LMFDB - Embedded newform 639.3.c.a.143.26

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [639,3,Mod(143,639)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(639, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("639.143");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 639 = 3^{2} \cdot 71 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	639.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.4114888926$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			143.26
Character	$\chi$	$=$	639.143
Dual form			639.3.c.a.143.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.432607i q^{2} +3.81285 q^{4} +3.81616i q^{5} +8.57190 q^{7} +3.37989i q^{8} +O(q^{10})$	\(q+0.432607i q^{2} +3.81285 q^{4} +3.81616i q^{5} +8.57190 q^{7} +3.37989i q^{8} -1.65090 q^{10} -13.2437i q^{11} +1.78075 q^{13} +3.70826i q^{14} +13.7892 q^{16} +12.1306i q^{17} +23.0495 q^{19} +14.5505i q^{20} +5.72932 q^{22} -8.79595i q^{23} +10.4369 q^{25} +0.770365i q^{26} +32.6834 q^{28} -40.9445i q^{29} -53.4571 q^{31} +19.4849i q^{32} -5.24779 q^{34} +32.7118i q^{35} +9.58036 q^{37} +9.97137i q^{38} -12.8982 q^{40} +62.7479i q^{41} -47.9613 q^{43} -50.4963i q^{44} +3.80519 q^{46} +80.0978i q^{47} +24.4775 q^{49} +4.51507i q^{50} +6.78974 q^{52} -77.5339i q^{53} +50.5402 q^{55} +28.9721i q^{56} +17.7129 q^{58} +2.47190i q^{59} +108.836 q^{61} -23.1259i q^{62} +46.7276 q^{64} +6.79564i q^{65} +37.5487 q^{67} +46.2523i q^{68} -14.1513 q^{70} +8.42615i q^{71} -121.563 q^{73} +4.14453i q^{74} +87.8843 q^{76} -113.524i q^{77} +139.824 q^{79} +52.6220i q^{80} -27.1452 q^{82} +145.438i q^{83} -46.2925 q^{85} -20.7484i q^{86} +44.7623 q^{88} +27.3203i q^{89} +15.2644 q^{91} -33.5376i q^{92} -34.6509 q^{94} +87.9607i q^{95} -16.4587 q^{97} +10.5891i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 96 q^{4}+O(q^{10}) $ 48 * q - 96 * q^4	$ 48 q - 96 q^{4} - 16 q^{10} - 8 q^{13} + 264 q^{16} + 40 q^{19} - 40 q^{22} - 360 q^{25} - 56 q^{28} - 8 q^{31} + 32 q^{34} + 104 q^{37} + 200 q^{40} - 64 q^{43} + 32 q^{46} + 448 q^{49} - 264 q^{52} + 104 q^{55} - 360 q^{58} - 24 q^{61} - 720 q^{64} + 96 q^{67} + 816 q^{70} - 248 q^{73} - 184 q^{76} + 416 q^{79} + 96 q^{85} - 200 q^{88} - 184 q^{91} + 448 q^{94} - 488 q^{97}+O(q^{100}) $ 48 * q - 96 * q^4 - 16 * q^10 - 8 * q^13 + 264 * q^16 + 40 * q^19 - 40 * q^22 - 360 * q^25 - 56 * q^28 - 8 * q^31 + 32 * q^34 + 104 * q^37 + 200 * q^40 - 64 * q^43 + 32 * q^46 + 448 * q^49 - 264 * q^52 + 104 * q^55 - 360 * q^58 - 24 * q^61 - 720 * q^64 + 96 * q^67 + 816 * q^70 - 248 * q^73 - 184 * q^76 + 416 * q^79 + 96 * q^85 - 200 * q^88 - 184 * q^91 + 448 * q^94 - 488 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/639\mathbb{Z}\right)^\times$.

$n$	$433$	$569$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	0.432607i	0.216304i	0.994134	+	0.108152i	$0.0344932\pi$
$2$	0.432607i	0.216304i	−0.994134	+	0.108152i	$0.965507\pi$
$3$	0	0
$3$	0	0
$4$	3.81285	0.953213
$5$	3.81616i	0.763233i	0.924321	+	0.381616i	$0.124632\pi$
$5$	3.81616i	0.763233i	−0.924321	+	0.381616i	$0.875368\pi$
$6$	0	0
$7$	8.57190	1.22456	0.612279	−	0.790642i	$-0.290253\pi$
$7$	8.57190	1.22456	0.612279	+	0.790642i	$0.290253\pi$
$8$	3.37989i	0.422487i
$9$	0	0
$10$	−1.65090	−0.165090
$11$	− 13.2437i	− 1.20397i	−0.798506	−	0.601987i	$-0.794376\pi$
$11$	− 13.2437i	− 1.20397i	0.798506	−	0.601987i	$-0.205624\pi$
$12$	0	0
$13$	1.78075	0.136981	0.0684904	−	0.997652i	$-0.478182\pi$
$13$	1.78075	0.136981	0.0684904	+	0.997652i	$0.478182\pi$
$14$	3.70826i	0.264876i
$15$	0	0
$16$	13.7892	0.861827
$17$	12.1306i	0.713566i	0.934187	+	0.356783i	$0.116126\pi$
$17$	12.1306i	0.713566i	−0.934187	+	0.356783i	$0.883874\pi$
$18$	0	0
$19$	23.0495	1.21313	0.606566	−	0.795033i	$-0.292547\pi$
$19$	23.0495	1.21313	0.606566	+	0.795033i	$0.292547\pi$
$20$	14.5505i	0.727523i
$21$	0	0
$22$	5.72932	0.260424
$23$	− 8.79595i	− 0.382432i	−0.981548	−	0.191216i	$-0.938757\pi$
$23$	− 8.79595i	− 0.382432i	0.981548	−	0.191216i	$-0.0612432\pi$
$24$	0	0
$25$	10.4369	0.417476
$26$	0.770365i	0.0296294i
$27$	0	0
$28$	32.6834	1.16726
$29$	− 40.9445i	− 1.41188i	−0.708272	−	0.705940i	$-0.750525\pi$
$29$	− 40.9445i	− 1.41188i	0.708272	−	0.705940i	$-0.249475\pi$
$30$	0	0
$31$	−53.4571	−1.72442	−0.862211	−	0.506550i	$-0.830921\pi$
$31$	−53.4571	−1.72442	−0.862211	+	0.506550i	$0.830921\pi$
$32$	19.4849i	0.608903i
$33$	0	0
$34$	−5.24779	−0.154347
$35$	32.7118i	0.934622i
$36$	0	0
$37$	9.58036	0.258929	0.129464	−	0.991584i	$-0.458674\pi$
$37$	9.58036	0.258929	0.129464	+	0.991584i	$0.458674\pi$
$38$	9.97137i	0.262405i
$39$	0	0
$40$	−12.8982	−0.322456
$41$	62.7479i	1.53044i	0.643771	+	0.765218i	$0.277369\pi$
$41$	62.7479i	1.53044i	−0.643771	+	0.765218i	$0.722631\pi$
$42$	0	0
$43$	−47.9613	−1.11538	−0.557690	−	0.830049i	$-0.688312\pi$
$43$	−47.9613	−1.11538	−0.557690	+	0.830049i	$0.688312\pi$
$44$	− 50.4963i	− 1.14764i
$45$	0	0
$46$	3.80519	0.0827215
$47$	80.0978i	1.70421i	0.523372	+	0.852104i	$0.324674\pi$
$47$	80.0978i	1.70421i	−0.523372	+	0.852104i	$0.675326\pi$
$48$	0	0
$49$	24.4775	0.499540
$50$	4.51507i	0.0903014i
$51$	0	0
$52$	6.78974	0.130572
$53$	− 77.5339i	− 1.46290i	−0.681893	−	0.731452i	$-0.738843\pi$
$53$	− 77.5339i	− 1.46290i	0.681893	−	0.731452i	$-0.261157\pi$
$54$	0	0
$55$	50.5402	0.918912
$56$	28.9721i	0.517359i
$57$	0	0
$58$	17.7129	0.305394
$59$	2.47190i	0.0418966i	0.999781	+	0.0209483i	$0.00666854\pi$
$59$	2.47190i	0.0418966i	−0.999781	+	0.0209483i	$0.993331\pi$
$60$	0	0
$61$	108.836	1.78420	0.892100	−	0.451839i	$-0.149232\pi$
$61$	108.836	1.78420	0.892100	+	0.451839i	$0.149232\pi$
$62$	− 23.1259i	− 0.372998i
$63$	0	0
$64$	46.7276	0.730119
$65$	6.79564i	0.104548i
$66$	0	0
$67$	37.5487	0.560428	0.280214	−	0.959938i	$-0.409595\pi$
$67$	37.5487	0.560428	0.280214	+	0.959938i	$0.409595\pi$
$68$	46.2523i	0.680180i
$69$	0	0
$70$	−14.1513	−0.202162
$71$	8.42615i	0.118678i
$71$	8.42615i	0.118678i
$72$	0	0
$73$	−121.563	−1.66524	−0.832621	−	0.553844i	$-0.813160\pi$
$73$	−121.563	−1.66524	−0.832621	+	0.553844i	$0.813160\pi$
$74$	4.14453i	0.0560072i
$75$	0	0
$76$	87.8843	1.15637
$77$	− 113.524i	− 1.47433i
$78$	0	0
$79$	139.824	1.76993	0.884964	−	0.465660i	$-0.154183\pi$
$79$	139.824	1.76993	0.884964	+	0.465660i	$0.154183\pi$
$80$	52.6220i	0.657775i
$81$	0	0
$82$	−27.1452	−0.331039
$83$	145.438i	1.75226i	0.482075	+	0.876130i	$0.339883\pi$
$83$	145.438i	1.75226i	−0.482075	+	0.876130i	$0.660117\pi$
$84$	0	0
$85$	−46.2925	−0.544617
$86$	− 20.7484i	− 0.241261i
$87$	0	0
$88$	44.7623	0.508663
$89$	27.3203i	0.306970i	0.988151	+	0.153485i	$0.0490496\pi$
$89$	27.3203i	0.306970i	−0.988151	+	0.153485i	$0.950950\pi$
$90$	0	0
$91$	15.2644	0.167741
$92$	− 33.5376i	− 0.364539i
$93$	0	0
$94$	−34.6509	−0.368626
$95$	87.9607i	0.925902i
$96$	0	0
$97$	−16.4587	−0.169677	−0.0848385	−	0.996395i	$-0.527037\pi$
$97$	−16.4587	−0.169677	−0.0848385	+	0.996395i	$0.527037\pi$
$98$	10.5891i	0.108052i
$99$	0	0
$100$	39.7943	0.397943
$101$	− 136.808i	− 1.35454i	−0.735735	−	0.677269i	$-0.763163\pi$
$101$	− 136.808i	− 1.35454i	0.735735	−	0.677269i	$-0.236837\pi$
$102$	0	0
$103$	−172.604	−1.67577	−0.837885	−	0.545847i	$-0.816208\pi$
$103$	−172.604	−1.67577	−0.837885	+	0.545847i	$0.816208\pi$
$104$	6.01875i	0.0578726i
$105$	0	0
$106$	33.5417	0.316431
$107$	75.6657i	0.707156i	0.935405	+	0.353578i	$0.115035\pi$
$107$	75.6657i	0.707156i	−0.935405	+	0.353578i	$0.884965\pi$
$108$	0	0
$109$	−52.0998	−0.477980	−0.238990	−	0.971022i	$-0.576816\pi$
$109$	−52.0998	−0.477980	−0.238990	+	0.971022i	$0.576816\pi$
$110$	21.8640i	0.198764i
$111$	0	0
$112$	118.200	1.05536
$113$	− 10.8718i	− 0.0962104i	−0.998842	−	0.0481052i	$-0.984682\pi$
$113$	− 10.8718i	− 0.0962104i	0.998842	−	0.0481052i	$-0.0153183\pi$
$114$	0	0
$115$	33.5668	0.291885
$116$	− 156.115i	− 1.34582i
$117$	0	0
$118$	−1.06936	−0.00906238
$119$	103.983i	0.873803i
$120$	0	0
$121$	−54.3957	−0.449551
$122$	47.0833i	0.385929i
$123$	0	0
$124$	−203.824	−1.64374
$125$	135.233i	1.08186i
$126$	0	0
$127$	91.1033	0.717349	0.358674	−	0.933463i	$-0.383229\pi$
$127$	91.1033	0.717349	0.358674	+	0.933463i	$0.383229\pi$
$128$	98.1543i	0.766831i
$129$	0	0
$130$	−2.93984	−0.0226142
$131$	− 157.746i	− 1.20417i	−0.798433	−	0.602084i	$-0.794337\pi$
$131$	− 157.746i	− 1.20417i	0.798433	−	0.602084i	$-0.205663\pi$
$132$	0	0
$133$	197.578	1.48555
$134$	16.2438i	0.121223i
$135$	0	0
$136$	−41.0002	−0.301472
$137$	− 134.798i	− 0.983925i	−0.870616	−	0.491962i	$-0.836280\pi$
$137$	− 134.798i	− 0.983925i	0.870616	−	0.491962i	$-0.163720\pi$
$138$	0	0
$139$	−59.4558	−0.427740	−0.213870	−	0.976862i	$-0.568607\pi$
$139$	−59.4558	−0.427740	−0.213870	+	0.976862i	$0.568607\pi$
$140$	124.725i	0.890894i
$141$	0	0
$142$	−3.64521	−0.0256705
$143$	− 23.5837i	− 0.164921i
$144$	0	0
$145$	156.251	1.07759
$146$	− 52.5888i	− 0.360198i
$147$	0	0
$148$	36.5285	0.246814
$149$	− 105.233i	− 0.706264i	−0.935574	−	0.353132i	$-0.885117\pi$
$149$	− 105.233i	− 0.706264i	0.935574	−	0.353132i	$-0.114883\pi$
$150$	0	0
$151$	−171.550	−1.13609	−0.568047	−	0.822996i	$-0.692301\pi$
$151$	−171.550	−1.13609	−0.568047	+	0.822996i	$0.692301\pi$
$152$	77.9049i	0.512532i
$153$	0	0
$154$	49.1112	0.318904
$155$	− 204.001i	− 1.31614i
$156$	0	0
$157$	82.6211	0.526249	0.263125	−	0.964762i	$-0.415247\pi$
$157$	82.6211	0.526249	0.263125	+	0.964762i	$0.415247\pi$
$158$	60.4890i	0.382841i
$159$	0	0
$160$	−74.3576	−0.464735
$161$	− 75.3980i	− 0.468310i
$162$	0	0
$163$	−258.481	−1.58577	−0.792886	−	0.609370i	$-0.791423\pi$
$163$	−258.481	−1.58577	−0.792886	+	0.609370i	$0.791423\pi$
$164$	239.248i	1.45883i
$165$	0	0
$166$	−62.9173	−0.379020
$167$	− 110.480i	− 0.661556i	−0.943709	−	0.330778i	$-0.892689\pi$
$167$	− 110.480i	− 0.661556i	0.943709	−	0.330778i	$-0.107311\pi$
$168$	0	0
$169$	−165.829	−0.981236
$170$	− 20.0264i	− 0.117803i
$171$	0	0
$172$	−182.869	−1.06319
$173$	− 98.6050i	− 0.569971i	−0.958532	−	0.284985i	$-0.908011\pi$
$173$	− 98.6050i	− 0.569971i	0.958532	−	0.284985i	$-0.0919888\pi$
$174$	0	0
$175$	89.4640	0.511223
$176$	− 182.621i	− 1.03762i
$177$	0	0
$178$	−11.8190	−0.0663986
$179$	− 38.7510i	− 0.216486i	−0.994124	−	0.108243i	$-0.965478\pi$
$179$	− 38.7510i	− 0.216486i	0.994124	−	0.108243i	$-0.0345224\pi$
$180$	0	0
$181$	−219.680	−1.21370	−0.606852	−	0.794815i	$-0.707568\pi$
$181$	−219.680	−1.21370	−0.606852	+	0.794815i	$0.707568\pi$
$182$	6.60350i	0.0362829i
$183$	0	0
$184$	29.7294	0.161573
$185$	36.5602i	0.197623i
$186$	0	0
$187$	160.654	0.859114
$188$	305.401i	1.62447i
$189$	0	0
$190$	−38.0524	−0.200276
$191$	0.131580i	0 0.000688902i	1.00000		0.000344451i	$0.000109642\pi$
$191$	0.131580i	0 0.000688902i	−1.00000		0.000344451i	$0.999890\pi$
$192$	0	0
$193$	84.8741	0.439762	0.219881	−	0.975527i	$-0.429433\pi$
$193$	84.8741	0.439762	0.219881	+	0.975527i	$0.429433\pi$
$194$	− 7.12014i	− 0.0367017i
$195$	0	0
$196$	93.3290	0.476168
$197$	188.386i	0.956277i	0.878285	+	0.478138i	$0.158688\pi$
$197$	188.386i	0.956277i	−0.878285	+	0.478138i	$0.841312\pi$
$198$	0	0
$199$	−94.4151	−0.474448	−0.237224	−	0.971455i	$-0.576238\pi$
$199$	−94.4151	−0.474448	−0.237224	+	0.971455i	$0.576238\pi$
$200$	35.2756i	0.176378i
$201$	0	0
$202$	59.1843	0.292992
$203$	− 350.972i	− 1.72893i
$204$	0	0
$205$	−239.456	−1.16808
$206$	− 74.6698i	− 0.362475i
$207$	0	0
$208$	24.5552	0.118054
$209$	− 305.261i	− 1.46058i
$210$	0	0
$211$	−330.134	−1.56462	−0.782309	−	0.622891i	$-0.785958\pi$
$211$	−330.134	−1.56462	−0.782309	+	0.622891i	$0.785958\pi$
$212$	− 295.625i	− 1.39446i
$213$	0	0
$214$	−32.7335	−0.152960
$215$	− 183.028i	− 0.851294i
$216$	0	0
$217$	−458.229	−2.11165
$218$	− 22.5387i	− 0.103389i
$219$	0	0
$220$	192.702	0.875919
$221$	21.6016i	0.0977449i
$222$	0	0
$223$	−356.595	−1.59908	−0.799541	−	0.600611i	$-0.794924\pi$
$223$	−356.595	−1.59908	−0.799541	+	0.600611i	$0.794924\pi$
$224$	167.023i	0.745637i
$225$	0	0
$226$	4.70321	0.0208106
$227$	− 179.201i	− 0.789432i	−0.918803	−	0.394716i	$-0.870843\pi$
$227$	− 179.201i	− 0.789432i	0.918803	−	0.394716i	$-0.129157\pi$
$228$	0	0
$229$	223.632	0.976559	0.488279	−	0.872687i	$-0.337625\pi$
$229$	223.632	0.976559	0.488279	+	0.872687i	$0.337625\pi$
$230$	14.5212i	0.0631358i
$231$	0	0
$232$	138.388	0.596500
$233$	− 115.673i	− 0.496453i	−0.968702	−	0.248226i	$-0.920152\pi$
$233$	− 115.673i	− 0.496453i	0.968702	−	0.248226i	$-0.0798477\pi$
$234$	0	0
$235$	−305.666	−1.30071
$236$	9.42498i	0.0399364i
$237$	0	0
$238$	−44.9836	−0.189007
$239$	− 218.101i	− 0.912557i	−0.889837	−	0.456279i	$-0.849182\pi$
$239$	− 218.101i	− 0.912557i	0.889837	−	0.456279i	$-0.150818\pi$
$240$	0	0
$241$	−238.886	−0.991229	−0.495614	−	0.868543i	$-0.665057\pi$
$241$	−238.886	−0.991229	−0.495614	+	0.868543i	$0.665057\pi$
$242$	− 23.5320i	− 0.0972395i
$243$	0	0
$244$	414.976	1.70072
$245$	93.4101i	0.381266i
$246$	0	0
$247$	41.0454	0.166176
$248$	− 180.679i	− 0.728545i
$249$	0	0
$250$	−58.5027	−0.234011
$251$	− 1.03278i	− 0.00411466i	−0.999998	−	0.00205733i	$-0.999345\pi$
$251$	− 1.03278i	− 0.00411466i	0.999998	−	0.00205733i	$-0.000654869\pi$
$252$	0	0
$253$	−116.491	−0.460438
$254$	39.4119i	0.155165i
$255$	0	0
$256$	144.448	0.564251
$257$	− 165.966i	− 0.645781i	−0.946436	−	0.322890i	$-0.895346\pi$
$257$	− 165.966i	− 0.645781i	0.946436	−	0.322890i	$-0.104654\pi$
$258$	0	0
$259$	82.1219	0.317073
$260$	25.9108i	0.0996568i
$261$	0	0
$262$	68.2420	0.260466
$263$	42.0246i	0.159789i	0.996803	+	0.0798946i	$0.0254584\pi$
$263$	42.0246i	0.159789i	−0.996803	+	0.0798946i	$0.974542\pi$
$264$	0	0
$265$	295.882	1.11654
$266$	85.4736i	0.321329i
$267$	0	0
$268$	143.167	0.534207
$269$	− 194.627i	− 0.723521i	−0.932271	−	0.361760i	$-0.882176\pi$
$269$	− 194.627i	− 0.723521i	0.932271	−	0.361760i	$-0.117824\pi$
$270$	0	0
$271$	267.255	0.986179	0.493090	−	0.869979i	$-0.335868\pi$
$271$	267.255	0.986179	0.493090	+	0.869979i	$0.335868\pi$
$272$	167.272i	0.614971i
$273$	0	0
$274$	58.3144	0.212826
$275$	− 138.223i	− 0.502629i
$276$	0	0
$277$	126.779	0.457687	0.228844	−	0.973463i	$-0.426506\pi$
$277$	126.779	0.457687	0.228844	+	0.973463i	$0.426506\pi$
$278$	− 25.7210i	− 0.0925216i
$279$	0	0
$280$	−110.562	−0.394866
$281$	− 82.4012i	− 0.293243i	−0.989193	−	0.146621i	$-0.953160\pi$
$281$	− 82.4012i	− 0.293243i	0.989193	−	0.146621i	$-0.0468399\pi$
$282$	0	0
$283$	−61.9678	−0.218967	−0.109484	−	0.993989i	$-0.534920\pi$
$283$	−61.9678	−0.218967	−0.109484	+	0.993989i	$0.534920\pi$
$284$	32.1277i	0.113126i
$285$	0	0
$286$	10.2025	0.0356730
$287$	537.869i	1.87411i
$288$	0	0
$289$	141.848	0.490823
$290$	67.5953i	0.233087i
$291$	0	0
$292$	−463.500	−1.58733
$293$	278.634i	0.950969i	0.879724	+	0.475485i	$0.157727\pi$
$293$	278.634i	0.950969i	−0.879724	+	0.475485i	$0.842273\pi$
$294$	0	0
$295$	−9.43317	−0.0319769
$296$	32.3806i	0.109394i
$297$	0	0
$298$	45.5247	0.152767
$299$	− 15.6634i	− 0.0523859i
$300$	0	0
$301$	−411.120	−1.36585
$302$	− 74.2139i	− 0.245741i
$303$	0	0
$304$	317.835	1.04551
$305$	415.337i	1.36176i
$306$	0	0
$307$	188.289	0.613319	0.306660	−	0.951819i	$-0.400789\pi$
$307$	188.289	0.613319	0.306660	+	0.951819i	$0.400789\pi$
$308$	− 432.849i	− 1.40535i
$309$	0	0
$310$	88.2523	0.284685
$311$	− 154.372i	− 0.496372i	−0.968712	−	0.248186i	$-0.920166\pi$
$311$	− 154.372i	− 0.496372i	0.968712	−	0.248186i	$-0.0798344\pi$
$312$	0	0
$313$	32.9855	0.105385	0.0526924	−	0.998611i	$-0.483220\pi$
$313$	32.9855	0.105385	0.0526924	+	0.998611i	$0.483220\pi$
$314$	35.7425i	0.113830i
$315$	0	0
$316$	533.129	1.68712
$317$	17.6296i	0.0556138i	0.999613	+	0.0278069i	$0.00885235\pi$
$317$	17.6296i	0.0556138i	−0.999613	+	0.0278069i	$0.991148\pi$
$318$	0	0
$319$	−542.257	−1.69986
$320$	178.320i	0.557251i
$321$	0	0
$322$	32.6177	0.101297
$323$	279.605i	0.865649i
$324$	0	0
$325$	18.5855	0.0571861
$326$	− 111.821i	− 0.343008i
$327$	0	0
$328$	−212.081	−0.646589
$329$	686.590i	2.08690i
$330$	0	0
$331$	235.325	0.710952	0.355476	−	0.934685i	$-0.384319\pi$
$331$	235.325	0.710952	0.355476	+	0.934685i	$0.384319\pi$
$332$	554.532i	1.67028i
$333$	0	0
$334$	47.7944	0.143097
$335$	143.292i	0.427737i
$336$	0	0
$337$	120.833	0.358554	0.179277	−	0.983799i	$-0.442624\pi$
$337$	120.833	0.358554	0.179277	+	0.983799i	$0.442624\pi$
$338$	− 71.7388i	− 0.212245i
$339$	0	0
$340$	−176.506	−0.519136
$341$	707.970i	2.07616i
$342$	0	0
$343$	−210.205	−0.612842
$344$	− 162.104i	− 0.471233i
$345$	0	0
$346$	42.6572	0.123287
$347$	506.913i	1.46084i	0.682996	+	0.730422i	$0.260676\pi$
$347$	506.913i	1.46084i	−0.682996	+	0.730422i	$0.739324\pi$
$348$	0	0
$349$	−341.842	−0.979490	−0.489745	−	0.871866i	$-0.662910\pi$
$349$	−341.842	−0.979490	−0.489745	+	0.871866i	$0.662910\pi$
$350$	38.7027i	0.110579i
$351$	0	0
$352$	258.052	0.733103
$353$	− 554.733i	− 1.57148i	−0.618555	−	0.785741i	$-0.712282\pi$
$353$	− 554.733i	− 1.57148i	0.618555	−	0.785741i	$-0.287718\pi$
$354$	0	0
$355$	−32.1556	−0.0905791
$356$	104.168i	0.292607i
$357$	0	0
$358$	16.7639	0.0468267
$359$	− 542.386i	− 1.51083i	−0.655249	−	0.755413i	$-0.727436\pi$
$359$	− 542.386i	− 1.51083i	0.655249	−	0.755413i	$-0.272564\pi$
$360$	0	0
$361$	170.279	0.471688
$362$	− 95.0352i	− 0.262528i
$363$	0	0
$364$	58.2010	0.159893
$365$	− 463.903i	− 1.27097i
$366$	0	0
$367$	−39.4734	−0.107557	−0.0537784	−	0.998553i	$-0.517126\pi$
$367$	−39.4734	−0.107557	−0.0537784	+	0.998553i	$0.517126\pi$
$368$	− 121.289i	− 0.329591i
$369$	0	0
$370$	−15.8162	−0.0427465
$371$	− 664.613i	− 1.79141i
$372$	0	0
$373$	693.318	1.85876	0.929381	−	0.369121i	$-0.120341\pi$
$373$	693.318	1.85876	0.929381	+	0.369121i	$0.120341\pi$
$374$	69.5002i	0.185829i
$375$	0	0
$376$	−270.722	−0.720006
$377$	− 72.9120i	− 0.193400i
$378$	0	0
$379$	−334.552	−0.882722	−0.441361	−	0.897330i	$-0.645504\pi$
$379$	−334.552	−0.882722	−0.441361	+	0.897330i	$0.645504\pi$
$380$	335.381i	0.882581i
$381$	0	0
$382$	−0.0569226	−0.000149012 0
$383$	− 656.324i	− 1.71364i	−0.515616	−	0.856820i	$-0.672437\pi$
$383$	− 656.324i	− 1.71364i	0.515616	−	0.856820i	$-0.327563\pi$
$384$	0	0
$385$	433.225	1.12526
$386$	36.7171i	0.0951221i
$387$	0	0
$388$	−62.7545	−0.161738
$389$	600.510i	1.54373i	0.635788	+	0.771864i	$0.280675\pi$
$389$	600.510i	1.54373i	−0.635788	+	0.771864i	$0.719325\pi$
$390$	0	0
$391$	106.700	0.272891
$392$	82.7313i	0.211049i
$393$	0	0
$394$	−81.4973	−0.206846
$395$	533.592i	1.35087i
$396$	0	0
$397$	−274.574	−0.691622	−0.345811	−	0.938304i	$-0.612396\pi$
$397$	−274.574	−0.691622	−0.345811	+	0.938304i	$0.612396\pi$
$398$	− 40.8446i	− 0.102625i
$399$	0	0
$400$	143.917	0.359792
$401$	− 324.381i	− 0.808930i	−0.914553	−	0.404465i	$-0.867458\pi$
$401$	− 324.381i	− 0.808930i	0.914553	−	0.404465i	$-0.132542\pi$
$402$	0	0
$403$	−95.1937	−0.236213
$404$	− 521.630i	− 1.29116i
$405$	0	0
$406$	151.833	0.373973
$407$	− 126.879i	− 0.311743i
$408$	0	0
$409$	−210.457	−0.514564	−0.257282	−	0.966336i	$-0.582827\pi$
$409$	−210.457	−0.514564	−0.257282	+	0.966336i	$0.582827\pi$
$410$	− 103.591i	− 0.252660i
$411$	0	0
$412$	−658.114	−1.59736
$413$	21.1889i	0.0513048i
$414$	0	0
$415$	−555.014	−1.33738
$416$	34.6978i	0.0834081i
$417$	0	0
$418$	132.058	0.315928
$419$	275.940i	0.658568i	0.944231	+	0.329284i	$0.106807\pi$
$419$	275.940i	0.658568i	−0.944231	+	0.329284i	$0.893193\pi$
$420$	0	0
$421$	1.83335	0.00435474	0.00217737	−	0.999998i	$-0.499307\pi$
$421$	1.83335	0.00435474	0.00217737	+	0.999998i	$0.499307\pi$
$422$	− 142.818i	− 0.338432i
$423$	0	0
$424$	262.056	0.618057
$425$	126.606i	0.297896i
$426$	0	0
$427$	932.933	2.18485
$428$	288.502i	0.674070i
$429$	0	0
$430$	79.1793	0.184138
$431$	826.317i	1.91721i	0.284743	+	0.958604i	$0.408092\pi$
$431$	826.317i	1.91721i	−0.284743	+	0.958604i	$0.591908\pi$
$432$	0	0
$433$	257.847	0.595490	0.297745	−	0.954645i	$-0.403765\pi$
$433$	257.847	0.595490	0.297745	+	0.954645i	$0.403765\pi$
$434$	− 198.233i	− 0.456758i
$435$	0	0
$436$	−198.649	−0.455616
$437$	− 202.742i	− 0.463941i
$438$	0	0
$439$	120.812	0.275198	0.137599	−	0.990488i	$-0.456062\pi$
$439$	120.812	0.275198	0.137599	+	0.990488i	$0.456062\pi$
$440$	170.820i	0.388228i
$441$	0	0
$442$	−9.34501	−0.0211426
$443$	− 204.462i	− 0.461540i	−0.973008	−	0.230770i	$-0.925875\pi$
$443$	− 204.462i	− 0.461540i	0.973008	−	0.230770i	$-0.0741246\pi$
$444$	0	0
$445$	−104.259	−0.234289
$446$	− 154.266i	− 0.345887i
$447$	0	0
$448$	400.545	0.894073
$449$	− 180.719i	− 0.402492i	−0.979541	−	0.201246i	$-0.935501\pi$
$449$	− 180.719i	− 0.402492i	0.979541	−	0.201246i	$-0.0644991\pi$
$450$	0	0
$451$	831.015	1.84260
$452$	− 41.4525i	− 0.0917090i
$453$	0	0
$454$	77.5236	0.170757
$455$	58.2515i	0.128025i
$456$	0	0
$457$	376.619	0.824111	0.412056	−	0.911159i	$-0.364811\pi$
$457$	376.619	0.824111	0.412056	+	0.911159i	$0.364811\pi$
$458$	96.7448i	0.211233i
$459$	0	0
$460$	127.985	0.278229
$461$	− 285.943i	− 0.620267i	−0.950693	−	0.310134i	$-0.899626\pi$
$461$	− 285.943i	− 0.620267i	0.950693	−	0.310134i	$-0.100374\pi$
$462$	0	0
$463$	−429.909	−0.928528	−0.464264	−	0.885697i	$-0.653681\pi$
$463$	−429.909	−0.928528	−0.464264	+	0.885697i	$0.653681\pi$
$464$	− 564.593i	− 1.21680i
$465$	0	0
$466$	50.0412	0.107384
$467$	− 24.0329i	− 0.0514623i	−0.999669	−	0.0257312i	$-0.991809\pi$
$467$	− 24.0329i	− 0.0514623i	0.999669	−	0.0257312i	$-0.00819139\pi$
$468$	0	0
$469$	321.863	0.686276
$470$	− 132.233i	− 0.281348i
$471$	0	0
$472$	−8.35476	−0.0177008
$473$	635.186i	1.34289i
$474$	0	0
$475$	240.565	0.506453
$476$	396.470i	0.832920i
$477$	0	0
$478$	94.3521	0.197389
$479$	191.327i	0.399429i	0.979854	+	0.199715i	$0.0640015\pi$
$479$	191.327i	0.399429i	−0.979854	+	0.199715i	$0.935998\pi$
$480$	0	0
$481$	17.0602	0.0354683
$482$	− 103.344i	− 0.214406i
$483$	0	0
$484$	−207.403	−0.428518
$485$	− 62.8090i	− 0.129503i
$486$	0	0
$487$	−459.721	−0.943986	−0.471993	−	0.881602i	$-0.656465\pi$
$487$	−459.721	−0.943986	−0.471993	+	0.881602i	$0.656465\pi$
$488$	367.855i	0.753801i
$489$	0	0
$490$	−40.4099	−0.0824691
$491$	483.232i	0.984179i	0.870545	+	0.492089i	$0.163767\pi$
$491$	483.232i	0.984179i	−0.870545	+	0.492089i	$0.836233\pi$
$492$	0	0
$493$	496.682	1.00747
$494$	17.7565i	0.0359444i
$495$	0	0
$496$	−737.132	−1.48615
$497$	72.2281i	0.145328i
$498$	0	0
$499$	407.984	0.817603	0.408802	−	0.912623i	$-0.365947\pi$
$499$	407.984	0.817603	0.408802	+	0.912623i	$0.365947\pi$
$500$	515.623i	1.03125i
$501$	0	0
$502$	0.446788	0.000890015 0
$503$	819.848i	1.62992i	0.579519	+	0.814959i	$0.303240\pi$
$503$	819.848i	1.62992i	−0.579519	+	0.814959i	$0.696760\pi$
$504$	0	0
$505$	522.083	1.03383
$506$	− 50.3948i	− 0.0995944i
$507$	0	0
$508$	347.363	0.683786
$509$	657.602i	1.29195i	0.763359	+	0.645975i	$0.223549\pi$
$509$	657.602i	1.29195i	−0.763359	+	0.645975i	$0.776451\pi$
$510$	0	0
$511$	−1042.02	−2.03918
$512$	455.107i	0.888880i
$513$	0	0
$514$	71.7979	0.139685
$515$	− 658.686i	− 1.27900i
$516$	0	0
$517$	1060.79	2.05182
$518$	35.5265i	0.0685840i
$519$	0	0
$520$	−22.9685	−0.0441703
$521$	706.722i	1.35647i	0.734844	+	0.678236i	$0.237255\pi$
$521$	706.722i	1.35647i	−0.734844	+	0.678236i	$0.762745\pi$
$522$	0	0
$523$	201.578	0.385426	0.192713	−	0.981255i	$-0.438271\pi$
$523$	201.578	0.385426	0.192713	+	0.981255i	$0.438271\pi$
$524$	− 601.462i	− 1.14783i
$525$	0	0
$526$	−18.1801	−0.0345630
$527$	− 648.468i	− 1.23049i
$528$	0	0
$529$	451.631	0.853745
$530$	128.001i	0.241511i
$531$	0	0
$532$	753.335	1.41604
$533$	111.738i	0.209641i
$534$	0	0
$535$	−288.753	−0.539724
$536$	126.911i	0.236773i
$537$	0	0
$538$	84.1971	0.156500
$539$	− 324.172i	− 0.601433i
$540$	0	0
$541$	−561.568	−1.03802	−0.519010	−	0.854768i	$-0.673699\pi$
$541$	−561.568	−1.03802	−0.519010	+	0.854768i	$0.673699\pi$
$542$	115.616i	0.213314i
$543$	0	0
$544$	−236.364	−0.434493
$545$	− 198.821i	− 0.364810i
$546$	0	0
$547$	92.6913	0.169454	0.0847270	−	0.996404i	$-0.472998\pi$
$547$	92.6913	0.169454	0.0847270	+	0.996404i	$0.472998\pi$
$548$	− 513.964i	− 0.937890i
$549$	0	0
$550$	59.7963	0.108720
$551$	− 943.750i	− 1.71280i
$552$	0	0
$553$	1198.56	2.16738
$554$	54.8457i	0.0989994i
$555$	0	0
$556$	−226.696	−0.407727
$557$	1075.26i	1.93045i	0.261429	+	0.965223i	$0.415806\pi$
$557$	1075.26i	1.93045i	−0.261429	+	0.965223i	$0.584194\pi$
$558$	0	0
$559$	−85.4072	−0.152786
$560$	451.071i	0.805483i
$561$	0	0
$562$	35.6473	0.0634294
$563$	639.292i	1.13551i	0.823198	+	0.567755i	$0.192188\pi$
$563$	639.292i	1.13551i	−0.823198	+	0.567755i	$0.807812\pi$
$564$	0	0
$565$	41.4885	0.0734309
$566$	− 26.8077i	− 0.0473634i
$567$	0	0
$568$	−28.4795	−0.0501400
$569$	− 409.770i	− 0.720159i	−0.932922	−	0.360079i	$-0.882750\pi$
$569$	− 409.770i	− 0.720159i	0.932922	−	0.360079i	$-0.117250\pi$
$570$	0	0
$571$	−189.524	−0.331916	−0.165958	−	0.986133i	$-0.553072\pi$
$571$	−189.524	−0.331916	−0.165958	+	0.986133i	$0.553072\pi$
$572$	− 89.9213i	− 0.157205i
$573$	0	0
$574$	−232.686	−0.405376
$575$	− 91.8023i	− 0.159656i
$576$	0	0
$577$	598.192	1.03673	0.518364	−	0.855160i	$-0.326541\pi$
$577$	598.192	1.03673	0.518364	+	0.855160i	$0.326541\pi$
$578$	61.3644i	0.106167i
$579$	0	0
$580$	595.762	1.02718
$581$	1246.68i	2.14574i
$582$	0	0
$583$	−1026.84	−1.76130
$584$	− 410.869i	− 0.703542i
$585$	0	0
$586$	−120.539	−0.205698
$587$	− 1053.04i	− 1.79393i	−0.442100	−	0.896966i	$-0.645766\pi$
$587$	− 1053.04i	− 1.79393i	0.442100	−	0.896966i	$-0.354234\pi$
$588$	0	0
$589$	−1232.16	−2.09195
$590$	− 4.08086i	− 0.00691671i
$591$	0	0
$592$	132.106	0.223152
$593$	− 181.041i	− 0.305296i	−0.988281	−	0.152648i	$-0.951220\pi$
$593$	− 181.041i	− 0.305296i	0.988281	−	0.152648i	$-0.0487801\pi$
$594$	0	0
$595$	−396.814	−0.666915
$596$	− 401.239i	− 0.673219i
$597$	0	0
$598$	6.77609	0.0113313
$599$	− 1001.22i	− 1.67148i	−0.549125	−	0.835740i	$-0.685039\pi$
$599$	− 1001.22i	− 1.67148i	0.549125	−	0.835740i	$-0.314961\pi$
$600$	0	0
$601$	−122.355	−0.203586	−0.101793	−	0.994806i	$-0.532458\pi$
$601$	−122.355	−0.203586	−0.101793	+	0.994806i	$0.532458\pi$
$602$	− 177.853i	− 0.295437i
$603$	0	0
$604$	−654.096	−1.08294
$605$	− 207.583i	− 0.343112i
$606$	0	0
$607$	−744.162	−1.22597	−0.612983	−	0.790096i	$-0.710031\pi$
$607$	−744.162	−1.22597	−0.612983	+	0.790096i	$0.710031\pi$
$608$	449.117i	0.738679i
$609$	0	0
$610$	−179.678	−0.294553
$611$	142.634i	0.233444i
$612$	0	0
$613$	1104.00	1.80098	0.900491	−	0.434874i	$-0.143207\pi$
$613$	1104.00	1.80098	0.900491	+	0.434874i	$0.143207\pi$
$614$	81.4552i	0.132663i
$615$	0	0
$616$	383.698	0.622887
$617$	170.252i	0.275936i	0.990437	+	0.137968i	$0.0440571\pi$
$617$	170.252i	0.275936i	−0.990437	+	0.137968i	$0.955943\pi$
$618$	0	0
$619$	43.1038	0.0696346	0.0348173	−	0.999394i	$-0.488915\pi$
$619$	43.1038	0.0696346	0.0348173	+	0.999394i	$0.488915\pi$
$620$	− 777.825i	− 1.25456i
$621$	0	0
$622$	66.7823	0.107367
$623$	234.187i	0.375902i
$624$	0	0
$625$	−255.149	−0.408239
$626$	14.2697i	0.0227951i
$627$	0	0
$628$	315.022	0.501627
$629$	116.216i	0.184763i
$630$	0	0
$631$	−377.868	−0.598840	−0.299420	−	0.954121i	$-0.596793\pi$
$631$	−377.868	−0.598840	−0.299420	+	0.954121i	$0.596793\pi$
$632$	472.591i	0.747771i
$633$	0	0
$634$	−7.62668	−0.0120295
$635$	347.665i	0.547504i
$636$	0	0
$637$	43.5883	0.0684274
$638$	− 234.584i	− 0.367687i
$639$	0	0
$640$	−374.573	−0.585270
$641$	− 704.531i	− 1.09911i	−0.835457	−	0.549556i	$-0.814797\pi$
$641$	− 704.531i	− 1.09911i	0.835457	−	0.549556i	$-0.185203\pi$
$642$	0	0
$643$	915.365	1.42359	0.711793	−	0.702390i	$-0.247884\pi$
$643$	915.365	1.42359	0.711793	+	0.702390i	$0.247884\pi$
$644$	− 287.481i	− 0.446399i
$645$	0	0
$646$	−120.959	−0.187243
$647$	493.484i	0.762727i	0.924425	+	0.381363i	$0.124545\pi$
$647$	493.484i	0.762727i	−0.924425	+	0.381363i	$0.875455\pi$
$648$	0	0
$649$	32.7371	0.0504424
$650$	8.04022i	0.0123696i
$651$	0	0
$652$	−985.549	−1.51158
$653$	552.129i	0.845527i	0.906240	+	0.422763i	$0.138940\pi$
$653$	552.129i	0.845527i	−0.906240	+	0.422763i	$0.861060\pi$
$654$	0	0
$655$	601.984	0.919060
$656$	865.246i	1.31897i
$657$	0	0
$658$	−297.024	−0.451404
$659$	− 939.344i	− 1.42541i	−0.701465	−	0.712704i	$-0.747470\pi$
$659$	− 939.344i	− 1.42541i	0.701465	−	0.712704i	$-0.252530\pi$
$660$	0	0
$661$	−835.558	−1.26408	−0.632041	−	0.774935i	$-0.717782\pi$
$661$	−835.558	−1.26408	−0.632041	+	0.774935i	$0.717782\pi$
$662$	101.803i	0.153781i
$663$	0	0
$664$	−491.564	−0.740307
$665$	753.990i	1.13382i
$666$	0	0
$667$	−360.146	−0.539948
$668$	− 421.243i	− 0.630604i
$669$	0	0
$670$	−61.9891	−0.0925210
$671$	− 1441.39i	− 2.14813i
$672$	0	0
$673$	−327.583	−0.486750	−0.243375	−	0.969932i	$-0.578255\pi$
$673$	−327.583	−0.486750	−0.243375	+	0.969932i	$0.578255\pi$
$674$	52.2730i	0.0775564i
$675$	0	0
$676$	−632.281	−0.935327
$677$	− 117.561i	− 0.173650i	−0.996224	−	0.0868250i	$-0.972328\pi$
$677$	− 117.561i	− 0.173650i	0.996224	−	0.0868250i	$-0.0276721\pi$
$678$	0	0
$679$	−141.082	−0.207779
$680$	− 156.464i	− 0.230094i
$681$	0	0
$682$	−306.273	−0.449080
$683$	− 183.810i	− 0.269122i	−0.990905	−	0.134561i	$-0.957038\pi$
$683$	− 183.810i	− 0.269122i	0.990905	−	0.134561i	$-0.0429624\pi$
$684$	0	0
$685$	514.410	0.750964
$686$	− 90.9360i	− 0.132560i
$687$	0	0
$688$	−661.350	−0.961265
$689$	− 138.069i	− 0.200390i
$690$	0	0
$691$	−192.972	−0.279264	−0.139632	−	0.990203i	$-0.544592\pi$
$691$	−192.972	−0.279264	−0.139632	+	0.990203i	$0.544592\pi$
$692$	− 375.966i	− 0.543304i
$693$	0	0
$694$	−219.294	−0.315986
$695$	− 226.893i	− 0.326465i
$696$	0	0
$697$	−761.171	−1.09207
$698$	− 147.883i	− 0.211867i
$699$	0	0
$700$	341.113	0.487304
$701$	− 1117.86i	− 1.59467i	−0.603538	−	0.797334i	$-0.706243\pi$
$701$	− 1117.86i	− 1.59467i	0.603538	−	0.797334i	$-0.293757\pi$
$702$	0	0
$703$	220.822	0.314114
$704$	− 618.847i	− 0.879044i
$705$	0	0
$706$	239.982	0.339917
$707$	− 1172.71i	− 1.65871i
$708$	0	0
$709$	809.105	1.14119	0.570596	−	0.821231i	$-0.306712\pi$
$709$	809.105	1.14119	0.570596	+	0.821231i	$0.306712\pi$
$710$	− 13.9107i	− 0.0195926i
$711$	0	0
$712$	−92.3397	−0.129691
$713$	470.205i	0.659475i
$714$	0	0
$715$	89.9994	0.125873
$716$	− 147.752i	− 0.206357i
$717$	0	0
$718$	234.640	0.326797
$719$	519.996i	0.723221i	0.932329	+	0.361611i	$0.117773\pi$
$719$	519.996i	0.723221i	−0.932329	+	0.361611i	$0.882227\pi$
$720$	0	0
$721$	−1479.55	−2.05208
$722$	73.6640i	0.102028i
$723$	0	0
$724$	−837.608	−1.15692
$725$	− 427.333i	− 0.589425i
$726$	0	0
$727$	1235.77	1.69982	0.849912	−	0.526925i	$-0.176655\pi$
$727$	1235.77	1.69982	0.849912	+	0.526925i	$0.176655\pi$
$728$	51.5921i	0.0708683i
$729$	0	0
$730$	200.688	0.274915
$731$	− 581.801i	− 0.795897i
$732$	0	0
$733$	636.648	0.868551	0.434276	−	0.900780i	$-0.357004\pi$
$733$	636.648	0.868551	0.434276	+	0.900780i	$0.357004\pi$
$734$	− 17.0765i	− 0.0232649i
$735$	0	0
$736$	171.388	0.232864
$737$	− 497.283i	− 0.674740i
$738$	0	0
$739$	−817.800	−1.10663	−0.553316	−	0.832972i	$-0.686638\pi$
$739$	−817.800	−1.10663	−0.553316	+	0.832972i	$0.686638\pi$
$740$	139.399i	0.188377i
$741$	0	0
$742$	287.516	0.387488
$743$	− 267.384i	− 0.359871i	−0.983678	−	0.179936i	$-0.942411\pi$
$743$	− 267.384i	− 0.359871i	0.983678	−	0.179936i	$-0.0575889\pi$
$744$	0	0
$745$	401.587	0.539044
$746$	299.934i	0.402057i
$747$	0	0
$748$	612.551	0.818919
$749$	648.598i	0.865953i
$750$	0	0
$751$	−309.695	−0.412377	−0.206189	−	0.978512i	$-0.566106\pi$
$751$	−309.695	−0.412377	−0.206189	+	0.978512i	$0.566106\pi$
$752$	1104.49i	1.46873i
$753$	0	0
$754$	31.5422	0.0418332
$755$	− 654.664i	− 0.867105i
$756$	0	0
$757$	−193.462	−0.255565	−0.127782	−	0.991802i	$-0.540786\pi$
$757$	−193.462	−0.255565	−0.127782	+	0.991802i	$0.540786\pi$
$758$	− 144.729i	− 0.190936i
$759$	0	0
$760$	−297.298	−0.391181
$761$	− 1381.74i	− 1.81570i	−0.419299	−	0.907848i	$-0.637724\pi$
$761$	− 1381.74i	− 1.81570i	0.419299	−	0.907848i	$-0.362276\pi$
$762$	0	0
$763$	−446.594	−0.585314
$764$	0.501696i	0 0.000656670i
$765$	0	0
$766$	283.930	0.370666
$767$	4.40184i	0.00573903i
$768$	0	0
$769$	−196.805	−0.255923	−0.127961	−	0.991779i	$-0.540843\pi$
$769$	−196.805	−0.255923	−0.127961	+	0.991779i	$0.540843\pi$
$770$	187.416i	0.243398i
$771$	0	0
$772$	323.612	0.419187
$773$	1150.23i	1.48800i	0.668177	+	0.744002i	$0.267075\pi$
$773$	1150.23i	1.48800i	−0.668177	+	0.744002i	$0.732925\pi$
$774$	0	0
$775$	−557.925	−0.719904
$776$	− 55.6286i	− 0.0716863i
$777$	0	0
$778$	−259.785	−0.333914
$779$	1446.31i	1.85662i
$780$	0	0
$781$	111.593	0.142885
$782$	46.1593i	0.0590272i
$783$	0	0
$784$	337.526	0.430517
$785$	315.296i	0.401651i
$786$	0	0
$787$	328.638	0.417583	0.208792	−	0.977960i	$-0.433047\pi$
$787$	328.638	0.417583	0.208792	+	0.977960i	$0.433047\pi$
$788$	718.290i	0.911535i
$789$	0	0
$790$	−230.836	−0.292197
$791$	− 93.1918i	− 0.117815i
$792$	0	0
$793$	193.810	0.244401
$794$	− 118.783i	− 0.149600i
$795$	0	0
$796$	−359.991	−0.452250
$797$	258.035i	0.323758i	0.986811	+	0.161879i	$0.0517554\pi$
$797$	258.035i	0.323758i	−0.986811	+	0.161879i	$0.948245\pi$
$798$	0	0
$799$	−971.636	−1.21607
$800$	203.362i	0.254202i
$801$	0	0
$802$	140.330	0.174974
$803$	1609.94i	2.00491i
$804$	0	0
$805$	287.731	0.357430
$806$	− 41.1815i	− 0.0510936i
$807$	0	0
$808$	462.398	0.572275
$809$	− 876.011i	− 1.08283i	−0.840755	−	0.541416i	$-0.817889\pi$
$809$	− 876.011i	− 1.08283i	0.840755	−	0.541416i	$-0.182111\pi$
$810$	0	0
$811$	−803.628	−0.990909	−0.495455	−	0.868634i	$-0.664999\pi$
$811$	−803.628	−0.990909	−0.495455	+	0.868634i	$0.664999\pi$
$812$	− 1338.20i	− 1.64804i
$813$	0	0
$814$	54.8889	0.0674311
$815$	− 986.406i	− 1.21031i
$816$	0	0
$817$	−1105.48	−1.35310
$818$	− 91.0451i	− 0.111302i
$819$	0	0
$820$	−913.011	−1.11343
$821$	1481.13i	1.80406i	0.431672	+	0.902030i	$0.357924\pi$
$821$	1481.13i	1.80406i	−0.431672	+	0.902030i	$0.642076\pi$
$822$	0	0
$823$	89.9870	0.109340	0.0546701	−	0.998504i	$-0.482589\pi$
$823$	89.9870	0.109340	0.0546701	+	0.998504i	$0.482589\pi$
$824$	− 583.384i	− 0.707990i
$825$	0	0
$826$	−9.16645	−0.0110974
$827$	475.668i	0.575173i	0.957755	+	0.287586i	$0.0928529\pi$
$827$	475.668i	0.575173i	−0.957755	+	0.287586i	$0.907147\pi$
$828$	0	0
$829$	1294.92	1.56203	0.781013	−	0.624515i	$-0.214703\pi$
$829$	1294.92	1.56203	0.781013	+	0.624515i	$0.214703\pi$
$830$	− 240.103i	− 0.289281i
$831$	0	0
$832$	83.2103	0.100012
$833$	296.927i	0.356455i
$834$	0	0
$835$	421.609	0.504921
$836$	− 1163.91i	− 1.39224i
$837$	0	0
$838$	−119.374	−0.142451
$839$	212.894i	0.253747i	0.991919	+	0.126874i	$0.0404942\pi$
$839$	212.894i	0.253747i	−0.991919	+	0.126874i	$0.959506\pi$
$840$	0	0
$841$	−835.452	−0.993403
$842$	0.793119i	0 0.000941946i
$843$	0	0
$844$	−1258.75	−1.49141
$845$	− 632.830i	− 0.748912i
$846$	0	0
$847$	−466.274	−0.550501
$848$	− 1069.13i	− 1.26077i
$849$	0	0
$850$	−54.7706	−0.0644360
$851$	− 84.2683i	− 0.0990227i
$852$	0	0
$853$	593.267	0.695506	0.347753	−	0.937586i	$-0.386945\pi$
$853$	593.267	0.695506	0.347753	+	0.937586i	$0.386945\pi$
$854$	403.593i	0.472592i
$855$	0	0
$856$	−255.742	−0.298764
$857$	718.925i	0.838885i	0.907782	+	0.419443i	$0.137774\pi$
$857$	718.925i	0.838885i	−0.907782	+	0.419443i	$0.862226\pi$
$858$	0	0
$859$	−5.71057	−0.00664793	−0.00332397	−	0.999994i	$-0.501058\pi$
$859$	−5.71057	−0.00664793	−0.00332397	+	0.999994i	$0.501058\pi$
$860$	− 697.860i	− 0.811465i
$861$	0	0
$862$	−357.470	−0.414699
$863$	726.458i	0.841782i	0.907111	+	0.420891i	$0.138283\pi$
$863$	726.458i	0.841782i	−0.907111	+	0.420891i	$0.861717\pi$
$864$	0	0
$865$	376.293	0.435021
$866$	111.546i	0.128807i
$867$	0	0
$868$	−1747.16	−2.01285
$869$	− 1851.79i	− 2.13094i
$870$	0	0
$871$	66.8648	0.0767679
$872$	− 176.092i	− 0.201940i
$873$	0	0
$874$	87.7077	0.100352
$875$	1159.20i	1.32480i
$876$	0	0
$877$	−1111.09	−1.26692	−0.633462	−	0.773773i	$-0.718367\pi$
$877$	−1111.09	−1.26692	−0.633462	+	0.773773i	$0.718367\pi$
$878$	52.2640i	0.0595262i
$879$	0	0
$880$	696.910	0.791943
$881$	1081.25i	1.22730i	0.789580	+	0.613648i	$0.210299\pi$
$881$	1081.25i	1.22730i	−0.789580	+	0.613648i	$0.789701\pi$
$882$	0	0
$883$	−57.3625	−0.0649632	−0.0324816	−	0.999472i	$-0.510341\pi$
$883$	−57.3625	−0.0649632	−0.0324816	+	0.999472i	$0.510341\pi$
$884$	82.3638i	0.0931717i
$885$	0	0
$886$	88.4519	0.0998328
$887$	− 1185.23i	− 1.33622i	−0.744063	−	0.668110i	$-0.767104\pi$
$887$	− 1185.23i	− 1.33622i	0.744063	−	0.668110i	$-0.232896\pi$
$888$	0	0
$889$	780.928	0.878434
$890$	− 45.1031i	− 0.0506776i
$891$	0	0
$892$	−1359.65	−1.52427
$893$	1846.21i	2.06743i
$894$	0	0
$895$	147.880	0.165229
$896$	841.369i	0.939028i
$897$	0	0
$898$	78.1803	0.0870604
$899$	2188.77i	2.43468i
$900$	0	0
$901$	940.535	1.04388
$902$	359.503i	0.398562i
$903$	0	0
$904$	36.7454	0.0406476
$905$	− 838.336i	− 0.926338i
$906$	0	0
$907$	−1071.47	−1.18133	−0.590666	−	0.806916i	$-0.701135\pi$
$907$	−1071.47	−1.18133	−0.590666	+	0.806916i	$0.701135\pi$
$908$	− 683.267i	− 0.752496i
$909$	0	0
$910$	−25.2000	−0.0276923
$911$	331.121i	0.363469i	0.983348	+	0.181735i	$0.0581712\pi$
$911$	331.121i	0.363469i	−0.983348	+	0.181735i	$0.941829\pi$
$912$	0	0
$913$	1926.13	2.10967
$914$	162.928i	0.178258i
$915$	0	0
$916$	852.675	0.930868
$917$	− 1352.18i	− 1.47457i
$918$	0	0
$919$	763.433	0.830722	0.415361	−	0.909657i	$-0.363655\pi$
$919$	763.433	0.830722	0.415361	+	0.909657i	$0.363655\pi$
$920$	113.452i	0.123318i
$921$	0	0
$922$	123.701	0.134166
$923$	15.0049i	0.0162566i
$924$	0	0
$925$	99.9891	0.108096
$926$	− 185.981i	− 0.200844i
$927$	0	0
$928$	797.800	0.859698
$929$	− 362.430i	− 0.390129i	−0.980790	−	0.195064i	$-0.937508\pi$
$929$	− 362.430i	− 0.390129i	0.980790	−	0.195064i	$-0.0624916\pi$
$930$	0	0
$931$	564.193	0.606008
$932$	− 441.046i	− 0.473225i
$933$	0	0
$934$	10.3968	0.0111315
$935$	613.084i	0.655704i
$936$	0	0
$937$	89.0434	0.0950303	0.0475151	−	0.998871i	$-0.484870\pi$
$937$	89.0434	0.0950303	0.0475151	+	0.998871i	$0.484870\pi$
$938$	139.240i	0.148444i
$939$	0	0
$940$	−1165.46	−1.23985
$941$	− 833.822i	− 0.886102i	−0.896496	−	0.443051i	$-0.853896\pi$
$941$	− 833.822i	− 0.886102i	0.896496	−	0.443051i	$-0.146104\pi$
$942$	0	0
$943$	551.927	0.585289
$944$	34.0856i	0.0361076i
$945$	0	0
$946$	−274.786	−0.290471
$947$	− 130.628i	− 0.137939i	−0.997619	−	0.0689694i	$-0.978029\pi$
$947$	− 130.628i	− 0.137939i	0.997619	−	0.0689694i	$-0.0219711\pi$
$948$	0	0
$949$	−216.473	−0.228106
$950$	104.070i	0.109547i
$951$	0	0
$952$	−351.450	−0.369170
$953$	499.664i	0.524306i	0.965026	+	0.262153i	$0.0844325\pi$
$953$	499.664i	0.524306i	−0.965026	+	0.262153i	$0.915567\pi$
$954$	0	0
$955$	−0.502132	−0.000525793 0
$956$	− 831.587i	− 0.869861i
$957$	0	0
$958$	−82.7693	−0.0863980
$959$	− 1155.47i	− 1.20487i
$960$	0	0
$961$	1896.66	1.97363
$962$	7.38038i	0.00767191i
$963$	0	0
$964$	−910.837	−0.944852
$965$	323.893i	0.335641i
$966$	0	0
$967$	1519.13	1.57097	0.785485	−	0.618881i	$-0.212414\pi$
$967$	1519.13	1.57097	0.785485	+	0.618881i	$0.212414\pi$
$968$	− 183.852i	− 0.189929i
$969$	0	0
$970$	27.1716	0.0280120
$971$	740.971i	0.763101i	0.924348	+	0.381550i	$0.124610\pi$
$971$	740.971i	0.763101i	−0.924348	+	0.381550i	$0.875390\pi$
$972$	0	0
$973$	−509.649	−0.523792
$974$	− 198.879i	− 0.204187i
$975$	0	0
$976$	1500.77	1.53767
$977$	− 206.907i	− 0.211778i	−0.994378	−	0.105889i	$-0.966231\pi$
$977$	− 206.907i	− 0.211778i	0.994378	−	0.105889i	$-0.0337688\pi$
$978$	0	0
$979$	361.822	0.369583
$980$	356.159i	0.363427i
$981$	0	0
$982$	−209.049	−0.212881
$983$	1309.58i	1.33223i	0.745848	+	0.666116i	$0.232045\pi$
$983$	1309.58i	1.33223i	−0.745848	+	0.666116i	$0.767955\pi$
$984$	0	0
$985$	−718.914	−0.729862
$986$	214.868i	0.217919i
$987$	0	0
$988$	156.500	0.158401
$989$	421.865i	0.426557i
$990$	0	0
$991$	1375.26	1.38775	0.693873	−	0.720097i	$-0.255903\pi$
$991$	1375.26	1.38775	0.693873	+	0.720097i	$0.255903\pi$
$992$	− 1041.61i	− 1.05001i
$993$	0	0
$994$	−31.2464	−0.0314350
$995$	− 360.304i	− 0.362114i
$996$	0	0
$997$	−1558.29	−1.56298	−0.781488	−	0.623920i	$-0.785539\pi$
$997$	−1558.29	−1.56298	−0.781488	+	0.623920i	$0.785539\pi$
$998$	176.497i	0.176850i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	639.3.c.a.143.26	yes	48
3.2	odd	2	inner	639.3.c.a.143.23	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
639.3.c.a.143.23	✓	48	3.2	odd	2	inner
639.3.c.a.143.26	yes	48	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 639 = 3^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	639.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.432607i q^{2} +3.81285 q^{4} +3.81616i q^{5} +8.57190 q^{7} +3.37989i q^{8} +O(q^{10})\)	\(q+0.432607i q^{2} +3.81285 q^{4} +3.81616i q^{5} +8.57190 q^{7} +3.37989i q^{8} -1.65090 q^{10} -13.2437i q^{11} +1.78075 q^{13} +3.70826i q^{14} +13.7892 q^{16} +12.1306i q^{17} +23.0495 q^{19} +14.5505i q^{20} +5.72932 q^{22} -8.79595i q^{23} +10.4369 q^{25} +0.770365i q^{26} +32.6834 q^{28} -40.9445i q^{29} -53.4571 q^{31} +19.4849i q^{32} -5.24779 q^{34} +32.7118i q^{35} +9.58036 q^{37} +9.97137i q^{38} -12.8982 q^{40} +62.7479i q^{41} -47.9613 q^{43} -50.4963i q^{44} +3.80519 q^{46} +80.0978i q^{47} +24.4775 q^{49} +4.51507i q^{50} +6.78974 q^{52} -77.5339i q^{53} +50.5402 q^{55} +28.9721i q^{56} +17.7129 q^{58} +2.47190i q^{59} +108.836 q^{61} -23.1259i q^{62} +46.7276 q^{64} +6.79564i q^{65} +37.5487 q^{67} +46.2523i q^{68} -14.1513 q^{70} +8.42615i q^{71} -121.563 q^{73} +4.14453i q^{74} +87.8843 q^{76} -113.524i q^{77} +139.824 q^{79} +52.6220i q^{80} -27.1452 q^{82} +145.438i q^{83} -46.2925 q^{85} -20.7484i q^{86} +44.7623 q^{88} +27.3203i q^{89} +15.2644 q^{91} -33.5376i q^{92} -34.6509 q^{94} +87.9607i q^{95} -16.4587 q^{97} +10.5891i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 96 q^{4}+O(q^{10}) \) 48 * q - 96 * q^4	\( 48 q - 96 q^{4} - 16 q^{10} - 8 q^{13} + 264 q^{16} + 40 q^{19} - 40 q^{22} - 360 q^{25} - 56 q^{28} - 8 q^{31} + 32 q^{34} + 104 q^{37} + 200 q^{40} - 64 q^{43} + 32 q^{46} + 448 q^{49} - 264 q^{52} + 104 q^{55} - 360 q^{58} - 24 q^{61} - 720 q^{64} + 96 q^{67} + 816 q^{70} - 248 q^{73} - 184 q^{76} + 416 q^{79} + 96 q^{85} - 200 q^{88} - 184 q^{91} + 448 q^{94} - 488 q^{97}+O(q^{100}) \) 48 * q - 96 * q^4 - 16 * q^10 - 8 * q^13 + 264 * q^16 + 40 * q^19 - 40 * q^22 - 360 * q^25 - 56 * q^28 - 8 * q^31 + 32 * q^34 + 104 * q^37 + 200 * q^40 - 64 * q^43 + 32 * q^46 + 448 * q^49 - 264 * q^52 + 104 * q^55 - 360 * q^58 - 24 * q^61 - 720 * q^64 + 96 * q^67 + 816 * q^70 - 248 * q^73 - 184 * q^76 + 416 * q^79 + 96 * q^85 - 200 * q^88 - 184 * q^91 + 448 * q^94 - 488 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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