LMFDB - Embedded newform 177.4.a.a.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,4,Mod(1,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("177.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.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:	$10.4433380710$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 34x^{5} + 25x^{4} + 315x^{3} - 146x^{2} - 736x + 512 $ x^7 - x^6 - 34x^5 + 25x^4 + 315x^3 - 146x^2 - 736*x + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.29817$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.29817 q^{2} +3.00000 q^{3} -6.31476 q^{4} +7.05496 q^{5} +3.89451 q^{6} -33.4497 q^{7} -18.5830 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+1.29817 q^{2} +3.00000 q^{3} -6.31476 q^{4} +7.05496 q^{5} +3.89451 q^{6} -33.4497 q^{7} -18.5830 q^{8} +9.00000 q^{9} +9.15853 q^{10} -27.5842 q^{11} -18.9443 q^{12} -55.6756 q^{13} -43.4234 q^{14} +21.1649 q^{15} +26.3942 q^{16} +108.437 q^{17} +11.6835 q^{18} -141.505 q^{19} -44.5504 q^{20} -100.349 q^{21} -35.8089 q^{22} +142.351 q^{23} -55.7489 q^{24} -75.2276 q^{25} -72.2763 q^{26} +27.0000 q^{27} +211.227 q^{28} -97.2883 q^{29} +27.4756 q^{30} -221.538 q^{31} +182.928 q^{32} -82.7525 q^{33} +140.770 q^{34} -235.987 q^{35} -56.8328 q^{36} +339.976 q^{37} -183.698 q^{38} -167.027 q^{39} -131.102 q^{40} -266.877 q^{41} -130.270 q^{42} -67.2505 q^{43} +174.187 q^{44} +63.4946 q^{45} +184.795 q^{46} -262.017 q^{47} +79.1827 q^{48} +775.886 q^{49} -97.6581 q^{50} +325.311 q^{51} +351.578 q^{52} +380.876 q^{53} +35.0506 q^{54} -194.605 q^{55} +621.596 q^{56} -424.516 q^{57} -126.297 q^{58} -59.0000 q^{59} -133.651 q^{60} -15.4226 q^{61} -287.593 q^{62} -301.048 q^{63} +26.3175 q^{64} -392.789 q^{65} -107.427 q^{66} -172.954 q^{67} -684.753 q^{68} +427.052 q^{69} -306.350 q^{70} -616.313 q^{71} -167.247 q^{72} -210.965 q^{73} +441.346 q^{74} -225.683 q^{75} +893.572 q^{76} +922.684 q^{77} -216.829 q^{78} +543.689 q^{79} +186.210 q^{80} +81.0000 q^{81} -346.452 q^{82} +350.751 q^{83} +633.681 q^{84} +765.019 q^{85} -87.3025 q^{86} -291.865 q^{87} +512.596 q^{88} +1447.29 q^{89} +82.4267 q^{90} +1862.34 q^{91} -898.911 q^{92} -664.613 q^{93} -340.143 q^{94} -998.315 q^{95} +548.784 q^{96} -625.674 q^{97} +1007.23 q^{98} -248.258 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9}+O(q^{10}) $ 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9	\( 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9} - 79 q^{10} - 131 q^{11} + 66 q^{12} - 123 q^{13} - 117 q^{14} - 84 q^{15} + 202 q^{16} - 235 q^{17} - 72 q^{18} - 80 q^{19} + 61 q^{20} - 177 q^{21} + 688 q^{22} - 274 q^{23} - 351 q^{24} + 193 q^{25} - 180 q^{26} + 189 q^{27} - 118 q^{28} - 406 q^{29} - 237 q^{30} - 346 q^{31} - 854 q^{32} - 393 q^{33} + 178 q^{34} - 424 q^{35} + 198 q^{36} - 157 q^{37} - 129 q^{38} - 369 q^{39} - 590 q^{40} - 825 q^{41} - 351 q^{42} - 815 q^{43} - 1690 q^{44} - 252 q^{45} + 1457 q^{46} - 1196 q^{47} + 606 q^{48} + 914 q^{49} + 713 q^{50} - 705 q^{51} + 1030 q^{52} - 900 q^{53} - 216 q^{54} - 1044 q^{55} + 2172 q^{56} - 240 q^{57} + 1242 q^{58} - 413 q^{59} + 183 q^{60} + 420 q^{61} + 646 q^{62} - 531 q^{63} + 3541 q^{64} + 190 q^{65} + 2064 q^{66} + 1316 q^{67} - 611 q^{68} - 822 q^{69} + 4658 q^{70} - 173 q^{71} - 1053 q^{72} - 418 q^{73} + 660 q^{74} + 579 q^{75} + 1540 q^{76} - 753 q^{77} - 540 q^{78} + 2635 q^{79} + 6155 q^{80} + 567 q^{81} - 125 q^{82} + 457 q^{83} - 354 q^{84} + 1270 q^{85} + 3482 q^{86} - 1218 q^{87} + 7685 q^{88} + 592 q^{89} - 711 q^{90} + 3179 q^{91} - 3500 q^{92} - 1038 q^{93} + 2064 q^{94} - 2250 q^{95} - 2562 q^{96} - 1906 q^{97} + 2994 q^{98} - 1179 q^{99}+O(q^{100}) \) 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9 - 79 * q^10 - 131 * q^11 + 66 * q^12 - 123 * q^13 - 117 * q^14 - 84 * q^15 + 202 * q^16 - 235 * q^17 - 72 * q^18 - 80 * q^19 + 61 * q^20 - 177 * q^21 + 688 * q^22 - 274 * q^23 - 351 * q^24 + 193 * q^25 - 180 * q^26 + 189 * q^27 - 118 * q^28 - 406 * q^29 - 237 * q^30 - 346 * q^31 - 854 * q^32 - 393 * q^33 + 178 * q^34 - 424 * q^35 + 198 * q^36 - 157 * q^37 - 129 * q^38 - 369 * q^39 - 590 * q^40 - 825 * q^41 - 351 * q^42 - 815 * q^43 - 1690 * q^44 - 252 * q^45 + 1457 * q^46 - 1196 * q^47 + 606 * q^48 + 914 * q^49 + 713 * q^50 - 705 * q^51 + 1030 * q^52 - 900 * q^53 - 216 * q^54 - 1044 * q^55 + 2172 * q^56 - 240 * q^57 + 1242 * q^58 - 413 * q^59 + 183 * q^60 + 420 * q^61 + 646 * q^62 - 531 * q^63 + 3541 * q^64 + 190 * q^65 + 2064 * q^66 + 1316 * q^67 - 611 * q^68 - 822 * q^69 + 4658 * q^70 - 173 * q^71 - 1053 * q^72 - 418 * q^73 + 660 * q^74 + 579 * q^75 + 1540 * q^76 - 753 * q^77 - 540 * q^78 + 2635 * q^79 + 6155 * q^80 + 567 * q^81 - 125 * q^82 + 457 * q^83 - 354 * q^84 + 1270 * q^85 + 3482 * q^86 - 1218 * q^87 + 7685 * q^88 + 592 * q^89 - 711 * q^90 + 3179 * q^91 - 3500 * q^92 - 1038 * q^93 + 2064 * q^94 - 2250 * q^95 - 2562 * q^96 - 1906 * q^97 + 2994 * q^98 - 1179 * 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 $

$2$	1.29817	0.458972	0.229486	−	0.973312i	$-0.426295\pi$
$2$	1.29817	0.458972	0.229486	+	0.973312i	$0.426295\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−6.31476	−0.789345
$5$	7.05496	0.631015	0.315507	−	0.948923i	$-0.397825\pi$
$5$	7.05496	0.631015	0.315507	+	0.948923i	$0.397825\pi$
$6$	3.89451	0.264988
$7$	−33.4497	−1.80612	−0.903058	−	0.429518i	$-0.858684\pi$
$7$	−33.4497	−1.80612	−0.903058	+	0.429518i	$0.858684\pi$
$8$	−18.5830	−0.821259
$9$	9.00000	0.333333
$10$	9.15853	0.289618
$11$	−27.5842	−0.756086	−0.378043	−	0.925788i	$-0.623403\pi$
$11$	−27.5842	−0.756086	−0.378043	+	0.925788i	$0.623403\pi$
$12$	−18.9443	−0.455728
$13$	−55.6756	−1.18782	−0.593909	−	0.804532i	$-0.702416\pi$
$13$	−55.6756	−1.18782	−0.593909	+	0.804532i	$0.702416\pi$
$14$	−43.4234	−0.828957
$15$	21.1649	0.364317
$16$	26.3942	0.412410
$17$	108.437	1.54705	0.773525	−	0.633766i	$-0.218492\pi$
$17$	108.437	1.54705	0.773525	+	0.633766i	$0.218492\pi$
$18$	11.6835	0.152991
$19$	−141.505	−1.70861	−0.854304	−	0.519773i	$-0.826017\pi$
$19$	−141.505	−1.70861	−0.854304	+	0.519773i	$0.826017\pi$
$20$	−44.5504	−0.498088
$21$	−100.349	−1.04276
$22$	−35.8089	−0.347022
$23$	142.351	1.29053	0.645265	−	0.763959i	$-0.276747\pi$
$23$	142.351	1.29053	0.645265	+	0.763959i	$0.276747\pi$
$24$	−55.7489	−0.474154
$25$	−75.2276	−0.601820
$26$	−72.2763	−0.545175
$27$	27.0000	0.192450
$28$	211.227	1.42565
$29$	−97.2883	−0.622965	−0.311482	−	0.950252i	$-0.600825\pi$
$29$	−97.2883	−0.622965	−0.311482	+	0.950252i	$0.600825\pi$
$30$	27.4756	0.167211
$31$	−221.538	−1.28353	−0.641763	−	0.766903i	$-0.721797\pi$
$31$	−221.538	−1.28353	−0.641763	+	0.766903i	$0.721797\pi$
$32$	182.928	1.01054
$33$	−82.7525	−0.436526
$34$	140.770	0.710052
$35$	−235.987	−1.13969
$36$	−56.8328	−0.263115
$37$	339.976	1.51059	0.755293	−	0.655387i	$-0.227494\pi$
$37$	339.976	1.51059	0.755293	+	0.655387i	$0.227494\pi$
$38$	−183.698	−0.784204
$39$	−167.027	−0.685787
$40$	−131.102	−0.518227
$41$	−266.877	−1.01657	−0.508283	−	0.861190i	$-0.669720\pi$
$41$	−266.877	−1.01657	−0.508283	+	0.861190i	$0.669720\pi$
$42$	−130.270	−0.478599
$43$	−67.2505	−0.238503	−0.119251	−	0.992864i	$-0.538049\pi$
$43$	−67.2505	−0.238503	−0.119251	+	0.992864i	$0.538049\pi$
$44$	174.187	0.596812
$45$	63.4946	0.210338
$46$	184.795	0.592317
$47$	−262.017	−0.813174	−0.406587	−	0.913612i	$-0.633281\pi$
$47$	−262.017	−0.813174	−0.406587	+	0.913612i	$0.633281\pi$
$48$	79.1827	0.238105
$49$	775.886	2.26206
$50$	−97.6581	−0.276219
$51$	325.311	0.893189
$52$	351.578	0.937598
$53$	380.876	0.987121	0.493560	−	0.869712i	$-0.335695\pi$
$53$	380.876	0.987121	0.493560	+	0.869712i	$0.335695\pi$
$54$	35.0506	0.0883292
$55$	−194.605	−0.477101
$56$	621.596	1.48329
$57$	−424.516	−0.986466
$58$	−126.297	−0.285923
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	−133.651	−0.287571
$61$	−15.4226	−0.0323715	−0.0161857	−	0.999869i	$-0.505152\pi$
$61$	−15.4226	−0.0323715	−0.0161857	+	0.999869i	$0.505152\pi$
$62$	−287.593	−0.589103
$63$	−301.048	−0.602039
$64$	26.3175	0.0514014
$65$	−392.789	−0.749531
$66$	−107.427	−0.200353
$67$	−172.954	−0.315369	−0.157685	−	0.987490i	$-0.550403\pi$
$67$	−172.954	−0.315369	−0.157685	+	0.987490i	$0.550403\pi$
$68$	−684.753	−1.22116
$69$	427.052	0.745088
$70$	−306.350	−0.523084
$71$	−616.313	−1.03018	−0.515091	−	0.857136i	$-0.672242\pi$
$71$	−616.313	−1.03018	−0.515091	+	0.857136i	$0.672242\pi$
$72$	−167.247	−0.273753
$73$	−210.965	−0.338242	−0.169121	−	0.985595i	$-0.554093\pi$
$73$	−210.965	−0.338242	−0.169121	+	0.985595i	$0.554093\pi$
$74$	441.346	0.693317
$75$	−225.683	−0.347461
$76$	893.572	1.34868
$77$	922.684	1.36558
$78$	−216.829	−0.314757
$79$	543.689	0.774302	0.387151	−	0.922016i	$-0.373459\pi$
$79$	543.689	0.774302	0.387151	+	0.922016i	$0.373459\pi$
$80$	186.210	0.260237
$81$	81.0000	0.111111
$82$	−346.452	−0.466575
$83$	350.751	0.463855	0.231927	−	0.972733i	$-0.425497\pi$
$83$	350.751	0.463855	0.231927	+	0.972733i	$0.425497\pi$
$84$	633.681	0.823099
$85$	765.019	0.976211
$86$	−87.3025	−0.109466
$87$	−291.865	−0.359669
$88$	512.596	0.620942
$89$	1447.29	1.72373	0.861867	−	0.507135i	$-0.169295\pi$
$89$	1447.29	1.72373	0.861867	+	0.507135i	$0.169295\pi$
$90$	82.4267	0.0965394
$91$	1862.34	2.14534
$92$	−898.911	−1.01867
$93$	−664.613	−0.741044
$94$	−340.143	−0.373224
$95$	−998.315	−1.07816
$96$	548.784	0.583438
$97$	−625.674	−0.654924	−0.327462	−	0.944864i	$-0.606193\pi$
$97$	−625.674	−0.654924	−0.327462	+	0.944864i	$0.606193\pi$
$98$	1007.23	1.03822
$99$	−248.258	−0.252029
$100$	475.044	0.475044
$101$	−902.589	−0.889218	−0.444609	−	0.895725i	$-0.646657\pi$
$101$	−902.589	−0.889218	−0.444609	+	0.895725i	$0.646657\pi$
$102$	422.309	0.409949
$103$	700.033	0.669673	0.334836	−	0.942276i	$-0.391319\pi$
$103$	700.033	0.669673	0.334836	+	0.942276i	$0.391319\pi$
$104$	1034.62	0.975507
$105$	−707.960	−0.657998
$106$	494.442	0.453061
$107$	1645.87	1.48703	0.743513	−	0.668721i	$-0.233158\pi$
$107$	1645.87	1.48703	0.743513	+	0.668721i	$0.233158\pi$
$108$	−170.498	−0.151909
$109$	−253.521	−0.222779	−0.111390	−	0.993777i	$-0.535530\pi$
$109$	−253.521	−0.222779	−0.111390	+	0.993777i	$0.535530\pi$
$110$	−252.630	−0.218976
$111$	1019.93	0.872138
$112$	−882.880	−0.744860
$113$	−947.504	−0.788793	−0.394397	−	0.918940i	$-0.629046\pi$
$113$	−947.504	−0.788793	−0.394397	+	0.918940i	$0.629046\pi$
$114$	−551.094	−0.452760
$115$	1004.28	0.814343
$116$	614.352	0.491734
$117$	−501.081	−0.395939
$118$	−76.5920	−0.0597531
$119$	−3627.19	−2.79415
$120$	−393.306	−0.299198
$121$	−570.113	−0.428335
$122$	−20.0211	−0.0148576
$123$	−800.631	−0.586915
$124$	1398.96	1.01314
$125$	−1412.60	−1.01077
$126$	−390.811	−0.276319
$127$	−2655.70	−1.85555	−0.927775	−	0.373139i	$-0.878281\pi$
$127$	−2655.70	−1.85555	−0.927775	+	0.373139i	$0.878281\pi$
$128$	−1429.26	−0.986952
$129$	−201.752	−0.137700
$130$	−509.907	−0.344014
$131$	−1677.06	−1.11851	−0.559256	−	0.828995i	$-0.688913\pi$
$131$	−1677.06	−1.11851	−0.559256	+	0.828995i	$0.688913\pi$
$132$	522.562	0.344570
$133$	4733.32	3.08595
$134$	−224.524	−0.144746
$135$	190.484	0.121439
$136$	−2015.08	−1.27053
$137$	687.255	0.428585	0.214293	−	0.976769i	$-0.431255\pi$
$137$	687.255	0.428585	0.214293	+	0.976769i	$0.431255\pi$
$138$	554.386	0.341974
$139$	−149.695	−0.0913451	−0.0456726	−	0.998956i	$-0.514543\pi$
$139$	−149.695	−0.0913451	−0.0456726	+	0.998956i	$0.514543\pi$
$140$	1490.20	0.899605
$141$	−786.052	−0.469486
$142$	−800.078	−0.472825
$143$	1535.77	0.898092
$144$	237.548	0.137470
$145$	−686.365	−0.393100
$146$	−273.869	−0.155243
$147$	2327.66	1.30600
$148$	−2146.87	−1.19237
$149$	221.901	0.122005	0.0610027	−	0.998138i	$-0.480570\pi$
$149$	221.901	0.122005	0.0610027	+	0.998138i	$0.480570\pi$
$150$	−292.974	−0.159475
$151$	−1262.38	−0.680340	−0.340170	−	0.940364i	$-0.610485\pi$
$151$	−1262.38	−0.680340	−0.340170	+	0.940364i	$0.610485\pi$
$152$	2629.59	1.40321
$153$	975.933	0.515683
$154$	1197.80	0.626762
$155$	−1562.94	−0.809924
$156$	1054.73	0.541323
$157$	−3446.33	−1.75189	−0.875947	−	0.482407i	$-0.839763\pi$
$157$	−3446.33	−1.75189	−0.875947	+	0.482407i	$0.839763\pi$
$158$	705.801	0.355383
$159$	1142.63	0.569914
$160$	1290.55	0.637668
$161$	−4761.60	−2.33085
$162$	105.152	0.0509969
$163$	−1915.37	−0.920388	−0.460194	−	0.887818i	$-0.652220\pi$
$163$	−1915.37	−0.920388	−0.460194	+	0.887818i	$0.652220\pi$
$164$	1685.26	0.802421
$165$	−583.816	−0.275454
$166$	455.334	0.212896
$167$	−3013.60	−1.39640	−0.698201	−	0.715902i	$-0.746016\pi$
$167$	−3013.60	−1.39640	−0.698201	+	0.715902i	$0.746016\pi$
$168$	1864.79	0.856378
$169$	902.774	0.410912
$170$	993.123	0.448053
$171$	−1273.55	−0.569536
$172$	424.671	0.188261
$173$	−1913.98	−0.841138	−0.420569	−	0.907261i	$-0.638170\pi$
$173$	−1913.98	−0.841138	−0.420569	+	0.907261i	$0.638170\pi$
$174$	−378.890	−0.165078
$175$	2516.34	1.08696
$176$	−728.063	−0.311817
$177$	−177.000	−0.0751646
$178$	1878.83	0.791146
$179$	1507.53	0.629485	0.314742	−	0.949177i	$-0.398082\pi$
$179$	1507.53	0.629485	0.314742	+	0.949177i	$0.398082\pi$
$180$	−400.953	−0.166029
$181$	−1970.30	−0.809123	−0.404562	−	0.914511i	$-0.632576\pi$
$181$	−1970.30	−0.809123	−0.404562	+	0.914511i	$0.632576\pi$
$182$	2417.63	0.984650
$183$	−46.2678	−0.0186897
$184$	−2645.30	−1.05986
$185$	2398.52	0.953202
$186$	−862.779	−0.340119
$187$	−2991.15	−1.16970
$188$	1654.58	0.641874
$189$	−903.143	−0.347587
$190$	−1295.98	−0.494844
$191$	−2132.62	−0.807911	−0.403955	−	0.914779i	$-0.632365\pi$
$191$	−2132.62	−0.807911	−0.403955	+	0.914779i	$0.632365\pi$
$192$	78.9526	0.0296766
$193$	1319.47	0.492112	0.246056	−	0.969256i	$-0.420865\pi$
$193$	1319.47	0.492112	0.246056	+	0.969256i	$0.420865\pi$
$194$	−812.231	−0.300592
$195$	−1178.37	−0.432742
$196$	−4899.53	−1.78554
$197$	187.057	0.0676511	0.0338255	−	0.999428i	$-0.489231\pi$
$197$	187.057	0.0676511	0.0338255	+	0.999428i	$0.489231\pi$
$198$	−322.280	−0.115674
$199$	3324.89	1.18440	0.592200	−	0.805791i	$-0.298260\pi$
$199$	3324.89	1.18440	0.592200	+	0.805791i	$0.298260\pi$
$200$	1397.95	0.494251
$201$	−518.863	−0.182078
$202$	−1171.71	−0.408126
$203$	3254.27	1.12515
$204$	−2054.26	−0.705034
$205$	−1882.81	−0.641468
$206$	908.761	0.307361
$207$	1281.16	0.430177
$208$	−1469.51	−0.489868
$209$	3903.31	1.29185
$210$	−919.051	−0.302003
$211$	4844.60	1.58065	0.790323	−	0.612691i	$-0.209913\pi$
$211$	4844.60	1.58065	0.790323	+	0.612691i	$0.209913\pi$
$212$	−2405.14	−0.779179
$213$	−1848.94	−0.594776
$214$	2136.61	0.682504
$215$	−474.450	−0.150499
$216$	−501.740	−0.158051
$217$	7410.38	2.31820
$218$	−329.114	−0.102249
$219$	−632.896	−0.195284
$220$	1228.88	0.376597
$221$	−6037.30	−1.83761
$222$	1324.04	0.400287
$223$	−4223.02	−1.26814	−0.634068	−	0.773277i	$-0.718616\pi$
$223$	−4223.02	−1.26814	−0.634068	+	0.773277i	$0.718616\pi$
$224$	−6118.89	−1.82516
$225$	−677.048	−0.200607
$226$	−1230.02	−0.362034
$227$	6199.87	1.81277	0.906387	−	0.422448i	$-0.138829\pi$
$227$	6199.87	1.81277	0.906387	+	0.422448i	$0.138829\pi$
$228$	2680.72	0.778662
$229$	−2502.28	−0.722075	−0.361037	−	0.932551i	$-0.617577\pi$
$229$	−2502.28	−0.722075	−0.361037	+	0.932551i	$0.617577\pi$
$230$	1303.72	0.373761
$231$	2768.05	0.788417
$232$	1807.91	0.511615
$233$	−5317.19	−1.49503	−0.747513	−	0.664247i	$-0.768752\pi$
$233$	−5317.19	−1.49503	−0.747513	+	0.664247i	$0.768752\pi$
$234$	−650.487	−0.181725
$235$	−1848.52	−0.513125
$236$	372.571	0.102764
$237$	1631.07	0.447043
$238$	−4708.71	−1.28244
$239$	6431.88	1.74077	0.870385	−	0.492372i	$-0.163870\pi$
$239$	6431.88	1.74077	0.870385	+	0.492372i	$0.163870\pi$
$240$	558.631	0.150248
$241$	2651.24	0.708638	0.354319	−	0.935125i	$-0.384713\pi$
$241$	2651.24	0.708638	0.354319	+	0.935125i	$0.384713\pi$
$242$	−740.103	−0.196594
$243$	243.000	0.0641500
$244$	97.3899	0.0255522
$245$	5473.84	1.42739
$246$	−1039.35	−0.269377
$247$	7878.40	2.02952
$248$	4116.83	1.05411
$249$	1052.25	0.267807
$250$	−1833.79	−0.463916
$251$	3189.73	0.802126	0.401063	−	0.916050i	$-0.368641\pi$
$251$	3189.73	0.802126	0.401063	+	0.916050i	$0.368641\pi$
$252$	1901.04	0.475216
$253$	−3926.63	−0.975751
$254$	−3447.54	−0.851646
$255$	2295.06	0.563616
$256$	−2065.96	−0.504385
$257$	6997.92	1.69851	0.849257	−	0.527980i	$-0.177050\pi$
$257$	6997.92	1.69851	0.849257	+	0.527980i	$0.177050\pi$
$258$	−261.908	−0.0632002
$259$	−11372.1	−2.72830
$260$	2480.37	0.591638
$261$	−875.594	−0.207655
$262$	−2177.10	−0.513366
$263$	−4833.10	−1.13316	−0.566582	−	0.824006i	$-0.691734\pi$
$263$	−4833.10	−1.13316	−0.566582	+	0.824006i	$0.691734\pi$
$264$	1537.79	0.358501
$265$	2687.07	0.622888
$266$	6144.65	1.41636
$267$	4341.87	0.995198
$268$	1092.16	0.248935
$269$	4142.13	0.938849	0.469424	−	0.882973i	$-0.344461\pi$
$269$	4142.13	0.938849	0.469424	+	0.882973i	$0.344461\pi$
$270$	247.280	0.0557370
$271$	−3931.65	−0.881295	−0.440648	−	0.897680i	$-0.645251\pi$
$271$	−3931.65	−0.881295	−0.440648	+	0.897680i	$0.645251\pi$
$272$	2862.11	0.638018
$273$	5587.01	1.23861
$274$	892.173	0.196709
$275$	2075.09	0.455028
$276$	−2696.73	−0.588131
$277$	−2635.02	−0.571564	−0.285782	−	0.958295i	$-0.592253\pi$
$277$	−2635.02	−0.571564	−0.285782	+	0.958295i	$0.592253\pi$
$278$	−194.329	−0.0419248
$279$	−1993.84	−0.427842
$280$	4385.33	0.935978
$281$	−2791.96	−0.592721	−0.296360	−	0.955076i	$-0.595773\pi$
$281$	−2791.96	−0.592721	−0.296360	+	0.955076i	$0.595773\pi$
$282$	−1020.43	−0.215481
$283$	−915.281	−0.192254	−0.0961268	−	0.995369i	$-0.530645\pi$
$283$	−915.281	−0.192254	−0.0961268	+	0.995369i	$0.530645\pi$
$284$	3891.87	0.813169
$285$	−2994.94	−0.622474
$286$	1993.68	0.412199
$287$	8926.97	1.83604
$288$	1646.35	0.336848
$289$	6845.59	1.39336
$290$	−891.017	−0.180422
$291$	−1877.02	−0.378120
$292$	1332.20	0.266989
$293$	2631.82	0.524752	0.262376	−	0.964966i	$-0.415494\pi$
$293$	2631.82	0.524752	0.262376	+	0.964966i	$0.415494\pi$
$294$	3021.69	0.599417
$295$	−416.243	−0.0821511
$296$	−6317.76	−1.24058
$297$	−744.773	−0.145509
$298$	288.065	0.0559971
$299$	−7925.47	−1.53291
$300$	1425.13	0.274267
$301$	2249.51	0.430763
$302$	−1638.79	−0.312257
$303$	−2707.77	−0.513390
$304$	−3734.93	−0.704647
$305$	−108.806	−0.0204269
$306$	1266.93	0.236684
$307$	−6778.17	−1.26010	−0.630050	−	0.776555i	$-0.716965\pi$
$307$	−6778.17	−1.26010	−0.630050	+	0.776555i	$0.716965\pi$
$308$	−5826.52	−1.07791
$309$	2100.10	0.386636
$310$	−2028.96	−0.371732
$311$	5915.03	1.07849	0.539245	−	0.842149i	$-0.318710\pi$
$311$	5915.03	1.07849	0.539245	+	0.842149i	$0.318710\pi$
$312$	3103.86	0.563209
$313$	6150.93	1.11077	0.555385	−	0.831593i	$-0.312571\pi$
$313$	6150.93	1.11077	0.555385	+	0.831593i	$0.312571\pi$
$314$	−4473.92	−0.804070
$315$	−2123.88	−0.379895
$316$	−3433.27	−0.611191
$317$	−3992.23	−0.707337	−0.353669	−	0.935371i	$-0.615066\pi$
$317$	−3992.23	−0.707337	−0.353669	+	0.935371i	$0.615066\pi$
$318$	1483.33	0.261575
$319$	2683.62	0.471015
$320$	185.669	0.0324350
$321$	4937.60	0.858535
$322$	−6181.36	−1.06979
$323$	−15344.4	−2.64330
$324$	−511.495	−0.0877050
$325$	4188.34	0.714853
$326$	−2486.47	−0.422432
$327$	−760.564	−0.128622
$328$	4959.37	0.834864
$329$	8764.42	1.46869
$330$	−757.891	−0.126426
$331$	−4224.03	−0.701431	−0.350715	−	0.936482i	$-0.614062\pi$
$331$	−4224.03	−0.701431	−0.350715	+	0.936482i	$0.614062\pi$
$332$	−2214.91	−0.366141
$333$	3059.78	0.503529
$334$	−3912.16	−0.640909
$335$	−1220.19	−0.199003
$336$	−2648.64	−0.430045
$337$	2266.93	0.366431	0.183216	−	0.983073i	$-0.441349\pi$
$337$	2266.93	0.366431	0.183216	+	0.983073i	$0.441349\pi$
$338$	1171.95	0.188597
$339$	−2842.51	−0.455410
$340$	−4830.91	−0.770567
$341$	6110.93	0.970456
$342$	−1653.28	−0.261401
$343$	−14479.9	−2.27942
$344$	1249.71	0.195872
$345$	3012.84	0.470161
$346$	−2484.66	−0.386059
$347$	−270.565	−0.0418579	−0.0209290	−	0.999781i	$-0.506662\pi$
$347$	−270.565	−0.0418579	−0.0209290	+	0.999781i	$0.506662\pi$
$348$	1843.06	0.283903
$349$	2623.43	0.402375	0.201188	−	0.979553i	$-0.435520\pi$
$349$	2623.43	0.402375	0.201188	+	0.979553i	$0.435520\pi$
$350$	3266.64	0.498883
$351$	−1503.24	−0.228596
$352$	−5045.92	−0.764058
$353$	−4779.31	−0.720615	−0.360307	−	0.932834i	$-0.617328\pi$
$353$	−4779.31	−0.720615	−0.360307	+	0.932834i	$0.617328\pi$
$354$	−229.776	−0.0344984
$355$	−4348.06	−0.650060
$356$	−9139.28	−1.36062
$357$	−10881.6	−1.61320
$358$	1957.02	0.288916
$359$	−5676.39	−0.834509	−0.417254	−	0.908790i	$-0.637008\pi$
$359$	−5676.39	−0.834509	−0.417254	+	0.908790i	$0.637008\pi$
$360$	−1179.92	−0.172742
$361$	13164.8	1.91934
$362$	−2557.78	−0.371365
$363$	−1710.34	−0.247299
$364$	−11760.2	−1.69341
$365$	−1488.35	−0.213435
$366$	−60.0634	−0.00857804
$367$	4096.80	0.582701	0.291350	−	0.956616i	$-0.405895\pi$
$367$	4096.80	0.582701	0.291350	+	0.956616i	$0.405895\pi$
$368$	3757.24	0.532227
$369$	−2401.89	−0.338855
$370$	3113.68	0.437493
$371$	−12740.2	−1.78286
$372$	4196.87	0.584939
$373$	11745.0	1.63038	0.815190	−	0.579194i	$-0.196633\pi$
$373$	11745.0	1.63038	0.815190	+	0.579194i	$0.196633\pi$
$374$	−3883.01	−0.536860
$375$	−4237.79	−0.583570
$376$	4869.06	0.667826
$377$	5416.58	0.739969
$378$	−1172.43	−0.159533
$379$	−973.770	−0.131977	−0.0659884	−	0.997820i	$-0.521020\pi$
$379$	−973.770	−0.131977	−0.0659884	+	0.997820i	$0.521020\pi$
$380$	6304.12	0.851038
$381$	−7967.09	−1.07130
$382$	−2768.50	−0.370808
$383$	−3020.77	−0.403013	−0.201506	−	0.979487i	$-0.564584\pi$
$383$	−3020.77	−0.403013	−0.201506	+	0.979487i	$0.564584\pi$
$384$	−4287.78	−0.569817
$385$	6509.50	0.861700
$386$	1712.90	0.225865
$387$	−605.255	−0.0795008
$388$	3950.98	0.516961
$389$	−327.769	−0.0427213	−0.0213606	−	0.999772i	$-0.506800\pi$
$389$	−327.769	−0.0427213	−0.0213606	+	0.999772i	$0.506800\pi$
$390$	−1529.72	−0.198616
$391$	15436.1	1.99651
$392$	−14418.3	−1.85774
$393$	−5031.17	−0.645774
$394$	242.832	0.0310499
$395$	3835.71	0.488596
$396$	1567.69	0.198937
$397$	5076.62	0.641785	0.320892	−	0.947116i	$-0.396017\pi$
$397$	5076.62	0.641785	0.320892	+	0.947116i	$0.396017\pi$
$398$	4316.27	0.543606
$399$	14200.0	1.78167
$400$	−1985.57	−0.248197
$401$	6277.21	0.781718	0.390859	−	0.920450i	$-0.372178\pi$
$401$	6277.21	0.781718	0.390859	+	0.920450i	$0.372178\pi$
$402$	−673.572	−0.0835689
$403$	12334.2	1.52460
$404$	5699.63	0.701899
$405$	571.452	0.0701127
$406$	4224.59	0.516411
$407$	−9377.96	−1.14213
$408$	−6045.25	−0.733540
$409$	−6562.26	−0.793357	−0.396679	−	0.917958i	$-0.629837\pi$
$409$	−6562.26	−0.793357	−0.396679	+	0.917958i	$0.629837\pi$
$410$	−2444.20	−0.294416
$411$	2061.77	0.247444
$412$	−4420.54	−0.528603
$413$	1973.54	0.235136
$414$	1663.16	0.197439
$415$	2474.54	0.292699
$416$	−10184.6	−1.20034
$417$	−449.085	−0.0527381
$418$	5067.16	0.592925
$419$	10923.8	1.27365	0.636827	−	0.771006i	$-0.280246\pi$
$419$	10923.8	1.27365	0.636827	+	0.771006i	$0.280246\pi$
$420$	4470.59	0.519387
$421$	1049.00	0.121437	0.0607187	−	0.998155i	$-0.480661\pi$
$421$	1049.00	0.121437	0.0607187	+	0.998155i	$0.480661\pi$
$422$	6289.11	0.725472
$423$	−2358.16	−0.271058
$424$	−7077.82	−0.810682
$425$	−8157.45	−0.931046
$426$	−2400.24	−0.272985
$427$	515.882	0.0584667
$428$	−10393.2	−1.17378
$429$	4607.30	0.518514
$430$	−615.916	−0.0690746
$431$	−8997.43	−1.00555	−0.502774	−	0.864418i	$-0.667687\pi$
$431$	−8997.43	−1.00555	−0.502774	+	0.864418i	$0.667687\pi$
$432$	712.644	0.0793683
$433$	−7942.35	−0.881490	−0.440745	−	0.897632i	$-0.645286\pi$
$433$	−7942.35	−0.881490	−0.440745	+	0.897632i	$0.645286\pi$
$434$	9619.92	1.06399
$435$	−2059.09	−0.226956
$436$	1600.93	0.175850
$437$	−20143.4	−2.20501
$438$	−821.606	−0.0896298
$439$	−3365.40	−0.365881	−0.182941	−	0.983124i	$-0.558562\pi$
$439$	−3365.40	−0.365881	−0.182941	+	0.983124i	$0.558562\pi$
$440$	3616.34	0.391824
$441$	6982.97	0.754019
$442$	−7837.43	−0.843413
$443$	−4056.81	−0.435090	−0.217545	−	0.976050i	$-0.569805\pi$
$443$	−4056.81	−0.435090	−0.217545	+	0.976050i	$0.569805\pi$
$444$	−6440.60	−0.688417
$445$	10210.6	1.08770
$446$	−5482.19	−0.582039
$447$	665.702	0.0704399
$448$	−880.315	−0.0928370
$449$	10892.7	1.14490	0.572449	−	0.819940i	$-0.305993\pi$
$449$	10892.7	1.14490	0.572449	+	0.819940i	$0.305993\pi$
$450$	−878.923	−0.0920729
$451$	7361.58	0.768611
$452$	5983.26	0.622630
$453$	−3787.15	−0.392794
$454$	8048.48	0.832013
$455$	13138.7	1.35374
$456$	7888.77	0.810144
$457$	9685.62	0.991409	0.495705	−	0.868491i	$-0.334910\pi$
$457$	9685.62	0.991409	0.495705	+	0.868491i	$0.334910\pi$
$458$	−3248.38	−0.331412
$459$	2927.80	0.297730
$460$	−6341.78	−0.642798
$461$	145.044	0.0146538	0.00732689	−	0.999973i	$-0.497668\pi$
$461$	145.044	0.0146538	0.00732689	+	0.999973i	$0.497668\pi$
$462$	3593.40	0.361861
$463$	−12516.4	−1.25634	−0.628169	−	0.778077i	$-0.716195\pi$
$463$	−12516.4	−1.25634	−0.628169	+	0.778077i	$0.716195\pi$
$464$	−2567.85	−0.256917
$465$	−4688.81	−0.467610
$466$	−6902.62	−0.686175
$467$	2625.49	0.260156	0.130078	−	0.991504i	$-0.458477\pi$
$467$	2625.49	0.260156	0.130078	+	0.991504i	$0.458477\pi$
$468$	3164.20	0.312533
$469$	5785.28	0.569593
$470$	−2399.69	−0.235510
$471$	−10339.0	−1.01146
$472$	1096.40	0.106919
$473$	1855.05	0.180328
$474$	2117.40	0.205180
$475$	10645.1	1.02828
$476$	22904.8	2.20555
$477$	3427.89	0.329040
$478$	8349.67	0.798964
$479$	−16253.8	−1.55043	−0.775216	−	0.631697i	$-0.782359\pi$
$479$	−16253.8	−1.55043	−0.775216	+	0.631697i	$0.782359\pi$
$480$	3871.65	0.368158
$481$	−18928.4	−1.79430
$482$	3441.76	0.325245
$483$	−14284.8	−1.34572
$484$	3600.13	0.338104
$485$	−4414.11	−0.413266
$486$	315.455	0.0294431
$487$	−1328.98	−0.123659	−0.0618296	−	0.998087i	$-0.519694\pi$
$487$	−1328.98	−0.123659	−0.0618296	+	0.998087i	$0.519694\pi$
$488$	286.597	0.0265854
$489$	−5746.11	−0.531386
$490$	7105.97	0.655133
$491$	14197.4	1.30493	0.652463	−	0.757820i	$-0.273736\pi$
$491$	14197.4	1.30493	0.652463	+	0.757820i	$0.273736\pi$
$492$	5055.79	0.463278
$493$	−10549.6	−0.963757
$494$	10227.5	0.931491
$495$	−1751.45	−0.159034
$496$	−5847.31	−0.529339
$497$	20615.5	1.86063
$498$	1366.00	0.122916
$499$	−7543.06	−0.676701	−0.338350	−	0.941020i	$-0.609869\pi$
$499$	−7543.06	−0.676701	−0.338350	+	0.941020i	$0.609869\pi$
$500$	8920.21	0.797848
$501$	−9040.79	−0.806213
$502$	4140.80	0.368154
$503$	−8035.31	−0.712280	−0.356140	−	0.934433i	$-0.615907\pi$
$503$	−8035.31	−0.712280	−0.356140	+	0.934433i	$0.615907\pi$
$504$	5594.36	0.494430
$505$	−6367.73	−0.561109
$506$	−5097.43	−0.447842
$507$	2708.32	0.237240
$508$	16770.1	1.46467
$509$	−2243.76	−0.195389	−0.0976946	−	0.995216i	$-0.531147\pi$
$509$	−2243.76	−0.195389	−0.0976946	+	0.995216i	$0.531147\pi$
$510$	2979.37	0.258684
$511$	7056.74	0.610904
$512$	8752.11	0.755453
$513$	−3820.65	−0.328822
$514$	9084.48	0.779570
$515$	4938.70	0.422573
$516$	1274.01	0.108692
$517$	7227.53	0.614829
$518$	−14762.9	−1.25221
$519$	−5741.93	−0.485631
$520$	7299.19	0.615559
$521$	5397.02	0.453834	0.226917	−	0.973914i	$-0.427135\pi$
$521$	5397.02	0.453834	0.226917	+	0.973914i	$0.427135\pi$
$522$	−1136.67	−0.0953078
$523$	−229.273	−0.0191690	−0.00958451	−	0.999954i	$-0.503051\pi$
$523$	−229.273	−0.0191690	−0.00958451	+	0.999954i	$0.503051\pi$
$524$	10590.2	0.882892
$525$	7549.03	0.627555
$526$	−6274.19	−0.520090
$527$	−24022.9	−1.98568
$528$	−2184.19	−0.180028
$529$	8096.74	0.665467
$530$	3488.27	0.285888
$531$	−531.000	−0.0433963
$532$	−29889.8	−2.43588
$533$	14858.5	1.20750
$534$	5636.48	0.456768
$535$	11611.5	0.938336
$536$	3214.01	0.259000
$537$	4522.58	0.363433
$538$	5377.19	0.430905
$539$	−21402.2	−1.71031
$540$	−1202.86	−0.0958571
$541$	21118.4	1.67829	0.839143	−	0.543911i	$-0.183057\pi$
$541$	21118.4	1.67829	0.839143	+	0.543911i	$0.183057\pi$
$542$	−5103.95	−0.404490
$543$	−5910.90	−0.467147
$544$	19836.2	1.56336
$545$	−1788.58	−0.140577
$546$	7252.88	0.568488
$547$	700.599	0.0547632	0.0273816	−	0.999625i	$-0.491283\pi$
$547$	700.599	0.0547632	0.0273816	+	0.999625i	$0.491283\pi$
$548$	−4339.85	−0.338302
$549$	−138.803	−0.0107905
$550$	2693.82	0.208845
$551$	13766.8	1.06440
$552$	−7935.90	−0.611910
$553$	−18186.3	−1.39848
$554$	−3420.70	−0.262332
$555$	7195.55	0.550332
$556$	945.288	0.0721028
$557$	13.7365	0.00104495	0.000522474	−	1.00000i	$-0.499834\pi$
$557$	13.7365	0.00104495	0.000522474		1.00000i	$0.499834\pi$
$558$	−2588.34	−0.196368
$559$	3744.21	0.283298
$560$	−6228.68	−0.470018
$561$	−8973.44	−0.675328
$562$	−3624.44	−0.272042
$563$	−20483.4	−1.53334	−0.766672	−	0.642039i	$-0.778089\pi$
$563$	−20483.4	−1.53334	−0.766672	+	0.642039i	$0.778089\pi$
$564$	4963.73	0.370586
$565$	−6684.60	−0.497740
$566$	−1188.19	−0.0882390
$567$	−2709.43	−0.200680
$568$	11452.9	0.846046
$569$	−16699.3	−1.23035	−0.615175	−	0.788390i	$-0.710915\pi$
$569$	−16699.3	−1.23035	−0.615175	+	0.788390i	$0.710915\pi$
$570$	−3887.94	−0.285698
$571$	−2261.27	−0.165729	−0.0828643	−	0.996561i	$-0.526407\pi$
$571$	−2261.27	−0.165729	−0.0828643	+	0.996561i	$0.526407\pi$
$572$	−9697.99	−0.708904
$573$	−6397.86	−0.466448
$574$	11588.7	0.842689
$575$	−10708.7	−0.776667
$576$	236.858	0.0171338
$577$	−12365.4	−0.892160	−0.446080	−	0.894993i	$-0.647180\pi$
$577$	−12365.4	−0.892160	−0.446080	+	0.894993i	$0.647180\pi$
$578$	8886.73	0.639514
$579$	3958.41	0.284121
$580$	4334.23	0.310291
$581$	−11732.5	−0.837776
$582$	−2436.69	−0.173547
$583$	−10506.2	−0.746348
$584$	3920.36	0.277784
$585$	−3535.10	−0.249844
$586$	3416.54	0.240847
$587$	17739.0	1.24730	0.623651	−	0.781703i	$-0.285649\pi$
$587$	17739.0	1.24730	0.623651	+	0.781703i	$0.285649\pi$
$588$	−14698.6	−1.03088
$589$	31348.8	2.19304
$590$	−540.353	−0.0377051
$591$	561.171	0.0390584
$592$	8973.40	0.622981
$593$	24070.9	1.66690	0.833451	−	0.552594i	$-0.186362\pi$
$593$	24070.9	1.66690	0.833451	+	0.552594i	$0.186362\pi$
$594$	−966.841	−0.0667844
$595$	−25589.7	−1.76315
$596$	−1401.25	−0.0963044
$597$	9974.68	0.683813
$598$	−10288.6	−0.703565
$599$	28779.3	1.96309	0.981544	−	0.191238i	$-0.0612501\pi$
$599$	28779.3	1.96309	0.981544	+	0.191238i	$0.0612501\pi$
$600$	4193.85	0.285356
$601$	−11612.0	−0.788128	−0.394064	−	0.919083i	$-0.628931\pi$
$601$	−11612.0	−0.788128	−0.394064	+	0.919083i	$0.628931\pi$
$602$	2920.25	0.197708
$603$	−1556.59	−0.105123
$604$	7971.65	0.537023
$605$	−4022.13	−0.270285
$606$	−3515.14	−0.235632
$607$	−821.129	−0.0549071	−0.0274535	−	0.999623i	$-0.508740\pi$
$607$	−821.129	−0.0549071	−0.0274535	+	0.999623i	$0.508740\pi$
$608$	−25885.3	−1.72662
$609$	9762.80	0.649604
$610$	−141.248	−0.00937536
$611$	14588.0	0.965903
$612$	−6162.78	−0.407052
$613$	7705.99	0.507736	0.253868	−	0.967239i	$-0.418297\pi$
$613$	7705.99	0.507736	0.253868	+	0.967239i	$0.418297\pi$
$614$	−8799.21	−0.578350
$615$	−5648.42	−0.370352
$616$	−17146.2	−1.12149
$617$	−447.613	−0.0292062	−0.0146031	−	0.999893i	$-0.504648\pi$
$617$	−447.613	−0.0292062	−0.0146031	+	0.999893i	$0.504648\pi$
$618$	2726.28	0.177455
$619$	−5073.90	−0.329462	−0.164731	−	0.986339i	$-0.552676\pi$
$619$	−5073.90	−0.329462	−0.164731	+	0.986339i	$0.552676\pi$
$620$	9869.58	0.639309
$621$	3843.47	0.248363
$622$	7678.70	0.494997
$623$	−48411.4	−3.11326
$624$	−4408.54	−0.282825
$625$	−562.371	−0.0359917
$626$	7984.94	0.509812
$627$	11709.9	0.745853
$628$	21762.8	1.38285
$629$	36866.0	2.33695
$630$	−2757.15	−0.174361
$631$	20851.5	1.31551	0.657753	−	0.753234i	$-0.271507\pi$
$631$	20851.5	1.31551	0.657753	+	0.753234i	$0.271507\pi$
$632$	−10103.4	−0.635902
$633$	14533.8	0.912586
$634$	−5182.59	−0.324648
$635$	−18735.8	−1.17088
$636$	−7215.43	−0.449859
$637$	−43197.9	−2.68691
$638$	3483.79	0.216183
$639$	−5546.82	−0.343394
$640$	−10083.4	−0.622781
$641$	−5938.37	−0.365915	−0.182957	−	0.983121i	$-0.558567\pi$
$641$	−5938.37	−0.365915	−0.182957	+	0.983121i	$0.558567\pi$
$642$	6409.84	0.394044
$643$	943.102	0.0578418	0.0289209	−	0.999582i	$-0.490793\pi$
$643$	943.102	0.0578418	0.0289209	+	0.999582i	$0.490793\pi$
$644$	30068.3	1.83984
$645$	−1423.35	−0.0868904
$646$	−19919.7	−1.21320
$647$	14236.8	0.865082	0.432541	−	0.901614i	$-0.357617\pi$
$647$	14236.8	0.865082	0.432541	+	0.901614i	$0.357617\pi$
$648$	−1505.22	−0.0912510
$649$	1627.47	0.0984340
$650$	5437.17	0.328098
$651$	22231.1	1.33841
$652$	12095.1	0.726503
$653$	8903.59	0.533575	0.266788	−	0.963755i	$-0.414038\pi$
$653$	8903.59	0.533575	0.266788	+	0.963755i	$0.414038\pi$
$654$	−987.341	−0.0590338
$655$	−11831.6	−0.705798
$656$	−7044.01	−0.419242
$657$	−1898.69	−0.112747
$658$	11377.7	0.674086
$659$	−4548.25	−0.268854	−0.134427	−	0.990924i	$-0.542919\pi$
$659$	−4548.25	−0.268854	−0.134427	+	0.990924i	$0.542919\pi$
$660$	3686.65	0.217429
$661$	−3357.62	−0.197574	−0.0987870	−	0.995109i	$-0.531496\pi$
$661$	−3357.62	−0.197574	−0.0987870	+	0.995109i	$0.531496\pi$
$662$	−5483.50	−0.321937
$663$	−18111.9	−1.06095
$664$	−6518.00	−0.380945
$665$	33393.4	1.94728
$666$	3972.12	0.231106
$667$	−13849.1	−0.803955
$668$	19030.1	1.10224
$669$	−12669.1	−0.732159
$670$	−1584.01	−0.0913366
$671$	425.419	0.0244756
$672$	−18356.7	−1.05376
$673$	−14707.2	−0.842382	−0.421191	−	0.906972i	$-0.638388\pi$
$673$	−14707.2	−0.842382	−0.421191	+	0.906972i	$0.638388\pi$
$674$	2942.85	0.168182
$675$	−2031.14	−0.115820
$676$	−5700.80	−0.324351
$677$	−7070.52	−0.401392	−0.200696	−	0.979654i	$-0.564320\pi$
$677$	−7070.52	−0.401392	−0.200696	+	0.979654i	$0.564320\pi$
$678$	−3690.06	−0.209020
$679$	20928.6	1.18287
$680$	−14216.3	−0.801722
$681$	18599.6	1.04661
$682$	7933.02	0.445412
$683$	16587.9	0.929308	0.464654	−	0.885492i	$-0.346179\pi$
$683$	16587.9	0.929308	0.464654	+	0.885492i	$0.346179\pi$
$684$	8042.15	0.449560
$685$	4848.56	0.270444
$686$	−18797.4	−1.04619
$687$	−7506.83	−0.416890
$688$	−1775.03	−0.0983608
$689$	−21205.5	−1.17252
$690$	3911.17	0.215791
$691$	−32722.2	−1.80147	−0.900733	−	0.434373i	$-0.856970\pi$
$691$	−32722.2	−1.80147	−0.900733	+	0.434373i	$0.856970\pi$
$692$	12086.3	0.663948
$693$	8304.15	0.455193
$694$	−351.239	−0.0192116
$695$	−1056.09	−0.0576401
$696$	5423.72	0.295381
$697$	−28939.4	−1.57268
$698$	3405.65	0.184679
$699$	−15951.6	−0.863154
$700$	−15890.1	−0.857984
$701$	−24968.5	−1.34529	−0.672645	−	0.739965i	$-0.734842\pi$
$701$	−24968.5	−1.34529	−0.672645	+	0.739965i	$0.734842\pi$
$702$	−1951.46	−0.104919
$703$	−48108.4	−2.58100
$704$	−725.947	−0.0388639
$705$	−5545.57	−0.296253
$706$	−6204.35	−0.330742
$707$	30191.4	1.60603
$708$	1117.71	0.0593308
$709$	33965.7	1.79916	0.899582	−	0.436752i	$-0.143871\pi$
$709$	33965.7	1.79916	0.899582	+	0.436752i	$0.143871\pi$
$710$	−5644.52	−0.298359
$711$	4893.20	0.258101
$712$	−26894.9	−1.41563
$713$	−31536.0	−1.65643
$714$	−14126.1	−0.740416
$715$	10834.8	0.566709
$716$	−9519.66	−0.496880
$717$	19295.6	1.00503
$718$	−7368.92	−0.383016
$719$	14477.7	0.750941	0.375470	−	0.926834i	$-0.377481\pi$
$719$	14477.7	0.750941	0.375470	+	0.926834i	$0.377481\pi$
$720$	1675.89	0.0867456
$721$	−23415.9	−1.20951
$722$	17090.1	0.880925
$723$	7953.73	0.409132
$724$	12442.0	0.638677
$725$	7318.76	0.374913
$726$	−2220.31	−0.113503
$727$	18345.6	0.935900	0.467950	−	0.883755i	$-0.344993\pi$
$727$	18345.6	0.935900	0.467950	+	0.883755i	$0.344993\pi$
$728$	−34607.7	−1.76188
$729$	729.000	0.0370370
$730$	−1932.13	−0.0979609
$731$	−7292.45	−0.368975
$732$	292.170	0.0147526
$733$	−24879.5	−1.25368	−0.626839	−	0.779149i	$-0.715652\pi$
$733$	−24879.5	−1.25368	−0.626839	+	0.779149i	$0.715652\pi$
$734$	5318.34	0.267443
$735$	16421.5	0.824105
$736$	26039.9	1.30414
$737$	4770.80	0.238446
$738$	−3118.06	−0.155525
$739$	522.214	0.0259945	0.0129973	−	0.999916i	$-0.495863\pi$
$739$	522.214	0.0259945	0.0129973	+	0.999916i	$0.495863\pi$
$740$	−15146.1	−0.752405
$741$	23635.2	1.17174
$742$	−16539.0	−0.818281
$743$	−1698.03	−0.0838422	−0.0419211	−	0.999121i	$-0.513348\pi$
$743$	−1698.03	−0.0838422	−0.0419211	+	0.999121i	$0.513348\pi$
$744$	12350.5	0.608589
$745$	1565.50	0.0769872
$746$	15247.0	0.748299
$747$	3156.76	0.154618
$748$	18888.4	0.923298
$749$	−55053.8	−2.68574
$750$	−5501.37	−0.267842
$751$	−21329.8	−1.03640	−0.518199	−	0.855260i	$-0.673397\pi$
$751$	−21329.8	−1.03640	−0.518199	+	0.855260i	$0.673397\pi$
$752$	−6915.75	−0.335361
$753$	9569.18	0.463108
$754$	7031.64	0.339625
$755$	−8906.06	−0.429304
$756$	5703.13	0.274366
$757$	−2570.95	−0.123438	−0.0617192	−	0.998094i	$-0.519658\pi$
$757$	−2570.95	−0.123438	−0.0617192	+	0.998094i	$0.519658\pi$
$758$	−1264.12	−0.0605736
$759$	−11779.9	−0.563350
$760$	18551.7	0.885447
$761$	−16345.4	−0.778606	−0.389303	−	0.921110i	$-0.627284\pi$
$761$	−16345.4	−0.778606	−0.389303	+	0.921110i	$0.627284\pi$
$762$	−10342.6	−0.491698
$763$	8480.23	0.402365
$764$	13467.0	0.637720
$765$	6885.17	0.325404
$766$	−3921.47	−0.184972
$767$	3284.86	0.154641
$768$	−6197.88	−0.291207
$769$	35378.7	1.65902	0.829512	−	0.558489i	$-0.188619\pi$
$769$	35378.7	1.65902	0.829512	+	0.558489i	$0.188619\pi$
$770$	8450.42	0.395496
$771$	20993.7	0.980637
$772$	−8332.13	−0.388446
$773$	−8622.56	−0.401206	−0.200603	−	0.979673i	$-0.564290\pi$
$773$	−8622.56	−0.401206	−0.200603	+	0.979673i	$0.564290\pi$
$774$	−785.723	−0.0364887
$775$	16665.7	0.772452
$776$	11626.9	0.537862
$777$	−34116.3	−1.57518
$778$	−425.500	−0.0196079
$779$	37764.6	1.73691
$780$	7441.11	0.341582
$781$	17000.5	0.778906
$782$	20038.7	0.916344
$783$	−2626.78	−0.119890
$784$	20478.9	0.932895
$785$	−24313.7	−1.10547
$786$	−6531.31	−0.296392
$787$	18664.1	0.845366	0.422683	−	0.906278i	$-0.361088\pi$
$787$	18664.1	0.845366	0.422683	+	0.906278i	$0.361088\pi$
$788$	−1181.22	−0.0534000
$789$	−14499.3	−0.654232
$790$	4979.39	0.224252
$791$	31693.8	1.42465
$792$	4613.36	0.206981
$793$	858.662	0.0384514
$794$	6590.32	0.294561
$795$	8061.20	0.359624
$796$	−20995.9	−0.934900
$797$	−41335.9	−1.83713	−0.918564	−	0.395272i	$-0.870650\pi$
$797$	−41335.9	−1.83713	−0.918564	+	0.395272i	$0.870650\pi$
$798$	18433.9	0.817738
$799$	−28412.4	−1.25802
$800$	−13761.2	−0.608166
$801$	13025.6	0.574578
$802$	8148.88	0.358787
$803$	5819.31	0.255740
$804$	3276.49	0.143723
$805$	−33592.9	−1.47080
$806$	16011.9	0.699747
$807$	12426.4	0.542045
$808$	16772.8	0.730278
$809$	19310.4	0.839206	0.419603	−	0.907708i	$-0.362169\pi$
$809$	19310.4	0.839206	0.419603	+	0.907708i	$0.362169\pi$
$810$	741.841	0.0321798
$811$	6287.79	0.272249	0.136125	−	0.990692i	$-0.456535\pi$
$811$	6287.79	0.272249	0.136125	+	0.990692i	$0.456535\pi$
$812$	−20549.9	−0.888129
$813$	−11795.0	−0.508816
$814$	−12174.2	−0.524207
$815$	−13512.8	−0.580778
$816$	8586.33	0.368360
$817$	9516.31	0.407508
$818$	−8518.93	−0.364129
$819$	16761.0	0.715113
$820$	11889.5	0.506339
$821$	−10235.6	−0.435109	−0.217554	−	0.976048i	$-0.569808\pi$
$821$	−10235.6	−0.435109	−0.217554	+	0.976048i	$0.569808\pi$
$822$	2676.52	0.113570
$823$	−7825.03	−0.331426	−0.165713	−	0.986174i	$-0.552992\pi$
$823$	−7825.03	−0.331426	−0.165713	+	0.986174i	$0.552992\pi$
$824$	−13008.7	−0.549975
$825$	6225.27	0.262710
$826$	2561.98	0.107921
$827$	15916.1	0.669233	0.334617	−	0.942354i	$-0.391393\pi$
$827$	15916.1	0.669233	0.334617	+	0.942354i	$0.391393\pi$
$828$	−8090.20	−0.339558
$829$	−26623.7	−1.11541	−0.557707	−	0.830038i	$-0.688319\pi$
$829$	−26623.7	−1.11541	−0.557707	+	0.830038i	$0.688319\pi$
$830$	3212.36	0.134341
$831$	−7905.07	−0.329993
$832$	−1465.24	−0.0610555
$833$	84134.7	3.49951
$834$	−582.988	−0.0242053
$835$	−21260.8	−0.881150
$836$	−24648.5	−1.01972
$837$	−5981.51	−0.247015
$838$	14180.9	0.584572
$839$	−19230.4	−0.791307	−0.395653	−	0.918400i	$-0.629482\pi$
$839$	−19230.4	−0.791307	−0.395653	+	0.918400i	$0.629482\pi$
$840$	13156.0	0.540387
$841$	−14924.0	−0.611915
$842$	1361.78	0.0557363
$843$	−8375.89	−0.342207
$844$	−30592.5	−1.24767
$845$	6369.04	0.259292
$846$	−3061.29	−0.124408
$847$	19070.1	0.773622
$848$	10052.9	0.407098
$849$	−2745.84	−0.110998
$850$	−10589.7	−0.427324
$851$	48395.8	1.94946
$852$	11675.6	0.469483
$853$	21069.1	0.845710	0.422855	−	0.906197i	$-0.361028\pi$
$853$	21069.1	0.845710	0.422855	+	0.906197i	$0.361028\pi$
$854$	669.701	0.0268346
$855$	−8984.83	−0.359386
$856$	−30585.1	−1.22123
$857$	−8066.10	−0.321508	−0.160754	−	0.986994i	$-0.551393\pi$
$857$	−8066.10	−0.321508	−0.160754	+	0.986994i	$0.551393\pi$
$858$	5981.05	0.237983
$859$	−17890.8	−0.710625	−0.355313	−	0.934748i	$-0.615626\pi$
$859$	−17890.8	−0.710625	−0.355313	+	0.934748i	$0.615626\pi$
$860$	2996.03	0.118795
$861$	26780.9	1.06004
$862$	−11680.2	−0.461518
$863$	−17900.4	−0.706069	−0.353035	−	0.935610i	$-0.614850\pi$
$863$	−17900.4	−0.706069	−0.353035	+	0.935610i	$0.614850\pi$
$864$	4939.05	0.194479
$865$	−13503.0	−0.530770
$866$	−10310.5	−0.404579
$867$	20536.8	0.804458
$868$	−46794.7	−1.82986
$869$	−14997.2	−0.585438
$870$	−2673.05	−0.104167
$871$	9629.34	0.374601
$872$	4711.18	0.182960
$873$	−5631.07	−0.218308
$874$	−26149.5	−1.01204
$875$	47251.0	1.82557
$876$	3996.59	0.154146
$877$	45119.0	1.73724	0.868620	−	0.495478i	$-0.165007\pi$
$877$	45119.0	1.73724	0.868620	+	0.495478i	$0.165007\pi$
$878$	−4368.86	−0.167929
$879$	7895.45	0.302966
$880$	−5136.45	−0.196761
$881$	−43192.5	−1.65175	−0.825876	−	0.563852i	$-0.809319\pi$
$881$	−43192.5	−1.65175	−0.825876	+	0.563852i	$0.809319\pi$
$882$	9065.08	0.346074
$883$	36353.7	1.38550	0.692751	−	0.721177i	$-0.256399\pi$
$883$	36353.7	1.38550	0.692751	+	0.721177i	$0.256399\pi$
$884$	38124.1	1.45051
$885$	−1248.73	−0.0474300
$886$	−5266.43	−0.199694
$887$	−4579.23	−0.173343	−0.0866717	−	0.996237i	$-0.527623\pi$
$887$	−4579.23	−0.173343	−0.0866717	+	0.996237i	$0.527623\pi$
$888$	−18953.3	−0.716251
$889$	88832.4	3.35134
$890$	13255.0	0.499224
$891$	−2234.32	−0.0840095
$892$	26667.3	1.00100
$893$	37076.9	1.38940
$894$	864.194	0.0323299
$895$	10635.5	0.397214
$896$	47808.4	1.78255
$897$	−23776.4	−0.885029
$898$	14140.6	0.525476
$899$	21553.0	0.799592
$900$	4275.39	0.158348
$901$	41301.1	1.52712
$902$	9556.58	0.352771
$903$	6748.54	0.248701
$904$	17607.4	0.647804
$905$	−13900.4	−0.510569
$906$	−4916.36	−0.180282
$907$	−47448.3	−1.73704	−0.868520	−	0.495654i	$-0.834929\pi$
$907$	−47448.3	−1.73704	−0.868520	+	0.495654i	$0.834929\pi$
$908$	−39150.7	−1.43090
$909$	−8123.30	−0.296406
$910$	17056.3	0.621329
$911$	−25809.9	−0.938660	−0.469330	−	0.883023i	$-0.655505\pi$
$911$	−25809.9	−0.938660	−0.469330	+	0.883023i	$0.655505\pi$
$912$	−11204.8	−0.406828
$913$	−9675.18	−0.350714
$914$	12573.6	0.455029
$915$	−326.417	−0.0117935
$916$	15801.3	0.569966
$917$	56097.2	2.02016
$918$	3800.78	0.136650
$919$	27429.5	0.984565	0.492283	−	0.870435i	$-0.336163\pi$
$919$	27429.5	0.984565	0.492283	+	0.870435i	$0.336163\pi$
$920$	−18662.5	−0.668787
$921$	−20334.5	−0.727519
$922$	188.292	0.00672567
$923$	34313.6	1.22367
$924$	−17479.6	−0.622333
$925$	−25575.6	−0.909102
$926$	−16248.3	−0.576624
$927$	6300.30	0.223224
$928$	−17796.7	−0.629533
$929$	−15216.0	−0.537375	−0.268688	−	0.963227i	$-0.586590\pi$
$929$	−15216.0	−0.537375	−0.268688	+	0.963227i	$0.586590\pi$
$930$	−6086.87	−0.214620
$931$	−109792.	−3.86497
$932$	33576.8	1.18009
$933$	17745.1	0.622666
$934$	3408.32	0.119404
$935$	−21102.4	−0.738099
$936$	9311.57	0.325169
$937$	44431.6	1.54911	0.774556	−	0.632506i	$-0.217973\pi$
$937$	44431.6	1.54911	0.774556	+	0.632506i	$0.217973\pi$
$938$	7510.27	0.261427
$939$	18452.8	0.641303
$940$	11673.0	0.405032
$941$	−5816.82	−0.201512	−0.100756	−	0.994911i	$-0.532126\pi$
$941$	−5816.82	−0.201512	−0.100756	+	0.994911i	$0.532126\pi$
$942$	−13421.8	−0.464230
$943$	−37990.2	−1.31191
$944$	−1557.26	−0.0536912
$945$	−6371.64	−0.219333
$946$	2408.17	0.0827657
$947$	19125.2	0.656267	0.328134	−	0.944631i	$-0.393580\pi$
$947$	19125.2	0.656267	0.328134	+	0.944631i	$0.393580\pi$
$948$	−10299.8	−0.352871
$949$	11745.6	0.401769
$950$	13819.1	0.471950
$951$	−11976.7	−0.408381
$952$	67404.0	2.29472
$953$	20408.0	0.693682	0.346841	−	0.937924i	$-0.387254\pi$
$953$	20408.0	0.693682	0.346841	+	0.937924i	$0.387254\pi$
$954$	4449.98	0.151020
$955$	−15045.5	−0.509804
$956$	−40615.8	−1.37407
$957$	8050.85	0.271940
$958$	−21100.2	−0.711605
$959$	−22988.5	−0.774075
$960$	557.007	0.0187264
$961$	19287.9	0.647440
$962$	−24572.2	−0.823535
$963$	14812.8	0.495676
$964$	−16742.0	−0.559359
$965$	9308.81	0.310530
$966$	−18544.1	−0.617646
$967$	−16417.9	−0.545981	−0.272991	−	0.962017i	$-0.588013\pi$
$967$	−16417.9	−0.545981	−0.272991	+	0.962017i	$0.588013\pi$
$968$	10594.4	0.351774
$969$	−46033.3	−1.52611
$970$	−5730.25	−0.189678
$971$	−567.686	−0.0187620	−0.00938101	−	0.999956i	$-0.502986\pi$
$971$	−567.686	−0.0187620	−0.00938101	+	0.999956i	$0.502986\pi$
$972$	−1534.49	−0.0506365
$973$	5007.26	0.164980
$974$	−1725.25	−0.0567561
$975$	12565.0	0.412721
$976$	−407.067	−0.0133503
$977$	906.220	0.0296751	0.0148375	−	0.999890i	$-0.495277\pi$
$977$	906.220	0.0296751	0.0148375	+	0.999890i	$0.495277\pi$
$978$	−7459.41	−0.243891
$979$	−39922.3	−1.30329
$980$	−34566.0	−1.12670
$981$	−2281.69	−0.0742598
$982$	18430.6	0.598925
$983$	54807.1	1.77831	0.889153	−	0.457609i	$-0.151294\pi$
$983$	54807.1	1.77831	0.889153	+	0.457609i	$0.151294\pi$
$984$	14878.1	0.482009
$985$	1319.68	0.0426888
$986$	−13695.2	−0.442338
$987$	26293.3	0.847947
$988$	−49750.2	−1.60199
$989$	−9573.16	−0.307795
$990$	−2273.67	−0.0729920
$991$	−22970.2	−0.736298	−0.368149	−	0.929767i	$-0.620008\pi$
$991$	−22970.2	−0.736298	−0.368149	+	0.929767i	$0.620008\pi$
$992$	−40525.4	−1.29706
$993$	−12672.1	−0.404971
$994$	26762.4	0.853976
$995$	23457.0	0.747374
$996$	−6644.73	−0.211392
$997$	−41803.1	−1.32790	−0.663951	−	0.747776i	$-0.731122\pi$
$997$	−41803.1	−1.32790	−0.663951	+	0.747776i	$0.731122\pi$
$998$	−9792.16	−0.310587
$999$	9179.35	0.290713

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.a.1.5	✓	7
3.2	odd	2		531.4.a.d.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.a.1.5	✓	7	1.1	even	1	trivial
531.4.a.d.1.3		7	3.2	odd	2

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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q+1.29817 q^{2} +3.00000 q^{3} -6.31476 q^{4} +7.05496 q^{5} +3.89451 q^{6} -33.4497 q^{7} -18.5830 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+1.29817 q^{2} +3.00000 q^{3} -6.31476 q^{4} +7.05496 q^{5} +3.89451 q^{6} -33.4497 q^{7} -18.5830 q^{8} +9.00000 q^{9} +9.15853 q^{10} -27.5842 q^{11} -18.9443 q^{12} -55.6756 q^{13} -43.4234 q^{14} +21.1649 q^{15} +26.3942 q^{16} +108.437 q^{17} +11.6835 q^{18} -141.505 q^{19} -44.5504 q^{20} -100.349 q^{21} -35.8089 q^{22} +142.351 q^{23} -55.7489 q^{24} -75.2276 q^{25} -72.2763 q^{26} +27.0000 q^{27} +211.227 q^{28} -97.2883 q^{29} +27.4756 q^{30} -221.538 q^{31} +182.928 q^{32} -82.7525 q^{33} +140.770 q^{34} -235.987 q^{35} -56.8328 q^{36} +339.976 q^{37} -183.698 q^{38} -167.027 q^{39} -131.102 q^{40} -266.877 q^{41} -130.270 q^{42} -67.2505 q^{43} +174.187 q^{44} +63.4946 q^{45} +184.795 q^{46} -262.017 q^{47} +79.1827 q^{48} +775.886 q^{49} -97.6581 q^{50} +325.311 q^{51} +351.578 q^{52} +380.876 q^{53} +35.0506 q^{54} -194.605 q^{55} +621.596 q^{56} -424.516 q^{57} -126.297 q^{58} -59.0000 q^{59} -133.651 q^{60} -15.4226 q^{61} -287.593 q^{62} -301.048 q^{63} +26.3175 q^{64} -392.789 q^{65} -107.427 q^{66} -172.954 q^{67} -684.753 q^{68} +427.052 q^{69} -306.350 q^{70} -616.313 q^{71} -167.247 q^{72} -210.965 q^{73} +441.346 q^{74} -225.683 q^{75} +893.572 q^{76} +922.684 q^{77} -216.829 q^{78} +543.689 q^{79} +186.210 q^{80} +81.0000 q^{81} -346.452 q^{82} +350.751 q^{83} +633.681 q^{84} +765.019 q^{85} -87.3025 q^{86} -291.865 q^{87} +512.596 q^{88} +1447.29 q^{89} +82.4267 q^{90} +1862.34 q^{91} -898.911 q^{92} -664.613 q^{93} -340.143 q^{94} -998.315 q^{95} +548.784 q^{96} -625.674 q^{97} +1007.23 q^{98} -248.258 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9}+O(q^{10}) \) 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9	\( 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9} - 79 q^{10} - 131 q^{11} + 66 q^{12} - 123 q^{13} - 117 q^{14} - 84 q^{15} + 202 q^{16} - 235 q^{17} - 72 q^{18} - 80 q^{19} + 61 q^{20} - 177 q^{21} + 688 q^{22} - 274 q^{23} - 351 q^{24} + 193 q^{25} - 180 q^{26} + 189 q^{27} - 118 q^{28} - 406 q^{29} - 237 q^{30} - 346 q^{31} - 854 q^{32} - 393 q^{33} + 178 q^{34} - 424 q^{35} + 198 q^{36} - 157 q^{37} - 129 q^{38} - 369 q^{39} - 590 q^{40} - 825 q^{41} - 351 q^{42} - 815 q^{43} - 1690 q^{44} - 252 q^{45} + 1457 q^{46} - 1196 q^{47} + 606 q^{48} + 914 q^{49} + 713 q^{50} - 705 q^{51} + 1030 q^{52} - 900 q^{53} - 216 q^{54} - 1044 q^{55} + 2172 q^{56} - 240 q^{57} + 1242 q^{58} - 413 q^{59} + 183 q^{60} + 420 q^{61} + 646 q^{62} - 531 q^{63} + 3541 q^{64} + 190 q^{65} + 2064 q^{66} + 1316 q^{67} - 611 q^{68} - 822 q^{69} + 4658 q^{70} - 173 q^{71} - 1053 q^{72} - 418 q^{73} + 660 q^{74} + 579 q^{75} + 1540 q^{76} - 753 q^{77} - 540 q^{78} + 2635 q^{79} + 6155 q^{80} + 567 q^{81} - 125 q^{82} + 457 q^{83} - 354 q^{84} + 1270 q^{85} + 3482 q^{86} - 1218 q^{87} + 7685 q^{88} + 592 q^{89} - 711 q^{90} + 3179 q^{91} - 3500 q^{92} - 1038 q^{93} + 2064 q^{94} - 2250 q^{95} - 2562 q^{96} - 1906 q^{97} + 2994 q^{98} - 1179 q^{99}+O(q^{100}) \) 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9 - 79 * q^10 - 131 * q^11 + 66 * q^12 - 123 * q^13 - 117 * q^14 - 84 * q^15 + 202 * q^16 - 235 * q^17 - 72 * q^18 - 80 * q^19 + 61 * q^20 - 177 * q^21 + 688 * q^22 - 274 * q^23 - 351 * q^24 + 193 * q^25 - 180 * q^26 + 189 * q^27 - 118 * q^28 - 406 * q^29 - 237 * q^30 - 346 * q^31 - 854 * q^32 - 393 * q^33 + 178 * q^34 - 424 * q^35 + 198 * q^36 - 157 * q^37 - 129 * q^38 - 369 * q^39 - 590 * q^40 - 825 * q^41 - 351 * q^42 - 815 * q^43 - 1690 * q^44 - 252 * q^45 + 1457 * q^46 - 1196 * q^47 + 606 * q^48 + 914 * q^49 + 713 * q^50 - 705 * q^51 + 1030 * q^52 - 900 * q^53 - 216 * q^54 - 1044 * q^55 + 2172 * q^56 - 240 * q^57 + 1242 * q^58 - 413 * q^59 + 183 * q^60 + 420 * q^61 + 646 * q^62 - 531 * q^63 + 3541 * q^64 + 190 * q^65 + 2064 * q^66 + 1316 * q^67 - 611 * q^68 - 822 * q^69 + 4658 * q^70 - 173 * q^71 - 1053 * q^72 - 418 * q^73 + 660 * q^74 + 579 * q^75 + 1540 * q^76 - 753 * q^77 - 540 * q^78 + 2635 * q^79 + 6155 * q^80 + 567 * q^81 - 125 * q^82 + 457 * q^83 - 354 * q^84 + 1270 * q^85 + 3482 * q^86 - 1218 * q^87 + 7685 * q^88 + 592 * q^89 - 711 * q^90 + 3179 * q^91 - 3500 * q^92 - 1038 * q^93 + 2064 * q^94 - 2250 * q^95 - 2562 * q^96 - 1906 * q^97 + 2994 * q^98 - 1179 * q^99

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