LMFDB - Embedded newform 4031.2.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, 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("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$1$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47917 q^{2} -2.68137 q^{3} +4.14631 q^{4} +1.95848 q^{5} +6.64759 q^{6} +0.410621 q^{7} -5.32107 q^{8} +4.18976 q^{9} +O(q^{10})$	\(q-2.47917 q^{2} -2.68137 q^{3} +4.14631 q^{4} +1.95848 q^{5} +6.64759 q^{6} +0.410621 q^{7} -5.32107 q^{8} +4.18976 q^{9} -4.85541 q^{10} -4.17004 q^{11} -11.1178 q^{12} -0.605821 q^{13} -1.01800 q^{14} -5.25141 q^{15} +4.89925 q^{16} -0.405624 q^{17} -10.3871 q^{18} -0.200900 q^{19} +8.12046 q^{20} -1.10103 q^{21} +10.3383 q^{22} +5.86909 q^{23} +14.2678 q^{24} -1.16436 q^{25} +1.50194 q^{26} -3.19018 q^{27} +1.70256 q^{28} +1.00000 q^{29} +13.0192 q^{30} +3.59476 q^{31} -1.50396 q^{32} +11.1814 q^{33} +1.00561 q^{34} +0.804193 q^{35} +17.3720 q^{36} +1.05032 q^{37} +0.498065 q^{38} +1.62443 q^{39} -10.4212 q^{40} -6.62556 q^{41} +2.72964 q^{42} -4.91951 q^{43} -17.2903 q^{44} +8.20555 q^{45} -14.5505 q^{46} +10.4750 q^{47} -13.1367 q^{48} -6.83139 q^{49} +2.88664 q^{50} +1.08763 q^{51} -2.51192 q^{52} -3.99437 q^{53} +7.90901 q^{54} -8.16694 q^{55} -2.18495 q^{56} +0.538687 q^{57} -2.47917 q^{58} +6.81153 q^{59} -21.7740 q^{60} -5.47783 q^{61} -8.91204 q^{62} +1.72040 q^{63} -6.06993 q^{64} -1.18649 q^{65} -27.7207 q^{66} +3.20916 q^{67} -1.68184 q^{68} -15.7372 q^{69} -1.99374 q^{70} +6.23817 q^{71} -22.2940 q^{72} -3.36284 q^{73} -2.60393 q^{74} +3.12207 q^{75} -0.832992 q^{76} -1.71231 q^{77} -4.02725 q^{78} -9.47977 q^{79} +9.59509 q^{80} -4.01522 q^{81} +16.4259 q^{82} +2.44323 q^{83} -4.56520 q^{84} -0.794407 q^{85} +12.1963 q^{86} -2.68137 q^{87} +22.1891 q^{88} -13.9122 q^{89} -20.3430 q^{90} -0.248763 q^{91} +24.3351 q^{92} -9.63889 q^{93} -25.9693 q^{94} -0.393458 q^{95} +4.03268 q^{96} -2.68812 q^{97} +16.9362 q^{98} -17.4714 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) $ 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9 - 18 * q^10 - 27 * q^11 - 8 * q^12 - 22 * q^13 - 24 * q^14 - 18 * q^15 + 5 * q^16 - 23 * q^17 + q^18 - 32 * q^19 - 14 * q^20 - 36 * q^21 - 6 * q^22 - 3 * q^23 - 18 * q^24 - 8 * q^25 - q^26 - 12 * q^27 - 9 * q^28 + 59 * q^29 - 18 * q^30 - 32 * q^31 - 39 * q^32 - 12 * q^33 - 18 * q^34 - 9 * q^35 + 10 * q^36 - 44 * q^37 + 5 * q^38 - 27 * q^39 - 68 * q^40 - 44 * q^41 - 25 * q^42 - 40 * q^43 - 56 * q^44 - 39 * q^45 - 40 * q^46 - 20 * q^47 - 9 * q^48 - 39 * q^49 - 21 * q^50 - 28 * q^51 - 49 * q^52 - 31 * q^53 - 32 * q^54 - 32 * q^55 - 48 * q^56 - 58 * q^57 - 5 * q^58 + 6 * q^59 - 44 * q^60 - 88 * q^61 + 35 * q^62 - 22 * q^63 - 10 * q^64 - 43 * q^65 - 31 * q^66 - 45 * q^67 - 29 * q^68 - 60 * q^69 - 14 * q^70 - 20 * q^71 - 4 * q^72 - 90 * q^73 - 25 * q^74 + 15 * q^75 - 64 * q^76 - 39 * q^77 - 28 * q^78 - 120 * q^79 + 24 * q^80 - 77 * q^81 - 71 * q^82 - 33 * q^83 - 14 * q^84 - 71 * q^85 - 61 * q^86 - 6 * q^87 - 34 * q^88 - 78 * q^89 - 88 * q^90 - 28 * q^91 - 31 * q^92 - 36 * q^93 - 4 * q^94 - 12 * q^95 - 29 * q^96 - 48 * q^97 - 4 * q^98 - 43 * 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 $

$2$	−2.47917	−1.75304	−0.876521	−	0.481364i	$-0.840141\pi$
$2$	−2.47917	−1.75304	−0.876521	+	0.481364i	$0.840141\pi$
$3$	−2.68137	−1.54809	−0.774045	−	0.633130i	$-0.781770\pi$
$3$	−2.68137	−1.54809	−0.774045	+	0.633130i	$0.781770\pi$
$4$	4.14631	2.07315
$5$	1.95848	0.875859	0.437929	−	0.899009i	$-0.355712\pi$
$5$	1.95848	0.875859	0.437929	+	0.899009i	$0.355712\pi$
$6$	6.64759	2.71387
$7$	0.410621	0.155200	0.0776001	−	0.996985i	$-0.475274\pi$
$7$	0.410621	0.155200	0.0776001	+	0.996985i	$0.475274\pi$
$8$	−5.32107	−1.88128
$9$	4.18976	1.39659
$10$	−4.85541	−1.53542
$11$	−4.17004	−1.25731	−0.628657	−	0.777683i	$-0.716395\pi$
$11$	−4.17004	−1.25731	−0.628657	+	0.777683i	$0.716395\pi$
$12$	−11.1178	−3.20943
$13$	−0.605821	−0.168025	−0.0840123	−	0.996465i	$-0.526774\pi$
$13$	−0.605821	−0.168025	−0.0840123	+	0.996465i	$0.526774\pi$
$14$	−1.01800	−0.272072
$15$	−5.25141	−1.35591
$16$	4.89925	1.22481
$17$	−0.405624	−0.0983783	−0.0491891	−	0.998789i	$-0.515664\pi$
$17$	−0.405624	−0.0983783	−0.0491891	+	0.998789i	$0.515664\pi$
$18$	−10.3871	−2.44827
$19$	−0.200900	−0.0460895	−0.0230448	−	0.999734i	$-0.507336\pi$
$19$	−0.200900	−0.0460895	−0.0230448	+	0.999734i	$0.507336\pi$
$20$	8.12046	1.81579
$21$	−1.10103	−0.240264
$22$	10.3383	2.20412
$23$	5.86909	1.22379	0.611895	−	0.790939i	$-0.290407\pi$
$23$	5.86909	1.22379	0.611895	+	0.790939i	$0.290407\pi$
$24$	14.2678	2.91240
$25$	−1.16436	−0.232871
$26$	1.50194	0.294554
$27$	−3.19018	−0.613950
$28$	1.70256	0.321754
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	13.0192	2.37696
$31$	3.59476	0.645638	0.322819	−	0.946461i	$-0.395369\pi$
$31$	3.59476	0.645638	0.322819	+	0.946461i	$0.395369\pi$
$32$	−1.50396	−0.265865
$33$	11.1814	1.94644
$34$	1.00561	0.172461
$35$	0.804193	0.135934
$36$	17.3720	2.89534
$37$	1.05032	0.172672	0.0863360	−	0.996266i	$-0.472484\pi$
$37$	1.05032	0.172672	0.0863360	+	0.996266i	$0.472484\pi$
$38$	0.498065	0.0807969
$39$	1.62443	0.260117
$40$	−10.4212	−1.64774
$41$	−6.62556	−1.03474	−0.517369	−	0.855762i	$-0.673089\pi$
$41$	−6.62556	−1.03474	−0.517369	+	0.855762i	$0.673089\pi$
$42$	2.72964	0.421193
$43$	−4.91951	−0.750219	−0.375109	−	0.926981i	$-0.622395\pi$
$43$	−4.91951	−0.750219	−0.375109	+	0.926981i	$0.622395\pi$
$44$	−17.2903	−2.60660
$45$	8.20555	1.22321
$46$	−14.5505	−2.14536
$47$	10.4750	1.52793	0.763965	−	0.645257i	$-0.223250\pi$
$47$	10.4750	1.52793	0.763965	+	0.645257i	$0.223250\pi$
$48$	−13.1367	−1.89612
$49$	−6.83139	−0.975913
$50$	2.88664	0.408233
$51$	1.08763	0.152299
$52$	−2.51192	−0.348341
$53$	−3.99437	−0.548669	−0.274335	−	0.961634i	$-0.588458\pi$
$53$	−3.99437	−0.548669	−0.274335	+	0.961634i	$0.588458\pi$
$54$	7.90901	1.07628
$55$	−8.16694	−1.10123
$56$	−2.18495	−0.291976
$57$	0.538687	0.0713508
$58$	−2.47917	−0.325532
$59$	6.81153	0.886785	0.443393	−	0.896328i	$-0.353775\pi$
$59$	6.81153	0.886785	0.443393	+	0.896328i	$0.353775\pi$
$60$	−21.7740	−2.81101
$61$	−5.47783	−0.701364	−0.350682	−	0.936495i	$-0.614050\pi$
$61$	−5.47783	−0.701364	−0.350682	+	0.936495i	$0.614050\pi$
$62$	−8.91204	−1.13183
$63$	1.72040	0.216750
$64$	−6.06993	−0.758741
$65$	−1.18649	−0.147166
$66$	−27.7207	−3.41218
$67$	3.20916	0.392061	0.196030	−	0.980598i	$-0.437195\pi$
$67$	3.20916	0.392061	0.196030	+	0.980598i	$0.437195\pi$
$68$	−1.68184	−0.203953
$69$	−15.7372	−1.89454
$70$	−1.99374	−0.238297
$71$	6.23817	0.740334	0.370167	−	0.928965i	$-0.379300\pi$
$71$	6.23817	0.740334	0.370167	+	0.928965i	$0.379300\pi$
$72$	−22.2940	−2.62737
$73$	−3.36284	−0.393591	−0.196795	−	0.980445i	$-0.563053\pi$
$73$	−3.36284	−0.393591	−0.196795	+	0.980445i	$0.563053\pi$
$74$	−2.60393	−0.302701
$75$	3.12207	0.360506
$76$	−0.832992	−0.0955507
$77$	−1.71231	−0.195135
$78$	−4.02725	−0.455996
$79$	−9.47977	−1.06656	−0.533279	−	0.845940i	$-0.679040\pi$
$79$	−9.47977	−1.06656	−0.533279	+	0.845940i	$0.679040\pi$
$80$	9.59509	1.07276
$81$	−4.01522	−0.446135
$82$	16.4259	1.81394
$83$	2.44323	0.268180	0.134090	−	0.990969i	$-0.457189\pi$
$83$	2.44323	0.268180	0.134090	+	0.990969i	$0.457189\pi$
$84$	−4.56520	−0.498104
$85$	−0.794407	−0.0861655
$86$	12.1963	1.31516
$87$	−2.68137	−0.287473
$88$	22.1891	2.36536
$89$	−13.9122	−1.47469	−0.737343	−	0.675519i	$-0.763920\pi$
$89$	−13.9122	−1.47469	−0.737343	+	0.675519i	$0.763920\pi$
$90$	−20.3430	−2.14434
$91$	−0.248763	−0.0260775
$92$	24.3351	2.53711
$93$	−9.63889	−0.999506
$94$	−25.9693	−2.67853
$95$	−0.393458	−0.0403679
$96$	4.03268	0.411583
$97$	−2.68812	−0.272937	−0.136469	−	0.990644i	$-0.543575\pi$
$97$	−2.68812	−0.272937	−0.136469	+	0.990644i	$0.543575\pi$
$98$	16.9362	1.71082
$99$	−17.4714	−1.75595
$100$	−4.82778	−0.482778
$101$	13.0089	1.29443	0.647216	−	0.762307i	$-0.275933\pi$
$101$	13.0089	1.29443	0.647216	+	0.762307i	$0.275933\pi$
$102$	−2.69642	−0.266986
$103$	1.46008	0.143866	0.0719329	−	0.997409i	$-0.477083\pi$
$103$	1.46008	0.143866	0.0719329	+	0.997409i	$0.477083\pi$
$104$	3.22362	0.316102
$105$	−2.15634	−0.210437
$106$	9.90274	0.961840
$107$	5.80780	0.561462	0.280731	−	0.959787i	$-0.409423\pi$
$107$	5.80780	0.561462	0.280731	+	0.959787i	$0.409423\pi$
$108$	−13.2275	−1.27281
$109$	11.0937	1.06259	0.531293	−	0.847188i	$-0.321706\pi$
$109$	11.0937	1.06259	0.531293	+	0.847188i	$0.321706\pi$
$110$	20.2473	1.93050
$111$	−2.81630	−0.267312
$112$	2.01174	0.190091
$113$	3.16529	0.297765	0.148883	−	0.988855i	$-0.452432\pi$
$113$	3.16529	0.297765	0.148883	+	0.988855i	$0.452432\pi$
$114$	−1.33550	−0.125081
$115$	11.4945	1.07187
$116$	4.14631	0.384975
$117$	−2.53824	−0.234661
$118$	−16.8870	−1.55457
$119$	−0.166558	−0.0152683
$120$	27.9432	2.55085
$121$	6.38921	0.580838
$122$	13.5805	1.22952
$123$	17.7656	1.60187
$124$	14.9050	1.33851
$125$	−12.0728	−1.07982
$126$	−4.26518	−0.379972
$127$	6.33258	0.561926	0.280963	−	0.959719i	$-0.409346\pi$
$127$	6.33258	0.561926	0.280963	+	0.959719i	$0.409346\pi$
$128$	18.0563	1.59597
$129$	13.1910	1.16141
$130$	2.94151	0.257988
$131$	−8.63688	−0.754608	−0.377304	−	0.926089i	$-0.623149\pi$
$131$	−8.63688	−0.754608	−0.377304	+	0.926089i	$0.623149\pi$
$132$	46.3616	4.03526
$133$	−0.0824936	−0.00715311
$134$	−7.95606	−0.687299
$135$	−6.24790	−0.537733
$136$	2.15836	0.185077
$137$	−2.61055	−0.223034	−0.111517	−	0.993763i	$-0.535571\pi$
$137$	−2.61055	−0.223034	−0.111517	+	0.993763i	$0.535571\pi$
$138$	39.0153	3.32121
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	3.33443	0.281811
$141$	−28.0873	−2.36538
$142$	−15.4655	−1.29784
$143$	2.52630	0.211260
$144$	20.5267	1.71056
$145$	1.95848	0.162643
$146$	8.33707	0.689981
$147$	18.3175	1.51080
$148$	4.35496	0.357975
$149$	−3.87330	−0.317313	−0.158657	−	0.987334i	$-0.550716\pi$
$149$	−3.87330	−0.317313	−0.158657	+	0.987334i	$0.550716\pi$
$150$	−7.74016	−0.631981
$151$	5.92328	0.482029	0.241015	−	0.970521i	$-0.422520\pi$
$151$	5.92328	0.482029	0.241015	+	0.970521i	$0.422520\pi$
$152$	1.06900	0.0867075
$153$	−1.69947	−0.137394
$154$	4.24511	0.342080
$155$	7.04027	0.565488
$156$	6.73540	0.539263
$157$	−2.43777	−0.194555	−0.0972774	−	0.995257i	$-0.531013\pi$
$157$	−2.43777	−0.194555	−0.0972774	+	0.995257i	$0.531013\pi$
$158$	23.5020	1.86972
$159$	10.7104	0.849390
$160$	−2.94548	−0.232860
$161$	2.40997	0.189933
$162$	9.95442	0.782093
$163$	5.59387	0.438146	0.219073	−	0.975709i	$-0.429697\pi$
$163$	5.59387	0.438146	0.219073	+	0.975709i	$0.429697\pi$
$164$	−27.4716	−2.14517
$165$	21.8986	1.70480
$166$	−6.05720	−0.470130
$167$	−0.892368	−0.0690535	−0.0345268	−	0.999404i	$-0.510992\pi$
$167$	−0.892368	−0.0690535	−0.0345268	+	0.999404i	$0.510992\pi$
$168$	5.85865	0.452005
$169$	−12.6330	−0.971768
$170$	1.96947	0.151052
$171$	−0.841720	−0.0643680
$172$	−20.3978	−1.55532
$173$	19.5849	1.48902	0.744508	−	0.667614i	$-0.232684\pi$
$173$	19.5849	1.48902	0.744508	+	0.667614i	$0.232684\pi$
$174$	6.64759	0.503952
$175$	−0.478109	−0.0361416
$176$	−20.4301	−1.53997
$177$	−18.2642	−1.37282
$178$	34.4907	2.58519
$179$	19.7817	1.47856	0.739279	−	0.673400i	$-0.235167\pi$
$179$	19.7817	1.47856	0.739279	+	0.673400i	$0.235167\pi$
$180$	34.0227	2.53591
$181$	−7.34346	−0.545835	−0.272917	−	0.962037i	$-0.587989\pi$
$181$	−7.34346	−0.545835	−0.272917	+	0.962037i	$0.587989\pi$
$182$	0.616727	0.0457149
$183$	14.6881	1.08577
$184$	−31.2299	−2.30230
$185$	2.05704	0.151236
$186$	23.8965	1.75218
$187$	1.69147	0.123692
$188$	43.4324	3.16764
$189$	−1.30995	−0.0952851
$190$	0.975451	0.0707667
$191$	6.33674	0.458510	0.229255	−	0.973366i	$-0.426371\pi$
$191$	6.33674	0.458510	0.229255	+	0.973366i	$0.426371\pi$
$192$	16.2757	1.17460
$193$	−14.2929	−1.02883	−0.514414	−	0.857542i	$-0.671990\pi$
$193$	−14.2929	−1.02883	−0.514414	+	0.857542i	$0.671990\pi$
$194$	6.66432	0.478470
$195$	3.18142	0.227826
$196$	−28.3250	−2.02322
$197$	8.24107	0.587152	0.293576	−	0.955936i	$-0.405155\pi$
$197$	8.24107	0.587152	0.293576	+	0.955936i	$0.405155\pi$
$198$	43.3147	3.07825
$199$	−13.7699	−0.976124	−0.488062	−	0.872809i	$-0.662296\pi$
$199$	−13.7699	−0.976124	−0.488062	+	0.872809i	$0.662296\pi$
$200$	6.19562	0.438096
$201$	−8.60494	−0.606946
$202$	−32.2513	−2.26919
$203$	0.410621	0.0288200
$204$	4.50964	0.315738
$205$	−12.9760	−0.906285
$206$	−3.61979	−0.252203
$207$	24.5901	1.70913
$208$	−2.96807	−0.205799
$209$	0.837759	0.0579490
$210$	5.34595	0.368905
$211$	−10.3373	−0.711647	−0.355824	−	0.934553i	$-0.615800\pi$
$211$	−10.3373	−0.711647	−0.355824	+	0.934553i	$0.615800\pi$
$212$	−16.5619	−1.13748
$213$	−16.7268	−1.14610
$214$	−14.3986	−0.984265
$215$	−9.63477	−0.657086
$216$	16.9752	1.15501
$217$	1.47608	0.100203
$218$	−27.5033	−1.86276
$219$	9.01703	0.609314
$220$	−33.8626	−2.28302
$221$	0.245736	0.0165300
$222$	6.98211	0.468609
$223$	7.94054	0.531738	0.265869	−	0.964009i	$-0.414341\pi$
$223$	7.94054	0.531738	0.265869	+	0.964009i	$0.414341\pi$
$224$	−0.617558	−0.0412623
$225$	−4.87836	−0.325224
$226$	−7.84731	−0.521995
$227$	−20.2152	−1.34173	−0.670866	−	0.741578i	$-0.734078\pi$
$227$	−20.2152	−1.34173	−0.670866	+	0.741578i	$0.734078\pi$
$228$	2.23356	0.147921
$229$	0.438133	0.0289526	0.0144763	−	0.999895i	$-0.495392\pi$
$229$	0.438133	0.0289526	0.0144763	+	0.999895i	$0.495392\pi$
$230$	−28.4969	−1.87903
$231$	4.59133	0.302087
$232$	−5.32107	−0.349346
$233$	10.1573	0.665428	0.332714	−	0.943028i	$-0.392036\pi$
$233$	10.1573	0.665428	0.332714	+	0.943028i	$0.392036\pi$
$234$	6.29275	0.411370
$235$	20.5150	1.33825
$236$	28.2427	1.83844
$237$	25.4188	1.65113
$238$	0.412926	0.0267660
$239$	−20.9555	−1.35550	−0.677751	−	0.735292i	$-0.737045\pi$
$239$	−20.9555	−1.35550	−0.677751	+	0.735292i	$0.737045\pi$
$240$	−25.7280	−1.66074
$241$	6.58572	0.424224	0.212112	−	0.977245i	$-0.431966\pi$
$241$	6.58572	0.424224	0.212112	+	0.977245i	$0.431966\pi$
$242$	−15.8400	−1.01823
$243$	20.3368	1.30461
$244$	−22.7128	−1.45404
$245$	−13.3791	−0.854762
$246$	−44.0440	−2.80814
$247$	0.121709	0.00774418
$248$	−19.1280	−1.21463
$249$	−6.55122	−0.415167
$250$	29.9305	1.89297
$251$	8.27323	0.522202	0.261101	−	0.965312i	$-0.415914\pi$
$251$	8.27323	0.522202	0.261101	+	0.965312i	$0.415914\pi$
$252$	7.13332	0.449357
$253$	−24.4743	−1.53869
$254$	−15.6996	−0.985079
$255$	2.13010	0.133392
$256$	−32.6249	−2.03906
$257$	−6.85899	−0.427852	−0.213926	−	0.976850i	$-0.568625\pi$
$257$	−6.85899	−0.427852	−0.213926	+	0.976850i	$0.568625\pi$
$258$	−32.7029	−2.03599
$259$	0.431285	0.0267987
$260$	−4.91955	−0.305097
$261$	4.18976	0.259339
$262$	21.4123	1.32286
$263$	13.2902	0.819506	0.409753	−	0.912197i	$-0.365615\pi$
$263$	13.2902	0.819506	0.409753	+	0.912197i	$0.365615\pi$
$264$	−59.4972	−3.66180
$265$	−7.82290	−0.480557
$266$	0.204516	0.0125397
$267$	37.3037	2.28295
$268$	13.3061	0.812802
$269$	15.6829	0.956205	0.478102	−	0.878304i	$-0.341325\pi$
$269$	15.6829	0.956205	0.478102	+	0.878304i	$0.341325\pi$
$270$	15.4896	0.942669
$271$	−12.5117	−0.760032	−0.380016	−	0.924980i	$-0.624082\pi$
$271$	−12.5117	−0.760032	−0.380016	+	0.924980i	$0.624082\pi$
$272$	−1.98725	−0.120495
$273$	0.667026	0.0403703
$274$	6.47200	0.390988
$275$	4.85541	0.292792
$276$	−65.2514	−3.92767
$277$	21.1091	1.26833	0.634163	−	0.773200i	$-0.281345\pi$
$277$	21.1091	1.26833	0.634163	+	0.773200i	$0.281345\pi$
$278$	−2.47917	−0.148691
$279$	15.0612	0.901689
$280$	−4.27917	−0.255729
$281$	−26.5155	−1.58178	−0.790891	−	0.611957i	$-0.790383\pi$
$281$	−26.5155	−1.58178	−0.790891	+	0.611957i	$0.790383\pi$
$282$	69.6333	4.14660
$283$	4.25904	0.253174	0.126587	−	0.991956i	$-0.459598\pi$
$283$	4.25904	0.253174	0.126587	+	0.991956i	$0.459598\pi$
$284$	25.8654	1.53483
$285$	1.05501	0.0624932
$286$	−6.26313	−0.370347
$287$	−2.72059	−0.160592
$288$	−6.30122	−0.371303
$289$	−16.8355	−0.990322
$290$	−4.85541	−0.285120
$291$	7.20785	0.422531
$292$	−13.9434	−0.815974
$293$	−8.73129	−0.510087	−0.255044	−	0.966930i	$-0.582090\pi$
$293$	−8.73129	−0.510087	−0.255044	+	0.966930i	$0.582090\pi$
$294$	−45.4123	−2.64850
$295$	13.3402	0.776699
$296$	−5.58884	−0.324845
$297$	13.3032	0.771927
$298$	9.60260	0.556263
$299$	−3.55562	−0.205627
$300$	12.9451	0.747384
$301$	−2.02006	−0.116434
$302$	−14.6848	−0.845017
$303$	−34.8816	−2.00390
$304$	−0.984258	−0.0564511
$305$	−10.7282	−0.614296
$306$	4.21327	0.240857
$307$	3.15764	0.180216	0.0901082	−	0.995932i	$-0.471279\pi$
$307$	3.15764	0.180216	0.0901082	+	0.995932i	$0.471279\pi$
$308$	−7.09975	−0.404546
$309$	−3.91501	−0.222717
$310$	−17.4541	−0.991324
$311$	6.61937	0.375350	0.187675	−	0.982231i	$-0.439905\pi$
$311$	6.61937	0.375350	0.187675	+	0.982231i	$0.439905\pi$
$312$	−8.64372	−0.489354
$313$	1.47411	0.0833216	0.0416608	−	0.999132i	$-0.486735\pi$
$313$	1.47411	0.0833216	0.0416608	+	0.999132i	$0.486735\pi$
$314$	6.04365	0.341063
$315$	3.36937	0.189843
$316$	−39.3060	−2.21114
$317$	−19.1324	−1.07458	−0.537292	−	0.843396i	$-0.680553\pi$
$317$	−19.1324	−1.07458	−0.537292	+	0.843396i	$0.680553\pi$
$318$	−26.5529	−1.48901
$319$	−4.17004	−0.233477
$320$	−11.8878	−0.664550
$321$	−15.5729	−0.869194
$322$	−5.97475	−0.332960
$323$	0.0814897	0.00453421
$324$	−16.6483	−0.924907
$325$	0.705391	0.0391281
$326$	−13.8682	−0.768087
$327$	−29.7464	−1.64498
$328$	35.2551	1.94664
$329$	4.30124	0.237135
$330$	−54.2904	−2.98859
$331$	29.4436	1.61837	0.809184	−	0.587555i	$-0.199909\pi$
$331$	29.4436	1.61837	0.809184	+	0.587555i	$0.199909\pi$
$332$	10.1304	0.555978
$333$	4.40059	0.241151
$334$	2.21234	0.121054
$335$	6.28507	0.343390
$336$	−5.39422	−0.294279
$337$	−29.0072	−1.58012	−0.790062	−	0.613027i	$-0.789952\pi$
$337$	−29.0072	−1.58012	−0.790062	+	0.613027i	$0.789952\pi$
$338$	31.3194	1.70355
$339$	−8.48732	−0.460968
$340$	−3.29385	−0.178634
$341$	−14.9903	−0.811770
$342$	2.08677	0.112840
$343$	−5.67946	−0.306662
$344$	26.1771	1.41137
$345$	−30.8210	−1.65935
$346$	−48.5545	−2.61031
$347$	−7.80164	−0.418814	−0.209407	−	0.977829i	$-0.567153\pi$
$347$	−7.80164	−0.418814	−0.209407	+	0.977829i	$0.567153\pi$
$348$	−11.1178	−0.595976
$349$	−4.03854	−0.216178	−0.108089	−	0.994141i	$-0.534473\pi$
$349$	−4.03854	−0.216178	−0.108089	+	0.994141i	$0.534473\pi$
$350$	1.18532	0.0633578
$351$	1.93268	0.103159
$352$	6.27157	0.334276
$353$	17.8272	0.948846	0.474423	−	0.880297i	$-0.342657\pi$
$353$	17.8272	0.948846	0.474423	+	0.880297i	$0.342657\pi$
$354$	45.2802	2.40662
$355$	12.2173	0.648428
$356$	−57.6841	−3.05725
$357$	0.446604	0.0236368
$358$	−49.0424	−2.59197
$359$	−30.8465	−1.62802	−0.814009	−	0.580852i	$-0.802719\pi$
$359$	−30.8465	−1.62802	−0.814009	+	0.580852i	$0.802719\pi$
$360$	−43.6623	−2.30121
$361$	−18.9596	−0.997876
$362$	18.2057	0.956871
$363$	−17.1319	−0.899189
$364$	−1.03145	−0.0540626
$365$	−6.58606	−0.344730
$366$	−36.4143	−1.90341
$367$	22.3586	1.16711	0.583555	−	0.812074i	$-0.301661\pi$
$367$	22.3586	1.16711	0.583555	+	0.812074i	$0.301661\pi$
$368$	28.7542	1.49891
$369$	−27.7595	−1.44510
$370$	−5.09975	−0.265123
$371$	−1.64017	−0.0851536
$372$	−39.9658	−2.07213
$373$	3.86359	0.200049	0.100025	−	0.994985i	$-0.468108\pi$
$373$	3.86359	0.200049	0.100025	+	0.994985i	$0.468108\pi$
$374$	−4.19344	−0.216838
$375$	32.3716	1.67166
$376$	−55.7381	−2.87447
$377$	−0.605821	−0.0312014
$378$	3.24761	0.167039
$379$	9.25235	0.475261	0.237631	−	0.971356i	$-0.423629\pi$
$379$	9.25235	0.475261	0.237631	+	0.971356i	$0.423629\pi$
$380$	−1.63140	−0.0836889
$381$	−16.9800	−0.869912
$382$	−15.7099	−0.803787
$383$	−10.7376	−0.548666	−0.274333	−	0.961635i	$-0.588457\pi$
$383$	−10.7376	−0.548666	−0.274333	+	0.961635i	$0.588457\pi$
$384$	−48.4157	−2.47071
$385$	−3.35352	−0.170911
$386$	35.4347	1.80358
$387$	−20.6116	−1.04774
$388$	−11.1458	−0.565841
$389$	−0.337312	−0.0171024	−0.00855121	−	0.999963i	$-0.502722\pi$
$389$	−0.337312	−0.0171024	−0.00855121	+	0.999963i	$0.502722\pi$
$390$	−7.88729	−0.399389
$391$	−2.38065	−0.120394
$392$	36.3503	1.83597
$393$	23.1587	1.16820
$394$	−20.4311	−1.02930
$395$	−18.5659	−0.934154
$396$	−72.4420	−3.64035
$397$	−27.7139	−1.39092	−0.695461	−	0.718564i	$-0.744800\pi$
$397$	−27.7139	−1.39092	−0.695461	+	0.718564i	$0.744800\pi$
$398$	34.1380	1.71119
$399$	0.221196	0.0110737
$400$	−5.70447	−0.285224
$401$	19.2179	0.959698	0.479849	−	0.877351i	$-0.340692\pi$
$401$	19.2179	0.959698	0.479849	+	0.877351i	$0.340692\pi$
$402$	21.3332	1.06400
$403$	−2.17778	−0.108483
$404$	53.9388	2.68355
$405$	−7.86372	−0.390751
$406$	−1.01800	−0.0505226
$407$	−4.37988	−0.217103
$408$	−5.78735	−0.286517
$409$	−36.1455	−1.78728	−0.893640	−	0.448784i	$-0.851857\pi$
$409$	−36.1455	−1.78728	−0.893640	+	0.448784i	$0.851857\pi$
$410$	32.1698	1.58876
$411$	6.99984	0.345277
$412$	6.05394	0.298256
$413$	2.79696	0.137629
$414$	−60.9631	−2.99617
$415$	4.78502	0.234888
$416$	0.911131	0.0446719
$417$	−2.68137	−0.131307
$418$	−2.07695	−0.101587
$419$	2.20805	0.107870	0.0539351	−	0.998544i	$-0.482824\pi$
$419$	2.20805	0.107870	0.0539351	+	0.998544i	$0.482824\pi$
$420$	−8.94086	−0.436269
$421$	7.58098	0.369474	0.184737	−	0.982788i	$-0.440857\pi$
$421$	7.58098	0.369474	0.184737	+	0.982788i	$0.440857\pi$
$422$	25.6279	1.24755
$423$	43.8875	2.13389
$424$	21.2543	1.03220
$425$	0.472291	0.0229095
$426$	41.4688	2.00917
$427$	−2.24931	−0.108852
$428$	24.0809	1.16400
$429$	−6.77394	−0.327049
$430$	23.8863	1.15190
$431$	−1.17713	−0.0567005	−0.0283503	−	0.999598i	$-0.509025\pi$
$431$	−1.17713	−0.0567005	−0.0283503	+	0.999598i	$0.509025\pi$
$432$	−15.6295	−0.751974
$433$	22.7661	1.09407	0.547035	−	0.837110i	$-0.315757\pi$
$433$	22.7661	1.09407	0.547035	+	0.837110i	$0.315757\pi$
$434$	−3.65947	−0.175660
$435$	−5.25141	−0.251786
$436$	45.9980	2.20291
$437$	−1.17910	−0.0564039
$438$	−22.3548	−1.06815
$439$	−13.0129	−0.621070	−0.310535	−	0.950562i	$-0.600508\pi$
$439$	−13.0129	−0.621070	−0.310535	+	0.950562i	$0.600508\pi$
$440$	43.4569	2.07172
$441$	−28.6219	−1.36295
$442$	−0.609222	−0.0289777
$443$	17.2192	0.818108	0.409054	−	0.912510i	$-0.365859\pi$
$443$	17.2192	0.818108	0.409054	+	0.912510i	$0.365859\pi$
$444$	−11.6773	−0.554179
$445$	−27.2467	−1.29162
$446$	−19.6860	−0.932159
$447$	10.3858	0.491230
$448$	−2.49244	−0.117757
$449$	13.0077	0.613871	0.306936	−	0.951730i	$-0.400696\pi$
$449$	13.0077	0.613871	0.306936	+	0.951730i	$0.400696\pi$
$450$	12.0943	0.570132
$451$	27.6288	1.30099
$452$	13.1243	0.617313
$453$	−15.8825	−0.746225
$454$	50.1171	2.35211
$455$	−0.487198	−0.0228402
$456$	−2.86639	−0.134231
$457$	15.1684	0.709548	0.354774	−	0.934952i	$-0.384558\pi$
$457$	15.1684	0.709548	0.354774	+	0.934952i	$0.384558\pi$
$458$	−1.08621	−0.0507551
$459$	1.29401	0.0603993
$460$	47.6598	2.22215
$461$	−12.1389	−0.565367	−0.282684	−	0.959213i	$-0.591225\pi$
$461$	−12.1389	−0.565367	−0.282684	+	0.959213i	$0.591225\pi$
$462$	−11.3827	−0.529571
$463$	−32.8069	−1.52467	−0.762334	−	0.647184i	$-0.775947\pi$
$463$	−32.8069	−1.52467	−0.762334	+	0.647184i	$0.775947\pi$
$464$	4.89925	0.227442
$465$	−18.8776	−0.875427
$466$	−25.1818	−1.16652
$467$	−2.71273	−0.125530	−0.0627651	−	0.998028i	$-0.519992\pi$
$467$	−2.71273	−0.125530	−0.0627651	+	0.998028i	$0.519992\pi$
$468$	−10.5243	−0.486488
$469$	1.31775	0.0608479
$470$	−50.8603	−2.34601
$471$	6.53656	0.301189
$472$	−36.2446	−1.66829
$473$	20.5146	0.943260
$474$	−63.0176	−2.89449
$475$	0.233919	0.0107329
$476$	−0.690600	−0.0316536
$477$	−16.7354	−0.766263
$478$	51.9524	2.37625
$479$	−32.7658	−1.49711	−0.748553	−	0.663075i	$-0.769251\pi$
$479$	−32.7658	−1.49711	−0.748553	+	0.663075i	$0.769251\pi$
$480$	7.89791	0.360489
$481$	−0.636308	−0.0290131
$482$	−16.3272	−0.743682
$483$	−6.46204	−0.294033
$484$	26.4916	1.20417
$485$	−5.26463	−0.239054
$486$	−50.4185	−2.28703
$487$	−8.33113	−0.377519	−0.188760	−	0.982023i	$-0.560447\pi$
$487$	−8.33113	−0.377519	−0.188760	+	0.982023i	$0.560447\pi$
$488$	29.1479	1.31946
$489$	−14.9992	−0.678289
$490$	33.1692	1.49843
$491$	−12.8513	−0.579970	−0.289985	−	0.957031i	$-0.593650\pi$
$491$	−12.8513	−0.579970	−0.289985	+	0.957031i	$0.593650\pi$
$492$	73.6616	3.32092
$493$	−0.405624	−0.0182684
$494$	−0.301739	−0.0135759
$495$	−34.2175	−1.53796
$496$	17.6116	0.790786
$497$	2.56152	0.114900
$498$	16.2416	0.727804
$499$	−22.6004	−1.01173	−0.505866	−	0.862612i	$-0.668827\pi$
$499$	−22.6004	−1.01173	−0.505866	+	0.862612i	$0.668827\pi$
$500$	−50.0574	−2.23864
$501$	2.39277	0.106901
$502$	−20.5108	−0.915441
$503$	−29.7640	−1.32711	−0.663555	−	0.748128i	$-0.730953\pi$
$503$	−29.7640	−1.32711	−0.663555	+	0.748128i	$0.730953\pi$
$504$	−9.15439	−0.407769
$505$	25.4776	1.13374
$506$	60.6762	2.69738
$507$	33.8737	1.50438
$508$	26.2568	1.16496
$509$	11.5983	0.514084	0.257042	−	0.966400i	$-0.417252\pi$
$509$	11.5983	0.514084	0.257042	+	0.966400i	$0.417252\pi$
$510$	−5.28089	−0.233842
$511$	−1.38085	−0.0610854
$512$	44.7703	1.97859
$513$	0.640905	0.0282967
$514$	17.0046	0.750042
$515$	2.85954	0.126006
$516$	54.6941	2.40777
$517$	−43.6810	−1.92109
$518$	−1.06923	−0.0469793
$519$	−52.5145	−2.30513
$520$	6.31339	0.276861
$521$	12.9918	0.569179	0.284590	−	0.958649i	$-0.408143\pi$
$521$	12.9918	0.569179	0.284590	+	0.958649i	$0.408143\pi$
$522$	−10.3871	−0.454633
$523$	−42.1879	−1.84475	−0.922374	−	0.386298i	$-0.873754\pi$
$523$	−42.1879	−1.84475	−0.922374	+	0.386298i	$0.873754\pi$
$524$	−35.8112	−1.56442
$525$	1.28199	0.0559506
$526$	−32.9486	−1.43663
$527$	−1.45812	−0.0635168
$528$	54.7806	2.38402
$529$	11.4463	0.497664
$530$	19.3943	0.842436
$531$	28.5386	1.23847
$532$	−0.342044	−0.0148295
$533$	4.01390	0.173862
$534$	−92.4823	−4.00210
$535$	11.3745	0.491761
$536$	−17.0762	−0.737577
$537$	−53.0422	−2.28894
$538$	−38.8807	−1.67627
$539$	28.4872	1.22703
$540$	−25.9057	−1.11480
$541$	−13.1731	−0.566354	−0.283177	−	0.959068i	$-0.591388\pi$
$541$	−13.1731	−0.566354	−0.283177	+	0.959068i	$0.591388\pi$
$542$	31.0187	1.33237
$543$	19.6905	0.845002
$544$	0.610042	0.0261553
$545$	21.7269	0.930676
$546$	−1.65367	−0.0707707
$547$	−15.9903	−0.683695	−0.341848	−	0.939755i	$-0.611053\pi$
$547$	−15.9903	−0.683695	−0.341848	+	0.939755i	$0.611053\pi$
$548$	−10.8241	−0.462384
$549$	−22.9508	−0.979514
$550$	−12.0374	−0.513277
$551$	−0.200900	−0.00855861
$552$	83.7389	3.56416
$553$	−3.89259	−0.165530
$554$	−52.3333	−2.22343
$555$	−5.51568	−0.234127
$556$	4.14631	0.175843
$557$	28.3216	1.20002	0.600012	−	0.799991i	$-0.295162\pi$
$557$	28.3216	1.20002	0.600012	+	0.799991i	$0.295162\pi$
$558$	−37.3393	−1.58070
$559$	2.98035	0.126055
$560$	3.93995	0.166493
$561$	−4.53545	−0.191487
$562$	65.7366	2.77293
$563$	21.7103	0.914980	0.457490	−	0.889215i	$-0.348749\pi$
$563$	21.7103	0.914980	0.457490	+	0.889215i	$0.348749\pi$
$564$	−116.459	−4.90379
$565$	6.19916	0.260800
$566$	−10.5589	−0.443824
$567$	−1.64873	−0.0692403
$568$	−33.1937	−1.39278
$569$	−24.1401	−1.01200	−0.506002	−	0.862532i	$-0.668877\pi$
$569$	−24.1401	−1.01200	−0.506002	+	0.862532i	$0.668877\pi$
$570$	−2.61555	−0.109553
$571$	−12.4077	−0.519247	−0.259624	−	0.965710i	$-0.583598\pi$
$571$	−12.4077	−0.519247	−0.259624	+	0.965710i	$0.583598\pi$
$572$	10.4748	0.437974
$573$	−16.9911	−0.709815
$574$	6.74483	0.281524
$575$	−6.83371	−0.284985
$576$	−25.4315	−1.05965
$577$	−12.6673	−0.527346	−0.263673	−	0.964612i	$-0.584934\pi$
$577$	−12.6673	−0.527346	−0.263673	+	0.964612i	$0.584934\pi$
$578$	41.7381	1.73607
$579$	38.3246	1.59272
$580$	8.12046	0.337184
$581$	1.00324	0.0416215
$582$	−17.8695	−0.740715
$583$	16.6567	0.689849
$584$	17.8939	0.740456
$585$	−4.97110	−0.205530
$586$	21.6464	0.894204
$587$	16.8527	0.695587	0.347793	−	0.937571i	$-0.386931\pi$
$587$	16.8527	0.695587	0.347793	+	0.937571i	$0.386931\pi$
$588$	75.9500	3.13212
$589$	−0.722186	−0.0297572
$590$	−33.0728	−1.36159
$591$	−22.0974	−0.908965
$592$	5.14579	0.211491
$593$	−40.2750	−1.65389	−0.826947	−	0.562279i	$-0.809925\pi$
$593$	−40.2750	−1.65389	−0.826947	+	0.562279i	$0.809925\pi$
$594$	−32.9808	−1.35322
$595$	−0.326200	−0.0133729
$596$	−16.0599	−0.657839
$597$	36.9223	1.51113
$598$	8.81501	0.360473
$599$	4.81758	0.196841	0.0984205	−	0.995145i	$-0.468621\pi$
$599$	4.81758	0.196841	0.0984205	+	0.995145i	$0.468621\pi$
$600$	−16.6128	−0.678213
$601$	20.0740	0.818836	0.409418	−	0.912347i	$-0.365732\pi$
$601$	20.0740	0.818836	0.409418	+	0.912347i	$0.365732\pi$
$602$	5.00807	0.204114
$603$	13.4456	0.547546
$604$	24.5597	0.999321
$605$	12.5131	0.508732
$606$	86.4776	3.51291
$607$	−20.2744	−0.822914	−0.411457	−	0.911429i	$-0.634980\pi$
$607$	−20.2744	−0.822914	−0.411457	+	0.911429i	$0.634980\pi$
$608$	0.302145	0.0122536
$609$	−1.10103	−0.0446159
$610$	26.5971	1.07689
$611$	−6.34596	−0.256730
$612$	−7.04651	−0.284838
$613$	2.47045	0.0997807	0.0498904	−	0.998755i	$-0.484113\pi$
$613$	2.47045	0.0997807	0.0498904	+	0.998755i	$0.484113\pi$
$614$	−7.82835	−0.315927
$615$	34.7936	1.40301
$616$	9.11130	0.367105
$617$	4.82057	0.194069	0.0970344	−	0.995281i	$-0.469064\pi$
$617$	4.82057	0.194069	0.0970344	+	0.995281i	$0.469064\pi$
$618$	9.70600	0.390433
$619$	−44.8087	−1.80101	−0.900506	−	0.434843i	$-0.856804\pi$
$619$	−44.8087	−1.80101	−0.900506	+	0.434843i	$0.856804\pi$
$620$	29.1911	1.17234
$621$	−18.7234	−0.751346
$622$	−16.4106	−0.658004
$623$	−5.71263	−0.228872
$624$	7.95850	0.318595
$625$	−17.8225	−0.712900
$626$	−3.65458	−0.146066
$627$	−2.24634	−0.0897103
$628$	−10.1077	−0.403342
$629$	−0.426036	−0.0169872
$630$	−8.35327	−0.332802
$631$	4.03271	0.160540	0.0802698	−	0.996773i	$-0.474422\pi$
$631$	4.03271	0.160540	0.0802698	+	0.996773i	$0.474422\pi$
$632$	50.4425	2.00650
$633$	27.7181	1.10169
$634$	47.4326	1.88379
$635$	12.4022	0.492167
$636$	44.4086	1.76092
$637$	4.13860	0.163977
$638$	10.3383	0.409295
$639$	26.1364	1.03394
$640$	35.3630	1.39784
$641$	31.5473	1.24604	0.623022	−	0.782204i	$-0.285905\pi$
$641$	31.5473	1.24604	0.623022	+	0.782204i	$0.285905\pi$
$642$	38.6079	1.52373
$643$	−9.19665	−0.362680	−0.181340	−	0.983420i	$-0.558044\pi$
$643$	−9.19665	−0.362680	−0.181340	+	0.983420i	$0.558044\pi$
$644$	9.99250	0.393759
$645$	25.8344	1.01723
$646$	−0.202027	−0.00794866
$647$	−40.3403	−1.58594	−0.792970	−	0.609260i	$-0.791466\pi$
$647$	−40.3403	−1.58594	−0.792970	+	0.609260i	$0.791466\pi$
$648$	21.3653	0.839307
$649$	−28.4043	−1.11497
$650$	−1.74879	−0.0685931
$651$	−3.95793	−0.155124
$652$	23.1939	0.908343
$653$	5.34350	0.209107	0.104554	−	0.994519i	$-0.466659\pi$
$653$	5.34350	0.209107	0.104554	+	0.994519i	$0.466659\pi$
$654$	73.7466	2.88372
$655$	−16.9152	−0.660930
$656$	−32.4603	−1.26736
$657$	−14.0895	−0.549683
$658$	−10.6635	−0.415708
$659$	31.9530	1.24471	0.622356	−	0.782734i	$-0.286175\pi$
$659$	31.9530	1.24471	0.622356	+	0.782734i	$0.286175\pi$
$660$	90.7983	3.53432
$661$	−10.6862	−0.415644	−0.207822	−	0.978167i	$-0.566637\pi$
$661$	−10.6862	−0.415644	−0.207822	+	0.978167i	$0.566637\pi$
$662$	−72.9959	−2.83707
$663$	−0.658909	−0.0255899
$664$	−13.0006	−0.504522
$665$	−0.161562	−0.00626511
$666$	−10.9098	−0.422748
$667$	5.86909	0.227252
$668$	−3.70003	−0.143159
$669$	−21.2915	−0.823179
$670$	−15.5818	−0.601977
$671$	22.8427	0.881834
$672$	1.65590	0.0638778
$673$	−20.4391	−0.787869	−0.393935	−	0.919138i	$-0.628886\pi$
$673$	−20.4391	−0.787869	−0.393935	+	0.919138i	$0.628886\pi$
$674$	71.9140	2.77002
$675$	3.71450	0.142971
$676$	−52.3802	−2.01462
$677$	−13.1318	−0.504698	−0.252349	−	0.967636i	$-0.581203\pi$
$677$	−13.1318	−0.504698	−0.252349	+	0.967636i	$0.581203\pi$
$678$	21.0415	0.808096
$679$	−1.10380	−0.0423599
$680$	4.22710	0.162102
$681$	54.2046	2.07712
$682$	37.1635	1.42307
$683$	1.14042	0.0436370	0.0218185	−	0.999762i	$-0.493054\pi$
$683$	1.14042	0.0436370	0.0218185	+	0.999762i	$0.493054\pi$
$684$	−3.49003	−0.133445
$685$	−5.11270	−0.195346
$686$	14.0804	0.537591
$687$	−1.17480	−0.0448213
$688$	−24.1019	−0.918878
$689$	2.41988	0.0921899
$690$	76.4107	2.90891
$691$	−34.7072	−1.32032	−0.660162	−	0.751123i	$-0.729512\pi$
$691$	−34.7072	−1.32032	−0.660162	+	0.751123i	$0.729512\pi$
$692$	81.2052	3.08696
$693$	−7.17414	−0.272523
$694$	19.3416	0.734198
$695$	1.95848	0.0742894
$696$	14.2678	0.540819
$697$	2.68749	0.101796
$698$	10.0122	0.378969
$699$	−27.2355	−1.03014
$700$	−1.98239	−0.0749272
$701$	−12.4755	−0.471192	−0.235596	−	0.971851i	$-0.575704\pi$
$701$	−12.4755	−0.471192	−0.235596	+	0.971851i	$0.575704\pi$
$702$	−4.79144	−0.180841
$703$	−0.211009	−0.00795837
$704$	25.3118	0.953975
$705$	−55.0084	−2.07174
$706$	−44.1967	−1.66337
$707$	5.34172	0.200896
$708$	−75.7291	−2.84608
$709$	−45.5508	−1.71070	−0.855349	−	0.518052i	$-0.826657\pi$
$709$	−45.5508	−1.71070	−0.855349	+	0.518052i	$0.826657\pi$
$710$	−30.2889	−1.13672
$711$	−39.7179	−1.48954
$712$	74.0276	2.77430
$713$	21.0980	0.790126
$714$	−1.10721	−0.0414362
$715$	4.94770	0.185034
$716$	82.0212	3.06528
$717$	56.1896	2.09844
$718$	76.4740	2.85398
$719$	−18.9813	−0.707883	−0.353941	−	0.935268i	$-0.615159\pi$
$719$	−18.9813	−0.707883	−0.353941	+	0.935268i	$0.615159\pi$
$720$	40.2011	1.49821
$721$	0.599539	0.0223280
$722$	47.0043	1.74932
$723$	−17.6588	−0.656737
$724$	−30.4482	−1.13160
$725$	−1.16436	−0.0432431
$726$	42.4729	1.57632
$727$	47.5236	1.76255	0.881276	−	0.472602i	$-0.156685\pi$
$727$	47.5236	1.76255	0.881276	+	0.472602i	$0.156685\pi$
$728$	1.32369	0.0490591
$729$	−42.4849	−1.57352
$730$	16.3280	0.604326
$731$	1.99547	0.0738052
$732$	60.9013	2.25098
$733$	11.7813	0.435154	0.217577	−	0.976043i	$-0.430185\pi$
$733$	11.7813	0.435154	0.217577	+	0.976043i	$0.430185\pi$
$734$	−55.4309	−2.04599
$735$	35.8745	1.32325
$736$	−8.82688	−0.325363
$737$	−13.3823	−0.492943
$738$	68.8206	2.53332
$739$	−5.08361	−0.187004	−0.0935018	−	0.995619i	$-0.529806\pi$
$739$	−5.08361	−0.187004	−0.0935018	+	0.995619i	$0.529806\pi$
$740$	8.52910	0.313536
$741$	−0.326348	−0.0119887
$742$	4.06628	0.149278
$743$	−35.1872	−1.29089	−0.645447	−	0.763805i	$-0.723329\pi$
$743$	−35.1872	−1.29089	−0.645447	+	0.763805i	$0.723329\pi$
$744$	51.2892	1.88035
$745$	−7.58579	−0.277922
$746$	−9.57852	−0.350695
$747$	10.2366	0.374536
$748$	7.01335	0.256433
$749$	2.38481	0.0871390
$750$	−80.2548	−2.93049
$751$	−7.25119	−0.264600	−0.132300	−	0.991210i	$-0.542236\pi$
$751$	−7.25119	−0.264600	−0.132300	+	0.991210i	$0.542236\pi$
$752$	51.3195	1.87143
$753$	−22.1836	−0.808416
$754$	1.50194	0.0546973
$755$	11.6006	0.422190
$756$	−5.43147	−0.197541
$757$	−30.1112	−1.09441	−0.547204	−	0.836999i	$-0.684308\pi$
$757$	−30.1112	−1.09441	−0.547204	+	0.836999i	$0.684308\pi$
$758$	−22.9382	−0.833153
$759$	65.6248	2.38203
$760$	2.09362	0.0759435
$761$	−29.6597	−1.07516	−0.537581	−	0.843212i	$-0.680662\pi$
$761$	−29.6597	−1.07516	−0.537581	+	0.843212i	$0.680662\pi$
$762$	42.0964	1.52499
$763$	4.55532	0.164914
$764$	26.2741	0.950562
$765$	−3.32837	−0.120337
$766$	26.6204	0.961834
$767$	−4.12657	−0.149002
$768$	87.4796	3.15665
$769$	−11.4839	−0.414119	−0.207059	−	0.978328i	$-0.566389\pi$
$769$	−11.4839	−0.414119	−0.207059	+	0.978328i	$0.566389\pi$
$770$	8.31395	0.299614
$771$	18.3915	0.662354
$772$	−59.2629	−2.13292
$773$	−4.63517	−0.166715	−0.0833577	−	0.996520i	$-0.526564\pi$
$773$	−4.63517	−0.166715	−0.0833577	+	0.996520i	$0.526564\pi$
$774$	51.0996	1.83674
$775$	−4.18558	−0.150350
$776$	14.3037	0.513472
$777$	−1.15643	−0.0414869
$778$	0.836256	0.0299812
$779$	1.33107	0.0476906
$780$	13.1911	0.472319
$781$	−26.0134	−0.930832
$782$	5.90204	0.211056
$783$	−3.19018	−0.114008
$784$	−33.4687	−1.19531
$785$	−4.77432	−0.170403
$786$	−57.4145	−2.04791
$787$	12.0519	0.429604	0.214802	−	0.976658i	$-0.431089\pi$
$787$	12.0519	0.429604	0.214802	+	0.976658i	$0.431089\pi$
$788$	34.1700	1.21726
$789$	−35.6358	−1.26867
$790$	46.0282	1.63761
$791$	1.29973	0.0462132
$792$	92.9668	3.30343
$793$	3.31858	0.117846
$794$	68.7076	2.43834
$795$	20.9761	0.743945
$796$	−57.0943	−2.02365
$797$	−15.1578	−0.536917	−0.268459	−	0.963291i	$-0.586514\pi$
$797$	−15.1578	−0.536917	−0.268459	+	0.963291i	$0.586514\pi$
$798$	−0.548384	−0.0194126
$799$	−4.24890	−0.150315
$800$	1.75114	0.0619123
$801$	−58.2885	−2.05952
$802$	−47.6446	−1.68239
$803$	14.0232	0.494867
$804$	−35.6787	−1.25829
$805$	4.71989	0.166354
$806$	5.39910	0.190175
$807$	−42.0517	−1.48029
$808$	−69.2211	−2.43519
$809$	19.4956	0.685428	0.342714	−	0.939440i	$-0.388654\pi$
$809$	19.4956	0.685428	0.342714	+	0.939440i	$0.388654\pi$
$810$	19.4955	0.685003
$811$	9.84254	0.345618	0.172809	−	0.984955i	$-0.444716\pi$
$811$	9.84254	0.345618	0.172809	+	0.984955i	$0.444716\pi$
$812$	1.70256	0.0597482
$813$	33.5485	1.17660
$814$	10.8585	0.380590
$815$	10.9555	0.383754
$816$	5.32857	0.186537
$817$	0.988328	0.0345772
$818$	89.6111	3.13318
$819$	−1.04226	−0.0364194
$820$	−53.8026	−1.87887
$821$	32.8380	1.14605	0.573027	−	0.819536i	$-0.305769\pi$
$821$	32.8380	1.14605	0.573027	+	0.819536i	$0.305769\pi$
$822$	−17.3538	−0.605284
$823$	−24.2639	−0.845788	−0.422894	−	0.906179i	$-0.638986\pi$
$823$	−24.2639	−0.845788	−0.422894	+	0.906179i	$0.638986\pi$
$824$	−7.76918	−0.270652
$825$	−13.0192	−0.453269
$826$	−6.93415	−0.241270
$827$	−27.6304	−0.960805	−0.480402	−	0.877048i	$-0.659509\pi$
$827$	−27.6304	−0.960805	−0.480402	+	0.877048i	$0.659509\pi$
$828$	101.958	3.54329
$829$	7.86292	0.273091	0.136545	−	0.990634i	$-0.456400\pi$
$829$	7.86292	0.273091	0.136545	+	0.990634i	$0.456400\pi$
$830$	−11.8629	−0.411768
$831$	−56.6015	−1.96348
$832$	3.67729	0.127487
$833$	2.77098	0.0960086
$834$	6.64759	0.230187
$835$	−1.74768	−0.0604811
$836$	3.47361	0.120137
$837$	−11.4679	−0.396389
$838$	−5.47414	−0.189101
$839$	50.1952	1.73293	0.866465	−	0.499238i	$-0.166387\pi$
$839$	50.1952	1.73293	0.866465	+	0.499238i	$0.166387\pi$
$840$	11.4741	0.395892
$841$	1.00000	0.0344828
$842$	−18.7946	−0.647704
$843$	71.0979	2.44874
$844$	−42.8615	−1.47535
$845$	−24.7414	−0.851131
$846$	−108.805	−3.74079
$847$	2.62355	0.0901461
$848$	−19.5694	−0.672017
$849$	−11.4201	−0.391936
$850$	−1.17089	−0.0401612
$851$	6.16444	0.211314
$852$	−69.3546	−2.37605
$853$	6.14227	0.210307	0.105154	−	0.994456i	$-0.466467\pi$
$853$	6.14227	0.210307	0.105154	+	0.994456i	$0.466467\pi$
$854$	5.57644	0.190822
$855$	−1.64849	−0.0563773
$856$	−30.9037	−1.05627
$857$	−8.10696	−0.276928	−0.138464	−	0.990367i	$-0.544217\pi$
$857$	−8.10696	−0.276928	−0.138464	+	0.990367i	$0.544217\pi$
$858$	16.7938	0.573331
$859$	−11.5917	−0.395504	−0.197752	−	0.980252i	$-0.563364\pi$
$859$	−11.5917	−0.395504	−0.197752	+	0.980252i	$0.563364\pi$
$860$	−39.9487	−1.36224
$861$	7.29493	0.248610
$862$	2.91832	0.0993984
$863$	−21.9370	−0.746743	−0.373371	−	0.927682i	$-0.621798\pi$
$863$	−21.9370	−0.746743	−0.373371	+	0.927682i	$0.621798\pi$
$864$	4.79790	0.163228
$865$	38.3567	1.30417
$866$	−56.4412	−1.91795
$867$	45.1422	1.53311
$868$	6.12030	0.207737
$869$	39.5310	1.34100
$870$	13.0192	0.441391
$871$	−1.94418	−0.0658759
$872$	−59.0306	−1.99903
$873$	−11.2626	−0.381180
$874$	2.92319	0.0988784
$875$	−4.95733	−0.167588
$876$	37.3874	1.26320
$877$	44.2606	1.49457	0.747287	−	0.664502i	$-0.231356\pi$
$877$	44.2606	1.49457	0.747287	+	0.664502i	$0.231356\pi$
$878$	32.2612	1.08876
$879$	23.4118	0.789662
$880$	−40.0119	−1.34880
$881$	−2.70455	−0.0911188	−0.0455594	−	0.998962i	$-0.514507\pi$
$881$	−2.70455	−0.0911188	−0.0455594	+	0.998962i	$0.514507\pi$
$882$	70.9586	2.38930
$883$	−56.0666	−1.88679	−0.943395	−	0.331671i	$-0.892388\pi$
$883$	−56.0666	−1.88679	−0.943395	+	0.331671i	$0.892388\pi$
$884$	1.01890	0.0342692
$885$	−35.7701	−1.20240
$886$	−42.6894	−1.43418
$887$	−41.7189	−1.40078	−0.700391	−	0.713759i	$-0.746991\pi$
$887$	−41.7189	−1.40078	−0.700391	+	0.713759i	$0.746991\pi$
$888$	14.9858	0.502889
$889$	2.60029	0.0872110
$890$	67.5493	2.26426
$891$	16.7436	0.560932
$892$	32.9239	1.10237
$893$	−2.10442	−0.0704216
$894$	−25.7481	−0.861146
$895$	38.7422	1.29501
$896$	7.41431	0.247695
$897$	9.53395	0.318329
$898$	−32.2484	−1.07614
$899$	3.59476	0.119892
$900$	−20.2272	−0.674240
$901$	1.62021	0.0539771
$902$	−68.4967	−2.28069
$903$	5.41652	0.180251
$904$	−16.8427	−0.560181
$905$	−14.3820	−0.478074
$906$	39.3755	1.30816
$907$	20.1459	0.668933	0.334467	−	0.942408i	$-0.391444\pi$
$907$	20.1459	0.668933	0.334467	+	0.942408i	$0.391444\pi$
$908$	−83.8186	−2.78162
$909$	54.5040	1.80778
$910$	1.20785	0.0400398
$911$	5.46076	0.180923	0.0904615	−	0.995900i	$-0.471166\pi$
$911$	5.46076	0.180923	0.0904615	+	0.995900i	$0.471166\pi$
$912$	2.63916	0.0873914
$913$	−10.1884	−0.337186
$914$	−37.6051	−1.24387
$915$	28.7663	0.950986
$916$	1.81663	0.0600232
$917$	−3.54649	−0.117115
$918$	−3.20808	−0.105883
$919$	53.8633	1.77679	0.888393	−	0.459084i	$-0.151822\pi$
$919$	53.8633	1.77679	0.888393	+	0.459084i	$0.151822\pi$
$920$	−61.1631	−2.01649
$921$	−8.46682	−0.278991
$922$	30.0946	0.991112
$923$	−3.77921	−0.124394
$924$	19.0371	0.626273
$925$	−1.22295	−0.0402103
$926$	81.3341	2.67281
$927$	6.11737	0.200921
$928$	−1.50396	−0.0493699
$929$	21.3794	0.701437	0.350718	−	0.936481i	$-0.385937\pi$
$929$	21.3794	0.701437	0.350718	+	0.936481i	$0.385937\pi$
$930$	46.8008	1.53466
$931$	1.37242	0.0449794
$932$	42.1154	1.37953
$933$	−17.7490	−0.581076
$934$	6.72533	0.220060
$935$	3.31271	0.108337
$936$	13.5062	0.441463
$937$	−32.0343	−1.04651	−0.523257	−	0.852175i	$-0.675283\pi$
$937$	−32.0343	−1.04651	−0.523257	+	0.852175i	$0.675283\pi$
$938$	−3.26693	−0.106669
$939$	−3.95264	−0.128989
$940$	85.0616	2.77440
$941$	−30.9660	−1.00946	−0.504731	−	0.863277i	$-0.668408\pi$
$941$	−30.9660	−1.00946	−0.504731	+	0.863277i	$0.668408\pi$
$942$	−16.2053	−0.527996
$943$	−38.8860	−1.26630
$944$	33.3714	1.08615
$945$	−2.56552	−0.0834563
$946$	−50.8592	−1.65357
$947$	6.18079	0.200849	0.100424	−	0.994945i	$-0.467980\pi$
$947$	6.18079	0.200849	0.100424	+	0.994945i	$0.467980\pi$
$948$	105.394	3.42304
$949$	2.03728	0.0661329
$950$	−0.579925	−0.0188153
$951$	51.3011	1.66355
$952$	0.886266	0.0287241
$953$	−35.1365	−1.13818	−0.569092	−	0.822274i	$-0.692705\pi$
$953$	−35.1365	−1.13818	−0.569092	+	0.822274i	$0.692705\pi$
$954$	41.4901	1.34329
$955$	12.4104	0.401590
$956$	−86.8881	−2.81016
$957$	11.1814	0.361444
$958$	81.2321	2.62449
$959$	−1.07195	−0.0346149
$960$	31.8757	1.02878
$961$	−18.0777	−0.583151
$962$	1.57752	0.0508612
$963$	24.3333	0.784129
$964$	27.3064	0.879481
$965$	−27.9924	−0.901107
$966$	16.0205	0.515452
$967$	1.43299	0.0460817	0.0230409	−	0.999735i	$-0.492665\pi$
$967$	1.43299	0.0460817	0.0230409	+	0.999735i	$0.492665\pi$
$968$	−33.9975	−1.09272
$969$	−0.218504	−0.00701937
$970$	13.0519	0.419072
$971$	16.3575	0.524936	0.262468	−	0.964941i	$-0.415464\pi$
$971$	16.3575	0.524936	0.262468	+	0.964941i	$0.415464\pi$
$972$	84.3227	2.70465
$973$	0.410621	0.0131639
$974$	20.6543	0.661807
$975$	−1.89142	−0.0605738
$976$	−26.8373	−0.859040
$977$	2.35906	0.0754730	0.0377365	−	0.999288i	$-0.487985\pi$
$977$	2.35906	0.0754730	0.0377365	+	0.999288i	$0.487985\pi$
$978$	37.1857	1.18907
$979$	58.0142	1.85414
$980$	−55.4740	−1.77205
$981$	46.4800	1.48399
$982$	31.8606	1.01671
$983$	1.65513	0.0527904	0.0263952	−	0.999652i	$-0.491597\pi$
$983$	1.65513	0.0527904	0.0263952	+	0.999652i	$0.491597\pi$
$984$	−94.5320	−3.01357
$985$	16.1400	0.514262
$986$	1.00561	0.0320252
$987$	−11.5332	−0.367107
$988$	0.504644	0.0160549
$989$	−28.8731	−0.918110
$990$	84.8311	2.69611
$991$	11.5386	0.366534	0.183267	−	0.983063i	$-0.441333\pi$
$991$	11.5386	0.366534	0.183267	+	0.983063i	$0.441333\pi$
$992$	−5.40637	−0.171653
$993$	−78.9494	−2.50538
$994$	−6.35046	−0.201424
$995$	−26.9681	−0.854947
$996$	−27.1634	−0.860704
$997$	10.6025	0.335786	0.167893	−	0.985805i	$-0.446304\pi$
$997$	10.6025	0.335786	0.167893	+	0.985805i	$0.446304\pi$
$998$	56.0302	1.77361
$999$	−3.35071	−0.106012

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.b.1.4	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.b.1.4	✓	59	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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.47917 q^{2} -2.68137 q^{3} +4.14631 q^{4} +1.95848 q^{5} +6.64759 q^{6} +0.410621 q^{7} -5.32107 q^{8} +4.18976 q^{9} +O(q^{10})\)	\(q-2.47917 q^{2} -2.68137 q^{3} +4.14631 q^{4} +1.95848 q^{5} +6.64759 q^{6} +0.410621 q^{7} -5.32107 q^{8} +4.18976 q^{9} -4.85541 q^{10} -4.17004 q^{11} -11.1178 q^{12} -0.605821 q^{13} -1.01800 q^{14} -5.25141 q^{15} +4.89925 q^{16} -0.405624 q^{17} -10.3871 q^{18} -0.200900 q^{19} +8.12046 q^{20} -1.10103 q^{21} +10.3383 q^{22} +5.86909 q^{23} +14.2678 q^{24} -1.16436 q^{25} +1.50194 q^{26} -3.19018 q^{27} +1.70256 q^{28} +1.00000 q^{29} +13.0192 q^{30} +3.59476 q^{31} -1.50396 q^{32} +11.1814 q^{33} +1.00561 q^{34} +0.804193 q^{35} +17.3720 q^{36} +1.05032 q^{37} +0.498065 q^{38} +1.62443 q^{39} -10.4212 q^{40} -6.62556 q^{41} +2.72964 q^{42} -4.91951 q^{43} -17.2903 q^{44} +8.20555 q^{45} -14.5505 q^{46} +10.4750 q^{47} -13.1367 q^{48} -6.83139 q^{49} +2.88664 q^{50} +1.08763 q^{51} -2.51192 q^{52} -3.99437 q^{53} +7.90901 q^{54} -8.16694 q^{55} -2.18495 q^{56} +0.538687 q^{57} -2.47917 q^{58} +6.81153 q^{59} -21.7740 q^{60} -5.47783 q^{61} -8.91204 q^{62} +1.72040 q^{63} -6.06993 q^{64} -1.18649 q^{65} -27.7207 q^{66} +3.20916 q^{67} -1.68184 q^{68} -15.7372 q^{69} -1.99374 q^{70} +6.23817 q^{71} -22.2940 q^{72} -3.36284 q^{73} -2.60393 q^{74} +3.12207 q^{75} -0.832992 q^{76} -1.71231 q^{77} -4.02725 q^{78} -9.47977 q^{79} +9.59509 q^{80} -4.01522 q^{81} +16.4259 q^{82} +2.44323 q^{83} -4.56520 q^{84} -0.794407 q^{85} +12.1963 q^{86} -2.68137 q^{87} +22.1891 q^{88} -13.9122 q^{89} -20.3430 q^{90} -0.248763 q^{91} +24.3351 q^{92} -9.63889 q^{93} -25.9693 q^{94} -0.393458 q^{95} +4.03268 q^{96} -2.68812 q^{97} +16.9362 q^{98} -17.4714 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9 - 18 * q^10 - 27 * q^11 - 8 * q^12 - 22 * q^13 - 24 * q^14 - 18 * q^15 + 5 * q^16 - 23 * q^17 + q^18 - 32 * q^19 - 14 * q^20 - 36 * q^21 - 6 * q^22 - 3 * q^23 - 18 * q^24 - 8 * q^25 - q^26 - 12 * q^27 - 9 * q^28 + 59 * q^29 - 18 * q^30 - 32 * q^31 - 39 * q^32 - 12 * q^33 - 18 * q^34 - 9 * q^35 + 10 * q^36 - 44 * q^37 + 5 * q^38 - 27 * q^39 - 68 * q^40 - 44 * q^41 - 25 * q^42 - 40 * q^43 - 56 * q^44 - 39 * q^45 - 40 * q^46 - 20 * q^47 - 9 * q^48 - 39 * q^49 - 21 * q^50 - 28 * q^51 - 49 * q^52 - 31 * q^53 - 32 * q^54 - 32 * q^55 - 48 * q^56 - 58 * q^57 - 5 * q^58 + 6 * q^59 - 44 * q^60 - 88 * q^61 + 35 * q^62 - 22 * q^63 - 10 * q^64 - 43 * q^65 - 31 * q^66 - 45 * q^67 - 29 * q^68 - 60 * q^69 - 14 * q^70 - 20 * q^71 - 4 * q^72 - 90 * q^73 - 25 * q^74 + 15 * q^75 - 64 * q^76 - 39 * q^77 - 28 * q^78 - 120 * q^79 + 24 * q^80 - 77 * q^81 - 71 * q^82 - 33 * q^83 - 14 * q^84 - 71 * q^85 - 61 * q^86 - 6 * q^87 - 34 * q^88 - 78 * q^89 - 88 * q^90 - 28 * q^91 - 31 * q^92 - 36 * q^93 - 4 * q^94 - 12 * q^95 - 29 * q^96 - 48 * q^97 - 4 * q^98 - 43 * q^99

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