LMFDB - Embedded newform 630.3.f.c.181.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [630,3,Mod(181,630)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("630.181"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(630, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	630.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,16,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$17.1662566547$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3317760000.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 7x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.4
Root			$1.72286 - 0.178197i$ of defining polynomial
Character	$\chi$	$=$	630.181
Dual form			630.3.f.c.181.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +(6.70141 - 2.02265i) q^{7} -2.82843 q^{8} -3.16228i q^{10} -11.9625 q^{11} +9.67294i q^{13} +(-9.47723 + 2.86045i) q^{14} +4.00000 q^{16} +7.09185i q^{17} +17.5508i q^{19} +4.47214i q^{20} +16.9175 q^{22} +2.84923 q^{23} -5.00000 q^{25} -13.6796i q^{26} +(13.4028 - 4.04529i) q^{28} +13.6929 q^{29} -19.9468i q^{31} -5.65685 q^{32} -10.0294i q^{34} +(4.52277 + 14.9848i) q^{35} -11.0672 q^{37} -24.8206i q^{38} -6.32456i q^{40} +28.2902i q^{41} -72.9666 q^{43} -23.9250 q^{44} -4.02942 q^{46} +28.3312i q^{47} +(40.8178 - 27.1092i) q^{49} +7.07107 q^{50} +19.3459i q^{52} +11.7110 q^{53} -26.7490i q^{55} +(-18.9545 + 5.72091i) q^{56} -19.3647 q^{58} +101.160i q^{59} +76.7055i q^{61} +28.2090i q^{62} +8.00000 q^{64} -21.6294 q^{65} -76.2407 q^{67} +14.1837i q^{68} +(-6.39617 - 21.1917i) q^{70} +95.4493 q^{71} +120.684i q^{73} +15.6513 q^{74} +35.1016i q^{76} +(-80.1657 + 24.1959i) q^{77} -14.8096 q^{79} +8.94427i q^{80} -40.0083i q^{82} -60.9570i q^{83} -15.8579 q^{85} +103.190 q^{86} +33.8351 q^{88} +88.6302i q^{89} +(19.5649 + 64.8223i) q^{91} +5.69845 q^{92} -40.0664i q^{94} -39.2448 q^{95} -18.8547i q^{97} +(-57.7251 + 38.3381i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} + 16 q^{11} - 32 q^{14} + 32 q^{16} + 96 q^{22} - 40 q^{25} + 144 q^{29} + 80 q^{35} - 48 q^{37} - 64 q^{43} + 32 q^{44} + 128 q^{46} - 24 q^{49} - 128 q^{53} - 64 q^{56} + 224 q^{58} + 64 q^{64}+ \cdots - 80 q^{95}+O(q^{100}) $ 8 * q + 16 * q^4 + 16 * q^11 - 32 * q^14 + 32 * q^16 + 96 * q^22 - 40 * q^25 + 144 * q^29 + 80 * q^35 - 48 * q^37 - 64 * q^43 + 32 * q^44 + 128 * q^46 - 24 * q^49 - 128 * q^53 - 64 * q^56 + 224 * q^58 + 64 * q^64 - 80 * q^65 - 192 * q^67 - 176 * q^71 + 160 * q^74 - 192 * q^77 - 288 * q^79 - 240 * q^85 + 64 * q^86 + 192 * q^88 + 64 * q^91 - 80 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/630\mathbb{Z}\right)^\times$.

$n$	$127$	$281$	$451$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$	6.70141	−	2.02265i	0.957344	−	0.288949i
$7$	6.70141	−	2.02265i	0.957344	−	0.288949i
$8$	−2.82843			−0.353553
$9$		0			0
$10$		−	3.16228i		−	0.316228i
$11$	−11.9625			−1.08750			−0.543750	−	0.839247i	$-0.682996\pi$
$11$	−11.9625			−1.08750			−0.543750	+	0.839247i	$0.682996\pi$
$12$		0			0
$13$			9.67294i			0.744072i	0.928218	+	0.372036i	$0.121340\pi$
$13$			9.67294i			0.744072i	−0.928218	+	0.372036i	$0.878660\pi$
$14$	−9.47723	+	2.86045i	−0.676945	+	0.204318i
$15$		0			0
$16$	4.00000			0.250000
$17$			7.09185i			0.417168i	0.978004	+	0.208584i	$0.0668854\pi$
$17$			7.09185i			0.417168i	−0.978004	+	0.208584i	$0.933115\pi$
$18$		0			0
$19$			17.5508i			0.923727i	0.886951	+	0.461864i	$0.152819\pi$
$19$			17.5508i			0.923727i	−0.886951	+	0.461864i	$0.847181\pi$
$20$			4.47214i			0.223607i
$21$		0			0
$22$	16.9175			0.768979
$23$	2.84923			0.123879			0.0619397	−	0.998080i	$-0.480271\pi$
$23$	2.84923			0.123879			0.0619397	+	0.998080i	$0.480271\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		−	13.6796i		−	0.526139i
$27$		0			0
$28$	13.4028	−	4.04529i	0.478672	−	0.144475i
$29$	13.6929			0.472170			0.236085	−	0.971732i	$-0.424136\pi$
$29$	13.6929			0.472170			0.236085	+	0.971732i	$0.424136\pi$
$30$		0			0
$31$		−	19.9468i		−	0.643445i	−0.946834	−	0.321723i	$-0.895738\pi$
$31$		−	19.9468i		−	0.643445i	0.946834	−	0.321723i	$-0.104262\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$		−	10.0294i		−	0.294982i
$35$	4.52277	+	14.9848i	0.129222	+	0.428137i
$36$		0			0
$37$	−11.0672			−0.299113			−0.149556	−	0.988753i	$-0.547785\pi$
$37$	−11.0672			−0.299113			−0.149556	+	0.988753i	$0.547785\pi$
$38$		−	24.8206i		−	0.653174i
$39$		0			0
$40$		−	6.32456i		−	0.158114i
$41$			28.2902i			0.690004i	0.938602	+	0.345002i	$0.112122\pi$
$41$			28.2902i			0.690004i	−0.938602	+	0.345002i	$0.887878\pi$
$42$		0			0
$43$	−72.9666			−1.69690			−0.848449	−	0.529278i	$-0.822463\pi$
$43$	−72.9666			−1.69690			−0.848449	+	0.529278i	$0.822463\pi$
$44$	−23.9250			−0.543750
$45$		0			0
$46$	−4.02942			−0.0875960
$47$			28.3312i			0.602792i	0.953499	+	0.301396i	$0.0974526\pi$
$47$			28.3312i			0.602792i	−0.953499	+	0.301396i	$0.902547\pi$
$48$		0			0
$49$	40.8178	−	27.1092i	0.833016	−	0.553248i
$50$	7.07107			0.141421
$51$		0			0
$52$			19.3459i			0.372036i
$53$	11.7110			0.220962			0.110481	−	0.993878i	$-0.464761\pi$
$53$	11.7110			0.220962			0.110481	+	0.993878i	$0.464761\pi$
$54$		0			0
$55$		−	26.7490i		−	0.486345i
$56$	−18.9545	+	5.72091i	−0.338472	+	0.102159i
$57$		0			0
$58$	−19.3647			−0.333875
$59$			101.160i			1.71458i	0.514833	+	0.857291i	$0.327854\pi$
$59$			101.160i			1.71458i	−0.514833	+	0.857291i	$0.672146\pi$
$60$		0			0
$61$			76.7055i			1.25747i	0.777621	+	0.628733i	$0.216426\pi$
$61$			76.7055i			1.25747i	−0.777621	+	0.628733i	$0.783574\pi$
$62$			28.2090i			0.454984i
$63$		0			0
$64$	8.00000			0.125000
$65$	−21.6294			−0.332759
$66$		0			0
$67$	−76.2407			−1.13792			−0.568960	−	0.822365i	$-0.692654\pi$
$67$	−76.2407			−1.13792			−0.568960	+	0.822365i	$0.692654\pi$
$68$			14.1837i			0.208584i
$69$		0			0
$70$	−6.39617	−	21.1917i	−0.0913738	−	0.302739i
$71$	95.4493			1.34436			0.672178	−	0.740389i	$-0.265359\pi$
$71$	95.4493			1.34436			0.672178	+	0.740389i	$0.265359\pi$
$72$		0			0
$73$			120.684i			1.65321i	0.562782	+	0.826605i	$0.309731\pi$
$73$			120.684i			1.65321i	−0.562782	+	0.826605i	$0.690269\pi$
$74$	15.6513			0.211505
$75$		0			0
$76$			35.1016i			0.461864i
$77$	−80.1657	+	24.1959i	−1.04111	+	0.314233i
$78$		0			0
$79$	−14.8096			−0.187463			−0.0937317	−	0.995597i	$-0.529880\pi$
$79$	−14.8096			−0.187463			−0.0937317	+	0.995597i	$0.529880\pi$
$80$			8.94427i			0.111803i
$81$		0			0
$82$		−	40.0083i		−	0.487906i
$83$		−	60.9570i		−	0.734422i	−0.930138	−	0.367211i	$-0.880313\pi$
$83$		−	60.9570i		−	0.734422i	0.930138	−	0.367211i	$-0.119687\pi$
$84$		0			0
$85$	−15.8579			−0.186563
$86$	103.190			1.19989
$87$		0			0
$88$	33.8351			0.384490
$89$			88.6302i			0.995845i	0.867222	+	0.497922i	$0.165904\pi$
$89$			88.6302i			0.995845i	−0.867222	+	0.497922i	$0.834096\pi$
$90$		0			0
$91$	19.5649	+	64.8223i	0.214999	+	0.712333i
$92$	5.69845			0.0619397
$93$		0			0
$94$		−	40.0664i		−	0.426238i
$95$	−39.2448			−0.413103
$96$		0			0
$97$		−	18.8547i		−	0.194378i	−0.995266	−	0.0971891i	$-0.969015\pi$
$97$		−	18.8547i		−	0.194378i	0.995266	−	0.0971891i	$-0.0309852\pi$
$98$	−57.7251	+	38.3381i	−0.589032	+	0.391206i
$99$		0			0
$100$	−10.0000			−0.100000
$101$		−	30.1594i		−	0.298608i	−0.988791	−	0.149304i	$-0.952297\pi$
$101$		−	30.1594i		−	0.298608i	0.988791	−	0.149304i	$-0.0477033\pi$
$102$		0			0
$103$			150.869i			1.46475i	0.680903	+	0.732374i	$0.261588\pi$
$103$			150.869i			1.46475i	−0.680903	+	0.732374i	$0.738412\pi$
$104$		−	27.3592i		−	0.263069i
$105$		0			0
$106$	−16.5618			−0.156243
$107$	−88.2689			−0.824943			−0.412472	−	0.910970i	$-0.635334\pi$
$107$	−88.2689			−0.824943			−0.412472	+	0.910970i	$0.635334\pi$
$108$		0			0
$109$	92.7216			0.850657			0.425328	−	0.905039i	$-0.360159\pi$
$109$	92.7216			0.850657			0.425328	+	0.905039i	$0.360159\pi$
$110$			37.8288i			0.343898i
$111$		0			0
$112$	26.8056	−	8.09058i	0.239336	−	0.0722374i
$113$	63.4942			0.561896			0.280948	−	0.959723i	$-0.409351\pi$
$113$	63.4942			0.561896			0.280948	+	0.959723i	$0.409351\pi$
$114$		0			0
$115$			6.37107i			0.0554006i
$116$	27.3859			0.236085
$117$		0			0
$118$		−	143.062i		−	1.21239i
$119$	14.3443	+	47.5254i	0.120540	+	0.399373i
$120$		0			0
$121$	22.1016			0.182658
$122$		−	108.478i		−	0.889163i
$123$		0			0
$124$		−	39.8936i		−	0.321723i
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−252.418			−1.98755			−0.993773	−	0.111423i	$-0.964459\pi$
$127$	−252.418			−1.98755			−0.993773	+	0.111423i	$0.964459\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$	30.5885			0.235296
$131$			206.558i			1.57678i	0.615174	+	0.788391i	$0.289086\pi$
$131$			206.558i			1.57678i	−0.615174	+	0.788391i	$0.710914\pi$
$132$		0			0
$133$	35.4991	+	117.615i	0.266910	+	0.884325i
$134$	107.821			0.804631
$135$		0			0
$136$		−	20.0588i		−	0.147491i
$137$	−179.062			−1.30702			−0.653512	−	0.756916i	$-0.726705\pi$
$137$	−179.062			−1.30702			−0.653512	+	0.756916i	$0.726705\pi$
$138$		0			0
$139$		−	29.8412i		−	0.214685i	−0.994222	−	0.107342i	$-0.965766\pi$
$139$		−	29.8412i		−	0.214685i	0.994222	−	0.107342i	$-0.0342341\pi$
$140$	9.04555	+	29.9696i	0.0646111	+	0.214069i
$141$		0			0
$142$	−134.986			−0.950604
$143$		−	115.713i		−	0.809179i
$144$		0			0
$145$			30.6184i			0.211161i
$146$		−	170.674i		−	1.16900i
$147$		0			0
$148$	−22.1343			−0.149556
$149$	−233.638			−1.56804			−0.784020	−	0.620736i	$-0.786834\pi$
$149$	−233.638			−1.56804			−0.784020	+	0.620736i	$0.786834\pi$
$150$		0			0
$151$	280.637			1.85852			0.929262	−	0.369421i	$-0.120444\pi$
$151$	280.637			1.85852			0.929262	+	0.369421i	$0.120444\pi$
$152$		−	49.6412i		−	0.326587i
$153$		0			0
$154$	113.371	−	34.2182i	0.736178	−	0.222196i
$155$	44.6024			0.287757
$156$		0			0
$157$			54.3182i			0.345976i	0.984924	+	0.172988i	$0.0553422\pi$
$157$			54.3182i			0.345976i	−0.984924	+	0.172988i	$0.944658\pi$
$158$	20.9440			0.132557
$159$		0			0
$160$		−	12.6491i		−	0.0790569i
$161$	19.0938	−	5.76298i	0.118595	−	0.0357949i
$162$		0			0
$163$	200.523			1.23020			0.615101	−	0.788448i	$-0.289115\pi$
$163$	200.523			1.23020			0.615101	+	0.788448i	$0.289115\pi$
$164$			56.5803i			0.345002i
$165$		0			0
$166$			86.2062i			0.519314i
$167$		−	282.815i		−	1.69351i	−0.531987	−	0.846753i	$-0.678554\pi$
$167$		−	282.815i		−	1.69351i	0.531987	−	0.846753i	$-0.321446\pi$
$168$		0			0
$169$	75.4342			0.446356
$170$	22.4264			0.131920
$171$		0			0
$172$	−145.933			−0.848449
$173$		−	127.279i		−	0.735714i	−0.929882	−	0.367857i	$-0.880092\pi$
$173$		−	127.279i		−	0.735714i	0.929882	−	0.367857i	$-0.119908\pi$
$174$		0			0
$175$	−33.5071	+	10.1132i	−0.191469	+	0.0577899i
$176$	−47.8500			−0.271875
$177$		0			0
$178$		−	125.342i		−	0.704168i
$179$	−122.342			−0.683472			−0.341736	−	0.939796i	$-0.611015\pi$
$179$	−122.342			−0.683472			−0.341736	+	0.939796i	$0.611015\pi$
$180$		0			0
$181$		−	97.3605i		−	0.537903i	−0.963154	−	0.268952i	$-0.913323\pi$
$181$		−	97.3605i		−	0.537903i	0.963154	−	0.268952i	$-0.0866772\pi$
$182$	−27.6690	−	91.6726i	−0.152027	−	0.503696i
$183$		0			0
$184$	−8.05883			−0.0437980
$185$		−	24.7469i		−	0.133767i
$186$		0			0
$187$		−	84.8363i		−	0.453670i
$188$			56.6624i			0.301396i
$189$		0			0
$190$	55.5005			0.292108
$191$	290.175			1.51924			0.759621	−	0.650367i	$-0.225385\pi$
$191$	290.175			1.51924			0.759621	+	0.650367i	$0.225385\pi$
$192$		0			0
$193$	90.7789			0.470357			0.235179	−	0.971952i	$-0.424433\pi$
$193$	90.7789			0.470357			0.235179	+	0.971952i	$0.424433\pi$
$194$			26.6645i			0.137446i
$195$		0			0
$196$	81.6356	−	54.2183i	0.416508	−	0.276624i
$197$	53.3711			0.270919			0.135460	−	0.990783i	$-0.456749\pi$
$197$	53.3711			0.270919			0.135460	+	0.990783i	$0.456749\pi$
$198$		0			0
$199$		−	301.563i		−	1.51539i	−0.652607	−	0.757696i	$-0.726325\pi$
$199$		−	301.563i		−	1.51539i	0.652607	−	0.757696i	$-0.273675\pi$
$200$	14.1421			0.0707107
$201$		0			0
$202$			42.6518i			0.211148i
$203$	91.7620	−	27.6960i	0.452030	−	0.136433i
$204$		0			0
$205$	−63.2587			−0.308579
$206$		−	213.361i		−	1.03573i
$207$		0			0
$208$			38.6918i			0.186018i
$209$		−	209.952i		−	1.00455i
$210$		0			0
$211$	−74.7911			−0.354460			−0.177230	−	0.984169i	$-0.556714\pi$
$211$	−74.7911			−0.354460			−0.177230	+	0.984169i	$0.556714\pi$
$212$	23.4219			0.110481
$213$		0			0
$214$	124.831			0.583323
$215$		−	163.158i		−	0.758876i
$216$		0			0
$217$	−40.3453	−	133.672i	−0.185923	−	0.615999i
$218$	−131.128			−0.601505
$219$		0			0
$220$		−	53.4980i		−	0.243173i
$221$	−68.5991			−0.310403
$222$		0			0
$223$		−	413.129i		−	1.85260i	−0.376789	−	0.926299i	$-0.622972\pi$
$223$		−	413.129i		−	1.85260i	0.376789	−	0.926299i	$-0.377028\pi$
$224$	−37.9089	+	11.4418i	−0.169236	+	0.0510795i
$225$		0			0
$226$	−89.7944			−0.397320
$227$		−	298.691i		−	1.31582i	−0.753097	−	0.657910i	$-0.771441\pi$
$227$		−	298.691i		−	1.31582i	0.753097	−	0.657910i	$-0.228559\pi$
$228$		0			0
$229$			323.323i			1.41189i	0.708265	+	0.705946i	$0.249478\pi$
$229$			323.323i			1.41189i	−0.708265	+	0.705946i	$0.750522\pi$
$230$		−	9.01005i		−	0.0391741i
$231$		0			0
$232$	−38.7295			−0.166937
$233$	323.951			1.39035			0.695174	−	0.718841i	$-0.255327\pi$
$233$	323.951			1.39035			0.695174	+	0.718841i	$0.255327\pi$
$234$		0			0
$235$	−63.3505			−0.269577
$236$			202.321i			0.857291i
$237$		0			0
$238$	−20.2859	−	67.2111i	−0.0852349	−	0.282400i
$239$	362.459			1.51657			0.758283	−	0.651926i	$-0.226039\pi$
$239$	362.459			1.51657			0.758283	+	0.651926i	$0.226039\pi$
$240$		0			0
$241$		−	124.863i		−	0.518102i	−0.965864	−	0.259051i	$-0.916590\pi$
$241$		−	124.863i		−	0.518102i	0.965864	−	0.259051i	$-0.0834098\pi$
$242$	−31.2563			−0.129158
$243$		0			0
$244$			153.411i			0.628733i
$245$	60.6179	+	91.2714i	0.247420	+	0.372536i
$246$		0			0
$247$	−169.768			−0.687320
$248$			56.4181i			0.227492i
$249$		0			0
$250$			15.8114i			0.0632456i
$251$			18.4515i			0.0735121i	0.999324	+	0.0367561i	$0.0117025\pi$
$251$			18.4515i			0.0735121i	−0.999324	+	0.0367561i	$0.988298\pi$
$252$		0			0
$253$	−34.0839			−0.134719
$254$	356.973			1.40541
$255$		0			0
$256$	16.0000			0.0625000
$257$			428.659i			1.66793i	0.551816	+	0.833966i	$0.313935\pi$
$257$			428.659i			1.66793i	−0.551816	+	0.833966i	$0.686065\pi$
$258$		0			0
$259$	−74.1657	+	22.3850i	−0.286354	+	0.0864285i
$260$	−43.2587			−0.166380
$261$		0			0
$262$		−	292.118i		−	1.11495i
$263$	133.575			0.507888			0.253944	−	0.967219i	$-0.418272\pi$
$263$	133.575			0.507888			0.253944	+	0.967219i	$0.418272\pi$
$264$		0			0
$265$			26.1865i			0.0988170i
$266$	−50.2033	−	166.333i	−0.188734	−	0.625312i
$267$		0			0
$268$	−152.481			−0.568960
$269$		−	493.257i		−	1.83367i	−0.399269	−	0.916834i	$-0.630736\pi$
$269$		−	493.257i		−	1.83367i	0.399269	−	0.916834i	$-0.369264\pi$
$270$		0			0
$271$		−	432.545i		−	1.59611i	−0.602586	−	0.798054i	$-0.705863\pi$
$271$		−	432.545i		−	1.59611i	0.602586	−	0.798054i	$-0.294137\pi$
$272$			28.3674i			0.104292i
$273$		0			0
$274$	253.232			0.924206
$275$	59.8125			0.217500
$276$		0			0
$277$	−171.941			−0.620724			−0.310362	−	0.950618i	$-0.600450\pi$
$277$	−171.941			−0.620724			−0.310362	+	0.950618i	$0.600450\pi$
$278$			42.2018i			0.151805i
$279$		0			0
$280$	−12.7923	−	42.3834i	−0.0456869	−	0.151369i
$281$	−74.7499			−0.266014			−0.133007	−	0.991115i	$-0.542463\pi$
$281$	−74.7499			−0.266014			−0.133007	+	0.991115i	$0.542463\pi$
$282$		0			0
$283$			165.904i			0.586232i	0.956077	+	0.293116i	$0.0946922\pi$
$283$			165.904i			0.586232i	−0.956077	+	0.293116i	$0.905308\pi$
$284$	190.899			0.672178
$285$		0			0
$286$			163.642i			0.572176i
$287$	57.2210	+	189.584i	0.199376	+	0.660571i
$288$		0			0
$289$	238.706			0.825971
$290$		−	43.3009i		−	0.149313i
$291$		0			0
$292$			241.369i			0.826605i
$293$			495.925i			1.69258i	0.532725	+	0.846289i	$0.321168\pi$
$293$			495.925i			1.69258i	−0.532725	+	0.846289i	$0.678832\pi$
$294$		0			0
$295$	−226.201			−0.766784
$296$	31.3027			0.105752
$297$		0			0
$298$	330.414			1.10877
$299$			27.5604i			0.0921753i
$300$		0			0
$301$	−488.979	+	147.586i	−1.62452	+	0.490318i
$302$	−396.881			−1.31418
$303$		0			0
$304$			70.2033i			0.230932i
$305$	−171.519			−0.562356
$306$		0			0
$307$		−	316.942i		−	1.03238i	−0.856473	−	0.516192i	$-0.827349\pi$
$307$		−	316.942i		−	1.03238i	0.856473	−	0.516192i	$-0.172651\pi$
$308$	−160.331	+	48.3918i	−0.520556	+	0.157116i
$309$		0			0
$310$	−63.0773			−0.203475
$311$		−	27.1325i		−	0.0872428i	−0.999048	−	0.0436214i	$-0.986110\pi$
$311$		−	27.1325i		−	0.0872428i	0.999048	−	0.0436214i	$-0.0138895\pi$
$312$		0			0
$313$			517.236i			1.65251i	0.563296	+	0.826255i	$0.309533\pi$
$313$			517.236i			1.65251i	−0.563296	+	0.826255i	$0.690467\pi$
$314$		−	76.8176i		−	0.244642i
$315$		0			0
$316$	−29.6192			−0.0937317
$317$	−213.041			−0.672054			−0.336027	−	0.941852i	$-0.609083\pi$
$317$	−213.041			−0.672054			−0.336027	+	0.941852i	$0.609083\pi$
$318$		0			0
$319$	−163.802			−0.513486
$320$			17.8885i			0.0559017i
$321$		0			0
$322$	−27.0028	+	8.15008i	−0.0838595	+	0.0253108i
$323$	−124.468			−0.385349
$324$		0			0
$325$		−	48.3647i		−	0.148814i
$326$	−283.582			−0.869884
$327$		0			0
$328$		−	80.0166i		−	0.243953i
$329$	57.3040	+	189.859i	0.174176	+	0.577079i
$330$		0			0
$331$	292.516			0.883735			0.441867	−	0.897080i	$-0.354316\pi$
$331$	292.516			0.883735			0.441867	+	0.897080i	$0.354316\pi$
$332$		−	121.914i		−	0.367211i
$333$		0			0
$334$			399.961i			1.19749i
$335$		−	170.479i		−	0.508894i
$336$		0			0
$337$	−122.881			−0.364633			−0.182316	−	0.983240i	$-0.558360\pi$
$337$	−122.881			−0.364633			−0.182316	+	0.983240i	$0.558360\pi$
$338$	−106.680			−0.315622
$339$		0			0
$340$	−31.7157			−0.0932816
$341$			238.614i			0.699747i
$342$		0			0
$343$	218.705	−	264.230i	0.637623	−	0.770349i
$344$	206.381			0.599944
$345$		0			0
$346$			179.999i			0.520228i
$347$	−207.406			−0.597713			−0.298856	−	0.954298i	$-0.596605\pi$
$347$	−207.406			−0.597713			−0.298856	+	0.954298i	$0.596605\pi$
$348$		0			0
$349$			490.264i			1.40477i	0.711798	+	0.702384i	$0.247881\pi$
$349$			490.264i			1.40477i	−0.711798	+	0.702384i	$0.752119\pi$
$350$	47.3861	−	14.3023i	0.135389	−	0.0408636i
$351$		0			0
$352$	67.6702			0.192245
$353$		−	454.199i		−	1.28668i	−0.765579	−	0.643342i	$-0.777547\pi$
$353$		−	454.199i		−	1.28668i	0.765579	−	0.643342i	$-0.222453\pi$
$354$		0			0
$355$			213.431i			0.601214i
$356$			177.260i			0.497922i
$357$		0			0
$358$	173.017			0.483288
$359$	−378.429			−1.05412			−0.527060	−	0.849828i	$-0.676706\pi$
$359$	−378.429			−1.05412			−0.527060	+	0.849828i	$0.676706\pi$
$360$		0			0
$361$	52.9689			0.146728
$362$			137.689i			0.380355i
$363$		0			0
$364$	39.1299	+	129.645i	0.107500	+	0.356167i
$365$	−269.859			−0.739338
$366$		0			0
$367$		−	181.636i		−	0.494920i	−0.968898	−	0.247460i	$-0.920404\pi$
$367$		−	181.636i		−	0.494920i	0.968898	−	0.247460i	$-0.0795959\pi$
$368$	11.3969			0.0309699
$369$		0			0
$370$			34.9975i			0.0945878i
$371$	78.4800	−	23.6871i	0.211536	−	0.0638467i
$372$		0			0
$373$	302.597			0.811253			0.405626	−	0.914039i	$-0.367053\pi$
$373$	302.597			0.811253			0.405626	+	0.914039i	$0.367053\pi$
$374$			119.977i			0.320793i
$375$		0			0
$376$		−	80.1328i		−	0.213119i
$377$			132.451i			0.351329i
$378$		0			0
$379$	706.225			1.86339			0.931695	−	0.363243i	$-0.118330\pi$
$379$	706.225			1.86339			0.931695	+	0.363243i	$0.118330\pi$
$380$	−78.4896			−0.206552
$381$		0			0
$382$	−410.369			−1.07427
$383$			238.607i			0.622995i	0.950247	+	0.311498i	$0.100831\pi$
$383$			238.607i			0.622995i	−0.950247	+	0.311498i	$0.899169\pi$
$384$		0			0
$385$	−54.1037	−	179.256i	−0.140529	−	0.465600i
$386$	−128.381			−0.332593
$387$		0			0
$388$		−	37.7094i		−	0.0971891i
$389$	523.592			1.34599			0.672997	−	0.739645i	$-0.265007\pi$
$389$	523.592			1.34599			0.672997	+	0.739645i	$0.265007\pi$
$390$		0			0
$391$			20.2063i			0.0516785i
$392$	−115.450	+	76.6763i	−0.294516	+	0.195603i
$393$		0			0
$394$	−75.4781			−0.191569
$395$		−	33.1153i		−	0.0838362i
$396$		0			0
$397$		−	219.940i		−	0.554004i	−0.960869	−	0.277002i	$-0.910659\pi$
$397$		−	219.940i		−	0.554004i	0.960869	−	0.277002i	$-0.0893408\pi$
$398$			426.475i			1.07154i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$	−559.885			−1.39622			−0.698111	−	0.715989i	$-0.745976\pi$
$401$	−559.885			−1.39622			−0.698111	+	0.715989i	$0.745976\pi$
$402$		0			0
$403$	192.944			0.478770
$404$		−	60.3188i		−	0.149304i
$405$		0			0
$406$	−129.771	+	39.1680i	−0.319633	+	0.0964730i
$407$	132.391			0.325285
$408$		0			0
$409$			477.443i			1.16734i	0.811990	+	0.583672i	$0.198384\pi$
$409$			477.443i			1.16734i	−0.811990	+	0.583672i	$0.801616\pi$
$410$	89.4613			0.218198
$411$		0			0
$412$			301.738i			0.732374i
$413$	204.611	+	677.917i	0.495427	+	1.64144i
$414$		0			0
$415$	136.304			0.328443
$416$		−	54.7184i		−	0.131535i
$417$		0			0
$418$			296.917i			0.710327i
$419$		−	618.800i		−	1.47685i	−0.674335	−	0.738425i	$-0.735570\pi$
$419$		−	618.800i		−	1.47685i	0.674335	−	0.738425i	$-0.264430\pi$
$420$		0			0
$421$	23.7665			0.0564524			0.0282262	−	0.999602i	$-0.491014\pi$
$421$	23.7665			0.0564524			0.0282262	+	0.999602i	$0.491014\pi$
$422$	105.771			0.250641
$423$		0			0
$424$	−33.1236			−0.0781217
$425$		−	35.4593i		−	0.0834336i
$426$		0			0
$427$	155.148	+	514.035i	0.363344	+	1.20383i
$428$	−176.538			−0.412472
$429$		0			0
$430$			230.741i			0.536606i
$431$	244.991			0.568424			0.284212	−	0.958762i	$-0.408268\pi$
$431$	244.991			0.568424			0.284212	+	0.958762i	$0.408268\pi$
$432$		0			0
$433$		−	630.500i		−	1.45612i	−0.685514	−	0.728060i	$-0.740422\pi$
$433$		−	630.500i		−	1.45612i	0.685514	−	0.728060i	$-0.259578\pi$
$434$	57.0569	+	189.040i	0.131468	+	0.435577i
$435$		0			0
$436$	185.443			0.425328
$437$			50.0063i			0.114431i
$438$		0			0
$439$		−	79.2321i		−	0.180483i	−0.995920	−	0.0902416i	$-0.971236\pi$
$439$		−	79.2321i		−	0.180483i	0.995920	−	0.0902416i	$-0.0287639\pi$
$440$			75.6575i			0.171949i
$441$		0			0
$442$	97.0137			0.219488
$443$	−494.991			−1.11736			−0.558680	−	0.829383i	$-0.688692\pi$
$443$	−494.991			−1.11736			−0.558680	+	0.829383i	$0.688692\pi$
$444$		0			0
$445$	−198.183			−0.445355
$446$			584.253i			1.30998i
$447$		0			0
$448$	53.6113	−	16.1812i	0.119668	−	0.0361187i
$449$	−539.070			−1.20060			−0.600301	−	0.799774i	$-0.704952\pi$
$449$	−539.070			−1.20060			−0.600301	+	0.799774i	$0.704952\pi$
$450$		0			0
$451$		−	338.421i		−	0.750379i
$452$	126.988			0.280948
$453$		0			0
$454$			422.413i			0.930425i
$455$	−144.947	+	43.7485i	−0.318565	+	0.0961506i
$456$		0			0
$457$	−265.432			−0.580814			−0.290407	−	0.956903i	$-0.593791\pi$
$457$	−265.432			−0.580814			−0.290407	+	0.956903i	$0.593791\pi$
$458$		−	457.248i		−	0.998359i
$459$		0			0
$460$			12.7421i			0.0277003i
$461$		−	358.881i		−	0.778484i	−0.921136	−	0.389242i	$-0.872737\pi$
$461$		−	358.881i		−	0.778484i	0.921136	−	0.389242i	$-0.127263\pi$
$462$		0			0
$463$	112.645			0.243294			0.121647	−	0.992573i	$-0.461182\pi$
$463$	112.645			0.243294			0.121647	+	0.992573i	$0.461182\pi$
$464$	54.7718			0.118043
$465$		0			0
$466$	−458.136			−0.983125
$467$			409.478i			0.876826i	0.898774	+	0.438413i	$0.144459\pi$
$467$			409.478i			0.876826i	−0.898774	+	0.438413i	$0.855541\pi$
$468$		0			0
$469$	−510.920	+	154.208i	−1.08938	+	0.328802i
$470$	89.5912			0.190619
$471$		0			0
$472$		−	286.125i		−	0.606196i
$473$	872.863			1.84538
$474$		0			0
$475$		−	87.7541i		−	0.184745i
$476$	28.6886	+	95.0508i	0.0602702	+	0.199687i
$477$		0			0
$478$	−512.595			−1.07237
$479$			99.5292i			0.207785i	0.994589	+	0.103893i	$0.0331299\pi$
$479$			99.5292i			0.207785i	−0.994589	+	0.103893i	$0.966870\pi$
$480$		0			0
$481$		−	107.052i		−	0.222561i
$482$			176.582i			0.366354i
$483$		0			0
$484$	44.2031			0.0913288
$485$	42.1603			0.0869286
$486$		0			0
$487$	−123.746			−0.254098			−0.127049	−	0.991896i	$-0.540551\pi$
$487$	−123.746			−0.254098			−0.127049	+	0.991896i	$0.540551\pi$
$488$		−	216.956i		−	0.444582i
$489$		0			0
$490$	−85.7267	−	129.077i	−0.174952	−	0.263423i
$491$	−42.2815			−0.0861131			−0.0430565	−	0.999073i	$-0.513710\pi$
$491$	−42.2815			−0.0861131			−0.0430565	+	0.999073i	$0.513710\pi$
$492$		0			0
$493$			97.1083i			0.196974i
$494$	240.088			0.486008
$495$		0			0
$496$		−	79.7872i		−	0.160861i
$497$	639.645	−	193.060i	1.28701	−	0.388451i
$498$		0			0
$499$	87.3026			0.174955			0.0874775	−	0.996166i	$-0.472119\pi$
$499$	87.3026			0.174955			0.0874775	+	0.996166i	$0.472119\pi$
$500$		−	22.3607i		−	0.0447214i
$501$		0			0
$502$		−	26.0944i		−	0.0519809i
$503$			452.879i			0.900356i	0.892939	+	0.450178i	$0.148639\pi$
$503$			452.879i			0.900356i	−0.892939	+	0.450178i	$0.851361\pi$
$504$		0			0
$505$	67.4385			0.133542
$506$	48.2019			0.0952607
$507$		0			0
$508$	−504.837			−0.993773
$509$			236.371i			0.464382i	0.972670	+	0.232191i	$0.0745895\pi$
$509$			236.371i			0.464382i	−0.972670	+	0.232191i	$0.925410\pi$
$510$		0			0
$511$	244.102	+	808.756i	0.477694	+	1.58269i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$		−	606.215i		−	1.17941i
$515$	−337.353			−0.655055
$516$		0			0
$517$		−	338.912i		−	0.655536i
$518$	104.886	−	31.6571i	0.202483	−	0.0611142i
$519$		0			0
$520$	61.1770			0.117648
$521$			446.261i			0.856548i	0.903649	+	0.428274i	$0.140878\pi$
$521$			446.261i			0.856548i	−0.903649	+	0.428274i	$0.859122\pi$
$522$		0			0
$523$		−	271.135i		−	0.518423i	−0.965821	−	0.259211i	$-0.916537\pi$
$523$		−	271.135i		−	0.518423i	0.965821	−	0.259211i	$-0.0834626\pi$
$524$			413.117i			0.788391i
$525$		0			0
$526$	−188.903			−0.359131
$527$	141.460			0.268425
$528$		0			0
$529$	−520.882			−0.984654
$530$		−	37.0333i		−	0.0698742i
$531$		0			0
$532$	70.9982	+	235.230i	0.133455	+	0.442162i
$533$	−273.649			−0.513413
$534$		0			0
$535$		−	197.375i		−	0.368926i
$536$	215.641			0.402316
$537$		0			0
$538$			697.570i			1.29660i
$539$	−488.283	+	324.294i	−0.905906	+	0.601658i
$540$		0			0
$541$	224.274			0.414555			0.207277	−	0.978282i	$-0.433540\pi$
$541$	224.274			0.414555			0.207277	+	0.978282i	$0.433540\pi$
$542$			611.712i			1.12862i
$543$		0			0
$544$		−	40.1176i		−	0.0737455i
$545$			207.332i			0.380425i
$546$		0			0
$547$	88.4533			0.161706			0.0808531	−	0.996726i	$-0.474236\pi$
$547$	88.4533			0.161706			0.0808531	+	0.996726i	$0.474236\pi$
$548$	−358.125			−0.653512
$549$		0			0
$550$	−84.5877			−0.153796
$551$			240.322i			0.436157i
$552$		0			0
$553$	−99.2453	+	29.9546i	−0.179467	+	0.0541675i
$554$	243.161			0.438918
$555$		0			0
$556$		−	59.6824i		−	0.107342i
$557$	54.4040			0.0976733			0.0488367	−	0.998807i	$-0.484449\pi$
$557$	54.4040			0.0976733			0.0488367	+	0.998807i	$0.484449\pi$
$558$		0			0
$559$		−	705.801i		−	1.26261i
$560$	18.0911	+	59.9392i	0.0323055	+	0.107034i
$561$		0			0
$562$	105.712			0.188100
$563$		−	484.965i		−	0.861394i	−0.902497	−	0.430697i	$-0.858268\pi$
$563$		−	484.965i		−	0.861394i	0.902497	−	0.430697i	$-0.141732\pi$
$564$		0			0
$565$			141.977i			0.251287i
$566$		−	234.623i		−	0.414529i
$567$		0			0
$568$	−269.971			−0.475302
$569$	306.635			0.538902			0.269451	−	0.963014i	$-0.413158\pi$
$569$	306.635			0.538902			0.269451	+	0.963014i	$0.413158\pi$
$570$		0			0
$571$	471.839			0.826339			0.413169	−	0.910654i	$-0.364422\pi$
$571$	471.839			0.826339			0.413169	+	0.910654i	$0.364422\pi$
$572$		−	231.425i		−	0.404590i
$573$		0			0
$574$	−80.9227	−	268.112i	−0.140980	−	0.467094i
$575$	−14.2461			−0.0247759
$576$		0			0
$577$			81.1961i			0.140721i	0.997522	+	0.0703606i	$0.0224150\pi$
$577$			81.1961i			0.140721i	−0.997522	+	0.0703606i	$0.977585\pi$
$578$	−337.581			−0.584050
$579$		0			0
$580$			61.2367i			0.105581i
$581$	−123.294	−	408.498i	−0.212211	−	0.703094i
$582$		0			0
$583$	−140.093			−0.240296
$584$		−	341.347i		−	0.584498i
$585$		0			0
$586$		−	701.344i		−	1.19683i
$587$		−	905.925i		−	1.54331i	−0.636039	−	0.771657i	$-0.719428\pi$
$587$		−	905.925i		−	1.54331i	0.636039	−	0.771657i	$-0.280572\pi$
$588$		0			0
$589$	350.083			0.594368
$590$	319.897			0.542198
$591$		0			0
$592$	−44.2687			−0.0747782
$593$		−	455.850i		−	0.768719i	−0.923184	−	0.384359i	$-0.874422\pi$
$593$		−	455.850i		−	0.768719i	0.923184	−	0.384359i	$-0.125578\pi$
$594$		0			0
$595$	−106.270	+	32.0748i	−0.178605	+	0.0539073i
$596$	−467.276			−0.784020
$597$		0			0
$598$		−	38.9763i		−	0.0651777i
$599$	−72.2734			−0.120657			−0.0603284	−	0.998179i	$-0.519215\pi$
$599$	−72.2734			−0.120657			−0.0603284	+	0.998179i	$0.519215\pi$
$600$		0			0
$601$			396.088i			0.659048i	0.944147	+	0.329524i	$0.106888\pi$
$601$			396.088i			0.659048i	−0.944147	+	0.329524i	$0.893112\pi$
$602$	691.521	−	208.718i	1.14871	−	0.346707i
$603$		0			0
$604$	561.274			0.929262
$605$			49.4206i			0.0816870i
$606$		0			0
$607$		−	6.14590i		−	0.0101250i	−0.999987	−	0.00506252i	$-0.998389\pi$
$607$		−	6.14590i		−	0.0101250i	0.999987	−	0.00506252i	$-0.00161146\pi$
$608$		−	99.2824i		−	0.163293i
$609$		0			0
$610$	242.564			0.397646
$611$	−274.046			−0.448521
$612$		0			0
$613$	−197.286			−0.321837			−0.160918	−	0.986968i	$-0.551446\pi$
$613$	−197.286			−0.321837			−0.160918	+	0.986968i	$0.551446\pi$
$614$			448.223i			0.730005i
$615$		0			0
$616$	226.743	−	68.4364i	0.368089	−	0.111098i
$617$	−152.122			−0.246551			−0.123276	−	0.992372i	$-0.539340\pi$
$617$	−152.122			−0.246551			−0.123276	+	0.992372i	$0.539340\pi$
$618$		0			0
$619$		−	700.262i		−	1.13128i	−0.824653	−	0.565639i	$-0.808629\pi$
$619$		−	700.262i		−	1.13128i	0.824653	−	0.565639i	$-0.191371\pi$
$620$	89.2048			0.143879
$621$		0			0
$622$			38.3712i			0.0616900i
$623$	179.267	+	593.947i	0.287749	+	0.953366i
$624$		0			0
$625$	25.0000			0.0400000
$626$		−	731.482i		−	1.16850i
$627$		0			0
$628$			108.636i			0.172988i
$629$		−	78.4867i		−	0.124780i
$630$		0			0
$631$	−133.714			−0.211908			−0.105954	−	0.994371i	$-0.533790\pi$
$631$	−133.714			−0.211908			−0.105954	+	0.994371i	$0.533790\pi$
$632$	41.8879			0.0662783
$633$		0			0
$634$	301.285			0.475214
$635$		−	564.425i		−	0.888858i
$636$		0			0
$637$	262.225	+	394.828i	0.411657	+	0.619824i
$638$	231.651			0.363089
$639$		0			0
$640$		−	25.2982i		−	0.0395285i
$641$	781.577			1.21931			0.609654	−	0.792667i	$-0.291308\pi$
$641$	781.577			1.21931			0.609654	+	0.792667i	$0.291308\pi$
$642$		0			0
$643$		−	346.920i		−	0.539533i	−0.962926	−	0.269766i	$-0.913054\pi$
$643$		−	346.920i		−	0.539533i	0.962926	−	0.269766i	$-0.0869465\pi$
$644$	38.1877	−	11.5260i	0.0592976	−	0.0178974i
$645$		0			0
$646$	176.024			0.272483
$647$		−	971.001i		−	1.50077i	−0.660998	−	0.750387i	$-0.729867\pi$
$647$		−	971.001i		−	1.50077i	0.660998	−	0.750387i	$-0.270133\pi$
$648$		0			0
$649$		−	1210.13i		−	1.86461i
$650$			68.3980i			0.105228i
$651$		0			0
$652$	401.046			0.615101
$653$	326.428			0.499890			0.249945	−	0.968260i	$-0.419587\pi$
$653$	326.428			0.499890			0.249945	+	0.968260i	$0.419587\pi$
$654$		0			0
$655$	−461.879			−0.705158
$656$			113.161i			0.172501i
$657$		0			0
$658$	−81.0401	−	268.501i	−0.123161	−	0.408057i
$659$	−305.606			−0.463743			−0.231871	−	0.972746i	$-0.574485\pi$
$659$	−305.606			−0.463743			−0.231871	+	0.972746i	$0.574485\pi$
$660$		0			0
$661$			232.398i			0.351586i	0.984427	+	0.175793i	$0.0562489\pi$
$661$			232.398i			0.351586i	−0.984427	+	0.175793i	$0.943751\pi$
$662$	−413.680			−0.624895
$663$		0			0
$664$			172.412i			0.259657i
$665$	−262.996	+	79.3784i	−0.395482	+	0.119366i
$666$		0			0
$667$	39.0143			0.0584922
$668$		−	565.631i		−	0.846753i
$669$		0			0
$670$			241.094i			0.359842i
$671$		−	917.589i		−	1.36750i
$672$		0			0
$673$	283.273			0.420911			0.210455	−	0.977603i	$-0.432505\pi$
$673$	283.273			0.420911			0.210455	+	0.977603i	$0.432505\pi$
$674$	173.780			0.257834
$675$		0			0
$676$	150.868			0.223178
$677$			425.511i			0.628524i	0.949336	+	0.314262i	$0.101757\pi$
$677$			425.511i			0.628524i	−0.949336	+	0.314262i	$0.898243\pi$
$678$		0			0
$679$	−38.1364	−	126.353i	−0.0561655	−	0.186087i
$680$	44.8528			0.0659600
$681$		0			0
$682$		−	337.451i		−	0.494796i
$683$	126.763			0.185597			0.0927986	−	0.995685i	$-0.470419\pi$
$683$	126.763			0.185597			0.0927986	+	0.995685i	$0.470419\pi$
$684$		0			0
$685$		−	400.395i		−	0.584519i
$686$	−309.295	+	373.677i	−0.450867	+	0.544719i
$687$		0			0
$688$	−291.866			−0.424224
$689$			113.279i			0.164411i
$690$		0			0
$691$		−	124.435i		−	0.180080i	−0.995938	−	0.0900401i	$-0.971300\pi$
$691$		−	124.435i		−	0.180080i	0.995938	−	0.0900401i	$-0.0286995\pi$
$692$		−	254.557i		−	0.367857i
$693$		0			0
$694$	293.317			0.422647
$695$	66.7270			0.0960100
$696$		0			0
$697$	−200.630			−0.287847
$698$		−	693.338i		−	0.993321i
$699$		0			0
$700$	−67.0141	+	20.2265i	−0.0957344	+	0.0288949i
$701$	−273.794			−0.390576			−0.195288	−	0.980746i	$-0.562564\pi$
$701$	−273.794			−0.390576			−0.195288	+	0.980746i	$0.562564\pi$
$702$		0			0
$703$		−	194.238i		−	0.276299i
$704$	−95.7001			−0.135938
$705$		0			0
$706$			642.335i			0.909823i
$707$	−61.0018	−	202.111i	−0.0862826	−	0.285871i
$708$		0			0
$709$	−462.551			−0.652399			−0.326200	−	0.945301i	$-0.605768\pi$
$709$	−462.551			−0.652399			−0.326200	+	0.945301i	$0.605768\pi$
$710$		−	301.837i		−	0.425123i
$711$		0			0
$712$		−	250.684i		−	0.352084i
$713$		−	56.8330i		−	0.0797096i
$714$		0			0
$715$	258.741			0.361876
$716$	−244.683			−0.341736
$717$		0			0
$718$	535.179			0.745375
$719$			773.673i			1.07604i	0.842932	+	0.538020i	$0.180828\pi$
$719$			773.673i			1.07604i	−0.842932	+	0.538020i	$0.819172\pi$
$720$		0			0
$721$	305.155	+	1011.04i	0.423238	+	1.40227i
$722$	−74.9093			−0.103753
$723$		0			0
$724$		−	194.721i		−	0.268952i
$725$	−68.4647			−0.0944341
$726$		0			0
$727$			1233.31i			1.69644i	0.529645	+	0.848219i	$0.322325\pi$
$727$			1233.31i			1.69644i	−0.529645	+	0.848219i	$0.677675\pi$
$728$	−55.3380	−	183.345i	−0.0760137	−	0.251848i
$729$		0			0
$730$	381.638			0.522791
$731$		−	517.468i		−	0.707891i
$732$		0			0
$733$			1421.58i			1.93939i	0.244309	+	0.969697i	$0.421439\pi$
$733$			1421.58i			1.93939i	−0.244309	+	0.969697i	$0.578561\pi$
$734$			256.872i			0.349961i
$735$		0			0
$736$	−16.1177			−0.0218990
$737$	912.030			1.23749
$738$		0			0
$739$	518.845			0.702091			0.351045	−	0.936358i	$-0.385826\pi$
$739$	518.845			0.702091			0.351045	+	0.936358i	$0.385826\pi$
$740$		−	49.4939i		−	0.0668836i
$741$		0			0
$742$	−110.987	+	33.4987i	−0.149579	+	0.0451465i
$743$	1012.05			1.36212			0.681058	−	0.732230i	$-0.261520\pi$
$743$	1012.05			1.36212			0.681058	+	0.732230i	$0.261520\pi$
$744$		0			0
$745$		−	522.430i		−	0.701248i
$746$	−427.937			−0.573642
$747$		0			0
$748$		−	169.673i		−	0.226835i
$749$	−591.526	+	178.537i	−0.789755	+	0.238367i
$750$		0			0
$751$	−1274.40			−1.69694			−0.848470	−	0.529243i	$-0.822476\pi$
$751$	−1274.40			−1.69694			−0.848470	+	0.529243i	$0.822476\pi$
$752$			113.325i			0.150698i
$753$		0			0
$754$		−	187.314i		−	0.248427i
$755$			627.524i			0.831157i
$756$		0			0
$757$	1145.83			1.51364			0.756822	−	0.653621i	$-0.226751\pi$
$757$	1145.83			1.51364			0.756822	+	0.653621i	$0.226751\pi$
$758$	−998.752			−1.31762
$759$		0			0
$760$	111.001			0.146054
$761$		−	580.236i		−	0.762465i	−0.924479	−	0.381233i	$-0.875500\pi$
$761$		−	580.236i		−	0.762465i	0.924479	−	0.381233i	$-0.124500\pi$
$762$		0			0
$763$	621.365	−	187.543i	0.814372	−	0.245797i
$764$	580.350			0.759621
$765$		0			0
$766$		−	337.441i		−	0.440524i
$767$	−978.517			−1.27577
$768$		0			0
$769$			58.9626i			0.0766744i	0.999265	+	0.0383372i	$0.0122061\pi$
$769$			58.9626i			0.0766744i	−0.999265	+	0.0383372i	$0.987794\pi$
$770$	76.5142	+	253.506i	0.0993691	+	0.329229i
$771$		0			0
$772$	181.558			0.235179
$773$			368.237i			0.476374i	0.971219	+	0.238187i	$0.0765531\pi$
$773$			368.237i			0.476374i	−0.971219	+	0.238187i	$0.923447\pi$
$774$		0			0
$775$			99.7340i			0.128689i
$776$			53.3291i			0.0687231i
$777$		0			0
$778$	−740.471			−0.951762
$779$	−496.515			−0.637375
$780$		0			0
$781$	−1141.81			−1.46199
$782$		−	28.5760i		−	0.0365422i
$783$		0			0
$784$	163.271	−	108.437i	0.208254	−	0.138312i
$785$	−121.459			−0.154725
$786$		0			0
$787$			1063.41i			1.35122i	0.737261	+	0.675608i	$0.236119\pi$
$787$			1063.41i			1.35122i	−0.737261	+	0.675608i	$0.763881\pi$
$788$	106.742			0.135460
$789$		0			0
$790$			46.8321i			0.0592811i
$791$	425.501	−	128.426i	0.537928	−	0.162359i
$792$		0			0
$793$	−741.967			−0.935646
$794$			311.041i			0.391740i
$795$		0			0
$796$		−	603.126i		−	0.757696i
$797$			274.031i			0.343828i	0.985112	+	0.171914i	$0.0549951\pi$
$797$			274.031i			0.343828i	−0.985112	+	0.171914i	$0.945005\pi$
$798$		0			0
$799$	−200.921			−0.251465
$800$	28.2843			0.0353553
$801$		0			0
$802$	791.797			0.987278
$803$		−	1443.69i		−	1.79787i
$804$		0			0
$805$	12.8864	+	42.6951i	0.0160080	+	0.0530374i
$806$	−272.864			−0.338541
$807$		0			0
$808$			85.3037i			0.105574i
$809$	−125.869			−0.155586			−0.0777930	−	0.996970i	$-0.524787\pi$
$809$	−125.869			−0.155586			−0.0777930	+	0.996970i	$0.524787\pi$
$810$		0			0
$811$		−	566.987i		−	0.699121i	−0.936914	−	0.349560i	$-0.886331\pi$
$811$		−	566.987i		−	0.699121i	0.936914	−	0.349560i	$-0.113669\pi$
$812$	183.524	−	55.3920i	0.226015	−	0.0682167i
$813$		0			0
$814$	−187.229			−0.230011
$815$			448.383i			0.550163i
$816$		0			0
$817$		−	1280.62i		−	1.56747i
$818$		−	675.207i		−	0.825436i
$819$		0			0
$820$	−126.517			−0.154290
$821$	444.471			0.541377			0.270689	−	0.962667i	$-0.412749\pi$
$821$	444.471			0.541377			0.270689	+	0.962667i	$0.412749\pi$
$822$		0			0
$823$	1028.74			1.24999			0.624995	−	0.780629i	$-0.285101\pi$
$823$	1028.74			1.24999			0.624995	+	0.780629i	$0.285101\pi$
$824$		−	426.722i		−	0.517867i
$825$		0			0
$826$	−289.364	−	958.719i	−0.350320	−	1.16068i
$827$	515.113			0.622870			0.311435	−	0.950267i	$-0.399190\pi$
$827$	515.113			0.622870			0.311435	+	0.950267i	$0.399190\pi$
$828$		0			0
$829$			1334.96i			1.61032i	0.593055	+	0.805162i	$0.297922\pi$
$829$			1334.96i			1.61032i	−0.593055	+	0.805162i	$0.702078\pi$
$830$	−192.763			−0.232244
$831$		0			0
$832$			77.3835i			0.0930090i
$833$	192.254	+	289.474i	0.230797	+	0.347508i
$834$		0			0
$835$	632.394			0.757359
$836$		−	419.903i		−	0.502277i
$837$		0			0
$838$			875.116i			1.04429i
$839$		−	2.13640i		−	0.00254637i	−0.999999	−	0.00127318i	$-0.999595\pi$
$839$		−	2.13640i		−	0.00254637i	0.999999	−	0.00127318i	$-0.000405267\pi$
$840$		0			0
$841$	−653.503			−0.777055
$842$	−33.6108			−0.0399179
$843$		0			0
$844$	−149.582			−0.177230
$845$			168.676i			0.199617i
$846$		0			0
$847$	148.112	−	44.7037i	0.174866	−	0.0527788i
$848$	46.8439			0.0552404
$849$		0			0
$850$			50.1470i			0.0589964i
$851$	−31.5329			−0.0370539
$852$		0			0
$853$		−	89.0278i		−	0.104370i	−0.998637	−	0.0521851i	$-0.983381\pi$
$853$		−	89.0278i		−	0.104370i	0.998637	−	0.0521851i	$-0.0166186\pi$
$854$	−219.412	−	726.955i	−0.256923	−	0.851235i
$855$		0			0
$856$	249.662			0.291662
$857$			820.566i			0.957486i	0.877955	+	0.478743i	$0.158907\pi$
$857$			820.566i			0.957486i	−0.877955	+	0.478743i	$0.841093\pi$
$858$		0			0
$859$		−	534.947i		−	0.622756i	−0.950286	−	0.311378i	$-0.899210\pi$
$859$		−	534.947i		−	0.622756i	0.950286	−	0.311378i	$-0.100790\pi$
$860$		−	326.317i		−	0.379438i
$861$		0			0
$862$	−346.469			−0.401936
$863$	276.972			0.320941			0.160471	−	0.987041i	$-0.448699\pi$
$863$	276.972			0.320941			0.160471	+	0.987041i	$0.448699\pi$
$864$		0			0
$865$	284.603			0.329021
$866$			891.661i			1.02963i
$867$		0			0
$868$	−80.6906	−	267.343i	−0.0929616	−	0.307999i
$869$	177.160			0.203867
$870$		0			0
$871$		−	737.472i		−	0.846695i
$872$	−262.256			−0.300753
$873$		0			0
$874$		−	70.7195i		−	0.0809148i
$875$	−22.6139	−	74.9240i	−0.0258444	−	0.0856275i
$876$		0			0
$877$	420.192			0.479125			0.239562	−	0.970881i	$-0.422996\pi$
$877$	420.192			0.479125			0.239562	+	0.970881i	$0.422996\pi$
$878$			112.051i			0.127621i
$879$		0			0
$880$		−	106.996i		−	0.121586i
$881$		−	897.207i		−	1.01840i	−0.860649	−	0.509198i	$-0.829942\pi$
$881$		−	897.207i		−	1.01840i	0.860649	−	0.509198i	$-0.170058\pi$
$882$		0			0
$883$	−1515.08			−1.71584			−0.857918	−	0.513786i	$-0.828243\pi$
$883$	−1515.08			−1.71584			−0.857918	+	0.513786i	$0.828243\pi$
$884$	−137.198			−0.155201
$885$		0			0
$886$	700.023			0.790093
$887$		−	976.074i		−	1.10042i	−0.835026	−	0.550211i	$-0.814547\pi$
$887$		−	976.074i		−	1.10042i	0.835026	−	0.550211i	$-0.185453\pi$
$888$		0			0
$889$	−1691.56	+	510.553i	−1.90277	+	0.574300i
$890$	280.273			0.314914
$891$		0			0
$892$		−	826.259i		−	0.926299i
$893$	−497.236			−0.556815
$894$		0			0
$895$		−	273.564i		−	0.305658i
$896$	−75.8178	+	22.8836i	−0.0846181	+	0.0255398i
$897$		0			0
$898$	762.360			0.848953
$899$		−	273.130i		−	0.303816i
$900$		0			0
$901$			83.0524i			0.0921781i
$902$			478.600i			0.530598i
$903$		0			0
$904$	−179.589			−0.198660
$905$	217.705			0.240558
$906$		0			0
$907$	1329.48			1.46580			0.732901	−	0.680335i	$-0.238166\pi$
$907$	1329.48			1.46580			0.732901	+	0.680335i	$0.238166\pi$
$908$		−	597.382i		−	0.657910i
$909$		0			0
$910$	204.986	−	61.8698i	0.225260	−	0.0679887i
$911$	1270.75			1.39489			0.697447	−	0.716637i	$-0.254319\pi$
$911$	1270.75			1.39489			0.697447	+	0.716637i	$0.254319\pi$
$912$		0			0
$913$			729.198i			0.798684i
$914$	375.377			0.410697
$915$		0			0
$916$			646.647i			0.705946i
$917$	417.795	+	1384.23i	0.455610	+	1.50952i
$918$		0			0
$919$	−1641.49			−1.78617			−0.893083	−	0.449893i	$-0.851462\pi$
$919$	−1641.49			−1.78617			−0.893083	+	0.449893i	$0.851462\pi$
$920$		−	18.0201i		−	0.0195871i
$921$		0			0
$922$			507.534i			0.550471i
$923$			923.275i			1.00030i
$924$		0			0
$925$	55.3359			0.0598225
$926$	−159.304			−0.172035
$927$		0			0
$928$	−77.4590			−0.0834687
$929$		−	1386.92i		−	1.49292i	−0.665433	−	0.746458i	$-0.731753\pi$
$929$		−	1386.92i		−	1.49292i	0.665433	−	0.746458i	$-0.268247\pi$
$930$		0			0
$931$	475.788	+	716.386i	0.511050	+	0.769480i
$932$	647.902			0.695174
$933$		0			0
$934$		−	579.089i		−	0.620010i
$935$	189.700			0.202888
$936$		0			0
$937$			359.625i			0.383805i	0.981414	+	0.191903i	$0.0614658\pi$
$937$			359.625i			0.383805i	−0.981414	+	0.191903i	$0.938534\pi$
$938$	722.550	−	218.083i	0.770309	−	0.232498i
$939$		0			0
$940$	−126.701			−0.134788
$941$			1575.46i			1.67424i	0.547020	+	0.837119i	$0.315762\pi$
$941$			1575.46i			1.67424i	−0.547020	+	0.837119i	$0.684238\pi$
$942$		0			0
$943$			80.6051i			0.0854773i
$944$			404.641i			0.428645i
$945$		0			0
$946$	−1234.42			−1.30488
$947$	1680.38			1.77442			0.887212	−	0.461361i	$-0.152639\pi$
$947$	1680.38			1.77442			0.887212	+	0.461361i	$0.152639\pi$
$948$		0			0
$949$	−1167.37			−1.23011
$950$			124.103i			0.130635i
$951$		0			0
$952$	−40.5718	−	134.422i	−0.0426175	−	0.141200i
$953$	−1419.84			−1.48986			−0.744931	−	0.667142i	$-0.767517\pi$
$953$	−1419.84			−1.48986			−0.744931	+	0.667142i	$0.767517\pi$
$954$		0			0
$955$			648.851i			0.679425i
$956$	724.918			0.758283
$957$		0			0
$958$		−	140.756i		−	0.146926i
$959$	−1199.97	+	362.180i	−1.25127	+	0.377664i
$960$		0			0
$961$	563.125			0.585978
$962$			151.394i			0.157375i
$963$		0			0
$964$		−	249.725i		−	0.259051i
$965$			202.988i			0.210350i
$966$		0			0
$967$	−1061.44			−1.09766			−0.548829	−	0.835934i	$-0.684926\pi$
$967$	−1061.44			−1.09766			−0.548829	+	0.835934i	$0.684926\pi$
$968$	−62.5127			−0.0645792
$969$		0			0
$970$	−59.6237			−0.0614678
$971$			1375.99i			1.41709i	0.705668	+	0.708543i	$0.250647\pi$
$971$			1375.99i			1.41709i	−0.705668	+	0.708543i	$0.749353\pi$
$972$		0			0
$973$	−60.3582	−	199.978i	−0.0620331	−	0.205527i
$974$	175.003			0.179675
$975$		0			0
$976$			306.822i			0.314367i
$977$	−541.316			−0.554059			−0.277030	−	0.960861i	$-0.589350\pi$
$977$	−541.316			−0.554059			−0.277030	+	0.960861i	$0.589350\pi$
$978$		0			0
$979$		−	1060.24i		−	1.08298i
$980$	121.236	+	182.543i	0.123710	+	0.186268i
$981$		0			0
$982$	59.7951			0.0608912
$983$			975.992i			0.992871i	0.868074	+	0.496435i	$0.165358\pi$
$983$			975.992i			0.992871i	−0.868074	+	0.496435i	$0.834642\pi$
$984$		0			0
$985$			119.341i			0.121159i
$986$		−	137.332i		−	0.139282i
$987$		0			0
$988$	−339.536			−0.343660
$989$	−207.898			−0.210211
$990$		0			0
$991$	459.246			0.463417			0.231708	−	0.972785i	$-0.425568\pi$
$991$	459.246			0.463417			0.231708	+	0.972785i	$0.425568\pi$
$992$			112.836i			0.113746i
$993$		0			0
$994$	−904.595	+	273.028i	−0.910055	+	0.274676i
$995$	674.316			0.677704
$996$		0			0
$997$			745.998i			0.748242i	0.927380	+	0.374121i	$0.122056\pi$
$997$			745.998i			0.748242i	−0.927380	+	0.374121i	$0.877944\pi$
$998$	−123.464			−0.123712
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	630.3.f.c.181.4		8
3.2	odd	2		210.3.f.a.181.7	yes	8
7.6	odd	2	inner	630.3.f.c.181.2		8
12.11	even	2		1680.3.s.a.1441.1		8
15.2	even	4		1050.3.h.b.349.15		16
15.8	even	4		1050.3.h.b.349.2		16
15.14	odd	2		1050.3.f.b.601.1		8
21.20	even	2		210.3.f.a.181.6	✓	8
84.83	odd	2		1680.3.s.a.1441.7		8
105.62	odd	4		1050.3.h.b.349.10		16
105.83	odd	4		1050.3.h.b.349.7		16
105.104	even	2		1050.3.f.b.601.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.f.a.181.6	✓	8	21.20	even	2
210.3.f.a.181.7	yes	8	3.2	odd	2
630.3.f.c.181.2		8	7.6	odd	2	inner
630.3.f.c.181.4		8	1.1	even	1	trivial
1050.3.f.b.601.1		8	15.14	odd	2
1050.3.f.b.601.3		8	105.104	even	2
1050.3.h.b.349.2		16	15.8	even	4
1050.3.h.b.349.7		16	105.83	odd	4
1050.3.h.b.349.10		16	105.62	odd	4
1050.3.h.b.349.15		16	15.2	even	4
1680.3.s.a.1441.1		8	12.11	even	2
1680.3.s.a.1441.7		8	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	630.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +(6.70141 - 2.02265i) q^{7} -2.82843 q^{8} -3.16228i q^{10} -11.9625 q^{11} +9.67294i q^{13} +(-9.47723 + 2.86045i) q^{14} +4.00000 q^{16} +7.09185i q^{17} +17.5508i q^{19} +4.47214i q^{20} +16.9175 q^{22} +2.84923 q^{23} -5.00000 q^{25} -13.6796i q^{26} +(13.4028 - 4.04529i) q^{28} +13.6929 q^{29} -19.9468i q^{31} -5.65685 q^{32} -10.0294i q^{34} +(4.52277 + 14.9848i) q^{35} -11.0672 q^{37} -24.8206i q^{38} -6.32456i q^{40} +28.2902i q^{41} -72.9666 q^{43} -23.9250 q^{44} -4.02942 q^{46} +28.3312i q^{47} +(40.8178 - 27.1092i) q^{49} +7.07107 q^{50} +19.3459i q^{52} +11.7110 q^{53} -26.7490i q^{55} +(-18.9545 + 5.72091i) q^{56} -19.3647 q^{58} +101.160i q^{59} +76.7055i q^{61} +28.2090i q^{62} +8.00000 q^{64} -21.6294 q^{65} -76.2407 q^{67} +14.1837i q^{68} +(-6.39617 - 21.1917i) q^{70} +95.4493 q^{71} +120.684i q^{73} +15.6513 q^{74} +35.1016i q^{76} +(-80.1657 + 24.1959i) q^{77} -14.8096 q^{79} +8.94427i q^{80} -40.0083i q^{82} -60.9570i q^{83} -15.8579 q^{85} +103.190 q^{86} +33.8351 q^{88} +88.6302i q^{89} +(19.5649 + 64.8223i) q^{91} +5.69845 q^{92} -40.0664i q^{94} -39.2448 q^{95} -18.8547i q^{97} +(-57.7251 + 38.3381i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} + 16 q^{11} - 32 q^{14} + 32 q^{16} + 96 q^{22} - 40 q^{25} + 144 q^{29} + 80 q^{35} - 48 q^{37} - 64 q^{43} + 32 q^{44} + 128 q^{46} - 24 q^{49} - 128 q^{53} - 64 q^{56} + 224 q^{58} + 64 q^{64}+ \cdots - 80 q^{95}+O(q^{100}) \) 8 * q + 16 * q^4 + 16 * q^11 - 32 * q^14 + 32 * q^16 + 96 * q^22 - 40 * q^25 + 144 * q^29 + 80 * q^35 - 48 * q^37 - 64 * q^43 + 32 * q^44 + 128 * q^46 - 24 * q^49 - 128 * q^53 - 64 * q^56 + 224 * q^58 + 64 * q^64 - 80 * q^65 - 192 * q^67 - 176 * q^71 + 160 * q^74 - 192 * q^77 - 288 * q^79 - 240 * q^85 + 64 * q^86 + 192 * q^88 + 64 * q^91 - 80 * q^95

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