LMFDB - Embedded newform 1849.2.a.o.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,2,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, 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("1849.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1849.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:	$14.7643393337$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 15 x^{16} + 106 x^{15} + 47 x^{14} - 897 x^{13} + 364 x^{12} + 3855 x^{11} + \cdots - 27 $ x^18 - 5x^17 - 15x^16 + 106x^15 + 47x^14 - 897x^13 + 364x^12 + 3855x^11 - 3223x^10 - 8851x^9 + 9909x^8 + 10400x^7 - 14483x^6 - 5118x^5 + 9945x^4 - 27x^3 - 2493x^2 + 513*x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.70036$ of defining polynomial
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70036 q^{2} -1.15587 q^{3} +0.891224 q^{4} +1.76341 q^{5} +1.96539 q^{6} -0.594565 q^{7} +1.88532 q^{8} -1.66397 q^{9} +O(q^{10})$	\(q-1.70036 q^{2} -1.15587 q^{3} +0.891224 q^{4} +1.76341 q^{5} +1.96539 q^{6} -0.594565 q^{7} +1.88532 q^{8} -1.66397 q^{9} -2.99843 q^{10} -4.34581 q^{11} -1.03014 q^{12} +3.24267 q^{13} +1.01098 q^{14} -2.03827 q^{15} -4.98817 q^{16} -5.54154 q^{17} +2.82935 q^{18} -2.71801 q^{19} +1.57159 q^{20} +0.687239 q^{21} +7.38944 q^{22} +5.29682 q^{23} -2.17918 q^{24} -1.89039 q^{25} -5.51371 q^{26} +5.39093 q^{27} -0.529891 q^{28} +9.86102 q^{29} +3.46578 q^{30} +3.73006 q^{31} +4.71104 q^{32} +5.02318 q^{33} +9.42261 q^{34} -1.04846 q^{35} -1.48297 q^{36} -1.34602 q^{37} +4.62159 q^{38} -3.74810 q^{39} +3.32458 q^{40} -11.2155 q^{41} -1.16855 q^{42} -3.87309 q^{44} -2.93426 q^{45} -9.00650 q^{46} +4.08771 q^{47} +5.76566 q^{48} -6.64649 q^{49} +3.21435 q^{50} +6.40528 q^{51} +2.88995 q^{52} -2.41097 q^{53} -9.16652 q^{54} -7.66343 q^{55} -1.12095 q^{56} +3.14166 q^{57} -16.7673 q^{58} -0.735135 q^{59} -1.81655 q^{60} -5.11518 q^{61} -6.34245 q^{62} +0.989339 q^{63} +1.96587 q^{64} +5.71815 q^{65} -8.54121 q^{66} -3.84908 q^{67} -4.93875 q^{68} -6.12242 q^{69} +1.78276 q^{70} +3.83751 q^{71} -3.13711 q^{72} +7.45576 q^{73} +2.28873 q^{74} +2.18505 q^{75} -2.42235 q^{76} +2.58387 q^{77} +6.37312 q^{78} +13.3934 q^{79} -8.79617 q^{80} -1.23929 q^{81} +19.0703 q^{82} +7.79301 q^{83} +0.612483 q^{84} -9.77199 q^{85} -11.3980 q^{87} -8.19324 q^{88} -4.50944 q^{89} +4.98929 q^{90} -1.92798 q^{91} +4.72065 q^{92} -4.31146 q^{93} -6.95057 q^{94} -4.79295 q^{95} -5.44534 q^{96} +0.184960 q^{97} +11.3014 q^{98} +7.23130 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 5 q^{2} + 5 q^{3} + 19 q^{4} + 11 q^{5} + 4 q^{6} + 6 q^{7} + 12 q^{8} + 15 q^{9}+O(q^{10}) $ 18 * q + 5 * q^2 + 5 * q^3 + 19 * q^4 + 11 * q^5 + 4 * q^6 + 6 * q^7 + 12 * q^8 + 15 * q^9	\( 18 q + 5 q^{2} + 5 q^{3} + 19 q^{4} + 11 q^{5} + 4 q^{6} + 6 q^{7} + 12 q^{8} + 15 q^{9} - 7 q^{10} - 2 q^{11} + 16 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 9 q^{16} - 11 q^{17} + 25 q^{18} + 31 q^{19} + 25 q^{20} + 19 q^{22} - 11 q^{23} + 18 q^{24} + 9 q^{25} + 27 q^{26} + 23 q^{27} + 20 q^{28} + 37 q^{29} - 17 q^{30} - 12 q^{31} + 39 q^{32} + 38 q^{33} + 14 q^{34} + 16 q^{35} + 47 q^{36} - 19 q^{37} + 56 q^{38} + 46 q^{39} + 6 q^{40} - 7 q^{41} - q^{42} + 7 q^{44} + 23 q^{45} - 47 q^{46} - q^{47} + 15 q^{48} - 6 q^{49} - 3 q^{50} + 38 q^{51} + 15 q^{52} + 3 q^{53} - 67 q^{54} + 28 q^{55} - 81 q^{56} + 46 q^{57} + 34 q^{58} + 17 q^{59} - 83 q^{60} + 28 q^{61} + 33 q^{62} + 26 q^{63} + 10 q^{64} + 16 q^{65} - 72 q^{66} + 18 q^{67} + 53 q^{68} + 7 q^{69} - 34 q^{70} + 86 q^{71} - 2 q^{72} - 27 q^{73} - 79 q^{74} + 31 q^{75} + 59 q^{76} + 43 q^{77} + 91 q^{78} + 17 q^{79} + 8 q^{80} - 10 q^{81} + 13 q^{82} - 12 q^{83} - 32 q^{84} + 28 q^{85} - 43 q^{87} - 23 q^{88} + 51 q^{89} + 10 q^{90} - 20 q^{91} + 18 q^{92} - 30 q^{93} - 15 q^{94} - q^{95} - 20 q^{96} - 19 q^{97} - 5 q^{98} + 38 q^{99}+O(q^{100}) \) 18 * q + 5 * q^2 + 5 * q^3 + 19 * q^4 + 11 * q^5 + 4 * q^6 + 6 * q^7 + 12 * q^8 + 15 * q^9 - 7 * q^10 - 2 * q^11 + 16 * q^12 - 7 * q^13 - 4 * q^14 + 3 * q^15 + 9 * q^16 - 11 * q^17 + 25 * q^18 + 31 * q^19 + 25 * q^20 + 19 * q^22 - 11 * q^23 + 18 * q^24 + 9 * q^25 + 27 * q^26 + 23 * q^27 + 20 * q^28 + 37 * q^29 - 17 * q^30 - 12 * q^31 + 39 * q^32 + 38 * q^33 + 14 * q^34 + 16 * q^35 + 47 * q^36 - 19 * q^37 + 56 * q^38 + 46 * q^39 + 6 * q^40 - 7 * q^41 - q^42 + 7 * q^44 + 23 * q^45 - 47 * q^46 - q^47 + 15 * q^48 - 6 * q^49 - 3 * q^50 + 38 * q^51 + 15 * q^52 + 3 * q^53 - 67 * q^54 + 28 * q^55 - 81 * q^56 + 46 * q^57 + 34 * q^58 + 17 * q^59 - 83 * q^60 + 28 * q^61 + 33 * q^62 + 26 * q^63 + 10 * q^64 + 16 * q^65 - 72 * q^66 + 18 * q^67 + 53 * q^68 + 7 * q^69 - 34 * q^70 + 86 * q^71 - 2 * q^72 - 27 * q^73 - 79 * q^74 + 31 * q^75 + 59 * q^76 + 43 * q^77 + 91 * q^78 + 17 * q^79 + 8 * q^80 - 10 * q^81 + 13 * q^82 - 12 * q^83 - 32 * q^84 + 28 * q^85 - 43 * q^87 - 23 * q^88 + 51 * q^89 + 10 * q^90 - 20 * q^91 + 18 * q^92 - 30 * q^93 - 15 * q^94 - q^95 - 20 * q^96 - 19 * q^97 - 5 * q^98 + 38 * 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 $

$2$	−1.70036	−1.20234	−0.601168	−	0.799123i	$-0.705298\pi$
$2$	−1.70036	−1.20234	−0.601168	+	0.799123i	$0.705298\pi$
$3$	−1.15587	−0.667340	−0.333670	−	0.942690i	$-0.608287\pi$
$3$	−1.15587	−0.667340	−0.333670	+	0.942690i	$0.608287\pi$
$4$	0.891224	0.445612
$5$	1.76341	0.788620	0.394310	−	0.918978i	$-0.370984\pi$
$5$	1.76341	0.788620	0.394310	+	0.918978i	$0.370984\pi$
$6$	1.96539	0.802367
$7$	−0.594565	−0.224725	−0.112362	−	0.993667i	$-0.535842\pi$
$7$	−0.594565	−0.224725	−0.112362	+	0.993667i	$0.535842\pi$
$8$	1.88532	0.666561
$9$	−1.66397	−0.554657
$10$	−2.99843	−0.948186
$11$	−4.34581	−1.31031	−0.655155	−	0.755494i	$-0.727397\pi$
$11$	−4.34581	−1.31031	−0.655155	+	0.755494i	$0.727397\pi$
$12$	−1.03014	−0.297375
$13$	3.24267	0.899355	0.449678	−	0.893191i	$-0.351539\pi$
$13$	3.24267	0.899355	0.449678	+	0.893191i	$0.351539\pi$
$14$	1.01098	0.270194
$15$	−2.03827	−0.526278
$16$	−4.98817	−1.24704
$17$	−5.54154	−1.34402	−0.672010	−	0.740542i	$-0.734569\pi$
$17$	−5.54154	−1.34402	−0.672010	+	0.740542i	$0.734569\pi$
$18$	2.82935	0.666884
$19$	−2.71801	−0.623554	−0.311777	−	0.950155i	$-0.600924\pi$
$19$	−2.71801	−0.623554	−0.311777	+	0.950155i	$0.600924\pi$
$20$	1.57159	0.351418
$21$	0.687239	0.149968
$22$	7.38944	1.57543
$23$	5.29682	1.10446	0.552232	−	0.833691i	$-0.313776\pi$
$23$	5.29682	1.10446	0.552232	+	0.833691i	$0.313776\pi$
$24$	−2.17918	−0.444823
$25$	−1.89039	−0.378079
$26$	−5.51371	−1.08133
$27$	5.39093	1.03749
$28$	−0.529891	−0.100140
$29$	9.86102	1.83115	0.915573	−	0.402152i	$-0.131738\pi$
$29$	9.86102	1.83115	0.915573	+	0.402152i	$0.131738\pi$
$30$	3.46578	0.632763
$31$	3.73006	0.669939	0.334970	−	0.942229i	$-0.391274\pi$
$31$	3.73006	0.669939	0.334970	+	0.942229i	$0.391274\pi$
$32$	4.71104	0.832803
$33$	5.02318	0.874423
$34$	9.42261	1.61596
$35$	−1.04846	−0.177222
$36$	−1.48297	−0.247162
$37$	−1.34602	−0.221285	−0.110643	−	0.993860i	$-0.535291\pi$
$37$	−1.34602	−0.221285	−0.110643	+	0.993860i	$0.535291\pi$
$38$	4.62159	0.749721
$39$	−3.74810	−0.600176
$40$	3.32458	0.525663
$41$	−11.2155	−1.75156	−0.875781	−	0.482710i	$-0.839653\pi$
$41$	−11.2155	−1.75156	−0.875781	+	0.482710i	$0.839653\pi$
$42$	−1.16855	−0.180312
$43$	0	0
$43$	0	0
$44$	−3.87309	−0.583890
$45$	−2.93426	−0.437413
$46$	−9.00650	−1.32794
$47$	4.08771	0.596253	0.298127	−	0.954526i	$-0.403638\pi$
$47$	4.08771	0.596253	0.298127	+	0.954526i	$0.403638\pi$
$48$	5.76566	0.832202
$49$	−6.64649	−0.949499
$50$	3.21435	0.454578
$51$	6.40528	0.896919
$52$	2.88995	0.400763
$53$	−2.41097	−0.331172	−0.165586	−	0.986195i	$-0.552952\pi$
$53$	−2.41097	−0.331172	−0.165586	+	0.986195i	$0.552952\pi$
$54$	−9.16652	−1.24741
$55$	−7.66343	−1.03334
$56$	−1.12095	−0.149793
$57$	3.14166	0.416123
$58$	−16.7673	−2.20165
$59$	−0.735135	−0.0957064	−0.0478532	−	0.998854i	$-0.515238\pi$
$59$	−0.735135	−0.0957064	−0.0478532	+	0.998854i	$0.515238\pi$
$60$	−1.81655	−0.234516
$61$	−5.11518	−0.654932	−0.327466	−	0.944863i	$-0.606195\pi$
$61$	−5.11518	−0.654932	−0.327466	+	0.944863i	$0.606195\pi$
$62$	−6.34245	−0.805492
$63$	0.989339	0.124645
$64$	1.96587	0.245733
$65$	5.71815	0.709249
$66$	−8.54121	−1.05135
$67$	−3.84908	−0.470240	−0.235120	−	0.971966i	$-0.575548\pi$
$67$	−3.84908	−0.470240	−0.235120	+	0.971966i	$0.575548\pi$
$68$	−4.93875	−0.598911
$69$	−6.12242	−0.737053
$70$	1.78276	0.213081
$71$	3.83751	0.455428	0.227714	−	0.973728i	$-0.426875\pi$
$71$	3.83751	0.455428	0.227714	+	0.973728i	$0.426875\pi$
$72$	−3.13711	−0.369712
$73$	7.45576	0.872631	0.436316	−	0.899794i	$-0.356283\pi$
$73$	7.45576	0.872631	0.436316	+	0.899794i	$0.356283\pi$
$74$	2.28873	0.266059
$75$	2.18505	0.252307
$76$	−2.42235	−0.277863
$77$	2.58387	0.294459
$78$	6.37312	0.721613
$79$	13.3934	1.50688	0.753440	−	0.657517i	$-0.228393\pi$
$79$	13.3934	1.50688	0.753440	+	0.657517i	$0.228393\pi$
$80$	−8.79617	−0.983442
$81$	−1.23929	−0.137699
$82$	19.0703	2.10596
$83$	7.79301	0.855394	0.427697	−	0.903922i	$-0.359325\pi$
$83$	7.79301	0.855394	0.427697	+	0.903922i	$0.359325\pi$
$84$	0.612483	0.0668274
$85$	−9.77199	−1.05992
$86$	0	0
$87$	−11.3980	−1.22200
$88$	−8.19324	−0.873402
$89$	−4.50944	−0.477999	−0.239000	−	0.971020i	$-0.576820\pi$
$89$	−4.50944	−0.477999	−0.239000	+	0.971020i	$0.576820\pi$
$90$	4.98929	0.525918
$91$	−1.92798	−0.202107
$92$	4.72065	0.492162
$93$	−4.31146	−0.447078
$94$	−6.95057	−0.716897
$95$	−4.79295	−0.491747
$96$	−5.44534	−0.555763
$97$	0.184960	0.0187798	0.00938992	−	0.999956i	$-0.497011\pi$
$97$	0.184960	0.0187798	0.00938992	+	0.999956i	$0.497011\pi$
$98$	11.3014	1.14162
$99$	7.23130	0.726773
$100$	−1.68476	−0.168476
$101$	4.12809	0.410760	0.205380	−	0.978682i	$-0.434157\pi$
$101$	4.12809	0.410760	0.205380	+	0.978682i	$0.434157\pi$
$102$	−10.8913	−1.07840
$103$	15.7336	1.55028	0.775141	−	0.631788i	$-0.217679\pi$
$103$	15.7336	1.55028	0.775141	+	0.631788i	$0.217679\pi$
$104$	6.11347	0.599475
$105$	1.21188	0.118268
$106$	4.09952	0.398180
$107$	0.0534833	0.00517043	0.00258521	−	0.999997i	$-0.499177\pi$
$107$	0.0534833	0.00517043	0.00258521	+	0.999997i	$0.499177\pi$
$108$	4.80453	0.462316
$109$	−5.25004	−0.502863	−0.251431	−	0.967875i	$-0.580901\pi$
$109$	−5.25004	−0.502863	−0.251431	+	0.967875i	$0.580901\pi$
$110$	13.0306	1.24242
$111$	1.55583	0.147673
$112$	2.96579	0.280241
$113$	−2.58205	−0.242899	−0.121450	−	0.992598i	$-0.538754\pi$
$113$	−2.58205	−0.242899	−0.121450	+	0.992598i	$0.538754\pi$
$114$	−5.34195	−0.500319
$115$	9.34045	0.871001
$116$	8.78838	0.815980
$117$	−5.39571	−0.498833
$118$	1.24999	0.115071
$119$	3.29481	0.302034
$120$	−3.84278	−0.350796
$121$	7.88606	0.716914
$122$	8.69765	0.787448
$123$	12.9636	1.16889
$124$	3.32432	0.298533
$125$	−12.1506	−1.08678
$126$	−1.68223	−0.149865
$127$	−13.0649	−1.15933	−0.579663	−	0.814856i	$-0.696816\pi$
$127$	−13.0649	−1.15933	−0.579663	+	0.814856i	$0.696816\pi$
$128$	−12.7648	−1.12826
$129$	0	0
$130$	−9.72291	−0.852756
$131$	10.7746	0.941385	0.470693	−	0.882297i	$-0.344004\pi$
$131$	10.7746	0.941385	0.470693	+	0.882297i	$0.344004\pi$
$132$	4.47678	0.389653
$133$	1.61603	0.140128
$134$	6.54482	0.565386
$135$	9.50641	0.818181
$136$	−10.4476	−0.895871
$137$	8.03721	0.686665	0.343333	−	0.939214i	$-0.388444\pi$
$137$	8.03721	0.686665	0.343333	+	0.939214i	$0.388444\pi$
$138$	10.4103	0.886185
$139$	21.8884	1.85655	0.928274	−	0.371896i	$-0.121292\pi$
$139$	21.8884	1.85655	0.928274	+	0.371896i	$0.121292\pi$
$140$	−0.934413	−0.0789723
$141$	−4.72485	−0.397904
$142$	−6.52514	−0.547578
$143$	−14.0920	−1.17843
$144$	8.30016	0.691680
$145$	17.3890	1.44408
$146$	−12.6775	−1.04920
$147$	7.68246	0.633639
$148$	−1.19961	−0.0986073
$149$	16.5094	1.35250	0.676250	−	0.736672i	$-0.263604\pi$
$149$	16.5094	1.35250	0.676250	+	0.736672i	$0.263604\pi$
$150$	−3.71536	−0.303358
$151$	−15.8693	−1.29142	−0.645711	−	0.763582i	$-0.723439\pi$
$151$	−15.8693	−1.29142	−0.645711	+	0.763582i	$0.723439\pi$
$152$	−5.12431	−0.415636
$153$	9.22095	0.745470
$154$	−4.39350	−0.354039
$155$	6.57762	0.528327
$156$	−3.34039	−0.267446
$157$	6.56626	0.524045	0.262022	−	0.965062i	$-0.415611\pi$
$157$	6.56626	0.524045	0.262022	+	0.965062i	$0.415611\pi$
$158$	−22.7737	−1.81178
$159$	2.78676	0.221005
$160$	8.30749	0.656764
$161$	−3.14931	−0.248200
$162$	2.10724	0.165561
$163$	12.7026	0.994944	0.497472	−	0.867480i	$-0.334262\pi$
$163$	12.7026	0.994944	0.497472	+	0.867480i	$0.334262\pi$
$164$	−9.99549	−0.780516
$165$	8.85791	0.689588
$166$	−13.2509	−1.02847
$167$	16.6713	1.29006	0.645032	−	0.764155i	$-0.276844\pi$
$167$	16.6713	1.29006	0.645032	+	0.764155i	$0.276844\pi$
$168$	1.29566	0.0999627
$169$	−2.48508	−0.191160
$170$	16.6159	1.27438
$171$	4.52268	0.345858
$172$	0	0
$173$	8.10129	0.615930	0.307965	−	0.951398i	$-0.400352\pi$
$173$	8.10129	0.615930	0.307965	+	0.951398i	$0.400352\pi$
$174$	19.3808	1.46925
$175$	1.12396	0.0849636
$176$	21.6776	1.63401
$177$	0.849718	0.0638687
$178$	7.66767	0.574716
$179$	17.4497	1.30425	0.652125	−	0.758111i	$-0.273878\pi$
$179$	17.4497	1.30425	0.652125	+	0.758111i	$0.273878\pi$
$180$	−2.61508	−0.194916
$181$	14.1146	1.04913	0.524567	−	0.851369i	$-0.324227\pi$
$181$	14.1146	1.04913	0.524567	+	0.851369i	$0.324227\pi$
$182$	3.27826	0.243001
$183$	5.91247	0.437063
$184$	9.98619	0.736192
$185$	−2.37359	−0.174510
$186$	7.33104	0.537538
$187$	24.0825	1.76108
$188$	3.64306	0.265698
$189$	−3.20526	−0.233148
$190$	8.14975	0.591245
$191$	18.9502	1.37119	0.685595	−	0.727983i	$-0.259542\pi$
$191$	18.9502	1.37119	0.685595	+	0.727983i	$0.259542\pi$
$192$	−2.27228	−0.163988
$193$	−21.1880	−1.52514	−0.762572	−	0.646903i	$-0.776064\pi$
$193$	−21.1880	−1.52514	−0.762572	+	0.646903i	$0.776064\pi$
$194$	−0.314499	−0.0225797
$195$	−6.60942	−0.473311
$196$	−5.92351	−0.423108
$197$	2.82273	0.201111	0.100556	−	0.994931i	$-0.467938\pi$
$197$	2.82273	0.201111	0.100556	+	0.994931i	$0.467938\pi$
$198$	−12.2958	−0.873825
$199$	0.0129904	0.000920863 0	0.000460431	−	1.00000i	$-0.499853\pi$
$199$	0.0129904	0.000920863 0	0.000460431		1.00000i	$0.499853\pi$
$200$	−3.56400	−0.252013
$201$	4.44902	0.313810
$202$	−7.01924	−0.493872
$203$	−5.86302	−0.411503
$204$	5.70854	0.399678
$205$	−19.7774	−1.38132
$206$	−26.7529	−1.86396
$207$	−8.81375	−0.612598
$208$	−16.1750	−1.12153
$209$	11.8119	0.817049
$210$	−2.06064	−0.142197
$211$	2.16836	0.149276	0.0746380	−	0.997211i	$-0.476220\pi$
$211$	2.16836	0.149276	0.0746380	+	0.997211i	$0.476220\pi$
$212$	−2.14871	−0.147574
$213$	−4.43565	−0.303926
$214$	−0.0909409	−0.00621659
$215$	0	0
$216$	10.1636	0.691547
$217$	−2.21777	−0.150552
$218$	8.92696	0.604610
$219$	−8.61787	−0.582342
$220$	−6.82983	−0.460467
$221$	−17.9694	−1.20875
$222$	−2.64546	−0.177552
$223$	−3.77604	−0.252862	−0.126431	−	0.991975i	$-0.540352\pi$
$223$	−3.77604	−0.252862	−0.126431	+	0.991975i	$0.540352\pi$
$224$	−2.80102	−0.187151
$225$	3.14556	0.209704
$226$	4.39042	0.292046
$227$	16.5799	1.10045	0.550224	−	0.835017i	$-0.314542\pi$
$227$	16.5799	1.10045	0.550224	+	0.835017i	$0.314542\pi$
$228$	2.79992	0.185429
$229$	−0.773978	−0.0511459	−0.0255730	−	0.999673i	$-0.508141\pi$
$229$	−0.773978	−0.0511459	−0.0255730	+	0.999673i	$0.508141\pi$
$230$	−15.8821	−1.04724
$231$	−2.98661	−0.196504
$232$	18.5912	1.22057
$233$	25.9761	1.70175	0.850876	−	0.525366i	$-0.176072\pi$
$233$	25.9761	1.70175	0.850876	+	0.525366i	$0.176072\pi$
$234$	9.17465	0.599765
$235$	7.20829	0.470217
$236$	−0.655169	−0.0426479
$237$	−15.4810	−1.00560
$238$	−5.60236	−0.363147
$239$	3.52561	0.228053	0.114026	−	0.993478i	$-0.463625\pi$
$239$	3.52561	0.228053	0.114026	+	0.993478i	$0.463625\pi$
$240$	10.1672	0.656291
$241$	19.5154	1.25710	0.628549	−	0.777770i	$-0.283649\pi$
$241$	19.5154	1.25710	0.628549	+	0.777770i	$0.283649\pi$
$242$	−13.4091	−0.861972
$243$	−14.7403	−0.945593
$244$	−4.55877	−0.291845
$245$	−11.7205	−0.748793
$246$	−22.0428	−1.40540
$247$	−8.81360	−0.560796
$248$	7.03236	0.446555
$249$	−9.00769	−0.570839
$250$	20.6603	1.30667
$251$	8.90689	0.562198	0.281099	−	0.959679i	$-0.409301\pi$
$251$	8.90689	0.562198	0.281099	+	0.959679i	$0.409301\pi$
$252$	0.881722	0.0555433
$253$	−23.0190	−1.44719
$254$	22.2151	1.39390
$255$	11.2951	0.707328
$256$	17.7730	1.11081
$257$	−0.0980769	−0.00611787	−0.00305893	−	0.999995i	$-0.500974\pi$
$257$	−0.0980769	−0.00611787	−0.00305893	+	0.999995i	$0.500974\pi$
$258$	0	0
$259$	0.800300	0.0497282
$260$	5.09615	0.316050
$261$	−16.4084	−1.01566
$262$	−18.3208	−1.13186
$263$	0.463866	0.0286032	0.0143016	−	0.999898i	$-0.495448\pi$
$263$	0.463866	0.0286032	0.0143016	+	0.999898i	$0.495448\pi$
$264$	9.47030	0.582856
$265$	−4.25152	−0.261169
$266$	−2.74784	−0.168481
$267$	5.21231	0.318988
$268$	−3.43039	−0.209544
$269$	21.4271	1.30643	0.653216	−	0.757171i	$-0.273419\pi$
$269$	21.4271	1.30643	0.653216	+	0.757171i	$0.273419\pi$
$270$	−16.1643	−0.983729
$271$	−9.45879	−0.574581	−0.287290	−	0.957844i	$-0.592754\pi$
$271$	−9.45879	−0.574581	−0.287290	+	0.957844i	$0.592754\pi$
$272$	27.6421	1.67605
$273$	2.22849	0.134874
$274$	−13.6662	−0.825603
$275$	8.21530	0.495401
$276$	−5.45645	−0.328440
$277$	−12.7119	−0.763785	−0.381893	−	0.924207i	$-0.624728\pi$
$277$	−12.7119	−0.763785	−0.381893	+	0.924207i	$0.624728\pi$
$278$	−37.2181	−2.23219
$279$	−6.20672	−0.371586
$280$	−1.97668	−0.118129
$281$	19.4455	1.16002	0.580011	−	0.814609i	$-0.303048\pi$
$281$	19.4455	1.16002	0.580011	+	0.814609i	$0.303048\pi$
$282$	8.03394	0.478414
$283$	2.46628	0.146605	0.0733026	−	0.997310i	$-0.476646\pi$
$283$	2.46628	0.146605	0.0733026	+	0.997310i	$0.476646\pi$
$284$	3.42008	0.202944
$285$	5.54002	0.328162
$286$	23.9615	1.41687
$287$	6.66833	0.393619
$288$	−7.83903	−0.461919
$289$	13.7086	0.806390
$290$	−29.5676	−1.73627
$291$	−0.213789	−0.0125325
$292$	6.64475	0.388855
$293$	−21.7907	−1.27303	−0.636515	−	0.771265i	$-0.719624\pi$
$293$	−21.7907	−1.27303	−0.636515	+	0.771265i	$0.719624\pi$
$294$	−13.0630	−0.761847
$295$	−1.29634	−0.0754759
$296$	−2.53769	−0.147500
$297$	−23.4280	−1.35943
$298$	−28.0719	−1.62616
$299$	17.1758	0.993305
$300$	1.94736	0.112431
$301$	0	0
$302$	26.9834	1.55272
$303$	−4.77153	−0.274117
$304$	13.5579	0.777598
$305$	−9.02015	−0.516492
$306$	−15.6789	−0.896305
$307$	−19.9355	−1.13778	−0.568890	−	0.822414i	$-0.692627\pi$
$307$	−19.9355	−1.13778	−0.568890	+	0.822414i	$0.692627\pi$
$308$	2.30280	0.131214
$309$	−18.1860	−1.03457
$310$	−11.1843	−0.635227
$311$	−5.89502	−0.334276	−0.167138	−	0.985934i	$-0.553453\pi$
$311$	−5.89502	−0.334276	−0.167138	+	0.985934i	$0.553453\pi$
$312$	−7.06636	−0.400054
$313$	−20.9619	−1.18484	−0.592418	−	0.805631i	$-0.701826\pi$
$313$	−20.9619	−1.18484	−0.592418	+	0.805631i	$0.701826\pi$
$314$	−11.1650	−0.630078
$315$	1.74461	0.0982975
$316$	11.9366	0.671484
$317$	21.7222	1.22004	0.610019	−	0.792387i	$-0.291162\pi$
$317$	21.7222	1.22004	0.610019	+	0.792387i	$0.291162\pi$
$318$	−4.73850	−0.265722
$319$	−42.8541	−2.39937
$320$	3.46663	0.193790
$321$	−0.0618196	−0.00345043
$322$	5.35495	0.298420
$323$	15.0619	0.838069
$324$	−1.10449	−0.0613604
$325$	−6.12993	−0.340027
$326$	−21.5990	−1.19626
$327$	6.06835	0.335581
$328$	−21.1447	−1.16752
$329$	−2.43041	−0.133993
$330$	−15.0616	−0.829116
$331$	9.48307	0.521237	0.260618	−	0.965442i	$-0.416074\pi$
$331$	9.48307	0.521237	0.260618	+	0.965442i	$0.416074\pi$
$332$	6.94532	0.381174
$333$	2.23975	0.122737
$334$	−28.3472	−1.55109
$335$	−6.78749	−0.370840
$336$	−3.42806	−0.187016
$337$	−24.9983	−1.36174	−0.680872	−	0.732402i	$-0.738399\pi$
$337$	−24.9983	−1.36174	−0.680872	+	0.732402i	$0.738399\pi$
$338$	4.22554	0.229839
$339$	2.98451	0.162096
$340$	−8.70902	−0.472313
$341$	−16.2102	−0.877829
$342$	−7.69019	−0.415838
$343$	8.11373	0.438100
$344$	0	0
$345$	−10.7963	−0.581255
$346$	−13.7751	−0.740555
$347$	10.2429	0.549867	0.274934	−	0.961463i	$-0.411344\pi$
$347$	10.2429	0.549867	0.274934	+	0.961463i	$0.411344\pi$
$348$	−10.1582	−0.544537
$349$	14.7987	0.792158	0.396079	−	0.918216i	$-0.370371\pi$
$349$	14.7987	0.792158	0.396079	+	0.918216i	$0.370371\pi$
$350$	−1.91114	−0.102155
$351$	17.4810	0.933068
$352$	−20.4733	−1.09123
$353$	−12.0266	−0.640114	−0.320057	−	0.947398i	$-0.603702\pi$
$353$	−12.0266	−0.640114	−0.320057	+	0.947398i	$0.603702\pi$
$354$	−1.44483	−0.0767917
$355$	6.76709	0.359160
$356$	−4.01892	−0.213002
$357$	−3.80836	−0.201560
$358$	−29.6707	−1.56815
$359$	7.70345	0.406573	0.203286	−	0.979119i	$-0.434838\pi$
$359$	7.70345	0.406573	0.203286	+	0.979119i	$0.434838\pi$
$360$	−5.53201	−0.291563
$361$	−11.6124	−0.611181
$362$	−24.0000	−1.26141
$363$	−9.11524	−0.478426
$364$	−1.71826	−0.0900614
$365$	13.1475	0.688174
$366$	−10.0533	−0.525496
$367$	3.21531	0.167838	0.0839190	−	0.996473i	$-0.473256\pi$
$367$	3.21531	0.167838	0.0839190	+	0.996473i	$0.473256\pi$
$368$	−26.4214	−1.37731
$369$	18.6622	0.971515
$370$	4.03596	0.209819
$371$	1.43348	0.0744226
$372$	−3.84248	−0.199223
$373$	17.6170	0.912173	0.456086	−	0.889936i	$-0.349251\pi$
$373$	17.6170	0.912173	0.456086	+	0.889936i	$0.349251\pi$
$374$	−40.9488	−2.11741
$375$	14.0445	0.725252
$376$	7.70663	0.397439
$377$	31.9761	1.64685
$378$	5.45010	0.280323
$379$	17.6797	0.908145	0.454072	−	0.890965i	$-0.349971\pi$
$379$	17.6797	0.908145	0.454072	+	0.890965i	$0.349971\pi$
$380$	−4.27159	−0.219128
$381$	15.1013	0.773665
$382$	−32.2222	−1.64863
$383$	−24.8944	−1.27205	−0.636023	−	0.771670i	$-0.719422\pi$
$383$	−24.8944	−1.27205	−0.636023	+	0.771670i	$0.719422\pi$
$384$	14.7544	0.752931
$385$	4.55641	0.232216
$386$	36.0272	1.83374
$387$	0	0
$388$	0.164841	0.00836852
$389$	6.72429	0.340935	0.170467	−	0.985363i	$-0.445472\pi$
$389$	6.72429	0.340935	0.170467	+	0.985363i	$0.445472\pi$
$390$	11.2384	0.569078
$391$	−29.3525	−1.48442
$392$	−12.5308	−0.632899
$393$	−12.4541	−0.628224
$394$	−4.79966	−0.241803
$395$	23.6181	1.18836
$396$	6.44470	0.323858
$397$	−20.3490	−1.02129	−0.510644	−	0.859792i	$-0.670593\pi$
$397$	−20.3490	−1.02129	−0.510644	+	0.859792i	$0.670593\pi$
$398$	−0.0220883	−0.00110719
$399$	−1.86792	−0.0935130
$400$	9.42961	0.471480
$401$	0.193055	0.00964071	0.00482036	−	0.999988i	$-0.498466\pi$
$401$	0.193055	0.00964071	0.00482036	+	0.999988i	$0.498466\pi$
$402$	−7.56494	−0.377305
$403$	12.0954	0.602513
$404$	3.67905	0.183040
$405$	−2.18538	−0.108592
$406$	9.96925	0.494765
$407$	5.84957	0.289952
$408$	12.0760	0.597851
$409$	0.492305	0.0243429	0.0121715	−	0.999926i	$-0.496126\pi$
$409$	0.492305	0.0243429	0.0121715	+	0.999926i	$0.496126\pi$
$410$	33.6287	1.66081
$411$	−9.28995	−0.458240
$412$	14.0222	0.690824
$413$	0.437086	0.0215076
$414$	14.9865	0.736549
$415$	13.7423	0.674581
$416$	15.2764	0.748985
$417$	−25.3001	−1.23895
$418$	−20.0846	−0.982367
$419$	16.0780	0.785463	0.392732	−	0.919653i	$-0.371530\pi$
$419$	16.0780	0.785463	0.392732	+	0.919653i	$0.371530\pi$
$420$	1.08006	0.0527014
$421$	36.4314	1.77556	0.887779	−	0.460271i	$-0.152248\pi$
$421$	36.4314	1.77556	0.887779	+	0.460271i	$0.152248\pi$
$422$	−3.68699	−0.179480
$423$	−6.80182	−0.330716
$424$	−4.54545	−0.220746
$425$	10.4757	0.508146
$426$	7.54220	0.365421
$427$	3.04131	0.147179
$428$	0.0476656	0.00230400
$429$	16.2885	0.786417
$430$	0	0
$431$	8.13149	0.391680	0.195840	−	0.980636i	$-0.437257\pi$
$431$	8.13149	0.391680	0.195840	+	0.980636i	$0.437257\pi$
$432$	−26.8909	−1.29379
$433$	27.5411	1.32354	0.661770	−	0.749707i	$-0.269806\pi$
$433$	27.5411	1.32354	0.661770	+	0.749707i	$0.269806\pi$
$434$	3.77100	0.181014
$435$	−20.0994	−0.963691
$436$	−4.67896	−0.224082
$437$	−14.3968	−0.688692
$438$	14.6535	0.700171
$439$	−37.2707	−1.77883	−0.889416	−	0.457099i	$-0.848889\pi$
$439$	−37.2707	−1.77883	−0.889416	+	0.457099i	$0.848889\pi$
$440$	−14.4480	−0.688782
$441$	11.0596	0.526646
$442$	30.5544	1.45333
$443$	−5.47448	−0.260101	−0.130050	−	0.991507i	$-0.541514\pi$
$443$	−5.47448	−0.260101	−0.130050	+	0.991507i	$0.541514\pi$
$444$	1.38659	0.0658046
$445$	−7.95198	−0.376960
$446$	6.42063	0.304026
$447$	−19.0826	−0.902578
$448$	−1.16884	−0.0552224
$449$	−22.7304	−1.07271	−0.536356	−	0.843992i	$-0.680200\pi$
$449$	−22.7304	−1.07271	−0.536356	+	0.843992i	$0.680200\pi$
$450$	−5.34859	−0.252135
$451$	48.7403	2.29509
$452$	−2.30119	−0.108239
$453$	18.3428	0.861818
$454$	−28.1918	−1.32311
$455$	−3.39981	−0.159386
$456$	5.92302	0.277371
$457$	33.1117	1.54890	0.774449	−	0.632636i	$-0.218027\pi$
$457$	33.1117	1.54890	0.774449	+	0.632636i	$0.218027\pi$
$458$	1.31604	0.0614946
$459$	−29.8740	−1.39440
$460$	8.32443	0.388129
$461$	−14.9822	−0.697790	−0.348895	−	0.937162i	$-0.613443\pi$
$461$	−14.9822	−0.697790	−0.348895	+	0.937162i	$0.613443\pi$
$462$	5.07831	0.236264
$463$	−18.2384	−0.847609	−0.423805	−	0.905754i	$-0.639306\pi$
$463$	−18.2384	−0.847609	−0.423805	+	0.905754i	$0.639306\pi$
$464$	−49.1884	−2.28352
$465$	−7.60286	−0.352574
$466$	−44.1688	−2.04608
$467$	−3.47982	−0.161027	−0.0805135	−	0.996754i	$-0.525656\pi$
$467$	−3.47982	−0.161027	−0.0805135	+	0.996754i	$0.525656\pi$
$468$	−4.80878	−0.222286
$469$	2.28853	0.105674
$470$	−12.2567	−0.565359
$471$	−7.58973	−0.349716
$472$	−1.38596	−0.0637941
$473$	0	0
$474$	26.3233	1.20907
$475$	5.13811	0.235753
$476$	2.93641	0.134590
$477$	4.01178	0.183687
$478$	−5.99481	−0.274196
$479$	23.9869	1.09599	0.547994	−	0.836482i	$-0.315391\pi$
$479$	23.9869	1.09599	0.547994	+	0.836482i	$0.315391\pi$
$480$	−9.60235	−0.438285
$481$	−4.36472	−0.199014
$482$	−33.1832	−1.51145
$483$	3.64018	0.165634
$484$	7.02824	0.319465
$485$	0.326160	0.0148102
$486$	25.0639	1.13692
$487$	−4.42005	−0.200291	−0.100146	−	0.994973i	$-0.531931\pi$
$487$	−4.42005	−0.200291	−0.100146	+	0.994973i	$0.531931\pi$
$488$	−9.64375	−0.436552
$489$	−14.6825	−0.663967
$490$	19.9290	0.900301
$491$	−15.6570	−0.706590	−0.353295	−	0.935512i	$-0.614939\pi$
$491$	−15.6570	−0.706590	−0.353295	+	0.935512i	$0.614939\pi$
$492$	11.5535	0.520870
$493$	−54.6452	−2.46110
$494$	14.9863	0.674265
$495$	12.7517	0.573147
$496$	−18.6062	−0.835443
$497$	−2.28165	−0.102346
$498$	15.3163	0.686341
$499$	−37.6600	−1.68589	−0.842947	−	0.537996i	$-0.819181\pi$
$499$	−37.6600	−1.68589	−0.842947	+	0.537996i	$0.819181\pi$
$500$	−10.8289	−0.484282
$501$	−19.2698	−0.860912
$502$	−15.1449	−0.675951
$503$	31.0430	1.38414	0.692070	−	0.721830i	$-0.256699\pi$
$503$	31.0430	1.38414	0.692070	+	0.721830i	$0.256699\pi$
$504$	1.86522	0.0830835
$505$	7.27951	0.323934
$506$	39.1405	1.74001
$507$	2.87243	0.127569
$508$	−11.6438	−0.516609
$509$	4.65215	0.206203	0.103101	−	0.994671i	$-0.467123\pi$
$509$	4.65215	0.206203	0.103101	+	0.994671i	$0.467123\pi$
$510$	−19.2058	−0.850446
$511$	−4.43294	−0.196102
$512$	−4.69090	−0.207310
$513$	−14.6526	−0.646928
$514$	0.166766	0.00735573
$515$	27.7448	1.22258
$516$	0	0
$517$	−17.7644	−0.781277
$518$	−1.36080	−0.0597900
$519$	−9.36402	−0.411035
$520$	10.7805	0.472758
$521$	7.07583	0.309998	0.154999	−	0.987915i	$-0.450463\pi$
$521$	7.07583	0.309998	0.154999	+	0.987915i	$0.450463\pi$
$522$	27.9003	1.22116
$523$	−4.41270	−0.192954	−0.0964770	−	0.995335i	$-0.530757\pi$
$523$	−4.41270	−0.192954	−0.0964770	+	0.995335i	$0.530757\pi$
$524$	9.60262	0.419492
$525$	−1.29915	−0.0566997
$526$	−0.788739	−0.0343906
$527$	−20.6703	−0.900412
$528$	−25.0565	−1.09044
$529$	5.05630	0.219839
$530$	7.22912	0.314013
$531$	1.22324	0.0530842
$532$	1.44025	0.0624426
$533$	−36.3681	−1.57528
$534$	−8.86281	−0.383531
$535$	0.0943128	0.00407750
$536$	−7.25674	−0.313443
$537$	−20.1695	−0.870379
$538$	−36.4338	−1.57077
$539$	28.8844	1.24414
$540$	8.47234	0.364591
$541$	−17.0661	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$541$	−17.0661	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$542$	16.0834	0.690839
$543$	−16.3147	−0.700129
$544$	−26.1064	−1.11930
$545$	−9.25796	−0.396567
$546$	−3.78923	−0.162164
$547$	10.2005	0.436140	0.218070	−	0.975933i	$-0.430024\pi$
$547$	10.2005	0.436140	0.218070	+	0.975933i	$0.430024\pi$
$548$	7.16295	0.305986
$549$	8.51151	0.363262
$550$	−13.9690	−0.595638
$551$	−26.8023	−1.14182
$552$	−11.5427	−0.491291
$553$	−7.96328	−0.338633
$554$	21.6148	0.918326
$555$	2.74356	0.116457
$556$	19.5074	0.827300
$557$	31.3958	1.33028	0.665142	−	0.746717i	$-0.268371\pi$
$557$	31.3958	1.33028	0.665142	+	0.746717i	$0.268371\pi$
$558$	10.5537	0.446772
$559$	0	0
$560$	5.22990	0.221004
$561$	−27.8361	−1.17524
$562$	−33.0644	−1.39474
$563$	−37.7263	−1.58998	−0.794988	−	0.606625i	$-0.792523\pi$
$563$	−37.7263	−1.58998	−0.794988	+	0.606625i	$0.792523\pi$
$564$	−4.21090	−0.177311
$565$	−4.55321	−0.191555
$566$	−4.19357	−0.176269
$567$	0.736841	0.0309444
$568$	7.23492	0.303571
$569$	3.61706	0.151635	0.0758175	−	0.997122i	$-0.475843\pi$
$569$	3.61706	0.151635	0.0758175	+	0.997122i	$0.475843\pi$
$570$	−9.42003	−0.394562
$571$	−15.3842	−0.643808	−0.321904	−	0.946772i	$-0.604323\pi$
$571$	−15.3842	−0.643808	−0.321904	+	0.946772i	$0.604323\pi$
$572$	−12.5592	−0.525124
$573$	−21.9039	−0.915050
$574$	−11.3386	−0.473262
$575$	−10.0131	−0.417574
$576$	−3.27115	−0.136298
$577$	−31.8469	−1.32580	−0.662902	−	0.748707i	$-0.730675\pi$
$577$	−31.8469	−1.32580	−0.662902	+	0.748707i	$0.730675\pi$
$578$	−23.3096	−0.969551
$579$	24.4905	1.01779
$580$	15.4975	0.643498
$581$	−4.63346	−0.192228
$582$	0.363519	0.0150683
$583$	10.4776	0.433939
$584$	14.0565	0.581662
$585$	−9.51483	−0.393390
$586$	37.0521	1.53061
$587$	−18.5935	−0.767436	−0.383718	−	0.923450i	$-0.625356\pi$
$587$	−18.5935	−0.767436	−0.383718	+	0.923450i	$0.625356\pi$
$588$	6.84679	0.282357
$589$	−10.1383	−0.417743
$590$	2.20425	0.0907474
$591$	−3.26271	−0.134210
$592$	6.71420	0.275952
$593$	−3.43324	−0.140986	−0.0704932	−	0.997512i	$-0.522457\pi$
$593$	−3.43324	−0.140986	−0.0704932	+	0.997512i	$0.522457\pi$
$594$	39.8360	1.63449
$595$	5.81008	0.238190
$596$	14.7135	0.602690
$597$	−0.0150151	−0.000614529 0
$598$	−29.2051	−1.19429
$599$	18.7285	0.765224	0.382612	−	0.923909i	$-0.375025\pi$
$599$	18.7285	0.765224	0.382612	+	0.923909i	$0.375025\pi$
$600$	4.11951	0.168178
$601$	4.01565	0.163802	0.0819010	−	0.996640i	$-0.473901\pi$
$601$	4.01565	0.163802	0.0819010	+	0.996640i	$0.473901\pi$
$602$	0	0
$603$	6.40475	0.260822
$604$	−14.1431	−0.575473
$605$	13.9063	0.565373
$606$	8.11331	0.329581
$607$	−25.5121	−1.03550	−0.517751	−	0.855531i	$-0.673231\pi$
$607$	−25.5121	−1.03550	−0.517751	+	0.855531i	$0.673231\pi$
$608$	−12.8046	−0.519297
$609$	6.77688	0.274613
$610$	15.3375	0.620997
$611$	13.2551	0.536243
$612$	8.21793	0.332190
$613$	−17.7223	−0.715797	−0.357899	−	0.933760i	$-0.616507\pi$
$613$	−17.7223	−0.715797	−0.357899	+	0.933760i	$0.616507\pi$
$614$	33.8975	1.36799
$615$	22.8601	0.921808
$616$	4.87141	0.196275
$617$	13.4716	0.542345	0.271172	−	0.962531i	$-0.412589\pi$
$617$	13.4716	0.542345	0.271172	+	0.962531i	$0.412589\pi$
$618$	30.9228	1.24390
$619$	30.3811	1.22112	0.610561	−	0.791970i	$-0.290944\pi$
$619$	30.3811	1.22112	0.610561	+	0.791970i	$0.290944\pi$
$620$	5.86213	0.235429
$621$	28.5548	1.14586
$622$	10.0237	0.401912
$623$	2.68116	0.107418
$624$	18.6961	0.748445
$625$	−11.9744	−0.478977
$626$	35.6427	1.42457
$627$	−13.6530	−0.545250
$628$	5.85201	0.233520
$629$	7.45905	0.297412
$630$	−2.96646	−0.118187
$631$	5.77209	0.229783	0.114892	−	0.993378i	$-0.463348\pi$
$631$	5.77209	0.229783	0.114892	+	0.993378i	$0.463348\pi$
$632$	25.2509	1.00443
$633$	−2.50634	−0.0996179
$634$	−36.9355	−1.46690
$635$	−23.0388	−0.914267
$636$	2.48363	0.0984823
$637$	−21.5524	−0.853937
$638$	72.8674	2.88485
$639$	−6.38549	−0.252606
$640$	−22.5095	−0.889765
$641$	12.0345	0.475333	0.237666	−	0.971347i	$-0.423617\pi$
$641$	12.0345	0.475333	0.237666	+	0.971347i	$0.423617\pi$
$642$	0.105116	0.00414858
$643$	−42.5459	−1.67785	−0.838923	−	0.544250i	$-0.816814\pi$
$643$	−42.5459	−1.67785	−0.838923	+	0.544250i	$0.816814\pi$
$644$	−2.80674	−0.110601
$645$	0	0
$646$	−25.6107	−1.00764
$647$	28.9612	1.13858	0.569291	−	0.822136i	$-0.307218\pi$
$647$	28.9612	1.13858	0.569291	+	0.822136i	$0.307218\pi$
$648$	−2.33646	−0.0917849
$649$	3.19476	0.125405
$650$	10.4231	0.408827
$651$	2.56345	0.100469
$652$	11.3209	0.443359
$653$	2.68809	0.105193	0.0525966	−	0.998616i	$-0.483250\pi$
$653$	2.68809	0.105193	0.0525966	+	0.998616i	$0.483250\pi$
$654$	−10.3184	−0.403481
$655$	19.0001	0.742395
$656$	55.9446	2.18427
$657$	−12.4062	−0.484011
$658$	4.13257	0.161104
$659$	7.80448	0.304019	0.152010	−	0.988379i	$-0.451426\pi$
$659$	7.80448	0.304019	0.152010	+	0.988379i	$0.451426\pi$
$660$	7.89438	0.307288
$661$	3.39429	0.132022	0.0660112	−	0.997819i	$-0.478973\pi$
$661$	3.39429	0.132022	0.0660112	+	0.997819i	$0.478973\pi$
$662$	−16.1246	−0.626701
$663$	20.7702	0.806649
$664$	14.6923	0.570172
$665$	2.84972	0.110508
$666$	−3.80837	−0.147571
$667$	52.2321	2.02243
$668$	14.8579	0.574868
$669$	4.36460	0.168745
$670$	11.5412	0.445875
$671$	22.2296	0.858164
$672$	3.23761	0.124894
$673$	16.2674	0.627062	0.313531	−	0.949578i	$-0.398488\pi$
$673$	16.2674	0.627062	0.313531	+	0.949578i	$0.398488\pi$
$674$	42.5061	1.63727
$675$	−10.1910	−0.392251
$676$	−2.21477	−0.0851833
$677$	−2.40528	−0.0924423	−0.0462211	−	0.998931i	$-0.514718\pi$
$677$	−2.40528	−0.0924423	−0.0462211	+	0.998931i	$0.514718\pi$
$678$	−5.07474	−0.194894
$679$	−0.109971	−0.00422029
$680$	−18.4233	−0.706502
$681$	−19.1642	−0.734373
$682$	27.5631	1.05545
$683$	4.78972	0.183273	0.0916367	−	0.995793i	$-0.470790\pi$
$683$	4.78972	0.183273	0.0916367	+	0.995793i	$0.470790\pi$
$684$	4.03072	0.154119
$685$	14.1729	0.541518
$686$	−13.7963	−0.526744
$687$	0.894617	0.0341318
$688$	0	0
$689$	−7.81798	−0.297841
$690$	18.3576	0.698863
$691$	17.7737	0.676143	0.338072	−	0.941120i	$-0.390225\pi$
$691$	17.7737	0.676143	0.338072	+	0.941120i	$0.390225\pi$
$692$	7.22006	0.274466
$693$	−4.29948	−0.163324
$694$	−17.4166	−0.661125
$695$	38.5981	1.46411
$696$	−21.4889	−0.814536
$697$	62.1509	2.35413
$698$	−25.1632	−0.952440
$699$	−30.0250	−1.13565
$700$	1.00170	0.0378608
$701$	2.59973	0.0981903	0.0490952	−	0.998794i	$-0.484366\pi$
$701$	2.59973	0.0981903	0.0490952	+	0.998794i	$0.484366\pi$
$702$	−29.7240	−1.12186
$703$	3.65851	0.137983
$704$	−8.54329	−0.321987
$705$	−8.33183	−0.313795
$706$	20.4496	0.769632
$707$	−2.45442	−0.0923080
$708$	0.757289	0.0284607
$709$	−38.4152	−1.44271	−0.721356	−	0.692564i	$-0.756481\pi$
$709$	−38.4152	−1.44271	−0.721356	+	0.692564i	$0.756481\pi$
$710$	−11.5065	−0.431831
$711$	−22.2863	−0.835801
$712$	−8.50173	−0.318616
$713$	19.7575	0.739923
$714$	6.47558	0.242343
$715$	−24.8500	−0.929337
$716$	15.5516	0.581190
$717$	−4.07514	−0.152189
$718$	−13.0986	−0.488837
$719$	33.3797	1.24485	0.622426	−	0.782679i	$-0.286147\pi$
$719$	33.3797	1.24485	0.622426	+	0.782679i	$0.286147\pi$
$720$	14.6366	0.545473
$721$	−9.35468	−0.348387
$722$	19.7453	0.734845
$723$	−22.5572	−0.838913
$724$	12.5793	0.467506
$725$	−18.6412	−0.692318
$726$	15.4992	0.575229
$727$	36.8788	1.36776	0.683880	−	0.729595i	$-0.260291\pi$
$727$	36.8788	1.36776	0.683880	+	0.729595i	$0.260291\pi$
$728$	−3.63486	−0.134717
$729$	20.7558	0.768732
$730$	−22.3556	−0.827416
$731$	0	0
$732$	5.26934	0.194760
$733$	26.6921	0.985894	0.492947	−	0.870059i	$-0.335920\pi$
$733$	26.6921	0.985894	0.492947	+	0.870059i	$0.335920\pi$
$734$	−5.46719	−0.201798
$735$	13.5473	0.499700
$736$	24.9535	0.919800
$737$	16.7274	0.616160
$738$	−31.7324	−1.16809
$739$	−50.9722	−1.87504	−0.937520	−	0.347930i	$-0.886885\pi$
$739$	−50.9722	−1.87504	−0.937520	+	0.347930i	$0.886885\pi$
$740$	−2.11540	−0.0777636
$741$	10.1874	0.374242
$742$	−2.43743	−0.0894809
$743$	20.7948	0.762886	0.381443	−	0.924392i	$-0.375427\pi$
$743$	20.7948	0.762886	0.381443	+	0.924392i	$0.375427\pi$
$744$	−8.12848	−0.298004
$745$	29.1127	1.06661
$746$	−29.9552	−1.09674
$747$	−12.9673	−0.474450
$748$	21.4629	0.784760
$749$	−0.0317993	−0.00116192
$750$	−23.8806	−0.871997
$751$	−10.7184	−0.391121	−0.195560	−	0.980692i	$-0.562653\pi$
$751$	−10.7184	−0.391121	−0.195560	+	0.980692i	$0.562653\pi$
$752$	−20.3902	−0.743553
$753$	−10.2952	−0.375177
$754$	−54.3708	−1.98007
$755$	−27.9840	−1.01844
$756$	−2.85660	−0.103894
$757$	3.34327	0.121513	0.0607566	−	0.998153i	$-0.480649\pi$
$757$	3.34327	0.121513	0.0607566	+	0.998153i	$0.480649\pi$
$758$	−30.0618	−1.09190
$759$	26.6069	0.965769
$760$	−9.03625	−0.327779
$761$	38.3661	1.39077	0.695384	−	0.718638i	$-0.255234\pi$
$761$	38.3661	1.39077	0.695384	+	0.718638i	$0.255234\pi$
$762$	−25.6777	−0.930205
$763$	3.12149	0.113006
$764$	16.8889	0.611018
$765$	16.2603	0.587892
$766$	42.3295	1.52943
$767$	−2.38380	−0.0860740
$768$	−20.5432	−0.741289
$769$	−0.652284	−0.0235220	−0.0117610	−	0.999931i	$-0.503744\pi$
$769$	−0.652284	−0.0235220	−0.0117610	+	0.999931i	$0.503744\pi$
$770$	−7.74754	−0.279202
$771$	0.113364	0.00408270
$772$	−18.8832	−0.679622
$773$	−43.3995	−1.56097	−0.780486	−	0.625174i	$-0.785028\pi$
$773$	−43.3995	−1.56097	−0.780486	+	0.625174i	$0.785028\pi$
$774$	0	0
$775$	−7.05130	−0.253290
$776$	0.348709	0.0125179
$777$	−0.925041	−0.0331856
$778$	−11.4337	−0.409918
$779$	30.4837	1.09219
$780$	−5.89047	−0.210913
$781$	−16.6771	−0.596752
$782$	49.9098	1.78477
$783$	53.1601	1.89979
$784$	33.1538	1.18406
$785$	11.5790	0.413272
$786$	21.1764	0.755337
$787$	−33.4430	−1.19211	−0.596057	−	0.802942i	$-0.703267\pi$
$787$	−33.4430	−1.19211	−0.596057	+	0.802942i	$0.703267\pi$
$788$	2.51569	0.0896176
$789$	−0.536167	−0.0190881
$790$	−40.1593	−1.42880
$791$	1.53520	0.0545854
$792$	13.6333	0.484438
$793$	−16.5869	−0.589016
$794$	34.6007	1.22793
$795$	4.91420	0.174289
$796$	0.0115773	0.000410347 0
$797$	−34.3345	−1.21619	−0.608096	−	0.793864i	$-0.708066\pi$
$797$	−34.3345	−1.21619	−0.608096	+	0.793864i	$0.708066\pi$
$798$	3.17614	0.112434
$799$	−22.6522	−0.801376
$800$	−8.90573	−0.314865
$801$	7.50357	0.265126
$802$	−0.328263	−0.0115914
$803$	−32.4013	−1.14342
$804$	3.96508	0.139837
$805$	−5.55351	−0.195735
$806$	−20.5665	−0.724424
$807$	−24.7669	−0.871835
$808$	7.78277	0.273797
$809$	46.6942	1.64168	0.820841	−	0.571157i	$-0.193505\pi$
$809$	46.6942	1.64168	0.820841	+	0.571157i	$0.193505\pi$
$810$	3.71593	0.130564
$811$	24.9497	0.876104	0.438052	−	0.898950i	$-0.355669\pi$
$811$	24.9497	0.876104	0.438052	+	0.898950i	$0.355669\pi$
$812$	−5.22526	−0.183371
$813$	10.9331	0.383441
$814$	−9.94637	−0.348620
$815$	22.3999	0.784633
$816$	−31.9506	−1.11850
$817$	0	0
$818$	−0.837096	−0.0292684
$819$	3.20810	0.112100
$820$	−17.6261	−0.615531
$821$	54.8527	1.91437	0.957186	−	0.289474i	$-0.0934804\pi$
$821$	54.8527	1.91437	0.957186	+	0.289474i	$0.0934804\pi$
$822$	15.7963	0.550958
$823$	27.1455	0.946234	0.473117	−	0.881000i	$-0.343129\pi$
$823$	27.1455	0.946234	0.473117	+	0.881000i	$0.343129\pi$
$824$	29.6629	1.03336
$825$	−9.49579	−0.330601
$826$	−0.743203	−0.0258593
$827$	−4.03843	−0.140430	−0.0702150	−	0.997532i	$-0.522369\pi$
$827$	−4.03843	−0.140430	−0.0702150	+	0.997532i	$0.522369\pi$
$828$	−7.85502	−0.272981
$829$	−12.3779	−0.429902	−0.214951	−	0.976625i	$-0.568959\pi$
$829$	−12.3779	−0.429902	−0.214951	+	0.976625i	$0.568959\pi$
$830$	−23.3668	−0.811073
$831$	14.6933	0.509705
$832$	6.37466	0.221002
$833$	36.8318	1.27615
$834$	43.0192	1.48963
$835$	29.3983	1.01737
$836$	10.5271	0.364087
$837$	20.1085	0.695052
$838$	−27.3384	−0.944391
$839$	−32.6896	−1.12857	−0.564285	−	0.825580i	$-0.690848\pi$
$839$	−32.6896	−1.12857	−0.564285	+	0.825580i	$0.690848\pi$
$840$	2.28478	0.0788325
$841$	68.2398	2.35310
$842$	−61.9465	−2.13482
$843$	−22.4764	−0.774130
$844$	1.93249	0.0665191
$845$	−4.38221	−0.150753
$846$	11.5655	0.397632
$847$	−4.68878	−0.161108
$848$	12.0263	0.412986
$849$	−2.85070	−0.0978356
$850$	−17.8124	−0.610962
$851$	−7.12965	−0.244401
$852$	−3.95315	−0.135433
$853$	9.16353	0.313753	0.156877	−	0.987618i	$-0.449857\pi$
$853$	9.16353	0.313753	0.156877	+	0.987618i	$0.449857\pi$
$854$	−5.17132	−0.176959
$855$	7.97533	0.272751
$856$	0.100833	0.00344640
$857$	−12.5475	−0.428616	−0.214308	−	0.976766i	$-0.568750\pi$
$857$	−12.5475	−0.428616	−0.214308	+	0.976766i	$0.568750\pi$
$858$	−27.6963	−0.945538
$859$	−13.4029	−0.457302	−0.228651	−	0.973508i	$-0.573432\pi$
$859$	−13.4029	−0.457302	−0.228651	+	0.973508i	$0.573432\pi$
$860$	0	0
$861$	−7.70770	−0.262678
$862$	−13.8265	−0.470931
$863$	−33.1267	−1.12765	−0.563823	−	0.825896i	$-0.690670\pi$
$863$	−33.1267	−1.12765	−0.563823	+	0.825896i	$0.690670\pi$
$864$	25.3969	0.864020
$865$	14.2859	0.485734
$866$	−46.8297	−1.59134
$867$	−15.8454	−0.538136
$868$	−1.97653	−0.0670877
$869$	−58.2053	−1.97448
$870$	34.1762	1.15868
$871$	−12.4813	−0.422913
$872$	−9.89800	−0.335189
$873$	−0.307768	−0.0104164
$874$	24.4797	0.828039
$875$	7.22431	0.244226
$876$	−7.68045	−0.259498
$877$	−0.274175	−0.00925825	−0.00462912	−	0.999989i	$-0.501474\pi$
$877$	−0.274175	−0.00925825	−0.00462912	+	0.999989i	$0.501474\pi$
$878$	63.3735	2.13875
$879$	25.1872	0.849544
$880$	38.2265	1.28861
$881$	−36.2982	−1.22292	−0.611458	−	0.791277i	$-0.709417\pi$
$881$	−36.2982	−1.22292	−0.611458	+	0.791277i	$0.709417\pi$
$882$	−18.8052	−0.633205
$883$	−2.16204	−0.0727583	−0.0363791	−	0.999338i	$-0.511582\pi$
$883$	−2.16204	−0.0727583	−0.0363791	+	0.999338i	$0.511582\pi$
$884$	−16.0147	−0.538634
$885$	1.49840	0.0503681
$886$	9.30859	0.312728
$887$	−36.5266	−1.22644	−0.613221	−	0.789911i	$-0.710127\pi$
$887$	−36.5266	−1.22644	−0.613221	+	0.789911i	$0.710127\pi$
$888$	2.93323	0.0984327
$889$	7.76796	0.260529
$890$	13.5212	0.453232
$891$	5.38573	0.180429
$892$	−3.36530	−0.112678
$893$	−11.1104	−0.371796
$894$	32.4474	1.08520
$895$	30.7709	1.02856
$896$	7.58949	0.253547
$897$	−19.8530	−0.662872
$898$	38.6498	1.28976
$899$	36.7823	1.22676
$900$	2.80340	0.0934466
$901$	13.3605	0.445102
$902$	−82.8760	−2.75947
$903$	0	0
$904$	−4.86799	−0.161907
$905$	24.8899	0.827367
$906$	−31.1893	−1.03619
$907$	23.9117	0.793976	0.396988	−	0.917824i	$-0.370056\pi$
$907$	23.9117	0.793976	0.396988	+	0.917824i	$0.370056\pi$
$908$	14.7764	0.490372
$909$	−6.86902	−0.227831
$910$	5.78091	0.191635
$911$	20.8468	0.690685	0.345343	−	0.938477i	$-0.387763\pi$
$911$	20.8468	0.690685	0.345343	+	0.938477i	$0.387763\pi$
$912$	−15.6711	−0.518922
$913$	−33.8669	−1.12083
$914$	−56.3018	−1.86230
$915$	10.4261	0.344676
$916$	−0.689788	−0.0227912
$917$	−6.40623	−0.211552
$918$	50.7966	1.67654
$919$	−26.0929	−0.860725	−0.430362	−	0.902656i	$-0.641614\pi$
$919$	−26.0929	−0.860725	−0.430362	+	0.902656i	$0.641614\pi$
$920$	17.6097	0.580575
$921$	23.0428	0.759287
$922$	25.4751	0.838978
$923$	12.4438	0.409592
$924$	−2.66174	−0.0875647
$925$	2.54452	0.0836633
$926$	31.0118	1.01911
$927$	−26.1803	−0.859875
$928$	46.4557	1.52498
$929$	−10.3828	−0.340648	−0.170324	−	0.985388i	$-0.554481\pi$
$929$	−10.3828	−0.340648	−0.170324	+	0.985388i	$0.554481\pi$
$930$	12.9276	0.423913
$931$	18.0652	0.592063
$932$	23.1505	0.758321
$933$	6.81386	0.223076
$934$	5.91695	0.193609
$935$	42.4672	1.38883
$936$	−10.1726	−0.332503
$937$	−23.6950	−0.774081	−0.387040	−	0.922063i	$-0.626503\pi$
$937$	−23.6950	−0.774081	−0.387040	+	0.922063i	$0.626503\pi$
$938$	−3.89132	−0.127056
$939$	24.2292	0.790689
$940$	6.42420	0.209534
$941$	−24.6205	−0.802607	−0.401303	−	0.915945i	$-0.631443\pi$
$941$	−24.6205	−0.802607	−0.401303	+	0.915945i	$0.631443\pi$
$942$	12.9053	0.420476
$943$	−59.4063	−1.93453
$944$	3.66698	0.119350
$945$	−5.65218	−0.183865
$946$	0	0
$947$	−6.75858	−0.219624	−0.109812	−	0.993952i	$-0.535025\pi$
$947$	−6.75858	−0.219624	−0.109812	+	0.993952i	$0.535025\pi$
$948$	−13.7971	−0.448108
$949$	24.1766	0.784805
$950$	−8.73663	−0.283454
$951$	−25.1079	−0.814181
$952$	6.21176	0.201324
$953$	−49.8456	−1.61466	−0.807329	−	0.590101i	$-0.799088\pi$
$953$	−49.8456	−1.61466	−0.807329	+	0.590101i	$0.799088\pi$
$954$	−6.82147	−0.220853
$955$	33.4170	1.08135
$956$	3.14211	0.101623
$957$	49.5337	1.60120
$958$	−40.7863	−1.31775
$959$	−4.77865	−0.154311
$960$	−4.00696	−0.129324
$961$	−17.0866	−0.551181
$962$	7.42159	0.239282
$963$	−0.0889946	−0.00286781
$964$	17.3926	0.560178
$965$	−37.3630	−1.20276
$966$	−6.18962	−0.199148
$967$	−22.6357	−0.727915	−0.363958	−	0.931416i	$-0.618575\pi$
$967$	−22.6357	−0.727915	−0.363958	+	0.931416i	$0.618575\pi$
$968$	14.8677	0.477867
$969$	−17.4096	−0.559277
$970$	−0.554589	−0.0178068
$971$	−59.9381	−1.92351	−0.961753	−	0.273920i	$-0.911680\pi$
$971$	−59.9381	−1.92351	−0.961753	+	0.273920i	$0.911680\pi$
$972$	−13.1369	−0.421367
$973$	−13.0141	−0.417212
$974$	7.51567	0.240818
$975$	7.08539	0.226914
$976$	25.5154	0.816727
$977$	−15.8508	−0.507113	−0.253556	−	0.967321i	$-0.581600\pi$
$977$	−15.8508	−0.507113	−0.253556	+	0.967321i	$0.581600\pi$
$978$	24.9656	0.798311
$979$	19.5972	0.626328
$980$	−10.4456	−0.333671
$981$	8.73591	0.278916
$982$	26.6225	0.849559
$983$	26.4577	0.843870	0.421935	−	0.906626i	$-0.361351\pi$
$983$	26.4577	0.843870	0.421935	+	0.906626i	$0.361351\pi$
$984$	24.4405	0.779135
$985$	4.97763	0.158600
$986$	92.9165	2.95906
$987$	2.80923	0.0894188
$988$	−7.85489	−0.249897
$989$	0	0
$990$	−21.6825	−0.689116
$991$	19.5746	0.621809	0.310904	−	0.950441i	$-0.399368\pi$
$991$	19.5746	0.621809	0.310904	+	0.950441i	$0.399368\pi$
$992$	17.5725	0.557927
$993$	−10.9612	−0.347842
$994$	3.87962	0.123054
$995$	0.0229073	0.000726211 0
$996$	−8.02787	−0.254373
$997$	−18.0624	−0.572041	−0.286020	−	0.958224i	$-0.592333\pi$
$997$	−18.0624	−0.572041	−0.286020	+	0.958224i	$0.592333\pi$
$998$	64.0356	2.02701
$999$	−7.25633	−0.229580

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.o.1.4		18
43.3	odd	42		43.2.g.a.9.3	✓	36
43.29	odd	42		43.2.g.a.24.3	yes	36
43.42	odd	2		1849.2.a.n.1.15		18
129.29	even	42		387.2.y.c.325.1		36
129.89	even	42		387.2.y.c.181.1		36
172.3	even	42		688.2.bg.c.353.2		36
172.115	even	42		688.2.bg.c.497.2		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.9.3	✓	36	43.3	odd	42
43.2.g.a.24.3	yes	36	43.29	odd	42
387.2.y.c.181.1		36	129.89	even	42
387.2.y.c.325.1		36	129.29	even	42
688.2.bg.c.353.2		36	172.3	even	42
688.2.bg.c.497.2		36	172.115	even	42
1849.2.a.n.1.15		18	43.42	odd	2
1849.2.a.o.1.4		18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q-1.70036 q^{2} -1.15587 q^{3} +0.891224 q^{4} +1.76341 q^{5} +1.96539 q^{6} -0.594565 q^{7} +1.88532 q^{8} -1.66397 q^{9} +O(q^{10})\)	\(q-1.70036 q^{2} -1.15587 q^{3} +0.891224 q^{4} +1.76341 q^{5} +1.96539 q^{6} -0.594565 q^{7} +1.88532 q^{8} -1.66397 q^{9} -2.99843 q^{10} -4.34581 q^{11} -1.03014 q^{12} +3.24267 q^{13} +1.01098 q^{14} -2.03827 q^{15} -4.98817 q^{16} -5.54154 q^{17} +2.82935 q^{18} -2.71801 q^{19} +1.57159 q^{20} +0.687239 q^{21} +7.38944 q^{22} +5.29682 q^{23} -2.17918 q^{24} -1.89039 q^{25} -5.51371 q^{26} +5.39093 q^{27} -0.529891 q^{28} +9.86102 q^{29} +3.46578 q^{30} +3.73006 q^{31} +4.71104 q^{32} +5.02318 q^{33} +9.42261 q^{34} -1.04846 q^{35} -1.48297 q^{36} -1.34602 q^{37} +4.62159 q^{38} -3.74810 q^{39} +3.32458 q^{40} -11.2155 q^{41} -1.16855 q^{42} -3.87309 q^{44} -2.93426 q^{45} -9.00650 q^{46} +4.08771 q^{47} +5.76566 q^{48} -6.64649 q^{49} +3.21435 q^{50} +6.40528 q^{51} +2.88995 q^{52} -2.41097 q^{53} -9.16652 q^{54} -7.66343 q^{55} -1.12095 q^{56} +3.14166 q^{57} -16.7673 q^{58} -0.735135 q^{59} -1.81655 q^{60} -5.11518 q^{61} -6.34245 q^{62} +0.989339 q^{63} +1.96587 q^{64} +5.71815 q^{65} -8.54121 q^{66} -3.84908 q^{67} -4.93875 q^{68} -6.12242 q^{69} +1.78276 q^{70} +3.83751 q^{71} -3.13711 q^{72} +7.45576 q^{73} +2.28873 q^{74} +2.18505 q^{75} -2.42235 q^{76} +2.58387 q^{77} +6.37312 q^{78} +13.3934 q^{79} -8.79617 q^{80} -1.23929 q^{81} +19.0703 q^{82} +7.79301 q^{83} +0.612483 q^{84} -9.77199 q^{85} -11.3980 q^{87} -8.19324 q^{88} -4.50944 q^{89} +4.98929 q^{90} -1.92798 q^{91} +4.72065 q^{92} -4.31146 q^{93} -6.95057 q^{94} -4.79295 q^{95} -5.44534 q^{96} +0.184960 q^{97} +11.3014 q^{98} +7.23130 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 5 q^{2} + 5 q^{3} + 19 q^{4} + 11 q^{5} + 4 q^{6} + 6 q^{7} + 12 q^{8} + 15 q^{9}+O(q^{10}) \) 18 * q + 5 * q^2 + 5 * q^3 + 19 * q^4 + 11 * q^5 + 4 * q^6 + 6 * q^7 + 12 * q^8 + 15 * q^9	\( 18 q + 5 q^{2} + 5 q^{3} + 19 q^{4} + 11 q^{5} + 4 q^{6} + 6 q^{7} + 12 q^{8} + 15 q^{9} - 7 q^{10} - 2 q^{11} + 16 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 9 q^{16} - 11 q^{17} + 25 q^{18} + 31 q^{19} + 25 q^{20} + 19 q^{22} - 11 q^{23} + 18 q^{24} + 9 q^{25} + 27 q^{26} + 23 q^{27} + 20 q^{28} + 37 q^{29} - 17 q^{30} - 12 q^{31} + 39 q^{32} + 38 q^{33} + 14 q^{34} + 16 q^{35} + 47 q^{36} - 19 q^{37} + 56 q^{38} + 46 q^{39} + 6 q^{40} - 7 q^{41} - q^{42} + 7 q^{44} + 23 q^{45} - 47 q^{46} - q^{47} + 15 q^{48} - 6 q^{49} - 3 q^{50} + 38 q^{51} + 15 q^{52} + 3 q^{53} - 67 q^{54} + 28 q^{55} - 81 q^{56} + 46 q^{57} + 34 q^{58} + 17 q^{59} - 83 q^{60} + 28 q^{61} + 33 q^{62} + 26 q^{63} + 10 q^{64} + 16 q^{65} - 72 q^{66} + 18 q^{67} + 53 q^{68} + 7 q^{69} - 34 q^{70} + 86 q^{71} - 2 q^{72} - 27 q^{73} - 79 q^{74} + 31 q^{75} + 59 q^{76} + 43 q^{77} + 91 q^{78} + 17 q^{79} + 8 q^{80} - 10 q^{81} + 13 q^{82} - 12 q^{83} - 32 q^{84} + 28 q^{85} - 43 q^{87} - 23 q^{88} + 51 q^{89} + 10 q^{90} - 20 q^{91} + 18 q^{92} - 30 q^{93} - 15 q^{94} - q^{95} - 20 q^{96} - 19 q^{97} - 5 q^{98} + 38 q^{99}+O(q^{100}) \) 18 * q + 5 * q^2 + 5 * q^3 + 19 * q^4 + 11 * q^5 + 4 * q^6 + 6 * q^7 + 12 * q^8 + 15 * q^9 - 7 * q^10 - 2 * q^11 + 16 * q^12 - 7 * q^13 - 4 * q^14 + 3 * q^15 + 9 * q^16 - 11 * q^17 + 25 * q^18 + 31 * q^19 + 25 * q^20 + 19 * q^22 - 11 * q^23 + 18 * q^24 + 9 * q^25 + 27 * q^26 + 23 * q^27 + 20 * q^28 + 37 * q^29 - 17 * q^30 - 12 * q^31 + 39 * q^32 + 38 * q^33 + 14 * q^34 + 16 * q^35 + 47 * q^36 - 19 * q^37 + 56 * q^38 + 46 * q^39 + 6 * q^40 - 7 * q^41 - q^42 + 7 * q^44 + 23 * q^45 - 47 * q^46 - q^47 + 15 * q^48 - 6 * q^49 - 3 * q^50 + 38 * q^51 + 15 * q^52 + 3 * q^53 - 67 * q^54 + 28 * q^55 - 81 * q^56 + 46 * q^57 + 34 * q^58 + 17 * q^59 - 83 * q^60 + 28 * q^61 + 33 * q^62 + 26 * q^63 + 10 * q^64 + 16 * q^65 - 72 * q^66 + 18 * q^67 + 53 * q^68 + 7 * q^69 - 34 * q^70 + 86 * q^71 - 2 * q^72 - 27 * q^73 - 79 * q^74 + 31 * q^75 + 59 * q^76 + 43 * q^77 + 91 * q^78 + 17 * q^79 + 8 * q^80 - 10 * q^81 + 13 * q^82 - 12 * q^83 - 32 * q^84 + 28 * q^85 - 43 * q^87 - 23 * q^88 + 51 * q^89 + 10 * q^90 - 20 * q^91 + 18 * q^92 - 30 * q^93 - 15 * q^94 - q^95 - 20 * q^96 - 19 * q^97 - 5 * q^98 + 38 * q^99

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