LMFDB - Embedded newform 1338.2.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1338,2,Mod(1,1338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1338, 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("1338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1338 = 2 \cdot 3 \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1338.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:	$10.6839837904$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.10273.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + x + 2 $ x^4 - 2x^3 - 5x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.38266$ of defining polynomial
Character	$\chi$	$=$	1338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.38266 q^{5} +1.00000 q^{6} +1.61326 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.38266 q^{5} +1.00000 q^{6} +1.61326 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.38266 q^{10} +1.29442 q^{11} -1.00000 q^{12} -6.29035 q^{13} -1.61326 q^{14} -2.38266 q^{15} +1.00000 q^{16} -6.73683 q^{17} -1.00000 q^{18} -4.10296 q^{19} +2.38266 q^{20} -1.61326 q^{21} -1.29442 q^{22} -8.82507 q^{23} +1.00000 q^{24} +0.677084 q^{25} +6.29035 q^{26} -1.00000 q^{27} +1.61326 q^{28} +3.74090 q^{29} +2.38266 q^{30} +2.10296 q^{31} -1.00000 q^{32} -1.29442 q^{33} +6.73683 q^{34} +3.84386 q^{35} +1.00000 q^{36} -9.37859 q^{37} +4.10296 q^{38} +6.29035 q^{39} -2.38266 q^{40} +8.42180 q^{41} +1.61326 q^{42} -3.32292 q^{43} +1.29442 q^{44} +2.38266 q^{45} +8.82507 q^{46} -1.48970 q^{47} -1.00000 q^{48} -4.39738 q^{49} -0.677084 q^{50} +6.73683 q^{51} -6.29035 q^{52} +7.88889 q^{53} +1.00000 q^{54} +3.08417 q^{55} -1.61326 q^{56} +4.10296 q^{57} -3.74090 q^{58} -11.0704 q^{59} -2.38266 q^{60} +4.37859 q^{61} -2.10296 q^{62} +1.61326 q^{63} +1.00000 q^{64} -14.9878 q^{65} +1.29442 q^{66} -3.61734 q^{67} -6.73683 q^{68} +8.82507 q^{69} -3.84386 q^{70} +2.01472 q^{71} -1.00000 q^{72} +7.03507 q^{73} +9.37859 q^{74} -0.677084 q^{75} -4.10296 q^{76} +2.08824 q^{77} -6.29035 q^{78} -16.2716 q^{79} +2.38266 q^{80} +1.00000 q^{81} -8.42180 q^{82} +4.90361 q^{83} -1.61326 q^{84} -16.0516 q^{85} +3.32292 q^{86} -3.74090 q^{87} -1.29442 q^{88} +18.3704 q^{89} -2.38266 q^{90} -10.1480 q^{91} -8.82507 q^{92} -2.10296 q^{93} +1.48970 q^{94} -9.77597 q^{95} +1.00000 q^{96} -15.7613 q^{97} +4.39738 q^{98} +1.29442 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} - 5 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 - 5 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} - 5 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} + 4 q^{11} - 4 q^{12} - 5 q^{13} + 5 q^{14} - 2 q^{15} + 4 q^{16} - 2 q^{17} - 4 q^{18} - 7 q^{19} + 2 q^{20} + 5 q^{21} - 4 q^{22} - 4 q^{23} + 4 q^{24} - 6 q^{25} + 5 q^{26} - 4 q^{27} - 5 q^{28} + 9 q^{29} + 2 q^{30} - q^{31} - 4 q^{32} - 4 q^{33} + 2 q^{34} + 4 q^{36} - 11 q^{37} + 7 q^{38} + 5 q^{39} - 2 q^{40} + 14 q^{41} - 5 q^{42} - 22 q^{43} + 4 q^{44} + 2 q^{45} + 4 q^{46} - 8 q^{47} - 4 q^{48} - 7 q^{49} + 6 q^{50} + 2 q^{51} - 5 q^{52} + 3 q^{53} + 4 q^{54} - 13 q^{55} + 5 q^{56} + 7 q^{57} - 9 q^{58} - 6 q^{59} - 2 q^{60} - 9 q^{61} + q^{62} - 5 q^{63} + 4 q^{64} - 3 q^{65} + 4 q^{66} - 22 q^{67} - 2 q^{68} + 4 q^{69} + 5 q^{71} - 4 q^{72} - 3 q^{73} + 11 q^{74} + 6 q^{75} - 7 q^{76} + 2 q^{77} - 5 q^{78} - 29 q^{79} + 2 q^{80} + 4 q^{81} - 14 q^{82} - 12 q^{83} + 5 q^{84} - 10 q^{85} + 22 q^{86} - 9 q^{87} - 4 q^{88} + 9 q^{89} - 2 q^{90} - 18 q^{91} - 4 q^{92} + q^{93} + 8 q^{94} - 2 q^{95} + 4 q^{96} - 29 q^{97} + 7 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 - 5 * q^7 - 4 * q^8 + 4 * q^9 - 2 * q^10 + 4 * q^11 - 4 * q^12 - 5 * q^13 + 5 * q^14 - 2 * q^15 + 4 * q^16 - 2 * q^17 - 4 * q^18 - 7 * q^19 + 2 * q^20 + 5 * q^21 - 4 * q^22 - 4 * q^23 + 4 * q^24 - 6 * q^25 + 5 * q^26 - 4 * q^27 - 5 * q^28 + 9 * q^29 + 2 * q^30 - q^31 - 4 * q^32 - 4 * q^33 + 2 * q^34 + 4 * q^36 - 11 * q^37 + 7 * q^38 + 5 * q^39 - 2 * q^40 + 14 * q^41 - 5 * q^42 - 22 * q^43 + 4 * q^44 + 2 * q^45 + 4 * q^46 - 8 * q^47 - 4 * q^48 - 7 * q^49 + 6 * q^50 + 2 * q^51 - 5 * q^52 + 3 * q^53 + 4 * q^54 - 13 * q^55 + 5 * q^56 + 7 * q^57 - 9 * q^58 - 6 * q^59 - 2 * q^60 - 9 * q^61 + q^62 - 5 * q^63 + 4 * q^64 - 3 * q^65 + 4 * q^66 - 22 * q^67 - 2 * q^68 + 4 * q^69 + 5 * q^71 - 4 * q^72 - 3 * q^73 + 11 * q^74 + 6 * q^75 - 7 * q^76 + 2 * q^77 - 5 * q^78 - 29 * q^79 + 2 * q^80 + 4 * q^81 - 14 * q^82 - 12 * q^83 + 5 * q^84 - 10 * q^85 + 22 * q^86 - 9 * q^87 - 4 * q^88 + 9 * q^89 - 2 * q^90 - 18 * q^91 - 4 * q^92 + q^93 + 8 * q^94 - 2 * q^95 + 4 * q^96 - 29 * q^97 + 7 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	2.38266	1.06556	0.532780	−	0.846254i	$-0.321147\pi$
$5$	2.38266	1.06556	0.532780	+	0.846254i	$0.321147\pi$
$6$	1.00000	0.408248
$7$	1.61326	0.609756	0.304878	−	0.952391i	$-0.401384\pi$
$7$	1.61326	0.609756	0.304878	+	0.952391i	$0.401384\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.38266	−0.753464
$11$	1.29442	0.390283	0.195141	−	0.980775i	$-0.437483\pi$
$11$	1.29442	0.390283	0.195141	+	0.980775i	$0.437483\pi$
$12$	−1.00000	−0.288675
$13$	−6.29035	−1.74463	−0.872314	−	0.488946i	$-0.837382\pi$
$13$	−6.29035	−1.74463	−0.872314	+	0.488946i	$0.837382\pi$
$14$	−1.61326	−0.431163
$15$	−2.38266	−0.615201
$16$	1.00000	0.250000
$17$	−6.73683	−1.63392	−0.816961	−	0.576693i	$-0.804343\pi$
$17$	−6.73683	−1.63392	−0.816961	+	0.576693i	$0.804343\pi$
$18$	−1.00000	−0.235702
$19$	−4.10296	−0.941283	−0.470642	−	0.882324i	$-0.655978\pi$
$19$	−4.10296	−0.941283	−0.470642	+	0.882324i	$0.655978\pi$
$20$	2.38266	0.532780
$21$	−1.61326	−0.352043
$22$	−1.29442	−0.275971
$23$	−8.82507	−1.84016	−0.920078	−	0.391736i	$-0.871875\pi$
$23$	−8.82507	−1.84016	−0.920078	+	0.391736i	$0.871875\pi$
$24$	1.00000	0.204124
$25$	0.677084	0.135417
$26$	6.29035	1.23364
$27$	−1.00000	−0.192450
$28$	1.61326	0.304878
$29$	3.74090	0.694669	0.347334	−	0.937741i	$-0.387087\pi$
$29$	3.74090	0.694669	0.347334	+	0.937741i	$0.387087\pi$
$30$	2.38266	0.435013
$31$	2.10296	0.377703	0.188851	−	0.982006i	$-0.439524\pi$
$31$	2.10296	0.377703	0.188851	+	0.982006i	$0.439524\pi$
$32$	−1.00000	−0.176777
$33$	−1.29442	−0.225330
$34$	6.73683	1.15536
$35$	3.84386	0.649732
$36$	1.00000	0.166667
$37$	−9.37859	−1.54183	−0.770915	−	0.636938i	$-0.780201\pi$
$37$	−9.37859	−1.54183	−0.770915	+	0.636938i	$0.780201\pi$
$38$	4.10296	0.665588
$39$	6.29035	1.00726
$40$	−2.38266	−0.376732
$41$	8.42180	1.31526	0.657632	−	0.753339i	$-0.271558\pi$
$41$	8.42180	1.31526	0.657632	+	0.753339i	$0.271558\pi$
$42$	1.61326	0.248932
$43$	−3.32292	−0.506740	−0.253370	−	0.967369i	$-0.581539\pi$
$43$	−3.32292	−0.506740	−0.253370	+	0.967369i	$0.581539\pi$
$44$	1.29442	0.195141
$45$	2.38266	0.355186
$46$	8.82507	1.30119
$47$	−1.48970	−0.217294	−0.108647	−	0.994080i	$-0.534652\pi$
$47$	−1.48970	−0.217294	−0.108647	+	0.994080i	$0.534652\pi$
$48$	−1.00000	−0.144338
$49$	−4.39738	−0.628197
$50$	−0.677084	−0.0957542
$51$	6.73683	0.943345
$52$	−6.29035	−0.872314
$53$	7.88889	1.08362	0.541812	−	0.840500i	$-0.317739\pi$
$53$	7.88889	1.08362	0.541812	+	0.840500i	$0.317739\pi$
$54$	1.00000	0.136083
$55$	3.08417	0.415869
$56$	−1.61326	−0.215581
$57$	4.10296	0.543450
$58$	−3.74090	−0.491205
$59$	−11.0704	−1.44124	−0.720621	−	0.693329i	$-0.756143\pi$
$59$	−11.0704	−1.44124	−0.720621	+	0.693329i	$0.756143\pi$
$60$	−2.38266	−0.307601
$61$	4.37859	0.560621	0.280311	−	0.959909i	$-0.409563\pi$
$61$	4.37859	0.560621	0.280311	+	0.959909i	$0.409563\pi$
$62$	−2.10296	−0.267076
$63$	1.61326	0.203252
$64$	1.00000	0.125000
$65$	−14.9878	−1.85901
$66$	1.29442	0.159332
$67$	−3.61734	−0.441928	−0.220964	−	0.975282i	$-0.570920\pi$
$67$	−3.61734	−0.441928	−0.220964	+	0.975282i	$0.570920\pi$
$68$	−6.73683	−0.816961
$69$	8.82507	1.06241
$70$	−3.84386	−0.459430
$71$	2.01472	0.239103	0.119551	−	0.992828i	$-0.461854\pi$
$71$	2.01472	0.239103	0.119551	+	0.992828i	$0.461854\pi$
$72$	−1.00000	−0.117851
$73$	7.03507	0.823392	0.411696	−	0.911321i	$-0.364937\pi$
$73$	7.03507	0.823392	0.411696	+	0.911321i	$0.364937\pi$
$74$	9.37859	1.09024
$75$	−0.677084	−0.0781830
$76$	−4.10296	−0.470642
$77$	2.08824	0.237977
$78$	−6.29035	−0.712242
$79$	−16.2716	−1.83069	−0.915347	−	0.402667i	$-0.868083\pi$
$79$	−16.2716	−1.83069	−0.915347	+	0.402667i	$0.868083\pi$
$80$	2.38266	0.266390
$81$	1.00000	0.111111
$82$	−8.42180	−0.930032
$83$	4.90361	0.538241	0.269121	−	0.963106i	$-0.413267\pi$
$83$	4.90361	0.538241	0.269121	+	0.963106i	$0.413267\pi$
$84$	−1.61326	−0.176021
$85$	−16.0516	−1.74104
$86$	3.32292	0.358319
$87$	−3.74090	−0.401067
$88$	−1.29442	−0.137986
$89$	18.3704	1.94726	0.973632	−	0.228126i	$-0.0732599\pi$
$89$	18.3704	1.94726	0.973632	+	0.228126i	$0.0732599\pi$
$90$	−2.38266	−0.251155
$91$	−10.1480	−1.06380
$92$	−8.82507	−0.920078
$93$	−2.10296	−0.218067
$94$	1.48970	0.153650
$95$	−9.77597	−1.00299
$96$	1.00000	0.102062
$97$	−15.7613	−1.60031	−0.800156	−	0.599791i	$-0.795250\pi$
$97$	−15.7613	−1.60031	−0.800156	+	0.599791i	$0.795250\pi$
$98$	4.39738	0.444202
$99$	1.29442	0.130094
$100$	0.677084	0.0677084
$101$	2.17267	0.216189	0.108094	−	0.994141i	$-0.465525\pi$
$101$	2.17267	0.216189	0.108094	+	0.994141i	$0.465525\pi$
$102$	−6.73683	−0.667046
$103$	4.33131	0.426776	0.213388	−	0.976968i	$-0.431550\pi$
$103$	4.33131	0.426776	0.213388	+	0.976968i	$0.431550\pi$
$104$	6.29035	0.616819
$105$	−3.84386	−0.375123
$106$	−7.88889	−0.766237
$107$	13.7531	1.32956	0.664782	−	0.747038i	$-0.268525\pi$
$107$	13.7531	1.32956	0.664782	+	0.747038i	$0.268525\pi$
$108$	−1.00000	−0.0962250
$109$	7.03125	0.673472	0.336736	−	0.941599i	$-0.390677\pi$
$109$	7.03125	0.673472	0.336736	+	0.941599i	$0.390677\pi$
$110$	−3.08417	−0.294064
$111$	9.37859	0.890176
$112$	1.61326	0.152439
$113$	8.71215	0.819570	0.409785	−	0.912182i	$-0.365604\pi$
$113$	8.71215	0.819570	0.409785	+	0.912182i	$0.365604\pi$
$114$	−4.10296	−0.384277
$115$	−21.0272	−1.96079
$116$	3.74090	0.347334
$117$	−6.29035	−0.581543
$118$	11.0704	1.01911
$119$	−10.8683	−0.996294
$120$	2.38266	0.217506
$121$	−9.32447	−0.847679
$122$	−4.37859	−0.396419
$123$	−8.42180	−0.759368
$124$	2.10296	0.188851
$125$	−10.3001	−0.921265
$126$	−1.61326	−0.143721
$127$	−9.49940	−0.842935	−0.421468	−	0.906843i	$-0.638485\pi$
$127$	−9.49940	−0.842935	−0.421468	+	0.906843i	$0.638485\pi$
$128$	−1.00000	−0.0883883
$129$	3.32292	0.292566
$130$	14.9878	1.31452
$131$	−14.3460	−1.25342	−0.626709	−	0.779253i	$-0.715598\pi$
$131$	−14.3460	−1.25342	−0.626709	+	0.779253i	$0.715598\pi$
$132$	−1.29442	−0.112665
$133$	−6.61916	−0.573954
$134$	3.61734	0.312490
$135$	−2.38266	−0.205067
$136$	6.73683	0.577679
$137$	−5.10296	−0.435975	−0.217988	−	0.975952i	$-0.569949\pi$
$137$	−5.10296	−0.435975	−0.217988	+	0.975952i	$0.569949\pi$
$138$	−8.82507	−0.751240
$139$	−5.63795	−0.478204	−0.239102	−	0.970994i	$-0.576853\pi$
$139$	−5.63795	−0.478204	−0.239102	+	0.970994i	$0.576853\pi$
$140$	3.84386	0.324866
$141$	1.48970	0.125455
$142$	−2.01472	−0.169071
$143$	−8.14236	−0.680898
$144$	1.00000	0.0833333
$145$	8.91332	0.740211
$146$	−7.03507	−0.582226
$147$	4.39738	0.362690
$148$	−9.37859	−0.770915
$149$	−5.67119	−0.464602	−0.232301	−	0.972644i	$-0.574625\pi$
$149$	−5.67119	−0.464602	−0.232301	+	0.972644i	$0.574625\pi$
$150$	0.677084	0.0552837
$151$	12.4343	1.01189	0.505943	−	0.862567i	$-0.331145\pi$
$151$	12.4343	1.01189	0.505943	+	0.862567i	$0.331145\pi$
$152$	4.10296	0.332794
$153$	−6.73683	−0.544641
$154$	−2.08824	−0.168275
$155$	5.01064	0.402465
$156$	6.29035	0.503631
$157$	8.23623	0.657323	0.328661	−	0.944448i	$-0.393402\pi$
$157$	8.23623	0.657323	0.328661	+	0.944448i	$0.393402\pi$
$158$	16.2716	1.29450
$159$	−7.88889	−0.625630
$160$	−2.38266	−0.188366
$161$	−14.2372	−1.12205
$162$	−1.00000	−0.0785674
$163$	−7.27000	−0.569430	−0.284715	−	0.958612i	$-0.591899\pi$
$163$	−7.27000	−0.569430	−0.284715	+	0.958612i	$0.591899\pi$
$164$	8.42180	0.657632
$165$	−3.08417	−0.240102
$166$	−4.90361	−0.380594
$167$	−9.90361	−0.766364	−0.383182	−	0.923673i	$-0.625172\pi$
$167$	−9.90361	−0.766364	−0.383182	+	0.923673i	$0.625172\pi$
$168$	1.61326	0.124466
$169$	26.5685	2.04373
$170$	16.0516	1.23110
$171$	−4.10296	−0.313761
$172$	−3.32292	−0.253370
$173$	23.2252	1.76578	0.882890	−	0.469580i	$-0.155595\pi$
$173$	23.2252	1.76578	0.882890	+	0.469580i	$0.155595\pi$
$174$	3.74090	0.283597
$175$	1.09232	0.0825713
$176$	1.29442	0.0975707
$177$	11.0704	0.832102
$178$	−18.3704	−1.37692
$179$	−21.2499	−1.58829	−0.794146	−	0.607727i	$-0.792081\pi$
$179$	−21.2499	−1.58829	−0.794146	+	0.607727i	$0.792081\pi$
$180$	2.38266	0.177593
$181$	−20.7569	−1.54285	−0.771425	−	0.636320i	$-0.780456\pi$
$181$	−20.7569	−1.54285	−0.771425	+	0.636320i	$0.780456\pi$
$182$	10.1480	0.752219
$183$	−4.37859	−0.323675
$184$	8.82507	0.650593
$185$	−22.3460	−1.64291
$186$	2.10296	0.154196
$187$	−8.72030	−0.637691
$188$	−1.48970	−0.108647
$189$	−1.61326	−0.117348
$190$	9.77597	0.709223
$191$	19.2018	1.38940	0.694698	−	0.719301i	$-0.255538\pi$
$191$	19.2018	1.38940	0.694698	+	0.719301i	$0.255538\pi$
$192$	−1.00000	−0.0721688
$193$	10.2433	0.737331	0.368665	−	0.929562i	$-0.379815\pi$
$193$	10.2433	0.737331	0.368665	+	0.929562i	$0.379815\pi$
$194$	15.7613	1.13159
$195$	14.9878	1.07330
$196$	−4.39738	−0.314099
$197$	9.76507	0.695732	0.347866	−	0.937544i	$-0.386906\pi$
$197$	9.76507	0.695732	0.347866	+	0.937544i	$0.386906\pi$
$198$	−1.29442	−0.0919905
$199$	−3.06564	−0.217317	−0.108659	−	0.994079i	$-0.534656\pi$
$199$	−3.06564	−0.217317	−0.108659	+	0.994079i	$0.534656\pi$
$200$	−0.677084	−0.0478771
$201$	3.61734	0.255147
$202$	−2.17267	−0.152869
$203$	6.03507	0.423579
$204$	6.73683	0.471673
$205$	20.0663	1.40149
$206$	−4.33131	−0.301776
$207$	−8.82507	−0.613385
$208$	−6.29035	−0.436157
$209$	−5.31096	−0.367367
$210$	3.84386	0.265252
$211$	0.240565	0.0165612	0.00828059	−	0.999966i	$-0.497364\pi$
$211$	0.240565	0.0165612	0.00828059	+	0.999966i	$0.497364\pi$
$212$	7.88889	0.541812
$213$	−2.01472	−0.138046
$214$	−13.7531	−0.940143
$215$	−7.91739	−0.539961
$216$	1.00000	0.0680414
$217$	3.39263	0.230307
$218$	−7.03125	−0.476217
$219$	−7.03507	−0.475386
$220$	3.08417	0.207935
$221$	42.3770	2.85059
$222$	−9.37859	−0.629450
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	−1.61326	−0.107791
$225$	0.677084	0.0451389
$226$	−8.71215	−0.579524
$227$	−14.8523	−0.985779	−0.492889	−	0.870092i	$-0.664059\pi$
$227$	−14.8523	−0.985779	−0.492889	+	0.870092i	$0.664059\pi$
$228$	4.10296	0.271725
$229$	5.82639	0.385019	0.192509	−	0.981295i	$-0.438337\pi$
$229$	5.82639	0.385019	0.192509	+	0.981295i	$0.438337\pi$
$230$	21.0272	1.38649
$231$	−2.08824	−0.137396
$232$	−3.74090	−0.245602
$233$	17.0770	1.11875	0.559375	−	0.828915i	$-0.311041\pi$
$233$	17.0770	1.11875	0.559375	+	0.828915i	$0.311041\pi$
$234$	6.29035	0.411213
$235$	−3.54944	−0.231540
$236$	−11.0704	−0.720621
$237$	16.2716	1.05695
$238$	10.8683	0.704486
$239$	−22.0091	−1.42365	−0.711824	−	0.702358i	$-0.752131\pi$
$239$	−22.0091	−1.42365	−0.711824	+	0.702358i	$0.752131\pi$
$240$	−2.38266	−0.153800
$241$	13.7948	0.888599	0.444299	−	0.895878i	$-0.353453\pi$
$241$	13.7948	0.888599	0.444299	+	0.895878i	$0.353453\pi$
$242$	9.32447	0.599400
$243$	−1.00000	−0.0641500
$244$	4.37859	0.280311
$245$	−10.4775	−0.669381
$246$	8.42180	0.536954
$247$	25.8090	1.64219
$248$	−2.10296	−0.133538
$249$	−4.90361	−0.310754
$250$	10.3001	0.651433
$251$	−12.2127	−0.770862	−0.385431	−	0.922737i	$-0.625947\pi$
$251$	−12.2127	−0.770862	−0.385431	+	0.922737i	$0.625947\pi$
$252$	1.61326	0.101626
$253$	−11.4234	−0.718181
$254$	9.49940	0.596045
$255$	16.0516	1.00519
$256$	1.00000	0.0625000
$257$	−15.2616	−0.951992	−0.475996	−	0.879447i	$-0.657912\pi$
$257$	−15.2616	−0.951992	−0.475996	+	0.879447i	$0.657912\pi$
$258$	−3.32292	−0.206876
$259$	−15.1301	−0.940141
$260$	−14.9878	−0.929503
$261$	3.74090	0.231556
$262$	14.3460	0.886300
$263$	29.3341	1.80882	0.904408	−	0.426669i	$-0.140313\pi$
$263$	29.3341	1.80882	0.904408	+	0.426669i	$0.140313\pi$
$264$	1.29442	0.0796661
$265$	18.7966	1.15466
$266$	6.61916	0.405846
$267$	−18.3704	−1.12425
$268$	−3.61734	−0.220964
$269$	5.04597	0.307658	0.153829	−	0.988097i	$-0.450839\pi$
$269$	5.04597	0.307658	0.153829	+	0.988097i	$0.450839\pi$
$270$	2.38266	0.145004
$271$	−17.5932	−1.06871	−0.534354	−	0.845261i	$-0.679445\pi$
$271$	−17.5932	−1.06871	−0.534354	+	0.845261i	$0.679445\pi$
$272$	−6.73683	−0.408480
$273$	10.1480	0.614184
$274$	5.10296	0.308281
$275$	0.876432	0.0528508
$276$	8.82507	0.531207
$277$	−21.1091	−1.26832	−0.634161	−	0.773201i	$-0.718654\pi$
$277$	−21.1091	−1.26832	−0.634161	+	0.773201i	$0.718654\pi$
$278$	5.63795	0.338141
$279$	2.10296	0.125901
$280$	−3.84386	−0.229715
$281$	24.8464	1.48221	0.741105	−	0.671389i	$-0.234302\pi$
$281$	24.8464	1.48221	0.741105	+	0.671389i	$0.234302\pi$
$282$	−1.48970	−0.0887101
$283$	15.2459	0.906277	0.453138	−	0.891440i	$-0.350304\pi$
$283$	15.2459	0.906277	0.453138	+	0.891440i	$0.350304\pi$
$284$	2.01472	0.119551
$285$	9.77597	0.579079
$286$	8.14236	0.481468
$287$	13.5866	0.801991
$288$	−1.00000	−0.0589256
$289$	28.3849	1.66970
$290$	−8.91332	−0.523408
$291$	15.7613	0.923941
$292$	7.03507	0.411696
$293$	16.2334	0.948363	0.474181	−	0.880427i	$-0.342744\pi$
$293$	16.2334	0.948363	0.474181	+	0.880427i	$0.342744\pi$
$294$	−4.39738	−0.256460
$295$	−26.3770	−1.53573
$296$	9.37859	0.545119
$297$	−1.29442	−0.0751099
$298$	5.67119	0.328523
$299$	55.5128	3.21039
$300$	−0.677084	−0.0390915
$301$	−5.36074	−0.308988
$302$	−12.4343	−0.715512
$303$	−2.17267	−0.124817
$304$	−4.10296	−0.235321
$305$	10.4327	0.597375
$306$	6.73683	0.385119
$307$	8.09889	0.462228	0.231114	−	0.972927i	$-0.425763\pi$
$307$	8.09889	0.462228	0.231114	+	0.972927i	$0.425763\pi$
$308$	2.08824	0.118989
$309$	−4.33131	−0.246399
$310$	−5.01064	−0.284585
$311$	−23.2534	−1.31858	−0.659291	−	0.751888i	$-0.729143\pi$
$311$	−23.2534	−1.31858	−0.659291	+	0.751888i	$0.729143\pi$
$312$	−6.29035	−0.356121
$313$	7.91176	0.447199	0.223599	−	0.974681i	$-0.428219\pi$
$313$	7.91176	0.447199	0.223599	+	0.974681i	$0.428219\pi$
$314$	−8.23623	−0.464797
$315$	3.84386	0.216577
$316$	−16.2716	−0.915347
$317$	−2.05975	−0.115687	−0.0578435	−	0.998326i	$-0.518422\pi$
$317$	−2.05975	−0.115687	−0.0578435	+	0.998326i	$0.518422\pi$
$318$	7.88889	0.442387
$319$	4.84231	0.271117
$320$	2.38266	0.133195
$321$	−13.7531	−0.767624
$322$	14.2372	0.793407
$323$	27.6409	1.53798
$324$	1.00000	0.0555556
$325$	−4.25910	−0.236252
$326$	7.27000	0.402648
$327$	−7.03125	−0.388829
$328$	−8.42180	−0.465016
$329$	−2.40327	−0.132497
$330$	3.08417	0.169778
$331$	6.59629	0.362565	0.181283	−	0.983431i	$-0.441975\pi$
$331$	6.59629	0.362565	0.181283	+	0.983431i	$0.441975\pi$
$332$	4.90361	0.269121
$333$	−9.37859	−0.513944
$334$	9.90361	0.541902
$335$	−8.61890	−0.470901
$336$	−1.61326	−0.0880107
$337$	13.5444	0.737812	0.368906	−	0.929467i	$-0.379732\pi$
$337$	13.5444	0.737812	0.368906	+	0.929467i	$0.379732\pi$
$338$	−26.5685	−1.44513
$339$	−8.71215	−0.473179
$340$	−16.0516	−0.870520
$341$	2.72211	0.147411
$342$	4.10296	0.221863
$343$	−18.3870	−0.992804
$344$	3.32292	0.179160
$345$	21.0272	1.13207
$346$	−23.2252	−1.24860
$347$	−11.3204	−0.607711	−0.303855	−	0.952718i	$-0.598274\pi$
$347$	−11.3204	−0.607711	−0.303855	+	0.952718i	$0.598274\pi$
$348$	−3.74090	−0.200534
$349$	−23.1517	−1.23928	−0.619641	−	0.784886i	$-0.712722\pi$
$349$	−23.1517	−1.23928	−0.619641	+	0.784886i	$0.712722\pi$
$350$	−1.09232	−0.0583867
$351$	6.29035	0.335754
$352$	−1.29442	−0.0689929
$353$	−31.6382	−1.68393	−0.841965	−	0.539531i	$-0.818601\pi$
$353$	−31.6382	−1.68393	−0.841965	+	0.539531i	$0.818601\pi$
$354$	−11.0704	−0.588385
$355$	4.80039	0.254778
$356$	18.3704	0.973632
$357$	10.8683	0.575211
$358$	21.2499	1.12309
$359$	−30.8685	−1.62918	−0.814589	−	0.580038i	$-0.803038\pi$
$359$	−30.8685	−1.62918	−0.814589	+	0.580038i	$0.803038\pi$
$360$	−2.38266	−0.125577
$361$	−2.16572	−0.113986
$362$	20.7569	1.09096
$363$	9.32447	0.489408
$364$	−10.1480	−0.531899
$365$	16.7622	0.877373
$366$	4.37859	0.228873
$367$	18.5835	0.970048	0.485024	−	0.874501i	$-0.338811\pi$
$367$	18.5835	0.970048	0.485024	+	0.874501i	$0.338811\pi$
$368$	−8.82507	−0.460039
$369$	8.42180	0.438422
$370$	22.3460	1.16171
$371$	12.7269	0.660746
$372$	−2.10296	−0.109033
$373$	4.22351	0.218685	0.109343	−	0.994004i	$-0.465125\pi$
$373$	4.22351	0.218685	0.109343	+	0.994004i	$0.465125\pi$
$374$	8.72030	0.450916
$375$	10.3001	0.531892
$376$	1.48970	0.0768252
$377$	−23.5316	−1.21194
$378$	1.61326	0.0829773
$379$	9.04753	0.464740	0.232370	−	0.972627i	$-0.425352\pi$
$379$	9.04753	0.464740	0.232370	+	0.972627i	$0.425352\pi$
$380$	−9.77597	−0.501497
$381$	9.49940	0.486669
$382$	−19.2018	−0.982452
$383$	14.8525	0.758928	0.379464	−	0.925207i	$-0.376108\pi$
$383$	14.8525	0.758928	0.379464	+	0.925207i	$0.376108\pi$
$384$	1.00000	0.0510310
$385$	4.97558	0.253579
$386$	−10.2433	−0.521371
$387$	−3.32292	−0.168913
$388$	−15.7613	−0.800156
$389$	−4.34446	−0.220273	−0.110137	−	0.993916i	$-0.535129\pi$
$389$	−4.34446	−0.220273	−0.110137	+	0.993916i	$0.535129\pi$
$390$	−14.9878	−0.758936
$391$	59.4530	3.00667
$392$	4.39738	0.222101
$393$	14.3460	0.723661
$394$	−9.76507	−0.491957
$395$	−38.7696	−1.95071
$396$	1.29442	0.0650471
$397$	−36.6902	−1.84143	−0.920715	−	0.390236i	$-0.872393\pi$
$397$	−36.6902	−1.84143	−0.920715	+	0.390236i	$0.872393\pi$
$398$	3.06564	0.153667
$399$	6.61916	0.331372
$400$	0.677084	0.0338542
$401$	7.96441	0.397724	0.198862	−	0.980028i	$-0.436275\pi$
$401$	7.96441	0.397724	0.198862	+	0.980028i	$0.436275\pi$
$402$	−3.61734	−0.180416
$403$	−13.2283	−0.658951
$404$	2.17267	0.108094
$405$	2.38266	0.118395
$406$	−6.03507	−0.299515
$407$	−12.1398	−0.601750
$408$	−6.73683	−0.333523
$409$	−5.89271	−0.291376	−0.145688	−	0.989331i	$-0.546540\pi$
$409$	−5.89271	−0.291376	−0.145688	+	0.989331i	$0.546540\pi$
$410$	−20.0663	−0.991005
$411$	5.10296	0.251710
$412$	4.33131	0.213388
$413$	−17.8595	−0.878807
$414$	8.82507	0.433729
$415$	11.6837	0.573528
$416$	6.29035	0.308410
$417$	5.63795	0.276091
$418$	5.31096	0.259767
$419$	−12.3962	−0.605593	−0.302797	−	0.953055i	$-0.597920\pi$
$419$	−12.3962	−0.605593	−0.302797	+	0.953055i	$0.597920\pi$
$420$	−3.84386	−0.187561
$421$	−27.5556	−1.34298	−0.671488	−	0.741015i	$-0.734345\pi$
$421$	−27.5556	−1.34298	−0.671488	+	0.741015i	$0.734345\pi$
$422$	−0.240565	−0.0117105
$423$	−1.48970	−0.0724315
$424$	−7.88889	−0.383119
$425$	−4.56140	−0.221261
$426$	2.01472	0.0976134
$427$	7.06382	0.341842
$428$	13.7531	0.664782
$429$	8.14236	0.393117
$430$	7.91739	0.381810
$431$	19.1468	0.922268	0.461134	−	0.887330i	$-0.347443\pi$
$431$	19.1468	0.922268	0.461134	+	0.887330i	$0.347443\pi$
$432$	−1.00000	−0.0481125
$433$	−30.9452	−1.48713	−0.743567	−	0.668662i	$-0.766868\pi$
$433$	−30.9452	−1.48713	−0.743567	+	0.668662i	$0.766868\pi$
$434$	−3.39263	−0.162851
$435$	−8.91332	−0.427361
$436$	7.03125	0.336736
$437$	36.2089	1.73211
$438$	7.03507	0.336148
$439$	22.6235	1.07976	0.539880	−	0.841742i	$-0.318470\pi$
$439$	22.6235	1.07976	0.539880	+	0.841742i	$0.318470\pi$
$440$	−3.08417	−0.147032
$441$	−4.39738	−0.209399
$442$	−42.3770	−2.01567
$443$	−0.982585	−0.0466840	−0.0233420	−	0.999728i	$-0.507431\pi$
$443$	−0.982585	−0.0466840	−0.0233420	+	0.999728i	$0.507431\pi$
$444$	9.37859	0.445088
$445$	43.7706	2.07492
$446$	1.00000	0.0473514
$447$	5.67119	0.268238
$448$	1.61326	0.0762195
$449$	5.66487	0.267342	0.133671	−	0.991026i	$-0.457324\pi$
$449$	5.66487	0.267342	0.133671	+	0.991026i	$0.457324\pi$
$450$	−0.677084	−0.0319181
$451$	10.9014	0.513325
$452$	8.71215	0.409785
$453$	−12.4343	−0.584213
$454$	14.8523	0.697051
$455$	−24.1792	−1.13354
$456$	−4.10296	−0.192139
$457$	−7.75756	−0.362883	−0.181442	−	0.983402i	$-0.558076\pi$
$457$	−7.75756	−0.362883	−0.181442	+	0.983402i	$0.558076\pi$
$458$	−5.82639	−0.272249
$459$	6.73683	0.314448
$460$	−21.0272	−0.980397
$461$	−28.0453	−1.30620	−0.653099	−	0.757272i	$-0.726532\pi$
$461$	−28.0453	−1.30620	−0.653099	+	0.757272i	$0.726532\pi$
$462$	2.08824	0.0971538
$463$	−2.07628	−0.0964931	−0.0482465	−	0.998835i	$-0.515363\pi$
$463$	−2.07628	−0.0964931	−0.0482465	+	0.998835i	$0.515363\pi$
$464$	3.74090	0.173667
$465$	−5.01064	−0.232363
$466$	−17.0770	−0.791075
$467$	23.7477	1.09891	0.549457	−	0.835522i	$-0.314835\pi$
$467$	23.7477	1.09891	0.549457	+	0.835522i	$0.314835\pi$
$468$	−6.29035	−0.290771
$469$	−5.83572	−0.269468
$470$	3.54944	0.163724
$471$	−8.23623	−0.379506
$472$	11.0704	0.509556
$473$	−4.30125	−0.197772
$474$	−16.2716	−0.747377
$475$	−2.77805	−0.127466
$476$	−10.8683	−0.498147
$477$	7.88889	0.361208
$478$	22.0091	1.00667
$479$	−16.7167	−0.763807	−0.381903	−	0.924202i	$-0.624731\pi$
$479$	−16.7167	−0.763807	−0.381903	+	0.924202i	$0.624731\pi$
$480$	2.38266	0.108753
$481$	58.9946	2.68992
$482$	−13.7948	−0.628334
$483$	14.2372	0.647814
$484$	−9.32447	−0.423840
$485$	−37.5538	−1.70523
$486$	1.00000	0.0453609
$487$	6.73683	0.305275	0.152637	−	0.988282i	$-0.451223\pi$
$487$	6.73683	0.305275	0.152637	+	0.988282i	$0.451223\pi$
$488$	−4.37859	−0.198209
$489$	7.27000	0.328761
$490$	10.4775	0.473324
$491$	17.0566	0.769754	0.384877	−	0.922968i	$-0.374244\pi$
$491$	17.0566	0.769754	0.384877	+	0.922968i	$0.374244\pi$
$492$	−8.42180	−0.379684
$493$	−25.2018	−1.13503
$494$	−25.8090	−1.16120
$495$	3.08417	0.138623
$496$	2.10296	0.0944257
$497$	3.25027	0.145795
$498$	4.90361	0.219736
$499$	35.2150	1.57644	0.788220	−	0.615394i	$-0.211003\pi$
$499$	35.2150	1.57644	0.788220	+	0.615394i	$0.211003\pi$
$500$	−10.3001	−0.460632
$501$	9.90361	0.442461
$502$	12.2127	0.545082
$503$	−5.47723	−0.244218	−0.122109	−	0.992517i	$-0.538966\pi$
$503$	−5.47723	−0.244218	−0.122109	+	0.992517i	$0.538966\pi$
$504$	−1.61326	−0.0718605
$505$	5.17674	0.230362
$506$	11.4234	0.507830
$507$	−26.5685	−1.17995
$508$	−9.49940	−0.421468
$509$	−37.9674	−1.68288	−0.841438	−	0.540354i	$-0.818290\pi$
$509$	−37.9674	−1.68288	−0.841438	+	0.540354i	$0.818290\pi$
$510$	−16.0516	−0.710777
$511$	11.3494	0.502069
$512$	−1.00000	−0.0441942
$513$	4.10296	0.181150
$514$	15.2616	0.673160
$515$	10.3200	0.454755
$516$	3.32292	0.146283
$517$	−1.92829	−0.0848062
$518$	15.1301	0.664780
$519$	−23.2252	−1.01947
$520$	14.9878	0.657258
$521$	−19.1242	−0.837848	−0.418924	−	0.908021i	$-0.637593\pi$
$521$	−19.1242	−0.837848	−0.418924	+	0.908021i	$0.637593\pi$
$522$	−3.74090	−0.163735
$523$	−43.0530	−1.88258	−0.941289	−	0.337602i	$-0.890384\pi$
$523$	−43.0530	−1.88258	−0.941289	+	0.337602i	$0.890384\pi$
$524$	−14.3460	−0.626709
$525$	−1.09232	−0.0476726
$526$	−29.3341	−1.27903
$527$	−14.1673	−0.617137
$528$	−1.29442	−0.0563324
$529$	54.8819	2.38617
$530$	−18.7966	−0.816471
$531$	−11.0704	−0.480414
$532$	−6.61916	−0.286977
$533$	−52.9761	−2.29465
$534$	18.3704	0.794967
$535$	32.7690	1.41673
$536$	3.61734	0.156245
$537$	21.2499	0.917001
$538$	−5.04597	−0.217547
$539$	−5.69206	−0.245174
$540$	−2.38266	−0.102534
$541$	−19.2809	−0.828950	−0.414475	−	0.910061i	$-0.636035\pi$
$541$	−19.2809	−0.828950	−0.414475	+	0.910061i	$0.636035\pi$
$542$	17.5932	0.755691
$543$	20.7569	0.890765
$544$	6.73683	0.288839
$545$	16.7531	0.717624
$546$	−10.1480	−0.434294
$547$	20.1958	0.863509	0.431755	−	0.901991i	$-0.357895\pi$
$547$	20.1958	0.863509	0.431755	+	0.901991i	$0.357895\pi$
$548$	−5.10296	−0.217988
$549$	4.37859	0.186874
$550$	−0.876432	−0.0373712
$551$	−15.3488	−0.653880
$552$	−8.82507	−0.375620
$553$	−26.2503	−1.11628
$554$	21.1091	0.896839
$555$	22.3460	0.948536
$556$	−5.63795	−0.239102
$557$	15.2462	0.646002	0.323001	−	0.946399i	$-0.395308\pi$
$557$	15.2462	0.646002	0.323001	+	0.946399i	$0.395308\pi$
$558$	−2.10296	−0.0890254
$559$	20.9023	0.884073
$560$	3.84386	0.162433
$561$	8.72030	0.368171
$562$	−24.8464	−1.04808
$563$	−12.8131	−0.540008	−0.270004	−	0.962859i	$-0.587025\pi$
$563$	−12.8131	−0.540008	−0.270004	+	0.962859i	$0.587025\pi$
$564$	1.48970	0.0627275
$565$	20.7581	0.873301
$566$	−15.2459	−0.640835
$567$	1.61326	0.0677507
$568$	−2.01472	−0.0845356
$569$	14.6814	0.615476	0.307738	−	0.951471i	$-0.400428\pi$
$569$	14.6814	0.615476	0.307738	+	0.951471i	$0.400428\pi$
$570$	−9.77597	−0.409470
$571$	10.8880	0.455647	0.227823	−	0.973702i	$-0.426839\pi$
$571$	10.8880	0.455647	0.227823	+	0.973702i	$0.426839\pi$
$572$	−8.14236	−0.340449
$573$	−19.2018	−0.802169
$574$	−13.5866	−0.567093
$575$	−5.97532	−0.249188
$576$	1.00000	0.0416667
$577$	26.2349	1.09217	0.546087	−	0.837729i	$-0.316117\pi$
$577$	26.2349	1.09217	0.546087	+	0.837729i	$0.316117\pi$
$578$	−28.3849	−1.18066
$579$	−10.2433	−0.425698
$580$	8.91332	0.370105
$581$	7.91082	0.328196
$582$	−15.7613	−0.653325
$583$	10.2116	0.422919
$584$	−7.03507	−0.291113
$585$	−14.9878	−0.619668
$586$	−16.2334	−0.670594
$587$	−10.5850	−0.436891	−0.218445	−	0.975849i	$-0.570099\pi$
$587$	−10.5850	−0.436891	−0.218445	+	0.975849i	$0.570099\pi$
$588$	4.39738	0.181345
$589$	−8.62836	−0.355525
$590$	26.3770	1.08592
$591$	−9.76507	−0.401681
$592$	−9.37859	−0.385458
$593$	−24.4252	−1.00302	−0.501512	−	0.865151i	$-0.667223\pi$
$593$	−24.4252	−1.00302	−0.501512	+	0.865151i	$0.667223\pi$
$594$	1.29442	0.0531107
$595$	−25.8955	−1.06161
$596$	−5.67119	−0.232301
$597$	3.06564	0.125468
$598$	−55.5128	−2.27009
$599$	39.2278	1.60281	0.801403	−	0.598125i	$-0.204087\pi$
$599$	39.2278	1.60281	0.801403	+	0.598125i	$0.204087\pi$
$600$	0.677084	0.0276418
$601$	34.6552	1.41361	0.706807	−	0.707407i	$-0.250135\pi$
$601$	34.6552	1.41361	0.706807	+	0.707407i	$0.250135\pi$
$602$	5.36074	0.218487
$603$	−3.61734	−0.147309
$604$	12.4343	0.505943
$605$	−22.2171	−0.903253
$606$	2.17267	0.0882587
$607$	1.28853	0.0522998	0.0261499	−	0.999658i	$-0.491675\pi$
$607$	1.28853	0.0522998	0.0261499	+	0.999658i	$0.491675\pi$
$608$	4.10296	0.166397
$609$	−6.03507	−0.244553
$610$	−10.4327	−0.422408
$611$	9.37070	0.379098
$612$	−6.73683	−0.272320
$613$	31.4868	1.27174	0.635870	−	0.771796i	$-0.280641\pi$
$613$	31.4868	1.27174	0.635870	+	0.771796i	$0.280641\pi$
$614$	−8.09889	−0.326844
$615$	−20.0663	−0.809152
$616$	−2.08824	−0.0841377
$617$	−20.5693	−0.828088	−0.414044	−	0.910257i	$-0.635884\pi$
$617$	−20.5693	−0.828088	−0.414044	+	0.910257i	$0.635884\pi$
$618$	4.33131	0.174231
$619$	−0.765762	−0.0307786	−0.0153893	−	0.999882i	$-0.504899\pi$
$619$	−0.765762	−0.0307786	−0.0153893	+	0.999882i	$0.504899\pi$
$620$	5.01064	0.201232
$621$	8.82507	0.354138
$622$	23.2534	0.932378
$623$	29.6364	1.18736
$624$	6.29035	0.251815
$625$	−27.9270	−1.11708
$626$	−7.91176	−0.316217
$627$	5.31096	0.212099
$628$	8.23623	0.328661
$629$	63.1820	2.51923
$630$	−3.84386	−0.153143
$631$	−25.8176	−1.02778	−0.513892	−	0.857855i	$-0.671797\pi$
$631$	−25.8176	−1.02778	−0.513892	+	0.857855i	$0.671797\pi$
$632$	16.2716	0.647248
$633$	−0.240565	−0.00956160
$634$	2.05975	0.0818030
$635$	−22.6339	−0.898198
$636$	−7.88889	−0.312815
$637$	27.6611	1.09597
$638$	−4.84231	−0.191709
$639$	2.01472	0.0797010
$640$	−2.38266	−0.0941830
$641$	23.2933	0.920029	0.460015	−	0.887911i	$-0.347844\pi$
$641$	23.2933	0.920029	0.460015	+	0.887911i	$0.347844\pi$
$642$	13.7531	0.542792
$643$	−34.0373	−1.34230	−0.671150	−	0.741321i	$-0.734199\pi$
$643$	−34.0373	−1.34230	−0.671150	+	0.741321i	$0.734199\pi$
$644$	−14.2372	−0.561023
$645$	7.91739	0.311747
$646$	−27.6409	−1.08752
$647$	−37.4626	−1.47281	−0.736404	−	0.676542i	$-0.763478\pi$
$647$	−37.4626	−1.47281	−0.736404	+	0.676542i	$0.763478\pi$
$648$	−1.00000	−0.0392837
$649$	−14.3297	−0.562492
$650$	4.25910	0.167055
$651$	−3.39263	−0.132968
$652$	−7.27000	−0.284715
$653$	36.2540	1.41873	0.709363	−	0.704843i	$-0.248983\pi$
$653$	36.2540	1.41873	0.709363	+	0.704843i	$0.248983\pi$
$654$	7.03125	0.274944
$655$	−34.1817	−1.33559
$656$	8.42180	0.328816
$657$	7.03507	0.274464
$658$	2.40327	0.0936893
$659$	−20.3589	−0.793071	−0.396535	−	0.918019i	$-0.629788\pi$
$659$	−20.3589	−0.793071	−0.396535	+	0.918019i	$0.629788\pi$
$660$	−3.08417	−0.120051
$661$	38.1758	1.48487	0.742434	−	0.669919i	$-0.233671\pi$
$661$	38.1758	1.48487	0.742434	+	0.669919i	$0.233671\pi$
$662$	−6.59629	−0.256372
$663$	−42.3770	−1.64579
$664$	−4.90361	−0.190297
$665$	−15.7712	−0.611582
$666$	9.37859	0.363413
$667$	−33.0138	−1.27830
$668$	−9.90361	−0.383182
$669$	1.00000	0.0386622
$670$	8.61890	0.332977
$671$	5.66774	0.218801
$672$	1.61326	0.0622330
$673$	20.4071	0.786635	0.393318	−	0.919403i	$-0.371327\pi$
$673$	20.4071	0.786635	0.393318	+	0.919403i	$0.371327\pi$
$674$	−13.5444	−0.521712
$675$	−0.677084	−0.0260610
$676$	26.5685	1.02186
$677$	−34.5293	−1.32707	−0.663535	−	0.748145i	$-0.730945\pi$
$677$	−34.5293	−1.32707	−0.663535	+	0.748145i	$0.730945\pi$
$678$	8.71215	0.334588
$679$	−25.4271	−0.975801
$680$	16.0516	0.615551
$681$	14.8523	0.569140
$682$	−2.72211	−0.104235
$683$	−7.23805	−0.276956	−0.138478	−	0.990365i	$-0.544221\pi$
$683$	−7.23805	−0.276956	−0.138478	+	0.990365i	$0.544221\pi$
$684$	−4.10296	−0.156881
$685$	−12.1586	−0.464558
$686$	18.3870	0.702018
$687$	−5.82639	−0.222291
$688$	−3.32292	−0.126685
$689$	−49.6239	−1.89052
$690$	−21.0272	−0.800491
$691$	−26.0593	−0.991343	−0.495671	−	0.868510i	$-0.665078\pi$
$691$	−26.0593	−0.991343	−0.495671	+	0.868510i	$0.665078\pi$
$692$	23.2252	0.882890
$693$	2.08824	0.0793258
$694$	11.3204	0.429717
$695$	−13.4333	−0.509555
$696$	3.74090	0.141799
$697$	−56.7363	−2.14904
$698$	23.1517	0.876304
$699$	−17.0770	−0.645910
$700$	1.09232	0.0412856
$701$	−3.75011	−0.141640	−0.0708198	−	0.997489i	$-0.522562\pi$
$701$	−3.75011	−0.141640	−0.0708198	+	0.997489i	$0.522562\pi$
$702$	−6.29035	−0.237414
$703$	38.4800	1.45130
$704$	1.29442	0.0487853
$705$	3.54944	0.133680
$706$	31.6382	1.19072
$707$	3.50509	0.131823
$708$	11.0704	0.416051
$709$	−24.7368	−0.929011	−0.464506	−	0.885570i	$-0.653768\pi$
$709$	−24.7368	−0.929011	−0.464506	+	0.885570i	$0.653768\pi$
$710$	−4.80039	−0.180156
$711$	−16.2716	−0.610231
$712$	−18.3704	−0.688461
$713$	−18.5588	−0.695031
$714$	−10.8683	−0.406735
$715$	−19.4005	−0.725537
$716$	−21.2499	−0.794146
$717$	22.0091	0.821944
$718$	30.8685	1.15200
$719$	23.8382	0.889015	0.444508	−	0.895775i	$-0.353379\pi$
$719$	23.8382	0.889015	0.444508	+	0.895775i	$0.353379\pi$
$720$	2.38266	0.0887966
$721$	6.98754	0.260229
$722$	2.16572	0.0805999
$723$	−13.7948	−0.513033
$724$	−20.7569	−0.771425
$725$	2.53291	0.0940698
$726$	−9.32447	−0.346064
$727$	−21.3209	−0.790749	−0.395375	−	0.918520i	$-0.629385\pi$
$727$	−21.3209	−0.790749	−0.395375	+	0.918520i	$0.629385\pi$
$728$	10.1480	0.376110
$729$	1.00000	0.0370370
$730$	−16.7622	−0.620397
$731$	22.3859	0.827973
$732$	−4.37859	−0.161837
$733$	41.0982	1.51800	0.758998	−	0.651093i	$-0.225689\pi$
$733$	41.0982	1.51800	0.758998	+	0.651093i	$0.225689\pi$
$734$	−18.5835	−0.685928
$735$	10.4775	0.386468
$736$	8.82507	0.325297
$737$	−4.68236	−0.172477
$738$	−8.42180	−0.310011
$739$	−15.0093	−0.552127	−0.276064	−	0.961139i	$-0.589030\pi$
$739$	−15.0093	−0.552127	−0.276064	+	0.961139i	$0.589030\pi$
$740$	−22.3460	−0.821456
$741$	−25.8090	−0.948119
$742$	−12.7269	−0.467218
$743$	37.6777	1.38226	0.691131	−	0.722729i	$-0.257113\pi$
$743$	37.6777	1.38226	0.691131	+	0.722729i	$0.257113\pi$
$744$	2.10296	0.0770982
$745$	−13.5125	−0.495061
$746$	−4.22351	−0.154634
$747$	4.90361	0.179414
$748$	−8.72030	−0.318846
$749$	22.1874	0.810710
$750$	−10.3001	−0.376105
$751$	0.0365249	0.00133281	0.000666406	−	1.00000i	$-0.499788\pi$
$751$	0.0365249	0.00133281	0.000666406		1.00000i	$0.499788\pi$
$752$	−1.48970	−0.0543236
$753$	12.2127	0.445057
$754$	23.5316	0.856970
$755$	29.6267	1.07822
$756$	−1.61326	−0.0586738
$757$	9.40958	0.341997	0.170999	−	0.985271i	$-0.445301\pi$
$757$	9.40958	0.341997	0.170999	+	0.985271i	$0.445301\pi$
$758$	−9.04753	−0.328621
$759$	11.4234	0.414642
$760$	9.77597	0.354612
$761$	4.60658	0.166988	0.0834941	−	0.996508i	$-0.473392\pi$
$761$	4.60658	0.166988	0.0834941	+	0.996508i	$0.473392\pi$
$762$	−9.49940	−0.344127
$763$	11.3433	0.410654
$764$	19.2018	0.694698
$765$	−16.0516	−0.580347
$766$	−14.8525	−0.536643
$767$	69.6366	2.51443
$768$	−1.00000	−0.0360844
$769$	53.1560	1.91685	0.958427	−	0.285337i	$-0.0921055\pi$
$769$	53.1560	1.91685	0.958427	+	0.285337i	$0.0921055\pi$
$770$	−4.97558	−0.179307
$771$	15.2616	0.549633
$772$	10.2433	0.368665
$773$	−38.6451	−1.38997	−0.694984	−	0.719025i	$-0.744589\pi$
$773$	−38.6451	−1.38997	−0.694984	+	0.719025i	$0.744589\pi$
$774$	3.32292	0.119440
$775$	1.42388	0.0511473
$776$	15.7613	0.565796
$777$	15.1301	0.542791
$778$	4.34446	0.155757
$779$	−34.5543	−1.23804
$780$	14.9878	0.536649
$781$	2.60789	0.0933177
$782$	−59.4530	−2.12604
$783$	−3.74090	−0.133689
$784$	−4.39738	−0.157049
$785$	19.6242	0.700417
$786$	−14.3460	−0.511706
$787$	49.8025	1.77527	0.887634	−	0.460550i	$-0.152348\pi$
$787$	49.8025	1.77527	0.887634	+	0.460550i	$0.152348\pi$
$788$	9.76507	0.347866
$789$	−29.3341	−1.04432
$790$	38.7696	1.37936
$791$	14.0550	0.499738
$792$	−1.29442	−0.0459952
$793$	−27.5429	−0.978075
$794$	36.6902	1.30209
$795$	−18.7966	−0.666646
$796$	−3.06564	−0.108659
$797$	−36.2893	−1.28543	−0.642716	−	0.766104i	$-0.722193\pi$
$797$	−36.2893	−1.28543	−0.642716	+	0.766104i	$0.722193\pi$
$798$	−6.61916	−0.234316
$799$	10.0358	0.355042
$800$	−0.677084	−0.0239385
$801$	18.3704	0.649088
$802$	−7.96441	−0.281233
$803$	9.10634	0.321356
$804$	3.61734	0.127574
$805$	−33.9224	−1.19561
$806$	13.2283	0.465949
$807$	−5.04597	−0.177627
$808$	−2.17267	−0.0764343
$809$	−21.1471	−0.743493	−0.371746	−	0.928334i	$-0.621241\pi$
$809$	−21.1471	−0.743493	−0.371746	+	0.928334i	$0.621241\pi$
$810$	−2.38266	−0.0837183
$811$	−33.5906	−1.17953	−0.589764	−	0.807576i	$-0.700779\pi$
$811$	−33.5906	−1.17953	−0.589764	+	0.807576i	$0.700779\pi$
$812$	6.03507	0.211789
$813$	17.5932	0.617019
$814$	12.1398	0.425501
$815$	−17.3220	−0.606762
$816$	6.73683	0.235836
$817$	13.6338	0.476986
$818$	5.89271	0.206034
$819$	−10.1480	−0.354599
$820$	20.0663	0.700746
$821$	−39.7957	−1.38888	−0.694439	−	0.719551i	$-0.744348\pi$
$821$	−39.7957	−1.38888	−0.694439	+	0.719551i	$0.744348\pi$
$822$	−5.10296	−0.177986
$823$	−33.5506	−1.16950	−0.584751	−	0.811213i	$-0.698808\pi$
$823$	−33.5506	−1.16950	−0.584751	+	0.811213i	$0.698808\pi$
$824$	−4.33131	−0.150888
$825$	−0.876432	−0.0305134
$826$	17.8595	0.621410
$827$	−37.2737	−1.29613	−0.648066	−	0.761584i	$-0.724422\pi$
$827$	−37.2737	−1.29613	−0.648066	+	0.761584i	$0.724422\pi$
$828$	−8.82507	−0.306693
$829$	1.16245	0.0403734	0.0201867	−	0.999796i	$-0.493574\pi$
$829$	1.16245	0.0403734	0.0201867	+	0.999796i	$0.493574\pi$
$830$	−11.6837	−0.405546
$831$	21.1091	0.732266
$832$	−6.29035	−0.218079
$833$	29.6244	1.02643
$834$	−5.63795	−0.195226
$835$	−23.5970	−0.816607
$836$	−5.31096	−0.183683
$837$	−2.10296	−0.0726889
$838$	12.3962	0.428219
$839$	−19.5397	−0.674587	−0.337293	−	0.941400i	$-0.609511\pi$
$839$	−19.5397	−0.674587	−0.337293	+	0.941400i	$0.609511\pi$
$840$	3.84386	0.132626
$841$	−15.0056	−0.517436
$842$	27.5556	0.949628
$843$	−24.8464	−0.855754
$844$	0.240565	0.00828059
$845$	63.3037	2.17771
$846$	1.48970	0.0512168
$847$	−15.0428	−0.516878
$848$	7.88889	0.270906
$849$	−15.2459	−0.523239
$850$	4.56140	0.156455
$851$	82.7668	2.83721
$852$	−2.01472	−0.0690231
$853$	19.5682	0.670004	0.335002	−	0.942217i	$-0.391263\pi$
$853$	19.5682	0.670004	0.335002	+	0.942217i	$0.391263\pi$
$854$	−7.06382	−0.241719
$855$	−9.77597	−0.334331
$856$	−13.7531	−0.470072
$857$	25.5547	0.872932	0.436466	−	0.899721i	$-0.356230\pi$
$857$	25.5547	0.872932	0.436466	+	0.899721i	$0.356230\pi$
$858$	−8.14236	−0.277976
$859$	−17.9939	−0.613945	−0.306973	−	0.951718i	$-0.599316\pi$
$859$	−17.9939	−0.613945	−0.306973	+	0.951718i	$0.599316\pi$
$860$	−7.91739	−0.269981
$861$	−13.5866	−0.463030
$862$	−19.1468	−0.652142
$863$	−25.7633	−0.876993	−0.438497	−	0.898733i	$-0.644489\pi$
$863$	−25.7633	−0.876993	−0.438497	+	0.898733i	$0.644489\pi$
$864$	1.00000	0.0340207
$865$	55.3379	1.88154
$866$	30.9452	1.05156
$867$	−28.3849	−0.964002
$868$	3.39263	0.115153
$869$	−21.0622	−0.714488
$870$	8.91332	0.302190
$871$	22.7543	0.771000
$872$	−7.03125	−0.238108
$873$	−15.7613	−0.533438
$874$	−36.2089	−1.22478
$875$	−16.6167	−0.561747
$876$	−7.03507	−0.237693
$877$	14.5703	0.492004	0.246002	−	0.969269i	$-0.420883\pi$
$877$	14.5703	0.492004	0.246002	+	0.969269i	$0.420883\pi$
$878$	−22.6235	−0.763505
$879$	−16.2334	−0.547538
$880$	3.08417	0.103967
$881$	22.2400	0.749286	0.374643	−	0.927169i	$-0.377765\pi$
$881$	22.2400	0.749286	0.374643	+	0.927169i	$0.377765\pi$
$882$	4.39738	0.148067
$883$	−36.4675	−1.22723	−0.613615	−	0.789606i	$-0.710285\pi$
$883$	−36.4675	−1.22723	−0.613615	+	0.789606i	$0.710285\pi$
$884$	42.3770	1.42529
$885$	26.3770	0.886654
$886$	0.982585	0.0330106
$887$	15.1332	0.508122	0.254061	−	0.967188i	$-0.418234\pi$
$887$	15.1332	0.508122	0.254061	+	0.967188i	$0.418234\pi$
$888$	−9.37859	−0.314725
$889$	−15.3250	−0.513985
$890$	−43.7706	−1.46719
$891$	1.29442	0.0433647
$892$	−1.00000	−0.0334825
$893$	6.11216	0.204536
$894$	−5.67119	−0.189673
$895$	−50.6313	−1.69242
$896$	−1.61326	−0.0538954
$897$	−55.5128	−1.85352
$898$	−5.66487	−0.189039
$899$	7.86697	0.262378
$900$	0.677084	0.0225695
$901$	−53.1462	−1.77056
$902$	−10.9014	−0.362975
$903$	5.36074	0.178394
$904$	−8.71215	−0.289762
$905$	−49.4568	−1.64400
$906$	12.4343	0.413101
$907$	50.7958	1.68665	0.843323	−	0.537407i	$-0.180596\pi$
$907$	50.7958	1.68665	0.843323	+	0.537407i	$0.180596\pi$
$908$	−14.8523	−0.492889
$909$	2.17267	0.0720630
$910$	24.1792	0.801534
$911$	50.6064	1.67667	0.838333	−	0.545159i	$-0.183531\pi$
$911$	50.6064	1.67667	0.838333	+	0.545159i	$0.183531\pi$
$912$	4.10296	0.135863
$913$	6.34734	0.210066
$914$	7.75756	0.256597
$915$	−10.4327	−0.344895
$916$	5.82639	0.192509
$917$	−23.1439	−0.764279
$918$	−6.73683	−0.222349
$919$	−1.34015	−0.0442074	−0.0221037	−	0.999756i	$-0.507036\pi$
$919$	−1.34015	−0.0442074	−0.0221037	+	0.999756i	$0.507036\pi$
$920$	21.0272	0.693246
$921$	−8.09889	−0.266867
$922$	28.0453	0.923622
$923$	−12.6733	−0.417146
$924$	−2.08824	−0.0686981
$925$	−6.35010	−0.208790
$926$	2.07628	0.0682309
$927$	4.33131	0.142259
$928$	−3.74090	−0.122801
$929$	−55.9142	−1.83448	−0.917242	−	0.398330i	$-0.869590\pi$
$929$	−55.9142	−1.83448	−0.917242	+	0.398330i	$0.869590\pi$
$930$	5.01064	0.164305
$931$	18.0423	0.591312
$932$	17.0770	0.559375
$933$	23.2534	0.761284
$934$	−23.7477	−0.777049
$935$	−20.7775	−0.679498
$936$	6.29035	0.205606
$937$	27.2834	0.891310	0.445655	−	0.895205i	$-0.352971\pi$
$937$	27.2834	0.891310	0.445655	+	0.895205i	$0.352971\pi$
$938$	5.83572	0.190543
$939$	−7.91176	−0.258190
$940$	−3.54944	−0.115770
$941$	22.9411	0.747859	0.373929	−	0.927457i	$-0.378010\pi$
$941$	22.9411	0.747859	0.373929	+	0.927457i	$0.378010\pi$
$942$	8.23623	0.268351
$943$	−74.3230	−2.42029
$944$	−11.0704	−0.360311
$945$	−3.84386	−0.125041
$946$	4.30125	0.139846
$947$	42.6806	1.38693	0.693467	−	0.720488i	$-0.256082\pi$
$947$	42.6806	1.38693	0.693467	+	0.720488i	$0.256082\pi$
$948$	16.2716	0.528476
$949$	−44.2530	−1.43651
$950$	2.77805	0.0901318
$951$	2.05975	0.0667919
$952$	10.8683	0.352243
$953$	−26.7188	−0.865507	−0.432753	−	0.901512i	$-0.642458\pi$
$953$	−26.7188	−0.865507	−0.432753	+	0.901512i	$0.642458\pi$
$954$	−7.88889	−0.255412
$955$	45.7515	1.48048
$956$	−22.0091	−0.711824
$957$	−4.84231	−0.156530
$958$	16.7167	0.540093
$959$	−8.23242	−0.265839
$960$	−2.38266	−0.0769001
$961$	−26.5776	−0.857341
$962$	−58.9946	−1.90206
$963$	13.7531	0.443188
$964$	13.7948	0.444299
$965$	24.4064	0.785669
$966$	−14.2372	−0.458073
$967$	−55.9573	−1.79946	−0.899732	−	0.436442i	$-0.856238\pi$
$967$	−55.9573	−1.79946	−0.899732	+	0.436442i	$0.856238\pi$
$968$	9.32447	0.299700
$969$	−27.6409	−0.887955
$970$	37.5538	1.20578
$971$	−5.50174	−0.176559	−0.0882796	−	0.996096i	$-0.528137\pi$
$971$	−5.50174	−0.176559	−0.0882796	+	0.996096i	$0.528137\pi$
$972$	−1.00000	−0.0320750
$973$	−9.09549	−0.291588
$974$	−6.73683	−0.215862
$975$	4.25910	0.136400
$976$	4.37859	0.140155
$977$	−15.0995	−0.483076	−0.241538	−	0.970391i	$-0.577652\pi$
$977$	−15.0995	−0.483076	−0.241538	+	0.970391i	$0.577652\pi$
$978$	−7.27000	−0.232469
$979$	23.7791	0.759983
$980$	−10.4775	−0.334691
$981$	7.03125	0.224491
$982$	−17.0566	−0.544298
$983$	48.5862	1.54966	0.774830	−	0.632170i	$-0.217836\pi$
$983$	48.5862	1.54966	0.774830	+	0.632170i	$0.217836\pi$
$984$	8.42180	0.268477
$985$	23.2669	0.741344
$986$	25.2018	0.802590
$987$	2.40327	0.0764970
$988$	25.8090	0.821095
$989$	29.3250	0.932480
$990$	−3.08417	−0.0980213
$991$	33.9605	1.07879	0.539395	−	0.842053i	$-0.318653\pi$
$991$	33.9605	1.07879	0.539395	+	0.842053i	$0.318653\pi$
$992$	−2.10296	−0.0667690
$993$	−6.59629	−0.209327
$994$	−3.25027	−0.103092
$995$	−7.30439	−0.231565
$996$	−4.90361	−0.155377
$997$	8.69148	0.275262	0.137631	−	0.990484i	$-0.456051\pi$
$997$	8.69148	0.275262	0.137631	+	0.990484i	$0.456051\pi$
$998$	−35.2150	−1.11471
$999$	9.37859	0.296725

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1338.2.a.g.1.4	✓	4
3.2	odd	2		4014.2.a.q.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.g.1.4	✓	4	1.1	even	1	trivial
4014.2.a.q.1.1		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1338 = 2 \cdot 3 \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1338.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.38266 q^{5} +1.00000 q^{6} +1.61326 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.38266 q^{5} +1.00000 q^{6} +1.61326 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.38266 q^{10} +1.29442 q^{11} -1.00000 q^{12} -6.29035 q^{13} -1.61326 q^{14} -2.38266 q^{15} +1.00000 q^{16} -6.73683 q^{17} -1.00000 q^{18} -4.10296 q^{19} +2.38266 q^{20} -1.61326 q^{21} -1.29442 q^{22} -8.82507 q^{23} +1.00000 q^{24} +0.677084 q^{25} +6.29035 q^{26} -1.00000 q^{27} +1.61326 q^{28} +3.74090 q^{29} +2.38266 q^{30} +2.10296 q^{31} -1.00000 q^{32} -1.29442 q^{33} +6.73683 q^{34} +3.84386 q^{35} +1.00000 q^{36} -9.37859 q^{37} +4.10296 q^{38} +6.29035 q^{39} -2.38266 q^{40} +8.42180 q^{41} +1.61326 q^{42} -3.32292 q^{43} +1.29442 q^{44} +2.38266 q^{45} +8.82507 q^{46} -1.48970 q^{47} -1.00000 q^{48} -4.39738 q^{49} -0.677084 q^{50} +6.73683 q^{51} -6.29035 q^{52} +7.88889 q^{53} +1.00000 q^{54} +3.08417 q^{55} -1.61326 q^{56} +4.10296 q^{57} -3.74090 q^{58} -11.0704 q^{59} -2.38266 q^{60} +4.37859 q^{61} -2.10296 q^{62} +1.61326 q^{63} +1.00000 q^{64} -14.9878 q^{65} +1.29442 q^{66} -3.61734 q^{67} -6.73683 q^{68} +8.82507 q^{69} -3.84386 q^{70} +2.01472 q^{71} -1.00000 q^{72} +7.03507 q^{73} +9.37859 q^{74} -0.677084 q^{75} -4.10296 q^{76} +2.08824 q^{77} -6.29035 q^{78} -16.2716 q^{79} +2.38266 q^{80} +1.00000 q^{81} -8.42180 q^{82} +4.90361 q^{83} -1.61326 q^{84} -16.0516 q^{85} +3.32292 q^{86} -3.74090 q^{87} -1.29442 q^{88} +18.3704 q^{89} -2.38266 q^{90} -10.1480 q^{91} -8.82507 q^{92} -2.10296 q^{93} +1.48970 q^{94} -9.77597 q^{95} +1.00000 q^{96} -15.7613 q^{97} +4.39738 q^{98} +1.29442 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} - 5 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 - 5 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} - 5 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} + 4 q^{11} - 4 q^{12} - 5 q^{13} + 5 q^{14} - 2 q^{15} + 4 q^{16} - 2 q^{17} - 4 q^{18} - 7 q^{19} + 2 q^{20} + 5 q^{21} - 4 q^{22} - 4 q^{23} + 4 q^{24} - 6 q^{25} + 5 q^{26} - 4 q^{27} - 5 q^{28} + 9 q^{29} + 2 q^{30} - q^{31} - 4 q^{32} - 4 q^{33} + 2 q^{34} + 4 q^{36} - 11 q^{37} + 7 q^{38} + 5 q^{39} - 2 q^{40} + 14 q^{41} - 5 q^{42} - 22 q^{43} + 4 q^{44} + 2 q^{45} + 4 q^{46} - 8 q^{47} - 4 q^{48} - 7 q^{49} + 6 q^{50} + 2 q^{51} - 5 q^{52} + 3 q^{53} + 4 q^{54} - 13 q^{55} + 5 q^{56} + 7 q^{57} - 9 q^{58} - 6 q^{59} - 2 q^{60} - 9 q^{61} + q^{62} - 5 q^{63} + 4 q^{64} - 3 q^{65} + 4 q^{66} - 22 q^{67} - 2 q^{68} + 4 q^{69} + 5 q^{71} - 4 q^{72} - 3 q^{73} + 11 q^{74} + 6 q^{75} - 7 q^{76} + 2 q^{77} - 5 q^{78} - 29 q^{79} + 2 q^{80} + 4 q^{81} - 14 q^{82} - 12 q^{83} + 5 q^{84} - 10 q^{85} + 22 q^{86} - 9 q^{87} - 4 q^{88} + 9 q^{89} - 2 q^{90} - 18 q^{91} - 4 q^{92} + q^{93} + 8 q^{94} - 2 q^{95} + 4 q^{96} - 29 q^{97} + 7 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 - 5 * q^7 - 4 * q^8 + 4 * q^9 - 2 * q^10 + 4 * q^11 - 4 * q^12 - 5 * q^13 + 5 * q^14 - 2 * q^15 + 4 * q^16 - 2 * q^17 - 4 * q^18 - 7 * q^19 + 2 * q^20 + 5 * q^21 - 4 * q^22 - 4 * q^23 + 4 * q^24 - 6 * q^25 + 5 * q^26 - 4 * q^27 - 5 * q^28 + 9 * q^29 + 2 * q^30 - q^31 - 4 * q^32 - 4 * q^33 + 2 * q^34 + 4 * q^36 - 11 * q^37 + 7 * q^38 + 5 * q^39 - 2 * q^40 + 14 * q^41 - 5 * q^42 - 22 * q^43 + 4 * q^44 + 2 * q^45 + 4 * q^46 - 8 * q^47 - 4 * q^48 - 7 * q^49 + 6 * q^50 + 2 * q^51 - 5 * q^52 + 3 * q^53 + 4 * q^54 - 13 * q^55 + 5 * q^56 + 7 * q^57 - 9 * q^58 - 6 * q^59 - 2 * q^60 - 9 * q^61 + q^62 - 5 * q^63 + 4 * q^64 - 3 * q^65 + 4 * q^66 - 22 * q^67 - 2 * q^68 + 4 * q^69 + 5 * q^71 - 4 * q^72 - 3 * q^73 + 11 * q^74 + 6 * q^75 - 7 * q^76 + 2 * q^77 - 5 * q^78 - 29 * q^79 + 2 * q^80 + 4 * q^81 - 14 * q^82 - 12 * q^83 + 5 * q^84 - 10 * q^85 + 22 * q^86 - 9 * q^87 - 4 * q^88 + 9 * q^89 - 2 * q^90 - 18 * q^91 - 4 * q^92 + q^93 + 8 * q^94 - 2 * q^95 + 4 * q^96 - 29 * q^97 + 7 * q^98 + 4 * q^99

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