LMFDB - Embedded newform 507.4.a.p.1.4

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} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 $ x^9 - 56x^7 - 27x^6 + 945x^5 + 763x^4 - 4139x^3 - 2478x^2 + 63*x + 27
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			1.4
Root			$-0.588238$ 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)

$f(q)$	$=$	$q-0.213700 q^{2} +3.00000 q^{3} -7.95433 q^{4} -15.3391 q^{5} -0.641100 q^{6} +32.3928 q^{7} +3.40944 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-0.213700 q^{2} +3.00000 q^{3} -7.95433 q^{4} -15.3391 q^{5} -0.641100 q^{6} +32.3928 q^{7} +3.40944 q^{8} +9.00000 q^{9} +3.27797 q^{10} -29.5925 q^{11} -23.8630 q^{12} -6.92233 q^{14} -46.0174 q^{15} +62.9061 q^{16} -78.1958 q^{17} -1.92330 q^{18} -10.6600 q^{19} +122.013 q^{20} +97.1783 q^{21} +6.32392 q^{22} -26.8789 q^{23} +10.2283 q^{24} +110.289 q^{25} +27.0000 q^{27} -257.663 q^{28} -190.785 q^{29} +9.83392 q^{30} +128.108 q^{31} -40.7185 q^{32} -88.7776 q^{33} +16.7104 q^{34} -496.877 q^{35} -71.5890 q^{36} +379.934 q^{37} +2.27805 q^{38} -52.2979 q^{40} +464.631 q^{41} -20.7670 q^{42} +322.758 q^{43} +235.389 q^{44} -138.052 q^{45} +5.74401 q^{46} +248.529 q^{47} +188.718 q^{48} +706.292 q^{49} -23.5688 q^{50} -234.587 q^{51} +740.167 q^{53} -5.76990 q^{54} +453.924 q^{55} +110.441 q^{56} -31.9801 q^{57} +40.7706 q^{58} +340.673 q^{59} +366.038 q^{60} -590.834 q^{61} -27.3767 q^{62} +291.535 q^{63} -494.547 q^{64} +18.9718 q^{66} +340.777 q^{67} +621.995 q^{68} -80.6366 q^{69} +106.183 q^{70} +36.2243 q^{71} +30.6850 q^{72} +164.572 q^{73} -81.1919 q^{74} +330.868 q^{75} +84.7935 q^{76} -958.584 q^{77} -327.242 q^{79} -964.925 q^{80} +81.0000 q^{81} -99.2916 q^{82} +1404.48 q^{83} -772.989 q^{84} +1199.46 q^{85} -68.9734 q^{86} -572.354 q^{87} -100.894 q^{88} -736.986 q^{89} +29.5018 q^{90} +213.803 q^{92} +384.324 q^{93} -53.1107 q^{94} +163.516 q^{95} -122.156 q^{96} -1494.87 q^{97} -150.935 q^{98} -266.333 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9}+O(q^{10}) $ 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9	\( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9} - 54 q^{10} + 85 q^{11} + 132 q^{12} + 158 q^{14} + 99 q^{15} + 216 q^{16} + 178 q^{17} + 54 q^{18} + 352 q^{19} + 402 q^{20} + 249 q^{21} - 630 q^{22} + 150 q^{23} + 261 q^{24} - 20 q^{25} + 243 q^{27} + 940 q^{28} - 97 q^{29} - 162 q^{30} + 717 q^{31} + 707 q^{32} + 255 q^{33} + 632 q^{34} - 418 q^{35} + 396 q^{36} + 1108 q^{37} - 660 q^{38} - 1506 q^{40} + 334 q^{41} + 474 q^{42} + 242 q^{43} - 307 q^{44} + 297 q^{45} + 979 q^{46} - 184 q^{47} + 648 q^{48} - 38 q^{49} - 2031 q^{50} + 534 q^{51} - 151 q^{53} + 162 q^{54} + 2064 q^{55} + 2276 q^{56} + 1056 q^{57} + 1161 q^{58} + 537 q^{59} + 1206 q^{60} - 1340 q^{61} + 347 q^{62} + 747 q^{63} + 893 q^{64} - 1890 q^{66} + 2308 q^{67} + 2785 q^{68} + 450 q^{69} - 1420 q^{70} + 96 q^{71} + 783 q^{72} + 2505 q^{73} - 1191 q^{74} - 60 q^{75} + 2409 q^{76} - 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 729 q^{81} + 1517 q^{82} + 1539 q^{83} + 2820 q^{84} + 4296 q^{85} - 3763 q^{86} - 291 q^{87} - 3716 q^{88} - 592 q^{89} - 486 q^{90} + 515 q^{92} + 2151 q^{93} - 692 q^{94} + 4158 q^{95} + 2121 q^{96} + 1445 q^{97} + 1457 q^{98} + 765 q^{99}+O(q^{100}) \) 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9 - 54 * q^10 + 85 * q^11 + 132 * q^12 + 158 * q^14 + 99 * q^15 + 216 * q^16 + 178 * q^17 + 54 * q^18 + 352 * q^19 + 402 * q^20 + 249 * q^21 - 630 * q^22 + 150 * q^23 + 261 * q^24 - 20 * q^25 + 243 * q^27 + 940 * q^28 - 97 * q^29 - 162 * q^30 + 717 * q^31 + 707 * q^32 + 255 * q^33 + 632 * q^34 - 418 * q^35 + 396 * q^36 + 1108 * q^37 - 660 * q^38 - 1506 * q^40 + 334 * q^41 + 474 * q^42 + 242 * q^43 - 307 * q^44 + 297 * q^45 + 979 * q^46 - 184 * q^47 + 648 * q^48 - 38 * q^49 - 2031 * q^50 + 534 * q^51 - 151 * q^53 + 162 * q^54 + 2064 * q^55 + 2276 * q^56 + 1056 * q^57 + 1161 * q^58 + 537 * q^59 + 1206 * q^60 - 1340 * q^61 + 347 * q^62 + 747 * q^63 + 893 * q^64 - 1890 * q^66 + 2308 * q^67 + 2785 * q^68 + 450 * q^69 - 1420 * q^70 + 96 * q^71 + 783 * q^72 + 2505 * q^73 - 1191 * q^74 - 60 * q^75 + 2409 * q^76 - 2142 * q^77 - 1591 * q^79 - 2671 * q^80 + 729 * q^81 + 1517 * q^82 + 1539 * q^83 + 2820 * q^84 + 4296 * q^85 - 3763 * q^86 - 291 * q^87 - 3716 * q^88 - 592 * q^89 - 486 * q^90 + 515 * q^92 + 2151 * q^93 - 692 * q^94 + 4158 * q^95 + 2121 * q^96 + 1445 * q^97 + 1457 * q^98 + 765 * 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 $

$2$	−0.213700	−0.0755543	−0.0377772	−	0.999286i	$-0.512028\pi$
$2$	−0.213700	−0.0755543	−0.0377772	+	0.999286i	$0.512028\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−7.95433	−0.994292
$5$	−15.3391	−1.37197	−0.685987	−	0.727614i	$-0.740629\pi$
$5$	−15.3391	−1.37197	−0.685987	+	0.727614i	$0.740629\pi$
$6$	−0.641100	−0.0436213
$7$	32.3928	1.74905	0.874523	−	0.484984i	$-0.161175\pi$
$7$	32.3928	1.74905	0.874523	+	0.484984i	$0.161175\pi$
$8$	3.40944	0.150677
$9$	9.00000	0.333333
$10$	3.27797	0.103659
$11$	−29.5925	−0.811135	−0.405567	−	0.914065i	$-0.632926\pi$
$11$	−29.5925	−0.811135	−0.405567	+	0.914065i	$0.632926\pi$
$12$	−23.8630	−0.574054
$13$	0	0
$13$	0	0
$14$	−6.92233	−0.132148
$15$	−46.0174	−0.792110
$16$	62.9061	0.982907
$17$	−78.1958	−1.11560	−0.557802	−	0.829974i	$-0.688355\pi$
$17$	−78.1958	−1.11560	−0.557802	+	0.829974i	$0.688355\pi$
$18$	−1.92330	−0.0251848
$19$	−10.6600	−0.128715	−0.0643574	−	0.997927i	$-0.520500\pi$
$19$	−10.6600	−0.128715	−0.0643574	+	0.997927i	$0.520500\pi$
$20$	122.013	1.36414
$21$	97.1783	1.00981
$22$	6.32392	0.0612847
$23$	−26.8789	−0.243680	−0.121840	−	0.992550i	$-0.538879\pi$
$23$	−26.8789	−0.243680	−0.121840	+	0.992550i	$0.538879\pi$
$24$	10.2283	0.0869936
$25$	110.289	0.882314
$26$	0	0
$27$	27.0000	0.192450
$28$	−257.663	−1.73906
$29$	−190.785	−1.22165	−0.610824	−	0.791766i	$-0.709162\pi$
$29$	−190.785	−1.22165	−0.610824	+	0.791766i	$0.709162\pi$
$30$	9.83392	0.0598473
$31$	128.108	0.742222	0.371111	−	0.928589i	$-0.378977\pi$
$31$	128.108	0.742222	0.371111	+	0.928589i	$0.378977\pi$
$32$	−40.7185	−0.224940
$33$	−88.7776	−0.468309
$34$	16.7104	0.0842887
$35$	−496.877	−2.39965
$36$	−71.5890	−0.331431
$37$	379.934	1.68813	0.844065	−	0.536241i	$-0.180156\pi$
$37$	379.934	1.68813	0.844065	+	0.536241i	$0.180156\pi$
$38$	2.27805	0.00972496
$39$	0	0
$40$	−52.2979	−0.206725
$41$	464.631	1.76983	0.884917	−	0.465749i	$-0.154215\pi$
$41$	464.631	1.76983	0.884917	+	0.465749i	$0.154215\pi$
$42$	−20.7670	−0.0762957
$43$	322.758	1.14466	0.572328	−	0.820025i	$-0.306041\pi$
$43$	322.758	1.14466	0.572328	+	0.820025i	$0.306041\pi$
$44$	235.389	0.806504
$45$	−138.052	−0.457325
$46$	5.74401	0.0184110
$47$	248.529	0.771314	0.385657	−	0.922642i	$-0.373975\pi$
$47$	248.529	0.771314	0.385657	+	0.922642i	$0.373975\pi$
$48$	188.718	0.567482
$49$	706.292	2.05916
$50$	−23.5688	−0.0666626
$51$	−234.587	−0.644094
$52$	0	0
$53$	740.167	1.91830	0.959148	−	0.282904i	$-0.0912976\pi$
$53$	740.167	1.91830	0.959148	+	0.282904i	$0.0912976\pi$
$54$	−5.76990	−0.0145404
$55$	453.924	1.11286
$56$	110.441	0.263542
$57$	−31.9801	−0.0743135
$58$	40.7706	0.0923008
$59$	340.673	0.751727	0.375864	−	0.926675i	$-0.377346\pi$
$59$	340.673	0.751727	0.375864	+	0.926675i	$0.377346\pi$
$60$	366.038	0.787588
$61$	−590.834	−1.24014	−0.620070	−	0.784547i	$-0.712896\pi$
$61$	−590.834	−1.24014	−0.620070	+	0.784547i	$0.712896\pi$
$62$	−27.3767	−0.0560781
$63$	291.535	0.583015
$64$	−494.547	−0.965912
$65$	0	0
$66$	18.9718	0.0353828
$67$	340.777	0.621382	0.310691	−	0.950511i	$-0.399440\pi$
$67$	340.777	0.621382	0.310691	+	0.950511i	$0.399440\pi$
$68$	621.995	1.10924
$69$	−80.6366	−0.140688
$70$	106.183	0.181304
$71$	36.2243	0.0605498	0.0302749	−	0.999542i	$-0.490362\pi$
$71$	36.2243	0.0605498	0.0302749	+	0.999542i	$0.490362\pi$
$72$	30.6850	0.0502258
$73$	164.572	0.263859	0.131929	−	0.991259i	$-0.457883\pi$
$73$	164.572	0.263859	0.131929	+	0.991259i	$0.457883\pi$
$74$	−81.1919	−0.127546
$75$	330.868	0.509404
$76$	84.7935	0.127980
$77$	−958.584	−1.41871
$78$	0	0
$79$	−327.242	−0.466046	−0.233023	−	0.972471i	$-0.574862\pi$
$79$	−327.242	−0.466046	−0.233023	+	0.972471i	$0.574862\pi$
$80$	−964.925	−1.34852
$81$	81.0000	0.111111
$82$	−99.2916	−0.133719
$83$	1404.48	1.85737	0.928684	−	0.370871i	$-0.120941\pi$
$83$	1404.48	1.85737	0.928684	+	0.370871i	$0.120941\pi$
$84$	−772.989	−1.00405
$85$	1199.46	1.53058
$86$	−68.9734	−0.0864837
$87$	−572.354	−0.705319
$88$	−100.894	−0.122220
$89$	−736.986	−0.877756	−0.438878	−	0.898547i	$-0.644624\pi$
$89$	−736.986	−0.877756	−0.438878	+	0.898547i	$0.644624\pi$
$90$	29.5018	0.0345529
$91$	0	0
$92$	213.803	0.242289
$93$	384.324	0.428522
$94$	−53.1107	−0.0582761
$95$	163.516	0.176593
$96$	−122.156	−0.129869
$97$	−1494.87	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$97$	−1494.87	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$98$	−150.935	−0.155578
$99$	−266.333	−0.270378
$100$	−877.277	−0.877277
$101$	−484.010	−0.476839	−0.238420	−	0.971162i	$-0.576629\pi$
$101$	−484.010	−0.476839	−0.238420	+	0.971162i	$0.576629\pi$
$102$	50.1313	0.0486641
$103$	214.049	0.204766	0.102383	−	0.994745i	$-0.467353\pi$
$103$	214.049	0.204766	0.102383	+	0.994745i	$0.467353\pi$
$104$	0	0
$105$	−1490.63	−1.38544
$106$	−158.174	−0.144936
$107$	148.288	0.133977	0.0669884	−	0.997754i	$-0.478661\pi$
$107$	148.288	0.133977	0.0669884	+	0.997754i	$0.478661\pi$
$108$	−214.767	−0.191351
$109$	504.881	0.443659	0.221829	−	0.975086i	$-0.428797\pi$
$109$	504.881	0.443659	0.221829	+	0.975086i	$0.428797\pi$
$110$	−97.0035	−0.0840811
$111$	1139.80	0.974642
$112$	2037.70	1.71915
$113$	−1136.49	−0.946120	−0.473060	−	0.881030i	$-0.656851\pi$
$113$	−1136.49	−0.946120	−0.473060	+	0.881030i	$0.656851\pi$
$114$	6.83415	0.00561471
$115$	412.299	0.334322
$116$	1517.56	1.21467
$117$	0	0
$118$	−72.8019	−0.0567962
$119$	−2532.98	−1.95124
$120$	−156.894	−0.119353
$121$	−455.282	−0.342060
$122$	126.261	0.0936979
$123$	1393.89	1.02181
$124$	−1019.01	−0.737985
$125$	225.651	0.161463
$126$	−62.3010	−0.0440493
$127$	−1286.84	−0.899122	−0.449561	−	0.893250i	$-0.648420\pi$
$127$	−1286.84	−0.899122	−0.449561	+	0.893250i	$0.648420\pi$
$128$	431.433	0.297919
$129$	968.275	0.660867
$130$	0	0
$131$	2229.88	1.48722	0.743610	−	0.668614i	$-0.233112\pi$
$131$	2229.88	1.48722	0.743610	+	0.668614i	$0.233112\pi$
$132$	706.166	0.465636
$133$	−345.308	−0.225128
$134$	−72.8241	−0.0469481
$135$	−414.157	−0.264037
$136$	−266.604	−0.168096
$137$	−474.745	−0.296060	−0.148030	−	0.988983i	$-0.547293\pi$
$137$	−474.745	−0.296060	−0.148030	+	0.988983i	$0.547293\pi$
$138$	17.2320	0.0106296
$139$	1927.02	1.17588	0.587941	−	0.808904i	$-0.299939\pi$
$139$	1927.02	1.17588	0.587941	+	0.808904i	$0.299939\pi$
$140$	3952.33	2.38595
$141$	745.588	0.445318
$142$	−7.74113	−0.00457480
$143$	0	0
$144$	566.155	0.327636
$145$	2926.47	1.67607
$146$	−35.1690	−0.0199357
$147$	2118.88	1.18886
$148$	−3022.12	−1.67849
$149$	1065.73	0.585959	0.292979	−	0.956119i	$-0.405353\pi$
$149$	1065.73	0.585959	0.292979	+	0.956119i	$0.405353\pi$
$150$	−70.7064	−0.0384877
$151$	877.888	0.473123	0.236561	−	0.971617i	$-0.423980\pi$
$151$	877.888	0.473123	0.236561	+	0.971617i	$0.423980\pi$
$152$	−36.3448	−0.0193944
$153$	−703.762	−0.371868
$154$	204.849	0.107190
$155$	−1965.07	−1.01831
$156$	0	0
$157$	2314.94	1.17676	0.588382	−	0.808583i	$-0.299765\pi$
$157$	2314.94	1.17676	0.588382	+	0.808583i	$0.299765\pi$
$158$	69.9317	0.0352118
$159$	2220.50	1.10753
$160$	624.587	0.308612
$161$	−870.681	−0.426207
$162$	−17.3097	−0.00839492
$163$	331.399	0.159246	0.0796232	−	0.996825i	$-0.474628\pi$
$163$	331.399	0.159246	0.0796232	+	0.996825i	$0.474628\pi$
$164$	−3695.83	−1.75973
$165$	1361.77	0.642508
$166$	−300.137	−0.140332
$167$	−2171.62	−1.00626	−0.503129	−	0.864212i	$-0.667818\pi$
$167$	−2171.62	−1.00626	−0.503129	+	0.864212i	$0.667818\pi$
$168$	331.324	0.152156
$169$	0	0
$170$	−256.324	−0.115642
$171$	−95.9404	−0.0429049
$172$	−2567.33	−1.13812
$173$	−2936.69	−1.29059	−0.645295	−	0.763933i	$-0.723266\pi$
$173$	−2936.69	−1.29059	−0.645295	+	0.763933i	$0.723266\pi$
$174$	122.312	0.0532899
$175$	3572.57	1.54321
$176$	−1861.55	−0.797270
$177$	1022.02	0.434010
$178$	157.494	0.0663183
$179$	1615.57	0.674599	0.337300	−	0.941397i	$-0.390486\pi$
$179$	1615.57	0.674599	0.337300	+	0.941397i	$0.390486\pi$
$180$	1098.11	0.454714
$181$	725.019	0.297736	0.148868	−	0.988857i	$-0.452437\pi$
$181$	725.019	0.297736	0.148868	+	0.988857i	$0.452437\pi$
$182$	0	0
$183$	−1772.50	−0.715995
$184$	−91.6419	−0.0367170
$185$	−5827.87	−2.31607
$186$	−82.1300	−0.0323767
$187$	2314.01	0.904905
$188$	−1976.89	−0.766911
$189$	874.605	0.336604
$190$	−34.9433	−0.0133424
$191$	1717.08	0.650490	0.325245	−	0.945630i	$-0.394553\pi$
$191$	1717.08	0.650490	0.325245	+	0.945630i	$0.394553\pi$
$192$	−1483.64	−0.557670
$193$	435.830	0.162548	0.0812740	−	0.996692i	$-0.474101\pi$
$193$	435.830	0.162548	0.0812740	+	0.996692i	$0.474101\pi$
$194$	319.454	0.118224
$195$	0	0
$196$	−5618.08	−2.04741
$197$	694.556	0.251193	0.125597	−	0.992081i	$-0.459915\pi$
$197$	694.556	0.251193	0.125597	+	0.992081i	$0.459915\pi$
$198$	56.9153	0.0204282
$199$	2899.24	1.03277	0.516386	−	0.856356i	$-0.327277\pi$
$199$	2899.24	1.03277	0.516386	+	0.856356i	$0.327277\pi$
$200$	376.024	0.132945
$201$	1022.33	0.358755
$202$	103.433	0.0360273
$203$	−6180.04	−2.13672
$204$	1865.99	0.640417
$205$	−7127.04	−2.42817
$206$	−45.7423	−0.0154709
$207$	−241.910	−0.0812265
$208$	0	0
$209$	315.457	0.104405
$210$	318.548	0.104676
$211$	5250.30	1.71301	0.856507	−	0.516136i	$-0.172630\pi$
$211$	5250.30	1.71301	0.856507	+	0.516136i	$0.172630\pi$
$212$	−5887.53	−1.90735
$213$	108.673	0.0349584
$214$	−31.6891	−0.0101225
$215$	−4950.84	−1.57044
$216$	92.0549	0.0289979
$217$	4149.77	1.29818
$218$	−107.893	−0.0335203
$219$	493.716	0.152339
$220$	−3610.66	−1.10650
$221$	0	0
$222$	−243.576	−0.0736384
$223$	−1382.46	−0.415141	−0.207570	−	0.978220i	$-0.566556\pi$
$223$	−1382.46	−0.415141	−0.207570	+	0.978220i	$0.566556\pi$
$224$	−1318.99	−0.393431
$225$	992.603	0.294105
$226$	242.867	0.0714835
$227$	1223.57	0.357760	0.178880	−	0.983871i	$-0.442753\pi$
$227$	1223.57	0.357760	0.178880	+	0.983871i	$0.442753\pi$
$228$	254.380	0.0738893
$229$	−4019.61	−1.15993	−0.579964	−	0.814642i	$-0.696933\pi$
$229$	−4019.61	−1.15993	−0.579964	+	0.814642i	$0.696933\pi$
$230$	−88.1082	−0.0252595
$231$	−2875.75	−0.819094
$232$	−650.468	−0.184075
$233$	1658.40	0.466290	0.233145	−	0.972442i	$-0.425098\pi$
$233$	1658.40	0.466290	0.233145	+	0.972442i	$0.425098\pi$
$234$	0	0
$235$	−3812.23	−1.05822
$236$	−2709.83	−0.747436
$237$	−981.727	−0.269072
$238$	541.297	0.147425
$239$	−618.554	−0.167410	−0.0837049	−	0.996491i	$-0.526675\pi$
$239$	−618.554	−0.167410	−0.0837049	+	0.996491i	$0.526675\pi$
$240$	−2894.77	−0.778570
$241$	−1135.25	−0.303434	−0.151717	−	0.988424i	$-0.548480\pi$
$241$	−1135.25	−0.303434	−0.151717	+	0.988424i	$0.548480\pi$
$242$	97.2938	0.0258441
$243$	243.000	0.0641500
$244$	4699.69	1.23306
$245$	−10833.9	−2.82512
$246$	−297.875	−0.0772025
$247$	0	0
$248$	436.777	0.111836
$249$	4213.44	1.07235
$250$	−48.2215	−0.0121992
$251$	2287.83	0.575326	0.287663	−	0.957732i	$-0.407122\pi$
$251$	2287.83	0.575326	0.287663	+	0.957732i	$0.407122\pi$
$252$	−2318.97	−0.579687
$253$	795.414	0.197657
$254$	274.997	0.0679326
$255$	3598.37	0.883681
$256$	3864.18	0.943403
$257$	−5113.49	−1.24113	−0.620566	−	0.784154i	$-0.713097\pi$
$257$	−5113.49	−1.24113	−0.620566	+	0.784154i	$0.713097\pi$
$258$	−206.920	−0.0499314
$259$	12307.1	2.95262
$260$	0	0
$261$	−1717.06	−0.407216
$262$	−476.526	−0.112366
$263$	2760.17	0.647145	0.323573	−	0.946203i	$-0.395116\pi$
$263$	2760.17	0.647145	0.323573	+	0.946203i	$0.395116\pi$
$264$	−302.682	−0.0705635
$265$	−11353.5	−2.63185
$266$	73.7923	0.0170094
$267$	−2210.96	−0.506773
$268$	−2710.66	−0.617834
$269$	−3310.32	−0.750312	−0.375156	−	0.926962i	$-0.622411\pi$
$269$	−3310.32	−0.750312	−0.375156	+	0.926962i	$0.622411\pi$
$270$	88.5053	0.0199491
$271$	2522.26	0.565374	0.282687	−	0.959212i	$-0.408774\pi$
$271$	2522.26	0.565374	0.282687	+	0.959212i	$0.408774\pi$
$272$	−4918.99	−1.09654
$273$	0	0
$274$	101.453	0.0223686
$275$	−3263.74	−0.715675
$276$	641.410	0.139885
$277$	−3455.48	−0.749530	−0.374765	−	0.927120i	$-0.622277\pi$
$277$	−3455.48	−0.749530	−0.374765	+	0.927120i	$0.622277\pi$
$278$	−411.804	−0.0888430
$279$	1152.97	0.247407
$280$	−1694.07	−0.361572
$281$	−981.649	−0.208400	−0.104200	−	0.994556i	$-0.533228\pi$
$281$	−981.649	−0.208400	−0.104200	+	0.994556i	$0.533228\pi$
$282$	−159.332	−0.0336457
$283$	5331.10	1.11979	0.559896	−	0.828563i	$-0.310841\pi$
$283$	5331.10	1.11979	0.559896	+	0.828563i	$0.310841\pi$
$284$	−288.140	−0.0602041
$285$	490.547	0.101956
$286$	0	0
$287$	15050.7	3.09552
$288$	−366.467	−0.0749801
$289$	1201.58	0.244572
$290$	−625.387	−0.126634
$291$	−4484.62	−0.903412
$292$	−1309.06	−0.262353
$293$	−2420.39	−0.482597	−0.241299	−	0.970451i	$-0.577573\pi$
$293$	−2420.39	−0.482597	−0.241299	+	0.970451i	$0.577573\pi$
$294$	−452.804	−0.0898233
$295$	−5225.64	−1.03135
$296$	1295.36	0.254363
$297$	−798.998	−0.156103
$298$	−227.746	−0.0442717
$299$	0	0
$300$	−2631.83	−0.506496
$301$	10455.0	2.00205
$302$	−187.605	−0.0357465
$303$	−1452.03	−0.275303
$304$	−670.581	−0.126515
$305$	9062.88	1.70144
$306$	150.394	0.0280962
$307$	−875.509	−0.162762	−0.0813810	−	0.996683i	$-0.525933\pi$
$307$	−875.509	−0.162762	−0.0813810	+	0.996683i	$0.525933\pi$
$308$	7624.90	1.41061
$309$	642.147	0.118222
$310$	419.935	0.0769377
$311$	6419.10	1.17040	0.585199	−	0.810890i	$-0.301016\pi$
$311$	6419.10	1.17040	0.585199	+	0.810890i	$0.301016\pi$
$312$	0	0
$313$	−5015.58	−0.905742	−0.452871	−	0.891576i	$-0.649600\pi$
$313$	−5015.58	−0.905742	−0.452871	+	0.891576i	$0.649600\pi$
$314$	−494.701	−0.0889096
$315$	−4471.90	−0.799882
$316$	2602.99	0.463386
$317$	5228.81	0.926433	0.463217	−	0.886245i	$-0.346695\pi$
$317$	5228.81	0.926433	0.463217	+	0.886245i	$0.346695\pi$
$318$	−474.521	−0.0836786
$319$	5645.80	0.990922
$320$	7585.92	1.32521
$321$	444.863	0.0773515
$322$	186.064	0.0322018
$323$	833.570	0.143595
$324$	−644.301	−0.110477
$325$	0	0
$326$	−70.8199	−0.0120318
$327$	1514.64	0.256146
$328$	1584.13	0.266674
$329$	8050.56	1.34906
$330$	−291.010	−0.0485442
$331$	−3186.81	−0.529193	−0.264596	−	0.964359i	$-0.585239\pi$
$331$	−3186.81	−0.529193	−0.264596	+	0.964359i	$0.585239\pi$
$332$	−11171.7	−1.84677
$333$	3419.41	0.562710
$334$	464.075	0.0760271
$335$	−5227.23	−0.852520
$336$	6113.11	0.992551
$337$	−5600.69	−0.905309	−0.452655	−	0.891686i	$-0.649523\pi$
$337$	−5600.69	−0.905309	−0.452655	+	0.891686i	$0.649523\pi$
$338$	0	0
$339$	−3409.46	−0.546243
$340$	−9540.87	−1.52184
$341$	−3791.04	−0.602042
$342$	20.5024	0.00324165
$343$	11768.0	1.85252
$344$	1100.42	0.172474
$345$	1236.90	0.193021
$346$	627.570	0.0975097
$347$	−6982.33	−1.08020	−0.540102	−	0.841599i	$-0.681614\pi$
$347$	−6982.33	−1.08020	−0.540102	+	0.841599i	$0.681614\pi$
$348$	4552.69	0.701293
$349$	2872.20	0.440531	0.220265	−	0.975440i	$-0.429308\pi$
$349$	2872.20	0.440531	0.220265	+	0.975440i	$0.429308\pi$
$350$	−763.459	−0.116596
$351$	0	0
$352$	1204.96	0.182457
$353$	−5104.55	−0.769655	−0.384827	−	0.922989i	$-0.625739\pi$
$353$	−5104.55	−0.769655	−0.384827	+	0.922989i	$0.625739\pi$
$354$	−218.406	−0.0327913
$355$	−555.650	−0.0830728
$356$	5862.23	0.872746
$357$	−7598.94	−1.12655
$358$	−345.247	−0.0509689
$359$	−10771.0	−1.58348	−0.791741	−	0.610857i	$-0.790825\pi$
$359$	−10771.0	−1.58348	−0.791741	+	0.610857i	$0.790825\pi$
$360$	−470.681	−0.0689085
$361$	−6745.36	−0.983433
$362$	−154.936	−0.0224952
$363$	−1365.85	−0.197489
$364$	0	0
$365$	−2524.39	−0.362008
$366$	378.783	0.0540965
$367$	−11380.5	−1.61868	−0.809340	−	0.587340i	$-0.800175\pi$
$367$	−11380.5	−1.61868	−0.809340	+	0.587340i	$0.800175\pi$
$368$	−1690.84	−0.239514
$369$	4181.68	0.589945
$370$	1245.41	0.174989
$371$	23976.1	3.35519
$372$	−3057.04	−0.426076
$373$	3196.75	0.443758	0.221879	−	0.975074i	$-0.428781\pi$
$373$	3196.75	0.443758	0.221879	+	0.975074i	$0.428781\pi$
$374$	−494.504	−0.0683695
$375$	676.952	0.0932204
$376$	847.346	0.116220
$377$	0	0
$378$	−186.903	−0.0254319
$379$	2050.75	0.277942	0.138971	−	0.990296i	$-0.455621\pi$
$379$	2050.75	0.277942	0.138971	+	0.990296i	$0.455621\pi$
$380$	−1300.66	−0.175585
$381$	−3860.52	−0.519108
$382$	−366.940	−0.0491473
$383$	−3507.54	−0.467955	−0.233978	−	0.972242i	$-0.575174\pi$
$383$	−3507.54	−0.467955	−0.233978	+	0.972242i	$0.575174\pi$
$384$	1294.30	0.172004
$385$	14703.9	1.94644
$386$	−93.1369	−0.0122812
$387$	2904.83	0.381552
$388$	11890.7	1.55582
$389$	−6572.66	−0.856676	−0.428338	−	0.903618i	$-0.640901\pi$
$389$	−6572.66	−0.856676	−0.428338	+	0.903618i	$0.640901\pi$
$390$	0	0
$391$	2101.81	0.271850
$392$	2408.06	0.310269
$393$	6689.65	0.858646
$394$	−148.427	−0.0189787
$395$	5019.62	0.639403
$396$	2118.50	0.268835
$397$	5285.98	0.668251	0.334126	−	0.942529i	$-0.391559\pi$
$397$	5285.98	0.668251	0.334126	+	0.942529i	$0.391559\pi$
$398$	−619.567	−0.0780304
$399$	−1035.92	−0.129978
$400$	6937.86	0.867233
$401$	919.541	0.114513	0.0572565	−	0.998360i	$-0.481765\pi$
$401$	919.541	0.114513	0.0572565	+	0.998360i	$0.481765\pi$
$402$	−218.472	−0.0271055
$403$	0	0
$404$	3849.97	0.474117
$405$	−1242.47	−0.152442
$406$	1320.67	0.161438
$407$	−11243.2	−1.36930
$408$	−799.811	−0.0970504
$409$	1539.17	0.186081	0.0930403	−	0.995662i	$-0.470341\pi$
$409$	1539.17	0.186081	0.0930403	+	0.995662i	$0.470341\pi$
$410$	1523.05	0.183458
$411$	−1424.23	−0.170930
$412$	−1702.62	−0.203597
$413$	11035.4	1.31481
$414$	51.6961	0.00613702
$415$	−21543.5	−2.54826
$416$	0	0
$417$	5781.06	0.678896
$418$	−67.4132	−0.00788825
$419$	10579.9	1.23357	0.616783	−	0.787133i	$-0.288436\pi$
$419$	10579.9	1.23357	0.616783	+	0.787133i	$0.288436\pi$
$420$	11857.0	1.37753
$421$	−74.9351	−0.00867485	−0.00433743	−	0.999991i	$-0.501381\pi$
$421$	−74.9351	−0.00867485	−0.00433743	+	0.999991i	$0.501381\pi$
$422$	−1121.99	−0.129426
$423$	2236.77	0.257105
$424$	2523.55	0.289044
$425$	−8624.15	−0.984313
$426$	−23.2234	−0.00264126
$427$	−19138.8	−2.16906
$428$	−1179.53	−0.133212
$429$	0	0
$430$	1057.99	0.118653
$431$	12165.6	1.35962	0.679810	−	0.733388i	$-0.262062\pi$
$431$	12165.6	1.35962	0.679810	+	0.733388i	$0.262062\pi$
$432$	1698.46	0.189161
$433$	−2869.23	−0.318445	−0.159222	−	0.987243i	$-0.550899\pi$
$433$	−2869.23	−0.318445	−0.159222	+	0.987243i	$0.550899\pi$
$434$	−886.806	−0.0980831
$435$	8779.41	0.967680
$436$	−4015.99	−0.441126
$437$	286.530	0.0313652
$438$	−105.507	−0.0115099
$439$	−3845.12	−0.418035	−0.209018	−	0.977912i	$-0.567027\pi$
$439$	−3845.12	−0.418035	−0.209018	+	0.977912i	$0.567027\pi$
$440$	1547.63	0.167682
$441$	6356.63	0.686387
$442$	0	0
$443$	−3858.30	−0.413799	−0.206900	−	0.978362i	$-0.566337\pi$
$443$	−3858.30	−0.413799	−0.206900	+	0.978362i	$0.566337\pi$
$444$	−9066.37	−0.969079
$445$	11304.7	1.20426
$446$	295.432	0.0313657
$447$	3197.18	0.338303
$448$	−16019.7	−1.68942
$449$	5550.51	0.583395	0.291698	−	0.956511i	$-0.405780\pi$
$449$	5550.51	0.583395	0.291698	+	0.956511i	$0.405780\pi$
$450$	−212.119	−0.0222209
$451$	−13749.6	−1.43557
$452$	9039.99	0.940719
$453$	2633.66	0.273157
$454$	−261.478	−0.0270303
$455$	0	0
$456$	−109.034	−0.0111974
$457$	−8102.42	−0.829355	−0.414677	−	0.909968i	$-0.636106\pi$
$457$	−8102.42	−0.829355	−0.414677	+	0.909968i	$0.636106\pi$
$458$	858.991	0.0876376
$459$	−2111.29	−0.214698
$460$	−3279.56	−0.332414
$461$	−9230.86	−0.932590	−0.466295	−	0.884629i	$-0.654412\pi$
$461$	−9230.86	−0.932590	−0.466295	+	0.884629i	$0.654412\pi$
$462$	614.548	0.0618861
$463$	13934.7	1.39870	0.699352	−	0.714777i	$-0.253472\pi$
$463$	13934.7	1.39870	0.699352	+	0.714777i	$0.253472\pi$
$464$	−12001.5	−1.20077
$465$	−5895.20	−0.587921
$466$	−354.400	−0.0352302
$467$	10918.0	1.08185	0.540926	−	0.841070i	$-0.318074\pi$
$467$	10918.0	1.08185	0.540926	+	0.841070i	$0.318074\pi$
$468$	0	0
$469$	11038.7	1.08682
$470$	814.673	0.0799533
$471$	6944.81	0.679405
$472$	1161.51	0.113268
$473$	−9551.23	−0.928470
$474$	209.795	0.0203295
$475$	−1175.69	−0.113567
$476$	20148.2	1.94010
$477$	6661.50	0.639432
$478$	132.185	0.0126485
$479$	11028.4	1.05198	0.525992	−	0.850490i	$-0.323694\pi$
$479$	11028.4	1.05198	0.525992	+	0.850490i	$0.323694\pi$
$480$	1873.76	0.178177
$481$	0	0
$482$	242.602	0.0229258
$483$	−2612.04	−0.246071
$484$	3621.47	0.340108
$485$	22930.0	2.14680
$486$	−51.9291	−0.00484681
$487$	1078.10	0.100315	0.0501576	−	0.998741i	$-0.484028\pi$
$487$	1078.10	0.100315	0.0501576	+	0.998741i	$0.484028\pi$
$488$	−2014.41	−0.186861
$489$	994.197	0.0919410
$490$	2315.21	0.213450
$491$	6572.88	0.604134	0.302067	−	0.953287i	$-0.402323\pi$
$491$	6572.88	0.604134	0.302067	+	0.953287i	$0.402323\pi$
$492$	−11087.5	−1.01598
$493$	14918.6	1.36288
$494$	0	0
$495$	4085.32	0.370952
$496$	8058.77	0.729535
$497$	1173.41	0.105904
$498$	−900.411	−0.0810208
$499$	9956.79	0.893241	0.446620	−	0.894724i	$-0.352627\pi$
$499$	9956.79	0.893241	0.446620	+	0.894724i	$0.352627\pi$
$500$	−1794.90	−0.160541
$501$	−6514.86	−0.580963
$502$	−488.910	−0.0434683
$503$	−11965.0	−1.06062	−0.530310	−	0.847804i	$-0.677924\pi$
$503$	−11965.0	−1.06062	−0.530310	+	0.847804i	$0.677924\pi$
$504$	993.971	0.0878472
$505$	7424.29	0.654211
$506$	−169.980	−0.0149338
$507$	0	0
$508$	10235.9	0.893990
$509$	−7160.44	−0.623538	−0.311769	−	0.950158i	$-0.600922\pi$
$509$	−7160.44	−0.623538	−0.311769	+	0.950158i	$0.600922\pi$
$510$	−768.971	−0.0667659
$511$	5330.94	0.461501
$512$	−4277.24	−0.369197
$513$	−287.821	−0.0247712
$514$	1092.75	0.0937729
$515$	−3283.33	−0.280933
$516$	−7701.98	−0.657095
$517$	−7354.61	−0.625639
$518$	−2630.03	−0.223083
$519$	−8810.06	−0.745123
$520$	0	0
$521$	−17213.1	−1.44745	−0.723725	−	0.690089i	$-0.757572\pi$
$521$	−17213.1	−1.44745	−0.723725	+	0.690089i	$0.757572\pi$
$522$	366.936	0.0307669
$523$	−10110.1	−0.845281	−0.422640	−	0.906297i	$-0.638897\pi$
$523$	−10110.1	−0.845281	−0.422640	+	0.906297i	$0.638897\pi$
$524$	−17737.2	−1.47873
$525$	10717.7	0.890971
$526$	−589.847	−0.0488946
$527$	−10017.5	−0.828026
$528$	−5584.65	−0.460304
$529$	−11444.5	−0.940620
$530$	2426.25	0.198848
$531$	3066.06	0.250576
$532$	2746.70	0.223843
$533$	0	0
$534$	472.481	0.0382889
$535$	−2274.61	−0.183813
$536$	1161.86	0.0936281
$537$	4846.71	0.389480
$538$	707.416	0.0566893
$539$	−20901.0	−1.67026
$540$	3294.34	0.262529
$541$	−5951.54	−0.472970	−0.236485	−	0.971635i	$-0.575995\pi$
$541$	−5951.54	−0.472970	−0.236485	+	0.971635i	$0.575995\pi$
$542$	−539.007	−0.0427165
$543$	2175.06	0.171898
$544$	3184.02	0.250944
$545$	−7744.43	−0.608688
$546$	0	0
$547$	5157.62	0.403152	0.201576	−	0.979473i	$-0.435394\pi$
$547$	5157.62	0.403152	0.201576	+	0.979473i	$0.435394\pi$
$548$	3776.28	0.294370
$549$	−5317.51	−0.413380
$550$	697.460	0.0540724
$551$	2033.77	0.157244
$552$	−274.926	−0.0211986
$553$	−10600.3	−0.815136
$554$	738.436	0.0566302
$555$	−17483.6	−1.33718
$556$	−15328.2	−1.16917
$557$	23314.4	1.77354	0.886772	−	0.462206i	$-0.152942\pi$
$557$	23314.4	1.77354	0.886772	+	0.462206i	$0.152942\pi$
$558$	−246.390	−0.0186927
$559$	0	0
$560$	−31256.6	−2.35863
$561$	6942.03	0.522447
$562$	209.778	0.0157455
$563$	2329.12	0.174353	0.0871764	−	0.996193i	$-0.472216\pi$
$563$	2329.12	0.174353	0.0871764	+	0.996193i	$0.472216\pi$
$564$	−5930.66	−0.442776
$565$	17432.7	1.29805
$566$	−1139.26	−0.0846051
$567$	2623.81	0.194338
$568$	123.505	0.00912348
$569$	−17446.6	−1.28541	−0.642706	−	0.766113i	$-0.722188\pi$
$569$	−17446.6	−1.28541	−0.642706	+	0.766113i	$0.722188\pi$
$570$	−104.830	−0.00770323
$571$	−413.526	−0.0303074	−0.0151537	−	0.999885i	$-0.504824\pi$
$571$	−413.526	−0.0303074	−0.0151537	+	0.999885i	$0.504824\pi$
$572$	0	0
$573$	5151.24	0.375560
$574$	−3216.33	−0.233880
$575$	−2964.45	−0.215002
$576$	−4450.92	−0.321971
$577$	21258.8	1.53382	0.766911	−	0.641754i	$-0.221793\pi$
$577$	21258.8	1.53382	0.766911	+	0.641754i	$0.221793\pi$
$578$	−256.778	−0.0184785
$579$	1307.49	0.0938471
$580$	−23278.1	−1.66650
$581$	45495.0	3.24862
$582$	958.362	0.0682567
$583$	−21903.4	−1.55600
$584$	561.098	0.0397575
$585$	0	0
$586$	517.238	0.0364623
$587$	−2965.69	−0.208530	−0.104265	−	0.994550i	$-0.533249\pi$
$587$	−2965.69	−0.208530	−0.104265	+	0.994550i	$0.533249\pi$
$588$	−16854.2	−1.18207
$589$	−1365.64	−0.0955349
$590$	1116.72	0.0779230
$591$	2083.67	0.145027
$592$	23900.2	1.65928
$593$	17786.8	1.23173	0.615865	−	0.787852i	$-0.288807\pi$
$593$	17786.8	1.23173	0.615865	+	0.787852i	$0.288807\pi$
$594$	170.746	0.0117943
$595$	38853.7	2.67705
$596$	−8477.16	−0.582614
$597$	8697.72	0.596271
$598$	0	0
$599$	21796.4	1.48677	0.743387	−	0.668861i	$-0.233218\pi$
$599$	21796.4	1.48677	0.743387	+	0.668861i	$0.233218\pi$
$600$	1128.07	0.0767557
$601$	20468.7	1.38924	0.694622	−	0.719375i	$-0.255572\pi$
$601$	20468.7	1.38924	0.694622	+	0.719375i	$0.255572\pi$
$602$	−2234.24	−0.151264
$603$	3066.99	0.207127
$604$	−6983.01	−0.470422
$605$	6983.64	0.469298
$606$	310.298	0.0208004
$607$	7852.53	0.525081	0.262541	−	0.964921i	$-0.415440\pi$
$607$	7852.53	0.525081	0.262541	+	0.964921i	$0.415440\pi$
$608$	434.061	0.0289531
$609$	−18540.1	−1.23364
$610$	−1936.74	−0.128551
$611$	0	0
$612$	5597.96	0.369745
$613$	2460.21	0.162099	0.0810497	−	0.996710i	$-0.474173\pi$
$613$	2460.21	0.162099	0.0810497	+	0.996710i	$0.474173\pi$
$614$	187.096	0.0122974
$615$	−21381.1	−1.40190
$616$	−3268.23	−0.213768
$617$	17829.3	1.16334	0.581670	−	0.813425i	$-0.302400\pi$
$617$	17829.3	1.16334	0.581670	+	0.813425i	$0.302400\pi$
$618$	−137.227	−0.00893215
$619$	−16901.7	−1.09748	−0.548739	−	0.835994i	$-0.684892\pi$
$619$	−16901.7	−1.09748	−0.548739	+	0.835994i	$0.684892\pi$
$620$	15630.8	1.01250
$621$	−725.729	−0.0468962
$622$	−1371.76	−0.0884286
$623$	−23873.0	−1.53524
$624$	0	0
$625$	−17247.4	−1.10384
$626$	1071.83	0.0684327
$627$	946.372	0.0602783
$628$	−18413.8	−1.17005
$629$	−29709.3	−1.88328
$630$	955.644	0.0604345
$631$	1833.82	0.115695	0.0578473	−	0.998325i	$-0.481576\pi$
$631$	1833.82	0.115695	0.0578473	+	0.998325i	$0.481576\pi$
$632$	−1115.71	−0.0702226
$633$	15750.9	0.989009
$634$	−1117.40	−0.0699960
$635$	19739.0	1.23357
$636$	−17662.6	−1.10121
$637$	0	0
$638$	−1206.51	−0.0748684
$639$	326.019	0.0201833
$640$	−6617.81	−0.408737
$641$	29172.1	1.79755	0.898773	−	0.438414i	$-0.144460\pi$
$641$	29172.1	1.79755	0.898773	+	0.438414i	$0.144460\pi$
$642$	−95.0672	−0.00584424
$643$	25103.8	1.53965	0.769826	−	0.638254i	$-0.220343\pi$
$643$	25103.8	1.53965	0.769826	+	0.638254i	$0.220343\pi$
$644$	6925.69	0.423774
$645$	−14852.5	−0.906693
$646$	−178.134	−0.0108492
$647$	−24471.6	−1.48698	−0.743492	−	0.668745i	$-0.766832\pi$
$647$	−24471.6	−1.48698	−0.743492	+	0.668745i	$0.766832\pi$
$648$	276.165	0.0167419
$649$	−10081.4	−0.609752
$650$	0	0
$651$	12449.3	0.749505
$652$	−2636.06	−0.158337
$653$	6632.68	0.397484	0.198742	−	0.980052i	$-0.436314\pi$
$653$	6632.68	0.397484	0.198742	+	0.980052i	$0.436314\pi$
$654$	−323.679	−0.0193530
$655$	−34204.5	−2.04043
$656$	29228.1	1.73958
$657$	1481.15	0.0879529
$658$	−1720.40	−0.101928
$659$	26436.9	1.56272	0.781361	−	0.624080i	$-0.214526\pi$
$659$	26436.9	1.56272	0.781361	+	0.624080i	$0.214526\pi$
$660$	−10832.0	−0.638840
$661$	27596.7	1.62389	0.811943	−	0.583737i	$-0.198410\pi$
$661$	27596.7	1.62389	0.811943	+	0.583737i	$0.198410\pi$
$662$	681.020	0.0399828
$663$	0	0
$664$	4788.49	0.279863
$665$	5296.73	0.308870
$666$	−730.727	−0.0425152
$667$	5128.07	0.297691
$668$	17273.8	1.00051
$669$	−4147.38	−0.239682
$670$	1117.06	0.0644115
$671$	17484.3	1.00592
$672$	−3956.96	−0.227147
$673$	18860.1	1.08024	0.540122	−	0.841587i	$-0.318378\pi$
$673$	18860.1	1.08024	0.540122	+	0.841587i	$0.318378\pi$
$674$	1196.87	0.0684000
$675$	2977.81	0.169801
$676$	0	0
$677$	−14705.5	−0.834825	−0.417412	−	0.908717i	$-0.637063\pi$
$677$	−14705.5	−0.834825	−0.417412	+	0.908717i	$0.637063\pi$
$678$	728.601	0.0412710
$679$	−48423.1	−2.73683
$680$	4089.47	0.230624
$681$	3670.72	0.206553
$682$	810.145	0.0454869
$683$	−17950.2	−1.00563	−0.502814	−	0.864395i	$-0.667702\pi$
$683$	−17950.2	−1.00563	−0.502814	+	0.864395i	$0.667702\pi$
$684$	763.141	0.0426600
$685$	7282.18	0.406186
$686$	−2514.83	−0.139966
$687$	−12058.8	−0.669685
$688$	20303.5	1.12509
$689$	0	0
$690$	−264.325	−0.0145836
$691$	−8692.38	−0.478544	−0.239272	−	0.970953i	$-0.576909\pi$
$691$	−8692.38	−0.478544	−0.239272	+	0.970953i	$0.576909\pi$
$692$	23359.4	1.28322
$693$	−8627.26	−0.472904
$694$	1492.12	0.0816141
$695$	−29558.8	−1.61328
$696$	−1951.41	−0.106276
$697$	−36332.2	−1.97443
$698$	−613.789	−0.0332840
$699$	4975.20	0.269212
$700$	−28417.4	−1.53440
$701$	23275.9	1.25409	0.627047	−	0.778982i	$-0.284264\pi$
$701$	23275.9	1.25409	0.627047	+	0.778982i	$0.284264\pi$
$702$	0	0
$703$	−4050.11	−0.217287
$704$	14634.9	0.783485
$705$	−11436.7	−0.610965
$706$	1090.84	0.0581507
$707$	−15678.4	−0.834014
$708$	−8129.49	−0.431532
$709$	−34164.7	−1.80971	−0.904854	−	0.425722i	$-0.860020\pi$
$709$	−34164.7	−1.80971	−0.904854	+	0.425722i	$0.860020\pi$
$710$	118.742	0.00627651
$711$	−2945.18	−0.155349
$712$	−2512.71	−0.132258
$713$	−3443.40	−0.180864
$714$	1623.89	0.0851157
$715$	0	0
$716$	−12850.8	−0.670748
$717$	−1855.66	−0.0966541
$718$	2301.75	0.119639
$719$	−8391.74	−0.435270	−0.217635	−	0.976030i	$-0.569834\pi$
$719$	−8391.74	−0.435270	−0.217635	+	0.976030i	$0.569834\pi$
$720$	−8684.32	−0.449508
$721$	6933.64	0.358145
$722$	1441.48	0.0743026
$723$	−3405.74	−0.175188
$724$	−5767.04	−0.296036
$725$	−21041.5	−1.07788
$726$	291.881	0.0149211
$727$	23500.4	1.19887	0.599436	−	0.800423i	$-0.295392\pi$
$727$	23500.4	1.19887	0.599436	+	0.800423i	$0.295392\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	539.462	0.0273512
$731$	−25238.3	−1.27698
$732$	14099.1	0.711908
$733$	1289.07	0.0649563	0.0324781	−	0.999472i	$-0.489660\pi$
$733$	1289.07	0.0649563	0.0324781	+	0.999472i	$0.489660\pi$
$734$	2432.01	0.122298
$735$	−32501.7	−1.63108
$736$	1094.47	0.0548133
$737$	−10084.5	−0.504024
$738$	−893.624	−0.0445729
$739$	−18197.8	−0.905844	−0.452922	−	0.891550i	$-0.649618\pi$
$739$	−18197.8	−0.905844	−0.452922	+	0.891550i	$0.649618\pi$
$740$	46356.8	2.30285
$741$	0	0
$742$	−5123.68	−0.253499
$743$	21277.2	1.05059	0.525294	−	0.850921i	$-0.323956\pi$
$743$	21277.2	1.05059	0.525294	+	0.850921i	$0.323956\pi$
$744$	1310.33	0.0645686
$745$	−16347.4	−0.803920
$746$	−683.146	−0.0335278
$747$	12640.3	0.619123
$748$	−18406.4	−0.899739
$749$	4803.45	0.234331
$750$	−144.665	−0.00704321
$751$	−25653.4	−1.24648	−0.623240	−	0.782030i	$-0.714184\pi$
$751$	−25653.4	−1.24648	−0.623240	+	0.782030i	$0.714184\pi$
$752$	15634.0	0.758130
$753$	6863.50	0.332164
$754$	0	0
$755$	−13466.0	−0.649112
$756$	−6956.90	−0.334682
$757$	26521.2	1.27336	0.636678	−	0.771130i	$-0.280308\pi$
$757$	26521.2	1.27336	0.636678	+	0.771130i	$0.280308\pi$
$758$	−438.245	−0.0209997
$759$	2386.24	0.114117
$760$	557.497	0.0266086
$761$	21607.2	1.02925	0.514625	−	0.857415i	$-0.327931\pi$
$761$	21607.2	1.02925	0.514625	+	0.857415i	$0.327931\pi$
$762$	824.992	0.0392209
$763$	16354.5	0.775979
$764$	−13658.2	−0.646776
$765$	10795.1	0.510193
$766$	749.560	0.0353560
$767$	0	0
$768$	11592.5	0.544674
$769$	−17179.2	−0.805591	−0.402795	−	0.915290i	$-0.631961\pi$
$769$	−17179.2	−0.805591	−0.402795	+	0.915290i	$0.631961\pi$
$770$	−3142.21	−0.147062
$771$	−15340.5	−0.716568
$772$	−3466.74	−0.161620
$773$	−12822.1	−0.596609	−0.298304	−	0.954471i	$-0.596421\pi$
$773$	−12822.1	−0.596609	−0.298304	+	0.954471i	$0.596421\pi$
$774$	−620.761	−0.0288279
$775$	14128.9	0.654873
$776$	−5096.67	−0.235773
$777$	36921.4	1.70469
$778$	1404.58	0.0647256
$779$	−4952.98	−0.227804
$780$	0	0
$781$	−1071.97	−0.0491140
$782$	−449.157	−0.0205394
$783$	−5151.18	−0.235106
$784$	44430.0	2.02396
$785$	−35509.1	−1.61449
$786$	−1429.58	−0.0648744
$787$	−35213.4	−1.59495	−0.797474	−	0.603354i	$-0.793831\pi$
$787$	−35213.4	−1.59495	−0.797474	+	0.603354i	$0.793831\pi$
$788$	−5524.73	−0.249759
$789$	8280.50	0.373629
$790$	−1072.69	−0.0483097
$791$	−36813.9	−1.65481
$792$	−908.045	−0.0407399
$793$	0	0
$794$	−1129.61	−0.0504893
$795$	−34060.6	−1.51950
$796$	−23061.5	−1.02688
$797$	7500.15	0.333336	0.166668	−	0.986013i	$-0.446699\pi$
$797$	7500.15	0.333336	0.166668	+	0.986013i	$0.446699\pi$
$798$	221.377	0.00982038
$799$	−19434.0	−0.860481
$800$	−4490.82	−0.198468
$801$	−6632.87	−0.292585
$802$	−196.506	−0.00865195
$803$	−4870.10	−0.214025
$804$	−8131.97	−0.356707
$805$	13355.5	0.584745
$806$	0	0
$807$	−9930.97	−0.433193
$808$	−1650.20	−0.0718489
$809$	−22020.5	−0.956982	−0.478491	−	0.878092i	$-0.658816\pi$
$809$	−22020.5	−0.956982	−0.478491	+	0.878092i	$0.658816\pi$
$810$	265.516	0.0115176
$811$	20444.6	0.885210	0.442605	−	0.896717i	$-0.354054\pi$
$811$	20444.6	0.885210	0.442605	+	0.896717i	$0.354054\pi$
$812$	49158.1	2.12452
$813$	7566.78	0.326419
$814$	2402.67	0.103457
$815$	−5083.37	−0.218482
$816$	−14757.0	−0.633085
$817$	−3440.62	−0.147334
$818$	−328.920	−0.0140592
$819$	0	0
$820$	56690.8	2.41431
$821$	40353.0	1.71538	0.857692	−	0.514164i	$-0.171898\pi$
$821$	40353.0	1.71538	0.857692	+	0.514164i	$0.171898\pi$
$822$	304.359	0.0129145
$823$	−33110.7	−1.40239	−0.701194	−	0.712971i	$-0.747349\pi$
$823$	−33110.7	−1.40239	−0.701194	+	0.712971i	$0.747349\pi$
$824$	729.787	0.0308536
$825$	−9791.21	−0.413195
$826$	−2358.26	−0.0993392
$827$	−43410.4	−1.82531	−0.912653	−	0.408735i	$-0.865970\pi$
$827$	−43410.4	−1.82531	−0.912653	+	0.408735i	$0.865970\pi$
$828$	1924.23	0.0807629
$829$	−5502.87	−0.230546	−0.115273	−	0.993334i	$-0.536774\pi$
$829$	−5502.87	−0.230546	−0.115273	+	0.993334i	$0.536774\pi$
$830$	4603.84	0.192532
$831$	−10366.4	−0.432741
$832$	0	0
$833$	−55229.1	−2.29721
$834$	−1235.41	−0.0512935
$835$	33310.8	1.38056
$836$	−2509.25	−0.103809
$837$	3458.92	0.142841
$838$	−2260.93	−0.0932013
$839$	−1698.27	−0.0698819	−0.0349409	−	0.999389i	$-0.511124\pi$
$839$	−1698.27	−0.0698819	−0.0349409	+	0.999389i	$0.511124\pi$
$840$	−5082.22	−0.208754
$841$	12009.8	0.492425
$842$	16.0136	0.000655423 0
$843$	−2944.95	−0.120320
$844$	−41762.7	−1.70323
$845$	0	0
$846$	−477.996	−0.0194254
$847$	−14747.9	−0.598279
$848$	46561.0	1.88551
$849$	15993.3	0.646512
$850$	1842.98	0.0743691
$851$	−10212.2	−0.411363
$852$	−864.421	−0.0347589
$853$	18686.8	0.750087	0.375044	−	0.927007i	$-0.377628\pi$
$853$	18686.8	0.750087	0.375044	+	0.927007i	$0.377628\pi$
$854$	4089.95	0.163882
$855$	1471.64	0.0588645
$856$	505.578	0.0201873
$857$	−9659.16	−0.385007	−0.192503	−	0.981296i	$-0.561661\pi$
$857$	−9659.16	−0.385007	−0.192503	+	0.981296i	$0.561661\pi$
$858$	0	0
$859$	−1559.90	−0.0619595	−0.0309798	−	0.999520i	$-0.509863\pi$
$859$	−1559.90	−0.0619595	−0.0309798	+	0.999520i	$0.509863\pi$
$860$	39380.6	1.56147
$861$	45152.1	1.78720
$862$	−2599.79	−0.102725
$863$	−11358.7	−0.448034	−0.224017	−	0.974585i	$-0.571917\pi$
$863$	−11358.7	−0.448034	−0.224017	+	0.974585i	$0.571917\pi$
$864$	−1099.40	−0.0432898
$865$	45046.3	1.77066
$866$	613.155	0.0240599
$867$	3604.74	0.141204
$868$	−33008.7	−1.29077
$869$	9683.93	0.378026
$870$	−1876.16	−0.0731124
$871$	0	0
$872$	1721.36	0.0668493
$873$	−13453.8	−0.521585
$874$	−61.2314	−0.00236977
$875$	7309.45	0.282405
$876$	−3927.18	−0.151469
$877$	−33301.6	−1.28223	−0.641115	−	0.767444i	$-0.721528\pi$
$877$	−33301.6	−1.28223	−0.641115	+	0.767444i	$0.721528\pi$
$878$	821.701	0.0315844
$879$	−7261.18	−0.278628
$880$	28554.6	1.09383
$881$	28912.3	1.10565	0.552826	−	0.833296i	$-0.313549\pi$
$881$	28912.3	1.10565	0.552826	+	0.833296i	$0.313549\pi$
$882$	−1358.41	−0.0518595
$883$	−49100.9	−1.87132	−0.935661	−	0.352901i	$-0.885195\pi$
$883$	−49100.9	−1.87132	−0.935661	+	0.352901i	$0.885195\pi$
$884$	0	0
$885$	−15676.9	−0.595451
$886$	824.517	0.0312643
$887$	4566.49	0.172861	0.0864304	−	0.996258i	$-0.472454\pi$
$887$	4566.49	0.172861	0.0864304	+	0.996258i	$0.472454\pi$
$888$	3886.09	0.146857
$889$	−41684.3	−1.57261
$890$	−2415.82	−0.0909870
$891$	−2396.99	−0.0901261
$892$	10996.5	0.412771
$893$	−2649.33	−0.0992795
$894$	−683.238	−0.0255603
$895$	−24781.4	−0.925533
$896$	13975.3	0.521074
$897$	0	0
$898$	−1186.14	−0.0440780
$899$	−24441.0	−0.906734
$900$	−7895.49	−0.292426
$901$	−57877.9	−2.14006
$902$	2938.29	0.108464
$903$	31365.1	1.15589
$904$	−3874.78	−0.142559
$905$	−11121.2	−0.408486
$906$	−562.814	−0.0206382
$907$	48873.3	1.78921	0.894604	−	0.446860i	$-0.147458\pi$
$907$	48873.3	1.78921	0.894604	+	0.446860i	$0.147458\pi$
$908$	−9732.72	−0.355718
$909$	−4356.09	−0.158946
$910$	0	0
$911$	31550.1	1.14742	0.573710	−	0.819058i	$-0.305503\pi$
$911$	31550.1	1.14742	0.573710	+	0.819058i	$0.305503\pi$
$912$	−2011.74	−0.0730433
$913$	−41562.1	−1.50658
$914$	1731.49	0.0626613
$915$	27188.7	0.982327
$916$	31973.3	1.15331
$917$	72232.1	2.60121
$918$	451.182	0.0162214
$919$	−11720.3	−0.420694	−0.210347	−	0.977627i	$-0.567459\pi$
$919$	−11720.3	−0.420694	−0.210347	+	0.977627i	$0.567459\pi$
$920$	1405.71	0.0503748
$921$	−2626.53	−0.0939707
$922$	1972.63	0.0704612
$923$	0	0
$924$	22874.7	0.814418
$925$	41902.7	1.48946
$926$	−2977.84	−0.105678
$927$	1926.44	0.0682553
$928$	7768.47	0.274798
$929$	−6911.23	−0.244080	−0.122040	−	0.992525i	$-0.538944\pi$
$929$	−6911.23	−0.244080	−0.122040	+	0.992525i	$0.538944\pi$
$930$	1259.80	0.0444200
$931$	−7529.10	−0.265044
$932$	−13191.5	−0.463628
$933$	19257.3	0.675730
$934$	−2333.18	−0.0817386
$935$	−35494.9	−1.24151
$936$	0	0
$937$	−28673.9	−0.999717	−0.499858	−	0.866107i	$-0.666615\pi$
$937$	−28673.9	−0.999717	−0.499858	+	0.866107i	$0.666615\pi$
$938$	−2358.97	−0.0821143
$939$	−15046.7	−0.522930
$940$	30323.7	1.05218
$941$	22463.6	0.778207	0.389103	−	0.921194i	$-0.372785\pi$
$941$	22463.6	0.778207	0.389103	+	0.921194i	$0.372785\pi$
$942$	−1484.10	−0.0513320
$943$	−12488.8	−0.431272
$944$	21430.4	0.738878
$945$	−13415.7	−0.461812
$946$	2041.10	0.0701499
$947$	36806.6	1.26299	0.631497	−	0.775379i	$-0.282441\pi$
$947$	36806.6	1.26299	0.631497	+	0.775379i	$0.282441\pi$
$948$	7808.98	0.267536
$949$	0	0
$950$	251.244	0.00858046
$951$	15686.4	0.534876
$952$	−8636.04	−0.294008
$953$	−1781.07	−0.0605400	−0.0302700	−	0.999542i	$-0.509637\pi$
$953$	−1781.07	−0.0605400	−0.0302700	+	0.999542i	$0.509637\pi$
$954$	−1423.56	−0.0483119
$955$	−26338.5	−0.892455
$956$	4920.19	0.166454
$957$	16937.4	0.572109
$958$	−2356.77	−0.0794819
$959$	−15378.3	−0.517822
$960$	22757.8	0.765108
$961$	−13379.3	−0.449107
$962$	0	0
$963$	1334.59	0.0446589
$964$	9030.12	0.301702
$965$	−6685.26	−0.223012
$966$	558.193	0.0185917
$967$	54777.2	1.82163	0.910815	−	0.412815i	$-0.135455\pi$
$967$	54777.2	1.82163	0.910815	+	0.412815i	$0.135455\pi$
$968$	−1552.26	−0.0515408
$969$	2500.71	0.0829044
$970$	−4900.15	−0.162200
$971$	−2391.54	−0.0790403	−0.0395201	−	0.999219i	$-0.512583\pi$
$971$	−2391.54	−0.0790403	−0.0395201	+	0.999219i	$0.512583\pi$
$972$	−1932.90	−0.0637838
$973$	62421.5	2.05667
$974$	−230.391	−0.00757925
$975$	0	0
$976$	−37167.0	−1.21894
$977$	−43738.5	−1.43226	−0.716130	−	0.697967i	$-0.754088\pi$
$977$	−43738.5	−1.43226	−0.716130	+	0.697967i	$0.754088\pi$
$978$	−212.460	−0.00694654
$979$	21809.3	0.711979
$980$	86176.5	2.80899
$981$	4543.92	0.147886
$982$	−1404.62	−0.0456449
$983$	27712.4	0.899174	0.449587	−	0.893236i	$-0.351571\pi$
$983$	27712.4	0.899174	0.449587	+	0.893236i	$0.351571\pi$
$984$	4752.39	0.153964
$985$	−10653.9	−0.344631
$986$	−3188.09	−0.102971
$987$	24151.7	0.778882
$988$	0	0
$989$	−8675.38	−0.278929
$990$	−873.031	−0.0280270
$991$	3550.23	0.113801	0.0569005	−	0.998380i	$-0.481878\pi$
$991$	3550.23	0.113801	0.0569005	+	0.998380i	$0.481878\pi$
$992$	−5216.37	−0.166956
$993$	−9560.42	−0.305529
$994$	−250.757	−0.00800153
$995$	−44471.8	−1.41694
$996$	−33515.1	−1.06623
$997$	−28338.4	−0.900188	−0.450094	−	0.892981i	$-0.648610\pi$
$997$	−28338.4	−0.900188	−0.450094	+	0.892981i	$0.648610\pi$
$998$	−2127.76	−0.0674882
$999$	10258.2	0.324881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.p.1.4	yes	9
3.2	odd	2		1521.4.a.bf.1.6		9
13.5	odd	4		507.4.b.k.337.10		18
13.8	odd	4		507.4.b.k.337.9		18
13.12	even	2		507.4.a.o.1.6	✓	9
39.38	odd	2		1521.4.a.bi.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.6	✓	9	13.12	even	2
507.4.a.p.1.4	yes	9	1.1	even	1	trivial
507.4.b.k.337.9		18	13.8	odd	4
507.4.b.k.337.10		18	13.5	odd	4
1521.4.a.bf.1.6		9	3.2	odd	2
1521.4.a.bi.1.4		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)

\(f(q)\)	\(=\)	\(q-0.213700 q^{2} +3.00000 q^{3} -7.95433 q^{4} -15.3391 q^{5} -0.641100 q^{6} +32.3928 q^{7} +3.40944 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-0.213700 q^{2} +3.00000 q^{3} -7.95433 q^{4} -15.3391 q^{5} -0.641100 q^{6} +32.3928 q^{7} +3.40944 q^{8} +9.00000 q^{9} +3.27797 q^{10} -29.5925 q^{11} -23.8630 q^{12} -6.92233 q^{14} -46.0174 q^{15} +62.9061 q^{16} -78.1958 q^{17} -1.92330 q^{18} -10.6600 q^{19} +122.013 q^{20} +97.1783 q^{21} +6.32392 q^{22} -26.8789 q^{23} +10.2283 q^{24} +110.289 q^{25} +27.0000 q^{27} -257.663 q^{28} -190.785 q^{29} +9.83392 q^{30} +128.108 q^{31} -40.7185 q^{32} -88.7776 q^{33} +16.7104 q^{34} -496.877 q^{35} -71.5890 q^{36} +379.934 q^{37} +2.27805 q^{38} -52.2979 q^{40} +464.631 q^{41} -20.7670 q^{42} +322.758 q^{43} +235.389 q^{44} -138.052 q^{45} +5.74401 q^{46} +248.529 q^{47} +188.718 q^{48} +706.292 q^{49} -23.5688 q^{50} -234.587 q^{51} +740.167 q^{53} -5.76990 q^{54} +453.924 q^{55} +110.441 q^{56} -31.9801 q^{57} +40.7706 q^{58} +340.673 q^{59} +366.038 q^{60} -590.834 q^{61} -27.3767 q^{62} +291.535 q^{63} -494.547 q^{64} +18.9718 q^{66} +340.777 q^{67} +621.995 q^{68} -80.6366 q^{69} +106.183 q^{70} +36.2243 q^{71} +30.6850 q^{72} +164.572 q^{73} -81.1919 q^{74} +330.868 q^{75} +84.7935 q^{76} -958.584 q^{77} -327.242 q^{79} -964.925 q^{80} +81.0000 q^{81} -99.2916 q^{82} +1404.48 q^{83} -772.989 q^{84} +1199.46 q^{85} -68.9734 q^{86} -572.354 q^{87} -100.894 q^{88} -736.986 q^{89} +29.5018 q^{90} +213.803 q^{92} +384.324 q^{93} -53.1107 q^{94} +163.516 q^{95} -122.156 q^{96} -1494.87 q^{97} -150.935 q^{98} -266.333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9}+O(q^{10}) \) 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9	\( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9} - 54 q^{10} + 85 q^{11} + 132 q^{12} + 158 q^{14} + 99 q^{15} + 216 q^{16} + 178 q^{17} + 54 q^{18} + 352 q^{19} + 402 q^{20} + 249 q^{21} - 630 q^{22} + 150 q^{23} + 261 q^{24} - 20 q^{25} + 243 q^{27} + 940 q^{28} - 97 q^{29} - 162 q^{30} + 717 q^{31} + 707 q^{32} + 255 q^{33} + 632 q^{34} - 418 q^{35} + 396 q^{36} + 1108 q^{37} - 660 q^{38} - 1506 q^{40} + 334 q^{41} + 474 q^{42} + 242 q^{43} - 307 q^{44} + 297 q^{45} + 979 q^{46} - 184 q^{47} + 648 q^{48} - 38 q^{49} - 2031 q^{50} + 534 q^{51} - 151 q^{53} + 162 q^{54} + 2064 q^{55} + 2276 q^{56} + 1056 q^{57} + 1161 q^{58} + 537 q^{59} + 1206 q^{60} - 1340 q^{61} + 347 q^{62} + 747 q^{63} + 893 q^{64} - 1890 q^{66} + 2308 q^{67} + 2785 q^{68} + 450 q^{69} - 1420 q^{70} + 96 q^{71} + 783 q^{72} + 2505 q^{73} - 1191 q^{74} - 60 q^{75} + 2409 q^{76} - 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 729 q^{81} + 1517 q^{82} + 1539 q^{83} + 2820 q^{84} + 4296 q^{85} - 3763 q^{86} - 291 q^{87} - 3716 q^{88} - 592 q^{89} - 486 q^{90} + 515 q^{92} + 2151 q^{93} - 692 q^{94} + 4158 q^{95} + 2121 q^{96} + 1445 q^{97} + 1457 q^{98} + 765 q^{99}+O(q^{100}) \) 9 * q + 6 * q^2 + 27 * q^3 + 44 * q^4 + 33 * q^5 + 18 * q^6 + 83 * q^7 + 87 * q^8 + 81 * q^9 - 54 * q^10 + 85 * q^11 + 132 * q^12 + 158 * q^14 + 99 * q^15 + 216 * q^16 + 178 * q^17 + 54 * q^18 + 352 * q^19 + 402 * q^20 + 249 * q^21 - 630 * q^22 + 150 * q^23 + 261 * q^24 - 20 * q^25 + 243 * q^27 + 940 * q^28 - 97 * q^29 - 162 * q^30 + 717 * q^31 + 707 * q^32 + 255 * q^33 + 632 * q^34 - 418 * q^35 + 396 * q^36 + 1108 * q^37 - 660 * q^38 - 1506 * q^40 + 334 * q^41 + 474 * q^42 + 242 * q^43 - 307 * q^44 + 297 * q^45 + 979 * q^46 - 184 * q^47 + 648 * q^48 - 38 * q^49 - 2031 * q^50 + 534 * q^51 - 151 * q^53 + 162 * q^54 + 2064 * q^55 + 2276 * q^56 + 1056 * q^57 + 1161 * q^58 + 537 * q^59 + 1206 * q^60 - 1340 * q^61 + 347 * q^62 + 747 * q^63 + 893 * q^64 - 1890 * q^66 + 2308 * q^67 + 2785 * q^68 + 450 * q^69 - 1420 * q^70 + 96 * q^71 + 783 * q^72 + 2505 * q^73 - 1191 * q^74 - 60 * q^75 + 2409 * q^76 - 2142 * q^77 - 1591 * q^79 - 2671 * q^80 + 729 * q^81 + 1517 * q^82 + 1539 * q^83 + 2820 * q^84 + 4296 * q^85 - 3763 * q^86 - 291 * q^87 - 3716 * q^88 - 592 * q^89 - 486 * q^90 + 515 * q^92 + 2151 * q^93 - 692 * q^94 + 4158 * q^95 + 2121 * q^96 + 1445 * q^97 + 1457 * q^98 + 765 * q^99

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