LMFDB - Embedded newform 475.3.d.c.474.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,3,Mod(474,475)]

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([1, 1]))

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

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

chi := DirichletCharacter("475.474");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	475.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.9428125571$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			474.6
Character	$\chi$	$=$	475.474
Dual form			475.3.d.c.474.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36559 q^{2} -3.55563 q^{3} +1.59600 q^{4} +8.41115 q^{6} -11.0785i q^{7} +5.68687 q^{8} +3.64250 q^{9} +O(q^{10})$	\(q-2.36559 q^{2} -3.55563 q^{3} +1.59600 q^{4} +8.41115 q^{6} -11.0785i q^{7} +5.68687 q^{8} +3.64250 q^{9} +16.7704 q^{11} -5.67479 q^{12} -14.3035 q^{13} +26.2070i q^{14} -19.8368 q^{16} +20.2160i q^{17} -8.61666 q^{18} +(16.4480 + 9.51127i) q^{19} +39.3909i q^{21} -39.6717 q^{22} -13.5882i q^{23} -20.2204 q^{24} +33.8362 q^{26} +19.0493 q^{27} -17.6812i q^{28} +30.3972i q^{29} -16.1256i q^{31} +24.1782 q^{32} -59.6292 q^{33} -47.8227i q^{34} +5.81344 q^{36} +51.2678 q^{37} +(-38.9091 - 22.4997i) q^{38} +50.8580 q^{39} -74.6652i q^{41} -93.1825i q^{42} +6.23197i q^{43} +26.7655 q^{44} +32.1441i q^{46} +44.0252i q^{47} +70.5323 q^{48} -73.7321 q^{49} -71.8806i q^{51} -22.8284 q^{52} +30.8058 q^{53} -45.0627 q^{54} -63.0017i q^{56} +(-58.4829 - 33.8185i) q^{57} -71.9073i q^{58} -73.0098i q^{59} -17.7828 q^{61} +38.1466i q^{62} -40.3533i q^{63} +22.1516 q^{64} +141.058 q^{66} -20.5719 q^{67} +32.2647i q^{68} +48.3147i q^{69} -7.48928i q^{71} +20.7144 q^{72} -29.2302i q^{73} -121.278 q^{74} +(26.2510 + 15.1800i) q^{76} -185.790i q^{77} -120.309 q^{78} -8.78894i q^{79} -100.515 q^{81} +176.627i q^{82} -63.7886i q^{83} +62.8679i q^{84} -14.7423i q^{86} -108.081i q^{87} +95.3708 q^{88} -162.908i q^{89} +158.461i q^{91} -21.6868i q^{92} +57.3367i q^{93} -104.145i q^{94} -85.9686 q^{96} +75.5974 q^{97} +174.420 q^{98} +61.0861 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9}+O(q^{10}) $ 24 * q + 24 * q^4 - 56 * q^6 + 96 * q^9	$ 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9} + 64 q^{11} - 88 q^{16} - 16 q^{19} - 200 q^{24} + 216 q^{26} - 160 q^{36} - 152 q^{39} + 512 q^{44} - 144 q^{49} + 152 q^{54} - 592 q^{61} - 376 q^{64} + 304 q^{66} - 272 q^{74} + 496 q^{76} - 744 q^{81} - 88 q^{96} + 624 q^{99}+O(q^{100}) $ 24 * q + 24 * q^4 - 56 * q^6 + 96 * q^9 + 64 * q^11 - 88 * q^16 - 16 * q^19 - 200 * q^24 + 216 * q^26 - 160 * q^36 - 152 * q^39 + 512 * q^44 - 144 * q^49 + 152 * q^54 - 592 * q^61 - 376 * q^64 + 304 * q^66 - 272 * q^74 + 496 * q^76 - 744 * q^81 - 88 * q^96 + 624 * 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 $

$2$	−2.36559			−1.18279			−0.591397	−	0.806381i	$-0.701423\pi$
$2$	−2.36559			−1.18279			−0.591397	+	0.806381i	$0.701423\pi$
$3$	−3.55563			−1.18521			−0.592605	−	0.805493i	$-0.701900\pi$
$3$	−3.55563			−1.18521			−0.592605	+	0.805493i	$0.701900\pi$
$4$	1.59600			0.399000
$5$		0			0
$5$		0			0
$6$	8.41115			1.40186
$7$		−	11.0785i		−	1.58264i	−0.611405	−	0.791318i	$-0.709395\pi$
$7$		−	11.0785i		−	1.58264i	0.611405	−	0.791318i	$-0.290605\pi$
$8$	5.68687			0.710859
$9$	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	$0.224076\pi$
$12$	−5.67479			−0.472899
$13$	−14.3035			−1.10027			−0.550136	−	0.835075i	$-0.685424\pi$
$13$	−14.3035			−1.10027			−0.550136	+	0.835075i	$0.685424\pi$
$14$			26.2070i			1.87193i
$15$		0			0
$16$	−19.8368			−1.23980
$17$			20.2160i			1.18918i	0.804031	+	0.594588i	$0.202685\pi$
$17$			20.2160i			1.18918i	−0.804031	+	0.594588i	$0.797315\pi$
$18$	−8.61666			−0.478703
$19$	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
$22$	−39.6717			−1.80326
$23$		−	13.5882i		−	0.590792i	−0.955375	−	0.295396i	$-0.904548\pi$
$23$		−	13.5882i		−	0.590792i	0.955375	−	0.295396i	$-0.0954516\pi$
$24$	−20.2204			−0.842517
$25$		0			0
$26$	33.8362			1.30139
$27$	19.0493			0.705529
$28$		−	17.6812i		−	0.631472i
$29$			30.3972i			1.04818i	0.851663	+	0.524090i	$0.175594\pi$
$29$			30.3972i			1.04818i	−0.851663	+	0.524090i	$0.824406\pi$
$30$		0			0
$31$		−	16.1256i		−	0.520181i	−0.965584	−	0.260091i	$-0.916248\pi$
$31$		−	16.1256i		−	0.520181i	0.965584	−	0.260091i	$-0.0837525\pi$
$32$	24.1782			0.755567
$33$	−59.6292			−1.80694
$34$		−	47.8227i		−	1.40655i
$35$		0			0
$36$	5.81344			0.161484
$37$	51.2678			1.38562			0.692808	−	0.721122i	$-0.256373\pi$
$37$	51.2678			1.38562			0.692808	+	0.721122i	$0.256373\pi$
$38$	−38.9091	−	22.4997i	−1.02392	−	0.592098i
$39$	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	$-0.364343\pi$
$42$		−	93.1825i		−	2.21863i
$43$			6.23197i			0.144929i	0.997371	+	0.0724647i	$0.0230865\pi$
$43$			6.23197i			0.144929i	−0.997371	+	0.0724647i	$0.976914\pi$
$44$	26.7655			0.608307
$45$		0			0
$46$			32.1441i			0.698785i
$47$			44.0252i			0.936706i	0.883541	+	0.468353i	$0.155152\pi$
$47$			44.0252i			0.936706i	−0.883541	+	0.468353i	$0.844848\pi$
$48$	70.5323			1.46942
$49$	−73.7321			−1.50474
$50$		0			0
$51$		−	71.8806i		−	1.40942i
$52$	−22.8284			−0.439008
$53$	30.8058			0.581241			0.290620	−	0.956838i	$-0.406138\pi$
$53$	30.8058			0.581241			0.290620	+	0.956838i	$0.406138\pi$
$54$	−45.0627			−0.834495
$55$		0			0
$56$		−	63.0017i		−	1.12503i
$57$	−58.4829	−	33.8185i	−1.02602	−	0.593308i
$58$		−	71.9073i		−	1.23978i
$59$		−	73.0098i		−	1.23745i	−0.785606	−	0.618727i	$-0.787649\pi$
$59$		−	73.0098i		−	1.23745i	0.785606	−	0.618727i	$-0.212351\pi$
$60$		0			0
$61$	−17.7828			−0.291522			−0.145761	−	0.989320i	$-0.546563\pi$
$61$	−17.7828			−0.291522			−0.145761	+	0.989320i	$0.546563\pi$
$62$			38.1466i			0.615267i
$63$		−	40.3533i		−	0.640528i
$64$	22.1516			0.346119
$65$		0			0
$66$	141.058			2.13724
$67$	−20.5719			−0.307044			−0.153522	−	0.988145i	$-0.549062\pi$
$67$	−20.5719			−0.307044			−0.153522	+	0.988145i	$0.549062\pi$
$68$			32.2647i			0.474481i
$69$			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	$-0.0167959\pi$
$72$	20.7144			0.287700
$73$		−	29.2302i		−	0.400414i	−0.979754	−	0.200207i	$-0.935838\pi$
$73$		−	29.2302i		−	0.400414i	0.979754	−	0.200207i	$-0.0641615\pi$
$74$	−121.278			−1.63890
$75$		0			0
$76$	26.2510	+	15.1800i	0.345408	+	0.199737i
$77$		−	185.790i		−	2.41285i
$78$	−120.309			−1.54242
$79$		−	8.78894i		−	0.111252i	−0.998452	−	0.0556262i	$-0.982284\pi$
$79$		−	8.78894i		−	0.111252i	0.998452	−	0.0556262i	$-0.0177155\pi$
$80$		0			0
$81$	−100.515			−1.24092
$82$			176.627i			2.15399i
$83$		−	63.7886i		−	0.768537i	−0.923221	−	0.384269i	$-0.874454\pi$
$83$		−	63.7886i		−	0.768537i	0.923221	−	0.384269i	$-0.125546\pi$
$84$			62.8679i			0.748427i
$85$		0			0
$86$		−	14.7423i		−	0.171422i
$87$		−	108.081i		−	1.24231i
$88$	95.3708			1.08376
$89$		−	162.908i		−	1.83043i	−0.402965	−	0.915215i	$-0.632020\pi$
$89$		−	162.908i		−	1.83043i	0.402965	−	0.915215i	$-0.367980\pi$
$90$		0			0
$91$			158.461i			1.74133i
$92$		−	21.6868i		−	0.235726i
$93$			57.3367i			0.616524i
$94$		−	104.145i		−	1.10793i
$95$		0			0
$96$	−85.9686			−0.895506
$97$	75.5974			0.779355			0.389677	−	0.920951i	$-0.372587\pi$
$97$	75.5974			0.779355			0.389677	+	0.920951i	$0.372587\pi$
$98$	174.420			1.77979
$99$	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	$0.475126\pi$
$102$			170.040i			1.66706i
$103$	7.15141			0.0694312			0.0347156	−	0.999397i	$-0.488947\pi$
$103$	7.15141			0.0694312			0.0347156	+	0.999397i	$0.488947\pi$
$104$	−81.3423			−0.782137
$105$		0			0
$106$	−72.8737			−0.687488
$107$	−85.9872			−0.803619			−0.401809	−	0.915723i	$-0.631619\pi$
$107$	−85.9872			−0.803619			−0.401809	+	0.915723i	$0.631619\pi$
$108$	30.4027			0.281506
$109$			95.0893i			0.872379i	0.899855	+	0.436190i	$0.143672\pi$
$109$			95.0893i			0.872379i	−0.899855	+	0.436190i	$0.856328\pi$
$110$		0			0
$111$	−182.289			−1.64225
$112$			219.761i			1.96215i
$113$	19.0355			0.168456			0.0842281	−	0.996447i	$-0.473158\pi$
$113$	19.0355			0.168456			0.0842281	+	0.996447i	$0.473158\pi$
$114$	138.346	+	80.0007i	1.21356	+	0.701761i
$115$		0			0
$116$			48.5140i			0.418224i
$117$	−52.1006			−0.445304
$118$			172.711i			1.46365i
$119$	223.962			1.88203
$120$		0			0
$121$	160.245			1.32434
$122$	42.0668			0.344810
$123$			265.482i			2.15839i
$124$		−	25.7365i		−	0.207552i
$125$		0			0
$126$			95.4592i			0.757613i
$127$	−93.4107			−0.735518			−0.367759	−	0.929921i	$-0.619875\pi$
$127$	−93.4107			−0.735518			−0.367759	+	0.929921i	$0.619875\pi$
$128$	−149.114			−1.16495
$129$		−	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	$0.339616\pi$
$132$	−95.1682			−0.720971
$133$	105.370	−	182.218i	0.792257	−	1.37006i
$134$	48.6647			0.363170
$135$		0			0
$136$			114.966i			0.845336i
$137$		−	93.8813i		−	0.685265i	−0.939470	−	0.342632i	$-0.888681\pi$
$137$		−	93.8813i		−	0.685265i	0.939470	−	0.342632i	$-0.111319\pi$
$138$		−	114.293i		−	0.828207i
$139$	−41.0986			−0.295673			−0.147837	−	0.989012i	$-0.547231\pi$
$139$	−41.0986			−0.295673			−0.147837	+	0.989012i	$0.547231\pi$
$140$		0			0
$141$		−	156.537i		−	1.11019i
$142$			17.7165i			0.124764i
$143$	−239.875			−1.67745
$144$	−72.2555			−0.501775
$145$		0			0
$146$			69.1467i			0.473607i
$147$	262.164			1.78343
$148$	81.8235			0.552861
$149$	−92.2911			−0.619404			−0.309702	−	0.950834i	$-0.600229\pi$
$149$	−92.2911			−0.619404			−0.309702	+	0.950834i	$0.600229\pi$
$150$		0			0
$151$		−	7.26853i		−	0.0481359i	−0.999710	−	0.0240680i	$-0.992338\pi$
$151$		−	7.26853i		−	0.0481359i	0.999710	−	0.0240680i	$-0.00766181\pi$
$152$	93.5375	+	54.0893i	0.615378	+	0.355851i
$153$			73.6368i			0.481286i
$154$			439.501i			2.85390i
$155$		0			0
$156$	81.1695			0.520317
$157$		−	229.214i		−	1.45996i	−0.683468	−	0.729981i	$-0.739529\pi$
$157$		−	229.214i		−	1.45996i	0.683468	−	0.729981i	$-0.260471\pi$
$158$			20.7910i			0.131589i
$159$	−109.534			−0.688892
$160$		0			0
$161$	−150.536			−0.935009
$162$	237.776			1.46775
$163$			34.0426i			0.208850i	0.994533	+	0.104425i	$0.0333002\pi$
$163$			34.0426i			0.208850i	−0.994533	+	0.104425i	$0.966700\pi$
$164$		−	119.166i		−	0.726621i
$165$		0			0
$166$			150.897i			0.909021i
$167$	−162.274			−0.971702			−0.485851	−	0.874042i	$-0.661490\pi$
$167$	−162.274			−0.971702			−0.485851	+	0.874042i	$0.661490\pi$
$168$			224.011i			1.33340i
$169$	35.5908			0.210597
$170$		0			0
$171$	59.9118	+	34.6448i	0.350361	+	0.202601i
$172$			9.94623i			0.0578269i
$173$	29.7956			0.172229			0.0861145	−	0.996285i	$-0.472555\pi$
$173$	29.7956			0.172229			0.0861145	+	0.996285i	$0.472555\pi$
$174$			255.676i			1.46940i
$175$		0			0
$176$	−332.670			−1.89017
$177$			259.596i			1.46664i
$178$			385.374i			2.16502i
$179$		−	59.2752i		−	0.331146i	−0.986197	−	0.165573i	$-0.947053\pi$
$179$		−	59.2752i		−	0.331146i	0.986197	−	0.165573i	$-0.0529474\pi$
$180$		0			0
$181$		−	205.700i		−	1.13647i	−0.822868	−	0.568233i	$-0.807627\pi$
$181$		−	205.700i		−	1.13647i	0.822868	−	0.568233i	$-0.192373\pi$
$182$		−	374.853i		−	2.05963i
$183$	63.2291			0.345514
$184$		−	77.2744i		−	0.419970i
$185$		0			0
$186$		−	135.635i		−	0.729221i
$187$			339.029i			1.81299i
$188$			70.2642i			0.373746i
$189$		−	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	$0.731180\pi$
$192$	−78.7629			−0.410223
$193$	−289.646			−1.50076			−0.750378	−	0.661009i	$-0.770128\pi$
$193$	−289.646			−1.50076			−0.750378	+	0.661009i	$0.770128\pi$
$194$	−178.832			−0.921816
$195$		0			0
$196$	−117.676			−0.600390
$197$		−	178.190i		−	0.904515i	−0.891887	−	0.452258i	$-0.850619\pi$
$197$		−	178.190i		−	0.904515i	0.891887	−	0.452258i	$-0.149381\pi$
$198$	−144.504			−0.729820
$199$	154.602			0.776893			0.388446	−	0.921471i	$-0.373012\pi$
$199$	154.602			0.776893			0.388446	+	0.921471i	$0.373012\pi$
$200$		0			0
$201$	73.1462			0.363912
$202$	−37.3035			−0.184671
$203$	336.754			1.65889
$204$		−	114.721i		−	0.562360i
$205$		0			0
$206$	−16.9173			−0.0821228
$207$		−	49.4951i		−	0.239107i
$208$	283.736			1.36412
$209$	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	$0.775220\pi$
$212$	49.1660			0.231915
$213$			26.6291i			0.125019i
$214$	203.410			0.950515
$215$		0			0
$216$	108.331			0.501531
$217$	−178.647			−0.823258
$218$		−	224.942i		−	1.03184i
$219$			103.932i			0.474575i
$220$		0			0
$221$		−	289.160i		−	1.30842i
$222$	431.221			1.94244
$223$	229.796			1.03048			0.515238	−	0.857047i	$-0.327704\pi$
$223$	229.796			1.03048			0.515238	+	0.857047i	$0.327704\pi$
$224$		−	267.857i		−	1.19579i
$225$		0			0
$226$	−45.0302			−0.199249
$227$	366.166			1.61306			0.806532	−	0.591190i	$-0.201342\pi$
$227$	366.166			1.61306			0.806532	+	0.591190i	$0.201342\pi$
$228$	−93.3387	−	53.9744i	−0.409380	−	0.236730i
$229$	−334.098			−1.45894			−0.729471	−	0.684012i	$-0.760234\pi$
$229$	−334.098			−1.45894			−0.729471	+	0.684012i	$0.760234\pi$
$230$		0			0
$231$			660.599i			2.85974i
$232$			172.865i			0.745108i
$233$			83.9900i			0.360472i	0.983623	+	0.180236i	$0.0576861\pi$
$233$			83.9900i			0.360472i	−0.983623	+	0.180236i	$0.942314\pi$
$234$	123.249			0.526703
$235$		0			0
$236$		−	116.524i		−	0.493745i
$237$			31.2502i			0.131857i
$238$	−529.801			−2.22606
$239$	110.020			0.460335			0.230168	−	0.973151i	$-0.426073\pi$
$239$	110.020			0.460335			0.230168	+	0.973151i	$0.426073\pi$
$240$		0			0
$241$		−	321.641i		−	1.33461i	−0.744785	−	0.667305i	$-0.767448\pi$
$241$		−	321.641i		−	1.33461i	0.744785	−	0.667305i	$-0.232552\pi$
$242$	−379.073			−1.56642
$243$	185.950			0.765225
$244$	−28.3814			−0.116317
$245$		0			0
$246$		−	628.021i		−	2.55293i
$247$	−235.264	−	136.045i	−0.952486	−	0.550788i
$248$		−	91.7043i		−	0.369775i
$249$			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	$0.407782\pi$
$252$		−	64.4039i		−	0.255571i
$253$		−	227.879i		−	0.900708i
$254$	220.971			0.869965
$255$		0			0
$256$	264.136			1.03178
$257$	171.202			0.666156			0.333078	−	0.942899i	$-0.391913\pi$
$257$	171.202			0.666156			0.333078	+	0.942899i	$0.391913\pi$
$258$			52.4180i			0.203171i
$259$		−	567.968i		−	2.19293i
$260$		0			0
$261$			110.722i			0.424222i
$262$	−299.238			−1.14213
$263$		−	184.993i		−	0.703395i	−0.936114	−	0.351697i	$-0.885605\pi$
$263$		−	184.993i		−	0.703395i	0.936114	−	0.351697i	$-0.114395\pi$
$264$	−339.103			−1.28448
$265$		0			0
$266$	−249.262	+	431.053i	−0.937076	+	1.62050i
$267$			579.242i			2.16944i
$268$	−32.8328			−0.122511
$269$		−	453.194i		−	1.68473i	−0.538904	−	0.842367i	$-0.681161\pi$
$269$		−	453.194i		−	1.68473i	0.538904	−	0.842367i	$-0.318839\pi$
$270$		0			0
$271$	336.795			1.24279			0.621393	−	0.783499i	$-0.286567\pi$
$271$	336.795			1.24279			0.621393	+	0.783499i	$0.286567\pi$
$272$		−	401.020i		−	1.47434i
$273$		−	563.428i		−	2.06384i
$274$			222.084i			0.810527i
$275$		0			0
$276$			77.1102i			0.279385i
$277$		−	270.449i		−	0.976351i	−0.872746	−	0.488175i	$-0.837663\pi$
$277$		−	270.449i		−	0.976351i	0.872746	−	0.488175i	$-0.162337\pi$
$278$	97.2223			0.349720
$279$		−	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	$0.913215\pi$
$282$			370.303i			1.31313i
$283$		−	437.891i		−	1.54732i	−0.633603	−	0.773658i	$-0.718425\pi$
$283$		−	437.891i		−	1.54732i	0.633603	−	0.773658i	$-0.281575\pi$
$284$		−	11.9529i		−	0.0420876i
$285$		0			0
$286$	567.446			1.98408
$287$	−827.175			−2.88214
$288$	88.0690			0.305795
$289$	−119.686			−0.414139
$290$		0			0
$291$	−268.796			−0.923699
$292$		−	46.6515i		−	0.159765i
$293$	−265.247			−0.905279			−0.452640	−	0.891694i	$-0.649518\pi$
$293$	−265.247			−0.905279			−0.452640	+	0.891694i	$0.649518\pi$
$294$	−620.172			−2.10943
$295$		0			0
$296$	291.553			0.984977
$297$	319.463			1.07563
$298$	218.323			0.732627
$299$			194.359i			0.650031i
$300$		0			0
$301$	69.0406			0.229371
$302$			17.1943i			0.0569349i
$303$	−56.0696			−0.185048
$304$	−326.275	−	188.673i	−1.07327	−	0.620635i
$305$		0			0
$306$		−	174.194i		−	0.569262i
$307$	−405.065			−1.31943			−0.659714	−	0.751517i	$-0.729323\pi$
$307$	−405.065			−1.31943			−0.659714	+	0.751517i	$0.729323\pi$
$308$		−	296.520i		−	0.962728i
$309$	−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	$0.668387\pi$
$312$	289.223			0.926997
$313$			513.442i			1.64039i	0.572085	+	0.820194i	$0.306135\pi$
$313$			513.442i			1.64039i	−0.572085	+	0.820194i	$0.693865\pi$
$314$			542.225i			1.72683i
$315$		0			0
$316$		−	14.0271i		−	0.0443897i
$317$	25.8051			0.0814042			0.0407021	−	0.999171i	$-0.487041\pi$
$317$	25.8051			0.0814042			0.0407021	+	0.999171i	$0.487041\pi$
$318$	259.112			0.814817
$319$			509.772i			1.59803i
$320$		0			0
$321$	305.739			0.952457
$322$	356.107			1.10592
$323$	−192.280	+	332.512i	−0.595293	+	1.02945i
$324$	−160.422			−0.495128
$325$		0			0
$326$		−	80.5307i		−	0.247027i
$327$		−	338.102i		−	1.03395i
$328$		−	424.611i		−	1.29455i
$329$	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	$0.916519\pi$
$332$		−	101.807i		−	0.306647i
$333$	186.743			0.560790
$334$	383.874			1.14932
$335$		0			0
$336$		−	781.388i		−	2.32556i
$337$	206.114			0.611613			0.305807	−	0.952094i	$-0.401074\pi$
$337$	206.114			0.611613			0.305807	+	0.952094i	$0.401074\pi$
$338$	−84.1932			−0.249092
$339$	−67.6833			−0.199656
$340$		0			0
$341$		−	270.432i		−	0.793057i
$342$	−141.727	−	81.9553i	−0.414405	−	0.239635i
$343$			273.994i			0.798815i
$344$			35.4404i			0.103024i
$345$		0			0
$346$	−70.4841			−0.203711
$347$			7.91893i			0.0228211i	0.999935	+	0.0114106i	$0.00363217\pi$
$347$			7.91893i			0.0228211i	−0.999935	+	0.0114106i	$0.996368\pi$
$348$		−	172.498i		−	0.495683i
$349$	−409.448			−1.17320			−0.586602	−	0.809876i	$-0.699535\pi$
$349$	−409.448			−1.17320			−0.586602	+	0.809876i	$0.699535\pi$
$350$		0			0
$351$	−272.472			−0.776273
$352$	405.476			1.15192
$353$		−	637.919i		−	1.80714i	−0.428445	−	0.903568i	$-0.640939\pi$
$353$		−	637.919i		−	1.80714i	0.428445	−	0.903568i	$-0.359061\pi$
$354$		−	614.097i		−	1.73474i
$355$		0			0
$356$		−	260.002i		−	0.730342i
$357$	−796.325			−2.23060
$358$			140.221i			0.391678i
$359$	237.272			0.660926			0.330463	−	0.943819i	$-0.392795\pi$
$359$	237.272			0.660926			0.330463	+	0.943819i	$0.392795\pi$
$360$		0			0
$361$	180.072	+	312.882i	0.498813	+	0.866710i
$362$			486.602i			1.34420i
$363$	−569.771			−1.56962
$364$			252.904i			0.694790i
$365$		0			0
$366$	−149.574			−0.408672
$367$		−	542.889i		−	1.47926i	−0.673013	−	0.739631i	$-0.735000\pi$
$367$		−	542.889i		−	1.47926i	0.673013	−	0.739631i	$-0.265000\pi$
$368$			269.546i			0.732463i
$369$		−	271.968i		−	0.737042i
$370$		0			0
$371$		−	341.280i		−	0.919892i
$372$			91.5095i			0.245993i
$373$	444.746			1.19235			0.596175	−	0.802855i	$-0.296687\pi$
$373$	444.746			1.19235			0.596175	+	0.802855i	$0.296687\pi$
$374$		−	802.003i		−	2.14439i
$375$		0			0
$376$			250.366i			0.665866i
$377$		−	434.787i		−	1.15328i
$378$			499.225i			1.32070i
$379$		−	150.221i		−	0.396362i	−0.980165	−	0.198181i	$-0.936497\pi$
$379$		−	150.221i		−	0.396362i	0.980165	−	0.198181i	$-0.0635035\pi$
$380$		0			0
$381$	332.134			0.871743
$382$	600.106			1.57096
$383$	85.8011			0.224024			0.112012	−	0.993707i	$-0.464271\pi$
$383$	85.8011			0.224024			0.112012	+	0.993707i	$0.464271\pi$
$384$	530.195			1.38072
$385$		0			0
$386$	685.182			1.77508
$387$			22.7000i			0.0586562i
$388$	120.654			0.310963
$389$	249.669			0.641822			0.320911	−	0.947109i	$-0.396011\pi$
$389$	249.669			0.641822			0.320911	+	0.947109i	$0.396011\pi$
$390$		0			0
$391$	274.699			0.702555
$392$	−419.305			−1.06966
$393$	−449.773			−1.14446
$394$			421.523i			1.06985i
$395$		0			0
$396$	97.4934			0.246195
$397$			340.476i			0.857622i	0.903394	+	0.428811i	$0.141067\pi$
$397$			340.476i			0.857622i	−0.903394	+	0.428811i	$0.858933\pi$
$398$	−365.724			−0.918904
$399$	−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	$0.790276\pi$
$402$	−173.034			−0.430432
$403$			230.653i			0.572341i
$404$	25.1677			0.0622964
$405$		0			0
$406$	−796.621			−1.96212
$407$	859.779			2.11248
$408$		−	408.775i		−	1.00190i
$409$		−	115.833i		−	0.283210i	−0.989923	−	0.141605i	$-0.954774\pi$
$409$		−	115.833i		−	0.283210i	0.989923	−	0.141605i	$-0.0452263\pi$
$410$		0			0
$411$			333.807i			0.812183i
$412$	11.4137			0.0277031
$413$	−808.836			−1.95844
$414$			117.085i			0.282814i
$415$		0			0
$416$	−345.833			−0.831329
$417$	146.131			0.350435
$418$	−652.519	−	377.328i	−1.56105	−	0.902700i
$419$	−81.4969			−0.194503			−0.0972516	−	0.995260i	$-0.531005\pi$
$419$	−81.4969			−0.194503			−0.0972516	+	0.995260i	$0.531005\pi$
$420$		0			0
$421$			221.652i			0.526490i	0.964729	+	0.263245i	$0.0847927\pi$
$421$			221.652i			0.526490i	−0.964729	+	0.263245i	$0.915207\pi$
$422$		−	647.806i		−	1.53508i
$423$			160.362i			0.379106i
$424$	175.188			0.413180
$425$		0			0
$426$		−	62.9934i		−	0.147872i
$427$			197.006i			0.461372i
$428$	−137.236			−0.320644
$429$	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	$-0.0790333\pi$
$432$	−377.876			−0.874714
$433$	389.534			0.899616			0.449808	−	0.893125i	$-0.351492\pi$
$433$	389.534			0.899616			0.449808	+	0.893125i	$0.351492\pi$
$434$	422.605			0.973744
$435$		0			0
$436$			151.763i			0.348079i
$437$	129.241	−	223.499i	0.295746	−	0.511438i
$438$		−	245.860i		−	0.561324i
$439$			465.691i			1.06080i	0.847748	+	0.530400i	$0.177958\pi$
$439$			465.691i			1.06080i	−0.847748	+	0.530400i	$0.822042\pi$
$440$		0			0
$441$	−268.569			−0.609001
$442$			684.033i			1.54759i
$443$		−	382.479i		−	0.863383i	−0.902021	−	0.431692i	$-0.857917\pi$
$443$		−	382.479i		−	0.863383i	0.902021	−	0.431692i	$-0.142083\pi$
$444$	−290.934			−0.655257
$445$		0			0
$446$	−543.603			−1.21884
$447$	328.153			0.734123
$448$		−	245.406i		−	0.547780i
$449$			666.393i			1.48417i	0.670305	+	0.742086i	$0.266163\pi$
$449$			666.393i			1.48417i	−0.670305	+	0.742086i	$0.733837\pi$
$450$		0			0
$451$		−	1252.16i		−	2.77641i
$452$	30.3807			0.0672140
$453$			25.8442i			0.0570512i
$454$	−866.196			−1.90792
$455$		0			0
$456$	−332.585	−	192.322i	−0.729352	−	0.421758i
$457$			627.200i			1.37243i	0.727399	+	0.686214i	$0.240729\pi$
$457$			627.200i			1.37243i	−0.727399	+	0.686214i	$0.759271\pi$
$458$	790.337			1.72563
$459$			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	$0.166762\pi$
$462$		−	1562.70i		−	3.38248i
$463$			791.786i			1.71012i	0.518528	+	0.855061i	$0.326480\pi$
$463$			791.786i			1.71012i	−0.518528	+	0.855061i	$0.673520\pi$
$464$		−	602.983i		−	1.29953i
$465$		0			0
$466$		−	198.686i		−	0.426364i
$467$			418.723i			0.896622i	0.893878	+	0.448311i	$0.147974\pi$
$467$			418.723i			0.896622i	−0.893878	+	0.448311i	$0.852026\pi$
$468$	−83.1526			−0.177677
$469$			227.905i			0.485939i
$470$		0			0
$471$			815.000i			1.73036i
$472$		−	415.197i		−	0.879655i
$473$			104.512i			0.220956i
$474$		−	73.9251i		−	0.155960i
$475$		0			0
$476$	357.443			0.750931
$477$	112.210			0.235241
$478$	−260.262			−0.544481
$479$	−63.6160			−0.132810			−0.0664050	−	0.997793i	$-0.521153\pi$
$479$	−63.6160			−0.132810			−0.0664050	+	0.997793i	$0.521153\pi$
$480$		0			0
$481$	−733.310			−1.52455
$482$			760.869i			1.57857i
$483$	535.252			1.10818
$484$	255.751			0.528411
$485$		0			0
$486$	−439.880			−0.905102
$487$	−556.997			−1.14373			−0.571866	−	0.820347i	$-0.693780\pi$
$487$	−556.997			−1.14373			−0.571866	+	0.820347i	$0.693780\pi$
$488$	−101.129			−0.207231
$489$		−	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	$0.162488\pi$
$492$			423.709i			0.861198i
$493$	−614.510			−1.24647
$494$	556.537	+	321.825i	1.12659	+	0.651469i
$495$		0			0
$496$			319.881i			0.644920i
$497$	−82.9696			−0.166941
$498$		−	536.536i		−	1.07738i
$499$	−761.632			−1.52632			−0.763158	−	0.646211i	$-0.776352\pi$
$499$	−761.632			−1.52632			−0.763158	+	0.646211i	$0.776352\pi$
$500$		0			0
$501$	576.987			1.15167
$502$	−339.248			−0.675792
$503$		−	34.7192i		−	0.0690242i	−0.999404	−	0.0345121i	$-0.989012\pi$
$503$		−	34.7192i		−	0.0690242i	0.999404	−	0.0345121i	$-0.0109877\pi$
$504$		−	229.484i		−	0.455325i
$505$		0			0
$506$			539.068i			1.06535i
$507$	−126.548			−0.249601
$508$	−149.084			−0.293472
$509$		−	9.31181i		−	0.0182943i	−0.999958	−	0.00914716i	$-0.997088\pi$
$509$		−	9.31181i		−	0.0182943i	0.999958	−	0.00914716i	$-0.00291167\pi$
$510$		0			0
$511$	−323.826			−0.633710
$512$	−28.3801			−0.0554300
$513$	313.322	+	181.183i	0.610764	+	0.353183i
$514$	−404.993			−0.787925
$515$		0			0
$516$		−	35.3651i		−	0.0685370i
$517$			738.318i			1.42808i
$518$			1343.58i			2.59378i
$519$	−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	$0.973059\pi$
$522$		−	261.922i		−	0.501767i
$523$	377.616			0.722019			0.361009	−	0.932562i	$-0.382432\pi$
$523$	377.616			0.722019			0.361009	+	0.932562i	$0.382432\pi$
$524$	201.888			0.385282
$525$		0			0
$526$			437.617i			0.831971i
$527$	325.995			0.618587
$528$	1182.85			2.24025
$529$	344.360			0.650965
$530$		0			0
$531$		−	265.938i		−	0.500826i
$532$	168.171	−	290.820i	0.316111	−	0.546654i
$533$			1067.98i			2.00371i
$534$		−	1370.25i		−	2.56600i
$535$		0			0
$536$	−116.990			−0.218265
$537$			210.761i			0.392478i
$538$			1072.07i			1.99269i
$539$	−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	$0.566906\pi$
$542$	−796.718			−1.46996
$543$			731.394i			1.34695i
$544$			488.785i			0.898502i
$545$		0			0
$546$			1332.84i			2.44110i
$547$	1050.58			1.92062			0.960309	−	0.278938i	$-0.0899823\pi$
$547$	1050.58			1.92062			0.960309	+	0.278938i	$0.0899823\pi$
$548$		−	149.835i		−	0.273421i
$549$	−64.7739			−0.117985
$550$		0			0
$551$	−289.116	+	499.973i	−0.524712	+	0.907391i
$552$			274.759i			0.497752i
$553$	−97.3678			−0.176072
$554$			639.771i			1.15482i
$555$		0			0
$556$	−65.5934			−0.117974
$557$		−	580.999i		−	1.04309i	−0.853225	−	0.521543i	$-0.825356\pi$
$557$		−	580.999i		−	1.04309i	0.853225	−	0.521543i	$-0.174644\pi$
$558$			138.949i			0.249012i
$559$		−	89.1391i		−	0.159462i
$560$		0			0
$561$		−	1205.46i		−	2.14877i
$562$		−	357.995i		−	0.637002i
$563$	796.409			1.41458			0.707291	−	0.706923i	$-0.249917\pi$
$563$	796.409			1.41458			0.707291	+	0.706923i	$0.249917\pi$
$564$		−	249.834i		−	0.442967i
$565$		0			0
$566$			1035.87i			1.83016i
$567$			1113.55i			1.96393i
$568$		−	42.5905i		−	0.0749833i
$569$		−	171.428i		−	0.301280i	−0.988589	−	0.150640i	$-0.951867\pi$
$569$		−	171.428i		−	0.301280i	0.988589	−	0.150640i	$-0.0481334\pi$
$570$		0			0
$571$	−673.960			−1.18031			−0.590157	−	0.807288i	$-0.700934\pi$
$571$	−673.960			−1.18031			−0.590157	+	0.807288i	$0.700934\pi$
$572$	−382.841			−0.669302
$573$	901.999			1.57417
$574$	1956.76			3.40898
$575$		0			0
$576$	80.6873			0.140082
$577$			795.916i			1.37940i	0.724094	+	0.689702i	$0.242258\pi$
$577$			795.916i			1.37940i	−0.724094	+	0.689702i	$0.757742\pi$
$578$	283.128			0.489840
$579$	1029.87			1.77871
$580$		0			0
$581$	−706.679			−1.21632
$582$	635.861			1.09254
$583$	516.623			0.886146
$584$		−	166.229i		−	0.284638i
$585$		0			0
$586$	627.464			1.07076
$587$			19.6929i			0.0335484i	0.999859	+	0.0167742i	$0.00533965\pi$
$587$			19.6929i			0.0335484i	−0.999859	+	0.0167742i	$0.994660\pi$
$588$	418.414			0.711589
$589$	153.375	−	265.234i	0.260399	−	0.450312i
$590$		0			0
$591$			633.576i			1.07204i
$592$	−1016.99			−1.71789
$593$		−	636.058i		−	1.07261i	−0.844024	−	0.536306i	$-0.819819\pi$
$593$		−	636.058i		−	1.07261i	0.844024	−	0.536306i	$-0.180181\pi$
$594$	−755.718			−1.27225
$595$		0			0
$596$	−147.297			−0.247142
$597$	−549.706			−0.920781
$598$		−	459.774i		−	0.768853i
$599$		−	936.525i		−	1.56348i	−0.623604	−	0.781741i	$-0.714332\pi$
$599$		−	936.525i		−	1.56348i	0.623604	−	0.781741i	$-0.285668\pi$
$600$		0			0
$601$		−	252.090i		−	0.419451i	−0.977760	−	0.209725i	$-0.932743\pi$
$601$		−	252.090i		−	0.419451i	0.977760	−	0.209725i	$-0.0672570\pi$
$602$	−163.321			−0.271298
$603$	−74.9334			−0.124268
$604$		−	11.6006i		−	0.0192062i
$605$		0			0
$606$	132.638			0.218874
$607$	498.526			0.821295			0.410648	−	0.911794i	$-0.365303\pi$
$607$	498.526			0.821295			0.410648	+	0.911794i	$0.365303\pi$
$608$	397.682	+	229.965i	0.654082	+	0.378232i
$609$	−1197.37			−1.96613
$610$		0			0
$611$		−	629.715i		−	1.03063i
$612$			117.524i			0.192033i
$613$		−	638.709i		−	1.04194i	−0.853575	−	0.520969i	$-0.825571\pi$
$613$		−	638.709i		−	1.04194i	0.853575	−	0.520969i	$-0.174429\pi$
$614$	958.215			1.56061
$615$		0			0
$616$		−	1056.56i		−	1.71520i
$617$		−	702.709i		−	1.13891i	−0.822021	−	0.569457i	$-0.807154\pi$
$617$		−	702.709i		−	1.13891i	0.822021	−	0.569457i	$-0.192846\pi$
$618$	60.1516			0.0973327
$619$	−195.693			−0.316144			−0.158072	−	0.987428i	$-0.550528\pi$
$619$	−195.693			−0.316144			−0.158072	+	0.987428i	$0.550528\pi$
$620$		0			0
$621$		−	258.846i		−	0.416821i
$622$	742.572			1.19385
$623$	−1804.77			−2.89691
$624$	−1008.86			−1.61676
$625$		0			0
$626$		−	1214.59i		−	1.94024i
$627$	−980.779	−	567.149i	−1.56424	−	0.904544i
$628$		−	365.826i		−	0.582525i
$629$			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	$0.818884\pi$
$632$		−	49.9815i		−	0.0790847i
$633$		−	973.694i		−	1.53822i
$634$	−61.0443			−0.0962843
$635$		0			0
$636$	−174.816			−0.274868
$637$	1054.63			1.65562
$638$		−	1205.91i		−	1.89014i
$639$		−	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	$0.802726\pi$
$642$	−723.251			−1.12656
$643$			195.660i			0.304293i	0.988358	+	0.152146i	$0.0486185\pi$
$643$			195.660i			0.304293i	−0.988358	+	0.152146i	$0.951382\pi$
$644$	−240.256			−0.373069
$645$		0			0
$646$	454.854	−	786.586i	0.704109	−	1.21763i
$647$			128.045i			0.197906i	0.995092	+	0.0989529i	$0.0315493\pi$
$647$			128.045i			0.197906i	−0.995092	+	0.0989529i	$0.968451\pi$
$648$	−571.614			−0.882120
$649$		−	1224.40i		−	1.88660i
$650$		0			0
$651$	635.202			0.975733
$652$			54.3320i			0.0833313i
$653$		−	717.897i		−	1.09938i	−0.835368	−	0.549692i	$-0.814745\pi$
$653$		−	717.897i		−	1.09938i	0.835368	−	0.549692i	$-0.185255\pi$
$654$			799.811i			1.22295i
$655$		0			0
$656$			1481.12i			2.25780i
$657$		−	106.471i		−	0.162057i
$658$	−1153.77			−1.75345
$659$		−	544.617i		−	0.826430i	−0.910633	−	0.413215i	$-0.864406\pi$
$659$		−	544.617i		−	0.826430i	0.910633	−	0.413215i	$-0.135594\pi$
$660$		0			0
$661$		−	435.420i		−	0.658729i	−0.944203	−	0.329365i	$-0.893165\pi$
$661$		−	435.420i		−	0.658729i	0.944203	−	0.329365i	$-0.106835\pi$
$662$		−	406.017i		−	0.613319i
$663$			1028.15i			1.55075i
$664$		−	362.757i		−	0.546321i
$665$		0			0
$666$	−441.757			−0.663299
$667$	413.044			0.619256
$668$	−258.990			−0.387709
$669$	−817.070			−1.22133
$670$		0			0
$671$	−298.224			−0.444447
$672$			952.399i			1.41726i
$673$	−397.296			−0.590336			−0.295168	−	0.955445i	$-0.595376\pi$
$673$	−397.296			−0.590336			−0.295168	+	0.955445i	$0.595376\pi$
$674$	−487.580			−0.723412
$675$		0			0
$676$	56.8030			0.0840281
$677$	−456.859			−0.674829			−0.337415	−	0.941356i	$-0.609552\pi$
$677$	−456.859			−0.674829			−0.337415	+	0.941356i	$0.609552\pi$
$678$	160.111			0.236152
$679$		−	837.502i		−	1.23343i
$680$		0			0
$681$	−1301.95			−1.91182
$682$			639.731i			0.938022i
$683$	880.553			1.28924			0.644621	−	0.764502i	$-0.277015\pi$
$683$	880.553			1.28924			0.644621	+	0.764502i	$0.277015\pi$
$684$	95.6192	+	55.2932i	0.139794	+	0.0808379i
$685$		0			0
$686$		−	648.155i		−	0.944833i
$687$	1187.93			1.72915
$688$		−	123.622i		−	0.179683i
$689$	−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	$0.507922\pi$
$692$	47.5538			0.0687194
$693$		−	676.739i		−	0.976535i
$694$		−	18.7329i		−	0.0269927i
$695$		0			0
$696$		−	614.644i		−	0.883109i
$697$	1509.43			2.16561
$698$	968.585			1.38766
$699$		−	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	$0.403318\pi$
$702$	644.556			0.918171
$703$	843.251	+	487.622i	1.19950	+	0.693630i
$704$	371.490			0.527685
$705$		0			0
$706$			1509.05i			2.13747i
$707$		−	174.699i		−	0.247099i
$708$			414.315i			0.585191i
$709$	−373.670			−0.527038			−0.263519	−	0.964654i	$-0.584883\pi$
$709$	−373.670			−0.527038			−0.263519	+	0.964654i	$0.584883\pi$
$710$		0			0
$711$		−	32.0137i		−	0.0450263i
$712$		−	926.438i		−	1.30118i
$713$	−219.118			−0.307319
$714$	1883.78			2.63834
$715$		0			0
$716$		−	94.6033i		−	0.132127i
$717$	−391.191			−0.545594
$718$	−561.289			−0.781739
$719$	328.089			0.456312			0.228156	−	0.973625i	$-0.426730\pi$
$719$	328.089			0.456312			0.228156	+	0.973625i	$0.426730\pi$
$720$		0			0
$721$		−	79.2266i		−	0.109884i
$722$	−425.975	−	740.150i	−0.589993	−	1.02514i
$723$			1143.64i			1.58179i
$724$		−	328.298i		−	0.453450i
$725$		0			0
$726$	1347.84			1.85653
$727$		−	1131.05i		−	1.55578i	−0.628402	−	0.777889i	$-0.716291\pi$
$727$		−	1131.05i		−	1.55578i	0.628402	−	0.777889i	$-0.283709\pi$
$728$			901.146i			1.23784i
$729$	243.465			0.333971
$730$		0			0
$731$	−125.985			−0.172347
$732$	100.914			0.137860
$733$		−	450.895i		−	0.615137i	−0.951526	−	0.307568i	$-0.900485\pi$
$733$		−	450.895i		−	0.615137i	0.951526	−	0.307568i	$-0.0995153\pi$
$734$			1284.25i			1.74966i
$735$		0			0
$736$		−	328.538i		−	0.446383i
$737$	−344.999			−0.468112
$738$			643.365i			0.871768i
$739$	195.478			0.264517			0.132258	−	0.991215i	$-0.457777\pi$
$739$	195.478			0.264517			0.132258	+	0.991215i	$0.457777\pi$
$740$		0			0
$741$	836.512	+	483.724i	1.12890	+	0.652800i
$742$			807.328i			1.08804i
$743$	194.450			0.261710			0.130855	−	0.991402i	$-0.458228\pi$
$743$	194.450			0.261710			0.130855	+	0.991402i	$0.458228\pi$
$744$			326.067i			0.438262i
$745$		0			0
$746$	−1052.09			−1.41030
$747$		−	232.350i		−	0.311044i
$748$			541.091i			0.723383i
$749$			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	$-0.216232\pi$
$752$		−	873.318i		−	1.16133i
$753$	−509.911			−0.677173
$754$			1028.53i			1.36409i
$755$		0			0
$756$		−	336.814i		−	0.445522i
$757$		−	891.082i		−	1.17712i	−0.808453	−	0.588561i	$-0.799695\pi$
$757$		−	891.082i		−	1.17712i	0.808453	−	0.588561i	$-0.200305\pi$
$758$			355.362i			0.468815i
$759$			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	$0.333738\pi$
$762$	−785.692			−1.03109
$763$	1053.44			1.38066
$764$	−404.876			−0.529943
$765$		0			0
$766$	−202.970			−0.264974
$767$			1044.30i			1.36154i
$768$	−939.170			−1.22288
$769$	−225.596			−0.293363			−0.146681	−	0.989184i	$-0.546859\pi$
$769$	−225.596			−0.293363			−0.146681	+	0.989184i	$0.546859\pi$
$770$		0			0
$771$	−608.731			−0.789535
$772$	−462.275			−0.598802
$773$	−1189.39			−1.53867			−0.769337	−	0.638843i	$-0.779413\pi$
$773$	−1189.39			−1.53867			−0.769337	+	0.638843i	$0.779413\pi$
$774$		−	53.6987i		−	0.0693782i
$775$		0			0
$776$	429.913			0.554011
$777$			2019.48i			2.59908i
$778$	−590.613			−0.759143
$779$	710.161	−	1228.09i	0.911632	−	1.57650i
$780$		0			0
$781$		−	125.598i		−	0.160817i
$782$	−649.825			−0.830978
$783$			579.045i			0.739521i
$784$	1462.61			1.86557
$785$		0			0
$786$	1063.98			1.35366
$787$	−254.533			−0.323422			−0.161711	−	0.986838i	$-0.551701\pi$
$787$	−254.533			−0.323422			−0.161711	+	0.986838i	$0.551701\pi$
$788$		−	284.391i		−	0.360902i
$789$			657.766i			0.833670i
$790$		0			0
$791$		−	210.884i		−	0.266605i
$792$	347.388			0.438622
$793$	254.357			0.320753
$794$		−	805.425i		−	1.01439i
$795$		0			0
$796$	246.744			0.309980
$797$	−719.295			−0.902503			−0.451251	−	0.892397i	$-0.649022\pi$
$797$	−719.295			−0.902503			−0.451251	+	0.892397i	$0.649022\pi$
$798$	886.284	−	1532.66i	1.11063	−	1.92063i
$799$	−890.013			−1.11391
$800$		0			0
$801$		−	593.394i		−	0.740816i
$802$		−	1161.51i		−	1.44826i
$803$		−	490.202i		−	0.610463i
$804$	116.741			0.145201
$805$		0			0
$806$		−	545.630i		−	0.676961i
$807$			1611.39i			1.99676i
$808$	89.6776			0.110987
$809$	−67.3018			−0.0831914			−0.0415957	−	0.999135i	$-0.513244\pi$
$809$	−67.3018			−0.0831914			−0.0415957	+	0.999135i	$0.513244\pi$
$810$		0			0
$811$			401.142i			0.494626i	0.968936	+	0.247313i	$0.0795477\pi$
$811$			401.142i			0.494626i	−0.968936	+	0.247313i	$0.920452\pi$
$812$	537.460			0.661896
$813$	−1197.52			−1.47296
$814$	−2033.88			−2.49863
$815$		0			0
$816$			1425.88i			1.74740i
$817$	−59.2739	+	102.503i	−0.0725507	+	0.125463i
$818$			274.013i			0.334979i
$819$			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	$0.519363\pi$
$822$		−	789.650i		−	0.960644i
$823$		−	93.5359i		−	0.113652i	−0.998384	−	0.0568262i	$-0.981902\pi$
$823$		−	93.5359i		−	0.113652i	0.998384	−	0.0568262i	$-0.0180981\pi$
$824$	40.6692			0.0493558
$825$		0			0
$826$	1913.37			2.31643
$827$	398.725			0.482135			0.241067	−	0.970508i	$-0.422503\pi$
$827$	398.725			0.482135			0.241067	+	0.970508i	$0.422503\pi$
$828$		−	78.9942i		−	0.0954036i
$829$		−	805.348i		−	0.971470i	−0.874106	−	0.485735i	$-0.838552\pi$
$829$		−	805.348i		−	0.971470i	0.874106	−	0.485735i	$-0.161448\pi$
$830$		0			0
$831$			961.617i			1.15718i
$832$	−316.846			−0.380825
$833$		−	1490.57i		−	1.78940i
$834$	−345.686			−0.414492
$835$		0			0
$836$	440.238	+	254.574i	0.526601	+	0.304514i
$837$		−	307.181i		−	0.367003i
$838$	192.788			0.230057
$839$			1547.08i			1.84396i	0.387235	+	0.921981i	$0.373430\pi$
$839$			1547.08i			1.84396i	−0.387235	+	0.921981i	$0.626570\pi$
$840$		0			0
$841$	−82.9912			−0.0986815
$842$		−	524.338i		−	0.622729i
$843$		−	538.090i		−	0.638304i
$844$			437.058i			0.517841i
$845$		0			0
$846$		−	379.350i		−	0.448404i
$847$		−	1775.26i		−	2.09594i
$848$	−611.087			−0.720622
$849$			1556.98i			1.83390i
$850$		0			0
$851$		−	696.638i		−	0.818611i
$852$			42.5001i			0.0498827i
$853$		−	252.303i		−	0.295784i	−0.989004	−	0.147892i	$-0.952751\pi$
$853$		−	252.303i		−	0.295784i	0.989004	−	0.147892i	$-0.0472487\pi$
$854$		−	466.035i		−	0.545708i
$855$		0			0
$856$	−488.998			−0.571259
$857$	−1397.35			−1.63051			−0.815255	−	0.579102i	$-0.803403\pi$
$857$	−1397.35			−1.63051			−0.815255	+	0.579102i	$0.803403\pi$
$858$	−2017.63			−2.35155
$859$	1472.96			1.71473			0.857366	−	0.514706i	$-0.172099\pi$
$859$	1472.96			1.71473			0.857366	+	0.514706i	$0.172099\pi$
$860$		0			0
$861$	2941.13			3.41595
$862$			501.112i			0.581336i
$863$	63.9533			0.0741058			0.0370529	−	0.999313i	$-0.488203\pi$
$863$	63.9533			0.0741058			0.0370529	+	0.999313i	$0.488203\pi$
$864$	460.576			0.533075
$865$		0			0
$866$	−921.476			−1.06406
$867$	425.559			0.490841
$868$	−285.121			−0.328480
$869$		−	147.394i		−	0.169613i
$870$		0			0
$871$	294.251			0.337832
$872$			540.761i			0.620138i
$873$	275.364			0.315422
$874$	−305.731	+	528.705i	−0.349807	+	0.604926i
$875$		0			0
$876$			165.875i			0.189356i
$877$	−399.492			−0.455522			−0.227761	−	0.973717i	$-0.573140\pi$
$877$	−399.492			−0.455522			−0.227761	+	0.973717i	$0.573140\pi$
$878$		−	1101.63i		−	1.25471i
$879$	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	$0.405740\pi$
$882$	635.324			0.720322
$883$			323.869i			0.366783i	0.983040	+	0.183392i	$0.0587076\pi$
$883$			323.869i			0.366783i	−0.983040	+	0.183392i	$0.941292\pi$
$884$		−	461.499i		−	0.522058i
$885$		0			0
$886$			904.787i			1.02120i
$887$	277.999			0.313415			0.156707	−	0.987645i	$-0.449912\pi$
$887$	277.999			0.313415			0.156707	+	0.987645i	$0.449912\pi$
$888$	−1036.66			−1.16740
$889$			1034.85i			1.16406i
$890$		0			0
$891$	−1685.67			−1.89188
$892$	366.755			0.411160
$893$	−418.735	+	724.125i	−0.468909	+	0.810890i
$894$	−776.275			−0.868316
$895$		0			0
$896$			1651.95i			1.84370i
$897$		−	691.070i		−	0.770424i
$898$		−	1576.41i		−	1.75547i
$899$	490.174			0.545244
$900$		0			0
$901$			622.769i			0.691197i
$902$			2962.10i			3.28392i
$903$	−245.483			−0.271852
$904$	108.253			0.119749
$905$		0			0
$906$		−	61.1367i		−	0.0674798i
$907$	−401.356			−0.442509			−0.221255	−	0.975216i	$-0.571015\pi$
$907$	−401.356			−0.442509			−0.221255	+	0.975216i	$0.571015\pi$
$908$	584.400			0.643613
$909$	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	$-0.0666551\pi$
$912$	1160.11	+	670.851i	1.27205	+	0.735583i
$913$		−	1069.76i		−	1.17170i
$914$		−	1483.70i		−	1.62330i
$915$		0			0
$916$	−533.220			−0.582118
$917$		−	1401.38i		−	1.52822i
$918$		−	910.987i		−	0.992361i
$919$	102.234			0.111244			0.0556222	−	0.998452i	$-0.482286\pi$
$919$	102.234			0.111244			0.0556222	+	0.998452i	$0.482286\pi$
$920$		0			0
$921$	1440.26			1.56380
$922$	−1888.54			−2.04831
$923$			107.123i			0.116060i
$924$			1054.32i			1.14103i
$925$		0			0
$926$		−	1873.04i		−	2.02272i
$927$	26.0490			0.0281004
$928$			734.949i			0.791971i
$929$	−1794.45			−1.93160			−0.965798	−	0.259295i	$-0.916510\pi$
$929$	−1794.45			−1.93160			−0.965798	+	0.259295i	$0.916510\pi$
$930$		0			0
$931$	−1212.74	−	701.286i	−1.30262	−	0.753261i
$932$			134.048i			0.143828i
$933$	1116.13			1.19629
$934$		−	990.525i		−	1.06052i
$935$		0			0
$936$	−296.289			−0.316549
$937$			823.194i			0.878542i	0.898355	+	0.439271i	$0.144763\pi$
$937$			823.194i			0.878542i	−0.898355	+	0.439271i	$0.855237\pi$
$938$		−	539.130i		−	0.574765i
$939$		−	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	$-0.0458409\pi$
$942$		−	1927.95i		−	2.04666i
$943$	−1014.57			−1.07589
$944$			1448.28i			1.53420i
$945$		0			0
$946$		−	247.233i		−	0.261346i
$947$			1412.14i			1.49118i	0.666407	+	0.745588i	$0.267831\pi$
$947$			1412.14i			1.49118i	−0.666407	+	0.745588i	$0.732169\pi$
$948$			49.8753i			0.0526111i
$949$			418.096i			0.440564i
$950$		0			0
$951$	−91.7535			−0.0964811
$952$	1273.64			1.33786
$953$	−432.598			−0.453933			−0.226966	−	0.973903i	$-0.572881\pi$
$953$	−432.598			−0.453933			−0.226966	+	0.973903i	$0.572881\pi$
$954$	−265.443			−0.278242
$955$		0			0
$956$	175.592			0.183674
$957$		−	1812.56i		−	1.89400i
$958$	150.489			0.157087
$959$	−1040.06			−1.08452
$960$		0			0
$961$	700.964			0.729411
$962$	1734.71			1.80323
$963$	−313.209			−0.325243
$964$		−	513.339i		−	0.532509i
$965$		0			0
$966$	−1266.18			−1.31075
$967$		−	10.0222i		−	0.0103642i	−0.999987	−	0.00518210i	$-0.998350\pi$
$967$		−	10.0222i		−	0.0103642i	0.999987	−	0.00518210i	$-0.00164952\pi$
$968$	911.291			0.941416
$969$	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	$-0.0559205\pi$
$972$	296.776			0.305325
$973$			455.309i			0.467943i
$974$	1317.63			1.35280
$975$		0			0
$976$	352.754			0.361428
$977$	−673.762			−0.689623			−0.344812	−	0.938672i	$-0.612057\pi$
$977$	−673.762			−0.689623			−0.344812	+	0.938672i	$0.612057\pi$
$978$			286.337i			0.292778i
$979$		−	2732.03i		−	2.79063i
$980$		0			0
$981$			346.363i			0.353071i
$982$	−2026.86			−2.06401
$983$	428.230			0.435636			0.217818	−	0.975989i	$-0.430106\pi$
$983$	428.230			0.435636			0.217818	+	0.975989i	$0.430106\pi$
$984$			1509.76i			1.53431i
$985$		0			0
$986$	1453.68			1.47432
$987$	−1734.19			−1.75703
$988$	−375.481	−	217.127i	−0.380042	−	0.219765i
$989$	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	$-0.230037\pi$
$992$		−	389.888i		−	0.393032i
$993$		−	610.270i		−	0.614572i
$994$	196.272			0.197457
$995$		0			0
$996$			361.987i			0.363441i
$997$			690.240i			0.692316i	0.938176	+	0.346158i	$0.112514\pi$
$997$			690.240i			0.692316i	−0.938176	+	0.346158i	$0.887486\pi$
$998$	1801.71			1.80532
$999$	976.615			0.977592

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.3.c.a.56.3	✓	12	5.2	odd	4
95.3.c.a.56.10	yes	12	95.37	even	4
475.3.c.g.151.3		12	95.18	even	4
475.3.c.g.151.10		12	5.3	odd	4
475.3.d.c.474.5		24	95.94	odd	2	inner
475.3.d.c.474.6		24	1.1	even	1	trivial
475.3.d.c.474.19		24	5.4	even	2	inner
475.3.d.c.474.20		24	19.18	odd	2	inner
855.3.e.a.721.3		12	285.227	odd	4
855.3.e.a.721.10		12	15.2	even	4
1520.3.h.a.721.4		12	20.7	even	4
1520.3.h.a.721.9		12	380.227	odd	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.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.36559 q^{2} -3.55563 q^{3} +1.59600 q^{4} +8.41115 q^{6} -11.0785i q^{7} +5.68687 q^{8} +3.64250 q^{9} +O(q^{10})\)	\(q-2.36559 q^{2} -3.55563 q^{3} +1.59600 q^{4} +8.41115 q^{6} -11.0785i q^{7} +5.68687 q^{8} +3.64250 q^{9} +16.7704 q^{11} -5.67479 q^{12} -14.3035 q^{13} +26.2070i q^{14} -19.8368 q^{16} +20.2160i q^{17} -8.61666 q^{18} +(16.4480 + 9.51127i) q^{19} +39.3909i q^{21} -39.6717 q^{22} -13.5882i q^{23} -20.2204 q^{24} +33.8362 q^{26} +19.0493 q^{27} -17.6812i q^{28} +30.3972i q^{29} -16.1256i q^{31} +24.1782 q^{32} -59.6292 q^{33} -47.8227i q^{34} +5.81344 q^{36} +51.2678 q^{37} +(-38.9091 - 22.4997i) q^{38} +50.8580 q^{39} -74.6652i q^{41} -93.1825i q^{42} +6.23197i q^{43} +26.7655 q^{44} +32.1441i q^{46} +44.0252i q^{47} +70.5323 q^{48} -73.7321 q^{49} -71.8806i q^{51} -22.8284 q^{52} +30.8058 q^{53} -45.0627 q^{54} -63.0017i q^{56} +(-58.4829 - 33.8185i) q^{57} -71.9073i q^{58} -73.0098i q^{59} -17.7828 q^{61} +38.1466i q^{62} -40.3533i q^{63} +22.1516 q^{64} +141.058 q^{66} -20.5719 q^{67} +32.2647i q^{68} +48.3147i q^{69} -7.48928i q^{71} +20.7144 q^{72} -29.2302i q^{73} -121.278 q^{74} +(26.2510 + 15.1800i) q^{76} -185.790i q^{77} -120.309 q^{78} -8.78894i q^{79} -100.515 q^{81} +176.627i q^{82} -63.7886i q^{83} +62.8679i q^{84} -14.7423i q^{86} -108.081i q^{87} +95.3708 q^{88} -162.908i q^{89} +158.461i q^{91} -21.6868i q^{92} +57.3367i q^{93} -104.145i q^{94} -85.9686 q^{96} +75.5974 q^{97} +174.420 q^{98} +61.0861 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9}+O(q^{10}) \) 24 * q + 24 * q^4 - 56 * q^6 + 96 * q^9	\( 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9} + 64 q^{11} - 88 q^{16} - 16 q^{19} - 200 q^{24} + 216 q^{26} - 160 q^{36} - 152 q^{39} + 512 q^{44} - 144 q^{49} + 152 q^{54} - 592 q^{61} - 376 q^{64} + 304 q^{66} - 272 q^{74} + 496 q^{76} - 744 q^{81} - 88 q^{96} + 624 q^{99}+O(q^{100}) \) 24 * q + 24 * q^4 - 56 * q^6 + 96 * q^9 + 64 * q^11 - 88 * q^16 - 16 * q^19 - 200 * q^24 + 216 * q^26 - 160 * q^36 - 152 * q^39 + 512 * q^44 - 144 * q^49 + 152 * q^54 - 592 * q^61 - 376 * q^64 + 304 * q^66 - 272 * q^74 + 496 * q^76 - 744 * q^81 - 88 * q^96 + 624 * q^99

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