LMFDB - Embedded newform 177.4.a.d.1.2

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			1.2
Root			$4.77847$ 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)

$f(q)$	$=$	$q-3.77847 q^{2} +3.00000 q^{3} +6.27681 q^{4} -5.74283 q^{5} -11.3354 q^{6} -24.4226 q^{7} +6.51103 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.77847 q^{2} +3.00000 q^{3} +6.27681 q^{4} -5.74283 q^{5} -11.3354 q^{6} -24.4226 q^{7} +6.51103 q^{8} +9.00000 q^{9} +21.6991 q^{10} +9.31943 q^{11} +18.8304 q^{12} +17.0721 q^{13} +92.2799 q^{14} -17.2285 q^{15} -74.8162 q^{16} +19.6787 q^{17} -34.0062 q^{18} -15.1270 q^{19} -36.0466 q^{20} -73.2678 q^{21} -35.2132 q^{22} +108.469 q^{23} +19.5331 q^{24} -92.0199 q^{25} -64.5065 q^{26} +27.0000 q^{27} -153.296 q^{28} +237.137 q^{29} +65.0973 q^{30} +184.628 q^{31} +230.602 q^{32} +27.9583 q^{33} -74.3553 q^{34} +140.255 q^{35} +56.4913 q^{36} +155.472 q^{37} +57.1569 q^{38} +51.2164 q^{39} -37.3917 q^{40} +261.745 q^{41} +276.840 q^{42} +127.647 q^{43} +58.4963 q^{44} -51.6855 q^{45} -409.848 q^{46} +555.816 q^{47} -224.448 q^{48} +253.463 q^{49} +347.694 q^{50} +59.0361 q^{51} +107.158 q^{52} -220.668 q^{53} -102.019 q^{54} -53.5199 q^{55} -159.016 q^{56} -45.3810 q^{57} -896.014 q^{58} +59.0000 q^{59} -108.140 q^{60} -436.694 q^{61} -697.610 q^{62} -219.803 q^{63} -272.793 q^{64} -98.0424 q^{65} -105.639 q^{66} +924.006 q^{67} +123.519 q^{68} +325.408 q^{69} -529.948 q^{70} -937.019 q^{71} +58.5992 q^{72} +13.0348 q^{73} -587.446 q^{74} -276.060 q^{75} -94.9493 q^{76} -227.605 q^{77} -193.519 q^{78} +140.714 q^{79} +429.657 q^{80} +81.0000 q^{81} -988.996 q^{82} -26.4265 q^{83} -459.888 q^{84} -113.011 q^{85} -482.310 q^{86} +711.411 q^{87} +60.6791 q^{88} -136.748 q^{89} +195.292 q^{90} -416.946 q^{91} +680.842 q^{92} +553.883 q^{93} -2100.13 q^{94} +86.8719 q^{95} +691.806 q^{96} -360.785 q^{97} -957.701 q^{98} +83.8749 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.77847	−1.33589	−0.667945	−	0.744211i	$-0.732826\pi$
$2$	−3.77847	−1.33589	−0.667945	+	0.744211i	$0.732826\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	6.27681	0.784601
$5$	−5.74283	−0.513655	−0.256827	−	0.966457i	$-0.582677\pi$
$5$	−5.74283	−0.513655	−0.256827	+	0.966457i	$0.582677\pi$
$6$	−11.3354	−0.771276
$7$	−24.4226	−1.31870	−0.659348	−	0.751838i	$-0.729168\pi$
$7$	−24.4226	−1.31870	−0.659348	+	0.751838i	$0.729168\pi$
$8$	6.51103	0.287749
$9$	9.00000	0.333333
$10$	21.6991	0.686186
$11$	9.31943	0.255447	0.127723	−	0.991810i	$-0.459233\pi$
$11$	9.31943	0.255447	0.127723	+	0.991810i	$0.459233\pi$
$12$	18.8304	0.452990
$13$	17.0721	0.364227	0.182114	−	0.983277i	$-0.441706\pi$
$13$	17.0721	0.364227	0.182114	+	0.983277i	$0.441706\pi$
$14$	92.2799	1.76163
$15$	−17.2285	−0.296559
$16$	−74.8162	−1.16900
$17$	19.6787	0.280752	0.140376	−	0.990098i	$-0.455169\pi$
$17$	19.6787	0.280752	0.140376	+	0.990098i	$0.455169\pi$
$18$	−34.0062	−0.445297
$19$	−15.1270	−0.182651	−0.0913256	−	0.995821i	$-0.529110\pi$
$19$	−15.1270	−0.182651	−0.0913256	+	0.995821i	$0.529110\pi$
$20$	−36.0466	−0.403014
$21$	−73.2678	−0.761349
$22$	−35.2132	−0.341249
$23$	108.469	0.983367	0.491683	−	0.870774i	$-0.336382\pi$
$23$	108.469	0.983367	0.491683	+	0.870774i	$0.336382\pi$
$24$	19.5331	0.166132
$25$	−92.0199	−0.736159
$26$	−64.5065	−0.486568
$27$	27.0000	0.192450
$28$	−153.296	−1.03465
$29$	237.137	1.51846	0.759228	−	0.650825i	$-0.225577\pi$
$29$	237.137	1.51846	0.759228	+	0.650825i	$0.225577\pi$
$30$	65.0973	0.396169
$31$	184.628	1.06968	0.534841	−	0.844953i	$-0.320372\pi$
$31$	184.628	1.06968	0.534841	+	0.844953i	$0.320372\pi$
$32$	230.602	1.27391
$33$	27.9583	0.147482
$34$	−74.3553	−0.375054
$35$	140.255	0.677354
$36$	56.4913	0.261534
$37$	155.472	0.690795	0.345398	−	0.938456i	$-0.387744\pi$
$37$	155.472	0.690795	0.345398	+	0.938456i	$0.387744\pi$
$38$	57.1569	0.244002
$39$	51.2164	0.210287
$40$	−37.3917	−0.147804
$41$	261.745	0.997019	0.498509	−	0.866884i	$-0.333881\pi$
$41$	261.745	0.997019	0.498509	+	0.866884i	$0.333881\pi$
$42$	276.840	1.01708
$43$	127.647	0.452697	0.226349	−	0.974046i	$-0.427321\pi$
$43$	127.647	0.452697	0.226349	+	0.974046i	$0.427321\pi$
$44$	58.4963	0.200424
$45$	−51.6855	−0.171218
$46$	−409.848	−1.31367
$47$	555.816	1.72498	0.862490	−	0.506075i	$-0.168904\pi$
$47$	555.816	1.72498	0.862490	+	0.506075i	$0.168904\pi$
$48$	−224.448	−0.674924
$49$	253.463	0.738959
$50$	347.694	0.983427
$51$	59.0361	0.162092
$52$	107.158	0.285773
$53$	−220.668	−0.571908	−0.285954	−	0.958243i	$-0.592311\pi$
$53$	−220.668	−0.571908	−0.285954	+	0.958243i	$0.592311\pi$
$54$	−102.019	−0.257092
$55$	−53.5199	−0.131211
$56$	−159.016	−0.379454
$57$	−45.3810	−0.105454
$58$	−896.014	−2.02849
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	−108.140	−0.232680
$61$	−436.694	−0.916606	−0.458303	−	0.888796i	$-0.651543\pi$
$61$	−436.694	−0.916606	−0.458303	+	0.888796i	$0.651543\pi$
$62$	−697.610	−1.42898
$63$	−219.803	−0.439565
$64$	−272.793	−0.532799
$65$	−98.0424	−0.187087
$66$	−105.639	−0.197020
$67$	924.006	1.68486	0.842428	−	0.538809i	$-0.181126\pi$
$67$	924.006	1.68486	0.842428	+	0.538809i	$0.181126\pi$
$68$	123.519	0.220278
$69$	325.408	0.567747
$70$	−529.948	−0.904870
$71$	−937.019	−1.56625	−0.783125	−	0.621865i	$-0.786375\pi$
$71$	−937.019	−1.56625	−0.783125	+	0.621865i	$0.786375\pi$
$72$	58.5992	0.0959165
$73$	13.0348	0.0208988	0.0104494	−	0.999945i	$-0.496674\pi$
$73$	13.0348	0.0208988	0.0104494	+	0.999945i	$0.496674\pi$
$74$	−587.446	−0.922826
$75$	−276.060	−0.425022
$76$	−94.9493	−0.143308
$77$	−227.605	−0.336857
$78$	−193.519	−0.280920
$79$	140.714	0.200400	0.100200	−	0.994967i	$-0.468052\pi$
$79$	140.714	0.200400	0.100200	+	0.994967i	$0.468052\pi$
$80$	429.657	0.600463
$81$	81.0000	0.111111
$82$	−988.996	−1.33191
$83$	−26.4265	−0.0349480	−0.0174740	−	0.999847i	$-0.505562\pi$
$83$	−26.4265	−0.0349480	−0.0174740	+	0.999847i	$0.505562\pi$
$84$	−459.888	−0.597355
$85$	−113.011	−0.144210
$86$	−482.310	−0.604753
$87$	711.411	0.876681
$88$	60.6791	0.0735047
$89$	−136.748	−0.162869	−0.0814343	−	0.996679i	$-0.525950\pi$
$89$	−136.748	−0.162869	−0.0814343	+	0.996679i	$0.525950\pi$
$90$	195.292	0.228729
$91$	−416.946	−0.480305
$92$	680.842	0.771551
$93$	553.883	0.617581
$94$	−2100.13	−2.30438
$95$	86.8719	0.0938196
$96$	691.806	0.735491
$97$	−360.785	−0.377651	−0.188826	−	0.982011i	$-0.560468\pi$
$97$	−360.785	−0.377651	−0.188826	+	0.982011i	$0.560468\pi$
$98$	−957.701	−0.987168
$99$	83.8749	0.0851489
$100$	−577.591	−0.577591
$101$	421.340	0.415098	0.207549	−	0.978225i	$-0.433451\pi$
$101$	421.340	0.415098	0.207549	+	0.978225i	$0.433451\pi$
$102$	−223.066	−0.216537
$103$	−1389.28	−1.32903	−0.664515	−	0.747275i	$-0.731362\pi$
$103$	−1389.28	−1.32903	−0.664515	+	0.747275i	$0.731362\pi$
$104$	111.157	0.104806
$105$	420.765	0.391071
$106$	833.788	0.764006
$107$	−339.530	−0.306763	−0.153382	−	0.988167i	$-0.549016\pi$
$107$	−339.530	−0.306763	−0.153382	+	0.988167i	$0.549016\pi$
$108$	169.474	0.150997
$109$	297.022	0.261005	0.130503	−	0.991448i	$-0.458341\pi$
$109$	297.022	0.261005	0.130503	+	0.991448i	$0.458341\pi$
$110$	202.223	0.175284
$111$	466.416	0.398831
$112$	1827.20	1.54156
$113$	1553.81	1.29354	0.646769	−	0.762686i	$-0.276120\pi$
$113$	1553.81	1.29354	0.646769	+	0.762686i	$0.276120\pi$
$114$	171.471	0.140875
$115$	−622.922	−0.505111
$116$	1488.46	1.19138
$117$	153.649	0.121409
$118$	−222.930	−0.173918
$119$	−480.605	−0.370227
$120$	−112.175	−0.0853346
$121$	−1244.15	−0.934747
$122$	1650.03	1.22448
$123$	785.236	0.575629
$124$	1158.87	0.839273
$125$	1246.31	0.891786
$126$	830.519	0.587211
$127$	1320.50	0.922642	0.461321	−	0.887233i	$-0.347376\pi$
$127$	1320.50	0.922642	0.461321	+	0.887233i	$0.347376\pi$
$128$	−814.078	−0.562148
$129$	382.941	0.261365
$130$	370.450	0.249928
$131$	−1738.41	−1.15943	−0.579716	−	0.814819i	$-0.696836\pi$
$131$	−1738.41	−1.15943	−0.579716	+	0.814819i	$0.696836\pi$
$132$	175.489	0.115715
$133$	369.441	0.240861
$134$	−3491.33	−2.25078
$135$	−155.056	−0.0988529
$136$	128.129	0.0807863
$137$	−776.721	−0.484378	−0.242189	−	0.970229i	$-0.577865\pi$
$137$	−776.721	−0.484378	−0.242189	+	0.970229i	$0.577865\pi$
$138$	−1229.54	−0.758447
$139$	855.546	0.522061	0.261030	−	0.965331i	$-0.415938\pi$
$139$	855.546	0.522061	0.261030	+	0.965331i	$0.415938\pi$
$140$	880.353	0.531453
$141$	1667.45	0.995917
$142$	3540.49	2.09234
$143$	159.103	0.0930407
$144$	−673.345	−0.389667
$145$	−1361.84	−0.779962
$146$	−49.2517	−0.0279185
$147$	760.389	0.426638
$148$	975.867	0.541999
$149$	2001.36	1.10039	0.550194	−	0.835037i	$-0.314554\pi$
$149$	2001.36	1.10039	0.550194	+	0.835037i	$0.314554\pi$
$150$	1043.08	0.567782
$151$	1656.52	0.892750	0.446375	−	0.894846i	$-0.352715\pi$
$151$	1656.52	0.892750	0.446375	+	0.894846i	$0.352715\pi$
$152$	−98.4924	−0.0525578
$153$	177.108	0.0935841
$154$	859.996	0.450003
$155$	−1060.29	−0.549447
$156$	321.475	0.164991
$157$	649.443	0.330135	0.165068	−	0.986282i	$-0.447216\pi$
$157$	649.443	0.330135	0.165068	+	0.986282i	$0.447216\pi$
$158$	−531.685	−0.267713
$159$	−662.005	−0.330191
$160$	−1324.31	−0.654349
$161$	−2649.10	−1.29676
$162$	−306.056	−0.148432
$163$	508.548	0.244372	0.122186	−	0.992507i	$-0.461010\pi$
$163$	508.548	0.244372	0.122186	+	0.992507i	$0.461010\pi$
$164$	1642.93	0.782262
$165$	−160.560	−0.0757549
$166$	99.8517	0.0466867
$167$	1419.46	0.657733	0.328867	−	0.944376i	$-0.393333\pi$
$167$	1419.46	0.657733	0.328867	+	0.944376i	$0.393333\pi$
$168$	−477.048	−0.219078
$169$	−1905.54	−0.867338
$170$	427.010	0.192648
$171$	−136.143	−0.0608837
$172$	801.215	0.355187
$173$	−929.169	−0.408343	−0.204172	−	0.978935i	$-0.565450\pi$
$173$	−929.169	−0.408343	−0.204172	+	0.978935i	$0.565450\pi$
$174$	−2688.04	−1.17115
$175$	2247.36	0.970770
$176$	−697.244	−0.298618
$177$	177.000	0.0751646
$178$	516.699	0.217574
$179$	−2212.72	−0.923948	−0.461974	−	0.886893i	$-0.652859\pi$
$179$	−2212.72	−0.923948	−0.461974	+	0.886893i	$0.652859\pi$
$180$	−324.420	−0.134338
$181$	−46.5937	−0.0191342	−0.00956709	−	0.999954i	$-0.503045\pi$
$181$	−46.5937	−0.0191342	−0.00956709	+	0.999954i	$0.503045\pi$
$182$	1575.41	0.641635
$183$	−1310.08	−0.529203
$184$	706.247	0.282963
$185$	−892.849	−0.354830
$186$	−2092.83	−0.825020
$187$	183.394	0.0717172
$188$	3488.75	1.35342
$189$	−659.410	−0.253783
$190$	−328.242	−0.125333
$191$	−2035.98	−0.771302	−0.385651	−	0.922645i	$-0.626023\pi$
$191$	−2035.98	−0.771302	−0.385651	+	0.922645i	$0.626023\pi$
$192$	−818.379	−0.307611
$193$	−1530.18	−0.570697	−0.285349	−	0.958424i	$-0.592109\pi$
$193$	−1530.18	−0.570697	−0.285349	+	0.958424i	$0.592109\pi$
$194$	1363.21	0.504500
$195$	−294.127	−0.108015
$196$	1590.94	0.579788
$197$	2461.81	0.890337	0.445169	−	0.895447i	$-0.353144\pi$
$197$	2461.81	0.890337	0.445169	+	0.895447i	$0.353144\pi$
$198$	−316.918	−0.113750
$199$	1442.27	0.513768	0.256884	−	0.966442i	$-0.417304\pi$
$199$	1442.27	0.513768	0.256884	+	0.966442i	$0.417304\pi$
$200$	−599.144	−0.211829
$201$	2772.02	0.972752
$202$	−1592.02	−0.554525
$203$	−5791.50	−2.00238
$204$	370.558	0.127178
$205$	−1503.16	−0.512123
$206$	5249.36	1.77544
$207$	976.225	0.327789
$208$	−1277.27	−0.425783
$209$	−140.975	−0.0466577
$210$	−1589.84	−0.522427
$211$	−732.346	−0.238942	−0.119471	−	0.992838i	$-0.538120\pi$
$211$	−732.346	−0.238942	−0.119471	+	0.992838i	$0.538120\pi$
$212$	−1385.09	−0.448720
$213$	−2811.06	−0.904274
$214$	1282.90	0.409802
$215$	−733.055	−0.232530
$216$	175.798	0.0553774
$217$	−4509.09	−1.41058
$218$	−1122.29	−0.348674
$219$	39.1045	0.0120659
$220$	−335.934	−0.102949
$221$	335.957	0.102258
$222$	−1762.34	−0.532794
$223$	2928.26	0.879332	0.439666	−	0.898161i	$-0.355097\pi$
$223$	2928.26	0.879332	0.439666	+	0.898161i	$0.355097\pi$
$224$	−5631.90	−1.67990
$225$	−828.179	−0.245386
$226$	−5871.00	−1.72802
$227$	3909.72	1.14316	0.571580	−	0.820546i	$-0.306331\pi$
$227$	3909.72	1.14316	0.571580	+	0.820546i	$0.306331\pi$
$228$	−284.848	−0.0827391
$229$	5580.68	1.61040	0.805200	−	0.593003i	$-0.202058\pi$
$229$	5580.68	1.61040	0.805200	+	0.593003i	$0.202058\pi$
$230$	2353.69	0.674772
$231$	−682.814	−0.194484
$232$	1544.01	0.436935
$233$	−3793.50	−1.06661	−0.533306	−	0.845923i	$-0.679050\pi$
$233$	−3793.50	−1.06661	−0.533306	+	0.845923i	$0.679050\pi$
$234$	−580.558	−0.162189
$235$	−3191.96	−0.886043
$236$	370.332	0.102146
$237$	422.143	0.115701
$238$	1815.95	0.494582
$239$	4995.54	1.35203	0.676013	−	0.736889i	$-0.263706\pi$
$239$	4995.54	1.35203	0.676013	+	0.736889i	$0.263706\pi$
$240$	1288.97	0.346678
$241$	1674.51	0.447572	0.223786	−	0.974638i	$-0.428158\pi$
$241$	1674.51	0.447572	0.223786	+	0.974638i	$0.428158\pi$
$242$	4700.97	1.24872
$243$	243.000	0.0641500
$244$	−2741.05	−0.719170
$245$	−1455.60	−0.379570
$246$	−2966.99	−0.768977
$247$	−258.250	−0.0665266
$248$	1202.12	0.307800
$249$	−79.2795	−0.0201773
$250$	−4709.14	−1.19133
$251$	4489.35	1.12895	0.564473	−	0.825452i	$-0.309080\pi$
$251$	4489.35	1.12895	0.564473	+	0.825452i	$0.309080\pi$
$252$	−1379.66	−0.344883
$253$	1010.87	0.251198
$254$	−4989.47	−1.23255
$255$	−339.034	−0.0832595
$256$	5258.31	1.28377
$257$	5099.27	1.23768	0.618840	−	0.785517i	$-0.287603\pi$
$257$	5099.27	1.23768	0.618840	+	0.785517i	$0.287603\pi$
$258$	−1446.93	−0.349154
$259$	−3797.03	−0.910949
$260$	−615.393	−0.146789
$261$	2134.23	0.506152
$262$	6568.52	1.54887
$263$	7949.30	1.86378	0.931892	−	0.362737i	$-0.118158\pi$
$263$	7949.30	1.86378	0.931892	+	0.362737i	$0.118158\pi$
$264$	182.037	0.0424379
$265$	1267.26	0.293763
$266$	−1395.92	−0.321764
$267$	−410.245	−0.0940322
$268$	5799.81	1.32194
$269$	−3831.43	−0.868426	−0.434213	−	0.900810i	$-0.642973\pi$
$269$	−3831.43	−0.868426	−0.434213	+	0.900810i	$0.642973\pi$
$270$	585.876	0.132056
$271$	−6824.86	−1.52982	−0.764909	−	0.644139i	$-0.777216\pi$
$271$	−6824.86	−1.52982	−0.764909	+	0.644139i	$0.777216\pi$
$272$	−1472.28	−0.328200
$273$	−1250.84	−0.277304
$274$	2934.82	0.647075
$275$	−857.573	−0.188049
$276$	2042.52	0.445455
$277$	−40.1056	−0.00869931	−0.00434966	−	0.999991i	$-0.501385\pi$
$277$	−40.1056	−0.00869931	−0.00434966	+	0.999991i	$0.501385\pi$
$278$	−3232.65	−0.697416
$279$	1661.65	0.356560
$280$	913.203	0.194908
$281$	−2405.83	−0.510746	−0.255373	−	0.966843i	$-0.582198\pi$
$281$	−2405.83	−0.510746	−0.255373	+	0.966843i	$0.582198\pi$
$282$	−6300.39	−1.33044
$283$	8232.15	1.72915	0.864577	−	0.502501i	$-0.167587\pi$
$283$	8232.15	1.72915	0.864577	+	0.502501i	$0.167587\pi$
$284$	−5881.49	−1.22888
$285$	260.616	0.0541668
$286$	−601.164	−0.124292
$287$	−6392.50	−1.31476
$288$	2075.42	0.424636
$289$	−4525.75	−0.921178
$290$	5145.66	1.04194
$291$	−1082.36	−0.218037
$292$	81.8171	0.0163972
$293$	1902.22	0.379280	0.189640	−	0.981854i	$-0.439268\pi$
$293$	1902.22	0.379280	0.189640	+	0.981854i	$0.439268\pi$
$294$	−2873.10	−0.569941
$295$	−338.827	−0.0668721
$296$	1012.28	0.198776
$297$	251.625	0.0491608
$298$	−7562.07	−1.47000
$299$	1851.80	0.358169
$300$	−1732.77	−0.333472
$301$	−3117.47	−0.596970
$302$	−6259.09	−1.19262
$303$	1264.02	0.239657
$304$	1131.74	0.213520
$305$	2507.86	0.470819
$306$	−669.198	−0.125018
$307$	9226.20	1.71520	0.857601	−	0.514315i	$-0.171954\pi$
$307$	9226.20	1.71520	0.857601	+	0.514315i	$0.171954\pi$
$308$	−1428.63	−0.264298
$309$	−4167.85	−0.767316
$310$	4006.26	0.734000
$311$	−2395.89	−0.436844	−0.218422	−	0.975854i	$-0.570091\pi$
$311$	−2395.89	−0.436844	−0.218422	+	0.975854i	$0.570091\pi$
$312$	333.471	0.0605099
$313$	−10071.9	−1.81884	−0.909421	−	0.415878i	$-0.863474\pi$
$313$	−10071.9	−1.81884	−0.909421	+	0.415878i	$0.863474\pi$
$314$	−2453.90	−0.441024
$315$	1262.29	0.225785
$316$	883.238	0.157234
$317$	1068.39	0.189296	0.0946478	−	0.995511i	$-0.469828\pi$
$317$	1068.39	0.189296	0.0946478	+	0.995511i	$0.469828\pi$
$318$	2501.36	0.441099
$319$	2209.98	0.387885
$320$	1566.60	0.273674
$321$	−1018.59	−0.177110
$322$	10009.6	1.73233
$323$	−297.680	−0.0512797
$324$	508.421	0.0871779
$325$	−1570.98	−0.268129
$326$	−1921.53	−0.326454
$327$	891.066	0.150691
$328$	1704.23	0.286892
$329$	−13574.5	−2.27472
$330$	606.670	0.101200
$331$	−9649.80	−1.60242	−0.801210	−	0.598384i	$-0.795810\pi$
$331$	−9649.80	−1.60242	−0.801210	+	0.598384i	$0.795810\pi$
$332$	−165.874	−0.0274203
$333$	1399.25	0.230265
$334$	−5363.40	−0.878659
$335$	−5306.41	−0.865434
$336$	5481.61	0.890019
$337$	−6617.95	−1.06974	−0.534871	−	0.844934i	$-0.679640\pi$
$337$	−6617.95	−1.06974	−0.534871	+	0.844934i	$0.679640\pi$
$338$	7200.03	1.15867
$339$	4661.42	0.746824
$340$	−709.351	−0.113147
$341$	1720.63	0.273247
$342$	514.412	0.0813340
$343$	2186.73	0.344234
$344$	831.113	0.130263
$345$	−1868.77	−0.291626
$346$	3510.83	0.545502
$347$	3888.20	0.601525	0.300763	−	0.953699i	$-0.402759\pi$
$347$	3888.20	0.601525	0.300763	+	0.953699i	$0.402759\pi$
$348$	4465.39	0.687845
$349$	−8651.21	−1.32690	−0.663451	−	0.748220i	$-0.730909\pi$
$349$	−8651.21	−1.32690	−0.663451	+	0.748220i	$0.730909\pi$
$350$	−8491.59	−1.29684
$351$	460.948	0.0700956
$352$	2149.08	0.325416
$353$	1996.55	0.301036	0.150518	−	0.988607i	$-0.451906\pi$
$353$	1996.55	0.301036	0.150518	+	0.988607i	$0.451906\pi$
$354$	−668.789	−0.100412
$355$	5381.14	0.804511
$356$	−858.343	−0.127787
$357$	−1441.81	−0.213751
$358$	8360.70	1.23429
$359$	11674.1	1.71626	0.858128	−	0.513436i	$-0.171627\pi$
$359$	11674.1	1.71626	0.858128	+	0.513436i	$0.171627\pi$
$360$	−336.526	−0.0492679
$361$	−6630.17	−0.966639
$362$	176.053	0.0255611
$363$	−3732.44	−0.539676
$364$	−2617.09	−0.376848
$365$	−74.8569	−0.0107348
$366$	4950.10	0.706956
$367$	11122.7	1.58201	0.791005	−	0.611810i	$-0.209558\pi$
$367$	11122.7	1.58201	0.791005	+	0.611810i	$0.209558\pi$
$368$	−8115.27	−1.14956
$369$	2355.71	0.332340
$370$	3373.60	0.474014
$371$	5389.30	0.754173
$372$	3476.62	0.484554
$373$	7447.39	1.03381	0.516905	−	0.856043i	$-0.327084\pi$
$373$	7447.39	1.03381	0.516905	+	0.856043i	$0.327084\pi$
$374$	−692.949	−0.0958063
$375$	3738.93	0.514873
$376$	3618.93	0.496362
$377$	4048.43	0.553063
$378$	2491.56	0.339026
$379$	−2736.87	−0.370933	−0.185467	−	0.982651i	$-0.559380\pi$
$379$	−2736.87	−0.370933	−0.185467	+	0.982651i	$0.559380\pi$
$380$	545.278	0.0736110
$381$	3961.51	0.532688
$382$	7692.90	1.03037
$383$	−3711.27	−0.495136	−0.247568	−	0.968871i	$-0.579631\pi$
$383$	−3711.27	−0.495136	−0.247568	+	0.968871i	$0.579631\pi$
$384$	−2442.23	−0.324556
$385$	1307.10	0.173028
$386$	5781.72	0.762388
$387$	1148.82	0.150899
$388$	−2264.58	−0.296306
$389$	68.7597	0.00896209	0.00448105	−	0.999990i	$-0.498574\pi$
$389$	68.7597	0.00896209	0.00448105	+	0.999990i	$0.498574\pi$
$390$	1111.35	0.144296
$391$	2134.54	0.276082
$392$	1650.30	0.212635
$393$	−5215.23	−0.669398
$394$	−9301.85	−1.18939
$395$	−808.100	−0.102936
$396$	526.466	0.0668079
$397$	−9361.13	−1.18343	−0.591715	−	0.806147i	$-0.701549\pi$
$397$	−9361.13	−1.18343	−0.591715	+	0.806147i	$0.701549\pi$
$398$	−5449.57	−0.686337
$399$	1108.32	0.139061
$400$	6884.57	0.860572
$401$	−95.5960	−0.0119048	−0.00595241	−	0.999982i	$-0.501895\pi$
$401$	−95.5960	−0.0119048	−0.00595241	+	0.999982i	$0.501895\pi$
$402$	−10474.0	−1.29949
$403$	3151.99	0.389607
$404$	2644.67	0.325686
$405$	−465.169	−0.0570727
$406$	21883.0	2.67496
$407$	1448.91	0.176461
$408$	384.386	0.0466420
$409$	−11054.7	−1.33648	−0.668240	−	0.743946i	$-0.732952\pi$
$409$	−11054.7	−1.33648	−0.668240	+	0.743946i	$0.732952\pi$
$410$	5679.64	0.684140
$411$	−2330.16	−0.279656
$412$	−8720.26	−1.04276
$413$	−1440.93	−0.171680
$414$	−3688.63	−0.437890
$415$	151.763	0.0179512
$416$	3936.87	0.463992
$417$	2566.64	0.301412
$418$	532.670	0.0623295
$419$	−4854.35	−0.565992	−0.282996	−	0.959121i	$-0.591328\pi$
$419$	−4854.35	−0.565992	−0.282996	+	0.959121i	$0.591328\pi$
$420$	2641.06	0.306834
$421$	−5480.81	−0.634485	−0.317242	−	0.948344i	$-0.602757\pi$
$421$	−5480.81	−0.634485	−0.317242	+	0.948344i	$0.602757\pi$
$422$	2767.14	0.319200
$423$	5002.34	0.574993
$424$	−1436.78	−0.164566
$425$	−1810.83	−0.206678
$426$	10621.5	1.20801
$427$	10665.2	1.20872
$428$	−2131.17	−0.240687
$429$	477.308	0.0537171
$430$	2769.82	0.310634
$431$	7366.56	0.823282	0.411641	−	0.911346i	$-0.364956\pi$
$431$	7366.56	0.823282	0.411641	+	0.911346i	$0.364956\pi$
$432$	−2020.04	−0.224975
$433$	228.362	0.0253450	0.0126725	−	0.999920i	$-0.495966\pi$
$433$	228.362	0.0253450	0.0126725	+	0.999920i	$0.495966\pi$
$434$	17037.4	1.88439
$435$	−4085.51	−0.450311
$436$	1864.35	0.204785
$437$	−1640.82	−0.179613
$438$	−147.755	−0.0161187
$439$	−16897.8	−1.83710	−0.918550	−	0.395305i	$-0.870639\pi$
$439$	−16897.8	−1.83710	−0.918550	+	0.395305i	$0.870639\pi$
$440$	−348.470	−0.0377560
$441$	2281.17	0.246320
$442$	−1269.40	−0.136605
$443$	8663.70	0.929176	0.464588	−	0.885527i	$-0.346202\pi$
$443$	8663.70	0.929176	0.464588	+	0.885527i	$0.346202\pi$
$444$	2927.60	0.312923
$445$	785.323	0.0836582
$446$	−11064.3	−1.17469
$447$	6004.08	0.635309
$448$	6662.31	0.702599
$449$	−12014.8	−1.26284	−0.631418	−	0.775442i	$-0.717527\pi$
$449$	−12014.8	−1.26284	−0.631418	+	0.775442i	$0.717527\pi$
$450$	3129.25	0.327809
$451$	2439.32	0.254685
$452$	9752.94	1.01491
$453$	4969.55	0.515430
$454$	−14772.8	−1.52714
$455$	2394.45	0.246711
$456$	−295.477	−0.0303443
$457$	15590.7	1.59585	0.797925	−	0.602756i	$-0.205931\pi$
$457$	15590.7	1.59585	0.797925	+	0.602756i	$0.205931\pi$
$458$	−21086.4	−2.15132
$459$	531.325	0.0540308
$460$	−3909.96	−0.396310
$461$	1475.12	0.149031	0.0745155	−	0.997220i	$-0.476259\pi$
$461$	1475.12	0.149031	0.0745155	+	0.997220i	$0.476259\pi$
$462$	2579.99	0.259809
$463$	1335.14	0.134015	0.0670077	−	0.997752i	$-0.478655\pi$
$463$	1335.14	0.134015	0.0670077	+	0.997752i	$0.478655\pi$
$464$	−17741.7	−1.77508
$465$	−3180.86	−0.317223
$466$	14333.6	1.42488
$467$	−13910.9	−1.37841	−0.689205	−	0.724566i	$-0.742040\pi$
$467$	−13910.9	−1.37841	−0.689205	+	0.724566i	$0.742040\pi$
$468$	964.426	0.0952577
$469$	−22566.6	−2.22181
$470$	12060.7	1.18366
$471$	1948.33	0.190604
$472$	384.151	0.0374618
$473$	1189.60	0.115640
$474$	−1595.05	−0.154564
$475$	1391.99	0.134460
$476$	−3016.66	−0.290480
$477$	−1986.02	−0.190636
$478$	−18875.5	−1.80616
$479$	15483.9	1.47699	0.738495	−	0.674259i	$-0.235537\pi$
$479$	15483.9	1.47699	0.738495	+	0.674259i	$0.235537\pi$
$480$	−3972.93	−0.377789
$481$	2654.24	0.251607
$482$	−6327.09	−0.597906
$483$	−7947.31	−0.748686
$484$	−7809.28	−0.733403
$485$	2071.93	0.193982
$486$	−918.167	−0.0856974
$487$	−12901.8	−1.20049	−0.600243	−	0.799817i	$-0.704930\pi$
$487$	−12901.8	−1.20049	−0.600243	+	0.799817i	$0.704930\pi$
$488$	−2843.33	−0.263753
$489$	1525.65	0.141088
$490$	5499.92	0.507063
$491$	−4348.11	−0.399649	−0.199824	−	0.979832i	$-0.564037\pi$
$491$	−4348.11	−0.399649	−0.199824	+	0.979832i	$0.564037\pi$
$492$	4928.78	0.451639
$493$	4666.55	0.426310
$494$	975.790	0.0888722
$495$	−481.679	−0.0437371
$496$	−13813.1	−1.25046
$497$	22884.4	2.06541
$498$	299.555	0.0269546
$499$	−12051.7	−1.08118	−0.540589	−	0.841287i	$-0.681798\pi$
$499$	−12051.7	−1.08118	−0.540589	+	0.841287i	$0.681798\pi$
$500$	7822.84	0.699696
$501$	4258.39	0.379742
$502$	−16962.9	−1.50815
$503$	16522.2	1.46459	0.732293	−	0.680990i	$-0.238450\pi$
$503$	16522.2	1.46459	0.732293	+	0.680990i	$0.238450\pi$
$504$	−1431.15	−0.126485
$505$	−2419.69	−0.213217
$506$	−3819.55	−0.335573
$507$	−5716.63	−0.500758
$508$	8288.53	0.723906
$509$	−15147.5	−1.31906	−0.659530	−	0.751678i	$-0.729245\pi$
$509$	−15147.5	−1.31906	−0.659530	+	0.751678i	$0.729245\pi$
$510$	1281.03	0.111225
$511$	−318.344	−0.0275592
$512$	−13355.7	−1.15282
$513$	−408.429	−0.0351512
$514$	−19267.4	−1.65340
$515$	7978.42	0.682662
$516$	2403.65	0.205067
$517$	5179.88	0.440640
$518$	14346.9	1.21693
$519$	−2787.51	−0.235757
$520$	−638.357	−0.0538342
$521$	14442.1	1.21443	0.607215	−	0.794538i	$-0.292287\pi$
$521$	14442.1	1.21443	0.607215	+	0.794538i	$0.292287\pi$
$522$	−8064.13	−0.676163
$523$	−4796.97	−0.401065	−0.200532	−	0.979687i	$-0.564267\pi$
$523$	−4796.97	−0.401065	−0.200532	+	0.979687i	$0.564267\pi$
$524$	−10911.7	−0.909691
$525$	6742.09	0.560474
$526$	−30036.2	−2.48981
$527$	3633.24	0.300315
$528$	−2091.73	−0.172407
$529$	−401.383	−0.0329895
$530$	−4788.31	−0.392435
$531$	531.000	0.0433963
$532$	2318.91	0.188980
$533$	4468.55	0.363142
$534$	1550.10	0.125617
$535$	1949.87	0.157570
$536$	6016.23	0.484816
$537$	−6638.17	−0.533442
$538$	14476.9	1.16012
$539$	2362.13	0.188765
$540$	−973.259	−0.0775600
$541$	−833.411	−0.0662313	−0.0331157	−	0.999452i	$-0.510543\pi$
$541$	−833.411	−0.0662313	−0.0331157	+	0.999452i	$0.510543\pi$
$542$	25787.5	2.04367
$543$	−139.781	−0.0110471
$544$	4537.95	0.357653
$545$	−1705.75	−0.134066
$546$	4726.24	0.370448
$547$	−19682.1	−1.53848	−0.769239	−	0.638961i	$-0.779365\pi$
$547$	−19682.1	−1.53848	−0.769239	+	0.638961i	$0.779365\pi$
$548$	−4875.33	−0.380043
$549$	−3930.25	−0.305535
$550$	3240.31	0.251213
$551$	−3587.17	−0.277348
$552$	2118.74	0.163369
$553$	−3436.61	−0.264267
$554$	151.538	0.0116213
$555$	−2678.55	−0.204861
$556$	5370.10	0.409609
$557$	−10287.7	−0.782594	−0.391297	−	0.920264i	$-0.627974\pi$
$557$	−10287.7	−0.782594	−0.391297	+	0.920264i	$0.627974\pi$
$558$	−6278.49	−0.476325
$559$	2179.21	0.164885
$560$	−10493.3	−0.791829
$561$	550.183	0.0414060
$562$	9090.34	0.682301
$563$	−8172.20	−0.611754	−0.305877	−	0.952071i	$-0.598950\pi$
$563$	−8172.20	−0.611754	−0.305877	+	0.952071i	$0.598950\pi$
$564$	10466.2	0.781398
$565$	−8923.25	−0.664431
$566$	−31104.9	−2.30996
$567$	−1978.23	−0.146522
$568$	−6100.96	−0.450687
$569$	9993.92	0.736322	0.368161	−	0.929762i	$-0.379988\pi$
$569$	9993.92	0.736322	0.368161	+	0.929762i	$0.379988\pi$
$570$	−984.727	−0.0723608
$571$	−8399.41	−0.615594	−0.307797	−	0.951452i	$-0.599592\pi$
$571$	−8399.41	−0.615594	−0.307797	+	0.951452i	$0.599592\pi$
$572$	998.656	0.0729998
$573$	−6107.95	−0.445311
$574$	24153.9	1.75638
$575$	−9981.34	−0.723914
$576$	−2455.14	−0.177600
$577$	−9807.97	−0.707645	−0.353822	−	0.935313i	$-0.615118\pi$
$577$	−9807.97	−0.707645	−0.353822	+	0.935313i	$0.615118\pi$
$578$	17100.4	1.23059
$579$	−4590.53	−0.329492
$580$	−8547.99	−0.611959
$581$	645.404	0.0460858
$582$	4089.64	0.291273
$583$	−2056.50	−0.146092
$584$	84.8702	0.00601362
$585$	−882.381	−0.0623624
$586$	−7187.48	−0.506676
$587$	−6955.54	−0.489073	−0.244537	−	0.969640i	$-0.578636\pi$
$587$	−6955.54	−0.489073	−0.244537	+	0.969640i	$0.578636\pi$
$588$	4772.81	0.334741
$589$	−2792.87	−0.195379
$590$	1280.25	0.0893338
$591$	7385.42	0.514036
$592$	−11631.8	−0.807542
$593$	−13988.6	−0.968710	−0.484355	−	0.874872i	$-0.660946\pi$
$593$	−13988.6	−0.968710	−0.484355	+	0.874872i	$0.660946\pi$
$594$	−950.755	−0.0656733
$595$	2760.03	0.190169
$596$	12562.1	0.863365
$597$	4326.81	0.296624
$598$	−6996.98	−0.478475
$599$	26538.4	1.81023	0.905117	−	0.425163i	$-0.139783\pi$
$599$	26538.4	1.81023	0.905117	+	0.425163i	$0.139783\pi$
$600$	−1797.43	−0.122300
$601$	−16540.3	−1.12261	−0.561307	−	0.827608i	$-0.689701\pi$
$601$	−16540.3	−1.12261	−0.561307	+	0.827608i	$0.689701\pi$
$602$	11779.3	0.797486
$603$	8316.06	0.561619
$604$	10397.6	0.700453
$605$	7144.93	0.480137
$606$	−4776.06	−0.320155
$607$	2733.68	0.182795	0.0913974	−	0.995814i	$-0.470867\pi$
$607$	2733.68	0.182795	0.0913974	+	0.995814i	$0.470867\pi$
$608$	−3488.32	−0.232681
$609$	−17374.5	−1.15608
$610$	−9475.87	−0.628962
$611$	9488.95	0.628285
$612$	1111.67	0.0734261
$613$	9935.81	0.654655	0.327328	−	0.944911i	$-0.393852\pi$
$613$	9935.81	0.654655	0.327328	+	0.944911i	$0.393852\pi$
$614$	−34860.9	−2.29132
$615$	−4509.48	−0.295674
$616$	−1481.94	−0.0969303
$617$	4126.63	0.269257	0.134629	−	0.990896i	$-0.457016\pi$
$617$	4126.63	0.269257	0.134629	+	0.990896i	$0.457016\pi$
$618$	15748.1	1.02505
$619$	13867.2	0.900436	0.450218	−	0.892919i	$-0.351346\pi$
$619$	13867.2	0.900436	0.450218	+	0.892919i	$0.351346\pi$
$620$	−6655.21	−0.431096
$621$	2928.67	0.189249
$622$	9052.79	0.583576
$623$	3339.75	0.214774
$624$	−3831.81	−0.245826
$625$	4345.14	0.278089
$626$	38056.3	2.42977
$627$	−422.925	−0.0269378
$628$	4076.43	0.259024
$629$	3059.49	0.193942
$630$	−4769.53	−0.301623
$631$	2950.74	0.186160	0.0930801	−	0.995659i	$-0.470329\pi$
$631$	2950.74	0.186160	0.0930801	+	0.995659i	$0.470329\pi$
$632$	916.196	0.0576651
$633$	−2197.04	−0.137953
$634$	−4036.87	−0.252878
$635$	−7583.42	−0.473919
$636$	−4155.28	−0.259068
$637$	4327.15	0.269149
$638$	−8350.34	−0.518171
$639$	−8433.17	−0.522083
$640$	4675.11	0.288750
$641$	−11795.4	−0.726820	−0.363410	−	0.931629i	$-0.618388\pi$
$641$	−11795.4	−0.726820	−0.363410	+	0.931629i	$0.618388\pi$
$642$	3848.71	0.236599
$643$	19304.7	1.18398	0.591992	−	0.805944i	$-0.298342\pi$
$643$	19304.7	1.18398	0.591992	+	0.805944i	$0.298342\pi$
$644$	−16627.9	−1.01744
$645$	−2199.16	−0.134251
$646$	1124.77	0.0685041
$647$	21361.3	1.29799	0.648995	−	0.760793i	$-0.275190\pi$
$647$	21361.3	1.29799	0.648995	+	0.760793i	$0.275190\pi$
$648$	527.393	0.0319722
$649$	549.846	0.0332563
$650$	5935.88	0.358191
$651$	−13527.3	−0.814401
$652$	3192.06	0.191734
$653$	5795.08	0.347288	0.173644	−	0.984809i	$-0.444446\pi$
$653$	5795.08	0.347288	0.173644	+	0.984809i	$0.444446\pi$
$654$	−3366.86	−0.201307
$655$	9983.39	0.595547
$656$	−19582.8	−1.16552
$657$	117.313	0.00696626
$658$	51290.6	3.03878
$659$	11950.9	0.706434	0.353217	−	0.935542i	$-0.385088\pi$
$659$	11950.9	0.706434	0.353217	+	0.935542i	$0.385088\pi$
$660$	−1007.80	−0.0594374
$661$	17906.5	1.05368	0.526838	−	0.849965i	$-0.323377\pi$
$661$	17906.5	1.05368	0.526838	+	0.849965i	$0.323377\pi$
$662$	36461.4	2.14065
$663$	1007.87	0.0590385
$664$	−172.064	−0.0100563
$665$	−2121.64	−0.123720
$666$	−5287.01	−0.307609
$667$	25722.1	1.49320
$668$	8909.70	0.516058
$669$	8784.79	0.507683
$670$	20050.1	1.15612
$671$	−4069.74	−0.234144
$672$	−16895.7	−0.969890
$673$	12825.6	0.734608	0.367304	−	0.930101i	$-0.380281\pi$
$673$	12825.6	0.734608	0.367304	+	0.930101i	$0.380281\pi$
$674$	25005.7	1.42906
$675$	−2484.54	−0.141674
$676$	−11960.7	−0.680514
$677$	11752.0	0.667156	0.333578	−	0.942722i	$-0.391744\pi$
$677$	11752.0	0.667156	0.333578	+	0.942722i	$0.391744\pi$
$678$	−17613.0	−0.997675
$679$	8811.31	0.498007
$680$	−735.821	−0.0414962
$681$	11729.2	0.660004
$682$	−6501.33	−0.365027
$683$	−10731.7	−0.601223	−0.300612	−	0.953747i	$-0.597191\pi$
$683$	−10731.7	−0.601223	−0.300612	+	0.953747i	$0.597191\pi$
$684$	−854.544	−0.0477694
$685$	4460.58	0.248803
$686$	−8262.47	−0.459858
$687$	16742.0	0.929765
$688$	−9550.05	−0.529204
$689$	−3767.28	−0.208305
$690$	7061.07	0.389580
$691$	28804.7	1.58579	0.792895	−	0.609358i	$-0.208573\pi$
$691$	28804.7	1.58579	0.792895	+	0.609358i	$0.208573\pi$
$692$	−5832.21	−0.320387
$693$	−2048.44	−0.112286
$694$	−14691.4	−0.803572
$695$	−4913.26	−0.268159
$696$	4632.02	0.252265
$697$	5150.81	0.279915
$698$	32688.3	1.77259
$699$	−11380.5	−0.615808
$700$	14106.3	0.761667
$701$	−1663.10	−0.0896070	−0.0448035	−	0.998996i	$-0.514266\pi$
$701$	−1663.10	−0.0896070	−0.0448035	+	0.998996i	$0.514266\pi$
$702$	−1741.67	−0.0936400
$703$	−2351.83	−0.126175
$704$	−2542.28	−0.136102
$705$	−9575.87	−0.511557
$706$	−7543.90	−0.402151
$707$	−10290.2	−0.547388
$708$	1110.99	0.0589742
$709$	32357.6	1.71399	0.856993	−	0.515328i	$-0.172330\pi$
$709$	32357.6	1.71399	0.856993	+	0.515328i	$0.172330\pi$
$710$	−20332.5	−1.07474
$711$	1266.43	0.0668001
$712$	−890.372	−0.0468653
$713$	20026.5	1.05189
$714$	5447.85	0.285547
$715$	−913.699	−0.0477908
$716$	−13888.8	−0.724931
$717$	14986.6	0.780593
$718$	−44110.2	−2.29273
$719$	−9360.97	−0.485543	−0.242771	−	0.970084i	$-0.578057\pi$
$719$	−9360.97	−0.485543	−0.242771	+	0.970084i	$0.578057\pi$
$720$	3866.91	0.200154
$721$	33929.9	1.75259
$722$	25051.9	1.29132
$723$	5023.54	0.258406
$724$	−292.460	−0.0150127
$725$	−21821.3	−1.11783
$726$	14102.9	0.720948
$727$	−18574.0	−0.947556	−0.473778	−	0.880644i	$-0.657110\pi$
$727$	−18574.0	−0.947556	−0.473778	+	0.880644i	$0.657110\pi$
$728$	−2714.74	−0.138208
$729$	729.000	0.0370370
$730$	282.844	0.0143405
$731$	2511.93	0.127096
$732$	−8223.14	−0.415213
$733$	26673.5	1.34408	0.672038	−	0.740516i	$-0.265419\pi$
$733$	26673.5	1.34408	0.672038	+	0.740516i	$0.265419\pi$
$734$	−42026.6	−2.11339
$735$	−4366.79	−0.219145
$736$	25013.3	1.25272
$737$	8611.21	0.430391
$738$	−8900.97	−0.443969
$739$	25484.9	1.26857	0.634287	−	0.773098i	$-0.281294\pi$
$739$	25484.9	1.26857	0.634287	+	0.773098i	$0.281294\pi$
$740$	−5604.24	−0.278400
$741$	−774.751	−0.0384091
$742$	−20363.3	−1.00749
$743$	−14039.9	−0.693237	−0.346619	−	0.938006i	$-0.612670\pi$
$743$	−14039.9	−0.693237	−0.346619	+	0.938006i	$0.612670\pi$
$744$	3606.35	0.177709
$745$	−11493.5	−0.565219
$746$	−28139.7	−1.38106
$747$	−237.839	−0.0116493
$748$	1151.13	0.0562694
$749$	8292.21	0.404527
$750$	−14127.4	−0.687813
$751$	34000.2	1.65205	0.826023	−	0.563637i	$-0.190598\pi$
$751$	34000.2	1.65205	0.826023	+	0.563637i	$0.190598\pi$
$752$	−41584.0	−2.01650
$753$	13468.1	0.651797
$754$	−15296.9	−0.738832
$755$	−9513.09	−0.458565
$756$	−4138.99	−0.199118
$757$	−37552.9	−1.80302	−0.901508	−	0.432763i	$-0.857539\pi$
$757$	−37552.9	−1.80302	−0.901508	+	0.432763i	$0.857539\pi$
$758$	10341.2	0.495526
$759$	3032.62	0.145029
$760$	565.625	0.0269966
$761$	−18976.0	−0.903917	−0.451959	−	0.892039i	$-0.649275\pi$
$761$	−18976.0	−0.903917	−0.451959	+	0.892039i	$0.649275\pi$
$762$	−14968.4	−0.711612
$763$	−7254.05	−0.344186
$764$	−12779.5	−0.605164
$765$	−1017.10	−0.0480699
$766$	14022.9	0.661447
$767$	1007.26	0.0474184
$768$	15774.9	0.741183
$769$	−14782.6	−0.693206	−0.346603	−	0.938012i	$-0.612665\pi$
$769$	−14782.6	−0.693206	−0.346603	+	0.938012i	$0.612665\pi$
$770$	−4938.82	−0.231146
$771$	15297.8	0.714575
$772$	−9604.62	−0.447769
$773$	−32518.7	−1.51309	−0.756543	−	0.653944i	$-0.773113\pi$
$773$	−32518.7	−1.51309	−0.756543	+	0.653944i	$0.773113\pi$
$774$	−4340.79	−0.201584
$775$	−16989.4	−0.787456
$776$	−2349.08	−0.108669
$777$	−11391.1	−0.525937
$778$	−259.806	−0.0119724
$779$	−3959.43	−0.182107
$780$	−1846.18	−0.0847485
$781$	−8732.48	−0.400093
$782$	−8065.28	−0.368816
$783$	6402.70	0.292227
$784$	−18963.1	−0.863845
$785$	−3729.64	−0.169575
$786$	19705.6	0.894242
$787$	−7602.30	−0.344336	−0.172168	−	0.985068i	$-0.555077\pi$
$787$	−7602.30	−0.344336	−0.172168	+	0.985068i	$0.555077\pi$
$788$	15452.3	0.698559
$789$	23847.9	1.07606
$790$	3053.38	0.137512
$791$	−37948.0	−1.70578
$792$	546.112	0.0245016
$793$	−7455.30	−0.333853
$794$	35370.7	1.58093
$795$	3801.79	0.169604
$796$	9052.85	0.403103
$797$	−7713.91	−0.342836	−0.171418	−	0.985198i	$-0.554835\pi$
$797$	−7713.91	−0.342836	−0.171418	+	0.985198i	$0.554835\pi$
$798$	−4187.76	−0.185771
$799$	10937.7	0.484292
$800$	−21220.0	−0.937799
$801$	−1230.74	−0.0542895
$802$	361.206	0.0159035
$803$	121.477	0.00533853
$804$	17399.4	0.763222
$805$	15213.4	0.666088
$806$	−11909.7	−0.520472
$807$	−11494.3	−0.501386
$808$	2743.36	0.119444
$809$	25690.3	1.11647	0.558234	−	0.829683i	$-0.311479\pi$
$809$	25690.3	1.11647	0.558234	+	0.829683i	$0.311479\pi$
$810$	1757.63	0.0762429
$811$	28467.8	1.23260	0.616302	−	0.787510i	$-0.288630\pi$
$811$	28467.8	1.23260	0.616302	+	0.787510i	$0.288630\pi$
$812$	−36352.1	−1.57107
$813$	−20474.6	−0.883241
$814$	−5474.66	−0.235733
$815$	−2920.51	−0.125523
$816$	−4416.85	−0.189486
$817$	−1930.92	−0.0826857
$818$	41769.8	1.78539
$819$	−3752.51	−0.160102
$820$	−9435.05	−0.401812
$821$	−38119.4	−1.62043	−0.810217	−	0.586130i	$-0.800651\pi$
$821$	−38119.4	−1.62043	−0.810217	+	0.586130i	$0.800651\pi$
$822$	8804.45	0.373589
$823$	−26656.3	−1.12902	−0.564508	−	0.825427i	$-0.690934\pi$
$823$	−26656.3	−1.12902	−0.564508	+	0.825427i	$0.690934\pi$
$824$	−9045.66	−0.382428
$825$	−2572.72	−0.108570
$826$	5444.52	0.229345
$827$	−19016.7	−0.799606	−0.399803	−	0.916601i	$-0.630922\pi$
$827$	−19016.7	−0.799606	−0.399803	+	0.916601i	$0.630922\pi$
$828$	6127.57	0.257184
$829$	−36260.4	−1.51915	−0.759575	−	0.650420i	$-0.774593\pi$
$829$	−36260.4	−1.51915	−0.759575	+	0.650420i	$0.774593\pi$
$830$	−573.432	−0.0239808
$831$	−120.317	−0.00502255
$832$	−4657.16	−0.194060
$833$	4987.82	0.207464
$834$	−9697.95	−0.402653
$835$	−8151.75	−0.337848
$836$	−884.873	−0.0366076
$837$	4984.95	0.205860
$838$	18342.0	0.756102
$839$	18034.3	0.742088	0.371044	−	0.928615i	$-0.379000\pi$
$839$	18034.3	0.742088	0.371044	+	0.928615i	$0.379000\pi$
$840$	2739.61	0.112530
$841$	31845.0	1.30571
$842$	20709.0	0.847602
$843$	−7217.49	−0.294880
$844$	−4596.79	−0.187474
$845$	10943.2	0.445512
$846$	−18901.2	−0.768127
$847$	30385.3	1.23265
$848$	16509.6	0.668562
$849$	24696.4	0.998327
$850$	6842.17	0.276099
$851$	16864.0	0.679305
$852$	−17644.5	−0.709494
$853$	−17485.1	−0.701851	−0.350926	−	0.936403i	$-0.614133\pi$
$853$	−17485.1	−0.701851	−0.350926	+	0.936403i	$0.614133\pi$
$854$	−40298.1	−1.61472
$855$	781.847	0.0312732
$856$	−2210.69	−0.0882709
$857$	3337.89	0.133046	0.0665228	−	0.997785i	$-0.478809\pi$
$857$	3337.89	0.133046	0.0665228	+	0.997785i	$0.478809\pi$
$858$	−1803.49	−0.0717601
$859$	−41566.7	−1.65103	−0.825517	−	0.564377i	$-0.809117\pi$
$859$	−41566.7	−1.65103	−0.825517	+	0.564377i	$0.809117\pi$
$860$	−4601.24	−0.182443
$861$	−19177.5	−0.759080
$862$	−27834.3	−1.09981
$863$	21883.2	0.863165	0.431583	−	0.902073i	$-0.357955\pi$
$863$	21883.2	0.863165	0.431583	+	0.902073i	$0.357955\pi$
$864$	6226.26	0.245164
$865$	5336.06	0.209747
$866$	−862.859	−0.0338581
$867$	−13577.2	−0.531842
$868$	−28302.7	−1.10675
$869$	1311.38	0.0511916
$870$	15437.0	0.601566
$871$	15774.8	0.613671
$872$	1933.92	0.0751041
$873$	−3247.07	−0.125884
$874$	6199.77	0.239943
$875$	−30438.1	−1.17599
$876$	245.451	0.00946693
$877$	−18985.8	−0.731019	−0.365509	−	0.930808i	$-0.619105\pi$
$877$	−18985.8	−0.731019	−0.365509	+	0.930808i	$0.619105\pi$
$878$	63847.7	2.45416
$879$	5706.66	0.218977
$880$	4004.16	0.153386
$881$	−47938.8	−1.83326	−0.916628	−	0.399742i	$-0.869100\pi$
$881$	−47938.8	−1.83326	−0.916628	+	0.399742i	$0.869100\pi$
$882$	−8619.31	−0.329056
$883$	45963.2	1.75174	0.875868	−	0.482550i	$-0.160290\pi$
$883$	45963.2	1.75174	0.875868	+	0.482550i	$0.160290\pi$
$884$	2108.74	0.0802314
$885$	−1016.48	−0.0386086
$886$	−32735.5	−1.24128
$887$	−25211.9	−0.954378	−0.477189	−	0.878801i	$-0.658344\pi$
$887$	−25211.9	−0.954378	−0.477189	+	0.878801i	$0.658344\pi$
$888$	3036.85	0.114763
$889$	−32250.1	−1.21668
$890$	−2967.32	−0.111758
$891$	754.874	0.0283830
$892$	18380.2	0.689925
$893$	−8407.83	−0.315070
$894$	−22686.2	−0.848703
$895$	12707.3	0.474590
$896$	19881.9	0.741303
$897$	5555.41	0.206789
$898$	45397.5	1.68701
$899$	43782.1	1.62426
$900$	−5198.32	−0.192530
$901$	−4342.47	−0.160565
$902$	−9216.88	−0.340231
$903$	−9352.41	−0.344661
$904$	10116.9	0.372215
$905$	267.580	0.00982836
$906$	−18777.3	−0.688557
$907$	−9607.36	−0.351717	−0.175858	−	0.984415i	$-0.556270\pi$
$907$	−9607.36	−0.351717	−0.175858	+	0.984415i	$0.556270\pi$
$908$	24540.6	0.896924
$909$	3792.06	0.138366
$910$	−9047.34	−0.329579
$911$	−44171.3	−1.60643	−0.803217	−	0.595686i	$-0.796880\pi$
$911$	−44171.3	−1.60643	−0.803217	+	0.595686i	$0.796880\pi$
$912$	3395.23	0.123276
$913$	−246.280	−0.00892736
$914$	−58909.1	−2.13188
$915$	7523.59	0.271827
$916$	35028.8	1.26352
$917$	42456.4	1.52894
$918$	−2007.59	−0.0721792
$919$	33636.3	1.20736	0.603678	−	0.797228i	$-0.293701\pi$
$919$	33636.3	1.20736	0.603678	+	0.797228i	$0.293701\pi$
$920$	−4055.86	−0.145345
$921$	27678.6	0.990273
$922$	−5573.70	−0.199089
$923$	−15996.9	−0.570471
$924$	−4285.89	−0.152593
$925$	−14306.5	−0.508535
$926$	−5044.78	−0.179030
$927$	−12503.5	−0.443010
$928$	54684.3	1.93437
$929$	11890.9	0.419945	0.209972	−	0.977707i	$-0.432663\pi$
$929$	11890.9	0.419945	0.209972	+	0.977707i	$0.432663\pi$
$930$	12018.8	0.423775
$931$	−3834.14	−0.134972
$932$	−23811.1	−0.836864
$933$	−7187.67	−0.252212
$934$	52561.7	1.84140
$935$	−1053.20	−0.0368379
$936$	1000.41	0.0349354
$937$	−18281.5	−0.637385	−0.318692	−	0.947858i	$-0.603244\pi$
$937$	−18281.5	−0.637385	−0.318692	+	0.947858i	$0.603244\pi$
$938$	85267.2	2.96810
$939$	−30215.7	−1.05011
$940$	−20035.3	−0.695190
$941$	40341.0	1.39753	0.698767	−	0.715349i	$-0.253732\pi$
$941$	40341.0	1.39753	0.698767	+	0.715349i	$0.253732\pi$
$942$	−7361.70	−0.254625
$943$	28391.4	0.980435
$944$	−4414.15	−0.152191
$945$	3786.88	0.130357
$946$	−4494.85	−0.154482
$947$	−44404.6	−1.52371	−0.761856	−	0.647746i	$-0.775712\pi$
$947$	−44404.6	−1.52371	−0.761856	+	0.647746i	$0.775712\pi$
$948$	2649.71	0.0907792
$949$	222.532	0.00761191
$950$	−5259.57	−0.179624
$951$	3205.17	0.109290
$952$	−3129.23	−0.106533
$953$	34851.1	1.18461	0.592307	−	0.805713i	$-0.298217\pi$
$953$	34851.1	1.18461	0.592307	+	0.805713i	$0.298217\pi$
$954$	7504.09	0.254669
$955$	11692.3	0.396183
$956$	31356.0	1.06080
$957$	6629.95	0.223945
$958$	−58505.5	−1.97310
$959$	18969.5	0.638747
$960$	4699.81	0.158006
$961$	4296.41	0.144218
$962$	−10028.9	−0.336119
$963$	−3055.77	−0.102254
$964$	10510.6	0.351165
$965$	8787.55	0.293141
$966$	30028.7	1.00016
$967$	2971.34	0.0988128	0.0494064	−	0.998779i	$-0.484267\pi$
$967$	2971.34	0.0988128	0.0494064	+	0.998779i	$0.484267\pi$
$968$	−8100.68	−0.268973
$969$	−893.040	−0.0296064
$970$	−7828.71	−0.259139
$971$	34741.0	1.14819	0.574094	−	0.818789i	$-0.305354\pi$
$971$	34741.0	1.14819	0.574094	+	0.818789i	$0.305354\pi$
$972$	1525.26	0.0503322
$973$	−20894.6	−0.688440
$974$	48749.1	1.60372
$975$	−4712.93	−0.154805
$976$	32671.8	1.07151
$977$	−31701.1	−1.03808	−0.519042	−	0.854749i	$-0.673711\pi$
$977$	−31701.1	−1.03808	−0.519042	+	0.854749i	$0.673711\pi$
$978$	−5764.60	−0.188478
$979$	−1274.42	−0.0416042
$980$	−9136.49	−0.297811
$981$	2673.20	0.0870017
$982$	16429.2	0.533887
$983$	−47493.5	−1.54100	−0.770502	−	0.637437i	$-0.779995\pi$
$983$	−47493.5	−1.54100	−0.770502	+	0.637437i	$0.779995\pi$
$984$	5112.70	0.165637
$985$	−14137.7	−0.457326
$986$	−17632.4	−0.569503
$987$	−40723.4	−1.31331
$988$	−1620.99	−0.0521968
$989$	13845.8	0.445167
$990$	1820.01	0.0584280
$991$	−48621.1	−1.55853	−0.779263	−	0.626697i	$-0.784406\pi$
$991$	−48621.1	−1.55853	−0.779263	+	0.626697i	$0.784406\pi$
$992$	42575.5	1.36268
$993$	−28949.4	−0.925157
$994$	−86468.0	−2.75915
$995$	−8282.71	−0.263899
$996$	−497.622	−0.0158311
$997$	37128.0	1.17939	0.589697	−	0.807625i	$-0.299247\pi$
$997$	37128.0	1.17939	0.589697	+	0.807625i	$0.299247\pi$
$998$	45536.9	1.44433
$999$	4197.74	0.132944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.d.1.2	✓	8
3.2	odd	2		531.4.a.e.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.d.1.2	✓	8	1.1	even	1	trivial
531.4.a.e.1.7		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)

\(f(q)\)	\(=\)	\(q-3.77847 q^{2} +3.00000 q^{3} +6.27681 q^{4} -5.74283 q^{5} -11.3354 q^{6} -24.4226 q^{7} +6.51103 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.77847 q^{2} +3.00000 q^{3} +6.27681 q^{4} -5.74283 q^{5} -11.3354 q^{6} -24.4226 q^{7} +6.51103 q^{8} +9.00000 q^{9} +21.6991 q^{10} +9.31943 q^{11} +18.8304 q^{12} +17.0721 q^{13} +92.2799 q^{14} -17.2285 q^{15} -74.8162 q^{16} +19.6787 q^{17} -34.0062 q^{18} -15.1270 q^{19} -36.0466 q^{20} -73.2678 q^{21} -35.2132 q^{22} +108.469 q^{23} +19.5331 q^{24} -92.0199 q^{25} -64.5065 q^{26} +27.0000 q^{27} -153.296 q^{28} +237.137 q^{29} +65.0973 q^{30} +184.628 q^{31} +230.602 q^{32} +27.9583 q^{33} -74.3553 q^{34} +140.255 q^{35} +56.4913 q^{36} +155.472 q^{37} +57.1569 q^{38} +51.2164 q^{39} -37.3917 q^{40} +261.745 q^{41} +276.840 q^{42} +127.647 q^{43} +58.4963 q^{44} -51.6855 q^{45} -409.848 q^{46} +555.816 q^{47} -224.448 q^{48} +253.463 q^{49} +347.694 q^{50} +59.0361 q^{51} +107.158 q^{52} -220.668 q^{53} -102.019 q^{54} -53.5199 q^{55} -159.016 q^{56} -45.3810 q^{57} -896.014 q^{58} +59.0000 q^{59} -108.140 q^{60} -436.694 q^{61} -697.610 q^{62} -219.803 q^{63} -272.793 q^{64} -98.0424 q^{65} -105.639 q^{66} +924.006 q^{67} +123.519 q^{68} +325.408 q^{69} -529.948 q^{70} -937.019 q^{71} +58.5992 q^{72} +13.0348 q^{73} -587.446 q^{74} -276.060 q^{75} -94.9493 q^{76} -227.605 q^{77} -193.519 q^{78} +140.714 q^{79} +429.657 q^{80} +81.0000 q^{81} -988.996 q^{82} -26.4265 q^{83} -459.888 q^{84} -113.011 q^{85} -482.310 q^{86} +711.411 q^{87} +60.6791 q^{88} -136.748 q^{89} +195.292 q^{90} -416.946 q^{91} +680.842 q^{92} +553.883 q^{93} -2100.13 q^{94} +86.8719 q^{95} +691.806 q^{96} -360.785 q^{97} -957.701 q^{98} +83.8749 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{2} + 24 q^{3} + 34 q^{4} + 42 q^{5} + 18 q^{6} + 53 q^{7} + 51 q^{8} + 72 q^{9}+O(q^{10}) \) 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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