LMFDB - Embedded newform 72.4.l.a.11.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,4,Mod(11,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1]))

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

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

chi := DirichletCharacter("72.11");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	72.l (of order $6$, degree $2$, 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:	$4.24813752041$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			11.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	72.11
Dual form			72.4.l.a.59.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.44949 - 1.41421i) q^{2} +(-3.72474 + 3.62302i) q^{3} +(4.00000 + 6.92820i) q^{4} +(14.2474 - 3.60697i) q^{6} -22.6274i q^{8} +(0.747449 - 26.9897i) q^{9} +O(q^{10})$	\(q+(-2.44949 - 1.41421i) q^{2} +(-3.72474 + 3.62302i) q^{3} +(4.00000 + 6.92820i) q^{4} +(14.2474 - 3.60697i) q^{6} -22.6274i q^{8} +(0.747449 - 26.9897i) q^{9} +(-44.1186 - 25.4719i) q^{11} +(-40.0000 - 11.3137i) q^{12} +(-32.0000 + 55.4256i) q^{16} -131.682i q^{17} +(-40.0000 + 65.0538i) q^{18} -57.2270 q^{19} +(72.0454 + 124.786i) q^{22} +(81.9796 + 84.2814i) q^{24} +(62.5000 - 108.253i) q^{25} +(95.0000 + 103.238i) q^{27} +(156.767 - 90.5097i) q^{32} +(256.616 - 64.9663i) q^{33} +(-186.227 + 322.555i) q^{34} +(189.980 - 102.780i) q^{36} +(140.177 + 80.9313i) q^{38} +(-415.995 + 240.175i) q^{41} +(-136.931 + 237.172i) q^{43} -407.550i q^{44} +(-81.6163 - 322.383i) q^{48} +(-171.500 - 297.047i) q^{49} +(-306.186 + 176.777i) q^{50} +(477.088 + 490.483i) q^{51} +(-86.7015 - 387.230i) q^{54} +(213.156 - 207.335i) q^{57} +(-493.654 + 285.011i) q^{59} -512.000 q^{64} +(-720.454 - 203.775i) q^{66} +(-491.476 - 851.262i) q^{67} +(912.322 - 526.730i) q^{68} +(-610.706 - 16.9128i) q^{72} +1229.09 q^{73} +(159.407 + 629.654i) q^{75} +(-228.908 - 396.481i) q^{76} +(-727.883 - 40.3468i) q^{81} +1358.63 q^{82} +(590.327 + 340.825i) q^{83} +(670.824 - 387.300i) q^{86} +(-576.363 + 998.290i) q^{88} +1329.36i q^{89} +(-256.000 + 905.097i) q^{96} +(499.545 - 865.238i) q^{97} +970.151i q^{98} +(-720.454 + 1171.71i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{3} + 16 q^{4} + 8 q^{6} - 46 q^{9}+O(q^{10}) $ 4 * q - 10 * q^3 + 16 * q^4 + 8 * q^6 - 46 * q^9	$ 4 q - 10 q^{3} + 16 q^{4} + 8 q^{6} - 46 q^{9} - 54 q^{11} - 160 q^{12} - 128 q^{16} - 160 q^{18} + 212 q^{19} + 200 q^{22} - 64 q^{24} + 250 q^{25} + 380 q^{27} + 370 q^{33} - 304 q^{34} + 368 q^{36} + 1080 q^{38} - 1566 q^{41} + 290 q^{43} - 640 q^{48} - 686 q^{49} + 1198 q^{51} + 584 q^{54} + 10 q^{57} - 2538 q^{59} - 2048 q^{64} - 2000 q^{66} - 70 q^{67} + 2160 q^{68} - 640 q^{72} + 860 q^{73} + 1250 q^{75} + 848 q^{76} - 658 q^{81} + 320 q^{82} + 4104 q^{86} - 1600 q^{88} - 1024 q^{96} + 1910 q^{97} - 2000 q^{99}+O(q^{100}) $ 4 * q - 10 * q^3 + 16 * q^4 + 8 * q^6 - 46 * q^9 - 54 * q^11 - 160 * q^12 - 128 * q^16 - 160 * q^18 + 212 * q^19 + 200 * q^22 - 64 * q^24 + 250 * q^25 + 380 * q^27 + 370 * q^33 - 304 * q^34 + 368 * q^36 + 1080 * q^38 - 1566 * q^41 + 290 * q^43 - 640 * q^48 - 686 * q^49 + 1198 * q^51 + 584 * q^54 + 10 * q^57 - 2538 * q^59 - 2048 * q^64 - 2000 * q^66 - 70 * q^67 + 2160 * q^68 - 640 * q^72 + 860 * q^73 + 1250 * q^75 + 848 * q^76 - 658 * q^81 + 320 * q^82 + 4104 * q^86 - 1600 * q^88 - 1024 * q^96 + 1910 * q^97 - 2000 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$	−2.44949	−	1.41421i	−0.866025	−	0.500000i
$2$	−2.44949	−	1.41421i	−0.866025	−	0.500000i
$3$	−3.72474	+	3.62302i	−0.716827	+	0.697251i
$3$	−3.72474	+	3.62302i	−0.716827	+	0.697251i
$4$	4.00000	+	6.92820i	0.500000	+	0.866025i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$	14.2474	−	3.60697i	0.969416	−	0.245423i
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$		−	22.6274i		−	1.00000i
$9$	0.747449	−	26.9897i	0.0276833	−	0.999617i
$10$		0			0
$11$	−44.1186	−	25.4719i	−1.20930	−	0.698188i	−0.246691	−	0.969094i	$-0.579343\pi$
$11$	−44.1186	−	25.4719i	−1.20930	−	0.698188i	−0.962606	+	0.270906i	$0.912677\pi$
$12$	−40.0000	−	11.3137i	−0.962250	−	0.272166i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$		0			0
$16$	−32.0000	+	55.4256i	−0.500000	+	0.866025i
$17$		−	131.682i		−	1.87869i	−0.342978	−	0.939343i	$-0.611436\pi$
$17$		−	131.682i		−	1.87869i	0.342978	−	0.939343i	$-0.388564\pi$
$18$	−40.0000	+	65.0538i	−0.523783	+	0.851852i
$19$	−57.2270			−0.690989			−0.345494	−	0.938421i	$-0.612289\pi$
$19$	−57.2270			−0.690989			−0.345494	+	0.938421i	$0.612289\pi$
$20$		0			0
$21$		0			0
$22$	72.0454	+	124.786i	0.698188	+	1.20930i
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$	81.9796	+	84.2814i	0.697251	+	0.716827i
$25$	62.5000	−	108.253i	0.500000	−	0.866025i
$26$		0			0
$27$	95.0000	+	103.238i	0.677139	+	0.735855i
$28$		0			0
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	156.767	−	90.5097i	0.866025	−	0.500000i
$33$	256.616	−	64.9663i	1.35367	−	0.342703i
$34$	−186.227	+	322.555i	−0.939343	+	1.62699i
$35$		0			0
$36$	189.980	−	102.780i	0.879535	−	0.475834i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	140.177	+	80.9313i	0.598414	+	0.345494i
$39$		0			0
$40$		0			0
$41$	−415.995	+	240.175i	−1.58457	+	0.914854i	−0.590394	+	0.807116i	$0.701027\pi$
$41$	−415.995	+	240.175i	−1.58457	+	0.914854i	−0.994179	+	0.107738i	$0.965639\pi$
$42$		0			0
$43$	−136.931	+	237.172i	−0.485624	+	0.841126i	−0.999864	−	0.0165210i	$-0.994741\pi$
$43$	−136.931	+	237.172i	−0.485624	+	0.841126i	0.514239	+	0.857647i	$0.328074\pi$
$44$		−	407.550i		−	1.39638i
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$	−81.6163	−	322.383i	−0.245423	−	0.969416i
$49$	−171.500	−	297.047i	−0.500000	−	0.866025i
$50$	−306.186	+	176.777i	−0.866025	+	0.500000i
$51$	477.088	+	490.483i	1.30992	+	1.34669i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−86.7015	−	387.230i	−0.218492	−	0.975839i
$55$		0			0
$56$		0			0
$57$	213.156	−	207.335i	0.495320	−	0.481792i
$58$		0			0
$59$	−493.654	+	285.011i	−1.08929	+	0.628904i	−0.933388	−	0.358868i	$-0.883163\pi$
$59$	−493.654	+	285.011i	−1.08929	+	0.628904i	−0.155905	+	0.987772i	$0.549829\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	−512.000			−1.00000
$65$		0			0
$66$	−720.454	−	203.775i	−1.34366	−	0.380045i
$67$	−491.476	−	851.262i	−0.896170	−	1.55221i	−0.832350	−	0.554250i	$-0.813005\pi$
$67$	−491.476	−	851.262i	−0.896170	−	1.55221i	−0.0638199	−	0.997961i	$-0.520328\pi$
$68$	912.322	−	526.730i	1.62699	−	0.939343i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−610.706	−	16.9128i	−0.999617	−	0.0276833i
$73$	1229.09			1.97060			0.985301	−	0.170827i	$-0.0546438\pi$
$73$	1229.09			1.97060			0.985301	+	0.170827i	$0.0546438\pi$
$74$		0			0
$75$	159.407	+	629.654i	0.245423	+	0.969416i
$76$	−228.908	−	396.481i	−0.345494	−	0.598414i
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	−727.883	−	40.3468i	−0.998467	−	0.0553454i
$82$	1358.63			1.82971
$83$	590.327	+	340.825i	0.780684	+	0.450728i	0.836673	−	0.547703i	$-0.184498\pi$
$83$	590.327	+	340.825i	0.780684	+	0.450728i	−0.0559884	+	0.998431i	$0.517831\pi$
$84$		0			0
$85$		0			0
$86$	670.824	−	387.300i	0.841126	−	0.485624i
$87$		0			0
$88$	−576.363	+	998.290i	−0.698188	+	1.20930i
$89$			1329.36i			1.58328i	0.610988	+	0.791640i	$0.290773\pi$
$89$			1329.36i			1.58328i	−0.610988	+	0.791640i	$0.709227\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−256.000	+	905.097i	−0.272166	+	0.962250i
$97$	499.545	−	865.238i	0.522898	−	0.905687i	−0.476746	−	0.879041i	$-0.658184\pi$
$97$	499.545	−	865.238i	0.522898	−	0.905687i	0.999645	−	0.0266459i	$-0.00848265\pi$
$98$			970.151i			1.00000i
$99$	−720.454	+	1171.71i	−0.731398	+	1.18951i
$100$	1000.00			1.00000
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$	−474.974	−	1876.14i	−0.461073	−	1.82123i
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	2183.77i		−	1.97302i	−0.163711	−	0.986508i	$-0.552346\pi$
$107$		−	2183.77i		−	1.97302i	0.163711	−	0.986508i	$-0.447654\pi$
$108$	−335.251	+	1071.13i	−0.298699	+	0.954347i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2067.37	+	1193.60i	−1.72108	+	0.993665i	−0.804345	+	0.594162i	$0.797484\pi$
$113$	−2067.37	+	1193.60i	−1.72108	+	0.993665i	−0.916732	+	0.399502i	$0.869183\pi$
$114$	−815.339	+	206.416i	−0.669855	+	0.169584i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	1612.27			1.25781
$119$		0			0
$120$		0			0
$121$	632.135	+	1094.89i	0.474933	+	0.822607i
$122$		0			0
$123$	679.317	−	2401.75i	0.497983	−	1.76064i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	1254.14	+	724.077i	0.866025	+	0.500000i
$129$	−349.245	−	1379.51i	−0.238367	−	0.941544i
$130$		0			0
$131$	2363.76	−	1364.72i	1.57651	−	0.910197i	0.581166	−	0.813785i	$-0.302597\pi$
$131$	2363.76	−	1364.72i	1.57651	−	0.910197i	0.995342	−	0.0964118i	$-0.0307366\pi$
$132$	1476.56	+	1518.02i	0.973624	+	1.00096i
$133$		0			0
$134$			2780.21i			1.79234i
$135$		0			0
$136$	−2979.63			−1.87869
$137$	697.906	+	402.936i	0.435227	+	0.251279i	0.701571	−	0.712599i	$-0.252482\pi$
$137$	697.906	+	402.936i	0.435227	+	0.251279i	−0.266344	+	0.963878i	$0.585816\pi$
$138$		0			0
$139$	−1636.11	−	2833.83i	−0.998368	−	1.72922i	−0.548645	−	0.836056i	$-0.684856\pi$
$139$	−1636.11	−	2833.83i	−0.998368	−	1.72922i	−0.449723	−	0.893168i	$-0.648477\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1472.00	+	905.097i	0.851852	+	0.523783i
$145$		0			0
$146$	−3010.64	−	1738.19i	−1.70659	−	0.985301i
$147$	1715.00	+	485.075i	0.962250	+	0.272166i
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$	500.000	−	1767.77i	0.272166	−	0.962250i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$			1294.90i			0.690989i
$153$	−3554.06	−	98.4258i	−1.87797	−	0.0520082i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	1725.88	+	1128.21i	0.837025	+	0.547164i
$163$	−970.000			−0.466112			−0.233056	−	0.972463i	$-0.574873\pi$
$163$	−970.000			−0.466112			−0.233056	+	0.972463i	$0.574873\pi$
$164$	−3327.96	−	1921.40i	−1.58457	−	0.914854i
$165$		0			0
$166$	−964.000	−	1669.70i	−0.450728	−	0.780684i
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	−1098.50	+	1902.66i	−0.500000	+	0.866025i
$170$		0			0
$171$	−42.7743	+	1544.54i	−0.0191288	+	0.690724i
$172$	−2190.90			−0.971248
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$	2823.59	−	1630.20i	1.20930	−	0.698188i
$177$	806.134	−	2850.11i	0.342332	−	1.21033i
$178$	1880.00	−	3256.26i	0.791640	−	1.37116i
$179$		−	2870.85i		−	1.19876i	−0.800465	−	0.599379i	$-0.795414\pi$
$179$		−	2870.85i		−	1.19876i	0.800465	−	0.599379i	$-0.204586\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−3354.20	+	5809.65i	−1.31168	+	2.27189i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$	1907.07	−	1854.99i	0.716827	−	0.697251i
$193$	2660.90	+	4608.82i	0.992415	+	1.71891i	0.602670	+	0.797990i	$0.294103\pi$
$193$	2660.90	+	4608.82i	0.992415	+	1.71891i	0.389745	+	0.920923i	$0.372563\pi$
$194$	−2447.26	+	1412.93i	−0.905687	+	0.522898i
$195$		0			0
$196$	1372.00	−	2376.37i	0.500000	−	0.866025i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	3421.79	−	1851.21i	1.22816	−	0.664443i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−2449.49	−	1414.21i	−0.866025	−	0.500000i
$201$	4914.76	+	1390.10i	1.72468	+	0.487813i
$202$		0			0
$203$		0			0
$204$	−1489.82	+	5267.30i	−0.511314	+	1.80777i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2524.78	+	1457.68i	0.835610	+	0.482440i
$210$		0			0
$211$	3059.00	+	5298.34i	0.998058	+	1.72869i	0.552975	+	0.833198i	$0.313493\pi$
$211$	3059.00	+	5298.34i	0.998058	+	1.72869i	0.445083	+	0.895489i	$0.353174\pi$
$212$		0			0
$213$		0			0
$214$	−3088.31	+	5349.12i	−0.986508	+	1.70868i
$215$		0			0
$216$	2336.00	−	2149.60i	0.735855	−	0.677139i
$217$		0			0
$218$		0			0
$219$	−4578.04	+	4453.01i	−1.41258	+	1.37400i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$	−2875.00	−	1767.77i	−0.851852	−	0.523783i
$226$	6752.00			1.98733
$227$	4103.25	+	2369.01i	1.19974	+	0.692673i	0.960498	−	0.278286i	$-0.0897663\pi$
$227$	4103.25	+	2369.01i	1.19974	+	0.692673i	0.239246	+	0.970959i	$0.423100\pi$
$228$	2289.08	+	647.450i	0.664904	+	0.188063i
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	7108.69i		−	1.99874i	−0.0355143	−	0.999369i	$-0.511307\pi$
$233$		−	7108.69i		−	1.99874i	0.0355143	−	0.999369i	$-0.488693\pi$
$234$		0			0
$235$		0			0
$236$	−3949.23	−	2280.09i	−1.08929	−	0.628904i
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	2891.08	−	5007.50i	0.772743	−	1.33843i	−0.163311	−	0.986575i	$-0.552217\pi$
$241$	2891.08	−	5007.50i	0.772743	−	1.33843i	0.936054	−	0.351856i	$-0.114449\pi$
$242$		−	3575.90i		−	0.949865i
$243$	2857.35	−	2486.85i	0.754318	−	0.656509i
$244$		0			0
$245$		0			0
$246$	−5060.56	+	4922.36i	−1.31158	+	1.27576i
$247$		0			0
$248$		0			0
$249$	−3433.64	+	869.279i	−0.873887	+	0.221238i
$250$		0			0
$251$		−	381.026i		−	0.0958174i	−0.998852	−	0.0479087i	$-0.984744\pi$
$251$		−	381.026i		−	0.0958174i	0.998852	−	0.0479087i	$-0.0152556\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−2048.00	−	3547.24i	−0.500000	−	0.866025i
$257$	247.544	−	142.919i	0.0600831	−	0.0346890i	−0.469658	−	0.882849i	$-0.655623\pi$
$257$	247.544	−	142.919i	0.0600831	−	0.0346890i	0.529741	+	0.848160i	$0.322289\pi$
$258$	−1095.45	+	3873.00i	−0.264340	+	0.934584i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−7720.00			−1.82039
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$	−1470.02	−	5806.55i	−0.342703	−	1.35367i
$265$		0			0
$266$		0			0
$267$	−4816.30	−	4951.53i	−1.10394	−	1.13494i
$268$	3931.81	−	6810.09i	0.896170	−	1.55221i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	7298.58	+	4213.84i	1.62699	+	0.939343i
$273$		0			0
$274$	−1139.68	−	1973.98i	−0.251279	−	0.435227i
$275$	−5514.83	+	3183.99i	−1.20930	+	0.698188i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$			9255.24i			1.99674i
$279$		0			0
$280$		0			0
$281$	1053.28	+	608.112i	0.223607	+	0.129099i	0.607619	−	0.794229i	$-0.292125\pi$
$281$	1053.28	+	608.112i	0.223607	+	0.129099i	−0.384013	+	0.923328i	$0.625458\pi$
$282$		0			0
$283$	−4015.00	−	6954.18i	−0.843346	−	1.46072i	−0.887050	−	0.461674i	$-0.847249\pi$
$283$	−4015.00	−	6954.18i	−0.843346	−	1.46072i	0.0437035	−	0.999045i	$-0.486084\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−2325.65	−	4298.75i	−0.475834	−	0.879535i
$289$	−12427.3			−2.52946
$290$		0			0
$291$	1274.10	+	5032.65i	0.256663	+	1.01381i
$292$	4916.36	+	8515.38i	0.985301	+	1.70659i
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$	−3514.87	−	3613.56i	−0.697251	−	0.716827i
$295$		0			0
$296$		0			0
$297$	−1561.61	−	6974.53i	−0.305097	−	1.36264i
$298$		0			0
$299$		0			0
$300$	−3724.74	+	3623.02i	−0.716827	+	0.697251i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	1831.27	−	3171.84i	0.345494	−	0.598414i
$305$		0			0
$306$	8566.44	+	5267.30i	1.60036	+	0.984024i
$307$	10233.9			1.90253			0.951265	−	0.308375i	$-0.0997851\pi$
$307$	10233.9			1.90253			0.951265	+	0.308375i	$0.0997851\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	5227.95	−	9055.07i	0.944093	−	1.63522i	0.186536	−	0.982448i	$-0.440274\pi$
$313$	5227.95	−	9055.07i	0.944093	−	1.63522i	0.757557	−	0.652769i	$-0.226393\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	7911.83	+	8133.98i	1.37569	+	1.41431i
$322$		0			0
$323$			7535.79i			1.29815i
$324$	−2632.00	−	5204.31i	−0.451303	−	0.892371i
$325$		0			0
$326$	2376.01	+	1371.79i	0.403665	+	0.233056i
$327$		0			0
$328$	5434.53	+	9412.89i	0.914854	+	1.58457i
$329$		0			0
$330$		0			0
$331$	4121.00	−	7137.78i	0.684322	−	1.18528i	−0.289327	−	0.957230i	$-0.593432\pi$
$331$	4121.00	−	7137.78i	0.684322	−	1.18528i	0.973649	−	0.228051i	$-0.0732351\pi$
$332$			5453.21i			0.901457i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−780.232	−	1351.40i	−0.126118	−	0.218444i	0.796051	−	0.605229i	$-0.206919\pi$
$337$	−780.232	−	1351.40i	−0.126118	−	0.218444i	−0.922170	+	0.386786i	$0.873585\pi$
$338$	5381.53	−	3107.03i	0.866025	−	0.500000i
$339$	3376.00	−	11936.0i	0.540882	−	1.91231i
$340$		0			0
$341$		0			0
$342$	2289.08	−	3722.84i	0.361928	−	0.588620i
$343$		0			0
$344$	5366.59	+	3098.40i	0.841126	+	0.485624i
$345$		0			0
$346$		0			0
$347$	−9474.14	+	5469.90i	−1.46570	+	0.846224i	−0.999265	−	0.0383308i	$-0.987796\pi$
$347$	−9474.14	+	5469.90i	−1.46570	+	0.846224i	−0.466437	+	0.884554i	$0.654463\pi$
$348$		0			0
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$	−9221.81			−1.39638
$353$	1781.98	+	1028.83i	0.268684	+	0.155125i	0.628289	−	0.777980i	$-0.283755\pi$
$353$	1781.98	+	1028.83i	0.268684	+	0.155125i	−0.359605	+	0.933104i	$0.617089\pi$
$354$	−6005.29	+	5841.28i	−0.901631	+	0.877007i
$355$		0			0
$356$	−9210.08	+	5317.44i	−1.37116	+	0.791640i
$357$		0			0
$358$	−4060.00	+	7032.13i	−0.599379	+	1.03815i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−3584.07			−0.522535
$362$		0			0
$363$	−6321.35	−	1787.95i	−0.914008	−	0.258521i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$	6171.30	+	11407.1i	0.870637	+	1.60929i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$	16432.2	−	9487.11i	2.27189	−	1.31168i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1993.12			0.270131			0.135066	−	0.990837i	$-0.456875\pi$
$379$	1993.12			0.270131			0.135066	+	0.990837i	$0.456875\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$	−7294.69	+	1846.77i	−0.969416	+	0.245423i
$385$		0			0
$386$		−	15052.3i		−	1.98483i
$387$	6298.84	+	3873.00i	0.827360	+	0.508723i
$388$	7992.73			1.04580
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$		0			0
$392$	−6721.40	+	3880.60i	−0.866025	+	0.500000i
$393$	−3860.00	+	13647.2i	−0.495448	+	1.75167i
$394$		0			0
$395$		0			0
$396$	−10999.6	−	304.623i	−1.39584	−	0.0386563i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	4000.00	+	6928.20i	0.500000	+	0.866025i
$401$	1006.95	−	581.361i	0.125398	−	0.0723984i	−0.435989	−	0.899952i	$-0.643601\pi$
$401$	1006.95	−	581.361i	0.125398	−	0.0723984i	0.561387	+	0.827554i	$0.310268\pi$
$402$	−10072.8	−	10355.6i	−1.24971	−	1.28480i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	11098.4	−	10795.3i	1.34669	−	1.30992i
$409$	2984.23	+	5168.84i	0.360784	+	0.624896i	0.988090	−	0.153877i	$-0.0491758\pi$
$409$	2984.23	+	5168.84i	0.360784	+	0.624896i	−0.627306	+	0.778773i	$0.715842\pi$
$410$		0			0
$411$	−4059.37	+	1027.69i	−0.487187	+	0.123339i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	16361.1	+	4627.62i	1.92136	+	0.543443i
$418$	−4122.95	−	7141.15i	−0.482440	−	0.835610i
$419$	3025.12	−	1746.55i	0.352713	−	0.203639i	−0.313166	−	0.949698i	$-0.601390\pi$
$419$	3025.12	−	1746.55i	0.352713	−	0.203639i	0.665880	+	0.746059i	$0.268056\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$		−	17304.3i		−	1.99612i
$423$		0			0
$424$		0			0
$425$	−14255.0	−	8230.15i	−1.62699	−	0.939343i
$426$		0			0
$427$		0			0
$428$	15129.6	−	8735.07i	1.70868	−	0.986508i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−8762.01	+	1961.83i	−0.975839	+	0.218492i
$433$	−17613.6			−1.95486			−0.977432	−	0.211252i	$-0.932246\pi$
$433$	−17613.6			−1.95486			−0.977432	+	0.211252i	$0.932246\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	17511.4	−	4433.28i	1.91033	−	0.483631i
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−8145.37	+	4406.70i	−0.879535	+	0.475834i
$442$		0			0
$443$	−15320.4	−	8845.23i	−1.64310	−	0.948645i	−0.979722	−	0.200361i	$-0.935789\pi$
$443$	−15320.4	−	8845.23i	−1.64310	−	0.948645i	−0.663378	−	0.748284i	$-0.730878\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			11448.2i			1.20328i	0.798767	+	0.601641i	$0.205486\pi$
$449$			11448.2i			1.20328i	−0.798767	+	0.601641i	$0.794514\pi$
$450$	4542.28	+	8395.99i	0.475834	+	0.879535i
$451$	24470.8			2.55496
$452$	−16539.0	−	9548.77i	−1.72108	−	0.993665i
$453$		0			0
$454$	−6700.57	−	11605.7i	−0.692673	−	1.19974i
$455$		0			0
$456$	−4691.45	−	4823.17i	−0.481792	−	0.495320i
$457$	−7736.13	+	13399.4i	−0.791862	+	1.37154i	0.132951	+	0.991123i	$0.457555\pi$
$457$	−7736.13	+	13399.4i	−0.791862	+	1.37154i	−0.924813	+	0.380422i	$0.875779\pi$
$458$		0			0
$459$	13594.6	−	12509.8i	1.38244	−	1.27213i
$460$		0			0
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$		0			0
$465$		0			0
$466$	−10053.2	+	17412.7i	−0.999369	+	1.73096i
$467$			6280.46i			0.622324i	0.950357	+	0.311162i	$0.100718\pi$
$467$			6280.46i			0.622324i	−0.950357	+	0.311162i	$0.899282\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	6449.07	+	11170.1i	0.628904	+	1.08929i
$473$	12082.4	−	6975.80i	1.17453	−	0.678114i
$474$		0			0
$475$	−3576.69	+	6195.01i	−0.345494	+	0.598414i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$	−14163.4	+	8177.22i	−1.33843	+	0.772743i
$483$		0			0
$484$	−5057.08	+	8759.12i	−0.474933	+	0.822607i
$485$		0			0
$486$	−10516.0	+	2050.61i	−0.981513	+	0.191394i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	3613.00	−	3514.33i	0.334122	−	0.324997i
$490$		0			0
$491$	1371.00	−	791.547i	0.126013	−	0.0727536i	−0.435669	−	0.900107i	$-0.643488\pi$
$491$	1371.00	−	791.547i	0.126013	−	0.0727536i	0.561681	+	0.827354i	$0.310155\pi$
$492$	19357.1	−	4900.55i	1.77375	−	0.449052i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	9640.00	+	2726.60i	0.867427	+	0.245345i
$499$	−10120.0	−	17528.3i	−0.907880	−	1.57249i	−0.817005	−	0.576631i	$-0.804367\pi$
$499$	−10120.0	−	17528.3i	−0.907880	−	1.57249i	−0.0908749	−	0.995862i	$-0.528966\pi$
$500$		0			0
$501$		0			0
$502$	−538.852	+	933.320i	−0.0479087	+	0.0829803i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2801.74	−	11066.8i	−0.245423	−	0.969416i
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$		0			0
$512$			11585.2i			1.00000i
$513$	−5436.57	−	5907.98i	−0.467895	−	0.508467i
$514$	−808.475			−0.0693780
$515$		0			0
$516$	8160.55	−	7937.68i	0.696217	−	0.677203i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	18908.8i		−	1.59004i	−0.606584	−	0.795020i	$-0.707460\pi$
$521$		−	18908.8i		−	1.59004i	0.606584	−	0.795020i	$-0.292540\pi$
$522$		0			0
$523$	−4750.00			−0.397138			−0.198569	−	0.980087i	$-0.563629\pi$
$523$	−4750.00			−0.397138			−0.198569	+	0.980087i	$0.563629\pi$
$524$	18910.1	+	10917.7i	1.57651	+	0.910197i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−4610.91	+	16302.0i	−0.380045	+	1.34366i
$529$	6083.50	−	10536.9i	0.500000	−	0.866025i
$530$		0			0
$531$	7323.38	+	13536.6i	0.598508	+	1.10629i
$532$		0			0
$533$		0			0
$534$	4794.96	+	18940.0i	0.388573	+	1.53486i
$535$		0			0
$536$	−19261.9	+	11120.8i	−1.55221	+	0.896170i
$537$	10401.2	+	10693.2i	0.835835	+	0.859303i
$538$		0			0
$539$			17473.7i			1.39638i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	−11918.5	−	20643.5i	−0.939343	−	1.62699i
$545$		0			0
$546$		0			0
$547$	−11227.4	+	19446.4i	−0.877601	+	1.52005i	−0.0236354	+	0.999721i	$0.507524\pi$
$547$	−11227.4	+	19446.4i	−0.877601	+	1.52005i	−0.853966	+	0.520329i	$0.825809\pi$
$548$			6446.98i			0.502557i
$549$		0			0
$550$	18011.4			1.39638
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	13088.9	−	22670.6i	0.998368	−	1.72922i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−8554.92	−	33791.8i	−0.643831	−	2.54312i
$562$	−1720.00	−	2979.13i	−0.129099	−	0.223607i
$563$	23118.1	−	13347.2i	1.73057	−	0.999146i	0.844629	−	0.535352i	$-0.179821\pi$
$563$	23118.1	−	13347.2i	1.73057	−	0.999146i	0.885943	−	0.463795i	$-0.153512\pi$
$564$		0			0
$565$		0			0
$566$			22712.3i			1.68669i
$567$		0			0
$568$		0			0
$569$	9913.06	+	5723.31i	0.730364	+	0.421676i	0.818555	−	0.574428i	$-0.194775\pi$
$569$	9913.06	+	5723.31i	0.730364	+	0.421676i	−0.0881913	+	0.996104i	$0.528109\pi$
$570$		0			0
$571$	5161.21	+	8939.47i	0.378266	+	0.655176i	0.990810	−	0.135261i	$-0.0431872\pi$
$571$	5161.21	+	8939.47i	0.378266	+	0.655176i	−0.612544	+	0.790436i	$0.709854\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−382.694	+	13818.7i	−0.0276833	+	0.999617i
$577$	7244.05			0.522658			0.261329	−	0.965250i	$-0.415839\pi$
$577$	7244.05			0.522658			0.261329	+	0.965250i	$0.415839\pi$
$578$	30440.4	+	17574.8i	2.19058	+	1.26473i
$579$	−26609.0	−	7526.17i	−1.90990	−	0.540202i
$580$		0			0
$581$		0			0
$582$	3996.36	−	14129.3i	0.284630	−	1.00632i
$583$		0			0
$584$		−	27811.1i		−	1.97060i
$585$		0			0
$586$		0			0
$587$	−3709.69	−	2141.79i	−0.260843	−	0.150598i	0.363876	−	0.931447i	$-0.381453\pi$
$587$	−3709.69	−	2141.79i	−0.260843	−	0.150598i	−0.624719	+	0.780849i	$0.714787\pi$
$588$	3499.30	+	13822.2i	0.245423	+	0.969416i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			12173.6i			0.843015i	0.906825	+	0.421507i	$0.138499\pi$
$593$			12173.6i			0.843015i	−0.906825	+	0.421507i	$0.861501\pi$
$594$	−6038.32	+	19292.5i	−0.417097	+	1.33263i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$	14247.4	−	3606.97i	0.969416	−	0.245423i
$601$	7533.43	−	13048.3i	0.511306	−	0.885608i	−0.488608	−	0.872503i	$-0.662495\pi$
$601$	7533.43	−	13048.3i	0.511306	−	0.885608i	0.999914	−	0.0131049i	$-0.00417154\pi$
$602$		0			0
$603$	−23342.6	+	12628.5i	−1.57643	+	0.852856i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	−8971.33	+	5179.60i	−0.598414	+	0.345494i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−13534.3	−	25017.0i	−0.893943	−	1.65237i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−25067.7	−	14472.8i	−1.64764	−	0.951265i
$615$		0			0
$616$		0			0
$617$	16545.1	−	9552.30i	1.07954	−	0.623275i	0.148771	−	0.988872i	$-0.452468\pi$
$617$	16545.1	−	9552.30i	1.07954	−	0.623275i	0.930774	+	0.365596i	$0.119135\pi$
$618$		0			0
$619$	−6629.34	+	11482.4i	−0.430462	+	0.745582i	−0.996913	−	0.0785136i	$-0.974983\pi$
$619$	−6629.34	+	11482.4i	−0.430462	+	0.745582i	0.566451	+	0.824095i	$0.308316\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7812.50	−	13531.6i	−0.500000	−	0.866025i
$626$	−25611.6	+	14786.9i	−1.63522	+	0.944093i
$627$	−14685.4	+	3717.83i	−0.935370	+	0.236804i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−30590.0	−	8652.16i	−1.92076	−	0.543274i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8745.38	−	5049.15i	−0.538880	−	0.311122i	0.205745	−	0.978606i	$-0.434038\pi$
$641$	−8745.38	−	5049.15i	−0.538880	−	0.311122i	−0.744625	+	0.667483i	$0.767372\pi$
$642$	−7876.78	−	31113.1i	−0.484224	−	1.91267i
$643$	314.446	+	544.637i	0.0192855	+	0.0334034i	0.875507	−	0.483205i	$-0.160528\pi$
$643$	314.446	+	544.637i	0.0192855	+	0.0334034i	−0.856222	+	0.516609i	$0.827194\pi$
$644$		0			0
$645$		0			0
$646$	10657.2	−	18458.8i	0.649076	−	1.12423i
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−912.943	+	16470.1i	−0.0553454	+	0.998467i
$649$	29039.1			1.75637
$650$		0			0
$651$		0			0
$652$	−3880.00	−	6720.36i	−0.233056	−	0.403665i
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$		−	30742.4i		−	1.82971i
$657$	918.681	−	33172.7i	0.0545527	−	1.96985i
$658$		0			0
$659$	13949.8	+	8053.95i	0.824596	+	0.476081i	0.851999	−	0.523544i	$-0.175390\pi$
$659$	13949.8	+	8053.95i	0.824596	+	0.476081i	−0.0274028	+	0.999624i	$0.508724\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$	−20188.7	+	11655.9i	−1.18528	+	0.684322i
$663$		0			0
$664$	7712.00	−	13357.6i	0.450728	−	0.780684i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−9595.00	+	16619.0i	−0.549569	+	0.951882i	0.448735	+	0.893665i	$0.351875\pi$
$673$	−9595.00	+	16619.0i	−0.549569	+	0.951882i	−0.998304	+	0.0582168i	$0.981459\pi$
$674$			4413.66i			0.252237i
$675$	17113.3	−	3831.70i	0.975839	−	0.218492i
$676$	−17576.0			−1.00000
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$	−25149.5	+	24462.6i	−1.42457	+	1.38567i
$679$		0			0
$680$		0			0
$681$	−23866.5	+	6042.19i	−1.34298	+	0.339996i
$682$		0			0
$683$		−	27182.7i		−	1.52287i	−0.648243	−	0.761434i	$-0.724496\pi$
$683$		−	27182.7i		−	1.52287i	0.648243	−	0.761434i	$-0.275504\pi$
$684$	−10872.0	+	5881.80i	−0.607749	+	0.328796i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−8763.61	−	15179.0i	−0.485624	−	0.841126i
$689$		0			0
$690$		0			0
$691$	989.000	−	1713.00i	0.0544477	−	0.0943061i	−0.837517	−	0.546411i	$-0.815994\pi$
$691$	989.000	−	1713.00i	0.0544477	−	0.0943061i	0.891965	+	0.452105i	$0.149327\pi$
$692$		0			0
$693$		0			0
$694$	30942.4			1.69245
$695$		0			0
$696$		0			0
$697$	31626.8	+	54779.2i	1.71872	+	2.97692i
$698$		0			0
$699$	25754.9	+	26478.1i	1.39362	+	1.43275i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	22588.7	+	13041.6i	1.20930	+	0.698188i
$705$		0			0
$706$	−2909.97	−	5040.21i	−0.155125	−	0.268684i
$707$		0			0
$708$	22970.7	−	5815.40i	1.21934	−	0.308695i
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$	30080.0			1.58328
$713$		0			0
$714$		0			0
$715$		0			0
$716$	19889.9	−	11483.4i	1.03815	−	0.599379i
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	8779.13	+	5068.63i	0.452528	+	0.261267i
$723$	7373.74	+	29126.1i	0.379298	+	1.49822i
$724$		0			0
$725$		0			0
$726$	12955.5	+	13319.3i	0.662294	+	0.680889i
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	−1633.00	+	19615.1i	−0.0829650	+	0.996552i
$730$		0			0
$731$	31231.4	+	18031.5i	1.58021	+	0.912336i
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			50075.3i			2.50278i
$738$	1015.51	−	36669.0i	0.0506523	−	1.82901i
$739$	−33358.6			−1.66051			−0.830253	−	0.557386i	$-0.811804\pi$
$739$	−33358.6			−1.66051			−0.830253	+	0.557386i	$0.811804\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	9640.00	−	15678.0i	0.472168	−	0.767908i
$748$	−53667.2			−2.62335
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$	1380.47	+	1419.23i	0.0668087	+	0.0686845i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−4882.13	−	2818.70i	−0.233940	−	0.135066i
$759$		0			0
$760$		0			0
$761$	−21114.6	+	12190.5i	−1.00579	+	0.580691i	−0.909955	−	0.414706i	$-0.863884\pi$
$761$	−21114.6	+	12190.5i	−1.00579	+	0.580691i	−0.0958314	+	0.995398i	$0.530551\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	20480.0	+	5792.62i	0.962250	+	0.272166i
$769$	−20053.0	−	34732.8i	−0.940351	−	1.62874i	−0.764803	−	0.644264i	$-0.777164\pi$
$769$	−20053.0	−	34732.8i	−0.940351	−	1.62874i	−0.175548	−	0.984471i	$-0.556170\pi$
$770$		0			0
$771$	−404.237	+	1429.19i	−0.0188823	+	0.0667590i
$772$	−21287.2	+	36870.6i	−0.992415	+	1.71891i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	−9951.70	−	18394.8i	−0.462153	−	0.854247i
$775$		0			0
$776$	−19578.1	−	11303.4i	−0.905687	−	0.522898i
$777$		0			0
$778$		0			0
$779$	23806.2	−	13744.5i	1.09492	−	0.632153i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	21952.0			1.00000
$785$		0			0
$786$	28755.0	−	27969.7i	1.30491	−	1.26927i
$787$	−3475.00	−	6018.88i	−0.157396	−	0.272617i	0.776533	−	0.630076i	$-0.216976\pi$
$787$	−3475.00	−	6018.88i	−0.157396	−	0.272617i	−0.933929	+	0.357459i	$0.883643\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	26512.7	+	16302.0i	1.18951	+	0.731398i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$		−	22627.4i		−	1.00000i
$801$	35879.0	+	993.629i	1.58267	+	0.0438304i
$802$	−3288.67			−0.144797
$803$	−54225.7	−	31307.2i	−2.38304	−	1.37585i
$804$	10028.1	+	39610.9i	0.439881	+	1.73752i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			9439.74i			0.410239i	0.978737	+	0.205120i	$0.0657584\pi$
$809$			9439.74i			0.410239i	−0.978737	+	0.205120i	$0.934242\pi$
$810$		0			0
$811$	−4466.91			−0.193408			−0.0967042	−	0.995313i	$-0.530830\pi$
$811$	−4466.91			−0.193408			−0.0967042	+	0.995313i	$0.530830\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−42452.2	+	10747.4i	−1.82123	+	0.461073i
$817$	7836.18	−	13572.7i	0.335561	−	0.581208i
$818$		−	16881.4i		−	0.721568i
$819$		0			0
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$	11396.8	+	3223.49i	0.483586	+	0.136779i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$	9005.68	−	31839.9i	0.380045	−	1.34366i
$826$		0			0
$827$		−	41360.1i		−	1.73909i	−0.493850	−	0.869547i	$-0.664411\pi$
$827$		−	41360.1i		−	1.73909i	0.493850	−	0.869547i	$-0.335589\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−39115.8	+	22583.5i	−1.62699	+	0.939343i
$834$	−33531.9	−	34473.4i	−1.39222	−	1.43131i
$835$		0			0
$836$			23322.9i			0.964880i
$837$		0			0
$838$	−9880.00			−0.407278
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	12194.5	+	21121.5i	0.500000	+	0.866025i
$842$		0			0
$843$	−6126.40	+	1551.00i	−0.250302	+	0.0633679i
$844$	−24472.0	+	42386.7i	−0.998058	+	1.72869i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	40150.0	+	11356.1i	1.62302	+	0.459060i
$850$	23278.4	+	40319.3i	0.939343	+	1.62699i
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$	−49413.0			−1.97302
$857$	−40348.0	−	23294.9i	−1.60824	−	0.928518i	−0.989764	−	0.142713i	$-0.954417\pi$
$857$	−40348.0	−	23294.9i	−1.60824	−	0.928518i	−0.618475	−	0.785804i	$-0.712249\pi$
$858$		0			0
$859$	−20615.5	−	35707.0i	−0.818848	−	1.41829i	−0.906532	−	0.422138i	$-0.861280\pi$
$859$	−20615.5	−	35707.0i	−0.818848	−	1.41829i	0.0876838	−	0.996148i	$-0.472053\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	24236.9	+	7585.86i	0.954347	+	0.298699i
$865$		0			0
$866$	43144.3	+	24909.4i	1.69296	+	0.977432i
$867$	46288.4	−	45024.2i	1.81319	−	1.76367i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−22979.1	−	14129.3i	−0.890864	−	0.547770i
$874$		0			0
$875$		0			0
$876$	−49163.6	−	13905.6i	−1.89621	−	0.536330i
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	31508.7i		−	1.20494i	−0.798141	−	0.602471i	$-0.794183\pi$
$881$		−	31508.7i		−	1.20494i	0.798141	−	0.602471i	$-0.205817\pi$
$882$	26184.0	+	725.138i	0.999617	+	0.0276833i
$883$	−42570.1			−1.62242			−0.811211	−	0.584753i	$-0.801191\pi$
$883$	−42570.1			−1.62242			−0.811211	+	0.584753i	$0.801191\pi$
$884$		0			0
$885$		0			0
$886$	25018.1	+	43332.6i	0.948645	+	1.64310i
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	31085.5	+	20320.6i	1.16880	+	0.764047i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	16190.2	−	28042.2i	0.601641	−	1.04207i
$899$		0			0
$900$	747.449	−	26989.7i	0.0276833	−	0.999617i
$901$		0			0
$902$	−59941.0	−	34607.0i	−2.21266	−	1.27748i
$903$		0			0
$904$	27008.0	+	46779.2i	0.993665	+	1.72108i
$905$		0			0
$906$		0			0
$907$	27308.3	−	47299.3i	0.999731	−	1.73158i	0.479762	−	0.877399i	$-0.340723\pi$
$907$	27308.3	−	47299.3i	0.999731	−	1.73158i	0.519968	−	0.854186i	$-0.325944\pi$
$908$			37904.2i			1.38535i
$909$		0			0
$910$		0			0
$911$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$911$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$912$	4670.66	+	18449.0i	0.169584	+	0.669855i
$913$	−17362.9	−	30073.5i	−0.629386	−	1.09013i
$914$	37899.1	−	21881.1i	1.37154	−	0.791862i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−50991.3	+	11417.1i	−1.83330	+	0.410479i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−38118.5	+	37077.4i	−1.36379	+	1.32654i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	48671.4	+	28100.4i	1.71890	+	0.992406i	0.920952	+	0.389676i	$0.127413\pi$
$929$	48671.4	+	28100.4i	1.71890	+	0.992406i	0.797945	+	0.602730i	$0.205920\pi$
$930$		0			0
$931$	9814.44	+	16999.1i	0.345494	+	0.598414i
$932$	49250.5	−	28434.8i	1.73096	−	0.999369i
$933$		0			0
$934$	8881.92	−	15383.9i	0.311162	−	0.538948i
$935$		0			0
$936$		0			0
$937$	56270.0			1.96186			0.980929	−	0.194367i	$-0.0622652\pi$
$937$	56270.0			1.96186			0.980929	+	0.194367i	$0.0622652\pi$
$938$		0			0
$939$	13333.9	+	52668.8i	0.463404	+	1.83044i
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		−	36481.5i		−	1.25781i
$945$		0			0
$946$	−39461.1			−1.35623
$947$	−42578.8	−	24582.9i	−1.46106	−	0.843544i	−0.462001	−	0.886879i	$-0.652868\pi$
$947$	−42578.8	−	24582.9i	−1.46106	−	0.843544i	−0.999061	+	0.0433353i	$0.986202\pi$
$948$		0			0
$949$		0			0
$950$	17522.1	−	10116.4i	0.598414	−	0.345494i
$951$		0			0
$952$		0			0
$953$		−	21477.7i		−	0.730042i	−0.930999	−	0.365021i	$-0.881062\pi$
$953$		−	21477.7i		−	0.730042i	0.930999	−	0.365021i	$-0.118938\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−14895.5	+	25799.8i	−0.500000	+	0.866025i
$962$		0			0
$963$	−58939.1	−	1632.25i	−1.97226	−	0.0546196i
$964$	46257.3			1.54549
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$	24774.5	−	14303.6i	0.822607	−	0.474933i
$969$	−27302.3	−	28068.9i	−0.905137	−	0.930550i
$970$		0			0
$971$			60514.2i			1.99999i	0.00267705	+	0.999996i	$0.499148\pi$
$971$			60514.2i			1.99999i	−0.00267705	+	0.999996i	$0.500852\pi$
$972$	28658.8	+	9848.92i	0.945712	+	0.325004i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−38382.2	+	22160.0i	−1.25686	+	0.725650i	−0.972463	−	0.233056i	$-0.925127\pi$
$977$	−38382.2	+	22160.0i	−1.25686	+	0.725650i	−0.284399	+	0.958706i	$0.591794\pi$
$978$	−13820.0	+	3498.76i	−0.451857	+	0.114395i
$979$	33861.3	−	58649.6i	1.10543	−	1.91466i
$980$		0			0
$981$		0			0
$982$	−4477.66			−0.145507
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$	−54345.3	−	15371.2i	−1.76064	−	0.497983i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	10510.7	+	41516.9i	0.335897	+	1.32679i
$994$		0			0
$995$		0			0
$996$	−19757.1	−	20311.8i	−0.628541	−	0.646189i
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$			57247.2i			1.81576i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.4.l.a.11.1	✓	4
3.2	odd	2		216.4.l.a.35.2		4
4.3	odd	2		288.4.p.a.47.2		4
8.3	odd	2	CM	72.4.l.a.11.1	✓	4
8.5	even	2		288.4.p.a.47.2		4
9.4	even	3		216.4.l.a.179.2		4
9.5	odd	6	inner	72.4.l.a.59.1	yes	4
12.11	even	2		864.4.p.a.143.1		4
24.5	odd	2		864.4.p.a.143.1		4
24.11	even	2		216.4.l.a.35.2		4
36.23	even	6		288.4.p.a.239.2		4
36.31	odd	6		864.4.p.a.719.1		4
72.5	odd	6		288.4.p.a.239.2		4
72.13	even	6		864.4.p.a.719.1		4
72.59	even	6	inner	72.4.l.a.59.1	yes	4
72.67	odd	6		216.4.l.a.179.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.l.a.11.1	✓	4	1.1	even	1	trivial
72.4.l.a.11.1	✓	4	8.3	odd	2	CM
72.4.l.a.59.1	yes	4	9.5	odd	6	inner
72.4.l.a.59.1	yes	4	72.59	even	6	inner
216.4.l.a.35.2		4	3.2	odd	2
216.4.l.a.35.2		4	24.11	even	2
216.4.l.a.179.2		4	9.4	even	3
216.4.l.a.179.2		4	72.67	odd	6
288.4.p.a.47.2		4	4.3	odd	2
288.4.p.a.47.2		4	8.5	even	2
288.4.p.a.239.2		4	36.23	even	6
288.4.p.a.239.2		4	72.5	odd	6
864.4.p.a.143.1		4	12.11	even	2
864.4.p.a.143.1		4	24.5	odd	2
864.4.p.a.719.1		4	36.31	odd	6
864.4.p.a.719.1		4	72.13	even	6

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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	72.l (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.44949 - 1.41421i) q^{2} +(-3.72474 + 3.62302i) q^{3} +(4.00000 + 6.92820i) q^{4} +(14.2474 - 3.60697i) q^{6} -22.6274i q^{8} +(0.747449 - 26.9897i) q^{9} +O(q^{10})\)	\(q+(-2.44949 - 1.41421i) q^{2} +(-3.72474 + 3.62302i) q^{3} +(4.00000 + 6.92820i) q^{4} +(14.2474 - 3.60697i) q^{6} -22.6274i q^{8} +(0.747449 - 26.9897i) q^{9} +(-44.1186 - 25.4719i) q^{11} +(-40.0000 - 11.3137i) q^{12} +(-32.0000 + 55.4256i) q^{16} -131.682i q^{17} +(-40.0000 + 65.0538i) q^{18} -57.2270 q^{19} +(72.0454 + 124.786i) q^{22} +(81.9796 + 84.2814i) q^{24} +(62.5000 - 108.253i) q^{25} +(95.0000 + 103.238i) q^{27} +(156.767 - 90.5097i) q^{32} +(256.616 - 64.9663i) q^{33} +(-186.227 + 322.555i) q^{34} +(189.980 - 102.780i) q^{36} +(140.177 + 80.9313i) q^{38} +(-415.995 + 240.175i) q^{41} +(-136.931 + 237.172i) q^{43} -407.550i q^{44} +(-81.6163 - 322.383i) q^{48} +(-171.500 - 297.047i) q^{49} +(-306.186 + 176.777i) q^{50} +(477.088 + 490.483i) q^{51} +(-86.7015 - 387.230i) q^{54} +(213.156 - 207.335i) q^{57} +(-493.654 + 285.011i) q^{59} -512.000 q^{64} +(-720.454 - 203.775i) q^{66} +(-491.476 - 851.262i) q^{67} +(912.322 - 526.730i) q^{68} +(-610.706 - 16.9128i) q^{72} +1229.09 q^{73} +(159.407 + 629.654i) q^{75} +(-228.908 - 396.481i) q^{76} +(-727.883 - 40.3468i) q^{81} +1358.63 q^{82} +(590.327 + 340.825i) q^{83} +(670.824 - 387.300i) q^{86} +(-576.363 + 998.290i) q^{88} +1329.36i q^{89} +(-256.000 + 905.097i) q^{96} +(499.545 - 865.238i) q^{97} +970.151i q^{98} +(-720.454 + 1171.71i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{3} + 16 q^{4} + 8 q^{6} - 46 q^{9}+O(q^{10}) \) 4 * q - 10 * q^3 + 16 * q^4 + 8 * q^6 - 46 * q^9	\( 4 q - 10 q^{3} + 16 q^{4} + 8 q^{6} - 46 q^{9} - 54 q^{11} - 160 q^{12} - 128 q^{16} - 160 q^{18} + 212 q^{19} + 200 q^{22} - 64 q^{24} + 250 q^{25} + 380 q^{27} + 370 q^{33} - 304 q^{34} + 368 q^{36} + 1080 q^{38} - 1566 q^{41} + 290 q^{43} - 640 q^{48} - 686 q^{49} + 1198 q^{51} + 584 q^{54} + 10 q^{57} - 2538 q^{59} - 2048 q^{64} - 2000 q^{66} - 70 q^{67} + 2160 q^{68} - 640 q^{72} + 860 q^{73} + 1250 q^{75} + 848 q^{76} - 658 q^{81} + 320 q^{82} + 4104 q^{86} - 1600 q^{88} - 1024 q^{96} + 1910 q^{97} - 2000 q^{99}+O(q^{100}) \) 4 * q - 10 * q^3 + 16 * q^4 + 8 * q^6 - 46 * q^9 - 54 * q^11 - 160 * q^12 - 128 * q^16 - 160 * q^18 + 212 * q^19 + 200 * q^22 - 64 * q^24 + 250 * q^25 + 380 * q^27 + 370 * q^33 - 304 * q^34 + 368 * q^36 + 1080 * q^38 - 1566 * q^41 + 290 * q^43 - 640 * q^48 - 686 * q^49 + 1198 * q^51 + 584 * q^54 + 10 * q^57 - 2538 * q^59 - 2048 * q^64 - 2000 * q^66 - 70 * q^67 + 2160 * q^68 - 640 * q^72 + 860 * q^73 + 1250 * q^75 + 848 * q^76 - 658 * q^81 + 320 * q^82 + 4104 * q^86 - 1600 * q^88 - 1024 * q^96 + 1910 * q^97 - 2000 * q^99

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