LMFDB - Embedded newform 507.4.b.k.337.15

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} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + \cdots + 729 $ x^18 + 112x^16 + 5026x^14 + 114847x^12 + 1397921x^10 + 8545747x^8 + 21033277x^6 + 6703200x^4 + 137781x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.15
Root			$4.83218i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.k.337.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.03025i q^{2} +3.00000 q^{3} -8.24289 q^{4} +8.08864i q^{5} +12.0907i q^{6} -5.95078i q^{7} -0.978887i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.03025i q^{2} +3.00000 q^{3} -8.24289 q^{4} +8.08864i q^{5} +12.0907i q^{6} -5.95078i q^{7} -0.978887i q^{8} +9.00000 q^{9} -32.5992 q^{10} -17.2359i q^{11} -24.7287 q^{12} +23.9831 q^{14} +24.2659i q^{15} -61.9979 q^{16} -92.9299 q^{17} +36.2722i q^{18} +13.3832i q^{19} -66.6738i q^{20} -17.8523i q^{21} +69.4648 q^{22} -219.710 q^{23} -2.93666i q^{24} +59.5738 q^{25} +27.0000 q^{27} +49.0516i q^{28} -199.485 q^{29} -97.7977 q^{30} +307.777i q^{31} -257.698i q^{32} -51.7076i q^{33} -374.531i q^{34} +48.1337 q^{35} -74.1860 q^{36} +333.777i q^{37} -53.9376 q^{38} +7.91787 q^{40} -200.689i q^{41} +71.9493 q^{42} -116.806 q^{43} +142.073i q^{44} +72.7978i q^{45} -885.487i q^{46} -338.610i q^{47} -185.994 q^{48} +307.588 q^{49} +240.097i q^{50} -278.790 q^{51} -26.6215 q^{53} +108.817i q^{54} +139.415 q^{55} -5.82514 q^{56} +40.1496i q^{57} -803.976i q^{58} +280.058i q^{59} -200.021i q^{60} -207.084 q^{61} -1240.42 q^{62} -53.5570i q^{63} +542.603 q^{64} +208.394 q^{66} -285.981i q^{67} +766.011 q^{68} -659.131 q^{69} +193.991i q^{70} -317.673i q^{71} -8.80999i q^{72} +63.0668i q^{73} -1345.20 q^{74} +178.721 q^{75} -110.316i q^{76} -102.567 q^{77} -623.835 q^{79} -501.479i q^{80} +81.0000 q^{81} +808.824 q^{82} +659.874i q^{83} +147.155i q^{84} -751.677i q^{85} -470.755i q^{86} -598.456 q^{87} -16.8720 q^{88} -1273.19i q^{89} -293.393 q^{90} +1811.05 q^{92} +923.331i q^{93} +1364.68 q^{94} -108.252 q^{95} -773.094i q^{96} +603.746i q^{97} +1239.66i q^{98} -155.123i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * 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 $

$2$	4.03025i	1.42491i	0.701719	+	0.712454i	$0.252416\pi$
$2$	4.03025i	1.42491i	−0.701719	+	0.712454i	$0.747584\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−8.24289	−1.03036
$5$	8.08864i	0.723470i	0.932281	+	0.361735i	$0.117816\pi$
$5$	8.08864i	0.723470i	−0.932281	+	0.361735i	$0.882184\pi$
$6$	12.0907i	0.822671i
$7$	− 5.95078i	− 0.321312i	−0.987010	−	0.160656i	$-0.948639\pi$
$7$	− 5.95078i	− 0.321312i	0.987010	−	0.160656i	$-0.0513610\pi$
$8$	− 0.978887i	− 0.0432611i
$9$	9.00000	0.333333
$10$	−32.5992	−1.03088
$11$	− 17.2359i	− 0.472437i	−0.971700	−	0.236219i	$-0.924092\pi$
$11$	− 17.2359i	− 0.472437i	0.971700	−	0.236219i	$-0.0759082\pi$
$12$	−24.7287	−0.594879
$13$	0	0
$13$	0	0
$14$	23.9831	0.457840
$15$	24.2659i	0.417696i
$16$	−61.9979	−0.968718
$17$	−92.9299	−1.32581	−0.662907	−	0.748702i	$-0.730677\pi$
$17$	−92.9299	−1.32581	−0.662907	+	0.748702i	$0.730677\pi$
$18$	36.2722i	0.474969i
$19$	13.3832i	0.161596i	0.996731	+	0.0807979i	$0.0257468\pi$
$19$	13.3832i	0.161596i	−0.996731	+	0.0807979i	$0.974253\pi$
$20$	− 66.6738i	− 0.745435i
$21$	− 17.8523i	− 0.185510i
$22$	69.4648	0.673179
$23$	−219.710	−1.99186	−0.995930	−	0.0901293i	$-0.971272\pi$
$23$	−219.710	−1.99186	−0.995930	+	0.0901293i	$0.971272\pi$
$24$	− 2.93666i	− 0.0249768i
$25$	59.5738	0.476591
$26$	0	0
$27$	27.0000	0.192450
$28$	49.0516i	0.331067i
$29$	−199.485	−1.27736	−0.638681	−	0.769471i	$-0.720520\pi$
$29$	−199.485	−1.27736	−0.638681	+	0.769471i	$0.720520\pi$
$30$	−97.7977	−0.595178
$31$	307.777i	1.78317i	0.452851	+	0.891586i	$0.350407\pi$
$31$	307.777i	1.78317i	−0.452851	+	0.891586i	$0.649593\pi$
$32$	− 257.698i	− 1.42359i
$33$	− 51.7076i	− 0.272762i
$34$	− 374.531i	− 1.88916i
$35$	48.1337	0.232460
$36$	−74.1860	−0.343454
$37$	333.777i	1.48304i	0.670930	+	0.741521i	$0.265895\pi$
$37$	333.777i	1.48304i	−0.670930	+	0.741521i	$0.734105\pi$
$38$	−53.9376	−0.230259
$39$	0	0
$40$	7.91787	0.0312981
$41$	− 200.689i	− 0.764446i	−0.924070	−	0.382223i	$-0.875159\pi$
$41$	− 200.689i	− 0.764446i	0.924070	−	0.382223i	$-0.124841\pi$
$42$	71.9493	0.264334
$43$	−116.806	−0.414248	−0.207124	−	0.978315i	$-0.566410\pi$
$43$	−116.806	−0.414248	−0.207124	+	0.978315i	$0.566410\pi$
$44$	142.073i	0.486781i
$45$	72.7978i	0.241157i
$46$	− 885.487i	− 2.83822i
$47$	− 338.610i	− 1.05088i	−0.850831	−	0.525440i	$-0.823901\pi$
$47$	− 338.610i	− 1.05088i	0.850831	−	0.525440i	$-0.176099\pi$
$48$	−185.994	−0.559289
$49$	307.588	0.896759
$50$	240.097i	0.679097i
$51$	−278.790	−0.765459
$52$	0	0
$53$	−26.6215	−0.0689951	−0.0344975	−	0.999405i	$-0.510983\pi$
$53$	−26.6215	−0.0689951	−0.0344975	+	0.999405i	$0.510983\pi$
$54$	108.817i	0.274224i
$55$	139.415	0.341794
$56$	−5.82514	−0.0139003
$57$	40.1496i	0.0932973i
$58$	− 803.976i	− 1.82012i
$59$	280.058i	0.617973i	0.951066	+	0.308987i	$0.0999899\pi$
$59$	280.058i	0.617973i	−0.951066	+	0.308987i	$0.900010\pi$
$60$	− 200.021i	− 0.430377i
$61$	−207.084	−0.434663	−0.217332	−	0.976098i	$-0.569735\pi$
$61$	−207.084	−0.434663	−0.217332	+	0.976098i	$0.569735\pi$
$62$	−1240.42	−2.54086
$63$	− 53.5570i	− 0.107104i
$64$	542.603	1.05977
$65$	0	0
$66$	208.394	0.388660
$67$	− 285.981i	− 0.521465i	−0.965411	−	0.260732i	$-0.916036\pi$
$67$	− 285.981i	− 0.521465i	0.965411	−	0.260732i	$-0.0839640\pi$
$68$	766.011	1.36607
$69$	−659.131	−1.15000
$70$	193.991i	0.331233i
$71$	− 317.673i	− 0.530998i	−0.964111	−	0.265499i	$-0.914463\pi$
$71$	− 317.673i	− 0.530998i	0.964111	−	0.265499i	$-0.0855368\pi$
$72$	− 8.80999i	− 0.0144204i
$73$	63.0668i	0.101115i	0.998721	+	0.0505576i	$0.0160999\pi$
$73$	63.0668i	0.101115i	−0.998721	+	0.0505576i	$0.983900\pi$
$74$	−1345.20	−2.11320
$75$	178.721	0.275160
$76$	− 110.316i	− 0.166502i
$77$	−102.567	−0.151800
$78$	0	0
$79$	−623.835	−0.888443	−0.444221	−	0.895917i	$-0.646520\pi$
$79$	−623.835	−0.888443	−0.444221	+	0.895917i	$0.646520\pi$
$80$	− 501.479i	− 0.700838i
$81$	81.0000	0.111111
$82$	808.824	1.08926
$83$	659.874i	0.872658i	0.899787	+	0.436329i	$0.143722\pi$
$83$	659.874i	0.872658i	−0.899787	+	0.436329i	$0.856278\pi$
$84$	147.155i	0.191142i
$85$	− 751.677i	− 0.959186i
$86$	− 470.755i	− 0.590265i
$87$	−598.456	−0.737486
$88$	−16.8720	−0.0204382
$89$	− 1273.19i	− 1.51638i	−0.652032	−	0.758191i	$-0.726083\pi$
$89$	− 1273.19i	− 1.51638i	0.652032	−	0.758191i	$-0.273917\pi$
$90$	−293.393	−0.343626
$91$	0	0
$92$	1811.05	2.05233
$93$	923.331i	1.02952i
$94$	1364.68	1.49741
$95$	−108.252	−0.116910
$96$	− 773.094i	− 0.821912i
$97$	603.746i	0.631970i	0.948764	+	0.315985i	$0.102335\pi$
$97$	603.746i	0.631970i	−0.948764	+	0.315985i	$0.897665\pi$
$98$	1239.66i	1.27780i
$99$	− 155.123i	− 0.157479i
$100$	−491.060	−0.491060
$101$	740.588	0.729616	0.364808	−	0.931083i	$-0.381135\pi$
$101$	740.588	0.729616	0.364808	+	0.931083i	$0.381135\pi$
$102$	− 1123.59i	− 1.09071i
$103$	1888.57	1.80666	0.903332	−	0.428942i	$-0.141113\pi$
$103$	1888.57	1.80666	0.903332	+	0.428942i	$0.141113\pi$
$104$	0	0
$105$	144.401	0.134211
$106$	− 107.291i	− 0.0983116i
$107$	−919.463	−0.830728	−0.415364	−	0.909655i	$-0.636346\pi$
$107$	−919.463	−0.830728	−0.415364	+	0.909655i	$0.636346\pi$
$108$	−222.558	−0.198293
$109$	1570.40i	1.37998i	0.723821	+	0.689988i	$0.242384\pi$
$109$	1570.40i	1.37998i	−0.723821	+	0.689988i	$0.757616\pi$
$110$	561.876i	0.487025i
$111$	1001.33i	0.856234i
$112$	368.936i	0.311260i
$113$	−1324.66	−1.10277	−0.551386	−	0.834250i	$-0.685901\pi$
$113$	−1324.66	−1.10277	−0.551386	+	0.834250i	$0.685901\pi$
$114$	−161.813	−0.132940
$115$	− 1777.16i	− 1.44105i
$116$	1644.34	1.31614
$117$	0	0
$118$	−1128.70	−0.880555
$119$	553.006i	0.426000i
$120$	23.7536	0.0180700
$121$	1033.92	0.776803
$122$	− 834.601i	− 0.619354i
$123$	− 602.066i	− 0.441353i
$124$	− 2536.97i	− 1.83731i
$125$	1492.95i	1.06827i
$126$	215.848	0.152613
$127$	2350.39	1.64223	0.821115	−	0.570763i	$-0.193352\pi$
$127$	2350.39	1.64223	0.821115	+	0.570763i	$0.193352\pi$
$128$	125.240i	0.0864824i
$129$	−350.417	−0.239166
$130$	0	0
$131$	2308.69	1.53978	0.769890	−	0.638177i	$-0.220311\pi$
$131$	2308.69	1.53978	0.769890	+	0.638177i	$0.220311\pi$
$132$	426.220i	0.281043i
$133$	79.6405	0.0519226
$134$	1152.57	0.743039
$135$	218.393i	0.139232i
$136$	90.9680i	0.0573562i
$137$	374.912i	0.233802i	0.993144	+	0.116901i	$0.0372961\pi$
$137$	374.912i	0.233802i	−0.993144	+	0.116901i	$0.962704\pi$
$138$	− 2656.46i	− 1.63864i
$139$	487.711	0.297605	0.148802	−	0.988867i	$-0.452458\pi$
$139$	487.711	0.297605	0.148802	+	0.988867i	$0.452458\pi$
$140$	−396.761	−0.239517
$141$	− 1015.83i	− 0.606726i
$142$	1280.30	0.756623
$143$	0	0
$144$	−557.981	−0.322906
$145$	− 1613.57i	− 0.924134i
$146$	−254.175	−0.144080
$147$	922.765	0.517744
$148$	− 2751.28i	− 1.52807i
$149$	1055.91i	0.580558i	0.956942	+	0.290279i	$0.0937481\pi$
$149$	1055.91i	0.580558i	−0.956942	+	0.290279i	$0.906252\pi$
$150$	720.292i	0.392077i
$151$	888.560i	0.478874i	0.970912	+	0.239437i	$0.0769629\pi$
$151$	888.560i	0.478874i	−0.970912	+	0.239437i	$0.923037\pi$
$152$	13.1007	0.00699081
$153$	−836.369	−0.441938
$154$	− 413.370i	− 0.216301i
$155$	−2489.50	−1.29007
$156$	0	0
$157$	−3648.56	−1.85469	−0.927346	−	0.374206i	$-0.877915\pi$
$157$	−3648.56	−1.85469	−0.927346	+	0.374206i	$0.877915\pi$
$158$	− 2514.21i	− 1.26595i
$159$	−79.8644	−0.0398343
$160$	2084.43	1.02993
$161$	1307.45i	0.640008i
$162$	326.450i	0.158323i
$163$	− 1386.49i	− 0.666249i	−0.942883	−	0.333125i	$-0.891897\pi$
$163$	− 1386.49i	− 0.666249i	0.942883	−	0.333125i	$-0.108103\pi$
$164$	1654.25i	0.787655i
$165$	418.244	0.197335
$166$	−2659.46	−1.24346
$167$	3376.19i	1.56442i	0.623017	+	0.782209i	$0.285907\pi$
$167$	3376.19i	1.56442i	−0.623017	+	0.782209i	$0.714093\pi$
$168$	−17.4754	−0.00802535
$169$	0	0
$170$	3029.44	1.36675
$171$	120.449i	0.0538652i
$172$	962.815	0.426825
$173$	−341.817	−0.150219	−0.0751095	−	0.997175i	$-0.523931\pi$
$173$	−341.817	−0.150219	−0.0751095	+	0.997175i	$0.523931\pi$
$174$	− 2411.93i	− 1.05085i
$175$	− 354.511i	− 0.153134i
$176$	1068.59i	0.457658i
$177$	840.174i	0.356787i
$178$	5131.28	2.16070
$179$	2885.55	1.20489	0.602446	−	0.798159i	$-0.294193\pi$
$179$	2885.55	1.20489	0.602446	+	0.798159i	$0.294193\pi$
$180$	− 600.064i	− 0.248478i
$181$	−3795.62	−1.55871	−0.779354	−	0.626584i	$-0.784453\pi$
$181$	−3795.62	−1.55871	−0.779354	+	0.626584i	$0.784453\pi$
$182$	0	0
$183$	−621.253	−0.250953
$184$	215.072i	0.0861701i
$185$	−2699.80	−1.07294
$186$	−3721.25	−1.46696
$187$	1601.73i	0.626363i
$188$	2791.12i	1.08278i
$189$	− 160.671i	− 0.0618365i
$190$	− 436.282i	− 0.166586i
$191$	−2805.90	−1.06297	−0.531486	−	0.847067i	$-0.678366\pi$
$191$	−2805.90	−1.06297	−0.531486	+	0.847067i	$0.678366\pi$
$192$	1627.81	0.611859
$193$	2485.64i	0.927048i	0.886084	+	0.463524i	$0.153415\pi$
$193$	2485.64i	0.927048i	−0.886084	+	0.463524i	$0.846585\pi$
$194$	−2433.24	−0.900499
$195$	0	0
$196$	−2535.41	−0.923985
$197$	2750.70i	0.994818i	0.867516	+	0.497409i	$0.165715\pi$
$197$	2750.70i	0.994818i	−0.867516	+	0.497409i	$0.834285\pi$
$198$	625.183	0.224393
$199$	−3998.53	−1.42436	−0.712182	−	0.701994i	$-0.752293\pi$
$199$	−3998.53	−1.42436	−0.712182	+	0.701994i	$0.752293\pi$
$200$	− 58.3161i	− 0.0206178i
$201$	− 857.943i	− 0.301068i
$202$	2984.75i	1.03964i
$203$	1187.09i	0.410432i
$204$	2298.03	0.788698
$205$	1623.30	0.553054
$206$	7611.41i	2.57433i
$207$	−1977.39	−0.663953
$208$	0	0
$209$	230.671	0.0763438
$210$	581.973i	0.191238i
$211$	−1375.80	−0.448882	−0.224441	−	0.974488i	$-0.572056\pi$
$211$	−1375.80	−0.448882	−0.224441	+	0.974488i	$0.572056\pi$
$212$	219.438	0.0710898
$213$	− 953.020i	− 0.306572i
$214$	− 3705.66i	− 1.18371i
$215$	− 944.799i	− 0.299696i
$216$	− 26.4300i	− 0.00832561i
$217$	1831.51	0.572955
$218$	−6329.12	−1.96634
$219$	189.200i	0.0583789i
$220$	−1149.18	−0.352171
$221$	0	0
$222$	−4035.61	−1.22005
$223$	1694.37i	0.508805i	0.967098	+	0.254403i	$0.0818788\pi$
$223$	1694.37i	0.508805i	−0.967098	+	0.254403i	$0.918121\pi$
$224$	−1533.50	−0.457418
$225$	536.164	0.158864
$226$	− 5338.70i	− 1.57135i
$227$	− 405.400i	− 0.118534i	−0.998242	−	0.0592672i	$-0.981124\pi$
$227$	− 405.400i	− 0.118534i	0.998242	−	0.0592672i	$-0.0188764\pi$
$228$	− 330.949i	− 0.0961299i
$229$	− 2359.54i	− 0.680884i	−0.940265	−	0.340442i	$-0.889423\pi$
$229$	− 2359.54i	− 0.680884i	0.940265	−	0.340442i	$-0.110577\pi$
$230$	7162.39	2.05337
$231$	−307.701	−0.0876416
$232$	195.274i	0.0552601i
$233$	−938.507	−0.263878	−0.131939	−	0.991258i	$-0.542120\pi$
$233$	−938.507	−0.263878	−0.131939	+	0.991258i	$0.542120\pi$
$234$	0	0
$235$	2738.90	0.760280
$236$	− 2308.49i	− 0.636736i
$237$	−1871.51	−0.512943
$238$	−2228.75	−0.607010
$239$	4664.25i	1.26237i	0.775634	+	0.631183i	$0.217430\pi$
$239$	4664.25i	1.26237i	−0.775634	+	0.631183i	$0.782570\pi$
$240$	− 1504.44i	− 0.404629i
$241$	3774.94i	1.00898i	0.863416	+	0.504492i	$0.168320\pi$
$241$	3774.94i	1.00898i	−0.863416	+	0.504492i	$0.831680\pi$
$242$	4166.97i	1.10687i
$243$	243.000	0.0641500
$244$	1706.97	0.447860
$245$	2487.97i	0.648778i
$246$	2426.47	0.628887
$247$	0	0
$248$	301.279	0.0771421
$249$	1979.62i	0.503829i
$250$	−6016.96	−1.52219
$251$	960.695	0.241588	0.120794	−	0.992678i	$-0.461456\pi$
$251$	960.695	0.241588	0.120794	+	0.992678i	$0.461456\pi$
$252$	441.464i	0.110356i
$253$	3786.90i	0.941029i
$254$	9472.64i	2.34003i
$255$	− 2255.03i	− 0.553787i
$256$	3836.08	0.936542
$257$	16.7302	0.00406070	0.00203035	−	0.999998i	$-0.499354\pi$
$257$	16.7302	0.00406070	0.00203035	+	0.999998i	$0.499354\pi$
$258$	− 1412.27i	− 0.340790i
$259$	1986.23	0.476519
$260$	0	0
$261$	−1795.37	−0.425788
$262$	9304.58i	2.19404i
$263$	−400.736	−0.0939560	−0.0469780	−	0.998896i	$-0.514959\pi$
$263$	−400.736	−0.0939560	−0.0469780	+	0.998896i	$0.514959\pi$
$264$	−50.6159	−0.0118000
$265$	− 215.332i	− 0.0499159i
$266$	320.971i	0.0739849i
$267$	− 3819.57i	− 0.875484i
$268$	2357.31i	0.537297i
$269$	2236.97	0.507027	0.253514	−	0.967332i	$-0.418414\pi$
$269$	2236.97	0.507027	0.253514	+	0.967332i	$0.418414\pi$
$270$	−880.179	−0.198393
$271$	4018.89i	0.900850i	0.892814	+	0.450425i	$0.148727\pi$
$271$	4018.89i	0.900850i	−0.892814	+	0.450425i	$0.851273\pi$
$272$	5761.46	1.28434
$273$	0	0
$274$	−1510.99	−0.333147
$275$	− 1026.81i	− 0.225159i
$276$	5433.14	1.18492
$277$	−6792.95	−1.47346	−0.736731	−	0.676186i	$-0.763632\pi$
$277$	−6792.95	−1.47346	−0.736731	+	0.676186i	$0.763632\pi$
$278$	1965.59i	0.424059i
$279$	2769.99i	0.594391i
$280$	− 47.1175i	− 0.0100565i
$281$	7286.80i	1.54695i	0.633824	+	0.773477i	$0.281484\pi$
$281$	7286.80i	1.54695i	−0.633824	+	0.773477i	$0.718516\pi$
$282$	4094.04	0.864528
$283$	2429.77	0.510369	0.255185	−	0.966892i	$-0.417864\pi$
$283$	2429.77	0.510369	0.255185	+	0.966892i	$0.417864\pi$
$284$	2618.54i	0.547120i
$285$	−324.756	−0.0674979
$286$	0	0
$287$	−1194.25	−0.245626
$288$	− 2319.28i	− 0.474531i
$289$	3722.97	0.757780
$290$	6503.07	1.31681
$291$	1811.24i	0.364868i
$292$	− 519.853i	− 0.104185i
$293$	− 4211.25i	− 0.839672i	−0.907600	−	0.419836i	$-0.862088\pi$
$293$	− 4211.25i	− 0.839672i	0.907600	−	0.419836i	$-0.137912\pi$
$294$	3718.97i	0.737737i
$295$	−2265.29	−0.447085
$296$	326.730	0.0641580
$297$	− 465.368i	− 0.0909206i
$298$	−4255.56	−0.827242
$299$	0	0
$300$	−1473.18	−0.283514
$301$	695.084i	0.133103i
$302$	−3581.12	−0.682351
$303$	2221.76	0.421244
$304$	− 829.731i	− 0.156541i
$305$	− 1675.03i	− 0.314466i
$306$	− 3370.78i	− 0.629720i
$307$	− 8212.27i	− 1.52671i	−0.645982	−	0.763353i	$-0.723552\pi$
$307$	− 8212.27i	− 1.52671i	0.645982	−	0.763353i	$-0.276448\pi$
$308$	845.447	0.156408
$309$	5665.71	1.04308
$310$	− 10033.3i	− 1.83823i
$311$	4238.55	0.772818	0.386409	−	0.922328i	$-0.373715\pi$
$311$	4238.55	0.772818	0.386409	+	0.922328i	$0.373715\pi$
$312$	0	0
$313$	3807.82	0.687638	0.343819	−	0.939036i	$-0.388279\pi$
$313$	3807.82	0.687638	0.343819	+	0.939036i	$0.388279\pi$
$314$	− 14704.6i	− 2.64276i
$315$	433.204	0.0774865
$316$	5142.20	0.915416
$317$	− 10497.7i	− 1.85996i	−0.367608	−	0.929981i	$-0.619823\pi$
$317$	− 10497.7i	− 1.85996i	0.367608	−	0.929981i	$-0.380177\pi$
$318$	− 321.873i	− 0.0567602i
$319$	3438.31i	0.603474i
$320$	4388.92i	0.766713i
$321$	−2758.39	−0.479621
$322$	−5269.34	−0.911953
$323$	− 1243.70i	− 0.214246i
$324$	−667.674	−0.114485
$325$	0	0
$326$	5587.91	0.949343
$327$	4711.21i	0.796730i
$328$	−196.452	−0.0330708
$329$	−2014.99	−0.337660
$330$	1685.63i	0.281184i
$331$	11905.2i	1.97695i	0.151389	+	0.988474i	$0.451625\pi$
$331$	11905.2i	1.97695i	−0.151389	+	0.988474i	$0.548375\pi$
$332$	− 5439.27i	− 0.899152i
$333$	3003.99i	0.494347i
$334$	−13606.9	−2.22915
$335$	2313.20	0.377264
$336$	1106.81i	0.179706i
$337$	−8981.18	−1.45174	−0.725869	−	0.687833i	$-0.758562\pi$
$337$	−8981.18	−1.45174	−0.725869	+	0.687833i	$0.758562\pi$
$338$	0	0
$339$	−3973.97	−0.636686
$340$	6195.99i	0.988308i
$341$	5304.80	0.842437
$342$	−485.439	−0.0767530
$343$	− 3871.51i	− 0.609451i
$344$	114.339i	0.0179208i
$345$	− 5331.48i	− 0.831992i
$346$	− 1377.61i	− 0.214048i
$347$	−5712.87	−0.883813	−0.441906	−	0.897061i	$-0.645698\pi$
$347$	−5712.87	−0.883813	−0.441906	+	0.897061i	$0.645698\pi$
$348$	4933.01	0.759876
$349$	− 9507.28i	− 1.45820i	−0.684406	−	0.729102i	$-0.739938\pi$
$349$	− 9507.28i	− 1.45820i	0.684406	−	0.729102i	$-0.260062\pi$
$350$	1428.77	0.218202
$351$	0	0
$352$	−4441.65	−0.672559
$353$	− 3972.96i	− 0.599035i	−0.954091	−	0.299518i	$-0.903174\pi$
$353$	− 3972.96i	− 0.599035i	0.954091	−	0.299518i	$-0.0968257\pi$
$354$	−3386.11	−0.508389
$355$	2569.55	0.384162
$356$	10494.8i	1.56242i
$357$	1659.02i	0.245951i
$358$	11629.5i	1.71686i
$359$	109.866i	0.0161519i	0.999967	+	0.00807594i	$0.00257068\pi$
$359$	109.866i	0.0161519i	−0.999967	+	0.00807594i	$0.997429\pi$
$360$	71.2609	0.0104327
$361$	6679.89	0.973887
$362$	− 15297.3i	− 2.22101i
$363$	3101.77	0.448487
$364$	0	0
$365$	−510.125	−0.0731539
$366$	− 2503.80i	− 0.357584i
$367$	7020.02	0.998480	0.499240	−	0.866464i	$-0.333613\pi$
$367$	7020.02	0.998480	0.499240	+	0.866464i	$0.333613\pi$
$368$	13621.6	1.92955
$369$	− 1806.20i	− 0.254815i
$370$	− 10880.9i	− 1.52884i
$371$	158.418i	0.0221689i
$372$	− 7610.91i	− 1.06077i
$373$	2227.18	0.309166	0.154583	−	0.987980i	$-0.450597\pi$
$373$	2227.18	0.309166	0.154583	+	0.987980i	$0.450597\pi$
$374$	−6455.36	−0.892510
$375$	4478.86i	0.616766i
$376$	−331.461	−0.0454622
$377$	0	0
$378$	647.544	0.0881113
$379$	− 11004.6i	− 1.49147i	−0.666244	−	0.745734i	$-0.732099\pi$
$379$	− 11004.6i	− 1.49147i	0.666244	−	0.745734i	$-0.267901\pi$
$380$	892.309	0.120459
$381$	7051.16	0.948142
$382$	− 11308.5i	− 1.51464i
$383$	− 4436.85i	− 0.591938i	−0.955198	−	0.295969i	$-0.904357\pi$
$383$	− 4436.85i	− 0.591938i	0.955198	−	0.295969i	$-0.0956426\pi$
$384$	375.720i	0.0499306i
$385$	− 829.627i	− 0.109823i
$386$	−10017.7	−1.32096
$387$	−1051.25	−0.138083
$388$	− 4976.61i	− 0.651157i
$389$	2561.58	0.333874	0.166937	−	0.985968i	$-0.446612\pi$
$389$	2561.58	0.333874	0.166937	+	0.985968i	$0.446612\pi$
$390$	0	0
$391$	20417.7	2.64083
$392$	− 301.094i	− 0.0387948i
$393$	6926.07	0.888992
$394$	−11086.0	−1.41752
$395$	− 5045.98i	− 0.642762i
$396$	1278.66i	0.162260i
$397$	− 2206.90i	− 0.278996i	−0.990222	−	0.139498i	$-0.955451\pi$
$397$	− 2206.90i	− 0.278996i	0.990222	−	0.139498i	$-0.0445488\pi$
$398$	− 16115.1i	− 2.02959i
$399$	238.922	0.0299775
$400$	−3693.45	−0.461682
$401$	8071.02i	1.00511i	0.864546	+	0.502553i	$0.167606\pi$
$401$	8071.02i	1.00511i	−0.864546	+	0.502553i	$0.832394\pi$
$402$	3457.72	0.428994
$403$	0	0
$404$	−6104.58	−0.751768
$405$	655.180i	0.0803856i
$406$	−4784.28	−0.584827
$407$	5752.93	0.700644
$408$	272.904i	0.0331146i
$409$	6298.36i	0.761452i	0.924688	+	0.380726i	$0.124326\pi$
$409$	6298.36i	0.761452i	−0.924688	+	0.380726i	$0.875674\pi$
$410$	6542.29i	0.788051i
$411$	1124.74i	0.134986i
$412$	−15567.3	−1.86152
$413$	1666.56	0.198562
$414$	− 7969.38i	− 0.946072i
$415$	−5337.49	−0.631342
$416$	0	0
$417$	1463.13	0.171822
$418$	929.662i	0.108783i
$419$	−1432.01	−0.166965	−0.0834824	−	0.996509i	$-0.526604\pi$
$419$	−1432.01	−0.166965	−0.0834824	+	0.996509i	$0.526604\pi$
$420$	−1190.28	−0.138285
$421$	9641.43i	1.11614i	0.829794	+	0.558069i	$0.188458\pi$
$421$	9641.43i	1.11614i	−0.829794	+	0.558069i	$0.811542\pi$
$422$	− 5544.82i	− 0.639615i
$423$	− 3047.49i	− 0.350293i
$424$	26.0594i	0.00298480i
$425$	−5536.19	−0.631870
$426$	3840.90	0.436837
$427$	1232.31i	0.139662i
$428$	7579.03	0.855949
$429$	0	0
$430$	3807.77	0.427040
$431$	3370.61i	0.376698i	0.982102	+	0.188349i	$0.0603136\pi$
$431$	3370.61i	0.376698i	−0.982102	+	0.188349i	$0.939686\pi$
$432$	−1673.94	−0.186430
$433$	−6248.91	−0.693541	−0.346770	−	0.937950i	$-0.612722\pi$
$433$	−6248.91	−0.693541	−0.346770	+	0.937950i	$0.612722\pi$
$434$	7381.45i	0.816407i
$435$	− 4840.70i	− 0.533549i
$436$	− 12944.7i	− 1.42187i
$437$	− 2940.43i	− 0.321876i
$438$	−762.524	−0.0831845
$439$	−2866.16	−0.311604	−0.155802	−	0.987788i	$-0.549796\pi$
$439$	−2866.16	−0.311604	−0.155802	+	0.987788i	$0.549796\pi$
$440$	− 136.471i	− 0.0147864i
$441$	2768.29	0.298920
$442$	0	0
$443$	10776.3	1.15575	0.577877	−	0.816124i	$-0.303881\pi$
$443$	10776.3	1.15575	0.577877	+	0.816124i	$0.303881\pi$
$444$	− 8253.85i	− 0.882230i
$445$	10298.4	1.09706
$446$	−6828.74	−0.725000
$447$	3167.72i	0.335185i
$448$	− 3228.91i	− 0.340517i
$449$	− 13834.9i	− 1.45414i	−0.686564	−	0.727069i	$-0.740882\pi$
$449$	− 13834.9i	− 1.45414i	0.686564	−	0.727069i	$-0.259118\pi$
$450$	2160.87i	0.226366i
$451$	−3459.04	−0.361153
$452$	10919.0	1.13625
$453$	2665.68i	0.276478i
$454$	1633.86	0.168901
$455$	0	0
$456$	39.3020	0.00403615
$457$	17233.0i	1.76395i	0.471295	+	0.881976i	$0.343787\pi$
$457$	17233.0i	1.76395i	−0.471295	+	0.881976i	$0.656213\pi$
$458$	9509.51	0.970197
$459$	−2509.11	−0.255153
$460$	14648.9i	1.48480i
$461$	11485.0i	1.16033i	0.814500	+	0.580164i	$0.197012\pi$
$461$	11485.0i	1.16033i	−0.814500	+	0.580164i	$0.802988\pi$
$462$	− 1240.11i	− 0.124881i
$463$	− 1613.61i	− 0.161967i	−0.996715	−	0.0809837i	$-0.974194\pi$
$463$	− 1613.61i	− 0.161967i	0.996715	−	0.0809837i	$-0.0258062\pi$
$464$	12367.7	1.23740
$465$	−7468.49	−0.744824
$466$	− 3782.41i	− 0.376002i
$467$	8149.20	0.807495	0.403747	−	0.914871i	$-0.367707\pi$
$467$	8149.20	0.807495	0.403747	+	0.914871i	$0.367707\pi$
$468$	0	0
$469$	−1701.81	−0.167553
$470$	11038.4i	1.08333i
$471$	−10945.7	−1.07081
$472$	274.145	0.0267342
$473$	2013.24i	0.195706i
$474$	− 7542.63i	− 0.730896i
$475$	797.289i	0.0770150i
$476$	− 4558.36i	− 0.438933i
$477$	−239.593	−0.0229984
$478$	−18798.1	−1.79875
$479$	− 1827.62i	− 0.174334i	−0.996194	−	0.0871670i	$-0.972219\pi$
$479$	− 1827.62i	− 0.174334i	0.996194	−	0.0871670i	$-0.0277814\pi$
$480$	6253.28	0.594629
$481$	0	0
$482$	−15213.9	−1.43771
$483$	3922.34i	0.369509i
$484$	−8522.52	−0.800387
$485$	−4883.49	−0.457212
$486$	979.350i	0.0914078i
$487$	− 7838.95i	− 0.729398i	−0.931125	−	0.364699i	$-0.881172\pi$
$487$	− 7838.95i	− 0.729398i	0.931125	−	0.364699i	$-0.118828\pi$
$488$	202.712i	0.0188040i
$489$	− 4159.48i	− 0.384659i
$490$	−10027.1	−0.924449
$491$	−17196.1	−1.58055	−0.790273	−	0.612755i	$-0.790061\pi$
$491$	−17196.1	−1.58055	−0.790273	+	0.612755i	$0.790061\pi$
$492$	4962.76i	0.454753i
$493$	18538.2	1.69354
$494$	0	0
$495$	1254.73	0.113931
$496$	− 19081.5i	− 1.72739i
$497$	−1890.40	−0.170616
$498$	−7978.37	−0.717910
$499$	5355.28i	0.480431i	0.970720	+	0.240216i	$0.0772182\pi$
$499$	5355.28i	0.480431i	−0.970720	+	0.240216i	$0.922782\pi$
$500$	− 12306.2i	− 1.10070i
$501$	10128.6i	0.903217i
$502$	3871.84i	0.344240i
$503$	2979.80	0.264141	0.132071	−	0.991240i	$-0.457837\pi$
$503$	2979.80	0.264141	0.132071	+	0.991240i	$0.457837\pi$
$504$	−52.4263	−0.00463344
$505$	5990.35i	0.527856i
$506$	−15262.1	−1.34088
$507$	0	0
$508$	−19374.0	−1.69209
$509$	15353.7i	1.33701i	0.743706	+	0.668507i	$0.233066\pi$
$509$	15353.7i	1.33701i	−0.743706	+	0.668507i	$0.766934\pi$
$510$	9088.33	0.789094
$511$	375.297	0.0324895
$512$	16462.3i	1.42097i
$513$	361.347i	0.0310991i
$514$	67.4267i	0.00578611i
$515$	15276.0i	1.30707i
$516$	2888.44	0.246428
$517$	−5836.24	−0.496475
$518$	8005.00i	0.678995i
$519$	−1025.45	−0.0867289
$520$	0	0
$521$	433.801	0.0364783	0.0182391	−	0.999834i	$-0.494194\pi$
$521$	433.801	0.0364783	0.0182391	+	0.999834i	$0.494194\pi$
$522$	− 7235.78i	− 0.606708i
$523$	18900.3	1.58021	0.790106	−	0.612970i	$-0.210026\pi$
$523$	18900.3	1.58021	0.790106	+	0.612970i	$0.210026\pi$
$524$	−19030.3	−1.58653
$525$	− 1063.53i	− 0.0884121i
$526$	− 1615.06i	− 0.133879i
$527$	− 28601.7i	− 2.36415i
$528$	3205.76i	0.264229i
$529$	36105.7	2.96751
$530$	867.839	0.0711255
$531$	2520.52i	0.205991i
$532$	−656.468	−0.0534990
$533$	0	0
$534$	15393.8	1.24748
$535$	− 7437.21i	− 0.601007i
$536$	−279.943	−0.0225592
$537$	8656.64	0.695645
$538$	9015.53i	0.722467i
$539$	− 5301.55i	− 0.423662i
$540$	− 1800.19i	− 0.143459i
$541$	− 13247.5i	− 1.05278i	−0.850243	−	0.526391i	$-0.823545\pi$
$541$	− 13247.5i	− 1.05278i	0.850243	−	0.526391i	$-0.176455\pi$
$542$	−16197.1	−1.28363
$543$	−11386.9	−0.899920
$544$	23947.9i	1.88742i
$545$	−12702.4	−0.998372
$546$	0	0
$547$	−5543.52	−0.433316	−0.216658	−	0.976248i	$-0.569516\pi$
$547$	−5543.52	−0.433316	−0.216658	+	0.976248i	$0.569516\pi$
$548$	− 3090.36i	− 0.240901i
$549$	−1863.76	−0.144888
$550$	4138.28	0.320831
$551$	− 2669.76i	− 0.206416i
$552$	645.215i	0.0497503i
$553$	3712.31i	0.285467i
$554$	− 27377.3i	− 2.09955i
$555$	−8099.40	−0.619460
$556$	−4020.14	−0.306640
$557$	− 6733.19i	− 0.512198i	−0.966651	−	0.256099i	$-0.917563\pi$
$557$	− 6733.19i	− 0.512198i	0.966651	−	0.256099i	$-0.0824374\pi$
$558$	−11163.7	−0.846952
$559$	0	0
$560$	−2984.19	−0.225188
$561$	4805.18i	0.361631i
$562$	−29367.6	−2.20427
$563$	−24282.4	−1.81773	−0.908863	−	0.417096i	$-0.863048\pi$
$563$	−24282.4	−1.81773	−0.908863	+	0.417096i	$0.863048\pi$
$564$	8373.37i	0.625146i
$565$	− 10714.7i	− 0.797823i
$566$	9792.55i	0.727229i
$567$	− 482.013i	− 0.0357013i
$568$	−310.966	−0.0229716
$569$	19876.2	1.46441	0.732207	−	0.681082i	$-0.238490\pi$
$569$	19876.2	1.46441	0.732207	+	0.681082i	$0.238490\pi$
$570$	− 1308.85i	− 0.0961782i
$571$	−225.014	−0.0164913	−0.00824567	−	0.999966i	$-0.502625\pi$
$571$	−225.014	−0.0164913	−0.00824567	+	0.999966i	$0.502625\pi$
$572$	0	0
$573$	−8417.70	−0.613707
$574$	− 4813.14i	− 0.349994i
$575$	−13089.0	−0.949302
$576$	4883.43	0.353257
$577$	− 12258.1i	− 0.884424i	−0.896910	−	0.442212i	$-0.854194\pi$
$577$	− 12258.1i	− 0.884424i	0.896910	−	0.442212i	$-0.145806\pi$
$578$	15004.5i	1.07977i
$579$	7456.92i	0.535231i
$580$	13300.4i	0.952191i
$581$	3926.77	0.280395
$582$	−7299.73	−0.519903
$583$	458.844i	0.0325958i
$584$	61.7353	0.00437436
$585$	0	0
$586$	16972.4	1.19645
$587$	10282.3i	0.722992i	0.932374	+	0.361496i	$0.117734\pi$
$587$	10282.3i	0.722992i	−0.932374	+	0.361496i	$0.882266\pi$
$588$	−7606.24	−0.533463
$589$	−4119.04	−0.288153
$590$	− 9129.67i	− 0.637055i
$591$	8252.10i	0.574359i
$592$	− 20693.5i	− 1.43665i
$593$	− 12355.1i	− 0.855586i	−0.903877	−	0.427793i	$-0.859291\pi$
$593$	− 12355.1i	− 0.855586i	0.903877	−	0.427793i	$-0.140709\pi$
$594$	1875.55	0.129553
$595$	−4473.07	−0.308198
$596$	− 8703.71i	− 0.598184i
$597$	−11995.6	−0.822357
$598$	0	0
$599$	13035.2	0.889157	0.444578	−	0.895740i	$-0.353354\pi$
$599$	13035.2	0.889157	0.444578	+	0.895740i	$0.353354\pi$
$600$	− 174.948i	− 0.0119037i
$601$	−5153.36	−0.349767	−0.174884	−	0.984589i	$-0.555955\pi$
$601$	−5153.36	−0.349767	−0.174884	+	0.984589i	$0.555955\pi$
$602$	−2801.36	−0.189659
$603$	− 2573.83i	− 0.173822i
$604$	− 7324.30i	− 0.493413i
$605$	8363.05i	0.561994i
$606$	8954.25i	0.600234i
$607$	−1406.55	−0.0940530	−0.0470265	−	0.998894i	$-0.514975\pi$
$607$	−1406.55	−0.0940530	−0.0470265	+	0.998894i	$0.514975\pi$
$608$	3448.83	0.230047
$609$	3561.28i	0.236963i
$610$	6750.79	0.448085
$611$	0	0
$612$	6894.10	0.455355
$613$	− 15984.3i	− 1.05318i	−0.850120	−	0.526590i	$-0.823470\pi$
$613$	− 15984.3i	− 1.05318i	0.850120	−	0.526590i	$-0.176530\pi$
$614$	33097.5	2.17541
$615$	4869.90	0.319306
$616$	100.401i	0.00656703i
$617$	− 10324.4i	− 0.673652i	−0.941567	−	0.336826i	$-0.890647\pi$
$617$	− 10324.4i	− 0.673652i	0.941567	−	0.336826i	$-0.109353\pi$
$618$	22834.2i	1.48629i
$619$	7423.00i	0.481996i	0.970526	+	0.240998i	$0.0774748\pi$
$619$	7423.00i	0.481996i	−0.970526	+	0.240998i	$0.922525\pi$
$620$	20520.6	1.32924
$621$	−5932.18	−0.383334
$622$	17082.4i	1.10119i
$623$	−7576.48	−0.487232
$624$	0	0
$625$	−4629.23	−0.296271
$626$	15346.4i	0.979820i
$627$	692.014	0.0440771
$628$	30074.6	1.91100
$629$	− 31017.8i	− 1.96624i
$630$	1745.92i	0.110411i
$631$	− 1190.30i	− 0.0750954i	−0.999295	−	0.0375477i	$-0.988045\pi$
$631$	− 1190.30i	− 0.0750954i	0.999295	−	0.0375477i	$-0.0119546\pi$
$632$	610.665i	0.0384350i
$633$	−4127.41	−0.259162
$634$	42308.2	2.65027
$635$	19011.5i	1.18810i
$636$	658.313	0.0410437
$637$	0	0
$638$	−13857.2	−0.859894
$639$	− 2859.06i	− 0.176999i
$640$	−1013.02	−0.0625674
$641$	−705.064	−0.0434451	−0.0217226	−	0.999764i	$-0.506915\pi$
$641$	−705.064	−0.0434451	−0.0217226	+	0.999764i	$0.506915\pi$
$642$	− 11117.0i	− 0.683415i
$643$	14641.4i	0.897976i	0.893538	+	0.448988i	$0.148215\pi$
$643$	14641.4i	0.897976i	−0.893538	+	0.448988i	$0.851785\pi$
$644$	− 10777.1i	− 0.659439i
$645$	− 2834.40i	− 0.173030i
$646$	5012.42	0.305280
$647$	−1520.83	−0.0924111	−0.0462056	−	0.998932i	$-0.514713\pi$
$647$	−1520.83	−0.0924111	−0.0462056	+	0.998932i	$0.514713\pi$
$648$	− 79.2899i	− 0.00480679i
$649$	4827.04	0.291954
$650$	0	0
$651$	5494.54	0.330796
$652$	11428.7i	0.686477i
$653$	−22054.4	−1.32168	−0.660838	−	0.750529i	$-0.729799\pi$
$653$	−22054.4	−1.32168	−0.660838	+	0.750529i	$0.729799\pi$
$654$	−18987.3	−1.13527
$655$	18674.2i	1.11398i
$656$	12442.3i	0.740532i
$657$	567.601i	0.0337051i
$658$	− 8120.92i	− 0.481134i
$659$	−12652.3	−0.747895	−0.373947	−	0.927450i	$-0.621996\pi$
$659$	−12652.3	−0.747895	−0.373947	+	0.927450i	$0.621996\pi$
$660$	−3447.54	−0.203326
$661$	10893.2i	0.640994i	0.947250	+	0.320497i	$0.103850\pi$
$661$	10893.2i	0.640994i	−0.947250	+	0.320497i	$0.896150\pi$
$662$	−47981.0	−2.81697
$663$	0	0
$664$	645.942	0.0377522
$665$	644.184i	0.0375645i
$666$	−12106.8	−0.704399
$667$	43829.0	2.54433
$668$	− 27829.6i	− 1.61191i
$669$	5083.12i	0.293759i
$670$	9322.76i	0.537567i
$671$	3569.28i	0.205351i
$672$	−4600.51	−0.264090
$673$	9019.75	0.516621	0.258310	−	0.966062i	$-0.416834\pi$
$673$	9019.75	0.516621	0.258310	+	0.966062i	$0.416834\pi$
$674$	− 36196.4i	− 2.06859i
$675$	1608.49	0.0917199
$676$	0	0
$677$	−24923.6	−1.41491	−0.707454	−	0.706760i	$-0.750156\pi$
$677$	−24923.6	−1.41491	−0.707454	+	0.706760i	$0.750156\pi$
$678$	− 16016.1i	− 0.907218i
$679$	3592.76	0.203060
$680$	−735.807	−0.0414955
$681$	− 1216.20i	− 0.0684359i
$682$	21379.7i	1.20039i
$683$	24634.2i	1.38009i	0.723767	+	0.690044i	$0.242409\pi$
$683$	24634.2i	1.38009i	−0.723767	+	0.690044i	$0.757591\pi$
$684$	− 992.846i	− 0.0555006i
$685$	−3032.53	−0.169149
$686$	15603.1	0.868411
$687$	− 7078.61i	− 0.393109i
$688$	7241.70	0.401290
$689$	0	0
$690$	21487.2	1.18551
$691$	− 10340.2i	− 0.569264i	−0.958637	−	0.284632i	$-0.908129\pi$
$691$	− 10340.2i	− 0.569264i	0.958637	−	0.284632i	$-0.0918714\pi$
$692$	2817.56	0.154780
$693$	−923.102	−0.0505999
$694$	− 23024.3i	− 1.25935i
$695$	3944.92i	0.215308i
$696$	585.821i	0.0319045i
$697$	18650.0i	1.01351i
$698$	38316.7	2.07780
$699$	−2815.52	−0.152350
$700$	2922.19i	0.157784i
$701$	20833.9	1.12252	0.561258	−	0.827641i	$-0.310317\pi$
$701$	20833.9	1.12252	0.561258	+	0.827641i	$0.310317\pi$
$702$	0	0
$703$	−4467.00	−0.239653
$704$	− 9352.23i	− 0.500676i
$705$	8216.69	0.438948
$706$	16012.0	0.853569
$707$	− 4407.07i	− 0.234434i
$708$	− 6925.46i	− 0.367619i
$709$	− 9189.56i	− 0.486772i	−0.969930	−	0.243386i	$-0.921742\pi$
$709$	− 9189.56i	− 0.486772i	0.969930	−	0.243386i	$-0.0782581\pi$
$710$	10355.9i	0.547395i
$711$	−5614.52	−0.296148
$712$	−1246.31	−0.0656004
$713$	− 67621.8i	− 3.55183i
$714$	−6686.25	−0.350457
$715$	0	0
$716$	−23785.2	−1.24147
$717$	13992.8i	0.728827i
$718$	−442.788	−0.0230149
$719$	−7503.22	−0.389183	−0.194592	−	0.980884i	$-0.562338\pi$
$719$	−7503.22	−0.389183	−0.194592	+	0.980884i	$0.562338\pi$
$720$	− 4513.31i	− 0.233613i
$721$	− 11238.5i	− 0.580503i
$722$	26921.6i	1.38770i
$723$	11324.8i	0.582537i
$724$	31286.8	1.60603
$725$	−11884.1	−0.608779
$726$	12500.9i	0.639053i
$727$	20727.2	1.05740	0.528700	−	0.848809i	$-0.322680\pi$
$727$	20727.2	1.05740	0.528700	+	0.848809i	$0.322680\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 2055.93i	− 0.104237i
$731$	10854.7	0.549216
$732$	5120.92	0.258572
$733$	17361.3i	0.874836i	0.899258	+	0.437418i	$0.144107\pi$
$733$	17361.3i	0.874836i	−0.899258	+	0.437418i	$0.855893\pi$
$734$	28292.4i	1.42274i
$735$	7463.92i	0.374572i
$736$	56618.9i	2.83560i
$737$	−4929.13	−0.246359
$738$	7279.42	0.363088
$739$	− 18093.8i	− 0.900663i	−0.892861	−	0.450331i	$-0.851306\pi$
$739$	− 18093.8i	− 0.900663i	0.892861	−	0.450331i	$-0.148694\pi$
$740$	22254.1	1.10551
$741$	0	0
$742$	−638.466	−0.0315887
$743$	− 14875.3i	− 0.734482i	−0.930126	−	0.367241i	$-0.880302\pi$
$743$	− 14875.3i	− 0.734482i	0.930126	−	0.367241i	$-0.119698\pi$
$744$	903.837	0.0445380
$745$	−8540.85	−0.420017
$746$	8976.07i	0.440533i
$747$	5938.87i	0.290886i
$748$	− 13202.9i	− 0.645380i
$749$	5471.52i	0.266923i
$750$	−18050.9	−0.878834
$751$	17750.9	0.862502	0.431251	−	0.902232i	$-0.358072\pi$
$751$	17750.9	0.862502	0.431251	+	0.902232i	$0.358072\pi$
$752$	20993.1i	1.01801i
$753$	2882.09	0.139481
$754$	0	0
$755$	−7187.25	−0.346451
$756$	1324.39i	0.0637139i
$757$	−10220.3	−0.490702	−0.245351	−	0.969434i	$-0.578903\pi$
$757$	−10220.3	−0.490702	−0.245351	+	0.969434i	$0.578903\pi$
$758$	44351.1	2.12520
$759$	11360.7i	0.543303i
$760$	105.967i	0.00505765i
$761$	14112.0i	0.672221i	0.941823	+	0.336110i	$0.109112\pi$
$761$	14112.0i	0.672221i	−0.941823	+	0.336110i	$0.890888\pi$
$762$	28417.9i	1.35101i
$763$	9345.13	0.443403
$764$	23128.7	1.09525
$765$	− 6765.10i	− 0.319729i
$766$	17881.6	0.843457
$767$	0	0
$768$	11508.2	0.540713
$769$	− 22736.7i	− 1.06620i	−0.846052	−	0.533100i	$-0.821027\pi$
$769$	− 22736.7i	− 1.06620i	0.846052	−	0.533100i	$-0.178973\pi$
$770$	3343.60	0.156487
$771$	50.1905	0.00234444
$772$	− 20488.8i	− 0.955194i
$773$	− 343.173i	− 0.0159678i	−0.999968	−	0.00798388i	$-0.997459\pi$
$773$	− 343.173i	− 0.0159678i	0.999968	−	0.00798388i	$-0.00254137\pi$
$774$	− 4236.80i	− 0.196755i
$775$	18335.4i	0.849844i
$776$	590.999	0.0273397
$777$	5958.69	0.275118
$778$	10323.8i	0.475740i
$779$	2685.86	0.123531
$780$	0	0
$781$	−5475.37	−0.250863
$782$	82288.3i	3.76294i
$783$	−5386.11	−0.245829
$784$	−19069.8	−0.868706
$785$	− 29511.9i	− 1.34181i
$786$	27913.8i	1.26673i
$787$	19086.4i	0.864496i	0.901755	+	0.432248i	$0.142279\pi$
$787$	19086.4i	0.864496i	−0.901755	+	0.432248i	$0.857721\pi$
$788$	− 22673.7i	− 1.02502i
$789$	−1202.21	−0.0542455
$790$	20336.6	0.915876
$791$	7882.74i	0.354334i
$792$	−151.848	−0.00681272
$793$	0	0
$794$	8894.36	0.397543
$795$	− 645.995i	− 0.0288190i
$796$	32959.5	1.46761
$797$	−12031.1	−0.534709	−0.267354	−	0.963598i	$-0.586149\pi$
$797$	−12031.1	−0.534709	−0.267354	+	0.963598i	$0.586149\pi$
$798$	962.913i	0.0427152i
$799$	31467.0i	1.39327i
$800$	− 15352.1i	− 0.678471i
$801$	− 11458.7i	− 0.505461i
$802$	−32528.2	−1.43218
$803$	1087.01	0.0477706
$804$	7071.93i	0.310208i
$805$	−10575.5	−0.463027
$806$	0	0
$807$	6710.91	0.292732
$808$	− 724.952i	− 0.0315640i
$809$	10175.4	0.442210	0.221105	−	0.975250i	$-0.429034\pi$
$809$	10175.4	0.442210	0.221105	+	0.975250i	$0.429034\pi$
$810$	−2640.54	−0.114542
$811$	26754.1i	1.15840i	0.815185	+	0.579201i	$0.196635\pi$
$811$	26754.1i	1.15840i	−0.815185	+	0.579201i	$0.803365\pi$
$812$	− 9785.08i	− 0.422893i
$813$	12056.7i	0.520106i
$814$	23185.7i	0.998353i
$815$	11214.9	0.482011
$816$	17284.4	0.741513
$817$	− 1563.23i	− 0.0669408i
$818$	−25383.9	−1.08500
$819$	0	0
$820$	−13380.7	−0.569845
$821$	− 15777.5i	− 0.670693i	−0.942095	−	0.335347i	$-0.891147\pi$
$821$	− 15777.5i	− 0.670693i	0.942095	−	0.335347i	$-0.108853\pi$
$822$	−4532.97	−0.192342
$823$	13863.9	0.587197	0.293599	−	0.955929i	$-0.405147\pi$
$823$	13863.9	0.587197	0.293599	+	0.955929i	$0.405147\pi$
$824$	− 1848.70i	− 0.0781583i
$825$	− 3080.42i	− 0.129996i
$826$	6716.66i	0.282933i
$827$	26835.0i	1.12835i	0.825655	+	0.564175i	$0.190806\pi$
$827$	26835.0i	1.12835i	−0.825655	+	0.564175i	$0.809194\pi$
$828$	16299.4	0.684111
$829$	−625.251	−0.0261953	−0.0130976	−	0.999914i	$-0.504169\pi$
$829$	−625.251	−0.0261953	−0.0130976	+	0.999914i	$0.504169\pi$
$830$	− 21511.4i	− 0.899604i
$831$	−20378.9	−0.850704
$832$	0	0
$833$	−28584.2	−1.18893
$834$	5896.78i	0.244831i
$835$	−27308.8	−1.13181
$836$	−1901.40	−0.0786617
$837$	8309.98i	0.343172i
$838$	− 5771.35i	− 0.237909i
$839$	27307.4i	1.12367i	0.827251	+	0.561833i	$0.189904\pi$
$839$	27307.4i	1.12367i	−0.827251	+	0.561833i	$0.810096\pi$
$840$	− 141.353i	− 0.00580610i
$841$	15405.4	0.631656
$842$	−38857.3	−1.59039
$843$	21860.4i	0.893135i
$844$	11340.6	0.462510
$845$	0	0
$846$	12282.1	0.499135
$847$	− 6152.66i	− 0.249596i
$848$	1650.48	0.0668368
$849$	7289.30	0.294662
$850$	− 22312.2i	− 0.900356i
$851$	− 73334.2i	− 2.95401i
$852$	7855.63i	0.315880i
$853$	− 44801.9i	− 1.79834i	−0.437595	−	0.899172i	$-0.644170\pi$
$853$	− 44801.9i	− 1.79834i	0.437595	−	0.899172i	$-0.355830\pi$
$854$	−4966.53	−0.199006
$855$	−974.268	−0.0389699
$856$	900.051i	0.0359382i
$857$	25167.1	1.00314	0.501571	−	0.865116i	$-0.332756\pi$
$857$	25167.1	1.00314	0.501571	+	0.865116i	$0.332756\pi$
$858$	0	0
$859$	4059.10	0.161228	0.0806138	−	0.996745i	$-0.474312\pi$
$859$	4059.10	0.161228	0.0806138	+	0.996745i	$0.474312\pi$
$860$	7787.87i	0.308795i
$861$	−3582.76	−0.141812
$862$	−13584.4	−0.536759
$863$	818.924i	0.0323019i	0.999870	+	0.0161509i	$0.00514122\pi$
$863$	818.924i	0.0323019i	−0.999870	+	0.0161509i	$0.994859\pi$
$864$	− 6957.85i	− 0.273971i
$865$	− 2764.84i	− 0.108679i
$866$	− 25184.6i	− 0.988231i
$867$	11168.9	0.437505
$868$	−15096.9	−0.590350
$869$	10752.3i	0.419733i
$870$	19509.2	0.760258
$871$	0	0
$872$	1537.25	0.0596993
$873$	5433.71i	0.210657i
$874$	11850.7	0.458644
$875$	8884.23	0.343248
$876$	− 1559.56i	− 0.0601513i
$877$	190.251i	0.00732533i	0.999993	+	0.00366267i	$0.00116587\pi$
$877$	190.251i	0.00732533i	−0.999993	+	0.00366267i	$0.998834\pi$
$878$	− 11551.3i	− 0.444007i
$879$	− 12633.8i	− 0.484785i
$880$	−8643.43	−0.331102
$881$	19803.4	0.757315	0.378657	−	0.925537i	$-0.376386\pi$
$881$	19803.4	0.757315	0.378657	+	0.925537i	$0.376386\pi$
$882$	11156.9i	0.425933i
$883$	−19652.1	−0.748974	−0.374487	−	0.927232i	$-0.622181\pi$
$883$	−19652.1	−0.748974	−0.374487	+	0.927232i	$0.622181\pi$
$884$	0	0
$885$	−6795.87	−0.258125
$886$	43431.2i	1.64684i
$887$	−26295.0	−0.995379	−0.497689	−	0.867355i	$-0.665818\pi$
$887$	−26295.0	−0.995379	−0.497689	+	0.867355i	$0.665818\pi$
$888$	980.189	0.0370417
$889$	− 13986.6i	− 0.527668i
$890$	41505.1i	1.56321i
$891$	− 1396.11i	− 0.0524930i
$892$	− 13966.5i	− 0.524253i
$893$	4531.69	0.169818
$894$	−12766.7	−0.477608
$895$	23340.2i	0.871704i
$896$	745.275	0.0277878
$897$	0	0
$898$	55758.0	2.07201
$899$	− 61397.0i	− 2.27776i
$900$	−4419.54	−0.163687
$901$	2473.93	0.0914746
$902$	− 13940.8i	− 0.514609i
$903$	2085.25i	0.0768470i
$904$	1296.69i	0.0477072i
$905$	− 30701.4i	− 1.12768i
$906$	−10743.3	−0.393956
$907$	−42417.3	−1.55286	−0.776429	−	0.630204i	$-0.782971\pi$
$907$	−42417.3	−1.55286	−0.776429	+	0.630204i	$0.782971\pi$
$908$	3341.66i	0.122133i
$909$	6665.29	0.243205
$910$	0	0
$911$	−872.245	−0.0317220	−0.0158610	−	0.999874i	$-0.505049\pi$
$911$	−872.245	−0.0317220	−0.0158610	+	0.999874i	$0.505049\pi$
$912$	− 2489.19i	− 0.0903788i
$913$	11373.5	0.412276
$914$	−69453.2	−2.51347
$915$	− 5025.10i	− 0.181557i
$916$	19449.4i	0.701557i
$917$	− 13738.5i	− 0.494749i
$918$	− 10112.3i	− 0.363569i
$919$	−17181.4	−0.616716	−0.308358	−	0.951270i	$-0.599779\pi$
$919$	−17181.4	−0.616716	−0.308358	+	0.951270i	$0.599779\pi$
$920$	−1739.64	−0.0623415
$921$	− 24636.8i	− 0.881444i
$922$	−46287.5	−1.65336
$923$	0	0
$924$	2536.34	0.0903025
$925$	19884.4i	0.706804i
$926$	6503.25	0.230788
$927$	16997.1	0.602221
$928$	51407.0i	1.81845i
$929$	− 56042.9i	− 1.97923i	−0.143728	−	0.989617i	$-0.545909\pi$
$929$	− 56042.9i	− 1.97923i	0.143728	−	0.989617i	$-0.454091\pi$
$930$	− 30099.9i	− 1.06130i
$931$	4116.52i	0.144912i
$932$	7736.00	0.271890
$933$	12715.7	0.446187
$934$	32843.3i	1.15060i
$935$	−12955.8	−0.453155
$936$	0	0
$937$	−36672.5	−1.27859	−0.639295	−	0.768961i	$-0.720774\pi$
$937$	−36672.5	−1.27859	−0.639295	+	0.768961i	$0.720774\pi$
$938$	− 6858.71i	− 0.238747i
$939$	11423.5	0.397008
$940$	−22576.4	−0.783363
$941$	21069.0i	0.729895i	0.931028	+	0.364947i	$0.118913\pi$
$941$	21069.0i	0.729895i	−0.931028	+	0.364947i	$0.881087\pi$
$942$	− 44113.7i	− 1.52580i
$943$	44093.4i	1.52267i
$944$	− 17363.0i	− 0.598642i
$945$	1299.61	0.0447369
$946$	−8113.87	−0.278863
$947$	− 1838.70i	− 0.0630938i	−0.999502	−	0.0315469i	$-0.989957\pi$
$947$	− 1838.70i	− 0.0630938i	0.999502	−	0.0315469i	$-0.0100434\pi$
$948$	15426.6	0.528516
$949$	0	0
$950$	−3213.27	−0.109739
$951$	− 31493.0i	− 1.07385i
$952$	541.330	0.0184292
$953$	−13599.8	−0.462266	−0.231133	−	0.972922i	$-0.574243\pi$
$953$	−13599.8	−0.462266	−0.231133	+	0.972922i	$0.574243\pi$
$954$	− 965.620i	− 0.0327705i
$955$	− 22695.9i	− 0.769029i
$956$	− 38446.9i	− 1.30069i
$957$	10314.9i	0.348416i
$958$	7365.75	0.248410
$959$	2231.02	0.0751235
$960$	13166.8i	0.442662i
$961$	−64935.6	−2.17971
$962$	0	0
$963$	−8275.17	−0.276909
$964$	− 31116.4i	− 1.03962i
$965$	−20105.5	−0.670692
$966$	−15808.0	−0.526516
$967$	− 2081.30i	− 0.0692141i	−0.999401	−	0.0346070i	$-0.988982\pi$
$967$	− 2081.30i	− 0.0692141i	0.999401	−	0.0346070i	$-0.0110180\pi$
$968$	− 1012.10i	− 0.0336054i
$969$	− 3731.10i	− 0.123695i
$970$	− 19681.6i	− 0.651484i
$971$	1636.62	0.0540904	0.0270452	−	0.999634i	$-0.491390\pi$
$971$	1636.62	0.0540904	0.0270452	+	0.999634i	$0.491390\pi$
$972$	−2003.02	−0.0660977
$973$	− 2902.26i	− 0.0956240i
$974$	31592.9	1.03932
$975$	0	0
$976$	12838.8	0.421066
$977$	29387.1i	0.962311i	0.876635	+	0.481156i	$0.159783\pi$
$977$	29387.1i	0.962311i	−0.876635	+	0.481156i	$0.840217\pi$
$978$	16763.7	0.548104
$979$	−21944.6	−0.716396
$980$	− 20508.1i	− 0.668476i
$981$	14133.6i	0.459992i
$982$	− 69304.4i	− 2.25213i
$983$	24084.4i	0.781457i	0.920506	+	0.390728i	$0.127777\pi$
$983$	24084.4i	0.781457i	−0.920506	+	0.390728i	$0.872223\pi$
$984$	−589.355	−0.0190934
$985$	−22249.4	−0.719722
$986$	74713.4i	2.41314i
$987$	−6044.98	−0.194948
$988$	0	0
$989$	25663.4	0.825125
$990$	5056.88i	0.162342i
$991$	1413.43	0.0453068	0.0226534	−	0.999743i	$-0.492789\pi$
$991$	1413.43	0.0453068	0.0226534	+	0.999743i	$0.492789\pi$
$992$	79313.5	2.53851
$993$	35715.7i	1.14139i
$994$	− 7618.79i	− 0.243112i
$995$	− 32342.7i	− 1.03049i
$996$	− 16317.8i	− 0.519126i
$997$	−33357.4	−1.05962	−0.529809	−	0.848117i	$-0.677737\pi$
$997$	−33357.4	−1.05962	−0.529809	+	0.848117i	$0.677737\pi$
$998$	−21583.1	−0.684570
$999$	9011.97i	0.285411i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.k.337.15		18
13.5	odd	4		507.4.a.p.1.7	yes	9
13.8	odd	4		507.4.a.o.1.3	✓	9
13.12	even	2	inner	507.4.b.k.337.4		18
39.5	even	4		1521.4.a.bf.1.3		9
39.8	even	4		1521.4.a.bi.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.3	✓	9	13.8	odd	4
507.4.a.p.1.7	yes	9	13.5	odd	4
507.4.b.k.337.4		18	13.12	even	2	inner
507.4.b.k.337.15		18	1.1	even	1	trivial
1521.4.a.bf.1.3		9	39.5	even	4
1521.4.a.bi.1.7		9	39.8	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)

\(f(q)\)	\(=\)	\(q+4.03025i q^{2} +3.00000 q^{3} -8.24289 q^{4} +8.08864i q^{5} +12.0907i q^{6} -5.95078i q^{7} -0.978887i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.03025i q^{2} +3.00000 q^{3} -8.24289 q^{4} +8.08864i q^{5} +12.0907i q^{6} -5.95078i q^{7} -0.978887i q^{8} +9.00000 q^{9} -32.5992 q^{10} -17.2359i q^{11} -24.7287 q^{12} +23.9831 q^{14} +24.2659i q^{15} -61.9979 q^{16} -92.9299 q^{17} +36.2722i q^{18} +13.3832i q^{19} -66.6738i q^{20} -17.8523i q^{21} +69.4648 q^{22} -219.710 q^{23} -2.93666i q^{24} +59.5738 q^{25} +27.0000 q^{27} +49.0516i q^{28} -199.485 q^{29} -97.7977 q^{30} +307.777i q^{31} -257.698i q^{32} -51.7076i q^{33} -374.531i q^{34} +48.1337 q^{35} -74.1860 q^{36} +333.777i q^{37} -53.9376 q^{38} +7.91787 q^{40} -200.689i q^{41} +71.9493 q^{42} -116.806 q^{43} +142.073i q^{44} +72.7978i q^{45} -885.487i q^{46} -338.610i q^{47} -185.994 q^{48} +307.588 q^{49} +240.097i q^{50} -278.790 q^{51} -26.6215 q^{53} +108.817i q^{54} +139.415 q^{55} -5.82514 q^{56} +40.1496i q^{57} -803.976i q^{58} +280.058i q^{59} -200.021i q^{60} -207.084 q^{61} -1240.42 q^{62} -53.5570i q^{63} +542.603 q^{64} +208.394 q^{66} -285.981i q^{67} +766.011 q^{68} -659.131 q^{69} +193.991i q^{70} -317.673i q^{71} -8.80999i q^{72} +63.0668i q^{73} -1345.20 q^{74} +178.721 q^{75} -110.316i q^{76} -102.567 q^{77} -623.835 q^{79} -501.479i q^{80} +81.0000 q^{81} +808.824 q^{82} +659.874i q^{83} +147.155i q^{84} -751.677i q^{85} -470.755i q^{86} -598.456 q^{87} -16.8720 q^{88} -1273.19i q^{89} -293.393 q^{90} +1811.05 q^{92} +923.331i q^{93} +1364.68 q^{94} -108.252 q^{95} -773.094i q^{96} +603.746i q^{97} +1239.66i q^{98} -155.123i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

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