LMFDB - Embedded newform 475.3.c.i.151.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,3,Mod(151,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("475.151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	475.c (of order $2$, degree $1$, 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:	$12.9428125571$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 42x^{12} + 677x^{10} + 5313x^{8} + 21125x^{6} + 40138x^{4} + 30565x^{2} + 3675 $ x^14 + 42x^12 + 677x^10 + 5313x^8 + 21125x^6 + 40138x^4 + 30565x^2 + 3675
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			151.9
Root			$1.40632i$ of defining polynomial
Character	$\chi$	$=$	475.151
Dual form			475.3.c.i.151.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.40632i q^{2} -2.91656i q^{3} +2.02225 q^{4} +4.10162 q^{6} +11.7208 q^{7} +8.46924i q^{8} +0.493705 q^{9} +O(q^{10})$	\(q+1.40632i q^{2} -2.91656i q^{3} +2.02225 q^{4} +4.10162 q^{6} +11.7208 q^{7} +8.46924i q^{8} +0.493705 q^{9} -6.36259 q^{11} -5.89801i q^{12} +17.7815i q^{13} +16.4833i q^{14} -3.82148 q^{16} +17.5868 q^{17} +0.694309i q^{18} +(-16.8066 + 8.86223i) q^{19} -34.1845i q^{21} -8.94786i q^{22} -1.53826 q^{23} +24.7010 q^{24} -25.0065 q^{26} -27.6889i q^{27} +23.7025 q^{28} -39.9749i q^{29} +9.24369i q^{31} +28.5027i q^{32} +18.5568i q^{33} +24.7328i q^{34} +0.998396 q^{36} +56.4183i q^{37} +(-12.4632 - 23.6355i) q^{38} +51.8607 q^{39} -39.5382i q^{41} +48.0744 q^{42} -18.4360 q^{43} -12.8668 q^{44} -2.16329i q^{46} +61.5720 q^{47} +11.1456i q^{48} +88.3780 q^{49} -51.2930i q^{51} +35.9587i q^{52} -66.5116i q^{53} +38.9396 q^{54} +99.2665i q^{56} +(25.8472 + 49.0173i) q^{57} +56.2177 q^{58} -9.42839i q^{59} -36.5472 q^{61} -12.9996 q^{62} +5.78663 q^{63} -55.3700 q^{64} -26.0969 q^{66} +2.55637i q^{67} +35.5650 q^{68} +4.48642i q^{69} -53.8192i q^{71} +4.18130i q^{72} +115.725 q^{73} -79.3424 q^{74} +(-33.9871 + 17.9217i) q^{76} -74.5748 q^{77} +72.9329i q^{78} +95.9222i q^{79} -76.3129 q^{81} +55.6035 q^{82} -143.935 q^{83} -69.1296i q^{84} -25.9270i q^{86} -116.589 q^{87} -53.8863i q^{88} -19.4615i q^{89} +208.414i q^{91} -3.11075 q^{92} +26.9597 q^{93} +86.5902i q^{94} +83.1297 q^{96} -163.156i q^{97} +124.288i q^{98} -3.14124 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 28 q^{4} - 4 q^{6} + 20 q^{7} - 36 q^{9}+O(q^{10}) $ 14 * q - 28 * q^4 - 4 * q^6 + 20 * q^7 - 36 * q^9	$ 14 q - 28 q^{4} - 4 q^{6} + 20 q^{7} - 36 q^{9} - 4 q^{11} + 36 q^{16} - 22 q^{17} + 39 q^{19} - 12 q^{23} - 44 q^{24} + 30 q^{26} - 98 q^{28} + 4 q^{36} - 37 q^{38} - 32 q^{39} - 250 q^{42} - 90 q^{43} - 52 q^{44} - 148 q^{47} + 234 q^{49} + 98 q^{54} + 195 q^{57} + 274 q^{58} + 222 q^{61} - 518 q^{62} - 198 q^{63} - 218 q^{64} + 92 q^{66} - 80 q^{68} + 228 q^{73} - 92 q^{74} - 351 q^{76} + 260 q^{77} + 402 q^{81} - 58 q^{82} + 280 q^{83} + 282 q^{87} - 302 q^{92} + 358 q^{93} + 190 q^{96} - 180 q^{99}+O(q^{100}) $ 14 * q - 28 * q^4 - 4 * q^6 + 20 * q^7 - 36 * q^9 - 4 * q^11 + 36 * q^16 - 22 * q^17 + 39 * q^19 - 12 * q^23 - 44 * q^24 + 30 * q^26 - 98 * q^28 + 4 * q^36 - 37 * q^38 - 32 * q^39 - 250 * q^42 - 90 * q^43 - 52 * q^44 - 148 * q^47 + 234 * q^49 + 98 * q^54 + 195 * q^57 + 274 * q^58 + 222 * q^61 - 518 * q^62 - 198 * q^63 - 218 * q^64 + 92 * q^66 - 80 * q^68 + 228 * q^73 - 92 * q^74 - 351 * q^76 + 260 * q^77 + 402 * q^81 - 58 * q^82 + 280 * q^83 + 282 * q^87 - 302 * q^92 + 358 * q^93 + 190 * q^96 - 180 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$1$	$-1$

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 $

$2$			1.40632i			0.703162i	0.936157	+	0.351581i	$0.114356\pi$
$2$			1.40632i			0.703162i	−0.936157	+	0.351581i	$0.885644\pi$
$3$		−	2.91656i		−	0.972185i	−0.873907	−	0.486093i	$-0.838422\pi$
$3$		−	2.91656i		−	0.972185i	0.873907	−	0.486093i	$-0.161578\pi$
$4$	2.02225			0.505563
$5$		0			0
$5$		0			0
$6$	4.10162			0.683604
$7$	11.7208			1.67441			0.837203	−	0.546893i	$-0.184190\pi$
$7$	11.7208			1.67441			0.837203	+	0.546893i	$0.184190\pi$
$8$			8.46924i			1.05865i
$9$	0.493705			0.0548561
$10$		0			0
$11$	−6.36259			−0.578417			−0.289209	−	0.957266i	$-0.593392\pi$
$11$	−6.36259			−0.578417			−0.289209	+	0.957266i	$0.593392\pi$
$12$		−	5.89801i		−	0.491501i
$13$			17.7815i			1.36781i	0.729573	+	0.683903i	$0.239719\pi$
$13$			17.7815i			1.36781i	−0.729573	+	0.683903i	$0.760281\pi$
$14$			16.4833i			1.17738i
$15$		0			0
$16$	−3.82148			−0.238842
$17$	17.5868			1.03452			0.517260	−	0.855828i	$-0.326952\pi$
$17$	17.5868			1.03452			0.517260	+	0.855828i	$0.326952\pi$
$18$			0.694309i			0.0385727i
$19$	−16.8066	+	8.86223i	−0.884557	+	0.466433i
$19$	−16.8066	+	8.86223i	−0.884557	+	0.466433i
$20$		0			0
$21$		−	34.1845i		−	1.62783i
$22$		−	8.94786i		−	0.406721i
$23$	−1.53826			−0.0668809			−0.0334404	−	0.999441i	$-0.510646\pi$
$23$	−1.53826			−0.0668809			−0.0334404	+	0.999441i	$0.510646\pi$
$24$	24.7010			1.02921
$25$		0			0
$26$	−25.0065			−0.961789
$27$		−	27.6889i		−	1.02552i
$28$	23.7025			0.846518
$29$		−	39.9749i		−	1.37845i	−0.724549	−	0.689223i	$-0.757952\pi$
$29$		−	39.9749i		−	1.37845i	0.724549	−	0.689223i	$-0.242048\pi$
$30$		0			0
$31$			9.24369i			0.298184i	0.988823	+	0.149092i	$0.0476350\pi$
$31$			9.24369i			0.298184i	−0.988823	+	0.149092i	$0.952365\pi$
$32$			28.5027i			0.890710i
$33$			18.5568i			0.562328i
$34$			24.7328i			0.727435i
$35$		0			0
$36$	0.998396			0.0277332
$37$			56.4183i			1.52482i	0.647095	+	0.762410i	$0.275984\pi$
$37$			56.4183i			1.52482i	−0.647095	+	0.762410i	$0.724016\pi$
$38$	−12.4632	−	23.6355i	−0.327978	−	0.621986i
$39$	51.8607			1.32976
$40$		0			0
$41$		−	39.5382i		−	0.964347i	−0.876076	−	0.482173i	$-0.839848\pi$
$41$		−	39.5382i		−	0.964347i	0.876076	−	0.482173i	$-0.160152\pi$
$42$	48.0744			1.14463
$43$	−18.4360			−0.428744			−0.214372	−	0.976752i	$-0.568770\pi$
$43$	−18.4360			−0.428744			−0.214372	+	0.976752i	$0.568770\pi$
$44$	−12.8668			−0.292426
$45$		0			0
$46$		−	2.16329i		−	0.0470281i
$47$	61.5720			1.31004			0.655021	−	0.755610i	$-0.272660\pi$
$47$	61.5720			1.31004			0.655021	+	0.755610i	$0.272660\pi$
$48$			11.1456i			0.232199i
$49$	88.3780			1.80363
$50$		0			0
$51$		−	51.2930i		−	1.00574i
$52$			35.9587i			0.691513i
$53$		−	66.5116i		−	1.25494i	−0.778642	−	0.627468i	$-0.784091\pi$
$53$		−	66.5116i		−	1.25494i	0.778642	−	0.627468i	$-0.215909\pi$
$54$	38.9396			0.721103
$55$		0			0
$56$			99.2665i			1.77262i
$57$	25.8472	+	49.0173i	0.453459	+	0.859953i
$58$	56.2177			0.969270
$59$		−	9.42839i		−	0.159803i	−0.996803	−	0.0799016i	$-0.974539\pi$
$59$		−	9.42839i		−	0.159803i	0.996803	−	0.0799016i	$-0.0254606\pi$
$60$		0			0
$61$	−36.5472			−0.599134			−0.299567	−	0.954075i	$-0.596842\pi$
$61$	−36.5472			−0.599134			−0.299567	+	0.954075i	$0.596842\pi$
$62$	−12.9996			−0.209671
$63$	5.78663			0.0918513
$64$	−55.3700			−0.865156
$65$		0			0
$66$	−26.0969			−0.395408
$67$			2.55637i			0.0381548i	0.999818	+	0.0190774i	$0.00607289\pi$
$67$			2.55637i			0.0381548i	−0.999818	+	0.0190774i	$0.993927\pi$
$68$	35.5650			0.523015
$69$			4.48642i			0.0650206i
$70$		0			0
$71$		−	53.8192i		−	0.758017i	−0.925393	−	0.379008i	$-0.876265\pi$
$71$		−	53.8192i		−	0.758017i	0.925393	−	0.379008i	$-0.123735\pi$
$72$			4.18130i			0.0580737i
$73$	115.725			1.58527			0.792636	−	0.609695i	$-0.208708\pi$
$73$	115.725			1.58527			0.792636	+	0.609695i	$0.208708\pi$
$74$	−79.3424			−1.07219
$75$		0			0
$76$	−33.9871	+	17.9217i	−0.447199	+	0.235811i
$77$	−74.5748			−0.968504
$78$			72.9329i			0.935037i
$79$			95.9222i			1.21421i	0.794623	+	0.607103i	$0.207668\pi$
$79$			95.9222i			1.21421i	−0.794623	+	0.607103i	$0.792332\pi$
$80$		0			0
$81$	−76.3129			−0.942135
$82$	55.6035			0.678092
$83$	−143.935			−1.73416			−0.867081	−	0.498168i	$-0.834006\pi$
$83$	−143.935			−1.73416			−0.867081	+	0.498168i	$0.834006\pi$
$84$		−	69.1296i		−	0.822972i
$85$		0			0
$86$		−	25.9270i		−	0.301476i
$87$	−116.589			−1.34010
$88$		−	53.8863i		−	0.612344i
$89$		−	19.4615i		−	0.218668i	−0.994005	−	0.109334i	$-0.965128\pi$
$89$		−	19.4615i		−	0.218668i	0.994005	−	0.109334i	$-0.0348718\pi$
$90$		0			0
$91$			208.414i			2.29026i
$92$	−3.11075			−0.0338125
$93$	26.9597			0.289890
$94$			86.5902i			0.921172i
$95$		0			0
$96$	83.1297			0.865935
$97$		−	163.156i		−	1.68202i	−0.541017	−	0.841012i	$-0.681960\pi$
$97$		−	163.156i		−	1.68202i	0.541017	−	0.841012i	$-0.318040\pi$
$98$			124.288i			1.26825i
$99$	−3.14124			−0.0317297
$100$		0			0
$101$	−57.2111			−0.566447			−0.283223	−	0.959054i	$-0.591404\pi$
$101$	−57.2111			−0.566447			−0.283223	+	0.959054i	$0.591404\pi$
$102$	72.1346			0.707202
$103$			22.5345i			0.218782i	0.993999	+	0.109391i	$0.0348900\pi$
$103$			22.5345i			0.218782i	−0.993999	+	0.109391i	$0.965110\pi$
$104$	−150.596			−1.44803
$105$		0			0
$106$	93.5369			0.882424
$107$			68.7799i			0.642803i	0.946943	+	0.321401i	$0.104154\pi$
$107$			68.7799i			0.642803i	−0.946943	+	0.321401i	$0.895846\pi$
$108$		−	55.9940i		−	0.518463i
$109$		−	79.5462i		−	0.729781i	−0.931050	−	0.364891i	$-0.881106\pi$
$109$		−	79.5462i		−	0.729781i	0.931050	−	0.364891i	$-0.118894\pi$
$110$		0			0
$111$	164.547			1.48241
$112$	−44.7909			−0.399919
$113$		−	188.929i		−	1.67193i	−0.548780	−	0.835967i	$-0.684907\pi$
$113$		−	188.929i		−	1.67193i	0.548780	−	0.835967i	$-0.315093\pi$
$114$	−68.9342	+	36.3495i	−0.604686	+	0.318855i
$115$		0			0
$116$		−	80.8394i		−	0.696891i
$117$			8.77880i			0.0750325i
$118$	13.2594			0.112368
$119$	206.132			1.73221
$120$		0			0
$121$	−80.5175			−0.665434
$122$		−	51.3972i		−	0.421288i
$123$	−115.315			−0.937524
$124$			18.6931i			0.150751i
$125$		0			0
$126$			8.13788i			0.0645864i
$127$			12.5720i			0.0989923i	0.998774	+	0.0494961i	$0.0157616\pi$
$127$			12.5720i			0.0989923i	−0.998774	+	0.0494961i	$0.984238\pi$
$128$			36.1427i			0.282365i
$129$			53.7695i			0.416818i
$130$		0			0
$131$	96.6711			0.737947			0.368974	−	0.929440i	$-0.379709\pi$
$131$	96.6711			0.737947			0.368974	+	0.929440i	$0.379709\pi$
$132$			37.5266i			0.284293i
$133$	−196.987	+	103.873i	−1.48111	+	0.780998i
$134$	−3.59508			−0.0268290
$135$		0			0
$136$			148.947i			1.09520i
$137$	−219.187			−1.59991			−0.799954	−	0.600061i	$-0.795143\pi$
$137$	−219.187			−1.59991			−0.799954	+	0.600061i	$0.795143\pi$
$138$	−6.30936			−0.0457200
$139$	−188.602			−1.35685			−0.678425	−	0.734669i	$-0.737337\pi$
$139$	−188.602			−1.35685			−0.678425	+	0.734669i	$0.737337\pi$
$140$		0			0
$141$		−	179.578i		−	1.27360i
$142$	75.6872			0.533008
$143$		−	113.136i		−	0.791163i
$144$	−1.88668			−0.0131020
$145$		0			0
$146$			162.747i			1.11470i
$147$		−	257.759i		−	1.75346i
$148$			114.092i			0.770893i
$149$	177.432			1.19082			0.595410	−	0.803422i	$-0.296990\pi$
$149$	177.432			1.19082			0.595410	+	0.803422i	$0.296990\pi$
$150$		0			0
$151$		−	9.49631i		−	0.0628895i	−0.999505	−	0.0314447i	$-0.989989\pi$
$151$		−	9.49631i		−	0.0628895i	0.999505	−	0.0314447i	$-0.0100108\pi$
$152$	−75.0563	−	142.339i	−0.493792	−	0.936440i
$153$	8.68271			0.0567497
$154$		−	104.876i		−	0.681016i
$155$		0			0
$156$	104.875			0.672278
$157$	−27.3090			−0.173943			−0.0869715	−	0.996211i	$-0.527719\pi$
$157$	−27.3090			−0.173943			−0.0869715	+	0.996211i	$0.527719\pi$
$158$	−134.898			−0.853783
$159$	−193.985			−1.22003
$160$		0			0
$161$	−18.0297			−0.111986
$162$		−	107.321i		−	0.662473i
$163$	−128.144			−0.786161			−0.393081	−	0.919504i	$-0.628591\pi$
$163$	−128.144			−0.786161			−0.393081	+	0.919504i	$0.628591\pi$
$164$		−	79.9563i		−	0.487538i
$165$		0			0
$166$		−	202.420i		−	1.21940i
$167$			83.5190i			0.500114i	0.968231	+	0.250057i	$0.0804493\pi$
$167$			83.5190i			0.500114i	−0.968231	+	0.250057i	$0.919551\pi$
$168$	289.516			1.72331
$169$	−147.181			−0.870894
$170$		0			0
$171$	−8.29749	+	4.37532i	−0.0485233	+	0.0255867i
$172$	−37.2822			−0.216757
$173$			125.617i			0.726110i	0.931768	+	0.363055i	$0.118266\pi$
$173$			125.617i			0.726110i	−0.931768	+	0.363055i	$0.881734\pi$
$174$		−	163.962i		−	0.942310i
$175$		0			0
$176$	24.3145			0.138151
$177$	−27.4984			−0.155358
$178$	27.3691			0.153759
$179$			86.1103i			0.481063i	0.970641	+	0.240532i	$0.0773218\pi$
$179$			86.1103i			0.481063i	−0.970641	+	0.240532i	$0.922678\pi$
$180$		0			0
$181$		−	299.094i		−	1.65246i	−0.563336	−	0.826228i	$-0.690482\pi$
$181$		−	299.094i		−	1.65246i	0.563336	−	0.826228i	$-0.309518\pi$
$182$	−293.097			−1.61043
$183$			106.592i			0.582469i
$184$		−	13.0279i		−	0.0708038i
$185$		0			0
$186$			37.9141i			0.203839i
$187$	−111.898			−0.598384
$188$	124.514			0.662309
$189$		−	324.537i		−	1.71713i
$190$		0			0
$191$	−6.80277			−0.0356166			−0.0178083	−	0.999841i	$-0.505669\pi$
$191$	−6.80277			−0.0356166			−0.0178083	+	0.999841i	$0.505669\pi$
$192$			161.490i			0.841092i
$193$		−	235.945i		−	1.22251i	−0.791433	−	0.611256i	$-0.790665\pi$
$193$		−	235.945i		−	1.22251i	0.791433	−	0.611256i	$-0.209335\pi$
$194$	229.451			1.18274
$195$		0			0
$196$	178.723			0.911850
$197$	−29.4014			−0.149245			−0.0746227	−	0.997212i	$-0.523775\pi$
$197$	−29.4014			−0.149245			−0.0746227	+	0.997212i	$0.523775\pi$
$198$		−	4.41760i		−	0.0223111i
$199$	−65.6370			−0.329834			−0.164917	−	0.986307i	$-0.552736\pi$
$199$	−65.6370			−0.329834			−0.164917	+	0.986307i	$0.552736\pi$
$200$		0			0
$201$	7.45579			0.0370935
$202$		−	80.4573i		−	0.398304i
$203$		−	468.539i		−	2.30808i
$204$		−	103.727i		−	0.508468i
$205$		0			0
$206$	−31.6909			−0.153839
$207$	−0.759447			−0.00366882
$208$		−	67.9516i		−	0.326690i
$209$	106.933	−	56.3867i	0.511643	−	0.269793i
$210$		0			0
$211$		−	45.4463i		−	0.215385i	−0.994184	−	0.107693i	$-0.965654\pi$
$211$		−	45.4463i		−	0.215385i	0.994184	−	0.107693i	$-0.0343463\pi$
$212$		−	134.503i		−	0.634450i
$213$	−156.967			−0.736933
$214$	−96.7268			−0.451995
$215$		0			0
$216$	234.504			1.08567
$217$			108.344i			0.499280i
$218$	111.868			0.513154
$219$		−	337.518i		−	1.54118i
$220$		0			0
$221$			312.720i			1.41502i
$222$			231.407i			1.04237i
$223$			236.447i			1.06030i	0.847904	+	0.530149i	$0.177864\pi$
$223$			236.447i			1.06030i	−0.847904	+	0.530149i	$0.822136\pi$
$224$			334.076i			1.49141i
$225$		0			0
$226$	265.695			1.17564
$227$			143.537i			0.632323i	0.948705	+	0.316162i	$0.102394\pi$
$227$			143.537i			0.632323i	−0.948705	+	0.316162i	$0.897606\pi$
$228$	52.2695	+	99.1254i	0.229252	+	0.434761i
$229$	−102.645			−0.448230			−0.224115	−	0.974563i	$-0.571949\pi$
$229$	−102.645			−0.448230			−0.224115	+	0.974563i	$0.571949\pi$
$230$		0			0
$231$			217.502i			0.941566i
$232$	338.557			1.45930
$233$	−170.820			−0.733132			−0.366566	−	0.930392i	$-0.619467\pi$
$233$	−170.820			−0.733132			−0.366566	+	0.930392i	$0.619467\pi$
$234$	−12.3458			−0.0527600
$235$		0			0
$236$		−	19.0666i		−	0.0807907i
$237$	279.762			1.18043
$238$			289.889i			1.21802i
$239$	194.352			0.813186			0.406593	−	0.913609i	$-0.366717\pi$
$239$	194.352			0.813186			0.406593	+	0.913609i	$0.366717\pi$
$240$		0			0
$241$			457.159i			1.89693i	0.316888	+	0.948463i	$0.397362\pi$
$241$			457.159i			1.89693i	−0.316888	+	0.948463i	$0.602638\pi$
$242$		−	113.234i		−	0.467908i
$243$		−	26.6294i		−	0.109586i
$244$	−73.9076			−0.302900
$245$		0			0
$246$		−	162.171i		−	0.659231i
$247$	−157.584	−	298.846i	−0.637990	−	1.20990i
$248$	−78.2870			−0.315674
$249$			419.796i			1.68593i
$250$		0			0
$251$	−345.070			−1.37478			−0.687391	−	0.726288i	$-0.741244\pi$
$251$	−345.070			−1.37478			−0.687391	+	0.726288i	$0.741244\pi$
$252$	11.7020			0.0464367
$253$	9.78732			0.0386850
$254$	−17.6803			−0.0696076
$255$		0			0
$256$	−272.308			−1.06370
$257$			286.104i			1.11324i	0.830766	+	0.556622i	$0.187903\pi$
$257$			286.104i			1.11324i	−0.830766	+	0.556622i	$0.812097\pi$
$258$	−75.6174			−0.293091
$259$			661.270i			2.55317i
$260$		0			0
$261$		−	19.7358i		−	0.0756161i
$262$			135.951i			0.518896i
$263$	−203.484			−0.773703			−0.386851	−	0.922142i	$-0.626437\pi$
$263$	−203.484			−0.773703			−0.386851	+	0.922142i	$0.626437\pi$
$264$	−157.162			−0.595312
$265$		0			0
$266$	−146.079	−	277.028i	−0.549168	−	1.04146i
$267$	−56.7604			−0.212586
$268$			5.16963i			0.0192896i
$269$			420.928i			1.56479i	0.622783	+	0.782395i	$0.286002\pi$
$269$			420.928i			1.56479i	−0.622783	+	0.782395i	$0.713998\pi$
$270$		0			0
$271$	−127.006			−0.468655			−0.234328	−	0.972158i	$-0.575289\pi$
$271$	−127.006			−0.468655			−0.234328	+	0.972158i	$0.575289\pi$
$272$	−67.2078			−0.247087
$273$	607.850			2.22656
$274$		−	308.249i		−	1.12499i
$275$		0			0
$276$			9.07268i			0.0328720i
$277$	342.553			1.23665			0.618327	−	0.785921i	$-0.287811\pi$
$277$	342.553			1.23665			0.618327	+	0.785921i	$0.287811\pi$
$278$		−	265.236i		−	0.954086i
$279$			4.56366i			0.0163572i
$280$		0			0
$281$		−	151.942i		−	0.540718i	−0.962760	−	0.270359i	$-0.912858\pi$
$281$		−	151.942i		−	0.540718i	0.962760	−	0.270359i	$-0.0871424\pi$
$282$	252.545			0.895550
$283$	240.982			0.851527			0.425764	−	0.904834i	$-0.360006\pi$
$283$	240.982			0.851527			0.425764	+	0.904834i	$0.360006\pi$
$284$		−	108.836i		−	0.383225i
$285$		0			0
$286$	159.106			0.556315
$287$		−	463.421i		−	1.61471i
$288$			14.0719i			0.0488609i
$289$	20.2970			0.0702317
$290$		0			0
$291$	−475.854			−1.63524
$292$	234.025			0.801455
$293$		−	7.05363i		−	0.0240738i	−0.999928	−	0.0120369i	$-0.996168\pi$
$293$		−	7.05363i		−	0.0240738i	0.999928	−	0.0120369i	$-0.00383156\pi$
$294$	362.493			1.23297
$295$		0			0
$296$	−477.820			−1.61426
$297$			176.173i			0.593176i
$298$			249.527i			0.837339i
$299$		−	27.3526i		−	0.0914801i
$300$		0			0
$301$	−216.085			−0.717891
$302$	13.3549			0.0442215
$303$			166.859i			0.550691i
$304$	64.2260	−	33.8668i	0.211270	−	0.111404i
$305$		0			0
$306$			12.2107i			0.0399042i
$307$			44.7370i			0.145723i	0.997342	+	0.0728616i	$0.0232131\pi$
$307$			44.7370i			0.145723i	−0.997342	+	0.0728616i	$0.976787\pi$
$308$	−150.809			−0.489640
$309$	65.7232			0.212696
$310$		0			0
$311$	−110.679			−0.355883			−0.177941	−	0.984041i	$-0.556944\pi$
$311$	−110.679			−0.355883			−0.177941	+	0.984041i	$0.556944\pi$
$312$			439.220i			1.40776i
$313$	131.862			0.421286			0.210643	−	0.977563i	$-0.432444\pi$
$313$	131.862			0.421286			0.210643	+	0.977563i	$0.432444\pi$
$314$		−	38.4054i		−	0.122310i
$315$		0			0
$316$			193.979i			0.613858i
$317$			309.486i			0.976295i	0.872761	+	0.488148i	$0.162327\pi$
$317$			309.486i			0.976295i	−0.872761	+	0.488148i	$0.837673\pi$
$318$		−	272.806i		−	0.857879i
$319$			254.344i			0.797316i
$320$		0			0
$321$	200.600			0.624923
$322$		−	25.3556i		−	0.0787441i
$323$	−295.575	+	155.859i	−0.915091	+	0.482534i
$324$	−154.324			−0.476309
$325$		0			0
$326$		−	180.212i		−	0.552799i
$327$	−232.001			−0.709483
$328$	334.859			1.02091
$329$	721.675			2.19354
$330$		0			0
$331$			200.092i			0.604508i	0.953227	+	0.302254i	$0.0977390\pi$
$331$			200.092i			0.604508i	−0.953227	+	0.302254i	$0.902261\pi$
$332$	−291.074			−0.876728
$333$			27.8540i			0.0836456i
$334$	−117.455			−0.351661
$335$		0			0
$336$			130.635i			0.388795i
$337$			425.356i			1.26218i	0.775708	+	0.631092i	$0.217393\pi$
$337$			425.356i			1.26218i	−0.775708	+	0.631092i	$0.782607\pi$
$338$		−	206.984i		−	0.612380i
$339$	−551.021			−1.62543
$340$		0			0
$341$		−	58.8138i		−	0.172475i
$342$	−6.15312	−	11.6690i	−0.0179916	−	0.0341198i
$343$	461.543			1.34561
$344$		−	156.139i		−	0.453892i
$345$		0			0
$346$	−176.658			−0.510573
$347$	553.742			1.59580			0.797900	−	0.602790i	$-0.205944\pi$
$347$	553.742			1.59580			0.797900	+	0.602790i	$0.205944\pi$
$348$	−235.773			−0.677508
$349$	273.499			0.783666			0.391833	−	0.920036i	$-0.371841\pi$
$349$	273.499			0.783666			0.391833	+	0.920036i	$0.371841\pi$
$350$		0			0
$351$	492.350			1.40271
$352$		−	181.351i		−	0.515202i
$353$	−545.758			−1.54606			−0.773028	−	0.634372i	$-0.781259\pi$
$353$	−545.758			−1.54606			−0.773028	+	0.634372i	$0.781259\pi$
$354$		−	38.6717i		−	0.109242i
$355$		0			0
$356$		−	39.3560i		−	0.110551i
$357$		−	601.197i		−	1.68402i
$358$	−121.099			−0.338265
$359$	−73.9842			−0.206084			−0.103042	−	0.994677i	$-0.532858\pi$
$359$	−73.9842			−0.206084			−0.103042	+	0.994677i	$0.532858\pi$
$360$		0			0
$361$	203.922	−	297.887i	0.564881	−	0.825173i
$362$	420.624			1.16194
$363$			234.834i			0.646925i
$364$			421.466i			1.15787i
$365$		0			0
$366$	−149.903			−0.409570
$367$	−397.670			−1.08357			−0.541785	−	0.840517i	$-0.682251\pi$
$367$	−397.670			−1.08357			−0.541785	+	0.840517i	$0.682251\pi$
$368$	5.87843			0.0159740
$369$		−	19.5202i		−	0.0529003i
$370$		0			0
$371$		−	779.572i		−	2.10127i
$372$	54.5194			0.146558
$373$		−	284.800i		−	0.763538i	−0.924258	−	0.381769i	$-0.875315\pi$
$373$		−	284.800i		−	0.763538i	0.924258	−	0.381769i	$-0.124685\pi$
$374$		−	157.365i		−	0.420761i
$375$		0			0
$376$			521.468i			1.38688i
$377$	710.813			1.88545
$378$	456.404			1.20742
$379$		−	543.499i		−	1.43403i	−0.697055	−	0.717017i	$-0.745507\pi$
$379$		−	543.499i		−	1.43403i	0.697055	−	0.717017i	$-0.254493\pi$
$380$		0			0
$381$	36.6670			0.0962388
$382$		−	9.56690i		−	0.0250442i
$383$			598.496i			1.56265i	0.624123	+	0.781326i	$0.285456\pi$
$383$			598.496i			1.56265i	−0.624123	+	0.781326i	$0.714544\pi$
$384$	105.412			0.274511
$385$		0			0
$386$	331.815			0.859624
$387$	−9.10193			−0.0235192
$388$		−	329.943i		−	0.850369i
$389$	714.756			1.83742			0.918709	−	0.394935i	$-0.129233\pi$
$389$	714.756			1.83742			0.918709	+	0.394935i	$0.129233\pi$
$390$		0			0
$391$	−27.0531			−0.0691896
$392$			748.494i			1.90942i
$393$		−	281.947i		−	0.717421i
$394$		−	41.3478i		−	0.104944i
$395$		0			0
$396$	−6.35238			−0.0160414
$397$	375.385			0.945555			0.472777	−	0.881182i	$-0.343251\pi$
$397$	375.385			0.945555			0.472777	+	0.881182i	$0.343251\pi$
$398$		−	92.3068i		−	0.231927i
$399$	302.950	+	574.524i	0.759274	+	1.43991i
$400$		0			0
$401$		−	261.219i		−	0.651419i	−0.945470	−	0.325709i	$-0.894397\pi$
$401$		−	261.219i		−	0.651419i	0.945470	−	0.325709i	$-0.105603\pi$
$402$			10.4853i			0.0260827i
$403$	−164.367			−0.407858
$404$	−115.695			−0.286375
$405$		0			0
$406$	658.918			1.62295
$407$		−	358.966i		−	0.881982i
$408$	434.413			1.06474
$409$		−	723.501i		−	1.76895i	−0.466587	−	0.884475i	$-0.654516\pi$
$409$		−	723.501i		−	1.76895i	0.466587	−	0.884475i	$-0.345484\pi$
$410$		0			0
$411$			639.272i			1.55541i
$412$			45.5705i			0.110608i
$413$		−	110.509i		−	0.267575i
$414$		−	1.06803i		−	0.00257978i
$415$		0			0
$416$	−506.821			−1.21832
$417$			550.069i			1.31911i
$418$	79.2980	+	150.383i	0.189708	+	0.359768i
$419$	−667.084			−1.59209			−0.796043	−	0.605240i	$-0.793077\pi$
$419$	−667.084			−1.59209			−0.796043	+	0.605240i	$0.793077\pi$
$420$		0			0
$421$		−	358.717i		−	0.852059i	−0.904709	−	0.426030i	$-0.859912\pi$
$421$		−	358.717i		−	0.852059i	0.904709	−	0.426030i	$-0.140088\pi$
$422$	63.9123			0.151451
$423$	30.3984			0.0718638
$424$	563.303			1.32854
$425$		0			0
$426$		−	220.746i		−	0.518183i
$427$	−428.363			−1.00319
$428$			139.090i			0.324978i
$429$	−329.968			−0.769156
$430$		0			0
$431$		−	742.801i		−	1.72344i	−0.507387	−	0.861718i	$-0.669389\pi$
$431$		−	742.801i		−	1.72344i	0.507387	−	0.861718i	$-0.330611\pi$
$432$			105.813i			0.244937i
$433$			802.697i			1.85380i	0.375304	+	0.926902i	$0.377538\pi$
$433$			802.697i			1.85380i	−0.375304	+	0.926902i	$0.622462\pi$
$434$	−152.366			−0.351075
$435$		0			0
$436$		−	160.862i		−	0.368951i
$437$	25.8529	−	13.6324i	0.0591599	−	0.0311955i
$438$	474.660			1.08370
$439$		−	89.6015i		−	0.204104i	−0.994779	−	0.102052i	$-0.967459\pi$
$439$		−	89.6015i		−	0.204104i	0.994779	−	0.102052i	$-0.0325408\pi$
$440$		0			0
$441$	43.6326			0.0989402
$442$	−439.786			−0.994990
$443$	−541.058			−1.22135			−0.610675	−	0.791881i	$-0.709102\pi$
$443$	−541.058			−1.22135			−0.610675	+	0.791881i	$0.709102\pi$
$444$	332.756			0.749450
$445$		0			0
$446$	−332.520			−0.745561
$447$		−	517.490i		−	1.15770i
$448$	−648.982			−1.44862
$449$		−	462.639i		−	1.03038i	−0.857077	−	0.515188i	$-0.827722\pi$
$449$		−	462.639i		−	1.03038i	0.857077	−	0.515188i	$-0.172278\pi$
$450$		0			0
$451$			251.565i			0.557795i
$452$		−	382.061i		−	0.845269i
$453$	−27.6965			−0.0611402
$454$	−201.860			−0.444626
$455$		0			0
$456$	−415.139	+	218.906i	−0.910393	+	0.480057i
$457$	668.815			1.46349			0.731745	−	0.681578i	$-0.238706\pi$
$457$	668.815			1.46349			0.731745	+	0.681578i	$0.238706\pi$
$458$		−	144.352i		−	0.315178i
$459$		−	486.961i		−	1.06092i
$460$		0			0
$461$	317.165			0.687993			0.343996	−	0.938971i	$-0.388219\pi$
$461$	317.165			0.687993			0.343996	+	0.938971i	$0.388219\pi$
$462$	−305.878			−0.662073
$463$	888.893			1.91985			0.959927	−	0.280250i	$-0.0904172\pi$
$463$	888.893			1.91985			0.959927	+	0.280250i	$0.0904172\pi$
$464$			152.763i			0.329231i
$465$		0			0
$466$		−	240.228i		−	0.515510i
$467$	−564.676			−1.20916			−0.604578	−	0.796546i	$-0.706658\pi$
$467$	−564.676			−1.20916			−0.604578	+	0.796546i	$0.706658\pi$
$468$			17.7530i			0.0379337i
$469$			29.9628i			0.0638865i
$470$		0			0
$471$			79.6483i			0.169105i
$472$	79.8513			0.169176
$473$	117.301			0.247993
$474$			393.437i			0.830035i
$475$		0			0
$476$	416.852			0.875740
$477$		−	32.8371i		−	0.0688409i
$478$			273.321i			0.571802i
$479$	−535.419			−1.11778			−0.558892	−	0.829240i	$-0.688773\pi$
$479$	−535.419			−1.11778			−0.558892	+	0.829240i	$0.688773\pi$
$480$		0			0
$481$	−1003.20			−2.08566
$482$	−642.914			−1.33385
$483$			52.5846i			0.108871i
$484$	−162.827			−0.336419
$485$		0			0
$486$	37.4496			0.0770567
$487$		−	396.046i		−	0.813236i	−0.913598	−	0.406618i	$-0.866708\pi$
$487$		−	396.046i		−	0.813236i	0.913598	−	0.406618i	$-0.133292\pi$
$488$		−	309.527i		−	0.634276i
$489$			373.740i			0.764294i
$490$		0			0
$491$	14.9147			0.0303763			0.0151881	−	0.999885i	$-0.495165\pi$
$491$	14.9147			0.0303763			0.0151881	+	0.999885i	$0.495165\pi$
$492$	−233.197			−0.473977
$493$		−	703.033i		−	1.42603i
$494$	420.274	−	221.613i	0.850757	−	0.448610i
$495$		0			0
$496$		−	35.3246i		−	0.0712189i
$497$		−	630.806i		−	1.26923i
$498$	−590.369			−1.18548
$499$	−437.113			−0.875978			−0.437989	−	0.898980i	$-0.644309\pi$
$499$	−437.113			−0.875978			−0.437989	+	0.898980i	$0.644309\pi$
$500$		0			0
$501$	243.588			0.486203
$502$		−	485.281i		−	0.966694i
$503$	−391.808			−0.778942			−0.389471	−	0.921039i	$-0.627342\pi$
$503$	−391.808			−0.778942			−0.389471	+	0.921039i	$0.627342\pi$
$504$			49.0084i			0.0972388i
$505$		0			0
$506$			13.7641i			0.0272019i
$507$			429.262i			0.846671i
$508$			25.4238i			0.0500469i
$509$			83.4859i			0.164020i	0.996632	+	0.0820098i	$0.0261339\pi$
$509$			83.4859i			0.164020i	−0.996632	+	0.0820098i	$0.973866\pi$
$510$		0			0
$511$	1356.39			2.65439
$512$		−	238.383i		−	0.465591i
$513$	245.385	+	465.356i	0.478334	+	0.907126i
$514$	−402.354			−0.782791
$515$		0			0
$516$			108.736i			0.210728i
$517$	−391.757			−0.757751
$518$	−929.959			−1.79529
$519$	366.369			0.705914
$520$		0			0
$521$			561.658i			1.07804i	0.842294	+	0.539019i	$0.181205\pi$
$521$			561.658i			1.07804i	−0.842294	+	0.539019i	$0.818795\pi$
$522$	27.7549			0.0531704
$523$		−	941.358i		−	1.79992i	−0.435973	−	0.899960i	$-0.643596\pi$
$523$		−	941.358i		−	1.79992i	0.435973	−	0.899960i	$-0.356404\pi$
$524$	195.493			0.373079
$525$		0			0
$526$		−	286.164i		−	0.544038i
$527$			162.567i			0.308477i
$528$		−	70.9146i		−	0.134308i
$529$	−526.634			−0.995527
$530$		0			0
$531$		−	4.65484i		−	0.00876618i
$532$	−398.358	+	210.057i	−0.748793	+	0.394844i
$533$	703.048			1.31904
$534$		−	79.8236i		−	0.149482i
$535$		0			0
$536$	−21.6505			−0.0403927
$537$	251.145			0.467682
$538$	−591.962			−1.10030
$539$	−562.313			−1.04325
$540$		0			0
$541$	890.685			1.64637			0.823184	−	0.567774i	$-0.192195\pi$
$541$	890.685			1.64637			0.823184	+	0.567774i	$0.192195\pi$
$542$		−	178.611i		−	0.329541i
$543$	−872.326			−1.60649
$544$			501.273i			0.921457i
$545$		0			0
$546$			854.835i			1.56563i
$547$			374.919i			0.685410i	0.939443	+	0.342705i	$0.111343\pi$
$547$			374.919i			0.685410i	−0.939443	+	0.342705i	$0.888657\pi$
$548$	−443.252			−0.808855
$549$	−18.0435			−0.0328662
$550$		0			0
$551$	354.267	+	671.841i	0.642952	+	1.21931i
$552$	−37.9966			−0.0688344
$553$			1124.29i			2.03307i
$554$			481.740i			0.869568i
$555$		0			0
$556$	−381.402			−0.685974
$557$	−93.1828			−0.167294			−0.0836470	−	0.996495i	$-0.526657\pi$
$557$	−93.1828			−0.167294			−0.0836470	+	0.996495i	$0.526657\pi$
$558$	−6.41798			−0.0115018
$559$		−	327.819i		−	0.586438i
$560$		0			0
$561$			326.356i			0.581740i
$562$	213.679			0.380213
$563$		−	526.142i		−	0.934533i	−0.884116	−	0.467267i	$-0.845239\pi$
$563$		−	526.142i		−	0.934533i	0.884116	−	0.467267i	$-0.154761\pi$
$564$		−	363.152i		−	0.643887i
$565$		0			0
$566$			338.899i			0.598762i
$567$	−894.451			−1.57752
$568$	455.807			0.802478
$569$			320.890i			0.563955i	0.959421	+	0.281977i	$0.0909903\pi$
$569$			320.890i			0.563955i	−0.959421	+	0.281977i	$0.909010\pi$
$570$		0			0
$571$	649.237			1.13702			0.568508	−	0.822677i	$-0.307521\pi$
$571$	649.237			1.13702			0.568508	+	0.822677i	$0.307521\pi$
$572$		−	228.790i		−	0.399983i
$573$			19.8407i			0.0346259i
$574$	651.720			1.13540
$575$		0			0
$576$	−27.3364			−0.0474591
$577$	49.2538			0.0853618			0.0426809	−	0.999089i	$-0.486410\pi$
$577$	49.2538			0.0853618			0.0426809	+	0.999089i	$0.486410\pi$
$578$			28.5441i			0.0493842i
$579$	−688.146			−1.18851
$580$		0			0
$581$	−1687.04			−2.90369
$582$		−	669.205i		−	1.14984i
$583$			423.186i			0.725877i
$584$			980.102i			1.67826i
$585$		0			0
$586$	9.91968			0.0169278
$587$	194.238			0.330900			0.165450	−	0.986218i	$-0.447092\pi$
$587$	194.238			0.330900			0.165450	+	0.986218i	$0.447092\pi$
$588$		−	521.254i		−	0.886487i
$589$	−81.9197	−	155.355i	−0.139083	−	0.263760i
$590$		0			0
$591$			85.7507i			0.145094i
$592$		−	215.601i		−	0.364192i
$593$	284.628			0.479980			0.239990	−	0.970775i	$-0.422856\pi$
$593$	284.628			0.479980			0.239990	+	0.970775i	$0.422856\pi$
$594$	−247.757			−0.417099
$595$		0			0
$596$	358.813			0.602035
$597$			191.434i			0.320660i
$598$	38.4665			0.0643253
$599$		−	723.173i		−	1.20730i	−0.797249	−	0.603650i	$-0.793712\pi$
$599$		−	723.173i		−	1.20730i	0.797249	−	0.603650i	$-0.206288\pi$
$600$		0			0
$601$			194.647i			0.323872i	0.986801	+	0.161936i	$0.0517739\pi$
$601$			194.647i			0.323872i	−0.986801	+	0.161936i	$0.948226\pi$
$602$		−	303.886i		−	0.504793i
$603$			1.26209i			0.00209302i
$604$		−	19.2039i		−	0.0317946i
$605$		0			0
$606$	−234.658			−0.387225
$607$		−	69.2294i		−	0.114052i	−0.998373	−	0.0570259i	$-0.981838\pi$
$607$		−	69.2294i		−	0.114052i	0.998373	−	0.0570259i	$-0.0181618\pi$
$608$	−252.598	−	479.033i	−0.415456	−	0.787883i
$609$	−1366.52			−2.24388
$610$		0			0
$611$			1094.84i			1.79188i
$612$	17.5586			0.0286906
$613$	220.182			0.359187			0.179593	−	0.983741i	$-0.442522\pi$
$613$	220.182			0.359187			0.179593	+	0.983741i	$0.442522\pi$
$614$	−62.9147			−0.102467
$615$		0			0
$616$		−	631.592i		−	1.02531i
$617$	−399.663			−0.647753			−0.323876	−	0.946099i	$-0.604986\pi$
$617$	−399.663			−0.647753			−0.323876	+	0.946099i	$0.604986\pi$
$618$			92.4281i			0.149560i
$619$	1061.05			1.71413			0.857067	−	0.515206i	$-0.172284\pi$
$619$	1061.05			1.71413			0.857067	+	0.515206i	$0.172284\pi$
$620$		0			0
$621$			42.5928i			0.0685874i
$622$		−	155.651i		−	0.250243i
$623$		−	228.105i		−	0.366139i
$624$	−198.185			−0.317603
$625$		0			0
$626$			185.441i			0.296232i
$627$	−164.455	−	311.877i	−0.262289	−	0.497411i
$628$	−55.2258			−0.0879392
$629$			992.220i			1.57746i
$630$		0			0
$631$	374.486			0.593480			0.296740	−	0.954958i	$-0.404101\pi$
$631$	374.486			0.593480			0.296740	+	0.954958i	$0.404101\pi$
$632$	−812.388			−1.28542
$633$	−132.547			−0.209395
$634$	−435.237			−0.686494
$635$		0			0
$636$	−392.286			−0.616803
$637$			1571.49i			2.46702i
$638$	−357.690			−0.560643
$639$		−	26.5708i		−	0.0415818i
$640$		0			0
$641$		−	1147.29i		−	1.78984i	−0.446227	−	0.894920i	$-0.647233\pi$
$641$		−	1147.29i		−	1.78984i	0.446227	−	0.894920i	$-0.352767\pi$
$642$			282.109i			0.439422i
$643$	824.922			1.28293			0.641463	−	0.767154i	$-0.278328\pi$
$643$	824.922			1.28293			0.641463	+	0.767154i	$0.278328\pi$
$644$	−36.4606			−0.0566159
$645$		0			0
$646$	−219.188	−	415.674i	−0.339300	−	0.643457i
$647$	−153.175			−0.236746			−0.118373	−	0.992969i	$-0.537768\pi$
$647$	−153.175			−0.236746			−0.118373	+	0.992969i	$0.537768\pi$
$648$		−	646.312i		−	0.997395i
$649$			59.9890i			0.0924329i
$650$		0			0
$651$	315.991			0.485393
$652$	−259.140			−0.397454
$653$	1120.33			1.71567			0.857833	−	0.513928i	$-0.171810\pi$
$653$	1120.33			1.71567			0.857833	+	0.513928i	$0.171810\pi$
$654$		−	326.268i		−	0.498881i
$655$		0			0
$656$			151.095i			0.230327i
$657$	57.1339			0.0869618
$658$			1014.91i			1.54242i
$659$		−	875.929i		−	1.32918i	−0.747208	−	0.664590i	$-0.768606\pi$
$659$		−	875.929i		−	1.32918i	0.747208	−	0.664590i	$-0.231394\pi$
$660$		0			0
$661$		−	283.488i		−	0.428878i	−0.976737	−	0.214439i	$-0.931208\pi$
$661$		−	283.488i		−	0.428878i	0.976737	−	0.214439i	$-0.0687923\pi$
$662$	−281.394			−0.425067
$663$	912.066			1.37566
$664$		−	1219.02i		−	1.83588i
$665$		0			0
$666$	−39.1717			−0.0588164
$667$			61.4918i			0.0921917i
$668$			168.897i			0.252839i
$669$	689.609			1.03081
$670$		0			0
$671$	232.535			0.346549
$672$	974.350			1.44993
$673$		−	373.637i		−	0.555182i	−0.960699	−	0.277591i	$-0.910464\pi$
$673$		−	373.637i		−	0.555182i	0.960699	−	0.277591i	$-0.0895360\pi$
$674$	−598.188			−0.887520
$675$		0			0
$676$	−297.638			−0.440292
$677$			215.956i			0.318990i	0.987199	+	0.159495i	$0.0509866\pi$
$677$			215.956i			0.318990i	−0.987199	+	0.159495i	$0.949013\pi$
$678$		−	774.913i		−	1.14294i
$679$		−	1912.33i		−	2.81639i
$680$		0			0
$681$	418.635			0.614735
$682$	82.7113			0.121278
$683$			469.374i			0.687224i	0.939112	+	0.343612i	$0.111650\pi$
$683$			469.374i			0.687224i	−0.939112	+	0.343612i	$0.888350\pi$
$684$	−16.7796	+	8.84801i	−0.0245316	+	0.0129357i
$685$		0			0
$686$			649.079i			0.946179i
$687$			299.369i			0.435762i
$688$	70.4527			0.102402
$689$	1182.68			1.71651
$690$		0			0
$691$	−80.4567			−0.116435			−0.0582176	−	0.998304i	$-0.518542\pi$
$691$	−80.4567			−0.116435			−0.0582176	+	0.998304i	$0.518542\pi$
$692$			254.030i			0.367095i
$693$	−36.8180			−0.0531284
$694$			778.741i			1.12211i
$695$		0			0
$696$		−	987.421i		−	1.41871i
$697$		−	695.352i		−	0.997636i
$698$			384.629i			0.551044i
$699$			498.205i			0.712740i
$700$		0			0
$701$	144.654			0.206353			0.103177	−	0.994663i	$-0.467099\pi$
$701$	144.654			0.206353			0.103177	+	0.994663i	$0.467099\pi$
$702$			692.404i			0.986330i
$703$	−499.992	−	948.199i	−0.711226	−	1.34879i
$704$	352.296			0.500421
$705$		0			0
$706$		−	767.512i		−	1.08713i
$707$	−670.562			−0.948461
$708$	−55.6088			−0.0785435
$709$	−108.558			−0.153115			−0.0765573	−	0.997065i	$-0.524393\pi$
$709$	−108.558			−0.153115			−0.0765573	+	0.997065i	$0.524393\pi$
$710$		0			0
$711$			47.3573i			0.0666066i
$712$	164.824			0.231494
$713$		−	14.2192i		−	0.0199428i
$714$	845.477			1.18414
$715$		0			0
$716$			174.137i			0.243208i
$717$		−	566.837i		−	0.790568i
$718$		−	104.046i		−	0.144911i
$719$	804.887			1.11945			0.559727	−	0.828677i	$-0.310906\pi$
$719$	804.887			1.11945			0.559727	+	0.828677i	$0.310906\pi$
$720$		0			0
$721$			264.124i			0.366329i
$722$	418.926	+	286.780i	0.580230	+	0.397202i
$723$	1333.33			1.84416
$724$		−	604.845i		−	0.835421i
$725$		0			0
$726$	−330.252			−0.454893
$727$	−242.633			−0.333746			−0.166873	−	0.985978i	$-0.553367\pi$
$727$	−242.633			−0.333746			−0.166873	+	0.985978i	$0.553367\pi$
$728$	−1765.11			−2.42460
$729$	−764.482			−1.04867
$730$		0			0
$731$	−324.231			−0.443544
$732$			215.556i			0.294475i
$733$	−577.444			−0.787782			−0.393891	−	0.919157i	$-0.628871\pi$
$733$	−577.444			−0.787782			−0.393891	+	0.919157i	$0.628871\pi$
$734$		−	559.253i		−	0.761925i
$735$		0			0
$736$		−	43.8446i		−	0.0595715i
$737$		−	16.2651i		−	0.0220694i
$738$	27.4517			0.0371975
$739$	−482.046			−0.652295			−0.326148	−	0.945319i	$-0.605751\pi$
$739$	−482.046			−0.652295			−0.326148	+	0.945319i	$0.605751\pi$
$740$		0			0
$741$	−871.600	+	459.601i	−1.17625	+	0.620244i
$742$	1096.33			1.47753
$743$		−	713.547i		−	0.960359i	−0.877170	−	0.480179i	$-0.840572\pi$
$743$		−	713.547i		−	0.960359i	0.877170	−	0.480179i	$-0.159428\pi$
$744$			228.328i			0.306893i
$745$		0			0
$746$	400.521			0.536891
$747$	−71.0616			−0.0951293
$748$	−226.286			−0.302521
$749$			806.158i			1.07631i
$750$		0			0
$751$			1049.66i			1.39768i	0.715276	+	0.698842i	$0.246301\pi$
$751$			1049.66i			1.39768i	−0.715276	+	0.698842i	$0.753699\pi$
$752$	−235.296			−0.312894
$753$			1006.42i			1.33654i
$754$			999.634i			1.32577i
$755$		0			0
$756$		−	656.296i		−	0.868117i
$757$	87.6677			0.115809			0.0579047	−	0.998322i	$-0.481558\pi$
$757$	87.6677			0.115809			0.0579047	+	0.998322i	$0.481558\pi$
$758$	764.336			1.00836
$759$		−	28.5453i		−	0.0376090i
$760$		0			0
$761$	706.869			0.928869			0.464435	−	0.885607i	$-0.346258\pi$
$761$	706.869			0.928869			0.464435	+	0.885607i	$0.346258\pi$
$762$			51.5657i			0.0676715i
$763$		−	932.348i		−	1.22195i
$764$	−13.7569			−0.0180064
$765$		0			0
$766$	−841.679			−1.09880
$767$	167.651			0.218580
$768$			794.202i			1.03412i
$769$	−69.6679			−0.0905955			−0.0452977	−	0.998974i	$-0.514424\pi$
$769$	−69.6679			−0.0905955			−0.0452977	+	0.998974i	$0.514424\pi$
$770$		0			0
$771$	834.437			1.08228
$772$		−	477.140i		−	0.618057i
$773$			1236.62i			1.59977i	0.600153	+	0.799886i	$0.295107\pi$
$773$			1236.62i			1.59977i	−0.600153	+	0.799886i	$0.704893\pi$
$774$		−	12.8003i		−	0.0165378i
$775$		0			0
$776$	1381.81			1.78068
$777$	1928.63			2.48215
$778$			1005.18i			1.29200i
$779$	350.397	+	664.502i	0.449803	+	0.853019i
$780$		0			0
$781$			342.429i			0.438450i
$782$		−	38.0455i		−	0.0486515i
$783$	−1106.86			−1.41362
$784$	−337.735			−0.430784
$785$		0			0
$786$	396.508			0.504463
$787$			962.806i			1.22339i	0.791095	+	0.611693i	$0.209511\pi$
$787$			962.806i			1.22339i	−0.791095	+	0.611693i	$0.790489\pi$
$788$	−59.4570			−0.0754530
$789$			593.472i			0.752182i
$790$		0			0
$791$		−	2214.40i		−	2.79950i
$792$		−	26.6039i		−	0.0335908i
$793$		−	649.863i		−	0.819499i
$794$			527.913i			0.664878i
$795$		0			0
$796$	−132.735			−0.166752
$797$		−	412.057i		−	0.517010i	−0.966010	−	0.258505i	$-0.916770\pi$
$797$		−	412.057i		−	0.517010i	0.966010	−	0.258505i	$-0.0832298\pi$
$798$	−807.966	+	426.047i	−1.01249	+	0.533893i
$799$	1082.86			1.35527
$800$		0			0
$801$		−	9.60822i		−	0.0119953i
$802$	367.358			0.458053
$803$	−736.310			−0.916949
$804$	15.0775			0.0187531
$805$		0			0
$806$		−	231.153i		−	0.286790i
$807$	1227.66			1.52126
$808$		−	484.535i		−	0.599671i
$809$	445.562			0.550756			0.275378	−	0.961336i	$-0.411197\pi$
$809$	445.562			0.550756			0.275378	+	0.961336i	$0.411197\pi$
$810$		0			0
$811$			558.857i			0.689096i	0.938769	+	0.344548i	$0.111968\pi$
$811$			558.857i			0.689096i	−0.938769	+	0.344548i	$0.888032\pi$
$812$		−	947.505i		−	1.16688i
$813$			370.419i			0.455620i
$814$	504.823			0.620176
$815$		0			0
$816$			196.015i			0.240215i
$817$	309.846	−	163.384i	0.379248	−	0.199980i
$818$	1017.48			1.24386
$819$			102.895i			0.125635i
$820$		0			0
$821$	−1469.94			−1.79043			−0.895215	−	0.445634i	$-0.852978\pi$
$821$	−1469.94			−1.79043			−0.895215	+	0.445634i	$0.852978\pi$
$822$	−899.024			−1.09370
$823$	165.659			0.201287			0.100643	−	0.994923i	$-0.467910\pi$
$823$	165.659			0.201287			0.100643	+	0.994923i	$0.467910\pi$
$824$	−190.850			−0.231614
$825$		0			0
$826$	155.411			0.188149
$827$			503.503i			0.608831i	0.952539	+	0.304415i	$0.0984610\pi$
$827$			503.503i			0.608831i	−0.952539	+	0.304415i	$0.901539\pi$
$828$	−1.53579			−0.00185482
$829$		−	156.053i		−	0.188242i	−0.995561	−	0.0941211i	$-0.969996\pi$
$829$		−	156.053i		−	0.188242i	0.995561	−	0.0941211i	$-0.0300041\pi$
$830$		0			0
$831$		−	999.075i		−	1.20226i
$832$		−	984.560i		−	1.18337i
$833$	1554.29			1.86589
$834$	−773.575			−0.927548
$835$		0			0
$836$	216.246	−	114.028i	0.258668	−	0.136397i
$837$	255.948			0.305792
$838$		−	938.136i		−	1.11949i
$839$			543.442i			0.647725i	0.946104	+	0.323863i	$0.104982\pi$
$839$			543.442i			0.647725i	−0.946104	+	0.323863i	$0.895018\pi$
$840$		0			0
$841$	−756.994			−0.900112
$842$	504.472			0.599136
$843$	−443.147			−0.525678
$844$		−	91.9040i		−	0.108891i
$845$		0			0
$846$			42.7500i			0.0505319i
$847$	−943.732			−1.11421
$848$			254.173i			0.299732i
$849$		−	702.838i		−	0.827842i
$850$		0			0
$851$		−	86.7861i		−	0.101981i
$852$	−317.426			−0.372566
$853$	788.764			0.924694			0.462347	−	0.886699i	$-0.347007\pi$
$853$	788.764			0.924694			0.462347	+	0.886699i	$0.347007\pi$
$854$		−	602.418i		−	0.705407i
$855$		0			0
$856$	−582.514			−0.680506
$857$			387.309i			0.451935i	0.974135	+	0.225968i	$0.0725544\pi$
$857$			387.309i			0.451935i	−0.974135	+	0.225968i	$0.927446\pi$
$858$		−	464.042i		−	0.540842i
$859$	−47.4552			−0.0552447			−0.0276224	−	0.999618i	$-0.508794\pi$
$859$	−47.4552			−0.0552447			−0.0276224	+	0.999618i	$0.508794\pi$
$860$		0			0
$861$	−1351.59			−1.56979
$862$	1044.62			1.21185
$863$			852.591i			0.987938i	0.869479	+	0.493969i	$0.164454\pi$
$863$			852.591i			0.987938i	−0.869479	+	0.493969i	$0.835546\pi$
$864$	789.209			0.913437
$865$		0			0
$866$	−1128.85			−1.30352
$867$		−	59.1972i		−	0.0682782i
$868$			219.099i			0.252418i
$869$		−	610.313i		−	0.702317i
$870$		0			0
$871$	−45.4560			−0.0521883
$872$	673.695			0.772587
$873$		−	80.5511i		−	0.0922693i
$874$	19.1716	+	36.3575i	0.0219355	+	0.0415990i
$875$		0			0
$876$		−	682.547i		−	0.779163i
$877$			832.003i			0.948692i	0.880339	+	0.474346i	$0.157315\pi$
$877$			832.003i			0.948692i	−0.880339	+	0.474346i	$0.842685\pi$
$878$	126.009			0.143518
$879$	−20.5723			−0.0234042
$880$		0			0
$881$	−450.264			−0.511083			−0.255541	−	0.966798i	$-0.582254\pi$
$881$	−450.264			−0.511083			−0.255541	+	0.966798i	$0.582254\pi$
$882$			61.3616i			0.0695710i
$883$	−1035.05			−1.17219			−0.586096	−	0.810241i	$-0.699336\pi$
$883$	−1035.05			−1.17219			−0.586096	+	0.810241i	$0.699336\pi$
$884$			632.399i			0.715384i
$885$		0			0
$886$		−	760.903i		−	0.858807i
$887$		−	296.394i		−	0.334153i	−0.985944	−	0.167077i	$-0.946567\pi$
$887$		−	296.394i		−	0.334153i	0.985944	−	0.167077i	$-0.0534327\pi$
$888$			1393.59i			1.56936i
$889$			147.355i			0.165753i
$890$		0			0
$891$	485.548			0.544947
$892$			478.155i			0.536048i
$893$	−1034.81	+	545.665i	−1.15881	+	0.611047i
$894$	727.759			0.814048
$895$		0			0
$896$			423.623i			0.472794i
$897$	−79.7752			−0.0889356
$898$	650.620			0.724521
$899$	369.516			0.411030
$900$		0			0
$901$		−	1169.73i		−	1.29826i
$902$	−353.782			−0.392220
$903$			630.224i			0.697922i
$904$	1600.08			1.77000
$905$		0			0
$906$		−	38.9503i		−	0.0429915i
$907$			775.691i			0.855227i	0.903962	+	0.427614i	$0.140646\pi$
$907$			775.691i			0.855227i	−0.903962	+	0.427614i	$0.859354\pi$
$908$			290.269i			0.319679i
$909$	−28.2454			−0.0310730
$910$		0			0
$911$		−	1211.86i		−	1.33025i	−0.746731	−	0.665126i	$-0.768378\pi$
$911$		−	1211.86i		−	1.33025i	0.746731	−	0.665126i	$-0.231622\pi$
$912$	−98.7745	−	187.319i	−0.108305	−	0.205393i
$913$	915.802			1.00307
$914$			940.570i			1.02907i
$915$		0			0
$916$	−207.573			−0.226609
$917$	1133.07			1.23562
$918$	684.824			0.745996
$919$	−1413.02			−1.53757			−0.768784	−	0.639509i	$-0.779138\pi$
$919$	−1413.02			−1.53757			−0.768784	+	0.639509i	$0.779138\pi$
$920$		0			0
$921$	130.478			0.141670
$922$			446.036i			0.483770i
$923$	956.985			1.03682
$924$			439.843i			0.476021i
$925$		0			0
$926$			1250.07i			1.34997i
$927$			11.1254i			0.0120015i
$928$	1139.39			1.22780
$929$	−775.131			−0.834371			−0.417185	−	0.908821i	$-0.636983\pi$
$929$	−775.131			−0.834371			−0.417185	+	0.908821i	$0.636983\pi$
$930$		0			0
$931$	−1485.33	+	783.226i	−1.59541	+	0.841274i
$932$	−345.441			−0.370644
$933$			322.803i			0.345984i
$934$		−	794.117i		−	0.850233i
$935$		0			0
$936$	−74.3498			−0.0794335
$937$	808.931			0.863320			0.431660	−	0.902036i	$-0.357928\pi$
$937$	808.931			0.863320			0.431660	+	0.902036i	$0.357928\pi$
$938$	−42.1374			−0.0449226
$939$		−	384.584i		−	0.409568i
$940$		0			0
$941$			1692.92i			1.79906i	0.436856	+	0.899532i	$0.356092\pi$
$941$			1692.92i			1.79906i	−0.436856	+	0.899532i	$0.643908\pi$
$942$	−112.011			−0.118908
$943$			60.8201i			0.0644964i
$944$			36.0304i			0.0381678i
$945$		0			0
$946$			164.963i			0.174379i
$947$	1511.42			1.59601			0.798003	−	0.602654i	$-0.205890\pi$
$947$	1511.42			1.59601			0.798003	+	0.602654i	$0.205890\pi$
$948$	565.750			0.596783
$949$			2057.76i			2.16835i
$950$		0			0
$951$	902.632			0.949140
$952$			1745.78i			1.83381i
$953$		−	305.820i		−	0.320902i	−0.987044	−	0.160451i	$-0.948705\pi$
$953$		−	305.820i		−	0.320902i	0.987044	−	0.160451i	$-0.0512949\pi$
$954$	46.1796			0.0484063
$955$		0			0
$956$	393.028			0.411117
$957$	741.808			0.775139
$958$		−	752.972i		−	0.785983i
$959$	−2569.06			−2.67889
$960$		0			0
$961$	875.554			0.911087
$962$		−	1410.83i		−	1.46656i
$963$			33.9570i			0.0352617i
$964$			924.491i			0.959016i
$965$		0			0
$966$	−73.9510			−0.0765538
$967$	−188.075			−0.194493			−0.0972466	−	0.995260i	$-0.531004\pi$
$967$	−188.075			−0.194493			−0.0972466	+	0.995260i	$0.531004\pi$
$968$		−	681.922i		−	0.704465i
$969$	454.570	+	862.059i	0.469113	+	0.889638i
$970$		0			0
$971$			1108.38i			1.14148i	0.821129	+	0.570742i	$0.193345\pi$
$971$			1108.38i			1.14148i	−0.821129	+	0.570742i	$0.806655\pi$
$972$		−	53.8514i		−	0.0554027i
$973$	−2210.58			−2.27192
$974$	556.969			0.571836
$975$		0			0
$976$	139.664			0.143099
$977$		−	559.011i		−	0.572171i	−0.958204	−	0.286086i	$-0.907646\pi$
$977$		−	559.011i		−	0.572171i	0.958204	−	0.286086i	$-0.0923542\pi$
$978$	−525.599			−0.537423
$979$			123.825i			0.126481i
$980$		0			0
$981$		−	39.2723i		−	0.0400330i
$982$			20.9750i			0.0213594i
$983$		−	1006.84i		−	1.02425i	−0.858910	−	0.512126i	$-0.828858\pi$
$983$		−	1006.84i		−	1.02425i	0.858910	−	0.512126i	$-0.171142\pi$
$984$		−	976.634i		−	0.992514i
$985$		0			0
$986$	988.692			1.00273
$987$		−	2104.81i		−	2.13253i
$988$	−318.674	−	604.342i	−0.322544	−	0.611682i
$989$	28.3593			0.0286748
$990$		0			0
$991$			999.726i			1.00881i	0.863468	+	0.504403i	$0.168287\pi$
$991$			999.726i			1.00881i	−0.863468	+	0.504403i	$0.831713\pi$
$992$	−263.470			−0.265595
$993$	583.580			0.587693
$994$	887.117			0.892472
$995$		0			0
$996$			848.933i			0.852342i
$997$	−1250.46			−1.25422			−0.627112	−	0.778929i	$-0.715763\pi$
$997$	−1250.46			−1.25422			−0.627112	+	0.778929i	$0.715763\pi$
$998$		−	614.722i		−	0.615954i
$999$	1562.16			1.56373

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.3.c.i.151.9	yes	14
5.2	odd	4		475.3.d.d.474.12		28
5.3	odd	4		475.3.d.d.474.17		28
5.4	even	2		475.3.c.h.151.6	✓	14
19.18	odd	2	inner	475.3.c.i.151.6	yes	14
95.18	even	4		475.3.d.d.474.11		28
95.37	even	4		475.3.d.d.474.18		28
95.94	odd	2		475.3.c.h.151.9	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.3.c.h.151.6	✓	14	5.4	even	2
475.3.c.h.151.9	yes	14	95.94	odd	2
475.3.c.i.151.6	yes	14	19.18	odd	2	inner
475.3.c.i.151.9	yes	14	1.1	even	1	trivial
475.3.d.d.474.11		28	95.18	even	4
475.3.d.d.474.12		28	5.2	odd	4
475.3.d.d.474.17		28	5.3	odd	4
475.3.d.d.474.18		28	95.37	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	475.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.40632i q^{2} -2.91656i q^{3} +2.02225 q^{4} +4.10162 q^{6} +11.7208 q^{7} +8.46924i q^{8} +0.493705 q^{9} +O(q^{10})\)	\(q+1.40632i q^{2} -2.91656i q^{3} +2.02225 q^{4} +4.10162 q^{6} +11.7208 q^{7} +8.46924i q^{8} +0.493705 q^{9} -6.36259 q^{11} -5.89801i q^{12} +17.7815i q^{13} +16.4833i q^{14} -3.82148 q^{16} +17.5868 q^{17} +0.694309i q^{18} +(-16.8066 + 8.86223i) q^{19} -34.1845i q^{21} -8.94786i q^{22} -1.53826 q^{23} +24.7010 q^{24} -25.0065 q^{26} -27.6889i q^{27} +23.7025 q^{28} -39.9749i q^{29} +9.24369i q^{31} +28.5027i q^{32} +18.5568i q^{33} +24.7328i q^{34} +0.998396 q^{36} +56.4183i q^{37} +(-12.4632 - 23.6355i) q^{38} +51.8607 q^{39} -39.5382i q^{41} +48.0744 q^{42} -18.4360 q^{43} -12.8668 q^{44} -2.16329i q^{46} +61.5720 q^{47} +11.1456i q^{48} +88.3780 q^{49} -51.2930i q^{51} +35.9587i q^{52} -66.5116i q^{53} +38.9396 q^{54} +99.2665i q^{56} +(25.8472 + 49.0173i) q^{57} +56.2177 q^{58} -9.42839i q^{59} -36.5472 q^{61} -12.9996 q^{62} +5.78663 q^{63} -55.3700 q^{64} -26.0969 q^{66} +2.55637i q^{67} +35.5650 q^{68} +4.48642i q^{69} -53.8192i q^{71} +4.18130i q^{72} +115.725 q^{73} -79.3424 q^{74} +(-33.9871 + 17.9217i) q^{76} -74.5748 q^{77} +72.9329i q^{78} +95.9222i q^{79} -76.3129 q^{81} +55.6035 q^{82} -143.935 q^{83} -69.1296i q^{84} -25.9270i q^{86} -116.589 q^{87} -53.8863i q^{88} -19.4615i q^{89} +208.414i q^{91} -3.11075 q^{92} +26.9597 q^{93} +86.5902i q^{94} +83.1297 q^{96} -163.156i q^{97} +124.288i q^{98} -3.14124 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 28 q^{4} - 4 q^{6} + 20 q^{7} - 36 q^{9}+O(q^{10}) \) 14 * q - 28 * q^4 - 4 * q^6 + 20 * q^7 - 36 * q^9	\( 14 q - 28 q^{4} - 4 q^{6} + 20 q^{7} - 36 q^{9} - 4 q^{11} + 36 q^{16} - 22 q^{17} + 39 q^{19} - 12 q^{23} - 44 q^{24} + 30 q^{26} - 98 q^{28} + 4 q^{36} - 37 q^{38} - 32 q^{39} - 250 q^{42} - 90 q^{43} - 52 q^{44} - 148 q^{47} + 234 q^{49} + 98 q^{54} + 195 q^{57} + 274 q^{58} + 222 q^{61} - 518 q^{62} - 198 q^{63} - 218 q^{64} + 92 q^{66} - 80 q^{68} + 228 q^{73} - 92 q^{74} - 351 q^{76} + 260 q^{77} + 402 q^{81} - 58 q^{82} + 280 q^{83} + 282 q^{87} - 302 q^{92} + 358 q^{93} + 190 q^{96} - 180 q^{99}+O(q^{100}) \) 14 * q - 28 * q^4 - 4 * q^6 + 20 * q^7 - 36 * q^9 - 4 * q^11 + 36 * q^16 - 22 * q^17 + 39 * q^19 - 12 * q^23 - 44 * q^24 + 30 * q^26 - 98 * q^28 + 4 * q^36 - 37 * q^38 - 32 * q^39 - 250 * q^42 - 90 * q^43 - 52 * q^44 - 148 * q^47 + 234 * q^49 + 98 * q^54 + 195 * q^57 + 274 * q^58 + 222 * q^61 - 518 * q^62 - 198 * q^63 - 218 * q^64 + 92 * q^66 - 80 * q^68 + 228 * q^73 - 92 * q^74 - 351 * q^76 + 260 * q^77 + 402 * q^81 - 58 * q^82 + 280 * q^83 + 282 * q^87 - 302 * q^92 + 358 * q^93 + 190 * q^96 - 180 * q^99

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