LMFDB - Embedded newform 475.3.d.d.474.7

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:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			474.7
Character	$\chi$	$=$	475.474
Dual form			475.3.d.d.474.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.34822 q^{2} +5.73703 q^{3} +1.51415 q^{4} -13.4718 q^{6} +9.56678i q^{7} +5.83734 q^{8} +23.9135 q^{9} +O(q^{10})$	\(q-2.34822 q^{2} +5.73703 q^{3} +1.51415 q^{4} -13.4718 q^{6} +9.56678i q^{7} +5.83734 q^{8} +23.9135 q^{9} +4.15630 q^{11} +8.68670 q^{12} +4.55775 q^{13} -22.4649i q^{14} -19.7639 q^{16} -27.9397i q^{17} -56.1541 q^{18} +(-7.04287 + 17.6465i) q^{19} +54.8849i q^{21} -9.75992 q^{22} +13.9626i q^{23} +33.4890 q^{24} -10.7026 q^{26} +85.5590 q^{27} +14.4855i q^{28} +14.6141i q^{29} +28.9385i q^{31} +23.0608 q^{32} +23.8448 q^{33} +65.6087i q^{34} +36.2085 q^{36} -30.8172 q^{37} +(16.5382 - 41.4378i) q^{38} +26.1479 q^{39} +44.2953i q^{41} -128.882i q^{42} -69.2593i q^{43} +6.29325 q^{44} -32.7872i q^{46} +30.1880i q^{47} -113.386 q^{48} -42.5233 q^{49} -160.291i q^{51} +6.90110 q^{52} +54.3939 q^{53} -200.912 q^{54} +55.8445i q^{56} +(-40.4051 + 101.238i) q^{57} -34.3171i q^{58} -31.3331i q^{59} +99.9646 q^{61} -67.9540i q^{62} +228.775i q^{63} +24.9039 q^{64} -55.9929 q^{66} +3.05833 q^{67} -42.3049i q^{68} +80.1037i q^{69} -23.0099i q^{71} +139.591 q^{72} -21.1499i q^{73} +72.3655 q^{74} +(-10.6639 + 26.7193i) q^{76} +39.7625i q^{77} -61.4012 q^{78} -77.7042i q^{79} +275.633 q^{81} -104.015i q^{82} -93.1651i q^{83} +83.1038i q^{84} +162.636i q^{86} +83.8415i q^{87} +24.2617 q^{88} +126.623i q^{89} +43.6030i q^{91} +21.1414i q^{92} +166.021i q^{93} -70.8882i q^{94} +132.300 q^{96} -77.2807 q^{97} +99.8542 q^{98} +99.3917 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 56 q^{4} - 8 q^{6} + 72 q^{9}+O(q^{10}) $ 28 * q + 56 * q^4 - 8 * q^6 + 72 * q^9	$ 28 q + 56 q^{4} - 8 q^{6} + 72 q^{9} - 8 q^{11} + 72 q^{16} - 78 q^{19} + 88 q^{24} + 60 q^{26} + 8 q^{36} + 64 q^{39} + 104 q^{44} - 468 q^{49} - 196 q^{54} + 444 q^{61} + 436 q^{64} + 184 q^{66} + 184 q^{74} - 702 q^{76} + 804 q^{81} + 380 q^{96} + 360 q^{99}+O(q^{100}) $ 28 * q + 56 * q^4 - 8 * q^6 + 72 * q^9 - 8 * q^11 + 72 * q^16 - 78 * q^19 + 88 * q^24 + 60 * q^26 + 8 * q^36 + 64 * q^39 + 104 * q^44 - 468 * q^49 - 196 * q^54 + 444 * q^61 + 436 * q^64 + 184 * q^66 + 184 * q^74 - 702 * q^76 + 804 * q^81 + 380 * q^96 + 360 * 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 $

$2$	−2.34822			−1.17411			−0.587055	−	0.809547i	$-0.699713\pi$
$2$	−2.34822			−1.17411			−0.587055	+	0.809547i	$0.699713\pi$
$3$	5.73703			1.91234			0.956171	−	0.292809i	$-0.0945898\pi$
$3$	5.73703			1.91234			0.956171	+	0.292809i	$0.0945898\pi$
$4$	1.51415			0.378537
$5$		0			0
$5$		0			0
$6$	−13.4718			−2.24530
$7$			9.56678i			1.36668i	0.730099	+	0.683342i	$0.239474\pi$
$7$			9.56678i			1.36668i	−0.730099	+	0.683342i	$0.760526\pi$
$8$	5.83734			0.729667
$9$	23.9135			2.65705
$10$		0			0
$11$	4.15630			0.377846			0.188923	−	0.981992i	$-0.439500\pi$
$11$	4.15630			0.377846			0.188923	+	0.981992i	$0.439500\pi$
$12$	8.68670			0.723891
$13$	4.55775			0.350596			0.175298	−	0.984515i	$-0.443911\pi$
$13$	4.55775			0.350596			0.175298	+	0.984515i	$0.443911\pi$
$14$		−	22.4649i		−	1.60464i
$15$		0			0
$16$	−19.7639			−1.23525
$17$		−	27.9397i		−	1.64351i	−0.569838	−	0.821757i	$-0.692994\pi$
$17$		−	27.9397i		−	1.64351i	0.569838	−	0.821757i	$-0.307006\pi$
$18$	−56.1541			−3.11967
$19$	−7.04287	+	17.6465i	−0.370677	+	0.928762i
$19$	−7.04287	+	17.6465i	−0.370677	+	0.928762i
$20$		0			0
$21$			54.8849i			2.61357i
$22$	−9.75992			−0.443633
$23$			13.9626i			0.607069i	0.952820	+	0.303534i	$0.0981667\pi$
$23$			13.9626i			0.607069i	−0.952820	+	0.303534i	$0.901833\pi$
$24$	33.4890			1.39537
$25$		0			0
$26$	−10.7026			−0.411639
$27$	85.5590			3.16885
$28$			14.4855i			0.517340i
$29$			14.6141i			0.503934i	0.967736	+	0.251967i	$0.0810775\pi$
$29$			14.6141i			0.503934i	−0.967736	+	0.251967i	$0.918923\pi$
$30$		0			0
$31$			28.9385i			0.933499i	0.884389	+	0.466750i	$0.154575\pi$
$31$			28.9385i			0.933499i	−0.884389	+	0.466750i	$0.845425\pi$
$32$	23.0608			0.720650
$33$	23.8448			0.722570
$34$			65.6087i			1.92967i
$35$		0			0
$36$	36.2085			1.00579
$37$	−30.8172			−0.832896			−0.416448	−	0.909160i	$-0.636725\pi$
$37$	−30.8172			−0.832896			−0.416448	+	0.909160i	$0.636725\pi$
$38$	16.5382	−	41.4378i	0.435216	−	1.09047i
$39$	26.1479			0.670460
$40$		0			0
$41$			44.2953i			1.08037i	0.841545	+	0.540187i	$0.181646\pi$
$41$			44.2953i			1.08037i	−0.841545	+	0.540187i	$0.818354\pi$
$42$		−	128.882i		−	3.06862i
$43$		−	69.2593i		−	1.61068i	−0.592813	−	0.805340i	$-0.701983\pi$
$43$		−	69.2593i		−	1.61068i	0.592813	−	0.805340i	$-0.298017\pi$
$44$	6.29325			0.143028
$45$		0			0
$46$		−	32.7872i		−	0.712766i
$47$			30.1880i			0.642299i	0.947029	+	0.321149i	$0.104069\pi$
$47$			30.1880i			0.642299i	−0.947029	+	0.321149i	$0.895931\pi$
$48$	−113.386			−2.36221
$49$	−42.5233			−0.867823
$50$		0			0
$51$		−	160.291i		−	3.14296i
$52$	6.90110			0.132714
$53$	54.3939			1.02630			0.513150	−	0.858299i	$-0.328478\pi$
$53$	54.3939			1.02630			0.513150	+	0.858299i	$0.328478\pi$
$54$	−200.912			−3.72058
$55$		0			0
$56$			55.8445i			0.997224i
$57$	−40.4051	+	101.238i	−0.708862	+	1.77611i
$58$		−	34.3171i		−	0.591675i
$59$		−	31.3331i		−	0.531069i	−0.964101	−	0.265534i	$-0.914452\pi$
$59$		−	31.3331i		−	0.531069i	0.964101	−	0.265534i	$-0.0855484\pi$
$60$		0			0
$61$	99.9646			1.63876			0.819382	−	0.573248i	$-0.194317\pi$
$61$	99.9646			1.63876			0.819382	+	0.573248i	$0.194317\pi$
$62$		−	67.9540i		−	1.09603i
$63$			228.775i			3.63135i
$64$	24.9039			0.389124
$65$		0			0
$66$	−55.9929			−0.848378
$67$	3.05833			0.0456468			0.0228234	−	0.999740i	$-0.492734\pi$
$67$	3.05833			0.0456468			0.0228234	+	0.999740i	$0.492734\pi$
$68$		−	42.3049i		−	0.622130i
$69$			80.1037i			1.16092i
$70$		0			0
$71$		−	23.0099i		−	0.324084i	−0.986784	−	0.162042i	$-0.948192\pi$
$71$		−	23.0099i		−	0.324084i	0.986784	−	0.162042i	$-0.0518079\pi$
$72$	139.591			1.93876
$73$		−	21.1499i		−	0.289724i	−0.989452	−	0.144862i	$-0.953726\pi$
$73$		−	21.1499i		−	0.289724i	0.989452	−	0.144862i	$-0.0462738\pi$
$74$	72.3655			0.977912
$75$		0			0
$76$	−10.6639	+	26.7193i	−0.140315	+	0.351570i
$77$			39.7625i			0.516395i
$78$	−61.4012			−0.787195
$79$		−	77.7042i		−	0.983597i	−0.870709	−	0.491799i	$-0.836340\pi$
$79$		−	77.7042i		−	0.983597i	0.870709	−	0.491799i	$-0.163660\pi$
$80$		0			0
$81$	275.633			3.40288
$82$		−	104.015i		−	1.26848i
$83$		−	93.1651i		−	1.12247i	−0.827656	−	0.561236i	$-0.810326\pi$
$83$		−	93.1651i		−	1.12247i	0.827656	−	0.561236i	$-0.189674\pi$
$84$			83.1038i			0.989330i
$85$		0			0
$86$			162.636i			1.89112i
$87$			83.8415i			0.963695i
$88$	24.2617			0.275702
$89$			126.623i			1.42273i	0.702821	+	0.711366i	$0.251923\pi$
$89$			126.623i			1.42273i	−0.702821	+	0.711366i	$0.748077\pi$
$90$		0			0
$91$			43.6030i			0.479154i
$92$			21.1414i			0.229798i
$93$			166.021i			1.78517i
$94$		−	70.8882i		−	0.754130i
$95$		0			0
$96$	132.300			1.37813
$97$	−77.2807			−0.796708			−0.398354	−	0.917232i	$-0.630418\pi$
$97$	−77.2807			−0.796708			−0.398354	+	0.917232i	$0.630418\pi$
$98$	99.8542			1.01892
$99$	99.3917			1.00396
$100$		0			0
$101$	−145.782			−1.44339			−0.721693	−	0.692213i	$-0.756636\pi$
$101$	−145.782			−1.44339			−0.721693	+	0.692213i	$0.756636\pi$
$102$			376.399i			3.69019i
$103$	−7.27524			−0.0706334			−0.0353167	−	0.999376i	$-0.511244\pi$
$103$	−7.27524			−0.0706334			−0.0353167	+	0.999376i	$0.511244\pi$
$104$	26.6051			0.255819
$105$		0			0
$106$	−127.729			−1.20499
$107$	128.949			1.20513			0.602567	−	0.798068i	$-0.294145\pi$
$107$	128.949			1.20513			0.602567	+	0.798068i	$0.294145\pi$
$108$	129.549			1.19953
$109$			74.4133i			0.682691i	0.939938	+	0.341346i	$0.110883\pi$
$109$			74.4133i			0.682691i	−0.939938	+	0.341346i	$0.889117\pi$
$110$		0			0
$111$	−176.799			−1.59278
$112$		−	189.077i		−	1.68819i
$113$	174.597			1.54511			0.772554	−	0.634949i	$-0.218979\pi$
$113$	174.597			1.54511			0.772554	+	0.634949i	$0.218979\pi$
$114$	94.8802	−	237.730i	0.832283	−	2.08535i
$115$		0			0
$116$			22.1279i			0.190758i
$117$	108.992			0.931553
$118$			73.5770i			0.623534i
$119$	267.293			2.24616
$120$		0			0
$121$	−103.725			−0.857233
$122$	−234.739			−1.92409
$123$			254.123i			2.06604i
$124$			43.8171i			0.353364i
$125$		0			0
$126$		−	537.215i		−	4.26361i
$127$	−81.1228			−0.638762			−0.319381	−	0.947626i	$-0.603475\pi$
$127$	−81.1228			−0.638762			−0.319381	+	0.947626i	$0.603475\pi$
$128$	−150.723			−1.17752
$129$		−	397.342i		−	3.08017i
$130$		0			0
$131$	−51.8609			−0.395885			−0.197942	−	0.980214i	$-0.563426\pi$
$131$	−51.8609			−0.395885			−0.197942	+	0.980214i	$0.563426\pi$
$132$	36.1046			0.273519
$133$	−168.820	−	67.3776i	−1.26932	−	0.506599i
$134$	−7.18165			−0.0535944
$135$		0			0
$136$		−	163.094i		−	1.19922i
$137$			11.6713i			0.0851920i	0.999092	+	0.0425960i	$0.0135628\pi$
$137$			11.6713i			0.0851920i	−0.999092	+	0.0425960i	$0.986437\pi$
$138$		−	188.101i		−	1.36305i
$139$	−115.309			−0.829564			−0.414782	−	0.909921i	$-0.636142\pi$
$139$	−115.309			−0.829564			−0.414782	+	0.909921i	$0.636142\pi$
$140$		0			0
$141$			173.190i			1.22830i
$142$			54.0324i			0.380510i
$143$	18.9434			0.132471
$144$	−472.625			−3.28212
$145$		0			0
$146$			49.6645i			0.340168i
$147$	−243.958			−1.65957
$148$	−46.6617			−0.315282
$149$	−149.512			−1.00343			−0.501717	−	0.865032i	$-0.667298\pi$
$149$	−149.512			−1.00343			−0.501717	+	0.865032i	$0.667298\pi$
$150$		0			0
$151$		−	262.436i		−	1.73799i	−0.494823	−	0.868994i	$-0.664767\pi$
$151$		−	262.436i		−	1.73799i	0.494823	−	0.868994i	$-0.335233\pi$
$152$	−41.1116	+	103.008i	−0.270471	+	0.677687i
$153$		−	668.136i		−	4.36690i
$154$		−	93.3711i		−	0.606306i
$155$		0			0
$156$	39.5918			0.253794
$157$		−	106.277i		−	0.676926i	−0.940980	−	0.338463i	$-0.890093\pi$
$157$		−	106.277i		−	0.676926i	0.940980	−	0.338463i	$-0.109907\pi$
$158$			182.467i			1.15485i
$159$	312.059			1.96264
$160$		0			0
$161$	−133.577			−0.829671
$162$	−647.247			−3.99535
$163$		−	173.109i		−	1.06202i	−0.847366	−	0.531009i	$-0.821813\pi$
$163$		−	173.109i		−	1.06202i	0.847366	−	0.531009i	$-0.178187\pi$
$164$			67.0696i			0.408961i
$165$		0			0
$166$			218.772i			1.31791i
$167$	−30.8456			−0.184704			−0.0923522	−	0.995726i	$-0.529439\pi$
$167$	−30.8456			−0.184704			−0.0923522	+	0.995726i	$0.529439\pi$
$168$			320.382i			1.90703i
$169$	−148.227			−0.877082
$170$		0			0
$171$	−168.419	+	421.988i	−0.984909	+	2.46777i
$172$		−	104.869i		−	0.609702i
$173$	14.7393			0.0851981			0.0425990	−	0.999092i	$-0.486436\pi$
$173$	14.7393			0.0851981			0.0425990	+	0.999092i	$0.486436\pi$
$174$		−	196.878i		−	1.13148i
$175$		0			0
$176$	−82.1450			−0.466733
$177$		−	179.759i		−	1.01559i
$178$		−	297.339i		−	1.67045i
$179$		−	166.782i		−	0.931745i	−0.884852	−	0.465873i	$-0.845741\pi$
$179$		−	166.782i		−	0.931745i	0.884852	−	0.465873i	$-0.154259\pi$
$180$		0			0
$181$			200.231i			1.10625i	0.833099	+	0.553124i	$0.186564\pi$
$181$			200.231i			1.10625i	−0.833099	+	0.553124i	$0.813436\pi$
$182$		−	102.390i		−	0.562580i
$183$	573.499			3.13388
$184$			81.5043i			0.442958i
$185$		0			0
$186$		−	389.854i		−	2.09599i
$187$		−	116.126i		−	0.620995i
$188$			45.7091i			0.243134i
$189$			818.524i			4.33082i
$190$		0			0
$191$	264.646			1.38558			0.692792	−	0.721138i	$-0.256381\pi$
$191$	264.646			1.38558			0.692792	+	0.721138i	$0.256381\pi$
$192$	142.875			0.744138
$193$	−170.535			−0.883600			−0.441800	−	0.897113i	$-0.645660\pi$
$193$	−170.535			−0.883600			−0.441800	+	0.897113i	$0.645660\pi$
$194$	181.472			0.935423
$195$		0			0
$196$	−64.3866			−0.328503
$197$		−	193.450i		−	0.981979i	−0.871166	−	0.490989i	$-0.836635\pi$
$197$		−	193.450i		−	0.981979i	0.871166	−	0.490989i	$-0.163365\pi$
$198$	−233.394			−1.17876
$199$	−247.455			−1.24349			−0.621747	−	0.783218i	$-0.713577\pi$
$199$	−247.455			−1.24349			−0.621747	+	0.783218i	$0.713577\pi$
$200$		0			0
$201$	17.5457			0.0872923
$202$	342.329			1.69470
$203$	−139.810			−0.688719
$204$		−	242.704i		−	1.18973i
$205$		0			0
$206$	17.0839			0.0829315
$207$			333.894i			1.61301i
$208$	−90.0792			−0.433073
$209$	−29.2723	+	73.3441i	−0.140059	+	0.350929i
$210$		0			0
$211$			103.882i			0.492331i	0.969228	+	0.246166i	$0.0791707\pi$
$211$			103.882i			0.492331i	−0.969228	+	0.246166i	$0.920829\pi$
$212$	82.3604			0.388492
$213$		−	132.009i		−	0.619759i
$214$	−302.802			−1.41496
$215$		0			0
$216$	499.437			2.31221
$217$	−276.848			−1.27580
$218$		−	174.739i		−	0.801555i
$219$		−	121.337i		−	0.554051i
$220$		0			0
$221$		−	127.342i		−	0.576210i
$222$	415.163			1.87010
$223$	68.3811			0.306642			0.153321	−	0.988176i	$-0.451003\pi$
$223$	68.3811			0.306642			0.153321	+	0.988176i	$0.451003\pi$
$224$			220.618i			0.984900i
$225$		0			0
$226$	−409.993			−1.81413
$227$	−199.560			−0.879121			−0.439560	−	0.898213i	$-0.644866\pi$
$227$	−199.560			−0.879121			−0.439560	+	0.898213i	$0.644866\pi$
$228$	−61.1793	+	153.290i	−0.268330	+	0.672323i
$229$	75.3592			0.329079			0.164540	−	0.986370i	$-0.447386\pi$
$229$	75.3592			0.329079			0.164540	+	0.986370i	$0.447386\pi$
$230$		0			0
$231$			228.118i			0.987525i
$232$			85.3074i			0.367704i
$233$		−	10.3384i		−	0.0443709i	−0.999754	−	0.0221854i	$-0.992938\pi$
$233$		−	10.3384i		−	0.0443709i	0.999754	−	0.0221854i	$-0.00706243\pi$
$234$	−255.937			−1.09375
$235$		0			0
$236$		−	47.4428i		−	0.201029i
$237$		−	445.791i		−	1.88097i
$238$	−627.664			−2.63724
$239$	360.813			1.50968			0.754840	−	0.655909i	$-0.227715\pi$
$239$	360.813			1.50968			0.754840	+	0.655909i	$0.227715\pi$
$240$		0			0
$241$		−	446.555i		−	1.85293i	−0.376384	−	0.926464i	$-0.622833\pi$
$241$		−	446.555i		−	1.85293i	0.376384	−	0.926464i	$-0.377167\pi$
$242$	243.570			1.00649
$243$	811.283			3.33861
$244$	151.361			0.620332
$245$		0			0
$246$		−	596.738i		−	2.42577i
$247$	−32.0997	+	80.4282i	−0.129958	+	0.325620i
$248$			168.924i			0.681144i
$249$		−	534.491i		−	2.14655i
$250$		0			0
$251$	−295.147			−1.17588			−0.587942	−	0.808903i	$-0.700062\pi$
$251$	−295.147			−1.17588			−0.587942	+	0.808903i	$0.700062\pi$
$252$			346.399i			1.37460i
$253$			58.0327i			0.229378i
$254$	190.494			0.749978
$255$		0			0
$256$	254.316			0.993420
$257$	−96.9364			−0.377185			−0.188592	−	0.982055i	$-0.560392\pi$
$257$	−96.9364			−0.377185			−0.188592	+	0.982055i	$0.560392\pi$
$258$			933.048i			3.61646i
$259$		−	294.821i		−	1.13831i
$260$		0			0
$261$			349.474i			1.33898i
$262$	121.781			0.464812
$263$		−	288.418i		−	1.09665i	−0.836267	−	0.548323i	$-0.815267\pi$
$263$		−	288.418i		−	1.09665i	0.836267	−	0.548323i	$-0.184733\pi$
$264$	139.190			0.527236
$265$		0			0
$266$	396.427	+	158.218i	1.49033	+	0.594803i
$267$			726.441i			2.72075i
$268$	4.63077			0.0172790
$269$			174.475i			0.648607i	0.945953	+	0.324304i	$0.105130\pi$
$269$			174.475i			0.648607i	−0.945953	+	0.324304i	$0.894870\pi$
$270$		0			0
$271$	127.454			0.470311			0.235155	−	0.971958i	$-0.424440\pi$
$271$	127.454			0.470311			0.235155	+	0.971958i	$0.424440\pi$
$272$			552.199i			2.03015i
$273$			250.152i			0.916307i
$274$		−	27.4068i		−	0.100025i
$275$		0			0
$276$			121.289i			0.439452i
$277$		−	424.528i		−	1.53259i	−0.642488	−	0.766295i	$-0.722098\pi$
$277$		−	424.528i		−	1.53259i	0.642488	−	0.766295i	$-0.277902\pi$
$278$	270.772			0.974001
$279$			692.020i			2.48036i
$280$		0			0
$281$		−	166.293i		−	0.591791i	−0.955220	−	0.295895i	$-0.904382\pi$
$281$		−	166.293i		−	0.591791i	0.955220	−	0.295895i	$-0.0956180\pi$
$282$		−	406.688i		−	1.44215i
$283$		−	221.482i		−	0.782621i	−0.920259	−	0.391310i	$-0.872022\pi$
$283$		−	221.482i		−	0.782621i	0.920259	−	0.391310i	$-0.127978\pi$
$284$		−	34.8404i		−	0.122677i
$285$		0			0
$286$	−44.4833			−0.155536
$287$	−423.764			−1.47653
$288$	551.464			1.91480
$289$	−491.629			−1.70114
$290$		0			0
$291$	−443.361			−1.52358
$292$		−	32.0240i		−	0.109671i
$293$	−38.0208			−0.129764			−0.0648819	−	0.997893i	$-0.520667\pi$
$293$	−38.0208			−0.129764			−0.0648819	+	0.997893i	$0.520667\pi$
$294$	572.866			1.94853
$295$		0			0
$296$	−179.890			−0.607737
$297$	355.609			1.19734
$298$	351.087			1.17814
$299$			63.6380i			0.212836i
$300$		0			0
$301$	662.588			2.20129
$302$			616.258i			2.04059i
$303$	−836.355			−2.76025
$304$	139.195	−	348.764i	0.457878	−	1.14725i
$305$		0			0
$306$			1568.93i			5.12723i
$307$	285.495			0.929950			0.464975	−	0.885324i	$-0.346063\pi$
$307$	285.495			0.929950			0.464975	+	0.885324i	$0.346063\pi$
$308$			60.2062i			0.195475i
$309$	−41.7382			−0.135075
$310$		0			0
$311$	−4.55195			−0.0146365			−0.00731825	−	0.999973i	$-0.502329\pi$
$311$	−4.55195			−0.0146365			−0.00731825	+	0.999973i	$0.502329\pi$
$312$	152.634			0.489213
$313$			298.019i			0.952136i	0.879408	+	0.476068i	$0.157938\pi$
$313$			298.019i			0.952136i	−0.879408	+	0.476068i	$0.842062\pi$
$314$			249.563i			0.794786i
$315$		0			0
$316$		−	117.655i		−	0.372327i
$317$	179.077			0.564910			0.282455	−	0.959280i	$-0.408851\pi$
$317$	179.077			0.564910			0.282455	+	0.959280i	$0.408851\pi$
$318$	−732.785			−2.30435
$319$			60.7406i			0.190409i
$320$		0			0
$321$	739.786			2.30463
$322$	313.668			0.974126
$323$	493.038	+	196.776i	1.52643	+	0.609213i
$324$	417.349			1.28811
$325$		0			0
$326$			406.499i			1.24693i
$327$			426.911i			1.30554i
$328$			258.567i			0.788313i
$329$	−288.802			−0.877819
$330$		0			0
$331$		−	330.043i		−	0.997109i	−0.866858	−	0.498554i	$-0.833864\pi$
$331$		−	330.043i		−	0.997109i	0.866858	−	0.498554i	$-0.166136\pi$
$332$		−	141.066i		−	0.424896i
$333$	−736.945			−2.21305
$334$	72.4324			0.216863
$335$		0			0
$336$		−	1084.74i		−	3.22840i
$337$	−7.17182			−0.0212814			−0.0106407	−	0.999943i	$-0.503387\pi$
$337$	−7.17182			−0.0212814			−0.0106407	+	0.999943i	$0.503387\pi$
$338$	348.070			1.02979
$339$	1001.67			2.95477
$340$		0			0
$341$			120.277i			0.352719i
$342$	395.486	−	990.923i	1.15639	−	2.89743i
$343$			61.9608i			0.180644i
$344$		−	404.290i		−	1.17526i
$345$		0			0
$346$	−34.6111			−0.100032
$347$		−	38.9167i		−	0.112152i	−0.998427	−	0.0560759i	$-0.982141\pi$
$347$		−	38.9167i		−	0.112152i	0.998427	−	0.0560759i	$-0.0178589\pi$
$348$			126.948i			0.364794i
$349$	−342.266			−0.980704			−0.490352	−	0.871525i	$-0.663132\pi$
$349$	−342.266			−0.980704			−0.490352	+	0.871525i	$0.663132\pi$
$350$		0			0
$351$	389.957			1.11099
$352$	95.8476			0.272294
$353$			17.5325i			0.0496671i	0.999692	+	0.0248335i	$0.00790558\pi$
$353$			17.5325i			0.0496671i	−0.999692	+	0.0248335i	$0.992094\pi$
$354$			422.113i			1.19241i
$355$		0			0
$356$			191.726i			0.538556i
$357$	1533.47			4.29543
$358$			391.642i			1.09397i
$359$	−452.975			−1.26177			−0.630885	−	0.775876i	$-0.717308\pi$
$359$	−452.975			−1.26177			−0.630885	+	0.775876i	$0.717308\pi$
$360$		0			0
$361$	−261.796	−	248.564i	−0.725197	−	0.688542i
$362$		−	470.186i		−	1.29886i
$363$	−595.074			−1.63932
$364$			66.0214i			0.181377i
$365$		0			0
$366$	−1346.70			−3.67952
$367$		−	290.156i		−	0.790615i	−0.918549	−	0.395308i	$-0.870638\pi$
$367$		−	290.156i		−	0.790615i	0.918549	−	0.395308i	$-0.129362\pi$
$368$		−	275.956i		−	0.749880i
$369$			1059.26i			2.87061i
$370$		0			0
$371$			520.375i			1.40263i
$372$			251.380i			0.675752i
$373$	−555.723			−1.48987			−0.744936	−	0.667136i	$-0.767520\pi$
$373$	−555.723			−1.48987			−0.744936	+	0.667136i	$0.767520\pi$
$374$			272.690i			0.729117i
$375$		0			0
$376$			176.218i			0.468664i
$377$			66.6074i			0.176678i
$378$		−	1922.08i		−	5.08486i
$379$			4.05619i			0.0107023i	0.999986	+	0.00535117i	$0.00170334\pi$
$379$			4.05619i			0.0107023i	−0.999986	+	0.00535117i	$0.998297\pi$
$380$		0			0
$381$	−465.404			−1.22153
$382$	−621.449			−1.62683
$383$	392.529			1.02488			0.512439	−	0.858723i	$-0.328742\pi$
$383$	392.529			1.02488			0.512439	+	0.858723i	$0.328742\pi$
$384$	−864.703			−2.25183
$385$		0			0
$386$	400.454			1.03744
$387$		−	1656.23i		−	4.27966i
$388$	−117.014			−0.301583
$389$	−0.819385			−0.00210639			−0.00105319	−	0.999999i	$-0.500335\pi$
$389$	−0.819385			−0.00210639			−0.00105319	+	0.999999i	$0.500335\pi$
$390$		0			0
$391$	390.111			0.997726
$392$	−248.223			−0.633222
$393$	−297.527			−0.757067
$394$			454.263i			1.15295i
$395$		0			0
$396$	150.494			0.380034
$397$			132.366i			0.333414i	0.986006	+	0.166707i	$0.0533135\pi$
$397$			132.366i			0.333414i	−0.986006	+	0.166707i	$0.946687\pi$
$398$	581.080			1.46000
$399$	−968.525	−	386.547i	−2.42738	−	0.968790i
$400$		0			0
$401$			50.7026i			0.126440i	0.998000	+	0.0632202i	$0.0201370\pi$
$401$			50.7026i			0.126440i	−0.998000	+	0.0632202i	$0.979863\pi$
$402$	−41.2013			−0.102491
$403$			131.894i			0.327281i
$404$	−220.735			−0.546375
$405$		0			0
$406$	328.305			0.808632
$407$	−128.085			−0.314706
$408$		−	935.673i		−	2.29332i
$409$			67.8086i			0.165791i	0.996558	+	0.0828956i	$0.0264168\pi$
$409$			67.8086i			0.165791i	−0.996558	+	0.0828956i	$0.973583\pi$
$410$		0			0
$411$			66.9586i			0.162916i
$412$	−11.0158			−0.0267373
$413$	299.757			0.725803
$414$		−	784.057i		−	1.89386i
$415$		0			0
$416$	105.105			0.252657
$417$	−661.533			−1.58641
$418$	68.7379	−	172.228i	0.164445	−	0.412029i
$419$	222.546			0.531135			0.265568	−	0.964092i	$-0.414441\pi$
$419$	222.546			0.531135			0.265568	+	0.964092i	$0.414441\pi$
$420$		0			0
$421$		−	205.736i		−	0.488684i	−0.969689	−	0.244342i	$-0.921428\pi$
$421$		−	205.736i		−	0.488684i	0.969689	−	0.244342i	$-0.0785719\pi$
$422$		−	243.938i		−	0.578051i
$423$			721.901i			1.70662i
$424$	317.516			0.748858
$425$		0			0
$426$			309.985i			0.727665i
$427$			956.339i			2.23967i
$428$	195.248			0.456187
$429$	108.679			0.253331
$430$		0			0
$431$		−	479.116i		−	1.11164i	−0.831303	−	0.555819i	$-0.812405\pi$
$431$		−	479.116i		−	1.11164i	0.831303	−	0.555819i	$-0.187595\pi$
$432$	−1690.98			−3.91431
$433$	566.880			1.30919			0.654596	−	0.755979i	$-0.272839\pi$
$433$	566.880			1.30919			0.654596	+	0.755979i	$0.272839\pi$
$434$	650.101			1.49793
$435$		0			0
$436$			112.673i			0.258424i
$437$	−246.390	−	98.3367i	−0.563822	−	0.225027i
$438$			284.927i			0.650518i
$439$			684.577i			1.55940i	0.626152	+	0.779701i	$0.284629\pi$
$439$			684.577i			1.55940i	−0.626152	+	0.779701i	$0.715371\pi$
$440$		0			0
$441$	−1016.88			−2.30585
$442$			299.028i			0.676534i
$443$		−	473.575i		−	1.06902i	−0.845163	−	0.534509i	$-0.820496\pi$
$443$		−	473.575i		−	1.06902i	0.845163	−	0.534509i	$-0.179504\pi$
$444$	−267.699			−0.602926
$445$		0			0
$446$	−160.574			−0.360031
$447$	−857.752			−1.91891
$448$			238.251i			0.531809i
$449$			315.053i			0.701677i	0.936436	+	0.350839i	$0.114103\pi$
$449$			315.053i			0.701677i	−0.936436	+	0.350839i	$0.885897\pi$
$450$		0			0
$451$			184.105i			0.408215i
$452$	264.366			0.584880
$453$		−	1505.60i		−	3.32363i
$454$	468.612			1.03219
$455$		0			0
$456$	−235.858	+	590.962i	−0.517233	+	1.29597i
$457$			259.302i			0.567401i	0.958913	+	0.283700i	$0.0915621\pi$
$457$			259.302i			0.567401i	−0.958913	+	0.283700i	$0.908438\pi$
$458$	−176.960			−0.386376
$459$		−	2390.50i		−	5.20805i
$460$		0			0
$461$	681.783			1.47892			0.739461	−	0.673200i	$-0.235081\pi$
$461$	681.783			1.47892			0.739461	+	0.673200i	$0.235081\pi$
$462$		−	535.672i		−	1.15946i
$463$			640.879i			1.38419i	0.721808	+	0.692094i	$0.243311\pi$
$463$			640.879i			1.38419i	−0.721808	+	0.692094i	$0.756689\pi$
$464$		−	288.832i		−	0.622483i
$465$		0			0
$466$			24.2769i			0.0520963i
$467$		−	142.309i		−	0.304730i	−0.988324	−	0.152365i	$-0.951311\pi$
$467$		−	142.309i		−	0.304730i	0.988324	−	0.152365i	$-0.0486889\pi$
$468$	165.029			0.352627
$469$			29.2584i			0.0623847i
$470$		0			0
$471$		−	609.716i		−	1.29451i
$472$		−	182.902i		−	0.387503i
$473$		−	287.863i		−	0.608589i
$474$			1046.82i			2.20847i
$475$		0			0
$476$	404.721			0.850255
$477$	1300.75			2.72693
$478$	−847.270			−1.77253
$479$	−303.821			−0.634283			−0.317141	−	0.948378i	$-0.602723\pi$
$479$	−303.821			−0.634283			−0.317141	+	0.948378i	$0.602723\pi$
$480$		0			0
$481$	−140.457			−0.292010
$482$			1048.61i			2.17554i
$483$	−766.335			−1.58661
$484$	−157.055			−0.324494
$485$		0			0
$486$	−1905.07			−3.91990
$487$	−0.541296			−0.00111149			−0.000555746	−	1.00000i	$-0.500177\pi$
$487$	−0.541296			−0.00111149			−0.000555746		1.00000i	$0.500177\pi$
$488$	583.527			1.19575
$489$		−	993.131i		−	2.03094i
$490$		0			0
$491$	463.173			0.943326			0.471663	−	0.881779i	$-0.343654\pi$
$491$	463.173			0.943326			0.471663	+	0.881779i	$0.343654\pi$
$492$			384.780i			0.782073i
$493$	408.314			0.828223
$494$	75.3771	−	188.863i	0.152585	−	0.382315i
$495$		0			0
$496$		−	571.939i		−	1.15310i
$497$	220.131			0.442920
$498$			1255.10i			2.52029i
$499$	−174.865			−0.350431			−0.175215	−	0.984530i	$-0.556062\pi$
$499$	−174.865			−0.350431			−0.175215	+	0.984530i	$0.556062\pi$
$500$		0			0
$501$	−176.962			−0.353218
$502$	693.070			1.38062
$503$		−	503.015i		−	1.00003i	−0.866017	−	0.500015i	$-0.833328\pi$
$503$		−	503.015i		−	1.00003i	0.866017	−	0.500015i	$-0.166672\pi$
$504$			1335.44i			2.64968i
$505$		0			0
$506$		−	136.274i		−	0.269316i
$507$	−850.382			−1.67728
$508$	−122.832			−0.241795
$509$		−	358.496i		−	0.704315i	−0.935941	−	0.352157i	$-0.885448\pi$
$509$		−	358.496i		−	0.704315i	0.935941	−	0.352157i	$-0.114552\pi$
$510$		0			0
$511$	202.336			0.395961
$512$	5.70303			0.0111387
$513$	−602.581	+	1509.81i	−1.17462	+	2.94311i
$514$	227.628			0.442857
$515$		0			0
$516$		−	601.634i		−	1.16596i
$517$			125.471i			0.242690i
$518$			692.305i			1.33650i
$519$	84.5596			0.162928
$520$		0			0
$521$			882.596i			1.69404i	0.531559	+	0.847022i	$0.321607\pi$
$521$			882.596i			1.69404i	−0.531559	+	0.847022i	$0.678393\pi$
$522$		−	820.642i		−	1.57211i
$523$	2.57808			0.00492941			0.00246471	−	0.999997i	$-0.499215\pi$
$523$	2.57808			0.00492941			0.00246471	+	0.999997i	$0.499215\pi$
$524$	−78.5250			−0.149857
$525$		0			0
$526$			677.269i			1.28758i
$527$	808.534			1.53422
$528$	−471.268			−0.892553
$529$	334.046			0.631467
$530$		0			0
$531$		−	749.282i		−	1.41108i
$532$	−255.618	−	102.020i	−0.480485	−	0.191766i
$533$			201.887i			0.378775i
$534$		−	1705.84i		−	3.19446i
$535$		0			0
$536$	17.8525			0.0333070
$537$		−	956.835i		−	1.78182i
$538$		−	409.707i		−	0.761537i
$539$	−176.740			−0.327903
$540$		0			0
$541$	382.112			0.706307			0.353154	−	0.935565i	$-0.385109\pi$
$541$	382.112			0.706307			0.353154	+	0.935565i	$0.385109\pi$
$542$	−299.291			−0.552197
$543$			1148.73i			2.11552i
$544$		−	644.312i		−	1.18440i
$545$		0			0
$546$		−	587.412i		−	1.07585i
$547$	−93.8141			−0.171506			−0.0857532	−	0.996316i	$-0.527330\pi$
$547$	−93.8141			−0.171506			−0.0857532	+	0.996316i	$0.527330\pi$
$548$			17.6721i			0.0322483i
$549$	2390.50			4.35428
$550$		0			0
$551$	−257.887	−	102.925i	−0.468035	−	0.186797i
$552$			467.592i			0.847088i
$553$	743.379			1.34427
$554$			996.885i			1.79943i
$555$		0			0
$556$	−174.595			−0.314020
$557$			799.561i			1.43548i	0.696313	+	0.717739i	$0.254823\pi$
$557$			799.561i			1.43548i	−0.696313	+	0.717739i	$0.745177\pi$
$558$		−	1625.02i		−	2.91221i
$559$		−	315.667i		−	0.564699i
$560$		0			0
$561$		−	666.218i		−	1.18755i
$562$			390.493i			0.694828i
$563$	676.517			1.20163			0.600814	−	0.799389i	$-0.294843\pi$
$563$	676.517			1.20163			0.600814	+	0.799389i	$0.294843\pi$
$564$			262.234i			0.464955i
$565$		0			0
$566$			520.088i			0.918884i
$567$			2636.92i			4.65065i
$568$		−	134.317i		−	0.236473i
$569$			710.658i			1.24896i	0.781041	+	0.624480i	$0.214689\pi$
$569$			710.658i			1.24896i	−0.781041	+	0.624480i	$0.785311\pi$
$570$		0			0
$571$	−812.896			−1.42363			−0.711817	−	0.702364i	$-0.752128\pi$
$571$	−812.896			−1.42363			−0.711817	+	0.702364i	$0.752128\pi$
$572$	28.6831			0.0501452
$573$	1518.28			2.64971
$574$	995.091			1.73361
$575$		0			0
$576$	595.540			1.03392
$577$			812.174i			1.40758i	0.710408	+	0.703790i	$0.248510\pi$
$577$			812.174i			1.40758i	−0.710408	+	0.703790i	$0.751490\pi$
$578$	1154.45			1.99733
$579$	−978.363			−1.68975
$580$		0			0
$581$	891.291			1.53406
$582$	1041.11			1.78885
$583$	226.078			0.387783
$584$		−	123.459i		−	0.211402i
$585$		0			0
$586$	89.2812			0.152357
$587$		−	332.138i		−	0.565823i	−0.959146	−	0.282912i	$-0.908700\pi$
$587$		−	332.138i		−	0.565823i	0.959146	−	0.282912i	$-0.0913003\pi$
$588$	−369.387			−0.628210
$589$	−510.662	−	203.810i	−0.866998	−	0.346027i
$590$		0			0
$591$		−	1109.83i		−	1.87788i
$592$	609.069			1.02883
$593$		−	381.895i		−	0.644005i	−0.946739	−	0.322003i	$-0.895644\pi$
$593$		−	381.895i		−	0.644005i	0.946739	−	0.322003i	$-0.104356\pi$
$594$	−835.049			−1.40581
$595$		0			0
$596$	−226.383			−0.379837
$597$	−1419.66			−2.37798
$598$		−	149.436i		−	0.249893i
$599$			391.286i			0.653232i	0.945157	+	0.326616i	$0.105908\pi$
$599$			391.286i			0.653232i	−0.945157	+	0.326616i	$0.894092\pi$
$600$		0			0
$601$		−	708.847i		−	1.17945i	−0.807606	−	0.589723i	$-0.799237\pi$
$601$		−	708.847i		−	1.17945i	0.807606	−	0.589723i	$-0.200763\pi$
$602$	−1555.90			−2.58456
$603$	73.1354			0.121286
$604$		−	397.367i		−	0.657892i
$605$		0			0
$606$	1963.95			3.24084
$607$	−256.683			−0.422872			−0.211436	−	0.977392i	$-0.567814\pi$
$607$	−256.683			−0.422872			−0.211436	+	0.977392i	$0.567814\pi$
$608$	−162.414	+	406.942i	−0.267129	+	0.669312i
$609$	−802.093			−1.31707
$610$		0			0
$611$			137.590i			0.225188i
$612$		−	1011.66i		−	1.65303i
$613$			730.296i			1.19135i	0.803226	+	0.595674i	$0.203115\pi$
$613$			730.296i			1.19135i	−0.803226	+	0.595674i	$0.796885\pi$
$614$	−670.405			−1.09186
$615$		0			0
$616$			232.107i			0.376797i
$617$			91.8108i			0.148802i	0.997228	+	0.0744009i	$0.0237045\pi$
$617$			91.8108i			0.148802i	−0.997228	+	0.0744009i	$0.976296\pi$
$618$	98.0107			0.158593
$619$	−28.4153			−0.0459052			−0.0229526	−	0.999737i	$-0.507307\pi$
$619$	−28.4153			−0.0459052			−0.0229526	+	0.999737i	$0.507307\pi$
$620$		0			0
$621$			1194.62i			1.92371i
$622$	10.6890			0.0171849
$623$	−1211.38			−1.94442
$624$	−516.787			−0.828184
$625$		0			0
$626$		−	699.814i		−	1.11791i
$627$	−167.936	+	420.777i	−0.267840	+	0.671096i
$628$		−	160.919i		−	0.256241i
$629$			861.023i			1.36888i
$630$		0			0
$631$	430.427			0.682134			0.341067	−	0.940039i	$-0.389212\pi$
$631$	430.427			0.682134			0.341067	+	0.940039i	$0.389212\pi$
$632$		−	453.585i		−	0.717698i
$633$			595.973i			0.941506i
$634$	−420.512			−0.663268
$635$		0			0
$636$	472.504			0.742930
$637$	−193.811			−0.304256
$638$		−	142.632i		−	0.223562i
$639$		−	550.247i		−	0.861107i
$640$		0			0
$641$			340.928i			0.531868i	0.963991	+	0.265934i	$0.0856804\pi$
$641$			340.928i			0.531868i	−0.963991	+	0.265934i	$0.914320\pi$
$642$	−1737.18			−2.70589
$643$		−	245.635i		−	0.382014i	−0.981589	−	0.191007i	$-0.938825\pi$
$643$		−	245.635i		−	0.382014i	0.981589	−	0.191007i	$-0.0611753\pi$
$644$	−202.255			−0.314061
$645$		0			0
$646$	−1157.76	−	462.074i	−1.79220	−	0.715284i
$647$		−	728.400i		−	1.12581i	−0.826521	−	0.562906i	$-0.809683\pi$
$647$		−	728.400i		−	1.12581i	0.826521	−	0.562906i	$-0.190317\pi$
$648$	1608.96			2.48297
$649$		−	130.230i		−	0.200662i
$650$		0			0
$651$	−1588.29			−2.43976
$652$		−	262.112i		−	0.402013i
$653$		−	394.958i		−	0.604836i	−0.953175	−	0.302418i	$-0.902206\pi$
$653$		−	394.958i		−	0.604836i	0.953175	−	0.302418i	$-0.0977938\pi$
$654$		−	1002.48i		−	1.53285i
$655$		0			0
$656$		−	875.450i		−	1.33453i
$657$		−	505.766i		−	0.769812i
$658$	678.172			1.03066
$659$			115.919i			0.175901i	0.996125	+	0.0879504i	$0.0280317\pi$
$659$			115.919i			0.175901i	−0.996125	+	0.0879504i	$0.971968\pi$
$660$		0			0
$661$			1088.21i			1.64631i	0.567818	+	0.823154i	$0.307788\pi$
$661$			1088.21i			1.64631i	−0.567818	+	0.823154i	$0.692212\pi$
$662$			775.014i			1.17072i
$663$		−	730.567i		−	1.10191i
$664$		−	543.836i		−	0.819030i
$665$		0			0
$666$	1730.51			2.59836
$667$	−204.051			−0.305923
$668$	−46.7048			−0.0699174
$669$	392.304			0.586404
$670$		0			0
$671$	415.483			0.619200
$672$			1265.69i			1.88347i
$673$	−1183.83			−1.75903			−0.879517	−	0.475868i	$-0.842134\pi$
$673$	−1183.83			−1.75903			−0.879517	+	0.475868i	$0.842134\pi$
$674$	16.8410			0.0249867
$675$		0			0
$676$	−224.437			−0.332008
$677$	−833.194			−1.23071			−0.615357	−	0.788248i	$-0.710988\pi$
$677$	−833.194			−1.23071			−0.615357	+	0.788248i	$0.710988\pi$
$678$	−2352.14			−3.46923
$679$		−	739.327i		−	1.08885i
$680$		0			0
$681$	−1144.88			−1.68118
$682$		−	282.437i		−	0.414131i
$683$	−979.686			−1.43439			−0.717193	−	0.696874i	$-0.754573\pi$
$683$	−979.686			−1.43439			−0.717193	+	0.696874i	$0.754573\pi$
$684$	−255.012	+	638.952i	−0.372824	+	0.934141i
$685$		0			0
$686$		−	145.498i		−	0.212096i
$687$	432.338			0.629312
$688$			1368.84i			1.98959i
$689$	247.914			0.359817
$690$		0			0
$691$	268.944			0.389211			0.194605	−	0.980882i	$-0.437657\pi$
$691$	268.944			0.389211			0.194605	+	0.980882i	$0.437657\pi$
$692$	22.3174			0.0322506
$693$			950.858i			1.37209i
$694$			91.3850i			0.131679i
$695$		0			0
$696$			489.411i			0.703176i
$697$	1237.60			1.77561
$698$	803.715			1.15145
$699$		−	59.3118i		−	0.0848523i
$700$		0			0
$701$	555.168			0.791966			0.395983	−	0.918258i	$-0.370404\pi$
$701$	555.168			0.791966			0.395983	+	0.918258i	$0.370404\pi$
$702$	−915.705			−1.30442
$703$	217.041	−	543.814i	0.308736	−	0.773562i
$704$	103.508			0.147029
$705$		0			0
$706$		−	41.1702i		−	0.0583147i
$707$		−	1394.66i		−	1.97265i
$708$		−	272.181i		−	0.384436i
$709$	409.398			0.577430			0.288715	−	0.957415i	$-0.406772\pi$
$709$	409.398			0.577430			0.288715	+	0.957415i	$0.406772\pi$
$710$		0			0
$711$		−	1858.18i		−	2.61347i
$712$			739.142i			1.03812i
$713$	−404.056			−0.566698
$714$	−3600.93			−5.04331
$715$		0			0
$716$		−	252.533i		−	0.352700i
$717$	2070.00			2.88702
$718$	1063.69			1.48146
$719$	−530.021			−0.737164			−0.368582	−	0.929595i	$-0.620157\pi$
$719$	−530.021			−0.737164			−0.368582	+	0.929595i	$0.620157\pi$
$720$		0			0
$721$		−	69.6006i		−	0.0965335i
$722$	614.755	+	583.683i	0.851461	+	0.808425i
$723$		−	2561.90i		−	3.54343i
$724$			303.179i			0.418755i
$725$		0			0
$726$	1397.37			1.92475
$727$			60.1205i			0.0826967i	0.999145	+	0.0413483i	$0.0131653\pi$
$727$			60.1205i			0.0826967i	−0.999145	+	0.0413483i	$0.986835\pi$
$728$			254.526i			0.349623i
$729$	2173.65			2.98169
$730$		0			0
$731$	−1935.09			−2.64718
$732$	868.362			1.18629
$733$			625.418i			0.853231i	0.904433	+	0.426616i	$0.140294\pi$
$733$			625.418i			0.853231i	−0.904433	+	0.426616i	$0.859706\pi$
$734$			681.350i			0.928270i
$735$		0			0
$736$			321.988i			0.437484i
$737$	12.7114			0.0172474
$738$		−	2487.37i		−	3.37042i
$739$	110.100			0.148985			0.0744923	−	0.997222i	$-0.476266\pi$
$739$	110.100			0.148985			0.0744923	+	0.997222i	$0.476266\pi$
$740$		0			0
$741$	−184.157	+	461.419i	−0.248524	+	0.622698i
$742$		−	1221.96i		−	1.64684i
$743$	−194.296			−0.261503			−0.130751	−	0.991415i	$-0.541739\pi$
$743$	−194.296			−0.261503			−0.130751	+	0.991415i	$0.541739\pi$
$744$			969.119i			1.30258i
$745$		0			0
$746$	1304.96			1.74928
$747$		−	2227.90i		−	2.98247i
$748$		−	175.832i		−	0.235069i
$749$			1233.63i			1.64704i
$750$		0			0
$751$			62.7172i			0.0835115i	0.999128	+	0.0417558i	$0.0132951\pi$
$751$			62.7172i			0.0835115i	−0.999128	+	0.0417558i	$0.986705\pi$
$752$		−	596.635i		−	0.793397i
$753$	−1693.27			−2.24869
$754$		−	156.409i		−	0.207439i
$755$		0			0
$756$			1239.37i			1.63937i
$757$		−	580.530i		−	0.766882i	−0.923565	−	0.383441i	$-0.874739\pi$
$757$		−	580.530i		−	0.766882i	0.923565	−	0.383441i	$-0.125261\pi$
$758$		−	9.52483i		−	0.0125657i
$759$			332.935i			0.438650i
$760$		0			0
$761$	599.456			0.787721			0.393861	−	0.919170i	$-0.371139\pi$
$761$	599.456			0.787721			0.393861	+	0.919170i	$0.371139\pi$
$762$	1092.87			1.43421
$763$	−711.896			−0.933023
$764$	400.713			0.524494
$765$		0			0
$766$	−921.744			−1.20332
$767$		−	142.808i		−	0.186191i
$768$	1459.02			1.89976
$769$	227.963			0.296441			0.148220	−	0.988954i	$-0.452646\pi$
$769$	227.963			0.296441			0.148220	+	0.988954i	$0.452646\pi$
$770$		0			0
$771$	−556.127			−0.721306
$772$	−258.215			−0.334475
$773$	−984.904			−1.27413			−0.637066	−	0.770809i	$-0.719852\pi$
$773$	−984.904			−1.27413			−0.637066	+	0.770809i	$0.719852\pi$
$774$			3889.19i			5.02480i
$775$		0			0
$776$	−451.113			−0.581331
$777$		−	1691.40i		−	2.17683i
$778$	1.92410			0.00247313
$779$	−781.656	−	311.966i	−1.00341	−	0.400470i
$780$		0			0
$781$		−	95.6363i		−	0.122454i
$782$	−916.067			−1.17144
$783$			1250.37i			1.59689i
$784$	840.429			1.07198
$785$		0			0
$786$	698.660			0.888880
$787$	−567.593			−0.721211			−0.360605	−	0.932718i	$-0.617430\pi$
$787$	−567.593			−0.721211			−0.360605	+	0.932718i	$0.617430\pi$
$788$		−	292.911i		−	0.371715i
$789$		−	1654.66i		−	2.09716i
$790$		0			0
$791$			1670.33i			2.11167i
$792$	580.183			0.732554
$793$	455.614			0.574544
$794$		−	310.824i		−	0.391466i
$795$		0			0
$796$	−374.683			−0.470708
$797$	1174.98			1.47425			0.737124	−	0.675757i	$-0.236183\pi$
$797$	1174.98			1.47425			0.737124	+	0.675757i	$0.236183\pi$
$798$	2274.31	+	907.698i	2.85001	+	1.13747i
$799$	843.446			1.05563
$800$		0			0
$801$			3028.00i			3.78028i
$802$		−	119.061i		−	0.148455i
$803$		−	87.9052i		−	0.109471i
$804$	26.5668			0.0330433
$805$		0			0
$806$		−	309.717i		−	0.384265i
$807$			1000.97i			1.24036i
$808$	−850.979			−1.05319
$809$	944.954			1.16805			0.584026	−	0.811735i	$-0.301477\pi$
$809$	944.954			1.16805			0.584026	+	0.811735i	$0.301477\pi$
$810$		0			0
$811$		−	502.942i		−	0.620151i	−0.950712	−	0.310075i	$-0.899646\pi$
$811$		−	502.942i		−	0.620151i	0.950712	−	0.310075i	$-0.100354\pi$
$812$	−211.693			−0.260705
$813$	731.208			0.899395
$814$	300.773			0.369500
$815$		0			0
$816$			3167.98i			3.88233i
$817$	1222.18	+	487.784i	1.49594	+	0.597043i
$818$		−	159.230i		−	0.194657i
$819$			1042.70i			1.27314i
$820$		0			0
$821$	−1622.74			−1.97654			−0.988268	−	0.152732i	$-0.951193\pi$
$821$	−1622.74			−1.97654			−0.988268	+	0.152732i	$0.951193\pi$
$822$		−	157.234i		−	0.191282i
$823$			525.320i			0.638299i	0.947704	+	0.319150i	$0.103397\pi$
$823$			525.320i			0.638299i	−0.947704	+	0.319150i	$0.896603\pi$
$824$	−42.4680			−0.0515389
$825$		0			0
$826$	−703.895			−0.852173
$827$	1529.88			1.84991			0.924956	−	0.380075i	$-0.124102\pi$
$827$	1529.88			1.84991			0.924956	+	0.380075i	$0.124102\pi$
$828$			505.564i			0.610585i
$829$		−	940.737i		−	1.13479i	−0.823447	−	0.567393i	$-0.807952\pi$
$829$		−	940.737i		−	1.13479i	0.823447	−	0.567393i	$-0.192048\pi$
$830$		0			0
$831$		−	2435.53i		−	2.93084i
$832$	113.506			0.136425
$833$			1188.09i			1.42628i
$834$	1553.43			1.86262
$835$		0			0
$836$	−44.3226	+	111.054i	−0.0530174	+	0.132839i
$837$			2475.95i			2.95812i
$838$	−522.586			−0.623611
$839$		−	584.666i		−	0.696861i	−0.937335	−	0.348430i	$-0.886715\pi$
$839$		−	584.666i		−	0.696861i	0.937335	−	0.348430i	$-0.113285\pi$
$840$		0			0
$841$	627.428			0.746050
$842$			483.113i			0.573769i
$843$		−	954.029i		−	1.13171i
$844$			157.292i			0.186365i
$845$		0			0
$846$		−	1695.18i		−	2.00376i
$847$		−	992.316i		−	1.17157i
$848$	−1075.04			−1.26773
$849$		−	1270.65i		−	1.49664i
$850$		0			0
$851$		−	430.287i		−	0.505625i
$852$		−	199.880i		−	0.234601i
$853$			1035.83i			1.21433i	0.794574	+	0.607167i	$0.207694\pi$
$853$			1035.83i			1.21433i	−0.794574	+	0.607167i	$0.792306\pi$
$854$		−	2245.70i		−	2.62962i
$855$		0			0
$856$	752.721			0.879347
$857$	−488.572			−0.570096			−0.285048	−	0.958513i	$-0.592010\pi$
$857$	−488.572			−0.570096			−0.285048	+	0.958513i	$0.592010\pi$
$858$	−255.202			−0.297438
$859$	−55.9552			−0.0651399			−0.0325699	−	0.999469i	$-0.510369\pi$
$859$	−55.9552			−0.0651399			−0.0325699	+	0.999469i	$0.510369\pi$
$860$		0			0
$861$	−2431.14			−2.82363
$862$			1125.07i			1.30519i
$863$	−740.616			−0.858188			−0.429094	−	0.903260i	$-0.641167\pi$
$863$	−740.616			−0.858188			−0.429094	+	0.903260i	$0.641167\pi$
$864$	1973.06			2.28363
$865$		0			0
$866$	−1331.16			−1.53714
$867$	−2820.49			−3.25316
$868$	−419.189			−0.482936
$869$		−	322.962i		−	0.371648i
$870$		0			0
$871$	13.9391			0.0160036
$872$			434.376i			0.498137i
$873$	−1848.05			−2.11689
$874$	578.579	+	230.916i	0.661990	+	0.264206i
$875$		0			0
$876$		−	183.722i		−	0.209729i
$877$	−1241.85			−1.41602			−0.708009	−	0.706204i	$-0.750406\pi$
$877$	−1241.85			−1.41602			−0.708009	+	0.706204i	$0.750406\pi$
$878$		−	1607.54i		−	1.83091i
$879$	−218.126			−0.248153
$880$		0			0
$881$	−812.952			−0.922761			−0.461380	−	0.887202i	$-0.652646\pi$
$881$	−812.952			−0.922761			−0.461380	+	0.887202i	$0.652646\pi$
$882$	2387.86			2.70733
$883$		−	512.623i		−	0.580547i	−0.956944	−	0.290274i	$-0.906254\pi$
$883$		−	512.623i		−	0.580547i	0.956944	−	0.290274i	$-0.0937463\pi$
$884$		−	192.815i		−	0.218117i
$885$		0			0
$886$			1112.06i			1.25515i
$887$	600.453			0.676948			0.338474	−	0.940976i	$-0.390089\pi$
$887$	600.453			0.676948			0.338474	+	0.940976i	$0.390089\pi$
$888$	−1032.03			−1.16220
$889$		−	776.085i		−	0.872986i
$890$		0			0
$891$	1145.61			1.28576
$892$	103.539			0.116075
$893$	−532.713	−	212.610i	−0.596543	−	0.238086i
$894$	2014.19			2.25301
$895$		0			0
$896$		−	1441.94i		−	1.60930i
$897$			365.093i			0.407015i
$898$		−	739.815i		−	0.823847i
$899$	−422.910			−0.470422
$900$		0			0
$901$		−	1519.75i		−	1.68674i
$902$		−	432.319i		−	0.479289i
$903$	3801.29			4.20962
$904$	1019.18			1.12741
$905$		0			0
$906$			3535.49i			3.90231i
$907$	139.480			0.153782			0.0768910	−	0.997040i	$-0.475501\pi$
$907$	139.480			0.153782			0.0768910	+	0.997040i	$0.475501\pi$
$908$	−302.164			−0.332779
$909$	−3486.15			−3.83515
$910$		0			0
$911$			897.235i			0.984890i	0.870344	+	0.492445i	$0.163897\pi$
$911$			897.235i			0.984890i	−0.870344	+	0.492445i	$0.836103\pi$
$912$	798.565	−	2000.87i	0.875619	−	2.19393i
$913$		−	387.223i		−	0.424121i
$914$		−	608.899i		−	0.666191i
$915$		0			0
$916$	114.105			0.124569
$917$		−	496.142i		−	0.541049i
$918$			5613.42i			6.11483i
$919$	79.2393			0.0862234			0.0431117	−	0.999070i	$-0.486273\pi$
$919$	79.2393			0.0862234			0.0431117	+	0.999070i	$0.486273\pi$
$920$		0			0
$921$	1637.89			1.77838
$922$	−1600.98			−1.73642
$923$		−	104.874i		−	0.113623i
$924$			345.404i			0.373814i
$925$		0			0
$926$		−	1504.93i		−	1.62519i
$927$	−173.976			−0.187677
$928$			337.013i			0.363160i
$929$	535.116			0.576013			0.288007	−	0.957628i	$-0.407007\pi$
$929$	535.116			0.576013			0.288007	+	0.957628i	$0.407007\pi$
$930$		0			0
$931$	299.486	−	750.387i	0.321682	−	0.806001i
$932$		−	15.6539i		−	0.0167960i
$933$	−26.1147			−0.0279900
$934$			334.173i			0.357787i
$935$		0			0
$936$	636.221			0.679723
$937$			1523.13i			1.62554i	0.582587	+	0.812768i	$0.302041\pi$
$937$			1523.13i			1.62554i	−0.582587	+	0.812768i	$0.697959\pi$
$938$		−	68.7053i		−	0.0732466i
$939$			1709.74i			1.82081i
$940$		0			0
$941$		−	283.647i		−	0.301432i	−0.988577	−	0.150716i	$-0.951842\pi$
$941$		−	283.647i		−	0.301432i	0.988577	−	0.150716i	$-0.0481578\pi$
$942$			1431.75i			1.51990i
$943$	−618.477			−0.655861
$944$			619.265i			0.656001i
$945$		0			0
$946$			675.965i			0.714551i
$947$		−	1604.67i		−	1.69448i	−0.531211	−	0.847239i	$-0.678263\pi$
$947$		−	1604.67i		−	1.69448i	0.531211	−	0.847239i	$-0.321737\pi$
$948$		−	674.993i		−	0.712018i
$949$		−	96.3958i		−	0.101576i
$950$		0			0
$951$	1027.37			1.08030
$952$	1560.28			1.63895
$953$	−490.181			−0.514356			−0.257178	−	0.966364i	$-0.582793\pi$
$953$	−490.181			−0.514356			−0.257178	+	0.966364i	$0.582793\pi$
$954$	−3054.44			−3.20172
$955$		0			0
$956$	546.324			0.571469
$957$			348.471i			0.364128i
$958$	713.440			0.744718
$959$	−111.657			−0.116431
$960$		0			0
$961$	123.564			0.128579
$962$	329.824			0.342853
$963$	3083.63			3.20211
$964$		−	676.150i		−	0.701401i
$965$		0			0
$966$	1799.52			1.86286
$967$			969.777i			1.00287i	0.865195	+	0.501436i	$0.167195\pi$
$967$			969.777i			1.00287i	−0.865195	+	0.501436i	$0.832805\pi$
$968$	−605.479			−0.625494
$969$	2828.57	+	1128.91i	2.91906	+	1.16502i
$970$		0			0
$971$			521.725i			0.537307i	0.963237	+	0.268653i	$0.0865786\pi$
$971$			521.725i			0.537307i	−0.963237	+	0.268653i	$0.913421\pi$
$972$	1228.40			1.26379
$973$		−	1103.14i		−	1.13375i
$974$	1.27108			0.00130501
$975$		0			0
$976$	−1975.69			−2.02428
$977$	631.913			0.646790			0.323395	−	0.946264i	$-0.395176\pi$
$977$	631.913			0.646790			0.323395	+	0.946264i	$0.395176\pi$
$978$			2332.09i			2.38455i
$979$			526.284i			0.537573i
$980$		0			0
$981$			1779.48i			1.81395i
$982$	−1087.63			−1.10757
$983$	903.596			0.919223			0.459611	−	0.888120i	$-0.347989\pi$
$983$	903.596			0.919223			0.459611	+	0.888120i	$0.347989\pi$
$984$			1483.40i			1.50752i
$985$		0			0
$986$	−958.812			−0.972426
$987$	−1656.87			−1.67869
$988$	−48.6036	+	121.780i	−0.0491939	+	0.123259i
$989$	967.038			0.977794
$990$		0			0
$991$			691.185i			0.697462i	0.937223	+	0.348731i	$0.113387\pi$
$991$			691.185i			0.697462i	−0.937223	+	0.348731i	$0.886613\pi$
$992$			667.344i			0.672726i
$993$		−	1893.47i		−	1.90681i
$994$	−516.917			−0.520037
$995$		0			0
$996$		−	809.297i		−	0.812547i
$997$			285.575i			0.286435i	0.989691	+	0.143217i	$0.0457448\pi$
$997$			285.575i			0.286435i	−0.989691	+	0.143217i	$0.954255\pi$
$998$	410.622			0.411445
$999$	−2636.69			−2.63932

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.3.d.d.474.7		28
5.2	odd	4		475.3.c.h.151.4	✓	14
5.3	odd	4		475.3.c.i.151.11	yes	14
5.4	even	2	inner	475.3.d.d.474.22		28
19.18	odd	2	inner	475.3.d.d.474.21		28
95.18	even	4		475.3.c.i.151.4	yes	14
95.37	even	4		475.3.c.h.151.11	yes	14
95.94	odd	2	inner	475.3.d.d.474.8		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.3.c.h.151.4	✓	14	5.2	odd	4
475.3.c.h.151.11	yes	14	95.37	even	4
475.3.c.i.151.4	yes	14	95.18	even	4
475.3.c.i.151.11	yes	14	5.3	odd	4
475.3.d.d.474.7		28	1.1	even	1	trivial
475.3.d.d.474.8		28	95.94	odd	2	inner
475.3.d.d.474.21		28	19.18	odd	2	inner
475.3.d.d.474.22		28	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.34822 q^{2} +5.73703 q^{3} +1.51415 q^{4} -13.4718 q^{6} +9.56678i q^{7} +5.83734 q^{8} +23.9135 q^{9} +O(q^{10})\)	\(q-2.34822 q^{2} +5.73703 q^{3} +1.51415 q^{4} -13.4718 q^{6} +9.56678i q^{7} +5.83734 q^{8} +23.9135 q^{9} +4.15630 q^{11} +8.68670 q^{12} +4.55775 q^{13} -22.4649i q^{14} -19.7639 q^{16} -27.9397i q^{17} -56.1541 q^{18} +(-7.04287 + 17.6465i) q^{19} +54.8849i q^{21} -9.75992 q^{22} +13.9626i q^{23} +33.4890 q^{24} -10.7026 q^{26} +85.5590 q^{27} +14.4855i q^{28} +14.6141i q^{29} +28.9385i q^{31} +23.0608 q^{32} +23.8448 q^{33} +65.6087i q^{34} +36.2085 q^{36} -30.8172 q^{37} +(16.5382 - 41.4378i) q^{38} +26.1479 q^{39} +44.2953i q^{41} -128.882i q^{42} -69.2593i q^{43} +6.29325 q^{44} -32.7872i q^{46} +30.1880i q^{47} -113.386 q^{48} -42.5233 q^{49} -160.291i q^{51} +6.90110 q^{52} +54.3939 q^{53} -200.912 q^{54} +55.8445i q^{56} +(-40.4051 + 101.238i) q^{57} -34.3171i q^{58} -31.3331i q^{59} +99.9646 q^{61} -67.9540i q^{62} +228.775i q^{63} +24.9039 q^{64} -55.9929 q^{66} +3.05833 q^{67} -42.3049i q^{68} +80.1037i q^{69} -23.0099i q^{71} +139.591 q^{72} -21.1499i q^{73} +72.3655 q^{74} +(-10.6639 + 26.7193i) q^{76} +39.7625i q^{77} -61.4012 q^{78} -77.7042i q^{79} +275.633 q^{81} -104.015i q^{82} -93.1651i q^{83} +83.1038i q^{84} +162.636i q^{86} +83.8415i q^{87} +24.2617 q^{88} +126.623i q^{89} +43.6030i q^{91} +21.1414i q^{92} +166.021i q^{93} -70.8882i q^{94} +132.300 q^{96} -77.2807 q^{97} +99.8542 q^{98} +99.3917 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 56 q^{4} - 8 q^{6} + 72 q^{9}+O(q^{10}) \) 28 * q + 56 * q^4 - 8 * q^6 + 72 * q^9	\( 28 q + 56 q^{4} - 8 q^{6} + 72 q^{9} - 8 q^{11} + 72 q^{16} - 78 q^{19} + 88 q^{24} + 60 q^{26} + 8 q^{36} + 64 q^{39} + 104 q^{44} - 468 q^{49} - 196 q^{54} + 444 q^{61} + 436 q^{64} + 184 q^{66} + 184 q^{74} - 702 q^{76} + 804 q^{81} + 380 q^{96} + 360 q^{99}+O(q^{100}) \) 28 * q + 56 * q^4 - 8 * q^6 + 72 * q^9 - 8 * q^11 + 72 * q^16 - 78 * q^19 + 88 * q^24 + 60 * q^26 + 8 * q^36 + 64 * q^39 + 104 * q^44 - 468 * q^49 - 196 * q^54 + 444 * q^61 + 436 * q^64 + 184 * q^66 + 184 * q^74 - 702 * q^76 + 804 * q^81 + 380 * q^96 + 360 * q^99

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