LMFDB - Embedded newform 402.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [402,2,Mod(1,402)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(402, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("402.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 402 = 2 \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	402.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$3.20998616126$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	402.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.710831 q^{5} +1.00000 q^{6} -1.81361 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.710831 q^{5} +1.00000 q^{6} -1.81361 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.710831 q^{10} +0.578337 q^{11} +1.00000 q^{12} +4.72999 q^{13} -1.81361 q^{14} +0.710831 q^{15} +1.00000 q^{16} +4.20555 q^{17} +1.00000 q^{18} -2.00000 q^{19} +0.710831 q^{20} -1.81361 q^{21} +0.578337 q^{22} -5.44082 q^{23} +1.00000 q^{24} -4.49472 q^{25} +4.72999 q^{26} +1.00000 q^{27} -1.81361 q^{28} -4.72999 q^{29} +0.710831 q^{30} -2.18639 q^{31} +1.00000 q^{32} +0.578337 q^{33} +4.20555 q^{34} -1.28917 q^{35} +1.00000 q^{36} -8.07306 q^{37} -2.00000 q^{38} +4.72999 q^{39} +0.710831 q^{40} +9.96526 q^{41} -1.81361 q^{42} -3.75971 q^{43} +0.578337 q^{44} +0.710831 q^{45} -5.44082 q^{46} -2.05390 q^{47} +1.00000 q^{48} -3.71083 q^{49} -4.49472 q^{50} +4.20555 q^{51} +4.72999 q^{52} +4.71083 q^{53} +1.00000 q^{54} +0.411100 q^{55} -1.81361 q^{56} -2.00000 q^{57} -4.72999 q^{58} +0.132494 q^{59} +0.710831 q^{60} -6.15165 q^{61} -2.18639 q^{62} -1.81361 q^{63} +1.00000 q^{64} +3.36222 q^{65} +0.578337 q^{66} +1.00000 q^{67} +4.20555 q^{68} -5.44082 q^{69} -1.28917 q^{70} +8.15165 q^{71} +1.00000 q^{72} -5.49472 q^{73} -8.07306 q^{74} -4.49472 q^{75} -2.00000 q^{76} -1.04888 q^{77} +4.72999 q^{78} +2.35720 q^{79} +0.710831 q^{80} +1.00000 q^{81} +9.96526 q^{82} -13.5925 q^{83} -1.81361 q^{84} +2.98944 q^{85} -3.75971 q^{86} -4.72999 q^{87} +0.578337 q^{88} -3.83276 q^{89} +0.710831 q^{90} -8.57834 q^{91} -5.44082 q^{92} -2.18639 q^{93} -2.05390 q^{94} -1.42166 q^{95} +1.00000 q^{96} +6.78389 q^{97} -3.71083 q^{98} +0.578337 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + q^7 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{10} + 3 q^{12} - 6 q^{13} + q^{14} + 3 q^{15} + 3 q^{16} - 2 q^{17} + 3 q^{18} - 6 q^{19} + 3 q^{20} + q^{21} + 3 q^{23} + 3 q^{24} + 2 q^{25} - 6 q^{26} + 3 q^{27} + q^{28} + 6 q^{29} + 3 q^{30} - 13 q^{31} + 3 q^{32} - 2 q^{34} - 3 q^{35} + 3 q^{36} - 7 q^{37} - 6 q^{38} - 6 q^{39} + 3 q^{40} + 5 q^{41} + q^{42} - q^{43} + 3 q^{45} + 3 q^{46} - 10 q^{47} + 3 q^{48} - 12 q^{49} + 2 q^{50} - 2 q^{51} - 6 q^{52} + 15 q^{53} + 3 q^{54} - 28 q^{55} + q^{56} - 6 q^{57} + 6 q^{58} + 3 q^{59} + 3 q^{60} - 13 q^{62} + q^{63} + 3 q^{64} - 8 q^{65} + 3 q^{67} - 2 q^{68} + 3 q^{69} - 3 q^{70} + 6 q^{71} + 3 q^{72} - q^{73} - 7 q^{74} + 2 q^{75} - 6 q^{76} + 8 q^{77} - 6 q^{78} - 26 q^{79} + 3 q^{80} + 3 q^{81} + 5 q^{82} - 3 q^{83} + q^{84} - 22 q^{85} - q^{86} + 6 q^{87} + 16 q^{89} + 3 q^{90} - 24 q^{91} + 3 q^{92} - 13 q^{93} - 10 q^{94} - 6 q^{95} + 3 q^{96} + 4 q^{97} - 12 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + q^7 + 3 * q^8 + 3 * q^9 + 3 * q^10 + 3 * q^12 - 6 * q^13 + q^14 + 3 * q^15 + 3 * q^16 - 2 * q^17 + 3 * q^18 - 6 * q^19 + 3 * q^20 + q^21 + 3 * q^23 + 3 * q^24 + 2 * q^25 - 6 * q^26 + 3 * q^27 + q^28 + 6 * q^29 + 3 * q^30 - 13 * q^31 + 3 * q^32 - 2 * q^34 - 3 * q^35 + 3 * q^36 - 7 * q^37 - 6 * q^38 - 6 * q^39 + 3 * q^40 + 5 * q^41 + q^42 - q^43 + 3 * q^45 + 3 * q^46 - 10 * q^47 + 3 * q^48 - 12 * q^49 + 2 * q^50 - 2 * q^51 - 6 * q^52 + 15 * q^53 + 3 * q^54 - 28 * q^55 + q^56 - 6 * q^57 + 6 * q^58 + 3 * q^59 + 3 * q^60 - 13 * q^62 + q^63 + 3 * q^64 - 8 * q^65 + 3 * q^67 - 2 * q^68 + 3 * q^69 - 3 * q^70 + 6 * q^71 + 3 * q^72 - q^73 - 7 * q^74 + 2 * q^75 - 6 * q^76 + 8 * q^77 - 6 * q^78 - 26 * q^79 + 3 * q^80 + 3 * q^81 + 5 * q^82 - 3 * q^83 + q^84 - 22 * q^85 - q^86 + 6 * q^87 + 16 * q^89 + 3 * q^90 - 24 * q^91 + 3 * q^92 - 13 * q^93 - 10 * q^94 - 6 * q^95 + 3 * q^96 + 4 * q^97 - 12 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0.710831	0.317893	0.158947	−	0.987287i	$-0.449190\pi$
$5$	0.710831	0.317893	0.158947	+	0.987287i	$0.449190\pi$
$6$	1.00000	0.408248
$7$	−1.81361	−0.685479	−0.342739	−	0.939431i	$-0.611355\pi$
$7$	−1.81361	−0.685479	−0.342739	+	0.939431i	$0.611355\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0.710831	0.224785
$11$	0.578337	0.174375	0.0871876	−	0.996192i	$-0.472212\pi$
$11$	0.578337	0.174375	0.0871876	+	0.996192i	$0.472212\pi$
$12$	1.00000	0.288675
$13$	4.72999	1.31186	0.655931	−	0.754821i	$-0.272276\pi$
$13$	4.72999	1.31186	0.655931	+	0.754821i	$0.272276\pi$
$14$	−1.81361	−0.484707
$15$	0.710831	0.183536
$16$	1.00000	0.250000
$17$	4.20555	1.02000	0.509998	−	0.860176i	$-0.329646\pi$
$17$	4.20555	1.02000	0.509998	+	0.860176i	$0.329646\pi$
$18$	1.00000	0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0.710831	0.158947
$21$	−1.81361	−0.395761
$22$	0.578337	0.123302
$23$	−5.44082	−1.13449	−0.567245	−	0.823549i	$-0.691991\pi$
$23$	−5.44082	−1.13449	−0.567245	+	0.823549i	$0.691991\pi$
$24$	1.00000	0.204124
$25$	−4.49472	−0.898944
$26$	4.72999	0.927627
$27$	1.00000	0.192450
$28$	−1.81361	−0.342739
$29$	−4.72999	−0.878337	−0.439168	−	0.898405i	$-0.644727\pi$
$29$	−4.72999	−0.878337	−0.439168	+	0.898405i	$0.644727\pi$
$30$	0.710831	0.129779
$31$	−2.18639	−0.392688	−0.196344	−	0.980535i	$-0.562907\pi$
$31$	−2.18639	−0.392688	−0.196344	+	0.980535i	$0.562907\pi$
$32$	1.00000	0.176777
$33$	0.578337	0.100676
$34$	4.20555	0.721246
$35$	−1.28917	−0.217909
$36$	1.00000	0.166667
$37$	−8.07306	−1.32720	−0.663601	−	0.748087i	$-0.730973\pi$
$37$	−8.07306	−1.32720	−0.663601	+	0.748087i	$0.730973\pi$
$38$	−2.00000	−0.324443
$39$	4.72999	0.757404
$40$	0.710831	0.112392
$41$	9.96526	1.55631	0.778156	−	0.628071i	$-0.216155\pi$
$41$	9.96526	1.55631	0.778156	+	0.628071i	$0.216155\pi$
$42$	−1.81361	−0.279846
$43$	−3.75971	−0.573350	−0.286675	−	0.958028i	$-0.592550\pi$
$43$	−3.75971	−0.573350	−0.286675	+	0.958028i	$0.592550\pi$
$44$	0.578337	0.0871876
$45$	0.710831	0.105964
$46$	−5.44082	−0.802205
$47$	−2.05390	−0.299592	−0.149796	−	0.988717i	$-0.547862\pi$
$47$	−2.05390	−0.299592	−0.149796	+	0.988717i	$0.547862\pi$
$48$	1.00000	0.144338
$49$	−3.71083	−0.530119
$50$	−4.49472	−0.635649
$51$	4.20555	0.588895
$52$	4.72999	0.655931
$53$	4.71083	0.647082	0.323541	−	0.946214i	$-0.395127\pi$
$53$	4.71083	0.647082	0.323541	+	0.946214i	$0.395127\pi$
$54$	1.00000	0.136083
$55$	0.411100	0.0554327
$56$	−1.81361	−0.242353
$57$	−2.00000	−0.264906
$58$	−4.72999	−0.621078
$59$	0.132494	0.0172493	0.00862465	−	0.999963i	$-0.497255\pi$
$59$	0.132494	0.0172493	0.00862465	+	0.999963i	$0.497255\pi$
$60$	0.710831	0.0917679
$61$	−6.15165	−0.787638	−0.393819	−	0.919188i	$-0.628846\pi$
$61$	−6.15165	−0.787638	−0.393819	+	0.919188i	$0.628846\pi$
$62$	−2.18639	−0.277672
$63$	−1.81361	−0.228493
$64$	1.00000	0.125000
$65$	3.36222	0.417033
$66$	0.578337	0.0711884
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	4.20555	0.509998
$69$	−5.44082	−0.654998
$70$	−1.28917	−0.154085
$71$	8.15165	0.967423	0.483711	−	0.875228i	$-0.339288\pi$
$71$	8.15165	0.967423	0.483711	+	0.875228i	$0.339288\pi$
$72$	1.00000	0.117851
$73$	−5.49472	−0.643108	−0.321554	−	0.946891i	$-0.604205\pi$
$73$	−5.49472	−0.643108	−0.321554	+	0.946891i	$0.604205\pi$
$74$	−8.07306	−0.938474
$75$	−4.49472	−0.519005
$76$	−2.00000	−0.229416
$77$	−1.04888	−0.119531
$78$	4.72999	0.535566
$79$	2.35720	0.265206	0.132603	−	0.991169i	$-0.457666\pi$
$79$	2.35720	0.265206	0.132603	+	0.991169i	$0.457666\pi$
$80$	0.710831	0.0794734
$81$	1.00000	0.111111
$82$	9.96526	1.10048
$83$	−13.5925	−1.49197	−0.745984	−	0.665964i	$-0.768020\pi$
$83$	−13.5925	−1.49197	−0.745984	+	0.665964i	$0.768020\pi$
$84$	−1.81361	−0.197881
$85$	2.98944	0.324250
$86$	−3.75971	−0.405420
$87$	−4.72999	−0.507108
$88$	0.578337	0.0616509
$89$	−3.83276	−0.406272	−0.203136	−	0.979151i	$-0.565113\pi$
$89$	−3.83276	−0.406272	−0.203136	+	0.979151i	$0.565113\pi$
$90$	0.710831	0.0749282
$91$	−8.57834	−0.899254
$92$	−5.44082	−0.567245
$93$	−2.18639	−0.226718
$94$	−2.05390	−0.211844
$95$	−1.42166	−0.145860
$96$	1.00000	0.102062
$97$	6.78389	0.688799	0.344400	−	0.938823i	$-0.388082\pi$
$97$	6.78389	0.688799	0.344400	+	0.938823i	$0.388082\pi$
$98$	−3.71083	−0.374851
$99$	0.578337	0.0581251
$100$	−4.49472	−0.449472
$101$	9.25443	0.920850	0.460425	−	0.887699i	$-0.347697\pi$
$101$	9.25443	0.920850	0.460425	+	0.887699i	$0.347697\pi$
$102$	4.20555	0.416412
$103$	−9.45998	−0.932119	−0.466060	−	0.884753i	$-0.654327\pi$
$103$	−9.45998	−0.932119	−0.466060	+	0.884753i	$0.654327\pi$
$104$	4.72999	0.463813
$105$	−1.28917	−0.125810
$106$	4.71083	0.457556
$107$	6.20555	0.599913	0.299957	−	0.953953i	$-0.403028\pi$
$107$	6.20555	0.599913	0.299957	+	0.953953i	$0.403028\pi$
$108$	1.00000	0.0962250
$109$	2.15165	0.206091	0.103045	−	0.994677i	$-0.467141\pi$
$109$	2.15165	0.206091	0.103045	+	0.994677i	$0.467141\pi$
$110$	0.411100	0.0391969
$111$	−8.07306	−0.766261
$112$	−1.81361	−0.171370
$113$	18.8816	1.77624	0.888118	−	0.459616i	$-0.152013\pi$
$113$	18.8816	1.77624	0.888118	+	0.459616i	$0.152013\pi$
$114$	−2.00000	−0.187317
$115$	−3.86751	−0.360647
$116$	−4.72999	−0.439168
$117$	4.72999	0.437288
$118$	0.132494	0.0121971
$119$	−7.62721	−0.699185
$120$	0.710831	0.0648897
$121$	−10.6655	−0.969593
$122$	−6.15165	−0.556944
$123$	9.96526	0.898537
$124$	−2.18639	−0.196344
$125$	−6.74914	−0.603662
$126$	−1.81361	−0.161569
$127$	5.04888	0.448015	0.224008	−	0.974587i	$-0.428086\pi$
$127$	5.04888	0.448015	0.224008	+	0.974587i	$0.428086\pi$
$128$	1.00000	0.0883883
$129$	−3.75971	−0.331024
$130$	3.36222	0.294887
$131$	14.3380	1.25272	0.626360	−	0.779534i	$-0.284544\pi$
$131$	14.3380	1.25272	0.626360	+	0.779534i	$0.284544\pi$
$132$	0.578337	0.0503378
$133$	3.62721	0.314519
$134$	1.00000	0.0863868
$135$	0.710831	0.0611786
$136$	4.20555	0.360623
$137$	13.9058	1.18805	0.594027	−	0.804445i	$-0.297537\pi$
$137$	13.9058	1.18805	0.594027	+	0.804445i	$0.297537\pi$
$138$	−5.44082	−0.463153
$139$	−15.3869	−1.30510	−0.652551	−	0.757745i	$-0.726301\pi$
$139$	−15.3869	−1.30510	−0.652551	+	0.757745i	$0.726301\pi$
$140$	−1.28917	−0.108955
$141$	−2.05390	−0.172970
$142$	8.15165	0.684071
$143$	2.73553	0.228756
$144$	1.00000	0.0833333
$145$	−3.36222	−0.279218
$146$	−5.49472	−0.454746
$147$	−3.71083	−0.306064
$148$	−8.07306	−0.663601
$149$	−2.93554	−0.240489	−0.120244	−	0.992744i	$-0.538368\pi$
$149$	−2.93554	−0.240489	−0.120244	+	0.992744i	$0.538368\pi$
$150$	−4.49472	−0.366992
$151$	−8.67609	−0.706050	−0.353025	−	0.935614i	$-0.614847\pi$
$151$	−8.67609	−0.706050	−0.353025	+	0.935614i	$0.614847\pi$
$152$	−2.00000	−0.162221
$153$	4.20555	0.339999
$154$	−1.04888	−0.0845208
$155$	−1.55416	−0.124833
$156$	4.72999	0.378702
$157$	−2.13249	−0.170192	−0.0850958	−	0.996373i	$-0.527120\pi$
$157$	−2.13249	−0.170192	−0.0850958	+	0.996373i	$0.527120\pi$
$158$	2.35720	0.187529
$159$	4.71083	0.373593
$160$	0.710831	0.0561962
$161$	9.86751	0.777668
$162$	1.00000	0.0785674
$163$	20.8222	1.63092	0.815460	−	0.578813i	$-0.196484\pi$
$163$	20.8222	1.63092	0.815460	+	0.578813i	$0.196484\pi$
$164$	9.96526	0.778156
$165$	0.411100	0.0320041
$166$	−13.5925	−1.05498
$167$	−1.40251	−0.108529	−0.0542646	−	0.998527i	$-0.517281\pi$
$167$	−1.40251	−0.108529	−0.0542646	+	0.998527i	$0.517281\pi$
$168$	−1.81361	−0.139923
$169$	9.37279	0.720984
$170$	2.98944	0.229279
$171$	−2.00000	−0.152944
$172$	−3.75971	−0.286675
$173$	13.1028	0.996186	0.498093	−	0.867124i	$-0.334034\pi$
$173$	13.1028	0.996186	0.498093	+	0.867124i	$0.334034\pi$
$174$	−4.72999	−0.358579
$175$	8.15165	0.616207
$176$	0.578337	0.0435938
$177$	0.132494	0.00995889
$178$	−3.83276	−0.287278
$179$	3.15667	0.235941	0.117970	−	0.993017i	$-0.462361\pi$
$179$	3.15667	0.235941	0.117970	+	0.993017i	$0.462361\pi$
$180$	0.710831	0.0529822
$181$	−0.975820	−0.0725321	−0.0362661	−	0.999342i	$-0.511546\pi$
$181$	−0.975820	−0.0725321	−0.0362661	+	0.999342i	$0.511546\pi$
$182$	−8.57834	−0.635869
$183$	−6.15165	−0.454743
$184$	−5.44082	−0.401103
$185$	−5.73858	−0.421909
$186$	−2.18639	−0.160314
$187$	2.43223	0.177862
$188$	−2.05390	−0.149796
$189$	−1.81361	−0.131920
$190$	−1.42166	−0.103138
$191$	−14.2056	−1.02788	−0.513939	−	0.857827i	$-0.671814\pi$
$191$	−14.2056	−1.02788	−0.513939	+	0.857827i	$0.671814\pi$
$192$	1.00000	0.0721688
$193$	25.2197	1.81535	0.907676	−	0.419671i	$-0.137855\pi$
$193$	25.2197	1.81535	0.907676	+	0.419671i	$0.137855\pi$
$194$	6.78389	0.487055
$195$	3.36222	0.240774
$196$	−3.71083	−0.265059
$197$	−14.5819	−1.03892	−0.519459	−	0.854495i	$-0.673866\pi$
$197$	−14.5819	−1.03892	−0.519459	+	0.854495i	$0.673866\pi$
$198$	0.578337	0.0411006
$199$	−15.2544	−1.08136	−0.540679	−	0.841229i	$-0.681833\pi$
$199$	−15.2544	−1.08136	−0.540679	+	0.841229i	$0.681833\pi$
$200$	−4.49472	−0.317825
$201$	1.00000	0.0705346
$202$	9.25443	0.651139
$203$	8.57834	0.602081
$204$	4.20555	0.294447
$205$	7.08362	0.494741
$206$	−9.45998	−0.659108
$207$	−5.44082	−0.378163
$208$	4.72999	0.327966
$209$	−1.15667	−0.0800088
$210$	−1.28917	−0.0889611
$211$	27.3522	1.88300	0.941501	−	0.337011i	$-0.109416\pi$
$211$	27.3522	1.88300	0.941501	+	0.337011i	$0.109416\pi$
$212$	4.71083	0.323541
$213$	8.15165	0.558542
$214$	6.20555	0.424203
$215$	−2.67252	−0.182264
$216$	1.00000	0.0680414
$217$	3.96526	0.269179
$218$	2.15165	0.145728
$219$	−5.49472	−0.371299
$220$	0.411100	0.0277164
$221$	19.8922	1.33809
$222$	−8.07306	−0.541828
$223$	14.5783	0.976238	0.488119	−	0.872777i	$-0.337683\pi$
$223$	14.5783	0.976238	0.488119	+	0.872777i	$0.337683\pi$
$224$	−1.81361	−0.121177
$225$	−4.49472	−0.299648
$226$	18.8816	1.25599
$227$	−24.2786	−1.61143	−0.805714	−	0.592305i	$-0.798218\pi$
$227$	−24.2786	−1.61143	−0.805714	+	0.592305i	$0.798218\pi$
$228$	−2.00000	−0.132453
$229$	15.6116	1.03165	0.515823	−	0.856695i	$-0.327486\pi$
$229$	15.6116	1.03165	0.515823	+	0.856695i	$0.327486\pi$
$230$	−3.86751	−0.255016
$231$	−1.04888	−0.0690110
$232$	−4.72999	−0.310539
$233$	5.49472	0.359971	0.179985	−	0.983669i	$-0.442395\pi$
$233$	5.49472	0.359971	0.179985	+	0.983669i	$0.442395\pi$
$234$	4.72999	0.309209
$235$	−1.45998	−0.0952383
$236$	0.132494	0.00862465
$237$	2.35720	0.153117
$238$	−7.62721	−0.494399
$239$	2.98944	0.193371	0.0966853	−	0.995315i	$-0.469176\pi$
$239$	2.98944	0.193371	0.0966853	+	0.995315i	$0.469176\pi$
$240$	0.710831	0.0458840
$241$	21.7003	1.39784	0.698919	−	0.715201i	$-0.253665\pi$
$241$	21.7003	1.39784	0.698919	+	0.715201i	$0.253665\pi$
$242$	−10.6655	−0.685606
$243$	1.00000	0.0641500
$244$	−6.15165	−0.393819
$245$	−2.63778	−0.168521
$246$	9.96526	0.635362
$247$	−9.45998	−0.601924
$248$	−2.18639	−0.138836
$249$	−13.5925	−0.861388
$250$	−6.74914	−0.426853
$251$	9.15667	0.577964	0.288982	−	0.957335i	$-0.406683\pi$
$251$	9.15667	0.577964	0.288982	+	0.957335i	$0.406683\pi$
$252$	−1.81361	−0.114246
$253$	−3.14663	−0.197827
$254$	5.04888	0.316795
$255$	2.98944	0.187206
$256$	1.00000	0.0625000
$257$	4.57834	0.285589	0.142794	−	0.989752i	$-0.454391\pi$
$257$	4.57834	0.285589	0.142794	+	0.989752i	$0.454391\pi$
$258$	−3.75971	−0.234069
$259$	14.6413	0.909769
$260$	3.36222	0.208516
$261$	−4.72999	−0.292779
$262$	14.3380	0.885807
$263$	0.921921	0.0568481	0.0284240	−	0.999596i	$-0.490951\pi$
$263$	0.921921	0.0568481	0.0284240	+	0.999596i	$0.490951\pi$
$264$	0.578337	0.0355942
$265$	3.34861	0.205703
$266$	3.62721	0.222399
$267$	−3.83276	−0.234561
$268$	1.00000	0.0610847
$269$	23.7194	1.44620	0.723099	−	0.690744i	$-0.242717\pi$
$269$	23.7194	1.44620	0.723099	+	0.690744i	$0.242717\pi$
$270$	0.710831	0.0432598
$271$	−27.8172	−1.68977	−0.844887	−	0.534945i	$-0.820332\pi$
$271$	−27.8172	−1.68977	−0.844887	+	0.534945i	$0.820332\pi$
$272$	4.20555	0.254999
$273$	−8.57834	−0.519185
$274$	13.9058	0.840081
$275$	−2.59946	−0.156753
$276$	−5.44082	−0.327499
$277$	−6.17081	−0.370768	−0.185384	−	0.982666i	$-0.559353\pi$
$277$	−6.17081	−0.370768	−0.185384	+	0.982666i	$0.559353\pi$
$278$	−15.3869	−0.922846
$279$	−2.18639	−0.130896
$280$	−1.28917	−0.0770426
$281$	−16.5089	−0.984836	−0.492418	−	0.870359i	$-0.663887\pi$
$281$	−16.5089	−0.984836	−0.492418	+	0.870359i	$0.663887\pi$
$282$	−2.05390	−0.122308
$283$	9.42166	0.560060	0.280030	−	0.959991i	$-0.409656\pi$
$283$	9.42166	0.560060	0.280030	+	0.959991i	$0.409656\pi$
$284$	8.15165	0.483711
$285$	−1.42166	−0.0842120
$286$	2.73553	0.161755
$287$	−18.0731	−1.06682
$288$	1.00000	0.0589256
$289$	0.686652	0.0403913
$290$	−3.36222	−0.197437
$291$	6.78389	0.397679
$292$	−5.49472	−0.321554
$293$	−0.583877	−0.0341104	−0.0170552	−	0.999855i	$-0.505429\pi$
$293$	−0.583877	−0.0341104	−0.0170552	+	0.999855i	$0.505429\pi$
$294$	−3.71083	−0.216420
$295$	0.0941812	0.00548344
$296$	−8.07306	−0.469237
$297$	0.578337	0.0335585
$298$	−2.93554	−0.170051
$299$	−25.7350	−1.48829
$300$	−4.49472	−0.259503
$301$	6.81863	0.393019
$302$	−8.67609	−0.499253
$303$	9.25443	0.531653
$304$	−2.00000	−0.114708
$305$	−4.37279	−0.250385
$306$	4.20555	0.240415
$307$	−6.67609	−0.381025	−0.190512	−	0.981685i	$-0.561015\pi$
$307$	−6.67609	−0.381025	−0.190512	+	0.981685i	$0.561015\pi$
$308$	−1.04888	−0.0597653
$309$	−9.45998	−0.538159
$310$	−1.55416	−0.0882702
$311$	−3.51941	−0.199568	−0.0997839	−	0.995009i	$-0.531815\pi$
$311$	−3.51941	−0.199568	−0.0997839	+	0.995009i	$0.531815\pi$
$312$	4.72999	0.267783
$313$	31.4600	1.77822	0.889111	−	0.457691i	$-0.151323\pi$
$313$	31.4600	1.77822	0.889111	+	0.457691i	$0.151323\pi$
$314$	−2.13249	−0.120344
$315$	−1.28917	−0.0726364
$316$	2.35720	0.132603
$317$	−0.799473	−0.0449029	−0.0224514	−	0.999748i	$-0.507147\pi$
$317$	−0.799473	−0.0449029	−0.0224514	+	0.999748i	$0.507147\pi$
$318$	4.71083	0.264170
$319$	−2.73553	−0.153160
$320$	0.710831	0.0397367
$321$	6.20555	0.346360
$322$	9.86751	0.549895
$323$	−8.41110	−0.468006
$324$	1.00000	0.0555556
$325$	−21.2600	−1.17929
$326$	20.8222	1.15324
$327$	2.15165	0.118987
$328$	9.96526	0.550239
$329$	3.72496	0.205364
$330$	0.411100	0.0226303
$331$	18.2686	1.00413	0.502065	−	0.864830i	$-0.332574\pi$
$331$	18.2686	1.00413	0.502065	+	0.864830i	$0.332574\pi$
$332$	−13.5925	−0.745984
$333$	−8.07306	−0.442401
$334$	−1.40251	−0.0767417
$335$	0.710831	0.0388369
$336$	−1.81361	−0.0989403
$337$	18.8222	1.02531	0.512655	−	0.858595i	$-0.328662\pi$
$337$	18.8222	1.02531	0.512655	+	0.858595i	$0.328662\pi$
$338$	9.37279	0.509812
$339$	18.8816	1.02551
$340$	2.98944	0.162125
$341$	−1.26447	−0.0684750
$342$	−2.00000	−0.108148
$343$	19.4252	1.04886
$344$	−3.75971	−0.202710
$345$	−3.86751	−0.208220
$346$	13.1028	0.704410
$347$	−8.98944	−0.482578	−0.241289	−	0.970453i	$-0.577570\pi$
$347$	−8.98944	−0.482578	−0.241289	+	0.970453i	$0.577570\pi$
$348$	−4.72999	−0.253554
$349$	−23.7003	−1.26865	−0.634323	−	0.773068i	$-0.718721\pi$
$349$	−23.7003	−1.26865	−0.634323	+	0.773068i	$0.718721\pi$
$350$	8.15165	0.435724
$351$	4.72999	0.252468
$352$	0.578337	0.0308255
$353$	−9.96526	−0.530397	−0.265199	−	0.964194i	$-0.585438\pi$
$353$	−9.96526	−0.530397	−0.265199	+	0.964194i	$0.585438\pi$
$354$	0.132494	0.00704200
$355$	5.79445	0.307537
$356$	−3.83276	−0.203136
$357$	−7.62721	−0.403675
$358$	3.15667	0.166835
$359$	−26.1864	−1.38206	−0.691032	−	0.722824i	$-0.742844\pi$
$359$	−26.1864	−1.38206	−0.691032	+	0.722824i	$0.742844\pi$
$360$	0.710831	0.0374641
$361$	−15.0000	−0.789474
$362$	−0.975820	−0.0512880
$363$	−10.6655	−0.559795
$364$	−8.57834	−0.449627
$365$	−3.90582	−0.204440
$366$	−6.15165	−0.321552
$367$	3.51388	0.183423	0.0917114	−	0.995786i	$-0.470766\pi$
$367$	3.51388	0.183423	0.0917114	+	0.995786i	$0.470766\pi$
$368$	−5.44082	−0.283622
$369$	9.96526	0.518771
$370$	−5.73858	−0.298335
$371$	−8.54359	−0.443561
$372$	−2.18639	−0.113359
$373$	−12.5728	−0.650995	−0.325497	−	0.945543i	$-0.605532\pi$
$373$	−12.5728	−0.650995	−0.325497	+	0.945543i	$0.605532\pi$
$374$	2.43223	0.125767
$375$	−6.74914	−0.348524
$376$	−2.05390	−0.105922
$377$	−22.3728	−1.15226
$378$	−1.81361	−0.0932819
$379$	−0.132494	−0.00680578	−0.00340289	−	0.999994i	$-0.501083\pi$
$379$	−0.132494	−0.00680578	−0.00340289	+	0.999994i	$0.501083\pi$
$380$	−1.42166	−0.0729298
$381$	5.04888	0.258662
$382$	−14.2056	−0.726819
$383$	−11.5577	−0.590572	−0.295286	−	0.955409i	$-0.595415\pi$
$383$	−11.5577	−0.590572	−0.295286	+	0.955409i	$0.595415\pi$
$384$	1.00000	0.0510310
$385$	−0.745574	−0.0379980
$386$	25.2197	1.28365
$387$	−3.75971	−0.191117
$388$	6.78389	0.344400
$389$	17.5522	0.889931	0.444966	−	0.895548i	$-0.353216\pi$
$389$	17.5522	0.889931	0.444966	+	0.895548i	$0.353216\pi$
$390$	3.36222	0.170253
$391$	−22.8816	−1.15717
$392$	−3.71083	−0.187425
$393$	14.3380	0.723259
$394$	−14.5819	−0.734626
$395$	1.67557	0.0843072
$396$	0.578337	0.0290625
$397$	4.61665	0.231703	0.115852	−	0.993267i	$-0.463040\pi$
$397$	4.61665	0.231703	0.115852	+	0.993267i	$0.463040\pi$
$398$	−15.2544	−0.764635
$399$	3.62721	0.181588
$400$	−4.49472	−0.224736
$401$	−6.65139	−0.332155	−0.166077	−	0.986113i	$-0.553110\pi$
$401$	−6.65139	−0.332155	−0.166077	+	0.986113i	$0.553110\pi$
$402$	1.00000	0.0498755
$403$	−10.3416	−0.515153
$404$	9.25443	0.460425
$405$	0.710831	0.0353215
$406$	8.57834	0.425736
$407$	−4.66895	−0.231431
$408$	4.20555	0.208206
$409$	−38.4494	−1.90120	−0.950601	−	0.310417i	$-0.899531\pi$
$409$	−38.4494	−1.90120	−0.950601	+	0.310417i	$0.899531\pi$
$410$	7.08362	0.349835
$411$	13.9058	0.685923
$412$	−9.45998	−0.466060
$413$	−0.240293	−0.0118240
$414$	−5.44082	−0.267402
$415$	−9.66196	−0.474287
$416$	4.72999	0.231907
$417$	−15.3869	−0.753501
$418$	−1.15667	−0.0565748
$419$	−11.4947	−0.561554	−0.280777	−	0.959773i	$-0.590592\pi$
$419$	−11.4947	−0.561554	−0.280777	+	0.959773i	$0.590592\pi$
$420$	−1.28917	−0.0629050
$421$	−5.19142	−0.253014	−0.126507	−	0.991966i	$-0.540377\pi$
$421$	−5.19142	−0.253014	−0.126507	+	0.991966i	$0.540377\pi$
$422$	27.3522	1.33148
$423$	−2.05390	−0.0998640
$424$	4.71083	0.228778
$425$	−18.9028	−0.916919
$426$	8.15165	0.394949
$427$	11.1567	0.539909
$428$	6.20555	0.299957
$429$	2.73553	0.132073
$430$	−2.67252	−0.128880
$431$	3.34307	0.161030	0.0805150	−	0.996753i	$-0.474344\pi$
$431$	3.34307	0.161030	0.0805150	+	0.996753i	$0.474344\pi$
$432$	1.00000	0.0481125
$433$	−37.2927	−1.79217	−0.896087	−	0.443878i	$-0.853602\pi$
$433$	−37.2927	−1.79217	−0.896087	+	0.443878i	$0.853602\pi$
$434$	3.96526	0.190338
$435$	−3.36222	−0.161206
$436$	2.15165	0.103045
$437$	10.8816	0.520539
$438$	−5.49472	−0.262548
$439$	2.61665	0.124886	0.0624430	−	0.998049i	$-0.480111\pi$
$439$	2.61665	0.124886	0.0624430	+	0.998049i	$0.480111\pi$
$440$	0.411100	0.0195984
$441$	−3.71083	−0.176706
$442$	19.8922	0.946176
$443$	14.5783	0.692638	0.346319	−	0.938117i	$-0.387432\pi$
$443$	14.5783	0.692638	0.346319	+	0.938117i	$0.387432\pi$
$444$	−8.07306	−0.383130
$445$	−2.72445	−0.129151
$446$	14.5783	0.690304
$447$	−2.93554	−0.138846
$448$	−1.81361	−0.0856849
$449$	−21.2927	−1.00487	−0.502433	−	0.864616i	$-0.667562\pi$
$449$	−21.2927	−1.00487	−0.502433	+	0.864616i	$0.667562\pi$
$450$	−4.49472	−0.211883
$451$	5.76328	0.271382
$452$	18.8816	0.888118
$453$	−8.67609	−0.407638
$454$	−24.2786	−1.13945
$455$	−6.09775	−0.285867
$456$	−2.00000	−0.0936586
$457$	−2.47054	−0.115567	−0.0577835	−	0.998329i	$-0.518403\pi$
$457$	−2.47054	−0.115567	−0.0577835	+	0.998329i	$0.518403\pi$
$458$	15.6116	0.729483
$459$	4.20555	0.196298
$460$	−3.86751	−0.180323
$461$	−15.3466	−0.714764	−0.357382	−	0.933958i	$-0.616331\pi$
$461$	−15.3466	−0.714764	−0.357382	+	0.933958i	$0.616331\pi$
$462$	−1.04888	−0.0487981
$463$	29.0680	1.35091	0.675453	−	0.737403i	$-0.263948\pi$
$463$	29.0680	1.35091	0.675453	+	0.737403i	$0.263948\pi$
$464$	−4.72999	−0.219584
$465$	−1.55416	−0.0720723
$466$	5.49472	0.254538
$467$	−22.9200	−1.06061	−0.530304	−	0.847807i	$-0.677922\pi$
$467$	−22.9200	−1.06061	−0.530304	+	0.847807i	$0.677922\pi$
$468$	4.72999	0.218644
$469$	−1.81361	−0.0837446
$470$	−1.45998	−0.0673437
$471$	−2.13249	−0.0982602
$472$	0.132494	0.00609855
$473$	−2.17438	−0.0999780
$474$	2.35720	0.108270
$475$	8.98944	0.412464
$476$	−7.62721	−0.349593
$477$	4.71083	0.215694
$478$	2.98944	0.136734
$479$	−17.0297	−0.778108	−0.389054	−	0.921215i	$-0.627198\pi$
$479$	−17.0297	−0.778108	−0.389054	+	0.921215i	$0.627198\pi$
$480$	0.710831	0.0324449
$481$	−38.1855	−1.74111
$482$	21.7003	0.988420
$483$	9.86751	0.448987
$484$	−10.6655	−0.484797
$485$	4.82220	0.218965
$486$	1.00000	0.0453609
$487$	−16.3919	−0.742790	−0.371395	−	0.928475i	$-0.621120\pi$
$487$	−16.3919	−0.742790	−0.371395	+	0.928475i	$0.621120\pi$
$488$	−6.15165	−0.278472
$489$	20.8222	0.941612
$490$	−2.63778	−0.119163
$491$	41.5230	1.87391	0.936953	−	0.349455i	$-0.113633\pi$
$491$	41.5230	1.87391	0.936953	+	0.349455i	$0.113633\pi$
$492$	9.96526	0.449269
$493$	−19.8922	−0.895900
$494$	−9.45998	−0.425624
$495$	0.411100	0.0184776
$496$	−2.18639	−0.0981720
$497$	−14.7839	−0.663148
$498$	−13.5925	−0.609093
$499$	8.63778	0.386680	0.193340	−	0.981132i	$-0.438068\pi$
$499$	8.63778	0.386680	0.193340	+	0.981132i	$0.438068\pi$
$500$	−6.74914	−0.301831
$501$	−1.40251	−0.0626594
$502$	9.15667	0.408682
$503$	−24.9300	−1.11157	−0.555787	−	0.831325i	$-0.687583\pi$
$503$	−24.9300	−1.11157	−0.555787	+	0.831325i	$0.687583\pi$
$504$	−1.81361	−0.0807845
$505$	6.57834	0.292732
$506$	−3.14663	−0.139885
$507$	9.37279	0.416260
$508$	5.04888	0.224008
$509$	27.3083	1.21042	0.605210	−	0.796066i	$-0.293089\pi$
$509$	27.3083	1.21042	0.605210	+	0.796066i	$0.293089\pi$
$510$	2.98944	0.132375
$511$	9.96526	0.440837
$512$	1.00000	0.0441942
$513$	−2.00000	−0.0883022
$514$	4.57834	0.201942
$515$	−6.72445	−0.296315
$516$	−3.75971	−0.165512
$517$	−1.18785	−0.0522414
$518$	14.6413	0.643304
$519$	13.1028	0.575148
$520$	3.36222	0.147443
$521$	26.1361	1.14504	0.572521	−	0.819890i	$-0.305966\pi$
$521$	26.1361	1.14504	0.572521	+	0.819890i	$0.305966\pi$
$522$	−4.72999	−0.207026
$523$	28.8222	1.26031	0.630153	−	0.776471i	$-0.282992\pi$
$523$	28.8222	1.26031	0.630153	+	0.776471i	$0.282992\pi$
$524$	14.3380	0.626360
$525$	8.15165	0.355767
$526$	0.921921	0.0401977
$527$	−9.19499	−0.400540
$528$	0.578337	0.0251689
$529$	6.60252	0.287066
$530$	3.34861	0.145454
$531$	0.132494	0.00574977
$532$	3.62721	0.157260
$533$	47.1355	2.04167
$534$	−3.83276	−0.165860
$535$	4.41110	0.190708
$536$	1.00000	0.0431934
$537$	3.15667	0.136221
$538$	23.7194	1.02262
$539$	−2.14611	−0.0924396
$540$	0.710831	0.0305893
$541$	−38.6705	−1.66258	−0.831288	−	0.555841i	$-0.812396\pi$
$541$	−38.6705	−1.66258	−0.831288	+	0.555841i	$0.812396\pi$
$542$	−27.8172	−1.19485
$543$	−0.975820	−0.0418765
$544$	4.20555	0.180311
$545$	1.52946	0.0655149
$546$	−8.57834	−0.367119
$547$	−1.85746	−0.0794192	−0.0397096	−	0.999211i	$-0.512643\pi$
$547$	−1.85746	−0.0794192	−0.0397096	+	0.999211i	$0.512643\pi$
$548$	13.9058	0.594027
$549$	−6.15165	−0.262546
$550$	−2.59946	−0.110841
$551$	9.45998	0.403009
$552$	−5.44082	−0.231577
$553$	−4.27504	−0.181793
$554$	−6.17081	−0.262172
$555$	−5.73858	−0.243589
$556$	−15.3869	−0.652551
$557$	17.5139	0.742087	0.371043	−	0.928616i	$-0.379000\pi$
$557$	17.5139	0.742087	0.371043	+	0.928616i	$0.379000\pi$
$558$	−2.18639	−0.0925574
$559$	−17.7834	−0.752156
$560$	−1.28917	−0.0544773
$561$	2.43223	0.102689
$562$	−16.5089	−0.696384
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	−2.05390	−0.0864848
$565$	13.4217	0.564654
$566$	9.42166	0.396022
$567$	−1.81361	−0.0761643
$568$	8.15165	0.342036
$569$	−31.6061	−1.32500	−0.662498	−	0.749064i	$-0.730504\pi$
$569$	−31.6061	−1.32500	−0.662498	+	0.749064i	$0.730504\pi$
$570$	−1.42166	−0.0595469
$571$	33.4217	1.39865	0.699327	−	0.714802i	$-0.253483\pi$
$571$	33.4217	1.39865	0.699327	+	0.714802i	$0.253483\pi$
$572$	2.73553	0.114378
$573$	−14.2056	−0.593445
$574$	−18.0731	−0.754355
$575$	24.4550	1.01984
$576$	1.00000	0.0416667
$577$	−5.66553	−0.235859	−0.117929	−	0.993022i	$-0.537626\pi$
$577$	−5.66553	−0.235859	−0.117929	+	0.993022i	$0.537626\pi$
$578$	0.686652	0.0285609
$579$	25.2197	1.04809
$580$	−3.36222	−0.139609
$581$	24.6514	1.02271
$582$	6.78389	0.281201
$583$	2.72445	0.112835
$584$	−5.49472	−0.227373
$585$	3.36222	0.139011
$586$	−0.583877	−0.0241197
$587$	47.6938	1.96853	0.984267	−	0.176689i	$-0.0565386\pi$
$587$	47.6938	1.96853	0.984267	+	0.176689i	$0.0565386\pi$
$588$	−3.71083	−0.153032
$589$	4.37279	0.180178
$590$	0.0941812	0.00387738
$591$	−14.5819	−0.599820
$592$	−8.07306	−0.331801
$593$	−11.8081	−0.484899	−0.242450	−	0.970164i	$-0.577951\pi$
$593$	−11.8081	−0.484899	−0.242450	+	0.970164i	$0.577951\pi$
$594$	0.578337	0.0237295
$595$	−5.42166	−0.222267
$596$	−2.93554	−0.120244
$597$	−15.2544	−0.624322
$598$	−25.7350	−1.05238
$599$	33.2333	1.35788	0.678938	−	0.734196i	$-0.262441\pi$
$599$	33.2333	1.35788	0.678938	+	0.734196i	$0.262441\pi$
$600$	−4.49472	−0.183496
$601$	29.8016	1.21563	0.607816	−	0.794078i	$-0.292046\pi$
$601$	29.8016	1.21563	0.607816	+	0.794078i	$0.292046\pi$
$602$	6.81863	0.277907
$603$	1.00000	0.0407231
$604$	−8.67609	−0.353025
$605$	−7.58139	−0.308227
$606$	9.25443	0.375935
$607$	29.7250	1.20650	0.603250	−	0.797552i	$-0.293872\pi$
$607$	29.7250	1.20650	0.603250	+	0.797552i	$0.293872\pi$
$608$	−2.00000	−0.0811107
$609$	8.57834	0.347612
$610$	−4.37279	−0.177049
$611$	−9.71492	−0.393024
$612$	4.20555	0.169999
$613$	5.41809	0.218835	0.109417	−	0.993996i	$-0.465102\pi$
$613$	5.41809	0.218835	0.109417	+	0.993996i	$0.465102\pi$
$614$	−6.67609	−0.269425
$615$	7.08362	0.285639
$616$	−1.04888	−0.0422604
$617$	0.313348	0.0126149	0.00630747	−	0.999980i	$-0.497992\pi$
$617$	0.313348	0.0126149	0.00630747	+	0.999980i	$0.497992\pi$
$618$	−9.45998	−0.380536
$619$	−35.4983	−1.42680	−0.713398	−	0.700759i	$-0.752845\pi$
$619$	−35.4983	−1.42680	−0.713398	+	0.700759i	$0.752845\pi$
$620$	−1.55416	−0.0624165
$621$	−5.44082	−0.218333
$622$	−3.51941	−0.141116
$623$	6.95112	0.278491
$624$	4.72999	0.189351
$625$	17.6761	0.707044
$626$	31.4600	1.25739
$627$	−1.15667	−0.0461931
$628$	−2.13249	−0.0850958
$629$	−33.9516	−1.35374
$630$	−1.28917	−0.0513617
$631$	−31.4827	−1.25331	−0.626653	−	0.779298i	$-0.715576\pi$
$631$	−31.4827	−1.25331	−0.626653	+	0.779298i	$0.715576\pi$
$632$	2.35720	0.0937644
$633$	27.3522	1.08715
$634$	−0.799473	−0.0317511
$635$	3.58890	0.142421
$636$	4.71083	0.186797
$637$	−17.5522	−0.695443
$638$	−2.73553	−0.108301
$639$	8.15165	0.322474
$640$	0.710831	0.0280981
$641$	−12.2686	−0.484579	−0.242289	−	0.970204i	$-0.577898\pi$
$641$	−12.2686	−0.484579	−0.242289	+	0.970204i	$0.577898\pi$
$642$	6.20555	0.244914
$643$	21.3239	0.840933	0.420466	−	0.907308i	$-0.361866\pi$
$643$	21.3239	0.840933	0.420466	+	0.907308i	$0.361866\pi$
$644$	9.86751	0.388834
$645$	−2.67252	−0.105230
$646$	−8.41110	−0.330930
$647$	12.3728	0.486424	0.243212	−	0.969973i	$-0.421799\pi$
$647$	12.3728	0.486424	0.243212	+	0.969973i	$0.421799\pi$
$648$	1.00000	0.0392837
$649$	0.0766264	0.00300785
$650$	−21.2600	−0.833884
$651$	3.96526	0.155411
$652$	20.8222	0.815460
$653$	47.5925	1.86244	0.931219	−	0.364461i	$-0.118747\pi$
$653$	47.5925	1.86244	0.931219	+	0.364461i	$0.118747\pi$
$654$	2.15165	0.0841362
$655$	10.1919	0.398232
$656$	9.96526	0.389078
$657$	−5.49472	−0.214369
$658$	3.72496	0.145214
$659$	−6.37636	−0.248388	−0.124194	−	0.992258i	$-0.539634\pi$
$659$	−6.37636	−0.248388	−0.124194	+	0.992258i	$0.539634\pi$
$660$	0.411100	0.0160021
$661$	−31.2005	−1.21356	−0.606780	−	0.794870i	$-0.707539\pi$
$661$	−31.2005	−1.21356	−0.606780	+	0.794870i	$0.707539\pi$
$662$	18.2686	0.710028
$663$	19.8922	0.772549
$664$	−13.5925	−0.527490
$665$	2.57834	0.0999836
$666$	−8.07306	−0.312825
$667$	25.7350	0.996464
$668$	−1.40251	−0.0542646
$669$	14.5783	0.563631
$670$	0.710831	0.0274618
$671$	−3.55773	−0.137345
$672$	−1.81361	−0.0699614
$673$	−7.31386	−0.281929	−0.140964	−	0.990015i	$-0.545020\pi$
$673$	−7.31386	−0.281929	−0.140964	+	0.990015i	$0.545020\pi$
$674$	18.8222	0.725004
$675$	−4.49472	−0.173002
$676$	9.37279	0.360492
$677$	−44.4842	−1.70966	−0.854832	−	0.518904i	$-0.826340\pi$
$677$	−44.4842	−1.70966	−0.854832	+	0.518904i	$0.826340\pi$
$678$	18.8816	0.725145
$679$	−12.3033	−0.472157
$680$	2.98944	0.114640
$681$	−24.2786	−0.930358
$682$	−1.26447	−0.0484192
$683$	8.67609	0.331981	0.165991	−	0.986127i	$-0.446918\pi$
$683$	8.67609	0.331981	0.165991	+	0.986127i	$0.446918\pi$
$684$	−2.00000	−0.0764719
$685$	9.88469	0.377675
$686$	19.4252	0.741659
$687$	15.6116	0.595621
$688$	−3.75971	−0.143337
$689$	22.2822	0.848883
$690$	−3.86751	−0.147233
$691$	34.7738	1.32286	0.661430	−	0.750007i	$-0.269950\pi$
$691$	34.7738	1.32286	0.661430	+	0.750007i	$0.269950\pi$
$692$	13.1028	0.498093
$693$	−1.04888	−0.0398435
$694$	−8.98944	−0.341234
$695$	−10.9375	−0.414883
$696$	−4.72999	−0.179290
$697$	41.9094	1.58743
$698$	−23.7003	−0.897068
$699$	5.49472	0.207829
$700$	8.15165	0.308103
$701$	24.0036	0.906602	0.453301	−	0.891357i	$-0.350246\pi$
$701$	24.0036	0.906602	0.453301	+	0.891357i	$0.350246\pi$
$702$	4.72999	0.178522
$703$	16.1461	0.608962
$704$	0.578337	0.0217969
$705$	−1.45998	−0.0549859
$706$	−9.96526	−0.375047
$707$	−16.7839	−0.631223
$708$	0.132494	0.00497944
$709$	−37.0141	−1.39009	−0.695047	−	0.718964i	$-0.744617\pi$
$709$	−37.0141	−1.39009	−0.695047	+	0.718964i	$0.744617\pi$
$710$	5.79445	0.217462
$711$	2.35720	0.0884019
$712$	−3.83276	−0.143639
$713$	11.8958	0.445500
$714$	−7.62721	−0.285441
$715$	1.94450	0.0727201
$716$	3.15667	0.117970
$717$	2.98944	0.111643
$718$	−26.1864	−0.977268
$719$	−16.2147	−0.604705	−0.302352	−	0.953196i	$-0.597772\pi$
$719$	−16.2147	−0.604705	−0.302352	+	0.953196i	$0.597772\pi$
$720$	0.710831	0.0264911
$721$	17.1567	0.638948
$722$	−15.0000	−0.558242
$723$	21.7003	0.807042
$724$	−0.975820	−0.0362661
$725$	21.2600	0.789575
$726$	−10.6655	−0.395835
$727$	−8.49974	−0.315238	−0.157619	−	0.987500i	$-0.550382\pi$
$727$	−8.49974	−0.315238	−0.157619	+	0.987500i	$0.550382\pi$
$728$	−8.57834	−0.317934
$729$	1.00000	0.0370370
$730$	−3.90582	−0.144561
$731$	−15.8116	−0.584815
$732$	−6.15165	−0.227372
$733$	1.10278	0.0407319	0.0203660	−	0.999793i	$-0.493517\pi$
$733$	1.10278	0.0407319	0.0203660	+	0.999793i	$0.493517\pi$
$734$	3.51388	0.129700
$735$	−2.63778	−0.0972958
$736$	−5.44082	−0.200551
$737$	0.578337	0.0213033
$738$	9.96526	0.366826
$739$	−44.4247	−1.63419	−0.817095	−	0.576503i	$-0.804417\pi$
$739$	−44.4247	−1.63419	−0.817095	+	0.576503i	$0.804417\pi$
$740$	−5.73858	−0.210954
$741$	−9.45998	−0.347521
$742$	−8.54359	−0.313645
$743$	44.6258	1.63716	0.818580	−	0.574392i	$-0.194762\pi$
$743$	44.6258	1.63716	0.818580	+	0.574392i	$0.194762\pi$
$744$	−2.18639	−0.0801571
$745$	−2.08667	−0.0764498
$746$	−12.5728	−0.460323
$747$	−13.5925	−0.497322
$748$	2.43223	0.0889310
$749$	−11.2544	−0.411228
$750$	−6.74914	−0.246444
$751$	−25.1184	−0.916582	−0.458291	−	0.888802i	$-0.651538\pi$
$751$	−25.1184	−0.916582	−0.458291	+	0.888802i	$0.651538\pi$
$752$	−2.05390	−0.0748980
$753$	9.15667	0.333688
$754$	−22.3728	−0.814769
$755$	−6.16724	−0.224449
$756$	−1.81361	−0.0659602
$757$	27.7194	1.00748	0.503740	−	0.863855i	$-0.331957\pi$
$757$	27.7194	1.00748	0.503740	+	0.863855i	$0.331957\pi$
$758$	−0.132494	−0.00481241
$759$	−3.14663	−0.114215
$760$	−1.42166	−0.0515691
$761$	−52.4011	−1.89954	−0.949768	−	0.312954i	$-0.898682\pi$
$761$	−52.4011	−1.89954	−0.949768	+	0.312954i	$0.898682\pi$
$762$	5.04888	0.182901
$763$	−3.90225	−0.141271
$764$	−14.2056	−0.513939
$765$	2.98944	0.108083
$766$	−11.5577	−0.417598
$767$	0.626697	0.0226287
$768$	1.00000	0.0360844
$769$	−18.7839	−0.677364	−0.338682	−	0.940901i	$-0.609981\pi$
$769$	−18.7839	−0.677364	−0.338682	+	0.940901i	$0.609981\pi$
$770$	−0.745574	−0.0268686
$771$	4.57834	0.164885
$772$	25.2197	0.907676
$773$	28.8378	1.03722	0.518612	−	0.855010i	$-0.326449\pi$
$773$	28.8378	1.03722	0.518612	+	0.855010i	$0.326449\pi$
$774$	−3.75971	−0.135140
$775$	9.82722	0.353004
$776$	6.78389	0.243527
$777$	14.6413	0.525255
$778$	17.5522	0.629276
$779$	−19.9305	−0.714085
$780$	3.36222	0.120387
$781$	4.71440	0.168695
$782$	−22.8816	−0.818246
$783$	−4.72999	−0.169036
$784$	−3.71083	−0.132530
$785$	−1.51584	−0.0541028
$786$	14.3380	0.511421
$787$	−22.0419	−0.785708	−0.392854	−	0.919601i	$-0.628512\pi$
$787$	−22.0419	−0.785708	−0.392854	+	0.919601i	$0.628512\pi$
$788$	−14.5819	−0.519459
$789$	0.921921	0.0328213
$790$	1.67557	0.0596142
$791$	−34.2439	−1.21757
$792$	0.578337	0.0205503
$793$	−29.0972	−1.03327
$794$	4.61665	0.163839
$795$	3.34861	0.118763
$796$	−15.2544	−0.540679
$797$	−40.3955	−1.43088	−0.715441	−	0.698673i	$-0.753774\pi$
$797$	−40.3955	−1.43088	−0.715441	+	0.698673i	$0.753774\pi$
$798$	3.62721	0.128402
$799$	−8.63778	−0.305583
$800$	−4.49472	−0.158912
$801$	−3.83276	−0.135424
$802$	−6.65139	−0.234869
$803$	−3.17780	−0.112142
$804$	1.00000	0.0352673
$805$	7.01413	0.247216
$806$	−10.3416	−0.364268
$807$	23.7194	0.834963
$808$	9.25443	0.325570
$809$	−19.1814	−0.674381	−0.337191	−	0.941436i	$-0.609477\pi$
$809$	−19.1814	−0.674381	−0.337191	+	0.941436i	$0.609477\pi$
$810$	0.710831	0.0249761
$811$	−3.98638	−0.139981	−0.0699904	−	0.997548i	$-0.522297\pi$
$811$	−3.98638	−0.139981	−0.0699904	+	0.997548i	$0.522297\pi$
$812$	8.57834	0.301041
$813$	−27.8172	−0.975591
$814$	−4.66895	−0.163647
$815$	14.8011	0.518459
$816$	4.20555	0.147224
$817$	7.51941	0.263071
$818$	−38.4494	−1.34435
$819$	−8.57834	−0.299751
$820$	7.08362	0.247371
$821$	−46.1588	−1.61095	−0.805476	−	0.592628i	$-0.798091\pi$
$821$	−46.1588	−1.61095	−0.805476	+	0.592628i	$0.798091\pi$
$822$	13.9058	0.485021
$823$	−2.35166	−0.0819738	−0.0409869	−	0.999160i	$-0.513050\pi$
$823$	−2.35166	−0.0819738	−0.0409869	+	0.999160i	$0.513050\pi$
$824$	−9.45998	−0.329554
$825$	−2.59946	−0.0905017
$826$	−0.240293	−0.00836085
$827$	2.73553	0.0951236	0.0475618	−	0.998868i	$-0.484855\pi$
$827$	2.73553	0.0951236	0.0475618	+	0.998868i	$0.484855\pi$
$828$	−5.44082	−0.189082
$829$	19.0177	0.660512	0.330256	−	0.943891i	$-0.392865\pi$
$829$	19.0177	0.660512	0.330256	+	0.943891i	$0.392865\pi$
$830$	−9.66196	−0.335371
$831$	−6.17081	−0.214063
$832$	4.72999	0.163983
$833$	−15.6061	−0.540719
$834$	−15.3869	−0.532805
$835$	−0.996946	−0.0345007
$836$	−1.15667	−0.0400044
$837$	−2.18639	−0.0755728
$838$	−11.4947	−0.397078
$839$	−36.5628	−1.26229	−0.631143	−	0.775666i	$-0.717414\pi$
$839$	−36.5628	−1.26229	−0.631143	+	0.775666i	$0.717414\pi$
$840$	−1.28917	−0.0444805
$841$	−6.62721	−0.228525
$842$	−5.19142	−0.178908
$843$	−16.5089	−0.568595
$844$	27.3522	0.941501
$845$	6.66247	0.229196
$846$	−2.05390	−0.0706145
$847$	19.3431	0.664636
$848$	4.71083	0.161771
$849$	9.42166	0.323351
$850$	−18.9028	−0.648359
$851$	43.9240	1.50570
$852$	8.15165	0.279271
$853$	−53.6344	−1.83641	−0.918203	−	0.396111i	$-0.870360\pi$
$853$	−53.6344	−1.83641	−0.918203	+	0.396111i	$0.870360\pi$
$854$	11.1567	0.381774
$855$	−1.42166	−0.0486198
$856$	6.20555	0.212101
$857$	28.1325	0.960988	0.480494	−	0.876998i	$-0.340457\pi$
$857$	28.1325	0.960988	0.480494	+	0.876998i	$0.340457\pi$
$858$	2.73553	0.0933894
$859$	−1.37330	−0.0468565	−0.0234283	−	0.999726i	$-0.507458\pi$
$859$	−1.37330	−0.0468565	−0.0234283	+	0.999726i	$0.507458\pi$
$860$	−2.67252	−0.0911321
$861$	−18.0731	−0.615928
$862$	3.34307	0.113865
$863$	−6.34358	−0.215938	−0.107969	−	0.994154i	$-0.534435\pi$
$863$	−6.34358	−0.215938	−0.107969	+	0.994154i	$0.534435\pi$
$864$	1.00000	0.0340207
$865$	9.31386	0.316681
$866$	−37.2927	−1.26726
$867$	0.686652	0.0233199
$868$	3.96526	0.134590
$869$	1.36326	0.0462453
$870$	−3.36222	−0.113990
$871$	4.72999	0.160270
$872$	2.15165	0.0728641
$873$	6.78389	0.229600
$874$	10.8816	0.368077
$875$	12.2403	0.413797
$876$	−5.49472	−0.185649
$877$	−19.6555	−0.663718	−0.331859	−	0.943329i	$-0.607676\pi$
$877$	−19.6555	−0.663718	−0.331859	+	0.943329i	$0.607676\pi$
$878$	2.61665	0.0883077
$879$	−0.583877	−0.0196937
$880$	0.411100	0.0138582
$881$	14.7527	0.497032	0.248516	−	0.968628i	$-0.420057\pi$
$881$	14.7527	0.497032	0.248516	+	0.968628i	$0.420057\pi$
$882$	−3.71083	−0.124950
$883$	13.7497	0.462713	0.231356	−	0.972869i	$-0.425684\pi$
$883$	13.7497	0.462713	0.231356	+	0.972869i	$0.425684\pi$
$884$	19.8922	0.669047
$885$	0.0941812	0.00316587
$886$	14.5783	0.489769
$887$	−45.2141	−1.51814	−0.759071	−	0.651008i	$-0.774347\pi$
$887$	−45.2141	−1.51814	−0.759071	+	0.651008i	$0.774347\pi$
$888$	−8.07306	−0.270914
$889$	−9.15667	−0.307105
$890$	−2.72445	−0.0913237
$891$	0.578337	0.0193750
$892$	14.5783	0.488119
$893$	4.10780	0.137462
$894$	−2.93554	−0.0981791
$895$	2.24386	0.0750041
$896$	−1.81361	−0.0605883
$897$	−25.7350	−0.859267
$898$	−21.2927	−0.710548
$899$	10.3416	0.344912
$900$	−4.49472	−0.149824
$901$	19.8116	0.660021
$902$	5.76328	0.191896
$903$	6.81863	0.226910
$904$	18.8816	0.627994
$905$	−0.693644	−0.0230575
$906$	−8.67609	−0.288244
$907$	58.0071	1.92610	0.963048	−	0.269331i	$-0.0868025\pi$
$907$	58.0071	1.92610	0.963048	+	0.269331i	$0.0868025\pi$
$908$	−24.2786	−0.805714
$909$	9.25443	0.306950
$910$	−6.09775	−0.202139
$911$	−20.2283	−0.670193	−0.335096	−	0.942184i	$-0.608769\pi$
$911$	−20.2283	−0.670193	−0.335096	+	0.942184i	$0.608769\pi$
$912$	−2.00000	−0.0662266
$913$	−7.86103	−0.260162
$914$	−2.47054	−0.0817182
$915$	−4.37279	−0.144560
$916$	15.6116	0.515823
$917$	−26.0036	−0.858714
$918$	4.20555	0.138804
$919$	−39.2736	−1.29552	−0.647758	−	0.761846i	$-0.724293\pi$
$919$	−39.2736	−1.29552	−0.647758	+	0.761846i	$0.724293\pi$
$920$	−3.86751	−0.127508
$921$	−6.67609	−0.219985
$922$	−15.3466	−0.505415
$923$	38.5572	1.26913
$924$	−1.04888	−0.0345055
$925$	36.2861	1.19308
$926$	29.0680	0.955235
$927$	−9.45998	−0.310706
$928$	−4.72999	−0.155269
$929$	26.9583	0.884472	0.442236	−	0.896899i	$-0.354185\pi$
$929$	26.9583	0.884472	0.442236	+	0.896899i	$0.354185\pi$
$930$	−1.55416	−0.0509628
$931$	7.42166	0.243235
$932$	5.49472	0.179985
$933$	−3.51941	−0.115220
$934$	−22.9200	−0.749964
$935$	1.72890	0.0565412
$936$	4.72999	0.154604
$937$	−39.9789	−1.30605	−0.653026	−	0.757335i	$-0.726501\pi$
$937$	−39.9789	−1.30605	−0.653026	+	0.757335i	$0.726501\pi$
$938$	−1.81361	−0.0592164
$939$	31.4600	1.02666
$940$	−1.45998	−0.0476192
$941$	43.0177	1.40234	0.701169	−	0.712996i	$-0.252662\pi$
$941$	43.0177	1.40234	0.701169	+	0.712996i	$0.252662\pi$
$942$	−2.13249	−0.0694804
$943$	−54.2192	−1.76562
$944$	0.132494	0.00431232
$945$	−1.28917	−0.0419367
$946$	−2.17438	−0.0706951
$947$	14.8953	0.484031	0.242015	−	0.970272i	$-0.422192\pi$
$947$	14.8953	0.484031	0.242015	+	0.970272i	$0.422192\pi$
$948$	2.35720	0.0765583
$949$	−25.9900	−0.843670
$950$	8.98944	0.291656
$951$	−0.799473	−0.0259247
$952$	−7.62721	−0.247199
$953$	−33.6272	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$953$	−33.6272	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$954$	4.71083	0.152519
$955$	−10.0978	−0.326756
$956$	2.98944	0.0966853
$957$	−2.73553	−0.0884270
$958$	−17.0297	−0.550205
$959$	−25.2197	−0.814386
$960$	0.710831	0.0229420
$961$	−26.2197	−0.845796
$962$	−38.1855	−1.23115
$963$	6.20555	0.199971
$964$	21.7003	0.698919
$965$	17.9269	0.577089
$966$	9.86751	0.317482
$967$	51.5194	1.65675	0.828376	−	0.560172i	$-0.189265\pi$
$967$	51.5194	1.65675	0.828376	+	0.560172i	$0.189265\pi$
$968$	−10.6655	−0.342803
$969$	−8.41110	−0.270203
$970$	4.82220	0.154832
$971$	41.1255	1.31978	0.659890	−	0.751362i	$-0.270603\pi$
$971$	41.1255	1.31978	0.659890	+	0.751362i	$0.270603\pi$
$972$	1.00000	0.0320750
$973$	27.9058	0.894619
$974$	−16.3919	−0.525232
$975$	−21.2600	−0.680864
$976$	−6.15165	−0.196910
$977$	10.4423	0.334078	0.167039	−	0.985950i	$-0.446579\pi$
$977$	10.4423	0.334078	0.167039	+	0.985950i	$0.446579\pi$
$978$	20.8222	0.665821
$979$	−2.21663	−0.0708438
$980$	−2.63778	−0.0842607
$981$	2.15165	0.0686969
$982$	41.5230	1.32505
$983$	25.9789	0.828597	0.414299	−	0.910141i	$-0.364027\pi$
$983$	25.9789	0.828597	0.414299	+	0.910141i	$0.364027\pi$
$984$	9.96526	0.317681
$985$	−10.3653	−0.330265
$986$	−19.8922	−0.633497
$987$	3.72496	0.118567
$988$	−9.45998	−0.300962
$989$	20.4559	0.650459
$990$	0.411100	0.0130656
$991$	31.8419	1.01149	0.505745	−	0.862683i	$-0.331218\pi$
$991$	31.8419	1.01149	0.505745	+	0.862683i	$0.331218\pi$
$992$	−2.18639	−0.0694181
$993$	18.2686	0.579735
$994$	−14.7839	−0.468916
$995$	−10.8433	−0.343757
$996$	−13.5925	−0.430694
$997$	−57.6591	−1.82608	−0.913040	−	0.407870i	$-0.866272\pi$
$997$	−57.6591	−1.82608	−0.913040	+	0.407870i	$0.866272\pi$
$998$	8.63778	0.273424
$999$	−8.07306	−0.255420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.2.a.g.1.2	✓	3
3.2	odd	2		1206.2.a.m.1.2		3
4.3	odd	2		3216.2.a.s.1.2		3
12.11	even	2		9648.2.a.bj.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.2.a.g.1.2	✓	3	1.1	even	1	trivial
1206.2.a.m.1.2		3	3.2	odd	2
3216.2.a.s.1.2		3	4.3	odd	2
9648.2.a.bj.1.2		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 402 = 2 \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	402.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.710831 q^{5} +1.00000 q^{6} -1.81361 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.710831 q^{5} +1.00000 q^{6} -1.81361 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.710831 q^{10} +0.578337 q^{11} +1.00000 q^{12} +4.72999 q^{13} -1.81361 q^{14} +0.710831 q^{15} +1.00000 q^{16} +4.20555 q^{17} +1.00000 q^{18} -2.00000 q^{19} +0.710831 q^{20} -1.81361 q^{21} +0.578337 q^{22} -5.44082 q^{23} +1.00000 q^{24} -4.49472 q^{25} +4.72999 q^{26} +1.00000 q^{27} -1.81361 q^{28} -4.72999 q^{29} +0.710831 q^{30} -2.18639 q^{31} +1.00000 q^{32} +0.578337 q^{33} +4.20555 q^{34} -1.28917 q^{35} +1.00000 q^{36} -8.07306 q^{37} -2.00000 q^{38} +4.72999 q^{39} +0.710831 q^{40} +9.96526 q^{41} -1.81361 q^{42} -3.75971 q^{43} +0.578337 q^{44} +0.710831 q^{45} -5.44082 q^{46} -2.05390 q^{47} +1.00000 q^{48} -3.71083 q^{49} -4.49472 q^{50} +4.20555 q^{51} +4.72999 q^{52} +4.71083 q^{53} +1.00000 q^{54} +0.411100 q^{55} -1.81361 q^{56} -2.00000 q^{57} -4.72999 q^{58} +0.132494 q^{59} +0.710831 q^{60} -6.15165 q^{61} -2.18639 q^{62} -1.81361 q^{63} +1.00000 q^{64} +3.36222 q^{65} +0.578337 q^{66} +1.00000 q^{67} +4.20555 q^{68} -5.44082 q^{69} -1.28917 q^{70} +8.15165 q^{71} +1.00000 q^{72} -5.49472 q^{73} -8.07306 q^{74} -4.49472 q^{75} -2.00000 q^{76} -1.04888 q^{77} +4.72999 q^{78} +2.35720 q^{79} +0.710831 q^{80} +1.00000 q^{81} +9.96526 q^{82} -13.5925 q^{83} -1.81361 q^{84} +2.98944 q^{85} -3.75971 q^{86} -4.72999 q^{87} +0.578337 q^{88} -3.83276 q^{89} +0.710831 q^{90} -8.57834 q^{91} -5.44082 q^{92} -2.18639 q^{93} -2.05390 q^{94} -1.42166 q^{95} +1.00000 q^{96} +6.78389 q^{97} -3.71083 q^{98} +0.578337 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{10} + 3 q^{12} - 6 q^{13} + q^{14} + 3 q^{15} + 3 q^{16} - 2 q^{17} + 3 q^{18} - 6 q^{19} + 3 q^{20} + q^{21} + 3 q^{23} + 3 q^{24} + 2 q^{25} - 6 q^{26} + 3 q^{27} + q^{28} + 6 q^{29} + 3 q^{30} - 13 q^{31} + 3 q^{32} - 2 q^{34} - 3 q^{35} + 3 q^{36} - 7 q^{37} - 6 q^{38} - 6 q^{39} + 3 q^{40} + 5 q^{41} + q^{42} - q^{43} + 3 q^{45} + 3 q^{46} - 10 q^{47} + 3 q^{48} - 12 q^{49} + 2 q^{50} - 2 q^{51} - 6 q^{52} + 15 q^{53} + 3 q^{54} - 28 q^{55} + q^{56} - 6 q^{57} + 6 q^{58} + 3 q^{59} + 3 q^{60} - 13 q^{62} + q^{63} + 3 q^{64} - 8 q^{65} + 3 q^{67} - 2 q^{68} + 3 q^{69} - 3 q^{70} + 6 q^{71} + 3 q^{72} - q^{73} - 7 q^{74} + 2 q^{75} - 6 q^{76} + 8 q^{77} - 6 q^{78} - 26 q^{79} + 3 q^{80} + 3 q^{81} + 5 q^{82} - 3 q^{83} + q^{84} - 22 q^{85} - q^{86} + 6 q^{87} + 16 q^{89} + 3 q^{90} - 24 q^{91} + 3 q^{92} - 13 q^{93} - 10 q^{94} - 6 q^{95} + 3 q^{96} + 4 q^{97} - 12 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + q^7 + 3 * q^8 + 3 * q^9 + 3 * q^10 + 3 * q^12 - 6 * q^13 + q^14 + 3 * q^15 + 3 * q^16 - 2 * q^17 + 3 * q^18 - 6 * q^19 + 3 * q^20 + q^21 + 3 * q^23 + 3 * q^24 + 2 * q^25 - 6 * q^26 + 3 * q^27 + q^28 + 6 * q^29 + 3 * q^30 - 13 * q^31 + 3 * q^32 - 2 * q^34 - 3 * q^35 + 3 * q^36 - 7 * q^37 - 6 * q^38 - 6 * q^39 + 3 * q^40 + 5 * q^41 + q^42 - q^43 + 3 * q^45 + 3 * q^46 - 10 * q^47 + 3 * q^48 - 12 * q^49 + 2 * q^50 - 2 * q^51 - 6 * q^52 + 15 * q^53 + 3 * q^54 - 28 * q^55 + q^56 - 6 * q^57 + 6 * q^58 + 3 * q^59 + 3 * q^60 - 13 * q^62 + q^63 + 3 * q^64 - 8 * q^65 + 3 * q^67 - 2 * q^68 + 3 * q^69 - 3 * q^70 + 6 * q^71 + 3 * q^72 - q^73 - 7 * q^74 + 2 * q^75 - 6 * q^76 + 8 * q^77 - 6 * q^78 - 26 * q^79 + 3 * q^80 + 3 * q^81 + 5 * q^82 - 3 * q^83 + q^84 - 22 * q^85 - q^86 + 6 * q^87 + 16 * q^89 + 3 * q^90 - 24 * q^91 + 3 * q^92 - 13 * q^93 - 10 * q^94 - 6 * q^95 + 3 * q^96 + 4 * q^97 - 12 * q^98

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