LMFDB - Embedded newform 385.2.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [385,2,Mod(1,385)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(385, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("385.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 385 = 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	385.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:	$3.07424047782$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11348.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + x + 2 $ x^4 - x^3 - 5*x^2 + x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.589216$ of defining polynomial
Character	$\chi$	$=$	385.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.80513 q^{2} -1.06361 q^{3} +5.86874 q^{4} +1.00000 q^{5} -2.98356 q^{6} -1.00000 q^{7} +10.8523 q^{8} -1.86874 q^{9} +O(q^{10})$	\(q+2.80513 q^{2} -1.06361 q^{3} +5.86874 q^{4} +1.00000 q^{5} -2.98356 q^{6} -1.00000 q^{7} +10.8523 q^{8} -1.86874 q^{9} +2.80513 q^{10} -1.00000 q^{11} -6.24204 q^{12} -1.88518 q^{13} -2.80513 q^{14} -1.06361 q^{15} +18.7046 q^{16} -3.06361 q^{17} -5.24204 q^{18} -4.78868 q^{19} +5.86874 q^{20} +1.06361 q^{21} -2.80513 q^{22} +4.43691 q^{23} -11.5426 q^{24} +1.00000 q^{25} -5.28816 q^{26} +5.17843 q^{27} -5.86874 q^{28} -4.35686 q^{29} -2.98356 q^{30} +9.41538 q^{31} +30.7642 q^{32} +1.06361 q^{33} -8.59381 q^{34} -1.00000 q^{35} -10.9671 q^{36} -6.04717 q^{37} -13.4329 q^{38} +2.00509 q^{39} +10.8523 q^{40} -1.67791 q^{41} +2.98356 q^{42} -5.91995 q^{43} -5.86874 q^{44} -1.86874 q^{45} +12.4461 q^{46} -11.5898 q^{47} -19.8944 q^{48} +1.00000 q^{49} +2.80513 q^{50} +3.25848 q^{51} -11.0636 q^{52} -9.88707 q^{53} +14.5262 q^{54} -1.00000 q^{55} -10.8523 q^{56} +5.09329 q^{57} -12.2216 q^{58} +10.6790 q^{59} -6.24204 q^{60} +0.983558 q^{61} +26.4113 q^{62} +1.86874 q^{63} +48.8882 q^{64} -1.88518 q^{65} +2.98356 q^{66} +8.91081 q^{67} -17.9795 q^{68} -4.71914 q^{69} -2.80513 q^{70} +8.00000 q^{71} -20.2801 q^{72} +1.42047 q^{73} -16.9631 q^{74} -1.06361 q^{75} -28.1035 q^{76} +1.00000 q^{77} +5.62454 q^{78} +0.901619 q^{79} +18.7046 q^{80} +0.0983806 q^{81} -4.70675 q^{82} +6.91590 q^{83} +6.24204 q^{84} -3.06361 q^{85} -16.6062 q^{86} +4.63400 q^{87} -10.8523 q^{88} -0.821569 q^{89} -5.24204 q^{90} +1.88518 q^{91} +26.0391 q^{92} -10.0143 q^{93} -32.5108 q^{94} -4.78868 q^{95} -32.7210 q^{96} -5.53529 q^{97} +2.80513 q^{98} +1.86874 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 2 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 + 2 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 + 12 * q^8 + 8 * q^9	\( 4 q + 2 q^{2} + 2 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} + 12 q^{8} + 8 q^{9} + 2 q^{10} - 4 q^{11} - 12 q^{12} - 8 q^{13} - 2 q^{14} + 2 q^{15} + 12 q^{16} - 6 q^{17} - 8 q^{18} + 6 q^{19} + 8 q^{20} - 2 q^{21} - 2 q^{22} + 14 q^{23} - 6 q^{24} + 4 q^{25} - 6 q^{26} + 14 q^{27} - 8 q^{28} - 4 q^{29} + 4 q^{30} + 10 q^{31} + 26 q^{32} - 2 q^{33} - 4 q^{35} - 12 q^{36} - 2 q^{37} - 22 q^{38} + 16 q^{39} + 12 q^{40} - 10 q^{41} - 4 q^{42} - 14 q^{43} - 8 q^{44} + 8 q^{45} - 16 q^{46} + 16 q^{47} - 18 q^{48} + 4 q^{49} + 2 q^{50} + 16 q^{51} - 38 q^{52} + 2 q^{53} + 2 q^{54} - 4 q^{55} - 12 q^{56} - 4 q^{57} + 8 q^{58} + 26 q^{59} - 12 q^{60} - 12 q^{61} + 50 q^{62} - 8 q^{63} + 36 q^{64} - 8 q^{65} - 4 q^{66} - 10 q^{67} - 28 q^{68} - 14 q^{69} - 2 q^{70} + 32 q^{71} - 10 q^{72} - 14 q^{73} - 8 q^{74} + 2 q^{75} - 6 q^{76} + 4 q^{77} - 50 q^{78} + 20 q^{79} + 12 q^{80} - 16 q^{81} - 26 q^{82} - 10 q^{83} + 12 q^{84} - 6 q^{85} - 20 q^{86} - 4 q^{87} - 12 q^{88} - 10 q^{89} - 8 q^{90} + 8 q^{91} + 26 q^{92} + 14 q^{93} - 38 q^{94} + 6 q^{95} - 84 q^{96} - 2 q^{97} + 2 q^{98} - 8 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 2 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 + 12 * q^8 + 8 * q^9 + 2 * q^10 - 4 * q^11 - 12 * q^12 - 8 * q^13 - 2 * q^14 + 2 * q^15 + 12 * q^16 - 6 * q^17 - 8 * q^18 + 6 * q^19 + 8 * q^20 - 2 * q^21 - 2 * q^22 + 14 * q^23 - 6 * q^24 + 4 * q^25 - 6 * q^26 + 14 * q^27 - 8 * q^28 - 4 * q^29 + 4 * q^30 + 10 * q^31 + 26 * q^32 - 2 * q^33 - 4 * q^35 - 12 * q^36 - 2 * q^37 - 22 * q^38 + 16 * q^39 + 12 * q^40 - 10 * q^41 - 4 * q^42 - 14 * q^43 - 8 * q^44 + 8 * q^45 - 16 * q^46 + 16 * q^47 - 18 * q^48 + 4 * q^49 + 2 * q^50 + 16 * q^51 - 38 * q^52 + 2 * q^53 + 2 * q^54 - 4 * q^55 - 12 * q^56 - 4 * q^57 + 8 * q^58 + 26 * q^59 - 12 * q^60 - 12 * q^61 + 50 * q^62 - 8 * q^63 + 36 * q^64 - 8 * q^65 - 4 * q^66 - 10 * q^67 - 28 * q^68 - 14 * q^69 - 2 * q^70 + 32 * q^71 - 10 * q^72 - 14 * q^73 - 8 * q^74 + 2 * q^75 - 6 * q^76 + 4 * q^77 - 50 * q^78 + 20 * q^79 + 12 * q^80 - 16 * q^81 - 26 * q^82 - 10 * q^83 + 12 * q^84 - 6 * q^85 - 20 * q^86 - 4 * q^87 - 12 * q^88 - 10 * q^89 - 8 * q^90 + 8 * q^91 + 26 * q^92 + 14 * q^93 - 38 * q^94 + 6 * q^95 - 84 * q^96 - 2 * q^97 + 2 * q^98 - 8 * 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 $

$2$	2.80513	1.98352	0.991762	−	0.128094i	$-0.0408859\pi$
$2$	2.80513	1.98352	0.991762	+	0.128094i	$0.0408859\pi$
$3$	−1.06361	−0.614075	−0.307037	−	0.951697i	$-0.599338\pi$
$3$	−1.06361	−0.614075	−0.307037	+	0.951697i	$0.599338\pi$
$4$	5.86874	2.93437
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.98356	−1.21803
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	10.8523	3.83687
$9$	−1.86874	−0.622912
$10$	2.80513	0.887059
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−6.24204	−1.80192
$13$	−1.88518	−0.522854	−0.261427	−	0.965223i	$-0.584193\pi$
$13$	−1.88518	−0.522854	−0.261427	+	0.965223i	$0.584193\pi$
$14$	−2.80513	−0.749702
$15$	−1.06361	−0.274623
$16$	18.7046	4.67615
$17$	−3.06361	−0.743034	−0.371517	−	0.928426i	$-0.621162\pi$
$17$	−3.06361	−0.743034	−0.371517	+	0.928426i	$0.621162\pi$
$18$	−5.24204	−1.23556
$19$	−4.78868	−1.09860	−0.549300	−	0.835625i	$-0.685106\pi$
$19$	−4.78868	−1.09860	−0.549300	+	0.835625i	$0.685106\pi$
$20$	5.86874	1.31229
$21$	1.06361	0.232099
$22$	−2.80513	−0.598055
$23$	4.43691	0.925160	0.462580	−	0.886577i	$-0.346924\pi$
$23$	4.43691	0.925160	0.462580	+	0.886577i	$0.346924\pi$
$24$	−11.5426	−2.35612
$25$	1.00000	0.200000
$26$	−5.28816	−1.03709
$27$	5.17843	0.996590
$28$	−5.86874	−1.10909
$29$	−4.35686	−0.809049	−0.404525	−	0.914527i	$-0.632563\pi$
$29$	−4.35686	−0.809049	−0.404525	+	0.914527i	$0.632563\pi$
$30$	−2.98356	−0.544721
$31$	9.41538	1.69105	0.845526	−	0.533934i	$-0.179287\pi$
$31$	9.41538	1.69105	0.845526	+	0.533934i	$0.179287\pi$
$32$	30.7642	5.43838
$33$	1.06361	0.185151
$34$	−8.59381	−1.47383
$35$	−1.00000	−0.169031
$36$	−10.9671	−1.82785
$37$	−6.04717	−0.994148	−0.497074	−	0.867708i	$-0.665592\pi$
$37$	−6.04717	−0.994148	−0.497074	+	0.867708i	$0.665592\pi$
$38$	−13.4329	−2.17910
$39$	2.00509	0.321072
$40$	10.8523	1.71590
$41$	−1.67791	−0.262045	−0.131023	−	0.991379i	$-0.541826\pi$
$41$	−1.67791	−0.262045	−0.131023	+	0.991379i	$0.541826\pi$
$42$	2.98356	0.460373
$43$	−5.91995	−0.902784	−0.451392	−	0.892326i	$-0.649072\pi$
$43$	−5.91995	−0.902784	−0.451392	+	0.892326i	$0.649072\pi$
$44$	−5.86874	−0.884745
$45$	−1.86874	−0.278575
$46$	12.4461	1.83508
$47$	−11.5898	−1.69054	−0.845271	−	0.534339i	$-0.820561\pi$
$47$	−11.5898	−1.69054	−0.845271	+	0.534339i	$0.820561\pi$
$48$	−19.8944	−2.87150
$49$	1.00000	0.142857
$50$	2.80513	0.396705
$51$	3.25848	0.456279
$52$	−11.0636	−1.53425
$53$	−9.88707	−1.35809	−0.679046	−	0.734095i	$-0.737607\pi$
$53$	−9.88707	−1.35809	−0.679046	+	0.734095i	$0.737607\pi$
$54$	14.5262	1.97676
$55$	−1.00000	−0.134840
$56$	−10.8523	−1.45020
$57$	5.09329	0.674623
$58$	−12.2216	−1.60477
$59$	10.6790	1.39028	0.695141	−	0.718874i	$-0.255342\pi$
$59$	10.6790	1.39028	0.695141	+	0.718874i	$0.255342\pi$
$60$	−6.24204	−0.805844
$61$	0.983558	0.125932	0.0629659	−	0.998016i	$-0.479944\pi$
$61$	0.983558	0.125932	0.0629659	+	0.998016i	$0.479944\pi$
$62$	26.4113	3.35424
$63$	1.86874	0.235439
$64$	48.8882	6.11102
$65$	−1.88518	−0.233828
$66$	2.98356	0.367251
$67$	8.91081	1.08863	0.544314	−	0.838881i	$-0.316790\pi$
$67$	8.91081	1.08863	0.544314	+	0.838881i	$0.316790\pi$
$68$	−17.9795	−2.18034
$69$	−4.71914	−0.568118
$70$	−2.80513	−0.335277
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−20.2801	−2.39003
$73$	1.42047	0.166254	0.0831268	−	0.996539i	$-0.473509\pi$
$73$	1.42047	0.166254	0.0831268	+	0.996539i	$0.473509\pi$
$74$	−16.9631	−1.97192
$75$	−1.06361	−0.122815
$76$	−28.1035	−3.22370
$77$	1.00000	0.113961
$78$	5.62454	0.636853
$79$	0.901619	0.101440	0.0507201	−	0.998713i	$-0.483848\pi$
$79$	0.901619	0.101440	0.0507201	+	0.998713i	$0.483848\pi$
$80$	18.7046	2.09124
$81$	0.0983806	0.0109312
$82$	−4.70675	−0.519773
$83$	6.91590	0.759119	0.379559	−	0.925167i	$-0.376076\pi$
$83$	6.91590	0.759119	0.379559	+	0.925167i	$0.376076\pi$
$84$	6.24204	0.681062
$85$	−3.06361	−0.332295
$86$	−16.6062	−1.79069
$87$	4.63400	0.496817
$88$	−10.8523	−1.15686
$89$	−0.821569	−0.0870861	−0.0435430	−	0.999052i	$-0.513865\pi$
$89$	−0.821569	−0.0870861	−0.0435430	+	0.999052i	$0.513865\pi$
$90$	−5.24204	−0.552560
$91$	1.88518	0.197620
$92$	26.0391	2.71476
$93$	−10.0143	−1.03843
$94$	−32.5108	−3.35323
$95$	−4.78868	−0.491309
$96$	−32.7210	−3.33958
$97$	−5.53529	−0.562024	−0.281012	−	0.959704i	$-0.590670\pi$
$97$	−5.53529	−0.562024	−0.281012	+	0.959704i	$0.590670\pi$
$98$	2.80513	0.283361
$99$	1.86874	0.187815
$100$	5.86874	0.586874
$101$	14.6359	1.45633	0.728163	−	0.685404i	$-0.240375\pi$
$101$	14.6359	1.45633	0.728163	+	0.685404i	$0.240375\pi$
$102$	9.14046	0.905040
$103$	−8.02968	−0.791188	−0.395594	−	0.918426i	$-0.629461\pi$
$103$	−8.02968	−0.791188	−0.395594	+	0.918426i	$0.629461\pi$
$104$	−20.4585	−2.00612
$105$	1.06361	0.103798
$106$	−27.7345	−2.69381
$107$	−5.48304	−0.530065	−0.265033	−	0.964239i	$-0.585383\pi$
$107$	−5.48304	−0.530065	−0.265033	+	0.964239i	$0.585383\pi$
$108$	30.3908	2.92436
$109$	0.127218	0.0121853	0.00609264	−	0.999981i	$-0.498061\pi$
$109$	0.127218	0.0121853	0.00609264	+	0.999981i	$0.498061\pi$
$110$	−2.80513	−0.267458
$111$	6.43182	0.610482
$112$	−18.7046	−1.76742
$113$	12.1272	1.14083	0.570416	−	0.821356i	$-0.306782\pi$
$113$	12.1272	1.14083	0.570416	+	0.821356i	$0.306782\pi$
$114$	14.2873	1.33813
$115$	4.43691	0.413744
$116$	−25.5693	−2.37405
$117$	3.52290	0.325692
$118$	29.9558	2.75766
$119$	3.06361	0.280841
$120$	−11.5426	−1.05369
$121$	1.00000	0.0909091
$122$	2.75901	0.249789
$123$	1.78464	0.160915
$124$	55.2564	4.96217
$125$	1.00000	0.0894427
$126$	5.24204	0.466998
$127$	−3.61025	−0.320358	−0.160179	−	0.987088i	$-0.551207\pi$
$127$	−3.61025	−0.320358	−0.160179	+	0.987088i	$0.551207\pi$
$128$	75.6092	6.68297
$129$	6.29651	0.554377
$130$	−5.28816	−0.463802
$131$	15.9342	1.39218	0.696090	−	0.717954i	$-0.254921\pi$
$131$	15.9342	1.39218	0.696090	+	0.717954i	$0.254921\pi$
$132$	6.24204	0.543300
$133$	4.78868	0.415232
$134$	24.9960	2.15932
$135$	5.17843	0.445688
$136$	−33.2472	−2.85092
$137$	2.20727	0.188580	0.0942899	−	0.995545i	$-0.469942\pi$
$137$	2.20727	0.188580	0.0942899	+	0.995545i	$0.469942\pi$
$138$	−13.2378	−1.12688
$139$	10.9909	0.932233	0.466116	−	0.884723i	$-0.345653\pi$
$139$	10.9909	0.932233	0.466116	+	0.884723i	$0.345653\pi$
$140$	−5.86874	−0.495999
$141$	12.3270	1.03812
$142$	22.4410	1.88321
$143$	1.88518	0.157646
$144$	−34.9539	−2.91283
$145$	−4.35686	−0.361818
$146$	3.98460	0.329768
$147$	−1.06361	−0.0877250
$148$	−35.4892	−2.91720
$149$	−17.0852	−1.39967	−0.699837	−	0.714303i	$-0.746744\pi$
$149$	−17.0852	−1.39967	−0.699837	+	0.714303i	$0.746744\pi$
$150$	−2.98356	−0.243607
$151$	−10.2493	−0.834080	−0.417040	−	0.908888i	$-0.636932\pi$
$151$	−10.2493	−0.834080	−0.417040	+	0.908888i	$0.636932\pi$
$152$	−51.9682	−4.21518
$153$	5.72508	0.462845
$154$	2.80513	0.226044
$155$	9.41538	0.756262
$156$	11.7674	0.942142
$157$	9.56009	0.762978	0.381489	−	0.924373i	$-0.375411\pi$
$157$	9.56009	0.762978	0.381489	+	0.924373i	$0.375411\pi$
$158$	2.52916	0.201209
$159$	10.5160	0.833971
$160$	30.7642	2.43212
$161$	−4.43691	−0.349678
$162$	0.275970	0.0216823
$163$	−0.572224	−0.0448201	−0.0224100	−	0.999749i	$-0.507134\pi$
$163$	−0.572224	−0.0448201	−0.0224100	+	0.999749i	$0.507134\pi$
$164$	−9.84720	−0.768937
$165$	1.06361	0.0828019
$166$	19.4000	1.50573
$167$	20.6286	1.59629	0.798144	−	0.602467i	$-0.205815\pi$
$167$	20.6286	1.59629	0.798144	+	0.602467i	$0.205815\pi$
$168$	11.5426	0.890531
$169$	−9.44611	−0.726623
$170$	−8.59381	−0.659115
$171$	8.94879	0.684331
$172$	−34.7426	−2.64910
$173$	−14.9283	−1.13498	−0.567489	−	0.823381i	$-0.692085\pi$
$173$	−14.9283	−1.13498	−0.567489	+	0.823381i	$0.692085\pi$
$174$	12.9990	0.985448
$175$	−1.00000	−0.0755929
$176$	−18.7046	−1.40991
$177$	−11.3582	−0.853737
$178$	−2.30460	−0.172737
$179$	−13.9919	−1.04580	−0.522902	−	0.852393i	$-0.675151\pi$
$179$	−13.9919	−1.04580	−0.522902	+	0.852393i	$0.675151\pi$
$180$	−10.9671	−0.817441
$181$	16.6863	1.24028	0.620140	−	0.784491i	$-0.287076\pi$
$181$	16.6863	1.24028	0.620140	+	0.784491i	$0.287076\pi$
$182$	5.28816	0.391985
$183$	−1.04612	−0.0773315
$184$	48.1507	3.54972
$185$	−6.04717	−0.444597
$186$	−28.0913	−2.05976
$187$	3.06361	0.224033
$188$	−68.0173	−4.96067
$189$	−5.17843	−0.376675
$190$	−13.4329	−0.974523
$191$	13.0933	0.947397	0.473699	−	0.880687i	$-0.342919\pi$
$191$	13.0933	0.947397	0.473699	+	0.880687i	$0.342919\pi$
$192$	−51.9979	−3.75262
$193$	−2.81648	−0.202734	−0.101367	−	0.994849i	$-0.532322\pi$
$193$	−2.81648	−0.202734	−0.101367	+	0.994849i	$0.532322\pi$
$194$	−15.5272	−1.11479
$195$	2.00509	0.143588
$196$	5.86874	0.419195
$197$	−5.42778	−0.386713	−0.193357	−	0.981129i	$-0.561937\pi$
$197$	−5.42778	−0.386713	−0.193357	+	0.981129i	$0.561937\pi$
$198$	5.24204	0.372536
$199$	15.2380	1.08019	0.540096	−	0.841603i	$-0.318388\pi$
$199$	15.2380	1.08019	0.540096	+	0.841603i	$0.318388\pi$
$200$	10.8523	0.767373
$201$	−9.47762	−0.668500
$202$	41.0555	2.88866
$203$	4.35686	0.305792
$204$	19.1232	1.33889
$205$	−1.67791	−0.117190
$206$	−22.5243	−1.56934
$207$	−8.29142	−0.576293
$208$	−35.2615	−2.44494
$209$	4.78868	0.331240
$210$	2.98356	0.205885
$211$	11.3558	0.781767	0.390883	−	0.920440i	$-0.372170\pi$
$211$	11.3558	0.781767	0.390883	+	0.920440i	$0.372170\pi$
$212$	−58.0246	−3.98514
$213$	−8.50887	−0.583018
$214$	−15.3806	−1.05140
$215$	−5.91995	−0.403737
$216$	56.1979	3.82378
$217$	−9.41538	−0.639158
$218$	0.356863	0.0241698
$219$	−1.51083	−0.102092
$220$	−5.86874	−0.395670
$221$	5.77545	0.388499
$222$	18.0421	1.21090
$223$	−20.7930	−1.39240	−0.696201	−	0.717847i	$-0.745128\pi$
$223$	−20.7930	−1.39240	−0.696201	+	0.717847i	$0.745128\pi$
$224$	−30.7642	−2.05552
$225$	−1.86874	−0.124582
$226$	34.0184	2.26287
$227$	0.863644	0.0573221	0.0286610	−	0.999589i	$-0.490876\pi$
$227$	0.863644	0.0573221	0.0286610	+	0.999589i	$0.490876\pi$
$228$	29.8912	1.97959
$229$	−6.89862	−0.455874	−0.227937	−	0.973676i	$-0.573198\pi$
$229$	−6.89862	−0.455874	−0.227937	+	0.973676i	$0.573198\pi$
$230$	12.4461	0.820672
$231$	−1.06361	−0.0699803
$232$	−47.2820	−3.10421
$233$	3.20622	0.210047	0.105023	−	0.994470i	$-0.466508\pi$
$233$	3.20622	0.210047	0.105023	+	0.994470i	$0.466508\pi$
$234$	9.88218	0.646018
$235$	−11.5898	−0.756033
$236$	62.6720	4.07960
$237$	−0.958971	−0.0622918
$238$	8.59381	0.557054
$239$	6.11703	0.395678	0.197839	−	0.980235i	$-0.436608\pi$
$239$	6.11703	0.395678	0.197839	+	0.980235i	$0.436608\pi$
$240$	−19.8944	−1.28418
$241$	−13.0585	−0.841173	−0.420587	−	0.907252i	$-0.638176\pi$
$241$	−13.0585	−0.841173	−0.420587	+	0.907252i	$0.638176\pi$
$242$	2.80513	0.180320
$243$	−15.6399	−1.00330
$244$	5.77224	0.369530
$245$	1.00000	0.0638877
$246$	5.00614	0.319180
$247$	9.02752	0.574407
$248$	102.178	6.48834
$249$	−7.35582	−0.466156
$250$	2.80513	0.177412
$251$	−29.3744	−1.85410	−0.927048	−	0.374943i	$-0.877662\pi$
$251$	−29.3744	−1.85410	−0.927048	+	0.374943i	$0.877662\pi$
$252$	10.9671	0.690863
$253$	−4.43691	−0.278946
$254$	−10.1272	−0.635438
$255$	3.25848	0.204054
$256$	114.317	7.14482
$257$	−7.56818	−0.472090	−0.236045	−	0.971742i	$-0.575851\pi$
$257$	−7.56818	−0.472090	−0.236045	+	0.971742i	$0.575851\pi$
$258$	17.6625	1.09962
$259$	6.04717	0.375753
$260$	−11.0636	−0.686136
$261$	8.14183	0.503966
$262$	44.6975	2.76142
$263$	15.8647	0.978259	0.489129	−	0.872211i	$-0.337315\pi$
$263$	15.8647	0.978259	0.489129	+	0.872211i	$0.337315\pi$
$264$	11.5426	0.710398
$265$	−9.88707	−0.607358
$266$	13.4329	0.823622
$267$	0.873828	0.0534774
$268$	52.2952	3.19444
$269$	16.7806	1.02313	0.511565	−	0.859244i	$-0.329066\pi$
$269$	16.7806	1.02313	0.511565	+	0.859244i	$0.329066\pi$
$270$	14.5262	0.884034
$271$	−16.5682	−1.00645	−0.503224	−	0.864156i	$-0.667853\pi$
$271$	−16.5682	−1.00645	−0.503224	+	0.864156i	$0.667853\pi$
$272$	−57.3035	−3.47454
$273$	−2.00509	−0.121354
$274$	6.19167	0.374052
$275$	−1.00000	−0.0603023
$276$	−27.6954	−1.66707
$277$	−9.30865	−0.559303	−0.279651	−	0.960102i	$-0.590219\pi$
$277$	−9.30865	−0.559303	−0.279651	+	0.960102i	$0.590219\pi$
$278$	30.8308	1.84911
$279$	−17.5949	−1.05338
$280$	−10.8523	−0.648549
$281$	26.0943	1.55666	0.778329	−	0.627857i	$-0.216068\pi$
$281$	26.0943	1.55666	0.778329	+	0.627857i	$0.216068\pi$
$282$	34.5787	2.05913
$283$	−30.7898	−1.83026	−0.915131	−	0.403156i	$-0.867913\pi$
$283$	−30.7898	−1.83026	−0.915131	+	0.403156i	$0.867913\pi$
$284$	46.9499	2.78596
$285$	5.09329	0.301700
$286$	5.28816	0.312696
$287$	1.67791	0.0990438
$288$	−57.4901	−3.38763
$289$	−7.61430	−0.447900
$290$	−12.2216	−0.717674
$291$	5.88739	0.345125
$292$	8.33638	0.487849
$293$	−24.7353	−1.44505	−0.722526	−	0.691344i	$-0.757019\pi$
$293$	−24.7353	−1.44505	−0.722526	+	0.691344i	$0.757019\pi$
$294$	−2.98356	−0.174005
$295$	10.6790	0.621753
$296$	−65.6256	−3.81441
$297$	−5.17843	−0.300483
$298$	−47.9261	−2.77629
$299$	−8.36437	−0.483724
$300$	−6.24204	−0.360384
$301$	5.91995	0.341220
$302$	−28.7507	−1.65442
$303$	−15.5669	−0.894293
$304$	−89.5704	−5.13721
$305$	0.983558	0.0563184
$306$	16.0596	0.918064
$307$	9.65233	0.550887	0.275444	−	0.961317i	$-0.411175\pi$
$307$	9.65233	0.550887	0.275444	+	0.961317i	$0.411175\pi$
$308$	5.86874	0.334402
$309$	8.54044	0.485849
$310$	26.4113	1.50006
$311$	5.66062	0.320985	0.160492	−	0.987037i	$-0.448692\pi$
$311$	5.66062	0.320985	0.160492	+	0.987037i	$0.448692\pi$
$312$	21.7598	1.23191
$313$	8.19676	0.463308	0.231654	−	0.972798i	$-0.425586\pi$
$313$	8.19676	0.463308	0.231654	+	0.972798i	$0.425586\pi$
$314$	26.8173	1.51339
$315$	1.86874	0.105291
$316$	5.29137	0.297663
$317$	5.85008	0.328573	0.164287	−	0.986413i	$-0.447468\pi$
$317$	5.85008	0.328573	0.164287	+	0.986413i	$0.447468\pi$
$318$	29.4986	1.65420
$319$	4.35686	0.243937
$320$	48.8882	2.73293
$321$	5.83181	0.325500
$322$	−12.4461	−0.693594
$323$	14.6707	0.816297
$324$	0.577370	0.0320761
$325$	−1.88518	−0.104571
$326$	−1.60516	−0.0889017
$327$	−0.135310	−0.00748268
$328$	−18.2092	−1.00543
$329$	11.5898	0.638964
$330$	2.98356	0.164239
$331$	−34.1558	−1.87737	−0.938686	−	0.344774i	$-0.887956\pi$
$331$	−34.1558	−1.87737	−0.938686	+	0.344774i	$0.887956\pi$
$332$	40.5876	2.22753
$333$	11.3006	0.619267
$334$	57.8658	3.16628
$335$	8.91081	0.486850
$336$	19.8944	1.08533
$337$	−17.0132	−0.926770	−0.463385	−	0.886157i	$-0.653365\pi$
$337$	−17.0132	−0.926770	−0.463385	+	0.886157i	$0.653365\pi$
$338$	−26.4975	−1.44128
$339$	−12.8986	−0.700557
$340$	−17.9795	−0.975076
$341$	−9.41538	−0.509871
$342$	25.1025	1.35739
$343$	−1.00000	−0.0539949
$344$	−64.2450	−3.46386
$345$	−4.71914	−0.254070
$346$	−41.8758	−2.25126
$347$	−26.9723	−1.44795	−0.723973	−	0.689828i	$-0.757686\pi$
$347$	−26.9723	−1.44795	−0.723973	+	0.689828i	$0.757686\pi$
$348$	27.1957	1.45784
$349$	−1.27897	−0.0684617	−0.0342309	−	0.999414i	$-0.510898\pi$
$349$	−1.27897	−0.0684617	−0.0342309	+	0.999414i	$0.510898\pi$
$350$	−2.80513	−0.149940
$351$	−9.76226	−0.521071
$352$	−30.7642	−1.63973
$353$	8.34668	0.444249	0.222124	−	0.975018i	$-0.428701\pi$
$353$	8.34668	0.444249	0.222124	+	0.975018i	$0.428701\pi$
$354$	−31.8613	−1.69341
$355$	8.00000	0.424596
$356$	−4.82157	−0.255543
$357$	−3.25848	−0.172457
$358$	−39.2491	−2.07438
$359$	14.0299	0.740469	0.370234	−	0.928938i	$-0.379277\pi$
$359$	14.0299	0.740469	0.370234	+	0.928938i	$0.379277\pi$
$360$	−20.2801	−1.06885
$361$	3.93150	0.206921
$362$	46.8071	2.46012
$363$	−1.06361	−0.0558250
$364$	11.0636	0.579891
$365$	1.42047	0.0743509
$366$	−2.93450	−0.153389
$367$	−1.45877	−0.0761473	−0.0380736	−	0.999275i	$-0.512122\pi$
$367$	−1.45877	−0.0761473	−0.0380736	+	0.999275i	$0.512122\pi$
$368$	82.9906	4.32619
$369$	3.13557	0.163231
$370$	−16.9631	−0.881868
$371$	9.88707	0.513311
$372$	−58.7712	−3.04714
$373$	−34.7060	−1.79701	−0.898503	−	0.438967i	$-0.855344\pi$
$373$	−34.7060	−1.79701	−0.898503	+	0.438967i	$0.855344\pi$
$374$	8.59381	0.444375
$375$	−1.06361	−0.0549245
$376$	−125.776	−6.48638
$377$	8.21346	0.423015
$378$	−14.5262	−0.747145
$379$	16.1045	0.827234	0.413617	−	0.910451i	$-0.364265\pi$
$379$	16.1045	0.827234	0.413617	+	0.910451i	$0.364265\pi$
$380$	−28.1035	−1.44168
$381$	3.83990	0.196724
$382$	36.7283	1.87919
$383$	17.8717	0.913200	0.456600	−	0.889672i	$-0.349067\pi$
$383$	17.8717	0.913200	0.456600	+	0.889672i	$0.349067\pi$
$384$	−80.4186	−4.10385
$385$	1.00000	0.0509647
$386$	−7.90057	−0.402129
$387$	11.0628	0.562355
$388$	−32.4852	−1.64919
$389$	−3.51801	−0.178370	−0.0891851	−	0.996015i	$-0.528426\pi$
$389$	−3.51801	−0.178370	−0.0891851	+	0.996015i	$0.528426\pi$
$390$	5.62454	0.284810
$391$	−13.5930	−0.687426
$392$	10.8523	0.548124
$393$	−16.9478	−0.854903
$394$	−15.2256	−0.767055
$395$	0.901619	0.0453654
$396$	10.9671	0.551118
$397$	−23.5024	−1.17955	−0.589776	−	0.807567i	$-0.700784\pi$
$397$	−23.5024	−1.17955	−0.589776	+	0.807567i	$0.700784\pi$
$398$	42.7445	2.14259
$399$	−5.09329	−0.254983
$400$	18.7046	0.935229
$401$	32.8000	1.63795	0.818976	−	0.573828i	$-0.194542\pi$
$401$	32.8000	1.63795	0.818976	+	0.573828i	$0.194542\pi$
$402$	−26.5859	−1.32599
$403$	−17.7497	−0.884174
$404$	85.8942	4.27339
$405$	0.0983806	0.00488857
$406$	12.2216	0.606545
$407$	6.04717	0.299747
$408$	35.3620	1.75068
$409$	−16.8133	−0.831363	−0.415681	−	0.909510i	$-0.636457\pi$
$409$	−16.8133	−0.831363	−0.415681	+	0.909510i	$0.636457\pi$
$410$	−4.70675	−0.232450
$411$	−2.34767	−0.115802
$412$	−47.1241	−2.32164
$413$	−10.6790	−0.525477
$414$	−23.2585	−1.14309
$415$	6.91590	0.339488
$416$	−57.9959	−2.84348
$417$	−11.6900	−0.572461
$418$	13.4329	0.657023
$419$	−1.71998	−0.0840267	−0.0420134	−	0.999117i	$-0.513377\pi$
$419$	−1.71998	−0.0840267	−0.0420134	+	0.999117i	$0.513377\pi$
$420$	6.24204	0.304580
$421$	−25.4370	−1.23972	−0.619861	−	0.784712i	$-0.712811\pi$
$421$	−25.4370	−1.23972	−0.619861	+	0.784712i	$0.712811\pi$
$422$	31.8545	1.55065
$423$	21.6582	1.05306
$424$	−107.297	−5.21082
$425$	−3.06361	−0.148607
$426$	−23.8685	−1.15643
$427$	−0.983558	−0.0475977
$428$	−32.1785	−1.55541
$429$	−2.00509	−0.0968068
$430$	−16.6062	−0.800822
$431$	−0.216461	−0.0104265	−0.00521327	−	0.999986i	$-0.501659\pi$
$431$	−0.216461	−0.0104265	−0.00521327	+	0.999986i	$0.501659\pi$
$432$	96.8604	4.66020
$433$	7.77245	0.373520	0.186760	−	0.982406i	$-0.440201\pi$
$433$	7.77245	0.373520	0.186760	+	0.982406i	$0.440201\pi$
$434$	−26.4113	−1.26778
$435$	4.63400	0.222183
$436$	0.746610	0.0357561
$437$	−21.2470	−1.01638
$438$	−4.23806	−0.202502
$439$	−31.5445	−1.50554	−0.752768	−	0.658286i	$-0.771282\pi$
$439$	−31.5445	−1.50554	−0.752768	+	0.658286i	$0.771282\pi$
$440$	−10.8523	−0.517363
$441$	−1.86874	−0.0889874
$442$	16.2009	0.770596
$443$	25.8440	1.22789	0.613943	−	0.789351i	$-0.289583\pi$
$443$	25.8440	1.22789	0.613943	+	0.789351i	$0.289583\pi$
$444$	37.7467	1.79138
$445$	−0.821569	−0.0389461
$446$	−58.3270	−2.76186
$447$	18.1720	0.859505
$448$	−48.8882	−2.30975
$449$	−3.70968	−0.175071	−0.0875353	−	0.996161i	$-0.527899\pi$
$449$	−3.70968	−0.175071	−0.0875353	+	0.996161i	$0.527899\pi$
$450$	−5.24204	−0.247112
$451$	1.67791	0.0790096
$452$	71.1714	3.34762
$453$	10.9013	0.512188
$454$	2.42263	0.113700
$455$	1.88518	0.0883785
$456$	55.2739	2.58844
$457$	24.4882	1.14551	0.572754	−	0.819727i	$-0.305875\pi$
$457$	24.4882	1.14551	0.572754	+	0.819727i	$0.305875\pi$
$458$	−19.3515	−0.904236
$459$	−15.8647	−0.740500
$460$	26.0391	1.21408
$461$	−26.2213	−1.22125	−0.610625	−	0.791920i	$-0.709082\pi$
$461$	−26.2213	−1.22125	−0.610625	+	0.791920i	$0.709082\pi$
$462$	−2.98356	−0.138808
$463$	26.9129	1.25075	0.625374	−	0.780325i	$-0.284946\pi$
$463$	26.9129	1.25075	0.625374	+	0.780325i	$0.284946\pi$
$464$	−81.4933	−3.78323
$465$	−10.0143	−0.464401
$466$	8.99386	0.416633
$467$	−39.9229	−1.84741	−0.923707	−	0.383101i	$-0.874856\pi$
$467$	−39.9229	−1.84741	−0.923707	+	0.383101i	$0.874856\pi$
$468$	20.6750	0.955700
$469$	−8.91081	−0.411463
$470$	−32.5108	−1.49961
$471$	−10.1682	−0.468526
$472$	115.891	5.33432
$473$	5.91995	0.272200
$474$	−2.69003	−0.123557
$475$	−4.78868	−0.219720
$476$	17.9795	0.824090
$477$	18.4763	0.845972
$478$	17.1591	0.784837
$479$	31.3056	1.43039	0.715196	−	0.698924i	$-0.246338\pi$
$479$	31.3056	1.43039	0.715196	+	0.698924i	$0.246338\pi$
$480$	−32.7210	−1.49350
$481$	11.4000	0.519795
$482$	−36.6308	−1.66849
$483$	4.71914	0.214728
$484$	5.86874	0.266761
$485$	−5.53529	−0.251345
$486$	−43.8720	−1.99007
$487$	−18.6337	−0.844372	−0.422186	−	0.906509i	$-0.638737\pi$
$487$	−18.6337	−0.844372	−0.422186	+	0.906509i	$0.638737\pi$
$488$	10.6739	0.483183
$489$	0.608623	0.0275229
$490$	2.80513	0.126723
$491$	33.8596	1.52806	0.764031	−	0.645179i	$-0.223217\pi$
$491$	33.8596	1.52806	0.764031	+	0.645179i	$0.223217\pi$
$492$	10.4736	0.472185
$493$	13.3477	0.601151
$494$	25.3233	1.13935
$495$	1.86874	0.0839934
$496$	176.111	7.90761
$497$	−8.00000	−0.358849
$498$	−20.6340	−0.924632
$499$	−9.12826	−0.408637	−0.204319	−	0.978904i	$-0.565498\pi$
$499$	−9.12826	−0.408637	−0.204319	+	0.978904i	$0.565498\pi$
$500$	5.86874	0.262458
$501$	−21.9407	−0.980240
$502$	−82.3989	−3.67764
$503$	−1.18862	−0.0529977	−0.0264989	−	0.999649i	$-0.508436\pi$
$503$	−1.18862	−0.0529977	−0.0264989	+	0.999649i	$0.508436\pi$
$504$	20.2801	0.903346
$505$	14.6359	0.651288
$506$	−12.4461	−0.553297
$507$	10.0470	0.446201
$508$	−21.1876	−0.940049
$509$	−16.9909	−0.753107	−0.376553	−	0.926395i	$-0.622891\pi$
$509$	−16.9909	−0.753107	−0.376553	+	0.926395i	$0.622891\pi$
$510$	9.14046	0.404746
$511$	−1.42047	−0.0628380
$512$	169.455	7.48894
$513$	−24.7979	−1.09485
$514$	−21.2297	−0.936402
$515$	−8.02968	−0.353830
$516$	36.9526	1.62675
$517$	11.5898	0.509717
$518$	16.9631	0.745315
$519$	15.8779	0.696961
$520$	−20.4585	−0.897165
$521$	13.4841	0.590751	0.295375	−	0.955381i	$-0.404555\pi$
$521$	13.4841	0.590751	0.295375	+	0.955381i	$0.404555\pi$
$522$	22.8389	0.999629
$523$	−9.92295	−0.433900	−0.216950	−	0.976183i	$-0.569611\pi$
$523$	−9.92295	−0.433900	−0.216950	+	0.976183i	$0.569611\pi$
$524$	93.5138	4.08517
$525$	1.06361	0.0464197
$526$	44.5025	1.94040
$527$	−28.8450	−1.25651
$528$	19.8944	0.865791
$529$	−3.31380	−0.144078
$530$	−27.7345	−1.20471
$531$	−19.9561	−0.866023
$532$	28.1035	1.21844
$533$	3.16316	0.137011
$534$	2.45120	0.106074
$535$	−5.48304	−0.237052
$536$	96.7027	4.17692
$537$	14.8819	0.642202
$538$	47.0717	2.02940
$539$	−1.00000	−0.0430730
$540$	30.3908	1.30781
$541$	28.5763	1.22859	0.614296	−	0.789076i	$-0.289440\pi$
$541$	28.5763	1.22859	0.614296	+	0.789076i	$0.289440\pi$
$542$	−46.4760	−1.99631
$543$	−17.7477	−0.761625
$544$	−94.2493	−4.04091
$545$	0.127218	0.00544943
$546$	−5.62454	−0.240708
$547$	3.62044	0.154799	0.0773994	−	0.997000i	$-0.475338\pi$
$547$	3.62044	0.154799	0.0773994	+	0.997000i	$0.475338\pi$
$548$	12.9539	0.553362
$549$	−1.83801	−0.0784444
$550$	−2.80513	−0.119611
$551$	20.8636	0.888821
$552$	−51.2135	−2.17979
$553$	−0.901619	−0.0383408
$554$	−26.1119	−1.10939
$555$	6.43182	0.273016
$556$	64.5025	2.73551
$557$	23.8542	1.01073	0.505367	−	0.862904i	$-0.331357\pi$
$557$	23.8542	1.01073	0.505367	+	0.862904i	$0.331357\pi$
$558$	−49.3558	−2.08940
$559$	11.1602	0.472024
$560$	−18.7046	−0.790413
$561$	−3.25848	−0.137573
$562$	73.1979	3.08767
$563$	−31.7989	−1.34016	−0.670082	−	0.742287i	$-0.733741\pi$
$563$	−31.7989	−1.34016	−0.670082	+	0.742287i	$0.733741\pi$
$564$	72.3438	3.04622
$565$	12.1272	0.510196
$566$	−86.3692	−3.63037
$567$	−0.0983806	−0.00413160
$568$	86.8184	3.64282
$569$	−25.7046	−1.07759	−0.538796	−	0.842436i	$-0.681121\pi$
$569$	−25.7046	−1.07759	−0.538796	+	0.842436i	$0.681121\pi$
$570$	14.2873	0.598430
$571$	−14.3715	−0.601427	−0.300714	−	0.953714i	$-0.597225\pi$
$571$	−14.3715	−0.601427	−0.300714	+	0.953714i	$0.597225\pi$
$572$	11.0636	0.462593
$573$	−13.9261	−0.581773
$574$	4.70675	0.196456
$575$	4.43691	0.185032
$576$	−91.3591	−3.80663
$577$	15.4227	0.642055	0.321027	−	0.947070i	$-0.395972\pi$
$577$	15.4227	0.642055	0.321027	+	0.947070i	$0.395972\pi$
$578$	−21.3591	−0.888420
$579$	2.99563	0.124494
$580$	−25.5693	−1.06171
$581$	−6.91590	−0.286920
$582$	16.5149	0.684564
$583$	9.88707	0.409480
$584$	15.4154	0.637893
$585$	3.52290	0.145654
$586$	−69.3857	−2.86630
$587$	5.05343	0.208577	0.104289	−	0.994547i	$-0.466743\pi$
$587$	5.05343	0.208577	0.104289	+	0.994547i	$0.466743\pi$
$588$	−6.24204	−0.257417
$589$	−45.0873	−1.85779
$590$	29.9558	1.23326
$591$	5.77303	0.237471
$592$	−113.110	−4.64878
$593$	−15.9218	−0.653831	−0.326916	−	0.945053i	$-0.606009\pi$
$593$	−15.9218	−0.653831	−0.326916	+	0.945053i	$0.606009\pi$
$594$	−14.5262	−0.596015
$595$	3.06361	0.125596
$596$	−100.269	−4.10716
$597$	−16.2073	−0.663319
$598$	−23.4631	−0.959478
$599$	45.2470	1.84874	0.924371	−	0.381495i	$-0.124591\pi$
$599$	45.2470	1.84874	0.924371	+	0.381495i	$0.124591\pi$
$600$	−11.5426	−0.471225
$601$	−14.0267	−0.572160	−0.286080	−	0.958206i	$-0.592352\pi$
$601$	−14.0267	−0.572160	−0.286080	+	0.958206i	$0.592352\pi$
$602$	16.6062	0.676818
$603$	−16.6520	−0.678120
$604$	−60.1507	−2.44750
$605$	1.00000	0.0406558
$606$	−43.6670	−1.77385
$607$	20.9488	0.850285	0.425143	−	0.905126i	$-0.360224\pi$
$607$	20.9488	0.850285	0.425143	+	0.905126i	$0.360224\pi$
$608$	−147.320	−5.97461
$609$	−4.63400	−0.187779
$610$	2.75901	0.111709
$611$	21.8488	0.883906
$612$	33.5990	1.35816
$613$	8.00410	0.323283	0.161641	−	0.986850i	$-0.448321\pi$
$613$	8.00410	0.323283	0.161641	+	0.986850i	$0.448321\pi$
$614$	27.0760	1.09270
$615$	1.78464	0.0719636
$616$	10.8523	0.437251
$617$	15.5302	0.625222	0.312611	−	0.949881i	$-0.398796\pi$
$617$	15.5302	0.625222	0.312611	+	0.949881i	$0.398796\pi$
$618$	23.9570	0.963693
$619$	−11.7469	−0.472146	−0.236073	−	0.971735i	$-0.575861\pi$
$619$	−11.7469	−0.472146	−0.236073	+	0.971735i	$0.575861\pi$
$620$	55.2564	2.21915
$621$	22.9763	0.922005
$622$	15.8788	0.636681
$623$	0.821569	0.0329154
$624$	37.5044	1.50138
$625$	1.00000	0.0400000
$626$	22.9930	0.918983
$627$	−5.09329	−0.203406
$628$	56.1056	2.23886
$629$	18.5262	0.738686
$630$	5.24204	0.208848
$631$	−35.0048	−1.39352	−0.696760	−	0.717304i	$-0.745376\pi$
$631$	−35.0048	−1.39352	−0.696760	+	0.717304i	$0.745376\pi$
$632$	9.78464	0.389212
$633$	−12.0782	−0.480063
$634$	16.4102	0.651733
$635$	−3.61025	−0.143269
$636$	61.7155	2.44718
$637$	−1.88518	−0.0746935
$638$	12.2216	0.483856
$639$	−14.9499	−0.591408
$640$	75.6092	2.98872
$641$	18.9960	0.750295	0.375148	−	0.926965i	$-0.377592\pi$
$641$	18.9960	0.750295	0.375148	+	0.926965i	$0.377592\pi$
$642$	16.3590	0.645636
$643$	21.6474	0.853692	0.426846	−	0.904324i	$-0.359625\pi$
$643$	21.6474	0.853692	0.426846	+	0.904324i	$0.359625\pi$
$644$	−26.0391	−1.02608
$645$	6.29651	0.247925
$646$	41.1531	1.61915
$647$	10.1067	0.397337	0.198668	−	0.980067i	$-0.436338\pi$
$647$	10.1067	0.397337	0.198668	+	0.980067i	$0.436338\pi$
$648$	1.06766	0.0419415
$649$	−10.6790	−0.419186
$650$	−5.28816	−0.207419
$651$	10.0143	0.392491
$652$	−3.35823	−0.131519
$653$	37.6388	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$653$	37.6388	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$654$	−0.379563	−0.0148421
$655$	15.9342	0.622602
$656$	−31.3846	−1.22536
$657$	−2.65449	−0.103561
$658$	32.5108	1.26740
$659$	−10.0397	−0.391090	−0.195545	−	0.980695i	$-0.562648\pi$
$659$	−10.0397	−0.391090	−0.195545	+	0.980695i	$0.562648\pi$
$660$	6.24204	0.242971
$661$	−14.9526	−0.581587	−0.290794	−	0.956786i	$-0.593919\pi$
$661$	−14.9526	−0.581587	−0.290794	+	0.956786i	$0.593919\pi$
$662$	−95.8113	−3.72381
$663$	−6.14282	−0.238567
$664$	75.0534	2.91264
$665$	4.78868	0.185697
$666$	31.6995	1.22833
$667$	−19.3310	−0.748500
$668$	121.064	4.68410
$669$	22.1156	0.855039
$670$	24.9960	0.965678
$671$	−0.983558	−0.0379698
$672$	32.7210	1.26224
$673$	12.1394	0.467940	0.233970	−	0.972244i	$-0.424828\pi$
$673$	12.1394	0.467940	0.233970	+	0.972244i	$0.424828\pi$
$674$	−47.7243	−1.83827
$675$	5.17843	0.199318
$676$	−55.4367	−2.13218
$677$	3.56329	0.136948	0.0684742	−	0.997653i	$-0.478187\pi$
$677$	3.56329	0.136948	0.0684742	+	0.997653i	$0.478187\pi$
$678$	−36.1823	−1.38957
$679$	5.53529	0.212425
$680$	−33.2472	−1.27497
$681$	−0.918579	−0.0352000
$682$	−26.4113	−1.01134
$683$	15.2019	0.581685	0.290842	−	0.956771i	$-0.406064\pi$
$683$	15.2019	0.581685	0.290842	+	0.956771i	$0.406064\pi$
$684$	52.5181	2.00808
$685$	2.20727	0.0843354
$686$	−2.80513	−0.107100
$687$	7.33743	0.279941
$688$	−110.730	−4.22155
$689$	18.6389	0.710085
$690$	−13.2378	−0.503954
$691$	15.1722	0.577179	0.288589	−	0.957453i	$-0.406814\pi$
$691$	15.1722	0.577179	0.288589	+	0.957453i	$0.406814\pi$
$692$	−87.6102	−3.33044
$693$	−1.86874	−0.0709874
$694$	−75.6606	−2.87204
$695$	10.9909	0.416907
$696$	50.2895	1.90622
$697$	5.14046	0.194709
$698$	−3.58767	−0.135795
$699$	−3.41017	−0.128984
$700$	−5.86874	−0.221817
$701$	38.5126	1.45460	0.727301	−	0.686318i	$-0.240774\pi$
$701$	38.5126	1.45460	0.727301	+	0.686318i	$0.240774\pi$
$702$	−27.3844	−1.03356
$703$	28.9580	1.09217
$704$	−48.8882	−1.84254
$705$	12.3270	0.464261
$706$	23.4135	0.881178
$707$	−14.6359	−0.550439
$708$	−66.6585	−2.50518
$709$	−26.3590	−0.989931	−0.494966	−	0.868913i	$-0.664819\pi$
$709$	−26.3590	−0.989931	−0.494966	+	0.868913i	$0.664819\pi$
$710$	22.4410	0.842196
$711$	−1.68489	−0.0631883
$712$	−8.91590	−0.334138
$713$	41.7752	1.56449
$714$	−9.14046	−0.342073
$715$	1.88518	0.0705016
$716$	−82.1148	−3.06877
$717$	−6.50613	−0.242976
$718$	39.3556	1.46874
$719$	−15.7529	−0.587483	−0.293741	−	0.955885i	$-0.594900\pi$
$719$	−15.7529	−0.587483	−0.293741	+	0.955885i	$0.594900\pi$
$720$	−34.9539	−1.30266
$721$	8.02968	0.299041
$722$	11.0284	0.410433
$723$	13.8892	0.516543
$724$	97.9272	3.63944
$725$	−4.35686	−0.161810
$726$	−2.98356	−0.110730
$727$	−9.41780	−0.349287	−0.174643	−	0.984632i	$-0.555877\pi$
$727$	−9.41780	−0.349287	−0.174643	+	0.984632i	$0.555877\pi$
$728$	20.4585	0.758243
$729$	16.3396	0.605172
$730$	3.98460	0.147477
$731$	18.1364	0.670799
$732$	−6.13941	−0.226919
$733$	−30.1526	−1.11371	−0.556855	−	0.830609i	$-0.687992\pi$
$733$	−30.1526	−1.11371	−0.556855	+	0.830609i	$0.687992\pi$
$734$	−4.09204	−0.151040
$735$	−1.06361	−0.0392318
$736$	136.498	5.03138
$737$	−8.91081	−0.328234
$738$	8.79566	0.323773
$739$	−7.47494	−0.274970	−0.137485	−	0.990504i	$-0.543902\pi$
$739$	−7.47494	−0.274970	−0.137485	+	0.990504i	$0.543902\pi$
$740$	−35.4892	−1.30461
$741$	−9.60175	−0.352729
$742$	27.7345	1.01816
$743$	22.5170	0.826067	0.413034	−	0.910716i	$-0.364469\pi$
$743$	22.5170	0.826067	0.413034	+	0.910716i	$0.364469\pi$
$744$	−108.678	−3.98433
$745$	−17.0852	−0.625953
$746$	−97.3546	−3.56441
$747$	−12.9240	−0.472864
$748$	17.9795	0.657396
$749$	5.48304	0.200346
$750$	−2.98356	−0.108944
$751$	30.3505	1.10750	0.553752	−	0.832682i	$-0.313196\pi$
$751$	30.3505	1.10750	0.553752	+	0.832682i	$0.313196\pi$
$752$	−216.782	−7.90522
$753$	31.2429	1.13855
$754$	23.0398	0.839060
$755$	−10.2493	−0.373012
$756$	−30.3908	−1.10530
$757$	−15.5078	−0.563642	−0.281821	−	0.959467i	$-0.590938\pi$
$757$	−15.5078	−0.563642	−0.281821	+	0.959467i	$0.590938\pi$
$758$	45.1752	1.64084
$759$	4.71914	0.171294
$760$	−51.9682	−1.88509
$761$	−14.9097	−0.540476	−0.270238	−	0.962794i	$-0.587102\pi$
$761$	−14.9097	−0.540476	−0.270238	+	0.962794i	$0.587102\pi$
$762$	10.7714	0.390207
$763$	−0.127218	−0.00460561
$764$	76.8411	2.78001
$765$	5.72508	0.206991
$766$	50.1323	1.81135
$767$	−20.1317	−0.726914
$768$	−121.589	−4.38745
$769$	−7.38040	−0.266144	−0.133072	−	0.991106i	$-0.542484\pi$
$769$	−7.38040	−0.266144	−0.133072	+	0.991106i	$0.542484\pi$
$770$	2.80513	0.101090
$771$	8.04958	0.289899
$772$	−16.5292	−0.594897
$773$	54.6921	1.96714	0.983570	−	0.180530i	$-0.0577812\pi$
$773$	54.6921	1.96714	0.983570	+	0.180530i	$0.0577812\pi$
$774$	31.0326	1.11544
$775$	9.41538	0.338210
$776$	−60.0706	−2.15641
$777$	−6.43182	−0.230740
$778$	−9.86847	−0.353802
$779$	8.03498	0.287883
$780$	11.7674	0.421339
$781$	−8.00000	−0.286263
$782$	−38.1300	−1.36353
$783$	−22.5617	−0.806290
$784$	18.7046	0.668021
$785$	9.56009	0.341214
$786$	−47.5407	−1.69572
$787$	−19.8647	−0.708100	−0.354050	−	0.935227i	$-0.615196\pi$
$787$	−19.8647	−0.708100	−0.354050	+	0.935227i	$0.615196\pi$
$788$	−31.8542	−1.13476
$789$	−16.8738	−0.600724
$790$	2.52916	0.0899834
$791$	−12.1272	−0.431194
$792$	20.2801	0.720621
$793$	−1.85418	−0.0658439
$794$	−65.9272	−2.33967
$795$	10.5160	0.372963
$796$	89.4278	3.16968
$797$	−1.72619	−0.0611447	−0.0305724	−	0.999533i	$-0.509733\pi$
$797$	−1.72619	−0.0611447	−0.0305724	+	0.999533i	$0.509733\pi$
$798$	−14.2873	−0.505766
$799$	35.5065	1.25613
$800$	30.7642	1.08768
$801$	1.53529	0.0542470
$802$	92.0081	3.24892
$803$	−1.42047	−0.0501274
$804$	−55.6216	−1.96162
$805$	−4.43691	−0.156381
$806$	−49.7901	−1.75378
$807$	−17.8480	−0.628279
$808$	158.833	5.58772
$809$	−22.7531	−0.799957	−0.399979	−	0.916524i	$-0.630982\pi$
$809$	−22.7531	−0.799957	−0.399979	+	0.916524i	$0.630982\pi$
$810$	0.275970	0.00969660
$811$	−0.618807	−0.0217292	−0.0108646	−	0.999941i	$-0.503458\pi$
$811$	−0.618807	−0.0217292	−0.0108646	+	0.999941i	$0.503458\pi$
$812$	25.5693	0.897306
$813$	17.6221	0.618035
$814$	16.9631	0.594555
$815$	−0.572224	−0.0200441
$816$	60.9486	2.13363
$817$	28.3488	0.991798
$818$	−47.1634	−1.64903
$819$	−3.52290	−0.123100
$820$	−9.84720	−0.343879
$821$	−16.2317	−0.566492	−0.283246	−	0.959047i	$-0.591411\pi$
$821$	−16.2317	−0.566492	−0.283246	+	0.959047i	$0.591411\pi$
$822$	−6.58552	−0.229696
$823$	22.9027	0.798339	0.399169	−	0.916877i	$-0.369299\pi$
$823$	22.9027	0.798339	0.399169	+	0.916877i	$0.369299\pi$
$824$	−87.1404	−3.03568
$825$	1.06361	0.0370301
$826$	−29.9558	−1.04230
$827$	3.70217	0.128737	0.0643686	−	0.997926i	$-0.479497\pi$
$827$	3.70217	0.128737	0.0643686	+	0.997926i	$0.479497\pi$
$828$	−48.6602	−1.69106
$829$	29.7294	1.03254	0.516272	−	0.856425i	$-0.327319\pi$
$829$	29.7294	1.03254	0.516272	+	0.856425i	$0.327319\pi$
$830$	19.4000	0.673383
$831$	9.90076	0.343454
$832$	−92.1629	−3.19517
$833$	−3.06361	−0.106148
$834$	−32.7919	−1.13549
$835$	20.6286	0.713882
$836$	28.1035	0.971981
$837$	48.7569	1.68529
$838$	−4.82477	−0.166669
$839$	50.7539	1.75222	0.876110	−	0.482112i	$-0.160130\pi$
$839$	50.7539	1.75222	0.876110	+	0.482112i	$0.160130\pi$
$840$	11.5426	0.398257
$841$	−10.0177	−0.345439
$842$	−71.3539	−2.45902
$843$	−27.7542	−0.955904
$844$	66.6443	2.29399
$845$	−9.44611	−0.324956
$846$	60.7540	2.08877
$847$	−1.00000	−0.0343604
$848$	−184.933	−6.35064
$849$	32.7483	1.12392
$850$	−8.59381	−0.294765
$851$	−26.8308	−0.919747
$852$	−49.9363	−1.71079
$853$	33.2696	1.13913	0.569564	−	0.821947i	$-0.307112\pi$
$853$	33.2696	1.13913	0.569564	+	0.821947i	$0.307112\pi$
$854$	−2.75901	−0.0944112
$855$	8.94879	0.306042
$856$	−59.5035	−2.03379
$857$	−42.4874	−1.45134	−0.725671	−	0.688042i	$-0.758470\pi$
$857$	−42.4874	−1.45134	−0.725671	+	0.688042i	$0.758470\pi$
$858$	−5.62454	−0.192019
$859$	24.8903	0.849247	0.424623	−	0.905370i	$-0.360406\pi$
$859$	24.8903	0.849247	0.424623	+	0.905370i	$0.360406\pi$
$860$	−34.7426	−1.18471
$861$	−1.78464	−0.0608203
$862$	−0.607200	−0.0206813
$863$	−32.0246	−1.09013	−0.545065	−	0.838394i	$-0.683495\pi$
$863$	−32.0246	−1.09013	−0.545065	+	0.838394i	$0.683495\pi$
$864$	159.310	5.41984
$865$	−14.9283	−0.507577
$866$	21.8027	0.740886
$867$	8.09864	0.275044
$868$	−55.2564	−1.87552
$869$	−0.901619	−0.0305853
$870$	12.9990	0.440706
$871$	−16.7985	−0.569194
$872$	1.38061	0.0467533
$873$	10.3440	0.350091
$874$	−59.6005	−2.01602
$875$	−1.00000	−0.0338062
$876$	−8.86664	−0.299576
$877$	−0.284904	−0.00962054	−0.00481027	−	0.999988i	$-0.501531\pi$
$877$	−0.284904	−0.00962054	−0.00481027	+	0.999988i	$0.501531\pi$
$878$	−88.4863	−2.98627
$879$	26.3087	0.887371
$880$	−18.7046	−0.630532
$881$	−34.4135	−1.15942	−0.579711	−	0.814822i	$-0.696835\pi$
$881$	−34.4135	−1.15942	−0.579711	+	0.814822i	$0.696835\pi$
$882$	−5.24204	−0.176509
$883$	9.61435	0.323549	0.161774	−	0.986828i	$-0.448278\pi$
$883$	9.61435	0.323549	0.161774	+	0.986828i	$0.448278\pi$
$884$	33.8946	1.14000
$885$	−11.3582	−0.381803
$886$	72.4957	2.43554
$887$	57.0771	1.91646	0.958231	−	0.285996i	$-0.0923244\pi$
$887$	57.0771	1.91646	0.958231	+	0.285996i	$0.0923244\pi$
$888$	69.8000	2.34234
$889$	3.61025	0.121084
$890$	−2.30460	−0.0772505
$891$	−0.0983806	−0.00329587
$892$	−122.029	−4.08582
$893$	55.4997	1.85723
$894$	50.9747	1.70485
$895$	−13.9919	−0.467698
$896$	−75.6092	−2.52593
$897$	8.89642	0.297043
$898$	−10.4061	−0.347257
$899$	−41.0215	−1.36814
$900$	−10.9671	−0.365571
$901$	30.2901	1.00911
$902$	4.70675	0.156718
$903$	−6.29651	−0.209535
$904$	131.608	4.37722
$905$	16.6863	0.554670
$906$	30.5795	1.01594
$907$	0.387331	0.0128611	0.00643056	−	0.999979i	$-0.497953\pi$
$907$	0.387331	0.0128611	0.00643056	+	0.999979i	$0.497953\pi$
$908$	5.06850	0.168204
$909$	−27.3506	−0.907162
$910$	5.28816	0.175301
$911$	3.47494	0.115130	0.0575650	−	0.998342i	$-0.481666\pi$
$911$	3.47494	0.115130	0.0575650	+	0.998342i	$0.481666\pi$
$912$	95.2679	3.15463
$913$	−6.91590	−0.228883
$914$	68.6924	2.27214
$915$	−1.04612	−0.0345837
$916$	−40.4862	−1.33770
$917$	−15.9342	−0.526195
$918$	−44.5025	−1.46880
$919$	36.6280	1.20825	0.604123	−	0.796891i	$-0.293524\pi$
$919$	36.6280	1.20825	0.604123	+	0.796891i	$0.293524\pi$
$920$	48.1507	1.58748
$921$	−10.2663	−0.338286
$922$	−73.5542	−2.42238
$923$	−15.0814	−0.496411
$924$	−6.24204	−0.205348
$925$	−6.04717	−0.198830
$926$	75.4941	2.48089
$927$	15.0053	0.492840
$928$	−134.035	−4.39992
$929$	2.32940	0.0764250	0.0382125	−	0.999270i	$-0.487834\pi$
$929$	2.32940	0.0764250	0.0382125	+	0.999270i	$0.487834\pi$
$930$	−28.0913	−0.921151
$931$	−4.78868	−0.156943
$932$	18.8165	0.616354
$933$	−6.02069	−0.197109
$934$	−111.989	−3.66439
$935$	3.06361	0.100191
$936$	38.2315	1.24964
$937$	32.5558	1.06355	0.531776	−	0.846885i	$-0.321525\pi$
$937$	32.5558	1.06355	0.531776	+	0.846885i	$0.321525\pi$
$938$	−24.9960	−0.816147
$939$	−8.71815	−0.284506
$940$	−68.0173	−2.21848
$941$	−51.3096	−1.67265	−0.836323	−	0.548237i	$-0.815299\pi$
$941$	−51.3096	−1.67265	−0.836323	+	0.548237i	$0.815299\pi$
$942$	−28.5231	−0.929332
$943$	−7.44474	−0.242434
$944$	199.745	6.50116
$945$	−5.17843	−0.168454
$946$	16.6062	0.539914
$947$	−34.9621	−1.13611	−0.568057	−	0.822989i	$-0.692305\pi$
$947$	−34.9621	−1.13611	−0.568057	+	0.822989i	$0.692305\pi$
$948$	−5.62794	−0.182787
$949$	−2.67784	−0.0869264
$950$	−13.4329	−0.435820
$951$	−6.22220	−0.201769
$952$	33.2472	1.07755
$953$	11.5896	0.375423	0.187711	−	0.982224i	$-0.439893\pi$
$953$	11.5896	0.375423	0.187711	+	0.982224i	$0.439893\pi$
$954$	51.8284	1.67801
$955$	13.0933	0.423689
$956$	35.8993	1.16106
$957$	−4.63400	−0.149796
$958$	87.8163	2.83722
$959$	−2.20727	−0.0712764
$960$	−51.9979	−1.67822
$961$	57.6494	1.85966
$962$	31.9784	1.03102
$963$	10.2463	0.330184
$964$	−76.6370	−2.46831
$965$	−2.81648	−0.0906656
$966$	13.2378	0.425919
$967$	−58.3469	−1.87631	−0.938154	−	0.346217i	$-0.887466\pi$
$967$	−58.3469	−1.87631	−0.938154	+	0.346217i	$0.887466\pi$
$968$	10.8523	0.348806
$969$	−15.6038	−0.501268
$970$	−15.5272	−0.498548
$971$	−29.3302	−0.941252	−0.470626	−	0.882333i	$-0.655972\pi$
$971$	−29.3302	−0.941252	−0.470626	+	0.882333i	$0.655972\pi$
$972$	−91.7866	−2.94406
$973$	−10.9909	−0.352351
$974$	−52.2698	−1.67483
$975$	2.00509	0.0642143
$976$	18.3971	0.588875
$977$	18.5186	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$977$	18.5186	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$978$	1.70726	0.0545923
$979$	0.821569	0.0262574
$980$	5.86874	0.187470
$981$	−0.237737	−0.00759036
$982$	94.9805	3.03095
$983$	10.2916	0.328252	0.164126	−	0.986439i	$-0.447520\pi$
$983$	10.2916	0.328252	0.164126	+	0.986439i	$0.447520\pi$
$984$	19.3674	0.617411
$985$	−5.42778	−0.172943
$986$	37.4421	1.19240
$987$	−12.3270	−0.392372
$988$	52.9801	1.68552
$989$	−26.2663	−0.835220
$990$	5.24204	0.166603
$991$	−10.5418	−0.334870	−0.167435	−	0.985883i	$-0.553548\pi$
$991$	−10.5418	−0.334870	−0.167435	+	0.985883i	$0.553548\pi$
$992$	289.656	9.19659
$993$	36.3284	1.15285
$994$	−22.4410	−0.711786
$995$	15.2380	0.483077
$996$	−43.1693	−1.36787
$997$	21.4205	0.678393	0.339197	−	0.940716i	$-0.389845\pi$
$997$	21.4205	0.678393	0.339197	+	0.940716i	$0.389845\pi$
$998$	−25.6059	−0.810542
$999$	−31.3148	−0.990758

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	385.2.a.h.1.4	✓	4
3.2	odd	2		3465.2.a.bk.1.1		4
4.3	odd	2		6160.2.a.br.1.3		4
5.2	odd	4		1925.2.b.p.1849.8		8
5.3	odd	4		1925.2.b.p.1849.1		8
5.4	even	2		1925.2.a.x.1.1		4
7.6	odd	2		2695.2.a.l.1.4		4
11.10	odd	2		4235.2.a.r.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.a.h.1.4	✓	4	1.1	even	1	trivial
1925.2.a.x.1.1		4	5.4	even	2
1925.2.b.p.1849.1		8	5.3	odd	4
1925.2.b.p.1849.8		8	5.2	odd	4
2695.2.a.l.1.4		4	7.6	odd	2
3465.2.a.bk.1.1		4	3.2	odd	2
4235.2.a.r.1.1		4	11.10	odd	2
6160.2.a.br.1.3		4	4.3	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:	\( N \)	\(=\)	\( 385 = 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	385.a (trivial)

\(f(q)\)	\(=\)	\(q+2.80513 q^{2} -1.06361 q^{3} +5.86874 q^{4} +1.00000 q^{5} -2.98356 q^{6} -1.00000 q^{7} +10.8523 q^{8} -1.86874 q^{9} +O(q^{10})\)	\(q+2.80513 q^{2} -1.06361 q^{3} +5.86874 q^{4} +1.00000 q^{5} -2.98356 q^{6} -1.00000 q^{7} +10.8523 q^{8} -1.86874 q^{9} +2.80513 q^{10} -1.00000 q^{11} -6.24204 q^{12} -1.88518 q^{13} -2.80513 q^{14} -1.06361 q^{15} +18.7046 q^{16} -3.06361 q^{17} -5.24204 q^{18} -4.78868 q^{19} +5.86874 q^{20} +1.06361 q^{21} -2.80513 q^{22} +4.43691 q^{23} -11.5426 q^{24} +1.00000 q^{25} -5.28816 q^{26} +5.17843 q^{27} -5.86874 q^{28} -4.35686 q^{29} -2.98356 q^{30} +9.41538 q^{31} +30.7642 q^{32} +1.06361 q^{33} -8.59381 q^{34} -1.00000 q^{35} -10.9671 q^{36} -6.04717 q^{37} -13.4329 q^{38} +2.00509 q^{39} +10.8523 q^{40} -1.67791 q^{41} +2.98356 q^{42} -5.91995 q^{43} -5.86874 q^{44} -1.86874 q^{45} +12.4461 q^{46} -11.5898 q^{47} -19.8944 q^{48} +1.00000 q^{49} +2.80513 q^{50} +3.25848 q^{51} -11.0636 q^{52} -9.88707 q^{53} +14.5262 q^{54} -1.00000 q^{55} -10.8523 q^{56} +5.09329 q^{57} -12.2216 q^{58} +10.6790 q^{59} -6.24204 q^{60} +0.983558 q^{61} +26.4113 q^{62} +1.86874 q^{63} +48.8882 q^{64} -1.88518 q^{65} +2.98356 q^{66} +8.91081 q^{67} -17.9795 q^{68} -4.71914 q^{69} -2.80513 q^{70} +8.00000 q^{71} -20.2801 q^{72} +1.42047 q^{73} -16.9631 q^{74} -1.06361 q^{75} -28.1035 q^{76} +1.00000 q^{77} +5.62454 q^{78} +0.901619 q^{79} +18.7046 q^{80} +0.0983806 q^{81} -4.70675 q^{82} +6.91590 q^{83} +6.24204 q^{84} -3.06361 q^{85} -16.6062 q^{86} +4.63400 q^{87} -10.8523 q^{88} -0.821569 q^{89} -5.24204 q^{90} +1.88518 q^{91} +26.0391 q^{92} -10.0143 q^{93} -32.5108 q^{94} -4.78868 q^{95} -32.7210 q^{96} -5.53529 q^{97} +2.80513 q^{98} +1.86874 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 2 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 + 2 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 + 12 * q^8 + 8 * q^9	\( 4 q + 2 q^{2} + 2 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} + 12 q^{8} + 8 q^{9} + 2 q^{10} - 4 q^{11} - 12 q^{12} - 8 q^{13} - 2 q^{14} + 2 q^{15} + 12 q^{16} - 6 q^{17} - 8 q^{18} + 6 q^{19} + 8 q^{20} - 2 q^{21} - 2 q^{22} + 14 q^{23} - 6 q^{24} + 4 q^{25} - 6 q^{26} + 14 q^{27} - 8 q^{28} - 4 q^{29} + 4 q^{30} + 10 q^{31} + 26 q^{32} - 2 q^{33} - 4 q^{35} - 12 q^{36} - 2 q^{37} - 22 q^{38} + 16 q^{39} + 12 q^{40} - 10 q^{41} - 4 q^{42} - 14 q^{43} - 8 q^{44} + 8 q^{45} - 16 q^{46} + 16 q^{47} - 18 q^{48} + 4 q^{49} + 2 q^{50} + 16 q^{51} - 38 q^{52} + 2 q^{53} + 2 q^{54} - 4 q^{55} - 12 q^{56} - 4 q^{57} + 8 q^{58} + 26 q^{59} - 12 q^{60} - 12 q^{61} + 50 q^{62} - 8 q^{63} + 36 q^{64} - 8 q^{65} - 4 q^{66} - 10 q^{67} - 28 q^{68} - 14 q^{69} - 2 q^{70} + 32 q^{71} - 10 q^{72} - 14 q^{73} - 8 q^{74} + 2 q^{75} - 6 q^{76} + 4 q^{77} - 50 q^{78} + 20 q^{79} + 12 q^{80} - 16 q^{81} - 26 q^{82} - 10 q^{83} + 12 q^{84} - 6 q^{85} - 20 q^{86} - 4 q^{87} - 12 q^{88} - 10 q^{89} - 8 q^{90} + 8 q^{91} + 26 q^{92} + 14 q^{93} - 38 q^{94} + 6 q^{95} - 84 q^{96} - 2 q^{97} + 2 q^{98} - 8 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 2 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 + 12 * q^8 + 8 * q^9 + 2 * q^10 - 4 * q^11 - 12 * q^12 - 8 * q^13 - 2 * q^14 + 2 * q^15 + 12 * q^16 - 6 * q^17 - 8 * q^18 + 6 * q^19 + 8 * q^20 - 2 * q^21 - 2 * q^22 + 14 * q^23 - 6 * q^24 + 4 * q^25 - 6 * q^26 + 14 * q^27 - 8 * q^28 - 4 * q^29 + 4 * q^30 + 10 * q^31 + 26 * q^32 - 2 * q^33 - 4 * q^35 - 12 * q^36 - 2 * q^37 - 22 * q^38 + 16 * q^39 + 12 * q^40 - 10 * q^41 - 4 * q^42 - 14 * q^43 - 8 * q^44 + 8 * q^45 - 16 * q^46 + 16 * q^47 - 18 * q^48 + 4 * q^49 + 2 * q^50 + 16 * q^51 - 38 * q^52 + 2 * q^53 + 2 * q^54 - 4 * q^55 - 12 * q^56 - 4 * q^57 + 8 * q^58 + 26 * q^59 - 12 * q^60 - 12 * q^61 + 50 * q^62 - 8 * q^63 + 36 * q^64 - 8 * q^65 - 4 * q^66 - 10 * q^67 - 28 * q^68 - 14 * q^69 - 2 * q^70 + 32 * q^71 - 10 * q^72 - 14 * q^73 - 8 * q^74 + 2 * q^75 - 6 * q^76 + 4 * q^77 - 50 * q^78 + 20 * q^79 + 12 * q^80 - 16 * q^81 - 26 * q^82 - 10 * q^83 + 12 * q^84 - 6 * q^85 - 20 * q^86 - 4 * q^87 - 12 * q^88 - 10 * q^89 - 8 * q^90 + 8 * q^91 + 26 * q^92 + 14 * q^93 - 38 * q^94 + 6 * q^95 - 84 * q^96 - 2 * q^97 + 2 * q^98 - 8 * q^99

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