LMFDB - Embedded newform 507.4.b.e.337.3

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

$q$-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	$0.259954\pi$
$2$	4.12311i	1.45774i	−0.684653	+	0.728869i	$0.740046\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−9.00000	−1.12500
$5$	− 13.4424i	− 1.20232i	−0.799128	−	0.601161i	$-0.794705\pi$
$5$	− 13.4424i	− 1.20232i	0.799128	−	0.601161i	$-0.205295\pi$
$6$	− 12.3693i	− 0.841625i
$7$	− 31.4219i	− 1.69662i	−0.529498	−	0.848311i	$-0.677620\pi$
$7$	− 31.4219i	− 1.69662i	0.529498	−	0.848311i	$-0.322380\pi$
$8$	− 4.12311i	− 0.182217i
$9$	9.00000	0.333333
$10$	55.4243	1.75267
$11$	40.4962i	1.11001i	0.831849	+	0.555003i	$0.187283\pi$
$11$	40.4962i	1.11001i	−0.831849	+	0.555003i	$0.812717\pi$
$12$	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	$0.0520760\pi$
$19$	26.9779i	0.325745i	−0.986647	+	0.162873i	$0.947924\pi$
$20$	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	$-0.648585\pi$
$31$	− 308.270i	− 1.78603i	0.450025	−	0.893016i	$-0.351415\pi$
$32$	− 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	$0.0308178\pi$
$37$	43.5116i	0.193331i	−0.995317	+	0.0966657i	$0.969182\pi$
$38$	−111.233	−0.474851
$39$	0	0
$40$	−55.4243	−0.219084
$41$	− 47.8384i	− 0.182222i	−0.995841	−	0.0911110i	$-0.970958\pi$
$41$	− 47.8384i	− 0.182222i	0.995841	−	0.0911110i	$-0.0290418\pi$
$42$	−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	$0.0664055\pi$
$47$	133.468i	0.414218i	−0.978318	+	0.207109i	$0.933594\pi$
$48$	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	$0.225883\pi$
$59$	590.553i	1.30311i	−0.758601	+	0.651555i	$0.774117\pi$
$60$	− 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	$0.0673493\pi$
$67$	230.345i	0.420018i	−0.977700	+	0.210009i	$0.932651\pi$
$68$	−388.273	−0.692426
$69$	−57.0300	−0.0995015
$70$	− 1741.54i	− 2.97362i
$71$	− 449.412i	− 0.751203i	−0.926781	−	0.375601i	$-0.877436\pi$
$71$	− 449.412i	− 0.751203i	0.926781	−	0.375601i	$-0.122564\pi$
$72$	− 37.1080i	− 0.0607391i
$73$	− 389.711i	− 0.624826i	−0.949946	−	0.312413i	$-0.898863\pi$
$73$	− 389.711i	− 0.624826i	0.949946	−	0.312413i	$-0.101137\pi$
$74$	−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	$0.329386\pi$
$83$	1300.24i	1.71952i	−0.510700	+	0.859759i	$0.670614\pi$
$84$	− 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	$-0.814297\pi$
$89$	− 925.045i	− 1.10174i	0.834592	−	0.550869i	$-0.185703\pi$
$90$	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	$0.304211\pi$
$97$	1560.49i	1.63344i	−0.577031	+	0.816722i	$0.695789\pi$
$98$	− 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	$0.0464689\pi$
$109$	331.084i	0.290937i	−0.989363	+	0.145468i	$0.953531\pi$
$110$	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	$-0.806622\pi$
$137$	− 1830.70i	− 1.14166i	0.821069	−	0.570829i	$-0.193378\pi$
$138$	− 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	$-0.986909\pi$
$149$	− 149.557i	− 0.0822293i	0.999154	−	0.0411147i	$-0.0130909\pi$
$150$	688.936i	0.375009i
$151$	− 800.032i	− 0.431163i	−0.976486	−	0.215582i	$-0.930835\pi$
$151$	− 800.032i	− 0.431163i	0.976486	−	0.215582i	$-0.0691647\pi$
$152$	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	$0.344990\pi$
$163$	3678.25i	1.76750i	−0.467959	+	0.883750i	$0.655010\pi$
$164$	430.546i	0.205000i
$165$	−1633.09	−0.770522
$166$	−5361.03	−2.50661
$167$	− 3223.11i	− 1.49348i	−0.665114	−	0.746742i	$-0.731617\pi$
$167$	− 3223.11i	− 1.49348i	0.665114	−	0.746742i	$-0.268383\pi$
$168$	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	$-0.869028\pi$
$193$	− 2144.72i	− 0.799898i	0.916537	−	0.399949i	$-0.130972\pi$
$194$	−6434.08	−2.38113
$195$	0	0
$196$	5799.01	2.11334
$197$	239.739i	0.0867040i	0.999060	+	0.0433520i	$0.0138037\pi$
$197$	239.739i	0.0867040i	−0.999060	+	0.0433520i	$0.986196\pi$
$198$	−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	$0.00268256\pi$
$223$	56.1283i	0.0168548i	−0.999964	+	0.00842742i	$0.997317\pi$
$224$	−8162.01	−2.43459
$225$	−501.274	−0.148526
$226$	2866.40i	0.843673i
$227$	667.390i	0.195137i	0.995229	+	0.0975687i	$0.0311066\pi$
$227$	667.390i	0.195137i	−0.995229	+	0.0975687i	$0.968893\pi$
$228$	728.404i	0.211578i
$229$	723.299i	0.208720i	0.994540	+	0.104360i	$0.0332795\pi$
$229$	723.299i	0.208720i	−0.994540	+	0.104360i	$0.966721\pi$
$230$	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	$0.0663579\pi$
$239$	1529.39i	0.413925i	−0.978349	+	0.206963i	$0.933642\pi$
$240$	− 2217.99i	− 0.596544i
$241$	975.526i	0.260743i	0.991465	+	0.130372i	$0.0416170\pi$
$241$	975.526i	0.260743i	−0.991465	+	0.130372i	$0.958383\pi$
$242$	− 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	$0.487483\pi$
$271$	8915.55i	1.99845i	−0.0393133	+	0.999227i	$0.512517\pi$
$272$	−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	$0.0626103\pi$
$281$	1841.12i	0.390860i	−0.980718	+	0.195430i	$0.937390\pi$
$282$	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	$-0.954983\pi$
$293$	− 1413.85i	− 0.281905i	0.990016	−	0.140953i	$-0.0450165\pi$
$294$	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	$-0.858526\pi$
$307$	− 4625.64i	− 0.859932i	0.902845	−	0.429966i	$-0.141474\pi$
$308$	−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	$-0.718045\pi$
$317$	− 8741.63i	− 1.54883i	0.632679	−	0.774414i	$-0.281955\pi$
$318$	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	$-0.802981\pi$
$331$	− 6987.76i	− 1.16037i	0.814485	−	0.580184i	$-0.197019\pi$
$332$	− 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	$-0.888917\pi$
$349$	− 4458.75i	− 0.683872i	0.939723	−	0.341936i	$-0.111083\pi$
$350$	−7215.88	−1.10201
$351$	0	0
$352$	10519.1	1.59281
$353$	2249.17i	0.339126i	0.985519	+	0.169563i	$0.0542356\pi$
$353$	2249.17i	0.339126i	−0.985519	+	0.169563i	$0.945764\pi$
$354$	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	$-0.804418\pi$
$359$	− 7842.79i	− 1.15300i	0.817098	−	0.576499i	$-0.195582\pi$
$360$	−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	$-0.694749\pi$
$379$	− 12079.9i	− 1.63721i	0.574360	−	0.818603i	$-0.305251\pi$
$380$	−3263.82	−0.440607
$381$	741.463	0.0997015
$382$	− 5647.59i	− 0.756429i
$383$	10567.4i	1.40984i	0.709287	+	0.704919i	$0.249017\pi$
$383$	10567.4i	1.40984i	−0.709287	+	0.704919i	$0.750983\pi$
$384$	− 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	$-0.645300\pi$
$397$	− 14200.5i	− 1.79522i	0.440786	−	0.897612i	$-0.354700\pi$
$398$	6555.48i	0.825620i
$399$	−2543.09	−0.319082
$400$	3063.34	0.382918
$401$	− 12676.4i	− 1.57863i	−0.613992	−	0.789313i	$-0.710437\pi$
$401$	− 12676.4i	− 1.57863i	0.613992	−	0.789313i	$-0.289563\pi$
$402$	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	$-0.970448\pi$
$409$	− 1533.68i	− 0.185417i	0.995693	−	0.0927083i	$-0.0295524\pi$
$410$	− 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	$-0.986462\pi$
$421$	− 734.575i	− 0.0850380i	0.999096	−	0.0425190i	$-0.0135383\pi$
$422$	− 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	$0.277792\pi$
$431$	13709.3i	1.53215i	−0.642754	+	0.766073i	$0.722208\pi$
$432$	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	$0.237655\pi$
$449$	12923.2i	1.35832i	−0.733992	+	0.679158i	$0.762345\pi$
$450$	− 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	$-0.858523\pi$
$457$	− 8401.26i	− 0.859944i	0.902842	−	0.429972i	$-0.141477\pi$
$458$	−2982.24	−0.304260
$459$	−1164.82	−0.118451
$460$	2299.85i	0.233111i
$461$	− 17627.3i	− 1.78088i	−0.455098	−	0.890441i	$-0.650396\pi$
$461$	− 17627.3i	− 1.78088i	0.455098	−	0.890441i	$-0.349604\pi$
$462$	− 15739.5i	− 1.58500i
$463$	− 5461.81i	− 0.548233i	−0.961697	−	0.274116i	$-0.911615\pi$
$463$	− 5461.81i	− 0.548233i	0.961697	−	0.274116i	$-0.0883853\pi$
$464$	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	$-0.976053\pi$
$479$	− 1575.87i	− 0.150320i	0.997171	−	0.0751601i	$-0.0239468\pi$
$480$	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	$0.199299\pi$
$487$	12595.7i	1.17200i	−0.810310	+	0.586001i	$0.800701\pi$
$488$	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	$0.0203215\pi$
$499$	1422.30i	0.127597i	−0.997963	+	0.0637985i	$0.979678\pi$
$500$	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	$-0.795928\pi$
$509$	− 13736.3i	− 1.19617i	0.801432	−	0.598086i	$-0.204072\pi$
$510$	−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	$0.259923\pi$
$541$	18341.5i	1.45761i	−0.684723	+	0.728803i	$0.740077\pi$
$542$	−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	$0.00503106\pi$
$557$	415.532i	0.0316098i	−0.999875	+	0.0158049i	$0.994969\pi$
$558$	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	$-0.743872\pi$
$577$	− 19974.7i	− 1.44117i	0.693364	−	0.720587i	$-0.256128\pi$
$578$	− 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	$0.186776\pi$
$587$	15748.7i	1.10735i	−0.832732	+	0.553677i	$0.813224\pi$
$588$	−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	$0.152562\pi$
$593$	13318.4i	0.922297i	−0.887323	+	0.461148i	$0.847438\pi$
$594$	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	$-0.759808\pi$
$613$	− 20792.3i	− 1.36998i	0.728555	−	0.684988i	$-0.240192\pi$
$614$	19072.0	1.25356
$615$	1929.18	0.126491
$616$	− 5246.51i	− 0.343162i
$617$	1562.78i	0.101969i	0.998699	+	0.0509846i	$0.0162359\pi$
$617$	1562.78i	0.101969i	−0.998699	+	0.0509846i	$0.983764\pi$
$618$	7856.41i	0.511377i
$619$	758.406i	0.0492454i	0.999697	+	0.0246227i	$0.00783844\pi$
$619$	758.406i	0.0492454i	−0.999697	+	0.0246227i	$0.992162\pi$
$620$	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	$-0.851427\pi$
$631$	− 14265.2i	− 0.899981i	0.893033	−	0.449990i	$-0.148573\pi$
$632$	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	$0.0804723\pi$
$643$	8156.55i	0.500254i	−0.968213	+	0.250127i	$0.919528\pi$
$644$	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	$0.151717\pi$
$661$	15593.6i	0.917581i	−0.888545	+	0.458790i	$0.848283\pi$
$662$	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	$0.328335\pi$
$683$	30632.5i	1.71614i	−0.513537	+	0.858068i	$0.671665\pi$
$684$	− 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	$-0.980925\pi$
$691$	− 2175.72i	− 0.119780i	0.998205	−	0.0598901i	$-0.0190750\pi$
$692$	−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	$-0.767174\pi$
$709$	− 25219.7i	− 1.33589i	0.744211	−	0.667945i	$-0.232826\pi$
$710$	− 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	$-0.867729\pi$
$733$	− 16022.5i	− 0.807371i	0.914898	−	0.403685i	$-0.132271\pi$
$734$	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	$0.0301198\pi$
$739$	3796.21i	0.188966i	−0.995526	+	0.0944830i	$0.969880\pi$
$740$	−5264.09	−0.261502
$741$	0	0
$742$	−56804.3	−2.81044
$743$	30329.3i	1.49754i	0.662827	+	0.748772i	$0.269356\pi$
$743$	30329.3i	1.49754i	−0.662827	+	0.748772i	$0.730644\pi$
$744$	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	$-0.731360\pi$
$761$	− 31375.4i	− 1.49456i	0.664512	−	0.747278i	$-0.268640\pi$
$762$	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	$-0.905688\pi$
$769$	− 12452.7i	− 0.583946i	0.956427	−	0.291973i	$-0.0943118\pi$
$770$	70525.5	3.30073
$771$	−5454.56	−0.254788
$772$	19302.5i	0.899885i
$773$	37449.8i	1.74253i	0.490815	+	0.871264i	$0.336699\pi$
$773$	37449.8i	1.74253i	−0.490815	+	0.871264i	$0.663301\pi$
$774$	− 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	$-0.795470\pi$
$787$	− 26460.2i	− 1.19848i	0.800570	−	0.599240i	$-0.204530\pi$
$788$	− 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	$0.148487\pi$
$811$	20774.6i	0.899499i	−0.893155	+	0.449749i	$0.851513\pi$
$812$	− 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	$-0.725070\pi$
$821$	− 35769.2i	− 1.52053i	0.649614	−	0.760264i	$-0.274930\pi$
$822$	−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	$0.122658\pi$
$827$	17878.6i	0.751753i	−0.926670	+	0.375877i	$0.877342\pi$
$828$	−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	$0.224800\pi$
$839$	31542.7i	1.29794i	−0.760813	+	0.648971i	$0.775200\pi$
$840$	− 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	$-0.855780\pi$
$853$	− 21810.6i	− 0.875476i	0.899103	−	0.437738i	$-0.144220\pi$
$854$	−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	$-0.818135\pi$
$863$	− 27419.2i	− 1.08153i	0.841174	−	0.540765i	$-0.181865\pi$
$864$	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	$-0.693859\pi$
$877$	− 42604.2i	− 1.64041i	0.572068	−	0.820206i	$-0.306141\pi$
$878$	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	$-0.883710\pi$
$929$	− 20232.1i	− 0.714524i	0.934004	−	0.357262i	$-0.116290\pi$
$930$	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	$0.00220968\pi$
$941$	400.765i	0.0138837i	−0.999976	+	0.00694185i	$0.997790\pi$
$942$	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	$0.122049\pi$
$947$	21804.4i	0.748201i	−0.927388	+	0.374101i	$0.877951\pi$
$948$	−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	$-0.870119\pi$
$967$	− 23864.1i	− 0.793608i	0.917903	−	0.396804i	$-0.129881\pi$
$968$	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	$0.136800\pi$
$977$	25448.2i	0.833326i	−0.909061	+	0.416663i	$0.863200\pi$
$978$	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	$0.324245\pi$
$983$	52479.9i	1.70280i	−0.524519	+	0.851399i	$0.675755\pi$
$984$	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.337.3		4
13.3	even	3		39.4.j.b.4.1	✓	4
13.4	even	6		39.4.j.b.10.1	yes	4
13.5	odd	4		507.4.a.k.1.3		4
13.8	odd	4		507.4.a.k.1.2		4
13.12	even	2	inner	507.4.b.e.337.2		4
39.5	even	4		1521.4.a.z.1.2		4
39.8	even	4		1521.4.a.z.1.3		4
39.17	odd	6		117.4.q.d.10.2		4
39.29	odd	6		117.4.q.d.82.2		4
52.3	odd	6		624.4.bv.c.433.1		4
52.43	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	13.3	even	3
39.4.j.b.10.1	yes	4	13.4	even	6
117.4.q.d.10.2		4	39.17	odd	6
117.4.q.d.82.2		4	39.29	odd	6
507.4.a.k.1.2		4	13.8	odd	4
507.4.a.k.1.3		4	13.5	odd	4
507.4.b.e.337.2		4	13.12	even	2	inner
507.4.b.e.337.3		4	1.1	even	1	trivial
624.4.bv.c.49.2		4	52.43	odd	6
624.4.bv.c.433.1		4	52.3	odd	6
1521.4.a.z.1.2		4	39.5	even	4
1521.4.a.z.1.3		4	39.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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