LMFDB - Embedded newform 177.4.a.d.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} - 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			1.1
Root			$5.26439$ 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)

$f(q)$	$=$	$q-4.26439 q^{2} +3.00000 q^{3} +10.1850 q^{4} +13.9771 q^{5} -12.7932 q^{6} +21.4669 q^{7} -9.31764 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.26439 q^{2} +3.00000 q^{3} +10.1850 q^{4} +13.9771 q^{5} -12.7932 q^{6} +21.4669 q^{7} -9.31764 q^{8} +9.00000 q^{9} -59.6039 q^{10} +50.2623 q^{11} +30.5550 q^{12} -31.1995 q^{13} -91.5433 q^{14} +41.9314 q^{15} -41.7459 q^{16} +2.00676 q^{17} -38.3795 q^{18} -59.2842 q^{19} +142.357 q^{20} +64.4008 q^{21} -214.338 q^{22} -28.6724 q^{23} -27.9529 q^{24} +70.3601 q^{25} +133.047 q^{26} +27.0000 q^{27} +218.640 q^{28} +92.9707 q^{29} -178.812 q^{30} -88.6315 q^{31} +252.562 q^{32} +150.787 q^{33} -8.55761 q^{34} +300.046 q^{35} +91.6649 q^{36} +231.603 q^{37} +252.811 q^{38} -93.5986 q^{39} -130.234 q^{40} +237.954 q^{41} -274.630 q^{42} +29.7378 q^{43} +511.921 q^{44} +125.794 q^{45} +122.270 q^{46} -433.646 q^{47} -125.238 q^{48} +117.829 q^{49} -300.042 q^{50} +6.02029 q^{51} -317.767 q^{52} +464.489 q^{53} -115.138 q^{54} +702.522 q^{55} -200.021 q^{56} -177.853 q^{57} -396.463 q^{58} +59.0000 q^{59} +427.071 q^{60} -328.206 q^{61} +377.959 q^{62} +193.202 q^{63} -743.054 q^{64} -436.080 q^{65} -643.013 q^{66} -841.628 q^{67} +20.4388 q^{68} -86.0172 q^{69} -1279.51 q^{70} +46.3476 q^{71} -83.8588 q^{72} -738.776 q^{73} -987.644 q^{74} +211.080 q^{75} -603.809 q^{76} +1078.98 q^{77} +399.140 q^{78} +734.298 q^{79} -583.488 q^{80} +81.0000 q^{81} -1014.73 q^{82} +353.300 q^{83} +655.921 q^{84} +28.0488 q^{85} -126.813 q^{86} +278.912 q^{87} -468.326 q^{88} -704.085 q^{89} -536.435 q^{90} -669.758 q^{91} -292.028 q^{92} -265.894 q^{93} +1849.23 q^{94} -828.623 q^{95} +757.685 q^{96} +1479.12 q^{97} -502.468 q^{98} +452.360 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.26439	−1.50769	−0.753844	−	0.657053i	$-0.771803\pi$
$2$	−4.26439	−1.50769	−0.753844	+	0.657053i	$0.771803\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	10.1850	1.27312
$5$	13.9771	1.25015	0.625076	−	0.780564i	$-0.285068\pi$
$5$	13.9771	1.25015	0.625076	+	0.780564i	$0.285068\pi$
$6$	−12.7932	−0.870464
$7$	21.4669	1.15910	0.579552	−	0.814935i	$-0.303227\pi$
$7$	21.4669	1.15910	0.579552	+	0.814935i	$0.303227\pi$
$8$	−9.31764	−0.411785
$9$	9.00000	0.333333
$10$	−59.6039	−1.88484
$11$	50.2623	1.37769	0.688847	−	0.724907i	$-0.258117\pi$
$11$	50.2623	1.37769	0.688847	+	0.724907i	$0.258117\pi$
$12$	30.5550	0.735038
$13$	−31.1995	−0.665630	−0.332815	−	0.942992i	$-0.607998\pi$
$13$	−31.1995	−0.665630	−0.332815	+	0.942992i	$0.607998\pi$
$14$	−91.5433	−1.74757
$15$	41.9314	0.721776
$16$	−41.7459	−0.652280
$17$	2.00676	0.0286301	0.0143150	−	0.999898i	$-0.495443\pi$
$17$	2.00676	0.0286301	0.0143150	+	0.999898i	$0.495443\pi$
$18$	−38.3795	−0.502563
$19$	−59.2842	−0.715828	−0.357914	−	0.933755i	$-0.616512\pi$
$19$	−59.2842	−0.715828	−0.357914	+	0.933755i	$0.616512\pi$
$20$	142.357	1.59160
$21$	64.4008	0.669210
$22$	−214.338	−2.07713
$23$	−28.6724	−0.259939	−0.129970	−	0.991518i	$-0.541488\pi$
$23$	−28.6724	−0.259939	−0.129970	+	0.991518i	$0.541488\pi$
$24$	−27.9529	−0.237744
$25$	70.3601	0.562881
$26$	133.047	1.00356
$27$	27.0000	0.192450
$28$	218.640	1.47568
$29$	92.9707	0.595318	0.297659	−	0.954672i	$-0.403794\pi$
$29$	92.9707	0.595318	0.297659	+	0.954672i	$0.403794\pi$
$30$	−178.812	−1.08821
$31$	−88.6315	−0.513506	−0.256753	−	0.966477i	$-0.582653\pi$
$31$	−88.6315	−0.513506	−0.256753	+	0.966477i	$0.582653\pi$
$32$	252.562	1.39522
$33$	150.787	0.795412
$34$	−8.55761	−0.0431652
$35$	300.046	1.44906
$36$	91.6649	0.424375
$37$	231.603	1.02906	0.514531	−	0.857472i	$-0.327966\pi$
$37$	231.603	1.02906	0.514531	+	0.857472i	$0.327966\pi$
$38$	252.811	1.07925
$39$	−93.5986	−0.384302
$40$	−130.234	−0.514794
$41$	237.954	0.906393	0.453196	−	0.891411i	$-0.350284\pi$
$41$	237.954	0.906393	0.453196	+	0.891411i	$0.350284\pi$
$42$	−274.630	−1.00896
$43$	29.7378	0.105464	0.0527322	−	0.998609i	$-0.483207\pi$
$43$	29.7378	0.105464	0.0527322	+	0.998609i	$0.483207\pi$
$44$	511.921	1.75398
$45$	125.794	0.416717
$46$	122.270	0.391908
$47$	−433.646	−1.34582	−0.672912	−	0.739722i	$-0.734957\pi$
$47$	−433.646	−1.34582	−0.672912	+	0.739722i	$0.734957\pi$
$48$	−125.238	−0.376594
$49$	117.829	0.343524
$50$	−300.042	−0.848648
$51$	6.02029	0.0165296
$52$	−317.767	−0.847429
$53$	464.489	1.20382	0.601910	−	0.798564i	$-0.294406\pi$
$53$	464.489	1.20382	0.601910	+	0.798564i	$0.294406\pi$
$54$	−115.138	−0.290155
$55$	702.522	1.72233
$56$	−200.021	−0.477302
$57$	−177.853	−0.413284
$58$	−396.463	−0.897554
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	427.071	0.918910
$61$	−328.206	−0.688894	−0.344447	−	0.938806i	$-0.611934\pi$
$61$	−328.206	−0.688894	−0.344447	+	0.938806i	$0.611934\pi$
$62$	377.959	0.774207
$63$	193.202	0.386368
$64$	−743.054	−1.45128
$65$	−436.080	−0.832139
$66$	−643.013	−1.19923
$67$	−841.628	−1.53464	−0.767322	−	0.641261i	$-0.778411\pi$
$67$	−841.628	−1.53464	−0.767322	+	0.641261i	$0.778411\pi$
$68$	20.4388	0.0364496
$69$	−86.0172	−0.150076
$70$	−1279.51	−2.18473
$71$	46.3476	0.0774711	0.0387355	−	0.999249i	$-0.487667\pi$
$71$	46.3476	0.0774711	0.0387355	+	0.999249i	$0.487667\pi$
$72$	−83.8588	−0.137262
$73$	−738.776	−1.18448	−0.592241	−	0.805761i	$-0.701757\pi$
$73$	−738.776	−1.18448	−0.592241	+	0.805761i	$0.701757\pi$
$74$	−987.644	−1.55150
$75$	211.080	0.324979
$76$	−603.809	−0.911338
$77$	1078.98	1.59689
$78$	399.140	0.579407
$79$	734.298	1.04576	0.522880	−	0.852407i	$-0.324858\pi$
$79$	734.298	1.04576	0.522880	+	0.852407i	$0.324858\pi$
$80$	−583.488	−0.815449
$81$	81.0000	0.111111
$82$	−1014.73	−1.36656
$83$	353.300	0.467225	0.233613	−	0.972330i	$-0.424945\pi$
$83$	353.300	0.467225	0.233613	+	0.972330i	$0.424945\pi$
$84$	655.921	0.851987
$85$	28.0488	0.0357920
$86$	−126.813	−0.159008
$87$	278.912	0.343707
$88$	−468.326	−0.567314
$89$	−704.085	−0.838572	−0.419286	−	0.907854i	$-0.637720\pi$
$89$	−704.085	−0.838572	−0.419286	+	0.907854i	$0.637720\pi$
$90$	−536.435	−0.628280
$91$	−669.758	−0.771535
$92$	−292.028	−0.330935
$93$	−265.894	−0.296473
$94$	1849.23	2.02908
$95$	−828.623	−0.894894
$96$	757.685	0.805531
$97$	1479.12	1.54826	0.774132	−	0.633024i	$-0.218186\pi$
$97$	1479.12	1.54826	0.774132	+	0.633024i	$0.218186\pi$
$98$	−502.468	−0.517928
$99$	452.360	0.459231
$100$	716.617	0.716617
$101$	−1542.67	−1.51982	−0.759910	−	0.650028i	$-0.774757\pi$
$101$	−1542.67	−1.51982	−0.759910	+	0.650028i	$0.774757\pi$
$102$	−25.6728	−0.0249215
$103$	1588.52	1.51963	0.759813	−	0.650141i	$-0.225290\pi$
$103$	1588.52	1.51963	0.759813	+	0.650141i	$0.225290\pi$
$104$	290.706	0.274097
$105$	900.138	0.836614
$106$	−1980.76	−1.81499
$107$	813.279	0.734791	0.367395	−	0.930065i	$-0.380250\pi$
$107$	813.279	0.734791	0.367395	+	0.930065i	$0.380250\pi$
$108$	274.995	0.245013
$109$	179.016	0.157309	0.0786544	−	0.996902i	$-0.474938\pi$
$109$	179.016	0.157309	0.0786544	+	0.996902i	$0.474938\pi$
$110$	−2995.82	−2.59673
$111$	694.808	0.594129
$112$	−896.156	−0.756061
$113$	−2091.75	−1.74137	−0.870686	−	0.491839i	$-0.836325\pi$
$113$	−2091.75	−1.74137	−0.870686	+	0.491839i	$0.836325\pi$
$114$	758.433	0.623103
$115$	−400.758	−0.324964
$116$	946.906	0.757914
$117$	−280.796	−0.221877
$118$	−251.599	−0.196284
$119$	43.0790	0.0331853
$120$	−390.701	−0.297217
$121$	1195.29	0.898042
$122$	1399.60	1.03864
$123$	713.861	0.523306
$124$	−902.711	−0.653757
$125$	−763.709	−0.546466
$126$	−823.889	−0.582523
$127$	−232.320	−0.162324	−0.0811618	−	0.996701i	$-0.525863\pi$
$127$	−232.320	−0.162324	−0.0811618	+	0.996701i	$0.525863\pi$
$128$	1148.17	0.792853
$129$	89.2134	0.0608899
$130$	1859.61	1.25461
$131$	2481.08	1.65475	0.827377	−	0.561647i	$-0.189832\pi$
$131$	2481.08	1.65475	0.827377	+	0.561647i	$0.189832\pi$
$132$	1535.76	1.01266
$133$	−1272.65	−0.829720
$134$	3589.03	2.31377
$135$	377.382	0.240592
$136$	−18.6983	−0.0117894
$137$	1729.83	1.07875	0.539377	−	0.842064i	$-0.318660\pi$
$137$	1729.83	1.07875	0.539377	+	0.842064i	$0.318660\pi$
$138$	366.810	0.226268
$139$	3180.68	1.94088	0.970439	−	0.241346i	$-0.0775888\pi$
$139$	3180.68	1.94088	0.970439	+	0.241346i	$0.0775888\pi$
$140$	3055.96	1.84483
$141$	−1300.94	−0.777012
$142$	−197.644	−0.116802
$143$	−1568.16	−0.917035
$144$	−375.713	−0.217427
$145$	1299.46	0.744238
$146$	3150.42	1.78583
$147$	353.487	0.198334
$148$	2358.87	1.31012
$149$	−2741.73	−1.50746	−0.753729	−	0.657185i	$-0.771747\pi$
$149$	−2741.73	−1.50746	−0.753729	+	0.657185i	$0.771747\pi$
$150$	−900.127	−0.489967
$151$	−3325.09	−1.79200	−0.896000	−	0.444054i	$-0.853540\pi$
$151$	−3325.09	−1.79200	−0.896000	+	0.444054i	$0.853540\pi$
$152$	552.389	0.294768
$153$	18.0609	0.00954336
$154$	−4601.17	−2.40762
$155$	−1238.81	−0.641961
$156$	−953.300	−0.489264
$157$	2216.16	1.12656	0.563278	−	0.826268i	$-0.309540\pi$
$157$	2216.16	1.12656	0.563278	+	0.826268i	$0.309540\pi$
$158$	−3131.33	−1.57668
$159$	1393.47	0.695026
$160$	3530.09	1.74424
$161$	−615.508	−0.301297
$162$	−345.415	−0.167521
$163$	−3322.16	−1.59639	−0.798196	−	0.602397i	$-0.794212\pi$
$163$	−3322.16	−1.59639	−0.798196	+	0.602397i	$0.794212\pi$
$164$	2423.56	1.15395
$165$	2107.57	0.994386
$166$	−1506.61	−0.704430
$167$	3401.05	1.57593	0.787967	−	0.615717i	$-0.211134\pi$
$167$	3401.05	1.57593	0.787967	+	0.615717i	$0.211134\pi$
$168$	−600.063	−0.275571
$169$	−1223.59	−0.556937
$170$	−119.611	−0.0539631
$171$	−533.558	−0.238609
$172$	302.879	0.134269
$173$	−2335.37	−1.02633	−0.513164	−	0.858290i	$-0.671527\pi$
$173$	−2335.37	−1.02633	−0.513164	+	0.858290i	$0.671527\pi$
$174$	−1189.39	−0.518203
$175$	1510.41	0.652438
$176$	−2098.24	−0.898642
$177$	177.000	0.0751646
$178$	3002.49	1.26430
$179$	−3089.58	−1.29009	−0.645044	−	0.764145i	$-0.723161\pi$
$179$	−3089.58	−1.29009	−0.645044	+	0.764145i	$0.723161\pi$
$180$	1281.21	0.530533
$181$	−968.584	−0.397758	−0.198879	−	0.980024i	$-0.563730\pi$
$181$	−968.584	−0.397758	−0.198879	+	0.980024i	$0.563730\pi$
$182$	2856.11	1.16323
$183$	−984.619	−0.397733
$184$	267.159	0.107039
$185$	3237.14	1.28648
$186$	1133.88	0.446989
$187$	100.864	0.0394435
$188$	−4416.68	−1.71340
$189$	579.607	0.223070
$190$	3533.57	1.34922
$191$	−3777.44	−1.43102	−0.715512	−	0.698600i	$-0.753807\pi$
$191$	−3777.44	−1.43102	−0.715512	+	0.698600i	$0.753807\pi$
$192$	−2229.16	−0.837895
$193$	−1579.07	−0.588933	−0.294466	−	0.955662i	$-0.595142\pi$
$193$	−1579.07	−0.588933	−0.294466	+	0.955662i	$0.595142\pi$
$194$	−6307.53	−2.33430
$195$	−1308.24	−0.480436
$196$	1200.09	0.437349
$197$	1937.93	0.700871	0.350435	−	0.936587i	$-0.386034\pi$
$197$	1937.93	0.700871	0.350435	+	0.936587i	$0.386034\pi$
$198$	−1929.04	−0.692378
$199$	−4431.22	−1.57850	−0.789248	−	0.614075i	$-0.789529\pi$
$199$	−4431.22	−1.57850	−0.789248	+	0.614075i	$0.789529\pi$
$200$	−655.590	−0.231786
$201$	−2524.88	−0.886028
$202$	6578.56	2.29142
$203$	1995.80	0.690036
$204$	61.3165	0.0210442
$205$	3325.91	1.13313
$206$	−6774.06	−2.29112
$207$	−258.052	−0.0866465
$208$	1302.45	0.434177
$209$	−2979.76	−0.986193
$210$	−3838.54	−1.26135
$211$	453.497	0.147962	0.0739810	−	0.997260i	$-0.476430\pi$
$211$	453.497	0.147962	0.0739810	+	0.997260i	$0.476430\pi$
$212$	4730.82	1.53261
$213$	139.043	0.0447279
$214$	−3468.13	−1.10784
$215$	415.649	0.131847
$216$	−251.576	−0.0792481
$217$	−1902.65	−0.595207
$218$	−763.395	−0.237173
$219$	−2216.33	−0.683861
$220$	7155.18	2.19274
$221$	−62.6100	−0.0190570
$222$	−2962.93	−0.895761
$223$	551.465	0.165600	0.0828001	−	0.996566i	$-0.473614\pi$
$223$	551.465	0.165600	0.0828001	+	0.996566i	$0.473614\pi$
$224$	5421.72	1.61721
$225$	633.241	0.187627
$226$	8920.02	2.62545
$227$	5742.83	1.67914	0.839570	−	0.543252i	$-0.182807\pi$
$227$	5742.83	1.67914	0.839570	+	0.543252i	$0.182807\pi$
$228$	−1811.43	−0.526161
$229$	−5428.07	−1.56636	−0.783181	−	0.621794i	$-0.786404\pi$
$229$	−5428.07	−1.56636	−0.783181	+	0.621794i	$0.786404\pi$
$230$	1708.99	0.489944
$231$	3236.93	0.921966
$232$	−866.268	−0.245143
$233$	1383.05	0.388869	0.194435	−	0.980915i	$-0.437713\pi$
$233$	1383.05	0.388869	0.194435	+	0.980915i	$0.437713\pi$
$234$	1197.42	0.334521
$235$	−6061.12	−1.68249
$236$	600.914	0.165747
$237$	2202.89	0.603769
$238$	−183.706	−0.0500330
$239$	−380.199	−0.102900	−0.0514499	−	0.998676i	$-0.516384\pi$
$239$	−380.199	−0.102900	−0.0514499	+	0.998676i	$0.516384\pi$
$240$	−1750.46	−0.470800
$241$	−178.971	−0.0478362	−0.0239181	−	0.999714i	$-0.507614\pi$
$241$	−178.971	−0.0478362	−0.0239181	+	0.999714i	$0.507614\pi$
$242$	−5097.20	−1.35397
$243$	243.000	0.0641500
$244$	−3342.78	−0.877047
$245$	1646.91	0.429458
$246$	−3044.18	−0.788983
$247$	1849.64	0.476477
$248$	825.836	0.211454
$249$	1059.90	0.269753
$250$	3256.75	0.823900
$251$	1119.05	0.281409	0.140704	−	0.990052i	$-0.455063\pi$
$251$	1119.05	0.281409	0.140704	+	0.990052i	$0.455063\pi$
$252$	1967.76	0.491895
$253$	−1441.14	−0.358117
$254$	990.703	0.244733
$255$	84.1463	0.0206645
$256$	1048.17	0.255902
$257$	−2474.60	−0.600627	−0.300314	−	0.953841i	$-0.597091\pi$
$257$	−2474.60	−0.600627	−0.300314	+	0.953841i	$0.597091\pi$
$258$	−380.440	−0.0918030
$259$	4971.80	1.19279
$260$	−4441.47	−1.05942
$261$	836.736	0.198439
$262$	−10580.3	−2.49485
$263$	4863.06	1.14019	0.570093	−	0.821580i	$-0.306907\pi$
$263$	4863.06	1.14019	0.570093	+	0.821580i	$0.306907\pi$
$264$	−1404.98	−0.327539
$265$	6492.22	1.50496
$266$	5427.07	1.25096
$267$	−2112.26	−0.484150
$268$	−8571.97	−1.95379
$269$	4289.91	0.972345	0.486172	−	0.873863i	$-0.338393\pi$
$269$	4289.91	0.972345	0.486172	+	0.873863i	$0.338393\pi$
$270$	−1609.30	−0.362738
$271$	−2447.41	−0.548596	−0.274298	−	0.961645i	$-0.588445\pi$
$271$	−2447.41	−0.548596	−0.274298	+	0.961645i	$0.588445\pi$
$272$	−83.7741	−0.0186748
$273$	−2009.27	−0.445446
$274$	−7376.66	−1.62643
$275$	3536.46	0.775477
$276$	−876.084	−0.191065
$277$	−505.181	−0.109579	−0.0547895	−	0.998498i	$-0.517449\pi$
$277$	−505.181	−0.109579	−0.0547895	+	0.998498i	$0.517449\pi$
$278$	−13563.7	−2.92624
$279$	−797.683	−0.171169
$280$	−2795.72	−0.596701
$281$	−2909.81	−0.617738	−0.308869	−	0.951105i	$-0.599951\pi$
$281$	−2909.81	−0.617738	−0.308869	+	0.951105i	$0.599951\pi$
$282$	5547.70	1.17149
$283$	978.711	0.205577	0.102789	−	0.994703i	$-0.467224\pi$
$283$	978.711	0.205577	0.102789	+	0.994703i	$0.467224\pi$
$284$	472.050	0.0986303
$285$	−2485.87	−0.516667
$286$	6687.23	1.38260
$287$	5108.13	1.05060
$288$	2273.06	0.465073
$289$	−4908.97	−0.999180
$290$	−5541.41	−1.12208
$291$	4437.35	0.893891
$292$	−7524.42	−1.50799
$293$	4827.48	0.962541	0.481271	−	0.876572i	$-0.340175\pi$
$293$	4827.48	0.962541	0.481271	+	0.876572i	$0.340175\pi$
$294$	−1507.40	−0.299026
$295$	824.650	0.162756
$296$	−2157.99	−0.423752
$297$	1357.08	0.265137
$298$	11691.8	2.27278
$299$	894.565	0.173024
$300$	2149.85	0.413739
$301$	638.379	0.122244
$302$	14179.5	2.70178
$303$	−4628.02	−0.877469
$304$	2474.87	0.466920
$305$	−4587.38	−0.861222
$306$	−77.0185	−0.0143884
$307$	−1687.14	−0.313649	−0.156825	−	0.987626i	$-0.550126\pi$
$307$	−1687.14	−0.313649	−0.156825	+	0.987626i	$0.550126\pi$
$308$	10989.4	2.03304
$309$	4765.56	0.877357
$310$	5282.78	0.967876
$311$	−6570.44	−1.19799	−0.598996	−	0.800752i	$-0.704433\pi$
$311$	−6570.44	−1.19799	−0.598996	+	0.800752i	$0.704433\pi$
$312$	872.118	0.158250
$313$	2380.42	0.429871	0.214935	−	0.976628i	$-0.431046\pi$
$313$	2380.42	0.429871	0.214935	+	0.976628i	$0.431046\pi$
$314$	−9450.58	−1.69849
$315$	2700.41	0.483019
$316$	7478.82	1.33138
$317$	−1913.91	−0.339104	−0.169552	−	0.985521i	$-0.554232\pi$
$317$	−1913.91	−0.339104	−0.169552	+	0.985521i	$0.554232\pi$
$318$	−5942.28	−1.04788
$319$	4672.92	0.820167
$320$	−10385.8	−1.81432
$321$	2439.84	0.424232
$322$	2624.76	0.454262
$323$	−118.969	−0.0204942
$324$	824.984	0.141458
$325$	−2195.20	−0.374670
$326$	14167.0	2.40686
$327$	537.049	0.0908223
$328$	−2217.17	−0.373239
$329$	−9309.04	−1.55995
$330$	−8987.47	−1.49922
$331$	−5860.84	−0.973236	−0.486618	−	0.873615i	$-0.661770\pi$
$331$	−5860.84	−0.973236	−0.486618	+	0.873615i	$0.661770\pi$
$332$	3598.35	0.594835
$333$	2084.43	0.343020
$334$	−14503.4	−2.37602
$335$	−11763.5	−1.91854
$336$	−2688.47	−0.436512
$337$	−9890.58	−1.59874	−0.799368	−	0.600841i	$-0.794832\pi$
$337$	−9890.58	−1.59874	−0.799368	+	0.600841i	$0.794832\pi$
$338$	5217.86	0.839687
$339$	−6275.24	−1.00538
$340$	285.676	0.0455676
$341$	−4454.82	−0.707454
$342$	2275.30	0.359749
$343$	−4833.73	−0.760924
$344$	−277.086	−0.0434287
$345$	−1202.27	−0.187618
$346$	9958.92	1.54738
$347$	12225.3	1.89132	0.945658	−	0.325164i	$-0.105420\pi$
$347$	12225.3	1.89132	0.945658	+	0.325164i	$0.105420\pi$
$348$	2840.72	0.437582
$349$	−6380.04	−0.978555	−0.489278	−	0.872128i	$-0.662740\pi$
$349$	−6380.04	−0.978555	−0.489278	+	0.872128i	$0.662740\pi$
$350$	−6440.99	−0.983672
$351$	−842.387	−0.128101
$352$	12694.3	1.92219
$353$	−9132.68	−1.37701	−0.688503	−	0.725233i	$-0.741732\pi$
$353$	−9132.68	−1.37701	−0.688503	+	0.725233i	$0.741732\pi$
$354$	−754.796	−0.113325
$355$	647.806	0.0968506
$356$	−7171.10	−1.06761
$357$	129.237	0.0191595
$358$	13175.1	1.94505
$359$	40.7954	0.00599749	0.00299875	−	0.999996i	$-0.499045\pi$
$359$	40.7954	0.00599749	0.00299875	+	0.999996i	$0.499045\pi$
$360$	−1172.10	−0.171598
$361$	−3344.38	−0.487590
$362$	4130.42	0.599696
$363$	3585.88	0.518485
$364$	−6821.48	−0.982260
$365$	−10326.0	−1.48078
$366$	4198.80	0.599658
$367$	11214.7	1.59510	0.797552	−	0.603250i	$-0.206128\pi$
$367$	11214.7	1.59510	0.797552	+	0.603250i	$0.206128\pi$
$368$	1196.95	0.169553
$369$	2141.58	0.302131
$370$	−13804.4	−1.93962
$371$	9971.15	1.39535
$372$	−2708.13	−0.377447
$373$	−2620.02	−0.363698	−0.181849	−	0.983326i	$-0.558208\pi$
$373$	−2620.02	−0.363698	−0.181849	+	0.983326i	$0.558208\pi$
$374$	−430.125	−0.0594685
$375$	−2291.13	−0.315502
$376$	4040.56	0.554191
$377$	−2900.64	−0.396262
$378$	−2471.67	−0.336320
$379$	405.076	0.0549007	0.0274503	−	0.999623i	$-0.491261\pi$
$379$	405.076	0.0549007	0.0274503	+	0.999623i	$0.491261\pi$
$380$	−8439.52	−1.13931
$381$	−696.961	−0.0937175
$382$	16108.4	2.15754
$383$	−382.637	−0.0510492	−0.0255246	−	0.999674i	$-0.508126\pi$
$383$	−382.637	−0.0510492	−0.0255246	+	0.999674i	$0.508126\pi$
$384$	3444.52	0.457754
$385$	15081.0	1.99636
$386$	6733.77	0.887927
$387$	267.640	0.0351548
$388$	15064.8	1.97113
$389$	−6688.89	−0.871826	−0.435913	−	0.899989i	$-0.643574\pi$
$389$	−6688.89	−0.871826	−0.435913	+	0.899989i	$0.643574\pi$
$390$	5578.84	0.724347
$391$	−57.5387	−0.00744209
$392$	−1097.89	−0.141458
$393$	7443.23	0.955372
$394$	−8264.06	−1.05669
$395$	10263.4	1.30736
$396$	4607.28	0.584659
$397$	920.669	0.116391	0.0581953	−	0.998305i	$-0.481465\pi$
$397$	920.669	0.116391	0.0581953	+	0.998305i	$0.481465\pi$
$398$	18896.4	2.37988
$399$	−3817.95	−0.479039
$400$	−2937.24	−0.367156
$401$	9189.09	1.14434	0.572171	−	0.820134i	$-0.306101\pi$
$401$	9189.09	1.14434	0.572171	+	0.820134i	$0.306101\pi$
$402$	10767.1	1.33585
$403$	2765.26	0.341805
$404$	−15712.1	−1.93492
$405$	1132.15	0.138906
$406$	−8510.84	−1.04036
$407$	11640.9	1.41773
$408$	−56.0949	−0.00680664
$409$	15208.0	1.83860	0.919302	−	0.393553i	$-0.128754\pi$
$409$	15208.0	1.83860	0.919302	+	0.393553i	$0.128754\pi$
$410$	−14183.0	−1.70841
$411$	5189.49	0.622819
$412$	16179.1	1.93467
$413$	1266.55	0.150903
$414$	1100.43	0.130636
$415$	4938.12	0.584103
$416$	−7879.81	−0.928700
$417$	9542.05	1.12057
$418$	12706.8	1.48687
$419$	−13021.4	−1.51822	−0.759111	−	0.650962i	$-0.774366\pi$
$419$	−13021.4	−1.51822	−0.759111	+	0.650962i	$0.774366\pi$
$420$	9167.89	1.06511
$421$	−3275.96	−0.379242	−0.189621	−	0.981857i	$-0.560726\pi$
$421$	−3275.96	−0.379242	−0.189621	+	0.981857i	$0.560726\pi$
$422$	−1933.88	−0.223081
$423$	−3902.81	−0.448608
$424$	−4327.94	−0.495716
$425$	141.196	0.0161153
$426$	−592.932	−0.0674358
$427$	−7045.58	−0.798500
$428$	8283.23	0.935479
$429$	−4704.47	−0.529450
$430$	−1772.49	−0.198784
$431$	7674.36	0.857682	0.428841	−	0.903380i	$-0.358922\pi$
$431$	7674.36	0.857682	0.428841	+	0.903380i	$0.358922\pi$
$432$	−1127.14	−0.125531
$433$	−12802.6	−1.42091	−0.710456	−	0.703741i	$-0.751511\pi$
$433$	−12802.6	−1.42091	−0.710456	+	0.703741i	$0.751511\pi$
$434$	8113.62	0.897387
$435$	3898.39	0.429686
$436$	1823.28	0.200274
$437$	1699.82	0.186072
$438$	9451.27	1.03105
$439$	−1638.69	−0.178156	−0.0890781	−	0.996025i	$-0.528392\pi$
$439$	−1638.69	−0.178156	−0.0890781	+	0.996025i	$0.528392\pi$
$440$	−6545.85	−0.709229
$441$	1060.46	0.114508
$442$	266.993	0.0287321
$443$	−7210.63	−0.773335	−0.386667	−	0.922219i	$-0.626374\pi$
$443$	−7210.63	−0.773335	−0.386667	+	0.922219i	$0.626374\pi$
$444$	7076.62	0.756399
$445$	−9841.09	−1.04834
$446$	−2351.66	−0.249674
$447$	−8225.19	−0.870331
$448$	−15951.1	−1.68218
$449$	−1210.22	−0.127202	−0.0636012	−	0.997975i	$-0.520259\pi$
$449$	−1210.22	−0.127202	−0.0636012	+	0.997975i	$0.520259\pi$
$450$	−2700.38	−0.282883
$451$	11960.1	1.24873
$452$	−21304.4	−2.21698
$453$	−9975.27	−1.03461
$454$	−24489.6	−2.53162
$455$	−9361.29	−0.964536
$456$	1657.17	0.170184
$457$	−14691.6	−1.50382	−0.751908	−	0.659268i	$-0.770866\pi$
$457$	−14691.6	−1.50382	−0.751908	+	0.659268i	$0.770866\pi$
$458$	23147.4	2.36158
$459$	54.1826	0.00550986
$460$	−4081.71	−0.413719
$461$	1906.98	0.192662	0.0963309	−	0.995349i	$-0.469289\pi$
$461$	1906.98	0.192662	0.0963309	+	0.995349i	$0.469289\pi$
$462$	−13803.5	−1.39004
$463$	−2607.30	−0.261710	−0.130855	−	0.991402i	$-0.541772\pi$
$463$	−2607.30	−0.261710	−0.130855	+	0.991402i	$0.541772\pi$
$464$	−3881.15	−0.388314
$465$	−3716.44	−0.370636
$466$	−5897.85	−0.586293
$467$	9821.65	0.973216	0.486608	−	0.873621i	$-0.338234\pi$
$467$	9821.65	0.973216	0.486608	+	0.873621i	$0.338234\pi$
$468$	−2859.90	−0.282476
$469$	−18067.2	−1.77881
$470$	25847.0	2.53666
$471$	6648.49	0.650417
$472$	−549.741	−0.0536099
$473$	1494.69	0.145298
$474$	−9393.99	−0.910296
$475$	−4171.24	−0.402926
$476$	438.759	0.0422489
$477$	4180.40	0.401273
$478$	1621.32	0.155141
$479$	−3910.32	−0.373000	−0.186500	−	0.982455i	$-0.559714\pi$
$479$	−3910.32	−0.373000	−0.186500	+	0.982455i	$0.559714\pi$
$480$	10590.3	1.00704
$481$	−7225.90	−0.684974
$482$	763.201	0.0721221
$483$	−1846.52	−0.173954
$484$	12174.1	1.14332
$485$	20673.8	1.93557
$486$	−1036.25	−0.0967182
$487$	17045.3	1.58603	0.793016	−	0.609200i	$-0.208509\pi$
$487$	17045.3	1.58603	0.793016	+	0.609200i	$0.208509\pi$
$488$	3058.11	0.283676
$489$	−9966.49	−0.921678
$490$	−7023.06	−0.647488
$491$	−15162.2	−1.39361	−0.696804	−	0.717262i	$-0.745395\pi$
$491$	−15162.2	−1.39361	−0.696804	+	0.717262i	$0.745395\pi$
$492$	7270.67	0.666234
$493$	186.570	0.0170440
$494$	−7887.58	−0.718378
$495$	6322.70	0.574109
$496$	3700.00	0.334950
$497$	994.940	0.0897971
$498$	−4519.82	−0.406703
$499$	18914.2	1.69682	0.848411	−	0.529339i	$-0.177560\pi$
$499$	18914.2	1.69682	0.848411	+	0.529339i	$0.177560\pi$
$500$	−7778.37	−0.695719
$501$	10203.1	0.909866
$502$	−4772.04	−0.424276
$503$	10331.8	0.915849	0.457924	−	0.888991i	$-0.348593\pi$
$503$	10331.8	0.915849	0.457924	+	0.888991i	$0.348593\pi$
$504$	−1800.19	−0.159101
$505$	−21562.2	−1.90001
$506$	6145.57	0.539929
$507$	−3670.77	−0.321548
$508$	−2366.18	−0.206658
$509$	15683.4	1.36573	0.682864	−	0.730546i	$-0.260734\pi$
$509$	15683.4	1.36573	0.682864	+	0.730546i	$0.260734\pi$
$510$	−358.832	−0.0311556
$511$	−15859.2	−1.37294
$512$	−13655.2	−1.17867
$513$	−1600.67	−0.137761
$514$	10552.6	0.905558
$515$	22202.9	1.89976
$516$	908.637	0.0775204
$517$	−21796.0	−1.85413
$518$	−21201.7	−1.79836
$519$	−7006.11	−0.592551
$520$	4063.23	0.342663
$521$	−7550.71	−0.634938	−0.317469	−	0.948269i	$-0.602833\pi$
$521$	−7550.71	−0.634938	−0.317469	+	0.948269i	$0.602833\pi$
$522$	−3568.17	−0.299185
$523$	−12867.2	−1.07580	−0.537901	−	0.843008i	$-0.680783\pi$
$523$	−12867.2	−1.07580	−0.537901	+	0.843008i	$0.680783\pi$
$524$	25269.7	2.10671
$525$	4531.24	0.376685
$526$	−20738.0	−1.71905
$527$	−177.862	−0.0147017
$528$	−6294.73	−0.518831
$529$	−11344.9	−0.932431
$530$	−27685.3	−2.26901
$531$	531.000	0.0433963
$532$	−12961.9	−1.05634
$533$	−7424.04	−0.603322
$534$	9007.48	0.729947
$535$	11367.3	0.918600
$536$	7841.99	0.631944
$537$	−9268.73	−0.744833
$538$	−18293.9	−1.46599
$539$	5922.34	0.473272
$540$	3843.64	0.306303
$541$	−6294.73	−0.500243	−0.250121	−	0.968214i	$-0.580471\pi$
$541$	−6294.73	−0.500243	−0.250121	+	0.968214i	$0.580471\pi$
$542$	10436.7	0.827111
$543$	−2905.75	−0.229646
$544$	506.831	0.0399452
$545$	2502.13	0.196660
$546$	8568.32	0.671594
$547$	−17015.5	−1.33003	−0.665017	−	0.746828i	$-0.731576\pi$
$547$	−17015.5	−1.33003	−0.665017	+	0.746828i	$0.731576\pi$
$548$	17618.3	1.37339
$549$	−2953.86	−0.229631
$550$	−15080.8	−1.16918
$551$	−5511.70	−0.426146
$552$	801.477	0.0617991
$553$	15763.1	1.21214
$554$	2154.29	0.165211
$555$	9711.43	0.742751
$556$	32395.2	2.47098
$557$	425.562	0.0323728	0.0161864	−	0.999869i	$-0.494847\pi$
$557$	425.562	0.0323728	0.0161864	+	0.999869i	$0.494847\pi$
$558$	3401.63	0.258069
$559$	−927.805	−0.0702003
$560$	−12525.7	−0.945191
$561$	302.593	0.0227727
$562$	12408.5	0.931357
$563$	22080.6	1.65291	0.826454	−	0.563004i	$-0.190354\pi$
$563$	22080.6	1.65291	0.826454	+	0.563004i	$0.190354\pi$
$564$	−13250.0	−0.989233
$565$	−29236.6	−2.17698
$566$	−4173.60	−0.309946
$567$	1738.82	0.128789
$568$	−431.850	−0.0319015
$569$	14807.4	1.09096	0.545482	−	0.838123i	$-0.316347\pi$
$569$	14807.4	1.09096	0.545482	+	0.838123i	$0.316347\pi$
$570$	10600.7	0.778973
$571$	8320.22	0.609790	0.304895	−	0.952386i	$-0.401379\pi$
$571$	8320.22	0.609790	0.304895	+	0.952386i	$0.401379\pi$
$572$	−15971.7	−1.16750
$573$	−11332.3	−0.826202
$574$	−21783.0	−1.58398
$575$	−2017.39	−0.146315
$576$	−6687.48	−0.483759
$577$	6428.58	0.463822	0.231911	−	0.972737i	$-0.425502\pi$
$577$	6428.58	0.463822	0.231911	+	0.972737i	$0.425502\pi$
$578$	20933.8	1.50645
$579$	−4737.21	−0.340020
$580$	13235.0	0.947508
$581$	7584.26	0.541563
$582$	−18922.6	−1.34771
$583$	23346.3	1.65850
$584$	6883.65	0.487752
$585$	−3924.72	−0.277380
$586$	−20586.3	−1.45121
$587$	9120.74	0.641318	0.320659	−	0.947195i	$-0.396096\pi$
$587$	9120.74	0.641318	0.320659	+	0.947195i	$0.396096\pi$
$588$	3600.26	0.252504
$589$	5254.45	0.367582
$590$	−3516.63	−0.245385
$591$	5813.78	0.404648
$592$	−9668.47	−0.671236
$593$	−21233.1	−1.47038	−0.735192	−	0.677859i	$-0.762908\pi$
$593$	−21233.1	−1.47038	−0.735192	+	0.677859i	$0.762908\pi$
$594$	−5787.12	−0.399745
$595$	602.121	0.0414866
$596$	−27924.5	−1.91918
$597$	−13293.7	−0.911345
$598$	−3814.77	−0.260866
$599$	−2232.18	−0.152261	−0.0761305	−	0.997098i	$-0.524257\pi$
$599$	−2232.18	−0.152261	−0.0761305	+	0.997098i	$0.524257\pi$
$600$	−1966.77	−0.133822
$601$	8970.12	0.608817	0.304408	−	0.952542i	$-0.401541\pi$
$601$	8970.12	0.608817	0.304408	+	0.952542i	$0.401541\pi$
$602$	−2722.29	−0.184306
$603$	−7574.65	−0.511548
$604$	−33866.0	−2.28144
$605$	16706.8	1.12269
$606$	19735.7	1.32295
$607$	14005.7	0.936531	0.468265	−	0.883588i	$-0.344879\pi$
$607$	14005.7	0.936531	0.468265	+	0.883588i	$0.344879\pi$
$608$	−14972.9	−0.998738
$609$	5987.39	0.398393
$610$	19562.4	1.29845
$611$	13529.5	0.895821
$612$	183.950	0.0121499
$613$	10352.6	0.682119	0.341059	−	0.940042i	$-0.389214\pi$
$613$	10352.6	0.682119	0.341059	+	0.940042i	$0.389214\pi$
$614$	7194.63	0.472885
$615$	9977.72	0.654212
$616$	−10053.5	−0.657577
$617$	17951.3	1.17130	0.585649	−	0.810564i	$-0.300840\pi$
$617$	17951.3	1.17130	0.585649	+	0.810564i	$0.300840\pi$
$618$	−20322.2	−1.32278
$619$	11187.1	0.726406	0.363203	−	0.931710i	$-0.381683\pi$
$619$	11187.1	0.726406	0.363203	+	0.931710i	$0.381683\pi$
$620$	−12617.3	−0.817295
$621$	−774.155	−0.0500254
$622$	28018.9	1.80620
$623$	−15114.5	−0.971993
$624$	3907.36	0.250672
$625$	−19469.5	−1.24605
$626$	−10151.1	−0.648111
$627$	−8939.28	−0.569379
$628$	22571.6	1.43424
$629$	464.772	0.0294621
$630$	−11515.6	−0.728242
$631$	−5810.20	−0.366562	−0.183281	−	0.983061i	$-0.558672\pi$
$631$	−5810.20	−0.366562	−0.183281	+	0.983061i	$0.558672\pi$
$632$	−6841.92	−0.430628
$633$	1360.49	0.0854259
$634$	8161.67	0.511264
$635$	−3247.17	−0.202929
$636$	14192.5	0.884854
$637$	−3676.20	−0.228660
$638$	−19927.1	−1.23656
$639$	417.128	0.0258237
$640$	16048.2	0.991187
$641$	13147.8	0.810152	0.405076	−	0.914283i	$-0.367245\pi$
$641$	13147.8	0.810152	0.405076	+	0.914283i	$0.367245\pi$
$642$	−10404.4	−0.639609
$643$	15035.7	0.922162	0.461081	−	0.887358i	$-0.347462\pi$
$643$	15035.7	0.922162	0.461081	+	0.887358i	$0.347462\pi$
$644$	−6268.94	−0.383588
$645$	1246.95	0.0761217
$646$	507.331	0.0308989
$647$	−14522.4	−0.882435	−0.441217	−	0.897400i	$-0.645453\pi$
$647$	−14522.4	−0.882435	−0.441217	+	0.897400i	$0.645453\pi$
$648$	−754.729	−0.0457539
$649$	2965.47	0.179361
$650$	9361.18	0.564886
$651$	−5707.94	−0.343643
$652$	−33836.2	−2.03241
$653$	−7321.61	−0.438770	−0.219385	−	0.975638i	$-0.570405\pi$
$653$	−7321.61	−0.438770	−0.219385	+	0.975638i	$0.570405\pi$
$654$	−2290.18	−0.136932
$655$	34678.3	2.06869
$656$	−9933.59	−0.591222
$657$	−6648.98	−0.394827
$658$	39697.4	2.35192
$659$	18730.0	1.10716	0.553580	−	0.832796i	$-0.313261\pi$
$659$	18730.0	1.10716	0.553580	+	0.832796i	$0.313261\pi$
$660$	21465.5	1.26598
$661$	23121.7	1.36056	0.680281	−	0.732952i	$-0.261858\pi$
$661$	23121.7	1.36056	0.680281	+	0.732952i	$0.261858\pi$
$662$	24992.9	1.46734
$663$	−187.830	−0.0110026
$664$	−3291.92	−0.192396
$665$	−17788.0	−1.03728
$666$	−8888.79	−0.517168
$667$	−2665.69	−0.154747
$668$	34639.7	2.00636
$669$	1654.40	0.0956093
$670$	50164.3	2.89256
$671$	−16496.4	−0.949085
$672$	16265.2	0.933694
$673$	−9783.94	−0.560391	−0.280195	−	0.959943i	$-0.590399\pi$
$673$	−9783.94	−0.560391	−0.280195	+	0.959943i	$0.590399\pi$
$674$	42177.2	2.41040
$675$	1899.72	0.108326
$676$	−12462.2	−0.709049
$677$	−7309.79	−0.414975	−0.207487	−	0.978238i	$-0.566529\pi$
$677$	−7309.79	−0.414975	−0.207487	+	0.978238i	$0.566529\pi$
$678$	26760.1	1.51580
$679$	31752.1	1.79460
$680$	−261.348	−0.0147386
$681$	17228.5	0.969452
$682$	18997.1	1.06662
$683$	−26062.1	−1.46009	−0.730043	−	0.683401i	$-0.760500\pi$
$683$	−26062.1	−1.46009	−0.730043	+	0.683401i	$0.760500\pi$
$684$	−5434.28	−0.303779
$685$	24178.1	1.34861
$686$	20612.9	1.14724
$687$	−16284.2	−0.904339
$688$	−1241.43	−0.0687923
$689$	−14491.8	−0.801299
$690$	5126.96	0.282869
$691$	−5703.45	−0.313993	−0.156997	−	0.987599i	$-0.550181\pi$
$691$	−5703.45	−0.313993	−0.156997	+	0.987599i	$0.550181\pi$
$692$	−23785.7	−1.30664
$693$	9710.78	0.532298
$694$	−52133.2	−2.85151
$695$	44456.8	2.42639
$696$	−2598.80	−0.141534
$697$	477.516	0.0259501
$698$	27207.0	1.47536
$699$	4149.14	0.224514
$700$	15383.6	0.830634
$701$	29032.2	1.56424	0.782120	−	0.623127i	$-0.214138\pi$
$701$	29032.2	1.56424	0.782120	+	0.623127i	$0.214138\pi$
$702$	3592.26	0.193136
$703$	−13730.4	−0.736631
$704$	−37347.6	−1.99942
$705$	−18183.4	−0.971383
$706$	38945.3	2.07610
$707$	−33116.5	−1.76163
$708$	1802.74	0.0956938
$709$	14314.4	0.758235	0.379117	−	0.925349i	$-0.376228\pi$
$709$	14314.4	0.758235	0.379117	+	0.925349i	$0.376228\pi$
$710$	−2762.50	−0.146021
$711$	6608.68	0.348586
$712$	6560.41	0.345312
$713$	2541.28	0.133480
$714$	−551.117	−0.0288866
$715$	−21918.3	−1.14643
$716$	−31467.3	−1.64244
$717$	−1140.60	−0.0594092
$718$	−173.967	−0.00904235
$719$	19598.6	1.01656	0.508278	−	0.861193i	$-0.330282\pi$
$719$	19598.6	1.01656	0.508278	+	0.861193i	$0.330282\pi$
$720$	−5251.39	−0.271816
$721$	34100.6	1.76141
$722$	14261.7	0.735134
$723$	−536.913	−0.0276183
$724$	−9865.02	−0.506396
$725$	6541.43	0.335093
$726$	−15291.6	−0.781713
$727$	25217.2	1.28646	0.643228	−	0.765675i	$-0.277595\pi$
$727$	25217.2	1.28646	0.643228	+	0.765675i	$0.277595\pi$
$728$	6240.56	0.317707
$729$	729.000	0.0370370
$730$	44033.9	2.23256
$731$	59.6767	0.00301946
$732$	−10028.3	−0.506363
$733$	−14826.4	−0.747101	−0.373550	−	0.927610i	$-0.621860\pi$
$733$	−14826.4	−0.747101	−0.373550	+	0.927610i	$0.621860\pi$
$734$	−47823.9	−2.40492
$735$	4940.73	0.247948
$736$	−7241.55	−0.362673
$737$	−42302.1	−2.11427
$738$	−9132.53	−0.455519
$739$	−29860.2	−1.48637	−0.743184	−	0.669087i	$-0.766686\pi$
$739$	−29860.2	−1.48637	−0.743184	+	0.669087i	$0.766686\pi$
$740$	32970.3	1.63785
$741$	5548.92	0.275094
$742$	−42520.8	−2.10376
$743$	12908.8	0.637388	0.318694	−	0.947858i	$-0.396756\pi$
$743$	12908.8	0.637388	0.318694	+	0.947858i	$0.396756\pi$
$744$	2477.51	0.122083
$745$	−38321.5	−1.88455
$746$	11172.8	0.548344
$747$	3179.70	0.155742
$748$	1027.30	0.0502164
$749$	17458.6	0.851700
$750$	9770.25	0.475679
$751$	6544.75	0.318005	0.159002	−	0.987278i	$-0.449172\pi$
$751$	6544.75	0.318005	0.159002	+	0.987278i	$0.449172\pi$
$752$	18102.9	0.877854
$753$	3357.14	0.162471
$754$	12369.5	0.597439
$755$	−46475.2	−2.24027
$756$	5903.29	0.283996
$757$	−4239.81	−0.203565	−0.101782	−	0.994807i	$-0.532455\pi$
$757$	−4239.81	−0.203565	−0.101782	+	0.994807i	$0.532455\pi$
$758$	−1727.40	−0.0827731
$759$	−4323.42	−0.206759
$760$	7720.81	0.368504
$761$	16068.6	0.765421	0.382711	−	0.923868i	$-0.374991\pi$
$761$	16068.6	0.765421	0.382711	+	0.923868i	$0.374991\pi$
$762$	2972.11	0.141297
$763$	3842.93	0.182337
$764$	−38473.1	−1.82187
$765$	252.439	0.0119307
$766$	1631.71	0.0769663
$767$	−1840.77	−0.0866577
$768$	3144.52	0.147745
$769$	−12692.7	−0.595201	−0.297601	−	0.954690i	$-0.596186\pi$
$769$	−12692.7	−0.595201	−0.297601	+	0.954690i	$0.596186\pi$
$770$	−64311.1	−3.00989
$771$	−7423.79	−0.346772
$772$	−16082.8	−0.749784
$773$	5663.38	0.263515	0.131758	−	0.991282i	$-0.457938\pi$
$773$	5663.38	0.263515	0.131758	+	0.991282i	$0.457938\pi$
$774$	−1141.32	−0.0530025
$775$	−6236.12	−0.289043
$776$	−13781.9	−0.637553
$777$	14915.4	0.688658
$778$	28524.0	1.31444
$779$	−14106.9	−0.648822
$780$	−13324.4	−0.611654
$781$	2329.53	0.106731
$782$	245.367	0.0112203
$783$	2510.21	0.114569
$784$	−4918.87	−0.224074
$785$	30975.6	1.40837
$786$	−31740.8	−1.44040
$787$	−2659.15	−0.120443	−0.0602214	−	0.998185i	$-0.519181\pi$
$787$	−2659.15	−0.120443	−0.0602214	+	0.998185i	$0.519181\pi$
$788$	19737.8	0.892295
$789$	14589.2	0.658287
$790$	−43767.0	−1.97109
$791$	−44903.4	−2.01843
$792$	−4214.93	−0.189105
$793$	10239.9	0.458549
$794$	−3926.09	−0.175481
$795$	19476.7	0.868888
$796$	−45131.9	−2.00962
$797$	22767.0	1.01186	0.505928	−	0.862576i	$-0.331150\pi$
$797$	22767.0	1.01186	0.505928	+	0.862576i	$0.331150\pi$
$798$	16281.2	0.722242
$799$	−870.224	−0.0385311
$800$	17770.3	0.785342
$801$	−6336.77	−0.279524
$802$	−39185.8	−1.72531
$803$	−37132.5	−1.63185
$804$	−25715.9	−1.12802
$805$	−8603.03	−0.376667
$806$	−11792.1	−0.515335
$807$	12869.7	0.561383
$808$	14374.1	0.625840
$809$	32981.9	1.43335	0.716676	−	0.697406i	$-0.245662\pi$
$809$	32981.9	1.43335	0.716676	+	0.697406i	$0.245662\pi$
$810$	−4827.91	−0.209427
$811$	20038.8	0.867642	0.433821	−	0.900999i	$-0.357165\pi$
$811$	20038.8	0.867642	0.433821	+	0.900999i	$0.357165\pi$
$812$	20327.2	0.878502
$813$	−7342.23	−0.316732
$814$	−49641.2	−2.13750
$815$	−46434.3	−1.99573
$816$	−251.322	−0.0107819
$817$	−1762.98	−0.0754944
$818$	−64852.9	−2.77204
$819$	−6027.82	−0.257178
$820$	33874.3	1.44261
$821$	−16024.0	−0.681172	−0.340586	−	0.940213i	$-0.610626\pi$
$821$	−16024.0	−0.681172	−0.340586	+	0.940213i	$0.610626\pi$
$822$	−22130.0	−0.939017
$823$	25375.3	1.07476	0.537379	−	0.843341i	$-0.319414\pi$
$823$	25375.3	1.07476	0.537379	+	0.843341i	$0.319414\pi$
$824$	−14801.3	−0.625760
$825$	10609.4	0.447722
$826$	−5401.05	−0.227514
$827$	15790.0	0.663931	0.331966	−	0.943291i	$-0.392288\pi$
$827$	15790.0	0.663931	0.331966	+	0.943291i	$0.392288\pi$
$828$	−2628.25	−0.110312
$829$	26581.6	1.11365	0.556826	−	0.830629i	$-0.312019\pi$
$829$	26581.6	1.11365	0.556826	+	0.830629i	$0.312019\pi$
$830$	−21058.0	−0.880645
$831$	−1515.54	−0.0632655
$832$	23182.9	0.966013
$833$	236.454	0.00983513
$834$	−40691.0	−1.68947
$835$	47536.9	1.97016
$836$	−30348.8	−1.25555
$837$	−2393.05	−0.0988243
$838$	55528.1	2.28900
$839$	−8481.14	−0.348989	−0.174494	−	0.984658i	$-0.555829\pi$
$839$	−8481.14	−0.348989	−0.174494	+	0.984658i	$0.555829\pi$
$840$	−8387.16	−0.344505
$841$	−15745.4	−0.645596
$842$	13970.0	0.571778
$843$	−8729.42	−0.356651
$844$	4618.86	0.188374
$845$	−17102.3	−0.696256
$846$	16643.1	0.676361
$847$	25659.3	1.04093
$848$	−19390.5	−0.785228
$849$	2936.13	0.118690
$850$	−602.114	−0.0242969
$851$	−6640.61	−0.267494
$852$	1416.15	0.0569442
$853$	1049.74	0.0421366	0.0210683	−	0.999778i	$-0.493293\pi$
$853$	1049.74	0.0421366	0.0210683	+	0.999778i	$0.493293\pi$
$854$	30045.1	1.20389
$855$	−7457.61	−0.298298
$856$	−7577.84	−0.302576
$857$	13685.0	0.545472	0.272736	−	0.962089i	$-0.412071\pi$
$857$	13685.0	0.545472	0.272736	+	0.962089i	$0.412071\pi$
$858$	20061.7	0.798246
$859$	2391.73	0.0949998	0.0474999	−	0.998871i	$-0.484875\pi$
$859$	2391.73	0.0949998	0.0474999	+	0.998871i	$0.484875\pi$
$860$	4233.38	0.167857
$861$	15324.4	0.606567
$862$	−32726.5	−1.29312
$863$	−16019.6	−0.631883	−0.315942	−	0.948779i	$-0.602320\pi$
$863$	−16019.6	−0.631883	−0.315942	+	0.948779i	$0.602320\pi$
$864$	6819.17	0.268510
$865$	−32641.8	−1.28307
$866$	54595.4	2.14229
$867$	−14726.9	−0.576877
$868$	−19378.4	−0.757773
$869$	36907.5	1.44074
$870$	−16624.2	−0.647833
$871$	26258.4	1.02151
$872$	−1668.01	−0.0647774
$873$	13312.1	0.516088
$874$	−7248.69	−0.280539
$875$	−16394.5	−0.633411
$876$	−22573.3	−0.870639
$877$	21231.4	0.817483	0.408742	−	0.912650i	$-0.365968\pi$
$877$	21231.4	0.817483	0.408742	+	0.912650i	$0.365968\pi$
$878$	6988.03	0.268604
$879$	14482.5	0.555724
$880$	−29327.4	−1.12344
$881$	−40809.0	−1.56060	−0.780301	−	0.625405i	$-0.784934\pi$
$881$	−40809.0	−1.56060	−0.780301	+	0.625405i	$0.784934\pi$
$882$	−4522.21	−0.172643
$883$	−24840.7	−0.946723	−0.473361	−	0.880868i	$-0.656960\pi$
$883$	−24840.7	−0.946723	−0.473361	+	0.880868i	$0.656960\pi$
$884$	−637.682	−0.0242620
$885$	2473.95	0.0939672
$886$	30748.9	1.16595
$887$	−7368.83	−0.278942	−0.139471	−	0.990226i	$-0.544540\pi$
$887$	−7368.83	−0.278942	−0.139471	+	0.990226i	$0.544540\pi$
$888$	−6473.97	−0.244654
$889$	−4987.20	−0.188150
$890$	41966.2	1.58057
$891$	4071.24	0.153077
$892$	5616.67	0.210830
$893$	25708.4	0.963379
$894$	35075.4	1.31219
$895$	−43183.4	−1.61281
$896$	24647.8	0.919000
$897$	2683.69	0.0998952
$898$	5160.85	0.191782
$899$	−8240.13	−0.305699
$900$	6449.55	0.238872
$901$	932.119	0.0344655
$902$	−51002.4	−1.88270
$903$	1915.14	0.0705778
$904$	19490.2	0.717072
$905$	−13538.0	−0.497259
$906$	42538.4	1.55987
$907$	13529.1	0.495289	0.247645	−	0.968851i	$-0.420343\pi$
$907$	13529.1	0.495289	0.247645	+	0.968851i	$0.420343\pi$
$908$	58490.6	2.13775
$909$	−13884.1	−0.506607
$910$	39920.2	1.45422
$911$	−12796.3	−0.465380	−0.232690	−	0.972551i	$-0.574753\pi$
$911$	−12796.3	−0.465380	−0.232690	+	0.972551i	$0.574753\pi$
$912$	7424.62	0.269577
$913$	17757.6	0.643694
$914$	62650.7	2.26729
$915$	−13762.2	−0.497227
$916$	−55284.8	−1.99417
$917$	53261.1	1.91803
$918$	−231.055	−0.00830715
$919$	1229.33	0.0441259	0.0220629	−	0.999757i	$-0.492977\pi$
$919$	1229.33	0.0441259	0.0220629	+	0.999757i	$0.492977\pi$
$920$	3734.12	0.133815
$921$	−5061.43	−0.181085
$922$	−8132.12	−0.290474
$923$	−1446.02	−0.0515671
$924$	32968.1	1.17378
$925$	16295.6	0.579239
$926$	11118.5	0.394577
$927$	14296.7	0.506542
$928$	23480.8	0.830600
$929$	−40507.0	−1.43056	−0.715280	−	0.698838i	$-0.753701\pi$
$929$	−40507.0	−1.43056	−0.715280	+	0.698838i	$0.753701\pi$
$930$	15848.3	0.558804
$931$	−6985.39	−0.245904
$932$	14086.3	0.495078
$933$	−19711.3	−0.691661
$934$	−41883.3	−1.46731
$935$	1409.79	0.0493104
$936$	2616.35	0.0913656
$937$	46898.8	1.63513	0.817565	−	0.575837i	$-0.195324\pi$
$937$	46898.8	1.63513	0.817565	+	0.575837i	$0.195324\pi$
$938$	77045.4	2.68190
$939$	7141.27	0.248186
$940$	−61732.5	−2.14201
$941$	−10886.3	−0.377134	−0.188567	−	0.982060i	$-0.560384\pi$
$941$	−10886.3	−0.377134	−0.188567	+	0.982060i	$0.560384\pi$
$942$	−28351.7	−0.980626
$943$	−6822.70	−0.235607
$944$	−2463.01	−0.0849196
$945$	8101.24	0.278871
$946$	−6373.93	−0.219064
$947$	−33388.0	−1.14569	−0.572843	−	0.819665i	$-0.694159\pi$
$947$	−33388.0	−1.14569	−0.572843	+	0.819665i	$0.694159\pi$
$948$	22436.5	0.768673
$949$	23049.4	0.788426
$950$	17787.8	0.607486
$951$	−5741.74	−0.195782
$952$	−401.395	−0.0136652
$953$	−25026.7	−0.850676	−0.425338	−	0.905035i	$-0.639845\pi$
$953$	−25026.7	−0.850676	−0.425338	+	0.905035i	$0.639845\pi$
$954$	−17826.8	−0.604995
$955$	−52797.7	−1.78900
$956$	−3872.32	−0.131004
$957$	14018.8	0.473523
$958$	16675.1	0.562368
$959$	37134.1	1.25039
$960$	−31157.3	−1.04750
$961$	−21935.5	−0.736312
$962$	30814.0	1.03273
$963$	7319.51	0.244930
$964$	−1822.82	−0.0609014
$965$	−22070.9	−0.736255
$966$	7874.29	0.262268
$967$	15589.4	0.518430	0.259215	−	0.965820i	$-0.416536\pi$
$967$	15589.4	0.518430	0.259215	+	0.965820i	$0.416536\pi$
$968$	−11137.3	−0.369801
$969$	−356.908	−0.0118323
$970$	−88161.2	−2.91823
$971$	−55818.4	−1.84480	−0.922398	−	0.386240i	$-0.873774\pi$
$971$	−55818.4	−1.84480	−0.922398	+	0.386240i	$0.873774\pi$
$972$	2474.95	0.0816709
$973$	68279.5	2.24968
$974$	−72687.9	−2.39124
$975$	−6585.60	−0.216316
$976$	13701.3	0.449352
$977$	11417.4	0.373873	0.186936	−	0.982372i	$-0.440144\pi$
$977$	11417.4	0.373873	0.186936	+	0.982372i	$0.440144\pi$
$978$	42501.0	1.38960
$979$	−35388.9	−1.15530
$980$	16773.7	0.546753
$981$	1611.15	0.0524363
$982$	64657.6	2.10113
$983$	13517.3	0.438593	0.219296	−	0.975658i	$-0.429624\pi$
$983$	13517.3	0.438593	0.219296	+	0.975658i	$0.429624\pi$
$984$	−6651.50	−0.215490
$985$	27086.6	0.876195
$986$	−795.607	−0.0256970
$987$	−27927.1	−0.900639
$988$	18838.6	0.606614
$989$	−852.654	−0.0274144
$990$	−26962.4	−0.865578
$991$	20867.5	0.668897	0.334448	−	0.942414i	$-0.391450\pi$
$991$	20867.5	0.668897	0.334448	+	0.942414i	$0.391450\pi$
$992$	−22384.9	−0.716454
$993$	−17582.5	−0.561898
$994$	−4242.81	−0.135386
$995$	−61935.7	−1.97336
$996$	10795.1	0.343428
$997$	−22265.9	−0.707290	−0.353645	−	0.935380i	$-0.615058\pi$
$997$	−22265.9	−0.707290	−0.353645	+	0.935380i	$0.615058\pi$
$998$	−80657.2	−2.55828
$999$	6253.28	0.198043

Twists

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

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

\(f(q)\)	\(=\)	\(q-4.26439 q^{2} +3.00000 q^{3} +10.1850 q^{4} +13.9771 q^{5} -12.7932 q^{6} +21.4669 q^{7} -9.31764 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.26439 q^{2} +3.00000 q^{3} +10.1850 q^{4} +13.9771 q^{5} -12.7932 q^{6} +21.4669 q^{7} -9.31764 q^{8} +9.00000 q^{9} -59.6039 q^{10} +50.2623 q^{11} +30.5550 q^{12} -31.1995 q^{13} -91.5433 q^{14} +41.9314 q^{15} -41.7459 q^{16} +2.00676 q^{17} -38.3795 q^{18} -59.2842 q^{19} +142.357 q^{20} +64.4008 q^{21} -214.338 q^{22} -28.6724 q^{23} -27.9529 q^{24} +70.3601 q^{25} +133.047 q^{26} +27.0000 q^{27} +218.640 q^{28} +92.9707 q^{29} -178.812 q^{30} -88.6315 q^{31} +252.562 q^{32} +150.787 q^{33} -8.55761 q^{34} +300.046 q^{35} +91.6649 q^{36} +231.603 q^{37} +252.811 q^{38} -93.5986 q^{39} -130.234 q^{40} +237.954 q^{41} -274.630 q^{42} +29.7378 q^{43} +511.921 q^{44} +125.794 q^{45} +122.270 q^{46} -433.646 q^{47} -125.238 q^{48} +117.829 q^{49} -300.042 q^{50} +6.02029 q^{51} -317.767 q^{52} +464.489 q^{53} -115.138 q^{54} +702.522 q^{55} -200.021 q^{56} -177.853 q^{57} -396.463 q^{58} +59.0000 q^{59} +427.071 q^{60} -328.206 q^{61} +377.959 q^{62} +193.202 q^{63} -743.054 q^{64} -436.080 q^{65} -643.013 q^{66} -841.628 q^{67} +20.4388 q^{68} -86.0172 q^{69} -1279.51 q^{70} +46.3476 q^{71} -83.8588 q^{72} -738.776 q^{73} -987.644 q^{74} +211.080 q^{75} -603.809 q^{76} +1078.98 q^{77} +399.140 q^{78} +734.298 q^{79} -583.488 q^{80} +81.0000 q^{81} -1014.73 q^{82} +353.300 q^{83} +655.921 q^{84} +28.0488 q^{85} -126.813 q^{86} +278.912 q^{87} -468.326 q^{88} -704.085 q^{89} -536.435 q^{90} -669.758 q^{91} -292.028 q^{92} -265.894 q^{93} +1849.23 q^{94} -828.623 q^{95} +757.685 q^{96} +1479.12 q^{97} -502.468 q^{98} +452.360 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{2} + 24 q^{3} + 34 q^{4} + 42 q^{5} + 18 q^{6} + 53 q^{7} + 51 q^{8} + 72 q^{9}+O(q^{10}) \) 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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