LMFDB - Embedded newform 1334.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1334,2,Mod(1,1334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1334, 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("1334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1334 = 2 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1334.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:	$10.6520436296$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 22x^{6} + 151x^{4} - 332x^{2} + 6 $ x^8 - 22x^6 + 151x^4 - 332*x^2 + 6
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			1.1
Root			$2.27484$ of defining polynomial
Character	$\chi$	$=$	1334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.52651 q^{3} +1.00000 q^{4} -2.27484 q^{5} -2.52651 q^{6} +4.02057 q^{7} +1.00000 q^{8} +3.38328 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.52651 q^{3} +1.00000 q^{4} -2.27484 q^{5} -2.52651 q^{6} +4.02057 q^{7} +1.00000 q^{8} +3.38328 q^{9} -2.27484 q^{10} -4.07336 q^{11} -2.52651 q^{12} -1.05221 q^{13} +4.02057 q^{14} +5.74741 q^{15} +1.00000 q^{16} +3.99144 q^{17} +3.38328 q^{18} -0.689507 q^{19} -2.27484 q^{20} -10.1580 q^{21} -4.07336 q^{22} -1.00000 q^{23} -2.52651 q^{24} +0.174882 q^{25} -1.05221 q^{26} -0.968356 q^{27} +4.02057 q^{28} -1.00000 q^{29} +5.74741 q^{30} +4.06312 q^{31} +1.00000 q^{32} +10.2914 q^{33} +3.99144 q^{34} -9.14614 q^{35} +3.38328 q^{36} -1.35844 q^{37} -0.689507 q^{38} +2.65843 q^{39} -2.27484 q^{40} +5.36868 q^{41} -10.1580 q^{42} +9.08525 q^{43} -4.07336 q^{44} -7.69640 q^{45} -1.00000 q^{46} +7.52935 q^{47} -2.52651 q^{48} +9.16498 q^{49} +0.174882 q^{50} -10.0844 q^{51} -1.05221 q^{52} +11.7594 q^{53} -0.968356 q^{54} +9.26622 q^{55} +4.02057 q^{56} +1.74205 q^{57} -1.00000 q^{58} +8.29105 q^{59} +5.74741 q^{60} -0.914342 q^{61} +4.06312 q^{62} +13.6027 q^{63} +1.00000 q^{64} +2.39361 q^{65} +10.2914 q^{66} +6.25512 q^{67} +3.99144 q^{68} +2.52651 q^{69} -9.14614 q^{70} -13.2015 q^{71} +3.38328 q^{72} +8.81901 q^{73} -1.35844 q^{74} -0.441841 q^{75} -0.689507 q^{76} -16.3772 q^{77} +2.65843 q^{78} +2.41240 q^{79} -2.27484 q^{80} -7.70327 q^{81} +5.36868 q^{82} -3.22619 q^{83} -10.1580 q^{84} -9.07988 q^{85} +9.08525 q^{86} +2.52651 q^{87} -4.07336 q^{88} +5.58500 q^{89} -7.69640 q^{90} -4.23050 q^{91} -1.00000 q^{92} -10.2655 q^{93} +7.52935 q^{94} +1.56851 q^{95} -2.52651 q^{96} +1.81675 q^{97} +9.16498 q^{98} -13.7813 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{7} + 8 q^{8} + 12 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^6 + 8 * q^7 + 8 * q^8 + 12 * q^9	$ 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{7} + 8 q^{8} + 12 q^{9} + 8 q^{11} + 4 q^{12} + 4 q^{13} + 8 q^{14} + 8 q^{16} + 8 q^{17} + 12 q^{18} + 16 q^{19} - 4 q^{21} + 8 q^{22} - 8 q^{23} + 4 q^{24} + 4 q^{25} + 4 q^{26} + 4 q^{27} + 8 q^{28} - 8 q^{29} + 8 q^{31} + 8 q^{32} + 12 q^{33} + 8 q^{34} + 4 q^{35} + 12 q^{36} + 8 q^{37} + 16 q^{38} - 16 q^{39} + 8 q^{41} - 4 q^{42} + 32 q^{43} + 8 q^{44} - 8 q^{46} + 16 q^{47} + 4 q^{48} + 4 q^{49} + 4 q^{50} + 12 q^{51} + 4 q^{52} + 4 q^{54} + 8 q^{56} + 8 q^{57} - 8 q^{58} - 4 q^{59} + 8 q^{61} + 8 q^{62} + 24 q^{63} + 8 q^{64} - 4 q^{65} + 12 q^{66} + 20 q^{67} + 8 q^{68} - 4 q^{69} + 4 q^{70} - 24 q^{71} + 12 q^{72} - 4 q^{73} + 8 q^{74} - 16 q^{75} + 16 q^{76} - 16 q^{77} - 16 q^{78} + 4 q^{79} - 8 q^{81} + 8 q^{82} - 4 q^{83} - 4 q^{84} - 44 q^{85} + 32 q^{86} - 4 q^{87} + 8 q^{88} + 20 q^{89} - 8 q^{92} - 4 q^{93} + 16 q^{94} - 8 q^{95} + 4 q^{96} + 4 q^{97} + 4 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^6 + 8 * q^7 + 8 * q^8 + 12 * q^9 + 8 * q^11 + 4 * q^12 + 4 * q^13 + 8 * q^14 + 8 * q^16 + 8 * q^17 + 12 * q^18 + 16 * q^19 - 4 * q^21 + 8 * q^22 - 8 * q^23 + 4 * q^24 + 4 * q^25 + 4 * q^26 + 4 * q^27 + 8 * q^28 - 8 * q^29 + 8 * q^31 + 8 * q^32 + 12 * q^33 + 8 * q^34 + 4 * q^35 + 12 * q^36 + 8 * q^37 + 16 * q^38 - 16 * q^39 + 8 * q^41 - 4 * q^42 + 32 * q^43 + 8 * q^44 - 8 * q^46 + 16 * q^47 + 4 * q^48 + 4 * q^49 + 4 * q^50 + 12 * q^51 + 4 * q^52 + 4 * q^54 + 8 * q^56 + 8 * q^57 - 8 * q^58 - 4 * q^59 + 8 * q^61 + 8 * q^62 + 24 * q^63 + 8 * q^64 - 4 * q^65 + 12 * q^66 + 20 * q^67 + 8 * q^68 - 4 * q^69 + 4 * q^70 - 24 * q^71 + 12 * q^72 - 4 * q^73 + 8 * q^74 - 16 * q^75 + 16 * q^76 - 16 * q^77 - 16 * q^78 + 4 * q^79 - 8 * q^81 + 8 * q^82 - 4 * q^83 - 4 * q^84 - 44 * q^85 + 32 * q^86 - 4 * q^87 + 8 * q^88 + 20 * q^89 - 8 * q^92 - 4 * q^93 + 16 * q^94 - 8 * q^95 + 4 * q^96 + 4 * q^97 + 4 * 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
$3$	−2.52651	−1.45868	−0.729342	−	0.684149i	$-0.760174\pi$
$3$	−2.52651	−1.45868	−0.729342	+	0.684149i	$0.760174\pi$
$4$	1.00000	0.500000
$5$	−2.27484	−1.01734	−0.508669	−	0.860962i	$-0.669862\pi$
$5$	−2.27484	−1.01734	−0.508669	+	0.860962i	$0.669862\pi$
$6$	−2.52651	−1.03145
$7$	4.02057	1.51963	0.759816	−	0.650138i	$-0.225289\pi$
$7$	4.02057	1.51963	0.759816	+	0.650138i	$0.225289\pi$
$8$	1.00000	0.353553
$9$	3.38328	1.12776
$10$	−2.27484	−0.719367
$11$	−4.07336	−1.22816	−0.614081	−	0.789243i	$-0.710473\pi$
$11$	−4.07336	−1.22816	−0.614081	+	0.789243i	$0.710473\pi$
$12$	−2.52651	−0.729342
$13$	−1.05221	−0.291832	−0.145916	−	0.989297i	$-0.546613\pi$
$13$	−1.05221	−0.291832	−0.145916	+	0.989297i	$0.546613\pi$
$14$	4.02057	1.07454
$15$	5.74741	1.48397
$16$	1.00000	0.250000
$17$	3.99144	0.968067	0.484034	−	0.875049i	$-0.339171\pi$
$17$	3.99144	0.968067	0.484034	+	0.875049i	$0.339171\pi$
$18$	3.38328	0.797446
$19$	−0.689507	−0.158184	−0.0790918	−	0.996867i	$-0.525202\pi$
$19$	−0.689507	−0.158184	−0.0790918	+	0.996867i	$0.525202\pi$
$20$	−2.27484	−0.508669
$21$	−10.1580	−2.21666
$22$	−4.07336	−0.868442
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−2.52651	−0.515723
$25$	0.174882	0.0349763
$26$	−1.05221	−0.206356
$27$	−0.968356	−0.186360
$28$	4.02057	0.759816
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	5.74741	1.04933
$31$	4.06312	0.729759	0.364879	−	0.931055i	$-0.381110\pi$
$31$	4.06312	0.729759	0.364879	+	0.931055i	$0.381110\pi$
$32$	1.00000	0.176777
$33$	10.2914	1.79150
$34$	3.99144	0.684527
$35$	−9.14614	−1.54598
$36$	3.38328	0.563880
$37$	−1.35844	−0.223327	−0.111663	−	0.993746i	$-0.535618\pi$
$37$	−1.35844	−0.223327	−0.111663	+	0.993746i	$0.535618\pi$
$38$	−0.689507	−0.111853
$39$	2.65843	0.425690
$40$	−2.27484	−0.359683
$41$	5.36868	0.838446	0.419223	−	0.907883i	$-0.362303\pi$
$41$	5.36868	0.838446	0.419223	+	0.907883i	$0.362303\pi$
$42$	−10.1580	−1.56742
$43$	9.08525	1.38549	0.692743	−	0.721184i	$-0.256402\pi$
$43$	9.08525	1.38549	0.692743	+	0.721184i	$0.256402\pi$
$44$	−4.07336	−0.614081
$45$	−7.69640	−1.14731
$46$	−1.00000	−0.147442
$47$	7.52935	1.09827	0.549134	−	0.835734i	$-0.314958\pi$
$47$	7.52935	1.09827	0.549134	+	0.835734i	$0.314958\pi$
$48$	−2.52651	−0.364671
$49$	9.16498	1.30928
$50$	0.174882	0.0247320
$51$	−10.0844	−1.41210
$52$	−1.05221	−0.145916
$53$	11.7594	1.61528	0.807640	−	0.589675i	$-0.200744\pi$
$53$	11.7594	1.61528	0.807640	+	0.589675i	$0.200744\pi$
$54$	−0.968356	−0.131777
$55$	9.26622	1.24946
$56$	4.02057	0.537271
$57$	1.74205	0.230740
$58$	−1.00000	−0.131306
$59$	8.29105	1.07940	0.539702	−	0.841856i	$-0.318537\pi$
$59$	8.29105	1.07940	0.539702	+	0.841856i	$0.318537\pi$
$60$	5.74741	0.741987
$61$	−0.914342	−0.117069	−0.0585347	−	0.998285i	$-0.518643\pi$
$61$	−0.914342	−0.117069	−0.0585347	+	0.998285i	$0.518643\pi$
$62$	4.06312	0.516017
$63$	13.6027	1.71378
$64$	1.00000	0.125000
$65$	2.39361	0.296891
$66$	10.2914	1.26678
$67$	6.25512	0.764184	0.382092	−	0.924124i	$-0.375204\pi$
$67$	6.25512	0.764184	0.382092	+	0.924124i	$0.375204\pi$
$68$	3.99144	0.484034
$69$	2.52651	0.304157
$70$	−9.14614	−1.09317
$71$	−13.2015	−1.56673	−0.783366	−	0.621561i	$-0.786499\pi$
$71$	−13.2015	−1.56673	−0.783366	+	0.621561i	$0.786499\pi$
$72$	3.38328	0.398723
$73$	8.81901	1.03219	0.516094	−	0.856532i	$-0.327386\pi$
$73$	8.81901	1.03219	0.516094	+	0.856532i	$0.327386\pi$
$74$	−1.35844	−0.157916
$75$	−0.441841	−0.0510194
$76$	−0.689507	−0.0790918
$77$	−16.3772	−1.86636
$78$	2.65843	0.301008
$79$	2.41240	0.271417	0.135708	−	0.990749i	$-0.456669\pi$
$79$	2.41240	0.271417	0.135708	+	0.990749i	$0.456669\pi$
$80$	−2.27484	−0.254334
$81$	−7.70327	−0.855918
$82$	5.36868	0.592871
$83$	−3.22619	−0.354120	−0.177060	−	0.984200i	$-0.556659\pi$
$83$	−3.22619	−0.354120	−0.177060	+	0.984200i	$0.556659\pi$
$84$	−10.1580	−1.10833
$85$	−9.07988	−0.984851
$86$	9.08525	0.979687
$87$	2.52651	0.270871
$88$	−4.07336	−0.434221
$89$	5.58500	0.592009	0.296005	−	0.955187i	$-0.404346\pi$
$89$	5.58500	0.592009	0.296005	+	0.955187i	$0.404346\pi$
$90$	−7.69640	−0.811272
$91$	−4.23050	−0.443477
$92$	−1.00000	−0.104257
$93$	−10.2655	−1.06449
$94$	7.52935	0.776593
$95$	1.56851	0.160926
$96$	−2.52651	−0.257861
$97$	1.81675	0.184463	0.0922317	−	0.995738i	$-0.470600\pi$
$97$	1.81675	0.184463	0.0922317	+	0.995738i	$0.470600\pi$
$98$	9.16498	0.925803
$99$	−13.7813	−1.38507
$100$	0.174882	0.0174882
$101$	5.18253	0.515681	0.257840	−	0.966188i	$-0.416989\pi$
$101$	5.18253	0.515681	0.257840	+	0.966188i	$0.416989\pi$
$102$	−10.0844	−0.998508
$103$	17.2723	1.70189	0.850944	−	0.525256i	$-0.176030\pi$
$103$	17.2723	1.70189	0.850944	+	0.525256i	$0.176030\pi$
$104$	−1.05221	−0.103178
$105$	23.1079	2.25510
$106$	11.7594	1.14218
$107$	−3.73052	−0.360643	−0.180322	−	0.983608i	$-0.557714\pi$
$107$	−3.73052	−0.360643	−0.180322	+	0.983608i	$0.557714\pi$
$108$	−0.968356	−0.0931801
$109$	1.84901	0.177103	0.0885513	−	0.996072i	$-0.471776\pi$
$109$	1.84901	0.177103	0.0885513	+	0.996072i	$0.471776\pi$
$110$	9.26622	0.883499
$111$	3.43213	0.325763
$112$	4.02057	0.379908
$113$	−14.8401	−1.39604	−0.698020	−	0.716079i	$-0.745935\pi$
$113$	−14.8401	−1.39604	−0.698020	+	0.716079i	$0.745935\pi$
$114$	1.74205	0.163158
$115$	2.27484	0.212130
$116$	−1.00000	−0.0928477
$117$	−3.55993	−0.329116
$118$	8.29105	0.763253
$119$	16.0479	1.47111
$120$	5.74741	0.524664
$121$	5.59223	0.508384
$122$	−0.914342	−0.0827806
$123$	−13.5640	−1.22303
$124$	4.06312	0.364879
$125$	10.9764	0.981755
$126$	13.6027	1.21183
$127$	3.38925	0.300747	0.150373	−	0.988629i	$-0.451952\pi$
$127$	3.38925	0.300747	0.150373	+	0.988629i	$0.451952\pi$
$128$	1.00000	0.0883883
$129$	−22.9540	−2.02099
$130$	2.39361	0.209934
$131$	4.51475	0.394455	0.197228	−	0.980358i	$-0.436806\pi$
$131$	4.51475	0.394455	0.197228	+	0.980358i	$0.436806\pi$
$132$	10.2914	0.895751
$133$	−2.77221	−0.240381
$134$	6.25512	0.540360
$135$	2.20285	0.189591
$136$	3.99144	0.342263
$137$	−5.07647	−0.433712	−0.216856	−	0.976204i	$-0.569580\pi$
$137$	−5.07647	−0.433712	−0.216856	+	0.976204i	$0.569580\pi$
$138$	2.52651	0.215071
$139$	13.4970	1.14480	0.572402	−	0.819973i	$-0.306012\pi$
$139$	13.4970	1.14480	0.572402	+	0.819973i	$0.306012\pi$
$140$	−9.14614	−0.772990
$141$	−19.0230	−1.60203
$142$	−13.2015	−1.10785
$143$	4.28604	0.358417
$144$	3.38328	0.281940
$145$	2.27484	0.188915
$146$	8.81901	0.729867
$147$	−23.1555	−1.90983
$148$	−1.35844	−0.111663
$149$	−21.2633	−1.74196	−0.870980	−	0.491319i	$-0.836515\pi$
$149$	−21.2633	−1.74196	−0.870980	+	0.491319i	$0.836515\pi$
$150$	−0.441841	−0.0360762
$151$	15.9617	1.29895	0.649473	−	0.760385i	$-0.274990\pi$
$151$	15.9617	1.29895	0.649473	+	0.760385i	$0.274990\pi$
$152$	−0.689507	−0.0559264
$153$	13.5042	1.09175
$154$	−16.3772	−1.31971
$155$	−9.24294	−0.742411
$156$	2.65843	0.212845
$157$	−12.8418	−1.02489	−0.512444	−	0.858721i	$-0.671260\pi$
$157$	−12.8418	−1.02489	−0.512444	+	0.858721i	$0.671260\pi$
$158$	2.41240	0.191921
$159$	−29.7104	−2.35618
$160$	−2.27484	−0.179842
$161$	−4.02057	−0.316865
$162$	−7.70327	−0.605226
$163$	21.5471	1.68770	0.843850	−	0.536579i	$-0.180284\pi$
$163$	21.5471	1.68770	0.843850	+	0.536579i	$0.180284\pi$
$164$	5.36868	0.419223
$165$	−23.4112	−1.82256
$166$	−3.22619	−0.250401
$167$	−19.2848	−1.49230	−0.746152	−	0.665775i	$-0.768101\pi$
$167$	−19.2848	−1.49230	−0.746152	+	0.665775i	$0.768101\pi$
$168$	−10.1580	−0.783709
$169$	−11.8928	−0.914834
$170$	−9.07988	−0.696395
$171$	−2.33279	−0.178393
$172$	9.08525	0.692743
$173$	−16.9817	−1.29110	−0.645548	−	0.763720i	$-0.723371\pi$
$173$	−16.9817	−1.29110	−0.645548	+	0.763720i	$0.723371\pi$
$174$	2.52651	0.191535
$175$	0.703124	0.0531512
$176$	−4.07336	−0.307041
$177$	−20.9475	−1.57451
$178$	5.58500	0.418614
$179$	−21.1702	−1.58233	−0.791166	−	0.611601i	$-0.790526\pi$
$179$	−21.1702	−1.58233	−0.791166	+	0.611601i	$0.790526\pi$
$180$	−7.69640	−0.573656
$181$	2.84275	0.211300	0.105650	−	0.994403i	$-0.466308\pi$
$181$	2.84275	0.211300	0.105650	+	0.994403i	$0.466308\pi$
$182$	−4.23050	−0.313586
$183$	2.31010	0.170767
$184$	−1.00000	−0.0737210
$185$	3.09024	0.227199
$186$	−10.2655	−0.752706
$187$	−16.2586	−1.18894
$188$	7.52935	0.549134
$189$	−3.89334	−0.283199
$190$	1.56851	0.113792
$191$	−2.92985	−0.211996	−0.105998	−	0.994366i	$-0.533804\pi$
$191$	−2.92985	−0.211996	−0.105998	+	0.994366i	$0.533804\pi$
$192$	−2.52651	−0.182336
$193$	−12.7704	−0.919232	−0.459616	−	0.888118i	$-0.652013\pi$
$193$	−12.7704	−0.919232	−0.459616	+	0.888118i	$0.652013\pi$
$194$	1.81675	0.130435
$195$	−6.04750	−0.433071
$196$	9.16498	0.654642
$197$	−19.9492	−1.42132	−0.710659	−	0.703537i	$-0.751603\pi$
$197$	−19.9492	−1.42132	−0.710659	+	0.703537i	$0.751603\pi$
$198$	−13.7813	−0.979394
$199$	14.5519	1.03156	0.515778	−	0.856722i	$-0.327503\pi$
$199$	14.5519	1.03156	0.515778	+	0.856722i	$0.327503\pi$
$200$	0.174882	0.0123660
$201$	−15.8037	−1.11470
$202$	5.18253	0.364641
$203$	−4.02057	−0.282189
$204$	−10.0844	−0.706052
$205$	−12.2129	−0.852983
$206$	17.2723	1.20342
$207$	−3.38328	−0.235154
$208$	−1.05221	−0.0729579
$209$	2.80861	0.194275
$210$	23.1079	1.59459
$211$	13.6891	0.942400	0.471200	−	0.882026i	$-0.343821\pi$
$211$	13.6891	0.942400	0.471200	+	0.882026i	$0.343821\pi$
$212$	11.7594	0.807640
$213$	33.3538	2.28537
$214$	−3.73052	−0.255013
$215$	−20.6674	−1.40951
$216$	−0.968356	−0.0658883
$217$	16.3361	1.10896
$218$	1.84901	0.125230
$219$	−22.2814	−1.50564
$220$	9.26622	0.624728
$221$	−4.19985	−0.282513
$222$	3.43213	0.230349
$223$	3.29448	0.220615	0.110307	−	0.993898i	$-0.464816\pi$
$223$	3.29448	0.220615	0.110307	+	0.993898i	$0.464816\pi$
$224$	4.02057	0.268636
$225$	0.591673	0.0394449
$226$	−14.8401	−0.987149
$227$	−0.814251	−0.0540437	−0.0270219	−	0.999635i	$-0.508602\pi$
$227$	−0.814251	−0.0540437	−0.0270219	+	0.999635i	$0.508602\pi$
$228$	1.74205	0.115370
$229$	19.2836	1.27429	0.637147	−	0.770742i	$-0.280114\pi$
$229$	19.2836	1.27429	0.637147	+	0.770742i	$0.280114\pi$
$230$	2.27484	0.149998
$231$	41.3773	2.72242
$232$	−1.00000	−0.0656532
$233$	22.5407	1.47669	0.738344	−	0.674425i	$-0.235608\pi$
$233$	22.5407	1.47669	0.738344	+	0.674425i	$0.235608\pi$
$234$	−3.55993	−0.232720
$235$	−17.1280	−1.11731
$236$	8.29105	0.539702
$237$	−6.09498	−0.395911
$238$	16.0479	1.04023
$239$	0.784794	0.0507641	0.0253821	−	0.999678i	$-0.491920\pi$
$239$	0.784794	0.0507641	0.0253821	+	0.999678i	$0.491920\pi$
$240$	5.74741	0.370994
$241$	5.84106	0.376256	0.188128	−	0.982145i	$-0.439758\pi$
$241$	5.84106	0.376256	0.188128	+	0.982145i	$0.439758\pi$
$242$	5.59223	0.359482
$243$	22.3675	1.43487
$244$	−0.914342	−0.0585347
$245$	−20.8488	−1.33198
$246$	−13.5640	−0.864812
$247$	0.725508	0.0461630
$248$	4.06312	0.258009
$249$	8.15101	0.516549
$250$	10.9764	0.694206
$251$	−22.4973	−1.42002	−0.710009	−	0.704193i	$-0.751309\pi$
$251$	−22.4973	−1.42002	−0.710009	+	0.704193i	$0.751309\pi$
$252$	13.6027	0.856890
$253$	4.07336	0.256090
$254$	3.38925	0.212660
$255$	22.9405	1.43659
$256$	1.00000	0.0625000
$257$	−27.5413	−1.71798	−0.858991	−	0.511991i	$-0.828908\pi$
$257$	−27.5413	−1.71798	−0.858991	+	0.511991i	$0.828908\pi$
$258$	−22.9540	−1.42905
$259$	−5.46172	−0.339374
$260$	2.39361	0.148446
$261$	−3.38328	−0.209420
$262$	4.51475	0.278922
$263$	29.5706	1.82340	0.911702	−	0.410852i	$-0.134769\pi$
$263$	29.5706	1.82340	0.911702	+	0.410852i	$0.134769\pi$
$264$	10.2914	0.633391
$265$	−26.7508	−1.64329
$266$	−2.77221	−0.169975
$267$	−14.1106	−0.863554
$268$	6.25512	0.382092
$269$	15.9807	0.974361	0.487180	−	0.873301i	$-0.338025\pi$
$269$	15.9807	0.974361	0.487180	+	0.873301i	$0.338025\pi$
$270$	2.20285	0.134061
$271$	3.88599	0.236057	0.118029	−	0.993010i	$-0.462343\pi$
$271$	3.88599	0.236057	0.118029	+	0.993010i	$0.462343\pi$
$272$	3.99144	0.242017
$273$	10.6884	0.646893
$274$	−5.07647	−0.306681
$275$	−0.712355	−0.0429566
$276$	2.52651	0.152078
$277$	−16.4468	−0.988193	−0.494096	−	0.869407i	$-0.664501\pi$
$277$	−16.4468	−0.988193	−0.494096	+	0.869407i	$0.664501\pi$
$278$	13.4970	0.809499
$279$	13.7467	0.822992
$280$	−9.14614	−0.546586
$281$	8.75934	0.522539	0.261269	−	0.965266i	$-0.415859\pi$
$281$	8.75934	0.522539	0.261269	+	0.965266i	$0.415859\pi$
$282$	−19.0230	−1.13280
$283$	1.60867	0.0956256	0.0478128	−	0.998856i	$-0.484775\pi$
$283$	1.60867	0.0956256	0.0478128	+	0.998856i	$0.484775\pi$
$284$	−13.2015	−0.783366
$285$	−3.96288	−0.234740
$286$	4.28604	0.253439
$287$	21.5851	1.27413
$288$	3.38328	0.199362
$289$	−1.06838	−0.0628461
$290$	2.27484	0.133583
$291$	−4.59006	−0.269074
$292$	8.81901	0.516094
$293$	2.90474	0.169697	0.0848484	−	0.996394i	$-0.472959\pi$
$293$	2.90474	0.169697	0.0848484	+	0.996394i	$0.472959\pi$
$294$	−23.1555	−1.35045
$295$	−18.8608	−1.09812
$296$	−1.35844	−0.0789579
$297$	3.94446	0.228881
$298$	−21.2633	−1.23175
$299$	1.05221	0.0608511
$300$	−0.441841	−0.0255097
$301$	36.5279	2.10543
$302$	15.9617	0.918494
$303$	−13.0937	−0.752215
$304$	−0.689507	−0.0395459
$305$	2.07998	0.119099
$306$	13.5042	0.771981
$307$	−5.78051	−0.329911	−0.164956	−	0.986301i	$-0.552748\pi$
$307$	−5.78051	−0.329911	−0.164956	+	0.986301i	$0.552748\pi$
$308$	−16.3772	−0.933178
$309$	−43.6387	−2.48252
$310$	−9.24294	−0.524964
$311$	−12.4524	−0.706109	−0.353054	−	0.935603i	$-0.614857\pi$
$311$	−12.4524	−0.706109	−0.353054	+	0.935603i	$0.614857\pi$
$312$	2.65843	0.150504
$313$	−25.0009	−1.41313	−0.706567	−	0.707646i	$-0.749757\pi$
$313$	−25.0009	−1.41313	−0.706567	+	0.707646i	$0.749757\pi$
$314$	−12.8418	−0.724705
$315$	−30.9439	−1.74349
$316$	2.41240	0.135708
$317$	11.2165	0.629979	0.314989	−	0.949095i	$-0.397999\pi$
$317$	11.2165	0.629979	0.314989	+	0.949095i	$0.397999\pi$
$318$	−29.7104	−1.66607
$319$	4.07336	0.228064
$320$	−2.27484	−0.127167
$321$	9.42522	0.526065
$322$	−4.02057	−0.224058
$323$	−2.75213	−0.153132
$324$	−7.70327	−0.427959
$325$	−0.184013	−0.0102072
$326$	21.5471	1.19338
$327$	−4.67154	−0.258337
$328$	5.36868	0.296436
$329$	30.2723	1.66896
$330$	−23.4112	−1.28875
$331$	22.3992	1.23117	0.615586	−	0.788070i	$-0.288920\pi$
$331$	22.3992	1.23117	0.615586	+	0.788070i	$0.288920\pi$
$332$	−3.22619	−0.177060
$333$	−4.59599	−0.251859
$334$	−19.2848	−1.05522
$335$	−14.2294	−0.777434
$336$	−10.1580	−0.554166
$337$	−31.8547	−1.73523	−0.867617	−	0.497233i	$-0.834350\pi$
$337$	−31.8547	−1.73523	−0.867617	+	0.497233i	$0.834350\pi$
$338$	−11.8928	−0.646886
$339$	37.4937	2.03638
$340$	−9.07988	−0.492426
$341$	−16.5505	−0.896262
$342$	−2.33279	−0.126143
$343$	8.70446	0.469997
$344$	9.08525	0.489844
$345$	−5.74741	−0.309430
$346$	−16.9817	−0.912943
$347$	29.6019	1.58911	0.794555	−	0.607192i	$-0.207704\pi$
$347$	29.6019	1.58911	0.794555	+	0.607192i	$0.207704\pi$
$348$	2.52651	0.135435
$349$	−4.33893	−0.232257	−0.116129	−	0.993234i	$-0.537049\pi$
$349$	−4.33893	−0.232257	−0.116129	+	0.993234i	$0.537049\pi$
$350$	0.703124	0.0375835
$351$	1.01892	0.0543858
$352$	−4.07336	−0.217111
$353$	8.06528	0.429271	0.214636	−	0.976694i	$-0.431144\pi$
$353$	8.06528	0.429271	0.214636	+	0.976694i	$0.431144\pi$
$354$	−20.9475	−1.11335
$355$	30.0313	1.59390
$356$	5.58500	0.296005
$357$	−40.5452	−2.14588
$358$	−21.1702	−1.11888
$359$	−1.81149	−0.0956067	−0.0478033	−	0.998857i	$-0.515222\pi$
$359$	−1.81149	−0.0956067	−0.0478033	+	0.998857i	$0.515222\pi$
$360$	−7.69640	−0.405636
$361$	−18.5246	−0.974978
$362$	2.84275	0.149411
$363$	−14.1288	−0.741572
$364$	−4.23050	−0.221738
$365$	−20.0618	−1.05008
$366$	2.31010	0.120751
$367$	13.9560	0.728497	0.364248	−	0.931302i	$-0.381326\pi$
$367$	13.9560	0.728497	0.364248	+	0.931302i	$0.381326\pi$
$368$	−1.00000	−0.0521286
$369$	18.1637	0.945565
$370$	3.09024	0.160654
$371$	47.2796	2.45463
$372$	−10.2655	−0.532244
$373$	34.2974	1.77585	0.887926	−	0.459985i	$-0.152145\pi$
$373$	34.2974	1.77585	0.887926	+	0.459985i	$0.152145\pi$
$374$	−16.2586	−0.840710
$375$	−27.7319	−1.43207
$376$	7.52935	0.388296
$377$	1.05221	0.0541918
$378$	−3.89334	−0.200252
$379$	−24.8293	−1.27539	−0.637697	−	0.770288i	$-0.720113\pi$
$379$	−24.8293	−1.27539	−0.637697	+	0.770288i	$0.720113\pi$
$380$	1.56851	0.0804631
$381$	−8.56298	−0.438695
$382$	−2.92985	−0.149904
$383$	−3.18958	−0.162980	−0.0814900	−	0.996674i	$-0.525968\pi$
$383$	−3.18958	−0.162980	−0.0814900	+	0.996674i	$0.525968\pi$
$384$	−2.52651	−0.128931
$385$	37.2555	1.89872
$386$	−12.7704	−0.649995
$387$	30.7379	1.56250
$388$	1.81675	0.0922317
$389$	32.1734	1.63126	0.815629	−	0.578576i	$-0.196391\pi$
$389$	32.1734	1.63126	0.815629	+	0.578576i	$0.196391\pi$
$390$	−6.04750	−0.306227
$391$	−3.99144	−0.201856
$392$	9.16498	0.462902
$393$	−11.4066	−0.575385
$394$	−19.9492	−1.00502
$395$	−5.48783	−0.276122
$396$	−13.7813	−0.692536
$397$	−29.4209	−1.47659	−0.738297	−	0.674476i	$-0.764370\pi$
$397$	−29.4209	−1.47659	−0.738297	+	0.674476i	$0.764370\pi$
$398$	14.5519	0.729420
$399$	7.00403	0.350640
$400$	0.174882	0.00874408
$401$	5.92449	0.295855	0.147927	−	0.988998i	$-0.452740\pi$
$401$	5.92449	0.295855	0.147927	+	0.988998i	$0.452740\pi$
$402$	−15.8037	−0.788214
$403$	−4.27528	−0.212967
$404$	5.18253	0.257840
$405$	17.5237	0.870758
$406$	−4.02057	−0.199538
$407$	5.53342	0.274282
$408$	−10.0844	−0.499254
$409$	2.13101	0.105372	0.0526859	−	0.998611i	$-0.483222\pi$
$409$	2.13101	0.105372	0.0526859	+	0.998611i	$0.483222\pi$
$410$	−12.2129	−0.603150
$411$	12.8258	0.632649
$412$	17.2723	0.850944
$413$	33.3348	1.64030
$414$	−3.38328	−0.166279
$415$	7.33905	0.360260
$416$	−1.05221	−0.0515890
$417$	−34.1005	−1.66991
$418$	2.80861	0.137373
$419$	−6.65029	−0.324888	−0.162444	−	0.986718i	$-0.551938\pi$
$419$	−6.65029	−0.324888	−0.162444	+	0.986718i	$0.551938\pi$
$420$	23.1079	1.12755
$421$	9.84556	0.479843	0.239922	−	0.970792i	$-0.422878\pi$
$421$	9.84556	0.479843	0.239922	+	0.970792i	$0.422878\pi$
$422$	13.6891	0.666377
$423$	25.4739	1.23858
$424$	11.7594	0.571088
$425$	0.698030	0.0338594
$426$	33.3538	1.61600
$427$	−3.67617	−0.177903
$428$	−3.73052	−0.180322
$429$	−10.8287	−0.522817
$430$	−20.6674	−0.996673
$431$	15.5478	0.748910	0.374455	−	0.927245i	$-0.377830\pi$
$431$	15.5478	0.748910	0.374455	+	0.927245i	$0.377830\pi$
$432$	−0.968356	−0.0465900
$433$	14.2095	0.682863	0.341431	−	0.939907i	$-0.389088\pi$
$433$	14.2095	0.682863	0.341431	+	0.939907i	$0.389088\pi$
$434$	16.3361	0.784157
$435$	−5.74741	−0.275567
$436$	1.84901	0.0885513
$437$	0.689507	0.0329836
$438$	−22.2814	−1.06464
$439$	26.3943	1.25973	0.629865	−	0.776704i	$-0.283110\pi$
$439$	26.3943	1.25973	0.629865	+	0.776704i	$0.283110\pi$
$440$	9.26622	0.441750
$441$	31.0077	1.47656
$442$	−4.19985	−0.199767
$443$	−9.69636	−0.460688	−0.230344	−	0.973109i	$-0.573985\pi$
$443$	−9.69636	−0.460688	−0.230344	+	0.973109i	$0.573985\pi$
$444$	3.43213	0.162882
$445$	−12.7050	−0.602273
$446$	3.29448	0.155998
$447$	53.7221	2.54097
$448$	4.02057	0.189954
$449$	2.70910	0.127850	0.0639251	−	0.997955i	$-0.479638\pi$
$449$	2.70910	0.127850	0.0639251	+	0.997955i	$0.479638\pi$
$450$	0.591673	0.0278917
$451$	−21.8685	−1.02975
$452$	−14.8401	−0.698020
$453$	−40.3275	−1.89475
$454$	−0.814251	−0.0382147
$455$	9.62370	0.451166
$456$	1.74205	0.0815789
$457$	−6.10816	−0.285728	−0.142864	−	0.989742i	$-0.545631\pi$
$457$	−6.10816	−0.285728	−0.142864	+	0.989742i	$0.545631\pi$
$458$	19.2836	0.901062
$459$	−3.86514	−0.180409
$460$	2.27484	0.106065
$461$	−23.0683	−1.07440	−0.537199	−	0.843456i	$-0.680518\pi$
$461$	−23.0683	−1.07440	−0.537199	+	0.843456i	$0.680518\pi$
$462$	41.3773	1.92504
$463$	−28.8606	−1.34126	−0.670632	−	0.741790i	$-0.733977\pi$
$463$	−28.8606	−1.34126	−0.670632	+	0.741790i	$0.733977\pi$
$464$	−1.00000	−0.0464238
$465$	23.3524	1.08294
$466$	22.5407	1.04418
$467$	−3.21695	−0.148863	−0.0744314	−	0.997226i	$-0.523714\pi$
$467$	−3.21695	−0.148863	−0.0744314	+	0.997226i	$0.523714\pi$
$468$	−3.55993	−0.164558
$469$	25.1491	1.16128
$470$	−17.1280	−0.790057
$471$	32.4450	1.49499
$472$	8.29105	0.381627
$473$	−37.0074	−1.70160
$474$	−6.09498	−0.279951
$475$	−0.120582	−0.00553268
$476$	16.0479	0.735553
$477$	39.7854	1.82165
$478$	0.784794	0.0358957
$479$	15.6438	0.714782	0.357391	−	0.933955i	$-0.383666\pi$
$479$	15.6438	0.714782	0.357391	+	0.933955i	$0.383666\pi$
$480$	5.74741	0.262332
$481$	1.42937	0.0651738
$482$	5.84106	0.266053
$483$	10.1580	0.462206
$484$	5.59223	0.254192
$485$	−4.13282	−0.187662
$486$	22.3675	1.01461
$487$	1.80307	0.0817051	0.0408525	−	0.999165i	$-0.486993\pi$
$487$	1.80307	0.0817051	0.0408525	+	0.999165i	$0.486993\pi$
$488$	−0.914342	−0.0413903
$489$	−54.4391	−2.46182
$490$	−20.8488	−0.941854
$491$	18.5655	0.837849	0.418924	−	0.908021i	$-0.362407\pi$
$491$	18.5655	0.837849	0.418924	+	0.908021i	$0.362407\pi$
$492$	−13.5640	−0.611514
$493$	−3.99144	−0.179766
$494$	0.725508	0.0326422
$495$	31.3502	1.40909
$496$	4.06312	0.182440
$497$	−53.0776	−2.38086
$498$	8.15101	0.365255
$499$	22.0728	0.988115	0.494058	−	0.869429i	$-0.335513\pi$
$499$	22.0728	0.988115	0.494058	+	0.869429i	$0.335513\pi$
$500$	10.9764	0.490878
$501$	48.7234	2.17680
$502$	−22.4973	−1.00410
$503$	−6.80660	−0.303491	−0.151746	−	0.988420i	$-0.548489\pi$
$503$	−6.80660	−0.303491	−0.151746	+	0.988420i	$0.548489\pi$
$504$	13.6027	0.605913
$505$	−11.7894	−0.524621
$506$	4.07336	0.181083
$507$	30.0475	1.33445
$508$	3.38925	0.150373
$509$	−41.8912	−1.85679	−0.928397	−	0.371589i	$-0.878813\pi$
$509$	−41.8912	−1.85679	−0.928397	+	0.371589i	$0.878813\pi$
$510$	22.9405	1.01582
$511$	35.4575	1.56855
$512$	1.00000	0.0441942
$513$	0.667688	0.0294791
$514$	−27.5413	−1.21480
$515$	−39.2916	−1.73140
$516$	−22.9540	−1.01049
$517$	−30.6697	−1.34885
$518$	−5.46172	−0.239974
$519$	42.9046	1.88330
$520$	2.39361	0.104967
$521$	−15.7045	−0.688026	−0.344013	−	0.938965i	$-0.611786\pi$
$521$	−15.7045	−0.688026	−0.344013	+	0.938965i	$0.611786\pi$
$522$	−3.38328	−0.148082
$523$	−2.31410	−0.101188	−0.0505942	−	0.998719i	$-0.516112\pi$
$523$	−2.31410	−0.101188	−0.0505942	+	0.998719i	$0.516112\pi$
$524$	4.51475	0.197228
$525$	−1.77645	−0.0775308
$526$	29.5706	1.28934
$527$	16.2177	0.706455
$528$	10.2914	0.447875
$529$	1.00000	0.0434783
$530$	−26.7508	−1.16198
$531$	28.0509	1.21731
$532$	−2.77221	−0.120191
$533$	−5.64900	−0.244685
$534$	−14.1106	−0.610625
$535$	8.48633	0.366896
$536$	6.25512	0.270180
$537$	53.4867	2.30812
$538$	15.9807	0.688977
$539$	−37.3322	−1.60801
$540$	2.20285	0.0947956
$541$	−36.0348	−1.54926	−0.774629	−	0.632416i	$-0.782064\pi$
$541$	−36.0348	−1.54926	−0.774629	+	0.632416i	$0.782064\pi$
$542$	3.88599	0.166918
$543$	−7.18224	−0.308219
$544$	3.99144	0.171132
$545$	−4.20619	−0.180173
$546$	10.6884	0.457422
$547$	−21.6288	−0.924782	−0.462391	−	0.886676i	$-0.653008\pi$
$547$	−21.6288	−0.924782	−0.462391	+	0.886676i	$0.653008\pi$
$548$	−5.07647	−0.216856
$549$	−3.09347	−0.132026
$550$	−0.712355	−0.0303749
$551$	0.689507	0.0293740
$552$	2.52651	0.107536
$553$	9.69924	0.412454
$554$	−16.4468	−0.698758
$555$	−7.80753	−0.331411
$556$	13.4970	0.572402
$557$	38.1450	1.61626	0.808128	−	0.589007i	$-0.200481\pi$
$557$	38.1450	1.61626	0.808128	+	0.589007i	$0.200481\pi$
$558$	13.7467	0.581943
$559$	−9.55962	−0.404329
$560$	−9.14614	−0.386495
$561$	41.0775	1.73429
$562$	8.75934	0.369491
$563$	−2.52486	−0.106410	−0.0532051	−	0.998584i	$-0.516944\pi$
$563$	−2.52486	−0.106410	−0.0532051	+	0.998584i	$0.516944\pi$
$564$	−19.0230	−0.801013
$565$	33.7588	1.42024
$566$	1.60867	0.0676175
$567$	−30.9715	−1.30068
$568$	−13.2015	−0.553923
$569$	7.49653	0.314271	0.157135	−	0.987577i	$-0.449774\pi$
$569$	7.49653	0.314271	0.157135	+	0.987577i	$0.449774\pi$
$570$	−3.96288	−0.165987
$571$	−16.1557	−0.676096	−0.338048	−	0.941129i	$-0.609767\pi$
$571$	−16.1557	−0.676096	−0.338048	+	0.941129i	$0.609767\pi$
$572$	4.28604	0.179208
$573$	7.40231	0.309236
$574$	21.5851	0.900946
$575$	−0.174882	−0.00729307
$576$	3.38328	0.140970
$577$	22.4322	0.933864	0.466932	−	0.884293i	$-0.345359\pi$
$577$	22.4322	0.933864	0.466932	+	0.884293i	$0.345359\pi$
$578$	−1.06838	−0.0444389
$579$	32.2646	1.34087
$580$	2.27484	0.0944574
$581$	−12.9711	−0.538132
$582$	−4.59006	−0.190264
$583$	−47.9003	−1.98383
$584$	8.81901	0.364933
$585$	8.09826	0.334822
$586$	2.90474	0.119994
$587$	27.1122	1.11904	0.559520	−	0.828817i	$-0.310985\pi$
$587$	27.1122	1.11904	0.559520	+	0.828817i	$0.310985\pi$
$588$	−23.1555	−0.954915
$589$	−2.80155	−0.115436
$590$	−18.8608	−0.776486
$591$	50.4018	2.07325
$592$	−1.35844	−0.0558317
$593$	0.429068	0.0176197	0.00880985	−	0.999961i	$-0.497196\pi$
$593$	0.429068	0.0176197	0.00880985	+	0.999961i	$0.497196\pi$
$594$	3.94446	0.161843
$595$	−36.5063	−1.49661
$596$	−21.2633	−0.870980
$597$	−36.7655	−1.50471
$598$	1.05221	0.0430282
$599$	−5.89072	−0.240688	−0.120344	−	0.992732i	$-0.538400\pi$
$599$	−5.89072	−0.240688	−0.120344	+	0.992732i	$0.538400\pi$
$600$	−0.441841	−0.0180381
$601$	−3.27175	−0.133457	−0.0667287	−	0.997771i	$-0.521256\pi$
$601$	−3.27175	−0.133457	−0.0667287	+	0.997771i	$0.521256\pi$
$602$	36.5279	1.48876
$603$	21.1628	0.861816
$604$	15.9617	0.649473
$605$	−12.7214	−0.517199
$606$	−13.0937	−0.531896
$607$	11.2539	0.456781	0.228391	−	0.973570i	$-0.426654\pi$
$607$	11.2539	0.456781	0.228391	+	0.973570i	$0.426654\pi$
$608$	−0.689507	−0.0279632
$609$	10.1580	0.411624
$610$	2.07998	0.0842158
$611$	−7.92248	−0.320509
$612$	13.5042	0.545873
$613$	−30.6985	−1.23990	−0.619951	−	0.784641i	$-0.712847\pi$
$613$	−30.6985	−1.23990	−0.619951	+	0.784641i	$0.712847\pi$
$614$	−5.78051	−0.233282
$615$	30.8560	1.24423
$616$	−16.3772	−0.659857
$617$	4.74705	0.191109	0.0955545	−	0.995424i	$-0.469538\pi$
$617$	4.74705	0.191109	0.0955545	+	0.995424i	$0.469538\pi$
$618$	−43.6387	−1.75540
$619$	20.3267	0.816999	0.408500	−	0.912758i	$-0.366052\pi$
$619$	20.3267	0.816999	0.408500	+	0.912758i	$0.366052\pi$
$620$	−9.24294	−0.371205
$621$	0.968356	0.0388588
$622$	−12.4524	−0.499294
$623$	22.4549	0.899636
$624$	2.65843	0.106423
$625$	−25.8438	−1.03375
$626$	−25.0009	−0.999237
$627$	−7.09598	−0.283386
$628$	−12.8418	−0.512444
$629$	−5.42215	−0.216195
$630$	−30.9439	−1.23284
$631$	−31.7367	−1.26342	−0.631710	−	0.775205i	$-0.717646\pi$
$631$	−31.7367	−1.26342	−0.631710	+	0.775205i	$0.717646\pi$
$632$	2.41240	0.0959603
$633$	−34.5858	−1.37466
$634$	11.2165	0.445462
$635$	−7.70998	−0.305961
$636$	−29.7104	−1.17809
$637$	−9.64352	−0.382090
$638$	4.07336	0.161266
$639$	−44.6644	−1.76690
$640$	−2.27484	−0.0899208
$641$	−7.71281	−0.304638	−0.152319	−	0.988331i	$-0.548674\pi$
$641$	−7.71281	−0.304638	−0.152319	+	0.988331i	$0.548674\pi$
$642$	9.42522	0.371984
$643$	20.2441	0.798351	0.399175	−	0.916875i	$-0.369296\pi$
$643$	20.2441	0.798351	0.399175	+	0.916875i	$0.369296\pi$
$644$	−4.02057	−0.158433
$645$	52.2166	2.05603
$646$	−2.75213	−0.108281
$647$	−2.25412	−0.0886187	−0.0443093	−	0.999018i	$-0.514109\pi$
$647$	−2.25412	−0.0886187	−0.0443093	+	0.999018i	$0.514109\pi$
$648$	−7.70327	−0.302613
$649$	−33.7724	−1.32568
$650$	−0.184013	−0.00721758
$651$	−41.2733	−1.61763
$652$	21.5471	0.843850
$653$	16.5593	0.648014	0.324007	−	0.946055i	$-0.394970\pi$
$653$	16.5593	0.648014	0.324007	+	0.946055i	$0.394970\pi$
$654$	−4.67154	−0.182672
$655$	−10.2703	−0.401294
$656$	5.36868	0.209612
$657$	29.8372	1.16406
$658$	30.2723	1.18014
$659$	35.5135	1.38341	0.691705	−	0.722180i	$-0.256860\pi$
$659$	35.5135	1.38341	0.691705	+	0.722180i	$0.256860\pi$
$660$	−23.4112	−0.911281
$661$	16.1777	0.629241	0.314620	−	0.949218i	$-0.398123\pi$
$661$	16.1777	0.629241	0.314620	+	0.949218i	$0.398123\pi$
$662$	22.3992	0.870569
$663$	10.6110	0.412097
$664$	−3.22619	−0.125200
$665$	6.30632	0.244549
$666$	−4.59599	−0.178091
$667$	1.00000	0.0387202
$668$	−19.2848	−0.746152
$669$	−8.32356	−0.321807
$670$	−14.2294	−0.549729
$671$	3.72444	0.143780
$672$	−10.1580	−0.391854
$673$	2.50055	0.0963892	0.0481946	−	0.998838i	$-0.484653\pi$
$673$	2.50055	0.0963892	0.0481946	+	0.998838i	$0.484653\pi$
$674$	−31.8547	−1.22700
$675$	−0.169348	−0.00651819
$676$	−11.8928	−0.457417
$677$	37.1307	1.42705	0.713524	−	0.700631i	$-0.247098\pi$
$677$	37.1307	1.42705	0.713524	+	0.700631i	$0.247098\pi$
$678$	37.4937	1.43994
$679$	7.30439	0.280317
$680$	−9.07988	−0.348198
$681$	2.05722	0.0788327
$682$	−16.5505	−0.633753
$683$	−24.7125	−0.945598	−0.472799	−	0.881170i	$-0.656756\pi$
$683$	−24.7125	−0.945598	−0.472799	+	0.881170i	$0.656756\pi$
$684$	−2.33279	−0.0891965
$685$	11.5481	0.441232
$686$	8.70446	0.332338
$687$	−48.7202	−1.85879
$688$	9.08525	0.346372
$689$	−12.3734	−0.471390
$690$	−5.74741	−0.218800
$691$	−24.7252	−0.940592	−0.470296	−	0.882509i	$-0.655853\pi$
$691$	−24.7252	−0.940592	−0.470296	+	0.882509i	$0.655853\pi$
$692$	−16.9817	−0.645548
$693$	−55.4086	−2.10480
$694$	29.6019	1.12367
$695$	−30.7036	−1.16465
$696$	2.52651	0.0957673
$697$	21.4288	0.811672
$698$	−4.33893	−0.164231
$699$	−56.9493	−2.15402
$700$	0.703124	0.0265756
$701$	−19.1423	−0.722993	−0.361497	−	0.932373i	$-0.617734\pi$
$701$	−19.1423	−0.722993	−0.361497	+	0.932373i	$0.617734\pi$
$702$	1.01892	0.0384566
$703$	0.936655	0.0353266
$704$	−4.07336	−0.153520
$705$	43.2742	1.62980
$706$	8.06528	0.303541
$707$	20.8367	0.783645
$708$	−20.9475	−0.787254
$709$	4.53425	0.170287	0.0851437	−	0.996369i	$-0.472865\pi$
$709$	4.53425	0.170287	0.0851437	+	0.996369i	$0.472865\pi$
$710$	30.0313	1.12705
$711$	8.16183	0.306093
$712$	5.58500	0.209307
$713$	−4.06312	−0.152165
$714$	−40.5452	−1.51737
$715$	−9.75004	−0.364631
$716$	−21.1702	−0.791166
$717$	−1.98279	−0.0740488
$718$	−1.81149	−0.0676041
$719$	42.7309	1.59359	0.796796	−	0.604248i	$-0.206527\pi$
$719$	42.7309	1.59359	0.796796	+	0.604248i	$0.206527\pi$
$720$	−7.69640	−0.286828
$721$	69.4444	2.58625
$722$	−18.5246	−0.689414
$723$	−14.7575	−0.548838
$724$	2.84275	0.105650
$725$	−0.174882	−0.00649494
$726$	−14.1288	−0.524371
$727$	−19.7517	−0.732549	−0.366274	−	0.930507i	$-0.619367\pi$
$727$	−19.7517	−0.732549	−0.366274	+	0.930507i	$0.619367\pi$
$728$	−4.23050	−0.156793
$729$	−33.4020	−1.23711
$730$	−20.0618	−0.742521
$731$	36.2632	1.34124
$732$	2.31010	0.0853837
$733$	−0.359746	−0.0132875	−0.00664375	−	0.999978i	$-0.502115\pi$
$733$	−0.359746	−0.0132875	−0.00664375	+	0.999978i	$0.502115\pi$
$734$	13.9560	0.515125
$735$	52.6749	1.94294
$736$	−1.00000	−0.0368605
$737$	−25.4793	−0.938543
$738$	18.1637	0.668616
$739$	28.7036	1.05588	0.527940	−	0.849282i	$-0.322965\pi$
$739$	28.7036	1.05588	0.527940	+	0.849282i	$0.322965\pi$
$740$	3.09024	0.113599
$741$	−1.83301	−0.0673372
$742$	47.2796	1.73569
$743$	−5.26224	−0.193053	−0.0965264	−	0.995330i	$-0.530773\pi$
$743$	−5.26224	−0.193053	−0.0965264	+	0.995330i	$0.530773\pi$
$744$	−10.2655	−0.376353
$745$	48.3706	1.77216
$746$	34.2974	1.25572
$747$	−10.9151	−0.399362
$748$	−16.2586	−0.594472
$749$	−14.9988	−0.548045
$750$	−27.7319	−1.01263
$751$	−11.0973	−0.404948	−0.202474	−	0.979288i	$-0.564898\pi$
$751$	−11.0973	−0.404948	−0.202474	+	0.979288i	$0.564898\pi$
$752$	7.52935	0.274567
$753$	56.8398	2.07136
$754$	1.05221	0.0383194
$755$	−36.3103	−1.32147
$756$	−3.89334	−0.141599
$757$	−26.3660	−0.958290	−0.479145	−	0.877736i	$-0.659053\pi$
$757$	−26.3660	−0.958290	−0.479145	+	0.877736i	$0.659053\pi$
$758$	−24.8293	−0.901839
$759$	−10.2914	−0.373554
$760$	1.56851	0.0568960
$761$	5.98128	0.216821	0.108411	−	0.994106i	$-0.465424\pi$
$761$	5.98128	0.216821	0.108411	+	0.994106i	$0.465424\pi$
$762$	−8.56298	−0.310204
$763$	7.43406	0.269131
$764$	−2.92985	−0.105998
$765$	−30.7198	−1.11068
$766$	−3.18958	−0.115244
$767$	−8.72396	−0.315004
$768$	−2.52651	−0.0911678
$769$	−8.69885	−0.313689	−0.156844	−	0.987623i	$-0.550132\pi$
$769$	−8.69885	−0.313689	−0.156844	+	0.987623i	$0.550132\pi$
$770$	37.2555	1.34259
$771$	69.5836	2.50599
$772$	−12.7704	−0.459616
$773$	40.9877	1.47422	0.737112	−	0.675770i	$-0.236189\pi$
$773$	40.9877	1.47422	0.737112	+	0.675770i	$0.236189\pi$
$774$	30.7379	1.10485
$775$	0.710566	0.0255243
$776$	1.81675	0.0652177
$777$	13.7991	0.495040
$778$	32.1734	1.15347
$779$	−3.70174	−0.132628
$780$	−6.04750	−0.216535
$781$	53.7745	1.92420
$782$	−3.99144	−0.142734
$783$	0.968356	0.0346062
$784$	9.16498	0.327321
$785$	29.2130	1.04266
$786$	−11.4066	−0.406859
$787$	−15.5011	−0.552555	−0.276277	−	0.961078i	$-0.589101\pi$
$787$	−15.5011	−0.552555	−0.276277	+	0.961078i	$0.589101\pi$
$788$	−19.9492	−0.710659
$789$	−74.7107	−2.65977
$790$	−5.48783	−0.195248
$791$	−59.6657	−2.12147
$792$	−13.7813	−0.489697
$793$	0.962083	0.0341646
$794$	−29.4209	−1.04411
$795$	67.5862	2.39704
$796$	14.5519	0.515778
$797$	33.1587	1.17454	0.587271	−	0.809391i	$-0.300202\pi$
$797$	33.1587	1.17454	0.587271	+	0.809391i	$0.300202\pi$
$798$	7.00403	0.247940
$799$	30.0530	1.06320
$800$	0.174882	0.00618300
$801$	18.8956	0.667644
$802$	5.92449	0.209201
$803$	−35.9230	−1.26769
$804$	−15.8037	−0.557352
$805$	9.14614	0.322359
$806$	−4.27528	−0.150590
$807$	−40.3755	−1.42128
$808$	5.18253	0.182321
$809$	21.2575	0.747372	0.373686	−	0.927555i	$-0.378094\pi$
$809$	21.2575	0.747372	0.373686	+	0.927555i	$0.378094\pi$
$810$	17.5237	0.615719
$811$	−2.22548	−0.0781471	−0.0390735	−	0.999236i	$-0.512441\pi$
$811$	−2.22548	−0.0781471	−0.0390735	+	0.999236i	$0.512441\pi$
$812$	−4.02057	−0.141094
$813$	−9.81802	−0.344333
$814$	5.53342	0.193946
$815$	−49.0162	−1.71696
$816$	−10.0844	−0.353026
$817$	−6.26434	−0.219161
$818$	2.13101	0.0745091
$819$	−14.3130	−0.500135
$820$	−12.2129	−0.426492
$821$	16.6110	0.579727	0.289864	−	0.957068i	$-0.406390\pi$
$821$	16.6110	0.579727	0.289864	+	0.957068i	$0.406390\pi$
$822$	12.8258	0.447350
$823$	−45.3385	−1.58040	−0.790200	−	0.612849i	$-0.790024\pi$
$823$	−45.3385	−1.58040	−0.790200	+	0.612849i	$0.790024\pi$
$824$	17.2723	0.601708
$825$	1.79978	0.0626601
$826$	33.3348	1.15986
$827$	−56.3247	−1.95860	−0.979301	−	0.202411i	$-0.935122\pi$
$827$	−56.3247	−1.95860	−0.979301	+	0.202411i	$0.935122\pi$
$828$	−3.38328	−0.117577
$829$	−13.2327	−0.459590	−0.229795	−	0.973239i	$-0.573805\pi$
$829$	−13.2327	−0.459590	−0.229795	+	0.973239i	$0.573805\pi$
$830$	7.33905	0.254742
$831$	41.5531	1.44146
$832$	−1.05221	−0.0364790
$833$	36.5815	1.26747
$834$	−34.1005	−1.18080
$835$	43.8698	1.51818
$836$	2.80861	0.0971376
$837$	−3.93455	−0.135998
$838$	−6.65029	−0.229730
$839$	−41.0303	−1.41652	−0.708262	−	0.705950i	$-0.750520\pi$
$839$	−41.0303	−1.41652	−0.708262	+	0.705950i	$0.750520\pi$
$840$	23.1079	0.797297
$841$	1.00000	0.0344828
$842$	9.84556	0.339300
$843$	−22.1306	−0.762219
$844$	13.6891	0.471200
$845$	27.0543	0.930696
$846$	25.4739	0.875810
$847$	22.4839	0.772557
$848$	11.7594	0.403820
$849$	−4.06433	−0.139488
$850$	0.698030	0.0239422
$851$	1.35844	0.0465668
$852$	33.3538	1.14268
$853$	−26.4252	−0.904781	−0.452391	−	0.891820i	$-0.649429\pi$
$853$	−26.4252	−0.904781	−0.452391	+	0.891820i	$0.649429\pi$
$854$	−3.67617	−0.125796
$855$	5.30672	0.181486
$856$	−3.73052	−0.127507
$857$	−49.4810	−1.69024	−0.845120	−	0.534577i	$-0.820471\pi$
$857$	−49.4810	−1.69024	−0.845120	+	0.534577i	$0.820471\pi$
$858$	−10.8287	−0.369687
$859$	−50.9114	−1.73707	−0.868537	−	0.495624i	$-0.834939\pi$
$859$	−50.9114	−1.73707	−0.868537	+	0.495624i	$0.834939\pi$
$860$	−20.6674	−0.704754
$861$	−54.5352	−1.85855
$862$	15.5478	0.529560
$863$	−32.9329	−1.12105	−0.560524	−	0.828138i	$-0.689400\pi$
$863$	−32.9329	−1.12105	−0.560524	+	0.828138i	$0.689400\pi$
$864$	−0.968356	−0.0329441
$865$	38.6306	1.31348
$866$	14.2095	0.482857
$867$	2.69929	0.0916726
$868$	16.3361	0.554482
$869$	−9.82658	−0.333344
$870$	−5.74741	−0.194855
$871$	−6.58172	−0.223013
$872$	1.84901	0.0626152
$873$	6.14658	0.208030
$874$	0.689507	0.0233229
$875$	44.1312	1.49191
$876$	−22.2814	−0.752818
$877$	−36.6807	−1.23862	−0.619310	−	0.785147i	$-0.712588\pi$
$877$	−36.6807	−1.23862	−0.619310	+	0.785147i	$0.712588\pi$
$878$	26.3943	0.890764
$879$	−7.33887	−0.247534
$880$	9.26622	0.312364
$881$	−25.0996	−0.845626	−0.422813	−	0.906217i	$-0.638957\pi$
$881$	−25.0996	−0.845626	−0.422813	+	0.906217i	$0.638957\pi$
$882$	31.0077	1.04408
$883$	7.85102	0.264208	0.132104	−	0.991236i	$-0.457827\pi$
$883$	7.85102	0.264208	0.132104	+	0.991236i	$0.457827\pi$
$884$	−4.19985	−0.141256
$885$	47.6521	1.60181
$886$	−9.69636	−0.325756
$887$	0.506084	0.0169926	0.00849632	−	0.999964i	$-0.497296\pi$
$887$	0.506084	0.0169926	0.00849632	+	0.999964i	$0.497296\pi$
$888$	3.43213	0.115175
$889$	13.6267	0.457025
$890$	−12.7050	−0.425871
$891$	31.3781	1.05121
$892$	3.29448	0.110307
$893$	−5.19153	−0.173728
$894$	53.7221	1.79674
$895$	48.1587	1.60977
$896$	4.02057	0.134318
$897$	−2.65843	−0.0887625
$898$	2.70910	0.0904038
$899$	−4.06312	−0.135513
$900$	0.591673	0.0197224
$901$	46.9371	1.56370
$902$	−21.8685	−0.728142
$903$	−92.2882	−3.07116
$904$	−14.8401	−0.493574
$905$	−6.46678	−0.214963
$906$	−40.3275	−1.33979
$907$	−9.14178	−0.303548	−0.151774	−	0.988415i	$-0.548499\pi$
$907$	−9.14178	−0.303548	−0.151774	+	0.988415i	$0.548499\pi$
$908$	−0.814251	−0.0270219
$909$	17.5339	0.581563
$910$	9.62370	0.319022
$911$	−34.8256	−1.15382	−0.576912	−	0.816806i	$-0.695742\pi$
$911$	−34.8256	−1.15382	−0.576912	+	0.816806i	$0.695742\pi$
$912$	1.74205	0.0576850
$913$	13.1414	0.434917
$914$	−6.10816	−0.202040
$915$	−5.25509	−0.173728
$916$	19.2836	0.637147
$917$	18.1518	0.599427
$918$	−3.86514	−0.127569
$919$	17.1860	0.566913	0.283456	−	0.958985i	$-0.408519\pi$
$919$	17.1860	0.566913	0.283456	+	0.958985i	$0.408519\pi$
$920$	2.27484	0.0749991
$921$	14.6045	0.481236
$922$	−23.0683	−0.759714
$923$	13.8908	0.457222
$924$	41.3773	1.36121
$925$	−0.237567	−0.00781115
$926$	−28.8606	−0.948417
$927$	58.4369	1.91932
$928$	−1.00000	−0.0328266
$929$	28.6870	0.941189	0.470594	−	0.882350i	$-0.344039\pi$
$929$	28.6870	0.941189	0.470594	+	0.882350i	$0.344039\pi$
$930$	23.3524	0.765756
$931$	−6.31931	−0.207107
$932$	22.5407	0.738344
$933$	31.4611	1.02999
$934$	−3.21695	−0.105262
$935$	36.9856	1.20956
$936$	−3.55993	−0.116360
$937$	18.3946	0.600926	0.300463	−	0.953793i	$-0.402859\pi$
$937$	18.3946	0.600926	0.300463	+	0.953793i	$0.402859\pi$
$938$	25.1491	0.821149
$939$	63.1652	2.06132
$940$	−17.1280	−0.558655
$941$	43.6462	1.42282	0.711412	−	0.702775i	$-0.248056\pi$
$941$	43.6462	1.42282	0.711412	+	0.702775i	$0.248056\pi$
$942$	32.4450	1.05712
$943$	−5.36868	−0.174828
$944$	8.29105	0.269851
$945$	8.85672	0.288109
$946$	−37.0074	−1.20322
$947$	47.7632	1.55210	0.776048	−	0.630674i	$-0.217221\pi$
$947$	47.7632	1.55210	0.776048	+	0.630674i	$0.217221\pi$
$948$	−6.09498	−0.197956
$949$	−9.27949	−0.301225
$950$	−0.120582	−0.00391220
$951$	−28.3385	−0.918940
$952$	16.0479	0.520115
$953$	20.8172	0.674334	0.337167	−	0.941445i	$-0.390531\pi$
$953$	20.8172	0.674334	0.337167	+	0.941445i	$0.390531\pi$
$954$	39.7854	1.28810
$955$	6.66493	0.215672
$956$	0.784794	0.0253821
$957$	−10.2914	−0.332674
$958$	15.6438	0.505427
$959$	−20.4103	−0.659083
$960$	5.74741	0.185497
$961$	−14.4910	−0.467453
$962$	1.42937	0.0460848
$963$	−12.6214	−0.406719
$964$	5.84106	0.188128
$965$	29.0505	0.935170
$966$	10.1580	0.326829
$967$	−52.0905	−1.67512	−0.837559	−	0.546346i	$-0.816018\pi$
$967$	−52.0905	−1.67512	−0.837559	+	0.546346i	$0.816018\pi$
$968$	5.59223	0.179741
$969$	6.95329	0.223372
$970$	−4.13282	−0.132697
$971$	54.9701	1.76407	0.882037	−	0.471181i	$-0.156172\pi$
$971$	54.9701	1.76407	0.882037	+	0.471181i	$0.156172\pi$
$972$	22.3675	0.717437
$973$	54.2658	1.73968
$974$	1.80307	0.0577742
$975$	0.464911	0.0148891
$976$	−0.914342	−0.0292674
$977$	−43.0002	−1.37570	−0.687848	−	0.725854i	$-0.741445\pi$
$977$	−43.0002	−1.37570	−0.687848	+	0.725854i	$0.741445\pi$
$978$	−54.4391	−1.74077
$979$	−22.7497	−0.727084
$980$	−20.8488	−0.665992
$981$	6.25570	0.199729
$982$	18.5655	0.592448
$983$	18.8670	0.601765	0.300882	−	0.953661i	$-0.402719\pi$
$983$	18.8670	0.601765	0.300882	+	0.953661i	$0.402719\pi$
$984$	−13.5640	−0.432406
$985$	45.3811	1.44596
$986$	−3.99144	−0.127113
$987$	−76.4833	−2.43449
$988$	0.725508	0.0230815
$989$	−9.08525	−0.288894
$990$	31.3502	0.996374
$991$	−30.0174	−0.953534	−0.476767	−	0.879030i	$-0.658191\pi$
$991$	−30.0174	−0.953534	−0.476767	+	0.879030i	$0.658191\pi$
$992$	4.06312	0.129004
$993$	−56.5919	−1.79589
$994$	−53.0776	−1.68352
$995$	−33.1031	−1.04944
$996$	8.15101	0.258275
$997$	−38.5098	−1.21962	−0.609808	−	0.792549i	$-0.708754\pi$
$997$	−38.5098	−1.21962	−0.609808	+	0.792549i	$0.708754\pi$
$998$	22.0728	0.698703
$999$	1.31546	0.0416192

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1334.2.a.i.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1334.2.a.i.1.1	✓	8	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:	\( N \)	\(=\)	\( 1334 = 2 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1334.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.52651 q^{3} +1.00000 q^{4} -2.27484 q^{5} -2.52651 q^{6} +4.02057 q^{7} +1.00000 q^{8} +3.38328 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.52651 q^{3} +1.00000 q^{4} -2.27484 q^{5} -2.52651 q^{6} +4.02057 q^{7} +1.00000 q^{8} +3.38328 q^{9} -2.27484 q^{10} -4.07336 q^{11} -2.52651 q^{12} -1.05221 q^{13} +4.02057 q^{14} +5.74741 q^{15} +1.00000 q^{16} +3.99144 q^{17} +3.38328 q^{18} -0.689507 q^{19} -2.27484 q^{20} -10.1580 q^{21} -4.07336 q^{22} -1.00000 q^{23} -2.52651 q^{24} +0.174882 q^{25} -1.05221 q^{26} -0.968356 q^{27} +4.02057 q^{28} -1.00000 q^{29} +5.74741 q^{30} +4.06312 q^{31} +1.00000 q^{32} +10.2914 q^{33} +3.99144 q^{34} -9.14614 q^{35} +3.38328 q^{36} -1.35844 q^{37} -0.689507 q^{38} +2.65843 q^{39} -2.27484 q^{40} +5.36868 q^{41} -10.1580 q^{42} +9.08525 q^{43} -4.07336 q^{44} -7.69640 q^{45} -1.00000 q^{46} +7.52935 q^{47} -2.52651 q^{48} +9.16498 q^{49} +0.174882 q^{50} -10.0844 q^{51} -1.05221 q^{52} +11.7594 q^{53} -0.968356 q^{54} +9.26622 q^{55} +4.02057 q^{56} +1.74205 q^{57} -1.00000 q^{58} +8.29105 q^{59} +5.74741 q^{60} -0.914342 q^{61} +4.06312 q^{62} +13.6027 q^{63} +1.00000 q^{64} +2.39361 q^{65} +10.2914 q^{66} +6.25512 q^{67} +3.99144 q^{68} +2.52651 q^{69} -9.14614 q^{70} -13.2015 q^{71} +3.38328 q^{72} +8.81901 q^{73} -1.35844 q^{74} -0.441841 q^{75} -0.689507 q^{76} -16.3772 q^{77} +2.65843 q^{78} +2.41240 q^{79} -2.27484 q^{80} -7.70327 q^{81} +5.36868 q^{82} -3.22619 q^{83} -10.1580 q^{84} -9.07988 q^{85} +9.08525 q^{86} +2.52651 q^{87} -4.07336 q^{88} +5.58500 q^{89} -7.69640 q^{90} -4.23050 q^{91} -1.00000 q^{92} -10.2655 q^{93} +7.52935 q^{94} +1.56851 q^{95} -2.52651 q^{96} +1.81675 q^{97} +9.16498 q^{98} -13.7813 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{7} + 8 q^{8} + 12 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^6 + 8 * q^7 + 8 * q^8 + 12 * q^9	\( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{7} + 8 q^{8} + 12 q^{9} + 8 q^{11} + 4 q^{12} + 4 q^{13} + 8 q^{14} + 8 q^{16} + 8 q^{17} + 12 q^{18} + 16 q^{19} - 4 q^{21} + 8 q^{22} - 8 q^{23} + 4 q^{24} + 4 q^{25} + 4 q^{26} + 4 q^{27} + 8 q^{28} - 8 q^{29} + 8 q^{31} + 8 q^{32} + 12 q^{33} + 8 q^{34} + 4 q^{35} + 12 q^{36} + 8 q^{37} + 16 q^{38} - 16 q^{39} + 8 q^{41} - 4 q^{42} + 32 q^{43} + 8 q^{44} - 8 q^{46} + 16 q^{47} + 4 q^{48} + 4 q^{49} + 4 q^{50} + 12 q^{51} + 4 q^{52} + 4 q^{54} + 8 q^{56} + 8 q^{57} - 8 q^{58} - 4 q^{59} + 8 q^{61} + 8 q^{62} + 24 q^{63} + 8 q^{64} - 4 q^{65} + 12 q^{66} + 20 q^{67} + 8 q^{68} - 4 q^{69} + 4 q^{70} - 24 q^{71} + 12 q^{72} - 4 q^{73} + 8 q^{74} - 16 q^{75} + 16 q^{76} - 16 q^{77} - 16 q^{78} + 4 q^{79} - 8 q^{81} + 8 q^{82} - 4 q^{83} - 4 q^{84} - 44 q^{85} + 32 q^{86} - 4 q^{87} + 8 q^{88} + 20 q^{89} - 8 q^{92} - 4 q^{93} + 16 q^{94} - 8 q^{95} + 4 q^{96} + 4 q^{97} + 4 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^6 + 8 * q^7 + 8 * q^8 + 12 * q^9 + 8 * q^11 + 4 * q^12 + 4 * q^13 + 8 * q^14 + 8 * q^16 + 8 * q^17 + 12 * q^18 + 16 * q^19 - 4 * q^21 + 8 * q^22 - 8 * q^23 + 4 * q^24 + 4 * q^25 + 4 * q^26 + 4 * q^27 + 8 * q^28 - 8 * q^29 + 8 * q^31 + 8 * q^32 + 12 * q^33 + 8 * q^34 + 4 * q^35 + 12 * q^36 + 8 * q^37 + 16 * q^38 - 16 * q^39 + 8 * q^41 - 4 * q^42 + 32 * q^43 + 8 * q^44 - 8 * q^46 + 16 * q^47 + 4 * q^48 + 4 * q^49 + 4 * q^50 + 12 * q^51 + 4 * q^52 + 4 * q^54 + 8 * q^56 + 8 * q^57 - 8 * q^58 - 4 * q^59 + 8 * q^61 + 8 * q^62 + 24 * q^63 + 8 * q^64 - 4 * q^65 + 12 * q^66 + 20 * q^67 + 8 * q^68 - 4 * q^69 + 4 * q^70 - 24 * q^71 + 12 * q^72 - 4 * q^73 + 8 * q^74 - 16 * q^75 + 16 * q^76 - 16 * q^77 - 16 * q^78 + 4 * q^79 - 8 * q^81 + 8 * q^82 - 4 * q^83 - 4 * q^84 - 44 * q^85 + 32 * q^86 - 4 * q^87 + 8 * q^88 + 20 * q^89 - 8 * q^92 - 4 * q^93 + 16 * q^94 - 8 * q^95 + 4 * q^96 + 4 * q^97 + 4 * q^98

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