LMFDB - Embedded newform 117.4.g.d.100.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(55,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("117.55");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.g (of order $3$, degree $2$, minimal)

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:	no
Analytic conductor:	$6.90322347067$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 5x^{2} + 4x + 16 $ x^4 - x^3 + 5x^2 + 4x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			100.2
Root			$-0.780776 - 1.35234i$ of defining polynomial
Character	$\chi$	$=$	117.100
Dual form			117.4.g.d.55.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.219224 + 0.379706i) q^{2} +(3.90388 + 6.76172i) q^{4} +17.8078 q^{5} +(-2.71922 - 4.70983i) q^{7} -6.93087 q^{8} +O(q^{10})$	\(q+(-0.219224 + 0.379706i) q^{2} +(3.90388 + 6.76172i) q^{4} +17.8078 q^{5} +(-2.71922 - 4.70983i) q^{7} -6.93087 q^{8} +(-3.90388 + 6.76172i) q^{10} +(-11.2116 + 19.4191i) q^{11} +(21.9730 - 41.4027i) q^{13} +2.38447 q^{14} +(-29.7116 + 51.4621i) q^{16} +(33.9924 + 58.8766i) q^{17} +(40.4039 + 69.9816i) q^{19} +(69.5194 + 120.411i) q^{20} +(-4.91571 - 8.51427i) q^{22} +(70.2656 - 121.704i) q^{23} +192.116 q^{25} +(10.9039 + 17.4198i) q^{26} +(21.2311 - 36.7733i) q^{28} +(-53.3466 + 92.3990i) q^{29} -276.155 q^{31} +(-40.7505 - 70.5819i) q^{32} -29.8078 q^{34} +(-48.4233 - 83.8716i) q^{35} +(2.14584 - 3.71670i) q^{37} -35.4299 q^{38} -123.423 q^{40} +(113.884 - 197.254i) q^{41} +(-13.7647 - 23.8411i) q^{43} -175.076 q^{44} +(30.8078 + 53.3606i) q^{46} -318.617 q^{47} +(156.712 - 271.433i) q^{49} +(-42.1165 + 72.9479i) q^{50} +(365.734 - 13.0560i) q^{52} +67.6562 q^{53} +(-199.654 + 345.811i) q^{55} +(18.8466 + 32.6432i) q^{56} +(-23.3897 - 40.5121i) q^{58} +(-145.557 - 252.113i) q^{59} +(-331.655 - 574.444i) q^{61} +(60.5398 - 104.858i) q^{62} -439.652 q^{64} +(391.290 - 737.290i) q^{65} +(212.551 - 368.149i) q^{67} +(-265.405 + 459.695i) q^{68} +42.4621 q^{70} +(-76.4815 - 132.470i) q^{71} +117.268 q^{73} +(0.940837 + 1.62958i) q^{74} +(-315.464 + 546.400i) q^{76} +121.948 q^{77} +202.462 q^{79} +(-529.098 + 916.425i) q^{80} +(49.9323 + 86.4853i) q^{82} -336.155 q^{83} +(605.329 + 1048.46i) q^{85} +12.0702 q^{86} +(77.7065 - 134.592i) q^{88} +(359.097 - 621.974i) q^{89} +(-254.750 + 9.09407i) q^{91} +1097.23 q^{92} +(69.8485 - 120.981i) q^{94} +(719.503 + 1246.22i) q^{95} +(-379.684 - 657.632i) q^{97} +(68.7098 + 119.009i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 5 q^{2} - 5 q^{4} + 30 q^{5} - 15 q^{7} + 30 q^{8}+O(q^{10}) $ 4 * q - 5 * q^2 - 5 * q^4 + 30 * q^5 - 15 * q^7 + 30 * q^8	$ 4 q - 5 q^{2} - 5 q^{4} + 30 q^{5} - 15 q^{7} + 30 q^{8} + 5 q^{10} + 17 q^{11} + 125 q^{13} + 92 q^{14} - 57 q^{16} + 70 q^{17} + 141 q^{19} + 175 q^{20} + 170 q^{22} + 145 q^{23} + 150 q^{25} + 23 q^{26} - 80 q^{28} + 34 q^{29} - 280 q^{31} + 105 q^{32} - 78 q^{34} - 70 q^{35} + 190 q^{37} - 620 q^{38} - 370 q^{40} + 538 q^{41} - 455 q^{43} - 1360 q^{44} + 82 q^{46} - 120 q^{47} + 565 q^{49} + 450 q^{50} - 310 q^{52} - 1090 q^{53} - 510 q^{55} - 172 q^{56} + 595 q^{58} - 809 q^{59} - 502 q^{61} - 500 q^{62} - 2542 q^{64} + 555 q^{65} + 475 q^{67} - 505 q^{68} - 160 q^{70} + 127 q^{71} + 1170 q^{73} + 849 q^{74} + 140 q^{76} - 510 q^{77} + 480 q^{79} - 1065 q^{80} + 1515 q^{82} - 520 q^{83} + 1205 q^{85} + 3924 q^{86} + 1020 q^{88} + 921 q^{89} - 1287 q^{91} + 2080 q^{92} - 1040 q^{94} + 1270 q^{95} + 415 q^{97} + 1285 q^{98}+O(q^{100}) $ 4 * q - 5 * q^2 - 5 * q^4 + 30 * q^5 - 15 * q^7 + 30 * q^8 + 5 * q^10 + 17 * q^11 + 125 * q^13 + 92 * q^14 - 57 * q^16 + 70 * q^17 + 141 * q^19 + 175 * q^20 + 170 * q^22 + 145 * q^23 + 150 * q^25 + 23 * q^26 - 80 * q^28 + 34 * q^29 - 280 * q^31 + 105 * q^32 - 78 * q^34 - 70 * q^35 + 190 * q^37 - 620 * q^38 - 370 * q^40 + 538 * q^41 - 455 * q^43 - 1360 * q^44 + 82 * q^46 - 120 * q^47 + 565 * q^49 + 450 * q^50 - 310 * q^52 - 1090 * q^53 - 510 * q^55 - 172 * q^56 + 595 * q^58 - 809 * q^59 - 502 * q^61 - 500 * q^62 - 2542 * q^64 + 555 * q^65 + 475 * q^67 - 505 * q^68 - 160 * q^70 + 127 * q^71 + 1170 * q^73 + 849 * q^74 + 140 * q^76 - 510 * q^77 + 480 * q^79 - 1065 * q^80 + 1515 * q^82 - 520 * q^83 + 1205 * q^85 + 3924 * q^86 + 1020 * q^88 + 921 * q^89 - 1287 * q^91 + 2080 * q^92 - 1040 * q^94 + 1270 * q^95 + 415 * q^97 + 1285 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	−0.219224	+	0.379706i	−0.0775072	+	0.134246i	−0.902174	−	0.431373i	$-0.858029\pi$
$2$	−0.219224	+	0.379706i	−0.0775072	+	0.134246i	0.824667	+	0.565619i	$0.191363\pi$
$3$		0			0
$3$		0			0
$4$	3.90388	+	6.76172i	0.487985	+	0.845215i
$5$	17.8078			1.59277			0.796387	−	0.604787i	$-0.206742\pi$
$5$	17.8078			1.59277			0.796387	+	0.604787i	$0.206742\pi$
$6$		0			0
$7$	−2.71922	−	4.70983i	−0.146824	−	0.254307i	0.783228	−	0.621735i	$-0.213572\pi$
$7$	−2.71922	−	4.70983i	−0.146824	−	0.254307i	−0.930052	+	0.367428i	$0.880239\pi$
$8$	−6.93087			−0.306304
$9$		0			0
$10$	−3.90388	+	6.76172i	−0.123452	+	0.213824i
$11$	−11.2116	+	19.4191i	−0.307313	+	0.532281i	−0.977774	−	0.209664i	$-0.932763\pi$
$11$	−11.2116	+	19.4191i	−0.307313	+	0.532281i	0.670461	+	0.741945i	$0.266096\pi$
$12$		0			0
$13$	21.9730	−	41.4027i	0.468786	−	0.883312i
$13$	21.9730	−	41.4027i	0.468786	−	0.883312i
$14$	2.38447			0.0455198
$15$		0			0
$16$	−29.7116	+	51.4621i	−0.464244	+	0.804095i
$17$	33.9924	+	58.8766i	0.484963	+	0.839981i	0.999851	−	0.0172769i	$-0.00549968\pi$
$17$	33.9924	+	58.8766i	0.484963	+	0.839981i	−0.514888	+	0.857258i	$0.672166\pi$
$18$		0			0
$19$	40.4039	+	69.9816i	0.487857	+	0.844993i	0.999902	−	0.0139650i	$-0.00444535\pi$
$19$	40.4039	+	69.9816i	0.487857	+	0.844993i	−0.512045	+	0.858958i	$0.671112\pi$
$20$	69.5194	+	120.411i	0.777251	+	1.34624i
$21$		0			0
$22$	−4.91571	−	8.51427i	−0.0476379	−	0.0825113i
$23$	70.2656	−	121.704i	0.637017	−	1.10335i	−0.349067	−	0.937098i	$-0.613501\pi$
$23$	70.2656	−	121.704i	0.637017	−	1.10335i	0.986084	−	0.166248i	$-0.0531653\pi$
$24$		0			0
$25$	192.116			1.53693
$26$	10.9039	+	17.4198i	0.0822472	+	0.131396i
$27$		0			0
$28$	21.2311	−	36.7733i	0.143296	−	0.248196i
$29$	−53.3466	+	92.3990i	−0.341594	+	0.591657i	−0.984729	−	0.174095i	$-0.944300\pi$
$29$	−53.3466	+	92.3990i	−0.341594	+	0.591657i	0.643135	+	0.765753i	$0.277633\pi$
$30$		0			0
$31$	−276.155			−1.59997			−0.799983	−	0.600023i	$-0.795158\pi$
$31$	−276.155			−1.59997			−0.799983	+	0.600023i	$0.795158\pi$
$32$	−40.7505	−	70.5819i	−0.225117	−	0.389913i
$33$		0			0
$34$	−29.8078			−0.150353
$35$	−48.4233	−	83.8716i	−0.233858	−	0.405054i
$36$		0			0
$37$	2.14584	−	3.71670i	0.00953442	−	0.0165141i	−0.861219	−	0.508234i	$-0.830298\pi$
$37$	2.14584	−	3.71670i	0.00953442	−	0.0165141i	0.870753	+	0.491720i	$0.163632\pi$
$38$	−35.4299			−0.151250
$39$		0			0
$40$	−123.423			−0.487873
$41$	113.884	−	197.254i	0.433799	−	0.751362i	−0.563398	−	0.826186i	$-0.690506\pi$
$41$	113.884	−	197.254i	0.433799	−	0.751362i	0.997197	+	0.0748237i	$0.0238394\pi$
$42$		0			0
$43$	−13.7647	−	23.8411i	−0.0488162	−	0.0845521i	0.840585	−	0.541680i	$-0.182212\pi$
$43$	−13.7647	−	23.8411i	−0.0488162	−	0.0845521i	−0.889401	+	0.457128i	$0.848878\pi$
$44$	−175.076			−0.599856
$45$		0			0
$46$	30.8078	+	53.3606i	0.0987469	+	0.171035i
$47$	−318.617			−0.988832			−0.494416	−	0.869225i	$-0.664618\pi$
$47$	−318.617			−0.988832			−0.494416	+	0.869225i	$0.664618\pi$
$48$		0			0
$49$	156.712	−	271.433i	0.456885	−	0.791348i
$50$	−42.1165	+	72.9479i	−0.119123	+	0.206328i
$51$		0			0
$52$	365.734	−	13.0560i	0.975349	−	0.0348181i
$53$	67.6562			0.175345			0.0876726	−	0.996149i	$-0.472057\pi$
$53$	67.6562			0.175345			0.0876726	+	0.996149i	$0.472057\pi$
$54$		0			0
$55$	−199.654	+	345.811i	−0.489480	+	0.847804i
$56$	18.8466	+	32.6432i	0.0449729	+	0.0778953i
$57$		0			0
$58$	−23.3897	−	40.5121i	−0.0529519	−	0.0917155i
$59$	−145.557	−	252.113i	−0.321186	−	0.556310i	0.659547	−	0.751663i	$-0.270748\pi$
$59$	−145.557	−	252.113i	−0.321186	−	0.556310i	−0.980733	+	0.195353i	$0.937415\pi$
$60$		0			0
$61$	−331.655	−	574.444i	−0.696133	−	1.20574i	−0.969797	−	0.243912i	$-0.921569\pi$
$61$	−331.655	−	574.444i	−0.696133	−	1.20574i	0.273664	−	0.961825i	$-0.411764\pi$
$62$	60.5398	−	104.858i	0.124009	−	0.214790i
$63$		0			0
$64$	−439.652			−0.858696
$65$	391.290	−	737.290i	0.746670	−	1.40692i
$66$		0			0
$67$	212.551	−	368.149i	0.387570	−	0.671291i	−0.604552	−	0.796566i	$-0.706648\pi$
$67$	212.551	−	368.149i	0.387570	−	0.671291i	0.992122	+	0.125275i	$0.0399812\pi$
$68$	−265.405	+	459.695i	−0.473310	+	0.819796i
$69$		0			0
$70$	42.4621			0.0725028
$71$	−76.4815	−	132.470i	−0.127841	−	0.221427i	0.794999	−	0.606611i	$-0.207471\pi$
$71$	−76.4815	−	132.470i	−0.127841	−	0.221427i	−0.922840	+	0.385184i	$0.874138\pi$
$72$		0			0
$73$	117.268			0.188016			0.0940081	−	0.995571i	$-0.470032\pi$
$73$	117.268			0.188016			0.0940081	+	0.995571i	$0.470032\pi$
$74$	0.940837	+	1.62958i	0.00147797	+	0.00255993i
$75$		0			0
$76$	−315.464	+	546.400i	−0.476134	+	0.824689i
$77$	121.948			0.180484
$78$		0			0
$79$	202.462			0.288339			0.144169	−	0.989553i	$-0.453949\pi$
$79$	202.462			0.288339			0.144169	+	0.989553i	$0.453949\pi$
$80$	−529.098	+	916.425i	−0.739437	+	1.28074i
$81$		0			0
$82$	49.9323	+	86.4853i	0.0672452	+	0.116472i
$83$	−336.155			−0.444552			−0.222276	−	0.974984i	$-0.571349\pi$
$83$	−336.155			−0.444552			−0.222276	+	0.974984i	$0.571349\pi$
$84$		0			0
$85$	605.329	+	1048.46i	0.772437	+	1.33790i
$86$	12.0702			0.0151344
$87$		0			0
$88$	77.7065	−	134.592i	0.0941311	−	0.163040i
$89$	359.097	−	621.974i	0.427688	−	0.740777i	−0.568979	−	0.822352i	$-0.692662\pi$
$89$	359.097	−	621.974i	0.427688	−	0.740777i	0.996667	+	0.0815748i	$0.0259949\pi$
$90$		0			0
$91$	−254.750	+	9.09407i	−0.293462	+	0.0104760i
$92$	1097.23			1.24342
$93$		0			0
$94$	69.8485	−	120.981i	0.0766417	−	0.132747i
$95$	719.503	+	1246.22i	0.777047	+	1.34588i
$96$		0			0
$97$	−379.684	−	657.632i	−0.397434	−	0.688376i	0.595975	−	0.803003i	$-0.296766\pi$
$97$	−379.684	−	657.632i	−0.397434	−	0.688376i	−0.993409	+	0.114628i	$0.963433\pi$
$98$	68.7098	+	119.009i	0.0708238	+	0.122670i
$99$		0			0
$100$	750.000	+	1299.04i	0.750000	+	1.29904i
$101$	−174.348	+	301.980i	−0.171766	+	0.297507i	−0.939037	−	0.343816i	$-0.888280\pi$
$101$	−174.348	+	301.980i	−0.171766	+	0.297507i	0.767272	+	0.641322i	$0.221614\pi$
$102$		0			0
$103$	−580.303			−0.555136			−0.277568	−	0.960706i	$-0.589528\pi$
$103$	−580.303			−0.555136			−0.277568	+	0.960706i	$0.589528\pi$
$104$	−152.292	+	286.957i	−0.143591	+	0.270562i
$105$		0			0
$106$	−14.8318	+	25.6895i	−0.0135905	+	0.0235395i
$107$	285.747	−	494.928i	0.258170	−	0.447163i	−0.707582	−	0.706631i	$-0.750214\pi$
$107$	285.747	−	494.928i	0.258170	−	0.447163i	0.965752	+	0.259468i	$0.0835473\pi$
$108$		0			0
$109$	176.004			0.154661			0.0773307	−	0.997005i	$-0.475360\pi$
$109$	176.004			0.154661			0.0773307	+	0.997005i	$0.475360\pi$
$110$	−87.5379	−	151.620i	−0.0758765	−	0.131422i
$111$		0			0
$112$	323.170			0.272649
$113$	632.441	+	1095.42i	0.526505	+	0.911933i	0.999523	+	0.0308807i	$0.00983120\pi$
$113$	632.441	+	1095.42i	0.526505	+	0.911933i	−0.473018	+	0.881053i	$0.656835\pi$
$114$		0			0
$115$	1251.27	−	2167.27i	1.01462	−	1.75738i
$116$	−833.035			−0.666770
$117$		0			0
$118$	127.638			0.0995768
$119$	184.866	−	320.197i	0.142409	−	0.246659i
$120$		0			0
$121$	414.098	+	717.239i	0.311118	+	0.538872i
$122$	290.827			0.215821
$123$		0			0
$124$	−1078.08	−	1867.29i	−0.780760	−	1.35232i
$125$	1195.19			0.855211
$126$		0			0
$127$	−1302.05	+	2255.22i	−0.909752	+	1.57574i	−0.0953448	+	0.995444i	$0.530395\pi$
$127$	−1302.05	+	2255.22i	−0.909752	+	1.57574i	−0.814408	+	0.580293i	$0.802938\pi$
$128$	422.386	−	731.594i	0.291672	−	0.505190i
$129$		0			0
$130$	194.174	+	310.207i	0.131001	+	0.209284i
$131$	−2131.70			−1.42174			−0.710870	−	0.703324i	$-0.751698\pi$
$131$	−2131.70			−1.42174			−0.710870	+	0.703324i	$0.751698\pi$
$132$		0			0
$133$	219.734	−	380.591i	0.143259	−	0.248131i
$134$	93.1922	+	161.414i	0.0600790	+	0.104060i
$135$		0			0
$136$	−235.597	−	408.066i	−0.148546	−	0.257290i
$137$	343.992	+	595.812i	0.214520	+	0.371560i	0.953124	−	0.302580i	$-0.0978479\pi$
$137$	343.992	+	595.812i	0.214520	+	0.371560i	−0.738604	+	0.674140i	$0.764515\pi$
$138$		0			0
$139$	339.790	+	588.534i	0.207343	+	0.359128i	0.950877	−	0.309570i	$-0.100185\pi$
$139$	339.790	+	588.534i	0.207343	+	0.359128i	−0.743534	+	0.668698i	$0.766852\pi$
$140$	378.078	−	654.850i	0.228239	−	0.395321i
$141$		0			0
$142$	67.0662			0.0396343
$143$	557.652	+	890.890i	0.326106	+	0.520979i
$144$		0			0
$145$	−949.983	+	1645.42i	−0.544082	+	0.942377i
$146$	−25.7079	+	44.5274i	−0.0145726	+	0.0252405i
$147$		0			0
$148$	33.5084			0.0186106
$149$	−987.731	−	1710.80i	−0.543074	−	0.940632i	−0.998725	−	0.0504739i	$-0.983927\pi$
$149$	−987.731	−	1710.80i	−0.543074	−	0.940632i	0.455651	−	0.890159i	$-0.349407\pi$
$150$		0			0
$151$	1803.24			0.971824			0.485912	−	0.874008i	$-0.338487\pi$
$151$	1803.24			0.971824			0.485912	+	0.874008i	$0.338487\pi$
$152$	−280.034	−	485.033i	−0.149433	−	0.258825i
$153$		0			0
$154$	−26.7339	+	46.3044i	−0.0139888	+	0.0242293i
$155$	−4917.71			−2.54839
$156$		0			0
$157$	−397.168			−0.201894			−0.100947	−	0.994892i	$-0.532187\pi$
$157$	−397.168			−0.201894			−0.100947	+	0.994892i	$0.532187\pi$
$158$	−44.3845	+	76.8762i	−0.0223483	+	0.0387085i
$159$		0			0
$160$	−725.675	−	1256.91i	−0.358560	−	0.621044i
$161$	−764.272			−0.374118
$162$		0			0
$163$	−470.696	−	815.270i	−0.226183	−	0.391760i	0.730491	−	0.682922i	$-0.239291\pi$
$163$	−470.696	−	815.270i	−0.226183	−	0.391760i	−0.956674	+	0.291162i	$0.905958\pi$
$164$	1778.37			0.846750
$165$		0			0
$166$	73.6932	−	127.640i	0.0344560	−	0.0596796i
$167$	1840.22	−	3187.35i	0.852696	−	1.47691i	−0.0260704	−	0.999660i	$-0.508299\pi$
$167$	1840.22	−	3187.35i	0.852696	−	1.47691i	0.878766	−	0.477252i	$-0.158367\pi$
$168$		0			0
$169$	−1231.37	−	1819.49i	−0.560479	−	0.828168i
$170$	−530.810			−0.239478
$171$		0			0
$172$	107.471	−	186.146i	0.0476431	−	0.0825203i
$173$	711.387	+	1232.16i	0.312634	+	0.541499i	0.978932	−	0.204187i	$-0.0654552\pi$
$173$	711.387	+	1232.16i	0.312634	+	0.541499i	−0.666297	+	0.745686i	$0.732122\pi$
$174$		0			0
$175$	−522.408	−	904.837i	−0.225659	−	0.390853i
$176$	−666.233	−	1153.95i	−0.285336	−	0.494217i
$177$		0			0
$178$	157.445	+	272.703i	0.0662978	+	0.114831i
$179$	583.946	−	1011.42i	0.243833	−	0.422331i	−0.717970	−	0.696074i	$-0.754928\pi$
$179$	583.946	−	1011.42i	0.243833	−	0.422331i	0.961803	+	0.273743i	$0.0882617\pi$
$180$		0			0
$181$	−1133.96			−0.465673			−0.232836	−	0.972516i	$-0.574801\pi$
$181$	−1133.96			−0.465673			−0.232836	+	0.972516i	$0.574801\pi$
$182$	52.3940	−	98.7237i	0.0213390	−	0.0402082i
$183$		0			0
$184$	−487.002	+	843.512i	−0.195121	+	0.337959i
$185$	38.2126	−	66.1861i	0.0151862	−	0.0263032i
$186$		0			0
$187$	−1524.44			−0.596141
$188$	−1243.84	−	2154.40i	−0.482536	−	0.835776i
$189$		0			0
$190$	−630.928			−0.240907
$191$	1341.06	+	2322.78i	0.508040	+	0.879952i	0.999957	+	0.00930919i	$0.00296325\pi$
$191$	1341.06	+	2322.78i	0.508040	+	0.879952i	−0.491916	+	0.870642i	$0.663703\pi$
$192$		0			0
$193$	985.333	−	1706.65i	0.367491	−	0.636514i	−0.621681	−	0.783270i	$-0.713550\pi$
$193$	985.333	−	1706.65i	0.367491	−	0.636514i	0.989173	+	0.146757i	$0.0468834\pi$
$194$	332.943			0.123216
$195$		0			0
$196$	2447.14			0.891813
$197$	−2008.02	+	3478.00i	−0.726222	+	1.25785i	0.232247	+	0.972657i	$0.425392\pi$
$197$	−2008.02	+	3478.00i	−0.726222	+	1.25785i	−0.958469	+	0.285197i	$0.907941\pi$
$198$		0			0
$199$	2113.03	+	3659.87i	0.752707	+	1.30373i	0.946506	+	0.322685i	$0.104585\pi$
$199$	2113.03	+	3659.87i	0.752707	+	1.30373i	−0.193800	+	0.981041i	$0.562081\pi$
$200$	−1331.53			−0.470768
$201$		0			0
$202$	−76.4426	−	132.402i	−0.0266261	−	0.0461178i
$203$	580.245			0.200617
$204$		0			0
$205$	2028.03	−	3512.65i	0.690944	−	1.19675i
$206$	127.216	−	220.345i	0.0430270	−	0.0745250i
$207$		0			0
$208$	1477.82	+	2360.92i	0.492635	+	0.787021i
$209$	−1811.98			−0.599699
$210$		0			0
$211$	−682.334	+	1181.84i	−0.222625	+	0.385597i	−0.955604	−	0.294654i	$-0.904796\pi$
$211$	−682.334	+	1181.84i	−0.222625	+	0.385597i	0.732980	+	0.680251i	$0.238129\pi$
$212$	264.122	+	457.473i	0.0855659	+	0.148204i
$213$		0			0
$214$	125.285	+	217.000i	0.0400201	+	0.0693168i
$215$	−245.118	−	424.557i	−0.0777532	−	0.134672i
$216$		0			0
$217$	750.928	+	1300.65i	0.234914	+	0.406883i
$218$	−38.5842	+	66.8297i	−0.0119874	+	0.0207628i
$219$		0			0
$220$	−3117.71			−0.955436
$221$	3184.57	−	113.683i	0.969309	−	0.0346025i
$222$		0			0
$223$	529.734	−	917.527i	0.159075	−	0.275525i	−0.775461	−	0.631396i	$-0.782482\pi$
$223$	529.734	−	917.527i	0.159075	−	0.275525i	0.934535	+	0.355871i	$0.115816\pi$
$224$	−221.619	+	383.856i	−0.0661052	+	0.114498i
$225$		0			0
$226$	−554.584			−0.163232
$227$	1732.10	+	3000.08i	0.506446	+	0.877190i	0.999972	+	0.00745930i	$0.00237439\pi$
$227$	1732.10	+	3000.08i	0.506446	+	0.877190i	−0.493526	+	0.869731i	$0.664292\pi$
$228$		0			0
$229$	−2324.64			−0.670815			−0.335407	−	0.942073i	$-0.608874\pi$
$229$	−2324.64			−0.670815			−0.335407	+	0.942073i	$0.608874\pi$
$230$	548.617	+	950.233i	0.157282	+	0.272420i
$231$		0			0
$232$	369.738	−	640.405i	0.104631	−	0.181227i
$233$	3731.01			1.04904			0.524521	−	0.851398i	$-0.324245\pi$
$233$	3731.01			1.04904			0.524521	+	0.851398i	$0.324245\pi$
$234$		0			0
$235$	−5673.86			−1.57499
$236$	1136.48	−	1968.44i	0.313468	−	0.542942i
$237$		0			0
$238$	81.0540	+	140.390i	0.0220754	+	0.0382357i
$239$	−6044.47			−1.63592			−0.817958	−	0.575278i	$-0.804894\pi$
$239$	−6044.47			−1.63592			−0.817958	+	0.575278i	$0.804894\pi$
$240$		0			0
$241$	2586.98	+	4480.78i	0.691461	+	1.19765i	0.971359	+	0.237616i	$0.0763659\pi$
$241$	2586.98	+	4480.78i	0.691461	+	1.19765i	−0.279898	+	0.960030i	$0.590301\pi$
$242$	−363.120			−0.0964556
$243$		0			0
$244$	2589.49	−	4485.12i	0.679405	−	1.17676i
$245$	2790.68	−	4833.61i	0.727715	−	1.26044i
$246$		0			0
$247$	3785.22	−	135.125i	0.975093	−	0.0348090i
$248$	1914.00			0.490076
$249$		0			0
$250$	−262.015	+	453.823i	−0.0662851	+	0.114809i
$251$	2810.37	+	4867.70i	0.706728	+	1.22409i	0.966064	+	0.258301i	$0.0831628\pi$
$251$	2810.37	+	4867.70i	0.706728	+	1.22409i	−0.259337	+	0.965787i	$0.583504\pi$
$252$		0			0
$253$	1575.59	+	2729.00i	0.391527	+	0.678144i
$254$	−570.882	−	988.796i	−0.141025	−	0.244262i
$255$		0			0
$256$	−1573.42	−	2725.24i	−0.384135	−	0.665341i
$257$	−837.070	+	1449.85i	−0.203171	+	0.351903i	−0.949549	−	0.313620i	$-0.898458\pi$
$257$	−837.070	+	1449.85i	−0.203171	+	0.351903i	0.746377	+	0.665523i	$0.231791\pi$
$258$		0			0
$259$	−23.3401			−0.00559954
$260$	6512.90	−	232.498i	1.55351	−	0.0554574i
$261$		0			0
$262$	467.320	−	809.422i	0.110195	−	0.190864i
$263$	−3154.59	+	5463.91i	−0.739622	+	1.28106i	0.213044	+	0.977043i	$0.431662\pi$
$263$	−3154.59	+	5463.91i	−0.739622	+	1.28106i	−0.952666	+	0.304020i	$0.901671\pi$
$264$		0			0
$265$	1204.81			0.279285
$266$	96.3419	+	166.869i	0.0222072	+	0.0384639i
$267$		0			0
$268$	3319.09			0.756514
$269$	−1241.37	−	2150.11i	−0.281366	−	0.487340i	0.690356	−	0.723470i	$-0.257454\pi$
$269$	−1241.37	−	2150.11i	−0.281366	−	0.487340i	−0.971721	+	0.236131i	$0.924121\pi$
$270$		0			0
$271$	−1417.86	+	2455.81i	−0.317819	+	0.550478i	−0.980033	−	0.198837i	$-0.936284\pi$
$271$	−1417.86	+	2455.81i	−0.317819	+	0.550478i	0.662214	+	0.749315i	$0.269617\pi$
$272$	−4039.88			−0.900566
$273$		0			0
$274$	−301.645			−0.0665075
$275$	−2153.94	+	3730.74i	−0.472318	+	0.818080i
$276$		0			0
$277$	1918.76	+	3323.38i	0.416198	+	0.720876i	0.995553	−	0.0941989i	$-0.0300290\pi$
$277$	1918.76	+	3323.38i	0.416198	+	0.720876i	−0.579355	+	0.815075i	$0.696696\pi$
$278$	−297.960			−0.0642822
$279$		0			0
$280$	335.616	+	581.303i	0.0716317	+	0.124070i
$281$	9122.13			1.93659			0.968293	−	0.249819i	$-0.0803712\pi$
$281$	9122.13			1.93659			0.968293	+	0.249819i	$0.0803712\pi$
$282$		0			0
$283$	−1063.92	+	1842.77i	−0.223476	+	0.387072i	−0.955861	−	0.293819i	$-0.905074\pi$
$283$	−1063.92	+	1842.77i	−0.223476	+	0.387072i	0.732385	+	0.680891i	$0.238407\pi$
$284$	597.150	−	1034.29i	0.124769	−	0.216106i
$285$		0			0
$286$	−460.527	+	16.4399i	−0.0952152	+	0.00339900i
$287$	−1238.71			−0.254769
$288$		0			0
$289$	145.530	−	252.066i	0.0296215	−	0.0513059i
$290$	−416.518	−	721.430i	−0.0843405	−	0.146082i
$291$		0			0
$292$	457.800	+	792.934i	0.0917491	+	0.158914i
$293$	4137.38	+	7166.16i	0.824944	+	1.42884i	0.901962	+	0.431815i	$0.142127\pi$
$293$	4137.38	+	7166.16i	0.824944	+	1.42884i	−0.0770183	+	0.997030i	$0.524540\pi$
$294$		0			0
$295$	−2592.05	−	4489.56i	−0.511576	−	0.886076i
$296$	−14.8725	+	25.7600i	−0.00292043	+	0.00505834i
$297$		0			0
$298$	866.136			0.168369
$299$	−3494.92	−	5583.38i	−0.675974	−	1.07992i
$300$		0			0
$301$	−74.8585	+	129.659i	−0.0143348	+	0.0248286i
$302$	−395.312	+	684.701i	−0.0753234	+	0.130464i
$303$		0			0
$304$	−4801.86			−0.905940
$305$	−5906.04	−	10229.6i	−1.10878	−	1.92047i
$306$		0			0
$307$	−3610.49			−0.671211			−0.335605	−	0.942003i	$-0.608941\pi$
$307$	−3610.49			−0.671211			−0.335605	+	0.942003i	$0.608941\pi$
$308$	476.070	+	824.578i	0.0880734	+	0.152548i
$309$		0			0
$310$	1078.08	−	1867.29i	0.197518	−	0.342112i
$311$	−3331.06			−0.607354			−0.303677	−	0.952775i	$-0.598214\pi$
$311$	−3331.06			−0.607354			−0.303677	+	0.952775i	$0.598214\pi$
$312$		0			0
$313$	−358.125			−0.0646724			−0.0323362	−	0.999477i	$-0.510295\pi$
$313$	−358.125			−0.0646724			−0.0323362	+	0.999477i	$0.510295\pi$
$314$	87.0685	−	150.807i	0.0156483	−	0.0271036i
$315$		0			0
$316$	790.388	+	1368.99i	0.140705	+	0.243708i
$317$	−3047.46			−0.539944			−0.269972	−	0.962868i	$-0.587014\pi$
$317$	−3047.46			−0.539944			−0.269972	+	0.962868i	$0.587014\pi$
$318$		0			0
$319$	−1196.21	−	2071.89i	−0.209952	−	0.363647i
$320$	−7829.23			−1.36771
$321$		0			0
$322$	167.546	−	290.199i	0.0289969	−	0.0502241i
$323$	−2746.85	+	4757.69i	−0.473185	+	0.819581i
$324$		0			0
$325$	4221.38	−	7954.15i	0.720492	−	1.35759i
$326$	412.751			0.0701232
$327$		0			0
$328$	−789.318	+	1367.14i	−0.132874	+	0.230145i
$329$	866.392	+	1500.63i	0.145185	+	0.251467i
$330$		0			0
$331$	−3847.39	−	6663.87i	−0.638887	−	1.10658i	−0.985677	−	0.168641i	$-0.946062\pi$
$331$	−3847.39	−	6663.87i	−0.638887	−	1.10658i	0.346791	−	0.937942i	$-0.387271\pi$
$332$	−1312.31	−	2272.99i	−0.216935	−	0.375742i
$333$		0			0
$334$	806.838	+	1397.48i	0.132180	+	0.228943i
$335$	3785.05	−	6555.90i	0.617312	−	1.06922i
$336$		0			0
$337$	4712.21			0.761693			0.380846	−	0.924638i	$-0.375633\pi$
$337$	4712.21			0.761693			0.380846	+	0.924638i	$0.375633\pi$
$338$	960.817	−	68.6862i	0.154620	−	0.0110534i
$339$		0			0
$340$	−4726.27	+	8186.13i	−0.753876	+	1.30575i
$341$	3096.16	−	5362.70i	0.491690	−	0.851632i
$342$		0			0
$343$	−3569.92			−0.561976
$344$	95.4013	+	165.240i	0.0149526	+	0.0258986i
$345$		0			0
$346$	−623.811			−0.0969257
$347$	−2630.99	−	4557.01i	−0.407029	−	0.704995i	0.587526	−	0.809205i	$-0.300102\pi$
$347$	−2630.99	−	4557.01i	−0.407029	−	0.704995i	−0.994555	+	0.104210i	$0.966768\pi$
$348$		0			0
$349$	−25.1672	+	43.5909i	−0.00386009	+	0.00668587i	−0.867949	−	0.496653i	$-0.834562\pi$
$349$	−25.1672	+	43.5909i	−0.00386009	+	0.00668587i	0.864089	+	0.503339i	$0.167895\pi$
$350$	458.096			0.0699608
$351$		0			0
$352$	1827.52			0.276725
$353$	4528.82	−	7844.14i	0.682846	−	1.18272i	−0.291263	−	0.956643i	$-0.594075\pi$
$353$	4528.82	−	7844.14i	0.682846	−	1.18272i	0.974109	−	0.226081i	$-0.0725914\pi$
$354$		0			0
$355$	−1361.96	−	2358.99i	−0.203621	−	0.352683i
$356$	5607.49			0.834821
$357$		0			0
$358$	256.029	+	443.456i	0.0377977	+	0.0654675i
$359$	−7177.86			−1.05525			−0.527623	−	0.849479i	$-0.676917\pi$
$359$	−7177.86			−1.05525			−0.527623	+	0.849479i	$0.676917\pi$
$360$		0			0
$361$	164.553	−	285.014i	0.0239908	−	0.0415532i
$362$	248.591	−	430.573i	0.0360930	−	0.0625150i
$363$		0			0
$364$	−1056.00	−	1687.04i	−0.152059	−	0.242926i
$365$	2088.28			0.299467
$366$		0			0
$367$	−2002.07	+	3467.69i	−0.284761	+	0.493221i	−0.972551	−	0.232689i	$-0.925248\pi$
$367$	−2002.07	+	3467.69i	−0.284761	+	0.493221i	0.687790	+	0.725910i	$0.258581\pi$
$368$	4175.41	+	7232.03i	0.591463	+	1.02444i
$369$		0			0
$370$	16.7542	+	29.0191i	0.00235408	+	0.00407738i
$371$	−183.972	−	318.649i	−0.0257449	−	0.0445915i
$372$		0			0
$373$	5007.09	+	8672.53i	0.695060	+	1.20388i	0.970161	+	0.242464i	$0.0779555\pi$
$373$	5007.09	+	8672.53i	0.695060	+	1.20388i	−0.275101	+	0.961415i	$0.588711\pi$
$374$	334.194	−	578.841i	0.0462053	−	0.0800299i
$375$		0			0
$376$	2208.30			0.302883
$377$	2653.39	+	4238.98i	0.362484	+	0.579094i
$378$		0			0
$379$	4084.56	−	7074.66i	0.553587	−	0.958842i	−0.444425	−	0.895816i	$-0.646592\pi$
$379$	4084.56	−	7074.66i	0.553587	−	0.958842i	0.998012	−	0.0630252i	$-0.0200749\pi$
$380$	−5617.71	+	9730.16i	−0.758375	+	1.31354i
$381$		0			0
$382$	−1175.97			−0.157507
$383$	−3655.12	−	6330.86i	−0.487645	−	0.844626i	0.512254	−	0.858834i	$-0.328811\pi$
$383$	−3655.12	−	6330.86i	−0.487645	−	0.844626i	−0.999899	+	0.0142079i	$0.995477\pi$
$384$		0			0
$385$	2171.62			0.287470
$386$	432.017	+	748.275i	0.0569665	+	0.0986689i
$387$		0			0
$388$	2964.48	−	5134.64i	0.387884	−	0.671834i
$389$	−8785.47			−1.14509			−0.572546	−	0.819872i	$-0.694044\pi$
$389$	−8785.47			−1.14509			−0.572546	+	0.819872i	$0.694044\pi$
$390$		0			0
$391$	9553.99			1.23572
$392$	−1086.15	+	1881.26i	−0.139946	+	0.242393i
$393$		0			0
$394$	−880.412	−	1524.92i	−0.112575	−	0.194986i
$395$	3605.40			0.459259
$396$		0			0
$397$	−5633.40	−	9757.33i	−0.712171	−	1.23352i	−0.964040	−	0.265756i	$-0.914379\pi$
$397$	−5633.40	−	9757.33i	−0.712171	−	1.23352i	0.251869	−	0.967761i	$-0.418955\pi$
$398$	−1852.90			−0.233361
$399$		0			0
$400$	−5708.10	+	9886.71i	−0.713512	+	1.23584i
$401$	788.117	−	1365.06i	0.0981464	−	0.169995i	−0.812771	−	0.582583i	$-0.802042\pi$
$401$	788.117	−	1365.06i	0.0981464	−	0.169995i	0.910917	+	0.412589i	$0.135375\pi$
$402$		0			0
$403$	−6067.96	+	11433.6i	−0.750042	+	1.41327i
$404$	−2722.54			−0.335276
$405$		0			0
$406$	−127.203	+	220.323i	−0.0155493	+	0.0269321i
$407$	48.1168	+	83.3407i	0.00586010	+	0.0101500i
$408$		0			0
$409$	−3377.89	−	5850.68i	−0.408377	−	0.707329i	0.586331	−	0.810071i	$-0.300572\pi$
$409$	−3377.89	−	5850.68i	−0.408377	−	0.707329i	−0.994708	+	0.102742i	$0.967238\pi$
$410$	889.183	+	1540.11i	0.107106	+	0.185514i
$411$		0			0
$412$	−2265.43	−	3923.85i	−0.270898	−	0.469209i
$413$	−791.606	+	1371.10i	−0.0943157	+	0.163360i
$414$		0			0
$415$	−5986.17			−0.708072
$416$	−3817.69	+	136.284i	−0.449947	+	0.0160622i
$417$		0			0
$418$	397.228	−	688.019i	0.0464810	−	0.0805074i
$419$	5378.09	−	9315.13i	0.627057	−	1.08610i	−0.361082	−	0.932534i	$-0.617592\pi$
$419$	5378.09	−	9315.13i	0.627057	−	1.08610i	0.988139	−	0.153561i	$-0.0490742\pi$
$420$		0			0
$421$	7886.03			0.912925			0.456463	−	0.889743i	$-0.349116\pi$
$421$	7886.03			0.912925			0.456463	+	0.889743i	$0.349116\pi$
$422$	−299.167	−	518.173i	−0.0345100	−	0.0597731i
$423$		0			0
$424$	−468.916			−0.0537089
$425$	6530.50	+	11311.2i	0.745355	+	1.29099i
$426$		0			0
$427$	−1803.69	+	3124.08i	−0.204418	+	0.354063i
$428$	4462.08			0.503932
$429$		0			0
$430$	214.943			0.0241057
$431$	−7042.31	+	12197.6i	−0.787044	+	1.36320i	0.140726	+	0.990049i	$0.455056\pi$
$431$	−7042.31	+	12197.6i	−0.787044	+	1.36320i	−0.927770	+	0.373152i	$0.878277\pi$
$432$		0			0
$433$	−932.072	−	1614.40i	−0.103447	−	0.179175i	0.809656	−	0.586905i	$-0.199654\pi$
$433$	−932.072	−	1614.40i	−0.103447	−	0.179175i	−0.913103	+	0.407730i	$0.866321\pi$
$434$	−658.485			−0.0728301
$435$		0			0
$436$	687.098	+	1190.09i	0.0754725	+	0.130722i
$437$	11356.0			1.24309
$438$		0			0
$439$	−3077.24	+	5329.94i	−0.334553	+	0.579463i	−0.983399	−	0.181457i	$-0.941919\pi$
$439$	−3077.24	+	5329.94i	−0.334553	+	0.579463i	0.648846	+	0.760920i	$0.275252\pi$
$440$	1383.78	−	2396.77i	0.149930	−	0.259686i
$441$		0			0
$442$	−654.966	+	1234.12i	−0.0704832	+	0.132808i
$443$	14539.3			1.55933			0.779663	−	0.626200i	$-0.215391\pi$
$443$	14539.3			1.55933			0.779663	+	0.626200i	$0.215391\pi$
$444$		0			0
$445$	6394.72	−	11076.0i	0.681210	−	1.17989i
$446$	232.261	+	402.287i	0.0246589	+	0.0427104i
$447$		0			0
$448$	1195.51	+	2070.69i	0.126077	+	0.218373i
$449$	−3521.93	−	6100.17i	−0.370179	−	0.641169i	0.619414	−	0.785065i	$-0.287370\pi$
$449$	−3521.93	−	6100.17i	−0.370179	−	0.641169i	−0.989593	+	0.143896i	$0.954037\pi$
$450$		0			0
$451$	2553.66	+	4423.08i	0.266624	+	0.461806i
$452$	−4937.95	+	8552.78i	−0.513853	+	0.890020i
$453$		0			0
$454$	−1518.87			−0.157013
$455$	−4536.52	+	161.945i	−0.467418	+	0.0166859i
$456$		0			0
$457$	−7049.43	+	12210.0i	−0.721572	+	1.24980i	0.238798	+	0.971069i	$0.423247\pi$
$457$	−7049.43	+	12210.0i	−0.721572	+	1.24980i	−0.960370	+	0.278730i	$0.910087\pi$
$458$	509.616	−	882.681i	0.0519930	−	0.0900545i
$459$		0			0
$460$	19539.3			1.98049
$461$	7224.85	+	12513.8i	0.729924	+	1.26426i	0.956915	+	0.290368i	$0.0937777\pi$
$461$	7224.85	+	12513.8i	0.729924	+	1.26426i	−0.226991	+	0.973897i	$0.572889\pi$
$462$		0			0
$463$	15806.5			1.58659			0.793293	−	0.608840i	$-0.208365\pi$
$463$	15806.5			1.58659			0.793293	+	0.608840i	$0.208365\pi$
$464$	−3170.03	−	5490.65i	−0.317166	−	0.549347i
$465$		0			0
$466$	−817.926	+	1416.69i	−0.0813083	+	0.140830i
$467$	15071.3			1.49340			0.746699	−	0.665162i	$-0.231638\pi$
$467$	15071.3			1.49340			0.746699	+	0.665162i	$0.231638\pi$
$468$		0			0
$469$	−2311.89			−0.227619
$470$	1243.84	−	2154.40i	0.122073	−	0.211437i
$471$		0			0
$472$	1008.84	+	1747.36i	0.0983804	+	0.170400i
$473$	617.299			0.0600073
$474$		0			0
$475$	7762.25	+	13444.6i	0.749803	+	1.29870i
$476$	2886.78			0.277973
$477$		0			0
$478$	1325.09	−	2295.12i	0.126795	−	0.219616i
$479$	−196.272	+	339.954i	−0.0187222	+	0.0324277i	−0.875235	−	0.483698i	$-0.839293\pi$
$479$	−196.272	+	339.954i	−0.0187222	+	0.0324277i	0.856513	+	0.516126i	$0.172626\pi$
$480$		0			0
$481$	−106.731	−	170.511i	−0.0101175	−	0.0161634i
$482$	−2268.51			−0.214373
$483$		0			0
$484$	−3233.18	+	5600.03i	−0.303642	+	0.525923i
$485$	−6761.33	−	11711.0i	−0.633023	−	1.09643i
$486$		0			0
$487$	−4748.94	−	8225.41i	−0.441879	−	0.765357i	0.555950	−	0.831216i	$-0.312355\pi$
$487$	−4748.94	−	8225.41i	−0.441879	−	0.765357i	−0.997829	+	0.0658588i	$0.979021\pi$
$488$	2298.66	+	3981.40i	0.213228	+	0.369322i
$489$		0			0
$490$	1223.57	+	2119.28i	0.112806	+	0.195386i
$491$	−946.912	+	1640.10i	−0.0870337	+	0.150747i	−0.906256	−	0.422729i	$-0.861072\pi$
$491$	−946.912	+	1640.10i	−0.0870337	+	0.150747i	0.819222	+	0.573476i	$0.194405\pi$
$492$		0			0
$493$	−7253.52			−0.662641
$494$	−778.502	+	1466.90i	−0.0709038	+	0.133601i
$495$		0			0
$496$	8205.03	−	14211.5i	0.742775	−	1.28652i
$497$	−415.941	+	720.430i	−0.0375402	+	0.0650216i
$498$		0			0
$499$	−13370.1			−1.19945			−0.599727	−	0.800205i	$-0.704724\pi$
$499$	−13370.1			−1.19945			−0.599727	+	0.800205i	$0.704724\pi$
$500$	4665.90	+	8081.57i	0.417330	+	0.722838i
$501$		0			0
$502$	−2464.39			−0.219106
$503$	2777.36	+	4810.52i	0.246195	+	0.426423i	0.962467	−	0.271399i	$-0.0874862\pi$
$503$	2777.36	+	4810.52i	0.246195	+	0.426423i	−0.716272	+	0.697822i	$0.754153\pi$
$504$		0			0
$505$	−3104.76	+	5377.60i	−0.273584	+	0.473861i
$506$	−1381.62			−0.121385
$507$		0			0
$508$	−20332.3			−1.77578
$509$	1098.78	−	1903.13i	0.0956824	−	0.165727i	−0.814211	−	0.580569i	$-0.802830\pi$
$509$	1098.78	−	1903.13i	0.0956824	−	0.165727i	0.909893	+	0.414843i	$0.136163\pi$
$510$		0			0
$511$	−318.878	−	552.313i	−0.0276053	−	0.0478139i
$512$	8137.89			0.702437
$513$		0			0
$514$	−367.011	−	635.682i	−0.0314945	−	0.0545500i
$515$	−10333.9			−0.884206
$516$		0			0
$517$	3572.23	−	6187.28i	0.303881	−	0.526337i
$518$	5.11669	−	8.86237i	0.000434005	−	0.000751718i
$519$		0			0
$520$	−2711.98	+	5110.06i	−0.228708	+	0.430944i
$521$	−17005.2			−1.42997			−0.714983	−	0.699142i	$-0.753565\pi$
$521$	−17005.2			−1.42997			−0.714983	+	0.699142i	$0.753565\pi$
$522$		0			0
$523$	7243.11	−	12545.4i	0.605581	−	1.04890i	−0.386378	−	0.922341i	$-0.626274\pi$
$523$	7243.11	−	12545.4i	0.605581	−	1.04890i	0.991959	−	0.126557i	$-0.0403928\pi$
$524$	−8321.92	−	14414.0i	−0.693788	−	1.20168i
$525$		0			0
$526$	−1383.12	−	2395.64i	−0.114652	−	0.198583i
$527$	−9387.19	−	16259.1i	−0.775925	−	1.34394i
$528$		0			0
$529$	−3791.02	−	6566.23i	−0.311582	−	0.539675i
$530$	−264.122	+	457.473i	−0.0216466	+	0.0374931i
$531$		0			0
$532$	3431.27			0.279632
$533$	−5664.46	−	9049.39i	−0.460328	−	0.735408i
$534$		0			0
$535$	5088.51	−	8813.56i	0.411206	−	0.712230i
$536$	−1473.16	+	2551.59i	−0.118714	+	0.205619i
$537$		0			0
$538$	1088.55			0.0872315
$539$	3513.99	+	6086.41i	0.280813	+	0.486383i
$540$		0			0
$541$	−15266.7			−1.21325			−0.606623	−	0.794990i	$-0.707476\pi$
$541$	−15266.7			−1.21325			−0.606623	+	0.794990i	$0.707476\pi$
$542$	−621.657	−	1076.74i	−0.0492665	−	0.0853321i
$543$		0			0
$544$	2770.41	−	4798.50i	0.218347	−	0.378187i
$545$	3134.23			0.246341
$546$		0			0
$547$	15260.5			1.19286			0.596430	−	0.802665i	$-0.296586\pi$
$547$	15260.5			1.19286			0.596430	+	0.802665i	$0.296586\pi$
$548$	−2685.81	+	4651.96i	−0.209365	+	0.362631i
$549$		0			0
$550$	−944.390	−	1635.73i	−0.0732162	−	0.126814i
$551$	−8621.64			−0.666595
$552$		0			0
$553$	−550.540	−	953.563i	−0.0423351	−	0.0733266i
$554$	−1682.55			−0.129033
$555$		0			0
$556$	−2653.00	+	4595.13i	−0.202360	+	0.350498i
$557$	−5221.05	+	9043.12i	−0.397169	+	0.687916i	−0.993375	−	0.114915i	$-0.963341\pi$
$557$	−5221.05	+	9043.12i	−0.397169	+	0.687916i	0.596207	+	0.802831i	$0.296674\pi$
$558$		0			0
$559$	−1289.54	+	46.0341i	−0.0975702	+	0.00348307i
$560$	5754.94			0.434269
$561$		0			0
$562$	−1999.79	+	3463.73i	−0.150099	+	0.259980i
$563$	3572.63	+	6187.98i	0.267440	+	0.463219i	0.968200	−	0.250178i	$-0.0804891\pi$
$563$	3572.63	+	6187.98i	0.267440	+	0.463219i	−0.700760	+	0.713397i	$0.747156\pi$
$564$		0			0
$565$	11262.4	+	19507.0i	0.838604	+	1.45250i
$566$	−466.475	−	807.958i	−0.0346420	−	0.0600018i
$567$		0			0
$568$	530.083	+	918.131i	0.0391581	+	0.0678238i
$569$	2219.43	−	3844.17i	0.163521	−	0.283226i	−0.772608	−	0.634883i	$-0.781048\pi$
$569$	2219.43	−	3844.17i	0.163521	−	0.283226i	0.936129	+	0.351657i	$0.114382\pi$
$570$		0			0
$571$	10117.3			0.741497			0.370748	−	0.928733i	$-0.379101\pi$
$571$	10117.3			0.741497			0.370748	+	0.928733i	$0.379101\pi$
$572$	−3846.94	+	7248.62i	−0.281204	+	0.529860i
$573$		0			0
$574$	271.554	−	470.346i	0.0197464	−	0.0342018i
$575$	13499.2	−	23381.3i	0.979052	−	1.69577i
$576$		0			0
$577$	3105.60			0.224069			0.112035	−	0.993704i	$-0.464263\pi$
$577$	3105.60			0.224069			0.112035	+	0.993704i	$0.464263\pi$
$578$	63.8074	+	110.518i	0.00459176	+	0.00795316i
$579$		0			0
$580$	−14834.5			−1.06202
$581$	914.081	+	1583.24i	0.0652711	+	0.113053i
$582$		0			0
$583$	−758.538	+	1313.83i	−0.0538858	+	0.0933329i
$584$	−812.769			−0.0575901
$585$		0			0
$586$	−3628.05			−0.255757
$587$	9831.16	−	17028.1i	0.691270	−	1.19731i	−0.280152	−	0.959956i	$-0.590385\pi$
$587$	9831.16	−	17028.1i	0.691270	−	1.19731i	0.971422	−	0.237359i	$-0.0762818\pi$
$588$		0			0
$589$	−11157.7	−	19325.8i	−0.780555	−	1.35196i
$590$	2272.95			0.158603
$591$		0			0
$592$	127.513	+	220.859i	0.00885261	+	0.0153332i
$593$	−6395.51			−0.442888			−0.221444	−	0.975173i	$-0.571077\pi$
$593$	−6395.51			−0.442888			−0.221444	+	0.975173i	$0.571077\pi$
$594$		0			0
$595$	3292.05	−	5702.00i	0.226825	−	0.392872i
$596$	7711.97	−	13357.5i	0.530025	−	0.918029i
$597$		0			0
$598$	2886.21	−	103.032i	0.197368	−	0.00704567i
$599$	−8878.48			−0.605618			−0.302809	−	0.953051i	$-0.597924\pi$
$599$	−8878.48			−0.605618			−0.302809	+	0.953051i	$0.597924\pi$
$600$		0			0
$601$	−9550.29	+	16541.6i	−0.648194	+	1.12270i	0.335360	+	0.942090i	$0.391142\pi$
$601$	−9550.29	+	16541.6i	−0.648194	+	1.12270i	−0.983554	+	0.180615i	$0.942191\pi$
$602$	−32.8215	−	56.8485i	−0.00222210	−	0.00384879i
$603$		0			0
$604$	7039.63	+	12193.0i	0.474236	+	0.821401i
$605$	7374.16	+	12772.4i	0.495541	+	0.858302i
$606$		0			0
$607$	−8297.88	−	14372.4i	−0.554861	−	0.961047i	−0.997914	−	0.0645522i	$-0.979438\pi$
$607$	−8297.88	−	14372.4i	−0.554861	−	0.961047i	0.443053	−	0.896495i	$-0.353895\pi$
$608$	3292.95	−	5703.56i	0.219650	−	0.380444i
$609$		0			0
$610$	5178.97			0.343755
$611$	−7000.98	+	13191.6i	−0.463551	+	0.873447i
$612$		0			0
$613$	−8234.58	+	14262.7i	−0.542564	+	0.939748i	0.456192	+	0.889881i	$0.349213\pi$
$613$	−8234.58	+	14262.7i	−0.542564	+	0.939748i	−0.998756	+	0.0498668i	$0.984120\pi$
$614$	791.505	−	1370.93i	0.0520237	−	0.0901077i
$615$		0			0
$616$	−845.205			−0.0552829
$617$	5057.99	+	8760.69i	0.330027	+	0.571624i	0.982517	−	0.186174i	$-0.0596087\pi$
$617$	5057.99	+	8760.69i	0.330027	+	0.571624i	−0.652489	+	0.757798i	$0.726275\pi$
$618$		0			0
$619$	18854.8			1.22430			0.612148	−	0.790743i	$-0.290306\pi$
$619$	18854.8			1.22430			0.612148	+	0.790743i	$0.290306\pi$
$620$	−19198.2	−	33252.2i	−1.24357	−	2.15393i
$621$		0			0
$622$	730.247	−	1264.83i	0.0470743	−	0.0815352i
$623$	−3905.86			−0.251180
$624$		0			0
$625$	−2730.82			−0.174773
$626$	78.5095	−	135.983i	0.00501258	−	0.00868204i
$627$		0			0
$628$	−1550.50	−	2685.54i	−0.0985215	−	0.170644i
$629$	291.769			0.0184954
$630$		0			0
$631$	−9473.12	−	16407.9i	−0.597653	−	1.03517i	−0.993167	−	0.116705i	$-0.962767\pi$
$631$	−9473.12	−	16407.9i	−0.597653	−	1.03517i	0.395514	−	0.918460i	$-0.370567\pi$
$632$	−1403.24			−0.0883194
$633$		0			0
$634$	668.074	−	1157.14i	0.0418496	−	0.0724856i
$635$	−23186.7	+	40160.5i	−1.44903	+	2.50979i
$636$		0			0
$637$	−7794.62	−	12452.5i	−0.484826	−	0.774545i
$638$	1048.95			0.0650912
$639$		0			0
$640$	7521.75	−	13028.1i	0.464568	−	0.804655i
$641$	−11793.5	−	20426.9i	−0.726698	−	1.25868i	−0.958271	−	0.285861i	$-0.907721\pi$
$641$	−11793.5	−	20426.9i	−0.726698	−	1.25868i	0.231573	−	0.972818i	$-0.425613\pi$
$642$		0			0
$643$	13576.5	+	23515.2i	0.832669	+	1.44222i	0.895915	+	0.444226i	$0.146521\pi$
$643$	13576.5	+	23515.2i	0.832669	+	1.44222i	−0.0632461	+	0.997998i	$0.520145\pi$
$644$	−2983.63	−	5167.79i	−0.182564	−	0.316211i
$645$		0			0
$646$	−1204.35	−	2085.99i	−0.0733506	−	0.127047i
$647$	3428.36	−	5938.09i	0.208319	−	0.360820i	−0.742866	−	0.669440i	$-0.766534\pi$
$647$	3428.36	−	5938.09i	0.208319	−	0.360820i	0.951185	+	0.308620i	$0.0998673\pi$
$648$		0			0
$649$	6527.75			0.394817
$650$	2094.82	+	3346.62i	0.126408	+	0.201947i
$651$		0			0
$652$	3675.09	−	6365.44i	0.220748	−	0.382346i
$653$	−4036.95	+	6992.20i	−0.241926	+	0.419029i	−0.961263	−	0.275633i	$-0.911113\pi$
$653$	−4036.95	+	6992.20i	−0.241926	+	0.419029i	0.719337	+	0.694662i	$0.244446\pi$
$654$		0			0
$655$	−37960.9			−2.26451
$656$	6767.39	+	11721.5i	0.402778	+	0.697632i
$657$		0			0
$658$	−759.734			−0.0450114
$659$	2652.86	+	4594.89i	0.156815	+	0.271611i	0.933718	−	0.358008i	$-0.116544\pi$
$659$	2652.86	+	4594.89i	0.156815	+	0.271611i	−0.776904	+	0.629620i	$0.783211\pi$
$660$		0			0
$661$	−12924.2	+	22385.3i	−0.760502	+	1.31723i	0.182091	+	0.983282i	$0.441714\pi$
$661$	−12924.2	+	22385.3i	−0.760502	+	1.31723i	−0.942592	+	0.333946i	$0.891620\pi$
$662$	3373.75			0.198073
$663$		0			0
$664$	2329.85			0.136168
$665$	3912.98	−	6777.48i	0.228179	−	0.395217i
$666$		0			0
$667$	7496.86	+	12984.9i	0.435202	+	0.753792i
$668$	28735.9			1.66441
$669$		0			0
$670$	1659.55	+	2874.42i	0.0956923	+	0.165744i
$671$	14873.6			0.855722
$672$		0			0
$673$	7264.55	−	12582.6i	0.416089	−	0.720687i	−0.579453	−	0.815005i	$-0.696734\pi$
$673$	7264.55	−	12582.6i	0.416089	−	0.720687i	0.995542	+	0.0943186i	$0.0300673\pi$
$674$	−1033.03	+	1789.26i	−0.0590367	+	0.102255i
$675$		0			0
$676$	7495.72	−	15429.3i	0.426475	−	0.877860i
$677$	−12058.1			−0.684535			−0.342267	−	0.939603i	$-0.611195\pi$
$677$	−12058.1			−0.684535			−0.342267	+	0.939603i	$0.611195\pi$
$678$		0			0
$679$	−2064.89	+	3576.50i	−0.116706	+	0.202140i
$680$	−4195.46	−	7266.74i	−0.236601	−	0.409804i
$681$		0			0
$682$	1357.50	+	2351.26i	0.0762190	+	0.132015i
$683$	15014.4	+	26005.7i	0.841156	+	1.45693i	0.888918	+	0.458067i	$0.151458\pi$
$683$	15014.4	+	26005.7i	0.841156	+	1.45693i	−0.0477615	+	0.998859i	$0.515209\pi$
$684$		0			0
$685$	6125.74	+	10610.1i	0.341682	+	0.591811i
$686$	782.611	−	1355.52i	0.0435572	−	0.0754433i
$687$		0			0
$688$	1635.89			0.0906505
$689$	1486.61	−	2801.15i	0.0821994	−	0.154884i
$690$		0			0
$691$	−224.848	+	389.448i	−0.0123786	+	0.0214404i	−0.872148	−	0.489241i	$-0.837274\pi$
$691$	−224.848	+	389.448i	−0.0123786	+	0.0214404i	0.859770	+	0.510682i	$0.170607\pi$
$692$	−5554.34	+	9620.40i	−0.305122	+	0.528487i
$693$		0			0
$694$	2307.10			0.126191
$695$	6050.90	+	10480.5i	0.330250	+	0.572010i
$696$		0			0
$697$	15484.8			0.841506
$698$	−11.0345	−	19.1123i	−0.000598370	−	0.00103641i
$699$		0			0
$700$	4078.84	−	7064.75i	0.220236	−	0.381461i
$701$	26986.0			1.45399			0.726994	−	0.686644i	$-0.240917\pi$
$701$	26986.0			1.45399			0.726994	+	0.686644i	$0.240917\pi$
$702$		0			0
$703$	346.801			0.0186057
$704$	4929.23	−	8537.67i	0.263888	−	0.457068i
$705$		0			0
$706$	1985.65	+	3439.24i	0.105851	+	0.183339i
$707$	1896.37			0.100877
$708$		0			0
$709$	4549.44	+	7879.85i	0.240984	+	0.417396i	0.960995	−	0.276566i	$-0.0891965\pi$
$709$	4549.44	+	7879.85i	0.240984	+	0.417396i	−0.720011	+	0.693963i	$0.755863\pi$
$710$	1194.30			0.0631285
$711$		0			0
$712$	−2488.85	+	4310.82i	−0.131002	+	0.226903i
$713$	−19404.2	+	33609.1i	−1.01921	+	1.76532i
$714$		0			0
$715$	9930.53	+	15864.8i	0.519414	+	0.829802i
$716$	9118.62			0.475948
$717$		0			0
$718$	1573.56	−	2725.48i	0.0817892	−	0.141663i
$719$	−3146.78	−	5450.38i	−0.163220	−	0.282705i	0.772802	−	0.634647i	$-0.218855\pi$
$719$	−3146.78	−	5450.38i	−0.163220	−	0.282705i	−0.936022	+	0.351942i	$0.885521\pi$
$720$		0			0
$721$	1577.97	+	2733.13i	0.0815074	+	0.141175i
$722$	72.1476	+	124.963i	0.00371892	+	0.00644135i
$723$		0			0
$724$	−4426.86	−	7667.54i	−0.227242	−	0.393594i
$725$	−10248.8	+	17751.4i	−0.525006	+	0.909337i
$726$		0			0
$727$	18070.7			0.921878			0.460939	−	0.887432i	$-0.347513\pi$
$727$	18070.7			0.921878			0.460939	+	0.887432i	$0.347513\pi$
$728$	1765.64	−	63.0298i	0.0898885	−	0.00320885i
$729$		0			0
$730$	−457.800	+	792.934i	−0.0232109	+	0.0402025i
$731$	935.790	−	1620.84i	0.0473481	−	0.0820093i
$732$		0			0
$733$	34771.5			1.75214			0.876068	−	0.482188i	$-0.160158\pi$
$733$	34771.5			1.75214			0.876068	+	0.482188i	$0.160158\pi$
$734$	−877.803	−	1520.40i	−0.0441421	−	0.0764563i
$735$		0			0
$736$	−11453.4			−0.573613
$737$	4766.09	+	8255.10i	0.238210	+	0.412592i
$738$		0			0
$739$	−11815.7	+	20465.4i	−0.588158	+	1.01872i	0.406316	+	0.913733i	$0.366813\pi$
$739$	−11815.7	+	20465.4i	−0.588158	+	1.01872i	−0.994474	+	0.104986i	$0.966520\pi$
$740$	596.710			0.0296425
$741$		0			0
$742$	161.324			0.00798167
$743$	16251.4	−	28148.3i	0.802431	−	1.38985i	−0.115581	−	0.993298i	$-0.536873\pi$
$743$	16251.4	−	28148.3i	0.802431	−	1.38985i	0.918012	−	0.396553i	$-0.129794\pi$
$744$		0			0
$745$	−17589.3	−	30465.5i	−0.864995	−	1.49822i
$746$	−4390.69			−0.215489
$747$		0			0
$748$	−5951.25	−	10307.9i	−0.290908	−	0.503868i
$749$	−3108.04			−0.151622
$750$		0			0
$751$	−1010.43	+	1750.12i	−0.0490960	+	0.0850368i	−0.889529	−	0.456879i	$-0.848967\pi$
$751$	−1010.43	+	1750.12i	−0.0490960	+	0.0850368i	0.840433	+	0.541915i	$0.182301\pi$
$752$	9466.65	−	16396.7i	0.459060	−	0.795115i
$753$		0			0
$754$	−2191.25	+	78.2235i	−0.105836	+	0.00377816i
$755$	32111.6			1.54790
$756$		0			0
$757$	−6284.11	+	10884.4i	−0.301717	+	0.522589i	−0.976525	−	0.215404i	$-0.930893\pi$
$757$	−6284.11	+	10884.4i	−0.301717	+	0.522589i	0.674808	+	0.737993i	$0.264226\pi$
$758$	1790.86	+	3101.87i	0.0858141	+	0.148634i
$759$		0			0
$760$	−4986.78	−	8637.36i	−0.238013	−	0.412250i
$761$	−4352.40	−	7538.59i	−0.207325	−	0.359098i	0.743546	−	0.668685i	$-0.233143\pi$
$761$	−4352.40	−	7538.59i	−0.207325	−	0.359098i	−0.950871	+	0.309587i	$0.899809\pi$
$762$		0			0
$763$	−478.593	−	828.948i	−0.0227081	−	0.0393315i
$764$	−10470.7	+	18135.8i	−0.495832	+	0.858807i
$765$		0			0
$766$	3205.16			0.151184
$767$	−13636.5	+	486.797i	−0.641962	+	0.0229168i
$768$		0			0
$769$	10957.9	−	18979.7i	0.513853	−	0.890020i	−0.486018	−	0.873949i	$-0.661551\pi$
$769$	10957.9	−	18979.7i	0.513853	−	0.890020i	0.999871	−	0.0160706i	$-0.00511566\pi$
$770$	−476.070	+	824.578i	−0.0222810	+	0.0385918i
$771$		0			0
$772$	15386.5			0.717322
$773$	−11538.8	−	19985.7i	−0.536896	−	0.929930i	−0.999069	−	0.0431408i	$-0.986264\pi$
$773$	−11538.8	−	19985.7i	−0.536896	−	0.929930i	0.462173	−	0.886790i	$-0.347070\pi$
$774$		0			0
$775$	−53054.0			−2.45904
$776$	2631.54	+	4557.96i	0.121736	+	0.210852i
$777$		0			0
$778$	1925.98	−	3335.90i	0.0887530	−	0.153725i
$779$	18405.5			0.846528
$780$		0			0

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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.219224 + 0.379706i) q^{2} +(3.90388 + 6.76172i) q^{4} +17.8078 q^{5} +(-2.71922 - 4.70983i) q^{7} -6.93087 q^{8} +O(q^{10})\)	\(q+(-0.219224 + 0.379706i) q^{2} +(3.90388 + 6.76172i) q^{4} +17.8078 q^{5} +(-2.71922 - 4.70983i) q^{7} -6.93087 q^{8} +(-3.90388 + 6.76172i) q^{10} +(-11.2116 + 19.4191i) q^{11} +(21.9730 - 41.4027i) q^{13} +2.38447 q^{14} +(-29.7116 + 51.4621i) q^{16} +(33.9924 + 58.8766i) q^{17} +(40.4039 + 69.9816i) q^{19} +(69.5194 + 120.411i) q^{20} +(-4.91571 - 8.51427i) q^{22} +(70.2656 - 121.704i) q^{23} +192.116 q^{25} +(10.9039 + 17.4198i) q^{26} +(21.2311 - 36.7733i) q^{28} +(-53.3466 + 92.3990i) q^{29} -276.155 q^{31} +(-40.7505 - 70.5819i) q^{32} -29.8078 q^{34} +(-48.4233 - 83.8716i) q^{35} +(2.14584 - 3.71670i) q^{37} -35.4299 q^{38} -123.423 q^{40} +(113.884 - 197.254i) q^{41} +(-13.7647 - 23.8411i) q^{43} -175.076 q^{44} +(30.8078 + 53.3606i) q^{46} -318.617 q^{47} +(156.712 - 271.433i) q^{49} +(-42.1165 + 72.9479i) q^{50} +(365.734 - 13.0560i) q^{52} +67.6562 q^{53} +(-199.654 + 345.811i) q^{55} +(18.8466 + 32.6432i) q^{56} +(-23.3897 - 40.5121i) q^{58} +(-145.557 - 252.113i) q^{59} +(-331.655 - 574.444i) q^{61} +(60.5398 - 104.858i) q^{62} -439.652 q^{64} +(391.290 - 737.290i) q^{65} +(212.551 - 368.149i) q^{67} +(-265.405 + 459.695i) q^{68} +42.4621 q^{70} +(-76.4815 - 132.470i) q^{71} +117.268 q^{73} +(0.940837 + 1.62958i) q^{74} +(-315.464 + 546.400i) q^{76} +121.948 q^{77} +202.462 q^{79} +(-529.098 + 916.425i) q^{80} +(49.9323 + 86.4853i) q^{82} -336.155 q^{83} +(605.329 + 1048.46i) q^{85} +12.0702 q^{86} +(77.7065 - 134.592i) q^{88} +(359.097 - 621.974i) q^{89} +(-254.750 + 9.09407i) q^{91} +1097.23 q^{92} +(69.8485 - 120.981i) q^{94} +(719.503 + 1246.22i) q^{95} +(-379.684 - 657.632i) q^{97} +(68.7098 + 119.009i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5 q^{2} - 5 q^{4} + 30 q^{5} - 15 q^{7} + 30 q^{8}+O(q^{10}) \) 4 * q - 5 * q^2 - 5 * q^4 + 30 * q^5 - 15 * q^7 + 30 * q^8	\( 4 q - 5 q^{2} - 5 q^{4} + 30 q^{5} - 15 q^{7} + 30 q^{8} + 5 q^{10} + 17 q^{11} + 125 q^{13} + 92 q^{14} - 57 q^{16} + 70 q^{17} + 141 q^{19} + 175 q^{20} + 170 q^{22} + 145 q^{23} + 150 q^{25} + 23 q^{26} - 80 q^{28} + 34 q^{29} - 280 q^{31} + 105 q^{32} - 78 q^{34} - 70 q^{35} + 190 q^{37} - 620 q^{38} - 370 q^{40} + 538 q^{41} - 455 q^{43} - 1360 q^{44} + 82 q^{46} - 120 q^{47} + 565 q^{49} + 450 q^{50} - 310 q^{52} - 1090 q^{53} - 510 q^{55} - 172 q^{56} + 595 q^{58} - 809 q^{59} - 502 q^{61} - 500 q^{62} - 2542 q^{64} + 555 q^{65} + 475 q^{67} - 505 q^{68} - 160 q^{70} + 127 q^{71} + 1170 q^{73} + 849 q^{74} + 140 q^{76} - 510 q^{77} + 480 q^{79} - 1065 q^{80} + 1515 q^{82} - 520 q^{83} + 1205 q^{85} + 3924 q^{86} + 1020 q^{88} + 921 q^{89} - 1287 q^{91} + 2080 q^{92} - 1040 q^{94} + 1270 q^{95} + 415 q^{97} + 1285 q^{98}+O(q^{100}) \) 4 * q - 5 * q^2 - 5 * q^4 + 30 * q^5 - 15 * q^7 + 30 * q^8 + 5 * q^10 + 17 * q^11 + 125 * q^13 + 92 * q^14 - 57 * q^16 + 70 * q^17 + 141 * q^19 + 175 * q^20 + 170 * q^22 + 145 * q^23 + 150 * q^25 + 23 * q^26 - 80 * q^28 + 34 * q^29 - 280 * q^31 + 105 * q^32 - 78 * q^34 - 70 * q^35 + 190 * q^37 - 620 * q^38 - 370 * q^40 + 538 * q^41 - 455 * q^43 - 1360 * q^44 + 82 * q^46 - 120 * q^47 + 565 * q^49 + 450 * q^50 - 310 * q^52 - 1090 * q^53 - 510 * q^55 - 172 * q^56 + 595 * q^58 - 809 * q^59 - 502 * q^61 - 500 * q^62 - 2542 * q^64 + 555 * q^65 + 475 * q^67 - 505 * q^68 - 160 * q^70 + 127 * q^71 + 1170 * q^73 + 849 * q^74 + 140 * q^76 - 510 * q^77 + 480 * q^79 - 1065 * q^80 + 1515 * q^82 - 520 * q^83 + 1205 * q^85 + 3924 * q^86 + 1020 * q^88 + 921 * q^89 - 1287 * q^91 + 2080 * q^92 - 1040 * q^94 + 1270 * q^95 + 415 * q^97 + 1285 * q^98

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