LMFDB - Embedded newform 177.4.a.c.1.1

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} - 49x^{6} + 89x^{5} + 648x^{4} - 1023x^{3} - 1476x^{2} + 1940x - 384 $ x^8 - 2x^7 - 49x^6 + 89x^5 + 648x^4 - 1023x^3 - 1476x^2 + 1940*x - 384
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.19624$ 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)

$f(q)$	$=$	$q-5.19624 q^{2} -3.00000 q^{3} +19.0009 q^{4} -8.30102 q^{5} +15.5887 q^{6} +21.5539 q^{7} -57.1634 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-5.19624 q^{2} -3.00000 q^{3} +19.0009 q^{4} -8.30102 q^{5} +15.5887 q^{6} +21.5539 q^{7} -57.1634 q^{8} +9.00000 q^{9} +43.1341 q^{10} -28.3746 q^{11} -57.0027 q^{12} -28.2089 q^{13} -111.999 q^{14} +24.9031 q^{15} +145.027 q^{16} +21.8166 q^{17} -46.7662 q^{18} -122.388 q^{19} -157.727 q^{20} -64.6617 q^{21} +147.441 q^{22} +82.1571 q^{23} +171.490 q^{24} -56.0930 q^{25} +146.580 q^{26} -27.0000 q^{27} +409.544 q^{28} +86.9209 q^{29} -129.402 q^{30} +131.178 q^{31} -296.290 q^{32} +85.1239 q^{33} -113.364 q^{34} -178.919 q^{35} +171.008 q^{36} +280.467 q^{37} +635.957 q^{38} +84.6266 q^{39} +474.515 q^{40} +381.788 q^{41} +335.998 q^{42} +452.501 q^{43} -539.144 q^{44} -74.7092 q^{45} -426.908 q^{46} -158.067 q^{47} -435.082 q^{48} +121.570 q^{49} +291.473 q^{50} -65.4499 q^{51} -535.994 q^{52} -162.922 q^{53} +140.298 q^{54} +235.539 q^{55} -1232.09 q^{56} +367.164 q^{57} -451.662 q^{58} -59.0000 q^{59} +473.181 q^{60} -368.344 q^{61} -681.632 q^{62} +193.985 q^{63} +379.374 q^{64} +234.162 q^{65} -442.324 q^{66} +177.950 q^{67} +414.536 q^{68} -246.471 q^{69} +929.708 q^{70} +58.9220 q^{71} -514.470 q^{72} +880.299 q^{73} -1457.37 q^{74} +168.279 q^{75} -2325.48 q^{76} -611.584 q^{77} -439.740 q^{78} -825.284 q^{79} -1203.88 q^{80} +81.0000 q^{81} -1983.86 q^{82} +1426.68 q^{83} -1228.63 q^{84} -181.100 q^{85} -2351.30 q^{86} -260.763 q^{87} +1621.99 q^{88} +1556.78 q^{89} +388.207 q^{90} -608.011 q^{91} +1561.06 q^{92} -393.534 q^{93} +821.356 q^{94} +1015.94 q^{95} +888.869 q^{96} +1811.25 q^{97} -631.708 q^{98} -255.372 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9} + 29 q^{10} - 27 q^{11} - 114 q^{12} + 89 q^{13} - 37 q^{14} + 36 q^{15} + 362 q^{16} + 79 q^{17} + 18 q^{18} + 288 q^{19} + 457 q^{20} - 159 q^{21} + 596 q^{22} + 202 q^{23} - 9 q^{24} + 264 q^{25} + 270 q^{26} - 216 q^{27} + 702 q^{28} - 114 q^{29} - 87 q^{30} + 538 q^{31} + 316 q^{32} + 81 q^{33} + 498 q^{34} - 196 q^{35} + 342 q^{36} + 395 q^{37} + 397 q^{38} - 267 q^{39} + 918 q^{40} - 39 q^{41} + 111 q^{42} + 527 q^{43} + 64 q^{44} - 108 q^{45} - 539 q^{46} + 860 q^{47} - 1086 q^{48} + 347 q^{49} - 591 q^{50} - 237 q^{51} - 644 q^{52} - 812 q^{53} - 54 q^{54} + 536 q^{55} - 2218 q^{56} - 864 q^{57} - 1154 q^{58} - 472 q^{59} - 1371 q^{60} - 460 q^{61} - 2014 q^{62} + 477 q^{63} - 451 q^{64} - 986 q^{65} - 1788 q^{66} + 1934 q^{67} - 69 q^{68} - 606 q^{69} - 1028 q^{70} - 1687 q^{71} + 27 q^{72} + 1980 q^{73} - 2400 q^{74} - 792 q^{75} - 940 q^{76} - 821 q^{77} - 810 q^{78} + 3319 q^{79} - 2119 q^{80} + 648 q^{81} + 429 q^{82} + 2057 q^{83} - 2106 q^{84} + 566 q^{85} - 6690 q^{86} + 342 q^{87} + 1189 q^{88} + 1668 q^{89} + 261 q^{90} + 2427 q^{91} - 980 q^{92} - 1614 q^{93} + 332 q^{94} + 2146 q^{95} - 948 q^{96} + 1956 q^{97} - 2026 q^{98} - 243 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9 + 29 * q^10 - 27 * q^11 - 114 * q^12 + 89 * q^13 - 37 * q^14 + 36 * q^15 + 362 * q^16 + 79 * q^17 + 18 * q^18 + 288 * q^19 + 457 * q^20 - 159 * q^21 + 596 * q^22 + 202 * q^23 - 9 * q^24 + 264 * q^25 + 270 * q^26 - 216 * q^27 + 702 * q^28 - 114 * q^29 - 87 * q^30 + 538 * q^31 + 316 * q^32 + 81 * q^33 + 498 * q^34 - 196 * q^35 + 342 * q^36 + 395 * q^37 + 397 * q^38 - 267 * q^39 + 918 * q^40 - 39 * q^41 + 111 * q^42 + 527 * q^43 + 64 * q^44 - 108 * q^45 - 539 * q^46 + 860 * q^47 - 1086 * q^48 + 347 * q^49 - 591 * q^50 - 237 * q^51 - 644 * q^52 - 812 * q^53 - 54 * q^54 + 536 * q^55 - 2218 * q^56 - 864 * q^57 - 1154 * q^58 - 472 * q^59 - 1371 * q^60 - 460 * q^61 - 2014 * q^62 + 477 * q^63 - 451 * q^64 - 986 * q^65 - 1788 * q^66 + 1934 * q^67 - 69 * q^68 - 606 * q^69 - 1028 * q^70 - 1687 * q^71 + 27 * q^72 + 1980 * q^73 - 2400 * q^74 - 792 * q^75 - 940 * q^76 - 821 * q^77 - 810 * q^78 + 3319 * q^79 - 2119 * q^80 + 648 * q^81 + 429 * q^82 + 2057 * q^83 - 2106 * q^84 + 566 * q^85 - 6690 * q^86 + 342 * q^87 + 1189 * q^88 + 1668 * q^89 + 261 * q^90 + 2427 * q^91 - 980 * q^92 - 1614 * q^93 + 332 * q^94 + 2146 * q^95 - 948 * q^96 + 1956 * q^97 - 2026 * q^98 - 243 * 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 $

$2$	−5.19624	−1.83715	−0.918574	−	0.395249i	$-0.870658\pi$
$2$	−5.19624	−1.83715	−0.918574	+	0.395249i	$0.870658\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	19.0009	2.37511
$5$	−8.30102	−0.742466	−0.371233	−	0.928540i	$-0.621065\pi$
$5$	−8.30102	−0.742466	−0.371233	+	0.928540i	$0.621065\pi$
$6$	15.5887	1.06068
$7$	21.5539	1.16380	0.581900	−	0.813260i	$-0.302309\pi$
$7$	21.5539	1.16380	0.581900	+	0.813260i	$0.302309\pi$
$8$	−57.1634	−2.52629
$9$	9.00000	0.333333
$10$	43.1341	1.36402
$11$	−28.3746	−0.777752	−0.388876	−	0.921290i	$-0.627137\pi$
$11$	−28.3746	−0.777752	−0.388876	+	0.921290i	$0.627137\pi$
$12$	−57.0027	−1.37127
$13$	−28.2089	−0.601826	−0.300913	−	0.953652i	$-0.597291\pi$
$13$	−28.2089	−0.601826	−0.300913	+	0.953652i	$0.597291\pi$
$14$	−111.999	−2.13807
$15$	24.9031	0.428663
$16$	145.027	2.26605
$17$	21.8166	0.311254	0.155627	−	0.987816i	$-0.450260\pi$
$17$	21.8166	0.311254	0.155627	+	0.987816i	$0.450260\pi$
$18$	−46.7662	−0.612383
$19$	−122.388	−1.47777	−0.738887	−	0.673830i	$-0.764648\pi$
$19$	−122.388	−1.47777	−0.738887	+	0.673830i	$0.764648\pi$
$20$	−157.727	−1.76344
$21$	−64.6617	−0.671921
$22$	147.441	1.42885
$23$	82.1571	0.744823	0.372412	−	0.928068i	$-0.378531\pi$
$23$	82.1571	0.744823	0.372412	+	0.928068i	$0.378531\pi$
$24$	171.490	1.45855
$25$	−56.0930	−0.448744
$26$	146.580	1.10564
$27$	−27.0000	−0.192450
$28$	409.544	2.76416
$29$	86.9209	0.556580	0.278290	−	0.960497i	$-0.410232\pi$
$29$	86.9209	0.556580	0.278290	+	0.960497i	$0.410232\pi$
$30$	−129.402	−0.787518
$31$	131.178	0.760009	0.380004	−	0.924985i	$-0.375922\pi$
$31$	131.178	0.760009	0.380004	+	0.924985i	$0.375922\pi$
$32$	−296.290	−1.63678
$33$	85.1239	0.449035
$34$	−113.364	−0.571819
$35$	−178.919	−0.864083
$36$	171.008	0.791705
$37$	280.467	1.24617	0.623087	−	0.782152i	$-0.285878\pi$
$37$	280.467	1.24617	0.623087	+	0.782152i	$0.285878\pi$
$38$	635.957	2.71489
$39$	84.6266	0.347464
$40$	474.515	1.87568
$41$	381.788	1.45427	0.727137	−	0.686493i	$-0.240851\pi$
$41$	381.788	1.45427	0.727137	+	0.686493i	$0.240851\pi$
$42$	335.998	1.23442
$43$	452.501	1.60478	0.802392	−	0.596798i	$-0.203560\pi$
$43$	452.501	1.60478	0.802392	+	0.596798i	$0.203560\pi$
$44$	−539.144	−1.84725
$45$	−74.7092	−0.247489
$46$	−426.908	−1.36835
$47$	−158.067	−0.490564	−0.245282	−	0.969452i	$-0.578881\pi$
$47$	−158.067	−0.490564	−0.245282	+	0.969452i	$0.578881\pi$
$48$	−435.082	−1.30831
$49$	121.570	0.354432
$50$	291.473	0.824409
$51$	−65.4499	−0.179702
$52$	−535.994	−1.42940
$53$	−162.922	−0.422245	−0.211123	−	0.977460i	$-0.567712\pi$
$53$	−162.922	−0.422245	−0.211123	+	0.977460i	$0.567712\pi$
$54$	140.298	0.353559
$55$	235.539	0.577455
$56$	−1232.09	−2.94010
$57$	367.164	0.853193
$58$	−451.662	−1.02252
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	473.181	1.01812
$61$	−368.344	−0.773141	−0.386570	−	0.922260i	$-0.626340\pi$
$61$	−368.344	−0.773141	−0.386570	+	0.922260i	$0.626340\pi$
$62$	−681.632	−1.39625
$63$	193.985	0.387934
$64$	379.374	0.740965
$65$	234.162	0.446835
$66$	−442.324	−0.824945
$67$	177.950	0.324478	0.162239	−	0.986752i	$-0.448128\pi$
$67$	177.950	0.324478	0.162239	+	0.986752i	$0.448128\pi$
$68$	414.536	0.739263
$69$	−246.471	−0.430024
$70$	929.708	1.58745
$71$	58.9220	0.0984894	0.0492447	−	0.998787i	$-0.484319\pi$
$71$	58.9220	0.0984894	0.0492447	+	0.998787i	$0.484319\pi$
$72$	−514.470	−0.842096
$73$	880.299	1.41139	0.705693	−	0.708518i	$-0.250636\pi$
$73$	880.299	1.41139	0.705693	+	0.708518i	$0.250636\pi$
$74$	−1457.37	−2.28941
$75$	168.279	0.259082
$76$	−2325.48	−3.50988
$77$	−611.584	−0.905148
$78$	−439.740	−0.638343
$79$	−825.284	−1.17534	−0.587669	−	0.809101i	$-0.699954\pi$
$79$	−825.284	−1.17534	−0.587669	+	0.809101i	$0.699954\pi$
$80$	−1203.88	−1.68247
$81$	81.0000	0.111111
$82$	−1983.86	−2.67172
$83$	1426.68	1.88673	0.943367	−	0.331752i	$-0.107640\pi$
$83$	1426.68	1.88673	0.943367	+	0.331752i	$0.107640\pi$
$84$	−1228.63	−1.59589
$85$	−181.100	−0.231095
$86$	−2351.30	−2.94823
$87$	−260.763	−0.321341
$88$	1621.99	1.96483
$89$	1556.78	1.85413	0.927067	−	0.374896i	$-0.122322\pi$
$89$	1556.78	1.85413	0.927067	+	0.374896i	$0.122322\pi$
$90$	388.207	0.454673
$91$	−608.011	−0.700405
$92$	1561.06	1.76904
$93$	−393.534	−0.438791
$94$	821.356	0.901238
$95$	1015.94	1.09720
$96$	888.869	0.944998
$97$	1811.25	1.89593	0.947964	−	0.318377i	$-0.103138\pi$
$97$	1811.25	1.89593	0.947964	+	0.318377i	$0.103138\pi$
$98$	−631.708	−0.651144
$99$	−255.372	−0.259251
$100$	−1065.82	−1.06582
$101$	1132.95	1.11617	0.558085	−	0.829784i	$-0.311536\pi$
$101$	1132.95	1.11617	0.558085	+	0.829784i	$0.311536\pi$
$102$	340.093	0.330140
$103$	1479.79	1.41561	0.707805	−	0.706408i	$-0.249685\pi$
$103$	1479.79	1.41561	0.707805	+	0.706408i	$0.249685\pi$
$104$	1612.51	1.52038
$105$	536.758	0.498878
$106$	846.580	0.775727
$107$	293.850	0.265491	0.132746	−	0.991150i	$-0.457621\pi$
$107$	293.850	0.265491	0.132746	+	0.991150i	$0.457621\pi$
$108$	−513.025	−0.457091
$109$	−801.451	−0.704267	−0.352133	−	0.935950i	$-0.614544\pi$
$109$	−801.451	−0.704267	−0.352133	+	0.935950i	$0.614544\pi$
$110$	−1223.91	−1.06087
$111$	−841.401	−0.719479
$112$	3125.90	2.63723
$113$	−1834.04	−1.52683	−0.763415	−	0.645908i	$-0.776479\pi$
$113$	−1834.04	−1.52683	−0.763415	+	0.645908i	$0.776479\pi$
$114$	−1907.87	−1.56744
$115$	−681.988	−0.553006
$116$	1651.58	1.32194
$117$	−253.880	−0.200609
$118$	306.578	0.239176
$119$	470.233	0.362237
$120$	−1423.54	−1.08293
$121$	−525.880	−0.395102
$122$	1914.00	1.42037
$123$	−1145.36	−0.839625
$124$	2492.50	1.80511
$125$	1503.26	1.07564
$126$	−1007.99	−0.712691
$127$	745.001	0.520537	0.260268	−	0.965536i	$-0.416189\pi$
$127$	745.001	0.520537	0.260268	+	0.965536i	$0.416189\pi$
$128$	398.999	0.275522
$129$	−1357.50	−0.926522
$130$	−1216.76	−0.820902
$131$	−2205.52	−1.47097	−0.735485	−	0.677541i	$-0.763046\pi$
$131$	−2205.52	−1.47097	−0.735485	+	0.677541i	$0.763046\pi$
$132$	1617.43	1.06651
$133$	−2637.93	−1.71983
$134$	−924.669	−0.596114
$135$	224.128	0.142888
$136$	−1247.11	−0.786316
$137$	744.233	0.464118	0.232059	−	0.972702i	$-0.425454\pi$
$137$	744.233	0.464118	0.232059	+	0.972702i	$0.425454\pi$
$138$	1280.72	0.790018
$139$	267.712	0.163360	0.0816801	−	0.996659i	$-0.473971\pi$
$139$	267.712	0.163360	0.0816801	+	0.996659i	$0.473971\pi$
$140$	−3399.63	−2.05229
$141$	474.202	0.283227
$142$	−306.173	−0.180940
$143$	800.416	0.468071
$144$	1305.25	0.755351
$145$	−721.533	−0.413242
$146$	−4574.24	−2.59292
$147$	−364.711	−0.204631
$148$	5329.13	2.95981
$149$	−2478.30	−1.36262	−0.681308	−	0.731997i	$-0.738589\pi$
$149$	−2478.30	−1.36262	−0.681308	+	0.731997i	$0.738589\pi$
$150$	−874.418	−0.475973
$151$	1902.48	1.02531	0.512655	−	0.858595i	$-0.328662\pi$
$151$	1902.48	1.02531	0.512655	+	0.858595i	$0.328662\pi$
$152$	6996.10	3.73328
$153$	196.350	0.103751
$154$	3177.94	1.66289
$155$	−1088.91	−0.564281
$156$	1607.98	0.825267
$157$	−3638.32	−1.84949	−0.924745	−	0.380588i	$-0.875722\pi$
$157$	−3638.32	−1.84949	−0.924745	+	0.380588i	$0.875722\pi$
$158$	4288.37	2.15927
$159$	488.765	0.243783
$160$	2459.51	1.21526
$161$	1770.81	0.866826
$162$	−420.895	−0.204128
$163$	1188.42	0.571068	0.285534	−	0.958369i	$-0.407829\pi$
$163$	1188.42	0.571068	0.285534	+	0.958369i	$0.407829\pi$
$164$	7254.31	3.45407
$165$	−706.616	−0.333394
$166$	−7413.39	−3.46621
$167$	1667.69	0.772752	0.386376	−	0.922341i	$-0.373727\pi$
$167$	1667.69	0.772752	0.386376	+	0.922341i	$0.373727\pi$
$168$	3696.28	1.69746
$169$	−1401.26	−0.637806
$170$	941.041	0.424556
$171$	−1101.49	−0.492591
$172$	8597.92	3.81154
$173$	1854.57	0.815032	0.407516	−	0.913198i	$-0.366395\pi$
$173$	1854.57	0.815032	0.407516	+	0.913198i	$0.366395\pi$
$174$	1354.99	0.590352
$175$	−1209.02	−0.522249
$176$	−4115.10	−1.76243
$177$	177.000	0.0751646
$178$	−8089.38	−3.40632
$179$	3650.51	1.52431	0.762155	−	0.647394i	$-0.224141\pi$
$179$	3650.51	1.52431	0.762155	+	0.647394i	$0.224141\pi$
$180$	−1419.54	−0.587814
$181$	912.195	0.374602	0.187301	−	0.982303i	$-0.440026\pi$
$181$	912.195	0.374602	0.187301	+	0.982303i	$0.440026\pi$
$182$	3159.37	1.28675
$183$	1105.03	0.446373
$184$	−4696.38	−1.88164
$185$	−2328.16	−0.925243
$186$	2044.90	0.806124
$187$	−619.039	−0.242078
$188$	−3003.42	−1.16514
$189$	−581.955	−0.223974
$190$	−5279.09	−2.01571
$191$	2060.65	0.780646	0.390323	−	0.920678i	$-0.372363\pi$
$191$	2060.65	0.780646	0.390323	+	0.920678i	$0.372363\pi$
$192$	−1138.12	−0.427796
$193$	−1917.94	−0.715318	−0.357659	−	0.933852i	$-0.616425\pi$
$193$	−1917.94	−0.715318	−0.357659	+	0.933852i	$0.616425\pi$
$194$	−9411.71	−3.48310
$195$	−702.487	−0.257980
$196$	2309.94	0.841817
$197$	−2490.32	−0.900651	−0.450325	−	0.892864i	$-0.648692\pi$
$197$	−2490.32	−0.900651	−0.450325	+	0.892864i	$0.648692\pi$
$198$	1326.97	0.476282
$199$	3412.12	1.21547	0.607735	−	0.794140i	$-0.292078\pi$
$199$	3412.12	1.21547	0.607735	+	0.794140i	$0.292078\pi$
$200$	3206.46	1.13366
$201$	−533.849	−0.187337
$202$	−5887.10	−2.05057
$203$	1873.48	0.647748
$204$	−1243.61	−0.426814
$205$	−3169.23	−1.07975
$206$	−7689.34	−2.60069
$207$	739.414	0.248274
$208$	−4091.06	−1.36377
$209$	3472.71	1.14934
$210$	−2789.12	−0.916514
$211$	892.077	0.291057	0.145529	−	0.989354i	$-0.453512\pi$
$211$	892.077	0.291057	0.145529	+	0.989354i	$0.453512\pi$
$212$	−3095.66	−1.00288
$213$	−176.766	−0.0568629
$214$	−1526.92	−0.487747
$215$	−3756.22	−1.19150
$216$	1543.41	0.486184
$217$	2827.40	0.884499
$218$	4164.53	1.29384
$219$	−2640.90	−0.814864
$220$	4475.45	1.37152
$221$	−615.423	−0.187320
$222$	4372.12	1.32179
$223$	−2891.35	−0.868246	−0.434123	−	0.900854i	$-0.642942\pi$
$223$	−2891.35	−0.868246	−0.434123	+	0.900854i	$0.642942\pi$
$224$	−6386.20	−1.90489
$225$	−504.837	−0.149581
$226$	9530.10	2.80501
$227$	−1746.45	−0.510643	−0.255321	−	0.966856i	$-0.582181\pi$
$227$	−1746.45	−0.510643	−0.255321	+	0.966856i	$0.582181\pi$
$228$	6976.44	2.02643
$229$	−1195.00	−0.344837	−0.172418	−	0.985024i	$-0.555158\pi$
$229$	−1195.00	−0.344837	−0.172418	+	0.985024i	$0.555158\pi$
$230$	3543.77	1.01595
$231$	1834.75	0.522588
$232$	−4968.69	−1.40608
$233$	−4741.93	−1.33328	−0.666640	−	0.745380i	$-0.732268\pi$
$233$	−4741.93	−1.33328	−0.666640	+	0.745380i	$0.732268\pi$
$234$	1319.22	0.368548
$235$	1312.12	0.364227
$236$	−1121.05	−0.309213
$237$	2475.85	0.678582
$238$	−2443.45	−0.665483
$239$	−2889.11	−0.781928	−0.390964	−	0.920406i	$-0.627858\pi$
$239$	−2889.11	−0.781928	−0.390964	+	0.920406i	$0.627858\pi$
$240$	3611.63	0.971373
$241$	2648.11	0.707799	0.353899	−	0.935284i	$-0.384856\pi$
$241$	2648.11	0.707799	0.353899	+	0.935284i	$0.384856\pi$
$242$	2732.60	0.725860
$243$	−243.000	−0.0641500
$244$	−6998.87	−1.83630
$245$	−1009.16	−0.263154
$246$	5951.58	1.54252
$247$	3452.42	0.889362
$248$	−7498.58	−1.92000
$249$	−4280.05	−1.08931
$250$	−7811.29	−1.97612
$251$	2327.49	0.585299	0.292650	−	0.956220i	$-0.405463\pi$
$251$	2327.49	0.585299	0.292650	+	0.956220i	$0.405463\pi$
$252$	3685.89	0.921386
$253$	−2331.18	−0.579288
$254$	−3871.21	−0.956303
$255$	543.301	0.133423
$256$	−5108.29	−1.24714
$257$	−4224.97	−1.02547	−0.512737	−	0.858546i	$-0.671368\pi$
$257$	−4224.97	−1.02547	−0.512737	+	0.858546i	$0.671368\pi$
$258$	7053.91	1.70216
$259$	6045.15	1.45030
$260$	4449.30	1.06128
$261$	782.288	0.185527
$262$	11460.4	2.70239
$263$	3245.27	0.760881	0.380441	−	0.924805i	$-0.375772\pi$
$263$	3245.27	0.760881	0.380441	+	0.924805i	$0.375772\pi$
$264$	−4865.97	−1.13439
$265$	1352.42	0.313503
$266$	13707.3	3.15959
$267$	−4670.33	−1.07048
$268$	3381.21	0.770672
$269$	−4695.98	−1.06438	−0.532191	−	0.846624i	$-0.678631\pi$
$269$	−4695.98	−1.06438	−0.532191	+	0.846624i	$0.678631\pi$
$270$	−1164.62	−0.262506
$271$	1442.96	0.323444	0.161722	−	0.986836i	$-0.448295\pi$
$271$	1442.96	0.323444	0.161722	+	0.986836i	$0.448295\pi$
$272$	3164.01	0.705317
$273$	1824.03	0.404379
$274$	−3867.21	−0.852653
$275$	1591.62	0.349012
$276$	−4683.18	−1.02136
$277$	6781.78	1.47104	0.735519	−	0.677504i	$-0.236938\pi$
$277$	6781.78	1.47104	0.735519	+	0.677504i	$0.236938\pi$
$278$	−1391.10	−0.300117
$279$	1180.60	0.253336
$280$	10227.6	2.18292
$281$	916.383	0.194544	0.0972719	−	0.995258i	$-0.468988\pi$
$281$	916.383	0.194544	0.0972719	+	0.995258i	$0.468988\pi$
$282$	−2464.07	−0.520330
$283$	−1693.18	−0.355651	−0.177825	−	0.984062i	$-0.556906\pi$
$283$	−1693.18	−0.355651	−0.177825	+	0.984062i	$0.556906\pi$
$284$	1119.57	0.233924
$285$	−3047.83	−0.633467
$286$	−4159.15	−0.859916
$287$	8229.01	1.69248
$288$	−2666.61	−0.545595
$289$	−4437.03	−0.903121
$290$	3749.26	0.759186
$291$	−5433.76	−1.09461
$292$	16726.5	3.35220
$293$	−4712.00	−0.939515	−0.469757	−	0.882796i	$-0.655659\pi$
$293$	−4712.00	−0.939515	−0.469757	+	0.882796i	$0.655659\pi$
$294$	1895.12	0.375938
$295$	489.760	0.0966609
$296$	−16032.4	−3.14820
$297$	766.115	0.149678
$298$	12877.8	2.50333
$299$	−2317.56	−0.448254
$300$	3197.45	0.615350
$301$	9753.15	1.86765
$302$	−9885.75	−1.88364
$303$	−3398.86	−0.644421
$304$	−17749.6	−3.34871
$305$	3057.63	0.574031
$306$	−1020.28	−0.190606
$307$	7720.48	1.43528	0.717640	−	0.696414i	$-0.245222\pi$
$307$	7720.48	1.43528	0.717640	+	0.696414i	$0.245222\pi$
$308$	−11620.6	−2.14983
$309$	−4439.37	−0.817303
$310$	5658.25	1.03667
$311$	5773.53	1.05269	0.526345	−	0.850271i	$-0.323562\pi$
$311$	5773.53	1.05269	0.526345	+	0.850271i	$0.323562\pi$
$312$	−4837.54	−0.877794
$313$	−9875.96	−1.78346	−0.891729	−	0.452570i	$-0.850507\pi$
$313$	−9875.96	−1.78346	−0.891729	+	0.452570i	$0.850507\pi$
$314$	18905.6	3.39779
$315$	−1610.27	−0.288028
$316$	−15681.1	−2.79156
$317$	−6005.53	−1.06405	−0.532025	−	0.846729i	$-0.678569\pi$
$317$	−6005.53	−1.06405	−0.532025	+	0.846729i	$0.678569\pi$
$318$	−2539.74	−0.447866
$319$	−2466.35	−0.432881
$320$	−3149.19	−0.550141
$321$	−881.551	−0.153282
$322$	−9201.53	−1.59249
$323$	−2670.09	−0.459962
$324$	1539.07	0.263902
$325$	1582.32	0.270066
$326$	−6175.31	−1.04914
$327$	2404.35	0.406609
$328$	−21824.3	−3.67391
$329$	−3406.97	−0.570918
$330$	3671.74	0.612493
$331$	6895.07	1.14498	0.572488	−	0.819913i	$-0.305978\pi$
$331$	6895.07	1.14498	0.572488	+	0.819913i	$0.305978\pi$
$332$	27108.3	4.48121
$333$	2524.20	0.415392
$334$	−8665.71	−1.41966
$335$	−1477.16	−0.240914
$336$	−9377.71	−1.52261
$337$	6764.00	1.09335	0.546675	−	0.837345i	$-0.315893\pi$
$337$	6764.00	1.09335	0.546675	+	0.837345i	$0.315893\pi$
$338$	7281.28	1.17174
$339$	5502.12	0.881516
$340$	−3441.07	−0.548878
$341$	−3722.13	−0.591098
$342$	5723.61	0.904963
$343$	−4772.67	−0.751312
$344$	−25866.5	−4.05415
$345$	2045.96	0.319278
$346$	−9636.81	−1.49734
$347$	5474.73	0.846971	0.423485	−	0.905903i	$-0.360807\pi$
$347$	5474.73	0.846971	0.423485	+	0.905903i	$0.360807\pi$
$348$	−4954.73	−0.763222
$349$	−6375.89	−0.977919	−0.488960	−	0.872306i	$-0.662623\pi$
$349$	−6375.89	−0.977919	−0.488960	+	0.872306i	$0.662623\pi$
$350$	6282.37	0.959448
$351$	761.639	0.115821
$352$	8407.11	1.27301
$353$	4971.14	0.749539	0.374770	−	0.927118i	$-0.377722\pi$
$353$	4971.14	0.749539	0.374770	+	0.927118i	$0.377722\pi$
$354$	−919.734	−0.138089
$355$	−489.113	−0.0731251
$356$	29580.2	4.40378
$357$	−1410.70	−0.209138
$358$	−18968.9	−2.80039
$359$	2073.46	0.304827	0.152414	−	0.988317i	$-0.451295\pi$
$359$	2073.46	0.304827	0.152414	+	0.988317i	$0.451295\pi$
$360$	4270.63	0.625228
$361$	8119.79	1.18382
$362$	−4739.98	−0.688199
$363$	1577.64	0.228112
$364$	−11552.8	−1.66354
$365$	−7307.38	−1.04791
$366$	−5742.01	−0.820053
$367$	−6485.59	−0.922466	−0.461233	−	0.887279i	$-0.652593\pi$
$367$	−6485.59	−0.922466	−0.461233	+	0.887279i	$0.652593\pi$
$368$	11915.0	1.68781
$369$	3436.09	0.484758
$370$	12097.7	1.69981
$371$	−3511.59	−0.491409
$372$	−7477.51	−1.04218
$373$	−2610.25	−0.362342	−0.181171	−	0.983452i	$-0.557989\pi$
$373$	−2610.25	−0.362342	−0.181171	+	0.983452i	$0.557989\pi$
$374$	3216.67	0.444733
$375$	−4509.77	−0.621023
$376$	9035.66	1.23931
$377$	−2451.94	−0.334964
$378$	3023.98	0.411473
$379$	−5609.34	−0.760244	−0.380122	−	0.924936i	$-0.624118\pi$
$379$	−5609.34	−0.760244	−0.380122	+	0.924936i	$0.624118\pi$
$380$	19303.9	2.60597
$381$	−2235.00	−0.300532
$382$	−10707.6	−1.43416
$383$	2978.80	0.397414	0.198707	−	0.980059i	$-0.436326\pi$
$383$	2978.80	0.397414	0.198707	+	0.980059i	$0.436326\pi$
$384$	−1197.00	−0.159073
$385$	5076.77	0.672042
$386$	9966.08	1.31415
$387$	4072.51	0.534928
$388$	34415.5	4.50305
$389$	10137.6	1.32132	0.660662	−	0.750683i	$-0.270276\pi$
$389$	10137.6	1.32132	0.660662	+	0.750683i	$0.270276\pi$
$390$	3650.29	0.473948
$391$	1792.39	0.231829
$392$	−6949.36	−0.895397
$393$	6616.56	0.849265
$394$	12940.3	1.65463
$395$	6850.70	0.872649
$396$	−4852.29	−0.615750
$397$	12921.9	1.63359	0.816793	−	0.576930i	$-0.195750\pi$
$397$	12921.9	1.63359	0.816793	+	0.576930i	$0.195750\pi$
$398$	−17730.2	−2.23300
$399$	7913.80	0.992947
$400$	−8135.02	−1.01688
$401$	−9556.68	−1.19012	−0.595060	−	0.803681i	$-0.702872\pi$
$401$	−9556.68	−1.19012	−0.595060	+	0.803681i	$0.702872\pi$
$402$	2774.01	0.344167
$403$	−3700.38	−0.457393
$404$	21527.1	2.65103
$405$	−672.383	−0.0824962
$406$	−9735.07	−1.19001
$407$	−7958.14	−0.969215
$408$	3741.34	0.453980
$409$	8381.64	1.01331	0.506657	−	0.862148i	$-0.330881\pi$
$409$	8381.64	1.01331	0.506657	+	0.862148i	$0.330881\pi$
$410$	16468.1	1.98366
$411$	−2232.70	−0.267958
$412$	28117.3	3.36224
$413$	−1271.68	−0.151514
$414$	−3842.17	−0.456117
$415$	−11842.9	−1.40084
$416$	8358.00	0.985059
$417$	−803.137	−0.0943161
$418$	−18045.0	−2.11151
$419$	−9005.40	−1.04998	−0.524991	−	0.851108i	$-0.675931\pi$
$419$	−9005.40	−1.04998	−0.524991	+	0.851108i	$0.675931\pi$
$420$	10198.9	1.18489
$421$	754.089	0.0872970	0.0436485	−	0.999047i	$-0.486102\pi$
$421$	754.089	0.0872970	0.0436485	+	0.999047i	$0.486102\pi$
$422$	−4635.45	−0.534716
$423$	−1422.61	−0.163521
$424$	9313.15	1.06671
$425$	−1223.76	−0.139673
$426$	918.518	0.104466
$427$	−7939.24	−0.899782
$428$	5583.42	0.630572
$429$	−2401.25	−0.270241
$430$	19518.2	2.18896
$431$	−2505.79	−0.280046	−0.140023	−	0.990148i	$-0.544718\pi$
$431$	−2505.79	−0.280046	−0.140023	+	0.990148i	$0.544718\pi$
$432$	−3915.74	−0.436102
$433$	−176.502	−0.0195893	−0.00979465	−	0.999952i	$-0.503118\pi$
$433$	−176.502	−0.0195893	−0.00979465	+	0.999952i	$0.503118\pi$
$434$	−14691.8	−1.62496
$435$	2164.60	0.238585
$436$	−15228.3	−1.67271
$437$	−10055.0	−1.10068
$438$	13722.7	1.49703
$439$	−2223.93	−0.241782	−0.120891	−	0.992666i	$-0.538575\pi$
$439$	−2223.93	−0.241782	−0.120891	+	0.992666i	$0.538575\pi$
$440$	−13464.2	−1.45882
$441$	1094.13	0.118144
$442$	3197.88	0.344135
$443$	−6065.11	−0.650479	−0.325239	−	0.945632i	$-0.605445\pi$
$443$	−6065.11	−0.650479	−0.325239	+	0.945632i	$0.605445\pi$
$444$	−15987.4	−1.70885
$445$	−12922.8	−1.37663
$446$	15024.1	1.59510
$447$	7434.89	0.786707
$448$	8176.99	0.862336
$449$	−127.588	−0.0134104	−0.00670518	−	0.999978i	$-0.502134\pi$
$449$	−127.588	−0.0134104	−0.00670518	+	0.999978i	$0.502134\pi$
$450$	2623.25	0.274803
$451$	−10833.1	−1.13106
$452$	−34848.4	−3.62640
$453$	−5707.44	−0.591962
$454$	9074.97	0.938126
$455$	5047.11	0.520027
$456$	−20988.3	−2.15541
$457$	15246.9	1.56065	0.780327	−	0.625372i	$-0.215053\pi$
$457$	15246.9	1.56065	0.780327	+	0.625372i	$0.215053\pi$
$458$	6209.49	0.633517
$459$	−589.049	−0.0599008
$460$	−12958.4	−1.31345
$461$	8148.20	0.823210	0.411605	−	0.911362i	$-0.364968\pi$
$461$	8148.20	0.823210	0.411605	+	0.911362i	$0.364968\pi$
$462$	−9533.81	−0.960071
$463$	6518.90	0.654340	0.327170	−	0.944966i	$-0.393905\pi$
$463$	6518.90	0.654340	0.327170	+	0.944966i	$0.393905\pi$
$464$	12605.9	1.26124
$465$	3266.74	0.325788
$466$	24640.2	2.44943
$467$	−263.751	−0.0261348	−0.0130674	−	0.999915i	$-0.504160\pi$
$467$	−263.751	−0.0261348	−0.0130674	+	0.999915i	$0.504160\pi$
$468$	−4823.95	−0.476468
$469$	3835.51	0.377628
$470$	−6818.10	−0.669139
$471$	10915.0	1.06780
$472$	3372.64	0.328895
$473$	−12839.5	−1.24812
$474$	−12865.1	−1.24666
$475$	6865.10	0.663142
$476$	8934.86	0.860354
$477$	−1466.29	−0.140748
$478$	15012.5	1.43652
$479$	3204.17	0.305642	0.152821	−	0.988254i	$-0.451164\pi$
$479$	3204.17	0.305642	0.152821	+	0.988254i	$0.451164\pi$
$480$	−7378.52	−0.701629
$481$	−7911.65	−0.749980
$482$	−13760.2	−1.30033
$483$	−5312.42	−0.500462
$484$	−9992.20	−0.938411
$485$	−15035.3	−1.40766
$486$	1262.69	0.117853
$487$	5211.90	0.484957	0.242478	−	0.970157i	$-0.422040\pi$
$487$	5211.90	0.484957	0.242478	+	0.970157i	$0.422040\pi$
$488$	21055.8	1.95318
$489$	−3565.26	−0.329706
$490$	5243.82	0.483453
$491$	−10189.9	−0.936589	−0.468295	−	0.883572i	$-0.655131\pi$
$491$	−10189.9	−0.936589	−0.468295	+	0.883572i	$0.655131\pi$
$492$	−21762.9	−1.99421
$493$	1896.32	0.173237
$494$	−17939.6	−1.63389
$495$	2119.85	0.192485
$496$	19024.4	1.72222
$497$	1270.00	0.114622
$498$	22240.2	2.00122
$499$	−20602.0	−1.84824	−0.924120	−	0.382103i	$-0.875200\pi$
$499$	−20602.0	−1.84824	−0.924120	+	0.382103i	$0.875200\pi$
$500$	28563.3	2.55478
$501$	−5003.07	−0.446149
$502$	−12094.2	−1.07528
$503$	−8471.66	−0.750959	−0.375480	−	0.926831i	$-0.622522\pi$
$503$	−8471.66	−0.750959	−0.375480	+	0.926831i	$0.622522\pi$
$504$	−11088.8	−0.980032
$505$	−9404.67	−0.828718
$506$	12113.4	1.06424
$507$	4203.78	0.368237
$508$	14155.7	1.23633
$509$	11199.1	0.975230	0.487615	−	0.873059i	$-0.337867\pi$
$509$	11199.1	0.975230	0.487615	+	0.873059i	$0.337867\pi$
$510$	−2823.12	−0.245118
$511$	18973.9	1.64257
$512$	23351.9	2.01566
$513$	3304.47	0.284398
$514$	21954.0	1.88395
$515$	−12283.8	−1.05104
$516$	−25793.8	−2.20060
$517$	4485.10	0.381537
$518$	−31412.1	−2.66441
$519$	−5563.72	−0.470559
$520$	−13385.5	−1.12883
$521$	−3200.10	−0.269096	−0.134548	−	0.990907i	$-0.542958\pi$
$521$	−3200.10	−0.269096	−0.134548	+	0.990907i	$0.542958\pi$
$522$	−4064.96	−0.340840
$523$	16516.2	1.38089	0.690443	−	0.723386i	$-0.257415\pi$
$523$	16516.2	1.38089	0.690443	+	0.723386i	$0.257415\pi$
$524$	−41906.9	−3.49372
$525$	3627.07	0.301520
$526$	−16863.2	−1.39785
$527$	2861.86	0.236555
$528$	12345.3	1.01754
$529$	−5417.21	−0.445238
$530$	−7027.48	−0.575951
$531$	−531.000	−0.0433963
$532$	−50123.2	−4.08480
$533$	−10769.8	−0.875219
$534$	24268.1	1.96664
$535$	−2439.26	−0.197118
$536$	−10172.2	−0.819724
$537$	−10951.5	−0.880061
$538$	24401.4	1.95543
$539$	−3449.51	−0.275660
$540$	4258.63	0.339375
$541$	20852.2	1.65713	0.828566	−	0.559892i	$-0.189157\pi$
$541$	20852.2	1.65713	0.828566	+	0.559892i	$0.189157\pi$
$542$	−7497.95	−0.594215
$543$	−2736.58	−0.216276
$544$	−6464.04	−0.509455
$545$	6652.86	0.522894
$546$	−9478.11	−0.742904
$547$	809.628	0.0632856	0.0316428	−	0.999499i	$-0.489926\pi$
$547$	809.628	0.0632856	0.0316428	+	0.999499i	$0.489926\pi$
$548$	14141.1	1.10233
$549$	−3315.09	−0.257714
$550$	−8270.43	−0.641186
$551$	−10638.1	−0.822499
$552$	14089.1	1.08636
$553$	−17788.1	−1.36786
$554$	−35239.7	−2.70251
$555$	6984.49	0.534189
$556$	5086.78	0.387999
$557$	−21643.2	−1.64641	−0.823206	−	0.567743i	$-0.807817\pi$
$557$	−21643.2	−1.64641	−0.823206	+	0.567743i	$0.807817\pi$
$558$	−6134.69	−0.465416
$559$	−12764.5	−0.965800
$560$	−25948.2	−1.95806
$561$	1857.12	0.139764
$562$	−4761.74	−0.357406
$563$	23646.1	1.77010	0.885050	−	0.465496i	$-0.154124\pi$
$563$	23646.1	1.77010	0.885050	+	0.465496i	$0.154124\pi$
$564$	9010.27	0.672697
$565$	15224.4	1.13362
$566$	8798.17	0.653383
$567$	1745.87	0.129311
$568$	−3368.18	−0.248813
$569$	−19407.6	−1.42990	−0.714948	−	0.699177i	$-0.753550\pi$
$569$	−19407.6	−1.42990	−0.714948	+	0.699177i	$0.753550\pi$
$570$	15837.3	1.16377
$571$	16659.0	1.22094	0.610470	−	0.792039i	$-0.290980\pi$
$571$	16659.0	1.22094	0.610470	+	0.792039i	$0.290980\pi$
$572$	15208.6	1.11172
$573$	−6181.95	−0.450706
$574$	−42759.9	−3.10935
$575$	−4608.44	−0.334235
$576$	3414.37	0.246988
$577$	−19454.1	−1.40361	−0.701805	−	0.712369i	$-0.747622\pi$
$577$	−19454.1	−1.40361	−0.701805	+	0.712369i	$0.747622\pi$
$578$	23055.9	1.65917
$579$	5753.82	0.412989
$580$	−13709.8	−0.981496
$581$	30750.6	2.19578
$582$	28235.1	2.01097
$583$	4622.84	0.328402
$584$	−50320.8	−3.56557
$585$	2107.46	0.148945
$586$	24484.7	1.72603
$587$	−2742.52	−0.192838	−0.0964190	−	0.995341i	$-0.530739\pi$
$587$	−2742.52	−0.192838	−0.0964190	+	0.995341i	$0.530739\pi$
$588$	−6929.83	−0.486023
$589$	−16054.6	−1.12312
$590$	−2544.91	−0.177580
$591$	7470.97	0.519991
$592$	40675.4	2.82390
$593$	9829.13	0.680664	0.340332	−	0.940305i	$-0.389460\pi$
$593$	9829.13	0.680664	0.340332	+	0.940305i	$0.389460\pi$
$594$	−3980.92	−0.274982
$595$	−3903.42	−0.268949
$596$	−47089.9	−3.23637
$597$	−10236.3	−0.701752
$598$	12042.6	0.823509
$599$	25261.4	1.72312	0.861562	−	0.507653i	$-0.169487\pi$
$599$	25261.4	1.72312	0.861562	+	0.507653i	$0.169487\pi$
$600$	−9619.39	−0.654517
$601$	23269.7	1.57935	0.789677	−	0.613523i	$-0.210248\pi$
$601$	23269.7	1.57935	0.789677	+	0.613523i	$0.210248\pi$
$602$	−50679.7	−3.43115
$603$	1601.55	0.108159
$604$	36148.9	2.43523
$605$	4365.35	0.293350
$606$	17661.3	1.18390
$607$	11725.1	0.784029	0.392015	−	0.919959i	$-0.371778\pi$
$607$	11725.1	0.784029	0.392015	+	0.919959i	$0.371778\pi$
$608$	36262.3	2.41880
$609$	−5620.45	−0.373977
$610$	−15888.2	−1.05458
$611$	4458.90	0.295234
$612$	3730.82	0.246421
$613$	−4075.52	−0.268530	−0.134265	−	0.990945i	$-0.542867\pi$
$613$	−4075.52	−0.268530	−0.134265	+	0.990945i	$0.542867\pi$
$614$	−40117.5	−2.63682
$615$	9507.69	0.623393
$616$	34960.2	2.28667
$617$	−2326.85	−0.151824	−0.0759121	−	0.997115i	$-0.524187\pi$
$617$	−2326.85	−0.151824	−0.0759121	+	0.997115i	$0.524187\pi$
$618$	23068.0	1.50151
$619$	27579.3	1.79080	0.895400	−	0.445263i	$-0.146890\pi$
$619$	27579.3	1.79080	0.895400	+	0.445263i	$0.146890\pi$
$620$	−20690.3	−1.34023
$621$	−2218.24	−0.143341
$622$	−30000.6	−1.93395
$623$	33554.6	2.15784
$624$	12273.2	0.787372
$625$	−5466.95	−0.349885
$626$	51317.8	3.27648
$627$	−10418.1	−0.663573
$628$	−69131.5	−4.39275
$629$	6118.84	0.387876
$630$	8367.37	0.529149
$631$	−4818.78	−0.304013	−0.152007	−	0.988379i	$-0.548574\pi$
$631$	−4818.78	−0.304013	−0.152007	+	0.988379i	$0.548574\pi$
$632$	47176.0	2.96924
$633$	−2676.23	−0.168042
$634$	31206.2	1.95482
$635$	−6184.27	−0.386481
$636$	9286.98	0.579014
$637$	−3429.36	−0.213306
$638$	12815.7	0.795266
$639$	530.298	0.0328298
$640$	−3312.10	−0.204566
$641$	−16896.8	−1.04116	−0.520580	−	0.853813i	$-0.674284\pi$
$641$	−16896.8	−1.04116	−0.520580	+	0.853813i	$0.674284\pi$
$642$	4580.75	0.281601
$643$	−30887.2	−1.89436	−0.947180	−	0.320702i	$-0.896081\pi$
$643$	−30887.2	−1.89436	−0.947180	+	0.320702i	$0.896081\pi$
$644$	33646.9	2.05881
$645$	11268.7	0.687911
$646$	13874.4	0.845019
$647$	−4782.21	−0.290585	−0.145292	−	0.989389i	$-0.546412\pi$
$647$	−4782.21	−0.290585	−0.145292	+	0.989389i	$0.546412\pi$
$648$	−4630.23	−0.280699
$649$	1674.10	0.101255
$650$	−8222.11	−0.496150
$651$	−8482.19	−0.510666
$652$	22581.0	1.35635
$653$	−33078.5	−1.98233	−0.991164	−	0.132639i	$-0.957655\pi$
$653$	−33078.5	−1.98233	−0.991164	+	0.132639i	$0.957655\pi$
$654$	−12493.6	−0.747000
$655$	18308.1	1.09215
$656$	55369.6	3.29546
$657$	7922.69	0.470462
$658$	17703.4	1.04886
$659$	10359.2	0.612351	0.306175	−	0.951975i	$-0.400951\pi$
$659$	10359.2	0.612351	0.306175	+	0.951975i	$0.400951\pi$
$660$	−13426.3	−0.791848
$661$	−14606.6	−0.859502	−0.429751	−	0.902947i	$-0.641399\pi$
$661$	−14606.6	−0.859502	−0.429751	+	0.902947i	$0.641399\pi$
$662$	−35828.4	−2.10349
$663$	1846.27	0.108149
$664$	−81554.0	−4.76643
$665$	21897.6	1.27692
$666$	−13116.4	−0.763136
$667$	7141.17	0.414553
$668$	31687.6	1.83538
$669$	8674.04	0.501282
$670$	7675.70	0.442594
$671$	10451.6	0.601312
$672$	19158.6	1.09979
$673$	4901.10	0.280719	0.140359	−	0.990101i	$-0.455174\pi$
$673$	4901.10	0.280719	0.140359	+	0.990101i	$0.455174\pi$
$674$	−35147.4	−2.00865
$675$	1514.51	0.0863608
$676$	−26625.2	−1.51486
$677$	−7783.32	−0.441857	−0.220929	−	0.975290i	$-0.570909\pi$
$677$	−7783.32	−0.441857	−0.220929	+	0.975290i	$0.570909\pi$
$678$	−28590.3	−1.61948
$679$	39039.6	2.20648
$680$	10352.3	0.583813
$681$	5239.35	0.294820
$682$	19341.1	1.08594
$683$	−5065.20	−0.283770	−0.141885	−	0.989883i	$-0.545316\pi$
$683$	−5065.20	−0.283770	−0.141885	+	0.989883i	$0.545316\pi$
$684$	−20929.3	−1.16996
$685$	−6177.89	−0.344592
$686$	24800.0	1.38027
$687$	3584.99	0.199092
$688$	65624.9	3.63652
$689$	4595.83	0.254118
$690$	−10631.3	−0.586562
$691$	−987.814	−0.0543824	−0.0271912	−	0.999630i	$-0.508656\pi$
$691$	−987.814	−0.0543824	−0.0271912	+	0.999630i	$0.508656\pi$
$692$	35238.6	1.93579
$693$	−5504.25	−0.301716
$694$	−28448.0	−1.55601
$695$	−2222.29	−0.121289
$696$	14906.1	0.811801
$697$	8329.32	0.452648
$698$	33130.7	1.79658
$699$	14225.8	0.769769
$700$	−22972.5	−1.24040
$701$	24301.9	1.30937	0.654685	−	0.755902i	$-0.272801\pi$
$701$	24301.9	1.30937	0.654685	+	0.755902i	$0.272801\pi$
$702$	−3957.66	−0.212781
$703$	−34325.7	−1.84156
$704$	−10764.6	−0.576287
$705$	−3936.36	−0.210287
$706$	−25831.3	−1.37701
$707$	24419.6	1.29900
$708$	3363.16	0.178524
$709$	19625.0	1.03954	0.519769	−	0.854307i	$-0.326018\pi$
$709$	19625.0	1.03954	0.519769	+	0.854307i	$0.326018\pi$
$710$	2541.55	0.134342
$711$	−7427.56	−0.391779
$712$	−88990.5	−4.68408
$713$	10777.2	0.566072
$714$	7330.34	0.384217
$715$	−6644.27	−0.347527
$716$	69362.9	3.62041
$717$	8667.32	0.451446
$718$	−10774.2	−0.560012
$719$	−12274.5	−0.636663	−0.318331	−	0.947979i	$-0.603123\pi$
$719$	−12274.5	−0.636663	−0.318331	+	0.947979i	$0.603123\pi$
$720$	−10834.9	−0.560822
$721$	31895.2	1.64749
$722$	−42192.4	−2.17484
$723$	−7944.32	−0.408648
$724$	17332.5	0.889722
$725$	−4875.65	−0.249762
$726$	−8197.80	−0.419076
$727$	9818.35	0.500884	0.250442	−	0.968132i	$-0.419424\pi$
$727$	9818.35	0.500884	0.250442	+	0.968132i	$0.419424\pi$
$728$	34755.9	1.76942
$729$	729.000	0.0370370
$730$	37970.9	1.92516
$731$	9872.04	0.499495
$732$	20996.6	1.06019
$733$	−36133.7	−1.82078	−0.910388	−	0.413756i	$-0.864217\pi$
$733$	−36133.7	−1.82078	−0.910388	+	0.413756i	$0.864217\pi$
$734$	33700.7	1.69471
$735$	3027.47	0.151932
$736$	−24342.3	−1.21912
$737$	−5049.26	−0.252363
$738$	−17854.7	−0.890572
$739$	15833.8	0.788166	0.394083	−	0.919075i	$-0.371062\pi$
$739$	15833.8	0.788166	0.394083	+	0.919075i	$0.371062\pi$
$740$	−44237.2	−2.19756
$741$	−10357.3	−0.513473
$742$	18247.1	0.902792
$743$	23888.6	1.17953	0.589764	−	0.807576i	$-0.299221\pi$
$743$	23888.6	1.17953	0.589764	+	0.807576i	$0.299221\pi$
$744$	22495.7	1.10851
$745$	20572.4	1.01170
$746$	13563.5	0.665676
$747$	12840.2	0.628911
$748$	−11762.3	−0.574963
$749$	6333.62	0.308979
$750$	23433.9	1.14091
$751$	−29824.3	−1.44914	−0.724571	−	0.689200i	$-0.757962\pi$
$751$	−29824.3	−1.44914	−0.724571	+	0.689200i	$0.757962\pi$
$752$	−22924.1	−1.11164
$753$	−6982.48	−0.337923
$754$	12740.9	0.615378
$755$	−15792.5	−0.761257
$756$	−11057.7	−0.531963
$757$	7136.54	0.342645	0.171322	−	0.985215i	$-0.445196\pi$
$757$	7136.54	0.342645	0.171322	+	0.985215i	$0.445196\pi$
$758$	29147.5	1.39668
$759$	6993.53	0.334452
$760$	−58074.8	−2.77184
$761$	−19820.8	−0.944157	−0.472079	−	0.881557i	$-0.656496\pi$
$761$	−19820.8	−0.944157	−0.472079	+	0.881557i	$0.656496\pi$
$762$	11613.6	0.552122
$763$	−17274.4	−0.819626
$764$	39154.2	1.85412
$765$	−1629.90	−0.0770318
$766$	−15478.5	−0.730108
$767$	1664.32	0.0783510
$768$	15324.9	0.720037
$769$	23969.6	1.12401	0.562006	−	0.827133i	$-0.310030\pi$
$769$	23969.6	1.12401	0.562006	+	0.827133i	$0.310030\pi$
$770$	−26380.1	−1.23464
$771$	12674.9	0.592057
$772$	−36442.6	−1.69896
$773$	41618.3	1.93649	0.968245	−	0.250004i	$-0.0804320\pi$
$773$	41618.3	1.93649	0.968245	+	0.250004i	$0.0804320\pi$
$774$	−21161.7	−0.982742
$775$	−7358.17	−0.341049
$776$	−103537.	−4.78966
$777$	−18135.5	−0.837331
$778$	−52677.2	−2.42747
$779$	−46726.2	−2.14909
$780$	−13347.9	−0.612733
$781$	−1671.89	−0.0766004
$782$	−9313.70	−0.425904
$783$	−2346.86	−0.107114
$784$	17631.0	0.803161
$785$	30201.8	1.37318
$786$	−34381.2	−1.56023
$787$	17130.9	0.775920	0.387960	−	0.921676i	$-0.373180\pi$
$787$	17130.9	0.775920	0.387960	+	0.921676i	$0.373180\pi$
$788$	−47318.4	−2.13915
$789$	−9735.80	−0.439295
$790$	−35597.9	−1.60319
$791$	−39530.7	−1.77693
$792$	14597.9	0.654942
$793$	10390.6	0.465296
$794$	−67145.5	−3.00114
$795$	−4057.25	−0.181001
$796$	64833.3	2.88688
$797$	−3545.01	−0.157554	−0.0787770	−	0.996892i	$-0.525102\pi$
$797$	−3545.01	−0.157554	−0.0787770	+	0.996892i	$0.525102\pi$
$798$	−41122.0	−1.82419
$799$	−3448.50	−0.152690
$800$	16619.8	0.734497
$801$	14011.0	0.618045
$802$	49658.8	2.18643
$803$	−24978.1	−1.09771
$804$	−10143.6	−0.444948
$805$	−14699.5	−0.643589
$806$	19228.1	0.840298
$807$	14087.9	0.614521
$808$	−64763.4	−2.81976
$809$	−11108.5	−0.482763	−0.241381	−	0.970430i	$-0.577600\pi$
$809$	−11108.5	−0.482763	−0.241381	+	0.970430i	$0.577600\pi$
$810$	3493.86	0.151558
$811$	−11416.7	−0.494320	−0.247160	−	0.968975i	$-0.579497\pi$
$811$	−11416.7	−0.494320	−0.247160	+	0.968975i	$0.579497\pi$
$812$	35597.9	1.53847
$813$	−4328.87	−0.186741
$814$	41352.4	1.78059
$815$	−9865.09	−0.423999
$816$	−9492.02	−0.407215
$817$	−55380.6	−2.37151
$818$	−43553.0	−1.86161
$819$	−5472.10	−0.233468
$820$	−60218.2	−2.56453
$821$	27265.7	1.15905	0.579524	−	0.814955i	$-0.303238\pi$
$821$	27265.7	1.15905	0.579524	+	0.814955i	$0.303238\pi$
$822$	11601.6	0.492279
$823$	18672.1	0.790849	0.395424	−	0.918499i	$-0.370598\pi$
$823$	18672.1	0.790849	0.395424	+	0.918499i	$0.370598\pi$
$824$	−84589.7	−3.57624
$825$	−4774.85	−0.201502
$826$	6607.95	0.278354
$827$	33292.9	1.39989	0.699945	−	0.714197i	$-0.253208\pi$
$827$	33292.9	1.39989	0.699945	+	0.714197i	$0.253208\pi$
$828$	14049.5	0.589680
$829$	21571.9	0.903768	0.451884	−	0.892077i	$-0.350752\pi$
$829$	21571.9	0.903768	0.451884	+	0.892077i	$0.350752\pi$
$830$	61538.7	2.57354
$831$	−20345.3	−0.849304
$832$	−10701.7	−0.445932
$833$	2652.25	0.110318
$834$	4173.29	0.173273
$835$	−13843.5	−0.573743
$836$	65984.7	2.72982
$837$	−3541.81	−0.146264
$838$	46794.2	1.92897
$839$	24952.4	1.02676	0.513381	−	0.858161i	$-0.328393\pi$
$839$	24952.4	1.02676	0.513381	+	0.858161i	$0.328393\pi$
$840$	−30682.9	−1.26031
$841$	−16833.8	−0.690219
$842$	−3918.43	−0.160378
$843$	−2749.15	−0.112320
$844$	16950.3	0.691294
$845$	11631.9	0.473549
$846$	7392.20	0.300413
$847$	−11334.8	−0.459820
$848$	−23628.1	−0.956830
$849$	5079.54	0.205335
$850$	6358.95	0.256600
$851$	23042.3	0.928180
$852$	−3358.71	−0.135056
$853$	22193.3	0.890836	0.445418	−	0.895323i	$-0.353055\pi$
$853$	22193.3	0.890836	0.445418	+	0.895323i	$0.353055\pi$
$854$	41254.2	1.65303
$855$	9143.50	0.365732
$856$	−16797.5	−0.670708
$857$	5155.43	0.205491	0.102746	−	0.994708i	$-0.467237\pi$
$857$	5155.43	0.205491	0.102746	+	0.994708i	$0.467237\pi$
$858$	12477.5	0.496473
$859$	−18059.6	−0.717327	−0.358664	−	0.933467i	$-0.616767\pi$
$859$	−18059.6	−0.717327	−0.358664	+	0.933467i	$0.616767\pi$
$860$	−71371.6	−2.82994
$861$	−24687.0	−0.977156
$862$	13020.7	0.514486
$863$	−3135.68	−0.123685	−0.0618423	−	0.998086i	$-0.519698\pi$
$863$	−3135.68	−0.123685	−0.0618423	+	0.998086i	$0.519698\pi$
$864$	7999.82	0.314999
$865$	−15394.9	−0.605134
$866$	917.149	0.0359884
$867$	13311.1	0.521417
$868$	53723.1	2.10078
$869$	23417.1	0.914122
$870$	−11247.8	−0.438316
$871$	−5019.76	−0.195279
$872$	45813.6	1.77918
$873$	16301.3	0.631976
$874$	52248.4	2.02211
$875$	32401.0	1.25183
$876$	−50179.4	−1.93539
$877$	24764.4	0.953516	0.476758	−	0.879035i	$-0.341812\pi$
$877$	24764.4	0.953516	0.476758	+	0.879035i	$0.341812\pi$
$878$	11556.1	0.444190
$879$	14136.0	0.542429
$880$	34159.5	1.30854
$881$	−24195.2	−0.925265	−0.462633	−	0.886550i	$-0.653095\pi$
$881$	−24195.2	−0.925265	−0.462633	+	0.886550i	$0.653095\pi$
$882$	−5685.37	−0.217048
$883$	−26387.8	−1.00569	−0.502843	−	0.864378i	$-0.667713\pi$
$883$	−26387.8	−1.00569	−0.502843	+	0.864378i	$0.667713\pi$
$884$	−11693.6	−0.444907
$885$	−1469.28	−0.0558072
$886$	31515.8	1.19503
$887$	45619.8	1.72690	0.863452	−	0.504431i	$-0.168298\pi$
$887$	45619.8	1.72690	0.863452	+	0.504431i	$0.168298\pi$
$888$	48097.3	1.81761
$889$	16057.7	0.605801
$890$	67150.1	2.52908
$891$	−2298.35	−0.0864169
$892$	−54938.2	−2.06218
$893$	19345.5	0.724942
$894$	−38633.5	−1.44530
$895$	−30302.9	−1.13175
$896$	8599.98	0.320653
$897$	6952.68	0.258799
$898$	662.978	0.0246368
$899$	11402.1	0.423005
$900$	−9592.36	−0.355273
$901$	−3554.40	−0.131425
$902$	56291.3	2.07793
$903$	−29259.4	−1.07829
$904$	104840.	3.85721
$905$	−7572.15	−0.278129
$906$	29657.2	1.08752
$907$	−28604.5	−1.04719	−0.523593	−	0.851968i	$-0.675409\pi$
$907$	−28604.5	−1.04719	−0.523593	+	0.851968i	$0.675409\pi$
$908$	−33184.1	−1.21283
$909$	10196.6	0.372056
$910$	−26226.0	−0.955367
$911$	23089.0	0.839706	0.419853	−	0.907592i	$-0.362082\pi$
$911$	23089.0	0.839706	0.419853	+	0.907592i	$0.362082\pi$
$912$	53248.7	1.93338
$913$	−40481.6	−1.46741
$914$	−79226.4	−2.86715
$915$	−9172.89	−0.331417
$916$	−22706.0	−0.819027
$917$	−47537.5	−1.71192
$918$	3060.84	0.110047
$919$	−14323.8	−0.514145	−0.257072	−	0.966392i	$-0.582758\pi$
$919$	−14323.8	−0.514145	−0.257072	+	0.966392i	$0.582758\pi$
$920$	38984.7	1.39705
$921$	−23161.4	−0.828659
$922$	−42340.0	−1.51236
$923$	−1662.12	−0.0592735
$924$	34861.9	1.24121
$925$	−15732.2	−0.559213
$926$	−33873.8	−1.20212
$927$	13318.1	0.471870
$928$	−25753.8	−0.911001
$929$	−34587.5	−1.22151	−0.610753	−	0.791821i	$-0.709133\pi$
$929$	−34587.5	−1.22151	−0.610753	+	0.791821i	$0.709133\pi$
$930$	−16974.7	−0.598520
$931$	−14878.7	−0.523770
$932$	−90101.0	−3.16669
$933$	−17320.6	−0.607771
$934$	1370.51	0.0480134
$935$	5138.66	0.179735
$936$	14512.6	0.506795
$937$	30905.7	1.07753	0.538765	−	0.842456i	$-0.318891\pi$
$937$	30905.7	1.07753	0.538765	+	0.842456i	$0.318891\pi$
$938$	−19930.2	−0.693758
$939$	29627.9	1.02968
$940$	24931.5	0.865081
$941$	−20269.1	−0.702184	−0.351092	−	0.936341i	$-0.614189\pi$
$941$	−20269.1	−0.702184	−0.351092	+	0.936341i	$0.614189\pi$
$942$	−56716.8	−1.96171
$943$	31366.6	1.08318
$944$	−8556.61	−0.295015
$945$	4830.82	0.166293
$946$	66717.3	2.29299
$947$	3806.38	0.130613	0.0653066	−	0.997865i	$-0.479197\pi$
$947$	3806.38	0.130613	0.0653066	+	0.997865i	$0.479197\pi$
$948$	47043.4	1.61171
$949$	−24832.2	−0.849408
$950$	−35672.7	−1.21829
$951$	18016.6	0.614330
$952$	−26880.1	−0.915115
$953$	−58087.5	−1.97444	−0.987219	−	0.159370i	$-0.949054\pi$
$953$	−58087.5	−1.97444	−0.987219	+	0.159370i	$0.949054\pi$
$954$	7619.22	0.258576
$955$	−17105.5	−0.579603
$956$	−54895.6	−1.85717
$957$	7399.05	0.249924
$958$	−16649.6	−0.561509
$959$	16041.1	0.540140
$960$	9447.58	0.317624
$961$	−12583.3	−0.422387
$962$	41110.8	1.37782
$963$	2644.65	0.0884971
$964$	50316.4	1.68110
$965$	15920.9	0.531100
$966$	27604.6	0.919423
$967$	−9873.60	−0.328349	−0.164175	−	0.986431i	$-0.552496\pi$
$967$	−9873.60	−0.328349	−0.164175	+	0.986431i	$0.552496\pi$
$968$	30061.1	0.998141
$969$	8010.27	0.265559
$970$	78126.9	2.58609
$971$	55181.6	1.82375	0.911875	−	0.410468i	$-0.134634\pi$
$971$	55181.6	1.82375	0.911875	+	0.410468i	$0.134634\pi$
$972$	−4617.22	−0.152364
$973$	5770.24	0.190119
$974$	−27082.3	−0.890937
$975$	−4746.96	−0.155922
$976$	−53419.9	−1.75198
$977$	14494.0	0.474620	0.237310	−	0.971434i	$-0.423734\pi$
$977$	14494.0	0.474620	0.237310	+	0.971434i	$0.423734\pi$
$978$	18525.9	0.605720
$979$	−44172.9	−1.44206
$980$	−19174.9	−0.625020
$981$	−7213.06	−0.234756
$982$	52949.4	1.72065
$983$	−38986.8	−1.26499	−0.632495	−	0.774564i	$-0.717969\pi$
$983$	−38986.8	−1.26499	−0.632495	+	0.774564i	$0.717969\pi$
$984$	65472.8	2.12114
$985$	20672.2	0.668703
$986$	−9853.74	−0.318263
$987$	10220.9	0.329620
$988$	65599.2	2.11234
$989$	37176.1	1.19528
$990$	−11015.2	−0.353623
$991$	−48817.5	−1.56482	−0.782412	−	0.622762i	$-0.786011\pi$
$991$	−48817.5	−1.56482	−0.782412	+	0.622762i	$0.786011\pi$
$992$	−38866.7	−1.24397
$993$	−20685.2	−0.661052
$994$	−6599.21	−0.210578
$995$	−28324.1	−0.902445
$996$	−81324.9	−2.58723
$997$	−14978.6	−0.475806	−0.237903	−	0.971289i	$-0.576460\pi$
$997$	−14978.6	−0.475806	−0.237903	+	0.971289i	$0.576460\pi$
$998$	107053.	3.39549
$999$	−7572.60	−0.239826

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.c.1.1	✓	8
3.2	odd	2		531.4.a.f.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.c.1.1	✓	8	1.1	even	1	trivial
531.4.a.f.1.8		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)

\(f(q)\)	\(=\)	\(q-5.19624 q^{2} -3.00000 q^{3} +19.0009 q^{4} -8.30102 q^{5} +15.5887 q^{6} +21.5539 q^{7} -57.1634 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-5.19624 q^{2} -3.00000 q^{3} +19.0009 q^{4} -8.30102 q^{5} +15.5887 q^{6} +21.5539 q^{7} -57.1634 q^{8} +9.00000 q^{9} +43.1341 q^{10} -28.3746 q^{11} -57.0027 q^{12} -28.2089 q^{13} -111.999 q^{14} +24.9031 q^{15} +145.027 q^{16} +21.8166 q^{17} -46.7662 q^{18} -122.388 q^{19} -157.727 q^{20} -64.6617 q^{21} +147.441 q^{22} +82.1571 q^{23} +171.490 q^{24} -56.0930 q^{25} +146.580 q^{26} -27.0000 q^{27} +409.544 q^{28} +86.9209 q^{29} -129.402 q^{30} +131.178 q^{31} -296.290 q^{32} +85.1239 q^{33} -113.364 q^{34} -178.919 q^{35} +171.008 q^{36} +280.467 q^{37} +635.957 q^{38} +84.6266 q^{39} +474.515 q^{40} +381.788 q^{41} +335.998 q^{42} +452.501 q^{43} -539.144 q^{44} -74.7092 q^{45} -426.908 q^{46} -158.067 q^{47} -435.082 q^{48} +121.570 q^{49} +291.473 q^{50} -65.4499 q^{51} -535.994 q^{52} -162.922 q^{53} +140.298 q^{54} +235.539 q^{55} -1232.09 q^{56} +367.164 q^{57} -451.662 q^{58} -59.0000 q^{59} +473.181 q^{60} -368.344 q^{61} -681.632 q^{62} +193.985 q^{63} +379.374 q^{64} +234.162 q^{65} -442.324 q^{66} +177.950 q^{67} +414.536 q^{68} -246.471 q^{69} +929.708 q^{70} +58.9220 q^{71} -514.470 q^{72} +880.299 q^{73} -1457.37 q^{74} +168.279 q^{75} -2325.48 q^{76} -611.584 q^{77} -439.740 q^{78} -825.284 q^{79} -1203.88 q^{80} +81.0000 q^{81} -1983.86 q^{82} +1426.68 q^{83} -1228.63 q^{84} -181.100 q^{85} -2351.30 q^{86} -260.763 q^{87} +1621.99 q^{88} +1556.78 q^{89} +388.207 q^{90} -608.011 q^{91} +1561.06 q^{92} -393.534 q^{93} +821.356 q^{94} +1015.94 q^{95} +888.869 q^{96} +1811.25 q^{97} -631.708 q^{98} -255.372 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9} + 29 q^{10} - 27 q^{11} - 114 q^{12} + 89 q^{13} - 37 q^{14} + 36 q^{15} + 362 q^{16} + 79 q^{17} + 18 q^{18} + 288 q^{19} + 457 q^{20} - 159 q^{21} + 596 q^{22} + 202 q^{23} - 9 q^{24} + 264 q^{25} + 270 q^{26} - 216 q^{27} + 702 q^{28} - 114 q^{29} - 87 q^{30} + 538 q^{31} + 316 q^{32} + 81 q^{33} + 498 q^{34} - 196 q^{35} + 342 q^{36} + 395 q^{37} + 397 q^{38} - 267 q^{39} + 918 q^{40} - 39 q^{41} + 111 q^{42} + 527 q^{43} + 64 q^{44} - 108 q^{45} - 539 q^{46} + 860 q^{47} - 1086 q^{48} + 347 q^{49} - 591 q^{50} - 237 q^{51} - 644 q^{52} - 812 q^{53} - 54 q^{54} + 536 q^{55} - 2218 q^{56} - 864 q^{57} - 1154 q^{58} - 472 q^{59} - 1371 q^{60} - 460 q^{61} - 2014 q^{62} + 477 q^{63} - 451 q^{64} - 986 q^{65} - 1788 q^{66} + 1934 q^{67} - 69 q^{68} - 606 q^{69} - 1028 q^{70} - 1687 q^{71} + 27 q^{72} + 1980 q^{73} - 2400 q^{74} - 792 q^{75} - 940 q^{76} - 821 q^{77} - 810 q^{78} + 3319 q^{79} - 2119 q^{80} + 648 q^{81} + 429 q^{82} + 2057 q^{83} - 2106 q^{84} + 566 q^{85} - 6690 q^{86} + 342 q^{87} + 1189 q^{88} + 1668 q^{89} + 261 q^{90} + 2427 q^{91} - 980 q^{92} - 1614 q^{93} + 332 q^{94} + 2146 q^{95} - 948 q^{96} + 1956 q^{97} - 2026 q^{98} - 243 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9 + 29 * q^10 - 27 * q^11 - 114 * q^12 + 89 * q^13 - 37 * q^14 + 36 * q^15 + 362 * q^16 + 79 * q^17 + 18 * q^18 + 288 * q^19 + 457 * q^20 - 159 * q^21 + 596 * q^22 + 202 * q^23 - 9 * q^24 + 264 * q^25 + 270 * q^26 - 216 * q^27 + 702 * q^28 - 114 * q^29 - 87 * q^30 + 538 * q^31 + 316 * q^32 + 81 * q^33 + 498 * q^34 - 196 * q^35 + 342 * q^36 + 395 * q^37 + 397 * q^38 - 267 * q^39 + 918 * q^40 - 39 * q^41 + 111 * q^42 + 527 * q^43 + 64 * q^44 - 108 * q^45 - 539 * q^46 + 860 * q^47 - 1086 * q^48 + 347 * q^49 - 591 * q^50 - 237 * q^51 - 644 * q^52 - 812 * q^53 - 54 * q^54 + 536 * q^55 - 2218 * q^56 - 864 * q^57 - 1154 * q^58 - 472 * q^59 - 1371 * q^60 - 460 * q^61 - 2014 * q^62 + 477 * q^63 - 451 * q^64 - 986 * q^65 - 1788 * q^66 + 1934 * q^67 - 69 * q^68 - 606 * q^69 - 1028 * q^70 - 1687 * q^71 + 27 * q^72 + 1980 * q^73 - 2400 * q^74 - 792 * q^75 - 940 * q^76 - 821 * q^77 - 810 * q^78 + 3319 * q^79 - 2119 * q^80 + 648 * q^81 + 429 * q^82 + 2057 * q^83 - 2106 * q^84 + 566 * q^85 - 6690 * q^86 + 342 * q^87 + 1189 * q^88 + 1668 * q^89 + 261 * q^90 + 2427 * q^91 - 980 * q^92 - 1614 * q^93 + 332 * q^94 + 2146 * q^95 - 948 * q^96 + 1956 * q^97 - 2026 * q^98 - 243 * q^99

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