LMFDB - Embedded newform 693.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.a (trivial)

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:	yes
Analytic conductor:	$40.8883236340$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 28x^{3} - 11x^{2} + 108x - 64 $ x^5 - 28x^3 - 11x^2 + 108*x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.36278$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.36278 q^{2} +20.7594 q^{4} +1.66863 q^{5} -7.00000 q^{7} -68.4261 q^{8} +O(q^{10})$	\(q-5.36278 q^{2} +20.7594 q^{4} +1.66863 q^{5} -7.00000 q^{7} -68.4261 q^{8} -8.94849 q^{10} -11.0000 q^{11} +0.996604 q^{13} +37.5395 q^{14} +200.879 q^{16} +99.5115 q^{17} +32.8695 q^{19} +34.6398 q^{20} +58.9906 q^{22} -72.6019 q^{23} -122.216 q^{25} -5.34457 q^{26} -145.316 q^{28} -45.0027 q^{29} -62.8102 q^{31} -529.860 q^{32} -533.659 q^{34} -11.6804 q^{35} -301.317 q^{37} -176.272 q^{38} -114.178 q^{40} +307.353 q^{41} +214.347 q^{43} -228.354 q^{44} +389.348 q^{46} +602.899 q^{47} +49.0000 q^{49} +655.416 q^{50} +20.6889 q^{52} -592.072 q^{53} -18.3549 q^{55} +478.983 q^{56} +241.339 q^{58} -695.030 q^{59} +442.815 q^{61} +336.837 q^{62} +1234.49 q^{64} +1.66296 q^{65} -555.143 q^{67} +2065.80 q^{68} +62.6394 q^{70} +153.352 q^{71} -147.489 q^{73} +1615.90 q^{74} +682.352 q^{76} +77.0000 q^{77} +676.959 q^{79} +335.192 q^{80} -1648.27 q^{82} +222.412 q^{83} +166.048 q^{85} -1149.50 q^{86} +752.687 q^{88} +1136.35 q^{89} -6.97623 q^{91} -1507.18 q^{92} -3233.22 q^{94} +54.8469 q^{95} +1010.60 q^{97} -262.776 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} - 60 q^{8}+O(q^{10}) $ 5 * q - 5 * q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 - 60 * q^8	$ 5 q - 5 q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} - 60 q^{8} + 55 q^{10} - 55 q^{11} + 111 q^{13} + 35 q^{14} + 201 q^{16} - 136 q^{17} + 111 q^{19} - 219 q^{20} + 55 q^{22} + 28 q^{23} + 190 q^{25} + q^{26} - 147 q^{28} - 61 q^{29} - 280 q^{31} - 535 q^{32} - 572 q^{34} + 49 q^{35} - 41 q^{37} - 267 q^{38} - 336 q^{40} - 426 q^{41} + 424 q^{43} - 231 q^{44} + 140 q^{46} - 75 q^{47} + 245 q^{49} - 490 q^{50} - 269 q^{52} - 1500 q^{53} + 77 q^{55} + 420 q^{56} - 1767 q^{58} - 757 q^{59} + 658 q^{61} + 568 q^{62} - 748 q^{64} - 537 q^{65} - 583 q^{67} + 1650 q^{68} - 385 q^{70} + 764 q^{71} + 875 q^{73} + 825 q^{74} + 213 q^{76} + 385 q^{77} - 244 q^{79} + 2577 q^{80} - 2006 q^{82} - 924 q^{83} - 1402 q^{85} - 1272 q^{86} + 660 q^{88} + 1110 q^{89} - 777 q^{91} + 2046 q^{92} - 3349 q^{94} - 1923 q^{95} - 852 q^{97} - 245 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 - 60 * q^8 + 55 * q^10 - 55 * q^11 + 111 * q^13 + 35 * q^14 + 201 * q^16 - 136 * q^17 + 111 * q^19 - 219 * q^20 + 55 * q^22 + 28 * q^23 + 190 * q^25 + q^26 - 147 * q^28 - 61 * q^29 - 280 * q^31 - 535 * q^32 - 572 * q^34 + 49 * q^35 - 41 * q^37 - 267 * q^38 - 336 * q^40 - 426 * q^41 + 424 * q^43 - 231 * q^44 + 140 * q^46 - 75 * q^47 + 245 * q^49 - 490 * q^50 - 269 * q^52 - 1500 * q^53 + 77 * q^55 + 420 * q^56 - 1767 * q^58 - 757 * q^59 + 658 * q^61 + 568 * q^62 - 748 * q^64 - 537 * q^65 - 583 * q^67 + 1650 * q^68 - 385 * q^70 + 764 * q^71 + 875 * q^73 + 825 * q^74 + 213 * q^76 + 385 * q^77 - 244 * q^79 + 2577 * q^80 - 2006 * q^82 - 924 * q^83 - 1402 * q^85 - 1272 * q^86 + 660 * q^88 + 1110 * q^89 - 777 * q^91 + 2046 * q^92 - 3349 * q^94 - 1923 * q^95 - 852 * q^97 - 245 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−5.36278	−1.89603	−0.948015	−	0.318226i	$-0.896913\pi$
$2$	−5.36278	−1.89603	−0.948015	+	0.318226i	$0.896913\pi$
$3$	0	0
$3$	0	0
$4$	20.7594	2.59493
$5$	1.66863	0.149247	0.0746233	−	0.997212i	$-0.476225\pi$
$5$	1.66863	0.149247	0.0746233	+	0.997212i	$0.476225\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−68.4261	−3.02403
$9$	0	0
$10$	−8.94849	−0.282976
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	0.996604	0.0212622	0.0106311	−	0.999943i	$-0.496616\pi$
$13$	0.996604	0.0212622	0.0106311	+	0.999943i	$0.496616\pi$
$14$	37.5395	0.716632
$15$	0	0
$16$	200.879	3.13873
$17$	99.5115	1.41971	0.709856	−	0.704347i	$-0.248760\pi$
$17$	99.5115	1.41971	0.709856	+	0.704347i	$0.248760\pi$
$18$	0	0
$19$	32.8695	0.396883	0.198441	−	0.980113i	$-0.436412\pi$
$19$	32.8695	0.396883	0.198441	+	0.980113i	$0.436412\pi$
$20$	34.6398	0.387284
$21$	0	0
$22$	58.9906	0.571675
$23$	−72.6019	−0.658198	−0.329099	−	0.944295i	$-0.606745\pi$
$23$	−72.6019	−0.658198	−0.329099	+	0.944295i	$0.606745\pi$
$24$	0	0
$25$	−122.216	−0.977725
$26$	−5.34457	−0.0403137
$27$	0	0
$28$	−145.316	−0.980791
$29$	−45.0027	−0.288165	−0.144082	−	0.989566i	$-0.546023\pi$
$29$	−45.0027	−0.288165	−0.144082	+	0.989566i	$0.546023\pi$
$30$	0	0
$31$	−62.8102	−0.363904	−0.181952	−	0.983307i	$-0.558242\pi$
$31$	−62.8102	−0.363904	−0.181952	+	0.983307i	$0.558242\pi$
$32$	−529.860	−2.92709
$33$	0	0
$34$	−533.659	−2.69182
$35$	−11.6804	−0.0564099
$36$	0	0
$37$	−301.317	−1.33882	−0.669408	−	0.742895i	$-0.733452\pi$
$37$	−301.317	−1.33882	−0.669408	+	0.742895i	$0.733452\pi$
$38$	−176.272	−0.752502
$39$	0	0
$40$	−114.178	−0.451327
$41$	307.353	1.17074	0.585371	−	0.810765i	$-0.300949\pi$
$41$	307.353	1.17074	0.585371	+	0.810765i	$0.300949\pi$
$42$	0	0
$43$	214.347	0.760178	0.380089	−	0.924950i	$-0.375893\pi$
$43$	214.347	0.760178	0.380089	+	0.924950i	$0.375893\pi$
$44$	−228.354	−0.782401
$45$	0	0
$46$	389.348	1.24796
$47$	602.899	1.87110	0.935552	−	0.353188i	$-0.114902\pi$
$47$	602.899	1.87110	0.935552	+	0.353188i	$0.114902\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	655.416	1.85380
$51$	0	0
$52$	20.6889	0.0551738
$53$	−592.072	−1.53448	−0.767239	−	0.641362i	$-0.778370\pi$
$53$	−592.072	−1.53448	−0.767239	+	0.641362i	$0.778370\pi$
$54$	0	0
$55$	−18.3549	−0.0449995
$56$	478.983	1.14298
$57$	0	0
$58$	241.339	0.546369
$59$	−695.030	−1.53365	−0.766824	−	0.641857i	$-0.778164\pi$
$59$	−695.030	−1.53365	−0.766824	+	0.641857i	$0.778164\pi$
$60$	0	0
$61$	442.815	0.929453	0.464726	−	0.885454i	$-0.346153\pi$
$61$	442.815	0.929453	0.464726	+	0.885454i	$0.346153\pi$
$62$	336.837	0.689974
$63$	0	0
$64$	1234.49	2.41112
$65$	1.66296	0.00317331
$66$	0	0
$67$	−555.143	−1.01226	−0.506131	−	0.862457i	$-0.668925\pi$
$67$	−555.143	−1.01226	−0.506131	+	0.862457i	$0.668925\pi$
$68$	2065.80	3.68405
$69$	0	0
$70$	62.6394	0.106955
$71$	153.352	0.256331	0.128166	−	0.991753i	$-0.459091\pi$
$71$	153.352	0.256331	0.128166	+	0.991753i	$0.459091\pi$
$72$	0	0
$73$	−147.489	−0.236470	−0.118235	−	0.992986i	$-0.537724\pi$
$73$	−147.489	−0.236470	−0.118235	+	0.992986i	$0.537724\pi$
$74$	1615.90	2.53843
$75$	0	0
$76$	682.352	1.02988
$77$	77.0000	0.113961
$78$	0	0
$79$	676.959	0.964099	0.482050	−	0.876144i	$-0.339893\pi$
$79$	676.959	0.964099	0.482050	+	0.876144i	$0.339893\pi$
$80$	335.192	0.468445
$81$	0	0
$82$	−1648.27	−2.21976
$83$	222.412	0.294131	0.147065	−	0.989127i	$-0.453017\pi$
$83$	222.412	0.294131	0.147065	+	0.989127i	$0.453017\pi$
$84$	0	0
$85$	166.048	0.211887
$86$	−1149.50	−1.44132
$87$	0	0
$88$	752.687	0.911781
$89$	1136.35	1.35340	0.676699	−	0.736260i	$-0.263410\pi$
$89$	1136.35	1.35340	0.676699	+	0.736260i	$0.263410\pi$
$90$	0	0
$91$	−6.97623	−0.00803634
$92$	−1507.18	−1.70798
$93$	0	0
$94$	−3233.22	−3.54767
$95$	54.8469	0.0592334
$96$	0	0
$97$	1010.60	1.05785	0.528923	−	0.848670i	$-0.322596\pi$
$97$	1010.60	1.05785	0.528923	+	0.848670i	$0.322596\pi$
$98$	−262.776	−0.270861
$99$	0	0
$100$	−2537.13	−2.53713
$101$	400.418	0.394486	0.197243	−	0.980355i	$-0.436801\pi$
$101$	400.418	0.394486	0.197243	+	0.980355i	$0.436801\pi$
$102$	0	0
$103$	−419.845	−0.401636	−0.200818	−	0.979629i	$-0.564360\pi$
$103$	−419.845	−0.401636	−0.200818	+	0.979629i	$0.564360\pi$
$104$	−68.1937	−0.0642975
$105$	0	0
$106$	3175.15	2.90941
$107$	−2131.99	−1.92623	−0.963116	−	0.269085i	$-0.913279\pi$
$107$	−2131.99	−1.92623	−0.963116	+	0.269085i	$0.913279\pi$
$108$	0	0
$109$	−14.0057	−0.0123074	−0.00615369	−	0.999981i	$-0.501959\pi$
$109$	−14.0057	−0.0123074	−0.00615369	+	0.999981i	$0.501959\pi$
$110$	98.4333	0.0853205
$111$	0	0
$112$	−1406.15	−1.18633
$113$	−1322.98	−1.10137	−0.550686	−	0.834712i	$-0.685634\pi$
$113$	−1322.98	−1.10137	−0.550686	+	0.834712i	$0.685634\pi$
$114$	0	0
$115$	−121.146	−0.0982338
$116$	−934.230	−0.747768
$117$	0	0
$118$	3727.30	2.90784
$119$	−696.581	−0.536601
$120$	0	0
$121$	121.000	0.0909091
$122$	−2374.72	−1.76227
$123$	0	0
$124$	−1303.90	−0.944306
$125$	−412.511	−0.295169
$126$	0	0
$127$	−2199.01	−1.53646	−0.768232	−	0.640172i	$-0.778863\pi$
$127$	−2199.01	−1.53646	−0.768232	+	0.640172i	$0.778863\pi$
$128$	−2381.45	−1.64447
$129$	0	0
$130$	−8.91810	−0.00601668
$131$	−1450.27	−0.967259	−0.483629	−	0.875273i	$-0.660682\pi$
$131$	−1450.27	−0.967259	−0.483629	+	0.875273i	$0.660682\pi$
$132$	0	0
$133$	−230.086	−0.150008
$134$	2977.11	1.91928
$135$	0	0
$136$	−6809.19	−4.29326
$137$	893.027	0.556909	0.278454	−	0.960449i	$-0.410178\pi$
$137$	893.027	0.556909	0.278454	+	0.960449i	$0.410178\pi$
$138$	0	0
$139$	−530.606	−0.323780	−0.161890	−	0.986809i	$-0.551759\pi$
$139$	−530.606	−0.323780	−0.161890	+	0.986809i	$0.551759\pi$
$140$	−242.478	−0.146380
$141$	0	0
$142$	−822.392	−0.486011
$143$	−10.9626	−0.00641079
$144$	0	0
$145$	−75.0927	−0.0430076
$146$	790.954	0.448355
$147$	0	0
$148$	−6255.17	−3.47413
$149$	−2958.97	−1.62690	−0.813451	−	0.581633i	$-0.802414\pi$
$149$	−2958.97	−1.62690	−0.813451	+	0.581633i	$0.802414\pi$
$150$	0	0
$151$	1668.28	0.899092	0.449546	−	0.893257i	$-0.351586\pi$
$151$	1668.28	0.899092	0.449546	+	0.893257i	$0.351586\pi$
$152$	−2249.13	−1.20019
$153$	0	0
$154$	−412.934	−0.216073
$155$	−104.807	−0.0543115
$156$	0	0
$157$	−1202.63	−0.611342	−0.305671	−	0.952137i	$-0.598881\pi$
$157$	−1202.63	−0.611342	−0.305671	+	0.952137i	$0.598881\pi$
$158$	−3630.38	−1.82796
$159$	0	0
$160$	−884.139	−0.436858
$161$	508.214	0.248775
$162$	0	0
$163$	−2534.83	−1.21806	−0.609029	−	0.793148i	$-0.708441\pi$
$163$	−2534.83	−1.21806	−0.609029	+	0.793148i	$0.708441\pi$
$164$	6380.47	3.03799
$165$	0	0
$166$	−1192.75	−0.557681
$167$	−2424.43	−1.12340	−0.561700	−	0.827341i	$-0.689853\pi$
$167$	−2424.43	−1.12340	−0.561700	+	0.827341i	$0.689853\pi$
$168$	0	0
$169$	−2196.01	−0.999548
$170$	−890.478	−0.401744
$171$	0	0
$172$	4449.73	1.97261
$173$	−1977.78	−0.869180	−0.434590	−	0.900628i	$-0.643107\pi$
$173$	−1977.78	−0.869180	−0.434590	+	0.900628i	$0.643107\pi$
$174$	0	0
$175$	855.510	0.369545
$176$	−2209.67	−0.946363
$177$	0	0
$178$	−6093.97	−2.56608
$179$	−1314.89	−0.549049	−0.274525	−	0.961580i	$-0.588520\pi$
$179$	−1314.89	−0.549049	−0.274525	+	0.961580i	$0.588520\pi$
$180$	0	0
$181$	2118.94	0.870163	0.435081	−	0.900391i	$-0.356720\pi$
$181$	2118.94	0.870163	0.435081	+	0.900391i	$0.356720\pi$
$182$	37.4120	0.0152372
$183$	0	0
$184$	4967.87	1.99041
$185$	−502.785	−0.199814
$186$	0	0
$187$	−1094.63	−0.428059
$188$	12515.9	4.85538
$189$	0	0
$190$	−294.132	−0.112308
$191$	−2136.61	−0.809421	−0.404711	−	0.914445i	$-0.632628\pi$
$191$	−2136.61	−0.809421	−0.404711	+	0.914445i	$0.632628\pi$
$192$	0	0
$193$	−910.852	−0.339713	−0.169856	−	0.985469i	$-0.554330\pi$
$193$	−910.852	−0.339713	−0.169856	+	0.985469i	$0.554330\pi$
$194$	−5419.64	−2.00571
$195$	0	0
$196$	1017.21	0.370704
$197$	2485.86	0.899038	0.449519	−	0.893271i	$-0.351595\pi$
$197$	2485.86	0.899038	0.449519	+	0.893271i	$0.351595\pi$
$198$	0	0
$199$	−384.145	−0.136841	−0.0684204	−	0.997657i	$-0.521796\pi$
$199$	−384.145	−0.136841	−0.0684204	+	0.997657i	$0.521796\pi$
$200$	8362.74	2.95668
$201$	0	0
$202$	−2147.36	−0.747958
$203$	315.019	0.108916
$204$	0	0
$205$	512.857	0.174729
$206$	2251.54	0.761515
$207$	0	0
$208$	200.196	0.0667362
$209$	−361.564	−0.119665
$210$	0	0
$211$	3875.31	1.26440	0.632198	−	0.774807i	$-0.282153\pi$
$211$	3875.31	1.26440	0.632198	+	0.774807i	$0.282153\pi$
$212$	−12291.1	−3.98186
$213$	0	0
$214$	11433.4	3.65219
$215$	357.666	0.113454
$216$	0	0
$217$	439.671	0.137543
$218$	75.1096	0.0233352
$219$	0	0
$220$	−381.037	−0.116771
$221$	99.1736	0.0301861
$222$	0	0
$223$	449.799	0.135071	0.0675354	−	0.997717i	$-0.478486\pi$
$223$	449.799	0.135071	0.0675354	+	0.997717i	$0.478486\pi$
$224$	3709.02	1.10634
$225$	0	0
$226$	7094.83	2.08823
$227$	−5873.32	−1.71729	−0.858647	−	0.512567i	$-0.828695\pi$
$227$	−5873.32	−1.71729	−0.858647	+	0.512567i	$0.828695\pi$
$228$	0	0
$229$	4207.68	1.21420	0.607099	−	0.794626i	$-0.292333\pi$
$229$	4207.68	1.21420	0.607099	+	0.794626i	$0.292333\pi$
$230$	649.677	0.186254
$231$	0	0
$232$	3079.36	0.871421
$233$	−5405.81	−1.51994	−0.759970	−	0.649958i	$-0.774787\pi$
$233$	−5405.81	−1.51994	−0.759970	+	0.649958i	$0.774787\pi$
$234$	0	0
$235$	1006.01	0.279256
$236$	−14428.4	−3.97971
$237$	0	0
$238$	3735.61	1.01741
$239$	5105.16	1.38170	0.690848	−	0.723000i	$-0.257237\pi$
$239$	5105.16	1.38170	0.690848	+	0.723000i	$0.257237\pi$
$240$	0	0
$241$	254.138	0.0679273	0.0339637	−	0.999423i	$-0.489187\pi$
$241$	254.138	0.0679273	0.0339637	+	0.999423i	$0.489187\pi$
$242$	−648.897	−0.172366
$243$	0	0
$244$	9192.58	2.41186
$245$	81.7627	0.0213209
$246$	0	0
$247$	32.7579	0.00843859
$248$	4297.85	1.10046
$249$	0	0
$250$	2212.21	0.559649
$251$	−7030.70	−1.76802	−0.884012	−	0.467464i	$-0.845168\pi$
$251$	−7030.70	−1.76802	−0.884012	+	0.467464i	$0.845168\pi$
$252$	0	0
$253$	798.621	0.198454
$254$	11792.8	2.91318
$255$	0	0
$256$	2895.22	0.706841
$257$	−6053.67	−1.46933	−0.734664	−	0.678431i	$-0.762660\pi$
$257$	−6053.67	−1.46933	−0.734664	+	0.678431i	$0.762660\pi$
$258$	0	0
$259$	2109.22	0.506025
$260$	34.5221	0.00823451
$261$	0	0
$262$	7777.50	1.83395
$263$	6106.79	1.43179	0.715895	−	0.698208i	$-0.246019\pi$
$263$	6106.79	1.43179	0.715895	+	0.698208i	$0.246019\pi$
$264$	0	0
$265$	−987.947	−0.229015
$266$	1233.90	0.284419
$267$	0	0
$268$	−11524.5	−2.62675
$269$	−4855.91	−1.10063	−0.550316	−	0.834956i	$-0.685493\pi$
$269$	−4855.91	−1.10063	−0.550316	+	0.834956i	$0.685493\pi$
$270$	0	0
$271$	−8501.09	−1.90555	−0.952775	−	0.303676i	$-0.901786\pi$
$271$	−8501.09	−1.90555	−0.952775	+	0.303676i	$0.901786\pi$
$272$	19989.7	4.45609
$273$	0	0
$274$	−4789.11	−1.05592
$275$	1344.37	0.294795
$276$	0	0
$277$	−6830.18	−1.48154	−0.740768	−	0.671761i	$-0.765538\pi$
$277$	−6830.18	−1.48154	−0.740768	+	0.671761i	$0.765538\pi$
$278$	2845.52	0.613896
$279$	0	0
$280$	799.243	0.170585
$281$	−4987.88	−1.05890	−0.529452	−	0.848340i	$-0.677603\pi$
$281$	−4987.88	−1.05890	−0.529452	+	0.848340i	$0.677603\pi$
$282$	0	0
$283$	−1952.88	−0.410200	−0.205100	−	0.978741i	$-0.565752\pi$
$283$	−1952.88	−0.410200	−0.205100	+	0.978741i	$0.565752\pi$
$284$	3183.50	0.665161
$285$	0	0
$286$	58.7903	0.0121550
$287$	−2151.47	−0.442499
$288$	0	0
$289$	4989.55	1.01558
$290$	402.706	0.0815438
$291$	0	0
$292$	−3061.80	−0.613624
$293$	3220.72	0.642172	0.321086	−	0.947050i	$-0.395952\pi$
$293$	3220.72	0.642172	0.321086	+	0.947050i	$0.395952\pi$
$294$	0	0
$295$	−1159.75	−0.228892
$296$	20617.9	4.04862
$297$	0	0
$298$	15868.3	3.08466
$299$	−72.3554	−0.0139947
$300$	0	0
$301$	−1500.43	−0.287320
$302$	−8946.64	−1.70471
$303$	0	0
$304$	6602.78	1.24571
$305$	738.893	0.138718
$306$	0	0
$307$	−5325.08	−0.989963	−0.494981	−	0.868904i	$-0.664825\pi$
$307$	−5325.08	−0.989963	−0.494981	+	0.868904i	$0.664825\pi$
$308$	1598.48	0.295720
$309$	0	0
$310$	562.056	0.102976
$311$	9625.36	1.75500	0.877498	−	0.479580i	$-0.159211\pi$
$311$	9625.36	1.75500	0.877498	+	0.479580i	$0.159211\pi$
$312$	0	0
$313$	−4315.55	−0.779327	−0.389664	−	0.920957i	$-0.627409\pi$
$313$	−4315.55	−0.779327	−0.389664	+	0.920957i	$0.627409\pi$
$314$	6449.46	1.15912
$315$	0	0
$316$	14053.3	2.50177
$317$	7399.84	1.31109	0.655546	−	0.755155i	$-0.272438\pi$
$317$	7399.84	1.31109	0.655546	+	0.755155i	$0.272438\pi$
$318$	0	0
$319$	495.029	0.0868850
$320$	2059.91	0.359852
$321$	0	0
$322$	−2725.44	−0.471686
$323$	3270.89	0.563459
$324$	0	0
$325$	−121.801	−0.0207886
$326$	13593.8	2.30947
$327$	0	0
$328$	−21031.0	−3.54037
$329$	−4220.30	−0.707211
$330$	0	0
$331$	−4899.29	−0.813562	−0.406781	−	0.913526i	$-0.633349\pi$
$331$	−4899.29	−0.813562	−0.406781	+	0.913526i	$0.633349\pi$
$332$	4617.14	0.763248
$333$	0	0
$334$	13001.7	2.13000
$335$	−926.327	−0.151077
$336$	0	0
$337$	−1445.02	−0.233577	−0.116788	−	0.993157i	$-0.537260\pi$
$337$	−1445.02	−0.233577	−0.116788	+	0.993157i	$0.537260\pi$
$338$	11776.7	1.89517
$339$	0	0
$340$	3447.06	0.549832
$341$	690.912	0.109721
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−14666.9	−2.29880
$345$	0	0
$346$	10606.4	1.64799
$347$	−4351.93	−0.673268	−0.336634	−	0.941636i	$-0.609289\pi$
$347$	−4351.93	−0.673268	−0.336634	+	0.941636i	$0.609289\pi$
$348$	0	0
$349$	7971.30	1.22262	0.611309	−	0.791392i	$-0.290643\pi$
$349$	7971.30	1.22262	0.611309	+	0.791392i	$0.290643\pi$
$350$	−4587.91	−0.700669
$351$	0	0
$352$	5828.46	0.882551
$353$	1169.88	0.176393	0.0881963	−	0.996103i	$-0.471890\pi$
$353$	1169.88	0.176393	0.0881963	+	0.996103i	$0.471890\pi$
$354$	0	0
$355$	255.887	0.0382565
$356$	23589.9	3.51197
$357$	0	0
$358$	7051.49	1.04101
$359$	−11998.8	−1.76400	−0.881998	−	0.471253i	$-0.843802\pi$
$359$	−11998.8	−1.76400	−0.881998	+	0.471253i	$0.843802\pi$
$360$	0	0
$361$	−5778.60	−0.842484
$362$	−11363.4	−1.64985
$363$	0	0
$364$	−144.823	−0.0208537
$365$	−246.105	−0.0352924
$366$	0	0
$367$	−8489.94	−1.20755	−0.603776	−	0.797154i	$-0.706338\pi$
$367$	−8489.94	−1.20755	−0.603776	+	0.797154i	$0.706338\pi$
$368$	−14584.2	−2.06590
$369$	0	0
$370$	2696.33	0.378853
$371$	4144.50	0.579978
$372$	0	0
$373$	−13306.1	−1.84708	−0.923542	−	0.383498i	$-0.874719\pi$
$373$	−13306.1	−1.84708	−0.923542	+	0.383498i	$0.874719\pi$
$374$	5870.25	0.811613
$375$	0	0
$376$	−41254.0	−5.65828
$377$	−44.8498	−0.00612701
$378$	0	0
$379$	−10645.0	−1.44274	−0.721368	−	0.692552i	$-0.756486\pi$
$379$	−10645.0	−1.44274	−0.721368	+	0.692552i	$0.756486\pi$
$380$	1138.59	0.153707
$381$	0	0
$382$	11458.2	1.53469
$383$	−2299.39	−0.306771	−0.153385	−	0.988166i	$-0.549018\pi$
$383$	−2299.39	−0.306771	−0.153385	+	0.988166i	$0.549018\pi$
$384$	0	0
$385$	128.484	0.0170082
$386$	4884.70	0.644105
$387$	0	0
$388$	20979.5	2.74504
$389$	3243.77	0.422791	0.211395	−	0.977401i	$-0.432199\pi$
$389$	3243.77	0.422791	0.211395	+	0.977401i	$0.432199\pi$
$390$	0	0
$391$	−7224.73	−0.934451
$392$	−3352.88	−0.432005
$393$	0	0
$394$	−13331.1	−1.70460
$395$	1129.59	0.143888
$396$	0	0
$397$	13850.8	1.75102	0.875508	−	0.483203i	$-0.160527\pi$
$397$	13850.8	1.75102	0.875508	+	0.483203i	$0.160527\pi$
$398$	2060.09	0.259454
$399$	0	0
$400$	−24550.5	−3.06882
$401$	−1238.44	−0.154227	−0.0771133	−	0.997022i	$-0.524570\pi$
$401$	−1238.44	−0.154227	−0.0771133	+	0.997022i	$0.524570\pi$
$402$	0	0
$403$	−62.5969	−0.00773740
$404$	8312.46	1.02366
$405$	0	0
$406$	−1689.38	−0.206508
$407$	3314.48	0.403668
$408$	0	0
$409$	−3951.51	−0.477726	−0.238863	−	0.971053i	$-0.576775\pi$
$409$	−3951.51	−0.477726	−0.238863	+	0.971053i	$0.576775\pi$
$410$	−2750.34	−0.331292
$411$	0	0
$412$	−8715.74	−1.04222
$413$	4865.21	0.579665
$414$	0	0
$415$	371.122	0.0438980
$416$	−528.061	−0.0622363
$417$	0	0
$418$	1938.99	0.226888
$419$	11603.6	1.35292	0.676462	−	0.736477i	$-0.263512\pi$
$419$	11603.6	1.35292	0.676462	+	0.736477i	$0.263512\pi$
$420$	0	0
$421$	−10498.9	−1.21541	−0.607703	−	0.794165i	$-0.707909\pi$
$421$	−10498.9	−1.21541	−0.607703	+	0.794165i	$0.707909\pi$
$422$	−20782.5	−2.39733
$423$	0	0
$424$	40513.1	4.64031
$425$	−12161.9	−1.38809
$426$	0	0
$427$	−3099.70	−0.351300
$428$	−44258.8	−4.99844
$429$	0	0
$430$	−1918.08	−0.215112
$431$	11109.2	1.24156	0.620778	−	0.783986i	$-0.286817\pi$
$431$	11109.2	1.24156	0.620778	+	0.783986i	$0.286817\pi$
$432$	0	0
$433$	12535.7	1.39129	0.695644	−	0.718387i	$-0.255119\pi$
$433$	12535.7	1.39129	0.695644	+	0.718387i	$0.255119\pi$
$434$	−2357.86	−0.260786
$435$	0	0
$436$	−290.751	−0.0319368
$437$	−2386.39	−0.261227
$438$	0	0
$439$	2227.80	0.242203	0.121102	−	0.992640i	$-0.461357\pi$
$439$	2227.80	0.242203	0.121102	+	0.992640i	$0.461357\pi$
$440$	1255.95	0.136080
$441$	0	0
$442$	−531.846	−0.0572338
$443$	833.252	0.0893657	0.0446828	−	0.999001i	$-0.485772\pi$
$443$	833.252	0.0893657	0.0446828	+	0.999001i	$0.485772\pi$
$444$	0	0
$445$	1896.14	0.201990
$446$	−2412.17	−0.256098
$447$	0	0
$448$	−8641.46	−0.911319
$449$	4903.89	0.515432	0.257716	−	0.966221i	$-0.417030\pi$
$449$	4903.89	0.515432	0.257716	+	0.966221i	$0.417030\pi$
$450$	0	0
$451$	−3380.88	−0.352992
$452$	−27464.2	−2.85798
$453$	0	0
$454$	31497.3	3.25604
$455$	−11.6407	−0.00119940
$456$	0	0
$457$	12100.7	1.23861	0.619307	−	0.785149i	$-0.287414\pi$
$457$	12100.7	1.23861	0.619307	+	0.785149i	$0.287414\pi$
$458$	−22564.9	−2.30216
$459$	0	0
$460$	−2514.91	−0.254910
$461$	1271.16	0.128425	0.0642125	−	0.997936i	$-0.479546\pi$
$461$	1271.16	0.128425	0.0642125	+	0.997936i	$0.479546\pi$
$462$	0	0
$463$	−16020.3	−1.60805	−0.804024	−	0.594596i	$-0.797312\pi$
$463$	−16020.3	−1.60805	−0.804024	+	0.594596i	$0.797312\pi$
$464$	−9040.08	−0.904472
$465$	0	0
$466$	28990.2	2.88185
$467$	10696.1	1.05986	0.529930	−	0.848042i	$-0.322218\pi$
$467$	10696.1	1.05986	0.529930	+	0.848042i	$0.322218\pi$
$468$	0	0
$469$	3886.00	0.382599
$470$	−5395.04	−0.529478
$471$	0	0
$472$	47558.2	4.63780
$473$	−2357.82	−0.229202
$474$	0	0
$475$	−4017.17	−0.388043
$476$	−14460.6	−1.39244
$477$	0	0
$478$	−27377.9	−2.61974
$479$	12055.2	1.14993	0.574963	−	0.818179i	$-0.305016\pi$
$479$	12055.2	1.14993	0.574963	+	0.818179i	$0.305016\pi$
$480$	0	0
$481$	−300.293	−0.0284661
$482$	−1362.89	−0.128792
$483$	0	0
$484$	2511.89	0.235903
$485$	1686.32	0.157880
$486$	0	0
$487$	−14273.2	−1.32809	−0.664045	−	0.747693i	$-0.731162\pi$
$487$	−14273.2	−1.32809	−0.664045	+	0.747693i	$0.731162\pi$
$488$	−30300.1	−2.81070
$489$	0	0
$490$	−438.476	−0.0404251
$491$	15616.7	1.43538	0.717691	−	0.696362i	$-0.245199\pi$
$491$	15616.7	1.43538	0.717691	+	0.696362i	$0.245199\pi$
$492$	0	0
$493$	−4478.28	−0.409111
$494$	−175.673	−0.0159998
$495$	0	0
$496$	−12617.2	−1.14220
$497$	−1073.46	−0.0968841
$498$	0	0
$499$	5820.03	0.522125	0.261062	−	0.965322i	$-0.415927\pi$
$499$	5820.03	0.522125	0.261062	+	0.965322i	$0.415927\pi$
$500$	−8563.49	−0.765942
$501$	0	0
$502$	37704.1	3.35223
$503$	1744.38	0.154629	0.0773143	−	0.997007i	$-0.475366\pi$
$503$	1744.38	0.154629	0.0773143	+	0.997007i	$0.475366\pi$
$504$	0	0
$505$	668.149	0.0588757
$506$	−4282.83	−0.376275
$507$	0	0
$508$	−45650.3	−3.98701
$509$	4561.13	0.397187	0.198594	−	0.980082i	$-0.436363\pi$
$509$	4561.13	0.397187	0.198594	+	0.980082i	$0.436363\pi$
$510$	0	0
$511$	1032.43	0.0893774
$512$	3525.13	0.304278
$513$	0	0
$514$	32464.5	2.78589
$515$	−700.565	−0.0599429
$516$	0	0
$517$	−6631.89	−0.564159
$518$	−11311.3	−0.959438
$519$	0	0
$520$	−113.790	−0.00959619
$521$	311.151	0.0261646	0.0130823	−	0.999914i	$-0.495836\pi$
$521$	311.151	0.0261646	0.0130823	+	0.999914i	$0.495836\pi$
$522$	0	0
$523$	18159.9	1.51831	0.759154	−	0.650911i	$-0.225613\pi$
$523$	18159.9	1.51831	0.759154	+	0.650911i	$0.225613\pi$
$524$	−30106.8	−2.50997
$525$	0	0
$526$	−32749.4	−2.71472
$527$	−6250.34	−0.516639
$528$	0	0
$529$	−6895.96	−0.566776
$530$	5298.14	0.434220
$531$	0	0
$532$	−4776.46	−0.389259
$533$	306.309	0.0248925
$534$	0	0
$535$	−3557.49	−0.287484
$536$	37986.3	3.06111
$537$	0	0
$538$	26041.2	2.08683
$539$	−539.000	−0.0430730
$540$	0	0
$541$	10858.8	0.862951	0.431476	−	0.902125i	$-0.357993\pi$
$541$	10858.8	0.862951	0.431476	+	0.902125i	$0.357993\pi$
$542$	45589.5	3.61298
$543$	0	0
$544$	−52727.2	−4.15562
$545$	−23.3703	−0.00183683
$546$	0	0
$547$	9560.10	0.747277	0.373638	−	0.927574i	$-0.378110\pi$
$547$	9560.10	0.747277	0.373638	+	0.927574i	$0.378110\pi$
$548$	18538.7	1.44514
$549$	0	0
$550$	−7209.58	−0.558941
$551$	−1479.21	−0.114368
$552$	0	0
$553$	−4738.71	−0.364395
$554$	36628.8	2.80904
$555$	0	0
$556$	−11015.1	−0.840186
$557$	961.471	0.0731397	0.0365699	−	0.999331i	$-0.488357\pi$
$557$	961.471	0.0731397	0.0365699	+	0.999331i	$0.488357\pi$
$558$	0	0
$559$	213.619	0.0161630
$560$	−2346.34	−0.177055
$561$	0	0
$562$	26748.9	2.00772
$563$	−26376.5	−1.97449	−0.987245	−	0.159206i	$-0.949106\pi$
$563$	−26376.5	−1.97449	−0.987245	+	0.159206i	$0.949106\pi$
$564$	0	0
$565$	−2207.55	−0.164376
$566$	10472.9	0.777751
$567$	0	0
$568$	−10493.3	−0.775154
$569$	−8256.23	−0.608294	−0.304147	−	0.952625i	$-0.598371\pi$
$569$	−8256.23	−0.608294	−0.304147	+	0.952625i	$0.598371\pi$
$570$	0	0
$571$	−25003.0	−1.83248	−0.916239	−	0.400632i	$-0.868791\pi$
$571$	−25003.0	−1.83248	−0.916239	+	0.400632i	$0.868791\pi$
$572$	−227.578	−0.0166355
$573$	0	0
$574$	11537.9	0.838992
$575$	8873.09	0.643537
$576$	0	0
$577$	−5560.17	−0.401166	−0.200583	−	0.979677i	$-0.564284\pi$
$577$	−5560.17	−0.401166	−0.200583	+	0.979677i	$0.564284\pi$
$578$	−26757.9	−1.92557
$579$	0	0
$580$	−1558.88	−0.111602
$581$	−1556.88	−0.111171
$582$	0	0
$583$	6512.79	0.462662
$584$	10092.1	0.715094
$585$	0	0
$586$	−17272.0	−1.21758
$587$	−3926.82	−0.276111	−0.138055	−	0.990425i	$-0.544085\pi$
$587$	−3926.82	−0.276111	−0.138055	+	0.990425i	$0.544085\pi$
$588$	0	0
$589$	−2064.54	−0.144427
$590$	6219.47	0.433986
$591$	0	0
$592$	−60528.1	−4.20218
$593$	14397.6	0.997027	0.498514	−	0.866882i	$-0.333879\pi$
$593$	14397.6	0.997027	0.498514	+	0.866882i	$0.333879\pi$
$594$	0	0
$595$	−1162.33	−0.0800858
$596$	−61426.6	−4.22170
$597$	0	0
$598$	388.026	0.0265344
$599$	14711.0	1.00347	0.501734	−	0.865022i	$-0.332696\pi$
$599$	14711.0	1.00347	0.501734	+	0.865022i	$0.332696\pi$
$600$	0	0
$601$	27436.8	1.86218	0.931090	−	0.364790i	$-0.118859\pi$
$601$	27436.8	1.86218	0.931090	+	0.364790i	$0.118859\pi$
$602$	8046.48	0.544768
$603$	0	0
$604$	34632.6	2.33308
$605$	201.904	0.0135679
$606$	0	0
$607$	6090.69	0.407271	0.203636	−	0.979047i	$-0.434724\pi$
$607$	6090.69	0.407271	0.203636	+	0.979047i	$0.434724\pi$
$608$	−17416.2	−1.16171
$609$	0	0
$610$	−3962.52	−0.263013
$611$	600.852	0.0397837
$612$	0	0
$613$	5175.61	0.341013	0.170506	−	0.985357i	$-0.445460\pi$
$613$	5175.61	0.341013	0.170506	+	0.985357i	$0.445460\pi$
$614$	28557.3	1.87700
$615$	0	0
$616$	−5268.81	−0.344621
$617$	10544.9	0.688044	0.344022	−	0.938962i	$-0.388211\pi$
$617$	10544.9	0.688044	0.344022	+	0.938962i	$0.388211\pi$
$618$	0	0
$619$	−8378.59	−0.544045	−0.272023	−	0.962291i	$-0.587693\pi$
$619$	−8378.59	−0.544045	−0.272023	+	0.962291i	$0.587693\pi$
$620$	−2175.73	−0.140934
$621$	0	0
$622$	−51618.7	−3.32753
$623$	−7954.42	−0.511536
$624$	0	0
$625$	14588.6	0.933673
$626$	23143.4	1.47763
$627$	0	0
$628$	−24966.0	−1.58639
$629$	−29984.5	−1.90073
$630$	0	0
$631$	−2651.03	−0.167251	−0.0836257	−	0.996497i	$-0.526650\pi$
$631$	−2651.03	−0.167251	−0.0836257	+	0.996497i	$0.526650\pi$
$632$	−46321.6	−2.91547
$633$	0	0
$634$	−39683.7	−2.48587
$635$	−3669.33	−0.229312
$636$	0	0
$637$	48.8336	0.00303745
$638$	−2654.73	−0.164737
$639$	0	0
$640$	−3973.75	−0.245431
$641$	6716.00	0.413831	0.206916	−	0.978359i	$-0.433657\pi$
$641$	6716.00	0.413831	0.206916	+	0.978359i	$0.433657\pi$
$642$	0	0
$643$	752.048	0.0461242	0.0230621	−	0.999734i	$-0.492658\pi$
$643$	752.048	0.0461242	0.0230621	+	0.999734i	$0.492658\pi$
$644$	10550.2	0.645555
$645$	0	0
$646$	−17541.1	−1.06834
$647$	10412.8	0.632717	0.316358	−	0.948640i	$-0.397540\pi$
$647$	10412.8	0.632717	0.316358	+	0.948640i	$0.397540\pi$
$648$	0	0
$649$	7645.33	0.462412
$650$	653.190	0.0394157
$651$	0	0
$652$	−52621.7	−3.16077
$653$	−15442.9	−0.925462	−0.462731	−	0.886499i	$-0.653130\pi$
$653$	−15442.9	−0.925462	−0.462731	+	0.886499i	$0.653130\pi$
$654$	0	0
$655$	−2419.96	−0.144360
$656$	61740.6	3.67464
$657$	0	0
$658$	22632.5	1.34089
$659$	5333.13	0.315249	0.157625	−	0.987499i	$-0.449616\pi$
$659$	5333.13	0.315249	0.157625	+	0.987499i	$0.449616\pi$
$660$	0	0
$661$	15118.3	0.889614	0.444807	−	0.895626i	$-0.353272\pi$
$661$	15118.3	0.889614	0.444807	+	0.895626i	$0.353272\pi$
$662$	26273.8	1.54254
$663$	0	0
$664$	−15218.8	−0.889461
$665$	−383.928	−0.0223881
$666$	0	0
$667$	3267.28	0.189670
$668$	−50329.8	−2.91514
$669$	0	0
$670$	4967.69	0.286446
$671$	−4870.96	−0.280241
$672$	0	0
$673$	−813.476	−0.0465931	−0.0232966	−	0.999729i	$-0.507416\pi$
$673$	−813.476	−0.0465931	−0.0232966	+	0.999729i	$0.507416\pi$
$674$	7749.33	0.442868
$675$	0	0
$676$	−45587.9	−2.59376
$677$	−10390.3	−0.589854	−0.294927	−	0.955520i	$-0.595295\pi$
$677$	−10390.3	−0.589854	−0.294927	+	0.955520i	$0.595295\pi$
$678$	0	0
$679$	−7074.22	−0.399828
$680$	−11362.0	−0.640754
$681$	0	0
$682$	−3705.21	−0.208035
$683$	−24315.7	−1.36225	−0.681124	−	0.732168i	$-0.738509\pi$
$683$	−24315.7	−1.36225	−0.681124	+	0.732168i	$0.738509\pi$
$684$	0	0
$685$	1490.13	0.0831167
$686$	1839.43	0.102376
$687$	0	0
$688$	43057.8	2.38599
$689$	−590.061	−0.0326263
$690$	0	0
$691$	−15379.7	−0.846701	−0.423351	−	0.905966i	$-0.639146\pi$
$691$	−15379.7	−0.846701	−0.423351	+	0.905966i	$0.639146\pi$
$692$	−41057.7	−2.25546
$693$	0	0
$694$	23338.5	1.27654
$695$	−885.384	−0.0483230
$696$	0	0
$697$	30585.2	1.66212
$698$	−42748.3	−2.31812
$699$	0	0
$700$	17759.9	0.958944
$701$	2397.08	0.129153	0.0645766	−	0.997913i	$-0.479430\pi$
$701$	2397.08	0.129153	0.0645766	+	0.997913i	$0.479430\pi$
$702$	0	0
$703$	−9904.13	−0.531353
$704$	−13579.4	−0.726981
$705$	0	0
$706$	−6273.83	−0.334446
$707$	−2802.93	−0.149102
$708$	0	0
$709$	17121.2	0.906912	0.453456	−	0.891279i	$-0.350191\pi$
$709$	17121.2	0.906912	0.453456	+	0.891279i	$0.350191\pi$
$710$	−1372.27	−0.0725355
$711$	0	0
$712$	−77755.6	−4.09272
$713$	4560.14	0.239521
$714$	0	0
$715$	−18.2926	−0.000956788 0
$716$	−27296.5	−1.42474
$717$	0	0
$718$	64347.2	3.34459
$719$	−34131.4	−1.77036	−0.885179	−	0.465251i	$-0.845964\pi$
$719$	−34131.4	−1.77036	−0.885179	+	0.465251i	$0.845964\pi$
$720$	0	0
$721$	2938.91	0.151804
$722$	30989.4	1.59737
$723$	0	0
$724$	43988.0	2.25801
$725$	5500.03	0.281746
$726$	0	0
$727$	−1143.65	−0.0583434	−0.0291717	−	0.999574i	$-0.509287\pi$
$727$	−1143.65	−0.0583434	−0.0291717	+	0.999574i	$0.509287\pi$
$728$	477.356	0.0243022
$729$	0	0
$730$	1319.81	0.0669154
$731$	21330.0	1.07923
$732$	0	0
$733$	24224.7	1.22068	0.610342	−	0.792138i	$-0.291032\pi$
$733$	24224.7	1.22068	0.610342	+	0.792138i	$0.291032\pi$
$734$	45529.7	2.28955
$735$	0	0
$736$	38468.9	1.92660
$737$	6106.57	0.305208
$738$	0	0
$739$	25815.9	1.28505	0.642525	−	0.766265i	$-0.277887\pi$
$739$	25815.9	1.28505	0.642525	+	0.766265i	$0.277887\pi$
$740$	−10437.5	−0.518502
$741$	0	0
$742$	−22226.1	−1.09966
$743$	17221.1	0.850312	0.425156	−	0.905120i	$-0.360219\pi$
$743$	17221.1	0.850312	0.425156	+	0.905120i	$0.360219\pi$
$744$	0	0
$745$	−4937.42	−0.242810
$746$	71357.5	3.50212
$747$	0	0
$748$	−22723.8	−1.11078
$749$	14923.9	0.728048
$750$	0	0
$751$	−15631.7	−0.759531	−0.379766	−	0.925083i	$-0.623995\pi$
$751$	−15631.7	−0.759531	−0.379766	+	0.925083i	$0.623995\pi$
$752$	121110.	5.87289
$753$	0	0
$754$	240.520	0.0116170
$755$	2783.74	0.134186
$756$	0	0
$757$	24756.7	1.18864	0.594319	−	0.804230i	$-0.297422\pi$
$757$	24756.7	1.18864	0.594319	+	0.804230i	$0.297422\pi$
$758$	57086.8	2.73547
$759$	0	0
$760$	−3752.96	−0.179124
$761$	9087.20	0.432866	0.216433	−	0.976297i	$-0.430558\pi$
$761$	9087.20	0.432866	0.216433	+	0.976297i	$0.430558\pi$
$762$	0	0
$763$	98.0400	0.00465175
$764$	−44354.8	−2.10039
$765$	0	0
$766$	12331.1	0.581647
$767$	−692.670	−0.0326087
$768$	0	0
$769$	10699.9	0.501753	0.250876	−	0.968019i	$-0.419281\pi$
$769$	10699.9	0.501753	0.250876	+	0.968019i	$0.419281\pi$
$770$	−689.033	−0.0322481
$771$	0	0
$772$	−18908.8	−0.881531
$773$	6586.45	0.306466	0.153233	−	0.988190i	$-0.451032\pi$
$773$	6586.45	0.306466	0.153233	+	0.988190i	$0.451032\pi$
$774$	0	0
$775$	7676.39	0.355799
$776$	−69151.5	−3.19896
$777$	0	0
$778$	−17395.6	−0.801623
$779$	10102.5	0.464648
$780$	0	0
$781$	−1686.87	−0.0772867
$782$	38744.7	1.77175
$783$	0	0
$784$	9843.06	0.448390
$785$	−2006.75	−0.0912406
$786$	0	0
$787$	−5581.26	−0.252796	−0.126398	−	0.991980i	$-0.540342\pi$
$787$	−5581.26	−0.252796	−0.126398	+	0.991980i	$0.540342\pi$
$788$	51605.1	2.33294
$789$	0	0
$790$	−6057.76	−0.272817
$791$	9260.83	0.416280
$792$	0	0
$793$	441.311	0.0197622
$794$	−74279.0	−3.31998
$795$	0	0
$796$	−7974.64	−0.355092
$797$	20336.2	0.903821	0.451911	−	0.892063i	$-0.350743\pi$
$797$	20336.2	0.903821	0.451911	+	0.892063i	$0.350743\pi$
$798$	0	0
$799$	59995.5	2.65643
$800$	64757.2	2.86189
$801$	0	0
$802$	6641.49	0.292418
$803$	1622.38	0.0712985
$804$	0	0
$805$	848.019	0.0371289
$806$	335.693	0.0146703
$807$	0	0
$808$	−27399.1	−1.19294
$809$	−5494.43	−0.238781	−0.119391	−	0.992847i	$-0.538094\pi$
$809$	−5494.43	−0.238781	−0.119391	+	0.992847i	$0.538094\pi$
$810$	0	0
$811$	33839.3	1.46518	0.732590	−	0.680671i	$-0.238312\pi$
$811$	33839.3	1.46518	0.732590	+	0.680671i	$0.238312\pi$
$812$	6539.61	0.282630
$813$	0	0
$814$	−17774.9	−0.765367
$815$	−4229.69	−0.181791
$816$	0	0
$817$	7045.48	0.301702
$818$	21191.1	0.905782
$819$	0	0
$820$	10646.6	0.453410
$821$	−35849.2	−1.52393	−0.761965	−	0.647618i	$-0.775765\pi$
$821$	−35849.2	−1.52393	−0.761965	+	0.647618i	$0.775765\pi$
$822$	0	0
$823$	−14873.5	−0.629961	−0.314981	−	0.949098i	$-0.601998\pi$
$823$	−14873.5	−0.629961	−0.314981	+	0.949098i	$0.601998\pi$
$824$	28728.3	1.21456
$825$	0	0
$826$	−26091.1	−1.09906
$827$	−5757.86	−0.242104	−0.121052	−	0.992646i	$-0.538627\pi$
$827$	−5757.86	−0.242104	−0.121052	+	0.992646i	$0.538627\pi$
$828$	0	0
$829$	−38718.4	−1.62213	−0.811064	−	0.584957i	$-0.801111\pi$
$829$	−38718.4	−1.62213	−0.811064	+	0.584957i	$0.801111\pi$
$830$	−1990.25	−0.0832319
$831$	0	0
$832$	1230.30	0.0512657
$833$	4876.07	0.202816
$834$	0	0
$835$	−4045.47	−0.167664
$836$	−7505.87	−0.310521
$837$	0	0
$838$	−62227.8	−2.56518
$839$	36011.2	1.48182	0.740908	−	0.671606i	$-0.234395\pi$
$839$	36011.2	1.48182	0.740908	+	0.671606i	$0.234395\pi$
$840$	0	0
$841$	−22363.8	−0.916961
$842$	56303.4	2.30445
$843$	0	0
$844$	80449.3	3.28102
$845$	−3664.32	−0.149179
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−118935.	−4.81631
$849$	0	0
$850$	65221.5	2.63186
$851$	21876.2	0.881205
$852$	0	0
$853$	−12086.6	−0.485157	−0.242578	−	0.970132i	$-0.577993\pi$
$853$	−12086.6	−0.485157	−0.242578	+	0.970132i	$0.577993\pi$
$854$	16623.0	0.666075
$855$	0	0
$856$	145883.	5.82499
$857$	8653.97	0.344941	0.172470	−	0.985015i	$-0.444825\pi$
$857$	8653.97	0.344941	0.172470	+	0.985015i	$0.444825\pi$
$858$	0	0
$859$	−46141.6	−1.83275	−0.916374	−	0.400324i	$-0.868898\pi$
$859$	−46141.6	−1.83275	−0.916374	+	0.400324i	$0.868898\pi$
$860$	7424.94	0.294405
$861$	0	0
$862$	−59576.1	−2.35403
$863$	−16324.1	−0.643893	−0.321946	−	0.946758i	$-0.604337\pi$
$863$	−16324.1	−0.643893	−0.321946	+	0.946758i	$0.604337\pi$
$864$	0	0
$865$	−3300.19	−0.129722
$866$	−67226.2	−2.63792
$867$	0	0
$868$	9127.32	0.356914
$869$	−7446.55	−0.290687
$870$	0	0
$871$	−553.258	−0.0215229
$872$	958.356	0.0372179
$873$	0	0
$874$	12797.7	0.495295
$875$	2887.58	0.111563
$876$	0	0
$877$	17297.6	0.666017	0.333008	−	0.942924i	$-0.391936\pi$
$877$	17297.6	0.666017	0.333008	+	0.942924i	$0.391936\pi$
$878$	−11947.2	−0.459225
$879$	0	0
$880$	−3687.11	−0.141241
$881$	−45762.2	−1.75002	−0.875009	−	0.484106i	$-0.839145\pi$
$881$	−45762.2	−1.75002	−0.875009	+	0.484106i	$0.839145\pi$
$882$	0	0
$883$	−1535.96	−0.0585380	−0.0292690	−	0.999572i	$-0.509318\pi$
$883$	−1535.96	−0.0585380	−0.0292690	+	0.999572i	$0.509318\pi$
$884$	2058.79	0.0783309
$885$	0	0
$886$	−4468.55	−0.169440
$887$	47975.2	1.81607	0.908033	−	0.418899i	$-0.137584\pi$
$887$	47975.2	1.81607	0.908033	+	0.418899i	$0.137584\pi$
$888$	0	0
$889$	15393.1	0.580729
$890$	−10168.6	−0.382979
$891$	0	0
$892$	9337.57	0.350499
$893$	19817.0	0.742610
$894$	0	0
$895$	−2194.07	−0.0819437
$896$	16670.1	0.621551
$897$	0	0
$898$	−26298.5	−0.977274
$899$	2826.62	0.104865
$900$	0	0
$901$	−58918.0	−2.17851
$902$	18130.9	0.669284
$903$	0	0
$904$	90526.0	3.33059
$905$	3535.72	0.129869
$906$	0	0
$907$	16461.0	0.602622	0.301311	−	0.953526i	$-0.402576\pi$
$907$	16461.0	0.602622	0.301311	+	0.953526i	$0.402576\pi$
$908$	−121927.	−4.45626
$909$	0	0
$910$	62.4267	0.00227409
$911$	4712.18	0.171374	0.0856869	−	0.996322i	$-0.472692\pi$
$911$	4712.18	0.171374	0.0856869	+	0.996322i	$0.472692\pi$
$912$	0	0
$913$	−2446.53	−0.0886837
$914$	−64893.4	−2.34845
$915$	0	0
$916$	87349.1	3.15076
$917$	10151.9	0.365589
$918$	0	0
$919$	16055.3	0.576294	0.288147	−	0.957586i	$-0.406961\pi$
$919$	16055.3	0.576294	0.288147	+	0.957586i	$0.406961\pi$
$920$	8289.52	0.297062
$921$	0	0
$922$	−6816.97	−0.243498
$923$	152.831	0.00545016
$924$	0	0
$925$	36825.6	1.30899
$926$	85913.4	3.04891
$927$	0	0
$928$	23845.1	0.843485
$929$	−9378.49	−0.331215	−0.165607	−	0.986192i	$-0.552958\pi$
$929$	−9378.49	−0.331215	−0.165607	+	0.986192i	$0.552958\pi$
$930$	0	0
$931$	1610.60	0.0566976
$932$	−112222.	−3.94414
$933$	0	0
$934$	−57360.6	−2.00953
$935$	−1826.52	−0.0638864
$936$	0	0
$937$	9100.65	0.317295	0.158647	−	0.987335i	$-0.449287\pi$
$937$	9100.65	0.317295	0.158647	+	0.987335i	$0.449287\pi$
$938$	−20839.8	−0.725419
$939$	0	0
$940$	20884.3	0.724650
$941$	−9738.84	−0.337383	−0.168691	−	0.985669i	$-0.553954\pi$
$941$	−9738.84	−0.337383	−0.168691	+	0.985669i	$0.553954\pi$
$942$	0	0
$943$	−22314.4	−0.770580
$944$	−139617.	−4.81371
$945$	0	0
$946$	12644.5	0.434574
$947$	−47225.8	−1.62052	−0.810260	−	0.586071i	$-0.800674\pi$
$947$	−47225.8	−1.62052	−0.810260	+	0.586071i	$0.800674\pi$
$948$	0	0
$949$	−146.989	−0.00502787
$950$	21543.2	0.735740
$951$	0	0
$952$	47664.3	1.62270
$953$	−972.630	−0.0330604	−0.0165302	−	0.999863i	$-0.505262\pi$
$953$	−972.630	−0.0330604	−0.0165302	+	0.999863i	$0.505262\pi$
$954$	0	0
$955$	−3565.20	−0.120803
$956$	105980.	3.58541
$957$	0	0
$958$	−64649.3	−2.18030
$959$	−6251.19	−0.210492
$960$	0	0
$961$	−25845.9	−0.867574
$962$	1610.41	0.0539726
$963$	0	0
$964$	5275.77	0.176267
$965$	−1519.87	−0.0507010
$966$	0	0
$967$	−21987.5	−0.731199	−0.365599	−	0.930772i	$-0.619136\pi$
$967$	−21987.5	−0.731199	−0.365599	+	0.930772i	$0.619136\pi$
$968$	−8279.56	−0.274912
$969$	0	0
$970$	−9043.36	−0.299345
$971$	18194.5	0.601326	0.300663	−	0.953730i	$-0.402792\pi$
$971$	18194.5	0.601326	0.300663	+	0.953730i	$0.402792\pi$
$972$	0	0
$973$	3714.24	0.122377
$974$	76544.0	2.51810
$975$	0	0
$976$	88952.0	2.91730
$977$	14356.0	0.470103	0.235051	−	0.971983i	$-0.424474\pi$
$977$	14356.0	0.470103	0.235051	+	0.971983i	$0.424474\pi$
$978$	0	0
$979$	−12499.8	−0.408065
$980$	1697.35	0.0553263
$981$	0	0
$982$	−83749.1	−2.72153
$983$	−25765.5	−0.836005	−0.418002	−	0.908446i	$-0.637270\pi$
$983$	−25765.5	−0.836005	−0.418002	+	0.908446i	$0.637270\pi$
$984$	0	0
$985$	4147.98	0.134178
$986$	24016.1	0.775687
$987$	0	0
$988$	680.035	0.0218976
$989$	−15562.0	−0.500347
$990$	0	0
$991$	7750.45	0.248437	0.124219	−	0.992255i	$-0.460358\pi$
$991$	7750.45	0.248437	0.124219	+	0.992255i	$0.460358\pi$
$992$	33280.6	1.06518
$993$	0	0
$994$	5756.75	0.183695
$995$	−640.995	−0.0204230
$996$	0	0
$997$	12105.1	0.384528	0.192264	−	0.981343i	$-0.438417\pi$
$997$	12105.1	0.384528	0.192264	+	0.981343i	$0.438417\pi$
$998$	−31211.5	−0.989964
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.n.1.1		5
3.2	odd	2		231.4.a.l.1.5	✓	5
21.20	even	2		1617.4.a.p.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.l.1.5	✓	5	3.2	odd	2
693.4.a.n.1.1		5	1.1	even	1	trivial
1617.4.a.p.1.5		5	21.20	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-5.36278 q^{2} +20.7594 q^{4} +1.66863 q^{5} -7.00000 q^{7} -68.4261 q^{8} +O(q^{10})\)	\(q-5.36278 q^{2} +20.7594 q^{4} +1.66863 q^{5} -7.00000 q^{7} -68.4261 q^{8} -8.94849 q^{10} -11.0000 q^{11} +0.996604 q^{13} +37.5395 q^{14} +200.879 q^{16} +99.5115 q^{17} +32.8695 q^{19} +34.6398 q^{20} +58.9906 q^{22} -72.6019 q^{23} -122.216 q^{25} -5.34457 q^{26} -145.316 q^{28} -45.0027 q^{29} -62.8102 q^{31} -529.860 q^{32} -533.659 q^{34} -11.6804 q^{35} -301.317 q^{37} -176.272 q^{38} -114.178 q^{40} +307.353 q^{41} +214.347 q^{43} -228.354 q^{44} +389.348 q^{46} +602.899 q^{47} +49.0000 q^{49} +655.416 q^{50} +20.6889 q^{52} -592.072 q^{53} -18.3549 q^{55} +478.983 q^{56} +241.339 q^{58} -695.030 q^{59} +442.815 q^{61} +336.837 q^{62} +1234.49 q^{64} +1.66296 q^{65} -555.143 q^{67} +2065.80 q^{68} +62.6394 q^{70} +153.352 q^{71} -147.489 q^{73} +1615.90 q^{74} +682.352 q^{76} +77.0000 q^{77} +676.959 q^{79} +335.192 q^{80} -1648.27 q^{82} +222.412 q^{83} +166.048 q^{85} -1149.50 q^{86} +752.687 q^{88} +1136.35 q^{89} -6.97623 q^{91} -1507.18 q^{92} -3233.22 q^{94} +54.8469 q^{95} +1010.60 q^{97} -262.776 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} - 60 q^{8}+O(q^{10}) \) 5 * q - 5 * q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 - 60 * q^8	\( 5 q - 5 q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} - 60 q^{8} + 55 q^{10} - 55 q^{11} + 111 q^{13} + 35 q^{14} + 201 q^{16} - 136 q^{17} + 111 q^{19} - 219 q^{20} + 55 q^{22} + 28 q^{23} + 190 q^{25} + q^{26} - 147 q^{28} - 61 q^{29} - 280 q^{31} - 535 q^{32} - 572 q^{34} + 49 q^{35} - 41 q^{37} - 267 q^{38} - 336 q^{40} - 426 q^{41} + 424 q^{43} - 231 q^{44} + 140 q^{46} - 75 q^{47} + 245 q^{49} - 490 q^{50} - 269 q^{52} - 1500 q^{53} + 77 q^{55} + 420 q^{56} - 1767 q^{58} - 757 q^{59} + 658 q^{61} + 568 q^{62} - 748 q^{64} - 537 q^{65} - 583 q^{67} + 1650 q^{68} - 385 q^{70} + 764 q^{71} + 875 q^{73} + 825 q^{74} + 213 q^{76} + 385 q^{77} - 244 q^{79} + 2577 q^{80} - 2006 q^{82} - 924 q^{83} - 1402 q^{85} - 1272 q^{86} + 660 q^{88} + 1110 q^{89} - 777 q^{91} + 2046 q^{92} - 3349 q^{94} - 1923 q^{95} - 852 q^{97} - 245 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 - 60 * q^8 + 55 * q^10 - 55 * q^11 + 111 * q^13 + 35 * q^14 + 201 * q^16 - 136 * q^17 + 111 * q^19 - 219 * q^20 + 55 * q^22 + 28 * q^23 + 190 * q^25 + q^26 - 147 * q^28 - 61 * q^29 - 280 * q^31 - 535 * q^32 - 572 * q^34 + 49 * q^35 - 41 * q^37 - 267 * q^38 - 336 * q^40 - 426 * q^41 + 424 * q^43 - 231 * q^44 + 140 * q^46 - 75 * q^47 + 245 * q^49 - 490 * q^50 - 269 * q^52 - 1500 * q^53 + 77 * q^55 + 420 * q^56 - 1767 * q^58 - 757 * q^59 + 658 * q^61 + 568 * q^62 - 748 * q^64 - 537 * q^65 - 583 * q^67 + 1650 * q^68 - 385 * q^70 + 764 * q^71 + 875 * q^73 + 825 * q^74 + 213 * q^76 + 385 * q^77 - 244 * q^79 + 2577 * q^80 - 2006 * q^82 - 924 * q^83 - 1402 * q^85 - 1272 * q^86 + 660 * q^88 + 1110 * q^89 - 777 * q^91 + 2046 * q^92 - 3349 * q^94 - 1923 * q^95 - 852 * q^97 - 245 * q^98

Introduction

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