LMFDB - Embedded newform 1150.4.a.s.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1150,4,Mod(1,1150)] 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("1150.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-10,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 38x^{3} + 38x^{2} + 202x + 101 $ x^5 - x^4 - 38x^3 + 38x^2 + 202*x + 101
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.55548$ of defining polynomial
Character	$\chi$	$=$	1150.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +10.0533 q^{3} +4.00000 q^{4} -20.1065 q^{6} +0.670353 q^{7} -8.00000 q^{8} +74.0682 q^{9} -15.7157 q^{11} +40.2131 q^{12} +50.2505 q^{13} -1.34071 q^{14} +16.0000 q^{16} +43.0881 q^{17} -148.136 q^{18} +86.7556 q^{19} +6.73924 q^{21} +31.4313 q^{22} -23.0000 q^{23} -80.4262 q^{24} -100.501 q^{26} +473.190 q^{27} +2.68141 q^{28} +147.078 q^{29} -121.171 q^{31} -32.0000 q^{32} -157.994 q^{33} -86.1762 q^{34} +296.273 q^{36} -241.489 q^{37} -173.511 q^{38} +505.181 q^{39} +232.308 q^{41} -13.4785 q^{42} -302.817 q^{43} -62.8627 q^{44} +46.0000 q^{46} -252.268 q^{47} +160.852 q^{48} -342.551 q^{49} +433.176 q^{51} +201.002 q^{52} +58.0364 q^{53} -946.380 q^{54} -5.36282 q^{56} +872.177 q^{57} -294.155 q^{58} -375.545 q^{59} +353.755 q^{61} +242.342 q^{62} +49.6518 q^{63} +64.0000 q^{64} +315.988 q^{66} +750.780 q^{67} +172.352 q^{68} -231.225 q^{69} +29.4436 q^{71} -592.546 q^{72} +80.5591 q^{73} +482.978 q^{74} +347.022 q^{76} -10.5350 q^{77} -1010.36 q^{78} -607.461 q^{79} +2757.26 q^{81} -464.616 q^{82} +1235.22 q^{83} +26.9569 q^{84} +605.635 q^{86} +1478.61 q^{87} +125.725 q^{88} +903.263 q^{89} +33.6855 q^{91} -92.0000 q^{92} -1218.17 q^{93} +504.536 q^{94} -321.705 q^{96} +1139.44 q^{97} +685.101 q^{98} -1164.03 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{2} + 12 q^{3} + 20 q^{4} - 24 q^{6} - 24 q^{7} - 40 q^{8} + 11 q^{9} - 54 q^{11} + 48 q^{12} + 36 q^{13} + 48 q^{14} + 80 q^{16} + 132 q^{17} - 22 q^{18} - 50 q^{19} - 158 q^{21} + 108 q^{22}+ \cdots - 1740 q^{99}+O(q^{100}) $ 5 * q - 10 * q^2 + 12 * q^3 + 20 * q^4 - 24 * q^6 - 24 * q^7 - 40 * q^8 + 11 * q^9 - 54 * q^11 + 48 * q^12 + 36 * q^13 + 48 * q^14 + 80 * q^16 + 132 * q^17 - 22 * q^18 - 50 * q^19 - 158 * q^21 + 108 * q^22 - 115 * q^23 - 96 * q^24 - 72 * q^26 + 489 * q^27 - 96 * q^28 - 104 * q^29 - 342 * q^31 - 160 * q^32 - 348 * q^33 - 264 * q^34 + 44 * q^36 + 380 * q^37 + 100 * q^38 + 193 * q^39 + 2 * q^41 + 316 * q^42 - 114 * q^43 - 216 * q^44 + 230 * q^46 + 966 * q^47 + 192 * q^48 - 441 * q^49 + 1200 * q^51 + 144 * q^52 + 308 * q^53 - 978 * q^54 + 192 * q^56 + 1410 * q^57 + 208 * q^58 - 277 * q^59 + 482 * q^61 + 684 * q^62 - 222 * q^63 + 320 * q^64 + 696 * q^66 + 2044 * q^67 + 528 * q^68 - 276 * q^69 + 208 * q^71 - 88 * q^72 + 646 * q^73 - 760 * q^74 - 200 * q^76 + 2124 * q^77 - 386 * q^78 - 3444 * q^79 + 2933 * q^81 - 4 * q^82 + 1218 * q^83 - 632 * q^84 + 228 * q^86 + 103 * q^87 + 432 * q^88 + 140 * q^89 + 1976 * q^91 - 460 * q^92 + 473 * q^93 - 1932 * q^94 - 384 * q^96 + 2816 * q^97 + 882 * q^98 - 1740 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	10.0533	1.93475	0.967376	−	0.253343i	$-0.0815303\pi$
$3$	10.0533	1.93475	0.967376	+	0.253343i	$0.0815303\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−20.1065	−1.36808
$7$	0.670353	0.0361956	0.0180978	−	0.999836i	$-0.494239\pi$
$7$	0.670353	0.0361956	0.0180978	+	0.999836i	$0.494239\pi$
$8$	−8.00000	−0.353553
$9$	74.0682	2.74327
$10$	0	0
$11$	−15.7157	−0.430768	−0.215384	−	0.976529i	$-0.569100\pi$
$11$	−15.7157	−0.430768	−0.215384	+	0.976529i	$0.569100\pi$
$12$	40.2131	0.967376
$13$	50.2505	1.07207	0.536037	−	0.844194i	$-0.319921\pi$
$13$	50.2505	1.07207	0.536037	+	0.844194i	$0.319921\pi$
$14$	−1.34071	−0.0255942
$15$	0	0
$16$	16.0000	0.250000
$17$	43.0881	0.614729	0.307365	−	0.951592i	$-0.400553\pi$
$17$	43.0881	0.614729	0.307365	+	0.951592i	$0.400553\pi$
$18$	−148.136	−1.93978
$19$	86.7556	1.04753	0.523766	−	0.851862i	$-0.324527\pi$
$19$	86.7556	1.04753	0.523766	+	0.851862i	$0.324527\pi$
$20$	0	0
$21$	6.73924	0.0700296
$22$	31.4313	0.304599
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−80.4262	−0.684038
$25$	0	0
$26$	−100.501	−0.758071
$27$	473.190	3.37279
$28$	2.68141	0.0180978
$29$	147.078	0.941780	0.470890	−	0.882192i	$-0.343933\pi$
$29$	147.078	0.941780	0.470890	+	0.882192i	$0.343933\pi$
$30$	0	0
$31$	−121.171	−0.702032	−0.351016	−	0.936370i	$-0.614164\pi$
$31$	−121.171	−0.702032	−0.351016	+	0.936370i	$0.614164\pi$
$32$	−32.0000	−0.176777
$33$	−157.994	−0.833430
$34$	−86.1762	−0.434679
$35$	0	0
$36$	296.273	1.37163
$37$	−241.489	−1.07299	−0.536493	−	0.843905i	$-0.680251\pi$
$37$	−241.489	−1.07299	−0.536493	+	0.843905i	$0.680251\pi$
$38$	−173.511	−0.740716
$39$	505.181	2.07420
$40$	0	0
$41$	232.308	0.884888	0.442444	−	0.896796i	$-0.354112\pi$
$41$	232.308	0.884888	0.442444	+	0.896796i	$0.354112\pi$
$42$	−13.4785	−0.0495184
$43$	−302.817	−1.07394	−0.536968	−	0.843603i	$-0.680430\pi$
$43$	−302.817	−1.07394	−0.536968	+	0.843603i	$0.680430\pi$
$44$	−62.8627	−0.215384
$45$	0	0
$46$	46.0000	0.147442
$47$	−252.268	−0.782916	−0.391458	−	0.920196i	$-0.628029\pi$
$47$	−252.268	−0.782916	−0.391458	+	0.920196i	$0.628029\pi$
$48$	160.852	0.483688
$49$	−342.551	−0.998690
$50$	0	0
$51$	433.176	1.18935
$52$	201.002	0.536037
$53$	58.0364	0.150413	0.0752067	−	0.997168i	$-0.476038\pi$
$53$	58.0364	0.150413	0.0752067	+	0.997168i	$0.476038\pi$
$54$	−946.380	−2.38493
$55$	0	0
$56$	−5.36282	−0.0127971
$57$	872.177	2.02671
$58$	−294.155	−0.665939
$59$	−375.545	−0.828675	−0.414337	−	0.910123i	$-0.635987\pi$
$59$	−375.545	−0.828675	−0.414337	+	0.910123i	$0.635987\pi$
$60$	0	0
$61$	353.755	0.742519	0.371259	−	0.928529i	$-0.378926\pi$
$61$	353.755	0.742519	0.371259	+	0.928529i	$0.378926\pi$
$62$	242.342	0.496411
$63$	49.6518	0.0992944
$64$	64.0000	0.125000
$65$	0	0
$66$	315.988	0.589324
$67$	750.780	1.36899	0.684495	−	0.729017i	$-0.260023\pi$
$67$	750.780	1.36899	0.684495	+	0.729017i	$0.260023\pi$
$68$	172.352	0.307365
$69$	−231.225	−0.403424
$70$	0	0
$71$	29.4436	0.0492156	0.0246078	−	0.999697i	$-0.492166\pi$
$71$	29.4436	0.0492156	0.0246078	+	0.999697i	$0.492166\pi$
$72$	−592.546	−0.969892
$73$	80.5591	0.129161	0.0645803	−	0.997913i	$-0.479429\pi$
$73$	80.5591	0.129161	0.0645803	+	0.997913i	$0.479429\pi$
$74$	482.978	0.758716
$75$	0	0
$76$	347.022	0.523766
$77$	−10.5350	−0.0155919
$78$	−1010.36	−1.46668
$79$	−607.461	−0.865122	−0.432561	−	0.901605i	$-0.642390\pi$
$79$	−607.461	−0.865122	−0.432561	+	0.901605i	$0.642390\pi$
$80$	0	0
$81$	2757.26	3.78225
$82$	−464.616	−0.625710
$83$	1235.22	1.63354	0.816768	−	0.576967i	$-0.195764\pi$
$83$	1235.22	1.63354	0.816768	+	0.576967i	$0.195764\pi$
$84$	26.9569	0.0350148
$85$	0	0
$86$	605.635	0.759387
$87$	1478.61	1.82211
$88$	125.725	0.152300
$89$	903.263	1.07579	0.537897	−	0.843010i	$-0.319219\pi$
$89$	903.263	1.07579	0.537897	+	0.843010i	$0.319219\pi$
$90$	0	0
$91$	33.6855	0.0388044
$92$	−92.0000	−0.104257
$93$	−1218.17	−1.35826
$94$	504.536	0.553606
$95$	0	0
$96$	−321.705	−0.342019
$97$	1139.44	1.19270	0.596351	−	0.802724i	$-0.296617\pi$
$97$	1139.44	1.19270	0.596351	+	0.802724i	$0.296617\pi$
$98$	685.101	0.706180
$99$	−1164.03	−1.18171
$100$	0	0
$101$	−241.262	−0.237688	−0.118844	−	0.992913i	$-0.537919\pi$
$101$	−241.262	−0.237688	−0.118844	+	0.992913i	$0.537919\pi$
$102$	−866.353	−0.840997
$103$	−304.633	−0.291421	−0.145711	−	0.989327i	$-0.546547\pi$
$103$	−304.633	−0.291421	−0.145711	+	0.989327i	$0.546547\pi$
$104$	−402.004	−0.379036
$105$	0	0
$106$	−116.073	−0.106358
$107$	1660.06	1.49985	0.749927	−	0.661521i	$-0.230089\pi$
$107$	1660.06	1.49985	0.749927	+	0.661521i	$0.230089\pi$
$108$	1892.76	1.68640
$109$	603.058	0.529931	0.264965	−	0.964258i	$-0.414639\pi$
$109$	603.058	0.529931	0.264965	+	0.964258i	$0.414639\pi$
$110$	0	0
$111$	−2427.75	−2.07596
$112$	10.7256	0.00904891
$113$	1231.01	1.02481	0.512404	−	0.858744i	$-0.328755\pi$
$113$	1231.01	1.02481	0.512404	+	0.858744i	$0.328755\pi$
$114$	−1744.35	−1.43310
$115$	0	0
$116$	588.310	0.470890
$117$	3721.96	2.94099
$118$	751.090	0.585961
$119$	28.8842	0.0222505
$120$	0	0
$121$	−1084.02	−0.814439
$122$	−707.510	−0.525040
$123$	2335.46	1.71204
$124$	−484.685	−0.351016
$125$	0	0
$126$	−99.3037	−0.0702117
$127$	−2793.94	−1.95214	−0.976071	−	0.217453i	$-0.930225\pi$
$127$	−2793.94	−1.95214	−0.976071	+	0.217453i	$0.930225\pi$
$128$	−128.000	−0.0883883
$129$	−3044.31	−2.07780
$130$	0	0
$131$	−1476.50	−0.984753	−0.492376	−	0.870382i	$-0.663872\pi$
$131$	−1476.50	−0.984753	−0.492376	+	0.870382i	$0.663872\pi$
$132$	−631.975	−0.416715
$133$	58.1568	0.0379161
$134$	−1501.56	−0.968022
$135$	0	0
$136$	−344.705	−0.217340
$137$	1955.43	1.21944	0.609720	−	0.792617i	$-0.291282\pi$
$137$	1955.43	1.21944	0.609720	+	0.792617i	$0.291282\pi$
$138$	462.450	0.285264
$139$	−3211.77	−1.95985	−0.979923	−	0.199375i	$-0.936109\pi$
$139$	−3211.77	−1.95985	−0.979923	+	0.199375i	$0.936109\pi$
$140$	0	0
$141$	−2536.12	−1.51475
$142$	−58.8871	−0.0348007
$143$	−789.719	−0.461816
$144$	1185.09	0.685817
$145$	0	0
$146$	−161.118	−0.0913303
$147$	−3443.75	−1.93222
$148$	−965.955	−0.536493
$149$	3016.92	1.65876	0.829382	−	0.558682i	$-0.188693\pi$
$149$	3016.92	1.65876	0.829382	+	0.558682i	$0.188693\pi$
$150$	0	0
$151$	−648.140	−0.349304	−0.174652	−	0.984630i	$-0.555880\pi$
$151$	−648.140	−0.349304	−0.174652	+	0.984630i	$0.555880\pi$
$152$	−694.044	−0.370358
$153$	3191.46	1.68637
$154$	21.0701	0.0110252
$155$	0	0
$156$	2020.73	1.03710
$157$	2846.40	1.44693	0.723463	−	0.690363i	$-0.242549\pi$
$157$	2846.40	1.44693	0.723463	+	0.690363i	$0.242549\pi$
$158$	1214.92	0.611734
$159$	583.456	0.291013
$160$	0	0
$161$	−15.4181	−0.00754731
$162$	−5514.53	−2.67446
$163$	592.664	0.284792	0.142396	−	0.989810i	$-0.454519\pi$
$163$	592.664	0.284792	0.142396	+	0.989810i	$0.454519\pi$
$164$	929.232	0.442444
$165$	0	0
$166$	−2470.45	−1.15508
$167$	−0.834417	−0.000386641 0	−0.000193321	−	1.00000i	$-0.500062\pi$
$167$	−0.834417	−0.000386641 0	−0.000193321		1.00000i	$0.500062\pi$
$168$	−53.9139	−0.0247592
$169$	328.109	0.149344
$170$	0	0
$171$	6425.83	2.87366
$172$	−1211.27	−0.536968
$173$	885.670	0.389227	0.194613	−	0.980880i	$-0.437655\pi$
$173$	885.670	0.389227	0.194613	+	0.980880i	$0.437655\pi$
$174$	−2957.22	−1.28843
$175$	0	0
$176$	−251.451	−0.107692
$177$	−3775.46	−1.60328
$178$	−1806.53	−0.760702
$179$	−3354.45	−1.40069	−0.700344	−	0.713805i	$-0.746970\pi$
$179$	−3354.45	−1.40069	−0.700344	+	0.713805i	$0.746970\pi$
$180$	0	0
$181$	79.3370	0.0325805	0.0162903	−	0.999867i	$-0.494814\pi$
$181$	79.3370	0.0325805	0.0162903	+	0.999867i	$0.494814\pi$
$182$	−67.3711	−0.0274389
$183$	3556.39	1.43659
$184$	184.000	0.0737210
$185$	0	0
$186$	2436.33	0.960433
$187$	−677.158	−0.264806
$188$	−1009.07	−0.391458
$189$	317.204	0.122080
$190$	0	0
$191$	−3240.80	−1.22773	−0.613863	−	0.789412i	$-0.710385\pi$
$191$	−3240.80	−1.22773	−0.613863	+	0.789412i	$0.710385\pi$
$192$	643.409	0.241844
$193$	−4713.15	−1.75782	−0.878912	−	0.476983i	$-0.841730\pi$
$193$	−4713.15	−1.75782	−0.878912	+	0.476983i	$0.841730\pi$
$194$	−2278.87	−0.843368
$195$	0	0
$196$	−1370.20	−0.499345
$197$	−1507.85	−0.545328	−0.272664	−	0.962109i	$-0.587905\pi$
$197$	−1507.85	−0.545328	−0.272664	+	0.962109i	$0.587905\pi$
$198$	2328.06	0.835597
$199$	−231.005	−0.0822891	−0.0411446	−	0.999153i	$-0.513100\pi$
$199$	−231.005	−0.0822891	−0.0411446	+	0.999153i	$0.513100\pi$
$200$	0	0
$201$	7547.79	2.64866
$202$	482.524	0.168071
$203$	98.5939	0.0340883
$204$	1732.71	0.594675
$205$	0	0
$206$	609.266	0.206066
$207$	−1703.57	−0.572011
$208$	804.007	0.268019
$209$	−1363.42	−0.451243
$210$	0	0
$211$	−995.976	−0.324957	−0.162478	−	0.986712i	$-0.551949\pi$
$211$	−995.976	−0.324957	−0.162478	+	0.986712i	$0.551949\pi$
$212$	232.146	0.0752067
$213$	296.004	0.0952200
$214$	−3320.13	−1.06056
$215$	0	0
$216$	−3785.52	−1.19246
$217$	−81.2274	−0.0254105
$218$	−1206.12	−0.374718
$219$	809.882	0.249894
$220$	0	0
$221$	2165.20	0.659036
$222$	4855.50	1.46793
$223$	5412.50	1.62533	0.812663	−	0.582734i	$-0.198017\pi$
$223$	5412.50	1.62533	0.812663	+	0.582734i	$0.198017\pi$
$224$	−21.4513	−0.00639855
$225$	0	0
$226$	−2462.01	−0.724649
$227$	3972.78	1.16160	0.580799	−	0.814047i	$-0.302740\pi$
$227$	3972.78	1.16160	0.580799	+	0.814047i	$0.302740\pi$
$228$	3488.71	1.01336
$229$	5209.62	1.50333	0.751663	−	0.659548i	$-0.229252\pi$
$229$	5209.62	1.50333	0.751663	+	0.659548i	$0.229252\pi$
$230$	0	0
$231$	−105.912	−0.0301665
$232$	−1176.62	−0.332970
$233$	−1183.13	−0.332658	−0.166329	−	0.986070i	$-0.553191\pi$
$233$	−1183.13	−0.332658	−0.166329	+	0.986070i	$0.553191\pi$
$234$	−7443.93	−2.07959
$235$	0	0
$236$	−1502.18	−0.414337
$237$	−6106.96	−1.67380
$238$	−57.7684	−0.0157335
$239$	3759.05	1.01737	0.508687	−	0.860951i	$-0.330131\pi$
$239$	3759.05	1.01737	0.508687	+	0.860951i	$0.330131\pi$
$240$	0	0
$241$	−5148.34	−1.37607	−0.688037	−	0.725675i	$-0.741527\pi$
$241$	−5148.34	−1.37607	−0.688037	+	0.725675i	$0.741527\pi$
$242$	2168.04	0.575895
$243$	14943.4	3.94493
$244$	1415.02	0.371259
$245$	0	0
$246$	−4670.91	−1.21059
$247$	4359.51	1.12303
$248$	969.369	0.248206
$249$	12418.0	3.16049
$250$	0	0
$251$	−4522.82	−1.13736	−0.568682	−	0.822558i	$-0.692546\pi$
$251$	−4522.82	−1.13736	−0.568682	+	0.822558i	$0.692546\pi$
$252$	198.607	0.0496472
$253$	361.460	0.0898214
$254$	5587.88	1.38037
$255$	0	0
$256$	256.000	0.0625000
$257$	−3690.30	−0.895698	−0.447849	−	0.894109i	$-0.647810\pi$
$257$	−3690.30	−0.895698	−0.447849	+	0.894109i	$0.647810\pi$
$258$	6088.61	1.46923
$259$	−161.883	−0.0388374
$260$	0	0
$261$	10893.8	2.58356
$262$	2953.00	0.696325
$263$	−6516.16	−1.52777	−0.763885	−	0.645352i	$-0.776711\pi$
$263$	−6516.16	−1.52777	−0.763885	+	0.645352i	$0.776711\pi$
$264$	1263.95	0.294662
$265$	0	0
$266$	−116.314	−0.0268107
$267$	9080.75	2.08140
$268$	3003.12	0.684495
$269$	−1546.27	−0.350474	−0.175237	−	0.984526i	$-0.556069\pi$
$269$	−1546.27	−0.350474	−0.175237	+	0.984526i	$0.556069\pi$
$270$	0	0
$271$	6420.51	1.43918	0.719590	−	0.694399i	$-0.244330\pi$
$271$	6420.51	1.43918	0.719590	+	0.694399i	$0.244330\pi$
$272$	689.410	0.153682
$273$	338.650	0.0750770
$274$	−3910.85	−0.862275
$275$	0	0
$276$	−924.901	−0.201712
$277$	5084.94	1.10298	0.551489	−	0.834182i	$-0.314060\pi$
$277$	5084.94	1.10298	0.551489	+	0.834182i	$0.314060\pi$
$278$	6423.54	1.38582
$279$	−8974.94	−1.92586
$280$	0	0
$281$	−1261.35	−0.267778	−0.133889	−	0.990996i	$-0.542747\pi$
$281$	−1261.35	−0.267778	−0.133889	+	0.990996i	$0.542747\pi$
$282$	5072.24	1.07109
$283$	8129.18	1.70753	0.853763	−	0.520662i	$-0.174315\pi$
$283$	8129.18	1.70753	0.853763	+	0.520662i	$0.174315\pi$
$284$	117.774	0.0246078
$285$	0	0
$286$	1579.44	0.326553
$287$	155.728	0.0320291
$288$	−2370.18	−0.484946
$289$	−3056.42	−0.622108
$290$	0	0
$291$	11455.1	2.30758
$292$	322.236	0.0645803
$293$	−6161.19	−1.22847	−0.614233	−	0.789124i	$-0.710535\pi$
$293$	−6161.19	−1.22847	−0.614233	+	0.789124i	$0.710535\pi$
$294$	6887.51	1.36628
$295$	0	0
$296$	1931.91	0.379358
$297$	−7436.49	−1.45289
$298$	−6033.84	−1.17292
$299$	−1155.76	−0.223543
$300$	0	0
$301$	−202.994	−0.0388718
$302$	1296.28	0.246995
$303$	−2425.47	−0.459867
$304$	1388.09	0.261883
$305$	0	0
$306$	−6382.92	−1.19244
$307$	1882.96	0.350052	0.175026	−	0.984564i	$-0.443999\pi$
$307$	1882.96	0.350052	0.175026	+	0.984564i	$0.443999\pi$
$308$	−42.1401	−0.00779597
$309$	−3062.56	−0.563828
$310$	0	0
$311$	2771.91	0.505403	0.252702	−	0.967544i	$-0.418681\pi$
$311$	2771.91	0.505403	0.252702	+	0.967544i	$0.418681\pi$
$312$	−4041.45	−0.733340
$313$	−6823.98	−1.23231	−0.616157	−	0.787624i	$-0.711311\pi$
$313$	−6823.98	−1.23231	−0.616157	+	0.787624i	$0.711311\pi$
$314$	−5692.80	−1.02313
$315$	0	0
$316$	−2429.84	−0.432561
$317$	8338.17	1.47735	0.738673	−	0.674064i	$-0.235453\pi$
$317$	8338.17	1.47735	0.738673	+	0.674064i	$0.235453\pi$
$318$	−1166.91	−0.205777
$319$	−2311.42	−0.405689
$320$	0	0
$321$	16689.1	2.90185
$322$	30.8362	0.00533676
$323$	3738.13	0.643948
$324$	11029.1	1.89113
$325$	0	0
$326$	−1185.33	−0.201378
$327$	6062.70	1.02528
$328$	−1858.46	−0.312855
$329$	−169.109	−0.0283382
$330$	0	0
$331$	1157.44	0.192202	0.0961009	−	0.995372i	$-0.469363\pi$
$331$	1157.44	0.192202	0.0961009	+	0.995372i	$0.469363\pi$
$332$	4940.90	0.816768
$333$	−17886.7	−2.94349
$334$	1.66883	0.000273397 0
$335$	0	0
$336$	107.828	0.0175074
$337$	−3991.75	−0.645236	−0.322618	−	0.946529i	$-0.604563\pi$
$337$	−3991.75	−0.645236	−0.322618	+	0.946529i	$0.604563\pi$
$338$	−656.218	−0.105602
$339$	12375.6	1.98275
$340$	0	0
$341$	1904.29	0.302413
$342$	−12851.7	−2.03198
$343$	−459.561	−0.0723439
$344$	2422.54	0.379693
$345$	0	0
$346$	−1771.34	−0.275225
$347$	6517.56	1.00830	0.504151	−	0.863615i	$-0.331805\pi$
$347$	6517.56	1.00830	0.504151	+	0.863615i	$0.331805\pi$
$348$	5914.44	0.911056
$349$	−6645.39	−1.01925	−0.509627	−	0.860395i	$-0.670217\pi$
$349$	−6645.39	−1.01925	−0.509627	+	0.860395i	$0.670217\pi$
$350$	0	0
$351$	23778.0	3.61589
$352$	502.901	0.0761498
$353$	3235.49	0.487841	0.243920	−	0.969795i	$-0.421566\pi$
$353$	3235.49	0.487841	0.243920	+	0.969795i	$0.421566\pi$
$354$	7550.91	1.13369
$355$	0	0
$356$	3613.05	0.537897
$357$	290.381	0.0430493
$358$	6708.90	0.990436
$359$	−7919.19	−1.16423	−0.582115	−	0.813106i	$-0.697775\pi$
$359$	−7919.19	−1.16423	−0.582115	+	0.813106i	$0.697775\pi$
$360$	0	0
$361$	667.527	0.0973213
$362$	−158.674	−0.0230379
$363$	−10897.9	−1.57574
$364$	134.742	0.0194022
$365$	0	0
$366$	−7112.78	−1.01582
$367$	−6956.40	−0.989430	−0.494715	−	0.869055i	$-0.664728\pi$
$367$	−6956.40	−0.989430	−0.494715	+	0.869055i	$0.664728\pi$
$368$	−368.000	−0.0521286
$369$	17206.6	2.42749
$370$	0	0
$371$	38.9049	0.00544431
$372$	−4872.67	−0.679129
$373$	−9288.00	−1.28932	−0.644658	−	0.764471i	$-0.723000\pi$
$373$	−9288.00	−1.28932	−0.644658	+	0.764471i	$0.723000\pi$
$374$	1354.32	0.187246
$375$	0	0
$376$	2018.14	0.276803
$377$	7390.72	1.00966
$378$	−634.408	−0.0863239
$379$	448.987	0.0608520	0.0304260	−	0.999537i	$-0.490314\pi$
$379$	448.987	0.0608520	0.0304260	+	0.999537i	$0.490314\pi$
$380$	0	0
$381$	−28088.2	−3.77691
$382$	6481.59	0.868134
$383$	−8724.18	−1.16393	−0.581965	−	0.813214i	$-0.697716\pi$
$383$	−8724.18	−1.16393	−0.581965	+	0.813214i	$0.697716\pi$
$384$	−1286.82	−0.171010
$385$	0	0
$386$	9426.31	1.24297
$387$	−22429.2	−2.94609
$388$	4557.74	0.596351
$389$	−5852.57	−0.762820	−0.381410	−	0.924406i	$-0.624561\pi$
$389$	−5852.57	−0.762820	−0.381410	+	0.924406i	$0.624561\pi$
$390$	0	0
$391$	−991.026	−0.128180
$392$	2740.41	0.353090
$393$	−14843.7	−1.90525
$394$	3015.69	0.385605
$395$	0	0
$396$	−4656.13	−0.590856
$397$	−6890.88	−0.871142	−0.435571	−	0.900154i	$-0.643454\pi$
$397$	−6890.88	−0.871142	−0.435571	+	0.900154i	$0.643454\pi$
$398$	462.011	0.0581872
$399$	584.666	0.0733582
$400$	0	0
$401$	10326.6	1.28600	0.642999	−	0.765867i	$-0.277690\pi$
$401$	10326.6	1.28600	0.642999	+	0.765867i	$0.277690\pi$
$402$	−15095.6	−1.87288
$403$	−6088.91	−0.752630
$404$	−965.048	−0.118844
$405$	0	0
$406$	−197.188	−0.0241041
$407$	3795.16	0.462209
$408$	−3465.41	−0.420499
$409$	6508.97	0.786914	0.393457	−	0.919343i	$-0.371279\pi$
$409$	6508.97	0.786914	0.393457	+	0.919343i	$0.371279\pi$
$410$	0	0
$411$	19658.4	2.35932
$412$	−1218.53	−0.145711
$413$	−251.748	−0.0299944
$414$	3407.14	0.404473
$415$	0	0
$416$	−1608.01	−0.189518
$417$	−32288.8	−3.79182
$418$	2726.84	0.319077
$419$	−12013.9	−1.40076	−0.700380	−	0.713770i	$-0.746986\pi$
$419$	−12013.9	−1.40076	−0.700380	+	0.713770i	$0.746986\pi$
$420$	0	0
$421$	−14560.1	−1.68555	−0.842774	−	0.538267i	$-0.819079\pi$
$421$	−14560.1	−1.68555	−0.842774	+	0.538267i	$0.819079\pi$
$422$	1991.95	0.229779
$423$	−18685.1	−2.14775
$424$	−464.291	−0.0531792
$425$	0	0
$426$	−592.008	−0.0673307
$427$	237.140	0.0268759
$428$	6640.25	0.749927
$429$	−7939.26	−0.893499
$430$	0	0
$431$	−2707.44	−0.302582	−0.151291	−	0.988489i	$-0.548343\pi$
$431$	−2707.44	−0.302582	−0.151291	+	0.988489i	$0.548343\pi$
$432$	7571.04	0.843198
$433$	−2044.10	−0.226866	−0.113433	−	0.993546i	$-0.536185\pi$
$433$	−2044.10	−0.226866	−0.113433	+	0.993546i	$0.536185\pi$
$434$	162.455	0.0179679
$435$	0	0
$436$	2412.23	0.264965
$437$	−1995.38	−0.218425
$438$	−1619.76	−0.176702
$439$	−15667.9	−1.70339	−0.851694	−	0.524040i	$-0.824424\pi$
$439$	−15667.9	−1.70339	−0.851694	+	0.524040i	$0.824424\pi$
$440$	0	0
$441$	−25372.1	−2.73967
$442$	−4330.39	−0.466009
$443$	−11116.6	−1.19225	−0.596124	−	0.802892i	$-0.703293\pi$
$443$	−11116.6	−1.19225	−0.596124	+	0.802892i	$0.703293\pi$
$444$	−9711.01	−1.03798
$445$	0	0
$446$	−10825.0	−1.14928
$447$	30329.9	3.20930
$448$	42.9026	0.00452445
$449$	14892.5	1.56530	0.782649	−	0.622463i	$-0.213868\pi$
$449$	14892.5	1.56530	0.782649	+	0.622463i	$0.213868\pi$
$450$	0	0
$451$	−3650.87	−0.381182
$452$	4924.03	0.512404
$453$	−6515.92	−0.675816
$454$	−7945.56	−0.821373
$455$	0	0
$456$	−6977.42	−0.716551
$457$	5764.57	0.590055	0.295027	−	0.955489i	$-0.404671\pi$
$457$	5764.57	0.590055	0.295027	+	0.955489i	$0.404671\pi$
$458$	−10419.2	−1.06301
$459$	20388.9	2.07336
$460$	0	0
$461$	−15874.1	−1.60375	−0.801877	−	0.597489i	$-0.796165\pi$
$461$	−15874.1	−1.60375	−0.801877	+	0.597489i	$0.796165\pi$
$462$	211.823	0.0213310
$463$	3812.03	0.382635	0.191318	−	0.981528i	$-0.438724\pi$
$463$	3812.03	0.382635	0.191318	+	0.981528i	$0.438724\pi$
$464$	2353.24	0.235445
$465$	0	0
$466$	2366.26	0.235225
$467$	13541.6	1.34182	0.670912	−	0.741537i	$-0.265903\pi$
$467$	13541.6	1.34182	0.670912	+	0.741537i	$0.265903\pi$
$468$	14887.9	1.47049
$469$	503.287	0.0495515
$470$	0	0
$471$	28615.6	2.79944
$472$	3004.36	0.292981
$473$	4758.98	0.462617
$474$	12213.9	1.18355
$475$	0	0
$476$	115.537	0.0111253
$477$	4298.65	0.412625
$478$	−7518.10	−0.719393
$479$	−14788.4	−1.41064	−0.705321	−	0.708888i	$-0.749197\pi$
$479$	−14788.4	−1.41064	−0.705321	+	0.708888i	$0.749197\pi$
$480$	0	0
$481$	−12134.9	−1.15032
$482$	10296.7	0.973031
$483$	−155.002	−0.0146022
$484$	−4336.07	−0.407219
$485$	0	0
$486$	−29886.8	−2.78949
$487$	−11307.8	−1.05217	−0.526084	−	0.850432i	$-0.676340\pi$
$487$	−11307.8	−1.05217	−0.526084	+	0.850432i	$0.676340\pi$
$488$	−2830.04	−0.262520
$489$	5958.21	0.551001
$490$	0	0
$491$	4126.13	0.379246	0.189623	−	0.981857i	$-0.439273\pi$
$491$	4126.13	0.379246	0.189623	+	0.981857i	$0.439273\pi$
$492$	9341.82	0.856020
$493$	6337.29	0.578940
$494$	−8719.01	−0.794103
$495$	0	0
$496$	−1938.74	−0.175508
$497$	19.7376	0.00178139
$498$	−24836.1	−2.23480
$499$	15160.9	1.36011	0.680055	−	0.733161i	$-0.261956\pi$
$499$	15160.9	1.36011	0.680055	+	0.733161i	$0.261956\pi$
$500$	0	0
$501$	−8.38862	−0.000748056 0
$502$	9045.65	0.804237
$503$	12971.8	1.14987	0.574935	−	0.818199i	$-0.305027\pi$
$503$	12971.8	1.14987	0.574935	+	0.818199i	$0.305027\pi$
$504$	−397.215	−0.0351059
$505$	0	0
$506$	−722.921	−0.0635133
$507$	3298.57	0.288944
$508$	−11175.8	−0.976071
$509$	−8093.24	−0.704767	−0.352384	−	0.935856i	$-0.614629\pi$
$509$	−8093.24	−0.704767	−0.352384	+	0.935856i	$0.614629\pi$
$510$	0	0
$511$	54.0030	0.00467505
$512$	−512.000	−0.0441942
$513$	41051.8	3.53311
$514$	7380.59	0.633354
$515$	0	0
$516$	−12177.2	−1.03890
$517$	3964.56	0.337256
$518$	323.765	0.0274622
$519$	8903.88	0.753058
$520$	0	0
$521$	−151.052	−0.0127019	−0.00635096	−	0.999980i	$-0.502022\pi$
$521$	−151.052	−0.0127019	−0.00635096	+	0.999980i	$0.502022\pi$
$522$	−21787.6	−1.82685
$523$	−3890.59	−0.325284	−0.162642	−	0.986685i	$-0.552002\pi$
$523$	−3890.59	−0.325284	−0.162642	+	0.986685i	$0.552002\pi$
$524$	−5906.01	−0.492376
$525$	0	0
$526$	13032.3	1.08030
$527$	−5221.04	−0.431560
$528$	−2527.90	−0.208358
$529$	529.000	0.0434783
$530$	0	0
$531$	−27816.0	−2.27328
$532$	232.627	0.0189580
$533$	11673.6	0.948666
$534$	−18161.5	−1.47177
$535$	0	0
$536$	−6006.24	−0.484011
$537$	−33723.2	−2.70999
$538$	3092.53	0.247822
$539$	5383.41	0.430204
$540$	0	0
$541$	−13618.3	−1.08225	−0.541125	−	0.840942i	$-0.682001\pi$
$541$	−13618.3	−1.08225	−0.541125	+	0.840942i	$0.682001\pi$
$542$	−12841.0	−1.01765
$543$	797.597	0.0630353
$544$	−1378.82	−0.108670
$545$	0	0
$546$	−677.299	−0.0530874
$547$	11813.5	0.923420	0.461710	−	0.887031i	$-0.347236\pi$
$547$	11813.5	0.923420	0.461710	+	0.887031i	$0.347236\pi$
$548$	7821.71	0.609720
$549$	26202.0	2.03693
$550$	0	0
$551$	12759.8	0.986544
$552$	1849.80	0.142632
$553$	−407.213	−0.0313136
$554$	−10169.9	−0.779923
$555$	0	0
$556$	−12847.1	−0.979923
$557$	15929.0	1.21173	0.605864	−	0.795568i	$-0.292827\pi$
$557$	15929.0	1.21173	0.605864	+	0.795568i	$0.292827\pi$
$558$	17949.9	1.36179
$559$	−15216.7	−1.15134
$560$	0	0
$561$	−6807.65	−0.512334
$562$	2522.69	0.189348
$563$	−10630.7	−0.795790	−0.397895	−	0.917431i	$-0.630259\pi$
$563$	−10630.7	−0.795790	−0.397895	+	0.917431i	$0.630259\pi$
$564$	−10144.5	−0.757375
$565$	0	0
$566$	−16258.4	−1.20740
$567$	1848.34	0.136901
$568$	−235.548	−0.0174003
$569$	16116.7	1.18743	0.593714	−	0.804676i	$-0.297661\pi$
$569$	16116.7	1.18743	0.593714	+	0.804676i	$0.297661\pi$
$570$	0	0
$571$	−13337.7	−0.977524	−0.488762	−	0.872417i	$-0.662551\pi$
$571$	−13337.7	−0.977524	−0.488762	+	0.872417i	$0.662551\pi$
$572$	−3158.88	−0.230908
$573$	−32580.6	−2.37535
$574$	−311.457	−0.0226480
$575$	0	0
$576$	4740.37	0.342909
$577$	−4935.92	−0.356127	−0.178063	−	0.984019i	$-0.556983\pi$
$577$	−4935.92	−0.356127	−0.178063	+	0.984019i	$0.556983\pi$
$578$	6112.83	0.439897
$579$	−47382.6	−3.40096
$580$	0	0
$581$	828.036	0.0591269
$582$	−22910.1	−1.63171
$583$	−912.081	−0.0647933
$584$	−644.472	−0.0456652
$585$	0	0
$586$	12322.4	0.868657
$587$	4386.29	0.308418	0.154209	−	0.988038i	$-0.450717\pi$
$587$	4386.29	0.308418	0.154209	+	0.988038i	$0.450717\pi$
$588$	−13775.0	−0.966109
$589$	−10512.3	−0.735400
$590$	0	0
$591$	−15158.8	−1.05507
$592$	−3863.82	−0.268247
$593$	−11004.4	−0.762053	−0.381026	−	0.924564i	$-0.624429\pi$
$593$	−11004.4	−0.762053	−0.381026	+	0.924564i	$0.624429\pi$
$594$	14873.0	1.02735
$595$	0	0
$596$	12067.7	0.829382
$597$	−2322.36	−0.159209
$598$	2311.52	0.158069
$599$	9742.23	0.664536	0.332268	−	0.943185i	$-0.392186\pi$
$599$	9742.23	0.664536	0.332268	+	0.943185i	$0.392186\pi$
$600$	0	0
$601$	−13053.0	−0.885929	−0.442964	−	0.896539i	$-0.646073\pi$
$601$	−13053.0	−0.885929	−0.442964	+	0.896539i	$0.646073\pi$
$602$	405.989	0.0274865
$603$	55608.9	3.75551
$604$	−2592.56	−0.174652
$605$	0	0
$606$	4850.94	0.325175
$607$	−16313.6	−1.09085	−0.545426	−	0.838159i	$-0.683632\pi$
$607$	−16313.6	−1.09085	−0.545426	+	0.838159i	$0.683632\pi$
$608$	−2776.18	−0.185179
$609$	991.191	0.0659525
$610$	0	0
$611$	−12676.6	−0.839345
$612$	12765.8	0.843184
$613$	−22248.4	−1.46592	−0.732958	−	0.680274i	$-0.761861\pi$
$613$	−22248.4	−1.46592	−0.732958	+	0.680274i	$0.761861\pi$
$614$	−3765.92	−0.247524
$615$	0	0
$616$	84.2803	0.00551258
$617$	−7012.62	−0.457565	−0.228782	−	0.973478i	$-0.573474\pi$
$617$	−7012.62	−0.457565	−0.228782	+	0.973478i	$0.573474\pi$
$618$	6125.12	0.398687
$619$	7635.90	0.495820	0.247910	−	0.968783i	$-0.420256\pi$
$619$	7635.90	0.495820	0.247910	+	0.968783i	$0.420256\pi$
$620$	0	0
$621$	−10883.4	−0.703276
$622$	−5543.82	−0.357374
$623$	605.505	0.0389391
$624$	8082.90	0.518550
$625$	0	0
$626$	13648.0	0.871377
$627$	−13706.8	−0.873044
$628$	11385.6	0.723463
$629$	−10405.3	−0.659596
$630$	0	0
$631$	16185.3	1.02112	0.510560	−	0.859842i	$-0.329438\pi$
$631$	16185.3	1.02112	0.510560	+	0.859842i	$0.329438\pi$
$632$	4859.68	0.305867
$633$	−10012.8	−0.628711
$634$	−16676.3	−1.04464
$635$	0	0
$636$	2333.82	0.145506
$637$	−17213.3	−1.07067
$638$	4622.84	0.286865
$639$	2180.83	0.135012
$640$	0	0
$641$	−4909.20	−0.302499	−0.151249	−	0.988496i	$-0.548330\pi$
$641$	−4909.20	−0.302499	−0.151249	+	0.988496i	$0.548330\pi$
$642$	−33378.1	−2.05191
$643$	−29722.2	−1.82291	−0.911453	−	0.411404i	$-0.865039\pi$
$643$	−29722.2	−1.82291	−0.911453	+	0.411404i	$0.865039\pi$
$644$	−61.6724	−0.00377366
$645$	0	0
$646$	−7476.26	−0.455340
$647$	6460.79	0.392581	0.196290	−	0.980546i	$-0.437110\pi$
$647$	6460.79	0.392581	0.196290	+	0.980546i	$0.437110\pi$
$648$	−22058.1	−1.33723
$649$	5901.94	0.356967
$650$	0	0
$651$	−816.601	−0.0491630
$652$	2370.66	0.142396
$653$	−14842.0	−0.889452	−0.444726	−	0.895667i	$-0.646699\pi$
$653$	−14842.0	−0.889452	−0.444726	+	0.895667i	$0.646699\pi$
$654$	−12125.4	−0.724986
$655$	0	0
$656$	3716.93	0.221222
$657$	5966.87	0.354322
$658$	338.217	0.0200381
$659$	−291.305	−0.0172195	−0.00860973	−	0.999963i	$-0.502741\pi$
$659$	−291.305	−0.0172195	−0.00860973	+	0.999963i	$0.502741\pi$
$660$	0	0
$661$	13045.5	0.767640	0.383820	−	0.923408i	$-0.374608\pi$
$661$	13045.5	0.767640	0.383820	+	0.923408i	$0.374608\pi$
$662$	−2314.88	−0.135907
$663$	21767.3	1.27507
$664$	−9881.79	−0.577542
$665$	0	0
$666$	35773.3	2.08136
$667$	−3382.78	−0.196375
$668$	−3.33767	−0.000193321 0
$669$	54413.3	3.14460
$670$	0	0
$671$	−5559.49	−0.319854
$672$	−215.656	−0.0123796
$673$	18521.1	1.06083	0.530414	−	0.847739i	$-0.322036\pi$
$673$	18521.1	1.06083	0.530414	+	0.847739i	$0.322036\pi$
$674$	7983.50	0.456251
$675$	0	0
$676$	1312.44	0.0746720
$677$	−2606.26	−0.147957	−0.0739783	−	0.997260i	$-0.523570\pi$
$677$	−2606.26	−0.147957	−0.0739783	+	0.997260i	$0.523570\pi$
$678$	−24751.3	−1.40202
$679$	763.824	0.0431706
$680$	0	0
$681$	39939.4	2.24740
$682$	−3808.57	−0.213838
$683$	7900.69	0.442623	0.221312	−	0.975203i	$-0.428966\pi$
$683$	7900.69	0.442623	0.221312	+	0.975203i	$0.428966\pi$
$684$	25703.3	1.43683
$685$	0	0
$686$	919.121	0.0511548
$687$	52373.7	2.90856
$688$	−4845.08	−0.268484
$689$	2916.36	0.161254
$690$	0	0
$691$	−4623.33	−0.254530	−0.127265	−	0.991869i	$-0.540620\pi$
$691$	−4623.33	−0.254530	−0.127265	+	0.991869i	$0.540620\pi$
$692$	3542.68	0.194613
$693$	−780.312	−0.0427729
$694$	−13035.1	−0.712977
$695$	0	0
$696$	−11828.9	−0.644214
$697$	10009.7	0.543967
$698$	13290.8	0.720721
$699$	−11894.3	−0.643611
$700$	0	0
$701$	−20427.9	−1.10064	−0.550322	−	0.834952i	$-0.685495\pi$
$701$	−20427.9	−1.10064	−0.550322	+	0.834952i	$0.685495\pi$
$702$	−47556.0	−2.55682
$703$	−20950.5	−1.12399
$704$	−1005.80	−0.0538460
$705$	0	0
$706$	−6470.98	−0.344956
$707$	−161.731	−0.00860326
$708$	−15101.8	−0.801640
$709$	−13724.3	−0.726978	−0.363489	−	0.931599i	$-0.618415\pi$
$709$	−13724.3	−0.726978	−0.363489	+	0.931599i	$0.618415\pi$
$710$	0	0
$711$	−44993.5	−2.37326
$712$	−7226.11	−0.380351
$713$	2786.94	0.146384
$714$	−580.762	−0.0304404
$715$	0	0
$716$	−13417.8	−0.700344
$717$	37790.7	1.96837
$718$	15838.4	0.823235
$719$	−1783.58	−0.0925125	−0.0462562	−	0.998930i	$-0.514729\pi$
$719$	−1783.58	−0.0925125	−0.0462562	+	0.998930i	$0.514729\pi$
$720$	0	0
$721$	−204.212	−0.0105482
$722$	−1335.05	−0.0688165
$723$	−51757.7	−2.66236
$724$	317.348	0.0162903
$725$	0	0
$726$	21795.9	1.11421
$727$	−22646.2	−1.15530	−0.577649	−	0.816285i	$-0.696030\pi$
$727$	−22646.2	−1.15530	−0.577649	+	0.816285i	$0.696030\pi$
$728$	−269.484	−0.0137194
$729$	75783.8	3.85021
$730$	0	0
$731$	−13047.8	−0.660180
$732$	14225.6	0.718295
$733$	28570.9	1.43969	0.719844	−	0.694136i	$-0.244213\pi$
$733$	28570.9	1.43969	0.719844	+	0.694136i	$0.244213\pi$
$734$	13912.8	0.699633
$735$	0	0
$736$	736.000	0.0368605
$737$	−11799.0	−0.589718
$738$	−34413.3	−1.71649
$739$	−618.299	−0.0307774	−0.0153887	−	0.999882i	$-0.504899\pi$
$739$	−618.299	−0.0307774	−0.0153887	+	0.999882i	$0.504899\pi$
$740$	0	0
$741$	43827.3	2.17279
$742$	−77.8097	−0.00384971
$743$	3799.48	0.187604	0.0938019	−	0.995591i	$-0.470098\pi$
$743$	3799.48	0.187604	0.0938019	+	0.995591i	$0.470098\pi$
$744$	9745.33	0.480217
$745$	0	0
$746$	18576.0	0.911683
$747$	91490.9	4.48123
$748$	−2708.63	−0.132403
$749$	1112.83	0.0542882
$750$	0	0
$751$	−35821.9	−1.74056	−0.870280	−	0.492557i	$-0.836062\pi$
$751$	−35821.9	−1.74056	−0.870280	+	0.492557i	$0.836062\pi$
$752$	−4036.29	−0.195729
$753$	−45469.2	−2.20052
$754$	−14781.4	−0.713936
$755$	0	0
$756$	1268.82	0.0610402
$757$	22756.9	1.09262	0.546310	−	0.837583i	$-0.316032\pi$
$757$	22756.9	1.09262	0.546310	+	0.837583i	$0.316032\pi$
$758$	−897.974	−0.0430289
$759$	3633.86	0.173782
$760$	0	0
$761$	−467.195	−0.0222547	−0.0111273	−	0.999938i	$-0.503542\pi$
$761$	−467.195	−0.0222547	−0.0111273	+	0.999938i	$0.503542\pi$
$762$	56176.4	2.67068
$763$	404.261	0.0191812
$764$	−12963.2	−0.613863
$765$	0	0
$766$	17448.4	0.823022
$767$	−18871.3	−0.888401
$768$	2573.64	0.120922
$769$	−25390.7	−1.19065	−0.595326	−	0.803484i	$-0.702977\pi$
$769$	−25390.7	−1.19065	−0.595326	+	0.803484i	$0.702977\pi$
$770$	0	0
$771$	−37099.5	−1.73295
$772$	−18852.6	−0.878912
$773$	−32898.5	−1.53076	−0.765380	−	0.643578i	$-0.777449\pi$
$773$	−32898.5	−1.53076	−0.765380	+	0.643578i	$0.777449\pi$
$774$	44858.3	2.08320
$775$	0	0
$776$	−9115.48	−0.421684
$777$	−1627.45	−0.0751409
$778$	11705.1	0.539395
$779$	20154.0	0.926948
$780$	0	0
$781$	−462.725	−0.0212005
$782$	1982.05	0.0906369
$783$	69595.6	3.17643
$784$	−5480.81	−0.249672
$785$	0	0
$786$	29687.4	1.34722
$787$	6114.83	0.276963	0.138482	−	0.990365i	$-0.455778\pi$
$787$	6114.83	0.276963	0.138482	+	0.990365i	$0.455778\pi$
$788$	−6031.39	−0.272664
$789$	−65508.7	−2.95586
$790$	0	0
$791$	825.209	0.0370936
$792$	9312.25	0.417799
$793$	17776.3	0.796036
$794$	13781.8	0.615991
$795$	0	0
$796$	−924.022	−0.0411446
$797$	39418.9	1.75193	0.875966	−	0.482373i	$-0.160225\pi$
$797$	39418.9	1.75193	0.875966	+	0.482373i	$0.160225\pi$
$798$	−1169.33	−0.0518721
$799$	−10869.8	−0.481282
$800$	0	0
$801$	66903.1	2.95119
$802$	−20653.2	−0.909338
$803$	−1266.04	−0.0556383
$804$	30191.2	1.32433
$805$	0	0
$806$	12177.8	0.532190
$807$	−15545.0	−0.678080
$808$	1930.10	0.0840353
$809$	−5365.35	−0.233171	−0.116586	−	0.993181i	$-0.537195\pi$
$809$	−5365.35	−0.233171	−0.116586	+	0.993181i	$0.537195\pi$
$810$	0	0
$811$	24016.0	1.03985	0.519924	−	0.854213i	$-0.325960\pi$
$811$	24016.0	1.03985	0.519924	+	0.854213i	$0.325960\pi$
$812$	394.375	0.0170442
$813$	64547.1	2.78446
$814$	−7590.31	−0.326831
$815$	0	0
$816$	6930.82	0.297337
$817$	−26271.1	−1.12498
$818$	−13017.9	−0.556433
$819$	2495.03	0.106451
$820$	0	0
$821$	2487.91	0.105760	0.0528799	−	0.998601i	$-0.483160\pi$
$821$	2487.91	0.105760	0.0528799	+	0.998601i	$0.483160\pi$
$822$	−39316.9	−1.66829
$823$	40471.5	1.71415	0.857077	−	0.515188i	$-0.172278\pi$
$823$	40471.5	1.71415	0.857077	+	0.515188i	$0.172278\pi$
$824$	2437.07	0.103033
$825$	0	0
$826$	503.495	0.0212092
$827$	18990.3	0.798499	0.399249	−	0.916842i	$-0.369271\pi$
$827$	18990.3	0.798499	0.399249	+	0.916842i	$0.369271\pi$
$828$	−6814.28	−0.286006
$829$	−5590.16	−0.234203	−0.117102	−	0.993120i	$-0.537360\pi$
$829$	−5590.16	−0.234203	−0.117102	+	0.993120i	$0.537360\pi$
$830$	0	0
$831$	51120.3	2.13399
$832$	3216.03	0.134009
$833$	−14759.9	−0.613924
$834$	64577.6	2.68122
$835$	0	0
$836$	−5453.68	−0.225622
$837$	−57337.0	−2.36781
$838$	24027.9	0.990487
$839$	41350.3	1.70151	0.850756	−	0.525560i	$-0.176144\pi$
$839$	41350.3	1.70151	0.850756	+	0.525560i	$0.176144\pi$
$840$	0	0
$841$	−2757.18	−0.113050
$842$	29120.2	1.19186
$843$	−12680.7	−0.518084
$844$	−3983.91	−0.162478
$845$	0	0
$846$	37370.1	1.51869
$847$	−726.674	−0.0294791
$848$	928.583	0.0376034
$849$	81724.9	3.30364
$850$	0	0
$851$	5554.24	0.223733
$852$	1184.02	0.0476100
$853$	−38767.6	−1.55613	−0.778064	−	0.628185i	$-0.783798\pi$
$853$	−38767.6	−1.55613	−0.778064	+	0.628185i	$0.783798\pi$
$854$	−474.281	−0.0190042
$855$	0	0
$856$	−13280.5	−0.530278
$857$	23232.3	0.926021	0.463010	−	0.886353i	$-0.346769\pi$
$857$	23232.3	0.926021	0.463010	+	0.886353i	$0.346769\pi$
$858$	15878.5	0.631799
$859$	14026.2	0.557121	0.278560	−	0.960419i	$-0.410143\pi$
$859$	14026.2	0.557121	0.278560	+	0.960419i	$0.410143\pi$
$860$	0	0
$861$	1565.58	0.0619684
$862$	5414.88	0.213958
$863$	−40191.4	−1.58532	−0.792661	−	0.609663i	$-0.791305\pi$
$863$	−40191.4	−1.58532	−0.792661	+	0.609663i	$0.791305\pi$
$864$	−15142.1	−0.596231
$865$	0	0
$866$	4088.20	0.160419
$867$	−30727.0	−1.20362
$868$	−324.910	−0.0127052
$869$	9546.64	0.372667
$870$	0	0
$871$	37727.0	1.46766
$872$	−4824.46	−0.187359
$873$	84396.0	3.27190
$874$	3990.76	0.154450
$875$	0	0
$876$	3239.53	0.124947
$877$	−26001.1	−1.00113	−0.500567	−	0.865698i	$-0.666875\pi$
$877$	−26001.1	−1.00113	−0.500567	+	0.865698i	$0.666875\pi$
$878$	31335.8	1.20448
$879$	−61940.1	−2.37678
$880$	0	0
$881$	48072.2	1.83836	0.919179	−	0.393839i	$-0.128853\pi$
$881$	48072.2	1.83836	0.919179	+	0.393839i	$0.128853\pi$
$882$	50744.2	1.93724
$883$	−21394.5	−0.815381	−0.407691	−	0.913120i	$-0.633666\pi$
$883$	−21394.5	−0.815381	−0.407691	+	0.913120i	$0.633666\pi$
$884$	8660.79	0.329518
$885$	0	0
$886$	22233.2	0.843047
$887$	−2470.47	−0.0935178	−0.0467589	−	0.998906i	$-0.514889\pi$
$887$	−2470.47	−0.0935178	−0.0467589	+	0.998906i	$0.514889\pi$
$888$	19422.0	0.733964
$889$	−1872.92	−0.0706590
$890$	0	0
$891$	−43332.2	−1.62927
$892$	21650.0	0.812663
$893$	−21885.7	−0.820129
$894$	−60659.8	−2.26932
$895$	0	0
$896$	−85.8051	−0.00319927
$897$	−11619.2	−0.432500
$898$	−29784.9	−1.10683
$899$	−17821.6	−0.661160
$900$	0	0
$901$	2500.68	0.0924636
$902$	7301.75	0.269536
$903$	−2040.76	−0.0752073
$904$	−9848.06	−0.362325
$905$	0	0
$906$	13031.8	0.477874
$907$	−46686.9	−1.70917	−0.854583	−	0.519314i	$-0.826187\pi$
$907$	−46686.9	−1.70917	−0.854583	+	0.519314i	$0.826187\pi$
$908$	15891.1	0.580799
$909$	−17869.8	−0.652041
$910$	0	0
$911$	8843.65	0.321628	0.160814	−	0.986985i	$-0.448588\pi$
$911$	8843.65	0.321628	0.160814	+	0.986985i	$0.448588\pi$
$912$	13954.8	0.506678
$913$	−19412.4	−0.703675
$914$	−11529.1	−0.417232
$915$	0	0
$916$	20838.5	0.751663
$917$	−989.777	−0.0356438
$918$	−40777.7	−1.46608
$919$	−37342.3	−1.34038	−0.670189	−	0.742190i	$-0.733787\pi$
$919$	−37342.3	−1.34038	−0.670189	+	0.742190i	$0.733787\pi$
$920$	0	0
$921$	18929.9	0.677265
$922$	31748.2	1.13403
$923$	1479.55	0.0527628
$924$	−423.646	−0.0150833
$925$	0	0
$926$	−7624.07	−0.270564
$927$	−22563.7	−0.799447
$928$	−4706.48	−0.166485
$929$	−8119.14	−0.286739	−0.143369	−	0.989669i	$-0.545794\pi$
$929$	−8119.14	−0.286739	−0.143369	+	0.989669i	$0.545794\pi$
$930$	0	0
$931$	−29718.2	−1.04616
$932$	−4732.51	−0.166329
$933$	27866.7	0.977831
$934$	−27083.2	−0.948812
$935$	0	0
$936$	−29775.7	−1.03980
$937$	7726.38	0.269381	0.134690	−	0.990888i	$-0.456996\pi$
$937$	7726.38	0.269381	0.134690	+	0.990888i	$0.456996\pi$
$938$	−1006.57	−0.0350382
$939$	−68603.3	−2.38422
$940$	0	0
$941$	−36044.0	−1.24867	−0.624337	−	0.781155i	$-0.714631\pi$
$941$	−36044.0	−1.24867	−0.624337	+	0.781155i	$0.714631\pi$
$942$	−57231.3	−1.97951
$943$	−5343.08	−0.184512
$944$	−6008.72	−0.207169
$945$	0	0
$946$	−9517.95	−0.327120
$947$	54162.7	1.85855	0.929277	−	0.369385i	$-0.120431\pi$
$947$	54162.7	1.85855	0.929277	+	0.369385i	$0.120431\pi$
$948$	−24427.9	−0.836899
$949$	4048.13	0.138470
$950$	0	0
$951$	83825.9	2.85830
$952$	−231.074	−0.00786675
$953$	50561.5	1.71862	0.859311	−	0.511453i	$-0.170893\pi$
$953$	50561.5	1.71862	0.859311	+	0.511453i	$0.170893\pi$
$954$	−8597.31	−0.291770
$955$	0	0
$956$	15036.2	0.508687
$957$	−23237.4	−0.784908
$958$	29576.7	0.997475
$959$	1310.83	0.0441384
$960$	0	0
$961$	−15108.5	−0.507151
$962$	24269.8	0.813400
$963$	122958.	4.11450
$964$	−20593.4	−0.688037
$965$	0	0
$966$	310.005	0.0103253
$967$	−50393.2	−1.67584	−0.837919	−	0.545794i	$-0.816228\pi$
$967$	−50393.2	−1.67584	−0.837919	+	0.545794i	$0.816228\pi$
$968$	8672.14	0.287948
$969$	37580.5	1.24588
$970$	0	0
$971$	−1810.68	−0.0598429	−0.0299215	−	0.999552i	$-0.509526\pi$
$971$	−1810.68	−0.0598429	−0.0299215	+	0.999552i	$0.509526\pi$
$972$	59773.5	1.97247
$973$	−2153.02	−0.0709379
$974$	22615.6	0.743996
$975$	0	0
$976$	5660.08	0.185630
$977$	−51110.8	−1.67367	−0.836837	−	0.547451i	$-0.815598\pi$
$977$	−51110.8	−1.67367	−0.836837	+	0.547451i	$0.815598\pi$
$978$	−11916.4	−0.389617
$979$	−14195.4	−0.463418
$980$	0	0
$981$	44667.4	1.45374
$982$	−8252.27	−0.268167
$983$	−21142.5	−0.686003	−0.343002	−	0.939335i	$-0.611444\pi$
$983$	−21142.5	−0.686003	−0.343002	+	0.939335i	$0.611444\pi$
$984$	−18683.6	−0.605297
$985$	0	0
$986$	−12674.6	−0.409372
$987$	−1700.09	−0.0548273
$988$	17438.0	0.561516
$989$	6964.80	0.223931
$990$	0	0
$991$	−39535.7	−1.26730	−0.633650	−	0.773620i	$-0.718444\pi$
$991$	−39535.7	−1.26730	−0.633650	+	0.773620i	$0.718444\pi$
$992$	3877.48	0.124103
$993$	11636.1	0.371863
$994$	−39.4751	−0.00125963
$995$	0	0
$996$	49672.2	1.58024
$997$	−9859.01	−0.313178	−0.156589	−	0.987664i	$-0.550050\pi$
$997$	−9859.01	−0.313178	−0.156589	+	0.987664i	$0.550050\pi$
$998$	−30321.8	−0.961743
$999$	−114270.	−3.61896

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.s.1.5	✓	5
5.2	odd	4		1150.4.b.p.599.1		10
5.3	odd	4		1150.4.b.p.599.10		10
5.4	even	2		1150.4.a.t.1.1	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.4.a.s.1.5	✓	5	1.1	even	1	trivial
1150.4.a.t.1.1	yes	5	5.4	even	2
1150.4.b.p.599.1		10	5.2	odd	4
1150.4.b.p.599.10		10	5.3	odd	4

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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +10.0533 q^{3} +4.00000 q^{4} -20.1065 q^{6} +0.670353 q^{7} -8.00000 q^{8} +74.0682 q^{9} -15.7157 q^{11} +40.2131 q^{12} +50.2505 q^{13} -1.34071 q^{14} +16.0000 q^{16} +43.0881 q^{17} -148.136 q^{18} +86.7556 q^{19} +6.73924 q^{21} +31.4313 q^{22} -23.0000 q^{23} -80.4262 q^{24} -100.501 q^{26} +473.190 q^{27} +2.68141 q^{28} +147.078 q^{29} -121.171 q^{31} -32.0000 q^{32} -157.994 q^{33} -86.1762 q^{34} +296.273 q^{36} -241.489 q^{37} -173.511 q^{38} +505.181 q^{39} +232.308 q^{41} -13.4785 q^{42} -302.817 q^{43} -62.8627 q^{44} +46.0000 q^{46} -252.268 q^{47} +160.852 q^{48} -342.551 q^{49} +433.176 q^{51} +201.002 q^{52} +58.0364 q^{53} -946.380 q^{54} -5.36282 q^{56} +872.177 q^{57} -294.155 q^{58} -375.545 q^{59} +353.755 q^{61} +242.342 q^{62} +49.6518 q^{63} +64.0000 q^{64} +315.988 q^{66} +750.780 q^{67} +172.352 q^{68} -231.225 q^{69} +29.4436 q^{71} -592.546 q^{72} +80.5591 q^{73} +482.978 q^{74} +347.022 q^{76} -10.5350 q^{77} -1010.36 q^{78} -607.461 q^{79} +2757.26 q^{81} -464.616 q^{82} +1235.22 q^{83} +26.9569 q^{84} +605.635 q^{86} +1478.61 q^{87} +125.725 q^{88} +903.263 q^{89} +33.6855 q^{91} -92.0000 q^{92} -1218.17 q^{93} +504.536 q^{94} -321.705 q^{96} +1139.44 q^{97} +685.101 q^{98} -1164.03 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{2} + 12 q^{3} + 20 q^{4} - 24 q^{6} - 24 q^{7} - 40 q^{8} + 11 q^{9} - 54 q^{11} + 48 q^{12} + 36 q^{13} + 48 q^{14} + 80 q^{16} + 132 q^{17} - 22 q^{18} - 50 q^{19} - 158 q^{21} + 108 q^{22}+ \cdots - 1740 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 + 12 * q^3 + 20 * q^4 - 24 * q^6 - 24 * q^7 - 40 * q^8 + 11 * q^9 - 54 * q^11 + 48 * q^12 + 36 * q^13 + 48 * q^14 + 80 * q^16 + 132 * q^17 - 22 * q^18 - 50 * q^19 - 158 * q^21 + 108 * q^22 - 115 * q^23 - 96 * q^24 - 72 * q^26 + 489 * q^27 - 96 * q^28 - 104 * q^29 - 342 * q^31 - 160 * q^32 - 348 * q^33 - 264 * q^34 + 44 * q^36 + 380 * q^37 + 100 * q^38 + 193 * q^39 + 2 * q^41 + 316 * q^42 - 114 * q^43 - 216 * q^44 + 230 * q^46 + 966 * q^47 + 192 * q^48 - 441 * q^49 + 1200 * q^51 + 144 * q^52 + 308 * q^53 - 978 * q^54 + 192 * q^56 + 1410 * q^57 + 208 * q^58 - 277 * q^59 + 482 * q^61 + 684 * q^62 - 222 * q^63 + 320 * q^64 + 696 * q^66 + 2044 * q^67 + 528 * q^68 - 276 * q^69 + 208 * q^71 - 88 * q^72 + 646 * q^73 - 760 * q^74 - 200 * q^76 + 2124 * q^77 - 386 * q^78 - 3444 * q^79 + 2933 * q^81 - 4 * q^82 + 1218 * q^83 - 632 * q^84 + 228 * q^86 + 103 * q^87 + 432 * q^88 + 140 * q^89 + 1976 * q^91 - 460 * q^92 + 473 * q^93 - 1932 * q^94 - 384 * q^96 + 2816 * q^97 + 882 * q^98 - 1740 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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