LMFDB - Embedded newform 672.4.i.c.209.59

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,4,Mod(209,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("672.209"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	672.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$39.6492835239$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.59
Character	$\chi$	$=$	672.209
Dual form			672.4.i.c.209.57

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(3.70908 + 3.63906i) q^{3} +9.02015i q^{5} +(-5.01951 + 17.8271i) q^{7} +(0.514533 + 26.9951i) q^{9} +8.71540 q^{11} -52.4602 q^{13} +(-32.8248 + 33.4564i) q^{15} -69.0503 q^{17} -129.296 q^{19} +(-83.4915 + 47.8557i) q^{21} -177.747i q^{23} +43.6369 q^{25} +(-96.3282 + 101.999i) q^{27} +247.372 q^{29} -52.1678i q^{31} +(32.3261 + 31.7158i) q^{33} +(-160.803 - 45.2767i) q^{35} +299.492i q^{37} +(-194.579 - 190.905i) q^{39} +185.754 q^{41} -330.296i q^{43} +(-243.500 + 4.64117i) q^{45} -81.9844 q^{47} +(-292.609 - 178.966i) q^{49} +(-256.113 - 251.278i) q^{51} +238.316 q^{53} +78.6142i q^{55} +(-479.570 - 470.517i) q^{57} -198.589i q^{59} +157.196 q^{61} +(-483.826 - 126.329i) q^{63} -473.198i q^{65} +527.615i q^{67} +(646.831 - 659.277i) q^{69} +245.212i q^{71} +745.090i q^{73} +(161.853 + 158.797i) q^{75} +(-43.7470 + 155.370i) q^{77} -598.112 q^{79} +(-728.471 + 27.7798i) q^{81} -3.93633i q^{83} -622.844i q^{85} +(917.522 + 900.200i) q^{87} -1258.73 q^{89} +(263.324 - 935.211i) q^{91} +(189.841 - 193.494i) q^{93} -1166.27i q^{95} +755.959i q^{97} +(4.48436 + 235.273i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 64 q^{7} + 104 q^{9} + 8 q^{15} - 976 q^{25} - 568 q^{39} - 4048 q^{49} - 1448 q^{57} + 2152 q^{63} - 4992 q^{79} + 1568 q^{81}+O(q^{100}) $ 80 * q - 64 * q^7 + 104 * q^9 + 8 * q^15 - 976 * q^25 - 568 * q^39 - 4048 * q^49 - 1448 * q^57 + 2152 * q^63 - 4992 * q^79 + 1568 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	3.70908	+	3.63906i	0.713813	+	0.700337i
$3$	3.70908	+	3.63906i	0.713813	+	0.700337i
$4$		0			0
$5$			9.02015i			0.806787i	0.915027	+	0.403393i	$0.132169\pi$
$5$			9.02015i			0.806787i	−0.915027	+	0.403393i	$0.867831\pi$
$6$		0			0
$7$	−5.01951	+	17.8271i	−0.271028	+	0.962571i
$7$	−5.01951	+	17.8271i	−0.271028	+	0.962571i
$8$		0			0
$9$	0.514533	+	26.9951i	0.0190568	+	0.999818i
$10$		0			0
$11$	8.71540			0.238890			0.119445	−	0.992841i	$-0.461888\pi$
$11$	8.71540			0.238890			0.119445	+	0.992841i	$0.461888\pi$
$12$		0			0
$13$	−52.4602			−1.11922			−0.559609	−	0.828757i	$-0.689049\pi$
$13$	−52.4602			−1.11922			−0.559609	+	0.828757i	$0.689049\pi$
$14$		0			0
$15$	−32.8248	+	33.4564i	−0.565022	+	0.575895i
$16$		0			0
$17$	−69.0503			−0.985127			−0.492563	−	0.870277i	$-0.663940\pi$
$17$	−69.0503			−0.985127			−0.492563	+	0.870277i	$0.663940\pi$
$18$		0			0
$19$	−129.296			−1.56119			−0.780595	−	0.625037i	$-0.785084\pi$
$19$	−129.296			−1.56119			−0.780595	+	0.625037i	$0.785084\pi$
$20$		0			0
$21$	−83.4915	+	47.8557i	−0.867587	+	0.497285i
$22$		0			0
$23$		−	177.747i		−	1.61143i	−0.592307	−	0.805713i	$-0.701783\pi$
$23$		−	177.747i		−	1.61143i	0.592307	−	0.805713i	$-0.298217\pi$
$24$		0			0
$25$	43.6369			0.349095
$26$		0			0
$27$	−96.3282	+	101.999i	−0.686607	+	0.727029i
$28$		0			0
$29$	247.372			1.58399			0.791996	−	0.610526i	$-0.209042\pi$
$29$	247.372			1.58399			0.791996	+	0.610526i	$0.209042\pi$
$30$		0			0
$31$		−	52.1678i		−	0.302245i	−0.988515	−	0.151123i	$-0.951711\pi$
$31$		−	52.1678i		−	0.302245i	0.988515	−	0.151123i	$-0.0482888\pi$
$32$		0			0
$33$	32.3261	+	31.7158i	0.170523	+	0.167304i
$34$		0			0
$35$	−160.803	−	45.2767i	−0.776590	−	0.218662i
$36$		0			0
$37$			299.492i			1.33071i	0.746528	+	0.665354i	$0.231719\pi$
$37$			299.492i			1.33071i	−0.746528	+	0.665354i	$0.768281\pi$
$38$		0			0
$39$	−194.579	−	190.905i	−0.798912	−	0.783829i
$40$		0			0
$41$	185.754			0.707559			0.353780	−	0.935329i	$-0.384896\pi$
$41$	185.754			0.707559			0.353780	+	0.935329i	$0.384896\pi$
$42$		0			0
$43$		−	330.296i		−	1.17139i	−0.810532	−	0.585695i	$-0.800822\pi$
$43$		−	330.296i		−	1.17139i	0.810532	−	0.585695i	$-0.199178\pi$
$44$		0			0
$45$	−243.500	+	4.64117i	−0.806640	+	0.0153748i
$46$		0			0
$47$	−81.9844			−0.254440			−0.127220	−	0.991875i	$-0.540605\pi$
$47$	−81.9844			−0.254440			−0.127220	+	0.991875i	$0.540605\pi$
$48$		0			0
$49$	−292.609	−	178.966i	−0.853088	−	0.521768i
$50$		0			0
$51$	−256.113	−	251.278i	−0.703196	−	0.689920i
$52$		0			0
$53$	238.316			0.617646			0.308823	−	0.951120i	$-0.400065\pi$
$53$	238.316			0.617646			0.308823	+	0.951120i	$0.400065\pi$
$54$		0			0
$55$			78.6142i			0.192733i
$56$		0			0
$57$	−479.570	−	470.517i	−1.11440	−	1.09336i
$58$		0			0
$59$		−	198.589i		−	0.438204i	−0.975702	−	0.219102i	$-0.929687\pi$
$59$		−	198.589i		−	0.438204i	0.975702	−	0.219102i	$-0.0703128\pi$
$60$		0			0
$61$	157.196			0.329950			0.164975	−	0.986298i	$-0.447246\pi$
$61$	157.196			0.329950			0.164975	+	0.986298i	$0.447246\pi$
$62$		0			0
$63$	−483.826	−	126.329i	−0.967562	−	0.252635i
$64$		0			0
$65$		−	473.198i		−	0.902970i
$66$		0			0
$67$			527.615i			0.962066i	0.876703	+	0.481033i	$0.159738\pi$
$67$			527.615i			0.962066i	−0.876703	+	0.481033i	$0.840262\pi$
$68$		0			0
$69$	646.831	−	659.277i	1.12854	−	1.15026i
$70$		0			0
$71$			245.212i			0.409878i	0.978775	+	0.204939i	$0.0656996\pi$
$71$			245.212i			0.409878i	−0.978775	+	0.204939i	$0.934300\pi$
$72$		0			0
$73$			745.090i			1.19461i	0.802016	+	0.597303i	$0.203761\pi$
$73$			745.090i			1.19461i	−0.802016	+	0.597303i	$0.796239\pi$
$74$		0			0
$75$	161.853	+	158.797i	0.249188	+	0.244484i
$76$		0			0
$77$	−43.7470	+	155.370i	−0.0647459	+	0.229949i
$78$		0			0
$79$	−598.112			−0.851808			−0.425904	−	0.904768i	$-0.640044\pi$
$79$	−598.112			−0.851808			−0.425904	+	0.904768i	$0.640044\pi$
$80$		0			0
$81$	−728.471	+	27.7798i	−0.999274	+	0.0381067i
$82$		0			0
$83$		−	3.93633i		−	0.00520564i	−0.999997	−	0.00260282i	$-0.999171\pi$
$83$		−	3.93633i		−	0.00520564i	0.999997	−	0.00260282i	$-0.000828505\pi$
$84$		0			0
$85$		−	622.844i		−	0.794787i
$86$		0			0
$87$	917.522	+	900.200i	1.13067	+	1.10933i
$88$		0			0
$89$	−1258.73			−1.49916			−0.749579	−	0.661915i	$-0.769744\pi$
$89$	−1258.73			−1.49916			−0.749579	+	0.661915i	$0.769744\pi$
$90$		0			0
$91$	263.324	−	935.211i	0.303339	−	1.07733i
$92$		0			0
$93$	189.841	−	193.494i	0.211674	−	0.215747i
$94$		0			0
$95$		−	1166.27i		−	1.25955i
$96$		0			0
$97$			755.959i			0.791300i	0.918401	+	0.395650i	$0.129481\pi$
$97$			755.959i			0.791300i	−0.918401	+	0.395650i	$0.870519\pi$
$98$		0			0
$99$	4.48436	+	235.273i	0.00455248	+	0.238847i
$100$		0			0
$101$			1045.75i			1.03026i	0.857113	+	0.515128i	$0.172256\pi$
$101$			1045.75i			1.03026i	−0.857113	+	0.515128i	$0.827744\pi$
$102$		0			0
$103$		−	1520.53i		−	1.45458i	−0.686328	−	0.727292i	$-0.740779\pi$
$103$		−	1520.53i		−	1.45458i	0.686328	−	0.727292i	$-0.259221\pi$
$104$		0			0
$105$	−431.666	−	753.106i	−0.401203	−	0.699958i
$106$		0			0
$107$	283.267			0.255930			0.127965	−	0.991779i	$-0.459156\pi$
$107$	283.267			0.255930			0.127965	+	0.991779i	$0.459156\pi$
$108$		0			0
$109$			1180.64i			1.03747i	0.854935	+	0.518736i	$0.173597\pi$
$109$			1180.64i			1.03747i	−0.854935	+	0.518736i	$0.826403\pi$
$110$		0			0
$111$	−1089.87	+	1110.84i	−0.931943	+	0.949876i
$112$		0			0
$113$		−	1867.56i		−	1.55473i	−0.629047	−	0.777367i	$-0.716555\pi$
$113$		−	1867.56i		−	1.55473i	0.629047	−	0.777367i	$-0.283445\pi$
$114$		0			0
$115$	1603.30			1.30008
$116$		0			0
$117$	−26.9925	−	1416.17i	−0.0213287	−	1.11901i
$118$		0			0
$119$	346.598	−	1230.96i	0.266997	−	0.948255i
$120$		0			0
$121$	−1255.04			−0.942932
$122$		0			0
$123$	688.977	+	675.970i	0.505065	+	0.495530i
$124$		0			0
$125$			1521.13i			1.08843i
$126$		0			0
$127$	−960.566			−0.671153			−0.335577	−	0.942013i	$-0.608931\pi$
$127$	−960.566			−0.671153			−0.335577	+	0.942013i	$0.608931\pi$
$128$		0			0
$129$	1201.97	−	1225.10i	0.820367	−	0.836152i
$130$		0			0
$131$			1477.86i			0.985659i	0.870126	+	0.492829i	$0.164037\pi$
$131$			1477.86i			0.985659i	−0.870126	+	0.492829i	$0.835963\pi$
$132$		0			0
$133$	649.004	−	2304.97i	0.423126	−	1.50276i
$134$		0			0
$135$	−920.050	−	868.895i	−0.586557	−	0.553945i
$136$		0			0
$137$			344.829i			0.215042i	0.994203	+	0.107521i	$0.0342913\pi$
$137$			344.829i			0.215042i	−0.994203	+	0.107521i	$0.965709\pi$
$138$		0			0
$139$	−1156.15			−0.705489			−0.352745	−	0.935720i	$-0.614752\pi$
$139$	−1156.15			−0.705489			−0.352745	+	0.935720i	$0.614752\pi$
$140$		0			0
$141$	−304.087	−	298.346i	−0.181622	−	0.178193i
$142$		0			0
$143$	−457.211			−0.267370
$144$		0			0
$145$			2231.33i			1.27794i
$146$		0			0
$147$	−434.042	−	1728.62i	−0.243532	−	0.969893i
$148$		0			0
$149$	−1849.45			−1.01687			−0.508433	−	0.861102i	$-0.669775\pi$
$149$	−1849.45			−1.01687			−0.508433	+	0.861102i	$0.669775\pi$
$150$		0			0
$151$	−2628.03			−1.41633			−0.708167	−	0.706045i	$-0.750477\pi$
$151$	−2628.03			−1.41633			−0.708167	+	0.706045i	$0.750477\pi$
$152$		0			0
$153$	−35.5287	−	1864.02i	−0.0187734	−	0.984948i
$154$		0			0
$155$	470.561			0.243848
$156$		0			0
$157$	−1909.32			−0.970577			−0.485288	−	0.874354i	$-0.661285\pi$
$157$	−1909.32			−0.970577			−0.485288	+	0.874354i	$0.661285\pi$
$158$		0			0
$159$	883.933	+	867.246i	0.440883	+	0.432560i
$160$		0			0
$161$	3168.71	+	892.202i	1.55111	+	0.436741i
$162$		0			0
$163$			3458.92i			1.66211i	0.556191	+	0.831054i	$0.312262\pi$
$163$			3458.92i			1.66211i	−0.556191	+	0.831054i	$0.687738\pi$
$164$		0			0
$165$	−286.081	+	291.586i	−0.134978	+	0.137575i
$166$		0			0
$167$	−2306.06			−1.06855			−0.534277	−	0.845310i	$-0.679416\pi$
$167$	−2306.06			−1.06855			−0.534277	+	0.845310i	$0.679416\pi$
$168$		0			0
$169$	555.068			0.252648
$170$		0			0
$171$	−66.5273	−	3490.37i	−0.0297513	−	1.56091i
$172$		0			0
$173$		−	18.1258i		−	0.00796576i	−0.999992	−	0.00398288i	$-0.998732\pi$
$173$		−	18.1258i		−	0.00796576i	0.999992	−	0.00398288i	$-0.00126779\pi$
$174$		0			0
$175$	−219.036	+	777.918i	−0.0946146	+	0.336029i
$176$		0			0
$177$	722.675	−	736.581i	0.306890	−	0.312796i
$178$		0			0
$179$	2596.01			1.08400			0.541998	−	0.840380i	$-0.317668\pi$
$179$	2596.01			1.08400			0.541998	+	0.840380i	$0.317668\pi$
$180$		0			0
$181$	−1200.67			−0.493069			−0.246534	−	0.969134i	$-0.579292\pi$
$181$	−1200.67			−0.493069			−0.246534	+	0.969134i	$0.579292\pi$
$182$		0			0
$183$	583.054	+	572.047i	0.235522	+	0.231076i
$184$		0			0
$185$	−2701.46			−1.07360
$186$		0			0
$187$	−601.801			−0.235337
$188$		0			0
$189$	−1334.83	−	2229.24i	−0.513728	−	0.857953i
$190$		0			0
$191$			1778.65i			0.673816i	0.941538	+	0.336908i	$0.109381\pi$
$191$			1778.65i			0.673816i	−0.941538	+	0.336908i	$0.890619\pi$
$192$		0			0
$193$	151.059			0.0563393			0.0281697	−	0.999603i	$-0.491032\pi$
$193$	151.059			0.0563393			0.0281697	+	0.999603i	$0.491032\pi$
$194$		0			0
$195$	1722.00	−	1755.13i	0.632383	−	0.644551i
$196$		0			0
$197$	−346.401			−0.125280			−0.0626398	−	0.998036i	$-0.519952\pi$
$197$	−346.401			−0.125280			−0.0626398	+	0.998036i	$0.519952\pi$
$198$		0			0
$199$		−	3780.44i		−	1.34668i	−0.739335	−	0.673338i	$-0.764860\pi$
$199$		−	3780.44i		−	1.34668i	0.739335	−	0.673338i	$-0.235140\pi$
$200$		0			0
$201$	−1920.02	+	1956.97i	−0.673770	+	0.686735i
$202$		0			0
$203$	−1241.69	+	4409.92i	−0.429306	+	1.52471i
$204$		0			0
$205$			1675.53i			0.570850i
$206$		0			0
$207$	4798.29	−	91.4567i	1.61113	−	0.0307086i
$208$		0			0
$209$	−1126.87			−0.372953
$210$		0			0
$211$		−	2168.92i		−	0.707652i	−0.935311	−	0.353826i	$-0.884881\pi$
$211$		−	2168.92i		−	0.707652i	0.935311	−	0.353826i	$-0.115119\pi$
$212$		0			0
$213$	−892.341	+	909.511i	−0.287053	+	0.292576i
$214$		0			0
$215$	2979.32			0.945061
$216$		0			0
$217$	929.998	+	261.857i	0.290933	+	0.0819169i
$218$		0			0
$219$	−2711.43	+	2763.60i	−0.836626	+	0.852725i
$220$		0			0
$221$	3622.39			1.10257
$222$		0			0
$223$			3892.25i			1.16881i	0.811462	+	0.584405i	$0.198672\pi$
$223$			3892.25i			1.16881i	−0.811462	+	0.584405i	$0.801328\pi$
$224$		0			0
$225$	22.4526	+	1177.98i	0.00665263	+	0.349032i
$226$		0			0
$227$			2544.85i			0.744085i	0.928216	+	0.372043i	$0.121342\pi$
$227$			2544.85i			0.744085i	−0.928216	+	0.372043i	$0.878658\pi$
$228$		0			0
$229$	775.951			0.223914			0.111957	−	0.993713i	$-0.464288\pi$
$229$	775.951			0.223914			0.111957	+	0.993713i	$0.464288\pi$
$230$		0			0
$231$	−727.661	+	417.082i	−0.207258	+	0.118796i
$232$		0			0
$233$		−	475.828i		−	0.133788i	−0.997760	−	0.0668938i	$-0.978691\pi$
$233$		−	475.828i		−	0.133788i	0.997760	−	0.0668938i	$-0.0213089\pi$
$234$		0			0
$235$		−	739.512i		−	0.205278i
$236$		0			0
$237$	−2218.44	−	2176.56i	−0.608031	−	0.596552i
$238$		0			0
$239$			295.661i			0.0800196i	0.999199	+	0.0400098i	$0.0127389\pi$
$239$			295.661i			0.0800196i	−0.999199	+	0.0400098i	$0.987261\pi$
$240$		0			0
$241$		−	3284.88i		−	0.877998i	−0.898488	−	0.438999i	$-0.855333\pi$
$241$		−	3284.88i		−	0.877998i	0.898488	−	0.438999i	$-0.144667\pi$
$242$		0			0
$243$	−2803.05	−	2547.91i	−0.739982	−	0.672627i
$244$		0			0
$245$	1614.30	−	2639.38i	0.420955	−	0.688260i
$246$		0			0
$247$	6782.90			1.74731
$248$		0			0
$249$	14.3245	−	14.6002i	0.00364570	−	0.00371585i
$250$		0			0
$251$		−	1029.64i		−	0.258926i	−0.991584	−	0.129463i	$-0.958675\pi$
$251$		−	1029.64i		−	0.258926i	0.991584	−	0.129463i	$-0.0413253\pi$
$252$		0			0
$253$		−	1549.13i		−	0.384953i
$254$		0			0
$255$	2266.56	−	2310.18i	0.556619	−	0.567329i
$256$		0			0
$257$	2451.85			0.595106			0.297553	−	0.954705i	$-0.403830\pi$
$257$	2451.85			0.595106			0.297553	+	0.954705i	$0.403830\pi$
$258$		0			0
$259$	−5339.06	−	1503.30i	−1.28090	−	0.360659i
$260$		0			0
$261$	127.281	+	6677.83i	0.0301858	+	1.58371i
$262$		0			0
$263$			4396.49i			1.03080i	0.856951	+	0.515398i	$0.172356\pi$
$263$			4396.49i			1.03080i	−0.856951	+	0.515398i	$0.827644\pi$
$264$		0			0
$265$			2149.65i			0.498308i
$266$		0			0
$267$	−4668.73	−	4580.59i	−1.07012	−	1.04992i
$268$		0			0
$269$			3402.69i			0.771248i	0.922656	+	0.385624i	$0.126014\pi$
$269$			3402.69i			0.771248i	−0.922656	+	0.385624i	$0.873986\pi$
$270$		0			0
$271$		−	4179.66i		−	0.936887i	−0.883493	−	0.468444i	$-0.844815\pi$
$271$		−	4179.66i		−	0.936887i	0.883493	−	0.468444i	$-0.155185\pi$
$272$		0			0
$273$	4379.98	−	2510.52i	0.971019	−	0.556570i
$274$		0			0
$275$	380.313			0.0833954
$276$		0			0
$277$			2605.51i			0.565163i	0.959243	+	0.282581i	$0.0911907\pi$
$277$			2605.51i			0.565163i	−0.959243	+	0.282581i	$0.908809\pi$
$278$		0			0
$279$	1408.27	−	26.8421i	0.302190	−	0.00575983i
$280$		0			0
$281$			7988.99i			1.69603i	0.529976	+	0.848013i	$0.322201\pi$
$281$			7988.99i			1.69603i	−0.529976	+	0.848013i	$0.677799\pi$
$282$		0			0
$283$	3622.36			0.760873			0.380437	−	0.924807i	$-0.375774\pi$
$283$	3622.36			0.760873			0.380437	+	0.924807i	$0.375774\pi$
$284$		0			0
$285$	4244.13	−	4325.79i	0.882107	−	0.899081i
$286$		0			0
$287$	−932.395	+	3311.45i	−0.191768	+	0.681077i
$288$		0			0
$289$	−145.058			−0.0295254
$290$		0			0
$291$	−2750.98	+	2803.91i	−0.554176	+	0.564840i
$292$		0			0
$293$			8260.81i			1.64710i	0.567241	+	0.823552i	$0.308011\pi$
$293$			8260.81i			1.64710i	−0.567241	+	0.823552i	$0.691989\pi$
$294$		0			0
$295$	1791.30			0.353537
$296$		0			0
$297$	−839.539	+	888.965i	−0.164024	+	0.173680i
$298$		0			0
$299$			9324.63i			1.80354i
$300$		0			0
$301$	5888.22	+	1657.93i	1.12755	+	0.317479i
$302$		0			0
$303$	−3805.54	+	3878.77i	−0.721527	+	0.735410i
$304$		0			0
$305$			1417.94i			0.266199i
$306$		0			0
$307$	6375.65			1.18527			0.592635	−	0.805471i	$-0.298088\pi$
$307$	6375.65			1.18527			0.592635	+	0.805471i	$0.298088\pi$
$308$		0			0
$309$	5533.29	−	5639.76i	1.01870	−	1.03830i
$310$		0			0
$311$	3566.99			0.650371			0.325186	−	0.945650i	$-0.394573\pi$
$311$	3566.99			0.650371			0.325186	+	0.945650i	$0.394573\pi$
$312$		0			0
$313$		−	2364.18i		−	0.426936i	−0.976950	−	0.213468i	$-0.931524\pi$
$313$		−	2364.18i		−	0.426936i	0.976950	−	0.213468i	$-0.0684760\pi$
$314$		0			0
$315$	1139.51	−	4364.19i	0.203823	−	0.780616i
$316$		0			0
$317$	5296.44			0.938415			0.469207	−	0.883088i	$-0.344540\pi$
$317$	5296.44			0.938415			0.469207	+	0.883088i	$0.344540\pi$
$318$		0			0
$319$	2155.94			0.378400
$320$		0			0
$321$	1050.66	+	1030.83i	0.182686	+	0.179237i
$322$		0			0
$323$	8927.95			1.53797
$324$		0			0
$325$	−2289.20			−0.390713
$326$		0			0
$327$	−4296.40	+	4379.07i	−0.726579	+	0.740560i
$328$		0			0
$329$	411.522	−	1461.54i	0.0689602	−	0.244916i
$330$		0			0
$331$		−	664.908i		−	0.110413i	−0.998475	−	0.0552064i	$-0.982418\pi$
$331$		−	664.908i		−	0.110413i	0.998475	−	0.0552064i	$-0.0175817\pi$
$332$		0			0
$333$	−8084.81	+	154.099i	−1.33047	+	0.0253590i
$334$		0			0
$335$	−4759.17			−0.776182
$336$		0			0
$337$	8618.15			1.39306			0.696529	−	0.717529i	$-0.254727\pi$
$337$	8618.15			1.39306			0.696529	+	0.717529i	$0.254727\pi$
$338$		0			0
$339$	6796.15	−	6926.92i	1.08884	−	1.10979i
$340$		0			0
$341$		−	454.663i		−	0.0722034i
$342$		0			0
$343$	4659.20	−	4318.04i	0.733449	−	0.679744i
$344$		0			0
$345$	5946.78	+	5834.51i	0.928011	+	0.910491i
$346$		0			0
$347$	−11525.2			−1.78301			−0.891507	−	0.453006i	$-0.850352\pi$
$347$	−11525.2			−1.78301			−0.891507	+	0.453006i	$0.850352\pi$
$348$		0			0
$349$	−232.968			−0.0357320			−0.0178660	−	0.999840i	$-0.505687\pi$
$349$	−232.968			−0.0357320			−0.0178660	+	0.999840i	$0.505687\pi$
$350$		0			0
$351$	5053.39	−	5350.90i	0.768462	−	0.813704i
$352$		0			0
$353$	10995.6			1.65789			0.828944	−	0.559331i	$-0.188942\pi$
$353$	10995.6			1.65789			0.828944	+	0.559331i	$0.188942\pi$
$354$		0			0
$355$	−2211.85			−0.330684
$356$		0			0
$357$	5765.11	−	3304.45i	0.854683	−	0.489888i
$358$		0			0
$359$		−	3732.36i		−	0.548709i	−0.961629	−	0.274354i	$-0.911536\pi$
$359$		−	3732.36i		−	0.548709i	0.961629	−	0.274354i	$-0.0884642\pi$
$360$		0			0
$361$	9858.53			1.43731
$362$		0			0
$363$	−4655.05	−	4567.17i	−0.673076	−	0.660370i
$364$		0			0
$365$	−6720.83			−0.963792
$366$		0			0
$367$			1187.53i			0.168907i	0.996427	+	0.0844534i	$0.0269144\pi$
$367$			1187.53i			0.168907i	−0.996427	+	0.0844534i	$0.973086\pi$
$368$		0			0
$369$	95.5768	+	5014.45i	0.0134838	+	0.707431i
$370$		0			0
$371$	−1196.23	+	4248.48i	−0.167399	+	0.594528i
$372$		0			0
$373$			4872.40i			0.676362i	0.941081	+	0.338181i	$0.109812\pi$
$373$			4872.40i			0.676362i	−0.941081	+	0.338181i	$0.890188\pi$
$374$		0			0
$375$	−5535.48	+	5641.99i	−0.762269	+	0.776937i
$376$		0			0
$377$	−12977.2			−1.77283
$378$		0			0
$379$		−	3969.71i		−	0.538022i	−0.963137	−	0.269011i	$-0.913303\pi$
$379$		−	3969.71i		−	0.538022i	0.963137	−	0.269011i	$-0.0866967\pi$
$380$		0			0
$381$	−3562.82	−	3495.55i	−0.479078	−	0.470033i
$382$		0			0
$383$	−2981.45			−0.397768			−0.198884	−	0.980023i	$-0.563732\pi$
$383$	−2981.45			−0.397768			−0.198884	+	0.980023i	$0.563732\pi$
$384$		0			0
$385$	−1401.46	−	394.605i	−0.185520	−	0.0522361i
$386$		0			0
$387$	8916.38	−	169.949i	1.17118	−	0.0223229i
$388$		0			0
$389$	477.774			0.0622728			0.0311364	−	0.999515i	$-0.490087\pi$
$389$	477.774			0.0622728			0.0311364	+	0.999515i	$0.490087\pi$
$390$		0			0
$391$			12273.5i			1.58746i
$392$		0			0
$393$	−5378.02	+	5481.50i	−0.690293	+	0.703576i
$394$		0			0
$395$		−	5395.06i		−	0.687227i
$396$		0			0
$397$	6740.60			0.852143			0.426072	−	0.904689i	$-0.359897\pi$
$397$	6740.60			0.852143			0.426072	+	0.904689i	$0.359897\pi$
$398$		0			0
$399$	10795.1	−	6187.57i	1.35447	−	0.776356i
$400$		0			0
$401$			10739.9i			1.33746i	0.743504	+	0.668732i	$0.233163\pi$
$401$			10739.9i			1.33746i	−0.743504	+	0.668732i	$0.766837\pi$
$402$		0			0
$403$			2736.73i			0.338278i
$404$		0			0
$405$	−250.578	−	6570.91i	−0.0307440	−	0.806201i
$406$		0			0
$407$			2610.19i			0.317893i
$408$		0			0
$409$		−	15374.7i		−	1.85875i	−0.369139	−	0.929374i	$-0.620347\pi$
$409$		−	15374.7i		−	1.85875i	0.369139	−	0.929374i	$-0.379653\pi$
$410$		0			0
$411$	−1254.85	+	1279.00i	−0.150602	+	0.153500i
$412$		0			0
$413$	3540.25	+	996.817i	0.421803	+	0.118766i
$414$		0			0
$415$	35.5063			0.00419984
$416$		0			0
$417$	−4288.23	−	4207.28i	−0.503587	−	0.494080i
$418$		0			0
$419$			1812.00i			0.211269i	0.994405	+	0.105635i	$0.0336874\pi$
$419$			1812.00i			0.211269i	−0.994405	+	0.105635i	$0.966313\pi$
$420$		0			0
$421$		−	1588.18i		−	0.183855i	−0.995766	−	0.0919276i	$-0.970697\pi$
$421$		−	1588.18i		−	0.183855i	0.995766	−	0.0919276i	$-0.0293028\pi$
$422$		0			0
$423$	−42.1837	−	2213.18i	−0.00484880	−	0.254393i
$424$		0			0
$425$	−3013.14			−0.343903
$426$		0			0
$427$	−789.049	+	2802.35i	−0.0894257	+	0.317600i
$428$		0			0
$429$	−1695.83	−	1663.82i	−0.190852	−	0.187249i
$430$		0			0
$431$		−	2266.42i		−	0.253294i	−0.991948	−	0.126647i	$-0.959578\pi$
$431$		−	2266.42i		−	0.253294i	0.991948	−	0.126647i	$-0.0404216\pi$
$432$		0			0
$433$			12900.1i			1.43173i	0.698238	+	0.715865i	$0.253968\pi$
$433$			12900.1i			1.43173i	−0.698238	+	0.715865i	$0.746032\pi$
$434$		0			0
$435$	−8119.94	+	8276.18i	−0.894992	+	0.912213i
$436$		0			0
$437$			22982.0i			2.51574i
$438$		0			0
$439$			3778.78i			0.410823i	0.978676	+	0.205412i	$0.0658533\pi$
$439$			3778.78i			0.410823i	−0.978676	+	0.205412i	$0.934147\pi$
$440$		0			0
$441$	4680.66	−	7991.09i	0.505416	−	0.862876i
$442$		0			0
$443$	372.016			0.0398985			0.0199492	−	0.999801i	$-0.493650\pi$
$443$	372.016			0.0398985			0.0199492	+	0.999801i	$0.493650\pi$
$444$		0			0
$445$		−	11353.9i		−	1.20950i
$446$		0			0
$447$	−6859.76	−	6730.26i	−0.725852	−	0.712149i
$448$		0			0
$449$		−	12340.3i		−	1.29705i	−0.761192	−	0.648526i	$-0.775386\pi$
$449$		−	12340.3i		−	1.29705i	0.761192	−	0.648526i	$-0.224614\pi$
$450$		0			0
$451$	1618.92			0.169029
$452$		0			0
$453$	−9747.58	−	9563.56i	−1.01100	−	0.991910i
$454$		0			0
$455$	8435.74	+	2375.22i	0.869173	+	0.244730i
$456$		0			0
$457$	−13249.9			−1.35625			−0.678124	−	0.734947i	$-0.737207\pi$
$457$	−13249.9			−1.35625			−0.678124	+	0.734947i	$0.737207\pi$
$458$		0			0
$459$	6651.49	−	7043.08i	0.676394	−	0.716216i
$460$		0			0
$461$		−	3328.58i		−	0.336285i	−0.985763	−	0.168143i	$-0.946223\pi$
$461$		−	3328.58i		−	0.336285i	0.985763	−	0.168143i	$-0.0537769\pi$
$462$		0			0
$463$	−4155.03			−0.417064			−0.208532	−	0.978016i	$-0.566869\pi$
$463$	−4155.03			−0.417064			−0.208532	+	0.978016i	$0.566869\pi$
$464$		0			0
$465$	1745.35	+	1712.40i	0.174061	+	0.170775i
$466$		0			0
$467$		−	497.911i		−	0.0493374i	−0.999696	−	0.0246687i	$-0.992147\pi$
$467$		−	497.911i		−	0.0493374i	0.999696	−	0.0246687i	$-0.00785309\pi$
$468$		0			0
$469$	−9405.83	−	2648.37i	−0.926057	−	0.260747i
$470$		0			0
$471$	−7081.83	−	6948.14i	−0.692810	−	0.679731i
$472$		0			0
$473$		−	2878.66i		−	0.279833i
$474$		0			0
$475$	−5642.09			−0.545004
$476$		0			0
$477$	122.622	+	6433.36i	0.0117703	+	0.617534i
$478$		0			0
$479$	20052.8			1.91281			0.956406	−	0.292041i	$-0.0943345\pi$
$479$	20052.8			1.91281			0.956406	+	0.292041i	$0.0943345\pi$
$480$		0			0
$481$		−	15711.4i		−	1.48935i
$482$		0			0
$483$	8506.21	+	14840.3i	0.801337	+	1.39805i
$484$		0			0
$485$	−6818.87			−0.638410
$486$		0			0
$487$	5918.14			0.550670			0.275335	−	0.961348i	$-0.411211\pi$
$487$	5918.14			0.550670			0.275335	+	0.961348i	$0.411211\pi$
$488$		0			0
$489$	−12587.2	+	12829.4i	−1.16404	+	1.18643i
$490$		0			0
$491$	2586.64			0.237746			0.118873	−	0.992909i	$-0.462072\pi$
$491$	2586.64			0.237746			0.118873	+	0.992909i	$0.462072\pi$
$492$		0			0
$493$	−17081.1			−1.56043
$494$		0			0
$495$	−2122.20	+	40.4496i	−0.192698	+	0.00367288i
$496$		0			0
$497$	−4371.41	−	1230.84i	−0.394537	−	0.111088i
$498$		0			0
$499$			18254.1i			1.63761i	0.574072	+	0.818805i	$0.305363\pi$
$499$			18254.1i			1.63761i	−0.574072	+	0.818805i	$0.694637\pi$
$500$		0			0
$501$	−8553.37	−	8391.89i	−0.762747	−	0.748347i
$502$		0			0
$503$	−10858.4			−0.962531			−0.481265	−	0.876575i	$-0.659823\pi$
$503$	−10858.4			−0.962531			−0.481265	+	0.876575i	$0.659823\pi$
$504$		0			0
$505$	−9432.81			−0.831197
$506$		0			0
$507$	2058.79	+	2019.92i	0.180343	+	0.176939i
$508$		0			0
$509$			5017.32i			0.436913i	0.975847	+	0.218456i	$0.0701022\pi$
$509$			5017.32i			0.436913i	−0.975847	+	0.218456i	$0.929898\pi$
$510$		0			0
$511$	−13282.8	−	3739.99i	−1.14989	−	0.323772i
$512$		0			0
$513$	12454.9	−	13188.1i	1.07192	−	1.13503i
$514$		0			0
$515$	13715.4			1.17354
$516$		0			0
$517$	−714.527			−0.0607831
$518$		0			0
$519$	65.9607	−	67.2299i	0.00557871	−	0.00568606i
$520$		0			0
$521$	13237.4			1.11313			0.556564	−	0.830805i	$-0.312119\pi$
$521$	13237.4			1.11313			0.556564	+	0.830805i	$0.312119\pi$
$522$		0			0
$523$	−17806.0			−1.48872			−0.744361	−	0.667777i	$-0.767246\pi$
$523$	−17806.0			−1.48872			−0.744361	+	0.667777i	$0.767246\pi$
$524$		0			0
$525$	−3643.31	+	2088.28i	−0.302871	+	0.173600i
$526$		0			0
$527$			3602.20i			0.297750i
$528$		0			0
$529$	−19426.9			−1.59669
$530$		0			0
$531$	5360.92	−	102.180i	0.438124	−	0.00835076i
$532$		0			0
$533$	−9744.70			−0.791913
$534$		0			0
$535$			2555.11i			0.206481i
$536$		0			0
$537$	9628.82	+	9447.04i	0.773770	+	0.759162i
$538$		0			0
$539$	−2550.20	−	1559.76i	−0.203794	−	0.124645i
$540$		0			0
$541$		−	7729.85i		−	0.614292i	−0.951662	−	0.307146i	$-0.900626\pi$
$541$		−	7729.85i		−	0.614292i	0.951662	−	0.307146i	$-0.0993740\pi$
$542$		0			0
$543$	−4453.40	−	4369.32i	−0.351959	−	0.345314i
$544$		0			0
$545$	−10649.5			−0.837018
$546$		0			0
$547$			9563.57i			0.747548i	0.927520	+	0.373774i	$0.121936\pi$
$547$			9563.57i			0.747548i	−0.927520	+	0.373774i	$0.878064\pi$
$548$		0			0
$549$	80.8828	+	4243.53i	0.00628779	+	0.329890i
$550$		0			0
$551$	−31984.3			−2.47291
$552$		0			0
$553$	3002.23	−	10662.6i	0.230864	−	0.819926i
$554$		0			0
$555$	−10019.9	−	9830.77i	−0.766347	−	0.751880i
$556$		0			0
$557$	16797.4			1.27779			0.638894	−	0.769295i	$-0.279392\pi$
$557$	16797.4			1.27779			0.638894	+	0.769295i	$0.279392\pi$
$558$		0			0
$559$			17327.4i			1.31104i
$560$		0			0
$561$	−2232.13	−	2189.99i	−0.167986	−	0.164815i
$562$		0			0
$563$		−	15342.1i		−	1.14848i	−0.818688	−	0.574239i	$-0.805298\pi$
$563$		−	15342.1i		−	1.14848i	0.818688	−	0.574239i	$-0.194702\pi$
$564$		0			0
$565$	16845.6			1.25434
$566$		0			0
$567$	3161.33	−	13125.9i	0.234151	−	0.972200i
$568$		0			0
$569$		−	23658.3i		−	1.74307i	−0.490329	−	0.871537i	$-0.663123\pi$
$569$		−	23658.3i		−	1.74307i	0.490329	−	0.871537i	$-0.336877\pi$
$570$		0			0
$571$		−	20556.7i		−	1.50661i	−0.657673	−	0.753304i	$-0.728459\pi$
$571$		−	20556.7i		−	1.50661i	0.657673	−	0.753304i	$-0.271541\pi$
$572$		0			0
$573$	−6472.62	+	6597.17i	−0.471898	+	0.480978i
$574$		0			0
$575$		−	7756.32i		−	0.562541i
$576$		0			0
$577$		−	1404.98i		−	0.101369i	−0.998715	−	0.0506847i	$-0.983860\pi$
$577$		−	1404.98i		−	0.101369i	0.998715	−	0.0506847i	$-0.0161404\pi$
$578$		0			0
$579$	560.291	+	549.713i	0.0402157	+	0.0394565i
$580$		0			0
$581$	70.1732	+	19.7584i	0.00501080	+	0.00141087i
$582$		0			0
$583$	2077.02			0.147549
$584$		0			0
$585$	12774.0	−	243.476i	0.902806	−	0.0172077i
$586$		0			0
$587$			14366.9i			1.01020i	0.863062	+	0.505098i	$0.168544\pi$
$587$			14366.9i			1.01020i	−0.863062	+	0.505098i	$0.831456\pi$
$588$		0			0
$589$			6745.10i			0.471862i
$590$		0			0
$591$	−1284.83	−	1260.57i	−0.0894262	−	0.0877379i
$592$		0			0
$593$	−5376.81			−0.372343			−0.186171	−	0.982517i	$-0.559608\pi$
$593$	−5376.81			−0.372343			−0.186171	+	0.982517i	$0.559608\pi$
$594$		0			0
$595$	11103.5	+	3126.37i	0.765039	+	0.215410i
$596$		0			0
$597$	13757.2	−	14022.0i	0.943127	−	0.961274i
$598$		0			0
$599$		−	15516.0i		−	1.05837i	−0.848505	−	0.529187i	$-0.822497\pi$
$599$		−	15516.0i		−	1.05837i	0.848505	−	0.529187i	$-0.177503\pi$
$600$		0			0
$601$			28358.0i			1.92470i	0.271811	+	0.962351i	$0.412378\pi$
$601$			28358.0i			1.92470i	−0.271811	+	0.962351i	$0.587622\pi$
$602$		0			0
$603$	−14243.0	+	271.476i	−0.961891	+	0.0183339i
$604$		0			0
$605$		−	11320.7i		−	0.760745i
$606$		0			0
$607$			16509.3i			1.10394i	0.833865	+	0.551969i	$0.186123\pi$
$607$			16509.3i			1.10394i	−0.833865	+	0.551969i	$0.813877\pi$
$608$		0			0
$609$	−20653.4	+	11838.2i	−1.37425	+	0.787696i
$610$		0			0
$611$	4300.92			0.284773
$612$		0			0
$613$			8227.79i			0.542117i	0.962563	+	0.271058i	$0.0873736\pi$
$613$			8227.79i			0.542117i	−0.962563	+	0.271058i	$0.912626\pi$
$614$		0			0
$615$	−6097.35	+	6214.68i	−0.399787	+	0.407480i
$616$		0			0
$617$		−	2275.68i		−	0.148485i	−0.997240	−	0.0742425i	$-0.976346\pi$
$617$		−	2275.68i		−	0.148485i	0.997240	−	0.0742425i	$-0.0236539\pi$
$618$		0			0
$619$	19550.8			1.26949			0.634743	−	0.772723i	$-0.281106\pi$
$619$	19550.8			1.26949			0.634743	+	0.772723i	$0.281106\pi$
$620$		0			0
$621$	18130.1	+	17122.0i	1.17155	+	1.10642i
$622$		0			0
$623$	6318.21	−	22439.5i	0.406314	−	1.44305i
$624$		0			0
$625$	−8266.21			−0.529037
$626$		0			0
$627$	−4179.64	−	4100.74i	−0.266218	−	0.261192i
$628$		0			0
$629$		−	20680.0i		−	1.31092i
$630$		0			0
$631$	−25354.7			−1.59961			−0.799807	−	0.600257i	$-0.795065\pi$
$631$	−25354.7			−1.59961			−0.799807	+	0.600257i	$0.795065\pi$
$632$		0			0
$633$	7892.82	−	8044.69i	0.495595	−	0.505131i
$634$		0			0
$635$		−	8664.45i		−	0.541477i
$636$		0			0
$637$	15350.3	+	9388.60i	0.954791	+	0.583972i
$638$		0			0
$639$	−6619.53	+	126.170i	−0.409803	+	0.00781096i
$640$		0			0
$641$			1867.33i			0.115063i	0.998344	+	0.0575313i	$0.0183229\pi$
$641$			1867.33i			0.115063i	−0.998344	+	0.0575313i	$0.981677\pi$
$642$		0			0
$643$	−5536.73			−0.339576			−0.169788	−	0.985481i	$-0.554308\pi$
$643$	−5536.73			−0.339576			−0.169788	+	0.985481i	$0.554308\pi$
$644$		0			0
$645$	11050.5	+	10841.9i	0.674597	+	0.661861i
$646$		0			0
$647$	7249.89			0.440529			0.220265	−	0.975440i	$-0.429308\pi$
$647$	7249.89			0.440529			0.220265	+	0.975440i	$0.429308\pi$
$648$		0			0
$649$		−	1730.78i		−	0.104683i
$650$		0			0
$651$	2496.53	+	4355.56i	0.150302	+	0.262224i
$652$		0			0
$653$	−5007.60			−0.300096			−0.150048	−	0.988679i	$-0.547943\pi$
$653$	−5007.60			−0.300096			−0.150048	+	0.988679i	$0.547943\pi$
$654$		0			0
$655$	−13330.5			−0.795216
$656$		0			0
$657$	−20113.8	+	383.374i	−1.19439	+	0.0227654i
$658$		0			0
$659$	20066.4			1.18616			0.593078	−	0.805145i	$-0.297912\pi$
$659$	20066.4			1.18616			0.593078	+	0.805145i	$0.297912\pi$
$660$		0			0
$661$	24092.4			1.41768			0.708840	−	0.705369i	$-0.249219\pi$
$661$	24092.4			1.41768			0.708840	+	0.705369i	$0.249219\pi$
$662$		0			0
$663$	13435.7	+	13182.1i	0.787029	+	0.772171i
$664$		0			0
$665$	20791.2	+	5854.11i	1.21240	+	0.341373i
$666$		0			0
$667$		−	43969.6i		−	2.55249i
$668$		0			0
$669$	−14164.1	+	14436.7i	−0.818560	+	0.834311i
$670$		0			0
$671$	1370.03			0.0788217
$672$		0			0
$673$	−17.0224			−0.000974985			−0.000487493	−	1.00000i	$-0.500155\pi$
$673$	−17.0224			−0.000974985			−0.000487493		1.00000i	$0.500155\pi$
$674$		0			0
$675$	−4203.46	+	4450.94i	−0.239691	+	0.253802i
$676$		0			0
$677$		−	25059.2i		−	1.42261i	−0.702885	−	0.711304i	$-0.748105\pi$
$677$		−	25059.2i		−	1.42261i	0.702885	−	0.711304i	$-0.251895\pi$
$678$		0			0
$679$	−13476.5	−	3794.55i	−0.761682	−	0.214464i
$680$		0			0
$681$	−9260.84	+	9439.04i	−0.521110	+	0.531137i
$682$		0			0
$683$	−22610.8			−1.26673			−0.633367	−	0.773851i	$-0.718328\pi$
$683$	−22610.8			−1.26673			−0.633367	+	0.773851i	$0.718328\pi$
$684$		0			0
$685$	−3110.41			−0.173493
$686$		0			0
$687$	2878.06	+	2823.73i	0.159832	+	0.156815i
$688$		0			0
$689$	−12502.1			−0.691280
$690$		0			0
$691$	22189.8			1.22162			0.610810	−	0.791777i	$-0.290844\pi$
$691$	22189.8			1.22162			0.610810	+	0.791777i	$0.290844\pi$
$692$		0			0
$693$	−4216.74	−	1101.01i	−0.231141	−	0.0603521i
$694$		0			0
$695$		−	10428.6i		−	0.569179i
$696$		0			0
$697$	−12826.4			−0.697036
$698$		0			0
$699$	1731.57	−	1764.88i	0.0936964	−	0.0954993i
$700$		0			0
$701$	9475.27			0.510522			0.255261	−	0.966872i	$-0.417839\pi$
$701$	9475.27			0.510522			0.255261	+	0.966872i	$0.417839\pi$
$702$		0			0
$703$		−	38723.2i		−	2.07749i
$704$		0			0
$705$	2691.13	−	2742.91i	0.143764	−	0.146530i
$706$		0			0
$707$	−18642.6	−	5249.15i	−0.991696	−	0.279228i
$708$		0			0
$709$		−	27170.6i		−	1.43923i	−0.694374	−	0.719614i	$-0.744319\pi$
$709$		−	27170.6i		−	1.43923i	0.694374	−	0.719614i	$-0.255681\pi$
$710$		0			0
$711$	−307.748	−	16146.1i	−0.0162327	−	0.851653i
$712$		0			0
$713$	−9272.65			−0.487046
$714$		0			0
$715$		−	4124.11i		−	0.215711i
$716$		0			0
$717$	−1075.93	+	1096.63i	−0.0560407	+	0.0571190i
$718$		0			0
$719$	387.852			0.0201174			0.0100587	−	0.999949i	$-0.496798\pi$
$719$	387.852			0.0201174			0.0100587	+	0.999949i	$0.496798\pi$
$720$		0			0
$721$	27106.6	+	7632.31i	1.40014	+	0.394233i
$722$		0			0
$723$	11953.8	−	12183.9i	0.614894	−	0.626726i
$724$		0			0
$725$	10794.5			0.552964
$726$		0			0
$727$			1084.95i			0.0553488i	0.999617	+	0.0276744i	$0.00881016\pi$
$727$			1084.95i			0.0553488i	−0.999617	+	0.0276744i	$0.991190\pi$
$728$		0			0
$729$	−1124.74	−	19650.8i	−0.0571427	−	0.998366i
$730$		0			0
$731$			22807.1i			1.15397i
$732$		0			0
$733$	−35437.5			−1.78570			−0.892848	−	0.450358i	$-0.851296\pi$
$733$	−35437.5			−1.78570			−0.892848	+	0.450358i	$0.851296\pi$
$734$		0			0
$735$	15592.4	−	3915.12i	0.782497	−	0.196478i
$736$		0			0
$737$			4598.37i			0.229828i
$738$		0			0
$739$			34674.6i			1.72601i	0.505192	+	0.863007i	$0.331422\pi$
$739$			34674.6i			1.72601i	−0.505192	+	0.863007i	$0.668578\pi$
$740$		0			0
$741$	25158.3	+	24683.4i	1.24725	+	1.22371i
$742$		0			0
$743$			22561.8i			1.11401i	0.830508	+	0.557007i	$0.188050\pi$
$743$			22561.8i			1.11401i	−0.830508	+	0.557007i	$0.811950\pi$
$744$		0			0
$745$		−	16682.3i		−	0.820394i
$746$		0			0
$747$	106.262	−	2.02537i	0.00520470	−	9.92029e-5i
$748$		0			0
$749$	−1421.86	+	5049.82i	−0.0693641	+	0.246351i
$750$		0			0
$751$	28928.9			1.40563			0.702816	−	0.711372i	$-0.251926\pi$
$751$	28928.9			1.40563			0.702816	+	0.711372i	$0.251926\pi$
$752$		0			0
$753$	3746.92	−	3819.02i	0.181335	−	0.184824i
$754$		0			0
$755$		−	23705.3i		−	1.14268i
$756$		0			0
$757$			19462.5i			0.934449i	0.884139	+	0.467224i	$0.154746\pi$
$757$			19462.5i			0.934449i	−0.884139	+	0.467224i	$0.845254\pi$
$758$		0			0
$759$	5637.39	−	5745.86i	0.269597	−	0.274785i
$760$		0			0
$761$	5992.80			0.285465			0.142732	−	0.989761i	$-0.454411\pi$
$761$	5992.80			0.285465			0.142732	+	0.989761i	$0.454411\pi$
$762$		0			0
$763$	−21047.3	−	5926.21i	−0.998640	−	0.281184i
$764$		0			0
$765$	16813.7	−	320.474i	0.794643	−	0.0151461i
$766$		0			0
$767$			10418.0i			0.490446i
$768$		0			0
$769$			15433.8i			0.723742i	0.932228	+	0.361871i	$0.117862\pi$
$769$			15433.8i			0.723742i	−0.932228	+	0.361871i	$0.882138\pi$
$770$		0			0
$771$	9094.10	+	8922.42i	0.424794	+	0.416774i
$772$		0			0
$773$		−	17828.8i		−	0.829569i	−0.909920	−	0.414785i	$-0.863857\pi$
$773$		−	17828.8i		−	0.829569i	0.909920	−	0.414785i	$-0.136143\pi$
$774$		0			0
$775$		−	2276.44i		−	0.105512i
$776$		0			0
$777$	−14332.4	−	25005.0i	−0.661740	−	1.15450i
$778$		0			0
$779$	−24017.3			−1.10463
$780$		0			0
$781$			2137.12i			0.0979157i
$782$		0			0
$783$	−23828.9	+	25231.8i	−1.08758	+	1.15161i
$784$		0			0
$785$		−	17222.4i		−	0.783049i
$786$		0			0
$787$	−30953.0			−1.40197			−0.700987	−	0.713174i	$-0.747257\pi$
$787$	−30953.0			−1.40197			−0.700987	+	0.713174i	$0.747257\pi$
$788$		0			0
$789$	−15999.1	+	16306.9i	−0.721904	+	0.735795i
$790$		0			0
$791$	33293.1	+	9374.22i	1.49654	+	0.421377i
$792$		0			0
$793$	−8246.55			−0.369286
$794$		0			0
$795$	−7822.69	+	7973.21i	−0.348984	+	0.355699i
$796$		0			0
$797$		−	7162.07i		−	0.318310i	−0.987254	−	0.159155i	$-0.949123\pi$
$797$		−	7162.07i		−	0.318310i	0.987254	−	0.159155i	$-0.0508770\pi$
$798$		0			0
$799$	5661.05			0.250655
$800$		0			0
$801$	−647.659	−	33979.5i	−0.0285692	−	1.49889i
$802$		0			0
$803$			6493.76i			0.285380i
$804$		0			0
$805$	−8047.79	+	28582.2i	−0.352357	+	1.25142i
$806$		0			0
$807$	−12382.6	+	12620.9i	−0.540134	+	0.550527i
$808$		0			0
$809$			21655.6i			0.941125i	0.882367	+	0.470562i	$0.155949\pi$
$809$			21655.6i			0.941125i	−0.882367	+	0.470562i	$0.844051\pi$
$810$		0			0
$811$	2523.86			0.109278			0.0546392	−	0.998506i	$-0.482599\pi$
$811$	2523.86			0.109278			0.0546392	+	0.998506i	$0.482599\pi$
$812$		0			0
$813$	15210.0	−	15502.7i	0.656137	−	0.668762i
$814$		0			0
$815$	−31200.0			−1.34097
$816$		0			0
$817$			42706.1i			1.82876i
$818$		0			0
$819$	25381.6	+	6627.26i	1.08291	+	0.282754i
$820$		0			0
$821$	−24747.6			−1.05200			−0.526002	−	0.850483i	$-0.676310\pi$
$821$	−24747.6			−1.05200			−0.526002	+	0.850483i	$0.676310\pi$
$822$		0			0
$823$	35614.1			1.50842			0.754210	−	0.656634i	$-0.228020\pi$
$823$	35614.1			1.50842			0.754210	+	0.656634i	$0.228020\pi$
$824$		0			0
$825$	1410.61	+	1383.98i	0.0595287	+	0.0584048i
$826$		0			0
$827$	−29256.6			−1.23017			−0.615085	−	0.788461i	$-0.710878\pi$
$827$	−29256.6			−1.23017			−0.615085	+	0.788461i	$0.710878\pi$
$828$		0			0
$829$	14291.2			0.598740			0.299370	−	0.954137i	$-0.403224\pi$
$829$	14291.2			0.598740			0.299370	+	0.954137i	$0.403224\pi$
$830$		0			0
$831$	−9481.61	+	9664.05i	−0.395804	+	0.403420i
$832$		0			0
$833$	20204.7	+	12357.7i	0.840399	+	0.514007i
$834$		0			0
$835$		−	20801.0i		−	0.862095i
$836$		0			0
$837$	5321.08	+	5025.23i	0.219741	+	0.207524i
$838$		0			0
$839$	−43247.8			−1.77959			−0.889796	−	0.456358i	$-0.849154\pi$
$839$	−43247.8			−1.77959			−0.889796	+	0.456358i	$0.849154\pi$
$840$		0			0
$841$	36803.8			1.50903
$842$		0			0
$843$	−29072.4	+	29631.8i	−1.18779	+	1.21064i
$844$		0			0
$845$			5006.79i			0.203833i
$846$		0			0
$847$	6299.69	−	22373.7i	0.255561	−	0.907639i
$848$		0			0
$849$	13435.6	+	13182.0i	0.543121	+	0.532868i
$850$		0			0
$851$	53233.7			2.14434
$852$		0			0
$853$	−27143.0			−1.08952			−0.544758	−	0.838593i	$-0.683378\pi$
$853$	−27143.0			−1.08952			−0.544758	+	0.838593i	$0.683378\pi$
$854$		0			0
$855$	31483.6	−	600.086i	1.25932	−	0.0240029i
$856$		0			0
$857$	41131.4			1.63947			0.819733	−	0.572746i	$-0.194122\pi$
$857$	41131.4			1.63947			0.819733	+	0.572746i	$0.194122\pi$
$858$		0			0
$859$	3353.69			0.133209			0.0666045	−	0.997779i	$-0.478783\pi$
$859$	3353.69			0.133209			0.0666045	+	0.997779i	$0.478783\pi$
$860$		0			0
$861$	−15508.9	+	8889.41i	−0.613870	+	0.351859i
$862$		0			0
$863$			32515.9i			1.28257i	0.767304	+	0.641283i	$0.221598\pi$
$863$			32515.9i			1.28257i	−0.767304	+	0.641283i	$0.778402\pi$
$864$		0			0
$865$	163.497			0.00642667
$866$		0			0
$867$	−538.033	−	527.876i	−0.0210756	−	0.0206777i
$868$		0			0
$869$	−5212.78			−0.203488
$870$		0			0
$871$		−	27678.8i		−	1.07676i
$872$		0			0
$873$	−20407.2	+	388.966i	−0.791156	+	0.0150796i
$874$		0			0
$875$	−27117.3	−	7635.33i	−1.04769	−	0.294996i
$876$		0			0
$877$		−	11908.4i		−	0.458517i	−0.973366	−	0.229259i	$-0.926370\pi$
$877$		−	11908.4i		−	0.458517i	0.973366	−	0.229259i	$-0.0736301\pi$
$878$		0			0
$879$	−30061.5	+	30640.0i	−1.15353	+	1.17572i
$880$		0			0
$881$	−36671.3			−1.40237			−0.701184	−	0.712980i	$-0.747345\pi$
$881$	−36671.3			−1.40237			−0.701184	+	0.712980i	$0.747345\pi$
$882$		0			0
$883$			37102.7i			1.41405i	0.707190	+	0.707024i	$0.249963\pi$
$883$			37102.7i			1.41405i	−0.707190	+	0.707024i	$0.750037\pi$
$884$		0			0
$885$	6644.07	+	6518.64i	0.252359	+	0.247595i
$886$		0			0
$887$	48774.0			1.84630			0.923152	−	0.384435i	$-0.125604\pi$
$887$	48774.0			1.84630			0.923152	+	0.384435i	$0.125604\pi$
$888$		0			0
$889$	4821.57	−	17124.1i	0.181901	−	0.646033i
$890$		0			0
$891$	−6348.91	+	242.112i	−0.238717	+	0.00910330i
$892$		0			0
$893$	10600.3			0.397228
$894$		0			0
$895$			23416.4i			0.874553i
$896$		0			0
$897$	−33932.8	+	34585.8i	−1.26308	+	1.28739i
$898$		0			0
$899$		−	12904.8i		−	0.478754i
$900$		0			0
$901$	−16455.8			−0.608459
$902$		0			0
$903$	15806.6	+	27576.9i	0.582514	+	1.01628i
$904$		0			0
$905$		−	10830.3i		−	0.397801i
$906$		0			0
$907$			1891.26i			0.0692372i	0.999401	+	0.0346186i	$0.0110216\pi$
$907$			1891.26i			0.0692372i	−0.999401	+	0.0346186i	$0.988978\pi$
$908$		0			0
$909$	−28230.1	+	538.073i	−1.03007	+	0.0196334i
$910$		0			0
$911$			674.242i			0.0245210i	0.999925	+	0.0122605i	$0.00390274\pi$
$911$			674.242i			0.0245210i	−0.999925	+	0.0122605i	$0.996097\pi$
$912$		0			0
$913$		−	34.3067i		−	0.00124358i
$914$		0			0
$915$	−5159.95	+	5259.23i	−0.186429	+	0.190016i
$916$		0			0
$917$	−26345.9	−	7418.13i	−0.948767	−	0.267141i
$918$		0			0
$919$	246.949			0.00886408			0.00443204	−	0.999990i	$-0.498589\pi$
$919$	246.949			0.00886408			0.00443204	+	0.999990i	$0.498589\pi$
$920$		0			0
$921$	23647.8	+	23201.4i	0.846060	+	0.830088i
$922$		0			0
$923$		−	12863.9i		−	0.458742i
$924$		0			0
$925$			13068.9i			0.464543i
$926$		0			0
$927$	41046.8	−	782.363i	1.45432	−	0.0277197i
$928$		0			0
$929$	−26737.6			−0.944275			−0.472138	−	0.881525i	$-0.656517\pi$
$929$	−26737.6			−0.944275			−0.472138	+	0.881525i	$0.656517\pi$
$930$		0			0
$931$	37833.3	+	23139.7i	1.33183	+	0.814578i
$932$		0			0
$933$	13230.3	+	12980.5i	0.464243	+	0.455479i
$934$		0			0
$935$		−	5428.33i		−	0.189867i
$936$		0			0
$937$		−	12780.7i		−	0.445602i	−0.974864	−	0.222801i	$-0.928480\pi$
$937$		−	12780.7i		−	0.445602i	0.974864	−	0.222801i	$-0.0715200\pi$
$938$		0			0
$939$	8603.37	−	8768.91i	0.298999	−	0.304753i
$940$		0			0
$941$		−	43874.1i		−	1.51993i	−0.649963	−	0.759966i	$-0.725216\pi$
$941$		−	43874.1i		−	1.51993i	0.649963	−	0.759966i	$-0.274784\pi$
$942$		0			0
$943$		−	33017.2i		−	1.14018i
$944$		0			0
$945$	20108.1	−	12040.4i	0.692185	−	0.414469i
$946$		0			0
$947$	−2594.12			−0.0890153			−0.0445077	−	0.999009i	$-0.514172\pi$
$947$	−2594.12			−0.0890153			−0.0445077	+	0.999009i	$0.514172\pi$
$948$		0			0
$949$		−	39087.6i		−	1.33702i
$950$		0			0
$951$	19644.9	+	19274.0i	0.669852	+	0.657206i
$952$		0			0
$953$			35759.6i			1.21550i	0.794130	+	0.607748i	$0.207927\pi$
$953$			35759.6i			1.21550i	−0.794130	+	0.607748i	$0.792073\pi$
$954$		0			0
$955$	−16043.7			−0.543626
$956$		0			0
$957$	7996.56	+	7845.60i	0.270107	+	0.265008i
$958$		0			0
$959$	−6147.30	−	1730.87i	−0.206993	−	0.0582824i
$960$		0			0
$961$	27069.5			0.908648
$962$		0			0
$963$	145.750	+	7646.82i	0.00487720	+	0.255883i
$964$		0			0
$965$			1362.58i			0.0454538i
$966$		0			0
$967$	−950.541			−0.0316105			−0.0158052	−	0.999875i	$-0.505031\pi$
$967$	−950.541			−0.0316105			−0.0158052	+	0.999875i	$0.505031\pi$
$968$		0			0
$969$	33114.5	+	32489.3i	1.09782	+	1.07710i
$970$		0			0
$971$		−	59078.8i		−	1.95255i	−0.216531	−	0.976276i	$-0.569474\pi$
$971$		−	59078.8i		−	1.95255i	0.216531	−	0.976276i	$-0.430526\pi$
$972$		0			0
$973$	5803.28	−	20610.7i	0.191207	−	0.679084i
$974$		0			0
$975$	−8490.82	−	8330.52i	−0.278896	−	0.273631i
$976$		0			0
$977$		−	14531.7i		−	0.475854i	−0.971283	−	0.237927i	$-0.923532\pi$
$977$		−	14531.7i		−	0.475854i	0.971283	−	0.237927i	$-0.0764678\pi$
$978$		0			0
$979$	−10970.3			−0.358134
$980$		0			0
$981$	−31871.4	+	607.477i	−1.03728	+	0.0197709i
$982$		0			0
$983$	58495.3			1.89798			0.948989	−	0.315310i	$-0.102109\pi$
$983$	58495.3			1.89798			0.948989	+	0.315310i	$0.102109\pi$
$984$		0			0
$985$		−	3124.59i		−	0.101074i
$986$		0			0
$987$	6845.00	−	3923.43i	0.220749	−	0.126529i
$988$		0			0
$989$	−58709.2			−1.88761
$990$		0			0
$991$	29846.4			0.956714			0.478357	−	0.878166i	$-0.341233\pi$
$991$	29846.4			0.956714			0.478357	+	0.878166i	$0.341233\pi$
$992$		0			0
$993$	2419.64	−	2466.20i	0.0773262	−	0.0788141i
$994$		0			0
$995$	34100.2			1.08648
$996$		0			0
$997$	14935.8			0.474444			0.237222	−	0.971455i	$-0.423763\pi$
$997$	14935.8			0.474444			0.237222	+	0.971455i	$0.423763\pi$
$998$		0			0
$999$	−30548.0	−	28849.5i	−0.967463	−	0.913672i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.i.c.209.59		80
3.2	odd	2	inner	672.4.i.c.209.58		80
4.3	odd	2		168.4.i.c.125.3	yes	80
7.6	odd	2	inner	672.4.i.c.209.21		80
8.3	odd	2		168.4.i.c.125.80	yes	80
8.5	even	2	inner	672.4.i.c.209.22		80
12.11	even	2		168.4.i.c.125.77	yes	80
21.20	even	2	inner	672.4.i.c.209.24		80
24.5	odd	2	inner	672.4.i.c.209.23		80
24.11	even	2		168.4.i.c.125.2	yes	80
28.27	even	2		168.4.i.c.125.4	yes	80
56.13	odd	2	inner	672.4.i.c.209.60		80
56.27	even	2		168.4.i.c.125.79	yes	80
84.83	odd	2		168.4.i.c.125.78	yes	80
168.83	odd	2		168.4.i.c.125.1	✓	80
168.125	even	2	inner	672.4.i.c.209.57		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.i.c.125.1	✓	80	168.83	odd	2
168.4.i.c.125.2	yes	80	24.11	even	2
168.4.i.c.125.3	yes	80	4.3	odd	2
168.4.i.c.125.4	yes	80	28.27	even	2
168.4.i.c.125.77	yes	80	12.11	even	2
168.4.i.c.125.78	yes	80	84.83	odd	2
168.4.i.c.125.79	yes	80	56.27	even	2
168.4.i.c.125.80	yes	80	8.3	odd	2
672.4.i.c.209.21		80	7.6	odd	2	inner
672.4.i.c.209.22		80	8.5	even	2	inner
672.4.i.c.209.23		80	24.5	odd	2	inner
672.4.i.c.209.24		80	21.20	even	2	inner
672.4.i.c.209.57		80	168.125	even	2	inner
672.4.i.c.209.58		80	3.2	odd	2	inner
672.4.i.c.209.59		80	1.1	even	1	trivial
672.4.i.c.209.60		80	56.13	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	672.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.70908 + 3.63906i) q^{3} +9.02015i q^{5} +(-5.01951 + 17.8271i) q^{7} +(0.514533 + 26.9951i) q^{9} +8.71540 q^{11} -52.4602 q^{13} +(-32.8248 + 33.4564i) q^{15} -69.0503 q^{17} -129.296 q^{19} +(-83.4915 + 47.8557i) q^{21} -177.747i q^{23} +43.6369 q^{25} +(-96.3282 + 101.999i) q^{27} +247.372 q^{29} -52.1678i q^{31} +(32.3261 + 31.7158i) q^{33} +(-160.803 - 45.2767i) q^{35} +299.492i q^{37} +(-194.579 - 190.905i) q^{39} +185.754 q^{41} -330.296i q^{43} +(-243.500 + 4.64117i) q^{45} -81.9844 q^{47} +(-292.609 - 178.966i) q^{49} +(-256.113 - 251.278i) q^{51} +238.316 q^{53} +78.6142i q^{55} +(-479.570 - 470.517i) q^{57} -198.589i q^{59} +157.196 q^{61} +(-483.826 - 126.329i) q^{63} -473.198i q^{65} +527.615i q^{67} +(646.831 - 659.277i) q^{69} +245.212i q^{71} +745.090i q^{73} +(161.853 + 158.797i) q^{75} +(-43.7470 + 155.370i) q^{77} -598.112 q^{79} +(-728.471 + 27.7798i) q^{81} -3.93633i q^{83} -622.844i q^{85} +(917.522 + 900.200i) q^{87} -1258.73 q^{89} +(263.324 - 935.211i) q^{91} +(189.841 - 193.494i) q^{93} -1166.27i q^{95} +755.959i q^{97} +(4.48436 + 235.273i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 64 q^{7} + 104 q^{9} + 8 q^{15} - 976 q^{25} - 568 q^{39} - 4048 q^{49} - 1448 q^{57} + 2152 q^{63} - 4992 q^{79} + 1568 q^{81}+O(q^{100}) \) 80 * q - 64 * q^7 + 104 * q^9 + 8 * q^15 - 976 * q^25 - 568 * q^39 - 4048 * q^49 - 1448 * q^57 + 2152 * q^63 - 4992 * q^79 + 1568 * q^81

Introduction

L-functions

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