LMFDB - Embedded newform 735.4.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,4,Mod(1,735)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, 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("735.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	735.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:	$43.3664038542$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.51264.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 15x^{2} + 16x + 46 $ x^4 - 2x^3 - 15x^2 + 16*x + 46
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.50184$ of defining polynomial
Character	$\chi$	$=$	735.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.912375 q^{2} -3.00000 q^{3} -7.16757 q^{4} +5.00000 q^{5} +2.73712 q^{6} +13.8385 q^{8} +9.00000 q^{9} -4.56187 q^{10} -21.7933 q^{11} +21.5027 q^{12} -34.4151 q^{13} -15.0000 q^{15} +44.7147 q^{16} +5.70794 q^{17} -8.21137 q^{18} +24.6906 q^{19} -35.8379 q^{20} +19.8837 q^{22} +92.6544 q^{23} -41.5155 q^{24} +25.0000 q^{25} +31.3995 q^{26} -27.0000 q^{27} -55.2788 q^{29} +13.6856 q^{30} +116.651 q^{31} -151.505 q^{32} +65.3800 q^{33} -5.20778 q^{34} -64.5082 q^{36} +90.6006 q^{37} -22.5271 q^{38} +103.245 q^{39} +69.1925 q^{40} +38.1301 q^{41} +229.920 q^{43} +156.205 q^{44} +45.0000 q^{45} -84.5355 q^{46} -19.6129 q^{47} -134.144 q^{48} -22.8094 q^{50} -17.1238 q^{51} +246.673 q^{52} -111.356 q^{53} +24.6341 q^{54} -108.967 q^{55} -74.0719 q^{57} +50.4350 q^{58} +402.344 q^{59} +107.514 q^{60} +359.954 q^{61} -106.429 q^{62} -219.488 q^{64} -172.075 q^{65} -59.6510 q^{66} -864.308 q^{67} -40.9120 q^{68} -277.963 q^{69} -1078.94 q^{71} +124.547 q^{72} +596.587 q^{73} -82.6617 q^{74} -75.0000 q^{75} -176.972 q^{76} -94.1984 q^{78} -723.054 q^{79} +223.573 q^{80} +81.0000 q^{81} -34.7889 q^{82} -1043.84 q^{83} +28.5397 q^{85} -209.773 q^{86} +165.836 q^{87} -301.587 q^{88} -243.768 q^{89} -41.0569 q^{90} -664.107 q^{92} -349.952 q^{93} +17.8943 q^{94} +123.453 q^{95} +454.514 q^{96} -609.252 q^{97} -196.140 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{2} - 12 q^{3} + 18 q^{4} + 20 q^{5} + 18 q^{6} - 30 q^{8} + 36 q^{9} - 30 q^{10} - 24 q^{11} - 54 q^{12} - 60 q^{15} - 46 q^{16} + 88 q^{17} - 54 q^{18} - 72 q^{19} + 90 q^{20} - 28 q^{22} + 90 q^{24}+ \cdots - 216 q^{99}+O(q^{100}) $ 4 * q - 6 * q^2 - 12 * q^3 + 18 * q^4 + 20 * q^5 + 18 * q^6 - 30 * q^8 + 36 * q^9 - 30 * q^10 - 24 * q^11 - 54 * q^12 - 60 * q^15 - 46 * q^16 + 88 * q^17 - 54 * q^18 - 72 * q^19 + 90 * q^20 - 28 * q^22 + 90 * q^24 + 100 * q^25 - 112 * q^26 - 108 * q^27 - 636 * q^29 + 90 * q^30 - 228 * q^31 - 126 * q^32 + 72 * q^33 + 48 * q^34 + 162 * q^36 - 68 * q^37 + 408 * q^38 - 150 * q^40 + 56 * q^41 + 20 * q^43 + 276 * q^44 + 180 * q^45 - 596 * q^46 + 744 * q^47 + 138 * q^48 - 150 * q^50 - 264 * q^51 + 768 * q^52 - 360 * q^53 + 162 * q^54 - 120 * q^55 + 216 * q^57 + 1352 * q^58 + 1828 * q^59 - 270 * q^60 - 1212 * q^61 + 1044 * q^62 - 718 * q^64 + 84 * q^66 - 1036 * q^67 + 1056 * q^68 - 876 * q^71 - 270 * q^72 - 420 * q^73 - 1584 * q^74 - 300 * q^75 - 2400 * q^76 + 336 * q^78 - 700 * q^79 - 230 * q^80 + 324 * q^81 - 492 * q^82 + 1020 * q^83 + 440 * q^85 + 636 * q^86 + 1908 * q^87 - 748 * q^88 - 1192 * q^89 - 270 * q^90 - 540 * q^92 + 684 * q^93 - 3732 * q^94 - 360 * q^95 + 378 * q^96 - 588 * q^97 - 216 * q^99

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 $

$2$	−0.912375	−0.322573	−0.161287	−	0.986908i	$-0.551564\pi$
$2$	−0.912375	−0.322573	−0.161287	+	0.986908i	$0.551564\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.16757	−0.895947
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	2.73712	0.186238
$7$	0	0
$7$	0	0
$8$	13.8385	0.611581
$9$	9.00000	0.333333
$10$	−4.56187	−0.144259
$11$	−21.7933	−0.597358	−0.298679	−	0.954354i	$-0.596546\pi$
$11$	−21.7933	−0.597358	−0.298679	+	0.954354i	$0.596546\pi$
$12$	21.5027	0.517275
$13$	−34.4151	−0.734233	−0.367117	−	0.930175i	$-0.619655\pi$
$13$	−34.4151	−0.734233	−0.367117	+	0.930175i	$0.619655\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	44.7147	0.698667
$17$	5.70794	0.0814340	0.0407170	−	0.999171i	$-0.487036\pi$
$17$	5.70794	0.0814340	0.0407170	+	0.999171i	$0.487036\pi$
$18$	−8.21137	−0.107524
$19$	24.6906	0.298127	0.149064	−	0.988828i	$-0.452374\pi$
$19$	24.6906	0.298127	0.149064	+	0.988828i	$0.452374\pi$
$20$	−35.8379	−0.400679
$21$	0	0
$22$	19.8837	0.192692
$23$	92.6544	0.839990	0.419995	−	0.907526i	$-0.362032\pi$
$23$	92.6544	0.839990	0.419995	+	0.907526i	$0.362032\pi$
$24$	−41.5155	−0.353097
$25$	25.0000	0.200000
$26$	31.3995	0.236844
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−55.2788	−0.353966	−0.176983	−	0.984214i	$-0.556634\pi$
$29$	−55.2788	−0.353966	−0.176983	+	0.984214i	$0.556634\pi$
$30$	13.6856	0.0832880
$31$	116.651	0.675841	0.337920	−	0.941175i	$-0.390277\pi$
$31$	116.651	0.675841	0.337920	+	0.941175i	$0.390277\pi$
$32$	−151.505	−0.836953
$33$	65.3800	0.344885
$34$	−5.20778	−0.0262684
$35$	0	0
$36$	−64.5082	−0.298649
$37$	90.6006	0.402558	0.201279	−	0.979534i	$-0.435490\pi$
$37$	90.6006	0.402558	0.201279	+	0.979534i	$0.435490\pi$
$38$	−22.5271	−0.0961679
$39$	103.245	0.423910
$40$	69.1925	0.273508
$41$	38.1301	0.145242	0.0726209	−	0.997360i	$-0.476864\pi$
$41$	38.1301	0.145242	0.0726209	+	0.997360i	$0.476864\pi$
$42$	0	0
$43$	229.920	0.815407	0.407703	−	0.913114i	$-0.366330\pi$
$43$	229.920	0.815407	0.407703	+	0.913114i	$0.366330\pi$
$44$	156.205	0.535201
$45$	45.0000	0.149071
$46$	−84.5355	−0.270958
$47$	−19.6129	−0.0608689	−0.0304345	−	0.999537i	$-0.509689\pi$
$47$	−19.6129	−0.0608689	−0.0304345	+	0.999537i	$0.509689\pi$
$48$	−134.144	−0.403375
$49$	0	0
$50$	−22.8094	−0.0645146
$51$	−17.1238	−0.0470159
$52$	246.673	0.657834
$53$	−111.356	−0.288603	−0.144302	−	0.989534i	$-0.546094\pi$
$53$	−111.356	−0.288603	−0.144302	+	0.989534i	$0.546094\pi$
$54$	24.6341	0.0620792
$55$	−108.967	−0.267147
$56$	0	0
$57$	−74.0719	−0.172124
$58$	50.4350	0.114180
$59$	402.344	0.887809	0.443905	−	0.896074i	$-0.353593\pi$
$59$	402.344	0.887809	0.443905	+	0.896074i	$0.353593\pi$
$60$	107.514	0.231332
$61$	359.954	0.755530	0.377765	−	0.925901i	$-0.376693\pi$
$61$	359.954	0.755530	0.377765	+	0.925901i	$0.376693\pi$
$62$	−106.429	−0.218008
$63$	0	0
$64$	−219.488	−0.428688
$65$	−172.075	−0.328359
$66$	−59.6510	−0.111251
$67$	−864.308	−1.57600	−0.788000	−	0.615675i	$-0.788883\pi$
$67$	−864.308	−1.57600	−0.788000	+	0.615675i	$0.788883\pi$
$68$	−40.9120	−0.0729605
$69$	−277.963	−0.484969
$70$	0	0
$71$	−1078.94	−1.80348	−0.901739	−	0.432280i	$-0.857709\pi$
$71$	−1078.94	−1.80348	−0.901739	+	0.432280i	$0.857709\pi$
$72$	124.547	0.203860
$73$	596.587	0.956509	0.478255	−	0.878221i	$-0.341270\pi$
$73$	596.587	0.956509	0.478255	+	0.878221i	$0.341270\pi$
$74$	−82.6617	−0.129854
$75$	−75.0000	−0.115470
$76$	−176.972	−0.267106
$77$	0	0
$78$	−94.1984	−0.136742
$79$	−723.054	−1.02975	−0.514873	−	0.857267i	$-0.672161\pi$
$79$	−723.054	−1.02975	−0.514873	+	0.857267i	$0.672161\pi$
$80$	223.573	0.312453
$81$	81.0000	0.111111
$82$	−34.7889	−0.0468511
$83$	−1043.84	−1.38044	−0.690219	−	0.723601i	$-0.742486\pi$
$83$	−1043.84	−1.38044	−0.690219	+	0.723601i	$0.742486\pi$
$84$	0	0
$85$	28.5397	0.0364184
$86$	−209.773	−0.263028
$87$	165.836	0.204362
$88$	−301.587	−0.365333
$89$	−243.768	−0.290329	−0.145165	−	0.989408i	$-0.546371\pi$
$89$	−243.768	−0.290329	−0.145165	+	0.989408i	$0.546371\pi$
$90$	−41.0569	−0.0480864
$91$	0	0
$92$	−664.107	−0.752586
$93$	−349.952	−0.390197
$94$	17.8943	0.0196347
$95$	123.453	0.133327
$96$	454.514	0.483215
$97$	−609.252	−0.637734	−0.318867	−	0.947800i	$-0.603302\pi$
$97$	−609.252	−0.637734	−0.318867	+	0.947800i	$0.603302\pi$
$98$	0	0
$99$	−196.140	−0.199119
$100$	−179.189	−0.179189
$101$	441.823	0.435278	0.217639	−	0.976029i	$-0.430164\pi$
$101$	441.823	0.435278	0.217639	+	0.976029i	$0.430164\pi$
$102$	15.6233	0.0151661
$103$	−806.414	−0.771440	−0.385720	−	0.922616i	$-0.626047\pi$
$103$	−806.414	−0.771440	−0.385720	+	0.922616i	$0.626047\pi$
$104$	−476.254	−0.449043
$105$	0	0
$106$	101.599	0.0930956
$107$	−618.019	−0.558375	−0.279188	−	0.960237i	$-0.590065\pi$
$107$	−618.019	−0.558375	−0.279188	+	0.960237i	$0.590065\pi$
$108$	193.524	0.172425
$109$	−2147.07	−1.88671	−0.943357	−	0.331779i	$-0.892351\pi$
$109$	−2147.07	−1.88671	−0.943357	+	0.331779i	$0.892351\pi$
$110$	99.4184	0.0861743
$111$	−271.802	−0.232417
$112$	0	0
$113$	−1235.34	−1.02842	−0.514209	−	0.857665i	$-0.671915\pi$
$113$	−1235.34	−1.02842	−0.514209	+	0.857665i	$0.671915\pi$
$114$	67.5813	0.0555226
$115$	463.272	0.375655
$116$	396.215	0.317135
$117$	−309.736	−0.244744
$118$	−367.089	−0.286383
$119$	0	0
$120$	−207.578	−0.157910
$121$	−856.051	−0.643164
$122$	−328.413	−0.243714
$123$	−114.390	−0.0838554
$124$	−836.102	−0.605517
$125$	125.000	0.0894427
$126$	0	0
$127$	2200.67	1.53762	0.768811	−	0.639475i	$-0.220848\pi$
$127$	2200.67	1.53762	0.768811	+	0.639475i	$0.220848\pi$
$128$	1412.29	0.975236
$129$	−689.760	−0.470775
$130$	156.997	0.105920
$131$	2656.55	1.77179	0.885893	−	0.463890i	$-0.153547\pi$
$131$	2656.55	1.77179	0.885893	+	0.463890i	$0.153547\pi$
$132$	−468.616	−0.308998
$133$	0	0
$134$	788.572	0.508375
$135$	−135.000	−0.0860663
$136$	78.9893	0.0498035
$137$	−2143.69	−1.33684	−0.668422	−	0.743782i	$-0.733030\pi$
$137$	−2143.69	−1.33684	−0.668422	+	0.743782i	$0.733030\pi$
$138$	253.607	0.156438
$139$	−2827.76	−1.72552	−0.862761	−	0.505612i	$-0.831267\pi$
$139$	−2827.76	−1.72552	−0.862761	+	0.505612i	$0.831267\pi$
$140$	0	0
$141$	58.8388	0.0351427
$142$	984.400	0.581754
$143$	750.020	0.438600
$144$	402.432	0.232889
$145$	−276.394	−0.158298
$146$	−544.311	−0.308544
$147$	0	0
$148$	−649.386	−0.360670
$149$	−922.724	−0.507332	−0.253666	−	0.967292i	$-0.581636\pi$
$149$	−922.724	−0.507332	−0.253666	+	0.967292i	$0.581636\pi$
$150$	68.4281	0.0372475
$151$	−835.324	−0.450183	−0.225092	−	0.974338i	$-0.572268\pi$
$151$	−835.324	−0.450183	−0.225092	+	0.974338i	$0.572268\pi$
$152$	341.682	0.182329
$153$	51.3714	0.0271447
$154$	0	0
$155$	583.253	0.302245
$156$	−740.018	−0.379800
$157$	−2055.05	−1.04465	−0.522327	−	0.852745i	$-0.674936\pi$
$157$	−2055.05	−1.04465	−0.522327	+	0.852745i	$0.674936\pi$
$158$	659.696	0.332168
$159$	334.069	0.166625
$160$	−757.523	−0.374297
$161$	0	0
$162$	−73.9023	−0.0358415
$163$	1361.53	0.654251	0.327126	−	0.944981i	$-0.393920\pi$
$163$	1361.53	0.654251	0.327126	+	0.944981i	$0.393920\pi$
$164$	−273.300	−0.130129
$165$	326.900	0.154237
$166$	952.373	0.445292
$167$	1042.90	0.483247	0.241623	−	0.970370i	$-0.422320\pi$
$167$	1042.90	0.483247	0.241623	+	0.970370i	$0.422320\pi$
$168$	0	0
$169$	−1012.60	−0.460902
$170$	−26.0389	−0.0117476
$171$	222.216	0.0993758
$172$	−1647.97	−0.730561
$173$	3397.23	1.49299	0.746494	−	0.665392i	$-0.231736\pi$
$173$	3397.23	1.49299	0.746494	+	0.665392i	$0.231736\pi$
$174$	−151.305	−0.0659218
$175$	0	0
$176$	−974.482	−0.417354
$177$	−1207.03	−0.512577
$178$	222.407	0.0936524
$179$	125.381	0.0523543	0.0261771	−	0.999657i	$-0.491667\pi$
$179$	125.381	0.0523543	0.0261771	+	0.999657i	$0.491667\pi$
$180$	−322.541	−0.133560
$181$	988.234	0.405828	0.202914	−	0.979197i	$-0.434959\pi$
$181$	988.234	0.405828	0.202914	+	0.979197i	$0.434959\pi$
$182$	0	0
$183$	−1079.86	−0.436206
$184$	1282.20	0.513722
$185$	453.003	0.180029
$186$	319.287	0.125867
$187$	−124.395	−0.0486452
$188$	140.577	0.0545353
$189$	0	0
$190$	−112.636	−0.0430076
$191$	−2669.26	−1.01121	−0.505604	−	0.862766i	$-0.668730\pi$
$191$	−2669.26	−1.01121	−0.505604	+	0.862766i	$0.668730\pi$
$192$	658.465	0.247503
$193$	−2228.29	−0.831067	−0.415534	−	0.909578i	$-0.636405\pi$
$193$	−2228.29	−0.831067	−0.415534	+	0.909578i	$0.636405\pi$
$194$	555.866	0.205716
$195$	516.226	0.189578
$196$	0	0
$197$	−365.525	−0.132196	−0.0660979	−	0.997813i	$-0.521055\pi$
$197$	−365.525	−0.132196	−0.0660979	+	0.997813i	$0.521055\pi$
$198$	178.953	0.0642305
$199$	−2789.70	−0.993751	−0.496875	−	0.867822i	$-0.665519\pi$
$199$	−2789.70	−0.993751	−0.496875	+	0.867822i	$0.665519\pi$
$200$	345.963	0.122316
$201$	2592.92	0.909904
$202$	−403.108	−0.140409
$203$	0	0
$204$	122.736	0.0421238
$205$	190.650	0.0649541
$206$	735.752	0.248846
$207$	833.890	0.279997
$208$	−1538.86	−0.512984
$209$	−538.091	−0.178089
$210$	0	0
$211$	−578.631	−0.188789	−0.0943947	−	0.995535i	$-0.530092\pi$
$211$	−578.631	−0.188789	−0.0943947	+	0.995535i	$0.530092\pi$
$212$	798.155	0.258573
$213$	3236.83	1.04124
$214$	563.865	0.180117
$215$	1149.60	0.364661
$216$	−373.640	−0.117699
$217$	0	0
$218$	1958.93	0.608603
$219$	−1789.76	−0.552241
$220$	781.026	0.239349
$221$	−196.439	−0.0597915
$222$	247.985	0.0749715
$223$	4664.97	1.40085	0.700424	−	0.713727i	$-0.252994\pi$
$223$	4664.97	1.40085	0.700424	+	0.713727i	$0.252994\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	1127.10	0.331740
$227$	2575.43	0.753028	0.376514	−	0.926411i	$-0.377123\pi$
$227$	2575.43	0.753028	0.376514	+	0.926411i	$0.377123\pi$
$228$	530.916	0.154214
$229$	−4924.10	−1.42093	−0.710467	−	0.703731i	$-0.751516\pi$
$229$	−4924.10	−1.42093	−0.710467	+	0.703731i	$0.751516\pi$
$230$	−422.678	−0.121176
$231$	0	0
$232$	−764.976	−0.216479
$233$	6013.17	1.69071	0.845356	−	0.534203i	$-0.179388\pi$
$233$	6013.17	1.69071	0.845356	+	0.534203i	$0.179388\pi$
$234$	282.595	0.0789479
$235$	−98.0646	−0.0272214
$236$	−2883.83	−0.795430
$237$	2169.16	0.594524
$238$	0	0
$239$	302.198	0.0817890	0.0408945	−	0.999163i	$-0.486979\pi$
$239$	302.198	0.0817890	0.0408945	+	0.999163i	$0.486979\pi$
$240$	−670.720	−0.180395
$241$	−4196.17	−1.12157	−0.560786	−	0.827961i	$-0.689501\pi$
$241$	−4196.17	−1.12157	−0.560786	+	0.827961i	$0.689501\pi$
$242$	781.039	0.207467
$243$	−243.000	−0.0641500
$244$	−2579.99	−0.676915
$245$	0	0
$246$	104.367	0.0270495
$247$	−849.731	−0.218895
$248$	1614.27	0.413332
$249$	3131.52	0.796996
$250$	−114.047	−0.0288518
$251$	7388.87	1.85809	0.929046	−	0.369964i	$-0.120630\pi$
$251$	7388.87	1.85809	0.929046	+	0.369964i	$0.120630\pi$
$252$	0	0
$253$	−2019.25	−0.501775
$254$	−2007.84	−0.495996
$255$	−85.6190	−0.0210262
$256$	467.368	0.114104
$257$	−290.777	−0.0705766	−0.0352883	−	0.999377i	$-0.511235\pi$
$257$	−290.777	−0.0705766	−0.0352883	+	0.999377i	$0.511235\pi$
$258$	629.320	0.151859
$259$	0	0
$260$	1233.36	0.294192
$261$	−497.509	−0.117989
$262$	−2423.77	−0.571530
$263$	1165.41	0.273241	0.136620	−	0.990623i	$-0.456376\pi$
$263$	1165.41	0.273241	0.136620	+	0.990623i	$0.456376\pi$
$264$	904.762	0.210925
$265$	−556.782	−0.129067
$266$	0	0
$267$	731.303	0.167622
$268$	6194.99	1.41201
$269$	−4355.80	−0.987278	−0.493639	−	0.869667i	$-0.664334\pi$
$269$	−4355.80	−0.987278	−0.493639	+	0.869667i	$0.664334\pi$
$270$	123.171	0.0277627
$271$	−4132.57	−0.926330	−0.463165	−	0.886272i	$-0.653286\pi$
$271$	−4132.57	−0.926330	−0.463165	+	0.886272i	$0.653286\pi$
$272$	255.229	0.0568952
$273$	0	0
$274$	1955.85	0.431230
$275$	−544.833	−0.119472
$276$	1992.32	0.434506
$277$	−2253.23	−0.488749	−0.244374	−	0.969681i	$-0.578583\pi$
$277$	−2253.23	−0.488749	−0.244374	+	0.969681i	$0.578583\pi$
$278$	2579.98	0.556607
$279$	1049.86	0.225280
$280$	0	0
$281$	−4130.62	−0.876912	−0.438456	−	0.898753i	$-0.644474\pi$
$281$	−4130.62	−0.876912	−0.438456	+	0.898753i	$0.644474\pi$
$282$	−53.6830	−0.0113361
$283$	−5688.49	−1.19486	−0.597430	−	0.801921i	$-0.703811\pi$
$283$	−5688.49	−1.19486	−0.597430	+	0.801921i	$0.703811\pi$
$284$	7733.40	1.61582
$285$	−370.360	−0.0769762
$286$	−684.299	−0.141481
$287$	0	0
$288$	−1363.54	−0.278984
$289$	−4880.42	−0.993369
$290$	252.175	0.0510628
$291$	1827.76	0.368196
$292$	−4276.08	−0.856981
$293$	4506.38	0.898518	0.449259	−	0.893402i	$-0.351688\pi$
$293$	4506.38	0.898518	0.449259	+	0.893402i	$0.351688\pi$
$294$	0	0
$295$	2011.72	0.397040
$296$	1253.78	0.246197
$297$	588.420	0.114962
$298$	841.870	0.163652
$299$	−3188.71	−0.616749
$300$	537.568	0.103455
$301$	0	0
$302$	762.128	0.145217
$303$	−1325.47	−0.251308
$304$	1104.03	0.208292
$305$	1799.77	0.337883
$306$	−46.8700	−0.00875614
$307$	5380.42	1.00025	0.500125	−	0.865953i	$-0.333287\pi$
$307$	5380.42	1.00025	0.500125	+	0.865953i	$0.333287\pi$
$308$	0	0
$309$	2419.24	0.445391
$310$	−532.145	−0.0974962
$311$	−1862.37	−0.339568	−0.169784	−	0.985481i	$-0.554307\pi$
$311$	−1862.37	−0.339568	−0.169784	+	0.985481i	$0.554307\pi$
$312$	1428.76	0.259255
$313$	−7756.59	−1.40073	−0.700364	−	0.713785i	$-0.746979\pi$
$313$	−7756.59	−1.40073	−0.700364	+	0.713785i	$0.746979\pi$
$314$	1874.97	0.336977
$315$	0	0
$316$	5182.54	0.922597
$317$	−4108.22	−0.727888	−0.363944	−	0.931421i	$-0.618570\pi$
$317$	−4108.22	−0.727888	−0.363944	+	0.931421i	$0.618570\pi$
$318$	−304.796	−0.0537488
$319$	1204.71	0.211444
$320$	−1097.44	−0.191715
$321$	1854.06	0.322378
$322$	0	0
$323$	140.933	0.0242777
$324$	−580.573	−0.0995496
$325$	−860.377	−0.146847
$326$	−1242.22	−0.211044
$327$	6441.21	1.08930
$328$	527.663	0.0888272
$329$	0	0
$330$	−298.255	−0.0497528
$331$	10738.0	1.78312	0.891561	−	0.452901i	$-0.149611\pi$
$331$	10738.0	1.78312	0.891561	+	0.452901i	$0.149611\pi$
$332$	7481.80	1.23680
$333$	815.405	0.134186
$334$	−951.518	−0.155882
$335$	−4321.54	−0.704809
$336$	0	0
$337$	6108.83	0.987446	0.493723	−	0.869619i	$-0.335636\pi$
$337$	6108.83	0.987446	0.493723	+	0.869619i	$0.335636\pi$
$338$	923.872	0.148675
$339$	3706.03	0.593758
$340$	−204.560	−0.0326289
$341$	−2542.21	−0.403719
$342$	−202.744	−0.0320560
$343$	0	0
$344$	3181.75	0.498688
$345$	−1389.82	−0.216885
$346$	−3099.55	−0.481598
$347$	4083.12	0.631680	0.315840	−	0.948812i	$-0.397714\pi$
$347$	4083.12	0.631680	0.315840	+	0.948812i	$0.397714\pi$
$348$	−1188.64	−0.183098
$349$	−6984.77	−1.07131	−0.535653	−	0.844438i	$-0.679935\pi$
$349$	−6984.77	−1.07131	−0.535653	+	0.844438i	$0.679935\pi$
$350$	0	0
$351$	929.207	0.141303
$352$	3301.79	0.499960
$353$	−5150.68	−0.776610	−0.388305	−	0.921531i	$-0.626939\pi$
$353$	−5150.68	−0.776610	−0.388305	+	0.921531i	$0.626939\pi$
$354$	1101.27	0.165344
$355$	−5394.71	−0.806540
$356$	1747.22	0.260120
$357$	0	0
$358$	−114.394	−0.0168881
$359$	−9534.37	−1.40169	−0.700843	−	0.713316i	$-0.747192\pi$
$359$	−9534.37	−1.40169	−0.700843	+	0.713316i	$0.747192\pi$
$360$	622.733	0.0911692
$361$	−6249.37	−0.911120
$362$	−901.640	−0.130909
$363$	2568.15	0.371331
$364$	0	0
$365$	2982.93	0.427764
$366$	985.238	0.140708
$367$	−2838.67	−0.403752	−0.201876	−	0.979411i	$-0.564704\pi$
$367$	−2838.67	−0.403752	−0.201876	+	0.979411i	$0.564704\pi$
$368$	4143.01	0.586873
$369$	343.170	0.0484139
$370$	−413.308	−0.0580727
$371$	0	0
$372$	2508.31	0.349596
$373$	5050.10	0.701030	0.350515	−	0.936557i	$-0.386007\pi$
$373$	5050.10	0.701030	0.350515	+	0.936557i	$0.386007\pi$
$374$	113.495	0.0156916
$375$	−375.000	−0.0516398
$376$	−271.414	−0.0372263
$377$	1902.42	0.259893
$378$	0	0
$379$	10250.5	1.38927	0.694633	−	0.719365i	$-0.255567\pi$
$379$	10250.5	1.38927	0.694633	+	0.719365i	$0.255567\pi$
$380$	−884.860	−0.119454
$381$	−6602.02	−0.887747
$382$	2435.36	0.326189
$383$	10788.3	1.43931	0.719656	−	0.694330i	$-0.244299\pi$
$383$	10788.3	1.43931	0.719656	+	0.694330i	$0.244299\pi$
$384$	−4236.88	−0.563053
$385$	0	0
$386$	2033.04	0.268080
$387$	2069.28	0.271802
$388$	4366.86	0.571375
$389$	13666.5	1.78129	0.890644	−	0.454701i	$-0.150254\pi$
$389$	13666.5	1.78129	0.890644	+	0.454701i	$0.150254\pi$
$390$	−470.992	−0.0611528
$391$	528.865	0.0684038
$392$	0	0
$393$	−7969.65	−1.02294
$394$	333.496	0.0426428
$395$	−3615.27	−0.460516
$396$	1405.85	0.178400
$397$	4595.66	0.580982	0.290491	−	0.956878i	$-0.406181\pi$
$397$	4595.66	0.580982	0.290491	+	0.956878i	$0.406181\pi$
$398$	2545.25	0.320557
$399$	0	0
$400$	1117.87	0.139733
$401$	743.095	0.0925396	0.0462698	−	0.998929i	$-0.485267\pi$
$401$	743.095	0.0925396	0.0462698	+	0.998929i	$0.485267\pi$
$402$	−2365.72	−0.293511
$403$	−4014.54	−0.496225
$404$	−3166.80	−0.389986
$405$	405.000	0.0496904
$406$	0	0
$407$	−1974.49	−0.240471
$408$	−236.968	−0.0287541
$409$	−10501.4	−1.26959	−0.634793	−	0.772683i	$-0.718914\pi$
$409$	−10501.4	−1.26959	−0.634793	+	0.772683i	$0.718914\pi$
$410$	−173.944	−0.0209525
$411$	6431.06	0.771827
$412$	5780.03	0.691169
$413$	0	0
$414$	−760.820	−0.0903194
$415$	−5219.20	−0.617350
$416$	5214.04	0.614518
$417$	8483.28	0.996230
$418$	490.941	0.0574467
$419$	−5007.49	−0.583847	−0.291924	−	0.956442i	$-0.594295\pi$
$419$	−5007.49	−0.583847	−0.291924	+	0.956442i	$0.594295\pi$
$420$	0	0
$421$	3845.77	0.445205	0.222603	−	0.974909i	$-0.428545\pi$
$421$	3845.77	0.445205	0.222603	+	0.974909i	$0.428545\pi$
$422$	527.928	0.0608984
$423$	−176.516	−0.0202896
$424$	−1541.01	−0.176504
$425$	142.698	0.0162868
$426$	−2953.20	−0.335876
$427$	0	0
$428$	4429.70	0.500275
$429$	−2250.06	−0.253226
$430$	−1048.87	−0.117630
$431$	2164.42	0.241894	0.120947	−	0.992659i	$-0.461407\pi$
$431$	2164.42	0.241894	0.120947	+	0.992659i	$0.461407\pi$
$432$	−1207.30	−0.134458
$433$	9645.06	1.07047	0.535233	−	0.844704i	$-0.320224\pi$
$433$	9645.06	1.07047	0.535233	+	0.844704i	$0.320224\pi$
$434$	0	0
$435$	829.182	0.0913936
$436$	15389.3	1.69040
$437$	2287.70	0.250424
$438$	1632.93	0.178138
$439$	−3346.93	−0.363873	−0.181936	−	0.983310i	$-0.558236\pi$
$439$	−3346.93	−0.363873	−0.181936	+	0.983310i	$0.558236\pi$
$440$	−1507.94	−0.163382
$441$	0	0
$442$	179.226	0.0192871
$443$	−1595.52	−0.171118	−0.0855592	−	0.996333i	$-0.527268\pi$
$443$	−1595.52	−0.171118	−0.0855592	+	0.996333i	$0.527268\pi$
$444$	1948.16	0.208233
$445$	−1218.84	−0.129839
$446$	−4256.20	−0.451876
$447$	2768.17	0.292909
$448$	0	0
$449$	−4120.69	−0.433112	−0.216556	−	0.976270i	$-0.569482\pi$
$449$	−4120.69	−0.433112	−0.216556	+	0.976270i	$0.569482\pi$
$450$	−205.284	−0.0215049
$451$	−830.981	−0.0867613
$452$	8854.41	0.921408
$453$	2505.97	0.259913
$454$	−2349.76	−0.242907
$455$	0	0
$456$	−1025.04	−0.105268
$457$	−2387.24	−0.244355	−0.122177	−	0.992508i	$-0.538988\pi$
$457$	−2387.24	−0.244355	−0.122177	+	0.992508i	$0.538988\pi$
$458$	4492.62	0.458355
$459$	−154.114	−0.0156720
$460$	−3320.54	−0.336567
$461$	5144.27	0.519724	0.259862	−	0.965646i	$-0.416323\pi$
$461$	5144.27	0.519724	0.259862	+	0.965646i	$0.416323\pi$
$462$	0	0
$463$	−5168.77	−0.518818	−0.259409	−	0.965768i	$-0.583528\pi$
$463$	−5168.77	−0.518818	−0.259409	+	0.965768i	$0.583528\pi$
$464$	−2471.77	−0.247304
$465$	−1749.76	−0.174501
$466$	−5486.26	−0.545378
$467$	13240.9	1.31202	0.656011	−	0.754751i	$-0.272243\pi$
$467$	13240.9	1.31202	0.656011	+	0.754751i	$0.272243\pi$
$468$	2220.05	0.219278
$469$	0	0
$470$	89.4717	0.00878090
$471$	6165.14	0.603131
$472$	5567.84	0.542968
$473$	−5010.73	−0.487090
$474$	−1979.09	−0.191777
$475$	617.266	0.0596255
$476$	0	0
$477$	−1002.21	−0.0962011
$478$	−275.718	−0.0263829
$479$	−5317.53	−0.507232	−0.253616	−	0.967305i	$-0.581620\pi$
$479$	−5317.53	−0.507232	−0.253616	+	0.967305i	$0.581620\pi$
$480$	2272.57	0.216100
$481$	−3118.03	−0.295571
$482$	3828.48	0.361789
$483$	0	0
$484$	6135.81	0.576240
$485$	−3046.26	−0.285203
$486$	221.707	0.0206931
$487$	−6179.43	−0.574983	−0.287491	−	0.957783i	$-0.592821\pi$
$487$	−6179.43	−0.574983	−0.287491	+	0.957783i	$0.592821\pi$
$488$	4981.22	0.462068
$489$	−4084.58	−0.377732
$490$	0	0
$491$	−14637.1	−1.34534	−0.672670	−	0.739943i	$-0.734852\pi$
$491$	−14637.1	−1.34534	−0.672670	+	0.739943i	$0.734852\pi$
$492$	819.900	0.0751300
$493$	−315.528	−0.0288249
$494$	775.273	0.0706096
$495$	−980.700	−0.0890489
$496$	5215.99	0.472188
$497$	0	0
$498$	−2857.12	−0.257089
$499$	−19822.8	−1.77833	−0.889167	−	0.457582i	$-0.848716\pi$
$499$	−19822.8	−1.77833	−0.889167	+	0.457582i	$0.848716\pi$
$500$	−895.947	−0.0801359
$501$	−3128.71	−0.279003
$502$	−6741.41	−0.599371
$503$	11114.5	0.985228	0.492614	−	0.870248i	$-0.336041\pi$
$503$	11114.5	0.985228	0.492614	+	0.870248i	$0.336041\pi$
$504$	0	0
$505$	2209.12	0.194662
$506$	1842.31	0.161859
$507$	3037.80	0.266102
$508$	−15773.5	−1.37763
$509$	−16458.2	−1.43320	−0.716599	−	0.697485i	$-0.754302\pi$
$509$	−16458.2	−1.43320	−0.716599	+	0.697485i	$0.754302\pi$
$510$	78.1166	0.00678248
$511$	0	0
$512$	−11724.8	−1.01204
$513$	−666.647	−0.0573747
$514$	265.298	0.0227661
$515$	−4032.07	−0.344999
$516$	4943.91	0.421790
$517$	427.431	0.0363605
$518$	0	0
$519$	−10191.7	−0.861977
$520$	−2381.27	−0.200818
$521$	−957.425	−0.0805097	−0.0402548	−	0.999189i	$-0.512817\pi$
$521$	−957.425	−0.0805097	−0.0402548	+	0.999189i	$0.512817\pi$
$522$	453.915	0.0380600
$523$	17562.5	1.46836	0.734181	−	0.678954i	$-0.237566\pi$
$523$	17562.5	1.46836	0.734181	+	0.678954i	$0.237566\pi$
$524$	−19041.0	−1.58743
$525$	0	0
$526$	−1063.29	−0.0881402
$527$	665.834	0.0550364
$528$	2923.45	0.240960
$529$	−3582.16	−0.294416
$530$	507.993	0.0416336
$531$	3621.10	0.295936
$532$	0	0
$533$	−1312.25	−0.106641
$534$	−667.222	−0.0540703
$535$	−3090.10	−0.249713
$536$	−11960.7	−0.963852
$537$	−376.143	−0.0302268
$538$	3974.12	0.318469
$539$	0	0
$540$	967.622	0.0771108
$541$	154.715	0.0122953	0.00614763	−	0.999981i	$-0.498043\pi$
$541$	154.715	0.0122953	0.00614763	+	0.999981i	$0.498043\pi$
$542$	3770.45	0.298809
$543$	−2964.70	−0.234305
$544$	−864.779	−0.0681564
$545$	−10735.3	−0.843764
$546$	0	0
$547$	−1178.60	−0.0921264	−0.0460632	−	0.998939i	$-0.514668\pi$
$547$	−1178.60	−0.0921264	−0.0460632	+	0.998939i	$0.514668\pi$
$548$	15365.0	1.19774
$549$	3239.58	0.251843
$550$	497.092	0.0385383
$551$	−1364.87	−0.105527
$552$	−3846.60	−0.296598
$553$	0	0
$554$	2055.79	0.157657
$555$	−1359.01	−0.103940
$556$	20268.2	1.54598
$557$	−22450.8	−1.70785	−0.853924	−	0.520398i	$-0.825784\pi$
$557$	−22450.8	−1.70785	−0.853924	+	0.520398i	$0.825784\pi$
$558$	−957.862	−0.0726694
$559$	−7912.72	−0.598699
$560$	0	0
$561$	373.185	0.0280853
$562$	3768.67	0.282868
$563$	−16467.4	−1.23271	−0.616357	−	0.787467i	$-0.711392\pi$
$563$	−16467.4	−1.23271	−0.616357	+	0.787467i	$0.711392\pi$
$564$	−421.731	−0.0314860
$565$	−6176.72	−0.459923
$566$	5190.03	0.385430
$567$	0	0
$568$	−14931.0	−1.10297
$569$	−18743.5	−1.38096	−0.690480	−	0.723351i	$-0.742601\pi$
$569$	−18743.5	−1.38096	−0.690480	+	0.723351i	$0.742601\pi$
$570$	337.907	0.0248304
$571$	−6688.98	−0.490236	−0.245118	−	0.969493i	$-0.578827\pi$
$571$	−6688.98	−0.490236	−0.245118	+	0.969493i	$0.578827\pi$
$572$	−5375.82	−0.392962
$573$	8007.77	0.583821
$574$	0	0
$575$	2316.36	0.167998
$576$	−1975.40	−0.142896
$577$	−6065.16	−0.437601	−0.218800	−	0.975770i	$-0.570214\pi$
$577$	−6065.16	−0.437601	−0.218800	+	0.975770i	$0.570214\pi$
$578$	4452.77	0.320434
$579$	6684.88	0.479817
$580$	1981.07	0.141827
$581$	0	0
$582$	−1667.60	−0.118770
$583$	2426.83	0.172399
$584$	8255.87	0.584983
$585$	−1548.68	−0.109453
$586$	−4111.51	−0.289838
$587$	−24343.0	−1.71166	−0.855828	−	0.517260i	$-0.826952\pi$
$587$	−24343.0	−1.71166	−0.855828	+	0.517260i	$0.826952\pi$
$588$	0	0
$589$	2880.18	0.201487
$590$	−1835.44	−0.128075
$591$	1096.57	0.0763233
$592$	4051.18	0.281254
$593$	13866.9	0.960278	0.480139	−	0.877193i	$-0.340586\pi$
$593$	13866.9	0.960278	0.480139	+	0.877193i	$0.340586\pi$
$594$	−536.859	−0.0370835
$595$	0	0
$596$	6613.69	0.454543
$597$	8369.09	0.573742
$598$	2909.30	0.198947
$599$	6098.42	0.415985	0.207992	−	0.978130i	$-0.433307\pi$
$599$	6098.42	0.415985	0.207992	+	0.978130i	$0.433307\pi$
$600$	−1037.89	−0.0706193
$601$	−17556.7	−1.19160	−0.595802	−	0.803131i	$-0.703166\pi$
$601$	−17556.7	−1.19160	−0.595802	+	0.803131i	$0.703166\pi$
$602$	0	0
$603$	−7778.77	−0.525333
$604$	5987.24	0.403340
$605$	−4280.25	−0.287631
$606$	1209.33	0.0810652
$607$	8367.17	0.559494	0.279747	−	0.960074i	$-0.409749\pi$
$607$	8367.17	0.559494	0.279747	+	0.960074i	$0.409749\pi$
$608$	−3740.75	−0.249519
$609$	0	0
$610$	−1642.06	−0.108992
$611$	674.981	0.0446920
$612$	−368.208	−0.0243202
$613$	−26940.0	−1.77503	−0.887516	−	0.460776i	$-0.847571\pi$
$613$	−26940.0	−1.77503	−0.887516	+	0.460776i	$0.847571\pi$
$614$	−4908.96	−0.322654
$615$	−571.951	−0.0375013
$616$	0	0
$617$	−13320.3	−0.869136	−0.434568	−	0.900639i	$-0.643099\pi$
$617$	−13320.3	−0.869136	−0.434568	+	0.900639i	$0.643099\pi$
$618$	−2207.26	−0.143671
$619$	7336.05	0.476350	0.238175	−	0.971222i	$-0.423451\pi$
$619$	7336.05	0.476350	0.238175	+	0.971222i	$0.423451\pi$
$620$	−4180.51	−0.270796
$621$	−2501.67	−0.161656
$622$	1699.18	0.109535
$623$	0	0
$624$	4616.58	0.296172
$625$	625.000	0.0400000
$626$	7076.91	0.451838
$627$	1614.27	0.102820
$628$	14729.7	0.935954
$629$	517.142	0.0327819
$630$	0	0
$631$	103.633	0.00653815	0.00326908	−	0.999995i	$-0.498959\pi$
$631$	103.633	0.00653815	0.00326908	+	0.999995i	$0.498959\pi$
$632$	−10006.0	−0.629773
$633$	1735.89	0.108998
$634$	3748.23	0.234797
$635$	11003.4	0.687646
$636$	−2394.46	−0.149287
$637$	0	0
$638$	−1099.15	−0.0682063
$639$	−9710.49	−0.601159
$640$	7061.46	0.436139
$641$	9950.30	0.613125	0.306562	−	0.951851i	$-0.400821\pi$
$641$	9950.30	0.613125	0.306562	+	0.951851i	$0.400821\pi$
$642$	−1691.60	−0.103991
$643$	−1176.12	−0.0721334	−0.0360667	−	0.999349i	$-0.511483\pi$
$643$	−1176.12	−0.0721334	−0.0360667	+	0.999349i	$0.511483\pi$
$644$	0	0
$645$	−3448.80	−0.210537
$646$	−128.583	−0.00783134
$647$	5590.22	0.339682	0.169841	−	0.985471i	$-0.445675\pi$
$647$	5590.22	0.339682	0.169841	+	0.985471i	$0.445675\pi$
$648$	1120.92	0.0679535
$649$	−8768.42	−0.530340
$650$	784.986	0.0473688
$651$	0	0
$652$	−9758.83	−0.586174
$653$	24889.8	1.49160	0.745799	−	0.666171i	$-0.232068\pi$
$653$	24889.8	1.49160	0.745799	+	0.666171i	$0.232068\pi$
$654$	−5876.79	−0.351377
$655$	13282.8	0.792367
$656$	1704.97	0.101476
$657$	5369.28	0.318836
$658$	0	0
$659$	30597.0	1.80863	0.904317	−	0.426861i	$-0.140381\pi$
$659$	30597.0	1.80863	0.904317	+	0.426861i	$0.140381\pi$
$660$	−2343.08	−0.138188
$661$	11335.6	0.667024	0.333512	−	0.942746i	$-0.391766\pi$
$661$	11335.6	0.667024	0.333512	+	0.942746i	$0.391766\pi$
$662$	−9797.07	−0.575187
$663$	589.317	0.0345207
$664$	−14445.2	−0.844250
$665$	0	0
$666$	−743.955	−0.0432848
$667$	−5121.82	−0.297328
$668$	−7475.08	−0.432963
$669$	−13994.9	−0.808780
$670$	3942.86	0.227352
$671$	−7844.59	−0.451322
$672$	0	0
$673$	−7919.80	−0.453619	−0.226810	−	0.973939i	$-0.572830\pi$
$673$	−7919.80	−0.453619	−0.226810	+	0.973939i	$0.572830\pi$
$674$	−5573.54	−0.318524
$675$	−675.000	−0.0384900
$676$	7257.89	0.412943
$677$	−1189.67	−0.0675373	−0.0337686	−	0.999430i	$-0.510751\pi$
$677$	−1189.67	−0.0675373	−0.0337686	+	0.999430i	$0.510751\pi$
$678$	−3381.29	−0.191530
$679$	0	0
$680$	394.947	0.0222728
$681$	−7726.29	−0.434761
$682$	2319.44	0.130229
$683$	−15705.9	−0.879895	−0.439947	−	0.898024i	$-0.645003\pi$
$683$	−15705.9	−0.879895	−0.439947	+	0.898024i	$0.645003\pi$
$684$	−1592.75	−0.0890354
$685$	−10718.4	−0.597855
$686$	0	0
$687$	14772.3	0.820376
$688$	10280.8	0.569698
$689$	3832.34	0.211902
$690$	1268.03	0.0699611
$691$	21910.8	1.20626	0.603131	−	0.797642i	$-0.293920\pi$
$691$	21910.8	1.20626	0.603131	+	0.797642i	$0.293920\pi$
$692$	−24349.9	−1.33764
$693$	0	0
$694$	−3725.33	−0.203763
$695$	−14138.8	−0.771677
$696$	2294.93	0.124984
$697$	217.644	0.0118276
$698$	6372.72	0.345575
$699$	−18039.5	−0.976133
$700$	0	0
$701$	−11148.3	−0.600666	−0.300333	−	0.953834i	$-0.597098\pi$
$701$	−11148.3	−0.600666	−0.300333	+	0.953834i	$0.597098\pi$
$702$	−847.785	−0.0455806
$703$	2236.99	0.120014
$704$	4783.39	0.256080
$705$	294.194	0.0157163
$706$	4699.35	0.250513
$707$	0	0
$708$	8651.49	0.459241
$709$	−32409.1	−1.71671	−0.858357	−	0.513053i	$-0.828514\pi$
$709$	−32409.1	−1.71671	−0.858357	+	0.513053i	$0.828514\pi$
$710$	4922.00	0.260168
$711$	−6507.48	−0.343249
$712$	−3373.38	−0.177560
$713$	10808.2	0.567700
$714$	0	0
$715$	3750.10	0.196148
$716$	−898.677	−0.0469066
$717$	−906.594	−0.0472209
$718$	8698.92	0.452146
$719$	6557.21	0.340115	0.170058	−	0.985434i	$-0.445605\pi$
$719$	6557.21	0.340115	0.170058	+	0.985434i	$0.445605\pi$
$720$	2012.16	0.104151
$721$	0	0
$722$	5701.77	0.293903
$723$	12588.5	0.647540
$724$	−7083.24	−0.363600
$725$	−1381.97	−0.0707932
$726$	−2343.12	−0.119781
$727$	−13303.4	−0.678673	−0.339337	−	0.940665i	$-0.610203\pi$
$727$	−13303.4	−0.678673	−0.339337	+	0.940665i	$0.610203\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−2721.55	−0.137985
$731$	1312.37	0.0664018
$732$	7739.98	0.390817
$733$	−4179.54	−0.210607	−0.105304	−	0.994440i	$-0.533581\pi$
$733$	−4179.54	−0.210607	−0.105304	+	0.994440i	$0.533581\pi$
$734$	2589.93	0.130240
$735$	0	0
$736$	−14037.6	−0.703032
$737$	18836.1	0.941436
$738$	−313.100	−0.0156170
$739$	15706.6	0.781835	0.390918	−	0.920426i	$-0.372158\pi$
$739$	15706.6	0.781835	0.390918	+	0.920426i	$0.372158\pi$
$740$	−3246.93	−0.161297
$741$	2549.19	0.126379
$742$	0	0
$743$	−4706.99	−0.232413	−0.116206	−	0.993225i	$-0.537073\pi$
$743$	−4706.99	−0.232413	−0.116206	+	0.993225i	$0.537073\pi$
$744$	−4842.81	−0.238637
$745$	−4613.62	−0.226886
$746$	−4607.58	−0.226134
$747$	−9394.56	−0.460146
$748$	891.610	0.0435835
$749$	0	0
$750$	342.140	0.0166576
$751$	10206.7	0.495934	0.247967	−	0.968768i	$-0.420238\pi$
$751$	10206.7	0.495934	0.247967	+	0.968768i	$0.420238\pi$
$752$	−876.986	−0.0425271
$753$	−22166.6	−1.07277
$754$	−1735.72	−0.0838346
$755$	−4176.62	−0.201328
$756$	0	0
$757$	−2215.43	−0.106369	−0.0531843	−	0.998585i	$-0.516937\pi$
$757$	−2215.43	−0.106369	−0.0531843	+	0.998585i	$0.516937\pi$
$758$	−9352.27	−0.448140
$759$	6057.74	0.289700
$760$	1708.41	0.0815401
$761$	13185.5	0.628085	0.314043	−	0.949409i	$-0.398317\pi$
$761$	13185.5	0.628085	0.314043	+	0.949409i	$0.398317\pi$
$762$	6023.51	0.286363
$763$	0	0
$764$	19132.1	0.905988
$765$	256.857	0.0121395
$766$	−9842.98	−0.464284
$767$	−13846.7	−0.651859
$768$	−1402.10	−0.0658777
$769$	−20178.7	−0.946247	−0.473123	−	0.880996i	$-0.656874\pi$
$769$	−20178.7	−0.946247	−0.473123	+	0.880996i	$0.656874\pi$
$770$	0	0
$771$	872.332	0.0407474
$772$	15971.4	0.744592
$773$	24715.2	1.14999	0.574996	−	0.818156i	$-0.305003\pi$
$773$	24715.2	1.14999	0.574996	+	0.818156i	$0.305003\pi$
$774$	−1887.96	−0.0876761
$775$	2916.27	0.135168
$776$	−8431.14	−0.390026
$777$	0	0
$778$	−12469.0	−0.574596
$779$	941.455	0.0433006
$780$	−3700.09	−0.169852
$781$	23513.8	1.07732
$782$	−482.523	−0.0220652
$783$	1492.53	0.0681208
$784$	0	0
$785$	−10275.2	−0.467183
$786$	7271.31	0.329973
$787$	−19544.5	−0.885244	−0.442622	−	0.896708i	$-0.645952\pi$
$787$	−19544.5	−0.885244	−0.442622	+	0.896708i	$0.645952\pi$
$788$	2619.93	0.118440
$789$	−3496.23	−0.157756
$790$	3298.48	0.148550
$791$	0	0
$792$	−2714.28	−0.121778
$793$	−12387.8	−0.554735
$794$	−4192.97	−0.187409
$795$	1670.34	0.0745170
$796$	19995.4	0.890348
$797$	−9401.10	−0.417822	−0.208911	−	0.977935i	$-0.566992\pi$
$797$	−9401.10	−0.417822	−0.208911	+	0.977935i	$0.566992\pi$
$798$	0	0
$799$	−111.949	−0.00495680
$800$	−3787.61	−0.167391
$801$	−2193.91	−0.0967764
$802$	−677.981	−0.0298508
$803$	−13001.6	−0.571378
$804$	−18585.0	−0.815225
$805$	0	0
$806$	3662.77	0.160069
$807$	13067.4	0.570005
$808$	6114.18	0.266208
$809$	−6808.77	−0.295901	−0.147950	−	0.988995i	$-0.547268\pi$
$809$	−6808.77	−0.295901	−0.147950	+	0.988995i	$0.547268\pi$
$810$	−369.512	−0.0160288
$811$	−23645.6	−1.02381	−0.511905	−	0.859042i	$-0.671060\pi$
$811$	−23645.6	−1.02381	−0.511905	+	0.859042i	$0.671060\pi$
$812$	0	0
$813$	12397.7	0.534817
$814$	1801.47	0.0775696
$815$	6807.63	0.292590
$816$	−765.686	−0.0328485
$817$	5676.88	0.243095
$818$	9581.20	0.409534
$819$	0	0
$820$	−1366.50	−0.0581954
$821$	−29.4211	−0.00125067	−0.000625337	−	1.00000i	$-0.500199\pi$
$821$	−29.4211	−0.00125067	−0.000625337		1.00000i	$0.500199\pi$
$822$	−5867.54	−0.248971
$823$	1489.88	0.0631031	0.0315516	−	0.999502i	$-0.489955\pi$
$823$	1489.88	0.0631031	0.0315516	+	0.999502i	$0.489955\pi$
$824$	−11159.6	−0.471799
$825$	1634.50	0.0689770
$826$	0	0
$827$	3903.20	0.164121	0.0820603	−	0.996627i	$-0.473850\pi$
$827$	3903.20	0.164121	0.0820603	+	0.996627i	$0.473850\pi$
$828$	−5976.96	−0.250862
$829$	15812.3	0.662464	0.331232	−	0.943549i	$-0.392536\pi$
$829$	15812.3	0.662464	0.331232	+	0.943549i	$0.392536\pi$
$830$	4761.87	0.199141
$831$	6759.69	0.282179
$832$	7553.72	0.314757
$833$	0	0
$834$	−7739.93	−0.321357
$835$	5214.51	0.216115
$836$	3856.81	0.159558
$837$	−3149.57	−0.130066
$838$	4568.71	0.188333
$839$	14730.3	0.606136	0.303068	−	0.952969i	$-0.401989\pi$
$839$	14730.3	0.606136	0.303068	+	0.952969i	$0.401989\pi$
$840$	0	0
$841$	−21333.3	−0.874708
$842$	−3508.78	−0.143611
$843$	12391.9	0.506285
$844$	4147.38	0.169145
$845$	−5063.01	−0.206122
$846$	161.049	0.00654489
$847$	0	0
$848$	−4979.26	−0.201637
$849$	17065.5	0.689853
$850$	−130.194	−0.00525368
$851$	8394.54	0.338145
$852$	−23200.2	−0.932894
$853$	12630.7	0.506996	0.253498	−	0.967336i	$-0.418419\pi$
$853$	12630.7	0.506996	0.253498	+	0.967336i	$0.418419\pi$
$854$	0	0
$855$	1111.08	0.0444422
$856$	−8552.46	−0.341492
$857$	11507.3	0.458671	0.229335	−	0.973347i	$-0.426345\pi$
$857$	11507.3	0.458671	0.229335	+	0.973347i	$0.426345\pi$
$858$	2052.90	0.0816838
$859$	−4264.27	−0.169377	−0.0846886	−	0.996407i	$-0.526990\pi$
$859$	−4264.27	−0.169377	−0.0846886	+	0.996407i	$0.526990\pi$
$860$	−8239.85	−0.326717
$861$	0	0
$862$	−1974.76	−0.0780284
$863$	17273.3	0.681333	0.340666	−	0.940184i	$-0.389347\pi$
$863$	17273.3	0.681333	0.340666	+	0.940184i	$0.389347\pi$
$864$	4090.62	0.161072
$865$	16986.2	0.667685
$866$	−8799.91	−0.345304
$867$	14641.3	0.573522
$868$	0	0
$869$	15757.8	0.615127
$870$	−756.524	−0.0294811
$871$	29745.2	1.15715
$872$	−29712.2	−1.15388
$873$	−5483.27	−0.212578
$874$	−2087.24	−0.0807801
$875$	0	0
$876$	12828.2	0.494778
$877$	−17372.6	−0.668906	−0.334453	−	0.942412i	$-0.608552\pi$
$877$	−17372.6	−0.668906	−0.334453	+	0.942412i	$0.608552\pi$
$878$	3053.65	0.117376
$879$	−13519.1	−0.518759
$880$	−4872.41	−0.186646
$881$	49105.7	1.87788	0.938941	−	0.344078i	$-0.111809\pi$
$881$	49105.7	1.87788	0.938941	+	0.344078i	$0.111809\pi$
$882$	0	0
$883$	−1915.84	−0.0730161	−0.0365081	−	0.999333i	$-0.511623\pi$
$883$	−1915.84	−0.0730161	−0.0365081	+	0.999333i	$0.511623\pi$
$884$	1407.99	0.0535700
$885$	−6035.16	−0.229231
$886$	1455.71	0.0551982
$887$	−175.503	−0.00664354	−0.00332177	−	0.999994i	$-0.501057\pi$
$887$	−175.503	−0.00664354	−0.00332177	+	0.999994i	$0.501057\pi$
$888$	−3761.33	−0.142142
$889$	0	0
$890$	1112.04	0.0418826
$891$	−1765.26	−0.0663731
$892$	−33436.5	−1.25509
$893$	−484.256	−0.0181467
$894$	−2525.61	−0.0944844
$895$	626.905	0.0234135
$896$	0	0
$897$	9566.13	0.356080
$898$	3759.61	0.139710
$899$	−6448.30	−0.239225
$900$	−1612.70	−0.0597298
$901$	−635.615	−0.0235021
$902$	758.166	0.0279869
$903$	0	0
$904$	−17095.3	−0.628962
$905$	4941.17	0.181492
$906$	−2286.38	−0.0838411
$907$	35638.4	1.30469	0.652346	−	0.757922i	$-0.273785\pi$
$907$	35638.4	1.30469	0.652346	+	0.757922i	$0.273785\pi$
$908$	−18459.6	−0.674673
$909$	3976.41	0.145093
$910$	0	0
$911$	−6181.19	−0.224799	−0.112400	−	0.993663i	$-0.535854\pi$
$911$	−6181.19	−0.224799	−0.112400	+	0.993663i	$0.535854\pi$
$912$	−3312.10	−0.120257
$913$	22748.8	0.824615
$914$	2178.05	0.0788223
$915$	−5399.31	−0.195077
$916$	35293.9	1.27308
$917$	0	0
$918$	140.610	0.00505536
$919$	−21118.6	−0.758040	−0.379020	−	0.925388i	$-0.623739\pi$
$919$	−21118.6	−0.758040	−0.379020	+	0.925388i	$0.623739\pi$
$920$	6410.99	0.229744
$921$	−16141.3	−0.577494
$922$	−4693.50	−0.167649
$923$	37131.9	1.32417
$924$	0	0
$925$	2265.01	0.0805116
$926$	4715.85	0.167357
$927$	−7257.73	−0.257147
$928$	8374.99	0.296253
$929$	−38221.8	−1.34986	−0.674928	−	0.737884i	$-0.735825\pi$
$929$	−38221.8	−1.34986	−0.674928	+	0.737884i	$0.735825\pi$
$930$	1596.44	0.0562895
$931$	0	0
$932$	−43099.8	−1.51479
$933$	5587.12	0.196049
$934$	−12080.6	−0.423223
$935$	−621.975	−0.0217548
$936$	−4286.28	−0.149681
$937$	51676.0	1.80169	0.900844	−	0.434142i	$-0.142948\pi$
$937$	51676.0	1.80169	0.900844	+	0.434142i	$0.142948\pi$
$938$	0	0
$939$	23269.8	0.808711
$940$	702.885	0.0243889
$941$	−37184.2	−1.28817	−0.644086	−	0.764953i	$-0.722762\pi$
$941$	−37184.2	−1.28817	−0.644086	+	0.764953i	$0.722762\pi$
$942$	−5624.92	−0.194554
$943$	3532.92	0.122002
$944$	17990.7	0.620283
$945$	0	0
$946$	4571.66	0.157122
$947$	31923.3	1.09543	0.547713	−	0.836666i	$-0.315498\pi$
$947$	31923.3	1.09543	0.547713	+	0.836666i	$0.315498\pi$
$948$	−15547.6	−0.532662
$949$	−20531.6	−0.702301
$950$	−563.178	−0.0192336
$951$	12324.6	0.420246
$952$	0	0
$953$	2088.54	0.0709912	0.0354956	−	0.999370i	$-0.488699\pi$
$953$	2088.54	0.0709912	0.0354956	+	0.999370i	$0.488699\pi$
$954$	914.388	0.0310319
$955$	−13346.3	−0.452226
$956$	−2166.03	−0.0732786
$957$	−3614.13	−0.122077
$958$	4851.58	0.163619
$959$	0	0
$960$	3292.33	0.110687
$961$	−16183.6	−0.543239
$962$	2844.81	0.0953434
$963$	−5562.17	−0.186125
$964$	30076.3	1.00487
$965$	−11141.5	−0.371664
$966$	0	0
$967$	46995.4	1.56285	0.781423	−	0.624002i	$-0.214494\pi$
$967$	46995.4	1.56285	0.781423	+	0.624002i	$0.214494\pi$
$968$	−11846.5	−0.393347
$969$	−422.798	−0.0140167
$970$	2779.33	0.0919989
$971$	27063.6	0.894452	0.447226	−	0.894421i	$-0.352412\pi$
$971$	27063.6	0.894452	0.447226	+	0.894421i	$0.352412\pi$
$972$	1741.72	0.0574750
$973$	0	0
$974$	5637.95	0.185474
$975$	2581.13	0.0847819
$976$	16095.2	0.527864
$977$	−15115.3	−0.494964	−0.247482	−	0.968892i	$-0.579603\pi$
$977$	−15115.3	−0.494964	−0.247482	+	0.968892i	$0.579603\pi$
$978$	3726.66	0.121846
$979$	5312.51	0.173431
$980$	0	0
$981$	−19323.6	−0.628905
$982$	13354.5	0.433970
$983$	−31800.7	−1.03182	−0.515912	−	0.856641i	$-0.672547\pi$
$983$	−31800.7	−1.03182	−0.515912	+	0.856641i	$0.672547\pi$
$984$	−1582.99	−0.0512844
$985$	−1827.62	−0.0591198
$986$	287.879	0.00929812
$987$	0	0
$988$	6090.51	0.196118
$989$	21303.1	0.684934
$990$	894.766	0.0287248
$991$	59005.2	1.89139	0.945693	−	0.325061i	$-0.105385\pi$
$991$	59005.2	1.89139	0.945693	+	0.325061i	$0.105385\pi$
$992$	−17673.1	−0.565647
$993$	−32214.0	−1.02949
$994$	0	0
$995$	−13948.5	−0.444419
$996$	−22445.4	−0.714066
$997$	−3071.30	−0.0975618	−0.0487809	−	0.998810i	$-0.515534\pi$
$997$	−3071.30	−0.0975618	−0.0487809	+	0.998810i	$0.515534\pi$
$998$	18085.8	0.573643
$999$	−2446.22	−0.0774723

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.4.a.t.1.3	✓	4
3.2	odd	2		2205.4.a.bp.1.2		4
7.6	odd	2		735.4.a.u.1.3	yes	4
21.20	even	2		2205.4.a.bq.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.4.a.t.1.3	✓	4	1.1	even	1	trivial
735.4.a.u.1.3	yes	4	7.6	odd	2
2205.4.a.bp.1.2		4	3.2	odd	2
2205.4.a.bq.1.2		4	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:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	735.a (trivial)

\(f(q)\)	\(=\)	\(q-0.912375 q^{2} -3.00000 q^{3} -7.16757 q^{4} +5.00000 q^{5} +2.73712 q^{6} +13.8385 q^{8} +9.00000 q^{9} -4.56187 q^{10} -21.7933 q^{11} +21.5027 q^{12} -34.4151 q^{13} -15.0000 q^{15} +44.7147 q^{16} +5.70794 q^{17} -8.21137 q^{18} +24.6906 q^{19} -35.8379 q^{20} +19.8837 q^{22} +92.6544 q^{23} -41.5155 q^{24} +25.0000 q^{25} +31.3995 q^{26} -27.0000 q^{27} -55.2788 q^{29} +13.6856 q^{30} +116.651 q^{31} -151.505 q^{32} +65.3800 q^{33} -5.20778 q^{34} -64.5082 q^{36} +90.6006 q^{37} -22.5271 q^{38} +103.245 q^{39} +69.1925 q^{40} +38.1301 q^{41} +229.920 q^{43} +156.205 q^{44} +45.0000 q^{45} -84.5355 q^{46} -19.6129 q^{47} -134.144 q^{48} -22.8094 q^{50} -17.1238 q^{51} +246.673 q^{52} -111.356 q^{53} +24.6341 q^{54} -108.967 q^{55} -74.0719 q^{57} +50.4350 q^{58} +402.344 q^{59} +107.514 q^{60} +359.954 q^{61} -106.429 q^{62} -219.488 q^{64} -172.075 q^{65} -59.6510 q^{66} -864.308 q^{67} -40.9120 q^{68} -277.963 q^{69} -1078.94 q^{71} +124.547 q^{72} +596.587 q^{73} -82.6617 q^{74} -75.0000 q^{75} -176.972 q^{76} -94.1984 q^{78} -723.054 q^{79} +223.573 q^{80} +81.0000 q^{81} -34.7889 q^{82} -1043.84 q^{83} +28.5397 q^{85} -209.773 q^{86} +165.836 q^{87} -301.587 q^{88} -243.768 q^{89} -41.0569 q^{90} -664.107 q^{92} -349.952 q^{93} +17.8943 q^{94} +123.453 q^{95} +454.514 q^{96} -609.252 q^{97} -196.140 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{2} - 12 q^{3} + 18 q^{4} + 20 q^{5} + 18 q^{6} - 30 q^{8} + 36 q^{9} - 30 q^{10} - 24 q^{11} - 54 q^{12} - 60 q^{15} - 46 q^{16} + 88 q^{17} - 54 q^{18} - 72 q^{19} + 90 q^{20} - 28 q^{22} + 90 q^{24}+ \cdots - 216 q^{99}+O(q^{100}) \) 4 * q - 6 * q^2 - 12 * q^3 + 18 * q^4 + 20 * q^5 + 18 * q^6 - 30 * q^8 + 36 * q^9 - 30 * q^10 - 24 * q^11 - 54 * q^12 - 60 * q^15 - 46 * q^16 + 88 * q^17 - 54 * q^18 - 72 * q^19 + 90 * q^20 - 28 * q^22 + 90 * q^24 + 100 * q^25 - 112 * q^26 - 108 * q^27 - 636 * q^29 + 90 * q^30 - 228 * q^31 - 126 * q^32 + 72 * q^33 + 48 * q^34 + 162 * q^36 - 68 * q^37 + 408 * q^38 - 150 * q^40 + 56 * q^41 + 20 * q^43 + 276 * q^44 + 180 * q^45 - 596 * q^46 + 744 * q^47 + 138 * q^48 - 150 * q^50 - 264 * q^51 + 768 * q^52 - 360 * q^53 + 162 * q^54 - 120 * q^55 + 216 * q^57 + 1352 * q^58 + 1828 * q^59 - 270 * q^60 - 1212 * q^61 + 1044 * q^62 - 718 * q^64 + 84 * q^66 - 1036 * q^67 + 1056 * q^68 - 876 * q^71 - 270 * q^72 - 420 * q^73 - 1584 * q^74 - 300 * q^75 - 2400 * q^76 + 336 * q^78 - 700 * q^79 - 230 * q^80 + 324 * q^81 - 492 * q^82 + 1020 * q^83 + 440 * q^85 + 636 * q^86 + 1908 * q^87 - 748 * q^88 - 1192 * q^89 - 270 * q^90 - 540 * q^92 + 684 * q^93 - 3732 * q^94 - 360 * q^95 + 378 * q^96 - 588 * q^97 - 216 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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