LMFDB - Embedded newform 165.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [165,2,Mod(1,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("165.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 165 = 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	165.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:	$1.31753163335$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	165.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} +2.41421 q^{6} +0.828427 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} +2.41421 q^{6} +0.828427 q^{7} -4.41421 q^{8} +1.00000 q^{9} +2.41421 q^{10} -1.00000 q^{11} -3.82843 q^{12} -5.65685 q^{13} -2.00000 q^{14} +1.00000 q^{15} +3.00000 q^{16} -1.17157 q^{17} -2.41421 q^{18} -6.82843 q^{19} -3.82843 q^{20} -0.828427 q^{21} +2.41421 q^{22} -4.00000 q^{23} +4.41421 q^{24} +1.00000 q^{25} +13.6569 q^{26} -1.00000 q^{27} +3.17157 q^{28} -4.82843 q^{29} -2.41421 q^{30} +1.58579 q^{32} +1.00000 q^{33} +2.82843 q^{34} -0.828427 q^{35} +3.82843 q^{36} +11.6569 q^{37} +16.4853 q^{38} +5.65685 q^{39} +4.41421 q^{40} +4.82843 q^{41} +2.00000 q^{42} -8.82843 q^{43} -3.82843 q^{44} -1.00000 q^{45} +9.65685 q^{46} -4.00000 q^{47} -3.00000 q^{48} -6.31371 q^{49} -2.41421 q^{50} +1.17157 q^{51} -21.6569 q^{52} +9.31371 q^{53} +2.41421 q^{54} +1.00000 q^{55} -3.65685 q^{56} +6.82843 q^{57} +11.6569 q^{58} -4.00000 q^{59} +3.82843 q^{60} -11.6569 q^{61} +0.828427 q^{63} -9.82843 q^{64} +5.65685 q^{65} -2.41421 q^{66} -5.65685 q^{67} -4.48528 q^{68} +4.00000 q^{69} +2.00000 q^{70} +2.34315 q^{71} -4.41421 q^{72} +11.3137 q^{73} -28.1421 q^{74} -1.00000 q^{75} -26.1421 q^{76} -0.828427 q^{77} -13.6569 q^{78} +8.48528 q^{79} -3.00000 q^{80} +1.00000 q^{81} -11.6569 q^{82} -10.0000 q^{83} -3.17157 q^{84} +1.17157 q^{85} +21.3137 q^{86} +4.82843 q^{87} +4.41421 q^{88} +3.65685 q^{89} +2.41421 q^{90} -4.68629 q^{91} -15.3137 q^{92} +9.65685 q^{94} +6.82843 q^{95} -1.58579 q^{96} +11.6569 q^{97} +15.2426 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} - 2 q^{12} - 4 q^{14} + 2 q^{15} + 6 q^{16} - 8 q^{17} - 2 q^{18} - 8 q^{19} - 2 q^{20} + 4 q^{21} + 2 q^{22} - 8 q^{23} + 6 q^{24} + 2 q^{25} + 16 q^{26} - 2 q^{27} + 12 q^{28} - 4 q^{29} - 2 q^{30} + 6 q^{32} + 2 q^{33} + 4 q^{35} + 2 q^{36} + 12 q^{37} + 16 q^{38} + 6 q^{40} + 4 q^{41} + 4 q^{42} - 12 q^{43} - 2 q^{44} - 2 q^{45} + 8 q^{46} - 8 q^{47} - 6 q^{48} + 10 q^{49} - 2 q^{50} + 8 q^{51} - 32 q^{52} - 4 q^{53} + 2 q^{54} + 2 q^{55} + 4 q^{56} + 8 q^{57} + 12 q^{58} - 8 q^{59} + 2 q^{60} - 12 q^{61} - 4 q^{63} - 14 q^{64} - 2 q^{66} + 8 q^{68} + 8 q^{69} + 4 q^{70} + 16 q^{71} - 6 q^{72} - 28 q^{74} - 2 q^{75} - 24 q^{76} + 4 q^{77} - 16 q^{78} - 6 q^{80} + 2 q^{81} - 12 q^{82} - 20 q^{83} - 12 q^{84} + 8 q^{85} + 20 q^{86} + 4 q^{87} + 6 q^{88} - 4 q^{89} + 2 q^{90} - 32 q^{91} - 8 q^{92} + 8 q^{94} + 8 q^{95} - 6 q^{96} + 12 q^{97} + 22 q^{98} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 + 2 * q^10 - 2 * q^11 - 2 * q^12 - 4 * q^14 + 2 * q^15 + 6 * q^16 - 8 * q^17 - 2 * q^18 - 8 * q^19 - 2 * q^20 + 4 * q^21 + 2 * q^22 - 8 * q^23 + 6 * q^24 + 2 * q^25 + 16 * q^26 - 2 * q^27 + 12 * q^28 - 4 * q^29 - 2 * q^30 + 6 * q^32 + 2 * q^33 + 4 * q^35 + 2 * q^36 + 12 * q^37 + 16 * q^38 + 6 * q^40 + 4 * q^41 + 4 * q^42 - 12 * q^43 - 2 * q^44 - 2 * q^45 + 8 * q^46 - 8 * q^47 - 6 * q^48 + 10 * q^49 - 2 * q^50 + 8 * q^51 - 32 * q^52 - 4 * q^53 + 2 * q^54 + 2 * q^55 + 4 * q^56 + 8 * q^57 + 12 * q^58 - 8 * q^59 + 2 * q^60 - 12 * q^61 - 4 * q^63 - 14 * q^64 - 2 * q^66 + 8 * q^68 + 8 * q^69 + 4 * q^70 + 16 * q^71 - 6 * q^72 - 28 * q^74 - 2 * q^75 - 24 * q^76 + 4 * q^77 - 16 * q^78 - 6 * q^80 + 2 * q^81 - 12 * q^82 - 20 * q^83 - 12 * q^84 + 8 * q^85 + 20 * q^86 + 4 * q^87 + 6 * q^88 - 4 * q^89 + 2 * q^90 - 32 * q^91 - 8 * q^92 + 8 * q^94 + 8 * q^95 - 6 * q^96 + 12 * q^97 + 22 * 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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.82843	1.91421
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.41421	0.985599
$7$	0.828427	0.313116	0.156558	−	0.987669i	$-0.449960\pi$
$7$	0.828427	0.313116	0.156558	+	0.987669i	$0.449960\pi$
$8$	−4.41421	−1.56066
$9$	1.00000	0.333333
$10$	2.41421	0.763441
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−3.82843	−1.10517
$13$	−5.65685	−1.56893	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	−5.65685	−1.56893	−0.784465	+	0.620174i	$0.787062\pi$
$14$	−2.00000	−0.534522
$15$	1.00000	0.258199
$16$	3.00000	0.750000
$17$	−1.17157	−0.284148	−0.142074	−	0.989856i	$-0.545377\pi$
$17$	−1.17157	−0.284148	−0.142074	+	0.989856i	$0.545377\pi$
$18$	−2.41421	−0.569036
$19$	−6.82843	−1.56655	−0.783274	−	0.621676i	$-0.786452\pi$
$19$	−6.82843	−1.56655	−0.783274	+	0.621676i	$0.786452\pi$
$20$	−3.82843	−0.856062
$21$	−0.828427	−0.180778
$22$	2.41421	0.514712
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	4.41421	0.901048
$25$	1.00000	0.200000
$26$	13.6569	2.67833
$27$	−1.00000	−0.192450
$28$	3.17157	0.599371
$29$	−4.82843	−0.896616	−0.448308	−	0.893879i	$-0.647973\pi$
$29$	−4.82843	−0.896616	−0.448308	+	0.893879i	$0.647973\pi$
$30$	−2.41421	−0.440773
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.58579	0.280330
$33$	1.00000	0.174078
$34$	2.82843	0.485071
$35$	−0.828427	−0.140030
$36$	3.82843	0.638071
$37$	11.6569	1.91638	0.958188	−	0.286141i	$-0.0923726\pi$
$37$	11.6569	1.91638	0.958188	+	0.286141i	$0.0923726\pi$
$38$	16.4853	2.67427
$39$	5.65685	0.905822
$40$	4.41421	0.697948
$41$	4.82843	0.754074	0.377037	−	0.926198i	$-0.376943\pi$
$41$	4.82843	0.754074	0.377037	+	0.926198i	$0.376943\pi$
$42$	2.00000	0.308607
$43$	−8.82843	−1.34632	−0.673161	−	0.739496i	$-0.735064\pi$
$43$	−8.82843	−1.34632	−0.673161	+	0.739496i	$0.735064\pi$
$44$	−3.82843	−0.577157
$45$	−1.00000	−0.149071
$46$	9.65685	1.42383
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	−3.00000	−0.433013
$49$	−6.31371	−0.901958
$50$	−2.41421	−0.341421
$51$	1.17157	0.164053
$52$	−21.6569	−3.00327
$53$	9.31371	1.27934	0.639668	−	0.768651i	$-0.279072\pi$
$53$	9.31371	1.27934	0.639668	+	0.768651i	$0.279072\pi$
$54$	2.41421	0.328533
$55$	1.00000	0.134840
$56$	−3.65685	−0.488668
$57$	6.82843	0.904447
$58$	11.6569	1.53062
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	3.82843	0.494248
$61$	−11.6569	−1.49251	−0.746254	−	0.665662i	$-0.768149\pi$
$61$	−11.6569	−1.49251	−0.746254	+	0.665662i	$0.768149\pi$
$62$	0	0
$63$	0.828427	0.104372
$64$	−9.82843	−1.22855
$65$	5.65685	0.701646
$66$	−2.41421	−0.297169
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	−4.48528	−0.543920
$69$	4.00000	0.481543
$70$	2.00000	0.239046
$71$	2.34315	0.278080	0.139040	−	0.990287i	$-0.455598\pi$
$71$	2.34315	0.278080	0.139040	+	0.990287i	$0.455598\pi$
$72$	−4.41421	−0.520220
$73$	11.3137	1.32417	0.662085	−	0.749429i	$-0.269672\pi$
$73$	11.3137	1.32417	0.662085	+	0.749429i	$0.269672\pi$
$74$	−28.1421	−3.27146
$75$	−1.00000	−0.115470
$76$	−26.1421	−2.99871
$77$	−0.828427	−0.0944080
$78$	−13.6569	−1.54633
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	−11.6569	−1.28728
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	−3.17157	−0.346047
$85$	1.17157	0.127075
$86$	21.3137	2.29832
$87$	4.82843	0.517662
$88$	4.41421	0.470557
$89$	3.65685	0.387626	0.193813	−	0.981039i	$-0.437915\pi$
$89$	3.65685	0.387626	0.193813	+	0.981039i	$0.437915\pi$
$90$	2.41421	0.254480
$91$	−4.68629	−0.491257
$92$	−15.3137	−1.59656
$93$	0	0
$94$	9.65685	0.996028
$95$	6.82843	0.700582
$96$	−1.58579	−0.161849
$97$	11.6569	1.18357	0.591787	−	0.806094i	$-0.298423\pi$
$97$	11.6569	1.18357	0.591787	+	0.806094i	$0.298423\pi$
$98$	15.2426	1.53974
$99$	−1.00000	−0.100504
$100$	3.82843	0.382843
$101$	−0.828427	−0.0824316	−0.0412158	−	0.999150i	$-0.513123\pi$
$101$	−0.828427	−0.0824316	−0.0412158	+	0.999150i	$0.513123\pi$
$102$	−2.82843	−0.280056
$103$	−3.31371	−0.326509	−0.163255	−	0.986584i	$-0.552199\pi$
$103$	−3.31371	−0.326509	−0.163255	+	0.986584i	$0.552199\pi$
$104$	24.9706	2.44857
$105$	0.828427	0.0808462
$106$	−22.4853	−2.18396
$107$	17.3137	1.67378	0.836890	−	0.547372i	$-0.184372\pi$
$107$	17.3137	1.67378	0.836890	+	0.547372i	$0.184372\pi$
$108$	−3.82843	−0.368391
$109$	17.3137	1.65835	0.829176	−	0.558987i	$-0.188810\pi$
$109$	17.3137	1.65835	0.829176	+	0.558987i	$0.188810\pi$
$110$	−2.41421	−0.230186
$111$	−11.6569	−1.10642
$112$	2.48528	0.234837
$113$	−18.9706	−1.78460	−0.892300	−	0.451442i	$-0.850910\pi$
$113$	−18.9706	−1.78460	−0.892300	+	0.451442i	$0.850910\pi$
$114$	−16.4853	−1.54399
$115$	4.00000	0.373002
$116$	−18.4853	−1.71632
$117$	−5.65685	−0.522976
$118$	9.65685	0.888985
$119$	−0.970563	−0.0889713
$120$	−4.41421	−0.402961
$121$	1.00000	0.0909091
$122$	28.1421	2.54787
$123$	−4.82843	−0.435365
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	−2.00000	−0.178174
$127$	−14.4853	−1.28536	−0.642680	−	0.766134i	$-0.722178\pi$
$127$	−14.4853	−1.28536	−0.642680	+	0.766134i	$0.722178\pi$
$128$	20.5563	1.81694
$129$	8.82843	0.777300
$130$	−13.6569	−1.19779
$131$	3.31371	0.289520	0.144760	−	0.989467i	$-0.453759\pi$
$131$	3.31371	0.289520	0.144760	+	0.989467i	$0.453759\pi$
$132$	3.82843	0.333222
$133$	−5.65685	−0.490511
$134$	13.6569	1.17977
$135$	1.00000	0.0860663
$136$	5.17157	0.443459
$137$	−13.3137	−1.13747	−0.568733	−	0.822522i	$-0.692566\pi$
$137$	−13.3137	−1.13747	−0.568733	+	0.822522i	$0.692566\pi$
$138$	−9.65685	−0.822046
$139$	0.485281	0.0411610	0.0205805	−	0.999788i	$-0.493449\pi$
$139$	0.485281	0.0411610	0.0205805	+	0.999788i	$0.493449\pi$
$140$	−3.17157	−0.268047
$141$	4.00000	0.336861
$142$	−5.65685	−0.474713
$143$	5.65685	0.473050
$144$	3.00000	0.250000
$145$	4.82843	0.400979
$146$	−27.3137	−2.26050
$147$	6.31371	0.520746
$148$	44.6274	3.66835
$149$	1.51472	0.124091	0.0620453	−	0.998073i	$-0.480238\pi$
$149$	1.51472	0.124091	0.0620453	+	0.998073i	$0.480238\pi$
$150$	2.41421	0.197120
$151$	16.4853	1.34155	0.670777	−	0.741659i	$-0.265961\pi$
$151$	16.4853	1.34155	0.670777	+	0.741659i	$0.265961\pi$
$152$	30.1421	2.44485
$153$	−1.17157	−0.0947161
$154$	2.00000	0.161165
$155$	0	0
$156$	21.6569	1.73394
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	−20.4853	−1.62972
$159$	−9.31371	−0.738625
$160$	−1.58579	−0.125367
$161$	−3.31371	−0.261157
$162$	−2.41421	−0.189679
$163$	−7.31371	−0.572854	−0.286427	−	0.958102i	$-0.592468\pi$
$163$	−7.31371	−0.572854	−0.286427	+	0.958102i	$0.592468\pi$
$164$	18.4853	1.44346
$165$	−1.00000	−0.0778499
$166$	24.1421	1.87379
$167$	−13.3137	−1.03025	−0.515123	−	0.857116i	$-0.672254\pi$
$167$	−13.3137	−1.03025	−0.515123	+	0.857116i	$0.672254\pi$
$168$	3.65685	0.282132
$169$	19.0000	1.46154
$170$	−2.82843	−0.216930
$171$	−6.82843	−0.522183
$172$	−33.7990	−2.57715
$173$	2.82843	0.215041	0.107521	−	0.994203i	$-0.465709\pi$
$173$	2.82843	0.215041	0.107521	+	0.994203i	$0.465709\pi$
$174$	−11.6569	−0.883704
$175$	0.828427	0.0626232
$176$	−3.00000	−0.226134
$177$	4.00000	0.300658
$178$	−8.82843	−0.661719
$179$	−17.6569	−1.31974	−0.659868	−	0.751382i	$-0.729388\pi$
$179$	−17.6569	−1.31974	−0.659868	+	0.751382i	$0.729388\pi$
$180$	−3.82843	−0.285354
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	11.3137	0.838628
$183$	11.6569	0.861699
$184$	17.6569	1.30168
$185$	−11.6569	−0.857029
$186$	0	0
$187$	1.17157	0.0856739
$188$	−15.3137	−1.11687
$189$	−0.828427	−0.0602592
$190$	−16.4853	−1.19597
$191$	5.65685	0.409316	0.204658	−	0.978834i	$-0.434392\pi$
$191$	5.65685	0.409316	0.204658	+	0.978834i	$0.434392\pi$
$192$	9.82843	0.709306
$193$	13.6569	0.983042	0.491521	−	0.870866i	$-0.336441\pi$
$193$	13.6569	0.983042	0.491521	+	0.870866i	$0.336441\pi$
$194$	−28.1421	−2.02049
$195$	−5.65685	−0.405096
$196$	−24.1716	−1.72654
$197$	−8.48528	−0.604551	−0.302276	−	0.953221i	$-0.597746\pi$
$197$	−8.48528	−0.604551	−0.302276	+	0.953221i	$0.597746\pi$
$198$	2.41421	0.171571
$199$	−21.6569	−1.53521	−0.767607	−	0.640921i	$-0.778553\pi$
$199$	−21.6569	−1.53521	−0.767607	+	0.640921i	$0.778553\pi$
$200$	−4.41421	−0.312132
$201$	5.65685	0.399004
$202$	2.00000	0.140720
$203$	−4.00000	−0.280745
$204$	4.48528	0.314033
$205$	−4.82843	−0.337232
$206$	8.00000	0.557386
$207$	−4.00000	−0.278019
$208$	−16.9706	−1.17670
$209$	6.82843	0.472332
$210$	−2.00000	−0.138013
$211$	1.17157	0.0806544	0.0403272	−	0.999187i	$-0.487160\pi$
$211$	1.17157	0.0806544	0.0403272	+	0.999187i	$0.487160\pi$
$212$	35.6569	2.44892
$213$	−2.34315	−0.160550
$214$	−41.7990	−2.85732
$215$	8.82843	0.602094
$216$	4.41421	0.300349
$217$	0	0
$218$	−41.7990	−2.83098
$219$	−11.3137	−0.764510
$220$	3.82843	0.258113
$221$	6.62742	0.445808
$222$	28.1421	1.88878
$223$	−6.34315	−0.424768	−0.212384	−	0.977186i	$-0.568123\pi$
$223$	−6.34315	−0.424768	−0.212384	+	0.977186i	$0.568123\pi$
$224$	1.31371	0.0877758
$225$	1.00000	0.0666667
$226$	45.7990	3.04650
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	26.1421	1.73131
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	−9.65685	−0.636754
$231$	0.828427	0.0545065
$232$	21.3137	1.39931
$233$	−18.8284	−1.23349	−0.616746	−	0.787163i	$-0.711549\pi$
$233$	−18.8284	−1.23349	−0.616746	+	0.787163i	$0.711549\pi$
$234$	13.6569	0.892776
$235$	4.00000	0.260931
$236$	−15.3137	−0.996838
$237$	−8.48528	−0.551178
$238$	2.34315	0.151884
$239$	17.6569	1.14213	0.571063	−	0.820906i	$-0.306531\pi$
$239$	17.6569	1.14213	0.571063	+	0.820906i	$0.306531\pi$
$240$	3.00000	0.193649
$241$	−12.3431	−0.795092	−0.397546	−	0.917582i	$-0.630138\pi$
$241$	−12.3431	−0.795092	−0.397546	+	0.917582i	$0.630138\pi$
$242$	−2.41421	−0.155192
$243$	−1.00000	−0.0641500
$244$	−44.6274	−2.85698
$245$	6.31371	0.403368
$246$	11.6569	0.743214
$247$	38.6274	2.45780
$248$	0	0
$249$	10.0000	0.633724
$250$	2.41421	0.152688
$251$	20.9706	1.32365	0.661825	−	0.749658i	$-0.269782\pi$
$251$	20.9706	1.32365	0.661825	+	0.749658i	$0.269782\pi$
$252$	3.17157	0.199790
$253$	4.00000	0.251478
$254$	34.9706	2.19425
$255$	−1.17157	−0.0733667
$256$	−29.9706	−1.87316
$257$	−16.3431	−1.01946	−0.509729	−	0.860335i	$-0.670254\pi$
$257$	−16.3431	−1.01946	−0.509729	+	0.860335i	$0.670254\pi$
$258$	−21.3137	−1.32693
$259$	9.65685	0.600048
$260$	21.6569	1.34310
$261$	−4.82843	−0.298872
$262$	−8.00000	−0.494242
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	−4.41421	−0.271676
$265$	−9.31371	−0.572137
$266$	13.6569	0.837355
$267$	−3.65685	−0.223796
$268$	−21.6569	−1.32290
$269$	20.6274	1.25768	0.628838	−	0.777536i	$-0.283531\pi$
$269$	20.6274	1.25768	0.628838	+	0.777536i	$0.283531\pi$
$270$	−2.41421	−0.146924
$271$	−11.7990	−0.716738	−0.358369	−	0.933580i	$-0.616667\pi$
$271$	−11.7990	−0.716738	−0.358369	+	0.933580i	$0.616667\pi$
$272$	−3.51472	−0.213111
$273$	4.68629	0.283627
$274$	32.1421	1.94178
$275$	−1.00000	−0.0603023
$276$	15.3137	0.921777
$277$	−2.34315	−0.140786	−0.0703930	−	0.997519i	$-0.522425\pi$
$277$	−2.34315	−0.140786	−0.0703930	+	0.997519i	$0.522425\pi$
$278$	−1.17157	−0.0702663
$279$	0	0
$280$	3.65685	0.218539
$281$	−11.1716	−0.666440	−0.333220	−	0.942849i	$-0.608135\pi$
$281$	−11.1716	−0.666440	−0.333220	+	0.942849i	$0.608135\pi$
$282$	−9.65685	−0.575057
$283$	−8.82843	−0.524796	−0.262398	−	0.964960i	$-0.584513\pi$
$283$	−8.82843	−0.524796	−0.262398	+	0.964960i	$0.584513\pi$
$284$	8.97056	0.532305
$285$	−6.82843	−0.404481
$286$	−13.6569	−0.807547
$287$	4.00000	0.236113
$288$	1.58579	0.0934434
$289$	−15.6274	−0.919260
$290$	−11.6569	−0.684514
$291$	−11.6569	−0.683337
$292$	43.3137	2.53474
$293$	−6.82843	−0.398921	−0.199460	−	0.979906i	$-0.563919\pi$
$293$	−6.82843	−0.398921	−0.199460	+	0.979906i	$0.563919\pi$
$294$	−15.2426	−0.888969
$295$	4.00000	0.232889
$296$	−51.4558	−2.99081
$297$	1.00000	0.0580259
$298$	−3.65685	−0.211836
$299$	22.6274	1.30858
$300$	−3.82843	−0.221034
$301$	−7.31371	−0.421555
$302$	−39.7990	−2.29017
$303$	0.828427	0.0475919
$304$	−20.4853	−1.17491
$305$	11.6569	0.667470
$306$	2.82843	0.161690
$307$	−3.17157	−0.181011	−0.0905056	−	0.995896i	$-0.528848\pi$
$307$	−3.17157	−0.181011	−0.0905056	+	0.995896i	$0.528848\pi$
$308$	−3.17157	−0.180717
$309$	3.31371	0.188510
$310$	0	0
$311$	−3.31371	−0.187903	−0.0939516	−	0.995577i	$-0.529950\pi$
$311$	−3.31371	−0.187903	−0.0939516	+	0.995577i	$0.529950\pi$
$312$	−24.9706	−1.41368
$313$	15.6569	0.884978	0.442489	−	0.896774i	$-0.354096\pi$
$313$	15.6569	0.884978	0.442489	+	0.896774i	$0.354096\pi$
$314$	−43.4558	−2.45236
$315$	−0.828427	−0.0466766
$316$	32.4853	1.82744
$317$	−26.2843	−1.47627	−0.738136	−	0.674652i	$-0.764294\pi$
$317$	−26.2843	−1.47627	−0.738136	+	0.674652i	$0.764294\pi$
$318$	22.4853	1.26091
$319$	4.82843	0.270340
$320$	9.82843	0.549426
$321$	−17.3137	−0.966357
$322$	8.00000	0.445823
$323$	8.00000	0.445132
$324$	3.82843	0.212690
$325$	−5.65685	−0.313786
$326$	17.6569	0.977923
$327$	−17.3137	−0.957450
$328$	−21.3137	−1.17685
$329$	−3.31371	−0.182691
$330$	2.41421	0.132898
$331$	6.34315	0.348651	0.174325	−	0.984688i	$-0.444226\pi$
$331$	6.34315	0.348651	0.174325	+	0.984688i	$0.444226\pi$
$332$	−38.2843	−2.10112
$333$	11.6569	0.638792
$334$	32.1421	1.75874
$335$	5.65685	0.309067
$336$	−2.48528	−0.135583
$337$	3.31371	0.180509	0.0902546	−	0.995919i	$-0.471232\pi$
$337$	3.31371	0.180509	0.0902546	+	0.995919i	$0.471232\pi$
$338$	−45.8701	−2.49500
$339$	18.9706	1.03034
$340$	4.48528	0.243249
$341$	0	0
$342$	16.4853	0.891422
$343$	−11.0294	−0.595534
$344$	38.9706	2.10115
$345$	−4.00000	−0.215353
$346$	−6.82843	−0.367099
$347$	−29.3137	−1.57364	−0.786821	−	0.617181i	$-0.788275\pi$
$347$	−29.3137	−1.57364	−0.786821	+	0.617181i	$0.788275\pi$
$348$	18.4853	0.990915
$349$	10.9706	0.587241	0.293620	−	0.955922i	$-0.405140\pi$
$349$	10.9706	0.587241	0.293620	+	0.955922i	$0.405140\pi$
$350$	−2.00000	−0.106904
$351$	5.65685	0.301941
$352$	−1.58579	−0.0845227
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	−9.65685	−0.513256
$355$	−2.34315	−0.124361
$356$	14.0000	0.741999
$357$	0.970563	0.0513676
$358$	42.6274	2.25293
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	4.41421	0.232649
$361$	27.6274	1.45407
$362$	33.7990	1.77644
$363$	−1.00000	−0.0524864
$364$	−17.9411	−0.940370
$365$	−11.3137	−0.592187
$366$	−28.1421	−1.47101
$367$	9.65685	0.504084	0.252042	−	0.967716i	$-0.418898\pi$
$367$	9.65685	0.504084	0.252042	+	0.967716i	$0.418898\pi$
$368$	−12.0000	−0.625543
$369$	4.82843	0.251358
$370$	28.1421	1.46304
$371$	7.71573	0.400581
$372$	0	0
$373$	10.6274	0.550267	0.275133	−	0.961406i	$-0.411278\pi$
$373$	10.6274	0.550267	0.275133	+	0.961406i	$0.411278\pi$
$374$	−2.82843	−0.146254
$375$	1.00000	0.0516398
$376$	17.6569	0.910583
$377$	27.3137	1.40673
$378$	2.00000	0.102869
$379$	−23.3137	−1.19754	−0.598772	−	0.800919i	$-0.704345\pi$
$379$	−23.3137	−1.19754	−0.598772	+	0.800919i	$0.704345\pi$
$380$	26.1421	1.34106
$381$	14.4853	0.742103
$382$	−13.6569	−0.698745
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	−20.5563	−1.04901
$385$	0.828427	0.0422206
$386$	−32.9706	−1.67816
$387$	−8.82843	−0.448774
$388$	44.6274	2.26561
$389$	−23.6569	−1.19945	−0.599725	−	0.800206i	$-0.704723\pi$
$389$	−23.6569	−1.19945	−0.599725	+	0.800206i	$0.704723\pi$
$390$	13.6569	0.691542
$391$	4.68629	0.236996
$392$	27.8701	1.40765
$393$	−3.31371	−0.167154
$394$	20.4853	1.03203
$395$	−8.48528	−0.426941
$396$	−3.82843	−0.192386
$397$	−14.9706	−0.751351	−0.375676	−	0.926751i	$-0.622589\pi$
$397$	−14.9706	−0.751351	−0.375676	+	0.926751i	$0.622589\pi$
$398$	52.2843	2.62077
$399$	5.65685	0.283197
$400$	3.00000	0.150000
$401$	−6.68629	−0.333897	−0.166949	−	0.985966i	$-0.553391\pi$
$401$	−6.68629	−0.333897	−0.166949	+	0.985966i	$0.553391\pi$
$402$	−13.6569	−0.681142
$403$	0	0
$404$	−3.17157	−0.157792
$405$	−1.00000	−0.0496904
$406$	9.65685	0.479262
$407$	−11.6569	−0.577809
$408$	−5.17157	−0.256031
$409$	−19.6569	−0.971969	−0.485984	−	0.873967i	$-0.661539\pi$
$409$	−19.6569	−0.971969	−0.485984	+	0.873967i	$0.661539\pi$
$410$	11.6569	0.575691
$411$	13.3137	0.656717
$412$	−12.6863	−0.625009
$413$	−3.31371	−0.163057
$414$	9.65685	0.474608
$415$	10.0000	0.490881
$416$	−8.97056	−0.439818
$417$	−0.485281	−0.0237643
$418$	−16.4853	−0.806321
$419$	−36.9706	−1.80613	−0.903065	−	0.429504i	$-0.858689\pi$
$419$	−36.9706	−1.80613	−0.903065	+	0.429504i	$0.858689\pi$
$420$	3.17157	0.154757
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	−2.82843	−0.137686
$423$	−4.00000	−0.194487
$424$	−41.1127	−1.99661
$425$	−1.17157	−0.0568296
$426$	5.65685	0.274075
$427$	−9.65685	−0.467328
$428$	66.2843	3.20397
$429$	−5.65685	−0.273115
$430$	−21.3137	−1.02784
$431$	21.6569	1.04317	0.521587	−	0.853198i	$-0.325340\pi$
$431$	21.6569	1.04317	0.521587	+	0.853198i	$0.325340\pi$
$432$	−3.00000	−0.144338
$433$	−15.6569	−0.752420	−0.376210	−	0.926534i	$-0.622773\pi$
$433$	−15.6569	−0.752420	−0.376210	+	0.926534i	$0.622773\pi$
$434$	0	0
$435$	−4.82843	−0.231505
$436$	66.2843	3.17444
$437$	27.3137	1.30659
$438$	27.3137	1.30510
$439$	−20.4853	−0.977709	−0.488855	−	0.872365i	$-0.662585\pi$
$439$	−20.4853	−0.977709	−0.488855	+	0.872365i	$0.662585\pi$
$440$	−4.41421	−0.210439
$441$	−6.31371	−0.300653
$442$	−16.0000	−0.761042
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−44.6274	−2.11792
$445$	−3.65685	−0.173352
$446$	15.3137	0.725125
$447$	−1.51472	−0.0716437
$448$	−8.14214	−0.384680
$449$	30.9706	1.46159	0.730796	−	0.682596i	$-0.239149\pi$
$449$	30.9706	1.46159	0.730796	+	0.682596i	$0.239149\pi$
$450$	−2.41421	−0.113807
$451$	−4.82843	−0.227362
$452$	−72.6274	−3.41611
$453$	−16.4853	−0.774546
$454$	33.7990	1.58627
$455$	4.68629	0.219697
$456$	−30.1421	−1.41153
$457$	−23.3137	−1.09057	−0.545285	−	0.838251i	$-0.683578\pi$
$457$	−23.3137	−1.09057	−0.545285	+	0.838251i	$0.683578\pi$
$458$	4.82843	0.225618
$459$	1.17157	0.0546843
$460$	15.3137	0.714005
$461$	0.142136	0.00661992	0.00330996	−	0.999995i	$-0.498946\pi$
$461$	0.142136	0.00661992	0.00330996	+	0.999995i	$0.498946\pi$
$462$	−2.00000	−0.0930484
$463$	4.97056	0.231002	0.115501	−	0.993307i	$-0.463153\pi$
$463$	4.97056	0.231002	0.115501	+	0.993307i	$0.463153\pi$
$464$	−14.4853	−0.672462
$465$	0	0
$466$	45.4558	2.10570
$467$	−22.6274	−1.04707	−0.523536	−	0.852004i	$-0.675387\pi$
$467$	−22.6274	−1.04707	−0.523536	+	0.852004i	$0.675387\pi$
$468$	−21.6569	−1.00109
$469$	−4.68629	−0.216393
$470$	−9.65685	−0.445437
$471$	−18.0000	−0.829396
$472$	17.6569	0.812723
$473$	8.82843	0.405932
$474$	20.4853	0.940920
$475$	−6.82843	−0.313310
$476$	−3.71573	−0.170310
$477$	9.31371	0.426445
$478$	−42.6274	−1.94973
$479$	36.9706	1.68923	0.844614	−	0.535376i	$-0.179830\pi$
$479$	36.9706	1.68923	0.844614	+	0.535376i	$0.179830\pi$
$480$	1.58579	0.0723809
$481$	−65.9411	−3.00666
$482$	29.7990	1.35731
$483$	3.31371	0.150779
$484$	3.82843	0.174019
$485$	−11.6569	−0.529310
$486$	2.41421	0.109511
$487$	−12.9706	−0.587752	−0.293876	−	0.955844i	$-0.594945\pi$
$487$	−12.9706	−0.587752	−0.293876	+	0.955844i	$0.594945\pi$
$488$	51.4558	2.32930
$489$	7.31371	0.330737
$490$	−15.2426	−0.688592
$491$	14.3431	0.647297	0.323649	−	0.946177i	$-0.395090\pi$
$491$	14.3431	0.647297	0.323649	+	0.946177i	$0.395090\pi$
$492$	−18.4853	−0.833381
$493$	5.65685	0.254772
$494$	−93.2548	−4.19573
$495$	1.00000	0.0449467
$496$	0	0
$497$	1.94113	0.0870714
$498$	−24.1421	−1.08183
$499$	−22.3431	−1.00022	−0.500108	−	0.865963i	$-0.666706\pi$
$499$	−22.3431	−1.00022	−0.500108	+	0.865963i	$0.666706\pi$
$500$	−3.82843	−0.171212
$501$	13.3137	0.594813
$502$	−50.6274	−2.25961
$503$	17.3137	0.771980	0.385990	−	0.922503i	$-0.373860\pi$
$503$	17.3137	0.771980	0.385990	+	0.922503i	$0.373860\pi$
$504$	−3.65685	−0.162889
$505$	0.828427	0.0368645
$506$	−9.65685	−0.429300
$507$	−19.0000	−0.843820
$508$	−55.4558	−2.46046
$509$	18.6863	0.828255	0.414128	−	0.910219i	$-0.364087\pi$
$509$	18.6863	0.828255	0.414128	+	0.910219i	$0.364087\pi$
$510$	2.82843	0.125245
$511$	9.37258	0.414619
$512$	31.2426	1.38074
$513$	6.82843	0.301482
$514$	39.4558	1.74032
$515$	3.31371	0.146019
$516$	33.7990	1.48792
$517$	4.00000	0.175920
$518$	−23.3137	−1.02435
$519$	−2.82843	−0.124154
$520$	−24.9706	−1.09503
$521$	−32.6274	−1.42943	−0.714717	−	0.699414i	$-0.753444\pi$
$521$	−32.6274	−1.42943	−0.714717	+	0.699414i	$0.753444\pi$
$522$	11.6569	0.510207
$523$	−9.51472	−0.416050	−0.208025	−	0.978124i	$-0.566703\pi$
$523$	−9.51472	−0.416050	−0.208025	+	0.978124i	$0.566703\pi$
$524$	12.6863	0.554203
$525$	−0.828427	−0.0361555
$526$	−43.4558	−1.89476
$527$	0	0
$528$	3.00000	0.130558
$529$	−7.00000	−0.304348
$530$	22.4853	0.976698
$531$	−4.00000	−0.173585
$532$	−21.6569	−0.938944
$533$	−27.3137	−1.18309
$534$	8.82843	0.382043
$535$	−17.3137	−0.748537
$536$	24.9706	1.07856
$537$	17.6569	0.761950
$538$	−49.7990	−2.14699
$539$	6.31371	0.271951
$540$	3.82843	0.164749
$541$	17.3137	0.744374	0.372187	−	0.928158i	$-0.378608\pi$
$541$	17.3137	0.744374	0.372187	+	0.928158i	$0.378608\pi$
$542$	28.4853	1.22355
$543$	14.0000	0.600798
$544$	−1.85786	−0.0796553
$545$	−17.3137	−0.741638
$546$	−11.3137	−0.484182
$547$	−8.14214	−0.348133	−0.174066	−	0.984734i	$-0.555691\pi$
$547$	−8.14214	−0.348133	−0.174066	+	0.984734i	$0.555691\pi$
$548$	−50.9706	−2.17735
$549$	−11.6569	−0.497502
$550$	2.41421	0.102942
$551$	32.9706	1.40459
$552$	−17.6569	−0.751526
$553$	7.02944	0.298922
$554$	5.65685	0.240337
$555$	11.6569	0.494806
$556$	1.85786	0.0787910
$557$	5.17157	0.219127	0.109563	−	0.993980i	$-0.465055\pi$
$557$	5.17157	0.219127	0.109563	+	0.993980i	$0.465055\pi$
$558$	0	0
$559$	49.9411	2.11228
$560$	−2.48528	−0.105022
$561$	−1.17157	−0.0494638
$562$	26.9706	1.13768
$563$	31.6569	1.33418	0.667089	−	0.744978i	$-0.267540\pi$
$563$	31.6569	1.33418	0.667089	+	0.744978i	$0.267540\pi$
$564$	15.3137	0.644823
$565$	18.9706	0.798098
$566$	21.3137	0.895882
$567$	0.828427	0.0347907
$568$	−10.3431	−0.433989
$569$	−35.4558	−1.48639	−0.743193	−	0.669077i	$-0.766690\pi$
$569$	−35.4558	−1.48639	−0.743193	+	0.669077i	$0.766690\pi$
$570$	16.4853	0.690492
$571$	−16.4853	−0.689888	−0.344944	−	0.938623i	$-0.612102\pi$
$571$	−16.4853	−0.689888	−0.344944	+	0.938623i	$0.612102\pi$
$572$	21.6569	0.905519
$573$	−5.65685	−0.236318
$574$	−9.65685	−0.403069
$575$	−4.00000	−0.166812
$576$	−9.82843	−0.409518
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	37.7279	1.56927
$579$	−13.6569	−0.567559
$580$	18.4853	0.767560
$581$	−8.28427	−0.343689
$582$	28.1421	1.16653
$583$	−9.31371	−0.385734
$584$	−49.9411	−2.06658
$585$	5.65685	0.233882
$586$	16.4853	0.681001
$587$	14.6274	0.603738	0.301869	−	0.953349i	$-0.402389\pi$
$587$	14.6274	0.603738	0.301869	+	0.953349i	$0.402389\pi$
$588$	24.1716	0.996819
$589$	0	0
$590$	−9.65685	−0.397566
$591$	8.48528	0.349038
$592$	34.9706	1.43728
$593$	−22.8284	−0.937451	−0.468726	−	0.883344i	$-0.655287\pi$
$593$	−22.8284	−0.937451	−0.468726	+	0.883344i	$0.655287\pi$
$594$	−2.41421	−0.0990564
$595$	0.970563	0.0397892
$596$	5.79899	0.237536
$597$	21.6569	0.886356
$598$	−54.6274	−2.23388
$599$	27.3137	1.11601	0.558004	−	0.829838i	$-0.311567\pi$
$599$	27.3137	1.11601	0.558004	+	0.829838i	$0.311567\pi$
$600$	4.41421	0.180210
$601$	−5.31371	−0.216751	−0.108375	−	0.994110i	$-0.534565\pi$
$601$	−5.31371	−0.216751	−0.108375	+	0.994110i	$0.534565\pi$
$602$	17.6569	0.719640
$603$	−5.65685	−0.230365
$604$	63.1127	2.56802
$605$	−1.00000	−0.0406558
$606$	−2.00000	−0.0812444
$607$	1.51472	0.0614805	0.0307403	−	0.999527i	$-0.490214\pi$
$607$	1.51472	0.0614805	0.0307403	+	0.999527i	$0.490214\pi$
$608$	−10.8284	−0.439151
$609$	4.00000	0.162088
$610$	−28.1421	−1.13944
$611$	22.6274	0.915407
$612$	−4.48528	−0.181307
$613$	−45.9411	−1.85554	−0.927772	−	0.373147i	$-0.878279\pi$
$613$	−45.9411	−1.85554	−0.927772	+	0.373147i	$0.878279\pi$
$614$	7.65685	0.309005
$615$	4.82843	0.194701
$616$	3.65685	0.147339
$617$	0.343146	0.0138145	0.00690726	−	0.999976i	$-0.497801\pi$
$617$	0.343146	0.0138145	0.00690726	+	0.999976i	$0.497801\pi$
$618$	−8.00000	−0.321807
$619$	−14.3431	−0.576500	−0.288250	−	0.957555i	$-0.593073\pi$
$619$	−14.3431	−0.576500	−0.288250	+	0.957555i	$0.593073\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	8.00000	0.320771
$623$	3.02944	0.121372
$624$	16.9706	0.679366
$625$	1.00000	0.0400000
$626$	−37.7990	−1.51075
$627$	−6.82843	−0.272701
$628$	68.9117	2.74988
$629$	−13.6569	−0.544534
$630$	2.00000	0.0796819
$631$	45.6569	1.81757	0.908785	−	0.417264i	$-0.137011\pi$
$631$	45.6569	1.81757	0.908785	+	0.417264i	$0.137011\pi$
$632$	−37.4558	−1.48991
$633$	−1.17157	−0.0465658
$634$	63.4558	2.52015
$635$	14.4853	0.574831
$636$	−35.6569	−1.41389
$637$	35.7157	1.41511
$638$	−11.6569	−0.461499
$639$	2.34315	0.0926934
$640$	−20.5563	−0.812561
$641$	6.97056	0.275321	0.137660	−	0.990479i	$-0.456042\pi$
$641$	6.97056	0.275321	0.137660	+	0.990479i	$0.456042\pi$
$642$	41.7990	1.64967
$643$	37.9411	1.49625	0.748126	−	0.663557i	$-0.230954\pi$
$643$	37.9411	1.49625	0.748126	+	0.663557i	$0.230954\pi$
$644$	−12.6863	−0.499910
$645$	−8.82843	−0.347619
$646$	−19.3137	−0.759888
$647$	4.68629	0.184237	0.0921186	−	0.995748i	$-0.470636\pi$
$647$	4.68629	0.184237	0.0921186	+	0.995748i	$0.470636\pi$
$648$	−4.41421	−0.173407
$649$	4.00000	0.157014
$650$	13.6569	0.535666
$651$	0	0
$652$	−28.0000	−1.09656
$653$	6.97056	0.272779	0.136390	−	0.990655i	$-0.456450\pi$
$653$	6.97056	0.272779	0.136390	+	0.990655i	$0.456450\pi$
$654$	41.7990	1.63447
$655$	−3.31371	−0.129477
$656$	14.4853	0.565555
$657$	11.3137	0.441390
$658$	8.00000	0.311872
$659$	−15.3137	−0.596537	−0.298269	−	0.954482i	$-0.596409\pi$
$659$	−15.3137	−0.596537	−0.298269	+	0.954482i	$0.596409\pi$
$660$	−3.82843	−0.149021
$661$	9.31371	0.362261	0.181131	−	0.983459i	$-0.442024\pi$
$661$	9.31371	0.362261	0.181131	+	0.983459i	$0.442024\pi$
$662$	−15.3137	−0.595184
$663$	−6.62742	−0.257388
$664$	44.1421	1.71305
$665$	5.65685	0.219363
$666$	−28.1421	−1.09049
$667$	19.3137	0.747830
$668$	−50.9706	−1.97211
$669$	6.34315	0.245240
$670$	−13.6569	−0.527610
$671$	11.6569	0.450008
$672$	−1.31371	−0.0506774
$673$	18.3431	0.707076	0.353538	−	0.935420i	$-0.384978\pi$
$673$	18.3431	0.707076	0.353538	+	0.935420i	$0.384978\pi$
$674$	−8.00000	−0.308148
$675$	−1.00000	−0.0384900
$676$	72.7401	2.79770
$677$	29.4558	1.13208	0.566040	−	0.824378i	$-0.308475\pi$
$677$	29.4558	1.13208	0.566040	+	0.824378i	$0.308475\pi$
$678$	−45.7990	−1.75890
$679$	9.65685	0.370596
$680$	−5.17157	−0.198321
$681$	14.0000	0.536481
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−26.1421	−0.999570
$685$	13.3137	0.508691
$686$	26.6274	1.01664
$687$	2.00000	0.0763048
$688$	−26.4853	−1.00974
$689$	−52.6863	−2.00719
$690$	9.65685	0.367630
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	10.8284	0.411635
$693$	−0.828427	−0.0314693
$694$	70.7696	2.68638
$695$	−0.485281	−0.0184078
$696$	−21.3137	−0.807894
$697$	−5.65685	−0.214269
$698$	−26.4853	−1.00248
$699$	18.8284	0.712157
$700$	3.17157	0.119874
$701$	36.1421	1.36507	0.682535	−	0.730853i	$-0.260878\pi$
$701$	36.1421	1.36507	0.682535	+	0.730853i	$0.260878\pi$
$702$	−13.6569	−0.515445
$703$	−79.5980	−3.00209
$704$	9.82843	0.370423
$705$	−4.00000	−0.150649
$706$	−62.7696	−2.36236
$707$	−0.686292	−0.0258106
$708$	15.3137	0.575524
$709$	6.68629	0.251109	0.125554	−	0.992087i	$-0.459929\pi$
$709$	6.68629	0.251109	0.125554	+	0.992087i	$0.459929\pi$
$710$	5.65685	0.212298
$711$	8.48528	0.318223
$712$	−16.1421	−0.604952
$713$	0	0
$714$	−2.34315	−0.0876900
$715$	−5.65685	−0.211554
$716$	−67.5980	−2.52626
$717$	−17.6569	−0.659407
$718$	−28.9706	−1.08117
$719$	−47.5980	−1.77511	−0.887553	−	0.460706i	$-0.847596\pi$
$719$	−47.5980	−1.77511	−0.887553	+	0.460706i	$0.847596\pi$
$720$	−3.00000	−0.111803
$721$	−2.74517	−0.102235
$722$	−66.6985	−2.48226
$723$	12.3431	0.459047
$724$	−53.5980	−1.99195
$725$	−4.82843	−0.179323
$726$	2.41421	0.0895999
$727$	33.9411	1.25881	0.629403	−	0.777079i	$-0.283299\pi$
$727$	33.9411	1.25881	0.629403	+	0.777079i	$0.283299\pi$
$728$	20.6863	0.766685
$729$	1.00000	0.0370370
$730$	27.3137	1.01093
$731$	10.3431	0.382555
$732$	44.6274	1.64948
$733$	−6.34315	−0.234289	−0.117145	−	0.993115i	$-0.537374\pi$
$733$	−6.34315	−0.234289	−0.117145	+	0.993115i	$0.537374\pi$
$734$	−23.3137	−0.860525
$735$	−6.31371	−0.232885
$736$	−6.34315	−0.233811
$737$	5.65685	0.208373
$738$	−11.6569	−0.429095
$739$	−15.1127	−0.555930	−0.277965	−	0.960591i	$-0.589660\pi$
$739$	−15.1127	−0.555930	−0.277965	+	0.960591i	$0.589660\pi$
$740$	−44.6274	−1.64054
$741$	−38.6274	−1.41901
$742$	−18.6274	−0.683834
$743$	36.3431	1.33330	0.666650	−	0.745371i	$-0.267727\pi$
$743$	36.3431	1.33330	0.666650	+	0.745371i	$0.267727\pi$
$744$	0	0
$745$	−1.51472	−0.0554950
$746$	−25.6569	−0.939364
$747$	−10.0000	−0.365881
$748$	4.48528	0.163998
$749$	14.3431	0.524087
$750$	−2.41421	−0.0881546
$751$	20.2843	0.740184	0.370092	−	0.928995i	$-0.379326\pi$
$751$	20.2843	0.740184	0.370092	+	0.928995i	$0.379326\pi$
$752$	−12.0000	−0.437595
$753$	−20.9706	−0.764210
$754$	−65.9411	−2.40143
$755$	−16.4853	−0.599961
$756$	−3.17157	−0.115349
$757$	−36.6274	−1.33125	−0.665623	−	0.746288i	$-0.731834\pi$
$757$	−36.6274	−1.33125	−0.665623	+	0.746288i	$0.731834\pi$
$758$	56.2843	2.04434
$759$	−4.00000	−0.145191
$760$	−30.1421	−1.09337
$761$	−28.8284	−1.04503	−0.522515	−	0.852630i	$-0.675006\pi$
$761$	−28.8284	−1.04503	−0.522515	+	0.852630i	$0.675006\pi$
$762$	−34.9706	−1.26685
$763$	14.3431	0.519257
$764$	21.6569	0.783517
$765$	1.17157	0.0423583
$766$	19.3137	0.697833
$767$	22.6274	0.817029
$768$	29.9706	1.08147
$769$	10.6863	0.385358	0.192679	−	0.981262i	$-0.438282\pi$
$769$	10.6863	0.385358	0.192679	+	0.981262i	$0.438282\pi$
$770$	−2.00000	−0.0720750
$771$	16.3431	0.588584
$772$	52.2843	1.88175
$773$	3.65685	0.131528	0.0657640	−	0.997835i	$-0.479052\pi$
$773$	3.65685	0.131528	0.0657640	+	0.997835i	$0.479052\pi$
$774$	21.3137	0.766105
$775$	0	0
$776$	−51.4558	−1.84716
$777$	−9.65685	−0.346438
$778$	57.1127	2.04759
$779$	−32.9706	−1.18129
$780$	−21.6569	−0.775440
$781$	−2.34315	−0.0838443
$782$	−11.3137	−0.404577
$783$	4.82843	0.172554
$784$	−18.9411	−0.676469
$785$	−18.0000	−0.642448
$786$	8.00000	0.285351
$787$	−20.1421	−0.717990	−0.358995	−	0.933340i	$-0.616880\pi$
$787$	−20.1421	−0.717990	−0.358995	+	0.933340i	$0.616880\pi$
$788$	−32.4853	−1.15724
$789$	−18.0000	−0.640817
$790$	20.4853	0.728834
$791$	−15.7157	−0.558787
$792$	4.41421	0.156852
$793$	65.9411	2.34164
$794$	36.1421	1.28264
$795$	9.31371	0.330323
$796$	−82.9117	−2.93873
$797$	34.9706	1.23872	0.619360	−	0.785107i	$-0.287392\pi$
$797$	34.9706	1.23872	0.619360	+	0.785107i	$0.287392\pi$
$798$	−13.6569	−0.483447
$799$	4.68629	0.165789
$800$	1.58579	0.0560660
$801$	3.65685	0.129209
$802$	16.1421	0.569999
$803$	−11.3137	−0.399252
$804$	21.6569	0.763778
$805$	3.31371	0.116793
$806$	0	0
$807$	−20.6274	−0.726119
$808$	3.65685	0.128648
$809$	28.4264	0.999419	0.499710	−	0.866193i	$-0.333440\pi$
$809$	28.4264	0.999419	0.499710	+	0.866193i	$0.333440\pi$
$810$	2.41421	0.0848268
$811$	0.485281	0.0170405	0.00852027	−	0.999964i	$-0.497288\pi$
$811$	0.485281	0.0170405	0.00852027	+	0.999964i	$0.497288\pi$
$812$	−15.3137	−0.537406
$813$	11.7990	0.413809
$814$	28.1421	0.986381
$815$	7.31371	0.256188
$816$	3.51472	0.123040
$817$	60.2843	2.10908
$818$	47.4558	1.65925
$819$	−4.68629	−0.163752
$820$	−18.4853	−0.645534
$821$	−12.8284	−0.447715	−0.223858	−	0.974622i	$-0.571865\pi$
$821$	−12.8284	−0.447715	−0.223858	+	0.974622i	$0.571865\pi$
$822$	−32.1421	−1.12109
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	14.6274	0.509570
$825$	1.00000	0.0348155
$826$	8.00000	0.278356
$827$	41.3137	1.43662	0.718309	−	0.695724i	$-0.244916\pi$
$827$	41.3137	1.43662	0.718309	+	0.695724i	$0.244916\pi$
$828$	−15.3137	−0.532188
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	−24.1421	−0.837986
$831$	2.34315	0.0812828
$832$	55.5980	1.92751
$833$	7.39697	0.256290
$834$	1.17157	0.0405683
$835$	13.3137	0.460740
$836$	26.1421	0.904145
$837$	0	0
$838$	89.2548	3.08326
$839$	−22.6274	−0.781185	−0.390593	−	0.920564i	$-0.627730\pi$
$839$	−22.6274	−0.781185	−0.390593	+	0.920564i	$0.627730\pi$
$840$	−3.65685	−0.126173
$841$	−5.68629	−0.196079
$842$	14.4853	0.499196
$843$	11.1716	0.384769
$844$	4.48528	0.154390
$845$	−19.0000	−0.653620
$846$	9.65685	0.332009
$847$	0.828427	0.0284651
$848$	27.9411	0.959502
$849$	8.82843	0.302991
$850$	2.82843	0.0970143
$851$	−46.6274	−1.59837
$852$	−8.97056	−0.307326
$853$	−8.68629	−0.297413	−0.148706	−	0.988881i	$-0.547511\pi$
$853$	−8.68629	−0.297413	−0.148706	+	0.988881i	$0.547511\pi$
$854$	23.3137	0.797779
$855$	6.82843	0.233527
$856$	−76.4264	−2.61220
$857$	−28.4853	−0.973039	−0.486519	−	0.873670i	$-0.661734\pi$
$857$	−28.4853	−0.973039	−0.486519	+	0.873670i	$0.661734\pi$
$858$	13.6569	0.466237
$859$	−52.9706	−1.80733	−0.903666	−	0.428238i	$-0.859135\pi$
$859$	−52.9706	−1.80733	−0.903666	+	0.428238i	$0.859135\pi$
$860$	33.7990	1.15254
$861$	−4.00000	−0.136320
$862$	−52.2843	−1.78081
$863$	−20.6863	−0.704170	−0.352085	−	0.935968i	$-0.614527\pi$
$863$	−20.6863	−0.704170	−0.352085	+	0.935968i	$0.614527\pi$
$864$	−1.58579	−0.0539496
$865$	−2.82843	−0.0961694
$866$	37.7990	1.28446
$867$	15.6274	0.530735
$868$	0	0
$869$	−8.48528	−0.287843
$870$	11.6569	0.395204
$871$	32.0000	1.08428
$872$	−76.4264	−2.58812
$873$	11.6569	0.394525
$874$	−65.9411	−2.23049
$875$	−0.828427	−0.0280059
$876$	−43.3137	−1.46343
$877$	2.62742	0.0887216	0.0443608	−	0.999016i	$-0.485875\pi$
$877$	2.62742	0.0887216	0.0443608	+	0.999016i	$0.485875\pi$
$878$	49.4558	1.66905
$879$	6.82843	0.230317
$880$	3.00000	0.101130
$881$	−46.9706	−1.58248	−0.791239	−	0.611507i	$-0.790564\pi$
$881$	−46.9706	−1.58248	−0.791239	+	0.611507i	$0.790564\pi$
$882$	15.2426	0.513246
$883$	−5.37258	−0.180802	−0.0904009	−	0.995905i	$-0.528815\pi$
$883$	−5.37258	−0.180802	−0.0904009	+	0.995905i	$0.528815\pi$
$884$	25.3726	0.853372
$885$	−4.00000	−0.134459
$886$	−28.9706	−0.973285
$887$	−15.6569	−0.525706	−0.262853	−	0.964836i	$-0.584663\pi$
$887$	−15.6569	−0.525706	−0.262853	+	0.964836i	$0.584663\pi$
$888$	51.4558	1.72675
$889$	−12.0000	−0.402467
$890$	8.82843	0.295930
$891$	−1.00000	−0.0335013
$892$	−24.2843	−0.813098
$893$	27.3137	0.914018
$894$	3.65685	0.122304
$895$	17.6569	0.590204
$896$	17.0294	0.568914
$897$	−22.6274	−0.755507
$898$	−74.7696	−2.49509
$899$	0	0
$900$	3.82843	0.127614
$901$	−10.9117	−0.363521
$902$	11.6569	0.388131
$903$	7.31371	0.243385
$904$	83.7401	2.78515
$905$	14.0000	0.465376
$906$	39.7990	1.32223
$907$	40.9706	1.36041	0.680203	−	0.733024i	$-0.261892\pi$
$907$	40.9706	1.36041	0.680203	+	0.733024i	$0.261892\pi$
$908$	−53.5980	−1.77871
$909$	−0.828427	−0.0274772
$910$	−11.3137	−0.375046
$911$	−48.9706	−1.62247	−0.811234	−	0.584722i	$-0.801203\pi$
$911$	−48.9706	−1.62247	−0.811234	+	0.584722i	$0.801203\pi$
$912$	20.4853	0.678335
$913$	10.0000	0.330952
$914$	56.2843	1.86172
$915$	−11.6569	−0.385364
$916$	−7.65685	−0.252990
$917$	2.74517	0.0906534
$918$	−2.82843	−0.0933520
$919$	11.5147	0.379836	0.189918	−	0.981800i	$-0.439178\pi$
$919$	11.5147	0.379836	0.189918	+	0.981800i	$0.439178\pi$
$920$	−17.6569	−0.582129
$921$	3.17157	0.104507
$922$	−0.343146	−0.0113009
$923$	−13.2548	−0.436288
$924$	3.17157	0.104337
$925$	11.6569	0.383275
$926$	−12.0000	−0.394344
$927$	−3.31371	−0.108836
$928$	−7.65685	−0.251349
$929$	−45.5980	−1.49602	−0.748011	−	0.663687i	$-0.768991\pi$
$929$	−45.5980	−1.49602	−0.748011	+	0.663687i	$0.768991\pi$
$930$	0	0
$931$	43.1127	1.41296
$932$	−72.0833	−2.36117
$933$	3.31371	0.108486
$934$	54.6274	1.78746
$935$	−1.17157	−0.0383145
$936$	24.9706	0.816188
$937$	11.0294	0.360316	0.180158	−	0.983638i	$-0.442339\pi$
$937$	11.0294	0.360316	0.180158	+	0.983638i	$0.442339\pi$
$938$	11.3137	0.369406
$939$	−15.6569	−0.510942
$940$	15.3137	0.499478
$941$	−34.7696	−1.13346	−0.566728	−	0.823905i	$-0.691791\pi$
$941$	−34.7696	−1.13346	−0.566728	+	0.823905i	$0.691791\pi$
$942$	43.4558	1.41587
$943$	−19.3137	−0.628941
$944$	−12.0000	−0.390567
$945$	0.828427	0.0269487
$946$	−21.3137	−0.692968
$947$	6.62742	0.215362	0.107681	−	0.994185i	$-0.465657\pi$
$947$	6.62742	0.215362	0.107681	+	0.994185i	$0.465657\pi$
$948$	−32.4853	−1.05507
$949$	−64.0000	−2.07753
$950$	16.4853	0.534853
$951$	26.2843	0.852326
$952$	4.28427	0.138854
$953$	−11.7990	−0.382207	−0.191103	−	0.981570i	$-0.561207\pi$
$953$	−11.7990	−0.382207	−0.191103	+	0.981570i	$0.561207\pi$
$954$	−22.4853	−0.727988
$955$	−5.65685	−0.183052
$956$	67.5980	2.18627
$957$	−4.82843	−0.156081
$958$	−89.2548	−2.88369
$959$	−11.0294	−0.356159
$960$	−9.82843	−0.317211
$961$	−31.0000	−1.00000
$962$	159.196	5.13268
$963$	17.3137	0.557926
$964$	−47.2548	−1.52198
$965$	−13.6569	−0.439630
$966$	−8.00000	−0.257396
$967$	11.4558	0.368395	0.184198	−	0.982889i	$-0.441031\pi$
$967$	11.4558	0.368395	0.184198	+	0.982889i	$0.441031\pi$
$968$	−4.41421	−0.141878
$969$	−8.00000	−0.256997
$970$	28.1421	0.903590
$971$	−34.6274	−1.11125	−0.555623	−	0.831434i	$-0.687520\pi$
$971$	−34.6274	−1.11125	−0.555623	+	0.831434i	$0.687520\pi$
$972$	−3.82843	−0.122797
$973$	0.402020	0.0128882
$974$	31.3137	1.00336
$975$	5.65685	0.181164
$976$	−34.9706	−1.11938
$977$	2.68629	0.0859421	0.0429710	−	0.999076i	$-0.486318\pi$
$977$	2.68629	0.0859421	0.0429710	+	0.999076i	$0.486318\pi$
$978$	−17.6569	−0.564604
$979$	−3.65685	−0.116874
$980$	24.1716	0.772133
$981$	17.3137	0.552784
$982$	−34.6274	−1.10501
$983$	−30.6274	−0.976863	−0.488431	−	0.872602i	$-0.662431\pi$
$983$	−30.6274	−0.976863	−0.488431	+	0.872602i	$0.662431\pi$
$984$	21.3137	0.679456
$985$	8.48528	0.270364
$986$	−13.6569	−0.434923
$987$	3.31371	0.105477
$988$	147.882	4.70476
$989$	35.3137	1.12291
$990$	−2.41421	−0.0767287
$991$	−30.6274	−0.972912	−0.486456	−	0.873705i	$-0.661711\pi$
$991$	−30.6274	−0.972912	−0.486456	+	0.873705i	$0.661711\pi$
$992$	0	0
$993$	−6.34315	−0.201294
$994$	−4.68629	−0.148640
$995$	21.6569	0.686568
$996$	38.2843	1.21308
$997$	−39.3137	−1.24508	−0.622539	−	0.782589i	$-0.713899\pi$
$997$	−39.3137	−1.24508	−0.622539	+	0.782589i	$0.713899\pi$
$998$	53.9411	1.70748
$999$	−11.6569	−0.368807

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.2.a.a.1.1	✓	2
3.2	odd	2		495.2.a.d.1.2		2
4.3	odd	2		2640.2.a.bb.1.1		2
5.2	odd	4		825.2.c.e.199.1		4
5.3	odd	4		825.2.c.e.199.4		4
5.4	even	2		825.2.a.g.1.2		2
7.6	odd	2		8085.2.a.ba.1.1		2
11.10	odd	2		1815.2.a.k.1.2		2
12.11	even	2		7920.2.a.cg.1.1		2
15.2	even	4		2475.2.c.m.199.4		4
15.8	even	4		2475.2.c.m.199.1		4
15.14	odd	2		2475.2.a.m.1.1		2
33.32	even	2		5445.2.a.m.1.1		2
55.54	odd	2		9075.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.a.1.1	✓	2	1.1	even	1	trivial
495.2.a.d.1.2		2	3.2	odd	2
825.2.a.g.1.2		2	5.4	even	2
825.2.c.e.199.1		4	5.2	odd	4
825.2.c.e.199.4		4	5.3	odd	4
1815.2.a.k.1.2		2	11.10	odd	2
2475.2.a.m.1.1		2	15.14	odd	2
2475.2.c.m.199.1		4	15.8	even	4
2475.2.c.m.199.4		4	15.2	even	4
2640.2.a.bb.1.1		2	4.3	odd	2
5445.2.a.m.1.1		2	33.32	even	2
7920.2.a.cg.1.1		2	12.11	even	2
8085.2.a.ba.1.1		2	7.6	odd	2
9075.2.a.v.1.1		2	55.54	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:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	165.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} +2.41421 q^{6} +0.828427 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} +2.41421 q^{6} +0.828427 q^{7} -4.41421 q^{8} +1.00000 q^{9} +2.41421 q^{10} -1.00000 q^{11} -3.82843 q^{12} -5.65685 q^{13} -2.00000 q^{14} +1.00000 q^{15} +3.00000 q^{16} -1.17157 q^{17} -2.41421 q^{18} -6.82843 q^{19} -3.82843 q^{20} -0.828427 q^{21} +2.41421 q^{22} -4.00000 q^{23} +4.41421 q^{24} +1.00000 q^{25} +13.6569 q^{26} -1.00000 q^{27} +3.17157 q^{28} -4.82843 q^{29} -2.41421 q^{30} +1.58579 q^{32} +1.00000 q^{33} +2.82843 q^{34} -0.828427 q^{35} +3.82843 q^{36} +11.6569 q^{37} +16.4853 q^{38} +5.65685 q^{39} +4.41421 q^{40} +4.82843 q^{41} +2.00000 q^{42} -8.82843 q^{43} -3.82843 q^{44} -1.00000 q^{45} +9.65685 q^{46} -4.00000 q^{47} -3.00000 q^{48} -6.31371 q^{49} -2.41421 q^{50} +1.17157 q^{51} -21.6569 q^{52} +9.31371 q^{53} +2.41421 q^{54} +1.00000 q^{55} -3.65685 q^{56} +6.82843 q^{57} +11.6569 q^{58} -4.00000 q^{59} +3.82843 q^{60} -11.6569 q^{61} +0.828427 q^{63} -9.82843 q^{64} +5.65685 q^{65} -2.41421 q^{66} -5.65685 q^{67} -4.48528 q^{68} +4.00000 q^{69} +2.00000 q^{70} +2.34315 q^{71} -4.41421 q^{72} +11.3137 q^{73} -28.1421 q^{74} -1.00000 q^{75} -26.1421 q^{76} -0.828427 q^{77} -13.6569 q^{78} +8.48528 q^{79} -3.00000 q^{80} +1.00000 q^{81} -11.6569 q^{82} -10.0000 q^{83} -3.17157 q^{84} +1.17157 q^{85} +21.3137 q^{86} +4.82843 q^{87} +4.41421 q^{88} +3.65685 q^{89} +2.41421 q^{90} -4.68629 q^{91} -15.3137 q^{92} +9.65685 q^{94} +6.82843 q^{95} -1.58579 q^{96} +11.6569 q^{97} +15.2426 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} - 2 q^{12} - 4 q^{14} + 2 q^{15} + 6 q^{16} - 8 q^{17} - 2 q^{18} - 8 q^{19} - 2 q^{20} + 4 q^{21} + 2 q^{22} - 8 q^{23} + 6 q^{24} + 2 q^{25} + 16 q^{26} - 2 q^{27} + 12 q^{28} - 4 q^{29} - 2 q^{30} + 6 q^{32} + 2 q^{33} + 4 q^{35} + 2 q^{36} + 12 q^{37} + 16 q^{38} + 6 q^{40} + 4 q^{41} + 4 q^{42} - 12 q^{43} - 2 q^{44} - 2 q^{45} + 8 q^{46} - 8 q^{47} - 6 q^{48} + 10 q^{49} - 2 q^{50} + 8 q^{51} - 32 q^{52} - 4 q^{53} + 2 q^{54} + 2 q^{55} + 4 q^{56} + 8 q^{57} + 12 q^{58} - 8 q^{59} + 2 q^{60} - 12 q^{61} - 4 q^{63} - 14 q^{64} - 2 q^{66} + 8 q^{68} + 8 q^{69} + 4 q^{70} + 16 q^{71} - 6 q^{72} - 28 q^{74} - 2 q^{75} - 24 q^{76} + 4 q^{77} - 16 q^{78} - 6 q^{80} + 2 q^{81} - 12 q^{82} - 20 q^{83} - 12 q^{84} + 8 q^{85} + 20 q^{86} + 4 q^{87} + 6 q^{88} - 4 q^{89} + 2 q^{90} - 32 q^{91} - 8 q^{92} + 8 q^{94} + 8 q^{95} - 6 q^{96} + 12 q^{97} + 22 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 + 2 * q^10 - 2 * q^11 - 2 * q^12 - 4 * q^14 + 2 * q^15 + 6 * q^16 - 8 * q^17 - 2 * q^18 - 8 * q^19 - 2 * q^20 + 4 * q^21 + 2 * q^22 - 8 * q^23 + 6 * q^24 + 2 * q^25 + 16 * q^26 - 2 * q^27 + 12 * q^28 - 4 * q^29 - 2 * q^30 + 6 * q^32 + 2 * q^33 + 4 * q^35 + 2 * q^36 + 12 * q^37 + 16 * q^38 + 6 * q^40 + 4 * q^41 + 4 * q^42 - 12 * q^43 - 2 * q^44 - 2 * q^45 + 8 * q^46 - 8 * q^47 - 6 * q^48 + 10 * q^49 - 2 * q^50 + 8 * q^51 - 32 * q^52 - 4 * q^53 + 2 * q^54 + 2 * q^55 + 4 * q^56 + 8 * q^57 + 12 * q^58 - 8 * q^59 + 2 * q^60 - 12 * q^61 - 4 * q^63 - 14 * q^64 - 2 * q^66 + 8 * q^68 + 8 * q^69 + 4 * q^70 + 16 * q^71 - 6 * q^72 - 28 * q^74 - 2 * q^75 - 24 * q^76 + 4 * q^77 - 16 * q^78 - 6 * q^80 + 2 * q^81 - 12 * q^82 - 20 * q^83 - 12 * q^84 + 8 * q^85 + 20 * q^86 + 4 * q^87 + 6 * q^88 - 4 * q^89 + 2 * q^90 - 32 * q^91 - 8 * q^92 + 8 * q^94 + 8 * q^95 - 6 * q^96 + 12 * q^97 + 22 * q^98 - 2 * q^99

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