LMFDB - Embedded newform 1003.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1003,2,Mod(1,1003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1003, 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("1003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1003 = 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1003.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:	$8.00899532273$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.820249$ of defining polynomial
Character	$\chi$	$=$	1003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.82025 q^{2} -1.50694 q^{3} +1.31331 q^{4} +2.32719 q^{5} +2.74301 q^{6} +4.23607 q^{7} +1.24995 q^{8} -0.729126 q^{9} +O(q^{10})$	\(q-1.82025 q^{2} -1.50694 q^{3} +1.31331 q^{4} +2.32719 q^{5} +2.74301 q^{6} +4.23607 q^{7} +1.24995 q^{8} -0.729126 q^{9} -4.23607 q^{10} -6.05632 q^{11} -1.97908 q^{12} -1.17975 q^{13} -7.71070 q^{14} -3.50694 q^{15} -4.90184 q^{16} -1.00000 q^{17} +1.32719 q^{18} -1.87657 q^{19} +3.05632 q^{20} -6.38351 q^{21} +11.0240 q^{22} +0.193635 q^{23} -1.88360 q^{24} +0.415819 q^{25} +2.14744 q^{26} +5.61958 q^{27} +5.56326 q^{28} -3.41582 q^{29} +6.38351 q^{30} +2.14744 q^{31} +6.42266 q^{32} +9.12652 q^{33} +1.82025 q^{34} +9.85814 q^{35} -0.957567 q^{36} -2.61649 q^{37} +3.41582 q^{38} +1.77782 q^{39} +2.90888 q^{40} -5.29488 q^{41} +11.6196 q^{42} +5.54938 q^{43} -7.95381 q^{44} -1.69682 q^{45} -0.352463 q^{46} -12.5633 q^{47} +7.38679 q^{48} +10.9443 q^{49} -0.756894 q^{50} +1.50694 q^{51} -1.54938 q^{52} -4.66827 q^{53} -10.2290 q^{54} -14.0942 q^{55} +5.29488 q^{56} +2.82788 q^{57} +6.21764 q^{58} -1.00000 q^{59} -4.60569 q^{60} -7.52083 q^{61} -3.90888 q^{62} -3.08863 q^{63} -1.88717 q^{64} -2.74551 q^{65} -16.6125 q^{66} -1.08863 q^{67} -1.31331 q^{68} -0.291796 q^{69} -17.9443 q^{70} +7.43983 q^{71} -0.911372 q^{72} +3.71524 q^{73} +4.76267 q^{74} -0.626615 q^{75} -2.46451 q^{76} -25.6550 q^{77} -3.23607 q^{78} -12.8024 q^{79} -11.4075 q^{80} -6.28100 q^{81} +9.63800 q^{82} +4.03789 q^{83} -8.38351 q^{84} -2.32719 q^{85} -10.1012 q^{86} +5.14744 q^{87} -7.57010 q^{88} -8.83037 q^{89} +3.08863 q^{90} -4.99750 q^{91} +0.254302 q^{92} -3.23607 q^{93} +22.8683 q^{94} -4.36713 q^{95} -9.67858 q^{96} -9.73162 q^{97} -19.9213 q^{98} +4.41582 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} - 2 q^{3} + 5 q^{4} + q^{5} - 2 q^{6} + 8 q^{7} - 12 q^{8} + 2 q^{9}+O(q^{10}) $ 4 * q - 3 * q^2 - 2 * q^3 + 5 * q^4 + q^5 - 2 * q^6 + 8 * q^7 - 12 * q^8 + 2 * q^9	\( 4 q - 3 q^{2} - 2 q^{3} + 5 q^{4} + q^{5} - 2 q^{6} + 8 q^{7} - 12 q^{8} + 2 q^{9} - 8 q^{10} - 11 q^{11} + 14 q^{12} - 9 q^{13} - q^{14} - 10 q^{15} + 11 q^{16} - 4 q^{17} - 3 q^{18} + 10 q^{19} - q^{20} - 4 q^{21} + 14 q^{22} - 3 q^{23} - 20 q^{24} - 3 q^{25} - 4 q^{26} - 8 q^{27} + 5 q^{28} - 9 q^{29} + 4 q^{30} - 4 q^{31} - 17 q^{32} + 2 q^{33} + 3 q^{34} - 3 q^{35} - 9 q^{36} - 32 q^{37} + 9 q^{38} + 8 q^{39} + 11 q^{40} + 4 q^{41} + 16 q^{42} + 13 q^{43} - 23 q^{44} + 15 q^{45} + 20 q^{46} - 33 q^{47} + 24 q^{48} + 8 q^{49} + 18 q^{50} + 2 q^{51} + 3 q^{52} + 6 q^{53} - 2 q^{54} - 5 q^{55} - 4 q^{56} - 12 q^{57} - 9 q^{58} - 4 q^{59} + 4 q^{60} - 18 q^{61} - 15 q^{62} - 16 q^{63} + 34 q^{64} + 5 q^{65} - 6 q^{66} - 8 q^{67} - 5 q^{68} - 28 q^{69} - 36 q^{70} - 5 q^{71} + 18 q^{73} + 11 q^{74} - 2 q^{75} - 11 q^{76} - 37 q^{77} - 4 q^{78} + 27 q^{79} + 6 q^{80} - 8 q^{81} + 33 q^{82} - 22 q^{83} - 12 q^{84} - q^{85} - 19 q^{86} + 8 q^{87} + 25 q^{88} - 9 q^{89} + 16 q^{90} - 23 q^{91} - 51 q^{92} - 4 q^{93} + 27 q^{94} - 7 q^{95} - 60 q^{96} - 31 q^{97} + 14 q^{98} + 13 q^{99}+O(q^{100}) \) 4 * q - 3 * q^2 - 2 * q^3 + 5 * q^4 + q^5 - 2 * q^6 + 8 * q^7 - 12 * q^8 + 2 * q^9 - 8 * q^10 - 11 * q^11 + 14 * q^12 - 9 * q^13 - q^14 - 10 * q^15 + 11 * q^16 - 4 * q^17 - 3 * q^18 + 10 * q^19 - q^20 - 4 * q^21 + 14 * q^22 - 3 * q^23 - 20 * q^24 - 3 * q^25 - 4 * q^26 - 8 * q^27 + 5 * q^28 - 9 * q^29 + 4 * q^30 - 4 * q^31 - 17 * q^32 + 2 * q^33 + 3 * q^34 - 3 * q^35 - 9 * q^36 - 32 * q^37 + 9 * q^38 + 8 * q^39 + 11 * q^40 + 4 * q^41 + 16 * q^42 + 13 * q^43 - 23 * q^44 + 15 * q^45 + 20 * q^46 - 33 * q^47 + 24 * q^48 + 8 * q^49 + 18 * q^50 + 2 * q^51 + 3 * q^52 + 6 * q^53 - 2 * q^54 - 5 * q^55 - 4 * q^56 - 12 * q^57 - 9 * q^58 - 4 * q^59 + 4 * q^60 - 18 * q^61 - 15 * q^62 - 16 * q^63 + 34 * q^64 + 5 * q^65 - 6 * q^66 - 8 * q^67 - 5 * q^68 - 28 * q^69 - 36 * q^70 - 5 * q^71 + 18 * q^73 + 11 * q^74 - 2 * q^75 - 11 * q^76 - 37 * q^77 - 4 * q^78 + 27 * q^79 + 6 * q^80 - 8 * q^81 + 33 * q^82 - 22 * q^83 - 12 * q^84 - q^85 - 19 * q^86 + 8 * q^87 + 25 * q^88 - 9 * q^89 + 16 * q^90 - 23 * q^91 - 51 * q^92 - 4 * q^93 + 27 * q^94 - 7 * q^95 - 60 * q^96 - 31 * q^97 + 14 * q^98 + 13 * 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 $

$2$	−1.82025	−1.28711	−0.643555	−	0.765400i	$-0.722541\pi$
$2$	−1.82025	−1.28711	−0.643555	+	0.765400i	$0.722541\pi$
$3$	−1.50694	−0.870033	−0.435017	−	0.900422i	$-0.643258\pi$
$3$	−1.50694	−0.870033	−0.435017	+	0.900422i	$0.643258\pi$
$4$	1.31331	0.656654
$5$	2.32719	1.04075	0.520376	−	0.853937i	$-0.325792\pi$
$5$	2.32719	1.04075	0.520376	+	0.853937i	$0.325792\pi$
$6$	2.74301	1.11983
$7$	4.23607	1.60108	0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607	1.60108	0.800542	+	0.599277i	$0.204545\pi$
$8$	1.24995	0.441925
$9$	−0.729126	−0.243042
$10$	−4.23607	−1.33956
$11$	−6.05632	−1.82605	−0.913024	−	0.407905i	$-0.866259\pi$
$11$	−6.05632	−1.82605	−0.913024	+	0.407905i	$0.866259\pi$
$12$	−1.97908	−0.571311
$13$	−1.17975	−0.327204	−0.163602	−	0.986526i	$-0.552311\pi$
$13$	−1.17975	−0.327204	−0.163602	+	0.986526i	$0.552311\pi$
$14$	−7.71070	−2.06077
$15$	−3.50694	−0.905489
$16$	−4.90184	−1.22546
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.32719	0.312822
$19$	−1.87657	−0.430514	−0.215257	−	0.976557i	$-0.569059\pi$
$19$	−1.87657	−0.430514	−0.215257	+	0.976557i	$0.569059\pi$
$20$	3.05632	0.683413
$21$	−6.38351	−1.39300
$22$	11.0240	2.35033
$23$	0.193635	0.0403756	0.0201878	−	0.999796i	$-0.493574\pi$
$23$	0.193635	0.0403756	0.0201878	+	0.999796i	$0.493574\pi$
$24$	−1.88360	−0.384489
$25$	0.415819	0.0831637
$26$	2.14744	0.421148
$27$	5.61958	1.08149
$28$	5.56326	1.05136
$29$	−3.41582	−0.634302	−0.317151	−	0.948375i	$-0.602726\pi$
$29$	−3.41582	−0.634302	−0.317151	+	0.948375i	$0.602726\pi$
$30$	6.38351	1.16546
$31$	2.14744	0.385692	0.192846	−	0.981229i	$-0.438228\pi$
$31$	2.14744	0.385692	0.192846	+	0.981229i	$0.438228\pi$
$32$	6.42266	1.13538
$33$	9.12652	1.58872
$34$	1.82025	0.312170
$35$	9.85814	1.66633
$36$	−0.957567	−0.159594
$37$	−2.61649	−0.430149	−0.215074	−	0.976598i	$-0.568999\pi$
$37$	−2.61649	−0.430149	−0.215074	+	0.976598i	$0.568999\pi$
$38$	3.41582	0.554119
$39$	1.77782	0.284678
$40$	2.90888	0.459934
$41$	−5.29488	−0.826921	−0.413461	−	0.910522i	$-0.635680\pi$
$41$	−5.29488	−0.826921	−0.413461	+	0.910522i	$0.635680\pi$
$42$	11.6196	1.79294
$43$	5.54938	0.846272	0.423136	−	0.906066i	$-0.360929\pi$
$43$	5.54938	0.846272	0.423136	+	0.906066i	$0.360929\pi$
$44$	−7.95381	−1.19908
$45$	−1.69682	−0.252946
$46$	−0.352463	−0.0519679
$47$	−12.5633	−1.83254	−0.916270	−	0.400562i	$-0.868815\pi$
$47$	−12.5633	−1.83254	−0.916270	+	0.400562i	$0.868815\pi$
$48$	7.38679	1.06619
$49$	10.9443	1.56347
$50$	−0.756894	−0.107041
$51$	1.50694	0.211014
$52$	−1.54938	−0.214860
$53$	−4.66827	−0.641236	−0.320618	−	0.947209i	$-0.603891\pi$
$53$	−4.66827	−0.641236	−0.320618	+	0.947209i	$0.603891\pi$
$54$	−10.2290	−1.39199
$55$	−14.0942	−1.90046
$56$	5.29488	0.707558
$57$	2.82788	0.374561
$58$	6.21764	0.816416
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	−4.60569	−0.594592
$61$	−7.52083	−0.962943	−0.481472	−	0.876462i	$-0.659898\pi$
$61$	−7.52083	−0.962943	−0.481472	+	0.876462i	$0.659898\pi$
$62$	−3.90888	−0.496428
$63$	−3.08863	−0.389130
$64$	−1.88717	−0.235897
$65$	−2.74551	−0.340538
$66$	−16.6125	−2.04486
$67$	−1.08863	−0.132997	−0.0664985	−	0.997787i	$-0.521183\pi$
$67$	−1.08863	−0.132997	−0.0664985	+	0.997787i	$0.521183\pi$
$68$	−1.31331	−0.159262
$69$	−0.291796	−0.0351281
$70$	−17.9443	−2.14475
$71$	7.43983	0.882945	0.441472	−	0.897275i	$-0.354456\pi$
$71$	7.43983	0.882945	0.441472	+	0.897275i	$0.354456\pi$
$72$	−0.911372	−0.107406
$73$	3.71524	0.434836	0.217418	−	0.976079i	$-0.430236\pi$
$73$	3.71524	0.434836	0.217418	+	0.976079i	$0.430236\pi$
$74$	4.76267	0.553649
$75$	−0.626615	−0.0723552
$76$	−2.46451	−0.282699
$77$	−25.6550	−2.92366
$78$	−3.23607	−0.366413
$79$	−12.8024	−1.44038	−0.720192	−	0.693775i	$-0.755946\pi$
$79$	−12.8024	−1.44038	−0.720192	+	0.693775i	$0.755946\pi$
$80$	−11.4075	−1.27540
$81$	−6.28100	−0.697889
$82$	9.63800	1.06434
$83$	4.03789	0.443216	0.221608	−	0.975136i	$-0.428869\pi$
$83$	4.03789	0.443216	0.221608	+	0.975136i	$0.428869\pi$
$84$	−8.38351	−0.914716
$85$	−2.32719	−0.252419
$86$	−10.1012	−1.08925
$87$	5.14744	0.551864
$88$	−7.57010	−0.806976
$89$	−8.83037	−0.936018	−0.468009	−	0.883724i	$-0.655028\pi$
$89$	−8.83037	−0.936018	−0.468009	+	0.883724i	$0.655028\pi$
$90$	3.08863	0.325570
$91$	−4.99750	−0.523881
$92$	0.254302	0.0265128
$93$	−3.23607	−0.335565
$94$	22.8683	2.35868
$95$	−4.36713	−0.448058
$96$	−9.67858	−0.987816
$97$	−9.73162	−0.988096	−0.494048	−	0.869434i	$-0.664483\pi$
$97$	−9.73162	−0.988096	−0.494048	+	0.869434i	$0.664483\pi$
$98$	−19.9213	−2.01236
$99$	4.41582	0.443806
$100$	0.546098	0.0546098
$101$	11.2063	1.11506	0.557532	−	0.830155i	$-0.311748\pi$
$101$	11.2063	1.11506	0.557532	+	0.830155i	$0.311748\pi$
$102$	−2.74301	−0.271598
$103$	8.06771	0.794935	0.397467	−	0.917616i	$-0.369889\pi$
$103$	8.06771	0.794935	0.397467	+	0.917616i	$0.369889\pi$
$104$	−1.47463	−0.144600
$105$	−14.8556	−1.44976
$106$	8.49741	0.825341
$107$	−8.57714	−0.829184	−0.414592	−	0.910007i	$-0.636076\pi$
$107$	−8.57714	−0.829184	−0.414592	+	0.910007i	$0.636076\pi$
$108$	7.38023	0.710163
$109$	−2.62661	−0.251584	−0.125792	−	0.992057i	$-0.540147\pi$
$109$	−2.62661	−0.251584	−0.125792	+	0.992057i	$0.540147\pi$
$110$	25.6550	2.44611
$111$	3.94290	0.374244
$112$	−20.7645	−1.96206
$113$	6.68919	0.629266	0.314633	−	0.949213i	$-0.398119\pi$
$113$	6.68919	0.629266	0.314633	+	0.949213i	$0.398119\pi$
$114$	−5.14744	−0.482102
$115$	0.450625	0.0420210
$116$	−4.48602	−0.416516
$117$	0.860187	0.0795243
$118$	1.82025	0.167568
$119$	−4.23607	−0.388320
$120$	−4.38351	−0.400158
$121$	25.6790	2.33445
$122$	13.6898	1.23941
$123$	7.97908	0.719449
$124$	2.82025	0.253266
$125$	−10.6683	−0.954199
$126$	5.62207	0.500854
$127$	15.4752	1.37320	0.686602	−	0.727034i	$-0.259102\pi$
$127$	15.4752	1.37320	0.686602	+	0.727034i	$0.259102\pi$
$128$	−9.41020	−0.831752
$129$	−8.36259	−0.736285
$130$	4.99750	0.438310
$131$	−9.09875	−0.794962	−0.397481	−	0.917611i	$-0.630115\pi$
$131$	−9.09875	−0.794962	−0.397481	+	0.917611i	$0.630115\pi$
$132$	11.9859	1.04324
$133$	−7.94926	−0.689289
$134$	1.98157	0.171182
$135$	13.0778	1.12556
$136$	−1.24995	−0.107182
$137$	−9.76203	−0.834026	−0.417013	−	0.908900i	$-0.636923\pi$
$137$	−9.76203	−0.834026	−0.417013	+	0.908900i	$0.636923\pi$
$138$	0.531142	0.0452138
$139$	17.8279	1.51214	0.756070	−	0.654490i	$-0.227117\pi$
$139$	17.8279	1.51214	0.756070	+	0.654490i	$0.227117\pi$
$140$	12.9468	1.09420
$141$	18.9321	1.59437
$142$	−13.5423	−1.13645
$143$	7.14494	0.597490
$144$	3.57406	0.297838
$145$	−7.94926	−0.660150
$146$	−6.76267	−0.559682
$147$	−16.4924	−1.36027
$148$	−3.43626	−0.282459
$149$	−20.9422	−1.71565	−0.857827	−	0.513939i	$-0.828186\pi$
$149$	−20.9422	−1.71565	−0.857827	+	0.513939i	$0.828186\pi$
$150$	1.14059	0.0931292
$151$	−14.6082	−1.18880	−0.594399	−	0.804170i	$-0.702610\pi$
$151$	−14.6082	−1.18880	−0.594399	+	0.804170i	$0.702610\pi$
$152$	−2.34562	−0.190255
$153$	0.729126	0.0589463
$154$	46.6984	3.76307
$155$	4.99750	0.401409
$156$	2.33482	0.186935
$157$	−14.8480	−1.18500	−0.592500	−	0.805570i	$-0.701859\pi$
$157$	−14.8480	−1.18500	−0.592500	+	0.805570i	$0.701859\pi$
$158$	23.3036	1.85393
$159$	7.03481	0.557896
$160$	14.9468	1.18165
$161$	0.820249	0.0646447
$162$	11.4330	0.898260
$163$	−10.0942	−0.790639	−0.395320	−	0.918544i	$-0.629366\pi$
$163$	−10.0942	−0.790639	−0.395320	+	0.918544i	$0.629366\pi$
$164$	−6.95381	−0.543001
$165$	21.2392	1.65347
$166$	−7.34997	−0.570468
$167$	6.33028	0.489851	0.244926	−	0.969542i	$-0.421236\pi$
$167$	6.33028	0.489851	0.244926	+	0.969542i	$0.421236\pi$
$168$	−7.97908	−0.615599
$169$	−11.6082	−0.892938
$170$	4.23607	0.324892
$171$	1.36825	0.104633
$172$	7.28804	0.555707
$173$	−24.5500	−1.86650	−0.933250	−	0.359229i	$-0.883040\pi$
$173$	−24.5500	−1.86650	−0.933250	+	0.359229i	$0.883040\pi$
$174$	−9.36962	−0.710309
$175$	1.76144	0.133152
$176$	29.6871	2.23775
$177$	1.50694	0.113269
$178$	16.0735	1.20476
$179$	−8.95566	−0.669377	−0.334689	−	0.942329i	$-0.608631\pi$
$179$	−8.95566	−0.669377	−0.334689	+	0.942329i	$0.608631\pi$
$180$	−2.22844	−0.166098
$181$	11.2151	0.833615	0.416808	−	0.908995i	$-0.363149\pi$
$181$	11.2151	0.833615	0.416808	+	0.908995i	$0.363149\pi$
$182$	9.09670	0.674293
$183$	11.3334	0.837793
$184$	0.242034	0.0178430
$185$	−6.08908	−0.447678
$186$	5.89045	0.431909
$187$	6.05632	0.442882
$188$	−16.4994	−1.20334
$189$	23.8049	1.73155
$190$	7.94926	0.576700
$191$	12.1157	0.876663	0.438331	−	0.898813i	$-0.355570\pi$
$191$	12.1157	0.876663	0.438331	+	0.898813i	$0.355570\pi$
$192$	2.84386	0.205238
$193$	−26.6467	−1.91807	−0.959035	−	0.283289i	$-0.908574\pi$
$193$	−26.6467	−1.91807	−0.959035	+	0.283289i	$0.908574\pi$
$194$	17.7140	1.27179
$195$	4.13732	0.296279
$196$	14.3732	1.02666
$197$	−11.5196	−0.820735	−0.410367	−	0.911920i	$-0.634600\pi$
$197$	−11.5196	−0.820735	−0.410367	+	0.911920i	$0.634600\pi$
$198$	−8.03789	−0.571228
$199$	21.6196	1.53257	0.766286	−	0.642500i	$-0.222103\pi$
$199$	21.6196	1.53257	0.766286	+	0.642500i	$0.222103\pi$
$200$	0.519753	0.0367521
$201$	1.64050	0.115712
$202$	−20.3982	−1.43521
$203$	−14.4696	−1.01557
$204$	1.97908	0.138563
$205$	−12.3222	−0.860620
$206$	−14.6852	−1.02317
$207$	−0.141184	−0.00981297
$208$	5.78295	0.400975
$209$	11.3651	0.786139
$210$	27.0410	1.86600
$211$	−10.6473	−0.732993	−0.366497	−	0.930419i	$-0.619443\pi$
$211$	−10.6473	−0.732993	−0.366497	+	0.930419i	$0.619443\pi$
$212$	−6.13087	−0.421070
$213$	−11.2114	−0.768191
$214$	15.6125	1.06725
$215$	12.9145	0.880759
$216$	7.02420	0.477936
$217$	9.09670	0.617524
$218$	4.78109	0.323817
$219$	−5.59865	−0.378322
$220$	−18.5100	−1.24795
$221$	1.17975	0.0793586
$222$	−7.17706	−0.481693
$223$	18.0371	1.20785	0.603927	−	0.797040i	$-0.293602\pi$
$223$	18.0371	1.20785	0.603927	+	0.797040i	$0.293602\pi$
$224$	27.2068	1.81783
$225$	−0.303184	−0.0202123
$226$	−12.1760	−0.809935
$227$	−6.74242	−0.447510	−0.223755	−	0.974645i	$-0.571832\pi$
$227$	−6.74242	−0.447510	−0.223755	+	0.974645i	$0.571832\pi$
$228$	3.71387	0.245957
$229$	−7.89172	−0.521499	−0.260750	−	0.965406i	$-0.583970\pi$
$229$	−7.89172	−0.521499	−0.260750	+	0.965406i	$0.583970\pi$
$230$	−0.820249	−0.0540856
$231$	38.6606	2.54368
$232$	−4.26961	−0.280314
$233$	−13.6960	−0.897257	−0.448629	−	0.893718i	$-0.648087\pi$
$233$	−13.6960	−0.897257	−0.448629	+	0.893718i	$0.648087\pi$
$234$	−1.56575	−0.102357
$235$	−29.2371	−1.90722
$236$	−1.31331	−0.0854890
$237$	19.2925	1.25318
$238$	7.71070	0.499810
$239$	−21.3442	−1.38064	−0.690319	−	0.723505i	$-0.742530\pi$
$239$	−21.3442	−1.38064	−0.690319	+	0.723505i	$0.742530\pi$
$240$	17.1905	1.10964
$241$	19.7904	1.27481	0.637407	−	0.770527i	$-0.280007\pi$
$241$	19.7904	1.27481	0.637407	+	0.770527i	$0.280007\pi$
$242$	−46.7421	−3.00470
$243$	−7.39363	−0.474302
$244$	−9.87716	−0.632320
$245$	25.4694	1.62718
$246$	−14.5239	−0.926011
$247$	2.21388	0.140866
$248$	2.68420	0.170447
$249$	−6.08487	−0.385613
$250$	19.4189	1.22816
$251$	−14.3949	−0.908598	−0.454299	−	0.890849i	$-0.650110\pi$
$251$	−14.3949	−0.908598	−0.454299	+	0.890849i	$0.650110\pi$
$252$	−4.05632	−0.255524
$253$	−1.17271	−0.0737278
$254$	−28.1688	−1.76747
$255$	3.50694	0.219613
$256$	20.9033	1.30645
$257$	25.3379	1.58053	0.790267	−	0.612762i	$-0.209942\pi$
$257$	25.3379	1.58053	0.790267	+	0.612762i	$0.209942\pi$
$258$	15.2220	0.947680
$259$	−11.0836	−0.688704
$260$	−3.60569	−0.223616
$261$	2.49056	0.154162
$262$	16.5620	1.02320
$263$	−8.15697	−0.502981	−0.251490	−	0.967860i	$-0.580921\pi$
$263$	−8.15697	−0.502981	−0.251490	+	0.967860i	$0.580921\pi$
$264$	11.4077	0.702096
$265$	−10.8639	−0.667367
$266$	14.4696	0.887191
$267$	13.3069	0.814366
$268$	−1.42970	−0.0873330
$269$	4.87031	0.296948	0.148474	−	0.988916i	$-0.452564\pi$
$269$	4.87031	0.296948	0.148474	+	0.988916i	$0.452564\pi$
$270$	−23.8049	−1.44872
$271$	27.6777	1.68130	0.840651	−	0.541578i	$-0.182173\pi$
$271$	27.6777	1.68130	0.840651	+	0.541578i	$0.182173\pi$
$272$	4.90184	0.297218
$273$	7.53095	0.455794
$274$	17.7693	1.07348
$275$	−2.51833	−0.151861
$276$	−0.383218	−0.0230670
$277$	21.7612	1.30751	0.653753	−	0.756708i	$-0.273193\pi$
$277$	21.7612	1.30751	0.653753	+	0.756708i	$0.273193\pi$
$278$	−32.4512	−1.94629
$279$	−1.56575	−0.0937393
$280$	12.3222	0.736392
$281$	17.2588	1.02957	0.514787	−	0.857318i	$-0.327871\pi$
$281$	17.2588	1.02957	0.514787	+	0.857318i	$0.327871\pi$
$282$	−34.4611	−2.05213
$283$	−2.53549	−0.150719	−0.0753597	−	0.997156i	$-0.524010\pi$
$283$	−2.53549	−0.150719	−0.0753597	+	0.997156i	$0.524010\pi$
$284$	9.77078	0.579789
$285$	6.58101	0.389825
$286$	−13.0056	−0.769036
$287$	−22.4295	−1.32397
$288$	−4.68293	−0.275944
$289$	1.00000	0.0588235
$290$	14.4696	0.849687
$291$	14.6650	0.859677
$292$	4.87925	0.285537
$293$	−0.256401	−0.0149791	−0.00748955	−	0.999972i	$-0.502384\pi$
$293$	−0.256401	−0.0149791	−0.00748955	+	0.999972i	$0.502384\pi$
$294$	30.0202	1.75082
$295$	−2.32719	−0.135494
$296$	−3.27049	−0.190093
$297$	−34.0339	−1.97485
$298$	38.1201	2.20824
$299$	−0.228441	−0.0132111
$300$	−0.822938	−0.0475123
$301$	23.5075	1.35495
$302$	26.5905	1.53011
$303$	−16.8872	−0.970143
$304$	9.19863	0.527577
$305$	−17.5024	−1.00218
$306$	−1.32719	−0.0758705
$307$	7.48431	0.427152	0.213576	−	0.976926i	$-0.431489\pi$
$307$	7.48431	0.427152	0.213576	+	0.976926i	$0.431489\pi$
$308$	−33.6929	−1.91983
$309$	−12.1576	−0.691620
$310$	−9.09670	−0.516658
$311$	17.3398	0.983250	0.491625	−	0.870807i	$-0.336403\pi$
$311$	17.3398	0.983250	0.491625	+	0.870807i	$0.336403\pi$
$312$	2.22218	0.125806
$313$	−21.1950	−1.19801	−0.599006	−	0.800744i	$-0.704438\pi$
$313$	−21.1950	−1.19801	−0.599006	+	0.800744i	$0.704438\pi$
$314$	27.0271	1.52523
$315$	−7.18783	−0.404988
$316$	−16.8135	−0.945833
$317$	7.61195	0.427530	0.213765	−	0.976885i	$-0.431427\pi$
$317$	7.61195	0.427530	0.213765	+	0.976885i	$0.431427\pi$
$318$	−12.8051	−0.718074
$319$	20.6873	1.15827
$320$	−4.39181	−0.245510
$321$	12.9253	0.721417
$322$	−1.49306	−0.0832049
$323$	1.87657	0.104415
$324$	−8.24888	−0.458271
$325$	−0.490562	−0.0272115
$326$	18.3740	1.01764
$327$	3.95816	0.218887
$328$	−6.61835	−0.365437
$329$	−53.2188	−2.93405
$330$	−38.6606	−2.12819
$331$	15.8112	0.869060	0.434530	−	0.900657i	$-0.356914\pi$
$331$	15.8112	0.869060	0.434530	+	0.900657i	$0.356914\pi$
$332$	5.30299	0.291039
$333$	1.90775	0.104544
$334$	−11.5227	−0.630493
$335$	−2.53344	−0.138417
$336$	31.2909	1.70706
$337$	23.5979	1.28546	0.642729	−	0.766094i	$-0.277802\pi$
$337$	23.5979	1.28546	0.642729	+	0.766094i	$0.277802\pi$
$338$	21.1298	1.14931
$339$	−10.0802	−0.547482
$340$	−3.05632	−0.165752
$341$	−13.0056	−0.704292
$342$	−2.49056	−0.134674
$343$	16.7082	0.902158
$344$	6.93645	0.373988
$345$	−0.679065	−0.0365596
$346$	44.6871	2.40239
$347$	−22.5504	−1.21057	−0.605286	−	0.796008i	$-0.706941\pi$
$347$	−22.5504	−1.21057	−0.605286	+	0.796008i	$0.706941\pi$
$348$	6.76017	0.362383
$349$	29.9366	1.60247	0.801234	−	0.598352i	$-0.204177\pi$
$349$	29.9366	1.60247	0.801234	+	0.598352i	$0.204177\pi$
$350$	−3.20625	−0.171381
$351$	−6.62970	−0.353867
$352$	−38.8977	−2.07325
$353$	12.6208	0.671740	0.335870	−	0.941908i	$-0.390970\pi$
$353$	12.6208	0.671740	0.335870	+	0.941908i	$0.390970\pi$
$354$	−2.74301	−0.145789
$355$	17.3139	0.918926
$356$	−11.5970	−0.614639
$357$	6.38351	0.337851
$358$	16.3015	0.861563
$359$	−6.10080	−0.321988	−0.160994	−	0.986955i	$-0.551470\pi$
$359$	−6.10080	−0.321988	−0.160994	+	0.986955i	$0.551470\pi$
$360$	−2.12094	−0.111783
$361$	−15.4785	−0.814658
$362$	−20.4144	−1.07295
$363$	−38.6967	−2.03105
$364$	−6.56326	−0.344008
$365$	8.64608	0.452556
$366$	−20.6297	−1.07833
$367$	−13.3601	−0.697391	−0.348696	−	0.937236i	$-0.613375\pi$
$367$	−13.3601	−0.697391	−0.348696	+	0.937236i	$0.613375\pi$
$368$	−0.949165	−0.0494787
$369$	3.86064	0.200977
$370$	11.0836	0.576211
$371$	−19.7751	−1.02667
$372$	−4.24995	−0.220350
$373$	0.911372	0.0471891	0.0235945	−	0.999722i	$-0.492489\pi$
$373$	0.911372	0.0471891	0.0235945	+	0.999722i	$0.492489\pi$
$374$	−11.0240	−0.570038
$375$	16.0765	0.830185
$376$	−15.7035	−0.809845
$377$	4.02981	0.207546
$378$	−43.3309	−2.22870
$379$	−10.9660	−0.563284	−0.281642	−	0.959520i	$-0.590879\pi$
$379$	−10.9660	−0.563284	−0.281642	+	0.959520i	$0.590879\pi$
$380$	−5.73538	−0.294219
$381$	−23.3203	−1.19473
$382$	−22.0536	−1.12836
$383$	31.8011	1.62496	0.812481	−	0.582987i	$-0.198116\pi$
$383$	31.8011	1.62496	0.812481	+	0.582987i	$0.198116\pi$
$384$	14.1806	0.723652
$385$	−59.7040	−3.04280
$386$	48.5036	2.46877
$387$	−4.04619	−0.205680
$388$	−12.7806	−0.648837
$389$	1.88737	0.0956932	0.0478466	−	0.998855i	$-0.484764\pi$
$389$	1.88737	0.0956932	0.0478466	+	0.998855i	$0.484764\pi$
$390$	−7.53095	−0.381344
$391$	−0.193635	−0.00979252
$392$	13.6798	0.690935
$393$	13.7113	0.691643
$394$	20.9685	1.05638
$395$	−29.7937	−1.49908
$396$	5.79933	0.291427
$397$	−25.2733	−1.26843	−0.634216	−	0.773156i	$-0.718677\pi$
$397$	−25.2733	−1.26843	−0.634216	+	0.773156i	$0.718677\pi$
$398$	−39.3530	−1.97259
$399$	11.9791	0.599704
$400$	−2.03828	−0.101914
$401$	17.3386	0.865847	0.432924	−	0.901431i	$-0.357482\pi$
$401$	17.3386	0.865847	0.432924	+	0.901431i	$0.357482\pi$
$402$	−2.98612	−0.148934
$403$	−2.53344	−0.126200
$404$	14.7173	0.732211
$405$	−14.6171	−0.726329
$406$	26.3384	1.30715
$407$	15.8463	0.785472
$408$	1.88360	0.0932523
$409$	11.3462	0.561032	0.280516	−	0.959849i	$-0.409494\pi$
$409$	11.3462	0.561032	0.280516	+	0.959849i	$0.409494\pi$
$410$	22.4295	1.10771
$411$	14.7108	0.725631
$412$	10.5954	0.521997
$413$	−4.23607	−0.208443
$414$	0.256990	0.0126304
$415$	9.39694	0.461278
$416$	−7.57714	−0.371500
$417$	−26.8656	−1.31561
$418$	−20.6873	−1.01185
$419$	−14.5030	−0.708517	−0.354259	−	0.935147i	$-0.615267\pi$
$419$	−14.5030	−0.708517	−0.354259	+	0.935147i	$0.615267\pi$
$420$	−19.5100	−0.951992
$421$	−28.8070	−1.40397	−0.701983	−	0.712194i	$-0.747702\pi$
$421$	−28.8070	−1.40397	−0.701983	+	0.712194i	$0.747702\pi$
$422$	19.3808	0.943444
$423$	9.16020	0.445384
$424$	−5.83511	−0.283378
$425$	−0.415819	−0.0201702
$426$	20.4075	0.988747
$427$	−31.8587	−1.54175
$428$	−11.2644	−0.544487
$429$	−10.7670	−0.519836
$430$	−23.5075	−1.13363
$431$	−3.83540	−0.184745	−0.0923723	−	0.995725i	$-0.529445\pi$
$431$	−3.83540	−0.184745	−0.0923723	+	0.995725i	$0.529445\pi$
$432$	−27.5463	−1.32532
$433$	−25.4361	−1.22238	−0.611190	−	0.791484i	$-0.709309\pi$
$433$	−25.4361	−1.22238	−0.611190	+	0.791484i	$0.709309\pi$
$434$	−16.5583	−0.794822
$435$	11.9791	0.574353
$436$	−3.44955	−0.165204
$437$	−0.363368	−0.0173823
$438$	10.1909	0.486942
$439$	9.34983	0.446243	0.223121	−	0.974791i	$-0.428375\pi$
$439$	9.34983	0.446243	0.223121	+	0.974791i	$0.428375\pi$
$440$	−17.6171	−0.839861
$441$	−7.97975	−0.379988
$442$	−2.14744	−0.102143
$443$	3.74110	0.177745	0.0888726	−	0.996043i	$-0.471674\pi$
$443$	3.74110	0.177745	0.0888726	+	0.996043i	$0.471674\pi$
$444$	5.17824	0.245748
$445$	−20.5500	−0.974162
$446$	−32.8320	−1.55464
$447$	31.5587	1.49268
$448$	−7.99419	−0.377690
$449$	35.8366	1.69124	0.845618	−	0.533789i	$-0.179232\pi$
$449$	35.8366	1.69124	0.845618	+	0.533789i	$0.179232\pi$
$450$	0.551871	0.0260154
$451$	32.0675	1.51000
$452$	8.78496	0.413210
$453$	22.0137	1.03429
$454$	12.2729	0.575995
$455$	−11.6301	−0.545230
$456$	3.53471	0.165528
$457$	−36.0784	−1.68768	−0.843838	−	0.536598i	$-0.819709\pi$
$457$	−36.0784	−1.68768	−0.843838	+	0.536598i	$0.819709\pi$
$458$	14.3649	0.671227
$459$	−5.61958	−0.262299
$460$	0.591809	0.0275932
$461$	−38.3187	−1.78468	−0.892340	−	0.451364i	$-0.850938\pi$
$461$	−38.3187	−1.78468	−0.892340	+	0.451364i	$0.850938\pi$
$462$	−70.3718	−3.27399
$463$	21.9577	1.02046	0.510230	−	0.860038i	$-0.329560\pi$
$463$	21.9577	1.02046	0.510230	+	0.860038i	$0.329560\pi$
$464$	16.7438	0.777311
$465$	−7.53095	−0.349239
$466$	24.9302	1.15487
$467$	19.2449	0.890549	0.445274	−	0.895394i	$-0.353106\pi$
$467$	19.2449	0.890549	0.445274	+	0.895394i	$0.353106\pi$
$468$	1.12969	0.0522199
$469$	−4.61150	−0.212939
$470$	53.2188	2.45480
$471$	22.3751	1.03099
$472$	−1.24995	−0.0575337
$473$	−33.6088	−1.54533
$474$	−35.1171	−1.61298
$475$	−0.780311	−0.0358031
$476$	−5.56326	−0.254992
$477$	3.40375	0.155847
$478$	38.8517	1.77703
$479$	−16.3000	−0.744767	−0.372383	−	0.928079i	$-0.621459\pi$
$479$	−16.3000	−0.744767	−0.372383	+	0.928079i	$0.621459\pi$
$480$	−22.5239	−1.02807
$481$	3.08681	0.140746
$482$	−36.0235	−1.64083
$483$	−1.23607	−0.0562430
$484$	33.7244	1.53293
$485$	−22.6473	−1.02836
$486$	13.4583	0.610479
$487$	34.1942	1.54949	0.774744	−	0.632275i	$-0.217879\pi$
$487$	34.1942	1.54949	0.774744	+	0.632275i	$0.217879\pi$
$488$	−9.40067	−0.425548
$489$	15.2114	0.687883
$490$	−46.3607	−2.09436
$491$	−15.4348	−0.696563	−0.348281	−	0.937390i	$-0.613235\pi$
$491$	−15.4348	−0.696563	−0.348281	+	0.937390i	$0.613235\pi$
$492$	10.4790	0.472429
$493$	3.41582	0.153841
$494$	−4.02981	−0.181310
$495$	10.2765	0.461892
$496$	−10.5264	−0.472650
$497$	31.5156	1.41367
$498$	11.0760	0.496326
$499$	27.2981	1.22203	0.611016	−	0.791619i	$-0.290761\pi$
$499$	27.2981	1.22203	0.611016	+	0.791619i	$0.290761\pi$
$500$	−14.0107	−0.626578
$501$	−9.53936	−0.426187
$502$	26.2023	1.16947
$503$	12.8981	0.575097	0.287548	−	0.957766i	$-0.407160\pi$
$503$	12.8981	0.575097	0.287548	+	0.957766i	$0.407160\pi$
$504$	−3.86064	−0.171966
$505$	26.0791	1.16050
$506$	2.13463	0.0948958
$507$	17.4929	0.776885
$508$	20.3237	0.901719
$509$	−23.5119	−1.04215	−0.521073	−	0.853512i	$-0.674468\pi$
$509$	−23.5119	−1.04215	−0.521073	+	0.853512i	$0.674468\pi$
$510$	−6.38351	−0.282667
$511$	15.7380	0.696209
$512$	−19.2287	−0.849798
$513$	−10.5455	−0.465596
$514$	−46.1213	−2.03432
$515$	18.7751	0.827329
$516$	−10.9826	−0.483484
$517$	76.0871	3.34631
$518$	20.1750	0.886438
$519$	36.9954	1.62392
$520$	−3.43175	−0.150492
$521$	9.45073	0.414044	0.207022	−	0.978336i	$-0.433623\pi$
$521$	9.45073	0.414044	0.207022	+	0.978336i	$0.433623\pi$
$522$	−4.53344	−0.198423
$523$	17.7873	0.777787	0.388893	−	0.921283i	$-0.372857\pi$
$523$	17.7873	0.777787	0.388893	+	0.921283i	$0.372857\pi$
$524$	−11.9495	−0.522014
$525$	−2.65438	−0.115847
$526$	14.8477	0.647392
$527$	−2.14744	−0.0935440
$528$	−44.7367	−1.94692
$529$	−22.9625	−0.998370
$530$	19.7751	0.858975
$531$	0.729126	0.0316414
$532$	−10.4398	−0.452624
$533$	6.24664	0.270572
$534$	−24.2218	−1.04818
$535$	−19.9607	−0.862974
$536$	−1.36073	−0.0587747
$537$	13.4957	0.582381
$538$	−8.86518	−0.382205
$539$	−66.2820	−2.85497
$540$	17.1752	0.739103
$541$	−10.3492	−0.444945	−0.222472	−	0.974939i	$-0.571413\pi$
$541$	−10.3492	−0.444945	−0.222472	+	0.974939i	$0.571413\pi$
$542$	−50.3803	−2.16402
$543$	−16.9006	−0.725273
$544$	−6.42266	−0.275369
$545$	−6.11263	−0.261836
$546$	−13.7082	−0.586657
$547$	−24.6935	−1.05582	−0.527908	−	0.849302i	$-0.677023\pi$
$547$	−24.6935	−1.05582	−0.527908	+	0.849302i	$0.677023\pi$
$548$	−12.8205	−0.547666
$549$	5.48363	0.234036
$550$	4.58399	0.195462
$551$	6.41001	0.273076
$552$	−0.364731	−0.0155240
$553$	−54.2319	−2.30617
$554$	−39.6109	−1.68291
$555$	9.17588	0.389495
$556$	23.4135	0.992953
$557$	43.2979	1.83459	0.917295	−	0.398208i	$-0.130368\pi$
$557$	43.2979	1.83459	0.917295	+	0.398208i	$0.130368\pi$
$558$	2.85006	0.120653
$559$	−6.54688	−0.276904
$560$	−48.3230	−2.04202
$561$	−9.12652	−0.385322
$562$	−31.4153	−1.32518
$563$	18.5025	0.779788	0.389894	−	0.920860i	$-0.372512\pi$
$563$	18.5025	0.779788	0.389894	+	0.920860i	$0.372512\pi$
$564$	24.8637	1.04695
$565$	15.5670	0.654909
$566$	4.61523	0.193992
$567$	−26.6067	−1.11738
$568$	9.29942	0.390195
$569$	22.6670	0.950252	0.475126	−	0.879918i	$-0.342403\pi$
$569$	22.6670	0.950252	0.475126	+	0.879918i	$0.342403\pi$
$570$	−11.9791	−0.501748
$571$	6.77279	0.283432	0.141716	−	0.989907i	$-0.454738\pi$
$571$	6.77279	0.283432	0.141716	+	0.989907i	$0.454738\pi$
$572$	9.38351	0.392344
$573$	−18.2577	−0.762726
$574$	40.8272	1.70410
$575$	0.0805169	0.00335779
$576$	1.37599	0.0573328
$577$	43.4466	1.80870	0.904352	−	0.426786i	$-0.140354\pi$
$577$	43.4466	1.80870	0.904352	+	0.426786i	$0.140354\pi$
$578$	−1.82025	−0.0757124
$579$	40.1550	1.66878
$580$	−10.4398	−0.433490
$581$	17.1048	0.709626
$582$	−26.6939	−1.10650
$583$	28.2725	1.17093
$584$	4.64387	0.192165
$585$	2.00182	0.0827651
$586$	0.466714	0.0192798
$587$	−7.44213	−0.307170	−0.153585	−	0.988135i	$-0.549082\pi$
$587$	−7.44213	−0.307170	−0.153585	+	0.988135i	$0.549082\pi$
$588$	−21.6596	−0.893225
$589$	−4.02981	−0.166046
$590$	4.23607	0.174396
$591$	17.3593	0.714067
$592$	12.8256	0.527130
$593$	−35.1592	−1.44382	−0.721908	−	0.691989i	$-0.756735\pi$
$593$	−35.1592	−1.44382	−0.721908	+	0.691989i	$0.756735\pi$
$594$	61.9502	2.54185
$595$	−9.85814	−0.404144
$596$	−27.5036	−1.12659
$597$	−32.5794	−1.33339
$598$	0.415819	0.0170041
$599$	−13.5801	−0.554869	−0.277434	−	0.960745i	$-0.589484\pi$
$599$	−13.5801	−0.554869	−0.277434	+	0.960745i	$0.589484\pi$
$600$	−0.783238	−0.0319756
$601$	−31.1511	−1.27068	−0.635340	−	0.772232i	$-0.719140\pi$
$601$	−31.1511	−1.27068	−0.635340	+	0.772232i	$0.719140\pi$
$602$	−42.7896	−1.74397
$603$	0.793747	0.0323239
$604$	−19.1850	−0.780628
$605$	59.7599	2.42959
$606$	30.7389	1.24868
$607$	12.9695	0.526418	0.263209	−	0.964739i	$-0.415219\pi$
$607$	12.9695	0.526418	0.263209	+	0.964739i	$0.415219\pi$
$608$	−12.0526	−0.488796
$609$	21.8049	0.883579
$610$	31.8587	1.28992
$611$	14.8215	0.599614
$612$	0.957567	0.0387073
$613$	−12.0543	−0.486867	−0.243434	−	0.969918i	$-0.578274\pi$
$613$	−12.0543	−0.486867	−0.243434	+	0.969918i	$0.578274\pi$
$614$	−13.6233	−0.549792
$615$	18.5688	0.748768
$616$	−32.0675	−1.29204
$617$	8.91274	0.358814	0.179407	−	0.983775i	$-0.442582\pi$
$617$	8.91274	0.358814	0.179407	+	0.983775i	$0.442582\pi$
$618$	22.1298	0.890191
$619$	2.85192	0.114628	0.0573141	−	0.998356i	$-0.481746\pi$
$619$	2.85192	0.114628	0.0573141	+	0.998356i	$0.481746\pi$
$620$	6.56326	0.263587
$621$	1.08814	0.0436657
$622$	−31.5628	−1.26555
$623$	−37.4061	−1.49864
$624$	−8.71457	−0.348862
$625$	−26.9062	−1.07625
$626$	38.5802	1.54197
$627$	−17.1265	−0.683967
$628$	−19.5000	−0.778135
$629$	2.61649	0.104326
$630$	13.0836	0.521265
$631$	−20.1765	−0.803216	−0.401608	−	0.915812i	$-0.631548\pi$
$631$	−20.1765	−0.803216	−0.401608	+	0.915812i	$0.631548\pi$
$632$	−16.0024	−0.636541
$633$	16.0449	0.637729
$634$	−13.8556	−0.550278
$635$	36.0138	1.42916
$636$	9.23886	0.366345
$637$	−12.9115	−0.511573
$638$	−37.6560	−1.49082
$639$	−5.42457	−0.214593
$640$	−21.8993	−0.865648
$641$	25.6827	1.01441	0.507204	−	0.861826i	$-0.330679\pi$
$641$	25.6827	1.01441	0.507204	+	0.861826i	$0.330679\pi$
$642$	−23.5272	−0.928544
$643$	−3.68063	−0.145150	−0.0725749	−	0.997363i	$-0.523122\pi$
$643$	−3.68063	−0.145150	−0.0725749	+	0.997363i	$0.523122\pi$
$644$	1.07724	0.0424492
$645$	−19.4613	−0.766289
$646$	−3.41582	−0.134394
$647$	−28.4237	−1.11745	−0.558726	−	0.829353i	$-0.688709\pi$
$647$	−28.4237	−1.11745	−0.558726	+	0.829353i	$0.688709\pi$
$648$	−7.85094	−0.308414
$649$	6.05632	0.237731
$650$	0.892946	0.0350242
$651$	−13.7082	−0.537267
$652$	−13.2568	−0.519176
$653$	−4.00019	−0.156540	−0.0782698	−	0.996932i	$-0.524940\pi$
$653$	−4.00019	−0.156540	−0.0782698	+	0.996932i	$0.524940\pi$
$654$	−7.20483	−0.281731
$655$	−21.1745	−0.827357
$656$	25.9547	1.01336
$657$	−2.70888	−0.105683
$658$	96.8715	3.77644
$659$	19.1543	0.746145	0.373073	−	0.927802i	$-0.378304\pi$
$659$	19.1543	0.746145	0.373073	+	0.927802i	$0.378304\pi$
$660$	27.8935	1.08575
$661$	−17.2924	−0.672594	−0.336297	−	0.941756i	$-0.609175\pi$
$661$	−17.2924	−0.672594	−0.336297	+	0.941756i	$0.609175\pi$
$662$	−28.7803	−1.11858
$663$	−1.77782	−0.0690446
$664$	5.04717	0.195868
$665$	−18.4995	−0.717378
$666$	−3.47258	−0.134560
$667$	−0.661421	−0.0256103
$668$	8.31360	0.321663
$669$	−27.1809	−1.05087
$670$	4.61150	0.178158
$671$	45.5485	1.75838
$672$	−40.9991	−1.58158
$673$	−18.6348	−0.718320	−0.359160	−	0.933276i	$-0.616937\pi$
$673$	−18.6348	−0.718320	−0.359160	+	0.933276i	$0.616937\pi$
$674$	−42.9540	−1.65453
$675$	2.33673	0.0899406
$676$	−15.2451	−0.586351
$677$	45.9267	1.76511	0.882553	−	0.470213i	$-0.155823\pi$
$677$	45.9267	1.76511	0.882553	+	0.470213i	$0.155823\pi$
$678$	18.3485	0.704670
$679$	−41.2238	−1.58202
$680$	−2.90888	−0.111550
$681$	10.1604	0.389349
$682$	23.6734	0.906501
$683$	35.0776	1.34221	0.671104	−	0.741363i	$-0.265820\pi$
$683$	35.0776	1.34221	0.671104	+	0.741363i	$0.265820\pi$
$684$	1.79694	0.0687076
$685$	−22.7181	−0.868014
$686$	−30.4131	−1.16118
$687$	11.8924	0.453722
$688$	−27.2021	−1.03707
$689$	5.50739	0.209815
$690$	1.23607	0.0470563
$691$	40.0384	1.52313	0.761567	−	0.648086i	$-0.224430\pi$
$691$	40.0384	1.52313	0.761567	+	0.648086i	$0.224430\pi$
$692$	−32.2416	−1.22564
$693$	18.7057	0.710571
$694$	41.0474	1.55814
$695$	41.4889	1.57376
$696$	6.43405	0.243882
$697$	5.29488	0.200558
$698$	−54.4920	−2.06255
$699$	20.6391	0.780643
$700$	2.31331	0.0874348
$701$	18.6281	0.703575	0.351787	−	0.936080i	$-0.385574\pi$
$701$	18.6281	0.703575	0.351787	+	0.936080i	$0.385574\pi$
$702$	12.0677	0.455466
$703$	4.91002	0.185185
$704$	11.4293	0.430759
$705$	44.0586	1.65934
$706$	−22.9731	−0.864603
$707$	47.4705	1.78531
$708$	1.97908	0.0743783
$709$	−46.0016	−1.72763	−0.863814	−	0.503811i	$-0.831931\pi$
$709$	−46.0016	−1.72763	−0.863814	+	0.503811i	$0.831931\pi$
$710$	−31.5156	−1.18276
$711$	9.33457	0.350074
$712$	−11.0375	−0.413649
$713$	0.415819	0.0155725
$714$	−11.6196	−0.434852
$715$	16.6277	0.621839
$716$	−11.7615	−0.439549
$717$	32.1644	1.20120
$718$	11.1050	0.414434
$719$	16.4778	0.614516	0.307258	−	0.951626i	$-0.400588\pi$
$719$	16.4778	0.614516	0.307258	+	0.951626i	$0.400588\pi$
$720$	8.31752	0.309976
$721$	34.1753	1.27276
$722$	28.1747	1.04855
$723$	−29.8230	−1.10913
$724$	14.7289	0.547396
$725$	−1.42036	−0.0527509
$726$	70.4377	2.61419
$727$	−30.0422	−1.11421	−0.557103	−	0.830444i	$-0.688087\pi$
$727$	−30.0422	−1.11421	−0.557103	+	0.830444i	$0.688087\pi$
$728$	−6.24664	−0.231516
$729$	29.9848	1.11055
$730$	−15.7380	−0.582490
$731$	−5.54938	−0.205251
$732$	14.8843	0.550140
$733$	23.0611	0.851782	0.425891	−	0.904775i	$-0.359961\pi$
$733$	23.0611	0.851782	0.425891	+	0.904775i	$0.359961\pi$
$734$	24.3187	0.897619
$735$	−38.3809	−1.41570
$736$	1.24365	0.0458415
$737$	6.59307	0.242859
$738$	−7.02732	−0.258679
$739$	15.6664	0.576296	0.288148	−	0.957586i	$-0.406960\pi$
$739$	15.6664	0.576296	0.288148	+	0.957586i	$0.406960\pi$
$740$	−7.99683	−0.293969
$741$	−3.33619	−0.122558
$742$	35.9956	1.32144
$743$	10.5899	0.388506	0.194253	−	0.980951i	$-0.437772\pi$
$743$	10.5899	0.388506	0.194253	+	0.980951i	$0.437772\pi$
$744$	−4.04493	−0.148294
$745$	−48.7366	−1.78557
$746$	−1.65893	−0.0607376
$747$	−2.94413	−0.107720
$748$	7.95381	0.290820
$749$	−36.3334	−1.32759
$750$	−29.2632	−1.06854
$751$	36.1591	1.31946	0.659732	−	0.751501i	$-0.270670\pi$
$751$	36.1591	1.31946	0.659732	+	0.751501i	$0.270670\pi$
$752$	61.5831	2.24570
$753$	21.6923	0.790510
$754$	−7.33527	−0.267135
$755$	−33.9960	−1.23724
$756$	31.2632	1.13703
$757$	−5.12529	−0.186282	−0.0931409	−	0.995653i	$-0.529691\pi$
$757$	−5.12529	−0.186282	−0.0931409	+	0.995653i	$0.529691\pi$
$758$	19.9608	0.725009
$759$	1.76721	0.0641456
$760$	−5.45870	−0.198008
$761$	45.8944	1.66367	0.831835	−	0.555023i	$-0.187291\pi$
$761$	45.8944	1.66367	0.831835	+	0.555023i	$0.187291\pi$
$762$	42.4487	1.53775
$763$	−11.1265	−0.402807
$764$	15.9117	0.575664
$765$	1.69682	0.0613485
$766$	−57.8860	−2.09151
$767$	1.17975	0.0425983
$768$	−31.5000	−1.13666
$769$	−37.0289	−1.33530	−0.667648	−	0.744477i	$-0.732699\pi$
$769$	−37.0289	−1.33530	−0.667648	+	0.744477i	$0.732699\pi$
$770$	108.676	3.91642
$771$	−38.1827	−1.37512
$772$	−34.9953	−1.25951
$773$	−21.6080	−0.777186	−0.388593	−	0.921410i	$-0.627039\pi$
$773$	−21.6080	−0.777186	−0.388593	+	0.921410i	$0.627039\pi$
$774$	7.36508	0.264732
$775$	0.892946	0.0320756
$776$	−12.1641	−0.436664
$777$	16.7024	0.599195
$778$	−3.43548	−0.123168
$779$	9.93620	0.356001
$780$	5.43357	0.194553
$781$	−45.0579	−1.61230
$782$	0.352463	0.0126041
$783$	−19.1955	−0.685990
$784$	−53.6471	−1.91597
$785$	−34.5542	−1.23329
$786$	−24.9580	−0.890221
$787$	−29.5662	−1.05392	−0.526960	−	0.849890i	$-0.676668\pi$
$787$	−29.5662	−1.05392	−0.526960	+	0.849890i	$0.676668\pi$
$788$	−15.1287	−0.538938
$789$	12.2921	0.437610
$790$	54.2319	1.92948
$791$	28.3359	1.00751
$792$	5.51956	0.196129
$793$	8.87270	0.315079
$794$	46.0037	1.63261
$795$	16.3713	0.580632
$796$	28.3931	1.00637
$797$	−9.85100	−0.348940	−0.174470	−	0.984662i	$-0.555821\pi$
$797$	−9.85100	−0.348940	−0.174470	+	0.984662i	$0.555821\pi$
$798$	−21.8049	−0.771885
$799$	12.5633	0.444456
$800$	2.67066	0.0944222
$801$	6.43845	0.227492
$802$	−31.5605	−1.11444
$803$	−22.5007	−0.794032
$804$	2.15448	0.0759826
$805$	1.90888	0.0672791
$806$	4.61150	0.162433
$807$	−7.33927	−0.258355
$808$	14.0073	0.492774
$809$	−46.8783	−1.64815	−0.824077	−	0.566478i	$-0.808305\pi$
$809$	−46.8783	−1.64815	−0.824077	+	0.566478i	$0.808305\pi$
$810$	26.6067	0.934865
$811$	1.29942	0.0456289	0.0228145	−	0.999740i	$-0.492737\pi$
$811$	1.29942	0.0456289	0.0228145	+	0.999740i	$0.492737\pi$
$812$	−19.0031	−0.666878
$813$	−41.7087	−1.46279
$814$	−28.8442	−1.01099
$815$	−23.4912	−0.822859
$816$	−7.38679	−0.258589
$817$	−10.4138	−0.364332
$818$	−20.6529	−0.722110
$819$	3.64381	0.127325
$820$	−16.1828	−0.565129
$821$	45.0119	1.57093	0.785463	−	0.618909i	$-0.212425\pi$
$821$	45.0119	1.57093	0.785463	+	0.618909i	$0.212425\pi$
$822$	−26.7773	−0.933967
$823$	24.2726	0.846089	0.423045	−	0.906109i	$-0.360961\pi$
$823$	24.2726	0.846089	0.423045	+	0.906109i	$0.360961\pi$
$824$	10.0842	0.351301
$825$	3.79498	0.132124
$826$	7.71070	0.268290
$827$	−3.75552	−0.130592	−0.0652962	−	0.997866i	$-0.520799\pi$
$827$	−3.75552	−0.130592	−0.0652962	+	0.997866i	$0.520799\pi$
$828$	−0.185418	−0.00644372
$829$	−48.0227	−1.66790	−0.833948	−	0.551843i	$-0.813925\pi$
$829$	−48.0227	−1.66790	−0.833948	+	0.551843i	$0.813925\pi$
$830$	−17.1048	−0.593716
$831$	−32.7929	−1.13757
$832$	2.22639	0.0771863
$833$	−10.9443	−0.379197
$834$	48.9020	1.69334
$835$	14.7318	0.509814
$836$	14.9258	0.516221
$837$	12.0677	0.417121
$838$	26.3991	0.911940
$839$	−17.8480	−0.616181	−0.308090	−	0.951357i	$-0.599690\pi$
$839$	−17.8480	−0.616181	−0.308090	+	0.951357i	$0.599690\pi$
$840$	−18.5688	−0.640686
$841$	−17.3322	−0.597661
$842$	52.4358	1.80706
$843$	−26.0080	−0.895764
$844$	−13.9832	−0.481323
$845$	−27.0145	−0.929326
$846$	−16.6738	−0.573259
$847$	108.778	3.73765
$848$	22.8831	0.785808
$849$	3.82084	0.131131
$850$	0.756894	0.0259612
$851$	−0.506643	−0.0173675
$852$	−14.7240	−0.504436
$853$	16.6744	0.570922	0.285461	−	0.958390i	$-0.407853\pi$
$853$	16.6744	0.570922	0.285461	+	0.958390i	$0.407853\pi$
$854$	57.9908	1.98441
$855$	3.18419	0.108897
$856$	−10.7210	−0.366437
$857$	13.0734	0.446580	0.223290	−	0.974752i	$-0.428320\pi$
$857$	13.0734	0.446580	0.223290	+	0.974752i	$0.428320\pi$
$858$	19.5987	0.669087
$859$	−0.219583	−0.00749206	−0.00374603	−	0.999993i	$-0.501192\pi$
$859$	−0.219583	−0.00749206	−0.00374603	+	0.999993i	$0.501192\pi$
$860$	16.9607	0.578353
$861$	33.7999	1.15190
$862$	6.98138	0.237787
$863$	−1.70299	−0.0579703	−0.0289851	−	0.999580i	$-0.509228\pi$
$863$	−1.70299	−0.0579703	−0.0289851	+	0.999580i	$0.509228\pi$
$864$	36.0927	1.22790
$865$	−57.1325	−1.94256
$866$	46.3000	1.57334
$867$	−1.50694	−0.0511784
$868$	11.9468	0.405500
$869$	77.5355	2.63021
$870$	−21.8049	−0.739256
$871$	1.28431	0.0435172
$872$	−3.28314	−0.111181
$873$	7.09558	0.240149
$874$	0.661421	0.0223729
$875$	−45.1915	−1.52775
$876$	−7.35275	−0.248427
$877$	34.6752	1.17090	0.585449	−	0.810709i	$-0.300918\pi$
$877$	34.6752	1.17090	0.585449	+	0.810709i	$0.300918\pi$
$878$	−17.0190	−0.574364
$879$	0.386381	0.0130323
$880$	69.0875	2.32894
$881$	2.02713	0.0682956	0.0341478	−	0.999417i	$-0.489128\pi$
$881$	2.02713	0.0682956	0.0341478	+	0.999417i	$0.489128\pi$
$882$	14.5251	0.489087
$883$	−51.9781	−1.74920	−0.874602	−	0.484842i	$-0.838877\pi$
$883$	−51.9781	−1.74920	−0.874602	+	0.484842i	$0.838877\pi$
$884$	1.54938	0.0521111
$885$	3.50694	0.117885
$886$	−6.80974	−0.228778
$887$	31.9302	1.07211	0.536056	−	0.844182i	$-0.319913\pi$
$887$	31.9302	1.07211	0.536056	+	0.844182i	$0.319913\pi$
$888$	4.92844	0.165387
$889$	65.5541	2.19861
$890$	37.4061	1.25385
$891$	38.0397	1.27438
$892$	23.6883	0.793142
$893$	23.5758	0.788934
$894$	−57.4447	−1.92124
$895$	−20.8415	−0.696656
$896$	−39.8623	−1.33170
$897$	0.344247	0.0114941
$898$	−65.2316	−2.17681
$899$	−7.33527	−0.244645
$900$	−0.398174	−0.0132725
$901$	4.66827	0.155522
$902$	−58.3708	−1.94354
$903$	−35.4245	−1.17885
$904$	8.36116	0.278088
$905$	26.0998	0.867586
$906$	−40.0704	−1.33125
$907$	11.6481	0.386770	0.193385	−	0.981123i	$-0.438053\pi$
$907$	11.6481	0.386770	0.193385	+	0.981123i	$0.438053\pi$
$908$	−8.85487	−0.293859
$909$	−8.17077	−0.271007
$910$	21.1698	0.701771
$911$	59.4007	1.96803	0.984017	−	0.178074i	$-0.0569867\pi$
$911$	59.4007	1.96803	0.984017	+	0.178074i	$0.0569867\pi$
$912$	−13.8618	−0.459010
$913$	−24.4547	−0.809334
$914$	65.6716	2.17223
$915$	26.3751	0.871934
$916$	−10.3642	−0.342444
$917$	−38.5429	−1.27280
$918$	10.2290	0.337608
$919$	−31.8535	−1.05075	−0.525375	−	0.850871i	$-0.676075\pi$
$919$	−31.8535	−1.05075	−0.525375	+	0.850871i	$0.676075\pi$
$920$	0.563259	0.0185701
$921$	−11.2784	−0.371636
$922$	69.7496	2.29708
$923$	−8.77714	−0.288903
$924$	50.7732	1.67032
$925$	−1.08799	−0.0357728
$926$	−39.9684	−1.31344
$927$	−5.88237	−0.193203
$928$	−21.9387	−0.720172
$929$	30.6175	1.00453	0.502264	−	0.864714i	$-0.332500\pi$
$929$	30.6175	1.00453	0.502264	+	0.864714i	$0.332500\pi$
$930$	13.7082	0.449510
$931$	−20.5377	−0.673094
$932$	−17.9871	−0.589187
$933$	−26.1301	−0.855461
$934$	−35.0306	−1.14624
$935$	14.0942	0.460930
$936$	1.07519	0.0351438
$937$	16.2592	0.531165	0.265583	−	0.964088i	$-0.414436\pi$
$937$	16.2592	0.531165	0.265583	+	0.964088i	$0.414436\pi$
$938$	8.39408	0.274076
$939$	31.9396	1.04231
$940$	−38.3973	−1.25238
$941$	30.6239	0.998311	0.499156	−	0.866512i	$-0.333644\pi$
$941$	30.6239	0.998311	0.499156	+	0.866512i	$0.333644\pi$
$942$	−40.7283	−1.32700
$943$	−1.02527	−0.0333875
$944$	4.90184	0.159541
$945$	55.3986	1.80212
$946$	61.1764	1.98901
$947$	−18.6434	−0.605830	−0.302915	−	0.953018i	$-0.597960\pi$
$947$	−18.6434	−0.605830	−0.302915	+	0.953018i	$0.597960\pi$
$948$	25.3370	0.822907
$949$	−4.38306	−0.142280
$950$	1.42036	0.0460826
$951$	−11.4708	−0.371965
$952$	−5.29488	−0.171608
$953$	19.0284	0.616389	0.308194	−	0.951323i	$-0.400275\pi$
$953$	19.0284	0.616389	0.308194	+	0.951323i	$0.400275\pi$
$954$	−6.19568	−0.200593
$955$	28.1956	0.912388
$956$	−28.0314	−0.906602
$957$	−31.1745	−1.00773
$958$	29.6701	0.958597
$959$	−41.3526	−1.33535
$960$	6.61821	0.213602
$961$	−26.3885	−0.851242
$962$	−5.61876	−0.181156
$963$	6.25382	0.201526
$964$	25.9909	0.837111
$965$	−62.0119	−1.99623
$966$	2.24995	0.0723910
$967$	21.7004	0.697837	0.348919	−	0.937153i	$-0.386549\pi$
$967$	21.7004	0.697837	0.348919	+	0.937153i	$0.386549\pi$
$968$	32.0975	1.03165
$969$	−2.82788	−0.0908445
$970$	41.2238	1.32362
$971$	12.2646	0.393590	0.196795	−	0.980445i	$-0.436947\pi$
$971$	12.2646	0.393590	0.196795	+	0.980445i	$0.436947\pi$
$972$	−9.71011	−0.311452
$973$	75.5201	2.42106
$974$	−62.2420	−1.99436
$975$	0.739249	0.0236749
$976$	36.8659	1.18005
$977$	27.5478	0.881333	0.440666	−	0.897671i	$-0.354742\pi$
$977$	27.5478	0.881333	0.440666	+	0.897671i	$0.354742\pi$
$978$	−27.6885	−0.885381
$979$	53.4795	1.70921
$980$	33.4492	1.06849
$981$	1.91513	0.0611455
$982$	28.0952	0.896553
$983$	25.8412	0.824205	0.412103	−	0.911137i	$-0.364794\pi$
$983$	25.8412	0.824205	0.412103	+	0.911137i	$0.364794\pi$
$984$	9.97346	0.317942
$985$	−26.8082	−0.854181
$986$	−6.21764	−0.198010
$987$	80.1977	2.55272
$988$	2.90751	0.0925001
$989$	1.07455	0.0341687
$990$	−18.7057	−0.594506
$991$	28.0960	0.892497	0.446249	−	0.894909i	$-0.352760\pi$
$991$	28.0960	0.892497	0.446249	+	0.894909i	$0.352760\pi$
$992$	13.7923	0.437906
$993$	−23.8265	−0.756111
$994$	−57.3663	−1.81955
$995$	50.3129	1.59503
$996$	−7.99130	−0.253214
$997$	23.8698	0.755964	0.377982	−	0.925813i	$-0.376618\pi$
$997$	23.8698	0.755964	0.377982	+	0.925813i	$0.376618\pi$
$998$	−49.6894	−1.57289
$999$	−14.7036	−0.465201

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1003.2.a.f.1.2	✓	4
3.2	odd	2		9027.2.a.i.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1003.2.a.f.1.2	✓	4	1.1	even	1	trivial
9027.2.a.i.1.3		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1003 = 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1003.a (trivial)

\(f(q)\)	\(=\)	\(q-1.82025 q^{2} -1.50694 q^{3} +1.31331 q^{4} +2.32719 q^{5} +2.74301 q^{6} +4.23607 q^{7} +1.24995 q^{8} -0.729126 q^{9} +O(q^{10})\)	\(q-1.82025 q^{2} -1.50694 q^{3} +1.31331 q^{4} +2.32719 q^{5} +2.74301 q^{6} +4.23607 q^{7} +1.24995 q^{8} -0.729126 q^{9} -4.23607 q^{10} -6.05632 q^{11} -1.97908 q^{12} -1.17975 q^{13} -7.71070 q^{14} -3.50694 q^{15} -4.90184 q^{16} -1.00000 q^{17} +1.32719 q^{18} -1.87657 q^{19} +3.05632 q^{20} -6.38351 q^{21} +11.0240 q^{22} +0.193635 q^{23} -1.88360 q^{24} +0.415819 q^{25} +2.14744 q^{26} +5.61958 q^{27} +5.56326 q^{28} -3.41582 q^{29} +6.38351 q^{30} +2.14744 q^{31} +6.42266 q^{32} +9.12652 q^{33} +1.82025 q^{34} +9.85814 q^{35} -0.957567 q^{36} -2.61649 q^{37} +3.41582 q^{38} +1.77782 q^{39} +2.90888 q^{40} -5.29488 q^{41} +11.6196 q^{42} +5.54938 q^{43} -7.95381 q^{44} -1.69682 q^{45} -0.352463 q^{46} -12.5633 q^{47} +7.38679 q^{48} +10.9443 q^{49} -0.756894 q^{50} +1.50694 q^{51} -1.54938 q^{52} -4.66827 q^{53} -10.2290 q^{54} -14.0942 q^{55} +5.29488 q^{56} +2.82788 q^{57} +6.21764 q^{58} -1.00000 q^{59} -4.60569 q^{60} -7.52083 q^{61} -3.90888 q^{62} -3.08863 q^{63} -1.88717 q^{64} -2.74551 q^{65} -16.6125 q^{66} -1.08863 q^{67} -1.31331 q^{68} -0.291796 q^{69} -17.9443 q^{70} +7.43983 q^{71} -0.911372 q^{72} +3.71524 q^{73} +4.76267 q^{74} -0.626615 q^{75} -2.46451 q^{76} -25.6550 q^{77} -3.23607 q^{78} -12.8024 q^{79} -11.4075 q^{80} -6.28100 q^{81} +9.63800 q^{82} +4.03789 q^{83} -8.38351 q^{84} -2.32719 q^{85} -10.1012 q^{86} +5.14744 q^{87} -7.57010 q^{88} -8.83037 q^{89} +3.08863 q^{90} -4.99750 q^{91} +0.254302 q^{92} -3.23607 q^{93} +22.8683 q^{94} -4.36713 q^{95} -9.67858 q^{96} -9.73162 q^{97} -19.9213 q^{98} +4.41582 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} - 2 q^{3} + 5 q^{4} + q^{5} - 2 q^{6} + 8 q^{7} - 12 q^{8} + 2 q^{9}+O(q^{10}) \) 4 * q - 3 * q^2 - 2 * q^3 + 5 * q^4 + q^5 - 2 * q^6 + 8 * q^7 - 12 * q^8 + 2 * q^9	\( 4 q - 3 q^{2} - 2 q^{3} + 5 q^{4} + q^{5} - 2 q^{6} + 8 q^{7} - 12 q^{8} + 2 q^{9} - 8 q^{10} - 11 q^{11} + 14 q^{12} - 9 q^{13} - q^{14} - 10 q^{15} + 11 q^{16} - 4 q^{17} - 3 q^{18} + 10 q^{19} - q^{20} - 4 q^{21} + 14 q^{22} - 3 q^{23} - 20 q^{24} - 3 q^{25} - 4 q^{26} - 8 q^{27} + 5 q^{28} - 9 q^{29} + 4 q^{30} - 4 q^{31} - 17 q^{32} + 2 q^{33} + 3 q^{34} - 3 q^{35} - 9 q^{36} - 32 q^{37} + 9 q^{38} + 8 q^{39} + 11 q^{40} + 4 q^{41} + 16 q^{42} + 13 q^{43} - 23 q^{44} + 15 q^{45} + 20 q^{46} - 33 q^{47} + 24 q^{48} + 8 q^{49} + 18 q^{50} + 2 q^{51} + 3 q^{52} + 6 q^{53} - 2 q^{54} - 5 q^{55} - 4 q^{56} - 12 q^{57} - 9 q^{58} - 4 q^{59} + 4 q^{60} - 18 q^{61} - 15 q^{62} - 16 q^{63} + 34 q^{64} + 5 q^{65} - 6 q^{66} - 8 q^{67} - 5 q^{68} - 28 q^{69} - 36 q^{70} - 5 q^{71} + 18 q^{73} + 11 q^{74} - 2 q^{75} - 11 q^{76} - 37 q^{77} - 4 q^{78} + 27 q^{79} + 6 q^{80} - 8 q^{81} + 33 q^{82} - 22 q^{83} - 12 q^{84} - q^{85} - 19 q^{86} + 8 q^{87} + 25 q^{88} - 9 q^{89} + 16 q^{90} - 23 q^{91} - 51 q^{92} - 4 q^{93} + 27 q^{94} - 7 q^{95} - 60 q^{96} - 31 q^{97} + 14 q^{98} + 13 q^{99}+O(q^{100}) \) 4 * q - 3 * q^2 - 2 * q^3 + 5 * q^4 + q^5 - 2 * q^6 + 8 * q^7 - 12 * q^8 + 2 * q^9 - 8 * q^10 - 11 * q^11 + 14 * q^12 - 9 * q^13 - q^14 - 10 * q^15 + 11 * q^16 - 4 * q^17 - 3 * q^18 + 10 * q^19 - q^20 - 4 * q^21 + 14 * q^22 - 3 * q^23 - 20 * q^24 - 3 * q^25 - 4 * q^26 - 8 * q^27 + 5 * q^28 - 9 * q^29 + 4 * q^30 - 4 * q^31 - 17 * q^32 + 2 * q^33 + 3 * q^34 - 3 * q^35 - 9 * q^36 - 32 * q^37 + 9 * q^38 + 8 * q^39 + 11 * q^40 + 4 * q^41 + 16 * q^42 + 13 * q^43 - 23 * q^44 + 15 * q^45 + 20 * q^46 - 33 * q^47 + 24 * q^48 + 8 * q^49 + 18 * q^50 + 2 * q^51 + 3 * q^52 + 6 * q^53 - 2 * q^54 - 5 * q^55 - 4 * q^56 - 12 * q^57 - 9 * q^58 - 4 * q^59 + 4 * q^60 - 18 * q^61 - 15 * q^62 - 16 * q^63 + 34 * q^64 + 5 * q^65 - 6 * q^66 - 8 * q^67 - 5 * q^68 - 28 * q^69 - 36 * q^70 - 5 * q^71 + 18 * q^73 + 11 * q^74 - 2 * q^75 - 11 * q^76 - 37 * q^77 - 4 * q^78 + 27 * q^79 + 6 * q^80 - 8 * q^81 + 33 * q^82 - 22 * q^83 - 12 * q^84 - q^85 - 19 * q^86 + 8 * q^87 + 25 * q^88 - 9 * q^89 + 16 * q^90 - 23 * q^91 - 51 * q^92 - 4 * q^93 + 27 * q^94 - 7 * q^95 - 60 * q^96 - 31 * q^97 + 14 * q^98 + 13 * q^99

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