LMFDB - Embedded newform 693.4.a.k.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
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			1.2
Root			$2.56155$ 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)

$f(q)$	$=$	\(q+3.56155 q^{2} +4.68466 q^{4} +15.6847 q^{5} +7.00000 q^{7} -11.8078 q^{8} +55.8617 q^{10} -11.0000 q^{11} +52.9157 q^{13} +24.9309 q^{14} -79.5312 q^{16} +77.1231 q^{17} +13.4081 q^{19} +73.4773 q^{20} -39.1771 q^{22} +59.0152 q^{23} +121.009 q^{25} +188.462 q^{26} +32.7926 q^{28} +69.3305 q^{29} +75.9697 q^{31} -188.793 q^{32} +274.678 q^{34} +109.793 q^{35} -335.948 q^{37} +47.7538 q^{38} -185.201 q^{40} +318.617 q^{41} -57.2614 q^{43} -51.5312 q^{44} +210.186 q^{46} +577.779 q^{47} +49.0000 q^{49} +430.978 q^{50} +247.892 q^{52} -315.555 q^{53} -172.531 q^{55} -82.6543 q^{56} +246.924 q^{58} +598.717 q^{59} -337.879 q^{61} +270.570 q^{62} -36.1449 q^{64} +829.965 q^{65} -107.779 q^{67} +361.295 q^{68} +391.032 q^{70} -405.390 q^{71} -133.300 q^{73} -1196.50 q^{74} +62.8125 q^{76} -77.0000 q^{77} -922.038 q^{79} -1247.42 q^{80} +1134.77 q^{82} +1221.17 q^{83} +1209.65 q^{85} -203.939 q^{86} +129.885 q^{88} -1580.19 q^{89} +370.410 q^{91} +276.466 q^{92} +2057.79 q^{94} +210.302 q^{95} -287.097 q^{97} +174.516 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} - 3 q^{4} + 19 q^{5} + 14 q^{7} - 3 q^{8} + 54 q^{10} - 22 q^{11} + 11 q^{13} + 21 q^{14} - 23 q^{16} + 146 q^{17} - 101 q^{19} + 48 q^{20} - 33 q^{22} + 184 q^{23} + 7 q^{25} + 212 q^{26}+ \cdots + 147 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 - 3 * q^4 + 19 * q^5 + 14 * q^7 - 3 * q^8 + 54 * q^10 - 22 * q^11 + 11 * q^13 + 21 * q^14 - 23 * q^16 + 146 * q^17 - 101 * q^19 + 48 * q^20 - 33 * q^22 + 184 * q^23 + 7 * q^25 + 212 * q^26 - 21 * q^28 + 217 * q^29 + 20 * q^31 - 291 * q^32 + 236 * q^34 + 133 * q^35 - 173 * q^37 + 112 * q^38 - 156 * q^40 + 60 * q^41 - 164 * q^43 + 33 * q^44 + 140 * q^46 + 467 * q^47 + 98 * q^49 + 495 * q^50 + 570 * q^52 - 326 * q^53 - 209 * q^55 - 21 * q^56 + 164 * q^58 + 781 * q^59 - 148 * q^61 + 302 * q^62 - 431 * q^64 + 691 * q^65 + 473 * q^67 - 168 * q^68 + 378 * q^70 + 756 * q^71 - 213 * q^73 - 1288 * q^74 + 942 * q^76 - 154 * q^77 + 168 * q^79 - 1060 * q^80 + 1280 * q^82 + 2096 * q^83 + 1438 * q^85 - 144 * q^86 + 33 * q^88 - 736 * q^89 + 77 * q^91 - 684 * q^92 + 2120 * q^94 - 169 * q^95 + 638 * q^97 + 147 * q^98

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 $

$2$	3.56155	1.25920	0.629600	−	0.776920i	$-0.283219\pi$
$2$	3.56155	1.25920	0.629600	+	0.776920i	$0.283219\pi$
$3$	0	0
$3$	0	0
$4$	4.68466	0.585582
$5$	15.6847	1.40288	0.701439	−	0.712729i	$-0.252541\pi$
$5$	15.6847	1.40288	0.701439	+	0.712729i	$0.252541\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−11.8078	−0.521834
$9$	0	0
$10$	55.8617	1.76650
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	52.9157	1.12894	0.564468	−	0.825455i	$-0.309081\pi$
$13$	52.9157	1.12894	0.564468	+	0.825455i	$0.309081\pi$
$14$	24.9309	0.475933
$15$	0	0
$16$	−79.5312	−1.24268
$17$	77.1231	1.10030	0.550150	−	0.835066i	$-0.314571\pi$
$17$	77.1231	1.10030	0.550150	+	0.835066i	$0.314571\pi$
$18$	0	0
$19$	13.4081	0.161897	0.0809484	−	0.996718i	$-0.474205\pi$
$19$	13.4081	0.161897	0.0809484	+	0.996718i	$0.474205\pi$
$20$	73.4773	0.821501
$21$	0	0
$22$	−39.1771	−0.379663
$23$	59.0152	0.535022	0.267511	−	0.963555i	$-0.413799\pi$
$23$	59.0152	0.535022	0.267511	+	0.963555i	$0.413799\pi$
$24$	0	0
$25$	121.009	0.968068
$26$	188.462	1.42156
$27$	0	0
$28$	32.7926	0.221329
$29$	69.3305	0.443943	0.221972	−	0.975053i	$-0.428751\pi$
$29$	69.3305	0.443943	0.221972	+	0.975053i	$0.428751\pi$
$30$	0	0
$31$	75.9697	0.440147	0.220074	−	0.975483i	$-0.429370\pi$
$31$	75.9697	0.440147	0.220074	+	0.975483i	$0.429370\pi$
$32$	−188.793	−1.04294
$33$	0	0
$34$	274.678	1.38550
$35$	109.793	0.530238
$36$	0	0
$37$	−335.948	−1.49269	−0.746344	−	0.665560i	$-0.768193\pi$
$37$	−335.948	−1.49269	−0.746344	+	0.665560i	$0.768193\pi$
$38$	47.7538	0.203860
$39$	0	0
$40$	−185.201	−0.732070
$41$	318.617	1.21365	0.606825	−	0.794835i	$-0.292443\pi$
$41$	318.617	1.21365	0.606825	+	0.794835i	$0.292443\pi$
$42$	0	0
$43$	−57.2614	−0.203076	−0.101538	−	0.994832i	$-0.532376\pi$
$43$	−57.2614	−0.203076	−0.101538	+	0.994832i	$0.532376\pi$
$44$	−51.5312	−0.176560
$45$	0	0
$46$	210.186	0.673699
$47$	577.779	1.79314	0.896572	−	0.442898i	$-0.146050\pi$
$47$	577.779	1.79314	0.896572	+	0.442898i	$0.146050\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	430.978	1.21899
$51$	0	0
$52$	247.892	0.661085
$53$	−315.555	−0.817826	−0.408913	−	0.912573i	$-0.634092\pi$
$53$	−315.555	−0.817826	−0.408913	+	0.912573i	$0.634092\pi$
$54$	0	0
$55$	−172.531	−0.422984
$56$	−82.6543	−0.197235
$57$	0	0
$58$	246.924	0.559013
$59$	598.717	1.32112	0.660562	−	0.750772i	$-0.270318\pi$
$59$	598.717	1.32112	0.660562	+	0.750772i	$0.270318\pi$
$60$	0	0
$61$	−337.879	−0.709196	−0.354598	−	0.935019i	$-0.615382\pi$
$61$	−337.879	−0.709196	−0.354598	+	0.935019i	$0.615382\pi$
$62$	270.570	0.554233
$63$	0	0
$64$	−36.1449	−0.0705955
$65$	829.965	1.58376
$66$	0	0
$67$	−107.779	−0.196527	−0.0982637	−	0.995160i	$-0.531329\pi$
$67$	−107.779	−0.196527	−0.0982637	+	0.995160i	$0.531329\pi$
$68$	361.295	0.644316
$69$	0	0
$70$	391.032	0.667675
$71$	−405.390	−0.677619	−0.338810	−	0.940855i	$-0.610024\pi$
$71$	−405.390	−0.677619	−0.338810	+	0.940855i	$0.610024\pi$
$72$	0	0
$73$	−133.300	−0.213721	−0.106860	−	0.994274i	$-0.534080\pi$
$73$	−133.300	−0.213721	−0.106860	+	0.994274i	$0.534080\pi$
$74$	−1196.50	−1.87959
$75$	0	0
$76$	62.8125	0.0948039
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−922.038	−1.31313	−0.656566	−	0.754269i	$-0.727991\pi$
$79$	−922.038	−1.31313	−0.656566	+	0.754269i	$0.727991\pi$
$80$	−1247.42	−1.74332
$81$	0	0
$82$	1134.77	1.52823
$83$	1221.17	1.61495	0.807475	−	0.589902i	$-0.200833\pi$
$83$	1221.17	1.61495	0.807475	+	0.589902i	$0.200833\pi$
$84$	0	0
$85$	1209.65	1.54359
$86$	−203.939	−0.255713
$87$	0	0
$88$	129.885	0.157339
$89$	−1580.19	−1.88202	−0.941012	−	0.338373i	$-0.890123\pi$
$89$	−1580.19	−1.88202	−0.941012	+	0.338373i	$0.890123\pi$
$90$	0	0
$91$	370.410	0.426698
$92$	276.466	0.313300
$93$	0	0
$94$	2057.79	2.25793
$95$	210.302	0.227121
$96$	0	0
$97$	−287.097	−0.300518	−0.150259	−	0.988647i	$-0.548011\pi$
$97$	−287.097	−0.300518	−0.150259	+	0.988647i	$0.548011\pi$
$98$	174.516	0.179886
$99$	0	0
$100$	566.884	0.566884
$101$	−780.867	−0.769299	−0.384650	−	0.923063i	$-0.625678\pi$
$101$	−780.867	−0.769299	−0.384650	+	0.923063i	$0.625678\pi$
$102$	0	0
$103$	1218.22	1.16538	0.582691	−	0.812694i	$-0.302000\pi$
$103$	1218.22	1.16538	0.582691	+	0.812694i	$0.302000\pi$
$104$	−624.816	−0.589118
$105$	0	0
$106$	−1123.87	−1.02981
$107$	873.196	0.788926	0.394463	−	0.918912i	$-0.370931\pi$
$107$	873.196	0.788926	0.394463	+	0.918912i	$0.370931\pi$
$108$	0	0
$109$	126.424	0.111094	0.0555470	−	0.998456i	$-0.482310\pi$
$109$	126.424	0.111094	0.0555470	+	0.998456i	$0.482310\pi$
$110$	−614.479	−0.532621
$111$	0	0
$112$	−556.719	−0.469687
$113$	−257.318	−0.214217	−0.107108	−	0.994247i	$-0.534159\pi$
$113$	−257.318	−0.214217	−0.107108	+	0.994247i	$0.534159\pi$
$114$	0	0
$115$	925.633	0.750571
$116$	324.790	0.259965
$117$	0	0
$118$	2132.36	1.66356
$119$	539.862	0.415874
$120$	0	0
$121$	121.000	0.0909091
$122$	−1203.37	−0.893019
$123$	0	0
$124$	355.892	0.257742
$125$	−62.6052	−0.0447966
$126$	0	0
$127$	74.9450	0.0523645	0.0261822	−	0.999657i	$-0.491665\pi$
$127$	74.9450	0.0523645	0.0261822	+	0.999657i	$0.491665\pi$
$128$	1381.61	0.954048
$129$	0	0
$130$	2955.96	1.99427
$131$	2636.30	1.75828	0.879141	−	0.476561i	$-0.158117\pi$
$131$	2636.30	1.75828	0.879141	+	0.476561i	$0.158117\pi$
$132$	0	0
$133$	93.8570	0.0611912
$134$	−383.862	−0.247467
$135$	0	0
$136$	−910.651	−0.574174
$137$	−1146.87	−0.715210	−0.357605	−	0.933873i	$-0.616407\pi$
$137$	−1146.87	−0.715210	−0.357605	+	0.933873i	$0.616407\pi$
$138$	0	0
$139$	−1013.00	−0.618139	−0.309070	−	0.951039i	$-0.600018\pi$
$139$	−1013.00	−0.618139	−0.309070	+	0.951039i	$0.600018\pi$
$140$	514.341	0.310498
$141$	0	0
$142$	−1443.82	−0.853257
$143$	−582.073	−0.340387
$144$	0	0
$145$	1087.43	0.622798
$146$	−474.756	−0.269117
$147$	0	0
$148$	−1573.80	−0.874092
$149$	2236.91	1.22990	0.614948	−	0.788568i	$-0.289177\pi$
$149$	2236.91	1.22990	0.614948	+	0.788568i	$0.289177\pi$
$150$	0	0
$151$	636.657	0.343115	0.171558	−	0.985174i	$-0.445120\pi$
$151$	636.657	0.343115	0.171558	+	0.985174i	$0.445120\pi$
$152$	−158.320	−0.0844833
$153$	0	0
$154$	−274.240	−0.143499
$155$	1191.56	0.617473
$156$	0	0
$157$	−3561.59	−1.81048	−0.905241	−	0.424899i	$-0.860310\pi$
$157$	−3561.59	−1.81048	−0.905241	+	0.424899i	$0.860310\pi$
$158$	−3283.89	−1.65349
$159$	0	0
$160$	−2961.15	−1.46312
$161$	413.106	0.202219
$162$	0	0
$163$	2540.60	1.22083	0.610414	−	0.792082i	$-0.291003\pi$
$163$	2540.60	1.22083	0.610414	+	0.792082i	$0.291003\pi$
$164$	1492.61	0.710692
$165$	0	0
$166$	4349.26	2.03354
$167$	−1737.09	−0.804913	−0.402456	−	0.915439i	$-0.631843\pi$
$167$	−1737.09	−0.804913	−0.402456	+	0.915439i	$0.631843\pi$
$168$	0	0
$169$	603.073	0.274498
$170$	4308.23	1.94368
$171$	0	0
$172$	−268.250	−0.118918
$173$	−4194.96	−1.84357	−0.921784	−	0.387704i	$-0.873268\pi$
$173$	−4194.96	−1.84357	−0.921784	+	0.387704i	$0.873268\pi$
$174$	0	0
$175$	847.060	0.365895
$176$	874.844	0.374681
$177$	0	0
$178$	−5627.94	−2.36984
$179$	−1349.85	−0.563645	−0.281822	−	0.959467i	$-0.590939\pi$
$179$	−1349.85	−0.563645	−0.281822	+	0.959467i	$0.590939\pi$
$180$	0	0
$181$	−1198.06	−0.491994	−0.245997	−	0.969271i	$-0.579115\pi$
$181$	−1198.06	−0.491994	−0.245997	+	0.969271i	$0.579115\pi$
$182$	1319.23	0.537298
$183$	0	0
$184$	−696.837	−0.279193
$185$	−5269.23	−2.09406
$186$	0	0
$187$	−848.354	−0.331753
$188$	2706.70	1.05003
$189$	0	0
$190$	749.002	0.285991
$191$	−764.873	−0.289761	−0.144880	−	0.989449i	$-0.546280\pi$
$191$	−764.873	−0.289761	−0.144880	+	0.989449i	$0.546280\pi$
$192$	0	0
$193$	−2019.52	−0.753204	−0.376602	−	0.926375i	$-0.622907\pi$
$193$	−2019.52	−0.753204	−0.376602	+	0.926375i	$0.622907\pi$
$194$	−1022.51	−0.378412
$195$	0	0
$196$	229.548	0.0836546
$197$	−1499.94	−0.542470	−0.271235	−	0.962513i	$-0.587432\pi$
$197$	−1499.94	−0.542470	−0.271235	+	0.962513i	$0.587432\pi$
$198$	0	0
$199$	−4574.41	−1.62950	−0.814752	−	0.579809i	$-0.803127\pi$
$199$	−4574.41	−1.62950	−0.814752	+	0.579809i	$0.803127\pi$
$200$	−1428.84	−0.505171
$201$	0	0
$202$	−2781.10	−0.968701
$203$	485.313	0.167795
$204$	0	0
$205$	4997.40	1.70260
$206$	4338.74	1.46745
$207$	0	0
$208$	−4208.45	−1.40290
$209$	−147.490	−0.0488137
$210$	0	0
$211$	−4456.96	−1.45417	−0.727084	−	0.686548i	$-0.759125\pi$
$211$	−4456.96	−1.45417	−0.727084	+	0.686548i	$0.759125\pi$
$212$	−1478.27	−0.478905
$213$	0	0
$214$	3109.93	0.993414
$215$	−898.125	−0.284891
$216$	0	0
$217$	531.788	0.166360
$218$	450.267	0.139890
$219$	0	0
$220$	−808.250	−0.247692
$221$	4081.02	1.24217
$222$	0	0
$223$	−3003.50	−0.901924	−0.450962	−	0.892543i	$-0.648919\pi$
$223$	−3003.50	−0.901924	−0.450962	+	0.892543i	$0.648919\pi$
$224$	−1321.55	−0.394195
$225$	0	0
$226$	−916.453	−0.269741
$227$	−1252.20	−0.366130	−0.183065	−	0.983101i	$-0.558602\pi$
$227$	−1252.20	−0.366130	−0.183065	+	0.983101i	$0.558602\pi$
$228$	0	0
$229$	2862.80	0.826110	0.413055	−	0.910706i	$-0.364462\pi$
$229$	2862.80	0.826110	0.413055	+	0.910706i	$0.364462\pi$
$230$	3296.69	0.945118
$231$	0	0
$232$	−818.638	−0.231665
$233$	−4969.92	−1.39738	−0.698692	−	0.715423i	$-0.746234\pi$
$233$	−4969.92	−1.39738	−0.698692	+	0.715423i	$0.746234\pi$
$234$	0	0
$235$	9062.27	2.51556
$236$	2804.78	0.773627
$237$	0	0
$238$	1922.75	0.523669
$239$	−4602.18	−1.24557	−0.622783	−	0.782395i	$-0.713998\pi$
$239$	−4602.18	−1.24557	−0.622783	+	0.782395i	$0.713998\pi$
$240$	0	0
$241$	−4147.29	−1.10851	−0.554254	−	0.832348i	$-0.686996\pi$
$241$	−4147.29	−1.10851	−0.554254	+	0.832348i	$0.686996\pi$
$242$	430.948	0.114473
$243$	0	0
$244$	−1582.85	−0.415292
$245$	768.548	0.200411
$246$	0	0
$247$	709.501	0.182771
$248$	−897.032	−0.229684
$249$	0	0
$250$	−222.972	−0.0564078
$251$	−988.383	−0.248551	−0.124275	−	0.992248i	$-0.539661\pi$
$251$	−988.383	−0.248551	−0.124275	+	0.992248i	$0.539661\pi$
$252$	0	0
$253$	−649.167	−0.161315
$254$	266.920	0.0659373
$255$	0	0
$256$	5209.83	1.27193
$257$	2856.10	0.693224	0.346612	−	0.938009i	$-0.387332\pi$
$257$	2856.10	0.693224	0.346612	+	0.938009i	$0.387332\pi$
$258$	0	0
$259$	−2351.64	−0.564183
$260$	3888.10	0.927423
$261$	0	0
$262$	9389.34	2.21403
$263$	−5772.45	−1.35340	−0.676701	−	0.736258i	$-0.736591\pi$
$263$	−5772.45	−1.35340	−0.676701	+	0.736258i	$0.736591\pi$
$264$	0	0
$265$	−4949.37	−1.14731
$266$	334.277	0.0770519
$267$	0	0
$268$	−504.909	−0.115083
$269$	1626.04	0.368555	0.184278	−	0.982874i	$-0.441005\pi$
$269$	1626.04	0.368555	0.184278	+	0.982874i	$0.441005\pi$
$270$	0	0
$271$	−1816.20	−0.407109	−0.203554	−	0.979064i	$-0.565249\pi$
$271$	−1816.20	−0.407109	−0.203554	+	0.979064i	$0.565249\pi$
$272$	−6133.70	−1.36732
$273$	0	0
$274$	−4084.64	−0.900592
$275$	−1331.09	−0.291884
$276$	0	0
$277$	−8091.98	−1.75523	−0.877617	−	0.479362i	$-0.840868\pi$
$277$	−8091.98	−1.75523	−0.877617	+	0.479362i	$0.840868\pi$
$278$	−3607.85	−0.778361
$279$	0	0
$280$	−1296.41	−0.276697
$281$	8041.32	1.70713	0.853567	−	0.520982i	$-0.174434\pi$
$281$	8041.32	1.70713	0.853567	+	0.520982i	$0.174434\pi$
$282$	0	0
$283$	644.465	0.135369	0.0676846	−	0.997707i	$-0.478439\pi$
$283$	644.465	0.135369	0.0676846	+	0.997707i	$0.478439\pi$
$284$	−1899.11	−0.396802
$285$	0	0
$286$	−2073.08	−0.428615
$287$	2230.32	0.458717
$288$	0	0
$289$	1034.97	0.210660
$290$	3872.92	0.784227
$291$	0	0
$292$	−624.466	−0.125151
$293$	6451.96	1.28644	0.643221	−	0.765681i	$-0.277598\pi$
$293$	6451.96	1.28644	0.643221	+	0.765681i	$0.277598\pi$
$294$	0	0
$295$	9390.67	1.85338
$296$	3966.79	0.778936
$297$	0	0
$298$	7966.86	1.54868
$299$	3122.83	0.604006
$300$	0	0
$301$	−400.830	−0.0767556
$302$	2267.49	0.432051
$303$	0	0
$304$	−1066.37	−0.201185
$305$	−5299.51	−0.994916
$306$	0	0
$307$	4175.92	0.776327	0.388164	−	0.921590i	$-0.373110\pi$
$307$	4175.92	0.776327	0.388164	+	0.921590i	$0.373110\pi$
$308$	−360.719	−0.0667333
$309$	0	0
$310$	4243.80	0.777521
$311$	8619.22	1.57155	0.785774	−	0.618514i	$-0.212265\pi$
$311$	8619.22	1.57155	0.785774	+	0.618514i	$0.212265\pi$
$312$	0	0
$313$	−10300.8	−1.86019	−0.930093	−	0.367324i	$-0.880274\pi$
$313$	−10300.8	−1.86019	−0.930093	+	0.367324i	$0.880274\pi$
$314$	−12684.8	−2.27976
$315$	0	0
$316$	−4319.43	−0.768946
$317$	−7696.26	−1.36361	−0.681806	−	0.731533i	$-0.738805\pi$
$317$	−7696.26	−1.36361	−0.681806	+	0.731533i	$0.738805\pi$
$318$	0	0
$319$	−762.635	−0.133854
$320$	−566.920	−0.0990369
$321$	0	0
$322$	1471.30	0.254634
$323$	1034.08	0.178135
$324$	0	0
$325$	6403.25	1.09289
$326$	9048.48	1.53727
$327$	0	0
$328$	−3762.16	−0.633325
$329$	4044.46	0.677745
$330$	0	0
$331$	6322.41	1.04988	0.524941	−	0.851138i	$-0.324087\pi$
$331$	6322.41	1.04988	0.524941	+	0.851138i	$0.324087\pi$
$332$	5720.77	0.945686
$333$	0	0
$334$	−6186.75	−1.01355
$335$	−1690.48	−0.275704
$336$	0	0
$337$	1576.06	0.254758	0.127379	−	0.991854i	$-0.459344\pi$
$337$	1576.06	0.254758	0.127379	+	0.991854i	$0.459344\pi$
$338$	2147.88	0.345648
$339$	0	0
$340$	5666.80	0.903897
$341$	−835.667	−0.132709
$342$	0	0
$343$	343.000	0.0539949
$344$	676.129	0.105972
$345$	0	0
$346$	−14940.6	−2.32142
$347$	−4989.29	−0.771870	−0.385935	−	0.922526i	$-0.626121\pi$
$347$	−4989.29	−0.771870	−0.385935	+	0.922526i	$0.626121\pi$
$348$	0	0
$349$	4348.98	0.667035	0.333518	−	0.942744i	$-0.391764\pi$
$349$	4348.98	0.667035	0.333518	+	0.942744i	$0.391764\pi$
$350$	3016.85	0.460735
$351$	0	0
$352$	2076.72	0.314459
$353$	3724.71	0.561604	0.280802	−	0.959766i	$-0.409399\pi$
$353$	3724.71	0.561604	0.280802	+	0.959766i	$0.409399\pi$
$354$	0	0
$355$	−6358.40	−0.950617
$356$	−7402.66	−1.10208
$357$	0	0
$358$	−4807.56	−0.709741
$359$	5815.53	0.854963	0.427481	−	0.904024i	$-0.359401\pi$
$359$	5815.53	0.854963	0.427481	+	0.904024i	$0.359401\pi$
$360$	0	0
$361$	−6679.22	−0.973789
$362$	−4266.94	−0.619518
$363$	0	0
$364$	1735.24	0.249867
$365$	−2090.77	−0.299824
$366$	0	0
$367$	9298.43	1.32255	0.661273	−	0.750146i	$-0.270017\pi$
$367$	9298.43	1.32255	0.661273	+	0.750146i	$0.270017\pi$
$368$	−4693.55	−0.664859
$369$	0	0
$370$	−18766.6	−2.63684
$371$	−2208.88	−0.309109
$372$	0	0
$373$	3697.85	0.513318	0.256659	−	0.966502i	$-0.417378\pi$
$373$	3697.85	0.513318	0.256659	+	0.966502i	$0.417378\pi$
$374$	−3021.46	−0.417743
$375$	0	0
$376$	−6822.28	−0.935724
$377$	3668.67	0.501184
$378$	0	0
$379$	2040.70	0.276580	0.138290	−	0.990392i	$-0.455839\pi$
$379$	2040.70	0.276580	0.138290	+	0.990392i	$0.455839\pi$
$380$	985.193	0.132998
$381$	0	0
$382$	−2724.14	−0.364866
$383$	3311.92	0.441857	0.220928	−	0.975290i	$-0.429091\pi$
$383$	3311.92	0.441857	0.220928	+	0.975290i	$0.429091\pi$
$384$	0	0
$385$	−1207.72	−0.159873
$386$	−7192.63	−0.948433
$387$	0	0
$388$	−1344.95	−0.175978
$389$	−13095.1	−1.70680	−0.853401	−	0.521255i	$-0.825464\pi$
$389$	−13095.1	−1.70680	−0.853401	+	0.521255i	$0.825464\pi$
$390$	0	0
$391$	4551.43	0.588685
$392$	−578.580	−0.0745478
$393$	0	0
$394$	−5342.13	−0.683077
$395$	−14461.8	−1.84216
$396$	0	0
$397$	15359.1	1.94169	0.970843	−	0.239716i	$-0.0770543\pi$
$397$	15359.1	1.94169	0.970843	+	0.239716i	$0.0770543\pi$
$398$	−16292.0	−2.05187
$399$	0	0
$400$	−9623.96	−1.20299
$401$	11767.6	1.46545	0.732726	−	0.680524i	$-0.238248\pi$
$401$	11767.6	1.46545	0.732726	+	0.680524i	$0.238248\pi$
$402$	0	0
$403$	4019.99	0.496898
$404$	−3658.10	−0.450488
$405$	0	0
$406$	1728.47	0.211287
$407$	3695.43	0.450063
$408$	0	0
$409$	5071.33	0.613108	0.306554	−	0.951853i	$-0.400824\pi$
$409$	5071.33	0.613108	0.306554	+	0.951853i	$0.400824\pi$
$410$	17798.5	2.14392
$411$	0	0
$412$	5706.93	0.682427
$413$	4191.02	0.499338
$414$	0	0
$415$	19153.6	2.26558
$416$	−9990.10	−1.17742
$417$	0	0
$418$	−525.292	−0.0614662
$419$	−5085.27	−0.592916	−0.296458	−	0.955046i	$-0.595806\pi$
$419$	−5085.27	−0.592916	−0.296458	+	0.955046i	$0.595806\pi$
$420$	0	0
$421$	15655.8	1.81239	0.906196	−	0.422859i	$-0.138973\pi$
$421$	15655.8	1.81239	0.906196	+	0.422859i	$0.138973\pi$
$422$	−15873.7	−1.83109
$423$	0	0
$424$	3726.00	0.426770
$425$	9332.55	1.06517
$426$	0	0
$427$	−2365.15	−0.268051
$428$	4090.62	0.461981
$429$	0	0
$430$	−3198.72	−0.358735
$431$	−5775.20	−0.645433	−0.322717	−	0.946496i	$-0.604596\pi$
$431$	−5775.20	−0.645433	−0.322717	+	0.946496i	$0.604596\pi$
$432$	0	0
$433$	−6756.97	−0.749929	−0.374964	−	0.927039i	$-0.622345\pi$
$433$	−6756.97	−0.749929	−0.374964	+	0.927039i	$0.622345\pi$
$434$	1893.99	0.209480
$435$	0	0
$436$	592.255	0.0650547
$437$	791.283	0.0866183
$438$	0	0
$439$	−9068.21	−0.985882	−0.492941	−	0.870063i	$-0.664078\pi$
$439$	−9068.21	−0.985882	−0.492941	+	0.870063i	$0.664078\pi$
$440$	2037.21	0.220727
$441$	0	0
$442$	14534.8	1.56414
$443$	7095.86	0.761026	0.380513	−	0.924776i	$-0.375748\pi$
$443$	7095.86	0.761026	0.380513	+	0.924776i	$0.375748\pi$
$444$	0	0
$445$	−24784.8	−2.64025
$446$	−10697.1	−1.13570
$447$	0	0
$448$	−253.014	−0.0266826
$449$	−630.680	−0.0662887	−0.0331443	−	0.999451i	$-0.510552\pi$
$449$	−630.680	−0.0662887	−0.0331443	+	0.999451i	$0.510552\pi$
$450$	0	0
$451$	−3504.79	−0.365929
$452$	−1205.45	−0.125441
$453$	0	0
$454$	−4459.78	−0.461030
$455$	5809.75	0.598605
$456$	0	0
$457$	12432.5	1.27258	0.636290	−	0.771450i	$-0.280468\pi$
$457$	12432.5	1.27258	0.636290	+	0.771450i	$0.280468\pi$
$458$	10196.0	1.04024
$459$	0	0
$460$	4336.27	0.439521
$461$	−13057.3	−1.31918	−0.659588	−	0.751627i	$-0.729269\pi$
$461$	−13057.3	−1.31918	−0.659588	+	0.751627i	$0.729269\pi$
$462$	0	0
$463$	8608.12	0.864046	0.432023	−	0.901863i	$-0.357800\pi$
$463$	8608.12	0.864046	0.432023	+	0.901863i	$0.357800\pi$
$464$	−5513.94	−0.551677
$465$	0	0
$466$	−17700.6	−1.75958
$467$	−14997.6	−1.48610	−0.743049	−	0.669237i	$-0.766621\pi$
$467$	−14997.6	−1.48610	−0.743049	+	0.669237i	$0.766621\pi$
$468$	0	0
$469$	−754.455	−0.0742804
$470$	32275.8	3.16760
$471$	0	0
$472$	−7069.51	−0.689408
$473$	629.875	0.0612298
$474$	0	0
$475$	1622.50	0.156727
$476$	2529.07	0.243529
$477$	0	0
$478$	−16390.9	−1.56841
$479$	20698.8	1.97443	0.987216	−	0.159386i	$-0.0509515\pi$
$479$	20698.8	1.97443	0.987216	+	0.159386i	$0.0509515\pi$
$480$	0	0
$481$	−17776.9	−1.68515
$482$	−14770.8	−1.39583
$483$	0	0
$484$	566.844	0.0532348
$485$	−4503.01	−0.421590
$486$	0	0
$487$	15390.1	1.43202	0.716008	−	0.698092i	$-0.245968\pi$
$487$	15390.1	1.43202	0.716008	+	0.698092i	$0.245968\pi$
$488$	3989.59	0.370083
$489$	0	0
$490$	2737.23	0.252358
$491$	−10771.4	−0.990030	−0.495015	−	0.868884i	$-0.664837\pi$
$491$	−10771.4	−0.990030	−0.495015	+	0.868884i	$0.664837\pi$
$492$	0	0
$493$	5346.98	0.488471
$494$	2526.93	0.230145
$495$	0	0
$496$	−6041.96	−0.546960
$497$	−2837.73	−0.256116
$498$	0	0
$499$	−5999.79	−0.538251	−0.269126	−	0.963105i	$-0.586735\pi$
$499$	−5999.79	−0.538251	−0.269126	+	0.963105i	$0.586735\pi$
$500$	−293.284	−0.0262321
$501$	0	0
$502$	−3520.18	−0.312975
$503$	−17610.5	−1.56106	−0.780528	−	0.625121i	$-0.785050\pi$
$503$	−17610.5	−1.56106	−0.780528	+	0.625121i	$0.785050\pi$
$504$	0	0
$505$	−12247.6	−1.07923
$506$	−2312.04	−0.203128
$507$	0	0
$508$	351.092	0.0306637
$509$	5920.42	0.515556	0.257778	−	0.966204i	$-0.417010\pi$
$509$	5920.42	0.515556	0.257778	+	0.966204i	$0.417010\pi$
$510$	0	0
$511$	−933.101	−0.0807788
$512$	7502.22	0.647567
$513$	0	0
$514$	10172.1	0.872907
$515$	19107.3	1.63489
$516$	0	0
$517$	−6355.57	−0.540653
$518$	−8375.47	−0.710419
$519$	0	0
$520$	−9800.03	−0.826461
$521$	15886.2	1.33587	0.667935	−	0.744220i	$-0.267179\pi$
$521$	15886.2	1.33587	0.667935	+	0.744220i	$0.267179\pi$
$522$	0	0
$523$	1728.58	0.144523	0.0722613	−	0.997386i	$-0.476978\pi$
$523$	1728.58	0.144523	0.0722613	+	0.997386i	$0.476978\pi$
$524$	12350.2	1.02962
$525$	0	0
$526$	−20558.9	−1.70420
$527$	5859.02	0.484294
$528$	0	0
$529$	−8684.21	−0.713751
$530$	−17627.4	−1.44469
$531$	0	0
$532$	439.688	0.0358325
$533$	16859.9	1.37013
$534$	0	0
$535$	13695.8	1.10677
$536$	1272.63	0.102555
$537$	0	0
$538$	5791.23	0.464085
$539$	−539.000	−0.0430730
$540$	0	0
$541$	1941.99	0.154331	0.0771653	−	0.997018i	$-0.475413\pi$
$541$	1941.99	0.154331	0.0771653	+	0.997018i	$0.475413\pi$
$542$	−6468.51	−0.512631
$543$	0	0
$544$	−14560.3	−1.14755
$545$	1982.92	0.155852
$546$	0	0
$547$	−1450.56	−0.113384	−0.0566922	−	0.998392i	$-0.518055\pi$
$547$	−1450.56	−0.113384	−0.0566922	+	0.998392i	$0.518055\pi$
$548$	−5372.70	−0.418814
$549$	0	0
$550$	−4740.76	−0.367539
$551$	929.593	0.0718729
$552$	0	0
$553$	−6454.26	−0.496317
$554$	−28820.0	−2.21019
$555$	0	0
$556$	−4745.55	−0.361972
$557$	−20821.8	−1.58393	−0.791965	−	0.610567i	$-0.790942\pi$
$557$	−20821.8	−1.58393	−0.791965	+	0.610567i	$0.790942\pi$
$558$	0	0
$559$	−3030.03	−0.229260
$560$	−8731.94	−0.658914
$561$	0	0
$562$	28639.6	2.14962
$563$	3694.11	0.276533	0.138267	−	0.990395i	$-0.455847\pi$
$563$	3694.11	0.276533	0.138267	+	0.990395i	$0.455847\pi$
$564$	0	0
$565$	−4035.95	−0.300520
$566$	2295.30	0.170457
$567$	0	0
$568$	4786.75	0.353605
$569$	16649.2	1.22666	0.613332	−	0.789825i	$-0.289829\pi$
$569$	16649.2	1.22666	0.613332	+	0.789825i	$0.289829\pi$
$570$	0	0
$571$	546.005	0.0400168	0.0200084	−	0.999800i	$-0.493631\pi$
$571$	546.005	0.0400168	0.0200084	+	0.999800i	$0.493631\pi$
$572$	−2726.81	−0.199325
$573$	0	0
$574$	7943.41	0.577616
$575$	7141.34	0.517938
$576$	0	0
$577$	7910.36	0.570732	0.285366	−	0.958419i	$-0.407885\pi$
$577$	7910.36	0.570732	0.285366	+	0.958419i	$0.407885\pi$
$578$	3686.11	0.265263
$579$	0	0
$580$	5094.22	0.364700
$581$	8548.19	0.610394
$582$	0	0
$583$	3471.10	0.246584
$584$	1573.98	0.111527
$585$	0	0
$586$	22979.0	1.61989
$587$	744.718	0.0523642	0.0261821	−	0.999657i	$-0.491665\pi$
$587$	744.718	0.0523642	0.0261821	+	0.999657i	$0.491665\pi$
$588$	0	0
$589$	1018.61	0.0712584
$590$	33445.4	2.33377
$591$	0	0
$592$	26718.4	1.85493
$593$	8167.66	0.565608	0.282804	−	0.959178i	$-0.408735\pi$
$593$	8167.66	0.565608	0.282804	+	0.959178i	$0.408735\pi$
$594$	0	0
$595$	8467.55	0.583421
$596$	10479.1	0.720205
$597$	0	0
$598$	11122.1	0.760564
$599$	−12115.4	−0.826413	−0.413206	−	0.910637i	$-0.635591\pi$
$599$	−12115.4	−0.826413	−0.413206	+	0.910637i	$0.635591\pi$
$600$	0	0
$601$	13606.0	0.923458	0.461729	−	0.887021i	$-0.347229\pi$
$601$	13606.0	0.923458	0.461729	+	0.887021i	$0.347229\pi$
$602$	−1427.58	−0.0966506
$603$	0	0
$604$	2982.52	0.200922
$605$	1897.84	0.127534
$606$	0	0
$607$	−11404.4	−0.762585	−0.381293	−	0.924454i	$-0.624521\pi$
$607$	−11404.4	−0.762585	−0.381293	+	0.924454i	$0.624521\pi$
$608$	−2531.36	−0.168849
$609$	0	0
$610$	−18874.5	−1.25280
$611$	30573.6	2.02435
$612$	0	0
$613$	15266.2	1.00587	0.502934	−	0.864325i	$-0.332254\pi$
$613$	15266.2	1.00587	0.502934	+	0.864325i	$0.332254\pi$
$614$	14872.8	0.977550
$615$	0	0
$616$	909.198	0.0594685
$617$	−23638.9	−1.54241	−0.771205	−	0.636586i	$-0.780346\pi$
$617$	−23638.9	−1.54241	−0.771205	+	0.636586i	$0.780346\pi$
$618$	0	0
$619$	27681.1	1.79741	0.898706	−	0.438551i	$-0.144508\pi$
$619$	27681.1	1.79741	0.898706	+	0.438551i	$0.144508\pi$
$620$	5582.05	0.361581
$621$	0	0
$622$	30697.8	1.97889
$623$	−11061.4	−0.711338
$624$	0	0
$625$	−16108.0	−1.03091
$626$	−36687.0	−2.34235
$627$	0	0
$628$	−16684.8	−1.06019
$629$	−25909.3	−1.64241
$630$	0	0
$631$	18208.7	1.14877	0.574386	−	0.818585i	$-0.305241\pi$
$631$	18208.7	1.14877	0.574386	+	0.818585i	$0.305241\pi$
$632$	10887.2	0.685237
$633$	0	0
$634$	−27410.6	−1.71706
$635$	1175.49	0.0734610
$636$	0	0
$637$	2592.87	0.161277
$638$	−2716.17	−0.168549
$639$	0	0
$640$	21670.1	1.33841
$641$	23373.8	1.44026	0.720131	−	0.693838i	$-0.244082\pi$
$641$	23373.8	1.44026	0.720131	+	0.693838i	$0.244082\pi$
$642$	0	0
$643$	6742.72	0.413541	0.206770	−	0.978389i	$-0.433705\pi$
$643$	6742.72	0.413541	0.206770	+	0.978389i	$0.433705\pi$
$644$	1935.26	0.118416
$645$	0	0
$646$	3682.92	0.224307
$647$	17592.4	1.06898	0.534490	−	0.845175i	$-0.320504\pi$
$647$	17592.4	1.06898	0.534490	+	0.845175i	$0.320504\pi$
$648$	0	0
$649$	−6585.89	−0.398334
$650$	22805.5	1.37616
$651$	0	0
$652$	11901.8	0.714896
$653$	−20074.0	−1.20300	−0.601498	−	0.798875i	$-0.705429\pi$
$653$	−20074.0	−1.20300	−0.601498	+	0.798875i	$0.705429\pi$
$654$	0	0
$655$	41349.5	2.46666
$656$	−25340.0	−1.50817
$657$	0	0
$658$	14404.5	0.853416
$659$	13778.4	0.814460	0.407230	−	0.913326i	$-0.366495\pi$
$659$	13778.4	0.814460	0.407230	+	0.913326i	$0.366495\pi$
$660$	0	0
$661$	−7724.16	−0.454516	−0.227258	−	0.973835i	$-0.572976\pi$
$661$	−7724.16	−0.454516	−0.227258	+	0.973835i	$0.572976\pi$
$662$	22517.6	1.32201
$663$	0	0
$664$	−14419.3	−0.842737
$665$	1472.11	0.0858438
$666$	0	0
$667$	4091.55	0.237519
$668$	−8137.69	−0.471343
$669$	0	0
$670$	−6020.74	−0.347166
$671$	3716.67	0.213831
$672$	0	0
$673$	−18081.2	−1.03563	−0.517816	−	0.855492i	$-0.673255\pi$
$673$	−18081.2	−1.03563	−0.517816	+	0.855492i	$0.673255\pi$
$674$	5613.21	0.320791
$675$	0	0
$676$	2825.19	0.160741
$677$	23199.8	1.31705	0.658523	−	0.752561i	$-0.271182\pi$
$677$	23199.8	1.31705	0.658523	+	0.752561i	$0.271182\pi$
$678$	0	0
$679$	−2009.68	−0.113585
$680$	−14283.3	−0.805497
$681$	0	0
$682$	−2976.27	−0.167107
$683$	−13339.6	−0.747329	−0.373665	−	0.927564i	$-0.621899\pi$
$683$	−13339.6	−0.747329	−0.373665	+	0.927564i	$0.621899\pi$
$684$	0	0
$685$	−17988.3	−1.00335
$686$	1221.61	0.0679904
$687$	0	0
$688$	4554.07	0.252358
$689$	−16697.8	−0.923274
$690$	0	0
$691$	−17221.4	−0.948097	−0.474048	−	0.880499i	$-0.657208\pi$
$691$	−17221.4	−0.948097	−0.474048	+	0.880499i	$0.657208\pi$
$692$	−19652.0	−1.07956
$693$	0	0
$694$	−17769.6	−0.971938
$695$	−15888.5	−0.867175
$696$	0	0
$697$	24572.8	1.33538
$698$	15489.1	0.839930
$699$	0	0
$700$	3968.18	0.214262
$701$	27196.5	1.46533	0.732665	−	0.680589i	$-0.238276\pi$
$701$	27196.5	1.46533	0.732665	+	0.680589i	$0.238276\pi$
$702$	0	0
$703$	−4504.44	−0.241661
$704$	397.594	0.0212853
$705$	0	0
$706$	13265.8	0.707172
$707$	−5466.07	−0.290768
$708$	0	0
$709$	22682.8	1.20151	0.600755	−	0.799434i	$-0.294867\pi$
$709$	22682.8	1.20151	0.600755	+	0.799434i	$0.294867\pi$
$710$	−22645.8	−1.19702
$711$	0	0
$712$	18658.5	0.982105
$713$	4483.36	0.235488
$714$	0	0
$715$	−9129.61	−0.477522
$716$	−6323.58	−0.330060
$717$	0	0
$718$	20712.3	1.07657
$719$	−10878.2	−0.564241	−0.282121	−	0.959379i	$-0.591038\pi$
$719$	−10878.2	−0.564241	−0.282121	+	0.959379i	$0.591038\pi$
$720$	0	0
$721$	8527.51	0.440473
$722$	−23788.4	−1.22619
$723$	0	0
$724$	−5612.49	−0.288103
$725$	8389.58	0.429767
$726$	0	0
$727$	−2793.35	−0.142503	−0.0712514	−	0.997458i	$-0.522699\pi$
$727$	−2793.35	−0.142503	−0.0712514	+	0.997458i	$0.522699\pi$
$728$	−4373.71	−0.222666
$729$	0	0
$730$	−7446.38	−0.377538
$731$	−4416.17	−0.223445
$732$	0	0
$733$	−7597.31	−0.382828	−0.191414	−	0.981509i	$-0.561307\pi$
$733$	−7597.31	−0.382828	−0.191414	+	0.981509i	$0.561307\pi$
$734$	33116.9	1.66535
$735$	0	0
$736$	−11141.6	−0.557997
$737$	1185.57	0.0592553
$738$	0	0
$739$	−14304.7	−0.712052	−0.356026	−	0.934476i	$-0.615868\pi$
$739$	−14304.7	−0.712052	−0.356026	+	0.934476i	$0.615868\pi$
$740$	−24684.5	−1.22625
$741$	0	0
$742$	−7867.06	−0.389230
$743$	−31212.2	−1.54114	−0.770569	−	0.637357i	$-0.780028\pi$
$743$	−31212.2	−1.54114	−0.770569	+	0.637357i	$0.780028\pi$
$744$	0	0
$745$	35085.1	1.72539
$746$	13170.1	0.646369
$747$	0	0
$748$	−3974.25	−0.194269
$749$	6112.37	0.298186
$750$	0	0
$751$	30743.1	1.49378	0.746891	−	0.664946i	$-0.231546\pi$
$751$	30743.1	1.49378	0.746891	+	0.664946i	$0.231546\pi$
$752$	−45951.5	−2.22830
$753$	0	0
$754$	13066.2	0.631090
$755$	9985.75	0.481349
$756$	0	0
$757$	13176.9	0.632657	0.316328	−	0.948650i	$-0.397550\pi$
$757$	13176.9	0.632657	0.316328	+	0.948650i	$0.397550\pi$
$758$	7268.07	0.348270
$759$	0	0
$760$	−2483.20	−0.118520
$761$	10696.4	0.509520	0.254760	−	0.967004i	$-0.418004\pi$
$761$	10696.4	0.509520	0.254760	+	0.967004i	$0.418004\pi$
$762$	0	0
$763$	884.970	0.0419896
$764$	−3583.17	−0.169679
$765$	0	0
$766$	11795.6	0.556385
$767$	31681.5	1.49147
$768$	0	0
$769$	−7724.46	−0.362225	−0.181113	−	0.983462i	$-0.557970\pi$
$769$	−7724.46	−0.362225	−0.181113	+	0.983462i	$0.557970\pi$
$770$	−4301.35	−0.201312
$771$	0	0
$772$	−9460.77	−0.441063
$773$	−28801.0	−1.34010	−0.670051	−	0.742315i	$-0.733728\pi$
$773$	−28801.0	−1.34010	−0.670051	+	0.742315i	$0.733728\pi$
$774$	0	0
$775$	9192.98	0.426092
$776$	3389.97	0.156821
$777$	0	0
$778$	−46638.7	−2.14920
$779$	4272.07	0.196486
$780$	0	0
$781$	4459.29	0.204310
$782$	16210.2	0.741272
$783$	0	0
$784$	−3897.03	−0.177525
$785$	−55862.3	−2.53989
$786$	0	0
$787$	−11812.0	−0.535007	−0.267504	−	0.963557i	$-0.586199\pi$
$787$	−11812.0	−0.535007	−0.267504	+	0.963557i	$0.586199\pi$
$788$	−7026.72	−0.317661
$789$	0	0
$790$	−51506.6	−2.31965
$791$	−1801.23	−0.0809662
$792$	0	0
$793$	−17879.1	−0.800637
$794$	54702.1	2.44497
$795$	0	0
$796$	−21429.6	−0.954209
$797$	26884.2	1.19484	0.597420	−	0.801928i	$-0.296192\pi$
$797$	26884.2	1.19484	0.597420	+	0.801928i	$0.296192\pi$
$798$	0	0
$799$	44560.1	1.97300
$800$	−22845.5	−1.00964
$801$	0	0
$802$	41910.9	1.84530
$803$	1466.30	0.0644392
$804$	0	0
$805$	6479.43	0.283689
$806$	14317.4	0.625694
$807$	0	0
$808$	9220.30	0.401447
$809$	18360.8	0.797936	0.398968	−	0.916965i	$-0.369368\pi$
$809$	18360.8	0.797936	0.398968	+	0.916965i	$0.369368\pi$
$810$	0	0
$811$	17033.7	0.737528	0.368764	−	0.929523i	$-0.379781\pi$
$811$	17033.7	0.737528	0.368764	+	0.929523i	$0.379781\pi$
$812$	2273.53	0.0982576
$813$	0	0
$814$	13161.5	0.566718
$815$	39848.4	1.71267
$816$	0	0
$817$	−767.768	−0.0328774
$818$	18061.8	0.772026
$819$	0	0
$820$	23411.1	0.997015
$821$	29675.1	1.26147	0.630735	−	0.775998i	$-0.282754\pi$
$821$	29675.1	1.26147	0.630735	+	0.775998i	$0.282754\pi$
$822$	0	0
$823$	−24677.2	−1.04519	−0.522596	−	0.852581i	$-0.675036\pi$
$823$	−24677.2	−1.04519	−0.522596	+	0.852581i	$0.675036\pi$
$824$	−14384.4	−0.608137
$825$	0	0
$826$	14926.5	0.628766
$827$	9168.43	0.385511	0.192755	−	0.981247i	$-0.438258\pi$
$827$	9168.43	0.385511	0.192755	+	0.981247i	$0.438258\pi$
$828$	0	0
$829$	8065.03	0.337889	0.168945	−	0.985626i	$-0.445964\pi$
$829$	8065.03	0.337889	0.168945	+	0.985626i	$0.445964\pi$
$830$	68216.7	2.85281
$831$	0	0
$832$	−1912.63	−0.0796979
$833$	3779.03	0.157186
$834$	0	0
$835$	−27245.7	−1.12919
$836$	−690.938	−0.0285844
$837$	0	0
$838$	−18111.5	−0.746599
$839$	29665.7	1.22071	0.610354	−	0.792129i	$-0.291027\pi$
$839$	29665.7	1.22071	0.610354	+	0.792129i	$0.291027\pi$
$840$	0	0
$841$	−19582.3	−0.802915
$842$	55758.9	2.28216
$843$	0	0
$844$	−20879.3	−0.851535
$845$	9458.99	0.385088
$846$	0	0
$847$	847.000	0.0343604
$848$	25096.5	1.01629
$849$	0	0
$850$	33238.4	1.34126
$851$	−19826.0	−0.798622
$852$	0	0
$853$	−33087.0	−1.32811	−0.664056	−	0.747683i	$-0.731166\pi$
$853$	−33087.0	−1.32811	−0.664056	+	0.747683i	$0.731166\pi$
$854$	−8423.61	−0.337529
$855$	0	0
$856$	−10310.5	−0.411689
$857$	8204.78	0.327036	0.163518	−	0.986540i	$-0.447716\pi$
$857$	8204.78	0.327036	0.163518	+	0.986540i	$0.447716\pi$
$858$	0	0
$859$	19406.0	0.770808	0.385404	−	0.922748i	$-0.374062\pi$
$859$	19406.0	0.770808	0.385404	+	0.922748i	$0.374062\pi$
$860$	−4207.41	−0.166827
$861$	0	0
$862$	−20568.7	−0.812729
$863$	3352.45	0.132235	0.0661174	−	0.997812i	$-0.478939\pi$
$863$	3352.45	0.132235	0.0661174	+	0.997812i	$0.478939\pi$
$864$	0	0
$865$	−65796.6	−2.58630
$866$	−24065.3	−0.944310
$867$	0	0
$868$	2491.24	0.0974174
$869$	10142.4	0.395924
$870$	0	0
$871$	−5703.22	−0.221867
$872$	−1492.79	−0.0579727
$873$	0	0
$874$	2818.20	0.109070
$875$	−438.236	−0.0169315
$876$	0	0
$877$	−20076.9	−0.773033	−0.386516	−	0.922283i	$-0.626322\pi$
$877$	−20076.9	−0.773033	−0.386516	+	0.922283i	$0.626322\pi$
$878$	−32296.9	−1.24142
$879$	0	0
$880$	13721.6	0.525632
$881$	14800.2	0.565982	0.282991	−	0.959123i	$-0.408673\pi$
$881$	14800.2	0.565982	0.282991	+	0.959123i	$0.408673\pi$
$882$	0	0
$883$	−19955.4	−0.760534	−0.380267	−	0.924877i	$-0.624168\pi$
$883$	−19955.4	−0.760534	−0.380267	+	0.924877i	$0.624168\pi$
$884$	19118.2	0.727392
$885$	0	0
$886$	25272.3	0.958283
$887$	9205.47	0.348466	0.174233	−	0.984704i	$-0.444255\pi$
$887$	9205.47	0.348466	0.174233	+	0.984704i	$0.444255\pi$
$888$	0	0
$889$	524.615	0.0197919
$890$	−88272.3	−3.32460
$891$	0	0
$892$	−14070.4	−0.528151
$893$	7746.94	0.290304
$894$	0	0
$895$	−21171.9	−0.790725
$896$	9671.26	0.360596
$897$	0	0
$898$	−2246.20	−0.0834706
$899$	5267.02	0.195400
$900$	0	0
$901$	−24336.6	−0.899854
$902$	−12482.5	−0.460778
$903$	0	0
$904$	3038.35	0.111786
$905$	−18791.1	−0.690207
$906$	0	0
$907$	6722.78	0.246115	0.123057	−	0.992400i	$-0.460730\pi$
$907$	6722.78	0.246115	0.123057	+	0.992400i	$0.460730\pi$
$908$	−5866.13	−0.214399
$909$	0	0
$910$	20691.7	0.753763
$911$	−51855.3	−1.88589	−0.942944	−	0.332952i	$-0.891955\pi$
$911$	−51855.3	−1.88589	−0.942944	+	0.332952i	$0.891955\pi$
$912$	0	0
$913$	−13432.9	−0.486926
$914$	44279.1	1.60243
$915$	0	0
$916$	13411.2	0.483755
$917$	18454.1	0.664568
$918$	0	0
$919$	−55119.5	−1.97848	−0.989240	−	0.146302i	$-0.953263\pi$
$919$	−55119.5	−1.97848	−0.989240	+	0.146302i	$0.953263\pi$
$920$	−10929.7	−0.391674
$921$	0	0
$922$	−46504.4	−1.66111
$923$	−21451.5	−0.764989
$924$	0	0
$925$	−40652.6	−1.44502
$926$	30658.3	1.08801
$927$	0	0
$928$	−13089.1	−0.463007
$929$	−31459.7	−1.11104	−0.555522	−	0.831502i	$-0.687482\pi$
$929$	−31459.7	−1.11104	−0.555522	+	0.831502i	$0.687482\pi$
$930$	0	0
$931$	656.999	0.0231281
$932$	−23282.4	−0.818283
$933$	0	0
$934$	−53414.9	−1.87129
$935$	−13306.1	−0.465409
$936$	0	0
$937$	−13745.6	−0.479242	−0.239621	−	0.970867i	$-0.577023\pi$
$937$	−13745.6	−0.479242	−0.239621	+	0.970867i	$0.577023\pi$
$938$	−2687.03	−0.0935338
$939$	0	0
$940$	42453.6	1.47307
$941$	29814.2	1.03285	0.516427	−	0.856331i	$-0.327262\pi$
$941$	29814.2	1.03285	0.516427	+	0.856331i	$0.327262\pi$
$942$	0	0
$943$	18803.3	0.649330
$944$	−47616.7	−1.64173
$945$	0	0
$946$	2243.33	0.0771005
$947$	−32410.4	−1.11214	−0.556070	−	0.831135i	$-0.687691\pi$
$947$	−32410.4	−1.11214	−0.556070	+	0.831135i	$0.687691\pi$
$948$	0	0
$949$	−7053.67	−0.241277
$950$	5778.61	0.197351
$951$	0	0
$952$	−6374.56	−0.217018
$953$	−8931.28	−0.303581	−0.151790	−	0.988413i	$-0.548504\pi$
$953$	−8931.28	−0.303581	−0.151790	+	0.988413i	$0.548504\pi$
$954$	0	0
$955$	−11996.8	−0.406499
$956$	−21559.6	−0.729381
$957$	0	0
$958$	73719.9	2.48620
$959$	−8028.10	−0.270324
$960$	0	0
$961$	−24019.6	−0.806271
$962$	−63313.4	−2.12194
$963$	0	0
$964$	−19428.6	−0.649122
$965$	−31675.5	−1.05665
$966$	0	0
$967$	28665.0	0.953262	0.476631	−	0.879104i	$-0.341858\pi$
$967$	28665.0	0.953262	0.476631	+	0.879104i	$0.341858\pi$
$968$	−1428.74	−0.0474395
$969$	0	0
$970$	−16037.7	−0.530866
$971$	40925.2	1.35258	0.676288	−	0.736637i	$-0.263587\pi$
$971$	40925.2	1.35258	0.676288	+	0.736637i	$0.263587\pi$
$972$	0	0
$973$	−7090.99	−0.233635
$974$	54812.6	1.80319
$975$	0	0
$976$	26871.9	0.881300
$977$	21288.8	0.697124	0.348562	−	0.937286i	$-0.386670\pi$
$977$	21288.8	0.697124	0.348562	+	0.937286i	$0.386670\pi$
$978$	0	0
$979$	17382.1	0.567452
$980$	3600.39	0.117357
$981$	0	0
$982$	−38362.8	−1.24664
$983$	53128.8	1.72385	0.861925	−	0.507035i	$-0.169259\pi$
$983$	53128.8	1.72385	0.861925	+	0.507035i	$0.169259\pi$
$984$	0	0
$985$	−23526.1	−0.761019
$986$	19043.6	0.615082
$987$	0	0
$988$	3323.77	0.107028
$989$	−3379.29	−0.108650
$990$	0	0
$991$	−17978.4	−0.576290	−0.288145	−	0.957587i	$-0.593039\pi$
$991$	−17978.4	−0.576290	−0.288145	+	0.957587i	$0.593039\pi$
$992$	−14342.5	−0.459048
$993$	0	0
$994$	−10106.7	−0.322501
$995$	−71748.1	−2.28600
$996$	0	0
$997$	10063.9	0.319687	0.159844	−	0.987142i	$-0.448901\pi$
$997$	10063.9	0.319687	0.159844	+	0.987142i	$0.448901\pi$
$998$	−21368.6	−0.677766
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.k.1.2		2
3.2	odd	2		231.4.a.f.1.1	✓	2
21.20	even	2		1617.4.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.f.1.1	✓	2	3.2	odd	2
693.4.a.k.1.2		2	1.1	even	1	trivial
1617.4.a.i.1.1		2	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)

\(f(q)\)	\(=\)	\(q+3.56155 q^{2} +4.68466 q^{4} +15.6847 q^{5} +7.00000 q^{7} -11.8078 q^{8} +55.8617 q^{10} -11.0000 q^{11} +52.9157 q^{13} +24.9309 q^{14} -79.5312 q^{16} +77.1231 q^{17} +13.4081 q^{19} +73.4773 q^{20} -39.1771 q^{22} +59.0152 q^{23} +121.009 q^{25} +188.462 q^{26} +32.7926 q^{28} +69.3305 q^{29} +75.9697 q^{31} -188.793 q^{32} +274.678 q^{34} +109.793 q^{35} -335.948 q^{37} +47.7538 q^{38} -185.201 q^{40} +318.617 q^{41} -57.2614 q^{43} -51.5312 q^{44} +210.186 q^{46} +577.779 q^{47} +49.0000 q^{49} +430.978 q^{50} +247.892 q^{52} -315.555 q^{53} -172.531 q^{55} -82.6543 q^{56} +246.924 q^{58} +598.717 q^{59} -337.879 q^{61} +270.570 q^{62} -36.1449 q^{64} +829.965 q^{65} -107.779 q^{67} +361.295 q^{68} +391.032 q^{70} -405.390 q^{71} -133.300 q^{73} -1196.50 q^{74} +62.8125 q^{76} -77.0000 q^{77} -922.038 q^{79} -1247.42 q^{80} +1134.77 q^{82} +1221.17 q^{83} +1209.65 q^{85} -203.939 q^{86} +129.885 q^{88} -1580.19 q^{89} +370.410 q^{91} +276.466 q^{92} +2057.79 q^{94} +210.302 q^{95} -287.097 q^{97} +174.516 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} - 3 q^{4} + 19 q^{5} + 14 q^{7} - 3 q^{8} + 54 q^{10} - 22 q^{11} + 11 q^{13} + 21 q^{14} - 23 q^{16} + 146 q^{17} - 101 q^{19} + 48 q^{20} - 33 q^{22} + 184 q^{23} + 7 q^{25} + 212 q^{26}+ \cdots + 147 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 - 3 * q^4 + 19 * q^5 + 14 * q^7 - 3 * q^8 + 54 * q^10 - 22 * q^11 + 11 * q^13 + 21 * q^14 - 23 * q^16 + 146 * q^17 - 101 * q^19 + 48 * q^20 - 33 * q^22 + 184 * q^23 + 7 * q^25 + 212 * q^26 - 21 * q^28 + 217 * q^29 + 20 * q^31 - 291 * q^32 + 236 * q^34 + 133 * q^35 - 173 * q^37 + 112 * q^38 - 156 * q^40 + 60 * q^41 - 164 * q^43 + 33 * q^44 + 140 * q^46 + 467 * q^47 + 98 * q^49 + 495 * q^50 + 570 * q^52 - 326 * q^53 - 209 * q^55 - 21 * q^56 + 164 * q^58 + 781 * q^59 - 148 * q^61 + 302 * q^62 - 431 * q^64 + 691 * q^65 + 473 * q^67 - 168 * q^68 + 378 * q^70 + 756 * q^71 - 213 * q^73 - 1288 * q^74 + 942 * q^76 - 154 * q^77 + 168 * q^79 - 1060 * q^80 + 1280 * q^82 + 2096 * q^83 + 1438 * q^85 - 144 * q^86 + 33 * q^88 - 736 * q^89 + 77 * q^91 - 684 * q^92 + 2120 * q^94 - 169 * q^95 + 638 * q^97 + 147 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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