LMFDB - Embedded newform 507.4.a.q.1.1

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 $ x^9 - x^8 - 48x^7 + 29x^6 + 772x^5 - 150x^4 - 4745x^3 - 966x^2 + 9428*x + 5144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.82618$ 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)

$f(q)$	$=$	$q-3.82618 q^{2} -3.00000 q^{3} +6.63963 q^{4} +0.275426 q^{5} +11.4785 q^{6} +0.0981245 q^{7} +5.20502 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.82618 q^{2} -3.00000 q^{3} +6.63963 q^{4} +0.275426 q^{5} +11.4785 q^{6} +0.0981245 q^{7} +5.20502 q^{8} +9.00000 q^{9} -1.05383 q^{10} -0.749942 q^{11} -19.9189 q^{12} -0.375442 q^{14} -0.826279 q^{15} -73.0324 q^{16} +53.7635 q^{17} -34.4356 q^{18} +145.488 q^{19} +1.82873 q^{20} -0.294374 q^{21} +2.86941 q^{22} +29.3945 q^{23} -15.6150 q^{24} -124.924 q^{25} -27.0000 q^{27} +0.651511 q^{28} -267.798 q^{29} +3.16149 q^{30} -51.5466 q^{31} +237.795 q^{32} +2.24983 q^{33} -205.709 q^{34} +0.0270261 q^{35} +59.7567 q^{36} -133.939 q^{37} -556.661 q^{38} +1.43360 q^{40} +430.228 q^{41} +1.12633 q^{42} -282.181 q^{43} -4.97934 q^{44} +2.47884 q^{45} -112.469 q^{46} -212.022 q^{47} +219.097 q^{48} -342.990 q^{49} +477.982 q^{50} -161.291 q^{51} +573.160 q^{53} +103.307 q^{54} -0.206554 q^{55} +0.510740 q^{56} -436.463 q^{57} +1024.64 q^{58} +495.767 q^{59} -5.48619 q^{60} -310.911 q^{61} +197.226 q^{62} +0.883121 q^{63} -325.585 q^{64} -8.60823 q^{66} -103.628 q^{67} +356.970 q^{68} -88.1836 q^{69} -0.103407 q^{70} -203.277 q^{71} +46.8451 q^{72} -685.571 q^{73} +512.474 q^{74} +374.772 q^{75} +965.984 q^{76} -0.0735877 q^{77} +636.019 q^{79} -20.1150 q^{80} +81.0000 q^{81} -1646.13 q^{82} +506.740 q^{83} -1.95453 q^{84} +14.8079 q^{85} +1079.67 q^{86} +803.393 q^{87} -3.90346 q^{88} +700.628 q^{89} -9.48447 q^{90} +195.169 q^{92} +154.640 q^{93} +811.232 q^{94} +40.0711 q^{95} -713.384 q^{96} +874.596 q^{97} +1312.34 q^{98} -6.74948 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 8 q^{2} - 27 q^{3} + 32 q^{4} + 41 q^{5} - 24 q^{6} + q^{7} + 111 q^{8} + 81 q^{9}+O(q^{10}) $ 9 * q + 8 * q^2 - 27 * q^3 + 32 * q^4 + 41 * q^5 - 24 * q^6 + q^7 + 111 * q^8 + 81 * q^9	\( 9 q + 8 q^{2} - 27 q^{3} + 32 q^{4} + 41 q^{5} - 24 q^{6} + q^{7} + 111 q^{8} + 81 q^{9} + 198 q^{10} + 37 q^{11} - 96 q^{12} + 98 q^{14} - 123 q^{15} + 32 q^{16} - 134 q^{17} + 72 q^{18} - 72 q^{19} + 356 q^{20} - 3 q^{21} + 274 q^{22} + 226 q^{23} - 333 q^{24} + 612 q^{25} - 243 q^{27} + 132 q^{28} - 547 q^{29} - 594 q^{30} - 521 q^{31} + 721 q^{32} - 111 q^{33} - 100 q^{34} + 138 q^{35} + 288 q^{36} + 584 q^{37} - 416 q^{38} + 1342 q^{40} + 482 q^{41} - 294 q^{42} + 158 q^{43} + 1453 q^{44} + 369 q^{45} + 1537 q^{46} + 1500 q^{47} - 96 q^{48} + 642 q^{49} + 2777 q^{50} + 402 q^{51} + 1399 q^{53} - 216 q^{54} - 1408 q^{55} - 616 q^{56} + 216 q^{57} + 1455 q^{58} + 1541 q^{59} - 1068 q^{60} + 2092 q^{61} - 293 q^{62} + 9 q^{63} + 2481 q^{64} - 822 q^{66} + 252 q^{67} - 1579 q^{68} - 678 q^{69} + 2492 q^{70} + 2352 q^{71} + 999 q^{72} + 903 q^{73} + 1037 q^{74} - 1836 q^{75} - 485 q^{76} - 1686 q^{77} - 115 q^{79} + 5701 q^{80} + 729 q^{81} - 5147 q^{82} + 1207 q^{83} - 396 q^{84} + 4308 q^{85} + 5691 q^{86} + 1641 q^{87} - 484 q^{88} + 2336 q^{89} + 1782 q^{90} + 2087 q^{92} + 1563 q^{93} - 468 q^{94} - 222 q^{95} - 2163 q^{96} + 2155 q^{97} + 5593 q^{98} + 333 q^{99}+O(q^{100}) \) 9 * q + 8 * q^2 - 27 * q^3 + 32 * q^4 + 41 * q^5 - 24 * q^6 + q^7 + 111 * q^8 + 81 * q^9 + 198 * q^10 + 37 * q^11 - 96 * q^12 + 98 * q^14 - 123 * q^15 + 32 * q^16 - 134 * q^17 + 72 * q^18 - 72 * q^19 + 356 * q^20 - 3 * q^21 + 274 * q^22 + 226 * q^23 - 333 * q^24 + 612 * q^25 - 243 * q^27 + 132 * q^28 - 547 * q^29 - 594 * q^30 - 521 * q^31 + 721 * q^32 - 111 * q^33 - 100 * q^34 + 138 * q^35 + 288 * q^36 + 584 * q^37 - 416 * q^38 + 1342 * q^40 + 482 * q^41 - 294 * q^42 + 158 * q^43 + 1453 * q^44 + 369 * q^45 + 1537 * q^46 + 1500 * q^47 - 96 * q^48 + 642 * q^49 + 2777 * q^50 + 402 * q^51 + 1399 * q^53 - 216 * q^54 - 1408 * q^55 - 616 * q^56 + 216 * q^57 + 1455 * q^58 + 1541 * q^59 - 1068 * q^60 + 2092 * q^61 - 293 * q^62 + 9 * q^63 + 2481 * q^64 - 822 * q^66 + 252 * q^67 - 1579 * q^68 - 678 * q^69 + 2492 * q^70 + 2352 * q^71 + 999 * q^72 + 903 * q^73 + 1037 * q^74 - 1836 * q^75 - 485 * q^76 - 1686 * q^77 - 115 * q^79 + 5701 * q^80 + 729 * q^81 - 5147 * q^82 + 1207 * q^83 - 396 * q^84 + 4308 * q^85 + 5691 * q^86 + 1641 * q^87 - 484 * q^88 + 2336 * q^89 + 1782 * q^90 + 2087 * q^92 + 1563 * q^93 - 468 * q^94 - 222 * q^95 - 2163 * q^96 + 2155 * q^97 + 5593 * q^98 + 333 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−3.82618	−1.35276	−0.676379	−	0.736554i	$-0.736452\pi$
$2$	−3.82618	−1.35276	−0.676379	+	0.736554i	$0.736452\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	6.63963	0.829954
$5$	0.275426	0.0246349	0.0123174	−	0.999924i	$-0.496079\pi$
$5$	0.275426	0.0246349	0.0123174	+	0.999924i	$0.496079\pi$
$6$	11.4785	0.781015
$7$	0.0981245	0.00529823	0.00264911	−	0.999996i	$-0.499157\pi$
$7$	0.0981245	0.00529823	0.00264911	+	0.999996i	$0.499157\pi$
$8$	5.20502	0.230031
$9$	9.00000	0.333333
$10$	−1.05383	−0.0333250
$11$	−0.749942	−0.0205560	−0.0102780	−	0.999947i	$-0.503272\pi$
$11$	−0.749942	−0.0205560	−0.0102780	+	0.999947i	$0.503272\pi$
$12$	−19.9189	−0.479174
$13$	0	0
$13$	0	0
$14$	−0.375442	−0.00716722
$15$	−0.826279	−0.0142230
$16$	−73.0324	−1.14113
$17$	53.7635	0.767033	0.383517	−	0.923534i	$-0.374713\pi$
$17$	53.7635	0.767033	0.383517	+	0.923534i	$0.374713\pi$
$18$	−34.4356	−0.450919
$19$	145.488	1.75669	0.878346	−	0.478026i	$-0.158648\pi$
$19$	145.488	1.75669	0.878346	+	0.478026i	$0.158648\pi$
$20$	1.82873	0.0204458
$21$	−0.294374	−0.00305893
$22$	2.86941	0.0278073
$23$	29.3945	0.266486	0.133243	−	0.991083i	$-0.457461\pi$
$23$	29.3945	0.266486	0.133243	+	0.991083i	$0.457461\pi$
$24$	−15.6150	−0.132809
$25$	−124.924	−0.999393
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0.651511	0.00439728
$29$	−267.798	−1.71479	−0.857393	−	0.514662i	$-0.827917\pi$
$29$	−267.798	−1.71479	−0.857393	+	0.514662i	$0.827917\pi$
$30$	3.16149	0.0192402
$31$	−51.5466	−0.298647	−0.149323	−	0.988788i	$-0.547710\pi$
$31$	−51.5466	−0.298647	−0.149323	+	0.988788i	$0.547710\pi$
$32$	237.795	1.31364
$33$	2.24983	0.0118680
$34$	−205.709	−1.03761
$35$	0.0270261	0.000130521 0
$36$	59.7567	0.276651
$37$	−133.939	−0.595120	−0.297560	−	0.954703i	$-0.596173\pi$
$37$	−133.939	−0.595120	−0.297560	+	0.954703i	$0.596173\pi$
$38$	−556.661	−2.37638
$39$	0	0
$40$	1.43360	0.00566679
$41$	430.228	1.63879	0.819394	−	0.573230i	$-0.194310\pi$
$41$	430.228	1.63879	0.819394	+	0.573230i	$0.194310\pi$
$42$	1.12633	0.00413800
$43$	−282.181	−1.00075	−0.500375	−	0.865809i	$-0.666804\pi$
$43$	−282.181	−1.00075	−0.500375	+	0.865809i	$0.666804\pi$
$44$	−4.97934	−0.0170605
$45$	2.47884	0.00821163
$46$	−112.469	−0.360491
$47$	−212.022	−0.658011	−0.329006	−	0.944328i	$-0.606714\pi$
$47$	−212.022	−0.658011	−0.329006	+	0.944328i	$0.606714\pi$
$48$	219.097	0.658832
$49$	−342.990	−0.999972
$50$	477.982	1.35194
$51$	−161.291	−0.442847
$52$	0	0
$53$	573.160	1.48546	0.742731	−	0.669590i	$-0.233530\pi$
$53$	573.160	1.48546	0.742731	+	0.669590i	$0.233530\pi$
$54$	103.307	0.260338
$55$	−0.206554	−0.000506395 0
$56$	0.510740	0.00121876
$57$	−436.463	−1.01423
$58$	1024.64	2.31969
$59$	495.767	1.09396	0.546978	−	0.837147i	$-0.315778\pi$
$59$	495.767	1.09396	0.546978	+	0.837147i	$0.315778\pi$
$60$	−5.48619	−0.0118044
$61$	−310.911	−0.652591	−0.326295	−	0.945268i	$-0.605800\pi$
$61$	−310.911	−0.652591	−0.326295	+	0.945268i	$0.605800\pi$
$62$	197.226	0.403997
$63$	0.883121	0.00176608
$64$	−325.585	−0.635909
$65$	0	0
$66$	−8.60823	−0.0160545
$67$	−103.628	−0.188958	−0.0944788	−	0.995527i	$-0.530118\pi$
$67$	−103.628	−0.188958	−0.0944788	+	0.995527i	$0.530118\pi$
$68$	356.970	0.636602
$69$	−88.1836	−0.153856
$70$	−0.103407	−0.000176564 0
$71$	−203.277	−0.339783	−0.169891	−	0.985463i	$-0.554342\pi$
$71$	−203.277	−0.339783	−0.169891	+	0.985463i	$0.554342\pi$
$72$	46.8451	0.0766771
$73$	−685.571	−1.09918	−0.549589	−	0.835435i	$-0.685216\pi$
$73$	−685.571	−1.09918	−0.549589	+	0.835435i	$0.685216\pi$
$74$	512.474	0.805053
$75$	374.772	0.577000
$76$	965.984	1.45797
$77$	−0.0735877	−0.000108910 0
$78$	0	0
$79$	636.019	0.905794	0.452897	−	0.891563i	$-0.350391\pi$
$79$	636.019	0.905794	0.452897	+	0.891563i	$0.350391\pi$
$80$	−20.1150	−0.0281116
$81$	81.0000	0.111111
$82$	−1646.13	−2.21688
$83$	506.740	0.670143	0.335072	−	0.942193i	$-0.391239\pi$
$83$	506.740	0.670143	0.335072	+	0.942193i	$0.391239\pi$
$84$	−1.95453	−0.00253877
$85$	14.8079	0.0188958
$86$	1079.67	1.35377
$87$	803.393	0.990032
$88$	−3.90346	−0.00472852
$89$	700.628	0.834454	0.417227	−	0.908802i	$-0.363002\pi$
$89$	700.628	0.834454	0.417227	+	0.908802i	$0.363002\pi$
$90$	−9.48447	−0.0111083
$91$	0	0
$92$	195.169	0.221171
$93$	154.640	0.172424
$94$	811.232	0.890130
$95$	40.0711	0.0432759
$96$	−713.384	−0.758431
$97$	874.596	0.915482	0.457741	−	0.889086i	$-0.348659\pi$
$97$	874.596	0.915482	0.457741	+	0.889086i	$0.348659\pi$
$98$	1312.34	1.35272
$99$	−6.74948	−0.00685200
$100$	−829.450	−0.829450
$101$	823.762	0.811558	0.405779	−	0.913971i	$-0.367000\pi$
$101$	823.762	0.811558	0.405779	+	0.913971i	$0.367000\pi$
$102$	617.126	0.599065
$103$	1762.66	1.68622	0.843109	−	0.537742i	$-0.180723\pi$
$103$	1762.66	1.68622	0.843109	+	0.537742i	$0.180723\pi$
$104$	0	0
$105$	−0.0810782	−7.53564e−5 0
$106$	−2193.01	−2.00947
$107$	623.782	0.563582	0.281791	−	0.959476i	$-0.409072\pi$
$107$	623.782	0.563582	0.281791	+	0.959476i	$0.409072\pi$
$108$	−179.270	−0.159725
$109$	2197.05	1.93063	0.965316	−	0.261086i	$-0.0840805\pi$
$109$	2197.05	1.93063	0.965316	+	0.261086i	$0.0840805\pi$
$110$	0.790311	0.000685029 0
$111$	401.817	0.343593
$112$	−7.16627	−0.00604597
$113$	1439.15	1.19809	0.599045	−	0.800715i	$-0.295547\pi$
$113$	1439.15	1.19809	0.599045	+	0.800715i	$0.295547\pi$
$114$	1669.98	1.37200
$115$	8.09602	0.00656485
$116$	−1778.08	−1.42319
$117$	0	0
$118$	−1896.89	−1.47986
$119$	5.27552	0.00406392
$120$	−4.30079	−0.00327173
$121$	−1330.44	−0.999577
$122$	1189.60	0.882797
$123$	−1290.68	−0.946155
$124$	−342.250	−0.247863
$125$	−68.8357	−0.0492548
$126$	−3.37898	−0.00238907
$127$	804.759	0.562290	0.281145	−	0.959665i	$-0.409286\pi$
$127$	804.759	0.562290	0.281145	+	0.959665i	$0.409286\pi$
$128$	−656.610	−0.453411
$129$	846.543	0.577783
$130$	0	0
$131$	294.199	0.196216	0.0981079	−	0.995176i	$-0.468721\pi$
$131$	294.199	0.196216	0.0981079	+	0.995176i	$0.468721\pi$
$132$	14.9380	0.00984990
$133$	14.2759	0.00930735
$134$	396.498	0.255614
$135$	−7.43651	−0.00474098
$136$	279.840	0.176442
$137$	2215.06	1.38135	0.690677	−	0.723164i	$-0.257313\pi$
$137$	2215.06	1.38135	0.690677	+	0.723164i	$0.257313\pi$
$138$	337.406	0.208130
$139$	−1701.48	−1.03826	−0.519128	−	0.854697i	$-0.673743\pi$
$139$	−1701.48	−1.03826	−0.519128	+	0.854697i	$0.673743\pi$
$140$	0.179443	0.000108327 0
$141$	636.065	0.379903
$142$	777.775	0.459644
$143$	0	0
$144$	−657.291	−0.380377
$145$	−73.7586	−0.0422435
$146$	2623.12	1.48692
$147$	1028.97	0.577334
$148$	−889.306	−0.493922
$149$	−1417.26	−0.779237	−0.389618	−	0.920976i	$-0.627393\pi$
$149$	−1417.26	−0.779237	−0.389618	+	0.920976i	$0.627393\pi$
$150$	−1433.95	−0.780541
$151$	−2841.58	−1.53142	−0.765710	−	0.643186i	$-0.777612\pi$
$151$	−2841.58	−1.53142	−0.765710	+	0.643186i	$0.777612\pi$
$152$	757.265	0.404094
$153$	483.872	0.255678
$154$	0.281560	0.000147329 0
$155$	−14.1973	−0.00735712
$156$	0	0
$157$	2278.92	1.15846	0.579228	−	0.815166i	$-0.303354\pi$
$157$	2278.92	1.15846	0.579228	+	0.815166i	$0.303354\pi$
$158$	−2433.52	−1.22532
$159$	−1719.48	−0.857632
$160$	65.4949	0.0323614
$161$	2.88432	0.00141190
$162$	−309.920	−0.150306
$163$	−1685.86	−0.810102	−0.405051	−	0.914294i	$-0.632746\pi$
$163$	−1685.86	−0.810102	−0.405051	+	0.914294i	$0.632746\pi$
$164$	2856.56	1.36012
$165$	0.619661	0.000292367 0
$166$	−1938.88	−0.906542
$167$	−2041.53	−0.945980	−0.472990	−	0.881068i	$-0.656825\pi$
$167$	−2041.53	−0.945980	−0.472990	+	0.881068i	$0.656825\pi$
$168$	−1.53222	−0.000703650 0
$169$	0	0
$170$	−56.6576	−0.0255614
$171$	1309.39	0.585564
$172$	−1873.58	−0.830576
$173$	2762.17	1.21390	0.606948	−	0.794741i	$-0.292394\pi$
$173$	2762.17	1.21390	0.606948	+	0.794741i	$0.292394\pi$
$174$	−3073.92	−1.33927
$175$	−12.2581	−0.00529501
$176$	54.7700	0.0234571
$177$	−1487.30	−0.631595
$178$	−2680.73	−1.12881
$179$	−3176.00	−1.32618	−0.663088	−	0.748541i	$-0.730755\pi$
$179$	−3176.00	−1.32618	−0.663088	+	0.748541i	$0.730755\pi$
$180$	16.4586	0.00681527
$181$	2101.42	0.862967	0.431483	−	0.902121i	$-0.357990\pi$
$181$	2101.42	0.862967	0.431483	+	0.902121i	$0.357990\pi$
$182$	0	0
$183$	932.732	0.376773
$184$	152.999	0.0613002
$185$	−36.8903	−0.0146607
$186$	−591.679	−0.233248
$187$	−40.3195	−0.0157671
$188$	−1407.74	−0.546119
$189$	−2.64936	−0.00101964
$190$	−153.319	−0.0585418
$191$	−9.87240	−0.00374001	−0.00187001	−	0.999998i	$-0.500595\pi$
$191$	−9.87240	−0.00374001	−0.00187001	+	0.999998i	$0.500595\pi$
$192$	976.756	0.367142
$193$	−744.664	−0.277731	−0.138865	−	0.990311i	$-0.544346\pi$
$193$	−744.664	−0.277731	−0.138865	+	0.990311i	$0.544346\pi$
$194$	−3346.36	−1.23843
$195$	0	0
$196$	−2277.33	−0.829930
$197$	3165.24	1.14474	0.572371	−	0.819995i	$-0.306024\pi$
$197$	3165.24	1.14474	0.572371	+	0.819995i	$0.306024\pi$
$198$	25.8247	0.00926910
$199$	2683.08	0.955770	0.477885	−	0.878423i	$-0.341404\pi$
$199$	2683.08	0.955770	0.477885	+	0.878423i	$0.341404\pi$
$200$	−650.232	−0.229892
$201$	310.883	0.109095
$202$	−3151.86	−1.09784
$203$	−26.2775	−0.00908533
$204$	−1070.91	−0.367543
$205$	118.496	0.0403714
$206$	−6744.27	−2.28105
$207$	264.551	0.0888287
$208$	0	0
$209$	−109.107	−0.0361106
$210$	0.310220	0.000101939 0
$211$	−3908.43	−1.27520	−0.637600	−	0.770367i	$-0.720073\pi$
$211$	−3908.43	−1.27520	−0.637600	+	0.770367i	$0.720073\pi$
$212$	3805.57	1.23287
$213$	609.832	0.196174
$214$	−2386.70	−0.762390
$215$	−77.7201	−0.0246533
$216$	−140.535	−0.0442696
$217$	−5.05799	−0.00158230
$218$	−8406.29	−2.61168
$219$	2056.71	0.634611
$220$	−1.37144	−0.000420284 0
$221$	0	0
$222$	−1537.42	−0.464798
$223$	5672.13	1.70329	0.851645	−	0.524119i	$-0.175605\pi$
$223$	5672.13	1.70329	0.851645	+	0.524119i	$0.175605\pi$
$224$	23.3335	0.00695997
$225$	−1124.32	−0.333131
$226$	−5506.45	−1.62073
$227$	1490.13	0.435698	0.217849	−	0.975983i	$-0.430096\pi$
$227$	1490.13	0.435698	0.217849	+	0.975983i	$0.430096\pi$
$228$	−2897.95	−0.841761
$229$	2495.16	0.720020	0.360010	−	0.932948i	$-0.382773\pi$
$229$	2495.16	0.720020	0.360010	+	0.932948i	$0.382773\pi$
$230$	−30.9768	−0.00888066
$231$	0.220763	6.28794e−5 0
$232$	−1393.89	−0.394455
$233$	−251.850	−0.0708122	−0.0354061	−	0.999373i	$-0.511272\pi$
$233$	−251.850	−0.0708122	−0.0354061	+	0.999373i	$0.511272\pi$
$234$	0	0
$235$	−58.3963	−0.0162100
$236$	3291.71	0.907932
$237$	−1908.06	−0.522960
$238$	−20.1851	−0.00549750
$239$	4345.94	1.17622	0.588108	−	0.808782i	$-0.299873\pi$
$239$	4345.94	1.17622	0.588108	+	0.808782i	$0.299873\pi$
$240$	60.3451	0.0162302
$241$	−3429.14	−0.916556	−0.458278	−	0.888809i	$-0.651534\pi$
$241$	−3429.14	−0.916556	−0.458278	+	0.888809i	$0.651534\pi$
$242$	5090.49	1.35219
$243$	−243.000	−0.0641500
$244$	−2064.33	−0.541620
$245$	−94.4686	−0.0246342
$246$	4938.39	1.27992
$247$	0	0
$248$	−268.301	−0.0686981
$249$	−1520.22	−0.386907
$250$	263.378	0.0666298
$251$	1831.03	0.460452	0.230226	−	0.973137i	$-0.426053\pi$
$251$	1831.03	0.460452	0.230226	+	0.973137i	$0.426053\pi$
$252$	5.86360	0.00146576
$253$	−22.0442	−0.00547789
$254$	−3079.15	−0.760642
$255$	−44.4237	−0.0109095
$256$	5116.99	1.24926
$257$	−1626.17	−0.394700	−0.197350	−	0.980333i	$-0.563234\pi$
$257$	−1626.17	−0.394700	−0.197350	+	0.980333i	$0.563234\pi$
$258$	−3239.02	−0.781600
$259$	−13.1427	−0.00315308
$260$	0	0
$261$	−2410.18	−0.571595
$262$	−1125.66	−0.265433
$263$	−6931.39	−1.62513	−0.812563	−	0.582874i	$-0.801928\pi$
$263$	−6931.39	−1.62513	−0.812563	+	0.582874i	$0.801928\pi$
$264$	11.7104	0.00273002
$265$	157.863	0.0365942
$266$	−54.6221	−0.0125906
$267$	−2101.88	−0.481772
$268$	−688.050	−0.156826
$269$	−2668.42	−0.604820	−0.302410	−	0.953178i	$-0.597791\pi$
$269$	−2668.42	−0.604820	−0.302410	+	0.953178i	$0.597791\pi$
$270$	28.4534	0.00641340
$271$	−1244.44	−0.278946	−0.139473	−	0.990226i	$-0.544541\pi$
$271$	−1244.44	−0.278946	−0.139473	+	0.990226i	$0.544541\pi$
$272$	−3926.48	−0.875285
$273$	0	0
$274$	−8475.21	−1.86864
$275$	93.6859	0.0205435
$276$	−585.506	−0.127693
$277$	2100.45	0.455610	0.227805	−	0.973707i	$-0.426845\pi$
$277$	2100.45	0.455610	0.227805	+	0.973707i	$0.426845\pi$
$278$	6510.16	1.40451
$279$	−463.920	−0.0995489
$280$	0.140671	3.00240e−5 0
$281$	567.729	0.120526	0.0602631	−	0.998183i	$-0.480806\pi$
$281$	567.729	0.120526	0.0602631	+	0.998183i	$0.480806\pi$
$282$	−2433.70	−0.513917
$283$	8149.26	1.71174	0.855871	−	0.517189i	$-0.173022\pi$
$283$	8149.26	1.71174	0.855871	+	0.517189i	$0.173022\pi$
$284$	−1349.69	−0.282004
$285$	−120.213	−0.0249853
$286$	0	0
$287$	42.2159	0.00868268
$288$	2140.15	0.437881
$289$	−2022.48	−0.411660
$290$	282.213	0.0571453
$291$	−2623.79	−0.528554
$292$	−4551.94	−0.912268
$293$	7135.52	1.42274	0.711368	−	0.702820i	$-0.248076\pi$
$293$	7135.52	1.42274	0.711368	+	0.702820i	$0.248076\pi$
$294$	−3937.03	−0.780993
$295$	136.547	0.0269495
$296$	−697.155	−0.136896
$297$	20.2484	0.00395600
$298$	5422.68	1.05412
$299$	0	0
$300$	2488.35	0.478883
$301$	−27.6889	−0.00530220
$302$	10872.4	2.07164
$303$	−2471.29	−0.468553
$304$	−10625.3	−2.00461
$305$	−85.6330	−0.0160765
$306$	−1851.38	−0.345870
$307$	9511.65	1.76827	0.884134	−	0.467233i	$-0.154749\pi$
$307$	9511.65	1.76827	0.884134	+	0.467233i	$0.154749\pi$
$308$	−0.488595	−9.03906e−5 0
$309$	−5287.99	−0.973539
$310$	54.3214	0.00995241
$311$	8311.19	1.51538	0.757692	−	0.652613i	$-0.226327\pi$
$311$	8311.19	1.51538	0.757692	+	0.652613i	$0.226327\pi$
$312$	0	0
$313$	7108.85	1.28376	0.641878	−	0.766806i	$-0.278155\pi$
$313$	7108.85	1.28376	0.641878	+	0.766806i	$0.278155\pi$
$314$	−8719.55	−1.56711
$315$	0.243235	4.35071e−5 0
$316$	4222.93	0.751767
$317$	7578.33	1.34272	0.671359	−	0.741133i	$-0.265711\pi$
$317$	7578.33	1.34272	0.671359	+	0.741133i	$0.265711\pi$
$318$	6579.03	1.16017
$319$	200.833	0.0352491
$320$	−89.6748	−0.0156655
$321$	−1871.35	−0.325384
$322$	−11.0359	−0.00190996
$323$	7821.92	1.34744
$324$	537.810	0.0922171
$325$	0	0
$326$	6450.40	1.09587
$327$	−6591.14	−1.11465
$328$	2239.34	0.376973
$329$	−20.8045	−0.00348629
$330$	−2.37093	−0.000395502 0
$331$	−8490.05	−1.40984	−0.704918	−	0.709289i	$-0.749016\pi$
$331$	−8490.05	−1.40984	−0.704918	+	0.709289i	$0.749016\pi$
$332$	3364.56	0.556188
$333$	−1205.45	−0.198373
$334$	7811.27	1.27968
$335$	−28.5418	−0.00465495
$336$	21.4988	0.00349064
$337$	−6045.38	−0.977189	−0.488595	−	0.872511i	$-0.662490\pi$
$337$	−6045.38	−0.977189	−0.488595	+	0.872511i	$0.662490\pi$
$338$	0	0
$339$	−4317.46	−0.691717
$340$	98.3189	0.0156826
$341$	38.6570	0.00613898
$342$	−5009.95	−0.792126
$343$	−67.3125	−0.0105963
$344$	−1468.76	−0.230204
$345$	−24.2881	−0.00379022
$346$	−10568.6	−1.64211
$347$	−10546.5	−1.63160	−0.815802	−	0.578332i	$-0.803704\pi$
$347$	−10546.5	−1.63160	−0.815802	+	0.578332i	$0.803704\pi$
$348$	5334.23	0.821681
$349$	5414.94	0.830531	0.415266	−	0.909700i	$-0.363689\pi$
$349$	5414.94	0.830531	0.415266	+	0.909700i	$0.363689\pi$
$350$	46.9018	0.00716287
$351$	0	0
$352$	−178.332	−0.0270032
$353$	−324.034	−0.0488572	−0.0244286	−	0.999702i	$-0.507777\pi$
$353$	−324.034	−0.0488572	−0.0244286	+	0.999702i	$0.507777\pi$
$354$	5690.68	0.854396
$355$	−55.9879	−0.00837051
$356$	4651.91	0.692558
$357$	−15.8266	−0.00234630
$358$	12151.9	1.79400
$359$	−1472.77	−0.216517	−0.108259	−	0.994123i	$-0.534527\pi$
$359$	−1472.77	−0.216517	−0.108259	+	0.994123i	$0.534527\pi$
$360$	12.9024	0.00188893
$361$	14307.6	2.08596
$362$	−8040.39	−1.16738
$363$	3991.31	0.577106
$364$	0	0
$365$	−188.824	−0.0270781
$366$	−3568.80	−0.509683
$367$	10920.1	1.55321	0.776603	−	0.629991i	$-0.216941\pi$
$367$	10920.1	1.55321	0.776603	+	0.629991i	$0.216941\pi$
$368$	−2146.75	−0.304095
$369$	3872.05	0.546263
$370$	141.149	0.0198324
$371$	56.2410	0.00787032
$372$	1026.75	0.143104
$373$	1703.93	0.236532	0.118266	−	0.992982i	$-0.462266\pi$
$373$	1703.93	0.236532	0.118266	+	0.992982i	$0.462266\pi$
$374$	154.270	0.0213291
$375$	206.507	0.0284373
$376$	−1103.58	−0.151363
$377$	0	0
$378$	10.1369	0.00137933
$379$	−4125.91	−0.559192	−0.279596	−	0.960118i	$-0.590201\pi$
$379$	−4125.91	−0.559192	−0.279596	+	0.960118i	$0.590201\pi$
$380$	266.057	0.0359170
$381$	−2414.28	−0.324638
$382$	37.7736	0.00505933
$383$	−7173.72	−0.957076	−0.478538	−	0.878067i	$-0.658833\pi$
$383$	−7173.72	−0.957076	−0.478538	+	0.878067i	$0.658833\pi$
$384$	1969.83	0.261777
$385$	−0.0202680	−2.68299e−6 0
$386$	2849.22	0.375703
$387$	−2539.63	−0.333583
$388$	5806.99	0.759808
$389$	−10036.1	−1.30810	−0.654049	−	0.756452i	$-0.726931\pi$
$389$	−10036.1	−1.30810	−0.654049	+	0.756452i	$0.726931\pi$
$390$	0	0
$391$	1580.35	0.204404
$392$	−1785.27	−0.230025
$393$	−882.597	−0.113285
$394$	−12110.8	−1.54856
$395$	175.176	0.0223141
$396$	−44.8140	−0.00568684
$397$	−923.392	−0.116735	−0.0583674	−	0.998295i	$-0.518589\pi$
$397$	−923.392	−0.116735	−0.0583674	+	0.998295i	$0.518589\pi$
$398$	−10265.9	−1.29292
$399$	−42.8277	−0.00537360
$400$	9123.50	1.14044
$401$	13550.7	1.68751	0.843754	−	0.536730i	$-0.180341\pi$
$401$	13550.7	1.68751	0.843754	+	0.536730i	$0.180341\pi$
$402$	−1189.50	−0.147579
$403$	0	0
$404$	5469.48	0.673556
$405$	22.3095	0.00273721
$406$	100.542	0.0122902
$407$	100.446	0.0122333
$408$	−839.520	−0.101869
$409$	896.260	0.108355	0.0541775	−	0.998531i	$-0.482746\pi$
$409$	896.260	0.108355	0.0541775	+	0.998531i	$0.482746\pi$
$410$	−453.387	−0.0546127
$411$	−6645.18	−0.797525
$412$	11703.4	1.39948
$413$	48.6469	0.00579602
$414$	−1012.22	−0.120164
$415$	139.569	0.0165089
$416$	0	0
$417$	5104.43	0.599437
$418$	417.464	0.0488488
$419$	6535.59	0.762015	0.381008	−	0.924572i	$-0.375577\pi$
$419$	6535.59	0.762015	0.381008	+	0.924572i	$0.375577\pi$
$420$	−0.538330	−6.25424e−5 0
$421$	−1942.00	−0.224816	−0.112408	−	0.993662i	$-0.535856\pi$
$421$	−1942.00	−0.224816	−0.112408	+	0.993662i	$0.535856\pi$
$422$	14954.3	1.72504
$423$	−1908.19	−0.219337
$424$	2983.30	0.341703
$425$	−6716.36	−0.766568
$426$	−2333.32	−0.265375
$427$	−30.5080	−0.00345757
$428$	4141.68	0.467747
$429$	0	0
$430$	297.371	0.0333500
$431$	3512.60	0.392566	0.196283	−	0.980547i	$-0.437113\pi$
$431$	3512.60	0.392566	0.196283	+	0.980547i	$0.437113\pi$
$432$	1971.87	0.219611
$433$	3053.92	0.338943	0.169471	−	0.985535i	$-0.445794\pi$
$433$	3053.92	0.338943	0.169471	+	0.985535i	$0.445794\pi$
$434$	19.3528	0.00214047
$435$	221.276	0.0243893
$436$	14587.6	1.60233
$437$	4276.54	0.468134
$438$	−7869.35	−0.858475
$439$	1054.77	0.114673	0.0573364	−	0.998355i	$-0.481739\pi$
$439$	1054.77	0.114673	0.0573364	+	0.998355i	$0.481739\pi$
$440$	−1.07512	−0.000116487 0
$441$	−3086.91	−0.333324
$442$	0	0
$443$	−10586.0	−1.13534	−0.567672	−	0.823255i	$-0.692156\pi$
$443$	−10586.0	−1.13534	−0.567672	+	0.823255i	$0.692156\pi$
$444$	2667.92	0.285166
$445$	192.971	0.0205567
$446$	−21702.6	−2.30414
$447$	4251.77	0.449893
$448$	−31.9479	−0.00336919
$449$	5452.20	0.573062	0.286531	−	0.958071i	$-0.407498\pi$
$449$	5452.20	0.573062	0.286531	+	0.958071i	$0.407498\pi$
$450$	4301.84	0.450646
$451$	−322.646	−0.0336869
$452$	9555.44	0.994359
$453$	8524.74	0.884166
$454$	−5701.50	−0.589394
$455$	0	0
$456$	−2271.80	−0.233304
$457$	10048.4	1.02855	0.514273	−	0.857627i	$-0.328062\pi$
$457$	10048.4	1.02855	0.514273	+	0.857627i	$0.328062\pi$
$458$	−9546.92	−0.974013
$459$	−1451.62	−0.147616
$460$	53.7546	0.00544853
$461$	18199.7	1.83871	0.919355	−	0.393430i	$-0.128712\pi$
$461$	18199.7	1.83871	0.919355	+	0.393430i	$0.128712\pi$
$462$	−0.844679	−8.50606e−5 0
$463$	5535.23	0.555602	0.277801	−	0.960639i	$-0.410394\pi$
$463$	5535.23	0.555602	0.277801	+	0.960639i	$0.410394\pi$
$464$	19557.9	1.95679
$465$	42.5919	0.00424764
$466$	963.623	0.0957918
$467$	−3878.15	−0.384281	−0.192141	−	0.981367i	$-0.561543\pi$
$467$	−3878.15	−0.384281	−0.192141	+	0.981367i	$0.561543\pi$
$468$	0	0
$469$	−10.1684	−0.00100114
$470$	223.435	0.0219282
$471$	−6836.76	−0.668835
$472$	2580.48	0.251644
$473$	211.619	0.0205714
$474$	7300.56	0.707439
$475$	−18174.9	−1.75563
$476$	35.0275	0.00337286
$477$	5158.44	0.495154
$478$	−16628.4	−1.59114
$479$	8361.36	0.797579	0.398789	−	0.917043i	$-0.369430\pi$
$479$	8361.36	0.797579	0.398789	+	0.917043i	$0.369430\pi$
$480$	−196.485	−0.0186839
$481$	0	0
$482$	13120.5	1.23988
$483$	−8.65297	−0.000815163 0
$484$	−8833.61	−0.829603
$485$	240.887	0.0225528
$486$	929.761	0.0867795
$487$	11891.3	1.10646	0.553228	−	0.833030i	$-0.313396\pi$
$487$	11891.3	1.10646	0.553228	+	0.833030i	$0.313396\pi$
$488$	−1618.29	−0.150116
$489$	5057.58	0.467713
$490$	361.454	0.0333241
$491$	3098.06	0.284752	0.142376	−	0.989813i	$-0.454526\pi$
$491$	3098.06	0.284752	0.142376	+	0.989813i	$0.454526\pi$
$492$	−8569.67	−0.785265
$493$	−14397.8	−1.31530
$494$	0	0
$495$	−1.85898	−0.000168798 0
$496$	3764.57	0.340795
$497$	−19.9465	−0.00180025
$498$	5816.63	0.523392
$499$	16.6409	0.00149288	0.000746442	−	1.00000i	$-0.499762\pi$
$499$	16.6409	0.00149288	0.000746442		1.00000i	$0.499762\pi$
$500$	−457.044	−0.0408792
$501$	6124.60	0.546162
$502$	−7005.84	−0.622880
$503$	−644.957	−0.0571714	−0.0285857	−	0.999591i	$-0.509100\pi$
$503$	−644.957	−0.0571714	−0.0285857	+	0.999591i	$0.509100\pi$
$504$	4.59666	0.000406253 0
$505$	226.886	0.0199926
$506$	84.3449	0.00741026
$507$	0	0
$508$	5343.30	0.466675
$509$	−10587.2	−0.921943	−0.460972	−	0.887415i	$-0.652499\pi$
$509$	−10587.2	−0.921943	−0.460972	+	0.887415i	$0.652499\pi$
$510$	169.973	0.0147579
$511$	−67.2714	−0.00582370
$512$	−14325.6	−1.23654
$513$	−3928.16	−0.338075
$514$	6222.03	0.533934
$515$	485.484	0.0415398
$516$	5620.73	0.479533
$517$	159.004	0.0135261
$518$	50.2863	0.00426536
$519$	−8286.52	−0.700843
$520$	0	0
$521$	−11107.3	−0.934010	−0.467005	−	0.884255i	$-0.654667\pi$
$521$	−11107.3	−0.934010	−0.467005	+	0.884255i	$0.654667\pi$
$522$	9221.77	0.773230
$523$	−14348.8	−1.19967	−0.599836	−	0.800123i	$-0.704767\pi$
$523$	−14348.8	−1.19967	−0.599836	+	0.800123i	$0.704767\pi$
$524$	1953.37	0.162850
$525$	36.7744	0.00305708
$526$	26520.7	2.19840
$527$	−2771.33	−0.229072
$528$	−164.310	−0.0135430
$529$	−11303.0	−0.928985
$530$	−604.013	−0.0495031
$531$	4461.90	0.364652
$532$	94.7867	0.00772467
$533$	0	0
$534$	8042.18	0.651721
$535$	171.806	0.0138838
$536$	−539.384	−0.0434662
$537$	9528.01	0.765668
$538$	10209.9	0.818175
$539$	257.223	0.0205554
$540$	−49.3757	−0.00393480
$541$	7628.01	0.606199	0.303100	−	0.952959i	$-0.401979\pi$
$541$	7628.01	0.606199	0.303100	+	0.952959i	$0.401979\pi$
$542$	4761.45	0.377346
$543$	−6304.25	−0.498234
$544$	12784.7	1.00761
$545$	605.124	0.0475609
$546$	0	0
$547$	18866.7	1.47473	0.737367	−	0.675492i	$-0.236069\pi$
$547$	18866.7	1.47473	0.737367	+	0.675492i	$0.236069\pi$
$548$	14707.2	1.14646
$549$	−2798.20	−0.217530
$550$	−358.459	−0.0277904
$551$	−38961.2	−3.01235
$552$	−458.997	−0.0353917
$553$	62.4091	0.00479910
$554$	−8036.70	−0.616330
$555$	110.671	0.00846436
$556$	−11297.2	−0.861704
$557$	−15379.3	−1.16992	−0.584959	−	0.811063i	$-0.698889\pi$
$557$	−15379.3	−1.16992	−0.584959	+	0.811063i	$0.698889\pi$
$558$	1775.04	0.134666
$559$	0	0
$560$	−1.97378	−0.000148942 0
$561$	120.959	0.00910316
$562$	−2172.23	−0.163043
$563$	3052.45	0.228500	0.114250	−	0.993452i	$-0.463554\pi$
$563$	3052.45	0.228500	0.114250	+	0.993452i	$0.463554\pi$
$564$	4223.23	0.315302
$565$	396.381	0.0295148
$566$	−31180.5	−2.31557
$567$	7.94809	0.000588692 0
$568$	−1058.06	−0.0781607
$569$	−22687.0	−1.67151	−0.835754	−	0.549104i	$-0.814969\pi$
$569$	−22687.0	−1.67151	−0.835754	+	0.549104i	$0.814969\pi$
$570$	459.957	0.0337991
$571$	−1072.04	−0.0785697	−0.0392848	−	0.999228i	$-0.512508\pi$
$571$	−1072.04	−0.0785697	−0.0392848	+	0.999228i	$0.512508\pi$
$572$	0	0
$573$	29.6172	0.00215930
$574$	−161.526	−0.0117456
$575$	−3672.09	−0.266324
$576$	−2930.27	−0.211970
$577$	19238.6	1.38806	0.694031	−	0.719945i	$-0.255833\pi$
$577$	19238.6	1.38806	0.694031	+	0.719945i	$0.255833\pi$
$578$	7738.38	0.556876
$579$	2233.99	0.160348
$580$	−489.730	−0.0350602
$581$	49.7236	0.00355057
$582$	10039.1	0.715005
$583$	−429.836	−0.0305352
$584$	−3568.41	−0.252846
$585$	0	0
$586$	−27301.8	−1.92462
$587$	25386.6	1.78504	0.892518	−	0.451011i	$-0.148936\pi$
$587$	25386.6	1.78504	0.892518	+	0.451011i	$0.148936\pi$
$588$	6831.99	0.479161
$589$	−7499.39	−0.524630
$590$	−522.454	−0.0364561
$591$	−9495.72	−0.660917
$592$	9781.88	0.679110
$593$	−14173.0	−0.981475	−0.490738	−	0.871307i	$-0.663273\pi$
$593$	−14173.0	−0.981475	−0.490738	+	0.871307i	$0.663273\pi$
$594$	−77.4741	−0.00535152
$595$	1.45302	0.000100114 0
$596$	−9410.07	−0.646731
$597$	−8049.23	−0.551814
$598$	0	0
$599$	−17528.8	−1.19567	−0.597837	−	0.801618i	$-0.703973\pi$
$599$	−17528.8	−1.19567	−0.597837	+	0.801618i	$0.703973\pi$
$600$	1950.70	0.132728
$601$	21328.2	1.44758	0.723790	−	0.690020i	$-0.242398\pi$
$601$	21328.2	1.44758	0.723790	+	0.690020i	$0.242398\pi$
$602$	105.943	0.00717259
$603$	−932.650	−0.0629858
$604$	−18867.0	−1.27101
$605$	−366.438	−0.0246245
$606$	9455.58	0.633839
$607$	−19009.5	−1.27112	−0.635562	−	0.772050i	$-0.719232\pi$
$607$	−19009.5	−1.27112	−0.635562	+	0.772050i	$0.719232\pi$
$608$	34596.2	2.30766
$609$	78.8326	0.00524542
$610$	327.647	0.0217476
$611$	0	0
$612$	3212.73	0.212201
$613$	−3117.55	−0.205410	−0.102705	−	0.994712i	$-0.532750\pi$
$613$	−3117.55	−0.205410	−0.102705	+	0.994712i	$0.532750\pi$
$614$	−36393.2	−2.39204
$615$	−355.488	−0.0233084
$616$	−0.383025	−2.50528e−5 0
$617$	−16339.3	−1.06612	−0.533058	−	0.846079i	$-0.678957\pi$
$617$	−16339.3	−1.06612	−0.533058	+	0.846079i	$0.678957\pi$
$618$	20232.8	1.31696
$619$	−25388.2	−1.64853	−0.824264	−	0.566206i	$-0.808411\pi$
$619$	−25388.2	−1.64853	−0.824264	+	0.566206i	$0.808411\pi$
$620$	−94.2648	−0.00610607
$621$	−793.652	−0.0512853
$622$	−31800.1	−2.04995
$623$	68.7488	0.00442113
$624$	0	0
$625$	15596.6	0.998180
$626$	−27199.7	−1.73661
$627$	327.322	0.0208484
$628$	15131.2	0.961464
$629$	−7201.03	−0.456477
$630$	−0.930659	−5.88545e−5 0
$631$	−23365.1	−1.47409	−0.737043	−	0.675845i	$-0.763779\pi$
$631$	−23365.1	−1.47409	−0.737043	+	0.675845i	$0.763779\pi$
$632$	3310.49	0.208361
$633$	11725.3	0.736237
$634$	−28996.0	−1.81637
$635$	221.652	0.0138519
$636$	−11416.7	−0.711795
$637$	0	0
$638$	−768.422	−0.0476836
$639$	−1829.50	−0.113261
$640$	−180.848	−0.0111697
$641$	2411.19	0.148575	0.0742873	−	0.997237i	$-0.476332\pi$
$641$	2411.19	0.148575	0.0742873	+	0.997237i	$0.476332\pi$
$642$	7160.10	0.440166
$643$	1454.22	0.0891897	0.0445949	−	0.999005i	$-0.485800\pi$
$643$	1454.22	0.0891897	0.0445949	+	0.999005i	$0.485800\pi$
$644$	19.1508	0.00117182
$645$	233.160	0.0142336
$646$	−29928.1	−1.82276
$647$	18364.7	1.11590	0.557951	−	0.829874i	$-0.311588\pi$
$647$	18364.7	1.11590	0.557951	+	0.829874i	$0.311588\pi$
$648$	421.606	0.0255590
$649$	−371.797	−0.0224873
$650$	0	0
$651$	15.1740	0.000913540 0
$652$	−11193.5	−0.672348
$653$	23875.0	1.43078	0.715391	−	0.698724i	$-0.246248\pi$
$653$	23875.0	1.43078	0.715391	+	0.698724i	$0.246248\pi$
$654$	25218.9	1.50785
$655$	81.0301	0.00483375
$656$	−31420.6	−1.87007
$657$	−6170.14	−0.366393
$658$	79.6018	0.00471611
$659$	8691.91	0.513792	0.256896	−	0.966439i	$-0.417300\pi$
$659$	8691.91	0.513792	0.256896	+	0.966439i	$0.417300\pi$
$660$	4.11432	0.000242651 0
$661$	−14072.8	−0.828094	−0.414047	−	0.910255i	$-0.635885\pi$
$661$	−14072.8	−0.828094	−0.414047	+	0.910255i	$0.635885\pi$
$662$	32484.5	1.90717
$663$	0	0
$664$	2637.59	0.154154
$665$	3.93196	0.000229285 0
$666$	4612.27	0.268351
$667$	−7871.79	−0.456967
$668$	−13555.0	−0.785120
$669$	−17016.4	−0.983395
$670$	109.206	0.00629701
$671$	233.165	0.0134147
$672$	−70.0005	−0.00401834
$673$	−22779.1	−1.30471	−0.652354	−	0.757914i	$-0.726219\pi$
$673$	−22779.1	−1.30471	−0.652354	+	0.757914i	$0.726219\pi$
$674$	23130.7	1.32190
$675$	3372.95	0.192333
$676$	0	0
$677$	7195.74	0.408500	0.204250	−	0.978919i	$-0.434524\pi$
$677$	7195.74	0.408500	0.204250	+	0.978919i	$0.434524\pi$
$678$	16519.4	0.935726
$679$	85.8193	0.00485043
$680$	77.0753	0.00434662
$681$	−4470.39	−0.251550
$682$	−147.908	−0.00830455
$683$	17462.2	0.978291	0.489146	−	0.872202i	$-0.337309\pi$
$683$	17462.2	0.978291	0.489146	+	0.872202i	$0.337309\pi$
$684$	8693.85	0.485991
$685$	610.086	0.0340295
$686$	257.550	0.0143342
$687$	−7485.47	−0.415704
$688$	20608.4	1.14199
$689$	0	0
$690$	92.9305	0.00512725
$691$	−27774.8	−1.52909	−0.764547	−	0.644567i	$-0.777037\pi$
$691$	−27774.8	−1.52909	−0.764547	+	0.644567i	$0.777037\pi$
$692$	18339.8	1.00748
$693$	−0.662289	−3.63035e−5 0
$694$	40352.8	2.20716
$695$	−468.632	−0.0255773
$696$	4181.67	0.227738
$697$	23130.6	1.25701
$698$	−20718.5	−1.12351
$699$	755.550	0.0408835
$700$	−81.3894	−0.00439462
$701$	21314.5	1.14841	0.574207	−	0.818710i	$-0.305310\pi$
$701$	21314.5	1.14841	0.574207	+	0.818710i	$0.305310\pi$
$702$	0	0
$703$	−19486.5	−1.04544
$704$	244.170	0.0130717
$705$	175.189	0.00935886
$706$	1239.81	0.0660919
$707$	80.8313	0.00429982
$708$	−9875.13	−0.524195
$709$	14395.5	0.762529	0.381265	−	0.924466i	$-0.375489\pi$
$709$	14395.5	0.762529	0.381265	+	0.924466i	$0.375489\pi$
$710$	214.220	0.0113233
$711$	5724.17	0.301931
$712$	3646.78	0.191951
$713$	−1515.19	−0.0795852
$714$	60.5552	0.00317398
$715$	0	0
$716$	−21087.5	−1.10067
$717$	−13037.8	−0.679089
$718$	5635.07	0.292896
$719$	−4262.00	−0.221065	−0.110533	−	0.993873i	$-0.535256\pi$
$719$	−4262.00	−0.221065	−0.110533	+	0.993873i	$0.535256\pi$
$720$	−181.035	−0.00937054
$721$	172.961	0.00893397
$722$	−54743.5	−2.82180
$723$	10287.4	0.529174
$724$	13952.6	0.716222
$725$	33454.4	1.71375
$726$	−15271.5	−0.780685
$727$	19342.4	0.986753	0.493377	−	0.869816i	$-0.335762\pi$
$727$	19342.4	0.986753	0.493377	+	0.869816i	$0.335762\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	722.476	0.0366302
$731$	−15171.1	−0.767608
$732$	6193.00	0.312704
$733$	17947.5	0.904374	0.452187	−	0.891923i	$-0.350644\pi$
$733$	17947.5	0.904374	0.452187	+	0.891923i	$0.350644\pi$
$734$	−41782.4	−2.10111
$735$	283.406	0.0142226
$736$	6989.86	0.350067
$737$	77.7149	0.00388421
$738$	−14815.2	−0.738962
$739$	−22472.6	−1.11863	−0.559315	−	0.828955i	$-0.688936\pi$
$739$	−22472.6	−1.11863	−0.559315	+	0.828955i	$0.688936\pi$
$740$	−244.938	−0.0121677
$741$	0	0
$742$	−215.188	−0.0106466
$743$	−14221.4	−0.702198	−0.351099	−	0.936338i	$-0.614192\pi$
$743$	−14221.4	−0.702198	−0.351099	+	0.936338i	$0.614192\pi$
$744$	804.903	0.0396629
$745$	−390.350	−0.0191964
$746$	−6519.56	−0.319970
$747$	4560.66	0.223381
$748$	−267.707	−0.0130860
$749$	61.2083	0.00298599
$750$	−790.133	−0.0384688
$751$	−846.560	−0.0411337	−0.0205669	−	0.999788i	$-0.506547\pi$
$751$	−846.560	−0.0411337	−0.0205669	+	0.999788i	$0.506547\pi$
$752$	15484.4	0.750877
$753$	−5493.08	−0.265842
$754$	0	0
$755$	−782.646	−0.0377264
$756$	−17.5908	−0.000846258 0
$757$	15481.9	0.743329	0.371665	−	0.928367i	$-0.378787\pi$
$757$	15481.9	0.743329	0.371665	+	0.928367i	$0.378787\pi$
$758$	15786.5	0.756452
$759$	66.1326	0.00316266
$760$	208.571	0.00995481
$761$	36354.3	1.73172	0.865862	−	0.500282i	$-0.166770\pi$
$761$	36354.3	1.73172	0.865862	+	0.500282i	$0.166770\pi$
$762$	9237.45	0.439157
$763$	215.584	0.0102289
$764$	−65.5491	−0.00310404
$765$	133.271	0.00629859
$766$	27447.9	1.29469
$767$	0	0
$768$	−15351.0	−0.721263
$769$	−19944.4	−0.935256	−0.467628	−	0.883925i	$-0.654891\pi$
$769$	−19944.4	−0.935256	−0.467628	+	0.883925i	$0.654891\pi$
$770$	0.0775489	3.62944e−6 0
$771$	4878.52	0.227880
$772$	−4944.29	−0.230504
$773$	7536.80	0.350685	0.175343	−	0.984507i	$-0.443897\pi$
$773$	7536.80	0.350685	0.175343	+	0.984507i	$0.443897\pi$
$774$	9717.07	0.451257
$775$	6439.42	0.298465
$776$	4552.28	0.210590
$777$	39.4281	0.00182043
$778$	38399.9	1.76954
$779$	62592.8	2.87885
$780$	0	0
$781$	152.446	0.00698457
$782$	−6046.71	−0.276509
$783$	7230.54	0.330011
$784$	25049.4	1.14110
$785$	627.674	0.0285384
$786$	3376.97	0.153248
$787$	−8513.21	−0.385595	−0.192797	−	0.981239i	$-0.561756\pi$
$787$	−8513.21	−0.385595	−0.192797	+	0.981239i	$0.561756\pi$
$788$	21016.0	0.950082
$789$	20794.2	0.938267
$790$	−670.256	−0.0301856
$791$	141.216	0.00634775
$792$	−35.1311	−0.00157617
$793$	0	0
$794$	3533.06	0.157914
$795$	−473.590	−0.0211277
$796$	17814.6	0.793245
$797$	167.610	0.00744926	0.00372463	−	0.999993i	$-0.498814\pi$
$797$	167.610	0.00744926	0.00372463	+	0.999993i	$0.498814\pi$
$798$	163.866	0.00726918
$799$	−11399.0	−0.504717
$800$	−29706.3	−1.31284
$801$	6305.65	0.278151
$802$	−51847.5	−2.28279
$803$	514.139	0.0225947
$804$	2064.15	0.0905435
$805$	0.794419	3.47821e−5 0
$806$	0	0
$807$	8005.27	0.349193
$808$	4287.70	0.186684
$809$	−21465.9	−0.932882	−0.466441	−	0.884552i	$-0.654464\pi$
$809$	−21465.9	−0.932882	−0.466441	+	0.884552i	$0.654464\pi$
$810$	−85.3602	−0.00370278
$811$	−7038.94	−0.304773	−0.152386	−	0.988321i	$-0.548696\pi$
$811$	−7038.94	−0.304773	−0.152386	+	0.988321i	$0.548696\pi$
$812$	−174.473	−0.00754040
$813$	3733.32	0.161050
$814$	−384.326	−0.0165487
$815$	−464.330	−0.0199568
$816$	11779.4	0.505346
$817$	−41053.8	−1.75801
$818$	−3429.25	−0.146578
$819$	0	0
$820$	786.771	0.0335064
$821$	42154.6	1.79197	0.895983	−	0.444088i	$-0.146472\pi$
$821$	42154.6	1.79197	0.895983	+	0.444088i	$0.146472\pi$
$822$	25425.6	1.07886
$823$	−31428.3	−1.33113	−0.665565	−	0.746340i	$-0.731809\pi$
$823$	−31428.3	−1.33113	−0.665565	+	0.746340i	$0.731809\pi$
$824$	9174.70	0.387883
$825$	−281.058	−0.0118608
$826$	−186.132	−0.00784062
$827$	−3327.36	−0.139908	−0.0699538	−	0.997550i	$-0.522285\pi$
$827$	−3327.36	−0.139908	−0.0699538	+	0.997550i	$0.522285\pi$
$828$	1756.52	0.0737237
$829$	−37345.6	−1.56462	−0.782308	−	0.622891i	$-0.785958\pi$
$829$	−37345.6	−1.56462	−0.782308	+	0.622891i	$0.785958\pi$
$830$	−534.017	−0.0223325
$831$	−6301.36	−0.263047
$832$	0	0
$833$	−18440.4	−0.767012
$834$	−19530.5	−0.810893
$835$	−562.292	−0.0233041
$836$	−724.432	−0.0299701
$837$	1391.76	0.0574746
$838$	−25006.3	−1.03082
$839$	−22457.2	−0.924088	−0.462044	−	0.886857i	$-0.652884\pi$
$839$	−22457.2	−0.924088	−0.462044	+	0.886857i	$0.652884\pi$
$840$	−0.422014	−1.73343e−5 0
$841$	47326.6	1.94049
$842$	7430.45	0.304121
$843$	−1703.19	−0.0695858
$844$	−25950.5	−1.05836
$845$	0	0
$846$	7301.09	0.296710
$847$	−130.549	−0.00529599
$848$	−41859.2	−1.69511
$849$	−24447.8	−0.988275
$850$	25698.0	1.03698
$851$	−3937.07	−0.158591
$852$	4049.06	0.162815
$853$	−2238.49	−0.0898528	−0.0449264	−	0.998990i	$-0.514305\pi$
$853$	−2238.49	−0.0898528	−0.0449264	+	0.998990i	$0.514305\pi$
$854$	116.729	0.00467726
$855$	360.640	0.0144253
$856$	3246.79	0.129642
$857$	−28925.5	−1.15295	−0.576473	−	0.817116i	$-0.695571\pi$
$857$	−28925.5	−1.15295	−0.576473	+	0.817116i	$0.695571\pi$
$858$	0	0
$859$	1447.84	0.0575082	0.0287541	−	0.999587i	$-0.490846\pi$
$859$	1447.84	0.0575082	0.0287541	+	0.999587i	$0.490846\pi$
$860$	−516.033	−0.0204611
$861$	−126.648	−0.00501295
$862$	−13439.8	−0.531047
$863$	−22008.3	−0.868101	−0.434050	−	0.900889i	$-0.642916\pi$
$863$	−22008.3	−0.868101	−0.434050	+	0.900889i	$0.642916\pi$
$864$	−6420.45	−0.252810
$865$	760.775	0.0299042
$866$	−11684.9	−0.458508
$867$	6067.45	0.237672
$868$	−33.5832	−0.00131323
$869$	−476.977	−0.0186195
$870$	−846.640	−0.0329928
$871$	0	0
$872$	11435.7	0.444106
$873$	7871.36	0.305161
$874$	−16362.8	−0.633272
$875$	−6.75447	−0.000260963 0
$876$	13655.8	0.526698
$877$	24124.5	0.928880	0.464440	−	0.885605i	$-0.346256\pi$
$877$	24124.5	0.928880	0.464440	+	0.885605i	$0.346256\pi$
$878$	−4035.73	−0.155124
$879$	−21406.6	−0.821417
$880$	15.0851	0.000577862 0
$881$	−27295.0	−1.04381	−0.521903	−	0.853005i	$-0.674778\pi$
$881$	−27295.0	−1.04381	−0.521903	+	0.853005i	$0.674778\pi$
$882$	11811.1	0.450907
$883$	12916.8	0.492282	0.246141	−	0.969234i	$-0.420837\pi$
$883$	12916.8	0.492282	0.246141	+	0.969234i	$0.420837\pi$
$884$	0	0
$885$	−409.642	−0.0155593
$886$	40504.0	1.53585
$887$	42854.0	1.62221	0.811104	−	0.584902i	$-0.198867\pi$
$887$	42854.0	1.62221	0.811104	+	0.584902i	$0.198867\pi$
$888$	2091.46	0.0790371
$889$	78.9666	0.00297914
$890$	−738.343	−0.0278082
$891$	−60.7453	−0.00228400
$892$	37660.8	1.41365
$893$	−30846.5	−1.15592
$894$	−16268.0	−0.608596
$895$	−874.755	−0.0326702
$896$	−64.4295	−0.00240228
$897$	0	0
$898$	−20861.1	−0.775215
$899$	13804.1	0.512115
$900$	−7465.05	−0.276483
$901$	30815.1	1.13940
$902$	1234.50	0.0455703
$903$	83.0667	0.00306122
$904$	7490.81	0.275598
$905$	578.785	0.0212591
$906$	−32617.2	−1.19606
$907$	38176.4	1.39760	0.698802	−	0.715315i	$-0.253717\pi$
$907$	38176.4	1.39760	0.698802	+	0.715315i	$0.253717\pi$
$908$	9893.91	0.361609
$909$	7413.86	0.270519
$910$	0	0
$911$	−20038.9	−0.728779	−0.364390	−	0.931247i	$-0.618722\pi$
$911$	−20038.9	−0.728779	−0.364390	+	0.931247i	$0.618722\pi$
$912$	31875.9	1.15736
$913$	−380.025	−0.0137755
$914$	−38447.0	−1.39137
$915$	256.899	0.00928177
$916$	16566.9	0.597584
$917$	28.8681	0.00103960
$918$	5554.14	0.199688
$919$	45857.3	1.64602	0.823011	−	0.568026i	$-0.192293\pi$
$919$	45857.3	1.64602	0.823011	+	0.568026i	$0.192293\pi$
$920$	42.1399	0.00151012
$921$	−28534.9	−1.02091
$922$	−69635.3	−2.48733
$923$	0	0
$924$	1.46579	5.21870e−5 0
$925$	16732.2	0.594759
$926$	−21178.8	−0.751595
$927$	15864.0	0.562073
$928$	−63680.9	−2.25261
$929$	27907.7	0.985601	0.492801	−	0.870142i	$-0.335973\pi$
$929$	27907.7	0.985601	0.492801	+	0.870142i	$0.335973\pi$
$930$	−162.964	−0.00574602
$931$	−49900.8	−1.75664
$932$	−1672.19	−0.0587709
$933$	−24933.6	−0.874907
$934$	14838.5	0.519839
$935$	−11.1051	−0.000388422 0
$936$	0	0
$937$	25288.8	0.881696	0.440848	−	0.897582i	$-0.354678\pi$
$937$	25288.8	0.881696	0.440848	+	0.897582i	$0.354678\pi$
$938$	38.9062	0.00135430
$939$	−21326.5	−0.741177
$940$	−387.730	−0.0134536
$941$	14302.7	0.495488	0.247744	−	0.968826i	$-0.420311\pi$
$941$	14302.7	0.495488	0.247744	+	0.968826i	$0.420311\pi$
$942$	26158.6	0.904771
$943$	12646.3	0.436715
$944$	−36207.0	−1.24835
$945$	−0.729704	−2.51188e−5 0
$946$	−809.694	−0.0278281
$947$	−20519.2	−0.704102	−0.352051	−	0.935981i	$-0.614516\pi$
$947$	−20519.2	−0.704102	−0.352051	+	0.935981i	$0.614516\pi$
$948$	−12668.8	−0.434033
$949$	0	0
$950$	69540.4	2.37494
$951$	−22735.0	−0.775218
$952$	27.4592	0.000934829 0
$953$	37273.6	1.26696	0.633479	−	0.773759i	$-0.281626\pi$
$953$	37273.6	1.26696	0.633479	+	0.773759i	$0.281626\pi$
$954$	−19737.1	−0.669824
$955$	−2.71912	−9.21347e−5 0
$956$	28855.5	0.976205
$957$	−602.498	−0.0203511
$958$	−31992.0	−1.07893
$959$	217.352	0.00731872
$960$	269.024	0.00904450
$961$	−27133.9	−0.910810
$962$	0	0
$963$	5614.04	0.187861
$964$	−22768.2	−0.760699
$965$	−205.100	−0.00684187
$966$	33.1078	0.00110272
$967$	−40622.2	−1.35090	−0.675450	−	0.737405i	$-0.736051\pi$
$967$	−40622.2	−1.35090	−0.675450	+	0.737405i	$0.736051\pi$
$968$	−6924.95	−0.229934
$969$	−23465.8	−0.777945
$970$	−921.675	−0.0305085
$971$	2397.74	0.0792454	0.0396227	−	0.999215i	$-0.487384\pi$
$971$	2397.74	0.0792454	0.0396227	+	0.999215i	$0.487384\pi$
$972$	−1613.43	−0.0532416
$973$	−166.957	−0.00550091
$974$	−45498.0	−1.49677
$975$	0	0
$976$	22706.5	0.744691
$977$	−22969.2	−0.752150	−0.376075	−	0.926589i	$-0.622726\pi$
$977$	−22969.2	−0.752150	−0.376075	+	0.926589i	$0.622726\pi$
$978$	−19351.2	−0.632702
$979$	−525.430	−0.0171530
$980$	−627.236	−0.0204452
$981$	19773.4	0.643544
$982$	−11853.7	−0.385201
$983$	−36918.1	−1.19787	−0.598934	−	0.800799i	$-0.704409\pi$
$983$	−36918.1	−1.19787	−0.598934	+	0.800799i	$0.704409\pi$
$984$	−6718.03	−0.217645
$985$	871.790	0.0282006
$986$	55088.3	1.77928
$987$	62.4136	0.00201281
$988$	0	0
$989$	−8294.58	−0.266686
$990$	7.11280	0.000228343 0
$991$	−26279.4	−0.842374	−0.421187	−	0.906974i	$-0.638386\pi$
$991$	−26279.4	−0.842374	−0.421187	+	0.906974i	$0.638386\pi$
$992$	−12257.5	−0.392315
$993$	25470.2	0.813969
$994$	76.3188	0.00243530
$995$	738.990	0.0235453
$996$	−10093.7	−0.321115
$997$	24865.7	0.789875	0.394937	−	0.918708i	$-0.370766\pi$
$997$	24865.7	0.789875	0.394937	+	0.918708i	$0.370766\pi$
$998$	−63.6710	−0.00201951
$999$	3616.35	0.114531

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.q.1.1	yes	9
3.2	odd	2		1521.4.a.be.1.9		9
13.5	odd	4		507.4.b.j.337.16		18
13.8	odd	4		507.4.b.j.337.3		18
13.12	even	2		507.4.a.n.1.9	✓	9
39.38	odd	2		1521.4.a.bj.1.1		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.9	✓	9	13.12	even	2
507.4.a.q.1.1	yes	9	1.1	even	1	trivial
507.4.b.j.337.3		18	13.8	odd	4
507.4.b.j.337.16		18	13.5	odd	4
1521.4.a.be.1.9		9	3.2	odd	2
1521.4.a.bj.1.1		9	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)

\(f(q)\)	\(=\)	\(q-3.82618 q^{2} -3.00000 q^{3} +6.63963 q^{4} +0.275426 q^{5} +11.4785 q^{6} +0.0981245 q^{7} +5.20502 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.82618 q^{2} -3.00000 q^{3} +6.63963 q^{4} +0.275426 q^{5} +11.4785 q^{6} +0.0981245 q^{7} +5.20502 q^{8} +9.00000 q^{9} -1.05383 q^{10} -0.749942 q^{11} -19.9189 q^{12} -0.375442 q^{14} -0.826279 q^{15} -73.0324 q^{16} +53.7635 q^{17} -34.4356 q^{18} +145.488 q^{19} +1.82873 q^{20} -0.294374 q^{21} +2.86941 q^{22} +29.3945 q^{23} -15.6150 q^{24} -124.924 q^{25} -27.0000 q^{27} +0.651511 q^{28} -267.798 q^{29} +3.16149 q^{30} -51.5466 q^{31} +237.795 q^{32} +2.24983 q^{33} -205.709 q^{34} +0.0270261 q^{35} +59.7567 q^{36} -133.939 q^{37} -556.661 q^{38} +1.43360 q^{40} +430.228 q^{41} +1.12633 q^{42} -282.181 q^{43} -4.97934 q^{44} +2.47884 q^{45} -112.469 q^{46} -212.022 q^{47} +219.097 q^{48} -342.990 q^{49} +477.982 q^{50} -161.291 q^{51} +573.160 q^{53} +103.307 q^{54} -0.206554 q^{55} +0.510740 q^{56} -436.463 q^{57} +1024.64 q^{58} +495.767 q^{59} -5.48619 q^{60} -310.911 q^{61} +197.226 q^{62} +0.883121 q^{63} -325.585 q^{64} -8.60823 q^{66} -103.628 q^{67} +356.970 q^{68} -88.1836 q^{69} -0.103407 q^{70} -203.277 q^{71} +46.8451 q^{72} -685.571 q^{73} +512.474 q^{74} +374.772 q^{75} +965.984 q^{76} -0.0735877 q^{77} +636.019 q^{79} -20.1150 q^{80} +81.0000 q^{81} -1646.13 q^{82} +506.740 q^{83} -1.95453 q^{84} +14.8079 q^{85} +1079.67 q^{86} +803.393 q^{87} -3.90346 q^{88} +700.628 q^{89} -9.48447 q^{90} +195.169 q^{92} +154.640 q^{93} +811.232 q^{94} +40.0711 q^{95} -713.384 q^{96} +874.596 q^{97} +1312.34 q^{98} -6.74948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 8 q^{2} - 27 q^{3} + 32 q^{4} + 41 q^{5} - 24 q^{6} + q^{7} + 111 q^{8} + 81 q^{9}+O(q^{10}) \) 9 * q + 8 * q^2 - 27 * q^3 + 32 * q^4 + 41 * q^5 - 24 * q^6 + q^7 + 111 * q^8 + 81 * q^9	\( 9 q + 8 q^{2} - 27 q^{3} + 32 q^{4} + 41 q^{5} - 24 q^{6} + q^{7} + 111 q^{8} + 81 q^{9} + 198 q^{10} + 37 q^{11} - 96 q^{12} + 98 q^{14} - 123 q^{15} + 32 q^{16} - 134 q^{17} + 72 q^{18} - 72 q^{19} + 356 q^{20} - 3 q^{21} + 274 q^{22} + 226 q^{23} - 333 q^{24} + 612 q^{25} - 243 q^{27} + 132 q^{28} - 547 q^{29} - 594 q^{30} - 521 q^{31} + 721 q^{32} - 111 q^{33} - 100 q^{34} + 138 q^{35} + 288 q^{36} + 584 q^{37} - 416 q^{38} + 1342 q^{40} + 482 q^{41} - 294 q^{42} + 158 q^{43} + 1453 q^{44} + 369 q^{45} + 1537 q^{46} + 1500 q^{47} - 96 q^{48} + 642 q^{49} + 2777 q^{50} + 402 q^{51} + 1399 q^{53} - 216 q^{54} - 1408 q^{55} - 616 q^{56} + 216 q^{57} + 1455 q^{58} + 1541 q^{59} - 1068 q^{60} + 2092 q^{61} - 293 q^{62} + 9 q^{63} + 2481 q^{64} - 822 q^{66} + 252 q^{67} - 1579 q^{68} - 678 q^{69} + 2492 q^{70} + 2352 q^{71} + 999 q^{72} + 903 q^{73} + 1037 q^{74} - 1836 q^{75} - 485 q^{76} - 1686 q^{77} - 115 q^{79} + 5701 q^{80} + 729 q^{81} - 5147 q^{82} + 1207 q^{83} - 396 q^{84} + 4308 q^{85} + 5691 q^{86} + 1641 q^{87} - 484 q^{88} + 2336 q^{89} + 1782 q^{90} + 2087 q^{92} + 1563 q^{93} - 468 q^{94} - 222 q^{95} - 2163 q^{96} + 2155 q^{97} + 5593 q^{98} + 333 q^{99}+O(q^{100}) \) 9 * q + 8 * q^2 - 27 * q^3 + 32 * q^4 + 41 * q^5 - 24 * q^6 + q^7 + 111 * q^8 + 81 * q^9 + 198 * q^10 + 37 * q^11 - 96 * q^12 + 98 * q^14 - 123 * q^15 + 32 * q^16 - 134 * q^17 + 72 * q^18 - 72 * q^19 + 356 * q^20 - 3 * q^21 + 274 * q^22 + 226 * q^23 - 333 * q^24 + 612 * q^25 - 243 * q^27 + 132 * q^28 - 547 * q^29 - 594 * q^30 - 521 * q^31 + 721 * q^32 - 111 * q^33 - 100 * q^34 + 138 * q^35 + 288 * q^36 + 584 * q^37 - 416 * q^38 + 1342 * q^40 + 482 * q^41 - 294 * q^42 + 158 * q^43 + 1453 * q^44 + 369 * q^45 + 1537 * q^46 + 1500 * q^47 - 96 * q^48 + 642 * q^49 + 2777 * q^50 + 402 * q^51 + 1399 * q^53 - 216 * q^54 - 1408 * q^55 - 616 * q^56 + 216 * q^57 + 1455 * q^58 + 1541 * q^59 - 1068 * q^60 + 2092 * q^61 - 293 * q^62 + 9 * q^63 + 2481 * q^64 - 822 * q^66 + 252 * q^67 - 1579 * q^68 - 678 * q^69 + 2492 * q^70 + 2352 * q^71 + 999 * q^72 + 903 * q^73 + 1037 * q^74 - 1836 * q^75 - 485 * q^76 - 1686 * q^77 - 115 * q^79 + 5701 * q^80 + 729 * q^81 - 5147 * q^82 + 1207 * q^83 - 396 * q^84 + 4308 * q^85 + 5691 * q^86 + 1641 * q^87 - 484 * q^88 + 2336 * q^89 + 1782 * q^90 + 2087 * q^92 + 1563 * q^93 - 468 * q^94 - 222 * q^95 - 2163 * q^96 + 2155 * q^97 + 5593 * q^98 + 333 * q^99

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