LMFDB - Embedded newform 177.4.a.d.1.8

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			1.8
Root			$-4.15242$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.15242 q^{2} +3.00000 q^{3} +18.5474 q^{4} +1.69104 q^{5} +15.4573 q^{6} -11.0943 q^{7} +54.3449 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+5.15242 q^{2} +3.00000 q^{3} +18.5474 q^{4} +1.69104 q^{5} +15.4573 q^{6} -11.0943 q^{7} +54.3449 q^{8} +9.00000 q^{9} +8.71294 q^{10} -10.1415 q^{11} +55.6423 q^{12} +29.5495 q^{13} -57.1625 q^{14} +5.07312 q^{15} +131.628 q^{16} -29.3698 q^{17} +46.3718 q^{18} -46.9309 q^{19} +31.3645 q^{20} -33.2829 q^{21} -52.2532 q^{22} +22.2505 q^{23} +163.035 q^{24} -122.140 q^{25} +152.252 q^{26} +27.0000 q^{27} -205.771 q^{28} -103.959 q^{29} +26.1388 q^{30} -52.5878 q^{31} +243.445 q^{32} -30.4245 q^{33} -151.326 q^{34} -18.7609 q^{35} +166.927 q^{36} +5.51956 q^{37} -241.808 q^{38} +88.6485 q^{39} +91.8993 q^{40} +201.493 q^{41} -171.487 q^{42} +479.341 q^{43} -188.099 q^{44} +15.2193 q^{45} +114.644 q^{46} -133.168 q^{47} +394.885 q^{48} -219.917 q^{49} -629.319 q^{50} -88.1094 q^{51} +548.068 q^{52} -484.443 q^{53} +139.115 q^{54} -17.1497 q^{55} -602.918 q^{56} -140.793 q^{57} -535.639 q^{58} +59.0000 q^{59} +94.0934 q^{60} +578.161 q^{61} -270.954 q^{62} -99.8486 q^{63} +201.305 q^{64} +49.9694 q^{65} -156.760 q^{66} +52.3009 q^{67} -544.735 q^{68} +66.7516 q^{69} -96.6639 q^{70} +399.262 q^{71} +489.104 q^{72} -1045.81 q^{73} +28.4391 q^{74} -366.421 q^{75} -870.449 q^{76} +112.513 q^{77} +456.755 q^{78} +269.263 q^{79} +222.588 q^{80} +81.0000 q^{81} +1038.18 q^{82} +174.118 q^{83} -617.312 q^{84} -49.6655 q^{85} +2469.77 q^{86} -311.876 q^{87} -551.138 q^{88} +1380.40 q^{89} +78.4165 q^{90} -327.831 q^{91} +412.691 q^{92} -157.763 q^{93} -686.140 q^{94} -79.3620 q^{95} +730.335 q^{96} +628.773 q^{97} -1133.10 q^{98} -91.2734 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	5.15242	1.82166	0.910828	−	0.412786i	$-0.135444\pi$
$2$	5.15242	1.82166	0.910828	+	0.412786i	$0.135444\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	18.5474	2.31843
$5$	1.69104	0.151251	0.0756256	−	0.997136i	$-0.475905\pi$
$5$	1.69104	0.151251	0.0756256	+	0.997136i	$0.475905\pi$
$6$	15.4573	1.05173
$7$	−11.0943	−0.599035	−0.299518	−	0.954091i	$-0.596826\pi$
$7$	−11.0943	−0.599035	−0.299518	+	0.954091i	$0.596826\pi$
$8$	54.3449	2.40173
$9$	9.00000	0.333333
$10$	8.71294	0.275528
$11$	−10.1415	−0.277980	−0.138990	−	0.990294i	$-0.544386\pi$
$11$	−10.1415	−0.277980	−0.138990	+	0.990294i	$0.544386\pi$
$12$	55.6423	1.33855
$13$	29.5495	0.630428	0.315214	−	0.949021i	$-0.397924\pi$
$13$	29.5495	0.630428	0.315214	+	0.949021i	$0.397924\pi$
$14$	−57.1625	−1.09124
$15$	5.07312	0.0873249
$16$	131.628	2.05669
$17$	−29.3698	−0.419013	−0.209507	−	0.977807i	$-0.567186\pi$
$17$	−29.3698	−0.419013	−0.209507	+	0.977807i	$0.567186\pi$
$18$	46.3718	0.607219
$19$	−46.9309	−0.566668	−0.283334	−	0.959021i	$-0.591441\pi$
$19$	−46.9309	−0.566668	−0.283334	+	0.959021i	$0.591441\pi$
$20$	31.3645	0.350665
$21$	−33.2829	−0.345853
$22$	−52.2532	−0.506383
$23$	22.2505	0.201720	0.100860	−	0.994901i	$-0.467841\pi$
$23$	22.2505	0.201720	0.100860	+	0.994901i	$0.467841\pi$
$24$	163.035	1.38664
$25$	−122.140	−0.977123
$26$	152.252	1.14842
$27$	27.0000	0.192450
$28$	−205.771	−1.38882
$29$	−103.959	−0.665677	−0.332838	−	0.942984i	$-0.608006\pi$
$29$	−103.959	−0.665677	−0.332838	+	0.942984i	$0.608006\pi$
$30$	26.1388	0.159076
$31$	−52.5878	−0.304679	−0.152339	−	0.988328i	$-0.548681\pi$
$31$	−52.5878	−0.304679	−0.152339	+	0.988328i	$0.548681\pi$
$32$	243.445	1.34486
$33$	−30.4245	−0.160492
$34$	−151.326	−0.763298
$35$	−18.7609	−0.0906048
$36$	166.927	0.772810
$37$	5.51956	0.0245246	0.0122623	−	0.999925i	$-0.496097\pi$
$37$	5.51956	0.0245246	0.0122623	+	0.999925i	$0.496097\pi$
$38$	−241.808	−1.03227
$39$	88.6485	0.363978
$40$	91.8993	0.363264
$41$	201.493	0.767512	0.383756	−	0.923435i	$-0.374630\pi$
$41$	201.493	0.767512	0.383756	+	0.923435i	$0.374630\pi$
$42$	−171.487	−0.630026
$43$	479.341	1.69997	0.849987	−	0.526804i	$-0.176610\pi$
$43$	479.341	1.69997	0.849987	+	0.526804i	$0.176610\pi$
$44$	−188.099	−0.644476
$45$	15.2193	0.0504170
$46$	114.644	0.367464
$47$	−133.168	−0.413290	−0.206645	−	0.978416i	$-0.566254\pi$
$47$	−133.168	−0.413290	−0.206645	+	0.978416i	$0.566254\pi$
$48$	394.885	1.18743
$49$	−219.917	−0.641157
$50$	−629.319	−1.77998
$51$	−88.1094	−0.241917
$52$	548.068	1.46160
$53$	−484.443	−1.25553	−0.627767	−	0.778401i	$-0.716031\pi$
$53$	−484.443	−1.25553	−0.627767	+	0.778401i	$0.716031\pi$
$54$	139.115	0.350578
$55$	−17.1497	−0.0420447
$56$	−602.918	−1.43872
$57$	−140.793	−0.327166
$58$	−535.639	−1.21263
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	94.0934	0.202457
$61$	578.161	1.21354	0.606770	−	0.794877i	$-0.292465\pi$
$61$	578.161	1.21354	0.606770	+	0.794877i	$0.292465\pi$
$62$	−270.954	−0.555020
$63$	−99.8486	−0.199678
$64$	201.305	0.393174
$65$	49.9694	0.0953529
$66$	−156.760	−0.292360
$67$	52.3009	0.0953668	0.0476834	−	0.998862i	$-0.484816\pi$
$67$	52.3009	0.0953668	0.0476834	+	0.998862i	$0.484816\pi$
$68$	−544.735	−0.971453
$69$	66.7516	0.116463
$70$	−96.6639	−0.165051
$71$	399.262	0.667375	0.333687	−	0.942684i	$-0.391707\pi$
$71$	399.262	0.667375	0.333687	+	0.942684i	$0.391707\pi$
$72$	489.104	0.800576
$73$	−1045.81	−1.67675	−0.838374	−	0.545096i	$-0.816493\pi$
$73$	−1045.81	−1.67675	−0.838374	+	0.545096i	$0.816493\pi$
$74$	28.4391	0.0446754
$75$	−366.421	−0.564142
$76$	−870.449	−1.31378
$77$	112.513	0.166520
$78$	456.755	0.663042
$79$	269.263	0.383474	0.191737	−	0.981446i	$-0.438588\pi$
$79$	269.263	0.383474	0.191737	+	0.981446i	$0.438588\pi$
$80$	222.588	0.311077
$81$	81.0000	0.111111
$82$	1038.18	1.39814
$83$	174.118	0.230265	0.115132	−	0.993350i	$-0.463271\pi$
$83$	174.118	0.230265	0.115132	+	0.993350i	$0.463271\pi$
$84$	−617.312	−0.801837
$85$	−49.6655	−0.0633762
$86$	2469.77	3.09677
$87$	−311.876	−0.384329
$88$	−551.138	−0.667631
$89$	1380.40	1.64407	0.822036	−	0.569435i	$-0.192838\pi$
$89$	1380.40	1.64407	0.822036	+	0.569435i	$0.192838\pi$
$90$	78.4165	0.0918425
$91$	−327.831	−0.377648
$92$	412.691	0.467674
$93$	−157.763	−0.175906
$94$	−686.140	−0.752871
$95$	−79.3620	−0.0857092
$96$	730.335	0.776453
$97$	628.773	0.658167	0.329084	−	0.944301i	$-0.393260\pi$
$97$	628.773	0.658167	0.329084	+	0.944301i	$0.393260\pi$
$98$	−1133.10	−1.16797
$99$	−91.2734	−0.0926598
$100$	−2265.39	−2.26539
$101$	553.980	0.545773	0.272887	−	0.962046i	$-0.412022\pi$
$101$	553.980	0.545773	0.272887	+	0.962046i	$0.412022\pi$
$102$	−453.977	−0.440690
$103$	401.263	0.383861	0.191930	−	0.981409i	$-0.438525\pi$
$103$	401.263	0.383861	0.191930	+	0.981409i	$0.438525\pi$
$104$	1605.86	1.51412
$105$	−56.2826	−0.0523107
$106$	−2496.05	−2.28715
$107$	−129.942	−0.117401	−0.0587007	−	0.998276i	$-0.518696\pi$
$107$	−129.942	−0.117401	−0.0587007	+	0.998276i	$0.518696\pi$
$108$	500.781	0.446182
$109$	−1248.43	−1.09705	−0.548524	−	0.836135i	$-0.684810\pi$
$109$	−1248.43	−1.09705	−0.548524	+	0.836135i	$0.684810\pi$
$110$	−88.3623	−0.0765910
$111$	16.5587	0.0141593
$112$	−1460.32	−1.23203
$113$	1175.90	0.978933	0.489466	−	0.872022i	$-0.337192\pi$
$113$	1175.90	0.978933	0.489466	+	0.872022i	$0.337192\pi$
$114$	−725.424	−0.595984
$115$	37.6265	0.0305104
$116$	−1928.17	−1.54333
$117$	265.946	0.210143
$118$	303.993	0.237159
$119$	325.837	0.251004
$120$	275.698	0.209731
$121$	−1228.15	−0.922727
$122$	2978.93	2.21065
$123$	604.480	0.443123
$124$	−975.369	−0.706377
$125$	−417.924	−0.299042
$126$	−514.462	−0.363745
$127$	−1993.40	−1.39280	−0.696399	−	0.717654i	$-0.745216\pi$
$127$	−1993.40	−1.39280	−0.696399	+	0.717654i	$0.745216\pi$
$128$	−910.351	−0.628629
$129$	1438.02	0.981480
$130$	257.463	0.173700
$131$	1519.30	1.01330	0.506650	−	0.862152i	$-0.330884\pi$
$131$	1519.30	1.01330	0.506650	+	0.862152i	$0.330884\pi$
$132$	−564.296	−0.372089
$133$	520.665	0.339454
$134$	269.476	0.173726
$135$	45.6580	0.0291083
$136$	−1596.10	−1.00636
$137$	1596.83	0.995812	0.497906	−	0.867231i	$-0.334102\pi$
$137$	1596.83	0.995812	0.497906	+	0.867231i	$0.334102\pi$
$138$	343.933	0.212156
$139$	666.080	0.406447	0.203224	−	0.979132i	$-0.434858\pi$
$139$	666.080	0.406447	0.203224	+	0.979132i	$0.434858\pi$
$140$	−347.966	−0.210061
$141$	−399.505	−0.238613
$142$	2057.16	1.21573
$143$	−299.676	−0.175246
$144$	1184.65	0.685564
$145$	−175.798	−0.100684
$146$	−5388.44	−3.05446
$147$	−659.750	−0.370172
$148$	102.374	0.0568586
$149$	1767.90	0.972029	0.486014	−	0.873951i	$-0.338450\pi$
$149$	1767.90	0.972029	0.486014	+	0.873951i	$0.338450\pi$
$150$	−1887.96	−1.02767
$151$	−1053.25	−0.567633	−0.283817	−	0.958879i	$-0.591601\pi$
$151$	−1053.25	−0.567633	−0.283817	+	0.958879i	$0.591601\pi$
$152$	−2550.46	−1.36098
$153$	−264.328	−0.139671
$154$	579.713	0.303341
$155$	−88.9280	−0.0460830
$156$	1644.20	0.843857
$157$	−47.5287	−0.0241605	−0.0120803	−	0.999927i	$-0.503845\pi$
$157$	−47.5287	−0.0241605	−0.0120803	+	0.999927i	$0.503845\pi$
$158$	1387.36	0.698558
$159$	−1453.33	−0.724883
$160$	411.675	0.203411
$161$	−246.854	−0.120837
$162$	417.346	0.202406
$163$	3084.59	1.48223	0.741115	−	0.671378i	$-0.234297\pi$
$163$	3084.59	1.48223	0.741115	+	0.671378i	$0.234297\pi$
$164$	3737.19	1.77942
$165$	−51.4490	−0.0242745
$166$	897.132	0.419463
$167$	3685.43	1.70771	0.853854	−	0.520513i	$-0.174259\pi$
$167$	3685.43	1.70771	0.853854	+	0.520513i	$0.174259\pi$
$168$	−1808.75	−0.830645
$169$	−1323.83	−0.602561
$170$	−255.898	−0.115450
$171$	−422.378	−0.188889
$172$	8890.56	3.94127
$173$	504.194	0.221579	0.110790	−	0.993844i	$-0.464662\pi$
$173$	504.194	0.221579	0.110790	+	0.993844i	$0.464662\pi$
$174$	−1606.92	−0.700115
$175$	1355.06	0.585331
$176$	−1334.91	−0.571718
$177$	177.000	0.0751646
$178$	7112.42	2.99494
$179$	2931.88	1.22424	0.612120	−	0.790765i	$-0.290317\pi$
$179$	2931.88	1.22424	0.612120	+	0.790765i	$0.290317\pi$
$180$	282.280	0.116888
$181$	4335.05	1.78023	0.890116	−	0.455734i	$-0.150623\pi$
$181$	4335.05	1.78023	0.890116	+	0.455734i	$0.150623\pi$
$182$	−1689.12	−0.687946
$183$	1734.48	0.700638
$184$	1209.20	0.484476
$185$	9.33380	0.00370937
$186$	−812.863	−0.320441
$187$	297.854	0.116477
$188$	−2469.93	−0.958183
$189$	−299.546	−0.115284
$190$	−408.907	−0.156133
$191$	2286.00	0.866016	0.433008	−	0.901390i	$-0.357452\pi$
$191$	2286.00	0.866016	0.433008	+	0.901390i	$0.357452\pi$
$192$	603.915	0.226999
$193$	−1685.13	−0.628487	−0.314244	−	0.949342i	$-0.601751\pi$
$193$	−1685.13	−0.628487	−0.314244	+	0.949342i	$0.601751\pi$
$194$	3239.70	1.19895
$195$	149.908	0.0550520
$196$	−4078.89	−1.48648
$197$	690.913	0.249876	0.124938	−	0.992165i	$-0.460127\pi$
$197$	690.913	0.249876	0.124938	+	0.992165i	$0.460127\pi$
$198$	−470.279	−0.168794
$199$	−1448.24	−0.515894	−0.257947	−	0.966159i	$-0.583046\pi$
$199$	−1448.24	−0.515894	−0.257947	+	0.966159i	$0.583046\pi$
$200$	−6637.71	−2.34678
$201$	156.903	0.0550601
$202$	2854.34	0.994211
$203$	1153.35	0.398764
$204$	−1634.21	−0.560869
$205$	340.733	0.116087
$206$	2067.48	0.699262
$207$	200.255	0.0672400
$208$	3889.55	1.29659
$209$	475.950	0.157522
$210$	−289.992	−0.0952921
$211$	−5007.79	−1.63389	−0.816944	−	0.576717i	$-0.804333\pi$
$211$	−5007.79	−1.63389	−0.816944	+	0.576717i	$0.804333\pi$
$212$	−8985.18	−2.91087
$213$	1197.78	0.385309
$214$	−669.515	−0.213865
$215$	810.585	0.257123
$216$	1467.31	0.462213
$217$	583.424	0.182513
$218$	−6432.46	−1.99844
$219$	−3137.42	−0.968071
$220$	−318.082	−0.0974778
$221$	−867.864	−0.264158
$222$	85.3173	0.0257934
$223$	1275.24	0.382943	0.191471	−	0.981498i	$-0.438674\pi$
$223$	1275.24	0.382943	0.191471	+	0.981498i	$0.438674\pi$
$224$	−2700.85	−0.805616
$225$	−1099.26	−0.325708
$226$	6058.73	1.78328
$227$	−1302.24	−0.380760	−0.190380	−	0.981711i	$-0.560972\pi$
$227$	−1302.24	−0.380760	−0.190380	+	0.981711i	$0.560972\pi$
$228$	−2611.35	−0.758512
$229$	−3359.40	−0.969413	−0.484706	−	0.874677i	$-0.661073\pi$
$229$	−3359.40	−0.969413	−0.484706	+	0.874677i	$0.661073\pi$
$230$	193.868	0.0555794
$231$	337.538	0.0961401
$232$	−5649.62	−1.59877
$233$	−2610.02	−0.733855	−0.366927	−	0.930250i	$-0.619590\pi$
$233$	−2610.02	−0.733855	−0.366927	+	0.930250i	$0.619590\pi$
$234$	1370.26	0.382807
$235$	−225.193	−0.0625105
$236$	1094.30	0.301834
$237$	807.789	0.221399
$238$	1678.85	0.457243
$239$	−4236.99	−1.14673	−0.573364	−	0.819301i	$-0.694362\pi$
$239$	−4236.99	−1.14673	−0.573364	+	0.819301i	$0.694362\pi$
$240$	667.765	0.179600
$241$	−3164.38	−0.845792	−0.422896	−	0.906178i	$-0.638986\pi$
$241$	−3164.38	−0.845792	−0.422896	+	0.906178i	$0.638986\pi$
$242$	−6327.95	−1.68089
$243$	243.000	0.0641500
$244$	10723.4	2.81351
$245$	−371.888	−0.0969756
$246$	3114.54	0.807218
$247$	−1386.79	−0.357243
$248$	−2857.88	−0.731756
$249$	522.355	0.132943
$250$	−2153.32	−0.544752
$251$	−6495.46	−1.63343	−0.816713	−	0.577044i	$-0.804206\pi$
$251$	−6495.46	−1.63343	−0.816713	+	0.577044i	$0.804206\pi$
$252$	−1851.94	−0.462941
$253$	−225.654	−0.0560740
$254$	−10270.8	−2.53720
$255$	−148.996	−0.0365903
$256$	−6300.95	−1.53832
$257$	−4426.51	−1.07439	−0.537195	−	0.843458i	$-0.680516\pi$
$257$	−4426.51	−1.07439	−0.537195	+	0.843458i	$0.680516\pi$
$258$	7409.30	1.78792
$259$	−61.2356	−0.0146911
$260$	926.804	0.221069
$261$	−935.627	−0.221892
$262$	7828.10	1.84588
$263$	6798.01	1.59385	0.796927	−	0.604076i	$-0.206458\pi$
$263$	6798.01	1.59385	0.796927	+	0.604076i	$0.206458\pi$
$264$	−1653.41	−0.385457
$265$	−819.212	−0.189901
$266$	2682.69	0.618369
$267$	4141.21	0.949206
$268$	970.049	0.221101
$269$	6894.51	1.56270	0.781349	−	0.624095i	$-0.214532\pi$
$269$	6894.51	1.56270	0.781349	+	0.624095i	$0.214532\pi$
$270$	235.250	0.0530253
$271$	2936.35	0.658195	0.329097	−	0.944296i	$-0.393256\pi$
$271$	2936.35	0.658195	0.329097	+	0.944296i	$0.393256\pi$
$272$	−3865.90	−0.861781
$273$	−983.493	−0.218035
$274$	8227.53	1.81403
$275$	1238.69	0.271620
$276$	1238.07	0.270012
$277$	−7457.80	−1.61767	−0.808837	−	0.588033i	$-0.799902\pi$
$277$	−7457.80	−1.61767	−0.808837	+	0.588033i	$0.799902\pi$
$278$	3431.92	0.740407
$279$	−473.290	−0.101560
$280$	−1019.56	−0.217608
$281$	−5450.35	−1.15708	−0.578542	−	0.815652i	$-0.696378\pi$
$281$	−5450.35	−1.15708	−0.578542	+	0.815652i	$0.696378\pi$
$282$	−2058.42	−0.434671
$283$	2984.42	0.626873	0.313437	−	0.949609i	$-0.398520\pi$
$283$	2984.42	0.626873	0.313437	+	0.949609i	$0.398520\pi$
$284$	7405.28	1.54726
$285$	−238.086	−0.0494842
$286$	−1544.06	−0.319238
$287$	−2235.43	−0.459767
$288$	2191.00	0.448285
$289$	−4050.41	−0.824428
$290$	−905.786	−0.183412
$291$	1886.32	0.379993
$292$	−19397.1	−3.88742
$293$	2189.77	0.436613	0.218307	−	0.975880i	$-0.429947\pi$
$293$	2189.77	0.436613	0.218307	+	0.975880i	$0.429947\pi$
$294$	−3399.31	−0.674326
$295$	99.7713	0.0196912
$296$	299.960	0.0589014
$297$	−273.820	−0.0534972
$298$	9108.98	1.77070
$299$	657.493	0.127170
$300$	−6796.18	−1.30792
$301$	−5317.95	−1.01834
$302$	−5426.81	−1.03403
$303$	1661.94	0.315102
$304$	−6177.43	−1.16546
$305$	977.693	0.183549
$306$	−1361.93	−0.254433
$307$	−8606.40	−1.59998	−0.799989	−	0.600014i	$-0.795162\pi$
$307$	−8606.40	−1.59998	−0.799989	+	0.600014i	$0.795162\pi$
$308$	2086.82	0.386064
$309$	1203.79	0.221622
$310$	−458.195	−0.0839474
$311$	757.395	0.138096	0.0690482	−	0.997613i	$-0.478004\pi$
$311$	757.395	0.138096	0.0690482	+	0.997613i	$0.478004\pi$
$312$	4817.59	0.874175
$313$	−2172.72	−0.392363	−0.196182	−	0.980568i	$-0.562854\pi$
$313$	−2172.72	−0.392363	−0.196182	+	0.980568i	$0.562854\pi$
$314$	−244.888	−0.0440122
$315$	−168.848	−0.0302016
$316$	4994.14	0.889058
$317$	−6786.82	−1.20248	−0.601239	−	0.799069i	$-0.705326\pi$
$317$	−6786.82	−1.20248	−0.601239	+	0.799069i	$0.705326\pi$
$318$	−7488.16	−1.32049
$319$	1054.30	0.185045
$320$	340.414	0.0594679
$321$	−389.825	−0.0677817
$322$	−1271.90	−0.220124
$323$	1378.35	0.237441
$324$	1502.34	0.257603
$325$	−3609.19	−0.616005
$326$	15893.1	2.70011
$327$	−3745.30	−0.633381
$328$	10950.1	1.84335
$329$	1477.41	0.247575
$330$	−265.087	−0.0442198
$331$	−2842.29	−0.471984	−0.235992	−	0.971755i	$-0.575834\pi$
$331$	−2842.29	−0.471984	−0.235992	+	0.971755i	$0.575834\pi$
$332$	3229.45	0.533853
$333$	49.6761	0.00817487
$334$	18988.9	3.11086
$335$	88.4429	0.0144243
$336$	−4380.96	−0.711313
$337$	−6547.58	−1.05837	−0.529183	−	0.848508i	$-0.677501\pi$
$337$	−6547.58	−1.05837	−0.529183	+	0.848508i	$0.677501\pi$
$338$	−6820.91	−1.09766
$339$	3527.70	0.565187
$340$	−921.168	−0.146933
$341$	533.319	0.0846945
$342$	−2176.27	−0.344091
$343$	6245.16	0.983111
$344$	26049.7	4.08287
$345$	112.880	0.0176152
$346$	2597.82	0.403641
$347$	−35.2935	−0.00546010	−0.00273005	−	0.999996i	$-0.500869\pi$
$347$	−35.2935	−0.00546010	−0.00273005	+	0.999996i	$0.500869\pi$
$348$	−5784.50	−0.891040
$349$	10692.0	1.63991	0.819956	−	0.572427i	$-0.193998\pi$
$349$	10692.0	1.63991	0.819956	+	0.572427i	$0.193998\pi$
$350$	6981.85	1.06627
$351$	797.837	0.121326
$352$	−2468.89	−0.373842
$353$	−8661.89	−1.30602	−0.653011	−	0.757348i	$-0.726495\pi$
$353$	−8661.89	−1.30602	−0.653011	+	0.757348i	$0.726495\pi$
$354$	911.979	0.136924
$355$	675.167	0.100941
$356$	25603.0	3.81167
$357$	977.512	0.144917
$358$	15106.3	2.23014
$359$	−5422.56	−0.797192	−0.398596	−	0.917127i	$-0.630502\pi$
$359$	−5422.56	−0.797192	−0.398596	+	0.917127i	$0.630502\pi$
$360$	827.094	0.121088
$361$	−4656.49	−0.678887
$362$	22336.0	3.24297
$363$	−3684.45	−0.532737
$364$	−6080.43	−0.875552
$365$	−1768.50	−0.253610
$366$	8936.79	1.27632
$367$	−1346.75	−0.191552	−0.0957762	−	0.995403i	$-0.530533\pi$
$367$	−1346.75	−0.191552	−0.0957762	+	0.995403i	$0.530533\pi$
$368$	2928.80	0.414876
$369$	1813.44	0.255837
$370$	48.0916	0.00675720
$371$	5374.55	0.752110
$372$	−2926.11	−0.407827
$373$	−4622.11	−0.641618	−0.320809	−	0.947144i	$-0.603955\pi$
$373$	−4622.11	−0.641618	−0.320809	+	0.947144i	$0.603955\pi$
$374$	1534.67	0.212181
$375$	−1253.77	−0.172652
$376$	−7237.02	−0.992609
$377$	−3071.93	−0.419661
$378$	−1543.39	−0.210009
$379$	8654.75	1.17299	0.586497	−	0.809952i	$-0.300507\pi$
$379$	8654.75	1.17299	0.586497	+	0.809952i	$0.300507\pi$
$380$	−1471.96	−0.198711
$381$	−5980.19	−0.804133
$382$	11778.4	1.57758
$383$	−2461.00	−0.328333	−0.164166	−	0.986433i	$-0.552493\pi$
$383$	−2461.00	−0.328333	−0.164166	+	0.986433i	$0.552493\pi$
$384$	−2731.05	−0.362939
$385$	190.263	0.0251863
$386$	−8682.48	−1.14489
$387$	4314.07	0.566658
$388$	11662.1	1.52592
$389$	−8258.56	−1.07642	−0.538208	−	0.842812i	$-0.680898\pi$
$389$	−8258.56	−1.07642	−0.538208	+	0.842812i	$0.680898\pi$
$390$	772.390	0.100286
$391$	−653.494	−0.0845234
$392$	−11951.3	−1.53988
$393$	4557.91	0.585029
$394$	3559.87	0.455187
$395$	455.334	0.0580009
$396$	−1692.89	−0.214825
$397$	−4331.00	−0.547523	−0.273761	−	0.961798i	$-0.588268\pi$
$397$	−4331.00	−0.547523	−0.273761	+	0.961798i	$0.588268\pi$
$398$	−7461.94	−0.939781
$399$	1562.00	0.195984
$400$	−16077.1	−2.00964
$401$	−2292.26	−0.285461	−0.142731	−	0.989762i	$-0.545588\pi$
$401$	−2292.26	−0.285461	−0.142731	+	0.989762i	$0.545588\pi$
$402$	808.429	0.100300
$403$	−1553.94	−0.192078
$404$	10274.9	1.26534
$405$	136.974	0.0168057
$406$	5942.53	0.726411
$407$	−55.9766	−0.00681734
$408$	−4788.30	−0.581020
$409$	−13032.9	−1.57563	−0.787817	−	0.615909i	$-0.788789\pi$
$409$	−13032.9	−1.57563	−0.787817	+	0.615909i	$0.788789\pi$
$410$	1755.60	0.211471
$411$	4790.49	0.574932
$412$	7442.41	0.889954
$413$	−654.563	−0.0779878
$414$	1031.80	0.122488
$415$	294.441	0.0348278
$416$	7193.68	0.847834
$417$	1998.24	0.234662
$418$	2452.29	0.286951
$419$	−3919.78	−0.457026	−0.228513	−	0.973541i	$-0.573386\pi$
$419$	−3919.78	−0.457026	−0.228513	+	0.973541i	$0.573386\pi$
$420$	−1043.90	−0.121279
$421$	4905.30	0.567862	0.283931	−	0.958845i	$-0.408361\pi$
$421$	4905.30	0.567862	0.283931	+	0.958845i	$0.408361\pi$
$422$	−25802.2	−2.97638
$423$	−1198.52	−0.137763
$424$	−26327.0	−3.01545
$425$	3587.24	0.409428
$426$	6171.49	0.701901
$427$	−6414.29	−0.726954
$428$	−2410.09	−0.272187
$429$	−899.028	−0.101178
$430$	4176.47	0.468389
$431$	13528.1	1.51189	0.755947	−	0.654633i	$-0.227177\pi$
$431$	13528.1	1.51189	0.755947	+	0.654633i	$0.227177\pi$
$432$	3553.96	0.395810
$433$	15378.7	1.70683	0.853413	−	0.521236i	$-0.174529\pi$
$433$	15378.7	1.70683	0.853413	+	0.521236i	$0.174529\pi$
$434$	3006.05	0.332477
$435$	−527.394	−0.0581301
$436$	−23155.3	−2.54343
$437$	−1044.24	−0.114308
$438$	−16165.3	−1.76349
$439$	2929.73	0.318515	0.159258	−	0.987237i	$-0.449090\pi$
$439$	2929.73	0.318515	0.159258	+	0.987237i	$0.449090\pi$
$440$	−931.996	−0.100980
$441$	−1979.25	−0.213719
$442$	−4471.60	−0.481204
$443$	−5825.07	−0.624734	−0.312367	−	0.949961i	$-0.601122\pi$
$443$	−5825.07	−0.624734	−0.312367	+	0.949961i	$0.601122\pi$
$444$	307.121	0.0328273
$445$	2334.32	0.248668
$446$	6570.56	0.697590
$447$	5303.71	0.561201
$448$	−2233.33	−0.235525
$449$	−9997.96	−1.05085	−0.525426	−	0.850839i	$-0.676094\pi$
$449$	−9997.96	−1.05085	−0.525426	+	0.850839i	$0.676094\pi$
$450$	−5663.87	−0.593327
$451$	−2043.44	−0.213353
$452$	21810.0	2.26959
$453$	−3159.76	−0.327723
$454$	−6709.67	−0.693613
$455$	−554.375	−0.0571198
$456$	−7651.37	−0.785763
$457$	−11063.7	−1.13246	−0.566232	−	0.824246i	$-0.691599\pi$
$457$	−11063.7	−1.13246	−0.566232	+	0.824246i	$0.691599\pi$
$458$	−17309.0	−1.76594
$459$	−792.985	−0.0806392
$460$	697.876	0.0707362
$461$	9657.22	0.975665	0.487832	−	0.872937i	$-0.337788\pi$
$461$	9657.22	0.975665	0.487832	+	0.872937i	$0.337788\pi$
$462$	1739.14	0.175134
$463$	2678.74	0.268880	0.134440	−	0.990922i	$-0.457076\pi$
$463$	2678.74	0.268880	0.134440	+	0.990922i	$0.457076\pi$
$464$	−13683.9	−1.36909
$465$	−266.784	−0.0266060
$466$	−13447.9	−1.33683
$467$	−1146.00	−0.113556	−0.0567781	−	0.998387i	$-0.518083\pi$
$467$	−1146.00	−0.113556	−0.0567781	+	0.998387i	$0.518083\pi$
$468$	4932.61	0.487201
$469$	−580.242	−0.0571281
$470$	−1160.29	−0.113873
$471$	−142.586	−0.0139491
$472$	3206.35	0.312678
$473$	−4861.24	−0.472558
$474$	4162.07	0.403313
$475$	5732.16	0.553704
$476$	6043.45	0.581935
$477$	−4359.99	−0.418512
$478$	−21830.7	−2.08894
$479$	18182.6	1.73441	0.867207	−	0.497948i	$-0.165913\pi$
$479$	18182.6	1.73441	0.867207	+	0.497948i	$0.165913\pi$
$480$	1235.02	0.117439
$481$	163.100	0.0154610
$482$	−16304.2	−1.54074
$483$	−740.562	−0.0697655
$484$	−22779.0	−2.13928
$485$	1063.28	0.0995485
$486$	1252.04	0.116859
$487$	3231.33	0.300668	0.150334	−	0.988635i	$-0.451965\pi$
$487$	3231.33	0.300668	0.150334	+	0.988635i	$0.451965\pi$
$488$	31420.1	2.91459
$489$	9253.76	0.855766
$490$	−1916.12	−0.176656
$491$	6518.84	0.599168	0.299584	−	0.954070i	$-0.403152\pi$
$491$	6518.84	0.599168	0.299584	+	0.954070i	$0.403152\pi$
$492$	11211.6	1.02735
$493$	3053.25	0.278927
$494$	−7145.31	−0.650774
$495$	−154.347	−0.0140149
$496$	−6922.04	−0.626630
$497$	−4429.52	−0.399781
$498$	2691.39	0.242177
$499$	14603.3	1.31009	0.655045	−	0.755590i	$-0.272650\pi$
$499$	14603.3	1.31009	0.655045	+	0.755590i	$0.272650\pi$
$500$	−7751.42	−0.693308
$501$	11056.3	0.985945
$502$	−33467.4	−2.97554
$503$	−1522.39	−0.134950	−0.0674751	−	0.997721i	$-0.521494\pi$
$503$	−1522.39	−0.134950	−0.0674751	+	0.997721i	$0.521494\pi$
$504$	−5426.26	−0.479573
$505$	936.802	0.0825488
$506$	−1162.66	−0.102148
$507$	−3971.48	−0.347889
$508$	−36972.4	−3.22911
$509$	33.4888	0.00291624	0.00145812	−	0.999999i	$-0.499536\pi$
$509$	33.4888	0.00291624	0.00145812	+	0.999999i	$0.499536\pi$
$510$	−767.693	−0.0666549
$511$	11602.5	1.00443
$512$	−25182.4	−2.17366
$513$	−1267.14	−0.109055
$514$	−22807.3	−1.95717
$515$	678.552	0.0580593
$516$	26671.7	2.27549
$517$	1350.53	0.114886
$518$	−315.512	−0.0267622
$519$	1512.58	0.127929
$520$	2715.58	0.229012
$521$	12883.7	1.08339	0.541694	−	0.840576i	$-0.317783\pi$
$521$	12883.7	1.08339	0.541694	+	0.840576i	$0.317783\pi$
$522$	−4820.75	−0.404211
$523$	14819.1	1.23899	0.619497	−	0.784999i	$-0.287337\pi$
$523$	14819.1	1.23899	0.619497	+	0.784999i	$0.287337\pi$
$524$	28179.2	2.34927
$525$	4065.18	0.337941
$526$	35026.2	2.90345
$527$	1544.49	0.127665
$528$	−4004.72	−0.330081
$529$	−11671.9	−0.959309
$530$	−4220.92	−0.345934
$531$	531.000	0.0433963
$532$	9657.01	0.787001
$533$	5954.03	0.483861
$534$	21337.3	1.72913
$535$	−219.737	−0.0177571
$536$	2842.29	0.229045
$537$	8795.64	0.706815
$538$	35523.4	2.84670
$539$	2230.28	0.178228
$540$	846.840	0.0674856
$541$	−14334.6	−1.13917	−0.569585	−	0.821932i	$-0.692896\pi$
$541$	−14334.6	−1.13917	−0.569585	+	0.821932i	$0.692896\pi$
$542$	15129.3	1.19900
$543$	13005.2	1.02782
$544$	−7149.93	−0.563512
$545$	−2111.15	−0.165930
$546$	−5067.37	−0.397186
$547$	23516.6	1.83820	0.919102	−	0.394019i	$-0.128915\pi$
$547$	23516.6	1.83820	0.919102	+	0.394019i	$0.128915\pi$
$548$	29617.1	2.30872
$549$	5203.45	0.404514
$550$	6382.23	0.494799
$551$	4878.87	0.377218
$552$	3627.61	0.279713
$553$	−2987.28	−0.229715
$554$	−38425.7	−2.94685
$555$	28.0014	0.00214161
$556$	12354.1	0.942320
$557$	−2834.56	−0.215627	−0.107813	−	0.994171i	$-0.534385\pi$
$557$	−2834.56	−0.215627	−0.107813	+	0.994171i	$0.534385\pi$
$558$	−2438.59	−0.185007
$559$	14164.3	1.07171
$560$	−2469.46	−0.186346
$561$	893.561	0.0672481
$562$	−28082.5	−2.10781
$563$	−15215.1	−1.13897	−0.569484	−	0.822002i	$-0.692857\pi$
$563$	−15215.1	−1.13897	−0.569484	+	0.822002i	$0.692857\pi$
$564$	−7409.80	−0.553207
$565$	1988.49	0.148065
$566$	15377.0	1.14195
$567$	−898.638	−0.0665595
$568$	21697.8	1.60285
$569$	−18473.2	−1.36105	−0.680526	−	0.732724i	$-0.738249\pi$
$569$	−18473.2	−1.36105	−0.680526	+	0.732724i	$0.738249\pi$
$570$	−1226.72	−0.0901432
$571$	2472.87	0.181237	0.0906185	−	0.995886i	$-0.471116\pi$
$571$	2472.87	0.181237	0.0906185	+	0.995886i	$0.471116\pi$
$572$	−5558.23	−0.406296
$573$	6858.00	0.499995
$574$	−11517.9	−0.837537
$575$	−2717.69	−0.197105
$576$	1811.74	0.131058
$577$	1987.70	0.143413	0.0717064	−	0.997426i	$-0.477156\pi$
$577$	1987.70	0.143413	0.0717064	+	0.997426i	$0.477156\pi$
$578$	−20869.4	−1.50182
$579$	−5055.38	−0.362857
$580$	−3260.60	−0.233430
$581$	−1931.72	−0.137937
$582$	9719.11	0.692217
$583$	4912.97	0.349013
$584$	−56834.3	−4.02709
$585$	449.724	0.0317843
$586$	11282.6	0.795359
$587$	1325.75	0.0932192	0.0466096	−	0.998913i	$-0.485158\pi$
$587$	1325.75	0.0932192	0.0466096	+	0.998913i	$0.485158\pi$
$588$	−12236.7	−0.858218
$589$	2467.99	0.172652
$590$	514.064	0.0358706
$591$	2072.74	0.144266
$592$	726.530	0.0504395
$593$	23945.6	1.65822	0.829112	−	0.559083i	$-0.188847\pi$
$593$	23945.6	1.65822	0.829112	+	0.559083i	$0.188847\pi$
$594$	−1410.84	−0.0974535
$595$	551.004	0.0379646
$596$	32790.1	2.25358
$597$	−4344.72	−0.297852
$598$	3387.68	0.231660
$599$	−21924.8	−1.49553	−0.747766	−	0.663962i	$-0.768874\pi$
$599$	−21924.8	−1.49553	−0.747766	+	0.663962i	$0.768874\pi$
$600$	−19913.1	−1.35492
$601$	−6361.22	−0.431746	−0.215873	−	0.976421i	$-0.569260\pi$
$601$	−6361.22	−0.431746	−0.215873	+	0.976421i	$0.569260\pi$
$602$	−27400.3	−1.85507
$603$	470.708	0.0317889
$604$	−19535.2	−1.31602
$605$	−2076.85	−0.139564
$606$	8563.02	0.574008
$607$	−4019.17	−0.268753	−0.134376	−	0.990930i	$-0.542903\pi$
$607$	−4019.17	−0.268753	−0.134376	+	0.990930i	$0.542903\pi$
$608$	−11425.1	−0.762087
$609$	3460.04	0.230227
$610$	5037.49	0.334364
$611$	−3935.06	−0.260549
$612$	−4902.62	−0.323818
$613$	7840.19	0.516578	0.258289	−	0.966068i	$-0.416841\pi$
$613$	7840.19	0.516578	0.258289	+	0.966068i	$0.416841\pi$
$614$	−44343.8	−2.91461
$615$	1022.20	0.0670229
$616$	6114.49	0.399935
$617$	1831.09	0.119476	0.0597382	−	0.998214i	$-0.480973\pi$
$617$	1831.09	0.119476	0.0597382	+	0.998214i	$0.480973\pi$
$618$	6202.43	0.403719
$619$	7868.38	0.510916	0.255458	−	0.966820i	$-0.417774\pi$
$619$	7868.38	0.510916	0.255458	+	0.966820i	$0.417774\pi$
$620$	−1649.39	−0.106840
$621$	600.765	0.0388210
$622$	3902.42	0.251564
$623$	−15314.6	−0.984858
$624$	11668.6	0.748589
$625$	14560.8	0.931893
$626$	−11194.8	−0.714751
$627$	1427.85	0.0909454
$628$	−881.536	−0.0560145
$629$	−162.109	−0.0102761
$630$	−869.975	−0.0550169
$631$	−4830.13	−0.304730	−0.152365	−	0.988324i	$-0.548689\pi$
$631$	−4830.13	−0.304730	−0.152365	+	0.988324i	$0.548689\pi$
$632$	14633.1	0.921000
$633$	−15023.4	−0.943325
$634$	−34968.5	−2.19050
$635$	−3370.91	−0.210662
$636$	−26955.5	−1.68059
$637$	−6498.43	−0.404203
$638$	5432.17	0.337088
$639$	3593.35	0.222458
$640$	−1539.44	−0.0950808
$641$	5637.67	0.347386	0.173693	−	0.984800i	$-0.444430\pi$
$641$	5637.67	0.347386	0.173693	+	0.984800i	$0.444430\pi$
$642$	−2008.54	−0.123475
$643$	−16485.8	−1.01110	−0.505548	−	0.862798i	$-0.668710\pi$
$643$	−16485.8	−1.01110	−0.505548	+	0.862798i	$0.668710\pi$
$644$	−4578.51	−0.280153
$645$	2431.75	0.148450
$646$	7101.85	0.432537
$647$	9709.68	0.589995	0.294998	−	0.955498i	$-0.404681\pi$
$647$	9709.68	0.589995	0.294998	+	0.955498i	$0.404681\pi$
$648$	4401.94	0.266859
$649$	−598.348	−0.0361899
$650$	−18596.1	−1.12215
$651$	1750.27	0.105374
$652$	57211.2	3.43645
$653$	−21099.7	−1.26446	−0.632231	−	0.774780i	$-0.717861\pi$
$653$	−21099.7	−1.26446	−0.632231	+	0.774780i	$0.717861\pi$
$654$	−19297.4	−1.15380
$655$	2569.20	0.153263
$656$	26522.2	1.57853
$657$	−9412.27	−0.558916
$658$	7612.23	0.450997
$659$	18800.6	1.11133	0.555665	−	0.831407i	$-0.312464\pi$
$659$	18800.6	1.11133	0.555665	+	0.831407i	$0.312464\pi$
$660$	−954.247	−0.0562788
$661$	−11341.7	−0.667384	−0.333692	−	0.942682i	$-0.608295\pi$
$661$	−11341.7	−0.667384	−0.333692	+	0.942682i	$0.608295\pi$
$662$	−14644.7	−0.859792
$663$	−2603.59	−0.152511
$664$	9462.45	0.553033
$665$	880.465	0.0513428
$666$	255.952	0.0148918
$667$	−2313.14	−0.134280
$668$	68355.3	3.95920
$669$	3825.71	0.221092
$670$	455.695	0.0262762
$671$	−5863.42	−0.337339
$672$	−8102.54	−0.465123
$673$	29906.1	1.71292	0.856460	−	0.516214i	$-0.172659\pi$
$673$	29906.1	1.71292	0.856460	+	0.516214i	$0.172659\pi$
$674$	−33735.9	−1.92798
$675$	−3297.79	−0.188047
$676$	−24553.6	−1.39700
$677$	7445.33	0.422670	0.211335	−	0.977414i	$-0.432219\pi$
$677$	7445.33	0.422670	0.211335	+	0.977414i	$0.432219\pi$
$678$	18176.2	1.02958
$679$	−6975.79	−0.394266
$680$	−2699.07	−0.152212
$681$	−3906.71	−0.219832
$682$	2747.88	0.154284
$683$	−6445.33	−0.361089	−0.180544	−	0.983567i	$-0.557786\pi$
$683$	−6445.33	−0.361089	−0.180544	+	0.983567i	$0.557786\pi$
$684$	−7834.04	−0.437927
$685$	2700.30	0.150618
$686$	32177.7	1.79089
$687$	−10078.2	−0.559691
$688$	63094.8	3.49632
$689$	−14315.1	−0.791524
$690$	581.603	0.0320888
$691$	−3347.76	−0.184305	−0.0921525	−	0.995745i	$-0.529375\pi$
$691$	−3347.76	−0.184305	−0.0921525	+	0.995745i	$0.529375\pi$
$692$	9351.52	0.513716
$693$	1012.61	0.0555065
$694$	−181.847	−0.00994643
$695$	1126.37	0.0614756
$696$	−16948.9	−0.923053
$697$	−5917.82	−0.321598
$698$	55089.6	2.98735
$699$	−7830.06	−0.423691
$700$	25132.9	1.35705
$701$	−4893.37	−0.263652	−0.131826	−	0.991273i	$-0.542084\pi$
$701$	−4893.37	−0.263652	−0.131826	+	0.991273i	$0.542084\pi$
$702$	4110.79	0.221014
$703$	−259.038	−0.0138973
$704$	−2041.53	−0.109294
$705$	−675.579	−0.0360905
$706$	−44629.7	−2.37912
$707$	−6146.02	−0.326938
$708$	3282.90	0.174264
$709$	11964.9	0.633782	0.316891	−	0.948462i	$-0.397361\pi$
$709$	11964.9	0.633782	0.316891	+	0.948462i	$0.397361\pi$
$710$	3478.74	0.183880
$711$	2423.37	0.127825
$712$	75017.9	3.94861
$713$	−1170.11	−0.0614598
$714$	5036.55	0.263989
$715$	−506.764	−0.0265061
$716$	54378.9	2.83831
$717$	−12711.0	−0.662063
$718$	−27939.3	−1.45221
$719$	25587.9	1.32722	0.663608	−	0.748081i	$-0.269024\pi$
$719$	25587.9	1.32722	0.663608	+	0.748081i	$0.269024\pi$
$720$	2003.30	0.103692
$721$	−4451.73	−0.229946
$722$	−23992.2	−1.23670
$723$	−9493.15	−0.488318
$724$	80404.2	4.12734
$725$	12697.5	0.650448
$726$	−18983.8	−0.970463
$727$	−18321.4	−0.934667	−0.467334	−	0.884081i	$-0.654785\pi$
$727$	−18321.4	−0.934667	−0.467334	+	0.884081i	$0.654785\pi$
$728$	−17815.9	−0.907009
$729$	729.000	0.0370370
$730$	−9112.07	−0.461990
$731$	−14078.2	−0.712312
$732$	32170.2	1.62438
$733$	−11139.1	−0.561298	−0.280649	−	0.959810i	$-0.590550\pi$
$733$	−11139.1	−0.561298	−0.280649	+	0.959810i	$0.590550\pi$
$734$	−6939.02	−0.348943
$735$	−1115.66	−0.0559889
$736$	5416.78	0.271284
$737$	−530.410	−0.0265100
$738$	9343.61	0.466047
$739$	−16786.9	−0.835610	−0.417805	−	0.908537i	$-0.637200\pi$
$739$	−16786.9	−0.835610	−0.417805	+	0.908537i	$0.637200\pi$
$740$	173.118	0.00859993
$741$	−4160.36	−0.206254
$742$	27692.0	1.37009
$743$	35364.4	1.74615	0.873077	−	0.487582i	$-0.162121\pi$
$743$	35364.4	1.74615	0.873077	+	0.487582i	$0.162121\pi$
$744$	−8573.63	−0.422479
$745$	2989.59	0.147020
$746$	−23815.0	−1.16881
$747$	1567.07	0.0767549
$748$	5524.43	0.270044
$749$	1441.61	0.0703276
$750$	−6459.96	−0.314513
$751$	−14920.5	−0.724977	−0.362488	−	0.931988i	$-0.618073\pi$
$751$	−14920.5	−0.724977	−0.362488	+	0.931988i	$0.618073\pi$
$752$	−17528.7	−0.850009
$753$	−19486.4	−0.943059
$754$	−15827.9	−0.764478
$755$	−1781.09	−0.0858551
$756$	−5555.81	−0.267279
$757$	−17934.5	−0.861083	−0.430541	−	0.902571i	$-0.641677\pi$
$757$	−17934.5	−0.861083	−0.430541	+	0.902571i	$0.641677\pi$
$758$	44592.9	2.13679
$759$	−676.961	−0.0323744
$760$	−4312.92	−0.205850
$761$	−11900.1	−0.566858	−0.283429	−	0.958993i	$-0.591472\pi$
$761$	−11900.1	−0.566858	−0.283429	+	0.958993i	$0.591472\pi$
$762$	−30812.5	−1.46485
$763$	13850.5	0.657171
$764$	42399.4	2.00780
$765$	−446.989	−0.0211254
$766$	−12680.1	−0.598109
$767$	1743.42	0.0820747
$768$	−18902.9	−0.888149
$769$	−3012.80	−0.141280	−0.0706400	−	0.997502i	$-0.522504\pi$
$769$	−3012.80	−0.141280	−0.0706400	+	0.997502i	$0.522504\pi$
$770$	980.317	0.0458807
$771$	−13279.5	−0.620299
$772$	−31254.8	−1.45710
$773$	24647.7	1.14685	0.573427	−	0.819257i	$-0.305614\pi$
$773$	24647.7	1.14685	0.573427	+	0.819257i	$0.305614\pi$
$774$	22227.9	1.03226
$775$	6423.09	0.297709
$776$	34170.6	1.58074
$777$	−183.707	−0.00848192
$778$	−42551.6	−1.96086
$779$	−9456.27	−0.434924
$780$	2780.41	0.127634
$781$	−4049.11	−0.185517
$782$	−3367.08	−0.153973
$783$	−2806.88	−0.128110
$784$	−28947.2	−1.31866
$785$	−80.3729	−0.00365431
$786$	23484.3	1.06572
$787$	16431.2	0.744230	0.372115	−	0.928187i	$-0.378633\pi$
$787$	16431.2	0.744230	0.372115	+	0.928187i	$0.378633\pi$
$788$	12814.7	0.579319
$789$	20394.0	0.920212
$790$	2346.07	0.105658
$791$	−13045.8	−0.586415
$792$	−4960.24	−0.222544
$793$	17084.4	0.765050
$794$	−22315.1	−0.997398
$795$	−2457.64	−0.109639
$796$	−26861.1	−1.19606
$797$	−2015.52	−0.0895776	−0.0447888	−	0.998996i	$-0.514261\pi$
$797$	−2015.52	−0.0895776	−0.0447888	+	0.998996i	$0.514261\pi$
$798$	8048.06	0.357015
$799$	3911.13	0.173174
$800$	−29734.4	−1.31409
$801$	12423.6	0.548024
$802$	−11810.7	−0.520012
$803$	10606.1	0.466102
$804$	2910.15	0.127653
$805$	−417.440	−0.0182768
$806$	−8006.57	−0.349900
$807$	20683.5	0.902224
$808$	30106.0	1.31080
$809$	5483.10	0.238289	0.119144	−	0.992877i	$-0.461985\pi$
$809$	5483.10	0.238289	0.119144	+	0.992877i	$0.461985\pi$
$810$	705.749	0.0306142
$811$	−12029.9	−0.520870	−0.260435	−	0.965491i	$-0.583866\pi$
$811$	−12029.9	−0.520870	−0.260435	+	0.965491i	$0.583866\pi$
$812$	21391.6	0.924507
$813$	8809.06	0.380009
$814$	−288.415	−0.0124188
$815$	5216.15	0.224189
$816$	−11597.7	−0.497549
$817$	−22495.9	−0.963321
$818$	−67150.9	−2.87026
$819$	−2950.48	−0.125883
$820$	6319.73	0.269140
$821$	13978.8	0.594230	0.297115	−	0.954842i	$-0.403975\pi$
$821$	13978.8	0.594230	0.297115	+	0.954842i	$0.403975\pi$
$822$	24682.6	1.04733
$823$	−7223.95	−0.305967	−0.152984	−	0.988229i	$-0.548888\pi$
$823$	−7223.95	−0.305967	−0.152984	+	0.988229i	$0.548888\pi$
$824$	21806.6	0.921928
$825$	3716.06	0.156820
$826$	−3372.59	−0.142067
$827$	−34864.4	−1.46597	−0.732983	−	0.680247i	$-0.761872\pi$
$827$	−34864.4	−1.46597	−0.732983	+	0.680247i	$0.761872\pi$
$828$	3714.22	0.155891
$829$	−14737.1	−0.617420	−0.308710	−	0.951156i	$-0.599897\pi$
$829$	−14737.1	−0.617420	−0.308710	+	0.951156i	$0.599897\pi$
$830$	1517.08	0.0634443
$831$	−22373.4	−0.933964
$832$	5948.46	0.247867
$833$	6458.91	0.268653
$834$	10295.8	0.427474
$835$	6232.20	0.258293
$836$	8827.65	0.365204
$837$	−1419.87	−0.0586355
$838$	−20196.4	−0.832544
$839$	−33895.2	−1.39475	−0.697374	−	0.716708i	$-0.745648\pi$
$839$	−33895.2	−1.39475	−0.697374	+	0.716708i	$0.745648\pi$
$840$	−3058.67	−0.125636
$841$	−13581.6	−0.556874
$842$	25274.2	1.03445
$843$	−16351.1	−0.668043
$844$	−92881.7	−3.78806
$845$	−2238.64	−0.0911380
$846$	−6175.26	−0.250957
$847$	13625.5	0.552746
$848$	−63766.4	−2.58225
$849$	8953.25	0.361926
$850$	18483.0	0.745836
$851$	122.813	0.00494710
$852$	22215.8	0.893313
$853$	−36174.1	−1.45202	−0.726012	−	0.687682i	$-0.758629\pi$
$853$	−36174.1	−1.45202	−0.726012	+	0.687682i	$0.758629\pi$
$854$	−33049.1	−1.32426
$855$	−714.258	−0.0285697
$856$	−7061.67	−0.281966
$857$	20528.3	0.818241	0.409120	−	0.912480i	$-0.365836\pi$
$857$	20528.3	0.818241	0.409120	+	0.912480i	$0.365836\pi$
$858$	−4632.17	−0.184312
$859$	1704.21	0.0676912	0.0338456	−	0.999427i	$-0.489225\pi$
$859$	1704.21	0.0676912	0.0338456	+	0.999427i	$0.489225\pi$
$860$	15034.3	0.596122
$861$	−6706.28	−0.265446
$862$	69702.6	2.75415
$863$	39538.0	1.55955	0.779773	−	0.626063i	$-0.215335\pi$
$863$	39538.0	1.55955	0.779773	+	0.626063i	$0.215335\pi$
$864$	6573.01	0.258818
$865$	852.612	0.0335141
$866$	79237.8	3.10925
$867$	−12151.2	−0.475984
$868$	10821.0	0.423145
$869$	−2730.73	−0.106598
$870$	−2717.36	−0.105893
$871$	1545.47	0.0601219
$872$	−67846.0	−2.63481
$873$	5658.96	0.219389
$874$	−5380.36	−0.208230
$875$	4636.57	0.179137
$876$	−58191.2	−2.24441
$877$	25216.9	0.970938	0.485469	−	0.874254i	$-0.338649\pi$
$877$	25216.9	0.970938	0.485469	+	0.874254i	$0.338649\pi$
$878$	15095.2	0.580226
$879$	6569.31	0.252079
$880$	−2257.38	−0.0864730
$881$	−35122.3	−1.34313	−0.671567	−	0.740944i	$-0.734378\pi$
$881$	−35122.3	−1.34313	−0.671567	+	0.740944i	$0.734378\pi$
$882$	−10197.9	−0.389322
$883$	20139.2	0.767539	0.383770	−	0.923429i	$-0.374626\pi$
$883$	20139.2	0.767539	0.383770	+	0.923429i	$0.374626\pi$
$884$	−16096.7	−0.612431
$885$	299.314	0.0113687
$886$	−30013.2	−1.13805
$887$	35541.7	1.34540	0.672702	−	0.739914i	$-0.265134\pi$
$887$	35541.7	1.34540	0.672702	+	0.739914i	$0.265134\pi$
$888$	899.880	0.0340068
$889$	22115.3	0.834336
$890$	12027.4	0.452987
$891$	−821.461	−0.0308866
$892$	23652.4	0.887826
$893$	6249.72	0.234198
$894$	27326.9	1.02232
$895$	4957.92	0.185168
$896$	10099.7	0.376571
$897$	1972.48	0.0734216
$898$	−51513.7	−1.91429
$899$	5466.95	0.202818
$900$	−20388.5	−0.755131
$901$	14228.0	0.526086
$902$	−10528.7	−0.388655
$903$	−15953.9	−0.587941
$904$	63904.2	2.35113
$905$	7330.74	0.269262
$906$	−16280.4	−0.596999
$907$	−5257.49	−0.192472	−0.0962360	−	0.995359i	$-0.530680\pi$
$907$	−5257.49	−0.192472	−0.0962360	+	0.995359i	$0.530680\pi$
$908$	−24153.1	−0.882765
$909$	4985.82	0.181924
$910$	−2856.37	−0.104053
$911$	−25638.3	−0.932419	−0.466210	−	0.884674i	$-0.654381\pi$
$911$	−25638.3	−0.932419	−0.466210	+	0.884674i	$0.654381\pi$
$912$	−18532.3	−0.672879
$913$	−1765.82	−0.0640089
$914$	−57004.6	−2.06296
$915$	2933.08	0.105972
$916$	−62308.3	−2.24752
$917$	−16855.6	−0.607002
$918$	−4085.79	−0.146897
$919$	40415.4	1.45069	0.725344	−	0.688387i	$-0.241681\pi$
$919$	40415.4	1.45069	0.725344	+	0.688387i	$0.241681\pi$
$920$	2044.81	0.0732776
$921$	−25819.2	−0.923748
$922$	49758.1	1.77733
$923$	11798.0	0.420732
$924$	6260.47	0.222894
$925$	−674.162	−0.0239636
$926$	13802.0	0.489807
$927$	3611.37	0.127954
$928$	−25308.2	−0.895239
$929$	22855.5	0.807176	0.403588	−	0.914941i	$-0.367763\pi$
$929$	22855.5	0.807176	0.403588	+	0.914941i	$0.367763\pi$
$930$	−1374.58	−0.0484671
$931$	10320.9	0.363323
$932$	−48409.2	−1.70139
$933$	2272.19	0.0797299
$934$	−5904.70	−0.206860
$935$	503.682	0.0176173
$936$	14452.8	0.504705
$937$	−17288.6	−0.602770	−0.301385	−	0.953503i	$-0.597449\pi$
$937$	−17288.6	−0.602770	−0.301385	+	0.953503i	$0.597449\pi$
$938$	−2989.65	−0.104068
$939$	−6518.17	−0.226531
$940$	−4176.75	−0.144926
$941$	31998.0	1.10851	0.554253	−	0.832348i	$-0.313004\pi$
$941$	31998.0	1.10851	0.554253	+	0.832348i	$0.313004\pi$
$942$	−734.664	−0.0254104
$943$	4483.34	0.154822
$944$	7766.06	0.267758
$945$	−506.544	−0.0174369
$946$	−25047.1	−0.860838
$947$	−13177.3	−0.452170	−0.226085	−	0.974108i	$-0.572593\pi$
$947$	−13177.3	−0.452170	−0.226085	+	0.974108i	$0.572593\pi$
$948$	14982.4	0.513298
$949$	−30903.1	−1.05707
$950$	29534.5	1.00866
$951$	−20360.5	−0.694251
$952$	17707.6	0.602843
$953$	35095.6	1.19292	0.596462	−	0.802641i	$-0.296573\pi$
$953$	35095.6	1.19292	0.596462	+	0.802641i	$0.296573\pi$
$954$	−22464.5	−0.762384
$955$	3865.71	0.130986
$956$	−78585.3	−2.65861
$957$	3162.89	0.106836
$958$	93684.4	3.15951
$959$	−17715.7	−0.596527
$960$	1021.24	0.0343338
$961$	−27025.5	−0.907171
$962$	840.362	0.0281646
$963$	−1169.48	−0.0391338
$964$	−58691.2	−1.96091
$965$	−2849.61	−0.0950594
$966$	−3815.69	−0.127089
$967$	−556.001	−0.0184900	−0.00924498	−	0.999957i	$-0.502943\pi$
$967$	−556.001	−0.0184900	−0.00924498	+	0.999957i	$0.502943\pi$
$968$	−66743.7	−2.21614
$969$	4135.06	0.137087
$970$	5478.47	0.181343
$971$	−46251.0	−1.52859	−0.764297	−	0.644865i	$-0.776914\pi$
$971$	−46251.0	−1.52859	−0.764297	+	0.644865i	$0.776914\pi$
$972$	4507.03	0.148727
$973$	−7389.68	−0.243476
$974$	16649.2	0.547714
$975$	−10827.6	−0.355651
$976$	76102.3	2.49588
$977$	29081.6	0.952306	0.476153	−	0.879362i	$-0.342031\pi$
$977$	29081.6	0.952306	0.476153	+	0.879362i	$0.342031\pi$
$978$	47679.3	1.55891
$979$	−13999.4	−0.457019
$980$	−6897.57	−0.224831
$981$	−11235.9	−0.365683
$982$	33587.8	1.09148
$983$	−43019.6	−1.39584	−0.697920	−	0.716175i	$-0.745891\pi$
$983$	−43019.6	−1.39584	−0.697920	+	0.716175i	$0.745891\pi$
$984$	32850.4	1.06426
$985$	1168.36	0.0377940
$986$	15731.6	0.508110
$987$	4432.23	0.142938
$988$	−25721.3	−0.828244
$989$	10665.6	0.342919
$990$	−795.260	−0.0255303
$991$	−19398.1	−0.621798	−0.310899	−	0.950443i	$-0.600630\pi$
$991$	−19398.1	−0.621798	−0.310899	+	0.950443i	$0.600630\pi$
$992$	−12802.2	−0.409749
$993$	−8526.88	−0.272500
$994$	−22822.8	−0.728264
$995$	−2449.03	−0.0780295
$996$	9688.36	0.308220
$997$	48069.8	1.52697	0.763484	−	0.645827i	$-0.223487\pi$
$997$	48069.8	1.52697	0.763484	+	0.645827i	$0.223487\pi$
$998$	75242.5	2.38653
$999$	149.028	0.00471976

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+5.15242 q^{2} +3.00000 q^{3} +18.5474 q^{4} +1.69104 q^{5} +15.4573 q^{6} -11.0943 q^{7} +54.3449 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+5.15242 q^{2} +3.00000 q^{3} +18.5474 q^{4} +1.69104 q^{5} +15.4573 q^{6} -11.0943 q^{7} +54.3449 q^{8} +9.00000 q^{9} +8.71294 q^{10} -10.1415 q^{11} +55.6423 q^{12} +29.5495 q^{13} -57.1625 q^{14} +5.07312 q^{15} +131.628 q^{16} -29.3698 q^{17} +46.3718 q^{18} -46.9309 q^{19} +31.3645 q^{20} -33.2829 q^{21} -52.2532 q^{22} +22.2505 q^{23} +163.035 q^{24} -122.140 q^{25} +152.252 q^{26} +27.0000 q^{27} -205.771 q^{28} -103.959 q^{29} +26.1388 q^{30} -52.5878 q^{31} +243.445 q^{32} -30.4245 q^{33} -151.326 q^{34} -18.7609 q^{35} +166.927 q^{36} +5.51956 q^{37} -241.808 q^{38} +88.6485 q^{39} +91.8993 q^{40} +201.493 q^{41} -171.487 q^{42} +479.341 q^{43} -188.099 q^{44} +15.2193 q^{45} +114.644 q^{46} -133.168 q^{47} +394.885 q^{48} -219.917 q^{49} -629.319 q^{50} -88.1094 q^{51} +548.068 q^{52} -484.443 q^{53} +139.115 q^{54} -17.1497 q^{55} -602.918 q^{56} -140.793 q^{57} -535.639 q^{58} +59.0000 q^{59} +94.0934 q^{60} +578.161 q^{61} -270.954 q^{62} -99.8486 q^{63} +201.305 q^{64} +49.9694 q^{65} -156.760 q^{66} +52.3009 q^{67} -544.735 q^{68} +66.7516 q^{69} -96.6639 q^{70} +399.262 q^{71} +489.104 q^{72} -1045.81 q^{73} +28.4391 q^{74} -366.421 q^{75} -870.449 q^{76} +112.513 q^{77} +456.755 q^{78} +269.263 q^{79} +222.588 q^{80} +81.0000 q^{81} +1038.18 q^{82} +174.118 q^{83} -617.312 q^{84} -49.6655 q^{85} +2469.77 q^{86} -311.876 q^{87} -551.138 q^{88} +1380.40 q^{89} +78.4165 q^{90} -327.831 q^{91} +412.691 q^{92} -157.763 q^{93} -686.140 q^{94} -79.3620 q^{95} +730.335 q^{96} +628.773 q^{97} -1133.10 q^{98} -91.2734 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{2} + 24 q^{3} + 34 q^{4} + 42 q^{5} + 18 q^{6} + 53 q^{7} + 51 q^{8} + 72 q^{9}+O(q^{10}) \) 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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