LMFDB - Embedded newform 109.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [109,2,Mod(1,109)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(109, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("109.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 109 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	109.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:	$0.870369382032$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 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			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.554958 q^{2} -1.44504 q^{3} -1.69202 q^{4} +1.04892 q^{5} +0.801938 q^{6} -4.85086 q^{7} +2.04892 q^{8} -0.911854 q^{9} +O(q^{10})$	\(q-0.554958 q^{2} -1.44504 q^{3} -1.69202 q^{4} +1.04892 q^{5} +0.801938 q^{6} -4.85086 q^{7} +2.04892 q^{8} -0.911854 q^{9} -0.582105 q^{10} -5.80194 q^{11} +2.44504 q^{12} +2.93900 q^{13} +2.69202 q^{14} -1.51573 q^{15} +2.24698 q^{16} +2.35690 q^{17} +0.506041 q^{18} +2.85086 q^{19} -1.77479 q^{20} +7.00969 q^{21} +3.21983 q^{22} -0.225209 q^{23} -2.96077 q^{24} -3.89977 q^{25} -1.63102 q^{26} +5.65279 q^{27} +8.20775 q^{28} -7.76271 q^{29} +0.841166 q^{30} +0.268750 q^{31} -5.34481 q^{32} +8.38404 q^{33} -1.30798 q^{34} -5.08815 q^{35} +1.54288 q^{36} -1.69202 q^{37} -1.58211 q^{38} -4.24698 q^{39} +2.14914 q^{40} -11.1468 q^{41} -3.89008 q^{42} +8.07606 q^{43} +9.81700 q^{44} -0.956459 q^{45} +0.124982 q^{46} -2.77479 q^{47} -3.24698 q^{48} +16.5308 q^{49} +2.16421 q^{50} -3.40581 q^{51} -4.97285 q^{52} +1.30798 q^{53} -3.13706 q^{54} -6.08575 q^{55} -9.93900 q^{56} -4.11960 q^{57} +4.30798 q^{58} -7.08815 q^{59} +2.56465 q^{60} -13.2620 q^{61} -0.149145 q^{62} +4.42327 q^{63} -1.52781 q^{64} +3.08277 q^{65} -4.65279 q^{66} -12.7017 q^{67} -3.98792 q^{68} +0.325437 q^{69} +2.82371 q^{70} +4.34481 q^{71} -1.86831 q^{72} +5.19806 q^{73} +0.939001 q^{74} +5.63533 q^{75} -4.82371 q^{76} +28.1444 q^{77} +2.35690 q^{78} +8.83877 q^{79} +2.35690 q^{80} -5.43296 q^{81} +6.18598 q^{82} +3.00969 q^{83} -11.8605 q^{84} +2.47219 q^{85} -4.48188 q^{86} +11.2174 q^{87} -11.8877 q^{88} -1.92394 q^{89} +0.530795 q^{90} -14.2567 q^{91} +0.381059 q^{92} -0.388355 q^{93} +1.53989 q^{94} +2.99031 q^{95} +7.72348 q^{96} +1.23490 q^{97} -9.17390 q^{98} +5.29052 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} - 4 q^{3} - 6 q^{5} - 2 q^{6} - q^{7} - 3 q^{8} + q^{9}+O(q^{10}) $ 3 * q - 2 * q^2 - 4 * q^3 - 6 * q^5 - 2 * q^6 - q^7 - 3 * q^8 + q^9	\( 3 q - 2 q^{2} - 4 q^{3} - 6 q^{5} - 2 q^{6} - q^{7} - 3 q^{8} + q^{9} + 4 q^{10} - 13 q^{11} + 7 q^{12} - q^{13} + 3 q^{14} + 8 q^{15} + 2 q^{16} + 3 q^{17} + 11 q^{18} - 5 q^{19} - 7 q^{20} - q^{21} + 11 q^{22} + q^{23} + 4 q^{24} + 11 q^{25} + 10 q^{26} - q^{27} + 7 q^{28} - 6 q^{29} + 11 q^{30} - 7 q^{31} + 7 q^{32} + 15 q^{33} - 9 q^{34} - 19 q^{35} - 14 q^{36} + q^{38} - 8 q^{39} + 20 q^{40} - 6 q^{41} - 11 q^{42} + 9 q^{43} - 9 q^{45} - 24 q^{46} - 10 q^{47} - 5 q^{48} + 12 q^{49} - 5 q^{50} + 3 q^{51} - 21 q^{52} + 9 q^{53} - 4 q^{54} + 19 q^{55} - 20 q^{56} + 9 q^{57} + 18 q^{58} - 25 q^{59} - 14 q^{60} - 10 q^{61} - 14 q^{62} + 16 q^{63} - 11 q^{64} + 16 q^{65} + 4 q^{66} - 11 q^{67} + 7 q^{68} + 22 q^{69} + q^{70} - 10 q^{71} - 8 q^{72} + 20 q^{73} - 7 q^{74} - 17 q^{75} - 7 q^{76} + 16 q^{77} + 3 q^{78} - 6 q^{79} + 3 q^{80} + 3 q^{81} + 4 q^{82} - 13 q^{83} + q^{85} + 15 q^{86} - 6 q^{87} + 6 q^{88} - 21 q^{89} - 36 q^{90} - 16 q^{91} + 35 q^{92} + 28 q^{93} + 30 q^{94} + 31 q^{95} - 7 q^{96} - 20 q^{97} + 6 q^{98} + 5 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 - 4 * q^3 - 6 * q^5 - 2 * q^6 - q^7 - 3 * q^8 + q^9 + 4 * q^10 - 13 * q^11 + 7 * q^12 - q^13 + 3 * q^14 + 8 * q^15 + 2 * q^16 + 3 * q^17 + 11 * q^18 - 5 * q^19 - 7 * q^20 - q^21 + 11 * q^22 + q^23 + 4 * q^24 + 11 * q^25 + 10 * q^26 - q^27 + 7 * q^28 - 6 * q^29 + 11 * q^30 - 7 * q^31 + 7 * q^32 + 15 * q^33 - 9 * q^34 - 19 * q^35 - 14 * q^36 + q^38 - 8 * q^39 + 20 * q^40 - 6 * q^41 - 11 * q^42 + 9 * q^43 - 9 * q^45 - 24 * q^46 - 10 * q^47 - 5 * q^48 + 12 * q^49 - 5 * q^50 + 3 * q^51 - 21 * q^52 + 9 * q^53 - 4 * q^54 + 19 * q^55 - 20 * q^56 + 9 * q^57 + 18 * q^58 - 25 * q^59 - 14 * q^60 - 10 * q^61 - 14 * q^62 + 16 * q^63 - 11 * q^64 + 16 * q^65 + 4 * q^66 - 11 * q^67 + 7 * q^68 + 22 * q^69 + q^70 - 10 * q^71 - 8 * q^72 + 20 * q^73 - 7 * q^74 - 17 * q^75 - 7 * q^76 + 16 * q^77 + 3 * q^78 - 6 * q^79 + 3 * q^80 + 3 * q^81 + 4 * q^82 - 13 * q^83 + q^85 + 15 * q^86 - 6 * q^87 + 6 * q^88 - 21 * q^89 - 36 * q^90 - 16 * q^91 + 35 * q^92 + 28 * q^93 + 30 * q^94 + 31 * q^95 - 7 * q^96 - 20 * q^97 + 6 * q^98 + 5 * 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 $

$2$	−0.554958	−0.392415	−0.196207	−	0.980562i	$-0.562863\pi$
$2$	−0.554958	−0.392415	−0.196207	+	0.980562i	$0.562863\pi$
$3$	−1.44504	−0.834295	−0.417148	−	0.908839i	$-0.636970\pi$
$3$	−1.44504	−0.834295	−0.417148	+	0.908839i	$0.636970\pi$
$4$	−1.69202	−0.846011
$5$	1.04892	0.469090	0.234545	−	0.972105i	$-0.424640\pi$
$5$	1.04892	0.469090	0.234545	+	0.972105i	$0.424640\pi$
$6$	0.801938	0.327390
$7$	−4.85086	−1.83345	−0.916725	−	0.399518i	$-0.869178\pi$
$7$	−4.85086	−1.83345	−0.916725	+	0.399518i	$0.869178\pi$
$8$	2.04892	0.724402
$9$	−0.911854	−0.303951
$10$	−0.582105	−0.184078
$11$	−5.80194	−1.74935	−0.874675	−	0.484710i	$-0.838925\pi$
$11$	−5.80194	−1.74935	−0.874675	+	0.484710i	$0.838925\pi$
$12$	2.44504	0.705823
$13$	2.93900	0.815132	0.407566	−	0.913176i	$-0.366378\pi$
$13$	2.93900	0.815132	0.407566	+	0.913176i	$0.366378\pi$
$14$	2.69202	0.719473
$15$	−1.51573	−0.391360
$16$	2.24698	0.561745
$17$	2.35690	0.571631	0.285816	−	0.958285i	$-0.407736\pi$
$17$	2.35690	0.571631	0.285816	+	0.958285i	$0.407736\pi$
$18$	0.506041	0.119275
$19$	2.85086	0.654031	0.327016	−	0.945019i	$-0.393957\pi$
$19$	2.85086	0.654031	0.327016	+	0.945019i	$0.393957\pi$
$20$	−1.77479	−0.396855
$21$	7.00969	1.52964
$22$	3.21983	0.686471
$23$	−0.225209	−0.0469594	−0.0234797	−	0.999724i	$-0.507475\pi$
$23$	−0.225209	−0.0469594	−0.0234797	+	0.999724i	$0.507475\pi$
$24$	−2.96077	−0.604365
$25$	−3.89977	−0.779954
$26$	−1.63102	−0.319870
$27$	5.65279	1.08788
$28$	8.20775	1.55112
$29$	−7.76271	−1.44150	−0.720749	−	0.693196i	$-0.756202\pi$
$29$	−7.76271	−1.44150	−0.720749	+	0.693196i	$0.756202\pi$
$30$	0.841166	0.153575
$31$	0.268750	0.0482689	0.0241345	−	0.999709i	$-0.492317\pi$
$31$	0.268750	0.0482689	0.0241345	+	0.999709i	$0.492317\pi$
$32$	−5.34481	−0.944839
$33$	8.38404	1.45947
$34$	−1.30798	−0.224316
$35$	−5.08815	−0.860054
$36$	1.54288	0.257146
$37$	−1.69202	−0.278167	−0.139083	−	0.990281i	$-0.544416\pi$
$37$	−1.69202	−0.278167	−0.139083	+	0.990281i	$0.544416\pi$
$38$	−1.58211	−0.256651
$39$	−4.24698	−0.680061
$40$	2.14914	0.339810
$41$	−11.1468	−1.74083	−0.870415	−	0.492319i	$-0.836150\pi$
$41$	−11.1468	−1.74083	−0.870415	+	0.492319i	$0.836150\pi$
$42$	−3.89008	−0.600253
$43$	8.07606	1.23159	0.615794	−	0.787907i	$-0.288835\pi$
$43$	8.07606	1.23159	0.615794	+	0.787907i	$0.288835\pi$
$44$	9.81700	1.47997
$45$	−0.956459	−0.142581
$46$	0.124982	0.0184276
$47$	−2.77479	−0.404745	−0.202372	−	0.979309i	$-0.564865\pi$
$47$	−2.77479	−0.404745	−0.202372	+	0.979309i	$0.564865\pi$
$48$	−3.24698	−0.468661
$49$	16.5308	2.36154
$50$	2.16421	0.306066
$51$	−3.40581	−0.476909
$52$	−4.97285	−0.689611
$53$	1.30798	0.179665	0.0898323	−	0.995957i	$-0.471367\pi$
$53$	1.30798	0.179665	0.0898323	+	0.995957i	$0.471367\pi$
$54$	−3.13706	−0.426900
$55$	−6.08575	−0.820603
$56$	−9.93900	−1.32815
$57$	−4.11960	−0.545655
$58$	4.30798	0.565665
$59$	−7.08815	−0.922798	−0.461399	−	0.887193i	$-0.652652\pi$
$59$	−7.08815	−0.922798	−0.461399	+	0.887193i	$0.652652\pi$
$60$	2.56465	0.331094
$61$	−13.2620	−1.69803	−0.849015	−	0.528368i	$-0.822804\pi$
$61$	−13.2620	−1.69803	−0.849015	+	0.528368i	$0.822804\pi$
$62$	−0.149145	−0.0189414
$63$	4.42327	0.557280
$64$	−1.52781	−0.190976
$65$	3.08277	0.382370
$66$	−4.65279	−0.572719
$67$	−12.7017	−1.55176	−0.775880	−	0.630880i	$-0.782694\pi$
$67$	−12.7017	−1.55176	−0.775880	+	0.630880i	$0.782694\pi$
$68$	−3.98792	−0.483606
$69$	0.325437	0.0391780
$70$	2.82371	0.337498
$71$	4.34481	0.515635	0.257817	−	0.966194i	$-0.416997\pi$
$71$	4.34481	0.515635	0.257817	+	0.966194i	$0.416997\pi$
$72$	−1.86831	−0.220183
$73$	5.19806	0.608387	0.304194	−	0.952610i	$-0.401613\pi$
$73$	5.19806	0.608387	0.304194	+	0.952610i	$0.401613\pi$
$74$	0.939001	0.109157
$75$	5.63533	0.650712
$76$	−4.82371	−0.553317
$77$	28.1444	3.20735
$78$	2.35690	0.266866
$79$	8.83877	0.994440	0.497220	−	0.867625i	$-0.334354\pi$
$79$	8.83877	0.994440	0.497220	+	0.867625i	$0.334354\pi$
$80$	2.35690	0.263509
$81$	−5.43296	−0.603662
$82$	6.18598	0.683127
$83$	3.00969	0.330356	0.165178	−	0.986264i	$-0.447180\pi$
$83$	3.00969	0.330356	0.165178	+	0.986264i	$0.447180\pi$
$84$	−11.8605	−1.29409
$85$	2.47219	0.268147
$86$	−4.48188	−0.483293
$87$	11.2174	1.20264
$88$	−11.8877	−1.26723
$89$	−1.92394	−0.203937	−0.101968	−	0.994788i	$-0.532514\pi$
$89$	−1.92394	−0.203937	−0.101968	+	0.994788i	$0.532514\pi$
$90$	0.530795	0.0559507
$91$	−14.2567	−1.49450
$92$	0.381059	0.0397282
$93$	−0.388355	−0.0402705
$94$	1.53989	0.158828
$95$	2.99031	0.306799
$96$	7.72348	0.788274
$97$	1.23490	0.125385	0.0626924	−	0.998033i	$-0.480031\pi$
$97$	1.23490	0.125385	0.0626924	+	0.998033i	$0.480031\pi$
$98$	−9.17390	−0.926704
$99$	5.29052	0.531717
$100$	6.59850	0.659850
$101$	5.81402	0.578517	0.289258	−	0.957251i	$-0.406591\pi$
$101$	5.81402	0.578517	0.289258	+	0.957251i	$0.406591\pi$
$102$	1.89008	0.187146
$103$	9.05429	0.892146	0.446073	−	0.894997i	$-0.352822\pi$
$103$	9.05429	0.892146	0.446073	+	0.894997i	$0.352822\pi$
$104$	6.02177	0.590483
$105$	7.35258	0.717539
$106$	−0.725873	−0.0705030
$107$	−4.25667	−0.411508	−0.205754	−	0.978604i	$-0.565965\pi$
$107$	−4.25667	−0.411508	−0.205754	+	0.978604i	$0.565965\pi$
$108$	−9.56465	−0.920359
$109$	−1.00000	−0.0957826
$109$	−1.00000	−0.0957826
$110$	3.37734	0.322017
$111$	2.44504	0.232073
$112$	−10.8998	−1.02993
$113$	−3.47219	−0.326636	−0.163318	−	0.986573i	$-0.552220\pi$
$113$	−3.47219	−0.326636	−0.163318	+	0.986573i	$0.552220\pi$
$114$	2.28621	0.214123
$115$	−0.236226	−0.0220282
$116$	13.1347	1.21952
$117$	−2.67994	−0.247761
$118$	3.93362	0.362119
$119$	−11.4330	−1.04806
$120$	−3.10560	−0.283502
$121$	22.6625	2.06023
$122$	7.35988	0.666332
$123$	16.1075	1.45237
$124$	−0.454731	−0.0408360
$125$	−9.33513	−0.834959
$126$	−2.45473	−0.218685
$127$	−6.37196	−0.565420	−0.282710	−	0.959205i	$-0.591233\pi$
$127$	−6.37196	−0.565420	−0.282710	+	0.959205i	$0.591233\pi$
$128$	11.5375	1.01978
$129$	−11.6703	−1.02751
$130$	−1.71081	−0.150048
$131$	−15.5375	−1.35752	−0.678759	−	0.734361i	$-0.737482\pi$
$131$	−15.5375	−1.35752	−0.678759	+	0.734361i	$0.737482\pi$
$132$	−14.1860	−1.23473
$133$	−13.8291	−1.19913
$134$	7.04892	0.608934
$135$	5.92931	0.510314
$136$	4.82908	0.414091
$137$	1.80731	0.154409	0.0772047	−	0.997015i	$-0.475401\pi$
$137$	1.80731	0.154409	0.0772047	+	0.997015i	$0.475401\pi$
$138$	−0.180604	−0.0153740
$139$	8.14675	0.690998	0.345499	−	0.938419i	$-0.387710\pi$
$139$	8.14675	0.690998	0.345499	+	0.938419i	$0.387710\pi$
$140$	8.60925	0.727615
$141$	4.00969	0.337677
$142$	−2.41119	−0.202343
$143$	−17.0519	−1.42595
$144$	−2.04892	−0.170743
$145$	−8.14244	−0.676193
$146$	−2.88471	−0.238740
$147$	−23.8877	−1.97022
$148$	2.86294	0.235332
$149$	−6.65279	−0.545018	−0.272509	−	0.962153i	$-0.587853\pi$
$149$	−6.65279	−0.545018	−0.272509	+	0.962153i	$0.587853\pi$
$150$	−3.12737	−0.255349
$151$	8.38835	0.682635	0.341317	−	0.939948i	$-0.389127\pi$
$151$	8.38835	0.682635	0.341317	+	0.939948i	$0.389127\pi$
$152$	5.84117	0.473781
$153$	−2.14914	−0.173748
$154$	−15.6189	−1.25861
$155$	0.281896	0.0226425
$156$	7.18598	0.575339
$157$	10.0151	0.799289	0.399645	−	0.916670i	$-0.369134\pi$
$157$	10.0151	0.799289	0.399645	+	0.916670i	$0.369134\pi$
$158$	−4.90515	−0.390233
$159$	−1.89008	−0.149893
$160$	−5.60627	−0.443214
$161$	1.09246	0.0860977
$162$	3.01507	0.236886
$163$	−12.0761	−0.945870	−0.472935	−	0.881097i	$-0.656806\pi$
$163$	−12.0761	−0.945870	−0.472935	+	0.881097i	$0.656806\pi$
$164$	18.8605	1.47276
$165$	8.79417	0.684625
$166$	−1.67025	−0.129637
$167$	−8.03684	−0.621909	−0.310955	−	0.950425i	$-0.600649\pi$
$167$	−8.03684	−0.621909	−0.310955	+	0.950425i	$0.600649\pi$
$168$	14.3623	1.10807
$169$	−4.36227	−0.335559
$170$	−1.37196	−0.105225
$171$	−2.59956	−0.198794
$172$	−13.6649	−1.04194
$173$	8.08575	0.614748	0.307374	−	0.951589i	$-0.400550\pi$
$173$	8.08575	0.614748	0.307374	+	0.951589i	$0.400550\pi$
$174$	−6.22521	−0.471932
$175$	18.9172	1.43001
$176$	−13.0368	−0.982688
$177$	10.2427	0.769886
$178$	1.06770	0.0800278
$179$	−8.71917	−0.651701	−0.325851	−	0.945421i	$-0.605651\pi$
$179$	−8.71917	−0.651701	−0.325851	+	0.945421i	$0.605651\pi$
$180$	1.61835	0.120625
$181$	−19.8605	−1.47622	−0.738111	−	0.674679i	$-0.764282\pi$
$181$	−19.8605	−1.47622	−0.738111	+	0.674679i	$0.764282\pi$
$182$	7.91185	0.586466
$183$	19.1642	1.41666
$184$	−0.461435	−0.0340175
$185$	−1.77479	−0.130485
$186$	0.215521	0.0158027
$187$	−13.6746	−0.999983
$188$	4.69501	0.342418
$189$	−27.4209	−1.99458
$190$	−1.65950	−0.120393
$191$	8.77346	0.634825	0.317413	−	0.948287i	$-0.397186\pi$
$191$	8.77346	0.634825	0.317413	+	0.948287i	$0.397186\pi$
$192$	2.20775	0.159331
$193$	−2.18300	−0.157136	−0.0785678	−	0.996909i	$-0.525035\pi$
$193$	−2.18300	−0.157136	−0.0785678	+	0.996909i	$0.525035\pi$
$194$	−0.685317	−0.0492029
$195$	−4.45473	−0.319010
$196$	−27.9705	−1.99789
$197$	27.0629	1.92815	0.964077	−	0.265625i	$-0.0855782\pi$
$197$	27.0629	1.92815	0.964077	+	0.265625i	$0.0855782\pi$
$198$	−2.93602	−0.208654
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−7.99031	−0.565000
$201$	18.3545	1.29463
$202$	−3.22654	−0.227018
$203$	37.6558	2.64292
$204$	5.76271	0.403470
$205$	−11.6920	−0.816606
$206$	−5.02475	−0.350091
$207$	0.205358	0.0142734
$208$	6.60388	0.457896
$209$	−16.5405	−1.14413
$210$	−4.08038	−0.281573
$211$	16.1371	1.11092	0.555461	−	0.831543i	$-0.312542\pi$
$211$	16.1371	1.11092	0.555461	+	0.831543i	$0.312542\pi$
$212$	−2.21313	−0.151998
$213$	−6.27844	−0.430192
$214$	2.36227	0.161482
$215$	8.47112	0.577726
$216$	11.5821	0.788062
$217$	−1.30367	−0.0884987
$218$	0.554958	0.0375865
$219$	−7.51142	−0.507575
$220$	10.2972	0.694239
$221$	6.92692	0.465955
$222$	−1.35690	−0.0910689
$223$	24.9312	1.66952	0.834759	−	0.550615i	$-0.185607\pi$
$223$	24.9312	1.66952	0.834759	+	0.550615i	$0.185607\pi$
$224$	25.9269	1.73232
$225$	3.55602	0.237068
$226$	1.92692	0.128177
$227$	−8.97285	−0.595549	−0.297775	−	0.954636i	$-0.596244\pi$
$227$	−8.97285	−0.595549	−0.297775	+	0.954636i	$0.596244\pi$
$228$	6.97046	0.461630
$229$	−10.9366	−0.722711	−0.361356	−	0.932428i	$-0.617686\pi$
$229$	−10.9366	−0.722711	−0.361356	+	0.932428i	$0.617686\pi$
$230$	0.131096	0.00864418
$231$	−40.6698	−2.67587
$232$	−15.9051	−1.04422
$233$	−5.33513	−0.349516	−0.174758	−	0.984611i	$-0.555914\pi$
$233$	−5.33513	−0.349516	−0.174758	+	0.984611i	$0.555914\pi$
$234$	1.48725	0.0972249
$235$	−2.91053	−0.189862
$236$	11.9933	0.780697
$237$	−12.7724	−0.829656
$238$	6.34481	0.411273
$239$	−24.5308	−1.58677	−0.793383	−	0.608723i	$-0.791682\pi$
$239$	−24.5308	−1.58677	−0.793383	+	0.608723i	$0.791682\pi$
$240$	−3.40581	−0.219844
$241$	17.7017	1.14027	0.570134	−	0.821552i	$-0.306891\pi$
$241$	17.7017	1.14027	0.570134	+	0.821552i	$0.306891\pi$
$242$	−12.5767	−0.808463
$243$	−9.10752	−0.584248
$244$	22.4397	1.43655
$245$	17.3394	1.10778
$246$	−8.93900	−0.569930
$247$	8.37867	0.533122
$248$	0.550646	0.0349661
$249$	−4.34913	−0.275615
$250$	5.18060	0.327650
$251$	−22.9976	−1.45160	−0.725798	−	0.687908i	$-0.758529\pi$
$251$	−22.9976	−1.45160	−0.725798	+	0.687908i	$0.758529\pi$
$252$	−7.48427	−0.471465
$253$	1.30665	0.0821484
$254$	3.53617	0.221879
$255$	−3.57242	−0.223713
$256$	−3.34721	−0.209200
$257$	9.93900	0.619978	0.309989	−	0.950740i	$-0.399675\pi$
$257$	9.93900	0.619978	0.309989	+	0.950740i	$0.399675\pi$
$258$	6.47650	0.403209
$259$	8.20775	0.510005
$260$	−5.21611	−0.323490
$261$	7.07846	0.438146
$262$	8.62266	0.532710
$263$	−9.03684	−0.557235	−0.278618	−	0.960402i	$-0.589876\pi$
$263$	−9.03684	−0.557235	−0.278618	+	0.960402i	$0.589876\pi$
$264$	17.1782	1.05725
$265$	1.37196	0.0842789
$266$	7.67456	0.470558
$267$	2.78017	0.170143
$268$	21.4916	1.31281
$269$	−8.37627	−0.510710	−0.255355	−	0.966847i	$-0.582192\pi$
$269$	−8.37627	−0.510710	−0.255355	+	0.966847i	$0.582192\pi$
$270$	−3.29052	−0.200255
$271$	26.3991	1.60363	0.801816	−	0.597571i	$-0.203867\pi$
$271$	26.3991	1.60363	0.801816	+	0.597571i	$0.203867\pi$
$272$	5.29590	0.321111
$273$	20.6015	1.24686
$274$	−1.00298	−0.0605925
$275$	22.6262	1.36441
$276$	−0.550646	−0.0331450
$277$	11.4862	0.690138	0.345069	−	0.938577i	$-0.387855\pi$
$277$	11.4862	0.690138	0.345069	+	0.938577i	$0.387855\pi$
$278$	−4.52111	−0.271158
$279$	−0.245061	−0.0146714
$280$	−10.4252	−0.623024
$281$	6.34183	0.378322	0.189161	−	0.981946i	$-0.439423\pi$
$281$	6.34183	0.378322	0.189161	+	0.981946i	$0.439423\pi$
$282$	−2.22521	−0.132509
$283$	−14.2078	−0.844563	−0.422282	−	0.906465i	$-0.638771\pi$
$283$	−14.2078	−0.844563	−0.422282	+	0.906465i	$0.638771\pi$
$284$	−7.35152	−0.436232
$285$	−4.32113	−0.255961
$286$	9.46309	0.559564
$287$	54.0713	3.19173
$288$	4.87369	0.287185
$289$	−11.4450	−0.673238
$290$	4.51871	0.265348
$291$	−1.78448	−0.104608
$292$	−8.79523	−0.514702
$293$	−27.0834	−1.58223	−0.791113	−	0.611670i	$-0.790498\pi$
$293$	−27.0834	−1.58223	−0.791113	+	0.611670i	$0.790498\pi$
$294$	13.2567	0.773145
$295$	−7.43488	−0.432875
$296$	−3.46681	−0.201504
$297$	−32.7972	−1.90308
$298$	3.69202	0.213873
$299$	−0.661890	−0.0382781
$300$	−9.53511	−0.550510
$301$	−39.1758	−2.25806
$302$	−4.65519	−0.267876
$303$	−8.40150	−0.482654
$304$	6.40581	0.367399
$305$	−13.9108	−0.796529
$306$	1.19269	0.0681813
$307$	−24.7972	−1.41525	−0.707624	−	0.706589i	$-0.750233\pi$
$307$	−24.7972	−1.41525	−0.707624	+	0.706589i	$0.750233\pi$
$308$	−47.6209	−2.71345
$309$	−13.0838	−0.744313
$310$	−0.156441	−0.00888523
$311$	15.2814	0.866530	0.433265	−	0.901267i	$-0.357361\pi$
$311$	15.2814	0.866530	0.433265	+	0.901267i	$0.357361\pi$
$312$	−8.70171	−0.492637
$313$	−7.32304	−0.413923	−0.206961	−	0.978349i	$-0.566357\pi$
$313$	−7.32304	−0.413923	−0.206961	+	0.978349i	$0.566357\pi$
$314$	−5.55794	−0.313653
$315$	4.63965	0.261414
$316$	−14.9554	−0.841307
$317$	−16.8974	−0.949051	−0.474526	−	0.880242i	$-0.657380\pi$
$317$	−16.8974	−0.949051	−0.474526	+	0.880242i	$0.657380\pi$
$318$	1.04892	0.0588204
$319$	45.0388	2.52169
$320$	−1.60255	−0.0895851
$321$	6.15106	0.343319
$322$	−0.606268	−0.0337860
$323$	6.71917	0.373865
$324$	9.19269	0.510705
$325$	−11.4614	−0.635766
$326$	6.70171	0.371173
$327$	1.44504	0.0799110
$328$	−22.8388	−1.26106
$329$	13.4601	0.742080
$330$	−4.88040	−0.268657
$331$	−12.4166	−0.682476	−0.341238	−	0.939977i	$-0.610846\pi$
$331$	−12.4166	−0.682476	−0.341238	+	0.939977i	$0.610846\pi$
$332$	−5.09246	−0.279485
$333$	1.54288	0.0845491
$334$	4.46011	0.244046
$335$	−13.3230	−0.727916
$336$	15.7506	0.859267
$337$	9.85756	0.536976	0.268488	−	0.963283i	$-0.413476\pi$
$337$	9.85756	0.536976	0.268488	+	0.963283i	$0.413476\pi$
$338$	2.42088	0.131678
$339$	5.01746	0.272511
$340$	−4.18300	−0.226855
$341$	−1.55927	−0.0844392
$342$	1.44265	0.0780095
$343$	−46.2325	−2.49632
$344$	16.5472	0.892165
$345$	0.341356	0.0183780
$346$	−4.48725	−0.241236
$347$	−28.6407	−1.53751	−0.768757	−	0.639541i	$-0.779124\pi$
$347$	−28.6407	−1.53751	−0.768757	+	0.639541i	$0.779124\pi$
$348$	−18.9801	−1.01744
$349$	−6.01208	−0.321819	−0.160910	−	0.986969i	$-0.551443\pi$
$349$	−6.01208	−0.321819	−0.160910	+	0.986969i	$0.551443\pi$
$350$	−10.4983	−0.561156
$351$	16.6136	0.886766
$352$	31.0103	1.65285
$353$	8.57002	0.456136	0.228068	−	0.973645i	$-0.426759\pi$
$353$	8.57002	0.456136	0.228068	+	0.973645i	$0.426759\pi$
$354$	−5.68425	−0.302115
$355$	4.55735	0.241879
$356$	3.25534	0.172533
$357$	16.5211	0.874390
$358$	4.83877	0.255737
$359$	32.4849	1.71449	0.857243	−	0.514913i	$-0.172176\pi$
$359$	32.4849	1.71449	0.857243	+	0.514913i	$0.172176\pi$
$360$	−1.95971	−0.103286
$361$	−10.8726	−0.572243
$362$	11.0218	0.579291
$363$	−32.7482	−1.71884
$364$	24.1226	1.26437
$365$	5.45234	0.285388
$366$	−10.6353	−0.555918
$367$	6.39612	0.333875	0.166937	−	0.985967i	$-0.446612\pi$
$367$	6.39612	0.333875	0.166937	+	0.985967i	$0.446612\pi$
$368$	−0.506041	−0.0263792
$369$	10.1642	0.529128
$370$	0.984935	0.0512043
$371$	−6.34481	−0.329406
$372$	0.657105	0.0340693
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	7.58881	0.392408
$375$	13.4896	0.696602
$376$	−5.68532	−0.293198
$377$	−22.8146	−1.17501
$378$	15.2174	0.782701
$379$	0.158834	0.00815873	0.00407937	−	0.999992i	$-0.498701\pi$
$379$	0.158834	0.00815873	0.00407937	+	0.999992i	$0.498701\pi$
$380$	−5.05967	−0.259556
$381$	9.20775	0.471727
$382$	−4.86890	−0.249115
$383$	−19.9433	−1.01906	−0.509528	−	0.860454i	$-0.670180\pi$
$383$	−19.9433	−1.01906	−0.509528	+	0.860454i	$0.670180\pi$
$384$	−16.6722	−0.850798
$385$	29.5211	1.50453
$386$	1.21147	0.0616623
$387$	−7.36419	−0.374343
$388$	−2.08947	−0.106077
$389$	19.8291	1.00537	0.502687	−	0.864468i	$-0.332345\pi$
$389$	19.8291	1.00537	0.502687	+	0.864468i	$0.332345\pi$
$390$	2.47219	0.125184
$391$	−0.530795	−0.0268435
$392$	33.8702	1.71071
$393$	22.4523	1.13257
$394$	−15.0188	−0.756636
$395$	9.27114	0.466482
$396$	−8.95167	−0.449839
$397$	15.2131	0.763525	0.381762	−	0.924260i	$-0.375317\pi$
$397$	15.2131	0.763525	0.381762	+	0.924260i	$0.375317\pi$
$398$	0	0
$399$	19.9836	1.00043
$400$	−8.76271	−0.438135
$401$	16.9530	0.846593	0.423296	−	0.905991i	$-0.360873\pi$
$401$	16.9530	0.846593	0.423296	+	0.905991i	$0.360873\pi$
$402$	−10.1860	−0.508031
$403$	0.789856	0.0393455
$404$	−9.83745	−0.489431
$405$	−5.69873	−0.283172
$406$	−20.8974	−1.03712
$407$	9.81700	0.486611
$408$	−6.97823	−0.345474
$409$	−14.4383	−0.713930	−0.356965	−	0.934118i	$-0.616188\pi$
$409$	−14.4383	−0.713930	−0.356965	+	0.934118i	$0.616188\pi$
$410$	6.48858	0.320448
$411$	−2.61165	−0.128823
$412$	−15.3201	−0.754765
$413$	34.3836	1.69190
$414$	−0.113965	−0.00560108
$415$	3.15691	0.154967
$416$	−15.7084	−0.770168
$417$	−11.7724	−0.576497
$418$	9.17928	0.448973
$419$	2.85517	0.139484	0.0697420	−	0.997565i	$-0.477782\pi$
$419$	2.85517	0.139484	0.0697420	+	0.997565i	$0.477782\pi$
$420$	−12.4407	−0.607045
$421$	4.52350	0.220462	0.110231	−	0.993906i	$-0.464841\pi$
$421$	4.52350	0.220462	0.110231	+	0.993906i	$0.464841\pi$
$422$	−8.95539	−0.435942
$423$	2.53020	0.123023
$424$	2.67994	0.130149
$425$	−9.19136	−0.445846
$426$	3.48427	0.168813
$427$	64.3323	3.11326
$428$	7.20237	0.348140
$429$	24.6407	1.18966
$430$	−4.70112	−0.226708
$431$	29.0925	1.40133	0.700667	−	0.713488i	$-0.252886\pi$
$431$	29.0925	1.40133	0.700667	+	0.713488i	$0.252886\pi$
$432$	12.7017	0.611111
$433$	−7.81940	−0.375776	−0.187888	−	0.982190i	$-0.560164\pi$
$433$	−7.81940	−0.375776	−0.187888	+	0.982190i	$0.560164\pi$
$434$	0.723480	0.0347282
$435$	11.7662	0.564145
$436$	1.69202	0.0810331
$437$	−0.642039	−0.0307129
$438$	4.16852	0.199180
$439$	−28.7211	−1.37078	−0.685391	−	0.728175i	$-0.740369\pi$
$439$	−28.7211	−1.37078	−0.685391	+	0.728175i	$0.740369\pi$
$440$	−12.4692	−0.594446
$441$	−15.0737	−0.717794
$442$	−3.84415	−0.182848
$443$	−40.7861	−1.93781	−0.968904	−	0.247437i	$-0.920412\pi$
$443$	−40.7861	−1.93781	−0.968904	+	0.247437i	$0.920412\pi$
$444$	−4.13706	−0.196336
$445$	−2.01805	−0.0956647
$446$	−13.8358	−0.655144
$447$	9.61356	0.454706
$448$	7.41119	0.350146
$449$	−27.3709	−1.29171	−0.645856	−	0.763459i	$-0.723499\pi$
$449$	−27.3709	−1.29171	−0.645856	+	0.763459i	$0.723499\pi$
$450$	−1.97344	−0.0930290
$451$	64.6728	3.04532
$452$	5.87502	0.276338
$453$	−12.1215	−0.569519
$454$	4.97956	0.233702
$455$	−14.9541	−0.701057
$456$	−8.44073	−0.395273
$457$	−24.6679	−1.15391	−0.576957	−	0.816775i	$-0.695760\pi$
$457$	−24.6679	−1.15391	−0.576957	+	0.816775i	$0.695760\pi$
$458$	6.06936	0.283603
$459$	13.3230	0.621866
$460$	0.399699	0.0186361
$461$	3.78017	0.176060	0.0880300	−	0.996118i	$-0.471943\pi$
$461$	3.78017	0.176060	0.0880300	+	0.996118i	$0.471943\pi$
$462$	22.5700	1.05005
$463$	−13.2301	−0.614855	−0.307428	−	0.951571i	$-0.599468\pi$
$463$	−13.2301	−0.614855	−0.307428	+	0.951571i	$0.599468\pi$
$464$	−17.4426	−0.809755
$465$	−0.407352	−0.0188905
$466$	2.96077	0.137155
$467$	−16.2687	−0.752828	−0.376414	−	0.926452i	$-0.622843\pi$
$467$	−16.2687	−0.752828	−0.376414	+	0.926452i	$0.622843\pi$
$468$	4.53452	0.209608
$469$	61.6142	2.84508
$470$	1.61522	0.0745045
$471$	−14.4722	−0.666843
$472$	−14.5230	−0.668476
$473$	−46.8568	−2.15448
$474$	7.08815	0.325569
$475$	−11.1177	−0.510114
$476$	19.3448	0.886668
$477$	−1.19269	−0.0546093
$478$	13.6136	0.622670
$479$	−2.00670	−0.0916887	−0.0458443	−	0.998949i	$-0.514598\pi$
$479$	−2.00670	−0.0916887	−0.0458443	+	0.998949i	$0.514598\pi$
$480$	8.10129	0.369772
$481$	−4.97285	−0.226743
$482$	−9.82371	−0.447458
$483$	−1.57865	−0.0718309
$484$	−38.3454	−1.74297
$485$	1.29531	0.0588168
$486$	5.05429	0.229267
$487$	10.1304	0.459050	0.229525	−	0.973303i	$-0.426283\pi$
$487$	10.1304	0.459050	0.229525	+	0.973303i	$0.426283\pi$
$488$	−27.1728	−1.23006
$489$	17.4504	0.789135
$490$	−9.62266	−0.434708
$491$	28.5284	1.28747	0.643734	−	0.765249i	$-0.277384\pi$
$491$	28.5284	1.28747	0.643734	+	0.765249i	$0.277384\pi$
$492$	−27.2543	−1.22872
$493$	−18.2959	−0.824006
$494$	−4.64981	−0.209205
$495$	5.54932	0.249423
$496$	0.603875	0.0271148
$497$	−21.0761	−0.945391
$498$	2.41358	0.108155
$499$	11.8931	0.532407	0.266203	−	0.963917i	$-0.414231\pi$
$499$	11.8931	0.532407	0.266203	+	0.963917i	$0.414231\pi$
$500$	15.7952	0.706384
$501$	11.6136	0.518856
$502$	12.7627	0.569628
$503$	34.2247	1.52601	0.763003	−	0.646395i	$-0.223724\pi$
$503$	34.2247	1.52601	0.763003	+	0.646395i	$0.223724\pi$
$504$	9.06292	0.403694
$505$	6.09843	0.271376
$506$	−0.725136	−0.0322362
$507$	6.30367	0.279956
$508$	10.7815	0.478352
$509$	2.55257	0.113140	0.0565702	−	0.998399i	$-0.481984\pi$
$509$	2.55257	0.113140	0.0565702	+	0.998399i	$0.481984\pi$
$510$	1.98254	0.0877884
$511$	−25.2150	−1.11545
$512$	−21.2174	−0.937687
$513$	16.1153	0.711508
$514$	−5.51573	−0.243288
$515$	9.49721	0.418497
$516$	19.7463	0.869283
$517$	16.0992	0.708040
$518$	−4.55496	−0.200133
$519$	−11.6843	−0.512882
$520$	6.31634	0.276990
$521$	−17.6799	−0.774572	−0.387286	−	0.921960i	$-0.626587\pi$
$521$	−17.6799	−0.774572	−0.387286	+	0.921960i	$0.626587\pi$
$522$	−3.92825	−0.171935
$523$	2.10215	0.0919205	0.0459602	−	0.998943i	$-0.485365\pi$
$523$	2.10215	0.0919205	0.0459602	+	0.998943i	$0.485365\pi$
$524$	26.2898	1.14847
$525$	−27.3362	−1.19305
$526$	5.01507	0.218667
$527$	0.633415	0.0275920
$528$	18.8388	0.819852
$529$	−22.9493	−0.997795
$530$	−0.761381	−0.0330723
$531$	6.46335	0.280486
$532$	23.3991	1.01448
$533$	−32.7603	−1.41901
$534$	−1.54288	−0.0667668
$535$	−4.46489	−0.193034
$536$	−26.0248	−1.12410
$537$	12.5996	0.543711
$538$	4.64848	0.200410
$539$	−95.9106	−4.13116
$540$	−10.0325	−0.431731
$541$	−17.1666	−0.738050	−0.369025	−	0.929419i	$-0.620308\pi$
$541$	−17.1666	−0.738050	−0.369025	+	0.929419i	$0.620308\pi$
$542$	−14.6504	−0.629289
$543$	28.6993	1.23161
$544$	−12.5972	−0.540099
$545$	−1.04892	−0.0449307
$546$	−11.4330	−0.489286
$547$	15.5821	0.666243	0.333121	−	0.942884i	$-0.391898\pi$
$547$	15.5821	0.666243	0.333121	+	0.942884i	$0.391898\pi$
$548$	−3.05802	−0.130632
$549$	12.0930	0.516119
$550$	−12.5566	−0.535416
$551$	−22.1304	−0.942785
$552$	0.666793	0.0283806
$553$	−42.8756	−1.82326
$554$	−6.37435	−0.270820
$555$	2.56465	0.108863
$556$	−13.7845	−0.584592
$557$	26.2717	1.11317	0.556585	−	0.830791i	$-0.312112\pi$
$557$	26.2717	1.11317	0.556585	+	0.830791i	$0.312112\pi$
$558$	0.135998	0.00575727
$559$	23.7356	1.00391
$560$	−11.4330	−0.483131
$561$	19.7603	0.834281
$562$	−3.51945	−0.148459
$563$	20.0597	0.845414	0.422707	−	0.906266i	$-0.361080\pi$
$563$	20.0597	0.845414	0.422707	+	0.906266i	$0.361080\pi$
$564$	−6.78448	−0.285678
$565$	−3.64204	−0.153222
$566$	7.88471	0.331419
$567$	26.3545	1.10679
$568$	8.90217	0.373527
$569$	−41.9124	−1.75706	−0.878531	−	0.477686i	$-0.841476\pi$
$569$	−41.9124	−1.75706	−0.878531	+	0.477686i	$0.841476\pi$
$570$	2.39804	0.100443
$571$	3.84415	0.160873	0.0804363	−	0.996760i	$-0.474369\pi$
$571$	3.84415	0.160873	0.0804363	+	0.996760i	$0.474369\pi$
$572$	28.8522	1.20637
$573$	−12.6780	−0.529632
$574$	−30.0073	−1.25248
$575$	0.878265	0.0366262
$576$	1.39314	0.0580475
$577$	−42.6795	−1.77677	−0.888385	−	0.459100i	$-0.848172\pi$
$577$	−42.6795	−1.77677	−0.888385	+	0.459100i	$0.848172\pi$
$578$	6.35152	0.264188
$579$	3.15452	0.131097
$580$	13.7772	0.572066
$581$	−14.5996	−0.605692
$582$	0.990311	0.0410497
$583$	−7.58881	−0.314296
$584$	10.6504	0.440717
$585$	−2.81104	−0.116222
$586$	15.0301	0.620889
$587$	−22.8552	−0.943334	−0.471667	−	0.881777i	$-0.656347\pi$
$587$	−22.8552	−0.943334	−0.471667	+	0.881777i	$0.656347\pi$
$588$	40.4185	1.66683
$589$	0.766167	0.0315694
$590$	4.12605	0.169867
$591$	−39.1070	−1.60865
$592$	−3.80194	−0.156259
$593$	9.91617	0.407208	0.203604	−	0.979053i	$-0.434734\pi$
$593$	9.91617	0.407208	0.203604	+	0.979053i	$0.434734\pi$
$594$	18.2010	0.746798
$595$	−11.9922	−0.491634
$596$	11.2567	0.461091
$597$	0	0
$598$	0.367322	0.0150209
$599$	3.52052	0.143844	0.0719222	−	0.997410i	$-0.477087\pi$
$599$	3.52052	0.143844	0.0719222	+	0.997410i	$0.477087\pi$
$600$	11.5463	0.471377
$601$	36.4379	1.48633	0.743166	−	0.669107i	$-0.233323\pi$
$601$	36.4379	1.48633	0.743166	+	0.669107i	$0.233323\pi$
$602$	21.7409	0.886095
$603$	11.5821	0.471660
$604$	−14.1933	−0.577516
$605$	23.7711	0.966431
$606$	4.66248	0.189400
$607$	27.9168	1.13311	0.566553	−	0.824025i	$-0.308277\pi$
$607$	27.9168	1.13311	0.566553	+	0.824025i	$0.308277\pi$
$608$	−15.2373	−0.617954
$609$	−54.4142	−2.20497
$610$	7.71991	0.312570
$611$	−8.15511	−0.329921
$612$	3.63640	0.146993
$613$	5.15106	0.208050	0.104025	−	0.994575i	$-0.466828\pi$
$613$	5.15106	0.208050	0.104025	+	0.994575i	$0.466828\pi$
$614$	13.7614	0.555364
$615$	16.8955	0.681291
$616$	57.6655	2.32341
$617$	−29.5526	−1.18974	−0.594871	−	0.803821i	$-0.702797\pi$
$617$	−29.5526	−1.18974	−0.594871	+	0.803821i	$0.702797\pi$
$618$	7.26098	0.292079
$619$	−45.0025	−1.80880	−0.904402	−	0.426682i	$-0.859682\pi$
$619$	−45.0025	−1.80880	−0.904402	+	0.426682i	$0.859682\pi$
$620$	−0.476975	−0.0191558
$621$	−1.27306	−0.0510862
$622$	−8.48055	−0.340039
$623$	9.33273	0.373908
$624$	−9.54288	−0.382021
$625$	9.70709	0.388283
$626$	4.06398	0.162429
$627$	23.9017	0.954542
$628$	−16.9457	−0.676207
$629$	−3.98792	−0.159009
$630$	−2.57481	−0.102583
$631$	1.14483	0.0455751	0.0227875	−	0.999740i	$-0.492746\pi$
$631$	1.14483	0.0455751	0.0227875	+	0.999740i	$0.492746\pi$
$632$	18.1099	0.720374
$633$	−23.3187	−0.926836
$634$	9.37734	0.372422
$635$	−6.68366	−0.265233
$636$	3.19806	0.126811
$637$	48.5840	1.92497
$638$	−24.9946	−0.989547
$639$	−3.96184	−0.156728
$640$	12.1019	0.478369
$641$	10.2209	0.403701	0.201851	−	0.979416i	$-0.435304\pi$
$641$	10.2209	0.403701	0.201851	+	0.979416i	$0.435304\pi$
$642$	−3.41358	−0.134723
$643$	7.08336	0.279340	0.139670	−	0.990198i	$-0.455396\pi$
$643$	7.08336	0.279340	0.139670	+	0.990198i	$0.455396\pi$
$644$	−1.84846	−0.0728396
$645$	−12.2411	−0.481994
$646$	−3.72886	−0.146710
$647$	−13.9554	−0.548643	−0.274322	−	0.961638i	$-0.588453\pi$
$647$	−13.9554	−0.548643	−0.274322	+	0.961638i	$0.588453\pi$
$648$	−11.1317	−0.437294
$649$	41.1250	1.61430
$650$	6.36062	0.249484
$651$	1.88385	0.0738340
$652$	20.4330	0.800216
$653$	31.1353	1.21842	0.609208	−	0.793010i	$-0.291487\pi$
$653$	31.1353	1.21842	0.609208	+	0.793010i	$0.291487\pi$
$654$	−0.801938	−0.0313582
$655$	−16.2976	−0.636798
$656$	−25.0465	−0.977902
$657$	−4.73987	−0.184920
$658$	−7.46980	−0.291203
$659$	26.5502	1.03425	0.517124	−	0.855911i	$-0.327003\pi$
$659$	26.5502	1.03425	0.517124	+	0.855911i	$0.327003\pi$
$660$	−14.8799	−0.579200
$661$	−19.7657	−0.768796	−0.384398	−	0.923167i	$-0.625591\pi$
$661$	−19.7657	−0.768796	−0.384398	+	0.923167i	$0.625591\pi$
$662$	6.89067	0.267814
$663$	−10.0097	−0.388744
$664$	6.16660	0.239311
$665$	−14.5056	−0.562502
$666$	−0.856232	−0.0331783
$667$	1.74823	0.0676919
$668$	13.5985	0.526142
$669$	−36.0267	−1.39287
$670$	7.39373	0.285645
$671$	76.9456	2.97045
$672$	−37.4655	−1.44526
$673$	32.2911	1.24473	0.622365	−	0.782727i	$-0.286172\pi$
$673$	32.2911	1.24473	0.622365	+	0.782727i	$0.286172\pi$
$674$	−5.47053	−0.210717
$675$	−22.0446	−0.848497
$676$	7.38106	0.283887
$677$	41.9705	1.61306	0.806528	−	0.591196i	$-0.201344\pi$
$677$	41.9705	1.61306	0.806528	+	0.591196i	$0.201344\pi$
$678$	−2.78448	−0.106937
$679$	−5.99031	−0.229887
$680$	5.06531	0.194246
$681$	12.9661	0.496864
$682$	0.865330	0.0331352
$683$	−37.6276	−1.43978	−0.719889	−	0.694089i	$-0.755807\pi$
$683$	−37.6276	−1.43978	−0.719889	+	0.694089i	$0.755807\pi$
$684$	4.39852	0.168182
$685$	1.89572	0.0724319
$686$	25.6571	0.979593
$687$	15.8039	0.602955
$688$	18.1468	0.691838
$689$	3.84415	0.146450
$690$	−0.189439	−0.00721180
$691$	5.71140	0.217272	0.108636	−	0.994082i	$-0.465352\pi$
$691$	5.71140	0.217272	0.108636	+	0.994082i	$0.465352\pi$
$692$	−13.6813	−0.520084
$693$	−25.6635	−0.974878
$694$	15.8944	0.603343
$695$	8.54527	0.324141
$696$	22.9836	0.871191
$697$	−26.2717	−0.995113
$698$	3.33645	0.126287
$699$	7.70948	0.291599
$700$	−32.0084	−1.20980
$701$	30.8866	1.16657	0.583286	−	0.812267i	$-0.301767\pi$
$701$	30.8866	1.16657	0.583286	+	0.812267i	$0.301767\pi$
$702$	−9.21983	−0.347980
$703$	−4.82371	−0.181930
$704$	8.86426	0.334085
$705$	4.20583	0.158401
$706$	−4.75600	−0.178995
$707$	−28.2030	−1.06068
$708$	−17.3308	−0.651332
$709$	−21.4504	−0.805587	−0.402794	−	0.915291i	$-0.631961\pi$
$709$	−21.4504	−0.805587	−0.402794	+	0.915291i	$0.631961\pi$
$710$	−2.52914	−0.0949169
$711$	−8.05967	−0.302261
$712$	−3.94198	−0.147732
$713$	−0.0605250	−0.00226668
$714$	−9.16852	−0.343123
$715$	−17.8860	−0.668900
$716$	14.7530	0.551346
$717$	35.4480	1.32383
$718$	−18.0277	−0.672789
$719$	−19.1306	−0.713452	−0.356726	−	0.934209i	$-0.616107\pi$
$719$	−19.1306	−0.713452	−0.356726	+	0.934209i	$0.616107\pi$
$720$	−2.14914	−0.0800939
$721$	−43.9211	−1.63571
$722$	6.03385	0.224557
$723$	−25.5797	−0.951320
$724$	33.6045	1.24890
$725$	30.2728	1.12430
$726$	18.1739	0.674497
$727$	−9.99031	−0.370520	−0.185260	−	0.982690i	$-0.559313\pi$
$727$	−9.99031	−0.370520	−0.185260	+	0.982690i	$0.559313\pi$
$728$	−29.2107	−1.08262
$729$	29.4596	1.09110
$730$	−3.02582	−0.111991
$731$	19.0344	0.704014
$732$	−32.4263	−1.19851
$733$	−22.0224	−0.813414	−0.406707	−	0.913559i	$-0.633323\pi$
$733$	−22.0224	−0.813414	−0.406707	+	0.913559i	$0.633323\pi$
$734$	−3.54958	−0.131017
$735$	−25.0562	−0.924212
$736$	1.20370	0.0443690
$737$	73.6945	2.71457
$738$	−5.64071	−0.207637
$739$	−5.80433	−0.213516	−0.106758	−	0.994285i	$-0.534047\pi$
$739$	−5.80433	−0.213516	−0.106758	+	0.994285i	$0.534047\pi$
$740$	3.00298	0.110392
$741$	−12.1075	−0.444781
$742$	3.52111	0.129264
$743$	20.5364	0.753409	0.376704	−	0.926334i	$-0.377057\pi$
$743$	20.5364	0.753409	0.376704	+	0.926334i	$0.377057\pi$
$744$	−0.795707	−0.0291720
$745$	−6.97823	−0.255663
$746$	−3.32975	−0.121911
$747$	−2.74440	−0.100412
$748$	23.1377	0.845996
$749$	20.6485	0.754479
$750$	−7.48619	−0.273357
$751$	−44.1976	−1.61279	−0.806396	−	0.591375i	$-0.798585\pi$
$751$	−44.1976	−1.61279	−0.806396	+	0.591375i	$0.798585\pi$
$752$	−6.23490	−0.227363
$753$	33.2325	1.21106
$754$	12.6612	0.461092
$755$	8.79869	0.320217
$756$	46.3967	1.68743
$757$	36.9724	1.34378	0.671892	−	0.740649i	$-0.265482\pi$
$757$	36.9724	1.34378	0.671892	+	0.740649i	$0.265482\pi$
$758$	−0.0881460	−0.00320161
$759$	−1.88816	−0.0685360
$760$	6.12690	0.222246
$761$	−48.0422	−1.74153	−0.870764	−	0.491700i	$-0.836376\pi$
$761$	−48.0422	−1.74153	−0.870764	+	0.491700i	$0.836376\pi$
$762$	−5.10992	−0.185113
$763$	4.85086	0.175613
$764$	−14.8449	−0.537069
$765$	−2.25428	−0.0815035
$766$	11.0677	0.399892
$767$	−20.8321	−0.752202
$768$	4.83685	0.174535
$769$	40.1631	1.44832	0.724160	−	0.689632i	$-0.242228\pi$
$769$	40.1631	1.44832	0.724160	+	0.689632i	$0.242228\pi$
$770$	−16.3830	−0.590402
$771$	−14.3623	−0.517245
$772$	3.69368	0.132938
$773$	−23.0476	−0.828964	−0.414482	−	0.910057i	$-0.636037\pi$
$773$	−23.0476	−0.828964	−0.414482	+	0.910057i	$0.636037\pi$
$774$	4.08682	0.146898
$775$	−1.04806	−0.0376475
$776$	2.53020	0.0908290
$777$	−11.8605	−0.425495
$778$	−11.0043	−0.394524
$779$	−31.7778	−1.13856
$780$	7.53750	0.269886
$781$	−25.2083	−0.902025
$782$	0.294569	0.0105338
$783$	−43.8810	−1.56818
$784$	37.1444	1.32658
$785$	10.5050	0.374939
$786$	−12.4601	−0.444437
$787$	3.70065	0.131914	0.0659569	−	0.997822i	$-0.478990\pi$
$787$	3.70065	0.131914	0.0659569	+	0.997822i	$0.478990\pi$
$788$	−45.7910	−1.63124
$789$	13.0586	0.464899
$790$	−5.14510	−0.183054
$791$	16.8431	0.598871
$792$	10.8398	0.385177
$793$	−38.9772	−1.38412
$794$	−8.44265	−0.299618
$795$	−1.98254	−0.0703135
$796$	0	0
$797$	42.4306	1.50297	0.751484	−	0.659751i	$-0.229338\pi$
$797$	42.4306	1.50297	0.751484	+	0.659751i	$0.229338\pi$
$798$	−11.0901	−0.392584
$799$	−6.53989	−0.231365
$800$	20.8436	0.736931
$801$	1.75435	0.0619869
$802$	−9.40821	−0.332215
$803$	−30.1588	−1.06428
$804$	−31.0562	−1.09527
$805$	1.14590	0.0403876
$806$	−0.438337	−0.0154398
$807$	12.1041	0.426083
$808$	11.9124	0.419078
$809$	39.4730	1.38780	0.693899	−	0.720072i	$-0.255891\pi$
$809$	39.4730	1.38780	0.693899	+	0.720072i	$0.255891\pi$
$810$	3.16255	0.111121
$811$	22.0040	0.772667	0.386333	−	0.922359i	$-0.373741\pi$
$811$	22.0040	0.772667	0.386333	+	0.922359i	$0.373741\pi$
$812$	−63.7144	−2.23594
$813$	−38.1478	−1.33790
$814$	−5.44803	−0.190953
$815$	−12.6668	−0.443698
$816$	−7.65279	−0.267901
$817$	23.0237	0.805497
$818$	8.01267	0.280157
$819$	13.0000	0.454257
$820$	19.7832	0.690858
$821$	37.4805	1.30808	0.654040	−	0.756460i	$-0.273073\pi$
$821$	37.4805	1.30808	0.654040	+	0.756460i	$0.273073\pi$
$822$	1.44935	0.0505520
$823$	12.9457	0.451259	0.225629	−	0.974213i	$-0.427556\pi$
$823$	12.9457	0.451259	0.225629	+	0.974213i	$0.427556\pi$
$824$	18.5515	0.646272
$825$	−32.6959	−1.13832
$826$	−19.0814	−0.663928
$827$	−11.4795	−0.399181	−0.199590	−	0.979879i	$-0.563961\pi$
$827$	−11.4795	−0.399181	−0.199590	+	0.979879i	$0.563961\pi$
$828$	−0.347470	−0.0120754
$829$	−18.5845	−0.645466	−0.322733	−	0.946490i	$-0.604602\pi$
$829$	−18.5845	−0.645466	−0.322733	+	0.946490i	$0.604602\pi$
$830$	−1.75196	−0.0608113
$831$	−16.5980	−0.575779
$832$	−4.49024	−0.155671
$833$	38.9614	1.34993
$834$	6.53319	0.226226
$835$	−8.42998	−0.291731
$836$	27.9869	0.967946
$837$	1.51919	0.0525108
$838$	−1.58450	−0.0547356
$839$	7.75600	0.267767	0.133884	−	0.990997i	$-0.457255\pi$
$839$	7.75600	0.267767	0.133884	+	0.990997i	$0.457255\pi$
$840$	15.0648	0.519786
$841$	31.2597	1.07792
$842$	−2.51035	−0.0865125
$843$	−9.16421	−0.315632
$844$	−27.3043	−0.939851
$845$	−4.57566	−0.157408
$846$	−1.40416	−0.0482759
$847$	−109.932	−3.77732
$848$	2.93900	0.100926
$849$	20.5308	0.704615
$850$	5.10082	0.174957
$851$	0.381059	0.0130625
$852$	10.6233	0.363947
$853$	48.4088	1.65749	0.828743	−	0.559630i	$-0.189057\pi$
$853$	48.4088	1.65749	0.828743	+	0.559630i	$0.189057\pi$
$854$	−35.7017	−1.22169
$855$	−2.72673	−0.0932521
$856$	−8.72156	−0.298097
$857$	51.8582	1.77144	0.885720	−	0.464219i	$-0.153665\pi$
$857$	51.8582	1.77144	0.885720	+	0.464219i	$0.153665\pi$
$858$	−13.6746	−0.466842
$859$	17.1758	0.586031	0.293016	−	0.956108i	$-0.405341\pi$
$859$	17.1758	0.586031	0.293016	+	0.956108i	$0.405341\pi$
$860$	−14.3333	−0.488762
$861$	−78.1353	−2.66284
$862$	−16.1451	−0.549904
$863$	−22.1769	−0.754910	−0.377455	−	0.926028i	$-0.623201\pi$
$863$	−22.1769	−0.754910	−0.377455	+	0.926028i	$0.623201\pi$
$864$	−30.2131	−1.02787
$865$	8.48129	0.288372
$866$	4.33944	0.147460
$867$	16.5386	0.561679
$868$	2.20583	0.0748708
$869$	−51.2820	−1.73962
$870$	−6.52973	−0.221379
$871$	−37.3303	−1.26489
$872$	−2.04892	−0.0693851
$873$	−1.12605	−0.0381109
$874$	0.356305	0.0120522
$875$	45.2833	1.53086
$876$	12.7095	0.429414
$877$	−43.1957	−1.45861	−0.729307	−	0.684187i	$-0.760157\pi$
$877$	−43.1957	−1.45861	−0.729307	+	0.684187i	$0.760157\pi$
$878$	15.9390	0.537915
$879$	39.1366	1.32004
$880$	−13.6746	−0.460969
$881$	35.9095	1.20982	0.604910	−	0.796294i	$-0.293209\pi$
$881$	35.9095	1.20982	0.604910	+	0.796294i	$0.293209\pi$
$882$	8.36526	0.281673
$883$	27.5609	0.927499	0.463750	−	0.885966i	$-0.346504\pi$
$883$	27.5609	0.927499	0.463750	+	0.885966i	$0.346504\pi$
$884$	−11.7205	−0.394203
$885$	10.7437	0.361146
$886$	22.6346	0.760424
$887$	−50.2331	−1.68666	−0.843331	−	0.537394i	$-0.819409\pi$
$887$	−50.2331	−1.68666	−0.843331	+	0.537394i	$0.819409\pi$
$888$	5.00969	0.168114
$889$	30.9095	1.03667
$890$	1.11993	0.0375402
$891$	31.5217	1.05602
$892$	−42.1842	−1.41243
$893$	−7.91053	−0.264716
$894$	−5.33513	−0.178433
$895$	−9.14569	−0.305707
$896$	−55.9667	−1.86972
$897$	0.956459	0.0319353
$898$	15.1897	0.506887
$899$	−2.08623	−0.0695796
$900$	−6.01687	−0.200562
$901$	3.08277	0.102702
$902$	−35.8907	−1.19503
$903$	56.6107	1.88389
$904$	−7.11423	−0.236616
$905$	−20.8321	−0.692481
$906$	6.72694	0.223488
$907$	30.9065	1.02623	0.513116	−	0.858319i	$-0.328491\pi$
$907$	30.9065	1.02623	0.513116	+	0.858319i	$0.328491\pi$
$908$	15.1823	0.503841
$909$	−5.30154	−0.175841
$910$	8.29888	0.275105
$911$	−39.2857	−1.30159	−0.650797	−	0.759252i	$-0.725565\pi$
$911$	−39.2857	−1.30159	−0.650797	+	0.759252i	$0.725565\pi$
$912$	−9.25667	−0.306519
$913$	−17.4620	−0.577909
$914$	13.6896	0.452813
$915$	20.1017	0.664541
$916$	18.5050	0.611422
$917$	75.3702	2.48894
$918$	−7.39373	−0.244030
$919$	−7.82264	−0.258045	−0.129023	−	0.991642i	$-0.541184\pi$
$919$	−7.82264	−0.258045	−0.129023	+	0.991642i	$0.541184\pi$
$920$	−0.484008	−0.0159573
$921$	35.8329	1.18073
$922$	−2.09783	−0.0690885
$923$	12.7694	0.420310
$924$	68.8141	2.26382
$925$	6.59850	0.216957
$926$	7.34216	0.241278
$927$	−8.25619	−0.271169
$928$	41.4902	1.36198
$929$	−7.67755	−0.251892	−0.125946	−	0.992037i	$-0.540197\pi$
$929$	−7.67755	−0.251892	−0.125946	+	0.992037i	$0.540197\pi$
$930$	0.226063	0.00741291
$931$	47.1269	1.54452
$932$	9.02715	0.295694
$933$	−22.0823	−0.722942
$934$	9.02848	0.295421
$935$	−14.3435	−0.469082
$936$	−5.49098	−0.179478
$937$	−10.7084	−0.349829	−0.174914	−	0.984584i	$-0.555965\pi$
$937$	−10.7084	−0.349829	−0.174914	+	0.984584i	$0.555965\pi$
$938$	−34.1933	−1.11645
$939$	10.5821	0.345334
$940$	4.92467	0.160625
$941$	−22.1420	−0.721807	−0.360904	−	0.932603i	$-0.617532\pi$
$941$	−22.1420	−0.721807	−0.360904	+	0.932603i	$0.617532\pi$
$942$	8.03146	0.261679
$943$	2.51035	0.0817483
$944$	−15.9269	−0.518377
$945$	−28.7622	−0.935636
$946$	26.0036	0.845449
$947$	58.9385	1.91524	0.957622	−	0.288027i	$-0.0929992\pi$
$947$	58.9385	1.91524	0.957622	+	0.288027i	$0.0929992\pi$
$948$	21.6112	0.701898
$949$	15.2771	0.495916
$950$	6.16985	0.200176
$951$	24.4174	0.791789
$952$	−23.4252	−0.759215
$953$	36.5808	1.18497	0.592484	−	0.805582i	$-0.298147\pi$
$953$	36.5808	1.18497	0.592484	+	0.805582i	$0.298147\pi$
$954$	0.661890	0.0214295
$955$	9.20264	0.297790
$956$	41.5066	1.34242
$957$	−65.0829	−2.10383
$958$	1.11364	0.0359800
$959$	−8.76702	−0.283102
$960$	2.31575	0.0747405
$961$	−30.9278	−0.997670
$962$	2.75973	0.0889771
$963$	3.88146	0.125078
$964$	−29.9517	−0.964678
$965$	−2.28978	−0.0737107
$966$	0.876083	0.0281875
$967$	−47.2674	−1.52002	−0.760009	−	0.649913i	$-0.774805\pi$
$967$	−47.2674	−1.52002	−0.760009	+	0.649913i	$0.774805\pi$
$968$	46.4336	1.49243
$969$	−9.70948	−0.311913
$970$	−0.718841	−0.0230806
$971$	−54.2747	−1.74176	−0.870879	−	0.491497i	$-0.836450\pi$
$971$	−54.2747	−1.74176	−0.870879	+	0.491497i	$0.836450\pi$
$972$	15.4101	0.494280
$973$	−39.5187	−1.26691
$974$	−5.62192	−0.180138
$975$	16.5623	0.530417
$976$	−29.7995	−0.953860
$977$	45.5472	1.45718	0.728592	−	0.684948i	$-0.240175\pi$
$977$	45.5472	1.45718	0.728592	+	0.684948i	$0.240175\pi$
$978$	−9.68425	−0.309668
$979$	11.1626	0.356757
$980$	−29.3387	−0.937190
$981$	0.911854	0.0291133
$982$	−15.8321	−0.505222
$983$	40.4922	1.29150	0.645750	−	0.763549i	$-0.276545\pi$
$983$	40.4922	1.29150	0.645750	+	0.763549i	$0.276545\pi$
$984$	33.0030	1.05210
$985$	28.3868	0.904478
$986$	10.1535	0.323352
$987$	−19.4504	−0.619114
$988$	−14.1769	−0.451027
$989$	−1.81881	−0.0578346
$990$	−3.07964	−0.0978774
$991$	−23.0793	−0.733138	−0.366569	−	0.930391i	$-0.619468\pi$
$991$	−23.0793	−0.733138	−0.366569	+	0.930391i	$0.619468\pi$
$992$	−1.43642	−0.0456063
$993$	17.9425	0.569387
$994$	11.6963	0.370985
$995$	0	0
$996$	7.35881	0.233173
$997$	−11.6614	−0.369321	−0.184660	−	0.982802i	$-0.559119\pi$
$997$	−11.6614	−0.369321	−0.184660	+	0.982802i	$0.559119\pi$
$998$	−6.60015	−0.208924
$999$	−9.56465	−0.302612

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	109.2.a.b.1.2	✓	3
3.2	odd	2		981.2.a.c.1.2		3
4.3	odd	2		1744.2.a.l.1.2		3
5.4	even	2		2725.2.a.h.1.2		3
7.6	odd	2		5341.2.a.d.1.2		3
8.3	odd	2		6976.2.a.s.1.2		3
8.5	even	2		6976.2.a.z.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
109.2.a.b.1.2	✓	3	1.1	even	1	trivial
981.2.a.c.1.2		3	3.2	odd	2
1744.2.a.l.1.2		3	4.3	odd	2
2725.2.a.h.1.2		3	5.4	even	2
5341.2.a.d.1.2		3	7.6	odd	2
6976.2.a.s.1.2		3	8.3	odd	2
6976.2.a.z.1.2		3	8.5	even	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:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	109.a (trivial)

\(f(q)\)	\(=\)	\(q-0.554958 q^{2} -1.44504 q^{3} -1.69202 q^{4} +1.04892 q^{5} +0.801938 q^{6} -4.85086 q^{7} +2.04892 q^{8} -0.911854 q^{9} +O(q^{10})\)	\(q-0.554958 q^{2} -1.44504 q^{3} -1.69202 q^{4} +1.04892 q^{5} +0.801938 q^{6} -4.85086 q^{7} +2.04892 q^{8} -0.911854 q^{9} -0.582105 q^{10} -5.80194 q^{11} +2.44504 q^{12} +2.93900 q^{13} +2.69202 q^{14} -1.51573 q^{15} +2.24698 q^{16} +2.35690 q^{17} +0.506041 q^{18} +2.85086 q^{19} -1.77479 q^{20} +7.00969 q^{21} +3.21983 q^{22} -0.225209 q^{23} -2.96077 q^{24} -3.89977 q^{25} -1.63102 q^{26} +5.65279 q^{27} +8.20775 q^{28} -7.76271 q^{29} +0.841166 q^{30} +0.268750 q^{31} -5.34481 q^{32} +8.38404 q^{33} -1.30798 q^{34} -5.08815 q^{35} +1.54288 q^{36} -1.69202 q^{37} -1.58211 q^{38} -4.24698 q^{39} +2.14914 q^{40} -11.1468 q^{41} -3.89008 q^{42} +8.07606 q^{43} +9.81700 q^{44} -0.956459 q^{45} +0.124982 q^{46} -2.77479 q^{47} -3.24698 q^{48} +16.5308 q^{49} +2.16421 q^{50} -3.40581 q^{51} -4.97285 q^{52} +1.30798 q^{53} -3.13706 q^{54} -6.08575 q^{55} -9.93900 q^{56} -4.11960 q^{57} +4.30798 q^{58} -7.08815 q^{59} +2.56465 q^{60} -13.2620 q^{61} -0.149145 q^{62} +4.42327 q^{63} -1.52781 q^{64} +3.08277 q^{65} -4.65279 q^{66} -12.7017 q^{67} -3.98792 q^{68} +0.325437 q^{69} +2.82371 q^{70} +4.34481 q^{71} -1.86831 q^{72} +5.19806 q^{73} +0.939001 q^{74} +5.63533 q^{75} -4.82371 q^{76} +28.1444 q^{77} +2.35690 q^{78} +8.83877 q^{79} +2.35690 q^{80} -5.43296 q^{81} +6.18598 q^{82} +3.00969 q^{83} -11.8605 q^{84} +2.47219 q^{85} -4.48188 q^{86} +11.2174 q^{87} -11.8877 q^{88} -1.92394 q^{89} +0.530795 q^{90} -14.2567 q^{91} +0.381059 q^{92} -0.388355 q^{93} +1.53989 q^{94} +2.99031 q^{95} +7.72348 q^{96} +1.23490 q^{97} -9.17390 q^{98} +5.29052 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} - 4 q^{3} - 6 q^{5} - 2 q^{6} - q^{7} - 3 q^{8} + q^{9}+O(q^{10}) \) 3 * q - 2 * q^2 - 4 * q^3 - 6 * q^5 - 2 * q^6 - q^7 - 3 * q^8 + q^9	\( 3 q - 2 q^{2} - 4 q^{3} - 6 q^{5} - 2 q^{6} - q^{7} - 3 q^{8} + q^{9} + 4 q^{10} - 13 q^{11} + 7 q^{12} - q^{13} + 3 q^{14} + 8 q^{15} + 2 q^{16} + 3 q^{17} + 11 q^{18} - 5 q^{19} - 7 q^{20} - q^{21} + 11 q^{22} + q^{23} + 4 q^{24} + 11 q^{25} + 10 q^{26} - q^{27} + 7 q^{28} - 6 q^{29} + 11 q^{30} - 7 q^{31} + 7 q^{32} + 15 q^{33} - 9 q^{34} - 19 q^{35} - 14 q^{36} + q^{38} - 8 q^{39} + 20 q^{40} - 6 q^{41} - 11 q^{42} + 9 q^{43} - 9 q^{45} - 24 q^{46} - 10 q^{47} - 5 q^{48} + 12 q^{49} - 5 q^{50} + 3 q^{51} - 21 q^{52} + 9 q^{53} - 4 q^{54} + 19 q^{55} - 20 q^{56} + 9 q^{57} + 18 q^{58} - 25 q^{59} - 14 q^{60} - 10 q^{61} - 14 q^{62} + 16 q^{63} - 11 q^{64} + 16 q^{65} + 4 q^{66} - 11 q^{67} + 7 q^{68} + 22 q^{69} + q^{70} - 10 q^{71} - 8 q^{72} + 20 q^{73} - 7 q^{74} - 17 q^{75} - 7 q^{76} + 16 q^{77} + 3 q^{78} - 6 q^{79} + 3 q^{80} + 3 q^{81} + 4 q^{82} - 13 q^{83} + q^{85} + 15 q^{86} - 6 q^{87} + 6 q^{88} - 21 q^{89} - 36 q^{90} - 16 q^{91} + 35 q^{92} + 28 q^{93} + 30 q^{94} + 31 q^{95} - 7 q^{96} - 20 q^{97} + 6 q^{98} + 5 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 - 4 * q^3 - 6 * q^5 - 2 * q^6 - q^7 - 3 * q^8 + q^9 + 4 * q^10 - 13 * q^11 + 7 * q^12 - q^13 + 3 * q^14 + 8 * q^15 + 2 * q^16 + 3 * q^17 + 11 * q^18 - 5 * q^19 - 7 * q^20 - q^21 + 11 * q^22 + q^23 + 4 * q^24 + 11 * q^25 + 10 * q^26 - q^27 + 7 * q^28 - 6 * q^29 + 11 * q^30 - 7 * q^31 + 7 * q^32 + 15 * q^33 - 9 * q^34 - 19 * q^35 - 14 * q^36 + q^38 - 8 * q^39 + 20 * q^40 - 6 * q^41 - 11 * q^42 + 9 * q^43 - 9 * q^45 - 24 * q^46 - 10 * q^47 - 5 * q^48 + 12 * q^49 - 5 * q^50 + 3 * q^51 - 21 * q^52 + 9 * q^53 - 4 * q^54 + 19 * q^55 - 20 * q^56 + 9 * q^57 + 18 * q^58 - 25 * q^59 - 14 * q^60 - 10 * q^61 - 14 * q^62 + 16 * q^63 - 11 * q^64 + 16 * q^65 + 4 * q^66 - 11 * q^67 + 7 * q^68 + 22 * q^69 + q^70 - 10 * q^71 - 8 * q^72 + 20 * q^73 - 7 * q^74 - 17 * q^75 - 7 * q^76 + 16 * q^77 + 3 * q^78 - 6 * q^79 + 3 * q^80 + 3 * q^81 + 4 * q^82 - 13 * q^83 + q^85 + 15 * q^86 - 6 * q^87 + 6 * q^88 - 21 * q^89 - 36 * q^90 - 16 * q^91 + 35 * q^92 + 28 * q^93 + 30 * q^94 + 31 * q^95 - 7 * q^96 - 20 * q^97 + 6 * q^98 + 5 * q^99

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