LMFDB - Embedded newform 8041.2.a.g.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, 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("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.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:	$64.2077082653$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.66399 q^{2} +1.84747 q^{3} +0.768874 q^{4} +1.24336 q^{5} -3.07417 q^{6} +0.0311850 q^{7} +2.04859 q^{8} +0.413132 q^{9} +O(q^{10})$	\(q-1.66399 q^{2} +1.84747 q^{3} +0.768874 q^{4} +1.24336 q^{5} -3.07417 q^{6} +0.0311850 q^{7} +2.04859 q^{8} +0.413132 q^{9} -2.06894 q^{10} -1.00000 q^{11} +1.42047 q^{12} +3.49994 q^{13} -0.0518916 q^{14} +2.29706 q^{15} -4.94658 q^{16} +1.00000 q^{17} -0.687449 q^{18} -5.25992 q^{19} +0.955986 q^{20} +0.0576132 q^{21} +1.66399 q^{22} -4.42189 q^{23} +3.78469 q^{24} -3.45406 q^{25} -5.82388 q^{26} -4.77915 q^{27} +0.0239773 q^{28} -2.78056 q^{29} -3.82229 q^{30} -0.289166 q^{31} +4.13391 q^{32} -1.84747 q^{33} -1.66399 q^{34} +0.0387741 q^{35} +0.317647 q^{36} +10.1101 q^{37} +8.75248 q^{38} +6.46602 q^{39} +2.54712 q^{40} +4.09151 q^{41} -0.0958679 q^{42} +1.00000 q^{43} -0.768874 q^{44} +0.513671 q^{45} +7.35799 q^{46} +13.1048 q^{47} -9.13864 q^{48} -6.99903 q^{49} +5.74754 q^{50} +1.84747 q^{51} +2.69102 q^{52} -5.95331 q^{53} +7.95248 q^{54} -1.24336 q^{55} +0.0638850 q^{56} -9.71753 q^{57} +4.62683 q^{58} +10.5664 q^{59} +1.76615 q^{60} -5.77116 q^{61} +0.481170 q^{62} +0.0128835 q^{63} +3.01437 q^{64} +4.35168 q^{65} +3.07417 q^{66} -13.2819 q^{67} +0.768874 q^{68} -8.16928 q^{69} -0.0645198 q^{70} -0.811957 q^{71} +0.846336 q^{72} -7.66046 q^{73} -16.8232 q^{74} -6.38126 q^{75} -4.04422 q^{76} -0.0311850 q^{77} -10.7594 q^{78} +3.36894 q^{79} -6.15037 q^{80} -10.0687 q^{81} -6.80825 q^{82} -0.511288 q^{83} +0.0442973 q^{84} +1.24336 q^{85} -1.66399 q^{86} -5.13699 q^{87} -2.04859 q^{88} -13.7370 q^{89} -0.854745 q^{90} +0.109146 q^{91} -3.39987 q^{92} -0.534225 q^{93} -21.8064 q^{94} -6.53996 q^{95} +7.63726 q^{96} -9.63387 q^{97} +11.6463 q^{98} -0.413132 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9}+O(q^{10}) $ 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9	\( 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9} - q^{10} - 69 q^{11} - 3 q^{12} - 28 q^{13} - 15 q^{14} - 45 q^{15} + 53 q^{16} + 69 q^{17} - 17 q^{18} - 32 q^{19} - 21 q^{20} - 38 q^{21} + 11 q^{22} - 41 q^{23} - 11 q^{24} + 67 q^{25} - 6 q^{26} - 3 q^{27} - 21 q^{28} - 22 q^{29} - 22 q^{30} - 27 q^{31} - 87 q^{32} + 3 q^{33} - 11 q^{34} - 44 q^{35} + 59 q^{36} + 24 q^{37} - 22 q^{38} - 59 q^{39} + q^{40} - 43 q^{41} - 15 q^{42} + 69 q^{43} - 65 q^{44} - 12 q^{45} - 21 q^{46} - 99 q^{47} + 2 q^{48} + 64 q^{49} - 78 q^{50} - 3 q^{51} - 57 q^{52} - 50 q^{53} + 20 q^{54} + 6 q^{55} - 59 q^{56} - 15 q^{57} + 22 q^{58} - 82 q^{59} - 86 q^{60} - 24 q^{61} + 15 q^{62} - 63 q^{63} + 63 q^{64} - 23 q^{65} + 10 q^{66} - 54 q^{67} + 65 q^{68} + 36 q^{69} + 9 q^{70} - 128 q^{71} - 69 q^{72} + 2 q^{73} - 58 q^{74} - 31 q^{75} - 76 q^{76} + 11 q^{77} - 19 q^{78} - 43 q^{79} - 19 q^{80} + 49 q^{81} - 2 q^{82} - 62 q^{83} - 82 q^{84} - 6 q^{85} - 11 q^{86} - 62 q^{87} + 33 q^{88} - 49 q^{89} - 37 q^{90} - 2 q^{91} - 96 q^{92} - 29 q^{93} - 75 q^{94} - 133 q^{95} - 86 q^{96} + 5 q^{97} - 72 q^{98} - 56 q^{99}+O(q^{100}) \) 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9 - q^10 - 69 * q^11 - 3 * q^12 - 28 * q^13 - 15 * q^14 - 45 * q^15 + 53 * q^16 + 69 * q^17 - 17 * q^18 - 32 * q^19 - 21 * q^20 - 38 * q^21 + 11 * q^22 - 41 * q^23 - 11 * q^24 + 67 * q^25 - 6 * q^26 - 3 * q^27 - 21 * q^28 - 22 * q^29 - 22 * q^30 - 27 * q^31 - 87 * q^32 + 3 * q^33 - 11 * q^34 - 44 * q^35 + 59 * q^36 + 24 * q^37 - 22 * q^38 - 59 * q^39 + q^40 - 43 * q^41 - 15 * q^42 + 69 * q^43 - 65 * q^44 - 12 * q^45 - 21 * q^46 - 99 * q^47 + 2 * q^48 + 64 * q^49 - 78 * q^50 - 3 * q^51 - 57 * q^52 - 50 * q^53 + 20 * q^54 + 6 * q^55 - 59 * q^56 - 15 * q^57 + 22 * q^58 - 82 * q^59 - 86 * q^60 - 24 * q^61 + 15 * q^62 - 63 * q^63 + 63 * q^64 - 23 * q^65 + 10 * q^66 - 54 * q^67 + 65 * q^68 + 36 * q^69 + 9 * q^70 - 128 * q^71 - 69 * q^72 + 2 * q^73 - 58 * q^74 - 31 * q^75 - 76 * q^76 + 11 * q^77 - 19 * q^78 - 43 * q^79 - 19 * q^80 + 49 * q^81 - 2 * q^82 - 62 * q^83 - 82 * q^84 - 6 * q^85 - 11 * q^86 - 62 * q^87 + 33 * q^88 - 49 * q^89 - 37 * q^90 - 2 * q^91 - 96 * q^92 - 29 * q^93 - 75 * q^94 - 133 * q^95 - 86 * q^96 + 5 * q^97 - 72 * q^98 - 56 * 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 $

$2$	−1.66399	−1.17662	−0.588311	−	0.808635i	$-0.700207\pi$
$2$	−1.66399	−1.17662	−0.588311	+	0.808635i	$0.700207\pi$
$3$	1.84747	1.06664	0.533318	−	0.845915i	$-0.320945\pi$
$3$	1.84747	1.06664	0.533318	+	0.845915i	$0.320945\pi$
$4$	0.768874	0.384437
$5$	1.24336	0.556046	0.278023	−	0.960574i	$-0.410321\pi$
$5$	1.24336	0.556046	0.278023	+	0.960574i	$0.410321\pi$
$6$	−3.07417	−1.25503
$7$	0.0311850	0.0117868	0.00589340	−	0.999983i	$-0.498124\pi$
$7$	0.0311850	0.0117868	0.00589340	+	0.999983i	$0.498124\pi$
$8$	2.04859	0.724284
$9$	0.413132	0.137711
$10$	−2.06894	−0.654256
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.42047	0.410054
$13$	3.49994	0.970709	0.485355	−	0.874317i	$-0.338691\pi$
$13$	3.49994	0.970709	0.485355	+	0.874317i	$0.338691\pi$
$14$	−0.0518916	−0.0138686
$15$	2.29706	0.593099
$16$	−4.94658	−1.23665
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−0.687449	−0.162033
$19$	−5.25992	−1.20671	−0.603354	−	0.797473i	$-0.706170\pi$
$19$	−5.25992	−1.20671	−0.603354	+	0.797473i	$0.706170\pi$
$20$	0.955986	0.213765
$21$	0.0576132	0.0125722
$22$	1.66399	0.354765
$23$	−4.42189	−0.922027	−0.461013	−	0.887393i	$-0.652514\pi$
$23$	−4.42189	−0.922027	−0.461013	+	0.887393i	$0.652514\pi$
$24$	3.78469	0.772547
$25$	−3.45406	−0.690812
$26$	−5.82388	−1.14216
$27$	−4.77915	−0.919748
$28$	0.0239773	0.00453129
$29$	−2.78056	−0.516336	−0.258168	−	0.966100i	$-0.583119\pi$
$29$	−2.78056	−0.516336	−0.258168	+	0.966100i	$0.583119\pi$
$30$	−3.82229	−0.697852
$31$	−0.289166	−0.0519358	−0.0259679	−	0.999663i	$-0.508267\pi$
$31$	−0.289166	−0.0519358	−0.0259679	+	0.999663i	$0.508267\pi$
$32$	4.13391	0.730779
$33$	−1.84747	−0.321603
$34$	−1.66399	−0.285373
$35$	0.0387741	0.00655401
$36$	0.317647	0.0529411
$37$	10.1101	1.66210	0.831049	−	0.556199i	$-0.187741\pi$
$37$	10.1101	1.66210	0.831049	+	0.556199i	$0.187741\pi$
$38$	8.75248	1.41984
$39$	6.46602	1.03539
$40$	2.54712	0.402736
$41$	4.09151	0.638987	0.319494	−	0.947588i	$-0.396487\pi$
$41$	4.09151	0.638987	0.319494	+	0.947588i	$0.396487\pi$
$42$	−0.0958679	−0.0147927
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−0.768874	−0.115912
$45$	0.513671	0.0765735
$46$	7.35799	1.08488
$47$	13.1048	1.91154	0.955769	−	0.294118i	$-0.0950257\pi$
$47$	13.1048	1.91154	0.955769	+	0.294118i	$0.0950257\pi$
$48$	−9.13864	−1.31905
$49$	−6.99903	−0.999861
$50$	5.74754	0.812824
$51$	1.84747	0.258697
$52$	2.69102	0.373177
$53$	−5.95331	−0.817750	−0.408875	−	0.912590i	$-0.634079\pi$
$53$	−5.95331	−0.817750	−0.408875	+	0.912590i	$0.634079\pi$
$54$	7.95248	1.08220
$55$	−1.24336	−0.167654
$56$	0.0638850	0.00853700
$57$	−9.71753	−1.28712
$58$	4.62683	0.607532
$59$	10.5664	1.37563	0.687815	−	0.725886i	$-0.258570\pi$
$59$	10.5664	1.37563	0.687815	+	0.725886i	$0.258570\pi$
$60$	1.76615	0.228009
$61$	−5.77116	−0.738921	−0.369460	−	0.929246i	$-0.620457\pi$
$61$	−5.77116	−0.738921	−0.369460	+	0.929246i	$0.620457\pi$
$62$	0.481170	0.0611087
$63$	0.0128835	0.00162317
$64$	3.01437	0.376796
$65$	4.35168	0.539759
$66$	3.07417	0.378404
$67$	−13.2819	−1.62264	−0.811319	−	0.584604i	$-0.801250\pi$
$67$	−13.2819	−1.62264	−0.811319	+	0.584604i	$0.801250\pi$
$68$	0.768874	0.0932397
$69$	−8.16928	−0.983466
$70$	−0.0645198	−0.00771159
$71$	−0.811957	−0.0963616	−0.0481808	−	0.998839i	$-0.515342\pi$
$71$	−0.811957	−0.0963616	−0.0481808	+	0.998839i	$0.515342\pi$
$72$	0.846336	0.0997416
$73$	−7.66046	−0.896589	−0.448295	−	0.893886i	$-0.647968\pi$
$73$	−7.66046	−0.896589	−0.448295	+	0.893886i	$0.647968\pi$
$74$	−16.8232	−1.95566
$75$	−6.38126	−0.736845
$76$	−4.04422	−0.463904
$77$	−0.0311850	−0.00355386
$78$	−10.7594	−1.21826
$79$	3.36894	0.379036	0.189518	−	0.981877i	$-0.439308\pi$
$79$	3.36894	0.379036	0.189518	+	0.981877i	$0.439308\pi$
$80$	−6.15037	−0.687632
$81$	−10.0687	−1.11875
$82$	−6.80825	−0.751846
$83$	−0.511288	−0.0561212	−0.0280606	−	0.999606i	$-0.508933\pi$
$83$	−0.511288	−0.0561212	−0.0280606	+	0.999606i	$0.508933\pi$
$84$	0.0442973	0.00483323
$85$	1.24336	0.134861
$86$	−1.66399	−0.179433
$87$	−5.13699	−0.550743
$88$	−2.04859	−0.218380
$89$	−13.7370	−1.45612	−0.728061	−	0.685513i	$-0.759578\pi$
$89$	−13.7370	−1.45612	−0.728061	+	0.685513i	$0.759578\pi$
$90$	−0.854745	−0.0900980
$91$	0.109146	0.0114416
$92$	−3.39987	−0.354461
$93$	−0.534225	−0.0553965
$94$	−21.8064	−2.24916
$95$	−6.53996	−0.670986
$96$	7.63726	0.779474
$97$	−9.63387	−0.978171	−0.489085	−	0.872236i	$-0.662669\pi$
$97$	−9.63387	−0.978171	−0.489085	+	0.872236i	$0.662669\pi$
$98$	11.6463	1.17646
$99$	−0.413132	−0.0415213
$100$	−2.65574	−0.265574
$101$	−17.1787	−1.70935	−0.854674	−	0.519166i	$-0.826243\pi$
$101$	−17.1787	−1.70935	−0.854674	+	0.519166i	$0.826243\pi$
$102$	−3.07417	−0.304388
$103$	0.863988	0.0851313	0.0425656	−	0.999094i	$-0.486447\pi$
$103$	0.863988	0.0851313	0.0425656	+	0.999094i	$0.486447\pi$
$104$	7.16993	0.703069
$105$	0.0716338	0.00699074
$106$	9.90626	0.962181
$107$	−11.5026	−1.11199	−0.555997	−	0.831184i	$-0.687663\pi$
$107$	−11.5026	−1.11199	−0.555997	+	0.831184i	$0.687663\pi$
$108$	−3.67457	−0.353585
$109$	3.39601	0.325279	0.162640	−	0.986686i	$-0.447999\pi$
$109$	3.39601	0.325279	0.162640	+	0.986686i	$0.447999\pi$
$110$	2.06894	0.197266
$111$	18.6782	1.77285
$112$	−0.154259	−0.0145761
$113$	3.92016	0.368777	0.184389	−	0.982853i	$-0.440970\pi$
$113$	3.92016	0.368777	0.184389	+	0.982853i	$0.440970\pi$
$114$	16.1699	1.51445
$115$	−5.49798	−0.512690
$116$	−2.13790	−0.198499
$117$	1.44594	0.133677
$118$	−17.5825	−1.61860
$119$	0.0311850	0.00285872
$120$	4.70573	0.429572
$121$	1.00000	0.0909091
$122$	9.60317	0.869430
$123$	7.55893	0.681566
$124$	−0.222332	−0.0199660
$125$	−10.5114	−0.940170
$126$	−0.0214381	−0.00190985
$127$	8.57189	0.760632	0.380316	−	0.924857i	$-0.375815\pi$
$127$	8.57189	0.760632	0.380316	+	0.924857i	$0.375815\pi$
$128$	−13.2837	−1.17412
$129$	1.84747	0.162660
$130$	−7.24117	−0.635092
$131$	−0.973771	−0.0850788	−0.0425394	−	0.999095i	$-0.513545\pi$
$131$	−0.973771	−0.0850788	−0.0425394	+	0.999095i	$0.513545\pi$
$132$	−1.42047	−0.123636
$133$	−0.164030	−0.0142232
$134$	22.1009	1.90923
$135$	−5.94219	−0.511423
$136$	2.04859	0.175665
$137$	−2.45605	−0.209834	−0.104917	−	0.994481i	$-0.533458\pi$
$137$	−2.45605	−0.209834	−0.104917	+	0.994481i	$0.533458\pi$
$138$	13.5936	1.15717
$139$	−3.68775	−0.312791	−0.156395	−	0.987695i	$-0.549987\pi$
$139$	−3.68775	−0.312791	−0.156395	+	0.987695i	$0.549987\pi$
$140$	0.0298124	0.00251961
$141$	24.2108	2.03891
$142$	1.35109	0.113381
$143$	−3.49994	−0.292680
$144$	−2.04359	−0.170299
$145$	−3.45723	−0.287107
$146$	12.7470	1.05495
$147$	−12.9305	−1.06649
$148$	7.77343	0.638972
$149$	−9.05753	−0.742022	−0.371011	−	0.928628i	$-0.620989\pi$
$149$	−9.05753	−0.742022	−0.371011	+	0.928628i	$0.620989\pi$
$150$	10.6184	0.866987
$151$	2.13425	0.173682	0.0868412	−	0.996222i	$-0.472323\pi$
$151$	2.13425	0.173682	0.0868412	+	0.996222i	$0.472323\pi$
$152$	−10.7754	−0.874000
$153$	0.413132	0.0333997
$154$	0.0518916	0.00418154
$155$	−0.359537	−0.0288787
$156$	4.97156	0.398043
$157$	4.54706	0.362895	0.181448	−	0.983401i	$-0.441922\pi$
$157$	4.54706	0.362895	0.181448	+	0.983401i	$0.441922\pi$
$158$	−5.60590	−0.445981
$159$	−10.9985	−0.872241
$160$	5.13993	0.406347
$161$	−0.137896	−0.0108678
$162$	16.7543	1.31634
$163$	23.2936	1.82449	0.912246	−	0.409642i	$-0.134346\pi$
$163$	23.2936	1.82449	0.912246	+	0.409642i	$0.134346\pi$
$164$	3.14586	0.245650
$165$	−2.29706	−0.178826
$166$	0.850780	0.0660334
$167$	6.92118	0.535577	0.267789	−	0.963478i	$-0.413707\pi$
$167$	6.92118	0.535577	0.267789	+	0.963478i	$0.413707\pi$
$168$	0.118025	0.00910586
$169$	−0.750407	−0.0577236
$170$	−2.06894	−0.158680
$171$	−2.17304	−0.166177
$172$	0.768874	0.0586261
$173$	5.14447	0.391127	0.195563	−	0.980691i	$-0.437346\pi$
$173$	5.14447	0.391127	0.195563	+	0.980691i	$0.437346\pi$
$174$	8.54791	0.648015
$175$	−0.107715	−0.00814247
$176$	4.94658	0.372863
$177$	19.5211	1.46730
$178$	22.8583	1.71330
$179$	6.24711	0.466931	0.233466	−	0.972365i	$-0.424993\pi$
$179$	6.24711	0.466931	0.233466	+	0.972365i	$0.424993\pi$
$180$	0.394948	0.0294377
$181$	3.60300	0.267809	0.133904	−	0.990994i	$-0.457248\pi$
$181$	3.60300	0.267809	0.133904	+	0.990994i	$0.457248\pi$
$182$	−0.181617	−0.0134624
$183$	−10.6620	−0.788159
$184$	−9.05861	−0.667810
$185$	12.5705	0.924204
$186$	0.888946	0.0651807
$187$	−1.00000	−0.0731272
$188$	10.0760	0.734866
$189$	−0.149038	−0.0108409
$190$	10.8825	0.789496
$191$	−11.2755	−0.815868	−0.407934	−	0.913011i	$-0.633751\pi$
$191$	−11.2755	−0.815868	−0.407934	+	0.913011i	$0.633751\pi$
$192$	5.56894	0.401904
$193$	−24.7900	−1.78443	−0.892213	−	0.451615i	$-0.850848\pi$
$193$	−24.7900	−1.78443	−0.892213	+	0.451615i	$0.850848\pi$
$194$	16.0307	1.15094
$195$	8.03958	0.575726
$196$	−5.38137	−0.384384
$197$	−14.5191	−1.03444	−0.517222	−	0.855851i	$-0.673034\pi$
$197$	−14.5191	−1.03444	−0.517222	+	0.855851i	$0.673034\pi$
$198$	0.687449	0.0488549
$199$	−12.3326	−0.874234	−0.437117	−	0.899405i	$-0.644001\pi$
$199$	−12.3326	−0.874234	−0.437117	+	0.899405i	$0.644001\pi$
$200$	−7.07594	−0.500345
$201$	−24.5378	−1.73076
$202$	28.5853	2.01125
$203$	−0.0867116	−0.00608596
$204$	1.42047	0.0994528
$205$	5.08721	0.355306
$206$	−1.43767	−0.100167
$207$	−1.82682	−0.126973
$208$	−17.3127	−1.20042
$209$	5.25992	0.363836
$210$	−0.119198	−0.00822545
$211$	−26.1271	−1.79866	−0.899331	−	0.437269i	$-0.855946\pi$
$211$	−26.1271	−1.79866	−0.899331	+	0.437269i	$0.855946\pi$
$212$	−4.57735	−0.314373
$213$	−1.50006	−0.102783
$214$	19.1402	1.30840
$215$	1.24336	0.0847963
$216$	−9.79050	−0.666159
$217$	−0.00901763	−0.000612157 0
$218$	−5.65094	−0.382730
$219$	−14.1524	−0.956333
$220$	−0.955986	−0.0644525
$221$	3.49994	0.235432
$222$	−31.0803	−2.08598
$223$	−12.1955	−0.816672	−0.408336	−	0.912832i	$-0.633891\pi$
$223$	−12.1955	−0.816672	−0.408336	+	0.912832i	$0.633891\pi$
$224$	0.128916	0.00861355
$225$	−1.42698	−0.0951322
$226$	−6.52311	−0.433911
$227$	−13.0764	−0.867910	−0.433955	−	0.900935i	$-0.642882\pi$
$227$	−13.0764	−0.867910	−0.433955	+	0.900935i	$0.642882\pi$
$228$	−7.47156	−0.494816
$229$	2.15431	0.142361	0.0711804	−	0.997463i	$-0.477323\pi$
$229$	2.15431	0.142361	0.0711804	+	0.997463i	$0.477323\pi$
$230$	9.14861	0.603242
$231$	−0.0576132	−0.00379067
$232$	−5.69621	−0.373974
$233$	5.39996	0.353763	0.176881	−	0.984232i	$-0.443399\pi$
$233$	5.39996	0.353763	0.176881	+	0.984232i	$0.443399\pi$
$234$	−2.40603	−0.157287
$235$	16.2940	1.06290
$236$	8.12425	0.528844
$237$	6.22401	0.404293
$238$	−0.0518916	−0.00336363
$239$	−2.67730	−0.173180	−0.0865900	−	0.996244i	$-0.527597\pi$
$239$	−2.67730	−0.173180	−0.0865900	+	0.996244i	$0.527597\pi$
$240$	−11.3626	−0.733453
$241$	26.7524	1.72327	0.861636	−	0.507526i	$-0.169440\pi$
$241$	26.7524	1.72327	0.861636	+	0.507526i	$0.169440\pi$
$242$	−1.66399	−0.106966
$243$	−4.26416	−0.273546
$244$	−4.43729	−0.284069
$245$	−8.70229	−0.555969
$246$	−12.5780	−0.801945
$247$	−18.4094	−1.17136
$248$	−0.592381	−0.0376162
$249$	−0.944588	−0.0598608
$250$	17.4909	1.10622
$251$	−7.99065	−0.504365	−0.252183	−	0.967680i	$-0.581148\pi$
$251$	−7.99065	−0.504365	−0.252183	+	0.967680i	$0.581148\pi$
$252$	0.00990579	0.000624006 0
$253$	4.42189	0.278002
$254$	−14.2636	−0.894976
$255$	2.29706	0.143848
$256$	16.0753	1.00470
$257$	0.550791	0.0343574	0.0171787	−	0.999852i	$-0.494532\pi$
$257$	0.550791	0.0343574	0.0171787	+	0.999852i	$0.494532\pi$
$258$	−3.07417	−0.191390
$259$	0.315285	0.0195908
$260$	3.34589	0.207504
$261$	−1.14874	−0.0711050
$262$	1.62035	0.100105
$263$	1.17884	0.0726904	0.0363452	−	0.999339i	$-0.488428\pi$
$263$	1.17884	0.0726904	0.0363452	+	0.999339i	$0.488428\pi$
$264$	−3.78469	−0.232932
$265$	−7.40209	−0.454707
$266$	0.272946	0.0167354
$267$	−25.3787	−1.55315
$268$	−10.2121	−0.623802
$269$	19.5036	1.18915	0.594577	−	0.804039i	$-0.297320\pi$
$269$	19.5036	1.18915	0.594577	+	0.804039i	$0.297320\pi$
$270$	9.88777	0.601751
$271$	30.6217	1.86013	0.930067	−	0.367391i	$-0.119749\pi$
$271$	30.6217	1.86013	0.930067	+	0.367391i	$0.119749\pi$
$272$	−4.94658	−0.299931
$273$	0.201643	0.0122040
$274$	4.08685	0.246895
$275$	3.45406	0.208288
$276$	−6.28115	−0.378081
$277$	−0.644936	−0.0387504	−0.0193752	−	0.999812i	$-0.506168\pi$
$277$	−0.644936	−0.0387504	−0.0193752	+	0.999812i	$0.506168\pi$
$278$	6.13639	0.368036
$279$	−0.119464	−0.00715211
$280$	0.0794319	0.00474697
$281$	−27.6827	−1.65141	−0.825706	−	0.564101i	$-0.809223\pi$
$281$	−27.6827	−1.65141	−0.825706	+	0.564101i	$0.809223\pi$
$282$	−40.2866	−2.39903
$283$	−21.6468	−1.28677	−0.643383	−	0.765544i	$-0.722470\pi$
$283$	−21.6468	−1.28677	−0.643383	+	0.765544i	$0.722470\pi$
$284$	−0.624293	−0.0370450
$285$	−12.0824	−0.715697
$286$	5.82388	0.344373
$287$	0.127594	0.00753162
$288$	1.70785	0.100636
$289$	1.00000	0.0588235
$290$	5.75280	0.337816
$291$	−17.7982	−1.04335
$292$	−5.88993	−0.344682
$293$	30.9671	1.80911	0.904557	−	0.426352i	$-0.140202\pi$
$293$	30.9671	1.80911	0.904557	+	0.426352i	$0.140202\pi$
$294$	21.5162	1.25485
$295$	13.1378	0.764915
$296$	20.7115	1.20383
$297$	4.77915	0.277315
$298$	15.0717	0.873079
$299$	−15.4763	−0.895020
$300$	−4.90639	−0.283271
$301$	0.0311850	0.00179747
$302$	−3.55137	−0.204358
$303$	−31.7371	−1.82325
$304$	26.0186	1.49227
$305$	−7.17561	−0.410874
$306$	−0.687449	−0.0392988
$307$	−23.6287	−1.34856	−0.674281	−	0.738475i	$-0.735546\pi$
$307$	−23.6287	−1.34856	−0.674281	+	0.738475i	$0.735546\pi$
$308$	−0.0239773	−0.00136623
$309$	1.59619	0.0908040
$310$	0.598267	0.0339793
$311$	9.27572	0.525978	0.262989	−	0.964799i	$-0.415292\pi$
$311$	9.27572	0.525978	0.262989	+	0.964799i	$0.415292\pi$
$312$	13.2462	0.749919
$313$	−23.1126	−1.30640	−0.653202	−	0.757184i	$-0.726575\pi$
$313$	−23.1126	−1.30640	−0.653202	+	0.757184i	$0.726575\pi$
$314$	−7.56629	−0.426990
$315$	0.0160188	0.000902557 0
$316$	2.59029	0.145715
$317$	24.3691	1.36870	0.684352	−	0.729152i	$-0.260085\pi$
$317$	24.3691	1.36870	0.684352	+	0.729152i	$0.260085\pi$
$318$	18.3015	1.02630
$319$	2.78056	0.155681
$320$	3.74793	0.209516
$321$	−21.2506	−1.18609
$322$	0.229459	0.0127872
$323$	−5.25992	−0.292670
$324$	−7.74158	−0.430088
$325$	−12.0890	−0.670578
$326$	−38.7603	−2.14674
$327$	6.27402	0.346954
$328$	8.38181	0.462808
$329$	0.408674	0.0225309
$330$	3.82229	0.210410
$331$	25.7679	1.41633	0.708166	−	0.706046i	$-0.249523\pi$
$331$	25.7679	1.41633	0.708166	+	0.706046i	$0.249523\pi$
$332$	−0.393116	−0.0215751
$333$	4.17683	0.228889
$334$	−11.5168	−0.630171
$335$	−16.5141	−0.902261
$336$	−0.284988	−0.0155474
$337$	1.60254	0.0872959	0.0436479	−	0.999047i	$-0.486102\pi$
$337$	1.60254	0.0872959	0.0436479	+	0.999047i	$0.486102\pi$
$338$	1.24867	0.0679188
$339$	7.24236	0.393351
$340$	0.955986	0.0518456
$341$	0.289166	0.0156592
$342$	3.61593	0.195527
$343$	−0.436559	−0.0235720
$344$	2.04859	0.110452
$345$	−10.1573	−0.546853
$346$	−8.56037	−0.460208
$347$	−36.1877	−1.94266	−0.971330	−	0.237735i	$-0.923595\pi$
$347$	−36.1877	−1.94266	−0.971330	+	0.237735i	$0.923595\pi$
$348$	−3.94970	−0.211726
$349$	2.74523	0.146949	0.0734743	−	0.997297i	$-0.476591\pi$
$349$	2.74523	0.146949	0.0734743	+	0.997297i	$0.476591\pi$
$350$	0.179237	0.00958060
$351$	−16.7268	−0.892808
$352$	−4.13391	−0.220338
$353$	−6.09982	−0.324661	−0.162330	−	0.986736i	$-0.551901\pi$
$353$	−6.09982	−0.324661	−0.162330	+	0.986736i	$0.551901\pi$
$354$	−32.4830	−1.72645
$355$	−1.00955	−0.0535815
$356$	−10.5620	−0.559787
$357$	0.0576132	0.00304921
$358$	−10.3952	−0.549401
$359$	−14.7446	−0.778189	−0.389094	−	0.921198i	$-0.627212\pi$
$359$	−14.7446	−0.778189	−0.389094	+	0.921198i	$0.627212\pi$
$360$	1.05230	0.0554610
$361$	8.66678	0.456146
$362$	−5.99537	−0.315110
$363$	1.84747	0.0969668
$364$	0.0839192	0.00439856
$365$	−9.52469	−0.498545
$366$	17.7415	0.927364
$367$	−28.4531	−1.48524	−0.742620	−	0.669713i	$-0.766417\pi$
$367$	−28.4531	−1.48524	−0.742620	+	0.669713i	$0.766417\pi$
$368$	21.8732	1.14022
$369$	1.69034	0.0879953
$370$	−20.9173	−1.08744
$371$	−0.185654	−0.00963866
$372$	−0.410752	−0.0212965
$373$	−4.70699	−0.243718	−0.121859	−	0.992547i	$-0.538886\pi$
$373$	−4.70699	−0.243718	−0.121859	+	0.992547i	$0.538886\pi$
$374$	1.66399	0.0860431
$375$	−19.4195	−1.00282
$376$	26.8464	1.38450
$377$	−9.73179	−0.501213
$378$	0.247998	0.0127556
$379$	15.9752	0.820591	0.410295	−	0.911953i	$-0.365426\pi$
$379$	15.9752	0.820591	0.410295	+	0.911953i	$0.365426\pi$
$380$	−5.02841	−0.257952
$381$	15.8363	0.811317
$382$	18.7624	0.959967
$383$	25.7909	1.31786	0.658928	−	0.752206i	$-0.271010\pi$
$383$	25.7909	1.31786	0.658928	+	0.752206i	$0.271010\pi$
$384$	−24.5412	−1.25236
$385$	−0.0387741	−0.00197611
$386$	41.2505	2.09959
$387$	0.413132	0.0210007
$388$	−7.40723	−0.376045
$389$	24.4107	1.23767	0.618836	−	0.785520i	$-0.287604\pi$
$389$	24.4107	1.23767	0.618836	+	0.785520i	$0.287604\pi$
$390$	−13.3778	−0.677412
$391$	−4.42189	−0.223624
$392$	−14.3381	−0.724184
$393$	−1.79901	−0.0907480
$394$	24.1597	1.21715
$395$	4.18880	0.210761
$396$	−0.317647	−0.0159623
$397$	−10.5607	−0.530026	−0.265013	−	0.964245i	$-0.585376\pi$
$397$	−10.5607	−0.530026	−0.265013	+	0.964245i	$0.585376\pi$
$398$	20.5214	1.02864
$399$	−0.303041	−0.0151710
$400$	17.0858	0.854290
$401$	13.9064	0.694454	0.347227	−	0.937781i	$-0.387123\pi$
$401$	13.9064	0.694454	0.347227	+	0.937781i	$0.387123\pi$
$402$	40.8307	2.03645
$403$	−1.01206	−0.0504145
$404$	−13.2083	−0.657137
$405$	−12.5190	−0.622075
$406$	0.144287	0.00716087
$407$	−10.1101	−0.501142
$408$	3.78469	0.187370
$409$	26.4549	1.30811	0.654055	−	0.756447i	$-0.273066\pi$
$409$	26.4549	1.30811	0.654055	+	0.756447i	$0.273066\pi$
$410$	−8.46509	−0.418061
$411$	−4.53746	−0.223817
$412$	0.664298	0.0327276
$413$	0.329513	0.0162143
$414$	3.03982	0.149399
$415$	−0.635714	−0.0312060
$416$	14.4684	0.709374
$417$	−6.81299	−0.333634
$418$	−8.75248	−0.428098
$419$	−7.68294	−0.375336	−0.187668	−	0.982233i	$-0.560093\pi$
$419$	−7.68294	−0.375336	−0.187668	+	0.982233i	$0.560093\pi$
$420$	0.0550774	0.00268750
$421$	23.1835	1.12989	0.564947	−	0.825127i	$-0.308897\pi$
$421$	23.1835	1.12989	0.564947	+	0.825127i	$0.308897\pi$
$422$	43.4753	2.11634
$423$	5.41403	0.263239
$424$	−12.1959	−0.592283
$425$	−3.45406	−0.167547
$426$	2.49610	0.120936
$427$	−0.179973	−0.00870952
$428$	−8.84402	−0.427492
$429$	−6.46602	−0.312183
$430$	−2.06894	−0.0997731
$431$	−22.0653	−1.06285	−0.531423	−	0.847107i	$-0.678343\pi$
$431$	−22.0653	−1.06285	−0.531423	+	0.847107i	$0.678343\pi$
$432$	23.6405	1.13740
$433$	−2.73527	−0.131449	−0.0657244	−	0.997838i	$-0.520936\pi$
$433$	−2.73527	−0.131449	−0.0657244	+	0.997838i	$0.520936\pi$
$434$	0.0150053	0.000720276 0
$435$	−6.38711	−0.306238
$436$	2.61111	0.125049
$437$	23.2588	1.11262
$438$	23.5496	1.12524
$439$	−25.9404	−1.23807	−0.619033	−	0.785365i	$-0.712475\pi$
$439$	−25.9404	−1.23807	−0.619033	+	0.785365i	$0.712475\pi$
$440$	−2.54712	−0.121429
$441$	−2.89152	−0.137692
$442$	−5.82388	−0.277014
$443$	−32.1727	−1.52857	−0.764285	−	0.644878i	$-0.776908\pi$
$443$	−32.1727	−1.52857	−0.764285	+	0.644878i	$0.776908\pi$
$444$	14.3612	0.681550
$445$	−17.0800	−0.809671
$446$	20.2933	0.960913
$447$	−16.7335	−0.791467
$448$	0.0940029	0.00444122
$449$	30.9338	1.45986	0.729929	−	0.683523i	$-0.239553\pi$
$449$	30.9338	1.45986	0.729929	+	0.683523i	$0.239553\pi$
$450$	2.37449	0.111935
$451$	−4.09151	−0.192662
$452$	3.01411	0.141772
$453$	3.94295	0.185256
$454$	21.7590	1.02120
$455$	0.135707	0.00636204
$456$	−19.9072	−0.932239
$457$	−26.3308	−1.23170	−0.615851	−	0.787863i	$-0.711188\pi$
$457$	−26.3308	−1.23170	−0.615851	+	0.787863i	$0.711188\pi$
$458$	−3.58476	−0.167505
$459$	−4.77915	−0.223072
$460$	−4.22726	−0.197097
$461$	−18.0636	−0.841304	−0.420652	−	0.907222i	$-0.638199\pi$
$461$	−18.0636	−0.841304	−0.420652	+	0.907222i	$0.638199\pi$
$462$	0.0958679	0.00446018
$463$	6.04264	0.280825	0.140413	−	0.990093i	$-0.455157\pi$
$463$	6.04264	0.280825	0.140413	+	0.990093i	$0.455157\pi$
$464$	13.7543	0.638525
$465$	−0.664232	−0.0308030
$466$	−8.98549	−0.416245
$467$	0.893753	0.0413579	0.0206790	−	0.999786i	$-0.493417\pi$
$467$	0.893753	0.0413579	0.0206790	+	0.999786i	$0.493417\pi$
$468$	1.11174	0.0513904
$469$	−0.414194	−0.0191257
$470$	−27.1131	−1.25064
$471$	8.40055	0.387077
$472$	21.6462	0.996348
$473$	−1.00000	−0.0459800
$474$	−10.3567	−0.475700
$475$	18.1681	0.833609
$476$	0.0239773	0.00109900
$477$	−2.45950	−0.112613
$478$	4.45501	0.203767
$479$	40.0823	1.83141	0.915703	−	0.401856i	$-0.131635\pi$
$479$	40.0823	1.83141	0.915703	+	0.401856i	$0.131635\pi$
$480$	9.49584	0.433424
$481$	35.3849	1.61341
$482$	−44.5158	−2.02764
$483$	−0.254759	−0.0115919
$484$	0.768874	0.0349488
$485$	−11.9783	−0.543908
$486$	7.09554	0.321860
$487$	−4.48907	−0.203419	−0.101710	−	0.994814i	$-0.532431\pi$
$487$	−4.48907	−0.203419	−0.101710	+	0.994814i	$0.532431\pi$
$488$	−11.8227	−0.535189
$489$	43.0341	1.94607
$490$	14.4806	0.654165
$491$	−33.8266	−1.52658	−0.763288	−	0.646059i	$-0.776416\pi$
$491$	−33.8266	−1.52658	−0.763288	+	0.646059i	$0.776416\pi$
$492$	5.81187	0.262019
$493$	−2.78056	−0.125230
$494$	30.6332	1.37825
$495$	−0.513671	−0.0230878
$496$	1.43038	0.0642261
$497$	−0.0253209	−0.00113580
$498$	1.57179	0.0704335
$499$	18.7854	0.840949	0.420475	−	0.907304i	$-0.361864\pi$
$499$	18.7854	0.840949	0.420475	+	0.907304i	$0.361864\pi$
$500$	−8.08196	−0.361436
$501$	12.7867	0.571266
$502$	13.2964	0.593447
$503$	3.78604	0.168811	0.0844056	−	0.996431i	$-0.473101\pi$
$503$	3.78604	0.168811	0.0844056	+	0.996431i	$0.473101\pi$
$504$	0.0263930	0.00117564
$505$	−21.3593	−0.950476
$506$	−7.35799	−0.327103
$507$	−1.38635	−0.0615700
$508$	6.59070	0.292415
$509$	25.2665	1.11992	0.559960	−	0.828520i	$-0.310817\pi$
$509$	25.2665	1.11992	0.559960	+	0.828520i	$0.310817\pi$
$510$	−3.82229	−0.169254
$511$	−0.238891	−0.0105679
$512$	−0.181728	−0.00803132
$513$	25.1380	1.10987
$514$	−0.916513	−0.0404257
$515$	1.07425	0.0473369
$516$	1.42047	0.0625327
$517$	−13.1048	−0.576351
$518$	−0.524632	−0.0230510
$519$	9.50424	0.417190
$520$	8.91478	0.390939
$521$	−44.2144	−1.93707	−0.968534	−	0.248879i	$-0.919938\pi$
$521$	−44.2144	−1.93707	−0.968534	+	0.248879i	$0.919938\pi$
$522$	1.91149	0.0836637
$523$	16.4916	0.721129	0.360564	−	0.932734i	$-0.382584\pi$
$523$	16.4916	0.721129	0.360564	+	0.932734i	$0.382584\pi$
$524$	−0.748707	−0.0327074
$525$	−0.198999	−0.00868505
$526$	−1.96158	−0.0855290
$527$	−0.289166	−0.0125963
$528$	9.13864	0.397708
$529$	−3.44693	−0.149866
$530$	12.3170	0.535018
$531$	4.36533	0.189439
$532$	−0.126119	−0.00546794
$533$	14.3201	0.620271
$534$	42.2300	1.82747
$535$	−14.3018	−0.618320
$536$	−27.2090	−1.17525
$537$	11.5413	0.498045
$538$	−32.4538	−1.39918
$539$	6.99903	0.301469
$540$	−4.56880	−0.196610
$541$	10.8329	0.465741	0.232871	−	0.972508i	$-0.425188\pi$
$541$	10.8329	0.465741	0.232871	+	0.972508i	$0.425188\pi$
$542$	−50.9542	−2.18867
$543$	6.65642	0.285654
$544$	4.13391	0.177240
$545$	4.22246	0.180870
$546$	−0.335532	−0.0143595
$547$	−38.6904	−1.65428	−0.827141	−	0.561995i	$-0.810034\pi$
$547$	−38.6904	−1.65428	−0.827141	+	0.561995i	$0.810034\pi$
$548$	−1.88839	−0.0806681
$549$	−2.38425	−0.101757
$550$	−5.74754	−0.245076
$551$	14.6255	0.623068
$552$	−16.7355	−0.712309
$553$	0.105060	0.00446762
$554$	1.07317	0.0455946
$555$	23.2236	0.985788
$556$	−2.83542	−0.120248
$557$	−15.2934	−0.648004	−0.324002	−	0.946056i	$-0.605028\pi$
$557$	−15.2934	−0.648004	−0.324002	+	0.946056i	$0.605028\pi$
$558$	0.198787	0.00841532
$559$	3.49994	0.148032
$560$	−0.191799	−0.00810499
$561$	−1.84747	−0.0780001
$562$	46.0638	1.94309
$563$	16.8693	0.710956	0.355478	−	0.934685i	$-0.384318\pi$
$563$	16.8693	0.710956	0.355478	+	0.934685i	$0.384318\pi$
$564$	18.6150	0.783834
$565$	4.87415	0.205057
$566$	36.0201	1.51404
$567$	−0.313993	−0.0131864
$568$	−1.66336	−0.0697932
$569$	−8.51834	−0.357107	−0.178554	−	0.983930i	$-0.557142\pi$
$569$	−8.51834	−0.357107	−0.178554	+	0.983930i	$0.557142\pi$
$570$	20.1050	0.842105
$571$	−29.7264	−1.24401	−0.622006	−	0.783013i	$-0.713682\pi$
$571$	−29.7264	−1.24401	−0.622006	+	0.783013i	$0.713682\pi$
$572$	−2.69102	−0.112517
$573$	−20.8311	−0.870233
$574$	−0.212315	−0.00886186
$575$	15.2735	0.636948
$576$	1.24533	0.0518888
$577$	−33.1391	−1.37960	−0.689799	−	0.724001i	$-0.742301\pi$
$577$	−33.1391	−1.37960	−0.689799	+	0.724001i	$0.742301\pi$
$578$	−1.66399	−0.0692130
$579$	−45.7988	−1.90333
$580$	−2.65817	−0.110375
$581$	−0.0159445	−0.000661490 0
$582$	29.6162	1.22763
$583$	5.95331	0.246561
$584$	−15.6931	−0.649385
$585$	1.79782	0.0743306
$586$	−51.5290	−2.12864
$587$	−34.9504	−1.44256	−0.721279	−	0.692645i	$-0.756445\pi$
$587$	−34.9504	−1.44256	−0.721279	+	0.692645i	$0.756445\pi$
$588$	−9.94190	−0.409997
$589$	1.52099	0.0626713
$590$	−21.8613	−0.900015
$591$	−26.8236	−1.10337
$592$	−50.0107	−2.05543
$593$	13.2137	0.542620	0.271310	−	0.962492i	$-0.412543\pi$
$593$	13.2137	0.542620	0.271310	+	0.962492i	$0.412543\pi$
$594$	−7.95248	−0.326294
$595$	0.0387741	0.00158958
$596$	−6.96411	−0.285261
$597$	−22.7841	−0.932489
$598$	25.7525	1.05310
$599$	−8.14703	−0.332879	−0.166439	−	0.986052i	$-0.553227\pi$
$599$	−8.14703	−0.332879	−0.166439	+	0.986052i	$0.553227\pi$
$600$	−13.0726	−0.533685
$601$	−21.6584	−0.883466	−0.441733	−	0.897147i	$-0.645636\pi$
$601$	−21.6584	−0.883466	−0.441733	+	0.897147i	$0.645636\pi$
$602$	−0.0518916	−0.00211494
$603$	−5.48716	−0.223454
$604$	1.64097	0.0667700
$605$	1.24336	0.0505497
$606$	52.8104	2.14527
$607$	−22.1221	−0.897908	−0.448954	−	0.893555i	$-0.648203\pi$
$607$	−22.1221	−0.897908	−0.448954	+	0.893555i	$0.648203\pi$
$608$	−21.7440	−0.881837
$609$	−0.160197	−0.00649150
$610$	11.9402	0.483443
$611$	45.8662	1.85555
$612$	0.317647	0.0128401
$613$	34.0633	1.37580	0.687902	−	0.725804i	$-0.258532\pi$
$613$	34.0633	1.37580	0.687902	+	0.725804i	$0.258532\pi$
$614$	39.3180	1.58675
$615$	9.39846	0.378982
$616$	−0.0638850	−0.00257400
$617$	29.6929	1.19539	0.597696	−	0.801723i	$-0.296083\pi$
$617$	29.6929	1.19539	0.597696	+	0.801723i	$0.296083\pi$
$618$	−2.65605	−0.106842
$619$	−9.46611	−0.380475	−0.190238	−	0.981738i	$-0.560926\pi$
$619$	−9.46611	−0.380475	−0.190238	+	0.981738i	$0.560926\pi$
$620$	−0.276439	−0.0111020
$621$	21.1329	0.848033
$622$	−15.4347	−0.618876
$623$	−0.428388	−0.0171630
$624$	−31.9847	−1.28041
$625$	4.20086	0.168034
$626$	38.4593	1.53714
$627$	9.71753	0.388081
$628$	3.49612	0.139510
$629$	10.1101	0.403118
$630$	−0.0266552	−0.00106197
$631$	−10.2705	−0.408862	−0.204431	−	0.978881i	$-0.565534\pi$
$631$	−10.2705	−0.408862	−0.204431	+	0.978881i	$0.565534\pi$
$632$	6.90157	0.274530
$633$	−48.2689	−1.91852
$634$	−40.5500	−1.61045
$635$	10.6579	0.422947
$636$	−8.45649	−0.335322
$637$	−24.4962	−0.970574
$638$	−4.62683	−0.183178
$639$	−0.335446	−0.0132700
$640$	−16.5164	−0.652868
$641$	3.75266	0.148221	0.0741106	−	0.997250i	$-0.476388\pi$
$641$	3.75266	0.148221	0.0741106	+	0.997250i	$0.476388\pi$
$642$	35.3608	1.39558
$643$	6.56660	0.258962	0.129481	−	0.991582i	$-0.458669\pi$
$643$	6.56660	0.258962	0.129481	+	0.991582i	$0.458669\pi$
$644$	−0.106025	−0.00417797
$645$	2.29706	0.0904467
$646$	8.75248	0.344362
$647$	−10.3694	−0.407665	−0.203832	−	0.979006i	$-0.565340\pi$
$647$	−10.3694	−0.407665	−0.203832	+	0.979006i	$0.565340\pi$
$648$	−20.6266	−0.810290
$649$	−10.5664	−0.414768
$650$	20.1160	0.789016
$651$	−0.0166598	−0.000652948 0
$652$	17.9098	0.701403
$653$	−9.90294	−0.387532	−0.193766	−	0.981048i	$-0.562070\pi$
$653$	−9.90294	−0.387532	−0.193766	+	0.981048i	$0.562070\pi$
$654$	−10.4399	−0.408233
$655$	−1.21075	−0.0473077
$656$	−20.2390	−0.790200
$657$	−3.16478	−0.123470
$658$	−0.680031	−0.0265104
$659$	−39.7314	−1.54772	−0.773859	−	0.633359i	$-0.781676\pi$
$659$	−39.7314	−1.54772	−0.773859	+	0.633359i	$0.781676\pi$
$660$	−1.76615	−0.0687473
$661$	18.5390	0.721085	0.360543	−	0.932743i	$-0.382591\pi$
$661$	18.5390	0.721085	0.360543	+	0.932743i	$0.382591\pi$
$662$	−42.8776	−1.66649
$663$	6.46602	0.251120
$664$	−1.04742	−0.0406477
$665$	−0.203948	−0.00790878
$666$	−6.95021	−0.269315
$667$	12.2953	0.476076
$668$	5.32152	0.205896
$669$	−22.5308	−0.871091
$670$	27.4793	1.06162
$671$	5.77116	0.222793
$672$	0.238168	0.00918751
$673$	6.87386	0.264968	0.132484	−	0.991185i	$-0.457705\pi$
$673$	6.87386	0.264968	0.132484	+	0.991185i	$0.457705\pi$
$674$	−2.66662	−0.102714
$675$	16.5075	0.635373
$676$	−0.576969	−0.0221911
$677$	−15.0860	−0.579802	−0.289901	−	0.957057i	$-0.593622\pi$
$677$	−15.0860	−0.579802	−0.289901	+	0.957057i	$0.593622\pi$
$678$	−12.0512	−0.462825
$679$	−0.300432	−0.0115295
$680$	2.54712	0.0976777
$681$	−24.1582	−0.925743
$682$	−0.481170	−0.0184250
$683$	−44.1030	−1.68756	−0.843778	−	0.536693i	$-0.819673\pi$
$683$	−44.1030	−1.68756	−0.843778	+	0.536693i	$0.819673\pi$
$684$	−1.67080	−0.0638845
$685$	−3.05374	−0.116678
$686$	0.726431	0.0277353
$687$	3.98002	0.151847
$688$	−4.94658	−0.188587
$689$	−20.8362	−0.793797
$690$	16.9018	0.643439
$691$	25.6350	0.975203	0.487602	−	0.873066i	$-0.337872\pi$
$691$	25.6350	0.975203	0.487602	+	0.873066i	$0.337872\pi$
$692$	3.95545	0.150364
$693$	−0.0128835	−0.000489404 0
$694$	60.2162	2.28577
$695$	−4.58519	−0.173926
$696$	−10.5236	−0.398894
$697$	4.09151	0.154977
$698$	−4.56804	−0.172903
$699$	9.97624	0.377336
$700$	−0.0828191	−0.00313027
$701$	−37.8442	−1.42935	−0.714677	−	0.699454i	$-0.753426\pi$
$701$	−37.8442	−1.42935	−0.714677	+	0.699454i	$0.753426\pi$
$702$	27.8332	1.05050
$703$	−53.1786	−2.00567
$704$	−3.01437	−0.113608
$705$	30.1026	1.13373
$706$	10.1501	0.382003
$707$	−0.535718	−0.0201477
$708$	15.0093	0.564083
$709$	48.5212	1.82225	0.911125	−	0.412129i	$-0.135215\pi$
$709$	48.5212	1.82225	0.911125	+	0.412129i	$0.135215\pi$
$710$	1.67989	0.0630452
$711$	1.39182	0.0521973
$712$	−28.1415	−1.05465
$713$	1.27866	0.0478862
$714$	−0.0958679	−0.00358777
$715$	−4.35168	−0.162744
$716$	4.80324	0.179506
$717$	−4.94622	−0.184720
$718$	24.5349	0.915633
$719$	−46.0088	−1.71584	−0.857920	−	0.513783i	$-0.828244\pi$
$719$	−46.0088	−1.71584	−0.857920	+	0.513783i	$0.828244\pi$
$720$	−2.54091	−0.0946943
$721$	0.0269434	0.00100343
$722$	−14.4215	−0.536711
$723$	49.4241	1.83810
$724$	2.77026	0.102956
$725$	9.60422	0.356692
$726$	−3.07417	−0.114093
$727$	27.4828	1.01928	0.509640	−	0.860388i	$-0.329778\pi$
$727$	27.4828	1.01928	0.509640	+	0.860388i	$0.329778\pi$
$728$	0.223594	0.00828694
$729$	22.3283	0.826972
$730$	15.8490	0.586599
$731$	1.00000	0.0369863
$732$	−8.19775	−0.302998
$733$	−11.3404	−0.418869	−0.209434	−	0.977823i	$-0.567162\pi$
$733$	−11.3404	−0.418869	−0.209434	+	0.977823i	$0.567162\pi$
$734$	47.3458	1.74757
$735$	−16.0772	−0.593016
$736$	−18.2797	−0.673798
$737$	13.2819	0.489243
$738$	−2.81271	−0.103537
$739$	7.67521	0.282337	0.141169	−	0.989986i	$-0.454914\pi$
$739$	7.67521	0.282337	0.141169	+	0.989986i	$0.454914\pi$
$740$	9.66516	0.355298
$741$	−34.0108	−1.24942
$742$	0.308926	0.0113410
$743$	−37.7938	−1.38652	−0.693260	−	0.720687i	$-0.743826\pi$
$743$	−37.7938	−1.38652	−0.693260	+	0.720687i	$0.743826\pi$
$744$	−1.09440	−0.0401228
$745$	−11.2618	−0.412599
$746$	7.83239	0.286764
$747$	−0.211230	−0.00772849
$748$	−0.768874	−0.0281128
$749$	−0.358707	−0.0131069
$750$	32.3139	1.17994
$751$	34.1652	1.24670	0.623352	−	0.781941i	$-0.285770\pi$
$751$	34.1652	1.24670	0.623352	+	0.781941i	$0.285770\pi$
$752$	−64.8242	−2.36389
$753$	−14.7625	−0.537974
$754$	16.1936	0.589737
$755$	2.65363	0.0965755
$756$	−0.114591	−0.00416764
$757$	37.4914	1.36265	0.681324	−	0.731982i	$-0.261404\pi$
$757$	37.4914	1.36265	0.681324	+	0.731982i	$0.261404\pi$
$758$	−26.5826	−0.965524
$759$	8.16928	0.296526
$760$	−13.3977	−0.485985
$761$	−12.5989	−0.456710	−0.228355	−	0.973578i	$-0.573335\pi$
$761$	−12.5989	−0.456710	−0.228355	+	0.973578i	$0.573335\pi$
$762$	−26.3515	−0.954612
$763$	0.105905	0.00383400
$764$	−8.66946	−0.313650
$765$	0.513671	0.0185718
$766$	−42.9160	−1.55062
$767$	36.9819	1.33534
$768$	29.6985	1.07165
$769$	−7.05586	−0.254441	−0.127220	−	0.991874i	$-0.540606\pi$
$769$	−7.05586	−0.254441	−0.127220	+	0.991874i	$0.540606\pi$
$770$	0.0645198	0.00232513
$771$	1.01757	0.0366468
$772$	−19.0604	−0.686000
$773$	17.8294	0.641279	0.320640	−	0.947201i	$-0.396102\pi$
$773$	17.8294	0.641279	0.320640	+	0.947201i	$0.396102\pi$
$774$	−0.687449	−0.0247098
$775$	0.998797	0.0358779
$776$	−19.7358	−0.708474
$777$	0.582478	0.0208963
$778$	−40.6193	−1.45627
$779$	−21.5210	−0.771071
$780$	6.18143	0.221331
$781$	0.811957	0.0290541
$782$	7.35799	0.263121
$783$	13.2887	0.474900
$784$	34.6213	1.23647
$785$	5.65363	0.201787
$786$	2.99354	0.106776
$787$	43.9202	1.56559	0.782793	−	0.622282i	$-0.213794\pi$
$787$	43.9202	1.56559	0.782793	+	0.622282i	$0.213794\pi$
$788$	−11.1634	−0.397679
$789$	2.17787	0.0775341
$790$	−6.97014	−0.247986
$791$	0.122250	0.00434671
$792$	−0.846336	−0.0300732
$793$	−20.1987	−0.717277
$794$	17.5729	0.623640
$795$	−13.6751	−0.485006
$796$	−9.48221	−0.336088
$797$	16.7168	0.592139	0.296069	−	0.955166i	$-0.404324\pi$
$797$	16.7168	0.592139	0.296069	+	0.955166i	$0.404324\pi$
$798$	0.504258	0.0178505
$799$	13.1048	0.463616
$800$	−14.2788	−0.504831
$801$	−5.67520	−0.200523
$802$	−23.1402	−0.817109
$803$	7.66046	0.270332
$804$	−18.8665	−0.665369
$805$	−0.171454	−0.00604297
$806$	1.68407	0.0593188
$807$	36.0322	1.26839
$808$	−35.1921	−1.23805
$809$	24.4091	0.858177	0.429088	−	0.903262i	$-0.358835\pi$
$809$	24.4091	0.858177	0.429088	+	0.903262i	$0.358835\pi$
$810$	20.8316	0.731946
$811$	8.29677	0.291339	0.145669	−	0.989333i	$-0.453466\pi$
$811$	8.29677	0.291339	0.145669	+	0.989333i	$0.453466\pi$
$812$	−0.0666703	−0.00233967
$813$	56.5725	1.98408
$814$	16.8232	0.589654
$815$	28.9622	1.01450
$816$	−9.13864	−0.319916
$817$	−5.25992	−0.184021
$818$	−44.0208	−1.53915
$819$	0.0450915	0.00157562
$820$	3.91143	0.136593
$821$	44.5956	1.55640	0.778199	−	0.628018i	$-0.216134\pi$
$821$	44.5956	1.55640	0.778199	+	0.628018i	$0.216134\pi$
$822$	7.55031	0.263347
$823$	29.0932	1.01413	0.507063	−	0.861909i	$-0.330731\pi$
$823$	29.0932	1.01413	0.507063	+	0.861909i	$0.330731\pi$
$824$	1.76995	0.0616592
$825$	6.38126	0.222167
$826$	−0.548308	−0.0190781
$827$	−10.3349	−0.359381	−0.179690	−	0.983723i	$-0.557510\pi$
$827$	−10.3349	−0.359381	−0.179690	+	0.983723i	$0.557510\pi$
$828$	−1.40460	−0.0488131
$829$	−6.82387	−0.237003	−0.118501	−	0.992954i	$-0.537809\pi$
$829$	−6.82387	−0.237003	−0.118501	+	0.992954i	$0.537809\pi$
$830$	1.05782	0.0367176
$831$	−1.19150	−0.0413326
$832$	10.5501	0.365759
$833$	−6.99903	−0.242502
$834$	11.3368	0.392560
$835$	8.60551	0.297806
$836$	4.04422	0.139872
$837$	1.38197	0.0477678
$838$	12.7844	0.441628
$839$	41.8740	1.44565	0.722825	−	0.691031i	$-0.242843\pi$
$839$	41.8740	1.44565	0.722825	+	0.691031i	$0.242843\pi$
$840$	0.146748	0.00506328
$841$	−21.2685	−0.733397
$842$	−38.5772	−1.32946
$843$	−51.1429	−1.76145
$844$	−20.0884	−0.691472
$845$	−0.933024	−0.0320970
$846$	−9.00891	−0.309733
$847$	0.0311850	0.00107153
$848$	29.4485	1.01127
$849$	−39.9917	−1.37251
$850$	5.74754	0.197139
$851$	−44.7059	−1.53250
$852$	−1.15336	−0.0395135
$853$	14.3131	0.490071	0.245035	−	0.969514i	$-0.421200\pi$
$853$	14.3131	0.490071	0.245035	+	0.969514i	$0.421200\pi$
$854$	0.299474	0.0102478
$855$	−2.70187	−0.0924019
$856$	−23.5640	−0.805400
$857$	−36.3183	−1.24061	−0.620305	−	0.784360i	$-0.712991\pi$
$857$	−36.3183	−1.24061	−0.620305	+	0.784360i	$0.712991\pi$
$858$	10.7594	0.367321
$859$	−6.63213	−0.226286	−0.113143	−	0.993579i	$-0.536092\pi$
$859$	−6.63213	−0.226286	−0.113143	+	0.993579i	$0.536092\pi$
$860$	0.955986	0.0325988
$861$	0.235725	0.00803349
$862$	36.7165	1.25057
$863$	4.38558	0.149287	0.0746434	−	0.997210i	$-0.476218\pi$
$863$	4.38558	0.149287	0.0746434	+	0.997210i	$0.476218\pi$
$864$	−19.7566	−0.672132
$865$	6.39642	0.217485
$866$	4.55147	0.154665
$867$	1.84747	0.0627432
$868$	−0.00693342	−0.000235336 0
$869$	−3.36894	−0.114284
$870$	10.6281	0.360327
$871$	−46.4857	−1.57511
$872$	6.95702	0.235594
$873$	−3.98006	−0.134705
$874$	−38.7024	−1.30913
$875$	−0.327798	−0.0110816
$876$	−10.8815	−0.367650
$877$	38.8413	1.31158	0.655789	−	0.754944i	$-0.272336\pi$
$877$	38.8413	1.31158	0.655789	+	0.754944i	$0.272336\pi$
$878$	43.1646	1.45673
$879$	57.2106	1.92967
$880$	6.15037	0.207329
$881$	32.5138	1.09542	0.547709	−	0.836669i	$-0.315500\pi$
$881$	32.5138	1.09542	0.547709	+	0.836669i	$0.315500\pi$
$882$	4.81147	0.162011
$883$	−32.7859	−1.10333	−0.551666	−	0.834065i	$-0.686008\pi$
$883$	−32.7859	−1.10333	−0.551666	+	0.834065i	$0.686008\pi$
$884$	2.69102	0.0905086
$885$	24.2717	0.815885
$886$	53.5351	1.79855
$887$	−22.5285	−0.756432	−0.378216	−	0.925717i	$-0.623462\pi$
$887$	−22.5285	−0.756432	−0.378216	+	0.925717i	$0.623462\pi$
$888$	38.2638	1.28405
$889$	0.267314	0.00896542
$890$	28.4211	0.952676
$891$	10.0687	0.337315
$892$	−9.37682	−0.313959
$893$	−68.9305	−2.30667
$894$	27.8444	0.931257
$895$	7.76739	0.259635
$896$	−0.414252	−0.0138392
$897$	−28.5920	−0.954660
$898$	−51.4737	−1.71770
$899$	0.804043	0.0268163
$900$	−1.09717	−0.0365724
$901$	−5.95331	−0.198333
$902$	6.80825	0.226690
$903$	0.0576132	0.00191725
$904$	8.03077	0.267099
$905$	4.47982	0.148914
$906$	−6.56104	−0.217976
$907$	−57.6261	−1.91344	−0.956721	−	0.291007i	$-0.906010\pi$
$907$	−57.6261	−1.91344	−0.956721	+	0.291007i	$0.906010\pi$
$908$	−10.0541	−0.333657
$909$	−7.09708	−0.235395
$910$	−0.225815	−0.00748571
$911$	−12.9954	−0.430557	−0.215278	−	0.976553i	$-0.569066\pi$
$911$	−12.9954	−0.430557	−0.215278	+	0.976553i	$0.569066\pi$
$912$	48.0685	1.59171
$913$	0.511288	0.0169212
$914$	43.8142	1.44925
$915$	−13.2567	−0.438253
$916$	1.65639	0.0547288
$917$	−0.0303670	−0.00100281
$918$	7.95248	0.262471
$919$	45.4946	1.50073	0.750365	−	0.661024i	$-0.229878\pi$
$919$	45.4946	1.50073	0.750365	+	0.661024i	$0.229878\pi$
$920$	−11.2631	−0.371333
$921$	−43.6532	−1.43842
$922$	30.0577	0.989896
$923$	−2.84180	−0.0935391
$924$	−0.0442973	−0.00145727
$925$	−34.9211	−1.14820
$926$	−10.0549	−0.330425
$927$	0.356941	0.0117235
$928$	−11.4946	−0.377328
$929$	31.0361	1.01826	0.509131	−	0.860689i	$-0.329967\pi$
$929$	31.0361	1.01826	0.509131	+	0.860689i	$0.329967\pi$
$930$	1.10528	0.0362435
$931$	36.8143	1.20654
$932$	4.15189	0.136000
$933$	17.1366	0.561026
$934$	−1.48720	−0.0486626
$935$	−1.24336	−0.0406621
$936$	2.96213	0.0968201
$937$	29.9531	0.978525	0.489262	−	0.872137i	$-0.337266\pi$
$937$	29.9531	0.978525	0.489262	+	0.872137i	$0.337266\pi$
$938$	0.689216	0.0225037
$939$	−42.6998	−1.39346
$940$	12.5280	0.408620
$941$	−16.7839	−0.547139	−0.273570	−	0.961852i	$-0.588204\pi$
$941$	−16.7839	−0.547139	−0.273570	+	0.961852i	$0.588204\pi$
$942$	−13.9785	−0.455443
$943$	−18.0922	−0.589163
$944$	−52.2677	−1.70117
$945$	−0.185307	−0.00602804
$946$	1.66399	0.0541011
$947$	−16.7225	−0.543409	−0.271704	−	0.962381i	$-0.587587\pi$
$947$	−16.7225	−0.543409	−0.271704	+	0.962381i	$0.587587\pi$
$948$	4.78548	0.155425
$949$	−26.8112	−0.870327
$950$	−30.2316	−0.980842
$951$	45.0210	1.45991
$952$	0.0638850	0.00207053
$953$	−23.2361	−0.752692	−0.376346	−	0.926479i	$-0.622820\pi$
$953$	−23.2361	−0.752692	−0.376346	+	0.926479i	$0.622820\pi$
$954$	4.09259	0.132503
$955$	−14.0195	−0.453660
$956$	−2.05851	−0.0665768
$957$	5.13699	0.166055
$958$	−66.6966	−2.15487
$959$	−0.0765917	−0.00247328
$960$	6.92418	0.223477
$961$	−30.9164	−0.997303
$962$	−58.8803	−1.89838
$963$	−4.75207	−0.153133
$964$	20.5692	0.662490
$965$	−30.8229	−0.992224
$966$	0.423917	0.0136393
$967$	−43.5139	−1.39931	−0.699656	−	0.714479i	$-0.746664\pi$
$967$	−43.5139	−1.39931	−0.699656	+	0.714479i	$0.746664\pi$
$968$	2.04859	0.0658440
$969$	−9.71753	−0.312172
$970$	19.9319	0.639974
$971$	−13.3176	−0.427383	−0.213691	−	0.976901i	$-0.568549\pi$
$971$	−13.3176	−0.427383	−0.213691	+	0.976901i	$0.568549\pi$
$972$	−3.27861	−0.105161
$973$	−0.115002	−0.00368680
$974$	7.46979	0.239347
$975$	−22.3341	−0.715262
$976$	28.5475	0.913783
$977$	41.2670	1.32025	0.660125	−	0.751156i	$-0.270503\pi$
$977$	41.2670	1.32025	0.660125	+	0.751156i	$0.270503\pi$
$978$	−71.6084	−2.28978
$979$	13.7370	0.439037
$980$	−6.69097	−0.213735
$981$	1.40300	0.0447944
$982$	56.2873	1.79620
$983$	−28.9531	−0.923459	−0.461730	−	0.887021i	$-0.652771\pi$
$983$	−28.9531	−0.923459	−0.461730	+	0.887021i	$0.652771\pi$
$984$	15.4851	0.493648
$985$	−18.0524	−0.575199
$986$	4.62683	0.147348
$987$	0.755012	0.0240323
$988$	−14.1545	−0.450316
$989$	−4.42189	−0.140608
$990$	0.854745	0.0271656
$991$	15.0446	0.477909	0.238954	−	0.971031i	$-0.423195\pi$
$991$	15.0446	0.477909	0.238954	+	0.971031i	$0.423195\pi$
$992$	−1.19539	−0.0379535
$993$	47.6053	1.51071
$994$	0.0421337	0.00133640
$995$	−15.3338	−0.486115
$996$	−0.726270	−0.0230127
$997$	−9.65579	−0.305802	−0.152901	−	0.988242i	$-0.548862\pi$
$997$	−9.65579	−0.305802	−0.152901	+	0.988242i	$0.548862\pi$
$998$	−31.2587	−0.989478
$999$	−48.3179	−1.52871

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.g.1.20	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.g.1.20	✓	69	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

\(f(q)\)	\(=\)	\(q-1.66399 q^{2} +1.84747 q^{3} +0.768874 q^{4} +1.24336 q^{5} -3.07417 q^{6} +0.0311850 q^{7} +2.04859 q^{8} +0.413132 q^{9} +O(q^{10})\)	\(q-1.66399 q^{2} +1.84747 q^{3} +0.768874 q^{4} +1.24336 q^{5} -3.07417 q^{6} +0.0311850 q^{7} +2.04859 q^{8} +0.413132 q^{9} -2.06894 q^{10} -1.00000 q^{11} +1.42047 q^{12} +3.49994 q^{13} -0.0518916 q^{14} +2.29706 q^{15} -4.94658 q^{16} +1.00000 q^{17} -0.687449 q^{18} -5.25992 q^{19} +0.955986 q^{20} +0.0576132 q^{21} +1.66399 q^{22} -4.42189 q^{23} +3.78469 q^{24} -3.45406 q^{25} -5.82388 q^{26} -4.77915 q^{27} +0.0239773 q^{28} -2.78056 q^{29} -3.82229 q^{30} -0.289166 q^{31} +4.13391 q^{32} -1.84747 q^{33} -1.66399 q^{34} +0.0387741 q^{35} +0.317647 q^{36} +10.1101 q^{37} +8.75248 q^{38} +6.46602 q^{39} +2.54712 q^{40} +4.09151 q^{41} -0.0958679 q^{42} +1.00000 q^{43} -0.768874 q^{44} +0.513671 q^{45} +7.35799 q^{46} +13.1048 q^{47} -9.13864 q^{48} -6.99903 q^{49} +5.74754 q^{50} +1.84747 q^{51} +2.69102 q^{52} -5.95331 q^{53} +7.95248 q^{54} -1.24336 q^{55} +0.0638850 q^{56} -9.71753 q^{57} +4.62683 q^{58} +10.5664 q^{59} +1.76615 q^{60} -5.77116 q^{61} +0.481170 q^{62} +0.0128835 q^{63} +3.01437 q^{64} +4.35168 q^{65} +3.07417 q^{66} -13.2819 q^{67} +0.768874 q^{68} -8.16928 q^{69} -0.0645198 q^{70} -0.811957 q^{71} +0.846336 q^{72} -7.66046 q^{73} -16.8232 q^{74} -6.38126 q^{75} -4.04422 q^{76} -0.0311850 q^{77} -10.7594 q^{78} +3.36894 q^{79} -6.15037 q^{80} -10.0687 q^{81} -6.80825 q^{82} -0.511288 q^{83} +0.0442973 q^{84} +1.24336 q^{85} -1.66399 q^{86} -5.13699 q^{87} -2.04859 q^{88} -13.7370 q^{89} -0.854745 q^{90} +0.109146 q^{91} -3.39987 q^{92} -0.534225 q^{93} -21.8064 q^{94} -6.53996 q^{95} +7.63726 q^{96} -9.63387 q^{97} +11.6463 q^{98} -0.413132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9}+O(q^{10}) \) 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9	\( 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9} - q^{10} - 69 q^{11} - 3 q^{12} - 28 q^{13} - 15 q^{14} - 45 q^{15} + 53 q^{16} + 69 q^{17} - 17 q^{18} - 32 q^{19} - 21 q^{20} - 38 q^{21} + 11 q^{22} - 41 q^{23} - 11 q^{24} + 67 q^{25} - 6 q^{26} - 3 q^{27} - 21 q^{28} - 22 q^{29} - 22 q^{30} - 27 q^{31} - 87 q^{32} + 3 q^{33} - 11 q^{34} - 44 q^{35} + 59 q^{36} + 24 q^{37} - 22 q^{38} - 59 q^{39} + q^{40} - 43 q^{41} - 15 q^{42} + 69 q^{43} - 65 q^{44} - 12 q^{45} - 21 q^{46} - 99 q^{47} + 2 q^{48} + 64 q^{49} - 78 q^{50} - 3 q^{51} - 57 q^{52} - 50 q^{53} + 20 q^{54} + 6 q^{55} - 59 q^{56} - 15 q^{57} + 22 q^{58} - 82 q^{59} - 86 q^{60} - 24 q^{61} + 15 q^{62} - 63 q^{63} + 63 q^{64} - 23 q^{65} + 10 q^{66} - 54 q^{67} + 65 q^{68} + 36 q^{69} + 9 q^{70} - 128 q^{71} - 69 q^{72} + 2 q^{73} - 58 q^{74} - 31 q^{75} - 76 q^{76} + 11 q^{77} - 19 q^{78} - 43 q^{79} - 19 q^{80} + 49 q^{81} - 2 q^{82} - 62 q^{83} - 82 q^{84} - 6 q^{85} - 11 q^{86} - 62 q^{87} + 33 q^{88} - 49 q^{89} - 37 q^{90} - 2 q^{91} - 96 q^{92} - 29 q^{93} - 75 q^{94} - 133 q^{95} - 86 q^{96} + 5 q^{97} - 72 q^{98} - 56 q^{99}+O(q^{100}) \) 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9 - q^10 - 69 * q^11 - 3 * q^12 - 28 * q^13 - 15 * q^14 - 45 * q^15 + 53 * q^16 + 69 * q^17 - 17 * q^18 - 32 * q^19 - 21 * q^20 - 38 * q^21 + 11 * q^22 - 41 * q^23 - 11 * q^24 + 67 * q^25 - 6 * q^26 - 3 * q^27 - 21 * q^28 - 22 * q^29 - 22 * q^30 - 27 * q^31 - 87 * q^32 + 3 * q^33 - 11 * q^34 - 44 * q^35 + 59 * q^36 + 24 * q^37 - 22 * q^38 - 59 * q^39 + q^40 - 43 * q^41 - 15 * q^42 + 69 * q^43 - 65 * q^44 - 12 * q^45 - 21 * q^46 - 99 * q^47 + 2 * q^48 + 64 * q^49 - 78 * q^50 - 3 * q^51 - 57 * q^52 - 50 * q^53 + 20 * q^54 + 6 * q^55 - 59 * q^56 - 15 * q^57 + 22 * q^58 - 82 * q^59 - 86 * q^60 - 24 * q^61 + 15 * q^62 - 63 * q^63 + 63 * q^64 - 23 * q^65 + 10 * q^66 - 54 * q^67 + 65 * q^68 + 36 * q^69 + 9 * q^70 - 128 * q^71 - 69 * q^72 + 2 * q^73 - 58 * q^74 - 31 * q^75 - 76 * q^76 + 11 * q^77 - 19 * q^78 - 43 * q^79 - 19 * q^80 + 49 * q^81 - 2 * q^82 - 62 * q^83 - 82 * q^84 - 6 * q^85 - 11 * q^86 - 62 * q^87 + 33 * q^88 - 49 * q^89 - 37 * q^90 - 2 * q^91 - 96 * q^92 - 29 * q^93 - 75 * q^94 - 133 * q^95 - 86 * q^96 + 5 * q^97 - 72 * q^98 - 56 * q^99

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