LMFDB - Embedded newform 507.4.b.j.337.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.b (of order $2$, degree $1$, not 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:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 97 x^{16} + 3906 x^{14} + 84743 x^{12} + 1077128 x^{10} + 8187552 x^{8} + 36483705 x^{6} + \cdots + 26460736 $ x^18 + 97x^16 + 3906x^14 + 84743x^12 + 1077128x^10 + 8187552x^8 + 36483705x^6 + 88861676x^4 + 98825392x^2 + 26460736
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.14
Root			$4.23649i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.j.337.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23649i q^{2} -3.00000 q^{3} -2.47490 q^{4} +13.5815i q^{5} -9.70948i q^{6} +1.42933i q^{7} +17.8820i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+3.23649i q^{2} -3.00000 q^{3} -2.47490 q^{4} +13.5815i q^{5} -9.70948i q^{6} +1.42933i q^{7} +17.8820i q^{8} +9.00000 q^{9} -43.9566 q^{10} +54.5673i q^{11} +7.42469 q^{12} -4.62601 q^{14} -40.7446i q^{15} -77.6741 q^{16} +114.413 q^{17} +29.1285i q^{18} +104.933i q^{19} -33.6129i q^{20} -4.28798i q^{21} -176.607 q^{22} +64.5438 q^{23} -53.6459i q^{24} -59.4583 q^{25} -27.0000 q^{27} -3.53743i q^{28} -60.8037 q^{29} +131.870 q^{30} +148.902i q^{31} -108.336i q^{32} -163.702i q^{33} +370.296i q^{34} -19.4125 q^{35} -22.2741 q^{36} -20.9326i q^{37} -339.615 q^{38} -242.865 q^{40} -371.761i q^{41} +13.8780 q^{42} -40.2951 q^{43} -135.048i q^{44} +122.234i q^{45} +208.896i q^{46} +639.802i q^{47} +233.022 q^{48} +340.957 q^{49} -192.436i q^{50} -343.238 q^{51} +102.124 q^{53} -87.3854i q^{54} -741.108 q^{55} -25.5592 q^{56} -314.799i q^{57} -196.791i q^{58} -704.586i q^{59} +100.839i q^{60} -819.087 q^{61} -481.919 q^{62} +12.8639i q^{63} -270.764 q^{64} +529.820 q^{66} -574.918i q^{67} -283.159 q^{68} -193.631 q^{69} -62.8283i q^{70} -365.790i q^{71} +160.938i q^{72} -965.233i q^{73} +67.7482 q^{74} +178.375 q^{75} -259.698i q^{76} -77.9945 q^{77} +580.173 q^{79} -1054.93i q^{80} +81.0000 q^{81} +1203.20 q^{82} -175.372i q^{83} +10.6123i q^{84} +1553.90i q^{85} -130.415i q^{86} +182.411 q^{87} -975.770 q^{88} -20.0351i q^{89} -395.609 q^{90} -159.739 q^{92} -446.705i q^{93} -2070.72 q^{94} -1425.15 q^{95} +325.008i q^{96} -1226.72i q^{97} +1103.51i q^{98} +491.106i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9}+O(q^{10}) $ 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9	$ 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9} - 396 q^{10} + 192 q^{12} + 196 q^{14} + 64 q^{16} + 268 q^{17} + 548 q^{22} - 452 q^{23} - 1224 q^{25} - 486 q^{27} - 1094 q^{29} + 1188 q^{30} + 276 q^{35} - 576 q^{36} + 832 q^{38} + 2684 q^{40} - 588 q^{42} - 316 q^{43} - 192 q^{48} - 1284 q^{49} - 804 q^{51} + 2798 q^{53} - 2816 q^{55} + 1232 q^{56} + 4184 q^{61} + 586 q^{62} - 4962 q^{64} - 1644 q^{66} - 3158 q^{68} + 1356 q^{69} + 2074 q^{74} + 3672 q^{75} + 3372 q^{77} - 230 q^{79} + 1458 q^{81} + 10294 q^{82} + 3282 q^{87} + 968 q^{88} - 3564 q^{90} + 4174 q^{92} - 936 q^{94} + 444 q^{95}+O(q^{100}) $ 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9 - 396 * q^10 + 192 * q^12 + 196 * q^14 + 64 * q^16 + 268 * q^17 + 548 * q^22 - 452 * q^23 - 1224 * q^25 - 486 * q^27 - 1094 * q^29 + 1188 * q^30 + 276 * q^35 - 576 * q^36 + 832 * q^38 + 2684 * q^40 - 588 * q^42 - 316 * q^43 - 192 * q^48 - 1284 * q^49 - 804 * q^51 + 2798 * q^53 - 2816 * q^55 + 1232 * q^56 + 4184 * q^61 + 586 * q^62 - 4962 * q^64 - 1644 * q^66 - 3158 * q^68 + 1356 * q^69 + 2074 * q^74 + 3672 * q^75 + 3372 * q^77 - 230 * q^79 + 1458 * q^81 + 10294 * q^82 + 3282 * q^87 + 968 * q^88 - 3564 * q^90 + 4174 * q^92 - 936 * q^94 + 444 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

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 $

$2$	3.23649i	1.14427i	0.820158	+	0.572137i	$0.193885\pi$
$2$	3.23649i	1.14427i	−0.820158	+	0.572137i	$0.806115\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−2.47490	−0.309362
$5$	13.5815i	1.21477i	0.794408	+	0.607385i	$0.207781\pi$
$5$	13.5815i	1.21477i	−0.794408	+	0.607385i	$0.792219\pi$
$6$	− 9.70948i	− 0.660647i
$7$	1.42933i	0.0771763i	0.999255	+	0.0385882i	$0.0122860\pi$
$7$	1.42933i	0.0771763i	−0.999255	+	0.0385882i	$0.987714\pi$
$8$	17.8820i	0.790279i
$9$	9.00000	0.333333
$10$	−43.9566	−1.39003
$11$	54.5673i	1.49570i	0.663870	+	0.747848i	$0.268913\pi$
$11$	54.5673i	1.49570i	−0.663870	+	0.747848i	$0.731087\pi$
$12$	7.42469	0.178610
$13$	0	0
$13$	0	0
$14$	−4.62601	−0.0883109
$15$	− 40.7446i	− 0.701348i
$16$	−77.6741	−1.21366
$17$	114.413	1.63230	0.816151	−	0.577839i	$-0.196104\pi$
$17$	114.413	1.63230	0.816151	+	0.577839i	$0.196104\pi$
$18$	29.1285i	0.381425i
$19$	104.933i	1.26701i	0.773737	+	0.633507i	$0.218385\pi$
$19$	104.933i	1.26701i	−0.773737	+	0.633507i	$0.781615\pi$
$20$	− 33.6129i	− 0.375804i
$21$	− 4.28798i	− 0.0445578i
$22$	−176.607	−1.71149
$23$	64.5438	0.585144	0.292572	−	0.956244i	$-0.405489\pi$
$23$	64.5438	0.585144	0.292572	+	0.956244i	$0.405489\pi$
$24$	− 53.6459i	− 0.456268i
$25$	−59.4583	−0.475666
$26$	0	0
$27$	−27.0000	−0.192450
$28$	− 3.53743i	− 0.0238754i
$29$	−60.8037	−0.389343	−0.194672	−	0.980868i	$-0.562364\pi$
$29$	−60.8037	−0.389343	−0.194672	+	0.980868i	$0.562364\pi$
$30$	131.870	0.802534
$31$	148.902i	0.862694i	0.902186	+	0.431347i	$0.141961\pi$
$31$	148.902i	0.862694i	−0.902186	+	0.431347i	$0.858039\pi$
$32$	− 108.336i	− 0.598477i
$33$	− 163.702i	− 0.863541i
$34$	370.296i	1.86780i
$35$	−19.4125	−0.0937515
$36$	−22.2741	−0.103121
$37$	− 20.9326i	− 0.0930079i	−0.998918	−	0.0465040i	$-0.985192\pi$
$37$	− 20.9326i	− 0.0930079i	0.998918	−	0.0465040i	$-0.0148080\pi$
$38$	−339.615	−1.44981
$39$	0	0
$40$	−242.865	−0.960007
$41$	− 371.761i	− 1.41608i	−0.706173	−	0.708040i	$-0.749580\pi$
$41$	− 371.761i	− 1.41608i	0.706173	−	0.708040i	$-0.250420\pi$
$42$	13.8780	0.0509863
$43$	−40.2951	−0.142906	−0.0714528	−	0.997444i	$-0.522764\pi$
$43$	−40.2951	−0.142906	−0.0714528	+	0.997444i	$0.522764\pi$
$44$	− 135.048i	− 0.462712i
$45$	122.234i	0.404923i
$46$	208.896i	0.669565i
$47$	639.802i	1.98563i	0.119652	+	0.992816i	$0.461822\pi$
$47$	639.802i	1.98563i	−0.119652	+	0.992816i	$0.538178\pi$
$48$	233.022	0.700705
$49$	340.957	0.994044
$50$	− 192.436i	− 0.544292i
$51$	−343.238	−0.942410
$52$	0	0
$53$	102.124	0.264676	0.132338	−	0.991205i	$-0.457752\pi$
$53$	102.124	0.264676	0.132338	+	0.991205i	$0.457752\pi$
$54$	− 87.3854i	− 0.220216i
$55$	−741.108	−1.81693
$56$	−25.5592	−0.0609908
$57$	− 314.799i	− 0.731511i
$58$	− 196.791i	− 0.445515i
$59$	− 704.586i	− 1.55473i	−0.629048	−	0.777367i	$-0.716555\pi$
$59$	− 704.586i	− 1.55473i	0.629048	−	0.777367i	$-0.283445\pi$
$60$	100.839i	0.216970i
$61$	−819.087	−1.71924	−0.859618	−	0.510938i	$-0.829298\pi$
$61$	−819.087	−1.71924	−0.859618	+	0.510938i	$0.829298\pi$
$62$	−481.919	−0.987158
$63$	12.8639i	0.0257254i
$64$	−270.764	−0.528835
$65$	0	0
$66$	529.820	0.988127
$67$	− 574.918i	− 1.04832i	−0.851620	−	0.524159i	$-0.824380\pi$
$67$	− 574.918i	− 1.04832i	0.851620	−	0.524159i	$-0.175620\pi$
$68$	−283.159	−0.504972
$69$	−193.631	−0.337833
$70$	− 62.8283i	− 0.107277i
$71$	− 365.790i	− 0.611427i	−0.952124	−	0.305714i	$-0.901105\pi$
$71$	− 365.790i	− 0.611427i	0.952124	−	0.305714i	$-0.0988950\pi$
$72$	160.938i	0.263426i
$73$	− 965.233i	− 1.54756i	−0.633454	−	0.773780i	$-0.718363\pi$
$73$	− 965.233i	− 1.54756i	0.633454	−	0.773780i	$-0.281637\pi$
$74$	67.7482	0.106427
$75$	178.375	0.274626
$76$	− 259.698i	− 0.391966i
$77$	−77.9945	−0.115432
$78$	0	0
$79$	580.173	0.826261	0.413130	−	0.910672i	$-0.364435\pi$
$79$	580.173	0.826261	0.413130	+	0.910672i	$0.364435\pi$
$80$	− 1054.93i	− 1.47431i
$81$	81.0000	0.111111
$82$	1203.20	1.62038
$83$	− 175.372i	− 0.231923i	−0.993254	−	0.115962i	$-0.963005\pi$
$83$	− 175.372i	− 0.231923i	0.993254	−	0.115962i	$-0.0369949\pi$
$84$	10.6123i	0.0137845i
$85$	1553.90i	1.98287i
$86$	− 130.415i	− 0.163523i
$87$	182.411	0.224787
$88$	−975.770	−1.18202
$89$	− 20.0351i	− 0.0238620i	−0.999929	−	0.0119310i	$-0.996202\pi$
$89$	− 20.0351i	− 0.0238620i	0.999929	−	0.0119310i	$-0.00379784\pi$
$90$	−395.609	−0.463343
$91$	0	0
$92$	−159.739	−0.181021
$93$	− 446.705i	− 0.498076i
$94$	−2070.72	−2.27211
$95$	−1425.15	−1.53913
$96$	325.008i	0.345531i
$97$	− 1226.72i	− 1.28406i	−0.766678	−	0.642031i	$-0.778092\pi$
$97$	− 1226.72i	− 1.28406i	0.766678	−	0.642031i	$-0.221908\pi$
$98$	1103.51i	1.13746i
$99$	491.106i	0.498565i
$100$	147.153	0.147153
$101$	57.9799	0.0571209	0.0285605	−	0.999592i	$-0.490908\pi$
$101$	57.9799	0.0571209	0.0285605	+	0.999592i	$0.490908\pi$
$102$	− 1110.89i	− 1.07837i
$103$	954.989	0.913571	0.456786	−	0.889577i	$-0.349001\pi$
$103$	954.989	0.913571	0.456786	+	0.889577i	$0.349001\pi$
$104$	0	0
$105$	58.2374	0.0541275
$106$	330.524i	0.302861i
$107$	1604.68	1.44982	0.724908	−	0.688846i	$-0.241882\pi$
$107$	1604.68	1.44982	0.724908	+	0.688846i	$0.241882\pi$
$108$	66.8222	0.0595368
$109$	1534.93i	1.34880i	0.738365	+	0.674401i	$0.235598\pi$
$109$	1534.93i	1.34880i	−0.738365	+	0.674401i	$0.764402\pi$
$110$	− 2398.59i	− 2.07906i
$111$	62.7977i	0.0536982i
$112$	− 111.022i	− 0.0936656i
$113$	−1789.20	−1.48951	−0.744753	−	0.667341i	$-0.767433\pi$
$113$	−1789.20	−1.48951	−0.744753	+	0.667341i	$0.767433\pi$
$114$	1018.84	0.837049
$115$	876.604i	0.710815i
$116$	150.483	0.120448
$117$	0	0
$118$	2280.39	1.77904
$119$	163.533i	0.125975i
$120$	728.594	0.554260
$121$	−1646.59	−1.23711
$122$	− 2650.97i	− 1.96728i
$123$	1115.28i	0.817574i
$124$	− 368.516i	− 0.266885i
$125$	890.158i	0.636945i
$126$	−41.6341	−0.0294370
$127$	−1648.30	−1.15168	−0.575838	−	0.817564i	$-0.695324\pi$
$127$	−1648.30	−1.15168	−0.575838	+	0.817564i	$0.695324\pi$
$128$	− 1743.01i	− 1.20361i
$129$	120.885	0.0825066
$130$	0	0
$131$	2278.88	1.51990	0.759948	−	0.649983i	$-0.225224\pi$
$131$	2278.88	1.51990	0.759948	+	0.649983i	$0.225224\pi$
$132$	405.145i	0.267147i
$133$	−149.983	−0.0977835
$134$	1860.72	1.19956
$135$	− 366.702i	− 0.233783i
$136$	2045.92i	1.28997i
$137$	− 719.324i	− 0.448584i	−0.974522	−	0.224292i	$-0.927993\pi$
$137$	− 719.324i	− 0.448584i	0.974522	−	0.224292i	$-0.0720070\pi$
$138$	− 626.687i	− 0.386573i
$139$	1777.65	1.08474	0.542369	−	0.840140i	$-0.317527\pi$
$139$	1777.65	1.08474	0.542369	+	0.840140i	$0.317527\pi$
$140$	48.0438	0.0290032
$141$	− 1919.41i	− 1.14641i
$142$	1183.88	0.699640
$143$	0	0
$144$	−699.067	−0.404552
$145$	− 825.807i	− 0.472963i
$146$	3123.97	1.77083
$147$	−1022.87	−0.573911
$148$	51.8060i	0.0287731i
$149$	− 266.172i	− 0.146347i	−0.997319	−	0.0731734i	$-0.976687\pi$
$149$	− 266.172i	− 0.146347i	0.997319	−	0.0731734i	$-0.0233127\pi$
$150$	577.309i	0.314247i
$151$	− 1417.27i	− 0.763812i	−0.924201	−	0.381906i	$-0.875268\pi$
$151$	− 1417.27i	− 0.763812i	0.924201	−	0.381906i	$-0.124732\pi$
$152$	−1876.41	−1.00129
$153$	1029.71	0.544101
$154$	− 252.429i	− 0.132086i
$155$	−2022.31	−1.04797
$156$	0	0
$157$	35.6073	0.0181005	0.00905023	−	0.999959i	$-0.497119\pi$
$157$	35.6073	0.0181005	0.00905023	+	0.999959i	$0.497119\pi$
$158$	1877.73i	0.945468i
$159$	−306.372	−0.152810
$160$	1471.37	0.727012
$161$	92.2541i	0.0451593i
$162$	262.156i	0.127142i
$163$	97.7228i	0.0469585i	0.999724	+	0.0234793i	$0.00747437\pi$
$163$	97.7228i	0.0469585i	−0.999724	+	0.0234793i	$0.992526\pi$
$164$	920.069i	0.438081i
$165$	2223.32	1.04900
$166$	567.592	0.265383
$167$	2715.91i	1.25846i	0.777218	+	0.629231i	$0.216630\pi$
$167$	2715.91i	1.25846i	−0.777218	+	0.629231i	$0.783370\pi$
$168$	76.6775	0.0352131
$169$	0	0
$170$	−5029.19	−2.26895
$171$	944.397i	0.422338i
$172$	99.7262	0.0442096
$173$	−2074.88	−0.911852	−0.455926	−	0.890018i	$-0.650692\pi$
$173$	−2074.88	−0.911852	−0.455926	+	0.890018i	$0.650692\pi$
$174$	590.372i	0.257218i
$175$	− 84.9853i	− 0.0367102i
$176$	− 4238.46i	− 1.81526i
$177$	2113.76i	0.897626i
$178$	64.8435	0.0273046
$179$	−1023.50	−0.427372	−0.213686	−	0.976902i	$-0.568547\pi$
$179$	−1023.50	−0.427372	−0.213686	+	0.976902i	$0.568547\pi$
$180$	− 302.516i	− 0.125268i
$181$	−1773.72	−0.728395	−0.364197	−	0.931322i	$-0.618657\pi$
$181$	−1773.72	−0.728395	−0.364197	+	0.931322i	$0.618657\pi$
$182$	0	0
$183$	2457.26	0.992601
$184$	1154.17i	0.462427i
$185$	284.297	0.112983
$186$	1445.76	0.569936
$187$	6243.18i	2.44143i
$188$	− 1583.44i	− 0.614279i
$189$	− 38.5918i	− 0.0148526i
$190$	− 4612.49i	− 1.76119i
$191$	2251.37	0.852896	0.426448	−	0.904512i	$-0.359765\pi$
$191$	2251.37	0.852896	0.426448	+	0.904512i	$0.359765\pi$
$192$	812.291	0.305323
$193$	3876.48i	1.44578i	0.690964	+	0.722889i	$0.257186\pi$
$193$	3876.48i	1.44578i	−0.690964	+	0.722889i	$0.742814\pi$
$194$	3970.26	1.46932
$195$	0	0
$196$	−843.834	−0.307520
$197$	756.708i	0.273671i	0.990594	+	0.136835i	$0.0436932\pi$
$197$	756.708i	0.273671i	−0.990594	+	0.136835i	$0.956307\pi$
$198$	−1589.46	−0.570495
$199$	1986.32	0.707572	0.353786	−	0.935326i	$-0.384894\pi$
$199$	1986.32	0.707572	0.353786	+	0.935326i	$0.384894\pi$
$200$	− 1063.23i	− 0.375909i
$201$	1724.75i	0.605247i
$202$	187.652i	0.0653620i
$203$	− 86.9082i	− 0.0300481i
$204$	849.478	0.291546
$205$	5049.08	1.72021
$206$	3090.82i	1.04538i
$207$	580.894	0.195048
$208$	0	0
$209$	−5725.91	−1.89507
$210$	188.485i	0.0619366i
$211$	−390.462	−0.127396	−0.0636980	−	0.997969i	$-0.520289\pi$
$211$	−390.462	−0.127396	−0.0636980	+	0.997969i	$0.520289\pi$
$212$	−252.746	−0.0818806
$213$	1097.37i	0.353008i
$214$	5193.54i	1.65899i
$215$	− 547.269i	− 0.173597i
$216$	− 482.813i	− 0.152089i
$217$	−212.829	−0.0665795
$218$	−4967.79	−1.54340
$219$	2895.70i	0.893485i
$220$	1834.17	0.562088
$221$	0	0
$222$	−203.244	−0.0614454
$223$	1814.65i	0.544922i	0.962167	+	0.272461i	$0.0878376\pi$
$223$	1814.65i	0.544922i	−0.962167	+	0.272461i	$0.912162\pi$
$224$	154.847	0.0461883
$225$	−535.124	−0.158555
$226$	− 5790.75i	− 1.70440i
$227$	1046.66i	0.306032i	0.988224	+	0.153016i	$0.0488986\pi$
$227$	1046.66i	0.306032i	−0.988224	+	0.153016i	$0.951101\pi$
$228$	779.095i	0.226302i
$229$	6018.10i	1.73662i	0.496018	+	0.868312i	$0.334795\pi$
$229$	6018.10i	1.73662i	−0.496018	+	0.868312i	$0.665205\pi$
$230$	−2837.12	−0.813367
$231$	233.983	0.0666449
$232$	− 1087.29i	− 0.307690i
$233$	4312.29	1.21248	0.606239	−	0.795282i	$-0.292677\pi$
$233$	4312.29	1.21248	0.606239	+	0.795282i	$0.292677\pi$
$234$	0	0
$235$	−8689.50	−2.41209
$236$	1743.78i	0.480976i
$237$	−1740.52	−0.477042
$238$	−529.273	−0.144150
$239$	− 3341.91i	− 0.904478i	−0.891897	−	0.452239i	$-0.850625\pi$
$239$	− 3341.91i	− 0.904478i	0.891897	−	0.452239i	$-0.149375\pi$
$240$	3164.80i	0.851196i
$241$	1981.50i	0.529625i	0.964300	+	0.264812i	$0.0853100\pi$
$241$	1981.50i	0.529625i	−0.964300	+	0.264812i	$0.914690\pi$
$242$	− 5329.18i	− 1.41559i
$243$	−243.000	−0.0641500
$244$	2027.16	0.531866
$245$	4630.72i	1.20753i
$246$	−3609.60	−0.935528
$247$	0	0
$248$	−2662.65	−0.681768
$249$	526.117i	0.133901i
$250$	−2880.99	−0.728839
$251$	−484.149	−0.121750	−0.0608749	−	0.998145i	$-0.519389\pi$
$251$	−484.149	−0.121750	−0.0608749	+	0.998145i	$0.519389\pi$
$252$	− 31.8369i	− 0.00795848i
$253$	3521.98i	0.875197i
$254$	− 5334.71i	− 1.31783i
$255$	− 4661.70i	− 1.14481i
$256$	3475.14	0.848423
$257$	−1964.83	−0.476897	−0.238448	−	0.971155i	$-0.576639\pi$
$257$	−1964.83	−0.476897	−0.238448	+	0.971155i	$0.576639\pi$
$258$	391.244i	0.0944101i
$259$	29.9195	0.00717801
$260$	0	0
$261$	−547.233	−0.129781
$262$	7375.58i	1.73918i
$263$	4886.27	1.14563	0.572814	−	0.819685i	$-0.305852\pi$
$263$	4886.27	1.14563	0.572814	+	0.819685i	$0.305852\pi$
$264$	2927.31	0.682438
$265$	1387.00i	0.321520i
$266$	− 485.420i	− 0.111891i
$267$	60.1053i	0.0137767i
$268$	1422.86i	0.324310i
$269$	−2379.98	−0.539443	−0.269722	−	0.962938i	$-0.586932\pi$
$269$	−2379.98	−0.539443	−0.269722	+	0.962938i	$0.586932\pi$
$270$	1186.83	0.267511
$271$	− 5468.73i	− 1.22584i	−0.790146	−	0.612919i	$-0.789995\pi$
$271$	− 5468.73i	− 1.22584i	0.790146	−	0.612919i	$-0.210005\pi$
$272$	−8886.89	−1.98105
$273$	0	0
$274$	2328.09	0.513303
$275$	− 3244.48i	− 0.711452i
$276$	479.218	0.104513
$277$	−5298.94	−1.14940	−0.574698	−	0.818366i	$-0.694880\pi$
$277$	−5298.94	−1.14940	−0.574698	+	0.818366i	$0.694880\pi$
$278$	5753.37i	1.24124i
$279$	1340.11i	0.287565i
$280$	− 347.133i	− 0.0740898i
$281$	6632.52i	1.40805i	0.710173	+	0.704027i	$0.248617\pi$
$281$	6632.52i	1.40805i	−0.710173	+	0.704027i	$0.751383\pi$
$282$	6212.15	1.31180
$283$	3693.82	0.775882	0.387941	−	0.921684i	$-0.373186\pi$
$283$	3693.82	0.775882	0.387941	+	0.921684i	$0.373186\pi$
$284$	905.294i	0.189152i
$285$	4275.45	0.888618
$286$	0	0
$287$	531.367	0.109288
$288$	− 975.024i	− 0.199492i
$289$	8177.24	1.66441
$290$	2672.72	0.541199
$291$	3680.15i	0.741354i
$292$	2388.85i	0.478757i
$293$	3051.86i	0.608503i	0.952592	+	0.304252i	$0.0984063\pi$
$293$	3051.86i	0.608503i	−0.952592	+	0.304252i	$0.901594\pi$
$294$	− 3310.52i	− 0.656712i
$295$	9569.36	1.88864
$296$	374.316	0.0735022
$297$	− 1473.32i	− 0.287847i
$298$	861.465	0.167461
$299$	0	0
$300$	−441.459	−0.0849589
$301$	− 57.5948i	− 0.0110289i
$302$	4586.98	0.874010
$303$	−173.940	−0.0329788
$304$	− 8150.57i	− 1.53772i
$305$	− 11124.5i	− 2.08848i
$306$	3332.66i	0.622600i
$307$	4737.05i	0.880643i	0.897840	+	0.440322i	$0.145136\pi$
$307$	4737.05i	0.880643i	−0.897840	+	0.440322i	$0.854864\pi$
$308$	193.028	0.0357104
$309$	−2864.97	−0.527451
$310$	− 6545.20i	− 1.19917i
$311$	10095.5	1.84072	0.920359	−	0.391076i	$-0.127897\pi$
$311$	10095.5	1.84072	0.920359	+	0.391076i	$0.127897\pi$
$312$	0	0
$313$	5235.36	0.945432	0.472716	−	0.881215i	$-0.343274\pi$
$313$	5235.36	0.945432	0.472716	+	0.881215i	$0.343274\pi$
$314$	115.243i	0.0207119i
$315$	−174.712	−0.0312505
$316$	−1435.87	−0.255614
$317$	− 6701.12i	− 1.18730i	−0.804725	−	0.593648i	$-0.797687\pi$
$317$	− 6701.12i	− 1.18730i	0.804725	−	0.593648i	$-0.202313\pi$
$318$	− 991.571i	− 0.174857i
$319$	− 3317.89i	− 0.582339i
$320$	− 3677.39i	− 0.642413i
$321$	−4814.04	−0.837052
$322$	−298.580	−0.0516746
$323$	12005.7i	2.06815i
$324$	−200.467	−0.0343736
$325$	0	0
$326$	−316.279	−0.0537334
$327$	− 4604.78i	− 0.778731i
$328$	6647.81	1.11910
$329$	−914.485	−0.153244
$330$	7195.78i	1.20035i
$331$	− 764.044i	− 0.126875i	−0.997986	−	0.0634376i	$-0.979794\pi$
$331$	− 764.044i	− 0.126875i	0.997986	−	0.0634376i	$-0.0202064\pi$
$332$	434.029i	0.0717482i
$333$	− 188.393i	− 0.0310026i
$334$	−8790.02	−1.44002
$335$	7808.27	1.27347
$336$	333.065i	0.0540779i
$337$	−3310.33	−0.535089	−0.267545	−	0.963545i	$-0.586212\pi$
$337$	−3310.33	−0.535089	−0.267545	+	0.963545i	$0.586212\pi$
$338$	0	0
$339$	5367.61	0.859966
$340$	− 3845.74i	− 0.613425i
$341$	−8125.15	−1.29033
$342$	−3056.53	−0.483270
$343$	977.598i	0.153893i
$344$	− 720.555i	− 0.112935i
$345$	− 2629.81i	− 0.410389i
$346$	− 6715.35i	− 1.04341i
$347$	869.809	0.134564	0.0672821	−	0.997734i	$-0.478567\pi$
$347$	869.809	0.134564	0.0672821	+	0.997734i	$0.478567\pi$
$348$	−451.448	−0.0695407
$349$	− 10694.9i	− 1.64036i	−0.572104	−	0.820181i	$-0.693873\pi$
$349$	− 10694.9i	− 1.64036i	0.572104	−	0.820181i	$-0.306127\pi$
$350$	275.054	0.0420065
$351$	0	0
$352$	5911.60	0.895140
$353$	1514.92i	0.228417i	0.993457	+	0.114208i	$0.0364332\pi$
$353$	1514.92i	0.228417i	−0.993457	+	0.114208i	$0.963567\pi$
$354$	−6841.16	−1.02713
$355$	4968.00	0.742744
$356$	49.5848i	0.00738200i
$357$	− 490.599i	− 0.0727317i
$358$	− 3312.54i	− 0.489031i
$359$	− 7006.81i	− 1.03010i	−0.857160	−	0.515049i	$-0.827774\pi$
$359$	− 7006.81i	− 1.03010i	0.857160	−	0.515049i	$-0.172226\pi$
$360$	−2185.78	−0.320002
$361$	−4151.92	−0.605325
$362$	− 5740.63i	− 0.833483i
$363$	4939.77	0.714244
$364$	0	0
$365$	13109.3	1.87993
$366$	7952.91i	1.13581i
$367$	−572.331	−0.0814045	−0.0407022	−	0.999171i	$-0.512960\pi$
$367$	−572.331	−0.0814045	−0.0407022	+	0.999171i	$0.512960\pi$
$368$	−5013.38	−0.710164
$369$	− 3345.85i	− 0.472027i
$370$	920.125i	0.129284i
$371$	145.968i	0.0204267i
$372$	1105.55i	0.154086i
$373$	2405.41	0.333907	0.166954	−	0.985965i	$-0.446607\pi$
$373$	2405.41	0.333907	0.166954	+	0.985965i	$0.446607\pi$
$374$	−20206.0	−2.79366
$375$	− 2670.47i	− 0.367740i
$376$	−11440.9	−1.56920
$377$	0	0
$378$	124.902	0.0169954
$379$	− 3161.53i	− 0.428488i	−0.976780	−	0.214244i	$-0.931271\pi$
$379$	− 3161.53i	− 0.428488i	0.976780	−	0.214244i	$-0.0687288\pi$
$380$	3527.10	0.476149
$381$	4944.89	0.664920
$382$	7286.53i	0.975946i
$383$	− 7568.71i	− 1.00977i	−0.863186	−	0.504887i	$-0.831534\pi$
$383$	− 7568.71i	− 1.00977i	0.863186	−	0.504887i	$-0.168466\pi$
$384$	5229.04i	0.694904i
$385$	− 1059.28i	− 0.140224i
$386$	−12546.2	−1.65437
$387$	−362.656	−0.0476352
$388$	3036.00i	0.397240i
$389$	6757.17	0.880726	0.440363	−	0.897820i	$-0.354850\pi$
$389$	6757.17	0.880726	0.440363	+	0.897820i	$0.354850\pi$
$390$	0	0
$391$	7384.62	0.955131
$392$	6096.98i	0.785572i
$393$	−6836.63	−0.877513
$394$	−2449.08	−0.313154
$395$	7879.65i	1.00372i
$396$	− 1215.44i	− 0.154237i
$397$	7313.92i	0.924623i	0.886718	+	0.462312i	$0.152980\pi$
$397$	7313.92i	0.924623i	−0.886718	+	0.462312i	$0.847020\pi$
$398$	6428.73i	0.809656i
$399$	449.950	0.0564553
$400$	4618.37	0.577296
$401$	− 5349.32i	− 0.666165i	−0.942898	−	0.333082i	$-0.891911\pi$
$401$	− 5349.32i	− 0.666165i	0.942898	−	0.333082i	$-0.108089\pi$
$402$	−5582.15	−0.692568
$403$	0	0
$404$	−143.494	−0.0176711
$405$	1100.10i	0.134974i
$406$	281.278	0.0343832
$407$	1142.23	0.139112
$408$	− 6137.77i	− 0.744766i
$409$	− 13444.2i	− 1.62537i	−0.582706	−	0.812683i	$-0.698006\pi$
$409$	− 13444.2i	− 1.62537i	0.582706	−	0.812683i	$-0.301994\pi$
$410$	16341.3i	1.96839i
$411$	2157.97i	0.258990i
$412$	−2363.50	−0.282624
$413$	1007.08	0.119989
$414$	1880.06i	0.223188i
$415$	2381.83	0.281733
$416$	0	0
$417$	−5332.96	−0.626274
$418$	− 18531.9i	− 2.16848i
$419$	−8313.65	−0.969328	−0.484664	−	0.874700i	$-0.661058\pi$
$419$	−8313.65	−0.969328	−0.484664	+	0.874700i	$0.661058\pi$
$420$	−144.131	−0.0167450
$421$	− 3491.02i	− 0.404138i	−0.979371	−	0.202069i	$-0.935234\pi$
$421$	− 3491.02i	− 0.404138i	0.979371	−	0.202069i	$-0.0647664\pi$
$422$	− 1263.73i	− 0.145776i
$423$	5758.22i	0.661877i
$424$	1826.18i	0.209167i
$425$	−6802.77	−0.776431
$426$	−3551.64	−0.403937
$427$	− 1170.74i	− 0.132684i
$428$	−3971.42	−0.448518
$429$	0	0
$430$	1771.23	0.198643
$431$	− 1269.80i	− 0.141912i	−0.997479	−	0.0709560i	$-0.977395\pi$
$431$	− 1269.80i	− 0.141912i	0.997479	−	0.0709560i	$-0.0226050\pi$
$432$	2097.20	0.233568
$433$	−6157.71	−0.683419	−0.341710	−	0.939806i	$-0.611006\pi$
$433$	−6157.71	−0.683419	−0.341710	+	0.939806i	$0.611006\pi$
$434$	− 688.819i	− 0.0761852i
$435$	2477.42i	0.273065i
$436$	− 3798.79i	− 0.417268i
$437$	6772.77i	0.741386i
$438$	−9371.91	−1.02239
$439$	2466.57	0.268162	0.134081	−	0.990970i	$-0.457192\pi$
$439$	2466.57	0.268162	0.134081	+	0.990970i	$0.457192\pi$
$440$	− 13252.5i	− 1.43588i
$441$	3068.61	0.331348
$442$	0	0
$443$	−1858.98	−0.199374	−0.0996871	−	0.995019i	$-0.531784\pi$
$443$	−1858.98	−0.199374	−0.0996871	+	0.995019i	$0.531784\pi$
$444$	− 155.418i	− 0.0166122i
$445$	272.108	0.0289868
$446$	−5873.09	−0.623540
$447$	798.516i	0.0844934i
$448$	− 387.010i	− 0.0408136i
$449$	− 5176.22i	− 0.544055i	−0.962289	−	0.272028i	$-0.912306\pi$
$449$	− 5176.22i	− 0.544055i	0.962289	−	0.272028i	$-0.0876942\pi$
$450$	− 1731.93i	− 0.181431i
$451$	20286.0	2.11802
$452$	4428.09	0.460797
$453$	4251.80i	0.440987i
$454$	−3387.51	−0.350184
$455$	0	0
$456$	5629.22	0.578098
$457$	− 3327.37i	− 0.340586i	−0.985393	−	0.170293i	$-0.945529\pi$
$457$	− 3327.37i	− 0.340586i	0.985393	−	0.170293i	$-0.0544714\pi$
$458$	−19477.5	−1.98717
$459$	−3089.14	−0.314137
$460$	− 2169.50i	− 0.219899i
$461$	8992.19i	0.908477i	0.890880	+	0.454238i	$0.150089\pi$
$461$	8992.19i	0.908477i	−0.890880	+	0.454238i	$0.849911\pi$
$462$	757.286i	0.0762600i
$463$	13143.4i	1.31928i	0.751582	+	0.659640i	$0.229291\pi$
$463$	13143.4i	1.31928i	−0.751582	+	0.659640i	$0.770709\pi$
$464$	4722.87	0.472529
$465$	6066.94	0.605048
$466$	13956.7i	1.38741i
$467$	−7798.23	−0.772718	−0.386359	−	0.922349i	$-0.626267\pi$
$467$	−7798.23	−0.772718	−0.386359	+	0.922349i	$0.626267\pi$
$468$	0	0
$469$	821.745	0.0809054
$470$	− 28123.5i	− 2.76009i
$471$	−106.822	−0.0104503
$472$	12599.4	1.22867
$473$	− 2198.79i	− 0.213743i
$474$	− 5633.18i	− 0.545866i
$475$	− 6239.13i	− 0.602676i
$476$	− 404.727i	− 0.0389719i
$477$	919.115	0.0882252
$478$	10816.1	1.03497
$479$	− 7341.94i	− 0.700338i	−0.936687	−	0.350169i	$-0.886124\pi$
$479$	− 7341.94i	− 0.700338i	0.936687	−	0.350169i	$-0.113876\pi$
$480$	−4414.11	−0.419741
$481$	0	0
$482$	−6413.11	−0.606036
$483$	− 276.762i	− 0.0260727i
$484$	4075.14	0.382714
$485$	16660.7	1.55984
$486$	− 786.468i	− 0.0734052i
$487$	− 2684.77i	− 0.249812i	−0.992169	−	0.124906i	$-0.960137\pi$
$487$	− 2684.77i	− 0.249812i	0.992169	−	0.124906i	$-0.0398629\pi$
$488$	− 14646.9i	− 1.35867i
$489$	− 293.168i	− 0.0271115i
$490$	−14987.3	−1.38175
$491$	9212.74	0.846772	0.423386	−	0.905949i	$-0.360841\pi$
$491$	9212.74	0.846772	0.423386	+	0.905949i	$0.360841\pi$
$492$	− 2760.21i	− 0.252926i
$493$	−6956.70	−0.635526
$494$	0	0
$495$	−6669.97	−0.605642
$496$	− 11565.8i	− 1.04701i
$497$	522.834	0.0471877
$498$	−1702.77	−0.153219
$499$	− 11401.9i	− 1.02289i	−0.859317	−	0.511443i	$-0.829111\pi$
$499$	− 11401.9i	− 1.02289i	0.859317	−	0.511443i	$-0.170889\pi$
$500$	− 2203.05i	− 0.197047i
$501$	− 8147.72i	− 0.726573i
$502$	− 1566.94i	− 0.139315i
$503$	1603.20	0.142114	0.0710570	−	0.997472i	$-0.477363\pi$
$503$	1603.20	0.142114	0.0710570	+	0.997472i	$0.477363\pi$
$504$	−230.032	−0.0203303
$505$	787.456i	0.0693888i
$506$	−11398.9	−1.00147
$507$	0	0
$508$	4079.37	0.356285
$509$	− 3427.55i	− 0.298475i	−0.988801	−	0.149237i	$-0.952318\pi$
$509$	− 3427.55i	− 0.298475i	0.988801	−	0.149237i	$-0.0476818\pi$
$510$	15087.6	1.30998
$511$	1379.63	0.119435
$512$	− 2696.83i	− 0.232781i
$513$	− 2833.19i	− 0.243837i
$514$	− 6359.15i	− 0.545701i
$515$	12970.2i	1.10978i
$516$	−299.178	−0.0255244
$517$	−34912.3	−2.96990
$518$	96.8342i	0.00821361i
$519$	6224.65	0.526458
$520$	0	0
$521$	−5244.77	−0.441032	−0.220516	−	0.975383i	$-0.570774\pi$
$521$	−5244.77	−0.441032	−0.220516	+	0.975383i	$0.570774\pi$
$522$	− 1771.12i	− 0.148505i
$523$	−15817.1	−1.32243	−0.661216	−	0.750196i	$-0.729959\pi$
$523$	−15817.1	−1.32243	−0.661216	+	0.750196i	$0.729959\pi$
$524$	−5639.99	−0.470198
$525$	254.956i	0.0211946i
$526$	15814.4i	1.31091i
$527$	17036.2i	1.40818i
$528$	12715.4i	1.04804i
$529$	−8001.10	−0.657607
$530$	−4489.02	−0.367907
$531$	− 6341.27i	− 0.518244i
$532$	371.193	0.0302505
$533$	0	0
$534$	−194.531	−0.0157643
$535$	21794.0i	1.76119i
$536$	10280.7	0.828464
$537$	3070.49	0.246743
$538$	− 7702.81i	− 0.617271i
$539$	18605.1i	1.48679i
$540$	907.549i	0.0723235i
$541$	5694.69i	0.452558i	0.974063	+	0.226279i	$0.0726561\pi$
$541$	5694.69i	0.452558i	−0.974063	+	0.226279i	$0.927344\pi$
$542$	17699.5	1.40269
$543$	5321.15	0.420539
$544$	− 12395.0i	− 0.976895i
$545$	−20846.7	−1.63848
$546$	0	0
$547$	−20598.9	−1.61014	−0.805070	−	0.593180i	$-0.797872\pi$
$547$	−20598.9	−1.61014	−0.805070	+	0.593180i	$0.797872\pi$
$548$	1780.25i	0.138775i
$549$	−7371.78	−0.573078
$550$	10500.7	0.814096
$551$	− 6380.31i	− 0.493304i
$552$	− 3462.51i	− 0.266982i
$553$	829.257i	0.0637678i
$554$	− 17150.0i	− 1.31522i
$555$	−852.890	−0.0652309
$556$	−4399.51	−0.335577
$557$	18421.7i	1.40135i	0.713482	+	0.700674i	$0.247117\pi$
$557$	18421.7i	1.40135i	−0.713482	+	0.700674i	$0.752883\pi$
$558$	−4337.27	−0.329053
$559$	0	0
$560$	1507.84	0.113782
$561$	− 18729.6i	− 1.40956i
$562$	−21466.1	−1.61120
$563$	6516.79	0.487833	0.243916	−	0.969796i	$-0.421568\pi$
$563$	6516.79	0.487833	0.243916	+	0.969796i	$0.421568\pi$
$564$	4750.33i	0.354654i
$565$	− 24300.1i	− 1.80941i
$566$	11955.0i	0.887822i
$567$	115.775i	0.00857515i
$568$	6541.05	0.483198
$569$	24233.0	1.78541	0.892706	−	0.450640i	$-0.148804\pi$
$569$	24233.0	1.78541	0.892706	+	0.450640i	$0.148804\pi$
$570$	13837.5i	1.01682i
$571$	−11060.7	−0.810640	−0.405320	−	0.914175i	$-0.632840\pi$
$571$	−11060.7	−0.810640	−0.405320	+	0.914175i	$0.632840\pi$
$572$	0	0
$573$	−6754.10	−0.492420
$574$	1719.77i	0.125055i
$575$	−3837.66	−0.278333
$576$	−2436.87	−0.176278
$577$	19118.6i	1.37941i	0.724092	+	0.689703i	$0.242259\pi$
$577$	19118.6i	1.37941i	−0.724092	+	0.689703i	$0.757741\pi$
$578$	26465.6i	1.90454i
$579$	− 11629.4i	− 0.834720i
$580$	2043.79i	0.146317i
$581$	250.664	0.0178990
$582$	−11910.8	−0.848312
$583$	5572.63i	0.395874i
$584$	17260.3	1.22300
$585$	0	0
$586$	−9877.32	−0.696294
$587$	21636.2i	1.52133i	0.649142	+	0.760667i	$0.275128\pi$
$587$	21636.2i	1.52133i	−0.649142	+	0.760667i	$0.724872\pi$
$588$	2531.50	0.177546
$589$	−15624.7	−1.09305
$590$	30971.2i	2.16112i
$591$	− 2270.12i	− 0.158004i
$592$	1625.92i	0.112880i
$593$	2148.58i	0.148789i	0.997229	+	0.0743943i	$0.0237023\pi$
$593$	2148.58i	0.148789i	−0.997229	+	0.0743943i	$0.976298\pi$
$594$	4768.38	0.329376
$595$	−2221.03	−0.153031
$596$	658.749i	0.0452742i
$597$	−5958.97	−0.408517
$598$	0	0
$599$	27957.0	1.90700	0.953499	−	0.301397i	$-0.0974529\pi$
$599$	27957.0	1.90700	0.953499	+	0.301397i	$0.0974529\pi$
$600$	3189.69i	0.217031i
$601$	−13217.8	−0.897111	−0.448555	−	0.893755i	$-0.648061\pi$
$601$	−13217.8	−0.897111	−0.448555	+	0.893755i	$0.648061\pi$
$602$	186.405	0.0126201
$603$	− 5174.26i	− 0.349440i
$604$	3507.59i	0.236295i
$605$	− 22363.2i	− 1.50280i
$606$	− 562.955i	− 0.0377368i
$607$	−23642.4	−1.58091	−0.790457	−	0.612518i	$-0.790157\pi$
$607$	−23642.4	−1.58091	−0.790457	+	0.612518i	$0.790157\pi$
$608$	11368.0	0.758279
$609$	260.725i	0.0173483i
$610$	36004.3	2.38979
$611$	0	0
$612$	−2548.43	−0.168324
$613$	8117.53i	0.534851i	0.963578	+	0.267426i	$0.0861730\pi$
$613$	8117.53i	0.534851i	−0.963578	+	0.267426i	$0.913827\pi$
$614$	−15331.4	−1.00770
$615$	−15147.2	−0.993164
$616$	− 1394.69i	− 0.0912237i
$617$	− 20928.0i	− 1.36553i	−0.730639	−	0.682764i	$-0.760778\pi$
$617$	− 20928.0i	− 1.36553i	0.730639	−	0.682764i	$-0.239222\pi$
$618$	− 9272.45i	− 0.603548i
$619$	− 3529.43i	− 0.229175i	−0.993413	−	0.114588i	$-0.963445\pi$
$619$	− 3529.43i	− 0.229175i	0.993413	−	0.114588i	$-0.0365547\pi$
$620$	5005.01	0.324204
$621$	−1742.68	−0.112611
$622$	32674.0i	2.10628i
$623$	28.6367	0.00184158
$624$	0	0
$625$	−19522.0	−1.24941
$626$	16944.2i	1.08183i
$627$	17177.7	1.09412
$628$	−88.1244	−0.00559959
$629$	− 2394.95i	− 0.151817i
$630$	− 565.455i	− 0.0357591i
$631$	10466.2i	0.660307i	0.943927	+	0.330154i	$0.107101\pi$
$631$	10466.2i	0.660307i	−0.943927	+	0.330154i	$0.892899\pi$
$632$	10374.6i	0.652976i
$633$	1171.39	0.0735521
$634$	21688.1	1.35859
$635$	− 22386.4i	− 1.39902i
$636$	758.239	0.0472738
$637$	0	0
$638$	10738.3	0.666356
$639$	− 3292.11i	− 0.203809i
$640$	23672.8	1.46211
$641$	26594.1	1.63870	0.819349	−	0.573295i	$-0.194335\pi$
$641$	26594.1	1.63870	0.819349	+	0.573295i	$0.194335\pi$
$642$	− 15580.6i	− 0.957816i
$643$	6965.66i	0.427214i	0.976920	+	0.213607i	$0.0685212\pi$
$643$	6965.66i	0.427214i	−0.976920	+	0.213607i	$0.931479\pi$
$644$	− 228.319i	− 0.0139706i
$645$	1641.81i	0.100227i
$646$	−38856.2	−2.36653
$647$	−3956.40	−0.240405	−0.120203	−	0.992749i	$-0.538354\pi$
$647$	−3956.40	−0.240405	−0.120203	+	0.992749i	$0.538354\pi$
$648$	1448.44i	0.0878087i
$649$	38447.3	2.32541
$650$	0	0
$651$	638.486	0.0384397
$652$	− 241.854i	− 0.0145272i
$653$	24311.1	1.45691	0.728457	−	0.685091i	$-0.240238\pi$
$653$	24311.1	1.45691	0.728457	+	0.685091i	$0.240238\pi$
$654$	14903.4	0.891082
$655$	30950.7i	1.84632i
$656$	28876.2i	1.71864i
$657$	− 8687.09i	− 0.515854i
$658$	− 2959.73i	− 0.175353i
$659$	8891.33	0.525580	0.262790	−	0.964853i	$-0.415357\pi$
$659$	8891.33	0.525580	0.262790	+	0.964853i	$0.415357\pi$
$660$	−5502.50	−0.324522
$661$	994.333i	0.0585099i	0.999572	+	0.0292550i	$0.00931347\pi$
$661$	994.333i	0.0585099i	−0.999572	+	0.0292550i	$0.990687\pi$
$662$	2472.82	0.145180
$663$	0	0
$664$	3136.00	0.183284
$665$	− 2037.01i	− 0.118785i
$666$	609.733	0.0354755
$667$	−3924.50	−0.227822
$668$	− 6721.59i	− 0.389321i
$669$	− 5443.94i	− 0.314611i
$670$	25271.4i	1.45719i
$671$	− 44695.4i	− 2.57145i
$672$	−464.542	−0.0266668
$673$	−5228.19	−0.299453	−0.149726	−	0.988727i	$-0.547839\pi$
$673$	−5228.19	−0.299453	−0.149726	+	0.988727i	$0.547839\pi$
$674$	− 10713.9i	− 0.612288i
$675$	1605.37	0.0915420
$676$	0	0
$677$	−10208.1	−0.579512	−0.289756	−	0.957101i	$-0.593574\pi$
$677$	−10208.1	−0.579512	−0.289756	+	0.957101i	$0.593574\pi$
$678$	17372.2i	0.984037i
$679$	1753.38	0.0990993
$680$	−27786.8	−1.56702
$681$	− 3139.98i	− 0.176688i
$682$	− 26297.0i	− 1.47649i
$683$	15514.5i	0.869172i	0.900630	+	0.434586i	$0.143105\pi$
$683$	15514.5i	0.869172i	−0.900630	+	0.434586i	$0.856895\pi$
$684$	− 2337.28i	− 0.130655i
$685$	9769.53	0.544927
$686$	−3163.99	−0.176096
$687$	− 18054.3i	− 1.00264i
$688$	3129.88	0.173438
$689$	0	0
$690$	8511.37	0.469598
$691$	17826.4i	0.981403i	0.871328	+	0.490701i	$0.163259\pi$
$691$	17826.4i	0.981403i	−0.871328	+	0.490701i	$0.836741\pi$
$692$	5135.12	0.282092
$693$	−701.950	−0.0384775
$694$	2815.13i	0.153978i
$695$	24143.3i	1.31771i
$696$	3261.87i	0.177645i
$697$	− 42534.1i	− 2.31147i
$698$	34614.1	1.87702
$699$	−12936.9	−0.700025
$700$	210.330i	0.0113567i
$701$	−9266.46	−0.499271	−0.249636	−	0.968340i	$-0.580311\pi$
$701$	−9266.46	−0.499271	−0.249636	+	0.968340i	$0.580311\pi$
$702$	0	0
$703$	2196.52	0.117842
$704$	− 14774.8i	− 0.790977i
$705$	26068.5	1.39262
$706$	−4903.04	−0.261371
$707$	82.8722i	0.00440839i
$708$	− 5231.33i	− 0.277691i
$709$	− 5844.49i	− 0.309583i	−0.987947	−	0.154792i	$-0.950529\pi$
$709$	− 5844.49i	− 0.309583i	0.987947	−	0.154792i	$-0.0494706\pi$
$710$	16078.9i	0.849902i
$711$	5221.56	0.275420
$712$	358.267	0.0188576
$713$	9610.67i	0.504800i
$714$	1587.82	0.0832250
$715$	0	0
$716$	2533.04	0.132213
$717$	10025.7i	0.522200i
$718$	22677.5	1.17872
$719$	−11776.3	−0.610825	−0.305412	−	0.952220i	$-0.598794\pi$
$719$	−11776.3	−0.610825	−0.305412	+	0.952220i	$0.598794\pi$
$720$	− 9494.40i	− 0.491438i
$721$	1364.99i	0.0705061i
$722$	− 13437.7i	− 0.692658i
$723$	− 5944.50i	− 0.305779i
$724$	4389.77	0.225338
$725$	3615.28	0.185197
$726$	15987.5i	0.817291i
$727$	−26770.6	−1.36570	−0.682851	−	0.730558i	$-0.739260\pi$
$727$	−26770.6	−1.36570	−0.682851	+	0.730558i	$0.739260\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	42428.3i	2.15115i
$731$	−4610.26	−0.233265
$732$	−6081.47	−0.307073
$733$	− 21653.8i	− 1.09113i	−0.838067	−	0.545567i	$-0.816314\pi$
$733$	− 21653.8i	− 1.09113i	0.838067	−	0.545567i	$-0.183686\pi$
$734$	− 1852.35i	− 0.0931490i
$735$	− 13892.2i	− 0.697170i
$736$	− 6992.41i	− 0.350195i
$737$	31371.7	1.56797
$738$	10828.8	0.540128
$739$	9986.74i	0.497115i	0.968617	+	0.248558i	$0.0799565\pi$
$739$	9986.74i	0.497115i	−0.968617	+	0.248558i	$0.920043\pi$
$740$	−703.605	−0.0349527
$741$	0	0
$742$	−472.426	−0.0233737
$743$	7505.34i	0.370584i	0.982683	+	0.185292i	$0.0593231\pi$
$743$	7505.34i	0.370584i	−0.982683	+	0.185292i	$0.940677\pi$
$744$	7987.96	0.393619
$745$	3615.03	0.177778
$746$	7785.10i	0.382081i
$747$	− 1578.35i	− 0.0773077i
$748$	− 15451.2i	− 0.755285i
$749$	2293.61i	0.111891i
$750$	8642.97	0.420796
$751$	−13441.9	−0.653134	−0.326567	−	0.945174i	$-0.605892\pi$
$751$	−13441.9	−0.653134	−0.326567	+	0.945174i	$0.605892\pi$
$752$	− 49696.0i	− 2.40988i
$753$	1452.45	0.0702922
$754$	0	0
$755$	19248.7	0.927856
$756$	95.5107i	0.00459483i
$757$	32951.1	1.58207	0.791035	−	0.611771i	$-0.209543\pi$
$757$	32951.1	1.58207	0.791035	+	0.611771i	$0.209543\pi$
$758$	10232.3	0.490308
$759$	− 10565.9i	− 0.505295i
$760$	− 25484.5i	− 1.21634i
$761$	9866.20i	0.469973i	0.971999	+	0.234986i	$0.0755046\pi$
$761$	9866.20i	0.469973i	−0.971999	+	0.234986i	$0.924495\pi$
$762$	16004.1i	0.760851i
$763$	−2193.91	−0.104096
$764$	−5571.90	−0.263854
$765$	13985.1i	0.660957i
$766$	24496.1	1.15546
$767$	0	0
$768$	−10425.4	−0.489837
$769$	− 31948.6i	− 1.49818i	−0.662471	−	0.749088i	$-0.730492\pi$
$769$	− 31948.6i	− 1.49818i	0.662471	−	0.749088i	$-0.269508\pi$
$770$	3428.37	0.160454
$771$	5894.48	0.275337
$772$	− 9593.89i	− 0.447269i
$773$	10227.7i	0.475893i	0.971278	+	0.237946i	$0.0764743\pi$
$773$	10227.7i	0.475893i	−0.971278	+	0.237946i	$0.923526\pi$
$774$	− 1173.73i	− 0.0545077i
$775$	− 8853.43i	− 0.410354i
$776$	21936.1	1.01477
$777$	−89.7584	−0.00414423
$778$	21869.6i	1.00779i
$779$	39009.9	1.79419
$780$	0	0
$781$	19960.2	0.914510
$782$	23900.3i	1.09293i
$783$	1641.70	0.0749292
$784$	−26483.5	−1.20643
$785$	483.602i	0.0219879i
$786$	− 22126.7i	− 1.00411i
$787$	18651.6i	0.844802i	0.906409	+	0.422401i	$0.138813\pi$
$787$	18651.6i	0.844802i	−0.906409	+	0.422401i	$0.861187\pi$
$788$	− 1872.77i	− 0.0846634i
$789$	−14658.8	−0.661429
$790$	−25502.4	−1.14853
$791$	− 2557.35i	− 0.114955i
$792$	−8781.93	−0.394006
$793$	0	0
$794$	−23671.5	−1.05802
$795$	− 4161.00i	− 0.185630i
$796$	−4915.95	−0.218896
$797$	−32566.8	−1.44740	−0.723698	−	0.690117i	$-0.757559\pi$
$797$	−32566.8	−1.44740	−0.723698	+	0.690117i	$0.757559\pi$
$798$	1456.26i	0.0646004i
$799$	73201.4i	3.24115i
$800$	6441.47i	0.284675i
$801$	− 180.316i	− 0.00795400i
$802$	17313.0	0.762275
$803$	52670.1	2.31468
$804$	− 4268.59i	− 0.187241i
$805$	−1252.95	−0.0548581
$806$	0	0
$807$	7139.95	0.311448
$808$	1036.79i	0.0451415i
$809$	23895.3	1.03846	0.519231	−	0.854634i	$-0.326219\pi$
$809$	23895.3	1.03846	0.519231	+	0.854634i	$0.326219\pi$
$810$	−3560.48	−0.154448
$811$	219.216i	0.00949165i	0.999989	+	0.00474582i	$0.00151065\pi$
$811$	219.216i	0.00949165i	−0.999989	+	0.00474582i	$0.998489\pi$
$812$	215.089i	0.00929574i
$813$	16406.2i	0.707738i
$814$	3696.83i	0.159182i
$815$	−1327.23	−0.0570438
$816$	26660.7	1.14376
$817$	− 4228.28i	− 0.181063i
$818$	43512.2	1.85986
$819$	0	0
$820$	−12496.0	−0.532168
$821$	7550.84i	0.320982i	0.987037	+	0.160491i	$0.0513078\pi$
$821$	7550.84i	0.320982i	−0.987037	+	0.160491i	$0.948692\pi$
$822$	−6984.27	−0.296356
$823$	14082.8	0.596471	0.298235	−	0.954492i	$-0.403602\pi$
$823$	14082.8	0.596471	0.298235	+	0.954492i	$0.403602\pi$
$824$	17077.1i	0.721976i
$825$	9733.43i	0.410757i
$826$	3259.42i	0.137300i
$827$	− 5424.18i	− 0.228074i	−0.993477	−	0.114037i	$-0.963622\pi$
$827$	− 5424.18i	− 0.228074i	0.993477	−	0.114037i	$-0.0363782\pi$
$828$	−1437.65	−0.0603404
$829$	−26888.3	−1.12650	−0.563251	−	0.826286i	$-0.690450\pi$
$829$	−26888.3	−1.12650	−0.563251	+	0.826286i	$0.690450\pi$
$830$	7708.77i	0.322380i
$831$	15896.8	0.663604
$832$	0	0
$833$	39009.8	1.62258
$834$	− 17260.1i	− 0.716629i
$835$	−36886.2	−1.52874
$836$	14171.0	0.586262
$837$	− 4020.34i	− 0.166025i
$838$	− 26907.1i	− 1.10918i
$839$	23680.9i	0.974441i	0.873279	+	0.487220i	$0.161989\pi$
$839$	23680.9i	0.974441i	−0.873279	+	0.487220i	$0.838011\pi$
$840$	1041.40i	0.0427758i
$841$	−20691.9	−0.848412
$842$	11298.7	0.462444
$843$	− 19897.6i	− 0.812941i
$844$	966.354	0.0394115
$845$	0	0
$846$	−18636.4	−0.757369
$847$	− 2353.51i	− 0.0954754i
$848$	−7932.38	−0.321225
$849$	−11081.5	−0.447956
$850$	− 22017.1i	− 0.888449i
$851$	− 1351.07i	− 0.0544230i
$852$	− 2715.88i	− 0.109207i
$853$	− 9653.82i	− 0.387504i	−0.981051	−	0.193752i	$-0.937934\pi$
$853$	− 9653.82i	− 0.387504i	0.981051	−	0.193752i	$-0.0620656\pi$
$854$	3789.10	0.151827
$855$	−12826.4	−0.513044
$856$	28694.8i	1.14576i
$857$	−41438.8	−1.65172	−0.825859	−	0.563877i	$-0.809309\pi$
$857$	−41438.8	−1.65172	−0.825859	+	0.563877i	$0.809309\pi$
$858$	0	0
$859$	−20696.9	−0.822085	−0.411042	−	0.911616i	$-0.634835\pi$
$859$	−20696.9	−0.822085	−0.411042	+	0.911616i	$0.634835\pi$
$860$	1354.43i	0.0537045i
$861$	−1594.10	−0.0630974
$862$	4109.70	0.162386
$863$	− 19513.8i	− 0.769707i	−0.922978	−	0.384854i	$-0.874252\pi$
$863$	− 19513.8i	− 0.769707i	0.922978	−	0.384854i	$-0.125748\pi$
$864$	2925.07i	0.115177i
$865$	− 28180.1i	− 1.10769i
$866$	− 19929.4i	− 0.782018i
$867$	−24531.7	−0.960947
$868$	526.729	0.0205972
$869$	31658.5i	1.23583i
$870$	−8018.16	−0.312461
$871$	0	0
$872$	−27447.5	−1.06593
$873$	− 11040.4i	− 0.428021i
$874$	−21920.0	−0.848348
$875$	−1272.33	−0.0491571
$876$	− 7166.55i	− 0.276410i
$877$	− 35595.5i	− 1.37055i	−0.728282	−	0.685277i	$-0.759681\pi$
$877$	− 35595.5i	− 1.37055i	0.728282	−	0.685277i	$-0.240319\pi$
$878$	7983.05i	0.306851i
$879$	− 9155.57i	− 0.351319i
$880$	57564.9	2.20513
$881$	−4277.22	−0.163568	−0.0817838	−	0.996650i	$-0.526062\pi$
$881$	−4277.22	−0.163568	−0.0817838	+	0.996650i	$0.526062\pi$
$882$	9931.55i	0.379153i
$883$	−355.428	−0.0135460	−0.00677300	−	0.999977i	$-0.502156\pi$
$883$	−355.428	−0.0135460	−0.00677300	+	0.999977i	$0.502156\pi$
$884$	0	0
$885$	−28708.1	−1.09041
$886$	− 6016.58i	− 0.228139i
$887$	−7133.87	−0.270047	−0.135024	−	0.990842i	$-0.543111\pi$
$887$	−7133.87	−0.270047	−0.135024	+	0.990842i	$0.543111\pi$
$888$	−1122.95	−0.0424365
$889$	− 2355.96i	− 0.0888821i
$890$	880.675i	0.0331689i
$891$	4419.95i	0.166188i
$892$	− 4491.06i	− 0.168578i
$893$	−67136.3	−2.51582
$894$	−2584.39	−0.0966835
$895$	− 13900.6i	− 0.519159i
$896$	2491.33	0.0928902
$897$	0	0
$898$	16752.8	0.622548
$899$	− 9053.76i	− 0.335884i
$900$	1324.38	0.0490510
$901$	11684.3	0.432030
$902$	65655.4i	2.42360i
$903$	172.784i	0.00636756i
$904$	− 31994.5i	− 1.17712i
$905$	− 24089.8i	− 0.884832i
$906$	−13760.9	−0.504610
$907$	21719.2	0.795122	0.397561	−	0.917576i	$-0.369857\pi$
$907$	21719.2	0.795122	0.397561	+	0.917576i	$0.369857\pi$
$908$	− 2590.38i	− 0.0946747i
$909$	521.819	0.0190403
$910$	0	0
$911$	−39331.5	−1.43042	−0.715209	−	0.698911i	$-0.753669\pi$
$911$	−39331.5	−1.43042	−0.715209	+	0.698911i	$0.753669\pi$
$912$	24451.7i	0.887804i
$913$	9569.59	0.346886
$914$	10769.0	0.389723
$915$	33373.4i	1.20578i
$916$	− 14894.2i	− 0.537246i
$917$	3257.26i	0.117300i
$918$	− 9997.98i	− 0.359458i
$919$	−33117.0	−1.18871	−0.594357	−	0.804202i	$-0.702593\pi$
$919$	−33117.0	−1.18871	−0.594357	+	0.804202i	$0.702593\pi$
$920$	−15675.4	−0.561742
$921$	− 14211.1i	− 0.508440i
$922$	−29103.2	−1.03955
$923$	0	0
$924$	−579.085	−0.0206174
$925$	1244.61i	0.0442407i
$926$	−42538.6	−1.50962
$927$	8594.90	0.304524
$928$	6587.22i	0.233013i
$929$	39917.7i	1.40975i	0.709332	+	0.704874i	$0.248997\pi$
$929$	39917.7i	1.40975i	−0.709332	+	0.704874i	$0.751003\pi$
$930$	19635.6i	0.692341i
$931$	35777.6i	1.25947i
$932$	−10672.5	−0.375095
$933$	−30286.5	−1.06274
$934$	− 25238.9i	− 0.884201i
$935$	−84792.1	−2.96577
$936$	0	0
$937$	−37219.5	−1.29766	−0.648830	−	0.760934i	$-0.724741\pi$
$937$	−37219.5	−1.29766	−0.648830	+	0.760934i	$0.724741\pi$
$938$	2659.57i	0.0925779i
$939$	−15706.1	−0.545845
$940$	21505.6	0.746208
$941$	− 53360.4i	− 1.84856i	−0.381710	−	0.924282i	$-0.624665\pi$
$941$	− 53360.4i	− 1.84856i	0.381710	−	0.924282i	$-0.375335\pi$
$942$	− 345.728i	− 0.0119580i
$943$	− 23994.8i	− 0.828610i
$944$	54728.0i	1.88691i
$945$	524.136	0.0180425
$946$	7116.38	0.244581
$947$	29333.8i	1.00657i	0.864121	+	0.503284i	$0.167875\pi$
$947$	29333.8i	1.00657i	−0.864121	+	0.503284i	$0.832125\pi$
$948$	4307.61	0.147579
$949$	0	0
$950$	20192.9	0.689626
$951$	20103.4i	0.685485i
$952$	−2924.29	−0.0995554
$953$	18645.0	0.633759	0.316879	−	0.948466i	$-0.397365\pi$
$953$	18645.0	0.633759	0.316879	+	0.948466i	$0.397365\pi$
$954$	2974.71i	0.100954i
$955$	30577.0i	1.03607i
$956$	8270.88i	0.279811i
$957$	9953.67i	0.336214i
$958$	23762.2	0.801378
$959$	1028.15	0.0346201
$960$	11032.2i	0.370898i
$961$	7619.34	0.255760
$962$	0	0
$963$	14442.1	0.483272
$964$	− 4904.01i	− 0.163846i
$965$	−52648.6	−1.75629
$966$	895.740	0.0298343
$967$	43847.6i	1.45816i	0.684427	+	0.729081i	$0.260052\pi$
$967$	43847.6i	1.45816i	−0.684427	+	0.729081i	$0.739948\pi$
$968$	− 29444.3i	− 0.977659i
$969$	− 36017.0i	− 1.19405i
$970$	53922.2i	1.78489i
$971$	11602.4	0.383459	0.191729	−	0.981448i	$-0.438590\pi$
$971$	11602.4	0.383459	0.191729	+	0.981448i	$0.438590\pi$
$972$	601.400	0.0198456
$973$	2540.85i	0.0837161i
$974$	8689.24	0.285853
$975$	0	0
$976$	63621.8	2.08656
$977$	− 11324.2i	− 0.370823i	−0.982661	−	0.185411i	$-0.940638\pi$
$977$	− 11324.2i	− 0.370823i	0.982661	−	0.185411i	$-0.0593618\pi$
$978$	948.837	0.0310230
$979$	1093.26	0.0356903
$980$	− 11460.6i	− 0.373565i
$981$	13814.4i	0.449601i
$982$	29817.0i	0.968939i
$983$	47443.8i	1.53939i	0.638410	+	0.769696i	$0.279592\pi$
$983$	47443.8i	1.53939i	−0.638410	+	0.769696i	$0.720408\pi$
$984$	−19943.4	−0.646111
$985$	−10277.3	−0.332447
$986$	− 22515.3i	− 0.727215i
$987$	2743.46	0.0884754
$988$	0	0
$989$	−2600.80	−0.0836203
$990$	− 21587.3i	− 0.693021i
$991$	51752.6	1.65891	0.829453	−	0.558577i	$-0.188653\pi$
$991$	51752.6	1.65891	0.829453	+	0.558577i	$0.188653\pi$
$992$	16131.4	0.516303
$993$	2292.13i	0.0732514i
$994$	1692.15i	0.0539957i
$995$	26977.3i	0.859537i
$996$	− 1302.09i	− 0.0414239i
$997$	16949.8	0.538420	0.269210	−	0.963082i	$-0.413237\pi$
$997$	16949.8	0.538420	0.269210	+	0.963082i	$0.413237\pi$
$998$	36902.3	1.17046
$999$	565.180i	0.0178994i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.j.337.14		18
13.5	odd	4		507.4.a.n.1.8	✓	9
13.8	odd	4		507.4.a.q.1.2	yes	9
13.12	even	2	inner	507.4.b.j.337.5		18
39.5	even	4		1521.4.a.bj.1.2		9
39.8	even	4		1521.4.a.be.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.8	✓	9	13.5	odd	4
507.4.a.q.1.2	yes	9	13.8	odd	4
507.4.b.j.337.5		18	13.12	even	2	inner
507.4.b.j.337.14		18	1.1	even	1	trivial
1521.4.a.be.1.8		9	39.8	even	4
1521.4.a.bj.1.2		9	39.5	even	4

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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.23649i q^{2} -3.00000 q^{3} -2.47490 q^{4} +13.5815i q^{5} -9.70948i q^{6} +1.42933i q^{7} +17.8820i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+3.23649i q^{2} -3.00000 q^{3} -2.47490 q^{4} +13.5815i q^{5} -9.70948i q^{6} +1.42933i q^{7} +17.8820i q^{8} +9.00000 q^{9} -43.9566 q^{10} +54.5673i q^{11} +7.42469 q^{12} -4.62601 q^{14} -40.7446i q^{15} -77.6741 q^{16} +114.413 q^{17} +29.1285i q^{18} +104.933i q^{19} -33.6129i q^{20} -4.28798i q^{21} -176.607 q^{22} +64.5438 q^{23} -53.6459i q^{24} -59.4583 q^{25} -27.0000 q^{27} -3.53743i q^{28} -60.8037 q^{29} +131.870 q^{30} +148.902i q^{31} -108.336i q^{32} -163.702i q^{33} +370.296i q^{34} -19.4125 q^{35} -22.2741 q^{36} -20.9326i q^{37} -339.615 q^{38} -242.865 q^{40} -371.761i q^{41} +13.8780 q^{42} -40.2951 q^{43} -135.048i q^{44} +122.234i q^{45} +208.896i q^{46} +639.802i q^{47} +233.022 q^{48} +340.957 q^{49} -192.436i q^{50} -343.238 q^{51} +102.124 q^{53} -87.3854i q^{54} -741.108 q^{55} -25.5592 q^{56} -314.799i q^{57} -196.791i q^{58} -704.586i q^{59} +100.839i q^{60} -819.087 q^{61} -481.919 q^{62} +12.8639i q^{63} -270.764 q^{64} +529.820 q^{66} -574.918i q^{67} -283.159 q^{68} -193.631 q^{69} -62.8283i q^{70} -365.790i q^{71} +160.938i q^{72} -965.233i q^{73} +67.7482 q^{74} +178.375 q^{75} -259.698i q^{76} -77.9945 q^{77} +580.173 q^{79} -1054.93i q^{80} +81.0000 q^{81} +1203.20 q^{82} -175.372i q^{83} +10.6123i q^{84} +1553.90i q^{85} -130.415i q^{86} +182.411 q^{87} -975.770 q^{88} -20.0351i q^{89} -395.609 q^{90} -159.739 q^{92} -446.705i q^{93} -2070.72 q^{94} -1425.15 q^{95} +325.008i q^{96} -1226.72i q^{97} +1103.51i q^{98} +491.106i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9}+O(q^{10}) \) 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9	\( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9} - 396 q^{10} + 192 q^{12} + 196 q^{14} + 64 q^{16} + 268 q^{17} + 548 q^{22} - 452 q^{23} - 1224 q^{25} - 486 q^{27} - 1094 q^{29} + 1188 q^{30} + 276 q^{35} - 576 q^{36} + 832 q^{38} + 2684 q^{40} - 588 q^{42} - 316 q^{43} - 192 q^{48} - 1284 q^{49} - 804 q^{51} + 2798 q^{53} - 2816 q^{55} + 1232 q^{56} + 4184 q^{61} + 586 q^{62} - 4962 q^{64} - 1644 q^{66} - 3158 q^{68} + 1356 q^{69} + 2074 q^{74} + 3672 q^{75} + 3372 q^{77} - 230 q^{79} + 1458 q^{81} + 10294 q^{82} + 3282 q^{87} + 968 q^{88} - 3564 q^{90} + 4174 q^{92} - 936 q^{94} + 444 q^{95}+O(q^{100}) \) 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9 - 396 * q^10 + 192 * q^12 + 196 * q^14 + 64 * q^16 + 268 * q^17 + 548 * q^22 - 452 * q^23 - 1224 * q^25 - 486 * q^27 - 1094 * q^29 + 1188 * q^30 + 276 * q^35 - 576 * q^36 + 832 * q^38 + 2684 * q^40 - 588 * q^42 - 316 * q^43 - 192 * q^48 - 1284 * q^49 - 804 * q^51 + 2798 * q^53 - 2816 * q^55 + 1232 * q^56 + 4184 * q^61 + 586 * q^62 - 4962 * q^64 - 1644 * q^66 - 3158 * q^68 + 1356 * q^69 + 2074 * q^74 + 3672 * q^75 + 3372 * q^77 - 230 * q^79 + 1458 * q^81 + 10294 * q^82 + 3282 * q^87 + 968 * q^88 - 3564 * q^90 + 4174 * q^92 - 936 * q^94 + 444 * q^95

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