LMFDB - Embedded newform 825.4.a.z.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [825,4,Mod(1,825)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("825.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,4,15,46] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$48.6765757547$
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} - 41x^{3} + 3x^{2} + 294x - 200 $ x^5 - x^4 - 41x^3 + 3x^2 + 294*x - 200
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-4.58039$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.58039 q^{2} +3.00000 q^{3} +23.1407 q^{4} +16.7412 q^{6} -34.4181 q^{7} +84.4912 q^{8} +9.00000 q^{9} +11.0000 q^{11} +69.4222 q^{12} +71.2887 q^{13} -192.066 q^{14} +286.368 q^{16} +22.3593 q^{17} +50.2235 q^{18} +88.1755 q^{19} -103.254 q^{21} +61.3843 q^{22} -21.5477 q^{23} +253.474 q^{24} +397.819 q^{26} +27.0000 q^{27} -796.460 q^{28} +118.280 q^{29} -33.5268 q^{31} +922.115 q^{32} +33.0000 q^{33} +124.774 q^{34} +208.267 q^{36} -364.750 q^{37} +492.054 q^{38} +213.866 q^{39} +48.9166 q^{41} -576.199 q^{42} -95.8534 q^{43} +254.548 q^{44} -120.244 q^{46} +132.383 q^{47} +859.104 q^{48} +841.605 q^{49} +67.0780 q^{51} +1649.67 q^{52} -300.685 q^{53} +150.671 q^{54} -2908.03 q^{56} +264.526 q^{57} +660.050 q^{58} -654.281 q^{59} -772.721 q^{61} -187.092 q^{62} -309.763 q^{63} +2854.82 q^{64} +184.153 q^{66} +112.087 q^{67} +517.412 q^{68} -64.6430 q^{69} +548.159 q^{71} +760.421 q^{72} -559.178 q^{73} -2035.45 q^{74} +2040.45 q^{76} -378.599 q^{77} +1193.46 q^{78} +48.5332 q^{79} +81.0000 q^{81} +272.974 q^{82} -447.844 q^{83} -2389.38 q^{84} -534.899 q^{86} +354.841 q^{87} +929.403 q^{88} -552.419 q^{89} -2453.62 q^{91} -498.629 q^{92} -100.580 q^{93} +738.746 q^{94} +2766.34 q^{96} +413.909 q^{97} +4696.48 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 15 q^{3} + 46 q^{4} + 12 q^{6} - 38 q^{7} + 72 q^{8} + 45 q^{9} + 55 q^{11} + 138 q^{12} + 43 q^{13} + 55 q^{14} + 634 q^{16} + 150 q^{17} + 36 q^{18} - 337 q^{19} - 114 q^{21} + 44 q^{22}+ \cdots + 495 q^{99}+O(q^{100}) $ 5 * q + 4 * q^2 + 15 * q^3 + 46 * q^4 + 12 * q^6 - 38 * q^7 + 72 * q^8 + 45 * q^9 + 55 * q^11 + 138 * q^12 + 43 * q^13 + 55 * q^14 + 634 * q^16 + 150 * q^17 + 36 * q^18 - 337 * q^19 - 114 * q^21 + 44 * q^22 - 201 * q^23 + 216 * q^24 + 359 * q^26 + 135 * q^27 - 441 * q^28 - 153 * q^29 + 263 * q^31 + 280 * q^32 + 165 * q^33 + 517 * q^34 + 414 * q^36 - 350 * q^37 + 786 * q^38 + 129 * q^39 - 88 * q^41 + 165 * q^42 + 161 * q^43 + 506 * q^44 + 436 * q^46 + 490 * q^47 + 1902 * q^48 + 2381 * q^49 + 450 * q^51 + 289 * q^52 + 558 * q^53 + 108 * q^54 + 211 * q^56 - 1011 * q^57 - 969 * q^58 + 410 * q^59 - 1130 * q^61 + 463 * q^62 - 342 * q^63 + 3170 * q^64 + 132 * q^66 + 1130 * q^67 + 3333 * q^68 - 603 * q^69 + 1249 * q^71 + 648 * q^72 - 44 * q^73 - 2985 * q^74 - 2556 * q^76 - 418 * q^77 + 1077 * q^78 + 2258 * q^79 + 405 * q^81 - 3737 * q^82 - 957 * q^83 - 1323 * q^84 - 1435 * q^86 - 459 * q^87 + 792 * q^88 + 347 * q^89 - 612 * q^91 - 6006 * q^92 + 789 * q^93 + 907 * q^94 + 840 * q^96 - 2355 * q^97 + 9417 * q^98 + 495 * q^99

Coefficient data

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

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

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

$2$	5.58039	1.97297	0.986483	−	0.163865i	$-0.0523963\pi$
$2$	5.58039	1.97297	0.986483	+	0.163865i	$0.0523963\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	23.1407	2.89259
$5$	0	0
$5$	0	0
$6$	16.7412	1.13909
$7$	−34.4181	−1.85840	−0.929201	−	0.369575i	$-0.879503\pi$
$7$	−34.4181	−1.85840	−0.929201	+	0.369575i	$0.879503\pi$
$8$	84.4912	3.73402
$9$	9.00000	0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	69.4222	1.67004
$13$	71.2887	1.52092	0.760459	−	0.649386i	$-0.224974\pi$
$13$	71.2887	1.52092	0.760459	+	0.649386i	$0.224974\pi$
$14$	−192.066	−3.66656
$15$	0	0
$16$	286.368	4.47450
$17$	22.3593	0.318996	0.159498	−	0.987198i	$-0.449012\pi$
$17$	22.3593	0.318996	0.159498	+	0.987198i	$0.449012\pi$
$18$	50.2235	0.657655
$19$	88.1755	1.06468	0.532338	−	0.846532i	$-0.321314\pi$
$19$	88.1755	1.06468	0.532338	+	0.846532i	$0.321314\pi$
$20$	0	0
$21$	−103.254	−1.07295
$22$	61.3843	0.594871
$23$	−21.5477	−0.195348	−0.0976738	−	0.995218i	$-0.531140\pi$
$23$	−21.5477	−0.195348	−0.0976738	+	0.995218i	$0.531140\pi$
$24$	253.474	2.15584
$25$	0	0
$26$	397.819	3.00072
$27$	27.0000	0.192450
$28$	−796.460	−5.37560
$29$	118.280	0.757383	0.378691	−	0.925523i	$-0.376374\pi$
$29$	118.280	0.757383	0.378691	+	0.925523i	$0.376374\pi$
$30$	0	0
$31$	−33.5268	−0.194245	−0.0971223	−	0.995272i	$-0.530964\pi$
$31$	−33.5268	−0.194245	−0.0971223	+	0.995272i	$0.530964\pi$
$32$	922.115	5.09401
$33$	33.0000	0.174078
$34$	124.774	0.629369
$35$	0	0
$36$	208.267	0.964198
$37$	−364.750	−1.62066	−0.810332	−	0.585970i	$-0.800713\pi$
$37$	−364.750	−1.62066	−0.810332	+	0.585970i	$0.800713\pi$
$38$	492.054	2.10057
$39$	213.866	0.878102
$40$	0	0
$41$	48.9166	0.186329	0.0931645	−	0.995651i	$-0.470302\pi$
$41$	48.9166	0.186329	0.0931645	+	0.995651i	$0.470302\pi$
$42$	−576.199	−2.11689
$43$	−95.8534	−0.339942	−0.169971	−	0.985449i	$-0.554367\pi$
$43$	−95.8534	−0.339942	−0.169971	+	0.985449i	$0.554367\pi$
$44$	254.548	0.872149
$45$	0	0
$46$	−120.244	−0.385414
$47$	132.383	0.410851	0.205425	−	0.978673i	$-0.434142\pi$
$47$	132.383	0.410851	0.205425	+	0.978673i	$0.434142\pi$
$48$	859.104	2.58335
$49$	841.605	2.45366
$50$	0	0
$51$	67.0780	0.184173
$52$	1649.67	4.39939
$53$	−300.685	−0.779288	−0.389644	−	0.920966i	$-0.627402\pi$
$53$	−300.685	−0.779288	−0.389644	+	0.920966i	$0.627402\pi$
$54$	150.671	0.379697
$55$	0	0
$56$	−2908.03	−6.93931
$57$	264.526	0.614691
$58$	660.050	1.49429
$59$	−654.281	−1.44373	−0.721865	−	0.692033i	$-0.756715\pi$
$59$	−654.281	−1.44373	−0.721865	+	0.692033i	$0.756715\pi$
$60$	0	0
$61$	−772.721	−1.62191	−0.810957	−	0.585106i	$-0.801053\pi$
$61$	−772.721	−1.62191	−0.810957	+	0.585106i	$0.801053\pi$
$62$	−187.092	−0.383238
$63$	−309.763	−0.619467
$64$	2854.82	5.57581
$65$	0	0
$66$	184.153	0.343449
$67$	112.087	0.204382	0.102191	−	0.994765i	$-0.467415\pi$
$67$	112.087	0.204382	0.102191	+	0.994765i	$0.467415\pi$
$68$	517.412	0.922726
$69$	−64.6430	−0.112784
$70$	0	0
$71$	548.159	0.916261	0.458130	−	0.888885i	$-0.348519\pi$
$71$	548.159	0.916261	0.458130	+	0.888885i	$0.348519\pi$
$72$	760.421	1.24467
$73$	−559.178	−0.896531	−0.448266	−	0.893900i	$-0.647958\pi$
$73$	−559.178	−0.896531	−0.448266	+	0.893900i	$0.647958\pi$
$74$	−2035.45	−3.19752
$75$	0	0
$76$	2040.45	3.07967
$77$	−378.599	−0.560329
$78$	1193.46	1.73247
$79$	48.5332	0.0691192	0.0345596	−	0.999403i	$-0.488997\pi$
$79$	48.5332	0.0691192	0.0345596	+	0.999403i	$0.488997\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	272.974	0.367621
$83$	−447.844	−0.592256	−0.296128	−	0.955148i	$-0.595696\pi$
$83$	−447.844	−0.592256	−0.296128	+	0.955148i	$0.595696\pi$
$84$	−2389.38	−3.10360
$85$	0	0
$86$	−534.899	−0.670694
$87$	354.841	0.437275
$88$	929.403	1.12585
$89$	−552.419	−0.657936	−0.328968	−	0.944341i	$-0.606701\pi$
$89$	−552.419	−0.657936	−0.328968	+	0.944341i	$0.606701\pi$
$90$	0	0
$91$	−2453.62	−2.82648
$92$	−498.629	−0.565061
$93$	−100.580	−0.112147
$94$	738.746	0.810594
$95$	0	0
$96$	2766.34	2.94103
$97$	413.909	0.433259	0.216630	−	0.976254i	$-0.430494\pi$
$97$	413.909	0.433259	0.216630	+	0.976254i	$0.430494\pi$
$98$	4696.48	4.84098
$99$	99.0000	0.100504
$100$	0	0
$101$	−439.536	−0.433025	−0.216512	−	0.976280i	$-0.569468\pi$
$101$	−439.536	−0.433025	−0.216512	+	0.976280i	$0.569468\pi$
$102$	374.322	0.363366
$103$	1112.25	1.06401	0.532005	−	0.846741i	$-0.321439\pi$
$103$	1112.25	1.06401	0.532005	+	0.846741i	$0.321439\pi$
$104$	6023.27	5.67914
$105$	0	0
$106$	−1677.94	−1.53751
$107$	−863.930	−0.780554	−0.390277	−	0.920698i	$-0.627621\pi$
$107$	−863.930	−0.780554	−0.390277	+	0.920698i	$0.627621\pi$
$108$	624.800	0.556680
$109$	−1040.80	−0.914588	−0.457294	−	0.889316i	$-0.651181\pi$
$109$	−1040.80	−0.914588	−0.457294	+	0.889316i	$0.651181\pi$
$110$	0	0
$111$	−1094.25	−0.935691
$112$	−9856.24	−8.31542
$113$	−1826.14	−1.52025	−0.760127	−	0.649774i	$-0.774863\pi$
$113$	−1826.14	−1.52025	−0.760127	+	0.649774i	$0.774863\pi$
$114$	1476.16	1.21276
$115$	0	0
$116$	2737.09	2.19080
$117$	641.598	0.506972
$118$	−3651.14	−2.84843
$119$	−769.566	−0.592823
$120$	0	0
$121$	121.000	0.0909091
$122$	−4312.08	−3.19998
$123$	146.750	0.107577
$124$	−775.834	−0.561871
$125$	0	0
$126$	−1728.60	−1.22219
$127$	574.281	0.401254	0.200627	−	0.979668i	$-0.435702\pi$
$127$	574.281	0.401254	0.200627	+	0.979668i	$0.435702\pi$
$128$	8554.06	5.90687
$129$	−287.560	−0.196266
$130$	0	0
$131$	−2684.56	−1.79047	−0.895234	−	0.445596i	$-0.852992\pi$
$131$	−2684.56	−1.79047	−0.895234	+	0.445596i	$0.852992\pi$
$132$	763.644	0.503536
$133$	−3034.83	−1.97860
$134$	625.489	0.403239
$135$	0	0
$136$	1889.17	1.19114
$137$	−3013.45	−1.87925	−0.939623	−	0.342212i	$-0.888824\pi$
$137$	−3013.45	−1.87925	−0.939623	+	0.342212i	$0.888824\pi$
$138$	−360.733	−0.222519
$139$	277.411	0.169278	0.0846391	−	0.996412i	$-0.473026\pi$
$139$	277.411	0.169278	0.0846391	+	0.996412i	$0.473026\pi$
$140$	0	0
$141$	397.148	0.237205
$142$	3058.94	1.80775
$143$	784.176	0.458574
$144$	2577.31	1.49150
$145$	0	0
$146$	−3120.43	−1.76883
$147$	2524.81	1.41662
$148$	−8440.59	−4.68792
$149$	1594.62	0.876755	0.438378	−	0.898791i	$-0.355553\pi$
$149$	1594.62	0.876755	0.438378	+	0.898791i	$0.355553\pi$
$150$	0	0
$151$	2213.29	1.19281	0.596407	−	0.802682i	$-0.296594\pi$
$151$	2213.29	1.19281	0.596407	+	0.802682i	$0.296594\pi$
$152$	7450.06	3.97552
$153$	201.234	0.106332
$154$	−2112.73	−1.10551
$155$	0	0
$156$	4949.02	2.53999
$157$	−1145.63	−0.582367	−0.291183	−	0.956667i	$-0.594049\pi$
$157$	−1145.63	−0.582367	−0.291183	+	0.956667i	$0.594049\pi$
$158$	270.834	0.136370
$159$	−902.055	−0.449922
$160$	0	0
$161$	741.629	0.363034
$162$	452.012	0.219218
$163$	−594.732	−0.285785	−0.142893	−	0.989738i	$-0.545640\pi$
$163$	−594.732	−0.285785	−0.142893	+	0.989738i	$0.545640\pi$
$164$	1131.97	0.538974
$165$	0	0
$166$	−2499.14	−1.16850
$167$	2927.43	1.35648	0.678239	−	0.734842i	$-0.262744\pi$
$167$	2927.43	1.35648	0.678239	+	0.734842i	$0.262744\pi$
$168$	−8724.08	−4.00641
$169$	2885.08	1.31319
$170$	0	0
$171$	793.579	0.354892
$172$	−2218.12	−0.983314
$173$	−1274.26	−0.560000	−0.280000	−	0.960000i	$-0.590334\pi$
$173$	−1274.26	−0.560000	−0.280000	+	0.960000i	$0.590334\pi$
$174$	1980.15	0.862729
$175$	0	0
$176$	3150.05	1.34911
$177$	−1962.84	−0.833538
$178$	−3082.71	−1.29809
$179$	1626.12	0.679005	0.339503	−	0.940605i	$-0.389741\pi$
$179$	1626.12	0.679005	0.339503	+	0.940605i	$0.389741\pi$
$180$	0	0
$181$	1638.75	0.672970	0.336485	−	0.941689i	$-0.390762\pi$
$181$	1638.75	0.672970	0.336485	+	0.941689i	$0.390762\pi$
$182$	−13692.2	−5.57654
$183$	−2318.16	−0.936412
$184$	−1820.59	−0.729432
$185$	0	0
$186$	−561.277	−0.221263
$187$	245.953	0.0961810
$188$	3063.43	1.18842
$189$	−929.288	−0.357650
$190$	0	0
$191$	3460.92	1.31112	0.655559	−	0.755144i	$-0.272433\pi$
$191$	3460.92	1.31112	0.655559	+	0.755144i	$0.272433\pi$
$192$	8564.45	3.21920
$193$	1725.78	0.643650	0.321825	−	0.946799i	$-0.395704\pi$
$193$	1725.78	0.643650	0.321825	+	0.946799i	$0.395704\pi$
$194$	2309.78	0.854805
$195$	0	0
$196$	19475.4	7.09743
$197$	−405.275	−0.146572	−0.0732859	−	0.997311i	$-0.523349\pi$
$197$	−405.275	−0.146572	−0.0732859	+	0.997311i	$0.523349\pi$
$198$	552.459	0.198290
$199$	3801.45	1.35416	0.677079	−	0.735910i	$-0.263245\pi$
$199$	3801.45	1.35416	0.677079	+	0.735910i	$0.263245\pi$
$200$	0	0
$201$	336.261	0.118000
$202$	−2452.78	−0.854343
$203$	−4070.98	−1.40752
$204$	1552.24	0.532736
$205$	0	0
$206$	6206.78	2.09926
$207$	−193.929	−0.0651159
$208$	20414.8	6.80534
$209$	969.930	0.321012
$210$	0	0
$211$	−1538.57	−0.501987	−0.250993	−	0.967989i	$-0.580757\pi$
$211$	−1538.57	−0.501987	−0.250993	+	0.967989i	$0.580757\pi$
$212$	−6958.07	−2.25416
$213$	1644.48	0.529003
$214$	−4821.07	−1.54001
$215$	0	0
$216$	2281.26	0.718612
$217$	1153.93	0.360985
$218$	−5808.04	−1.80445
$219$	−1677.53	−0.517613
$220$	0	0
$221$	1593.97	0.485167
$222$	−6106.35	−1.84609
$223$	2558.19	0.768201	0.384101	−	0.923291i	$-0.374512\pi$
$223$	2558.19	0.768201	0.384101	+	0.923291i	$0.374512\pi$
$224$	−31737.4	−9.46672
$225$	0	0
$226$	−10190.6	−2.99941
$227$	5834.09	1.70582	0.852912	−	0.522055i	$-0.174834\pi$
$227$	5834.09	1.70582	0.852912	+	0.522055i	$0.174834\pi$
$228$	6121.34	1.77805
$229$	−1308.94	−0.377718	−0.188859	−	0.982004i	$-0.560479\pi$
$229$	−1308.94	−0.377718	−0.188859	+	0.982004i	$0.560479\pi$
$230$	0	0
$231$	−1135.80	−0.323506
$232$	9993.65	2.82808
$233$	−2360.58	−0.663719	−0.331859	−	0.943329i	$-0.607676\pi$
$233$	−2360.58	−0.663719	−0.331859	+	0.943329i	$0.607676\pi$
$234$	3580.37	1.00024
$235$	0	0
$236$	−15140.5	−4.17613
$237$	145.600	0.0399060
$238$	−4294.48	−1.16962
$239$	5766.38	1.56065	0.780327	−	0.625372i	$-0.215053\pi$
$239$	5766.38	1.56065	0.780327	+	0.625372i	$0.215053\pi$
$240$	0	0
$241$	−2644.73	−0.706896	−0.353448	−	0.935454i	$-0.614991\pi$
$241$	−2644.73	−0.706896	−0.353448	+	0.935454i	$0.614991\pi$
$242$	675.227	0.179360
$243$	243.000	0.0641500
$244$	−17881.3	−4.69154
$245$	0	0
$246$	818.921	0.212246
$247$	6285.92	1.61928
$248$	−2832.72	−0.725313
$249$	−1343.53	−0.341939
$250$	0	0
$251$	−2435.59	−0.612483	−0.306241	−	0.951954i	$-0.599071\pi$
$251$	−2435.59	−0.612483	−0.306241	+	0.951954i	$0.599071\pi$
$252$	−7168.14	−1.79187
$253$	−237.024	−0.0588995
$254$	3204.71	0.791660
$255$	0	0
$256$	24896.5	6.07824
$257$	−5522.51	−1.34041	−0.670204	−	0.742177i	$-0.733793\pi$
$257$	−5522.51	−1.34041	−0.670204	+	0.742177i	$0.733793\pi$
$258$	−1604.70	−0.387225
$259$	12554.0	3.01185
$260$	0	0
$261$	1064.52	0.252461
$262$	−14980.9	−3.53253
$263$	2699.88	0.633011	0.316505	−	0.948591i	$-0.397490\pi$
$263$	2699.88	0.633011	0.316505	+	0.948591i	$0.397490\pi$
$264$	2788.21	0.650009
$265$	0	0
$266$	−16935.5	−3.90370
$267$	−1657.26	−0.379860
$268$	2593.77	0.591194
$269$	1593.94	0.361279	0.180639	−	0.983549i	$-0.442183\pi$
$269$	1593.94	0.361279	0.180639	+	0.983549i	$0.442183\pi$
$270$	0	0
$271$	1826.60	0.409440	0.204720	−	0.978821i	$-0.434372\pi$
$271$	1826.60	0.409440	0.204720	+	0.978821i	$0.434372\pi$
$272$	6403.00	1.42735
$273$	−7360.86	−1.63187
$274$	−16816.2	−3.70769
$275$	0	0
$276$	−1495.89	−0.326238
$277$	2702.41	0.586181	0.293091	−	0.956085i	$-0.405316\pi$
$277$	2702.41	0.586181	0.293091	+	0.956085i	$0.405316\pi$
$278$	1548.06	0.333980
$279$	−301.741	−0.0647482
$280$	0	0
$281$	−2961.96	−0.628810	−0.314405	−	0.949289i	$-0.601805\pi$
$281$	−2961.96	−0.628810	−0.314405	+	0.949289i	$0.601805\pi$
$282$	2216.24	0.467997
$283$	−3504.45	−0.736105	−0.368052	−	0.929805i	$-0.619975\pi$
$283$	−3504.45	−0.736105	−0.368052	+	0.929805i	$0.619975\pi$
$284$	12684.8	2.65037
$285$	0	0
$286$	4376.01	0.904750
$287$	−1683.62	−0.346274
$288$	8299.03	1.69800
$289$	−4413.06	−0.898241
$290$	0	0
$291$	1241.73	0.250142
$292$	−12939.8	−2.59330
$293$	−1004.75	−0.200335	−0.100168	−	0.994971i	$-0.531938\pi$
$293$	−1004.75	−0.200335	−0.100168	+	0.994971i	$0.531938\pi$
$294$	14089.4	2.79494
$295$	0	0
$296$	−30818.2	−6.05159
$297$	297.000	0.0580259
$298$	8898.61	1.72981
$299$	−1536.10	−0.297108
$300$	0	0
$301$	3299.09	0.631749
$302$	12351.0	2.35338
$303$	−1318.61	−0.250007
$304$	25250.6	4.76389
$305$	0	0
$306$	1122.96	0.209790
$307$	−6690.01	−1.24371	−0.621856	−	0.783132i	$-0.713621\pi$
$307$	−6690.01	−1.24371	−0.621856	+	0.783132i	$0.713621\pi$
$308$	−8761.06	−1.62080
$309$	3336.74	0.614307
$310$	0	0
$311$	−1700.37	−0.310030	−0.155015	−	0.987912i	$-0.549543\pi$
$311$	−1700.37	−0.310030	−0.155015	+	0.987912i	$0.549543\pi$
$312$	18069.8	3.27885
$313$	−439.046	−0.0792855	−0.0396428	−	0.999214i	$-0.512622\pi$
$313$	−439.046	−0.0792855	−0.0396428	+	0.999214i	$0.512622\pi$
$314$	−6393.09	−1.14899
$315$	0	0
$316$	1123.10	0.199934
$317$	5152.91	0.912985	0.456492	−	0.889727i	$-0.349106\pi$
$317$	5152.91	0.912985	0.456492	+	0.889727i	$0.349106\pi$
$318$	−5033.82	−0.887681
$319$	1301.08	0.228360
$320$	0	0
$321$	−2591.79	−0.450653
$322$	4138.58	0.716255
$323$	1971.55	0.339628
$324$	1874.40	0.321399
$325$	0	0
$326$	−3318.84	−0.563844
$327$	−3122.39	−0.528038
$328$	4133.02	0.695756
$329$	−4556.36	−0.763526
$330$	0	0
$331$	−3246.99	−0.539187	−0.269593	−	0.962974i	$-0.586889\pi$
$331$	−3246.99	−0.539187	−0.269593	+	0.962974i	$0.586889\pi$
$332$	−10363.4	−1.71316
$333$	−3282.75	−0.540222
$334$	16336.2	2.67628
$335$	0	0
$336$	−29568.7	−4.80091
$337$	−1338.15	−0.216301	−0.108151	−	0.994135i	$-0.534493\pi$
$337$	−1338.15	−0.216301	−0.108151	+	0.994135i	$0.534493\pi$
$338$	16099.9	2.59088
$339$	−5478.42	−0.877719
$340$	0	0
$341$	−368.794	−0.0585670
$342$	4428.48	0.700190
$343$	−17161.0	−2.70148
$344$	−8098.77	−1.26935
$345$	0	0
$346$	−7110.85	−1.10486
$347$	−3421.40	−0.529309	−0.264655	−	0.964343i	$-0.585258\pi$
$347$	−3421.40	−0.529309	−0.264655	+	0.964343i	$0.585258\pi$
$348$	8211.28	1.26486
$349$	−7537.81	−1.15613	−0.578066	−	0.815990i	$-0.696192\pi$
$349$	−7537.81	−1.15613	−0.578066	+	0.815990i	$0.696192\pi$
$350$	0	0
$351$	1924.79	0.292701
$352$	10143.3	1.53590
$353$	−5803.16	−0.874988	−0.437494	−	0.899221i	$-0.644134\pi$
$353$	−5803.16	−0.874988	−0.437494	+	0.899221i	$0.644134\pi$
$354$	−10953.4	−1.64454
$355$	0	0
$356$	−12783.4	−1.90314
$357$	−2308.70	−0.342267
$358$	9074.39	1.33965
$359$	5558.49	0.817175	0.408587	−	0.912719i	$-0.366022\pi$
$359$	5558.49	0.817175	0.408587	+	0.912719i	$0.366022\pi$
$360$	0	0
$361$	915.919	0.133535
$362$	9144.88	1.32775
$363$	363.000	0.0524864
$364$	−56778.6	−8.17584
$365$	0	0
$366$	−12936.2	−1.84751
$367$	−2853.78	−0.405902	−0.202951	−	0.979189i	$-0.565053\pi$
$367$	−2853.78	−0.405902	−0.202951	+	0.979189i	$0.565053\pi$
$368$	−6170.56	−0.874083
$369$	440.249	0.0621097
$370$	0	0
$371$	10349.0	1.44823
$372$	−2327.50	−0.324396
$373$	−7500.89	−1.04124	−0.520618	−	0.853789i	$-0.674299\pi$
$373$	−7500.89	−1.04124	−0.520618	+	0.853789i	$0.674299\pi$
$374$	1372.51	0.189762
$375$	0	0
$376$	11185.2	1.53412
$377$	8432.05	1.15192
$378$	−5185.79	−0.705630
$379$	12965.1	1.75718	0.878591	−	0.477575i	$-0.158484\pi$
$379$	12965.1	1.75718	0.878591	+	0.477575i	$0.158484\pi$
$380$	0	0
$381$	1722.84	0.231664
$382$	19313.3	2.58679
$383$	−4928.95	−0.657591	−0.328796	−	0.944401i	$-0.606643\pi$
$383$	−4928.95	−0.657591	−0.328796	+	0.944401i	$0.606643\pi$
$384$	25662.2	3.41033
$385$	0	0
$386$	9630.53	1.26990
$387$	−862.681	−0.113314
$388$	9578.17	1.25324
$389$	13182.9	1.71825	0.859123	−	0.511769i	$-0.171010\pi$
$389$	13182.9	1.71825	0.859123	+	0.511769i	$0.171010\pi$
$390$	0	0
$391$	−481.791	−0.0623152
$392$	71108.2	9.16201
$393$	−8053.69	−1.03373
$394$	−2261.59	−0.289181
$395$	0	0
$396$	2290.93	0.290716
$397$	−13815.5	−1.74655	−0.873277	−	0.487223i	$-0.838010\pi$
$397$	−13815.5	−1.74655	−0.873277	+	0.487223i	$0.838010\pi$
$398$	21213.6	2.67171
$399$	−9104.50	−1.14234
$400$	0	0
$401$	4812.84	0.599357	0.299678	−	0.954040i	$-0.403121\pi$
$401$	4812.84	0.599357	0.299678	+	0.954040i	$0.403121\pi$
$402$	1876.47	0.232810
$403$	−2390.08	−0.295430
$404$	−10171.2	−1.25256
$405$	0	0
$406$	−22717.7	−2.77699
$407$	−4012.25	−0.488649
$408$	5667.51	0.687704
$409$	−9776.38	−1.18193	−0.590967	−	0.806696i	$-0.701254\pi$
$409$	−9776.38	−1.18193	−0.590967	+	0.806696i	$0.701254\pi$
$410$	0	0
$411$	−9040.36	−1.08498
$412$	25738.2	3.07775
$413$	22519.1	2.68303
$414$	−1082.20	−0.128471
$415$	0	0
$416$	65736.4	7.74757
$417$	832.233	0.0977329
$418$	5412.59	0.633345
$419$	14077.8	1.64139	0.820697	−	0.571363i	$-0.193585\pi$
$419$	14077.8	1.64139	0.820697	+	0.571363i	$0.193585\pi$
$420$	0	0
$421$	17100.8	1.97967	0.989837	−	0.142204i	$-0.0454188\pi$
$421$	17100.8	1.97967	0.989837	+	0.142204i	$0.0454188\pi$
$422$	−8585.79	−0.990402
$423$	1191.44	0.136950
$424$	−25405.2	−2.90988
$425$	0	0
$426$	9176.82	1.04371
$427$	26595.6	3.01417
$428$	−19992.0	−2.25782
$429$	2352.53	0.264758
$430$	0	0
$431$	−4374.76	−0.488920	−0.244460	−	0.969659i	$-0.578611\pi$
$431$	−4374.76	−0.488920	−0.244460	+	0.969659i	$0.578611\pi$
$432$	7731.93	0.861118
$433$	5388.62	0.598061	0.299031	−	0.954244i	$-0.403337\pi$
$433$	5388.62	0.598061	0.299031	+	0.954244i	$0.403337\pi$
$434$	6439.36	0.712210
$435$	0	0
$436$	−24084.8	−2.64553
$437$	−1899.98	−0.207982
$438$	−9361.29	−1.02123
$439$	5378.90	0.584786	0.292393	−	0.956298i	$-0.405549\pi$
$439$	5378.90	0.584786	0.292393	+	0.956298i	$0.405549\pi$
$440$	0	0
$441$	7574.44	0.817886
$442$	8894.96	0.957218
$443$	6872.36	0.737055	0.368528	−	0.929617i	$-0.379862\pi$
$443$	6872.36	0.737055	0.368528	+	0.929617i	$0.379862\pi$
$444$	−25321.8	−2.70657
$445$	0	0
$446$	14275.7	1.51563
$447$	4783.87	0.506195
$448$	−98257.3	−10.3621
$449$	8196.97	0.861557	0.430778	−	0.902458i	$-0.358239\pi$
$449$	8196.97	0.861557	0.430778	+	0.902458i	$0.358239\pi$
$450$	0	0
$451$	538.083	0.0561803
$452$	−42258.2	−4.39748
$453$	6639.87	0.688672
$454$	32556.5	3.36553
$455$	0	0
$456$	22350.2	2.29527
$457$	3727.32	0.381525	0.190762	−	0.981636i	$-0.438904\pi$
$457$	3727.32	0.381525	0.190762	+	0.981636i	$0.438904\pi$
$458$	−7304.42	−0.745225
$459$	603.702	0.0613909
$460$	0	0
$461$	5801.06	0.586079	0.293039	−	0.956100i	$-0.405333\pi$
$461$	5801.06	0.586079	0.293039	+	0.956100i	$0.405333\pi$
$462$	−6338.19	−0.638267
$463$	−12402.6	−1.24492	−0.622461	−	0.782651i	$-0.713867\pi$
$463$	−12402.6	−1.24492	−0.622461	+	0.782651i	$0.713867\pi$
$464$	33871.7	3.38891
$465$	0	0
$466$	−13172.9	−1.30949
$467$	−11343.7	−1.12404	−0.562018	−	0.827125i	$-0.689975\pi$
$467$	−11343.7	−1.12404	−0.562018	+	0.827125i	$0.689975\pi$
$468$	14847.1	1.46646
$469$	−3857.82	−0.379824
$470$	0	0
$471$	−3436.90	−0.336230
$472$	−55281.0	−5.39092
$473$	−1054.39	−0.102496
$474$	812.503	0.0787331
$475$	0	0
$476$	−17808.3	−1.71480
$477$	−2706.16	−0.259763
$478$	32178.6	3.07912
$479$	−2383.99	−0.227405	−0.113703	−	0.993515i	$-0.536271\pi$
$479$	−2383.99	−0.227405	−0.113703	+	0.993515i	$0.536271\pi$
$480$	0	0
$481$	−26002.6	−2.46490
$482$	−14758.6	−1.39468
$483$	2224.89	0.209598
$484$	2800.03	0.262963
$485$	0	0
$486$	1356.03	0.126566
$487$	−11186.7	−1.04090	−0.520451	−	0.853891i	$-0.674236\pi$
$487$	−11186.7	−1.04090	−0.520451	+	0.853891i	$0.674236\pi$
$488$	−65288.1	−6.05626
$489$	−1784.20	−0.164998
$490$	0	0
$491$	−9326.27	−0.857207	−0.428604	−	0.903493i	$-0.640994\pi$
$491$	−9326.27	−0.857207	−0.428604	+	0.903493i	$0.640994\pi$
$492$	3395.90	0.311177
$493$	2644.67	0.241602
$494$	35077.9	3.19479
$495$	0	0
$496$	−9600.99	−0.869148
$497$	−18866.6	−1.70278
$498$	−7497.43	−0.674635
$499$	19118.7	1.71517	0.857587	−	0.514339i	$-0.171963\pi$
$499$	19118.7	1.71517	0.857587	+	0.514339i	$0.171963\pi$
$500$	0	0
$501$	8782.30	0.783163
$502$	−13591.5	−1.20841
$503$	−8523.10	−0.755519	−0.377760	−	0.925904i	$-0.623305\pi$
$503$	−8523.10	−0.755519	−0.377760	+	0.925904i	$0.623305\pi$
$504$	−26172.2	−2.31310
$505$	0	0
$506$	−1322.69	−0.116207
$507$	8655.24	0.758171
$508$	13289.3	1.16066
$509$	−10064.2	−0.876397	−0.438199	−	0.898878i	$-0.644383\pi$
$509$	−10064.2	−0.876397	−0.438199	+	0.898878i	$0.644383\pi$
$510$	0	0
$511$	19245.8	1.66612
$512$	70499.5	6.08529
$513$	2380.74	0.204897
$514$	−30817.7	−2.64458
$515$	0	0
$516$	−6654.36	−0.567716
$517$	1456.21	0.123876
$518$	70056.3	5.94227
$519$	−3822.77	−0.323316
$520$	0	0
$521$	20495.7	1.72348	0.861742	−	0.507347i	$-0.169374\pi$
$521$	20495.7	1.72348	0.861742	+	0.507347i	$0.169374\pi$
$522$	5940.45	0.498097
$523$	−8079.94	−0.675547	−0.337773	−	0.941227i	$-0.609674\pi$
$523$	−8079.94	−0.675547	−0.337773	+	0.941227i	$0.609674\pi$
$524$	−62122.8	−5.17910
$525$	0	0
$526$	15066.4	1.24891
$527$	−749.637	−0.0619633
$528$	9450.14	0.778910
$529$	−11702.7	−0.961839
$530$	0	0
$531$	−5888.53	−0.481244
$532$	−70228.3	−5.72327
$533$	3487.20	0.283391
$534$	−9248.14	−0.749450
$535$	0	0
$536$	9470.36	0.763167
$537$	4878.36	0.392024
$538$	8894.78	0.712791
$539$	9257.65	0.739806
$540$	0	0
$541$	−17460.5	−1.38759	−0.693793	−	0.720174i	$-0.744062\pi$
$541$	−17460.5	−1.38759	−0.693793	+	0.720174i	$0.744062\pi$
$542$	10193.2	0.807811
$543$	4916.26	0.388539
$544$	20617.9	1.62497
$545$	0	0
$546$	−41076.5	−3.21962
$547$	6652.51	0.520001	0.260001	−	0.965608i	$-0.416277\pi$
$547$	6652.51	0.520001	0.260001	+	0.965608i	$0.416277\pi$
$548$	−69733.5	−5.43589
$549$	−6954.49	−0.540638
$550$	0	0
$551$	10429.4	0.806368
$552$	−5461.76	−0.421138
$553$	−1670.42	−0.128451
$554$	15080.5	1.15652
$555$	0	0
$556$	6419.49	0.489653
$557$	−5887.04	−0.447831	−0.223916	−	0.974609i	$-0.571884\pi$
$557$	−5887.04	−0.447831	−0.223916	+	0.974609i	$0.571884\pi$
$558$	−1683.83	−0.127746
$559$	−6833.26	−0.517024
$560$	0	0
$561$	737.858	0.0555301
$562$	−16528.9	−1.24062
$563$	3208.71	0.240197	0.120099	−	0.992762i	$-0.461679\pi$
$563$	3208.71	0.240197	0.120099	+	0.992762i	$0.461679\pi$
$564$	9190.29	0.686137
$565$	0	0
$566$	−19556.2	−1.45231
$567$	−2787.86	−0.206489
$568$	46314.6	3.42134
$569$	17083.4	1.25865	0.629326	−	0.777142i	$-0.283331\pi$
$569$	17083.4	1.25865	0.629326	+	0.777142i	$0.283331\pi$
$570$	0	0
$571$	−16469.8	−1.20707	−0.603536	−	0.797336i	$-0.706242\pi$
$571$	−16469.8	−1.20707	−0.603536	+	0.797336i	$0.706242\pi$
$572$	18146.4	1.32647
$573$	10382.8	0.756974
$574$	−9395.23	−0.683187
$575$	0	0
$576$	25693.3	1.85860
$577$	12436.4	0.897287	0.448643	−	0.893711i	$-0.351907\pi$
$577$	12436.4	0.897287	0.448643	+	0.893711i	$0.351907\pi$
$578$	−24626.6	−1.77220
$579$	5177.34	0.371611
$580$	0	0
$581$	15413.9	1.10065
$582$	6929.33	0.493522
$583$	−3307.53	−0.234964
$584$	−47245.6	−3.34767
$585$	0	0
$586$	−5606.91	−0.395255
$587$	−15445.5	−1.08604	−0.543019	−	0.839720i	$-0.682719\pi$
$587$	−15445.5	−1.08604	−0.543019	+	0.839720i	$0.682719\pi$
$588$	58426.1	4.09770
$589$	−2956.24	−0.206808
$590$	0	0
$591$	−1215.83	−0.0846233
$592$	−104453.	−7.25166
$593$	−22999.6	−1.59272	−0.796360	−	0.604824i	$-0.793244\pi$
$593$	−22999.6	−1.59272	−0.796360	+	0.604824i	$0.793244\pi$
$594$	1657.38	0.114483
$595$	0	0
$596$	36900.7	2.53610
$597$	11404.3	0.781824
$598$	−8572.06	−0.586183
$599$	5960.87	0.406602	0.203301	−	0.979116i	$-0.434833\pi$
$599$	5960.87	0.406602	0.203301	+	0.979116i	$0.434833\pi$
$600$	0	0
$601$	−4282.39	−0.290653	−0.145326	−	0.989384i	$-0.546423\pi$
$601$	−4282.39	−0.290653	−0.145326	+	0.989384i	$0.546423\pi$
$602$	18410.2	1.24642
$603$	1008.78	0.0681273
$604$	51217.2	3.45033
$605$	0	0
$606$	−7358.35	−0.493255
$607$	−8197.97	−0.548180	−0.274090	−	0.961704i	$-0.588377\pi$
$607$	−8197.97	−0.548180	−0.274090	+	0.961704i	$0.588377\pi$
$608$	81307.9	5.42347
$609$	−12212.9	−0.812633
$610$	0	0
$611$	9437.38	0.624870
$612$	4656.71	0.307575
$613$	4185.87	0.275801	0.137900	−	0.990446i	$-0.455965\pi$
$613$	4185.87	0.275801	0.137900	+	0.990446i	$0.455965\pi$
$614$	−37332.9	−2.45380
$615$	0	0
$616$	−31988.3	−2.09228
$617$	21993.7	1.43506	0.717530	−	0.696528i	$-0.245273\pi$
$617$	21993.7	1.43506	0.717530	+	0.696528i	$0.245273\pi$
$618$	18620.3	1.21201
$619$	−25378.3	−1.64788	−0.823941	−	0.566676i	$-0.808229\pi$
$619$	−25378.3	−1.64788	−0.823941	+	0.566676i	$0.808229\pi$
$620$	0	0
$621$	−581.787	−0.0375947
$622$	−9488.75	−0.611679
$623$	19013.2	1.22271
$624$	61244.4	3.92907
$625$	0	0
$626$	−2450.05	−0.156428
$627$	2909.79	0.185336
$628$	−26510.8	−1.68455
$629$	−8155.58	−0.516986
$630$	0	0
$631$	17172.2	1.08338	0.541691	−	0.840578i	$-0.317784\pi$
$631$	17172.2	1.08338	0.541691	+	0.840578i	$0.317784\pi$
$632$	4100.63	0.258092
$633$	−4615.70	−0.289822
$634$	28755.2	1.80129
$635$	0	0
$636$	−20874.2	−1.30144
$637$	59996.9	3.73181
$638$	7260.55	0.450545
$639$	4933.43	0.305420
$640$	0	0
$641$	31171.2	1.92073	0.960364	−	0.278749i	$-0.0899198\pi$
$641$	31171.2	1.92073	0.960364	+	0.278749i	$0.0899198\pi$
$642$	−14463.2	−0.889123
$643$	20847.3	1.27860	0.639298	−	0.768959i	$-0.279225\pi$
$643$	20847.3	1.27860	0.639298	+	0.768959i	$0.279225\pi$
$644$	17161.8	1.05011
$645$	0	0
$646$	11002.0	0.670074
$647$	−5380.69	−0.326950	−0.163475	−	0.986547i	$-0.552270\pi$
$647$	−5380.69	−0.326950	−0.163475	+	0.986547i	$0.552270\pi$
$648$	6843.79	0.414891
$649$	−7197.09	−0.435301
$650$	0	0
$651$	3461.78	0.208415
$652$	−13762.5	−0.826660
$653$	26506.6	1.58849	0.794245	−	0.607597i	$-0.207867\pi$
$653$	26506.6	1.58849	0.794245	+	0.607597i	$0.207867\pi$
$654$	−17424.1	−1.04180
$655$	0	0
$656$	14008.1	0.833729
$657$	−5032.60	−0.298844
$658$	−25426.2	−1.50641
$659$	−25165.6	−1.48758	−0.743789	−	0.668414i	$-0.766973\pi$
$659$	−25165.6	−1.48758	−0.743789	+	0.668414i	$0.766973\pi$
$660$	0	0
$661$	2365.09	0.139170	0.0695849	−	0.997576i	$-0.477833\pi$
$661$	2365.09	0.139170	0.0695849	+	0.997576i	$0.477833\pi$
$662$	−18119.5	−1.06380
$663$	4781.91	0.280111
$664$	−37838.9	−2.21150
$665$	0	0
$666$	−18319.0	−1.06584
$667$	−2548.66	−0.147953
$668$	67743.0	3.92374
$669$	7674.56	0.443521
$670$	0	0
$671$	−8499.93	−0.489025
$672$	−95212.3	−5.46561
$673$	9349.93	0.535532	0.267766	−	0.963484i	$-0.413715\pi$
$673$	9349.93	0.535532	0.267766	+	0.963484i	$0.413715\pi$
$674$	−7467.38	−0.426755
$675$	0	0
$676$	66762.9	3.79852
$677$	767.511	0.0435714	0.0217857	−	0.999763i	$-0.493065\pi$
$677$	767.511	0.0435714	0.0217857	+	0.999763i	$0.493065\pi$
$678$	−30571.7	−1.73171
$679$	−14246.0	−0.805170
$680$	0	0
$681$	17502.3	0.984858
$682$	−2058.02	−0.115551
$683$	6514.91	0.364987	0.182494	−	0.983207i	$-0.441583\pi$
$683$	6514.91	0.364987	0.182494	+	0.983207i	$0.441583\pi$
$684$	18364.0	1.02656
$685$	0	0
$686$	−95765.1	−5.32993
$687$	−3926.83	−0.218076
$688$	−27449.3	−1.52107
$689$	−21435.4	−1.18523
$690$	0	0
$691$	30296.0	1.66789	0.833946	−	0.551846i	$-0.186076\pi$
$691$	30296.0	1.66789	0.833946	+	0.551846i	$0.186076\pi$
$692$	−29487.2	−1.61985
$693$	−3407.39	−0.186776
$694$	−19092.7	−1.04431
$695$	0	0
$696$	29980.9	1.63279
$697$	1093.74	0.0594383
$698$	−42063.9	−2.28101
$699$	−7081.73	−0.383198
$700$	0	0
$701$	24513.3	1.32076	0.660381	−	0.750931i	$-0.270395\pi$
$701$	24513.3	1.32076	0.660381	+	0.750931i	$0.270395\pi$
$702$	10741.1	0.577488
$703$	−32162.1	−1.72548
$704$	31403.0	1.68117
$705$	0	0
$706$	−32383.9	−1.72632
$707$	15128.0	0.804734
$708$	−45421.6	−2.41109
$709$	−36554.1	−1.93628	−0.968138	−	0.250419i	$-0.919432\pi$
$709$	−36554.1	−1.93628	−0.968138	+	0.250419i	$0.919432\pi$
$710$	0	0
$711$	436.799	0.0230397
$712$	−46674.6	−2.45675
$713$	722.423	0.0379452
$714$	−12883.4	−0.675280
$715$	0	0
$716$	37629.6	1.96409
$717$	17299.1	0.901044
$718$	31018.5	1.61226
$719$	−24860.8	−1.28950	−0.644752	−	0.764392i	$-0.723039\pi$
$719$	−24860.8	−1.28950	−0.644752	+	0.764392i	$0.723039\pi$
$720$	0	0
$721$	−38281.5	−1.97736
$722$	5111.18	0.263461
$723$	−7934.18	−0.408126
$724$	37922.0	1.94663
$725$	0	0
$726$	2025.68	0.103554
$727$	11414.7	0.582324	0.291162	−	0.956674i	$-0.405958\pi$
$727$	11414.7	0.582324	0.291162	+	0.956674i	$0.405958\pi$
$728$	−207309.	−10.5541
$729$	729.000	0.0370370
$730$	0	0
$731$	−2143.22	−0.108440
$732$	−53644.0	−2.70866
$733$	−11746.4	−0.591902	−0.295951	−	0.955203i	$-0.595636\pi$
$733$	−11746.4	−0.591902	−0.295951	+	0.955203i	$0.595636\pi$
$734$	−15925.2	−0.800831
$735$	0	0
$736$	−19869.4	−0.995103
$737$	1232.96	0.0616235
$738$	2456.76	0.122540
$739$	−5642.98	−0.280894	−0.140447	−	0.990088i	$-0.544854\pi$
$739$	−5642.98	−0.280894	−0.140447	+	0.990088i	$0.544854\pi$
$740$	0	0
$741$	18857.7	0.934894
$742$	57751.5	2.85731
$743$	3718.47	0.183604	0.0918018	−	0.995777i	$-0.470737\pi$
$743$	3718.47	0.183604	0.0918018	+	0.995777i	$0.470737\pi$
$744$	−8498.15	−0.418760
$745$	0	0
$746$	−41857.9	−2.05432
$747$	−4030.60	−0.197419
$748$	5691.53	0.278212
$749$	29734.8	1.45058
$750$	0	0
$751$	−11732.6	−0.570080	−0.285040	−	0.958516i	$-0.592007\pi$
$751$	−11732.6	−0.570080	−0.285040	+	0.958516i	$0.592007\pi$
$752$	37910.1	1.83835
$753$	−7306.77	−0.353617
$754$	47054.1	2.27269
$755$	0	0
$756$	−21504.4	−1.03453
$757$	−11048.3	−0.530457	−0.265229	−	0.964186i	$-0.585447\pi$
$757$	−11048.3	−0.530457	−0.265229	+	0.964186i	$0.585447\pi$
$758$	72350.2	3.46686
$759$	−711.073	−0.0340057
$760$	0	0
$761$	16924.6	0.806198	0.403099	−	0.915156i	$-0.367933\pi$
$761$	16924.6	0.806198	0.403099	+	0.915156i	$0.367933\pi$
$762$	9614.14	0.457065
$763$	35822.2	1.69967
$764$	80088.3	3.79253
$765$	0	0
$766$	−27505.4	−1.29740
$767$	−46642.8	−2.19580
$768$	74689.4	3.50927
$769$	26448.0	1.24023	0.620116	−	0.784510i	$-0.287086\pi$
$769$	26448.0	1.24023	0.620116	+	0.784510i	$0.287086\pi$
$770$	0	0
$771$	−16567.5	−0.773884
$772$	39935.8	1.86182
$773$	33612.4	1.56398	0.781988	−	0.623294i	$-0.214206\pi$
$773$	33612.4	1.56398	0.781988	+	0.623294i	$0.214206\pi$
$774$	−4814.09	−0.223565
$775$	0	0
$776$	34971.7	1.61780
$777$	37662.0	1.73889
$778$	73565.5	3.39004
$779$	4313.25	0.198380
$780$	0	0
$781$	6029.75	0.276263
$782$	−2688.58	−0.122946
$783$	3193.57	0.145758
$784$	241009.	10.9789
$785$	0	0
$786$	−44942.7	−2.03951
$787$	19267.2	0.872684	0.436342	−	0.899781i	$-0.356274\pi$
$787$	19267.2	0.872684	0.436342	+	0.899781i	$0.356274\pi$
$788$	−9378.37	−0.423973
$789$	8099.64	0.365469
$790$	0	0
$791$	62852.2	2.82524
$792$	8364.63	0.375283
$793$	−55086.3	−2.46680
$794$	−77096.1	−3.44589
$795$	0	0
$796$	87968.3	3.91703
$797$	31120.9	1.38314	0.691568	−	0.722312i	$-0.256920\pi$
$797$	31120.9	1.38314	0.691568	+	0.722312i	$0.256920\pi$
$798$	−50806.6	−2.25380
$799$	2959.99	0.131060
$800$	0	0
$801$	−4971.77	−0.219312
$802$	26857.5	1.18251
$803$	−6150.95	−0.270314
$804$	7781.32	0.341326
$805$	0	0
$806$	−13337.6	−0.582873
$807$	4781.81	0.208584
$808$	−37136.9	−1.61692
$809$	18201.5	0.791015	0.395508	−	0.918463i	$-0.370569\pi$
$809$	18201.5	0.791015	0.395508	+	0.918463i	$0.370569\pi$
$810$	0	0
$811$	−3221.28	−0.139475	−0.0697376	−	0.997565i	$-0.522216\pi$
$811$	−3221.28	−0.139475	−0.0697376	+	0.997565i	$0.522216\pi$
$812$	−94205.5	−4.07139
$813$	5479.81	0.236390
$814$	−22389.9	−0.964087
$815$	0	0
$816$	19209.0	0.824080
$817$	−8451.92	−0.361928
$818$	−54556.0	−2.33191
$819$	−22082.6	−0.942159
$820$	0	0
$821$	6884.62	0.292661	0.146331	−	0.989236i	$-0.453254\pi$
$821$	6884.62	0.292661	0.146331	+	0.989236i	$0.453254\pi$
$822$	−50448.7	−2.14063
$823$	19720.3	0.835246	0.417623	−	0.908620i	$-0.362863\pi$
$823$	19720.3	0.835246	0.417623	+	0.908620i	$0.362863\pi$
$824$	93975.2	3.97304
$825$	0	0
$826$	125665.	5.29353
$827$	−17844.0	−0.750297	−0.375148	−	0.926965i	$-0.622408\pi$
$827$	−17844.0	−0.750297	−0.375148	+	0.926965i	$0.622408\pi$
$828$	−4487.66	−0.188354
$829$	−31143.1	−1.30476	−0.652379	−	0.757893i	$-0.726229\pi$
$829$	−31143.1	−1.30476	−0.652379	+	0.757893i	$0.726229\pi$
$830$	0	0
$831$	8107.24	0.338432
$832$	203516.	8.48035
$833$	18817.7	0.782708
$834$	4644.18	0.192824
$835$	0	0
$836$	22444.9	0.928557
$837$	−905.223	−0.0373824
$838$	78559.5	3.23842
$839$	33261.2	1.36866	0.684329	−	0.729174i	$-0.260095\pi$
$839$	33261.2	1.36866	0.684329	+	0.729174i	$0.260095\pi$
$840$	0	0
$841$	−10398.8	−0.426371
$842$	95429.2	3.90583
$843$	−8885.87	−0.363043
$844$	−35603.5	−1.45204
$845$	0	0
$846$	6648.72	0.270198
$847$	−4164.59	−0.168946
$848$	−86106.5	−3.48692
$849$	−10513.3	−0.424990
$850$	0	0
$851$	7859.52	0.316593
$852$	38054.4	1.53019
$853$	419.167	0.0168253	0.00841266	−	0.999965i	$-0.497322\pi$
$853$	419.167	0.0168253	0.00841266	+	0.999965i	$0.497322\pi$
$854$	148414.	5.94685
$855$	0	0
$856$	−72994.5	−2.91460
$857$	−8915.18	−0.355352	−0.177676	−	0.984089i	$-0.556858\pi$
$857$	−8915.18	−0.355352	−0.177676	+	0.984089i	$0.556858\pi$
$858$	13128.0	0.522358
$859$	4449.47	0.176733	0.0883667	−	0.996088i	$-0.471835\pi$
$859$	4449.47	0.176733	0.0883667	+	0.996088i	$0.471835\pi$
$860$	0	0
$861$	−5050.85	−0.199922
$862$	−24412.8	−0.964623
$863$	25151.7	0.992089	0.496044	−	0.868297i	$-0.334785\pi$
$863$	25151.7	0.992089	0.496044	+	0.868297i	$0.334785\pi$
$864$	24897.1	0.980343
$865$	0	0
$866$	30070.6	1.17995
$867$	−13239.2	−0.518600
$868$	26702.7	1.04418
$869$	533.866	0.0208402
$870$	0	0
$871$	7990.53	0.310848
$872$	−87938.0	−3.41509
$873$	3725.18	0.144420
$874$	−10602.6	−0.410341
$875$	0	0
$876$	−38819.4	−1.49724
$877$	16936.9	0.652129	0.326065	−	0.945348i	$-0.394277\pi$
$877$	16936.9	0.652129	0.326065	+	0.945348i	$0.394277\pi$
$878$	30016.4	1.15376
$879$	−3014.26	−0.115664
$880$	0	0
$881$	−32920.0	−1.25891	−0.629456	−	0.777036i	$-0.716722\pi$
$881$	−32920.0	−1.25891	−0.629456	+	0.777036i	$0.716722\pi$
$882$	42268.3	1.61366
$883$	−23658.9	−0.901683	−0.450842	−	0.892604i	$-0.648876\pi$
$883$	−23658.9	−0.901683	−0.450842	+	0.892604i	$0.648876\pi$
$884$	36885.6	1.40339
$885$	0	0
$886$	38350.4	1.45418
$887$	−19319.2	−0.731312	−0.365656	−	0.930750i	$-0.619155\pi$
$887$	−19319.2	−0.731312	−0.365656	+	0.930750i	$0.619155\pi$
$888$	−92454.6	−3.49389
$889$	−19765.7	−0.745691
$890$	0	0
$891$	891.000	0.0335013
$892$	59198.3	2.22209
$893$	11672.9	0.437423
$894$	26695.8	0.998705
$895$	0	0
$896$	−294414.	−10.9773
$897$	−4608.31	−0.171535
$898$	45742.3	1.69982
$899$	−3965.56	−0.147118
$900$	0	0
$901$	−6723.12	−0.248590
$902$	3002.71	0.110842
$903$	9897.27	0.364740
$904$	−154293.	−5.67666
$905$	0	0
$906$	37053.1	1.35873
$907$	13020.2	0.476657	0.238329	−	0.971185i	$-0.423400\pi$
$907$	13020.2	0.476657	0.238329	+	0.971185i	$0.423400\pi$
$908$	135005.	4.93425
$909$	−3955.83	−0.144342
$910$	0	0
$911$	260.135	0.00946065	0.00473032	−	0.999989i	$-0.498494\pi$
$911$	260.135	0.00946065	0.00473032	+	0.999989i	$0.498494\pi$
$912$	75751.9	2.75043
$913$	−4926.29	−0.178572
$914$	20799.9	0.752735
$915$	0	0
$916$	−30289.9	−1.09258
$917$	92397.5	3.32741
$918$	3368.89	0.121122
$919$	23631.7	0.848247	0.424123	−	0.905604i	$-0.360582\pi$
$919$	23631.7	0.848247	0.424123	+	0.905604i	$0.360582\pi$
$920$	0	0
$921$	−20070.0	−0.718057
$922$	32372.2	1.15631
$923$	39077.5	1.39356
$924$	−26283.2	−0.935772
$925$	0	0
$926$	−69211.4	−2.45619
$927$	10010.2	0.354670
$928$	109068.	3.85812
$929$	19536.7	0.689966	0.344983	−	0.938609i	$-0.387885\pi$
$929$	19536.7	0.689966	0.344983	+	0.938609i	$0.387885\pi$
$930$	0	0
$931$	74208.9	2.61235
$932$	−54625.5	−1.91987
$933$	−5101.12	−0.178996
$934$	−63302.4	−2.21769
$935$	0	0
$936$	54209.4	1.89305
$937$	−17072.3	−0.595227	−0.297614	−	0.954686i	$-0.596191\pi$
$937$	−17072.3	−0.595227	−0.297614	+	0.954686i	$0.596191\pi$
$938$	−21528.1	−0.749380
$939$	−1317.14	−0.0457755
$940$	0	0
$941$	−24908.1	−0.862890	−0.431445	−	0.902139i	$-0.641996\pi$
$941$	−24908.1	−0.862890	−0.431445	+	0.902139i	$0.641996\pi$
$942$	−19179.3	−0.663370
$943$	−1054.04	−0.0363989
$944$	−187365.	−6.45997
$945$	0	0
$946$	−5883.89	−0.202222
$947$	42208.2	1.44835	0.724173	−	0.689619i	$-0.242222\pi$
$947$	42208.2	1.44835	0.724173	+	0.689619i	$0.242222\pi$
$948$	3369.29	0.115432
$949$	−39863.0	−1.36355
$950$	0	0
$951$	15458.7	0.527112
$952$	−65021.6	−2.21361
$953$	38261.2	1.30053	0.650263	−	0.759709i	$-0.274659\pi$
$953$	38261.2	1.30053	0.650263	+	0.759709i	$0.274659\pi$
$954$	−15101.5	−0.512503
$955$	0	0
$956$	133438.	4.51433
$957$	3903.25	0.131843
$958$	−13303.6	−0.448663
$959$	103717.	3.49239
$960$	0	0
$961$	−28667.0	−0.962269
$962$	−145105.	−4.86316
$963$	−7775.37	−0.260185
$964$	−61200.9	−2.04476
$965$	0	0
$966$	12415.7	0.413530
$967$	−2841.88	−0.0945074	−0.0472537	−	0.998883i	$-0.515047\pi$
$967$	−2841.88	−0.0945074	−0.0472537	+	0.998883i	$0.515047\pi$
$968$	10223.4	0.339456
$969$	5914.64	0.196084
$970$	0	0
$971$	−9655.22	−0.319105	−0.159552	−	0.987189i	$-0.551005\pi$
$971$	−9655.22	−0.319105	−0.159552	+	0.987189i	$0.551005\pi$
$972$	5623.20	0.185560
$973$	−9547.95	−0.314587
$974$	−62426.3	−2.05366
$975$	0	0
$976$	−221282.	−7.25725
$977$	−20818.5	−0.681722	−0.340861	−	0.940114i	$-0.610719\pi$
$977$	−20818.5	−0.681722	−0.340861	+	0.940114i	$0.610719\pi$
$978$	−9956.51	−0.325536
$979$	−6076.61	−0.198375
$980$	0	0
$981$	−9367.16	−0.304863
$982$	−52044.2	−1.69124
$983$	5762.06	0.186959	0.0934797	−	0.995621i	$-0.470201\pi$
$983$	5762.06	0.186959	0.0934797	+	0.995621i	$0.470201\pi$
$984$	12399.1	0.401695
$985$	0	0
$986$	14758.3	0.476673
$987$	−13669.1	−0.440822
$988$	145461.	4.68393
$989$	2065.42	0.0664069
$990$	0	0
$991$	48292.2	1.54798	0.773992	−	0.633195i	$-0.218257\pi$
$991$	48292.2	1.54798	0.773992	+	0.633195i	$0.218257\pi$
$992$	−30915.5	−0.989485
$993$	−9740.98	−0.311300
$994$	−105283.	−3.35953
$995$	0	0
$996$	−31090.3	−0.989091
$997$	−42833.0	−1.36062	−0.680308	−	0.732926i	$-0.738154\pi$
$997$	−42833.0	−1.36062	−0.680308	+	0.732926i	$0.738154\pi$
$998$	106690.	3.38398
$999$	−9848.26	−0.311897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.a.z.1.5	yes	5
3.2	odd	2		2475.4.a.bf.1.1		5
5.2	odd	4		825.4.c.r.199.10		10
5.3	odd	4		825.4.c.r.199.1		10
5.4	even	2		825.4.a.w.1.1	✓	5
15.14	odd	2		2475.4.a.bm.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.w.1.1	✓	5	5.4	even	2
825.4.a.z.1.5	yes	5	1.1	even	1	trivial
825.4.c.r.199.1		10	5.3	odd	4
825.4.c.r.199.10		10	5.2	odd	4
2475.4.a.bf.1.1		5	3.2	odd	2
2475.4.a.bm.1.5		5	15.14	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+5.58039 q^{2} +3.00000 q^{3} +23.1407 q^{4} +16.7412 q^{6} -34.4181 q^{7} +84.4912 q^{8} +9.00000 q^{9} +11.0000 q^{11} +69.4222 q^{12} +71.2887 q^{13} -192.066 q^{14} +286.368 q^{16} +22.3593 q^{17} +50.2235 q^{18} +88.1755 q^{19} -103.254 q^{21} +61.3843 q^{22} -21.5477 q^{23} +253.474 q^{24} +397.819 q^{26} +27.0000 q^{27} -796.460 q^{28} +118.280 q^{29} -33.5268 q^{31} +922.115 q^{32} +33.0000 q^{33} +124.774 q^{34} +208.267 q^{36} -364.750 q^{37} +492.054 q^{38} +213.866 q^{39} +48.9166 q^{41} -576.199 q^{42} -95.8534 q^{43} +254.548 q^{44} -120.244 q^{46} +132.383 q^{47} +859.104 q^{48} +841.605 q^{49} +67.0780 q^{51} +1649.67 q^{52} -300.685 q^{53} +150.671 q^{54} -2908.03 q^{56} +264.526 q^{57} +660.050 q^{58} -654.281 q^{59} -772.721 q^{61} -187.092 q^{62} -309.763 q^{63} +2854.82 q^{64} +184.153 q^{66} +112.087 q^{67} +517.412 q^{68} -64.6430 q^{69} +548.159 q^{71} +760.421 q^{72} -559.178 q^{73} -2035.45 q^{74} +2040.45 q^{76} -378.599 q^{77} +1193.46 q^{78} +48.5332 q^{79} +81.0000 q^{81} +272.974 q^{82} -447.844 q^{83} -2389.38 q^{84} -534.899 q^{86} +354.841 q^{87} +929.403 q^{88} -552.419 q^{89} -2453.62 q^{91} -498.629 q^{92} -100.580 q^{93} +738.746 q^{94} +2766.34 q^{96} +413.909 q^{97} +4696.48 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 15 q^{3} + 46 q^{4} + 12 q^{6} - 38 q^{7} + 72 q^{8} + 45 q^{9} + 55 q^{11} + 138 q^{12} + 43 q^{13} + 55 q^{14} + 634 q^{16} + 150 q^{17} + 36 q^{18} - 337 q^{19} - 114 q^{21} + 44 q^{22}+ \cdots + 495 q^{99}+O(q^{100}) \) 5 * q + 4 * q^2 + 15 * q^3 + 46 * q^4 + 12 * q^6 - 38 * q^7 + 72 * q^8 + 45 * q^9 + 55 * q^11 + 138 * q^12 + 43 * q^13 + 55 * q^14 + 634 * q^16 + 150 * q^17 + 36 * q^18 - 337 * q^19 - 114 * q^21 + 44 * q^22 - 201 * q^23 + 216 * q^24 + 359 * q^26 + 135 * q^27 - 441 * q^28 - 153 * q^29 + 263 * q^31 + 280 * q^32 + 165 * q^33 + 517 * q^34 + 414 * q^36 - 350 * q^37 + 786 * q^38 + 129 * q^39 - 88 * q^41 + 165 * q^42 + 161 * q^43 + 506 * q^44 + 436 * q^46 + 490 * q^47 + 1902 * q^48 + 2381 * q^49 + 450 * q^51 + 289 * q^52 + 558 * q^53 + 108 * q^54 + 211 * q^56 - 1011 * q^57 - 969 * q^58 + 410 * q^59 - 1130 * q^61 + 463 * q^62 - 342 * q^63 + 3170 * q^64 + 132 * q^66 + 1130 * q^67 + 3333 * q^68 - 603 * q^69 + 1249 * q^71 + 648 * q^72 - 44 * q^73 - 2985 * q^74 - 2556 * q^76 - 418 * q^77 + 1077 * q^78 + 2258 * q^79 + 405 * q^81 - 3737 * q^82 - 957 * q^83 - 1323 * q^84 - 1435 * q^86 - 459 * q^87 + 792 * q^88 + 347 * q^89 - 612 * q^91 - 6006 * q^92 + 789 * q^93 + 907 * q^94 + 840 * q^96 - 2355 * q^97 + 9417 * q^98 + 495 * q^99

Introduction

L-functions

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