LMFDB - Embedded newform 4009.2.a.e.1.15

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:	$0$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q-1.89977 q^{2} -2.93667 q^{3} +1.60912 q^{4} -0.296578 q^{5} +5.57898 q^{6} +2.59029 q^{7} +0.742589 q^{8} +5.62400 q^{9} +O(q^{10})$	\(q-1.89977 q^{2} -2.93667 q^{3} +1.60912 q^{4} -0.296578 q^{5} +5.57898 q^{6} +2.59029 q^{7} +0.742589 q^{8} +5.62400 q^{9} +0.563430 q^{10} -3.83827 q^{11} -4.72543 q^{12} -6.51339 q^{13} -4.92096 q^{14} +0.870951 q^{15} -4.62898 q^{16} -6.32060 q^{17} -10.6843 q^{18} +1.00000 q^{19} -0.477229 q^{20} -7.60683 q^{21} +7.29181 q^{22} +0.665939 q^{23} -2.18074 q^{24} -4.91204 q^{25} +12.3739 q^{26} -7.70582 q^{27} +4.16808 q^{28} -3.30802 q^{29} -1.65461 q^{30} -7.60037 q^{31} +7.30880 q^{32} +11.2717 q^{33} +12.0077 q^{34} -0.768225 q^{35} +9.04967 q^{36} +6.95495 q^{37} -1.89977 q^{38} +19.1277 q^{39} -0.220236 q^{40} +3.26454 q^{41} +14.4512 q^{42} -12.6758 q^{43} -6.17621 q^{44} -1.66796 q^{45} -1.26513 q^{46} -11.0932 q^{47} +13.5938 q^{48} -0.290374 q^{49} +9.33174 q^{50} +18.5615 q^{51} -10.4808 q^{52} +12.6953 q^{53} +14.6393 q^{54} +1.13835 q^{55} +1.92352 q^{56} -2.93667 q^{57} +6.28447 q^{58} -9.11722 q^{59} +1.40146 q^{60} -2.32040 q^{61} +14.4389 q^{62} +14.5678 q^{63} -4.62707 q^{64} +1.93173 q^{65} -21.4136 q^{66} -9.67150 q^{67} -10.1706 q^{68} -1.95564 q^{69} +1.45945 q^{70} +10.4087 q^{71} +4.17632 q^{72} -7.68676 q^{73} -13.2128 q^{74} +14.4250 q^{75} +1.60912 q^{76} -9.94224 q^{77} -36.3381 q^{78} +15.9584 q^{79} +1.37285 q^{80} +5.75742 q^{81} -6.20186 q^{82} -0.340445 q^{83} -12.2403 q^{84} +1.87455 q^{85} +24.0810 q^{86} +9.71455 q^{87} -2.85025 q^{88} +2.60962 q^{89} +3.16873 q^{90} -16.8716 q^{91} +1.07157 q^{92} +22.3197 q^{93} +21.0745 q^{94} -0.296578 q^{95} -21.4635 q^{96} -12.8515 q^{97} +0.551643 q^{98} -21.5864 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) $ 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.89977	−1.34334	−0.671669	−	0.740851i	$-0.734422\pi$
$2$	−1.89977	−1.34334	−0.671669	+	0.740851i	$0.734422\pi$
$3$	−2.93667	−1.69548	−0.847742	−	0.530408i	$-0.822039\pi$
$3$	−2.93667	−1.69548	−0.847742	+	0.530408i	$0.822039\pi$
$4$	1.60912	0.804558
$5$	−0.296578	−0.132634	−0.0663169	−	0.997799i	$-0.521125\pi$
$5$	−0.296578	−0.132634	−0.0663169	+	0.997799i	$0.521125\pi$
$6$	5.57898	2.27761
$7$	2.59029	0.979039	0.489520	−	0.871992i	$-0.337172\pi$
$7$	2.59029	0.979039	0.489520	+	0.871992i	$0.337172\pi$
$8$	0.742589	0.262545
$9$	5.62400	1.87467
$10$	0.563430	0.178172
$11$	−3.83827	−1.15728	−0.578640	−	0.815583i	$-0.696416\pi$
$11$	−3.83827	−1.15728	−0.578640	+	0.815583i	$0.696416\pi$
$12$	−4.72543	−1.36412
$13$	−6.51339	−1.80649	−0.903245	−	0.429125i	$-0.858822\pi$
$13$	−6.51339	−1.80649	−0.903245	+	0.429125i	$0.858822\pi$
$14$	−4.92096	−1.31518
$15$	0.870951	0.224879
$16$	−4.62898	−1.15724
$17$	−6.32060	−1.53297	−0.766485	−	0.642262i	$-0.777996\pi$
$17$	−6.32060	−1.53297	−0.766485	+	0.642262i	$0.777996\pi$
$18$	−10.6843	−2.51831
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.477229	−0.106712
$21$	−7.60683	−1.65995
$22$	7.29181	1.55462
$23$	0.665939	0.138858	0.0694290	−	0.997587i	$-0.477882\pi$
$23$	0.665939	0.138858	0.0694290	+	0.997587i	$0.477882\pi$
$24$	−2.18074	−0.445141
$25$	−4.91204	−0.982408
$26$	12.3739	2.42673
$27$	−7.70582	−1.48299
$28$	4.16808	0.787694
$29$	−3.30802	−0.614284	−0.307142	−	0.951664i	$-0.599373\pi$
$29$	−3.30802	−0.614284	−0.307142	+	0.951664i	$0.599373\pi$
$30$	−1.65461	−0.302088
$31$	−7.60037	−1.36507	−0.682533	−	0.730854i	$-0.739122\pi$
$31$	−7.60037	−1.36507	−0.682533	+	0.730854i	$0.739122\pi$
$32$	7.30880	1.29203
$33$	11.2717	1.96215
$34$	12.0077	2.05930
$35$	−0.768225	−0.129854
$36$	9.04967	1.50828
$37$	6.95495	1.14339	0.571694	−	0.820467i	$-0.306287\pi$
$37$	6.95495	1.14339	0.571694	+	0.820467i	$0.306287\pi$
$38$	−1.89977	−0.308183
$39$	19.1277	3.06288
$40$	−0.220236	−0.0348223
$41$	3.26454	0.509835	0.254917	−	0.966963i	$-0.417952\pi$
$41$	3.26454	0.509835	0.254917	+	0.966963i	$0.417952\pi$
$42$	14.4512	2.22987
$43$	−12.6758	−1.93304	−0.966519	−	0.256594i	$-0.917400\pi$
$43$	−12.6758	−1.93304	−0.966519	+	0.256594i	$0.917400\pi$
$44$	−6.17621	−0.931099
$45$	−1.66796	−0.248645
$46$	−1.26513	−0.186533
$47$	−11.0932	−1.61811	−0.809053	−	0.587735i	$-0.800020\pi$
$47$	−11.0932	−1.61811	−0.809053	+	0.587735i	$0.800020\pi$
$48$	13.5938	1.96209
$49$	−0.290374	−0.0414820
$50$	9.33174	1.31971
$51$	18.5615	2.59913
$52$	−10.4808	−1.45343
$53$	12.6953	1.74383	0.871915	−	0.489658i	$-0.162878\pi$
$53$	12.6953	1.74383	0.871915	+	0.489658i	$0.162878\pi$
$54$	14.6393	1.99215
$55$	1.13835	0.153495
$56$	1.92352	0.257042
$57$	−2.93667	−0.388971
$58$	6.28447	0.825191
$59$	−9.11722	−1.18696	−0.593480	−	0.804849i	$-0.702246\pi$
$59$	−9.11722	−1.18696	−0.593480	+	0.804849i	$0.702246\pi$
$60$	1.40146	0.180928
$61$	−2.32040	−0.297097	−0.148548	−	0.988905i	$-0.547460\pi$
$61$	−2.32040	−0.297097	−0.148548	+	0.988905i	$0.547460\pi$
$62$	14.4389	1.83375
$63$	14.5678	1.83537
$64$	−4.62707	−0.578384
$65$	1.93173	0.239602
$66$	−21.4136	−2.63583
$67$	−9.67150	−1.18156	−0.590781	−	0.806832i	$-0.701180\pi$
$67$	−9.67150	−1.18156	−0.590781	+	0.806832i	$0.701180\pi$
$68$	−10.1706	−1.23336
$69$	−1.95564	−0.235431
$70$	1.45945	0.174438
$71$	10.4087	1.23529	0.617645	−	0.786457i	$-0.288087\pi$
$71$	10.4087	1.23529	0.617645	+	0.786457i	$0.288087\pi$
$72$	4.17632	0.492185
$73$	−7.68676	−0.899667	−0.449833	−	0.893113i	$-0.648517\pi$
$73$	−7.68676	−0.899667	−0.449833	+	0.893113i	$0.648517\pi$
$74$	−13.2128	−1.53596
$75$	14.4250	1.66566
$76$	1.60912	0.184578
$77$	−9.94224	−1.13302
$78$	−36.3381	−4.11448
$79$	15.9584	1.79546	0.897730	−	0.440547i	$-0.145215\pi$
$79$	15.9584	1.79546	0.897730	+	0.440547i	$0.145215\pi$
$80$	1.37285	0.153490
$81$	5.75742	0.639713
$82$	−6.20186	−0.684881
$83$	−0.340445	−0.0373687	−0.0186843	−	0.999825i	$-0.505948\pi$
$83$	−0.340445	−0.0373687	−0.0186843	+	0.999825i	$0.505948\pi$
$84$	−12.2403	−1.33552
$85$	1.87455	0.203324
$86$	24.0810	2.59672
$87$	9.71455	1.04151
$88$	−2.85025	−0.303838
$89$	2.60962	0.276620	0.138310	−	0.990389i	$-0.455833\pi$
$89$	2.60962	0.276620	0.138310	+	0.990389i	$0.455833\pi$
$90$	3.16873	0.334014
$91$	−16.8716	−1.76863
$92$	1.07157	0.111719
$93$	22.3197	2.31445
$94$	21.0745	2.17366
$95$	−0.296578	−0.0304283
$96$	−21.4635	−2.19061
$97$	−12.8515	−1.30487	−0.652436	−	0.757844i	$-0.726253\pi$
$97$	−12.8515	−1.30487	−0.652436	+	0.757844i	$0.726253\pi$
$98$	0.551643	0.0557243
$99$	−21.5864	−2.16952
$100$	−7.90404	−0.790404
$101$	−5.15211	−0.512654	−0.256327	−	0.966590i	$-0.582512\pi$
$101$	−5.15211	−0.512654	−0.256327	+	0.966590i	$0.582512\pi$
$102$	−35.2625	−3.49151
$103$	−5.50164	−0.542093	−0.271047	−	0.962566i	$-0.587370\pi$
$103$	−5.50164	−0.542093	−0.271047	+	0.962566i	$0.587370\pi$
$104$	−4.83678	−0.474285
$105$	2.25602	0.220165
$106$	−24.1181	−2.34255
$107$	3.96620	0.383427	0.191714	−	0.981451i	$-0.438596\pi$
$107$	3.96620	0.383427	0.191714	+	0.981451i	$0.438596\pi$
$108$	−12.3996	−1.19315
$109$	−16.6994	−1.59951	−0.799756	−	0.600325i	$-0.795038\pi$
$109$	−16.6994	−1.59951	−0.799756	+	0.600325i	$0.795038\pi$
$110$	−2.16259	−0.206195
$111$	−20.4244	−1.93860
$112$	−11.9904	−1.13299
$113$	−11.7878	−1.10890	−0.554452	−	0.832216i	$-0.687072\pi$
$113$	−11.7878	−1.10890	−0.554452	+	0.832216i	$0.687072\pi$
$114$	5.57898	0.522519
$115$	−0.197503	−0.0184173
$116$	−5.32299	−0.494227
$117$	−36.6314	−3.38657
$118$	17.3206	1.59449
$119$	−16.3722	−1.50084
$120$	0.646759	0.0590408
$121$	3.73228	0.339298
$122$	4.40822	0.399101
$123$	−9.58685	−0.864417
$124$	−12.2299	−1.09828
$125$	2.93970	0.262934
$126$	−27.6755	−2.46553
$127$	1.63833	0.145378	0.0726892	−	0.997355i	$-0.476842\pi$
$127$	1.63833	0.145378	0.0726892	+	0.997355i	$0.476842\pi$
$128$	−5.82725	−0.515061
$129$	37.2245	3.27744
$130$	−3.66984	−0.321866
$131$	3.92639	0.343050	0.171525	−	0.985180i	$-0.445130\pi$
$131$	3.92639	0.343050	0.171525	+	0.985180i	$0.445130\pi$
$132$	18.1375	1.57866
$133$	2.59029	0.224607
$134$	18.3736	1.58724
$135$	2.28538	0.196694
$136$	−4.69361	−0.402474
$137$	16.2212	1.38587	0.692936	−	0.720999i	$-0.256317\pi$
$137$	16.2212	1.38587	0.692936	+	0.720999i	$0.256317\pi$
$138$	3.71526	0.316264
$139$	−21.4216	−1.81695	−0.908476	−	0.417937i	$-0.862753\pi$
$139$	−21.4216	−1.81695	−0.908476	+	0.417937i	$0.862753\pi$
$140$	−1.23616	−0.104475
$141$	32.5770	2.74347
$142$	−19.7742	−1.65941
$143$	25.0001	2.09062
$144$	−26.0334	−2.16945
$145$	0.981088	0.0814749
$146$	14.6030	1.20856
$147$	0.852731	0.0703320
$148$	11.1913	0.919921
$149$	−7.84894	−0.643010	−0.321505	−	0.946908i	$-0.604189\pi$
$149$	−7.84894	−0.643010	−0.321505	+	0.946908i	$0.604189\pi$
$150$	−27.4042	−2.23754
$151$	15.5837	1.26819	0.634094	−	0.773256i	$-0.281373\pi$
$151$	15.5837	1.26819	0.634094	+	0.773256i	$0.281373\pi$
$152$	0.742589	0.0602319
$153$	−35.5471	−2.87381
$154$	18.8879	1.52203
$155$	2.25411	0.181054
$156$	30.7786	2.46426
$157$	−24.0972	−1.92316	−0.961582	−	0.274518i	$-0.911482\pi$
$157$	−24.0972	−1.92316	−0.961582	+	0.274518i	$0.911482\pi$
$158$	−30.3172	−2.41191
$159$	−37.2818	−2.95664
$160$	−2.16763	−0.171366
$161$	1.72498	0.135947
$162$	−10.9378	−0.859351
$163$	8.03243	0.629149	0.314574	−	0.949233i	$-0.398138\pi$
$163$	8.03243	0.629149	0.314574	+	0.949233i	$0.398138\pi$
$164$	5.25302	0.410192
$165$	−3.34294	−0.260248
$166$	0.646766	0.0501988
$167$	−12.3851	−0.958389	−0.479194	−	0.877709i	$-0.659071\pi$
$167$	−12.3851	−0.958389	−0.479194	+	0.877709i	$0.659071\pi$
$168$	−5.64875	−0.435810
$169$	29.4243	2.26341
$170$	−3.56122	−0.273133
$171$	5.62400	0.430078
$172$	−20.3968	−1.55524
$173$	−6.42176	−0.488238	−0.244119	−	0.969745i	$-0.578499\pi$
$173$	−6.42176	−0.488238	−0.244119	+	0.969745i	$0.578499\pi$
$174$	−18.4554	−1.39910
$175$	−12.7236	−0.961816
$176$	17.7672	1.33926
$177$	26.7742	2.01247
$178$	−4.95768	−0.371594
$179$	5.49309	0.410573	0.205287	−	0.978702i	$-0.434187\pi$
$179$	5.49309	0.410573	0.205287	+	0.978702i	$0.434187\pi$
$180$	−2.68394	−0.200049
$181$	2.05465	0.152721	0.0763605	−	0.997080i	$-0.475670\pi$
$181$	2.05465	0.152721	0.0763605	+	0.997080i	$0.475670\pi$
$182$	32.0521	2.37586
$183$	6.81423	0.503723
$184$	0.494519	0.0364564
$185$	−2.06269	−0.151652
$186$	−42.4023	−3.10909
$187$	24.2601	1.77408
$188$	−17.8502	−1.30186
$189$	−19.9604	−1.45190
$190$	0.563430	0.0408755
$191$	−18.9224	−1.36917	−0.684587	−	0.728931i	$-0.740018\pi$
$191$	−18.9224	−1.36917	−0.684587	+	0.728931i	$0.740018\pi$
$192$	13.5882	0.980641
$193$	−22.3365	−1.60782	−0.803909	−	0.594752i	$-0.797250\pi$
$193$	−22.3365	−1.60782	−0.803909	+	0.594752i	$0.797250\pi$
$194$	24.4149	1.75289
$195$	−5.67285	−0.406241
$196$	−0.467245	−0.0333746
$197$	−21.4395	−1.52750	−0.763751	−	0.645511i	$-0.776644\pi$
$197$	−21.4395	−1.52750	−0.763751	+	0.645511i	$0.776644\pi$
$198$	41.0092	2.91440
$199$	3.99481	0.283184	0.141592	−	0.989925i	$-0.454778\pi$
$199$	3.99481	0.283184	0.141592	+	0.989925i	$0.454778\pi$
$200$	−3.64763	−0.257926
$201$	28.4020	2.00332
$202$	9.78781	0.688668
$203$	−8.56875	−0.601408
$204$	29.8676	2.09115
$205$	−0.968191	−0.0676214
$206$	10.4518	0.728215
$207$	3.74525	0.260313
$208$	30.1504	2.09055
$209$	−3.83827	−0.265498
$210$	−4.28591	−0.295756
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	20.4282	1.40301
$213$	−30.5670	−2.09441
$214$	−7.53486	−0.515073
$215$	3.75936	0.256386
$216$	−5.72226	−0.389351
$217$	−19.6872	−1.33645
$218$	31.7250	2.14869
$219$	22.5734	1.52537
$220$	1.83173	0.123495
$221$	41.1686	2.76930
$222$	38.8015	2.60419
$223$	11.6931	0.783028	0.391514	−	0.920172i	$-0.371952\pi$
$223$	11.6931	0.783028	0.391514	+	0.920172i	$0.371952\pi$
$224$	18.9320	1.26494
$225$	−27.6253	−1.84169
$226$	22.3941	1.48963
$227$	6.75956	0.448648	0.224324	−	0.974515i	$-0.427983\pi$
$227$	6.75956	0.448648	0.224324	+	0.974515i	$0.427983\pi$
$228$	−4.72543	−0.312950
$229$	6.83156	0.451442	0.225721	−	0.974192i	$-0.427526\pi$
$229$	6.83156	0.451442	0.225721	+	0.974192i	$0.427526\pi$
$230$	0.375210	0.0247406
$231$	29.1970	1.92102
$232$	−2.45650	−0.161277
$233$	25.9208	1.69813	0.849065	−	0.528289i	$-0.177166\pi$
$233$	25.9208	1.69813	0.849065	+	0.528289i	$0.177166\pi$
$234$	69.5911	4.54931
$235$	3.29000	0.214616
$236$	−14.6707	−0.954978
$237$	−46.8645	−3.04417
$238$	31.1034	2.01613
$239$	14.9931	0.969822	0.484911	−	0.874564i	$-0.338852\pi$
$239$	14.9931	0.969822	0.484911	+	0.874564i	$0.338852\pi$
$240$	−4.03162	−0.260240
$241$	20.8805	1.34503	0.672515	−	0.740083i	$-0.265214\pi$
$241$	20.8805	1.34503	0.672515	+	0.740083i	$0.265214\pi$
$242$	−7.09047	−0.455793
$243$	6.20987	0.398363
$244$	−3.73379	−0.239031
$245$	0.0861186	0.00550191
$246$	18.2128	1.16120
$247$	−6.51339	−0.414437
$248$	−5.64395	−0.358391
$249$	0.999772	0.0633580
$250$	−5.58474	−0.353210
$251$	3.97722	0.251040	0.125520	−	0.992091i	$-0.459940\pi$
$251$	3.97722	0.251040	0.125520	+	0.992091i	$0.459940\pi$
$252$	23.4413	1.47666
$253$	−2.55605	−0.160698
$254$	−3.11245	−0.195292
$255$	−5.50494	−0.344732
$256$	20.3246	1.27028
$257$	−0.149127	−0.00930231	−0.00465116	−	0.999989i	$-0.501481\pi$
$257$	−0.149127	−0.00930231	−0.00465116	+	0.999989i	$0.501481\pi$
$258$	−70.7179	−4.40271
$259$	18.0154	1.11942
$260$	3.10838	0.192774
$261$	−18.6043	−1.15158
$262$	−7.45923	−0.460833
$263$	−11.3600	−0.700487	−0.350244	−	0.936659i	$-0.613901\pi$
$263$	−11.3600	−0.700487	−0.350244	+	0.936659i	$0.613901\pi$
$264$	8.37024	0.515153
$265$	−3.76514	−0.231291
$266$	−4.92096	−0.301723
$267$	−7.66359	−0.469004
$268$	−15.5626	−0.950635
$269$	−21.8187	−1.33031	−0.665153	−	0.746707i	$-0.731634\pi$
$269$	−21.8187	−1.33031	−0.665153	+	0.746707i	$0.731634\pi$
$270$	−4.34169	−0.264227
$271$	−9.59849	−0.583067	−0.291533	−	0.956561i	$-0.594165\pi$
$271$	−9.59849	−0.583067	−0.291533	+	0.956561i	$0.594165\pi$
$272$	29.2579	1.77402
$273$	49.5463	2.99868
$274$	−30.8165	−1.86169
$275$	18.8537	1.13692
$276$	−3.14685	−0.189418
$277$	−9.09522	−0.546479	−0.273239	−	0.961946i	$-0.588095\pi$
$277$	−9.09522	−0.546479	−0.273239	+	0.961946i	$0.588095\pi$
$278$	40.6960	2.44078
$279$	−42.7445	−2.55905
$280$	−0.570476	−0.0340924
$281$	23.3677	1.39400	0.697001	−	0.717070i	$-0.254517\pi$
$281$	23.3677	1.39400	0.697001	+	0.717070i	$0.254517\pi$
$282$	−61.8886	−3.68541
$283$	2.21315	0.131558	0.0657791	−	0.997834i	$-0.479047\pi$
$283$	2.21315	0.131558	0.0657791	+	0.997834i	$0.479047\pi$
$284$	16.7489	0.993862
$285$	0.870951	0.0515907
$286$	−47.4944	−2.80841
$287$	8.45611	0.499148
$288$	41.1047	2.42212
$289$	22.9500	1.35000
$290$	−1.86384	−0.109448
$291$	37.7406	2.21239
$292$	−12.3689	−0.723834
$293$	16.2579	0.949797	0.474898	−	0.880041i	$-0.342485\pi$
$293$	16.2579	0.949797	0.474898	+	0.880041i	$0.342485\pi$
$294$	−1.61999	−0.0944797
$295$	2.70397	0.157431
$296$	5.16467	0.300190
$297$	29.5770	1.71623
$298$	14.9112	0.863781
$299$	−4.33752	−0.250846
$300$	23.2115	1.34012
$301$	−32.8340	−1.89252
$302$	−29.6055	−1.70360
$303$	15.1300	0.869197
$304$	−4.62898	−0.265490
$305$	0.688180	0.0394051
$306$	67.5312	3.86050
$307$	−4.31347	−0.246183	−0.123091	−	0.992395i	$-0.539281\pi$
$307$	−4.31347	−0.246183	−0.123091	+	0.992395i	$0.539281\pi$
$308$	−15.9982	−0.911583
$309$	16.1565	0.919111
$310$	−4.28228	−0.243217
$311$	31.3438	1.77734	0.888671	−	0.458545i	$-0.151629\pi$
$311$	31.3438	1.77734	0.888671	+	0.458545i	$0.151629\pi$
$312$	14.2040	0.804143
$313$	−15.2019	−0.859260	−0.429630	−	0.903005i	$-0.641356\pi$
$313$	−15.2019	−0.859260	−0.429630	+	0.903005i	$0.641356\pi$
$314$	45.7790	2.58346
$315$	−4.32050	−0.243433
$316$	25.6789	1.44455
$317$	−6.16881	−0.346475	−0.173237	−	0.984880i	$-0.555423\pi$
$317$	−6.16881	−0.346475	−0.173237	+	0.984880i	$0.555423\pi$
$318$	70.8267	3.97176
$319$	12.6971	0.710899
$320$	1.37229	0.0767133
$321$	−11.6474	−0.650095
$322$	−3.27706	−0.182623
$323$	−6.32060	−0.351688
$324$	9.26435	0.514686
$325$	31.9941	1.77471
$326$	−15.2598	−0.845160
$327$	49.0406	2.71195
$328$	2.42421	0.133855
$329$	−28.7346	−1.58419
$330$	6.35081	0.349601
$331$	−5.77173	−0.317243	−0.158621	−	0.987339i	$-0.550705\pi$
$331$	−5.77173	−0.317243	−0.158621	+	0.987339i	$0.550705\pi$
$332$	−0.547815	−0.0300653
$333$	39.1147	2.14347
$334$	23.5288	1.28744
$335$	2.86836	0.156715
$336$	35.2118	1.92096
$337$	−5.83902	−0.318072	−0.159036	−	0.987273i	$-0.550839\pi$
$337$	−5.83902	−0.318072	−0.159036	+	0.987273i	$0.550839\pi$
$338$	−55.8993	−3.04052
$339$	34.6168	1.88013
$340$	3.01637	0.163586
$341$	29.1722	1.57977
$342$	−10.6843	−0.577741
$343$	−18.8842	−1.01965
$344$	−9.41290	−0.507509
$345$	0.580001	0.0312262
$346$	12.1999	0.655868
$347$	−22.0287	−1.18256	−0.591282	−	0.806465i	$-0.701378\pi$
$347$	−22.0287	−1.18256	−0.591282	+	0.806465i	$0.701378\pi$
$348$	15.6318	0.837955
$349$	−10.4808	−0.561025	−0.280512	−	0.959850i	$-0.590504\pi$
$349$	−10.4808	−0.561025	−0.280512	+	0.959850i	$0.590504\pi$
$350$	24.1719	1.29204
$351$	50.1911	2.67900
$352$	−28.0531	−1.49524
$353$	−2.89207	−0.153929	−0.0769647	−	0.997034i	$-0.524523\pi$
$353$	−2.89207	−0.153929	−0.0769647	+	0.997034i	$0.524523\pi$
$354$	−50.8648	−2.70343
$355$	−3.08700	−0.163841
$356$	4.19919	0.222556
$357$	48.0797	2.54465
$358$	−10.4356	−0.551539
$359$	15.3170	0.808401	0.404200	−	0.914670i	$-0.367550\pi$
$359$	15.3170	0.808401	0.404200	+	0.914670i	$0.367550\pi$
$360$	−1.23861	−0.0652803
$361$	1.00000	0.0526316
$362$	−3.90336	−0.205156
$363$	−10.9605	−0.575275
$364$	−27.1484	−1.42296
$365$	2.27973	0.119326
$366$	−12.9455	−0.676670
$367$	−6.26185	−0.326866	−0.163433	−	0.986554i	$-0.552257\pi$
$367$	−6.26185	−0.326866	−0.163433	+	0.986554i	$0.552257\pi$
$368$	−3.08262	−0.160693
$369$	18.3598	0.955771
$370$	3.91863	0.203720
$371$	32.8845	1.70728
$372$	35.9151	1.86211
$373$	−35.3482	−1.83026	−0.915129	−	0.403161i	$-0.867912\pi$
$373$	−35.3482	−1.83026	−0.915129	+	0.403161i	$0.867912\pi$
$374$	−46.0886	−2.38319
$375$	−8.63291	−0.445801
$376$	−8.23767	−0.424826
$377$	21.5464	1.10970
$378$	37.9200	1.95040
$379$	−18.0093	−0.925075	−0.462537	−	0.886600i	$-0.653061\pi$
$379$	−18.0093	−0.925075	−0.462537	+	0.886600i	$0.653061\pi$
$380$	−0.477229	−0.0244813
$381$	−4.81123	−0.246487
$382$	35.9481	1.83927
$383$	−19.7138	−1.00733	−0.503664	−	0.863900i	$-0.668015\pi$
$383$	−19.7138	−1.00733	−0.503664	+	0.863900i	$0.668015\pi$
$384$	17.1127	0.873278
$385$	2.94865	0.150277
$386$	42.4342	2.15984
$387$	−71.2886	−3.62381
$388$	−20.6796	−1.04985
$389$	−27.2885	−1.38358	−0.691792	−	0.722097i	$-0.743178\pi$
$389$	−27.2885	−1.38358	−0.691792	+	0.722097i	$0.743178\pi$
$390$	10.7771	0.545719
$391$	−4.20914	−0.212865
$392$	−0.215628	−0.0108909
$393$	−11.5305	−0.581637
$394$	40.7301	2.05195
$395$	−4.73291	−0.238139
$396$	−34.7351	−1.74550
$397$	−7.56630	−0.379742	−0.189871	−	0.981809i	$-0.560807\pi$
$397$	−7.56630	−0.379742	−0.189871	+	0.981809i	$0.560807\pi$
$398$	−7.58921	−0.380413
$399$	−7.60683	−0.380818
$400$	22.7377	1.13689
$401$	−16.1443	−0.806209	−0.403104	−	0.915154i	$-0.632069\pi$
$401$	−16.1443	−0.806209	−0.403104	+	0.915154i	$0.632069\pi$
$402$	−53.9571	−2.69114
$403$	49.5042	2.46598
$404$	−8.29034	−0.412460
$405$	−1.70752	−0.0848476
$406$	16.2786	0.807895
$407$	−26.6950	−1.32322
$408$	13.7836	0.682388
$409$	−1.54823	−0.0765553	−0.0382776	−	0.999267i	$-0.512187\pi$
$409$	−1.54823	−0.0765553	−0.0382776	+	0.999267i	$0.512187\pi$
$410$	1.83934	0.0908384
$411$	−47.6363	−2.34972
$412$	−8.85278	−0.436145
$413$	−23.6163	−1.16208
$414$	−7.11509	−0.349688
$415$	0.100969	0.00495635
$416$	−47.6051	−2.33403
$417$	62.9079	3.08062
$418$	7.29181	0.356654
$419$	4.69179	0.229209	0.114604	−	0.993411i	$-0.463440\pi$
$419$	4.69179	0.229209	0.114604	+	0.993411i	$0.463440\pi$
$420$	3.63020	0.177136
$421$	16.6576	0.811843	0.405922	−	0.913908i	$-0.366951\pi$
$421$	16.6576	0.811843	0.405922	+	0.913908i	$0.366951\pi$
$422$	1.89977	0.0924792
$423$	−62.3881	−3.03341
$424$	9.42737	0.457833
$425$	31.0470	1.50600
$426$	58.0701	2.81351
$427$	−6.01052	−0.290869
$428$	6.38208	0.308489
$429$	−73.4170	−3.54461
$430$	−7.14191	−0.344414
$431$	22.7598	1.09630	0.548151	−	0.836379i	$-0.315332\pi$
$431$	22.7598	1.09630	0.548151	+	0.836379i	$0.315332\pi$
$432$	35.6701	1.71618
$433$	−22.9198	−1.10146	−0.550728	−	0.834685i	$-0.685650\pi$
$433$	−22.9198	−1.10146	−0.550728	+	0.834685i	$0.685650\pi$
$434$	37.4011	1.79531
$435$	−2.88113	−0.138139
$436$	−26.8713	−1.28690
$437$	0.665939	0.0318562
$438$	−42.8843	−2.04909
$439$	−12.3668	−0.590233	−0.295117	−	0.955461i	$-0.595358\pi$
$439$	−12.3668	−0.590233	−0.295117	+	0.955461i	$0.595358\pi$
$440$	0.845324	0.0402992
$441$	−1.63306	−0.0777649
$442$	−78.2107	−3.72010
$443$	1.54446	0.0733794	0.0366897	−	0.999327i	$-0.488319\pi$
$443$	1.54446	0.0733794	0.0366897	+	0.999327i	$0.488319\pi$
$444$	−32.8652	−1.55971
$445$	−0.773958	−0.0366891
$446$	−22.2142	−1.05187
$447$	23.0497	1.09021
$448$	−11.9855	−0.566260
$449$	0.547669	0.0258461	0.0129231	−	0.999916i	$-0.495886\pi$
$449$	0.547669	0.0258461	0.0129231	+	0.999916i	$0.495886\pi$
$450$	52.4817	2.47401
$451$	−12.5302	−0.590022
$452$	−18.9679	−0.892177
$453$	−45.7642	−2.15019
$454$	−12.8416	−0.602686
$455$	5.00375	0.234580
$456$	−2.18074	−0.102122
$457$	38.7043	1.81051	0.905254	−	0.424870i	$-0.139680\pi$
$457$	38.7043	1.81051	0.905254	+	0.424870i	$0.139680\pi$
$458$	−12.9784	−0.606439
$459$	48.7054	2.27338
$460$	−0.317805	−0.0148178
$461$	−8.18136	−0.381044	−0.190522	−	0.981683i	$-0.561018\pi$
$461$	−8.18136	−0.381044	−0.190522	+	0.981683i	$0.561018\pi$
$462$	−55.4676	−2.58058
$463$	31.8216	1.47888	0.739439	−	0.673224i	$-0.235091\pi$
$463$	31.8216	1.47888	0.739439	+	0.673224i	$0.235091\pi$
$464$	15.3128	0.710877
$465$	−6.61955	−0.306974
$466$	−49.2435	−2.28116
$467$	10.3578	0.479302	0.239651	−	0.970859i	$-0.422967\pi$
$467$	10.3578	0.479302	0.239651	+	0.970859i	$0.422967\pi$
$468$	−58.9441	−2.72469
$469$	−25.0520	−1.15680
$470$	−6.25023	−0.288302
$471$	70.7653	3.26069
$472$	−6.77034	−0.311630
$473$	48.6530	2.23707
$474$	89.0316	4.08936
$475$	−4.91204	−0.225380
$476$	−26.3448	−1.20751
$477$	71.3982	3.26910
$478$	−28.4834	−1.30280
$479$	4.58645	0.209560	0.104780	−	0.994495i	$-0.466586\pi$
$479$	4.58645	0.209560	0.104780	+	0.994495i	$0.466586\pi$
$480$	6.36561	0.290549
$481$	−45.3003	−2.06552
$482$	−39.6681	−1.80683
$483$	−5.06569	−0.230497
$484$	6.00567	0.272985
$485$	3.81148	0.173070
$486$	−11.7973	−0.535137
$487$	−22.4696	−1.01819	−0.509097	−	0.860709i	$-0.670021\pi$
$487$	−22.4696	−1.01819	−0.509097	+	0.860709i	$0.670021\pi$
$488$	−1.72310	−0.0780012
$489$	−23.5886	−1.06671
$490$	−0.163605	−0.00739093
$491$	3.25391	0.146847	0.0734236	−	0.997301i	$-0.476608\pi$
$491$	3.25391	0.146847	0.0734236	+	0.997301i	$0.476608\pi$
$492$	−15.4263	−0.695474
$493$	20.9087	0.941680
$494$	12.3739	0.556730
$495$	6.40207	0.287751
$496$	35.1820	1.57972
$497$	26.9617	1.20940
$498$	−1.89933	−0.0851112
$499$	−27.0665	−1.21166	−0.605831	−	0.795593i	$-0.707159\pi$
$499$	−27.0665	−1.21166	−0.605831	+	0.795593i	$0.707159\pi$
$500$	4.73031	0.211546
$501$	36.3709	1.62493
$502$	−7.55579	−0.337232
$503$	−14.0133	−0.624820	−0.312410	−	0.949947i	$-0.601136\pi$
$503$	−14.0133	−0.624820	−0.312410	+	0.949947i	$0.601136\pi$
$504$	10.8179	0.481868
$505$	1.52800	0.0679953
$506$	4.85590	0.215871
$507$	−86.4094	−3.83757
$508$	2.63627	0.116965
$509$	9.79344	0.434087	0.217043	−	0.976162i	$-0.430359\pi$
$509$	9.79344	0.434087	0.217043	+	0.976162i	$0.430359\pi$
$510$	10.4581	0.463092
$511$	−19.9110	−0.880809
$512$	−26.9574	−1.19136
$513$	−7.70582	−0.340220
$514$	0.283307	0.0124962
$515$	1.63167	0.0718999
$516$	59.8986	2.63689
$517$	42.5786	1.87260
$518$	−34.2250	−1.50376
$519$	18.8586	0.827799
$520$	1.43448	0.0629062
$521$	6.19917	0.271591	0.135795	−	0.990737i	$-0.456641\pi$
$521$	6.19917	0.271591	0.135795	+	0.990737i	$0.456641\pi$
$522$	35.3439	1.54696
$523$	24.4872	1.07075	0.535374	−	0.844615i	$-0.320171\pi$
$523$	24.4872	1.07075	0.535374	+	0.844615i	$0.320171\pi$
$524$	6.31802	0.276004
$525$	37.3651	1.63074
$526$	21.5813	0.940991
$527$	48.0389	2.09261
$528$	−52.1765	−2.27069
$529$	−22.5565	−0.980718
$530$	7.15289	0.310702
$531$	−51.2753	−2.22516
$532$	4.16808	0.180709
$533$	−21.2632	−0.921012
$534$	14.5590	0.630031
$535$	−1.17629	−0.0508554
$536$	−7.18195	−0.310213
$537$	−16.1314	−0.696120
$538$	41.4504	1.78705
$539$	1.11453	0.0480063
$540$	3.67744	0.158252
$541$	11.8089	0.507705	0.253852	−	0.967243i	$-0.418302\pi$
$541$	11.8089	0.507705	0.253852	+	0.967243i	$0.418302\pi$
$542$	18.2349	0.783256
$543$	−6.03382	−0.258936
$544$	−46.1960	−1.98064
$545$	4.95268	0.212150
$546$	−94.1264	−4.02824
$547$	14.6771	0.627549	0.313774	−	0.949498i	$-0.398406\pi$
$547$	14.6771	0.627549	0.313774	+	0.949498i	$0.398406\pi$
$548$	26.1018	1.11501
$549$	−13.0499	−0.556958
$550$	−35.8177	−1.52727
$551$	−3.30802	−0.140926
$552$	−1.45224	−0.0618113
$553$	41.3369	1.75783
$554$	17.2788	0.734106
$555$	6.05743	0.257123
$556$	−34.4698	−1.46184
$557$	−41.5198	−1.75925	−0.879624	−	0.475669i	$-0.842206\pi$
$557$	−41.5198	−1.75925	−0.879624	+	0.475669i	$0.842206\pi$
$558$	81.2046	3.43767
$559$	82.5624	3.49202
$560$	3.55610	0.150273
$561$	−71.2439	−3.00792
$562$	−44.3933	−1.87262
$563$	22.3767	0.943066	0.471533	−	0.881848i	$-0.343701\pi$
$563$	22.3767	0.943066	0.471533	+	0.881848i	$0.343701\pi$
$564$	52.4201	2.20728
$565$	3.49601	0.147078
$566$	−4.20447	−0.176727
$567$	14.9134	0.626304
$568$	7.72941	0.324319
$569$	−6.73736	−0.282445	−0.141222	−	0.989978i	$-0.545103\pi$
$569$	−6.73736	−0.282445	−0.141222	+	0.989978i	$0.545103\pi$
$570$	−1.65461	−0.0693038
$571$	13.6906	0.572932	0.286466	−	0.958090i	$-0.407519\pi$
$571$	13.6906	0.572932	0.286466	+	0.958090i	$0.407519\pi$
$572$	40.2281	1.68202
$573$	55.5687	2.32142
$574$	−16.0646	−0.670525
$575$	−3.27112	−0.136415
$576$	−26.0227	−1.08428
$577$	−9.01376	−0.375248	−0.187624	−	0.982241i	$-0.560079\pi$
$577$	−9.01376	−0.375248	−0.187624	+	0.982241i	$0.560079\pi$
$578$	−43.5996	−1.81351
$579$	65.5949	2.72603
$580$	1.57868	0.0655513
$581$	−0.881852	−0.0365854
$582$	−71.6983	−2.97199
$583$	−48.7278	−2.01810
$584$	−5.70810	−0.236203
$585$	10.8641	0.449174
$586$	−30.8862	−1.27590
$587$	−17.2825	−0.713327	−0.356664	−	0.934233i	$-0.616086\pi$
$587$	−17.2825	−0.713327	−0.356664	+	0.934233i	$0.616086\pi$
$588$	1.37214	0.0565862
$589$	−7.60037	−0.313168
$590$	−5.13691	−0.211483
$591$	62.9607	2.58986
$592$	−32.1943	−1.32318
$593$	22.0485	0.905421	0.452711	−	0.891657i	$-0.350457\pi$
$593$	22.0485	0.905421	0.452711	+	0.891657i	$0.350457\pi$
$594$	−56.1894	−2.30548
$595$	4.85565	0.199062
$596$	−12.6299	−0.517339
$597$	−11.7314	−0.480135
$598$	8.24029	0.336970
$599$	25.0232	1.02242	0.511211	−	0.859455i	$-0.329197\pi$
$599$	25.0232	1.02242	0.511211	+	0.859455i	$0.329197\pi$
$600$	10.7119	0.437310
$601$	−22.4314	−0.914995	−0.457498	−	0.889211i	$-0.651254\pi$
$601$	−22.4314	−0.914995	−0.457498	+	0.889211i	$0.651254\pi$
$602$	62.3770	2.54230
$603$	−54.3926	−2.21504
$604$	25.0761	1.02033
$605$	−1.10691	−0.0450025
$606$	−28.7435	−1.16763
$607$	−18.1327	−0.735984	−0.367992	−	0.929829i	$-0.619955\pi$
$607$	−18.1327	−0.735984	−0.367992	+	0.929829i	$0.619955\pi$
$608$	7.30880	0.296411
$609$	25.1636	1.01968
$610$	−1.30738	−0.0529343
$611$	72.2543	2.92309
$612$	−57.1994	−2.31215
$613$	27.5528	1.11285	0.556423	−	0.830899i	$-0.312174\pi$
$613$	27.5528	1.11285	0.556423	+	0.830899i	$0.312174\pi$
$614$	8.19458	0.330706
$615$	2.84325	0.114651
$616$	−7.38300	−0.297469
$617$	−42.9468	−1.72898	−0.864488	−	0.502654i	$-0.832357\pi$
$617$	−42.9468	−1.72898	−0.864488	+	0.502654i	$0.832357\pi$
$618$	−30.6936	−1.23468
$619$	15.6126	0.627523	0.313761	−	0.949502i	$-0.398411\pi$
$619$	15.6126	0.627523	0.313761	+	0.949502i	$0.398411\pi$
$620$	3.62712	0.145669
$621$	−5.13161	−0.205924
$622$	−59.5459	−2.38757
$623$	6.75969	0.270821
$624$	−88.5415	−3.54450
$625$	23.6884	0.947534
$626$	28.8800	1.15428
$627$	11.2717	0.450148
$628$	−38.7751	−1.54730
$629$	−43.9595	−1.75278
$630$	8.20795	0.327013
$631$	19.3402	0.769921	0.384961	−	0.922933i	$-0.374215\pi$
$631$	19.3402	0.769921	0.384961	+	0.922933i	$0.374215\pi$
$632$	11.8505	0.471389
$633$	2.93667	0.116722
$634$	11.7193	0.465433
$635$	−0.485894	−0.0192821
$636$	−59.9907	−2.37878
$637$	1.89132	0.0749368
$638$	−24.1215	−0.954978
$639$	58.5388	2.31576
$640$	1.72824	0.0683146
$641$	21.9733	0.867892	0.433946	−	0.900939i	$-0.357121\pi$
$641$	21.9733	0.867892	0.433946	+	0.900939i	$0.357121\pi$
$642$	22.1274	0.873298
$643$	45.2299	1.78369	0.891846	−	0.452340i	$-0.149410\pi$
$643$	45.2299	1.78369	0.891846	+	0.452340i	$0.149410\pi$
$644$	2.77569	0.109378
$645$	−11.0400	−0.434699
$646$	12.0077	0.472435
$647$	48.0934	1.89075	0.945374	−	0.325989i	$-0.105697\pi$
$647$	48.0934	1.89075	0.945374	+	0.325989i	$0.105697\pi$
$648$	4.27539	0.167953
$649$	34.9943	1.37365
$650$	−60.7813	−2.38404
$651$	57.8147	2.26594
$652$	12.9251	0.506187
$653$	4.97432	0.194660	0.0973302	−	0.995252i	$-0.468970\pi$
$653$	4.97432	0.194660	0.0973302	+	0.995252i	$0.468970\pi$
$654$	−93.1657	−3.64307
$655$	−1.16448	−0.0455001
$656$	−15.1115	−0.590003
$657$	−43.2304	−1.68658
$658$	54.5891	2.12810
$659$	42.2810	1.64703	0.823517	−	0.567291i	$-0.192009\pi$
$659$	42.2810	1.64703	0.823517	+	0.567291i	$0.192009\pi$
$660$	−5.37918	−0.209384
$661$	20.4001	0.793471	0.396736	−	0.917933i	$-0.370143\pi$
$661$	20.4001	0.793471	0.396736	+	0.917933i	$0.370143\pi$
$662$	10.9649	0.426165
$663$	−120.898	−4.69530
$664$	−0.252810	−0.00981095
$665$	−0.768225	−0.0297905
$666$	−74.3088	−2.87941
$667$	−2.20294	−0.0852982
$668$	−19.9291	−0.771079
$669$	−34.3387	−1.32761
$670$	−5.44921	−0.210521
$671$	8.90631	0.343824
$672$	−55.5968	−2.14469
$673$	−17.6593	−0.680715	−0.340357	−	0.940296i	$-0.610548\pi$
$673$	−17.6593	−0.680715	−0.340357	+	0.940296i	$0.610548\pi$
$674$	11.0928	0.427278
$675$	37.8513	1.45690
$676$	47.3471	1.82104
$677$	37.7849	1.45219	0.726096	−	0.687594i	$-0.241333\pi$
$677$	37.7849	1.45219	0.726096	+	0.687594i	$0.241333\pi$
$678$	−65.7639	−2.52565
$679$	−33.2892	−1.27752
$680$	1.39202	0.0533816
$681$	−19.8506	−0.760676
$682$	−55.4205	−2.12216
$683$	−31.6536	−1.21119	−0.605596	−	0.795772i	$-0.707065\pi$
$683$	−31.6536	−1.21119	−0.605596	+	0.795772i	$0.707065\pi$
$684$	9.04967	0.346023
$685$	−4.81086	−0.183814
$686$	35.8756	1.36974
$687$	−20.0620	−0.765413
$688$	58.6759	2.23700
$689$	−82.6893	−3.15021
$690$	−1.10187	−0.0419473
$691$	−29.1948	−1.11062	−0.555311	−	0.831643i	$-0.687401\pi$
$691$	−29.1948	−1.11062	−0.555311	+	0.831643i	$0.687401\pi$
$692$	−10.3334	−0.392815
$693$	−55.9152	−2.12404
$694$	41.8495	1.58858
$695$	6.35317	0.240989
$696$	7.21392	0.273443
$697$	−20.6338	−0.781562
$698$	19.9111	0.753646
$699$	−76.1208	−2.87915
$700$	−20.4738	−0.773837
$701$	4.37747	0.165335	0.0826675	−	0.996577i	$-0.473656\pi$
$701$	4.37747	0.165335	0.0826675	+	0.996577i	$0.473656\pi$
$702$	−95.3514	−3.59881
$703$	6.95495	0.262311
$704$	17.7599	0.669352
$705$	−9.66162	−0.363878
$706$	5.49426	0.206779
$707$	−13.3455	−0.501908
$708$	43.0828	1.61915
$709$	−17.0757	−0.641291	−0.320646	−	0.947199i	$-0.603900\pi$
$709$	−17.0757	−0.641291	−0.320646	+	0.947199i	$0.603900\pi$
$710$	5.86459	0.220094
$711$	89.7501	3.36589
$712$	1.93788	0.0726251
$713$	−5.06139	−0.189550
$714$	−91.3403	−3.41832
$715$	−7.41450	−0.277287
$716$	8.83902	0.330330
$717$	−44.0297	−1.64432
$718$	−29.0988	−1.08596
$719$	20.9166	0.780058	0.390029	−	0.920803i	$-0.372465\pi$
$719$	20.9166	0.780058	0.390029	+	0.920803i	$0.372465\pi$
$720$	7.72094	0.287743
$721$	−14.2509	−0.530731
$722$	−1.89977	−0.0707020
$723$	−61.3190	−2.28048
$724$	3.30617	0.122873
$725$	16.2491	0.603478
$726$	20.8223	0.772789
$727$	53.4040	1.98064	0.990322	−	0.138786i	$-0.0443201\pi$
$727$	53.4040	1.98064	0.990322	+	0.138786i	$0.0443201\pi$
$728$	−12.5287	−0.464344
$729$	−35.5086	−1.31513
$730$	−4.33095	−0.160296
$731$	80.1185	2.96329
$732$	10.9649	0.405274
$733$	−7.75826	−0.286558	−0.143279	−	0.989682i	$-0.545765\pi$
$733$	−7.75826	−0.286558	−0.143279	+	0.989682i	$0.545765\pi$
$734$	11.8961	0.439091
$735$	−0.252901	−0.00932841
$736$	4.86722	0.179408
$737$	37.1218	1.36740
$738$	−34.8793	−1.28392
$739$	−8.78443	−0.323141	−0.161570	−	0.986861i	$-0.551656\pi$
$739$	−8.78443	−0.323141	−0.161570	+	0.986861i	$0.551656\pi$
$740$	−3.31910	−0.122013
$741$	19.1277	0.702672
$742$	−62.4729	−2.29345
$743$	−20.2223	−0.741885	−0.370943	−	0.928656i	$-0.620965\pi$
$743$	−20.2223	−0.741885	−0.370943	+	0.928656i	$0.620965\pi$
$744$	16.5744	0.607647
$745$	2.32783	0.0852850
$746$	67.1533	2.45866
$747$	−1.91466	−0.0700538
$748$	39.0374	1.42735
$749$	10.2736	0.375390
$750$	16.4005	0.598862
$751$	−9.25172	−0.337600	−0.168800	−	0.985650i	$-0.553989\pi$
$751$	−9.25172	−0.337600	−0.168800	+	0.985650i	$0.553989\pi$
$752$	51.3501	1.87254
$753$	−11.6798	−0.425634
$754$	−40.9332	−1.49070
$755$	−4.62180	−0.168205
$756$	−32.1185	−1.16814
$757$	−27.0867	−0.984482	−0.492241	−	0.870459i	$-0.663822\pi$
$757$	−27.0867	−0.984482	−0.492241	+	0.870459i	$0.663822\pi$
$758$	34.2135	1.24269
$759$	7.50627	0.272460
$760$	−0.220236	−0.00798879
$761$	13.4380	0.487127	0.243564	−	0.969885i	$-0.421684\pi$
$761$	13.4380	0.487127	0.243564	+	0.969885i	$0.421684\pi$
$762$	9.14022	0.331115
$763$	−43.2564	−1.56599
$764$	−30.4483	−1.10158
$765$	10.5425	0.381165
$766$	37.4516	1.35318
$767$	59.3840	2.14423
$768$	−59.6864	−2.15375
$769$	0.803969	0.0289919	0.0144959	−	0.999895i	$-0.495386\pi$
$769$	0.803969	0.0289919	0.0144959	+	0.999895i	$0.495386\pi$
$770$	−5.60175	−0.201873
$771$	0.437937	0.0157719
$772$	−35.9421	−1.29358
$773$	−36.5587	−1.31493	−0.657463	−	0.753486i	$-0.728371\pi$
$773$	−36.5587	−1.31493	−0.657463	+	0.753486i	$0.728371\pi$
$774$	135.432	4.86800
$775$	37.3333	1.34105
$776$	−9.54339	−0.342588
$777$	−52.9051	−1.89796
$778$	51.8419	1.85862
$779$	3.26454	0.116964
$780$	−9.12827	−0.326845
$781$	−39.9515	−1.42958
$782$	7.99638	0.285950
$783$	25.4910	0.910975
$784$	1.34413	0.0480048
$785$	7.14670	0.255077
$786$	21.9053	0.781335
$787$	30.3112	1.08048	0.540240	−	0.841511i	$-0.318334\pi$
$787$	30.3112	1.08048	0.540240	+	0.841511i	$0.318334\pi$
$788$	−34.4987	−1.22896
$789$	33.3605	1.18767
$790$	8.99143	0.319901
$791$	−30.5339	−1.08566
$792$	−16.0298	−0.569596
$793$	15.1137	0.536702
$794$	14.3742	0.510122
$795$	11.0570	0.392150
$796$	6.42811	0.227838
$797$	16.2570	0.575854	0.287927	−	0.957652i	$-0.407034\pi$
$797$	16.2570	0.575854	0.287927	+	0.957652i	$0.407034\pi$
$798$	14.4512	0.511567
$799$	70.1156	2.48051
$800$	−35.9011	−1.26930
$801$	14.6765	0.518570
$802$	30.6704	1.08301
$803$	29.5038	1.04117
$804$	45.7020	1.61179
$805$	−0.511591	−0.0180312
$806$	−94.0465	−3.31265
$807$	64.0741	2.25551
$808$	−3.82590	−0.134595
$809$	30.0789	1.05752	0.528758	−	0.848773i	$-0.322658\pi$
$809$	30.0789	1.05752	0.528758	+	0.848773i	$0.322658\pi$
$810$	3.24390	0.113979
$811$	−12.8411	−0.450914	−0.225457	−	0.974253i	$-0.572387\pi$
$811$	−12.8411	−0.450914	−0.225457	+	0.974253i	$0.572387\pi$
$812$	−13.7881	−0.483868
$813$	28.1876	0.988581
$814$	50.7142	1.77753
$815$	−2.38225	−0.0834464
$816$	−85.9207	−3.00783
$817$	−12.6758	−0.443469
$818$	2.94128	0.102840
$819$	−94.8860	−3.31559
$820$	−1.55793	−0.0544053
$821$	44.0980	1.53903	0.769516	−	0.638627i	$-0.220497\pi$
$821$	44.0980	1.53903	0.769516	+	0.638627i	$0.220497\pi$
$822$	90.4979	3.15648
$823$	−10.6195	−0.370173	−0.185086	−	0.982722i	$-0.559256\pi$
$823$	−10.6195	−0.370173	−0.185086	+	0.982722i	$0.559256\pi$
$824$	−4.08546	−0.142324
$825$	−55.3671	−1.92763
$826$	44.8654	1.56107
$827$	−4.05662	−0.141062	−0.0705312	−	0.997510i	$-0.522469\pi$
$827$	−4.05662	−0.141062	−0.0705312	+	0.997510i	$0.522469\pi$
$828$	6.02653	0.209437
$829$	−28.8885	−1.00334	−0.501669	−	0.865060i	$-0.667280\pi$
$829$	−28.8885	−1.00334	−0.501669	+	0.865060i	$0.667280\pi$
$830$	−0.191817	−0.00665806
$831$	26.7096	0.926546
$832$	30.1379	1.04484
$833$	1.83534	0.0635906
$834$	−119.510	−4.13831
$835$	3.67316	0.127115
$836$	−6.17621	−0.213609
$837$	58.5671	2.02438
$838$	−8.91330	−0.307905
$839$	36.6258	1.26446	0.632232	−	0.774779i	$-0.282139\pi$
$839$	36.6258	1.26446	0.632232	+	0.774779i	$0.282139\pi$
$840$	1.67530	0.0578032
$841$	−18.0570	−0.622655
$842$	−31.6456	−1.09058
$843$	−68.6232	−2.36351
$844$	−1.60912	−0.0553880
$845$	−8.72661	−0.300205
$846$	118.523	4.07490
$847$	9.66771	0.332186
$848$	−58.7661	−2.01804
$849$	−6.49929	−0.223055
$850$	−58.9822	−2.02307
$851$	4.63158	0.158768
$852$	−49.1858	−1.68508
$853$	−12.1916	−0.417431	−0.208716	−	0.977976i	$-0.566928\pi$
$853$	−12.1916	−0.417431	−0.208716	+	0.977976i	$0.566928\pi$
$854$	11.4186	0.390736
$855$	−1.66796	−0.0570430
$856$	2.94526	0.100667
$857$	−1.97626	−0.0675079	−0.0337539	−	0.999430i	$-0.510746\pi$
$857$	−1.97626	−0.0675079	−0.0337539	+	0.999430i	$0.510746\pi$
$858$	139.475	4.76161
$859$	41.0468	1.40050	0.700249	−	0.713899i	$-0.253072\pi$
$859$	41.0468	1.40050	0.700249	+	0.713899i	$0.253072\pi$
$860$	6.04925	0.206278
$861$	−24.8328	−0.846298
$862$	−43.2384	−1.47270
$863$	−32.8154	−1.11705	−0.558524	−	0.829488i	$-0.688632\pi$
$863$	−32.8154	−1.11705	−0.558524	+	0.829488i	$0.688632\pi$
$864$	−56.3204	−1.91606
$865$	1.90456	0.0647568
$866$	43.5423	1.47963
$867$	−67.3964	−2.28890
$868$	−31.6790	−1.07525
$869$	−61.2525	−2.07785
$870$	5.47347	0.185568
$871$	62.9943	2.13448
$872$	−12.4008	−0.419944
$873$	−72.2769	−2.44620
$874$	−1.26513	−0.0427936
$875$	7.61468	0.257423
$876$	36.3233	1.22725
$877$	−23.5769	−0.796135	−0.398067	−	0.917356i	$-0.630319\pi$
$877$	−23.5769	−0.796135	−0.398067	+	0.917356i	$0.630319\pi$
$878$	23.4940	0.792883
$879$	−47.7440	−1.61037
$880$	−5.26938	−0.177631
$881$	−18.9171	−0.637333	−0.318666	−	0.947867i	$-0.603235\pi$
$881$	−18.9171	−0.637333	−0.318666	+	0.947867i	$0.603235\pi$
$882$	3.10244	0.104465
$883$	35.2931	1.18771	0.593854	−	0.804573i	$-0.297606\pi$
$883$	35.2931	1.18771	0.593854	+	0.804573i	$0.297606\pi$
$884$	66.2450	2.22806
$885$	−7.94065	−0.266922
$886$	−2.93411	−0.0985734
$887$	20.9667	0.703991	0.351996	−	0.936002i	$-0.385503\pi$
$887$	20.9667	0.703991	0.351996	+	0.936002i	$0.385503\pi$
$888$	−15.1669	−0.508968
$889$	4.24376	0.142331
$890$	1.47034	0.0492859
$891$	−22.0985	−0.740327
$892$	18.8155	0.629991
$893$	−11.0932	−0.371219
$894$	−43.7891	−1.46453
$895$	−1.62913	−0.0544559
$896$	−15.0943	−0.504265
$897$	12.7379	0.425305
$898$	−1.04044	−0.0347201
$899$	25.1422	0.838539
$900$	−44.4524	−1.48175
$901$	−80.2417	−2.67324
$902$	23.8044	0.792599
$903$	96.4225	3.20874
$904$	−8.75350	−0.291137
$905$	−0.609365	−0.0202560
$906$	86.9414	2.88844
$907$	24.0537	0.798688	0.399344	−	0.916801i	$-0.369238\pi$
$907$	24.0537	0.798688	0.399344	+	0.916801i	$0.369238\pi$
$908$	10.8769	0.360963
$909$	−28.9755	−0.961056
$910$	−9.50597	−0.315120
$911$	−37.8287	−1.25332	−0.626660	−	0.779293i	$-0.715578\pi$
$911$	−37.8287	−1.25332	−0.626660	+	0.779293i	$0.715578\pi$
$912$	13.5938	0.450134
$913$	1.30672	0.0432460
$914$	−73.5291	−2.43213
$915$	−2.02095	−0.0668107
$916$	10.9928	0.363211
$917$	10.1705	0.335860
$918$	−92.5290	−3.05391
$919$	23.8837	0.787851	0.393926	−	0.919142i	$-0.371117\pi$
$919$	23.8837	0.787851	0.393926	+	0.919142i	$0.371117\pi$
$920$	−0.146664	−0.00483536
$921$	12.6672	0.417399
$922$	15.5427	0.511871
$923$	−67.7962	−2.23154
$924$	46.9814	1.54557
$925$	−34.1630	−1.12327
$926$	−60.4537	−1.98663
$927$	−30.9413	−1.01624
$928$	−24.1777	−0.793671
$929$	−6.31467	−0.207178	−0.103589	−	0.994620i	$-0.533033\pi$
$929$	−6.31467	−0.207178	−0.103589	+	0.994620i	$0.533033\pi$
$930$	12.5756	0.412371
$931$	−0.290374	−0.00951661
$932$	41.7096	1.36624
$933$	−92.0462	−3.01346
$934$	−19.6774	−0.643864
$935$	−7.19503	−0.235303
$936$	−27.2020	−0.889127
$937$	−3.97927	−0.129997	−0.0649986	−	0.997885i	$-0.520704\pi$
$937$	−3.97927	−0.129997	−0.0649986	+	0.997885i	$0.520704\pi$
$938$	47.5930	1.55397
$939$	44.6428	1.45686
$940$	5.29399	0.172671
$941$	32.8542	1.07102	0.535508	−	0.844530i	$-0.320120\pi$
$941$	32.8542	1.07102	0.535508	+	0.844530i	$0.320120\pi$
$942$	−134.438	−4.38022
$943$	2.17398	0.0707946
$944$	42.2034	1.37360
$945$	5.91981	0.192571
$946$	−92.4294	−3.00514
$947$	−48.4407	−1.57411	−0.787056	−	0.616882i	$-0.788396\pi$
$947$	−48.4407	−1.57411	−0.787056	+	0.616882i	$0.788396\pi$
$948$	−75.4103	−2.44921
$949$	50.0669	1.62524
$950$	9.33174	0.302761
$951$	18.1157	0.587443
$952$	−12.1578	−0.394037
$953$	22.4903	0.728533	0.364267	−	0.931295i	$-0.381320\pi$
$953$	22.4903	0.728533	0.364267	+	0.931295i	$0.381320\pi$
$954$	−135.640	−4.39151
$955$	5.61197	0.181599
$956$	24.1256	0.780278
$957$	−37.2870	−1.20532
$958$	−8.71318	−0.281510
$959$	42.0177	1.35682
$960$	−4.02995	−0.130066
$961$	26.7656	0.863408
$962$	86.0601	2.77469
$963$	22.3059	0.718799
$964$	33.5991	1.08216
$965$	6.62453	0.213251
$966$	9.62362	0.309635
$967$	38.3375	1.23285	0.616425	−	0.787414i	$-0.288580\pi$
$967$	38.3375	1.23285	0.616425	+	0.787414i	$0.288580\pi$
$968$	2.77155	0.0890810
$969$	18.5615	0.596281
$970$	−7.24092	−0.232492
$971$	−31.9105	−1.02406	−0.512028	−	0.858969i	$-0.671106\pi$
$971$	−31.9105	−1.02406	−0.512028	+	0.858969i	$0.671106\pi$
$972$	9.99240	0.320506
$973$	−55.4881	−1.77887
$974$	42.6870	1.36778
$975$	−93.9559	−3.00900
$976$	10.7411	0.343813
$977$	−9.54722	−0.305443	−0.152721	−	0.988269i	$-0.548804\pi$
$977$	−9.54722	−0.305443	−0.152721	+	0.988269i	$0.548804\pi$
$978$	44.8128	1.43296
$979$	−10.0164	−0.320126
$980$	0.138575	0.00442661
$981$	−93.9175	−2.99856
$982$	−6.18168	−0.197265
$983$	−16.8738	−0.538192	−0.269096	−	0.963113i	$-0.586725\pi$
$983$	−16.8738	−0.538192	−0.269096	+	0.963113i	$0.586725\pi$
$984$	−7.11909	−0.226948
$985$	6.35850	0.202599
$986$	−39.7216	−1.26499
$987$	84.3839	2.68597
$988$	−10.4808	−0.333439
$989$	−8.44130	−0.268418
$990$	−12.1624	−0.386548
$991$	28.6389	0.909744	0.454872	−	0.890557i	$-0.349685\pi$
$991$	28.6389	0.909744	0.454872	+	0.890557i	$0.349685\pi$
$992$	−55.5496	−1.76370
$993$	16.9496	0.537880
$994$	−51.2209	−1.62463
$995$	−1.18477	−0.0375599
$996$	1.60875	0.0509752
$997$	36.0667	1.14224	0.571122	−	0.820866i	$-0.306508\pi$
$997$	36.0667	1.14224	0.571122	+	0.820866i	$0.306508\pi$
$998$	51.4200	1.62767
$999$	−53.5936	−1.69563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.e.1.15	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.e.1.15	✓	82	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)

\(f(q)\)	\(=\)	\(q-1.89977 q^{2} -2.93667 q^{3} +1.60912 q^{4} -0.296578 q^{5} +5.57898 q^{6} +2.59029 q^{7} +0.742589 q^{8} +5.62400 q^{9} +O(q^{10})\)	\(q-1.89977 q^{2} -2.93667 q^{3} +1.60912 q^{4} -0.296578 q^{5} +5.57898 q^{6} +2.59029 q^{7} +0.742589 q^{8} +5.62400 q^{9} +0.563430 q^{10} -3.83827 q^{11} -4.72543 q^{12} -6.51339 q^{13} -4.92096 q^{14} +0.870951 q^{15} -4.62898 q^{16} -6.32060 q^{17} -10.6843 q^{18} +1.00000 q^{19} -0.477229 q^{20} -7.60683 q^{21} +7.29181 q^{22} +0.665939 q^{23} -2.18074 q^{24} -4.91204 q^{25} +12.3739 q^{26} -7.70582 q^{27} +4.16808 q^{28} -3.30802 q^{29} -1.65461 q^{30} -7.60037 q^{31} +7.30880 q^{32} +11.2717 q^{33} +12.0077 q^{34} -0.768225 q^{35} +9.04967 q^{36} +6.95495 q^{37} -1.89977 q^{38} +19.1277 q^{39} -0.220236 q^{40} +3.26454 q^{41} +14.4512 q^{42} -12.6758 q^{43} -6.17621 q^{44} -1.66796 q^{45} -1.26513 q^{46} -11.0932 q^{47} +13.5938 q^{48} -0.290374 q^{49} +9.33174 q^{50} +18.5615 q^{51} -10.4808 q^{52} +12.6953 q^{53} +14.6393 q^{54} +1.13835 q^{55} +1.92352 q^{56} -2.93667 q^{57} +6.28447 q^{58} -9.11722 q^{59} +1.40146 q^{60} -2.32040 q^{61} +14.4389 q^{62} +14.5678 q^{63} -4.62707 q^{64} +1.93173 q^{65} -21.4136 q^{66} -9.67150 q^{67} -10.1706 q^{68} -1.95564 q^{69} +1.45945 q^{70} +10.4087 q^{71} +4.17632 q^{72} -7.68676 q^{73} -13.2128 q^{74} +14.4250 q^{75} +1.60912 q^{76} -9.94224 q^{77} -36.3381 q^{78} +15.9584 q^{79} +1.37285 q^{80} +5.75742 q^{81} -6.20186 q^{82} -0.340445 q^{83} -12.2403 q^{84} +1.87455 q^{85} +24.0810 q^{86} +9.71455 q^{87} -2.85025 q^{88} +2.60962 q^{89} +3.16873 q^{90} -16.8716 q^{91} +1.07157 q^{92} +22.3197 q^{93} +21.0745 q^{94} -0.296578 q^{95} -21.4635 q^{96} -12.8515 q^{97} +0.551643 q^{98} -21.5864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * q^99

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