LMFDB - Embedded newform 507.4.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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, 0]))

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

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

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 25x^{2} + 24x + 78 $ x^4 - 2x^3 - 25x^2 + 24*x + 78
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.36176$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.36176 q^{2} -3.00000 q^{3} -2.42208 q^{4} +6.42208 q^{5} -7.08529 q^{6} -29.4938 q^{7} -24.6145 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.36176 q^{2} -3.00000 q^{3} -2.42208 q^{4} +6.42208 q^{5} -7.08529 q^{6} -29.4938 q^{7} -24.6145 q^{8} +9.00000 q^{9} +15.1674 q^{10} -0.624715 q^{11} +7.26623 q^{12} -69.6575 q^{14} -19.2662 q^{15} -38.7569 q^{16} +87.7291 q^{17} +21.2559 q^{18} +82.8018 q^{19} -15.5548 q^{20} +88.4815 q^{21} -1.47543 q^{22} -74.7977 q^{23} +73.8434 q^{24} -83.7569 q^{25} -27.0000 q^{27} +71.4364 q^{28} +226.329 q^{29} -45.5023 q^{30} +173.660 q^{31} +105.381 q^{32} +1.87415 q^{33} +207.195 q^{34} -189.412 q^{35} -21.7987 q^{36} +112.020 q^{37} +195.558 q^{38} -158.076 q^{40} -267.011 q^{41} +208.972 q^{42} +383.450 q^{43} +1.51311 q^{44} +57.7987 q^{45} -176.654 q^{46} +337.380 q^{47} +116.271 q^{48} +526.887 q^{49} -197.814 q^{50} -263.187 q^{51} -146.354 q^{53} -63.7676 q^{54} -4.01197 q^{55} +725.975 q^{56} -248.406 q^{57} +534.536 q^{58} -529.173 q^{59} +46.6643 q^{60} +203.272 q^{61} +410.144 q^{62} -265.445 q^{63} +558.941 q^{64} +4.42629 q^{66} +121.497 q^{67} -212.487 q^{68} +224.393 q^{69} -447.346 q^{70} -661.314 q^{71} -221.530 q^{72} +167.341 q^{73} +264.565 q^{74} +251.271 q^{75} -200.552 q^{76} +18.4253 q^{77} -101.399 q^{79} -248.900 q^{80} +81.0000 q^{81} -630.617 q^{82} -506.985 q^{83} -214.309 q^{84} +563.403 q^{85} +905.617 q^{86} -678.988 q^{87} +15.3770 q^{88} +1402.33 q^{89} +136.507 q^{90} +181.166 q^{92} -520.981 q^{93} +796.810 q^{94} +531.760 q^{95} -316.143 q^{96} +1902.89 q^{97} +1244.38 q^{98} -5.62244 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 12 q^{3} + 22 q^{4} - 6 q^{5} - 6 q^{6} - 14 q^{7} + 54 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 12 * q^3 + 22 * q^4 - 6 * q^5 - 6 * q^6 - 14 * q^7 + 54 * q^8 + 36 * q^9	\( 4 q + 2 q^{2} - 12 q^{3} + 22 q^{4} - 6 q^{5} - 6 q^{6} - 14 q^{7} + 54 q^{8} + 36 q^{9} - 62 q^{10} + 40 q^{11} - 66 q^{12} + 40 q^{14} + 18 q^{15} + 122 q^{16} + 98 q^{17} + 18 q^{18} + 124 q^{19} - 466 q^{20} + 42 q^{21} + 220 q^{22} + 104 q^{23} - 162 q^{24} - 58 q^{25} - 108 q^{27} - 144 q^{28} + 194 q^{29} + 186 q^{30} + 26 q^{31} + 654 q^{32} - 120 q^{33} + 1062 q^{34} + 88 q^{35} + 198 q^{36} + 102 q^{37} + 332 q^{38} - 998 q^{40} - 1054 q^{41} - 120 q^{42} + 450 q^{43} - 44 q^{44} - 54 q^{45} - 172 q^{46} - 96 q^{47} - 366 q^{48} + 1070 q^{49} + 996 q^{50} - 294 q^{51} + 262 q^{53} - 54 q^{54} + 204 q^{55} + 2164 q^{56} - 372 q^{57} + 722 q^{58} + 308 q^{59} + 1398 q^{60} - 928 q^{61} + 2780 q^{62} - 126 q^{63} + 1026 q^{64} - 660 q^{66} - 1134 q^{67} + 1786 q^{68} - 312 q^{69} - 2324 q^{70} + 1064 q^{71} + 486 q^{72} + 952 q^{73} + 1158 q^{74} + 174 q^{75} - 1708 q^{76} + 2508 q^{77} - 746 q^{79} - 2922 q^{80} + 324 q^{81} + 1734 q^{82} - 404 q^{83} + 432 q^{84} - 1394 q^{85} + 3168 q^{86} - 582 q^{87} + 3060 q^{88} + 1620 q^{89} - 558 q^{90} + 332 q^{92} - 78 q^{93} - 772 q^{94} + 2204 q^{95} - 1962 q^{96} + 2166 q^{97} - 1906 q^{98} + 360 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 12 * q^3 + 22 * q^4 - 6 * q^5 - 6 * q^6 - 14 * q^7 + 54 * q^8 + 36 * q^9 - 62 * q^10 + 40 * q^11 - 66 * q^12 + 40 * q^14 + 18 * q^15 + 122 * q^16 + 98 * q^17 + 18 * q^18 + 124 * q^19 - 466 * q^20 + 42 * q^21 + 220 * q^22 + 104 * q^23 - 162 * q^24 - 58 * q^25 - 108 * q^27 - 144 * q^28 + 194 * q^29 + 186 * q^30 + 26 * q^31 + 654 * q^32 - 120 * q^33 + 1062 * q^34 + 88 * q^35 + 198 * q^36 + 102 * q^37 + 332 * q^38 - 998 * q^40 - 1054 * q^41 - 120 * q^42 + 450 * q^43 - 44 * q^44 - 54 * q^45 - 172 * q^46 - 96 * q^47 - 366 * q^48 + 1070 * q^49 + 996 * q^50 - 294 * q^51 + 262 * q^53 - 54 * q^54 + 204 * q^55 + 2164 * q^56 - 372 * q^57 + 722 * q^58 + 308 * q^59 + 1398 * q^60 - 928 * q^61 + 2780 * q^62 - 126 * q^63 + 1026 * q^64 - 660 * q^66 - 1134 * q^67 + 1786 * q^68 - 312 * q^69 - 2324 * q^70 + 1064 * q^71 + 486 * q^72 + 952 * q^73 + 1158 * q^74 + 174 * q^75 - 1708 * q^76 + 2508 * q^77 - 746 * q^79 - 2922 * q^80 + 324 * q^81 + 1734 * q^82 - 404 * q^83 + 432 * q^84 - 1394 * q^85 + 3168 * q^86 - 582 * q^87 + 3060 * q^88 + 1620 * q^89 - 558 * q^90 + 332 * q^92 - 78 * q^93 - 772 * q^94 + 2204 * q^95 - 1962 * q^96 + 2166 * q^97 - 1906 * q^98 + 360 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.36176	0.835009	0.417505	−	0.908675i	$-0.362905\pi$
$2$	2.36176	0.835009	0.417505	+	0.908675i	$0.362905\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−2.42208	−0.302760
$5$	6.42208	0.574408	0.287204	−	0.957869i	$-0.407274\pi$
$5$	6.42208	0.574408	0.287204	+	0.957869i	$0.407274\pi$
$6$	−7.08529	−0.482093
$7$	−29.4938	−1.59252	−0.796259	−	0.604956i	$-0.793191\pi$
$7$	−29.4938	−1.59252	−0.796259	+	0.604956i	$0.793191\pi$
$8$	−24.6145	−1.08782
$9$	9.00000	0.333333
$10$	15.1674	0.479636
$11$	−0.624715	−0.0171235	−0.00856176	−	0.999963i	$-0.502725\pi$
$11$	−0.624715	−0.0171235	−0.00856176	+	0.999963i	$0.502725\pi$
$12$	7.26623	0.174798
$13$	0	0
$13$	0	0
$14$	−69.6575	−1.32977
$15$	−19.2662	−0.331635
$16$	−38.7569	−0.605577
$17$	87.7291	1.25161	0.625807	−	0.779978i	$-0.284770\pi$
$17$	87.7291	1.25161	0.625807	+	0.779978i	$0.284770\pi$
$18$	21.2559	0.278336
$19$	82.8018	0.999792	0.499896	−	0.866085i	$-0.333372\pi$
$19$	82.8018	0.999792	0.499896	+	0.866085i	$0.333372\pi$
$20$	−15.5548	−0.173908
$21$	88.4815	0.919441
$22$	−1.47543	−0.0142983
$23$	−74.7977	−0.678104	−0.339052	−	0.940768i	$-0.610106\pi$
$23$	−74.7977	−0.678104	−0.339052	+	0.940768i	$0.610106\pi$
$24$	73.8434	0.628051
$25$	−83.7569	−0.670055
$26$	0	0
$27$	−27.0000	−0.192450
$28$	71.4364	0.482150
$29$	226.329	1.44925	0.724625	−	0.689143i	$-0.242013\pi$
$29$	226.329	1.44925	0.724625	+	0.689143i	$0.242013\pi$
$30$	−45.5023	−0.276918
$31$	173.660	1.00614	0.503070	−	0.864246i	$-0.332204\pi$
$31$	173.660	1.00614	0.503070	+	0.864246i	$0.332204\pi$
$32$	105.381	0.582154
$33$	1.87415	0.00988627
$34$	207.195	1.04511
$35$	−189.412	−0.914755
$36$	−21.7987	−0.100920
$37$	112.020	0.497730	0.248865	−	0.968538i	$-0.419942\pi$
$37$	112.020	0.497730	0.248865	+	0.968538i	$0.419942\pi$
$38$	195.558	0.834835
$39$	0	0
$40$	−158.076	−0.624850
$41$	−267.011	−1.01708	−0.508538	−	0.861040i	$-0.669814\pi$
$41$	−267.011	−1.01708	−0.508538	+	0.861040i	$0.669814\pi$
$42$	208.972	0.767741
$43$	383.450	1.35990	0.679948	−	0.733260i	$-0.262002\pi$
$43$	383.450	1.35990	0.679948	+	0.733260i	$0.262002\pi$
$44$	1.51311	0.00518431
$45$	57.7987	0.191469
$46$	−176.654	−0.566223
$47$	337.380	1.04706	0.523530	−	0.852007i	$-0.324615\pi$
$47$	337.380	1.04706	0.523530	+	0.852007i	$0.324615\pi$
$48$	116.271	0.349630
$49$	526.887	1.53611
$50$	−197.814	−0.559502
$51$	−263.187	−0.722619
$52$	0	0
$53$	−146.354	−0.379308	−0.189654	−	0.981851i	$-0.560737\pi$
$53$	−146.354	−0.379308	−0.189654	+	0.981851i	$0.560737\pi$
$54$	−63.7676	−0.160698
$55$	−4.01197	−0.00983589
$56$	725.975	1.73237
$57$	−248.406	−0.577230
$58$	534.536	1.21014
$59$	−529.173	−1.16767	−0.583834	−	0.811873i	$-0.698448\pi$
$59$	−529.173	−1.16767	−0.583834	+	0.811873i	$0.698448\pi$
$60$	46.6643	0.100406
$61$	203.272	0.426660	0.213330	−	0.976980i	$-0.431569\pi$
$61$	203.272	0.426660	0.213330	+	0.976980i	$0.431569\pi$
$62$	410.144	0.840135
$63$	−265.445	−0.530839
$64$	558.941	1.09168
$65$	0	0
$66$	4.42629	0.00825513
$67$	121.497	0.221540	0.110770	−	0.993846i	$-0.464668\pi$
$67$	121.497	0.221540	0.110770	+	0.993846i	$0.464668\pi$
$68$	−212.487	−0.378938
$69$	224.393	0.391504
$70$	−447.346	−0.763829
$71$	−661.314	−1.10540	−0.552701	−	0.833380i	$-0.686403\pi$
$71$	−661.314	−1.10540	−0.552701	+	0.833380i	$0.686403\pi$
$72$	−221.530	−0.362605
$73$	167.341	0.268299	0.134150	−	0.990961i	$-0.457170\pi$
$73$	167.341	0.268299	0.134150	+	0.990961i	$0.457170\pi$
$74$	264.565	0.415609
$75$	251.271	0.386857
$76$	−200.552	−0.302697
$77$	18.4253	0.0272695
$78$	0	0
$79$	−101.399	−0.144408	−0.0722042	−	0.997390i	$-0.523003\pi$
$79$	−101.399	−0.144408	−0.0722042	+	0.997390i	$0.523003\pi$
$80$	−248.900	−0.347848
$81$	81.0000	0.111111
$82$	−630.617	−0.849268
$83$	−506.985	−0.670468	−0.335234	−	0.942135i	$-0.608815\pi$
$83$	−506.985	−0.670468	−0.335234	+	0.942135i	$0.608815\pi$
$84$	−214.309	−0.278370
$85$	563.403	0.718937
$86$	905.617	1.13553
$87$	−678.988	−0.836725
$88$	15.3770	0.0186272
$89$	1402.33	1.67019	0.835095	−	0.550106i	$-0.185413\pi$
$89$	1402.33	1.67019	0.835095	+	0.550106i	$0.185413\pi$
$90$	136.507	0.159879
$91$	0	0
$92$	181.166	0.205303
$93$	−520.981	−0.580895
$94$	796.810	0.874306
$95$	531.760	0.574289
$96$	−316.143	−0.336107
$97$	1902.89	1.99185	0.995924	−	0.0901969i	$-0.0287497\pi$
$97$	1902.89	1.99185	0.995924	+	0.0901969i	$0.0287497\pi$
$98$	1244.38	1.28267
$99$	−5.62244	−0.00570784
$100$	202.866	0.202866
$101$	1833.09	1.80594	0.902968	−	0.429708i	$-0.141383\pi$
$101$	1833.09	1.80594	0.902968	+	0.429708i	$0.141383\pi$
$102$	−621.586	−0.603394
$103$	1446.99	1.38423	0.692115	−	0.721787i	$-0.256679\pi$
$103$	1446.99	1.38423	0.692115	+	0.721787i	$0.256679\pi$
$104$	0	0
$105$	568.235	0.528134
$106$	−345.654	−0.316725
$107$	−369.286	−0.333647	−0.166823	−	0.985987i	$-0.553351\pi$
$107$	−369.286	−0.333647	−0.166823	+	0.985987i	$0.553351\pi$
$108$	65.3961	0.0582661
$109$	−815.694	−0.716782	−0.358391	−	0.933572i	$-0.616675\pi$
$109$	−815.694	−0.716782	−0.358391	+	0.933572i	$0.616675\pi$
$110$	−9.47532	−0.00821306
$111$	−336.061	−0.287365
$112$	1143.09	0.964392
$113$	1790.56	1.49064	0.745319	−	0.666708i	$-0.232297\pi$
$113$	1790.56	1.49064	0.745319	+	0.666708i	$0.232297\pi$
$114$	−586.675	−0.481992
$115$	−480.357	−0.389509
$116$	−548.187	−0.438775
$117$	0	0
$118$	−1249.78	−0.975014
$119$	−2587.47	−1.99322
$120$	474.228	0.360758
$121$	−1330.61	−0.999707
$122$	480.079	0.356265
$123$	801.033	0.587209
$124$	−420.619	−0.304618
$125$	−1340.65	−0.959293
$126$	−626.917	−0.443256
$127$	−45.2900	−0.0316444	−0.0158222	−	0.999875i	$-0.505037\pi$
$127$	−45.2900	−0.0316444	−0.0158222	+	0.999875i	$0.505037\pi$
$128$	477.036	0.329410
$129$	−1150.35	−0.785137
$130$	0	0
$131$	1051.82	0.701511	0.350756	−	0.936467i	$-0.385925\pi$
$131$	1051.82	0.701511	0.350756	+	0.936467i	$0.385925\pi$
$132$	−4.53933	−0.00299316
$133$	−2442.14	−1.59219
$134$	286.946	0.184988
$135$	−173.396	−0.110545
$136$	−2159.41	−1.36153
$137$	−1542.94	−0.962208	−0.481104	−	0.876664i	$-0.659764\pi$
$137$	−1542.94	−0.962208	−0.481104	+	0.876664i	$0.659764\pi$
$138$	529.963	0.326909
$139$	37.8644	0.0231052	0.0115526	−	0.999933i	$-0.496323\pi$
$139$	37.8644	0.0231052	0.0115526	+	0.999933i	$0.496323\pi$
$140$	458.770	0.276951
$141$	−1012.14	−0.604521
$142$	−1561.87	−0.923020
$143$	0	0
$144$	−348.812	−0.201859
$145$	1453.50	0.832461
$146$	395.221	0.224032
$147$	−1580.66	−0.886875
$148$	−271.322	−0.150693
$149$	−1822.40	−1.00199	−0.500995	−	0.865450i	$-0.667033\pi$
$149$	−1822.40	−1.00199	−0.500995	+	0.865450i	$0.667033\pi$
$150$	593.442	0.323029
$151$	−3239.36	−1.74580	−0.872900	−	0.487899i	$-0.837763\pi$
$151$	−3239.36	−1.74580	−0.872900	+	0.487899i	$0.837763\pi$
$152$	−2038.12	−1.08759
$153$	789.562	0.417205
$154$	43.5161	0.0227703
$155$	1115.26	0.577934
$156$	0	0
$157$	830.565	0.422206	0.211103	−	0.977464i	$-0.432295\pi$
$157$	830.565	0.422206	0.211103	+	0.977464i	$0.432295\pi$
$158$	−239.480	−0.120582
$159$	439.063	0.218993
$160$	676.766	0.334394
$161$	2206.07	1.07989
$162$	191.303	0.0927788
$163$	−2079.90	−0.999451	−0.499725	−	0.866184i	$-0.666566\pi$
$163$	−2079.90	−0.999451	−0.499725	+	0.866184i	$0.666566\pi$
$164$	646.721	0.307930
$165$	12.0359	0.00567875
$166$	−1197.38	−0.559847
$167$	85.9790	0.0398398	0.0199199	−	0.999802i	$-0.493659\pi$
$167$	85.9790	0.0398398	0.0199199	+	0.999802i	$0.493659\pi$
$168$	−2177.93	−1.00018
$169$	0	0
$170$	1330.62	0.600319
$171$	745.217	0.333264
$172$	−928.745	−0.411722
$173$	2706.47	1.18942	0.594708	−	0.803942i	$-0.297268\pi$
$173$	2706.47	1.18942	0.594708	+	0.803942i	$0.297268\pi$
$174$	−1603.61	−0.698673
$175$	2470.31	1.06708
$176$	24.2120	0.0103696
$177$	1587.52	0.674154
$178$	3311.97	1.39462
$179$	−4402.10	−1.83815	−0.919074	−	0.394085i	$-0.871062\pi$
$179$	−4402.10	−1.83815	−0.919074	+	0.394085i	$0.871062\pi$
$180$	−139.993	−0.0579692
$181$	−1673.98	−0.687435	−0.343718	−	0.939073i	$-0.611686\pi$
$181$	−1673.98	−0.687435	−0.343718	+	0.939073i	$0.611686\pi$
$182$	0	0
$183$	−609.815	−0.246332
$184$	1841.11	0.737653
$185$	719.403	0.285900
$186$	−1230.43	−0.485052
$187$	−54.8057	−0.0214320
$188$	−817.159	−0.317008
$189$	796.334	0.306480
$190$	1255.89	0.479536
$191$	−290.117	−0.109907	−0.0549533	−	0.998489i	$-0.517501\pi$
$191$	−290.117	−0.109907	−0.0549533	+	0.998489i	$0.517501\pi$
$192$	−1676.82	−0.630282
$193$	1039.50	0.387693	0.193847	−	0.981032i	$-0.437904\pi$
$193$	1039.50	0.387693	0.193847	+	0.981032i	$0.437904\pi$
$194$	4494.18	1.66321
$195$	0	0
$196$	−1276.16	−0.465073
$197$	1418.70	0.513089	0.256544	−	0.966532i	$-0.417416\pi$
$197$	1418.70	0.513089	0.256544	+	0.966532i	$0.417416\pi$
$198$	−13.2789	−0.00476610
$199$	−2388.39	−0.850795	−0.425398	−	0.905007i	$-0.639866\pi$
$199$	−2388.39	−0.850795	−0.425398	+	0.905007i	$0.639866\pi$
$200$	2061.63	0.728897
$201$	−364.490	−0.127906
$202$	4329.33	1.50797
$203$	−6675.32	−2.30796
$204$	637.460	0.218780
$205$	−1714.77	−0.584217
$206$	3417.44	1.15584
$207$	−673.179	−0.226035
$208$	0	0
$209$	−51.7276	−0.0171200
$210$	1342.04	0.440997
$211$	4341.45	1.41648	0.708241	−	0.705971i	$-0.249489\pi$
$211$	4341.45	1.41648	0.708241	+	0.705971i	$0.249489\pi$
$212$	354.481	0.114839
$213$	1983.94	0.638204
$214$	−872.165	−0.278598
$215$	2462.54	0.781136
$216$	664.591	0.209350
$217$	−5121.91	−1.60229
$218$	−1926.47	−0.598520
$219$	−502.024	−0.154903
$220$	9.71730	0.00297791
$221$	0	0
$222$	−793.696	−0.239952
$223$	4615.37	1.38596	0.692978	−	0.720959i	$-0.256298\pi$
$223$	4615.37	1.38596	0.692978	+	0.720959i	$0.256298\pi$
$224$	−3108.09	−0.927091
$225$	−753.812	−0.223352
$226$	4228.89	1.24470
$227$	−2163.44	−0.632565	−0.316283	−	0.948665i	$-0.602435\pi$
$227$	−2163.44	−0.632565	−0.316283	+	0.948665i	$0.602435\pi$
$228$	601.657	0.174762
$229$	1859.48	0.536584	0.268292	−	0.963338i	$-0.413541\pi$
$229$	1859.48	0.536584	0.268292	+	0.963338i	$0.413541\pi$
$230$	−1134.49	−0.325243
$231$	−55.2758	−0.0157441
$232$	−5570.97	−1.57652
$233$	2866.87	0.806073	0.403037	−	0.915184i	$-0.367955\pi$
$233$	2866.87	0.806073	0.403037	+	0.915184i	$0.367955\pi$
$234$	0	0
$235$	2166.68	0.601440
$236$	1281.70	0.353523
$237$	304.197	0.0833743
$238$	−6110.99	−1.66435
$239$	1893.55	0.512485	0.256242	−	0.966613i	$-0.417516\pi$
$239$	1893.55	0.512485	0.256242	+	0.966613i	$0.417516\pi$
$240$	746.700	0.200830
$241$	−1813.58	−0.484741	−0.242371	−	0.970184i	$-0.577925\pi$
$241$	−1813.58	−0.484741	−0.242371	+	0.970184i	$0.577925\pi$
$242$	−3142.58	−0.834764
$243$	−243.000	−0.0641500
$244$	−492.340	−0.129176
$245$	3383.71	0.882356
$246$	1891.85	0.490325
$247$	0	0
$248$	−4274.56	−1.09449
$249$	1520.96	0.387095
$250$	−3166.30	−0.801019
$251$	4162.32	1.04671	0.523353	−	0.852116i	$-0.324681\pi$
$251$	4162.32	1.04671	0.523353	+	0.852116i	$0.324681\pi$
$252$	642.927	0.160717
$253$	46.7273	0.0116115
$254$	−106.964	−0.0264234
$255$	−1690.21	−0.415078
$256$	−3344.88	−0.816621
$257$	5985.30	1.45274	0.726368	−	0.687306i	$-0.241207\pi$
$257$	5985.30	1.45274	0.726368	+	0.687306i	$0.241207\pi$
$258$	−2716.85	−0.655596
$259$	−3303.91	−0.792645
$260$	0	0
$261$	2036.96	0.483084
$262$	2484.15	0.585768
$263$	574.618	0.134724	0.0673621	−	0.997729i	$-0.478542\pi$
$263$	574.618	0.134724	0.0673621	+	0.997729i	$0.478542\pi$
$264$	−46.1311	−0.0107544
$265$	−939.899	−0.217877
$266$	−5767.77	−1.32949
$267$	−4206.99	−0.964284
$268$	−294.275	−0.0670734
$269$	6348.61	1.43896	0.719482	−	0.694511i	$-0.244379\pi$
$269$	6348.61	1.43896	0.719482	+	0.694511i	$0.244379\pi$
$270$	−409.520	−0.0923060
$271$	−3278.38	−0.734861	−0.367431	−	0.930051i	$-0.619762\pi$
$271$	−3278.38	−0.734861	−0.367431	+	0.930051i	$0.619762\pi$
$272$	−3400.11	−0.757948
$273$	0	0
$274$	−3644.06	−0.803452
$275$	52.3242	0.0114737
$276$	−543.498	−0.118532
$277$	3952.17	0.857267	0.428633	−	0.903478i	$-0.358995\pi$
$277$	3952.17	0.857267	0.428633	+	0.903478i	$0.358995\pi$
$278$	89.4267	0.0192930
$279$	1562.94	0.335380
$280$	4662.27	0.995085
$281$	−411.389	−0.0873360	−0.0436680	−	0.999046i	$-0.513904\pi$
$281$	−411.389	−0.0873360	−0.0436680	+	0.999046i	$0.513904\pi$
$282$	−2390.43	−0.504781
$283$	−5872.78	−1.23357	−0.616785	−	0.787131i	$-0.711565\pi$
$283$	−5872.78	−1.23357	−0.616785	+	0.787131i	$0.711565\pi$
$284$	1601.75	0.334671
$285$	−1595.28	−0.331566
$286$	0	0
$287$	7875.18	1.61971
$288$	948.430	0.194051
$289$	2783.39	0.566537
$290$	3432.83	0.695113
$291$	−5708.67	−1.14999
$292$	−405.314	−0.0812301
$293$	−500.957	−0.0998847	−0.0499423	−	0.998752i	$-0.515904\pi$
$293$	−500.957	−0.0998847	−0.0499423	+	0.998752i	$0.515904\pi$
$294$	−3733.14	−0.740549
$295$	−3398.39	−0.670718
$296$	−2757.32	−0.541439
$297$	16.8673	0.00329542
$298$	−4304.07	−0.836671
$299$	0	0
$300$	−608.597	−0.117125
$301$	−11309.4	−2.16566
$302$	−7650.61	−1.45776
$303$	−5499.28	−1.04266
$304$	−3209.14	−0.605451
$305$	1305.43	0.245077
$306$	1864.76	0.348370
$307$	−5975.57	−1.11089	−0.555446	−	0.831553i	$-0.687452\pi$
$307$	−5975.57	−1.11089	−0.555446	+	0.831553i	$0.687452\pi$
$308$	−44.6274	−0.00825611
$309$	−4340.96	−0.799185
$310$	2633.98	0.482581
$311$	44.4925	0.00811234	0.00405617	−	0.999992i	$-0.498709\pi$
$311$	44.4925	0.00811234	0.00405617	+	0.999992i	$0.498709\pi$
$312$	0	0
$313$	9957.78	1.79823	0.899117	−	0.437709i	$-0.144210\pi$
$313$	9957.78	1.79823	0.899117	+	0.437709i	$0.144210\pi$
$314$	1961.60	0.352546
$315$	−1704.71	−0.304918
$316$	245.596	0.0437211
$317$	7752.29	1.37354	0.686770	−	0.726875i	$-0.259028\pi$
$317$	7752.29	1.37354	0.686770	+	0.726875i	$0.259028\pi$
$318$	1036.96	0.182862
$319$	−141.391	−0.0248163
$320$	3589.56	0.627070
$321$	1107.86	0.192631
$322$	5210.22	0.901721
$323$	7264.13	1.25135
$324$	−196.188	−0.0336400
$325$	0	0
$326$	−4912.23	−0.834550
$327$	2447.08	0.413834
$328$	6572.34	1.10639
$329$	−9950.62	−1.66746
$330$	28.4260	0.00474181
$331$	−1338.85	−0.222326	−0.111163	−	0.993802i	$-0.535458\pi$
$331$	−1338.85	−0.222326	−0.111163	+	0.993802i	$0.535458\pi$
$332$	1227.96	0.202991
$333$	1008.18	0.165910
$334$	203.062	0.0332666
$335$	780.261	0.127254
$336$	−3429.27	−0.556792
$337$	3788.95	0.612454	0.306227	−	0.951958i	$-0.400933\pi$
$337$	3788.95	0.612454	0.306227	+	0.951958i	$0.400933\pi$
$338$	0	0
$339$	−5371.69	−0.860621
$340$	−1364.61	−0.217665
$341$	−108.488	−0.0172286
$342$	1760.02	0.278278
$343$	−5423.53	−0.853770
$344$	−9438.42	−1.47932
$345$	1441.07	0.224883
$346$	6392.03	0.993173
$347$	7795.15	1.20595	0.602977	−	0.797759i	$-0.293981\pi$
$347$	7795.15	1.20595	0.602977	+	0.797759i	$0.293981\pi$
$348$	1644.56	0.253327
$349$	134.533	0.0206344	0.0103172	−	0.999947i	$-0.496716\pi$
$349$	134.533	0.0206344	0.0103172	+	0.999947i	$0.496716\pi$
$350$	5834.29	0.891018
$351$	0	0
$352$	−65.8332	−0.00996853
$353$	−2973.71	−0.448370	−0.224185	−	0.974547i	$-0.571972\pi$
$353$	−2973.71	−0.448370	−0.224185	+	0.974547i	$0.571972\pi$
$354$	3749.34	0.562925
$355$	−4247.01	−0.634952
$356$	−3396.56	−0.505666
$357$	7762.40	1.15078
$358$	−10396.7	−1.53487
$359$	8671.60	1.27485	0.637423	−	0.770514i	$-0.280001\pi$
$359$	8671.60	1.27485	0.637423	+	0.770514i	$0.280001\pi$
$360$	−1422.68	−0.208283
$361$	−2.85364	−0.000416043 0
$362$	−3953.54	−0.574015
$363$	3991.83	0.577181
$364$	0	0
$365$	1074.68	0.154113
$366$	−1440.24	−0.205690
$367$	4514.32	0.642086	0.321043	−	0.947065i	$-0.395967\pi$
$367$	4514.32	0.642086	0.321043	+	0.947065i	$0.395967\pi$
$368$	2898.93	0.410644
$369$	−2403.10	−0.339025
$370$	1699.06	0.238729
$371$	4316.55	0.604054
$372$	1261.86	0.175871
$373$	6570.60	0.912099	0.456049	−	0.889955i	$-0.349264\pi$
$373$	6570.60	0.912099	0.456049	+	0.889955i	$0.349264\pi$
$374$	−129.438	−0.0178959
$375$	4021.96	0.553848
$376$	−8304.42	−1.13901
$377$	0	0
$378$	1880.75	0.255914
$379$	−5490.38	−0.744121	−0.372060	−	0.928209i	$-0.621349\pi$
$379$	−5490.38	−0.744121	−0.372060	+	0.928209i	$0.621349\pi$
$380$	−1287.96	−0.173871
$381$	135.870	0.0182699
$382$	−685.188	−0.0917730
$383$	10187.3	1.35912	0.679562	−	0.733618i	$-0.262170\pi$
$383$	10187.3	1.35912	0.679562	+	0.733618i	$0.262170\pi$
$384$	−1431.11	−0.190185
$385$	118.328	0.0156638
$386$	2455.05	0.323728
$387$	3451.05	0.453299
$388$	−4608.95	−0.603051
$389$	4883.97	0.636573	0.318287	−	0.947995i	$-0.396893\pi$
$389$	4883.97	0.636573	0.318287	+	0.947995i	$0.396893\pi$
$390$	0	0
$391$	−6561.94	−0.848725
$392$	−12969.0	−1.67101
$393$	−3155.46	−0.405018
$394$	3350.64	0.428434
$395$	−651.192	−0.0829494
$396$	13.6180	0.00172810
$397$	−2115.04	−0.267382	−0.133691	−	0.991023i	$-0.542683\pi$
$397$	−2115.04	−0.267382	−0.133691	+	0.991023i	$0.542683\pi$
$398$	−5640.80	−0.710422
$399$	7326.43	0.919249
$400$	3246.16	0.405770
$401$	−674.254	−0.0839667	−0.0419834	−	0.999118i	$-0.513368\pi$
$401$	−674.254	−0.0839667	−0.0419834	+	0.999118i	$0.513368\pi$
$402$	−860.839	−0.106803
$403$	0	0
$404$	−4439.89	−0.546765
$405$	520.188	0.0638231
$406$	−15765.5	−1.92717
$407$	−69.9808	−0.00852290
$408$	6478.22	0.786077
$409$	4673.33	0.564991	0.282496	−	0.959269i	$-0.408838\pi$
$409$	4673.33	0.564991	0.282496	+	0.959269i	$0.408838\pi$
$410$	−4049.87	−0.487826
$411$	4628.83	0.555531
$412$	−3504.71	−0.419089
$413$	15607.3	1.85953
$414$	−1589.89	−0.188741
$415$	−3255.90	−0.385122
$416$	0	0
$417$	−113.593	−0.0133398
$418$	−122.168	−0.0142953
$419$	−256.853	−0.0299477	−0.0149739	−	0.999888i	$-0.504767\pi$
$419$	−256.853	−0.0299477	−0.0149739	+	0.999888i	$0.504767\pi$
$420$	−1376.31	−0.159898
$421$	−8746.82	−1.01257	−0.506287	−	0.862365i	$-0.668982\pi$
$421$	−8746.82	−1.01257	−0.506287	+	0.862365i	$0.668982\pi$
$422$	10253.5	1.18278
$423$	3036.42	0.349020
$424$	3602.43	0.412617
$425$	−7347.92	−0.838650
$426$	4685.60	0.532906
$427$	−5995.26	−0.679464
$428$	894.439	0.101015
$429$	0	0
$430$	5815.95	0.652255
$431$	8762.72	0.979316	0.489658	−	0.871914i	$-0.337122\pi$
$431$	8762.72	0.979316	0.489658	+	0.871914i	$0.337122\pi$
$432$	1046.44	0.116543
$433$	6425.50	0.713140	0.356570	−	0.934269i	$-0.383946\pi$
$433$	6425.50	0.713140	0.356570	+	0.934269i	$0.383946\pi$
$434$	−12096.7	−1.33793
$435$	−4360.51	−0.480622
$436$	1975.67	0.217013
$437$	−6193.39	−0.677963
$438$	−1185.66	−0.129345
$439$	6820.28	0.741490	0.370745	−	0.928735i	$-0.379102\pi$
$439$	6820.28	0.741490	0.370745	+	0.928735i	$0.379102\pi$
$440$	98.7525	0.0106996
$441$	4741.98	0.512038
$442$	0	0
$443$	5062.48	0.542948	0.271474	−	0.962446i	$-0.412489\pi$
$443$	5062.48	0.542948	0.271474	+	0.962446i	$0.412489\pi$
$444$	813.966	0.0870025
$445$	9005.88	0.959370
$446$	10900.4	1.15729
$447$	5467.19	0.578500
$448$	−16485.3	−1.73852
$449$	−6590.82	−0.692740	−0.346370	−	0.938098i	$-0.612586\pi$
$449$	−6590.82	−0.692740	−0.346370	+	0.938098i	$0.612586\pi$
$450$	−1780.33	−0.186501
$451$	166.806	0.0174159
$452$	−4336.89	−0.451305
$453$	9718.09	1.00794
$454$	−5109.52	−0.528198
$455$	0	0
$456$	6114.37	0.627920
$457$	−2254.75	−0.230793	−0.115397	−	0.993319i	$-0.536814\pi$
$457$	−2254.75	−0.230793	−0.115397	+	0.993319i	$0.536814\pi$
$458$	4391.65	0.448053
$459$	−2368.69	−0.240873
$460$	1163.46	0.117928
$461$	7358.79	0.743456	0.371728	−	0.928342i	$-0.378765\pi$
$461$	7358.79	0.743456	0.371728	+	0.928342i	$0.378765\pi$
$462$	−130.548	−0.0131464
$463$	11598.3	1.16419	0.582096	−	0.813120i	$-0.302233\pi$
$463$	11598.3	1.16419	0.582096	+	0.813120i	$0.302233\pi$
$464$	−8771.82	−0.877633
$465$	−3345.78	−0.333671
$466$	6770.87	0.673079
$467$	−302.150	−0.0299397	−0.0149698	−	0.999888i	$-0.504765\pi$
$467$	−302.150	−0.0299397	−0.0149698	+	0.999888i	$0.504765\pi$
$468$	0	0
$469$	−3583.41	−0.352807
$470$	5117.18	0.502208
$471$	−2491.69	−0.243760
$472$	13025.3	1.27021
$473$	−239.547	−0.0232862
$474$	718.441	0.0696183
$475$	−6935.23	−0.669916
$476$	6267.05	0.603466
$477$	−1317.19	−0.126436
$478$	4472.12	0.427929
$479$	−3146.15	−0.300107	−0.150053	−	0.988678i	$-0.547945\pi$
$479$	−3146.15	−0.300107	−0.150053	+	0.988678i	$0.547945\pi$
$480$	−2030.30	−0.193062
$481$	0	0
$482$	−4283.24	−0.404764
$483$	−6618.22	−0.623477
$484$	3222.84	0.302671
$485$	12220.5	1.14413
$486$	−573.908	−0.0535659
$487$	3068.16	0.285486	0.142743	−	0.989760i	$-0.454408\pi$
$487$	3068.16	0.285486	0.142743	+	0.989760i	$0.454408\pi$
$488$	−5003.43	−0.464128
$489$	6239.70	0.577033
$490$	7991.51	0.736775
$491$	−100.558	−0.00924262	−0.00462131	−	0.999989i	$-0.501471\pi$
$491$	−100.558	−0.00924262	−0.00462131	+	0.999989i	$0.501471\pi$
$492$	−1940.16	−0.177783
$493$	19855.7	1.81390
$494$	0	0
$495$	−36.1077	−0.00327863
$496$	−6730.54	−0.609295
$497$	19504.7	1.76037
$498$	3592.14	0.323228
$499$	3616.55	0.324447	0.162223	−	0.986754i	$-0.448133\pi$
$499$	3616.55	0.324447	0.162223	+	0.986754i	$0.448133\pi$
$500$	3247.17	0.290435
$501$	−257.937	−0.0230015
$502$	9830.40	0.874009
$503$	−5372.64	−0.476251	−0.238125	−	0.971234i	$-0.576533\pi$
$503$	−5372.64	−0.476251	−0.238125	+	0.971234i	$0.576533\pi$
$504$	6533.78	0.577456
$505$	11772.3	1.03734
$506$	110.359	0.00969574
$507$	0	0
$508$	109.696	0.00958065
$509$	−11314.7	−0.985298	−0.492649	−	0.870228i	$-0.663971\pi$
$509$	−11314.7	−0.985298	−0.492649	+	0.870228i	$0.663971\pi$
$510$	−3991.87	−0.346594
$511$	−4935.54	−0.427271
$512$	−11716.1	−1.01130
$513$	−2235.65	−0.192410
$514$	14135.9	1.21305
$515$	9292.65	0.795113
$516$	2786.24	0.237708
$517$	−210.766	−0.0179294
$518$	−7803.05	−0.661865
$519$	−8119.40	−0.686709
$520$	0	0
$521$	−18470.9	−1.55321	−0.776606	−	0.629986i	$-0.783061\pi$
$521$	−18470.9	−1.55321	−0.776606	+	0.629986i	$0.783061\pi$
$522$	4810.82	0.403379
$523$	10891.0	0.910576	0.455288	−	0.890344i	$-0.349536\pi$
$523$	10891.0	0.910576	0.455288	+	0.890344i	$0.349536\pi$
$524$	−2547.59	−0.212389
$525$	−7410.94	−0.616076
$526$	1357.11	0.112496
$527$	15235.1	1.25930
$528$	−72.6361	−0.00598690
$529$	−6572.30	−0.540174
$530$	−2219.82	−0.181930
$531$	−4762.56	−0.389223
$532$	5915.06	0.482050
$533$	0	0
$534$	−9935.92	−0.805186
$535$	−2371.58	−0.191649
$536$	−2990.58	−0.240995
$537$	13206.3	1.06126
$538$	14993.9	1.20155
$539$	−329.154	−0.0263037
$540$	419.979	0.0334685
$541$	−13416.3	−1.06620	−0.533099	−	0.846053i	$-0.678973\pi$
$541$	−13416.3	−1.06620	−0.533099	+	0.846053i	$0.678973\pi$
$542$	−7742.75	−0.613616
$543$	5021.93	0.396891
$544$	9244.99	0.728632
$545$	−5238.45	−0.411726
$546$	0	0
$547$	−17849.0	−1.39519	−0.697593	−	0.716495i	$-0.745745\pi$
$547$	−17849.0	−1.39519	−0.697593	+	0.716495i	$0.745745\pi$
$548$	3737.13	0.291318
$549$	1829.45	0.142220
$550$	123.577	0.00958065
$551$	18740.5	1.44895
$552$	−5523.32	−0.425884
$553$	2990.64	0.229973
$554$	9334.09	0.715826
$555$	−2158.21	−0.165065
$556$	−91.7105	−0.00699531
$557$	−20131.4	−1.53141	−0.765703	−	0.643195i	$-0.777609\pi$
$557$	−20131.4	−1.53141	−0.765703	+	0.643195i	$0.777609\pi$
$558$	3691.30	0.280045
$559$	0	0
$560$	7341.02	0.553955
$561$	164.417	0.0123738
$562$	−971.603	−0.0729263
$563$	−22345.8	−1.67276	−0.836380	−	0.548150i	$-0.815332\pi$
$563$	−22345.8	−1.67276	−0.836380	+	0.548150i	$0.815332\pi$
$564$	2451.48	0.183025
$565$	11499.1	0.856235
$566$	−13870.1	−1.03004
$567$	−2389.00	−0.176946
$568$	16277.9	1.20247
$569$	−8455.72	−0.622992	−0.311496	−	0.950248i	$-0.600830\pi$
$569$	−8455.72	−0.622992	−0.311496	+	0.950248i	$0.600830\pi$
$570$	−3767.67	−0.276860
$571$	12813.0	0.939069	0.469534	−	0.882914i	$-0.344422\pi$
$571$	12813.0	0.939069	0.469534	+	0.882914i	$0.344422\pi$
$572$	0	0
$573$	870.352	0.0634546
$574$	18599.3	1.35247
$575$	6264.83	0.454367
$576$	5030.47	0.363894
$577$	1971.59	0.142251	0.0711253	−	0.997467i	$-0.477341\pi$
$577$	1971.59	0.142251	0.0711253	+	0.997467i	$0.477341\pi$
$578$	6573.72	0.473063
$579$	−3118.50	−0.223835
$580$	−3520.50	−0.252036
$581$	14952.9	1.06773
$582$	−13482.5	−0.960255
$583$	91.4298	0.00649508
$584$	−4119.02	−0.291860
$585$	0	0
$586$	−1183.14	−0.0834046
$587$	8585.69	0.603696	0.301848	−	0.953356i	$-0.402397\pi$
$587$	8585.69	0.603696	0.301848	+	0.953356i	$0.402397\pi$
$588$	3828.48	0.268510
$589$	14379.4	1.00593
$590$	−8026.19	−0.560056
$591$	−4256.11	−0.296232
$592$	−4341.56	−0.301414
$593$	−1746.73	−0.120961	−0.0604803	−	0.998169i	$-0.519263\pi$
$593$	−1746.73	−0.120961	−0.0604803	+	0.998169i	$0.519263\pi$
$594$	39.8366	0.00275171
$595$	−16616.9	−1.14492
$596$	4413.99	0.303362
$597$	7165.16	0.491207
$598$	0	0
$599$	27531.1	1.87794	0.938972	−	0.343994i	$-0.111780\pi$
$599$	27531.1	1.87794	0.938972	+	0.343994i	$0.111780\pi$
$600$	−6184.90	−0.420829
$601$	−17539.1	−1.19041	−0.595203	−	0.803575i	$-0.702928\pi$
$601$	−17539.1	−1.19041	−0.595203	+	0.803575i	$0.702928\pi$
$602$	−26710.1	−1.80835
$603$	1093.47	0.0738467
$604$	7845.99	0.528558
$605$	−8545.28	−0.574240
$606$	−12988.0	−0.870629
$607$	−22691.1	−1.51730	−0.758651	−	0.651497i	$-0.774141\pi$
$607$	−22691.1	−1.51730	−0.758651	+	0.651497i	$0.774141\pi$
$608$	8725.75	0.582033
$609$	20026.0	1.33250
$610$	3083.11	0.204642
$611$	0	0
$612$	−1912.38	−0.126313
$613$	7215.30	0.475405	0.237702	−	0.971338i	$-0.423606\pi$
$613$	7215.30	0.475405	0.237702	+	0.971338i	$0.423606\pi$
$614$	−14112.9	−0.927605
$615$	5144.30	0.337298
$616$	−453.528	−0.0296642
$617$	−16870.4	−1.10077	−0.550387	−	0.834909i	$-0.685520\pi$
$617$	−16870.4	−1.10077	−0.550387	+	0.834909i	$0.685520\pi$
$618$	−10252.3	−0.667327
$619$	2244.53	0.145743	0.0728717	−	0.997341i	$-0.476784\pi$
$619$	2244.53	0.145743	0.0728717	+	0.997341i	$0.476784\pi$
$620$	−2701.25	−0.174975
$621$	2019.54	0.130501
$622$	105.081	0.00677388
$623$	−41360.1	−2.65981
$624$	0	0
$625$	1859.84	0.119030
$626$	23517.9	1.50154
$627$	155.183	0.00988421
$628$	−2011.69	−0.127827
$629$	9827.44	0.622966
$630$	−4026.11	−0.254610
$631$	−3669.22	−0.231488	−0.115744	−	0.993279i	$-0.536925\pi$
$631$	−3669.22	−0.231488	−0.115744	+	0.993279i	$0.536925\pi$
$632$	2495.88	0.157090
$633$	−13024.3	−0.817806
$634$	18309.1	1.14692
$635$	−290.856	−0.0181768
$636$	−1063.44	−0.0663024
$637$	0	0
$638$	−333.933	−0.0207218
$639$	−5951.82	−0.368467
$640$	3063.56	0.189215
$641$	14678.6	0.904477	0.452239	−	0.891897i	$-0.350626\pi$
$641$	14678.6	0.904477	0.452239	+	0.891897i	$0.350626\pi$
$642$	2616.50	0.160849
$643$	5519.72	0.338533	0.169266	−	0.985570i	$-0.445860\pi$
$643$	5519.72	0.338533	0.169266	+	0.985570i	$0.445860\pi$
$644$	−5343.28	−0.326948
$645$	−7387.63	−0.450989
$646$	17156.2	1.04489
$647$	−11326.9	−0.688261	−0.344131	−	0.938922i	$-0.611826\pi$
$647$	−11326.9	−0.688261	−0.344131	+	0.938922i	$0.611826\pi$
$648$	−1993.77	−0.120868
$649$	330.582	0.0199946
$650$	0	0
$651$	15365.7	0.925085
$652$	5037.68	0.302593
$653$	3902.89	0.233893	0.116946	−	0.993138i	$-0.462689\pi$
$653$	3902.89	0.233893	0.116946	+	0.993138i	$0.462689\pi$
$654$	5779.42	0.345556
$655$	6754.87	0.402954
$656$	10348.5	0.615918
$657$	1506.07	0.0894330
$658$	−23501.0	−1.39235
$659$	8022.47	0.474220	0.237110	−	0.971483i	$-0.423800\pi$
$659$	8022.47	0.474220	0.237110	+	0.971483i	$0.423800\pi$
$660$	−29.1519	−0.00171930
$661$	−5168.76	−0.304147	−0.152074	−	0.988369i	$-0.548595\pi$
$661$	−5168.76	−0.304147	−0.152074	+	0.988369i	$0.548595\pi$
$662$	−3162.05	−0.185645
$663$	0	0
$664$	12479.2	0.729346
$665$	−15683.6	−0.914565
$666$	2381.09	0.138536
$667$	−16928.9	−0.982743
$668$	−208.248	−0.0120619
$669$	−13846.1	−0.800182
$670$	1842.79	0.106259
$671$	−126.987	−0.00730593
$672$	9324.28	0.535256
$673$	−6654.10	−0.381124	−0.190562	−	0.981675i	$-0.561031\pi$
$673$	−6654.10	−0.381124	−0.190562	+	0.981675i	$0.561031\pi$
$674$	8948.59	0.511405
$675$	2261.44	0.128952
$676$	0	0
$677$	−20649.4	−1.17226	−0.586130	−	0.810217i	$-0.699349\pi$
$677$	−20649.4	−1.17226	−0.586130	+	0.810217i	$0.699349\pi$
$678$	−12686.7	−0.718626
$679$	−56123.6	−3.17205
$680$	−13867.9	−0.782071
$681$	6490.31	0.365212
$682$	−256.223	−0.0143861
$683$	28475.6	1.59530	0.797649	−	0.603122i	$-0.206077\pi$
$683$	28475.6	1.59530	0.797649	+	0.603122i	$0.206077\pi$
$684$	−1804.97	−0.100899
$685$	−9908.90	−0.552700
$686$	−12809.1	−0.712905
$687$	−5578.43	−0.309797
$688$	−14861.3	−0.823522
$689$	0	0
$690$	3403.47	0.187779
$691$	10610.8	0.584160	0.292080	−	0.956394i	$-0.405653\pi$
$691$	10610.8	0.584160	0.292080	+	0.956394i	$0.405653\pi$
$692$	−6555.27	−0.360107
$693$	165.827	0.00908984
$694$	18410.3	1.00698
$695$	243.168	0.0132718
$696$	16712.9	0.910203
$697$	−23424.6	−1.27299
$698$	317.736	0.0172299
$699$	−8600.62	−0.465387
$700$	−5983.29	−0.323067
$701$	−13518.9	−0.728390	−0.364195	−	0.931323i	$-0.618656\pi$
$701$	−13518.9	−0.728390	−0.364195	+	0.931323i	$0.618656\pi$
$702$	0	0
$703$	9275.49	0.497627
$704$	−349.179	−0.0186934
$705$	−6500.03	−0.347242
$706$	−7023.19	−0.374393
$707$	−54064.9	−2.87599
$708$	−3845.09	−0.204107
$709$	14845.4	0.786361	0.393180	−	0.919461i	$-0.371375\pi$
$709$	14845.4	0.786361	0.393180	+	0.919461i	$0.371375\pi$
$710$	−10030.4	−0.530190
$711$	−912.590	−0.0481362
$712$	−34517.7	−1.81686
$713$	−12989.4	−0.682267
$714$	18333.0	0.960915
$715$	0	0
$716$	10662.2	0.556517
$717$	−5680.66	−0.295883
$718$	20480.3	1.06451
$719$	−35900.1	−1.86210	−0.931049	−	0.364893i	$-0.881106\pi$
$719$	−35900.1	−1.86210	−0.931049	+	0.364893i	$0.881106\pi$
$720$	−2240.10	−0.115949
$721$	−42677.2	−2.20441
$722$	−6.73962	−0.000347400 0
$723$	5440.73	0.279866
$724$	4054.50	0.208128
$725$	−18956.6	−0.971078
$726$	9427.75	0.481951
$727$	12951.4	0.660715	0.330357	−	0.943856i	$-0.392831\pi$
$727$	12951.4	0.660715	0.330357	+	0.943856i	$0.392831\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	2538.14	0.128686
$731$	33639.7	1.70206
$732$	1477.02	0.0745795
$733$	−1105.36	−0.0556989	−0.0278494	−	0.999612i	$-0.508866\pi$
$733$	−1105.36	−0.0556989	−0.0278494	+	0.999612i	$0.508866\pi$
$734$	10661.8	0.536148
$735$	−10151.1	−0.509428
$736$	−7882.27	−0.394761
$737$	−75.9009	−0.00379355
$738$	−5675.55	−0.283089
$739$	13638.1	0.678869	0.339434	−	0.940630i	$-0.389764\pi$
$739$	13638.1	0.678869	0.339434	+	0.940630i	$0.389764\pi$
$740$	−1742.45	−0.0865591
$741$	0	0
$742$	10194.7	0.504391
$743$	10154.9	0.501408	0.250704	−	0.968064i	$-0.419338\pi$
$743$	10154.9	0.501408	0.250704	+	0.968064i	$0.419338\pi$
$744$	12823.7	0.631907
$745$	−11703.6	−0.575552
$746$	15518.2	0.761611
$747$	−4562.87	−0.223489
$748$	132.744	0.00648876
$749$	10891.7	0.531338
$750$	9498.91	0.462468
$751$	26579.5	1.29148	0.645738	−	0.763559i	$-0.276550\pi$
$751$	26579.5	1.29148	0.645738	+	0.763559i	$0.276550\pi$
$752$	−13075.8	−0.634076
$753$	−12487.0	−0.604316
$754$	0	0
$755$	−20803.5	−1.00280
$756$	−1928.78	−0.0927898
$757$	−13677.7	−0.656705	−0.328352	−	0.944555i	$-0.606493\pi$
$757$	−13677.7	−0.656705	−0.328352	+	0.944555i	$0.606493\pi$
$758$	−12967.0	−0.621348
$759$	−140.182	−0.00670392
$760$	−13089.0	−0.624720
$761$	17996.7	0.857268	0.428634	−	0.903478i	$-0.358995\pi$
$761$	17996.7	0.857268	0.428634	+	0.903478i	$0.358995\pi$
$762$	320.893	0.0152555
$763$	24057.9	1.14149
$764$	702.687	0.0332753
$765$	5070.63	0.239646
$766$	24059.9	1.13488
$767$	0	0
$768$	10034.6	0.471476
$769$	2995.04	0.140447	0.0702236	−	0.997531i	$-0.477629\pi$
$769$	2995.04	0.140447	0.0702236	+	0.997531i	$0.477629\pi$
$770$	279.464	0.0130794
$771$	−17955.9	−0.838737
$772$	−2517.75	−0.117378
$773$	−26055.4	−1.21235	−0.606177	−	0.795330i	$-0.707298\pi$
$773$	−26055.4	−1.21235	−0.606177	+	0.795330i	$0.707298\pi$
$774$	8150.56	0.378509
$775$	−14545.3	−0.674169
$776$	−46838.6	−2.16676
$777$	9911.73	0.457634
$778$	11534.8	0.531545
$779$	−22109.0	−1.01686
$780$	0	0
$781$	413.133	0.0189284
$782$	−15497.7	−0.708693
$783$	−6110.89	−0.278908
$784$	−20420.5	−0.930235
$785$	5333.95	0.242518
$786$	−7452.45	−0.338193
$787$	−22992.0	−1.04139	−0.520697	−	0.853741i	$-0.674328\pi$
$787$	−22992.0	−1.04139	−0.520697	+	0.853741i	$0.674328\pi$
$788$	−3436.21	−0.155343
$789$	−1723.85	−0.0777830
$790$	−1537.96	−0.0692635
$791$	−52810.6	−2.37387
$792$	138.393	0.00620908
$793$	0	0
$794$	−4995.22	−0.223267
$795$	2819.70	0.125792
$796$	5784.86	0.257587
$797$	−25826.3	−1.14782	−0.573910	−	0.818918i	$-0.694574\pi$
$797$	−25826.3	−1.14782	−0.573910	+	0.818918i	$0.694574\pi$
$798$	17303.3	0.767582
$799$	29598.0	1.31052
$800$	−8826.40	−0.390075
$801$	12621.0	0.556730
$802$	−1592.43	−0.0701130
$803$	−104.541	−0.00459423
$804$	882.824	0.0387249
$805$	14167.6	0.620299
$806$	0	0
$807$	−19045.8	−0.830787
$808$	−45120.6	−1.96453
$809$	−28495.8	−1.23839	−0.619195	−	0.785237i	$-0.712541\pi$
$809$	−28495.8	−1.23839	−0.619195	+	0.785237i	$0.712541\pi$
$810$	1228.56	0.0532929
$811$	6992.41	0.302758	0.151379	−	0.988476i	$-0.451629\pi$
$811$	6992.41	0.302758	0.151379	+	0.988476i	$0.451629\pi$
$812$	16168.1	0.698757
$813$	9835.14	0.424272
$814$	−165.278	−0.00711670
$815$	−13357.3	−0.574093
$816$	10200.3	0.437602
$817$	31750.4	1.35961
$818$	11037.3	0.471773
$819$	0	0
$820$	4153.29	0.176877
$821$	31512.7	1.33959	0.669793	−	0.742547i	$-0.266383\pi$
$821$	31512.7	1.33959	0.669793	+	0.742547i	$0.266383\pi$
$822$	10932.2	0.463873
$823$	39159.6	1.65859	0.829293	−	0.558814i	$-0.188743\pi$
$823$	39159.6	1.65859	0.829293	+	0.558814i	$0.188743\pi$
$824$	−35616.8	−1.50579
$825$	−156.973	−0.00662435
$826$	36860.8	1.55273
$827$	−36557.6	−1.53716	−0.768581	−	0.639752i	$-0.779037\pi$
$827$	−36557.6	−1.53716	−0.768581	+	0.639752i	$0.779037\pi$
$828$	1630.49	0.0684342
$829$	14675.2	0.614825	0.307413	−	0.951576i	$-0.400537\pi$
$829$	14675.2	0.614825	0.307413	+	0.951576i	$0.400537\pi$
$830$	−7689.66	−0.321581
$831$	−11856.5	−0.494943
$832$	0	0
$833$	46223.3	1.92262
$834$	−268.280	−0.0111388
$835$	552.164	0.0228843
$836$	125.288	0.00518323
$837$	−4688.83	−0.193632
$838$	−606.626	−0.0250066
$839$	−8515.56	−0.350405	−0.175202	−	0.984532i	$-0.556058\pi$
$839$	−8515.56	−0.350405	−0.175202	+	0.984532i	$0.556058\pi$
$840$	−13986.8	−0.574513
$841$	26835.9	1.10033
$842$	−20657.9	−0.845509
$843$	1234.17	0.0504234
$844$	−10515.3	−0.428854
$845$	0	0
$846$	7171.29	0.291435
$847$	39244.8	1.59205
$848$	5672.24	0.229700
$849$	17618.3	0.712202
$850$	−17354.0	−0.700281
$851$	−8378.86	−0.337513
$852$	−4805.26	−0.193222
$853$	9645.93	0.387187	0.193593	−	0.981082i	$-0.437986\pi$
$853$	9645.93	0.387187	0.193593	+	0.981082i	$0.437986\pi$
$854$	−14159.4	−0.567359
$855$	4785.84	0.191430
$856$	9089.77	0.362946
$857$	36139.6	1.44050	0.720248	−	0.693717i	$-0.244028\pi$
$857$	36139.6	1.44050	0.720248	+	0.693717i	$0.244028\pi$
$858$	0	0
$859$	−7108.04	−0.282332	−0.141166	−	0.989986i	$-0.545085\pi$
$859$	−7108.04	−0.282332	−0.141166	+	0.989986i	$0.545085\pi$
$860$	−5964.47	−0.236496
$861$	−23625.5	−0.935141
$862$	20695.5	0.817738
$863$	16225.8	0.640016	0.320008	−	0.947415i	$-0.396314\pi$
$863$	16225.8	0.640016	0.320008	+	0.947415i	$0.396314\pi$
$864$	−2845.29	−0.112036
$865$	17381.1	0.683210
$866$	15175.5	0.595478
$867$	−8350.18	−0.327090
$868$	12405.7	0.485110
$869$	63.3455	0.00247278
$870$	−10298.5	−0.401324
$871$	0	0
$872$	20077.9	0.779727
$873$	17126.0	0.663949
$874$	−14627.3	−0.566106
$875$	39541.0	1.52769
$876$	1215.94	0.0468982
$877$	−30983.0	−1.19295	−0.596477	−	0.802630i	$-0.703433\pi$
$877$	−30983.0	−1.19295	−0.596477	+	0.802630i	$0.703433\pi$
$878$	16107.9	0.619151
$879$	1502.87	0.0576684
$880$	155.492	0.00595639
$881$	−7670.76	−0.293342	−0.146671	−	0.989185i	$-0.546856\pi$
$881$	−7670.76	−0.293342	−0.146671	+	0.989185i	$0.546856\pi$
$882$	11199.4	0.427556
$883$	−34340.6	−1.30878	−0.654390	−	0.756157i	$-0.727075\pi$
$883$	−34340.6	−1.30878	−0.654390	+	0.756157i	$0.727075\pi$
$884$	0	0
$885$	10195.2	0.387239
$886$	11956.4	0.453366
$887$	−19208.3	−0.727118	−0.363559	−	0.931571i	$-0.618438\pi$
$887$	−19208.3	−0.727118	−0.363559	+	0.931571i	$0.618438\pi$
$888$	8271.96	0.312600
$889$	1335.78	0.0503943
$890$	21269.8	0.801083
$891$	−50.6019	−0.00190261
$892$	−11178.8	−0.419611
$893$	27935.7	1.04684
$894$	12912.2	0.483052
$895$	−28270.6	−1.05585
$896$	−14069.6	−0.524591
$897$	0	0
$898$	−15566.0	−0.578444
$899$	39304.4	1.45815
$900$	1825.79	0.0676219
$901$	−12839.5	−0.474747
$902$	393.956	0.0145425
$903$	33928.2	1.25034
$904$	−44073.8	−1.62154
$905$	−10750.4	−0.394868
$906$	22951.8	0.841637
$907$	−46481.0	−1.70163	−0.850813	−	0.525468i	$-0.823890\pi$
$907$	−46481.0	−1.70163	−0.850813	+	0.525468i	$0.823890\pi$
$908$	5240.01	0.191515
$909$	16497.8	0.601979
$910$	0	0
$911$	−34109.3	−1.24050	−0.620248	−	0.784406i	$-0.712968\pi$
$911$	−34109.3	−1.24050	−0.620248	+	0.784406i	$0.712968\pi$
$912$	9627.43	0.349557
$913$	316.721	0.0114808
$914$	−5325.18	−0.192715
$915$	−3916.28	−0.141495
$916$	−4503.80	−0.162456
$917$	−31022.2	−1.11717
$918$	−5594.27	−0.201131
$919$	−37533.6	−1.34725	−0.673623	−	0.739075i	$-0.735263\pi$
$919$	−37533.6	−1.34725	−0.673623	+	0.739075i	$0.735263\pi$
$920$	11823.7	0.423714
$921$	17926.7	0.641373
$922$	17379.7	0.620793
$923$	0	0
$924$	133.882	0.00476667
$925$	−9382.48	−0.333507
$926$	27392.5	0.972111
$927$	13022.9	0.461410
$928$	23850.8	0.843687
$929$	4966.75	0.175408	0.0877038	−	0.996147i	$-0.472047\pi$
$929$	4966.75	0.175408	0.0877038	+	0.996147i	$0.472047\pi$
$930$	−7901.94	−0.278618
$931$	43627.2	1.53579
$932$	−6943.79	−0.244047
$933$	−133.477	−0.00468366
$934$	−713.606	−0.0249999
$935$	−351.966	−0.0123107
$936$	0	0
$937$	−5096.90	−0.177704	−0.0888519	−	0.996045i	$-0.528320\pi$
$937$	−5096.90	−0.177704	−0.0888519	+	0.996045i	$0.528320\pi$
$938$	−8463.15	−0.294597
$939$	−29873.3	−1.03821
$940$	−5247.86	−0.182092
$941$	54774.8	1.89756	0.948781	−	0.315933i	$-0.102318\pi$
$941$	54774.8	1.89756	0.948781	+	0.315933i	$0.102318\pi$
$942$	−5884.79	−0.203542
$943$	19971.8	0.689684
$944$	20509.1	0.707113
$945$	5114.12	0.176045
$946$	−565.753	−0.0194442
$947$	−13768.5	−0.472456	−0.236228	−	0.971698i	$-0.575911\pi$
$947$	−13768.5	−0.472456	−0.236228	+	0.971698i	$0.575911\pi$
$948$	−736.788	−0.0252424
$949$	0	0
$950$	−16379.4	−0.559386
$951$	−23256.9	−0.793013
$952$	63689.2	2.16825
$953$	35866.6	1.21913	0.609566	−	0.792735i	$-0.291344\pi$
$953$	35866.6	1.21913	0.609566	+	0.792735i	$0.291344\pi$
$954$	−3110.89	−0.105575
$955$	−1863.16	−0.0631312
$956$	−4586.33	−0.155160
$957$	424.174	0.0143277
$958$	−7430.46	−0.250592
$959$	45507.3	1.53233
$960$	−10768.7	−0.362039
$961$	366.899	0.0123158
$962$	0	0
$963$	−3323.57	−0.111216
$964$	4392.62	0.146760
$965$	6675.75	0.222694
$966$	−15630.7	−0.520609
$967$	−24476.5	−0.813972	−0.406986	−	0.913434i	$-0.633420\pi$
$967$	−24476.5	−0.813972	−0.406986	+	0.913434i	$0.633420\pi$
$968$	32752.3	1.08750
$969$	−21792.4	−0.722469
$970$	28861.9	0.955362
$971$	−8976.07	−0.296659	−0.148329	−	0.988938i	$-0.547390\pi$
$971$	−8976.07	−0.296659	−0.148329	+	0.988938i	$0.547390\pi$
$972$	588.565	0.0194220
$973$	−1116.77	−0.0367954
$974$	7246.26	0.238383
$975$	0	0
$976$	−7878.18	−0.258376
$977$	−42002.8	−1.37542	−0.687711	−	0.725984i	$-0.741385\pi$
$977$	−42002.8	−1.37542	−0.687711	+	0.725984i	$0.741385\pi$
$978$	14736.7	0.481828
$979$	−876.058	−0.0285995
$980$	−8195.60	−0.267142
$981$	−7341.24	−0.238927
$982$	−237.495	−0.00771767
$983$	43240.7	1.40301	0.701507	−	0.712662i	$-0.252511\pi$
$983$	43240.7	1.40301	0.701507	+	0.712662i	$0.252511\pi$
$984$	−19717.0	−0.638776
$985$	9111.03	0.294722
$986$	46894.3	1.51462
$987$	29851.9	0.962710
$988$	0	0
$989$	−28681.2	−0.922152
$990$	−85.2779	−0.00273769
$991$	37916.5	1.21540	0.607698	−	0.794168i	$-0.292093\pi$
$991$	37916.5	1.21540	0.607698	+	0.794168i	$0.292093\pi$
$992$	18300.5	0.585728
$993$	4016.56	0.128360
$994$	46065.4	1.46993
$995$	−15338.4	−0.488704
$996$	−3683.87	−0.117197
$997$	6634.86	0.210760	0.105380	−	0.994432i	$-0.466394\pi$
$997$	6634.86	0.210760	0.105380	+	0.994432i	$0.466394\pi$
$998$	8541.43	0.270916
$999$	−3024.55	−0.0957883

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.m.1.3		4
3.2	odd	2		1521.4.a.v.1.2		4
13.3	even	3		39.4.e.c.22.2	yes	8
13.5	odd	4		507.4.b.h.337.3		8
13.8	odd	4		507.4.b.h.337.6		8
13.9	even	3		39.4.e.c.16.2	✓	8
13.12	even	2		507.4.a.i.1.2		4
39.29	odd	6		117.4.g.e.100.3		8
39.35	odd	6		117.4.g.e.55.3		8
39.38	odd	2		1521.4.a.bb.1.3		4
52.3	odd	6		624.4.q.i.529.3		8
52.35	odd	6		624.4.q.i.289.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.c.16.2	✓	8	13.9	even	3
39.4.e.c.22.2	yes	8	13.3	even	3
117.4.g.e.55.3		8	39.35	odd	6
117.4.g.e.100.3		8	39.29	odd	6
507.4.a.i.1.2		4	13.12	even	2
507.4.a.m.1.3		4	1.1	even	1	trivial
507.4.b.h.337.3		8	13.5	odd	4
507.4.b.h.337.6		8	13.8	odd	4
624.4.q.i.289.3		8	52.35	odd	6
624.4.q.i.529.3		8	52.3	odd	6
1521.4.a.v.1.2		4	3.2	odd	2
1521.4.a.bb.1.3		4	39.38	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q+2.36176 q^{2} -3.00000 q^{3} -2.42208 q^{4} +6.42208 q^{5} -7.08529 q^{6} -29.4938 q^{7} -24.6145 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.36176 q^{2} -3.00000 q^{3} -2.42208 q^{4} +6.42208 q^{5} -7.08529 q^{6} -29.4938 q^{7} -24.6145 q^{8} +9.00000 q^{9} +15.1674 q^{10} -0.624715 q^{11} +7.26623 q^{12} -69.6575 q^{14} -19.2662 q^{15} -38.7569 q^{16} +87.7291 q^{17} +21.2559 q^{18} +82.8018 q^{19} -15.5548 q^{20} +88.4815 q^{21} -1.47543 q^{22} -74.7977 q^{23} +73.8434 q^{24} -83.7569 q^{25} -27.0000 q^{27} +71.4364 q^{28} +226.329 q^{29} -45.5023 q^{30} +173.660 q^{31} +105.381 q^{32} +1.87415 q^{33} +207.195 q^{34} -189.412 q^{35} -21.7987 q^{36} +112.020 q^{37} +195.558 q^{38} -158.076 q^{40} -267.011 q^{41} +208.972 q^{42} +383.450 q^{43} +1.51311 q^{44} +57.7987 q^{45} -176.654 q^{46} +337.380 q^{47} +116.271 q^{48} +526.887 q^{49} -197.814 q^{50} -263.187 q^{51} -146.354 q^{53} -63.7676 q^{54} -4.01197 q^{55} +725.975 q^{56} -248.406 q^{57} +534.536 q^{58} -529.173 q^{59} +46.6643 q^{60} +203.272 q^{61} +410.144 q^{62} -265.445 q^{63} +558.941 q^{64} +4.42629 q^{66} +121.497 q^{67} -212.487 q^{68} +224.393 q^{69} -447.346 q^{70} -661.314 q^{71} -221.530 q^{72} +167.341 q^{73} +264.565 q^{74} +251.271 q^{75} -200.552 q^{76} +18.4253 q^{77} -101.399 q^{79} -248.900 q^{80} +81.0000 q^{81} -630.617 q^{82} -506.985 q^{83} -214.309 q^{84} +563.403 q^{85} +905.617 q^{86} -678.988 q^{87} +15.3770 q^{88} +1402.33 q^{89} +136.507 q^{90} +181.166 q^{92} -520.981 q^{93} +796.810 q^{94} +531.760 q^{95} -316.143 q^{96} +1902.89 q^{97} +1244.38 q^{98} -5.62244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 12 q^{3} + 22 q^{4} - 6 q^{5} - 6 q^{6} - 14 q^{7} + 54 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 12 * q^3 + 22 * q^4 - 6 * q^5 - 6 * q^6 - 14 * q^7 + 54 * q^8 + 36 * q^9	\( 4 q + 2 q^{2} - 12 q^{3} + 22 q^{4} - 6 q^{5} - 6 q^{6} - 14 q^{7} + 54 q^{8} + 36 q^{9} - 62 q^{10} + 40 q^{11} - 66 q^{12} + 40 q^{14} + 18 q^{15} + 122 q^{16} + 98 q^{17} + 18 q^{18} + 124 q^{19} - 466 q^{20} + 42 q^{21} + 220 q^{22} + 104 q^{23} - 162 q^{24} - 58 q^{25} - 108 q^{27} - 144 q^{28} + 194 q^{29} + 186 q^{30} + 26 q^{31} + 654 q^{32} - 120 q^{33} + 1062 q^{34} + 88 q^{35} + 198 q^{36} + 102 q^{37} + 332 q^{38} - 998 q^{40} - 1054 q^{41} - 120 q^{42} + 450 q^{43} - 44 q^{44} - 54 q^{45} - 172 q^{46} - 96 q^{47} - 366 q^{48} + 1070 q^{49} + 996 q^{50} - 294 q^{51} + 262 q^{53} - 54 q^{54} + 204 q^{55} + 2164 q^{56} - 372 q^{57} + 722 q^{58} + 308 q^{59} + 1398 q^{60} - 928 q^{61} + 2780 q^{62} - 126 q^{63} + 1026 q^{64} - 660 q^{66} - 1134 q^{67} + 1786 q^{68} - 312 q^{69} - 2324 q^{70} + 1064 q^{71} + 486 q^{72} + 952 q^{73} + 1158 q^{74} + 174 q^{75} - 1708 q^{76} + 2508 q^{77} - 746 q^{79} - 2922 q^{80} + 324 q^{81} + 1734 q^{82} - 404 q^{83} + 432 q^{84} - 1394 q^{85} + 3168 q^{86} - 582 q^{87} + 3060 q^{88} + 1620 q^{89} - 558 q^{90} + 332 q^{92} - 78 q^{93} - 772 q^{94} + 2204 q^{95} - 1962 q^{96} + 2166 q^{97} - 1906 q^{98} + 360 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 12 * q^3 + 22 * q^4 - 6 * q^5 - 6 * q^6 - 14 * q^7 + 54 * q^8 + 36 * q^9 - 62 * q^10 + 40 * q^11 - 66 * q^12 + 40 * q^14 + 18 * q^15 + 122 * q^16 + 98 * q^17 + 18 * q^18 + 124 * q^19 - 466 * q^20 + 42 * q^21 + 220 * q^22 + 104 * q^23 - 162 * q^24 - 58 * q^25 - 108 * q^27 - 144 * q^28 + 194 * q^29 + 186 * q^30 + 26 * q^31 + 654 * q^32 - 120 * q^33 + 1062 * q^34 + 88 * q^35 + 198 * q^36 + 102 * q^37 + 332 * q^38 - 998 * q^40 - 1054 * q^41 - 120 * q^42 + 450 * q^43 - 44 * q^44 - 54 * q^45 - 172 * q^46 - 96 * q^47 - 366 * q^48 + 1070 * q^49 + 996 * q^50 - 294 * q^51 + 262 * q^53 - 54 * q^54 + 204 * q^55 + 2164 * q^56 - 372 * q^57 + 722 * q^58 + 308 * q^59 + 1398 * q^60 - 928 * q^61 + 2780 * q^62 - 126 * q^63 + 1026 * q^64 - 660 * q^66 - 1134 * q^67 + 1786 * q^68 - 312 * q^69 - 2324 * q^70 + 1064 * q^71 + 486 * q^72 + 952 * q^73 + 1158 * q^74 + 174 * q^75 - 1708 * q^76 + 2508 * q^77 - 746 * q^79 - 2922 * q^80 + 324 * q^81 + 1734 * q^82 - 404 * q^83 + 432 * q^84 - 1394 * q^85 + 3168 * q^86 - 582 * q^87 + 3060 * q^88 + 1620 * q^89 - 558 * q^90 + 332 * q^92 - 78 * q^93 - 772 * q^94 + 2204 * q^95 - 1962 * q^96 + 2166 * q^97 - 1906 * q^98 + 360 * q^99

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