LMFDB - Embedded newform 693.4.a.t.1.7

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:	$8$
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			1.7
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	$-0.287392\pi$
$2$	3.50363	1.23872	0.619361	+	0.785107i	$0.287392\pi$
$3$	0	0
$3$	0	0
$4$	4.27544	0.534430
$5$	−15.3058	−1.36899	−0.684495	−	0.729017i	$-0.739977\pi$
$5$	−15.3058	−1.36899	−0.684495	+	0.729017i	$0.739977\pi$
$6$	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	$-0.325089\pi$
$17$	73.2134	1.04452	0.522260	+	0.852786i	$0.325089\pi$
$18$	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	$-0.168758\pi$
$23$	190.324	1.72544	0.862722	+	0.505678i	$0.168758\pi$
$24$	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	$-0.132790\pi$
$29$	285.553	1.82848	0.914238	+	0.405177i	$0.132790\pi$
$30$	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	$-0.508443\pi$
$41$	−13.9250	−0.0530420	−0.0265210	+	0.999648i	$0.508443\pi$
$42$	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	$-0.546990\pi$
$47$	−94.7888	−0.294178	−0.147089	+	0.989123i	$0.546990\pi$
$48$	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	$-0.543758\pi$
$53$	−105.750	−0.274074	−0.137037	+	0.990566i	$0.543758\pi$
$54$	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	$-0.565138\pi$
$59$	−184.187	−0.406424	−0.203212	+	0.979135i	$0.565138\pi$
$60$	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	$-0.596237\pi$
$71$	−356.265	−0.595505	−0.297752	+	0.954643i	$0.596237\pi$
$72$	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	$-0.179462\pi$
$83$	1278.27	1.69047	0.845233	+	0.534399i	$0.179462\pi$
$84$	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	$-0.548547\pi$
$89$	−255.120	−0.303850	−0.151925	+	0.988392i	$0.548547\pi$
$90$	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	$-0.309486\pi$
$101$	1143.78	1.12684	0.563419	+	0.826171i	$0.309486\pi$
$102$	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	$-0.250566\pi$
$107$	1562.49	1.41170	0.705849	+	0.708363i	$0.250566\pi$
$108$	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	$-0.454591\pi$
$113$	341.558	0.284345	0.142173	+	0.989842i	$0.454591\pi$
$114$	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	$-0.256735\pi$
$131$	2075.09	1.38398	0.691989	+	0.721908i	$0.256735\pi$
$132$	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	$-0.480230\pi$
$137$	199.066	0.124141	0.0620707	+	0.998072i	$0.480230\pi$
$138$	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	$-0.554232\pi$
$149$	−616.754	−0.339104	−0.169552	+	0.985521i	$0.554232\pi$
$150$	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	$-0.514261\pi$
$167$	−193.315	−0.0895759	−0.0447880	+	0.998997i	$0.514261\pi$
$168$	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	$-0.309809\pi$
$173$	2560.26	1.12516	0.562580	+	0.826743i	$0.309809\pi$
$174$	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	$-0.302620\pi$
$179$	2783.34	1.16221	0.581107	+	0.813827i	$0.302620\pi$
$180$	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	$-0.297146\pi$
$191$	3141.28	1.19003	0.595014	+	0.803715i	$0.297146\pi$
$192$	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	$-0.686775\pi$
$197$	−3061.85	−1.10735	−0.553675	+	0.832733i	$0.686775\pi$
$198$	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	$-0.476841\pi$
$227$	497.228	0.145384	0.0726921	+	0.997354i	$0.476841\pi$
$228$	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	$-0.261187\pi$
$233$	4849.94	1.36365	0.681825	+	0.731516i	$0.261187\pi$
$234$	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	$-0.0329819\pi$
$239$	7350.07	1.98927	0.994637	+	0.103430i	$0.0329819\pi$
$240$	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	$-0.684246\pi$
$251$	−4350.71	−1.09408	−0.547041	+	0.837106i	$0.684246\pi$
$252$	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	$-0.667066\pi$
$257$	−4128.98	−1.00217	−0.501087	+	0.865397i	$0.667066\pi$
$258$	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	$-0.797007\pi$
$263$	−6853.69	−1.60691	−0.803454	+	0.595367i	$0.797007\pi$
$264$	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	$-0.824693\pi$
$269$	−7519.13	−1.70427	−0.852136	+	0.523320i	$0.824693\pi$
$270$	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	$-0.448064\pi$
$281$	1530.30	0.324876	0.162438	+	0.986719i	$0.448064\pi$
$282$	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	$-0.348362\pi$
$293$	4599.78	0.917140	0.458570	+	0.888658i	$0.348362\pi$
$294$	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	$-0.448708\pi$
$311$	1759.91	0.320886	0.160443	+	0.987045i	$0.448708\pi$
$312$	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	$-0.446407\pi$
$317$	1891.58	0.335147	0.167573	+	0.985860i	$0.446407\pi$
$318$	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	$-0.297553\pi$
$347$	7678.94	1.18797	0.593987	+	0.804475i	$0.297553\pi$
$348$	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	$-0.464460\pi$
$353$	1477.92	0.222838	0.111419	+	0.993774i	$0.464460\pi$
$354$	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	$-0.468250\pi$
$359$	1354.69	0.199158	0.0995791	+	0.995030i	$0.468250\pi$
$360$	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	$-0.622858\pi$
$383$	−5643.47	−0.752918	−0.376459	+	0.926433i	$0.622858\pi$
$384$	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	$-0.841769\pi$
$389$	−13487.4	−1.75794	−0.878970	+	0.476877i	$0.841769\pi$
$390$	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	$-0.750942\pi$
$401$	−11389.7	−1.41839	−0.709196	+	0.705011i	$0.750942\pi$
$402$	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	$-0.340343\pi$
$419$	8247.54	0.961620	0.480810	+	0.876825i	$0.340343\pi$
$420$	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	$-0.698024\pi$
$431$	−10428.7	−1.16551	−0.582753	+	0.812649i	$0.698024\pi$
$432$	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	$-0.337142\pi$
$443$	9130.18	0.979206	0.489603	+	0.871946i	$0.337142\pi$
$444$	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	$-0.389718\pi$
$449$	6461.47	0.679143	0.339572	+	0.940580i	$0.389718\pi$
$450$	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	$-0.378156\pi$
$461$	7393.95	0.747007	0.373504	+	0.927629i	$0.378156\pi$
$462$	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	$-0.677579\pi$
$467$	−10685.1	−1.05878	−0.529389	+	0.848379i	$0.677579\pi$
$468$	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	$-0.834753\pi$
$479$	−18204.4	−1.73649	−0.868247	+	0.496133i	$0.834753\pi$
$480$	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	$-0.214043\pi$
$491$	17022.7	1.56461	0.782306	+	0.622894i	$0.214043\pi$
$492$	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	$-0.176708\pi$
$503$	19173.9	1.69965	0.849824	+	0.527067i	$0.176708\pi$
$504$	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	$-0.456620\pi$
$509$	3120.30	0.271719	0.135859	+	0.990728i	$0.456620\pi$
$510$	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	$-0.608240\pi$
$521$	−7932.70	−0.667059	−0.333530	+	0.942740i	$0.608240\pi$
$522$	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	$-0.472215\pi$
$557$	2292.04	0.174357	0.0871786	+	0.996193i	$0.472215\pi$
$558$	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	$-0.592890\pi$
$563$	−7686.51	−0.575395	−0.287698	+	0.957721i	$0.592890\pi$
$564$	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	$-0.512251\pi$
$569$	−1044.55	−0.0769592	−0.0384796	+	0.999259i	$0.512251\pi$
$570$	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	$-0.552760\pi$
$587$	−4693.01	−0.329985	−0.164993	+	0.986295i	$0.552760\pi$
$588$	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	$-0.341105\pi$
$593$	13825.6	0.957418	0.478709	+	0.877974i	$0.341105\pi$
$594$	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	$-0.346266\pi$
$599$	13616.7	0.928823	0.464411	+	0.885620i	$0.346266\pi$
$600$	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	$-0.324162\pi$
$617$	16084.3	1.04948	0.524742	+	0.851261i	$0.324162\pi$
$618$	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	$-0.400911\pi$
$641$	9941.56	0.612586	0.306293	+	0.951937i	$0.400911\pi$
$642$	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	$-0.424147\pi$
$647$	7769.45	0.472100	0.236050	+	0.971741i	$0.424147\pi$
$648$	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	$-0.672800\pi$
$653$	−17240.5	−1.03319	−0.516594	+	0.856230i	$0.672800\pi$
$654$	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	$-0.791330\pi$
$659$	−26820.8	−1.58542	−0.792709	+	0.609600i	$0.791330\pi$
$660$	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	$-0.268905\pi$
$677$	23388.8	1.32778	0.663889	+	0.747831i	$0.268905\pi$
$678$	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	$-0.420109\pi$
$683$	8866.19	0.496713	0.248357	+	0.968669i	$0.420109\pi$
$684$	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	$-0.325692\pi$
$701$	19326.3	1.04129	0.520644	+	0.853774i	$0.325692\pi$
$702$	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	$-0.529074\pi$
$719$	−3517.02	−0.182424	−0.0912119	+	0.995832i	$0.529074\pi$
$720$	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	$-0.388835\pi$
$743$	13860.1	0.684357	0.342178	+	0.939635i	$0.388835\pi$
$744$	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	$-0.608389\pi$
$761$	−14022.2	−0.667942	−0.333971	+	0.942583i	$0.608389\pi$
$762$	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	$-0.960298\pi$
$773$	−42649.3	−1.98446	−0.992232	+	0.124405i	$0.960298\pi$
$774$	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	$-0.746113\pi$
$797$	−31429.2	−1.39684	−0.698419	+	0.715690i	$0.746113\pi$
$798$	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	$-0.634249\pi$
$809$	−18839.2	−0.818727	−0.409363	+	0.912371i	$0.634249\pi$
$810$	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	$-0.735399\pi$
$821$	−31707.7	−1.34788	−0.673938	+	0.738788i	$0.735399\pi$
$822$	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	$-0.432506\pi$
$827$	10010.3	0.420909	0.210455	+	0.977604i	$0.432506\pi$
$828$	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	$-0.543042\pi$
$839$	−6552.24	−0.269617	−0.134808	+	0.990872i	$0.543042\pi$
$840$	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	$-0.531544\pi$
$857$	−4964.30	−0.197873	−0.0989365	+	0.995094i	$0.531544\pi$
$858$	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	$-0.429118\pi$
$863$	11197.9	0.441692	0.220846	+	0.975309i	$0.429118\pi$
$864$	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	$-0.740713\pi$
$881$	−35886.4	−1.37236	−0.686178	+	0.727434i	$0.740713\pi$
$882$	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	$-0.657799\pi$
$887$	−25132.3	−0.951365	−0.475682	+	0.879617i	$0.657799\pi$
$888$	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	$-0.785835\pi$
$911$	−43008.3	−1.56414	−0.782068	+	0.623193i	$0.785835\pi$
$912$	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	$-0.662269\pi$
$929$	−27635.2	−0.975975	−0.487988	+	0.872851i	$0.662269\pi$
$930$	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	$-0.375731\pi$
$941$	21970.4	0.761122	0.380561	+	0.924756i	$0.375731\pi$
$942$	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	$-0.396825\pi$
$947$	18563.0	0.636976	0.318488	+	0.947927i	$0.396825\pi$
$948$	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	$-0.503475\pi$
$953$	−642.345	−0.0218338	−0.0109169	+	0.999940i	$0.503475\pi$
$954$	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	$-0.158993\pi$
$971$	53121.2	1.75565	0.877827	+	0.478978i	$0.158993\pi$
$972$	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	$-0.559727\pi$
$977$	−11393.0	−0.373075	−0.186537	+	0.982448i	$0.559727\pi$
$978$	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	$-0.196212\pi$
$983$	50295.2	1.63191	0.815955	+	0.578115i	$0.196212\pi$
$984$	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.t.1.7	yes	8
3.2	odd	2		693.4.a.s.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.s.1.2	✓	8	3.2	odd	2
693.4.a.t.1.7	yes	8	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:	\( k \)	\(=\)	\( 4 \)
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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