LMFDB - Embedded newform 735.4.a.z.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [735,4,Mod(1,735)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	735.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,3,-15,25,-25,-9,0,21,45,-15,43] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$43.3664038542$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 30x^{3} + 22x^{2} + 153x - 84 $ x^5 - 2x^4 - 30x^3 + 22x^2 + 153x - 84
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-4.10571$ 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)

$f(q)$	$=$	\(q+5.10571 q^{2} -3.00000 q^{3} +18.0682 q^{4} -5.00000 q^{5} -15.3171 q^{6} +51.4055 q^{8} +9.00000 q^{9} -25.5285 q^{10} -40.7354 q^{11} -54.2047 q^{12} -70.9263 q^{13} +15.0000 q^{15} +117.915 q^{16} -22.6701 q^{17} +45.9514 q^{18} -77.6588 q^{19} -90.3412 q^{20} -207.983 q^{22} -90.0622 q^{23} -154.217 q^{24} +25.0000 q^{25} -362.129 q^{26} -27.0000 q^{27} +213.332 q^{29} +76.5856 q^{30} +175.729 q^{31} +190.798 q^{32} +122.206 q^{33} -115.747 q^{34} +162.614 q^{36} -354.933 q^{37} -396.503 q^{38} +212.779 q^{39} -257.028 q^{40} -249.001 q^{41} +297.920 q^{43} -736.018 q^{44} -45.0000 q^{45} -459.831 q^{46} -168.771 q^{47} -353.746 q^{48} +127.643 q^{50} +68.0102 q^{51} -1281.51 q^{52} -102.473 q^{53} -137.854 q^{54} +203.677 q^{55} +232.976 q^{57} +1089.21 q^{58} +23.4797 q^{59} +271.024 q^{60} +874.628 q^{61} +897.220 q^{62} +30.8343 q^{64} +354.631 q^{65} +623.949 q^{66} -810.771 q^{67} -409.609 q^{68} +270.187 q^{69} -632.487 q^{71} +462.650 q^{72} -914.872 q^{73} -1812.18 q^{74} -75.0000 q^{75} -1403.16 q^{76} +1086.39 q^{78} -520.886 q^{79} -589.577 q^{80} +81.0000 q^{81} -1271.33 q^{82} +1027.67 q^{83} +113.350 q^{85} +1521.09 q^{86} -639.996 q^{87} -2094.02 q^{88} -1003.75 q^{89} -229.757 q^{90} -1627.27 q^{92} -527.187 q^{93} -861.694 q^{94} +388.294 q^{95} -572.394 q^{96} +639.620 q^{97} -366.619 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} - 15 q^{3} + 25 q^{4} - 25 q^{5} - 9 q^{6} + 21 q^{8} + 45 q^{9} - 15 q^{10} + 43 q^{11} - 75 q^{12} - 123 q^{13} + 75 q^{15} + 161 q^{16} - 124 q^{17} + 27 q^{18} - 37 q^{19} - 125 q^{20}+ \cdots + 387 q^{99}+O(q^{100}) $ 5 * q + 3 * q^2 - 15 * q^3 + 25 * q^4 - 25 * q^5 - 9 * q^6 + 21 * q^8 + 45 * q^9 - 15 * q^10 + 43 * q^11 - 75 * q^12 - 123 * q^13 + 75 * q^15 + 161 * q^16 - 124 * q^17 + 27 * q^18 - 37 * q^19 - 125 * q^20 - 221 * q^22 + 77 * q^23 - 63 * q^24 + 125 * q^25 + 79 * q^26 - 135 * q^27 + 360 * q^29 + 45 * q^30 - 314 * q^31 - 59 * q^32 - 129 * q^33 - 176 * q^34 + 225 * q^36 + 225 * q^37 - 759 * q^38 + 369 * q^39 - 105 * q^40 - 341 * q^41 + 32 * q^43 + 679 * q^44 - 225 * q^45 - 331 * q^46 - 25 * q^47 - 483 * q^48 + 75 * q^50 + 372 * q^51 - 2299 * q^52 - 317 * q^53 - 81 * q^54 - 215 * q^55 + 111 * q^57 + 8 * q^58 - 676 * q^59 + 375 * q^60 + 188 * q^61 + 348 * q^62 - 1103 * q^64 + 615 * q^65 + 663 * q^66 - 1776 * q^67 - 1280 * q^68 - 231 * q^69 - 6 * q^71 + 189 * q^72 - 2006 * q^73 - 2729 * q^74 - 375 * q^75 - 1417 * q^76 - 237 * q^78 + 200 * q^79 - 805 * q^80 + 405 * q^81 + 539 * q^82 + 332 * q^83 + 620 * q^85 + 4262 * q^86 - 1080 * q^87 - 4529 * q^88 - 894 * q^89 - 135 * q^90 - 3687 * q^92 + 942 * q^93 - 4233 * q^94 + 185 * q^95 + 177 * q^96 + 576 * q^97 + 387 * 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 $

$2$	5.10571	1.80514	0.902570	−	0.430543i	$-0.141678\pi$
$2$	5.10571	1.80514	0.902570	+	0.430543i	$0.141678\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	18.0682	2.25853
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−15.3171	−1.04220
$7$	0	0
$7$	0	0
$8$	51.4055	2.27182
$9$	9.00000	0.333333
$10$	−25.5285	−0.807283
$11$	−40.7354	−1.11656	−0.558281	−	0.829652i	$-0.688539\pi$
$11$	−40.7354	−1.11656	−0.558281	+	0.829652i	$0.688539\pi$
$12$	−54.2047	−1.30396
$13$	−70.9263	−1.51319	−0.756593	−	0.653887i	$-0.773137\pi$
$13$	−70.9263	−1.51319	−0.756593	+	0.653887i	$0.773137\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	117.915	1.84243
$17$	−22.6701	−0.323430	−0.161715	−	0.986838i	$-0.551702\pi$
$17$	−22.6701	−0.323430	−0.161715	+	0.986838i	$0.551702\pi$
$18$	45.9514	0.601713
$19$	−77.6588	−0.937692	−0.468846	−	0.883280i	$-0.655330\pi$
$19$	−77.6588	−0.937692	−0.468846	+	0.883280i	$0.655330\pi$
$20$	−90.3412	−1.01005
$21$	0	0
$22$	−207.983	−2.01555
$23$	−90.0622	−0.816490	−0.408245	−	0.912872i	$-0.633859\pi$
$23$	−90.0622	−0.816490	−0.408245	+	0.912872i	$0.633859\pi$
$24$	−154.217	−1.31164
$25$	25.0000	0.200000
$26$	−362.129	−2.73151
$27$	−27.0000	−0.192450
$28$	0	0
$29$	213.332	1.36603	0.683013	−	0.730406i	$-0.260669\pi$
$29$	213.332	1.36603	0.683013	+	0.730406i	$0.260669\pi$
$30$	76.5856	0.466085
$31$	175.729	1.01812	0.509062	−	0.860730i	$-0.329992\pi$
$31$	175.729	1.01812	0.509062	+	0.860730i	$0.329992\pi$
$32$	190.798	1.05402
$33$	122.206	0.644648
$34$	−115.747	−0.583836
$35$	0	0
$36$	162.614	0.752844
$37$	−354.933	−1.57704	−0.788521	−	0.615007i	$-0.789153\pi$
$37$	−354.933	−1.57704	−0.788521	+	0.615007i	$0.789153\pi$
$38$	−396.503	−1.69267
$39$	212.779	0.873638
$40$	−257.028	−1.01599
$41$	−249.001	−0.948475	−0.474238	−	0.880397i	$-0.657276\pi$
$41$	−249.001	−0.948475	−0.474238	+	0.880397i	$0.657276\pi$
$42$	0	0
$43$	297.920	1.05657	0.528284	−	0.849068i	$-0.322836\pi$
$43$	297.920	1.05657	0.528284	+	0.849068i	$0.322836\pi$
$44$	−736.018	−2.52179
$45$	−45.0000	−0.149071
$46$	−459.831	−1.47388
$47$	−168.771	−0.523782	−0.261891	−	0.965097i	$-0.584346\pi$
$47$	−168.771	−0.523782	−0.261891	+	0.965097i	$0.584346\pi$
$48$	−353.746	−1.06373
$49$	0	0
$50$	127.643	0.361028
$51$	68.0102	0.186732
$52$	−1281.51	−3.41757
$53$	−102.473	−0.265581	−0.132790	−	0.991144i	$-0.542394\pi$
$53$	−102.473	−0.265581	−0.132790	+	0.991144i	$0.542394\pi$
$54$	−137.854	−0.347399
$55$	203.677	0.499342
$56$	0	0
$57$	232.976	0.541377
$58$	1089.21	2.46587
$59$	23.4797	0.0518102	0.0259051	−	0.999664i	$-0.491753\pi$
$59$	23.4797	0.0518102	0.0259051	+	0.999664i	$0.491753\pi$
$60$	271.024	0.583150
$61$	874.628	1.83581	0.917907	−	0.396796i	$-0.129878\pi$
$61$	874.628	1.83581	0.917907	+	0.396796i	$0.129878\pi$
$62$	897.220	1.83786
$63$	0	0
$64$	30.8343	0.0602232
$65$	354.631	0.676717
$66$	623.949	1.16368
$67$	−810.771	−1.47838	−0.739190	−	0.673497i	$-0.764792\pi$
$67$	−810.771	−1.47838	−0.739190	+	0.673497i	$0.764792\pi$
$68$	−409.609	−0.730476
$69$	270.187	0.471401
$70$	0	0
$71$	−632.487	−1.05722	−0.528608	−	0.848866i	$-0.677286\pi$
$71$	−632.487	−1.05722	−0.528608	+	0.848866i	$0.677286\pi$
$72$	462.650	0.757275
$73$	−914.872	−1.46682	−0.733409	−	0.679788i	$-0.762072\pi$
$73$	−914.872	−1.46682	−0.733409	+	0.679788i	$0.762072\pi$
$74$	−1812.18	−2.84678
$75$	−75.0000	−0.115470
$76$	−1403.16	−2.11781
$77$	0	0
$78$	1086.39	1.57704
$79$	−520.886	−0.741827	−0.370913	−	0.928668i	$-0.620955\pi$
$79$	−520.886	−0.741827	−0.370913	+	0.928668i	$0.620955\pi$
$80$	−589.577	−0.823960
$81$	81.0000	0.111111
$82$	−1271.33	−1.71213
$83$	1027.67	1.35905	0.679527	−	0.733651i	$-0.262185\pi$
$83$	1027.67	1.35905	0.679527	+	0.733651i	$0.262185\pi$
$84$	0	0
$85$	113.350	0.144642
$86$	1521.09	1.90725
$87$	−639.996	−0.788676
$88$	−2094.02	−2.53663
$89$	−1003.75	−1.19548	−0.597738	−	0.801692i	$-0.703934\pi$
$89$	−1003.75	−1.19548	−0.597738	+	0.801692i	$0.703934\pi$
$90$	−229.757	−0.269094
$91$	0	0
$92$	−1627.27	−1.84407
$93$	−527.187	−0.587814
$94$	−861.694	−0.945500
$95$	388.294	0.419349
$96$	−572.394	−0.608538
$97$	639.620	0.669522	0.334761	−	0.942303i	$-0.391344\pi$
$97$	639.620	0.669522	0.334761	+	0.942303i	$0.391344\pi$
$98$	0	0
$99$	−366.619	−0.372188
$100$	451.706	0.451706
$101$	633.869	0.624479	0.312239	−	0.950003i	$-0.398921\pi$
$101$	633.869	0.624479	0.312239	+	0.950003i	$0.398921\pi$
$102$	347.240	0.337078
$103$	387.578	0.370769	0.185385	−	0.982666i	$-0.440647\pi$
$103$	387.578	0.370769	0.185385	+	0.982666i	$0.440647\pi$
$104$	−3646.00	−3.43769
$105$	0	0
$106$	−523.198	−0.479410
$107$	138.730	0.125341	0.0626707	−	0.998034i	$-0.480038\pi$
$107$	138.730	0.125341	0.0626707	+	0.998034i	$0.480038\pi$
$108$	−487.843	−0.434654
$109$	−1712.97	−1.50526	−0.752628	−	0.658447i	$-0.771214\pi$
$109$	−1712.97	−1.50526	−0.752628	+	0.658447i	$0.771214\pi$
$110$	1039.92	0.901382
$111$	1064.80	0.910506
$112$	0	0
$113$	2083.35	1.73438	0.867192	−	0.497975i	$-0.165923\pi$
$113$	2083.35	1.73438	0.867192	+	0.497975i	$0.165923\pi$
$114$	1189.51	0.977261
$115$	450.311	0.365145
$116$	3854.54	3.08521
$117$	−638.336	−0.504395
$118$	119.881	0.0935246
$119$	0	0
$120$	771.083	0.586582
$121$	328.375	0.246713
$122$	4465.59	3.31390
$123$	747.004	0.547602
$124$	3175.11	2.29946
$125$	−125.000	−0.0894427
$126$	0	0
$127$	432.085	0.301900	0.150950	−	0.988541i	$-0.451767\pi$
$127$	432.085	0.301900	0.150950	+	0.988541i	$0.451767\pi$
$128$	−1368.95	−0.945308
$129$	−893.760	−0.610009
$130$	1810.64	1.22157
$131$	657.499	0.438519	0.219259	−	0.975667i	$-0.429636\pi$
$131$	657.499	0.438519	0.219259	+	0.975667i	$0.429636\pi$
$132$	2208.05	1.45596
$133$	0	0
$134$	−4139.56	−2.66868
$135$	135.000	0.0860663
$136$	−1165.37	−0.734775
$137$	292.515	0.182418	0.0912090	−	0.995832i	$-0.470927\pi$
$137$	292.515	0.182418	0.0912090	+	0.995832i	$0.470927\pi$
$138$	1379.49	0.850944
$139$	1673.02	1.02089	0.510445	−	0.859910i	$-0.329481\pi$
$139$	1673.02	1.02089	0.510445	+	0.859910i	$0.329481\pi$
$140$	0	0
$141$	506.312	0.302406
$142$	−3229.29	−1.90842
$143$	2889.21	1.68957
$144$	1061.24	0.614143
$145$	−1066.66	−0.610906
$146$	−4671.07	−2.64781
$147$	0	0
$148$	−6413.01	−3.56180
$149$	2521.01	1.38610	0.693050	−	0.720890i	$-0.256266\pi$
$149$	2521.01	1.38610	0.693050	+	0.720890i	$0.256266\pi$
$150$	−382.928	−0.208440
$151$	1379.32	0.743361	0.371680	−	0.928361i	$-0.378782\pi$
$151$	1379.32	0.743361	0.371680	+	0.928361i	$0.378782\pi$
$152$	−3992.09	−2.13027
$153$	−204.031	−0.107810
$154$	0	0
$155$	−878.644	−0.455319
$156$	3844.54	1.97314
$157$	−1992.73	−1.01298	−0.506488	−	0.862247i	$-0.669057\pi$
$157$	−1992.73	−1.01298	−0.506488	+	0.862247i	$0.669057\pi$
$158$	−2659.49	−1.33910
$159$	307.420	0.153333
$160$	−953.990	−0.471372
$161$	0	0
$162$	413.562	0.200571
$163$	1902.94	0.914418	0.457209	−	0.889359i	$-0.348849\pi$
$163$	1902.94	0.914418	0.457209	+	0.889359i	$0.348849\pi$
$164$	−4499.02	−2.14216
$165$	−611.031	−0.288295
$166$	5246.98	2.45328
$167$	378.184	0.175238	0.0876189	−	0.996154i	$-0.472074\pi$
$167$	378.184	0.175238	0.0876189	+	0.996154i	$0.472074\pi$
$168$	0	0
$169$	2833.54	1.28973
$170$	578.734	0.261099
$171$	−698.929	−0.312564
$172$	5382.89	2.38629
$173$	664.037	0.291825	0.145913	−	0.989297i	$-0.453388\pi$
$173$	664.037	0.291825	0.145913	+	0.989297i	$0.453388\pi$
$174$	−3267.63	−1.42367
$175$	0	0
$176$	−4803.34	−2.05719
$177$	−70.4392	−0.0299126
$178$	−5124.86	−2.15800
$179$	−334.999	−0.139883	−0.0699414	−	0.997551i	$-0.522281\pi$
$179$	−334.999	−0.139883	−0.0699414	+	0.997551i	$0.522281\pi$
$180$	−813.071	−0.336682
$181$	−1225.58	−0.503298	−0.251649	−	0.967819i	$-0.580973\pi$
$181$	−1225.58	−0.503298	−0.251649	+	0.967819i	$0.580973\pi$
$182$	0	0
$183$	−2623.88	−1.05991
$184$	−4629.69	−1.85492
$185$	1774.66	0.705275
$186$	−2691.66	−1.06109
$187$	923.475	0.361129
$188$	−3049.39	−1.18298
$189$	0	0
$190$	1982.51	0.756983
$191$	−1581.40	−0.599088	−0.299544	−	0.954082i	$-0.596835\pi$
$191$	−1581.40	−0.599088	−0.299544	+	0.954082i	$0.596835\pi$
$192$	−92.5029	−0.0347699
$193$	−3894.63	−1.45255	−0.726273	−	0.687406i	$-0.758749\pi$
$193$	−3894.63	−1.45255	−0.726273	+	0.687406i	$0.758749\pi$
$194$	3265.71	1.20858
$195$	−1063.89	−0.390703
$196$	0	0
$197$	993.858	0.359439	0.179719	−	0.983718i	$-0.442481\pi$
$197$	993.858	0.359439	0.179719	+	0.983718i	$0.442481\pi$
$198$	−1871.85	−0.671851
$199$	3079.88	1.09712	0.548560	−	0.836111i	$-0.315176\pi$
$199$	3079.88	1.09712	0.548560	+	0.836111i	$0.315176\pi$
$200$	1285.14	0.454365
$201$	2432.31	0.853543
$202$	3236.35	1.12727
$203$	0	0
$204$	1228.83	0.421740
$205$	1245.01	0.424171
$206$	1978.86	0.669290
$207$	−810.560	−0.272163
$208$	−8363.31	−2.78794
$209$	3163.46	1.04699
$210$	0	0
$211$	2114.72	0.689968	0.344984	−	0.938609i	$-0.387884\pi$
$211$	2114.72	0.689968	0.344984	+	0.938609i	$0.387884\pi$
$212$	−1851.51	−0.599822
$213$	1897.46	0.610384
$214$	708.315	0.226259
$215$	−1489.60	−0.472511
$216$	−1387.95	−0.437213
$217$	0	0
$218$	−8745.92	−2.71720
$219$	2744.62	0.846867
$220$	3680.09	1.12778
$221$	1607.90	0.489409
$222$	5436.55	1.64359
$223$	−2578.44	−0.774282	−0.387141	−	0.922021i	$-0.626537\pi$
$223$	−2578.44	−0.774282	−0.387141	+	0.922021i	$0.626537\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	10637.0	3.13080
$227$	−2925.92	−0.855509	−0.427754	−	0.903895i	$-0.640695\pi$
$227$	−2925.92	−0.855509	−0.427754	+	0.903895i	$0.640695\pi$
$228$	4209.47	1.22272
$229$	1047.88	0.302383	0.151191	−	0.988505i	$-0.451689\pi$
$229$	1047.88	0.302383	0.151191	+	0.988505i	$0.451689\pi$
$230$	2299.16	0.659139
$231$	0	0
$232$	10966.4	3.10337
$233$	−3869.99	−1.08812	−0.544058	−	0.839047i	$-0.683113\pi$
$233$	−3869.99	−1.08812	−0.544058	+	0.839047i	$0.683113\pi$
$234$	−3259.16	−0.910504
$235$	843.854	0.234242
$236$	424.237	0.117015
$237$	1562.66	0.428294
$238$	0	0
$239$	−6060.77	−1.64033	−0.820164	−	0.572128i	$-0.806118\pi$
$239$	−6060.77	−1.64033	−0.820164	+	0.572128i	$0.806118\pi$
$240$	1768.73	0.475713
$241$	4184.12	1.11835	0.559176	−	0.829049i	$-0.311118\pi$
$241$	4184.12	1.11835	0.559176	+	0.829049i	$0.311118\pi$
$242$	1676.58	0.445351
$243$	−243.000	−0.0641500
$244$	15803.0	4.14624
$245$	0	0
$246$	3813.98	0.988499
$247$	5508.05	1.41890
$248$	9033.43	2.31300
$249$	−3083.01	−0.784650
$250$	−638.213	−0.161457
$251$	2631.65	0.661786	0.330893	−	0.943668i	$-0.392650\pi$
$251$	2631.65	0.661786	0.330893	+	0.943668i	$0.392650\pi$
$252$	0	0
$253$	3668.72	0.911663
$254$	2206.10	0.544972
$255$	−340.051	−0.0835092
$256$	−7236.14	−1.76664
$257$	−3288.77	−0.798240	−0.399120	−	0.916899i	$-0.630684\pi$
$257$	−3288.77	−0.798240	−0.399120	+	0.916899i	$0.630684\pi$
$258$	−4563.28	−1.10115
$259$	0	0
$260$	6407.57	1.52839
$261$	1919.99	0.455342
$262$	3357.00	0.791588
$263$	−5408.90	−1.26816	−0.634081	−	0.773266i	$-0.718622\pi$
$263$	−5408.90	−1.26816	−0.634081	+	0.773266i	$0.718622\pi$
$264$	6282.07	1.46453
$265$	512.366	0.118771
$266$	0	0
$267$	3011.25	0.690208
$268$	−14649.2	−3.33897
$269$	−539.788	−0.122348	−0.0611738	−	0.998127i	$-0.519484\pi$
$269$	−539.788	−0.122348	−0.0611738	+	0.998127i	$0.519484\pi$
$270$	689.270	0.155362
$271$	−4091.14	−0.917045	−0.458522	−	0.888683i	$-0.651621\pi$
$271$	−4091.14	−0.917045	−0.458522	+	0.888683i	$0.651621\pi$
$272$	−2673.15	−0.595896
$273$	0	0
$274$	1493.50	0.329290
$275$	−1018.39	−0.223313
$276$	4881.80	1.06467
$277$	3091.34	0.670543	0.335271	−	0.942122i	$-0.391172\pi$
$277$	3091.34	0.670543	0.335271	+	0.942122i	$0.391172\pi$
$278$	8541.95	1.84285
$279$	1581.56	0.339375
$280$	0	0
$281$	7215.73	1.53187	0.765933	−	0.642920i	$-0.222277\pi$
$281$	7215.73	1.53187	0.765933	+	0.642920i	$0.222277\pi$
$282$	2585.08	0.545884
$283$	−2964.72	−0.622735	−0.311368	−	0.950290i	$-0.600787\pi$
$283$	−2964.72	−0.622735	−0.311368	+	0.950290i	$0.600787\pi$
$284$	−11427.9	−2.38776
$285$	−1164.88	−0.242111
$286$	14751.5	3.04990
$287$	0	0
$288$	1717.18	0.351340
$289$	−4399.07	−0.895393
$290$	−5446.05	−1.10277
$291$	−1918.86	−0.386548
$292$	−16530.1	−3.31285
$293$	7924.10	1.57997	0.789984	−	0.613127i	$-0.210089\pi$
$293$	7924.10	1.57997	0.789984	+	0.613127i	$0.210089\pi$
$294$	0	0
$295$	−117.399	−0.0231702
$296$	−18245.5	−3.58276
$297$	1099.86	0.214883
$298$	12871.5	2.50210
$299$	6387.78	1.23550
$300$	−1355.12	−0.260793
$301$	0	0
$302$	7042.41	1.34187
$303$	−1901.61	−0.360543
$304$	−9157.17	−1.72763
$305$	−4373.14	−0.821001
$306$	−1041.72	−0.194612
$307$	4067.00	0.756079	0.378039	−	0.925790i	$-0.376598\pi$
$307$	4067.00	0.756079	0.378039	+	0.925790i	$0.376598\pi$
$308$	0	0
$309$	−1162.73	−0.214064
$310$	−4486.10	−0.821914
$311$	−7652.58	−1.39530	−0.697650	−	0.716439i	$-0.745771\pi$
$311$	−7652.58	−1.39530	−0.697650	+	0.716439i	$0.745771\pi$
$312$	10938.0	1.98475
$313$	−1822.12	−0.329049	−0.164525	−	0.986373i	$-0.552609\pi$
$313$	−1822.12	−0.329049	−0.164525	+	0.986373i	$0.552609\pi$
$314$	−10174.3	−1.82856
$315$	0	0
$316$	−9411.50	−1.67544
$317$	3148.86	0.557910	0.278955	−	0.960304i	$-0.410012\pi$
$317$	3148.86	0.557910	0.278955	+	0.960304i	$0.410012\pi$
$318$	1569.59	0.276788
$319$	−8690.17	−1.52525
$320$	−154.172	−0.0269327
$321$	−416.190	−0.0723659
$322$	0	0
$323$	1760.53	0.303277
$324$	1463.53	0.250948
$325$	−1773.16	−0.302637
$326$	9715.87	1.65065
$327$	5138.91	0.869059
$328$	−12800.0	−2.15477
$329$	0	0
$330$	−3119.75	−0.520413
$331$	−5856.12	−0.972452	−0.486226	−	0.873833i	$-0.661627\pi$
$331$	−5856.12	−0.972452	−0.486226	+	0.873833i	$0.661627\pi$
$332$	18568.2	3.06946
$333$	−3194.39	−0.525681
$334$	1930.89	0.316329
$335$	4053.86	0.661152
$336$	0	0
$337$	−7065.69	−1.14212	−0.571058	−	0.820910i	$-0.693467\pi$
$337$	−7065.69	−1.14212	−0.571058	+	0.820910i	$0.693467\pi$
$338$	14467.2	2.32814
$339$	−6250.06	−1.00135
$340$	2048.04	0.326679
$341$	−7158.39	−1.13680
$342$	−3568.53	−0.564222
$343$	0	0
$344$	15314.7	2.40033
$345$	−1350.93	−0.210817
$346$	3390.38	0.526785
$347$	9897.45	1.53119	0.765595	−	0.643323i	$-0.222445\pi$
$347$	9897.45	1.53119	0.765595	+	0.643323i	$0.222445\pi$
$348$	−11563.6	−1.78125
$349$	−11832.6	−1.81486	−0.907431	−	0.420201i	$-0.861960\pi$
$349$	−11832.6	−1.81486	−0.907431	+	0.420201i	$0.861960\pi$
$350$	0	0
$351$	1915.01	0.291213
$352$	−7772.23	−1.17688
$353$	−7007.38	−1.05656	−0.528279	−	0.849071i	$-0.677163\pi$
$353$	−7007.38	−1.05656	−0.528279	+	0.849071i	$0.677163\pi$
$354$	−359.642	−0.0539965
$355$	3162.43	0.472802
$356$	−18136.0	−2.70002
$357$	0	0
$358$	−1710.41	−0.252508
$359$	2814.39	0.413754	0.206877	−	0.978367i	$-0.433670\pi$
$359$	2814.39	0.413754	0.206877	+	0.978367i	$0.433670\pi$
$360$	−2313.25	−0.338663
$361$	−828.115	−0.120734
$362$	−6257.47	−0.908523
$363$	−985.124	−0.142440
$364$	0	0
$365$	4574.36	0.655981
$366$	−13396.8	−1.91328
$367$	−2349.86	−0.334227	−0.167114	−	0.985938i	$-0.553445\pi$
$367$	−2349.86	−0.334227	−0.167114	+	0.985938i	$0.553445\pi$
$368$	−10619.7	−1.50433
$369$	−2241.01	−0.316158
$370$	9060.91	1.27312
$371$	0	0
$372$	−9525.34	−1.32760
$373$	−1659.30	−0.230336	−0.115168	−	0.993346i	$-0.536741\pi$
$373$	−1659.30	−0.230336	−0.115168	+	0.993346i	$0.536741\pi$
$374$	4714.99	0.651889
$375$	375.000	0.0516398
$376$	−8675.75	−1.18994
$377$	−15130.8	−2.06705
$378$	0	0
$379$	5327.80	0.722087	0.361043	−	0.932549i	$-0.382421\pi$
$379$	5327.80	0.722087	0.361043	+	0.932549i	$0.382421\pi$
$380$	7015.79	0.947111
$381$	−1296.25	−0.174302
$382$	−8074.15	−1.08144
$383$	−6239.75	−0.832471	−0.416235	−	0.909257i	$-0.636651\pi$
$383$	−6239.75	−0.832471	−0.416235	+	0.909257i	$0.636651\pi$
$384$	4106.86	0.545774
$385$	0	0
$386$	−19884.8	−2.62205
$387$	2681.28	0.352189
$388$	11556.8	1.51213
$389$	9485.21	1.23630	0.618148	−	0.786062i	$-0.287883\pi$
$389$	9485.21	1.23630	0.618148	+	0.786062i	$0.287883\pi$
$390$	−5431.93	−0.705273
$391$	2041.72	0.264077
$392$	0	0
$393$	−1972.50	−0.253179
$394$	5074.35	0.648837
$395$	2604.43	0.331755
$396$	−6624.16	−0.840597
$397$	−7892.74	−0.997796	−0.498898	−	0.866661i	$-0.666262\pi$
$397$	−7892.74	−0.997796	−0.498898	+	0.866661i	$0.666262\pi$
$398$	15725.0	1.98046
$399$	0	0
$400$	2947.89	0.368486
$401$	−12459.7	−1.55163	−0.775817	−	0.630958i	$-0.782662\pi$
$401$	−12459.7	−1.55163	−0.775817	+	0.630958i	$0.782662\pi$
$402$	12418.7	1.54077
$403$	−12463.8	−1.54061
$404$	11452.9	1.41040
$405$	−405.000	−0.0496904
$406$	0	0
$407$	14458.3	1.76087
$408$	3496.10	0.424223
$409$	−3403.62	−0.411487	−0.205744	−	0.978606i	$-0.565961\pi$
$409$	−3403.62	−0.411487	−0.205744	+	0.978606i	$0.565961\pi$
$410$	6356.64	0.765688
$411$	−877.546	−0.105319
$412$	7002.86	0.837393
$413$	0	0
$414$	−4138.48	−0.491293
$415$	−5138.35	−0.607787
$416$	−13532.6	−1.59493
$417$	−5019.06	−0.589411
$418$	16151.7	1.88997
$419$	1376.83	0.160532	0.0802658	−	0.996773i	$-0.474423\pi$
$419$	1376.83	0.160532	0.0802658	+	0.996773i	$0.474423\pi$
$420$	0	0
$421$	−15757.3	−1.82414	−0.912069	−	0.410036i	$-0.865516\pi$
$421$	−15757.3	−1.82414	−0.912069	+	0.410036i	$0.865516\pi$
$422$	10797.1	1.24549
$423$	−1518.94	−0.174594
$424$	−5267.69	−0.603353
$425$	−566.752	−0.0646859
$426$	9687.88	1.10183
$427$	0	0
$428$	2506.61	0.283088
$429$	−8667.63	−0.975472
$430$	−7605.46	−0.852949
$431$	5083.70	0.568152	0.284076	−	0.958802i	$-0.408313\pi$
$431$	5083.70	0.568152	0.284076	+	0.958802i	$0.408313\pi$
$432$	−3183.72	−0.354576
$433$	−4631.44	−0.514025	−0.257013	−	0.966408i	$-0.582738\pi$
$433$	−4631.44	−0.514025	−0.257013	+	0.966408i	$0.582738\pi$
$434$	0	0
$435$	3199.98	0.352706
$436$	−30950.4	−3.39966
$437$	6994.12	0.765616
$438$	14013.2	1.52871
$439$	2338.87	0.254279	0.127139	−	0.991885i	$-0.459420\pi$
$439$	2338.87	0.254279	0.127139	+	0.991885i	$0.459420\pi$
$440$	10470.1	1.13442
$441$	0	0
$442$	8209.49	0.883451
$443$	11811.5	1.26677	0.633387	−	0.773835i	$-0.281664\pi$
$443$	11811.5	1.26677	0.633387	+	0.773835i	$0.281664\pi$
$444$	19239.0	2.05641
$445$	5018.75	0.534633
$446$	−13164.7	−1.39769
$447$	−7563.02	−0.800265
$448$	0	0
$449$	9582.29	1.00716	0.503582	−	0.863948i	$-0.332015\pi$
$449$	9582.29	1.00716	0.503582	+	0.863948i	$0.332015\pi$
$450$	1148.78	0.120343
$451$	10143.2	1.05903
$452$	37642.5	3.91716
$453$	−4137.96	−0.429180
$454$	−14938.9	−1.54431
$455$	0	0
$456$	11976.3	1.22991
$457$	−6571.95	−0.672697	−0.336349	−	0.941738i	$-0.609192\pi$
$457$	−6571.95	−0.672697	−0.336349	+	0.941738i	$0.609192\pi$
$458$	5350.15	0.545843
$459$	612.092	0.0622440
$460$	8136.33	0.824692
$461$	4197.59	0.424081	0.212041	−	0.977261i	$-0.431989\pi$
$461$	4197.59	0.424081	0.212041	+	0.977261i	$0.431989\pi$
$462$	0	0
$463$	−6125.75	−0.614876	−0.307438	−	0.951568i	$-0.599472\pi$
$463$	−6125.75	−0.614876	−0.307438	+	0.951568i	$0.599472\pi$
$464$	25155.2	2.51681
$465$	2635.93	0.262878
$466$	−19759.0	−1.96420
$467$	−3228.08	−0.319867	−0.159934	−	0.987128i	$-0.551128\pi$
$467$	−3228.08	−0.319867	−0.159934	+	0.987128i	$0.551128\pi$
$468$	−11533.6	−1.13919
$469$	0	0
$470$	4308.47	0.422840
$471$	5978.19	0.584842
$472$	1206.99	0.117704
$473$	−12135.9	−1.17972
$474$	7978.48	0.773130
$475$	−1941.47	−0.187538
$476$	0	0
$477$	−922.259	−0.0885269
$478$	−30944.5	−2.96102
$479$	−3126.31	−0.298214	−0.149107	−	0.988821i	$-0.547640\pi$
$479$	−3126.31	−0.298214	−0.149107	+	0.988821i	$0.547640\pi$
$480$	2861.97	0.272147
$481$	25174.1	2.38636
$482$	21362.9	2.01878
$483$	0	0
$484$	5933.15	0.557208
$485$	−3198.10	−0.299419
$486$	−1240.69	−0.115800
$487$	−869.038	−0.0808621	−0.0404311	−	0.999182i	$-0.512873\pi$
$487$	−869.038	−0.0808621	−0.0404311	+	0.999182i	$0.512873\pi$
$488$	44960.7	4.17065
$489$	−5708.83	−0.527939
$490$	0	0
$491$	6617.71	0.608254	0.304127	−	0.952631i	$-0.401635\pi$
$491$	6617.71	0.608254	0.304127	+	0.952631i	$0.401635\pi$
$492$	13497.1	1.23678
$493$	−4836.25	−0.441813
$494$	28122.5	2.56132
$495$	1833.09	0.166447
$496$	20721.2	1.87582
$497$	0	0
$498$	−15740.9	−1.41640
$499$	−888.154	−0.0796778	−0.0398389	−	0.999206i	$-0.512684\pi$
$499$	−888.154	−0.0796778	−0.0398389	+	0.999206i	$0.512684\pi$
$500$	−2258.53	−0.202009
$501$	−1134.55	−0.101174
$502$	13436.4	1.19462
$503$	10757.4	0.953572	0.476786	−	0.879019i	$-0.341802\pi$
$503$	10757.4	0.953572	0.476786	+	0.879019i	$0.341802\pi$
$504$	0	0
$505$	−3169.35	−0.279275
$506$	18731.4	1.64568
$507$	−8500.61	−0.744626
$508$	7807.01	0.681850
$509$	−8860.03	−0.771540	−0.385770	−	0.922595i	$-0.626064\pi$
$509$	−8860.03	−0.771540	−0.385770	+	0.922595i	$0.626064\pi$
$510$	−1736.20	−0.150746
$511$	0	0
$512$	−25994.0	−2.24372
$513$	2096.79	0.180459
$514$	−16791.5	−1.44093
$515$	−1937.89	−0.165813
$516$	−16148.7	−1.37772
$517$	6874.95	0.584835
$518$	0	0
$519$	−1992.11	−0.168485
$520$	18230.0	1.53738
$521$	75.3875	0.00633932	0.00316966	−	0.999995i	$-0.498991\pi$
$521$	75.3875	0.00633932	0.00316966	+	0.999995i	$0.498991\pi$
$522$	9802.90	0.821956
$523$	−8666.59	−0.724596	−0.362298	−	0.932062i	$-0.618008\pi$
$523$	−8666.59	−0.724596	−0.362298	+	0.932062i	$0.618008\pi$
$524$	11879.9	0.990408
$525$	0	0
$526$	−27616.2	−2.28921
$527$	−3983.79	−0.329291
$528$	14410.0	1.18772
$529$	−4055.80	−0.333344
$530$	2615.99	0.214399
$531$	211.318	0.0172701
$532$	0	0
$533$	17660.7	1.43522
$534$	15374.6	1.24592
$535$	−693.650	−0.0560544
$536$	−41678.1	−3.35862
$537$	1005.00	0.0807614
$538$	−2756.00	−0.220854
$539$	0	0
$540$	2439.21	0.194383
$541$	12284.8	0.976272	0.488136	−	0.872768i	$-0.337677\pi$
$541$	12284.8	0.976272	0.488136	+	0.872768i	$0.337677\pi$
$542$	−20888.2	−1.65539
$543$	3676.75	0.290579
$544$	−4325.40	−0.340901
$545$	8564.85	0.673171
$546$	0	0
$547$	−2906.85	−0.227218	−0.113609	−	0.993526i	$-0.536241\pi$
$547$	−2906.85	−0.227218	−0.113609	+	0.993526i	$0.536241\pi$
$548$	5285.24	0.411997
$549$	7871.65	0.611938
$550$	−5199.58	−0.403110
$551$	−16567.1	−1.28091
$552$	13889.1	1.07094
$553$	0	0
$554$	15783.5	1.21042
$555$	−5323.99	−0.407191
$556$	30228.5	2.30571
$557$	3873.82	0.294684	0.147342	−	0.989086i	$-0.452928\pi$
$557$	3873.82	0.294684	0.147342	+	0.989086i	$0.452928\pi$
$558$	8074.98	0.612619
$559$	−21130.4	−1.59878
$560$	0	0
$561$	−2770.43	−0.208498
$562$	36841.4	2.76523
$563$	5461.09	0.408805	0.204403	−	0.978887i	$-0.434475\pi$
$563$	5461.09	0.408805	0.204403	+	0.978887i	$0.434475\pi$
$564$	9148.18	0.682992
$565$	−10416.8	−0.775640
$566$	−15137.0	−1.12412
$567$	0	0
$568$	−32513.3	−2.40181
$569$	13105.0	0.965534	0.482767	−	0.875749i	$-0.339632\pi$
$569$	13105.0	0.965534	0.482767	+	0.875749i	$0.339632\pi$
$570$	−5947.54	−0.437044
$571$	3439.26	0.252064	0.126032	−	0.992026i	$-0.459776\pi$
$571$	3439.26	0.252064	0.126032	+	0.992026i	$0.459776\pi$
$572$	52203.0	3.81594
$573$	4744.19	0.345884
$574$	0	0
$575$	−2251.56	−0.163298
$576$	277.509	0.0200744
$577$	−17792.2	−1.28371	−0.641853	−	0.766828i	$-0.721834\pi$
$577$	−17792.2	−1.28371	−0.641853	+	0.766828i	$0.721834\pi$
$578$	−22460.3	−1.61631
$579$	11683.9	0.838628
$580$	−19272.7	−1.37975
$581$	0	0
$582$	−9797.14	−0.697774
$583$	4174.29	0.296538
$584$	−47029.4	−3.33235
$585$	3191.68	0.225572
$586$	40458.1	2.85206
$587$	−14416.6	−1.01369	−0.506846	−	0.862037i	$-0.669189\pi$
$587$	−14416.6	−1.01369	−0.506846	+	0.862037i	$0.669189\pi$
$588$	0	0
$589$	−13646.9	−0.954686
$590$	−599.403	−0.0418255
$591$	−2981.57	−0.207522
$592$	−41852.1	−2.90559
$593$	−8172.03	−0.565911	−0.282955	−	0.959133i	$-0.591315\pi$
$593$	−8172.03	−0.565911	−0.282955	+	0.959133i	$0.591315\pi$
$594$	5615.54	0.387893
$595$	0	0
$596$	45550.2	3.13055
$597$	−9239.65	−0.633423
$598$	32614.1	2.23025
$599$	−2492.18	−0.169996	−0.0849980	−	0.996381i	$-0.527088\pi$
$599$	−2492.18	−0.169996	−0.0849980	+	0.996381i	$0.527088\pi$
$600$	−3855.41	−0.262328
$601$	13835.4	0.939031	0.469516	−	0.882924i	$-0.344429\pi$
$601$	13835.4	0.939031	0.469516	+	0.882924i	$0.344429\pi$
$602$	0	0
$603$	−7296.94	−0.492794
$604$	24921.9	1.67890
$605$	−1641.87	−0.110333
$606$	−9709.05	−0.650830
$607$	−7335.90	−0.490536	−0.245268	−	0.969455i	$-0.578876\pi$
$607$	−7335.90	−0.490536	−0.245268	+	0.969455i	$0.578876\pi$
$608$	−14817.1	−0.988345
$609$	0	0
$610$	−22328.0	−1.48202
$611$	11970.3	0.792579
$612$	−3686.48	−0.243492
$613$	6012.04	0.396124	0.198062	−	0.980189i	$-0.436535\pi$
$613$	6012.04	0.396124	0.198062	+	0.980189i	$0.436535\pi$
$614$	20764.9	1.36483
$615$	−3735.02	−0.244895
$616$	0	0
$617$	−22927.4	−1.49598	−0.747991	−	0.663708i	$-0.768982\pi$
$617$	−22927.4	−1.49598	−0.747991	+	0.663708i	$0.768982\pi$
$618$	−5936.58	−0.386415
$619$	−17921.5	−1.16369	−0.581847	−	0.813298i	$-0.697670\pi$
$619$	−17921.5	−1.16369	−0.581847	+	0.813298i	$0.697670\pi$
$620$	−15875.6	−1.02835
$621$	2431.68	0.157134
$622$	−39071.8	−2.51871
$623$	0	0
$624$	25089.9	1.60962
$625$	625.000	0.0400000
$626$	−9303.22	−0.593980
$627$	−9490.39	−0.604481
$628$	−36005.1	−2.28784
$629$	8046.35	0.510062
$630$	0	0
$631$	19270.7	1.21578	0.607888	−	0.794023i	$-0.292017\pi$
$631$	19270.7	1.21578	0.607888	+	0.794023i	$0.292017\pi$
$632$	−26776.4	−1.68530
$633$	−6344.16	−0.398353
$634$	16077.1	1.00711
$635$	−2160.42	−0.135014
$636$	5554.53	0.346308
$637$	0	0
$638$	−44369.5	−2.75330
$639$	−5692.38	−0.352405
$640$	6844.76	0.422755
$641$	28238.6	1.74003	0.870013	−	0.493029i	$-0.164110\pi$
$641$	28238.6	1.74003	0.870013	+	0.493029i	$0.164110\pi$
$642$	−2124.94	−0.130631
$643$	19055.3	1.16869	0.584346	−	0.811505i	$-0.301351\pi$
$643$	19055.3	1.16869	0.584346	+	0.811505i	$0.301351\pi$
$644$	0	0
$645$	4468.80	0.272804
$646$	8988.75	0.547458
$647$	16211.3	0.985057	0.492529	−	0.870296i	$-0.336073\pi$
$647$	16211.3	0.985057	0.492529	+	0.870296i	$0.336073\pi$
$648$	4163.85	0.252425
$649$	−956.457	−0.0578493
$650$	−9053.22	−0.546302
$651$	0	0
$652$	34382.9	2.06524
$653$	−26532.1	−1.59001	−0.795007	−	0.606600i	$-0.792533\pi$
$653$	−26532.1	−1.59001	−0.795007	+	0.606600i	$0.792533\pi$
$654$	26237.8	1.56877
$655$	−3287.50	−0.196112
$656$	−29361.1	−1.74750
$657$	−8233.85	−0.488939
$658$	0	0
$659$	1883.19	0.111318	0.0556592	−	0.998450i	$-0.482274\pi$
$659$	1883.19	0.111318	0.0556592	+	0.998450i	$0.482274\pi$
$660$	−11040.3	−0.651124
$661$	−4674.27	−0.275050	−0.137525	−	0.990498i	$-0.543915\pi$
$661$	−4674.27	−0.275050	−0.137525	+	0.990498i	$0.543915\pi$
$662$	−29899.6	−1.75541
$663$	−4823.71	−0.282560
$664$	52827.9	3.08753
$665$	0	0
$666$	−16309.6	−0.948928
$667$	−19213.2	−1.11535
$668$	6833.11	0.395780
$669$	7735.31	0.447032
$670$	20697.8	1.19347
$671$	−35628.3	−2.04980
$672$	0	0
$673$	−23344.3	−1.33709	−0.668543	−	0.743674i	$-0.733082\pi$
$673$	−23344.3	−1.33709	−0.668543	+	0.743674i	$0.733082\pi$
$674$	−36075.4	−2.06168
$675$	−675.000	−0.0384900
$676$	51197.0	2.91289
$677$	−1399.49	−0.0794490	−0.0397245	−	0.999211i	$-0.512648\pi$
$677$	−1399.49	−0.0794490	−0.0397245	+	0.999211i	$0.512648\pi$
$678$	−31911.0	−1.80757
$679$	0	0
$680$	5826.83	0.328601
$681$	8777.77	0.493928
$682$	−36548.6	−2.05208
$683$	33020.7	1.84993	0.924964	−	0.380054i	$-0.124095\pi$
$683$	33020.7	1.84993	0.924964	+	0.380054i	$0.124095\pi$
$684$	−12628.4	−0.705935
$685$	−1462.58	−0.0815798
$686$	0	0
$687$	−3143.63	−0.174581
$688$	35129.4	1.94665
$689$	7268.04	0.401873
$690$	−6897.47	−0.380554
$691$	−720.176	−0.0396480	−0.0198240	−	0.999803i	$-0.506311\pi$
$691$	−720.176	−0.0396480	−0.0198240	+	0.999803i	$0.506311\pi$
$692$	11998.0	0.659096
$693$	0	0
$694$	50533.5	2.76401
$695$	−8365.10	−0.456556
$696$	−32899.3	−1.79173
$697$	5644.88	0.306765
$698$	−60414.0	−3.27608
$699$	11610.0	0.628224
$700$	0	0
$701$	11675.1	0.629048	0.314524	−	0.949249i	$-0.398155\pi$
$701$	11675.1	0.629048	0.314524	+	0.949249i	$0.398155\pi$
$702$	9777.48	0.525680
$703$	27563.6	1.47878
$704$	−1256.05	−0.0672430
$705$	−2531.56	−0.135240
$706$	−35777.6	−1.90724
$707$	0	0
$708$	−1272.71	−0.0675586
$709$	−26771.7	−1.41810	−0.709050	−	0.705158i	$-0.750876\pi$
$709$	−26771.7	−1.41810	−0.709050	+	0.705158i	$0.750876\pi$
$710$	16146.5	0.853473
$711$	−4687.98	−0.247276
$712$	−51598.3	−2.71591
$713$	−15826.5	−0.831288
$714$	0	0
$715$	−14446.1	−0.755597
$716$	−6052.85	−0.315930
$717$	18182.3	0.947044
$718$	14369.4	0.746884
$719$	−34783.5	−1.80418	−0.902089	−	0.431550i	$-0.857967\pi$
$719$	−34783.5	−1.80418	−0.902089	+	0.431550i	$0.857967\pi$
$720$	−5306.20	−0.274653
$721$	0	0
$722$	−4228.11	−0.217942
$723$	−12552.4	−0.645680
$724$	−22144.1	−1.13671
$725$	5333.30	0.273205
$726$	−5029.75	−0.257124
$727$	−25368.1	−1.29415	−0.647077	−	0.762425i	$-0.724009\pi$
$727$	−25368.1	−1.29415	−0.647077	+	0.762425i	$0.724009\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	23355.3	1.18414
$731$	−6753.87	−0.341725
$732$	−47409.0	−2.39383
$733$	−31691.5	−1.59694	−0.798468	−	0.602038i	$-0.794356\pi$
$733$	−31691.5	−1.59694	−0.798468	+	0.602038i	$0.794356\pi$
$734$	−11997.7	−0.603327
$735$	0	0
$736$	−17183.7	−0.860596
$737$	33027.1	1.65070
$738$	−11442.0	−0.570710
$739$	−13196.0	−0.656862	−0.328431	−	0.944528i	$-0.606520\pi$
$739$	−13196.0	−0.656862	−0.328431	+	0.944528i	$0.606520\pi$
$740$	32065.1	1.59289
$741$	−16524.1	−0.819203
$742$	0	0
$743$	8244.34	0.407073	0.203537	−	0.979067i	$-0.434756\pi$
$743$	8244.34	0.407073	0.203537	+	0.979067i	$0.434756\pi$
$744$	−27100.3	−1.33541
$745$	−12605.0	−0.619883
$746$	−8471.90	−0.415788
$747$	9249.03	0.453018
$748$	16685.6	0.815622
$749$	0	0
$750$	1914.64	0.0932170
$751$	31769.8	1.54367	0.771834	−	0.635824i	$-0.219339\pi$
$751$	31769.8	1.54367	0.771834	+	0.635824i	$0.219339\pi$
$752$	−19900.7	−0.965031
$753$	−7894.95	−0.382082
$754$	−77253.7	−3.73132
$755$	−6896.60	−0.332441
$756$	0	0
$757$	−3019.09	−0.144954	−0.0724772	−	0.997370i	$-0.523090\pi$
$757$	−3019.09	−0.144954	−0.0724772	+	0.997370i	$0.523090\pi$
$758$	27202.2	1.30347
$759$	−11006.2	−0.526349
$760$	19960.4	0.952686
$761$	−24096.3	−1.14782	−0.573910	−	0.818918i	$-0.694574\pi$
$761$	−24096.3	−1.14782	−0.573910	+	0.818918i	$0.694574\pi$
$762$	−6618.29	−0.314640
$763$	0	0
$764$	−28573.1	−1.35306
$765$	1020.15	0.0482140
$766$	−31858.3	−1.50273
$767$	−1665.33	−0.0783984
$768$	21708.4	1.01997
$769$	−7954.16	−0.372996	−0.186498	−	0.982455i	$-0.559714\pi$
$769$	−7954.16	−0.372996	−0.186498	+	0.982455i	$0.559714\pi$
$770$	0	0
$771$	9866.30	0.460864
$772$	−70369.1	−3.28062
$773$	−36239.4	−1.68621	−0.843105	−	0.537749i	$-0.819275\pi$
$773$	−36239.4	−1.68621	−0.843105	+	0.537749i	$0.819275\pi$
$774$	13689.8	0.635751
$775$	4393.22	0.203625
$776$	32880.0	1.52104
$777$	0	0
$778$	48428.7	2.23169
$779$	19337.1	0.889377
$780$	−19222.7	−0.882414
$781$	25764.6	1.18045
$782$	10424.4	0.476696
$783$	−5759.96	−0.262892
$784$	0	0
$785$	9963.65	0.453017
$786$	−10071.0	−0.457024
$787$	9684.13	0.438630	0.219315	−	0.975654i	$-0.429618\pi$
$787$	9684.13	0.438630	0.219315	+	0.975654i	$0.429618\pi$
$788$	17957.3	0.811803
$789$	16226.7	0.732174
$790$	13297.5	0.598864
$791$	0	0
$792$	−18846.2	−0.845545
$793$	−62034.1	−2.77793
$794$	−40298.0	−1.80116
$795$	−1537.10	−0.0685727
$796$	55648.1	2.47788
$797$	−6717.98	−0.298573	−0.149287	−	0.988794i	$-0.547698\pi$
$797$	−6717.98	−0.298573	−0.149287	+	0.988794i	$0.547698\pi$
$798$	0	0
$799$	3826.05	0.169407
$800$	4769.95	0.210804
$801$	−9033.75	−0.398492
$802$	−63615.3	−2.80092
$803$	37267.7	1.63779
$804$	43947.7	1.92775
$805$	0	0
$806$	−63636.5	−2.78102
$807$	1619.37	0.0706374
$808$	32584.4	1.41871
$809$	−1210.73	−0.0526167	−0.0263083	−	0.999654i	$-0.508375\pi$
$809$	−1210.73	−0.0526167	−0.0263083	+	0.999654i	$0.508375\pi$
$810$	−2067.81	−0.0896981
$811$	−34825.1	−1.50786	−0.753930	−	0.656955i	$-0.771844\pi$
$811$	−34825.1	−1.50786	−0.753930	+	0.656955i	$0.771844\pi$
$812$	0	0
$813$	12273.4	0.529456
$814$	73820.0	3.17861
$815$	−9514.72	−0.408940
$816$	8019.46	0.344041
$817$	−23136.1	−0.990734
$818$	−17377.9	−0.742792
$819$	0	0
$820$	22495.1	0.958003
$821$	30169.6	1.28249	0.641247	−	0.767334i	$-0.278417\pi$
$821$	30169.6	1.28249	0.641247	+	0.767334i	$0.278417\pi$
$822$	−4480.49	−0.190116
$823$	−26016.4	−1.10191	−0.550956	−	0.834534i	$-0.685737\pi$
$823$	−26016.4	−1.10191	−0.550956	+	0.834534i	$0.685737\pi$
$824$	19923.7	0.842322
$825$	3055.16	0.128930
$826$	0	0
$827$	15602.4	0.656045	0.328022	−	0.944670i	$-0.393618\pi$
$827$	15602.4	0.656045	0.328022	+	0.944670i	$0.393618\pi$
$828$	−14645.4	−0.614689
$829$	19506.0	0.817215	0.408607	−	0.912710i	$-0.366015\pi$
$829$	19506.0	0.817215	0.408607	+	0.912710i	$0.366015\pi$
$830$	−26234.9	−1.09714
$831$	−9274.01	−0.387138
$832$	−2186.96	−0.0911289
$833$	0	0
$834$	−25625.8	−1.06397
$835$	−1890.92	−0.0783687
$836$	57158.2	2.36466
$837$	−4744.68	−0.195938
$838$	7029.71	0.289782
$839$	12231.2	0.503300	0.251650	−	0.967818i	$-0.419027\pi$
$839$	12231.2	0.503300	0.251650	+	0.967818i	$0.419027\pi$
$840$	0	0
$841$	21121.6	0.866028
$842$	−80452.0	−3.29283
$843$	−21647.2	−0.884423
$844$	38209.3	1.55831
$845$	−14167.7	−0.576784
$846$	−7755.25	−0.315167
$847$	0	0
$848$	−12083.2	−0.489314
$849$	8894.15	0.359536
$850$	−2893.67	−0.116767
$851$	31966.0	1.28764
$852$	34283.8	1.37857
$853$	−1234.83	−0.0495660	−0.0247830	−	0.999693i	$-0.507889\pi$
$853$	−1234.83	−0.0495660	−0.0247830	+	0.999693i	$0.507889\pi$
$854$	0	0
$855$	3494.64	0.139783
$856$	7131.49	0.284754
$857$	−24101.4	−0.960662	−0.480331	−	0.877087i	$-0.659483\pi$
$857$	−24101.4	−0.960662	−0.480331	+	0.877087i	$0.659483\pi$
$858$	−44254.4	−1.76086
$859$	−13362.9	−0.530777	−0.265389	−	0.964142i	$-0.585500\pi$
$859$	−13362.9	−0.530777	−0.265389	+	0.964142i	$0.585500\pi$
$860$	−26914.5	−1.06718
$861$	0	0
$862$	25955.9	1.02559
$863$	−21934.1	−0.865174	−0.432587	−	0.901592i	$-0.642399\pi$
$863$	−21934.1	−0.865174	−0.432587	+	0.901592i	$0.642399\pi$
$864$	−5151.54	−0.202846
$865$	−3320.18	−0.130508
$866$	−23646.8	−0.927888
$867$	13197.2	0.516956
$868$	0	0
$869$	21218.5	0.828296
$870$	16338.2	0.636685
$871$	57505.0	2.23706
$872$	−88056.1	−3.41967
$873$	5756.58	0.223174
$874$	35709.9	1.38204
$875$	0	0
$876$	49590.4	1.91268
$877$	−770.318	−0.0296600	−0.0148300	−	0.999890i	$-0.504721\pi$
$877$	−770.318	−0.0296600	−0.0148300	+	0.999890i	$0.504721\pi$
$878$	11941.6	0.459009
$879$	−23772.3	−0.912195
$880$	24016.7	0.920003
$881$	−17887.9	−0.684064	−0.342032	−	0.939688i	$-0.611115\pi$
$881$	−17887.9	−0.684064	−0.342032	+	0.939688i	$0.611115\pi$
$882$	0	0
$883$	583.389	0.0222340	0.0111170	−	0.999938i	$-0.496461\pi$
$883$	583.389	0.0222340	0.0111170	+	0.999938i	$0.496461\pi$
$884$	29052.0	1.10534
$885$	352.196	0.0133773
$886$	60306.0	2.28670
$887$	42443.8	1.60668	0.803340	−	0.595521i	$-0.203054\pi$
$887$	42443.8	1.60668	0.803340	+	0.595521i	$0.203054\pi$
$888$	54736.5	2.06851
$889$	0	0
$890$	25624.3	0.965087
$891$	−3299.57	−0.124063
$892$	−46587.8	−1.74874
$893$	13106.5	0.491146
$894$	−38614.6	−1.44459
$895$	1675.00	0.0625575
$896$	0	0
$897$	−19163.3	−0.713317
$898$	48924.4	1.81807
$899$	37488.6	1.39078
$900$	4065.35	0.150569
$901$	2323.08	0.0858967
$902$	51788.1	1.91170
$903$	0	0
$904$	107096.	3.94021
$905$	6127.92	0.225082
$906$	−21127.2	−0.774729
$907$	−27717.8	−1.01472	−0.507362	−	0.861733i	$-0.669379\pi$
$907$	−27717.8	−1.01472	−0.507362	+	0.861733i	$0.669379\pi$
$908$	−52866.3	−1.93219
$909$	5704.82	0.208160
$910$	0	0
$911$	20446.7	0.743609	0.371805	−	0.928311i	$-0.378739\pi$
$911$	20446.7	0.743609	0.371805	+	0.928311i	$0.378739\pi$
$912$	27471.5	0.997448
$913$	−41862.6	−1.51747
$914$	−33554.4	−1.21431
$915$	13119.4	0.474005
$916$	18933.3	0.682941
$917$	0	0
$918$	3125.16	0.112359
$919$	17455.5	0.626556	0.313278	−	0.949662i	$-0.398573\pi$
$919$	17455.5	0.626556	0.313278	+	0.949662i	$0.398573\pi$
$920$	23148.5	0.829546
$921$	−12201.0	−0.436522
$922$	21431.7	0.765526
$923$	44859.9	1.59976
$924$	0	0
$925$	−8873.32	−0.315409
$926$	−31276.3	−1.10994
$927$	3488.20	0.123590
$928$	40703.3	1.43982
$929$	−20748.8	−0.732773	−0.366387	−	0.930463i	$-0.619405\pi$
$929$	−20748.8	−0.732773	−0.366387	+	0.930463i	$0.619405\pi$
$930$	13458.3	0.474532
$931$	0	0
$932$	−69923.8	−2.45754
$933$	22957.7	0.805576
$934$	−16481.7	−0.577405
$935$	−4617.38	−0.161502
$936$	−32814.0	−1.14590
$937$	−12135.5	−0.423106	−0.211553	−	0.977366i	$-0.567852\pi$
$937$	−12135.5	−0.423106	−0.211553	+	0.977366i	$0.567852\pi$
$938$	0	0
$939$	5466.36	0.189977
$940$	15247.0	0.529044
$941$	25266.0	0.875292	0.437646	−	0.899147i	$-0.355812\pi$
$941$	25266.0	0.875292	0.437646	+	0.899147i	$0.355812\pi$
$942$	30522.9	1.05572
$943$	22425.6	0.774421
$944$	2768.62	0.0954566
$945$	0	0
$946$	−61962.3	−2.12957
$947$	−42768.7	−1.46758	−0.733789	−	0.679377i	$-0.762250\pi$
$947$	−42768.7	−1.46758	−0.733789	+	0.679377i	$0.762250\pi$
$948$	28234.5	0.967315
$949$	64888.4	2.21957
$950$	−9912.57	−0.338533
$951$	−9446.57	−0.322110
$952$	0	0
$953$	−25855.5	−0.878846	−0.439423	−	0.898280i	$-0.644817\pi$
$953$	−25855.5	−0.878846	−0.439423	+	0.898280i	$0.644817\pi$
$954$	−4708.78	−0.159803
$955$	7906.98	0.267920
$956$	−109507.	−3.70473
$957$	26070.5	0.880606
$958$	−15962.0	−0.538318
$959$	0	0
$960$	462.515	0.0155496
$961$	1089.64	0.0365761
$962$	128531.	4.30771
$963$	1248.57	0.0417805
$964$	75599.6	2.52583
$965$	19473.1	0.649598
$966$	0	0
$967$	−20336.0	−0.676278	−0.338139	−	0.941096i	$-0.609797\pi$
$967$	−20336.0	−0.676278	−0.338139	+	0.941096i	$0.609797\pi$
$968$	16880.3	0.560488
$969$	−5281.59	−0.175097
$970$	−16328.6	−0.540493
$971$	31123.7	1.02864	0.514319	−	0.857599i	$-0.328045\pi$
$971$	31123.7	1.02864	0.514319	+	0.857599i	$0.328045\pi$
$972$	−4390.58	−0.144885
$973$	0	0
$974$	−4437.05	−0.145967
$975$	5319.47	0.174728
$976$	103132.	3.38236
$977$	31317.3	1.02552	0.512758	−	0.858533i	$-0.328624\pi$
$977$	31317.3	1.02552	0.512758	+	0.858533i	$0.328624\pi$
$978$	−29147.6	−0.953004
$979$	40888.2	1.33482
$980$	0	0
$981$	−15416.7	−0.501752
$982$	33788.1	1.09798
$983$	4002.12	0.129855	0.0649277	−	0.997890i	$-0.479318\pi$
$983$	4002.12	0.129855	0.0649277	+	0.997890i	$0.479318\pi$
$984$	38400.1	1.24406
$985$	−4969.29	−0.160746
$986$	−24692.5	−0.797535
$987$	0	0
$988$	99520.7	3.20463
$989$	−26831.3	−0.862677
$990$	9359.24	0.300461
$991$	10042.5	0.321906	0.160953	−	0.986962i	$-0.448543\pi$
$991$	10042.5	0.321906	0.160953	+	0.986962i	$0.448543\pi$
$992$	33528.7	1.07312
$993$	17568.4	0.561445
$994$	0	0
$995$	−15399.4	−0.490647
$996$	−55704.6	−1.77216
$997$	9407.44	0.298833	0.149417	−	0.988774i	$-0.452260\pi$
$997$	9407.44	0.298833	0.149417	+	0.988774i	$0.452260\pi$
$998$	−4534.65	−0.143830
$999$	9583.18	0.303502

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.4.a.z.1.5		5
3.2	odd	2		2205.4.a.bs.1.1		5
7.3	odd	6		105.4.i.d.16.1	✓	10
7.5	odd	6		105.4.i.d.46.1	yes	10
7.6	odd	2		735.4.a.ba.1.5		5
21.5	even	6		315.4.j.h.46.5		10
21.17	even	6		315.4.j.h.226.5		10
21.20	even	2		2205.4.a.br.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.i.d.16.1	✓	10	7.3	odd	6
105.4.i.d.46.1	yes	10	7.5	odd	6
315.4.j.h.46.5		10	21.5	even	6
315.4.j.h.226.5		10	21.17	even	6
735.4.a.z.1.5		5	1.1	even	1	trivial
735.4.a.ba.1.5		5	7.6	odd	2
2205.4.a.br.1.1		5	21.20	even	2
2205.4.a.bs.1.1		5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	735.a (trivial)

\(f(q)\)	\(=\)	\(q+5.10571 q^{2} -3.00000 q^{3} +18.0682 q^{4} -5.00000 q^{5} -15.3171 q^{6} +51.4055 q^{8} +9.00000 q^{9} -25.5285 q^{10} -40.7354 q^{11} -54.2047 q^{12} -70.9263 q^{13} +15.0000 q^{15} +117.915 q^{16} -22.6701 q^{17} +45.9514 q^{18} -77.6588 q^{19} -90.3412 q^{20} -207.983 q^{22} -90.0622 q^{23} -154.217 q^{24} +25.0000 q^{25} -362.129 q^{26} -27.0000 q^{27} +213.332 q^{29} +76.5856 q^{30} +175.729 q^{31} +190.798 q^{32} +122.206 q^{33} -115.747 q^{34} +162.614 q^{36} -354.933 q^{37} -396.503 q^{38} +212.779 q^{39} -257.028 q^{40} -249.001 q^{41} +297.920 q^{43} -736.018 q^{44} -45.0000 q^{45} -459.831 q^{46} -168.771 q^{47} -353.746 q^{48} +127.643 q^{50} +68.0102 q^{51} -1281.51 q^{52} -102.473 q^{53} -137.854 q^{54} +203.677 q^{55} +232.976 q^{57} +1089.21 q^{58} +23.4797 q^{59} +271.024 q^{60} +874.628 q^{61} +897.220 q^{62} +30.8343 q^{64} +354.631 q^{65} +623.949 q^{66} -810.771 q^{67} -409.609 q^{68} +270.187 q^{69} -632.487 q^{71} +462.650 q^{72} -914.872 q^{73} -1812.18 q^{74} -75.0000 q^{75} -1403.16 q^{76} +1086.39 q^{78} -520.886 q^{79} -589.577 q^{80} +81.0000 q^{81} -1271.33 q^{82} +1027.67 q^{83} +113.350 q^{85} +1521.09 q^{86} -639.996 q^{87} -2094.02 q^{88} -1003.75 q^{89} -229.757 q^{90} -1627.27 q^{92} -527.187 q^{93} -861.694 q^{94} +388.294 q^{95} -572.394 q^{96} +639.620 q^{97} -366.619 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} - 15 q^{3} + 25 q^{4} - 25 q^{5} - 9 q^{6} + 21 q^{8} + 45 q^{9} - 15 q^{10} + 43 q^{11} - 75 q^{12} - 123 q^{13} + 75 q^{15} + 161 q^{16} - 124 q^{17} + 27 q^{18} - 37 q^{19} - 125 q^{20}+ \cdots + 387 q^{99}+O(q^{100}) \) 5 * q + 3 * q^2 - 15 * q^3 + 25 * q^4 - 25 * q^5 - 9 * q^6 + 21 * q^8 + 45 * q^9 - 15 * q^10 + 43 * q^11 - 75 * q^12 - 123 * q^13 + 75 * q^15 + 161 * q^16 - 124 * q^17 + 27 * q^18 - 37 * q^19 - 125 * q^20 - 221 * q^22 + 77 * q^23 - 63 * q^24 + 125 * q^25 + 79 * q^26 - 135 * q^27 + 360 * q^29 + 45 * q^30 - 314 * q^31 - 59 * q^32 - 129 * q^33 - 176 * q^34 + 225 * q^36 + 225 * q^37 - 759 * q^38 + 369 * q^39 - 105 * q^40 - 341 * q^41 + 32 * q^43 + 679 * q^44 - 225 * q^45 - 331 * q^46 - 25 * q^47 - 483 * q^48 + 75 * q^50 + 372 * q^51 - 2299 * q^52 - 317 * q^53 - 81 * q^54 - 215 * q^55 + 111 * q^57 + 8 * q^58 - 676 * q^59 + 375 * q^60 + 188 * q^61 + 348 * q^62 - 1103 * q^64 + 615 * q^65 + 663 * q^66 - 1776 * q^67 - 1280 * q^68 - 231 * q^69 - 6 * q^71 + 189 * q^72 - 2006 * q^73 - 2729 * q^74 - 375 * q^75 - 1417 * q^76 - 237 * q^78 + 200 * q^79 - 805 * q^80 + 405 * q^81 + 539 * q^82 + 332 * q^83 + 620 * q^85 + 4262 * q^86 - 1080 * q^87 - 4529 * q^88 - 894 * q^89 - 135 * q^90 - 3687 * q^92 + 942 * q^93 - 4233 * q^94 + 185 * q^95 + 177 * q^96 + 576 * q^97 + 387 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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