LMFDB - Embedded newform 105.4.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [105,4,Mod(1,105)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(105, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("105.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 105 = 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	105.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:	$6.19520055060$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	105.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.82843 q^{2} -3.00000 q^{3} -4.65685 q^{4} +5.00000 q^{5} -5.48528 q^{6} -7.00000 q^{7} -23.1421 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+1.82843 q^{2} -3.00000 q^{3} -4.65685 q^{4} +5.00000 q^{5} -5.48528 q^{6} -7.00000 q^{7} -23.1421 q^{8} +9.00000 q^{9} +9.14214 q^{10} -64.5685 q^{11} +13.9706 q^{12} -32.3431 q^{13} -12.7990 q^{14} -15.0000 q^{15} -5.05887 q^{16} -56.3431 q^{17} +16.4558 q^{18} -2.74517 q^{19} -23.2843 q^{20} +21.0000 q^{21} -118.059 q^{22} +88.1665 q^{23} +69.4264 q^{24} +25.0000 q^{25} -59.1371 q^{26} -27.0000 q^{27} +32.5980 q^{28} +246.735 q^{29} -27.4264 q^{30} -110.912 q^{31} +175.887 q^{32} +193.706 q^{33} -103.019 q^{34} -35.0000 q^{35} -41.9117 q^{36} +120.676 q^{37} -5.01934 q^{38} +97.0294 q^{39} -115.711 q^{40} -176.274 q^{41} +38.3970 q^{42} -443.362 q^{43} +300.686 q^{44} +45.0000 q^{45} +161.206 q^{46} -345.206 q^{47} +15.1766 q^{48} +49.0000 q^{49} +45.7107 q^{50} +169.029 q^{51} +150.617 q^{52} +260.981 q^{53} -49.3675 q^{54} -322.843 q^{55} +161.995 q^{56} +8.23550 q^{57} +451.137 q^{58} +628.999 q^{59} +69.8528 q^{60} -115.206 q^{61} -202.794 q^{62} -63.0000 q^{63} +362.068 q^{64} -161.716 q^{65} +354.177 q^{66} -951.480 q^{67} +262.382 q^{68} -264.500 q^{69} -63.9949 q^{70} +356.264 q^{71} -208.279 q^{72} -656.754 q^{73} +220.648 q^{74} -75.0000 q^{75} +12.7838 q^{76} +451.980 q^{77} +177.411 q^{78} +440.195 q^{79} -25.2944 q^{80} +81.0000 q^{81} -322.304 q^{82} -54.4121 q^{83} -97.7939 q^{84} -281.716 q^{85} -810.656 q^{86} -740.205 q^{87} +1494.25 q^{88} -1018.78 q^{89} +82.2792 q^{90} +226.402 q^{91} -410.579 q^{92} +332.735 q^{93} -631.184 q^{94} -13.7258 q^{95} -527.662 q^{96} -724.108 q^{97} +89.5929 q^{98} -581.117 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 6 q^{3} + 2 q^{4} + 10 q^{5} + 6 q^{6} - 14 q^{7} - 18 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 6 * q^3 + 2 * q^4 + 10 * q^5 + 6 * q^6 - 14 * q^7 - 18 * q^8 + 18 * q^9	\( 2 q - 2 q^{2} - 6 q^{3} + 2 q^{4} + 10 q^{5} + 6 q^{6} - 14 q^{7} - 18 q^{8} + 18 q^{9} - 10 q^{10} - 16 q^{11} - 6 q^{12} - 76 q^{13} + 14 q^{14} - 30 q^{15} - 78 q^{16} - 124 q^{17} - 18 q^{18} - 96 q^{19} + 10 q^{20} + 42 q^{21} - 304 q^{22} - 16 q^{23} + 54 q^{24} + 50 q^{25} + 108 q^{26} - 54 q^{27} - 14 q^{28} + 188 q^{29} + 30 q^{30} - 120 q^{31} + 414 q^{32} + 48 q^{33} + 156 q^{34} - 70 q^{35} + 18 q^{36} - 132 q^{37} + 352 q^{38} + 228 q^{39} - 90 q^{40} + 100 q^{41} - 42 q^{42} - 536 q^{43} + 624 q^{44} + 90 q^{45} + 560 q^{46} - 928 q^{47} + 234 q^{48} + 98 q^{49} - 50 q^{50} + 372 q^{51} - 140 q^{52} + 884 q^{53} + 54 q^{54} - 80 q^{55} + 126 q^{56} + 288 q^{57} + 676 q^{58} + 104 q^{59} - 30 q^{60} - 468 q^{61} - 168 q^{62} - 126 q^{63} + 34 q^{64} - 380 q^{65} + 912 q^{66} - 1688 q^{67} - 188 q^{68} + 48 q^{69} + 70 q^{70} - 136 q^{71} - 162 q^{72} + 508 q^{73} + 1188 q^{74} - 150 q^{75} - 608 q^{76} + 112 q^{77} - 324 q^{78} - 432 q^{79} - 390 q^{80} + 162 q^{81} - 1380 q^{82} - 584 q^{83} + 42 q^{84} - 620 q^{85} - 456 q^{86} - 564 q^{87} + 1744 q^{88} - 1404 q^{89} - 90 q^{90} + 532 q^{91} - 1104 q^{92} + 360 q^{93} + 1600 q^{94} - 480 q^{95} - 1242 q^{96} - 1188 q^{97} - 98 q^{98} - 144 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 6 * q^3 + 2 * q^4 + 10 * q^5 + 6 * q^6 - 14 * q^7 - 18 * q^8 + 18 * q^9 - 10 * q^10 - 16 * q^11 - 6 * q^12 - 76 * q^13 + 14 * q^14 - 30 * q^15 - 78 * q^16 - 124 * q^17 - 18 * q^18 - 96 * q^19 + 10 * q^20 + 42 * q^21 - 304 * q^22 - 16 * q^23 + 54 * q^24 + 50 * q^25 + 108 * q^26 - 54 * q^27 - 14 * q^28 + 188 * q^29 + 30 * q^30 - 120 * q^31 + 414 * q^32 + 48 * q^33 + 156 * q^34 - 70 * q^35 + 18 * q^36 - 132 * q^37 + 352 * q^38 + 228 * q^39 - 90 * q^40 + 100 * q^41 - 42 * q^42 - 536 * q^43 + 624 * q^44 + 90 * q^45 + 560 * q^46 - 928 * q^47 + 234 * q^48 + 98 * q^49 - 50 * q^50 + 372 * q^51 - 140 * q^52 + 884 * q^53 + 54 * q^54 - 80 * q^55 + 126 * q^56 + 288 * q^57 + 676 * q^58 + 104 * q^59 - 30 * q^60 - 468 * q^61 - 168 * q^62 - 126 * q^63 + 34 * q^64 - 380 * q^65 + 912 * q^66 - 1688 * q^67 - 188 * q^68 + 48 * q^69 + 70 * q^70 - 136 * q^71 - 162 * q^72 + 508 * q^73 + 1188 * q^74 - 150 * q^75 - 608 * q^76 + 112 * q^77 - 324 * q^78 - 432 * q^79 - 390 * q^80 + 162 * q^81 - 1380 * q^82 - 584 * q^83 + 42 * q^84 - 620 * q^85 - 456 * q^86 - 564 * q^87 + 1744 * q^88 - 1404 * q^89 - 90 * q^90 + 532 * q^91 - 1104 * q^92 + 360 * q^93 + 1600 * q^94 - 480 * q^95 - 1242 * q^96 - 1188 * q^97 - 98 * q^98 - 144 * 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 $

$2$	1.82843	0.646447	0.323223	−	0.946323i	$-0.395234\pi$
$2$	1.82843	0.646447	0.323223	+	0.946323i	$0.395234\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−4.65685	−0.582107
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−5.48528	−0.373226
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−23.1421	−1.02275
$9$	9.00000	0.333333
$10$	9.14214	0.289100
$11$	−64.5685	−1.76983	−0.884916	−	0.465751i	$-0.845784\pi$
$11$	−64.5685	−1.76983	−0.884916	+	0.465751i	$0.845784\pi$
$12$	13.9706	0.336080
$13$	−32.3431	−0.690029	−0.345014	−	0.938597i	$-0.612126\pi$
$13$	−32.3431	−0.690029	−0.345014	+	0.938597i	$0.612126\pi$
$14$	−12.7990	−0.244334
$15$	−15.0000	−0.258199
$16$	−5.05887	−0.0790449
$17$	−56.3431	−0.803836	−0.401918	−	0.915676i	$-0.631656\pi$
$17$	−56.3431	−0.803836	−0.401918	+	0.915676i	$0.631656\pi$
$18$	16.4558	0.215482
$19$	−2.74517	−0.0331465	−0.0165733	−	0.999863i	$-0.505276\pi$
$19$	−2.74517	−0.0331465	−0.0165733	+	0.999863i	$0.505276\pi$
$20$	−23.2843	−0.260326
$21$	21.0000	0.218218
$22$	−118.059	−1.14410
$23$	88.1665	0.799304	0.399652	−	0.916667i	$-0.369131\pi$
$23$	88.1665	0.799304	0.399652	+	0.916667i	$0.369131\pi$
$24$	69.4264	0.590484
$25$	25.0000	0.200000
$26$	−59.1371	−0.446067
$27$	−27.0000	−0.192450
$28$	32.5980	0.220016
$29$	246.735	1.57992	0.789958	−	0.613161i	$-0.210102\pi$
$29$	246.735	1.57992	0.789958	+	0.613161i	$0.210102\pi$
$30$	−27.4264	−0.166912
$31$	−110.912	−0.642591	−0.321296	−	0.946979i	$-0.604118\pi$
$31$	−110.912	−0.642591	−0.321296	+	0.946979i	$0.604118\pi$
$32$	175.887	0.971649
$33$	193.706	1.02181
$34$	−103.019	−0.519637
$35$	−35.0000	−0.169031
$36$	−41.9117	−0.194036
$37$	120.676	0.536190	0.268095	−	0.963392i	$-0.413606\pi$
$37$	120.676	0.536190	0.268095	+	0.963392i	$0.413606\pi$
$38$	−5.01934	−0.0214275
$39$	97.0294	0.398388
$40$	−115.711	−0.457387
$41$	−176.274	−0.671449	−0.335724	−	0.941960i	$-0.608981\pi$
$41$	−176.274	−0.671449	−0.335724	+	0.941960i	$0.608981\pi$
$42$	38.3970	0.141066
$43$	−443.362	−1.57238	−0.786188	−	0.617988i	$-0.787948\pi$
$43$	−443.362	−1.57238	−0.786188	+	0.617988i	$0.787948\pi$
$44$	300.686	1.03023
$45$	45.0000	0.149071
$46$	161.206	0.516707
$47$	−345.206	−1.07135	−0.535675	−	0.844424i	$-0.679943\pi$
$47$	−345.206	−1.07135	−0.535675	+	0.844424i	$0.679943\pi$
$48$	15.1766	0.0456366
$49$	49.0000	0.142857
$50$	45.7107	0.129289
$51$	169.029	0.464095
$52$	150.617	0.401670
$53$	260.981	0.676386	0.338193	−	0.941077i	$-0.390184\pi$
$53$	260.981	0.676386	0.338193	+	0.941077i	$0.390184\pi$
$54$	−49.3675	−0.124409
$55$	−322.843	−0.791493
$56$	161.995	0.386562
$57$	8.23550	0.0191372
$58$	451.137	1.02133
$59$	628.999	1.38794	0.693972	−	0.720002i	$-0.255859\pi$
$59$	628.999	1.38794	0.693972	+	0.720002i	$0.255859\pi$
$60$	69.8528	0.150299
$61$	−115.206	−0.241814	−0.120907	−	0.992664i	$-0.538580\pi$
$61$	−115.206	−0.241814	−0.120907	+	0.992664i	$0.538580\pi$
$62$	−202.794	−0.415401
$63$	−63.0000	−0.125988
$64$	362.068	0.707164
$65$	−161.716	−0.308590
$66$	354.177	0.660547
$67$	−951.480	−1.73495	−0.867476	−	0.497479i	$-0.834259\pi$
$67$	−951.480	−1.73495	−0.867476	+	0.497479i	$0.834259\pi$
$68$	262.382	0.467919
$69$	−264.500	−0.461478
$70$	−63.9949	−0.109269
$71$	356.264	0.595504	0.297752	−	0.954643i	$-0.403763\pi$
$71$	356.264	0.595504	0.297752	+	0.954643i	$0.403763\pi$
$72$	−208.279	−0.340916
$73$	−656.754	−1.05298	−0.526488	−	0.850183i	$-0.676491\pi$
$73$	−656.754	−1.05298	−0.526488	+	0.850183i	$0.676491\pi$
$74$	220.648	0.346618
$75$	−75.0000	−0.115470
$76$	12.7838	0.0192948
$77$	451.980	0.668933
$78$	177.411	0.257537
$79$	440.195	0.626909	0.313455	−	0.949603i	$-0.398514\pi$
$79$	440.195	0.626909	0.313455	+	0.949603i	$0.398514\pi$
$80$	−25.2944	−0.0353500
$81$	81.0000	0.111111
$82$	−322.304	−0.434056
$83$	−54.4121	−0.0719579	−0.0359790	−	0.999353i	$-0.511455\pi$
$83$	−54.4121	−0.0719579	−0.0359790	+	0.999353i	$0.511455\pi$
$84$	−97.7939	−0.127026
$85$	−281.716	−0.359487
$86$	−810.656	−1.01646
$87$	−740.205	−0.912165
$88$	1494.25	1.81009
$89$	−1018.78	−1.21338	−0.606690	−	0.794938i	$-0.707503\pi$
$89$	−1018.78	−1.21338	−0.606690	+	0.794938i	$0.707503\pi$
$90$	82.2792	0.0963666
$91$	226.402	0.260806
$92$	−410.579	−0.465280
$93$	332.735	0.371000
$94$	−631.184	−0.692571
$95$	−13.7258	−0.0148236
$96$	−527.662	−0.560982
$97$	−724.108	−0.757959	−0.378979	−	0.925405i	$-0.623725\pi$
$97$	−724.108	−0.757959	−0.378979	+	0.925405i	$0.623725\pi$
$98$	89.5929	0.0923495
$99$	−581.117	−0.589944
$100$	−116.421	−0.116421
$101$	268.725	0.264744	0.132372	−	0.991200i	$-0.457741\pi$
$101$	268.725	0.264744	0.132372	+	0.991200i	$0.457741\pi$
$102$	309.058	0.300013
$103$	−1840.63	−1.76080	−0.880399	−	0.474233i	$-0.842725\pi$
$103$	−1840.63	−1.76080	−0.880399	+	0.474233i	$0.842725\pi$
$104$	748.489	0.705725
$105$	105.000	0.0975900
$106$	477.184	0.437247
$107$	−243.087	−0.219627	−0.109813	−	0.993952i	$-0.535025\pi$
$107$	−243.087	−0.219627	−0.109813	+	0.993952i	$0.535025\pi$
$108$	125.735	0.112027
$109$	−405.176	−0.356044	−0.178022	−	0.984027i	$-0.556970\pi$
$109$	−405.176	−0.356044	−0.178022	+	0.984027i	$0.556970\pi$
$110$	−590.294	−0.511658
$111$	−362.029	−0.309570
$112$	35.4121	0.0298762
$113$	−28.1766	−0.0234569	−0.0117285	−	0.999931i	$-0.503733\pi$
$113$	−28.1766	−0.0234569	−0.0117285	+	0.999931i	$0.503733\pi$
$114$	15.0580	0.0123712
$115$	440.833	0.357460
$116$	−1149.01	−0.919680
$117$	−291.088	−0.230010
$118$	1150.08	0.897232
$119$	394.402	0.303822
$120$	347.132	0.264072
$121$	2838.10	2.13230
$122$	−210.646	−0.156320
$123$	528.823	0.387661
$124$	516.500	0.374057
$125$	125.000	0.0894427
$126$	−115.191	−0.0814446
$127$	−2740.90	−1.91508	−0.957541	−	0.288298i	$-0.906911\pi$
$127$	−2740.90	−1.91508	−0.957541	+	0.288298i	$0.906911\pi$
$128$	−745.083	−0.514505
$129$	1330.09	0.907811
$130$	−295.685	−0.199487
$131$	−1832.04	−1.22188	−0.610938	−	0.791678i	$-0.709208\pi$
$131$	−1832.04	−1.22188	−0.610938	+	0.791678i	$0.709208\pi$
$132$	−902.059	−0.594804
$133$	19.2162	0.0125282
$134$	−1739.71	−1.12155
$135$	−135.000	−0.0860663
$136$	1303.90	0.822122
$137$	382.747	0.238688	0.119344	−	0.992853i	$-0.461921\pi$
$137$	382.747	0.238688	0.119344	+	0.992853i	$0.461921\pi$
$138$	−483.618	−0.298321
$139$	3053.60	1.86333	0.931667	−	0.363314i	$-0.118355\pi$
$139$	3053.60	1.86333	0.931667	+	0.363314i	$0.118355\pi$
$140$	162.990	0.0983940
$141$	1035.62	0.618545
$142$	651.403	0.384961
$143$	2088.35	1.22123
$144$	−45.5299	−0.0263483
$145$	1233.68	0.706560
$146$	−1200.83	−0.680692
$147$	−147.000	−0.0824786
$148$	−561.971	−0.312120
$149$	3560.60	1.95769	0.978843	−	0.204611i	$-0.0655929\pi$
$149$	3560.60	1.95769	0.978843	+	0.204611i	$0.0655929\pi$
$150$	−137.132	−0.0746452
$151$	3261.80	1.75789	0.878945	−	0.476923i	$-0.158248\pi$
$151$	3261.80	1.75789	0.878945	+	0.476923i	$0.158248\pi$
$152$	63.5290	0.0339005
$153$	−507.088	−0.267945
$154$	826.412	0.432430
$155$	−554.558	−0.287376
$156$	−451.852	−0.231905
$157$	2878.46	1.46322	0.731611	−	0.681723i	$-0.238769\pi$
$157$	2878.46	1.46322	0.731611	+	0.681723i	$0.238769\pi$
$158$	804.865	0.405263
$159$	−782.942	−0.390512
$160$	879.437	0.434535
$161$	−617.166	−0.302108
$162$	148.103	0.0718274
$163$	−927.537	−0.445708	−0.222854	−	0.974852i	$-0.571537\pi$
$163$	−927.537	−0.445708	−0.222854	+	0.974852i	$0.571537\pi$
$164$	820.883	0.390855
$165$	968.528	0.456969
$166$	−99.4886	−0.0465169
$167$	1094.52	0.507164	0.253582	−	0.967314i	$-0.418391\pi$
$167$	1094.52	0.507164	0.253582	+	0.967314i	$0.418391\pi$
$168$	−485.985	−0.223182
$169$	−1150.92	−0.523860
$170$	−515.097	−0.232389
$171$	−24.7065	−0.0110488
$172$	2064.67	0.915290
$173$	−1713.25	−0.752926	−0.376463	−	0.926432i	$-0.622860\pi$
$173$	−1713.25	−0.752926	−0.376463	+	0.926432i	$0.622860\pi$
$174$	−1353.41	−0.589666
$175$	−175.000	−0.0755929
$176$	326.644	0.139896
$177$	−1887.00	−0.801330
$178$	−1862.77	−0.784386
$179$	−4065.58	−1.69763	−0.848816	−	0.528689i	$-0.822684\pi$
$179$	−4065.58	−1.69763	−0.848816	+	0.528689i	$0.822684\pi$
$180$	−209.558	−0.0867754
$181$	−2791.40	−1.14631	−0.573157	−	0.819445i	$-0.694282\pi$
$181$	−2791.40	−1.14631	−0.573157	+	0.819445i	$0.694282\pi$
$182$	413.960	0.168597
$183$	345.618	0.139611
$184$	−2040.36	−0.817486
$185$	603.381	0.239792
$186$	608.382	0.239832
$187$	3637.99	1.42266
$188$	1607.57	0.623640
$189$	189.000	0.0727393
$190$	−25.0967	−0.00958266
$191$	−634.185	−0.240251	−0.120126	−	0.992759i	$-0.538330\pi$
$191$	−634.185	−0.240251	−0.120126	+	0.992759i	$0.538330\pi$
$192$	−1086.20	−0.408281
$193$	−254.999	−0.0951049	−0.0475524	−	0.998869i	$-0.515142\pi$
$193$	−254.999	−0.0951049	−0.0475524	+	0.998869i	$0.515142\pi$
$194$	−1323.98	−0.489980
$195$	485.147	0.178165
$196$	−228.186	−0.0831581
$197$	4172.37	1.50898	0.754490	−	0.656311i	$-0.227884\pi$
$197$	4172.37	1.50898	0.754490	+	0.656311i	$0.227884\pi$
$198$	−1062.53	−0.381367
$199$	−4626.48	−1.64805	−0.824026	−	0.566552i	$-0.808277\pi$
$199$	−4626.48	−1.64805	−0.824026	+	0.566552i	$0.808277\pi$
$200$	−578.553	−0.204550
$201$	2854.44	1.00168
$202$	491.344	0.171143
$203$	−1727.15	−0.597152
$204$	−787.145	−0.270153
$205$	−881.371	−0.300281
$206$	−3365.45	−1.13826
$207$	793.499	0.266435
$208$	163.620	0.0545433
$209$	177.251	0.0586638
$210$	191.985	0.0630867
$211$	−1562.64	−0.509843	−0.254921	−	0.966962i	$-0.582050\pi$
$211$	−1562.64	−0.509843	−0.254921	+	0.966962i	$0.582050\pi$
$212$	−1215.35	−0.393729
$213$	−1068.79	−0.343814
$214$	−444.466	−0.141977
$215$	−2216.81	−0.703188
$216$	624.838	0.196828
$217$	776.382	0.242877
$218$	−740.834	−0.230163
$219$	1970.26	0.607935
$220$	1503.43	0.460733
$221$	1822.31	0.554670
$222$	−661.943	−0.200120
$223$	−1236.39	−0.371278	−0.185639	−	0.982618i	$-0.559436\pi$
$223$	−1236.39	−0.371278	−0.185639	+	0.982618i	$0.559436\pi$
$224$	−1231.21	−0.367249
$225$	225.000	0.0666667
$226$	−51.5189	−0.0151637
$227$	4181.82	1.22272	0.611359	−	0.791353i	$-0.290623\pi$
$227$	4181.82	1.22272	0.611359	+	0.791353i	$0.290623\pi$
$228$	−38.3515	−0.0111399
$229$	−484.774	−0.139890	−0.0699449	−	0.997551i	$-0.522282\pi$
$229$	−484.774	−0.139890	−0.0699449	+	0.997551i	$0.522282\pi$
$230$	806.030	0.231079
$231$	−1355.94	−0.386209
$232$	−5709.98	−1.61585
$233$	2080.54	0.584982	0.292491	−	0.956268i	$-0.405516\pi$
$233$	2080.54	0.584982	0.292491	+	0.956268i	$0.405516\pi$
$234$	−532.234	−0.148689
$235$	−1726.03	−0.479123
$236$	−2929.16	−0.807932
$237$	−1320.59	−0.361946
$238$	721.135	0.196404
$239$	6814.10	1.84422	0.922108	−	0.386933i	$-0.126466\pi$
$239$	6814.10	1.84422	0.922108	+	0.386933i	$0.126466\pi$
$240$	75.8831	0.0204093
$241$	−3921.84	−1.04825	−0.524125	−	0.851642i	$-0.675607\pi$
$241$	−3921.84	−1.04825	−0.524125	+	0.851642i	$0.675607\pi$
$242$	5189.25	1.37842
$243$	−243.000	−0.0641500
$244$	536.498	0.140761
$245$	245.000	0.0638877
$246$	966.913	0.250602
$247$	88.7873	0.0228721
$248$	2566.73	0.657209
$249$	163.236	0.0415449
$250$	228.553	0.0578199
$251$	−5219.10	−1.31246	−0.656228	−	0.754562i	$-0.727849\pi$
$251$	−5219.10	−1.31246	−0.656228	+	0.754562i	$0.727849\pi$
$252$	293.382	0.0733386
$253$	−5692.78	−1.41463
$254$	−5011.53	−1.23800
$255$	845.147	0.207550
$256$	−4258.88	−1.03976
$257$	6975.71	1.69312	0.846562	−	0.532289i	$-0.178668\pi$
$257$	6975.71	1.69312	0.846562	+	0.532289i	$0.178668\pi$
$258$	2431.97	0.586852
$259$	−844.733	−0.202661
$260$	753.087	0.179632
$261$	2220.62	0.526639
$262$	−3349.75	−0.789878
$263$	−3607.36	−0.845776	−0.422888	−	0.906182i	$-0.638984\pi$
$263$	−3607.36	−0.845776	−0.422888	+	0.906182i	$0.638984\pi$
$264$	−4482.76	−1.04506
$265$	1304.90	0.302489
$266$	35.1354	0.00809882
$267$	3056.35	0.700546
$268$	4430.90	1.00993
$269$	5.88572	0.00133405	0.000667023	−	1.00000i	$-0.499788\pi$
$269$	5.88572	0.00133405	0.000667023		1.00000i	$0.499788\pi$
$270$	−246.838	−0.0556373
$271$	6916.32	1.55032	0.775160	−	0.631765i	$-0.217669\pi$
$271$	6916.32	1.55032	0.775160	+	0.631765i	$0.217669\pi$
$272$	285.033	0.0635392
$273$	−679.206	−0.150577
$274$	699.825	0.154299
$275$	−1614.21	−0.353966
$276$	1231.74	0.268630
$277$	−2119.46	−0.459733	−0.229867	−	0.973222i	$-0.573829\pi$
$277$	−2119.46	−0.459733	−0.229867	+	0.973222i	$0.573829\pi$
$278$	5583.29	1.20455
$279$	−998.205	−0.214197
$280$	809.975	0.172876
$281$	−239.917	−0.0509334	−0.0254667	−	0.999676i	$-0.508107\pi$
$281$	−239.917	−0.0509334	−0.0254667	+	0.999676i	$0.508107\pi$
$282$	1893.55	0.399856
$283$	−4542.12	−0.954067	−0.477034	−	0.878885i	$-0.658288\pi$
$283$	−4542.12	−0.954067	−0.477034	+	0.878885i	$0.658288\pi$
$284$	−1659.07	−0.346647
$285$	41.1775	0.00855840
$286$	3818.40	0.789463
$287$	1233.92	0.253784
$288$	1582.99	0.323883
$289$	−1738.45	−0.353847
$290$	2255.69	0.456753
$291$	2172.32	0.437608
$292$	3058.41	0.612944
$293$	−2171.70	−0.433010	−0.216505	−	0.976281i	$-0.569466\pi$
$293$	−2171.70	−0.433010	−0.216505	+	0.976281i	$0.569466\pi$
$294$	−268.779	−0.0533180
$295$	3145.00	0.620708
$296$	−2792.70	−0.548387
$297$	1743.35	0.340604
$298$	6510.29	1.26554
$299$	−2851.58	−0.551543
$300$	349.264	0.0672159
$301$	3103.54	0.594302
$302$	5963.96	1.13638
$303$	−806.175	−0.152850
$304$	13.8875	0.00262007
$305$	−576.030	−0.108142
$306$	−927.174	−0.173212
$307$	−3508.64	−0.652276	−0.326138	−	0.945322i	$-0.605747\pi$
$307$	−3508.64	−0.652276	−0.326138	+	0.945322i	$0.605747\pi$
$308$	−2104.80	−0.389391
$309$	5521.88	1.01660
$310$	−1013.97	−0.185773
$311$	−3133.25	−0.571287	−0.285643	−	0.958336i	$-0.592207\pi$
$311$	−3133.25	−0.571287	−0.285643	+	0.958336i	$0.592207\pi$
$312$	−2245.47	−0.407451
$313$	6389.59	1.15387	0.576935	−	0.816790i	$-0.304249\pi$
$313$	6389.59	1.15387	0.576935	+	0.816790i	$0.304249\pi$
$314$	5263.05	0.945895
$315$	−315.000	−0.0563436
$316$	−2049.92	−0.364928
$317$	1634.44	0.289587	0.144794	−	0.989462i	$-0.453748\pi$
$317$	1634.44	0.289587	0.144794	+	0.989462i	$0.453748\pi$
$318$	−1431.55	−0.252445
$319$	−15931.3	−2.79618
$320$	1810.34	0.316253
$321$	729.260	0.126802
$322$	−1128.44	−0.195297
$323$	154.671	0.0266444
$324$	−377.205	−0.0646785
$325$	−808.579	−0.138006
$326$	−1695.93	−0.288126
$327$	1215.53	0.205562
$328$	4079.36	0.686723
$329$	2416.44	0.404932
$330$	1770.88	0.295406
$331$	4386.17	0.728355	0.364177	−	0.931330i	$-0.381350\pi$
$331$	4386.17	0.728355	0.364177	+	0.931330i	$0.381350\pi$
$332$	253.389	0.0418872
$333$	1086.09	0.178730
$334$	2001.25	0.327854
$335$	−4757.40	−0.775894
$336$	−106.236	−0.0172490
$337$	1713.98	0.277051	0.138526	−	0.990359i	$-0.455764\pi$
$337$	1713.98	0.277051	0.138526	+	0.990359i	$0.455764\pi$
$338$	−2104.38	−0.338648
$339$	84.5299	0.0135429
$340$	1311.91	0.209260
$341$	7161.41	1.13728
$342$	−45.1740	−0.00714249
$343$	−343.000	−0.0539949
$344$	10260.4	1.60814
$345$	−1322.50	−0.206379
$346$	−3132.56	−0.486727
$347$	−1744.83	−0.269935	−0.134967	−	0.990850i	$-0.543093\pi$
$347$	−1744.83	−0.269935	−0.134967	+	0.990850i	$0.543093\pi$
$348$	3447.03	0.530977
$349$	7046.78	1.08082	0.540409	−	0.841403i	$-0.318270\pi$
$349$	7046.78	1.08082	0.540409	+	0.841403i	$0.318270\pi$
$350$	−319.975	−0.0488668
$351$	873.265	0.132796
$352$	−11356.8	−1.71966
$353$	−12668.5	−1.91013	−0.955064	−	0.296400i	$-0.904214\pi$
$353$	−12668.5	−1.91013	−0.955064	+	0.296400i	$0.904214\pi$
$354$	−3450.24	−0.518017
$355$	1781.32	0.266317
$356$	4744.33	0.706317
$357$	−1183.21	−0.175411
$358$	−7433.62	−1.09743
$359$	37.7844	0.00555483	0.00277742	−	0.999996i	$-0.499116\pi$
$359$	37.7844	0.00555483	0.00277742	+	0.999996i	$0.499116\pi$
$360$	−1041.40	−0.152462
$361$	−6851.46	−0.998901
$362$	−5103.87	−0.741031
$363$	−8514.29	−1.23109
$364$	−1054.32	−0.151817
$365$	−3283.77	−0.470905
$366$	631.938	0.0902511
$367$	−759.829	−0.108073	−0.0540364	−	0.998539i	$-0.517209\pi$
$367$	−759.829	−0.108073	−0.0540364	+	0.998539i	$0.517209\pi$
$368$	−446.023	−0.0631809
$369$	−1586.47	−0.223816
$370$	1103.24	0.155012
$371$	−1826.86	−0.255650
$372$	−1549.50	−0.215962
$373$	719.320	0.0998525	0.0499263	−	0.998753i	$-0.484101\pi$
$373$	719.320	0.0998525	0.0499263	+	0.998753i	$0.484101\pi$
$374$	6651.81	0.919671
$375$	−375.000	−0.0516398
$376$	7988.81	1.09572
$377$	−7980.19	−1.09019
$378$	345.573	0.0470221
$379$	572.559	0.0775999	0.0388000	−	0.999247i	$-0.487646\pi$
$379$	572.559	0.0775999	0.0388000	+	0.999247i	$0.487646\pi$
$380$	63.9192	0.00862891
$381$	8222.69	1.10567
$382$	−1159.56	−0.155310
$383$	4513.18	0.602122	0.301061	−	0.953605i	$-0.402659\pi$
$383$	4513.18	0.602122	0.301061	+	0.953605i	$0.402659\pi$
$384$	2235.25	0.297050
$385$	2259.90	0.299156
$386$	−466.247	−0.0614802
$387$	−3990.26	−0.524125
$388$	3372.06	0.441213
$389$	−6902.13	−0.899619	−0.449810	−	0.893124i	$-0.648508\pi$
$389$	−6902.13	−0.899619	−0.449810	+	0.893124i	$0.648508\pi$
$390$	887.056	0.115174
$391$	−4967.58	−0.642510
$392$	−1133.96	−0.146107
$393$	5496.11	0.705451
$394$	7628.88	0.975475
$395$	2200.98	0.280362
$396$	2706.18	0.343410
$397$	4124.58	0.521427	0.260714	−	0.965416i	$-0.416042\pi$
$397$	4124.58	0.521427	0.260714	+	0.965416i	$0.416042\pi$
$398$	−8459.18	−1.06538
$399$	−57.6485	−0.00723317
$400$	−126.472	−0.0158090
$401$	−1002.50	−0.124844	−0.0624219	−	0.998050i	$-0.519882\pi$
$401$	−1002.50	−0.124844	−0.0624219	+	0.998050i	$0.519882\pi$
$402$	5219.14	0.647530
$403$	3587.23	0.443406
$404$	−1251.41	−0.154109
$405$	405.000	0.0496904
$406$	−3157.96	−0.386027
$407$	−7791.89	−0.948967
$408$	−3911.70	−0.474652
$409$	10335.0	1.24947	0.624736	−	0.780836i	$-0.285206\pi$
$409$	10335.0	1.24947	0.624736	+	0.780836i	$0.285206\pi$
$410$	−1611.52	−0.194116
$411$	−1148.24	−0.137807
$412$	8571.53	1.02497
$413$	−4402.99	−0.524594
$414$	1450.85	0.172236
$415$	−272.061	−0.0321806
$416$	−5688.75	−0.670466
$417$	−9160.81	−1.07580
$418$	324.091	0.0379230
$419$	3183.21	0.371145	0.185573	−	0.982631i	$-0.440586\pi$
$419$	3183.21	0.371145	0.185573	+	0.982631i	$0.440586\pi$
$420$	−488.970	−0.0568078
$421$	−6944.34	−0.803911	−0.401956	−	0.915659i	$-0.631669\pi$
$421$	−6944.34	−0.803911	−0.401956	+	0.915659i	$0.631669\pi$
$422$	−2857.18	−0.329586
$423$	−3106.85	−0.357117
$424$	−6039.65	−0.691772
$425$	−1408.58	−0.160767
$426$	−1954.21	−0.222258
$427$	806.442	0.0913969
$428$	1132.02	0.127846
$429$	−6265.05	−0.705080
$430$	−4053.28	−0.454573
$431$	3868.41	0.432331	0.216166	−	0.976357i	$-0.430645\pi$
$431$	3868.41	0.432331	0.216166	+	0.976357i	$0.430645\pi$
$432$	136.590	0.0152122
$433$	−6132.96	−0.680673	−0.340336	−	0.940304i	$-0.610541\pi$
$433$	−6132.96	−0.680673	−0.340336	+	0.940304i	$0.610541\pi$
$434$	1419.56	0.157007
$435$	−3701.03	−0.407932
$436$	1886.84	0.207256
$437$	−242.032	−0.0264942
$438$	3602.48	0.392998
$439$	−4090.14	−0.444673	−0.222337	−	0.974970i	$-0.571368\pi$
$439$	−4090.14	−0.444673	−0.222337	+	0.974970i	$0.571368\pi$
$440$	7471.27	0.809497
$441$	441.000	0.0476190
$442$	3331.97	0.358565
$443$	12434.5	1.33359	0.666795	−	0.745241i	$-0.267666\pi$
$443$	12434.5	1.33359	0.666795	+	0.745241i	$0.267666\pi$
$444$	1685.91	0.180203
$445$	−5093.92	−0.542640
$446$	−2260.66	−0.240012
$447$	−10681.8	−1.13027
$448$	−2534.48	−0.267283
$449$	883.046	0.0928141	0.0464071	−	0.998923i	$-0.485223\pi$
$449$	883.046	0.0928141	0.0464071	+	0.998923i	$0.485223\pi$
$450$	411.396	0.0430964
$451$	11381.8	1.18835
$452$	131.214	0.0136544
$453$	−9785.40	−1.01492
$454$	7646.15	0.790422
$455$	1132.01	0.116636
$456$	−190.587	−0.0195725
$457$	−9068.44	−0.928235	−0.464118	−	0.885774i	$-0.653629\pi$
$457$	−9068.44	−0.928235	−0.464118	+	0.885774i	$0.653629\pi$
$458$	−886.373	−0.0904312
$459$	1521.26	0.154698
$460$	−2052.89	−0.208080
$461$	12508.9	1.26377	0.631885	−	0.775063i	$-0.282282\pi$
$461$	12508.9	1.26377	0.631885	+	0.775063i	$0.282282\pi$
$462$	−2479.24	−0.249663
$463$	12688.7	1.27363	0.636817	−	0.771015i	$-0.280251\pi$
$463$	12688.7	1.27363	0.636817	+	0.771015i	$0.280251\pi$
$464$	−1248.20	−0.124884
$465$	1663.68	0.165916
$466$	3804.12	0.378160
$467$	−10136.5	−1.00442	−0.502208	−	0.864747i	$-0.667479\pi$
$467$	−10136.5	−1.00442	−0.502208	+	0.864747i	$0.667479\pi$
$468$	1355.56	0.133890
$469$	6660.36	0.655750
$470$	−3155.92	−0.309727
$471$	−8635.37	−0.844791
$472$	−14556.4	−1.41952
$473$	28627.3	2.78284
$474$	−2414.59	−0.233979
$475$	−68.6292	−0.00662931
$476$	−1836.67	−0.176857
$477$	2348.83	0.225462
$478$	12459.1	1.19219
$479$	−11361.1	−1.08372	−0.541861	−	0.840468i	$-0.682280\pi$
$479$	−11361.1	−1.08372	−0.541861	+	0.840468i	$0.682280\pi$
$480$	−2638.31	−0.250879
$481$	−3903.05	−0.369987
$482$	−7170.80	−0.677637
$483$	1851.50	0.174422
$484$	−13216.6	−1.24123
$485$	−3620.54	−0.338969
$486$	−444.308	−0.0414696
$487$	7929.53	0.737826	0.368913	−	0.929464i	$-0.379730\pi$
$487$	7929.53	0.737826	0.368913	+	0.929464i	$0.379730\pi$
$488$	2666.11	0.247314
$489$	2782.61	0.257329
$490$	447.965	0.0413000
$491$	8111.51	0.745555	0.372777	−	0.927921i	$-0.378405\pi$
$491$	8111.51	0.745555	0.372777	+	0.927921i	$0.378405\pi$
$492$	−2462.65	−0.225660
$493$	−13901.8	−1.26999
$494$	162.341	0.0147856
$495$	−2905.58	−0.263831
$496$	561.088	0.0507936
$497$	−2493.85	−0.225079
$498$	298.466	0.0268566
$499$	16816.6	1.50865	0.754324	−	0.656502i	$-0.227965\pi$
$499$	16816.6	1.50865	0.754324	+	0.656502i	$0.227965\pi$
$500$	−582.107	−0.0520652
$501$	−3283.55	−0.292811
$502$	−9542.74	−0.848433
$503$	−17764.6	−1.57472	−0.787362	−	0.616491i	$-0.788554\pi$
$503$	−17764.6	−1.57472	−0.787362	+	0.616491i	$0.788554\pi$
$504$	1457.95	0.128854
$505$	1343.62	0.118397
$506$	−10408.8	−0.914485
$507$	3452.76	0.302451
$508$	12764.0	1.11478
$509$	13908.8	1.21120	0.605598	−	0.795771i	$-0.292934\pi$
$509$	13908.8	1.21120	0.605598	+	0.795771i	$0.292934\pi$
$510$	1545.29	0.134170
$511$	4597.27	0.397987
$512$	−1826.38	−0.157647
$513$	74.1195	0.00637905
$514$	12754.6	1.09451
$515$	−9203.13	−0.787453
$516$	−6194.02	−0.528443
$517$	22289.5	1.89611
$518$	−1544.53	−0.131009
$519$	5139.76	0.434702
$520$	3742.45	0.315610
$521$	−8639.68	−0.726510	−0.363255	−	0.931690i	$-0.618335\pi$
$521$	−8639.68	−0.726510	−0.363255	+	0.931690i	$0.618335\pi$
$522$	4060.23	0.340444
$523$	23242.2	1.94323	0.971617	−	0.236561i	$-0.0760203\pi$
$523$	23242.2	1.94323	0.971617	+	0.236561i	$0.0760203\pi$
$524$	8531.53	0.711263
$525$	525.000	0.0436436
$526$	−6595.79	−0.546749
$527$	6249.11	0.516538
$528$	−979.932	−0.0807691
$529$	−4393.66	−0.361113
$530$	2385.92	0.195543
$531$	5660.99	0.462648
$532$	−89.4869	−0.00729276
$533$	5701.26	0.463319
$534$	5588.32	0.452865
$535$	−1215.43	−0.0982201
$536$	22019.3	1.77442
$537$	12196.8	0.980128
$538$	10.7616	0.000862390 0
$539$	−3163.86	−0.252833
$540$	628.675	0.0500998
$541$	11395.2	0.905577	0.452789	−	0.891618i	$-0.350429\pi$
$541$	11395.2	0.905577	0.452789	+	0.891618i	$0.350429\pi$
$542$	12646.0	1.00220
$543$	8374.19	0.661825
$544$	−9910.04	−0.781047
$545$	−2025.88	−0.159228
$546$	−1241.88	−0.0973398
$547$	−7870.21	−0.615184	−0.307592	−	0.951518i	$-0.599523\pi$
$547$	−7870.21	−0.615184	−0.307592	+	0.951518i	$0.599523\pi$
$548$	−1782.40	−0.138942
$549$	−1036.85	−0.0806045
$550$	−2951.47	−0.228820
$551$	−677.329	−0.0523687
$552$	6121.08	0.471976
$553$	−3081.37	−0.236949
$554$	−3875.28	−0.297193
$555$	−1810.14	−0.138444
$556$	−14220.2	−1.08466
$557$	17769.8	1.35176	0.675880	−	0.737012i	$-0.263764\pi$
$557$	17769.8	1.35176	0.675880	+	0.737012i	$0.263764\pi$
$558$	−1825.15	−0.138467
$559$	14339.7	1.08498
$560$	177.061	0.0133610
$561$	−10914.0	−0.821370
$562$	−438.672	−0.0329257
$563$	15192.8	1.13730	0.568651	−	0.822579i	$-0.307466\pi$
$563$	15192.8	1.13730	0.568651	+	0.822579i	$0.307466\pi$
$564$	−4822.72	−0.360059
$565$	−140.883	−0.0104903
$566$	−8304.93	−0.616753
$567$	−567.000	−0.0419961
$568$	−8244.71	−0.609050
$569$	−23300.0	−1.71667	−0.858335	−	0.513090i	$-0.828501\pi$
$569$	−23300.0	−1.71667	−0.858335	+	0.513090i	$0.828501\pi$
$570$	75.2900	0.00553255
$571$	10638.2	0.779673	0.389837	−	0.920884i	$-0.372531\pi$
$571$	10638.2	0.779673	0.389837	+	0.920884i	$0.372531\pi$
$572$	−9725.14	−0.710889
$573$	1902.55	0.138709
$574$	2256.13	0.164058
$575$	2204.16	0.159861
$576$	3258.61	0.235721
$577$	−897.258	−0.0647372	−0.0323686	−	0.999476i	$-0.510305\pi$
$577$	−897.258	−0.0647372	−0.0323686	+	0.999476i	$0.510305\pi$
$578$	−3178.63	−0.228743
$579$	764.997	0.0549088
$580$	−5745.05	−0.411293
$581$	380.885	0.0271975
$582$	3971.93	0.282890
$583$	−16851.1	−1.19709
$584$	15198.7	1.07693
$585$	−1455.44	−0.102863
$586$	−3970.79	−0.279918
$587$	−14712.9	−1.03452	−0.517261	−	0.855828i	$-0.673048\pi$
$587$	−14712.9	−1.03452	−0.517261	+	0.855828i	$0.673048\pi$
$588$	684.558	0.0480114
$589$	304.471	0.0212997
$590$	5750.40	0.401254
$591$	−12517.1	−0.871210
$592$	−610.486	−0.0423831
$593$	−7216.29	−0.499726	−0.249863	−	0.968281i	$-0.580386\pi$
$593$	−7216.29	−0.499726	−0.249863	+	0.968281i	$0.580386\pi$
$594$	3187.59	0.220182
$595$	1972.01	0.135873
$596$	−16581.2	−1.13958
$597$	13879.4	0.951503
$598$	−5213.91	−0.356543
$599$	−20885.5	−1.42464	−0.712320	−	0.701855i	$-0.752355\pi$
$599$	−20885.5	−1.42464	−0.712320	+	0.701855i	$0.752355\pi$
$600$	1735.66	0.118097
$601$	−11047.7	−0.749823	−0.374911	−	0.927061i	$-0.622327\pi$
$601$	−11047.7	−0.749823	−0.374911	+	0.927061i	$0.622327\pi$
$602$	5674.59	0.384185
$603$	−8563.32	−0.578317
$604$	−15189.7	−1.02328
$605$	14190.5	0.953595
$606$	−1474.03	−0.0988093
$607$	−9434.94	−0.630894	−0.315447	−	0.948943i	$-0.602154\pi$
$607$	−9434.94	−0.630894	−0.315447	+	0.948943i	$0.602154\pi$
$608$	−482.840	−0.0322068
$609$	5181.44	0.344766
$610$	−1053.23	−0.0699082
$611$	11165.0	0.739263
$612$	2361.44	0.155973
$613$	−17662.6	−1.16376	−0.581881	−	0.813274i	$-0.697683\pi$
$613$	−17662.6	−1.16376	−0.581881	+	0.813274i	$0.697683\pi$
$614$	−6415.30	−0.421662
$615$	2644.11	0.173367
$616$	−10459.8	−0.684150
$617$	−10817.4	−0.705820	−0.352910	−	0.935657i	$-0.614808\pi$
$617$	−10817.4	−0.705820	−0.352910	+	0.935657i	$0.614808\pi$
$618$	10096.3	0.657176
$619$	29073.9	1.88785	0.943926	−	0.330158i	$-0.107102\pi$
$619$	29073.9	1.88785	0.943926	+	0.330158i	$0.107102\pi$
$620$	2582.50	0.167283
$621$	−2380.50	−0.153826
$622$	−5728.92	−0.369306
$623$	7131.49	0.458615
$624$	−490.860	−0.0314906
$625$	625.000	0.0400000
$626$	11682.9	0.745915
$627$	−531.754	−0.0338696
$628$	−13404.5	−0.851751
$629$	−6799.28	−0.431009
$630$	−575.955	−0.0364231
$631$	−2203.17	−0.138996	−0.0694981	−	0.997582i	$-0.522140\pi$
$631$	−2203.17	−0.138996	−0.0694981	+	0.997582i	$0.522140\pi$
$632$	−10187.1	−0.641170
$633$	4687.93	0.294358
$634$	2988.45	0.187203
$635$	−13704.5	−0.856451
$636$	3646.05	0.227319
$637$	−1584.81	−0.0985755
$638$	−29129.3	−1.80758
$639$	3206.38	0.198501
$640$	−3725.42	−0.230094
$641$	22466.5	1.38436	0.692180	−	0.721725i	$-0.256651\pi$
$641$	22466.5	1.38436	0.692180	+	0.721725i	$0.256651\pi$
$642$	1333.40	0.0819705
$643$	−12347.0	−0.757257	−0.378629	−	0.925549i	$-0.623604\pi$
$643$	−12347.0	−0.757257	−0.378629	+	0.925549i	$0.623604\pi$
$644$	2874.05	0.175859
$645$	6650.44	0.405986
$646$	282.805	0.0172242
$647$	−24114.0	−1.46525	−0.732626	−	0.680631i	$-0.761706\pi$
$647$	−24114.0	−1.46525	−0.732626	+	0.680631i	$0.761706\pi$
$648$	−1874.51	−0.113639
$649$	−40613.6	−2.45643
$650$	−1478.43	−0.0892134
$651$	−2329.15	−0.140225
$652$	4319.41	0.259449
$653$	7843.33	0.470035	0.235018	−	0.971991i	$-0.424485\pi$
$653$	7843.33	0.470035	0.235018	+	0.971991i	$0.424485\pi$
$654$	2222.50	0.132885
$655$	−9160.18	−0.546440
$656$	891.749	0.0530746
$657$	−5910.78	−0.350992
$658$	4418.29	0.261767
$659$	21242.8	1.25569	0.627846	−	0.778338i	$-0.283937\pi$
$659$	21242.8	1.25569	0.627846	+	0.778338i	$0.283937\pi$
$660$	−4510.29	−0.266005
$661$	−22221.7	−1.30760	−0.653801	−	0.756667i	$-0.726827\pi$
$661$	−22221.7	−1.30760	−0.653801	+	0.756667i	$0.726827\pi$
$662$	8019.79	0.470843
$663$	−5466.94	−0.320239
$664$	1259.21	0.0735948
$665$	96.0808	0.00560279
$666$	1985.83	0.115539
$667$	21753.8	1.26283
$668$	−5097.01	−0.295223
$669$	3709.18	0.214358
$670$	−8698.56	−0.501574
$671$	7438.69	0.427969
$672$	3693.63	0.212031
$673$	−3787.85	−0.216955	−0.108478	−	0.994099i	$-0.534598\pi$
$673$	−3787.85	−0.216955	−0.108478	+	0.994099i	$0.534598\pi$
$674$	3133.88	0.179099
$675$	−675.000	−0.0384900
$676$	5359.67	0.304943
$677$	−11296.8	−0.641314	−0.320657	−	0.947195i	$-0.603904\pi$
$677$	−11296.8	−0.641314	−0.320657	+	0.947195i	$0.603904\pi$
$678$	154.557	0.00875474
$679$	5068.75	0.286481
$680$	6519.50	0.367664
$681$	−12545.5	−0.705937
$682$	13094.1	0.735190
$683$	4807.14	0.269312	0.134656	−	0.990892i	$-0.457007\pi$
$683$	4807.14	0.269312	0.134656	+	0.990892i	$0.457007\pi$
$684$	115.055	0.00643161
$685$	1913.73	0.106745
$686$	−627.151	−0.0349048
$687$	1454.32	0.0807654
$688$	2242.92	0.124288
$689$	−8440.94	−0.466726
$690$	−2418.09	−0.133413
$691$	5393.47	0.296928	0.148464	−	0.988918i	$-0.452567\pi$
$691$	5393.47	0.296928	0.148464	+	0.988918i	$0.452567\pi$
$692$	7978.37	0.438283
$693$	4067.82	0.222978
$694$	−3190.29	−0.174498
$695$	15268.0	0.833308
$696$	17129.9	0.932914
$697$	9931.84	0.539735
$698$	12884.5	0.698691
$699$	−6241.63	−0.337740
$700$	814.949	0.0440031
$701$	2404.77	0.129568	0.0647838	−	0.997899i	$-0.479364\pi$
$701$	2404.77	0.129568	0.0647838	+	0.997899i	$0.479364\pi$
$702$	1596.70	0.0858456
$703$	−331.276	−0.0177729
$704$	−23378.2	−1.25156
$705$	5178.09	0.276622
$706$	−23163.4	−1.23480
$707$	−1881.07	−0.100064
$708$	8787.47	0.466460
$709$	−21617.3	−1.14507	−0.572535	−	0.819881i	$-0.694040\pi$
$709$	−21617.3	−1.14507	−0.572535	+	0.819881i	$0.694040\pi$
$710$	3257.01	0.172160
$711$	3961.76	0.208970
$712$	23576.8	1.24098
$713$	−9778.70	−0.513626
$714$	−2163.41	−0.113394
$715$	10441.7	0.546153
$716$	18932.8	0.988203
$717$	−20442.3	−1.06476
$718$	69.0860	0.00359090
$719$	−18228.6	−0.945498	−0.472749	−	0.881197i	$-0.656738\pi$
$719$	−18228.6	−0.945498	−0.472749	+	0.881197i	$0.656738\pi$
$720$	−227.649	−0.0117833
$721$	12884.4	0.665519
$722$	−12527.4	−0.645736
$723$	11765.5	0.605207
$724$	12999.1	0.667278
$725$	6168.38	0.315983
$726$	−15567.8	−0.795832
$727$	20196.5	1.03033	0.515164	−	0.857092i	$-0.327731\pi$
$727$	20196.5	1.03033	0.515164	+	0.857092i	$0.327731\pi$
$728$	−5239.43	−0.266739
$729$	729.000	0.0370370
$730$	−6004.13	−0.304415
$731$	24980.4	1.26393
$732$	−1609.49	−0.0812686
$733$	−15264.9	−0.769196	−0.384598	−	0.923084i	$-0.625660\pi$
$733$	−15264.9	−0.769196	−0.384598	+	0.923084i	$0.625660\pi$
$734$	−1389.29	−0.0698633
$735$	−735.000	−0.0368856
$736$	15507.4	0.776643
$737$	61435.7	3.07057
$738$	−2900.74	−0.144685
$739$	13906.0	0.692207	0.346103	−	0.938196i	$-0.387505\pi$
$739$	13906.0	0.692207	0.346103	+	0.938196i	$0.387505\pi$
$740$	−2809.86	−0.139584
$741$	−266.362	−0.0132052
$742$	−3340.29	−0.165264
$743$	4592.87	0.226778	0.113389	−	0.993551i	$-0.463829\pi$
$743$	4592.87	0.226778	0.113389	+	0.993551i	$0.463829\pi$
$744$	−7700.20	−0.379440
$745$	17803.0	0.875504
$746$	1315.22	0.0645493
$747$	−489.709	−0.0239860
$748$	−16941.6	−0.828137
$749$	1701.61	0.0830111
$750$	−685.660	−0.0333824
$751$	−8390.80	−0.407702	−0.203851	−	0.979002i	$-0.565346\pi$
$751$	−8390.80	−0.407702	−0.203851	+	0.979002i	$0.565346\pi$
$752$	1746.35	0.0846848
$753$	15657.3	0.757747
$754$	−14591.2	−0.704748
$755$	16309.0	0.786153
$756$	−880.145	−0.0423420
$757$	−1368.89	−0.0657240	−0.0328620	−	0.999460i	$-0.510462\pi$
$757$	−1368.89	−0.0657240	−0.0328620	+	0.999460i	$0.510462\pi$
$758$	1046.88	0.0501642
$759$	17078.4	0.816739
$760$	317.645	0.0151608
$761$	−1623.77	−0.0773478	−0.0386739	−	0.999252i	$-0.512313\pi$
$761$	−1623.77	−0.0773478	−0.0386739	+	0.999252i	$0.512313\pi$
$762$	15034.6	0.714759
$763$	2836.23	0.134572
$764$	2953.31	0.139852
$765$	−2535.44	−0.119829
$766$	8252.03	0.389240
$767$	−20343.8	−0.957722
$768$	12776.6	0.600308
$769$	−26842.5	−1.25873	−0.629366	−	0.777109i	$-0.716685\pi$
$769$	−26842.5	−1.25873	−0.629366	+	0.777109i	$0.716685\pi$
$770$	4132.06	0.193388
$771$	−20927.1	−0.977526
$772$	1187.49	0.0553612
$773$	−20961.4	−0.975330	−0.487665	−	0.873031i	$-0.662151\pi$
$773$	−20961.4	−0.975330	−0.487665	+	0.873031i	$0.662151\pi$
$774$	−7295.90	−0.338819
$775$	−2772.79	−0.128518
$776$	16757.4	0.775200
$777$	2534.20	0.117006
$778$	−12620.0	−0.581556
$779$	483.902	0.0222562
$780$	−2259.26	−0.103711
$781$	−23003.5	−1.05394
$782$	−9082.86	−0.415348
$783$	−6661.85	−0.304055
$784$	−247.885	−0.0112921
$785$	14392.3	0.654373
$786$	10049.2	0.456036
$787$	35333.2	1.60037	0.800187	−	0.599751i	$-0.204734\pi$
$787$	35333.2	1.60037	0.800187	+	0.599751i	$0.204734\pi$
$788$	−19430.1	−0.878388
$789$	10822.1	0.488309
$790$	4024.32	0.181239
$791$	197.236	0.00886589
$792$	13448.3	0.603364
$793$	3726.13	0.166858
$794$	7541.49	0.337075
$795$	−3914.71	−0.174642
$796$	21544.8	0.959342
$797$	−8137.04	−0.361642	−0.180821	−	0.983516i	$-0.557875\pi$
$797$	−8137.04	−0.361642	−0.180821	+	0.983516i	$0.557875\pi$
$798$	−105.406	−0.00467586
$799$	19450.0	0.861191
$800$	4397.18	0.194330
$801$	−9169.05	−0.404460
$802$	−1832.99	−0.0807049
$803$	42405.6	1.86359
$804$	−13292.7	−0.583082
$805$	−3085.83	−0.135107
$806$	6558.99	0.286639
$807$	−17.6572	−0.000770212 0
$808$	−6218.87	−0.270766
$809$	−36281.0	−1.57673	−0.788364	−	0.615209i	$-0.789072\pi$
$809$	−36281.0	−1.57673	−0.788364	+	0.615209i	$0.789072\pi$
$810$	740.513	0.0321222
$811$	−34237.2	−1.48240	−0.741202	−	0.671282i	$-0.765744\pi$
$811$	−34237.2	−1.48240	−0.741202	+	0.671282i	$0.765744\pi$
$812$	8043.06	0.347606
$813$	−20749.0	−0.895077
$814$	−14246.9	−0.613456
$815$	−4637.69	−0.199326
$816$	−855.099	−0.0366844
$817$	1217.10	0.0521188
$818$	18896.8	0.807717
$819$	2037.62	0.0869355
$820$	4104.42	0.174796
$821$	−23247.8	−0.988250	−0.494125	−	0.869391i	$-0.664511\pi$
$821$	−23247.8	−0.988250	−0.494125	+	0.869391i	$0.664511\pi$
$822$	−2099.47	−0.0890846
$823$	42934.0	1.81845	0.909225	−	0.416306i	$-0.136675\pi$
$823$	42934.0	1.81845	0.909225	+	0.416306i	$0.136675\pi$
$824$	42596.0	1.80085
$825$	4842.64	0.204363
$826$	−8050.55	−0.339122
$827$	−781.391	−0.0328557	−0.0164278	−	0.999865i	$-0.505229\pi$
$827$	−781.391	−0.0328557	−0.0164278	+	0.999865i	$0.505229\pi$
$828$	−3695.21	−0.155093
$829$	−33493.3	−1.40322	−0.701611	−	0.712561i	$-0.747535\pi$
$829$	−33493.3	−1.40322	−0.701611	+	0.712561i	$0.747535\pi$
$830$	−497.443	−0.0208030
$831$	6358.39	0.265427
$832$	−11710.4	−0.487964
$833$	−2760.81	−0.114834
$834$	−16749.9	−0.695445
$835$	5472.59	0.226811
$836$	−825.434	−0.0341486
$837$	2994.62	0.123667
$838$	5820.27	0.239926
$839$	15155.7	0.623639	0.311819	−	0.950141i	$-0.399062\pi$
$839$	15155.7	0.623639	0.311819	+	0.950141i	$0.399062\pi$
$840$	−2429.92	−0.0998099
$841$	36489.2	1.49613
$842$	−12697.2	−0.519686
$843$	719.752	0.0294064
$844$	7277.01	0.296783
$845$	−5754.60	−0.234277
$846$	−5680.66	−0.230857
$847$	−19866.7	−0.805935
$848$	−1320.27	−0.0534649
$849$	13626.4	0.550831
$850$	−2575.48	−0.103927
$851$	10639.6	0.428579
$852$	4977.21	0.200137
$853$	−2917.48	−0.117107	−0.0585537	−	0.998284i	$-0.518649\pi$
$853$	−2917.48	−0.117107	−0.0585537	+	0.998284i	$0.518649\pi$
$854$	1474.52	0.0590832
$855$	−123.532	−0.00494119
$856$	5625.54	0.224623
$857$	−31560.7	−1.25799	−0.628993	−	0.777411i	$-0.716532\pi$
$857$	−31560.7	−1.25799	−0.628993	+	0.777411i	$0.716532\pi$
$858$	−11455.2	−0.455797
$859$	−1404.81	−0.0557991	−0.0278995	−	0.999611i	$-0.508882\pi$
$859$	−1404.81	−0.0557991	−0.0278995	+	0.999611i	$0.508882\pi$
$860$	10323.4	0.409330
$861$	−3701.76	−0.146522
$862$	7073.11	0.279479
$863$	−9808.24	−0.386879	−0.193439	−	0.981112i	$-0.561964\pi$
$863$	−9808.24	−0.386879	−0.193439	+	0.981112i	$0.561964\pi$
$864$	−4748.96	−0.186994
$865$	−8566.27	−0.336719
$866$	−11213.7	−0.440018
$867$	5215.35	0.204294
$868$	−3615.50	−0.141380
$869$	−28422.8	−1.10952
$870$	−6767.06	−0.263707
$871$	30773.9	1.19717
$872$	9376.63	0.364143
$873$	−6516.97	−0.252653
$874$	−442.537	−0.0171271
$875$	−875.000	−0.0338062
$876$	−9175.22	−0.353883
$877$	−7196.05	−0.277073	−0.138537	−	0.990357i	$-0.544240\pi$
$877$	−7196.05	−0.277073	−0.138537	+	0.990357i	$0.544240\pi$
$878$	−7478.52	−0.287458
$879$	6515.10	0.249999
$880$	1633.22	0.0625635
$881$	−3183.27	−0.121733	−0.0608667	−	0.998146i	$-0.519386\pi$
$881$	−3183.27	−0.121733	−0.0608667	+	0.998146i	$0.519386\pi$
$882$	806.336	0.0307832
$883$	25392.5	0.967751	0.483876	−	0.875137i	$-0.339229\pi$
$883$	25392.5	0.967751	0.483876	+	0.875137i	$0.339229\pi$
$884$	−8486.25	−0.322877
$885$	−9434.99	−0.358366
$886$	22735.6	0.862095
$887$	−30634.2	−1.15964	−0.579818	−	0.814746i	$-0.696876\pi$
$887$	−30634.2	−1.15964	−0.579818	+	0.814746i	$0.696876\pi$
$888$	8378.11	0.316612
$889$	19186.3	0.723833
$890$	−9313.86	−0.350788
$891$	−5230.05	−0.196648
$892$	5757.71	0.216124
$893$	947.648	0.0355116
$894$	−19530.9	−0.730660
$895$	−20327.9	−0.759204
$896$	5215.58	0.194465
$897$	8554.75	0.318433
$898$	1614.59	0.0599994
$899$	−27365.8	−1.01524
$900$	−1047.79	−0.0388071
$901$	−14704.5	−0.543704
$902$	20810.7	0.768206
$903$	−9310.61	−0.343120
$904$	652.067	0.0239905
$905$	−13957.0	−0.512648
$906$	−17891.9	−0.656091
$907$	−28089.9	−1.02834	−0.514172	−	0.857687i	$-0.671901\pi$
$907$	−28089.9	−1.02834	−0.514172	+	0.857687i	$0.671901\pi$
$908$	−19474.1	−0.711753
$909$	2418.52	0.0882480
$910$	2069.80	0.0753990
$911$	−36102.7	−1.31299	−0.656495	−	0.754330i	$-0.727962\pi$
$911$	−36102.7	−1.31299	−0.656495	+	0.754330i	$0.727962\pi$
$912$	−41.6624	−0.00151270
$913$	3513.31	0.127353
$914$	−16581.0	−0.600055
$915$	1728.09	0.0624360
$916$	2257.52	0.0814308
$917$	12824.3	0.461826
$918$	2781.52	0.100004
$919$	−14533.8	−0.521682	−0.260841	−	0.965382i	$-0.584000\pi$
$919$	−14533.8	−0.521682	−0.260841	+	0.965382i	$0.584000\pi$
$920$	−10201.8	−0.365591
$921$	10525.9	0.376592
$922$	22871.6	0.816959
$923$	−11522.7	−0.410915
$924$	6314.41	0.224815
$925$	3016.90	0.107238
$926$	23200.3	0.823336
$927$	−16565.6	−0.586933
$928$	43397.6	1.53512
$929$	16539.6	0.584118	0.292059	−	0.956400i	$-0.405660\pi$
$929$	16539.6	0.584118	0.292059	+	0.956400i	$0.405660\pi$
$930$	3041.91	0.107256
$931$	−134.513	−0.00473522
$932$	−9688.79	−0.340522
$933$	9399.74	0.329833
$934$	−18533.9	−0.649301
$935$	18190.0	0.636231
$936$	6736.41	0.235242
$937$	30212.3	1.05335	0.526677	−	0.850065i	$-0.323438\pi$
$937$	30212.3	1.05335	0.526677	+	0.850065i	$0.323438\pi$
$938$	12178.0	0.423908
$939$	−19168.8	−0.666187
$940$	8037.87	0.278900
$941$	−26414.4	−0.915074	−0.457537	−	0.889191i	$-0.651268\pi$
$941$	−26414.4	−0.915074	−0.457537	+	0.889191i	$0.651268\pi$
$942$	−15789.1	−0.546112
$943$	−15541.5	−0.536692
$944$	−3182.03	−0.109710
$945$	945.000	0.0325300
$946$	52342.9	1.79896
$947$	10187.3	0.349570	0.174785	−	0.984607i	$-0.444077\pi$
$947$	10187.3	0.349570	0.174785	+	0.984607i	$0.444077\pi$
$948$	6149.77	0.210691
$949$	21241.5	0.726583
$950$	−125.483	−0.00428549
$951$	−4903.31	−0.167193
$952$	−9127.31	−0.310733
$953$	2211.39	0.0751669	0.0375834	−	0.999293i	$-0.488034\pi$
$953$	2211.39	0.0751669	0.0375834	+	0.999293i	$0.488034\pi$
$954$	4294.66	0.145749
$955$	−3170.92	−0.107444
$956$	−31732.3	−1.07353
$957$	47794.0	1.61438
$958$	−20773.0	−0.700569
$959$	−2679.23	−0.0902156
$960$	−5431.02	−0.182589
$961$	−17489.6	−0.587077
$962$	−7136.44	−0.239177
$963$	−2187.78	−0.0732089
$964$	18263.4	0.610193
$965$	−1275.00	−0.0425322
$966$	3385.33	0.112755
$967$	7955.89	0.264575	0.132287	−	0.991211i	$-0.457768\pi$
$967$	7955.89	0.264575	0.132287	+	0.991211i	$0.457768\pi$
$968$	−65679.6	−2.18081
$969$	−464.014	−0.0153832
$970$	−6619.89	−0.219126
$971$	53071.2	1.75400	0.877001	−	0.480488i	$-0.159541\pi$
$971$	53071.2	1.75400	0.877001	+	0.480488i	$0.159541\pi$
$972$	1131.62	0.0373422
$973$	−21375.2	−0.704274
$974$	14498.6	0.476965
$975$	2425.74	0.0796777
$976$	582.813	0.0191141
$977$	−22448.2	−0.735089	−0.367545	−	0.930006i	$-0.619801\pi$
$977$	−22448.2	−0.735089	−0.367545	+	0.930006i	$0.619801\pi$
$978$	5087.80	0.166350
$979$	65781.4	2.14748
$980$	−1140.93	−0.0371894
$981$	−3646.58	−0.118681
$982$	14831.3	0.481961
$983$	21712.8	0.704509	0.352254	−	0.935904i	$-0.385415\pi$
$983$	21712.8	0.704509	0.352254	+	0.935904i	$0.385415\pi$
$984$	−12238.1	−0.396479
$985$	20861.9	0.674837
$986$	−25418.5	−0.820983
$987$	−7249.33	−0.233788
$988$	−413.470	−0.0133140
$989$	−39089.7	−1.25681
$990$	−5312.65	−0.170553
$991$	−37849.2	−1.21324	−0.606620	−	0.794992i	$-0.707475\pi$
$991$	−37849.2	−1.21324	−0.606620	+	0.794992i	$0.707475\pi$
$992$	−19508.0	−0.624373
$993$	−13158.5	−0.420516
$994$	−4559.82	−0.145502
$995$	−23132.4	−0.737031
$996$	−760.168	−0.0241836
$997$	−39573.1	−1.25707	−0.628533	−	0.777783i	$-0.716344\pi$
$997$	−39573.1	−1.25707	−0.628533	+	0.777783i	$0.716344\pi$
$998$	30748.0	0.975261
$999$	−3258.26	−0.103190

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.a.e.1.2	✓	2
3.2	odd	2		315.4.a.k.1.1		2
4.3	odd	2		1680.4.a.bo.1.2		2
5.2	odd	4		525.4.d.l.274.3		4
5.3	odd	4		525.4.d.l.274.2		4
5.4	even	2		525.4.a.l.1.1		2
7.6	odd	2		735.4.a.o.1.2		2
15.14	odd	2		1575.4.a.q.1.2		2
21.20	even	2		2205.4.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.e.1.2	✓	2	1.1	even	1	trivial
315.4.a.k.1.1		2	3.2	odd	2
525.4.a.l.1.1		2	5.4	even	2
525.4.d.l.274.2		4	5.3	odd	4
525.4.d.l.274.3		4	5.2	odd	4
735.4.a.o.1.2		2	7.6	odd	2
1575.4.a.q.1.2		2	15.14	odd	2
1680.4.a.bo.1.2		2	4.3	odd	2
2205.4.a.bb.1.1		2	21.20	even	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:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.a (trivial)

\(f(q)\)	\(=\)	\(q+1.82843 q^{2} -3.00000 q^{3} -4.65685 q^{4} +5.00000 q^{5} -5.48528 q^{6} -7.00000 q^{7} -23.1421 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+1.82843 q^{2} -3.00000 q^{3} -4.65685 q^{4} +5.00000 q^{5} -5.48528 q^{6} -7.00000 q^{7} -23.1421 q^{8} +9.00000 q^{9} +9.14214 q^{10} -64.5685 q^{11} +13.9706 q^{12} -32.3431 q^{13} -12.7990 q^{14} -15.0000 q^{15} -5.05887 q^{16} -56.3431 q^{17} +16.4558 q^{18} -2.74517 q^{19} -23.2843 q^{20} +21.0000 q^{21} -118.059 q^{22} +88.1665 q^{23} +69.4264 q^{24} +25.0000 q^{25} -59.1371 q^{26} -27.0000 q^{27} +32.5980 q^{28} +246.735 q^{29} -27.4264 q^{30} -110.912 q^{31} +175.887 q^{32} +193.706 q^{33} -103.019 q^{34} -35.0000 q^{35} -41.9117 q^{36} +120.676 q^{37} -5.01934 q^{38} +97.0294 q^{39} -115.711 q^{40} -176.274 q^{41} +38.3970 q^{42} -443.362 q^{43} +300.686 q^{44} +45.0000 q^{45} +161.206 q^{46} -345.206 q^{47} +15.1766 q^{48} +49.0000 q^{49} +45.7107 q^{50} +169.029 q^{51} +150.617 q^{52} +260.981 q^{53} -49.3675 q^{54} -322.843 q^{55} +161.995 q^{56} +8.23550 q^{57} +451.137 q^{58} +628.999 q^{59} +69.8528 q^{60} -115.206 q^{61} -202.794 q^{62} -63.0000 q^{63} +362.068 q^{64} -161.716 q^{65} +354.177 q^{66} -951.480 q^{67} +262.382 q^{68} -264.500 q^{69} -63.9949 q^{70} +356.264 q^{71} -208.279 q^{72} -656.754 q^{73} +220.648 q^{74} -75.0000 q^{75} +12.7838 q^{76} +451.980 q^{77} +177.411 q^{78} +440.195 q^{79} -25.2944 q^{80} +81.0000 q^{81} -322.304 q^{82} -54.4121 q^{83} -97.7939 q^{84} -281.716 q^{85} -810.656 q^{86} -740.205 q^{87} +1494.25 q^{88} -1018.78 q^{89} +82.2792 q^{90} +226.402 q^{91} -410.579 q^{92} +332.735 q^{93} -631.184 q^{94} -13.7258 q^{95} -527.662 q^{96} -724.108 q^{97} +89.5929 q^{98} -581.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 6 q^{3} + 2 q^{4} + 10 q^{5} + 6 q^{6} - 14 q^{7} - 18 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 6 * q^3 + 2 * q^4 + 10 * q^5 + 6 * q^6 - 14 * q^7 - 18 * q^8 + 18 * q^9	\( 2 q - 2 q^{2} - 6 q^{3} + 2 q^{4} + 10 q^{5} + 6 q^{6} - 14 q^{7} - 18 q^{8} + 18 q^{9} - 10 q^{10} - 16 q^{11} - 6 q^{12} - 76 q^{13} + 14 q^{14} - 30 q^{15} - 78 q^{16} - 124 q^{17} - 18 q^{18} - 96 q^{19} + 10 q^{20} + 42 q^{21} - 304 q^{22} - 16 q^{23} + 54 q^{24} + 50 q^{25} + 108 q^{26} - 54 q^{27} - 14 q^{28} + 188 q^{29} + 30 q^{30} - 120 q^{31} + 414 q^{32} + 48 q^{33} + 156 q^{34} - 70 q^{35} + 18 q^{36} - 132 q^{37} + 352 q^{38} + 228 q^{39} - 90 q^{40} + 100 q^{41} - 42 q^{42} - 536 q^{43} + 624 q^{44} + 90 q^{45} + 560 q^{46} - 928 q^{47} + 234 q^{48} + 98 q^{49} - 50 q^{50} + 372 q^{51} - 140 q^{52} + 884 q^{53} + 54 q^{54} - 80 q^{55} + 126 q^{56} + 288 q^{57} + 676 q^{58} + 104 q^{59} - 30 q^{60} - 468 q^{61} - 168 q^{62} - 126 q^{63} + 34 q^{64} - 380 q^{65} + 912 q^{66} - 1688 q^{67} - 188 q^{68} + 48 q^{69} + 70 q^{70} - 136 q^{71} - 162 q^{72} + 508 q^{73} + 1188 q^{74} - 150 q^{75} - 608 q^{76} + 112 q^{77} - 324 q^{78} - 432 q^{79} - 390 q^{80} + 162 q^{81} - 1380 q^{82} - 584 q^{83} + 42 q^{84} - 620 q^{85} - 456 q^{86} - 564 q^{87} + 1744 q^{88} - 1404 q^{89} - 90 q^{90} + 532 q^{91} - 1104 q^{92} + 360 q^{93} + 1600 q^{94} - 480 q^{95} - 1242 q^{96} - 1188 q^{97} - 98 q^{98} - 144 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 6 * q^3 + 2 * q^4 + 10 * q^5 + 6 * q^6 - 14 * q^7 - 18 * q^8 + 18 * q^9 - 10 * q^10 - 16 * q^11 - 6 * q^12 - 76 * q^13 + 14 * q^14 - 30 * q^15 - 78 * q^16 - 124 * q^17 - 18 * q^18 - 96 * q^19 + 10 * q^20 + 42 * q^21 - 304 * q^22 - 16 * q^23 + 54 * q^24 + 50 * q^25 + 108 * q^26 - 54 * q^27 - 14 * q^28 + 188 * q^29 + 30 * q^30 - 120 * q^31 + 414 * q^32 + 48 * q^33 + 156 * q^34 - 70 * q^35 + 18 * q^36 - 132 * q^37 + 352 * q^38 + 228 * q^39 - 90 * q^40 + 100 * q^41 - 42 * q^42 - 536 * q^43 + 624 * q^44 + 90 * q^45 + 560 * q^46 - 928 * q^47 + 234 * q^48 + 98 * q^49 - 50 * q^50 + 372 * q^51 - 140 * q^52 + 884 * q^53 + 54 * q^54 - 80 * q^55 + 126 * q^56 + 288 * q^57 + 676 * q^58 + 104 * q^59 - 30 * q^60 - 468 * q^61 - 168 * q^62 - 126 * q^63 + 34 * q^64 - 380 * q^65 + 912 * q^66 - 1688 * q^67 - 188 * q^68 + 48 * q^69 + 70 * q^70 - 136 * q^71 - 162 * q^72 + 508 * q^73 + 1188 * q^74 - 150 * q^75 - 608 * q^76 + 112 * q^77 - 324 * q^78 - 432 * q^79 - 390 * q^80 + 162 * q^81 - 1380 * q^82 - 584 * q^83 + 42 * q^84 - 620 * q^85 - 456 * q^86 - 564 * q^87 + 1744 * q^88 - 1404 * q^89 - 90 * q^90 + 532 * q^91 - 1104 * q^92 + 360 * q^93 + 1600 * q^94 - 480 * q^95 - 1242 * q^96 - 1188 * q^97 - 98 * q^98 - 144 * q^99

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