LMFDB - Embedded newform 177.4.a.a.1.7

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			1.7
Root			$-4.44426$ 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)

$f(q)$	$=$	$q+3.44426 q^{2} +3.00000 q^{3} +3.86292 q^{4} -20.9057 q^{5} +10.3328 q^{6} -30.0572 q^{7} -14.2492 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+3.44426 q^{2} +3.00000 q^{3} +3.86292 q^{4} -20.9057 q^{5} +10.3328 q^{6} -30.0572 q^{7} -14.2492 q^{8} +9.00000 q^{9} -72.0046 q^{10} +51.2171 q^{11} +11.5888 q^{12} -22.3086 q^{13} -103.525 q^{14} -62.7171 q^{15} -79.9812 q^{16} -89.1524 q^{17} +30.9983 q^{18} +96.6448 q^{19} -80.7571 q^{20} -90.1715 q^{21} +176.405 q^{22} +76.1937 q^{23} -42.7475 q^{24} +312.048 q^{25} -76.8365 q^{26} +27.0000 q^{27} -116.108 q^{28} -71.5601 q^{29} -216.014 q^{30} -129.288 q^{31} -161.483 q^{32} +153.651 q^{33} -307.064 q^{34} +628.366 q^{35} +34.7663 q^{36} -108.382 q^{37} +332.870 q^{38} -66.9257 q^{39} +297.889 q^{40} -357.190 q^{41} -310.574 q^{42} -237.327 q^{43} +197.848 q^{44} -188.151 q^{45} +262.431 q^{46} +97.2732 q^{47} -239.944 q^{48} +560.433 q^{49} +1074.77 q^{50} -267.457 q^{51} -86.1763 q^{52} -705.515 q^{53} +92.9950 q^{54} -1070.73 q^{55} +428.290 q^{56} +289.934 q^{57} -246.471 q^{58} -59.0000 q^{59} -242.271 q^{60} -549.755 q^{61} -445.302 q^{62} -270.514 q^{63} +83.6615 q^{64} +466.376 q^{65} +529.215 q^{66} +652.131 q^{67} -344.389 q^{68} +228.581 q^{69} +2164.25 q^{70} -37.3589 q^{71} -128.243 q^{72} -600.093 q^{73} -373.297 q^{74} +936.144 q^{75} +373.331 q^{76} -1539.44 q^{77} -230.510 q^{78} +1221.13 q^{79} +1672.06 q^{80} +81.0000 q^{81} -1230.26 q^{82} -577.079 q^{83} -348.325 q^{84} +1863.79 q^{85} -817.417 q^{86} -214.680 q^{87} -729.802 q^{88} -375.999 q^{89} -648.042 q^{90} +670.532 q^{91} +294.330 q^{92} -387.864 q^{93} +335.034 q^{94} -2020.43 q^{95} -484.448 q^{96} +322.400 q^{97} +1930.28 q^{98} +460.954 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.44426	1.21773	0.608865	−	0.793274i	$-0.291625\pi$
$2$	3.44426	1.21773	0.608865	+	0.793274i	$0.291625\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	3.86292	0.482865
$5$	−20.9057	−1.86986	−0.934931	−	0.354830i	$-0.884539\pi$
$5$	−20.9057	−1.86986	−0.934931	+	0.354830i	$0.884539\pi$
$6$	10.3328	0.703056
$7$	−30.0572	−1.62293	−0.811467	−	0.584398i	$-0.801331\pi$
$7$	−30.0572	−1.62293	−0.811467	+	0.584398i	$0.801331\pi$
$8$	−14.2492	−0.629730
$9$	9.00000	0.333333
$10$	−72.0046	−2.27699
$11$	51.2171	1.40387	0.701934	−	0.712242i	$-0.252320\pi$
$11$	51.2171	1.40387	0.701934	+	0.712242i	$0.252320\pi$
$12$	11.5888	0.278782
$13$	−22.3086	−0.475945	−0.237973	−	0.971272i	$-0.576483\pi$
$13$	−22.3086	−0.475945	−0.237973	+	0.971272i	$0.576483\pi$
$14$	−103.525	−1.97629
$15$	−62.7171	−1.07957
$16$	−79.9812	−1.24971
$17$	−89.1524	−1.27192	−0.635960	−	0.771722i	$-0.719395\pi$
$17$	−89.1524	−1.27192	−0.635960	+	0.771722i	$0.719395\pi$
$18$	30.9983	0.405910
$19$	96.6448	1.16694	0.583469	−	0.812135i	$-0.301695\pi$
$19$	96.6448	1.16694	0.583469	+	0.812135i	$0.301695\pi$
$20$	−80.7571	−0.902891
$21$	−90.1715	−0.937001
$22$	176.405	1.70953
$23$	76.1937	0.690760	0.345380	−	0.938463i	$-0.387750\pi$
$23$	76.1937	0.690760	0.345380	+	0.938463i	$0.387750\pi$
$24$	−42.7475	−0.363575
$25$	312.048	2.49638
$26$	−76.8365	−0.579572
$27$	27.0000	0.192450
$28$	−116.108	−0.783658
$29$	−71.5601	−0.458220	−0.229110	−	0.973401i	$-0.573581\pi$
$29$	−71.5601	−0.458220	−0.229110	+	0.973401i	$0.573581\pi$
$30$	−216.014	−1.31462
$31$	−129.288	−0.749059	−0.374530	−	0.927215i	$-0.622196\pi$
$31$	−129.288	−0.749059	−0.374530	+	0.927215i	$0.622196\pi$
$32$	−161.483	−0.892074
$33$	153.651	0.810524
$34$	−307.064	−1.54885
$35$	628.366	3.03466
$36$	34.7663	0.160955
$37$	−108.382	−0.481567	−0.240783	−	0.970579i	$-0.577404\pi$
$37$	−108.382	−0.481567	−0.240783	+	0.970579i	$0.577404\pi$
$38$	332.870	1.42102
$39$	−66.9257	−0.274787
$40$	297.889	1.17751
$41$	−357.190	−1.36058	−0.680290	−	0.732943i	$-0.738146\pi$
$41$	−357.190	−1.36058	−0.680290	+	0.732943i	$0.738146\pi$
$42$	−310.574	−1.14101
$43$	−237.327	−0.841676	−0.420838	−	0.907136i	$-0.638264\pi$
$43$	−237.327	−0.841676	−0.420838	+	0.907136i	$0.638264\pi$
$44$	197.848	0.677879
$45$	−188.151	−0.623287
$46$	262.431	0.841159
$47$	97.2732	0.301889	0.150944	−	0.988542i	$-0.451769\pi$
$47$	97.2732	0.301889	0.150944	+	0.988542i	$0.451769\pi$
$48$	−239.944	−0.721518
$49$	560.433	1.63392
$50$	1074.77	3.03992
$51$	−267.457	−0.734343
$52$	−86.1763	−0.229817
$53$	−705.515	−1.82849	−0.914245	−	0.405162i	$-0.867215\pi$
$53$	−705.515	−1.82849	−0.914245	+	0.405162i	$0.867215\pi$
$54$	92.9950	0.234352
$55$	−1070.73	−2.62504
$56$	428.290	1.02201
$57$	289.934	0.673733
$58$	−246.471	−0.557988
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	−242.271	−0.521285
$61$	−549.755	−1.15392	−0.576958	−	0.816774i	$-0.695760\pi$
$61$	−549.755	−1.15392	−0.576958	+	0.816774i	$0.695760\pi$
$62$	−445.302	−0.912152
$63$	−270.514	−0.540978
$64$	83.6615	0.163401
$65$	466.376	0.889951
$66$	529.215	0.986999
$67$	652.131	1.18911	0.594555	−	0.804055i	$-0.297328\pi$
$67$	652.131	1.18911	0.594555	+	0.804055i	$0.297328\pi$
$68$	−344.389	−0.614166
$69$	228.581	0.398811
$70$	2164.25	3.69540
$71$	−37.3589	−0.0624463	−0.0312232	−	0.999512i	$-0.509940\pi$
$71$	−37.3589	−0.0624463	−0.0312232	+	0.999512i	$0.509940\pi$
$72$	−128.243	−0.209910
$73$	−600.093	−0.962131	−0.481066	−	0.876685i	$-0.659750\pi$
$73$	−600.093	−0.962131	−0.481066	+	0.876685i	$0.659750\pi$
$74$	−373.297	−0.586418
$75$	936.144	1.44129
$76$	373.331	0.563474
$77$	−1539.44	−2.27839
$78$	−230.510	−0.334616
$79$	1221.13	1.73909	0.869546	−	0.493851i	$-0.164411\pi$
$79$	1221.13	1.73909	0.869546	+	0.493851i	$0.164411\pi$
$80$	1672.06	2.33678
$81$	81.0000	0.111111
$82$	−1230.26	−1.65682
$83$	−577.079	−0.763164	−0.381582	−	0.924335i	$-0.624621\pi$
$83$	−577.079	−0.763164	−0.381582	+	0.924335i	$0.624621\pi$
$84$	−348.325	−0.452445
$85$	1863.79	2.37831
$86$	−817.417	−1.02493
$87$	−214.680	−0.264553
$88$	−729.802	−0.884058
$89$	−375.999	−0.447818	−0.223909	−	0.974610i	$-0.571882\pi$
$89$	−375.999	−0.447818	−0.223909	+	0.974610i	$0.571882\pi$
$90$	−648.042	−0.758995
$91$	670.532	0.772427
$92$	294.330	0.333544
$93$	−387.864	−0.432470
$94$	335.034	0.367619
$95$	−2020.43	−2.18201
$96$	−484.448	−0.515039
$97$	322.400	0.337472	0.168736	−	0.985661i	$-0.446032\pi$
$97$	322.400	0.337472	0.168736	+	0.985661i	$0.446032\pi$
$98$	1930.28	1.98967
$99$	460.954	0.467956
$100$	1205.42	1.20542
$101$	657.772	0.648027	0.324014	−	0.946052i	$-0.394968\pi$
$101$	657.772	0.648027	0.324014	+	0.946052i	$0.394968\pi$
$102$	−921.192	−0.894231
$103$	1003.17	0.959667	0.479834	−	0.877359i	$-0.340697\pi$
$103$	1003.17	0.959667	0.479834	+	0.877359i	$0.340697\pi$
$104$	317.879	0.299717
$105$	1885.10	1.75206
$106$	−2429.98	−2.22661
$107$	−882.607	−0.797428	−0.398714	−	0.917075i	$-0.630543\pi$
$107$	−882.607	−0.797428	−0.398714	+	0.917075i	$0.630543\pi$
$108$	104.299	0.0929275
$109$	608.066	0.534332	0.267166	−	0.963651i	$-0.413913\pi$
$109$	608.066	0.534332	0.267166	+	0.963651i	$0.413913\pi$
$110$	−3687.87	−3.19659
$111$	−325.147	−0.278033
$112$	2404.01	2.02819
$113$	373.130	0.310629	0.155315	−	0.987865i	$-0.450361\pi$
$113$	373.130	0.310629	0.155315	+	0.987865i	$0.450361\pi$
$114$	998.609	0.820424
$115$	−1592.88	−1.29163
$116$	−276.431	−0.221258
$117$	−200.777	−0.158648
$118$	−203.211	−0.158535
$119$	2679.67	2.06424
$120$	893.666	0.679835
$121$	1292.20	0.970846
$122$	−1893.50	−1.40516
$123$	−1071.57	−0.785531
$124$	−499.430	−0.361695
$125$	−3910.37	−2.79803
$126$	−931.722	−0.658765
$127$	626.337	0.437625	0.218813	−	0.975767i	$-0.429782\pi$
$127$	626.337	0.437625	0.218813	+	0.975767i	$0.429782\pi$
$128$	1580.01	1.09105
$129$	−711.982	−0.485942
$130$	1606.32	1.08372
$131$	1191.46	0.794641	0.397320	−	0.917680i	$-0.369940\pi$
$131$	1191.46	0.794641	0.397320	+	0.917680i	$0.369940\pi$
$132$	593.543	0.391374
$133$	−2904.87	−1.89387
$134$	2246.11	1.44802
$135$	−564.454	−0.359855
$136$	1270.35	0.800966
$137$	−487.325	−0.303905	−0.151952	−	0.988388i	$-0.548556\pi$
$137$	−487.325	−0.303905	−0.151952	+	0.988388i	$0.548556\pi$
$138$	787.293	0.485644
$139$	552.716	0.337272	0.168636	−	0.985678i	$-0.446064\pi$
$139$	552.716	0.337272	0.168636	+	0.985678i	$0.446064\pi$
$140$	2427.33	1.46533
$141$	291.820	0.174295
$142$	−128.674	−0.0760427
$143$	−1142.58	−0.668164
$144$	−719.831	−0.416569
$145$	1496.01	0.856807
$146$	−2066.88	−1.17162
$147$	1681.30	0.943341
$148$	−418.673	−0.232532
$149$	−3509.24	−1.92945	−0.964725	−	0.263260i	$-0.915202\pi$
$149$	−3509.24	−1.92945	−0.964725	+	0.263260i	$0.915202\pi$
$150$	3224.32	1.75510
$151$	3175.51	1.71138	0.855692	−	0.517485i	$-0.173132\pi$
$151$	3175.51	1.71138	0.855692	+	0.517485i	$0.173132\pi$
$152$	−1377.11	−0.734857
$153$	−802.371	−0.423973
$154$	−5302.24	−2.77446
$155$	2702.86	1.40064
$156$	−258.529	−0.132685
$157$	719.754	0.365877	0.182938	−	0.983124i	$-0.441439\pi$
$157$	719.754	0.365877	0.182938	+	0.983124i	$0.441439\pi$
$158$	4205.90	2.11774
$159$	−2116.55	−1.05568
$160$	3375.91	1.66806
$161$	−2290.17	−1.12106
$162$	278.985	0.135303
$163$	−1106.29	−0.531602	−0.265801	−	0.964028i	$-0.585636\pi$
$163$	−1106.29	−0.531602	−0.265801	+	0.964028i	$0.585636\pi$
$164$	−1379.80	−0.656977
$165$	−3212.19	−1.51557
$166$	−1987.61	−0.929327
$167$	−821.206	−0.380520	−0.190260	−	0.981734i	$-0.560933\pi$
$167$	−821.206	−0.380520	−0.190260	+	0.981734i	$0.560933\pi$
$168$	1284.87	0.590058
$169$	−1699.33	−0.773476
$170$	6419.38	2.89614
$171$	869.803	0.388980
$172$	−916.777	−0.406416
$173$	−4010.92	−1.76269	−0.881343	−	0.472476i	$-0.843360\pi$
$173$	−4010.92	−1.76269	−0.881343	+	0.472476i	$0.843360\pi$
$174$	−739.414	−0.322154
$175$	−9379.28	−4.05147
$176$	−4096.41	−1.75442
$177$	−177.000	−0.0751646
$178$	−1295.04	−0.545321
$179$	−716.188	−0.299053	−0.149526	−	0.988758i	$-0.547775\pi$
$179$	−716.188	−0.299053	−0.149526	+	0.988758i	$0.547775\pi$
$180$	−726.814	−0.300964
$181$	1786.44	0.733619	0.366809	−	0.930296i	$-0.380450\pi$
$181$	1786.44	0.733619	0.366809	+	0.930296i	$0.380450\pi$
$182$	2309.49	0.940608
$183$	−1649.26	−0.666214
$184$	−1085.70	−0.434993
$185$	2265.81	0.900463
$186$	−1335.91	−0.526631
$187$	−4566.13	−1.78561
$188$	375.759	0.145771
$189$	−811.543	−0.312334
$190$	−6958.87	−2.65710
$191$	853.280	0.323252	0.161626	−	0.986852i	$-0.448326\pi$
$191$	853.280	0.323252	0.161626	+	0.986852i	$0.448326\pi$
$192$	250.985	0.0943398
$193$	−3478.84	−1.29747	−0.648737	−	0.761013i	$-0.724702\pi$
$193$	−3478.84	−1.29747	−0.648737	+	0.761013i	$0.724702\pi$
$194$	1110.43	0.410949
$195$	1399.13	0.513814
$196$	2164.91	0.788961
$197$	−337.299	−0.121988	−0.0609938	−	0.998138i	$-0.519427\pi$
$197$	−337.299	−0.121988	−0.0609938	+	0.998138i	$0.519427\pi$
$198$	1587.65	0.569844
$199$	−2613.64	−0.931035	−0.465518	−	0.885039i	$-0.654132\pi$
$199$	−2613.64	−0.931035	−0.465518	+	0.885039i	$0.654132\pi$
$200$	−4446.42	−1.57205
$201$	1956.39	0.686533
$202$	2265.54	0.789122
$203$	2150.89	0.743660
$204$	−1033.17	−0.354589
$205$	7467.31	2.54410
$206$	3455.19	1.16862
$207$	685.744	0.230253
$208$	1784.27	0.594792
$209$	4949.87	1.63823
$210$	6492.76	2.13354
$211$	4988.17	1.62749	0.813743	−	0.581224i	$-0.197426\pi$
$211$	4988.17	1.62749	0.813743	+	0.581224i	$0.197426\pi$
$212$	−2725.35	−0.882914
$213$	−112.077	−0.0360534
$214$	−3039.93	−0.971052
$215$	4961.49	1.57382
$216$	−384.728	−0.121192
$217$	3886.04	1.21567
$218$	2094.34	0.650672
$219$	−1800.28	−0.555487
$220$	−4136.15	−1.26754
$221$	1988.86	0.605364
$222$	−1119.89	−0.338569
$223$	1752.62	0.526296	0.263148	−	0.964755i	$-0.415239\pi$
$223$	1752.62	0.526296	0.263148	+	0.964755i	$0.415239\pi$
$224$	4853.71	1.44778
$225$	2808.43	0.832128
$226$	1285.16	0.378262
$227$	−2983.27	−0.872276	−0.436138	−	0.899880i	$-0.643654\pi$
$227$	−2983.27	−0.872276	−0.436138	+	0.899880i	$0.643654\pi$
$228$	1119.99	0.325322
$229$	1672.34	0.482583	0.241292	−	0.970453i	$-0.422429\pi$
$229$	1672.34	0.482583	0.241292	+	0.970453i	$0.422429\pi$
$230$	−5486.30	−1.57285
$231$	−4618.33	−1.31543
$232$	1019.67	0.288555
$233$	508.127	0.142869	0.0714345	−	0.997445i	$-0.477242\pi$
$233$	508.127	0.142869	0.0714345	+	0.997445i	$0.477242\pi$
$234$	−691.529	−0.193191
$235$	−2033.56	−0.564490
$236$	−227.912	−0.0628637
$237$	3663.40	1.00407
$238$	9229.47	2.51369
$239$	−2497.08	−0.675828	−0.337914	−	0.941177i	$-0.609721\pi$
$239$	−2497.08	−0.675828	−0.337914	+	0.941177i	$0.609721\pi$
$240$	5016.19	1.34914
$241$	−4476.15	−1.19641	−0.598204	−	0.801344i	$-0.704119\pi$
$241$	−4476.15	−1.19641	−0.598204	+	0.801344i	$0.704119\pi$
$242$	4450.66	1.18223
$243$	243.000	0.0641500
$244$	−2123.66	−0.557186
$245$	−11716.2	−3.05520
$246$	−3690.77	−0.956564
$247$	−2156.01	−0.555399
$248$	1842.25	0.471705
$249$	−1731.24	−0.440613
$250$	−13468.3	−3.40724
$251$	−5524.04	−1.38914	−0.694571	−	0.719425i	$-0.744406\pi$
$251$	−5524.04	−1.38914	−0.694571	+	0.719425i	$0.744406\pi$
$252$	−1044.98	−0.261219
$253$	3902.43	0.969737
$254$	2157.27	0.532909
$255$	5591.38	1.37312
$256$	4772.68	1.16521
$257$	7709.96	1.87134	0.935669	−	0.352879i	$-0.114797\pi$
$257$	7709.96	1.87134	0.935669	+	0.352879i	$0.114797\pi$
$258$	−2452.25	−0.591746
$259$	3257.67	0.781551
$260$	1801.57	0.429727
$261$	−644.041	−0.152740
$262$	4103.68	0.967657
$263$	−2619.57	−0.614181	−0.307091	−	0.951680i	$-0.599356\pi$
$263$	−2619.57	−0.614181	−0.307091	+	0.951680i	$0.599356\pi$
$264$	−2189.41	−0.510411
$265$	14749.3	3.41902
$266$	−10005.1	−2.30622
$267$	−1128.00	−0.258548
$268$	2519.13	0.574180
$269$	3075.82	0.697160	0.348580	−	0.937279i	$-0.386664\pi$
$269$	3075.82	0.697160	0.348580	+	0.937279i	$0.386664\pi$
$270$	−1944.12	−0.438206
$271$	−4028.15	−0.902926	−0.451463	−	0.892290i	$-0.649098\pi$
$271$	−4028.15	−0.902926	−0.451463	+	0.892290i	$0.649098\pi$
$272$	7130.51	1.58953
$273$	2011.60	0.445961
$274$	−1678.47	−0.370074
$275$	15982.2	3.50459
$276$	882.991	0.192572
$277$	−3905.75	−0.847197	−0.423599	−	0.905850i	$-0.639233\pi$
$277$	−3905.75	−0.847197	−0.423599	+	0.905850i	$0.639233\pi$
$278$	1903.70	0.410706
$279$	−1163.59	−0.249686
$280$	−8953.69	−1.91102
$281$	−3502.01	−0.743461	−0.371730	−	0.928341i	$-0.621235\pi$
$281$	−3502.01	−0.743461	−0.371730	+	0.928341i	$0.621235\pi$
$282$	1005.10	0.212245
$283$	−6544.11	−1.37458	−0.687291	−	0.726382i	$-0.741200\pi$
$283$	−6544.11	−1.37458	−0.687291	+	0.726382i	$0.741200\pi$
$284$	−144.315	−0.0301532
$285$	−6061.28	−1.25979
$286$	−3935.35	−0.813643
$287$	10736.1	2.20813
$288$	−1453.34	−0.297358
$289$	3035.14	0.617778
$290$	5152.65	1.04336
$291$	967.200	0.194839
$292$	−2318.11	−0.464580
$293$	6261.31	1.24843	0.624215	−	0.781253i	$-0.285419\pi$
$293$	6261.31	1.24843	0.624215	+	0.781253i	$0.285419\pi$
$294$	5790.83	1.14873
$295$	1233.44	0.243435
$296$	1544.36	0.303257
$297$	1382.86	0.270175
$298$	−12086.7	−2.34955
$299$	−1699.77	−0.328764
$300$	3616.25	0.695948
$301$	7133.38	1.36598
$302$	10937.3	2.08400
$303$	1973.32	0.374139
$304$	−7729.77	−1.45833
$305$	11493.0	2.15766
$306$	−2763.57	−0.516284
$307$	−3929.77	−0.730566	−0.365283	−	0.930896i	$-0.619028\pi$
$307$	−3929.77	−0.730566	−0.365283	+	0.930896i	$0.619028\pi$
$308$	−5946.74	−1.10015
$309$	3009.52	0.554064
$310$	9309.35	1.70560
$311$	−6515.56	−1.18798	−0.593992	−	0.804471i	$-0.702449\pi$
$311$	−6515.56	−1.18798	−0.593992	+	0.804471i	$0.702449\pi$
$312$	953.636	0.173042
$313$	−61.8440	−0.0111681	−0.00558407	−	0.999984i	$-0.501777\pi$
$313$	−61.8440	−0.0111681	−0.00558407	+	0.999984i	$0.501777\pi$
$314$	2479.02	0.445539
$315$	5655.29	1.01155
$316$	4717.15	0.839747
$317$	4490.24	0.795574	0.397787	−	0.917478i	$-0.369778\pi$
$317$	4490.24	0.795574	0.397787	+	0.917478i	$0.369778\pi$
$318$	−7289.93	−1.28553
$319$	−3665.10	−0.643280
$320$	−1749.00	−0.305538
$321$	−2647.82	−0.460395
$322$	−7887.93	−1.36515
$323$	−8616.11	−1.48425
$324$	312.897	0.0536517
$325$	−6961.35	−1.18814
$326$	−3810.34	−0.647348
$327$	1824.20	0.308497
$328$	5089.67	0.856798
$329$	−2923.76	−0.489945
$330$	−11063.6	−1.84555
$331$	7049.54	1.17063	0.585314	−	0.810807i	$-0.300971\pi$
$331$	7049.54	1.17063	0.585314	+	0.810807i	$0.300971\pi$
$332$	−2229.21	−0.368505
$333$	−975.442	−0.160522
$334$	−2828.45	−0.463371
$335$	−13633.2	−2.22347
$336$	7212.02	1.17098
$337$	4495.19	0.726613	0.363307	−	0.931670i	$-0.381648\pi$
$337$	4495.19	0.726613	0.363307	+	0.931670i	$0.381648\pi$
$338$	−5852.92	−0.941885
$339$	1119.39	0.179342
$340$	7199.68	1.14840
$341$	−6621.77	−1.05158
$342$	2995.83	0.473672
$343$	−6535.42	−1.02880
$344$	3381.72	0.530029
$345$	−4778.65	−0.745721
$346$	−13814.7	−2.14648
$347$	9082.61	1.40513	0.702565	−	0.711620i	$-0.252038\pi$
$347$	9082.61	1.40513	0.702565	+	0.711620i	$0.252038\pi$
$348$	−829.293	−0.127744
$349$	12020.9	1.84374	0.921870	−	0.387498i	$-0.126661\pi$
$349$	12020.9	1.84374	0.921870	+	0.387498i	$0.126661\pi$
$350$	−32304.7	−4.93359
$351$	−602.332	−0.0915957
$352$	−8270.68	−1.25235
$353$	−5538.86	−0.835138	−0.417569	−	0.908645i	$-0.637118\pi$
$353$	−5538.86	−0.835138	−0.417569	+	0.908645i	$0.637118\pi$
$354$	−609.634	−0.0915302
$355$	781.014	0.116766
$356$	−1452.45	−0.216236
$357$	8039.00	1.19179
$358$	−2466.74	−0.364165
$359$	−10116.0	−1.48719	−0.743595	−	0.668630i	$-0.766881\pi$
$359$	−10116.0	−1.48719	−0.743595	+	0.668630i	$0.766881\pi$
$360$	2681.00	0.392503
$361$	2481.22	0.361747
$362$	6152.96	0.893349
$363$	3876.59	0.560518
$364$	2590.21	0.372978
$365$	12545.4	1.79905
$366$	−5680.49	−0.811268
$367$	−5324.58	−0.757332	−0.378666	−	0.925533i	$-0.623617\pi$
$367$	−5324.58	−0.757332	−0.378666	+	0.925533i	$0.623617\pi$
$368$	−6094.07	−0.863248
$369$	−3214.71	−0.453527
$370$	7804.04	1.09652
$371$	21205.8	2.96752
$372$	−1498.29	−0.208825
$373$	−7097.68	−0.985266	−0.492633	−	0.870237i	$-0.663965\pi$
$373$	−7097.68	−0.985266	−0.492633	+	0.870237i	$0.663965\pi$
$374$	−15726.9	−2.17439
$375$	−11731.1	−1.61544
$376$	−1386.06	−0.190108
$377$	1596.40	0.218087
$378$	−2795.17	−0.380338
$379$	−8367.60	−1.13408	−0.567038	−	0.823692i	$-0.691911\pi$
$379$	−8367.60	−1.13408	−0.567038	+	0.823692i	$0.691911\pi$
$380$	−7804.75	−1.05362
$381$	1879.01	0.252663
$382$	2938.92	0.393634
$383$	−10657.5	−1.42186	−0.710928	−	0.703265i	$-0.751725\pi$
$383$	−10657.5	−1.42186	−0.710928	+	0.703265i	$0.751725\pi$
$384$	4740.04	0.629920
$385$	32183.1	4.26027
$386$	−11982.0	−1.57997
$387$	−2135.95	−0.280559
$388$	1245.41	0.162953
$389$	1280.56	0.166907	0.0834534	−	0.996512i	$-0.473405\pi$
$389$	1280.56	0.166907	0.0834534	+	0.996512i	$0.473405\pi$
$390$	4818.96	0.625686
$391$	−6792.85	−0.878591
$392$	−7985.70	−1.02893
$393$	3574.37	0.458786
$394$	−1161.75	−0.148548
$395$	−25528.7	−3.25186
$396$	1780.63	0.225960
$397$	831.103	0.105068	0.0525338	−	0.998619i	$-0.483270\pi$
$397$	831.103	0.105068	0.0525338	+	0.998619i	$0.483270\pi$
$398$	−9002.05	−1.13375
$399$	−8714.61	−1.09342
$400$	−24958.0	−3.11975
$401$	4713.15	0.586941	0.293471	−	0.955968i	$-0.405190\pi$
$401$	4713.15	0.586941	0.293471	+	0.955968i	$0.405190\pi$
$402$	6738.32	0.836012
$403$	2884.23	0.356511
$404$	2540.92	0.312910
$405$	−1693.36	−0.207762
$406$	7408.23	0.905577
$407$	−5551.04	−0.676056
$408$	3811.04	0.462438
$409$	10700.4	1.29364	0.646820	−	0.762643i	$-0.276098\pi$
$409$	10700.4	1.29364	0.646820	+	0.762643i	$0.276098\pi$
$410$	25719.4	3.09802
$411$	−1461.97	−0.175460
$412$	3875.19	0.463390
$413$	1773.37	0.211288
$414$	2361.88	0.280386
$415$	12064.2	1.42701
$416$	3602.45	0.424578
$417$	1658.15	0.194724
$418$	17048.6	1.99492
$419$	3995.59	0.465864	0.232932	−	0.972493i	$-0.425168\pi$
$419$	3995.59	0.465864	0.232932	+	0.972493i	$0.425168\pi$
$420$	7281.98	0.846010
$421$	5484.63	0.634928	0.317464	−	0.948270i	$-0.397169\pi$
$421$	5484.63	0.634928	0.317464	+	0.948270i	$0.397169\pi$
$422$	17180.5	1.98184
$423$	875.459	0.100630
$424$	10053.0	1.15146
$425$	−27819.8	−3.17520
$426$	−386.021	−0.0439033
$427$	16524.1	1.87273
$428$	−3409.44	−0.385050
$429$	−3427.74	−0.385765
$430$	17088.7	1.91648
$431$	−9116.24	−1.01883	−0.509413	−	0.860522i	$-0.670137\pi$
$431$	−9116.24	−1.01883	−0.509413	+	0.860522i	$0.670137\pi$
$432$	−2159.49	−0.240506
$433$	−13162.3	−1.46083	−0.730415	−	0.683004i	$-0.760673\pi$
$433$	−13162.3	−1.46083	−0.730415	+	0.683004i	$0.760673\pi$
$434$	13384.5	1.48036
$435$	4488.04	0.494678
$436$	2348.91	0.258010
$437$	7363.73	0.806075
$438$	−6200.63	−0.676433
$439$	−2279.58	−0.247833	−0.123916	−	0.992293i	$-0.539545\pi$
$439$	−2279.58	−0.247833	−0.123916	+	0.992293i	$0.539545\pi$
$440$	15257.0	1.65307
$441$	5043.90	0.544638
$442$	6850.16	0.737169
$443$	13194.2	1.41507	0.707536	−	0.706678i	$-0.249807\pi$
$443$	13194.2	1.41507	0.707536	+	0.706678i	$0.249807\pi$
$444$	−1256.02	−0.134252
$445$	7860.51	0.837357
$446$	6036.47	0.640886
$447$	−10527.7	−1.11397
$448$	−2514.63	−0.265190
$449$	3287.96	0.345587	0.172794	−	0.984958i	$-0.444721\pi$
$449$	3287.96	0.345587	0.172794	+	0.984958i	$0.444721\pi$
$450$	9672.97	1.01331
$451$	−18294.3	−1.91007
$452$	1441.37	0.149992
$453$	9526.52	0.988068
$454$	−10275.2	−1.06220
$455$	−14017.9	−1.44433
$456$	−4131.32	−0.424270
$457$	−14476.7	−1.48182	−0.740908	−	0.671606i	$-0.765605\pi$
$457$	−14476.7	−1.48182	−0.740908	+	0.671606i	$0.765605\pi$
$458$	5759.99	0.587656
$459$	−2407.11	−0.244781
$460$	−6153.18	−0.623682
$461$	−4765.61	−0.481468	−0.240734	−	0.970591i	$-0.577388\pi$
$461$	−4765.61	−0.481468	−0.240734	+	0.970591i	$0.577388\pi$
$462$	−15906.7	−1.60183
$463$	6286.30	0.630992	0.315496	−	0.948927i	$-0.397829\pi$
$463$	6286.30	0.630992	0.315496	+	0.948927i	$0.397829\pi$
$464$	5723.46	0.572640
$465$	8108.58	0.808658
$466$	1750.12	0.173976
$467$	5623.77	0.557253	0.278626	−	0.960400i	$-0.410121\pi$
$467$	5623.77	0.557253	0.278626	+	0.960400i	$0.410121\pi$
$468$	−775.587	−0.0766058
$469$	−19601.2	−1.92985
$470$	−7004.12	−0.687396
$471$	2159.26	0.211239
$472$	840.701	0.0819839
$473$	−12155.2	−1.18160
$474$	12617.7	1.22268
$475$	30157.8	2.91313
$476$	10351.3	0.996750
$477$	−6349.64	−0.609497
$478$	−8600.61	−0.822976
$479$	10680.0	1.01875	0.509376	−	0.860544i	$-0.329876\pi$
$479$	10680.0	1.01875	0.509376	+	0.860544i	$0.329876\pi$
$480$	10127.7	0.963052
$481$	2417.86	0.229199
$482$	−15417.0	−1.45690
$483$	−6870.50	−0.647244
$484$	4991.65	0.468788
$485$	−6739.99	−0.631025
$486$	836.955	0.0781174
$487$	8071.47	0.751034	0.375517	−	0.926816i	$-0.377465\pi$
$487$	8071.47	0.751034	0.375517	+	0.926816i	$0.377465\pi$
$488$	7833.55	0.726656
$489$	−3318.86	−0.306921
$490$	−40353.8	−3.72040
$491$	−11495.5	−1.05659	−0.528295	−	0.849061i	$-0.677168\pi$
$491$	−11495.5	−1.05659	−0.528295	+	0.849061i	$0.677168\pi$
$492$	−4139.40	−0.379306
$493$	6379.75	0.582818
$494$	−7425.85	−0.676326
$495$	−9636.57	−0.875013
$496$	10340.6	0.936104
$497$	1122.90	0.101346
$498$	−5962.83	−0.536547
$499$	−340.307	−0.0305295	−0.0152648	−	0.999883i	$-0.504859\pi$
$499$	−340.307	−0.0305295	−0.0152648	+	0.999883i	$0.504859\pi$
$500$	−15105.4	−1.35107
$501$	−2463.62	−0.219693
$502$	−19026.2	−1.69160
$503$	−6818.69	−0.604434	−0.302217	−	0.953239i	$-0.597727\pi$
$503$	−6818.69	−0.604434	−0.302217	+	0.953239i	$0.597727\pi$
$504$	3854.61	0.340670
$505$	−13751.2	−1.21172
$506$	13441.0	1.18088
$507$	−5097.98	−0.446567
$508$	2419.49	0.211314
$509$	19771.2	1.72170	0.860849	−	0.508860i	$-0.169933\pi$
$509$	19771.2	1.72170	0.860849	+	0.508860i	$0.169933\pi$
$510$	19258.1	1.67209
$511$	18037.1	1.56148
$512$	3798.25	0.327853
$513$	2609.41	0.224578
$514$	26555.1	2.27878
$515$	−20972.1	−1.79445
$516$	−2750.33	−0.234644
$517$	4982.06	0.423812
$518$	11220.3	0.951718
$519$	−12032.8	−1.01769
$520$	−6645.47	−0.560429
$521$	20826.6	1.75130	0.875652	−	0.482943i	$-0.160432\pi$
$521$	20826.6	1.75130	0.875652	+	0.482943i	$0.160432\pi$
$522$	−2218.24	−0.185996
$523$	5693.85	0.476051	0.238026	−	0.971259i	$-0.423500\pi$
$523$	5693.85	0.476051	0.238026	+	0.971259i	$0.423500\pi$
$524$	4602.50	0.383704
$525$	−28137.8	−2.33912
$526$	−9022.49	−0.747907
$527$	11526.3	0.952743
$528$	−12289.2	−1.01292
$529$	−6361.52	−0.522850
$530$	50800.4	4.16345
$531$	−531.000	−0.0433963
$532$	−11221.3	−0.914482
$533$	7968.41	0.647561
$534$	−3885.11	−0.314841
$535$	18451.5	1.49108
$536$	−9292.32	−0.748819
$537$	−2148.56	−0.172658
$538$	10593.9	0.848952
$539$	28703.8	2.29380
$540$	−2180.44	−0.173762
$541$	1565.72	0.124428	0.0622140	−	0.998063i	$-0.480184\pi$
$541$	1565.72	0.124428	0.0622140	+	0.998063i	$0.480184\pi$
$542$	−13874.0	−1.09952
$543$	5359.32	0.423555
$544$	14396.6	1.13465
$545$	−12712.0	−0.999127
$546$	6928.46	0.543060
$547$	2599.10	0.203162	0.101581	−	0.994827i	$-0.467610\pi$
$547$	2599.10	0.203162	0.101581	+	0.994827i	$0.467610\pi$
$548$	−1882.50	−0.146745
$549$	−4947.79	−0.384639
$550$	55046.9	4.26765
$551$	−6915.91	−0.534714
$552$	−3257.09	−0.251143
$553$	−36703.8	−2.82243
$554$	−13452.4	−1.03166
$555$	6797.43	0.519883
$556$	2135.10	0.162857
$557$	14567.3	1.10814	0.554072	−	0.832469i	$-0.313073\pi$
$557$	14567.3	1.10814	0.554072	+	0.832469i	$0.313073\pi$
$558$	−4007.72	−0.304051
$559$	5294.43	0.400591
$560$	−50257.5	−3.79244
$561$	−13698.4	−1.03092
$562$	−12061.8	−0.905334
$563$	11318.1	0.847251	0.423626	−	0.905837i	$-0.360757\pi$
$563$	11318.1	0.847251	0.423626	+	0.905837i	$0.360757\pi$
$564$	1127.28	0.0841612
$565$	−7800.54	−0.580834
$566$	−22539.6	−1.67387
$567$	−2434.63	−0.180326
$568$	532.334	0.0393243
$569$	−16128.1	−1.18827	−0.594134	−	0.804366i	$-0.702505\pi$
$569$	−16128.1	−1.18827	−0.594134	+	0.804366i	$0.702505\pi$
$570$	−20876.6	−1.53408
$571$	−13301.5	−0.974867	−0.487433	−	0.873160i	$-0.662067\pi$
$571$	−13301.5	−0.974867	−0.487433	+	0.873160i	$0.662067\pi$
$572$	−4413.70	−0.322633
$573$	2559.84	0.186630
$574$	36978.0	2.68891
$575$	23776.1	1.72440
$576$	752.954	0.0544671
$577$	−19302.8	−1.39270	−0.696348	−	0.717704i	$-0.745193\pi$
$577$	−19302.8	−1.39270	−0.696348	+	0.717704i	$0.745193\pi$
$578$	10453.8	0.752287
$579$	−10436.5	−0.749096
$580$	5778.98	0.413723
$581$	17345.3	1.23857
$582$	3331.29	0.237262
$583$	−36134.5	−2.56696
$584$	8550.83	0.605883
$585$	4197.39	0.296650
$586$	21565.6	1.52025
$587$	17933.2	1.26096	0.630480	−	0.776205i	$-0.282858\pi$
$587$	17933.2	1.26096	0.630480	+	0.776205i	$0.282858\pi$
$588$	6494.73	0.455507
$589$	−12495.0	−0.874107
$590$	4248.27	0.296438
$591$	−1011.90	−0.0704296
$592$	8668.56	0.601817
$593$	−26500.4	−1.83515	−0.917573	−	0.397567i	$-0.869855\pi$
$593$	−26500.4	−1.83515	−0.917573	+	0.397567i	$0.869855\pi$
$594$	4762.94	0.329000
$595$	−56020.3	−3.85985
$596$	−13555.9	−0.931664
$597$	−7840.92	−0.537533
$598$	−5854.46	−0.400346
$599$	14006.2	0.955388	0.477694	−	0.878526i	$-0.341473\pi$
$599$	14006.2	0.955388	0.477694	+	0.878526i	$0.341473\pi$
$600$	−13339.3	−0.907623
$601$	8209.99	0.557225	0.278613	−	0.960404i	$-0.410125\pi$
$601$	8209.99	0.557225	0.278613	+	0.960404i	$0.410125\pi$
$602$	24569.2	1.66340
$603$	5869.17	0.396370
$604$	12266.7	0.826368
$605$	−27014.2	−1.81535
$606$	6796.61	0.455600
$607$	150.423	0.0100585	0.00502924	−	0.999987i	$-0.498399\pi$
$607$	150.423	0.0100585	0.00502924	+	0.999987i	$0.498399\pi$
$608$	−15606.5	−1.04100
$609$	6452.68	0.429352
$610$	39584.9	2.62745
$611$	−2170.03	−0.143682
$612$	−3099.50	−0.204722
$613$	1790.10	0.117947	0.0589734	−	0.998260i	$-0.481217\pi$
$613$	1790.10	0.117947	0.0589734	+	0.998260i	$0.481217\pi$
$614$	−13535.1	−0.889632
$615$	22401.9	1.46883
$616$	21935.8	1.43477
$617$	12232.0	0.798122	0.399061	−	0.916924i	$-0.369336\pi$
$617$	12232.0	0.798122	0.399061	+	0.916924i	$0.369336\pi$
$618$	10365.6	0.674700
$619$	−5439.47	−0.353200	−0.176600	−	0.984283i	$-0.556510\pi$
$619$	−5439.47	−0.353200	−0.176600	+	0.984283i	$0.556510\pi$
$620$	10440.9	0.676319
$621$	2057.23	0.132937
$622$	−22441.3	−1.44664
$623$	11301.5	0.726779
$624$	5352.80	0.343403
$625$	42742.9	2.73555
$626$	−213.007	−0.0135998
$627$	14849.6	0.945832
$628$	2780.35	0.176669
$629$	9662.56	0.612514
$630$	19478.3	1.23180
$631$	12962.2	0.817776	0.408888	−	0.912585i	$-0.365917\pi$
$631$	12962.2	0.817776	0.408888	+	0.912585i	$0.365917\pi$
$632$	−17400.1	−1.09516
$633$	14964.5	0.939630
$634$	15465.5	0.968794
$635$	−13094.0	−0.818299
$636$	−8176.05	−0.509751
$637$	−12502.5	−0.777654
$638$	−12623.6	−0.783341
$639$	−336.230	−0.0208154
$640$	−33031.3	−2.04012
$641$	−13358.5	−0.823131	−0.411566	−	0.911380i	$-0.635018\pi$
$641$	−13358.5	−0.823131	−0.411566	+	0.911380i	$0.635018\pi$
$642$	−9119.78	−0.560637
$643$	−23023.9	−1.41209	−0.706046	−	0.708166i	$-0.749523\pi$
$643$	−23023.9	−1.41209	−0.706046	+	0.708166i	$0.749523\pi$
$644$	−8846.74	−0.541320
$645$	14884.5	0.908644
$646$	−29676.1	−1.80742
$647$	29405.1	1.78676	0.893379	−	0.449303i	$-0.148328\pi$
$647$	29405.1	1.78676	0.893379	+	0.449303i	$0.148328\pi$
$648$	−1154.18	−0.0699700
$649$	−3021.81	−0.182768
$650$	−23976.7	−1.44683
$651$	11658.1	0.701870
$652$	−4273.50	−0.256692
$653$	16468.8	0.986944	0.493472	−	0.869762i	$-0.335728\pi$
$653$	16468.8	0.986944	0.493472	+	0.869762i	$0.335728\pi$
$654$	6283.01	0.375666
$655$	−24908.2	−1.48587
$656$	28568.5	1.70033
$657$	−5400.84	−0.320710
$658$	−10070.2	−0.596621
$659$	−377.335	−0.0223048	−0.0111524	−	0.999938i	$-0.503550\pi$
$659$	−377.335	−0.0223048	−0.0111524	+	0.999938i	$0.503550\pi$
$660$	−12408.4	−0.731815
$661$	−15474.3	−0.910559	−0.455280	−	0.890348i	$-0.650461\pi$
$661$	−15474.3	−0.910559	−0.455280	+	0.890348i	$0.650461\pi$
$662$	24280.4	1.42551
$663$	5966.59	0.349507
$664$	8222.89	0.480588
$665$	60728.3	3.54127
$666$	−3359.68	−0.195473
$667$	−5452.43	−0.316520
$668$	−3172.26	−0.183740
$669$	5257.86	0.303857
$670$	−46956.4	−2.70759
$671$	−28156.9	−1.61995
$672$	14561.1	0.835875
$673$	15634.5	0.895490	0.447745	−	0.894161i	$-0.352227\pi$
$673$	15634.5	0.895490	0.447745	+	0.894161i	$0.352227\pi$
$674$	15482.6	0.884818
$675$	8425.29	0.480429
$676$	−6564.37	−0.373485
$677$	−26000.2	−1.47603	−0.738014	−	0.674786i	$-0.764236\pi$
$677$	−26000.2	−1.47603	−0.738014	+	0.674786i	$0.764236\pi$
$678$	3855.47	0.218390
$679$	−9690.42	−0.547694
$680$	−26557.5	−1.49770
$681$	−8949.81	−0.503609
$682$	−22807.1	−1.28054
$683$	−19549.4	−1.09522	−0.547612	−	0.836733i	$-0.684463\pi$
$683$	−19549.4	−1.09522	−0.547612	+	0.836733i	$0.684463\pi$
$684$	3359.98	0.187825
$685$	10187.9	0.568260
$686$	−22509.7	−1.25280
$687$	5017.03	0.278620
$688$	18981.7	1.05185
$689$	15739.0	0.870261
$690$	−16458.9	−0.908087
$691$	−28184.4	−1.55164	−0.775821	−	0.630952i	$-0.782664\pi$
$691$	−28184.4	−1.55164	−0.775821	+	0.630952i	$0.782664\pi$
$692$	−15493.9	−0.851140
$693$	−13855.0	−0.759462
$694$	31282.9	1.71107
$695$	−11554.9	−0.630652
$696$	3059.01	0.166597
$697$	31844.4	1.73055
$698$	41403.2	2.24518
$699$	1524.38	0.0824855
$700$	−36231.4	−1.95631
$701$	2313.19	0.124633	0.0623167	−	0.998056i	$-0.480151\pi$
$701$	2313.19	0.124633	0.0623167	+	0.998056i	$0.480151\pi$
$702$	−2074.59	−0.111539
$703$	−10474.6	−0.561959
$704$	4284.90	0.229394
$705$	−6100.69	−0.325908
$706$	−19077.3	−1.01697
$707$	−19770.8	−1.05171
$708$	−683.737	−0.0362944
$709$	−16383.6	−0.867841	−0.433920	−	0.900951i	$-0.642870\pi$
$709$	−16383.6	−0.867841	−0.433920	+	0.900951i	$0.642870\pi$
$710$	2690.02	0.142189
$711$	10990.2	0.579698
$712$	5357.67	0.282004
$713$	−9850.95	−0.517421
$714$	27688.4	1.45128
$715$	23886.5	1.24937
$716$	−2766.58	−0.144402
$717$	−7491.25	−0.390190
$718$	−34842.1	−1.81100
$719$	−30186.8	−1.56575	−0.782876	−	0.622177i	$-0.786248\pi$
$719$	−30186.8	−1.56575	−0.782876	+	0.622177i	$0.786248\pi$
$720$	15048.6	0.778926
$721$	−30152.6	−1.55748
$722$	8545.96	0.440510
$723$	−13428.5	−0.690746
$724$	6900.88	0.354239
$725$	−22330.2	−1.14389
$726$	13352.0	0.682559
$727$	16487.2	0.841095	0.420547	−	0.907271i	$-0.361838\pi$
$727$	16487.2	0.841095	0.420547	+	0.907271i	$0.361838\pi$
$728$	−9554.53	−0.486421
$729$	729.000	0.0370370
$730$	43209.5	2.19076
$731$	21158.3	1.07054
$732$	−6370.98	−0.321691
$733$	−13859.4	−0.698372	−0.349186	−	0.937053i	$-0.613542\pi$
$733$	−13859.4	−0.698372	−0.349186	+	0.937053i	$0.613542\pi$
$734$	−18339.2	−0.922226
$735$	−35148.7	−1.76392
$736$	−12304.0	−0.616210
$737$	33400.3	1.66935
$738$	−11072.3	−0.552273
$739$	36238.9	1.80388	0.901942	−	0.431856i	$-0.142141\pi$
$739$	36238.9	1.80388	0.901942	+	0.431856i	$0.142141\pi$
$740$	8752.65	0.434803
$741$	−6468.02	−0.320660
$742$	73038.2	3.61364
$743$	11663.4	0.575893	0.287946	−	0.957647i	$-0.407028\pi$
$743$	11663.4	0.575893	0.287946	+	0.957647i	$0.407028\pi$
$744$	5526.75	0.272339
$745$	73363.1	3.60780
$746$	−24446.3	−1.19979
$747$	−5193.71	−0.254388
$748$	−17638.6	−0.862207
$749$	26528.6	1.29417
$750$	−40405.0	−1.96717
$751$	4619.42	0.224454	0.112227	−	0.993683i	$-0.464202\pi$
$751$	4619.42	0.224454	0.112227	+	0.993683i	$0.464202\pi$
$752$	−7780.03	−0.377272
$753$	−16572.1	−0.802021
$754$	5498.43	0.265571
$755$	−66386.2	−3.20005
$756$	−3134.93	−0.150815
$757$	−17625.2	−0.846233	−0.423116	−	0.906075i	$-0.639064\pi$
$757$	−17625.2	−0.846233	−0.423116	+	0.906075i	$0.639064\pi$
$758$	−28820.2	−1.38100
$759$	11707.3	0.559878
$760$	28789.4	1.37408
$761$	−3247.66	−0.154701	−0.0773506	−	0.997004i	$-0.524646\pi$
$761$	−3247.66	−0.154701	−0.0773506	+	0.997004i	$0.524646\pi$
$762$	6471.80	0.307675
$763$	−18276.7	−0.867185
$764$	3296.16	0.156087
$765$	16774.1	0.792771
$766$	−36707.1	−1.73144
$767$	1316.21	0.0619628
$768$	14318.0	0.672732
$769$	−3993.05	−0.187247	−0.0936236	−	0.995608i	$-0.529845\pi$
$769$	−3993.05	−0.187247	−0.0936236	+	0.995608i	$0.529845\pi$
$770$	110847.	5.18785
$771$	23129.9	1.08042
$772$	−13438.5	−0.626505
$773$	9486.90	0.441423	0.220712	−	0.975339i	$-0.429162\pi$
$773$	9486.90	0.441423	0.220712	+	0.975339i	$0.429162\pi$
$774$	−7356.75	−0.341645
$775$	−40344.1	−1.86994
$776$	−4593.93	−0.212516
$777$	9773.01	0.451229
$778$	4410.57	0.203247
$779$	−34520.6	−1.58771
$780$	5404.72	0.248103
$781$	−1913.42	−0.0876664
$782$	−23396.3	−1.06989
$783$	−1932.12	−0.0881844
$784$	−44824.1	−2.04191
$785$	−15047.0	−0.684139
$786$	12311.0	0.558677
$787$	9310.51	0.421708	0.210854	−	0.977518i	$-0.432376\pi$
$787$	9310.51	0.421708	0.210854	+	0.977518i	$0.432376\pi$
$788$	−1302.96	−0.0589036
$789$	−7858.72	−0.354598
$790$	−87927.3	−3.95989
$791$	−11215.2	−0.504131
$792$	−6568.22	−0.294686
$793$	12264.2	0.549200
$794$	2862.53	0.127944
$795$	44247.9	1.97397
$796$	−10096.3	−0.449565
$797$	1402.21	0.0623196	0.0311598	−	0.999514i	$-0.490080\pi$
$797$	1402.21	0.0623196	0.0311598	+	0.999514i	$0.490080\pi$
$798$	−30015.4	−1.33149
$799$	−8672.14	−0.383978
$800$	−50390.3	−2.22696
$801$	−3383.99	−0.149273
$802$	16233.3	0.714735
$803$	−30735.1	−1.35071
$804$	7557.39	0.331503
$805$	47877.5	2.09623
$806$	9934.05	0.434134
$807$	9227.45	0.402505
$808$	−9372.71	−0.408082
$809$	33015.3	1.43480	0.717402	−	0.696659i	$-0.245331\pi$
$809$	33015.3	1.43480	0.717402	+	0.696659i	$0.245331\pi$
$810$	−5832.37	−0.252998
$811$	35812.1	1.55060	0.775298	−	0.631596i	$-0.217600\pi$
$811$	35812.1	1.55060	0.775298	+	0.631596i	$0.217600\pi$
$812$	8308.73	0.359088
$813$	−12084.5	−0.521304
$814$	−19119.2	−0.823254
$815$	23127.7	0.994023
$816$	21391.5	0.917713
$817$	−22936.4	−0.982184
$818$	36854.8	1.57530
$819$	6034.79	0.257476
$820$	28845.6	1.22846
$821$	−1940.92	−0.0825076	−0.0412538	−	0.999149i	$-0.513135\pi$
$821$	−1940.92	−0.0825076	−0.0412538	+	0.999149i	$0.513135\pi$
$822$	−5035.42	−0.213662
$823$	−1861.48	−0.0788424	−0.0394212	−	0.999223i	$-0.512551\pi$
$823$	−1861.48	−0.0788424	−0.0394212	+	0.999223i	$0.512551\pi$
$824$	−14294.4	−0.604332
$825$	47946.6	2.02338
$826$	6107.95	0.257292
$827$	−7336.67	−0.308490	−0.154245	−	0.988033i	$-0.549294\pi$
$827$	−7336.67	−0.308490	−0.154245	+	0.988033i	$0.549294\pi$
$828$	2648.97	0.111181
$829$	30106.6	1.26133	0.630667	−	0.776053i	$-0.282781\pi$
$829$	30106.6	1.26133	0.630667	+	0.776053i	$0.282781\pi$
$830$	41552.3	1.73771
$831$	−11717.2	−0.489130
$832$	−1866.37	−0.0777701
$833$	−49963.9	−2.07821
$834$	5711.09	0.237121
$835$	17167.9	0.711520
$836$	19121.0	0.791044
$837$	−3490.78	−0.144157
$838$	13761.8	0.567297
$839$	−27636.8	−1.13722	−0.568611	−	0.822607i	$-0.692519\pi$
$839$	−27636.8	−1.13722	−0.568611	+	0.822607i	$0.692519\pi$
$840$	−26861.1	−1.10333
$841$	−19268.2	−0.790035
$842$	18890.5	0.773170
$843$	−10506.0	−0.429237
$844$	19268.9	0.785857
$845$	35525.6	1.44629
$846$	3015.31	0.122540
$847$	−38839.7	−1.57562
$848$	56428.0	2.28508
$849$	−19632.3	−0.793615
$850$	−95818.7	−3.86653
$851$	−8258.07	−0.332647
$852$	−432.944	−0.0174089
$853$	−23055.8	−0.925457	−0.462728	−	0.886500i	$-0.653129\pi$
$853$	−23055.8	−0.925457	−0.462728	+	0.886500i	$0.653129\pi$
$854$	56913.1	2.28048
$855$	−18183.8	−0.727338
$856$	12576.4	0.502165
$857$	−44339.3	−1.76733	−0.883665	−	0.468119i	$-0.844932\pi$
$857$	−44339.3	−1.76733	−0.883665	+	0.468119i	$0.844932\pi$
$858$	−11806.0	−0.469757
$859$	−5864.49	−0.232938	−0.116469	−	0.993194i	$-0.537158\pi$
$859$	−5864.49	−0.232938	−0.116469	+	0.993194i	$0.537158\pi$
$860$	19165.8	0.759942
$861$	32208.4	1.27487
$862$	−31398.7	−1.24065
$863$	8819.67	0.347886	0.173943	−	0.984756i	$-0.444349\pi$
$863$	8819.67	0.347886	0.173943	+	0.984756i	$0.444349\pi$
$864$	−4360.03	−0.171680
$865$	83851.1	3.29598
$866$	−45334.3	−1.77890
$867$	9105.43	0.356674
$868$	15011.5	0.587007
$869$	62543.0	2.44146
$870$	15458.0	0.602384
$871$	−14548.1	−0.565951
$872$	−8664.44	−0.336485
$873$	2901.60	0.112491
$874$	25362.6	0.981582
$875$	117535.	4.54102
$876$	−6954.34	−0.268225
$877$	−16591.0	−0.638814	−0.319407	−	0.947618i	$-0.603484\pi$
$877$	−16591.0	−0.638814	−0.319407	+	0.947618i	$0.603484\pi$
$878$	−7851.47	−0.301793
$879$	18783.9	0.720781
$880$	85638.3	3.28053
$881$	−28452.1	−1.08806	−0.544028	−	0.839067i	$-0.683101\pi$
$881$	−28452.1	−1.08806	−0.544028	+	0.839067i	$0.683101\pi$
$882$	17372.5	0.663222
$883$	−6177.75	−0.235445	−0.117722	−	0.993047i	$-0.537559\pi$
$883$	−6177.75	−0.235445	−0.117722	+	0.993047i	$0.537559\pi$
$884$	7682.82	0.292309
$885$	3700.31	0.140547
$886$	45444.3	1.72317
$887$	−13155.6	−0.497996	−0.248998	−	0.968504i	$-0.580101\pi$
$887$	−13155.6	−0.497996	−0.248998	+	0.968504i	$0.580101\pi$
$888$	4633.08	0.175086
$889$	−18825.9	−0.710237
$890$	27073.6	1.01967
$891$	4148.59	0.155985
$892$	6770.23	0.254130
$893$	9400.95	0.352285
$894$	−36260.2	−1.35651
$895$	14972.4	0.559187
$896$	−47490.7	−1.77071
$897$	−5099.32	−0.189812
$898$	11324.6	0.420832
$899$	9251.87	0.343234
$900$	10848.8	0.401806
$901$	62898.4	2.32569
$902$	−63010.2	−2.32595
$903$	21400.1	0.788652
$904$	−5316.79	−0.195613
$905$	−37346.8	−1.37177
$906$	32811.8	1.20320
$907$	11767.5	0.430796	0.215398	−	0.976526i	$-0.430895\pi$
$907$	11767.5	0.430796	0.215398	+	0.976526i	$0.430895\pi$
$908$	−11524.1	−0.421192
$909$	5919.95	0.216009
$910$	−48281.4	−1.75881
$911$	8944.57	0.325298	0.162649	−	0.986684i	$-0.447996\pi$
$911$	8944.57	0.325298	0.162649	+	0.986684i	$0.447996\pi$
$912$	−23189.3	−0.841968
$913$	−29556.3	−1.07138
$914$	−49861.4	−1.80445
$915$	34479.0	1.24573
$916$	6460.14	0.233023
$917$	−35811.8	−1.28965
$918$	−8290.72	−0.298077
$919$	38367.6	1.37718	0.688591	−	0.725150i	$-0.258230\pi$
$919$	38367.6	1.37718	0.688591	+	0.725150i	$0.258230\pi$
$920$	22697.3	0.813376
$921$	−11789.3	−0.421793
$922$	−16414.0	−0.586298
$923$	833.424	0.0297210
$924$	−17840.2	−0.635174
$925$	−33820.5	−1.20218
$926$	21651.7	0.768378
$927$	9028.57	0.319889
$928$	11555.7	0.408766
$929$	−47966.0	−1.69399	−0.846993	−	0.531604i	$-0.821590\pi$
$929$	−47966.0	−1.69399	−0.846993	+	0.531604i	$0.821590\pi$
$930$	27928.0	0.984727
$931$	54162.9	1.90668
$932$	1962.85	0.0689865
$933$	−19546.7	−0.685883
$934$	19369.7	0.678583
$935$	95458.1	3.33884
$936$	2860.91	0.0999057
$937$	72.7929	0.00253793	0.00126896	−	0.999999i	$-0.499596\pi$
$937$	72.7929	0.00253793	0.00126896	+	0.999999i	$0.499596\pi$
$938$	−67511.6	−2.35003
$939$	−185.532	−0.00644793
$940$	−7855.50	−0.272573
$941$	−1145.95	−0.0396992	−0.0198496	−	0.999803i	$-0.506319\pi$
$941$	−1145.95	−0.0396992	−0.0198496	+	0.999803i	$0.506319\pi$
$942$	7437.06	0.257232
$943$	−27215.7	−0.939835
$944$	4718.89	0.162698
$945$	16965.9	0.584021
$946$	−41865.7	−1.43887
$947$	−52363.7	−1.79682	−0.898412	−	0.439153i	$-0.855279\pi$
$947$	−52363.7	−1.79682	−0.898412	+	0.439153i	$0.855279\pi$
$948$	14151.4	0.484828
$949$	13387.2	0.457922
$950$	103871.	3.54740
$951$	13470.7	0.459325
$952$	−38183.0	−1.29992
$953$	−18197.6	−0.618549	−0.309274	−	0.950973i	$-0.600086\pi$
$953$	−18197.6	−0.618549	−0.309274	+	0.950973i	$0.600086\pi$
$954$	−21869.8	−0.742202
$955$	−17838.4	−0.604437
$956$	−9646.04	−0.326334
$957$	−10995.3	−0.371398
$958$	36784.7	1.24056
$959$	14647.6	0.493217
$960$	−5247.00	−0.176402
$961$	−13075.6	−0.438910
$962$	8327.73	0.279103
$963$	−7943.46	−0.265809
$964$	−17291.0	−0.577704
$965$	72727.5	2.42610
$966$	−23663.8	−0.788168
$967$	−15190.1	−0.505149	−0.252575	−	0.967577i	$-0.581277\pi$
$967$	−15190.1	−0.505149	−0.252575	+	0.967577i	$0.581277\pi$
$968$	−18412.7	−0.611371
$969$	−25848.3	−0.856933
$970$	−23214.3	−0.768418
$971$	−33558.7	−1.10911	−0.554557	−	0.832146i	$-0.687112\pi$
$971$	−33558.7	−1.10911	−0.554557	+	0.832146i	$0.687112\pi$
$972$	938.690	0.0309758
$973$	−16613.1	−0.547370
$974$	27800.2	0.914556
$975$	−20884.0	−0.685974
$976$	43970.0	1.44206
$977$	13843.4	0.453317	0.226658	−	0.973974i	$-0.427220\pi$
$977$	13843.4	0.453317	0.226658	+	0.973974i	$0.427220\pi$
$978$	−11431.0	−0.373746
$979$	−19257.6	−0.628677
$980$	−45258.9	−1.47525
$981$	5472.60	0.178111
$982$	−39593.5	−1.28664
$983$	−10695.3	−0.347027	−0.173514	−	0.984831i	$-0.555512\pi$
$983$	−10695.3	−0.347027	−0.173514	+	0.984831i	$0.555512\pi$
$984$	15269.0	0.494673
$985$	7051.47	0.228100
$986$	21973.5	0.709715
$987$	−8771.27	−0.282870
$988$	−8328.49	−0.268183
$989$	−18082.8	−0.581396
$990$	−33190.8	−1.06553
$991$	14749.0	0.472773	0.236386	−	0.971659i	$-0.424037\pi$
$991$	14749.0	0.472773	0.236386	+	0.971659i	$0.424037\pi$
$992$	20877.8	0.668216
$993$	21148.6	0.675862
$994$	3867.57	0.123412
$995$	54640.0	1.74091
$996$	−6687.63	−0.212757
$997$	30262.1	0.961294	0.480647	−	0.876914i	$-0.340402\pi$
$997$	30262.1	0.961294	0.480647	+	0.876914i	$0.340402\pi$
$998$	−1172.11	−0.0371767
$999$	−2926.33	−0.0926776

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.a.1.7	✓	7
3.2	odd	2		531.4.a.d.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.a.1.7	✓	7	1.1	even	1	trivial
531.4.a.d.1.1		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)

\(f(q)\)	\(=\)	\(q+3.44426 q^{2} +3.00000 q^{3} +3.86292 q^{4} -20.9057 q^{5} +10.3328 q^{6} -30.0572 q^{7} -14.2492 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+3.44426 q^{2} +3.00000 q^{3} +3.86292 q^{4} -20.9057 q^{5} +10.3328 q^{6} -30.0572 q^{7} -14.2492 q^{8} +9.00000 q^{9} -72.0046 q^{10} +51.2171 q^{11} +11.5888 q^{12} -22.3086 q^{13} -103.525 q^{14} -62.7171 q^{15} -79.9812 q^{16} -89.1524 q^{17} +30.9983 q^{18} +96.6448 q^{19} -80.7571 q^{20} -90.1715 q^{21} +176.405 q^{22} +76.1937 q^{23} -42.7475 q^{24} +312.048 q^{25} -76.8365 q^{26} +27.0000 q^{27} -116.108 q^{28} -71.5601 q^{29} -216.014 q^{30} -129.288 q^{31} -161.483 q^{32} +153.651 q^{33} -307.064 q^{34} +628.366 q^{35} +34.7663 q^{36} -108.382 q^{37} +332.870 q^{38} -66.9257 q^{39} +297.889 q^{40} -357.190 q^{41} -310.574 q^{42} -237.327 q^{43} +197.848 q^{44} -188.151 q^{45} +262.431 q^{46} +97.2732 q^{47} -239.944 q^{48} +560.433 q^{49} +1074.77 q^{50} -267.457 q^{51} -86.1763 q^{52} -705.515 q^{53} +92.9950 q^{54} -1070.73 q^{55} +428.290 q^{56} +289.934 q^{57} -246.471 q^{58} -59.0000 q^{59} -242.271 q^{60} -549.755 q^{61} -445.302 q^{62} -270.514 q^{63} +83.6615 q^{64} +466.376 q^{65} +529.215 q^{66} +652.131 q^{67} -344.389 q^{68} +228.581 q^{69} +2164.25 q^{70} -37.3589 q^{71} -128.243 q^{72} -600.093 q^{73} -373.297 q^{74} +936.144 q^{75} +373.331 q^{76} -1539.44 q^{77} -230.510 q^{78} +1221.13 q^{79} +1672.06 q^{80} +81.0000 q^{81} -1230.26 q^{82} -577.079 q^{83} -348.325 q^{84} +1863.79 q^{85} -817.417 q^{86} -214.680 q^{87} -729.802 q^{88} -375.999 q^{89} -648.042 q^{90} +670.532 q^{91} +294.330 q^{92} -387.864 q^{93} +335.034 q^{94} -2020.43 q^{95} -484.448 q^{96} +322.400 q^{97} +1930.28 q^{98} +460.954 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9}+O(q^{10}) \) 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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