LMFDB - Embedded newform 847.4.a.n.1.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [847,4,Mod(1,847)] 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("847.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(847, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	847.a (trivial)

Newform invariants

comment:select newform

sage:traces = [14,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.9746177749$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 67 x^{12} + 380 x^{11} + 1774 x^{10} - 8872 x^{9} - 23849 x^{8} + 93880 x^{7} + \cdots - 1952 $ x^14 - 6x^13 - 67x^12 + 380x^11 + 1774x^10 - 8872x^9 - 23849x^8 + 93880x^7 + 172119x^6 - 447614x^5 - 618482x^4 + 795744x^3 + 877632x^2 - 118560*x - 1952
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-4.41395$ of defining polynomial
Character	$\chi$	$=$	847.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.41395 q^{2} +8.77003 q^{3} +21.3108 q^{4} -13.8023 q^{5} +47.4805 q^{6} -7.00000 q^{7} +72.0641 q^{8} +49.9135 q^{9} -74.7248 q^{10} +186.897 q^{12} +30.0089 q^{13} -37.8976 q^{14} -121.046 q^{15} +219.665 q^{16} +106.800 q^{17} +270.229 q^{18} -52.2797 q^{19} -294.138 q^{20} -61.3902 q^{21} +86.9878 q^{23} +632.004 q^{24} +65.5030 q^{25} +162.467 q^{26} +200.952 q^{27} -149.176 q^{28} -187.342 q^{29} -655.339 q^{30} +180.697 q^{31} +612.739 q^{32} +578.208 q^{34} +96.6160 q^{35} +1063.70 q^{36} -364.432 q^{37} -283.040 q^{38} +263.179 q^{39} -994.649 q^{40} +128.976 q^{41} -332.363 q^{42} -48.2610 q^{43} -688.920 q^{45} +470.948 q^{46} -455.085 q^{47} +1926.46 q^{48} +49.0000 q^{49} +354.630 q^{50} +936.636 q^{51} +639.515 q^{52} -389.192 q^{53} +1087.94 q^{54} -504.449 q^{56} -458.495 q^{57} -1014.26 q^{58} +146.448 q^{59} -2579.60 q^{60} -194.654 q^{61} +978.285 q^{62} -349.394 q^{63} +1560.02 q^{64} -414.192 q^{65} -302.780 q^{67} +2275.99 q^{68} +762.886 q^{69} +523.074 q^{70} -395.902 q^{71} +3596.97 q^{72} +552.678 q^{73} -1973.01 q^{74} +574.463 q^{75} -1114.12 q^{76} +1424.84 q^{78} -824.060 q^{79} -3031.87 q^{80} +414.690 q^{81} +698.270 q^{82} +1041.32 q^{83} -1308.28 q^{84} -1474.08 q^{85} -261.282 q^{86} -1642.99 q^{87} -806.869 q^{89} -3729.77 q^{90} -210.063 q^{91} +1853.78 q^{92} +1584.72 q^{93} -2463.81 q^{94} +721.579 q^{95} +5373.74 q^{96} +524.575 q^{97} +265.283 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 8 q^{2} + 4 q^{3} + 60 q^{4} + 6 q^{5} + 34 q^{6} - 98 q^{7} + 96 q^{8} + 58 q^{9} + 4 q^{10} - 14 q^{12} + 164 q^{13} - 56 q^{14} - 240 q^{15} + 356 q^{16} + 276 q^{17} + 516 q^{18} - 12 q^{19}+ \cdots + 392 q^{98}+O(q^{100}) $ 14 * q + 8 * q^2 + 4 * q^3 + 60 * q^4 + 6 * q^5 + 34 * q^6 - 98 * q^7 + 96 * q^8 + 58 * q^9 + 4 * q^10 - 14 * q^12 + 164 * q^13 - 56 * q^14 - 240 * q^15 + 356 * q^16 + 276 * q^17 + 516 * q^18 - 12 * q^19 - 122 * q^20 - 28 * q^21 - 416 * q^23 + 836 * q^24 + 308 * q^25 + 186 * q^26 - 320 * q^27 - 420 * q^28 + 544 * q^29 + 40 * q^30 + 280 * q^31 + 1516 * q^32 + 488 * q^34 - 42 * q^35 + 830 * q^36 - 734 * q^37 - 570 * q^38 + 948 * q^39 + 1182 * q^40 + 1584 * q^41 - 238 * q^42 + 136 * q^43 + 706 * q^45 - 568 * q^46 - 356 * q^47 + 4008 * q^48 + 686 * q^49 + 640 * q^50 + 1056 * q^51 + 3060 * q^52 + 318 * q^53 - 2064 * q^54 - 672 * q^56 + 412 * q^57 - 1222 * q^58 - 944 * q^59 - 1976 * q^60 + 1808 * q^61 + 3788 * q^62 - 406 * q^63 + 3726 * q^64 + 916 * q^65 - 1156 * q^67 + 1712 * q^68 + 664 * q^69 - 28 * q^70 - 1968 * q^71 + 2972 * q^72 + 3948 * q^73 + 3396 * q^74 + 668 * q^75 - 5626 * q^76 + 1460 * q^78 - 1596 * q^79 - 340 * q^80 + 1250 * q^81 + 3916 * q^82 + 2456 * q^83 + 98 * q^84 + 4456 * q^85 - 1226 * q^86 - 2972 * q^87 + 1330 * q^89 - 4290 * q^90 - 1148 * q^91 - 3142 * q^92 - 1532 * q^93 + 1286 * q^94 + 6092 * q^95 + 13718 * q^96 - 2514 * q^97 + 392 * q^98

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 $

$2$	5.41395	1.91412	0.957060	−	0.289891i	$-0.0936191\pi$
$2$	5.41395	1.91412	0.957060	+	0.289891i	$0.0936191\pi$
$3$	8.77003	1.68779	0.843897	−	0.536506i	$-0.180256\pi$
$3$	8.77003	1.68779	0.843897	+	0.536506i	$0.180256\pi$
$4$	21.3108	2.66385
$5$	−13.8023	−1.23451	−0.617257	−	0.786762i	$-0.711756\pi$
$5$	−13.8023	−1.23451	−0.617257	+	0.786762i	$0.711756\pi$
$6$	47.4805	3.23064
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	72.0641	3.18481
$9$	49.9135	1.84865
$10$	−74.7248	−2.36301
$11$	0	0
$11$	0	0
$12$	186.897	4.49603
$13$	30.0089	0.640229	0.320115	−	0.947379i	$-0.396279\pi$
$13$	30.0089	0.640229	0.320115	+	0.947379i	$0.396279\pi$
$14$	−37.8976	−0.723469
$15$	−121.046	−2.08360
$16$	219.665	3.43226
$17$	106.800	1.52369	0.761845	−	0.647760i	$-0.224294\pi$
$17$	106.800	1.52369	0.761845	+	0.647760i	$0.224294\pi$
$18$	270.229	3.53853
$19$	−52.2797	−0.631252	−0.315626	−	0.948884i	$-0.602214\pi$
$19$	−52.2797	−0.631252	−0.315626	+	0.948884i	$0.602214\pi$
$20$	−294.138	−3.28856
$21$	−61.3902	−0.637926
$22$	0	0
$23$	86.9878	0.788618	0.394309	−	0.918978i	$-0.370984\pi$
$23$	86.9878	0.788618	0.394309	+	0.918978i	$0.370984\pi$
$24$	632.004	5.37531
$25$	65.5030	0.524024
$26$	162.467	1.22548
$27$	200.952	1.43234
$28$	−149.176	−1.00684
$29$	−187.342	−1.19960	−0.599802	−	0.800149i	$-0.704754\pi$
$29$	−187.342	−1.19960	−0.599802	+	0.800149i	$0.704754\pi$
$30$	−655.339	−3.98827
$31$	180.697	1.04691	0.523454	−	0.852054i	$-0.324643\pi$
$31$	180.697	1.04691	0.523454	+	0.852054i	$0.324643\pi$
$32$	612.739	3.38494
$33$	0	0
$34$	578.208	2.91652
$35$	96.6160	0.466602
$36$	1063.70	4.92452
$37$	−364.432	−1.61925	−0.809624	−	0.586949i	$-0.800329\pi$
$37$	−364.432	−1.61925	−0.809624	+	0.586949i	$0.800329\pi$
$38$	−283.040	−1.20829
$39$	263.179	1.08058
$40$	−994.649	−3.93169
$41$	128.976	0.491285	0.245643	−	0.969360i	$-0.421001\pi$
$41$	128.976	0.491285	0.245643	+	0.969360i	$0.421001\pi$
$42$	−332.363	−1.22107
$43$	−48.2610	−0.171157	−0.0855783	−	0.996331i	$-0.527274\pi$
$43$	−48.2610	−0.171157	−0.0855783	+	0.996331i	$0.527274\pi$
$44$	0	0
$45$	−688.920	−2.28218
$46$	470.948	1.50951
$47$	−455.085	−1.41236	−0.706181	−	0.708032i	$-0.749583\pi$
$47$	−455.085	−1.41236	−0.706181	+	0.708032i	$0.749583\pi$
$48$	1926.46	5.79294
$49$	49.0000	0.142857
$50$	354.630	1.00304
$51$	936.636	2.57167
$52$	639.515	1.70548
$53$	−389.192	−1.00867	−0.504337	−	0.863507i	$-0.668263\pi$
$53$	−389.192	−1.00867	−0.504337	+	0.863507i	$0.668263\pi$
$54$	1087.94	2.74167
$55$	0	0
$56$	−504.449	−1.20375
$57$	−458.495	−1.06542
$58$	−1014.26	−2.29618
$59$	146.448	0.323152	0.161576	−	0.986860i	$-0.448342\pi$
$59$	146.448	0.323152	0.161576	+	0.986860i	$0.448342\pi$
$60$	−2579.60	−5.55041
$61$	−194.654	−0.408573	−0.204286	−	0.978911i	$-0.565487\pi$
$61$	−194.654	−0.408573	−0.204286	+	0.978911i	$0.565487\pi$
$62$	978.285	2.00391
$63$	−349.394	−0.698723
$64$	1560.02	3.04692
$65$	−414.192	−0.790372
$66$	0	0
$67$	−302.780	−0.552096	−0.276048	−	0.961144i	$-0.589025\pi$
$67$	−302.780	−0.552096	−0.276048	+	0.961144i	$0.589025\pi$
$68$	2275.99	4.05888
$69$	762.886	1.33102
$70$	523.074	0.893132
$71$	−395.902	−0.661760	−0.330880	−	0.943673i	$-0.607345\pi$
$71$	−395.902	−0.661760	−0.330880	+	0.943673i	$0.607345\pi$
$72$	3596.97	5.88759
$73$	552.678	0.886111	0.443055	−	0.896494i	$-0.353895\pi$
$73$	552.678	0.886111	0.443055	+	0.896494i	$0.353895\pi$
$74$	−1973.01	−3.09943
$75$	574.463	0.884444
$76$	−1114.12	−1.68156
$77$	0	0
$78$	1424.84	2.06835
$79$	−824.060	−1.17359	−0.586797	−	0.809734i	$-0.699611\pi$
$79$	−824.060	−1.17359	−0.586797	+	0.809734i	$0.699611\pi$
$80$	−3031.87	−4.23717
$81$	414.690	0.568847
$82$	698.270	0.940379
$83$	1041.32	1.37710	0.688550	−	0.725188i	$-0.258247\pi$
$83$	1041.32	1.37710	0.688550	+	0.725188i	$0.258247\pi$
$84$	−1308.28	−1.69934
$85$	−1474.08	−1.88101
$86$	−261.282	−0.327614
$87$	−1642.99	−2.02468
$88$	0	0
$89$	−806.869	−0.960988	−0.480494	−	0.876998i	$-0.659543\pi$
$89$	−806.869	−0.960988	−0.480494	+	0.876998i	$0.659543\pi$
$90$	−3729.77	−4.36836
$91$	−210.063	−0.241984
$92$	1853.78	2.10076
$93$	1584.72	1.76697
$94$	−2463.81	−2.70343
$95$	721.579	0.779289
$96$	5373.74	5.71308
$97$	524.575	0.549098	0.274549	−	0.961573i	$-0.411471\pi$
$97$	524.575	0.549098	0.274549	+	0.961573i	$0.411471\pi$
$98$	265.283	0.273446
$99$	0	0
$100$	1395.92	1.39592
$101$	800.638	0.788777	0.394388	−	0.918944i	$-0.370956\pi$
$101$	800.638	0.788777	0.394388	+	0.918944i	$0.370956\pi$
$102$	5070.90	4.92249
$103$	−35.8170	−0.0342636	−0.0171318	−	0.999853i	$-0.505453\pi$
$103$	−35.8170	−0.0342636	−0.0171318	+	0.999853i	$0.505453\pi$
$104$	2162.57	2.03901
$105$	847.325	0.787528
$106$	−2107.07	−1.93072
$107$	−151.852	−0.137197	−0.0685987	−	0.997644i	$-0.521853\pi$
$107$	−151.852	−0.137197	−0.0685987	+	0.997644i	$0.521853\pi$
$108$	4282.45	3.81554
$109$	1722.37	1.51352	0.756759	−	0.653694i	$-0.226782\pi$
$109$	1722.37	1.51352	0.756759	+	0.653694i	$0.226782\pi$
$110$	0	0
$111$	−3196.08	−2.73296
$112$	−1537.65	−1.29727
$113$	−1851.03	−1.54097	−0.770487	−	0.637455i	$-0.779987\pi$
$113$	−1851.03	−1.54097	−0.770487	+	0.637455i	$0.779987\pi$
$114$	−2482.27	−2.03935
$115$	−1200.63	−0.973560
$116$	−3992.41	−3.19557
$117$	1497.85	1.18356
$118$	792.864	0.618551
$119$	−747.597	−0.575900
$120$	−8723.10	−6.63589
$121$	0	0
$122$	−1053.85	−0.782057
$123$	1131.13	0.829188
$124$	3850.81	2.78881
$125$	821.195	0.587599
$126$	−1891.60	−1.33744
$127$	−468.009	−0.327001	−0.163500	−	0.986543i	$-0.552279\pi$
$127$	−468.009	−0.327001	−0.163500	+	0.986543i	$0.552279\pi$
$128$	3543.96	2.44723
$129$	−423.250	−0.288877
$130$	−2242.41	−1.51287
$131$	−1581.15	−1.05455	−0.527274	−	0.849695i	$-0.676786\pi$
$131$	−1581.15	−1.05455	−0.527274	+	0.849695i	$0.676786\pi$
$132$	0	0
$133$	365.958	0.238591
$134$	−1639.23	−1.05678
$135$	−2773.59	−1.76824
$136$	7696.42	4.85266
$137$	1236.62	0.771177	0.385589	−	0.922671i	$-0.373998\pi$
$137$	1236.62	0.771177	0.385589	+	0.922671i	$0.373998\pi$
$138$	4130.22	2.54774
$139$	−1064.09	−0.649314	−0.324657	−	0.945832i	$-0.605249\pi$
$139$	−1064.09	−0.649314	−0.324657	+	0.945832i	$0.605249\pi$
$140$	2058.97	1.24296
$141$	−3991.11	−2.38377
$142$	−2143.39	−1.26669
$143$	0	0
$144$	10964.2	6.34503
$145$	2585.74	1.48093
$146$	2992.17	1.69612
$147$	429.732	0.241113
$148$	−7766.34	−4.31344
$149$	−2314.77	−1.27270	−0.636352	−	0.771398i	$-0.719558\pi$
$149$	−2314.77	−1.27270	−0.636352	+	0.771398i	$0.719558\pi$
$150$	3110.11	1.69293
$151$	1822.45	0.982178	0.491089	−	0.871109i	$-0.336599\pi$
$151$	1822.45	0.982178	0.491089	+	0.871109i	$0.336599\pi$
$152$	−3767.49	−2.01042
$153$	5330.74	2.81676
$154$	0	0
$155$	−2494.03	−1.29242
$156$	5608.57	2.87849
$157$	−2697.54	−1.37125	−0.685627	−	0.727953i	$-0.740472\pi$
$157$	−2697.54	−1.37125	−0.685627	+	0.727953i	$0.740472\pi$
$158$	−4461.41	−2.24640
$159$	−3413.23	−1.70243
$160$	−8457.20	−4.17875
$161$	−608.915	−0.298070
$162$	2245.11	1.08884
$163$	742.028	0.356565	0.178283	−	0.983979i	$-0.442946\pi$
$163$	742.028	0.356565	0.178283	+	0.983979i	$0.442946\pi$
$164$	2748.59	1.30871
$165$	0	0
$166$	5637.63	2.63594
$167$	−2732.03	−1.26593	−0.632966	−	0.774179i	$-0.718163\pi$
$167$	−2732.03	−1.26593	−0.632966	+	0.774179i	$0.718163\pi$
$168$	−4424.03	−2.03167
$169$	−1296.46	−0.590106
$170$	−7980.58	−3.60049
$171$	−2609.46	−1.16696
$172$	−1028.48	−0.455936
$173$	−230.194	−0.101164	−0.0505818	−	0.998720i	$-0.516108\pi$
$173$	−230.194	−0.101164	−0.0505818	+	0.998720i	$0.516108\pi$
$174$	−8895.08	−3.87548
$175$	−458.521	−0.198062
$176$	0	0
$177$	1284.36	0.545414
$178$	−4368.35	−1.83945
$179$	−3463.69	−1.44630	−0.723151	−	0.690690i	$-0.757307\pi$
$179$	−3463.69	−1.44630	−0.723151	+	0.690690i	$0.757307\pi$
$180$	−14681.4	−6.07939
$181$	838.524	0.344348	0.172174	−	0.985067i	$-0.444921\pi$
$181$	838.524	0.344348	0.172174	+	0.985067i	$0.444921\pi$
$182$	−1137.27	−0.463186
$183$	−1707.13	−0.689587
$184$	6268.70	2.51160
$185$	5029.99	1.99898
$186$	8579.59	3.38218
$187$	0	0
$188$	−9698.23	−3.76232
$189$	−1406.66	−0.541374
$190$	3906.59	1.49165
$191$	−3240.80	−1.22773	−0.613865	−	0.789411i	$-0.710386\pi$
$191$	−3240.80	−1.22773	−0.613865	+	0.789411i	$0.710386\pi$
$192$	13681.4	5.14257
$193$	2849.42	1.06273	0.531363	−	0.847144i	$-0.321680\pi$
$193$	2849.42	1.06273	0.531363	+	0.847144i	$0.321680\pi$
$194$	2840.02	1.05104
$195$	−3632.48	−1.33398
$196$	1044.23	0.380550
$197$	3620.43	1.30936	0.654682	−	0.755904i	$-0.272802\pi$
$197$	3620.43	1.30936	0.654682	+	0.755904i	$0.272802\pi$
$198$	0	0
$199$	1456.55	0.518855	0.259428	−	0.965763i	$-0.416466\pi$
$199$	1456.55	0.518855	0.259428	+	0.965763i	$0.416466\pi$
$200$	4720.41	1.66892
$201$	−2655.39	−0.931825
$202$	4334.61	1.50981
$203$	1311.39	0.453408
$204$	19960.5	6.85056
$205$	−1780.17	−0.606498
$206$	−193.911	−0.0655846
$207$	4341.86	1.45788
$208$	6591.90	2.19743
$209$	0	0
$210$	4587.37	1.50742
$211$	4057.90	1.32397	0.661984	−	0.749518i	$-0.269715\pi$
$211$	4057.90	1.32397	0.661984	+	0.749518i	$0.269715\pi$
$212$	−8294.01	−2.68696
$213$	−3472.07	−1.11691
$214$	−822.121	−0.262612
$215$	666.112	0.211295
$216$	14481.4	4.56173
$217$	−1264.88	−0.395694
$218$	9324.84	2.89705
$219$	4847.00	1.49557
$220$	0	0
$221$	3204.94	0.975511
$222$	−17303.4	−5.23120
$223$	1589.64	0.477354	0.238677	−	0.971099i	$-0.423286\pi$
$223$	1589.64	0.477354	0.238677	+	0.971099i	$0.423286\pi$
$224$	−4289.18	−1.27939
$225$	3269.48	0.968735
$226$	−10021.4	−2.94961
$227$	577.950	0.168986	0.0844932	−	0.996424i	$-0.473073\pi$
$227$	577.950	0.168986	0.0844932	+	0.996424i	$0.473073\pi$
$228$	−9770.90	−2.83813
$229$	337.007	0.0972492	0.0486246	−	0.998817i	$-0.484516\pi$
$229$	337.007	0.0972492	0.0486246	+	0.998817i	$0.484516\pi$
$230$	−6500.15	−1.86351
$231$	0	0
$232$	−13500.6	−3.82051
$233$	5367.41	1.50915	0.754573	−	0.656217i	$-0.227844\pi$
$233$	5367.41	1.50915	0.754573	+	0.656217i	$0.227844\pi$
$234$	8109.28	2.26547
$235$	6281.21	1.74358
$236$	3120.94	0.860829
$237$	−7227.03	−1.98078
$238$	−4047.45	−1.10234
$239$	4056.64	1.09792	0.548959	−	0.835849i	$-0.315024\pi$
$239$	4056.64	1.09792	0.548959	+	0.835849i	$0.315024\pi$
$240$	−26589.6	−7.15147
$241$	1544.94	0.412939	0.206469	−	0.978453i	$-0.433803\pi$
$241$	1544.94	0.412939	0.206469	+	0.978453i	$0.433803\pi$
$242$	0	0
$243$	−1788.86	−0.472243
$244$	−4148.25	−1.08838
$245$	−676.312	−0.176359
$246$	6123.85	1.58716
$247$	−1568.86	−0.404146
$248$	13021.8	3.33421
$249$	9132.38	2.32426
$250$	4445.90	1.12473
$251$	5397.44	1.35730	0.678652	−	0.734460i	$-0.262565\pi$
$251$	5397.44	1.35730	0.678652	+	0.734460i	$0.262565\pi$
$252$	−7445.88	−1.86129
$253$	0	0
$254$	−2533.78	−0.625919
$255$	−12927.7	−3.17476
$256$	6706.65	1.63737
$257$	2509.96	0.609209	0.304605	−	0.952479i	$-0.401476\pi$
$257$	2509.96	0.609209	0.304605	+	0.952479i	$0.401476\pi$
$258$	−2291.46	−0.552945
$259$	2551.02	0.612018
$260$	−8826.77	−2.10543
$261$	−9350.88	−2.21764
$262$	−8560.26	−2.01853
$263$	2918.55	0.684280	0.342140	−	0.939649i	$-0.388848\pi$
$263$	2918.55	0.684280	0.342140	+	0.939649i	$0.388848\pi$
$264$	0	0
$265$	5371.74	1.24522
$266$	1981.28	0.456691
$267$	−7076.27	−1.62195
$268$	−6452.49	−1.47070
$269$	−4066.05	−0.921604	−0.460802	−	0.887503i	$-0.652438\pi$
$269$	−4066.05	−0.921604	−0.460802	+	0.887503i	$0.652438\pi$
$270$	−15016.1	−3.38463
$271$	3310.65	0.742094	0.371047	−	0.928614i	$-0.378999\pi$
$271$	3310.65	0.742094	0.371047	+	0.928614i	$0.378999\pi$
$272$	23460.1	5.22969
$273$	−1842.26	−0.408419
$274$	6694.98	1.47613
$275$	0	0
$276$	16257.7	3.54565
$277$	−78.4692	−0.0170208	−0.00851039	−	0.999964i	$-0.502709\pi$
$277$	−78.4692	−0.0170208	−0.00851039	+	0.999964i	$0.502709\pi$
$278$	−5760.91	−1.24286
$279$	9019.22	1.93536
$280$	6962.54	1.48604
$281$	4810.68	1.02128	0.510642	−	0.859793i	$-0.329408\pi$
$281$	4810.68	1.02128	0.510642	+	0.859793i	$0.329408\pi$
$282$	−21607.7	−4.56283
$283$	1378.13	0.289475	0.144738	−	0.989470i	$-0.453766\pi$
$283$	1378.13	0.289475	0.144738	+	0.989470i	$0.453766\pi$
$284$	−8437.00	−1.76283
$285$	6328.27	1.31528
$286$	0	0
$287$	−902.833	−0.185688
$288$	30583.9	6.25756
$289$	6493.16	1.32163
$290$	13999.1	2.83467
$291$	4600.54	0.926764
$292$	11778.0	2.36047
$293$	−5549.00	−1.10640	−0.553201	−	0.833048i	$-0.686594\pi$
$293$	−5549.00	−1.10640	−0.553201	+	0.833048i	$0.686594\pi$
$294$	2326.54	0.461520
$295$	−2021.32	−0.398935
$296$	−26262.4	−5.15700
$297$	0	0
$298$	−12532.0	−2.43611
$299$	2610.41	0.504897
$300$	12242.3	2.35603
$301$	337.827	0.0646911
$302$	9866.64	1.88001
$303$	7021.62	1.33129
$304$	−11484.0	−2.16662
$305$	2686.68	0.504389
$306$	28860.3	5.39162
$307$	8345.50	1.55148	0.775738	−	0.631055i	$-0.217378\pi$
$307$	8345.50	1.55148	0.775738	+	0.631055i	$0.217378\pi$
$308$	0	0
$309$	−314.116	−0.0578299
$310$	−13502.6	−2.47385
$311$	7032.19	1.28218	0.641092	−	0.767464i	$-0.278482\pi$
$311$	7032.19	1.28218	0.641092	+	0.767464i	$0.278482\pi$
$312$	18965.8	3.44143
$313$	−4620.55	−0.834406	−0.417203	−	0.908813i	$-0.636990\pi$
$313$	−4620.55	−0.834406	−0.417203	+	0.908813i	$0.636990\pi$
$314$	−14604.3	−2.62474
$315$	4822.44	0.862583
$316$	−17561.4	−3.12628
$317$	11013.6	1.95137	0.975687	−	0.219170i	$-0.0703348\pi$
$317$	11013.6	1.95137	0.975687	+	0.219170i	$0.0703348\pi$
$318$	−18479.0	−3.25866
$319$	0	0
$320$	−21531.9	−3.76146
$321$	−1331.75	−0.231561
$322$	−3296.63	−0.570541
$323$	−5583.45	−0.961832
$324$	8837.38	1.51532
$325$	1965.67	0.335495
$326$	4017.30	0.682508
$327$	15105.3	2.55451
$328$	9294.55	1.56465
$329$	3185.60	0.533822
$330$	0	0
$331$	−11304.7	−1.87722	−0.938612	−	0.344974i	$-0.887888\pi$
$331$	−11304.7	−1.87722	−0.938612	+	0.344974i	$0.887888\pi$
$332$	22191.3	3.66839
$333$	−18190.0	−2.99342
$334$	−14791.1	−2.42315
$335$	4179.05	0.681570
$336$	−13485.3	−2.18953
$337$	−5454.98	−0.881756	−0.440878	−	0.897567i	$-0.645333\pi$
$337$	−5454.98	−0.881756	−0.440878	+	0.897567i	$0.645333\pi$
$338$	−7018.98	−1.12953
$339$	−16233.6	−2.60085
$340$	−31413.8	−5.01075
$341$	0	0
$342$	−14127.5	−2.23370
$343$	−343.000	−0.0539949
$344$	−3477.88	−0.545101
$345$	−10529.6	−1.64317
$346$	−1246.26	−0.193639
$347$	2943.67	0.455402	0.227701	−	0.973731i	$-0.426879\pi$
$347$	2943.67	0.455402	0.227701	+	0.973731i	$0.426879\pi$
$348$	−35013.5	−5.39346
$349$	−5541.62	−0.849960	−0.424980	−	0.905203i	$-0.639719\pi$
$349$	−5541.62	−0.849960	−0.424980	+	0.905203i	$0.639719\pi$
$350$	−2482.41	−0.379115
$351$	6030.35	0.917026
$352$	0	0
$353$	10219.3	1.54085	0.770426	−	0.637530i	$-0.220044\pi$
$353$	10219.3	1.54085	0.770426	+	0.637530i	$0.220044\pi$
$354$	6953.44	1.04399
$355$	5464.35	0.816951
$356$	−17195.0	−2.55993
$357$	−6556.45	−0.972001
$358$	−18752.2	−2.76839
$359$	−969.299	−0.142500	−0.0712502	−	0.997458i	$-0.522699\pi$
$359$	−969.299	−0.142500	−0.0712502	+	0.997458i	$0.522699\pi$
$360$	−49646.3	−7.26831
$361$	−4125.83	−0.601521
$362$	4539.72	0.659123
$363$	0	0
$364$	−4476.61	−0.644610
$365$	−7628.22	−1.09392
$366$	−9242.29	−1.31995
$367$	12219.9	1.73807	0.869037	−	0.494747i	$-0.164739\pi$
$367$	12219.9	1.73807	0.869037	+	0.494747i	$0.164739\pi$
$368$	19108.1	2.70674
$369$	6437.65	0.908213
$370$	27232.1	3.82629
$371$	2724.35	0.381243
$372$	33771.7	4.70694
$373$	−8149.03	−1.13121	−0.565604	−	0.824677i	$-0.691357\pi$
$373$	−8149.03	−1.13121	−0.565604	+	0.824677i	$0.691357\pi$
$374$	0	0
$375$	7201.90	0.991746
$376$	−32795.3	−4.49810
$377$	−5621.93	−0.768022
$378$	−7615.59	−1.03625
$379$	−10012.2	−1.35697	−0.678487	−	0.734612i	$-0.737364\pi$
$379$	−10012.2	−1.35697	−0.678487	+	0.734612i	$0.737364\pi$
$380$	15377.4	2.07591
$381$	−4104.46	−0.551910
$382$	−17545.5	−2.35002
$383$	−324.732	−0.0433238	−0.0216619	−	0.999765i	$-0.506896\pi$
$383$	−324.732	−0.0433238	−0.0216619	+	0.999765i	$0.506896\pi$
$384$	31080.7	4.13041
$385$	0	0
$386$	15426.6	2.03418
$387$	−2408.87	−0.316408
$388$	11179.1	1.46272
$389$	−2755.25	−0.359117	−0.179559	−	0.983747i	$-0.557467\pi$
$389$	−2755.25	−0.359117	−0.179559	+	0.983747i	$0.557467\pi$
$390$	−19666.0	−2.55341
$391$	9290.27	1.20161
$392$	3531.14	0.454973
$393$	−13866.7	−1.77986
$394$	19600.8	2.50628
$395$	11373.9	1.44882
$396$	0	0
$397$	−13014.6	−1.64529	−0.822647	−	0.568552i	$-0.807504\pi$
$397$	−13014.6	−1.64529	−0.822647	+	0.568552i	$0.807504\pi$
$398$	7885.70	0.993151
$399$	3209.46	0.402692
$400$	14388.7	1.79858
$401$	7492.11	0.933013	0.466507	−	0.884518i	$-0.345512\pi$
$401$	7492.11	0.933013	0.466507	+	0.884518i	$0.345512\pi$
$402$	−14376.1	−1.78362
$403$	5422.53	0.670262
$404$	17062.3	2.10118
$405$	−5723.66	−0.702250
$406$	7099.81	0.867876
$407$	0	0
$408$	67497.8	8.19029
$409$	228.102	0.0275769	0.0137884	−	0.999905i	$-0.495611\pi$
$409$	228.102	0.0275769	0.0137884	+	0.999905i	$0.495611\pi$
$410$	−9637.72	−1.16091
$411$	10845.2	1.30159
$412$	−763.289	−0.0912732
$413$	−1025.14	−0.122140
$414$	23506.6	2.79055
$415$	−14372.5	−1.70005
$416$	18387.7	2.16714
$417$	−9332.08	−1.09591
$418$	0	0
$419$	5590.55	0.651829	0.325914	−	0.945399i	$-0.394328\pi$
$419$	5590.55	0.651829	0.325914	+	0.945399i	$0.394328\pi$
$420$	18057.2	2.09786
$421$	13801.0	1.59768	0.798838	−	0.601546i	$-0.205448\pi$
$421$	13801.0	1.59768	0.798838	+	0.601546i	$0.205448\pi$
$422$	21969.2	2.53423
$423$	−22714.9	−2.61096
$424$	−28046.8	−3.21244
$425$	6995.69	0.798449
$426$	−18797.6	−2.13791
$427$	1362.58	0.154426
$428$	−3236.10	−0.365474
$429$	0	0
$430$	3606.29	0.404444
$431$	10274.4	1.14827	0.574133	−	0.818762i	$-0.305339\pi$
$431$	10274.4	1.14827	0.574133	+	0.818762i	$0.305339\pi$
$432$	44142.0	4.91616
$433$	−1446.07	−0.160494	−0.0802470	−	0.996775i	$-0.525571\pi$
$433$	−1446.07	−0.160494	−0.0802470	+	0.996775i	$0.525571\pi$
$434$	−6848.00	−0.757406
$435$	22677.1	2.49950
$436$	36705.2	4.03179
$437$	−4547.70	−0.497817
$438$	26241.4	2.86270
$439$	−638.674	−0.0694356	−0.0347178	−	0.999397i	$-0.511053\pi$
$439$	−638.674	−0.0694356	−0.0347178	+	0.999397i	$0.511053\pi$
$440$	0	0
$441$	2445.76	0.264092
$442$	17351.4	1.86724
$443$	−1275.25	−0.136769	−0.0683847	−	0.997659i	$-0.521785\pi$
$443$	−1275.25	−0.136769	−0.0683847	+	0.997659i	$0.521785\pi$
$444$	−68111.0	−7.28019
$445$	11136.6	1.18635
$446$	8606.20	0.913712
$447$	−20300.6	−2.14806
$448$	−10920.2	−1.15163
$449$	958.015	0.100694	0.0503469	−	0.998732i	$-0.483967\pi$
$449$	958.015	0.100694	0.0503469	+	0.998732i	$0.483967\pi$
$450$	17700.8	1.85427
$451$	0	0
$452$	−39446.9	−4.10493
$453$	15982.9	1.65771
$454$	3128.99	0.323460
$455$	2899.34	0.298733
$456$	−33041.0	−3.39317
$457$	−5761.44	−0.589734	−0.294867	−	0.955538i	$-0.595275\pi$
$457$	−5761.44	−0.589734	−0.294867	+	0.955538i	$0.595275\pi$
$458$	1824.54	0.186147
$459$	21461.6	2.18244
$460$	−25586.4	−2.59342
$461$	6714.16	0.678329	0.339165	−	0.940727i	$-0.389856\pi$
$461$	6714.16	0.678329	0.339165	+	0.940727i	$0.389856\pi$
$462$	0	0
$463$	−8184.02	−0.821477	−0.410738	−	0.911753i	$-0.634729\pi$
$463$	−8184.02	−0.821477	−0.410738	+	0.911753i	$0.634729\pi$
$464$	−41152.4	−4.11735
$465$	−21872.8	−2.18134
$466$	29058.9	2.88868
$467$	1573.83	0.155949	0.0779745	−	0.996955i	$-0.475155\pi$
$467$	1573.83	0.155949	0.0779745	+	0.996955i	$0.475155\pi$
$468$	31920.4	3.15282
$469$	2119.46	0.208673
$470$	34006.1	3.33742
$471$	−23657.5	−2.31439
$472$	10553.7	1.02918
$473$	0	0
$474$	−39126.7	−3.79146
$475$	−3424.48	−0.330791
$476$	−15931.9	−1.53411
$477$	−19425.9	−1.86468
$478$	21962.5	2.10155
$479$	11620.5	1.10846	0.554231	−	0.832363i	$-0.313012\pi$
$479$	11620.5	1.10846	0.554231	+	0.832363i	$0.313012\pi$
$480$	−74169.9	−7.05287
$481$	−10936.2	−1.03669
$482$	8364.22	0.790414
$483$	−5340.20	−0.503080
$484$	0	0
$485$	−7240.33	−0.677869
$486$	−9684.77	−0.903930
$487$	14664.8	1.36452	0.682262	−	0.731107i	$-0.260996\pi$
$487$	14664.8	1.36452	0.682262	+	0.731107i	$0.260996\pi$
$488$	−14027.6	−1.30123
$489$	6507.61	0.601808
$490$	−3661.52	−0.337572
$491$	5909.66	0.543176	0.271588	−	0.962414i	$-0.412451\pi$
$491$	5909.66	0.543176	0.271588	+	0.962414i	$0.412451\pi$
$492$	24105.2	2.20883
$493$	−20008.0	−1.82782
$494$	−8493.72	−0.773584
$495$	0	0
$496$	39692.8	3.59326
$497$	2771.31	0.250122
$498$	49442.2	4.44891
$499$	−13150.1	−1.17972	−0.589859	−	0.807506i	$-0.700817\pi$
$499$	−13150.1	−1.17972	−0.589859	+	0.807506i	$0.700817\pi$
$500$	17500.3	1.56528
$501$	−23960.0	−2.13663
$502$	29221.4	2.59804
$503$	−17658.9	−1.56535	−0.782676	−	0.622429i	$-0.786146\pi$
$503$	−17658.9	−1.56535	−0.782676	+	0.622429i	$0.786146\pi$
$504$	−25178.8	−2.22530
$505$	−11050.6	−0.973756
$506$	0	0
$507$	−11370.0	−0.995977
$508$	−9973.67	−0.871082
$509$	2013.01	0.175295	0.0876473	−	0.996152i	$-0.472065\pi$
$509$	2013.01	0.175295	0.0876473	+	0.996152i	$0.472065\pi$
$510$	−69990.0	−6.07688
$511$	−3868.75	−0.334918
$512$	7957.75	0.686887
$513$	−10505.7	−0.904167
$514$	13588.8	1.16610
$515$	494.356	0.0422989
$516$	−9019.81	−0.769525
$517$	0	0
$518$	13811.1	1.17148
$519$	−2018.81	−0.170743
$520$	−29848.4	−2.51719
$521$	22999.9	1.93406	0.967029	−	0.254667i	$-0.0819659\pi$
$521$	22999.9	1.93406	0.967029	+	0.254667i	$0.0819659\pi$
$522$	−50625.2	−4.24483
$523$	−3324.73	−0.277974	−0.138987	−	0.990294i	$-0.544385\pi$
$523$	−3324.73	−0.277974	−0.138987	+	0.990294i	$0.544385\pi$
$524$	−33695.6	−2.80916
$525$	−4021.24	−0.334288
$526$	15800.9	1.30979
$527$	19298.4	1.59516
$528$	0	0
$529$	−4600.12	−0.378081
$530$	29082.3	2.38350
$531$	7309.75	0.597394
$532$	7798.86	0.635571
$533$	3870.44	0.314535
$534$	−38310.5	−3.10461
$535$	2095.91	0.169372
$536$	−21819.6	−1.75832
$537$	−30376.6	−2.44106
$538$	−22013.4	−1.76406
$539$	0	0
$540$	−59107.5	−4.71034
$541$	12088.8	0.960698	0.480349	−	0.877078i	$-0.340510\pi$
$541$	12088.8	0.960698	0.480349	+	0.877078i	$0.340510\pi$
$542$	17923.7	1.42046
$543$	7353.88	0.581188
$544$	65440.3	5.15759
$545$	−23772.7	−1.86846
$546$	−9973.87	−0.781763
$547$	−7.61110	−0.000594930 0	−0.000297465	−	1.00000i	$-0.500095\pi$
$547$	−7.61110	−0.000594930 0	−0.000297465		1.00000i	$0.500095\pi$
$548$	26353.3	2.05430
$549$	−9715.88	−0.755307
$550$	0	0
$551$	9794.18	0.757252
$552$	54976.7	4.23906
$553$	5768.42	0.443577
$554$	−424.828	−0.0325798
$555$	44113.1	3.37387
$556$	−22676.6	−1.72968
$557$	−16354.5	−1.24410	−0.622048	−	0.782979i	$-0.713699\pi$
$557$	−16354.5	−1.24410	−0.622048	+	0.782979i	$0.713699\pi$
$558$	48829.6	3.70452
$559$	−1448.26	−0.109579
$560$	21223.1	1.60150
$561$	0	0
$562$	26044.8	1.95486
$563$	15181.3	1.13644	0.568220	−	0.822877i	$-0.307632\pi$
$563$	15181.3	1.13644	0.568220	+	0.822877i	$0.307632\pi$
$564$	−85053.8	−6.35002
$565$	25548.4	1.90235
$566$	7461.14	0.554091
$567$	−2902.83	−0.215004
$568$	−28530.3	−2.10758
$569$	−15609.4	−1.15005	−0.575027	−	0.818134i	$-0.695008\pi$
$569$	−15609.4	−1.15005	−0.575027	+	0.818134i	$0.695008\pi$
$570$	34260.9	2.51760
$571$	−19353.4	−1.41842	−0.709209	−	0.704998i	$-0.750948\pi$
$571$	−19353.4	−1.41842	−0.709209	+	0.704998i	$0.750948\pi$
$572$	0	0
$573$	−28422.0	−2.07215
$574$	−4887.89	−0.355430
$575$	5697.96	0.413255
$576$	77866.1	5.63268
$577$	6005.46	0.433294	0.216647	−	0.976250i	$-0.430488\pi$
$577$	6005.46	0.433294	0.216647	+	0.976250i	$0.430488\pi$
$578$	35153.6	2.52975
$579$	24989.5	1.79366
$580$	55104.3	3.94497
$581$	−7289.22	−0.520495
$582$	24907.1	1.77394
$583$	0	0
$584$	39828.2	2.82210
$585$	−20673.7	−1.46112
$586$	−30042.0	−2.11779
$587$	−2321.88	−0.163261	−0.0816306	−	0.996663i	$-0.526013\pi$
$587$	−2321.88	−0.163261	−0.0816306	+	0.996663i	$0.526013\pi$
$588$	9157.93	0.642290
$589$	−9446.80	−0.660863
$590$	−10943.3	−0.763610
$591$	31751.3	2.20994
$592$	−80052.7	−5.55768
$593$	−20556.4	−1.42352	−0.711761	−	0.702422i	$-0.752102\pi$
$593$	−20556.4	−1.42352	−0.711761	+	0.702422i	$0.752102\pi$
$594$	0	0
$595$	10318.5	0.710957
$596$	−49329.6	−3.39030
$597$	12774.0	0.875721
$598$	14132.6	0.966432
$599$	−16020.4	−1.09278	−0.546391	−	0.837530i	$-0.683999\pi$
$599$	−16020.4	−1.09278	−0.546391	+	0.837530i	$0.683999\pi$
$600$	41398.2	2.81679
$601$	1426.54	0.0968216	0.0484108	−	0.998828i	$-0.484584\pi$
$601$	1426.54	0.0968216	0.0484108	+	0.998828i	$0.484584\pi$
$602$	1828.98	0.123826
$603$	−15112.8	−1.02063
$604$	38837.9	2.61638
$605$	0	0
$606$	38014.7	2.54825
$607$	−23151.1	−1.54807	−0.774033	−	0.633146i	$-0.781763\pi$
$607$	−23151.1	−1.54807	−0.774033	+	0.633146i	$0.781763\pi$
$608$	−32033.8	−2.13675
$609$	11501.0	0.765258
$610$	14545.5	0.965460
$611$	−13656.6	−0.904235
$612$	113602.	7.50344
$613$	7127.41	0.469614	0.234807	−	0.972042i	$-0.424554\pi$
$613$	7127.41	0.469614	0.234807	+	0.972042i	$0.424554\pi$
$614$	45182.1	2.96971
$615$	−15612.1	−1.02364
$616$	0	0
$617$	18914.5	1.23415	0.617074	−	0.786905i	$-0.288318\pi$
$617$	18914.5	1.23415	0.617074	+	0.786905i	$0.288318\pi$
$618$	−1700.61	−0.110693
$619$	−6655.53	−0.432162	−0.216081	−	0.976375i	$-0.569328\pi$
$619$	−6655.53	−0.432162	−0.216081	+	0.976375i	$0.569328\pi$
$620$	−53149.9	−3.44283
$621$	17480.4	1.12957
$622$	38071.9	2.45425
$623$	5648.09	0.363220
$624$	57811.2	3.70881
$625$	−19522.2	−1.24942
$626$	−25015.4	−1.59715
$627$	0	0
$628$	−57486.7	−3.65282
$629$	−38921.2	−2.46723
$630$	26108.4	1.65109
$631$	−16295.1	−1.02805	−0.514024	−	0.857776i	$-0.671846\pi$
$631$	−16295.1	−1.02805	−0.514024	+	0.857776i	$0.671846\pi$
$632$	−59385.1	−3.73768
$633$	35587.9	2.23458
$634$	59627.0	3.73516
$635$	6459.60	0.403687
$636$	−72738.7	−4.53503
$637$	1470.44	0.0914614
$638$	0	0
$639$	−19760.8	−1.22336
$640$	−48914.8	−3.02114
$641$	−2897.30	−0.178528	−0.0892641	−	0.996008i	$-0.528452\pi$
$641$	−2897.30	−0.178528	−0.0892641	+	0.996008i	$0.528452\pi$
$642$	−7210.03	−0.443235
$643$	−24686.0	−1.51403	−0.757016	−	0.653397i	$-0.773343\pi$
$643$	−24686.0	−1.51403	−0.757016	+	0.653397i	$0.773343\pi$
$644$	−12976.5	−0.794014
$645$	5841.82	0.356622
$646$	−30228.5	−1.84106
$647$	15318.3	0.930794	0.465397	−	0.885102i	$-0.345912\pi$
$647$	15318.3	0.930794	0.465397	+	0.885102i	$0.345912\pi$
$648$	29884.2	1.81167
$649$	0	0
$650$	10642.1	0.642178
$651$	−11093.0	−0.667850
$652$	15813.2	0.949837
$653$	9002.33	0.539492	0.269746	−	0.962932i	$-0.413060\pi$
$653$	9002.33	0.539492	0.269746	+	0.962932i	$0.413060\pi$
$654$	81779.1	4.88963
$655$	21823.5	1.30185
$656$	28331.5	1.68622
$657$	27586.1	1.63811
$658$	17246.6	1.02180
$659$	−21842.6	−1.29115	−0.645574	−	0.763698i	$-0.723382\pi$
$659$	−21842.6	−1.29115	−0.645574	+	0.763698i	$0.723382\pi$
$660$	0	0
$661$	11027.6	0.648901	0.324450	−	0.945903i	$-0.394821\pi$
$661$	11027.6	0.648901	0.324450	+	0.945903i	$0.394821\pi$
$662$	−61202.9	−3.59323
$663$	28107.5	1.64646
$664$	75041.5	4.38581
$665$	−5051.05	−0.294544
$666$	−98479.9	−5.72976
$667$	−16296.5	−0.946029
$668$	−58221.8	−3.37226
$669$	13941.2	0.805674
$670$	22625.2	1.30461
$671$	0	0
$672$	−37616.2	−2.15934
$673$	−1991.58	−0.114071	−0.0570354	−	0.998372i	$-0.518165\pi$
$673$	−1991.58	−0.114071	−0.0570354	+	0.998372i	$0.518165\pi$
$674$	−29533.0	−1.68779
$675$	13162.9	0.750580
$676$	−27628.7	−1.57196
$677$	24272.2	1.37793	0.688964	−	0.724795i	$-0.258066\pi$
$677$	24272.2	1.37793	0.688964	+	0.724795i	$0.258066\pi$
$678$	−87887.7	−4.97833
$679$	−3672.02	−0.207539
$680$	−106228.	−5.99068
$681$	5068.64	0.285214
$682$	0	0
$683$	2994.98	0.167789	0.0838944	−	0.996475i	$-0.473264\pi$
$683$	2994.98	0.167789	0.0838944	+	0.996475i	$0.473264\pi$
$684$	−55609.7	−3.10861
$685$	−17068.1	−0.952029
$686$	−1856.98	−0.103353
$687$	2955.56	0.164137
$688$	−10601.2	−0.587453
$689$	−11679.3	−0.645782
$690$	−57006.5	−3.14522
$691$	32733.9	1.80211	0.901053	−	0.433710i	$-0.142796\pi$
$691$	32733.9	1.80211	0.901053	+	0.433710i	$0.142796\pi$
$692$	−4905.62	−0.269485
$693$	0	0
$694$	15936.9	0.871693
$695$	14686.8	0.801587
$696$	−118401.	−6.44824
$697$	13774.6	0.748566
$698$	−30002.0	−1.62693
$699$	47072.4	2.54712
$700$	−9771.45	−0.527609
$701$	−29409.5	−1.58457	−0.792283	−	0.610154i	$-0.791107\pi$
$701$	−29409.5	−1.58457	−0.792283	+	0.610154i	$0.791107\pi$
$702$	32648.0	1.75530
$703$	19052.4	1.02215
$704$	0	0
$705$	55086.4	2.94280
$706$	55327.0	2.94937
$707$	−5604.47	−0.298130
$708$	27370.7	1.45290
$709$	31423.9	1.66453	0.832264	−	0.554379i	$-0.187044\pi$
$709$	31423.9	1.66453	0.832264	+	0.554379i	$0.187044\pi$
$710$	29583.7	1.56374
$711$	−41131.7	−2.16956
$712$	−58146.3	−3.06057
$713$	15718.5	0.825611
$714$	−35496.3	−1.86053
$715$	0	0
$716$	−73814.0	−3.85273
$717$	35576.9	1.85306
$718$	−5247.74	−0.272763
$719$	−19292.3	−1.00067	−0.500335	−	0.865832i	$-0.666790\pi$
$719$	−19292.3	−1.00067	−0.500335	+	0.865832i	$0.666790\pi$
$720$	−151331.	−7.83303
$721$	250.719	0.0129504
$722$	−22337.0	−1.15138
$723$	13549.2	0.696955
$724$	17869.6	0.917292
$725$	−12271.4	−0.628621
$726$	0	0
$727$	−13262.9	−0.676609	−0.338304	−	0.941037i	$-0.609853\pi$
$727$	−13262.9	−0.676609	−0.338304	+	0.941037i	$0.609853\pi$
$728$	−15138.0	−0.770674
$729$	−26884.9	−1.36590
$730$	−41298.8	−2.09389
$731$	−5154.26	−0.260789
$732$	−36380.3	−1.83696
$733$	−6096.90	−0.307222	−0.153611	−	0.988131i	$-0.549090\pi$
$733$	−6096.90	−0.307222	−0.153611	+	0.988131i	$0.549090\pi$
$734$	66157.9	3.32688
$735$	−5931.28	−0.297658
$736$	53300.9	2.66942
$737$	0	0
$738$	34853.1	1.73843
$739$	34913.3	1.73790	0.868949	−	0.494901i	$-0.164796\pi$
$739$	34913.3	1.73790	0.868949	+	0.494901i	$0.164796\pi$
$740$	107193.	5.32500
$741$	−13758.9	−0.682115
$742$	14749.5	0.729744
$743$	−3720.35	−0.183696	−0.0918482	−	0.995773i	$-0.529277\pi$
$743$	−3720.35	−0.183696	−0.0918482	+	0.995773i	$0.529277\pi$
$744$	114201.	5.62746
$745$	31949.1	1.57117
$746$	−44118.4	−2.16527
$747$	51975.7	2.54577
$748$	0	0
$749$	1062.97	0.0518558
$750$	38990.7	1.89832
$751$	23548.6	1.14421	0.572105	−	0.820181i	$-0.306127\pi$
$751$	23548.6	1.14421	0.572105	+	0.820181i	$0.306127\pi$
$752$	−99966.0	−4.84759
$753$	47335.7	2.29085
$754$	−30436.8	−1.47008
$755$	−25154.0	−1.21251
$756$	−29977.1	−1.44214
$757$	33683.8	1.61725	0.808625	−	0.588324i	$-0.200212\pi$
$757$	33683.8	1.61725	0.808625	+	0.588324i	$0.200212\pi$
$758$	−54205.6	−2.59741
$759$	0	0
$760$	51999.9	2.48189
$761$	18322.8	0.872802	0.436401	−	0.899752i	$-0.356253\pi$
$761$	18322.8	0.872802	0.436401	+	0.899752i	$0.356253\pi$
$762$	−22221.3	−1.05642
$763$	−12056.6	−0.572056
$764$	−69064.2	−3.27049
$765$	−73576.3	−3.47733
$766$	−1758.08	−0.0829270
$767$	4394.76	0.206891
$768$	58817.5	2.76354
$769$	−2175.73	−0.102027	−0.0510134	−	0.998698i	$-0.516245\pi$
$769$	−2175.73	−0.102027	−0.0510134	+	0.998698i	$0.516245\pi$
$770$	0	0
$771$	22012.4	1.02822
$772$	60723.6	2.83094
$773$	−22514.7	−1.04760	−0.523802	−	0.851840i	$-0.675487\pi$
$773$	−22514.7	−1.04760	−0.523802	+	0.851840i	$0.675487\pi$
$774$	−13041.5	−0.605643
$775$	11836.2	0.548605
$776$	37803.0	1.74877
$777$	22372.5	1.03296
$778$	−14916.8	−0.687394
$779$	−6742.84	−0.310125
$780$	−77411.0	−3.55354
$781$	0	0
$782$	50297.0	2.30002
$783$	−37646.7	−1.71824
$784$	10763.6	0.490323
$785$	37232.2	1.69283
$786$	−75073.8	−3.40686
$787$	−202.840	−0.00918738	−0.00459369	−	0.999989i	$-0.501462\pi$
$787$	−202.840	−0.00918738	−0.00459369	+	0.999989i	$0.501462\pi$
$788$	77154.3	3.48795
$789$	25595.8	1.15492
$790$	61577.7	2.77321
$791$	12957.2	0.582434
$792$	0	0
$793$	−5841.37	−0.261580
$794$	−70460.1	−3.14929
$795$	47110.4	2.10168
$796$	31040.3	1.38215
$797$	35765.0	1.58954	0.794768	−	0.606913i	$-0.207592\pi$
$797$	35765.0	1.58954	0.794768	+	0.606913i	$0.207592\pi$
$798$	17375.9	0.770800
$799$	−48602.9	−2.15200
$800$	40136.3	1.77379
$801$	−40273.6	−1.77653
$802$	40561.9	1.78590
$803$	0	0
$804$	−56588.5	−2.48224
$805$	8404.41	0.367971
$806$	29357.3	1.28296
$807$	−35659.4	−1.55548
$808$	57697.2	2.51211
$809$	27890.8	1.21210	0.606049	−	0.795427i	$-0.292754\pi$
$809$	27890.8	1.21210	0.606049	+	0.795427i	$0.292754\pi$
$810$	−30987.6	−1.34419
$811$	4519.06	0.195667	0.0978333	−	0.995203i	$-0.468809\pi$
$811$	4519.06	0.195667	0.0978333	+	0.995203i	$0.468809\pi$
$812$	27946.9	1.20781
$813$	29034.5	1.25250
$814$	0	0
$815$	−10241.7	−0.440185
$816$	205746.	8.82664
$817$	2523.07	0.108043
$818$	1234.93	0.0527854
$819$	−10484.9	−0.447343
$820$	−37936.8	−1.61562
$821$	−28198.0	−1.19868	−0.599341	−	0.800494i	$-0.704571\pi$
$821$	−28198.0	−1.19868	−0.599341	+	0.800494i	$0.704571\pi$
$822$	58715.2	2.49139
$823$	−21369.8	−0.905110	−0.452555	−	0.891736i	$-0.649487\pi$
$823$	−21369.8	−0.905110	−0.452555	+	0.891736i	$0.649487\pi$
$824$	−2581.12	−0.109123
$825$	0	0
$826$	−5550.05	−0.233790
$827$	5051.86	0.212419	0.106210	−	0.994344i	$-0.466129\pi$
$827$	5051.86	0.212419	0.106210	+	0.994344i	$0.466129\pi$
$828$	92528.7	3.88357
$829$	−17871.6	−0.748742	−0.374371	−	0.927279i	$-0.622141\pi$
$829$	−17871.6	−0.748742	−0.374371	+	0.927279i	$0.622141\pi$
$830$	−77812.2	−3.25410
$831$	−688.177	−0.0287276
$832$	46814.6	1.95073
$833$	5233.18	0.217670
$834$	−50523.4	−2.09770
$835$	37708.2	1.56281
$836$	0	0
$837$	36311.4	1.49953
$838$	30266.9	1.24768
$839$	22145.7	0.911267	0.455633	−	0.890167i	$-0.349413\pi$
$839$	22145.7	0.911267	0.455633	+	0.890167i	$0.349413\pi$
$840$	61061.7	2.50813
$841$	10708.0	0.439049
$842$	74718.1	3.05814
$843$	42189.8	1.72372
$844$	86477.1	3.52685
$845$	17894.2	0.728494
$846$	−122977.	−4.99768
$847$	0	0
$848$	−85491.8	−3.46203
$849$	12086.3	0.488575
$850$	37874.3	1.52833
$851$	−31701.1	−1.27697
$852$	−73992.7	−2.97529
$853$	−38049.2	−1.52729	−0.763645	−	0.645636i	$-0.776592\pi$
$853$	−38049.2	−1.52729	−0.763645	+	0.645636i	$0.776592\pi$
$854$	7376.94	0.295590
$855$	36016.5	1.44063
$856$	−10943.1	−0.436948
$857$	13673.1	0.544999	0.272500	−	0.962156i	$-0.412150\pi$
$857$	13673.1	0.544999	0.272500	+	0.962156i	$0.412150\pi$
$858$	0	0
$859$	20654.5	0.820399	0.410199	−	0.911996i	$-0.365459\pi$
$859$	20654.5	0.820399	0.410199	+	0.911996i	$0.365459\pi$
$860$	14195.4	0.562859
$861$	−7917.88	−0.313404
$862$	55625.3	2.19792
$863$	27735.2	1.09399	0.546997	−	0.837135i	$-0.315771\pi$
$863$	27735.2	1.09399	0.546997	+	0.837135i	$0.315771\pi$
$864$	123131.	4.84838
$865$	3177.20	0.124888
$866$	−7828.97	−0.307205
$867$	56945.2	2.23064
$868$	−26955.6	−1.05407
$869$	0	0
$870$	122772.	4.78434
$871$	−9086.11	−0.353468
$872$	124121.	4.82027
$873$	26183.3	1.01509
$874$	−24621.0	−0.952881
$875$	−5748.36	−0.222092
$876$	103294.	3.98398
$877$	−36791.3	−1.41660	−0.708298	−	0.705913i	$-0.750537\pi$
$877$	−36791.3	−1.41660	−0.708298	+	0.705913i	$0.750537\pi$
$878$	−3457.75	−0.132908
$879$	−48664.9	−1.86738
$880$	0	0
$881$	−15223.9	−0.582188	−0.291094	−	0.956694i	$-0.594019\pi$
$881$	−15223.9	−0.582188	−0.291094	+	0.956694i	$0.594019\pi$
$882$	13241.2	0.505504
$883$	24566.9	0.936288	0.468144	−	0.883652i	$-0.344923\pi$
$883$	24566.9	0.936288	0.468144	+	0.883652i	$0.344923\pi$
$884$	68300.0	2.59862
$885$	−17727.1	−0.673321
$886$	−6904.12	−0.261793
$887$	1549.80	0.0586664	0.0293332	−	0.999570i	$-0.490662\pi$
$887$	1549.80	0.0586664	0.0293332	+	0.999570i	$0.490662\pi$
$888$	−230322.	−8.70395
$889$	3276.07	0.123595
$890$	60293.2	2.27082
$891$	0	0
$892$	33876.4	1.27160
$893$	23791.7	0.891556
$894$	−109906.	−4.11165
$895$	47806.8	1.78548
$896$	−24807.7	−0.924965
$897$	22893.4	0.852161
$898$	5186.64	0.192740
$899$	−33852.1	−1.25588
$900$	69675.3	2.58057
$901$	−41565.6	−1.53690
$902$	0	0
$903$	2962.75	0.109185
$904$	−133393.	−4.90771
$905$	−11573.5	−0.425102
$906$	86530.8	3.17306
$907$	−29601.6	−1.08369	−0.541844	−	0.840479i	$-0.682273\pi$
$907$	−29601.6	−1.08369	−0.541844	+	0.840479i	$0.682273\pi$
$908$	12316.6	0.450155
$909$	39962.6	1.45817
$910$	15696.9	0.571810
$911$	11921.1	0.433550	0.216775	−	0.976222i	$-0.430446\pi$
$911$	11921.1	0.433550	0.216775	+	0.976222i	$0.430446\pi$
$912$	−100715.	−3.65681
$913$	0	0
$914$	−31192.1	−1.12882
$915$	23562.2	0.851304
$916$	7181.90	0.259057
$917$	11068.1	0.398581
$918$	116192.	4.17745
$919$	−7688.00	−0.275956	−0.137978	−	0.990435i	$-0.544060\pi$
$919$	−7688.00	−0.275956	−0.137978	+	0.990435i	$0.544060\pi$
$920$	−86522.3	−3.10061
$921$	73190.3	2.61857
$922$	36350.1	1.29840
$923$	−11880.6	−0.423678
$924$	0	0
$925$	−23871.4	−0.848525
$926$	−44307.9	−1.57240
$927$	−1787.75	−0.0633413
$928$	−114792.	−4.06059
$929$	14582.9	0.515014	0.257507	−	0.966276i	$-0.417099\pi$
$929$	14582.9	0.515014	0.257507	+	0.966276i	$0.417099\pi$
$930$	−118418.	−4.17535
$931$	−2561.71	−0.0901788
$932$	114384.	4.02014
$933$	61672.6	2.16406
$934$	8520.64	0.298505
$935$	0	0
$936$	107941.	3.76941
$937$	30266.7	1.05525	0.527625	−	0.849478i	$-0.323083\pi$
$937$	30266.7	1.05525	0.527625	+	0.849478i	$0.323083\pi$
$938$	11474.6	0.399425
$939$	−40522.4	−1.40830
$940$	133858.	4.64464
$941$	−31355.3	−1.08624	−0.543120	−	0.839655i	$-0.682757\pi$
$941$	−31355.3	−1.08624	−0.543120	+	0.839655i	$0.682757\pi$
$942$	−128080.	−4.43003
$943$	11219.4	0.387436
$944$	32169.5	1.10914
$945$	19415.1	0.668333
$946$	0	0
$947$	15016.2	0.515271	0.257635	−	0.966242i	$-0.417057\pi$
$947$	15016.2	0.515271	0.257635	+	0.966242i	$0.417057\pi$
$948$	−154014.	−5.27652
$949$	16585.3	0.567314
$950$	−18539.9	−0.633173
$951$	96589.6	3.29352
$952$	−53874.9	−1.83413
$953$	31774.7	1.08005	0.540023	−	0.841650i	$-0.318415\pi$
$953$	31774.7	1.08005	0.540023	+	0.841650i	$0.318415\pi$
$954$	−105171.	−3.56922
$955$	44730.5	1.51565
$956$	86450.4	2.92469
$957$	0	0
$958$	62912.7	2.12173
$959$	−8656.32	−0.291478
$960$	−188835.	−6.34857
$961$	2860.48	0.0960183
$962$	−59208.0	−1.98435
$963$	−7579.48	−0.253630
$964$	32923.9	1.10001
$965$	−39328.6	−1.31195
$966$	−28911.6	−0.962955
$967$	37330.5	1.24143	0.620717	−	0.784034i	$-0.286841\pi$
$967$	37330.5	1.24143	0.620717	+	0.784034i	$0.286841\pi$
$968$	0	0
$969$	−48967.1	−1.62337
$970$	−39198.7	−1.29752
$971$	36461.3	1.20505	0.602523	−	0.798102i	$-0.294162\pi$
$971$	36461.3	1.20505	0.602523	+	0.798102i	$0.294162\pi$
$972$	−38122.0	−1.25799
$973$	7448.61	0.245418
$974$	79394.2	2.61186
$975$	17239.0	0.566247
$976$	−42758.7	−1.40233
$977$	−56203.4	−1.84043	−0.920217	−	0.391408i	$-0.871988\pi$
$977$	−56203.4	−1.84043	−0.920217	+	0.391408i	$0.871988\pi$
$978$	35231.9	1.15193
$979$	0	0
$980$	−14412.8	−0.469795
$981$	85969.6	2.79796
$982$	31994.6	1.03970
$983$	32452.2	1.05297	0.526483	−	0.850186i	$-0.323510\pi$
$983$	32452.2	1.05297	0.526483	+	0.850186i	$0.323510\pi$
$984$	81513.5	2.64081
$985$	−49970.2	−1.61643
$986$	−108322.	−3.49867
$987$	27937.8	0.900982
$988$	−33433.7	−1.07659
$989$	−4198.12	−0.134977
$990$	0	0
$991$	−2393.27	−0.0767152	−0.0383576	−	0.999264i	$-0.512213\pi$
$991$	−2393.27	−0.0767152	−0.0383576	+	0.999264i	$0.512213\pi$
$992$	110720.	3.54372
$993$	−99142.4	−3.16837
$994$	15003.8	0.478763
$995$	−20103.7	−0.640534
$996$	194619.	6.19149
$997$	−57755.5	−1.83464	−0.917319	−	0.398152i	$-0.869652\pi$
$997$	−57755.5	−1.83464	−0.917319	+	0.398152i	$0.869652\pi$
$998$	−71193.9	−2.25812
$999$	−73233.2	−2.31931

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.4.a.n.1.13	yes	14
11.10	odd	2		847.4.a.m.1.2	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.4.a.m.1.2	✓	14	11.10	odd	2
847.4.a.n.1.13	yes	14	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q+5.41395 q^{2} +8.77003 q^{3} +21.3108 q^{4} -13.8023 q^{5} +47.4805 q^{6} -7.00000 q^{7} +72.0641 q^{8} +49.9135 q^{9} -74.7248 q^{10} +186.897 q^{12} +30.0089 q^{13} -37.8976 q^{14} -121.046 q^{15} +219.665 q^{16} +106.800 q^{17} +270.229 q^{18} -52.2797 q^{19} -294.138 q^{20} -61.3902 q^{21} +86.9878 q^{23} +632.004 q^{24} +65.5030 q^{25} +162.467 q^{26} +200.952 q^{27} -149.176 q^{28} -187.342 q^{29} -655.339 q^{30} +180.697 q^{31} +612.739 q^{32} +578.208 q^{34} +96.6160 q^{35} +1063.70 q^{36} -364.432 q^{37} -283.040 q^{38} +263.179 q^{39} -994.649 q^{40} +128.976 q^{41} -332.363 q^{42} -48.2610 q^{43} -688.920 q^{45} +470.948 q^{46} -455.085 q^{47} +1926.46 q^{48} +49.0000 q^{49} +354.630 q^{50} +936.636 q^{51} +639.515 q^{52} -389.192 q^{53} +1087.94 q^{54} -504.449 q^{56} -458.495 q^{57} -1014.26 q^{58} +146.448 q^{59} -2579.60 q^{60} -194.654 q^{61} +978.285 q^{62} -349.394 q^{63} +1560.02 q^{64} -414.192 q^{65} -302.780 q^{67} +2275.99 q^{68} +762.886 q^{69} +523.074 q^{70} -395.902 q^{71} +3596.97 q^{72} +552.678 q^{73} -1973.01 q^{74} +574.463 q^{75} -1114.12 q^{76} +1424.84 q^{78} -824.060 q^{79} -3031.87 q^{80} +414.690 q^{81} +698.270 q^{82} +1041.32 q^{83} -1308.28 q^{84} -1474.08 q^{85} -261.282 q^{86} -1642.99 q^{87} -806.869 q^{89} -3729.77 q^{90} -210.063 q^{91} +1853.78 q^{92} +1584.72 q^{93} -2463.81 q^{94} +721.579 q^{95} +5373.74 q^{96} +524.575 q^{97} +265.283 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 8 q^{2} + 4 q^{3} + 60 q^{4} + 6 q^{5} + 34 q^{6} - 98 q^{7} + 96 q^{8} + 58 q^{9} + 4 q^{10} - 14 q^{12} + 164 q^{13} - 56 q^{14} - 240 q^{15} + 356 q^{16} + 276 q^{17} + 516 q^{18} - 12 q^{19}+ \cdots + 392 q^{98}+O(q^{100}) \) 14 * q + 8 * q^2 + 4 * q^3 + 60 * q^4 + 6 * q^5 + 34 * q^6 - 98 * q^7 + 96 * q^8 + 58 * q^9 + 4 * q^10 - 14 * q^12 + 164 * q^13 - 56 * q^14 - 240 * q^15 + 356 * q^16 + 276 * q^17 + 516 * q^18 - 12 * q^19 - 122 * q^20 - 28 * q^21 - 416 * q^23 + 836 * q^24 + 308 * q^25 + 186 * q^26 - 320 * q^27 - 420 * q^28 + 544 * q^29 + 40 * q^30 + 280 * q^31 + 1516 * q^32 + 488 * q^34 - 42 * q^35 + 830 * q^36 - 734 * q^37 - 570 * q^38 + 948 * q^39 + 1182 * q^40 + 1584 * q^41 - 238 * q^42 + 136 * q^43 + 706 * q^45 - 568 * q^46 - 356 * q^47 + 4008 * q^48 + 686 * q^49 + 640 * q^50 + 1056 * q^51 + 3060 * q^52 + 318 * q^53 - 2064 * q^54 - 672 * q^56 + 412 * q^57 - 1222 * q^58 - 944 * q^59 - 1976 * q^60 + 1808 * q^61 + 3788 * q^62 - 406 * q^63 + 3726 * q^64 + 916 * q^65 - 1156 * q^67 + 1712 * q^68 + 664 * q^69 - 28 * q^70 - 1968 * q^71 + 2972 * q^72 + 3948 * q^73 + 3396 * q^74 + 668 * q^75 - 5626 * q^76 + 1460 * q^78 - 1596 * q^79 - 340 * q^80 + 1250 * q^81 + 3916 * q^82 + 2456 * q^83 + 98 * q^84 + 4456 * q^85 - 1226 * q^86 - 2972 * q^87 + 1330 * q^89 - 4290 * q^90 - 1148 * q^91 - 3142 * q^92 - 1532 * q^93 + 1286 * q^94 + 6092 * q^95 + 13718 * q^96 - 2514 * q^97 + 392 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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