LMFDB - Embedded newform 4009.2.a.c.1.12

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			1.12
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)

$f(q)$	$=$	$q-2.25185 q^{2} -1.69642 q^{3} +3.07082 q^{4} -1.12007 q^{5} +3.82008 q^{6} -4.02650 q^{7} -2.41131 q^{8} -0.122162 q^{9} +O(q^{10})$	\(q-2.25185 q^{2} -1.69642 q^{3} +3.07082 q^{4} -1.12007 q^{5} +3.82008 q^{6} -4.02650 q^{7} -2.41131 q^{8} -0.122162 q^{9} +2.52222 q^{10} -2.48572 q^{11} -5.20939 q^{12} +4.38293 q^{13} +9.06706 q^{14} +1.90010 q^{15} -0.711723 q^{16} -3.94588 q^{17} +0.275090 q^{18} +1.00000 q^{19} -3.43952 q^{20} +6.83063 q^{21} +5.59746 q^{22} -3.68526 q^{23} +4.09060 q^{24} -3.74545 q^{25} -9.86969 q^{26} +5.29650 q^{27} -12.3646 q^{28} +3.83383 q^{29} -4.27874 q^{30} -4.15303 q^{31} +6.42532 q^{32} +4.21682 q^{33} +8.88552 q^{34} +4.50995 q^{35} -0.375136 q^{36} +6.52957 q^{37} -2.25185 q^{38} -7.43529 q^{39} +2.70083 q^{40} -4.13530 q^{41} -15.3815 q^{42} -4.23897 q^{43} -7.63319 q^{44} +0.136829 q^{45} +8.29865 q^{46} +7.04135 q^{47} +1.20738 q^{48} +9.21269 q^{49} +8.43418 q^{50} +6.69387 q^{51} +13.4592 q^{52} +2.97967 q^{53} -11.9269 q^{54} +2.78417 q^{55} +9.70915 q^{56} -1.69642 q^{57} -8.63319 q^{58} -11.9334 q^{59} +5.83487 q^{60} +1.19622 q^{61} +9.35199 q^{62} +0.491884 q^{63} -13.0454 q^{64} -4.90918 q^{65} -9.49564 q^{66} +8.71052 q^{67} -12.1171 q^{68} +6.25175 q^{69} -10.1557 q^{70} +9.49978 q^{71} +0.294570 q^{72} +11.9836 q^{73} -14.7036 q^{74} +6.35385 q^{75} +3.07082 q^{76} +10.0087 q^{77} +16.7431 q^{78} +5.78566 q^{79} +0.797177 q^{80} -8.61859 q^{81} +9.31206 q^{82} -14.2548 q^{83} +20.9756 q^{84} +4.41965 q^{85} +9.54552 q^{86} -6.50378 q^{87} +5.99385 q^{88} +14.2616 q^{89} -0.308119 q^{90} -17.6479 q^{91} -11.3168 q^{92} +7.04528 q^{93} -15.8560 q^{94} -1.12007 q^{95} -10.9000 q^{96} +2.21368 q^{97} -20.7456 q^{98} +0.303660 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.25185	−1.59230	−0.796148	−	0.605102i	$-0.793132\pi$
$2$	−2.25185	−1.59230	−0.796148	+	0.605102i	$0.793132\pi$
$3$	−1.69642	−0.979428	−0.489714	−	0.871883i	$-0.662899\pi$
$3$	−1.69642	−0.979428	−0.489714	+	0.871883i	$0.662899\pi$
$4$	3.07082	1.53541
$5$	−1.12007	−0.500909	−0.250455	−	0.968128i	$-0.580580\pi$
$5$	−1.12007	−0.500909	−0.250455	+	0.968128i	$0.580580\pi$
$6$	3.82008	1.55954
$7$	−4.02650	−1.52187	−0.760937	−	0.648826i	$-0.775260\pi$
$7$	−4.02650	−1.52187	−0.760937	+	0.648826i	$0.775260\pi$
$8$	−2.41131	−0.852528
$9$	−0.122162	−0.0407206
$10$	2.52222	0.797596
$11$	−2.48572	−0.749473	−0.374736	−	0.927131i	$-0.622267\pi$
$11$	−2.48572	−0.749473	−0.374736	+	0.927131i	$0.622267\pi$
$12$	−5.20939	−1.50382
$13$	4.38293	1.21561	0.607803	−	0.794088i	$-0.292051\pi$
$13$	4.38293	1.21561	0.607803	+	0.794088i	$0.292051\pi$
$14$	9.06706	2.42327
$15$	1.90010	0.490604
$16$	−0.711723	−0.177931
$17$	−3.94588	−0.957016	−0.478508	−	0.878083i	$-0.658822\pi$
$17$	−3.94588	−0.957016	−0.478508	+	0.878083i	$0.658822\pi$
$18$	0.275090	0.0648393
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.43952	−0.769100
$21$	6.83063	1.49057
$22$	5.59746	1.19338
$23$	−3.68526	−0.768430	−0.384215	−	0.923244i	$-0.625528\pi$
$23$	−3.68526	−0.768430	−0.384215	+	0.923244i	$0.625528\pi$
$24$	4.09060	0.834990
$25$	−3.74545	−0.749090
$26$	−9.86969	−1.93561
$27$	5.29650	1.01931
$28$	−12.3646	−2.33670
$29$	3.83383	0.711924	0.355962	−	0.934500i	$-0.384153\pi$
$29$	3.83383	0.711924	0.355962	+	0.934500i	$0.384153\pi$
$30$	−4.27874	−0.781188
$31$	−4.15303	−0.745906	−0.372953	−	0.927850i	$-0.621655\pi$
$31$	−4.15303	−0.745906	−0.372953	+	0.927850i	$0.621655\pi$
$32$	6.42532	1.13585
$33$	4.21682	0.734055
$34$	8.88552	1.52385
$35$	4.50995	0.762320
$36$	−0.375136	−0.0625227
$37$	6.52957	1.07346	0.536728	−	0.843756i	$-0.319660\pi$
$37$	6.52957	1.07346	0.536728	+	0.843756i	$0.319660\pi$
$38$	−2.25185	−0.365298
$39$	−7.43529	−1.19060
$40$	2.70083	0.427039
$41$	−4.13530	−0.645825	−0.322913	−	0.946429i	$-0.604662\pi$
$41$	−4.13530	−0.645825	−0.322913	+	0.946429i	$0.604662\pi$
$42$	−15.3815	−2.37342
$43$	−4.23897	−0.646437	−0.323219	−	0.946324i	$-0.604765\pi$
$43$	−4.23897	−0.646437	−0.323219	+	0.946324i	$0.604765\pi$
$44$	−7.63319	−1.15075
$45$	0.136829	0.0203973
$46$	8.29865	1.22357
$47$	7.04135	1.02709	0.513543	−	0.858064i	$-0.328333\pi$
$47$	7.04135	1.02709	0.513543	+	0.858064i	$0.328333\pi$
$48$	1.20738	0.174270
$49$	9.21269	1.31610
$50$	8.43418	1.19277
$51$	6.69387	0.937329
$52$	13.4592	1.86645
$53$	2.97967	0.409289	0.204644	−	0.978836i	$-0.434396\pi$
$53$	2.97967	0.409289	0.204644	+	0.978836i	$0.434396\pi$
$54$	−11.9269	−1.62305
$55$	2.78417	0.375418
$56$	9.70915	1.29744
$57$	−1.69642	−0.224696
$58$	−8.63319	−1.13359
$59$	−11.9334	−1.55360	−0.776801	−	0.629746i	$-0.783159\pi$
$59$	−11.9334	−1.55360	−0.776801	+	0.629746i	$0.783159\pi$
$60$	5.83487	0.753278
$61$	1.19622	0.153160	0.0765799	−	0.997063i	$-0.475600\pi$
$61$	1.19622	0.153160	0.0765799	+	0.997063i	$0.475600\pi$
$62$	9.35199	1.18770
$63$	0.491884	0.0619716
$64$	−13.0454	−1.63067
$65$	−4.90918	−0.608909
$66$	−9.49564	−1.16883
$67$	8.71052	1.06416	0.532080	−	0.846694i	$-0.321411\pi$
$67$	8.71052	1.06416	0.532080	+	0.846694i	$0.321411\pi$
$68$	−12.1171	−1.46941
$69$	6.25175	0.752622
$70$	−10.1557	−1.21384
$71$	9.49978	1.12742	0.563708	−	0.825974i	$-0.309374\pi$
$71$	9.49978	1.12742	0.563708	+	0.825974i	$0.309374\pi$
$72$	0.294570	0.0347154
$73$	11.9836	1.40258	0.701288	−	0.712878i	$-0.252609\pi$
$73$	11.9836	1.40258	0.701288	+	0.712878i	$0.252609\pi$
$74$	−14.7036	−1.70926
$75$	6.35385	0.733680
$76$	3.07082	0.352247
$77$	10.0087	1.14060
$78$	16.7431	1.89579
$79$	5.78566	0.650938	0.325469	−	0.945553i	$-0.394478\pi$
$79$	5.78566	0.650938	0.325469	+	0.945553i	$0.394478\pi$
$80$	0.797177	0.0891271
$81$	−8.61859	−0.957621
$82$	9.31206	1.02835
$83$	−14.2548	−1.56467	−0.782334	−	0.622859i	$-0.785971\pi$
$83$	−14.2548	−1.56467	−0.782334	+	0.622859i	$0.785971\pi$
$84$	20.9756	2.28863
$85$	4.41965	0.479378
$86$	9.54552	1.02932
$87$	−6.50378	−0.697278
$88$	5.99385	0.638947
$89$	14.2616	1.51173	0.755865	−	0.654727i	$-0.227217\pi$
$89$	14.2616	1.51173	0.755865	+	0.654727i	$0.227217\pi$
$90$	−0.308119	−0.0324786
$91$	−17.6479	−1.85000
$92$	−11.3168	−1.17985
$93$	7.04528	0.730562
$94$	−15.8560	−1.63542
$95$	−1.12007	−0.114916
$96$	−10.9000	−1.11248
$97$	2.21368	0.224765	0.112383	−	0.993665i	$-0.464152\pi$
$97$	2.21368	0.224765	0.112383	+	0.993665i	$0.464152\pi$
$98$	−20.7456	−2.09562
$99$	0.303660	0.0305190
$100$	−11.5016	−1.15016
$101$	−4.48942	−0.446714	−0.223357	−	0.974737i	$-0.571702\pi$
$101$	−4.48942	−0.446714	−0.223357	+	0.974737i	$0.571702\pi$
$102$	−15.0736	−1.49251
$103$	17.9596	1.76961	0.884805	−	0.465961i	$-0.154291\pi$
$103$	17.9596	1.76961	0.884805	+	0.465961i	$0.154291\pi$
$104$	−10.5686	−1.03634
$105$	−7.65076	−0.746638
$106$	−6.70975	−0.651709
$107$	−5.37537	−0.519656	−0.259828	−	0.965655i	$-0.583666\pi$
$107$	−5.37537	−0.519656	−0.259828	+	0.965655i	$0.583666\pi$
$108$	16.2646	1.56506
$109$	−1.59754	−0.153017	−0.0765085	−	0.997069i	$-0.524377\pi$
$109$	−1.59754	−0.153017	−0.0765085	+	0.997069i	$0.524377\pi$
$110$	−6.26953	−0.597776
$111$	−11.0769	−1.05137
$112$	2.86575	0.270788
$113$	−4.95133	−0.465782	−0.232891	−	0.972503i	$-0.574818\pi$
$113$	−4.95133	−0.465782	−0.232891	+	0.972503i	$0.574818\pi$
$114$	3.82008	0.357783
$115$	4.12774	0.384914
$116$	11.7730	1.09309
$117$	−0.535427	−0.0495002
$118$	26.8723	2.47380
$119$	15.8881	1.45646
$120$	−4.58174	−0.418254
$121$	−4.82119	−0.438290
$122$	−2.69370	−0.243876
$123$	7.01520	0.632539
$124$	−12.7532	−1.14527
$125$	9.79549	0.876135
$126$	−1.10765	−0.0986771
$127$	15.9156	1.41228	0.706142	−	0.708071i	$-0.250434\pi$
$127$	15.9156	1.41228	0.706142	+	0.708071i	$0.250434\pi$
$128$	16.5256	1.46067
$129$	7.19108	0.633139
$130$	11.0547	0.969563
$131$	−3.44142	−0.300678	−0.150339	−	0.988634i	$-0.548037\pi$
$131$	−3.44142	−0.300678	−0.150339	+	0.988634i	$0.548037\pi$
$132$	12.9491	1.12707
$133$	−4.02650	−0.349142
$134$	−19.6148	−1.69446
$135$	−5.93243	−0.510582
$136$	9.51475	0.815883
$137$	−11.7639	−1.00505	−0.502527	−	0.864561i	$-0.667596\pi$
$137$	−11.7639	−1.00505	−0.502527	+	0.864561i	$0.667596\pi$
$138$	−14.0780	−1.19840
$139$	−4.31817	−0.366262	−0.183131	−	0.983089i	$-0.558623\pi$
$139$	−4.31817	−0.366262	−0.183131	+	0.983089i	$0.558623\pi$
$140$	13.8492	1.17047
$141$	−11.9451	−1.00596
$142$	−21.3921	−1.79518
$143$	−10.8947	−0.911064
$144$	0.0869453	0.00724544
$145$	−4.29414	−0.356609
$146$	−26.9853	−2.23332
$147$	−15.6286	−1.28902
$148$	20.0511	1.64819
$149$	21.3018	1.74511	0.872557	−	0.488512i	$-0.162460\pi$
$149$	21.3018	1.74511	0.872557	+	0.488512i	$0.162460\pi$
$150$	−14.3079	−1.16824
$151$	−9.06852	−0.737986	−0.368993	−	0.929432i	$-0.620297\pi$
$151$	−9.06852	−0.737986	−0.368993	+	0.929432i	$0.620297\pi$
$152$	−2.41131	−0.195583
$153$	0.482036	0.0389703
$154$	−22.5382	−1.81618
$155$	4.65167	0.373631
$156$	−22.8324	−1.82806
$157$	10.0060	0.798564	0.399282	−	0.916828i	$-0.369259\pi$
$157$	10.0060	0.798564	0.399282	+	0.916828i	$0.369259\pi$
$158$	−13.0284	−1.03649
$159$	−5.05476	−0.400869
$160$	−7.19678	−0.568956
$161$	14.8387	1.16945
$162$	19.4077	1.52482
$163$	−11.4133	−0.893956	−0.446978	−	0.894545i	$-0.647500\pi$
$163$	−11.4133	−0.893956	−0.446978	+	0.894545i	$0.647500\pi$
$164$	−12.6987	−0.991605
$165$	−4.72312	−0.367695
$166$	32.0997	2.49142
$167$	−8.18730	−0.633552	−0.316776	−	0.948500i	$-0.602600\pi$
$167$	−8.18730	−0.633552	−0.316776	+	0.948500i	$0.602600\pi$
$168$	−16.4708	−1.27075
$169$	6.21010	0.477700
$170$	−9.95237	−0.763312
$171$	−0.122162	−0.00934195
$172$	−13.0171	−0.992545
$173$	16.8621	1.28200	0.641002	−	0.767539i	$-0.278519\pi$
$173$	16.8621	1.28200	0.641002	+	0.767539i	$0.278519\pi$
$174$	14.6455	1.11027
$175$	15.0811	1.14002
$176$	1.76914	0.133354
$177$	20.2441	1.52164
$178$	−32.1150	−2.40712
$179$	−9.04301	−0.675906	−0.337953	−	0.941163i	$-0.609735\pi$
$179$	−9.04301	−0.675906	−0.337953	+	0.941163i	$0.609735\pi$
$180$	0.420178	0.0313182
$181$	1.49721	0.111287	0.0556433	−	0.998451i	$-0.482279\pi$
$181$	1.49721	0.111287	0.0556433	+	0.998451i	$0.482279\pi$
$182$	39.7403	2.94575
$183$	−2.02929	−0.150009
$184$	8.88632	0.655108
$185$	−7.31356	−0.537703
$186$	−15.8649	−1.16327
$187$	9.80835	0.717258
$188$	21.6227	1.57700
$189$	−21.3263	−1.55126
$190$	2.52222	0.182981
$191$	18.2691	1.32191	0.660954	−	0.750427i	$-0.270152\pi$
$191$	18.2691	1.32191	0.660954	+	0.750427i	$0.270152\pi$
$192$	22.1304	1.59713
$193$	19.5292	1.40574	0.702870	−	0.711319i	$-0.251902\pi$
$193$	19.5292	1.40574	0.702870	+	0.711319i	$0.251902\pi$
$194$	−4.98487	−0.357893
$195$	8.32802	0.596382
$196$	28.2905	2.02075
$197$	6.85772	0.488592	0.244296	−	0.969701i	$-0.421443\pi$
$197$	6.85772	0.488592	0.244296	+	0.969701i	$0.421443\pi$
$198$	−0.683796	−0.0485953
$199$	3.97573	0.281832	0.140916	−	0.990022i	$-0.454995\pi$
$199$	3.97573	0.281832	0.140916	+	0.990022i	$0.454995\pi$
$200$	9.03145	0.638620
$201$	−14.7767	−1.04227
$202$	10.1095	0.711302
$203$	−15.4369	−1.08346
$204$	20.5556	1.43918
$205$	4.63181	0.323500
$206$	−40.4422	−2.81774
$207$	0.450198	0.0312909
$208$	−3.11943	−0.216294
$209$	−2.48572	−0.171941
$210$	17.2283	1.18887
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	9.15001	0.628425
$213$	−16.1156	−1.10422
$214$	12.1045	0.827447
$215$	4.74793	0.323806
$216$	−12.7715	−0.868991
$217$	16.7222	1.13517
$218$	3.59743	0.243648
$219$	−20.3292	−1.37372
$220$	8.54968	0.576419
$221$	−17.2945	−1.16336
$222$	24.9435	1.67410
$223$	29.1951	1.95505	0.977524	−	0.210823i	$-0.0676144\pi$
$223$	29.1951	1.95505	0.977524	+	0.210823i	$0.0676144\pi$
$224$	−25.8715	−1.72861
$225$	0.457551	0.0305034
$226$	11.1496	0.741663
$227$	−9.29434	−0.616887	−0.308443	−	0.951243i	$-0.599808\pi$
$227$	−9.29434	−0.616887	−0.308443	+	0.951243i	$0.599808\pi$
$228$	−5.20939	−0.345000
$229$	−13.4718	−0.890238	−0.445119	−	0.895471i	$-0.646839\pi$
$229$	−13.4718	−0.890238	−0.445119	+	0.895471i	$0.646839\pi$
$230$	−9.29504	−0.612897
$231$	−16.9790	−1.11714
$232$	−9.24456	−0.606935
$233$	4.47067	0.292883	0.146442	−	0.989219i	$-0.453218\pi$
$233$	4.47067	0.292883	0.146442	+	0.989219i	$0.453218\pi$
$234$	1.20570	0.0788191
$235$	−7.88678	−0.514477
$236$	−36.6454	−2.38541
$237$	−9.81491	−0.637547
$238$	−35.7775	−2.31911
$239$	12.5033	0.808774	0.404387	−	0.914588i	$-0.367485\pi$
$239$	12.5033	0.808774	0.404387	+	0.914588i	$0.367485\pi$
$240$	−1.35235	−0.0872936
$241$	13.4302	0.865117	0.432559	−	0.901606i	$-0.357611\pi$
$241$	13.4302	0.865117	0.432559	+	0.901606i	$0.357611\pi$
$242$	10.8566	0.697888
$243$	−1.26874	−0.0813899
$244$	3.67336	0.235163
$245$	−10.3188	−0.659246
$246$	−15.7972	−1.00719
$247$	4.38293	0.278879
$248$	10.0143	0.635906
$249$	24.1821	1.53248
$250$	−22.0579	−1.39507
$251$	12.3915	0.782145	0.391072	−	0.920360i	$-0.372104\pi$
$251$	12.3915	0.782145	0.391072	+	0.920360i	$0.372104\pi$
$252$	1.51049	0.0951517
$253$	9.16053	0.575918
$254$	−35.8396	−2.24877
$255$	−7.49758	−0.469516
$256$	−11.1223	−0.695145
$257$	14.7048	0.917258	0.458629	−	0.888628i	$-0.348341\pi$
$257$	14.7048	0.917258	0.458629	+	0.888628i	$0.348341\pi$
$258$	−16.1932	−1.00815
$259$	−26.2913	−1.63366
$260$	−15.0752	−0.934923
$261$	−0.468347	−0.0289900
$262$	7.74955	0.478769
$263$	−18.3385	−1.13080	−0.565400	−	0.824817i	$-0.691278\pi$
$263$	−18.3385	−1.13080	−0.565400	+	0.824817i	$0.691278\pi$
$264$	−10.1681	−0.625802
$265$	−3.33743	−0.205016
$266$	9.06706	0.555937
$267$	−24.1937	−1.48063
$268$	26.7484	1.63392
$269$	−22.3961	−1.36551	−0.682756	−	0.730646i	$-0.739219\pi$
$269$	−22.3961	−1.36551	−0.682756	+	0.730646i	$0.739219\pi$
$270$	13.3589	0.812998
$271$	4.00112	0.243051	0.121525	−	0.992588i	$-0.461221\pi$
$271$	4.00112	0.243051	0.121525	+	0.992588i	$0.461221\pi$
$272$	2.80837	0.170283
$273$	29.9382	1.81194
$274$	26.4904	1.60034
$275$	9.31014	0.561423
$276$	19.1980	1.15558
$277$	22.0830	1.32684	0.663420	−	0.748248i	$-0.269104\pi$
$277$	22.0830	1.32684	0.663420	+	0.748248i	$0.269104\pi$
$278$	9.72385	0.583198
$279$	0.507342	0.0303738
$280$	−10.8749	−0.649899
$281$	−21.6769	−1.29314	−0.646569	−	0.762855i	$-0.723797\pi$
$281$	−21.6769	−1.29314	−0.646569	+	0.762855i	$0.723797\pi$
$282$	26.8985	1.60178
$283$	4.33848	0.257896	0.128948	−	0.991651i	$-0.458840\pi$
$283$	4.33848	0.257896	0.128948	+	0.991651i	$0.458840\pi$
$284$	29.1721	1.73104
$285$	1.90010	0.112552
$286$	24.5333	1.45068
$287$	16.6508	0.982864
$288$	−0.784928	−0.0462523
$289$	−1.43003	−0.0841196
$290$	9.66975	0.567827
$291$	−3.75533	−0.220142
$292$	36.7995	2.15353
$293$	−10.6239	−0.620654	−0.310327	−	0.950630i	$-0.600439\pi$
$293$	−10.6239	−0.620654	−0.310327	+	0.950630i	$0.600439\pi$
$294$	35.1932	2.05251
$295$	13.3663	0.778214
$296$	−15.7448	−0.915151
$297$	−13.1656	−0.763946
$298$	−47.9685	−2.77874
$299$	−16.1523	−0.934109
$300$	19.5115	1.12650
$301$	17.0682	0.983796
$302$	20.4209	1.17509
$303$	7.61595	0.437525
$304$	−0.711723	−0.0408201
$305$	−1.33984	−0.0767192
$306$	−1.08547	−0.0620522
$307$	−15.4552	−0.882074	−0.441037	−	0.897489i	$-0.645389\pi$
$307$	−15.4552	−0.882074	−0.441037	+	0.897489i	$0.645389\pi$
$308$	30.7350	1.75129
$309$	−30.4670	−1.73321
$310$	−10.4749	−0.594932
$311$	−34.1974	−1.93916	−0.969578	−	0.244784i	$-0.921283\pi$
$311$	−34.1974	−1.93916	−0.969578	+	0.244784i	$0.921283\pi$
$312$	17.9288	1.01502
$313$	−6.22729	−0.351987	−0.175994	−	0.984391i	$-0.556314\pi$
$313$	−6.22729	−0.351987	−0.175994	+	0.984391i	$0.556314\pi$
$314$	−22.5319	−1.27155
$315$	−0.550943	−0.0310421
$316$	17.7667	0.999455
$317$	23.8962	1.34215	0.671073	−	0.741392i	$-0.265834\pi$
$317$	23.8962	1.34215	0.671073	+	0.741392i	$0.265834\pi$
$318$	11.3826	0.638302
$319$	−9.52982	−0.533567
$320$	14.6117	0.816819
$321$	9.11887	0.508966
$322$	−33.4145	−1.86212
$323$	−3.94588	−0.219555
$324$	−26.4661	−1.47034
$325$	−16.4161	−0.910599
$326$	25.7009	1.42344
$327$	2.71010	0.149869
$328$	9.97150	0.550584
$329$	−28.3520	−1.56309
$330$	10.6358	0.585479
$331$	−30.0572	−1.65209	−0.826046	−	0.563603i	$-0.809415\pi$
$331$	−30.0572	−1.65209	−0.826046	+	0.563603i	$0.809415\pi$
$332$	−43.7739	−2.40240
$333$	−0.797664	−0.0437117
$334$	18.4365	1.00880
$335$	−9.75636	−0.533047
$336$	−4.86151	−0.265217
$337$	−25.9760	−1.41500	−0.707501	−	0.706712i	$-0.750178\pi$
$337$	−25.9760	−1.41500	−0.707501	+	0.706712i	$0.750178\pi$
$338$	−13.9842	−0.760640
$339$	8.39953	0.456200
$340$	13.5719	0.736041
$341$	10.3233	0.559037
$342$	0.275090	0.0148751
$343$	−8.90939	−0.481062
$344$	10.2215	0.551106
$345$	−7.00238	−0.376995
$346$	−37.9709	−2.04133
$347$	−2.34371	−0.125817	−0.0629084	−	0.998019i	$-0.520038\pi$
$347$	−2.34371	−0.125817	−0.0629084	+	0.998019i	$0.520038\pi$
$348$	−19.9719	−1.07061
$349$	−0.786662	−0.0421090	−0.0210545	−	0.999778i	$-0.506702\pi$
$349$	−0.786662	−0.0421090	−0.0210545	+	0.999778i	$0.506702\pi$
$350$	−33.9602	−1.81525
$351$	23.2142	1.23908
$352$	−15.9715	−0.851286
$353$	−26.6744	−1.41974	−0.709868	−	0.704335i	$-0.751245\pi$
$353$	−26.6744	−1.41974	−0.709868	+	0.704335i	$0.751245\pi$
$354$	−45.5867	−2.42290
$355$	−10.6404	−0.564733
$356$	43.7949	2.32112
$357$	−26.9528	−1.42650
$358$	20.3635	1.07624
$359$	−23.9133	−1.26209	−0.631047	−	0.775745i	$-0.717374\pi$
$359$	−23.9133	−1.26209	−0.631047	+	0.775745i	$0.717374\pi$
$360$	−0.329938	−0.0173893
$361$	1.00000	0.0526316
$362$	−3.37148	−0.177201
$363$	8.17877	0.429274
$364$	−54.1934	−2.84050
$365$	−13.4224	−0.702563
$366$	4.56964	0.238859
$367$	−2.07338	−0.108230	−0.0541149	−	0.998535i	$-0.517234\pi$
$367$	−2.07338	−0.108230	−0.0541149	+	0.998535i	$0.517234\pi$
$368$	2.62288	0.136727
$369$	0.505176	0.0262984
$370$	16.4690	0.856183
$371$	−11.9976	−0.622886
$372$	21.6348	1.12171
$373$	−10.0392	−0.519809	−0.259905	−	0.965634i	$-0.583691\pi$
$373$	−10.0392	−0.519809	−0.259905	+	0.965634i	$0.583691\pi$
$374$	−22.0869	−1.14209
$375$	−16.6173	−0.858111
$376$	−16.9789	−0.875619
$377$	16.8034	0.865419
$378$	48.0236	2.47007
$379$	−18.4413	−0.947267	−0.473633	−	0.880722i	$-0.657058\pi$
$379$	−18.4413	−0.947267	−0.473633	+	0.880722i	$0.657058\pi$
$380$	−3.43952	−0.176444
$381$	−26.9996	−1.38323
$382$	−41.1393	−2.10487
$383$	−22.7069	−1.16027	−0.580133	−	0.814522i	$-0.697000\pi$
$383$	−22.7069	−1.16027	−0.580133	+	0.814522i	$0.697000\pi$
$384$	−28.0343	−1.43062
$385$	−11.2105	−0.571338
$386$	−43.9767	−2.23835
$387$	0.517841	0.0263233
$388$	6.79781	0.345107
$389$	−31.1974	−1.58177	−0.790885	−	0.611965i	$-0.790379\pi$
$389$	−31.1974	−1.58177	−0.790885	+	0.611965i	$0.790379\pi$
$390$	−18.7534	−0.949617
$391$	14.5416	0.735400
$392$	−22.2147	−1.12201
$393$	5.83809	0.294493
$394$	−15.4425	−0.777984
$395$	−6.48033	−0.326061
$396$	0.932484	0.0468591
$397$	9.21080	0.462277	0.231138	−	0.972921i	$-0.425755\pi$
$397$	9.21080	0.462277	0.231138	+	0.972921i	$0.425755\pi$
$398$	−8.95274	−0.448760
$399$	6.83063	0.341959
$400$	2.66572	0.133286
$401$	25.0044	1.24866	0.624331	−	0.781160i	$-0.285372\pi$
$401$	25.0044	1.24866	0.624331	+	0.781160i	$0.285372\pi$
$402$	33.2749	1.65960
$403$	−18.2025	−0.906729
$404$	−13.7862	−0.685889
$405$	9.65340	0.479681
$406$	34.7615	1.72519
$407$	−16.2307	−0.804526
$408$	−16.1410	−0.799099
$409$	−22.2158	−1.09850	−0.549250	−	0.835658i	$-0.685087\pi$
$409$	−22.2158	−1.09850	−0.549250	+	0.835658i	$0.685087\pi$
$410$	−10.4301	−0.515107
$411$	19.9564	0.984378
$412$	55.1506	2.71707
$413$	48.0500	2.36439
$414$	−1.01378	−0.0498244
$415$	15.9663	0.783757
$416$	28.1617	1.38074
$417$	7.32542	0.358727
$418$	5.59746	0.273781
$419$	−27.8399	−1.36007	−0.680033	−	0.733182i	$-0.738035\pi$
$419$	−27.8399	−1.36007	−0.680033	+	0.733182i	$0.738035\pi$
$420$	−23.4941	−1.14639
$421$	−6.78808	−0.330831	−0.165415	−	0.986224i	$-0.552896\pi$
$421$	−6.78808	−0.330831	−0.165415	+	0.986224i	$0.552896\pi$
$422$	−2.25185	−0.109618
$423$	−0.860183	−0.0418235
$424$	−7.18491	−0.348930
$425$	14.7791	0.716892
$426$	36.2899	1.75825
$427$	−4.81657	−0.233090
$428$	−16.5068	−0.797884
$429$	18.4821	0.892322
$430$	−10.6916	−0.515596
$431$	−5.23567	−0.252193	−0.126097	−	0.992018i	$-0.540245\pi$
$431$	−5.23567	−0.252193	−0.126097	+	0.992018i	$0.540245\pi$
$432$	−3.76964	−0.181367
$433$	18.7965	0.903303	0.451651	−	0.892195i	$-0.350835\pi$
$433$	18.7965	0.903303	0.451651	+	0.892195i	$0.350835\pi$
$434$	−37.6558	−1.80753
$435$	7.28466	0.349273
$436$	−4.90576	−0.234943
$437$	−3.68526	−0.176290
$438$	45.7783	2.18737
$439$	−18.1490	−0.866206	−0.433103	−	0.901345i	$-0.642581\pi$
$439$	−18.1490	−0.866206	−0.433103	+	0.901345i	$0.642581\pi$
$440$	−6.71351	−0.320054
$441$	−1.12544	−0.0535923
$442$	38.9446	1.85241
$443$	−38.4985	−1.82912	−0.914560	−	0.404451i	$-0.867463\pi$
$443$	−38.4985	−1.82912	−0.914560	+	0.404451i	$0.867463\pi$
$444$	−34.0151	−1.61429
$445$	−15.9740	−0.757240
$446$	−65.7429	−3.11302
$447$	−36.1368	−1.70921
$448$	52.5272	2.48168
$449$	−7.60038	−0.358684	−0.179342	−	0.983787i	$-0.557397\pi$
$449$	−7.60038	−0.358684	−0.179342	+	0.983787i	$0.557397\pi$
$450$	−1.03033	−0.0485704
$451$	10.2792	0.484029
$452$	−15.2046	−0.715165
$453$	15.3840	0.722804
$454$	20.9294	0.982267
$455$	19.7668	0.926682
$456$	4.09060	0.191560
$457$	−3.86933	−0.181000	−0.0904999	−	0.995896i	$-0.528846\pi$
$457$	−3.86933	−0.181000	−0.0904999	+	0.995896i	$0.528846\pi$
$458$	30.3363	1.41752
$459$	−20.8993	−0.975497
$460$	12.6755	0.590999
$461$	−37.6000	−1.75121	−0.875604	−	0.483030i	$-0.839536\pi$
$461$	−37.6000	−1.75121	−0.875604	+	0.483030i	$0.839536\pi$
$462$	38.2342	1.77882
$463$	34.8066	1.61760	0.808800	−	0.588083i	$-0.200117\pi$
$463$	34.8066	1.61760	0.808800	+	0.588083i	$0.200117\pi$
$464$	−2.72862	−0.126673
$465$	−7.89118	−0.365945
$466$	−10.0673	−0.466357
$467$	20.4310	0.945436	0.472718	−	0.881214i	$-0.343273\pi$
$467$	20.4310	0.945436	0.472718	+	0.881214i	$0.343273\pi$
$468$	−1.64420	−0.0760031
$469$	−35.0729	−1.61952
$470$	17.7598	0.819199
$471$	−16.9743	−0.782136
$472$	28.7753	1.32449
$473$	10.5369	0.484487
$474$	22.1017	1.01516
$475$	−3.74545	−0.171853
$476$	48.7894	2.23626
$477$	−0.364001	−0.0166665
$478$	−28.1556	−1.28781
$479$	−19.2739	−0.880646	−0.440323	−	0.897840i	$-0.645136\pi$
$479$	−19.2739	−0.880646	−0.440323	+	0.897840i	$0.645136\pi$
$480$	12.2088	0.557251
$481$	28.6187	1.30490
$482$	−30.2428	−1.37752
$483$	−25.1727	−1.14540
$484$	−14.8050	−0.672954
$485$	−2.47947	−0.112587
$486$	2.85701	0.129597
$487$	8.21480	0.372248	0.186124	−	0.982526i	$-0.440407\pi$
$487$	8.21480	0.372248	0.186124	+	0.982526i	$0.440407\pi$
$488$	−2.88445	−0.130573
$489$	19.3617	0.875566
$490$	23.2364	1.04971
$491$	38.6937	1.74622	0.873112	−	0.487520i	$-0.162098\pi$
$491$	38.6937	1.74622	0.873112	+	0.487520i	$0.162098\pi$
$492$	21.5424	0.971206
$493$	−15.1278	−0.681323
$494$	−9.86969	−0.444059
$495$	−0.340120	−0.0152872
$496$	2.95581	0.132720
$497$	−38.2509	−1.71579
$498$	−54.4545	−2.44016
$499$	−30.7776	−1.37779	−0.688897	−	0.724859i	$-0.741905\pi$
$499$	−30.7776	−1.37779	−0.688897	+	0.724859i	$0.741905\pi$
$500$	30.0801	1.34522
$501$	13.8891	0.620519
$502$	−27.9038	−1.24541
$503$	−7.64072	−0.340683	−0.170341	−	0.985385i	$-0.554487\pi$
$503$	−7.64072	−0.340683	−0.170341	+	0.985385i	$0.554487\pi$
$504$	−1.18609	−0.0528325
$505$	5.02846	0.223763
$506$	−20.6281	−0.917031
$507$	−10.5349	−0.467873
$508$	48.8740	2.16843
$509$	32.5693	1.44361	0.721805	−	0.692097i	$-0.243313\pi$
$509$	32.5693	1.44361	0.721805	+	0.692097i	$0.243313\pi$
$510$	16.8834	0.747609
$511$	−48.2520	−2.13454
$512$	−8.00541	−0.353793
$513$	5.29650	0.233846
$514$	−33.1129	−1.46055
$515$	−20.1159	−0.886414
$516$	22.0825	0.972127
$517$	−17.5028	−0.769773
$518$	59.2040	2.60128
$519$	−28.6052	−1.25563
$520$	11.8376	0.519112
$521$	1.36055	0.0596066	0.0298033	−	0.999556i	$-0.490512\pi$
$521$	1.36055	0.0596066	0.0298033	+	0.999556i	$0.490512\pi$
$522$	1.05465	0.0461606
$523$	−14.7344	−0.644290	−0.322145	−	0.946690i	$-0.604404\pi$
$523$	−14.7344	−0.644290	−0.322145	+	0.946690i	$0.604404\pi$
$524$	−10.5680	−0.461664
$525$	−25.5838	−1.11657
$526$	41.2955	1.80057
$527$	16.3874	0.713845
$528$	−3.00121	−0.130611
$529$	−9.41885	−0.409515
$530$	7.51537	0.326447
$531$	1.45781	0.0632636
$532$	−12.3646	−0.536075
$533$	−18.1247	−0.785070
$534$	54.4805	2.35760
$535$	6.02077	0.260301
$536$	−21.0038	−0.907226
$537$	15.3407	0.662001
$538$	50.4325	2.17430
$539$	−22.9002	−0.986380
$540$	−18.2174	−0.783952
$541$	−16.8589	−0.724820	−0.362410	−	0.932019i	$-0.618046\pi$
$541$	−16.8589	−0.724820	−0.362410	+	0.932019i	$0.618046\pi$
$542$	−9.00990	−0.387008
$543$	−2.53989	−0.108997
$544$	−25.3535	−1.08702
$545$	1.78936	0.0766476
$546$	−67.4162	−2.88515
$547$	45.3508	1.93906	0.969529	−	0.244976i	$-0.0787800\pi$
$547$	45.3508	1.93906	0.969529	+	0.244976i	$0.0787800\pi$
$548$	−36.1246	−1.54317
$549$	−0.146132	−0.00623676
$550$	−20.9650	−0.893951
$551$	3.83383	0.163326
$552$	−15.0749	−0.641631
$553$	−23.2960	−0.990645
$554$	−49.7276	−2.11272
$555$	12.4069	0.526642
$556$	−13.2603	−0.562362
$557$	4.93287	0.209013	0.104506	−	0.994524i	$-0.466674\pi$
$557$	4.93287	0.209013	0.104506	+	0.994524i	$0.466674\pi$
$558$	−1.14246	−0.0483640
$559$	−18.5791	−0.785814
$560$	−3.20983	−0.135640
$561$	−16.6391	−0.702503
$562$	48.8132	2.05906
$563$	29.9198	1.26097	0.630484	−	0.776202i	$-0.282856\pi$
$563$	29.9198	1.26097	0.630484	+	0.776202i	$0.282856\pi$
$564$	−36.6811	−1.54455
$565$	5.54582	0.233314
$566$	−9.76960	−0.410647
$567$	34.7027	1.45738
$568$	−22.9069	−0.961154
$569$	−1.35117	−0.0566440	−0.0283220	−	0.999599i	$-0.509016\pi$
$569$	−1.35117	−0.0566440	−0.0283220	+	0.999599i	$0.509016\pi$
$570$	−4.27874	−0.179217
$571$	15.7296	0.658265	0.329132	−	0.944284i	$-0.393244\pi$
$571$	15.7296	0.658265	0.329132	+	0.944284i	$0.393244\pi$
$572$	−33.4558	−1.39886
$573$	−30.9921	−1.29471
$574$	−37.4950	−1.56501
$575$	13.8030	0.575623
$576$	1.59365	0.0664020
$577$	4.21519	0.175481	0.0877404	−	0.996143i	$-0.472035\pi$
$577$	4.21519	0.175481	0.0877404	+	0.996143i	$0.472035\pi$
$578$	3.22022	0.133943
$579$	−33.1296	−1.37682
$580$	−13.1865	−0.547540
$581$	57.3970	2.38123
$582$	8.45644	0.350531
$583$	−7.40662	−0.306751
$584$	−28.8962	−1.19573
$585$	0.599714	0.0247951
$586$	23.9234	0.988266
$587$	−42.4642	−1.75269	−0.876343	−	0.481687i	$-0.840024\pi$
$587$	−42.4642	−1.75269	−0.876343	+	0.481687i	$0.840024\pi$
$588$	−47.9925	−1.97918
$589$	−4.15303	−0.171123
$590$	−30.0988	−1.23915
$591$	−11.6336	−0.478541
$592$	−4.64724	−0.191001
$593$	−8.01253	−0.329035	−0.164518	−	0.986374i	$-0.552607\pi$
$593$	−8.01253	−0.329035	−0.164518	+	0.986374i	$0.552607\pi$
$594$	29.6469	1.21643
$595$	−17.7957	−0.729553
$596$	65.4140	2.67946
$597$	−6.74451	−0.276034
$598$	36.3724	1.48738
$599$	13.1507	0.537323	0.268661	−	0.963235i	$-0.413419\pi$
$599$	13.1507	0.537323	0.268661	+	0.963235i	$0.413419\pi$
$600$	−15.3211	−0.625483
$601$	21.4682	0.875705	0.437852	−	0.899047i	$-0.355739\pi$
$601$	21.4682	0.875705	0.437852	+	0.899047i	$0.355739\pi$
$602$	−38.4350	−1.56649
$603$	−1.06409	−0.0433332
$604$	−27.8478	−1.13311
$605$	5.40006	0.219544
$606$	−17.1499	−0.696669
$607$	−0.756094	−0.0306889	−0.0153445	−	0.999882i	$-0.504884\pi$
$607$	−0.756094	−0.0306889	−0.0153445	+	0.999882i	$0.504884\pi$
$608$	6.42532	0.260581
$609$	26.1874	1.06117
$610$	3.01712	0.122160
$611$	30.8617	1.24853
$612$	1.48024	0.0598353
$613$	16.2818	0.657614	0.328807	−	0.944397i	$-0.393353\pi$
$613$	16.2818	0.657614	0.328807	+	0.944397i	$0.393353\pi$
$614$	34.8027	1.40452
$615$	−7.85749	−0.316845
$616$	−24.1342	−0.972396
$617$	−12.8861	−0.518776	−0.259388	−	0.965773i	$-0.583521\pi$
$617$	−12.8861	−0.518776	−0.259388	+	0.965773i	$0.583521\pi$
$618$	68.6070	2.75978
$619$	45.7103	1.83725	0.918627	−	0.395126i	$-0.129299\pi$
$619$	45.7103	1.83725	0.918627	+	0.395126i	$0.129299\pi$
$620$	14.2844	0.573676
$621$	−19.5190	−0.783269
$622$	77.0073	3.08771
$623$	−57.4245	−2.30066
$624$	5.29186	0.211844
$625$	7.75565	0.310226
$626$	14.0229	0.560468
$627$	4.21682	0.168404
$628$	30.7265	1.22612
$629$	−25.7649	−1.02731
$630$	1.24064	0.0494283
$631$	24.8898	0.990847	0.495423	−	0.868652i	$-0.335013\pi$
$631$	24.8898	0.990847	0.495423	+	0.868652i	$0.335013\pi$
$632$	−13.9510	−0.554943
$633$	−1.69642	−0.0674266
$634$	−53.8106	−2.13709
$635$	−17.8266	−0.707426
$636$	−15.5222	−0.615497
$637$	40.3786	1.59986
$638$	21.4597	0.849598
$639$	−1.16051	−0.0459091
$640$	−18.5098	−0.731662
$641$	−23.4241	−0.925197	−0.462598	−	0.886568i	$-0.653083\pi$
$641$	−23.4241	−0.925197	−0.462598	+	0.886568i	$0.653083\pi$
$642$	−20.5343	−0.810425
$643$	−22.3000	−0.879427	−0.439714	−	0.898138i	$-0.644920\pi$
$643$	−22.3000	−0.879427	−0.439714	+	0.898138i	$0.644920\pi$
$644$	45.5669	1.79559
$645$	−8.05449	−0.317145
$646$	8.88552	0.349596
$647$	−47.3142	−1.86011	−0.930056	−	0.367418i	$-0.880242\pi$
$647$	−47.3142	−1.86011	−0.930056	+	0.367418i	$0.880242\pi$
$648$	20.7821	0.816399
$649$	29.6632	1.16438
$650$	36.9665	1.44994
$651$	−28.3678	−1.11182
$652$	−35.0481	−1.37259
$653$	−18.8337	−0.737019	−0.368510	−	0.929624i	$-0.620132\pi$
$653$	−18.8337	−0.737019	−0.368510	+	0.929624i	$0.620132\pi$
$654$	−6.10274	−0.238636
$655$	3.85462	0.150613
$656$	2.94319	0.114912
$657$	−1.46394	−0.0571137
$658$	63.8443	2.48891
$659$	6.66147	0.259494	0.129747	−	0.991547i	$-0.458583\pi$
$659$	6.66147	0.259494	0.129747	+	0.991547i	$0.458583\pi$
$660$	−14.5038	−0.564561
$661$	6.19938	0.241128	0.120564	−	0.992706i	$-0.461530\pi$
$661$	6.19938	0.241128	0.120564	+	0.992706i	$0.461530\pi$
$662$	67.6842	2.63062
$663$	29.3388	1.13942
$664$	34.3728	1.33392
$665$	4.50995	0.174888
$666$	1.79622	0.0696020
$667$	−14.1287	−0.547064
$668$	−25.1417	−0.972761
$669$	−49.5271	−1.91483
$670$	21.9698	0.848769
$671$	−2.97346	−0.114789
$672$	43.8890	1.69305
$673$	22.8410	0.880458	0.440229	−	0.897886i	$-0.354897\pi$
$673$	22.8410	0.880458	0.440229	+	0.897886i	$0.354897\pi$
$674$	58.4940	2.25310
$675$	−19.8378	−0.763556
$676$	19.0701	0.733464
$677$	48.8853	1.87881	0.939407	−	0.342804i	$-0.111377\pi$
$677$	48.8853	1.87881	0.939407	+	0.342804i	$0.111377\pi$
$678$	−18.9145	−0.726405
$679$	−8.91339	−0.342064
$680$	−10.6572	−0.408683
$681$	15.7671	0.604196
$682$	−23.2464	−0.890152
$683$	13.3277	0.509968	0.254984	−	0.966945i	$-0.417930\pi$
$683$	13.3277	0.509968	0.254984	+	0.966945i	$0.417930\pi$
$684$	−0.375136	−0.0143437
$685$	13.1763	0.503441
$686$	20.0626	0.765993
$687$	22.8537	0.871925
$688$	3.01697	0.115021
$689$	13.0597	0.497534
$690$	15.7683	0.600288
$691$	−3.26772	−0.124310	−0.0621550	−	0.998067i	$-0.519797\pi$
$691$	−3.26772	−0.124310	−0.0621550	+	0.998067i	$0.519797\pi$
$692$	51.7805	1.96840
$693$	−1.22269	−0.0464460
$694$	5.27767	0.200338
$695$	4.83664	0.183464
$696$	15.6826	0.594449
$697$	16.3174	0.618065
$698$	1.77144	0.0670501
$699$	−7.58412	−0.286858
$700$	46.3111	1.75040
$701$	44.7460	1.69003	0.845017	−	0.534739i	$-0.179590\pi$
$701$	44.7460	1.69003	0.845017	+	0.534739i	$0.179590\pi$
$702$	−52.2748	−1.97298
$703$	6.52957	0.246268
$704$	32.4272	1.22215
$705$	13.3793	0.503893
$706$	60.0667	2.26064
$707$	18.0767	0.679843
$708$	62.1660	2.33634
$709$	5.55058	0.208456	0.104228	−	0.994553i	$-0.466763\pi$
$709$	5.55058	0.208456	0.104228	+	0.994553i	$0.466763\pi$
$710$	23.9605	0.899223
$711$	−0.706787	−0.0265066
$712$	−34.3893	−1.28879
$713$	15.3050	0.573177
$714$	60.6937	2.27140
$715$	12.2028	0.456360
$716$	−27.7694	−1.03779
$717$	−21.2109	−0.792136
$718$	53.8490	2.00963
$719$	−12.7436	−0.475255	−0.237628	−	0.971356i	$-0.576370\pi$
$719$	−12.7436	−0.475255	−0.237628	+	0.971356i	$0.576370\pi$
$720$	−0.0973845	−0.00362931
$721$	−72.3142	−2.69312
$722$	−2.25185	−0.0838051
$723$	−22.7833	−0.847320
$724$	4.59765	0.170870
$725$	−14.3594	−0.533295
$726$	−18.4173	−0.683531
$727$	−2.71026	−0.100518	−0.0502589	−	0.998736i	$-0.516005\pi$
$727$	−2.71026	−0.100518	−0.0502589	+	0.998736i	$0.516005\pi$
$728$	42.5545	1.57718
$729$	28.0081	1.03734
$730$	30.2253	1.11869
$731$	16.7265	0.618651
$732$	−6.23156	−0.230325
$733$	20.5859	0.760357	0.380179	−	0.924913i	$-0.375863\pi$
$733$	20.5859	0.760357	0.380179	+	0.924913i	$0.375863\pi$
$734$	4.66894	0.172334
$735$	17.5051	0.645684
$736$	−23.6790	−0.872818
$737$	−21.6519	−0.797559
$738$	−1.13758	−0.0418748
$739$	15.1848	0.558584	0.279292	−	0.960206i	$-0.409900\pi$
$739$	15.1848	0.558584	0.279292	+	0.960206i	$0.409900\pi$
$740$	−22.4586	−0.825594
$741$	−7.43529	−0.273142
$742$	27.0168	0.991819
$743$	−12.0796	−0.443159	−0.221580	−	0.975142i	$-0.571121\pi$
$743$	−12.0796	−0.443159	−0.221580	+	0.975142i	$0.571121\pi$
$744$	−16.9884	−0.622824
$745$	−23.8595	−0.874144
$746$	22.6067	0.827690
$747$	1.74139	0.0637142
$748$	30.1196	1.10128
$749$	21.6439	0.790851
$750$	37.4195	1.36637
$751$	−12.9709	−0.473314	−0.236657	−	0.971593i	$-0.576052\pi$
$751$	−12.9709	−0.473314	−0.236657	+	0.971593i	$0.576052\pi$
$752$	−5.01148	−0.182750
$753$	−21.0212	−0.766054
$754$	−37.8387	−1.37800
$755$	10.1574	0.369664
$756$	−65.4892	−2.38182
$757$	37.8880	1.37706	0.688532	−	0.725206i	$-0.258256\pi$
$757$	37.8880	1.37706	0.688532	+	0.725206i	$0.258256\pi$
$758$	41.5270	1.50833
$759$	−15.5401	−0.564070
$760$	2.70083	0.0979695
$761$	30.8382	1.11788	0.558942	−	0.829206i	$-0.311207\pi$
$761$	30.8382	1.11788	0.558942	+	0.829206i	$0.311207\pi$
$762$	60.7989	2.20251
$763$	6.43251	0.232872
$764$	56.1011	2.02967
$765$	−0.539912	−0.0195206
$766$	51.1324	1.84749
$767$	−52.3035	−1.88857
$768$	18.8681	0.680844
$769$	8.31375	0.299801	0.149901	−	0.988701i	$-0.452105\pi$
$769$	8.31375	0.299801	0.149901	+	0.988701i	$0.452105\pi$
$770$	25.2443	0.909740
$771$	−24.9454	−0.898388
$772$	59.9704	2.15838
$773$	−43.7621	−1.57401	−0.787007	−	0.616944i	$-0.788371\pi$
$773$	−43.7621	−1.57401	−0.787007	+	0.616944i	$0.788371\pi$
$774$	−1.16610	−0.0419145
$775$	15.5550	0.558751
$776$	−5.33788	−0.191619
$777$	44.6011	1.60006
$778$	70.2517	2.51865
$779$	−4.13530	−0.148162
$780$	25.5738	0.915690
$781$	−23.6138	−0.844968
$782$	−32.7455	−1.17098
$783$	20.3058	0.725672
$784$	−6.55688	−0.234174
$785$	−11.2074	−0.400008
$786$	−13.1465	−0.468920
$787$	−13.4709	−0.480185	−0.240092	−	0.970750i	$-0.577178\pi$
$787$	−13.4709	−0.480185	−0.240092	+	0.970750i	$0.577178\pi$
$788$	21.0588	0.750189
$789$	31.1098	1.10754
$790$	14.5927	0.519185
$791$	19.9365	0.708861
$792$	−0.732220	−0.0260183
$793$	5.24294	0.186182
$794$	−20.7413	−0.736081
$795$	5.66167	0.200799
$796$	12.2087	0.432727
$797$	−18.9963	−0.672884	−0.336442	−	0.941704i	$-0.609224\pi$
$797$	−18.9963	−0.672884	−0.336442	+	0.941704i	$0.609224\pi$
$798$	−15.3815	−0.544500
$799$	−27.7843	−0.982938
$800$	−24.0657	−0.850851
$801$	−1.74223	−0.0615586
$802$	−56.3062	−1.98824
$803$	−29.7879	−1.05119
$804$	−45.3765	−1.60031
$805$	−16.6203	−0.585790
$806$	40.9891	1.44378
$807$	37.9931	1.33742
$808$	10.8254	0.380837
$809$	53.5150	1.88149	0.940744	−	0.339118i	$-0.110128\pi$
$809$	53.5150	1.88149	0.940744	+	0.339118i	$0.110128\pi$
$810$	−21.7380	−0.763795
$811$	27.9357	0.980955	0.490477	−	0.871454i	$-0.336822\pi$
$811$	27.9357	0.980955	0.490477	+	0.871454i	$0.336822\pi$
$812$	−47.4039	−1.66355
$813$	−6.78757	−0.238051
$814$	36.5490	1.28104
$815$	12.7836	0.447791
$816$	−4.76418	−0.166779
$817$	−4.23897	−0.148303
$818$	50.0266	1.74914
$819$	2.15590	0.0753331
$820$	14.2234	0.496704
$821$	−21.3327	−0.744515	−0.372258	−	0.928129i	$-0.621416\pi$
$821$	−21.3327	−0.744515	−0.372258	+	0.928129i	$0.621416\pi$
$822$	−44.9388	−1.56742
$823$	−20.0414	−0.698599	−0.349300	−	0.937011i	$-0.613580\pi$
$823$	−20.0414	−0.698599	−0.349300	+	0.937011i	$0.613580\pi$
$824$	−43.3062	−1.50864
$825$	−15.7939	−0.549873
$826$	−108.201	−3.76480
$827$	3.61644	0.125756	0.0628780	−	0.998021i	$-0.479972\pi$
$827$	3.61644	0.125756	0.0628780	+	0.998021i	$0.479972\pi$
$828$	1.38248	0.0480443
$829$	27.5824	0.957978	0.478989	−	0.877821i	$-0.341003\pi$
$829$	27.5824	0.957978	0.478989	+	0.877821i	$0.341003\pi$
$830$	−35.9538	−1.24797
$831$	−37.4620	−1.29954
$832$	−57.1771	−1.98226
$833$	−36.3522	−1.25953
$834$	−16.4957	−0.571200
$835$	9.17032	0.317352
$836$	−7.63319	−0.263999
$837$	−21.9965	−0.760310
$838$	62.6911	2.16563
$839$	−23.0648	−0.796285	−0.398142	−	0.917324i	$-0.630345\pi$
$839$	−23.0648	−0.796285	−0.398142	+	0.917324i	$0.630345\pi$
$840$	18.4484	0.636530
$841$	−14.3018	−0.493165
$842$	15.2857	0.526781
$843$	36.7732	1.26654
$844$	3.07082	0.105702
$845$	−6.95573	−0.239284
$846$	1.93700	0.0665955
$847$	19.4125	0.667022
$848$	−2.12070	−0.0728250
$849$	−7.35988	−0.252591
$850$	−33.2803	−1.14150
$851$	−24.0632	−0.824875
$852$	−49.4881	−1.69543
$853$	−37.9284	−1.29864	−0.649322	−	0.760514i	$-0.724947\pi$
$853$	−37.9284	−1.29864	−0.649322	+	0.760514i	$0.724947\pi$
$854$	10.8462	0.371148
$855$	0.136829	0.00467947
$856$	12.9617	0.443022
$857$	−17.8439	−0.609537	−0.304769	−	0.952426i	$-0.598579\pi$
$857$	−17.8439	−0.609537	−0.304769	+	0.952426i	$0.598579\pi$
$858$	−41.6188	−1.42084
$859$	18.5329	0.632334	0.316167	−	0.948704i	$-0.397604\pi$
$859$	18.5329	0.632334	0.316167	+	0.948704i	$0.397604\pi$
$860$	14.5800	0.497175
$861$	−28.2467	−0.962645
$862$	11.7899	0.401566
$863$	38.7756	1.31994	0.659968	−	0.751294i	$-0.270570\pi$
$863$	38.7756	1.31994	0.659968	+	0.751294i	$0.270570\pi$
$864$	34.0317	1.15778
$865$	−18.8867	−0.642167
$866$	−42.3269	−1.43833
$867$	2.42594	0.0823891
$868$	51.3507	1.74296
$869$	−14.3815	−0.487860
$870$	−16.4039	−0.556146
$871$	38.1776	1.29360
$872$	3.85218	0.130451
$873$	−0.270427	−0.00915258
$874$	8.29865	0.280706
$875$	−39.4415	−1.33337
$876$	−62.4273	−2.10922
$877$	1.77452	0.0599211	0.0299606	−	0.999551i	$-0.490462\pi$
$877$	1.77452	0.0599211	0.0299606	+	0.999551i	$0.490462\pi$
$878$	40.8688	1.37926
$879$	18.0226	0.607886
$880$	−1.98156	−0.0667983
$881$	−2.26884	−0.0764392	−0.0382196	−	0.999269i	$-0.512169\pi$
$881$	−2.26884	−0.0764392	−0.0382196	+	0.999269i	$0.512169\pi$
$882$	2.53432	0.0853349
$883$	−44.5844	−1.50039	−0.750193	−	0.661219i	$-0.770039\pi$
$883$	−44.5844	−1.50039	−0.750193	+	0.661219i	$0.770039\pi$
$884$	−53.1083	−1.78623
$885$	−22.6748	−0.762204
$886$	86.6928	2.91250
$887$	8.29649	0.278569	0.139284	−	0.990252i	$-0.455520\pi$
$887$	8.29649	0.278569	0.139284	+	0.990252i	$0.455520\pi$
$888$	26.7099	0.896324
$889$	−64.0842	−2.14932
$890$	35.9710	1.20575
$891$	21.4234	0.717711
$892$	89.6528	3.00180
$893$	7.04135	0.235630
$894$	81.3746	2.72158
$895$	10.1288	0.338568
$896$	−66.5402	−2.22295
$897$	27.4010	0.914893
$898$	17.1149	0.571131
$899$	−15.9220	−0.531028
$900$	1.40505	0.0468352
$901$	−11.7574	−0.391696
$902$	−23.1472	−0.770717
$903$	−28.9549	−0.963557
$904$	11.9392	0.397092
$905$	−1.67697	−0.0557444
$906$	−34.6425	−1.15092
$907$	−5.64883	−0.187566	−0.0937832	−	0.995593i	$-0.529896\pi$
$907$	−5.64883	−0.187566	−0.0937832	+	0.995593i	$0.529896\pi$
$908$	−28.5412	−0.947173
$909$	0.548436	0.0181905
$910$	−44.5118	−1.47555
$911$	−25.9750	−0.860589	−0.430294	−	0.902689i	$-0.641590\pi$
$911$	−25.9750	−0.860589	−0.430294	+	0.902689i	$0.641590\pi$
$912$	1.20738	0.0399803
$913$	35.4335	1.17268
$914$	8.71315	0.288205
$915$	2.27294	0.0751409
$916$	−41.3693	−1.36688
$917$	13.8569	0.457594
$918$	47.0621	1.55328
$919$	7.83511	0.258456	0.129228	−	0.991615i	$-0.458750\pi$
$919$	7.83511	0.258456	0.129228	+	0.991615i	$0.458750\pi$
$920$	−9.95327	−0.328150
$921$	26.2185	0.863928
$922$	84.6695	2.78844
$923$	41.6369	1.37050
$924$	−52.1395	−1.71526
$925$	−24.4562	−0.804115
$926$	−78.3792	−2.57570
$927$	−2.19397	−0.0720596
$928$	24.6335	0.808636
$929$	19.8932	0.652675	0.326338	−	0.945253i	$-0.394185\pi$
$929$	19.8932	0.652675	0.326338	+	0.945253i	$0.394185\pi$
$930$	17.7697	0.582693
$931$	9.21269	0.301934
$932$	13.7286	0.449695
$933$	58.0131	1.89926
$934$	−46.0076	−1.50541
$935$	−10.9860	−0.359281
$936$	1.29108	0.0422003
$937$	7.08249	0.231375	0.115687	−	0.993286i	$-0.463093\pi$
$937$	7.08249	0.231375	0.115687	+	0.993286i	$0.463093\pi$
$938$	78.9788	2.57875
$939$	10.5641	0.344746
$940$	−24.2188	−0.789931
$941$	−22.4772	−0.732736	−0.366368	−	0.930470i	$-0.619399\pi$
$941$	−22.4772	−0.732736	−0.366368	+	0.930470i	$0.619399\pi$
$942$	38.2236	1.24539
$943$	15.2397	0.496272
$944$	8.49330	0.276433
$945$	23.8869	0.777041
$946$	−23.7275	−0.771448
$947$	−43.3377	−1.40829	−0.704143	−	0.710058i	$-0.748669\pi$
$947$	−43.3377	−1.40829	−0.704143	+	0.710058i	$0.748669\pi$
$948$	−30.1398	−0.978895
$949$	52.5234	1.70498
$950$	8.43418	0.273641
$951$	−40.5380	−1.31453
$952$	−38.3111	−1.24167
$953$	19.1723	0.621052	0.310526	−	0.950565i	$-0.399495\pi$
$953$	19.1723	0.621052	0.310526	+	0.950565i	$0.399495\pi$
$954$	0.819676	0.0265380
$955$	−20.4626	−0.662155
$956$	38.3955	1.24180
$957$	16.1666	0.522591
$958$	43.4018	1.40225
$959$	47.3672	1.52957
$960$	−24.7876	−0.800016
$961$	−13.7523	−0.443624
$962$	−64.4449	−2.07779
$963$	0.656664	0.0211607
$964$	41.2418	1.32831
$965$	−21.8740	−0.704148
$966$	56.6850	1.82381
$967$	37.2765	1.19873	0.599366	−	0.800475i	$-0.295419\pi$
$967$	37.2765	1.19873	0.599366	+	0.800475i	$0.295419\pi$
$968$	11.6254	0.373655
$969$	6.69387	0.215038
$970$	5.58339	0.179272
$971$	48.7357	1.56400	0.782002	−	0.623276i	$-0.214199\pi$
$971$	48.7357	1.56400	0.782002	+	0.623276i	$0.214199\pi$
$972$	−3.89607	−0.124967
$973$	17.3871	0.557405
$974$	−18.4985	−0.592729
$975$	27.8485	0.891866
$976$	−0.851375	−0.0272518
$977$	−0.607532	−0.0194367	−0.00971833	−	0.999953i	$-0.503093\pi$
$977$	−0.607532	−0.0194367	−0.00971833	+	0.999953i	$0.503093\pi$
$978$	−43.5996	−1.39416
$979$	−35.4504	−1.13300
$980$	−31.6872	−1.01221
$981$	0.195159	0.00623094
$982$	−87.1324	−2.78051
$983$	−18.7593	−0.598330	−0.299165	−	0.954201i	$-0.596708\pi$
$983$	−18.7593	−0.598330	−0.299165	+	0.954201i	$0.596708\pi$
$984$	−16.9158	−0.539257
$985$	−7.68111	−0.244740
$986$	34.0655	1.08487
$987$	48.0968	1.53094
$988$	13.4592	0.428194
$989$	15.6217	0.496742
$990$	0.765897	0.0243418
$991$	−28.4052	−0.902321	−0.451161	−	0.892443i	$-0.648990\pi$
$991$	−28.4052	−0.902321	−0.451161	+	0.892443i	$0.648990\pi$
$992$	−26.6845	−0.847235
$993$	50.9896	1.61811
$994$	86.1351	2.73204
$995$	−4.45309	−0.141172
$996$	74.2589	2.35298
$997$	−34.1448	−1.08138	−0.540689	−	0.841223i	$-0.681836\pi$
$997$	−34.1448	−1.08138	−0.540689	+	0.841223i	$0.681836\pi$
$998$	69.3064	2.19386
$999$	34.5839	1.09418

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.12	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.12	✓	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)

\(f(q)\)	\(=\)	\(q-2.25185 q^{2} -1.69642 q^{3} +3.07082 q^{4} -1.12007 q^{5} +3.82008 q^{6} -4.02650 q^{7} -2.41131 q^{8} -0.122162 q^{9} +O(q^{10})\)	\(q-2.25185 q^{2} -1.69642 q^{3} +3.07082 q^{4} -1.12007 q^{5} +3.82008 q^{6} -4.02650 q^{7} -2.41131 q^{8} -0.122162 q^{9} +2.52222 q^{10} -2.48572 q^{11} -5.20939 q^{12} +4.38293 q^{13} +9.06706 q^{14} +1.90010 q^{15} -0.711723 q^{16} -3.94588 q^{17} +0.275090 q^{18} +1.00000 q^{19} -3.43952 q^{20} +6.83063 q^{21} +5.59746 q^{22} -3.68526 q^{23} +4.09060 q^{24} -3.74545 q^{25} -9.86969 q^{26} +5.29650 q^{27} -12.3646 q^{28} +3.83383 q^{29} -4.27874 q^{30} -4.15303 q^{31} +6.42532 q^{32} +4.21682 q^{33} +8.88552 q^{34} +4.50995 q^{35} -0.375136 q^{36} +6.52957 q^{37} -2.25185 q^{38} -7.43529 q^{39} +2.70083 q^{40} -4.13530 q^{41} -15.3815 q^{42} -4.23897 q^{43} -7.63319 q^{44} +0.136829 q^{45} +8.29865 q^{46} +7.04135 q^{47} +1.20738 q^{48} +9.21269 q^{49} +8.43418 q^{50} +6.69387 q^{51} +13.4592 q^{52} +2.97967 q^{53} -11.9269 q^{54} +2.78417 q^{55} +9.70915 q^{56} -1.69642 q^{57} -8.63319 q^{58} -11.9334 q^{59} +5.83487 q^{60} +1.19622 q^{61} +9.35199 q^{62} +0.491884 q^{63} -13.0454 q^{64} -4.90918 q^{65} -9.49564 q^{66} +8.71052 q^{67} -12.1171 q^{68} +6.25175 q^{69} -10.1557 q^{70} +9.49978 q^{71} +0.294570 q^{72} +11.9836 q^{73} -14.7036 q^{74} +6.35385 q^{75} +3.07082 q^{76} +10.0087 q^{77} +16.7431 q^{78} +5.78566 q^{79} +0.797177 q^{80} -8.61859 q^{81} +9.31206 q^{82} -14.2548 q^{83} +20.9756 q^{84} +4.41965 q^{85} +9.54552 q^{86} -6.50378 q^{87} +5.99385 q^{88} +14.2616 q^{89} -0.308119 q^{90} -17.6479 q^{91} -11.3168 q^{92} +7.04528 q^{93} -15.8560 q^{94} -1.12007 q^{95} -10.9000 q^{96} +2.21368 q^{97} -20.7456 q^{98} +0.303660 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 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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