LMFDB - Embedded newform 735.4.a.m.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,-4,-6,2,10,12,0,-12,18,-20,-92] 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:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	735.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.236068 q^{2} -3.00000 q^{3} -7.94427 q^{4} +5.00000 q^{5} -0.708204 q^{6} -3.76393 q^{8} +9.00000 q^{9} +1.18034 q^{10} -50.4721 q^{11} +23.8328 q^{12} +80.9706 q^{13} -15.0000 q^{15} +62.6656 q^{16} -76.3870 q^{17} +2.12461 q^{18} -4.13777 q^{19} -39.7214 q^{20} -11.9149 q^{22} -204.721 q^{23} +11.2918 q^{24} +25.0000 q^{25} +19.1146 q^{26} -27.0000 q^{27} -91.1672 q^{29} -3.54102 q^{30} -198.079 q^{31} +44.9048 q^{32} +151.416 q^{33} -18.0325 q^{34} -71.4984 q^{36} +155.666 q^{37} -0.976794 q^{38} -242.912 q^{39} -18.8197 q^{40} +156.885 q^{41} +354.217 q^{43} +400.964 q^{44} +45.0000 q^{45} -48.3282 q^{46} +175.659 q^{47} -187.997 q^{48} +5.90170 q^{50} +229.161 q^{51} -643.252 q^{52} +200.302 q^{53} -6.37384 q^{54} -252.361 q^{55} +12.4133 q^{57} -21.5217 q^{58} +312.498 q^{59} +119.164 q^{60} +154.170 q^{61} -46.7601 q^{62} -490.724 q^{64} +404.853 q^{65} +35.7446 q^{66} +734.715 q^{67} +606.839 q^{68} +614.164 q^{69} -678.577 q^{71} -33.8754 q^{72} +60.8003 q^{73} +36.7477 q^{74} -75.0000 q^{75} +32.8715 q^{76} -57.3437 q^{78} -1286.26 q^{79} +313.328 q^{80} +81.0000 q^{81} +37.0356 q^{82} -116.170 q^{83} -381.935 q^{85} +83.6192 q^{86} +273.502 q^{87} +189.974 q^{88} +916.440 q^{89} +10.6231 q^{90} +1626.36 q^{92} +594.237 q^{93} +41.4676 q^{94} -20.6888 q^{95} -134.714 q^{96} +1416.30 q^{97} -454.249 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 6 q^{3} + 2 q^{4} + 10 q^{5} + 12 q^{6} - 12 q^{8} + 18 q^{9} - 20 q^{10} - 92 q^{11} - 6 q^{12} - 8 q^{13} - 30 q^{15} + 18 q^{16} + 44 q^{17} - 36 q^{18} + 108 q^{19} + 10 q^{20} + 164 q^{22}+ \cdots - 828 q^{99}+O(q^{100}) $ 2 * q - 4 * q^2 - 6 * q^3 + 2 * q^4 + 10 * q^5 + 12 * q^6 - 12 * q^8 + 18 * q^9 - 20 * q^10 - 92 * q^11 - 6 * q^12 - 8 * q^13 - 30 * q^15 + 18 * q^16 + 44 * q^17 - 36 * q^18 + 108 * q^19 + 10 * q^20 + 164 * q^22 - 320 * q^23 + 36 * q^24 + 50 * q^25 + 396 * q^26 - 54 * q^27 - 236 * q^29 + 60 * q^30 + 60 * q^31 + 300 * q^32 + 276 * q^33 - 528 * q^34 + 18 * q^36 + 204 * q^37 - 476 * q^38 + 24 * q^39 - 60 * q^40 - 44 * q^41 + 136 * q^43 - 12 * q^44 + 90 * q^45 + 440 * q^46 - 400 * q^47 - 54 * q^48 - 100 * q^50 - 132 * q^51 - 1528 * q^52 + 16 * q^53 + 108 * q^54 - 460 * q^55 - 324 * q^57 + 592 * q^58 + 464 * q^59 - 30 * q^60 + 684 * q^61 - 1140 * q^62 - 1214 * q^64 - 40 * q^65 - 492 * q^66 + 736 * q^67 + 1804 * q^68 + 960 * q^69 - 740 * q^71 - 108 * q^72 - 424 * q^73 - 168 * q^74 - 150 * q^75 + 1148 * q^76 - 1188 * q^78 - 408 * q^79 + 90 * q^80 + 162 * q^81 + 888 * q^82 - 608 * q^83 + 220 * q^85 + 1008 * q^86 + 708 * q^87 + 532 * q^88 + 1332 * q^89 - 180 * q^90 + 480 * q^92 - 180 * q^93 + 2480 * q^94 + 540 * q^95 - 900 * q^96 + 2448 * q^97 - 828 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.236068	0.0834626	0.0417313	−	0.999129i	$-0.486713\pi$
$2$	0.236068	0.0834626	0.0417313	+	0.999129i	$0.486713\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.94427	−0.993034
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−0.708204	−0.0481872
$7$	0	0
$7$	0	0
$8$	−3.76393	−0.166344
$9$	9.00000	0.333333
$10$	1.18034	0.0373256
$11$	−50.4721	−1.38345	−0.691724	−	0.722162i	$-0.743148\pi$
$11$	−50.4721	−1.38345	−0.691724	+	0.722162i	$0.743148\pi$
$12$	23.8328	0.573328
$13$	80.9706	1.72748	0.863738	−	0.503940i	$-0.168117\pi$
$13$	80.9706	1.72748	0.863738	+	0.503940i	$0.168117\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	62.6656	0.979150
$17$	−76.3870	−1.08980	−0.544899	−	0.838502i	$-0.683432\pi$
$17$	−76.3870	−1.08980	−0.544899	+	0.838502i	$0.683432\pi$
$18$	2.12461	0.0278209
$19$	−4.13777	−0.0499615	−0.0249808	−	0.999688i	$-0.507952\pi$
$19$	−4.13777	−0.0499615	−0.0249808	+	0.999688i	$0.507952\pi$
$20$	−39.7214	−0.444098
$21$	0	0
$22$	−11.9149	−0.115466
$23$	−204.721	−1.85597	−0.927986	−	0.372615i	$-0.878461\pi$
$23$	−204.721	−1.85597	−0.927986	+	0.372615i	$0.878461\pi$
$24$	11.2918	0.0960387
$25$	25.0000	0.200000
$26$	19.1146	0.144180
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−91.1672	−0.583770	−0.291885	−	0.956453i	$-0.594282\pi$
$29$	−91.1672	−0.583770	−0.291885	+	0.956453i	$0.594282\pi$
$30$	−3.54102	−0.0215500
$31$	−198.079	−1.14761	−0.573807	−	0.818991i	$-0.694534\pi$
$31$	−198.079	−1.14761	−0.573807	+	0.818991i	$0.694534\pi$
$32$	44.9048	0.248066
$33$	151.416	0.798734
$34$	−18.0325	−0.0909574
$35$	0	0
$36$	−71.4984	−0.331011
$37$	155.666	0.691656	0.345828	−	0.938298i	$-0.387598\pi$
$37$	155.666	0.691656	0.345828	+	0.938298i	$0.387598\pi$
$38$	−0.976794	−0.00416992
$39$	−242.912	−0.997359
$40$	−18.8197	−0.0743912
$41$	156.885	0.597595	0.298797	−	0.954317i	$-0.403415\pi$
$41$	156.885	0.597595	0.298797	+	0.954317i	$0.403415\pi$
$42$	0	0
$43$	354.217	1.25622	0.628111	−	0.778124i	$-0.283828\pi$
$43$	354.217	1.25622	0.628111	+	0.778124i	$0.283828\pi$
$44$	400.964	1.37381
$45$	45.0000	0.149071
$46$	−48.3282	−0.154904
$47$	175.659	0.545161	0.272580	−	0.962133i	$-0.412123\pi$
$47$	175.659	0.545161	0.272580	+	0.962133i	$0.412123\pi$
$48$	−187.997	−0.565313
$49$	0	0
$50$	5.90170	0.0166925
$51$	229.161	0.629195
$52$	−643.252	−1.71544
$53$	200.302	0.519124	0.259562	−	0.965726i	$-0.416422\pi$
$53$	200.302	0.519124	0.259562	+	0.965726i	$0.416422\pi$
$54$	−6.37384	−0.0160624
$55$	−252.361	−0.618696
$56$	0	0
$57$	12.4133	0.0288453
$58$	−21.5217	−0.0487230
$59$	312.498	0.689556	0.344778	−	0.938684i	$-0.387954\pi$
$59$	312.498	0.689556	0.344778	+	0.938684i	$0.387954\pi$
$60$	119.164	0.256400
$61$	154.170	0.323598	0.161799	−	0.986824i	$-0.448270\pi$
$61$	154.170	0.323598	0.161799	+	0.986824i	$0.448270\pi$
$62$	−46.7601	−0.0957829
$63$	0	0
$64$	−490.724	−0.958446
$65$	404.853	0.772551
$66$	35.7446	0.0666644
$67$	734.715	1.33970	0.669849	−	0.742498i	$-0.266359\pi$
$67$	734.715	1.33970	0.669849	+	0.742498i	$0.266359\pi$
$68$	606.839	1.08221
$69$	614.164	1.07155
$70$	0	0
$71$	−678.577	−1.13426	−0.567129	−	0.823629i	$-0.691946\pi$
$71$	−678.577	−1.13426	−0.567129	+	0.823629i	$0.691946\pi$
$72$	−33.8754	−0.0554480
$73$	60.8003	0.0974813	0.0487407	−	0.998811i	$-0.484479\pi$
$73$	60.8003	0.0974813	0.0487407	+	0.998811i	$0.484479\pi$
$74$	36.7477	0.0577274
$75$	−75.0000	−0.115470
$76$	32.8715	0.0496135
$77$	0	0
$78$	−57.3437	−0.0832422
$79$	−1286.26	−1.83184	−0.915919	−	0.401363i	$-0.868537\pi$
$79$	−1286.26	−1.83184	−0.915919	+	0.401363i	$0.868537\pi$
$80$	313.328	0.437889
$81$	81.0000	0.111111
$82$	37.0356	0.0498768
$83$	−116.170	−0.153631	−0.0768153	−	0.997045i	$-0.524475\pi$
$83$	−116.170	−0.153631	−0.0768153	+	0.997045i	$0.524475\pi$
$84$	0	0
$85$	−381.935	−0.487372
$86$	83.6192	0.104848
$87$	273.502	0.337040
$88$	189.974	0.230128
$89$	916.440	1.09149	0.545744	−	0.837952i	$-0.316247\pi$
$89$	916.440	1.09149	0.545744	+	0.837952i	$0.316247\pi$
$90$	10.6231	0.0124419
$91$	0	0
$92$	1626.36	1.84304
$93$	594.237	0.662575
$94$	41.4676	0.0455006
$95$	−20.6888	−0.0223435
$96$	−134.714	−0.143221
$97$	1416.30	1.48251	0.741256	−	0.671222i	$-0.234230\pi$
$97$	1416.30	1.48251	0.741256	+	0.671222i	$0.234230\pi$
$98$	0	0
$99$	−454.249	−0.461149
$100$	−198.607	−0.198607
$101$	1379.19	1.35876	0.679381	−	0.733785i	$-0.262248\pi$
$101$	1379.19	1.35876	0.679381	+	0.733785i	$0.262248\pi$
$102$	54.0976	0.0525143
$103$	1308.43	1.25169	0.625844	−	0.779949i	$-0.284755\pi$
$103$	1308.43	1.25169	0.625844	+	0.779949i	$0.284755\pi$
$104$	−304.768	−0.287355
$105$	0	0
$106$	47.2849	0.0433275
$107$	1265.65	1.14351	0.571754	−	0.820425i	$-0.306263\pi$
$107$	1265.65	1.14351	0.571754	+	0.820425i	$0.306263\pi$
$108$	214.495	0.191109
$109$	2069.32	1.81839	0.909196	−	0.416368i	$-0.136697\pi$
$109$	2069.32	1.81839	0.909196	+	0.416368i	$0.136697\pi$
$110$	−59.5743	−0.0516380
$111$	−466.997	−0.399328
$112$	0	0
$113$	−1953.89	−1.62661	−0.813303	−	0.581840i	$-0.802333\pi$
$113$	−1953.89	−1.62661	−0.813303	+	0.581840i	$0.802333\pi$
$114$	2.93038	0.00240750
$115$	−1023.61	−0.830016
$116$	724.257	0.579703
$117$	728.735	0.575826
$118$	73.7709	0.0575522
$119$	0	0
$120$	56.4590	0.0429498
$121$	1216.44	0.913927
$122$	36.3947	0.0270083
$123$	−470.656	−0.345022
$124$	1573.59	1.13962
$125$	125.000	0.0894427
$126$	0	0
$127$	224.251	0.156685	0.0783426	−	0.996926i	$-0.475037\pi$
$127$	224.251	0.156685	0.0783426	+	0.996926i	$0.475037\pi$
$128$	−475.083	−0.328061
$129$	−1062.65	−0.725280
$130$	95.5728	0.0644792
$131$	490.898	0.327404	0.163702	−	0.986510i	$-0.447656\pi$
$131$	490.898	0.327404	0.163702	+	0.986510i	$0.447656\pi$
$132$	−1202.89	−0.793170
$133$	0	0
$134$	173.443	0.111815
$135$	−135.000	−0.0860663
$136$	287.515	0.181281
$137$	1831.12	1.14192	0.570961	−	0.820977i	$-0.306571\pi$
$137$	1831.12	1.14192	0.570961	+	0.820977i	$0.306571\pi$
$138$	144.984	0.0894340
$139$	3050.84	1.86165	0.930823	−	0.365469i	$-0.119091\pi$
$139$	3050.84	1.86165	0.930823	+	0.365469i	$0.119091\pi$
$140$	0	0
$141$	−526.978	−0.314749
$142$	−160.190	−0.0946682
$143$	−4086.76	−2.38987
$144$	563.991	0.326383
$145$	−455.836	−0.261070
$146$	14.3530	0.00813605
$147$	0	0
$148$	−1236.65	−0.686838
$149$	2246.55	1.23520	0.617599	−	0.786493i	$-0.288105\pi$
$149$	2246.55	1.23520	0.617599	+	0.786493i	$0.288105\pi$
$150$	−17.7051	−0.00963743
$151$	−1311.53	−0.706826	−0.353413	−	0.935467i	$-0.614979\pi$
$151$	−1311.53	−0.706826	−0.353413	+	0.935467i	$0.614979\pi$
$152$	15.5743	0.00831079
$153$	−687.483	−0.363266
$154$	0	0
$155$	−990.395	−0.513228
$156$	1929.76	0.990412
$157$	−1790.94	−0.910398	−0.455199	−	0.890390i	$-0.650432\pi$
$157$	−1790.94	−0.910398	−0.455199	+	0.890390i	$0.650432\pi$
$158$	−303.644	−0.152890
$159$	−600.906	−0.299716
$160$	224.524	0.110939
$161$	0	0
$162$	19.1215	0.00927363
$163$	−491.108	−0.235991	−0.117996	−	0.993014i	$-0.537647\pi$
$163$	−491.108	−0.235991	−0.117996	+	0.993014i	$0.537647\pi$
$164$	−1246.34	−0.593432
$165$	757.082	0.357205
$166$	−27.4241	−0.0128224
$167$	826.059	0.382769	0.191384	−	0.981515i	$-0.438702\pi$
$167$	826.059	0.382769	0.191384	+	0.981515i	$0.438702\pi$
$168$	0	0
$169$	4359.24	1.98418
$170$	−90.1626	−0.0406774
$171$	−37.2399	−0.0166538
$172$	−2813.99	−1.24747
$173$	−2918.00	−1.28238	−0.641190	−	0.767382i	$-0.721559\pi$
$173$	−2918.00	−1.28238	−0.641190	+	0.767382i	$0.721559\pi$
$174$	64.5650	0.0281302
$175$	0	0
$176$	−3162.87	−1.35460
$177$	−937.495	−0.398116
$178$	216.342	0.0910984
$179$	955.745	0.399082	0.199541	−	0.979889i	$-0.436055\pi$
$179$	955.745	0.399082	0.199541	+	0.979889i	$0.436055\pi$
$180$	−357.492	−0.148033
$181$	−206.080	−0.0846289	−0.0423145	−	0.999104i	$-0.513473\pi$
$181$	−206.080	−0.0846289	−0.0423145	+	0.999104i	$0.513473\pi$
$182$	0	0
$183$	−462.511	−0.186829
$184$	770.557	0.308730
$185$	778.328	0.309318
$186$	140.280	0.0553003
$187$	3855.41	1.50768
$188$	−1395.49	−0.541363
$189$	0	0
$190$	−4.88397	−0.00186485
$191$	−2018.63	−0.764727	−0.382364	−	0.924012i	$-0.624890\pi$
$191$	−2018.63	−0.764727	−0.382364	+	0.924012i	$0.624890\pi$
$192$	1472.17	0.553359
$193$	1031.69	0.384781	0.192390	−	0.981318i	$-0.438376\pi$
$193$	1031.69	0.384781	0.192390	+	0.981318i	$0.438376\pi$
$194$	334.344	0.123734
$195$	−1214.56	−0.446033
$196$	0	0
$197$	−205.955	−0.0744857	−0.0372429	−	0.999306i	$-0.511858\pi$
$197$	−205.955	−0.0744857	−0.0372429	+	0.999306i	$0.511858\pi$
$198$	−107.234	−0.0384887
$199$	1831.38	0.652376	0.326188	−	0.945305i	$-0.394236\pi$
$199$	1831.38	0.652376	0.326188	+	0.945305i	$0.394236\pi$
$200$	−94.0983	−0.0332688
$201$	−2204.15	−0.773475
$202$	325.584	0.113406
$203$	0	0
$204$	−1820.52	−0.624812
$205$	784.427	0.267253
$206$	308.879	0.104469
$207$	−1842.49	−0.618657
$208$	5074.07	1.69146
$209$	208.842	0.0691191
$210$	0	0
$211$	1030.19	0.336119	0.168060	−	0.985777i	$-0.446250\pi$
$211$	1030.19	0.336119	0.168060	+	0.985777i	$0.446250\pi$
$212$	−1591.25	−0.515508
$213$	2035.73	0.654864
$214$	298.780	0.0954402
$215$	1771.08	0.561800
$216$	101.626	0.0320129
$217$	0	0
$218$	488.500	0.151768
$219$	−182.401	−0.0562809
$220$	2004.82	0.614387
$221$	−6185.10	−1.88260
$222$	−110.243	−0.0333289
$223$	−5368.67	−1.61217	−0.806083	−	0.591803i	$-0.798416\pi$
$223$	−5368.67	−1.61217	−0.806083	+	0.591803i	$0.798416\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	−461.251	−0.135761
$227$	932.121	0.272542	0.136271	−	0.990672i	$-0.456488\pi$
$227$	932.121	0.272542	0.136271	+	0.990672i	$0.456488\pi$
$228$	−98.6146	−0.0286444
$229$	−3163.05	−0.912752	−0.456376	−	0.889787i	$-0.650853\pi$
$229$	−3163.05	−0.912752	−0.456376	+	0.889787i	$0.650853\pi$
$230$	−241.641	−0.0692753
$231$	0	0
$232$	343.147	0.0971065
$233$	436.562	0.122747	0.0613737	−	0.998115i	$-0.480452\pi$
$233$	436.562	0.122747	0.0613737	+	0.998115i	$0.480452\pi$
$234$	172.031	0.0480599
$235$	878.297	0.243803
$236$	−2482.57	−0.684753
$237$	3858.77	1.05761
$238$	0	0
$239$	−1980.82	−0.536103	−0.268051	−	0.963405i	$-0.586380\pi$
$239$	−1980.82	−0.536103	−0.268051	+	0.963405i	$0.586380\pi$
$240$	−939.984	−0.252816
$241$	−5303.55	−1.41756	−0.708780	−	0.705430i	$-0.750754\pi$
$241$	−5303.55	−1.41756	−0.708780	+	0.705430i	$0.750754\pi$
$242$	287.162	0.0762787
$243$	−243.000	−0.0641500
$244$	−1224.77	−0.321344
$245$	0	0
$246$	−111.107	−0.0287964
$247$	−335.037	−0.0863074
$248$	745.556	0.190899
$249$	348.511	0.0886987
$250$	29.5085	0.00746512
$251$	2996.04	0.753420	0.376710	−	0.926331i	$-0.377055\pi$
$251$	2996.04	0.753420	0.376710	+	0.926331i	$0.377055\pi$
$252$	0	0
$253$	10332.7	2.56764
$254$	52.9384	0.0130774
$255$	1145.80	0.281385
$256$	3813.64	0.931065
$257$	−968.861	−0.235159	−0.117580	−	0.993063i	$-0.537513\pi$
$257$	−968.861	−0.235159	−0.117580	+	0.993063i	$0.537513\pi$
$258$	−250.858	−0.0605338
$259$	0	0
$260$	−3216.26	−0.767170
$261$	−820.505	−0.194590
$262$	115.885	0.0273260
$263$	−4830.18	−1.13248	−0.566239	−	0.824241i	$-0.691602\pi$
$263$	−4830.18	−1.13248	−0.566239	+	0.824241i	$0.691602\pi$
$264$	−569.921	−0.132864
$265$	1001.51	0.232159
$266$	0	0
$267$	−2749.32	−0.630171
$268$	−5836.78	−1.33037
$269$	4774.97	1.08229	0.541143	−	0.840930i	$-0.317992\pi$
$269$	4774.97	1.08229	0.541143	+	0.840930i	$0.317992\pi$
$270$	−31.8692	−0.00718332
$271$	−141.909	−0.0318094	−0.0159047	−	0.999874i	$-0.505063\pi$
$271$	−141.909	−0.0318094	−0.0159047	+	0.999874i	$0.505063\pi$
$272$	−4786.84	−1.06708
$273$	0	0
$274$	432.269	0.0953078
$275$	−1261.80	−0.276689
$276$	−4879.09	−1.06408
$277$	−3621.13	−0.785460	−0.392730	−	0.919654i	$-0.628469\pi$
$277$	−3621.13	−0.785460	−0.392730	+	0.919654i	$0.628469\pi$
$278$	720.206	0.155378
$279$	−1782.71	−0.382538
$280$	0	0
$281$	5790.87	1.22937	0.614687	−	0.788771i	$-0.289282\pi$
$281$	5790.87	1.22937	0.614687	+	0.788771i	$0.289282\pi$
$282$	−124.403	−0.0262698
$283$	−526.043	−0.110495	−0.0552474	−	0.998473i	$-0.517595\pi$
$283$	−526.043	−0.110495	−0.0552474	+	0.998473i	$0.517595\pi$
$284$	5390.80	1.12636
$285$	62.0665	0.0129000
$286$	−964.753	−0.199465
$287$	0	0
$288$	404.143	0.0826888
$289$	921.972	0.187660
$290$	−107.608	−0.0217896
$291$	−4248.91	−0.855929
$292$	−483.014	−0.0968023
$293$	−1914.88	−0.381803	−0.190901	−	0.981609i	$-0.561141\pi$
$293$	−1914.88	−0.381803	−0.190901	+	0.981609i	$0.561141\pi$
$294$	0	0
$295$	1562.49	0.308379
$296$	−585.915	−0.115053
$297$	1362.75	0.266245
$298$	530.339	0.103093
$299$	−16576.4	−3.20615
$300$	595.820	0.114666
$301$	0	0
$302$	−309.610	−0.0589936
$303$	−4137.58	−0.784482
$304$	−259.296	−0.0489199
$305$	770.851	0.144717
$306$	−162.293	−0.0303191
$307$	6244.17	1.16083	0.580413	−	0.814322i	$-0.302891\pi$
$307$	6244.17	1.16083	0.580413	+	0.814322i	$0.302891\pi$
$308$	0	0
$309$	−3925.30	−0.722662
$310$	−233.800	−0.0428354
$311$	9658.78	1.76109	0.880545	−	0.473962i	$-0.157177\pi$
$311$	9658.78	1.76109	0.880545	+	0.473962i	$0.157177\pi$
$312$	914.303	0.165905
$313$	−2198.34	−0.396988	−0.198494	−	0.980102i	$-0.563605\pi$
$313$	−2198.34	−0.396988	−0.198494	+	0.980102i	$0.563605\pi$
$314$	−422.783	−0.0759842
$315$	0	0
$316$	10218.4	1.81908
$317$	3030.78	0.536990	0.268495	−	0.963281i	$-0.413474\pi$
$317$	3030.78	0.536990	0.268495	+	0.963281i	$0.413474\pi$
$318$	−141.855	−0.0250151
$319$	4601.40	0.807615
$320$	−2453.62	−0.428630
$321$	−3796.96	−0.660204
$322$	0	0
$323$	316.072	0.0544480
$324$	−643.486	−0.110337
$325$	2024.26	0.345495
$326$	−115.935	−0.0196965
$327$	−6207.96	−1.04985
$328$	−590.506	−0.0994062
$329$	0	0
$330$	178.723	0.0298132
$331$	4753.74	0.789393	0.394696	−	0.918812i	$-0.370850\pi$
$331$	4753.74	0.789393	0.394696	+	0.918812i	$0.370850\pi$
$332$	922.888	0.152560
$333$	1400.99	0.230552
$334$	195.006	0.0319469
$335$	3673.58	0.599131
$336$	0	0
$337$	8824.40	1.42640	0.713199	−	0.700962i	$-0.247246\pi$
$337$	8824.40	1.42640	0.713199	+	0.700962i	$0.247246\pi$
$338$	1029.08	0.165605
$339$	5861.67	0.939122
$340$	3034.20	0.483977
$341$	9997.47	1.58766
$342$	−8.79115	−0.00138997
$343$	0	0
$344$	−1333.25	−0.208965
$345$	3070.82	0.479210
$346$	−688.847	−0.107031
$347$	−3413.97	−0.528160	−0.264080	−	0.964501i	$-0.585068\pi$
$347$	−3413.97	−0.528160	−0.264080	+	0.964501i	$0.585068\pi$
$348$	−2172.77	−0.334692
$349$	5676.32	0.870621	0.435310	−	0.900280i	$-0.356639\pi$
$349$	5676.32	0.870621	0.435310	+	0.900280i	$0.356639\pi$
$350$	0	0
$351$	−2186.21	−0.332453
$352$	−2266.44	−0.343187
$353$	−6225.80	−0.938713	−0.469357	−	0.883009i	$-0.655514\pi$
$353$	−6225.80	−0.938713	−0.469357	+	0.883009i	$0.655514\pi$
$354$	−221.313	−0.0332278
$355$	−3392.89	−0.507256
$356$	−7280.45	−1.08388
$357$	0	0
$358$	225.621	0.0333084
$359$	−4907.73	−0.721505	−0.360752	−	0.932662i	$-0.617480\pi$
$359$	−4907.73	−0.721505	−0.360752	+	0.932662i	$0.617480\pi$
$360$	−169.377	−0.0247971
$361$	−6841.88	−0.997504
$362$	−48.6490	−0.00706335
$363$	−3649.31	−0.527656
$364$	0	0
$365$	304.001	0.0435950
$366$	−109.184	−0.0155933
$367$	3906.48	0.555631	0.277816	−	0.960634i	$-0.410390\pi$
$367$	3906.48	0.555631	0.277816	+	0.960634i	$0.410390\pi$
$368$	−12829.0	−1.81728
$369$	1411.97	0.199198
$370$	183.738	0.0258165
$371$	0	0
$372$	−4720.78	−0.657960
$373$	−2102.52	−0.291862	−0.145931	−	0.989295i	$-0.546618\pi$
$373$	−2102.52	−0.291862	−0.145931	+	0.989295i	$0.546618\pi$
$374$	910.140	0.125835
$375$	−375.000	−0.0516398
$376$	−661.170	−0.0906842
$377$	−7381.86	−1.00845
$378$	0	0
$379$	−6612.76	−0.896239	−0.448120	−	0.893974i	$-0.647906\pi$
$379$	−6612.76	−0.896239	−0.448120	+	0.893974i	$0.647906\pi$
$380$	164.358	0.0221878
$381$	−672.752	−0.0904623
$382$	−476.534	−0.0638262
$383$	−2.37457	−0.000316801 0	−0.000158401	−	1.00000i	$-0.500050\pi$
$383$	−2.37457	−0.000316801 0	−0.000158401		1.00000i	$0.500050\pi$
$384$	1425.25	0.189406
$385$	0	0
$386$	243.549	0.0321148
$387$	3187.95	0.418741
$388$	−11251.5	−1.47218
$389$	7716.98	1.00583	0.502913	−	0.864337i	$-0.332261\pi$
$389$	7716.98	1.00583	0.502913	+	0.864337i	$0.332261\pi$
$390$	−286.718	−0.0372271
$391$	15638.0	2.02263
$392$	0	0
$393$	−1472.69	−0.189027
$394$	−48.6194	−0.00621678
$395$	−6431.28	−0.819223
$396$	3608.68	0.457937
$397$	−6403.95	−0.809584	−0.404792	−	0.914409i	$-0.632656\pi$
$397$	−6403.95	−0.809584	−0.404792	+	0.914409i	$0.632656\pi$
$398$	432.329	0.0544490
$399$	0	0
$400$	1566.64	0.195830
$401$	−10969.9	−1.36611	−0.683054	−	0.730368i	$-0.739349\pi$
$401$	−10969.9	−1.36611	−0.683054	+	0.730368i	$0.739349\pi$
$402$	−520.328	−0.0645562
$403$	−16038.6	−1.98248
$404$	−10956.7	−1.34930
$405$	405.000	0.0496904
$406$	0	0
$407$	−7856.78	−0.956870
$408$	−862.546	−0.104663
$409$	400.353	0.0484015	0.0242007	−	0.999707i	$-0.492296\pi$
$409$	400.353	0.0484015	0.0242007	+	0.999707i	$0.492296\pi$
$410$	185.178	0.0223056
$411$	−5493.37	−0.659289
$412$	−10394.6	−1.24297
$413$	0	0
$414$	−434.953	−0.0516348
$415$	−580.851	−0.0687057
$416$	3635.97	0.428529
$417$	−9152.52	−1.07482
$418$	49.3009	0.00576887
$419$	15815.4	1.84399	0.921995	−	0.387201i	$-0.126558\pi$
$419$	15815.4	1.84399	0.921995	+	0.387201i	$0.126558\pi$
$420$	0	0
$421$	1936.53	0.224182	0.112091	−	0.993698i	$-0.464245\pi$
$421$	1936.53	0.224182	0.112091	+	0.993698i	$0.464245\pi$
$422$	243.195	0.0280534
$423$	1580.93	0.181720
$424$	−753.923	−0.0863531
$425$	−1909.67	−0.217960
$426$	480.571	0.0546567
$427$	0	0
$428$	−10054.7	−1.13554
$429$	12260.3	1.37979
$430$	418.096	0.0468893
$431$	2030.91	0.226973	0.113487	−	0.993540i	$-0.463798\pi$
$431$	2030.91	0.226973	0.113487	+	0.993540i	$0.463798\pi$
$432$	−1691.97	−0.188438
$433$	10784.1	1.19689	0.598443	−	0.801165i	$-0.295786\pi$
$433$	10784.1	1.19689	0.598443	+	0.801165i	$0.295786\pi$
$434$	0	0
$435$	1367.51	0.150729
$436$	−16439.2	−1.80573
$437$	847.089	0.0927272
$438$	−43.0590	−0.00469735
$439$	6304.19	0.685382	0.342691	−	0.939448i	$-0.388662\pi$
$439$	6304.19	0.685382	0.342691	+	0.939448i	$0.388662\pi$
$440$	949.868	0.102916
$441$	0	0
$442$	−1460.10	−0.157127
$443$	15494.8	1.66181	0.830905	−	0.556414i	$-0.187823\pi$
$443$	15494.8	1.66181	0.830905	+	0.556414i	$0.187823\pi$
$444$	3709.95	0.396546
$445$	4582.20	0.488128
$446$	−1267.37	−0.134556
$447$	−6739.65	−0.713142
$448$	0	0
$449$	−242.018	−0.0254377	−0.0127189	−	0.999919i	$-0.504049\pi$
$449$	−242.018	−0.0254377	−0.0127189	+	0.999919i	$0.504049\pi$
$450$	53.1153	0.00556418
$451$	−7918.34	−0.826741
$452$	15522.2	1.61528
$453$	3934.59	0.408086
$454$	220.044	0.0227471
$455$	0	0
$456$	−46.7228	−0.00479824
$457$	−11670.4	−1.19457	−0.597283	−	0.802030i	$-0.703753\pi$
$457$	−11670.4	−1.19457	−0.597283	+	0.802030i	$0.703753\pi$
$458$	−746.695	−0.0761807
$459$	2062.45	0.209732
$460$	8131.81	0.824234
$461$	−12128.1	−1.22530	−0.612649	−	0.790355i	$-0.709896\pi$
$461$	−12128.1	−1.22530	−0.612649	+	0.790355i	$0.709896\pi$
$462$	0	0
$463$	5161.64	0.518103	0.259051	−	0.965864i	$-0.416590\pi$
$463$	5161.64	0.518103	0.259051	+	0.965864i	$0.416590\pi$
$464$	−5713.05	−0.571598
$465$	2971.18	0.296313
$466$	103.058	0.0102448
$467$	−11680.2	−1.15738	−0.578691	−	0.815547i	$-0.696436\pi$
$467$	−11680.2	−1.15738	−0.578691	+	0.815547i	$0.696436\pi$
$468$	−5789.27	−0.571814
$469$	0	0
$470$	207.338	0.0203485
$471$	5372.82	0.525619
$472$	−1176.22	−0.114703
$473$	−17878.1	−1.73792
$474$	910.932	0.0882711
$475$	−103.444	−0.00999230
$476$	0	0
$477$	1802.72	0.173041
$478$	−467.608	−0.0447445
$479$	18458.7	1.76075	0.880373	−	0.474282i	$-0.157292\pi$
$479$	18458.7	1.76075	0.880373	+	0.474282i	$0.157292\pi$
$480$	−673.572	−0.0640505
$481$	12604.3	1.19482
$482$	−1252.00	−0.118313
$483$	0	0
$484$	−9663.70	−0.907560
$485$	7081.51	0.663000
$486$	−57.3645	−0.00535413
$487$	8630.11	0.803014	0.401507	−	0.915856i	$-0.368487\pi$
$487$	8630.11	0.803014	0.401507	+	0.915856i	$0.368487\pi$
$488$	−580.286	−0.0538286
$489$	1473.33	0.136250
$490$	0	0
$491$	17801.6	1.63621	0.818103	−	0.575072i	$-0.195026\pi$
$491$	17801.6	1.63621	0.818103	+	0.575072i	$0.195026\pi$
$492$	3739.02	0.342618
$493$	6963.99	0.636191
$494$	−79.0916	−0.00720344
$495$	−2271.25	−0.206232
$496$	−12412.7	−1.12369
$497$	0	0
$498$	82.2723	0.00740303
$499$	200.167	0.0179574	0.00897868	−	0.999960i	$-0.497142\pi$
$499$	200.167	0.0179574	0.00897868	+	0.999960i	$0.497142\pi$
$500$	−993.034	−0.0888197
$501$	−2478.18	−0.220992
$502$	707.269	0.0628824
$503$	−16400.4	−1.45379	−0.726896	−	0.686747i	$-0.759038\pi$
$503$	−16400.4	−1.45379	−0.726896	+	0.686747i	$0.759038\pi$
$504$	0	0
$505$	6895.97	0.607657
$506$	2439.23	0.214302
$507$	−13077.7	−1.14556
$508$	−1781.51	−0.155594
$509$	15006.6	1.30679	0.653394	−	0.757018i	$-0.273344\pi$
$509$	15006.6	1.30679	0.653394	+	0.757018i	$0.273344\pi$
$510$	270.488	0.0234851
$511$	0	0
$512$	4700.94	0.405770
$513$	111.720	0.00961510
$514$	−228.717	−0.0196270
$515$	6542.17	0.559772
$516$	8441.98	0.720228
$517$	−8865.91	−0.754201
$518$	0	0
$519$	8754.01	0.740382
$520$	−1523.84	−0.128509
$521$	−7113.18	−0.598146	−0.299073	−	0.954230i	$-0.596677\pi$
$521$	−7113.18	−0.598146	−0.299073	+	0.954230i	$0.596677\pi$
$522$	−193.695	−0.0162410
$523$	8888.46	0.743146	0.371573	−	0.928404i	$-0.378819\pi$
$523$	8888.46	0.743146	0.371573	+	0.928404i	$0.378819\pi$
$524$	−3899.83	−0.325123
$525$	0	0
$526$	−1140.25	−0.0945196
$527$	15130.7	1.25067
$528$	9488.60	0.782081
$529$	29743.8	2.44463
$530$	236.424	0.0193766
$531$	2812.49	0.229852
$532$	0	0
$533$	12703.1	1.03233
$534$	−649.026	−0.0525957
$535$	6328.27	0.511392
$536$	−2765.42	−0.222850
$537$	−2867.23	−0.230410
$538$	1127.22	0.0903305
$539$	0	0
$540$	1072.48	0.0854668
$541$	−653.827	−0.0519597	−0.0259799	−	0.999662i	$-0.508271\pi$
$541$	−653.827	−0.0519597	−0.0259799	+	0.999662i	$0.508271\pi$
$542$	−33.5001	−0.00265489
$543$	618.241	0.0488605
$544$	−3430.14	−0.270342
$545$	10346.6	0.813210
$546$	0	0
$547$	1138.52	0.0889940	0.0444970	−	0.999010i	$-0.485831\pi$
$547$	1138.52	0.0889940	0.0444970	+	0.999010i	$0.485831\pi$
$548$	−14546.9	−1.13397
$549$	1387.53	0.107866
$550$	−297.871	−0.0230932
$551$	377.229	0.0291660
$552$	−2311.67	−0.178245
$553$	0	0
$554$	−854.832	−0.0655566
$555$	−2334.98	−0.178585
$556$	−24236.7	−1.84868
$557$	−19804.8	−1.50657	−0.753283	−	0.657696i	$-0.771531\pi$
$557$	−19804.8	−1.50657	−0.753283	+	0.657696i	$0.771531\pi$
$558$	−420.841	−0.0319276
$559$	28681.1	2.17009
$560$	0	0
$561$	−11566.2	−0.870458
$562$	1367.04	0.102607
$563$	−10276.3	−0.769265	−0.384632	−	0.923070i	$-0.625672\pi$
$563$	−10276.3	−0.769265	−0.384632	+	0.923070i	$0.625672\pi$
$564$	4186.46	0.312556
$565$	−9769.45	−0.727440
$566$	−124.182	−0.00922219
$567$	0	0
$568$	2554.12	0.188677
$569$	4139.03	0.304951	0.152475	−	0.988307i	$-0.451276\pi$
$569$	4139.03	0.304951	0.152475	+	0.988307i	$0.451276\pi$
$570$	14.6519	0.00107667
$571$	−4486.81	−0.328839	−0.164420	−	0.986390i	$-0.552575\pi$
$571$	−4486.81	−0.328839	−0.164420	+	0.986390i	$0.552575\pi$
$572$	32466.3	2.37323
$573$	6055.89	0.441516
$574$	0	0
$575$	−5118.03	−0.371194
$576$	−4416.52	−0.319482
$577$	1104.77	0.0797093	0.0398547	−	0.999205i	$-0.487311\pi$
$577$	1104.77	0.0797093	0.0398547	+	0.999205i	$0.487311\pi$
$578$	217.648	0.0156626
$579$	−3095.07	−0.222153
$580$	3621.28	0.259251
$581$	0	0
$582$	−1003.03	−0.0714381
$583$	−10109.7	−0.718181
$584$	−228.848	−0.0162154
$585$	3643.68	0.257517
$586$	−452.041	−0.0318663
$587$	10413.2	0.732199	0.366100	−	0.930576i	$-0.380693\pi$
$587$	10413.2	0.732199	0.366100	+	0.930576i	$0.380693\pi$
$588$	0	0
$589$	819.605	0.0573365
$590$	368.854	0.0257381
$591$	617.865	0.0430044
$592$	9754.89	0.677235
$593$	3235.16	0.224034	0.112017	−	0.993706i	$-0.464269\pi$
$593$	3235.16	0.224034	0.112017	+	0.993706i	$0.464269\pi$
$594$	321.701	0.0222215
$595$	0	0
$596$	−17847.2	−1.22659
$597$	−5494.13	−0.376649
$598$	−3913.16	−0.267594
$599$	569.048	0.0388158	0.0194079	−	0.999812i	$-0.493822\pi$
$599$	569.048	0.0388158	0.0194079	+	0.999812i	$0.493822\pi$
$600$	282.295	0.0192077
$601$	−3760.89	−0.255258	−0.127629	−	0.991822i	$-0.540737\pi$
$601$	−3760.89	−0.255258	−0.127629	+	0.991822i	$0.540737\pi$
$602$	0	0
$603$	6612.44	0.446566
$604$	10419.1	0.701902
$605$	6082.18	0.408720
$606$	−976.751	−0.0654749
$607$	2224.05	0.148717	0.0743585	−	0.997232i	$-0.476309\pi$
$607$	2224.05	0.148717	0.0743585	+	0.997232i	$0.476309\pi$
$608$	−185.806	−0.0123938
$609$	0	0
$610$	181.973	0.0120785
$611$	14223.2	0.941753
$612$	5461.55	0.360735
$613$	5914.50	0.389697	0.194849	−	0.980833i	$-0.437578\pi$
$613$	5914.50	0.389697	0.194849	+	0.980833i	$0.437578\pi$
$614$	1474.05	0.0968856
$615$	−2353.28	−0.154298
$616$	0	0
$617$	−18591.2	−1.21306	−0.606528	−	0.795062i	$-0.707438\pi$
$617$	−18591.2	−1.21306	−0.606528	+	0.795062i	$0.707438\pi$
$618$	−926.638	−0.0603153
$619$	−5125.97	−0.332844	−0.166422	−	0.986055i	$-0.553221\pi$
$619$	−5125.97	−0.332844	−0.166422	+	0.986055i	$0.553221\pi$
$620$	7867.96	0.509653
$621$	5527.48	0.357182
$622$	2280.13	0.146985
$623$	0	0
$624$	−15222.2	−0.976565
$625$	625.000	0.0400000
$626$	−518.957	−0.0331337
$627$	−626.526	−0.0399060
$628$	14227.7	0.904056
$629$	−11890.8	−0.753765
$630$	0	0
$631$	−10649.0	−0.671839	−0.335919	−	0.941891i	$-0.609047\pi$
$631$	−10649.0	−0.671839	−0.335919	+	0.941891i	$0.609047\pi$
$632$	4841.38	0.304715
$633$	−3090.57	−0.194058
$634$	715.471	0.0448186
$635$	1121.25	0.0700718
$636$	4773.76	0.297629
$637$	0	0
$638$	1086.24	0.0674056
$639$	−6107.20	−0.378086
$640$	−2375.41	−0.146713
$641$	24025.7	1.48044	0.740218	−	0.672367i	$-0.234722\pi$
$641$	24025.7	1.48044	0.740218	+	0.672367i	$0.234722\pi$
$642$	−896.341	−0.0551024
$643$	−14929.3	−0.915638	−0.457819	−	0.889045i	$-0.651369\pi$
$643$	−14929.3	−0.915638	−0.457819	+	0.889045i	$0.651369\pi$
$644$	0	0
$645$	−5313.25	−0.324355
$646$	74.6144	0.00454437
$647$	−14479.1	−0.879801	−0.439901	−	0.898046i	$-0.644986\pi$
$647$	−14479.1	−0.879801	−0.439901	+	0.898046i	$0.644986\pi$
$648$	−304.878	−0.0184827
$649$	−15772.5	−0.953965
$650$	477.864	0.0288360
$651$	0	0
$652$	3901.50	0.234347
$653$	898.168	0.0538255	0.0269127	−	0.999638i	$-0.491432\pi$
$653$	898.168	0.0538255	0.0269127	+	0.999638i	$0.491432\pi$
$654$	−1465.50	−0.0876232
$655$	2454.49	0.146420
$656$	9831.33	0.585135
$657$	547.203	0.0324938
$658$	0	0
$659$	−30198.0	−1.78505	−0.892526	−	0.450997i	$-0.851069\pi$
$659$	−30198.0	−1.78505	−0.892526	+	0.450997i	$0.851069\pi$
$660$	−6014.47	−0.354716
$661$	19337.8	1.13790	0.568952	−	0.822371i	$-0.307349\pi$
$661$	19337.8	1.13790	0.568952	+	0.822371i	$0.307349\pi$
$662$	1122.20	0.0658848
$663$	18555.3	1.08692
$664$	437.257	0.0255555
$665$	0	0
$666$	330.729	0.0192425
$667$	18663.9	1.08346
$668$	−6562.44	−0.380102
$669$	16106.0	0.930784
$670$	867.214	0.0500051
$671$	−7781.30	−0.447681
$672$	0	0
$673$	10132.2	0.580336	0.290168	−	0.956976i	$-0.406289\pi$
$673$	10132.2	0.580336	0.290168	+	0.956976i	$0.406289\pi$
$674$	2083.16	0.119051
$675$	−675.000	−0.0384900
$676$	−34631.0	−1.97035
$677$	33177.3	1.88347	0.941733	−	0.336361i	$-0.109196\pi$
$677$	33177.3	1.88347	0.941733	+	0.336361i	$0.109196\pi$
$678$	1383.75	0.0783816
$679$	0	0
$680$	1437.58	0.0810714
$681$	−2796.36	−0.157352
$682$	2360.08	0.132511
$683$	−11423.6	−0.639987	−0.319994	−	0.947420i	$-0.603681\pi$
$683$	−11423.6	−0.639987	−0.319994	+	0.947420i	$0.603681\pi$
$684$	295.844	0.0165378
$685$	9155.61	0.510683
$686$	0	0
$687$	9489.15	0.526978
$688$	22197.2	1.23003
$689$	16218.6	0.896775
$690$	724.922	0.0399961
$691$	19737.4	1.08661	0.543304	−	0.839536i	$-0.317173\pi$
$691$	19737.4	1.08661	0.543304	+	0.839536i	$0.317173\pi$
$692$	23181.4	1.27345
$693$	0	0
$694$	−805.929	−0.0440816
$695$	15254.2	0.832554
$696$	−1029.44	−0.0560645
$697$	−11984.0	−0.651258
$698$	1340.00	0.0726643
$699$	−1309.69	−0.0708682
$700$	0	0
$701$	11307.8	0.609258	0.304629	−	0.952471i	$-0.401468\pi$
$701$	11307.8	0.609258	0.304629	+	0.952471i	$0.401468\pi$
$702$	−516.093	−0.0277474
$703$	−644.108	−0.0345562
$704$	24767.9	1.32596
$705$	−2634.89	−0.140760
$706$	−1469.71	−0.0783475
$707$	0	0
$708$	7447.72	0.395342
$709$	30859.2	1.63461	0.817307	−	0.576202i	$-0.195466\pi$
$709$	30859.2	1.63461	0.817307	+	0.576202i	$0.195466\pi$
$710$	−800.952	−0.0423369
$711$	−11576.3	−0.610613
$712$	−3449.42	−0.181562
$713$	40551.0	2.12994
$714$	0	0
$715$	−20433.8	−1.06878
$716$	−7592.69	−0.396302
$717$	5942.46	0.309519
$718$	−1158.56	−0.0602187
$719$	−33152.4	−1.71958	−0.859789	−	0.510650i	$-0.829405\pi$
$719$	−33152.4	−1.71958	−0.859789	+	0.510650i	$0.829405\pi$
$720$	2819.95	0.145963
$721$	0	0
$722$	−1615.15	−0.0832543
$723$	15910.7	0.818428
$724$	1637.16	0.0840394
$725$	−2279.18	−0.116754
$726$	−861.485	−0.0440395
$727$	16743.0	0.854146	0.427073	−	0.904217i	$-0.359545\pi$
$727$	16743.0	0.854146	0.427073	+	0.904217i	$0.359545\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	71.7650	0.00363855
$731$	−27057.5	−1.36903
$732$	3674.31	0.185528
$733$	8827.55	0.444820	0.222410	−	0.974953i	$-0.428608\pi$
$733$	8827.55	0.444820	0.222410	+	0.974953i	$0.428608\pi$
$734$	922.195	0.0463744
$735$	0	0
$736$	−9192.97	−0.460404
$737$	−37082.6	−1.85340
$738$	333.321	0.0166256
$739$	36154.0	1.79966	0.899829	−	0.436243i	$-0.143691\pi$
$739$	36154.0	1.79966	0.899829	+	0.436243i	$0.143691\pi$
$740$	−6183.25	−0.307163
$741$	1005.11	0.0498296
$742$	0	0
$743$	−1820.69	−0.0898987	−0.0449494	−	0.998989i	$-0.514313\pi$
$743$	−1820.69	−0.0898987	−0.0449494	+	0.998989i	$0.514313\pi$
$744$	−2236.67	−0.110215
$745$	11232.8	0.552398
$746$	−496.338	−0.0243596
$747$	−1045.53	−0.0512102
$748$	−30628.5	−1.49718
$749$	0	0
$750$	−88.5255	−0.00430999
$751$	27764.4	1.34905	0.674526	−	0.738251i	$-0.264348\pi$
$751$	27764.4	1.34905	0.674526	+	0.738251i	$0.264348\pi$
$752$	11007.8	0.533795
$753$	−8988.12	−0.434987
$754$	−1742.62	−0.0841678
$755$	−6557.65	−0.316102
$756$	0	0
$757$	−13518.3	−0.649050	−0.324525	−	0.945877i	$-0.605205\pi$
$757$	−13518.3	−0.649050	−0.324525	+	0.945877i	$0.605205\pi$
$758$	−1561.06	−0.0748025
$759$	−30998.2	−1.48243
$760$	77.8714	0.00371670
$761$	−30695.2	−1.46216	−0.731078	−	0.682294i	$-0.760983\pi$
$761$	−30695.2	−1.46216	−0.731078	+	0.682294i	$0.760983\pi$
$762$	−158.815	−0.00755022
$763$	0	0
$764$	16036.5	0.759400
$765$	−3437.41	−0.162457
$766$	−0.560560	−2.64410e−5 0
$767$	25303.2	1.19119
$768$	−11440.9	−0.537551
$769$	−4536.39	−0.212726	−0.106363	−	0.994327i	$-0.533921\pi$
$769$	−4536.39	−0.212726	−0.106363	+	0.994327i	$0.533921\pi$
$770$	0	0
$771$	2906.58	0.135769
$772$	−8196.03	−0.382100
$773$	−31238.9	−1.45354	−0.726768	−	0.686883i	$-0.758978\pi$
$773$	−31238.9	−1.45354	−0.726768	+	0.686883i	$0.758978\pi$
$774$	752.573	0.0349492
$775$	−4951.97	−0.229523
$776$	−5330.86	−0.246607
$777$	0	0
$778$	1821.73	0.0839490
$779$	−649.155	−0.0298567
$780$	9648.78	0.442926
$781$	34249.2	1.56919
$782$	3691.64	0.168814
$783$	2461.51	0.112347
$784$	0	0
$785$	−8954.70	−0.407143
$786$	−347.656	−0.0157767
$787$	−39597.2	−1.79350	−0.896752	−	0.442534i	$-0.854079\pi$
$787$	−39597.2	−1.79350	−0.896752	+	0.442534i	$0.854079\pi$
$788$	1636.16	0.0739669
$789$	14490.5	0.653836
$790$	−1518.22	−0.0683745
$791$	0	0
$792$	1709.76	0.0767093
$793$	12483.3	0.559008
$794$	−1511.77	−0.0675700
$795$	−3004.53	−0.134037
$796$	−14549.0	−0.647832
$797$	16567.0	0.736302	0.368151	−	0.929766i	$-0.379991\pi$
$797$	16567.0	0.736302	0.368151	+	0.929766i	$0.379991\pi$
$798$	0	0
$799$	−13418.1	−0.594115
$800$	1122.62	0.0496133
$801$	8247.96	0.363829
$802$	−2589.63	−0.114019
$803$	−3068.72	−0.134860
$804$	17510.3	0.768087
$805$	0	0
$806$	−3786.19	−0.165463
$807$	−14324.9	−0.624859
$808$	−5191.20	−0.226022
$809$	−12141.6	−0.527657	−0.263828	−	0.964570i	$-0.584985\pi$
$809$	−12141.6	−0.527657	−0.263828	+	0.964570i	$0.584985\pi$
$810$	95.6075	0.00414729
$811$	−30295.6	−1.31174	−0.655870	−	0.754873i	$-0.727698\pi$
$811$	−30295.6	−1.31174	−0.655870	+	0.754873i	$0.727698\pi$
$812$	0	0
$813$	425.726	0.0183651
$814$	−1854.73	−0.0798629
$815$	−2455.54	−0.105538
$816$	14360.5	0.616077
$817$	−1465.67	−0.0627628
$818$	94.5106	0.00403971
$819$	0	0
$820$	−6231.70	−0.265391
$821$	−14914.8	−0.634018	−0.317009	−	0.948423i	$-0.602678\pi$
$821$	−14914.8	−0.634018	−0.317009	+	0.948423i	$0.602678\pi$
$822$	−1296.81	−0.0550260
$823$	−31077.6	−1.31628	−0.658138	−	0.752897i	$-0.728656\pi$
$823$	−31077.6	−1.31628	−0.658138	+	0.752897i	$0.728656\pi$
$824$	−4924.85	−0.208210
$825$	3785.41	0.159747
$826$	0	0
$827$	15527.8	0.652908	0.326454	−	0.945213i	$-0.394146\pi$
$827$	15527.8	0.652908	0.326454	+	0.945213i	$0.394146\pi$
$828$	14637.3	0.614348
$829$	40221.5	1.68510	0.842551	−	0.538617i	$-0.181053\pi$
$829$	40221.5	1.68510	0.842551	+	0.538617i	$0.181053\pi$
$830$	−137.120	−0.00573436
$831$	10863.4	0.453486
$832$	−39734.2	−1.65569
$833$	0	0
$834$	−2160.62	−0.0897075
$835$	4130.29	0.171179
$836$	−1659.10	−0.0686377
$837$	5348.13	0.220858
$838$	3733.51	0.153904
$839$	21153.7	0.870448	0.435224	−	0.900322i	$-0.356669\pi$
$839$	21153.7	0.870448	0.435224	+	0.900322i	$0.356669\pi$
$840$	0	0
$841$	−16077.5	−0.659213
$842$	457.152	0.0187108
$843$	−17372.6	−0.709780
$844$	−8184.10	−0.333778
$845$	21796.2	0.887351
$846$	373.208	0.0151669
$847$	0	0
$848$	12552.0	0.508301
$849$	1578.13	0.0637942
$850$	−450.813	−0.0181915
$851$	−31868.1	−1.28369
$852$	−16172.4	−0.650302
$853$	−636.075	−0.0255320	−0.0127660	−	0.999919i	$-0.504064\pi$
$853$	−636.075	−0.0255320	−0.0127660	+	0.999919i	$0.504064\pi$
$854$	0	0
$855$	−186.200	−0.00744782
$856$	−4763.83	−0.190215
$857$	3941.46	0.157104	0.0785518	−	0.996910i	$-0.474970\pi$
$857$	3941.46	0.157104	0.0785518	+	0.996910i	$0.474970\pi$
$858$	2894.26	0.115161
$859$	21781.6	0.865165	0.432583	−	0.901594i	$-0.357602\pi$
$859$	21781.6	0.865165	0.432583	+	0.901594i	$0.357602\pi$
$860$	−14070.0	−0.557886
$861$	0	0
$862$	479.432	0.0189438
$863$	−44697.9	−1.76308	−0.881538	−	0.472112i	$-0.843492\pi$
$863$	−44697.9	−1.76308	−0.881538	+	0.472112i	$0.843492\pi$
$864$	−1212.43	−0.0477404
$865$	−14590.0	−0.573498
$866$	2545.79	0.0998953
$867$	−2765.92	−0.108345
$868$	0	0
$869$	64920.1	2.53425
$870$	322.825	0.0125802
$871$	59490.3	2.31430
$872$	−7788.78	−0.302478
$873$	12746.7	0.494171
$874$	199.971	0.00773926
$875$	0	0
$876$	1449.04	0.0558888
$877$	−20171.7	−0.776682	−0.388341	−	0.921516i	$-0.626952\pi$
$877$	−20171.7	−0.776682	−0.388341	+	0.921516i	$0.626952\pi$
$878$	1488.22	0.0572038
$879$	5744.63	0.220434
$880$	−15814.3	−0.605797
$881$	−11577.6	−0.442744	−0.221372	−	0.975189i	$-0.571054\pi$
$881$	−11577.6	−0.442744	−0.221372	+	0.975189i	$0.571054\pi$
$882$	0	0
$883$	−35388.3	−1.34871	−0.674355	−	0.738407i	$-0.735578\pi$
$883$	−35388.3	−1.34871	−0.674355	+	0.738407i	$0.735578\pi$
$884$	49136.1	1.86949
$885$	−4687.48	−0.178043
$886$	3657.83	0.138699
$887$	41705.0	1.57871	0.789356	−	0.613936i	$-0.210415\pi$
$887$	41705.0	1.57871	0.789356	+	0.613936i	$0.210415\pi$
$888$	1757.74	0.0664257
$889$	0	0
$890$	1081.71	0.0407405
$891$	−4088.24	−0.153716
$892$	42650.2	1.60093
$893$	−726.838	−0.0272371
$894$	−1591.02	−0.0595207
$895$	4778.72	0.178475
$896$	0	0
$897$	49729.2	1.85107
$898$	−57.1328	−0.00212310
$899$	18058.3	0.669942
$900$	−1787.46	−0.0662023
$901$	−15300.5	−0.565740
$902$	−1869.27	−0.0690020
$903$	0	0
$904$	7354.31	0.270576
$905$	−1030.40	−0.0378472
$906$	928.830	0.0340600
$907$	−28645.1	−1.04867	−0.524336	−	0.851511i	$-0.675687\pi$
$907$	−28645.1	−1.04867	−0.524336	+	0.851511i	$0.675687\pi$
$908$	−7405.02	−0.270643
$909$	12412.8	0.452921
$910$	0	0
$911$	13337.8	0.485074	0.242537	−	0.970142i	$-0.422020\pi$
$911$	13337.8	0.485074	0.242537	+	0.970142i	$0.422020\pi$
$912$	777.887	0.0282439
$913$	5863.36	0.212540
$914$	−2755.00	−0.0997016
$915$	−2312.55	−0.0835527
$916$	25128.1	0.906394
$917$	0	0
$918$	486.878	0.0175048
$919$	−28911.0	−1.03774	−0.518871	−	0.854853i	$-0.673647\pi$
$919$	−28911.0	−1.03774	−0.518871	+	0.854853i	$0.673647\pi$
$920$	3852.79	0.138068
$921$	−18732.5	−0.670203
$922$	−2863.06	−0.102267
$923$	−54944.8	−1.95940
$924$	0	0
$925$	3891.64	0.138331
$926$	1218.50	0.0432422
$927$	11775.9	0.417229
$928$	−4093.84	−0.144814
$929$	−7093.88	−0.250530	−0.125265	−	0.992123i	$-0.539978\pi$
$929$	−7093.88	−0.250530	−0.125265	+	0.992123i	$0.539978\pi$
$930$	701.401	0.0247310
$931$	0	0
$932$	−3468.17	−0.121892
$933$	−28976.3	−1.01677
$934$	−2757.33	−0.0965981
$935$	19277.1	0.674254
$936$	−2742.91	−0.0957851
$937$	19271.1	0.671888	0.335944	−	0.941882i	$-0.390945\pi$
$937$	19271.1	0.671888	0.335944	+	0.941882i	$0.390945\pi$
$938$	0	0
$939$	6595.01	0.229201
$940$	−6977.43	−0.242105
$941$	18115.2	0.627563	0.313782	−	0.949495i	$-0.398404\pi$
$941$	18115.2	0.627563	0.313782	+	0.949495i	$0.398404\pi$
$942$	1268.35	0.0438695
$943$	−32117.8	−1.10912
$944$	19582.9	0.675180
$945$	0	0
$946$	−4220.44	−0.145051
$947$	−2475.55	−0.0849468	−0.0424734	−	0.999098i	$-0.513524\pi$
$947$	−2475.55	−0.0849468	−0.0424734	+	0.999098i	$0.513524\pi$
$948$	−30655.1	−1.05024
$949$	4923.04	0.168397
$950$	−24.4199	−0.000833984 0
$951$	−9092.35	−0.310031
$952$	0	0
$953$	12866.6	0.437344	0.218672	−	0.975798i	$-0.429827\pi$
$953$	12866.6	0.437344	0.218672	+	0.975798i	$0.429827\pi$
$954$	425.564	0.0144425
$955$	−10093.2	−0.341997
$956$	15736.2	0.532368
$957$	−13804.2	−0.466277
$958$	4357.50	0.146957
$959$	0	0
$960$	7360.87	0.247470
$961$	9444.26	0.317017
$962$	2975.48	0.0997228
$963$	11390.9	0.381169
$964$	42132.9	1.40768
$965$	5158.45	0.172079
$966$	0	0
$967$	−2142.86	−0.0712614	−0.0356307	−	0.999365i	$-0.511344\pi$
$967$	−2142.86	−0.0712614	−0.0356307	+	0.999365i	$0.511344\pi$
$968$	−4578.58	−0.152026
$969$	−948.215	−0.0314356
$970$	1671.72	0.0553357
$971$	−2879.06	−0.0951529	−0.0475765	−	0.998868i	$-0.515150\pi$
$971$	−2879.06	−0.0951529	−0.0475765	+	0.998868i	$0.515150\pi$
$972$	1930.46	0.0637032
$973$	0	0
$974$	2037.29	0.0670217
$975$	−6072.79	−0.199472
$976$	9661.18	0.316851
$977$	48741.1	1.59607	0.798037	−	0.602608i	$-0.205872\pi$
$977$	48741.1	1.59607	0.798037	+	0.602608i	$0.205872\pi$
$978$	347.805	0.0113718
$979$	−46254.7	−1.51002
$980$	0	0
$981$	18623.9	0.606131
$982$	4202.40	0.136562
$983$	45756.8	1.48466	0.742328	−	0.670037i	$-0.233722\pi$
$983$	45756.8	1.48466	0.742328	+	0.670037i	$0.233722\pi$
$984$	1771.52	0.0573922
$985$	−1029.78	−0.0333110
$986$	1643.97	0.0530982
$987$	0	0
$988$	2661.63	0.0857062
$989$	−72515.7	−2.33151
$990$	−536.168	−0.0172127
$991$	−51552.1	−1.65248	−0.826240	−	0.563319i	$-0.809524\pi$
$991$	−51552.1	−1.65248	−0.826240	+	0.563319i	$0.809524\pi$
$992$	−8894.70	−0.284684
$993$	−14261.2	−0.455756
$994$	0	0
$995$	9156.88	0.291751
$996$	−2768.67	−0.0880808
$997$	−25565.3	−0.812097	−0.406048	−	0.913852i	$-0.633094\pi$
$997$	−25565.3	−0.812097	−0.406048	+	0.913852i	$0.633094\pi$
$998$	47.2531	0.00149877
$999$	−4202.97	−0.133109

Twists

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

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

\(f(q)\)	\(=\)	\(q+0.236068 q^{2} -3.00000 q^{3} -7.94427 q^{4} +5.00000 q^{5} -0.708204 q^{6} -3.76393 q^{8} +9.00000 q^{9} +1.18034 q^{10} -50.4721 q^{11} +23.8328 q^{12} +80.9706 q^{13} -15.0000 q^{15} +62.6656 q^{16} -76.3870 q^{17} +2.12461 q^{18} -4.13777 q^{19} -39.7214 q^{20} -11.9149 q^{22} -204.721 q^{23} +11.2918 q^{24} +25.0000 q^{25} +19.1146 q^{26} -27.0000 q^{27} -91.1672 q^{29} -3.54102 q^{30} -198.079 q^{31} +44.9048 q^{32} +151.416 q^{33} -18.0325 q^{34} -71.4984 q^{36} +155.666 q^{37} -0.976794 q^{38} -242.912 q^{39} -18.8197 q^{40} +156.885 q^{41} +354.217 q^{43} +400.964 q^{44} +45.0000 q^{45} -48.3282 q^{46} +175.659 q^{47} -187.997 q^{48} +5.90170 q^{50} +229.161 q^{51} -643.252 q^{52} +200.302 q^{53} -6.37384 q^{54} -252.361 q^{55} +12.4133 q^{57} -21.5217 q^{58} +312.498 q^{59} +119.164 q^{60} +154.170 q^{61} -46.7601 q^{62} -490.724 q^{64} +404.853 q^{65} +35.7446 q^{66} +734.715 q^{67} +606.839 q^{68} +614.164 q^{69} -678.577 q^{71} -33.8754 q^{72} +60.8003 q^{73} +36.7477 q^{74} -75.0000 q^{75} +32.8715 q^{76} -57.3437 q^{78} -1286.26 q^{79} +313.328 q^{80} +81.0000 q^{81} +37.0356 q^{82} -116.170 q^{83} -381.935 q^{85} +83.6192 q^{86} +273.502 q^{87} +189.974 q^{88} +916.440 q^{89} +10.6231 q^{90} +1626.36 q^{92} +594.237 q^{93} +41.4676 q^{94} -20.6888 q^{95} -134.714 q^{96} +1416.30 q^{97} -454.249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 6 q^{3} + 2 q^{4} + 10 q^{5} + 12 q^{6} - 12 q^{8} + 18 q^{9} - 20 q^{10} - 92 q^{11} - 6 q^{12} - 8 q^{13} - 30 q^{15} + 18 q^{16} + 44 q^{17} - 36 q^{18} + 108 q^{19} + 10 q^{20} + 164 q^{22}+ \cdots - 828 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 6 * q^3 + 2 * q^4 + 10 * q^5 + 12 * q^6 - 12 * q^8 + 18 * q^9 - 20 * q^10 - 92 * q^11 - 6 * q^12 - 8 * q^13 - 30 * q^15 + 18 * q^16 + 44 * q^17 - 36 * q^18 + 108 * q^19 + 10 * q^20 + 164 * q^22 - 320 * q^23 + 36 * q^24 + 50 * q^25 + 396 * q^26 - 54 * q^27 - 236 * q^29 + 60 * q^30 + 60 * q^31 + 300 * q^32 + 276 * q^33 - 528 * q^34 + 18 * q^36 + 204 * q^37 - 476 * q^38 + 24 * q^39 - 60 * q^40 - 44 * q^41 + 136 * q^43 - 12 * q^44 + 90 * q^45 + 440 * q^46 - 400 * q^47 - 54 * q^48 - 100 * q^50 - 132 * q^51 - 1528 * q^52 + 16 * q^53 + 108 * q^54 - 460 * q^55 - 324 * q^57 + 592 * q^58 + 464 * q^59 - 30 * q^60 + 684 * q^61 - 1140 * q^62 - 1214 * q^64 - 40 * q^65 - 492 * q^66 + 736 * q^67 + 1804 * q^68 + 960 * q^69 - 740 * q^71 - 108 * q^72 - 424 * q^73 - 168 * q^74 - 150 * q^75 + 1148 * q^76 - 1188 * q^78 - 408 * q^79 + 90 * q^80 + 162 * q^81 + 888 * q^82 - 608 * q^83 + 220 * q^85 + 1008 * q^86 + 708 * q^87 + 532 * q^88 + 1332 * q^89 - 180 * q^90 + 480 * q^92 - 180 * q^93 + 2480 * q^94 + 540 * q^95 - 900 * q^96 + 2448 * q^97 - 828 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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