LMFDB - Embedded newform 475.3.c.g.151.10

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 30x^{10} + 329x^{8} + 1620x^{6} + 3479x^{4} + 2470x^{2} + 55 $ x^12 + 30x^10 + 329x^8 + 1620x^6 + 3479x^4 + 2470*x^2 + 55
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			151.10
Root			$2.36559i$ of defining polynomial
Character	$\chi$	$=$	475.151
Dual form			475.3.c.g.151.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.36559i q^{2} -3.55563i q^{3} -1.59600 q^{4} +8.41115 q^{6} -11.0785 q^{7} +5.68687i q^{8} -3.64250 q^{9} +O(q^{10})$	\(q+2.36559i q^{2} -3.55563i q^{3} -1.59600 q^{4} +8.41115 q^{6} -11.0785 q^{7} +5.68687i q^{8} -3.64250 q^{9} +16.7704 q^{11} +5.67479i q^{12} -14.3035i q^{13} -26.2070i q^{14} -19.8368 q^{16} +20.2160 q^{17} -8.61666i q^{18} +(-16.4480 - 9.51127i) q^{19} +39.3909i q^{21} +39.6717i q^{22} +13.5882 q^{23} +20.2204 q^{24} +33.8362 q^{26} -19.0493i q^{27} +17.6812 q^{28} -30.3972i q^{29} -16.1256i q^{31} -24.1782i q^{32} -59.6292i q^{33} +47.8227i q^{34} +5.81344 q^{36} -51.2678i q^{37} +(22.4997 - 38.9091i) q^{38} -50.8580 q^{39} -74.6652i q^{41} -93.1825 q^{42} -6.23197 q^{43} -26.7655 q^{44} +32.1441i q^{46} +44.0252 q^{47} +70.5323i q^{48} +73.7321 q^{49} -71.8806i q^{51} +22.8284i q^{52} +30.8058i q^{53} +45.0627 q^{54} -63.0017i q^{56} +(-33.8185 + 58.4829i) q^{57} +71.9073 q^{58} +73.0098i q^{59} -17.7828 q^{61} +38.1466 q^{62} +40.3533 q^{63} -22.1516 q^{64} +141.058 q^{66} +20.5719i q^{67} -32.2647 q^{68} -48.3147i q^{69} -7.48928i q^{71} -20.7144i q^{72} +29.2302 q^{73} +121.278 q^{74} +(26.2510 + 15.1800i) q^{76} -185.790 q^{77} -120.309i q^{78} +8.78894i q^{79} -100.515 q^{81} +176.627 q^{82} +63.7886 q^{83} -62.8679i q^{84} -14.7423i q^{86} -108.081 q^{87} +95.3708i q^{88} +162.908i q^{89} +158.461i q^{91} -21.6868 q^{92} -57.3367 q^{93} +104.145i q^{94} -85.9686 q^{96} -75.5974i q^{97} +174.420i q^{98} -61.0861 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{4} - 28 q^{6} - 20 q^{7} - 48 q^{9}+O(q^{10}) $ 12 * q - 12 * q^4 - 28 * q^6 - 20 * q^7 - 48 * q^9	$ 12 q - 12 q^{4} - 28 q^{6} - 20 q^{7} - 48 q^{9} + 32 q^{11} - 44 q^{16} + 44 q^{17} + 8 q^{19} - 36 q^{23} + 100 q^{24} + 108 q^{26} + 36 q^{28} - 80 q^{36} + 44 q^{38} + 76 q^{39} - 100 q^{42} - 320 q^{43} - 256 q^{44} + 56 q^{47} + 72 q^{49} - 76 q^{54} - 60 q^{57} - 68 q^{58} - 296 q^{61} + 376 q^{62} + 96 q^{63} + 188 q^{64} + 152 q^{66} + 340 q^{68} + 244 q^{73} + 136 q^{74} + 248 q^{76} + 200 q^{77} - 372 q^{81} - 424 q^{82} + 160 q^{83} - 444 q^{87} - 716 q^{92} - 296 q^{93} - 44 q^{96} - 312 q^{99}+O(q^{100}) $ 12 * q - 12 * q^4 - 28 * q^6 - 20 * q^7 - 48 * q^9 + 32 * q^11 - 44 * q^16 + 44 * q^17 + 8 * q^19 - 36 * q^23 + 100 * q^24 + 108 * q^26 + 36 * q^28 - 80 * q^36 + 44 * q^38 + 76 * q^39 - 100 * q^42 - 320 * q^43 - 256 * q^44 + 56 * q^47 + 72 * q^49 - 76 * q^54 - 60 * q^57 - 68 * q^58 - 296 * q^61 + 376 * q^62 + 96 * q^63 + 188 * q^64 + 152 * q^66 + 340 * q^68 + 244 * q^73 + 136 * q^74 + 248 * q^76 + 200 * q^77 - 372 * q^81 - 424 * q^82 + 160 * q^83 - 444 * q^87 - 716 * q^92 - 296 * q^93 - 44 * q^96 - 312 * 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 $

$2$			2.36559i			1.18279i	0.806381	+	0.591397i	$0.201423\pi$
$2$			2.36559i			1.18279i	−0.806381	+	0.591397i	$0.798577\pi$
$3$		−	3.55563i		−	1.18521i	−0.805493	−	0.592605i	$-0.798100\pi$
$3$		−	3.55563i		−	1.18521i	0.805493	−	0.592605i	$-0.201900\pi$
$4$	−1.59600			−0.399000
$5$		0			0
$5$		0			0
$6$	8.41115			1.40186
$7$	−11.0785			−1.58264			−0.791318	−	0.611405i	$-0.790605\pi$
$7$	−11.0785			−1.58264			−0.791318	+	0.611405i	$0.790605\pi$
$8$			5.68687i			0.710859i
$9$	−3.64250			−0.404723
$10$		0			0
$11$	16.7704			1.52458			0.762289	−	0.647237i	$-0.224076\pi$
$11$	16.7704			1.52458			0.762289	+	0.647237i	$0.224076\pi$
$12$			5.67479i			0.472899i
$13$		−	14.3035i		−	1.10027i	−0.835075	−	0.550136i	$-0.814576\pi$
$13$		−	14.3035i		−	1.10027i	0.835075	−	0.550136i	$-0.185424\pi$
$14$		−	26.2070i		−	1.87193i
$15$		0			0
$16$	−19.8368			−1.23980
$17$	20.2160			1.18918			0.594588	−	0.804031i	$-0.297315\pi$
$17$	20.2160			1.18918			0.594588	+	0.804031i	$0.297315\pi$
$18$		−	8.61666i		−	0.478703i
$19$	−16.4480	−	9.51127i	−0.865683	−	0.500593i
$19$	−16.4480	−	9.51127i	−0.865683	−	0.500593i
$20$		0			0
$21$			39.3909i			1.87576i
$22$			39.6717i			1.80326i
$23$	13.5882			0.590792			0.295396	−	0.955375i	$-0.404548\pi$
$23$	13.5882			0.590792			0.295396	+	0.955375i	$0.404548\pi$
$24$	20.2204			0.842517
$25$		0			0
$26$	33.8362			1.30139
$27$		−	19.0493i		−	0.705529i
$28$	17.6812			0.631472
$29$		−	30.3972i		−	1.04818i	−0.851663	−	0.524090i	$-0.824406\pi$
$29$		−	30.3972i		−	1.04818i	0.851663	−	0.524090i	$-0.175594\pi$
$30$		0			0
$31$		−	16.1256i		−	0.520181i	−0.965584	−	0.260091i	$-0.916248\pi$
$31$		−	16.1256i		−	0.520181i	0.965584	−	0.260091i	$-0.0837525\pi$
$32$		−	24.1782i		−	0.755567i
$33$		−	59.6292i		−	1.80694i
$34$			47.8227i			1.40655i
$35$		0			0
$36$	5.81344			0.161484
$37$		−	51.2678i		−	1.38562i	−0.721122	−	0.692808i	$-0.756373\pi$
$37$		−	51.2678i		−	1.38562i	0.721122	−	0.692808i	$-0.243627\pi$
$38$	22.4997	−	38.9091i	0.592098	−	1.02392i
$39$	−50.8580			−1.30405
$40$		0			0
$41$		−	74.6652i		−	1.82110i	−0.413395	−	0.910552i	$-0.635657\pi$
$41$		−	74.6652i		−	1.82110i	0.413395	−	0.910552i	$-0.364343\pi$
$42$	−93.1825			−2.21863
$43$	−6.23197			−0.144929			−0.0724647	−	0.997371i	$-0.523086\pi$
$43$	−6.23197			−0.144929			−0.0724647	+	0.997371i	$0.523086\pi$
$44$	−26.7655			−0.608307
$45$		0			0
$46$			32.1441i			0.698785i
$47$	44.0252			0.936706			0.468353	−	0.883541i	$-0.344848\pi$
$47$	44.0252			0.936706			0.468353	+	0.883541i	$0.344848\pi$
$48$			70.5323i			1.46942i
$49$	73.7321			1.50474
$50$		0			0
$51$		−	71.8806i		−	1.40942i
$52$			22.8284i			0.439008i
$53$			30.8058i			0.581241i	0.956838	+	0.290620i	$0.0938617\pi$
$53$			30.8058i			0.581241i	−0.956838	+	0.290620i	$0.906138\pi$
$54$	45.0627			0.834495
$55$		0			0
$56$		−	63.0017i		−	1.12503i
$57$	−33.8185	+	58.4829i	−0.593308	+	1.02602i
$58$	71.9073			1.23978
$59$			73.0098i			1.23745i	0.785606	+	0.618727i	$0.212351\pi$
$59$			73.0098i			1.23745i	−0.785606	+	0.618727i	$0.787649\pi$
$60$		0			0
$61$	−17.7828			−0.291522			−0.145761	−	0.989320i	$-0.546563\pi$
$61$	−17.7828			−0.291522			−0.145761	+	0.989320i	$0.546563\pi$
$62$	38.1466			0.615267
$63$	40.3533			0.640528
$64$	−22.1516			−0.346119
$65$		0			0
$66$	141.058			2.13724
$67$			20.5719i			0.307044i	0.988145	+	0.153522i	$0.0490616\pi$
$67$			20.5719i			0.307044i	−0.988145	+	0.153522i	$0.950938\pi$
$68$	−32.2647			−0.474481
$69$		−	48.3147i		−	0.700212i
$70$		0			0
$71$		−	7.48928i		−	0.105483i	−0.998608	−	0.0527414i	$-0.983204\pi$
$71$		−	7.48928i		−	0.105483i	0.998608	−	0.0527414i	$-0.0167959\pi$
$72$		−	20.7144i		−	0.287700i
$73$	29.2302			0.400414			0.200207	−	0.979754i	$-0.435838\pi$
$73$	29.2302			0.400414			0.200207	+	0.979754i	$0.435838\pi$
$74$	121.278			1.63890
$75$		0			0
$76$	26.2510	+	15.1800i	0.345408	+	0.199737i
$77$	−185.790			−2.41285
$78$		−	120.309i		−	1.54242i
$79$			8.78894i			0.111252i	0.998452	+	0.0556262i	$0.0177155\pi$
$79$			8.78894i			0.111252i	−0.998452	+	0.0556262i	$0.982284\pi$
$80$		0			0
$81$	−100.515			−1.24092
$82$	176.627			2.15399
$83$	63.7886			0.768537			0.384269	−	0.923221i	$-0.374454\pi$
$83$	63.7886			0.768537			0.384269	+	0.923221i	$0.374454\pi$
$84$		−	62.8679i		−	0.748427i
$85$		0			0
$86$		−	14.7423i		−	0.171422i
$87$	−108.081			−1.24231
$88$			95.3708i			1.08376i
$89$			162.908i			1.83043i	0.402965	+	0.915215i	$0.367980\pi$
$89$			162.908i			1.83043i	−0.402965	+	0.915215i	$0.632020\pi$
$90$		0			0
$91$			158.461i			1.74133i
$92$	−21.6868			−0.235726
$93$	−57.3367			−0.616524
$94$			104.145i			1.10793i
$95$		0			0
$96$	−85.9686			−0.895506
$97$		−	75.5974i		−	0.779355i	−0.920951	−	0.389677i	$-0.872587\pi$
$97$		−	75.5974i		−	0.779355i	0.920951	−	0.389677i	$-0.127413\pi$
$98$			174.420i			1.77979i
$99$	−61.0861			−0.617031
$100$		0			0
$101$	15.7692			0.156131			0.0780656	−	0.996948i	$-0.475126\pi$
$101$	15.7692			0.156131			0.0780656	+	0.996948i	$0.475126\pi$
$102$	170.040			1.66706
$103$			7.15141i			0.0694312i	0.999397	+	0.0347156i	$0.0110525\pi$
$103$			7.15141i			0.0694312i	−0.999397	+	0.0347156i	$0.988947\pi$
$104$	81.3423			0.782137
$105$		0			0
$106$	−72.8737			−0.687488
$107$			85.9872i			0.803619i	0.915723	+	0.401809i	$0.131619\pi$
$107$			85.9872i			0.803619i	−0.915723	+	0.401809i	$0.868381\pi$
$108$			30.4027i			0.281506i
$109$		−	95.0893i		−	0.872379i	−0.899855	−	0.436190i	$-0.856328\pi$
$109$		−	95.0893i		−	0.872379i	0.899855	−	0.436190i	$-0.143672\pi$
$110$		0			0
$111$	−182.289			−1.64225
$112$	219.761			1.96215
$113$			19.0355i			0.168456i	0.996447	+	0.0842281i	$0.0268424\pi$
$113$			19.0355i			0.168456i	−0.996447	+	0.0842281i	$0.973158\pi$
$114$	−138.346	−	80.0007i	−1.21356	−	0.701761i
$115$		0			0
$116$			48.5140i			0.418224i
$117$			52.1006i			0.445304i
$118$	−172.711			−1.46365
$119$	−223.962			−1.88203
$120$		0			0
$121$	160.245			1.32434
$122$		−	42.0668i		−	0.344810i
$123$	−265.482			−2.15839
$124$			25.7365i			0.207552i
$125$		0			0
$126$			95.4592i			0.757613i
$127$			93.4107i			0.735518i	0.929921	+	0.367759i	$0.119875\pi$
$127$			93.4107i			0.735518i	−0.929921	+	0.367759i	$0.880125\pi$
$128$		−	149.114i		−	1.16495i
$129$			22.1586i			0.171772i
$130$		0			0
$131$	126.496			0.965619			0.482810	−	0.875725i	$-0.339616\pi$
$131$	126.496			0.965619			0.482810	+	0.875725i	$0.339616\pi$
$132$			95.1682i			0.720971i
$133$	182.218	+	105.370i	1.37006	+	0.792257i
$134$	−48.6647			−0.363170
$135$		0			0
$136$			114.966i			0.845336i
$137$	−93.8813			−0.685265			−0.342632	−	0.939470i	$-0.611319\pi$
$137$	−93.8813			−0.685265			−0.342632	+	0.939470i	$0.611319\pi$
$138$	114.293			0.828207
$139$	41.0986			0.295673			0.147837	−	0.989012i	$-0.452769\pi$
$139$	41.0986			0.295673			0.147837	+	0.989012i	$0.452769\pi$
$140$		0			0
$141$		−	156.537i		−	1.11019i
$142$	17.7165			0.124764
$143$		−	239.875i		−	1.67745i
$144$	72.2555			0.501775
$145$		0			0
$146$			69.1467i			0.473607i
$147$		−	262.164i		−	1.78343i
$148$			81.8235i			0.552861i
$149$	92.2911			0.619404			0.309702	−	0.950834i	$-0.399771\pi$
$149$	92.2911			0.619404			0.309702	+	0.950834i	$0.399771\pi$
$150$		0			0
$151$		−	7.26853i		−	0.0481359i	−0.999710	−	0.0240680i	$-0.992338\pi$
$151$		−	7.26853i		−	0.0481359i	0.999710	−	0.0240680i	$-0.00766181\pi$
$152$	54.0893	−	93.5375i	0.355851	−	0.615378i
$153$	−73.6368			−0.481286
$154$		−	439.501i		−	2.85390i
$155$		0			0
$156$	81.1695			0.520317
$157$	−229.214			−1.45996			−0.729981	−	0.683468i	$-0.760471\pi$
$157$	−229.214			−1.45996			−0.729981	+	0.683468i	$0.760471\pi$
$158$	−20.7910			−0.131589
$159$	109.534			0.688892
$160$		0			0
$161$	−150.536			−0.935009
$162$		−	237.776i		−	1.46775i
$163$	−34.0426			−0.208850			−0.104425	−	0.994533i	$-0.533300\pi$
$163$	−34.0426			−0.208850			−0.104425	+	0.994533i	$0.533300\pi$
$164$			119.166i			0.726621i
$165$		0			0
$166$			150.897i			0.909021i
$167$			162.274i			0.971702i	0.874042	+	0.485851i	$0.161490\pi$
$167$			162.274i			0.971702i	−0.874042	+	0.485851i	$0.838510\pi$
$168$	−224.011			−1.33340
$169$	−35.5908			−0.210597
$170$		0			0
$171$	59.9118	+	34.6448i	0.350361	+	0.202601i
$172$	9.94623			0.0578269
$173$			29.7956i			0.172229i	0.996285	+	0.0861145i	$0.0274451\pi$
$173$			29.7956i			0.172229i	−0.996285	+	0.0861145i	$0.972555\pi$
$174$		−	255.676i		−	1.46940i
$175$		0			0
$176$	−332.670			−1.89017
$177$	259.596			1.46664
$178$	−385.374			−2.16502
$179$			59.2752i			0.331146i	0.986197	+	0.165573i	$0.0529474\pi$
$179$			59.2752i			0.331146i	−0.986197	+	0.165573i	$0.947053\pi$
$180$		0			0
$181$		−	205.700i		−	1.13647i	−0.822868	−	0.568233i	$-0.807627\pi$
$181$		−	205.700i		−	1.13647i	0.822868	−	0.568233i	$-0.192373\pi$
$182$	−374.853			−2.05963
$183$			63.2291i			0.345514i
$184$			77.2744i			0.419970i
$185$		0			0
$186$		−	135.635i		−	0.729221i
$187$	339.029			1.81299
$188$	−70.2642			−0.373746
$189$			211.037i			1.11660i
$190$		0			0
$191$	−253.682			−1.32818			−0.664089	−	0.747654i	$-0.731180\pi$
$191$	−253.682			−1.32818			−0.664089	+	0.747654i	$0.731180\pi$
$192$			78.7629i			0.410223i
$193$		−	289.646i		−	1.50076i	−0.661009	−	0.750378i	$-0.729872\pi$
$193$		−	289.646i		−	1.50076i	0.661009	−	0.750378i	$-0.270128\pi$
$194$	178.832			0.921816
$195$		0			0
$196$	−117.676			−0.600390
$197$	−178.190			−0.904515			−0.452258	−	0.891887i	$-0.649381\pi$
$197$	−178.190			−0.904515			−0.452258	+	0.891887i	$0.649381\pi$
$198$		−	144.504i		−	0.729820i
$199$	−154.602			−0.776893			−0.388446	−	0.921471i	$-0.626988\pi$
$199$	−154.602			−0.776893			−0.388446	+	0.921471i	$0.626988\pi$
$200$		0			0
$201$	73.1462			0.363912
$202$			37.3035i			0.184671i
$203$			336.754i			1.65889i
$204$			114.721i			0.562360i
$205$		0			0
$206$	−16.9173			−0.0821228
$207$	−49.4951			−0.239107
$208$			283.736i			1.36412i
$209$	−275.838	−	159.507i	−1.31980	−	0.763193i
$210$		0			0
$211$			273.846i			1.29785i	0.760854	+	0.648923i	$0.224780\pi$
$211$			273.846i			1.29785i	−0.760854	+	0.648923i	$0.775220\pi$
$212$		−	49.1660i		−	0.231915i
$213$	−26.6291			−0.125019
$214$	−203.410			−0.950515
$215$		0			0
$216$	108.331			0.501531
$217$			178.647i			0.823258i
$218$	224.942			1.03184
$219$		−	103.932i		−	0.474575i
$220$		0			0
$221$		−	289.160i		−	1.30842i
$222$		−	431.221i		−	1.94244i
$223$			229.796i			1.03048i	0.857047	+	0.515238i	$0.172296\pi$
$223$			229.796i			1.03048i	−0.857047	+	0.515238i	$0.827704\pi$
$224$			267.857i			1.19579i
$225$		0			0
$226$	−45.0302			−0.199249
$227$		−	366.166i		−	1.61306i	−0.591190	−	0.806532i	$-0.701342\pi$
$227$		−	366.166i		−	1.61306i	0.591190	−	0.806532i	$-0.298658\pi$
$228$	53.9744	−	93.3387i	0.236730	−	0.409380i
$229$	334.098			1.45894			0.729471	−	0.684012i	$-0.239766\pi$
$229$	334.098			1.45894			0.729471	+	0.684012i	$0.239766\pi$
$230$		0			0
$231$			660.599i			2.85974i
$232$	172.865			0.745108
$233$	−83.9900			−0.360472			−0.180236	−	0.983623i	$-0.557686\pi$
$233$	−83.9900			−0.360472			−0.180236	+	0.983623i	$0.557686\pi$
$234$	−123.249			−0.526703
$235$		0			0
$236$		−	116.524i		−	0.493745i
$237$	31.2502			0.131857
$238$		−	529.801i		−	2.22606i
$239$	−110.020			−0.460335			−0.230168	−	0.973151i	$-0.573927\pi$
$239$	−110.020			−0.460335			−0.230168	+	0.973151i	$0.573927\pi$
$240$		0			0
$241$		−	321.641i		−	1.33461i	−0.744785	−	0.667305i	$-0.767448\pi$
$241$		−	321.641i		−	1.33461i	0.744785	−	0.667305i	$-0.232552\pi$
$242$			379.073i			1.56642i
$243$			185.950i			0.765225i
$244$	28.3814			0.116317
$245$		0			0
$246$		−	628.021i		−	2.55293i
$247$	−136.045	+	235.264i	−0.550788	+	0.952486i
$248$	91.7043			0.369775
$249$		−	226.809i		−	0.910878i
$250$		0			0
$251$	143.410			0.571353			0.285676	−	0.958326i	$-0.407782\pi$
$251$	143.410			0.571353			0.285676	+	0.958326i	$0.407782\pi$
$252$	−64.4039			−0.255571
$253$	227.879			0.900708
$254$	−220.971			−0.869965
$255$		0			0
$256$	264.136			1.03178
$257$		−	171.202i		−	0.666156i	−0.942899	−	0.333078i	$-0.891913\pi$
$257$		−	171.202i		−	0.666156i	0.942899	−	0.333078i	$-0.108087\pi$
$258$	−52.4180			−0.203171
$259$			567.968i			2.19293i
$260$		0			0
$261$			110.722i			0.424222i
$262$			299.238i			1.14213i
$263$	184.993			0.703395			0.351697	−	0.936114i	$-0.385605\pi$
$263$	184.993			0.703395			0.351697	+	0.936114i	$0.385605\pi$
$264$	339.103			1.28448
$265$		0			0
$266$	−249.262	+	431.053i	−0.937076	+	1.62050i
$267$	579.242			2.16944
$268$		−	32.8328i		−	0.122511i
$269$			453.194i			1.68473i	0.538904	+	0.842367i	$0.318839\pi$
$269$			453.194i			1.68473i	−0.538904	+	0.842367i	$0.681161\pi$
$270$		0			0
$271$	336.795			1.24279			0.621393	−	0.783499i	$-0.286567\pi$
$271$	336.795			1.24279			0.621393	+	0.783499i	$0.286567\pi$
$272$	−401.020			−1.47434
$273$	563.428			2.06384
$274$		−	222.084i		−	0.810527i
$275$		0			0
$276$			77.1102i			0.279385i
$277$	−270.449			−0.976351			−0.488175	−	0.872746i	$-0.662337\pi$
$277$	−270.449			−0.976351			−0.488175	+	0.872746i	$0.662337\pi$
$278$			97.2223i			0.349720i
$279$			58.7376i			0.210529i
$280$		0			0
$281$			151.335i			0.538558i	0.963062	+	0.269279i	$0.0867853\pi$
$281$			151.335i			0.538558i	−0.963062	+	0.269279i	$0.913215\pi$
$282$	370.303			1.31313
$283$	437.891			1.54732			0.773658	−	0.633603i	$-0.218425\pi$
$283$	437.891			1.54732			0.773658	+	0.633603i	$0.218425\pi$
$284$			11.9529i			0.0420876i
$285$		0			0
$286$	567.446			1.98408
$287$			827.175i			2.88214i
$288$			88.0690i			0.305795i
$289$	119.686			0.414139
$290$		0			0
$291$	−268.796			−0.923699
$292$	−46.6515			−0.159765
$293$		−	265.247i		−	0.905279i	−0.891694	−	0.452640i	$-0.850482\pi$
$293$		−	265.247i		−	0.905279i	0.891694	−	0.452640i	$-0.149518\pi$
$294$	620.172			2.10943
$295$		0			0
$296$	291.553			0.984977
$297$		−	319.463i		−	1.07563i
$298$			218.323i			0.732627i
$299$		−	194.359i		−	0.650031i
$300$		0			0
$301$	69.0406			0.229371
$302$	17.1943			0.0569349
$303$		−	56.0696i		−	0.185048i
$304$	326.275	+	188.673i	1.07327	+	0.620635i
$305$		0			0
$306$		−	174.194i		−	0.569262i
$307$			405.065i			1.31943i	0.751517	+	0.659714i	$0.229323\pi$
$307$			405.065i			1.31943i	−0.751517	+	0.659714i	$0.770677\pi$
$308$	296.520			0.962728
$309$	25.4278			0.0822906
$310$		0			0
$311$	−313.906			−1.00934			−0.504672	−	0.863311i	$-0.668387\pi$
$311$	−313.906			−1.00934			−0.504672	+	0.863311i	$0.668387\pi$
$312$		−	289.223i		−	0.926997i
$313$	−513.442			−1.64039			−0.820194	−	0.572085i	$-0.806135\pi$
$313$	−513.442			−1.64039			−0.820194	+	0.572085i	$0.806135\pi$
$314$		−	542.225i		−	1.72683i
$315$		0			0
$316$		−	14.0271i		−	0.0443897i
$317$		−	25.8051i		−	0.0814042i	−0.999171	−	0.0407021i	$-0.987041\pi$
$317$		−	25.8051i		−	0.0814042i	0.999171	−	0.0407021i	$-0.0129595\pi$
$318$			259.112i			0.814817i
$319$		−	509.772i		−	1.59803i
$320$		0			0
$321$	305.739			0.952457
$322$		−	356.107i		−	1.10592i
$323$	−332.512	−	192.280i	−1.02945	−	0.595293i
$324$	160.422			0.495128
$325$		0			0
$326$		−	80.5307i		−	0.247027i
$327$	−338.102			−1.03395
$328$	424.611			1.29455
$329$	−487.731			−1.48247
$330$		0			0
$331$			171.635i			0.518534i	0.965806	+	0.259267i	$0.0834810\pi$
$331$			171.635i			0.518534i	−0.965806	+	0.259267i	$0.916519\pi$
$332$	−101.807			−0.306647
$333$			186.743i			0.560790i
$334$	−383.874			−1.14932
$335$		0			0
$336$		−	781.388i		−	2.32556i
$337$		−	206.114i		−	0.611613i	−0.952094	−	0.305807i	$-0.901074\pi$
$337$		−	206.114i		−	0.611613i	0.952094	−	0.305807i	$-0.0989261\pi$
$338$		−	84.1932i		−	0.249092i
$339$	67.6833			0.199656
$340$		0			0
$341$		−	270.432i		−	0.793057i
$342$	−81.9553	+	141.727i	−0.239635	+	0.414405i
$343$	−273.994			−0.798815
$344$		−	35.4404i		−	0.103024i
$345$		0			0
$346$	−70.4841			−0.203711
$347$	7.91893			0.0228211			0.0114106	−	0.999935i	$-0.496368\pi$
$347$	7.91893			0.0228211			0.0114106	+	0.999935i	$0.496368\pi$
$348$	172.498			0.495683
$349$	409.448			1.17320			0.586602	−	0.809876i	$-0.300465\pi$
$349$	409.448			1.17320			0.586602	+	0.809876i	$0.300465\pi$
$350$		0			0
$351$	−272.472			−0.776273
$352$		−	405.476i		−	1.15192i
$353$	637.919			1.80714			0.903568	−	0.428445i	$-0.140939\pi$
$353$	637.919			1.80714			0.903568	+	0.428445i	$0.140939\pi$
$354$			614.097i			1.73474i
$355$		0			0
$356$		−	260.002i		−	0.730342i
$357$			796.325i			2.23060i
$358$	−140.221			−0.391678
$359$	−237.272			−0.660926			−0.330463	−	0.943819i	$-0.607205\pi$
$359$	−237.272			−0.660926			−0.330463	+	0.943819i	$0.607205\pi$
$360$		0			0
$361$	180.072	+	312.882i	0.498813	+	0.866710i
$362$	486.602			1.34420
$363$		−	569.771i		−	1.56962i
$364$		−	252.904i		−	0.694790i
$365$		0			0
$366$	−149.574			−0.408672
$367$	−542.889			−1.47926			−0.739631	−	0.673013i	$-0.765000\pi$
$367$	−542.889			−1.47926			−0.739631	+	0.673013i	$0.765000\pi$
$368$	−269.546			−0.732463
$369$			271.968i			0.737042i
$370$		0			0
$371$		−	341.280i		−	0.919892i
$372$	91.5095			0.245993
$373$			444.746i			1.19235i	0.802855	+	0.596175i	$0.203313\pi$
$373$			444.746i			1.19235i	−0.802855	+	0.596175i	$0.796687\pi$
$374$			802.003i			2.14439i
$375$		0			0
$376$			250.366i			0.665866i
$377$	−434.787			−1.15328
$378$	−499.225			−1.32070
$379$			150.221i			0.396362i	0.980165	+	0.198181i	$0.0635035\pi$
$379$			150.221i			0.396362i	−0.980165	+	0.198181i	$0.936497\pi$
$380$		0			0
$381$	332.134			0.871743
$382$		−	600.106i		−	1.57096i
$383$			85.8011i			0.224024i	0.993707	+	0.112012i	$0.0357295\pi$
$383$			85.8011i			0.224024i	−0.993707	+	0.112012i	$0.964271\pi$
$384$	−530.195			−1.38072
$385$		0			0
$386$	685.182			1.77508
$387$	22.7000			0.0586562
$388$			120.654i			0.310963i
$389$	−249.669			−0.641822			−0.320911	−	0.947109i	$-0.603989\pi$
$389$	−249.669			−0.641822			−0.320911	+	0.947109i	$0.603989\pi$
$390$		0			0
$391$	274.699			0.702555
$392$			419.305i			1.06966i
$393$		−	449.773i		−	1.14446i
$394$		−	421.523i		−	1.06985i
$395$		0			0
$396$	97.4934			0.246195
$397$	340.476			0.857622			0.428811	−	0.903394i	$-0.358933\pi$
$397$	340.476			0.857622			0.428811	+	0.903394i	$0.358933\pi$
$398$		−	365.724i		−	0.918904i
$399$	374.657	−	647.900i	0.938990	−	1.62381i
$400$		0			0
$401$			491.002i			1.22444i	0.790686	+	0.612221i	$0.209724\pi$
$401$			491.002i			1.22444i	−0.790686	+	0.612221i	$0.790276\pi$
$402$			173.034i			0.430432i
$403$	−230.653			−0.572341
$404$	−25.1677			−0.0622964
$405$		0			0
$406$	−796.621			−1.96212
$407$		−	859.779i		−	2.11248i
$408$	408.775			1.00190
$409$			115.833i			0.283210i	0.989923	+	0.141605i	$0.0452263\pi$
$409$			115.833i			0.283210i	−0.989923	+	0.141605i	$0.954774\pi$
$410$		0			0
$411$			333.807i			0.812183i
$412$		−	11.4137i		−	0.0277031i
$413$		−	808.836i		−	1.95844i
$414$		−	117.085i		−	0.282814i
$415$		0			0
$416$	−345.833			−0.831329
$417$		−	146.131i		−	0.350435i
$418$	377.328	−	652.519i	0.902700	−	1.56105i
$419$	81.4969			0.194503			0.0972516	−	0.995260i	$-0.468995\pi$
$419$	81.4969			0.194503			0.0972516	+	0.995260i	$0.468995\pi$
$420$		0			0
$421$			221.652i			0.526490i	0.964729	+	0.263245i	$0.0847927\pi$
$421$			221.652i			0.526490i	−0.964729	+	0.263245i	$0.915207\pi$
$422$	−647.806			−1.53508
$423$	−160.362			−0.379106
$424$	−175.188			−0.413180
$425$		0			0
$426$		−	62.9934i		−	0.147872i
$427$	197.006			0.461372
$428$		−	137.236i		−	0.320644i
$429$	−852.907			−1.98813
$430$		0			0
$431$		−	211.834i		−	0.491494i	−0.969334	−	0.245747i	$-0.920967\pi$
$431$		−	211.834i		−	0.491494i	0.969334	−	0.245747i	$-0.0790333\pi$
$432$			377.876i			0.874714i
$433$			389.534i			0.899616i	0.893125	+	0.449808i	$0.148508\pi$
$433$			389.534i			0.899616i	−0.893125	+	0.449808i	$0.851492\pi$
$434$	−422.605			−0.973744
$435$		0			0
$436$			151.763i			0.348079i
$437$	−223.499	−	129.241i	−0.511438	−	0.295746i
$438$	245.860			0.561324
$439$		−	465.691i		−	1.06080i	−0.847748	−	0.530400i	$-0.822042\pi$
$439$		−	465.691i		−	1.06080i	0.847748	−	0.530400i	$-0.177958\pi$
$440$		0			0
$441$	−268.569			−0.609001
$442$	684.033			1.54759
$443$	382.479			0.863383			0.431692	−	0.902021i	$-0.357917\pi$
$443$	382.479			0.863383			0.431692	+	0.902021i	$0.357917\pi$
$444$	290.934			0.655257
$445$		0			0
$446$	−543.603			−1.21884
$447$		−	328.153i		−	0.734123i
$448$	245.406			0.547780
$449$		−	666.393i		−	1.48417i	−0.670305	−	0.742086i	$-0.733837\pi$
$449$		−	666.393i		−	1.48417i	0.670305	−	0.742086i	$-0.266163\pi$
$450$		0			0
$451$		−	1252.16i		−	2.77641i
$452$		−	30.3807i		−	0.0672140i
$453$	−25.8442			−0.0570512
$454$	866.196			1.90792
$455$		0			0
$456$	−332.585	−	192.322i	−0.729352	−	0.421758i
$457$	627.200			1.37243			0.686214	−	0.727399i	$-0.259271\pi$
$457$	627.200			1.37243			0.686214	+	0.727399i	$0.259271\pi$
$458$			790.337i			1.72563i
$459$		−	385.100i		−	0.838998i
$460$		0			0
$461$	798.338			1.73175			0.865876	−	0.500258i	$-0.166762\pi$
$461$	798.338			1.73175			0.865876	+	0.500258i	$0.166762\pi$
$462$	−1562.70			−3.38248
$463$	−791.786			−1.71012			−0.855061	−	0.518528i	$-0.826480\pi$
$463$	−791.786			−1.71012			−0.855061	+	0.518528i	$0.826480\pi$
$464$			602.983i			1.29953i
$465$		0			0
$466$		−	198.686i		−	0.426364i
$467$	418.723			0.896622			0.448311	−	0.893878i	$-0.352026\pi$
$467$	418.723			0.896622			0.448311	+	0.893878i	$0.352026\pi$
$468$		−	83.1526i		−	0.177677i
$469$		−	227.905i		−	0.485939i
$470$		0			0
$471$			815.000i			1.73036i
$472$	−415.197			−0.879655
$473$	−104.512			−0.220956
$474$			73.9251i			0.155960i
$475$		0			0
$476$	357.443			0.750931
$477$		−	112.210i		−	0.235241i
$478$		−	260.262i		−	0.544481i
$479$	63.6160			0.132810			0.0664050	−	0.997793i	$-0.478847\pi$
$479$	63.6160			0.132810			0.0664050	+	0.997793i	$0.478847\pi$
$480$		0			0
$481$	−733.310			−1.52455
$482$	760.869			1.57857
$483$			535.252i			1.10818i
$484$	−255.751			−0.528411
$485$		0			0
$486$	−439.880			−0.905102
$487$			556.997i			1.14373i	0.820347	+	0.571866i	$0.193780\pi$
$487$			556.997i			1.14373i	−0.820347	+	0.571866i	$0.806220\pi$
$488$		−	101.129i		−	0.207231i
$489$			121.043i			0.247531i
$490$		0			0
$491$	856.809			1.74503			0.872514	−	0.488589i	$-0.162488\pi$
$491$	856.809			1.74503			0.872514	+	0.488589i	$0.162488\pi$
$492$	423.709			0.861198
$493$		−	614.510i		−	1.24647i
$494$	−556.537	−	321.825i	−1.12659	−	0.651469i
$495$		0			0
$496$			319.881i			0.644920i
$497$			82.9696i			0.166941i
$498$	536.536			1.07738
$499$	761.632			1.52632			0.763158	−	0.646211i	$-0.223648\pi$
$499$	761.632			1.52632			0.763158	+	0.646211i	$0.223648\pi$
$500$		0			0
$501$	576.987			1.15167
$502$			339.248i			0.675792i
$503$	34.7192			0.0690242			0.0345121	−	0.999404i	$-0.489012\pi$
$503$	34.7192			0.0690242			0.0345121	+	0.999404i	$0.489012\pi$
$504$			229.484i			0.455325i
$505$		0			0
$506$			539.068i			1.06535i
$507$			126.548i			0.249601i
$508$		−	149.084i		−	0.293472i
$509$			9.31181i			0.0182943i	0.999958	+	0.00914716i	$0.00291167\pi$
$509$			9.31181i			0.0182943i	−0.999958	+	0.00914716i	$0.997088\pi$
$510$		0			0
$511$	−323.826			−0.633710
$512$			28.3801i			0.0554300i
$513$	−181.183	+	313.322i	−0.353183	+	0.610764i
$514$	404.993			0.787925
$515$		0			0
$516$		−	35.3651i		−	0.0685370i
$517$	738.318			1.42808
$518$	−1343.58			−2.59378
$519$	105.942			0.204128
$520$		0			0
$521$			88.0858i			0.169071i	0.996420	+	0.0845353i	$0.0269406\pi$
$521$			88.0858i			0.169071i	−0.996420	+	0.0845353i	$0.973059\pi$
$522$	−261.922			−0.501767
$523$			377.616i			0.722019i	0.932562	+	0.361009i	$0.117568\pi$
$523$			377.616i			0.722019i	−0.932562	+	0.361009i	$0.882432\pi$
$524$	−201.888			−0.385282
$525$		0			0
$526$			437.617i			0.831971i
$527$		−	325.995i		−	0.618587i
$528$			1182.85i			2.24025i
$529$	−344.360			−0.650965
$530$		0			0
$531$		−	265.938i		−	0.500826i
$532$	−290.820	−	168.171i	−0.546654	−	0.316111i
$533$	−1067.98			−2.00371
$534$			1370.25i			2.56600i
$535$		0			0
$536$	−116.990			−0.218265
$537$	210.761			0.392478
$538$	−1072.07			−1.99269
$539$	1236.51			2.29409
$540$		0			0
$541$	−225.756			−0.417293			−0.208647	−	0.977991i	$-0.566906\pi$
$541$	−225.756			−0.417293			−0.208647	+	0.977991i	$0.566906\pi$
$542$			796.718i			1.46996i
$543$	−731.394			−1.34695
$544$		−	488.785i		−	0.898502i
$545$		0			0
$546$			1332.84i			2.44110i
$547$		−	1050.58i		−	1.92062i	−0.278938	−	0.960309i	$-0.589982\pi$
$547$		−	1050.58i		−	1.92062i	0.278938	−	0.960309i	$-0.410018\pi$
$548$	149.835			0.273421
$549$	64.7739			0.117985
$550$		0			0
$551$	−289.116	+	499.973i	−0.524712	+	0.907391i
$552$	274.759			0.497752
$553$		−	97.3678i		−	0.176072i
$554$		−	639.771i		−	1.15482i
$555$		0			0
$556$	−65.5934			−0.117974
$557$	−580.999			−1.04309			−0.521543	−	0.853225i	$-0.674644\pi$
$557$	−580.999			−1.04309			−0.521543	+	0.853225i	$0.674644\pi$
$558$	−138.949			−0.249012
$559$			89.1391i			0.159462i
$560$		0			0
$561$		−	1205.46i		−	2.14877i
$562$	−357.995			−0.637002
$563$			796.409i			1.41458i	0.706923	+	0.707291i	$0.250083\pi$
$563$			796.409i			1.41458i	−0.706923	+	0.707291i	$0.749917\pi$
$564$			249.834i			0.442967i
$565$		0			0
$566$			1035.87i			1.83016i
$567$	1113.55			1.96393
$568$	42.5905			0.0749833
$569$			171.428i			0.301280i	0.988589	+	0.150640i	$0.0481334\pi$
$569$			171.428i			0.301280i	−0.988589	+	0.150640i	$0.951867\pi$
$570$		0			0
$571$	−673.960			−1.18031			−0.590157	−	0.807288i	$-0.700934\pi$
$571$	−673.960			−1.18031			−0.590157	+	0.807288i	$0.700934\pi$
$572$			382.841i			0.669302i
$573$			901.999i			1.57417i
$574$	−1956.76			−3.40898
$575$		0			0
$576$	80.6873			0.140082
$577$	795.916			1.37940			0.689702	−	0.724094i	$-0.257742\pi$
$577$	795.916			1.37940			0.689702	+	0.724094i	$0.257742\pi$
$578$			283.128i			0.489840i
$579$	−1029.87			−1.77871
$580$		0			0
$581$	−706.679			−1.21632
$582$		−	635.861i		−	1.09254i
$583$			516.623i			0.886146i
$584$			166.229i			0.284638i
$585$		0			0
$586$	627.464			1.07076
$587$	19.6929			0.0335484			0.0167742	−	0.999859i	$-0.494660\pi$
$587$	19.6929			0.0335484			0.0167742	+	0.999859i	$0.494660\pi$
$588$			418.414i			0.711589i
$589$	−153.375	+	265.234i	−0.260399	+	0.450312i
$590$		0			0
$591$			633.576i			1.07204i
$592$			1016.99i			1.71789i
$593$	636.058			1.07261			0.536306	−	0.844024i	$-0.319819\pi$
$593$	636.058			1.07261			0.536306	+	0.844024i	$0.319819\pi$
$594$	755.718			1.27225
$595$		0			0
$596$	−147.297			−0.247142
$597$			549.706i			0.920781i
$598$	459.774			0.768853
$599$			936.525i			1.56348i	0.623604	+	0.781741i	$0.285668\pi$
$599$			936.525i			1.56348i	−0.623604	+	0.781741i	$0.714332\pi$
$600$		0			0
$601$		−	252.090i		−	0.419451i	−0.977760	−	0.209725i	$-0.932743\pi$
$601$		−	252.090i		−	0.419451i	0.977760	−	0.209725i	$-0.0672570\pi$
$602$			163.321i			0.271298i
$603$		−	74.9334i		−	0.124268i
$604$			11.6006i			0.0192062i
$605$		0			0
$606$	132.638			0.218874
$607$		−	498.526i		−	0.821295i	−0.911794	−	0.410648i	$-0.865303\pi$
$607$		−	498.526i		−	0.821295i	0.911794	−	0.410648i	$-0.134697\pi$
$608$	−229.965	+	397.682i	−0.378232	+	0.654082i
$609$	1197.37			1.96613
$610$		0			0
$611$		−	629.715i		−	1.03063i
$612$	117.524			0.192033
$613$	638.709			1.04194			0.520969	−	0.853575i	$-0.325571\pi$
$613$	638.709			1.04194			0.520969	+	0.853575i	$0.325571\pi$
$614$	−958.215			−1.56061
$615$		0			0
$616$		−	1056.56i		−	1.71520i
$617$	−702.709			−1.13891			−0.569457	−	0.822021i	$-0.692846\pi$
$617$	−702.709			−1.13891			−0.569457	+	0.822021i	$0.692846\pi$
$618$			60.1516i			0.0973327i
$619$	195.693			0.316144			0.158072	−	0.987428i	$-0.449472\pi$
$619$	195.693			0.316144			0.158072	+	0.987428i	$0.449472\pi$
$620$		0			0
$621$		−	258.846i		−	0.416821i
$622$		−	742.572i		−	1.19385i
$623$		−	1804.77i		−	2.89691i
$624$	1008.86			1.61676
$625$		0			0
$626$		−	1214.59i		−	1.94024i
$627$	−567.149	+	980.779i	−0.904544	+	1.56424i
$628$	365.826			0.582525
$629$		−	1036.43i		−	1.64774i
$630$		0			0
$631$	−1063.16			−1.68489			−0.842444	−	0.538784i	$-0.818884\pi$
$631$	−1063.16			−1.68489			−0.842444	+	0.538784i	$0.818884\pi$
$632$	−49.9815			−0.0790847
$633$	973.694			1.53822
$634$	61.0443			0.0962843
$635$		0			0
$636$	−174.816			−0.274868
$637$		−	1054.63i		−	1.65562i
$638$	1205.91			1.89014
$639$			27.2797i			0.0426913i
$640$		0			0
$641$			744.630i			1.16167i	0.814021	+	0.580835i	$0.197274\pi$
$641$			744.630i			1.16167i	−0.814021	+	0.580835i	$0.802726\pi$
$642$			723.251i			1.12656i
$643$	−195.660			−0.304293			−0.152146	−	0.988358i	$-0.548618\pi$
$643$	−195.660			−0.304293			−0.152146	+	0.988358i	$0.548618\pi$
$644$	240.256			0.373069
$645$		0			0
$646$	454.854	−	786.586i	0.704109	−	1.21763i
$647$	128.045			0.197906			0.0989529	−	0.995092i	$-0.468451\pi$
$647$	128.045			0.197906			0.0989529	+	0.995092i	$0.468451\pi$
$648$		−	571.614i		−	0.882120i
$649$			1224.40i			1.88660i
$650$		0			0
$651$	635.202			0.975733
$652$	54.3320			0.0833313
$653$	717.897			1.09938			0.549692	−	0.835368i	$-0.314745\pi$
$653$	717.897			1.09938			0.549692	+	0.835368i	$0.314745\pi$
$654$		−	799.811i		−	1.22295i
$655$		0			0
$656$			1481.12i			2.25780i
$657$	−106.471			−0.162057
$658$		−	1153.77i		−	1.75345i
$659$			544.617i			0.826430i	0.910633	+	0.413215i	$0.135594\pi$
$659$			544.617i			0.826430i	−0.910633	+	0.413215i	$0.864406\pi$
$660$		0			0
$661$		−	435.420i		−	0.658729i	−0.944203	−	0.329365i	$-0.893165\pi$
$661$		−	435.420i		−	0.658729i	0.944203	−	0.329365i	$-0.106835\pi$
$662$	−406.017			−0.613319
$663$	−1028.15			−1.55075
$664$			362.757i			0.546321i
$665$		0			0
$666$	−441.757			−0.663299
$667$		−	413.044i		−	0.619256i
$668$		−	258.990i		−	0.387709i
$669$	817.070			1.22133
$670$		0			0
$671$	−298.224			−0.444447
$672$	952.399			1.41726
$673$		−	397.296i		−	0.590336i	−0.955445	−	0.295168i	$-0.904624\pi$
$673$		−	397.296i		−	0.590336i	0.955445	−	0.295168i	$-0.0953756\pi$
$674$	487.580			0.723412
$675$		0			0
$676$	56.8030			0.0840281
$677$			456.859i			0.674829i	0.941356	+	0.337415i	$0.109552\pi$
$677$			456.859i			0.674829i	−0.941356	+	0.337415i	$0.890448\pi$
$678$			160.111i			0.236152i
$679$			837.502i			1.23343i
$680$		0			0
$681$	−1301.95			−1.91182
$682$	639.731			0.938022
$683$			880.553i			1.28924i	0.764502	+	0.644621i	$0.222985\pi$
$683$			880.553i			1.28924i	−0.764502	+	0.644621i	$0.777015\pi$
$684$	−95.6192	−	55.2932i	−0.139794	−	0.0808379i
$685$		0			0
$686$		−	648.155i		−	0.944833i
$687$		−	1187.93i		−	1.72915i
$688$	123.622			0.179683
$689$	440.631			0.639522
$690$		0			0
$691$	−34.3920			−0.0497713			−0.0248857	−	0.999690i	$-0.507922\pi$
$691$	−34.3920			−0.0497713			−0.0248857	+	0.999690i	$0.507922\pi$
$692$		−	47.5538i		−	0.0687194i
$693$	676.739			0.976535
$694$			18.7329i			0.0269927i
$695$		0			0
$696$		−	614.644i		−	0.883109i
$697$		−	1509.43i		−	2.16561i
$698$			968.585i			1.38766i
$699$			298.637i			0.427235i
$700$		0			0
$701$	419.321			0.598175			0.299088	−	0.954226i	$-0.403318\pi$
$701$	419.321			0.598175			0.299088	+	0.954226i	$0.403318\pi$
$702$		−	644.556i		−	0.918171i
$703$	−487.622	+	843.251i	−0.693630	+	1.19950i
$704$	−371.490			−0.527685
$705$		0			0
$706$			1509.05i			2.13747i
$707$	−174.699			−0.247099
$708$	−414.315			−0.585191
$709$	373.670			0.527038			0.263519	−	0.964654i	$-0.415117\pi$
$709$	373.670			0.527038			0.263519	+	0.964654i	$0.415117\pi$
$710$		0			0
$711$		−	32.0137i		−	0.0450263i
$712$	−926.438			−1.30118
$713$		−	219.118i		−	0.307319i
$714$	−1883.78			−2.63834
$715$		0			0
$716$		−	94.6033i		−	0.132127i
$717$			391.191i			0.545594i
$718$		−	561.289i		−	0.781739i
$719$	−328.089			−0.456312			−0.228156	−	0.973625i	$-0.573270\pi$
$719$	−328.089			−0.456312			−0.228156	+	0.973625i	$0.573270\pi$
$720$		0			0
$721$		−	79.2266i		−	0.109884i
$722$	−740.150	+	425.975i	−1.02514	+	0.589993i
$723$	−1143.64			−1.58179
$724$			328.298i			0.453450i
$725$		0			0
$726$	1347.84			1.85653
$727$	−1131.05			−1.55578			−0.777889	−	0.628402i	$-0.783709\pi$
$727$	−1131.05			−1.55578			−0.777889	+	0.628402i	$0.783709\pi$
$728$	−901.146			−1.23784
$729$	−243.465			−0.333971
$730$		0			0
$731$	−125.985			−0.172347
$732$		−	100.914i		−	0.137860i
$733$	450.895			0.615137			0.307568	−	0.951526i	$-0.400485\pi$
$733$	450.895			0.615137			0.307568	+	0.951526i	$0.400485\pi$
$734$		−	1284.25i		−	1.74966i
$735$		0			0
$736$		−	328.538i		−	0.446383i
$737$			344.999i			0.468112i
$738$	−643.365			−0.871768
$739$	−195.478			−0.264517			−0.132258	−	0.991215i	$-0.542223\pi$
$739$	−195.478			−0.264517			−0.132258	+	0.991215i	$0.542223\pi$
$740$		0			0
$741$	836.512	+	483.724i	1.12890	+	0.652800i
$742$	807.328			1.08804
$743$			194.450i			0.261710i	0.991402	+	0.130855i	$0.0417722\pi$
$743$			194.450i			0.261710i	−0.991402	+	0.130855i	$0.958228\pi$
$744$		−	326.067i		−	0.438262i
$745$		0			0
$746$	−1052.09			−1.41030
$747$	−232.350			−0.311044
$748$	−541.091			−0.723383
$749$		−	952.605i		−	1.27184i
$750$		0			0
$751$		−	943.646i		−	1.25652i	−0.778004	−	0.628259i	$-0.783768\pi$
$751$		−	943.646i		−	1.25652i	0.778004	−	0.628259i	$-0.216232\pi$
$752$	−873.318			−1.16133
$753$		−	509.911i		−	0.677173i
$754$		−	1028.53i		−	1.36409i
$755$		0			0
$756$		−	336.814i		−	0.445522i
$757$	−891.082			−1.17712			−0.588561	−	0.808453i	$-0.700305\pi$
$757$	−891.082			−1.17712			−0.588561	+	0.808453i	$0.700305\pi$
$758$	−355.362			−0.468815
$759$		−	810.254i		−	1.06753i
$760$		0			0
$761$	759.322			0.997795			0.498898	−	0.866661i	$-0.333738\pi$
$761$	759.322			0.997795			0.498898	+	0.866661i	$0.333738\pi$
$762$			785.692i			1.03109i
$763$			1053.44i			1.38066i
$764$	404.876			0.529943
$765$		0			0
$766$	−202.970			−0.264974
$767$	1044.30			1.36154
$768$		−	939.170i		−	1.22288i
$769$	225.596			0.293363			0.146681	−	0.989184i	$-0.453141\pi$
$769$	225.596			0.293363			0.146681	+	0.989184i	$0.453141\pi$
$770$		0			0
$771$	−608.731			−0.789535
$772$			462.275i			0.598802i
$773$		−	1189.39i		−	1.53867i	−0.638843	−	0.769337i	$-0.720587\pi$
$773$		−	1189.39i		−	1.53867i	0.638843	−	0.769337i	$-0.279413\pi$
$774$			53.6987i			0.0693782i
$775$		0			0
$776$	429.913			0.554011
$777$	2019.48			2.59908
$778$		−	590.613i		−	0.759143i
$779$	−710.161	+	1228.09i	−0.911632	+	1.57650i
$780$		0			0
$781$		−	125.598i		−	0.160817i
$782$			649.825i			0.830978i
$783$	−579.045			−0.739521
$784$	−1462.61			−1.86557
$785$		0			0
$786$	1063.98			1.35366
$787$			254.533i			0.323422i	0.986838	+	0.161711i	$0.0517013\pi$
$787$			254.533i			0.323422i	−0.986838	+	0.161711i	$0.948299\pi$
$788$	284.391			0.360902
$789$		−	657.766i		−	0.833670i
$790$		0			0
$791$		−	210.884i		−	0.266605i
$792$		−	347.388i		−	0.438622i
$793$			254.357i			0.320753i
$794$			805.425i			1.01439i
$795$		0			0
$796$	246.744			0.309980
$797$			719.295i			0.902503i	0.892397	+	0.451251i	$0.149022\pi$
$797$			719.295i			0.902503i	−0.892397	+	0.451251i	$0.850978\pi$
$798$	1532.66	+	886.284i	1.92063	+	1.11063i
$799$	890.013			1.11391
$800$		0			0
$801$		−	593.394i		−	0.740816i
$802$	−1161.51			−1.44826
$803$	490.202			0.610463
$804$	−116.741			−0.145201
$805$		0			0
$806$		−	545.630i		−	0.676961i
$807$	1611.39			1.99676
$808$			89.6776i			0.110987i
$809$	67.3018			0.0831914			0.0415957	−	0.999135i	$-0.486756\pi$
$809$	67.3018			0.0831914			0.0415957	+	0.999135i	$0.486756\pi$
$810$		0			0
$811$			401.142i			0.494626i	0.968936	+	0.247313i	$0.0795477\pi$
$811$			401.142i			0.494626i	−0.968936	+	0.247313i	$0.920452\pi$
$812$		−	537.460i		−	0.661896i
$813$		−	1197.52i		−	1.47296i
$814$	2033.88			2.49863
$815$		0			0
$816$			1425.88i			1.74740i
$817$	102.503	+	59.2739i	0.125463	+	0.0725507i
$818$	−274.013			−0.334979
$819$		−	577.194i		−	0.704755i
$820$		0			0
$821$	−99.8212			−0.121585			−0.0607925	−	0.998150i	$-0.519363\pi$
$821$	−99.8212			−0.121585			−0.0607925	+	0.998150i	$0.519363\pi$
$822$	−789.650			−0.960644
$823$	93.5359			0.113652			0.0568262	−	0.998384i	$-0.481902\pi$
$823$	93.5359			0.113652			0.0568262	+	0.998384i	$0.481902\pi$
$824$	−40.6692			−0.0493558
$825$		0			0
$826$	1913.37			2.31643
$827$		−	398.725i		−	0.482135i	−0.970508	−	0.241067i	$-0.922503\pi$
$827$		−	398.725i		−	0.482135i	0.970508	−	0.241067i	$-0.0774975\pi$
$828$	78.9942			0.0954036
$829$			805.348i			0.971470i	0.874106	+	0.485735i	$0.161448\pi$
$829$			805.348i			0.971470i	−0.874106	+	0.485735i	$0.838552\pi$
$830$		0			0
$831$			961.617i			1.15718i
$832$			316.846i			0.380825i
$833$	1490.57			1.78940
$834$	345.686			0.414492
$835$		0			0
$836$	440.238	+	254.574i	0.526601	+	0.304514i
$837$	−307.181			−0.367003
$838$			192.788i			0.230057i
$839$		−	1547.08i		−	1.84396i	−0.387235	−	0.921981i	$-0.626570\pi$
$839$		−	1547.08i		−	1.84396i	0.387235	−	0.921981i	$-0.373430\pi$
$840$		0			0
$841$	−82.9912			−0.0986815
$842$	−524.338			−0.622729
$843$	538.090			0.638304
$844$		−	437.058i		−	0.517841i
$845$		0			0
$846$		−	379.350i		−	0.448404i
$847$	−1775.26			−2.09594
$848$		−	611.087i		−	0.720622i
$849$		−	1556.98i		−	1.83390i
$850$		0			0
$851$		−	696.638i		−	0.818611i
$852$	42.5001			0.0498827
$853$	252.303			0.295784			0.147892	−	0.989004i	$-0.452751\pi$
$853$	252.303			0.295784			0.147892	+	0.989004i	$0.452751\pi$
$854$			466.035i			0.545708i
$855$		0			0
$856$	−488.998			−0.571259
$857$			1397.35i			1.63051i	0.579102	+	0.815255i	$0.303403\pi$
$857$			1397.35i			1.63051i	−0.579102	+	0.815255i	$0.696597\pi$
$858$		−	2017.63i		−	2.35155i
$859$	−1472.96			−1.71473			−0.857366	−	0.514706i	$-0.827901\pi$
$859$	−1472.96			−1.71473			−0.857366	+	0.514706i	$0.827901\pi$
$860$		0			0
$861$	2941.13			3.41595
$862$	501.112			0.581336
$863$			63.9533i			0.0741058i	0.999313	+	0.0370529i	$0.0117970\pi$
$863$			63.9533i			0.0741058i	−0.999313	+	0.0370529i	$0.988203\pi$
$864$	−460.576			−0.533075
$865$		0			0
$866$	−921.476			−1.06406
$867$		−	425.559i		−	0.490841i
$868$		−	285.121i		−	0.328480i
$869$			147.394i			0.169613i
$870$		0			0
$871$	294.251			0.337832
$872$	540.761			0.620138
$873$			275.364i			0.315422i
$874$	305.731	−	528.705i	0.349807	−	0.604926i
$875$		0			0
$876$			165.875i			0.189356i
$877$			399.492i			0.455522i	0.973717	+	0.227761i	$0.0731404\pi$
$877$			399.492i			0.455522i	−0.973717	+	0.227761i	$0.926860\pi$
$878$	1101.63			1.25471
$879$	−943.119			−1.07295
$880$		0			0
$881$	514.180			0.583632			0.291816	−	0.956474i	$-0.405740\pi$
$881$	514.180			0.583632			0.291816	+	0.956474i	$0.405740\pi$
$882$		−	635.324i		−	0.720322i
$883$	−323.869			−0.366783			−0.183392	−	0.983040i	$-0.558708\pi$
$883$	−323.869			−0.366783			−0.183392	+	0.983040i	$0.558708\pi$
$884$			461.499i			0.522058i
$885$		0			0
$886$			904.787i			1.02120i
$887$		−	277.999i		−	0.313415i	−0.987645	−	0.156707i	$-0.949912\pi$
$887$		−	277.999i		−	0.313415i	0.987645	−	0.156707i	$-0.0500879\pi$
$888$		−	1036.66i		−	1.16740i
$889$		−	1034.85i		−	1.16406i
$890$		0			0
$891$	−1685.67			−1.89188
$892$		−	366.755i		−	0.411160i
$893$	−724.125	−	418.735i	−0.810890	−	0.468909i
$894$	776.275			0.868316
$895$		0			0
$896$			1651.95i			1.84370i
$897$	−691.070			−0.770424
$898$	1576.41			1.75547
$899$	−490.174			−0.545244
$900$		0			0
$901$			622.769i			0.691197i
$902$	2962.10			3.28392
$903$		−	245.483i		−	0.271852i
$904$	−108.253			−0.119749
$905$		0			0
$906$		−	61.1367i		−	0.0674798i
$907$			401.356i			0.442509i	0.975216	+	0.221255i	$0.0710152\pi$
$907$			401.356i			0.442509i	−0.975216	+	0.221255i	$0.928985\pi$
$908$			584.400i			0.643613i
$909$	−57.4395			−0.0631898
$910$		0			0
$911$		−	378.750i		−	0.415752i	−0.978155	−	0.207876i	$-0.933345\pi$
$911$		−	378.750i		−	0.415752i	0.978155	−	0.207876i	$-0.0666551\pi$
$912$	670.851	−	1160.11i	0.735583	−	1.27205i
$913$	1069.76			1.17170
$914$			1483.70i			1.62330i
$915$		0			0
$916$	−533.220			−0.582118
$917$	−1401.38			−1.52822
$918$	910.987			0.992361
$919$	−102.234			−0.111244			−0.0556222	−	0.998452i	$-0.517714\pi$
$919$	−102.234			−0.111244			−0.0556222	+	0.998452i	$0.517714\pi$
$920$		0			0
$921$	1440.26			1.56380
$922$			1888.54i			2.04831i
$923$	−107.123			−0.116060
$924$		−	1054.32i		−	1.14103i
$925$		0			0
$926$		−	1873.04i		−	2.02272i
$927$		−	26.0490i		−	0.0281004i
$928$	−734.949			−0.791971
$929$	1794.45			1.93160			0.965798	−	0.259295i	$-0.0834903\pi$
$929$	1794.45			1.93160			0.965798	+	0.259295i	$0.0834903\pi$
$930$		0			0
$931$	−1212.74	−	701.286i	−1.30262	−	0.753261i
$932$	134.048			0.143828
$933$			1116.13i			1.19629i
$934$			990.525i			1.06052i
$935$		0			0
$936$	−296.289			−0.316549
$937$	823.194			0.878542			0.439271	−	0.898355i	$-0.355237\pi$
$937$	823.194			0.878542			0.439271	+	0.898355i	$0.355237\pi$
$938$	539.130			0.574765
$939$			1825.61i			1.94421i
$940$		0			0
$941$		−	270.097i		−	0.287032i	−0.989648	−	0.143516i	$-0.954159\pi$
$941$		−	270.097i		−	0.287032i	0.989648	−	0.143516i	$-0.0458409\pi$
$942$	−1927.95			−2.04666
$943$		−	1014.57i		−	1.07589i
$944$		−	1448.28i		−	1.53420i
$945$		0			0
$946$		−	247.233i		−	0.261346i
$947$	1412.14			1.49118			0.745588	−	0.666407i	$-0.232169\pi$
$947$	1412.14			1.49118			0.745588	+	0.666407i	$0.232169\pi$
$948$	−49.8753			−0.0526111
$949$		−	418.096i		−	0.440564i
$950$		0			0
$951$	−91.7535			−0.0964811
$952$		−	1273.64i		−	1.33786i
$953$		−	432.598i		−	0.453933i	−0.973903	−	0.226966i	$-0.927119\pi$
$953$		−	432.598i		−	0.453933i	0.973903	−	0.226966i	$-0.0728808\pi$
$954$	265.443			0.278242
$955$		0			0
$956$	175.592			0.183674
$957$	−1812.56			−1.89400
$958$			150.489i			0.157087i
$959$	1040.06			1.08452
$960$		0			0
$961$	700.964			0.729411
$962$		−	1734.71i		−	1.80323i
$963$		−	313.209i		−	0.325243i
$964$			513.339i			0.532509i
$965$		0			0
$966$	−1266.18			−1.31075
$967$	−10.0222			−0.0103642			−0.00518210	−	0.999987i	$-0.501650\pi$
$967$	−10.0222			−0.0103642			−0.00518210	+	0.999987i	$0.501650\pi$
$968$			911.291i			0.941416i
$969$	−683.675	+	1182.29i	−0.705547	+	1.22011i
$970$		0			0
$971$		−	339.417i		−	0.349554i	−0.984608	−	0.174777i	$-0.944080\pi$
$971$		−	339.417i		−	0.349554i	0.984608	−	0.174777i	$-0.0559205\pi$
$972$		−	296.776i		−	0.305325i
$973$	−455.309			−0.467943
$974$	−1317.63			−1.35280
$975$		0			0
$976$	352.754			0.361428
$977$			673.762i			0.689623i	0.938672	+	0.344812i	$0.112057\pi$
$977$			673.762i			0.689623i	−0.938672	+	0.344812i	$0.887943\pi$
$978$	−286.337			−0.292778
$979$			2732.03i			2.79063i
$980$		0			0
$981$			346.363i			0.353071i
$982$			2026.86i			2.06401i
$983$			428.230i			0.435636i	0.975989	+	0.217818i	$0.0698940\pi$
$983$			428.230i			0.435636i	−0.975989	+	0.217818i	$0.930106\pi$
$984$		−	1509.76i		−	1.53431i
$985$		0			0
$986$	1453.68			1.47432
$987$			1734.19i			1.75703i
$988$	217.127	−	375.481i	0.219765	−	0.380042i
$989$	−84.6813			−0.0856232
$990$		0			0
$991$		−	1310.89i		−	1.32280i	−0.750033	−	0.661400i	$-0.769963\pi$
$991$		−	1310.89i		−	1.32280i	0.750033	−	0.661400i	$-0.230037\pi$
$992$	−389.888			−0.393032
$993$	610.270			0.614572
$994$	−196.272			−0.197457
$995$		0			0
$996$			361.987i			0.363441i
$997$	690.240			0.692316			0.346158	−	0.938176i	$-0.387486\pi$
$997$	690.240			0.692316			0.346158	+	0.938176i	$0.387486\pi$
$998$			1801.71i			1.80532i
$999$	−976.615			−0.977592

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.3.c.g.151.10		12
5.2	odd	4		475.3.d.c.474.6		24
5.3	odd	4		475.3.d.c.474.19		24
5.4	even	2		95.3.c.a.56.3	✓	12
15.14	odd	2		855.3.e.a.721.10		12
19.18	odd	2	inner	475.3.c.g.151.3		12
20.19	odd	2		1520.3.h.a.721.4		12
95.18	even	4		475.3.d.c.474.5		24
95.37	even	4		475.3.d.c.474.20		24
95.94	odd	2		95.3.c.a.56.10	yes	12
285.284	even	2		855.3.e.a.721.3		12
380.379	even	2		1520.3.h.a.721.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.3.c.a.56.3	✓	12	5.4	even	2
95.3.c.a.56.10	yes	12	95.94	odd	2
475.3.c.g.151.3		12	19.18	odd	2	inner
475.3.c.g.151.10		12	1.1	even	1	trivial
475.3.d.c.474.5		24	95.18	even	4
475.3.d.c.474.6		24	5.2	odd	4
475.3.d.c.474.19		24	5.3	odd	4
475.3.d.c.474.20		24	95.37	even	4
855.3.e.a.721.3		12	285.284	even	2
855.3.e.a.721.10		12	15.14	odd	2
1520.3.h.a.721.4		12	20.19	odd	2
1520.3.h.a.721.9		12	380.379	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	475.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.36559i q^{2} -3.55563i q^{3} -1.59600 q^{4} +8.41115 q^{6} -11.0785 q^{7} +5.68687i q^{8} -3.64250 q^{9} +O(q^{10})\)	\(q+2.36559i q^{2} -3.55563i q^{3} -1.59600 q^{4} +8.41115 q^{6} -11.0785 q^{7} +5.68687i q^{8} -3.64250 q^{9} +16.7704 q^{11} +5.67479i q^{12} -14.3035i q^{13} -26.2070i q^{14} -19.8368 q^{16} +20.2160 q^{17} -8.61666i q^{18} +(-16.4480 - 9.51127i) q^{19} +39.3909i q^{21} +39.6717i q^{22} +13.5882 q^{23} +20.2204 q^{24} +33.8362 q^{26} -19.0493i q^{27} +17.6812 q^{28} -30.3972i q^{29} -16.1256i q^{31} -24.1782i q^{32} -59.6292i q^{33} +47.8227i q^{34} +5.81344 q^{36} -51.2678i q^{37} +(22.4997 - 38.9091i) q^{38} -50.8580 q^{39} -74.6652i q^{41} -93.1825 q^{42} -6.23197 q^{43} -26.7655 q^{44} +32.1441i q^{46} +44.0252 q^{47} +70.5323i q^{48} +73.7321 q^{49} -71.8806i q^{51} +22.8284i q^{52} +30.8058i q^{53} +45.0627 q^{54} -63.0017i q^{56} +(-33.8185 + 58.4829i) q^{57} +71.9073 q^{58} +73.0098i q^{59} -17.7828 q^{61} +38.1466 q^{62} +40.3533 q^{63} -22.1516 q^{64} +141.058 q^{66} +20.5719i q^{67} -32.2647 q^{68} -48.3147i q^{69} -7.48928i q^{71} -20.7144i q^{72} +29.2302 q^{73} +121.278 q^{74} +(26.2510 + 15.1800i) q^{76} -185.790 q^{77} -120.309i q^{78} +8.78894i q^{79} -100.515 q^{81} +176.627 q^{82} +63.7886 q^{83} -62.8679i q^{84} -14.7423i q^{86} -108.081 q^{87} +95.3708i q^{88} +162.908i q^{89} +158.461i q^{91} -21.6868 q^{92} -57.3367 q^{93} +104.145i q^{94} -85.9686 q^{96} -75.5974i q^{97} +174.420i q^{98} -61.0861 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{4} - 28 q^{6} - 20 q^{7} - 48 q^{9}+O(q^{10}) \) 12 * q - 12 * q^4 - 28 * q^6 - 20 * q^7 - 48 * q^9	\( 12 q - 12 q^{4} - 28 q^{6} - 20 q^{7} - 48 q^{9} + 32 q^{11} - 44 q^{16} + 44 q^{17} + 8 q^{19} - 36 q^{23} + 100 q^{24} + 108 q^{26} + 36 q^{28} - 80 q^{36} + 44 q^{38} + 76 q^{39} - 100 q^{42} - 320 q^{43} - 256 q^{44} + 56 q^{47} + 72 q^{49} - 76 q^{54} - 60 q^{57} - 68 q^{58} - 296 q^{61} + 376 q^{62} + 96 q^{63} + 188 q^{64} + 152 q^{66} + 340 q^{68} + 244 q^{73} + 136 q^{74} + 248 q^{76} + 200 q^{77} - 372 q^{81} - 424 q^{82} + 160 q^{83} - 444 q^{87} - 716 q^{92} - 296 q^{93} - 44 q^{96} - 312 q^{99}+O(q^{100}) \) 12 * q - 12 * q^4 - 28 * q^6 - 20 * q^7 - 48 * q^9 + 32 * q^11 - 44 * q^16 + 44 * q^17 + 8 * q^19 - 36 * q^23 + 100 * q^24 + 108 * q^26 + 36 * q^28 - 80 * q^36 + 44 * q^38 + 76 * q^39 - 100 * q^42 - 320 * q^43 - 256 * q^44 + 56 * q^47 + 72 * q^49 - 76 * q^54 - 60 * q^57 - 68 * q^58 - 296 * q^61 + 376 * q^62 + 96 * q^63 + 188 * q^64 + 152 * q^66 + 340 * q^68 + 244 * q^73 + 136 * q^74 + 248 * q^76 + 200 * q^77 - 372 * q^81 - 424 * q^82 + 160 * q^83 - 444 * q^87 - 716 * q^92 - 296 * q^93 - 44 * q^96 - 312 * q^99

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