LMFDB - Embedded newform 507.4.b.e.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 17x^{2} + 289 $ x^4 - 17*x^2 + 289
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$-3.57071 - 2.06155i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.e.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} +13.4424i q^{5} +12.3693i q^{6} +31.4219i q^{7} +4.12311i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} +13.4424i q^{5} +12.3693i q^{6} +31.4219i q^{7} +4.12311i q^{8} +9.00000 q^{9} +55.4243 q^{10} -40.4962i q^{11} +27.0000 q^{12} +129.556 q^{14} -40.3271i q^{15} -55.0000 q^{16} +43.1414 q^{17} -37.1080i q^{18} -26.9779i q^{19} -120.981i q^{20} -94.2656i q^{21} -166.970 q^{22} +19.0100 q^{23} -12.3693i q^{24} -55.6971 q^{25} -27.0000 q^{27} -282.797i q^{28} -154.111 q^{29} -166.273 q^{30} +308.270i q^{31} +259.756i q^{32} +121.489i q^{33} -177.877i q^{34} -422.384 q^{35} -81.0000 q^{36} -43.5116i q^{37} -111.233 q^{38} -55.4243 q^{40} +47.8384i q^{41} -388.667 q^{42} -342.121 q^{43} +364.466i q^{44} +120.981i q^{45} -78.3802i q^{46} -133.468i q^{47} +165.000 q^{48} -644.334 q^{49} +229.645i q^{50} -129.424 q^{51} -438.454 q^{53} +111.324i q^{54} +544.364 q^{55} -129.556 q^{56} +80.9338i q^{57} +635.418i q^{58} -590.553i q^{59} +362.944i q^{60} -541.304 q^{61} +1271.03 q^{62} +282.797i q^{63} +631.000 q^{64} +500.910 q^{66} -230.345i q^{67} -388.273 q^{68} -57.0300 q^{69} +1741.54i q^{70} +449.412i q^{71} +37.1080i q^{72} +389.711i q^{73} -179.403 q^{74} +167.091 q^{75} +242.801i q^{76} +1272.47 q^{77} -897.820 q^{79} -739.330i q^{80} +81.0000 q^{81} +197.243 q^{82} -1300.24i q^{83} +848.391i q^{84} +579.923i q^{85} +1410.60i q^{86} +462.334 q^{87} +166.970 q^{88} +925.045i q^{89} +498.819 q^{90} -171.090 q^{92} -924.811i q^{93} -550.301 q^{94} +362.647 q^{95} -779.267i q^{96} -1560.49i q^{97} +2656.66i q^{98} -364.466i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} - 36 q^{4} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 - 36 * q^4 + 36 * q^9	$ 4 q - 12 q^{3} - 36 q^{4} + 36 q^{9} + 136 q^{10} + 108 q^{12} + 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} + 120 q^{25} - 108 q^{27} + 12 q^{29} - 408 q^{30} - 804 q^{35} - 324 q^{36} + 612 q^{38} - 136 q^{40} - 612 q^{42} - 940 q^{43} + 660 q^{48} - 692 q^{49} - 432 q^{51} - 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} + 2524 q^{64} + 204 q^{66} - 1296 q^{68} - 828 q^{69} - 3060 q^{74} - 360 q^{75} + 2976 q^{77} + 8 q^{79} + 324 q^{81} - 68 q^{82} - 36 q^{87} + 68 q^{88} + 1224 q^{90} - 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100}) $ 4 * q - 12 * q^3 - 36 * q^4 + 36 * q^9 + 136 * q^10 + 108 * q^12 + 204 * q^14 - 220 * q^16 + 144 * q^17 - 68 * q^22 + 276 * q^23 + 120 * q^25 - 108 * q^27 + 12 * q^29 - 408 * q^30 - 804 * q^35 - 324 * q^36 + 612 * q^38 - 136 * q^40 - 612 * q^42 - 940 * q^43 + 660 * q^48 - 692 * q^49 - 432 * q^51 - 2268 * q^53 + 892 * q^55 - 204 * q^56 + 320 * q^61 + 2856 * q^62 + 2524 * q^64 + 204 * q^66 - 1296 * q^68 - 828 * q^69 - 3060 * q^74 - 360 * q^75 + 2976 * q^77 + 8 * q^79 + 324 * q^81 - 68 * q^82 - 36 * q^87 + 68 * q^88 + 1224 * q^90 - 2484 * q^92 - 5372 * q^94 + 108 * q^95

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	− 4.12311i	− 1.45774i	−0.684653	−	0.728869i	$-0.740046\pi$
$2$	− 4.12311i	− 1.45774i	0.684653	−	0.728869i	$-0.259954\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−9.00000	−1.12500
$5$	13.4424i	1.20232i	0.799128	+	0.601161i	$0.205295\pi$
$5$	13.4424i	1.20232i	−0.799128	+	0.601161i	$0.794705\pi$
$6$	12.3693i	0.841625i
$7$	31.4219i	1.69662i	0.529498	+	0.848311i	$0.322380\pi$
$7$	31.4219i	1.69662i	−0.529498	+	0.848311i	$0.677620\pi$
$8$	4.12311i	0.182217i
$9$	9.00000	0.333333
$10$	55.4243	1.75267
$11$	− 40.4962i	− 1.11001i	−0.831849	−	0.555003i	$-0.812717\pi$
$11$	− 40.4962i	− 1.11001i	0.831849	−	0.555003i	$-0.187283\pi$
$12$	27.0000	0.649519
$13$	0	0
$13$	0	0
$14$	129.556	2.47323
$15$	− 40.3271i	− 0.694161i
$16$	−55.0000	−0.859375
$17$	43.1414	0.615490	0.307745	−	0.951469i	$-0.400426\pi$
$17$	43.1414	0.615490	0.307745	+	0.951469i	$0.400426\pi$
$18$	− 37.1080i	− 0.485913i
$19$	− 26.9779i	− 0.325745i	−0.986647	−	0.162873i	$-0.947924\pi$
$19$	− 26.9779i	− 0.325745i	0.986647	−	0.162873i	$-0.0520760\pi$
$20$	− 120.981i	− 1.35261i
$21$	− 94.2656i	− 0.979545i
$22$	−166.970	−1.61810
$23$	19.0100	0.172342	0.0861709	−	0.996280i	$-0.472537\pi$
$23$	19.0100	0.172342	0.0861709	+	0.996280i	$0.472537\pi$
$24$	− 12.3693i	− 0.105203i
$25$	−55.6971	−0.445577
$26$	0	0
$27$	−27.0000	−0.192450
$28$	− 282.797i	− 1.90870i
$29$	−154.111	−0.986820	−0.493410	−	0.869797i	$-0.664250\pi$
$29$	−154.111	−0.986820	−0.493410	+	0.869797i	$0.664250\pi$
$30$	−166.273	−1.01190
$31$	308.270i	1.78603i	0.450025	+	0.893016i	$0.351415\pi$
$31$	308.270i	1.78603i	−0.450025	+	0.893016i	$0.648585\pi$
$32$	259.756i	1.43496i
$33$	121.489i	0.640862i
$34$	− 177.877i	− 0.897223i
$35$	−422.384	−2.03988
$36$	−81.0000	−0.375000
$37$	− 43.5116i	− 0.193331i	−0.995317	−	0.0966657i	$-0.969182\pi$
$37$	− 43.5116i	− 0.193331i	0.995317	−	0.0966657i	$-0.0308178\pi$
$38$	−111.233	−0.474851
$39$	0	0
$40$	−55.4243	−0.219084
$41$	47.8384i	0.182222i	0.995841	+	0.0911110i	$0.0290418\pi$
$41$	47.8384i	0.182222i	−0.995841	+	0.0911110i	$0.970958\pi$
$42$	−388.667	−1.42792
$43$	−342.121	−1.21333	−0.606663	−	0.794959i	$-0.707492\pi$
$43$	−342.121	−1.21333	−0.606663	+	0.794959i	$0.707492\pi$
$44$	364.466i	1.24876i
$45$	120.981i	0.400774i
$46$	− 78.3802i	− 0.251229i
$47$	− 133.468i	− 0.414218i	−0.978318	−	0.207109i	$-0.933594\pi$
$47$	− 133.468i	− 0.414218i	0.978318	−	0.207109i	$-0.0664055\pi$
$48$	165.000	0.496160
$49$	−644.334	−1.87853
$50$	229.645i	0.649535i
$51$	−129.424	−0.355353
$52$	0	0
$53$	−438.454	−1.13635	−0.568173	−	0.822909i	$-0.692350\pi$
$53$	−438.454	−1.13635	−0.568173	+	0.822909i	$0.692350\pi$
$54$	111.324i	0.280542i
$55$	544.364	1.33458
$56$	−129.556	−0.309154
$57$	80.9338i	0.188069i
$58$	635.418i	1.43852i
$59$	− 590.553i	− 1.30311i	−0.758601	−	0.651555i	$-0.774117\pi$
$59$	− 590.553i	− 1.30311i	0.758601	−	0.651555i	$-0.225883\pi$
$60$	362.944i	0.780931i
$61$	−541.304	−1.13618	−0.568089	−	0.822967i	$-0.692317\pi$
$61$	−541.304	−1.13618	−0.568089	+	0.822967i	$0.692317\pi$
$62$	1271.03	2.60357
$63$	282.797i	0.565541i
$64$	631.000	1.23242
$65$	0	0
$66$	500.910	0.934208
$67$	− 230.345i	− 0.420018i	−0.977700	−	0.210009i	$-0.932651\pi$
$67$	− 230.345i	− 0.420018i	0.977700	−	0.210009i	$-0.0673493\pi$
$68$	−388.273	−0.692426
$69$	−57.0300	−0.0995015
$70$	1741.54i	2.97362i
$71$	449.412i	0.751203i	0.926781	+	0.375601i	$0.122564\pi$
$71$	449.412i	0.751203i	−0.926781	+	0.375601i	$0.877436\pi$
$72$	37.1080i	0.0607391i
$73$	389.711i	0.624826i	0.949946	+	0.312413i	$0.101137\pi$
$73$	389.711i	0.624826i	−0.949946	+	0.312413i	$0.898863\pi$
$74$	−179.403	−0.281826
$75$	167.091	0.257254
$76$	242.801i	0.366464i
$77$	1272.47	1.88326
$78$	0	0
$79$	−897.820	−1.27864	−0.639321	−	0.768940i	$-0.720784\pi$
$79$	−897.820	−1.27864	−0.639321	+	0.768940i	$0.720784\pi$
$80$	− 739.330i	− 1.03325i
$81$	81.0000	0.111111
$82$	197.243	0.265632
$83$	− 1300.24i	− 1.71952i	−0.510700	−	0.859759i	$-0.670614\pi$
$83$	− 1300.24i	− 1.71952i	0.510700	−	0.859759i	$-0.329386\pi$
$84$	848.391i	1.10199i
$85$	579.923i	0.740017i
$86$	1410.60i	1.76871i
$87$	462.334	0.569741
$88$	166.970	0.202262
$89$	925.045i	1.10174i	0.834592	+	0.550869i	$0.185703\pi$
$89$	925.045i	1.10174i	−0.834592	+	0.550869i	$0.814297\pi$
$90$	498.819	0.584223
$91$	0	0
$92$	−171.090	−0.193884
$93$	− 924.811i	− 1.03117i
$94$	−550.301	−0.603822
$95$	362.647	0.391651
$96$	− 779.267i	− 0.828475i
$97$	− 1560.49i	− 1.63344i	−0.577031	−	0.816722i	$-0.695789\pi$
$97$	− 1560.49i	− 1.63344i	0.577031	−	0.816722i	$-0.304211\pi$
$98$	2656.66i	2.73840i
$99$	− 364.466i	− 0.370002i
$100$	501.274	0.501274
$101$	−958.840	−0.944635	−0.472318	−	0.881428i	$-0.656582\pi$
$101$	−958.840	−0.944635	−0.472318	+	0.881428i	$0.656582\pi$
$102$	533.630i	0.518012i
$103$	−635.153	−0.607606	−0.303803	−	0.952735i	$-0.598257\pi$
$103$	−635.153	−0.607606	−0.303803	+	0.952735i	$0.598257\pi$
$104$	0	0
$105$	1267.15	1.17773
$106$	1807.79i	1.65649i
$107$	−1448.32	−1.30855	−0.654275	−	0.756257i	$-0.727026\pi$
$107$	−1448.32	−1.30855	−0.654275	+	0.756257i	$0.727026\pi$
$108$	243.000	0.216506
$109$	− 331.084i	− 0.290937i	−0.989363	−	0.145468i	$-0.953531\pi$
$109$	− 331.084i	− 0.290937i	0.989363	−	0.145468i	$-0.0464689\pi$
$110$	− 2244.47i	− 1.94547i
$111$	130.535i	0.111620i
$112$	− 1728.20i	− 1.45803i
$113$	695.204	0.578755	0.289378	−	0.957215i	$-0.406552\pi$
$113$	695.204	0.578755	0.289378	+	0.957215i	$0.406552\pi$
$114$	333.699	0.274156
$115$	255.539i	0.207210i
$116$	1387.00	1.11017
$117$	0	0
$118$	−2434.91	−1.89959
$119$	1355.58i	1.04425i
$120$	166.273	0.126488
$121$	−308.940	−0.232111
$122$	2231.85i	1.65625i
$123$	− 143.515i	− 0.105206i
$124$	− 2774.43i	− 2.00929i
$125$	931.594i	0.666595i
$126$	1166.00	0.824410
$127$	−247.154	−0.172688	−0.0863441	−	0.996265i	$-0.527518\pi$
$127$	−247.154	−0.172688	−0.0863441	+	0.996265i	$0.527518\pi$
$128$	− 523.634i	− 0.361587i
$129$	1026.36	0.700514
$130$	0	0
$131$	472.243	0.314962	0.157481	−	0.987522i	$-0.449663\pi$
$131$	472.243	0.314962	0.157481	+	0.987522i	$0.449663\pi$
$132$	− 1093.40i	− 0.720969i
$133$	847.697	0.552667
$134$	−949.739	−0.612275
$135$	− 362.944i	− 0.231387i
$136$	177.877i	0.112153i
$137$	1830.70i	1.14166i	0.821069	+	0.570829i	$0.193378\pi$
$137$	1830.70i	1.14166i	−0.821069	+	0.570829i	$0.806622\pi$
$138$	235.141i	0.145047i
$139$	−100.000	−0.0610208	−0.0305104	−	0.999534i	$-0.509713\pi$
$139$	−100.000	−0.0610208	−0.0305104	+	0.999534i	$0.509713\pi$
$140$	3801.46	2.29487
$141$	400.403i	0.239149i
$142$	1852.97	1.09506
$143$	0	0
$144$	−495.000	−0.286458
$145$	− 2071.62i	− 1.18647i
$146$	1606.82	0.910832
$147$	1933.00	1.08457
$148$	391.604i	0.217498i
$149$	149.557i	0.0822293i	0.999154	+	0.0411147i	$0.0130909\pi$
$149$	149.557i	0.0822293i	−0.999154	+	0.0411147i	$0.986909\pi$
$150$	− 688.936i	− 0.375009i
$151$	800.032i	0.431163i	0.976486	+	0.215582i	$0.0691647\pi$
$151$	800.032i	0.431163i	−0.976486	+	0.215582i	$0.930835\pi$
$152$	111.233	0.0593564
$153$	388.273	0.205163
$154$	− 5246.51i	− 2.74530i
$155$	−4143.88	−2.14739
$156$	0	0
$157$	−2706.16	−1.37564	−0.687818	−	0.725884i	$-0.741431\pi$
$157$	−2706.16	−1.37564	−0.687818	+	0.725884i	$0.741431\pi$
$158$	3701.81i	1.86392i
$159$	1315.36	0.656070
$160$	−3491.73	−1.72528
$161$	597.330i	0.292399i
$162$	− 333.972i	− 0.161971i
$163$	− 3678.25i	− 1.76750i	−0.467959	−	0.883750i	$-0.655010\pi$
$163$	− 3678.25i	− 1.76750i	0.467959	−	0.883750i	$-0.344990\pi$
$164$	− 430.546i	− 0.205000i
$165$	−1633.09	−0.770522
$166$	−5361.03	−2.50661
$167$	3223.11i	1.49348i	0.665114	+	0.746742i	$0.268383\pi$
$167$	3223.11i	1.49348i	−0.665114	+	0.746742i	$0.731617\pi$
$168$	388.667	0.178490
$169$	0	0
$170$	2391.08	1.07875
$171$	− 242.801i	− 0.108582i
$172$	3079.09	1.36499
$173$	2689.54	1.18198	0.590988	−	0.806680i	$-0.298738\pi$
$173$	2689.54	1.18198	0.590988	+	0.806680i	$0.298738\pi$
$174$	− 1906.25i	− 0.830533i
$175$	− 1750.11i	− 0.755976i
$176$	2227.29i	0.953911i
$177$	1771.66i	0.752351i
$178$	3814.06	1.60604
$179$	1524.04	0.636381	0.318191	−	0.948027i	$-0.396925\pi$
$179$	1524.04	0.636381	0.318191	+	0.948027i	$0.396925\pi$
$180$	− 1088.83i	− 0.450871i
$181$	−476.881	−0.195836	−0.0979180	−	0.995194i	$-0.531218\pi$
$181$	−476.881	−0.195836	−0.0979180	+	0.995194i	$0.531218\pi$
$182$	0	0
$183$	1623.91	0.655973
$184$	78.3802i	0.0314036i
$185$	584.899	0.232446
$186$	−3813.09	−1.50317
$187$	− 1747.06i	− 0.683197i
$188$	1201.21i	0.465996i
$189$	− 848.391i	− 0.326515i
$190$	− 1495.23i	− 0.570924i
$191$	−1369.74	−0.518906	−0.259453	−	0.965756i	$-0.583542\pi$
$191$	−1369.74	−0.518906	−0.259453	+	0.965756i	$0.583542\pi$
$192$	−1893.00	−0.711539
$193$	2144.72i	0.799898i	0.916537	+	0.399949i	$0.130972\pi$
$193$	2144.72i	0.799898i	−0.916537	+	0.399949i	$0.869028\pi$
$194$	−6434.08	−2.38113
$195$	0	0
$196$	5799.01	2.11334
$197$	− 239.739i	− 0.0867040i	−0.999060	−	0.0433520i	$-0.986196\pi$
$197$	− 239.739i	− 0.0867040i	0.999060	−	0.0433520i	$-0.0138037\pi$
$198$	−1502.73	−0.539365
$199$	1589.94	0.566371	0.283185	−	0.959065i	$-0.408609\pi$
$199$	1589.94	0.566371	0.283185	+	0.959065i	$0.408609\pi$
$200$	− 229.645i	− 0.0811918i
$201$	691.036i	0.242497i
$202$	3953.40i	1.37703i
$203$	− 4842.47i	− 1.67426i
$204$	1164.82	0.399773
$205$	−643.061	−0.219090
$206$	2618.80i	0.885731i
$207$	171.090	0.0574472
$208$	0	0
$209$	−1092.50	−0.361579
$210$	− 5224.61i	− 1.71682i
$211$	−1872.85	−0.611055	−0.305527	−	0.952183i	$-0.598833\pi$
$211$	−1872.85	−0.611055	−0.305527	+	0.952183i	$0.598833\pi$
$212$	3946.09	1.27839
$213$	− 1348.24i	− 0.433707i
$214$	5971.59i	1.90752i
$215$	− 4598.92i	− 1.45881i
$216$	− 111.324i	− 0.0350677i
$217$	−9686.43	−3.03022
$218$	−1365.09	−0.424109
$219$	− 1169.13i	− 0.360743i
$220$	−4899.28	−1.50141
$221$	0	0
$222$	538.209	0.162713
$223$	− 56.1283i	− 0.0168548i	−0.999964	−	0.00842742i	$-0.997317\pi$
$223$	− 56.1283i	− 0.0168548i	0.999964	−	0.00842742i	$-0.00268256\pi$
$224$	−8162.01	−2.43459
$225$	−501.274	−0.148526
$226$	− 2866.40i	− 0.843673i
$227$	− 667.390i	− 0.195137i	−0.995229	−	0.0975687i	$-0.968893\pi$
$227$	− 667.390i	− 0.195137i	0.995229	−	0.0975687i	$-0.0311066\pi$
$228$	− 728.404i	− 0.211578i
$229$	− 723.299i	− 0.208720i	−0.994540	−	0.104360i	$-0.966721\pi$
$229$	− 723.299i	− 0.208720i	0.994540	−	0.104360i	$-0.0332795\pi$
$230$	1053.62	0.302058
$231$	−3817.40	−1.08730
$232$	− 635.418i	− 0.179816i
$233$	275.451	0.0774482	0.0387241	−	0.999250i	$-0.487671\pi$
$233$	275.451	0.0774482	0.0387241	+	0.999250i	$0.487671\pi$
$234$	0	0
$235$	1794.12	0.498024
$236$	5314.98i	1.46600i
$237$	2693.46	0.738224
$238$	5589.22	1.52225
$239$	− 1529.39i	− 0.413925i	−0.978349	−	0.206963i	$-0.933642\pi$
$239$	− 1529.39i	− 0.413925i	0.978349	−	0.206963i	$-0.0663579\pi$
$240$	2217.99i	0.596544i
$241$	− 975.526i	− 0.260743i	−0.991465	−	0.130372i	$-0.958383\pi$
$241$	− 975.526i	− 0.260743i	0.991465	−	0.130372i	$-0.0416170\pi$
$242$	1273.79i	0.338357i
$243$	−243.000	−0.0641500
$244$	4871.74	1.27820
$245$	− 8661.38i	− 2.25859i
$246$	−591.729	−0.153363
$247$	0	0
$248$	−1271.03	−0.325446
$249$	3900.72i	0.992765i
$250$	3841.06	0.971720
$251$	1874.14	0.471294	0.235647	−	0.971839i	$-0.424279\pi$
$251$	1874.14	0.471294	0.235647	+	0.971839i	$0.424279\pi$
$252$	− 2545.17i	− 0.636233i
$253$	− 769.832i	− 0.191300i
$254$	1019.04i	0.251734i
$255$	− 1739.77i	− 0.427249i
$256$	2889.00	0.705322
$257$	1818.19	0.441305	0.220653	−	0.975352i	$-0.429181\pi$
$257$	1818.19	0.441305	0.220653	+	0.975352i	$0.429181\pi$
$258$	− 4231.81i	− 1.02117i
$259$	1367.22	0.328010
$260$	0	0
$261$	−1387.00	−0.328940
$262$	− 1947.11i	− 0.459132i
$263$	673.799	0.157978	0.0789890	−	0.996875i	$-0.474831\pi$
$263$	673.799	0.157978	0.0789890	+	0.996875i	$0.474831\pi$
$264$	−500.910	−0.116776
$265$	− 5893.86i	− 1.36625i
$266$	− 3495.14i	− 0.805643i
$267$	− 2775.14i	− 0.636088i
$268$	2073.11i	0.472520i
$269$	−3356.40	−0.760756	−0.380378	−	0.924831i	$-0.624206\pi$
$269$	−3356.40	−0.760756	−0.380378	+	0.924831i	$0.624206\pi$
$270$	−1496.46	−0.337301
$271$	− 8915.55i	− 1.99845i	−0.0393133	−	0.999227i	$-0.512517\pi$
$271$	− 8915.55i	− 1.99845i	0.0393133	−	0.999227i	$-0.487483\pi$
$272$	−2372.78	−0.528937
$273$	0	0
$274$	7548.16	1.66424
$275$	2255.52i	0.494593i
$276$	513.270	0.111939
$277$	−4017.31	−0.871396	−0.435698	−	0.900093i	$-0.643498\pi$
$277$	−4017.31	−0.871396	−0.435698	+	0.900093i	$0.643498\pi$
$278$	412.311i	0.0889523i
$279$	2774.43i	0.595344i
$280$	− 1741.54i	− 0.371702i
$281$	− 1841.12i	− 0.390860i	−0.980718	−	0.195430i	$-0.937390\pi$
$281$	− 1841.12i	− 0.390860i	0.980718	−	0.195430i	$-0.0626103\pi$
$282$	1650.90	0.348617
$283$	−4849.40	−1.01861	−0.509305	−	0.860586i	$-0.670098\pi$
$283$	−4849.40	−1.01861	−0.509305	+	0.860586i	$0.670098\pi$
$284$	− 4044.71i	− 0.845103i
$285$	−1087.94	−0.226120
$286$	0	0
$287$	−1503.17	−0.309162
$288$	2337.80i	0.478320i
$289$	−3051.82	−0.621172
$290$	−8541.52	−1.72957
$291$	4681.48i	0.943070i
$292$	− 3507.40i	− 0.702929i
$293$	1413.85i	0.281905i	0.990016	+	0.140953i	$0.0450165\pi$
$293$	1413.85i	0.281905i	−0.990016	+	0.140953i	$0.954983\pi$
$294$	− 7969.97i	− 1.58101i
$295$	7938.43	1.56676
$296$	179.403	0.0352283
$297$	1093.40i	0.213621i
$298$	616.639	0.119869
$299$	0	0
$300$	−1503.82	−0.289411
$301$	− 10750.1i	− 2.05856i
$302$	3298.62	0.628523
$303$	2876.52	0.545385
$304$	1483.79i	0.279937i
$305$	− 7276.41i	− 1.36605i
$306$	− 1600.89i	− 0.299074i
$307$	4625.64i	0.859932i	0.902845	+	0.429966i	$0.141474\pi$
$307$	4625.64i	0.859932i	−0.902845	+	0.429966i	$0.858526\pi$
$308$	−11452.2	−2.11867
$309$	1905.46	0.350802
$310$	17085.7i	3.13032i
$311$	6060.79	1.10507	0.552534	−	0.833490i	$-0.313661\pi$
$311$	6060.79	1.10507	0.552534	+	0.833490i	$0.313661\pi$
$312$	0	0
$313$	969.946	0.175158	0.0875792	−	0.996158i	$-0.472087\pi$
$313$	969.946	0.175158	0.0875792	+	0.996158i	$0.472087\pi$
$314$	11157.8i	2.00532i
$315$	−3801.46	−0.679962
$316$	8080.38	1.43847
$317$	8741.63i	1.54883i	0.632679	+	0.774414i	$0.281955\pi$
$317$	8741.63i	1.54883i	−0.632679	+	0.774414i	$0.718045\pi$
$318$	− 5423.38i	− 0.956378i
$319$	6240.92i	1.09537i
$320$	8482.13i	1.48177i
$321$	4344.97	0.755491
$322$	2462.85	0.426241
$323$	− 1163.87i	− 0.200493i
$324$	−729.000	−0.125000
$325$	0	0
$326$	−15165.8	−2.57655
$327$	993.252i	0.167972i
$328$	−197.243	−0.0332040
$329$	4193.81	0.702772
$330$	6733.41i	1.12322i
$331$	6987.76i	1.16037i	0.814485	+	0.580184i	$0.197019\pi$
$331$	6987.76i	1.16037i	−0.814485	+	0.580184i	$0.802981\pi$
$332$	11702.2i	1.93446i
$333$	− 391.604i	− 0.0644438i
$334$	13289.2	2.17711
$335$	3096.39	0.504996
$336$	5184.61i	0.841797i
$337$	4156.59	0.671881	0.335940	−	0.941883i	$-0.390946\pi$
$337$	4156.59	0.671881	0.335940	+	0.941883i	$0.390946\pi$
$338$	0	0
$339$	−2085.61	−0.334144
$340$	− 5219.30i	− 0.832519i
$341$	12483.8	1.98250
$342$	−1001.10	−0.158284
$343$	− 9468.49i	− 1.49053i
$344$	− 1410.60i	− 0.221089i
$345$	− 766.618i	− 0.119633i
$346$	− 11089.3i	− 1.72301i
$347$	−312.513	−0.0483475	−0.0241737	−	0.999708i	$-0.507695\pi$
$347$	−312.513	−0.0483475	−0.0241737	+	0.999708i	$0.507695\pi$
$348$	−4161.01	−0.640958
$349$	4458.75i	0.683872i	0.939723	+	0.341936i	$0.111083\pi$
$349$	4458.75i	0.683872i	−0.939723	+	0.341936i	$0.888917\pi$
$350$	−7215.88	−1.10201
$351$	0	0
$352$	10519.1	1.59281
$353$	− 2249.17i	− 0.339126i	−0.985519	−	0.169563i	$-0.945764\pi$
$353$	− 2249.17i	− 0.339126i	0.985519	−	0.169563i	$-0.0542356\pi$
$354$	7304.74	1.09673
$355$	−6041.16	−0.903187
$356$	− 8325.41i	− 1.23945i
$357$	− 4066.75i	− 0.602900i
$358$	− 6283.78i	− 0.927677i
$359$	7842.79i	1.15300i	0.817098	+	0.576499i	$0.195582\pi$
$359$	7842.79i	1.15300i	−0.817098	+	0.576499i	$0.804418\pi$
$360$	−498.819	−0.0730279
$361$	6131.19	0.893890
$362$	1966.23i	0.285478i
$363$	926.820	0.134009
$364$	0	0
$365$	−5238.64	−0.751241
$366$	− 6695.56i	− 0.956237i
$367$	6660.24	0.947307	0.473653	−	0.880711i	$-0.342935\pi$
$367$	6660.24	0.947307	0.473653	+	0.880711i	$0.342935\pi$
$368$	−1045.55	−0.148106
$369$	430.546i	0.0607407i
$370$	− 2411.60i	− 0.338846i
$371$	− 13777.1i	− 1.92795i
$372$	8323.30i	1.16006i
$373$	−36.9873	−0.00513439	−0.00256720	−	0.999997i	$-0.500817\pi$
$373$	−36.9873	−0.00513439	−0.00256720	+	0.999997i	$0.500817\pi$
$374$	−7203.32	−0.995923
$375$	− 2794.78i	− 0.384859i
$376$	550.301	0.0754777
$377$	0	0
$378$	−3498.00	−0.475973
$379$	12079.9i	1.63721i	0.574360	+	0.818603i	$0.305251\pi$
$379$	12079.9i	1.63721i	−0.574360	+	0.818603i	$0.694749\pi$
$380$	−3263.82	−0.440607
$381$	741.463	0.0997015
$382$	5647.59i	0.756429i
$383$	− 10567.4i	− 1.40984i	−0.709287	−	0.704919i	$-0.750983\pi$
$383$	− 10567.4i	− 1.40984i	0.709287	−	0.704919i	$-0.249017\pi$
$384$	1570.90i	0.208763i
$385$	17104.9i	2.26428i
$386$	8842.90	1.16604
$387$	−3079.09	−0.404442
$388$	14044.4i	1.83763i
$389$	9757.49	1.27179	0.635893	−	0.771778i	$-0.280632\pi$
$389$	9757.49	1.27179	0.635893	+	0.771778i	$0.280632\pi$
$390$	0	0
$391$	820.119	0.106075
$392$	− 2656.66i	− 0.342300i
$393$	−1416.73	−0.181844
$394$	−988.469	−0.126392
$395$	− 12068.8i	− 1.53734i
$396$	3280.19i	0.416252i
$397$	14200.5i	1.79522i	0.440786	+	0.897612i	$0.354700\pi$
$397$	14200.5i	1.79522i	−0.440786	+	0.897612i	$0.645300\pi$
$398$	− 6555.48i	− 0.825620i
$399$	−2543.09	−0.319082
$400$	3063.34	0.382918
$401$	12676.4i	1.57863i	0.613992	+	0.789313i	$0.289563\pi$
$401$	12676.4i	1.57863i	−0.613992	+	0.789313i	$0.710437\pi$
$402$	2849.22	0.353497
$403$	0	0
$404$	8629.56	1.06271
$405$	1088.83i	0.133591i
$406$	−19966.0	−2.44063
$407$	−1762.05	−0.214599
$408$	− 533.630i	− 0.0647515i
$409$	1533.68i	0.185417i	0.995693	+	0.0927083i	$0.0295524\pi$
$409$	1533.68i	0.185417i	−0.995693	+	0.0927083i	$0.970448\pi$
$410$	2651.41i	0.319375i
$411$	− 5492.09i	− 0.659136i
$412$	5716.38	0.683557
$413$	18556.3	2.21089
$414$	− 705.422i	− 0.0837430i
$415$	17478.3	2.06741
$416$	0	0
$417$	300.000	0.0352304
$418$	4504.50i	0.527087i
$419$	−2165.89	−0.252532	−0.126266	−	0.991996i	$-0.540299\pi$
$419$	−2165.89	−0.252532	−0.126266	+	0.991996i	$0.540299\pi$
$420$	−11404.4	−1.32494
$421$	734.575i	0.0850380i	0.999096	+	0.0425190i	$0.0135383\pi$
$421$	734.575i	0.0850380i	−0.999096	+	0.0425190i	$0.986462\pi$
$422$	7721.98i	0.890758i
$423$	− 1201.21i	− 0.138073i
$424$	− 1807.79i	− 0.207062i
$425$	−2402.85	−0.274248
$426$	−5558.92	−0.632231
$427$	− 17008.8i	− 1.92767i
$428$	13034.9	1.47212
$429$	0	0
$430$	−18961.8	−2.12656
$431$	− 13709.3i	− 1.53215i	−0.642754	−	0.766073i	$-0.722208\pi$
$431$	− 13709.3i	− 1.53215i	0.642754	−	0.766073i	$-0.277792\pi$
$432$	1485.00	0.165387
$433$	−10049.9	−1.11540	−0.557701	−	0.830042i	$-0.688316\pi$
$433$	−10049.9	−1.11540	−0.557701	+	0.830042i	$0.688316\pi$
$434$	39938.2i	4.41727i
$435$	6214.87i	0.685011i
$436$	2979.76i	0.327304i
$437$	− 512.850i	− 0.0561395i
$438$	−4820.46	−0.525869
$439$	8133.47	0.884258	0.442129	−	0.896951i	$-0.354223\pi$
$439$	8133.47	0.884258	0.442129	+	0.896951i	$0.354223\pi$
$440$	2244.47i	0.243184i
$441$	−5799.01	−0.626175
$442$	0	0
$443$	−2370.78	−0.254264	−0.127132	−	0.991886i	$-0.540577\pi$
$443$	−2370.78	−0.254264	−0.127132	+	0.991886i	$0.540577\pi$
$444$	− 1174.81i	− 0.125572i
$445$	−12434.8	−1.32464
$446$	−231.423	−0.0245699
$447$	− 448.670i	− 0.0474751i
$448$	19827.2i	2.09095i
$449$	− 12923.2i	− 1.35832i	−0.733992	−	0.679158i	$-0.762345\pi$
$449$	− 12923.2i	− 1.35832i	0.733992	−	0.679158i	$-0.237655\pi$
$450$	2066.81i	0.216512i
$451$	1937.27	0.202267
$452$	−6256.84	−0.651099
$453$	− 2400.10i	− 0.248932i
$454$	−2751.72	−0.284459
$455$	0	0
$456$	−333.699	−0.0342694
$457$	8401.26i	0.859944i	0.902842	+	0.429972i	$0.141477\pi$
$457$	8401.26i	0.859944i	−0.902842	+	0.429972i	$0.858523\pi$
$458$	−2982.24	−0.304260
$459$	−1164.82	−0.118451
$460$	− 2299.85i	− 0.233111i
$461$	17627.3i	1.78088i	0.455098	+	0.890441i	$0.349604\pi$
$461$	17627.3i	1.78088i	−0.455098	+	0.890441i	$0.650396\pi$
$462$	15739.5i	1.58500i
$463$	5461.81i	0.548233i	0.961697	+	0.274116i	$0.0883853\pi$
$463$	5461.81i	0.548233i	−0.961697	+	0.274116i	$0.911615\pi$
$464$	8476.13	0.848048
$465$	12431.6	1.23979
$466$	− 1135.72i	− 0.112899i
$467$	−8262.19	−0.818691	−0.409345	−	0.912379i	$-0.634243\pi$
$467$	−8262.19	−0.818691	−0.409345	+	0.912379i	$0.634243\pi$
$468$	0	0
$469$	7237.89	0.712611
$470$	− 7397.35i	− 0.725988i
$471$	8118.47	0.794223
$472$	2434.91	0.237449
$473$	13854.6i	1.34680i
$474$	− 11105.4i	− 1.07614i
$475$	1502.59i	0.145145i
$476$	− 12200.3i	− 1.17479i
$477$	−3946.09	−0.378782
$478$	−6305.84	−0.603395
$479$	1575.87i	0.150320i	0.997171	+	0.0751601i	$0.0239468\pi$
$479$	1575.87i	0.150320i	−0.997171	+	0.0751601i	$0.976053\pi$
$480$	10475.2	0.996093
$481$	0	0
$482$	−4022.20	−0.380095
$483$	− 1791.99i	− 0.168816i
$484$	2780.46	0.261125
$485$	20976.7	1.96393
$486$	1001.91i	0.0935139i
$487$	− 12595.7i	− 1.17200i	−0.810310	−	0.586001i	$-0.800701\pi$
$487$	− 12595.7i	− 1.17200i	0.810310	−	0.586001i	$-0.199299\pi$
$488$	− 2231.85i	− 0.207031i
$489$	11034.7i	1.02047i
$490$	−35711.8	−3.29244
$491$	1071.21	0.0984586	0.0492293	−	0.998788i	$-0.484323\pi$
$491$	1071.21	0.0984586	0.0492293	+	0.998788i	$0.484323\pi$
$492$	1291.64i	0.118357i
$493$	−6648.59	−0.607378
$494$	0	0
$495$	4899.28	0.444861
$496$	− 16954.9i	− 1.53487i
$497$	−14121.4	−1.27451
$498$	16083.1	1.44719
$499$	− 1422.30i	− 0.127597i	−0.997963	−	0.0637985i	$-0.979678\pi$
$499$	− 1422.30i	− 0.127597i	0.997963	−	0.0637985i	$-0.0203215\pi$
$500$	− 8384.35i	− 0.749919i
$501$	− 9669.33i	− 0.862263i
$502$	− 7727.28i	− 0.687022i
$503$	−9349.34	−0.828760	−0.414380	−	0.910104i	$-0.636002\pi$
$503$	−9349.34	−0.828760	−0.414380	+	0.910104i	$0.636002\pi$
$504$	−1166.00	−0.103051
$505$	− 12889.1i	− 1.13576i
$506$	−3174.10	−0.278866
$507$	0	0
$508$	2224.39	0.194274
$509$	13736.3i	1.19617i	0.801432	+	0.598086i	$0.204072\pi$
$509$	13736.3i	1.19617i	−0.801432	+	0.598086i	$0.795928\pi$
$510$	−7173.25	−0.622817
$511$	−12245.5	−1.06009
$512$	− 16100.7i	− 1.38976i
$513$	728.404i	0.0626897i
$514$	− 7496.58i	− 0.643307i
$515$	− 8537.96i	− 0.730538i
$516$	−9237.28	−0.788079
$517$	−5404.93	−0.459785
$518$	− 5637.17i	− 0.478153i
$519$	−8068.62	−0.682414
$520$	0	0
$521$	11052.3	0.929386	0.464693	−	0.885472i	$-0.346165\pi$
$521$	11052.3	0.929386	0.464693	+	0.885472i	$0.346165\pi$
$522$	5718.76i	0.479508i
$523$	6477.04	0.541532	0.270766	−	0.962645i	$-0.412723\pi$
$523$	6477.04	0.541532	0.270766	+	0.962645i	$0.412723\pi$
$524$	−4250.19	−0.354332
$525$	5250.33i	0.436463i
$526$	− 2778.14i	− 0.230290i
$527$	13299.2i	1.09929i
$528$	− 6681.87i	− 0.550741i
$529$	−11805.6	−0.970298
$530$	−24301.0	−1.99164
$531$	− 5314.98i	− 0.434370i
$532$	−7629.27	−0.621750
$533$	0	0
$534$	−11442.2	−0.927250
$535$	− 19468.9i	− 1.57330i
$536$	949.739	0.0765344
$537$	−4572.12	−0.367415
$538$	13838.8i	1.10898i
$539$	26093.1i	2.08517i
$540$	3266.49i	0.260310i
$541$	− 18341.5i	− 1.45761i	−0.684723	−	0.728803i	$-0.740077\pi$
$541$	− 18341.5i	− 1.45761i	0.684723	−	0.728803i	$-0.259923\pi$
$542$	−36759.7	−2.91322
$543$	1430.64	0.113066
$544$	11206.2i	0.883204i
$545$	4450.55	0.349799
$546$	0	0
$547$	−18943.1	−1.48071	−0.740356	−	0.672215i	$-0.765343\pi$
$547$	−18943.1	−1.48071	−0.740356	+	0.672215i	$0.765343\pi$
$548$	− 16476.3i	− 1.28436i
$549$	−4871.74	−0.378726
$550$	9299.75	0.720987
$551$	4157.61i	0.321452i
$552$	− 235.141i	− 0.0181309i
$553$	− 28211.2i	− 2.16937i
$554$	16563.8i	1.27027i
$555$	−1754.70	−0.134203
$556$	900.000	0.0686484
$557$	− 415.532i	− 0.0316098i	−0.999875	−	0.0158049i	$-0.994969\pi$
$557$	− 415.532i	− 0.0316098i	0.999875	−	0.0158049i	$-0.00503106\pi$
$558$	11439.3	0.867856
$559$	0	0
$560$	23231.1	1.75303
$561$	5241.19i	0.394444i
$562$	−7591.12	−0.569772
$563$	−18291.8	−1.36929	−0.684643	−	0.728879i	$-0.740042\pi$
$563$	−18291.8	−1.36929	−0.684643	+	0.728879i	$0.740042\pi$
$564$	− 3603.63i	− 0.269043i
$565$	9345.19i	0.695850i
$566$	19994.6i	1.48487i
$567$	2545.17i	0.188514i
$568$	−1852.97	−0.136882
$569$	−4347.47	−0.320308	−0.160154	−	0.987092i	$-0.551199\pi$
$569$	−4347.47	−0.320308	−0.160154	+	0.987092i	$0.551199\pi$
$570$	4485.70i	0.329623i
$571$	16756.0	1.22805	0.614024	−	0.789288i	$-0.289550\pi$
$571$	16756.0	1.22805	0.614024	+	0.789288i	$0.289550\pi$
$572$	0	0
$573$	4109.23	0.299591
$574$	6197.74i	0.450677i
$575$	−1058.80	−0.0767915
$576$	5679.00	0.410807
$577$	19974.7i	1.44117i	0.693364	+	0.720587i	$0.256128\pi$
$577$	19974.7i	1.44117i	−0.693364	+	0.720587i	$0.743872\pi$
$578$	12583.0i	0.905506i
$579$	− 6434.16i	− 0.461821i
$580$	18644.6i	1.33478i
$581$	40856.0	2.91737
$582$	19302.2	1.37475
$583$	17755.7i	1.26135i
$584$	−1606.82	−0.113854
$585$	0	0
$586$	5829.47	0.410944
$587$	− 15748.7i	− 1.10735i	−0.832732	−	0.553677i	$-0.813224\pi$
$587$	− 15748.7i	− 1.10735i	0.832732	−	0.553677i	$-0.186776\pi$
$588$	−17397.0	−1.22014
$589$	8316.50	0.581792
$590$	− 32731.0i	− 2.28392i
$591$	719.217i	0.0500586i
$592$	2393.14i	0.166144i
$593$	− 13318.4i	− 0.922297i	−0.887323	−	0.461148i	$-0.847438\pi$
$593$	− 13318.4i	− 0.922297i	0.887323	−	0.461148i	$-0.152562\pi$
$594$	4508.19	0.311403
$595$	−18222.3	−1.25553
$596$	− 1346.01i	− 0.0925080i
$597$	−4769.82	−0.326994
$598$	0	0
$599$	2970.80	0.202644	0.101322	−	0.994854i	$-0.467693\pi$
$599$	2970.80	0.202644	0.101322	+	0.994854i	$0.467693\pi$
$600$	688.936i	0.0468761i
$601$	10632.6	0.721654	0.360827	−	0.932633i	$-0.382495\pi$
$601$	10632.6	0.721654	0.360827	+	0.932633i	$0.382495\pi$
$602$	−44323.8	−3.00083
$603$	− 2073.11i	− 0.140006i
$604$	− 7200.29i	− 0.485059i
$605$	− 4152.88i	− 0.279072i
$606$	− 11860.2i	− 0.795029i
$607$	11587.9	0.774856	0.387428	−	0.921900i	$-0.373364\pi$
$607$	11587.9	0.774856	0.387428	+	0.921900i	$0.373364\pi$
$608$	7007.67	0.467432
$609$	14527.4i	0.966634i
$610$	−30001.4	−1.99135
$611$	0	0
$612$	−3494.46	−0.230809
$613$	20792.3i	1.36998i	0.728555	+	0.684988i	$0.240192\pi$
$613$	20792.3i	1.36998i	−0.728555	+	0.684988i	$0.759808\pi$
$614$	19072.0	1.25356
$615$	1929.18	0.126491
$616$	5246.51i	0.343162i
$617$	− 1562.78i	− 0.101969i	−0.998699	−	0.0509846i	$-0.983764\pi$
$617$	− 1562.78i	− 0.101969i	0.998699	−	0.0509846i	$-0.0162359\pi$
$618$	− 7856.41i	− 0.511377i
$619$	− 758.406i	− 0.0492454i	−0.999697	−	0.0246227i	$-0.992162\pi$
$619$	− 758.406i	− 0.0492454i	0.999697	−	0.0246227i	$-0.00783844\pi$
$620$	37294.9	2.41581
$621$	−513.270	−0.0331672
$622$	− 24989.3i	− 1.61090i
$623$	−29066.7	−1.86923
$624$	0	0
$625$	−19485.0	−1.24704
$626$	− 3999.19i	− 0.255335i
$627$	3277.51	0.208758
$628$	24355.4	1.54759
$629$	− 1877.15i	− 0.118994i
$630$	15673.8i	0.991206i
$631$	14265.2i	0.899981i	0.893033	+	0.449990i	$0.148573\pi$
$631$	14265.2i	0.899981i	−0.893033	+	0.449990i	$0.851427\pi$
$632$	− 3701.81i	− 0.232990i
$633$	5618.56	0.352793
$634$	36042.7	2.25779
$635$	− 3322.34i	− 0.207627i
$636$	−11838.3	−0.738078
$637$	0	0
$638$	25732.0	1.59677
$639$	4044.71i	0.250401i
$640$	7038.88	0.434744
$641$	−3985.64	−0.245590	−0.122795	−	0.992432i	$-0.539186\pi$
$641$	−3985.64	−0.245590	−0.122795	+	0.992432i	$0.539186\pi$
$642$	− 17914.8i	− 1.10131i
$643$	− 8156.55i	− 0.500254i	−0.968213	−	0.250127i	$-0.919528\pi$
$643$	− 8156.55i	− 0.500254i	0.968213	−	0.250127i	$-0.0804723\pi$
$644$	− 5375.97i	− 0.328949i
$645$	13796.8i	0.842243i
$646$	−4798.74	−0.292266
$647$	−11279.2	−0.685368	−0.342684	−	0.939451i	$-0.611336\pi$
$647$	−11279.2	−0.685368	−0.342684	+	0.939451i	$0.611336\pi$
$648$	333.972i	0.0202464i
$649$	−23915.2	−1.44646
$650$	0	0
$651$	29059.3	1.74950
$652$	33104.2i	1.98844i
$653$	6565.75	0.393473	0.196736	−	0.980456i	$-0.436966\pi$
$653$	6565.75	0.393473	0.196736	+	0.980456i	$0.436966\pi$
$654$	4095.28	0.244860
$655$	6348.06i	0.378686i
$656$	− 2631.11i	− 0.156597i
$657$	3507.40i	0.208275i
$658$	− 17291.5i	− 1.02446i
$659$	4799.35	0.283696	0.141848	−	0.989888i	$-0.454696\pi$
$659$	4799.35	0.283696	0.141848	+	0.989888i	$0.454696\pi$
$660$	14697.8	0.866837
$661$	− 15593.6i	− 0.917581i	−0.888545	−	0.458790i	$-0.848283\pi$
$661$	− 15593.6i	− 0.917581i	0.888545	−	0.458790i	$-0.151717\pi$
$662$	28811.3	1.69151
$663$	0	0
$664$	5361.03	0.313326
$665$	11395.1i	0.664483i
$666$	−1614.63	−0.0939422
$667$	−2929.66	−0.170070
$668$	− 29008.0i	− 1.68017i
$669$	168.385i	0.00973115i
$670$	− 12766.7i	− 0.736152i
$671$	21920.8i	1.26116i
$672$	24486.0	1.40561
$673$	2205.54	0.126326	0.0631630	−	0.998003i	$-0.479881\pi$
$673$	2205.54	0.126326	0.0631630	+	0.998003i	$0.479881\pi$
$674$	− 17138.1i	− 0.979426i
$675$	1503.82	0.0857514
$676$	0	0
$677$	−15046.4	−0.854182	−0.427091	−	0.904209i	$-0.640462\pi$
$677$	−15046.4	−0.854182	−0.427091	+	0.904209i	$0.640462\pi$
$678$	8599.20i	0.487095i
$679$	49033.6	2.77134
$680$	−2391.08	−0.134844
$681$	2002.17i	0.112663i
$682$	− 51471.9i	− 2.88997i
$683$	− 30632.5i	− 1.71614i	−0.513537	−	0.858068i	$-0.671665\pi$
$683$	− 30632.5i	− 1.71614i	0.513537	−	0.858068i	$-0.328335\pi$
$684$	2185.21i	0.122155i
$685$	−24608.9	−1.37264
$686$	−39039.6	−2.17280
$687$	2169.90i	0.120505i
$688$	18816.7	1.04270
$689$	0	0
$690$	−3160.85	−0.174393
$691$	2175.72i	0.119780i	0.998205	+	0.0598901i	$0.0190750\pi$
$691$	2175.72i	0.119780i	−0.998205	+	0.0598901i	$0.980925\pi$
$692$	−24205.9	−1.32972
$693$	11452.2	0.627753
$694$	1288.52i	0.0704780i
$695$	− 1344.24i	− 0.0733666i
$696$	1906.25i	0.103817i
$697$	2063.82i	0.112156i
$698$	18383.9	0.996906
$699$	−826.354	−0.0447147
$700$	15751.0i	0.850473i
$701$	32718.2	1.76284	0.881419	−	0.472335i	$-0.156589\pi$
$701$	32718.2	1.76284	0.881419	+	0.472335i	$0.156589\pi$
$702$	0	0
$703$	−1173.85	−0.0629768
$704$	− 25553.1i	− 1.36799i
$705$	−5382.36	−0.287534
$706$	−9273.58	−0.494356
$707$	− 30128.6i	− 1.60269i
$708$	− 15944.9i	− 0.846395i
$709$	25219.7i	1.33589i	0.744211	+	0.667945i	$0.232826\pi$
$709$	25219.7i	1.33589i	−0.744211	+	0.667945i	$0.767174\pi$
$710$	24908.3i	1.31661i
$711$	−8080.38	−0.426214
$712$	−3814.06	−0.200756
$713$	5860.22i	0.307808i
$714$	−16767.7	−0.878871
$715$	0	0
$716$	−13716.4	−0.715929
$717$	4588.18i	0.238980i
$718$	32336.6	1.68077
$719$	35466.2	1.83959	0.919796	−	0.392398i	$-0.128354\pi$
$719$	35466.2	1.83959	0.919796	+	0.392398i	$0.128354\pi$
$720$	− 6653.97i	− 0.344415i
$721$	− 19957.7i	− 1.03088i
$722$	− 25279.5i	− 1.30306i
$723$	2926.58i	0.150540i
$724$	4291.93	0.220315
$725$	8583.57	0.439704
$726$	− 3821.38i	− 0.195351i
$727$	14262.2	0.727588	0.363794	−	0.931479i	$-0.381481\pi$
$727$	14262.2	0.727588	0.363794	+	0.931479i	$0.381481\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	21599.5i	1.09511i
$731$	−14759.6	−0.746790
$732$	−14615.2	−0.737970
$733$	16022.5i	0.807371i	0.914898	+	0.403685i	$0.132271\pi$
$733$	16022.5i	0.807371i	−0.914898	+	0.403685i	$0.867729\pi$
$734$	− 27460.9i	− 1.38093i
$735$	25984.1i	1.30400i
$736$	4937.96i	0.247304i
$737$	−9328.11	−0.466222
$738$	1775.19	0.0885440
$739$	− 3796.21i	− 0.188966i	−0.995526	−	0.0944830i	$-0.969880\pi$
$739$	− 3796.21i	− 0.188966i	0.995526	−	0.0944830i	$-0.0301198\pi$
$740$	−5264.09	−0.261502
$741$	0	0
$742$	−56804.3	−2.81044
$743$	− 30329.3i	− 1.49754i	−0.662827	−	0.748772i	$-0.730644\pi$
$743$	− 30329.3i	− 1.49754i	0.662827	−	0.748772i	$-0.269356\pi$
$744$	3813.09	0.187896
$745$	−2010.40	−0.0988661
$746$	152.502i	0.00748460i
$747$	− 11702.2i	− 0.573173i
$748$	15723.6i	0.768597i
$749$	− 45509.1i	− 2.22011i
$750$	−11523.2	−0.561023
$751$	−21551.9	−1.04719	−0.523595	−	0.851967i	$-0.675409\pi$
$751$	−21551.9	−1.04719	−0.523595	+	0.851967i	$0.675409\pi$
$752$	7340.72i	0.355969i
$753$	−5622.42	−0.272101
$754$	0	0
$755$	−10754.3	−0.518397
$756$	7635.52i	0.367329i
$757$	−20417.6	−0.980306	−0.490153	−	0.871637i	$-0.663059\pi$
$757$	−20417.6	−0.980306	−0.490153	+	0.871637i	$0.663059\pi$
$758$	49806.5	2.38662
$759$	2309.50i	0.110447i
$760$	1495.23i	0.0713655i
$761$	31375.4i	1.49456i	0.664512	+	0.747278i	$0.268640\pi$
$761$	31375.4i	1.49456i	−0.664512	+	0.747278i	$0.731360\pi$
$762$	− 3057.13i	− 0.145339i
$763$	10403.3	0.493609
$764$	12327.7	0.583770
$765$	5219.30i	0.246672i
$766$	−43570.5	−2.05518
$767$	0	0
$768$	−8667.00	−0.407218
$769$	12452.7i	0.583946i	0.956427	+	0.291973i	$0.0943118\pi$
$769$	12452.7i	0.583946i	−0.956427	+	0.291973i	$0.905688\pi$
$770$	70525.5	3.30073
$771$	−5454.56	−0.254788
$772$	− 19302.5i	− 0.899885i
$773$	− 37449.8i	− 1.74253i	−0.490815	−	0.871264i	$-0.663301\pi$
$773$	− 37449.8i	− 1.74253i	0.490815	−	0.871264i	$-0.336699\pi$
$774$	12695.4i	0.589571i
$775$	− 17169.8i	− 0.795815i
$776$	6434.08	0.297642
$777$	−4101.65	−0.189377
$778$	− 40231.2i	− 1.85393i
$779$	1290.58	0.0593580
$780$	0	0
$781$	18199.5	0.833839
$782$	− 3381.44i	− 0.154629i
$783$	4161.01	0.189914
$784$	35438.4	1.61436
$785$	− 36377.1i	− 1.65396i
$786$	5841.32i	0.265080i
$787$	26460.2i	1.19848i	0.800570	+	0.599240i	$0.204530\pi$
$787$	26460.2i	1.19848i	−0.800570	+	0.599240i	$0.795470\pi$
$788$	2157.65i	0.0975420i
$789$	−2021.40	−0.0912086
$790$	−49761.0	−2.24104
$791$	21844.6i	0.981928i
$792$	1502.73	0.0674207
$793$	0	0
$794$	58550.3	2.61697
$795$	17681.6i	0.788807i
$796$	−14309.4	−0.637167
$797$	−4749.47	−0.211085	−0.105543	−	0.994415i	$-0.533658\pi$
$797$	−4749.47	−0.211085	−0.105543	+	0.994415i	$0.533658\pi$
$798$	10485.4i	0.465138i
$799$	− 5757.99i	− 0.254947i
$800$	− 14467.6i	− 0.639386i
$801$	8325.41i	0.367246i
$802$	52266.1	2.30122
$803$	15781.8	0.693560
$804$	− 6219.33i	− 0.272809i
$805$	−8029.53	−0.351557
$806$	0	0
$807$	10069.2	0.439223
$808$	− 3953.40i	− 0.172129i
$809$	−4464.04	−0.194002	−0.0970009	−	0.995284i	$-0.530925\pi$
$809$	−4464.04	−0.194002	−0.0970009	+	0.995284i	$0.530925\pi$
$810$	4489.37	0.194741
$811$	− 20774.6i	− 0.899499i	−0.893155	−	0.449749i	$-0.851513\pi$
$811$	− 20774.6i	− 0.899499i	0.893155	−	0.449749i	$-0.148487\pi$
$812$	43582.2i	1.88354i
$813$	26746.6i	1.15381i
$814$	7265.13i	0.312829i
$815$	49444.3	2.12510
$816$	7118.34	0.305382
$817$	9229.73i	0.395235i
$818$	6323.51	0.270289
$819$	0	0
$820$	5787.55	0.246476
$821$	35769.2i	1.52053i	0.649614	+	0.760264i	$0.274930\pi$
$821$	35769.2i	1.52053i	−0.649614	+	0.760264i	$0.725070\pi$
$822$	−22644.5	−0.960848
$823$	−2945.44	−0.124753	−0.0623764	−	0.998053i	$-0.519868\pi$
$823$	−2945.44	−0.124753	−0.0623764	+	0.998053i	$0.519868\pi$
$824$	− 2618.80i	− 0.110716i
$825$	− 6766.56i	− 0.285553i
$826$	− 76509.6i	− 3.22289i
$827$	− 17878.6i	− 0.751753i	−0.926670	−	0.375877i	$-0.877342\pi$
$827$	− 17878.6i	− 0.751753i	0.926670	−	0.375877i	$-0.122658\pi$
$828$	−1539.81	−0.0646281
$829$	−13423.8	−0.562398	−0.281199	−	0.959649i	$-0.590732\pi$
$829$	−13423.8	−0.562398	−0.281199	+	0.959649i	$0.590732\pi$
$830$	− 72065.0i	− 3.01375i
$831$	12051.9	0.503101
$832$	0	0
$833$	−27797.5	−1.15621
$834$	− 1236.93i	− 0.0513566i
$835$	−43326.2	−1.79565
$836$	9832.53	0.406776
$837$	− 8323.30i	− 0.343722i
$838$	8930.21i	0.368125i
$839$	− 31542.7i	− 1.29794i	−0.760813	−	0.648971i	$-0.775200\pi$
$839$	− 31542.7i	− 1.29794i	0.760813	−	0.648971i	$-0.224800\pi$
$840$	5224.61i	0.214602i
$841$	−638.669	−0.0261867
$842$	3028.73	0.123963
$843$	5523.35i	0.225663i
$844$	16855.7	0.687437
$845$	0	0
$846$	−4952.71	−0.201274
$847$	− 9707.47i	− 0.393805i
$848$	24115.0	0.976547
$849$	14548.2	0.588095
$850$	9907.22i	0.399782i
$851$	− 827.155i	− 0.0333191i
$852$	12134.1i	0.487920i
$853$	21810.6i	0.875476i	0.899103	+	0.437738i	$0.144220\pi$
$853$	21810.6i	0.875476i	−0.899103	+	0.437738i	$0.855780\pi$
$854$	−70129.1	−2.81003
$855$	3263.82	0.130550
$856$	− 5971.59i	− 0.238440i
$857$	−33234.6	−1.32470	−0.662352	−	0.749193i	$-0.730442\pi$
$857$	−33234.6	−1.32470	−0.662352	+	0.749193i	$0.730442\pi$
$858$	0	0
$859$	42697.7	1.69596	0.847978	−	0.530032i	$-0.177820\pi$
$859$	42697.7	1.69596	0.847978	+	0.530032i	$0.177820\pi$
$860$	41390.3i	1.64116i
$861$	4509.52	0.178495
$862$	−56525.0	−2.23347
$863$	27419.2i	1.08153i	0.841174	+	0.540765i	$0.181865\pi$
$863$	27419.2i	1.08153i	−0.841174	+	0.540765i	$0.818135\pi$
$864$	− 7013.40i	− 0.276158i
$865$	36153.8i	1.42112i
$866$	41437.0i	1.62596i
$867$	9155.45	0.358634
$868$	87177.9	3.40900
$869$	36358.3i	1.41930i
$870$	25624.5	0.998567
$871$	0	0
$872$	1365.09	0.0530137
$873$	− 14044.4i	− 0.544482i
$874$	−2114.54	−0.0818367
$875$	−29272.4	−1.13096
$876$	10522.2i	0.405836i
$877$	42604.2i	1.64041i	0.572068	+	0.820206i	$0.306141\pi$
$877$	42604.2i	1.64041i	−0.572068	+	0.820206i	$0.693859\pi$
$878$	− 33535.2i	− 1.28902i
$879$	− 4241.56i	− 0.162758i
$880$	−29940.0	−1.14691
$881$	4699.40	0.179712	0.0898562	−	0.995955i	$-0.471359\pi$
$881$	4699.40	0.179712	0.0898562	+	0.995955i	$0.471359\pi$
$882$	23909.9i	0.912799i
$883$	−22233.5	−0.847358	−0.423679	−	0.905812i	$-0.639262\pi$
$883$	−22233.5	−0.847358	−0.423679	+	0.905812i	$0.639262\pi$
$884$	0	0
$885$	−23815.3	−0.904568
$886$	9774.98i	0.370651i
$887$	−8852.46	−0.335103	−0.167552	−	0.985863i	$-0.553586\pi$
$887$	−8852.46	−0.335103	−0.167552	+	0.985863i	$0.553586\pi$
$888$	−538.209	−0.0203391
$889$	− 7766.05i	− 0.292986i
$890$	51270.0i	1.93098i
$891$	− 3280.19i	− 0.123334i
$892$	505.155i	0.0189617i
$893$	−3600.68	−0.134930
$894$	−1849.92	−0.0692063
$895$	20486.7i	0.765135i
$896$	16453.6	0.613477
$897$	0	0
$898$	−53283.7	−1.98007
$899$	− 47508.0i	− 1.76249i
$900$	4511.47	0.167091
$901$	−18915.5	−0.699410
$902$	− 7987.58i	− 0.294853i
$903$	32250.3i	1.18851i
$904$	2866.40i	0.105459i
$905$	− 6410.41i	− 0.235458i
$906$	−9895.85	−0.362878
$907$	−37172.4	−1.36085	−0.680424	−	0.732818i	$-0.738205\pi$
$907$	−37172.4	−1.36085	−0.680424	+	0.732818i	$0.738205\pi$
$908$	6006.51i	0.219530i
$909$	−8629.56	−0.314878
$910$	0	0
$911$	38035.1	1.38327	0.691635	−	0.722247i	$-0.256891\pi$
$911$	38035.1	1.38327	0.691635	+	0.722247i	$0.256891\pi$
$912$	− 4451.36i	− 0.161622i
$913$	−52654.8	−1.90867
$914$	34639.3	1.25357
$915$	21829.2i	0.788691i
$916$	6509.70i	0.234810i
$917$	14838.8i	0.534372i
$918$	4802.67i	0.172671i
$919$	−8352.27	−0.299800	−0.149900	−	0.988701i	$-0.547895\pi$
$919$	−8352.27	−0.299800	−0.149900	+	0.988701i	$0.547895\pi$
$920$	−1053.62	−0.0377573
$921$	− 13876.9i	− 0.496482i
$922$	72679.4	2.59606
$923$	0	0
$924$	34356.6	1.22321
$925$	2423.47i	0.0861440i
$926$	22519.6	0.799179
$927$	−5716.38	−0.202535
$928$	− 40031.3i	− 1.41605i
$929$	20232.1i	0.714524i	0.934004	+	0.357262i	$0.116290\pi$
$929$	20232.1i	0.714524i	−0.934004	+	0.357262i	$0.883710\pi$
$930$	− 51257.0i	− 1.80729i
$931$	17382.8i	0.611921i
$932$	−2479.06	−0.0871292
$933$	−18182.4	−0.638011
$934$	34065.9i	1.19344i
$935$	23484.7	0.821423
$936$	0	0
$937$	−27766.9	−0.968095	−0.484048	−	0.875042i	$-0.660834\pi$
$937$	−27766.9	−0.968095	−0.484048	+	0.875042i	$0.660834\pi$
$938$	− 29842.6i	− 1.03880i
$939$	−2909.84	−0.101128
$940$	−16147.1	−0.560277
$941$	− 400.765i	− 0.0138837i	−0.999976	−	0.00694185i	$-0.997790\pi$
$941$	− 400.765i	− 0.0138837i	0.999976	−	0.00694185i	$-0.00220968\pi$
$942$	− 33473.3i	− 1.15777i
$943$	909.408i	0.0314045i
$944$	32480.4i	1.11986i
$945$	11404.4	0.392576
$946$	57124.0	1.96328
$947$	− 21804.4i	− 0.748201i	−0.927388	−	0.374101i	$-0.877951\pi$
$947$	− 21804.4i	− 0.748201i	0.927388	−	0.374101i	$-0.122049\pi$
$948$	−24241.1	−0.830502
$949$	0	0
$950$	6195.35	0.211583
$951$	− 26224.9i	− 0.894217i
$952$	−5589.22	−0.190281
$953$	−30480.4	−1.03605	−0.518026	−	0.855365i	$-0.673333\pi$
$953$	−30480.4	−1.03605	−0.518026	+	0.855365i	$0.673333\pi$
$954$	16270.1i	0.552165i
$955$	− 18412.6i	− 0.623892i
$956$	13764.5i	0.465666i
$957$	− 18722.8i	− 0.632415i
$958$	6497.48	0.219127
$959$	−57524.0	−1.93696
$960$	− 25446.4i	− 0.855499i
$961$	−65239.6	−2.18991
$962$	0	0
$963$	−13034.9	−0.436183
$964$	8779.73i	0.293336i
$965$	−28830.1	−0.961734
$966$	−7388.56	−0.246090
$967$	23864.1i	0.793608i	0.917903	+	0.396804i	$0.129881\pi$
$967$	23864.1i	0.793608i	−0.917903	+	0.396804i	$0.870119\pi$
$968$	− 1273.79i	− 0.0422947i
$969$	3491.60i	0.115755i
$970$	− 86489.2i	− 2.86289i
$971$	−14010.3	−0.463040	−0.231520	−	0.972830i	$-0.574370\pi$
$971$	−14010.3	−0.463040	−0.231520	+	0.972830i	$0.574370\pi$
$972$	2187.00	0.0721688
$973$	− 3142.19i	− 0.103529i
$974$	−51933.3	−1.70847
$975$	0	0
$976$	29771.7	0.976404
$977$	− 25448.2i	− 0.833326i	−0.909061	−	0.416663i	$-0.863200\pi$
$977$	− 25448.2i	− 0.833326i	0.909061	−	0.416663i	$-0.136800\pi$
$978$	45497.4	1.48757
$979$	37460.8	1.22293
$980$	77952.4i	2.54092i
$981$	− 2979.76i	− 0.0969789i
$982$	− 4416.72i	− 0.143527i
$983$	− 52479.9i	− 1.70280i	−0.524519	−	0.851399i	$-0.675755\pi$
$983$	− 52479.9i	− 1.70280i	0.524519	−	0.851399i	$-0.324245\pi$
$984$	591.729	0.0191703
$985$	3222.66	0.104246
$986$	27412.8i	0.885398i
$987$	−12581.4	−0.405746
$988$	0	0
$989$	−6503.73	−0.209107
$990$	− 20200.2i	− 0.648491i
$991$	38048.5	1.21963	0.609814	−	0.792545i	$-0.291244\pi$
$991$	38048.5	1.21963	0.609814	+	0.792545i	$0.291244\pi$
$992$	−80075.0	−2.56289
$993$	− 20963.3i	− 0.669939i
$994$	58223.9i	1.85790i
$995$	21372.5i	0.680960i
$996$	− 35106.5i	− 1.11686i
$997$	−31236.6	−0.992251	−0.496125	−	0.868251i	$-0.665244\pi$
$997$	−31236.6	−0.992251	−0.496125	+	0.868251i	$0.665244\pi$
$998$	−5864.30	−0.186003
$999$	1174.81i	0.0372066i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.e.337.2		4
13.5	odd	4		507.4.a.k.1.2		4
13.8	odd	4		507.4.a.k.1.3		4
13.9	even	3		39.4.j.b.10.1	yes	4
13.10	even	6		39.4.j.b.4.1	✓	4
13.12	even	2	inner	507.4.b.e.337.3		4
39.5	even	4		1521.4.a.z.1.3		4
39.8	even	4		1521.4.a.z.1.2		4
39.23	odd	6		117.4.q.d.82.2		4
39.35	odd	6		117.4.q.d.10.2		4
52.23	odd	6		624.4.bv.c.433.1		4
52.35	odd	6		624.4.bv.c.49.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.b.4.1	✓	4	13.10	even	6
39.4.j.b.10.1	yes	4	13.9	even	3
117.4.q.d.10.2		4	39.35	odd	6
117.4.q.d.82.2		4	39.23	odd	6
507.4.a.k.1.2		4	13.5	odd	4
507.4.a.k.1.3		4	13.8	odd	4
507.4.b.e.337.2		4	1.1	even	1	trivial
507.4.b.e.337.3		4	13.12	even	2	inner
624.4.bv.c.49.2		4	52.35	odd	6
624.4.bv.c.433.1		4	52.23	odd	6
1521.4.a.z.1.2		4	39.8	even	4
1521.4.a.z.1.3		4	39.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} +13.4424i q^{5} +12.3693i q^{6} +31.4219i q^{7} +4.12311i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} +13.4424i q^{5} +12.3693i q^{6} +31.4219i q^{7} +4.12311i q^{8} +9.00000 q^{9} +55.4243 q^{10} -40.4962i q^{11} +27.0000 q^{12} +129.556 q^{14} -40.3271i q^{15} -55.0000 q^{16} +43.1414 q^{17} -37.1080i q^{18} -26.9779i q^{19} -120.981i q^{20} -94.2656i q^{21} -166.970 q^{22} +19.0100 q^{23} -12.3693i q^{24} -55.6971 q^{25} -27.0000 q^{27} -282.797i q^{28} -154.111 q^{29} -166.273 q^{30} +308.270i q^{31} +259.756i q^{32} +121.489i q^{33} -177.877i q^{34} -422.384 q^{35} -81.0000 q^{36} -43.5116i q^{37} -111.233 q^{38} -55.4243 q^{40} +47.8384i q^{41} -388.667 q^{42} -342.121 q^{43} +364.466i q^{44} +120.981i q^{45} -78.3802i q^{46} -133.468i q^{47} +165.000 q^{48} -644.334 q^{49} +229.645i q^{50} -129.424 q^{51} -438.454 q^{53} +111.324i q^{54} +544.364 q^{55} -129.556 q^{56} +80.9338i q^{57} +635.418i q^{58} -590.553i q^{59} +362.944i q^{60} -541.304 q^{61} +1271.03 q^{62} +282.797i q^{63} +631.000 q^{64} +500.910 q^{66} -230.345i q^{67} -388.273 q^{68} -57.0300 q^{69} +1741.54i q^{70} +449.412i q^{71} +37.1080i q^{72} +389.711i q^{73} -179.403 q^{74} +167.091 q^{75} +242.801i q^{76} +1272.47 q^{77} -897.820 q^{79} -739.330i q^{80} +81.0000 q^{81} +197.243 q^{82} -1300.24i q^{83} +848.391i q^{84} +579.923i q^{85} +1410.60i q^{86} +462.334 q^{87} +166.970 q^{88} +925.045i q^{89} +498.819 q^{90} -171.090 q^{92} -924.811i q^{93} -550.301 q^{94} +362.647 q^{95} -779.267i q^{96} -1560.49i q^{97} +2656.66i q^{98} -364.466i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} - 36 q^{4} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 - 36 * q^4 + 36 * q^9	\( 4 q - 12 q^{3} - 36 q^{4} + 36 q^{9} + 136 q^{10} + 108 q^{12} + 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} + 120 q^{25} - 108 q^{27} + 12 q^{29} - 408 q^{30} - 804 q^{35} - 324 q^{36} + 612 q^{38} - 136 q^{40} - 612 q^{42} - 940 q^{43} + 660 q^{48} - 692 q^{49} - 432 q^{51} - 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} + 2524 q^{64} + 204 q^{66} - 1296 q^{68} - 828 q^{69} - 3060 q^{74} - 360 q^{75} + 2976 q^{77} + 8 q^{79} + 324 q^{81} - 68 q^{82} - 36 q^{87} + 68 q^{88} + 1224 q^{90} - 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100}) \) 4 * q - 12 * q^3 - 36 * q^4 + 36 * q^9 + 136 * q^10 + 108 * q^12 + 204 * q^14 - 220 * q^16 + 144 * q^17 - 68 * q^22 + 276 * q^23 + 120 * q^25 - 108 * q^27 + 12 * q^29 - 408 * q^30 - 804 * q^35 - 324 * q^36 + 612 * q^38 - 136 * q^40 - 612 * q^42 - 940 * q^43 + 660 * q^48 - 692 * q^49 - 432 * q^51 - 2268 * q^53 + 892 * q^55 - 204 * q^56 + 320 * q^61 + 2856 * q^62 + 2524 * q^64 + 204 * q^66 - 1296 * q^68 - 828 * q^69 - 3060 * q^74 - 360 * q^75 + 2976 * q^77 + 8 * q^79 + 324 * q^81 - 68 * q^82 - 36 * q^87 + 68 * q^88 + 1224 * q^90 - 2484 * q^92 - 5372 * q^94 + 108 * q^95

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