LMFDB - Embedded newform 177.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,2,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, 2, names="a")

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

chi := DirichletCharacter("177.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
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:	$1.41335211578$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} -3.00000 q^{5} -0.381966 q^{6} -4.61803 q^{7} +1.47214 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} -3.00000 q^{5} -0.381966 q^{6} -4.61803 q^{7} +1.47214 q^{8} +1.00000 q^{9} +1.14590 q^{10} -5.47214 q^{11} -1.85410 q^{12} +6.70820 q^{13} +1.76393 q^{14} -3.00000 q^{15} +3.14590 q^{16} -0.381966 q^{17} -0.381966 q^{18} +0.854102 q^{19} +5.56231 q^{20} -4.61803 q^{21} +2.09017 q^{22} -3.61803 q^{23} +1.47214 q^{24} +4.00000 q^{25} -2.56231 q^{26} +1.00000 q^{27} +8.56231 q^{28} -6.61803 q^{29} +1.14590 q^{30} -4.61803 q^{31} -4.14590 q^{32} -5.47214 q^{33} +0.145898 q^{34} +13.8541 q^{35} -1.85410 q^{36} -6.09017 q^{37} -0.326238 q^{38} +6.70820 q^{39} -4.41641 q^{40} +5.09017 q^{41} +1.76393 q^{42} -0.708204 q^{43} +10.1459 q^{44} -3.00000 q^{45} +1.38197 q^{46} +1.85410 q^{47} +3.14590 q^{48} +14.3262 q^{49} -1.52786 q^{50} -0.381966 q^{51} -12.4377 q^{52} -3.47214 q^{53} -0.381966 q^{54} +16.4164 q^{55} -6.79837 q^{56} +0.854102 q^{57} +2.52786 q^{58} +1.00000 q^{59} +5.56231 q^{60} +0.909830 q^{61} +1.76393 q^{62} -4.61803 q^{63} -4.70820 q^{64} -20.1246 q^{65} +2.09017 q^{66} +0.236068 q^{67} +0.708204 q^{68} -3.61803 q^{69} -5.29180 q^{70} -0.236068 q^{71} +1.47214 q^{72} -3.09017 q^{73} +2.32624 q^{74} +4.00000 q^{75} -1.58359 q^{76} +25.2705 q^{77} -2.56231 q^{78} -3.00000 q^{79} -9.43769 q^{80} +1.00000 q^{81} -1.94427 q^{82} -7.85410 q^{83} +8.56231 q^{84} +1.14590 q^{85} +0.270510 q^{86} -6.61803 q^{87} -8.05573 q^{88} -5.09017 q^{89} +1.14590 q^{90} -30.9787 q^{91} +6.70820 q^{92} -4.61803 q^{93} -0.708204 q^{94} -2.56231 q^{95} -4.14590 q^{96} -0.527864 q^{97} -5.47214 q^{98} -5.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 6 q^{5} - 3 q^{6} - 7 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 6 * q^5 - 3 * q^6 - 7 * q^7 - 6 * q^8 + 2 * q^9	\( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 6 q^{5} - 3 q^{6} - 7 q^{7} - 6 q^{8} + 2 q^{9} + 9 q^{10} - 2 q^{11} + 3 q^{12} + 8 q^{14} - 6 q^{15} + 13 q^{16} - 3 q^{17} - 3 q^{18} - 5 q^{19} - 9 q^{20} - 7 q^{21} - 7 q^{22} - 5 q^{23} - 6 q^{24} + 8 q^{25} + 15 q^{26} + 2 q^{27} - 3 q^{28} - 11 q^{29} + 9 q^{30} - 7 q^{31} - 15 q^{32} - 2 q^{33} + 7 q^{34} + 21 q^{35} + 3 q^{36} - q^{37} + 15 q^{38} + 18 q^{40} - q^{41} + 8 q^{42} + 12 q^{43} + 27 q^{44} - 6 q^{45} + 5 q^{46} - 3 q^{47} + 13 q^{48} + 13 q^{49} - 12 q^{50} - 3 q^{51} - 45 q^{52} + 2 q^{53} - 3 q^{54} + 6 q^{55} + 11 q^{56} - 5 q^{57} + 14 q^{58} + 2 q^{59} - 9 q^{60} + 13 q^{61} + 8 q^{62} - 7 q^{63} + 4 q^{64} - 7 q^{66} - 4 q^{67} - 12 q^{68} - 5 q^{69} - 24 q^{70} + 4 q^{71} - 6 q^{72} + 5 q^{73} - 11 q^{74} + 8 q^{75} - 30 q^{76} + 17 q^{77} + 15 q^{78} - 6 q^{79} - 39 q^{80} + 2 q^{81} + 14 q^{82} - 9 q^{83} - 3 q^{84} + 9 q^{85} - 33 q^{86} - 11 q^{87} - 34 q^{88} + q^{89} + 9 q^{90} - 15 q^{91} - 7 q^{93} + 12 q^{94} + 15 q^{95} - 15 q^{96} - 10 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 6 * q^5 - 3 * q^6 - 7 * q^7 - 6 * q^8 + 2 * q^9 + 9 * q^10 - 2 * q^11 + 3 * q^12 + 8 * q^14 - 6 * q^15 + 13 * q^16 - 3 * q^17 - 3 * q^18 - 5 * q^19 - 9 * q^20 - 7 * q^21 - 7 * q^22 - 5 * q^23 - 6 * q^24 + 8 * q^25 + 15 * q^26 + 2 * q^27 - 3 * q^28 - 11 * q^29 + 9 * q^30 - 7 * q^31 - 15 * q^32 - 2 * q^33 + 7 * q^34 + 21 * q^35 + 3 * q^36 - q^37 + 15 * q^38 + 18 * q^40 - q^41 + 8 * q^42 + 12 * q^43 + 27 * q^44 - 6 * q^45 + 5 * q^46 - 3 * q^47 + 13 * q^48 + 13 * q^49 - 12 * q^50 - 3 * q^51 - 45 * q^52 + 2 * q^53 - 3 * q^54 + 6 * q^55 + 11 * q^56 - 5 * q^57 + 14 * q^58 + 2 * q^59 - 9 * q^60 + 13 * q^61 + 8 * q^62 - 7 * q^63 + 4 * q^64 - 7 * q^66 - 4 * q^67 - 12 * q^68 - 5 * q^69 - 24 * q^70 + 4 * q^71 - 6 * q^72 + 5 * q^73 - 11 * q^74 + 8 * q^75 - 30 * q^76 + 17 * q^77 + 15 * q^78 - 6 * q^79 - 39 * q^80 + 2 * q^81 + 14 * q^82 - 9 * q^83 - 3 * q^84 + 9 * q^85 - 33 * q^86 - 11 * q^87 - 34 * q^88 + q^89 + 9 * q^90 - 15 * q^91 - 7 * q^93 + 12 * q^94 + 15 * q^95 - 15 * q^96 - 10 * q^97 - 2 * q^98 - 2 * 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 $

$2$	−0.381966	−0.270091	−0.135045	−	0.990839i	$-0.543118\pi$
$2$	−0.381966	−0.270091	−0.135045	+	0.990839i	$0.543118\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.85410	−0.927051
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	−0.381966	−0.155937
$7$	−4.61803	−1.74545	−0.872726	−	0.488210i	$-0.837650\pi$
$7$	−4.61803	−1.74545	−0.872726	+	0.488210i	$0.837650\pi$
$8$	1.47214	0.520479
$9$	1.00000	0.333333
$10$	1.14590	0.362365
$11$	−5.47214	−1.64991	−0.824956	−	0.565198i	$-0.808800\pi$
$11$	−5.47214	−1.64991	−0.824956	+	0.565198i	$0.808800\pi$
$12$	−1.85410	−0.535233
$13$	6.70820	1.86052	0.930261	−	0.366900i	$-0.119581\pi$
$13$	6.70820	1.86052	0.930261	+	0.366900i	$0.119581\pi$
$14$	1.76393	0.471431
$15$	−3.00000	−0.774597
$16$	3.14590	0.786475
$17$	−0.381966	−0.0926404	−0.0463202	−	0.998927i	$-0.514749\pi$
$17$	−0.381966	−0.0926404	−0.0463202	+	0.998927i	$0.514749\pi$
$18$	−0.381966	−0.0900303
$19$	0.854102	0.195944	0.0979722	−	0.995189i	$-0.468764\pi$
$19$	0.854102	0.195944	0.0979722	+	0.995189i	$0.468764\pi$
$20$	5.56231	1.24377
$21$	−4.61803	−1.00774
$22$	2.09017	0.445626
$23$	−3.61803	−0.754412	−0.377206	−	0.926129i	$-0.623115\pi$
$23$	−3.61803	−0.754412	−0.377206	+	0.926129i	$0.623115\pi$
$24$	1.47214	0.300498
$25$	4.00000	0.800000
$26$	−2.56231	−0.502510
$27$	1.00000	0.192450
$28$	8.56231	1.61812
$29$	−6.61803	−1.22894	−0.614469	−	0.788941i	$-0.710630\pi$
$29$	−6.61803	−1.22894	−0.614469	+	0.788941i	$0.710630\pi$
$30$	1.14590	0.209211
$31$	−4.61803	−0.829423	−0.414712	−	0.909953i	$-0.636118\pi$
$31$	−4.61803	−0.829423	−0.414712	+	0.909953i	$0.636118\pi$
$32$	−4.14590	−0.732898
$33$	−5.47214	−0.952577
$34$	0.145898	0.0250213
$35$	13.8541	2.34177
$36$	−1.85410	−0.309017
$37$	−6.09017	−1.00122	−0.500609	−	0.865674i	$-0.666891\pi$
$37$	−6.09017	−1.00122	−0.500609	+	0.865674i	$0.666891\pi$
$38$	−0.326238	−0.0529228
$39$	6.70820	1.07417
$40$	−4.41641	−0.698295
$41$	5.09017	0.794951	0.397475	−	0.917613i	$-0.369886\pi$
$41$	5.09017	0.794951	0.397475	+	0.917613i	$0.369886\pi$
$42$	1.76393	0.272181
$43$	−0.708204	−0.108000	−0.0540000	−	0.998541i	$-0.517197\pi$
$43$	−0.708204	−0.108000	−0.0540000	+	0.998541i	$0.517197\pi$
$44$	10.1459	1.52955
$45$	−3.00000	−0.447214
$46$	1.38197	0.203760
$47$	1.85410	0.270449	0.135224	−	0.990815i	$-0.456825\pi$
$47$	1.85410	0.270449	0.135224	+	0.990815i	$0.456825\pi$
$48$	3.14590	0.454071
$49$	14.3262	2.04661
$50$	−1.52786	−0.216073
$51$	−0.381966	−0.0534859
$52$	−12.4377	−1.72480
$53$	−3.47214	−0.476935	−0.238467	−	0.971151i	$-0.576645\pi$
$53$	−3.47214	−0.476935	−0.238467	+	0.971151i	$0.576645\pi$
$54$	−0.381966	−0.0519790
$55$	16.4164	2.21359
$56$	−6.79837	−0.908471
$57$	0.854102	0.113129
$58$	2.52786	0.331925
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	5.56231	0.718091
$61$	0.909830	0.116492	0.0582459	−	0.998302i	$-0.481449\pi$
$61$	0.909830	0.116492	0.0582459	+	0.998302i	$0.481449\pi$
$62$	1.76393	0.224020
$63$	−4.61803	−0.581818
$64$	−4.70820	−0.588525
$65$	−20.1246	−2.49615
$66$	2.09017	0.257282
$67$	0.236068	0.0288403	0.0144201	−	0.999896i	$-0.495410\pi$
$67$	0.236068	0.0288403	0.0144201	+	0.999896i	$0.495410\pi$
$68$	0.708204	0.0858823
$69$	−3.61803	−0.435560
$70$	−5.29180	−0.632491
$71$	−0.236068	−0.0280161	−0.0140081	−	0.999902i	$-0.504459\pi$
$71$	−0.236068	−0.0280161	−0.0140081	+	0.999902i	$0.504459\pi$
$72$	1.47214	0.173493
$73$	−3.09017	−0.361677	−0.180839	−	0.983513i	$-0.557881\pi$
$73$	−3.09017	−0.361677	−0.180839	+	0.983513i	$0.557881\pi$
$74$	2.32624	0.270420
$75$	4.00000	0.461880
$76$	−1.58359	−0.181650
$77$	25.2705	2.87984
$78$	−2.56231	−0.290124
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	−9.43769	−1.05517
$81$	1.00000	0.111111
$82$	−1.94427	−0.214709
$83$	−7.85410	−0.862100	−0.431050	−	0.902328i	$-0.641857\pi$
$83$	−7.85410	−0.862100	−0.431050	+	0.902328i	$0.641857\pi$
$84$	8.56231	0.934224
$85$	1.14590	0.124290
$86$	0.270510	0.0291698
$87$	−6.61803	−0.709528
$88$	−8.05573	−0.858743
$89$	−5.09017	−0.539557	−0.269778	−	0.962922i	$-0.586950\pi$
$89$	−5.09017	−0.539557	−0.269778	+	0.962922i	$0.586950\pi$
$90$	1.14590	0.120788
$91$	−30.9787	−3.24745
$92$	6.70820	0.699379
$93$	−4.61803	−0.478868
$94$	−0.708204	−0.0730457
$95$	−2.56231	−0.262887
$96$	−4.14590	−0.423139
$97$	−0.527864	−0.0535965	−0.0267982	−	0.999641i	$-0.508531\pi$
$97$	−0.527864	−0.0535965	−0.0267982	+	0.999641i	$0.508531\pi$
$98$	−5.47214	−0.552769
$99$	−5.47214	−0.549970
$100$	−7.41641	−0.741641
$101$	−11.2361	−1.11803	−0.559015	−	0.829157i	$-0.688821\pi$
$101$	−11.2361	−1.11803	−0.559015	+	0.829157i	$0.688821\pi$
$102$	0.145898	0.0144461
$103$	−14.7639	−1.45473	−0.727367	−	0.686249i	$-0.759256\pi$
$103$	−14.7639	−1.45473	−0.727367	+	0.686249i	$0.759256\pi$
$104$	9.87539	0.968361
$105$	13.8541	1.35202
$106$	1.32624	0.128816
$107$	19.0344	1.84013	0.920064	−	0.391767i	$-0.128136\pi$
$107$	19.0344	1.84013	0.920064	+	0.391767i	$0.128136\pi$
$108$	−1.85410	−0.178411
$109$	6.85410	0.656504	0.328252	−	0.944590i	$-0.393540\pi$
$109$	6.85410	0.656504	0.328252	+	0.944590i	$0.393540\pi$
$110$	−6.27051	−0.597870
$111$	−6.09017	−0.578053
$112$	−14.5279	−1.37275
$113$	10.2361	0.962928	0.481464	−	0.876466i	$-0.340105\pi$
$113$	10.2361	0.962928	0.481464	+	0.876466i	$0.340105\pi$
$114$	−0.326238	−0.0305550
$115$	10.8541	1.01215
$116$	12.2705	1.13929
$117$	6.70820	0.620174
$118$	−0.381966	−0.0351628
$119$	1.76393	0.161699
$120$	−4.41641	−0.403161
$121$	18.9443	1.72221
$122$	−0.347524	−0.0314634
$123$	5.09017	0.458965
$124$	8.56231	0.768918
$125$	3.00000	0.268328
$126$	1.76393	0.157144
$127$	−11.4721	−1.01799	−0.508994	−	0.860770i	$-0.669982\pi$
$127$	−11.4721	−1.01799	−0.508994	+	0.860770i	$0.669982\pi$
$128$	10.0902	0.891853
$129$	−0.708204	−0.0623539
$130$	7.68692	0.674187
$131$	11.2361	0.981700	0.490850	−	0.871244i	$-0.336686\pi$
$131$	11.2361	0.981700	0.490850	+	0.871244i	$0.336686\pi$
$132$	10.1459	0.883087
$133$	−3.94427	−0.342012
$134$	−0.0901699	−0.00778950
$135$	−3.00000	−0.258199
$136$	−0.562306	−0.0482173
$137$	−13.4721	−1.15100	−0.575501	−	0.817801i	$-0.695193\pi$
$137$	−13.4721	−1.15100	−0.575501	+	0.817801i	$0.695193\pi$
$138$	1.38197	0.117641
$139$	−5.29180	−0.448844	−0.224422	−	0.974492i	$-0.572049\pi$
$139$	−5.29180	−0.448844	−0.224422	+	0.974492i	$0.572049\pi$
$140$	−25.6869	−2.17094
$141$	1.85410	0.156144
$142$	0.0901699	0.00756689
$143$	−36.7082	−3.06969
$144$	3.14590	0.262158
$145$	19.8541	1.64879
$146$	1.18034	0.0976856
$147$	14.3262	1.18161
$148$	11.2918	0.928180
$149$	7.03444	0.576284	0.288142	−	0.957588i	$-0.406962\pi$
$149$	7.03444	0.576284	0.288142	+	0.957588i	$0.406962\pi$
$150$	−1.52786	−0.124750
$151$	2.14590	0.174631	0.0873154	−	0.996181i	$-0.472171\pi$
$151$	2.14590	0.174631	0.0873154	+	0.996181i	$0.472171\pi$
$152$	1.25735	0.101985
$153$	−0.381966	−0.0308801
$154$	−9.65248	−0.777819
$155$	13.8541	1.11279
$156$	−12.4377	−0.995812
$157$	−9.94427	−0.793639	−0.396820	−	0.917897i	$-0.629886\pi$
$157$	−9.94427	−0.793639	−0.396820	+	0.917897i	$0.629886\pi$
$158$	1.14590	0.0911628
$159$	−3.47214	−0.275358
$160$	12.4377	0.983286
$161$	16.7082	1.31679
$162$	−0.381966	−0.0300101
$163$	−7.38197	−0.578200	−0.289100	−	0.957299i	$-0.593356\pi$
$163$	−7.38197	−0.578200	−0.289100	+	0.957299i	$0.593356\pi$
$164$	−9.43769	−0.736960
$165$	16.4164	1.27802
$166$	3.00000	0.232845
$167$	−20.3262	−1.57289	−0.786446	−	0.617659i	$-0.788081\pi$
$167$	−20.3262	−1.57289	−0.786446	+	0.617659i	$0.788081\pi$
$168$	−6.79837	−0.524506
$169$	32.0000	2.46154
$170$	−0.437694	−0.0335696
$171$	0.854102	0.0653148
$172$	1.31308	0.100122
$173$	16.6180	1.26345	0.631723	−	0.775194i	$-0.282348\pi$
$173$	16.6180	1.26345	0.631723	+	0.775194i	$0.282348\pi$
$174$	2.52786	0.191637
$175$	−18.4721	−1.39636
$176$	−17.2148	−1.29761
$177$	1.00000	0.0751646
$178$	1.94427	0.145729
$179$	9.18034	0.686171	0.343085	−	0.939304i	$-0.388528\pi$
$179$	9.18034	0.686171	0.343085	+	0.939304i	$0.388528\pi$
$180$	5.56231	0.414590
$181$	11.5623	0.859419	0.429710	−	0.902967i	$-0.358616\pi$
$181$	11.5623	0.859419	0.429710	+	0.902967i	$0.358616\pi$
$182$	11.8328	0.877107
$183$	0.909830	0.0672566
$184$	−5.32624	−0.392655
$185$	18.2705	1.34327
$186$	1.76393	0.129338
$187$	2.09017	0.152848
$188$	−3.43769	−0.250720
$189$	−4.61803	−0.335913
$190$	0.978714	0.0710034
$191$	−5.29180	−0.382901	−0.191450	−	0.981502i	$-0.561319\pi$
$191$	−5.29180	−0.382901	−0.191450	+	0.981502i	$0.561319\pi$
$192$	−4.70820	−0.339785
$193$	−27.4164	−1.97348	−0.986738	−	0.162320i	$-0.948102\pi$
$193$	−27.4164	−1.97348	−0.986738	+	0.162320i	$0.948102\pi$
$194$	0.201626	0.0144759
$195$	−20.1246	−1.44115
$196$	−26.5623	−1.89731
$197$	−12.1803	−0.867813	−0.433907	−	0.900958i	$-0.642865\pi$
$197$	−12.1803	−0.867813	−0.433907	+	0.900958i	$0.642865\pi$
$198$	2.09017	0.148542
$199$	24.5066	1.73723	0.868613	−	0.495492i	$-0.165012\pi$
$199$	24.5066	1.73723	0.868613	+	0.495492i	$0.165012\pi$
$200$	5.88854	0.416383
$201$	0.236068	0.0166510
$202$	4.29180	0.301970
$203$	30.5623	2.14505
$204$	0.708204	0.0495842
$205$	−15.2705	−1.06654
$206$	5.63932	0.392910
$207$	−3.61803	−0.251471
$208$	21.1033	1.46325
$209$	−4.67376	−0.323291
$210$	−5.29180	−0.365169
$211$	2.38197	0.163981	0.0819907	−	0.996633i	$-0.473872\pi$
$211$	2.38197	0.163981	0.0819907	+	0.996633i	$0.473872\pi$
$212$	6.43769	0.442143
$213$	−0.236068	−0.0161751
$214$	−7.27051	−0.497002
$215$	2.12461	0.144897
$216$	1.47214	0.100166
$217$	21.3262	1.44772
$218$	−2.61803	−0.177316
$219$	−3.09017	−0.208814
$220$	−30.4377	−2.05211
$221$	−2.56231	−0.172359
$222$	2.32624	0.156127
$223$	11.4164	0.764499	0.382250	−	0.924059i	$-0.375149\pi$
$223$	11.4164	0.764499	0.382250	+	0.924059i	$0.375149\pi$
$224$	19.1459	1.27924
$225$	4.00000	0.266667
$226$	−3.90983	−0.260078
$227$	−11.6180	−0.771116	−0.385558	−	0.922684i	$-0.625991\pi$
$227$	−11.6180	−0.771116	−0.385558	+	0.922684i	$0.625991\pi$
$228$	−1.58359	−0.104876
$229$	−18.9787	−1.25415	−0.627074	−	0.778959i	$-0.715748\pi$
$229$	−18.9787	−1.25415	−0.627074	+	0.778959i	$0.715748\pi$
$230$	−4.14590	−0.273372
$231$	25.2705	1.66268
$232$	−9.74265	−0.639636
$233$	0.763932	0.0500469	0.0250234	−	0.999687i	$-0.492034\pi$
$233$	0.763932	0.0500469	0.0250234	+	0.999687i	$0.492034\pi$
$234$	−2.56231	−0.167503
$235$	−5.56231	−0.362845
$236$	−1.85410	−0.120692
$237$	−3.00000	−0.194871
$238$	−0.673762	−0.0436735
$239$	−18.7082	−1.21013	−0.605067	−	0.796175i	$-0.706854\pi$
$239$	−18.7082	−1.21013	−0.605067	+	0.796175i	$0.706854\pi$
$240$	−9.43769	−0.609201
$241$	2.47214	0.159244	0.0796221	−	0.996825i	$-0.474629\pi$
$241$	2.47214	0.159244	0.0796221	+	0.996825i	$0.474629\pi$
$242$	−7.23607	−0.465152
$243$	1.00000	0.0641500
$244$	−1.68692	−0.107994
$245$	−42.9787	−2.74581
$246$	−1.94427	−0.123962
$247$	5.72949	0.364559
$248$	−6.79837	−0.431697
$249$	−7.85410	−0.497733
$250$	−1.14590	−0.0724730
$251$	11.9443	0.753916	0.376958	−	0.926230i	$-0.376970\pi$
$251$	11.9443	0.753916	0.376958	+	0.926230i	$0.376970\pi$
$252$	8.56231	0.539375
$253$	19.7984	1.24471
$254$	4.38197	0.274949
$255$	1.14590	0.0717589
$256$	5.56231	0.347644
$257$	−21.8885	−1.36537	−0.682685	−	0.730713i	$-0.739188\pi$
$257$	−21.8885	−1.36537	−0.682685	+	0.730713i	$0.739188\pi$
$258$	0.270510	0.0168412
$259$	28.1246	1.74758
$260$	37.3131	2.31406
$261$	−6.61803	−0.409646
$262$	−4.29180	−0.265148
$263$	−1.38197	−0.0852157	−0.0426078	−	0.999092i	$-0.513567\pi$
$263$	−1.38197	−0.0852157	−0.0426078	+	0.999092i	$0.513567\pi$
$264$	−8.05573	−0.495796
$265$	10.4164	0.639875
$266$	1.50658	0.0923742
$267$	−5.09017	−0.311513
$268$	−0.437694	−0.0267364
$269$	27.1803	1.65721	0.828607	−	0.559830i	$-0.189134\pi$
$269$	27.1803	1.65721	0.828607	+	0.559830i	$0.189134\pi$
$270$	1.14590	0.0697371
$271$	−21.1803	−1.28661	−0.643307	−	0.765608i	$-0.722438\pi$
$271$	−21.1803	−1.28661	−0.643307	+	0.765608i	$0.722438\pi$
$272$	−1.20163	−0.0728593
$273$	−30.9787	−1.87492
$274$	5.14590	0.310875
$275$	−21.8885	−1.31993
$276$	6.70820	0.403786
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	2.02129	0.121229
$279$	−4.61803	−0.276474
$280$	20.3951	1.21884
$281$	23.1246	1.37950	0.689749	−	0.724048i	$-0.257721\pi$
$281$	23.1246	1.37950	0.689749	+	0.724048i	$0.257721\pi$
$282$	−0.708204	−0.0421729
$283$	5.14590	0.305892	0.152946	−	0.988235i	$-0.451124\pi$
$283$	5.14590	0.305892	0.152946	+	0.988235i	$0.451124\pi$
$284$	0.437694	0.0259724
$285$	−2.56231	−0.151778
$286$	14.0213	0.829096
$287$	−23.5066	−1.38755
$288$	−4.14590	−0.244299
$289$	−16.8541	−0.991418
$290$	−7.58359	−0.445324
$291$	−0.527864	−0.0309439
$292$	5.72949	0.335293
$293$	−20.7426	−1.21180	−0.605899	−	0.795541i	$-0.707187\pi$
$293$	−20.7426	−1.21180	−0.605899	+	0.795541i	$0.707187\pi$
$294$	−5.47214	−0.319141
$295$	−3.00000	−0.174667
$296$	−8.96556	−0.521112
$297$	−5.47214	−0.317526
$298$	−2.68692	−0.155649
$299$	−24.2705	−1.40360
$300$	−7.41641	−0.428187
$301$	3.27051	0.188509
$302$	−0.819660	−0.0471661
$303$	−11.2361	−0.645495
$304$	2.68692	0.154105
$305$	−2.72949	−0.156290
$306$	0.145898	0.00834044
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	−46.8541	−2.66976
$309$	−14.7639	−0.839891
$310$	−5.29180	−0.300554
$311$	−15.9787	−0.906070	−0.453035	−	0.891493i	$-0.649659\pi$
$311$	−15.9787	−0.906070	−0.453035	+	0.891493i	$0.649659\pi$
$312$	9.87539	0.559084
$313$	−7.43769	−0.420403	−0.210202	−	0.977658i	$-0.567412\pi$
$313$	−7.43769	−0.420403	−0.210202	+	0.977658i	$0.567412\pi$
$314$	3.79837	0.214355
$315$	13.8541	0.780590
$316$	5.56231	0.312904
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	1.32624	0.0743717
$319$	36.2148	2.02764
$320$	14.1246	0.789590
$321$	19.0344	1.06240
$322$	−6.38197	−0.355653
$323$	−0.326238	−0.0181524
$324$	−1.85410	−0.103006
$325$	26.8328	1.48842
$326$	2.81966	0.156167
$327$	6.85410	0.379033
$328$	7.49342	0.413755
$329$	−8.56231	−0.472055
$330$	−6.27051	−0.345180
$331$	19.2361	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$331$	19.2361	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$332$	14.5623	0.799210
$333$	−6.09017	−0.333739
$334$	7.76393	0.424823
$335$	−0.708204	−0.0386933
$336$	−14.5279	−0.792560
$337$	−15.4377	−0.840945	−0.420472	−	0.907305i	$-0.638136\pi$
$337$	−15.4377	−0.840945	−0.420472	+	0.907305i	$0.638136\pi$
$338$	−12.2229	−0.664839
$339$	10.2361	0.555947
$340$	−2.12461	−0.115223
$341$	25.2705	1.36847
$342$	−0.326238	−0.0176409
$343$	−33.8328	−1.82680
$344$	−1.04257	−0.0562117
$345$	10.8541	0.584365
$346$	−6.34752	−0.341245
$347$	27.7426	1.48930	0.744652	−	0.667453i	$-0.232616\pi$
$347$	27.7426	1.48930	0.744652	+	0.667453i	$0.232616\pi$
$348$	12.2705	0.657768
$349$	23.0000	1.23116	0.615581	−	0.788074i	$-0.288921\pi$
$349$	23.0000	1.23116	0.615581	+	0.788074i	$0.288921\pi$
$350$	7.05573	0.377145
$351$	6.70820	0.358057
$352$	22.6869	1.20922
$353$	18.2705	0.972441	0.486221	−	0.873836i	$-0.338375\pi$
$353$	18.2705	0.972441	0.486221	+	0.873836i	$0.338375\pi$
$354$	−0.381966	−0.0203013
$355$	0.708204	0.0375876
$356$	9.43769	0.500197
$357$	1.76393	0.0933572
$358$	−3.50658	−0.185328
$359$	5.88854	0.310785	0.155393	−	0.987853i	$-0.450336\pi$
$359$	5.88854	0.310785	0.155393	+	0.987853i	$0.450336\pi$
$360$	−4.41641	−0.232765
$361$	−18.2705	−0.961606
$362$	−4.41641	−0.232121
$363$	18.9443	0.994316
$364$	57.4377	3.01055
$365$	9.27051	0.485241
$366$	−0.347524	−0.0181654
$367$	−22.4164	−1.17013	−0.585063	−	0.810987i	$-0.698930\pi$
$367$	−22.4164	−1.17013	−0.585063	+	0.810987i	$0.698930\pi$
$368$	−11.3820	−0.593326
$369$	5.09017	0.264984
$370$	−6.97871	−0.362806
$371$	16.0344	0.832467
$372$	8.56231	0.443935
$373$	4.56231	0.236227	0.118114	−	0.993000i	$-0.462315\pi$
$373$	4.56231	0.236227	0.118114	+	0.993000i	$0.462315\pi$
$374$	−0.798374	−0.0412829
$375$	3.00000	0.154919
$376$	2.72949	0.140763
$377$	−44.3951	−2.28647
$378$	1.76393	0.0907269
$379$	−20.4164	−1.04872	−0.524360	−	0.851497i	$-0.675695\pi$
$379$	−20.4164	−1.04872	−0.524360	+	0.851497i	$0.675695\pi$
$380$	4.75078	0.243710
$381$	−11.4721	−0.587735
$382$	2.02129	0.103418
$383$	20.0557	1.02480	0.512400	−	0.858747i	$-0.328757\pi$
$383$	20.0557	1.02480	0.512400	+	0.858747i	$0.328757\pi$
$384$	10.0902	0.514912
$385$	−75.8115	−3.86371
$386$	10.4721	0.533018
$387$	−0.708204	−0.0360000
$388$	0.978714	0.0496867
$389$	23.8885	1.21120	0.605599	−	0.795770i	$-0.292934\pi$
$389$	23.8885	1.21120	0.605599	+	0.795770i	$0.292934\pi$
$390$	7.68692	0.389242
$391$	1.38197	0.0698890
$392$	21.0902	1.06521
$393$	11.2361	0.566785
$394$	4.65248	0.234388
$395$	9.00000	0.452839
$396$	10.1459	0.509851
$397$	−34.4164	−1.72731	−0.863655	−	0.504083i	$-0.831830\pi$
$397$	−34.4164	−1.72731	−0.863655	+	0.504083i	$0.831830\pi$
$398$	−9.36068	−0.469208
$399$	−3.94427	−0.197461
$400$	12.5836	0.629180
$401$	−26.5623	−1.32646	−0.663229	−	0.748416i	$-0.730815\pi$
$401$	−26.5623	−1.32646	−0.663229	+	0.748416i	$0.730815\pi$
$402$	−0.0901699	−0.00449727
$403$	−30.9787	−1.54316
$404$	20.8328	1.03647
$405$	−3.00000	−0.149071
$406$	−11.6738	−0.579359
$407$	33.3262	1.65192
$408$	−0.562306	−0.0278383
$409$	−1.41641	−0.0700369	−0.0350184	−	0.999387i	$-0.511149\pi$
$409$	−1.41641	−0.0700369	−0.0350184	+	0.999387i	$0.511149\pi$
$410$	5.83282	0.288062
$411$	−13.4721	−0.664531
$412$	27.3738	1.34861
$413$	−4.61803	−0.227239
$414$	1.38197	0.0679199
$415$	23.5623	1.15663
$416$	−27.8115	−1.36357
$417$	−5.29180	−0.259140
$418$	1.78522	0.0873179
$419$	−2.94427	−0.143837	−0.0719185	−	0.997411i	$-0.522912\pi$
$419$	−2.94427	−0.143837	−0.0719185	+	0.997411i	$0.522912\pi$
$420$	−25.6869	−1.25339
$421$	10.4164	0.507665	0.253832	−	0.967248i	$-0.418309\pi$
$421$	10.4164	0.507665	0.253832	+	0.967248i	$0.418309\pi$
$422$	−0.909830	−0.0442898
$423$	1.85410	0.0901495
$424$	−5.11146	−0.248234
$425$	−1.52786	−0.0741123
$426$	0.0901699	0.00436875
$427$	−4.20163	−0.203331
$428$	−35.2918	−1.70589
$429$	−36.7082	−1.77229
$430$	−0.811529	−0.0391354
$431$	14.5066	0.698757	0.349379	−	0.936982i	$-0.386393\pi$
$431$	14.5066	0.698757	0.349379	+	0.936982i	$0.386393\pi$
$432$	3.14590	0.151357
$433$	16.8541	0.809956	0.404978	−	0.914326i	$-0.367279\pi$
$433$	16.8541	0.809956	0.404978	+	0.914326i	$0.367279\pi$
$434$	−8.14590	−0.391016
$435$	19.8541	0.951931
$436$	−12.7082	−0.608613
$437$	−3.09017	−0.147823
$438$	1.18034	0.0563988
$439$	1.14590	0.0546907	0.0273454	−	0.999626i	$-0.491295\pi$
$439$	1.14590	0.0546907	0.0273454	+	0.999626i	$0.491295\pi$
$440$	24.1672	1.15213
$441$	14.3262	0.682202
$442$	0.978714	0.0465527
$443$	−20.5623	−0.976945	−0.488472	−	0.872579i	$-0.662446\pi$
$443$	−20.5623	−0.976945	−0.488472	+	0.872579i	$0.662446\pi$
$444$	11.2918	0.535885
$445$	15.2705	0.723892
$446$	−4.36068	−0.206484
$447$	7.03444	0.332718
$448$	21.7426	1.02724
$449$	17.0689	0.805530	0.402765	−	0.915303i	$-0.368049\pi$
$449$	17.0689	0.805530	0.402765	+	0.915303i	$0.368049\pi$
$450$	−1.52786	−0.0720242
$451$	−27.8541	−1.31160
$452$	−18.9787	−0.892684
$453$	2.14590	0.100823
$454$	4.43769	0.208271
$455$	92.9361	4.35691
$456$	1.25735	0.0588810
$457$	7.85410	0.367399	0.183700	−	0.982982i	$-0.441193\pi$
$457$	7.85410	0.367399	0.183700	+	0.982982i	$0.441193\pi$
$458$	7.24922	0.338734
$459$	−0.381966	−0.0178286
$460$	−20.1246	−0.938315
$461$	2.67376	0.124530	0.0622648	−	0.998060i	$-0.480168\pi$
$461$	2.67376	0.124530	0.0622648	+	0.998060i	$0.480168\pi$
$462$	−9.65248	−0.449074
$463$	11.0344	0.512814	0.256407	−	0.966569i	$-0.417461\pi$
$463$	11.0344	0.512814	0.256407	+	0.966569i	$0.417461\pi$
$464$	−20.8197	−0.966528
$465$	13.8541	0.642469
$466$	−0.291796	−0.0135172
$467$	7.05573	0.326500	0.163250	−	0.986585i	$-0.447802\pi$
$467$	7.05573	0.326500	0.163250	+	0.986585i	$0.447802\pi$
$468$	−12.4377	−0.574933
$469$	−1.09017	−0.0503394
$470$	2.12461	0.0980010
$471$	−9.94427	−0.458208
$472$	1.47214	0.0677605
$473$	3.87539	0.178191
$474$	1.14590	0.0526328
$475$	3.41641	0.156756
$476$	−3.27051	−0.149904
$477$	−3.47214	−0.158978
$478$	7.14590	0.326846
$479$	−23.5066	−1.07404	−0.537021	−	0.843569i	$-0.680451\pi$
$479$	−23.5066	−1.07404	−0.537021	+	0.843569i	$0.680451\pi$
$480$	12.4377	0.567700
$481$	−40.8541	−1.86279
$482$	−0.944272	−0.0430104
$483$	16.7082	0.760250
$484$	−35.1246	−1.59657
$485$	1.58359	0.0719072
$486$	−0.381966	−0.0173263
$487$	−8.56231	−0.387995	−0.193998	−	0.981002i	$-0.562145\pi$
$487$	−8.56231	−0.387995	−0.193998	+	0.981002i	$0.562145\pi$
$488$	1.33939	0.0606315
$489$	−7.38197	−0.333824
$490$	16.4164	0.741618
$491$	−39.9230	−1.80170	−0.900850	−	0.434131i	$-0.857055\pi$
$491$	−39.9230	−1.80170	−0.900850	+	0.434131i	$0.857055\pi$
$492$	−9.43769	−0.425484
$493$	2.52786	0.113849
$494$	−2.18847	−0.0984639
$495$	16.4164	0.737863
$496$	−14.5279	−0.652320
$497$	1.09017	0.0489008
$498$	3.00000	0.134433
$499$	5.00000	0.223831	0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000	0.223831	0.111915	+	0.993718i	$0.464301\pi$
$500$	−5.56231	−0.248754
$501$	−20.3262	−0.908109
$502$	−4.56231	−0.203626
$503$	−24.9098	−1.11067	−0.555337	−	0.831625i	$-0.687411\pi$
$503$	−24.9098	−1.11067	−0.555337	+	0.831625i	$0.687411\pi$
$504$	−6.79837	−0.302824
$505$	33.7082	1.50000
$506$	−7.56231	−0.336186
$507$	32.0000	1.42117
$508$	21.2705	0.943726
$509$	8.21478	0.364114	0.182057	−	0.983288i	$-0.441724\pi$
$509$	8.21478	0.364114	0.182057	+	0.983288i	$0.441724\pi$
$510$	−0.437694	−0.0193814
$511$	14.2705	0.631290
$512$	−22.3050	−0.985749
$513$	0.854102	0.0377095
$514$	8.36068	0.368774
$515$	44.2918	1.95173
$516$	1.31308	0.0578052
$517$	−10.1459	−0.446216
$518$	−10.7426	−0.472005
$519$	16.6180	0.729451
$520$	−29.6262	−1.29919
$521$	−12.2148	−0.535139	−0.267570	−	0.963539i	$-0.586221\pi$
$521$	−12.2148	−0.535139	−0.267570	+	0.963539i	$0.586221\pi$
$522$	2.52786	0.110642
$523$	−41.3607	−1.80858	−0.904288	−	0.426923i	$-0.859598\pi$
$523$	−41.3607	−1.80858	−0.904288	+	0.426923i	$0.859598\pi$
$524$	−20.8328	−0.910086
$525$	−18.4721	−0.806190
$526$	0.527864	0.0230160
$527$	1.76393	0.0768381
$528$	−17.2148	−0.749177
$529$	−9.90983	−0.430862
$530$	−3.97871	−0.172824
$531$	1.00000	0.0433963
$532$	7.31308	0.317062
$533$	34.1459	1.47902
$534$	1.94427	0.0841369
$535$	−57.1033	−2.46879
$536$	0.347524	0.0150108
$537$	9.18034	0.396161
$538$	−10.3820	−0.447598
$539$	−78.3951	−3.37672
$540$	5.56231	0.239364
$541$	−10.7082	−0.460382	−0.230191	−	0.973146i	$-0.573935\pi$
$541$	−10.7082	−0.460382	−0.230191	+	0.973146i	$0.573935\pi$
$542$	8.09017	0.347503
$543$	11.5623	0.496186
$544$	1.58359	0.0678960
$545$	−20.5623	−0.880792
$546$	11.8328	0.506398
$547$	23.9787	1.02526	0.512628	−	0.858611i	$-0.328672\pi$
$547$	23.9787	1.02526	0.512628	+	0.858611i	$0.328672\pi$
$548$	24.9787	1.06704
$549$	0.909830	0.0388306
$550$	8.36068	0.356501
$551$	−5.65248	−0.240804
$552$	−5.32624	−0.226700
$553$	13.8541	0.589136
$554$	−1.90983	−0.0811409
$555$	18.2705	0.775540
$556$	9.81153	0.416102
$557$	−9.05573	−0.383704	−0.191852	−	0.981424i	$-0.561449\pi$
$557$	−9.05573	−0.383704	−0.191852	+	0.981424i	$0.561449\pi$
$558$	1.76393	0.0746732
$559$	−4.75078	−0.200936
$560$	43.5836	1.84174
$561$	2.09017	0.0882470
$562$	−8.83282	−0.372590
$563$	−33.4721	−1.41068	−0.705341	−	0.708868i	$-0.749206\pi$
$563$	−33.4721	−1.41068	−0.705341	+	0.708868i	$0.749206\pi$
$564$	−3.43769	−0.144753
$565$	−30.7082	−1.29190
$566$	−1.96556	−0.0826186
$567$	−4.61803	−0.193939
$568$	−0.347524	−0.0145818
$569$	−40.2148	−1.68589	−0.842946	−	0.537999i	$-0.819181\pi$
$569$	−40.2148	−1.68589	−0.842946	+	0.537999i	$0.819181\pi$
$570$	0.978714	0.0409938
$571$	33.8328	1.41586	0.707930	−	0.706283i	$-0.249629\pi$
$571$	33.8328	1.41586	0.707930	+	0.706283i	$0.249629\pi$
$572$	68.0608	2.84576
$573$	−5.29180	−0.221068
$574$	8.97871	0.374764
$575$	−14.4721	−0.603530
$576$	−4.70820	−0.196175
$577$	−21.3607	−0.889257	−0.444628	−	0.895715i	$-0.646664\pi$
$577$	−21.3607	−0.889257	−0.444628	+	0.895715i	$0.646664\pi$
$578$	6.43769	0.267773
$579$	−27.4164	−1.13939
$580$	−36.8115	−1.52852
$581$	36.2705	1.50475
$582$	0.201626	0.00835767
$583$	19.0000	0.786900
$584$	−4.54915	−0.188245
$585$	−20.1246	−0.832050
$586$	7.92299	0.327296
$587$	43.0689	1.77764	0.888822	−	0.458254i	$-0.151525\pi$
$587$	43.0689	1.77764	0.888822	+	0.458254i	$0.151525\pi$
$588$	−26.5623	−1.09541
$589$	−3.94427	−0.162521
$590$	1.14590	0.0471759
$591$	−12.1803	−0.501032
$592$	−19.1591	−0.787432
$593$	17.6738	0.725774	0.362887	−	0.931833i	$-0.381791\pi$
$593$	17.6738	0.725774	0.362887	+	0.931833i	$0.381791\pi$
$594$	2.09017	0.0857607
$595$	−5.29180	−0.216942
$596$	−13.0426	−0.534245
$597$	24.5066	1.00299
$598$	9.27051	0.379099
$599$	14.1803	0.579393	0.289696	−	0.957119i	$-0.406446\pi$
$599$	14.1803	0.579393	0.289696	+	0.957119i	$0.406446\pi$
$600$	5.88854	0.240399
$601$	−33.0344	−1.34750	−0.673751	−	0.738958i	$-0.735318\pi$
$601$	−33.0344	−1.34750	−0.673751	+	0.738958i	$0.735318\pi$
$602$	−1.24922	−0.0509146
$603$	0.236068	0.00961343
$604$	−3.97871	−0.161892
$605$	−56.8328	−2.31058
$606$	4.29180	0.174342
$607$	−47.6525	−1.93415	−0.967077	−	0.254483i	$-0.918095\pi$
$607$	−47.6525	−1.93415	−0.967077	+	0.254483i	$0.918095\pi$
$608$	−3.54102	−0.143607
$609$	30.5623	1.23845
$610$	1.04257	0.0422125
$611$	12.4377	0.503175
$612$	0.708204	0.0286274
$613$	39.7082	1.60380	0.801900	−	0.597459i	$-0.203823\pi$
$613$	39.7082	1.60380	0.801900	+	0.597459i	$0.203823\pi$
$614$	4.58359	0.184979
$615$	−15.2705	−0.615766
$616$	37.2016	1.49890
$617$	30.6312	1.23317	0.616583	−	0.787290i	$-0.288517\pi$
$617$	30.6312	1.23317	0.616583	+	0.787290i	$0.288517\pi$
$618$	5.63932	0.226847
$619$	−33.0689	−1.32915	−0.664575	−	0.747221i	$-0.731388\pi$
$619$	−33.0689	−1.32915	−0.664575	+	0.747221i	$0.731388\pi$
$620$	−25.6869	−1.03161
$621$	−3.61803	−0.145187
$622$	6.10333	0.244721
$623$	23.5066	0.941771
$624$	21.1033	0.844809
$625$	−29.0000	−1.16000
$626$	2.84095	0.113547
$627$	−4.67376	−0.186652
$628$	18.4377	0.735744
$629$	2.32624	0.0927532
$630$	−5.29180	−0.210830
$631$	22.4164	0.892383	0.446192	−	0.894937i	$-0.352780\pi$
$631$	22.4164	0.892383	0.446192	+	0.894937i	$0.352780\pi$
$632$	−4.41641	−0.175675
$633$	2.38197	0.0946746
$634$	−1.14590	−0.0455094
$635$	34.4164	1.36577
$636$	6.43769	0.255271
$637$	96.1033	3.80775
$638$	−13.8328	−0.547646
$639$	−0.236068	−0.00933870
$640$	−30.2705	−1.19655
$641$	30.5967	1.20850	0.604249	−	0.796795i	$-0.293473\pi$
$641$	30.5967	1.20850	0.604249	+	0.796795i	$0.293473\pi$
$642$	−7.27051	−0.286944
$643$	8.38197	0.330552	0.165276	−	0.986247i	$-0.447148\pi$
$643$	8.38197	0.330552	0.165276	+	0.986247i	$0.447148\pi$
$644$	−30.9787	−1.22073
$645$	2.12461	0.0836565
$646$	0.124612	0.00490279
$647$	10.4721	0.411702	0.205851	−	0.978583i	$-0.434004\pi$
$647$	10.4721	0.411702	0.205851	+	0.978583i	$0.434004\pi$
$648$	1.47214	0.0578310
$649$	−5.47214	−0.214800
$650$	−10.2492	−0.402008
$651$	21.3262	0.835841
$652$	13.6869	0.536021
$653$	17.6738	0.691628	0.345814	−	0.938303i	$-0.387603\pi$
$653$	17.6738	0.691628	0.345814	+	0.938303i	$0.387603\pi$
$654$	−2.61803	−0.102373
$655$	−33.7082	−1.31709
$656$	16.0132	0.625209
$657$	−3.09017	−0.120559
$658$	3.27051	0.127498
$659$	−2.09017	−0.0814215	−0.0407107	−	0.999171i	$-0.512962\pi$
$659$	−2.09017	−0.0814215	−0.0407107	+	0.999171i	$0.512962\pi$
$660$	−30.4377	−1.18479
$661$	29.5623	1.14984	0.574920	−	0.818209i	$-0.305033\pi$
$661$	29.5623	1.14984	0.574920	+	0.818209i	$0.305033\pi$
$662$	−7.34752	−0.285570
$663$	−2.56231	−0.0995117
$664$	−11.5623	−0.448704
$665$	11.8328	0.458857
$666$	2.32624	0.0901399
$667$	23.9443	0.927126
$668$	37.6869	1.45815
$669$	11.4164	0.441384
$670$	0.270510	0.0104507
$671$	−4.97871	−0.192201
$672$	19.1459	0.738569
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	5.89667	0.227131
$675$	4.00000	0.153960
$676$	−59.3313	−2.28197
$677$	−5.81966	−0.223668	−0.111834	−	0.993727i	$-0.535672\pi$
$677$	−5.81966	−0.223668	−0.111834	+	0.993727i	$0.535672\pi$
$678$	−3.90983	−0.150156
$679$	2.43769	0.0935501
$680$	1.68692	0.0646903
$681$	−11.6180	−0.445204
$682$	−9.65248	−0.369612
$683$	6.81966	0.260947	0.130474	−	0.991452i	$-0.458350\pi$
$683$	6.81966	0.260947	0.130474	+	0.991452i	$0.458350\pi$
$684$	−1.58359	−0.0605502
$685$	40.4164	1.54423
$686$	12.9230	0.493402
$687$	−18.9787	−0.724083
$688$	−2.22794	−0.0849393
$689$	−23.2918	−0.887347
$690$	−4.14590	−0.157832
$691$	13.3475	0.507764	0.253882	−	0.967235i	$-0.418293\pi$
$691$	13.3475	0.507764	0.253882	+	0.967235i	$0.418293\pi$
$692$	−30.8115	−1.17128
$693$	25.2705	0.959947
$694$	−10.5967	−0.402247
$695$	15.8754	0.602188
$696$	−9.74265	−0.369294
$697$	−1.94427	−0.0736445
$698$	−8.78522	−0.332525
$699$	0.763932	0.0288946
$700$	34.2492	1.29450
$701$	−48.8673	−1.84569	−0.922845	−	0.385171i	$-0.874143\pi$
$701$	−48.8673	−1.84569	−0.922845	+	0.385171i	$0.874143\pi$
$702$	−2.56231	−0.0967080
$703$	−5.20163	−0.196183
$704$	25.7639	0.971015
$705$	−5.56231	−0.209489
$706$	−6.97871	−0.262647
$707$	51.8885	1.95147
$708$	−1.85410	−0.0696814
$709$	−2.29180	−0.0860702	−0.0430351	−	0.999074i	$-0.513703\pi$
$709$	−2.29180	−0.0860702	−0.0430351	+	0.999074i	$0.513703\pi$
$710$	−0.270510	−0.0101521
$711$	−3.00000	−0.112509
$712$	−7.49342	−0.280828
$713$	16.7082	0.625727
$714$	−0.673762	−0.0252149
$715$	110.125	4.11843
$716$	−17.0213	−0.636115
$717$	−18.7082	−0.698671
$718$	−2.24922	−0.0839403
$719$	26.5623	0.990607	0.495303	−	0.868720i	$-0.335057\pi$
$719$	26.5623	0.990607	0.495303	+	0.868720i	$0.335057\pi$
$720$	−9.43769	−0.351722
$721$	68.1803	2.53917
$722$	6.97871	0.259721
$723$	2.47214	0.0919397
$724$	−21.4377	−0.796726
$725$	−26.4721	−0.983150
$726$	−7.23607	−0.268556
$727$	−21.4164	−0.794291	−0.397145	−	0.917756i	$-0.629999\pi$
$727$	−21.4164	−0.794291	−0.397145	+	0.917756i	$0.629999\pi$
$728$	−45.6049	−1.69023
$729$	1.00000	0.0370370
$730$	−3.54102	−0.131059
$731$	0.270510	0.0100052
$732$	−1.68692	−0.0623503
$733$	15.9443	0.588915	0.294458	−	0.955665i	$-0.404861\pi$
$733$	15.9443	0.588915	0.294458	+	0.955665i	$0.404861\pi$
$734$	8.56231	0.316040
$735$	−42.9787	−1.58529
$736$	15.0000	0.552907
$737$	−1.29180	−0.0475839
$738$	−1.94427	−0.0715696
$739$	24.8541	0.914273	0.457136	−	0.889397i	$-0.348875\pi$
$739$	24.8541	0.914273	0.457136	+	0.889397i	$0.348875\pi$
$740$	−33.8754	−1.24528
$741$	5.72949	0.210478
$742$	−6.12461	−0.224842
$743$	−3.79837	−0.139349	−0.0696744	−	0.997570i	$-0.522196\pi$
$743$	−3.79837	−0.139349	−0.0696744	+	0.997570i	$0.522196\pi$
$744$	−6.79837	−0.249240
$745$	−21.1033	−0.773166
$746$	−1.74265	−0.0638028
$747$	−7.85410	−0.287367
$748$	−3.87539	−0.141698
$749$	−87.9017	−3.21186
$750$	−1.14590	−0.0418423
$751$	28.5410	1.04148	0.520738	−	0.853716i	$-0.325657\pi$
$751$	28.5410	1.04148	0.520738	+	0.853716i	$0.325657\pi$
$752$	5.83282	0.212701
$753$	11.9443	0.435273
$754$	16.9574	0.617553
$755$	−6.43769	−0.234292
$756$	8.56231	0.311408
$757$	−22.4508	−0.815990	−0.407995	−	0.912984i	$-0.633772\pi$
$757$	−22.4508	−0.815990	−0.407995	+	0.912984i	$0.633772\pi$
$758$	7.79837	0.283250
$759$	19.7984	0.718635
$760$	−3.77206	−0.136827
$761$	−32.8885	−1.19221	−0.596104	−	0.802907i	$-0.703286\pi$
$761$	−32.8885	−1.19221	−0.596104	+	0.802907i	$0.703286\pi$
$762$	4.38197	0.158742
$763$	−31.6525	−1.14590
$764$	9.81153	0.354969
$765$	1.14590	0.0414300
$766$	−7.66061	−0.276789
$767$	6.70820	0.242219
$768$	5.56231	0.200712
$769$	−10.2705	−0.370364	−0.185182	−	0.982704i	$-0.559287\pi$
$769$	−10.2705	−0.370364	−0.185182	+	0.982704i	$0.559287\pi$
$770$	28.9574	1.04355
$771$	−21.8885	−0.788297
$772$	50.8328	1.82951
$773$	18.6525	0.670883	0.335441	−	0.942061i	$-0.391115\pi$
$773$	18.6525	0.670883	0.335441	+	0.942061i	$0.391115\pi$
$774$	0.270510	0.00972328
$775$	−18.4721	−0.663539
$776$	−0.777088	−0.0278958
$777$	28.1246	1.00896
$778$	−9.12461	−0.327133
$779$	4.34752	0.155766
$780$	37.3131	1.33602
$781$	1.29180	0.0462241
$782$	−0.527864	−0.0188764
$783$	−6.61803	−0.236509
$784$	45.0689	1.60960
$785$	29.8328	1.06478
$786$	−4.29180	−0.153083
$787$	2.29180	0.0816937	0.0408469	−	0.999165i	$-0.486994\pi$
$787$	2.29180	0.0816937	0.0408469	+	0.999165i	$0.486994\pi$
$788$	22.5836	0.804507
$789$	−1.38197	−0.0491993
$790$	−3.43769	−0.122308
$791$	−47.2705	−1.68075
$792$	−8.05573	−0.286248
$793$	6.10333	0.216735
$794$	13.1459	0.466530
$795$	10.4164	0.369432
$796$	−45.4377	−1.61050
$797$	26.0689	0.923407	0.461704	−	0.887034i	$-0.347238\pi$
$797$	26.0689	0.923407	0.461704	+	0.887034i	$0.347238\pi$
$798$	1.50658	0.0533323
$799$	−0.708204	−0.0250545
$800$	−16.5836	−0.586319
$801$	−5.09017	−0.179852
$802$	10.1459	0.358264
$803$	16.9098	0.596735
$804$	−0.437694	−0.0154363
$805$	−50.1246	−1.76666
$806$	11.8328	0.416793
$807$	27.1803	0.956793
$808$	−16.5410	−0.581911
$809$	−4.74265	−0.166743	−0.0833713	−	0.996519i	$-0.526569\pi$
$809$	−4.74265	−0.166743	−0.0833713	+	0.996519i	$0.526569\pi$
$810$	1.14590	0.0402628
$811$	42.5410	1.49382	0.746909	−	0.664927i	$-0.231537\pi$
$811$	42.5410	1.49382	0.746909	+	0.664927i	$0.231537\pi$
$812$	−56.6656	−1.98857
$813$	−21.1803	−0.742827
$814$	−12.7295	−0.446168
$815$	22.1459	0.775737
$816$	−1.20163	−0.0420653
$817$	−0.604878	−0.0211620
$818$	0.541020	0.0189163
$819$	−30.9787	−1.08248
$820$	28.3131	0.988736
$821$	−53.6312	−1.87174	−0.935871	−	0.352344i	$-0.885385\pi$
$821$	−53.6312	−1.87174	−0.935871	+	0.352344i	$0.885385\pi$
$822$	5.14590	0.179484
$823$	27.0689	0.943562	0.471781	−	0.881716i	$-0.343611\pi$
$823$	27.0689	0.943562	0.471781	+	0.881716i	$0.343611\pi$
$824$	−21.7345	−0.757158
$825$	−21.8885	−0.762061
$826$	1.76393	0.0613750
$827$	−3.00000	−0.104320	−0.0521601	−	0.998639i	$-0.516611\pi$
$827$	−3.00000	−0.104320	−0.0521601	+	0.998639i	$0.516611\pi$
$828$	6.70820	0.233126
$829$	47.2705	1.64177	0.820886	−	0.571092i	$-0.193480\pi$
$829$	47.2705	1.64177	0.820886	+	0.571092i	$0.193480\pi$
$830$	−9.00000	−0.312395
$831$	5.00000	0.173448
$832$	−31.5836	−1.09496
$833$	−5.47214	−0.189598
$834$	2.02129	0.0699914
$835$	60.9787	2.11026
$836$	8.66563	0.299707
$837$	−4.61803	−0.159623
$838$	1.12461	0.0388491
$839$	−15.0000	−0.517858	−0.258929	−	0.965896i	$-0.583369\pi$
$839$	−15.0000	−0.517858	−0.258929	+	0.965896i	$0.583369\pi$
$840$	20.3951	0.703699
$841$	14.7984	0.510289
$842$	−3.97871	−0.137116
$843$	23.1246	0.796454
$844$	−4.41641	−0.152019
$845$	−96.0000	−3.30250
$846$	−0.708204	−0.0243486
$847$	−87.4853	−3.00603
$848$	−10.9230	−0.375097
$849$	5.14590	0.176607
$850$	0.583592	0.0200170
$851$	22.0344	0.755331
$852$	0.437694	0.0149952
$853$	38.3820	1.31417	0.657087	−	0.753815i	$-0.271789\pi$
$853$	38.3820	1.31417	0.657087	+	0.753815i	$0.271789\pi$
$854$	1.60488	0.0549178
$855$	−2.56231	−0.0876290
$856$	28.0213	0.957748
$857$	0.347524	0.0118712	0.00593560	−	0.999982i	$-0.498111\pi$
$857$	0.347524	0.0118712	0.00593560	+	0.999982i	$0.498111\pi$
$858$	14.0213	0.478679
$859$	−23.5279	−0.802760	−0.401380	−	0.915912i	$-0.631469\pi$
$859$	−23.5279	−0.802760	−0.401380	+	0.915912i	$0.631469\pi$
$860$	−3.93925	−0.134327
$861$	−23.5066	−0.801102
$862$	−5.54102	−0.188728
$863$	21.0344	0.716021	0.358010	−	0.933718i	$-0.383455\pi$
$863$	21.0344	0.716021	0.358010	+	0.933718i	$0.383455\pi$
$864$	−4.14590	−0.141046
$865$	−49.8541	−1.69509
$866$	−6.43769	−0.218762
$867$	−16.8541	−0.572395
$868$	−39.5410	−1.34211
$869$	16.4164	0.556888
$870$	−7.58359	−0.257108
$871$	1.58359	0.0536580
$872$	10.0902	0.341696
$873$	−0.527864	−0.0178655
$874$	1.18034	0.0399256
$875$	−13.8541	−0.468354
$876$	5.72949	0.193582
$877$	−17.4164	−0.588110	−0.294055	−	0.955788i	$-0.595005\pi$
$877$	−17.4164	−0.588110	−0.294055	+	0.955788i	$0.595005\pi$
$878$	−0.437694	−0.0147715
$879$	−20.7426	−0.699632
$880$	51.6443	1.74093
$881$	2.65248	0.0893642	0.0446821	−	0.999001i	$-0.485773\pi$
$881$	2.65248	0.0893642	0.0446821	+	0.999001i	$0.485773\pi$
$882$	−5.47214	−0.184256
$883$	−49.8885	−1.67888	−0.839442	−	0.543450i	$-0.817118\pi$
$883$	−49.8885	−1.67888	−0.839442	+	0.543450i	$0.817118\pi$
$884$	4.75078	0.159786
$885$	−3.00000	−0.100844
$886$	7.85410	0.263864
$887$	−45.7639	−1.53660	−0.768301	−	0.640088i	$-0.778898\pi$
$887$	−45.7639	−1.53660	−0.768301	+	0.640088i	$0.778898\pi$
$888$	−8.96556	−0.300864
$889$	52.9787	1.77685
$890$	−5.83282	−0.195516
$891$	−5.47214	−0.183323
$892$	−21.1672	−0.708730
$893$	1.58359	0.0529929
$894$	−2.68692	−0.0898640
$895$	−27.5410	−0.920595
$896$	−46.5967	−1.55669
$897$	−24.2705	−0.810369
$898$	−6.51973	−0.217566
$899$	30.5623	1.01931
$900$	−7.41641	−0.247214
$901$	1.32624	0.0441834
$902$	10.6393	0.354251
$903$	3.27051	0.108836
$904$	15.0689	0.501184
$905$	−34.6869	−1.15303
$906$	−0.819660	−0.0272314
$907$	−34.4164	−1.14278	−0.571389	−	0.820679i	$-0.693595\pi$
$907$	−34.4164	−1.14278	−0.571389	+	0.820679i	$0.693595\pi$
$908$	21.5410	0.714864
$909$	−11.2361	−0.372677
$910$	−35.4984	−1.17676
$911$	44.0344	1.45893	0.729463	−	0.684020i	$-0.239770\pi$
$911$	44.0344	1.45893	0.729463	+	0.684020i	$0.239770\pi$
$912$	2.68692	0.0889727
$913$	42.9787	1.42239
$914$	−3.00000	−0.0992312
$915$	−2.72949	−0.0902342
$916$	35.1885	1.16266
$917$	−51.8885	−1.71351
$918$	0.145898	0.00481535
$919$	45.7082	1.50777	0.753887	−	0.657004i	$-0.228176\pi$
$919$	45.7082	1.50777	0.753887	+	0.657004i	$0.228176\pi$
$920$	15.9787	0.526803
$921$	−12.0000	−0.395413
$922$	−1.02129	−0.0336343
$923$	−1.58359	−0.0521246
$924$	−46.8541	−1.54139
$925$	−24.3607	−0.800974
$926$	−4.21478	−0.138506
$927$	−14.7639	−0.484911
$928$	27.4377	0.900686
$929$	−41.3607	−1.35700	−0.678500	−	0.734600i	$-0.737370\pi$
$929$	−41.3607	−1.35700	−0.678500	+	0.734600i	$0.737370\pi$
$930$	−5.29180	−0.173525
$931$	12.2361	0.401021
$932$	−1.41641	−0.0463960
$933$	−15.9787	−0.523120
$934$	−2.69505	−0.0881847
$935$	−6.27051	−0.205068
$936$	9.87539	0.322787
$937$	−15.9443	−0.520877	−0.260438	−	0.965490i	$-0.583867\pi$
$937$	−15.9443	−0.520877	−0.260438	+	0.965490i	$0.583867\pi$
$938$	0.416408	0.0135962
$939$	−7.43769	−0.242720
$940$	10.3131	0.336376
$941$	−8.23607	−0.268488	−0.134244	−	0.990948i	$-0.542861\pi$
$941$	−8.23607	−0.268488	−0.134244	+	0.990948i	$0.542861\pi$
$942$	3.79837	0.123758
$943$	−18.4164	−0.599721
$944$	3.14590	0.102390
$945$	13.8541	0.450674
$946$	−1.48027	−0.0481276
$947$	49.4508	1.60694	0.803468	−	0.595347i	$-0.202986\pi$
$947$	49.4508	1.60694	0.803468	+	0.595347i	$0.202986\pi$
$948$	5.56231	0.180655
$949$	−20.7295	−0.672908
$950$	−1.30495	−0.0423382
$951$	3.00000	0.0972817
$952$	2.59675	0.0841611
$953$	−13.5279	−0.438211	−0.219105	−	0.975701i	$-0.570314\pi$
$953$	−13.5279	−0.438211	−0.219105	+	0.975701i	$0.570314\pi$
$954$	1.32624	0.0429385
$955$	15.8754	0.513715
$956$	34.6869	1.12186
$957$	36.2148	1.17066
$958$	8.97871	0.290089
$959$	62.2148	2.00902
$960$	14.1246	0.455870
$961$	−9.67376	−0.312057
$962$	15.6049	0.503121
$963$	19.0344	0.613376
$964$	−4.58359	−0.147628
$965$	82.2492	2.64770
$966$	−6.38197	−0.205336
$967$	−3.97871	−0.127947	−0.0639734	−	0.997952i	$-0.520377\pi$
$967$	−3.97871	−0.127947	−0.0639734	+	0.997952i	$0.520377\pi$
$968$	27.8885	0.896372
$969$	−0.326238	−0.0104803
$970$	−0.604878	−0.0194215
$971$	16.9230	0.543084	0.271542	−	0.962427i	$-0.412466\pi$
$971$	16.9230	0.543084	0.271542	+	0.962427i	$0.412466\pi$
$972$	−1.85410	−0.0594703
$973$	24.4377	0.783437
$974$	3.27051	0.104794
$975$	26.8328	0.859338
$976$	2.86223	0.0916178
$977$	−37.0689	−1.18594	−0.592969	−	0.805225i	$-0.702044\pi$
$977$	−37.0689	−1.18594	−0.592969	+	0.805225i	$0.702044\pi$
$978$	2.81966	0.0901628
$979$	27.8541	0.890221
$980$	79.6869	2.54551
$981$	6.85410	0.218835
$982$	15.2492	0.486622
$983$	−0.167184	−0.00533235	−0.00266618	−	0.999996i	$-0.500849\pi$
$983$	−0.167184	−0.00533235	−0.00266618	+	0.999996i	$0.500849\pi$
$984$	7.49342	0.238882
$985$	36.5410	1.16429
$986$	−0.965558	−0.0307496
$987$	−8.56231	−0.272541
$988$	−10.6231	−0.337965
$989$	2.56231	0.0814766
$990$	−6.27051	−0.199290
$991$	16.8541	0.535388	0.267694	−	0.963504i	$-0.413738\pi$
$991$	16.8541	0.535388	0.267694	+	0.963504i	$0.413738\pi$
$992$	19.1459	0.607883
$993$	19.2361	0.610438
$994$	−0.416408	−0.0132077
$995$	−73.5197	−2.33073
$996$	14.5623	0.461424
$997$	−25.2705	−0.800325	−0.400163	−	0.916444i	$-0.631046\pi$
$997$	−25.2705	−0.800325	−0.400163	+	0.916444i	$0.631046\pi$
$998$	−1.90983	−0.0604546
$999$	−6.09017	−0.192684

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.a.a.1.2	✓	2
3.2	odd	2		531.2.a.c.1.1		2
4.3	odd	2		2832.2.a.h.1.2		2
5.4	even	2		4425.2.a.u.1.1		2
7.6	odd	2		8673.2.a.j.1.2		2
12.11	even	2		8496.2.a.bg.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.a.a.1.2	✓	2	1.1	even	1	trivial
531.2.a.c.1.1		2	3.2	odd	2
2832.2.a.h.1.2		2	4.3	odd	2
4425.2.a.u.1.1		2	5.4	even	2
8496.2.a.bg.1.2		2	12.11	even	2
8673.2.a.j.1.2		2	7.6	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} -3.00000 q^{5} -0.381966 q^{6} -4.61803 q^{7} +1.47214 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} -3.00000 q^{5} -0.381966 q^{6} -4.61803 q^{7} +1.47214 q^{8} +1.00000 q^{9} +1.14590 q^{10} -5.47214 q^{11} -1.85410 q^{12} +6.70820 q^{13} +1.76393 q^{14} -3.00000 q^{15} +3.14590 q^{16} -0.381966 q^{17} -0.381966 q^{18} +0.854102 q^{19} +5.56231 q^{20} -4.61803 q^{21} +2.09017 q^{22} -3.61803 q^{23} +1.47214 q^{24} +4.00000 q^{25} -2.56231 q^{26} +1.00000 q^{27} +8.56231 q^{28} -6.61803 q^{29} +1.14590 q^{30} -4.61803 q^{31} -4.14590 q^{32} -5.47214 q^{33} +0.145898 q^{34} +13.8541 q^{35} -1.85410 q^{36} -6.09017 q^{37} -0.326238 q^{38} +6.70820 q^{39} -4.41641 q^{40} +5.09017 q^{41} +1.76393 q^{42} -0.708204 q^{43} +10.1459 q^{44} -3.00000 q^{45} +1.38197 q^{46} +1.85410 q^{47} +3.14590 q^{48} +14.3262 q^{49} -1.52786 q^{50} -0.381966 q^{51} -12.4377 q^{52} -3.47214 q^{53} -0.381966 q^{54} +16.4164 q^{55} -6.79837 q^{56} +0.854102 q^{57} +2.52786 q^{58} +1.00000 q^{59} +5.56231 q^{60} +0.909830 q^{61} +1.76393 q^{62} -4.61803 q^{63} -4.70820 q^{64} -20.1246 q^{65} +2.09017 q^{66} +0.236068 q^{67} +0.708204 q^{68} -3.61803 q^{69} -5.29180 q^{70} -0.236068 q^{71} +1.47214 q^{72} -3.09017 q^{73} +2.32624 q^{74} +4.00000 q^{75} -1.58359 q^{76} +25.2705 q^{77} -2.56231 q^{78} -3.00000 q^{79} -9.43769 q^{80} +1.00000 q^{81} -1.94427 q^{82} -7.85410 q^{83} +8.56231 q^{84} +1.14590 q^{85} +0.270510 q^{86} -6.61803 q^{87} -8.05573 q^{88} -5.09017 q^{89} +1.14590 q^{90} -30.9787 q^{91} +6.70820 q^{92} -4.61803 q^{93} -0.708204 q^{94} -2.56231 q^{95} -4.14590 q^{96} -0.527864 q^{97} -5.47214 q^{98} -5.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 6 q^{5} - 3 q^{6} - 7 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 6 * q^5 - 3 * q^6 - 7 * q^7 - 6 * q^8 + 2 * q^9	\( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 6 q^{5} - 3 q^{6} - 7 q^{7} - 6 q^{8} + 2 q^{9} + 9 q^{10} - 2 q^{11} + 3 q^{12} + 8 q^{14} - 6 q^{15} + 13 q^{16} - 3 q^{17} - 3 q^{18} - 5 q^{19} - 9 q^{20} - 7 q^{21} - 7 q^{22} - 5 q^{23} - 6 q^{24} + 8 q^{25} + 15 q^{26} + 2 q^{27} - 3 q^{28} - 11 q^{29} + 9 q^{30} - 7 q^{31} - 15 q^{32} - 2 q^{33} + 7 q^{34} + 21 q^{35} + 3 q^{36} - q^{37} + 15 q^{38} + 18 q^{40} - q^{41} + 8 q^{42} + 12 q^{43} + 27 q^{44} - 6 q^{45} + 5 q^{46} - 3 q^{47} + 13 q^{48} + 13 q^{49} - 12 q^{50} - 3 q^{51} - 45 q^{52} + 2 q^{53} - 3 q^{54} + 6 q^{55} + 11 q^{56} - 5 q^{57} + 14 q^{58} + 2 q^{59} - 9 q^{60} + 13 q^{61} + 8 q^{62} - 7 q^{63} + 4 q^{64} - 7 q^{66} - 4 q^{67} - 12 q^{68} - 5 q^{69} - 24 q^{70} + 4 q^{71} - 6 q^{72} + 5 q^{73} - 11 q^{74} + 8 q^{75} - 30 q^{76} + 17 q^{77} + 15 q^{78} - 6 q^{79} - 39 q^{80} + 2 q^{81} + 14 q^{82} - 9 q^{83} - 3 q^{84} + 9 q^{85} - 33 q^{86} - 11 q^{87} - 34 q^{88} + q^{89} + 9 q^{90} - 15 q^{91} - 7 q^{93} + 12 q^{94} + 15 q^{95} - 15 q^{96} - 10 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 6 * q^5 - 3 * q^6 - 7 * q^7 - 6 * q^8 + 2 * q^9 + 9 * q^10 - 2 * q^11 + 3 * q^12 + 8 * q^14 - 6 * q^15 + 13 * q^16 - 3 * q^17 - 3 * q^18 - 5 * q^19 - 9 * q^20 - 7 * q^21 - 7 * q^22 - 5 * q^23 - 6 * q^24 + 8 * q^25 + 15 * q^26 + 2 * q^27 - 3 * q^28 - 11 * q^29 + 9 * q^30 - 7 * q^31 - 15 * q^32 - 2 * q^33 + 7 * q^34 + 21 * q^35 + 3 * q^36 - q^37 + 15 * q^38 + 18 * q^40 - q^41 + 8 * q^42 + 12 * q^43 + 27 * q^44 - 6 * q^45 + 5 * q^46 - 3 * q^47 + 13 * q^48 + 13 * q^49 - 12 * q^50 - 3 * q^51 - 45 * q^52 + 2 * q^53 - 3 * q^54 + 6 * q^55 + 11 * q^56 - 5 * q^57 + 14 * q^58 + 2 * q^59 - 9 * q^60 + 13 * q^61 + 8 * q^62 - 7 * q^63 + 4 * q^64 - 7 * q^66 - 4 * q^67 - 12 * q^68 - 5 * q^69 - 24 * q^70 + 4 * q^71 - 6 * q^72 + 5 * q^73 - 11 * q^74 + 8 * q^75 - 30 * q^76 + 17 * q^77 + 15 * q^78 - 6 * q^79 - 39 * q^80 + 2 * q^81 + 14 * q^82 - 9 * q^83 - 3 * q^84 + 9 * q^85 - 33 * q^86 - 11 * q^87 - 34 * q^88 + q^89 + 9 * q^90 - 15 * q^91 - 7 * q^93 + 12 * q^94 + 15 * q^95 - 15 * q^96 - 10 * q^97 - 2 * q^98 - 2 * q^99

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