LMFDB - Embedded newform 4009.2.a.c.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30773 q^{2} -2.77619 q^{3} +3.32561 q^{4} +2.46750 q^{5} +6.40670 q^{6} -0.00709873 q^{7} -3.05915 q^{8} +4.70724 q^{9} +O(q^{10})$	\(q-2.30773 q^{2} -2.77619 q^{3} +3.32561 q^{4} +2.46750 q^{5} +6.40670 q^{6} -0.00709873 q^{7} -3.05915 q^{8} +4.70724 q^{9} -5.69433 q^{10} +0.0942316 q^{11} -9.23254 q^{12} +2.65123 q^{13} +0.0163820 q^{14} -6.85027 q^{15} +0.408469 q^{16} -0.103040 q^{17} -10.8630 q^{18} +1.00000 q^{19} +8.20596 q^{20} +0.0197074 q^{21} -0.217461 q^{22} +6.73094 q^{23} +8.49279 q^{24} +1.08858 q^{25} -6.11831 q^{26} -4.73963 q^{27} -0.0236076 q^{28} +1.19143 q^{29} +15.8086 q^{30} -3.37678 q^{31} +5.17567 q^{32} -0.261605 q^{33} +0.237788 q^{34} -0.0175162 q^{35} +15.6545 q^{36} -2.20219 q^{37} -2.30773 q^{38} -7.36031 q^{39} -7.54847 q^{40} -4.74460 q^{41} -0.0454794 q^{42} +0.284048 q^{43} +0.313378 q^{44} +11.6151 q^{45} -15.5332 q^{46} -5.74737 q^{47} -1.13399 q^{48} -6.99995 q^{49} -2.51214 q^{50} +0.286059 q^{51} +8.81694 q^{52} -8.09152 q^{53} +10.9378 q^{54} +0.232517 q^{55} +0.0217161 q^{56} -2.77619 q^{57} -2.74949 q^{58} -15.0926 q^{59} -22.7813 q^{60} +6.40394 q^{61} +7.79269 q^{62} -0.0334155 q^{63} -12.7610 q^{64} +6.54191 q^{65} +0.603713 q^{66} -12.4217 q^{67} -0.342671 q^{68} -18.6864 q^{69} +0.0404225 q^{70} +5.05126 q^{71} -14.4002 q^{72} +8.19008 q^{73} +5.08206 q^{74} -3.02210 q^{75} +3.32561 q^{76} -0.000668925 q^{77} +16.9856 q^{78} -13.2669 q^{79} +1.00790 q^{80} -0.963597 q^{81} +10.9492 q^{82} +2.51249 q^{83} +0.0655393 q^{84} -0.254252 q^{85} -0.655507 q^{86} -3.30763 q^{87} -0.288269 q^{88} -14.9772 q^{89} -26.8046 q^{90} -0.0188203 q^{91} +22.3845 q^{92} +9.37459 q^{93} +13.2634 q^{94} +2.46750 q^{95} -14.3686 q^{96} +13.5456 q^{97} +16.1540 q^{98} +0.443571 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * 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 $

$2$	−2.30773	−1.63181	−0.815905	−	0.578186i	$-0.803761\pi$
$2$	−2.30773	−1.63181	−0.815905	+	0.578186i	$0.803761\pi$
$3$	−2.77619	−1.60284	−0.801418	−	0.598105i	$-0.795920\pi$
$3$	−2.77619	−1.60284	−0.801418	+	0.598105i	$0.795920\pi$
$4$	3.32561	1.66281
$5$	2.46750	1.10350	0.551751	−	0.834009i	$-0.313960\pi$
$5$	2.46750	1.10350	0.551751	+	0.834009i	$0.313960\pi$
$6$	6.40670	2.61552
$7$	−0.00709873	−0.00268307	−0.00134153	−	0.999999i	$-0.500427\pi$
$7$	−0.00709873	−0.00268307	−0.00134153	+	0.999999i	$0.500427\pi$
$8$	−3.05915	−1.08157
$9$	4.70724	1.56908
$10$	−5.69433	−1.80071
$11$	0.0942316	0.0284119	0.0142059	−	0.999899i	$-0.495478\pi$
$11$	0.0942316	0.0284119	0.0142059	+	0.999899i	$0.495478\pi$
$12$	−9.23254	−2.66520
$13$	2.65123	0.735318	0.367659	−	0.929961i	$-0.380159\pi$
$13$	2.65123	0.735318	0.367659	+	0.929961i	$0.380159\pi$
$14$	0.0163820	0.00437826
$15$	−6.85027	−1.76873
$16$	0.408469	0.102117
$17$	−0.103040	−0.0249909	−0.0124954	−	0.999922i	$-0.503978\pi$
$17$	−0.103040	−0.0249909	−0.0124954	+	0.999922i	$0.503978\pi$
$18$	−10.8630	−2.56044
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	8.20596	1.83491
$21$	0.0197074	0.00430052
$22$	−0.217461	−0.0463628
$23$	6.73094	1.40350	0.701749	−	0.712424i	$-0.252403\pi$
$23$	6.73094	1.40350	0.701749	+	0.712424i	$0.252403\pi$
$24$	8.49279	1.73358
$25$	1.08858	0.217715
$26$	−6.11831	−1.19990
$27$	−4.73963	−0.912143
$28$	−0.0236076	−0.00446142
$29$	1.19143	0.221242	0.110621	−	0.993863i	$-0.464716\pi$
$29$	1.19143	0.221242	0.110621	+	0.993863i	$0.464716\pi$
$30$	15.8086	2.88623
$31$	−3.37678	−0.606488	−0.303244	−	0.952913i	$-0.598070\pi$
$31$	−3.37678	−0.606488	−0.303244	+	0.952913i	$0.598070\pi$
$32$	5.17567	0.914938
$33$	−0.261605	−0.0455396
$34$	0.237788	0.0407804
$35$	−0.0175162	−0.00296077
$36$	15.6545	2.60908
$37$	−2.20219	−0.362038	−0.181019	−	0.983480i	$-0.557940\pi$
$37$	−2.20219	−0.362038	−0.181019	+	0.983480i	$0.557940\pi$
$38$	−2.30773	−0.374363
$39$	−7.36031	−1.17859
$40$	−7.54847	−1.19352
$41$	−4.74460	−0.740982	−0.370491	−	0.928836i	$-0.620811\pi$
$41$	−4.74460	−0.740982	−0.370491	+	0.928836i	$0.620811\pi$
$42$	−0.0454794	−0.00701763
$43$	0.284048	0.0433170	0.0216585	−	0.999765i	$-0.493105\pi$
$43$	0.284048	0.0433170	0.0216585	+	0.999765i	$0.493105\pi$
$44$	0.313378	0.0472435
$45$	11.6151	1.73148
$46$	−15.5332	−2.29024
$47$	−5.74737	−0.838339	−0.419170	−	0.907908i	$-0.637679\pi$
$47$	−5.74737	−0.838339	−0.419170	+	0.907908i	$0.637679\pi$
$48$	−1.13399	−0.163677
$49$	−6.99995	−0.999993
$50$	−2.51214	−0.355270
$51$	0.286059	0.0400563
$52$	8.81694	1.22269
$53$	−8.09152	−1.11146	−0.555728	−	0.831364i	$-0.687560\pi$
$53$	−8.09152	−1.11146	−0.555728	+	0.831364i	$0.687560\pi$
$54$	10.9378	1.48844
$55$	0.232517	0.0313526
$56$	0.0217161	0.00290194
$57$	−2.77619	−0.367716
$58$	−2.74949	−0.361026
$59$	−15.0926	−1.96489	−0.982447	−	0.186544i	$-0.940271\pi$
$59$	−15.0926	−1.96489	−0.982447	+	0.186544i	$0.940271\pi$
$60$	−22.7813	−2.94106
$61$	6.40394	0.819940	0.409970	−	0.912099i	$-0.365539\pi$
$61$	6.40394	0.819940	0.409970	+	0.912099i	$0.365539\pi$
$62$	7.79269	0.989673
$63$	−0.0334155	−0.00420995
$64$	−12.7610	−1.59512
$65$	6.54191	0.811424
$66$	0.603713	0.0743120
$67$	−12.4217	−1.51755	−0.758777	−	0.651351i	$-0.774203\pi$
$67$	−12.4217	−1.51755	−0.758777	+	0.651351i	$0.774203\pi$
$68$	−0.342671	−0.0415550
$69$	−18.6864	−2.24958
$70$	0.0404225	0.00483142
$71$	5.05126	0.599474	0.299737	−	0.954022i	$-0.403101\pi$
$71$	5.05126	0.599474	0.299737	+	0.954022i	$0.403101\pi$
$72$	−14.4002	−1.69708
$73$	8.19008	0.958576	0.479288	−	0.877658i	$-0.340895\pi$
$73$	8.19008	0.958576	0.479288	+	0.877658i	$0.340895\pi$
$74$	5.08206	0.590778
$75$	−3.02210	−0.348962
$76$	3.32561	0.381474
$77$	−0.000668925 0	−7.62311e−5 0
$78$	16.9856	1.92324
$79$	−13.2669	−1.49264	−0.746322	−	0.665585i	$-0.768182\pi$
$79$	−13.2669	−1.49264	−0.746322	+	0.665585i	$0.768182\pi$
$80$	1.00790	0.112686
$81$	−0.963597	−0.107066
$82$	10.9492	1.20914
$83$	2.51249	0.275782	0.137891	−	0.990447i	$-0.455968\pi$
$83$	2.51249	0.275782	0.137891	+	0.990447i	$0.455968\pi$
$84$	0.0655393	0.00715093
$85$	−0.254252	−0.0275775
$86$	−0.655507	−0.0706851
$87$	−3.30763	−0.354615
$88$	−0.288269	−0.0307295
$89$	−14.9772	−1.58757	−0.793787	−	0.608195i	$-0.791894\pi$
$89$	−14.9772	−1.58757	−0.793787	+	0.608195i	$0.791894\pi$
$90$	−26.8046	−2.82545
$91$	−0.0188203	−0.00197291
$92$	22.3845	2.33375
$93$	9.37459	0.972100
$94$	13.2634	1.36801
$95$	2.46750	0.253161
$96$	−14.3686	−1.46649
$97$	13.5456	1.37534	0.687672	−	0.726022i	$-0.258633\pi$
$97$	13.5456	1.37534	0.687672	+	0.726022i	$0.258633\pi$
$98$	16.1540	1.63180
$99$	0.443571	0.0445806
$100$	3.62018	0.362018
$101$	1.87558	0.186627	0.0933137	−	0.995637i	$-0.470254\pi$
$101$	1.87558	0.186627	0.0933137	+	0.995637i	$0.470254\pi$
$102$	−0.660147	−0.0653642
$103$	−16.9679	−1.67189	−0.835947	−	0.548810i	$-0.815081\pi$
$103$	−16.9679	−1.67189	−0.835947	+	0.548810i	$0.815081\pi$
$104$	−8.11050	−0.795300
$105$	0.0486282	0.00474563
$106$	18.6730	1.81368
$107$	−18.0512	−1.74507	−0.872536	−	0.488550i	$-0.837526\pi$
$107$	−18.0512	−1.74507	−0.872536	+	0.488550i	$0.837526\pi$
$108$	−15.7622	−1.51672
$109$	12.1388	1.16268	0.581341	−	0.813660i	$-0.302528\pi$
$109$	12.1388	1.16268	0.581341	+	0.813660i	$0.302528\pi$
$110$	−0.536586	−0.0511614
$111$	6.11371	0.580287
$112$	−0.00289961	−0.000273987 0
$113$	10.5627	0.993652	0.496826	−	0.867850i	$-0.334499\pi$
$113$	10.5627	0.993652	0.496826	+	0.867850i	$0.334499\pi$
$114$	6.40670	0.600042
$115$	16.6086	1.54876
$116$	3.96222	0.367883
$117$	12.4800	1.15377
$118$	34.8297	3.20633
$119$	0.000731454 0	6.70523e−5 0
$120$	20.9560	1.91301
$121$	−10.9911	−0.999193
$122$	−14.7786	−1.33799
$123$	13.1719	1.18767
$124$	−11.2299	−1.00847
$125$	−9.65145	−0.863252
$126$	0.0771138	0.00686984
$127$	20.6097	1.82882	0.914409	−	0.404792i	$-0.132656\pi$
$127$	20.6097	1.82882	0.914409	+	0.404792i	$0.132656\pi$
$128$	19.0975	1.68800
$129$	−0.788573	−0.0694300
$130$	−15.0969	−1.32409
$131$	−8.80506	−0.769302	−0.384651	−	0.923062i	$-0.625678\pi$
$131$	−8.80506	−0.769302	−0.384651	+	0.923062i	$0.625678\pi$
$132$	−0.869997	−0.0757235
$133$	−0.00709873	−0.000615538 0
$134$	28.6659	2.47636
$135$	−11.6951	−1.00655
$136$	0.315215	0.0270295
$137$	18.4343	1.57495	0.787476	−	0.616345i	$-0.211387\pi$
$137$	18.4343	1.57495	0.787476	+	0.616345i	$0.211387\pi$
$138$	43.1231	3.67088
$139$	7.83167	0.664274	0.332137	−	0.943231i	$-0.392230\pi$
$139$	7.83167	0.664274	0.332137	+	0.943231i	$0.392230\pi$
$140$	−0.0582519	−0.00492319
$141$	15.9558	1.34372
$142$	−11.6569	−0.978229
$143$	0.249829	0.0208918
$144$	1.92276	0.160230
$145$	2.93985	0.244141
$146$	−18.9005	−1.56421
$147$	19.4332	1.60282
$148$	−7.32363	−0.601999
$149$	−14.6002	−1.19609	−0.598047	−	0.801461i	$-0.704056\pi$
$149$	−14.6002	−1.19609	−0.598047	+	0.801461i	$0.704056\pi$
$150$	6.97418	0.569439
$151$	−4.10746	−0.334260	−0.167130	−	0.985935i	$-0.553450\pi$
$151$	−4.10746	−0.334260	−0.167130	+	0.985935i	$0.553450\pi$
$152$	−3.05915	−0.248130
$153$	−0.485035	−0.0392127
$154$	0.00154370	0.000124395 0
$155$	−8.33222	−0.669260
$156$	−24.4775	−1.95977
$157$	13.9117	1.11027	0.555136	−	0.831760i	$-0.312666\pi$
$157$	13.9117	1.11027	0.555136	+	0.831760i	$0.312666\pi$
$158$	30.6164	2.43571
$159$	22.4636	1.78148
$160$	12.7710	1.00963
$161$	−0.0477812	−0.00376568
$162$	2.22372	0.174712
$163$	−6.72695	−0.526895	−0.263448	−	0.964674i	$-0.584860\pi$
$163$	−6.72695	−0.526895	−0.263448	+	0.964674i	$0.584860\pi$
$164$	−15.7787	−1.23211
$165$	−0.645511	−0.0502530
$166$	−5.79815	−0.450023
$167$	−11.9187	−0.922293	−0.461147	−	0.887324i	$-0.652562\pi$
$167$	−11.9187	−0.922293	−0.461147	+	0.887324i	$0.652562\pi$
$168$	−0.0602881	−0.00465133
$169$	−5.97101	−0.459308
$170$	0.586744	0.0450012
$171$	4.70724	0.359972
$172$	0.944635	0.0720277
$173$	3.85339	0.292968	0.146484	−	0.989213i	$-0.453204\pi$
$173$	3.85339	0.292968	0.146484	+	0.989213i	$0.453204\pi$
$174$	7.63311	0.578664
$175$	−0.00772751	−0.000584145 0
$176$	0.0384906	0.00290134
$177$	41.9000	3.14940
$178$	34.5632	2.59062
$179$	10.2772	0.768156	0.384078	−	0.923301i	$-0.374519\pi$
$179$	10.2772	0.768156	0.384078	+	0.923301i	$0.374519\pi$
$180$	38.6274	2.87912
$181$	9.36529	0.696116	0.348058	−	0.937473i	$-0.386841\pi$
$181$	9.36529	0.696116	0.348058	+	0.937473i	$0.386841\pi$
$182$	0.0434322	0.00321941
$183$	−17.7786	−1.31423
$184$	−20.5910	−1.51799
$185$	−5.43392	−0.399510
$186$	−21.6340	−1.58628
$187$	−0.00970963	−0.000710038 0
$188$	−19.1135	−1.39400
$189$	0.0336454	0.00244734
$190$	−5.69433	−0.413110
$191$	−13.7017	−0.991423	−0.495712	−	0.868487i	$-0.665093\pi$
$191$	−13.7017	−0.991423	−0.495712	+	0.868487i	$0.665093\pi$
$192$	35.4269	2.55672
$193$	−10.5585	−0.760021	−0.380010	−	0.924982i	$-0.624080\pi$
$193$	−10.5585	−0.760021	−0.380010	+	0.924982i	$0.624080\pi$
$194$	−31.2595	−2.24430
$195$	−18.1616	−1.30058
$196$	−23.2791	−1.66279
$197$	22.7222	1.61889	0.809443	−	0.587198i	$-0.199769\pi$
$197$	22.7222	1.61889	0.809443	+	0.587198i	$0.199769\pi$
$198$	−1.02364	−0.0727470
$199$	1.60359	0.113675	0.0568377	−	0.998383i	$-0.481898\pi$
$199$	1.60359	0.113675	0.0568377	+	0.998383i	$0.481898\pi$
$200$	−3.33012	−0.235475
$201$	34.4851	2.43239
$202$	−4.32833	−0.304541
$203$	−0.00845762	−0.000593608 0
$204$	0.951321	0.0666058
$205$	−11.7073	−0.817675
$206$	39.1572	2.72821
$207$	31.6842	2.20220
$208$	1.08294	0.0750885
$209$	0.0942316	0.00651814
$210$	−0.112221	−0.00774396
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−26.9092	−1.84813
$213$	−14.0233	−0.960859
$214$	41.6572	2.84763
$215$	0.700891	0.0478004
$216$	14.4993	0.986549
$217$	0.0239709	0.00162725
$218$	−28.0130	−1.89728
$219$	−22.7372	−1.53644
$220$	0.773261	0.0521332
$221$	−0.273182	−0.0183762
$222$	−14.1088	−0.946919
$223$	4.01739	0.269024	0.134512	−	0.990912i	$-0.457053\pi$
$223$	4.01739	0.269024	0.134512	+	0.990912i	$0.457053\pi$
$224$	−0.0367407	−0.00245484
$225$	5.12419	0.341613
$226$	−24.3758	−1.62145
$227$	−15.0161	−0.996651	−0.498326	−	0.866990i	$-0.666052\pi$
$227$	−15.0161	−0.996651	−0.498326	+	0.866990i	$0.666052\pi$
$228$	−9.23254	−0.611440
$229$	−5.95353	−0.393421	−0.196710	−	0.980462i	$-0.563026\pi$
$229$	−5.95353	−0.393421	−0.196710	+	0.980462i	$0.563026\pi$
$230$	−38.3282	−2.52729
$231$	0.00185706	0.000122186 0
$232$	−3.64475	−0.239290
$233$	−17.7642	−1.16377	−0.581885	−	0.813271i	$-0.697685\pi$
$233$	−17.7642	−1.16377	−0.581885	+	0.813271i	$0.697685\pi$
$234$	−28.8004	−1.88274
$235$	−14.1816	−0.925109
$236$	−50.1922	−3.26724
$237$	36.8315	2.39246
$238$	−0.00168800	−0.000109417 0
$239$	7.79067	0.503936	0.251968	−	0.967736i	$-0.418922\pi$
$239$	7.79067	0.503936	0.251968	+	0.967736i	$0.418922\pi$
$240$	−2.79812	−0.180618
$241$	15.4883	0.997687	0.498844	−	0.866692i	$-0.333758\pi$
$241$	15.4883	0.997687	0.498844	+	0.866692i	$0.333758\pi$
$242$	25.3645	1.63049
$243$	16.8940	1.08375
$244$	21.2970	1.36340
$245$	−17.2724	−1.10349
$246$	−30.3972	−1.93806
$247$	2.65123	0.168693
$248$	10.3301	0.655961
$249$	−6.97516	−0.442033
$250$	22.2729	1.40866
$251$	−30.1430	−1.90261	−0.951305	−	0.308253i	$-0.900256\pi$
$251$	−30.1430	−1.90261	−0.951305	+	0.308253i	$0.900256\pi$
$252$	−0.111127	−0.00700033
$253$	0.634267	0.0398761
$254$	−47.5617	−2.98428
$255$	0.705852	0.0442021
$256$	−18.5500	−1.15937
$257$	13.6903	0.853975	0.426988	−	0.904257i	$-0.359575\pi$
$257$	13.6903	0.853975	0.426988	+	0.904257i	$0.359575\pi$
$258$	1.81981	0.113297
$259$	0.0156328	0.000971373 0
$260$	21.7558	1.34924
$261$	5.60833	0.347147
$262$	20.3197	1.25536
$263$	−5.82562	−0.359223	−0.179612	−	0.983738i	$-0.557484\pi$
$263$	−5.82562	−0.359223	−0.179612	+	0.983738i	$0.557484\pi$
$264$	0.800289	0.0492544
$265$	−19.9659	−1.22649
$266$	0.0163820	0.00100444
$267$	41.5795	2.54462
$268$	−41.3098	−2.52340
$269$	6.07542	0.370425	0.185212	−	0.982699i	$-0.440703\pi$
$269$	6.07542	0.370425	0.185212	+	0.982699i	$0.440703\pi$
$270$	26.9890	1.64250
$271$	16.5306	1.00416	0.502082	−	0.864820i	$-0.332568\pi$
$271$	16.5306	1.00416	0.502082	+	0.864820i	$0.332568\pi$
$272$	−0.0420886	−0.00255200
$273$	0.0522489	0.00316225
$274$	−42.5415	−2.57002
$275$	0.102578	0.00618570
$276$	−62.1437	−3.74061
$277$	3.99850	0.240247	0.120123	−	0.992759i	$-0.461671\pi$
$277$	3.99850	0.240247	0.120123	+	0.992759i	$0.461671\pi$
$278$	−18.0734	−1.08397
$279$	−15.8953	−0.951628
$280$	0.0535846	0.00320229
$281$	−0.886828	−0.0529037	−0.0264519	−	0.999650i	$-0.508421\pi$
$281$	−0.886828	−0.0529037	−0.0264519	+	0.999650i	$0.508421\pi$
$282$	−36.8216	−2.19270
$283$	14.3953	0.855710	0.427855	−	0.903847i	$-0.359269\pi$
$283$	14.3953	0.855710	0.427855	+	0.903847i	$0.359269\pi$
$284$	16.7985	0.996809
$285$	−6.85027	−0.405775
$286$	−0.576538	−0.0340914
$287$	0.0336807	0.00198811
$288$	24.3631	1.43561
$289$	−16.9894	−0.999375
$290$	−6.78437	−0.398392
$291$	−37.6051	−2.20445
$292$	27.2370	1.59393
$293$	−0.185756	−0.0108520	−0.00542599	−	0.999985i	$-0.501727\pi$
$293$	−0.185756	−0.0108520	−0.00542599	+	0.999985i	$0.501727\pi$
$294$	−44.8466	−2.61550
$295$	−37.2411	−2.16826
$296$	6.73684	0.391571
$297$	−0.446623	−0.0259157
$298$	33.6932	1.95180
$299$	17.8452	1.03202
$300$	−10.0503	−0.580255
$301$	−0.00201638	−0.000116222 0
$302$	9.47889	0.545449
$303$	−5.20698	−0.299133
$304$	0.408469	0.0234273
$305$	15.8017	0.904805
$306$	1.11933	0.0639877
$307$	−5.93099	−0.338500	−0.169250	−	0.985573i	$-0.554134\pi$
$307$	−5.93099	−0.338500	−0.169250	+	0.985573i	$0.554134\pi$
$308$	−0.00222458	−0.000126757 0
$309$	47.1061	2.67977
$310$	19.2285	1.09211
$311$	24.6039	1.39516	0.697580	−	0.716506i	$-0.254260\pi$
$311$	24.6039	1.39516	0.697580	+	0.716506i	$0.254260\pi$
$312$	22.5163	1.27473
$313$	−17.1223	−0.967808	−0.483904	−	0.875121i	$-0.660782\pi$
$313$	−17.1223	−0.967808	−0.483904	+	0.875121i	$0.660782\pi$
$314$	−32.1043	−1.81175
$315$	−0.0824528	−0.00464569
$316$	−44.1206	−2.48198
$317$	−32.2130	−1.80926	−0.904630	−	0.426197i	$-0.859853\pi$
$317$	−32.2130	−1.80926	−0.904630	+	0.426197i	$0.859853\pi$
$318$	−51.8399	−2.90704
$319$	0.112270	0.00628591
$320$	−31.4878	−1.76022
$321$	50.1135	2.79706
$322$	0.110266	0.00614488
$323$	−0.103040	−0.00573330
$324$	−3.20455	−0.178030
$325$	2.88606	0.160090
$326$	15.5240	0.859793
$327$	−33.6995	−1.86359
$328$	14.5144	0.801426
$329$	0.0407990	0.00224932
$330$	1.48967	0.0820034
$331$	−20.9500	−1.15152	−0.575759	−	0.817619i	$-0.695293\pi$
$331$	−20.9500	−1.15152	−0.575759	+	0.817619i	$0.695293\pi$
$332$	8.35557	0.458571
$333$	−10.3663	−0.568067
$334$	27.5050	1.50501
$335$	−30.6506	−1.67462
$336$	0.00804987	0.000439157 0
$337$	17.7372	0.966205	0.483102	−	0.875564i	$-0.339510\pi$
$337$	17.7372	0.966205	0.483102	+	0.875564i	$0.339510\pi$
$338$	13.7795	0.749504
$339$	−29.3240	−1.59266
$340$	−0.845543	−0.0458560
$341$	−0.318199	−0.0172315
$342$	−10.8630	−0.587406
$343$	0.0993819	0.00536612
$344$	−0.868947	−0.0468505
$345$	−46.1087	−2.48241
$346$	−8.89257	−0.478068
$347$	−22.3755	−1.20118	−0.600589	−	0.799558i	$-0.705067\pi$
$347$	−22.3755	−1.20118	−0.600589	+	0.799558i	$0.705067\pi$
$348$	−10.9999	−0.589656
$349$	−20.4619	−1.09530	−0.547651	−	0.836707i	$-0.684478\pi$
$349$	−20.4619	−1.09530	−0.547651	+	0.836707i	$0.684478\pi$
$350$	0.0178330	0.000953214 0
$351$	−12.5658	−0.670715
$352$	0.487711	0.0259951
$353$	1.97517	0.105128	0.0525640	−	0.998618i	$-0.483261\pi$
$353$	1.97517	0.105128	0.0525640	+	0.998618i	$0.483261\pi$
$354$	−96.6939	−5.13922
$355$	12.4640	0.661521
$356$	−49.8082	−2.63983
$357$	−0.00203066	−0.000107474 0
$358$	−23.7170	−1.25348
$359$	−5.28651	−0.279011	−0.139506	−	0.990221i	$-0.544551\pi$
$359$	−5.28651	−0.279011	−0.139506	+	0.990221i	$0.544551\pi$
$360$	−35.5325	−1.87273
$361$	1.00000	0.0526316
$362$	−21.6125	−1.13593
$363$	30.5135	1.60154
$364$	−0.0625891	−0.00328056
$365$	20.2091	1.05779
$366$	41.0281	2.14457
$367$	−8.17698	−0.426835	−0.213418	−	0.976961i	$-0.568459\pi$
$367$	−8.17698	−0.426835	−0.213418	+	0.976961i	$0.568459\pi$
$368$	2.74938	0.143321
$369$	−22.3340	−1.16266
$370$	12.5400	0.651924
$371$	0.0574395	0.00298211
$372$	31.1762	1.61641
$373$	32.5326	1.68447	0.842236	−	0.539108i	$-0.181239\pi$
$373$	32.5326	1.68447	0.842236	+	0.539108i	$0.181239\pi$
$374$	0.0224072	0.00115865
$375$	26.7943	1.38365
$376$	17.5821	0.906726
$377$	3.15874	0.162683
$378$	−0.0776444	−0.00399360
$379$	−21.6289	−1.11100	−0.555501	−	0.831516i	$-0.687473\pi$
$379$	−21.6289	−1.11100	−0.555501	+	0.831516i	$0.687473\pi$
$380$	8.20596	0.420957
$381$	−57.2166	−2.93129
$382$	31.6199	1.61782
$383$	−22.0712	−1.12779	−0.563893	−	0.825848i	$-0.690697\pi$
$383$	−22.0712	−1.12779	−0.563893	+	0.825848i	$0.690697\pi$
$384$	−53.0184	−2.70558
$385$	−0.00165058	−8.41211e−5 0
$386$	24.3663	1.24021
$387$	1.33709	0.0679679
$388$	45.0473	2.28693
$389$	0.268461	0.0136115	0.00680575	−	0.999977i	$-0.497834\pi$
$389$	0.268461	0.0136115	0.00680575	+	0.999977i	$0.497834\pi$
$390$	41.9120	2.12230
$391$	−0.693557	−0.0350747
$392$	21.4139	1.08157
$393$	24.4446	1.23306
$394$	−52.4366	−2.64172
$395$	−32.7362	−1.64713
$396$	1.47514	0.0741288
$397$	22.7082	1.13969	0.569845	−	0.821752i	$-0.307003\pi$
$397$	22.7082	1.13969	0.569845	+	0.821752i	$0.307003\pi$
$398$	−3.70065	−0.185497
$399$	0.0197074	0.000986606 0
$400$	0.444649	0.0222325
$401$	3.20323	0.159962	0.0799808	−	0.996796i	$-0.474514\pi$
$401$	3.20323	0.159962	0.0799808	+	0.996796i	$0.474514\pi$
$402$	−79.5822	−3.96920
$403$	−8.95260	−0.445961
$404$	6.23746	0.310325
$405$	−2.37768	−0.118148
$406$	0.0195179	0.000968656 0
$407$	−0.207516	−0.0102862
$408$	−0.875098	−0.0433238
$409$	−3.19210	−0.157839	−0.0789195	−	0.996881i	$-0.525147\pi$
$409$	−3.19210	−0.157839	−0.0789195	+	0.996881i	$0.525147\pi$
$410$	27.0173	1.33429
$411$	−51.1773	−2.52439
$412$	−56.4285	−2.78003
$413$	0.107139	0.00527194
$414$	−73.1185	−3.59358
$415$	6.19958	0.304325
$416$	13.7219	0.672770
$417$	−21.7422	−1.06472
$418$	−0.217461	−0.0106364
$419$	−11.0482	−0.539742	−0.269871	−	0.962897i	$-0.586981\pi$
$419$	−11.0482	−0.539742	−0.269871	+	0.962897i	$0.586981\pi$
$420$	0.161719	0.00789106
$421$	19.8039	0.965185	0.482592	−	0.875845i	$-0.339695\pi$
$421$	19.8039	0.965185	0.482592	+	0.875845i	$0.339695\pi$
$422$	−2.30773	−0.112338
$423$	−27.0542	−1.31542
$424$	24.7532	1.20212
$425$	−0.112167	−0.00544090
$426$	32.3619	1.56794
$427$	−0.0454599	−0.00219996
$428$	−60.0311	−2.90171
$429$	−0.693574	−0.0334861
$430$	−1.61747	−0.0780011
$431$	−17.0291	−0.820261	−0.410131	−	0.912027i	$-0.634517\pi$
$431$	−17.0291	−0.820261	−0.410131	+	0.912027i	$0.634517\pi$
$432$	−1.93599	−0.0931454
$433$	−3.50404	−0.168394	−0.0841968	−	0.996449i	$-0.526832\pi$
$433$	−3.50404	−0.168394	−0.0841968	+	0.996449i	$0.526832\pi$
$434$	−0.0553182	−0.00265536
$435$	−8.16159	−0.391318
$436$	40.3688	1.93331
$437$	6.73094	0.321985
$438$	52.4714	2.50718
$439$	34.1640	1.63056	0.815279	−	0.579069i	$-0.196584\pi$
$439$	34.1640	1.63056	0.815279	+	0.579069i	$0.196584\pi$
$440$	−0.711304	−0.0339101
$441$	−32.9505	−1.56907
$442$	0.630431	0.0299865
$443$	3.97717	0.188961	0.0944804	−	0.995527i	$-0.469881\pi$
$443$	3.97717	0.188961	0.0944804	+	0.995527i	$0.469881\pi$
$444$	20.3318	0.964905
$445$	−36.9562	−1.75189
$446$	−9.27105	−0.438997
$447$	40.5329	1.91714
$448$	0.0905868	0.00427982
$449$	12.1845	0.575020	0.287510	−	0.957778i	$-0.407172\pi$
$449$	12.1845	0.575020	0.287510	+	0.957778i	$0.407172\pi$
$450$	−11.8252	−0.557447
$451$	−0.447091	−0.0210527
$452$	35.1273	1.65225
$453$	11.4031	0.535764
$454$	34.6530	1.62635
$455$	−0.0464393	−0.00217711
$456$	8.49279	0.397711
$457$	−9.15968	−0.428472	−0.214236	−	0.976782i	$-0.568726\pi$
$457$	−9.15968	−0.428472	−0.214236	+	0.976782i	$0.568726\pi$
$458$	13.7391	0.641988
$459$	0.488372	0.0227953
$460$	55.2338	2.57529
$461$	−5.04330	−0.234890	−0.117445	−	0.993079i	$-0.537470\pi$
$461$	−5.04330	−0.234890	−0.117445	+	0.993079i	$0.537470\pi$
$462$	−0.00428560	−0.000199384 0
$463$	−14.5978	−0.678415	−0.339208	−	0.940712i	$-0.610159\pi$
$463$	−14.5978	−0.678415	−0.339208	+	0.940712i	$0.610159\pi$
$464$	0.486660	0.0225926
$465$	23.1318	1.07271
$466$	40.9949	1.89905
$467$	−19.8244	−0.917366	−0.458683	−	0.888600i	$-0.651679\pi$
$467$	−19.8244	−0.917366	−0.458683	+	0.888600i	$0.651679\pi$
$468$	41.5035	1.91850
$469$	0.0881784	0.00407170
$470$	32.7274	1.50960
$471$	−38.6214	−1.77958
$472$	46.1706	2.12518
$473$	0.0267663	0.00123072
$474$	−84.9971	−3.90405
$475$	1.08858	0.0499473
$476$	0.00243253	0.000111495 0
$477$	−38.0887	−1.74396
$478$	−17.9788	−0.822329
$479$	−31.5237	−1.44036	−0.720178	−	0.693789i	$-0.755940\pi$
$479$	−31.5237	−1.44036	−0.720178	+	0.693789i	$0.755940\pi$
$480$	−35.4547	−1.61828
$481$	−5.83851	−0.266213
$482$	−35.7427	−1.62804
$483$	0.132650	0.00603577
$484$	−36.5522	−1.66146
$485$	33.4237	1.51769
$486$	−38.9868	−1.76848
$487$	−10.6169	−0.481097	−0.240548	−	0.970637i	$-0.577327\pi$
$487$	−10.6169	−0.481097	−0.240548	+	0.970637i	$0.577327\pi$
$488$	−19.5906	−0.886826
$489$	18.6753	0.844526
$490$	39.8600	1.80069
$491$	0.124979	0.00564020	0.00282010	−	0.999996i	$-0.499102\pi$
$491$	0.124979	0.00564020	0.00282010	+	0.999996i	$0.499102\pi$
$492$	43.8047	1.97487
$493$	−0.122765	−0.00552904
$494$	−6.11831	−0.275276
$495$	1.09451	0.0491947
$496$	−1.37931	−0.0619328
$497$	−0.0358576	−0.00160843
$498$	16.0968	0.721313
$499$	37.5379	1.68043	0.840214	−	0.542255i	$-0.182429\pi$
$499$	37.5379	1.68043	0.840214	+	0.542255i	$0.182429\pi$
$500$	−32.0970	−1.43542
$501$	33.0885	1.47828
$502$	69.5619	3.10470
$503$	−6.71478	−0.299397	−0.149699	−	0.988732i	$-0.547830\pi$
$503$	−6.71478	−0.299397	−0.149699	+	0.988732i	$0.547830\pi$
$504$	0.102223	0.00455337
$505$	4.62801	0.205944
$506$	−1.46372	−0.0650702
$507$	16.5767	0.736195
$508$	68.5399	3.04097
$509$	1.79275	0.0794623	0.0397311	−	0.999210i	$-0.487350\pi$
$509$	1.79275	0.0794623	0.0397311	+	0.999210i	$0.487350\pi$
$510$	−1.62891	−0.0721295
$511$	−0.0581392	−0.00257193
$512$	4.61323	0.203878
$513$	−4.73963	−0.209260
$514$	−31.5934	−1.39353
$515$	−41.8683	−1.84494
$516$	−2.62249	−0.115449
$517$	−0.541583	−0.0238188
$518$	−0.0360762	−0.00158510
$519$	−10.6977	−0.469579
$520$	−20.0127	−0.877614
$521$	−44.5121	−1.95011	−0.975056	−	0.221959i	$-0.928755\pi$
$521$	−44.5121	−1.95011	−0.975056	+	0.221959i	$0.928755\pi$
$522$	−12.9425	−0.566478
$523$	−25.7186	−1.12460	−0.562299	−	0.826934i	$-0.690083\pi$
$523$	−25.7186	−1.12460	−0.562299	+	0.826934i	$0.690083\pi$
$524$	−29.2822	−1.27920
$525$	0.0214531	0.000936288 0
$526$	13.4440	0.586184
$527$	0.347944	0.0151567
$528$	−0.106857	−0.00465037
$529$	22.3056	0.969808
$530$	46.0758	2.00140
$531$	−71.0447	−3.08308
$532$	−0.0236076	−0.00102352
$533$	−12.5790	−0.544857
$534$	−95.9541	−4.15234
$535$	−44.5413	−1.92569
$536$	37.9999	1.64135
$537$	−28.5315	−1.23123
$538$	−14.0204	−0.604463
$539$	−0.659616	−0.0284117
$540$	−38.8932	−1.67370
$541$	−0.404887	−0.0174074	−0.00870372	−	0.999962i	$-0.502771\pi$
$541$	−0.404887	−0.0174074	−0.00870372	+	0.999962i	$0.502771\pi$
$542$	−38.1482	−1.63860
$543$	−25.9998	−1.11576
$544$	−0.533301	−0.0228651
$545$	29.9524	1.28302
$546$	−0.120576	−0.00516019
$547$	11.5896	0.495537	0.247769	−	0.968819i	$-0.420303\pi$
$547$	11.5896	0.495537	0.247769	+	0.968819i	$0.420303\pi$
$548$	61.3055	2.61884
$549$	30.1449	1.28655
$550$	−0.236723	−0.0100939
$551$	1.19143	0.0507565
$552$	57.1645	2.43308
$553$	0.0941783	0.00400487
$554$	−9.22745	−0.392037
$555$	15.0856	0.640348
$556$	26.0451	1.10456
$557$	−36.2871	−1.53754	−0.768768	−	0.639528i	$-0.779130\pi$
$557$	−36.2871	−1.53754	−0.768768	+	0.639528i	$0.779130\pi$
$558$	36.6821	1.55288
$559$	0.753076	0.0318517
$560$	−0.00715480	−0.000302345 0
$561$	0.0269558	0.00113807
$562$	2.04656	0.0863288
$563$	−2.38673	−0.100589	−0.0502943	−	0.998734i	$-0.516016\pi$
$563$	−2.38673	−0.100589	−0.0502943	+	0.998734i	$0.516016\pi$
$564$	53.0628	2.23435
$565$	26.0634	1.09650
$566$	−33.2204	−1.39636
$567$	0.00684032	0.000287266 0
$568$	−15.4526	−0.648375
$569$	−18.6287	−0.780957	−0.390478	−	0.920612i	$-0.627690\pi$
$569$	−18.6287	−0.780957	−0.390478	+	0.920612i	$0.627690\pi$
$570$	15.8086	0.662147
$571$	5.68741	0.238011	0.119005	−	0.992894i	$-0.462029\pi$
$571$	5.68741	0.238011	0.119005	+	0.992894i	$0.462029\pi$
$572$	0.830835	0.0347389
$573$	38.0387	1.58909
$574$	−0.0777258	−0.00324421
$575$	7.32714	0.305563
$576$	−60.0690	−2.50288
$577$	4.53220	0.188678	0.0943391	−	0.995540i	$-0.469926\pi$
$577$	4.53220	0.188678	0.0943391	+	0.995540i	$0.469926\pi$
$578$	39.2069	1.63079
$579$	29.3126	1.21819
$580$	9.77680	0.405959
$581$	−0.0178355	−0.000739941 0
$582$	86.7823	3.59724
$583$	−0.762477	−0.0315786
$584$	−25.0547	−1.03677
$585$	30.7943	1.27319
$586$	0.428675	0.0177084
$587$	−16.6321	−0.686482	−0.343241	−	0.939247i	$-0.611525\pi$
$587$	−16.6321	−0.686482	−0.343241	+	0.939247i	$0.611525\pi$
$588$	64.6273	2.66518
$589$	−3.37678	−0.139138
$590$	85.9424	3.53819
$591$	−63.0811	−2.59481
$592$	−0.899526	−0.0369703
$593$	−0.445303	−0.0182864	−0.00914321	−	0.999958i	$-0.502910\pi$
$593$	−0.445303	−0.0182864	−0.00914321	+	0.999958i	$0.502910\pi$
$594$	1.03068	0.0422895
$595$	0.00180487	7.39923e−5 0
$596$	−48.5545	−1.98887
$597$	−4.45187	−0.182203
$598$	−41.1820	−1.68406
$599$	−26.0030	−1.06245	−0.531226	−	0.847230i	$-0.678269\pi$
$599$	−26.0030	−1.06245	−0.531226	+	0.847230i	$0.678269\pi$
$600$	9.24505	0.377428
$601$	−7.63101	−0.311275	−0.155638	−	0.987814i	$-0.549743\pi$
$601$	−7.63101	−0.311275	−0.155638	+	0.987814i	$0.549743\pi$
$602$	0.00465327	0.000189653 0
$603$	−58.4720	−2.38116
$604$	−13.6598	−0.555810
$605$	−27.1206	−1.10261
$606$	12.0163	0.488128
$607$	−12.9441	−0.525387	−0.262693	−	0.964879i	$-0.584611\pi$
$607$	−12.9441	−0.525387	−0.262693	+	0.964879i	$0.584611\pi$
$608$	5.17567	0.209901
$609$	0.0234800	0.000951456 0
$610$	−36.4661	−1.47647
$611$	−15.2376	−0.616446
$612$	−1.61304	−0.0652031
$613$	28.5776	1.15424	0.577119	−	0.816660i	$-0.304177\pi$
$613$	28.5776	1.15424	0.577119	+	0.816660i	$0.304177\pi$
$614$	13.6871	0.552367
$615$	32.5018	1.31060
$616$	0.00204634	8.24495e−5 0
$617$	11.6101	0.467406	0.233703	−	0.972308i	$-0.424916\pi$
$617$	11.6101	0.467406	0.233703	+	0.972308i	$0.424916\pi$
$618$	−108.708	−4.37288
$619$	6.03758	0.242671	0.121335	−	0.992612i	$-0.461282\pi$
$619$	6.03758	0.242671	0.121335	+	0.992612i	$0.461282\pi$
$620$	−27.7097	−1.11285
$621$	−31.9022	−1.28019
$622$	−56.7792	−2.27664
$623$	0.106319	0.00425957
$624$	−3.00646	−0.120355
$625$	−29.2579	−1.17032
$626$	39.5136	1.57928
$627$	−0.261605	−0.0104475
$628$	46.2648	1.84617
$629$	0.226914	0.00904765
$630$	0.190279	0.00758088
$631$	24.0925	0.959106	0.479553	−	0.877513i	$-0.340799\pi$
$631$	24.0925	0.959106	0.479553	+	0.877513i	$0.340799\pi$
$632$	40.5855	1.61440
$633$	−2.77619	−0.110344
$634$	74.3388	2.95237
$635$	50.8546	2.01810
$636$	74.7052	2.96226
$637$	−18.5584	−0.735312
$638$	−0.259089	−0.0102574
$639$	23.7775	0.940624
$640$	47.1232	1.86271
$641$	23.1153	0.913001	0.456501	−	0.889723i	$-0.349103\pi$
$641$	23.1153	0.913001	0.456501	+	0.889723i	$0.349103\pi$
$642$	−115.648	−4.56427
$643$	20.4235	0.805423	0.402712	−	0.915327i	$-0.368068\pi$
$643$	20.4235	0.805423	0.402712	+	0.915327i	$0.368068\pi$
$644$	−0.158902	−0.00626160
$645$	−1.94581	−0.0766161
$646$	0.237788	0.00935566
$647$	39.3756	1.54802	0.774008	−	0.633175i	$-0.218249\pi$
$647$	39.3756	1.54802	0.774008	+	0.633175i	$0.218249\pi$
$648$	2.94779	0.115800
$649$	−1.42220	−0.0558263
$650$	−6.66024	−0.261236
$651$	−0.0665477	−0.00260821
$652$	−22.3712	−0.876125
$653$	40.6583	1.59108	0.795542	−	0.605899i	$-0.207186\pi$
$653$	40.6583	1.59108	0.795542	+	0.605899i	$0.207186\pi$
$654$	77.7694	3.04102
$655$	−21.7265	−0.848926
$656$	−1.93802	−0.0756670
$657$	38.5527	1.50408
$658$	−0.0941531	−0.00367047
$659$	2.19351	0.0854472	0.0427236	−	0.999087i	$-0.486397\pi$
$659$	2.19351	0.0854472	0.0427236	+	0.999087i	$0.486397\pi$
$660$	−2.14672	−0.0835610
$661$	−37.3415	−1.45242	−0.726208	−	0.687475i	$-0.758719\pi$
$661$	−37.3415	−1.45242	−0.726208	+	0.687475i	$0.758719\pi$
$662$	48.3470	1.87906
$663$	0.758407	0.0294541
$664$	−7.68609	−0.298278
$665$	−0.0175162	−0.000679247 0
$666$	23.9225	0.926978
$667$	8.01942	0.310513
$668$	−39.6368	−1.53359
$669$	−11.1530	−0.431202
$670$	70.7333	2.73267
$671$	0.603453	0.0232961
$672$	0.101999	0.00393471
$673$	−1.01672	−0.0391918	−0.0195959	−	0.999808i	$-0.506238\pi$
$673$	−1.01672	−0.0391918	−0.0195959	+	0.999808i	$0.506238\pi$
$674$	−40.9326	−1.57666
$675$	−5.15945	−0.198587
$676$	−19.8572	−0.763740
$677$	28.9019	1.11079	0.555395	−	0.831586i	$-0.312567\pi$
$677$	28.9019	1.11079	0.555395	+	0.831586i	$0.312567\pi$
$678$	67.6718	2.59892
$679$	−0.0961563	−0.00369014
$680$	0.777795	0.0298271
$681$	41.6875	1.59747
$682$	0.734318	0.0281185
$683$	−10.9535	−0.419123	−0.209561	−	0.977795i	$-0.567204\pi$
$683$	−10.9535	−0.419123	−0.209561	+	0.977795i	$0.567204\pi$
$684$	15.6545	0.598563
$685$	45.4868	1.73796
$686$	−0.229346	−0.00875649
$687$	16.5282	0.630588
$688$	0.116025	0.00442341
$689$	−21.4524	−0.817273
$690$	106.406	4.05082
$691$	−1.00609	−0.0382735	−0.0191368	−	0.999817i	$-0.506092\pi$
$691$	−1.00609	−0.0382735	−0.0191368	+	0.999817i	$0.506092\pi$
$692$	12.8149	0.487148
$693$	−0.00314879	−0.000119613 0
$694$	51.6365	1.96009
$695$	19.3247	0.733027
$696$	10.1185	0.383542
$697$	0.488884	0.0185178
$698$	47.2206	1.78733
$699$	49.3168	1.86533
$700$	−0.0256987	−0.000971320 0
$701$	−30.8253	−1.16426	−0.582128	−	0.813097i	$-0.697780\pi$
$701$	−30.8253	−1.16426	−0.582128	+	0.813097i	$0.697780\pi$
$702$	28.9985	1.09448
$703$	−2.20219	−0.0830572
$704$	−1.20249	−0.0453204
$705$	39.3710	1.48280
$706$	−4.55817	−0.171549
$707$	−0.0133143	−0.000500734 0
$708$	139.343	5.23684
$709$	−29.4238	−1.10503	−0.552517	−	0.833501i	$-0.686333\pi$
$709$	−29.4238	−1.10503	−0.552517	+	0.833501i	$0.686333\pi$
$710$	−28.7635	−1.07948
$711$	−62.4506	−2.34208
$712$	45.8174	1.71708
$713$	−22.7289	−0.851204
$714$	0.00468620	0.000175377 0
$715$	0.616454	0.0230541
$716$	34.1781	1.27729
$717$	−21.6284	−0.807727
$718$	12.1998	0.455294
$719$	16.7108	0.623207	0.311604	−	0.950212i	$-0.399134\pi$
$719$	16.7108	0.623207	0.311604	+	0.950212i	$0.399134\pi$
$720$	4.74442	0.176814
$721$	0.120450	0.00448581
$722$	−2.30773	−0.0858848
$723$	−42.9984	−1.59913
$724$	31.1453	1.15751
$725$	1.29696	0.0481678
$726$	−70.4168	−2.61341
$727$	−32.8965	−1.22007	−0.610033	−	0.792376i	$-0.708844\pi$
$727$	−32.8965	−1.22007	−0.610033	+	0.792376i	$0.708844\pi$
$728$	0.0575743	0.00213384
$729$	−44.0103	−1.63001
$730$	−46.6370	−1.72611
$731$	−0.0292684	−0.00108253
$732$	−59.1246	−2.18531
$733$	−1.18341	−0.0437101	−0.0218550	−	0.999761i	$-0.506957\pi$
$733$	−1.18341	−0.0437101	−0.0218550	+	0.999761i	$0.506957\pi$
$734$	18.8703	0.696514
$735$	47.9515	1.76872
$736$	34.8371	1.28411
$737$	−1.17052	−0.0431166
$738$	51.5408	1.89724
$739$	38.2843	1.40831	0.704155	−	0.710046i	$-0.251326\pi$
$739$	38.2843	1.40831	0.704155	+	0.710046i	$0.251326\pi$
$740$	−18.0711	−0.664307
$741$	−7.36031	−0.270388
$742$	−0.132555	−0.00486624
$743$	4.13006	0.151517	0.0757586	−	0.997126i	$-0.475862\pi$
$743$	4.13006	0.151517	0.0757586	+	0.997126i	$0.475862\pi$
$744$	−28.6783	−1.05140
$745$	−36.0260	−1.31989
$746$	−75.0764	−2.74874
$747$	11.8269	0.432724
$748$	−0.0322905	−0.00118066
$749$	0.128140	0.00468215
$750$	−61.8340	−2.25786
$751$	39.9230	1.45681	0.728406	−	0.685146i	$-0.240261\pi$
$751$	39.9230	1.45681	0.728406	+	0.685146i	$0.240261\pi$
$752$	−2.34762	−0.0856088
$753$	83.6827	3.04957
$754$	−7.28951	−0.265468
$755$	−10.1352	−0.368856
$756$	0.111891	0.00406945
$757$	40.7776	1.48209	0.741044	−	0.671456i	$-0.234331\pi$
$757$	40.7776	1.48209	0.741044	+	0.671456i	$0.234331\pi$
$758$	49.9136	1.81294
$759$	−1.76085	−0.0639147
$760$	−7.54847	−0.273812
$761$	−2.37731	−0.0861775	−0.0430887	−	0.999071i	$-0.513720\pi$
$761$	−2.37731	−0.0861775	−0.0430887	+	0.999071i	$0.513720\pi$
$762$	132.040	4.78332
$763$	−0.0861698	−0.00311956
$764$	−45.5667	−1.64854
$765$	−1.19682	−0.0432713
$766$	50.9344	1.84033
$767$	−40.0140	−1.44482
$768$	51.4983	1.85828
$769$	−43.4133	−1.56553	−0.782763	−	0.622320i	$-0.786190\pi$
$769$	−43.4133	−1.56553	−0.782763	+	0.622320i	$0.786190\pi$
$770$	0.00380908	0.000137270 0
$771$	−38.0068	−1.36878
$772$	−35.1136	−1.26377
$773$	47.8041	1.71939	0.859697	−	0.510805i	$-0.170653\pi$
$773$	47.8041	1.71939	0.859697	+	0.510805i	$0.170653\pi$
$774$	−3.08563	−0.110911
$775$	−3.67588	−0.132042
$776$	−41.4379	−1.48753
$777$	−0.0433996	−0.00155695
$778$	−0.619534	−0.0222114
$779$	−4.74460	−0.169993
$780$	−60.3984	−2.16261
$781$	0.475988	0.0170322
$782$	1.60054	0.0572352
$783$	−5.64692	−0.201805
$784$	−2.85926	−0.102116
$785$	34.3271	1.22519
$786$	−56.4114	−2.01213
$787$	3.04033	0.108376	0.0541880	−	0.998531i	$-0.482743\pi$
$787$	3.04033	0.108376	0.0541880	+	0.998531i	$0.482743\pi$
$788$	75.5650	2.69189
$789$	16.1731	0.575776
$790$	75.5462	2.68781
$791$	−0.0749816	−0.00266604
$792$	−1.35695	−0.0482171
$793$	16.9783	0.602916
$794$	−52.4043	−1.85976
$795$	55.4290	1.96587
$796$	5.33291	0.189020
$797$	17.6721	0.625976	0.312988	−	0.949757i	$-0.398670\pi$
$797$	17.6721	0.625976	0.312988	+	0.949757i	$0.398670\pi$
$798$	−0.0454794	−0.00160995
$799$	0.592209	0.0209508
$800$	5.63411	0.199196
$801$	−70.5011	−2.49103
$802$	−7.39218	−0.261027
$803$	0.771764	0.0272350
$804$	114.684	4.04459
$805$	−0.117900	−0.00415544
$806$	20.6602	0.727724
$807$	−16.8665	−0.593730
$808$	−5.73769	−0.201851
$809$	−6.89589	−0.242447	−0.121223	−	0.992625i	$-0.538682\pi$
$809$	−6.89589	−0.242447	−0.121223	+	0.992625i	$0.538682\pi$
$810$	5.48704	0.192795
$811$	23.3139	0.818662	0.409331	−	0.912386i	$-0.365762\pi$
$811$	23.3139	0.818662	0.409331	+	0.912386i	$0.365762\pi$
$812$	−0.0281267	−0.000987055 0
$813$	−45.8922	−1.60951
$814$	0.478891	0.0167851
$815$	−16.5988	−0.581430
$816$	0.116846	0.00409043
$817$	0.284048	0.00993760
$818$	7.36649	0.257563
$819$	−0.0885919	−0.00309565
$820$	−38.9340	−1.35963
$821$	−40.5426	−1.41495	−0.707474	−	0.706740i	$-0.750165\pi$
$821$	−40.5426	−1.41495	−0.707474	+	0.706740i	$0.750165\pi$
$822$	118.103	4.11933
$823$	50.2726	1.75239	0.876196	−	0.481955i	$-0.160073\pi$
$823$	50.2726	1.75239	0.876196	+	0.481955i	$0.160073\pi$
$824$	51.9073	1.80828
$825$	−0.284777	−0.00991466
$826$	−0.247247	−0.00860281
$827$	−35.8507	−1.24665	−0.623326	−	0.781962i	$-0.714219\pi$
$827$	−35.8507	−1.24665	−0.623326	+	0.781962i	$0.714219\pi$
$828$	105.369	3.66184
$829$	34.7719	1.20768	0.603839	−	0.797106i	$-0.293637\pi$
$829$	34.7719	1.20768	0.603839	+	0.797106i	$0.293637\pi$
$830$	−14.3069	−0.496601
$831$	−11.1006	−0.385076
$832$	−33.8322	−1.17292
$833$	0.721275	0.0249907
$834$	50.1751	1.73742
$835$	−29.4093	−1.01775
$836$	0.313378	0.0108384
$837$	16.0047	0.553203
$838$	25.4963	0.880756
$839$	38.1379	1.31667	0.658333	−	0.752726i	$-0.271262\pi$
$839$	38.1379	1.31667	0.658333	+	0.752726i	$0.271262\pi$
$840$	−0.148761	−0.00513274
$841$	−27.5805	−0.951052
$842$	−45.7021	−1.57500
$843$	2.46200	0.0847959
$844$	3.32561	0.114472
$845$	−14.7335	−0.506847
$846$	62.4338	2.14652
$847$	0.0780230	0.00268090
$848$	−3.30513	−0.113499
$849$	−39.9640	−1.37156
$850$	0.258851	0.00887851
$851$	−14.8228	−0.508120
$852$	−46.6359	−1.59772
$853$	−28.5963	−0.979119	−0.489560	−	0.871970i	$-0.662842\pi$
$853$	−28.5963	−0.979119	−0.489560	+	0.871970i	$0.662842\pi$
$854$	0.104909	0.00358991
$855$	11.6151	0.397229
$856$	55.2212	1.88742
$857$	34.4331	1.17621	0.588107	−	0.808783i	$-0.299873\pi$
$857$	34.4331	1.17621	0.588107	+	0.808783i	$0.299873\pi$
$858$	1.60058	0.0546429
$859$	−34.6922	−1.18368	−0.591841	−	0.806055i	$-0.701599\pi$
$859$	−34.6922	−1.18368	−0.591841	+	0.806055i	$0.701599\pi$
$860$	2.33089	0.0794827
$861$	−0.0935040	−0.00318661
$862$	39.2985	1.33851
$863$	−52.4408	−1.78510	−0.892552	−	0.450944i	$-0.851088\pi$
$863$	−52.4408	−1.78510	−0.892552	+	0.450944i	$0.851088\pi$
$864$	−24.5308	−0.834554
$865$	9.50825	0.323290
$866$	8.08638	0.274787
$867$	47.1658	1.60183
$868$	0.0797178	0.00270580
$869$	−1.25016	−0.0424088
$870$	18.8347	0.638557
$871$	−32.9328	−1.11588
$872$	−37.1343	−1.25753
$873$	63.7622	2.15802
$874$	−15.5332	−0.525418
$875$	0.0685131	0.00231617
$876$	−75.6152	−2.55480
$877$	−35.1235	−1.18604	−0.593019	−	0.805189i	$-0.702064\pi$
$877$	−35.1235	−1.18604	−0.593019	+	0.805189i	$0.702064\pi$
$878$	−78.8412	−2.66076
$879$	0.515694	0.0173939
$880$	0.0949758	0.00320163
$881$	37.2891	1.25630	0.628151	−	0.778091i	$-0.283812\pi$
$881$	37.2891	1.25630	0.628151	+	0.778091i	$0.283812\pi$
$882$	76.0407	2.56042
$883$	14.7557	0.496569	0.248284	−	0.968687i	$-0.420133\pi$
$883$	14.7557	0.496569	0.248284	+	0.968687i	$0.420133\pi$
$884$	−0.908498	−0.0305561
$885$	103.389	3.47537
$886$	−9.17822	−0.308348
$887$	−4.52761	−0.152022	−0.0760111	−	0.997107i	$-0.524218\pi$
$887$	−4.52761	−0.152022	−0.0760111	+	0.997107i	$0.524218\pi$
$888$	−18.7028	−0.627623
$889$	−0.146303	−0.00490684
$890$	85.2848	2.85875
$891$	−0.0908012	−0.00304196
$892$	13.3603	0.447335
$893$	−5.74737	−0.192328
$894$	−93.5389	−3.12841
$895$	25.3591	0.847661
$896$	−0.135568	−0.00452902
$897$	−49.5418	−1.65415
$898$	−28.1184	−0.938324
$899$	−4.02318	−0.134181
$900$	17.0411	0.568036
$901$	0.833750	0.0277763
$902$	1.03177	0.0343540
$903$	0.00559787	0.000186285 0
$904$	−32.3128	−1.07471
$905$	23.1089	0.768165
$906$	−26.3152	−0.874265
$907$	−41.5394	−1.37929	−0.689646	−	0.724147i	$-0.742234\pi$
$907$	−41.5394	−1.37929	−0.689646	+	0.724147i	$0.742234\pi$
$908$	−49.9376	−1.65724
$909$	8.82882	0.292833
$910$	0.107169	0.00355263
$911$	−4.23885	−0.140439	−0.0702197	−	0.997532i	$-0.522370\pi$
$911$	−4.23885	−0.140439	−0.0702197	+	0.997532i	$0.522370\pi$
$912$	−1.13399	−0.0375501
$913$	0.236756	0.00783548
$914$	21.1380	0.699185
$915$	−43.8687	−1.45025
$916$	−19.7991	−0.654182
$917$	0.0625048	0.00206409
$918$	−1.12703	−0.0371975
$919$	−52.8433	−1.74314	−0.871570	−	0.490270i	$-0.836898\pi$
$919$	−52.8433	−1.74314	−0.871570	+	0.490270i	$0.836898\pi$
$920$	−50.8083	−1.67510
$921$	16.4656	0.542559
$922$	11.6386	0.383296
$923$	13.3920	0.440804
$924$	0.00617587	0.000203171 0
$925$	−2.39725	−0.0788212
$926$	33.6877	1.10705
$927$	−79.8719	−2.62334
$928$	6.16643	0.202423
$929$	−27.6191	−0.906152	−0.453076	−	0.891472i	$-0.649673\pi$
$929$	−27.6191	−0.906152	−0.453076	+	0.891472i	$0.649673\pi$
$930$	−53.3820	−1.75046
$931$	−6.99995	−0.229414
$932$	−59.0768	−1.93512
$933$	−68.3052	−2.23621
$934$	45.7494	1.49697
$935$	−0.0239585	−0.000783528 0
$936$	−38.1781	−1.24789
$937$	38.6693	1.26327	0.631635	−	0.775266i	$-0.282384\pi$
$937$	38.6693	1.26327	0.631635	+	0.775266i	$0.282384\pi$
$938$	−0.203492	−0.00664425
$939$	47.5347	1.55124
$940$	−47.1626	−1.53828
$941$	27.5414	0.897824	0.448912	−	0.893576i	$-0.351812\pi$
$941$	27.5414	0.897824	0.448912	+	0.893576i	$0.351812\pi$
$942$	89.1278	2.90394
$943$	−31.9356	−1.03997
$944$	−6.16487	−0.200649
$945$	0.0830201	0.00270065
$946$	−0.0617695	−0.00200830
$947$	−11.3735	−0.369589	−0.184794	−	0.982777i	$-0.559162\pi$
$947$	−11.3735	−0.369589	−0.184794	+	0.982777i	$0.559162\pi$
$948$	122.487	3.97820
$949$	21.7137	0.704858
$950$	−2.51214	−0.0815045
$951$	89.4294	2.89995
$952$	−0.00223763	−7.25220e−5 0
$953$	−36.7727	−1.19118	−0.595592	−	0.803287i	$-0.703083\pi$
$953$	−36.7727	−1.19118	−0.595592	+	0.803287i	$0.703083\pi$
$954$	87.8985	2.84582
$955$	−33.8091	−1.09404
$956$	25.9087	0.837948
$957$	−0.311683	−0.0100753
$958$	72.7482	2.35039
$959$	−0.130861	−0.00422571
$960$	87.4161	2.82134
$961$	−19.5974	−0.632173
$962$	13.4737	0.434409
$963$	−84.9712	−2.73816
$964$	51.5080	1.65896
$965$	−26.0533	−0.838684
$966$	−0.306120	−0.00984923
$967$	−53.0647	−1.70645	−0.853223	−	0.521546i	$-0.825355\pi$
$967$	−53.0647	−1.70645	−0.853223	+	0.521546i	$0.825355\pi$
$968$	33.6235	1.08070
$969$	0.286059	0.00918954
$970$	−77.1329	−2.47659
$971$	13.2999	0.426814	0.213407	−	0.976963i	$-0.431544\pi$
$971$	13.2999	0.426814	0.213407	+	0.976963i	$0.431544\pi$
$972$	56.1830	1.80207
$973$	−0.0555949	−0.00178229
$974$	24.5009	0.785059
$975$	−8.01226	−0.256598
$976$	2.61581	0.0837300
$977$	−45.5049	−1.45583	−0.727915	−	0.685667i	$-0.759511\pi$
$977$	−45.5049	−1.45583	−0.727915	+	0.685667i	$0.759511\pi$
$978$	−43.0975	−1.37811
$979$	−1.41132	−0.0451060
$980$	−57.4413	−1.83490
$981$	57.1401	1.82434
$982$	−0.288417	−0.00920375
$983$	15.8820	0.506558	0.253279	−	0.967393i	$-0.418491\pi$
$983$	15.8820	0.506558	0.253279	+	0.967393i	$0.418491\pi$
$984$	−40.2949	−1.28455
$985$	56.0670	1.78644
$986$	0.283307	0.00902235
$987$	−0.113266	−0.00360529
$988$	8.81694	0.280504
$989$	1.91191	0.0607953
$990$	−2.52584	−0.0802764
$991$	17.7974	0.565354	0.282677	−	0.959215i	$-0.408777\pi$
$991$	17.7974	0.565354	0.282677	+	0.959215i	$0.408777\pi$
$992$	−17.4771	−0.554898
$993$	58.1613	1.84569
$994$	0.0827495	0.00262465
$995$	3.95686	0.125441
$996$	−23.1967	−0.735014
$997$	−42.0476	−1.33166	−0.665830	−	0.746103i	$-0.731923\pi$
$997$	−42.0476	−1.33166	−0.665830	+	0.746103i	$0.731923\pi$
$998$	−86.6273	−2.74214
$999$	10.4376	0.330230

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.10	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.10	✓	71	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30773 q^{2} -2.77619 q^{3} +3.32561 q^{4} +2.46750 q^{5} +6.40670 q^{6} -0.00709873 q^{7} -3.05915 q^{8} +4.70724 q^{9} +O(q^{10})\)	\(q-2.30773 q^{2} -2.77619 q^{3} +3.32561 q^{4} +2.46750 q^{5} +6.40670 q^{6} -0.00709873 q^{7} -3.05915 q^{8} +4.70724 q^{9} -5.69433 q^{10} +0.0942316 q^{11} -9.23254 q^{12} +2.65123 q^{13} +0.0163820 q^{14} -6.85027 q^{15} +0.408469 q^{16} -0.103040 q^{17} -10.8630 q^{18} +1.00000 q^{19} +8.20596 q^{20} +0.0197074 q^{21} -0.217461 q^{22} +6.73094 q^{23} +8.49279 q^{24} +1.08858 q^{25} -6.11831 q^{26} -4.73963 q^{27} -0.0236076 q^{28} +1.19143 q^{29} +15.8086 q^{30} -3.37678 q^{31} +5.17567 q^{32} -0.261605 q^{33} +0.237788 q^{34} -0.0175162 q^{35} +15.6545 q^{36} -2.20219 q^{37} -2.30773 q^{38} -7.36031 q^{39} -7.54847 q^{40} -4.74460 q^{41} -0.0454794 q^{42} +0.284048 q^{43} +0.313378 q^{44} +11.6151 q^{45} -15.5332 q^{46} -5.74737 q^{47} -1.13399 q^{48} -6.99995 q^{49} -2.51214 q^{50} +0.286059 q^{51} +8.81694 q^{52} -8.09152 q^{53} +10.9378 q^{54} +0.232517 q^{55} +0.0217161 q^{56} -2.77619 q^{57} -2.74949 q^{58} -15.0926 q^{59} -22.7813 q^{60} +6.40394 q^{61} +7.79269 q^{62} -0.0334155 q^{63} -12.7610 q^{64} +6.54191 q^{65} +0.603713 q^{66} -12.4217 q^{67} -0.342671 q^{68} -18.6864 q^{69} +0.0404225 q^{70} +5.05126 q^{71} -14.4002 q^{72} +8.19008 q^{73} +5.08206 q^{74} -3.02210 q^{75} +3.32561 q^{76} -0.000668925 q^{77} +16.9856 q^{78} -13.2669 q^{79} +1.00790 q^{80} -0.963597 q^{81} +10.9492 q^{82} +2.51249 q^{83} +0.0655393 q^{84} -0.254252 q^{85} -0.655507 q^{86} -3.30763 q^{87} -0.288269 q^{88} -14.9772 q^{89} -26.8046 q^{90} -0.0188203 q^{91} +22.3845 q^{92} +9.37459 q^{93} +13.2634 q^{94} +2.46750 q^{95} -14.3686 q^{96} +13.5456 q^{97} +16.1540 q^{98} +0.443571 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

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