LMFDB - Embedded newform 693.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,6,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, 6, names="a")

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

chi := DirichletCharacter("693.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
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:	$111.145987130$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
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)

$f(q)$

$=$

\(q+2.00000 q^{2} -28.0000 q^{4} +76.0000 q^{5} +49.0000 q^{7} -120.000 q^{8} +152.000 q^{10} -121.000 q^{11} -402.000 q^{13} +98.0000 q^{14} +656.000 q^{16} -376.000 q^{17} +2038.00 q^{19} -2128.00 q^{20} -242.000 q^{22} -2100.00 q^{23} +2651.00 q^{25} -804.000 q^{26} -1372.00 q^{28} -1478.00 q^{29} -3874.00 q^{31} +5152.00 q^{32} -752.000 q^{34} +3724.00 q^{35} -2862.00 q^{37} +4076.00 q^{38} -9120.00 q^{40} -1328.00 q^{41} -8776.00 q^{43} +3388.00 q^{44} -4200.00 q^{46} +3454.00 q^{47} +2401.00 q^{49} +5302.00 q^{50} +11256.0 q^{52} -37134.0 q^{53} -9196.00 q^{55} -5880.00 q^{56} -2956.00 q^{58} +37336.0 q^{59} -22426.0 q^{61} -7748.00 q^{62} -10688.0 q^{64} -30552.0 q^{65} +12852.0 q^{67} +10528.0 q^{68} +7448.00 q^{70} +33048.0 q^{71} +45848.0 q^{73} -5724.00 q^{74} -57064.0 q^{76} -5929.00 q^{77} -5048.00 q^{79} +49856.0 q^{80} -2656.00 q^{82} +2794.00 q^{83} -28576.0 q^{85} -17552.0 q^{86} +14520.0 q^{88} +46038.0 q^{89} -19698.0 q^{91} +58800.0 q^{92} +6908.00 q^{94} +154888. q^{95} -103558. q^{97} +4802.00 q^{98} +O(q^{100})\)

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 $

$2$	2.00000	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	2.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	0	0
$3$	0	0
$4$	−28.0000	−0.875000
$5$	76.0000	1.35953	0.679765	−	0.733430i	$-0.262082\pi$
$5$	76.0000	1.35953	0.679765	+	0.733430i	$0.262082\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	−120.000	−0.662913
$9$	0	0
$10$	152.000	0.480666
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−402.000	−0.659732	−0.329866	−	0.944028i	$-0.607004\pi$
$13$	−402.000	−0.659732	−0.329866	+	0.944028i	$0.607004\pi$
$14$	98.0000	0.133631
$15$	0	0
$16$	656.000	0.640625
$17$	−376.000	−0.315548	−0.157774	−	0.987475i	$-0.550432\pi$
$17$	−376.000	−0.315548	−0.157774	+	0.987475i	$0.550432\pi$
$18$	0	0
$19$	2038.00	1.29515	0.647575	−	0.762002i	$-0.275783\pi$
$19$	2038.00	1.29515	0.647575	+	0.762002i	$0.275783\pi$
$20$	−2128.00	−1.18959
$21$	0	0
$22$	−242.000	−0.106600
$23$	−2100.00	−0.827751	−0.413875	−	0.910333i	$-0.635825\pi$
$23$	−2100.00	−0.827751	−0.413875	+	0.910333i	$0.635825\pi$
$24$	0	0
$25$	2651.00	0.848320
$26$	−804.000	−0.233251
$27$	0	0
$28$	−1372.00	−0.330719
$29$	−1478.00	−0.326347	−0.163173	−	0.986597i	$-0.552173\pi$
$29$	−1478.00	−0.326347	−0.163173	+	0.986597i	$0.552173\pi$
$30$	0	0
$31$	−3874.00	−0.724028	−0.362014	−	0.932173i	$-0.617911\pi$
$31$	−3874.00	−0.724028	−0.362014	+	0.932173i	$0.617911\pi$
$32$	5152.00	0.889408
$33$	0	0
$34$	−752.000	−0.111563
$35$	3724.00	0.513854
$36$	0	0
$37$	−2862.00	−0.343689	−0.171844	−	0.985124i	$-0.554973\pi$
$37$	−2862.00	−0.343689	−0.171844	+	0.985124i	$0.554973\pi$
$38$	4076.00	0.457905
$39$	0	0
$40$	−9120.00	−0.901249
$41$	−1328.00	−0.123378	−0.0616891	−	0.998095i	$-0.519649\pi$
$41$	−1328.00	−0.123378	−0.0616891	+	0.998095i	$0.519649\pi$
$42$	0	0
$43$	−8776.00	−0.723811	−0.361906	−	0.932215i	$-0.617874\pi$
$43$	−8776.00	−0.723811	−0.361906	+	0.932215i	$0.617874\pi$
$44$	3388.00	0.263822
$45$	0	0
$46$	−4200.00	−0.292654
$47$	3454.00	0.228075	0.114038	−	0.993476i	$-0.463622\pi$
$47$	3454.00	0.228075	0.114038	+	0.993476i	$0.463622\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	5302.00	0.299926
$51$	0	0
$52$	11256.0	0.577266
$53$	−37134.0	−1.81586	−0.907929	−	0.419123i	$-0.862338\pi$
$53$	−37134.0	−1.81586	−0.907929	+	0.419123i	$0.862338\pi$
$54$	0	0
$55$	−9196.00	−0.409914
$56$	−5880.00	−0.250557
$57$	0	0
$58$	−2956.00	−0.115381
$59$	37336.0	1.39636	0.698181	−	0.715922i	$-0.253993\pi$
$59$	37336.0	1.39636	0.698181	+	0.715922i	$0.253993\pi$
$60$	0	0
$61$	−22426.0	−0.771662	−0.385831	−	0.922569i	$-0.626085\pi$
$61$	−22426.0	−0.771662	−0.385831	+	0.922569i	$0.626085\pi$
$62$	−7748.00	−0.255983
$63$	0	0
$64$	−10688.0	−0.326172
$65$	−30552.0	−0.896925
$66$	0	0
$67$	12852.0	0.349771	0.174885	−	0.984589i	$-0.444044\pi$
$67$	12852.0	0.349771	0.174885	+	0.984589i	$0.444044\pi$
$68$	10528.0	0.276105
$69$	0	0
$70$	7448.00	0.181675
$71$	33048.0	0.778035	0.389018	−	0.921230i	$-0.372815\pi$
$71$	33048.0	0.778035	0.389018	+	0.921230i	$0.372815\pi$
$72$	0	0
$73$	45848.0	1.00696	0.503481	−	0.864006i	$-0.332052\pi$
$73$	45848.0	1.00696	0.503481	+	0.864006i	$0.332052\pi$
$74$	−5724.00	−0.121512
$75$	0	0
$76$	−57064.0	−1.13326
$77$	−5929.00	−0.113961
$78$	0	0
$79$	−5048.00	−0.0910021	−0.0455011	−	0.998964i	$-0.514488\pi$
$79$	−5048.00	−0.0910021	−0.0455011	+	0.998964i	$0.514488\pi$
$80$	49856.0	0.870948
$81$	0	0
$82$	−2656.00	−0.0436208
$83$	2794.00	0.0445175	0.0222588	−	0.999752i	$-0.492914\pi$
$83$	2794.00	0.0445175	0.0222588	+	0.999752i	$0.492914\pi$
$84$	0	0
$85$	−28576.0	−0.428997
$86$	−17552.0	−0.255906
$87$	0	0
$88$	14520.0	0.199876
$89$	46038.0	0.616086	0.308043	−	0.951372i	$-0.400326\pi$
$89$	46038.0	0.616086	0.308043	+	0.951372i	$0.400326\pi$
$90$	0	0
$91$	−19698.0	−0.249355
$92$	58800.0	0.724282
$93$	0	0
$94$	6908.00	0.0806367
$95$	154888.	1.76079
$96$	0	0
$97$	−103558.	−1.11752	−0.558759	−	0.829330i	$-0.688722\pi$
$97$	−103558.	−1.11752	−0.558759	+	0.829330i	$0.688722\pi$
$98$	4802.00	0.0505076
$99$	0	0
$100$	−74228.0	−0.742280
$101$	−136146.	−1.32801	−0.664005	−	0.747728i	$-0.731145\pi$
$101$	−136146.	−1.32801	−0.664005	+	0.747728i	$0.731145\pi$
$102$	0	0
$103$	−127614.	−1.18524	−0.592619	−	0.805483i	$-0.701906\pi$
$103$	−127614.	−1.18524	−0.592619	+	0.805483i	$0.701906\pi$
$104$	48240.0	0.437345
$105$	0	0
$106$	−74268.0	−0.642003
$107$	−175344.	−1.48058	−0.740289	−	0.672288i	$-0.765311\pi$
$107$	−175344.	−1.48058	−0.740289	+	0.672288i	$0.765311\pi$
$108$	0	0
$109$	90322.0	0.728161	0.364080	−	0.931368i	$-0.381383\pi$
$109$	90322.0	0.728161	0.364080	+	0.931368i	$0.381383\pi$
$110$	−18392.0	−0.144926
$111$	0	0
$112$	32144.0	0.242133
$113$	224994.	1.65758	0.828791	−	0.559559i	$-0.189029\pi$
$113$	224994.	1.65758	0.828791	+	0.559559i	$0.189029\pi$
$114$	0	0
$115$	−159600.	−1.12535
$116$	41384.0	0.285554
$117$	0	0
$118$	74672.0	0.493688
$119$	−18424.0	−0.119266
$120$	0	0
$121$	14641.0	0.0909091
$122$	−44852.0	−0.272824
$123$	0	0
$124$	108472.	0.633525
$125$	−36024.0	−0.206213
$126$	0	0
$127$	−155368.	−0.854776	−0.427388	−	0.904068i	$-0.640566\pi$
$127$	−155368.	−0.854776	−0.427388	+	0.904068i	$0.640566\pi$
$128$	−186240.	−1.00473
$129$	0	0
$130$	−61104.0	−0.317111
$131$	−316562.	−1.61169	−0.805843	−	0.592129i	$-0.798288\pi$
$131$	−316562.	−1.61169	−0.805843	+	0.592129i	$0.798288\pi$
$132$	0	0
$133$	99862.0	0.489521
$134$	25704.0	0.123663
$135$	0	0
$136$	45120.0	0.209181
$137$	−125670.	−0.572045	−0.286023	−	0.958223i	$-0.592333\pi$
$137$	−125670.	−0.572045	−0.286023	+	0.958223i	$0.592333\pi$
$138$	0	0
$139$	−303278.	−1.33139	−0.665693	−	0.746226i	$-0.731864\pi$
$139$	−303278.	−1.33139	−0.665693	+	0.746226i	$0.731864\pi$
$140$	−104272.	−0.449622
$141$	0	0
$142$	66096.0	0.275077
$143$	48642.0	0.198917
$144$	0	0
$145$	−112328.	−0.443678
$146$	91696.0	0.356015
$147$	0	0
$148$	80136.0	0.300728
$149$	31670.0	0.116864	0.0584322	−	0.998291i	$-0.481390\pi$
$149$	31670.0	0.116864	0.0584322	+	0.998291i	$0.481390\pi$
$150$	0	0
$151$	333392.	1.18991	0.594953	−	0.803760i	$-0.297171\pi$
$151$	333392.	1.18991	0.594953	+	0.803760i	$0.297171\pi$
$152$	−244560.	−0.858571
$153$	0	0
$154$	−11858.0	−0.0402911
$155$	−294424.	−0.984337
$156$	0	0
$157$	−565712.	−1.83167	−0.915833	−	0.401559i	$-0.868468\pi$
$157$	−565712.	−1.83167	−0.915833	+	0.401559i	$0.868468\pi$
$158$	−10096.0	−0.0321741
$159$	0	0
$160$	391552.	1.20918
$161$	−102900.	−0.312860
$162$	0	0
$163$	−430212.	−1.26828	−0.634138	−	0.773220i	$-0.718645\pi$
$163$	−430212.	−1.26828	−0.634138	+	0.773220i	$0.718645\pi$
$164$	37184.0	0.107956
$165$	0	0
$166$	5588.00	0.0157393
$167$	−176636.	−0.490104	−0.245052	−	0.969510i	$-0.578805\pi$
$167$	−176636.	−0.490104	−0.245052	+	0.969510i	$0.578805\pi$
$168$	0	0
$169$	−209689.	−0.564753
$170$	−57152.0	−0.151673
$171$	0	0
$172$	245728.	0.633335
$173$	−416782.	−1.05875	−0.529375	−	0.848388i	$-0.677574\pi$
$173$	−416782.	−1.05875	−0.529375	+	0.848388i	$0.677574\pi$
$174$	0	0
$175$	129899.	0.320635
$176$	−79376.0	−0.193156
$177$	0	0
$178$	92076.0	0.217819
$179$	89964.0	0.209863	0.104932	−	0.994479i	$-0.466538\pi$
$179$	89964.0	0.209863	0.104932	+	0.994479i	$0.466538\pi$
$180$	0	0
$181$	−578140.	−1.31171	−0.655853	−	0.754889i	$-0.727691\pi$
$181$	−578140.	−1.31171	−0.655853	+	0.754889i	$0.727691\pi$
$182$	−39396.0	−0.0881604
$183$	0	0
$184$	252000.	0.548727
$185$	−217512.	−0.467255
$186$	0	0
$187$	45496.0	0.0951413
$188$	−96712.0	−0.199566
$189$	0	0
$190$	309776.	0.622535
$191$	−283164.	−0.561636	−0.280818	−	0.959761i	$-0.590606\pi$
$191$	−283164.	−0.561636	−0.280818	+	0.959761i	$0.590606\pi$
$192$	0	0
$193$	−659974.	−1.27536	−0.637681	−	0.770300i	$-0.720106\pi$
$193$	−659974.	−1.27536	−0.637681	+	0.770300i	$0.720106\pi$
$194$	−207116.	−0.395102
$195$	0	0
$196$	−67228.0	−0.125000
$197$	−452294.	−0.830339	−0.415169	−	0.909744i	$-0.636278\pi$
$197$	−452294.	−0.830339	−0.415169	+	0.909744i	$0.636278\pi$
$198$	0	0
$199$	−389774.	−0.697718	−0.348859	−	0.937175i	$-0.613431\pi$
$199$	−389774.	−0.697718	−0.348859	+	0.937175i	$0.613431\pi$
$200$	−318120.	−0.562362
$201$	0	0
$202$	−272292.	−0.469523
$203$	−72422.0	−0.123348
$204$	0	0
$205$	−100928.	−0.167736
$206$	−255228.	−0.419045
$207$	0	0
$208$	−263712.	−0.422641
$209$	−246598.	−0.390503
$210$	0	0
$211$	92280.0	0.142693	0.0713463	−	0.997452i	$-0.477270\pi$
$211$	92280.0	0.142693	0.0713463	+	0.997452i	$0.477270\pi$
$212$	1.03975e6	1.58888
$213$	0	0
$214$	−350688.	−0.523464
$215$	−666976.	−0.984043
$216$	0	0
$217$	−189826.	−0.273657
$218$	180644.	0.257444
$219$	0	0
$220$	257488.	0.358674
$221$	151152.	0.208177
$222$	0	0
$223$	117558.	0.158303	0.0791517	−	0.996863i	$-0.474779\pi$
$223$	117558.	0.158303	0.0791517	+	0.996863i	$0.474779\pi$
$224$	252448.	0.336165
$225$	0	0
$226$	449988.	0.586043
$227$	465414.	0.599480	0.299740	−	0.954021i	$-0.403100\pi$
$227$	465414.	0.599480	0.299740	+	0.954021i	$0.403100\pi$
$228$	0	0
$229$	1.48848e6	1.87567	0.937833	−	0.347087i	$-0.112829\pi$
$229$	1.48848e6	1.87567	0.937833	+	0.347087i	$0.112829\pi$
$230$	−319200.	−0.397872
$231$	0	0
$232$	177360.	0.216339
$233$	312026.	0.376531	0.188266	−	0.982118i	$-0.439713\pi$
$233$	312026.	0.376531	0.188266	+	0.982118i	$0.439713\pi$
$234$	0	0
$235$	262504.	0.310075
$236$	−1.04541e6	−1.22182
$237$	0	0
$238$	−36848.0	−0.0421669
$239$	1.08720e6	1.23116	0.615580	−	0.788074i	$-0.288922\pi$
$239$	1.08720e6	1.23116	0.615580	+	0.788074i	$0.288922\pi$
$240$	0	0
$241$	−131080.	−0.145376	−0.0726882	−	0.997355i	$-0.523158\pi$
$241$	−131080.	−0.145376	−0.0726882	+	0.997355i	$0.523158\pi$
$242$	29282.0	0.0321412
$243$	0	0
$244$	627928.	0.675204
$245$	182476.	0.194218
$246$	0	0
$247$	−819276.	−0.854452
$248$	464880.	0.479967
$249$	0	0
$250$	−72048.0	−0.0729074
$251$	908712.	0.910420	0.455210	−	0.890384i	$-0.349564\pi$
$251$	908712.	0.910420	0.455210	+	0.890384i	$0.349564\pi$
$252$	0	0
$253$	254100.	0.249576
$254$	−310736.	−0.302209
$255$	0	0
$256$	−30464.0	−0.0290527
$257$	−430314.	−0.406399	−0.203200	−	0.979137i	$-0.565134\pi$
$257$	−430314.	−0.406399	−0.203200	+	0.979137i	$0.565134\pi$
$258$	0	0
$259$	−140238.	−0.129902
$260$	855456.	0.784810
$261$	0	0
$262$	−633124.	−0.569817
$263$	1.84066e6	1.64090	0.820452	−	0.571715i	$-0.193722\pi$
$263$	1.84066e6	1.64090	0.820452	+	0.571715i	$0.193722\pi$
$264$	0	0
$265$	−2.82218e6	−2.46871
$266$	199724.	0.173072
$267$	0	0
$268$	−359856.	−0.306050
$269$	672740.	0.566848	0.283424	−	0.958995i	$-0.408530\pi$
$269$	672740.	0.566848	0.283424	+	0.958995i	$0.408530\pi$
$270$	0	0
$271$	−466004.	−0.385449	−0.192724	−	0.981253i	$-0.561732\pi$
$271$	−466004.	−0.385449	−0.192724	+	0.981253i	$0.561732\pi$
$272$	−246656.	−0.202148
$273$	0	0
$274$	−251340.	−0.202248
$275$	−320771.	−0.255778
$276$	0	0
$277$	2.42209e6	1.89667	0.948333	−	0.317277i	$-0.102768\pi$
$277$	2.42209e6	1.89667	0.948333	+	0.317277i	$0.102768\pi$
$278$	−606556.	−0.470716
$279$	0	0
$280$	−446880.	−0.340640
$281$	1.36659e6	1.03246	0.516231	−	0.856450i	$-0.327335\pi$
$281$	1.36659e6	1.03246	0.516231	+	0.856450i	$0.327335\pi$
$282$	0	0
$283$	−2.40264e6	−1.78329	−0.891646	−	0.452734i	$-0.850449\pi$
$283$	−2.40264e6	−1.78329	−0.891646	+	0.452734i	$0.850449\pi$
$284$	−925344.	−0.680781
$285$	0	0
$286$	97284.0	0.0703277
$287$	−65072.0	−0.0466326
$288$	0	0
$289$	−1.27848e6	−0.900429
$290$	−224656.	−0.156864
$291$	0	0
$292$	−1.28374e6	−0.881092
$293$	1.94441e6	1.32318	0.661590	−	0.749866i	$-0.269882\pi$
$293$	1.94441e6	1.32318	0.661590	+	0.749866i	$0.269882\pi$
$294$	0	0
$295$	2.83754e6	1.89839
$296$	343440.	0.227836
$297$	0	0
$298$	63340.0	0.0413178
$299$	844200.	0.546094
$300$	0	0
$301$	−430024.	−0.273575
$302$	666784.	0.420695
$303$	0	0
$304$	1.33693e6	0.829706
$305$	−1.70438e6	−1.04910
$306$	0	0
$307$	−1.51973e6	−0.920279	−0.460139	−	0.887847i	$-0.652201\pi$
$307$	−1.51973e6	−0.920279	−0.460139	+	0.887847i	$0.652201\pi$
$308$	166012.	0.0997155
$309$	0	0
$310$	−588848.	−0.348016
$311$	1.69750e6	0.995195	0.497598	−	0.867408i	$-0.334216\pi$
$311$	1.69750e6	0.995195	0.497598	+	0.867408i	$0.334216\pi$
$312$	0	0
$313$	−2.48311e6	−1.43263	−0.716316	−	0.697776i	$-0.754173\pi$
$313$	−2.48311e6	−1.43263	−0.716316	+	0.697776i	$0.754173\pi$
$314$	−1.13142e6	−0.647592
$315$	0	0
$316$	141344.	0.0796269
$317$	−1.31161e6	−0.733087	−0.366543	−	0.930401i	$-0.619459\pi$
$317$	−1.31161e6	−0.733087	−0.366543	+	0.930401i	$0.619459\pi$
$318$	0	0
$319$	178838.	0.0983973
$320$	−812288.	−0.443440
$321$	0	0
$322$	−205800.	−0.110613
$323$	−766288.	−0.408682
$324$	0	0
$325$	−1.06570e6	−0.559664
$326$	−860424.	−0.448403
$327$	0	0
$328$	159360.	0.0817890
$329$	169246.	0.0862042
$330$	0	0
$331$	−633596.	−0.317865	−0.158932	−	0.987289i	$-0.550805\pi$
$331$	−633596.	−0.317865	−0.158932	+	0.987289i	$0.550805\pi$
$332$	−78232.0	−0.0389528
$333$	0	0
$334$	−353272.	−0.173278
$335$	976752.	0.475524
$336$	0	0
$337$	−235282.	−0.112853	−0.0564266	−	0.998407i	$-0.517971\pi$
$337$	−235282.	−0.112853	−0.0564266	+	0.998407i	$0.517971\pi$
$338$	−419378.	−0.199670
$339$	0	0
$340$	800128.	0.375372
$341$	468754.	0.218303
$342$	0	0
$343$	117649.	0.0539949
$344$	1.05312e6	0.479824
$345$	0	0
$346$	−833564.	−0.374325
$347$	−1.39186e6	−0.620545	−0.310272	−	0.950648i	$-0.600420\pi$
$347$	−1.39186e6	−0.620545	−0.310272	+	0.950648i	$0.600420\pi$
$348$	0	0
$349$	−1.06241e6	−0.466903	−0.233452	−	0.972368i	$-0.575002\pi$
$349$	−1.06241e6	−0.466903	−0.233452	+	0.972368i	$0.575002\pi$
$350$	259798.	0.113362
$351$	0	0
$352$	−623392.	−0.268167
$353$	269846.	0.115260	0.0576301	−	0.998338i	$-0.481646\pi$
$353$	269846.	0.115260	0.0576301	+	0.998338i	$0.481646\pi$
$354$	0	0
$355$	2.51165e6	1.05776
$356$	−1.28906e6	−0.539075
$357$	0	0
$358$	179928.	0.0741978
$359$	−1.08900e6	−0.445956	−0.222978	−	0.974824i	$-0.571578\pi$
$359$	−1.08900e6	−0.445956	−0.222978	+	0.974824i	$0.571578\pi$
$360$	0	0
$361$	1.67735e6	0.677414
$362$	−1.15628e6	−0.463758
$363$	0	0
$364$	551544.	0.218186
$365$	3.48445e6	1.36899
$366$	0	0
$367$	−1.18306e6	−0.458502	−0.229251	−	0.973367i	$-0.573628\pi$
$367$	−1.18306e6	−0.458502	−0.229251	+	0.973367i	$0.573628\pi$
$368$	−1.37760e6	−0.530278
$369$	0	0
$370$	−435024.	−0.165200
$371$	−1.81957e6	−0.686330
$372$	0	0
$373$	−2.62457e6	−0.976756	−0.488378	−	0.872632i	$-0.662411\pi$
$373$	−2.62457e6	−0.976756	−0.488378	+	0.872632i	$0.662411\pi$
$374$	90992.0	0.0336375
$375$	0	0
$376$	−414480.	−0.151194
$377$	594156.	0.215302
$378$	0	0
$379$	−1.92967e6	−0.690056	−0.345028	−	0.938592i	$-0.612131\pi$
$379$	−1.92967e6	−0.690056	−0.345028	+	0.938592i	$0.612131\pi$
$380$	−4.33686e6	−1.54070
$381$	0	0
$382$	−566328.	−0.198568
$383$	−3.83604e6	−1.33624	−0.668122	−	0.744051i	$-0.732902\pi$
$383$	−3.83604e6	−1.33624	−0.668122	+	0.744051i	$0.732902\pi$
$384$	0	0
$385$	−450604.	−0.154933
$386$	−1.31995e6	−0.450909
$387$	0	0
$388$	2.89962e6	0.977828
$389$	1.41377e6	0.473703	0.236851	−	0.971546i	$-0.423885\pi$
$389$	1.41377e6	0.473703	0.236851	+	0.971546i	$0.423885\pi$
$390$	0	0
$391$	789600.	0.261195
$392$	−288120.	−0.0947018
$393$	0	0
$394$	−904588.	−0.293569
$395$	−383648.	−0.123720
$396$	0	0
$397$	−2.28410e6	−0.727341	−0.363671	−	0.931528i	$-0.618477\pi$
$397$	−2.28410e6	−0.727341	−0.363671	+	0.931528i	$0.618477\pi$
$398$	−779548.	−0.246681
$399$	0	0
$400$	1.73906e6	0.543455
$401$	3.25611e6	1.01120	0.505602	−	0.862767i	$-0.331270\pi$
$401$	3.25611e6	1.01120	0.505602	+	0.862767i	$0.331270\pi$
$402$	0	0
$403$	1.55735e6	0.477665
$404$	3.81209e6	1.16201
$405$	0	0
$406$	−144844.	−0.0436099
$407$	346302.	0.103626
$408$	0	0
$409$	1.22338e6	0.361621	0.180810	−	0.983518i	$-0.442128\pi$
$409$	1.22338e6	0.361621	0.180810	+	0.983518i	$0.442128\pi$
$410$	−201856.	−0.0593037
$411$	0	0
$412$	3.57319e6	1.03708
$413$	1.82946e6	0.527775
$414$	0	0
$415$	212344.	0.0605229
$416$	−2.07110e6	−0.586771
$417$	0	0
$418$	−493196.	−0.138063
$419$	3.12546e6	0.869719	0.434859	−	0.900498i	$-0.356798\pi$
$419$	3.12546e6	0.869719	0.434859	+	0.900498i	$0.356798\pi$
$420$	0	0
$421$	4.88641e6	1.34364	0.671822	−	0.740713i	$-0.265512\pi$
$421$	4.88641e6	1.34364	0.671822	+	0.740713i	$0.265512\pi$
$422$	184560.	0.0504495
$423$	0	0
$424$	4.45608e6	1.20376
$425$	−996776.	−0.267686
$426$	0	0
$427$	−1.09887e6	−0.291661
$428$	4.90963e6	1.29551
$429$	0	0
$430$	−1.33395e6	−0.347912
$431$	−2.95178e6	−0.765405	−0.382703	−	0.923872i	$-0.625007\pi$
$431$	−2.95178e6	−0.765405	−0.382703	+	0.923872i	$0.625007\pi$
$432$	0	0
$433$	3.94967e6	1.01238	0.506188	−	0.862423i	$-0.331054\pi$
$433$	3.94967e6	1.01238	0.506188	+	0.862423i	$0.331054\pi$
$434$	−379652.	−0.0967523
$435$	0	0
$436$	−2.52902e6	−0.637141
$437$	−4.27980e6	−1.07206
$438$	0	0
$439$	393260.	0.0973909	0.0486955	−	0.998814i	$-0.484494\pi$
$439$	393260.	0.0973909	0.0486955	+	0.998814i	$0.484494\pi$
$440$	1.10352e6	0.271737
$441$	0	0
$442$	302304.	0.0736018
$443$	−5.03867e6	−1.21985	−0.609925	−	0.792459i	$-0.708800\pi$
$443$	−5.03867e6	−1.21985	−0.609925	+	0.792459i	$0.708800\pi$
$444$	0	0
$445$	3.49889e6	0.837587
$446$	235116.	0.0559687
$447$	0	0
$448$	−523712.	−0.123281
$449$	4.15309e6	0.972200	0.486100	−	0.873903i	$-0.338419\pi$
$449$	4.15309e6	0.972200	0.486100	+	0.873903i	$0.338419\pi$
$450$	0	0
$451$	160688.	0.0371999
$452$	−6.29983e6	−1.45038
$453$	0	0
$454$	930828.	0.211948
$455$	−1.49705e6	−0.339006
$456$	0	0
$457$	1.23958e6	0.277641	0.138820	−	0.990318i	$-0.455669\pi$
$457$	1.23958e6	0.277641	0.138820	+	0.990318i	$0.455669\pi$
$458$	2.97697e6	0.663148
$459$	0	0
$460$	4.46880e6	0.984683
$461$	4.21302e6	0.923297	0.461648	−	0.887063i	$-0.347258\pi$
$461$	4.21302e6	0.923297	0.461648	+	0.887063i	$0.347258\pi$
$462$	0	0
$463$	−2.85934e6	−0.619889	−0.309944	−	0.950755i	$-0.600310\pi$
$463$	−2.85934e6	−0.619889	−0.309944	+	0.950755i	$0.600310\pi$
$464$	−969568.	−0.209066
$465$	0	0
$466$	624052.	0.133124
$467$	−949956.	−0.201563	−0.100782	−	0.994909i	$-0.532134\pi$
$467$	−949956.	−0.201563	−0.100782	+	0.994909i	$0.532134\pi$
$468$	0	0
$469$	629748.	0.132201
$470$	525008.	0.109628
$471$	0	0
$472$	−4.48032e6	−0.925665
$473$	1.06190e6	0.218237
$474$	0	0
$475$	5.40274e6	1.09870
$476$	515872.	0.104358
$477$	0	0
$478$	2.17440e6	0.435281
$479$	2.71673e6	0.541013	0.270506	−	0.962718i	$-0.412809\pi$
$479$	2.71673e6	0.541013	0.270506	+	0.962718i	$0.412809\pi$
$480$	0	0
$481$	1.15052e6	0.226743
$482$	−262160.	−0.0513983
$483$	0	0
$484$	−409948.	−0.0795455
$485$	−7.87041e6	−1.51930
$486$	0	0
$487$	8.65216e6	1.65311	0.826556	−	0.562854i	$-0.190297\pi$
$487$	8.65216e6	1.65311	0.826556	+	0.562854i	$0.190297\pi$
$488$	2.69112e6	0.511545
$489$	0	0
$490$	364952.	0.0686666
$491$	−1.98423e6	−0.371439	−0.185720	−	0.982603i	$-0.559462\pi$
$491$	−1.98423e6	−0.371439	−0.185720	+	0.982603i	$0.559462\pi$
$492$	0	0
$493$	555728.	0.102978
$494$	−1.63855e6	−0.302095
$495$	0	0
$496$	−2.54134e6	−0.463830
$497$	1.61935e6	0.294070
$498$	0	0
$499$	5.02934e6	0.904190	0.452095	−	0.891970i	$-0.350677\pi$
$499$	5.02934e6	0.904190	0.452095	+	0.891970i	$0.350677\pi$
$500$	1.00867e6	0.180437
$501$	0	0
$502$	1.81742e6	0.321882
$503$	−2.75478e6	−0.485475	−0.242738	−	0.970092i	$-0.578045\pi$
$503$	−2.75478e6	−0.485475	−0.242738	+	0.970092i	$0.578045\pi$
$504$	0	0
$505$	−1.03471e7	−1.80547
$506$	508200.	0.0882386
$507$	0	0
$508$	4.35030e6	0.747929
$509$	7.42547e6	1.27037	0.635184	−	0.772361i	$-0.280924\pi$
$509$	7.42547e6	1.27037	0.635184	+	0.772361i	$0.280924\pi$
$510$	0	0
$511$	2.24655e6	0.380596
$512$	5.89875e6	0.994455
$513$	0	0
$514$	−860628.	−0.143684
$515$	−9.69866e6	−1.61136
$516$	0	0
$517$	−417934.	−0.0687672
$518$	−280476.	−0.0459273
$519$	0	0
$520$	3.66624e6	0.594583
$521$	7.04761e6	1.13749	0.568745	−	0.822514i	$-0.307429\pi$
$521$	7.04761e6	1.13749	0.568745	+	0.822514i	$0.307429\pi$
$522$	0	0
$523$	4.89180e6	0.782014	0.391007	−	0.920388i	$-0.372127\pi$
$523$	4.89180e6	0.782014	0.391007	+	0.920388i	$0.372127\pi$
$524$	8.86374e6	1.41023
$525$	0	0
$526$	3.68131e6	0.580147
$527$	1.45662e6	0.228466
$528$	0	0
$529$	−2.02634e6	−0.314828
$530$	−5.64437e6	−0.872822
$531$	0	0
$532$	−2.79614e6	−0.428331
$533$	533856.	0.0813966
$534$	0	0
$535$	−1.33261e7	−2.01289
$536$	−1.54224e6	−0.231868
$537$	0	0
$538$	1.34548e6	0.200411
$539$	−290521.	−0.0430730
$540$	0	0
$541$	1.04634e7	1.53703	0.768513	−	0.639834i	$-0.220997\pi$
$541$	1.04634e7	1.53703	0.768513	+	0.639834i	$0.220997\pi$
$542$	−932008.	−0.136277
$543$	0	0
$544$	−1.93715e6	−0.280651
$545$	6.86447e6	0.989956
$546$	0	0
$547$	5.18065e6	0.740314	0.370157	−	0.928969i	$-0.379304\pi$
$547$	5.18065e6	0.740314	0.370157	+	0.928969i	$0.379304\pi$
$548$	3.51876e6	0.500539
$549$	0	0
$550$	−641542.	−0.0904312
$551$	−3.01216e6	−0.422668
$552$	0	0
$553$	−247352.	−0.0343956
$554$	4.84418e6	0.670573
$555$	0	0
$556$	8.49178e6	1.16496
$557$	4.15034e6	0.566820	0.283410	−	0.958999i	$-0.408534\pi$
$557$	4.15034e6	0.566820	0.283410	+	0.958999i	$0.408534\pi$
$558$	0	0
$559$	3.52795e6	0.477522
$560$	2.44294e6	0.329188
$561$	0	0
$562$	2.73319e6	0.365030
$563$	7.30678e6	0.971528	0.485764	−	0.874090i	$-0.338542\pi$
$563$	7.30678e6	0.971528	0.485764	+	0.874090i	$0.338542\pi$
$564$	0	0
$565$	1.70995e7	2.25353
$566$	−4.80528e6	−0.630489
$567$	0	0
$568$	−3.96576e6	−0.515769
$569$	−5.69210e6	−0.737042	−0.368521	−	0.929620i	$-0.620136\pi$
$569$	−5.69210e6	−0.737042	−0.368521	+	0.929620i	$0.620136\pi$
$570$	0	0
$571$	6.84144e6	0.878127	0.439064	−	0.898456i	$-0.355310\pi$
$571$	6.84144e6	0.878127	0.439064	+	0.898456i	$0.355310\pi$
$572$	−1.36198e6	−0.174052
$573$	0	0
$574$	−130144.	−0.0164871
$575$	−5.56710e6	−0.702198
$576$	0	0
$577$	−5.89572e6	−0.737221	−0.368610	−	0.929584i	$-0.620166\pi$
$577$	−5.89572e6	−0.737221	−0.368610	+	0.929584i	$0.620166\pi$
$578$	−2.55696e6	−0.318350
$579$	0	0
$580$	3.14518e6	0.388218
$581$	136906.	0.0168260
$582$	0	0
$583$	4.49321e6	0.547502
$584$	−5.50176e6	−0.667528
$585$	0	0
$586$	3.88882e6	0.467815
$587$	−3.35712e6	−0.402135	−0.201067	−	0.979577i	$-0.564441\pi$
$587$	−3.35712e6	−0.402135	−0.201067	+	0.979577i	$0.564441\pi$
$588$	0	0
$589$	−7.89521e6	−0.937725
$590$	5.67507e6	0.671184
$591$	0	0
$592$	−1.87747e6	−0.220176
$593$	6.76650e6	0.790183	0.395091	−	0.918642i	$-0.370713\pi$
$593$	6.76650e6	0.790183	0.395091	+	0.918642i	$0.370713\pi$
$594$	0	0
$595$	−1.40022e6	−0.162146
$596$	−886760.	−0.102256
$597$	0	0
$598$	1.68840e6	0.193073
$599$	4.56950e6	0.520357	0.260178	−	0.965561i	$-0.416219\pi$
$599$	4.56950e6	0.520357	0.260178	+	0.965561i	$0.416219\pi$
$600$	0	0
$601$	−1.62371e7	−1.83368	−0.916840	−	0.399255i	$-0.869269\pi$
$601$	−1.62371e7	−1.83368	−0.916840	+	0.399255i	$0.869269\pi$
$602$	−860048.	−0.0967234
$603$	0	0
$604$	−9.33498e6	−1.04117
$605$	1.11272e6	0.123594
$606$	0	0
$607$	1.63693e7	1.80326	0.901629	−	0.432511i	$-0.142372\pi$
$607$	1.63693e7	1.80326	0.901629	+	0.432511i	$0.142372\pi$
$608$	1.04998e7	1.15192
$609$	0	0
$610$	−3.40875e6	−0.370912
$611$	−1.38851e6	−0.150468
$612$	0	0
$613$	8.25460e6	0.887247	0.443624	−	0.896213i	$-0.353693\pi$
$613$	8.25460e6	0.887247	0.443624	+	0.896213i	$0.353693\pi$
$614$	−3.03945e6	−0.325368
$615$	0	0
$616$	711480.	0.0755459
$617$	−8.71255e6	−0.921366	−0.460683	−	0.887565i	$-0.652395\pi$
$617$	−8.71255e6	−0.921366	−0.460683	+	0.887565i	$0.652395\pi$
$618$	0	0
$619$	−1.74335e7	−1.82877	−0.914383	−	0.404850i	$-0.867324\pi$
$619$	−1.74335e7	−1.82877	−0.914383	+	0.404850i	$0.867324\pi$
$620$	8.24387e6	0.861295
$621$	0	0
$622$	3.39500e6	0.351855
$623$	2.25586e6	0.232859
$624$	0	0
$625$	−1.10222e7	−1.12867
$626$	−4.96621e6	−0.506512
$627$	0	0
$628$	1.58399e7	1.60271
$629$	1.07611e6	0.108450
$630$	0	0
$631$	1.42676e6	0.142651	0.0713257	−	0.997453i	$-0.477277\pi$
$631$	1.42676e6	0.142651	0.0713257	+	0.997453i	$0.477277\pi$
$632$	605760.	0.0603265
$633$	0	0
$634$	−2.62321e6	−0.259185
$635$	−1.18080e7	−1.16209
$636$	0	0
$637$	−965202.	−0.0942475
$638$	357676.	0.0347887
$639$	0	0
$640$	−1.41542e7	−1.36596
$641$	1.24282e7	1.19471	0.597354	−	0.801977i	$-0.296219\pi$
$641$	1.24282e7	1.19471	0.597354	+	0.801977i	$0.296219\pi$
$642$	0	0
$643$	−9.59562e6	−0.915262	−0.457631	−	0.889142i	$-0.651302\pi$
$643$	−9.59562e6	−0.915262	−0.457631	+	0.889142i	$0.651302\pi$
$644$	2.88120e6	0.273753
$645$	0	0
$646$	−1.53258e6	−0.144491
$647$	−3.55154e6	−0.333546	−0.166773	−	0.985995i	$-0.553335\pi$
$647$	−3.55154e6	−0.333546	−0.166773	+	0.985995i	$0.553335\pi$
$648$	0	0
$649$	−4.51766e6	−0.421019
$650$	−2.13140e6	−0.197871
$651$	0	0
$652$	1.20459e7	1.10974
$653$	1.20925e7	1.10977	0.554885	−	0.831927i	$-0.312762\pi$
$653$	1.20925e7	1.10977	0.554885	+	0.831927i	$0.312762\pi$
$654$	0	0
$655$	−2.40587e7	−2.19113
$656$	−871168.	−0.0790392
$657$	0	0
$658$	338492.	0.0304778
$659$	−1.36481e7	−1.22421	−0.612107	−	0.790775i	$-0.709678\pi$
$659$	−1.36481e7	−1.22421	−0.612107	+	0.790775i	$0.709678\pi$
$660$	0	0
$661$	−758596.	−0.0675316	−0.0337658	−	0.999430i	$-0.510750\pi$
$661$	−758596.	−0.0675316	−0.0337658	+	0.999430i	$0.510750\pi$
$662$	−1.26719e6	−0.112382
$663$	0	0
$664$	−335280.	−0.0295112
$665$	7.58951e6	0.665518
$666$	0	0
$667$	3.10380e6	0.270134
$668$	4.94581e6	0.428841
$669$	0	0
$670$	1.95350e6	0.168123
$671$	2.71355e6	0.232665
$672$	0	0
$673$	8.87005e6	0.754898	0.377449	−	0.926030i	$-0.376801\pi$
$673$	8.87005e6	0.754898	0.377449	+	0.926030i	$0.376801\pi$
$674$	−470564.	−0.0398996
$675$	0	0
$676$	5.87129e6	0.494159
$677$	1.82823e7	1.53306	0.766531	−	0.642207i	$-0.221981\pi$
$677$	1.82823e7	1.53306	0.766531	+	0.642207i	$0.221981\pi$
$678$	0	0
$679$	−5.07434e6	−0.422382
$680$	3.42912e6	0.284387
$681$	0	0
$682$	937508.	0.0771816
$683$	−1.04250e7	−0.855112	−0.427556	−	0.903989i	$-0.640625\pi$
$683$	−1.04250e7	−0.855112	−0.427556	+	0.903989i	$0.640625\pi$
$684$	0	0
$685$	−9.55092e6	−0.777712
$686$	235298.	0.0190901
$687$	0	0
$688$	−5.75706e6	−0.463692
$689$	1.49279e7	1.19798
$690$	0	0
$691$	−1.78231e7	−1.42000	−0.710001	−	0.704201i	$-0.751305\pi$
$691$	−1.78231e7	−1.42000	−0.710001	+	0.704201i	$0.751305\pi$
$692$	1.16699e7	0.926407
$693$	0	0
$694$	−2.78373e6	−0.219396
$695$	−2.30491e7	−1.81006
$696$	0	0
$697$	499328.	0.0389318
$698$	−2.12481e6	−0.165075
$699$	0	0
$700$	−3.63717e6	−0.280555
$701$	−5.02509e6	−0.386232	−0.193116	−	0.981176i	$-0.561859\pi$
$701$	−5.02509e6	−0.386232	−0.193116	+	0.981176i	$0.561859\pi$
$702$	0	0
$703$	−5.83276e6	−0.445129
$704$	1.29325e6	0.0983445
$705$	0	0
$706$	539692.	0.0407506
$707$	−6.67115e6	−0.501941
$708$	0	0
$709$	−2.14778e7	−1.60463	−0.802313	−	0.596903i	$-0.796398\pi$
$709$	−2.14778e7	−1.60463	−0.802313	+	0.596903i	$0.796398\pi$
$710$	5.02330e6	0.373975
$711$	0	0
$712$	−5.52456e6	−0.408411
$713$	8.13540e6	0.599315
$714$	0	0
$715$	3.69679e6	0.270433
$716$	−2.51899e6	−0.183630
$717$	0	0
$718$	−2.17800e6	−0.157669
$719$	327962.	0.0236593	0.0118296	−	0.999930i	$-0.496234\pi$
$719$	327962.	0.0236593	0.0118296	+	0.999930i	$0.496234\pi$
$720$	0	0
$721$	−6.25309e6	−0.447978
$722$	3.35469e6	0.239502
$723$	0	0
$724$	1.61879e7	1.14774
$725$	−3.91818e6	−0.276847
$726$	0	0
$727$	−1.88334e6	−0.132158	−0.0660788	−	0.997814i	$-0.521049\pi$
$727$	−1.88334e6	−0.132158	−0.0660788	+	0.997814i	$0.521049\pi$
$728$	2.36376e6	0.165301
$729$	0	0
$730$	6.96890e6	0.484013
$731$	3.29978e6	0.228397
$732$	0	0
$733$	4.01201e6	0.275805	0.137903	−	0.990446i	$-0.455964\pi$
$733$	4.01201e6	0.275805	0.137903	+	0.990446i	$0.455964\pi$
$734$	−2.36612e6	−0.162105
$735$	0	0
$736$	−1.08192e7	−0.736208
$737$	−1.55509e6	−0.105460
$738$	0	0
$739$	−1.16719e6	−0.0786194	−0.0393097	−	0.999227i	$-0.512516\pi$
$739$	−1.16719e6	−0.0786194	−0.0393097	+	0.999227i	$0.512516\pi$
$740$	6.09034e6	0.408848
$741$	0	0
$742$	−3.63913e6	−0.242654
$743$	−6.22826e6	−0.413900	−0.206950	−	0.978352i	$-0.566354\pi$
$743$	−6.22826e6	−0.413900	−0.206950	+	0.978352i	$0.566354\pi$
$744$	0	0
$745$	2.40692e6	0.158881
$746$	−5.24914e6	−0.345335
$747$	0	0
$748$	−1.27389e6	−0.0832487
$749$	−8.59186e6	−0.559606
$750$	0	0
$751$	1.22860e7	0.794895	0.397448	−	0.917625i	$-0.369896\pi$
$751$	1.22860e7	0.794895	0.397448	+	0.917625i	$0.369896\pi$
$752$	2.26582e6	0.146111
$753$	0	0
$754$	1.18831e6	0.0761206
$755$	2.53378e7	1.61771
$756$	0	0
$757$	6.22812e6	0.395018	0.197509	−	0.980301i	$-0.436715\pi$
$757$	6.22812e6	0.395018	0.197509	+	0.980301i	$0.436715\pi$
$758$	−3.85934e6	−0.243972
$759$	0	0
$760$	−1.85866e7	−1.16725
$761$	−6.57535e6	−0.411583	−0.205791	−	0.978596i	$-0.565977\pi$
$761$	−6.57535e6	−0.411583	−0.205791	+	0.978596i	$0.565977\pi$
$762$	0	0
$763$	4.42578e6	0.275219
$764$	7.92859e6	0.491431
$765$	0	0
$766$	−7.67208e6	−0.472434
$767$	−1.50091e7	−0.921224
$768$	0	0
$769$	2.33241e7	1.42229	0.711146	−	0.703044i	$-0.248176\pi$
$769$	2.33241e7	1.42229	0.711146	+	0.703044i	$0.248176\pi$
$770$	−901208.	−0.0547770
$771$	0	0
$772$	1.84793e7	1.11594
$773$	−1.46746e6	−0.0883317	−0.0441658	−	0.999024i	$-0.514063\pi$
$773$	−1.46746e6	−0.0883317	−0.0441658	+	0.999024i	$0.514063\pi$
$774$	0	0
$775$	−1.02700e7	−0.614207
$776$	1.24270e7	0.740816
$777$	0	0
$778$	2.82755e6	0.167479
$779$	−2.70646e6	−0.159793
$780$	0	0
$781$	−3.99881e6	−0.234586
$782$	1.57920e6	0.0923465
$783$	0	0
$784$	1.57506e6	0.0915179
$785$	−4.29941e7	−2.49020
$786$	0	0
$787$	1.53995e7	0.886275	0.443138	−	0.896454i	$-0.353865\pi$
$787$	1.53995e7	0.886275	0.443138	+	0.896454i	$0.353865\pi$
$788$	1.26642e7	0.726546
$789$	0	0
$790$	−767296.	−0.0437417
$791$	1.10247e7	0.626507
$792$	0	0
$793$	9.01525e6	0.509090
$794$	−4.56819e6	−0.257154
$795$	0	0
$796$	1.09137e7	0.610504
$797$	−1.80152e7	−1.00460	−0.502300	−	0.864693i	$-0.667513\pi$
$797$	−1.80152e7	−1.00460	−0.502300	+	0.864693i	$0.667513\pi$
$798$	0	0
$799$	−1.29870e6	−0.0719686
$800$	1.36580e7	0.754502
$801$	0	0
$802$	6.51223e6	0.357515
$803$	−5.54761e6	−0.303611
$804$	0	0
$805$	−7.82040e6	−0.425343
$806$	3.11470e6	0.168880
$807$	0	0
$808$	1.63375e7	0.880355
$809$	−2.03686e7	−1.09418	−0.547091	−	0.837073i	$-0.684265\pi$
$809$	−2.03686e7	−1.09418	−0.547091	+	0.837073i	$0.684265\pi$
$810$	0	0
$811$	−3.30384e6	−0.176387	−0.0881935	−	0.996103i	$-0.528109\pi$
$811$	−3.30384e6	−0.176387	−0.0881935	+	0.996103i	$0.528109\pi$
$812$	2.02782e6	0.107929
$813$	0	0
$814$	692604.	0.0366373
$815$	−3.26961e7	−1.72426
$816$	0	0
$817$	−1.78855e7	−0.937445
$818$	2.44676e6	0.127852
$819$	0	0
$820$	2.82598e6	0.146769
$821$	−5.43182e6	−0.281247	−0.140623	−	0.990063i	$-0.544911\pi$
$821$	−5.43182e6	−0.281247	−0.140623	+	0.990063i	$0.544911\pi$
$822$	0	0
$823$	3.34427e7	1.72109	0.860543	−	0.509378i	$-0.170125\pi$
$823$	3.34427e7	1.72109	0.860543	+	0.509378i	$0.170125\pi$
$824$	1.53137e7	0.785709
$825$	0	0
$826$	3.65893e6	0.186597
$827$	−2.02603e7	−1.03011	−0.515054	−	0.857158i	$-0.672228\pi$
$827$	−2.02603e7	−1.03011	−0.515054	+	0.857158i	$0.672228\pi$
$828$	0	0
$829$	2.43344e7	1.22980	0.614899	−	0.788606i	$-0.289197\pi$
$829$	2.43344e7	1.22980	0.614899	+	0.788606i	$0.289197\pi$
$830$	424688.	0.0213981
$831$	0	0
$832$	4.29658e6	0.215186
$833$	−902776.	−0.0450783
$834$	0	0
$835$	−1.34243e7	−0.666310
$836$	6.90474e6	0.341690
$837$	0	0
$838$	6.25092e6	0.307492
$839$	−8.35510e6	−0.409776	−0.204888	−	0.978785i	$-0.565683\pi$
$839$	−8.35510e6	−0.409776	−0.204888	+	0.978785i	$0.565683\pi$
$840$	0	0
$841$	−1.83267e7	−0.893498
$842$	9.77281e6	0.475050
$843$	0	0
$844$	−2.58384e6	−0.124856
$845$	−1.59364e7	−0.767799
$846$	0	0
$847$	717409.	0.0343604
$848$	−2.43599e7	−1.16328
$849$	0	0
$850$	−1.99355e6	−0.0946412
$851$	6.01020e6	0.284489
$852$	0	0
$853$	3.86356e7	1.81809	0.909044	−	0.416699i	$-0.136813\pi$
$853$	3.86356e7	1.81809	0.909044	+	0.416699i	$0.136813\pi$
$854$	−2.19775e6	−0.103118
$855$	0	0
$856$	2.10413e7	0.981494
$857$	−1.18601e7	−0.551615	−0.275808	−	0.961213i	$-0.588945\pi$
$857$	−1.18601e7	−0.551615	−0.275808	+	0.961213i	$0.588945\pi$
$858$	0	0
$859$	2.99518e7	1.38497	0.692486	−	0.721432i	$-0.256516\pi$
$859$	2.99518e7	1.38497	0.692486	+	0.721432i	$0.256516\pi$
$860$	1.86753e7	0.861038
$861$	0	0
$862$	−5.90357e6	−0.270612
$863$	3.56309e7	1.62855	0.814273	−	0.580482i	$-0.197136\pi$
$863$	3.56309e7	1.62855	0.814273	+	0.580482i	$0.197136\pi$
$864$	0	0
$865$	−3.16754e7	−1.43940
$866$	7.89935e6	0.357929
$867$	0	0
$868$	5.31513e6	0.239450
$869$	610808.	0.0274382
$870$	0	0
$871$	−5.16650e6	−0.230755
$872$	−1.08386e7	−0.482707
$873$	0	0
$874$	−8.55960e6	−0.379031
$875$	−1.76518e6	−0.0779413
$876$	0	0
$877$	−8.65017e6	−0.379775	−0.189887	−	0.981806i	$-0.560812\pi$
$877$	−8.65017e6	−0.379775	−0.189887	+	0.981806i	$0.560812\pi$
$878$	786520.	0.0344329
$879$	0	0
$880$	−6.03258e6	−0.262601
$881$	4.05850e7	1.76167	0.880836	−	0.473421i	$-0.156981\pi$
$881$	4.05850e7	1.76167	0.880836	+	0.473421i	$0.156981\pi$
$882$	0	0
$883$	−7.00806e6	−0.302480	−0.151240	−	0.988497i	$-0.548327\pi$
$883$	−7.00806e6	−0.302480	−0.151240	+	0.988497i	$0.548327\pi$
$884$	−4.23226e6	−0.182155
$885$	0	0
$886$	−1.00773e7	−0.431282
$887$	2.24071e7	0.956260	0.478130	−	0.878289i	$-0.341315\pi$
$887$	2.24071e7	0.956260	0.478130	+	0.878289i	$0.341315\pi$
$888$	0	0
$889$	−7.61303e6	−0.323075
$890$	6.99778e6	0.296132
$891$	0	0
$892$	−3.29162e6	−0.138515
$893$	7.03925e6	0.295391
$894$	0	0
$895$	6.83726e6	0.285315
$896$	−9.12576e6	−0.379751
$897$	0	0
$898$	8.30618e6	0.343725
$899$	5.72577e6	0.236284
$900$	0	0
$901$	1.39624e7	0.572991
$902$	321376.	0.0131522
$903$	0	0
$904$	−2.69993e7	−1.09883
$905$	−4.39386e7	−1.78330
$906$	0	0
$907$	1.06793e6	0.0431048	0.0215524	−	0.999768i	$-0.493139\pi$
$907$	1.06793e6	0.0431048	0.0215524	+	0.999768i	$0.493139\pi$
$908$	−1.30316e7	−0.524545
$909$	0	0
$910$	−2.99410e6	−0.119857
$911$	−2.05308e7	−0.819617	−0.409808	−	0.912172i	$-0.634404\pi$
$911$	−2.05308e7	−0.819617	−0.409808	+	0.912172i	$0.634404\pi$
$912$	0	0
$913$	−338074.	−0.0134225
$914$	2.47916e6	0.0981609
$915$	0	0
$916$	−4.16776e7	−1.64121
$917$	−1.55115e7	−0.609160
$918$	0	0
$919$	−4.00194e7	−1.56308	−0.781541	−	0.623854i	$-0.785566\pi$
$919$	−4.00194e7	−1.56308	−0.781541	+	0.623854i	$0.785566\pi$
$920$	1.91520e7	0.746010
$921$	0	0
$922$	8.42604e6	0.326435
$923$	−1.32853e7	−0.513295
$924$	0	0
$925$	−7.58716e6	−0.291558
$926$	−5.71869e6	−0.219164
$927$	0	0
$928$	−7.61466e6	−0.290255
$929$	5.13696e7	1.95284	0.976421	−	0.215877i	$-0.0692609\pi$
$929$	5.13696e7	1.95284	0.976421	+	0.215877i	$0.0692609\pi$
$930$	0	0
$931$	4.89324e6	0.185021
$932$	−8.73673e6	−0.329465
$933$	0	0
$934$	−1.89991e6	−0.0712634
$935$	3.45770e6	0.129347
$936$	0	0
$937$	−131296.	−0.00488543	−0.00244271	−	0.999997i	$-0.500778\pi$
$937$	−131296.	−0.00488543	−0.00244271	+	0.999997i	$0.500778\pi$
$938$	1.25950e6	0.0467401
$939$	0	0
$940$	−7.35011e6	−0.271315
$941$	−2.95063e7	−1.08628	−0.543139	−	0.839643i	$-0.682764\pi$
$941$	−2.95063e7	−1.08628	−0.543139	+	0.839643i	$0.682764\pi$
$942$	0	0
$943$	2.78880e6	0.102126
$944$	2.44924e7	0.894544
$945$	0	0
$946$	2.12379e6	0.0771586
$947$	4.75662e6	0.172355	0.0861774	−	0.996280i	$-0.472535\pi$
$947$	4.75662e6	0.172355	0.0861774	+	0.996280i	$0.472535\pi$
$948$	0	0
$949$	−1.84309e7	−0.664325
$950$	1.08055e7	0.388450
$951$	0	0
$952$	2.21088e6	0.0790629
$953$	2.80443e7	1.00026	0.500130	−	0.865950i	$-0.333286\pi$
$953$	2.80443e7	1.00026	0.500130	+	0.865950i	$0.333286\pi$
$954$	0	0
$955$	−2.15205e7	−0.763560
$956$	−3.04416e7	−1.07727
$957$	0	0
$958$	5.43346e6	0.191277
$959$	−6.15783e6	−0.216213
$960$	0	0
$961$	−1.36213e7	−0.475783
$962$	2.30105e6	0.0801656
$963$	0	0
$964$	3.67024e6	0.127204
$965$	−5.01580e7	−1.73389
$966$	0	0
$967$	1.18514e7	0.407571	0.203786	−	0.979016i	$-0.434675\pi$
$967$	1.18514e7	0.407571	0.203786	+	0.979016i	$0.434675\pi$
$968$	−1.75692e6	−0.0602648
$969$	0	0
$970$	−1.57408e7	−0.537153
$971$	1.46447e6	0.0498462	0.0249231	−	0.999689i	$-0.492066\pi$
$971$	1.46447e6	0.0498462	0.0249231	+	0.999689i	$0.492066\pi$
$972$	0	0
$973$	−1.48606e7	−0.503217
$974$	1.73043e7	0.584463
$975$	0	0
$976$	−1.47115e7	−0.494346
$977$	−4.44734e7	−1.49061	−0.745304	−	0.666725i	$-0.767696\pi$
$977$	−4.44734e7	−1.49061	−0.745304	+	0.666725i	$0.767696\pi$
$978$	0	0
$979$	−5.57060e6	−0.185757
$980$	−5.10933e6	−0.169941
$981$	0	0
$982$	−3.96846e6	−0.131324
$983$	−3.71293e7	−1.22555	−0.612777	−	0.790256i	$-0.709948\pi$
$983$	−3.71293e7	−1.22555	−0.612777	+	0.790256i	$0.709948\pi$
$984$	0	0
$985$	−3.43743e7	−1.12887
$986$	1.11146e6	0.0364083
$987$	0	0
$988$	2.29397e7	0.747646
$989$	1.84296e7	0.599136
$990$	0	0
$991$	−3.46054e7	−1.11933	−0.559667	−	0.828718i	$-0.689071\pi$
$991$	−3.46054e7	−1.11933	−0.559667	+	0.828718i	$0.689071\pi$
$992$	−1.99588e7	−0.643956
$993$	0	0
$994$	3.23870e6	0.103969
$995$	−2.96228e7	−0.948569
$996$	0	0
$997$	−9.68545e6	−0.308590	−0.154295	−	0.988025i	$-0.549311\pi$
$997$	−9.68545e6	−0.308590	−0.154295	+	0.988025i	$0.549311\pi$
$998$	1.00587e7	0.319679
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.6.a.b.1.1		1
3.2	odd	2		231.6.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.6.a.a.1.1	✓	1	3.2	odd	2
693.6.a.b.1.1		1	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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