LMFDB - Embedded newform 735.4.a.o.1.2

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 = [2,-2,6,2,-10,-6,0,-18,18,10,-16] 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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
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			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	\(q+1.82843 q^{2} +3.00000 q^{3} -4.65685 q^{4} -5.00000 q^{5} +5.48528 q^{6} -23.1421 q^{8} +9.00000 q^{9} -9.14214 q^{10} -64.5685 q^{11} -13.9706 q^{12} +32.3431 q^{13} -15.0000 q^{15} -5.05887 q^{16} +56.3431 q^{17} +16.4558 q^{18} +2.74517 q^{19} +23.2843 q^{20} -118.059 q^{22} +88.1665 q^{23} -69.4264 q^{24} +25.0000 q^{25} +59.1371 q^{26} +27.0000 q^{27} +246.735 q^{29} -27.4264 q^{30} +110.912 q^{31} +175.887 q^{32} -193.706 q^{33} +103.019 q^{34} -41.9117 q^{36} +120.676 q^{37} +5.01934 q^{38} +97.0294 q^{39} +115.711 q^{40} +176.274 q^{41} -443.362 q^{43} +300.686 q^{44} -45.0000 q^{45} +161.206 q^{46} +345.206 q^{47} -15.1766 q^{48} +45.7107 q^{50} +169.029 q^{51} -150.617 q^{52} +260.981 q^{53} +49.3675 q^{54} +322.843 q^{55} +8.23550 q^{57} +451.137 q^{58} -628.999 q^{59} +69.8528 q^{60} +115.206 q^{61} +202.794 q^{62} +362.068 q^{64} -161.716 q^{65} -354.177 q^{66} -951.480 q^{67} -262.382 q^{68} +264.500 q^{69} +356.264 q^{71} -208.279 q^{72} +656.754 q^{73} +220.648 q^{74} +75.0000 q^{75} -12.7838 q^{76} +177.411 q^{78} +440.195 q^{79} +25.2944 q^{80} +81.0000 q^{81} +322.304 q^{82} +54.4121 q^{83} -281.716 q^{85} -810.656 q^{86} +740.205 q^{87} +1494.25 q^{88} +1018.78 q^{89} -82.2792 q^{90} -410.579 q^{92} +332.735 q^{93} +631.184 q^{94} -13.7258 q^{95} +527.662 q^{96} +724.108 q^{97} -581.117 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 6 q^{6} - 18 q^{8} + 18 q^{9} + 10 q^{10} - 16 q^{11} + 6 q^{12} + 76 q^{13} - 30 q^{15} - 78 q^{16} + 124 q^{17} - 18 q^{18} + 96 q^{19} - 10 q^{20} - 304 q^{22}+ \cdots - 144 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 6 * q^3 + 2 * q^4 - 10 * q^5 - 6 * q^6 - 18 * q^8 + 18 * q^9 + 10 * q^10 - 16 * q^11 + 6 * q^12 + 76 * q^13 - 30 * q^15 - 78 * q^16 + 124 * q^17 - 18 * q^18 + 96 * q^19 - 10 * q^20 - 304 * q^22 - 16 * q^23 - 54 * q^24 + 50 * q^25 - 108 * q^26 + 54 * q^27 + 188 * q^29 + 30 * q^30 + 120 * q^31 + 414 * q^32 - 48 * q^33 - 156 * q^34 + 18 * q^36 - 132 * q^37 - 352 * q^38 + 228 * q^39 + 90 * q^40 - 100 * q^41 - 536 * q^43 + 624 * q^44 - 90 * q^45 + 560 * q^46 + 928 * q^47 - 234 * q^48 - 50 * q^50 + 372 * q^51 + 140 * q^52 + 884 * q^53 - 54 * q^54 + 80 * q^55 + 288 * q^57 + 676 * q^58 - 104 * q^59 - 30 * q^60 + 468 * q^61 + 168 * q^62 + 34 * q^64 - 380 * q^65 - 912 * q^66 - 1688 * q^67 + 188 * q^68 - 48 * q^69 - 136 * q^71 - 162 * q^72 - 508 * q^73 + 1188 * q^74 + 150 * q^75 + 608 * q^76 - 324 * q^78 - 432 * q^79 + 390 * q^80 + 162 * q^81 + 1380 * q^82 + 584 * q^83 - 620 * q^85 - 456 * q^86 + 564 * q^87 + 1744 * q^88 + 1404 * q^89 + 90 * q^90 - 1104 * q^92 + 360 * q^93 - 1600 * q^94 - 480 * q^95 + 1242 * q^96 + 1188 * q^97 - 144 * 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 $

$2$	1.82843	0.646447	0.323223	−	0.946323i	$-0.395234\pi$
$2$	1.82843	0.646447	0.323223	+	0.946323i	$0.395234\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−4.65685	−0.582107
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	5.48528	0.373226
$7$	0	0
$7$	0	0
$8$	−23.1421	−1.02275
$9$	9.00000	0.333333
$10$	−9.14214	−0.289100
$11$	−64.5685	−1.76983	−0.884916	−	0.465751i	$-0.845784\pi$
$11$	−64.5685	−1.76983	−0.884916	+	0.465751i	$0.845784\pi$
$12$	−13.9706	−0.336080
$13$	32.3431	0.690029	0.345014	−	0.938597i	$-0.387874\pi$
$13$	32.3431	0.690029	0.345014	+	0.938597i	$0.387874\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	−5.05887	−0.0790449
$17$	56.3431	0.803836	0.401918	−	0.915676i	$-0.368344\pi$
$17$	56.3431	0.803836	0.401918	+	0.915676i	$0.368344\pi$
$18$	16.4558	0.215482
$19$	2.74517	0.0331465	0.0165733	−	0.999863i	$-0.494724\pi$
$19$	2.74517	0.0331465	0.0165733	+	0.999863i	$0.494724\pi$
$20$	23.2843	0.260326
$21$	0	0
$22$	−118.059	−1.14410
$23$	88.1665	0.799304	0.399652	−	0.916667i	$-0.369131\pi$
$23$	88.1665	0.799304	0.399652	+	0.916667i	$0.369131\pi$
$24$	−69.4264	−0.590484
$25$	25.0000	0.200000
$26$	59.1371	0.446067
$27$	27.0000	0.192450
$28$	0	0
$29$	246.735	1.57992	0.789958	−	0.613161i	$-0.210102\pi$
$29$	246.735	1.57992	0.789958	+	0.613161i	$0.210102\pi$
$30$	−27.4264	−0.166912
$31$	110.912	0.642591	0.321296	−	0.946979i	$-0.395882\pi$
$31$	110.912	0.642591	0.321296	+	0.946979i	$0.395882\pi$
$32$	175.887	0.971649
$33$	−193.706	−1.02181
$34$	103.019	0.519637
$35$	0	0
$36$	−41.9117	−0.194036
$37$	120.676	0.536190	0.268095	−	0.963392i	$-0.413606\pi$
$37$	120.676	0.536190	0.268095	+	0.963392i	$0.413606\pi$
$38$	5.01934	0.0214275
$39$	97.0294	0.398388
$40$	115.711	0.457387
$41$	176.274	0.671449	0.335724	−	0.941960i	$-0.391019\pi$
$41$	176.274	0.671449	0.335724	+	0.941960i	$0.391019\pi$
$42$	0	0
$43$	−443.362	−1.57238	−0.786188	−	0.617988i	$-0.787948\pi$
$43$	−443.362	−1.57238	−0.786188	+	0.617988i	$0.787948\pi$
$44$	300.686	1.03023
$45$	−45.0000	−0.149071
$46$	161.206	0.516707
$47$	345.206	1.07135	0.535675	−	0.844424i	$-0.320057\pi$
$47$	345.206	1.07135	0.535675	+	0.844424i	$0.320057\pi$
$48$	−15.1766	−0.0456366
$49$	0	0
$50$	45.7107	0.129289
$51$	169.029	0.464095
$52$	−150.617	−0.401670
$53$	260.981	0.676386	0.338193	−	0.941077i	$-0.390184\pi$
$53$	260.981	0.676386	0.338193	+	0.941077i	$0.390184\pi$
$54$	49.3675	0.124409
$55$	322.843	0.791493
$56$	0	0
$57$	8.23550	0.0191372
$58$	451.137	1.02133
$59$	−628.999	−1.38794	−0.693972	−	0.720002i	$-0.744141\pi$
$59$	−628.999	−1.38794	−0.693972	+	0.720002i	$0.744141\pi$
$60$	69.8528	0.150299
$61$	115.206	0.241814	0.120907	−	0.992664i	$-0.461420\pi$
$61$	115.206	0.241814	0.120907	+	0.992664i	$0.461420\pi$
$62$	202.794	0.415401
$63$	0	0
$64$	362.068	0.707164
$65$	−161.716	−0.308590
$66$	−354.177	−0.660547
$67$	−951.480	−1.73495	−0.867476	−	0.497479i	$-0.834259\pi$
$67$	−951.480	−1.73495	−0.867476	+	0.497479i	$0.834259\pi$
$68$	−262.382	−0.467919
$69$	264.500	0.461478
$70$	0	0
$71$	356.264	0.595504	0.297752	−	0.954643i	$-0.403763\pi$
$71$	356.264	0.595504	0.297752	+	0.954643i	$0.403763\pi$
$72$	−208.279	−0.340916
$73$	656.754	1.05298	0.526488	−	0.850183i	$-0.323509\pi$
$73$	656.754	1.05298	0.526488	+	0.850183i	$0.323509\pi$
$74$	220.648	0.346618
$75$	75.0000	0.115470
$76$	−12.7838	−0.0192948
$77$	0	0
$78$	177.411	0.257537
$79$	440.195	0.626909	0.313455	−	0.949603i	$-0.398514\pi$
$79$	440.195	0.626909	0.313455	+	0.949603i	$0.398514\pi$
$80$	25.2944	0.0353500
$81$	81.0000	0.111111
$82$	322.304	0.434056
$83$	54.4121	0.0719579	0.0359790	−	0.999353i	$-0.488545\pi$
$83$	54.4121	0.0719579	0.0359790	+	0.999353i	$0.488545\pi$
$84$	0	0
$85$	−281.716	−0.359487
$86$	−810.656	−1.01646
$87$	740.205	0.912165
$88$	1494.25	1.81009
$89$	1018.78	1.21338	0.606690	−	0.794938i	$-0.292497\pi$
$89$	1018.78	1.21338	0.606690	+	0.794938i	$0.292497\pi$
$90$	−82.2792	−0.0963666
$91$	0	0
$92$	−410.579	−0.465280
$93$	332.735	0.371000
$94$	631.184	0.692571
$95$	−13.7258	−0.0148236
$96$	527.662	0.560982
$97$	724.108	0.757959	0.378979	−	0.925405i	$-0.376275\pi$
$97$	724.108	0.757959	0.378979	+	0.925405i	$0.376275\pi$
$98$	0	0
$99$	−581.117	−0.589944
$100$	−116.421	−0.116421
$101$	−268.725	−0.264744	−0.132372	−	0.991200i	$-0.542259\pi$
$101$	−268.725	−0.264744	−0.132372	+	0.991200i	$0.542259\pi$
$102$	309.058	0.300013
$103$	1840.63	1.76080	0.880399	−	0.474233i	$-0.157275\pi$
$103$	1840.63	1.76080	0.880399	+	0.474233i	$0.157275\pi$
$104$	−748.489	−0.705725
$105$	0	0
$106$	477.184	0.437247
$107$	−243.087	−0.219627	−0.109813	−	0.993952i	$-0.535025\pi$
$107$	−243.087	−0.219627	−0.109813	+	0.993952i	$0.535025\pi$
$108$	−125.735	−0.112027
$109$	−405.176	−0.356044	−0.178022	−	0.984027i	$-0.556970\pi$
$109$	−405.176	−0.356044	−0.178022	+	0.984027i	$0.556970\pi$
$110$	590.294	0.511658
$111$	362.029	0.309570
$112$	0	0
$113$	−28.1766	−0.0234569	−0.0117285	−	0.999931i	$-0.503733\pi$
$113$	−28.1766	−0.0234569	−0.0117285	+	0.999931i	$0.503733\pi$
$114$	15.0580	0.0123712
$115$	−440.833	−0.357460
$116$	−1149.01	−0.919680
$117$	291.088	0.230010
$118$	−1150.08	−0.897232
$119$	0	0
$120$	347.132	0.264072
$121$	2838.10	2.13230
$122$	210.646	0.156320
$123$	528.823	0.387661
$124$	−516.500	−0.374057
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2740.90	−1.91508	−0.957541	−	0.288298i	$-0.906911\pi$
$127$	−2740.90	−1.91508	−0.957541	+	0.288298i	$0.906911\pi$
$128$	−745.083	−0.514505
$129$	−1330.09	−0.907811
$130$	−295.685	−0.199487
$131$	1832.04	1.22188	0.610938	−	0.791678i	$-0.290792\pi$
$131$	1832.04	1.22188	0.610938	+	0.791678i	$0.290792\pi$
$132$	902.059	0.594804
$133$	0	0
$134$	−1739.71	−1.12155
$135$	−135.000	−0.0860663
$136$	−1303.90	−0.822122
$137$	382.747	0.238688	0.119344	−	0.992853i	$-0.461921\pi$
$137$	382.747	0.238688	0.119344	+	0.992853i	$0.461921\pi$
$138$	483.618	0.298321
$139$	−3053.60	−1.86333	−0.931667	−	0.363314i	$-0.881645\pi$
$139$	−3053.60	−1.86333	−0.931667	+	0.363314i	$0.881645\pi$
$140$	0	0
$141$	1035.62	0.618545
$142$	651.403	0.384961
$143$	−2088.35	−1.22123
$144$	−45.5299	−0.0263483
$145$	−1233.68	−0.706560
$146$	1200.83	0.680692
$147$	0	0
$148$	−561.971	−0.312120
$149$	3560.60	1.95769	0.978843	−	0.204611i	$-0.0655929\pi$
$149$	3560.60	1.95769	0.978843	+	0.204611i	$0.0655929\pi$
$150$	137.132	0.0746452
$151$	3261.80	1.75789	0.878945	−	0.476923i	$-0.158248\pi$
$151$	3261.80	1.75789	0.878945	+	0.476923i	$0.158248\pi$
$152$	−63.5290	−0.0339005
$153$	507.088	0.267945
$154$	0	0
$155$	−554.558	−0.287376
$156$	−451.852	−0.231905
$157$	−2878.46	−1.46322	−0.731611	−	0.681723i	$-0.761231\pi$
$157$	−2878.46	−1.46322	−0.731611	+	0.681723i	$0.761231\pi$
$158$	804.865	0.405263
$159$	782.942	0.390512
$160$	−879.437	−0.434535
$161$	0	0
$162$	148.103	0.0718274
$163$	−927.537	−0.445708	−0.222854	−	0.974852i	$-0.571537\pi$
$163$	−927.537	−0.445708	−0.222854	+	0.974852i	$0.571537\pi$
$164$	−820.883	−0.390855
$165$	968.528	0.456969
$166$	99.4886	0.0465169
$167$	−1094.52	−0.507164	−0.253582	−	0.967314i	$-0.581609\pi$
$167$	−1094.52	−0.507164	−0.253582	+	0.967314i	$0.581609\pi$
$168$	0	0
$169$	−1150.92	−0.523860
$170$	−515.097	−0.232389
$171$	24.7065	0.0110488
$172$	2064.67	0.915290
$173$	1713.25	0.752926	0.376463	−	0.926432i	$-0.377140\pi$
$173$	1713.25	0.752926	0.376463	+	0.926432i	$0.377140\pi$
$174$	1353.41	0.589666
$175$	0	0
$176$	326.644	0.139896
$177$	−1887.00	−0.801330
$178$	1862.77	0.784386
$179$	−4065.58	−1.69763	−0.848816	−	0.528689i	$-0.822684\pi$
$179$	−4065.58	−1.69763	−0.848816	+	0.528689i	$0.822684\pi$
$180$	209.558	0.0867754
$181$	2791.40	1.14631	0.573157	−	0.819445i	$-0.305718\pi$
$181$	2791.40	1.14631	0.573157	+	0.819445i	$0.305718\pi$
$182$	0	0
$183$	345.618	0.139611
$184$	−2040.36	−0.817486
$185$	−603.381	−0.239792
$186$	608.382	0.239832
$187$	−3637.99	−1.42266
$188$	−1607.57	−0.623640
$189$	0	0
$190$	−25.0967	−0.00958266
$191$	−634.185	−0.240251	−0.120126	−	0.992759i	$-0.538330\pi$
$191$	−634.185	−0.240251	−0.120126	+	0.992759i	$0.538330\pi$
$192$	1086.20	0.408281
$193$	−254.999	−0.0951049	−0.0475524	−	0.998869i	$-0.515142\pi$
$193$	−254.999	−0.0951049	−0.0475524	+	0.998869i	$0.515142\pi$
$194$	1323.98	0.489980
$195$	−485.147	−0.178165
$196$	0	0
$197$	4172.37	1.50898	0.754490	−	0.656311i	$-0.227884\pi$
$197$	4172.37	1.50898	0.754490	+	0.656311i	$0.227884\pi$
$198$	−1062.53	−0.381367
$199$	4626.48	1.64805	0.824026	−	0.566552i	$-0.191723\pi$
$199$	4626.48	1.64805	0.824026	+	0.566552i	$0.191723\pi$
$200$	−578.553	−0.204550
$201$	−2854.44	−1.00168
$202$	−491.344	−0.171143
$203$	0	0
$204$	−787.145	−0.270153
$205$	−881.371	−0.300281
$206$	3365.45	1.13826
$207$	793.499	0.266435
$208$	−163.620	−0.0545433
$209$	−177.251	−0.0586638
$210$	0	0
$211$	−1562.64	−0.509843	−0.254921	−	0.966962i	$-0.582050\pi$
$211$	−1562.64	−0.509843	−0.254921	+	0.966962i	$0.582050\pi$
$212$	−1215.35	−0.393729
$213$	1068.79	0.343814
$214$	−444.466	−0.141977
$215$	2216.81	0.703188
$216$	−624.838	−0.196828
$217$	0	0
$218$	−740.834	−0.230163
$219$	1970.26	0.607935
$220$	−1503.43	−0.460733
$221$	1822.31	0.554670
$222$	661.943	0.200120
$223$	1236.39	0.371278	0.185639	−	0.982618i	$-0.440564\pi$
$223$	1236.39	0.371278	0.185639	+	0.982618i	$0.440564\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	−51.5189	−0.0151637
$227$	−4181.82	−1.22272	−0.611359	−	0.791353i	$-0.709377\pi$
$227$	−4181.82	−1.22272	−0.611359	+	0.791353i	$0.709377\pi$
$228$	−38.3515	−0.0111399
$229$	484.774	0.139890	0.0699449	−	0.997551i	$-0.477718\pi$
$229$	484.774	0.139890	0.0699449	+	0.997551i	$0.477718\pi$
$230$	−806.030	−0.231079
$231$	0	0
$232$	−5709.98	−1.61585
$233$	2080.54	0.584982	0.292491	−	0.956268i	$-0.405516\pi$
$233$	2080.54	0.584982	0.292491	+	0.956268i	$0.405516\pi$
$234$	532.234	0.148689
$235$	−1726.03	−0.479123
$236$	2929.16	0.807932
$237$	1320.59	0.361946
$238$	0	0
$239$	6814.10	1.84422	0.922108	−	0.386933i	$-0.126466\pi$
$239$	6814.10	1.84422	0.922108	+	0.386933i	$0.126466\pi$
$240$	75.8831	0.0204093
$241$	3921.84	1.04825	0.524125	−	0.851642i	$-0.324393\pi$
$241$	3921.84	1.04825	0.524125	+	0.851642i	$0.324393\pi$
$242$	5189.25	1.37842
$243$	243.000	0.0641500
$244$	−536.498	−0.140761
$245$	0	0
$246$	966.913	0.250602
$247$	88.7873	0.0228721
$248$	−2566.73	−0.657209
$249$	163.236	0.0415449
$250$	−228.553	−0.0578199
$251$	5219.10	1.31246	0.656228	−	0.754562i	$-0.272151\pi$
$251$	5219.10	1.31246	0.656228	+	0.754562i	$0.272151\pi$
$252$	0	0
$253$	−5692.78	−1.41463
$254$	−5011.53	−1.23800
$255$	−845.147	−0.207550
$256$	−4258.88	−1.03976
$257$	−6975.71	−1.69312	−0.846562	−	0.532289i	$-0.821332\pi$
$257$	−6975.71	−1.69312	−0.846562	+	0.532289i	$0.821332\pi$
$258$	−2431.97	−0.586852
$259$	0	0
$260$	753.087	0.179632
$261$	2220.62	0.526639
$262$	3349.75	0.789878
$263$	−3607.36	−0.845776	−0.422888	−	0.906182i	$-0.638984\pi$
$263$	−3607.36	−0.845776	−0.422888	+	0.906182i	$0.638984\pi$
$264$	4482.76	1.04506
$265$	−1304.90	−0.302489
$266$	0	0
$267$	3056.35	0.700546
$268$	4430.90	1.00993
$269$	−5.88572	−0.00133405	−0.000667023	−	1.00000i	$-0.500212\pi$
$269$	−5.88572	−0.00133405	−0.000667023		1.00000i	$0.500212\pi$
$270$	−246.838	−0.0556373
$271$	−6916.32	−1.55032	−0.775160	−	0.631765i	$-0.782331\pi$
$271$	−6916.32	−1.55032	−0.775160	+	0.631765i	$0.782331\pi$
$272$	−285.033	−0.0635392
$273$	0	0
$274$	699.825	0.154299
$275$	−1614.21	−0.353966
$276$	−1231.74	−0.268630
$277$	−2119.46	−0.459733	−0.229867	−	0.973222i	$-0.573829\pi$
$277$	−2119.46	−0.459733	−0.229867	+	0.973222i	$0.573829\pi$
$278$	−5583.29	−1.20455
$279$	998.205	0.214197
$280$	0	0
$281$	−239.917	−0.0509334	−0.0254667	−	0.999676i	$-0.508107\pi$
$281$	−239.917	−0.0509334	−0.0254667	+	0.999676i	$0.508107\pi$
$282$	1893.55	0.399856
$283$	4542.12	0.954067	0.477034	−	0.878885i	$-0.341712\pi$
$283$	4542.12	0.954067	0.477034	+	0.878885i	$0.341712\pi$
$284$	−1659.07	−0.346647
$285$	−41.1775	−0.00855840
$286$	−3818.40	−0.789463
$287$	0	0
$288$	1582.99	0.323883
$289$	−1738.45	−0.353847
$290$	−2255.69	−0.456753
$291$	2172.32	0.437608
$292$	−3058.41	−0.612944
$293$	2171.70	0.433010	0.216505	−	0.976281i	$-0.430534\pi$
$293$	2171.70	0.433010	0.216505	+	0.976281i	$0.430534\pi$
$294$	0	0
$295$	3145.00	0.620708
$296$	−2792.70	−0.548387
$297$	−1743.35	−0.340604
$298$	6510.29	1.26554
$299$	2851.58	0.551543
$300$	−349.264	−0.0672159
$301$	0	0
$302$	5963.96	1.13638
$303$	−806.175	−0.152850
$304$	−13.8875	−0.00262007
$305$	−576.030	−0.108142
$306$	927.174	0.173212
$307$	3508.64	0.652276	0.326138	−	0.945322i	$-0.394253\pi$
$307$	3508.64	0.652276	0.326138	+	0.945322i	$0.394253\pi$
$308$	0	0
$309$	5521.88	1.01660
$310$	−1013.97	−0.185773
$311$	3133.25	0.571287	0.285643	−	0.958336i	$-0.407793\pi$
$311$	3133.25	0.571287	0.285643	+	0.958336i	$0.407793\pi$
$312$	−2245.47	−0.407451
$313$	−6389.59	−1.15387	−0.576935	−	0.816790i	$-0.695751\pi$
$313$	−6389.59	−1.15387	−0.576935	+	0.816790i	$0.695751\pi$
$314$	−5263.05	−0.945895
$315$	0	0
$316$	−2049.92	−0.364928
$317$	1634.44	0.289587	0.144794	−	0.989462i	$-0.453748\pi$
$317$	1634.44	0.289587	0.144794	+	0.989462i	$0.453748\pi$
$318$	1431.55	0.252445
$319$	−15931.3	−2.79618
$320$	−1810.34	−0.316253
$321$	−729.260	−0.126802
$322$	0	0
$323$	154.671	0.0266444
$324$	−377.205	−0.0646785
$325$	808.579	0.138006
$326$	−1695.93	−0.288126
$327$	−1215.53	−0.205562
$328$	−4079.36	−0.686723
$329$	0	0
$330$	1770.88	0.295406
$331$	4386.17	0.728355	0.364177	−	0.931330i	$-0.381350\pi$
$331$	4386.17	0.728355	0.364177	+	0.931330i	$0.381350\pi$
$332$	−253.389	−0.0418872
$333$	1086.09	0.178730
$334$	−2001.25	−0.327854
$335$	4757.40	0.775894
$336$	0	0
$337$	1713.98	0.277051	0.138526	−	0.990359i	$-0.455764\pi$
$337$	1713.98	0.277051	0.138526	+	0.990359i	$0.455764\pi$
$338$	−2104.38	−0.338648
$339$	−84.5299	−0.0135429
$340$	1311.91	0.209260
$341$	−7161.41	−1.13728
$342$	45.1740	0.00714249
$343$	0	0
$344$	10260.4	1.60814
$345$	−1322.50	−0.206379
$346$	3132.56	0.486727
$347$	−1744.83	−0.269935	−0.134967	−	0.990850i	$-0.543093\pi$
$347$	−1744.83	−0.269935	−0.134967	+	0.990850i	$0.543093\pi$
$348$	−3447.03	−0.530977
$349$	−7046.78	−1.08082	−0.540409	−	0.841403i	$-0.681730\pi$
$349$	−7046.78	−1.08082	−0.540409	+	0.841403i	$0.681730\pi$
$350$	0	0
$351$	873.265	0.132796
$352$	−11356.8	−1.71966
$353$	12668.5	1.91013	0.955064	−	0.296400i	$-0.0957863\pi$
$353$	12668.5	1.91013	0.955064	+	0.296400i	$0.0957863\pi$
$354$	−3450.24	−0.518017
$355$	−1781.32	−0.266317
$356$	−4744.33	−0.706317
$357$	0	0
$358$	−7433.62	−1.09743
$359$	37.7844	0.00555483	0.00277742	−	0.999996i	$-0.499116\pi$
$359$	37.7844	0.00555483	0.00277742	+	0.999996i	$0.499116\pi$
$360$	1041.40	0.152462
$361$	−6851.46	−0.998901
$362$	5103.87	0.741031
$363$	8514.29	1.23109
$364$	0	0
$365$	−3283.77	−0.470905
$366$	631.938	0.0902511
$367$	759.829	0.108073	0.0540364	−	0.998539i	$-0.482791\pi$
$367$	759.829	0.108073	0.0540364	+	0.998539i	$0.482791\pi$
$368$	−446.023	−0.0631809
$369$	1586.47	0.223816
$370$	−1103.24	−0.155012
$371$	0	0
$372$	−1549.50	−0.215962
$373$	719.320	0.0998525	0.0499263	−	0.998753i	$-0.484101\pi$
$373$	719.320	0.0998525	0.0499263	+	0.998753i	$0.484101\pi$
$374$	−6651.81	−0.919671
$375$	−375.000	−0.0516398
$376$	−7988.81	−1.09572
$377$	7980.19	1.09019
$378$	0	0
$379$	572.559	0.0775999	0.0388000	−	0.999247i	$-0.487646\pi$
$379$	572.559	0.0775999	0.0388000	+	0.999247i	$0.487646\pi$
$380$	63.9192	0.00862891
$381$	−8222.69	−1.10567
$382$	−1159.56	−0.155310
$383$	−4513.18	−0.602122	−0.301061	−	0.953605i	$-0.597341\pi$
$383$	−4513.18	−0.602122	−0.301061	+	0.953605i	$0.597341\pi$
$384$	−2235.25	−0.297050
$385$	0	0
$386$	−466.247	−0.0614802
$387$	−3990.26	−0.524125
$388$	−3372.06	−0.441213
$389$	−6902.13	−0.899619	−0.449810	−	0.893124i	$-0.648508\pi$
$389$	−6902.13	−0.899619	−0.449810	+	0.893124i	$0.648508\pi$
$390$	−887.056	−0.115174
$391$	4967.58	0.642510
$392$	0	0
$393$	5496.11	0.705451
$394$	7628.88	0.975475
$395$	−2200.98	−0.280362
$396$	2706.18	0.343410
$397$	−4124.58	−0.521427	−0.260714	−	0.965416i	$-0.583958\pi$
$397$	−4124.58	−0.521427	−0.260714	+	0.965416i	$0.583958\pi$
$398$	8459.18	1.06538
$399$	0	0
$400$	−126.472	−0.0158090
$401$	−1002.50	−0.124844	−0.0624219	−	0.998050i	$-0.519882\pi$
$401$	−1002.50	−0.124844	−0.0624219	+	0.998050i	$0.519882\pi$
$402$	−5219.14	−0.647530
$403$	3587.23	0.443406
$404$	1251.41	0.154109
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−7791.89	−0.948967
$408$	−3911.70	−0.474652
$409$	−10335.0	−1.24947	−0.624736	−	0.780836i	$-0.714794\pi$
$409$	−10335.0	−1.24947	−0.624736	+	0.780836i	$0.714794\pi$
$410$	−1611.52	−0.194116
$411$	1148.24	0.137807
$412$	−8571.53	−1.02497
$413$	0	0
$414$	1450.85	0.172236
$415$	−272.061	−0.0321806
$416$	5688.75	0.670466
$417$	−9160.81	−1.07580
$418$	−324.091	−0.0379230
$419$	−3183.21	−0.371145	−0.185573	−	0.982631i	$-0.559414\pi$
$419$	−3183.21	−0.371145	−0.185573	+	0.982631i	$0.559414\pi$
$420$	0	0
$421$	−6944.34	−0.803911	−0.401956	−	0.915659i	$-0.631669\pi$
$421$	−6944.34	−0.803911	−0.401956	+	0.915659i	$0.631669\pi$
$422$	−2857.18	−0.329586
$423$	3106.85	0.357117
$424$	−6039.65	−0.691772
$425$	1408.58	0.160767
$426$	1954.21	0.222258
$427$	0	0
$428$	1132.02	0.127846
$429$	−6265.05	−0.705080
$430$	4053.28	0.454573
$431$	3868.41	0.432331	0.216166	−	0.976357i	$-0.430645\pi$
$431$	3868.41	0.432331	0.216166	+	0.976357i	$0.430645\pi$
$432$	−136.590	−0.0152122
$433$	6132.96	0.680673	0.340336	−	0.940304i	$-0.389459\pi$
$433$	6132.96	0.680673	0.340336	+	0.940304i	$0.389459\pi$
$434$	0	0
$435$	−3701.03	−0.407932
$436$	1886.84	0.207256
$437$	242.032	0.0264942
$438$	3602.48	0.392998
$439$	4090.14	0.444673	0.222337	−	0.974970i	$-0.428632\pi$
$439$	4090.14	0.444673	0.222337	+	0.974970i	$0.428632\pi$
$440$	−7471.27	−0.809497
$441$	0	0
$442$	3331.97	0.358565
$443$	12434.5	1.33359	0.666795	−	0.745241i	$-0.267666\pi$
$443$	12434.5	1.33359	0.666795	+	0.745241i	$0.267666\pi$
$444$	−1685.91	−0.180203
$445$	−5093.92	−0.542640
$446$	2260.66	0.240012
$447$	10681.8	1.13027
$448$	0	0
$449$	883.046	0.0928141	0.0464071	−	0.998923i	$-0.485223\pi$
$449$	883.046	0.0928141	0.0464071	+	0.998923i	$0.485223\pi$
$450$	411.396	0.0430964
$451$	−11381.8	−1.18835
$452$	131.214	0.0136544
$453$	9785.40	1.01492
$454$	−7646.15	−0.790422
$455$	0	0
$456$	−190.587	−0.0195725
$457$	−9068.44	−0.928235	−0.464118	−	0.885774i	$-0.653629\pi$
$457$	−9068.44	−0.928235	−0.464118	+	0.885774i	$0.653629\pi$
$458$	886.373	0.0904312
$459$	1521.26	0.154698
$460$	2052.89	0.208080
$461$	−12508.9	−1.26377	−0.631885	−	0.775063i	$-0.717718\pi$
$461$	−12508.9	−1.26377	−0.631885	+	0.775063i	$0.717718\pi$
$462$	0	0
$463$	12688.7	1.27363	0.636817	−	0.771015i	$-0.280251\pi$
$463$	12688.7	1.27363	0.636817	+	0.771015i	$0.280251\pi$
$464$	−1248.20	−0.124884
$465$	−1663.68	−0.165916
$466$	3804.12	0.378160
$467$	10136.5	1.00442	0.502208	−	0.864747i	$-0.332521\pi$
$467$	10136.5	1.00442	0.502208	+	0.864747i	$0.332521\pi$
$468$	−1355.56	−0.133890
$469$	0	0
$470$	−3155.92	−0.309727
$471$	−8635.37	−0.844791
$472$	14556.4	1.41952
$473$	28627.3	2.78284
$474$	2414.59	0.233979
$475$	68.6292	0.00662931
$476$	0	0
$477$	2348.83	0.225462
$478$	12459.1	1.19219
$479$	11361.1	1.08372	0.541861	−	0.840468i	$-0.317720\pi$
$479$	11361.1	1.08372	0.541861	+	0.840468i	$0.317720\pi$
$480$	−2638.31	−0.250879
$481$	3903.05	0.369987
$482$	7170.80	0.677637
$483$	0	0
$484$	−13216.6	−1.24123
$485$	−3620.54	−0.338969
$486$	444.308	0.0414696
$487$	7929.53	0.737826	0.368913	−	0.929464i	$-0.379730\pi$
$487$	7929.53	0.737826	0.368913	+	0.929464i	$0.379730\pi$
$488$	−2666.11	−0.247314
$489$	−2782.61	−0.257329
$490$	0	0
$491$	8111.51	0.745555	0.372777	−	0.927921i	$-0.378405\pi$
$491$	8111.51	0.745555	0.372777	+	0.927921i	$0.378405\pi$
$492$	−2462.65	−0.225660
$493$	13901.8	1.26999
$494$	162.341	0.0147856
$495$	2905.58	0.263831
$496$	−561.088	−0.0507936
$497$	0	0
$498$	298.466	0.0268566
$499$	16816.6	1.50865	0.754324	−	0.656502i	$-0.227965\pi$
$499$	16816.6	1.50865	0.754324	+	0.656502i	$0.227965\pi$
$500$	582.107	0.0520652
$501$	−3283.55	−0.292811
$502$	9542.74	0.848433
$503$	17764.6	1.57472	0.787362	−	0.616491i	$-0.211446\pi$
$503$	17764.6	1.57472	0.787362	+	0.616491i	$0.211446\pi$
$504$	0	0
$505$	1343.62	0.118397
$506$	−10408.8	−0.914485
$507$	−3452.76	−0.302451
$508$	12764.0	1.11478
$509$	−13908.8	−1.21120	−0.605598	−	0.795771i	$-0.707066\pi$
$509$	−13908.8	−1.21120	−0.605598	+	0.795771i	$0.707066\pi$
$510$	−1545.29	−0.134170
$511$	0	0
$512$	−1826.38	−0.157647
$513$	74.1195	0.00637905
$514$	−12754.6	−1.09451
$515$	−9203.13	−0.787453
$516$	6194.02	0.528443
$517$	−22289.5	−1.89611
$518$	0	0
$519$	5139.76	0.434702
$520$	3742.45	0.315610
$521$	8639.68	0.726510	0.363255	−	0.931690i	$-0.381665\pi$
$521$	8639.68	0.726510	0.363255	+	0.931690i	$0.381665\pi$
$522$	4060.23	0.340444
$523$	−23242.2	−1.94323	−0.971617	−	0.236561i	$-0.923980\pi$
$523$	−23242.2	−1.94323	−0.971617	+	0.236561i	$0.923980\pi$
$524$	−8531.53	−0.711263
$525$	0	0
$526$	−6595.79	−0.546749
$527$	6249.11	0.516538
$528$	979.932	0.0807691
$529$	−4393.66	−0.361113
$530$	−2385.92	−0.195543
$531$	−5660.99	−0.462648
$532$	0	0
$533$	5701.26	0.463319
$534$	5588.32	0.452865
$535$	1215.43	0.0982201
$536$	22019.3	1.77442
$537$	−12196.8	−0.980128
$538$	−10.7616	−0.000862390 0
$539$	0	0
$540$	628.675	0.0500998
$541$	11395.2	0.905577	0.452789	−	0.891618i	$-0.350429\pi$
$541$	11395.2	0.905577	0.452789	+	0.891618i	$0.350429\pi$
$542$	−12646.0	−1.00220
$543$	8374.19	0.661825
$544$	9910.04	0.781047
$545$	2025.88	0.159228
$546$	0	0
$547$	−7870.21	−0.615184	−0.307592	−	0.951518i	$-0.599523\pi$
$547$	−7870.21	−0.615184	−0.307592	+	0.951518i	$0.599523\pi$
$548$	−1782.40	−0.138942
$549$	1036.85	0.0806045
$550$	−2951.47	−0.228820
$551$	677.329	0.0523687
$552$	−6121.08	−0.471976
$553$	0	0
$554$	−3875.28	−0.297193
$555$	−1810.14	−0.138444
$556$	14220.2	1.08466
$557$	17769.8	1.35176	0.675880	−	0.737012i	$-0.263764\pi$
$557$	17769.8	1.35176	0.675880	+	0.737012i	$0.263764\pi$
$558$	1825.15	0.138467
$559$	−14339.7	−1.08498
$560$	0	0
$561$	−10914.0	−0.821370
$562$	−438.672	−0.0329257
$563$	−15192.8	−1.13730	−0.568651	−	0.822579i	$-0.692534\pi$
$563$	−15192.8	−1.13730	−0.568651	+	0.822579i	$0.692534\pi$
$564$	−4822.72	−0.360059
$565$	140.883	0.0104903
$566$	8304.93	0.616753
$567$	0	0
$568$	−8244.71	−0.609050
$569$	−23300.0	−1.71667	−0.858335	−	0.513090i	$-0.828501\pi$
$569$	−23300.0	−1.71667	−0.858335	+	0.513090i	$0.828501\pi$
$570$	−75.2900	−0.00553255
$571$	10638.2	0.779673	0.389837	−	0.920884i	$-0.372531\pi$
$571$	10638.2	0.779673	0.389837	+	0.920884i	$0.372531\pi$
$572$	9725.14	0.710889
$573$	−1902.55	−0.138709
$574$	0	0
$575$	2204.16	0.159861
$576$	3258.61	0.235721
$577$	897.258	0.0647372	0.0323686	−	0.999476i	$-0.489695\pi$
$577$	897.258	0.0647372	0.0323686	+	0.999476i	$0.489695\pi$
$578$	−3178.63	−0.228743
$579$	−764.997	−0.0549088
$580$	5745.05	0.411293
$581$	0	0
$582$	3971.93	0.282890
$583$	−16851.1	−1.19709
$584$	−15198.7	−1.07693
$585$	−1455.44	−0.102863
$586$	3970.79	0.279918
$587$	14712.9	1.03452	0.517261	−	0.855828i	$-0.326952\pi$
$587$	14712.9	1.03452	0.517261	+	0.855828i	$0.326952\pi$
$588$	0	0
$589$	304.471	0.0212997
$590$	5750.40	0.401254
$591$	12517.1	0.871210
$592$	−610.486	−0.0423831
$593$	7216.29	0.499726	0.249863	−	0.968281i	$-0.419614\pi$
$593$	7216.29	0.499726	0.249863	+	0.968281i	$0.419614\pi$
$594$	−3187.59	−0.220182
$595$	0	0
$596$	−16581.2	−1.13958
$597$	13879.4	0.951503
$598$	5213.91	0.356543
$599$	−20885.5	−1.42464	−0.712320	−	0.701855i	$-0.752355\pi$
$599$	−20885.5	−1.42464	−0.712320	+	0.701855i	$0.752355\pi$
$600$	−1735.66	−0.118097
$601$	11047.7	0.749823	0.374911	−	0.927061i	$-0.377673\pi$
$601$	11047.7	0.749823	0.374911	+	0.927061i	$0.377673\pi$
$602$	0	0
$603$	−8563.32	−0.578317
$604$	−15189.7	−1.02328
$605$	−14190.5	−0.953595
$606$	−1474.03	−0.0988093
$607$	9434.94	0.630894	0.315447	−	0.948943i	$-0.397846\pi$
$607$	9434.94	0.630894	0.315447	+	0.948943i	$0.397846\pi$
$608$	482.840	0.0322068
$609$	0	0
$610$	−1053.23	−0.0699082
$611$	11165.0	0.739263
$612$	−2361.44	−0.155973
$613$	−17662.6	−1.16376	−0.581881	−	0.813274i	$-0.697683\pi$
$613$	−17662.6	−1.16376	−0.581881	+	0.813274i	$0.697683\pi$
$614$	6415.30	0.421662
$615$	−2644.11	−0.173367
$616$	0	0
$617$	−10817.4	−0.705820	−0.352910	−	0.935657i	$-0.614808\pi$
$617$	−10817.4	−0.705820	−0.352910	+	0.935657i	$0.614808\pi$
$618$	10096.3	0.657176
$619$	−29073.9	−1.88785	−0.943926	−	0.330158i	$-0.892898\pi$
$619$	−29073.9	−1.88785	−0.943926	+	0.330158i	$0.892898\pi$
$620$	2582.50	0.167283
$621$	2380.50	0.153826
$622$	5728.92	0.369306
$623$	0	0
$624$	−490.860	−0.0314906
$625$	625.000	0.0400000
$626$	−11682.9	−0.745915
$627$	−531.754	−0.0338696
$628$	13404.5	0.851751
$629$	6799.28	0.431009
$630$	0	0
$631$	−2203.17	−0.138996	−0.0694981	−	0.997582i	$-0.522140\pi$
$631$	−2203.17	−0.138996	−0.0694981	+	0.997582i	$0.522140\pi$
$632$	−10187.1	−0.641170
$633$	−4687.93	−0.294358
$634$	2988.45	0.187203
$635$	13704.5	0.856451
$636$	−3646.05	−0.227319
$637$	0	0
$638$	−29129.3	−1.80758
$639$	3206.38	0.198501
$640$	3725.42	0.230094
$641$	22466.5	1.38436	0.692180	−	0.721725i	$-0.256651\pi$
$641$	22466.5	1.38436	0.692180	+	0.721725i	$0.256651\pi$
$642$	−1333.40	−0.0819705
$643$	12347.0	0.757257	0.378629	−	0.925549i	$-0.376396\pi$
$643$	12347.0	0.757257	0.378629	+	0.925549i	$0.376396\pi$
$644$	0	0
$645$	6650.44	0.405986
$646$	282.805	0.0172242
$647$	24114.0	1.46525	0.732626	−	0.680631i	$-0.238294\pi$
$647$	24114.0	1.46525	0.732626	+	0.680631i	$0.238294\pi$
$648$	−1874.51	−0.113639
$649$	40613.6	2.45643
$650$	1478.43	0.0892134
$651$	0	0
$652$	4319.41	0.259449
$653$	7843.33	0.470035	0.235018	−	0.971991i	$-0.424485\pi$
$653$	7843.33	0.470035	0.235018	+	0.971991i	$0.424485\pi$
$654$	−2222.50	−0.132885
$655$	−9160.18	−0.546440
$656$	−891.749	−0.0530746
$657$	5910.78	0.350992
$658$	0	0
$659$	21242.8	1.25569	0.627846	−	0.778338i	$-0.283937\pi$
$659$	21242.8	1.25569	0.627846	+	0.778338i	$0.283937\pi$
$660$	−4510.29	−0.266005
$661$	22221.7	1.30760	0.653801	−	0.756667i	$-0.273173\pi$
$661$	22221.7	1.30760	0.653801	+	0.756667i	$0.273173\pi$
$662$	8019.79	0.470843
$663$	5466.94	0.320239
$664$	−1259.21	−0.0735948
$665$	0	0
$666$	1985.83	0.115539
$667$	21753.8	1.26283
$668$	5097.01	0.295223
$669$	3709.18	0.214358
$670$	8698.56	0.501574
$671$	−7438.69	−0.427969
$672$	0	0
$673$	−3787.85	−0.216955	−0.108478	−	0.994099i	$-0.534598\pi$
$673$	−3787.85	−0.216955	−0.108478	+	0.994099i	$0.534598\pi$
$674$	3133.88	0.179099
$675$	675.000	0.0384900
$676$	5359.67	0.304943
$677$	11296.8	0.641314	0.320657	−	0.947195i	$-0.396096\pi$
$677$	11296.8	0.641314	0.320657	+	0.947195i	$0.396096\pi$
$678$	−154.557	−0.00875474
$679$	0	0
$680$	6519.50	0.367664
$681$	−12545.5	−0.705937
$682$	−13094.1	−0.735190
$683$	4807.14	0.269312	0.134656	−	0.990892i	$-0.457007\pi$
$683$	4807.14	0.269312	0.134656	+	0.990892i	$0.457007\pi$
$684$	−115.055	−0.00643161
$685$	−1913.73	−0.106745
$686$	0	0
$687$	1454.32	0.0807654
$688$	2242.92	0.124288
$689$	8440.94	0.466726
$690$	−2418.09	−0.133413
$691$	−5393.47	−0.296928	−0.148464	−	0.988918i	$-0.547433\pi$
$691$	−5393.47	−0.296928	−0.148464	+	0.988918i	$0.547433\pi$
$692$	−7978.37	−0.438283
$693$	0	0
$694$	−3190.29	−0.174498
$695$	15268.0	0.833308
$696$	−17129.9	−0.932914
$697$	9931.84	0.539735
$698$	−12884.5	−0.698691
$699$	6241.63	0.337740
$700$	0	0
$701$	2404.77	0.129568	0.0647838	−	0.997899i	$-0.479364\pi$
$701$	2404.77	0.129568	0.0647838	+	0.997899i	$0.479364\pi$
$702$	1596.70	0.0858456
$703$	331.276	0.0177729
$704$	−23378.2	−1.25156
$705$	−5178.09	−0.276622
$706$	23163.4	1.23480
$707$	0	0
$708$	8787.47	0.466460
$709$	−21617.3	−1.14507	−0.572535	−	0.819881i	$-0.694040\pi$
$709$	−21617.3	−1.14507	−0.572535	+	0.819881i	$0.694040\pi$
$710$	−3257.01	−0.172160
$711$	3961.76	0.208970
$712$	−23576.8	−1.24098
$713$	9778.70	0.513626
$714$	0	0
$715$	10441.7	0.546153
$716$	18932.8	0.988203
$717$	20442.3	1.06476
$718$	69.0860	0.00359090
$719$	18228.6	0.945498	0.472749	−	0.881197i	$-0.343262\pi$
$719$	18228.6	0.945498	0.472749	+	0.881197i	$0.343262\pi$
$720$	227.649	0.0117833
$721$	0	0
$722$	−12527.4	−0.645736
$723$	11765.5	0.605207
$724$	−12999.1	−0.667278
$725$	6168.38	0.315983
$726$	15567.8	0.795832
$727$	−20196.5	−1.03033	−0.515164	−	0.857092i	$-0.672269\pi$
$727$	−20196.5	−1.03033	−0.515164	+	0.857092i	$0.672269\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−6004.13	−0.304415
$731$	−24980.4	−1.26393
$732$	−1609.49	−0.0812686
$733$	15264.9	0.769196	0.384598	−	0.923084i	$-0.374340\pi$
$733$	15264.9	0.769196	0.384598	+	0.923084i	$0.374340\pi$
$734$	1389.29	0.0698633
$735$	0	0
$736$	15507.4	0.776643
$737$	61435.7	3.07057
$738$	2900.74	0.144685
$739$	13906.0	0.692207	0.346103	−	0.938196i	$-0.387505\pi$
$739$	13906.0	0.692207	0.346103	+	0.938196i	$0.387505\pi$
$740$	2809.86	0.139584
$741$	266.362	0.0132052
$742$	0	0
$743$	4592.87	0.226778	0.113389	−	0.993551i	$-0.463829\pi$
$743$	4592.87	0.226778	0.113389	+	0.993551i	$0.463829\pi$
$744$	−7700.20	−0.379440
$745$	−17803.0	−0.875504
$746$	1315.22	0.0645493
$747$	489.709	0.0239860
$748$	16941.6	0.828137
$749$	0	0
$750$	−685.660	−0.0333824
$751$	−8390.80	−0.407702	−0.203851	−	0.979002i	$-0.565346\pi$
$751$	−8390.80	−0.407702	−0.203851	+	0.979002i	$0.565346\pi$
$752$	−1746.35	−0.0846848
$753$	15657.3	0.757747
$754$	14591.2	0.704748
$755$	−16309.0	−0.786153
$756$	0	0
$757$	−1368.89	−0.0657240	−0.0328620	−	0.999460i	$-0.510462\pi$
$757$	−1368.89	−0.0657240	−0.0328620	+	0.999460i	$0.510462\pi$
$758$	1046.88	0.0501642
$759$	−17078.4	−0.816739
$760$	317.645	0.0151608
$761$	1623.77	0.0773478	0.0386739	−	0.999252i	$-0.487687\pi$
$761$	1623.77	0.0773478	0.0386739	+	0.999252i	$0.487687\pi$
$762$	−15034.6	−0.714759
$763$	0	0
$764$	2953.31	0.139852
$765$	−2535.44	−0.119829
$766$	−8252.03	−0.389240
$767$	−20343.8	−0.957722
$768$	−12776.6	−0.600308
$769$	26842.5	1.25873	0.629366	−	0.777109i	$-0.283315\pi$
$769$	26842.5	1.25873	0.629366	+	0.777109i	$0.283315\pi$
$770$	0	0
$771$	−20927.1	−0.977526
$772$	1187.49	0.0553612
$773$	20961.4	0.975330	0.487665	−	0.873031i	$-0.337849\pi$
$773$	20961.4	0.975330	0.487665	+	0.873031i	$0.337849\pi$
$774$	−7295.90	−0.338819
$775$	2772.79	0.128518
$776$	−16757.4	−0.775200
$777$	0	0
$778$	−12620.0	−0.581556
$779$	483.902	0.0222562
$780$	2259.26	0.103711
$781$	−23003.5	−1.05394
$782$	9082.86	0.415348
$783$	6661.85	0.304055
$784$	0	0
$785$	14392.3	0.654373
$786$	10049.2	0.456036
$787$	−35333.2	−1.60037	−0.800187	−	0.599751i	$-0.795266\pi$
$787$	−35333.2	−1.60037	−0.800187	+	0.599751i	$0.795266\pi$
$788$	−19430.1	−0.878388
$789$	−10822.1	−0.488309
$790$	−4024.32	−0.181239
$791$	0	0
$792$	13448.3	0.603364
$793$	3726.13	0.166858
$794$	−7541.49	−0.337075
$795$	−3914.71	−0.174642
$796$	−21544.8	−0.959342
$797$	8137.04	0.361642	0.180821	−	0.983516i	$-0.442125\pi$
$797$	8137.04	0.361642	0.180821	+	0.983516i	$0.442125\pi$
$798$	0	0
$799$	19450.0	0.861191
$800$	4397.18	0.194330
$801$	9169.05	0.404460
$802$	−1832.99	−0.0807049
$803$	−42405.6	−1.86359
$804$	13292.7	0.583082
$805$	0	0
$806$	6558.99	0.286639
$807$	−17.6572	−0.000770212 0
$808$	6218.87	0.270766
$809$	−36281.0	−1.57673	−0.788364	−	0.615209i	$-0.789072\pi$
$809$	−36281.0	−1.57673	−0.788364	+	0.615209i	$0.789072\pi$
$810$	−740.513	−0.0321222
$811$	34237.2	1.48240	0.741202	−	0.671282i	$-0.234256\pi$
$811$	34237.2	1.48240	0.741202	+	0.671282i	$0.234256\pi$
$812$	0	0
$813$	−20749.0	−0.895077
$814$	−14246.9	−0.613456
$815$	4637.69	0.199326
$816$	−855.099	−0.0366844
$817$	−1217.10	−0.0521188
$818$	−18896.8	−0.807717
$819$	0	0
$820$	4104.42	0.174796
$821$	−23247.8	−0.988250	−0.494125	−	0.869391i	$-0.664511\pi$
$821$	−23247.8	−0.988250	−0.494125	+	0.869391i	$0.664511\pi$
$822$	2099.47	0.0890846
$823$	42934.0	1.81845	0.909225	−	0.416306i	$-0.136675\pi$
$823$	42934.0	1.81845	0.909225	+	0.416306i	$0.136675\pi$
$824$	−42596.0	−1.80085
$825$	−4842.64	−0.204363
$826$	0	0
$827$	−781.391	−0.0328557	−0.0164278	−	0.999865i	$-0.505229\pi$
$827$	−781.391	−0.0328557	−0.0164278	+	0.999865i	$0.505229\pi$
$828$	−3695.21	−0.155093
$829$	33493.3	1.40322	0.701611	−	0.712561i	$-0.252465\pi$
$829$	33493.3	1.40322	0.701611	+	0.712561i	$0.252465\pi$
$830$	−497.443	−0.0208030
$831$	−6358.39	−0.265427
$832$	11710.4	0.487964
$833$	0	0
$834$	−16749.9	−0.695445
$835$	5472.59	0.226811
$836$	825.434	0.0341486
$837$	2994.62	0.123667
$838$	−5820.27	−0.239926
$839$	−15155.7	−0.623639	−0.311819	−	0.950141i	$-0.600938\pi$
$839$	−15155.7	−0.623639	−0.311819	+	0.950141i	$0.600938\pi$
$840$	0	0
$841$	36489.2	1.49613
$842$	−12697.2	−0.519686
$843$	−719.752	−0.0294064
$844$	7277.01	0.296783
$845$	5754.60	0.234277
$846$	5680.66	0.230857
$847$	0	0
$848$	−1320.27	−0.0534649
$849$	13626.4	0.550831
$850$	2575.48	0.103927
$851$	10639.6	0.428579
$852$	−4977.21	−0.200137
$853$	2917.48	0.117107	0.0585537	−	0.998284i	$-0.481351\pi$
$853$	2917.48	0.117107	0.0585537	+	0.998284i	$0.481351\pi$
$854$	0	0
$855$	−123.532	−0.00494119
$856$	5625.54	0.224623
$857$	31560.7	1.25799	0.628993	−	0.777411i	$-0.283468\pi$
$857$	31560.7	1.25799	0.628993	+	0.777411i	$0.283468\pi$
$858$	−11455.2	−0.455797
$859$	1404.81	0.0557991	0.0278995	−	0.999611i	$-0.491118\pi$
$859$	1404.81	0.0557991	0.0278995	+	0.999611i	$0.491118\pi$
$860$	−10323.4	−0.409330
$861$	0	0
$862$	7073.11	0.279479
$863$	−9808.24	−0.386879	−0.193439	−	0.981112i	$-0.561964\pi$
$863$	−9808.24	−0.386879	−0.193439	+	0.981112i	$0.561964\pi$
$864$	4748.96	0.186994
$865$	−8566.27	−0.336719
$866$	11213.7	0.440018
$867$	−5215.35	−0.204294
$868$	0	0
$869$	−28422.8	−1.10952
$870$	−6767.06	−0.263707
$871$	−30773.9	−1.19717
$872$	9376.63	0.364143
$873$	6516.97	0.252653
$874$	442.537	0.0171271
$875$	0	0
$876$	−9175.22	−0.353883
$877$	−7196.05	−0.277073	−0.138537	−	0.990357i	$-0.544240\pi$
$877$	−7196.05	−0.277073	−0.138537	+	0.990357i	$0.544240\pi$
$878$	7478.52	0.287458
$879$	6515.10	0.249999
$880$	−1633.22	−0.0625635
$881$	3183.27	0.121733	0.0608667	−	0.998146i	$-0.480614\pi$
$881$	3183.27	0.121733	0.0608667	+	0.998146i	$0.480614\pi$
$882$	0	0
$883$	25392.5	0.967751	0.483876	−	0.875137i	$-0.339229\pi$
$883$	25392.5	0.967751	0.483876	+	0.875137i	$0.339229\pi$
$884$	−8486.25	−0.322877
$885$	9434.99	0.358366
$886$	22735.6	0.862095
$887$	30634.2	1.15964	0.579818	−	0.814746i	$-0.303124\pi$
$887$	30634.2	1.15964	0.579818	+	0.814746i	$0.303124\pi$
$888$	−8378.11	−0.316612
$889$	0	0
$890$	−9313.86	−0.350788
$891$	−5230.05	−0.196648
$892$	−5757.71	−0.216124
$893$	947.648	0.0355116
$894$	19530.9	0.730660
$895$	20327.9	0.759204
$896$	0	0
$897$	8554.75	0.318433
$898$	1614.59	0.0599994
$899$	27365.8	1.01524
$900$	−1047.79	−0.0388071
$901$	14704.5	0.543704
$902$	−20810.7	−0.768206
$903$	0	0
$904$	652.067	0.0239905
$905$	−13957.0	−0.512648
$906$	17891.9	0.656091
$907$	−28089.9	−1.02834	−0.514172	−	0.857687i	$-0.671901\pi$
$907$	−28089.9	−1.02834	−0.514172	+	0.857687i	$0.671901\pi$
$908$	19474.1	0.711753
$909$	−2418.52	−0.0882480
$910$	0	0
$911$	−36102.7	−1.31299	−0.656495	−	0.754330i	$-0.727962\pi$
$911$	−36102.7	−1.31299	−0.656495	+	0.754330i	$0.727962\pi$
$912$	−41.6624	−0.00151270
$913$	−3513.31	−0.127353
$914$	−16581.0	−0.600055
$915$	−1728.09	−0.0624360
$916$	−2257.52	−0.0814308
$917$	0	0
$918$	2781.52	0.100004
$919$	−14533.8	−0.521682	−0.260841	−	0.965382i	$-0.584000\pi$
$919$	−14533.8	−0.521682	−0.260841	+	0.965382i	$0.584000\pi$
$920$	10201.8	0.365591
$921$	10525.9	0.376592
$922$	−22871.6	−0.816959
$923$	11522.7	0.410915
$924$	0	0
$925$	3016.90	0.107238
$926$	23200.3	0.823336
$927$	16565.6	0.586933
$928$	43397.6	1.53512
$929$	−16539.6	−0.584118	−0.292059	−	0.956400i	$-0.594340\pi$
$929$	−16539.6	−0.584118	−0.292059	+	0.956400i	$0.594340\pi$
$930$	−3041.91	−0.107256
$931$	0	0
$932$	−9688.79	−0.340522
$933$	9399.74	0.329833
$934$	18533.9	0.649301
$935$	18190.0	0.636231
$936$	−6736.41	−0.235242
$937$	−30212.3	−1.05335	−0.526677	−	0.850065i	$-0.676562\pi$
$937$	−30212.3	−1.05335	−0.526677	+	0.850065i	$0.676562\pi$
$938$	0	0
$939$	−19168.8	−0.666187
$940$	8037.87	0.278900
$941$	26414.4	0.915074	0.457537	−	0.889191i	$-0.348732\pi$
$941$	26414.4	0.915074	0.457537	+	0.889191i	$0.348732\pi$
$942$	−15789.1	−0.546112
$943$	15541.5	0.536692
$944$	3182.03	0.109710
$945$	0	0
$946$	52342.9	1.79896
$947$	10187.3	0.349570	0.174785	−	0.984607i	$-0.444077\pi$
$947$	10187.3	0.349570	0.174785	+	0.984607i	$0.444077\pi$
$948$	−6149.77	−0.210691
$949$	21241.5	0.726583
$950$	125.483	0.00428549
$951$	4903.31	0.167193
$952$	0	0
$953$	2211.39	0.0751669	0.0375834	−	0.999293i	$-0.488034\pi$
$953$	2211.39	0.0751669	0.0375834	+	0.999293i	$0.488034\pi$
$954$	4294.66	0.145749
$955$	3170.92	0.107444
$956$	−31732.3	−1.07353
$957$	−47794.0	−1.61438
$958$	20773.0	0.700569
$959$	0	0
$960$	−5431.02	−0.182589
$961$	−17489.6	−0.587077
$962$	7136.44	0.239177
$963$	−2187.78	−0.0732089
$964$	−18263.4	−0.610193
$965$	1275.00	0.0425322
$966$	0	0
$967$	7955.89	0.264575	0.132287	−	0.991211i	$-0.457768\pi$
$967$	7955.89	0.264575	0.132287	+	0.991211i	$0.457768\pi$
$968$	−65679.6	−2.18081
$969$	464.014	0.0153832
$970$	−6619.89	−0.219126
$971$	−53071.2	−1.75400	−0.877001	−	0.480488i	$-0.840459\pi$
$971$	−53071.2	−1.75400	−0.877001	+	0.480488i	$0.840459\pi$
$972$	−1131.62	−0.0373422
$973$	0	0
$974$	14498.6	0.476965
$975$	2425.74	0.0796777
$976$	−582.813	−0.0191141
$977$	−22448.2	−0.735089	−0.367545	−	0.930006i	$-0.619801\pi$
$977$	−22448.2	−0.735089	−0.367545	+	0.930006i	$0.619801\pi$
$978$	−5087.80	−0.166350
$979$	−65781.4	−2.14748
$980$	0	0
$981$	−3646.58	−0.118681
$982$	14831.3	0.481961
$983$	−21712.8	−0.704509	−0.352254	−	0.935904i	$-0.614585\pi$
$983$	−21712.8	−0.704509	−0.352254	+	0.935904i	$0.614585\pi$
$984$	−12238.1	−0.396479
$985$	−20861.9	−0.674837
$986$	25418.5	0.820983
$987$	0	0
$988$	−413.470	−0.0133140
$989$	−39089.7	−1.25681
$990$	5312.65	0.170553
$991$	−37849.2	−1.21324	−0.606620	−	0.794992i	$-0.707475\pi$
$991$	−37849.2	−1.21324	−0.606620	+	0.794992i	$0.707475\pi$
$992$	19508.0	0.624373
$993$	13158.5	0.420516
$994$	0	0
$995$	−23132.4	−0.737031
$996$	−760.168	−0.0241836
$997$	39573.1	1.25707	0.628533	−	0.777783i	$-0.283656\pi$
$997$	39573.1	1.25707	0.628533	+	0.777783i	$0.283656\pi$
$998$	30748.0	0.975261
$999$	3258.26	0.103190

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.4.a.o.1.2		2
3.2	odd	2		2205.4.a.bb.1.1		2
7.6	odd	2		105.4.a.e.1.2	✓	2
21.20	even	2		315.4.a.k.1.1		2
28.27	even	2		1680.4.a.bo.1.2		2
35.13	even	4		525.4.d.l.274.2		4
35.27	even	4		525.4.d.l.274.3		4
35.34	odd	2		525.4.a.l.1.1		2
105.104	even	2		1575.4.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.e.1.2	✓	2	7.6	odd	2
315.4.a.k.1.1		2	21.20	even	2
525.4.a.l.1.1		2	35.34	odd	2
525.4.d.l.274.2		4	35.13	even	4
525.4.d.l.274.3		4	35.27	even	4
735.4.a.o.1.2		2	1.1	even	1	trivial
1575.4.a.q.1.2		2	105.104	even	2
1680.4.a.bo.1.2		2	28.27	even	2
2205.4.a.bb.1.1		2	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)

\(f(q)\)	\(=\)	\(q+1.82843 q^{2} +3.00000 q^{3} -4.65685 q^{4} -5.00000 q^{5} +5.48528 q^{6} -23.1421 q^{8} +9.00000 q^{9} -9.14214 q^{10} -64.5685 q^{11} -13.9706 q^{12} +32.3431 q^{13} -15.0000 q^{15} -5.05887 q^{16} +56.3431 q^{17} +16.4558 q^{18} +2.74517 q^{19} +23.2843 q^{20} -118.059 q^{22} +88.1665 q^{23} -69.4264 q^{24} +25.0000 q^{25} +59.1371 q^{26} +27.0000 q^{27} +246.735 q^{29} -27.4264 q^{30} +110.912 q^{31} +175.887 q^{32} -193.706 q^{33} +103.019 q^{34} -41.9117 q^{36} +120.676 q^{37} +5.01934 q^{38} +97.0294 q^{39} +115.711 q^{40} +176.274 q^{41} -443.362 q^{43} +300.686 q^{44} -45.0000 q^{45} +161.206 q^{46} +345.206 q^{47} -15.1766 q^{48} +45.7107 q^{50} +169.029 q^{51} -150.617 q^{52} +260.981 q^{53} +49.3675 q^{54} +322.843 q^{55} +8.23550 q^{57} +451.137 q^{58} -628.999 q^{59} +69.8528 q^{60} +115.206 q^{61} +202.794 q^{62} +362.068 q^{64} -161.716 q^{65} -354.177 q^{66} -951.480 q^{67} -262.382 q^{68} +264.500 q^{69} +356.264 q^{71} -208.279 q^{72} +656.754 q^{73} +220.648 q^{74} +75.0000 q^{75} -12.7838 q^{76} +177.411 q^{78} +440.195 q^{79} +25.2944 q^{80} +81.0000 q^{81} +322.304 q^{82} +54.4121 q^{83} -281.716 q^{85} -810.656 q^{86} +740.205 q^{87} +1494.25 q^{88} +1018.78 q^{89} -82.2792 q^{90} -410.579 q^{92} +332.735 q^{93} +631.184 q^{94} -13.7258 q^{95} +527.662 q^{96} +724.108 q^{97} -581.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 6 q^{6} - 18 q^{8} + 18 q^{9} + 10 q^{10} - 16 q^{11} + 6 q^{12} + 76 q^{13} - 30 q^{15} - 78 q^{16} + 124 q^{17} - 18 q^{18} + 96 q^{19} - 10 q^{20} - 304 q^{22}+ \cdots - 144 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 6 * q^3 + 2 * q^4 - 10 * q^5 - 6 * q^6 - 18 * q^8 + 18 * q^9 + 10 * q^10 - 16 * q^11 + 6 * q^12 + 76 * q^13 - 30 * q^15 - 78 * q^16 + 124 * q^17 - 18 * q^18 + 96 * q^19 - 10 * q^20 - 304 * q^22 - 16 * q^23 - 54 * q^24 + 50 * q^25 - 108 * q^26 + 54 * q^27 + 188 * q^29 + 30 * q^30 + 120 * q^31 + 414 * q^32 - 48 * q^33 - 156 * q^34 + 18 * q^36 - 132 * q^37 - 352 * q^38 + 228 * q^39 + 90 * q^40 - 100 * q^41 - 536 * q^43 + 624 * q^44 - 90 * q^45 + 560 * q^46 + 928 * q^47 - 234 * q^48 - 50 * q^50 + 372 * q^51 + 140 * q^52 + 884 * q^53 - 54 * q^54 + 80 * q^55 + 288 * q^57 + 676 * q^58 - 104 * q^59 - 30 * q^60 + 468 * q^61 + 168 * q^62 + 34 * q^64 - 380 * q^65 - 912 * q^66 - 1688 * q^67 + 188 * q^68 - 48 * q^69 - 136 * q^71 - 162 * q^72 - 508 * q^73 + 1188 * q^74 + 150 * q^75 + 608 * q^76 - 324 * q^78 - 432 * q^79 + 390 * q^80 + 162 * q^81 + 1380 * q^82 + 584 * q^83 - 620 * q^85 - 456 * q^86 + 564 * q^87 + 1744 * q^88 + 1404 * q^89 + 90 * q^90 - 1104 * q^92 + 360 * q^93 - 1600 * q^94 - 480 * q^95 + 1242 * q^96 + 1188 * q^97 - 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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