LMFDB - Embedded newform 2738.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2738,2,Mod(1,2738)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2738, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2738.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2738 = 2 \cdot 37^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2738.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:	$21.8630400734$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	2738.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.347296 q^{3} +1.00000 q^{4} -0.120615 q^{5} -0.347296 q^{6} +1.34730 q^{7} -1.00000 q^{8} -2.87939 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.347296 q^{3} +1.00000 q^{4} -0.120615 q^{5} -0.347296 q^{6} +1.34730 q^{7} -1.00000 q^{8} -2.87939 q^{9} +0.120615 q^{10} +4.53209 q^{11} +0.347296 q^{12} -0.773318 q^{13} -1.34730 q^{14} -0.0418891 q^{15} +1.00000 q^{16} -3.00000 q^{17} +2.87939 q^{18} -2.04189 q^{19} -0.120615 q^{20} +0.467911 q^{21} -4.53209 q^{22} +0.0564370 q^{23} -0.347296 q^{24} -4.98545 q^{25} +0.773318 q^{26} -2.04189 q^{27} +1.34730 q^{28} -5.78106 q^{29} +0.0418891 q^{30} -3.34730 q^{31} -1.00000 q^{32} +1.57398 q^{33} +3.00000 q^{34} -0.162504 q^{35} -2.87939 q^{36} +2.04189 q^{38} -0.268571 q^{39} +0.120615 q^{40} -7.14796 q^{41} -0.467911 q^{42} +9.31315 q^{43} +4.53209 q^{44} +0.347296 q^{45} -0.0564370 q^{46} +8.51754 q^{47} +0.347296 q^{48} -5.18479 q^{49} +4.98545 q^{50} -1.04189 q^{51} -0.773318 q^{52} +2.78106 q^{53} +2.04189 q^{54} -0.546637 q^{55} -1.34730 q^{56} -0.709141 q^{57} +5.78106 q^{58} -12.9932 q^{59} -0.0418891 q^{60} -11.5175 q^{61} +3.34730 q^{62} -3.87939 q^{63} +1.00000 q^{64} +0.0932736 q^{65} -1.57398 q^{66} +5.12061 q^{67} -3.00000 q^{68} +0.0196004 q^{69} +0.162504 q^{70} +13.4338 q^{71} +2.87939 q^{72} -8.71688 q^{73} -1.73143 q^{75} -2.04189 q^{76} +6.10607 q^{77} +0.268571 q^{78} +4.14796 q^{79} -0.120615 q^{80} +7.92902 q^{81} +7.14796 q^{82} -5.92127 q^{83} +0.467911 q^{84} +0.361844 q^{85} -9.31315 q^{86} -2.00774 q^{87} -4.53209 q^{88} +7.61856 q^{89} -0.347296 q^{90} -1.04189 q^{91} +0.0564370 q^{92} -1.16250 q^{93} -8.51754 q^{94} +0.246282 q^{95} -0.347296 q^{96} -10.2412 q^{97} +5.18479 q^{98} -13.0496 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^9	$ 3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{9} + 6 q^{10} + 9 q^{11} - 9 q^{13} - 3 q^{14} + 3 q^{15} + 3 q^{16} - 9 q^{17} + 3 q^{18} - 3 q^{19} - 6 q^{20} + 6 q^{21} - 9 q^{22} + 15 q^{23} + 3 q^{25} + 9 q^{26} - 3 q^{27} + 3 q^{28} - 3 q^{30} - 9 q^{31} - 3 q^{32} - 3 q^{33} + 9 q^{34} - 3 q^{35} - 3 q^{36} + 3 q^{38} + 9 q^{39} + 6 q^{40} - 6 q^{41} - 6 q^{42} + 6 q^{43} + 9 q^{44} - 15 q^{46} + 3 q^{47} - 12 q^{49} - 3 q^{50} - 9 q^{52} - 9 q^{53} + 3 q^{54} - 15 q^{55} - 3 q^{56} - 18 q^{57} + 3 q^{59} + 3 q^{60} - 12 q^{61} + 9 q^{62} - 6 q^{63} + 3 q^{64} + 27 q^{65} + 3 q^{66} + 21 q^{67} - 9 q^{68} + 3 q^{69} + 3 q^{70} + 24 q^{71} + 3 q^{72} - 18 q^{73} - 15 q^{75} - 3 q^{76} + 6 q^{77} - 9 q^{78} - 3 q^{79} - 6 q^{80} - 9 q^{81} + 6 q^{82} - 9 q^{83} + 6 q^{84} + 18 q^{85} - 6 q^{86} + 18 q^{87} - 9 q^{88} + 3 q^{89} + 15 q^{92} - 6 q^{93} - 3 q^{94} - 3 q^{95} - 42 q^{97} + 12 q^{98} - 12 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^9 + 6 * q^10 + 9 * q^11 - 9 * q^13 - 3 * q^14 + 3 * q^15 + 3 * q^16 - 9 * q^17 + 3 * q^18 - 3 * q^19 - 6 * q^20 + 6 * q^21 - 9 * q^22 + 15 * q^23 + 3 * q^25 + 9 * q^26 - 3 * q^27 + 3 * q^28 - 3 * q^30 - 9 * q^31 - 3 * q^32 - 3 * q^33 + 9 * q^34 - 3 * q^35 - 3 * q^36 + 3 * q^38 + 9 * q^39 + 6 * q^40 - 6 * q^41 - 6 * q^42 + 6 * q^43 + 9 * q^44 - 15 * q^46 + 3 * q^47 - 12 * q^49 - 3 * q^50 - 9 * q^52 - 9 * q^53 + 3 * q^54 - 15 * q^55 - 3 * q^56 - 18 * q^57 + 3 * q^59 + 3 * q^60 - 12 * q^61 + 9 * q^62 - 6 * q^63 + 3 * q^64 + 27 * q^65 + 3 * q^66 + 21 * q^67 - 9 * q^68 + 3 * q^69 + 3 * q^70 + 24 * q^71 + 3 * q^72 - 18 * q^73 - 15 * q^75 - 3 * q^76 + 6 * q^77 - 9 * q^78 - 3 * q^79 - 6 * q^80 - 9 * q^81 + 6 * q^82 - 9 * q^83 + 6 * q^84 + 18 * q^85 - 6 * q^86 + 18 * q^87 - 9 * q^88 + 3 * q^89 + 15 * q^92 - 6 * q^93 - 3 * q^94 - 3 * q^95 - 42 * q^97 + 12 * q^98 - 12 * 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
$3$	0.347296	0.200512	0.100256	−	0.994962i	$-0.468034\pi$
$3$	0.347296	0.200512	0.100256	+	0.994962i	$0.468034\pi$
$4$	1.00000	0.500000
$5$	−0.120615	−0.0539406	−0.0269703	−	0.999636i	$-0.508586\pi$
$5$	−0.120615	−0.0539406	−0.0269703	+	0.999636i	$0.508586\pi$
$6$	−0.347296	−0.141783
$7$	1.34730	0.509230	0.254615	−	0.967042i	$-0.418051\pi$
$7$	1.34730	0.509230	0.254615	+	0.967042i	$0.418051\pi$
$8$	−1.00000	−0.353553
$9$	−2.87939	−0.959795
$10$	0.120615	0.0381417
$11$	4.53209	1.36648	0.683238	−	0.730196i	$-0.260571\pi$
$11$	4.53209	1.36648	0.683238	+	0.730196i	$0.260571\pi$
$12$	0.347296	0.100256
$13$	−0.773318	−0.214480	−0.107240	−	0.994233i	$-0.534201\pi$
$13$	−0.773318	−0.214480	−0.107240	+	0.994233i	$0.534201\pi$
$14$	−1.34730	−0.360080
$15$	−0.0418891	−0.0108157
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	2.87939	0.678678
$19$	−2.04189	−0.468441	−0.234221	−	0.972183i	$-0.575254\pi$
$19$	−2.04189	−0.468441	−0.234221	+	0.972183i	$0.575254\pi$
$20$	−0.120615	−0.0269703
$21$	0.467911	0.102107
$22$	−4.53209	−0.966245
$23$	0.0564370	0.0117679	0.00588396	−	0.999983i	$-0.498127\pi$
$23$	0.0564370	0.0117679	0.00588396	+	0.999983i	$0.498127\pi$
$24$	−0.347296	−0.0708916
$25$	−4.98545	−0.997090
$26$	0.773318	0.151660
$27$	−2.04189	−0.392962
$28$	1.34730	0.254615
$29$	−5.78106	−1.07352	−0.536758	−	0.843736i	$-0.680351\pi$
$29$	−5.78106	−1.07352	−0.536758	+	0.843736i	$0.680351\pi$
$30$	0.0418891	0.00764786
$31$	−3.34730	−0.601192	−0.300596	−	0.953752i	$-0.597186\pi$
$31$	−3.34730	−0.601192	−0.300596	+	0.953752i	$0.597186\pi$
$32$	−1.00000	−0.176777
$33$	1.57398	0.273994
$34$	3.00000	0.514496
$35$	−0.162504	−0.0274682
$36$	−2.87939	−0.479898
$37$	0	0
$37$	0	0
$38$	2.04189	0.331238
$39$	−0.268571	−0.0430057
$40$	0.120615	0.0190709
$41$	−7.14796	−1.11632	−0.558162	−	0.829732i	$-0.688493\pi$
$41$	−7.14796	−1.11632	−0.558162	+	0.829732i	$0.688493\pi$
$42$	−0.467911	−0.0722003
$43$	9.31315	1.42024	0.710121	−	0.704080i	$-0.248640\pi$
$43$	9.31315	1.42024	0.710121	+	0.704080i	$0.248640\pi$
$44$	4.53209	0.683238
$45$	0.347296	0.0517719
$46$	−0.0564370	−0.00832118
$47$	8.51754	1.24241	0.621206	−	0.783648i	$-0.286643\pi$
$47$	8.51754	1.24241	0.621206	+	0.783648i	$0.286643\pi$
$48$	0.347296	0.0501279
$49$	−5.18479	−0.740685
$50$	4.98545	0.705049
$51$	−1.04189	−0.145894
$52$	−0.773318	−0.107240
$53$	2.78106	0.382008	0.191004	−	0.981589i	$-0.438826\pi$
$53$	2.78106	0.382008	0.191004	+	0.981589i	$0.438826\pi$
$54$	2.04189	0.277866
$55$	−0.546637	−0.0737085
$56$	−1.34730	−0.180040
$57$	−0.709141	−0.0939280
$58$	5.78106	0.759090
$59$	−12.9932	−1.69157	−0.845785	−	0.533524i	$-0.820867\pi$
$59$	−12.9932	−1.69157	−0.845785	+	0.533524i	$0.820867\pi$
$60$	−0.0418891	−0.00540786
$61$	−11.5175	−1.47467	−0.737335	−	0.675527i	$-0.763916\pi$
$61$	−11.5175	−1.47467	−0.737335	+	0.675527i	$0.763916\pi$
$62$	3.34730	0.425107
$63$	−3.87939	−0.488757
$64$	1.00000	0.125000
$65$	0.0932736	0.0115692
$66$	−1.57398	−0.193743
$67$	5.12061	0.625583	0.312791	−	0.949822i	$-0.398736\pi$
$67$	5.12061	0.625583	0.312791	+	0.949822i	$0.398736\pi$
$68$	−3.00000	−0.363803
$69$	0.0196004	0.00235961
$70$	0.162504	0.0194229
$71$	13.4338	1.59429	0.797147	−	0.603785i	$-0.206341\pi$
$71$	13.4338	1.59429	0.797147	+	0.603785i	$0.206341\pi$
$72$	2.87939	0.339339
$73$	−8.71688	−1.02023	−0.510117	−	0.860105i	$-0.670398\pi$
$73$	−8.71688	−1.02023	−0.510117	+	0.860105i	$0.670398\pi$
$74$	0	0
$75$	−1.73143	−0.199928
$76$	−2.04189	−0.234221
$77$	6.10607	0.695851
$78$	0.268571	0.0304096
$79$	4.14796	0.466682	0.233341	−	0.972395i	$-0.425034\pi$
$79$	4.14796	0.466682	0.233341	+	0.972395i	$0.425034\pi$
$80$	−0.120615	−0.0134851
$81$	7.92902	0.881002
$82$	7.14796	0.789360
$83$	−5.92127	−0.649944	−0.324972	−	0.945724i	$-0.605355\pi$
$83$	−5.92127	−0.649944	−0.324972	+	0.945724i	$0.605355\pi$
$84$	0.467911	0.0510533
$85$	0.361844	0.0392475
$86$	−9.31315	−1.00426
$87$	−2.00774	−0.215252
$88$	−4.53209	−0.483122
$89$	7.61856	0.807565	0.403783	−	0.914855i	$-0.367695\pi$
$89$	7.61856	0.807565	0.403783	+	0.914855i	$0.367695\pi$
$90$	−0.347296	−0.0366083
$91$	−1.04189	−0.109220
$92$	0.0564370	0.00588396
$93$	−1.16250	−0.120546
$94$	−8.51754	−0.878517
$95$	0.246282	0.0252680
$96$	−0.347296	−0.0354458
$97$	−10.2412	−1.03984	−0.519920	−	0.854215i	$-0.674038\pi$
$97$	−10.2412	−1.03984	−0.519920	+	0.854215i	$0.674038\pi$
$98$	5.18479	0.523743
$99$	−13.0496	−1.31154
$100$	−4.98545	−0.498545
$101$	−16.1138	−1.60338	−0.801692	−	0.597737i	$-0.796067\pi$
$101$	−16.1138	−1.60338	−0.801692	+	0.597737i	$0.796067\pi$
$102$	1.04189	0.103162
$103$	−19.6313	−1.93433	−0.967167	−	0.254141i	$-0.918207\pi$
$103$	−19.6313	−1.93433	−0.967167	+	0.254141i	$0.918207\pi$
$104$	0.773318	0.0758301
$105$	−0.0564370	−0.00550769
$106$	−2.78106	−0.270120
$107$	−0.638156	−0.0616928	−0.0308464	−	0.999524i	$-0.509820\pi$
$107$	−0.638156	−0.0616928	−0.0308464	+	0.999524i	$0.509820\pi$
$108$	−2.04189	−0.196481
$109$	7.25671	0.695067	0.347533	−	0.937668i	$-0.387019\pi$
$109$	7.25671	0.695067	0.347533	+	0.937668i	$0.387019\pi$
$110$	0.546637	0.0521198
$111$	0	0
$112$	1.34730	0.127308
$113$	−13.8229	−1.30035	−0.650177	−	0.759783i	$-0.725305\pi$
$113$	−13.8229	−1.30035	−0.650177	+	0.759783i	$0.725305\pi$
$114$	0.709141	0.0664171
$115$	−0.00680713	−0.000634768 0
$116$	−5.78106	−0.536758
$117$	2.22668	0.205857
$118$	12.9932	1.19612
$119$	−4.04189	−0.370519
$120$	0.0418891	0.00382393
$121$	9.53983	0.867257
$122$	11.5175	1.04275
$123$	−2.48246	−0.223836
$124$	−3.34730	−0.300596
$125$	1.20439	0.107724
$126$	3.87939	0.345603
$127$	12.1138	1.07493	0.537463	−	0.843287i	$-0.319383\pi$
$127$	12.1138	1.07493	0.537463	+	0.843287i	$0.319383\pi$
$128$	−1.00000	−0.0883883
$129$	3.23442	0.284775
$130$	−0.0932736	−0.00818064
$131$	21.6459	1.89121	0.945605	−	0.325316i	$-0.105471\pi$
$131$	21.6459	1.89121	0.945605	+	0.325316i	$0.105471\pi$
$132$	1.57398	0.136997
$133$	−2.75103	−0.238545
$134$	−5.12061	−0.442354
$135$	0.246282	0.0211966
$136$	3.00000	0.257248
$137$	8.63816	0.738007	0.369004	−	0.929428i	$-0.379699\pi$
$137$	8.63816	0.738007	0.369004	+	0.929428i	$0.379699\pi$
$138$	−0.0196004	−0.00166849
$139$	−13.4047	−1.13697	−0.568485	−	0.822694i	$-0.692470\pi$
$139$	−13.4047	−1.13697	−0.568485	+	0.822694i	$0.692470\pi$
$140$	−0.162504	−0.0137341
$141$	2.95811	0.249118
$142$	−13.4338	−1.12734
$143$	−3.50475	−0.293082
$144$	−2.87939	−0.239949
$145$	0.697281	0.0579060
$146$	8.71688	0.721414
$147$	−1.80066	−0.148516
$148$	0	0
$149$	4.39693	0.360210	0.180105	−	0.983647i	$-0.442356\pi$
$149$	4.39693	0.360210	0.180105	+	0.983647i	$0.442356\pi$
$150$	1.73143	0.141371
$151$	14.7442	1.19987	0.599934	−	0.800050i	$-0.295194\pi$
$151$	14.7442	1.19987	0.599934	+	0.800050i	$0.295194\pi$
$152$	2.04189	0.165619
$153$	8.63816	0.698353
$154$	−6.10607	−0.492041
$155$	0.403733	0.0324286
$156$	−0.268571	−0.0215029
$157$	−15.6527	−1.24922	−0.624611	−	0.780936i	$-0.714742\pi$
$157$	−15.6527	−1.24922	−0.624611	+	0.780936i	$0.714742\pi$
$158$	−4.14796	−0.329994
$159$	0.965852	0.0765970
$160$	0.120615	0.00953543
$161$	0.0760373	0.00599258
$162$	−7.92902	−0.622962
$163$	−4.78106	−0.374481	−0.187241	−	0.982314i	$-0.559954\pi$
$163$	−4.78106	−0.374481	−0.187241	+	0.982314i	$0.559954\pi$
$164$	−7.14796	−0.558162
$165$	−0.189845	−0.0147794
$166$	5.92127	0.459580
$167$	2.00505	0.155156	0.0775778	−	0.996986i	$-0.475281\pi$
$167$	2.00505	0.155156	0.0775778	+	0.996986i	$0.475281\pi$
$168$	−0.467911	−0.0361001
$169$	−12.4020	−0.953998
$170$	−0.361844	−0.0277522
$171$	5.87939	0.449608
$172$	9.31315	0.710121
$173$	−13.7811	−1.04775	−0.523877	−	0.851794i	$-0.675515\pi$
$173$	−13.7811	−1.04775	−0.523877	+	0.851794i	$0.675515\pi$
$174$	2.00774	0.152206
$175$	−6.71688	−0.507749
$176$	4.53209	0.341619
$177$	−4.51249	−0.339179
$178$	−7.61856	−0.571035
$179$	−14.7246	−1.10057	−0.550285	−	0.834977i	$-0.685481\pi$
$179$	−14.7246	−1.10057	−0.550285	+	0.834977i	$0.685481\pi$
$180$	0.347296	0.0258859
$181$	−10.8990	−0.810115	−0.405058	−	0.914291i	$-0.632749\pi$
$181$	−10.8990	−0.810115	−0.405058	+	0.914291i	$0.632749\pi$
$182$	1.04189	0.0772300
$183$	−4.00000	−0.295689
$184$	−0.0564370	−0.00416059
$185$	0	0
$186$	1.16250	0.0852389
$187$	−13.5963	−0.994257
$188$	8.51754	0.621206
$189$	−2.75103	−0.200108
$190$	−0.246282	−0.0178672
$191$	9.05737	0.655368	0.327684	−	0.944787i	$-0.393732\pi$
$191$	9.05737	0.655368	0.327684	+	0.944787i	$0.393732\pi$
$192$	0.347296	0.0250640
$193$	12.5895	0.906209	0.453105	−	0.891457i	$-0.350316\pi$
$193$	12.5895	0.906209	0.453105	+	0.891457i	$0.350316\pi$
$194$	10.2412	0.735277
$195$	0.0323936	0.00231975
$196$	−5.18479	−0.370342
$197$	−11.9368	−0.850459	−0.425229	−	0.905086i	$-0.639807\pi$
$197$	−11.9368	−0.850459	−0.425229	+	0.905086i	$0.639807\pi$
$198$	13.0496	0.927397
$199$	2.29086	0.162395	0.0811974	−	0.996698i	$-0.474126\pi$
$199$	2.29086	0.162395	0.0811974	+	0.996698i	$0.474126\pi$
$200$	4.98545	0.352525
$201$	1.77837	0.125437
$202$	16.1138	1.13376
$203$	−7.78880	−0.546667
$204$	−1.04189	−0.0729468
$205$	0.862149	0.0602151
$206$	19.6313	1.36778
$207$	−0.162504	−0.0112948
$208$	−0.773318	−0.0536200
$209$	−9.25402	−0.640114
$210$	0.0564370	0.00389452
$211$	0.955423	0.0657740	0.0328870	−	0.999459i	$-0.489530\pi$
$211$	0.955423	0.0657740	0.0328870	+	0.999459i	$0.489530\pi$
$212$	2.78106	0.191004
$213$	4.66550	0.319675
$214$	0.638156	0.0436234
$215$	−1.12330	−0.0766086
$216$	2.04189	0.138933
$217$	−4.50980	−0.306145
$218$	−7.25671	−0.491486
$219$	−3.02734	−0.204569
$220$	−0.546637	−0.0368542
$221$	2.31996	0.156057
$222$	0	0
$223$	−17.8648	−1.19632	−0.598159	−	0.801377i	$-0.704101\pi$
$223$	−17.8648	−1.19632	−0.598159	+	0.801377i	$0.704101\pi$
$224$	−1.34730	−0.0900200
$225$	14.3550	0.957002
$226$	13.8229	0.919489
$227$	8.85029	0.587414	0.293707	−	0.955895i	$-0.405111\pi$
$227$	8.85029	0.587414	0.293707	+	0.955895i	$0.405111\pi$
$228$	−0.709141	−0.0469640
$229$	−17.4115	−1.15058	−0.575291	−	0.817949i	$-0.695111\pi$
$229$	−17.4115	−1.15058	−0.575291	+	0.817949i	$0.695111\pi$
$230$	0.00680713	0.000448849 0
$231$	2.12061	0.139526
$232$	5.78106	0.379545
$233$	−4.19253	−0.274662	−0.137331	−	0.990525i	$-0.543852\pi$
$233$	−4.19253	−0.274662	−0.137331	+	0.990525i	$0.543852\pi$
$234$	−2.22668	−0.145563
$235$	−1.02734	−0.0670163
$236$	−12.9932	−0.845785
$237$	1.44057	0.0935751
$238$	4.04189	0.261997
$239$	8.57667	0.554778	0.277389	−	0.960758i	$-0.410531\pi$
$239$	8.57667	0.554778	0.277389	+	0.960758i	$0.410531\pi$
$240$	−0.0418891	−0.00270393
$241$	−20.9145	−1.34722	−0.673610	−	0.739087i	$-0.735257\pi$
$241$	−20.9145	−1.34722	−0.673610	+	0.739087i	$0.735257\pi$
$242$	−9.53983	−0.613243
$243$	8.87939	0.569613
$244$	−11.5175	−0.737335
$245$	0.625362	0.0399529
$246$	2.48246	0.158276
$247$	1.57903	0.100471
$248$	3.34730	0.212554
$249$	−2.05644	−0.130321
$250$	−1.20439	−0.0761725
$251$	6.79561	0.428935	0.214467	−	0.976731i	$-0.431198\pi$
$251$	6.79561	0.428935	0.214467	+	0.976731i	$0.431198\pi$
$252$	−3.87939	−0.244378
$253$	0.255777	0.0160806
$254$	−12.1138	−0.760088
$255$	0.125667	0.00786959
$256$	1.00000	0.0625000
$257$	−9.88207	−0.616427	−0.308213	−	0.951317i	$-0.599731\pi$
$257$	−9.88207	−0.616427	−0.308213	+	0.951317i	$0.599731\pi$
$258$	−3.23442	−0.201366
$259$	0	0
$260$	0.0932736	0.00578458
$261$	16.6459	1.03036
$262$	−21.6459	−1.33729
$263$	−20.6509	−1.27339	−0.636696	−	0.771115i	$-0.719699\pi$
$263$	−20.6509	−1.27339	−0.636696	+	0.771115i	$0.719699\pi$
$264$	−1.57398	−0.0968716
$265$	−0.335437	−0.0206057
$266$	2.75103	0.168676
$267$	2.64590	0.161926
$268$	5.12061	0.312791
$269$	27.6263	1.68441	0.842203	−	0.539161i	$-0.181259\pi$
$269$	27.6263	1.68441	0.842203	+	0.539161i	$0.181259\pi$
$270$	−0.246282	−0.0149882
$271$	6.84255	0.415655	0.207828	−	0.978165i	$-0.433361\pi$
$271$	6.84255	0.415655	0.207828	+	0.978165i	$0.433361\pi$
$272$	−3.00000	−0.181902
$273$	−0.361844	−0.0218998
$274$	−8.63816	−0.521850
$275$	−22.5945	−1.36250
$276$	0.0196004	0.00117980
$277$	−23.8331	−1.43199	−0.715995	−	0.698106i	$-0.754027\pi$
$277$	−23.8331	−1.43199	−0.715995	+	0.698106i	$0.754027\pi$
$278$	13.4047	0.803959
$279$	9.63816	0.577021
$280$	0.162504	0.00971146
$281$	31.7520	1.89416	0.947082	−	0.320993i	$-0.104017\pi$
$281$	31.7520	1.89416	0.947082	+	0.320993i	$0.104017\pi$
$282$	−2.95811	−0.176153
$283$	−5.90941	−0.351278	−0.175639	−	0.984455i	$-0.556199\pi$
$283$	−5.90941	−0.351278	−0.175639	+	0.984455i	$0.556199\pi$
$284$	13.4338	0.797147
$285$	0.0855328	0.00506653
$286$	3.50475	0.207240
$287$	−9.63041	−0.568465
$288$	2.87939	0.169669
$289$	−8.00000	−0.470588
$290$	−0.697281	−0.0409458
$291$	−3.55674	−0.208500
$292$	−8.71688	−0.510117
$293$	−1.69459	−0.0989992	−0.0494996	−	0.998774i	$-0.515763\pi$
$293$	−1.69459	−0.0989992	−0.0494996	+	0.998774i	$0.515763\pi$
$294$	1.80066	0.105017
$295$	1.56717	0.0912442
$296$	0	0
$297$	−9.25402	−0.536973
$298$	−4.39693	−0.254707
$299$	−0.0436438	−0.00252398
$300$	−1.73143	−0.0999641
$301$	12.5476	0.723230
$302$	−14.7442	−0.848435
$303$	−5.59627	−0.321497
$304$	−2.04189	−0.117110
$305$	1.38919	0.0795445
$306$	−8.63816	−0.493810
$307$	−12.2267	−0.697814	−0.348907	−	0.937157i	$-0.613447\pi$
$307$	−12.2267	−0.697814	−0.348907	+	0.937157i	$0.613447\pi$
$308$	6.10607	0.347925
$309$	−6.81790	−0.387857
$310$	−0.403733	−0.0229305
$311$	−1.22163	−0.0692722	−0.0346361	−	0.999400i	$-0.511027\pi$
$311$	−1.22163	−0.0692722	−0.0346361	+	0.999400i	$0.511027\pi$
$312$	0.268571	0.0152048
$313$	−17.4115	−0.984155	−0.492077	−	0.870551i	$-0.663762\pi$
$313$	−17.4115	−0.984155	−0.492077	+	0.870551i	$0.663762\pi$
$314$	15.6527	0.883333
$315$	0.467911	0.0263638
$316$	4.14796	0.233341
$317$	−8.69728	−0.488488	−0.244244	−	0.969714i	$-0.578540\pi$
$317$	−8.69728	−0.488488	−0.244244	+	0.969714i	$0.578540\pi$
$318$	−0.965852	−0.0541623
$319$	−26.2003	−1.46693
$320$	−0.120615	−0.00674257
$321$	−0.221629	−0.0123701
$322$	−0.0760373	−0.00423740
$323$	6.12567	0.340841
$324$	7.92902	0.440501
$325$	3.85534	0.213856
$326$	4.78106	0.264798
$327$	2.52023	0.139369
$328$	7.14796	0.394680
$329$	11.4757	0.632673
$330$	0.189845	0.0104506
$331$	−8.24216	−0.453030	−0.226515	−	0.974008i	$-0.572733\pi$
$331$	−8.24216	−0.453030	−0.226515	+	0.974008i	$0.572733\pi$
$332$	−5.92127	−0.324972
$333$	0	0
$334$	−2.00505	−0.109712
$335$	−0.617622	−0.0337443
$336$	0.467911	0.0255266
$337$	14.0273	0.764118	0.382059	−	0.924138i	$-0.375215\pi$
$337$	14.0273	0.764118	0.382059	+	0.924138i	$0.375215\pi$
$338$	12.4020	0.674579
$339$	−4.80066	−0.260736
$340$	0.361844	0.0196238
$341$	−15.1702	−0.821515
$342$	−5.87939	−0.317921
$343$	−16.4165	−0.886409
$344$	−9.31315	−0.502131
$345$	−0.00236409	−0.000127278 0
$346$	13.7811	0.740874
$347$	19.8452	1.06535	0.532674	−	0.846320i	$-0.321187\pi$
$347$	19.8452	1.06535	0.532674	+	0.846320i	$0.321187\pi$
$348$	−2.00774	−0.107626
$349$	17.1976	0.920566	0.460283	−	0.887772i	$-0.347748\pi$
$349$	17.1976	0.920566	0.460283	+	0.887772i	$0.347748\pi$
$350$	6.71688	0.359032
$351$	1.57903	0.0842824
$352$	−4.53209	−0.241561
$353$	16.8753	0.898180	0.449090	−	0.893487i	$-0.351748\pi$
$353$	16.8753	0.898180	0.449090	+	0.893487i	$0.351748\pi$
$354$	4.51249	0.239836
$355$	−1.62031	−0.0859971
$356$	7.61856	0.403783
$357$	−1.40373	−0.0742934
$358$	14.7246	0.778220
$359$	−16.1138	−0.850454	−0.425227	−	0.905087i	$-0.639806\pi$
$359$	−16.1138	−0.850454	−0.425227	+	0.905087i	$0.639806\pi$
$360$	−0.347296	−0.0183041
$361$	−14.8307	−0.780563
$362$	10.8990	0.572838
$363$	3.31315	0.173895
$364$	−1.04189	−0.0546098
$365$	1.05138	0.0550320
$366$	4.00000	0.209083
$367$	−35.0823	−1.83128	−0.915642	−	0.401995i	$-0.868317\pi$
$367$	−35.0823	−1.83128	−0.915642	+	0.401995i	$0.868317\pi$
$368$	0.0564370	0.00294198
$369$	20.5817	1.07144
$370$	0	0
$371$	3.74691	0.194530
$372$	−1.16250	−0.0602730
$373$	−29.6955	−1.53758	−0.768788	−	0.639504i	$-0.779140\pi$
$373$	−29.6955	−1.53758	−0.768788	+	0.639504i	$0.779140\pi$
$374$	13.5963	0.703046
$375$	0.418281	0.0216000
$376$	−8.51754	−0.439259
$377$	4.47060	0.230248
$378$	2.75103	0.141498
$379$	0.419215	0.0215336	0.0107668	−	0.999942i	$-0.496573\pi$
$379$	0.419215	0.0215336	0.0107668	+	0.999942i	$0.496573\pi$
$380$	0.246282	0.0126340
$381$	4.20708	0.215535
$382$	−9.05737	−0.463415
$383$	−6.42839	−0.328475	−0.164238	−	0.986421i	$-0.552516\pi$
$383$	−6.42839	−0.328475	−0.164238	+	0.986421i	$0.552516\pi$
$384$	−0.347296	−0.0177229
$385$	−0.736482	−0.0375346
$386$	−12.5895	−0.640787
$387$	−26.8161	−1.36314
$388$	−10.2412	−0.519920
$389$	−3.11381	−0.157876	−0.0789382	−	0.996880i	$-0.525153\pi$
$389$	−3.11381	−0.157876	−0.0789382	+	0.996880i	$0.525153\pi$
$390$	−0.0323936	−0.00164031
$391$	−0.169311	−0.00856242
$392$	5.18479	0.261872
$393$	7.51754	0.379210
$394$	11.9368	0.601365
$395$	−0.500305	−0.0251731
$396$	−13.0496	−0.655769
$397$	19.7074	0.989085	0.494543	−	0.869153i	$-0.335336\pi$
$397$	19.7074	0.989085	0.494543	+	0.869153i	$0.335336\pi$
$398$	−2.29086	−0.114830
$399$	−0.955423	−0.0478310
$400$	−4.98545	−0.249273
$401$	18.9162	0.944631	0.472316	−	0.881430i	$-0.343418\pi$
$401$	18.9162	0.944631	0.472316	+	0.881430i	$0.343418\pi$
$402$	−1.77837	−0.0886971
$403$	2.58853	0.128944
$404$	−16.1138	−0.801692
$405$	−0.956356	−0.0475217
$406$	7.78880	0.386552
$407$	0	0
$408$	1.04189	0.0515812
$409$	6.37464	0.315206	0.157603	−	0.987503i	$-0.449623\pi$
$409$	6.37464	0.315206	0.157603	+	0.987503i	$0.449623\pi$
$410$	−0.862149	−0.0425785
$411$	3.00000	0.147979
$412$	−19.6313	−0.967167
$413$	−17.5057	−0.861398
$414$	0.162504	0.00798663
$415$	0.714193	0.0350584
$416$	0.773318	0.0379151
$417$	−4.65539	−0.227976
$418$	9.25402	0.452629
$419$	20.8854	1.02032	0.510159	−	0.860080i	$-0.329587\pi$
$419$	20.8854	1.02032	0.510159	+	0.860080i	$0.329587\pi$
$420$	−0.0564370	−0.00275384
$421$	−5.80840	−0.283084	−0.141542	−	0.989932i	$-0.545206\pi$
$421$	−5.80840	−0.283084	−0.141542	+	0.989932i	$0.545206\pi$
$422$	−0.955423	−0.0465092
$423$	−24.5253	−1.19246
$424$	−2.78106	−0.135060
$425$	14.9564	0.725490
$426$	−4.66550	−0.226044
$427$	−15.5175	−0.750946
$428$	−0.638156	−0.0308464
$429$	−1.21719	−0.0587663
$430$	1.12330	0.0541705
$431$	11.3601	0.547196	0.273598	−	0.961844i	$-0.411786\pi$
$431$	11.3601	0.547196	0.273598	+	0.961844i	$0.411786\pi$
$432$	−2.04189	−0.0982404
$433$	35.6049	1.71106	0.855532	−	0.517750i	$-0.173230\pi$
$433$	35.6049	1.71106	0.855532	+	0.517750i	$0.173230\pi$
$434$	4.50980	0.216477
$435$	0.242163	0.0116108
$436$	7.25671	0.347533
$437$	−0.115238	−0.00551258
$438$	3.02734	0.144652
$439$	22.5039	1.07405	0.537027	−	0.843565i	$-0.319547\pi$
$439$	22.5039	1.07405	0.537027	+	0.843565i	$0.319547\pi$
$440$	0.546637	0.0260599
$441$	14.9290	0.710905
$442$	−2.31996	−0.110349
$443$	−26.0942	−1.23977	−0.619887	−	0.784691i	$-0.712821\pi$
$443$	−26.0942	−1.23977	−0.619887	+	0.784691i	$0.712821\pi$
$444$	0	0
$445$	−0.918910	−0.0435605
$446$	17.8648	0.845925
$447$	1.52704	0.0722263
$448$	1.34730	0.0636538
$449$	−6.52528	−0.307947	−0.153974	−	0.988075i	$-0.549207\pi$
$449$	−6.52528	−0.307947	−0.153974	+	0.988075i	$0.549207\pi$
$450$	−14.3550	−0.676703
$451$	−32.3952	−1.52543
$452$	−13.8229	−0.650177
$453$	5.12061	0.240587
$454$	−8.85029	−0.415365
$455$	0.125667	0.00589137
$456$	0.709141	0.0332086
$457$	−32.7442	−1.53171	−0.765855	−	0.643013i	$-0.777684\pi$
$457$	−32.7442	−1.53171	−0.765855	+	0.643013i	$0.777684\pi$
$458$	17.4115	0.813585
$459$	6.12567	0.285922
$460$	−0.00680713	−0.000317384 0
$461$	19.5449	0.910296	0.455148	−	0.890416i	$-0.349586\pi$
$461$	19.5449	0.910296	0.455148	+	0.890416i	$0.349586\pi$
$462$	−2.12061	−0.0986599
$463$	42.8803	1.99282	0.996409	−	0.0846740i	$-0.0269849\pi$
$463$	42.8803	1.99282	0.996409	+	0.0846740i	$0.0269849\pi$
$464$	−5.78106	−0.268379
$465$	0.140215	0.00650232
$466$	4.19253	0.194215
$467$	21.0888	0.975875	0.487937	−	0.872879i	$-0.337749\pi$
$467$	21.0888	0.975875	0.487937	+	0.872879i	$0.337749\pi$
$468$	2.22668	0.102928
$469$	6.89899	0.318566
$470$	1.02734	0.0473877
$471$	−5.43613	−0.250484
$472$	12.9932	0.598060
$473$	42.2080	1.94073
$474$	−1.44057	−0.0661676
$475$	10.1797	0.467079
$476$	−4.04189	−0.185260
$477$	−8.00774	−0.366649
$478$	−8.57667	−0.392288
$479$	−24.2026	−1.10585	−0.552923	−	0.833232i	$-0.686488\pi$
$479$	−24.2026	−1.10585	−0.552923	+	0.833232i	$0.686488\pi$
$480$	0.0418891	0.00191197
$481$	0	0
$482$	20.9145	0.952628
$483$	0.0264075	0.00120158
$484$	9.53983	0.433629
$485$	1.23524	0.0560895
$486$	−8.87939	−0.402777
$487$	28.9682	1.31268	0.656338	−	0.754467i	$-0.272105\pi$
$487$	28.9682	1.31268	0.656338	+	0.754467i	$0.272105\pi$
$488$	11.5175	0.521375
$489$	−1.66044	−0.0750879
$490$	−0.625362	−0.0282510
$491$	−28.1516	−1.27046	−0.635231	−	0.772322i	$-0.719095\pi$
$491$	−28.1516	−1.27046	−0.635231	+	0.772322i	$0.719095\pi$
$492$	−2.48246	−0.111918
$493$	17.3432	0.781097
$494$	−1.57903	−0.0710439
$495$	1.57398	0.0707450
$496$	−3.34730	−0.150298
$497$	18.0993	0.811863
$498$	2.05644	0.0921511
$499$	−4.68510	−0.209734	−0.104867	−	0.994486i	$-0.533442\pi$
$499$	−4.68510	−0.209734	−0.104867	+	0.994486i	$0.533442\pi$
$500$	1.20439	0.0538621
$501$	0.696347	0.0311105
$502$	−6.79561	−0.303303
$503$	25.2831	1.12732	0.563659	−	0.826007i	$-0.309393\pi$
$503$	25.2831	1.12732	0.563659	+	0.826007i	$0.309393\pi$
$504$	3.87939	0.172802
$505$	1.94356	0.0864874
$506$	−0.255777	−0.0113707
$507$	−4.30716	−0.191288
$508$	12.1138	0.537463
$509$	21.3746	0.947414	0.473707	−	0.880682i	$-0.342916\pi$
$509$	21.3746	0.947414	0.473707	+	0.880682i	$0.342916\pi$
$510$	−0.125667	−0.00556464
$511$	−11.7442	−0.519534
$512$	−1.00000	−0.0441942
$513$	4.16931	0.184080
$514$	9.88207	0.435880
$515$	2.36783	0.104339
$516$	3.23442	0.142388
$517$	38.6023	1.69773
$518$	0	0
$519$	−4.78611	−0.210087
$520$	−0.0932736	−0.00409032
$521$	−19.8844	−0.871153	−0.435577	−	0.900152i	$-0.643455\pi$
$521$	−19.8844	−0.871153	−0.435577	+	0.900152i	$0.643455\pi$
$522$	−16.6459	−0.728571
$523$	−16.9837	−0.742645	−0.371323	−	0.928504i	$-0.621096\pi$
$523$	−16.9837	−0.742645	−0.371323	+	0.928504i	$0.621096\pi$
$524$	21.6459	0.945605
$525$	−2.33275	−0.101809
$526$	20.6509	0.900424
$527$	10.0419	0.437432
$528$	1.57398	0.0684986
$529$	−22.9968	−0.999862
$530$	0.335437	0.0145704
$531$	37.4124	1.62356
$532$	−2.75103	−0.119272
$533$	5.52765	0.239429
$534$	−2.64590	−0.114499
$535$	0.0769710	0.00332775
$536$	−5.12061	−0.221177
$537$	−5.11381	−0.220677
$538$	−27.6263	−1.19105
$539$	−23.4979	−1.01213
$540$	0.246282	0.0105983
$541$	29.2695	1.25839	0.629197	−	0.777246i	$-0.283384\pi$
$541$	29.2695	1.25839	0.629197	+	0.777246i	$0.283384\pi$
$542$	−6.84255	−0.293913
$543$	−3.78518	−0.162438
$544$	3.00000	0.128624
$545$	−0.875266	−0.0374923
$546$	0.361844	0.0154855
$547$	−1.33368	−0.0570241	−0.0285121	−	0.999593i	$-0.509077\pi$
$547$	−1.33368	−0.0570241	−0.0285121	+	0.999593i	$0.509077\pi$
$548$	8.63816	0.369004
$549$	33.1634	1.41538
$550$	22.5945	0.963433
$551$	11.8043	0.502879
$552$	−0.0196004	−0.000834247 0
$553$	5.58853	0.237648
$554$	23.8331	1.01257
$555$	0	0
$556$	−13.4047	−0.568485
$557$	32.3286	1.36981	0.684904	−	0.728633i	$-0.259844\pi$
$557$	32.3286	1.36981	0.684904	+	0.728633i	$0.259844\pi$
$558$	−9.63816	−0.408016
$559$	−7.20203	−0.304613
$560$	−0.162504	−0.00686704
$561$	−4.72193	−0.199360
$562$	−31.7520	−1.33938
$563$	−16.6878	−0.703306	−0.351653	−	0.936130i	$-0.614380\pi$
$563$	−16.6878	−0.703306	−0.351653	+	0.936130i	$0.614380\pi$
$564$	2.95811	0.124559
$565$	1.66725	0.0701418
$566$	5.90941	0.248391
$567$	10.6827	0.448633
$568$	−13.4338	−0.563668
$569$	−0.426022	−0.0178598	−0.00892989	−	0.999960i	$-0.502843\pi$
$569$	−0.426022	−0.0178598	−0.00892989	+	0.999960i	$0.502843\pi$
$570$	−0.0855328	−0.00358258
$571$	13.9513	0.583844	0.291922	−	0.956442i	$-0.405705\pi$
$571$	13.9513	0.583844	0.291922	+	0.956442i	$0.405705\pi$
$572$	−3.50475	−0.146541
$573$	3.14559	0.131409
$574$	9.63041	0.401966
$575$	−0.281364	−0.0117337
$576$	−2.87939	−0.119974
$577$	−10.9632	−0.456402	−0.228201	−	0.973614i	$-0.573284\pi$
$577$	−10.9632	−0.456402	−0.228201	+	0.973614i	$0.573284\pi$
$578$	8.00000	0.332756
$579$	4.37227	0.181705
$580$	0.697281	0.0289530
$581$	−7.97771	−0.330971
$582$	3.55674	0.147432
$583$	12.6040	0.522005
$584$	8.71688	0.360707
$585$	−0.268571	−0.0111040
$586$	1.69459	0.0700030
$587$	−12.7365	−0.525691	−0.262845	−	0.964838i	$-0.584661\pi$
$587$	−12.7365	−0.525691	−0.262845	+	0.964838i	$0.584661\pi$
$588$	−1.80066	−0.0742579
$589$	6.83481	0.281623
$590$	−1.56717	−0.0645194
$591$	−4.14559	−0.170527
$592$	0	0
$593$	33.6786	1.38301	0.691507	−	0.722369i	$-0.256947\pi$
$593$	33.6786	1.38301	0.691507	+	0.722369i	$0.256947\pi$
$594$	9.25402	0.379697
$595$	0.487511	0.0199860
$596$	4.39693	0.180105
$597$	0.795607	0.0325620
$598$	0.0436438	0.00178473
$599$	−13.6186	−0.556439	−0.278220	−	0.960518i	$-0.589744\pi$
$599$	−13.6186	−0.556439	−0.278220	+	0.960518i	$0.589744\pi$
$600$	1.73143	0.0706853
$601$	16.1821	0.660082	0.330041	−	0.943967i	$-0.392937\pi$
$601$	16.1821	0.660082	0.330041	+	0.943967i	$0.392937\pi$
$602$	−12.5476	−0.511401
$603$	−14.7442	−0.600431
$604$	14.7442	0.599934
$605$	−1.15064	−0.0467803
$606$	5.59627	0.227333
$607$	−41.3046	−1.67650	−0.838251	−	0.545285i	$-0.816421\pi$
$607$	−41.3046	−1.67650	−0.838251	+	0.545285i	$0.816421\pi$
$608$	2.04189	0.0828095
$609$	−2.70502	−0.109613
$610$	−1.38919	−0.0562465
$611$	−6.58677	−0.266472
$612$	8.63816	0.349177
$613$	−31.1121	−1.25660	−0.628302	−	0.777970i	$-0.716250\pi$
$613$	−31.1121	−1.25660	−0.628302	+	0.777970i	$0.716250\pi$
$614$	12.2267	0.493429
$615$	0.299421	0.0120738
$616$	−6.10607	−0.246020
$617$	19.1070	0.769219	0.384609	−	0.923079i	$-0.374336\pi$
$617$	19.1070	0.769219	0.384609	+	0.923079i	$0.374336\pi$
$618$	6.81790	0.274256
$619$	38.9273	1.56462	0.782309	−	0.622890i	$-0.214042\pi$
$619$	38.9273	1.56462	0.782309	+	0.622890i	$0.214042\pi$
$620$	0.403733	0.0162143
$621$	−0.115238	−0.00462434
$622$	1.22163	0.0489829
$623$	10.2645	0.411237
$624$	−0.268571	−0.0107514
$625$	24.7820	0.991280
$626$	17.4115	0.695902
$627$	−3.21389	−0.128350
$628$	−15.6527	−0.624611
$629$	0	0
$630$	−0.467911	−0.0186420
$631$	−0.675926	−0.0269082	−0.0134541	−	0.999909i	$-0.504283\pi$
$631$	−0.675926	−0.0269082	−0.0134541	+	0.999909i	$0.504283\pi$
$632$	−4.14796	−0.164997
$633$	0.331815	0.0131885
$634$	8.69728	0.345413
$635$	−1.46110	−0.0579821
$636$	0.965852	0.0382985
$637$	4.00950	0.158862
$638$	26.2003	1.03728
$639$	−38.6810	−1.53020
$640$	0.120615	0.00476772
$641$	18.8203	0.743356	0.371678	−	0.928362i	$-0.378783\pi$
$641$	18.8203	0.743356	0.371678	+	0.928362i	$0.378783\pi$
$642$	0.221629	0.00874701
$643$	45.3164	1.78711	0.893553	−	0.448958i	$-0.148205\pi$
$643$	45.3164	1.78711	0.893553	+	0.448958i	$0.148205\pi$
$644$	0.0760373	0.00299629
$645$	−0.390119	−0.0153609
$646$	−6.12567	−0.241011
$647$	20.2044	0.794317	0.397158	−	0.917750i	$-0.369996\pi$
$647$	20.2044	0.794317	0.397158	+	0.917750i	$0.369996\pi$
$648$	−7.92902	−0.311481
$649$	−58.8863	−2.31149
$650$	−3.85534	−0.151219
$651$	−1.56624	−0.0613857
$652$	−4.78106	−0.187241
$653$	−17.5689	−0.687525	−0.343763	−	0.939057i	$-0.611702\pi$
$653$	−17.5689	−0.687525	−0.343763	+	0.939057i	$0.611702\pi$
$654$	−2.52023	−0.0985488
$655$	−2.61081	−0.102013
$656$	−7.14796	−0.279081
$657$	25.0993	0.979215
$658$	−11.4757	−0.447367
$659$	32.5594	1.26834	0.634168	−	0.773196i	$-0.281343\pi$
$659$	32.5594	1.26834	0.634168	+	0.773196i	$0.281343\pi$
$660$	−0.189845	−0.00738971
$661$	23.4757	0.913097	0.456549	−	0.889699i	$-0.349085\pi$
$661$	23.4757	0.913097	0.456549	+	0.889699i	$0.349085\pi$
$662$	8.24216	0.320341
$663$	0.805712	0.0312913
$664$	5.92127	0.229790
$665$	0.331815	0.0128672
$666$	0	0
$667$	−0.326266	−0.0126331
$668$	2.00505	0.0775778
$669$	−6.20439	−0.239876
$670$	0.617622	0.0238608
$671$	−52.1985	−2.01510
$672$	−0.467911	−0.0180501
$673$	13.2445	0.510539	0.255270	−	0.966870i	$-0.417836\pi$
$673$	13.2445	0.510539	0.255270	+	0.966870i	$0.417836\pi$
$674$	−14.0273	−0.540313
$675$	10.1797	0.391818
$676$	−12.4020	−0.476999
$677$	5.38507	0.206965	0.103482	−	0.994631i	$-0.467001\pi$
$677$	5.38507	0.206965	0.103482	+	0.994631i	$0.467001\pi$
$678$	4.80066	0.184368
$679$	−13.7980	−0.529518
$680$	−0.361844	−0.0138761
$681$	3.07367	0.117783
$682$	15.1702	0.580899
$683$	25.1352	0.961770	0.480885	−	0.876784i	$-0.340315\pi$
$683$	25.1352	0.961770	0.480885	+	0.876784i	$0.340315\pi$
$684$	5.87939	0.224804
$685$	−1.04189	−0.0398085
$686$	16.4165	0.626786
$687$	−6.04694	−0.230705
$688$	9.31315	0.355060
$689$	−2.15064	−0.0819330
$690$	0.00236409	8.99995e−5 0
$691$	41.1269	1.56454	0.782271	−	0.622938i	$-0.214061\pi$
$691$	41.1269	1.56454	0.782271	+	0.622938i	$0.214061\pi$
$692$	−13.7811	−0.523877
$693$	−17.5817	−0.667874
$694$	−19.8452	−0.753315
$695$	1.61680	0.0613287
$696$	2.00774	0.0761032
$697$	21.4439	0.812244
$698$	−17.1976	−0.650938
$699$	−1.45605	−0.0550729
$700$	−6.71688	−0.253874
$701$	31.0232	1.17173	0.585865	−	0.810408i	$-0.300755\pi$
$701$	31.0232	1.17173	0.585865	+	0.810408i	$0.300755\pi$
$702$	−1.57903	−0.0595967
$703$	0	0
$704$	4.53209	0.170810
$705$	−0.356792	−0.0134376
$706$	−16.8753	−0.635109
$707$	−21.7101	−0.816491
$708$	−4.51249	−0.169590
$709$	−19.3773	−0.727731	−0.363865	−	0.931452i	$-0.618543\pi$
$709$	−19.3773	−0.727731	−0.363865	+	0.931452i	$0.618543\pi$
$710$	1.62031	0.0608092
$711$	−11.9436	−0.447919
$712$	−7.61856	−0.285517
$713$	−0.188911	−0.00707478
$714$	1.40373	0.0525334
$715$	0.422724	0.0158090
$716$	−14.7246	−0.550285
$717$	2.97864	0.111240
$718$	16.1138	0.601362
$719$	−30.5604	−1.13971	−0.569855	−	0.821746i	$-0.693000\pi$
$719$	−30.5604	−1.13971	−0.569855	+	0.821746i	$0.693000\pi$
$720$	0.347296	0.0129430
$721$	−26.4492	−0.985021
$722$	14.8307	0.551941
$723$	−7.26352	−0.270133
$724$	−10.8990	−0.405058
$725$	28.8212	1.07039
$726$	−3.31315	−0.122962
$727$	52.1958	1.93584	0.967918	−	0.251266i	$-0.0808468\pi$
$727$	52.1958	1.93584	0.967918	+	0.251266i	$0.0808468\pi$
$728$	1.04189	0.0386150
$729$	−20.7033	−0.766788
$730$	−1.05138	−0.0389135
$731$	−27.9394	−1.03338
$732$	−4.00000	−0.147844
$733$	1.21987	0.0450571	0.0225285	−	0.999746i	$-0.492828\pi$
$733$	1.21987	0.0450571	0.0225285	+	0.999746i	$0.492828\pi$
$734$	35.0823	1.29491
$735$	0.217186	0.00801103
$736$	−0.0564370	−0.00208029
$737$	23.2071	0.854844
$738$	−20.5817	−0.757624
$739$	−2.61318	−0.0961273	−0.0480637	−	0.998844i	$-0.515305\pi$
$739$	−2.61318	−0.0961273	−0.0480637	+	0.998844i	$0.515305\pi$
$740$	0	0
$741$	0.548392	0.0201457
$742$	−3.74691	−0.137553
$743$	−22.5594	−0.827625	−0.413813	−	0.910362i	$-0.635803\pi$
$743$	−22.5594	−0.827625	−0.413813	+	0.910362i	$0.635803\pi$
$744$	1.16250	0.0426195
$745$	−0.530334	−0.0194299
$746$	29.6955	1.08723
$747$	17.0496	0.623813
$748$	−13.5963	−0.497129
$749$	−0.859785	−0.0314159
$750$	−0.418281	−0.0152735
$751$	42.6691	1.55702	0.778509	−	0.627633i	$-0.215976\pi$
$751$	42.6691	1.55702	0.778509	+	0.627633i	$0.215976\pi$
$752$	8.51754	0.310603
$753$	2.36009	0.0860064
$754$	−4.47060	−0.162810
$755$	−1.77837	−0.0647215
$756$	−2.75103	−0.100054
$757$	−23.3158	−0.847428	−0.423714	−	0.905796i	$-0.639274\pi$
$757$	−23.3158	−0.847428	−0.423714	+	0.905796i	$0.639274\pi$
$758$	−0.419215	−0.0152266
$759$	0.0888306	0.00322435
$760$	−0.246282	−0.00893359
$761$	−4.11650	−0.149223	−0.0746114	−	0.997213i	$-0.523772\pi$
$761$	−4.11650	−0.149223	−0.0746114	+	0.997213i	$0.523772\pi$
$762$	−4.20708	−0.152406
$763$	9.77694	0.353949
$764$	9.05737	0.327684
$765$	−1.04189	−0.0376696
$766$	6.42839	0.232267
$767$	10.0479	0.362808
$768$	0.347296	0.0125320
$769$	6.18035	0.222869	0.111435	−	0.993772i	$-0.464455\pi$
$769$	6.18035	0.222869	0.111435	+	0.993772i	$0.464455\pi$
$770$	0.736482	0.0265410
$771$	−3.43201	−0.123601
$772$	12.5895	0.453105
$773$	18.1156	0.651571	0.325786	−	0.945444i	$-0.394371\pi$
$773$	18.1156	0.651571	0.325786	+	0.945444i	$0.394371\pi$
$774$	26.8161	0.963886
$775$	16.6878	0.599443
$776$	10.2412	0.367639
$777$	0	0
$778$	3.11381	0.111635
$779$	14.5953	0.522932
$780$	0.0323936	0.00115988
$781$	60.8830	2.17857
$782$	0.169311	0.00605455
$783$	11.8043	0.421851
$784$	−5.18479	−0.185171
$785$	1.88795	0.0673837
$786$	−7.51754	−0.268142
$787$	−27.9205	−0.995257	−0.497628	−	0.867390i	$-0.665796\pi$
$787$	−27.9205	−0.995257	−0.497628	+	0.867390i	$0.665796\pi$
$788$	−11.9368	−0.425229
$789$	−7.17200	−0.255330
$790$	0.500305	0.0178000
$791$	−18.6236	−0.662179
$792$	13.0496	0.463698
$793$	8.90673	0.316287
$794$	−19.7074	−0.699389
$795$	−0.116496	−0.00413169
$796$	2.29086	0.0811974
$797$	−1.97266	−0.0698752	−0.0349376	−	0.999389i	$-0.511123\pi$
$797$	−1.97266	−0.0698752	−0.0349376	+	0.999389i	$0.511123\pi$
$798$	0.955423	0.0338216
$799$	−25.5526	−0.903987
$800$	4.98545	0.176262
$801$	−21.9368	−0.775097
$802$	−18.9162	−0.667955
$803$	−39.5057	−1.39413
$804$	1.77837	0.0627183
$805$	−0.00917123	−0.000323243 0
$806$	−2.58853	−0.0911769
$807$	9.59451	0.337743
$808$	16.1138	0.566882
$809$	15.7041	0.552126	0.276063	−	0.961139i	$-0.410970\pi$
$809$	15.7041	0.552126	0.276063	+	0.961139i	$0.410970\pi$
$810$	0.956356	0.0336029
$811$	−40.5435	−1.42367	−0.711837	−	0.702345i	$-0.752136\pi$
$811$	−40.5435	−1.42367	−0.711837	+	0.702345i	$0.752136\pi$
$812$	−7.78880	−0.273333
$813$	2.37639	0.0833437
$814$	0	0
$815$	0.576666	0.0201997
$816$	−1.04189	−0.0364734
$817$	−19.0164	−0.665300
$818$	−6.37464	−0.222884
$819$	3.00000	0.104828
$820$	0.862149	0.0301075
$821$	−9.97359	−0.348081	−0.174040	−	0.984739i	$-0.555682\pi$
$821$	−9.97359	−0.348081	−0.174040	+	0.984739i	$0.555682\pi$
$822$	−3.00000	−0.104637
$823$	−10.6023	−0.369571	−0.184786	−	0.982779i	$-0.559159\pi$
$823$	−10.6023	−0.369571	−0.184786	+	0.982779i	$0.559159\pi$
$824$	19.6313	0.683890
$825$	−7.84699	−0.273197
$826$	17.5057	0.609101
$827$	25.2199	0.876981	0.438490	−	0.898736i	$-0.355513\pi$
$827$	25.2199	0.876981	0.438490	+	0.898736i	$0.355513\pi$
$828$	−0.162504	−0.00564740
$829$	5.47390	0.190116	0.0950582	−	0.995472i	$-0.469696\pi$
$829$	5.47390	0.190116	0.0950582	+	0.995472i	$0.469696\pi$
$830$	−0.714193	−0.0247900
$831$	−8.27713	−0.287131
$832$	−0.773318	−0.0268100
$833$	15.5544	0.538927
$834$	4.65539	0.161203
$835$	−0.241839	−0.00836918
$836$	−9.25402	−0.320057
$837$	6.83481	0.236246
$838$	−20.8854	−0.721473
$839$	13.5199	0.466759	0.233379	−	0.972386i	$-0.425022\pi$
$839$	13.5199	0.466759	0.233379	+	0.972386i	$0.425022\pi$
$840$	0.0564370	0.00194726
$841$	4.42065	0.152436
$842$	5.80840	0.200171
$843$	11.0273	0.379802
$844$	0.955423	0.0328870
$845$	1.49586	0.0514592
$846$	24.5253	0.843197
$847$	12.8530	0.441634
$848$	2.78106	0.0955020
$849$	−2.05232	−0.0704354
$850$	−14.9564	−0.512999
$851$	0	0
$852$	4.66550	0.159837
$853$	−22.6212	−0.774537	−0.387268	−	0.921967i	$-0.626581\pi$
$853$	−22.6212	−0.774537	−0.387268	+	0.921967i	$0.626581\pi$
$854$	15.5175	0.530999
$855$	−0.709141	−0.0242521
$856$	0.638156	0.0218117
$857$	−28.4148	−0.970630	−0.485315	−	0.874339i	$-0.661295\pi$
$857$	−28.4148	−0.970630	−0.485315	+	0.874339i	$0.661295\pi$
$858$	1.21719	0.0415540
$859$	22.5235	0.768493	0.384246	−	0.923231i	$-0.374461\pi$
$859$	22.5235	0.768493	0.384246	+	0.923231i	$0.374461\pi$
$860$	−1.12330	−0.0383043
$861$	−3.34461	−0.113984
$862$	−11.3601	−0.386926
$863$	−9.95306	−0.338806	−0.169403	−	0.985547i	$-0.554184\pi$
$863$	−9.95306	−0.338806	−0.169403	+	0.985547i	$0.554184\pi$
$864$	2.04189	0.0694665
$865$	1.66220	0.0565165
$866$	−35.6049	−1.20991
$867$	−2.77837	−0.0943584
$868$	−4.50980	−0.153073
$869$	18.7989	0.637709
$870$	−0.242163	−0.00821010
$871$	−3.95987	−0.134175
$872$	−7.25671	−0.245743
$873$	29.4884	0.998033
$874$	0.115238	0.00389799
$875$	1.62267	0.0548564
$876$	−3.02734	−0.102284
$877$	44.0042	1.48592	0.742959	−	0.669337i	$-0.233422\pi$
$877$	44.0042	1.48592	0.742959	+	0.669337i	$0.233422\pi$
$878$	−22.5039	−0.759471
$879$	−0.588526	−0.0198505
$880$	−0.546637	−0.0184271
$881$	−12.3158	−0.414931	−0.207466	−	0.978242i	$-0.566521\pi$
$881$	−12.3158	−0.414931	−0.207466	+	0.978242i	$0.566521\pi$
$882$	−14.9290	−0.502686
$883$	21.7246	0.731092	0.365546	−	0.930793i	$-0.380882\pi$
$883$	21.7246	0.731092	0.365546	+	0.930793i	$0.380882\pi$
$884$	2.31996	0.0780285
$885$	0.544273	0.0182955
$886$	26.0942	0.876652
$887$	10.7145	0.359758	0.179879	−	0.983689i	$-0.442429\pi$
$887$	10.7145	0.359758	0.179879	+	0.983689i	$0.442429\pi$
$888$	0	0
$889$	16.3209	0.547385
$890$	0.918910	0.0308019
$891$	35.9350	1.20387
$892$	−17.8648	−0.598159
$893$	−17.3919	−0.581997
$894$	−1.52704	−0.0510717
$895$	1.77601	0.0593654
$896$	−1.34730	−0.0450100
$897$	−0.0151573	−0.000506088 0
$898$	6.52528	0.217751
$899$	19.3509	0.645389
$900$	14.3550	0.478501
$901$	−8.34318	−0.277952
$902$	32.3952	1.07864
$903$	4.35773	0.145016
$904$	13.8229	0.459744
$905$	1.31458	0.0436981
$906$	−5.12061	−0.170121
$907$	18.2104	0.604666	0.302333	−	0.953202i	$-0.402235\pi$
$907$	18.2104	0.604666	0.302333	+	0.953202i	$0.402235\pi$
$908$	8.85029	0.293707
$909$	46.3979	1.53892
$910$	−0.125667	−0.00416583
$911$	20.6144	0.682987	0.341493	−	0.939884i	$-0.389067\pi$
$911$	20.6144	0.682987	0.341493	+	0.939884i	$0.389067\pi$
$912$	−0.709141	−0.0234820
$913$	−26.8357	−0.888133
$914$	32.7442	1.08308
$915$	0.482459	0.0159496
$916$	−17.4115	−0.575291
$917$	29.1634	0.963062
$918$	−6.12567	−0.202177
$919$	−25.1462	−0.829497	−0.414748	−	0.909936i	$-0.636130\pi$
$919$	−25.1462	−0.829497	−0.414748	+	0.909936i	$0.636130\pi$
$920$	0.00680713	0.000224425 0
$921$	−4.24628	−0.139920
$922$	−19.5449	−0.643676
$923$	−10.3886	−0.341944
$924$	2.12061	0.0697631
$925$	0	0
$926$	−42.8803	−1.40913
$927$	56.5262	1.85656
$928$	5.78106	0.189773
$929$	−30.2739	−0.993256	−0.496628	−	0.867963i	$-0.665429\pi$
$929$	−30.2739	−0.993256	−0.496628	+	0.867963i	$0.665429\pi$
$930$	−0.140215	−0.00459783
$931$	10.5868	0.346967
$932$	−4.19253	−0.137331
$933$	−0.424267	−0.0138899
$934$	−21.0888	−0.690048
$935$	1.63991	0.0536308
$936$	−2.22668	−0.0727814
$937$	−16.1421	−0.527339	−0.263669	−	0.964613i	$-0.584933\pi$
$937$	−16.1421	−0.527339	−0.263669	+	0.964613i	$0.584933\pi$
$938$	−6.89899	−0.225260
$939$	−6.04694	−0.197334
$940$	−1.02734	−0.0335082
$941$	9.83212	0.320518	0.160259	−	0.987075i	$-0.448767\pi$
$941$	9.83212	0.320518	0.160259	+	0.987075i	$0.448767\pi$
$942$	5.43613	0.177119
$943$	−0.403409	−0.0131368
$944$	−12.9932	−0.422892
$945$	0.331815	0.0107939
$946$	−42.2080	−1.37230
$947$	−53.1935	−1.72856	−0.864278	−	0.503015i	$-0.832224\pi$
$947$	−53.1935	−1.72856	−0.864278	+	0.503015i	$0.832224\pi$
$948$	1.44057	0.0467875
$949$	6.74092	0.218820
$950$	−10.1797	−0.330274
$951$	−3.02053	−0.0979475
$952$	4.04189	0.130998
$953$	48.8458	1.58227	0.791136	−	0.611640i	$-0.209490\pi$
$953$	48.8458	1.58227	0.791136	+	0.611640i	$0.209490\pi$
$954$	8.00774	0.259260
$955$	−1.09245	−0.0353509
$956$	8.57667	0.277389
$957$	−9.09926	−0.294137
$958$	24.2026	0.781952
$959$	11.6382	0.375816
$960$	−0.0418891	−0.00135196
$961$	−19.7956	−0.638568
$962$	0	0
$963$	1.83750	0.0592125
$964$	−20.9145	−0.673610
$965$	−1.51847	−0.0488814
$966$	−0.0264075	−0.000849647 0
$967$	8.38743	0.269722	0.134861	−	0.990865i	$-0.456941\pi$
$967$	8.38743	0.269722	0.134861	+	0.990865i	$0.456941\pi$
$968$	−9.53983	−0.306622
$969$	2.12742	0.0683426
$970$	−1.23524	−0.0396613
$971$	41.3515	1.32703	0.663517	−	0.748161i	$-0.269063\pi$
$971$	41.3515	1.32703	0.663517	+	0.748161i	$0.269063\pi$
$972$	8.87939	0.284806
$973$	−18.0601	−0.578979
$974$	−28.9682	−0.928202
$975$	1.33895	0.0428806
$976$	−11.5175	−0.368668
$977$	48.3542	1.54699	0.773494	−	0.633803i	$-0.218507\pi$
$977$	48.3542	1.54699	0.773494	+	0.633803i	$0.218507\pi$
$978$	1.66044	0.0530952
$979$	34.5280	1.10352
$980$	0.625362	0.0199765
$981$	−20.8949	−0.667122
$982$	28.1516	0.898353
$983$	54.3715	1.73418	0.867090	−	0.498152i	$-0.165988\pi$
$983$	54.3715	1.73418	0.867090	+	0.498152i	$0.165988\pi$
$984$	2.48246	0.0791379
$985$	1.43975	0.0458742
$986$	−17.3432	−0.552319
$987$	3.98545	0.126858
$988$	1.57903	0.0502356
$989$	0.525606	0.0167133
$990$	−1.57398	−0.0500243
$991$	−31.8871	−1.01293	−0.506464	−	0.862261i	$-0.669048\pi$
$991$	−31.8871	−1.01293	−0.506464	+	0.862261i	$0.669048\pi$
$992$	3.34730	0.106277
$993$	−2.86247	−0.0908378
$994$	−18.0993	−0.574074
$995$	−0.276311	−0.00875966
$996$	−2.05644	−0.0651607
$997$	−0.521984	−0.0165314	−0.00826570	−	0.999966i	$-0.502631\pi$
$997$	−0.521984	−0.0165314	−0.00826570	+	0.999966i	$0.502631\pi$
$998$	4.68510	0.148304
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2738.2.a.m.1.2		3
37.9	even	9		74.2.f.a.7.1	✓	6
37.33	even	9		74.2.f.a.53.1	yes	6
37.36	even	2		2738.2.a.p.1.2		3
111.83	odd	18		666.2.x.c.451.1		6
111.107	odd	18		666.2.x.c.127.1		6
148.83	odd	18		592.2.bc.b.81.1		6
148.107	odd	18		592.2.bc.b.497.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.f.a.7.1	✓	6	37.9	even	9
74.2.f.a.53.1	yes	6	37.33	even	9
592.2.bc.b.81.1		6	148.83	odd	18
592.2.bc.b.497.1		6	148.107	odd	18
666.2.x.c.127.1		6	111.107	odd	18
666.2.x.c.451.1		6	111.83	odd	18
2738.2.a.m.1.2		3	1.1	even	1	trivial
2738.2.a.p.1.2		3	37.36	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2738 = 2 \cdot 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2738.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.347296 q^{3} +1.00000 q^{4} -0.120615 q^{5} -0.347296 q^{6} +1.34730 q^{7} -1.00000 q^{8} -2.87939 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.347296 q^{3} +1.00000 q^{4} -0.120615 q^{5} -0.347296 q^{6} +1.34730 q^{7} -1.00000 q^{8} -2.87939 q^{9} +0.120615 q^{10} +4.53209 q^{11} +0.347296 q^{12} -0.773318 q^{13} -1.34730 q^{14} -0.0418891 q^{15} +1.00000 q^{16} -3.00000 q^{17} +2.87939 q^{18} -2.04189 q^{19} -0.120615 q^{20} +0.467911 q^{21} -4.53209 q^{22} +0.0564370 q^{23} -0.347296 q^{24} -4.98545 q^{25} +0.773318 q^{26} -2.04189 q^{27} +1.34730 q^{28} -5.78106 q^{29} +0.0418891 q^{30} -3.34730 q^{31} -1.00000 q^{32} +1.57398 q^{33} +3.00000 q^{34} -0.162504 q^{35} -2.87939 q^{36} +2.04189 q^{38} -0.268571 q^{39} +0.120615 q^{40} -7.14796 q^{41} -0.467911 q^{42} +9.31315 q^{43} +4.53209 q^{44} +0.347296 q^{45} -0.0564370 q^{46} +8.51754 q^{47} +0.347296 q^{48} -5.18479 q^{49} +4.98545 q^{50} -1.04189 q^{51} -0.773318 q^{52} +2.78106 q^{53} +2.04189 q^{54} -0.546637 q^{55} -1.34730 q^{56} -0.709141 q^{57} +5.78106 q^{58} -12.9932 q^{59} -0.0418891 q^{60} -11.5175 q^{61} +3.34730 q^{62} -3.87939 q^{63} +1.00000 q^{64} +0.0932736 q^{65} -1.57398 q^{66} +5.12061 q^{67} -3.00000 q^{68} +0.0196004 q^{69} +0.162504 q^{70} +13.4338 q^{71} +2.87939 q^{72} -8.71688 q^{73} -1.73143 q^{75} -2.04189 q^{76} +6.10607 q^{77} +0.268571 q^{78} +4.14796 q^{79} -0.120615 q^{80} +7.92902 q^{81} +7.14796 q^{82} -5.92127 q^{83} +0.467911 q^{84} +0.361844 q^{85} -9.31315 q^{86} -2.00774 q^{87} -4.53209 q^{88} +7.61856 q^{89} -0.347296 q^{90} -1.04189 q^{91} +0.0564370 q^{92} -1.16250 q^{93} -8.51754 q^{94} +0.246282 q^{95} -0.347296 q^{96} -10.2412 q^{97} +5.18479 q^{98} -13.0496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{9} + 6 q^{10} + 9 q^{11} - 9 q^{13} - 3 q^{14} + 3 q^{15} + 3 q^{16} - 9 q^{17} + 3 q^{18} - 3 q^{19} - 6 q^{20} + 6 q^{21} - 9 q^{22} + 15 q^{23} + 3 q^{25} + 9 q^{26} - 3 q^{27} + 3 q^{28} - 3 q^{30} - 9 q^{31} - 3 q^{32} - 3 q^{33} + 9 q^{34} - 3 q^{35} - 3 q^{36} + 3 q^{38} + 9 q^{39} + 6 q^{40} - 6 q^{41} - 6 q^{42} + 6 q^{43} + 9 q^{44} - 15 q^{46} + 3 q^{47} - 12 q^{49} - 3 q^{50} - 9 q^{52} - 9 q^{53} + 3 q^{54} - 15 q^{55} - 3 q^{56} - 18 q^{57} + 3 q^{59} + 3 q^{60} - 12 q^{61} + 9 q^{62} - 6 q^{63} + 3 q^{64} + 27 q^{65} + 3 q^{66} + 21 q^{67} - 9 q^{68} + 3 q^{69} + 3 q^{70} + 24 q^{71} + 3 q^{72} - 18 q^{73} - 15 q^{75} - 3 q^{76} + 6 q^{77} - 9 q^{78} - 3 q^{79} - 6 q^{80} - 9 q^{81} + 6 q^{82} - 9 q^{83} + 6 q^{84} + 18 q^{85} - 6 q^{86} + 18 q^{87} - 9 q^{88} + 3 q^{89} + 15 q^{92} - 6 q^{93} - 3 q^{94} - 3 q^{95} - 42 q^{97} + 12 q^{98} - 12 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^9 + 6 * q^10 + 9 * q^11 - 9 * q^13 - 3 * q^14 + 3 * q^15 + 3 * q^16 - 9 * q^17 + 3 * q^18 - 3 * q^19 - 6 * q^20 + 6 * q^21 - 9 * q^22 + 15 * q^23 + 3 * q^25 + 9 * q^26 - 3 * q^27 + 3 * q^28 - 3 * q^30 - 9 * q^31 - 3 * q^32 - 3 * q^33 + 9 * q^34 - 3 * q^35 - 3 * q^36 + 3 * q^38 + 9 * q^39 + 6 * q^40 - 6 * q^41 - 6 * q^42 + 6 * q^43 + 9 * q^44 - 15 * q^46 + 3 * q^47 - 12 * q^49 - 3 * q^50 - 9 * q^52 - 9 * q^53 + 3 * q^54 - 15 * q^55 - 3 * q^56 - 18 * q^57 + 3 * q^59 + 3 * q^60 - 12 * q^61 + 9 * q^62 - 6 * q^63 + 3 * q^64 + 27 * q^65 + 3 * q^66 + 21 * q^67 - 9 * q^68 + 3 * q^69 + 3 * q^70 + 24 * q^71 + 3 * q^72 - 18 * q^73 - 15 * q^75 - 3 * q^76 + 6 * q^77 - 9 * q^78 - 3 * q^79 - 6 * q^80 - 9 * q^81 + 6 * q^82 - 9 * q^83 + 6 * q^84 + 18 * q^85 - 6 * q^86 + 18 * q^87 - 9 * q^88 + 3 * q^89 + 15 * q^92 - 6 * q^93 - 3 * q^94 - 3 * q^95 - 42 * q^97 + 12 * q^98 - 12 * q^99

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