LMFDB - Embedded newform 693.4.a.s.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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 45x^{6} + 77x^{5} + 540x^{4} - 915x^{3} - 1452x^{2} + 2660x - 672 $ x^8 - 2x^7 - 45x^6 + 77x^5 + 540x^4 - 915x^3 - 1452x^2 + 2660*x - 672
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.50363$ 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)

$f(q)$	$=$	\(q-3.50363 q^{2} +4.27544 q^{4} +15.3058 q^{5} -7.00000 q^{7} +13.0495 q^{8} -53.6258 q^{10} +11.0000 q^{11} +5.90915 q^{13} +24.5254 q^{14} -79.9241 q^{16} -73.2134 q^{17} +80.3384 q^{19} +65.4390 q^{20} -38.5400 q^{22} -190.324 q^{23} +109.267 q^{25} -20.7035 q^{26} -29.9281 q^{28} -285.553 q^{29} +85.3709 q^{31} +175.629 q^{32} +256.513 q^{34} -107.140 q^{35} -149.061 q^{37} -281.476 q^{38} +199.732 q^{40} +13.9250 q^{41} -148.052 q^{43} +47.0299 q^{44} +666.824 q^{46} +94.7888 q^{47} +49.0000 q^{49} -382.831 q^{50} +25.2642 q^{52} +105.750 q^{53} +168.364 q^{55} -91.3463 q^{56} +1000.47 q^{58} +184.187 q^{59} -189.128 q^{61} -299.108 q^{62} +24.0535 q^{64} +90.4441 q^{65} -193.756 q^{67} -313.020 q^{68} +375.381 q^{70} +356.265 q^{71} +655.740 q^{73} +522.254 q^{74} +343.482 q^{76} -77.0000 q^{77} +110.189 q^{79} -1223.30 q^{80} -48.7881 q^{82} -1278.27 q^{83} -1120.59 q^{85} +518.721 q^{86} +143.544 q^{88} +255.120 q^{89} -41.3640 q^{91} -813.718 q^{92} -332.105 q^{94} +1229.64 q^{95} +372.537 q^{97} -171.678 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8} - 13 q^{10} + 88 q^{11} - 148 q^{13} + 14 q^{14} + 266 q^{16} - 114 q^{17} + 58 q^{19} - 291 q^{20} - 22 q^{22} - 246 q^{23} + 244 q^{25} - 305 q^{26}+ \cdots - 98 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 30 * q^4 - 10 * q^5 - 56 * q^7 - 15 * q^8 - 13 * q^10 + 88 * q^11 - 148 * q^13 + 14 * q^14 + 266 * q^16 - 114 * q^17 + 58 * q^19 - 291 * q^20 - 22 * q^22 - 246 * q^23 + 244 * q^25 - 305 * q^26 - 210 * q^28 + 72 * q^29 + 252 * q^31 - 1272 * q^32 + 630 * q^34 + 70 * q^35 - 80 * q^37 - 1885 * q^38 - 342 * q^40 - 682 * q^41 - 106 * q^43 + 330 * q^44 + 120 * q^46 - 828 * q^47 + 392 * q^49 - 801 * q^50 - 1681 * q^52 - 462 * q^53 - 110 * q^55 + 105 * q^56 - 1087 * q^58 - 626 * q^59 - 854 * q^61 - 1350 * q^62 + 2997 * q^64 - 22 * q^65 + 130 * q^67 - 2202 * q^68 + 91 * q^70 - 326 * q^71 - 390 * q^73 + 359 * q^74 + 2041 * q^76 - 616 * q^77 - 508 * q^79 - 4391 * q^80 + 1528 * q^82 - 1596 * q^83 - 880 * q^85 + 414 * q^86 - 165 * q^88 - 4324 * q^89 + 1036 * q^91 - 2092 * q^92 - 1685 * q^94 - 1076 * q^95 - 964 * q^97 - 98 * 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 $

$2$	−3.50363	−1.23872	−0.619361	−	0.785107i	$-0.712608\pi$
$2$	−3.50363	−1.23872	−0.619361	+	0.785107i	$0.712608\pi$
$3$	0	0
$3$	0	0
$4$	4.27544	0.534430
$5$	15.3058	1.36899	0.684495	−	0.729017i	$-0.260023\pi$
$5$	15.3058	1.36899	0.684495	+	0.729017i	$0.260023\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	13.0495	0.576711
$9$	0	0
$10$	−53.6258	−1.69580
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	5.90915	0.126069	0.0630347	−	0.998011i	$-0.479922\pi$
$13$	5.90915	0.126069	0.0630347	+	0.998011i	$0.479922\pi$
$14$	24.5254	0.468193
$15$	0	0
$16$	−79.9241	−1.24881
$17$	−73.2134	−1.04452	−0.522260	−	0.852786i	$-0.674911\pi$
$17$	−73.2134	−1.04452	−0.522260	+	0.852786i	$0.674911\pi$
$18$	0	0
$19$	80.3384	0.970046	0.485023	−	0.874501i	$-0.338811\pi$
$19$	80.3384	0.970046	0.485023	+	0.874501i	$0.338811\pi$
$20$	65.4390	0.731630
$21$	0	0
$22$	−38.5400	−0.373489
$23$	−190.324	−1.72544	−0.862722	−	0.505678i	$-0.831242\pi$
$23$	−190.324	−1.72544	−0.862722	+	0.505678i	$0.831242\pi$
$24$	0	0
$25$	109.267	0.874134
$26$	−20.7035	−0.156165
$27$	0	0
$28$	−29.9281	−0.201996
$29$	−285.553	−1.82848	−0.914238	−	0.405177i	$-0.867210\pi$
$29$	−285.553	−1.82848	−0.914238	+	0.405177i	$0.867210\pi$
$30$	0	0
$31$	85.3709	0.494615	0.247307	−	0.968937i	$-0.420454\pi$
$31$	85.3709	0.494615	0.247307	+	0.968937i	$0.420454\pi$
$32$	175.629	0.970222
$33$	0	0
$34$	256.513	1.29387
$35$	−107.140	−0.517430
$36$	0	0
$37$	−149.061	−0.662309	−0.331154	−	0.943577i	$-0.607438\pi$
$37$	−149.061	−0.662309	−0.331154	+	0.943577i	$0.607438\pi$
$38$	−281.476	−1.20162
$39$	0	0
$40$	199.732	0.789512
$41$	13.9250	0.0530420	0.0265210	−	0.999648i	$-0.491557\pi$
$41$	13.9250	0.0530420	0.0265210	+	0.999648i	$0.491557\pi$
$42$	0	0
$43$	−148.052	−0.525065	−0.262532	−	0.964923i	$-0.584558\pi$
$43$	−148.052	−0.525065	−0.262532	+	0.964923i	$0.584558\pi$
$44$	47.0299	0.161137
$45$	0	0
$46$	666.824	2.13735
$47$	94.7888	0.294178	0.147089	−	0.989123i	$-0.453010\pi$
$47$	94.7888	0.294178	0.147089	+	0.989123i	$0.453010\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−382.831	−1.08281
$51$	0	0
$52$	25.2642	0.0673754
$53$	105.750	0.274074	0.137037	−	0.990566i	$-0.456242\pi$
$53$	105.750	0.274074	0.137037	+	0.990566i	$0.456242\pi$
$54$	0	0
$55$	168.364	0.412766
$56$	−91.3463	−0.217976
$57$	0	0
$58$	1000.47	2.26497
$59$	184.187	0.406424	0.203212	−	0.979135i	$-0.434862\pi$
$59$	184.187	0.406424	0.203212	+	0.979135i	$0.434862\pi$
$60$	0	0
$61$	−189.128	−0.396972	−0.198486	−	0.980104i	$-0.563602\pi$
$61$	−189.128	−0.396972	−0.198486	+	0.980104i	$0.563602\pi$
$62$	−299.108	−0.612690
$63$	0	0
$64$	24.0535	0.0469795
$65$	90.4441	0.172588
$66$	0	0
$67$	−193.756	−0.353299	−0.176649	−	0.984274i	$-0.556526\pi$
$67$	−193.756	−0.353299	−0.176649	+	0.984274i	$0.556526\pi$
$68$	−313.020	−0.558224
$69$	0	0
$70$	375.381	0.640951
$71$	356.265	0.595505	0.297752	−	0.954643i	$-0.403763\pi$
$71$	356.265	0.595505	0.297752	+	0.954643i	$0.403763\pi$
$72$	0	0
$73$	655.740	1.05135	0.525675	−	0.850686i	$-0.323813\pi$
$73$	655.740	1.05135	0.525675	+	0.850686i	$0.323813\pi$
$74$	522.254	0.820416
$75$	0	0
$76$	343.482	0.518422
$77$	−77.0000	−0.113961
$78$	0	0
$79$	110.189	0.156927	0.0784634	−	0.996917i	$-0.474999\pi$
$79$	110.189	0.156927	0.0784634	+	0.996917i	$0.474999\pi$
$80$	−1223.30	−1.70961
$81$	0	0
$82$	−48.7881	−0.0657042
$83$	−1278.27	−1.69047	−0.845233	−	0.534399i	$-0.820538\pi$
$83$	−1278.27	−1.69047	−0.845233	+	0.534399i	$0.820538\pi$
$84$	0	0
$85$	−1120.59	−1.42994
$86$	518.721	0.650409
$87$	0	0
$88$	143.544	0.173885
$89$	255.120	0.303850	0.151925	−	0.988392i	$-0.451453\pi$
$89$	255.120	0.303850	0.151925	+	0.988392i	$0.451453\pi$
$90$	0	0
$91$	−41.3640	−0.0476498
$92$	−813.718	−0.922130
$93$	0	0
$94$	−332.105	−0.364405
$95$	1229.64	1.32798
$96$	0	0
$97$	372.537	0.389953	0.194976	−	0.980808i	$-0.437537\pi$
$97$	372.537	0.389953	0.194976	+	0.980808i	$0.437537\pi$
$98$	−171.678	−0.176960
$99$	0	0
$100$	467.164	0.467164
$101$	−1143.78	−1.12684	−0.563419	−	0.826171i	$-0.690514\pi$
$101$	−1143.78	−1.12684	−0.563419	+	0.826171i	$0.690514\pi$
$102$	0	0
$103$	471.721	0.451263	0.225631	−	0.974213i	$-0.427556\pi$
$103$	471.721	0.451263	0.225631	+	0.974213i	$0.427556\pi$
$104$	77.1113	0.0727056
$105$	0	0
$106$	−370.510	−0.339501
$107$	−1562.49	−1.41170	−0.705849	−	0.708363i	$-0.749434\pi$
$107$	−1562.49	−1.41170	−0.705849	+	0.708363i	$0.749434\pi$
$108$	0	0
$109$	−895.894	−0.787258	−0.393629	−	0.919269i	$-0.628781\pi$
$109$	−895.894	−0.787258	−0.393629	+	0.919269i	$0.628781\pi$
$110$	−589.884	−0.511302
$111$	0	0
$112$	559.469	0.472008
$113$	−341.558	−0.284345	−0.142173	−	0.989842i	$-0.545409\pi$
$113$	−341.558	−0.284345	−0.142173	+	0.989842i	$0.545409\pi$
$114$	0	0
$115$	−2913.05	−2.36212
$116$	−1220.87	−0.977194
$117$	0	0
$118$	−645.322	−0.503447
$119$	512.493	0.394792
$120$	0	0
$121$	121.000	0.0909091
$122$	662.634	0.491738
$123$	0	0
$124$	364.998	0.264337
$125$	−240.809	−0.172309
$126$	0	0
$127$	−2446.85	−1.70963	−0.854813	−	0.518935i	$-0.826329\pi$
$127$	−2446.85	−1.70963	−0.854813	+	0.518935i	$0.826329\pi$
$128$	−1489.31	−1.02842
$129$	0	0
$130$	−316.883	−0.213788
$131$	−2075.09	−1.38398	−0.691989	−	0.721908i	$-0.743265\pi$
$131$	−2075.09	−1.38398	−0.691989	+	0.721908i	$0.743265\pi$
$132$	0	0
$133$	−562.368	−0.366643
$134$	678.849	0.437639
$135$	0	0
$136$	−955.396	−0.602386
$137$	−199.066	−0.124141	−0.0620707	−	0.998072i	$-0.519770\pi$
$137$	−199.066	−0.124141	−0.0620707	+	0.998072i	$0.519770\pi$
$138$	0	0
$139$	2358.20	1.43899	0.719495	−	0.694497i	$-0.244373\pi$
$139$	2358.20	1.43899	0.719495	+	0.694497i	$0.244373\pi$
$140$	−458.073	−0.276530
$141$	0	0
$142$	−1248.22	−0.737664
$143$	65.0006	0.0380114
$144$	0	0
$145$	−4370.61	−2.50317
$146$	−2297.47	−1.30233
$147$	0	0
$148$	−637.300	−0.353958
$149$	616.754	0.339104	0.169552	−	0.985521i	$-0.445768\pi$
$149$	616.754	0.339104	0.169552	+	0.985521i	$0.445768\pi$
$150$	0	0
$151$	−997.331	−0.537494	−0.268747	−	0.963211i	$-0.586610\pi$
$151$	−997.331	−0.537494	−0.268747	+	0.963211i	$0.586610\pi$
$152$	1048.37	0.559436
$153$	0	0
$154$	269.780	0.141165
$155$	1306.67	0.677123
$156$	0	0
$157$	3181.76	1.61740	0.808701	−	0.588220i	$-0.200171\pi$
$157$	3181.76	1.61740	0.808701	+	0.588220i	$0.200171\pi$
$158$	−386.061	−0.194388
$159$	0	0
$160$	2688.14	1.32822
$161$	1332.27	0.652157
$162$	0	0
$163$	−2715.81	−1.30502	−0.652511	−	0.757779i	$-0.726284\pi$
$163$	−2715.81	−1.30502	−0.652511	+	0.757779i	$0.726284\pi$
$164$	59.5356	0.0283473
$165$	0	0
$166$	4478.60	2.09402
$167$	193.315	0.0895759	0.0447880	−	0.998997i	$-0.485739\pi$
$167$	193.315	0.0895759	0.0447880	+	0.998997i	$0.485739\pi$
$168$	0	0
$169$	−2162.08	−0.984106
$170$	3926.13	1.77129
$171$	0	0
$172$	−632.990	−0.280611
$173$	−2560.26	−1.12516	−0.562580	−	0.826743i	$-0.690191\pi$
$173$	−2560.26	−1.12516	−0.562580	+	0.826743i	$0.690191\pi$
$174$	0	0
$175$	−764.868	−0.330392
$176$	−879.165	−0.376532
$177$	0	0
$178$	−893.847	−0.376386
$179$	−2783.34	−1.16221	−0.581107	−	0.813827i	$-0.697380\pi$
$179$	−2783.34	−1.16221	−0.581107	+	0.813827i	$0.697380\pi$
$180$	0	0
$181$	3055.72	1.25486	0.627430	−	0.778673i	$-0.284107\pi$
$181$	3055.72	1.25486	0.627430	+	0.778673i	$0.284107\pi$
$182$	144.924	0.0590248
$183$	0	0
$184$	−2483.62	−0.995083
$185$	−2281.49	−0.906694
$186$	0	0
$187$	−805.347	−0.314935
$188$	405.264	0.157218
$189$	0	0
$190$	−4308.21	−1.64500
$191$	−3141.28	−1.19003	−0.595014	−	0.803715i	$-0.702854\pi$
$191$	−3141.28	−1.19003	−0.595014	+	0.803715i	$0.702854\pi$
$192$	0	0
$193$	−1695.74	−0.632446	−0.316223	−	0.948685i	$-0.602415\pi$
$193$	−1695.74	−0.632446	−0.316223	+	0.948685i	$0.602415\pi$
$194$	−1305.23	−0.483043
$195$	0	0
$196$	209.497	0.0763472
$197$	3061.85	1.10735	0.553675	−	0.832733i	$-0.313225\pi$
$197$	3061.85	1.10735	0.553675	+	0.832733i	$0.313225\pi$
$198$	0	0
$199$	66.9619	0.0238533	0.0119266	−	0.999929i	$-0.496204\pi$
$199$	66.9619	0.0238533	0.0119266	+	0.999929i	$0.496204\pi$
$200$	1425.87	0.504123
$201$	0	0
$202$	4007.40	1.39584
$203$	1998.87	0.691099
$204$	0	0
$205$	213.133	0.0726140
$206$	−1652.74	−0.558989
$207$	0	0
$208$	−472.284	−0.157437
$209$	883.722	0.292480
$210$	0	0
$211$	−913.636	−0.298092	−0.149046	−	0.988830i	$-0.547620\pi$
$211$	−913.636	−0.298092	−0.149046	+	0.988830i	$0.547620\pi$
$212$	452.130	0.146474
$213$	0	0
$214$	5474.39	1.74870
$215$	−2266.06	−0.718809
$216$	0	0
$217$	−597.596	−0.186947
$218$	3138.89	0.975193
$219$	0	0
$220$	719.829	0.220595
$221$	−432.629	−0.131682
$222$	0	0
$223$	153.648	0.0461391	0.0230695	−	0.999734i	$-0.492656\pi$
$223$	153.648	0.0461391	0.0230695	+	0.999734i	$0.492656\pi$
$224$	−1229.40	−0.366710
$225$	0	0
$226$	1196.69	0.352225
$227$	−497.228	−0.145384	−0.0726921	−	0.997354i	$-0.523159\pi$
$227$	−497.228	−0.145384	−0.0726921	+	0.997354i	$0.523159\pi$
$228$	0	0
$229$	3374.79	0.973854	0.486927	−	0.873443i	$-0.338118\pi$
$229$	3374.79	0.973854	0.486927	+	0.873443i	$0.338118\pi$
$230$	10206.3	2.92600
$231$	0	0
$232$	−3726.32	−1.05450
$233$	−4849.94	−1.36365	−0.681825	−	0.731516i	$-0.738813\pi$
$233$	−4849.94	−1.36365	−0.681825	+	0.731516i	$0.738813\pi$
$234$	0	0
$235$	1450.82	0.402727
$236$	787.479	0.217206
$237$	0	0
$238$	−1795.59	−0.489037
$239$	−7350.07	−1.98927	−0.994637	−	0.103430i	$-0.967018\pi$
$239$	−7350.07	−1.98927	−0.994637	+	0.103430i	$0.967018\pi$
$240$	0	0
$241$	919.171	0.245680	0.122840	−	0.992426i	$-0.460800\pi$
$241$	919.171	0.245680	0.122840	+	0.992426i	$0.460800\pi$
$242$	−423.940	−0.112611
$243$	0	0
$244$	−808.604	−0.212154
$245$	749.983	0.195570
$246$	0	0
$247$	474.731	0.122293
$248$	1114.05	0.285250
$249$	0	0
$250$	843.706	0.213443
$251$	4350.71	1.09408	0.547041	−	0.837106i	$-0.315754\pi$
$251$	4350.71	1.09408	0.547041	+	0.837106i	$0.315754\pi$
$252$	0	0
$253$	−2093.56	−0.520241
$254$	8572.86	2.11775
$255$	0	0
$256$	5025.56	1.22694
$257$	4128.98	1.00217	0.501087	−	0.865397i	$-0.332934\pi$
$257$	4128.98	1.00217	0.501087	+	0.865397i	$0.332934\pi$
$258$	0	0
$259$	1043.42	0.250329
$260$	386.689	0.0922362
$261$	0	0
$262$	7270.34	1.71436
$263$	6853.69	1.60691	0.803454	−	0.595367i	$-0.202993\pi$
$263$	6853.69	1.60691	0.803454	+	0.595367i	$0.202993\pi$
$264$	0	0
$265$	1618.59	0.375205
$266$	1970.33	0.454169
$267$	0	0
$268$	−828.392	−0.188814
$269$	7519.13	1.70427	0.852136	−	0.523320i	$-0.175307\pi$
$269$	7519.13	1.70427	0.852136	+	0.523320i	$0.175307\pi$
$270$	0	0
$271$	−5354.31	−1.20019	−0.600095	−	0.799929i	$-0.704870\pi$
$271$	−5354.31	−1.20019	−0.600095	+	0.799929i	$0.704870\pi$
$272$	5851.51	1.30441
$273$	0	0
$274$	697.455	0.153777
$275$	1201.93	0.263561
$276$	0	0
$277$	6529.96	1.41642	0.708208	−	0.706004i	$-0.249504\pi$
$277$	6529.96	1.41642	0.708208	+	0.706004i	$0.249504\pi$
$278$	−8262.26	−1.78251
$279$	0	0
$280$	−1398.13	−0.298407
$281$	−1530.30	−0.324876	−0.162438	−	0.986719i	$-0.551936\pi$
$281$	−1530.30	−0.324876	−0.162438	+	0.986719i	$0.551936\pi$
$282$	0	0
$283$	−5563.44	−1.16860	−0.584298	−	0.811540i	$-0.698630\pi$
$283$	−5563.44	−1.16860	−0.584298	+	0.811540i	$0.698630\pi$
$284$	1523.19	0.318256
$285$	0	0
$286$	−227.738	−0.0470855
$287$	−97.4751	−0.0200480
$288$	0	0
$289$	447.195	0.0910228
$290$	15313.0	3.10073
$291$	0	0
$292$	2803.58	0.561873
$293$	−4599.78	−0.917140	−0.458570	−	0.888658i	$-0.651638\pi$
$293$	−4599.78	−0.917140	−0.458570	+	0.888658i	$0.651638\pi$
$294$	0	0
$295$	2819.12	0.556391
$296$	−1945.16	−0.381961
$297$	0	0
$298$	−2160.88	−0.420055
$299$	−1124.65	−0.217526
$300$	0	0
$301$	1036.37	0.198456
$302$	3494.28	0.665806
$303$	0	0
$304$	−6420.97	−1.21141
$305$	−2894.74	−0.543451
$306$	0	0
$307$	5902.91	1.09738	0.548692	−	0.836025i	$-0.315126\pi$
$307$	5902.91	1.09738	0.548692	+	0.836025i	$0.315126\pi$
$308$	−329.209	−0.0609040
$309$	0	0
$310$	−4578.08	−0.838766
$311$	−1759.91	−0.320886	−0.160443	−	0.987045i	$-0.551292\pi$
$311$	−1759.91	−0.320886	−0.160443	+	0.987045i	$0.551292\pi$
$312$	0	0
$313$	−2376.13	−0.429095	−0.214548	−	0.976714i	$-0.568828\pi$
$313$	−2376.13	−0.429095	−0.214548	+	0.976714i	$0.568828\pi$
$314$	−11147.7	−2.00351
$315$	0	0
$316$	471.106	0.0838664
$317$	−1891.58	−0.335147	−0.167573	−	0.985860i	$-0.553593\pi$
$317$	−1891.58	−0.335147	−0.167573	+	0.985860i	$0.553593\pi$
$318$	0	0
$319$	−3141.08	−0.551307
$320$	368.158	0.0643145
$321$	0	0
$322$	−4667.77	−0.807841
$323$	−5881.84	−1.01323
$324$	0	0
$325$	645.674	0.110202
$326$	9515.20	1.61656
$327$	0	0
$328$	181.714	0.0305899
$329$	−663.522	−0.111189
$330$	0	0
$331$	−8905.72	−1.47886	−0.739430	−	0.673234i	$-0.764905\pi$
$331$	−8905.72	−1.47886	−0.739430	+	0.673234i	$0.764905\pi$
$332$	−5465.18	−0.903436
$333$	0	0
$334$	−677.306	−0.110960
$335$	−2965.58	−0.483663
$336$	0	0
$337$	10164.2	1.64297	0.821484	−	0.570232i	$-0.193147\pi$
$337$	10164.2	1.64297	0.821484	+	0.570232i	$0.193147\pi$
$338$	7575.14	1.21903
$339$	0	0
$340$	−4791.01	−0.764203
$341$	939.079	0.149132
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−1932.01	−0.302811
$345$	0	0
$346$	8970.20	1.39376
$347$	−7678.94	−1.18797	−0.593987	−	0.804475i	$-0.702447\pi$
$347$	−7678.94	−1.18797	−0.593987	+	0.804475i	$0.702447\pi$
$348$	0	0
$349$	8494.69	1.30289	0.651447	−	0.758694i	$-0.274162\pi$
$349$	8494.69	1.30289	0.651447	+	0.758694i	$0.274162\pi$
$350$	2679.82	0.409263
$351$	0	0
$352$	1931.92	0.292533
$353$	−1477.92	−0.222838	−0.111419	−	0.993774i	$-0.535540\pi$
$353$	−1477.92	−0.222838	−0.111419	+	0.993774i	$0.535540\pi$
$354$	0	0
$355$	5452.91	0.815240
$356$	1090.75	0.162387
$357$	0	0
$358$	9751.78	1.43966
$359$	−1354.69	−0.199158	−0.0995791	−	0.995030i	$-0.531750\pi$
$359$	−1354.69	−0.199158	−0.0995791	+	0.995030i	$0.531750\pi$
$360$	0	0
$361$	−404.749	−0.0590100
$362$	−10706.1	−1.55442
$363$	0	0
$364$	−176.850	−0.0254655
$365$	10036.6	1.43929
$366$	0	0
$367$	−2054.98	−0.292286	−0.146143	−	0.989263i	$-0.546686\pi$
$367$	−2054.98	−0.292286	−0.146143	+	0.989263i	$0.546686\pi$
$368$	15211.5	2.15476
$369$	0	0
$370$	7993.50	1.12314
$371$	−740.253	−0.103590
$372$	0	0
$373$	6632.28	0.920661	0.460330	−	0.887748i	$-0.347731\pi$
$373$	6632.28	0.920661	0.460330	+	0.887748i	$0.347731\pi$
$374$	2821.64	0.390116
$375$	0	0
$376$	1236.94	0.169656
$377$	−1687.37	−0.230515
$378$	0	0
$379$	−578.193	−0.0783636	−0.0391818	−	0.999232i	$-0.512475\pi$
$379$	−578.193	−0.0783636	−0.0391818	+	0.999232i	$0.512475\pi$
$380$	5257.26	0.709715
$381$	0	0
$382$	11005.9	1.47411
$383$	5643.47	0.752918	0.376459	−	0.926433i	$-0.377142\pi$
$383$	5643.47	0.752918	0.376459	+	0.926433i	$0.377142\pi$
$384$	0	0
$385$	−1178.54	−0.156011
$386$	5941.25	0.783424
$387$	0	0
$388$	1592.76	0.208403
$389$	13487.4	1.75794	0.878970	−	0.476877i	$-0.158231\pi$
$389$	13487.4	1.75794	0.878970	+	0.476877i	$0.158231\pi$
$390$	0	0
$391$	13934.2	1.80226
$392$	639.424	0.0823873
$393$	0	0
$394$	−10727.6	−1.37170
$395$	1686.53	0.214831
$396$	0	0
$397$	−15505.0	−1.96014	−0.980068	−	0.198664i	$-0.936340\pi$
$397$	−15505.0	−1.96014	−0.980068	+	0.198664i	$0.936340\pi$
$398$	−234.610	−0.0295476
$399$	0	0
$400$	−8733.05	−1.09163
$401$	11389.7	1.41839	0.709196	−	0.705011i	$-0.249058\pi$
$401$	11389.7	1.41839	0.709196	+	0.705011i	$0.249058\pi$
$402$	0	0
$403$	504.469	0.0623558
$404$	−4890.18	−0.602217
$405$	0	0
$406$	−7003.31	−0.856079
$407$	−1639.67	−0.199694
$408$	0	0
$409$	13512.3	1.63360	0.816798	−	0.576924i	$-0.195747\pi$
$409$	13512.3	1.63360	0.816798	+	0.576924i	$0.195747\pi$
$410$	−746.740	−0.0899485
$411$	0	0
$412$	2016.82	0.241169
$413$	−1289.31	−0.153614
$414$	0	0
$415$	−19564.9	−2.31423
$416$	1037.82	0.122315
$417$	0	0
$418$	−3096.24	−0.362301
$419$	−8247.54	−0.961620	−0.480810	−	0.876825i	$-0.659657\pi$
$419$	−8247.54	−0.961620	−0.480810	+	0.876825i	$0.659657\pi$
$420$	0	0
$421$	2630.75	0.304549	0.152275	−	0.988338i	$-0.451340\pi$
$421$	2630.75	0.304549	0.152275	+	0.988338i	$0.451340\pi$
$422$	3201.05	0.369252
$423$	0	0
$424$	1379.99	0.158062
$425$	−7999.79	−0.913051
$426$	0	0
$427$	1323.89	0.150041
$428$	−6680.34	−0.754454
$429$	0	0
$430$	7939.43	0.890404
$431$	10428.7	1.16551	0.582753	−	0.812649i	$-0.301976\pi$
$431$	10428.7	1.16551	0.582753	+	0.812649i	$0.301976\pi$
$432$	0	0
$433$	−8822.44	−0.979167	−0.489583	−	0.871956i	$-0.662851\pi$
$433$	−8822.44	−0.979167	−0.489583	+	0.871956i	$0.662851\pi$
$434$	2093.76	0.231575
$435$	0	0
$436$	−3830.35	−0.420735
$437$	−15290.3	−1.67376
$438$	0	0
$439$	−16899.6	−1.83730	−0.918652	−	0.395068i	$-0.870721\pi$
$439$	−16899.6	−1.83730	−0.918652	+	0.395068i	$0.870721\pi$
$440$	2197.06	0.238047
$441$	0	0
$442$	1515.77	0.163117
$443$	−9130.18	−0.979206	−0.489603	−	0.871946i	$-0.662858\pi$
$443$	−9130.18	−0.979206	−0.489603	+	0.871946i	$0.662858\pi$
$444$	0	0
$445$	3904.81	0.415968
$446$	−538.325	−0.0571534
$447$	0	0
$448$	−168.375	−0.0177566
$449$	−6461.47	−0.679143	−0.339572	−	0.940580i	$-0.610282\pi$
$449$	−6461.47	−0.679143	−0.339572	+	0.940580i	$0.610282\pi$
$450$	0	0
$451$	153.175	0.0159928
$452$	−1460.31	−0.151963
$453$	0	0
$454$	1742.11	0.180090
$455$	−633.109	−0.0652321
$456$	0	0
$457$	−9239.47	−0.945743	−0.472871	−	0.881132i	$-0.656782\pi$
$457$	−9239.47	−0.945743	−0.472871	+	0.881132i	$0.656782\pi$
$458$	−11824.0	−1.20633
$459$	0	0
$460$	−12454.6	−1.26239
$461$	−7393.95	−0.747007	−0.373504	−	0.927629i	$-0.621844\pi$
$461$	−7393.95	−0.747007	−0.373504	+	0.927629i	$0.621844\pi$
$462$	0	0
$463$	−3360.31	−0.337293	−0.168646	−	0.985677i	$-0.553940\pi$
$463$	−3360.31	−0.337293	−0.168646	+	0.985677i	$0.553940\pi$
$464$	22822.6	2.28343
$465$	0	0
$466$	16992.4	1.68918
$467$	10685.1	1.05878	0.529389	−	0.848379i	$-0.322421\pi$
$467$	10685.1	1.05878	0.529389	+	0.848379i	$0.322421\pi$
$468$	0	0
$469$	1356.29	0.133534
$470$	−5083.13	−0.498866
$471$	0	0
$472$	2403.54	0.234389
$473$	−1628.58	−0.158313
$474$	0	0
$475$	8778.31	0.847951
$476$	2191.14	0.210989
$477$	0	0
$478$	25751.9	2.46416
$479$	18204.4	1.73649	0.868247	−	0.496133i	$-0.165247\pi$
$479$	18204.4	1.73649	0.868247	+	0.496133i	$0.165247\pi$
$480$	0	0
$481$	−880.821	−0.0834969
$482$	−3220.44	−0.304330
$483$	0	0
$484$	517.329	0.0485846
$485$	5701.97	0.533841
$486$	0	0
$487$	−6213.98	−0.578197	−0.289099	−	0.957299i	$-0.593356\pi$
$487$	−6213.98	−0.578197	−0.289099	+	0.957299i	$0.593356\pi$
$488$	−2468.02	−0.228938
$489$	0	0
$490$	−2627.67	−0.242257
$491$	−17022.7	−1.56461	−0.782306	−	0.622894i	$-0.785957\pi$
$491$	−17022.7	−1.56461	−0.782306	+	0.622894i	$0.785957\pi$
$492$	0	0
$493$	20906.3	1.90988
$494$	−1663.28	−0.151487
$495$	0	0
$496$	−6823.19	−0.617682
$497$	−2493.85	−0.225080
$498$	0	0
$499$	−20595.2	−1.84763	−0.923814	−	0.382843i	$-0.874945\pi$
$499$	−20595.2	−1.84763	−0.923814	+	0.382843i	$0.874945\pi$
$500$	−1029.56	−0.0920871
$501$	0	0
$502$	−15243.3	−1.35526
$503$	−19173.9	−1.69965	−0.849824	−	0.527067i	$-0.823292\pi$
$503$	−19173.9	−1.69965	−0.849824	+	0.527067i	$0.823292\pi$
$504$	0	0
$505$	−17506.5	−1.54263
$506$	7335.07	0.644434
$507$	0	0
$508$	−10461.4	−0.913677
$509$	−3120.30	−0.271719	−0.135859	−	0.990728i	$-0.543380\pi$
$509$	−3120.30	−0.271719	−0.135859	+	0.990728i	$0.543380\pi$
$510$	0	0
$511$	−4590.18	−0.397373
$512$	−5693.25	−0.491423
$513$	0	0
$514$	−14466.4	−1.24141
$515$	7220.06	0.617774
$516$	0	0
$517$	1042.68	0.0886980
$518$	−3655.78	−0.310088
$519$	0	0
$520$	1180.25	0.0995333
$521$	7932.70	0.667059	0.333530	−	0.942740i	$-0.391760\pi$
$521$	7932.70	0.667059	0.333530	+	0.942740i	$0.391760\pi$
$522$	0	0
$523$	−4156.12	−0.347484	−0.173742	−	0.984791i	$-0.555586\pi$
$523$	−4156.12	−0.347484	−0.173742	+	0.984791i	$0.555586\pi$
$524$	−8871.91	−0.739640
$525$	0	0
$526$	−24012.8	−1.99051
$527$	−6250.29	−0.516635
$528$	0	0
$529$	24056.1	1.97716
$530$	−5670.95	−0.464774
$531$	0	0
$532$	−2404.37	−0.195945
$533$	82.2850	0.00668698
$534$	0	0
$535$	−23915.1	−1.93260
$536$	−2528.41	−0.203751
$537$	0	0
$538$	−26344.3	−2.11112
$539$	539.000	0.0430730
$540$	0	0
$541$	−21965.6	−1.74561	−0.872804	−	0.488070i	$-0.837701\pi$
$541$	−21965.6	−1.74561	−0.872804	+	0.488070i	$0.837701\pi$
$542$	18759.5	1.48670
$543$	0	0
$544$	−12858.4	−1.01342
$545$	−13712.4	−1.07775
$546$	0	0
$547$	11875.0	0.928224	0.464112	−	0.885776i	$-0.346373\pi$
$547$	11875.0	0.928224	0.464112	+	0.885776i	$0.346373\pi$
$548$	−851.096	−0.0663449
$549$	0	0
$550$	−4211.14	−0.326479
$551$	−22940.8	−1.77371
$552$	0	0
$553$	−771.322	−0.0593127
$554$	−22878.6	−1.75455
$555$	0	0
$556$	10082.3	0.769040
$557$	−2292.04	−0.174357	−0.0871786	−	0.996193i	$-0.527785\pi$
$557$	−2292.04	−0.174357	−0.0871786	+	0.996193i	$0.527785\pi$
$558$	0	0
$559$	−874.864	−0.0661946
$560$	8563.11	0.646174
$561$	0	0
$562$	5361.62	0.402431
$563$	7686.51	0.575395	0.287698	−	0.957721i	$-0.407110\pi$
$563$	7686.51	0.575395	0.287698	+	0.957721i	$0.407110\pi$
$564$	0	0
$565$	−5227.80	−0.389266
$566$	19492.3	1.44756
$567$	0	0
$568$	4649.07	0.343434
$569$	1044.55	0.0769592	0.0384796	−	0.999259i	$-0.487749\pi$
$569$	1044.55	0.0769592	0.0384796	+	0.999259i	$0.487749\pi$
$570$	0	0
$571$	−1085.27	−0.0795399	−0.0397700	−	0.999209i	$-0.512663\pi$
$571$	−1085.27	−0.0795399	−0.0397700	+	0.999209i	$0.512663\pi$
$572$	277.907	0.0203144
$573$	0	0
$574$	341.517	0.0248339
$575$	−20796.1	−1.50827
$576$	0	0
$577$	−12880.9	−0.929354	−0.464677	−	0.885480i	$-0.653830\pi$
$577$	−12880.9	−0.929354	−0.464677	+	0.885480i	$0.653830\pi$
$578$	−1566.81	−0.112752
$579$	0	0
$580$	−18686.3	−1.33777
$581$	8947.91	0.638936
$582$	0	0
$583$	1163.25	0.0826365
$584$	8557.06	0.606325
$585$	0	0
$586$	16115.9	1.13608
$587$	4693.01	0.329985	0.164993	−	0.986295i	$-0.447240\pi$
$587$	4693.01	0.329985	0.164993	+	0.986295i	$0.447240\pi$
$588$	0	0
$589$	6858.55	0.479799
$590$	−9877.15	−0.689213
$591$	0	0
$592$	11913.5	0.827101
$593$	−13825.6	−0.957418	−0.478709	−	0.877974i	$-0.658895\pi$
$593$	−13825.6	−0.957418	−0.478709	+	0.877974i	$0.658895\pi$
$594$	0	0
$595$	7844.11	0.540466
$596$	2636.90	0.181227
$597$	0	0
$598$	3940.36	0.269454
$599$	−13616.7	−0.928823	−0.464411	−	0.885620i	$-0.653734\pi$
$599$	−13616.7	−0.928823	−0.464411	+	0.885620i	$0.653734\pi$
$600$	0	0
$601$	25768.9	1.74898	0.874490	−	0.485044i	$-0.161196\pi$
$601$	25768.9	1.74898	0.874490	+	0.485044i	$0.161196\pi$
$602$	−3631.05	−0.245831
$603$	0	0
$604$	−4264.03	−0.287253
$605$	1852.00	0.124454
$606$	0	0
$607$	19103.1	1.27738	0.638691	−	0.769463i	$-0.279476\pi$
$607$	19103.1	1.27738	0.638691	+	0.769463i	$0.279476\pi$
$608$	14109.7	0.941161
$609$	0	0
$610$	10142.1	0.673184
$611$	560.121	0.0370869
$612$	0	0
$613$	−6915.62	−0.455659	−0.227830	−	0.973701i	$-0.573163\pi$
$613$	−6915.62	−0.455659	−0.227830	+	0.973701i	$0.573163\pi$
$614$	−20681.6	−1.35935
$615$	0	0
$616$	−1004.81	−0.0657223
$617$	−16084.3	−1.04948	−0.524742	−	0.851261i	$-0.675838\pi$
$617$	−16084.3	−1.04948	−0.524742	+	0.851261i	$0.675838\pi$
$618$	0	0
$619$	12041.5	0.781891	0.390945	−	0.920414i	$-0.372148\pi$
$619$	12041.5	0.781891	0.390945	+	0.920414i	$0.372148\pi$
$620$	5586.58	0.361875
$621$	0	0
$622$	6166.09	0.397488
$623$	−1785.84	−0.114845
$624$	0	0
$625$	−17344.1	−1.11002
$626$	8325.09	0.531530
$627$	0	0
$628$	13603.4	0.864389
$629$	10913.2	0.691795
$630$	0	0
$631$	19654.5	1.23999	0.619994	−	0.784607i	$-0.287135\pi$
$631$	19654.5	1.23999	0.619994	+	0.784607i	$0.287135\pi$
$632$	1437.91	0.0905014
$633$	0	0
$634$	6627.39	0.415153
$635$	−37450.9	−2.34046
$636$	0	0
$637$	289.548	0.0180099
$638$	11005.2	0.682915
$639$	0	0
$640$	−22795.0	−1.40789
$641$	−9941.56	−0.612586	−0.306293	−	0.951937i	$-0.599089\pi$
$641$	−9941.56	−0.612586	−0.306293	+	0.951937i	$0.599089\pi$
$642$	0	0
$643$	15162.4	0.929930	0.464965	−	0.885329i	$-0.346067\pi$
$643$	15162.4	0.929930	0.464965	+	0.885329i	$0.346067\pi$
$644$	5696.03	0.348532
$645$	0	0
$646$	20607.8	1.25511
$647$	−7769.45	−0.472100	−0.236050	−	0.971741i	$-0.575853\pi$
$647$	−7769.45	−0.472100	−0.236050	+	0.971741i	$0.575853\pi$
$648$	0	0
$649$	2026.05	0.122542
$650$	−2262.20	−0.136509
$651$	0	0
$652$	−11611.3	−0.697443
$653$	17240.5	1.03319	0.516594	−	0.856230i	$-0.327200\pi$
$653$	17240.5	1.03319	0.516594	+	0.856230i	$0.327200\pi$
$654$	0	0
$655$	−31760.8	−1.89465
$656$	−1112.94	−0.0662396
$657$	0	0
$658$	2324.74	0.137732
$659$	26820.8	1.58542	0.792709	−	0.609600i	$-0.208670\pi$
$659$	26820.8	1.58542	0.792709	+	0.609600i	$0.208670\pi$
$660$	0	0
$661$	33944.1	1.99738	0.998692	−	0.0511251i	$-0.0162807\pi$
$661$	33944.1	1.99738	0.998692	+	0.0511251i	$0.0162807\pi$
$662$	31202.4	1.83189
$663$	0	0
$664$	−16680.8	−0.974910
$665$	−8607.49	−0.501931
$666$	0	0
$667$	54347.5	3.15494
$668$	826.508	0.0478721
$669$	0	0
$670$	10390.3	0.599123
$671$	−2080.40	−0.119692
$672$	0	0
$673$	−9225.59	−0.528411	−0.264205	−	0.964466i	$-0.585110\pi$
$673$	−9225.59	−0.528411	−0.264205	+	0.964466i	$0.585110\pi$
$674$	−35611.7	−2.03518
$675$	0	0
$676$	−9243.86	−0.525936
$677$	−23388.8	−1.32778	−0.663889	−	0.747831i	$-0.731095\pi$
$677$	−23388.8	−1.32778	−0.663889	+	0.747831i	$0.731095\pi$
$678$	0	0
$679$	−2607.76	−0.147388
$680$	−14623.1	−0.824661
$681$	0	0
$682$	−3290.19	−0.184733
$683$	−8866.19	−0.496713	−0.248357	−	0.968669i	$-0.579891\pi$
$683$	−8866.19	−0.496713	−0.248357	+	0.968669i	$0.579891\pi$
$684$	0	0
$685$	−3046.86	−0.169948
$686$	1201.75	0.0668847
$687$	0	0
$688$	11833.0	0.655709
$689$	624.895	0.0345524
$690$	0	0
$691$	26611.8	1.46507	0.732534	−	0.680731i	$-0.238338\pi$
$691$	26611.8	1.46507	0.732534	+	0.680731i	$0.238338\pi$
$692$	−10946.2	−0.601320
$693$	0	0
$694$	26904.2	1.47157
$695$	36094.0	1.96996
$696$	0	0
$697$	−1019.50	−0.0554034
$698$	−29762.3	−1.61392
$699$	0	0
$700$	−3270.15	−0.176571
$701$	−19326.3	−1.04129	−0.520644	−	0.853774i	$-0.674308\pi$
$701$	−19326.3	−1.04129	−0.520644	+	0.853774i	$0.674308\pi$
$702$	0	0
$703$	−11975.3	−0.642470
$704$	264.589	0.0141649
$705$	0	0
$706$	5178.09	0.276034
$707$	8006.48	0.425905
$708$	0	0
$709$	24749.2	1.31097	0.655484	−	0.755209i	$-0.272465\pi$
$709$	24749.2	1.31097	0.655484	+	0.755209i	$0.272465\pi$
$710$	−19105.0	−1.00986
$711$	0	0
$712$	3329.18	0.175234
$713$	−16248.1	−0.853430
$714$	0	0
$715$	994.885	0.0520372
$716$	−11900.0	−0.621122
$717$	0	0
$718$	4746.34	0.246702
$719$	3517.02	0.182424	0.0912119	−	0.995832i	$-0.470926\pi$
$719$	3517.02	0.182424	0.0912119	+	0.995832i	$0.470926\pi$
$720$	0	0
$721$	−3302.05	−0.170561
$722$	1418.09	0.0730969
$723$	0	0
$724$	13064.5	0.670635
$725$	−31201.4	−1.59833
$726$	0	0
$727$	27443.7	1.40004	0.700021	−	0.714123i	$-0.253174\pi$
$727$	27443.7	1.40004	0.700021	+	0.714123i	$0.253174\pi$
$728$	−539.779	−0.0274801
$729$	0	0
$730$	−35164.6	−1.78288
$731$	10839.4	0.548441
$732$	0	0
$733$	−24925.8	−1.25601	−0.628005	−	0.778210i	$-0.716128\pi$
$733$	−24925.8	−1.25601	−0.628005	+	0.778210i	$0.716128\pi$
$734$	7199.88	0.362061
$735$	0	0
$736$	−33426.4	−1.67406
$737$	−2131.31	−0.106524
$738$	0	0
$739$	37245.9	1.85401	0.927004	−	0.375053i	$-0.122375\pi$
$739$	37245.9	1.85401	0.927004	+	0.375053i	$0.122375\pi$
$740$	−9754.38	−0.484565
$741$	0	0
$742$	2593.57	0.128319
$743$	−13860.1	−0.684357	−0.342178	−	0.939635i	$-0.611165\pi$
$743$	−13860.1	−0.684357	−0.342178	+	0.939635i	$0.611165\pi$
$744$	0	0
$745$	9439.90	0.464230
$746$	−23237.1	−1.14044
$747$	0	0
$748$	−3443.22	−0.168311
$749$	10937.4	0.533572
$750$	0	0
$751$	20219.8	0.982467	0.491234	−	0.871028i	$-0.336546\pi$
$751$	20219.8	0.982467	0.491234	+	0.871028i	$0.336546\pi$
$752$	−7575.91	−0.367374
$753$	0	0
$754$	5911.94	0.285544
$755$	−15264.9	−0.735825
$756$	0	0
$757$	4381.34	0.210360	0.105180	−	0.994453i	$-0.466458\pi$
$757$	4381.34	0.210360	0.105180	+	0.994453i	$0.466458\pi$
$758$	2025.78	0.0970706
$759$	0	0
$760$	16046.2	0.765863
$761$	14022.2	0.667942	0.333971	−	0.942583i	$-0.391611\pi$
$761$	14022.2	0.667942	0.333971	+	0.942583i	$0.391611\pi$
$762$	0	0
$763$	6271.26	0.297556
$764$	−13430.4	−0.635987
$765$	0	0
$766$	−19772.6	−0.932656
$767$	1088.39	0.0512377
$768$	0	0
$769$	10002.3	0.469042	0.234521	−	0.972111i	$-0.424648\pi$
$769$	10002.3	0.469042	0.234521	+	0.972111i	$0.424648\pi$
$770$	4129.19	0.193254
$771$	0	0
$772$	−7250.04	−0.337998
$773$	42649.3	1.98446	0.992232	−	0.124405i	$-0.0397021\pi$
$773$	42649.3	1.98446	0.992232	+	0.124405i	$0.0397021\pi$
$774$	0	0
$775$	9328.20	0.432360
$776$	4861.41	0.224890
$777$	0	0
$778$	−47254.9	−2.17760
$779$	1118.71	0.0514532
$780$	0	0
$781$	3918.91	0.179551
$782$	−48820.4	−2.23250
$783$	0	0
$784$	−3916.28	−0.178402
$785$	48699.3	2.21421
$786$	0	0
$787$	9129.62	0.413514	0.206757	−	0.978392i	$-0.433709\pi$
$787$	9129.62	0.413514	0.206757	+	0.978392i	$0.433709\pi$
$788$	13090.8	0.591801
$789$	0	0
$790$	−5908.97	−0.266116
$791$	2390.90	0.107472
$792$	0	0
$793$	−1117.58	−0.0500461
$794$	54323.8	2.42806
$795$	0	0
$796$	286.292	0.0127479
$797$	31429.2	1.39684	0.698419	−	0.715690i	$-0.253887\pi$
$797$	31429.2	1.39684	0.698419	+	0.715690i	$0.253887\pi$
$798$	0	0
$799$	−6939.81	−0.307275
$800$	19190.4	0.848105
$801$	0	0
$802$	−39905.4	−1.75699
$803$	7213.14	0.316994
$804$	0	0
$805$	20391.4	0.892796
$806$	−1767.47	−0.0772415
$807$	0	0
$808$	−14925.8	−0.649860
$809$	18839.2	0.818727	0.409363	−	0.912371i	$-0.365751\pi$
$809$	18839.2	0.818727	0.409363	+	0.912371i	$0.365751\pi$
$810$	0	0
$811$	10325.1	0.447059	0.223529	−	0.974697i	$-0.428242\pi$
$811$	10325.1	0.447059	0.223529	+	0.974697i	$0.428242\pi$
$812$	8546.06	0.369345
$813$	0	0
$814$	5744.79	0.247365
$815$	−41567.6	−1.78656
$816$	0	0
$817$	−11894.3	−0.509337
$818$	−47342.2	−2.02357
$819$	0	0
$820$	911.239	0.0388071
$821$	31707.7	1.34788	0.673938	−	0.738788i	$-0.264601\pi$
$821$	31707.7	1.34788	0.673938	+	0.738788i	$0.264601\pi$
$822$	0	0
$823$	−27144.9	−1.14971	−0.574856	−	0.818254i	$-0.694942\pi$
$823$	−27144.9	−1.14971	−0.574856	+	0.818254i	$0.694942\pi$
$824$	6155.72	0.260248
$825$	0	0
$826$	4517.25	0.190285
$827$	−10010.3	−0.420909	−0.210455	−	0.977604i	$-0.567494\pi$
$827$	−10010.3	−0.420909	−0.210455	+	0.977604i	$0.567494\pi$
$828$	0	0
$829$	−36122.6	−1.51338	−0.756689	−	0.653776i	$-0.773184\pi$
$829$	−36122.6	−1.51338	−0.756689	+	0.653776i	$0.773184\pi$
$830$	68548.4	2.86669
$831$	0	0
$832$	142.136	0.00592269
$833$	−3587.45	−0.149217
$834$	0	0
$835$	2958.84	0.122629
$836$	3778.30	0.156310
$837$	0	0
$838$	28896.4	1.19118
$839$	6552.24	0.269617	0.134808	−	0.990872i	$-0.456958\pi$
$839$	6552.24	0.269617	0.134808	+	0.990872i	$0.456958\pi$
$840$	0	0
$841$	57151.4	2.34333
$842$	−9217.20	−0.377251
$843$	0	0
$844$	−3906.20	−0.159309
$845$	−33092.3	−1.34723
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−8452.01	−0.342268
$849$	0	0
$850$	28028.3	1.13102
$851$	28369.8	1.14278
$852$	0	0
$853$	−29048.8	−1.16601	−0.583007	−	0.812467i	$-0.698124\pi$
$853$	−29048.8	−1.16601	−0.583007	+	0.812467i	$0.698124\pi$
$854$	−4638.43	−0.185859
$855$	0	0
$856$	−20389.7	−0.814141
$857$	4964.30	0.197873	0.0989365	−	0.995094i	$-0.468456\pi$
$857$	4964.30	0.197873	0.0989365	+	0.995094i	$0.468456\pi$
$858$	0	0
$859$	21708.3	0.862255	0.431127	−	0.902291i	$-0.358116\pi$
$859$	21708.3	0.862255	0.431127	+	0.902291i	$0.358116\pi$
$860$	−9688.40	−0.384153
$861$	0	0
$862$	−36538.3	−1.44374
$863$	−11197.9	−0.441692	−0.220846	−	0.975309i	$-0.570882\pi$
$863$	−11197.9	−0.441692	−0.220846	+	0.975309i	$0.570882\pi$
$864$	0	0
$865$	−39186.7	−1.54033
$866$	30910.6	1.21291
$867$	0	0
$868$	−2554.99	−0.0999101
$869$	1212.08	0.0473152
$870$	0	0
$871$	−1144.93	−0.0445402
$872$	−11691.0	−0.454020
$873$	0	0
$874$	53571.6	2.07332
$875$	1685.66	0.0651266
$876$	0	0
$877$	−20949.4	−0.806626	−0.403313	−	0.915062i	$-0.632141\pi$
$877$	−20949.4	−0.806626	−0.403313	+	0.915062i	$0.632141\pi$
$878$	59210.2	2.27591
$879$	0	0
$880$	−13456.3	−0.515468
$881$	35886.4	1.37236	0.686178	−	0.727434i	$-0.259287\pi$
$881$	35886.4	1.37236	0.686178	+	0.727434i	$0.259287\pi$
$882$	0	0
$883$	−7181.51	−0.273700	−0.136850	−	0.990592i	$-0.543698\pi$
$883$	−7181.51	−0.273700	−0.136850	+	0.990592i	$0.543698\pi$
$884$	−1849.68	−0.0703749
$885$	0	0
$886$	31988.8	1.21296
$887$	25132.3	0.951365	0.475682	−	0.879617i	$-0.342201\pi$
$887$	25132.3	0.951365	0.475682	+	0.879617i	$0.342201\pi$
$888$	0	0
$889$	17127.9	0.646178
$890$	−13681.0	−0.515268
$891$	0	0
$892$	656.912	0.0246581
$893$	7615.18	0.285366
$894$	0	0
$895$	−42601.1	−1.59106
$896$	10425.1	0.388705
$897$	0	0
$898$	22638.6	0.841269
$899$	−24377.9	−0.904392
$900$	0	0
$901$	−7742.34	−0.286276
$902$	−536.669	−0.0198106
$903$	0	0
$904$	−4457.15	−0.163985
$905$	46770.1	1.71789
$906$	0	0
$907$	−9475.36	−0.346884	−0.173442	−	0.984844i	$-0.555489\pi$
$907$	−9475.36	−0.346884	−0.173442	+	0.984844i	$0.555489\pi$
$908$	−2125.87	−0.0776977
$909$	0	0
$910$	2218.18	0.0808044
$911$	43008.3	1.56414	0.782068	−	0.623193i	$-0.214165\pi$
$911$	43008.3	1.56414	0.782068	+	0.623193i	$0.214165\pi$
$912$	0	0
$913$	−14061.0	−0.509694
$914$	32371.7	1.17151
$915$	0	0
$916$	14428.7	0.520457
$917$	14525.6	0.523094
$918$	0	0
$919$	23979.4	0.860727	0.430363	−	0.902656i	$-0.358385\pi$
$919$	23979.4	0.860727	0.430363	+	0.902656i	$0.358385\pi$
$920$	−38013.8	−1.36226
$921$	0	0
$922$	25905.7	0.925334
$923$	2105.22	0.0750750
$924$	0	0
$925$	−16287.4	−0.578947
$926$	11773.3	0.417812
$927$	0	0
$928$	−50151.4	−1.77403
$929$	27635.2	0.975975	0.487988	−	0.872851i	$-0.337731\pi$
$929$	27635.2	0.975975	0.487988	+	0.872851i	$0.337731\pi$
$930$	0	0
$931$	3936.58	0.138578
$932$	−20735.7	−0.728776
$933$	0	0
$934$	−37436.8	−1.31153
$935$	−12326.5	−0.431143
$936$	0	0
$937$	−5950.96	−0.207481	−0.103740	−	0.994604i	$-0.533081\pi$
$937$	−5950.96	−0.207481	−0.103740	+	0.994604i	$0.533081\pi$
$938$	−4751.94	−0.165412
$939$	0	0
$940$	6202.88	0.215230
$941$	−21970.4	−0.761122	−0.380561	−	0.924756i	$-0.624269\pi$
$941$	−21970.4	−0.761122	−0.380561	+	0.924756i	$0.624269\pi$
$942$	0	0
$943$	−2650.26	−0.0915210
$944$	−14720.9	−0.507549
$945$	0	0
$946$	5705.94	0.196106
$947$	−18563.0	−0.636976	−0.318488	−	0.947927i	$-0.603175\pi$
$947$	−18563.0	−0.636976	−0.318488	+	0.947927i	$0.603175\pi$
$948$	0	0
$949$	3874.86	0.132543
$950$	−30756.0	−1.05037
$951$	0	0
$952$	6687.77	0.227681
$953$	642.345	0.0218338	0.0109169	−	0.999940i	$-0.496525\pi$
$953$	642.345	0.0218338	0.0109169	+	0.999940i	$0.496525\pi$
$954$	0	0
$955$	−48079.8	−1.62914
$956$	−31424.8	−1.06313
$957$	0	0
$958$	−63781.5	−2.15103
$959$	1393.46	0.0469210
$960$	0	0
$961$	−22502.8	−0.755356
$962$	3086.07	0.103429
$963$	0	0
$964$	3929.86	0.131299
$965$	−25954.6	−0.865812
$966$	0	0
$967$	27334.8	0.909026	0.454513	−	0.890740i	$-0.349813\pi$
$967$	27334.8	0.909026	0.454513	+	0.890740i	$0.349813\pi$
$968$	1578.99	0.0524283
$969$	0	0
$970$	−19977.6	−0.661281
$971$	−53121.2	−1.75565	−0.877827	−	0.478978i	$-0.841007\pi$
$971$	−53121.2	−1.75565	−0.877827	+	0.478978i	$0.841007\pi$
$972$	0	0
$973$	−16507.4	−0.543887
$974$	21771.5	0.716225
$975$	0	0
$976$	15115.9	0.495745
$977$	11393.0	0.373075	0.186537	−	0.982448i	$-0.440273\pi$
$977$	11393.0	0.373075	0.186537	+	0.982448i	$0.440273\pi$
$978$	0	0
$979$	2806.32	0.0916143
$980$	3206.51	0.104519
$981$	0	0
$982$	59641.4	1.93812
$983$	−50295.2	−1.63191	−0.815955	−	0.578115i	$-0.803788\pi$
$983$	−50295.2	−1.63191	−0.815955	+	0.578115i	$0.803788\pi$
$984$	0	0
$985$	46864.0	1.51595
$986$	−73247.9	−2.36581
$987$	0	0
$988$	2029.69	0.0653572
$989$	28177.9	0.905970
$990$	0	0
$991$	−4361.21	−0.139797	−0.0698983	−	0.997554i	$-0.522267\pi$
$991$	−4361.21	−0.139797	−0.0698983	+	0.997554i	$0.522267\pi$
$992$	14993.6	0.479886
$993$	0	0
$994$	8737.54	0.278811
$995$	1024.90	0.0326549
$996$	0	0
$997$	−25946.3	−0.824199	−0.412100	−	0.911139i	$-0.635204\pi$
$997$	−25946.3	−0.824199	−0.412100	+	0.911139i	$0.635204\pi$
$998$	72157.9	2.28870
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.s.1.2	✓	8
3.2	odd	2		693.4.a.t.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.s.1.2	✓	8	1.1	even	1	trivial
693.4.a.t.1.7	yes	8	3.2	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)

\(f(q)\)	\(=\)	\(q-3.50363 q^{2} +4.27544 q^{4} +15.3058 q^{5} -7.00000 q^{7} +13.0495 q^{8} -53.6258 q^{10} +11.0000 q^{11} +5.90915 q^{13} +24.5254 q^{14} -79.9241 q^{16} -73.2134 q^{17} +80.3384 q^{19} +65.4390 q^{20} -38.5400 q^{22} -190.324 q^{23} +109.267 q^{25} -20.7035 q^{26} -29.9281 q^{28} -285.553 q^{29} +85.3709 q^{31} +175.629 q^{32} +256.513 q^{34} -107.140 q^{35} -149.061 q^{37} -281.476 q^{38} +199.732 q^{40} +13.9250 q^{41} -148.052 q^{43} +47.0299 q^{44} +666.824 q^{46} +94.7888 q^{47} +49.0000 q^{49} -382.831 q^{50} +25.2642 q^{52} +105.750 q^{53} +168.364 q^{55} -91.3463 q^{56} +1000.47 q^{58} +184.187 q^{59} -189.128 q^{61} -299.108 q^{62} +24.0535 q^{64} +90.4441 q^{65} -193.756 q^{67} -313.020 q^{68} +375.381 q^{70} +356.265 q^{71} +655.740 q^{73} +522.254 q^{74} +343.482 q^{76} -77.0000 q^{77} +110.189 q^{79} -1223.30 q^{80} -48.7881 q^{82} -1278.27 q^{83} -1120.59 q^{85} +518.721 q^{86} +143.544 q^{88} +255.120 q^{89} -41.3640 q^{91} -813.718 q^{92} -332.105 q^{94} +1229.64 q^{95} +372.537 q^{97} -171.678 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8} - 13 q^{10} + 88 q^{11} - 148 q^{13} + 14 q^{14} + 266 q^{16} - 114 q^{17} + 58 q^{19} - 291 q^{20} - 22 q^{22} - 246 q^{23} + 244 q^{25} - 305 q^{26}+ \cdots - 98 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 30 * q^4 - 10 * q^5 - 56 * q^7 - 15 * q^8 - 13 * q^10 + 88 * q^11 - 148 * q^13 + 14 * q^14 + 266 * q^16 - 114 * q^17 + 58 * q^19 - 291 * q^20 - 22 * q^22 - 246 * q^23 + 244 * q^25 - 305 * q^26 - 210 * q^28 + 72 * q^29 + 252 * q^31 - 1272 * q^32 + 630 * q^34 + 70 * q^35 - 80 * q^37 - 1885 * q^38 - 342 * q^40 - 682 * q^41 - 106 * q^43 + 330 * q^44 + 120 * q^46 - 828 * q^47 + 392 * q^49 - 801 * q^50 - 1681 * q^52 - 462 * q^53 - 110 * q^55 + 105 * q^56 - 1087 * q^58 - 626 * q^59 - 854 * q^61 - 1350 * q^62 + 2997 * q^64 - 22 * q^65 + 130 * q^67 - 2202 * q^68 + 91 * q^70 - 326 * q^71 - 390 * q^73 + 359 * q^74 + 2041 * q^76 - 616 * q^77 - 508 * q^79 - 4391 * q^80 + 1528 * q^82 - 1596 * q^83 - 880 * q^85 + 414 * q^86 - 165 * q^88 - 4324 * q^89 + 1036 * q^91 - 2092 * q^92 - 1685 * q^94 - 1076 * q^95 - 964 * q^97 - 98 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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