LMFDB - Embedded newform 1338.2.a.h.1.3

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:	$0$
Dimension:	$5$
Coefficient field:	5.5.356173.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 9x^{2} + 14x - 4 $ x^5 - 2x^4 - 7x^3 + 9x^2 + 14x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.85688$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.43386 q^{5} -1.00000 q^{6} -1.87103 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.43386 q^{5} -1.00000 q^{6} -1.87103 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.43386 q^{10} +5.50771 q^{11} +1.00000 q^{12} +5.81260 q^{13} +1.87103 q^{14} +1.43386 q^{15} +1.00000 q^{16} -3.16177 q^{17} -1.00000 q^{18} -0.890321 q^{19} +1.43386 q^{20} -1.87103 q^{21} -5.50771 q^{22} +1.68749 q^{23} -1.00000 q^{24} -2.94404 q^{25} -5.81260 q^{26} +1.00000 q^{27} -1.87103 q^{28} -6.78000 q^{29} -1.43386 q^{30} +2.18675 q^{31} -1.00000 q^{32} +5.50771 q^{33} +3.16177 q^{34} -2.68279 q^{35} +1.00000 q^{36} +9.53205 q^{37} +0.890321 q^{38} +5.81260 q^{39} -1.43386 q^{40} +5.50256 q^{41} +1.87103 q^{42} -9.08158 q^{43} +5.50771 q^{44} +1.43386 q^{45} -1.68749 q^{46} +4.43633 q^{47} +1.00000 q^{48} -3.49926 q^{49} +2.94404 q^{50} -3.16177 q^{51} +5.81260 q^{52} +9.50010 q^{53} -1.00000 q^{54} +7.89729 q^{55} +1.87103 q^{56} -0.890321 q^{57} +6.78000 q^{58} +14.8655 q^{59} +1.43386 q^{60} -12.3227 q^{61} -2.18675 q^{62} -1.87103 q^{63} +1.00000 q^{64} +8.33446 q^{65} -5.50771 q^{66} -0.993472 q^{67} -3.16177 q^{68} +1.68749 q^{69} +2.68279 q^{70} +4.42624 q^{71} -1.00000 q^{72} +2.32096 q^{73} -9.53205 q^{74} -2.94404 q^{75} -0.890321 q^{76} -10.3051 q^{77} -5.81260 q^{78} -12.5579 q^{79} +1.43386 q^{80} +1.00000 q^{81} -5.50256 q^{82} +15.1636 q^{83} -1.87103 q^{84} -4.53354 q^{85} +9.08158 q^{86} -6.78000 q^{87} -5.50771 q^{88} -3.41540 q^{89} -1.43386 q^{90} -10.8755 q^{91} +1.68749 q^{92} +2.18675 q^{93} -4.43633 q^{94} -1.27660 q^{95} -1.00000 q^{96} -13.6143 q^{97} +3.49926 q^{98} +5.50771 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} + 5 q^{5} - 5 q^{6} - q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 + 5 * q^5 - 5 * q^6 - q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} + 5 q^{5} - 5 q^{6} - q^{7} - 5 q^{8} + 5 q^{9} - 5 q^{10} + 9 q^{11} + 5 q^{12} + q^{14} + 5 q^{15} + 5 q^{16} + 6 q^{17} - 5 q^{18} - 4 q^{19} + 5 q^{20} - q^{21} - 9 q^{22} + 16 q^{23} - 5 q^{24} + 8 q^{25} + 5 q^{27} - q^{28} + 8 q^{29} - 5 q^{30} - q^{31} - 5 q^{32} + 9 q^{33} - 6 q^{34} + 22 q^{35} + 5 q^{36} - 2 q^{37} + 4 q^{38} - 5 q^{40} + 4 q^{41} + q^{42} + 3 q^{43} + 9 q^{44} + 5 q^{45} - 16 q^{46} + 18 q^{47} + 5 q^{48} + 2 q^{49} - 8 q^{50} + 6 q^{51} + 26 q^{53} - 5 q^{54} + q^{55} + q^{56} - 4 q^{57} - 8 q^{58} + 21 q^{59} + 5 q^{60} - 20 q^{61} + q^{62} - q^{63} + 5 q^{64} - 3 q^{65} - 9 q^{66} - 5 q^{67} + 6 q^{68} + 16 q^{69} - 22 q^{70} + 17 q^{71} - 5 q^{72} + 5 q^{73} + 2 q^{74} + 8 q^{75} - 4 q^{76} + 2 q^{77} - 21 q^{79} + 5 q^{80} + 5 q^{81} - 4 q^{82} + 11 q^{83} - q^{84} - 12 q^{85} - 3 q^{86} + 8 q^{87} - 9 q^{88} - 5 q^{89} - 5 q^{90} - 10 q^{91} + 16 q^{92} - q^{93} - 18 q^{94} + 10 q^{95} - 5 q^{96} - 11 q^{97} - 2 q^{98} + 9 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 + 5 * q^5 - 5 * q^6 - q^7 - 5 * q^8 + 5 * q^9 - 5 * q^10 + 9 * q^11 + 5 * q^12 + q^14 + 5 * q^15 + 5 * q^16 + 6 * q^17 - 5 * q^18 - 4 * q^19 + 5 * q^20 - q^21 - 9 * q^22 + 16 * q^23 - 5 * q^24 + 8 * q^25 + 5 * q^27 - q^28 + 8 * q^29 - 5 * q^30 - q^31 - 5 * q^32 + 9 * q^33 - 6 * q^34 + 22 * q^35 + 5 * q^36 - 2 * q^37 + 4 * q^38 - 5 * q^40 + 4 * q^41 + q^42 + 3 * q^43 + 9 * q^44 + 5 * q^45 - 16 * q^46 + 18 * q^47 + 5 * q^48 + 2 * q^49 - 8 * q^50 + 6 * q^51 + 26 * q^53 - 5 * q^54 + q^55 + q^56 - 4 * q^57 - 8 * q^58 + 21 * q^59 + 5 * q^60 - 20 * q^61 + q^62 - q^63 + 5 * q^64 - 3 * q^65 - 9 * q^66 - 5 * q^67 + 6 * q^68 + 16 * q^69 - 22 * q^70 + 17 * q^71 - 5 * q^72 + 5 * q^73 + 2 * q^74 + 8 * q^75 - 4 * q^76 + 2 * q^77 - 21 * q^79 + 5 * q^80 + 5 * q^81 - 4 * q^82 + 11 * q^83 - q^84 - 12 * q^85 - 3 * q^86 + 8 * q^87 - 9 * q^88 - 5 * q^89 - 5 * q^90 - 10 * q^91 + 16 * q^92 - q^93 - 18 * q^94 + 10 * q^95 - 5 * q^96 - 11 * q^97 - 2 * q^98 + 9 * 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
$5$	1.43386	0.641242	0.320621	−	0.947208i	$-0.396108\pi$
$5$	1.43386	0.641242	0.320621	+	0.947208i	$0.396108\pi$
$6$	−1.00000	−0.408248
$7$	−1.87103	−0.707182	−0.353591	−	0.935400i	$-0.615039\pi$
$7$	−1.87103	−0.707182	−0.353591	+	0.935400i	$0.615039\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.43386	−0.453427
$11$	5.50771	1.66064	0.830319	−	0.557288i	$-0.188158\pi$
$11$	5.50771	1.66064	0.830319	+	0.557288i	$0.188158\pi$
$12$	1.00000	0.288675
$13$	5.81260	1.61213	0.806063	−	0.591830i	$-0.201594\pi$
$13$	5.81260	1.61213	0.806063	+	0.591830i	$0.201594\pi$
$14$	1.87103	0.500053
$15$	1.43386	0.370221
$16$	1.00000	0.250000
$17$	−3.16177	−0.766841	−0.383421	−	0.923574i	$-0.625254\pi$
$17$	−3.16177	−0.766841	−0.383421	+	0.923574i	$0.625254\pi$
$18$	−1.00000	−0.235702
$19$	−0.890321	−0.204254	−0.102127	−	0.994771i	$-0.532565\pi$
$19$	−0.890321	−0.204254	−0.102127	+	0.994771i	$0.532565\pi$
$20$	1.43386	0.320621
$21$	−1.87103	−0.408291
$22$	−5.50771	−1.17425
$23$	1.68749	0.351867	0.175933	−	0.984402i	$-0.443706\pi$
$23$	1.68749	0.351867	0.175933	+	0.984402i	$0.443706\pi$
$24$	−1.00000	−0.204124
$25$	−2.94404	−0.588809
$26$	−5.81260	−1.13994
$27$	1.00000	0.192450
$28$	−1.87103	−0.353591
$29$	−6.78000	−1.25901	−0.629507	−	0.776995i	$-0.716743\pi$
$29$	−6.78000	−1.25901	−0.629507	+	0.776995i	$0.716743\pi$
$30$	−1.43386	−0.261786
$31$	2.18675	0.392753	0.196376	−	0.980529i	$-0.437083\pi$
$31$	2.18675	0.392753	0.196376	+	0.980529i	$0.437083\pi$
$32$	−1.00000	−0.176777
$33$	5.50771	0.958770
$34$	3.16177	0.542239
$35$	−2.68279	−0.453475
$36$	1.00000	0.166667
$37$	9.53205	1.56706	0.783530	−	0.621354i	$-0.213417\pi$
$37$	9.53205	1.56706	0.783530	+	0.621354i	$0.213417\pi$
$38$	0.890321	0.144429
$39$	5.81260	0.930761
$40$	−1.43386	−0.226713
$41$	5.50256	0.859356	0.429678	−	0.902982i	$-0.358627\pi$
$41$	5.50256	0.859356	0.429678	+	0.902982i	$0.358627\pi$
$42$	1.87103	0.288706
$43$	−9.08158	−1.38493	−0.692464	−	0.721453i	$-0.743475\pi$
$43$	−9.08158	−1.38493	−0.692464	+	0.721453i	$0.743475\pi$
$44$	5.50771	0.830319
$45$	1.43386	0.213747
$46$	−1.68749	−0.248808
$47$	4.43633	0.647105	0.323553	−	0.946210i	$-0.395123\pi$
$47$	4.43633	0.647105	0.323553	+	0.946210i	$0.395123\pi$
$48$	1.00000	0.144338
$49$	−3.49926	−0.499894
$50$	2.94404	0.416351
$51$	−3.16177	−0.442736
$52$	5.81260	0.806063
$53$	9.50010	1.30494	0.652469	−	0.757815i	$-0.273733\pi$
$53$	9.50010	1.30494	0.652469	+	0.757815i	$0.273733\pi$
$54$	−1.00000	−0.136083
$55$	7.89729	1.06487
$56$	1.87103	0.250026
$57$	−0.890321	−0.117926
$58$	6.78000	0.890257
$59$	14.8655	1.93533	0.967663	−	0.252246i	$-0.0811691\pi$
$59$	14.8655	1.93533	0.967663	+	0.252246i	$0.0811691\pi$
$60$	1.43386	0.185111
$61$	−12.3227	−1.57776	−0.788880	−	0.614547i	$-0.789339\pi$
$61$	−12.3227	−1.57776	−0.788880	+	0.614547i	$0.789339\pi$
$62$	−2.18675	−0.277718
$63$	−1.87103	−0.235727
$64$	1.00000	0.125000
$65$	8.33446	1.03376
$66$	−5.50771	−0.677953
$67$	−0.993472	−0.121372	−0.0606860	−	0.998157i	$-0.519329\pi$
$67$	−0.993472	−0.121372	−0.0606860	+	0.998157i	$0.519329\pi$
$68$	−3.16177	−0.383421
$69$	1.68749	0.203150
$70$	2.68279	0.320655
$71$	4.42624	0.525298	0.262649	−	0.964891i	$-0.415404\pi$
$71$	4.42624	0.525298	0.262649	+	0.964891i	$0.415404\pi$
$72$	−1.00000	−0.117851
$73$	2.32096	0.271648	0.135824	−	0.990733i	$-0.456632\pi$
$73$	2.32096	0.271648	0.135824	+	0.990733i	$0.456632\pi$
$74$	−9.53205	−1.10808
$75$	−2.94404	−0.339949
$76$	−0.890321	−0.102127
$77$	−10.3051	−1.17437
$78$	−5.81260	−0.658147
$79$	−12.5579	−1.41287	−0.706436	−	0.707777i	$-0.749698\pi$
$79$	−12.5579	−1.41287	−0.706436	+	0.707777i	$0.749698\pi$
$80$	1.43386	0.160310
$81$	1.00000	0.111111
$82$	−5.50256	−0.607657
$83$	15.1636	1.66442	0.832210	−	0.554460i	$-0.187075\pi$
$83$	15.1636	1.66442	0.832210	+	0.554460i	$0.187075\pi$
$84$	−1.87103	−0.204146
$85$	−4.53354	−0.491731
$86$	9.08158	0.979292
$87$	−6.78000	−0.726892
$88$	−5.50771	−0.587124
$89$	−3.41540	−0.362032	−0.181016	−	0.983480i	$-0.557939\pi$
$89$	−3.41540	−0.362032	−0.181016	+	0.983480i	$0.557939\pi$
$90$	−1.43386	−0.151142
$91$	−10.8755	−1.14007
$92$	1.68749	0.175933
$93$	2.18675	0.226756
$94$	−4.43633	−0.457573
$95$	−1.27660	−0.130976
$96$	−1.00000	−0.102062
$97$	−13.6143	−1.38232	−0.691160	−	0.722702i	$-0.742900\pi$
$97$	−13.6143	−1.38232	−0.691160	+	0.722702i	$0.742900\pi$
$98$	3.49926	0.353479
$99$	5.50771	0.553546
$100$	−2.94404	−0.294404
$101$	−9.77079	−0.972230	−0.486115	−	0.873895i	$-0.661586\pi$
$101$	−9.77079	−0.972230	−0.486115	+	0.873895i	$0.661586\pi$
$102$	3.16177	0.313062
$103$	7.03849	0.693523	0.346761	−	0.937953i	$-0.387281\pi$
$103$	7.03849	0.693523	0.346761	+	0.937953i	$0.387281\pi$
$104$	−5.81260	−0.569972
$105$	−2.68279	−0.261814
$106$	−9.50010	−0.922731
$107$	12.3178	1.19081	0.595405	−	0.803425i	$-0.296991\pi$
$107$	12.3178	1.19081	0.595405	+	0.803425i	$0.296991\pi$
$108$	1.00000	0.0962250
$109$	15.5040	1.48502	0.742509	−	0.669836i	$-0.233636\pi$
$109$	15.5040	1.48502	0.742509	+	0.669836i	$0.233636\pi$
$110$	−7.89729	−0.752977
$111$	9.53205	0.904743
$112$	−1.87103	−0.176795
$113$	8.44083	0.794047	0.397023	−	0.917808i	$-0.370043\pi$
$113$	8.44083	0.794047	0.397023	+	0.917808i	$0.370043\pi$
$114$	0.890321	0.0833862
$115$	2.41963	0.225632
$116$	−6.78000	−0.629507
$117$	5.81260	0.537375
$118$	−14.8655	−1.36848
$119$	5.91575	0.542296
$120$	−1.43386	−0.130893
$121$	19.3349	1.75772
$122$	12.3227	1.11564
$123$	5.50256	0.496150
$124$	2.18675	0.196376
$125$	−11.3907	−1.01881
$126$	1.87103	0.166684
$127$	12.7041	1.12731	0.563654	−	0.826011i	$-0.309395\pi$
$127$	12.7041	1.12731	0.563654	+	0.826011i	$0.309395\pi$
$128$	−1.00000	−0.0883883
$129$	−9.08158	−0.799588
$130$	−8.33446	−0.730980
$131$	−12.8692	−1.12439	−0.562193	−	0.827006i	$-0.690042\pi$
$131$	−12.8692	−1.12439	−0.562193	+	0.827006i	$0.690042\pi$
$132$	5.50771	0.479385
$133$	1.66581	0.144444
$134$	0.993472	0.0858229
$135$	1.43386	0.123407
$136$	3.16177	0.271119
$137$	−5.08158	−0.434149	−0.217074	−	0.976155i	$-0.569651\pi$
$137$	−5.08158	−0.434149	−0.217074	+	0.976155i	$0.569651\pi$
$138$	−1.68749	−0.143649
$139$	−6.73928	−0.571618	−0.285809	−	0.958287i	$-0.592262\pi$
$139$	−6.73928	−0.571618	−0.285809	+	0.958287i	$0.592262\pi$
$140$	−2.68279	−0.226737
$141$	4.43633	0.373606
$142$	−4.42624	−0.371442
$143$	32.0141	2.67716
$144$	1.00000	0.0833333
$145$	−9.72157	−0.807332
$146$	−2.32096	−0.192084
$147$	−3.49926	−0.288614
$148$	9.53205	0.783530
$149$	2.81176	0.230349	0.115174	−	0.993345i	$-0.463257\pi$
$149$	2.81176	0.230349	0.115174	+	0.993345i	$0.463257\pi$
$150$	2.94404	0.240380
$151$	−7.74113	−0.629965	−0.314982	−	0.949098i	$-0.601999\pi$
$151$	−7.74113	−0.629965	−0.314982	+	0.949098i	$0.601999\pi$
$152$	0.890321	0.0722145
$153$	−3.16177	−0.255614
$154$	10.3051	0.830407
$155$	3.13550	0.251850
$156$	5.81260	0.465380
$157$	−19.2202	−1.53394	−0.766969	−	0.641684i	$-0.778236\pi$
$157$	−19.2202	−1.53394	−0.766969	+	0.641684i	$0.778236\pi$
$158$	12.5579	0.999051
$159$	9.50010	0.753407
$160$	−1.43386	−0.113357
$161$	−3.15735	−0.248834
$162$	−1.00000	−0.0785674
$163$	4.23627	0.331810	0.165905	−	0.986142i	$-0.446945\pi$
$163$	4.23627	0.331810	0.165905	+	0.986142i	$0.446945\pi$
$164$	5.50256	0.429678
$165$	7.89729	0.614803
$166$	−15.1636	−1.17692
$167$	24.8132	1.92011	0.960053	−	0.279818i	$-0.0902743\pi$
$167$	24.8132	1.92011	0.960053	+	0.279818i	$0.0902743\pi$
$168$	1.87103	0.144353
$169$	20.7863	1.59895
$170$	4.53354	0.347706
$171$	−0.890321	−0.0680845
$172$	−9.08158	−0.692464
$173$	3.90530	0.296915	0.148457	−	0.988919i	$-0.452569\pi$
$173$	3.90530	0.296915	0.148457	+	0.988919i	$0.452569\pi$
$174$	6.78000	0.513990
$175$	5.50838	0.416395
$176$	5.50771	0.415160
$177$	14.8655	1.11736
$178$	3.41540	0.255995
$179$	6.73708	0.503553	0.251776	−	0.967785i	$-0.418985\pi$
$179$	6.73708	0.503553	0.251776	+	0.967785i	$0.418985\pi$
$180$	1.43386	0.106874
$181$	−3.43219	−0.255113	−0.127556	−	0.991831i	$-0.540713\pi$
$181$	−3.43219	−0.255113	−0.127556	+	0.991831i	$0.540713\pi$
$182$	10.8755	0.806148
$183$	−12.3227	−0.910920
$184$	−1.68749	−0.124404
$185$	13.6676	1.00486
$186$	−2.18675	−0.160341
$187$	−17.4141	−1.27345
$188$	4.43633	0.323553
$189$	−1.87103	−0.136097
$190$	1.27660	0.0926140
$191$	−19.5410	−1.41393	−0.706967	−	0.707247i	$-0.749937\pi$
$191$	−19.5410	−1.41393	−0.706967	+	0.707247i	$0.749937\pi$
$192$	1.00000	0.0721688
$193$	−5.55650	−0.399965	−0.199983	−	0.979799i	$-0.564089\pi$
$193$	−5.55650	−0.399965	−0.199983	+	0.979799i	$0.564089\pi$
$194$	13.6143	0.977448
$195$	8.33446	0.596843
$196$	−3.49926	−0.249947
$197$	−21.9757	−1.56571	−0.782853	−	0.622207i	$-0.786236\pi$
$197$	−21.9757	−1.56571	−0.782853	+	0.622207i	$0.786236\pi$
$198$	−5.50771	−0.391416
$199$	4.07733	0.289034	0.144517	−	0.989502i	$-0.453837\pi$
$199$	4.07733	0.289034	0.144517	+	0.989502i	$0.453837\pi$
$200$	2.94404	0.208175
$201$	−0.993472	−0.0700741
$202$	9.77079	0.687470
$203$	12.6856	0.890351
$204$	−3.16177	−0.221368
$205$	7.88991	0.551055
$206$	−7.03849	−0.490395
$207$	1.68749	0.117289
$208$	5.81260	0.403031
$209$	−4.90363	−0.339191
$210$	2.68279	0.185130
$211$	−10.0288	−0.690409	−0.345205	−	0.938527i	$-0.612190\pi$
$211$	−10.0288	−0.690409	−0.345205	+	0.938527i	$0.612190\pi$
$212$	9.50010	0.652469
$213$	4.42624	0.303281
$214$	−12.3178	−0.842030
$215$	−13.0217	−0.888074
$216$	−1.00000	−0.0680414
$217$	−4.09148	−0.277747
$218$	−15.5040	−1.05007
$219$	2.32096	0.156836
$220$	7.89729	0.532435
$221$	−18.3781	−1.23624
$222$	−9.53205	−0.639750
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	1.87103	0.125013
$225$	−2.94404	−0.196270
$226$	−8.44083	−0.561476
$227$	−20.0364	−1.32986	−0.664930	−	0.746906i	$-0.731539\pi$
$227$	−20.0364	−1.32986	−0.664930	+	0.746906i	$0.731539\pi$
$228$	−0.890321	−0.0589629
$229$	−18.4886	−1.22176	−0.610881	−	0.791723i	$-0.709185\pi$
$229$	−18.4886	−1.22176	−0.610881	+	0.791723i	$0.709185\pi$
$230$	−2.41963	−0.159546
$231$	−10.3051	−0.678024
$232$	6.78000	0.445129
$233$	6.03324	0.395251	0.197625	−	0.980278i	$-0.436677\pi$
$233$	6.03324	0.395251	0.197625	+	0.980278i	$0.436677\pi$
$234$	−5.81260	−0.379982
$235$	6.36108	0.414951
$236$	14.8655	0.967663
$237$	−12.5579	−0.815722
$238$	−5.91575	−0.383461
$239$	−24.6088	−1.59181	−0.795906	−	0.605420i	$-0.793005\pi$
$239$	−24.6088	−1.59181	−0.795906	+	0.605420i	$0.793005\pi$
$240$	1.43386	0.0925553
$241$	12.3387	0.794804	0.397402	−	0.917645i	$-0.369912\pi$
$241$	12.3387	0.794804	0.397402	+	0.917645i	$0.369912\pi$
$242$	−19.3349	−1.24289
$243$	1.00000	0.0641500
$244$	−12.3227	−0.788880
$245$	−5.01745	−0.320553
$246$	−5.50256	−0.350831
$247$	−5.17508	−0.329282
$248$	−2.18675	−0.138859
$249$	15.1636	0.960954
$250$	11.3907	0.720408
$251$	−13.9475	−0.880361	−0.440180	−	0.897909i	$-0.645086\pi$
$251$	−13.9475	−0.880361	−0.440180	+	0.897909i	$0.645086\pi$
$252$	−1.87103	−0.117864
$253$	9.29424	0.584324
$254$	−12.7041	−0.797127
$255$	−4.53354	−0.283901
$256$	1.00000	0.0625000
$257$	20.0207	1.24885	0.624427	−	0.781083i	$-0.285332\pi$
$257$	20.0207	1.24885	0.624427	+	0.781083i	$0.285332\pi$
$258$	9.08158	0.565394
$259$	−17.8347	−1.10820
$260$	8.33446	0.516881
$261$	−6.78000	−0.419671
$262$	12.8692	0.795061
$263$	−2.19289	−0.135219	−0.0676097	−	0.997712i	$-0.521537\pi$
$263$	−2.19289	−0.135219	−0.0676097	+	0.997712i	$0.521537\pi$
$264$	−5.50771	−0.338976
$265$	13.6218	0.836781
$266$	−1.66581	−0.102138
$267$	−3.41540	−0.209019
$268$	−0.993472	−0.0606860
$269$	−26.2148	−1.59835	−0.799174	−	0.601100i	$-0.794729\pi$
$269$	−26.2148	−1.59835	−0.799174	+	0.601100i	$0.794729\pi$
$270$	−1.43386	−0.0872620
$271$	8.88235	0.539564	0.269782	−	0.962921i	$-0.413048\pi$
$271$	8.88235	0.539564	0.269782	+	0.962921i	$0.413048\pi$
$272$	−3.16177	−0.191710
$273$	−10.8755	−0.658217
$274$	5.08158	0.306989
$275$	−16.2149	−0.977798
$276$	1.68749	0.101575
$277$	−25.3250	−1.52163	−0.760815	−	0.648968i	$-0.775201\pi$
$277$	−25.3250	−1.52163	−0.760815	+	0.648968i	$0.775201\pi$
$278$	6.73928	0.404195
$279$	2.18675	0.130918
$280$	2.68279	0.160327
$281$	20.0533	1.19628	0.598139	−	0.801392i	$-0.295907\pi$
$281$	20.0533	1.19628	0.598139	+	0.801392i	$0.295907\pi$
$282$	−4.43633	−0.264180
$283$	21.6602	1.28757	0.643783	−	0.765208i	$-0.277364\pi$
$283$	21.6602	1.28757	0.643783	+	0.765208i	$0.277364\pi$
$284$	4.42624	0.262649
$285$	−1.27660	−0.0756190
$286$	−32.0141	−1.89304
$287$	−10.2954	−0.607721
$288$	−1.00000	−0.0589256
$289$	−7.00322	−0.411954
$290$	9.72157	0.570870
$291$	−13.6143	−0.798083
$292$	2.32096	0.135824
$293$	−18.1963	−1.06304	−0.531519	−	0.847046i	$-0.678379\pi$
$293$	−18.1963	−1.06304	−0.531519	+	0.847046i	$0.678379\pi$
$294$	3.49926	0.204081
$295$	21.3151	1.24101
$296$	−9.53205	−0.554039
$297$	5.50771	0.319590
$298$	−2.81176	−0.162881
$299$	9.80873	0.567254
$300$	−2.94404	−0.169974
$301$	16.9919	0.979395
$302$	7.74113	0.445452
$303$	−9.77079	−0.561317
$304$	−0.890321	−0.0510634
$305$	−17.6690	−1.01173
$306$	3.16177	0.180746
$307$	−9.63403	−0.549843	−0.274921	−	0.961467i	$-0.588652\pi$
$307$	−9.63403	−0.549843	−0.274921	+	0.961467i	$0.588652\pi$
$308$	−10.3051	−0.587186
$309$	7.03849	0.400406
$310$	−3.13550	−0.178084
$311$	−7.31701	−0.414909	−0.207455	−	0.978245i	$-0.566518\pi$
$311$	−7.31701	−0.414909	−0.207455	+	0.978245i	$0.566518\pi$
$312$	−5.81260	−0.329074
$313$	20.3220	1.14867	0.574333	−	0.818622i	$-0.305261\pi$
$313$	20.3220	1.14867	0.574333	+	0.818622i	$0.305261\pi$
$314$	19.2202	1.08466
$315$	−2.68279	−0.151158
$316$	−12.5579	−0.706436
$317$	−34.6867	−1.94820	−0.974099	−	0.226121i	$-0.927395\pi$
$317$	−34.6867	−1.94820	−0.974099	+	0.226121i	$0.927395\pi$
$318$	−9.50010	−0.532739
$319$	−37.3423	−2.09077
$320$	1.43386	0.0801552
$321$	12.3178	0.687515
$322$	3.15735	0.175952
$323$	2.81499	0.156630
$324$	1.00000	0.0555556
$325$	−17.1125	−0.949233
$326$	−4.23627	−0.234625
$327$	15.5040	0.857375
$328$	−5.50256	−0.303828
$329$	−8.30049	−0.457621
$330$	−7.89729	−0.434732
$331$	20.8200	1.14437	0.572184	−	0.820125i	$-0.306096\pi$
$331$	20.8200	1.14437	0.572184	+	0.820125i	$0.306096\pi$
$332$	15.1636	0.832210
$333$	9.53205	0.522353
$334$	−24.8132	−1.35772
$335$	−1.42450	−0.0778288
$336$	−1.87103	−0.102073
$337$	17.8513	0.972421	0.486211	−	0.873842i	$-0.338379\pi$
$337$	17.8513	0.972421	0.486211	+	0.873842i	$0.338379\pi$
$338$	−20.7863	−1.13063
$339$	8.44083	0.458443
$340$	−4.53354	−0.245865
$341$	12.0440	0.652220
$342$	0.890321	0.0481430
$343$	19.6444	1.06070
$344$	9.08158	0.489646
$345$	2.41963	0.130269
$346$	−3.90530	−0.209950
$347$	16.4617	0.883709	0.441855	−	0.897087i	$-0.354321\pi$
$347$	16.4617	0.883709	0.441855	+	0.897087i	$0.354321\pi$
$348$	−6.78000	−0.363446
$349$	−15.7220	−0.841581	−0.420791	−	0.907158i	$-0.638247\pi$
$349$	−15.7220	−0.841581	−0.420791	+	0.907158i	$0.638247\pi$
$350$	−5.50838	−0.294435
$351$	5.81260	0.310254
$352$	−5.50771	−0.293562
$353$	23.2166	1.23569	0.617847	−	0.786298i	$-0.288005\pi$
$353$	23.2166	1.23569	0.617847	+	0.786298i	$0.288005\pi$
$354$	−14.8655	−0.790094
$355$	6.34662	0.336843
$356$	−3.41540	−0.181016
$357$	5.91575	0.313095
$358$	−6.73708	−0.356066
$359$	−15.3354	−0.809372	−0.404686	−	0.914456i	$-0.632619\pi$
$359$	−15.3354	−0.809372	−0.404686	+	0.914456i	$0.632619\pi$
$360$	−1.43386	−0.0755711
$361$	−18.2073	−0.958280
$362$	3.43219	0.180392
$363$	19.3349	1.01482
$364$	−10.8755	−0.570033
$365$	3.32793	0.174192
$366$	12.3227	0.644118
$367$	−1.82214	−0.0951152	−0.0475576	−	0.998868i	$-0.515144\pi$
$367$	−1.82214	−0.0951152	−0.0475576	+	0.998868i	$0.515144\pi$
$368$	1.68749	0.0879667
$369$	5.50256	0.286452
$370$	−13.6676	−0.710547
$371$	−17.7749	−0.922828
$372$	2.18675	0.113378
$373$	−24.9826	−1.29355	−0.646776	−	0.762680i	$-0.723883\pi$
$373$	−24.9826	−1.29355	−0.646776	+	0.762680i	$0.723883\pi$
$374$	17.4141	0.900462
$375$	−11.3907	−0.588211
$376$	−4.43633	−0.228786
$377$	−39.4094	−2.02969
$378$	1.87103	0.0962352
$379$	−9.99010	−0.513157	−0.256579	−	0.966523i	$-0.582595\pi$
$379$	−9.99010	−0.513157	−0.256579	+	0.966523i	$0.582595\pi$
$380$	−1.27660	−0.0654880
$381$	12.7041	0.650852
$382$	19.5410	0.999802
$383$	−14.7719	−0.754808	−0.377404	−	0.926049i	$-0.623183\pi$
$383$	−14.7719	−0.754808	−0.377404	+	0.926049i	$0.623183\pi$
$384$	−1.00000	−0.0510310
$385$	−14.7760	−0.753057
$386$	5.55650	0.282818
$387$	−9.08158	−0.461643
$388$	−13.6143	−0.691160
$389$	26.0548	1.32103	0.660516	−	0.750812i	$-0.270337\pi$
$389$	26.0548	1.32103	0.660516	+	0.750812i	$0.270337\pi$
$390$	−8.33446	−0.422032
$391$	−5.33547	−0.269826
$392$	3.49926	0.176739
$393$	−12.8692	−0.649165
$394$	21.9757	1.10712
$395$	−18.0062	−0.905993
$396$	5.50771	0.276773
$397$	−25.3046	−1.27000	−0.635000	−	0.772512i	$-0.719000\pi$
$397$	−25.3046	−1.27000	−0.635000	+	0.772512i	$0.719000\pi$
$398$	−4.07733	−0.204378
$399$	1.66581	0.0833950
$400$	−2.94404	−0.147202
$401$	16.9955	0.848716	0.424358	−	0.905495i	$-0.360500\pi$
$401$	16.9955	0.848716	0.424358	+	0.905495i	$0.360500\pi$
$402$	0.993472	0.0495499
$403$	12.7107	0.633167
$404$	−9.77079	−0.486115
$405$	1.43386	0.0712491
$406$	−12.6856	−0.629573
$407$	52.4998	2.60232
$408$	3.16177	0.156531
$409$	−30.6720	−1.51663	−0.758315	−	0.651888i	$-0.773977\pi$
$409$	−30.6720	−1.51663	−0.758315	+	0.651888i	$0.773977\pi$
$410$	−7.88991	−0.389655
$411$	−5.08158	−0.250656
$412$	7.03849	0.346761
$413$	−27.8138	−1.36863
$414$	−1.68749	−0.0829358
$415$	21.7425	1.06730
$416$	−5.81260	−0.284986
$417$	−6.73928	−0.330024
$418$	4.90363	0.239844
$419$	17.4958	0.854724	0.427362	−	0.904081i	$-0.359443\pi$
$419$	17.4958	0.854724	0.427362	+	0.904081i	$0.359443\pi$
$420$	−2.68279	−0.130907
$421$	−12.6330	−0.615695	−0.307847	−	0.951436i	$-0.599609\pi$
$421$	−12.6330	−0.615695	−0.307847	+	0.951436i	$0.599609\pi$
$422$	10.0288	0.488193
$423$	4.43633	0.215702
$424$	−9.50010	−0.461365
$425$	9.30838	0.451523
$426$	−4.42624	−0.214452
$427$	23.0561	1.11576
$428$	12.3178	0.595405
$429$	32.0141	1.54566
$430$	13.0217	0.627963
$431$	14.6767	0.706953	0.353477	−	0.935443i	$-0.384999\pi$
$431$	14.6767	0.706953	0.353477	+	0.935443i	$0.384999\pi$
$432$	1.00000	0.0481125
$433$	−22.5184	−1.08216	−0.541082	−	0.840970i	$-0.681985\pi$
$433$	−22.5184	−1.08216	−0.541082	+	0.840970i	$0.681985\pi$
$434$	4.09148	0.196397
$435$	−9.72157	−0.466114
$436$	15.5040	0.742509
$437$	−1.50241	−0.0718701
$438$	−2.32096	−0.110900
$439$	−8.52689	−0.406966	−0.203483	−	0.979078i	$-0.565226\pi$
$439$	−8.52689	−0.406966	−0.203483	+	0.979078i	$0.565226\pi$
$440$	−7.89729	−0.376489
$441$	−3.49926	−0.166631
$442$	18.3781	0.874157
$443$	−10.6999	−0.508368	−0.254184	−	0.967156i	$-0.581807\pi$
$443$	−10.6999	−0.508368	−0.254184	+	0.967156i	$0.581807\pi$
$444$	9.53205	0.452371
$445$	−4.89721	−0.232150
$446$	1.00000	0.0473514
$447$	2.81176	0.132992
$448$	−1.87103	−0.0883977
$449$	−42.0615	−1.98501	−0.992503	−	0.122224i	$-0.960997\pi$
$449$	−42.0615	−1.98501	−0.992503	+	0.122224i	$0.960997\pi$
$450$	2.94404	0.138784
$451$	30.3065	1.42708
$452$	8.44083	0.397023
$453$	−7.74113	−0.363710
$454$	20.0364	0.940353
$455$	−15.5940	−0.731058
$456$	0.890321	0.0416931
$457$	−8.63916	−0.404123	−0.202061	−	0.979373i	$-0.564764\pi$
$457$	−8.63916	−0.404123	−0.202061	+	0.979373i	$0.564764\pi$
$458$	18.4886	0.863916
$459$	−3.16177	−0.147579
$460$	2.41963	0.112816
$461$	−12.8499	−0.598480	−0.299240	−	0.954178i	$-0.596733\pi$
$461$	−12.8499	−0.598480	−0.299240	+	0.954178i	$0.596733\pi$
$462$	10.3051	0.479436
$463$	−14.9786	−0.696115	−0.348058	−	0.937473i	$-0.613159\pi$
$463$	−14.9786	−0.696115	−0.348058	+	0.937473i	$0.613159\pi$
$464$	−6.78000	−0.314753
$465$	3.13550	0.145405
$466$	−6.03324	−0.279484
$467$	6.56107	0.303610	0.151805	−	0.988410i	$-0.451491\pi$
$467$	6.56107	0.303610	0.151805	+	0.988410i	$0.451491\pi$
$468$	5.81260	0.268688
$469$	1.85881	0.0858320
$470$	−6.36108	−0.293415
$471$	−19.2202	−0.885619
$472$	−14.8655	−0.684241
$473$	−50.0187	−2.29986
$474$	12.5579	0.576802
$475$	2.62114	0.120266
$476$	5.91575	0.271148
$477$	9.50010	0.434979
$478$	24.6088	1.12558
$479$	−13.4894	−0.616347	−0.308173	−	0.951330i	$-0.599718\pi$
$479$	−13.4894	−0.616347	−0.308173	+	0.951330i	$0.599718\pi$
$480$	−1.43386	−0.0654465
$481$	55.4060	2.52630
$482$	−12.3387	−0.562011
$483$	−3.15735	−0.143664
$484$	19.3349	0.878859
$485$	−19.5210	−0.886402
$486$	−1.00000	−0.0453609
$487$	−32.9194	−1.49172	−0.745861	−	0.666102i	$-0.767962\pi$
$487$	−32.9194	−1.49172	−0.745861	+	0.666102i	$0.767962\pi$
$488$	12.3227	0.557822
$489$	4.23627	0.191571
$490$	5.01745	0.226665
$491$	−28.1030	−1.26827	−0.634135	−	0.773222i	$-0.718644\pi$
$491$	−28.1030	−1.26827	−0.634135	+	0.773222i	$0.718644\pi$
$492$	5.50256	0.248075
$493$	21.4368	0.965464
$494$	5.17508	0.232838
$495$	7.89729	0.354957
$496$	2.18675	0.0981882
$497$	−8.28162	−0.371481
$498$	−15.1636	−0.679497
$499$	35.1778	1.57478	0.787388	−	0.616457i	$-0.211433\pi$
$499$	35.1778	1.57478	0.787388	+	0.616457i	$0.211433\pi$
$500$	−11.3907	−0.509405
$501$	24.8132	1.10857
$502$	13.9475	0.622509
$503$	20.6999	0.922964	0.461482	−	0.887150i	$-0.347318\pi$
$503$	20.6999	0.922964	0.461482	+	0.887150i	$0.347318\pi$
$504$	1.87103	0.0833421
$505$	−14.0100	−0.623435
$506$	−9.29424	−0.413179
$507$	20.7863	0.923153
$508$	12.7041	0.563654
$509$	19.1836	0.850299	0.425150	−	0.905123i	$-0.360222\pi$
$509$	19.1836	0.850299	0.425150	+	0.905123i	$0.360222\pi$
$510$	4.53354	0.200748
$511$	−4.34258	−0.192104
$512$	−1.00000	−0.0441942
$513$	−0.890321	−0.0393086
$514$	−20.0207	−0.883074
$515$	10.0922	0.444716
$516$	−9.08158	−0.399794
$517$	24.4340	1.07461
$518$	17.8347	0.783613
$519$	3.90530	0.171424
$520$	−8.33446	−0.365490
$521$	−11.0551	−0.484331	−0.242165	−	0.970235i	$-0.577858\pi$
$521$	−11.0551	−0.484331	−0.242165	+	0.970235i	$0.577858\pi$
$522$	6.78000	0.296752
$523$	−32.4419	−1.41859	−0.709293	−	0.704913i	$-0.750986\pi$
$523$	−32.4419	−1.41859	−0.709293	+	0.704913i	$0.750986\pi$
$524$	−12.8692	−0.562193
$525$	5.50838	0.240406
$526$	2.19289	0.0956146
$527$	−6.91401	−0.301179
$528$	5.50771	0.239692
$529$	−20.1524	−0.876190
$530$	−13.6218	−0.591694
$531$	14.8655	0.645109
$532$	1.66581	0.0722222
$533$	31.9842	1.38539
$534$	3.41540	0.147799
$535$	17.6621	0.763598
$536$	0.993472	0.0429115
$537$	6.73708	0.290726
$538$	26.2148	1.13020
$539$	−19.2729	−0.830143
$540$	1.43386	0.0617035
$541$	19.6356	0.844200	0.422100	−	0.906549i	$-0.361293\pi$
$541$	19.6356	0.844200	0.422100	+	0.906549i	$0.361293\pi$
$542$	−8.88235	−0.381530
$543$	−3.43219	−0.147289
$544$	3.16177	0.135560
$545$	22.2306	0.952255
$546$	10.8755	0.465430
$547$	−18.3055	−0.782688	−0.391344	−	0.920245i	$-0.627990\pi$
$547$	−18.3055	−0.782688	−0.391344	+	0.920245i	$0.627990\pi$
$548$	−5.08158	−0.217074
$549$	−12.3227	−0.525920
$550$	16.2149	0.691408
$551$	6.03637	0.257158
$552$	−1.68749	−0.0718245
$553$	23.4961	0.999157
$554$	25.3250	1.07596
$555$	13.6676	0.580159
$556$	−6.73928	−0.285809
$557$	29.8604	1.26523	0.632614	−	0.774467i	$-0.281982\pi$
$557$	29.8604	1.26523	0.632614	+	0.774467i	$0.281982\pi$
$558$	−2.18675	−0.0925727
$559$	−52.7876	−2.23268
$560$	−2.68279	−0.113369
$561$	−17.4141	−0.735224
$562$	−20.0533	−0.845896
$563$	2.85418	0.120290	0.0601448	−	0.998190i	$-0.480844\pi$
$563$	2.85418	0.120290	0.0601448	+	0.998190i	$0.480844\pi$
$564$	4.43633	0.186803
$565$	12.1030	0.509176
$566$	−21.6602	−0.910447
$567$	−1.87103	−0.0785757
$568$	−4.42624	−0.185721
$569$	21.5560	0.903673	0.451837	−	0.892101i	$-0.350769\pi$
$569$	21.5560	0.903673	0.451837	+	0.892101i	$0.350769\pi$
$570$	1.27660	0.0534707
$571$	−31.3753	−1.31302	−0.656508	−	0.754319i	$-0.727967\pi$
$571$	−31.3753	−1.31302	−0.656508	+	0.754319i	$0.727967\pi$
$572$	32.0141	1.33858
$573$	−19.5410	−0.816335
$574$	10.2954	0.429724
$575$	−4.96806	−0.207182
$576$	1.00000	0.0416667
$577$	−13.1804	−0.548707	−0.274354	−	0.961629i	$-0.588464\pi$
$577$	−13.1804	−0.548707	−0.274354	+	0.961629i	$0.588464\pi$
$578$	7.00322	0.291296
$579$	−5.55650	−0.230920
$580$	−9.72157	−0.403666
$581$	−28.3715	−1.17705
$582$	13.6143	0.564330
$583$	52.3238	2.16703
$584$	−2.32096	−0.0960420
$585$	8.33446	0.344587
$586$	18.1963	0.751681
$587$	−25.8281	−1.06604	−0.533020	−	0.846103i	$-0.678943\pi$
$587$	−25.8281	−1.06604	−0.533020	+	0.846103i	$0.678943\pi$
$588$	−3.49926	−0.144307
$589$	−1.94691	−0.0802211
$590$	−21.3151	−0.877528
$591$	−21.9757	−0.903961
$592$	9.53205	0.391765
$593$	−7.80648	−0.320574	−0.160287	−	0.987070i	$-0.551242\pi$
$593$	−7.80648	−0.320574	−0.160287	+	0.987070i	$0.551242\pi$
$594$	−5.50771	−0.225984
$595$	8.48236	0.347743
$596$	2.81176	0.115174
$597$	4.07733	0.166874
$598$	−9.80873	−0.401109
$599$	12.2236	0.499442	0.249721	−	0.968318i	$-0.419661\pi$
$599$	12.2236	0.499442	0.249721	+	0.968318i	$0.419661\pi$
$600$	2.94404	0.120190
$601$	−18.5172	−0.755333	−0.377666	−	0.925942i	$-0.623273\pi$
$601$	−18.5172	−0.755333	−0.377666	+	0.925942i	$0.623273\pi$
$602$	−16.9919	−0.692537
$603$	−0.993472	−0.0404573
$604$	−7.74113	−0.314982
$605$	27.7236	1.12712
$606$	9.77079	0.396911
$607$	38.9871	1.58244	0.791219	−	0.611533i	$-0.209447\pi$
$607$	38.9871	1.58244	0.791219	+	0.611533i	$0.209447\pi$
$608$	0.890321	0.0361073
$609$	12.6856	0.514045
$610$	17.6690	0.715398
$611$	25.7866	1.04321
$612$	−3.16177	−0.127807
$613$	−27.2786	−1.10177	−0.550886	−	0.834580i	$-0.685710\pi$
$613$	−27.2786	−1.10177	−0.550886	+	0.834580i	$0.685710\pi$
$614$	9.63403	0.388798
$615$	7.88991	0.318152
$616$	10.3051	0.415203
$617$	−29.6233	−1.19259	−0.596295	−	0.802765i	$-0.703361\pi$
$617$	−29.6233	−1.19259	−0.596295	+	0.802765i	$0.703361\pi$
$618$	−7.03849	−0.283129
$619$	22.0162	0.884908	0.442454	−	0.896791i	$-0.354108\pi$
$619$	22.0162	0.884908	0.442454	+	0.896791i	$0.354108\pi$
$620$	3.13550	0.125925
$621$	1.68749	0.0677168
$622$	7.31701	0.293385
$623$	6.39031	0.256022
$624$	5.81260	0.232690
$625$	−1.61239	−0.0644956
$626$	−20.3220	−0.812230
$627$	−4.90363	−0.195832
$628$	−19.2202	−0.766969
$629$	−30.1381	−1.20169
$630$	2.68279	0.106885
$631$	28.6537	1.14069	0.570344	−	0.821406i	$-0.306810\pi$
$631$	28.6537	1.14069	0.570344	+	0.821406i	$0.306810\pi$
$632$	12.5579	0.499525
$633$	−10.0288	−0.398608
$634$	34.6867	1.37758
$635$	18.2159	0.722877
$636$	9.50010	0.376703
$637$	−20.3398	−0.805892
$638$	37.3423	1.47839
$639$	4.42624	0.175099
$640$	−1.43386	−0.0566783
$641$	0.654574	0.0258541	0.0129270	−	0.999916i	$-0.495885\pi$
$641$	0.654574	0.0258541	0.0129270	+	0.999916i	$0.495885\pi$
$642$	−12.3178	−0.486147
$643$	37.4692	1.47764	0.738821	−	0.673902i	$-0.235383\pi$
$643$	37.4692	1.47764	0.738821	+	0.673902i	$0.235383\pi$
$644$	−3.15735	−0.124417
$645$	−13.0217	−0.512730
$646$	−2.81499	−0.110754
$647$	−24.4525	−0.961327	−0.480663	−	0.876905i	$-0.659604\pi$
$647$	−24.4525	−0.961327	−0.480663	+	0.876905i	$0.659604\pi$
$648$	−1.00000	−0.0392837
$649$	81.8750	3.21388
$650$	17.1125	0.671209
$651$	−4.09148	−0.160358
$652$	4.23627	0.165905
$653$	−7.10115	−0.277889	−0.138945	−	0.990300i	$-0.544371\pi$
$653$	−7.10115	−0.277889	−0.138945	+	0.990300i	$0.544371\pi$
$654$	−15.5040	−0.606256
$655$	−18.4526	−0.721004
$656$	5.50256	0.214839
$657$	2.32096	0.0905492
$658$	8.30049	0.323587
$659$	27.0572	1.05400	0.527000	−	0.849865i	$-0.323317\pi$
$659$	27.0572	1.05400	0.527000	+	0.849865i	$0.323317\pi$
$660$	7.89729	0.307402
$661$	−1.87803	−0.0730471	−0.0365235	−	0.999333i	$-0.511628\pi$
$661$	−1.87803	−0.0730471	−0.0365235	+	0.999333i	$0.511628\pi$
$662$	−20.8200	−0.809190
$663$	−18.3781	−0.713746
$664$	−15.1636	−0.588462
$665$	2.38854	0.0926238
$666$	−9.53205	−0.369360
$667$	−11.4412	−0.443005
$668$	24.8132	0.960053
$669$	−1.00000	−0.0386622
$670$	1.42450	0.0550333
$671$	−67.8699	−2.62009
$672$	1.87103	0.0721764
$673$	27.8003	1.07162	0.535811	−	0.844338i	$-0.320006\pi$
$673$	27.8003	1.07162	0.535811	+	0.844338i	$0.320006\pi$
$674$	−17.8513	−0.687606
$675$	−2.94404	−0.113316
$676$	20.7863	0.799474
$677$	−5.33651	−0.205099	−0.102549	−	0.994728i	$-0.532700\pi$
$677$	−5.33651	−0.205099	−0.102549	+	0.994728i	$0.532700\pi$
$678$	−8.44083	−0.324168
$679$	25.4727	0.977552
$680$	4.53354	0.173853
$681$	−20.0364	−0.767795
$682$	−12.0440	−0.461189
$683$	29.6960	1.13629	0.568144	−	0.822929i	$-0.307662\pi$
$683$	29.6960	1.13629	0.568144	+	0.822929i	$0.307662\pi$
$684$	−0.890321	−0.0340423
$685$	−7.28628	−0.278394
$686$	−19.6444	−0.750026
$687$	−18.4886	−0.705384
$688$	−9.08158	−0.346232
$689$	55.2203	2.10372
$690$	−2.41963	−0.0921138
$691$	0.856810	0.0325946	0.0162973	−	0.999867i	$-0.494812\pi$
$691$	0.856810	0.0325946	0.0162973	+	0.999867i	$0.494812\pi$
$692$	3.90530	0.148457
$693$	−10.3051	−0.391458
$694$	−16.4617	−0.624877
$695$	−9.66318	−0.366545
$696$	6.78000	0.256995
$697$	−17.3978	−0.658990
$698$	15.7220	0.595088
$699$	6.03324	0.228198
$700$	5.50838	0.208197
$701$	5.75969	0.217541	0.108770	−	0.994067i	$-0.465309\pi$
$701$	5.75969	0.217541	0.108770	+	0.994067i	$0.465309\pi$
$702$	−5.81260	−0.219382
$703$	−8.48658	−0.320078
$704$	5.50771	0.207580
$705$	6.36108	0.239572
$706$	−23.2166	−0.873768
$707$	18.2814	0.687543
$708$	14.8655	0.558681
$709$	18.1581	0.681942	0.340971	−	0.940074i	$-0.389244\pi$
$709$	18.1581	0.681942	0.340971	+	0.940074i	$0.389244\pi$
$710$	−6.34662	−0.238184
$711$	−12.5579	−0.470957
$712$	3.41540	0.127998
$713$	3.69014	0.138197
$714$	−5.91575	−0.221391
$715$	45.9038	1.71671
$716$	6.73708	0.251776
$717$	−24.6088	−0.919033
$718$	15.3354	0.572312
$719$	35.6878	1.33093	0.665466	−	0.746428i	$-0.268233\pi$
$719$	35.6878	1.33093	0.665466	+	0.746428i	$0.268233\pi$
$720$	1.43386	0.0534368
$721$	−13.1692	−0.490446
$722$	18.2073	0.677607
$723$	12.3387	0.458880
$724$	−3.43219	−0.127556
$725$	19.9606	0.741318
$726$	−19.3349	−0.717586
$727$	37.3071	1.38365	0.691823	−	0.722068i	$-0.256808\pi$
$727$	37.3071	1.38365	0.691823	+	0.722068i	$0.256808\pi$
$728$	10.8755	0.403074
$729$	1.00000	0.0370370
$730$	−3.32793	−0.123172
$731$	28.7138	1.06202
$732$	−12.3227	−0.455460
$733$	36.6158	1.35244	0.676218	−	0.736701i	$-0.263618\pi$
$733$	36.6158	1.35244	0.676218	+	0.736701i	$0.263618\pi$
$734$	1.82214	0.0672566
$735$	−5.01745	−0.185071
$736$	−1.68749	−0.0622019
$737$	−5.47176	−0.201555
$738$	−5.50256	−0.202552
$739$	13.4619	0.495204	0.247602	−	0.968862i	$-0.420357\pi$
$739$	13.4619	0.495204	0.247602	+	0.968862i	$0.420357\pi$
$740$	13.6676	0.502432
$741$	−5.17508	−0.190111
$742$	17.7749	0.652538
$743$	−8.49550	−0.311670	−0.155835	−	0.987783i	$-0.549807\pi$
$743$	−8.49550	−0.311670	−0.155835	+	0.987783i	$0.549807\pi$
$744$	−2.18675	−0.0801703
$745$	4.03168	0.147709
$746$	24.9826	0.914679
$747$	15.1636	0.554807
$748$	−17.4141	−0.636723
$749$	−23.0470	−0.842120
$750$	11.3907	0.415928
$751$	−26.1796	−0.955309	−0.477654	−	0.878548i	$-0.658513\pi$
$751$	−26.1796	−0.955309	−0.477654	+	0.878548i	$0.658513\pi$
$752$	4.43633	0.161776
$753$	−13.9475	−0.508277
$754$	39.4094	1.43521
$755$	−11.0997	−0.403960
$756$	−1.87103	−0.0680486
$757$	−6.61208	−0.240320	−0.120160	−	0.992755i	$-0.538341\pi$
$757$	−6.61208	−0.240320	−0.120160	+	0.992755i	$0.538341\pi$
$758$	9.99010	0.362857
$759$	9.29424	0.337359
$760$	1.27660	0.0463070
$761$	−32.5736	−1.18079	−0.590396	−	0.807114i	$-0.701028\pi$
$761$	−32.5736	−1.18079	−0.590396	+	0.807114i	$0.701028\pi$
$762$	−12.7041	−0.460222
$763$	−29.0085	−1.05018
$764$	−19.5410	−0.706967
$765$	−4.53354	−0.163910
$766$	14.7719	0.533730
$767$	86.4074	3.11999
$768$	1.00000	0.0360844
$769$	−11.9619	−0.431356	−0.215678	−	0.976464i	$-0.569196\pi$
$769$	−11.9619	−0.431356	−0.215678	+	0.976464i	$0.569196\pi$
$770$	14.7760	0.532492
$771$	20.0207	0.721027
$772$	−5.55650	−0.199983
$773$	34.8936	1.25504	0.627518	−	0.778602i	$-0.284071\pi$
$773$	34.8936	1.25504	0.627518	+	0.778602i	$0.284071\pi$
$774$	9.08158	0.326431
$775$	−6.43790	−0.231256
$776$	13.6143	0.488724
$777$	−17.8347	−0.639817
$778$	−26.0548	−0.934111
$779$	−4.89905	−0.175527
$780$	8.33446	0.298422
$781$	24.3785	0.872330
$782$	5.33547	0.190796
$783$	−6.78000	−0.242297
$784$	−3.49926	−0.124974
$785$	−27.5591	−0.983625
$786$	12.8692	0.459029
$787$	−8.06959	−0.287650	−0.143825	−	0.989603i	$-0.545940\pi$
$787$	−8.06959	−0.287650	−0.143825	+	0.989603i	$0.545940\pi$
$788$	−21.9757	−0.782853
$789$	−2.19289	−0.0780690
$790$	18.0062	0.640633
$791$	−15.7930	−0.561535
$792$	−5.50771	−0.195708
$793$	−71.6269	−2.54355
$794$	25.3046	0.898026
$795$	13.6218	0.483116
$796$	4.07733	0.144517
$797$	8.25052	0.292248	0.146124	−	0.989266i	$-0.453320\pi$
$797$	8.25052	0.292248	0.146124	+	0.989266i	$0.453320\pi$
$798$	−1.66581	−0.0589692
$799$	−14.0266	−0.496227
$800$	2.94404	0.104088
$801$	−3.41540	−0.120677
$802$	−16.9955	−0.600133
$803$	12.7832	0.451108
$804$	−0.993472	−0.0350371
$805$	−4.52720	−0.159563
$806$	−12.7107	−0.447716
$807$	−26.2148	−0.922806
$808$	9.77079	0.343735
$809$	12.2391	0.430305	0.215152	−	0.976581i	$-0.430975\pi$
$809$	12.2391	0.430305	0.215152	+	0.976581i	$0.430975\pi$
$810$	−1.43386	−0.0503807
$811$	52.3012	1.83654	0.918272	−	0.395950i	$-0.129585\pi$
$811$	52.3012	1.83654	0.918272	+	0.395950i	$0.129585\pi$
$812$	12.6856	0.445176
$813$	8.88235	0.311518
$814$	−52.4998	−1.84012
$815$	6.07421	0.212770
$816$	−3.16177	−0.110684
$817$	8.08552	0.282876
$818$	30.6720	1.07242
$819$	−10.8755	−0.380022
$820$	7.88991	0.275528
$821$	−7.67481	−0.267853	−0.133926	−	0.990991i	$-0.542759\pi$
$821$	−7.67481	−0.267853	−0.133926	+	0.990991i	$0.542759\pi$
$822$	5.08158	0.177240
$823$	47.9518	1.67149	0.835747	−	0.549115i	$-0.185035\pi$
$823$	47.9518	1.67149	0.835747	+	0.549115i	$0.185035\pi$
$824$	−7.03849	−0.245197
$825$	−16.2149	−0.564532
$826$	27.8138	0.967766
$827$	−28.3630	−0.986278	−0.493139	−	0.869951i	$-0.664151\pi$
$827$	−28.3630	−0.986278	−0.493139	+	0.869951i	$0.664151\pi$
$828$	1.68749	0.0586445
$829$	14.6558	0.509016	0.254508	−	0.967071i	$-0.418087\pi$
$829$	14.6558	0.509016	0.254508	+	0.967071i	$0.418087\pi$
$830$	−21.7425	−0.754692
$831$	−25.3250	−0.878514
$832$	5.81260	0.201516
$833$	11.0638	0.383340
$834$	6.73928	0.233362
$835$	35.5787	1.23125
$836$	−4.90363	−0.169596
$837$	2.18675	0.0755853
$838$	−17.4958	−0.604381
$839$	−29.2760	−1.01072	−0.505361	−	0.862908i	$-0.668641\pi$
$839$	−29.2760	−1.01072	−0.505361	+	0.862908i	$0.668641\pi$
$840$	2.68279	0.0925651
$841$	16.9683	0.585115
$842$	12.6330	0.435362
$843$	20.0533	0.690671
$844$	−10.0288	−0.345205
$845$	29.8047	1.02531
$846$	−4.43633	−0.152524
$847$	−36.1761	−1.24303
$848$	9.50010	0.326235
$849$	21.6602	0.743377
$850$	−9.30838	−0.319275
$851$	16.0853	0.551397
$852$	4.42624	0.151641
$853$	−53.1026	−1.81820	−0.909099	−	0.416581i	$-0.863228\pi$
$853$	−53.1026	−1.81820	−0.909099	+	0.416581i	$0.863228\pi$
$854$	−23.0561	−0.788964
$855$	−1.27660	−0.0436587
$856$	−12.3178	−0.421015
$857$	23.0183	0.786292	0.393146	−	0.919476i	$-0.371387\pi$
$857$	23.0183	0.786292	0.393146	+	0.919476i	$0.371387\pi$
$858$	−32.0141	−1.09294
$859$	49.5835	1.69177	0.845884	−	0.533368i	$-0.179074\pi$
$859$	49.5835	1.69177	0.845884	+	0.533368i	$0.179074\pi$
$860$	−13.0217	−0.444037
$861$	−10.2954	−0.350868
$862$	−14.6767	−0.499891
$863$	4.09527	0.139405	0.0697023	−	0.997568i	$-0.477795\pi$
$863$	4.09527	0.139405	0.0697023	+	0.997568i	$0.477795\pi$
$864$	−1.00000	−0.0340207
$865$	5.59966	0.190394
$866$	22.5184	0.765206
$867$	−7.00322	−0.237842
$868$	−4.09148	−0.138874
$869$	−69.1652	−2.34627
$870$	9.72157	0.329592
$871$	−5.77466	−0.195667
$872$	−15.5040	−0.525033
$873$	−13.6143	−0.460773
$874$	1.50241	0.0508198
$875$	21.3122	0.720484
$876$	2.32096	0.0784179
$877$	−6.30454	−0.212889	−0.106445	−	0.994319i	$-0.533947\pi$
$877$	−6.30454	−0.212889	−0.106445	+	0.994319i	$0.533947\pi$
$878$	8.52689	0.287769
$879$	−18.1963	−0.613745
$880$	7.89729	0.266218
$881$	−20.8997	−0.704128	−0.352064	−	0.935976i	$-0.614520\pi$
$881$	−20.8997	−0.704128	−0.352064	+	0.935976i	$0.614520\pi$
$882$	3.49926	0.117826
$883$	0.269804	0.00907964	0.00453982	−	0.999990i	$-0.498555\pi$
$883$	0.269804	0.00907964	0.00453982	+	0.999990i	$0.498555\pi$
$884$	−18.3781	−0.618122
$885$	21.3151	0.716499
$886$	10.6999	0.359471
$887$	−40.5158	−1.36039	−0.680193	−	0.733033i	$-0.738104\pi$
$887$	−40.5158	−1.36039	−0.680193	+	0.733033i	$0.738104\pi$
$888$	−9.53205	−0.319875
$889$	−23.7697	−0.797212
$890$	4.89721	0.164155
$891$	5.50771	0.184515
$892$	−1.00000	−0.0334825
$893$	−3.94976	−0.132174
$894$	−2.81176	−0.0940395
$895$	9.66003	0.322899
$896$	1.87103	0.0625066
$897$	9.80873	0.327504
$898$	42.0615	1.40361
$899$	−14.8262	−0.494481
$900$	−2.94404	−0.0981348
$901$	−30.0371	−1.00068
$902$	−30.3065	−1.00910
$903$	16.9919	0.565454
$904$	−8.44083	−0.280738
$905$	−4.92128	−0.163589
$906$	7.74113	0.257182
$907$	−18.9554	−0.629405	−0.314703	−	0.949190i	$-0.601905\pi$
$907$	−18.9554	−0.629405	−0.314703	+	0.949190i	$0.601905\pi$
$908$	−20.0364	−0.664930
$909$	−9.77079	−0.324077
$910$	15.5940	0.516936
$911$	3.74007	0.123914	0.0619571	−	0.998079i	$-0.480266\pi$
$911$	3.74007	0.123914	0.0619571	+	0.998079i	$0.480266\pi$
$912$	−0.890321	−0.0294815
$913$	83.5167	2.76400
$914$	8.63916	0.285758
$915$	−17.6690	−0.584120
$916$	−18.4886	−0.610881
$917$	24.0786	0.795145
$918$	3.16177	0.104354
$919$	13.1779	0.434700	0.217350	−	0.976094i	$-0.430259\pi$
$919$	13.1779	0.434700	0.217350	+	0.976094i	$0.430259\pi$
$920$	−2.41963	−0.0797729
$921$	−9.63403	−0.317452
$922$	12.8499	0.423189
$923$	25.7280	0.846847
$924$	−10.3051	−0.339012
$925$	−28.0628	−0.922699
$926$	14.9786	0.492228
$927$	7.03849	0.231174
$928$	6.78000	0.222564
$929$	−46.8151	−1.53595	−0.767976	−	0.640478i	$-0.778736\pi$
$929$	−46.8151	−1.53595	−0.767976	+	0.640478i	$0.778736\pi$
$930$	−3.13550	−0.102817
$931$	3.11546	0.102105
$932$	6.03324	0.197625
$933$	−7.31701	−0.239548
$934$	−6.56107	−0.214685
$935$	−24.9694	−0.816587
$936$	−5.81260	−0.189991
$937$	25.6778	0.838859	0.419429	−	0.907788i	$-0.362230\pi$
$937$	25.6778	0.838859	0.419429	+	0.907788i	$0.362230\pi$
$938$	−1.85881	−0.0606924
$939$	20.3220	0.663183
$940$	6.36108	0.207476
$941$	15.3764	0.501255	0.250628	−	0.968084i	$-0.419363\pi$
$941$	15.3764	0.501255	0.250628	+	0.968084i	$0.419363\pi$
$942$	19.2202	0.626227
$943$	9.28555	0.302379
$944$	14.8655	0.483832
$945$	−2.68279	−0.0872712
$946$	50.0187	1.62625
$947$	27.2853	0.886653	0.443326	−	0.896360i	$-0.353798\pi$
$947$	27.2853	0.886653	0.443326	+	0.896360i	$0.353798\pi$
$948$	−12.5579	−0.407861
$949$	13.4908	0.437930
$950$	−2.62114	−0.0850411
$951$	−34.6867	−1.12479
$952$	−5.91575	−0.191731
$953$	−41.1242	−1.33214	−0.666072	−	0.745887i	$-0.732026\pi$
$953$	−41.1242	−1.33214	−0.666072	+	0.745887i	$0.732026\pi$
$954$	−9.50010	−0.307577
$955$	−28.0190	−0.906674
$956$	−24.6088	−0.795906
$957$	−37.3423	−1.20710
$958$	13.4894	0.435823
$959$	9.50777	0.307022
$960$	1.43386	0.0462777
$961$	−26.2181	−0.845745
$962$	−55.4060	−1.78636
$963$	12.3178	0.396937
$964$	12.3387	0.397402
$965$	−7.96724	−0.256475
$966$	3.15735	0.101586
$967$	43.0261	1.38363	0.691813	−	0.722077i	$-0.256812\pi$
$967$	43.0261	1.38363	0.691813	+	0.722077i	$0.256812\pi$
$968$	−19.3349	−0.621447
$969$	2.81499	0.0904304
$970$	19.5210	0.626781
$971$	45.7690	1.46880	0.734398	−	0.678719i	$-0.237465\pi$
$971$	45.7690	1.46880	0.734398	+	0.678719i	$0.237465\pi$
$972$	1.00000	0.0320750
$973$	12.6094	0.404238
$974$	32.9194	1.05481
$975$	−17.1125	−0.548040
$976$	−12.3227	−0.394440
$977$	37.1672	1.18908	0.594542	−	0.804064i	$-0.297333\pi$
$977$	37.1672	1.18908	0.594542	+	0.804064i	$0.297333\pi$
$978$	−4.23627	−0.135461
$979$	−18.8111	−0.601204
$980$	−5.01745	−0.160277
$981$	15.5040	0.495006
$982$	28.1030	0.896802
$983$	−35.8574	−1.14367	−0.571837	−	0.820367i	$-0.693769\pi$
$983$	−35.8574	−1.14367	−0.571837	+	0.820367i	$0.693769\pi$
$984$	−5.50256	−0.175415
$985$	−31.5101	−1.00400
$986$	−21.4368	−0.682686
$987$	−8.30049	−0.264208
$988$	−5.17508	−0.164641
$989$	−15.3251	−0.487310
$990$	−7.89729	−0.250992
$991$	−6.53157	−0.207482	−0.103741	−	0.994604i	$-0.533081\pi$
$991$	−6.53157	−0.207482	−0.103741	+	0.994604i	$0.533081\pi$
$992$	−2.18675	−0.0694295
$993$	20.8200	0.660701
$994$	8.28162	0.262677
$995$	5.84632	0.185341
$996$	15.1636	0.480477
$997$	−49.4426	−1.56586	−0.782932	−	0.622107i	$-0.786277\pi$
$997$	−49.4426	−1.56586	−0.782932	+	0.622107i	$0.786277\pi$
$998$	−35.1778	−1.11354
$999$	9.53205	0.301581

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1338.2.a.h.1.3	✓	5
3.2	odd	2		4014.2.a.r.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.h.1.3	✓	5	1.1	even	1	trivial
4014.2.a.r.1.3		5	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.43386 q^{5} -1.00000 q^{6} -1.87103 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.43386 q^{5} -1.00000 q^{6} -1.87103 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.43386 q^{10} +5.50771 q^{11} +1.00000 q^{12} +5.81260 q^{13} +1.87103 q^{14} +1.43386 q^{15} +1.00000 q^{16} -3.16177 q^{17} -1.00000 q^{18} -0.890321 q^{19} +1.43386 q^{20} -1.87103 q^{21} -5.50771 q^{22} +1.68749 q^{23} -1.00000 q^{24} -2.94404 q^{25} -5.81260 q^{26} +1.00000 q^{27} -1.87103 q^{28} -6.78000 q^{29} -1.43386 q^{30} +2.18675 q^{31} -1.00000 q^{32} +5.50771 q^{33} +3.16177 q^{34} -2.68279 q^{35} +1.00000 q^{36} +9.53205 q^{37} +0.890321 q^{38} +5.81260 q^{39} -1.43386 q^{40} +5.50256 q^{41} +1.87103 q^{42} -9.08158 q^{43} +5.50771 q^{44} +1.43386 q^{45} -1.68749 q^{46} +4.43633 q^{47} +1.00000 q^{48} -3.49926 q^{49} +2.94404 q^{50} -3.16177 q^{51} +5.81260 q^{52} +9.50010 q^{53} -1.00000 q^{54} +7.89729 q^{55} +1.87103 q^{56} -0.890321 q^{57} +6.78000 q^{58} +14.8655 q^{59} +1.43386 q^{60} -12.3227 q^{61} -2.18675 q^{62} -1.87103 q^{63} +1.00000 q^{64} +8.33446 q^{65} -5.50771 q^{66} -0.993472 q^{67} -3.16177 q^{68} +1.68749 q^{69} +2.68279 q^{70} +4.42624 q^{71} -1.00000 q^{72} +2.32096 q^{73} -9.53205 q^{74} -2.94404 q^{75} -0.890321 q^{76} -10.3051 q^{77} -5.81260 q^{78} -12.5579 q^{79} +1.43386 q^{80} +1.00000 q^{81} -5.50256 q^{82} +15.1636 q^{83} -1.87103 q^{84} -4.53354 q^{85} +9.08158 q^{86} -6.78000 q^{87} -5.50771 q^{88} -3.41540 q^{89} -1.43386 q^{90} -10.8755 q^{91} +1.68749 q^{92} +2.18675 q^{93} -4.43633 q^{94} -1.27660 q^{95} -1.00000 q^{96} -13.6143 q^{97} +3.49926 q^{98} +5.50771 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} + 5 q^{5} - 5 q^{6} - q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 + 5 * q^5 - 5 * q^6 - q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} + 5 q^{5} - 5 q^{6} - q^{7} - 5 q^{8} + 5 q^{9} - 5 q^{10} + 9 q^{11} + 5 q^{12} + q^{14} + 5 q^{15} + 5 q^{16} + 6 q^{17} - 5 q^{18} - 4 q^{19} + 5 q^{20} - q^{21} - 9 q^{22} + 16 q^{23} - 5 q^{24} + 8 q^{25} + 5 q^{27} - q^{28} + 8 q^{29} - 5 q^{30} - q^{31} - 5 q^{32} + 9 q^{33} - 6 q^{34} + 22 q^{35} + 5 q^{36} - 2 q^{37} + 4 q^{38} - 5 q^{40} + 4 q^{41} + q^{42} + 3 q^{43} + 9 q^{44} + 5 q^{45} - 16 q^{46} + 18 q^{47} + 5 q^{48} + 2 q^{49} - 8 q^{50} + 6 q^{51} + 26 q^{53} - 5 q^{54} + q^{55} + q^{56} - 4 q^{57} - 8 q^{58} + 21 q^{59} + 5 q^{60} - 20 q^{61} + q^{62} - q^{63} + 5 q^{64} - 3 q^{65} - 9 q^{66} - 5 q^{67} + 6 q^{68} + 16 q^{69} - 22 q^{70} + 17 q^{71} - 5 q^{72} + 5 q^{73} + 2 q^{74} + 8 q^{75} - 4 q^{76} + 2 q^{77} - 21 q^{79} + 5 q^{80} + 5 q^{81} - 4 q^{82} + 11 q^{83} - q^{84} - 12 q^{85} - 3 q^{86} + 8 q^{87} - 9 q^{88} - 5 q^{89} - 5 q^{90} - 10 q^{91} + 16 q^{92} - q^{93} - 18 q^{94} + 10 q^{95} - 5 q^{96} - 11 q^{97} - 2 q^{98} + 9 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 + 5 * q^5 - 5 * q^6 - q^7 - 5 * q^8 + 5 * q^9 - 5 * q^10 + 9 * q^11 + 5 * q^12 + q^14 + 5 * q^15 + 5 * q^16 + 6 * q^17 - 5 * q^18 - 4 * q^19 + 5 * q^20 - q^21 - 9 * q^22 + 16 * q^23 - 5 * q^24 + 8 * q^25 + 5 * q^27 - q^28 + 8 * q^29 - 5 * q^30 - q^31 - 5 * q^32 + 9 * q^33 - 6 * q^34 + 22 * q^35 + 5 * q^36 - 2 * q^37 + 4 * q^38 - 5 * q^40 + 4 * q^41 + q^42 + 3 * q^43 + 9 * q^44 + 5 * q^45 - 16 * q^46 + 18 * q^47 + 5 * q^48 + 2 * q^49 - 8 * q^50 + 6 * q^51 + 26 * q^53 - 5 * q^54 + q^55 + q^56 - 4 * q^57 - 8 * q^58 + 21 * q^59 + 5 * q^60 - 20 * q^61 + q^62 - q^63 + 5 * q^64 - 3 * q^65 - 9 * q^66 - 5 * q^67 + 6 * q^68 + 16 * q^69 - 22 * q^70 + 17 * q^71 - 5 * q^72 + 5 * q^73 + 2 * q^74 + 8 * q^75 - 4 * q^76 + 2 * q^77 - 21 * q^79 + 5 * q^80 + 5 * q^81 - 4 * q^82 + 11 * q^83 - q^84 - 12 * q^85 - 3 * q^86 + 8 * q^87 - 9 * q^88 - 5 * q^89 - 5 * q^90 - 10 * q^91 + 16 * q^92 - q^93 - 18 * q^94 + 10 * q^95 - 5 * q^96 - 11 * q^97 - 2 * q^98 + 9 * q^99

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