LMFDB - Embedded newform 177.4.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,4,Mod(1,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("177.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.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:	$10.4433380710$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 45x^{6} + 47x^{5} + 654x^{4} - 157x^{3} - 2898x^{2} + 96x + 2432 $ x^8 - 2x^7 - 45x^6 + 47x^5 + 654x^4 - 157x^3 - 2898x^2 + 96*x + 2432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.26905$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.26905 q^{2} +3.00000 q^{3} -6.38952 q^{4} +14.8820 q^{5} -3.80714 q^{6} +22.4903 q^{7} +18.2610 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.26905 q^{2} +3.00000 q^{3} -6.38952 q^{4} +14.8820 q^{5} -3.80714 q^{6} +22.4903 q^{7} +18.2610 q^{8} +9.00000 q^{9} -18.8859 q^{10} -70.2303 q^{11} -19.1686 q^{12} -0.125351 q^{13} -28.5413 q^{14} +44.6459 q^{15} +27.9422 q^{16} +57.2801 q^{17} -11.4214 q^{18} +40.8191 q^{19} -95.0887 q^{20} +67.4710 q^{21} +89.1255 q^{22} +190.065 q^{23} +54.7829 q^{24} +96.4733 q^{25} +0.159076 q^{26} +27.0000 q^{27} -143.702 q^{28} +133.399 q^{29} -56.6578 q^{30} +129.061 q^{31} -181.548 q^{32} -210.691 q^{33} -72.6911 q^{34} +334.701 q^{35} -57.5057 q^{36} -364.329 q^{37} -51.8013 q^{38} -0.376052 q^{39} +271.759 q^{40} +195.632 q^{41} -85.6238 q^{42} +31.6882 q^{43} +448.738 q^{44} +133.938 q^{45} -241.201 q^{46} +479.330 q^{47} +83.8265 q^{48} +162.815 q^{49} -122.429 q^{50} +171.840 q^{51} +0.800930 q^{52} -402.892 q^{53} -34.2642 q^{54} -1045.17 q^{55} +410.695 q^{56} +122.457 q^{57} -169.290 q^{58} +59.0000 q^{59} -285.266 q^{60} -209.849 q^{61} -163.785 q^{62} +202.413 q^{63} +6.85497 q^{64} -1.86547 q^{65} +267.376 q^{66} -455.024 q^{67} -365.992 q^{68} +570.195 q^{69} -424.750 q^{70} +203.497 q^{71} +164.349 q^{72} -177.712 q^{73} +462.350 q^{74} +289.420 q^{75} -260.815 q^{76} -1579.50 q^{77} +0.477227 q^{78} -491.575 q^{79} +415.835 q^{80} +81.0000 q^{81} -248.266 q^{82} -717.862 q^{83} -431.107 q^{84} +852.441 q^{85} -40.2137 q^{86} +400.197 q^{87} -1282.47 q^{88} +1306.23 q^{89} -169.973 q^{90} -2.81918 q^{91} -1214.42 q^{92} +387.184 q^{93} -608.291 q^{94} +607.469 q^{95} -544.643 q^{96} +538.879 q^{97} -206.620 q^{98} -632.073 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{2} + 24 q^{3} + 34 q^{4} + 42 q^{5} + 18 q^{6} + 53 q^{7} + 51 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q + 6 * q^2 + 24 * q^3 + 34 * q^4 + 42 * q^5 + 18 * q^6 + 53 * q^7 + 51 * q^8 + 72 * q^9	\( 8 q + 6 q^{2} + 24 q^{3} + 34 q^{4} + 42 q^{5} + 18 q^{6} + 53 q^{7} + 51 q^{8} + 72 q^{9} + 21 q^{10} + 67 q^{11} + 102 q^{12} + 33 q^{13} + 79 q^{14} + 126 q^{15} - 30 q^{16} + 139 q^{17} + 54 q^{18} + 64 q^{19} + 117 q^{20} + 159 q^{21} - 84 q^{22} + 226 q^{23} + 153 q^{24} + 96 q^{25} + 24 q^{26} + 216 q^{27} + 34 q^{28} + 456 q^{29} + 63 q^{30} + 124 q^{31} + 174 q^{32} + 201 q^{33} - 114 q^{34} + 556 q^{35} + 306 q^{36} + 127 q^{37} + 237 q^{38} + 99 q^{39} - 188 q^{40} + 425 q^{41} + 237 q^{42} - 115 q^{43} + 510 q^{44} + 378 q^{45} - 711 q^{46} + 420 q^{47} - 90 q^{48} + 171 q^{49} - 137 q^{50} + 417 q^{51} - 922 q^{52} + 98 q^{53} + 162 q^{54} - 616 q^{55} - 412 q^{56} + 192 q^{57} - 1548 q^{58} + 472 q^{59} + 351 q^{60} - 1254 q^{61} - 766 q^{62} + 477 q^{63} - 2019 q^{64} - 734 q^{65} - 252 q^{66} - 1010 q^{67} - 503 q^{68} + 678 q^{69} - 2956 q^{70} - 17 q^{71} + 459 q^{72} - 1180 q^{73} - 1228 q^{74} + 288 q^{75} - 2008 q^{76} + 441 q^{77} + 72 q^{78} - 873 q^{79} - 865 q^{80} + 648 q^{81} - 3645 q^{82} + 759 q^{83} + 102 q^{84} - 850 q^{85} - 1226 q^{86} + 1368 q^{87} - 3047 q^{88} + 988 q^{89} + 189 q^{90} - 2111 q^{91} - 1062 q^{92} + 372 q^{93} - 2240 q^{94} + 1822 q^{95} + 522 q^{96} - 668 q^{97} - 1368 q^{98} + 603 q^{99}+O(q^{100}) \) 8 * q + 6 * q^2 + 24 * q^3 + 34 * q^4 + 42 * q^5 + 18 * q^6 + 53 * q^7 + 51 * q^8 + 72 * q^9 + 21 * q^10 + 67 * q^11 + 102 * q^12 + 33 * q^13 + 79 * q^14 + 126 * q^15 - 30 * q^16 + 139 * q^17 + 54 * q^18 + 64 * q^19 + 117 * q^20 + 159 * q^21 - 84 * q^22 + 226 * q^23 + 153 * q^24 + 96 * q^25 + 24 * q^26 + 216 * q^27 + 34 * q^28 + 456 * q^29 + 63 * q^30 + 124 * q^31 + 174 * q^32 + 201 * q^33 - 114 * q^34 + 556 * q^35 + 306 * q^36 + 127 * q^37 + 237 * q^38 + 99 * q^39 - 188 * q^40 + 425 * q^41 + 237 * q^42 - 115 * q^43 + 510 * q^44 + 378 * q^45 - 711 * q^46 + 420 * q^47 - 90 * q^48 + 171 * q^49 - 137 * q^50 + 417 * q^51 - 922 * q^52 + 98 * q^53 + 162 * q^54 - 616 * q^55 - 412 * q^56 + 192 * q^57 - 1548 * q^58 + 472 * q^59 + 351 * q^60 - 1254 * q^61 - 766 * q^62 + 477 * q^63 - 2019 * q^64 - 734 * q^65 - 252 * q^66 - 1010 * q^67 - 503 * q^68 + 678 * q^69 - 2956 * q^70 - 17 * q^71 + 459 * q^72 - 1180 * q^73 - 1228 * q^74 + 288 * q^75 - 2008 * q^76 + 441 * q^77 + 72 * q^78 - 873 * q^79 - 865 * q^80 + 648 * q^81 - 3645 * q^82 + 759 * q^83 + 102 * q^84 - 850 * q^85 - 1226 * q^86 + 1368 * q^87 - 3047 * q^88 + 988 * q^89 + 189 * q^90 - 2111 * q^91 - 1062 * q^92 + 372 * q^93 - 2240 * q^94 + 1822 * q^95 + 522 * q^96 - 668 * q^97 - 1368 * q^98 + 603 * 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 $

$2$	−1.26905	−0.448676	−0.224338	−	0.974511i	$-0.572022\pi$
$2$	−1.26905	−0.448676	−0.224338	+	0.974511i	$0.572022\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−6.38952	−0.798690
$5$	14.8820	1.33108	0.665542	−	0.746360i	$-0.268200\pi$
$5$	14.8820	1.33108	0.665542	+	0.746360i	$0.268200\pi$
$6$	−3.80714	−0.259043
$7$	22.4903	1.21436	0.607182	−	0.794563i	$-0.292300\pi$
$7$	22.4903	1.21436	0.607182	+	0.794563i	$0.292300\pi$
$8$	18.2610	0.807028
$9$	9.00000	0.333333
$10$	−18.8859	−0.597225
$11$	−70.2303	−1.92502	−0.962510	−	0.271244i	$-0.912565\pi$
$11$	−70.2303	−1.92502	−0.962510	+	0.271244i	$0.912565\pi$
$12$	−19.1686	−0.461124
$13$	−0.125351	−0.00267431	−0.00133715	−	0.999999i	$-0.500426\pi$
$13$	−0.125351	−0.00267431	−0.00133715	+	0.999999i	$0.500426\pi$
$14$	−28.5413	−0.544855
$15$	44.6459	0.768502
$16$	27.9422	0.436596
$17$	57.2801	0.817203	0.408602	−	0.912713i	$-0.366017\pi$
$17$	57.2801	0.817203	0.408602	+	0.912713i	$0.366017\pi$
$18$	−11.4214	−0.149559
$19$	40.8191	0.492871	0.246435	−	0.969159i	$-0.420741\pi$
$19$	40.8191	0.492871	0.246435	+	0.969159i	$0.420741\pi$
$20$	−95.0887	−1.06312
$21$	67.4710	0.701113
$22$	89.1255	0.863710
$23$	190.065	1.72310	0.861550	−	0.507673i	$-0.169494\pi$
$23$	190.065	1.72310	0.861550	+	0.507673i	$0.169494\pi$
$24$	54.7829	0.465938
$25$	96.4733	0.771787
$26$	0.159076	0.00119990
$27$	27.0000	0.192450
$28$	−143.702	−0.969900
$29$	133.399	0.854193	0.427096	−	0.904206i	$-0.359537\pi$
$29$	133.399	0.854193	0.427096	+	0.904206i	$0.359537\pi$
$30$	−56.6578	−0.344808
$31$	129.061	0.747745	0.373872	−	0.927480i	$-0.378030\pi$
$31$	129.061	0.747745	0.373872	+	0.927480i	$0.378030\pi$
$32$	−181.548	−1.00292
$33$	−210.691	−1.11141
$34$	−72.6911	−0.366659
$35$	334.701	1.61642
$36$	−57.5057	−0.266230
$37$	−364.329	−1.61879	−0.809396	−	0.587263i	$-0.800205\pi$
$37$	−364.329	−1.61879	−0.809396	+	0.587263i	$0.800205\pi$
$38$	−51.8013	−0.221139
$39$	−0.376052	−0.00154401
$40$	271.759	1.07422
$41$	195.632	0.745184	0.372592	−	0.927995i	$-0.378469\pi$
$41$	195.632	0.745184	0.372592	+	0.927995i	$0.378469\pi$
$42$	−85.6238	−0.314572
$43$	31.6882	0.112381	0.0561907	−	0.998420i	$-0.482105\pi$
$43$	31.6882	0.112381	0.0561907	+	0.998420i	$0.482105\pi$
$44$	448.738	1.53750
$45$	133.938	0.443695
$46$	−241.201	−0.773113
$47$	479.330	1.48760	0.743802	−	0.668400i	$-0.233020\pi$
$47$	479.330	1.48760	0.743802	+	0.668400i	$0.233020\pi$
$48$	83.8265	0.252069
$49$	162.815	0.474679
$50$	−122.429	−0.346282
$51$	171.840	0.471813
$52$	0.800930	0.00213594
$53$	−402.892	−1.04418	−0.522090	−	0.852891i	$-0.674847\pi$
$53$	−402.892	−1.04418	−0.522090	+	0.852891i	$0.674847\pi$
$54$	−34.2642	−0.0863476
$55$	−1045.17	−2.56237
$56$	410.695	0.980026
$57$	122.457	0.284559
$58$	−169.290	−0.383255
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	−285.266	−0.613795
$61$	−209.849	−0.440465	−0.220232	−	0.975447i	$-0.570682\pi$
$61$	−209.849	−0.440465	−0.220232	+	0.975447i	$0.570682\pi$
$62$	−163.785	−0.335495
$63$	202.413	0.404788
$64$	6.85497	0.0133886
$65$	−1.86547	−0.00355973
$66$	267.376	0.498663
$67$	−455.024	−0.829702	−0.414851	−	0.909889i	$-0.636166\pi$
$67$	−455.024	−0.829702	−0.414851	+	0.909889i	$0.636166\pi$
$68$	−365.992	−0.652692
$69$	570.195	0.994832
$70$	−424.750	−0.725248
$71$	203.497	0.340151	0.170075	−	0.985431i	$-0.445599\pi$
$71$	203.497	0.340151	0.170075	+	0.985431i	$0.445599\pi$
$72$	164.349	0.269009
$73$	−177.712	−0.284927	−0.142463	−	0.989800i	$-0.545502\pi$
$73$	−177.712	−0.284927	−0.142463	+	0.989800i	$0.545502\pi$
$74$	462.350	0.726313
$75$	289.420	0.445591
$76$	−260.815	−0.393651
$77$	−1579.50	−2.33768
$78$	0.477227	0.000692761 0
$79$	−491.575	−0.700082	−0.350041	−	0.936734i	$-0.613832\pi$
$79$	−491.575	−0.700082	−0.350041	+	0.936734i	$0.613832\pi$
$80$	415.835	0.581147
$81$	81.0000	0.111111
$82$	−248.266	−0.334346
$83$	−717.862	−0.949345	−0.474673	−	0.880162i	$-0.657433\pi$
$83$	−717.862	−0.949345	−0.474673	+	0.880162i	$0.657433\pi$
$84$	−431.107	−0.559972
$85$	852.441	1.08777
$86$	−40.2137	−0.0504228
$87$	400.197	0.493168
$88$	−1282.47	−1.55355
$89$	1306.23	1.55573	0.777865	−	0.628431i	$-0.216303\pi$
$89$	1306.23	1.55573	0.777865	+	0.628431i	$0.216303\pi$
$90$	−169.973	−0.199075
$91$	−2.81918	−0.00324758
$92$	−1214.42	−1.37622
$93$	387.184	0.431711
$94$	−608.291	−0.667452
$95$	607.469	0.656053
$96$	−544.643	−0.579035
$97$	538.879	0.564071	0.282036	−	0.959404i	$-0.408990\pi$
$97$	538.879	0.564071	0.282036	+	0.959404i	$0.408990\pi$
$98$	−206.620	−0.212977
$99$	−632.073	−0.641674
$100$	−616.419	−0.616419
$101$	−1585.19	−1.56171	−0.780854	−	0.624714i	$-0.785216\pi$
$101$	−1585.19	−1.56171	−0.780854	+	0.624714i	$0.785216\pi$
$102$	−218.073	−0.211691
$103$	−512.174	−0.489961	−0.244980	−	0.969528i	$-0.578782\pi$
$103$	−512.174	−0.489961	−0.244980	+	0.969528i	$0.578782\pi$
$104$	−2.28902	−0.00215824
$105$	1004.10	0.933241
$106$	511.289	0.468498
$107$	90.3234	0.0816065	0.0408033	−	0.999167i	$-0.487008\pi$
$107$	90.3234	0.0816065	0.0408033	+	0.999167i	$0.487008\pi$
$108$	−172.517	−0.153708
$109$	−2241.55	−1.96974	−0.984870	−	0.173297i	$-0.944558\pi$
$109$	−2241.55	−1.96974	−0.984870	+	0.173297i	$0.944558\pi$
$110$	1326.36	1.14967
$111$	−1092.99	−0.934610
$112$	628.428	0.530187
$113$	−1538.31	−1.28064	−0.640319	−	0.768109i	$-0.721198\pi$
$113$	−1538.31	−1.28064	−0.640319	+	0.768109i	$0.721198\pi$
$114$	−155.404	−0.127675
$115$	2828.54	2.29359
$116$	−852.356	−0.682235
$117$	−1.12816	−0.000891436 0
$118$	−74.8737	−0.0584126
$119$	1288.25	0.992382
$120$	815.278	0.620203
$121$	3601.29	2.70571
$122$	266.308	0.197626
$123$	586.896	0.430232
$124$	−824.640	−0.597216
$125$	−424.533	−0.303771
$126$	−256.871	−0.181618
$127$	1230.53	0.859780	0.429890	−	0.902881i	$-0.358552\pi$
$127$	1230.53	0.859780	0.429890	+	0.902881i	$0.358552\pi$
$128$	1443.68	0.996911
$129$	95.0645	0.0648834
$130$	2.36736	0.00159716
$131$	972.455	0.648578	0.324289	−	0.945958i	$-0.394875\pi$
$131$	972.455	0.648578	0.324289	+	0.945958i	$0.394875\pi$
$132$	1346.21	0.887673
$133$	918.035	0.598525
$134$	577.447	0.372267
$135$	401.813	0.256167
$136$	1045.99	0.659506
$137$	−335.709	−0.209355	−0.104677	−	0.994506i	$-0.533381\pi$
$137$	−335.709	−0.209355	−0.104677	+	0.994506i	$0.533381\pi$
$138$	−723.604	−0.446357
$139$	−2222.42	−1.35614	−0.678068	−	0.734999i	$-0.737183\pi$
$139$	−2222.42	−1.35614	−0.678068	+	0.734999i	$0.737183\pi$
$140$	−2138.58	−1.29102
$141$	1437.99	0.858869
$142$	−258.248	−0.152617
$143$	8.80341	0.00514810
$144$	251.480	0.145532
$145$	1985.24	1.13700
$146$	225.525	0.127840
$147$	488.444	0.274056
$148$	2327.89	1.29291
$149$	1467.58	0.806906	0.403453	−	0.915000i	$-0.367810\pi$
$149$	1467.58	0.806906	0.403453	+	0.915000i	$0.367810\pi$
$150$	−367.287	−0.199926
$151$	−2400.31	−1.29361	−0.646804	−	0.762656i	$-0.723895\pi$
$151$	−2400.31	−1.29361	−0.646804	+	0.762656i	$0.723895\pi$
$152$	745.397	0.397761
$153$	515.521	0.272401
$154$	2004.46	1.04886
$155$	1920.69	0.995312
$156$	2.40279	0.00123319
$157$	−3571.25	−1.81539	−0.907697	−	0.419627i	$-0.862161\pi$
$157$	−3571.25	−1.81539	−0.907697	+	0.419627i	$0.862161\pi$
$158$	623.831	0.314110
$159$	−1208.68	−0.602857
$160$	−2701.79	−1.33497
$161$	4274.62	2.09247
$162$	−102.793	−0.0498528
$163$	−1305.57	−0.627365	−0.313683	−	0.949528i	$-0.601563\pi$
$163$	−1305.57	−0.627365	−0.313683	+	0.949528i	$0.601563\pi$
$164$	−1249.99	−0.595172
$165$	−3135.50	−1.47938
$166$	911.000	0.425948
$167$	156.555	0.0725423	0.0362711	−	0.999342i	$-0.488452\pi$
$167$	156.555	0.0725423	0.0362711	+	0.999342i	$0.488452\pi$
$168$	1232.09	0.565818
$169$	−2196.98	−0.999993
$170$	−1081.79	−0.488054
$171$	367.372	0.164290
$172$	−202.472	−0.0897579
$173$	357.266	0.157008	0.0785042	−	0.996914i	$-0.474986\pi$
$173$	357.266	0.157008	0.0785042	+	0.996914i	$0.474986\pi$
$174$	−507.869	−0.221273
$175$	2169.72	0.937230
$176$	−1962.39	−0.840457
$177$	177.000	0.0751646
$178$	−1657.66	−0.698018
$179$	1581.04	0.660183	0.330092	−	0.943949i	$-0.392920\pi$
$179$	1581.04	0.660183	0.330092	+	0.943949i	$0.392920\pi$
$180$	−855.799	−0.354375
$181$	1674.78	0.687766	0.343883	−	0.939012i	$-0.388258\pi$
$181$	1674.78	0.687766	0.343883	+	0.939012i	$0.388258\pi$
$182$	3.57766	0.00145711
$183$	−629.546	−0.254303
$184$	3470.77	1.39059
$185$	−5421.94	−2.15475
$186$	−491.354	−0.193698
$187$	−4022.80	−1.57313
$188$	−3062.69	−1.18814
$189$	607.239	0.233704
$190$	−770.907	−0.294355
$191$	5015.44	1.90002	0.950011	−	0.312217i	$-0.101071\pi$
$191$	5015.44	1.90002	0.950011	+	0.312217i	$0.101071\pi$
$192$	20.5649	0.00772992
$193$	1296.37	0.483496	0.241748	−	0.970339i	$-0.422279\pi$
$193$	1296.37	0.483496	0.241748	+	0.970339i	$0.422279\pi$
$194$	−683.863	−0.253085
$195$	−5.59640	−0.00205521
$196$	−1040.31	−0.379121
$197$	−2894.70	−1.04690	−0.523449	−	0.852057i	$-0.675355\pi$
$197$	−2894.70	−1.04690	−0.523449	+	0.852057i	$0.675355\pi$
$198$	802.129	0.287903
$199$	−2835.15	−1.00994	−0.504971	−	0.863136i	$-0.668497\pi$
$199$	−2835.15	−1.00994	−0.504971	+	0.863136i	$0.668497\pi$
$200$	1761.70	0.622854
$201$	−1365.07	−0.479029
$202$	2011.68	0.700700
$203$	3000.19	1.03730
$204$	−1097.98	−0.376832
$205$	2911.39	0.991904
$206$	649.972	0.219833
$207$	1710.58	0.574366
$208$	−3.50257	−0.00116759
$209$	−2866.74	−0.948787
$210$	−1274.25	−0.418722
$211$	894.249	0.291766	0.145883	−	0.989302i	$-0.453398\pi$
$211$	894.249	0.291766	0.145883	+	0.989302i	$0.453398\pi$
$212$	2574.29	0.833976
$213$	610.492	0.196386
$214$	−114.625	−0.0366148
$215$	471.583	0.149589
$216$	493.046	0.155313
$217$	2902.63	0.908034
$218$	2844.63	0.883774
$219$	−533.137	−0.164503
$220$	6678.11	2.04654
$221$	−7.18009	−0.00218545
$222$	1387.05	0.419337
$223$	2418.78	0.726338	0.363169	−	0.931723i	$-0.381695\pi$
$223$	2418.78	0.726338	0.363169	+	0.931723i	$0.381695\pi$
$224$	−4083.07	−1.21791
$225$	868.260	0.257262
$226$	1952.19	0.574591
$227$	−1116.89	−0.326565	−0.163283	−	0.986579i	$-0.552208\pi$
$227$	−1116.89	−0.326565	−0.163283	+	0.986579i	$0.552208\pi$
$228$	−782.444	−0.227275
$229$	−4003.70	−1.15534	−0.577668	−	0.816272i	$-0.696037\pi$
$229$	−4003.70	−1.15534	−0.577668	+	0.816272i	$0.696037\pi$
$230$	−3589.55	−1.02908
$231$	−4738.51	−1.34966
$232$	2436.00	0.689358
$233$	−3831.54	−1.07731	−0.538654	−	0.842527i	$-0.681067\pi$
$233$	−3831.54	−1.07731	−0.538654	+	0.842527i	$0.681067\pi$
$234$	1.43168	0.000399966 0
$235$	7133.37	1.98013
$236$	−376.982	−0.103981
$237$	−1474.72	−0.404193
$238$	−1634.85	−0.445258
$239$	−665.336	−0.180071	−0.0900356	−	0.995939i	$-0.528698\pi$
$239$	−665.336	−0.180071	−0.0900356	+	0.995939i	$0.528698\pi$
$240$	1247.50	0.335525
$241$	6021.52	1.60946	0.804730	−	0.593640i	$-0.202310\pi$
$241$	6021.52	1.60946	0.804730	+	0.593640i	$0.202310\pi$
$242$	−4570.21	−1.21398
$243$	243.000	0.0641500
$244$	1340.83	0.351795
$245$	2423.01	0.631838
$246$	−744.797	−0.193035
$247$	−5.11670	−0.00131809
$248$	2356.78	0.603451
$249$	−2153.59	−0.548105
$250$	538.752	0.136295
$251$	−1854.47	−0.466347	−0.233173	−	0.972435i	$-0.574911\pi$
$251$	−1854.47	−0.466347	−0.233173	+	0.972435i	$0.574911\pi$
$252$	−1293.32	−0.323300
$253$	−13348.3	−3.31700
$254$	−1561.60	−0.385762
$255$	2557.32	0.628023
$256$	−1886.94	−0.460678
$257$	−3407.16	−0.826976	−0.413488	−	0.910510i	$-0.635690\pi$
$257$	−3407.16	−0.826976	−0.413488	+	0.910510i	$0.635690\pi$
$258$	−120.641	−0.0291116
$259$	−8193.88	−1.96580
$260$	11.9194	0.00284312
$261$	1200.59	0.284731
$262$	−1234.09	−0.291001
$263$	−2085.60	−0.488986	−0.244493	−	0.969651i	$-0.578622\pi$
$263$	−2085.60	−0.488986	−0.244493	+	0.969651i	$0.578622\pi$
$264$	−3847.42	−0.896940
$265$	−5995.83	−1.38989
$266$	−1165.03	−0.268543
$267$	3918.69	0.898201
$268$	2907.39	0.662675
$269$	−6495.06	−1.47216	−0.736080	−	0.676895i	$-0.763325\pi$
$269$	−6495.06	−1.47216	−0.736080	+	0.676895i	$0.763325\pi$
$270$	−509.920	−0.114936
$271$	−3252.61	−0.729085	−0.364543	−	0.931187i	$-0.618775\pi$
$271$	−3252.61	−0.729085	−0.364543	+	0.931187i	$0.618775\pi$
$272$	1600.53	0.356788
$273$	−8.45753	−0.00187499
$274$	426.031	0.0939323
$275$	−6775.35	−1.48571
$276$	−3643.27	−0.794562
$277$	5905.21	1.28090	0.640451	−	0.767999i	$-0.278747\pi$
$277$	5905.21	1.28090	0.640451	+	0.767999i	$0.278747\pi$
$278$	2820.35	0.608465
$279$	1161.55	0.249248
$280$	6111.96	1.30450
$281$	6504.05	1.38078	0.690390	−	0.723437i	$-0.257439\pi$
$281$	6504.05	1.38078	0.690390	+	0.723437i	$0.257439\pi$
$282$	−1824.87	−0.385353
$283$	6587.83	1.38377	0.691883	−	0.722010i	$-0.256782\pi$
$283$	6587.83	1.38377	0.691883	+	0.722010i	$0.256782\pi$
$284$	−1300.25	−0.271675
$285$	1822.41	0.378772
$286$	−11.1719	−0.00230983
$287$	4399.82	0.904925
$288$	−1633.93	−0.334306
$289$	−1631.99	−0.332179
$290$	−2519.36	−0.510145
$291$	1616.64	0.325667
$292$	1135.50	0.227568
$293$	4655.68	0.928285	0.464142	−	0.885761i	$-0.346363\pi$
$293$	4655.68	0.928285	0.464142	+	0.885761i	$0.346363\pi$
$294$	−619.859	−0.122962
$295$	878.037	0.173292
$296$	−6653.00	−1.30641
$297$	−1896.22	−0.370470
$298$	−1862.43	−0.362039
$299$	−23.8248	−0.00460810
$300$	−1849.26	−0.355889
$301$	712.677	0.136472
$302$	3046.11	0.580410
$303$	−4755.58	−0.901652
$304$	1140.57	0.215186
$305$	−3122.96	−0.586296
$306$	−654.219	−0.122220
$307$	−1381.12	−0.256758	−0.128379	−	0.991725i	$-0.540977\pi$
$307$	−1381.12	−0.256758	−0.128379	+	0.991725i	$0.540977\pi$
$308$	10092.3	1.86708
$309$	−1536.52	−0.282879
$310$	−2437.44	−0.446572
$311$	2269.90	0.413873	0.206936	−	0.978354i	$-0.433651\pi$
$311$	2269.90	0.413873	0.206936	+	0.978354i	$0.433651\pi$
$312$	−6.86707	−0.00124606
$313$	−487.625	−0.0880581	−0.0440290	−	0.999030i	$-0.514019\pi$
$313$	−487.625	−0.0880581	−0.0440290	+	0.999030i	$0.514019\pi$
$314$	4532.08	0.814523
$315$	3012.31	0.538807
$316$	3140.93	0.559149
$317$	8517.00	1.50903	0.754515	−	0.656283i	$-0.227872\pi$
$317$	8517.00	1.50903	0.754515	+	0.656283i	$0.227872\pi$
$318$	1533.87	0.270487
$319$	−9368.66	−1.64434
$320$	102.015	0.0178214
$321$	270.970	0.0471155
$322$	−5424.69	−0.938840
$323$	2338.12	0.402776
$324$	−517.551	−0.0887434
$325$	−12.0930	−0.00206400
$326$	1656.83	0.281483
$327$	−6724.65	−1.13723
$328$	3572.43	0.601385
$329$	10780.3	1.80649
$330$	3979.09	0.663763
$331$	10338.9	1.71685	0.858423	−	0.512942i	$-0.171444\pi$
$331$	10338.9	1.71685	0.858423	+	0.512942i	$0.171444\pi$
$332$	4586.80	0.758233
$333$	−3278.96	−0.539597
$334$	−198.675	−0.0325480
$335$	−6771.66	−1.10440
$336$	1885.29	0.306103
$337$	−344.378	−0.0556660	−0.0278330	−	0.999613i	$-0.508861\pi$
$337$	−344.378	−0.0556660	−0.0278330	+	0.999613i	$0.508861\pi$
$338$	2788.07	0.448672
$339$	−4614.94	−0.739377
$340$	−5446.69	−0.868789
$341$	−9064.01	−1.43942
$342$	−466.212	−0.0737131
$343$	−4052.42	−0.637931
$344$	578.656	0.0906949
$345$	8485.63	1.32421
$346$	−453.387	−0.0704458
$347$	4032.71	0.623883	0.311942	−	0.950101i	$-0.399021\pi$
$347$	4032.71	0.623883	0.311942	+	0.950101i	$0.399021\pi$
$348$	−2557.07	−0.393889
$349$	6299.32	0.966175	0.483087	−	0.875572i	$-0.339515\pi$
$349$	6299.32	0.966175	0.483087	+	0.875572i	$0.339515\pi$
$350$	−2753.47	−0.420512
$351$	−3.38447	−0.000514671 0
$352$	12750.1	1.93064
$353$	−8240.96	−1.24256	−0.621278	−	0.783590i	$-0.713386\pi$
$353$	−8240.96	−1.24256	−0.621278	+	0.783590i	$0.713386\pi$
$354$	−224.621	−0.0337245
$355$	3028.44	0.452769
$356$	−8346.18	−1.24255
$357$	3864.74	0.572952
$358$	−2006.42	−0.296208
$359$	−6605.66	−0.971124	−0.485562	−	0.874202i	$-0.661385\pi$
$359$	−6605.66	−0.971124	−0.485562	+	0.874202i	$0.661385\pi$
$360$	2445.83	0.358074
$361$	−5192.80	−0.757078
$362$	−2125.38	−0.308584
$363$	10803.9	1.56214
$364$	18.0132	0.00259381
$365$	−2644.71	−0.379262
$366$	798.923	0.114099
$367$	10563.8	1.50253	0.751263	−	0.660003i	$-0.229445\pi$
$367$	10563.8	1.50253	0.751263	+	0.660003i	$0.229445\pi$
$368$	5310.83	0.752299
$369$	1760.69	0.248395
$370$	6880.69	0.966784
$371$	−9061.18	−1.26801
$372$	−2473.92	−0.344803
$373$	−9588.55	−1.33104	−0.665518	−	0.746382i	$-0.731789\pi$
$373$	−9588.55	−1.33104	−0.665518	+	0.746382i	$0.731789\pi$
$374$	5105.11	0.705827
$375$	−1273.60	−0.175382
$376$	8753.02	1.20054
$377$	−16.7217	−0.00228437
$378$	−770.614	−0.104857
$379$	8045.04	1.09036	0.545180	−	0.838319i	$-0.316461\pi$
$379$	8045.04	1.09036	0.545180	+	0.838319i	$0.316461\pi$
$380$	−3881.44	−0.523983
$381$	3691.59	0.496394
$382$	−6364.82	−0.852493
$383$	−13181.7	−1.75863	−0.879314	−	0.476242i	$-0.841999\pi$
$383$	−13181.7	−1.75863	−0.879314	+	0.476242i	$0.841999\pi$
$384$	4331.05	0.575567
$385$	−23506.1	−3.11164
$386$	−1645.15	−0.216933
$387$	285.193	0.0374605
$388$	−3443.18	−0.450518
$389$	1629.56	0.212396	0.106198	−	0.994345i	$-0.466132\pi$
$389$	1629.56	0.212396	0.106198	+	0.994345i	$0.466132\pi$
$390$	7.10208	0.000922123 0
$391$	10886.9	1.40812
$392$	2973.16	0.383079
$393$	2917.36	0.374457
$394$	3673.51	0.469718
$395$	−7315.61	−0.931869
$396$	4038.64	0.512498
$397$	2405.86	0.304148	0.152074	−	0.988369i	$-0.451405\pi$
$397$	2405.86	0.304148	0.152074	+	0.988369i	$0.451405\pi$
$398$	3597.94	0.453136
$399$	2754.11	0.345558
$400$	2695.67	0.336959
$401$	4383.33	0.545868	0.272934	−	0.962033i	$-0.412006\pi$
$401$	4383.33	0.545868	0.272934	+	0.962033i	$0.412006\pi$
$402$	1732.34	0.214928
$403$	−16.1779	−0.00199970
$404$	10128.6	1.24732
$405$	1205.44	0.147898
$406$	−3807.38	−0.465411
$407$	25586.9	3.11621
$408$	3137.97	0.380766
$409$	−8567.44	−1.03578	−0.517888	−	0.855448i	$-0.673282\pi$
$409$	−8567.44	−1.03578	−0.517888	+	0.855448i	$0.673282\pi$
$410$	−3694.69	−0.445043
$411$	−1007.13	−0.120871
$412$	3272.54	0.391327
$413$	1326.93	0.158097
$414$	−2170.81	−0.257704
$415$	−10683.2	−1.26366
$416$	22.7571	0.00268211
$417$	−6667.25	−0.782966
$418$	3638.02	0.425698
$419$	7908.61	0.922102	0.461051	−	0.887374i	$-0.347472\pi$
$419$	7908.61	0.922102	0.461051	+	0.887374i	$0.347472\pi$
$420$	−6415.73	−0.745370
$421$	4262.15	0.493407	0.246703	−	0.969091i	$-0.420653\pi$
$421$	4262.15	0.493407	0.246703	+	0.969091i	$0.420653\pi$
$422$	−1134.84	−0.130908
$423$	4313.97	0.495868
$424$	−7357.20	−0.842682
$425$	5526.00	0.630707
$426$	−774.743	−0.0881136
$427$	−4719.56	−0.534885
$428$	−577.124	−0.0651783
$429$	26.4102	0.00297226
$430$	−598.460	−0.0671170
$431$	12851.4	1.43627	0.718134	−	0.695905i	$-0.244997\pi$
$431$	12851.4	1.43627	0.718134	+	0.695905i	$0.244997\pi$
$432$	754.439	0.0840230
$433$	−10952.4	−1.21556	−0.607782	−	0.794104i	$-0.707940\pi$
$433$	−10952.4	−1.21556	−0.607782	+	0.794104i	$0.707940\pi$
$434$	−3683.57	−0.407413
$435$	5955.73	0.656449
$436$	14322.4	1.57321
$437$	7758.28	0.849266
$438$	676.576	0.0738083
$439$	−12153.2	−1.32128	−0.660638	−	0.750705i	$-0.729714\pi$
$439$	−12153.2	−1.32128	−0.660638	+	0.750705i	$0.729714\pi$
$440$	−19085.7	−2.06790
$441$	1465.33	0.158226
$442$	9.11187	0.000980560 0
$443$	11624.3	1.24670	0.623348	−	0.781944i	$-0.285772\pi$
$443$	11624.3	1.24670	0.623348	+	0.781944i	$0.285772\pi$
$444$	6983.66	0.746464
$445$	19439.3	2.07081
$446$	−3069.54	−0.325890
$447$	4402.75	0.465867
$448$	154.170	0.0162586
$449$	−9152.69	−0.962009	−0.481005	−	0.876718i	$-0.659728\pi$
$449$	−9152.69	−0.962009	−0.481005	+	0.876718i	$0.659728\pi$
$450$	−1101.86	−0.115427
$451$	−13739.3	−1.43450
$452$	9829.08	1.02283
$453$	−7200.94	−0.746865
$454$	1417.38	0.146522
$455$	−41.9549	−0.00432281
$456$	2236.19	0.229647
$457$	−1864.25	−0.190822	−0.0954112	−	0.995438i	$-0.530417\pi$
$457$	−1864.25	−0.190822	−0.0954112	+	0.995438i	$0.530417\pi$
$458$	5080.88	0.518371
$459$	1546.56	0.157271
$460$	−18073.0	−1.83187
$461$	−19215.7	−1.94136	−0.970679	−	0.240377i	$-0.922729\pi$
$461$	−19215.7	−1.94136	−0.970679	+	0.240377i	$0.922729\pi$
$462$	6013.38	0.605558
$463$	7787.00	0.781626	0.390813	−	0.920470i	$-0.372194\pi$
$463$	7787.00	0.781626	0.390813	+	0.920470i	$0.372194\pi$
$464$	3727.46	0.372937
$465$	5762.06	0.574643
$466$	4862.41	0.483362
$467$	−9537.66	−0.945076	−0.472538	−	0.881310i	$-0.656662\pi$
$467$	−9537.66	−0.945076	−0.472538	+	0.881310i	$0.656662\pi$
$468$	7.20837	0.000711981 0
$469$	−10233.6	−1.00756
$470$	−9052.58	−0.888435
$471$	−10713.7	−1.04812
$472$	1077.40	0.105066
$473$	−2225.47	−0.216336
$474$	1871.49	0.181351
$475$	3937.96	0.380391
$476$	−8231.29	−0.792606
$477$	−3626.03	−0.348060
$478$	844.343	0.0807936
$479$	−2560.30	−0.244223	−0.122112	−	0.992516i	$-0.538967\pi$
$479$	−2560.30	−0.244223	−0.122112	+	0.992516i	$0.538967\pi$
$480$	−8105.36	−0.770745
$481$	45.6689	0.00432915
$482$	−7641.58	−0.722126
$483$	12823.9	1.20809
$484$	−23010.5	−2.16102
$485$	8019.59	0.750827
$486$	−308.378	−0.0287825
$487$	4855.22	0.451768	0.225884	−	0.974154i	$-0.427473\pi$
$487$	4855.22	0.451768	0.225884	+	0.974154i	$0.427473\pi$
$488$	−3832.04	−0.355468
$489$	−3916.72	−0.362209
$490$	−3074.91	−0.283490
$491$	18773.3	1.72551	0.862755	−	0.505622i	$-0.168737\pi$
$491$	18773.3	1.72551	0.862755	+	0.505622i	$0.168737\pi$
$492$	−3749.98	−0.343622
$493$	7641.11	0.698049
$494$	6.49333	0.000591394 0
$495$	−9406.49	−0.854122
$496$	3606.25	0.326463
$497$	4576.72	0.413067
$498$	2733.00	0.245921
$499$	−8138.08	−0.730081	−0.365040	−	0.930992i	$-0.618945\pi$
$499$	−8138.08	−0.730081	−0.365040	+	0.930992i	$0.618945\pi$
$500$	2712.56	0.242619
$501$	469.664	0.0418823
$502$	2353.41	0.209238
$503$	7732.32	0.685422	0.342711	−	0.939441i	$-0.388655\pi$
$503$	7732.32	0.685422	0.342711	+	0.939441i	$0.388655\pi$
$504$	3696.26	0.326675
$505$	−23590.8	−2.07877
$506$	16939.6	1.48826
$507$	−6590.95	−0.577346
$508$	−7862.51	−0.686698
$509$	1365.77	0.118933	0.0594665	−	0.998230i	$-0.481060\pi$
$509$	1365.77	0.118933	0.0594665	+	0.998230i	$0.481060\pi$
$510$	−3245.36	−0.281778
$511$	−3996.81	−0.346005
$512$	−9154.84	−0.790216
$513$	1102.12	0.0948531
$514$	4323.84	0.371044
$515$	−7622.16	−0.652179
$516$	−607.417	−0.0518217
$517$	−33663.5	−2.86367
$518$	10398.4	0.882008
$519$	1071.80	0.0906488
$520$	−34.0652	−0.00287280
$521$	−5123.71	−0.430852	−0.215426	−	0.976520i	$-0.569114\pi$
$521$	−5123.71	−0.430852	−0.215426	+	0.976520i	$0.569114\pi$
$522$	−1523.61	−0.127752
$523$	20522.1	1.71581	0.857907	−	0.513805i	$-0.171765\pi$
$523$	20522.1	1.71581	0.857907	+	0.513805i	$0.171765\pi$
$524$	−6213.52	−0.518013
$525$	6509.15	0.541110
$526$	2646.72	0.219396
$527$	7392.64	0.611060
$528$	−5887.16	−0.485238
$529$	23957.7	1.96907
$530$	7608.99	0.623610
$531$	531.000	0.0433963
$532$	−5865.81	−0.478036
$533$	−24.5226	−0.00199285
$534$	−4972.99	−0.403001
$535$	1344.19	0.108625
$536$	−8309.18	−0.669593
$537$	4743.13	0.381157
$538$	8242.53	0.660522
$539$	−11434.5	−0.913767
$540$	−2567.40	−0.204598
$541$	−17106.3	−1.35944	−0.679720	−	0.733471i	$-0.737899\pi$
$541$	−17106.3	−1.35944	−0.679720	+	0.733471i	$0.737899\pi$
$542$	4127.71	0.327123
$543$	5024.35	0.397082
$544$	−10399.1	−0.819588
$545$	−33358.7	−2.62189
$546$	10.7330	0.000841263 0
$547$	4023.62	0.314511	0.157255	−	0.987558i	$-0.449735\pi$
$547$	4023.62	0.314511	0.157255	+	0.987558i	$0.449735\pi$
$548$	2145.02	0.167209
$549$	−1888.64	−0.146822
$550$	8598.23	0.666600
$551$	5445.23	0.421007
$552$	10412.3	0.802857
$553$	−11055.7	−0.850154
$554$	−7493.99	−0.574709
$555$	−16265.8	−1.24405
$556$	14200.2	1.08313
$557$	18019.0	1.37072	0.685360	−	0.728204i	$-0.259645\pi$
$557$	18019.0	1.37072	0.685360	+	0.728204i	$0.259645\pi$
$558$	−1474.06	−0.111832
$559$	−3.97213	−0.000300542 0
$560$	9352.26	0.705723
$561$	−12068.4	−0.908249
$562$	−8253.94	−0.619522
$563$	25706.0	1.92430	0.962148	−	0.272526i	$-0.0878591\pi$
$563$	25706.0	1.92430	0.962148	+	0.272526i	$0.0878591\pi$
$564$	−9188.06	−0.685970
$565$	−22893.1	−1.70464
$566$	−8360.25	−0.620862
$567$	1821.72	0.134929
$568$	3716.06	0.274511
$569$	−3090.70	−0.227713	−0.113857	−	0.993497i	$-0.536320\pi$
$569$	−3090.70	−0.227713	−0.113857	+	0.993497i	$0.536320\pi$
$570$	−2312.72	−0.169946
$571$	7296.29	0.534746	0.267373	−	0.963593i	$-0.413844\pi$
$571$	7296.29	0.534746	0.267373	+	0.963593i	$0.413844\pi$
$572$	−56.2496	−0.00411174
$573$	15046.3	1.09698
$574$	−5583.58	−0.406018
$575$	18336.2	1.32987
$576$	61.6947	0.00446287
$577$	−1681.44	−0.121316	−0.0606579	−	0.998159i	$-0.519320\pi$
$577$	−1681.44	−0.121316	−0.0606579	+	0.998159i	$0.519320\pi$
$578$	2071.07	0.149040
$579$	3889.11	0.279147
$580$	−12684.7	−0.908113
$581$	−16145.0	−1.15285
$582$	−2051.59	−0.146119
$583$	28295.2	2.01007
$584$	−3245.20	−0.229944
$585$	−16.7892	−0.00118658
$586$	−5908.27	−0.416499
$587$	−6885.79	−0.484169	−0.242084	−	0.970255i	$-0.577831\pi$
$587$	−6885.79	−0.484169	−0.242084	+	0.970255i	$0.577831\pi$
$588$	−3120.93	−0.218886
$589$	5268.17	0.368542
$590$	−1114.27	−0.0777521
$591$	−8684.11	−0.604427
$592$	−10180.1	−0.706759
$593$	−24880.0	−1.72293	−0.861467	−	0.507813i	$-0.830454\pi$
$593$	−24880.0	−1.72293	−0.861467	+	0.507813i	$0.830454\pi$
$594$	2406.39	0.166221
$595$	19171.7	1.32094
$596$	−9377.15	−0.644468
$597$	−8505.45	−0.583090
$598$	30.2347	0.00206754
$599$	5068.39	0.345724	0.172862	−	0.984946i	$-0.444698\pi$
$599$	5068.39	0.345724	0.172862	+	0.984946i	$0.444698\pi$
$600$	5285.09	0.359605
$601$	−5800.47	−0.393687	−0.196844	−	0.980435i	$-0.563069\pi$
$601$	−5800.47	−0.393687	−0.196844	+	0.980435i	$0.563069\pi$
$602$	−904.420	−0.0612316
$603$	−4095.22	−0.276567
$604$	15336.9	1.03319
$605$	53594.4	3.60152
$606$	6035.05	0.404549
$607$	8659.84	0.579064	0.289532	−	0.957168i	$-0.406500\pi$
$607$	8659.84	0.579064	0.289532	+	0.957168i	$0.406500\pi$
$608$	−7410.61	−0.494309
$609$	9000.57	0.598886
$610$	3963.18	0.263057
$611$	−60.0843	−0.00397831
$612$	−3293.93	−0.217564
$613$	−3303.27	−0.217647	−0.108824	−	0.994061i	$-0.534708\pi$
$613$	−3303.27	−0.217647	−0.108824	+	0.994061i	$0.534708\pi$
$614$	1752.70	0.115201
$615$	8734.17	0.572676
$616$	−28843.2	−1.88657
$617$	−9356.43	−0.610495	−0.305248	−	0.952273i	$-0.598739\pi$
$617$	−9356.43	−0.610495	−0.305248	+	0.952273i	$0.598739\pi$
$618$	1949.92	0.126921
$619$	1106.92	0.0718756	0.0359378	−	0.999354i	$-0.488558\pi$
$619$	1106.92	0.0718756	0.0359378	+	0.999354i	$0.488558\pi$
$620$	−12272.3	−0.794946
$621$	5131.75	0.331611
$622$	−2880.61	−0.185695
$623$	29377.5	1.88922
$624$	−10.5077	−0.000674110 0
$625$	−18377.1	−1.17613
$626$	618.818	0.0395095
$627$	−8600.22	−0.547782
$628$	22818.6	1.44994
$629$	−20868.8	−1.32288
$630$	−3822.75	−0.241749
$631$	23600.6	1.48895	0.744473	−	0.667653i	$-0.232701\pi$
$631$	23600.6	1.48895	0.744473	+	0.667653i	$0.232701\pi$
$632$	−8976.63	−0.564986
$633$	2682.75	0.168451
$634$	−10808.5	−0.677065
$635$	18312.7	1.14444
$636$	7722.87	0.481496
$637$	−20.4089	−0.00126944
$638$	11889.3	0.737774
$639$	1831.48	0.113384
$640$	21484.8	1.32697
$641$	−3922.67	−0.241710	−0.120855	−	0.992670i	$-0.538564\pi$
$641$	−3922.67	−0.241710	−0.120855	+	0.992670i	$0.538564\pi$
$642$	−343.874	−0.0211396
$643$	−11078.8	−0.679482	−0.339741	−	0.940519i	$-0.610339\pi$
$643$	−11078.8	−0.679482	−0.339741	+	0.940519i	$0.610339\pi$
$644$	−27312.8	−1.67123
$645$	1414.75	0.0863653
$646$	−2967.18	−0.180716
$647$	−18371.0	−1.11629	−0.558143	−	0.829745i	$-0.688486\pi$
$647$	−18371.0	−1.11629	−0.558143	+	0.829745i	$0.688486\pi$
$648$	1479.14	0.0896698
$649$	−4143.59	−0.250616
$650$	15.3466	0.000926064 0
$651$	8707.89	0.524254
$652$	8342.00	0.501070
$653$	12179.0	0.729862	0.364931	−	0.931035i	$-0.381093\pi$
$653$	12179.0	0.729862	0.364931	+	0.931035i	$0.381093\pi$
$654$	8533.89	0.510247
$655$	14472.1	0.863313
$656$	5466.38	0.325345
$657$	−1599.41	−0.0949757
$658$	−13680.7	−0.810529
$659$	−18245.7	−1.07853	−0.539266	−	0.842135i	$-0.681298\pi$
$659$	−18245.7	−1.07853	−0.539266	+	0.842135i	$0.681298\pi$
$660$	20034.3	1.18157
$661$	20360.6	1.19809	0.599043	−	0.800717i	$-0.295548\pi$
$661$	20360.6	1.19809	0.599043	+	0.800717i	$0.295548\pi$
$662$	−13120.5	−0.770307
$663$	−21.5403	−0.00126177
$664$	−13108.9	−0.766148
$665$	13662.2	0.796687
$666$	4161.15	0.242104
$667$	25354.5	1.47186
$668$	−1000.31	−0.0579388
$669$	7256.33	0.419351
$670$	8593.55	0.495519
$671$	14737.7	0.847904
$672$	−12249.2	−0.703159
$673$	−3894.49	−0.223063	−0.111531	−	0.993761i	$-0.535576\pi$
$673$	−3894.49	−0.223063	−0.111531	+	0.993761i	$0.535576\pi$
$674$	437.031	0.0249760
$675$	2604.78	0.148530
$676$	14037.7	0.798685
$677$	23571.6	1.33815	0.669077	−	0.743194i	$-0.266690\pi$
$677$	23571.6	1.33815	0.669077	+	0.743194i	$0.266690\pi$
$678$	5856.57	0.331740
$679$	12119.6	0.684987
$680$	15566.4	0.877859
$681$	−3350.66	−0.188542
$682$	11502.6	0.645834
$683$	−7789.31	−0.436383	−0.218192	−	0.975906i	$-0.570016\pi$
$683$	−7789.31	−0.436383	−0.218192	+	0.975906i	$0.570016\pi$
$684$	−2347.33	−0.131217
$685$	−4996.02	−0.278669
$686$	5142.71	0.286224
$687$	−12011.1	−0.667034
$688$	885.436	0.0490653
$689$	50.5028	0.00279246
$690$	−10768.7	−0.594139
$691$	−8889.84	−0.489414	−0.244707	−	0.969597i	$-0.578692\pi$
$691$	−8889.84	−0.489414	−0.244707	+	0.969597i	$0.578692\pi$
$692$	−2282.76	−0.125401
$693$	−14215.5	−0.779225
$694$	−5117.70	−0.279921
$695$	−33074.0	−1.80513
$696$	7307.99	0.398001
$697$	11205.8	0.608967
$698$	−7994.13	−0.433499
$699$	−11494.6	−0.621984
$700$	−13863.5	−0.748556
$701$	32472.7	1.74961	0.874804	−	0.484477i	$-0.160990\pi$
$701$	32472.7	1.74961	0.874804	+	0.484477i	$0.160990\pi$
$702$	4.29504	0.000230920 0
$703$	−14871.6	−0.797856
$704$	−481.426	−0.0257734
$705$	21400.1	1.14323
$706$	10458.2	0.557504
$707$	−35651.5	−1.89648
$708$	−1130.95	−0.0600332
$709$	4596.06	0.243454	0.121727	−	0.992564i	$-0.461157\pi$
$709$	4596.06	0.243454	0.121727	+	0.992564i	$0.461157\pi$
$710$	−3843.24	−0.203147
$711$	−4424.17	−0.233361
$712$	23853.0	1.25552
$713$	24530.0	1.28844
$714$	−4904.54	−0.257070
$715$	131.012	0.00685256
$716$	−10102.1	−0.527282
$717$	−1996.01	−0.103964
$718$	8382.89	0.435720
$719$	−81.6045	−0.00423273	−0.00211637	−	0.999998i	$-0.500674\pi$
$719$	−81.6045	−0.00423273	−0.00211637	+	0.999998i	$0.500674\pi$
$720$	3742.51	0.193716
$721$	−11519.0	−0.594991
$722$	6589.90	0.339682
$723$	18064.6	0.929223
$724$	−10701.1	−0.549312
$725$	12869.5	0.659254
$726$	−13710.6	−0.700894
$727$	−5240.27	−0.267333	−0.133666	−	0.991026i	$-0.542675\pi$
$727$	−5240.27	−0.267333	−0.133666	+	0.991026i	$0.542675\pi$
$728$	−51.4809	−0.00262089
$729$	729.000	0.0370370
$730$	3356.26	0.170166
$731$	1815.10	0.0918384
$732$	4022.50	0.203109
$733$	27362.1	1.37877	0.689386	−	0.724394i	$-0.257880\pi$
$733$	27362.1	1.37877	0.689386	+	0.724394i	$0.257880\pi$
$734$	−13406.0	−0.674147
$735$	7269.02	0.364792
$736$	−34505.8	−1.72813
$737$	31956.5	1.59719
$738$	−2234.39	−0.111449
$739$	15458.7	0.769497	0.384748	−	0.923021i	$-0.374288\pi$
$739$	15458.7	0.769497	0.384748	+	0.923021i	$0.374288\pi$
$740$	34643.6	1.72098
$741$	−15.3501	−0.000760999 0
$742$	11499.1	0.568927
$743$	−33376.7	−1.64801	−0.824006	−	0.566581i	$-0.808266\pi$
$743$	−33376.7	−1.64801	−0.824006	+	0.566581i	$0.808266\pi$
$744$	7070.35	0.348403
$745$	21840.5	1.07406
$746$	12168.3	0.597203
$747$	−6460.76	−0.316448
$748$	25703.7	1.25645
$749$	2031.40	0.0991000
$750$	1616.26	0.0786898
$751$	−8832.20	−0.429150	−0.214575	−	0.976708i	$-0.568837\pi$
$751$	−8832.20	−0.429150	−0.214575	+	0.976708i	$0.568837\pi$
$752$	13393.5	0.649483
$753$	−5563.41	−0.269245
$754$	21.2205	0.00102494
$755$	−35721.4	−1.72190
$756$	−3879.97	−0.186657
$757$	−4971.26	−0.238684	−0.119342	−	0.992853i	$-0.538078\pi$
$757$	−4971.26	−0.238684	−0.119342	+	0.992853i	$0.538078\pi$
$758$	−10209.5	−0.489218
$759$	−40045.0	−1.91507
$760$	11093.0	0.529453
$761$	25414.3	1.21060	0.605300	−	0.795997i	$-0.293053\pi$
$761$	25414.3	1.21060	0.605300	+	0.795997i	$0.293053\pi$
$762$	−4684.80	−0.222720
$763$	−50413.2	−2.39198
$764$	−32046.2	−1.51753
$765$	7671.97	0.362589
$766$	16728.2	0.789054
$767$	−7.39569	−0.000348165 0
$768$	−5660.82	−0.265973
$769$	30483.0	1.42945	0.714725	−	0.699406i	$-0.246552\pi$
$769$	30483.0	1.42945	0.714725	+	0.699406i	$0.246552\pi$
$770$	29830.4	1.39612
$771$	−10221.5	−0.477455
$772$	−8283.19	−0.386164
$773$	18113.2	0.842802	0.421401	−	0.906874i	$-0.361538\pi$
$773$	18113.2	0.842802	0.421401	+	0.906874i	$0.361538\pi$
$774$	−361.924	−0.0168076
$775$	12451.0	0.577099
$776$	9840.46	0.455221
$777$	−24581.6	−1.13496
$778$	−2067.99	−0.0952969
$779$	7985.52	0.367280
$780$	35.7583	0.00164148
$781$	−14291.7	−0.654797
$782$	−13816.0	−0.631790
$783$	3601.77	0.164389
$784$	4549.40	0.207243
$785$	−53147.3	−2.41644
$786$	−3702.27	−0.168010
$787$	3681.38	0.166743	0.0833717	−	0.996519i	$-0.473431\pi$
$787$	3681.38	0.166743	0.0833717	+	0.996519i	$0.473431\pi$
$788$	18495.8	0.836148
$789$	−6256.79	−0.282316
$790$	9283.84	0.418107
$791$	−34597.1	−1.55516
$792$	−11542.3	−0.517849
$793$	26.3046	0.00117794
$794$	−3053.15	−0.136464
$795$	−17987.5	−0.802454
$796$	18115.3	0.806631
$797$	−40435.4	−1.79711	−0.898554	−	0.438863i	$-0.855381\pi$
$797$	−40435.4	−1.79711	−0.898554	+	0.438863i	$0.855381\pi$
$798$	−3495.09	−0.155044
$799$	27456.0	1.21568
$800$	−17514.5	−0.774039
$801$	11756.1	0.518577
$802$	−5562.65	−0.244918
$803$	12480.8	0.548490
$804$	8722.16	0.382596
$805$	63614.9	2.78525
$806$	20.5305	0.000897216 0
$807$	−19485.2	−0.849952
$808$	−28947.1	−1.26034
$809$	−36431.5	−1.58327	−0.791633	−	0.610997i	$-0.790769\pi$
$809$	−36431.5	−1.58327	−0.791633	+	0.610997i	$0.790769\pi$
$810$	−1529.76	−0.0663584
$811$	−30923.2	−1.33891	−0.669457	−	0.742851i	$-0.733473\pi$
$811$	−30923.2	−1.33891	−0.669457	+	0.742851i	$0.733473\pi$
$812$	−19169.8	−0.828482
$813$	−9757.84	−0.420938
$814$	−32471.0	−1.39817
$815$	−19429.5	−0.835076
$816$	4801.59	0.205992
$817$	1293.48	0.0553895
$818$	10872.5	0.464728
$819$	−25.3726	−0.00108253
$820$	−18602.4	−0.792224
$821$	3516.20	0.149472	0.0747359	−	0.997203i	$-0.476189\pi$
$821$	3516.20	0.149472	0.0747359	+	0.997203i	$0.476189\pi$
$822$	1278.09	0.0542318
$823$	39234.9	1.66178	0.830889	−	0.556439i	$-0.187833\pi$
$823$	39234.9	1.66178	0.830889	+	0.556439i	$0.187833\pi$
$824$	−9352.78	−0.395412
$825$	−20326.1	−0.857773
$826$	−1683.93	−0.0709341
$827$	20321.9	0.854486	0.427243	−	0.904137i	$-0.359485\pi$
$827$	20321.9	0.854486	0.427243	+	0.904137i	$0.359485\pi$
$828$	−10929.8	−0.458741
$829$	−36098.3	−1.51236	−0.756180	−	0.654364i	$-0.772936\pi$
$829$	−36098.3	−1.51236	−0.756180	+	0.654364i	$0.772936\pi$
$830$	13557.5	0.566973
$831$	17715.6	0.739529
$832$	−0.859274	−3.58053e−5 0
$833$	9326.04	0.387909
$834$	8461.05	0.351298
$835$	2329.84	0.0965600
$836$	18317.1	0.757787
$837$	3484.65	0.143904
$838$	−10036.4	−0.413725
$839$	19751.4	0.812746	0.406373	−	0.913707i	$-0.366793\pi$
$839$	19751.4	0.812746	0.406373	+	0.913707i	$0.366793\pi$
$840$	18335.9	0.753152
$841$	−6593.69	−0.270355
$842$	−5408.86	−0.221380
$843$	19512.2	0.797194
$844$	−5713.82	−0.233031
$845$	−32695.5	−1.33108
$846$	−5474.62	−0.222484
$847$	80994.3	3.28571
$848$	−11257.7	−0.455885
$849$	19763.5	0.798917
$850$	−7012.75	−0.282983
$851$	−69246.2	−2.78934
$852$	−3900.75	−0.156852
$853$	13812.9	0.554448	0.277224	−	0.960805i	$-0.410586\pi$
$853$	13812.9	0.554448	0.277224	+	0.960805i	$0.410586\pi$
$854$	5989.34	0.239990
$855$	5467.22	0.218684
$856$	1649.39	0.0658588
$857$	10194.0	0.406323	0.203162	−	0.979145i	$-0.434878\pi$
$857$	10194.0	0.406323	0.203162	+	0.979145i	$0.434878\pi$
$858$	−33.5158	−0.00133358
$859$	−11608.7	−0.461099	−0.230549	−	0.973061i	$-0.574052\pi$
$859$	−11608.7	−0.461099	−0.230549	+	0.973061i	$0.574052\pi$
$860$	−3013.19	−0.119475
$861$	13199.5	0.522459
$862$	−16309.0	−0.644418
$863$	−45095.9	−1.77877	−0.889387	−	0.457155i	$-0.848868\pi$
$863$	−45095.9	−1.77877	−0.889387	+	0.457155i	$0.848868\pi$
$864$	−4901.79	−0.193012
$865$	5316.83	0.208991
$866$	13899.1	0.545393
$867$	−4895.98	−0.191783
$868$	−18546.4	−0.725238
$869$	34523.5	1.34767
$870$	−7558.09	−0.294533
$871$	57.0376	0.00221888
$872$	−40932.9	−1.58964
$873$	4849.91	0.188024
$874$	−9845.62	−0.381045
$875$	−9547.89	−0.368889
$876$	3406.49	0.131387
$877$	38676.9	1.48920	0.744599	−	0.667512i	$-0.232641\pi$
$877$	38676.9	1.48920	0.744599	+	0.667512i	$0.232641\pi$
$878$	15423.0	0.592824
$879$	13967.0	0.535946
$880$	−29204.2	−1.11872
$881$	−34130.6	−1.30521	−0.652604	−	0.757699i	$-0.726323\pi$
$881$	−34130.6	−1.30521	−0.652604	+	0.757699i	$0.726323\pi$
$882$	−1859.58	−0.0709923
$883$	−39285.3	−1.49723	−0.748615	−	0.663005i	$-0.769281\pi$
$883$	−39285.3	−1.49723	−0.748615	+	0.663005i	$0.769281\pi$
$884$	45.8774	0.00174550
$885$	2634.11	0.100050
$886$	−14751.8	−0.559362
$887$	−27255.1	−1.03172	−0.515860	−	0.856673i	$-0.672528\pi$
$887$	−27255.1	−1.03172	−0.515860	+	0.856673i	$0.672528\pi$
$888$	−19959.0	−0.754257
$889$	27675.1	1.04409
$890$	−24669.3	−0.929121
$891$	−5688.65	−0.213891
$892$	−15454.8	−0.580119
$893$	19565.8	0.733197
$894$	−5587.29	−0.209023
$895$	23529.1	0.878760
$896$	32468.9	1.21061
$897$	−71.4743	−0.00266049
$898$	11615.2	0.431630
$899$	17216.6	0.638718
$900$	−5547.77	−0.205473
$901$	−23077.7	−0.853307
$902$	17435.8	0.643623
$903$	2138.03	0.0787920
$904$	−28091.1	−1.03351
$905$	24924.1	0.915475
$906$	9138.33	0.335100
$907$	−40763.1	−1.49230	−0.746151	−	0.665777i	$-0.768100\pi$
$907$	−40763.1	−1.49230	−0.746151	+	0.665777i	$0.768100\pi$
$908$	7136.36	0.260824
$909$	−14266.7	−0.520569
$910$	53.2427	0.00193954
$911$	9175.77	0.333707	0.166853	−	0.985982i	$-0.446639\pi$
$911$	9175.77	0.333707	0.166853	+	0.985982i	$0.446639\pi$
$912$	3421.72	0.124238
$913$	50415.7	1.82751
$914$	2365.82	0.0856174
$915$	−9368.89	−0.338498
$916$	25581.7	0.922756
$917$	21870.8	0.787610
$918$	−1962.66	−0.0705636
$919$	−33814.3	−1.21374	−0.606871	−	0.794800i	$-0.707576\pi$
$919$	−33814.3	−1.21374	−0.606871	+	0.794800i	$0.707576\pi$
$920$	51651.9	1.85099
$921$	−4143.36	−0.148239
$922$	24385.7	0.871040
$923$	−25.5085	−0.000909668 0
$924$	30276.8	1.07796
$925$	−35148.0	−1.24936
$926$	−9882.07	−0.350696
$927$	−4609.56	−0.163320
$928$	−24218.3	−0.856685
$929$	428.503	0.0151332	0.00756659	−	0.999971i	$-0.497591\pi$
$929$	428.503	0.0151332	0.00756659	+	0.999971i	$0.497591\pi$
$930$	−7312.32	−0.257828
$931$	6645.96	0.233955
$932$	24481.7	0.860435
$933$	6809.71	0.238950
$934$	12103.7	0.424032
$935$	−59867.2	−2.09397
$936$	−20.6012	−0.000719414 0
$937$	32339.2	1.12751	0.563755	−	0.825942i	$-0.309356\pi$
$937$	32339.2	1.12751	0.563755	+	0.825942i	$0.309356\pi$
$938$	12987.0	0.452068
$939$	−1462.87	−0.0508403
$940$	−45578.8	−1.58151
$941$	3308.12	0.114603	0.0573016	−	0.998357i	$-0.481750\pi$
$941$	3308.12	0.114603	0.0573016	+	0.998357i	$0.481750\pi$
$942$	13596.2	0.470265
$943$	37182.8	1.28403
$944$	1648.59	0.0568400
$945$	9036.92	0.311080
$946$	2824.22	0.0970649
$947$	2319.50	0.0795921	0.0397961	−	0.999208i	$-0.487329\pi$
$947$	2319.50	0.0795921	0.0397961	+	0.999208i	$0.487329\pi$
$948$	9422.79	0.322825
$949$	22.2764	0.000761982 0
$950$	−4997.45	−0.170672
$951$	25551.0	0.871239
$952$	23524.6	0.800880
$953$	8790.44	0.298793	0.149397	−	0.988777i	$-0.452267\pi$
$953$	8790.44	0.298793	0.149397	+	0.988777i	$0.452267\pi$
$954$	4601.60	0.156166
$955$	74639.6	2.52909
$956$	4251.18	0.143821
$957$	−28106.0	−0.949359
$958$	3249.13	0.109577
$959$	−7550.21	−0.254233
$960$	306.046	0.0102892
$961$	−13134.2	−0.440878
$962$	−57.9559	−0.00194238
$963$	812.911	0.0272022
$964$	−38474.6	−1.28546
$965$	19292.6	0.643575
$966$	−16274.1	−0.542039
$967$	−20867.5	−0.693953	−0.346976	−	0.937874i	$-0.612792\pi$
$967$	−20867.5	−0.693953	−0.346976	+	0.937874i	$0.612792\pi$
$968$	65763.1	2.18358
$969$	7014.37	0.232543
$970$	−10177.2	−0.336877
$971$	2174.66	0.0718726	0.0359363	−	0.999354i	$-0.488559\pi$
$971$	2174.66	0.0718726	0.0359363	+	0.999354i	$0.488559\pi$
$972$	−1552.65	−0.0512360
$973$	−49982.9	−1.64684
$974$	−6161.49	−0.202697
$975$	−36.2790	−0.00119165
$976$	−5863.62	−0.192305
$977$	−6656.04	−0.217959	−0.108979	−	0.994044i	$-0.534758\pi$
$977$	−6656.04	−0.217959	−0.108979	+	0.994044i	$0.534758\pi$
$978$	4970.50	0.162514
$979$	−91736.8	−2.99481
$980$	−15481.9	−0.504643
$981$	−20174.0	−0.656580
$982$	−23824.1	−0.774194
$983$	52028.5	1.68815	0.844075	−	0.536224i	$-0.180150\pi$
$983$	52028.5	1.68815	0.844075	+	0.536224i	$0.180150\pi$
$984$	10717.3	0.347210
$985$	−43078.9	−1.39351
$986$	−9696.92	−0.313198
$987$	32340.8	1.04298
$988$	32.6933	0.00105274
$989$	6022.81	0.193644
$990$	11937.3	0.383224
$991$	9152.99	0.293395	0.146697	−	0.989181i	$-0.453136\pi$
$991$	9152.99	0.293395	0.146697	+	0.989181i	$0.453136\pi$
$992$	−23430.8	−0.749927
$993$	31016.6	0.991222
$994$	−5808.07	−0.185333
$995$	−42192.7	−1.34432
$996$	13760.4	0.437766
$997$	−43711.9	−1.38854	−0.694268	−	0.719717i	$-0.744272\pi$
$997$	−43711.9	−1.38854	−0.694268	+	0.719717i	$0.744272\pi$
$998$	10327.6	0.327569
$999$	−9836.88	−0.311537

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.d.1.3	✓	8
3.2	odd	2		531.4.a.e.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.d.1.3	✓	8	1.1	even	1	trivial
531.4.a.e.1.6		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q-1.26905 q^{2} +3.00000 q^{3} -6.38952 q^{4} +14.8820 q^{5} -3.80714 q^{6} +22.4903 q^{7} +18.2610 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.26905 q^{2} +3.00000 q^{3} -6.38952 q^{4} +14.8820 q^{5} -3.80714 q^{6} +22.4903 q^{7} +18.2610 q^{8} +9.00000 q^{9} -18.8859 q^{10} -70.2303 q^{11} -19.1686 q^{12} -0.125351 q^{13} -28.5413 q^{14} +44.6459 q^{15} +27.9422 q^{16} +57.2801 q^{17} -11.4214 q^{18} +40.8191 q^{19} -95.0887 q^{20} +67.4710 q^{21} +89.1255 q^{22} +190.065 q^{23} +54.7829 q^{24} +96.4733 q^{25} +0.159076 q^{26} +27.0000 q^{27} -143.702 q^{28} +133.399 q^{29} -56.6578 q^{30} +129.061 q^{31} -181.548 q^{32} -210.691 q^{33} -72.6911 q^{34} +334.701 q^{35} -57.5057 q^{36} -364.329 q^{37} -51.8013 q^{38} -0.376052 q^{39} +271.759 q^{40} +195.632 q^{41} -85.6238 q^{42} +31.6882 q^{43} +448.738 q^{44} +133.938 q^{45} -241.201 q^{46} +479.330 q^{47} +83.8265 q^{48} +162.815 q^{49} -122.429 q^{50} +171.840 q^{51} +0.800930 q^{52} -402.892 q^{53} -34.2642 q^{54} -1045.17 q^{55} +410.695 q^{56} +122.457 q^{57} -169.290 q^{58} +59.0000 q^{59} -285.266 q^{60} -209.849 q^{61} -163.785 q^{62} +202.413 q^{63} +6.85497 q^{64} -1.86547 q^{65} +267.376 q^{66} -455.024 q^{67} -365.992 q^{68} +570.195 q^{69} -424.750 q^{70} +203.497 q^{71} +164.349 q^{72} -177.712 q^{73} +462.350 q^{74} +289.420 q^{75} -260.815 q^{76} -1579.50 q^{77} +0.477227 q^{78} -491.575 q^{79} +415.835 q^{80} +81.0000 q^{81} -248.266 q^{82} -717.862 q^{83} -431.107 q^{84} +852.441 q^{85} -40.2137 q^{86} +400.197 q^{87} -1282.47 q^{88} +1306.23 q^{89} -169.973 q^{90} -2.81918 q^{91} -1214.42 q^{92} +387.184 q^{93} -608.291 q^{94} +607.469 q^{95} -544.643 q^{96} +538.879 q^{97} -206.620 q^{98} -632.073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{2} + 24 q^{3} + 34 q^{4} + 42 q^{5} + 18 q^{6} + 53 q^{7} + 51 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q + 6 * q^2 + 24 * q^3 + 34 * q^4 + 42 * q^5 + 18 * q^6 + 53 * q^7 + 51 * q^8 + 72 * q^9	\( 8 q + 6 q^{2} + 24 q^{3} + 34 q^{4} + 42 q^{5} + 18 q^{6} + 53 q^{7} + 51 q^{8} + 72 q^{9} + 21 q^{10} + 67 q^{11} + 102 q^{12} + 33 q^{13} + 79 q^{14} + 126 q^{15} - 30 q^{16} + 139 q^{17} + 54 q^{18} + 64 q^{19} + 117 q^{20} + 159 q^{21} - 84 q^{22} + 226 q^{23} + 153 q^{24} + 96 q^{25} + 24 q^{26} + 216 q^{27} + 34 q^{28} + 456 q^{29} + 63 q^{30} + 124 q^{31} + 174 q^{32} + 201 q^{33} - 114 q^{34} + 556 q^{35} + 306 q^{36} + 127 q^{37} + 237 q^{38} + 99 q^{39} - 188 q^{40} + 425 q^{41} + 237 q^{42} - 115 q^{43} + 510 q^{44} + 378 q^{45} - 711 q^{46} + 420 q^{47} - 90 q^{48} + 171 q^{49} - 137 q^{50} + 417 q^{51} - 922 q^{52} + 98 q^{53} + 162 q^{54} - 616 q^{55} - 412 q^{56} + 192 q^{57} - 1548 q^{58} + 472 q^{59} + 351 q^{60} - 1254 q^{61} - 766 q^{62} + 477 q^{63} - 2019 q^{64} - 734 q^{65} - 252 q^{66} - 1010 q^{67} - 503 q^{68} + 678 q^{69} - 2956 q^{70} - 17 q^{71} + 459 q^{72} - 1180 q^{73} - 1228 q^{74} + 288 q^{75} - 2008 q^{76} + 441 q^{77} + 72 q^{78} - 873 q^{79} - 865 q^{80} + 648 q^{81} - 3645 q^{82} + 759 q^{83} + 102 q^{84} - 850 q^{85} - 1226 q^{86} + 1368 q^{87} - 3047 q^{88} + 988 q^{89} + 189 q^{90} - 2111 q^{91} - 1062 q^{92} + 372 q^{93} - 2240 q^{94} + 1822 q^{95} + 522 q^{96} - 668 q^{97} - 1368 q^{98} + 603 q^{99}+O(q^{100}) \) 8 * q + 6 * q^2 + 24 * q^3 + 34 * q^4 + 42 * q^5 + 18 * q^6 + 53 * q^7 + 51 * q^8 + 72 * q^9 + 21 * q^10 + 67 * q^11 + 102 * q^12 + 33 * q^13 + 79 * q^14 + 126 * q^15 - 30 * q^16 + 139 * q^17 + 54 * q^18 + 64 * q^19 + 117 * q^20 + 159 * q^21 - 84 * q^22 + 226 * q^23 + 153 * q^24 + 96 * q^25 + 24 * q^26 + 216 * q^27 + 34 * q^28 + 456 * q^29 + 63 * q^30 + 124 * q^31 + 174 * q^32 + 201 * q^33 - 114 * q^34 + 556 * q^35 + 306 * q^36 + 127 * q^37 + 237 * q^38 + 99 * q^39 - 188 * q^40 + 425 * q^41 + 237 * q^42 - 115 * q^43 + 510 * q^44 + 378 * q^45 - 711 * q^46 + 420 * q^47 - 90 * q^48 + 171 * q^49 - 137 * q^50 + 417 * q^51 - 922 * q^52 + 98 * q^53 + 162 * q^54 - 616 * q^55 - 412 * q^56 + 192 * q^57 - 1548 * q^58 + 472 * q^59 + 351 * q^60 - 1254 * q^61 - 766 * q^62 + 477 * q^63 - 2019 * q^64 - 734 * q^65 - 252 * q^66 - 1010 * q^67 - 503 * q^68 + 678 * q^69 - 2956 * q^70 - 17 * q^71 + 459 * q^72 - 1180 * q^73 - 1228 * q^74 + 288 * q^75 - 2008 * q^76 + 441 * q^77 + 72 * q^78 - 873 * q^79 - 865 * q^80 + 648 * q^81 - 3645 * q^82 + 759 * q^83 + 102 * q^84 - 850 * q^85 - 1226 * q^86 + 1368 * q^87 - 3047 * q^88 + 988 * q^89 + 189 * q^90 - 2111 * q^91 - 1062 * q^92 + 372 * q^93 - 2240 * q^94 + 1822 * q^95 + 522 * q^96 - 668 * q^97 - 1368 * q^98 + 603 * q^99

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