.64	0.0550894
$727$	−3460.05	−0.176515	−0.0882574	−	0.996098i	$-0.528130\pi$
$727$	−3460.05	−0.176515	−0.0882574	+	0.996098i	$0.528130\pi$
$728$	582.296	0.0296447
$729$	−19450.2	−0.988171
$730$	62587.8	3.17326
$731$	0	0
$732$	−233.002	−0.0117650
$733$	20446.6	1.03030	0.515151	−	0.857099i	$-0.327736\pi$
$733$	20446.6	1.03030	0.515151	+	0.857099i	$0.327736\pi$
$734$	−5507.16	−0.276939
$735$	−1335.37	−0.0670149
$736$	24435.8	1.22380
$737$	2441.47	0.122025
$738$	−25451.6	−1.26949
$739$	14368.1	0.715207	0.357603	−	0.933874i	$-0.383594\pi$
$739$	14368.1	0.715207	0.357603	+	0.933874i	$0.383594\pi$
$740$	−2839.04	−0.141034
$741$	−469.004	−0.0232514
$742$	518.707	0.0256635
$743$	10210.8	0.504167	0.252084	−	0.967705i	$-0.418884\pi$
$743$	10210.8	0.504167	0.252084	+	0.967705i	$0.418884\pi$
$744$	714.852	0.0352255
$745$	−19624.3	−0.965074
$746$	−6563.10	−0.322107
$747$	−27911.5	−1.36711
$748$	−3032.94	−0.148256
$749$	−1833.09	−0.0894254
$750$	−504.982	−0.0245858
$751$	−18487.6	−0.898297	−0.449149	−	0.893457i	$-0.648273\pi$
$751$	−18487.6	−0.898297	−0.449149	+	0.893457i	$0.648273\pi$
$752$	1759.30	0.0853124
$753$	−1084.43	−0.0524817
$754$	31539.1	1.52333
$755$	44026.3	2.12223
$756$	59.6659	0.00287041
$757$	29571.9	1.41983	0.709914	−	0.704288i	$-0.248734\pi$
$757$	29571.9	1.41983	0.709914	+	0.704288i	$0.248734\pi$
$758$	−5042.11	−0.241607
$759$	179.835	0.00860028
$760$	6157.04	0.293867
$761$	−38487.4	−1.83333	−0.916666	−	0.399655i	$-0.869130\pi$
$761$	−38487.4	−1.83333	−0.916666	+	0.399655i	$0.869130\pi$
$762$	−323.530	−0.0153809
$763$	1546.77	0.0733905
$764$	16129.4	0.763796
$765$	42403.7	2.00407
$766$	5266.79	0.248430
$767$	−14493.9	−0.682325
$768$	1229.40	0.0577632
$769$	6790.10	0.318410	0.159205	−	0.987246i	$-0.449107\pi$
$769$	6790.10	0.318410	0.159205	+	0.987246i	$0.449107\pi$
$770$	368.570	0.0172498
$771$	−215.881	−0.0100840
$772$	−12443.1	−0.580100
$773$	−33655.4	−1.56598	−0.782989	−	0.622036i	$-0.786306\pi$
$773$	−33655.4	−1.56598	−0.782989	+	0.622036i	$0.786306\pi$
$774$	0	0
$775$	47253.8	2.19020
$776$	−4019.25	−0.185931
$777$	7.17309	0.000331188 0
$778$	−20198.1	−0.930765
$779$	−9028.27	−0.415239
$780$	−1166.66	−0.0535554
$781$	2687.76	0.123144
$782$	40967.4	1.87339
$783$	−1848.82	−0.0843826
$784$	26963.6	1.22830
$785$	−31335.7	−1.42474
$786$	2187.73	0.0992797
$787$	−8767.38	−0.397107	−0.198554	−	0.980090i	$-0.563624\pi$
$787$	−8767.38	−0.397107	−0.198554	+	0.980090i	$0.563624\pi$
$788$	−10167.1	−0.459627
$789$	−1648.13	−0.0743664
$790$	71382.3	3.21477
$791$	40.9371	0.00184015
$792$	−1817.12	−0.0815261
$793$	11647.1	0.521562
$794$	30545.1	1.36524
$795$	594.548	0.0265238
$796$	−10840.3	−0.482693
$797$	−32490.3	−1.44400	−0.721999	−	0.691894i	$-0.756777\pi$
$797$	−32490.3	−1.44400	−0.721999	+	0.691894i	$0.756777\pi$
$798$	27.1921	0.00120625
$799$	2077.93	0.0920048
$800$	32293.4	1.42718
$801$	−28895.5	−1.27462
$802$	−12988.6	−0.571874
$803$	−6552.28	−0.287951
$804$	448.066	0.0196543
$805$	−1935.57	−0.0847452
$806$	62461.0	2.72965
$807$	−227.972	−0.00994421
$808$	−12055.3	−0.524880
$809$	35307.2	1.53441	0.767203	−	0.641404i	$-0.221648\pi$
$809$	35307.2	1.53441	0.767203	+	0.641404i	$0.221648\pi$
$810$	−44319.9	−1.92252
$811$	−20743.5	−0.898152	−0.449076	−	0.893493i	$-0.648247\pi$
$811$	−20743.5	−0.898152	−0.449076	+	0.893493i	$0.648247\pi$
$812$	−710.935	−0.0307253
$813$	1090.94	0.0470615
$814$	764.470	0.0329173
$815$	−37528.8	−1.61298
$816$	1694.09	0.0726776
$817$	0	0
$818$	−43212.1	−1.84704
$819$	−1489.78	−0.0635620
$820$	−22458.1	−0.956428
$821$	24557.5	1.04393	0.521963	−	0.852968i	$-0.325200\pi$
$821$	24557.5	1.04393	0.521963	+	0.852968i	$0.325200\pi$
$822$	−234.495	−0.00995007
$823$	−10118.0	−0.428546	−0.214273	−	0.976774i	$-0.568738\pi$
$823$	−10118.0	−0.428546	−0.214273	+	0.976774i	$0.568738\pi$
$824$	−8761.62	−0.370419
$825$	237.663	0.0100295
$826$	840.331	0.0353981
$827$	4459.70	0.187520	0.0937599	−	0.995595i	$-0.470111\pi$
$827$	4459.70	0.187520	0.0937599	+	0.995595i	$0.470111\pi$
$828$	−16680.5	−0.700106
$829$	19699.9	0.825337	0.412669	−	0.910881i	$-0.364597\pi$
$829$	19699.9	0.825337	0.412669	+	0.910881i	$0.364597\pi$
$830$	−63347.3	−2.64918
$831$	−1457.29	−0.0608339
$832$	5652.34	0.235528
$833$	31847.0	1.32465
$834$	2559.43	0.106266
$835$	26700.2	1.10659
$836$	1126.71	0.0466124
$837$	−3661.46	−0.151205
$838$	49550.8	2.04261
$839$	−37378.6	−1.53809	−0.769043	−	0.639197i	$-0.779267\pi$
$839$	−37378.6	−1.53809	−0.769043	+	0.639197i	$0.779267\pi$
$840$	−38.6968	−0.00158948
$841$	−2359.77	−0.0967556
$842$	−14973.3	−0.612842
$843$	−1198.79	−0.0489782
$844$	14999.0	0.611714
$845$	21178.9	0.862219
$846$	−2176.14	−0.0884365
$847$	1214.25	0.0492589
$848$	−12005.0	−0.486147
$849$	−1693.97	−0.0684769
$850$	54140.9	2.18473
$851$	−4014.66	−0.161717
$852$	493.265	0.0198345
$853$	−16465.7	−0.660932	−0.330466	−	0.943818i	$-0.607206\pi$
$853$	−16465.7	−0.660932	−0.330466	+	0.943818i	$0.607206\pi$
$854$	−675.277	−0.0270580
$855$	−15752.6	−0.630089
$856$	20511.3	0.818996
$857$	15994.4	0.637523	0.318761	−	0.947835i	$-0.396733\pi$
$857$	15994.4	0.637523	0.318761	+	0.947835i	$0.396733\pi$
$858$	314.148	0.0124998
$859$	−31869.5	−1.26586	−0.632929	−	0.774210i	$-0.718147\pi$
$859$	−31869.5	−1.26586	−0.632929	+	0.774210i	$0.718147\pi$
$860$	0	0
$861$	56.7424	0.00224597
$862$	1844.73	0.0728905
$863$	−45117.2	−1.77961	−0.889807	−	0.456338i	$-0.849161\pi$
$863$	−45117.2	−1.77961	−0.889807	+	0.456338i	$0.849161\pi$
$864$	−2502.26	−0.0985283
$865$	61894.0	2.43290
$866$	−4042.86	−0.158640
$867$	866.480	0.0339414
$868$	−1407.96	−0.0550566
$869$	−7472.98	−0.291719
$870$	−2095.95	−0.0816774
$871$	−22397.4	−0.871307
$872$	−17307.5	−0.672141
$873$	10283.1	0.398660
$874$	−15219.0	−0.589004
$875$	−569.001	−0.0219837
$876$	−1202.50	−0.0463796
$877$	−1949.36	−0.0750572	−0.0375286	−	0.999296i	$-0.511949\pi$
$877$	−1949.36	−0.0750572	−0.0375286	+	0.999296i	$0.511949\pi$
$878$	−12511.5	−0.480914
$879$	552.441	0.0211984
$880$	−8530.22	−0.326765
$881$	6112.63	0.233757	0.116878	−	0.993146i	$-0.462711\pi$
$881$	6112.63	0.233757	0.116878	+	0.993146i	$0.462711\pi$
$882$	−33352.3	−1.27328
$883$	−32217.6	−1.22787	−0.613934	−	0.789358i	$-0.710414\pi$
$883$	−32217.6	−1.22787	−0.613934	+	0.789358i	$0.710414\pi$
$884$	27823.5	1.05860
$885$	963.197	0.0365848
$886$	−43372.0	−1.64459
$887$	4595.87	0.173973	0.0869865	−	0.996209i	$-0.472276\pi$
$887$	4595.87	0.173973	0.0869865	+	0.996209i	$0.472276\pi$
$888$	−80.2629	−0.00303316
$889$	−364.545	−0.0137530
$890$	−65580.6	−2.46996
$891$	4639.83	0.174456
$892$	−2628.13	−0.0986506
$893$	−771.929	−0.0289268
$894$	969.784	0.0362801
$895$	−11886.1	−0.443919
$896$	1184.95	0.0441814
$897$	−1649.77	−0.0614093
$898$	25063.4	0.931378
$899$	43627.3	1.61852
$900$	−22044.3	−0.816454
$901$	−14179.3	−0.524284
$902$	6047.31	0.223230
$903$	0	0
$904$	−458.064	−0.0168528
$905$	2045.30	0.0751248
$906$	−2175.66	−0.0797810
$907$	25405.7	0.930078	0.465039	−	0.885290i	$-0.346040\pi$
$907$	25405.7	0.930078	0.465039	+	0.885290i	$0.346040\pi$
$908$	15241.9	0.557069
$909$	30843.0	1.12541
$910$	−3381.18	−0.123170
$911$	−24097.8	−0.876394	−0.438197	−	0.898879i	$-0.644383\pi$
$911$	−24097.8	−0.876394	−0.438197	+	0.898879i	$0.644383\pi$
$912$	−629.336	−0.0228502
$913$	6631.80	0.240395
$914$	18167.4	0.657467
$915$	−774.011	−0.0279650
$916$	16118.3	0.581402
$917$	2465.08	0.0887723
$918$	−4195.11	−0.150827
$919$	259.370	0.00930994	0.00465497	−	0.999989i	$-0.498518\pi$
$919$	259.370	0.00930994	0.00465497	+	0.999989i	$0.498518\pi$
$920$	21658.0	0.776133
$921$	−225.281	−0.00805999
$922$	24936.5	0.890716
$923$	−24656.8	−0.879296
$924$	−7.08132	−0.000252119 0
$925$	−5305.61	−0.188592
$926$	−14012.4	−0.497273
$927$	22416.3	0.794226
$928$	29815.0	1.05466
$929$	−33002.9	−1.16554	−0.582771	−	0.812636i	$-0.698032\pi$
$929$	−33002.9	−1.16554	−0.582771	+	0.812636i	$0.698032\pi$
$930$	−4150.88	−0.146358
$931$	−11830.9	−0.416478
$932$	−21044.3	−0.739624
$933$	1561.24	0.0547831
$934$	−53948.1	−1.88997
$935$	−10075.2	−0.352399
$936$	16669.9	0.582128
$937$	29081.5	1.01393	0.506964	−	0.861967i	$-0.330768\pi$
$937$	29081.5	1.01393	0.506964	+	0.861967i	$0.330768\pi$
$938$	1298.57	0.0452023
$939$	−2016.13	−0.0700679
$940$	−1920.20	−0.0666275
$941$	25697.4	0.890234	0.445117	−	0.895472i	$-0.353162\pi$
$941$	25697.4	0.890234	0.445117	+	0.895472i	$0.353162\pi$
$942$	1548.53	0.0535604
$943$	−31757.8	−1.09669
$944$	−19448.7	−0.670551
$945$	198.205	0.00682285
$946$	0	0
$947$	−18483.5	−0.634249	−0.317124	−	0.948384i	$-0.602717\pi$
$947$	−18483.5	−0.634249	−0.317124	+	0.948384i	$0.602717\pi$
$948$	−1371.46	−0.0469864
$949$	60109.1	2.05608
$950$	−20112.8	−0.686889
$951$	−1749.33	−0.0596486
$952$	922.873	0.0314186
$953$	8314.35	0.282611	0.141306	−	0.989966i	$-0.454870\pi$
$953$	8314.35	0.282611	0.141306	+	0.989966i	$0.454870\pi$
$954$	14849.4	0.503950
$955$	53580.3	1.81552
$956$	−16570.4	−0.560592
$957$	219.424	0.00741167
$958$	17996.8	0.606941
$959$	−264.223	−0.00889698
$960$	−375.629	−0.0126285
$961$	56609.9	1.90023
$962$	−7013.07	−0.235042
$963$	−52477.3	−1.75603
$964$	−9491.24	−0.317108
$965$	−41334.9	−1.37888
$966$	95.6509	0.00318583
$967$	35433.0	1.17833	0.589167	−	0.808011i	$-0.299456\pi$
$967$	35433.0	1.17833	0.589167	+	0.808011i	$0.299456\pi$
$968$	−13586.8	−0.451134
$969$	−743.318	−0.0246427
$970$	23338.3	0.772522
$971$	−15532.7	−0.513356	−0.256678	−	0.966497i	$-0.582628\pi$
$971$	−15532.7	−0.513356	−0.256678	+	0.966497i	$0.582628\pi$
$972$	2563.00	0.0845763
$973$	2883.89	0.0950189
$974$	−26215.1	−0.862409
$975$	−2180.26	−0.0716147
$976$	15628.7	0.512563
$977$	−6315.48	−0.206807	−0.103403	−	0.994640i	$-0.532973\pi$
$977$	−6315.48	−0.206807	−0.103403	+	0.994640i	$0.532973\pi$
$978$	1854.58	0.0606369
$979$	6865.60	0.224132
$980$	−29429.6	−0.959279
$981$	44280.7	1.44116
$982$	38654.2	1.25611
$983$	−3450.09	−0.111944	−0.0559720	−	0.998432i	$-0.517826\pi$
$983$	−3450.09	−0.111944	−0.0559720	+	0.998432i	$0.517826\pi$
$984$	−634.917	−0.0205695
$985$	−33774.0	−1.09252
$986$	49985.9	1.61448
$987$	4.85155	0.000156461 0
$988$	−10336.1	−0.332831
$989$	0	0
$990$	10551.4	0.338731
$991$	48936.6	1.56864	0.784320	−	0.620357i	$-0.213012\pi$
$991$	48936.6	1.56864	0.784320	+	0.620357i	$0.213012\pi$
$992$	59046.6	1.88985
$993$	2268.19	0.0724863
$994$	1429.56	0.0456167
$995$	−36010.4	−1.14734
$996$	1217.09	0.0387198
$997$	−57265.0	−1.81906	−0.909528	−	0.415642i	$-0.863557\pi$
$997$	−57265.0	−1.81906	−0.909528	+	0.415642i	$0.863557\pi$
$998$	72363.6	2.29522
$999$	411.106	0.0130198

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.k.1.12		60
43.12	odd	42		43.4.g.a.15.9	✓	120
43.18	odd	42		43.4.g.a.23.9	yes	120
43.42	odd	2		1849.4.a.l.1.49		60

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.15.9	✓	120	43.12	odd	42
43.4.g.a.23.9	yes	120	43.18	odd	42
1849.4.a.k.1.12		60	1.1	even	1	trivial
1849.4.a.l.1.49		60	43.42	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q-3.61784 q^{2} +0.230904 q^{3} +5.08877 q^{4} +16.9044 q^{5} -0.835373 q^{6} -0.941277 q^{7} +10.5324 q^{8} -26.9467 q^{9} +O(q^{10})\)	\(q-3.61784 q^{2} +0.230904 q^{3} +5.08877 q^{4} +16.9044 q^{5} -0.835373 q^{6} -0.941277 q^{7} +10.5324 q^{8} -26.9467 q^{9} -61.1575 q^{10} +6.40255 q^{11} +1.17502 q^{12} -58.7355 q^{13} +3.40539 q^{14} +3.90330 q^{15} -78.8146 q^{16} -93.0890 q^{17} +97.4888 q^{18} +34.5816 q^{19} +86.0227 q^{20} -0.217344 q^{21} -23.1634 q^{22} +121.644 q^{23} +2.43196 q^{24} +160.760 q^{25} +212.496 q^{26} -12.4565 q^{27} -4.78994 q^{28} +148.422 q^{29} -14.1215 q^{30} +293.940 q^{31} +200.880 q^{32} +1.47837 q^{33} +336.781 q^{34} -15.9117 q^{35} -137.125 q^{36} -33.0033 q^{37} -125.111 q^{38} -13.5622 q^{39} +178.044 q^{40} -261.072 q^{41} +0.786317 q^{42} +32.5811 q^{44} -455.518 q^{45} -440.089 q^{46} -22.3220 q^{47} -18.1986 q^{48} -342.114 q^{49} -581.603 q^{50} -21.4946 q^{51} -298.891 q^{52} +152.319 q^{53} +45.0656 q^{54} +108.231 q^{55} -9.91388 q^{56} +7.98503 q^{57} -536.969 q^{58} +246.765 q^{59} +19.8630 q^{60} -198.297 q^{61} -1063.43 q^{62} +25.3643 q^{63} -96.2338 q^{64} -992.890 q^{65} -5.34852 q^{66} +381.327 q^{67} -473.708 q^{68} +28.0881 q^{69} +57.5662 q^{70} +419.795 q^{71} -283.812 q^{72} -1023.39 q^{73} +119.401 q^{74} +37.1200 q^{75} +175.978 q^{76} -6.02657 q^{77} +49.0661 q^{78} -1167.19 q^{79} -1332.32 q^{80} +724.684 q^{81} +944.515 q^{82} +1035.81 q^{83} -1.10602 q^{84} -1573.62 q^{85} +34.2713 q^{87} +67.4340 q^{88} +1072.32 q^{89} +1647.99 q^{90} +55.2864 q^{91} +619.019 q^{92} +67.8719 q^{93} +80.7573 q^{94} +584.582 q^{95} +46.3839 q^{96} -381.609 q^{97} +1237.71 q^{98} -172.528 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 15 q^{2} - 37 q^{3} + 213 q^{4} - 51 q^{5} + 22 q^{6} - 54 q^{7} - 204 q^{8} + 387 q^{9}+O(q^{10}) \) 60 * q - 15 * q^2 - 37 * q^3 + 213 * q^4 - 51 * q^5 + 22 * q^6 - 54 * q^7 - 204 * q^8 + 387 * q^9	\( 60 q - 15 q^{2} - 37 q^{3} + 213 q^{4} - 51 q^{5} + 22 q^{6} - 54 q^{7} - 204 q^{8} + 387 q^{9} + 27 q^{10} + 43 q^{11} - 432 q^{12} + 13 q^{13} + 96 q^{14} + 65 q^{15} + 717 q^{16} - 138 q^{17} - 625 q^{18} - 610 q^{19} - 345 q^{20} + 611 q^{21} - 118 q^{22} - 243 q^{23} + 258 q^{24} + 899 q^{25} - 1071 q^{26} - 1609 q^{27} - 46 q^{28} - 773 q^{29} - 375 q^{30} - 97 q^{31} - 1967 q^{32} - 500 q^{33} - 217 q^{34} + 247 q^{35} + 175 q^{36} - 228 q^{37} + 1253 q^{38} - 1493 q^{39} + 2220 q^{40} - 951 q^{41} - 2643 q^{42} - 1378 q^{44} - 1086 q^{45} + 565 q^{46} - 2 q^{47} - 2303 q^{48} + 1264 q^{49} - 3273 q^{50} - 3076 q^{51} - 2825 q^{52} - 39 q^{53} + 5201 q^{54} - 1306 q^{55} + 3683 q^{56} + 1342 q^{57} + 2588 q^{58} - 1065 q^{59} + 2803 q^{60} - 2999 q^{61} - 5569 q^{62} - 2377 q^{63} + 2082 q^{64} - 5578 q^{65} + 3338 q^{66} + 961 q^{67} - 3754 q^{68} - 1817 q^{69} - 2738 q^{70} - 8003 q^{71} - 1412 q^{72} + 1011 q^{73} - 1413 q^{74} - 7457 q^{75} - 5516 q^{76} - 4052 q^{77} + 1091 q^{78} - 4422 q^{79} - 1610 q^{80} + 2108 q^{81} - 4676 q^{82} - 297 q^{83} - 54 q^{84} - 4333 q^{85} + 1377 q^{87} - 3652 q^{88} - 2480 q^{89} - 1414 q^{90} - 4551 q^{91} - 3286 q^{92} - 4 q^{93} - 4609 q^{94} - 835 q^{95} + 5864 q^{96} + 3785 q^{97} - 753 q^{98} + 5072 q^{99}+O(q^{100}) \) 60 * q - 15 * q^2 - 37 * q^3 + 213 * q^4 - 51 * q^5 + 22 * q^6 - 54 * q^7 - 204 * q^8 + 387 * q^9 + 27 * q^10 + 43 * q^11 - 432 * q^12 + 13 * q^13 + 96 * q^14 + 65 * q^15 + 717 * q^16 - 138 * q^17 - 625 * q^18 - 610 * q^19 - 345 * q^20 + 611 * q^21 - 118 * q^22 - 243 * q^23 + 258 * q^24 + 899 * q^25 - 1071 * q^26 - 1609 * q^27 - 46 * q^28 - 773 * q^29 - 375 * q^30 - 97 * q^31 - 1967 * q^32 - 500 * q^33 - 217 * q^34 + 247 * q^35 + 175 * q^36 - 228 * q^37 + 1253 * q^38 - 1493 * q^39 + 2220 * q^40 - 951 * q^41 - 2643 * q^42 - 1378 * q^44 - 1086 * q^45 + 565 * q^46 - 2 * q^47 - 2303 * q^48 + 1264 * q^49 - 3273 * q^50 - 3076 * q^51 - 2825 * q^52 - 39 * q^53 + 5201 * q^54 - 1306 * q^55 + 3683 * q^56 + 1342 * q^57 + 2588 * q^58 - 1065 * q^59 + 2803 * q^60 - 2999 * q^61 - 5569 * q^62 - 2377 * q^63 + 2082 * q^64 - 5578 * q^65 + 3338 * q^66 + 961 * q^67 - 3754 * q^68 - 1817 * q^69 - 2738 * q^70 - 8003 * q^71 - 1412 * q^72 + 1011 * q^73 - 1413 * q^74 - 7457 * q^75 - 5516 * q^76 - 4052 * q^77 + 1091 * q^78 - 4422 * q^79 - 1610 * q^80 + 2108 * q^81 - 4676 * q^82 - 297 * q^83 - 54 * q^84 - 4333 * q^85 + 1377 * q^87 - 3652 * q^88 - 2480 * q^89 - 1414 * q^90 - 4551 * q^91 - 3286 * q^92 - 4 * q^93 - 4609 * q^94 - 835 * q^95 + 5864 * q^96 + 3785 * q^97 - 753 * q^98 + 5072 * q^99

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