LMFDB - Embedded newform 693.4.a.b.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-3.00000 q^{2} +1.00000 q^{4} -12.0000 q^{5} +7.00000 q^{7} +21.0000 q^{8} +O(q^{10})$

\(q-3.00000 q^{2} +1.00000 q^{4} -12.0000 q^{5} +7.00000 q^{7} +21.0000 q^{8} +36.0000 q^{10} -11.0000 q^{11} +38.0000 q^{13} -21.0000 q^{14} -71.0000 q^{16} +48.0000 q^{17} -70.0000 q^{19} -12.0000 q^{20} +33.0000 q^{22} -12.0000 q^{23} +19.0000 q^{25} -114.000 q^{26} +7.00000 q^{28} -126.000 q^{29} -70.0000 q^{31} +45.0000 q^{32} -144.000 q^{34} -84.0000 q^{35} -358.000 q^{37} +210.000 q^{38} -252.000 q^{40} +216.000 q^{41} +344.000 q^{43} -11.0000 q^{44} +36.0000 q^{46} -390.000 q^{47} +49.0000 q^{49} -57.0000 q^{50} +38.0000 q^{52} -438.000 q^{53} +132.000 q^{55} +147.000 q^{56} +378.000 q^{58} +552.000 q^{59} +830.000 q^{61} +210.000 q^{62} +433.000 q^{64} -456.000 q^{65} -196.000 q^{67} +48.0000 q^{68} +252.000 q^{70} -648.000 q^{71} -16.0000 q^{73} +1074.00 q^{74} -70.0000 q^{76} -77.0000 q^{77} +1352.00 q^{79} +852.000 q^{80} -648.000 q^{82} -90.0000 q^{83} -576.000 q^{85} -1032.00 q^{86} -231.000 q^{88} -1146.00 q^{89} +266.000 q^{91} -12.0000 q^{92} +1170.00 q^{94} +840.000 q^{95} -70.0000 q^{97} -147.000 q^{98} +O(q^{100})\)

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 $

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	−12.0000	−1.07331	−0.536656	−	0.843801i	$-0.680313\pi$
$5$	−12.0000	−1.07331	−0.536656	+	0.843801i	$0.680313\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	21.0000	0.928078
$9$	0	0
$10$	36.0000	1.13842
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	38.0000	0.810716	0.405358	−	0.914158i	$-0.367147\pi$
$13$	38.0000	0.810716	0.405358	+	0.914158i	$0.367147\pi$
$14$	−21.0000	−0.400892
$15$	0	0
$16$	−71.0000	−1.10938
$17$	48.0000	0.684806	0.342403	−	0.939553i	$-0.388759\pi$
$17$	48.0000	0.684806	0.342403	+	0.939553i	$0.388759\pi$
$18$	0	0
$19$	−70.0000	−0.845216	−0.422608	−	0.906313i	$-0.638885\pi$
$19$	−70.0000	−0.845216	−0.422608	+	0.906313i	$0.638885\pi$
$20$	−12.0000	−0.134164
$21$	0	0
$22$	33.0000	0.319801
$23$	−12.0000	−0.108790	−0.0543951	−	0.998519i	$-0.517323\pi$
$23$	−12.0000	−0.108790	−0.0543951	+	0.998519i	$0.517323\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	−114.000	−0.859894
$27$	0	0
$28$	7.00000	0.0472456
$29$	−126.000	−0.806814	−0.403407	−	0.915021i	$-0.632174\pi$
$29$	−126.000	−0.806814	−0.403407	+	0.915021i	$0.632174\pi$
$30$	0	0
$31$	−70.0000	−0.405560	−0.202780	−	0.979224i	$-0.564998\pi$
$31$	−70.0000	−0.405560	−0.202780	+	0.979224i	$0.564998\pi$
$32$	45.0000	0.248592
$33$	0	0
$34$	−144.000	−0.726347
$35$	−84.0000	−0.405674
$36$	0	0
$37$	−358.000	−1.59067	−0.795336	−	0.606169i	$-0.792705\pi$
$37$	−358.000	−1.59067	−0.795336	+	0.606169i	$0.792705\pi$
$38$	210.000	0.896487
$39$	0	0
$40$	−252.000	−0.996117
$41$	216.000	0.822769	0.411385	−	0.911462i	$-0.365045\pi$
$41$	216.000	0.822769	0.411385	+	0.911462i	$0.365045\pi$
$42$	0	0
$43$	344.000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	344.000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−11.0000	−0.0376889
$45$	0	0
$46$	36.0000	0.115389
$47$	−390.000	−1.21037	−0.605185	−	0.796085i	$-0.706901\pi$
$47$	−390.000	−1.21037	−0.605185	+	0.796085i	$0.706901\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−57.0000	−0.161220
$51$	0	0
$52$	38.0000	0.101339
$53$	−438.000	−1.13517	−0.567584	−	0.823315i	$-0.692122\pi$
$53$	−438.000	−1.13517	−0.567584	+	0.823315i	$0.692122\pi$
$54$	0	0
$55$	132.000	0.323616
$56$	147.000	0.350780
$57$	0	0
$58$	378.000	0.855756
$59$	552.000	1.21804	0.609019	−	0.793155i	$-0.291563\pi$
$59$	552.000	1.21804	0.609019	+	0.793155i	$0.291563\pi$
$60$	0	0
$61$	830.000	1.74214	0.871071	−	0.491158i	$-0.163426\pi$
$61$	830.000	1.74214	0.871071	+	0.491158i	$0.163426\pi$
$62$	210.000	0.430162
$63$	0	0
$64$	433.000	0.845703
$65$	−456.000	−0.870151
$66$	0	0
$67$	−196.000	−0.357391	−0.178696	−	0.983904i	$-0.557188\pi$
$67$	−196.000	−0.357391	−0.178696	+	0.983904i	$0.557188\pi$
$68$	48.0000	0.0856008
$69$	0	0
$70$	252.000	0.430282
$71$	−648.000	−1.08315	−0.541574	−	0.840653i	$-0.682171\pi$
$71$	−648.000	−1.08315	−0.541574	+	0.840653i	$0.682171\pi$
$72$	0	0
$73$	−16.0000	−0.0256529	−0.0128264	−	0.999918i	$-0.504083\pi$
$73$	−16.0000	−0.0256529	−0.0128264	+	0.999918i	$0.504083\pi$
$74$	1074.00	1.68716
$75$	0	0
$76$	−70.0000	−0.105652
$77$	−77.0000	−0.113961
$78$	0	0
$79$	1352.00	1.92547	0.962733	−	0.270452i	$-0.0871732\pi$
$79$	1352.00	1.92547	0.962733	+	0.270452i	$0.0871732\pi$
$80$	852.000	1.19071
$81$	0	0
$82$	−648.000	−0.872678
$83$	−90.0000	−0.119021	−0.0595107	−	0.998228i	$-0.518954\pi$
$83$	−90.0000	−0.119021	−0.0595107	+	0.998228i	$0.518954\pi$
$84$	0	0
$85$	−576.000	−0.735011
$86$	−1032.00	−1.29399
$87$	0	0
$88$	−231.000	−0.279826
$89$	−1146.00	−1.36490	−0.682448	−	0.730934i	$-0.739085\pi$
$89$	−1146.00	−1.36490	−0.682448	+	0.730934i	$0.739085\pi$
$90$	0	0
$91$	266.000	0.306422
$92$	−12.0000	−0.0135988
$93$	0	0
$94$	1170.00	1.28379
$95$	840.000	0.907181
$96$	0	0
$97$	−70.0000	−0.0732724	−0.0366362	−	0.999329i	$-0.511664\pi$
$97$	−70.0000	−0.0732724	−0.0366362	+	0.999329i	$0.511664\pi$
$98$	−147.000	−0.151523
$99$	0	0
$100$	19.0000	0.0190000
$101$	1254.00	1.23542	0.617711	−	0.786405i	$-0.288060\pi$
$101$	1254.00	1.23542	0.617711	+	0.786405i	$0.288060\pi$
$102$	0	0
$103$	−682.000	−0.652422	−0.326211	−	0.945297i	$-0.605772\pi$
$103$	−682.000	−0.652422	−0.326211	+	0.945297i	$0.605772\pi$
$104$	798.000	0.752407
$105$	0	0
$106$	1314.00	1.20403
$107$	384.000	0.346941	0.173470	−	0.984839i	$-0.444502\pi$
$107$	384.000	0.346941	0.173470	+	0.984839i	$0.444502\pi$
$108$	0	0
$109$	−646.000	−0.567666	−0.283833	−	0.958874i	$-0.591606\pi$
$109$	−646.000	−0.567666	−0.283833	+	0.958874i	$0.591606\pi$
$110$	−396.000	−0.343247
$111$	0	0
$112$	−497.000	−0.419304
$113$	1314.00	1.09390	0.546950	−	0.837165i	$-0.315789\pi$
$113$	1314.00	1.09390	0.546950	+	0.837165i	$0.315789\pi$
$114$	0	0
$115$	144.000	0.116766
$116$	−126.000	−0.100852
$117$	0	0
$118$	−1656.00	−1.29193
$119$	336.000	0.258833
$120$	0	0
$121$	121.000	0.0909091
$122$	−2490.00	−1.84782
$123$	0	0
$124$	−70.0000	−0.0506950
$125$	1272.00	0.910169
$126$	0	0
$127$	344.000	0.240355	0.120177	−	0.992752i	$-0.461654\pi$
$127$	344.000	0.240355	0.120177	+	0.992752i	$0.461654\pi$
$128$	−1659.00	−1.14560
$129$	0	0
$130$	1368.00	0.922935
$131$	258.000	0.172073	0.0860365	−	0.996292i	$-0.472580\pi$
$131$	258.000	0.172073	0.0860365	+	0.996292i	$0.472580\pi$
$132$	0	0
$133$	−490.000	−0.319462
$134$	588.000	0.379071
$135$	0	0
$136$	1008.00	0.635554
$137$	2730.00	1.70248	0.851240	−	0.524777i	$-0.175851\pi$
$137$	2730.00	1.70248	0.851240	+	0.524777i	$0.175851\pi$
$138$	0	0
$139$	1838.00	1.12156	0.560781	−	0.827964i	$-0.310501\pi$
$139$	1838.00	1.12156	0.560781	+	0.827964i	$0.310501\pi$
$140$	−84.0000	−0.0507093
$141$	0	0
$142$	1944.00	1.14885
$143$	−418.000	−0.244440
$144$	0	0
$145$	1512.00	0.865964
$146$	48.0000	0.0272090
$147$	0	0
$148$	−358.000	−0.198834
$149$	510.000	0.280408	0.140204	−	0.990123i	$-0.455224\pi$
$149$	510.000	0.280408	0.140204	+	0.990123i	$0.455224\pi$
$150$	0	0
$151$	2864.00	1.54350	0.771752	−	0.635924i	$-0.219381\pi$
$151$	2864.00	1.54350	0.771752	+	0.635924i	$0.219381\pi$
$152$	−1470.00	−0.784426
$153$	0	0
$154$	231.000	0.120873
$155$	840.000	0.435293
$156$	0	0
$157$	−2968.00	−1.50874	−0.754370	−	0.656449i	$-0.772058\pi$
$157$	−2968.00	−1.50874	−0.754370	+	0.656449i	$0.772058\pi$
$158$	−4056.00	−2.04227
$159$	0	0
$160$	−540.000	−0.266817
$161$	−84.0000	−0.0411188
$162$	0	0
$163$	1604.00	0.770767	0.385383	−	0.922757i	$-0.374069\pi$
$163$	1604.00	0.770767	0.385383	+	0.922757i	$0.374069\pi$
$164$	216.000	0.102846
$165$	0	0
$166$	270.000	0.126241
$167$	−180.000	−0.0834061	−0.0417030	−	0.999130i	$-0.513278\pi$
$167$	−180.000	−0.0834061	−0.0417030	+	0.999130i	$0.513278\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	1728.00	0.779597
$171$	0	0
$172$	344.000	0.152499
$173$	1626.00	0.714581	0.357290	−	0.933993i	$-0.383701\pi$
$173$	1626.00	0.714581	0.357290	+	0.933993i	$0.383701\pi$
$174$	0	0
$175$	133.000	0.0574506
$176$	781.000	0.334489
$177$	0	0
$178$	3438.00	1.44769
$179$	3252.00	1.35791	0.678955	−	0.734180i	$-0.262433\pi$
$179$	3252.00	1.35791	0.678955	+	0.734180i	$0.262433\pi$
$180$	0	0
$181$	1820.00	0.747401	0.373700	−	0.927549i	$-0.378089\pi$
$181$	1820.00	0.747401	0.373700	+	0.927549i	$0.378089\pi$
$182$	−798.000	−0.325009
$183$	0	0
$184$	−252.000	−0.100966
$185$	4296.00	1.70729
$186$	0	0
$187$	−528.000	−0.206477
$188$	−390.000	−0.151296
$189$	0	0
$190$	−2520.00	−0.962211
$191$	1212.00	0.459148	0.229574	−	0.973291i	$-0.426267\pi$
$191$	1212.00	0.459148	0.229574	+	0.973291i	$0.426267\pi$
$192$	0	0
$193$	2522.00	0.940609	0.470304	−	0.882504i	$-0.344144\pi$
$193$	2522.00	0.940609	0.470304	+	0.882504i	$0.344144\pi$
$194$	210.000	0.0777171
$195$	0	0
$196$	49.0000	0.0178571
$197$	3474.00	1.25641	0.628204	−	0.778049i	$-0.283790\pi$
$197$	3474.00	1.25641	0.628204	+	0.778049i	$0.283790\pi$
$198$	0	0
$199$	−2842.00	−1.01238	−0.506191	−	0.862421i	$-0.668947\pi$
$199$	−2842.00	−1.01238	−0.506191	+	0.862421i	$0.668947\pi$
$200$	399.000	0.141068
$201$	0	0
$202$	−3762.00	−1.31036
$203$	−882.000	−0.304947
$204$	0	0
$205$	−2592.00	−0.883088
$206$	2046.00	0.691998
$207$	0	0
$208$	−2698.00	−0.899388
$209$	770.000	0.254842
$210$	0	0
$211$	5528.00	1.80362	0.901809	−	0.432136i	$-0.142240\pi$
$211$	5528.00	1.80362	0.901809	+	0.432136i	$0.142240\pi$
$212$	−438.000	−0.141896
$213$	0	0
$214$	−1152.00	−0.367986
$215$	−4128.00	−1.30943
$216$	0	0
$217$	−490.000	−0.153287
$218$	1938.00	0.602101
$219$	0	0
$220$	132.000	0.0404520
$221$	1824.00	0.555183
$222$	0	0
$223$	4034.00	1.21137	0.605687	−	0.795703i	$-0.292898\pi$
$223$	4034.00	1.21137	0.605687	+	0.795703i	$0.292898\pi$
$224$	315.000	0.0939590
$225$	0	0
$226$	−3942.00	−1.16026
$227$	−726.000	−0.212275	−0.106137	−	0.994351i	$-0.533848\pi$
$227$	−726.000	−0.212275	−0.106137	+	0.994351i	$0.533848\pi$
$228$	0	0
$229$	−2788.00	−0.804525	−0.402263	−	0.915524i	$-0.631776\pi$
$229$	−2788.00	−0.804525	−0.402263	+	0.915524i	$0.631776\pi$
$230$	−432.000	−0.123849
$231$	0	0
$232$	−2646.00	−0.748786
$233$	−2694.00	−0.757467	−0.378733	−	0.925506i	$-0.623640\pi$
$233$	−2694.00	−0.757467	−0.378733	+	0.925506i	$0.623640\pi$
$234$	0	0
$235$	4680.00	1.29910
$236$	552.000	0.152255
$237$	0	0
$238$	−1008.00	−0.274533
$239$	−6480.00	−1.75379	−0.876896	−	0.480680i	$-0.840390\pi$
$239$	−6480.00	−1.75379	−0.876896	+	0.480680i	$0.840390\pi$
$240$	0	0
$241$	−2320.00	−0.620101	−0.310050	−	0.950720i	$-0.600346\pi$
$241$	−2320.00	−0.620101	−0.310050	+	0.950720i	$0.600346\pi$
$242$	−363.000	−0.0964237
$243$	0	0
$244$	830.000	0.217768
$245$	−588.000	−0.153330
$246$	0	0
$247$	−2660.00	−0.685230
$248$	−1470.00	−0.376392
$249$	0	0
$250$	−3816.00	−0.965380
$251$	−2088.00	−0.525073	−0.262537	−	0.964922i	$-0.584559\pi$
$251$	−2088.00	−0.525073	−0.262537	+	0.964922i	$0.584559\pi$
$252$	0	0
$253$	132.000	0.0328015
$254$	−1032.00	−0.254935
$255$	0	0
$256$	1513.00	0.369385
$257$	4182.00	1.01504	0.507521	−	0.861639i	$-0.330562\pi$
$257$	4182.00	1.01504	0.507521	+	0.861639i	$0.330562\pi$
$258$	0	0
$259$	−2506.00	−0.601217
$260$	−456.000	−0.108769
$261$	0	0
$262$	−774.000	−0.182511
$263$	−3696.00	−0.866559	−0.433280	−	0.901260i	$-0.642644\pi$
$263$	−3696.00	−0.866559	−0.433280	+	0.901260i	$0.642644\pi$
$264$	0	0
$265$	5256.00	1.21839
$266$	1470.00	0.338840
$267$	0	0
$268$	−196.000	−0.0446739
$269$	6060.00	1.37355	0.686775	−	0.726870i	$-0.259026\pi$
$269$	6060.00	1.37355	0.686775	+	0.726870i	$0.259026\pi$
$270$	0	0
$271$	−8764.00	−1.96448	−0.982242	−	0.187619i	$-0.939923\pi$
$271$	−8764.00	−1.96448	−0.982242	+	0.187619i	$0.939923\pi$
$272$	−3408.00	−0.759707
$273$	0	0
$274$	−8190.00	−1.80575
$275$	−209.000	−0.0458297
$276$	0	0
$277$	5186.00	1.12490	0.562449	−	0.826832i	$-0.309859\pi$
$277$	5186.00	1.12490	0.562449	+	0.826832i	$0.309859\pi$
$278$	−5514.00	−1.18960
$279$	0	0
$280$	−1764.00	−0.376497
$281$	−3006.00	−0.638160	−0.319080	−	0.947728i	$-0.603374\pi$
$281$	−3006.00	−0.638160	−0.319080	+	0.947728i	$0.603374\pi$
$282$	0	0
$283$	−3922.00	−0.823812	−0.411906	−	0.911226i	$-0.635137\pi$
$283$	−3922.00	−0.823812	−0.411906	+	0.911226i	$0.635137\pi$
$284$	−648.000	−0.135393
$285$	0	0
$286$	1254.00	0.259268
$287$	1512.00	0.310977
$288$	0	0
$289$	−2609.00	−0.531040
$290$	−4536.00	−0.918493
$291$	0	0
$292$	−16.0000	−0.00320661
$293$	5778.00	1.15206	0.576031	−	0.817428i	$-0.304601\pi$
$293$	5778.00	1.15206	0.576031	+	0.817428i	$0.304601\pi$
$294$	0	0
$295$	−6624.00	−1.30734
$296$	−7518.00	−1.47627
$297$	0	0
$298$	−1530.00	−0.297418
$299$	−456.000	−0.0881979
$300$	0	0
$301$	2408.00	0.461112
$302$	−8592.00	−1.63713
$303$	0	0
$304$	4970.00	0.937661
$305$	−9960.00	−1.86986
$306$	0	0
$307$	−610.000	−0.113402	−0.0567012	−	0.998391i	$-0.518058\pi$
$307$	−610.000	−0.113402	−0.0567012	+	0.998391i	$0.518058\pi$
$308$	−77.0000	−0.0142451
$309$	0	0
$310$	−2520.00	−0.461698
$311$	−6882.00	−1.25480	−0.627399	−	0.778698i	$-0.715881\pi$
$311$	−6882.00	−1.25480	−0.627399	+	0.778698i	$0.715881\pi$
$312$	0	0
$313$	10334.0	1.86617	0.933087	−	0.359652i	$-0.117105\pi$
$313$	10334.0	1.86617	0.933087	+	0.359652i	$0.117105\pi$
$314$	8904.00	1.60026
$315$	0	0
$316$	1352.00	0.240683
$317$	−5934.00	−1.05138	−0.525689	−	0.850677i	$-0.676192\pi$
$317$	−5934.00	−1.05138	−0.525689	+	0.850677i	$0.676192\pi$
$318$	0	0
$319$	1386.00	0.243264
$320$	−5196.00	−0.907704
$321$	0	0
$322$	252.000	0.0436131
$323$	−3360.00	−0.578809
$324$	0	0
$325$	722.000	0.123229
$326$	−4812.00	−0.817522
$327$	0	0
$328$	4536.00	0.763594
$329$	−2730.00	−0.457477
$330$	0	0
$331$	−3220.00	−0.534705	−0.267352	−	0.963599i	$-0.586149\pi$
$331$	−3220.00	−0.534705	−0.267352	+	0.963599i	$0.586149\pi$
$332$	−90.0000	−0.0148777
$333$	0	0
$334$	540.000	0.0884655
$335$	2352.00	0.383592
$336$	0	0
$337$	−6658.00	−1.07621	−0.538107	−	0.842876i	$-0.680861\pi$
$337$	−6658.00	−1.07621	−0.538107	+	0.842876i	$0.680861\pi$
$338$	2259.00	0.363531
$339$	0	0
$340$	−576.000	−0.0918764
$341$	770.000	0.122281
$342$	0	0
$343$	343.000	0.0539949
$344$	7224.00	1.13224
$345$	0	0
$346$	−4878.00	−0.757927
$347$	6888.00	1.06561	0.532806	−	0.846238i	$-0.321138\pi$
$347$	6888.00	1.06561	0.532806	+	0.846238i	$0.321138\pi$
$348$	0	0
$349$	−6190.00	−0.949407	−0.474704	−	0.880146i	$-0.657445\pi$
$349$	−6190.00	−0.949407	−0.474704	+	0.880146i	$0.657445\pi$
$350$	−399.000	−0.0609356
$351$	0	0
$352$	−495.000	−0.0749534
$353$	3990.00	0.601604	0.300802	−	0.953687i	$-0.402746\pi$
$353$	3990.00	0.601604	0.300802	+	0.953687i	$0.402746\pi$
$354$	0	0
$355$	7776.00	1.16256
$356$	−1146.00	−0.170612
$357$	0	0
$358$	−9756.00	−1.44028
$359$	7656.00	1.12554	0.562769	−	0.826614i	$-0.309736\pi$
$359$	7656.00	1.12554	0.562769	+	0.826614i	$0.309736\pi$
$360$	0	0
$361$	−1959.00	−0.285610
$362$	−5460.00	−0.792738
$363$	0	0
$364$	266.000	0.0383027
$365$	192.000	0.0275335
$366$	0	0
$367$	8426.00	1.19846	0.599228	−	0.800578i	$-0.295474\pi$
$367$	8426.00	1.19846	0.599228	+	0.800578i	$0.295474\pi$
$368$	852.000	0.120689
$369$	0	0
$370$	−12888.0	−1.81085
$371$	−3066.00	−0.429053
$372$	0	0
$373$	11918.0	1.65440	0.827199	−	0.561909i	$-0.189933\pi$
$373$	11918.0	1.65440	0.827199	+	0.561909i	$0.189933\pi$
$374$	1584.00	0.219002
$375$	0	0
$376$	−8190.00	−1.12332
$377$	−4788.00	−0.654097
$378$	0	0
$379$	3908.00	0.529658	0.264829	−	0.964295i	$-0.414684\pi$
$379$	3908.00	0.529658	0.264829	+	0.964295i	$0.414684\pi$
$380$	840.000	0.113398
$381$	0	0
$382$	−3636.00	−0.487000
$383$	3246.00	0.433062	0.216531	−	0.976276i	$-0.430526\pi$
$383$	3246.00	0.433062	0.216531	+	0.976276i	$0.430526\pi$
$384$	0	0
$385$	924.000	0.122315
$386$	−7566.00	−0.997666
$387$	0	0
$388$	−70.0000	−0.00915905
$389$	8166.00	1.06435	0.532176	−	0.846634i	$-0.321375\pi$
$389$	8166.00	1.06435	0.532176	+	0.846634i	$0.321375\pi$
$390$	0	0
$391$	−576.000	−0.0745002
$392$	1029.00	0.132583
$393$	0	0
$394$	−10422.0	−1.33262
$395$	−16224.0	−2.06663
$396$	0	0
$397$	−2824.00	−0.357009	−0.178504	−	0.983939i	$-0.557126\pi$
$397$	−2824.00	−0.357009	−0.178504	+	0.983939i	$0.557126\pi$
$398$	8526.00	1.07379
$399$	0	0
$400$	−1349.00	−0.168625
$401$	10482.0	1.30535	0.652676	−	0.757637i	$-0.273646\pi$
$401$	10482.0	1.30535	0.652676	+	0.757637i	$0.273646\pi$
$402$	0	0
$403$	−2660.00	−0.328794
$404$	1254.00	0.154428
$405$	0	0
$406$	2646.00	0.323445
$407$	3938.00	0.479605
$408$	0	0
$409$	8156.00	0.986035	0.493017	−	0.870019i	$-0.335894\pi$
$409$	8156.00	0.986035	0.493017	+	0.870019i	$0.335894\pi$
$410$	7776.00	0.936657
$411$	0	0
$412$	−682.000	−0.0815527
$413$	3864.00	0.460375
$414$	0	0
$415$	1080.00	0.127747
$416$	1710.00	0.201538
$417$	0	0
$418$	−2310.00	−0.270301
$419$	11052.0	1.28861	0.644303	−	0.764771i	$-0.277148\pi$
$419$	11052.0	1.28861	0.644303	+	0.764771i	$0.277148\pi$
$420$	0	0
$421$	5006.00	0.579519	0.289760	−	0.957099i	$-0.406425\pi$
$421$	5006.00	0.579519	0.289760	+	0.957099i	$0.406425\pi$
$422$	−16584.0	−1.91302
$423$	0	0
$424$	−9198.00	−1.05352
$425$	912.000	0.104091
$426$	0	0
$427$	5810.00	0.658467
$428$	384.000	0.0433676
$429$	0	0
$430$	12384.0	1.38886
$431$	9480.00	1.05948	0.529740	−	0.848160i	$-0.322290\pi$
$431$	9480.00	1.05948	0.529740	+	0.848160i	$0.322290\pi$
$432$	0	0
$433$	−1942.00	−0.215535	−0.107767	−	0.994176i	$-0.534370\pi$
$433$	−1942.00	−0.215535	−0.107767	+	0.994176i	$0.534370\pi$
$434$	1470.00	0.162586
$435$	0	0
$436$	−646.000	−0.0709582
$437$	840.000	0.0919511
$438$	0	0
$439$	−13660.0	−1.48509	−0.742547	−	0.669794i	$-0.766382\pi$
$439$	−13660.0	−1.48509	−0.742547	+	0.669794i	$0.766382\pi$
$440$	2772.00	0.300341
$441$	0	0
$442$	−5472.00	−0.588861
$443$	−3828.00	−0.410550	−0.205275	−	0.978704i	$-0.565809\pi$
$443$	−3828.00	−0.410550	−0.205275	+	0.978704i	$0.565809\pi$
$444$	0	0
$445$	13752.0	1.46496
$446$	−12102.0	−1.28486
$447$	0	0
$448$	3031.00	0.319646
$449$	−18270.0	−1.92030	−0.960150	−	0.279486i	$-0.909836\pi$
$449$	−18270.0	−1.92030	−0.960150	+	0.279486i	$0.909836\pi$
$450$	0	0
$451$	−2376.00	−0.248074
$452$	1314.00	0.136738
$453$	0	0
$454$	2178.00	0.225151
$455$	−3192.00	−0.328886
$456$	0	0
$457$	10154.0	1.03935	0.519676	−	0.854363i	$-0.326053\pi$
$457$	10154.0	1.03935	0.519676	+	0.854363i	$0.326053\pi$
$458$	8364.00	0.853328
$459$	0	0
$460$	144.000	0.0145957
$461$	17190.0	1.73670	0.868349	−	0.495953i	$-0.165181\pi$
$461$	17190.0	1.73670	0.868349	+	0.495953i	$0.165181\pi$
$462$	0	0
$463$	4448.00	0.446471	0.223236	−	0.974765i	$-0.428338\pi$
$463$	4448.00	0.446471	0.223236	+	0.974765i	$0.428338\pi$
$464$	8946.00	0.895060
$465$	0	0
$466$	8082.00	0.803415
$467$	−11100.0	−1.09989	−0.549943	−	0.835202i	$-0.685351\pi$
$467$	−11100.0	−1.09989	−0.549943	+	0.835202i	$0.685351\pi$
$468$	0	0
$469$	−1372.00	−0.135081
$470$	−14040.0	−1.37791
$471$	0	0
$472$	11592.0	1.13043
$473$	−3784.00	−0.367840
$474$	0	0
$475$	−1330.00	−0.128473
$476$	336.000	0.0323541
$477$	0	0
$478$	19440.0	1.86018
$479$	15816.0	1.50867	0.754333	−	0.656491i	$-0.227960\pi$
$479$	15816.0	1.50867	0.754333	+	0.656491i	$0.227960\pi$
$480$	0	0
$481$	−13604.0	−1.28958
$482$	6960.00	0.657716
$483$	0	0
$484$	121.000	0.0113636
$485$	840.000	0.0786442
$486$	0	0
$487$	−1924.00	−0.179024	−0.0895121	−	0.995986i	$-0.528531\pi$
$487$	−1924.00	−0.179024	−0.0895121	+	0.995986i	$0.528531\pi$
$488$	17430.0	1.61684
$489$	0	0
$490$	1764.00	0.162631
$491$	−13068.0	−1.20112	−0.600561	−	0.799579i	$-0.705056\pi$
$491$	−13068.0	−1.20112	−0.600561	+	0.799579i	$0.705056\pi$
$492$	0	0
$493$	−6048.00	−0.552512
$494$	7980.00	0.726796
$495$	0	0
$496$	4970.00	0.449919
$497$	−4536.00	−0.409391
$498$	0	0
$499$	17876.0	1.60369	0.801843	−	0.597534i	$-0.203853\pi$
$499$	17876.0	1.60369	0.801843	+	0.597534i	$0.203853\pi$
$500$	1272.00	0.113771
$501$	0	0
$502$	6264.00	0.556924
$503$	3852.00	0.341456	0.170728	−	0.985318i	$-0.445388\pi$
$503$	3852.00	0.341456	0.170728	+	0.985318i	$0.445388\pi$
$504$	0	0
$505$	−15048.0	−1.32599
$506$	−396.000	−0.0347912
$507$	0	0
$508$	344.000	0.0300444
$509$	−15132.0	−1.31771	−0.658855	−	0.752270i	$-0.728959\pi$
$509$	−15132.0	−1.31771	−0.658855	+	0.752270i	$0.728959\pi$
$510$	0	0
$511$	−112.000	−0.00969587
$512$	8733.00	0.753804
$513$	0	0
$514$	−12546.0	−1.07662
$515$	8184.00	0.700253
$516$	0	0
$517$	4290.00	0.364940
$518$	7518.00	0.637687
$519$	0	0
$520$	−9576.00	−0.807568
$521$	3054.00	0.256810	0.128405	−	0.991722i	$-0.459014\pi$
$521$	3054.00	0.256810	0.128405	+	0.991722i	$0.459014\pi$
$522$	0	0
$523$	−11770.0	−0.984065	−0.492033	−	0.870577i	$-0.663746\pi$
$523$	−11770.0	−0.984065	−0.492033	+	0.870577i	$0.663746\pi$
$524$	258.000	0.0215091
$525$	0	0
$526$	11088.0	0.919125
$527$	−3360.00	−0.277730
$528$	0	0
$529$	−12023.0	−0.988165
$530$	−15768.0	−1.29230
$531$	0	0
$532$	−490.000	−0.0399327
$533$	8208.00	0.667032
$534$	0	0
$535$	−4608.00	−0.372376
$536$	−4116.00	−0.331687
$537$	0	0
$538$	−18180.0	−1.45687
$539$	−539.000	−0.0430730
$540$	0	0
$541$	10694.0	0.849854	0.424927	−	0.905228i	$-0.360300\pi$
$541$	10694.0	0.849854	0.424927	+	0.905228i	$0.360300\pi$
$542$	26292.0	2.08365
$543$	0	0
$544$	2160.00	0.170238
$545$	7752.00	0.609283
$546$	0	0
$547$	5636.00	0.440545	0.220272	−	0.975438i	$-0.429305\pi$
$547$	5636.00	0.440545	0.220272	+	0.975438i	$0.429305\pi$
$548$	2730.00	0.212810
$549$	0	0
$550$	627.000	0.0486098
$551$	8820.00	0.681932
$552$	0	0
$553$	9464.00	0.727758
$554$	−15558.0	−1.19313
$555$	0	0
$556$	1838.00	0.140195
$557$	−15126.0	−1.15064	−0.575322	−	0.817927i	$-0.695123\pi$
$557$	−15126.0	−1.15064	−0.575322	+	0.817927i	$0.695123\pi$
$558$	0	0
$559$	13072.0	0.989064
$560$	5964.00	0.450045
$561$	0	0
$562$	9018.00	0.676871
$563$	−3246.00	−0.242989	−0.121494	−	0.992592i	$-0.538769\pi$
$563$	−3246.00	−0.242989	−0.121494	+	0.992592i	$0.538769\pi$
$564$	0	0
$565$	−15768.0	−1.17410
$566$	11766.0	0.873784
$567$	0	0
$568$	−13608.0	−1.00524
$569$	1050.00	0.0773608	0.0386804	−	0.999252i	$-0.487685\pi$
$569$	1050.00	0.0773608	0.0386804	+	0.999252i	$0.487685\pi$
$570$	0	0
$571$	6860.00	0.502771	0.251385	−	0.967887i	$-0.419114\pi$
$571$	6860.00	0.502771	0.251385	+	0.967887i	$0.419114\pi$
$572$	−418.000	−0.0305550
$573$	0	0
$574$	−4536.00	−0.329841
$575$	−228.000	−0.0165361
$576$	0	0
$577$	−12634.0	−0.911543	−0.455771	−	0.890097i	$-0.650637\pi$
$577$	−12634.0	−0.911543	−0.455771	+	0.890097i	$0.650637\pi$
$578$	7827.00	0.563253
$579$	0	0
$580$	1512.00	0.108245
$581$	−630.000	−0.0449859
$582$	0	0
$583$	4818.00	0.342266
$584$	−336.000	−0.0238078
$585$	0	0
$586$	−17334.0	−1.22195
$587$	−18144.0	−1.27578	−0.637890	−	0.770127i	$-0.720193\pi$
$587$	−18144.0	−1.27578	−0.637890	+	0.770127i	$0.720193\pi$
$588$	0	0
$589$	4900.00	0.342786
$590$	19872.0	1.38664
$591$	0	0
$592$	25418.0	1.76465
$593$	10896.0	0.754545	0.377272	−	0.926102i	$-0.376862\pi$
$593$	10896.0	0.754545	0.377272	+	0.926102i	$0.376862\pi$
$594$	0	0
$595$	−4032.00	−0.277808
$596$	510.000	0.0350510
$597$	0	0
$598$	1368.00	0.0935480
$599$	20280.0	1.38334	0.691668	−	0.722216i	$-0.256876\pi$
$599$	20280.0	1.38334	0.691668	+	0.722216i	$0.256876\pi$
$600$	0	0
$601$	12332.0	0.836993	0.418496	−	0.908218i	$-0.362557\pi$
$601$	12332.0	0.836993	0.418496	+	0.908218i	$0.362557\pi$
$602$	−7224.00	−0.489083
$603$	0	0
$604$	2864.00	0.192938
$605$	−1452.00	−0.0975739
$606$	0	0
$607$	21800.0	1.45772	0.728859	−	0.684664i	$-0.240051\pi$
$607$	21800.0	1.45772	0.728859	+	0.684664i	$0.240051\pi$
$608$	−3150.00	−0.210114
$609$	0	0
$610$	29880.0	1.98329
$611$	−14820.0	−0.981265
$612$	0	0
$613$	18542.0	1.22170	0.610852	−	0.791745i	$-0.290827\pi$
$613$	18542.0	1.22170	0.610852	+	0.791745i	$0.290827\pi$
$614$	1830.00	0.120281
$615$	0	0
$616$	−1617.00	−0.105764
$617$	−10098.0	−0.658882	−0.329441	−	0.944176i	$-0.606860\pi$
$617$	−10098.0	−0.658882	−0.329441	+	0.944176i	$0.606860\pi$
$618$	0	0
$619$	−124.000	−0.00805167	−0.00402583	−	0.999992i	$-0.501281\pi$
$619$	−124.000	−0.00805167	−0.00402583	+	0.999992i	$0.501281\pi$
$620$	840.000	0.0544116
$621$	0	0
$622$	20646.0	1.33092
$623$	−8022.00	−0.515882
$624$	0	0
$625$	−17639.0	−1.12890
$626$	−31002.0	−1.97938
$627$	0	0
$628$	−2968.00	−0.188593
$629$	−17184.0	−1.08930
$630$	0	0
$631$	−14308.0	−0.902682	−0.451341	−	0.892351i	$-0.649054\pi$
$631$	−14308.0	−0.902682	−0.451341	+	0.892351i	$0.649054\pi$
$632$	28392.0	1.78698
$633$	0	0
$634$	17802.0	1.11515
$635$	−4128.00	−0.257976
$636$	0	0
$637$	1862.00	0.115817
$638$	−4158.00	−0.258020
$639$	0	0
$640$	19908.0	1.22958
$641$	678.000	0.0417775	0.0208888	−	0.999782i	$-0.493350\pi$
$641$	678.000	0.0417775	0.0208888	+	0.999782i	$0.493350\pi$
$642$	0	0
$643$	17408.0	1.06766	0.533829	−	0.845592i	$-0.320752\pi$
$643$	17408.0	1.06766	0.533829	+	0.845592i	$0.320752\pi$
$644$	−84.0000	−0.00513985
$645$	0	0
$646$	10080.0	0.613920
$647$	28686.0	1.74306	0.871532	−	0.490338i	$-0.163127\pi$
$647$	28686.0	1.74306	0.871532	+	0.490338i	$0.163127\pi$
$648$	0	0
$649$	−6072.00	−0.367252
$650$	−2166.00	−0.130704
$651$	0	0
$652$	1604.00	0.0963458
$653$	−9858.00	−0.590771	−0.295385	−	0.955378i	$-0.595448\pi$
$653$	−9858.00	−0.590771	−0.295385	+	0.955378i	$0.595448\pi$
$654$	0	0
$655$	−3096.00	−0.184688
$656$	−15336.0	−0.912759
$657$	0	0
$658$	8190.00	0.485227
$659$	22824.0	1.34916	0.674580	−	0.738201i	$-0.264325\pi$
$659$	22824.0	1.34916	0.674580	+	0.738201i	$0.264325\pi$
$660$	0	0
$661$	24212.0	1.42472	0.712358	−	0.701816i	$-0.247627\pi$
$661$	24212.0	1.42472	0.712358	+	0.701816i	$0.247627\pi$
$662$	9660.00	0.567140
$663$	0	0
$664$	−1890.00	−0.110461
$665$	5880.00	0.342882
$666$	0	0
$667$	1512.00	0.0877734
$668$	−180.000	−0.0104258
$669$	0	0
$670$	−7056.00	−0.406861
$671$	−9130.00	−0.525275
$672$	0	0
$673$	−17458.0	−0.999935	−0.499968	−	0.866044i	$-0.666655\pi$
$673$	−17458.0	−0.999935	−0.499968	+	0.866044i	$0.666655\pi$
$674$	19974.0	1.14150
$675$	0	0
$676$	−753.000	−0.0428425
$677$	−14574.0	−0.827362	−0.413681	−	0.910422i	$-0.635757\pi$
$677$	−14574.0	−0.827362	−0.413681	+	0.910422i	$0.635757\pi$
$678$	0	0
$679$	−490.000	−0.0276944
$680$	−12096.0	−0.682148
$681$	0	0
$682$	−2310.00	−0.129699
$683$	27588.0	1.54557	0.772786	−	0.634667i	$-0.218863\pi$
$683$	27588.0	1.54557	0.772786	+	0.634667i	$0.218863\pi$
$684$	0	0
$685$	−32760.0	−1.82729
$686$	−1029.00	−0.0572703
$687$	0	0
$688$	−24424.0	−1.35342
$689$	−16644.0	−0.920299
$690$	0	0
$691$	10424.0	0.573875	0.286938	−	0.957949i	$-0.407363\pi$
$691$	10424.0	0.573875	0.286938	+	0.957949i	$0.407363\pi$
$692$	1626.00	0.0893226
$693$	0	0
$694$	−20664.0	−1.13025
$695$	−22056.0	−1.20379
$696$	0	0
$697$	10368.0	0.563438
$698$	18570.0	1.00700
$699$	0	0
$700$	133.000	0.00718132
$701$	−3978.00	−0.214332	−0.107166	−	0.994241i	$-0.534178\pi$
$701$	−3978.00	−0.214332	−0.107166	+	0.994241i	$0.534178\pi$
$702$	0	0
$703$	25060.0	1.34446
$704$	−4763.00	−0.254989
$705$	0	0
$706$	−11970.0	−0.638098
$707$	8778.00	0.466946
$708$	0	0
$709$	18794.0	0.995520	0.497760	−	0.867315i	$-0.334156\pi$
$709$	18794.0	0.995520	0.497760	+	0.867315i	$0.334156\pi$
$710$	−23328.0	−1.23308
$711$	0	0
$712$	−24066.0	−1.26673
$713$	840.000	0.0441210
$714$	0	0
$715$	5016.00	0.262361
$716$	3252.00	0.169739
$717$	0	0
$718$	−22968.0	−1.19381
$719$	−33906.0	−1.75867	−0.879333	−	0.476208i	$-0.842011\pi$
$719$	−33906.0	−1.75867	−0.879333	+	0.476208i	$0.842011\pi$
$720$	0	0
$721$	−4774.00	−0.246592
$722$	5877.00	0.302935
$723$	0	0
$724$	1820.00	0.0934251
$725$	−2394.00	−0.122636
$726$	0	0
$727$	−2446.00	−0.124783	−0.0623914	−	0.998052i	$-0.519873\pi$
$727$	−2446.00	−0.124783	−0.0623914	+	0.998052i	$0.519873\pi$
$728$	5586.00	0.284383
$729$	0	0
$730$	−576.000	−0.0292037
$731$	16512.0	0.835456
$732$	0	0
$733$	−20410.0	−1.02846	−0.514230	−	0.857653i	$-0.671922\pi$
$733$	−20410.0	−1.02846	−0.514230	+	0.857653i	$0.671922\pi$
$734$	−25278.0	−1.27116
$735$	0	0
$736$	−540.000	−0.0270444
$737$	2156.00	0.107758
$738$	0	0
$739$	14564.0	0.724960	0.362480	−	0.931992i	$-0.381930\pi$
$739$	14564.0	0.724960	0.362480	+	0.931992i	$0.381930\pi$
$740$	4296.00	0.213411
$741$	0	0
$742$	9198.00	0.455080
$743$	7416.00	0.366173	0.183087	−	0.983097i	$-0.441391\pi$
$743$	7416.00	0.366173	0.183087	+	0.983097i	$0.441391\pi$
$744$	0	0
$745$	−6120.00	−0.300966
$746$	−35754.0	−1.75475
$747$	0	0
$748$	−528.000	−0.0258096
$749$	2688.00	0.131131
$750$	0	0
$751$	−17980.0	−0.873635	−0.436817	−	0.899550i	$-0.643894\pi$
$751$	−17980.0	−0.873635	−0.436817	+	0.899550i	$0.643894\pi$
$752$	27690.0	1.34275
$753$	0	0
$754$	14364.0	0.693775
$755$	−34368.0	−1.65666
$756$	0	0
$757$	3170.00	0.152200	0.0761001	−	0.997100i	$-0.475753\pi$
$757$	3170.00	0.152200	0.0761001	+	0.997100i	$0.475753\pi$
$758$	−11724.0	−0.561787
$759$	0	0
$760$	17640.0	0.841934
$761$	27492.0	1.30957	0.654786	−	0.755814i	$-0.272759\pi$
$761$	27492.0	1.30957	0.654786	+	0.755814i	$0.272759\pi$
$762$	0	0
$763$	−4522.00	−0.214558
$764$	1212.00	0.0573935
$765$	0	0
$766$	−9738.00	−0.459332
$767$	20976.0	0.987483
$768$	0	0
$769$	2108.00	0.0988510	0.0494255	−	0.998778i	$-0.484261\pi$
$769$	2108.00	0.0988510	0.0494255	+	0.998778i	$0.484261\pi$
$770$	−2772.00	−0.129735
$771$	0	0
$772$	2522.00	0.117576
$773$	−32280.0	−1.50198	−0.750990	−	0.660313i	$-0.770423\pi$
$773$	−32280.0	−1.50198	−0.750990	+	0.660313i	$0.770423\pi$
$774$	0	0
$775$	−1330.00	−0.0616452
$776$	−1470.00	−0.0680025
$777$	0	0
$778$	−24498.0	−1.12891
$779$	−15120.0	−0.695417
$780$	0	0
$781$	7128.00	0.326581
$782$	1728.00	0.0790194
$783$	0	0
$784$	−3479.00	−0.158482
$785$	35616.0	1.61935
$786$	0	0
$787$	9578.00	0.433823	0.216912	−	0.976191i	$-0.430402\pi$
$787$	9578.00	0.433823	0.216912	+	0.976191i	$0.430402\pi$
$788$	3474.00	0.157051
$789$	0	0
$790$	48672.0	2.19199
$791$	9198.00	0.413455
$792$	0	0
$793$	31540.0	1.41238
$794$	8472.00	0.378665
$795$	0	0
$796$	−2842.00	−0.126548
$797$	−11952.0	−0.531194	−0.265597	−	0.964084i	$-0.585569\pi$
$797$	−11952.0	−0.531194	−0.265597	+	0.964084i	$0.585569\pi$
$798$	0	0
$799$	−18720.0	−0.828869
$800$	855.000	0.0377860
$801$	0	0
$802$	−31446.0	−1.38453
$803$	176.000	0.00773463
$804$	0	0
$805$	1008.00	0.0441333
$806$	7980.00	0.348739
$807$	0	0
$808$	26334.0	1.14657
$809$	9030.00	0.392433	0.196216	−	0.980561i	$-0.437135\pi$
$809$	9030.00	0.392433	0.196216	+	0.980561i	$0.437135\pi$
$810$	0	0
$811$	−37762.0	−1.63502	−0.817511	−	0.575913i	$-0.804647\pi$
$811$	−37762.0	−1.63502	−0.817511	+	0.575913i	$0.804647\pi$
$812$	−882.000	−0.0381184
$813$	0	0
$814$	−11814.0	−0.508698
$815$	−19248.0	−0.827274
$816$	0	0
$817$	−24080.0	−1.03115
$818$	−24468.0	−1.04585
$819$	0	0
$820$	−2592.00	−0.110386
$821$	14334.0	0.609330	0.304665	−	0.952460i	$-0.401455\pi$
$821$	14334.0	0.609330	0.304665	+	0.952460i	$0.401455\pi$
$822$	0	0
$823$	13988.0	0.592456	0.296228	−	0.955117i	$-0.404271\pi$
$823$	13988.0	0.592456	0.296228	+	0.955117i	$0.404271\pi$
$824$	−14322.0	−0.605498
$825$	0	0
$826$	−11592.0	−0.488302
$827$	22284.0	0.936990	0.468495	−	0.883466i	$-0.344796\pi$
$827$	22284.0	0.936990	0.468495	+	0.883466i	$0.344796\pi$
$828$	0	0
$829$	−12868.0	−0.539112	−0.269556	−	0.962985i	$-0.586877\pi$
$829$	−12868.0	−0.539112	−0.269556	+	0.962985i	$0.586877\pi$
$830$	−3240.00	−0.135496
$831$	0	0
$832$	16454.0	0.685625
$833$	2352.00	0.0978295
$834$	0	0
$835$	2160.00	0.0895208
$836$	770.000	0.0318553
$837$	0	0
$838$	−33156.0	−1.36677
$839$	−2826.00	−0.116286	−0.0581432	−	0.998308i	$-0.518518\pi$
$839$	−2826.00	−0.116286	−0.0581432	+	0.998308i	$0.518518\pi$
$840$	0	0
$841$	−8513.00	−0.349051
$842$	−15018.0	−0.614673
$843$	0	0
$844$	5528.00	0.225452
$845$	9036.00	0.367867
$846$	0	0
$847$	847.000	0.0343604
$848$	31098.0	1.25933
$849$	0	0
$850$	−2736.00	−0.110405
$851$	4296.00	0.173049
$852$	0	0
$853$	−17962.0	−0.720993	−0.360497	−	0.932761i	$-0.617393\pi$
$853$	−17962.0	−0.720993	−0.360497	+	0.932761i	$0.617393\pi$
$854$	−17430.0	−0.698410
$855$	0	0
$856$	8064.00	0.321988
$857$	−47148.0	−1.87928	−0.939641	−	0.342161i	$-0.888841\pi$
$857$	−47148.0	−1.87928	−0.939641	+	0.342161i	$0.888841\pi$
$858$	0	0
$859$	34904.0	1.38639	0.693195	−	0.720750i	$-0.256202\pi$
$859$	34904.0	1.38639	0.693195	+	0.720750i	$0.256202\pi$
$860$	−4128.00	−0.163679
$861$	0	0
$862$	−28440.0	−1.12375
$863$	44052.0	1.73760	0.868799	−	0.495164i	$-0.164892\pi$
$863$	44052.0	1.73760	0.868799	+	0.495164i	$0.164892\pi$
$864$	0	0
$865$	−19512.0	−0.766969
$866$	5826.00	0.228609
$867$	0	0
$868$	−490.000	−0.0191609
$869$	−14872.0	−0.580550
$870$	0	0
$871$	−7448.00	−0.289743
$872$	−13566.0	−0.526838
$873$	0	0
$874$	−2520.00	−0.0975289
$875$	8904.00	0.344012
$876$	0	0
$877$	−36214.0	−1.39437	−0.697184	−	0.716893i	$-0.745564\pi$
$877$	−36214.0	−1.39437	−0.697184	+	0.716893i	$0.745564\pi$
$878$	40980.0	1.57518
$879$	0	0
$880$	−9372.00	−0.359011
$881$	−17046.0	−0.651866	−0.325933	−	0.945393i	$-0.605678\pi$
$881$	−17046.0	−0.651866	−0.325933	+	0.945393i	$0.605678\pi$
$882$	0	0
$883$	41276.0	1.57310	0.786550	−	0.617526i	$-0.211865\pi$
$883$	41276.0	1.57310	0.786550	+	0.617526i	$0.211865\pi$
$884$	1824.00	0.0693979
$885$	0	0
$886$	11484.0	0.435454
$887$	−6456.00	−0.244387	−0.122193	−	0.992506i	$-0.538993\pi$
$887$	−6456.00	−0.244387	−0.122193	+	0.992506i	$0.538993\pi$
$888$	0	0
$889$	2408.00	0.0908456
$890$	−41256.0	−1.55383
$891$	0	0
$892$	4034.00	0.151422
$893$	27300.0	1.02302
$894$	0	0
$895$	−39024.0	−1.45746
$896$	−11613.0	−0.432995
$897$	0	0
$898$	54810.0	2.03679
$899$	8820.00	0.327212
$900$	0	0
$901$	−21024.0	−0.777371
$902$	7128.00	0.263122
$903$	0	0
$904$	27594.0	1.01522
$905$	−21840.0	−0.802195
$906$	0	0
$907$	−11500.0	−0.421005	−0.210502	−	0.977593i	$-0.567510\pi$
$907$	−11500.0	−0.421005	−0.210502	+	0.977593i	$0.567510\pi$
$908$	−726.000	−0.0265343
$909$	0	0
$910$	9576.00	0.348837
$911$	−27396.0	−0.996345	−0.498172	−	0.867078i	$-0.665995\pi$
$911$	−27396.0	−0.996345	−0.498172	+	0.867078i	$0.665995\pi$
$912$	0	0
$913$	990.000	0.0358863
$914$	−30462.0	−1.10240
$915$	0	0
$916$	−2788.00	−0.100566
$917$	1806.00	0.0650375
$918$	0	0
$919$	8840.00	0.317307	0.158653	−	0.987334i	$-0.449285\pi$
$919$	8840.00	0.317307	0.158653	+	0.987334i	$0.449285\pi$
$920$	3024.00	0.108368
$921$	0	0
$922$	−51570.0	−1.84205
$923$	−24624.0	−0.878124
$924$	0	0
$925$	−6802.00	−0.241782
$926$	−13344.0	−0.473554
$927$	0	0
$928$	−5670.00	−0.200568
$929$	−2874.00	−0.101499	−0.0507497	−	0.998711i	$-0.516161\pi$
$929$	−2874.00	−0.101499	−0.0507497	+	0.998711i	$0.516161\pi$
$930$	0	0
$931$	−3430.00	−0.120745
$932$	−2694.00	−0.0946834
$933$	0	0
$934$	33300.0	1.16661
$935$	6336.00	0.221614
$936$	0	0
$937$	7832.00	0.273063	0.136532	−	0.990636i	$-0.456404\pi$
$937$	7832.00	0.273063	0.136532	+	0.990636i	$0.456404\pi$
$938$	4116.00	0.143275
$939$	0	0
$940$	4680.00	0.162388
$941$	−16926.0	−0.586368	−0.293184	−	0.956056i	$-0.594715\pi$
$941$	−16926.0	−0.586368	−0.293184	+	0.956056i	$0.594715\pi$
$942$	0	0
$943$	−2592.00	−0.0895092
$944$	−39192.0	−1.35126
$945$	0	0
$946$	11352.0	0.390154
$947$	17988.0	0.617245	0.308623	−	0.951185i	$-0.400132\pi$
$947$	17988.0	0.617245	0.308623	+	0.951185i	$0.400132\pi$
$948$	0	0
$949$	−608.000	−0.0207972
$950$	3990.00	0.136266
$951$	0	0
$952$	7056.00	0.240217
$953$	29142.0	0.990558	0.495279	−	0.868734i	$-0.335066\pi$
$953$	29142.0	0.990558	0.495279	+	0.868734i	$0.335066\pi$
$954$	0	0
$955$	−14544.0	−0.492809
$956$	−6480.00	−0.219224
$957$	0	0
$958$	−47448.0	−1.60018
$959$	19110.0	0.643477
$960$	0	0
$961$	−24891.0	−0.835521
$962$	40812.0	1.36781
$963$	0	0
$964$	−2320.00	−0.0775126
$965$	−30264.0	−1.00957
$966$	0	0
$967$	31160.0	1.03623	0.518117	−	0.855310i	$-0.326633\pi$
$967$	31160.0	1.03623	0.518117	+	0.855310i	$0.326633\pi$
$968$	2541.00	0.0843707
$969$	0	0
$970$	−2520.00	−0.0834148
$971$	33036.0	1.09184	0.545920	−	0.837838i	$-0.316180\pi$
$971$	33036.0	1.09184	0.545920	+	0.837838i	$0.316180\pi$
$972$	0	0
$973$	12866.0	0.423911
$974$	5772.00	0.189884
$975$	0	0
$976$	−58930.0	−1.93269
$977$	−12786.0	−0.418690	−0.209345	−	0.977842i	$-0.567133\pi$
$977$	−12786.0	−0.418690	−0.209345	+	0.977842i	$0.567133\pi$
$978$	0	0
$979$	12606.0	0.411532
$980$	−588.000	−0.0191663
$981$	0	0
$982$	39204.0	1.27398
$983$	−24918.0	−0.808505	−0.404253	−	0.914647i	$-0.632468\pi$
$983$	−24918.0	−0.808505	−0.404253	+	0.914647i	$0.632468\pi$
$984$	0	0
$985$	−41688.0	−1.34852
$986$	18144.0	0.586027
$987$	0	0
$988$	−2660.00	−0.0856537
$989$	−4128.00	−0.132723
$990$	0	0
$991$	−7648.00	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$991$	−7648.00	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$992$	−3150.00	−0.100819
$993$	0	0
$994$	13608.0	0.434225
$995$	34104.0	1.08660
$996$	0	0
$997$	−31750.0	−1.00856	−0.504279	−	0.863541i	$-0.668242\pi$
$997$	−31750.0	−1.00856	−0.504279	+	0.863541i	$0.668242\pi$
$998$	−53628.0	−1.70097
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.b.1.1		1
3.2	odd	2		77.4.a.a.1.1	✓	1
12.11	even	2		1232.4.a.d.1.1		1
15.14	odd	2		1925.4.a.a.1.1		1
21.20	even	2		539.4.a.a.1.1		1
33.32	even	2		847.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.a.1.1	✓	1	3.2	odd	2
539.4.a.a.1.1		1	21.20	even	2
693.4.a.b.1.1		1	1.1	even	1	trivial
847.4.a.a.1.1		1	33.32	even	2
1232.4.a.d.1.1		1	12.11	even	2
1925.4.a.a.1.1		1	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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