LMFDB - Embedded newform 847.4.a.m.1.2

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:	$1$
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			1.2
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	$-0.906381\pi$
$2$	−5.41395	−1.91412	−0.957060	+	0.289891i	$0.906381\pi$
$3$	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	$-0.603721\pi$
$13$	−30.0089	−0.640229	−0.320115	+	0.947379i	$0.603721\pi$
$14$	−37.8976	−0.723469
$15$	−121.046	−2.08360
$16$	219.665	3.43226
$17$	−106.800	−1.52369	−0.761845	−	0.647760i	$-0.775706\pi$
$17$	−106.800	−1.52369	−0.761845	+	0.647760i	$0.775706\pi$
$18$	−270.229	−3.53853
$19$	52.2797	0.631252	0.315626	−	0.948884i	$-0.397786\pi$
$19$	52.2797	0.631252	0.315626	+	0.948884i	$0.397786\pi$
$20$	−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	$-0.295246\pi$
$29$	187.342	1.19960	0.599802	+	0.800149i	$0.295246\pi$
$30$	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	$-0.578999\pi$
$41$	−128.976	−0.491285	−0.245643	+	0.969360i	$0.578999\pi$
$42$	−332.363	−1.22107
$43$	48.2610	0.171157	0.0855783	−	0.996331i	$-0.472726\pi$
$43$	48.2610	0.171157	0.0855783	+	0.996331i	$0.472726\pi$
$44$	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	$-0.434513\pi$
$61$	194.654	0.408573	0.204286	+	0.978911i	$0.434513\pi$
$62$	−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	$-0.646105\pi$
$73$	−552.678	−0.886111	−0.443055	+	0.896494i	$0.646105\pi$
$74$	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	$-0.300389\pi$
$79$	824.060	1.17359	0.586797	+	0.809734i	$0.300389\pi$
$80$	−3031.87	−4.23717
$81$	414.690	0.568847
$82$	698.270	0.940379
$83$	−1041.32	−1.37710	−0.688550	−	0.725188i	$-0.741753\pi$
$83$	−1041.32	−1.37710	−0.688550	+	0.725188i	$0.741753\pi$
$84$	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	$-0.629044\pi$
$101$	−800.638	−0.788777	−0.394388	+	0.918944i	$0.629044\pi$
$102$	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	$-0.478147\pi$
$107$	151.852	0.137197	0.0685987	+	0.997644i	$0.478147\pi$
$108$	4282.45	3.81554
$109$	−1722.37	−1.51352	−0.756759	−	0.653694i	$-0.773218\pi$
$109$	−1722.37	−1.51352	−0.756759	+	0.653694i	$0.773218\pi$
$110$	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	$-0.447721\pi$
$127$	468.009	0.327001	0.163500	+	0.986543i	$0.447721\pi$
$128$	−3543.96	−2.44723
$129$	423.250	0.288877
$130$	−2242.41	−1.51287
$131$	1581.15	1.05455	0.527274	−	0.849695i	$-0.323214\pi$
$131$	1581.15	1.05455	0.527274	+	0.849695i	$0.323214\pi$
$132$	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	$-0.394751\pi$
$139$	1064.09	0.649314	0.324657	+	0.945832i	$0.394751\pi$
$140$	−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	$-0.280442\pi$
$149$	2314.77	1.27270	0.636352	+	0.771398i	$0.280442\pi$
$150$	−3110.11	−1.69293
$151$	−1822.45	−0.982178	−0.491089	−	0.871109i	$-0.663401\pi$
$151$	−1822.45	−0.982178	−0.491089	+	0.871109i	$0.663401\pi$
$152$	−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	$-0.281837\pi$
$167$	2732.03	1.26593	0.632966	+	0.774179i	$0.281837\pi$
$168$	−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	$-0.483892\pi$
$173$	230.194	0.101164	0.0505818	+	0.998720i	$0.483892\pi$
$174$	−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	$-0.678320\pi$
$193$	−2849.42	−1.06273	−0.531363	+	0.847144i	$0.678320\pi$
$194$	−2840.02	−1.05104
$195$	3632.48	1.33398
$196$	1044.23	0.380550
$197$	−3620.43	−1.30936	−0.654682	−	0.755904i	$-0.727198\pi$
$197$	−3620.43	−1.30936	−0.654682	+	0.755904i	$0.727198\pi$
$198$	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	$-0.730285\pi$
$211$	−4057.90	−1.32397	−0.661984	+	0.749518i	$0.730285\pi$
$212$	−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	$-0.526927\pi$
$227$	−577.950	−0.168986	−0.0844932	+	0.996424i	$0.526927\pi$
$228$	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	$-0.772156\pi$
$233$	−5367.41	−1.50915	−0.754573	+	0.656217i	$0.772156\pi$
$234$	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	$-0.684976\pi$
$239$	−4056.64	−1.09792	−0.548959	+	0.835849i	$0.684976\pi$
$240$	−26589.6	−7.15147
$241$	−1544.94	−0.412939	−0.206469	−	0.978453i	$-0.566197\pi$
$241$	−1544.94	−0.412939	−0.206469	+	0.978453i	$0.566197\pi$
$242$	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	$-0.611152\pi$
$263$	−2918.55	−0.684280	−0.342140	+	0.939649i	$0.611152\pi$
$264$	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	$-0.621001\pi$
$271$	−3310.65	−0.742094	−0.371047	+	0.928614i	$0.621001\pi$
$272$	−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	$-0.497291\pi$
$277$	78.4692	0.0170208	0.00851039	+	0.999964i	$0.497291\pi$
$278$	−5760.91	−1.24286
$279$	9019.22	1.93536
$280$	6962.54	1.48604
$281$	−4810.68	−1.02128	−0.510642	−	0.859793i	$-0.670592\pi$
$281$	−4810.68	−1.02128	−0.510642	+	0.859793i	$0.670592\pi$
$282$	21607.7	4.56283
$283$	−1378.13	−0.289475	−0.144738	−	0.989470i	$-0.546234\pi$
$283$	−1378.13	−0.289475	−0.144738	+	0.989470i	$0.546234\pi$
$284$	−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	$-0.313406\pi$
$293$	5549.00	1.10640	0.553201	+	0.833048i	$0.313406\pi$
$294$	−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	$-0.782622\pi$
$307$	−8345.50	−1.55148	−0.775738	+	0.631055i	$0.782622\pi$
$308$	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	$-0.354667\pi$
$337$	5454.98	0.881756	0.440878	+	0.897567i	$0.354667\pi$
$338$	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	$-0.573121\pi$
$347$	−2943.67	−0.455402	−0.227701	+	0.973731i	$0.573121\pi$
$348$	35013.5	5.39346
$349$	5541.62	0.849960	0.424980	−	0.905203i	$-0.360281\pi$
$349$	5541.62	0.849960	0.424980	+	0.905203i	$0.360281\pi$
$350$	−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	$-0.477301\pi$
$359$	969.299	0.142500	0.0712502	+	0.997458i	$0.477301\pi$
$360$	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	$-0.308643\pi$
$373$	8149.03	1.13121	0.565604	+	0.824677i	$0.308643\pi$
$374$	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	$-0.504389\pi$
$409$	−228.102	−0.0275769	−0.0137884	+	0.999905i	$0.504389\pi$
$410$	−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	$-0.694661\pi$
$431$	−10274.4	−1.14827	−0.574133	+	0.818762i	$0.694661\pi$
$432$	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	$-0.488947\pi$
$439$	638.674	0.0694356	0.0347178	+	0.999397i	$0.488947\pi$
$440$	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	$-0.404725\pi$
$457$	5761.44	0.589734	0.294867	+	0.955538i	$0.404725\pi$
$458$	−1824.54	−0.186147
$459$	−21461.6	−2.18244
$460$	−25586.4	−2.59342
$461$	−6714.16	−0.678329	−0.339165	−	0.940727i	$-0.610144\pi$
$461$	−6714.16	−0.678329	−0.339165	+	0.940727i	$0.610144\pi$
$462$	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	$-0.686988\pi$
$479$	−11620.5	−1.10846	−0.554231	+	0.832363i	$0.686988\pi$
$480$	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	$-0.587549\pi$
$491$	−5909.66	−0.543176	−0.271588	+	0.962414i	$0.587549\pi$
$492$	−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	$-0.213854\pi$
$503$	17658.9	1.56535	0.782676	+	0.622429i	$0.213854\pi$
$504$	−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	$-0.455615\pi$
$523$	3324.73	0.277974	0.138987	+	0.990294i	$0.455615\pi$
$524$	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	$-0.659490\pi$
$541$	−12088.8	−0.960698	−0.480349	+	0.877078i	$0.659490\pi$
$542$	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	$-0.499905\pi$
$547$	7.61110	0.000594930 0	0.000297465		1.00000i	$0.499905\pi$
$548$	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	$-0.286301\pi$
$557$	16354.5	1.24410	0.622048	+	0.782979i	$0.286301\pi$
$558$	−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	$-0.692368\pi$
$563$	−15181.3	−1.13644	−0.568220	+	0.822877i	$0.692368\pi$
$564$	−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	$-0.304992\pi$
$569$	15609.4	1.15005	0.575027	+	0.818134i	$0.304992\pi$
$570$	34260.9	2.51760
$571$	19353.4	1.41842	0.709209	−	0.704998i	$-0.249052\pi$
$571$	19353.4	1.41842	0.709209	+	0.704998i	$0.249052\pi$
$572$	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	$-0.247898\pi$
$593$	20556.4	1.42352	0.711761	+	0.702422i	$0.247898\pi$
$594$	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	$-0.515416\pi$
$601$	−1426.54	−0.0968216	−0.0484108	+	0.998828i	$0.515416\pi$
$602$	−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	$-0.218237\pi$
$607$	23151.1	1.54807	0.774033	+	0.633146i	$0.218237\pi$
$608$	−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	$-0.575446\pi$
$613$	−7127.41	−0.469614	−0.234807	+	0.972042i	$0.575446\pi$
$614$	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	$-0.276618\pi$
$659$	21842.6	1.29115	0.645574	+	0.763698i	$0.276618\pi$
$660$	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	$-0.481835\pi$
$673$	1991.58	0.114071	0.0570354	+	0.998372i	$0.481835\pi$
$674$	−29533.0	−1.68779
$675$	13162.9	0.750580
$676$	−27628.7	−1.57196
$677$	−24272.2	−1.37793	−0.688964	−	0.724795i	$-0.741934\pi$
$677$	−24272.2	−1.37793	−0.688964	+	0.724795i	$0.741934\pi$
$678$	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	$-0.208893\pi$
$701$	29409.5	1.58457	0.792283	+	0.610154i	$0.208893\pi$
$702$	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	$-0.450910\pi$
$733$	6096.90	0.307222	0.153611	+	0.988131i	$0.450910\pi$
$734$	−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	$-0.835204\pi$
$739$	−34913.3	−1.73790	−0.868949	+	0.494901i	$0.835204\pi$
$740$	107193.	5.32500
$741$	−13758.9	−0.682115
$742$	14749.5	0.729744
$743$	3720.35	0.183696	0.0918482	−	0.995773i	$-0.470723\pi$
$743$	3720.35	0.183696	0.0918482	+	0.995773i	$0.470723\pi$
$744$	−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	$-0.643747\pi$
$761$	−18322.8	−0.872802	−0.436401	+	0.899752i	$0.643747\pi$
$762$	−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	$-0.483755\pi$
$769$	2175.73	0.102027	0.0510134	+	0.998698i	$0.483755\pi$
$770$	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	$-0.498538\pi$
$787$	202.840	0.00918738	0.00459369	+	0.999989i	$0.498538\pi$
$788$	−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	$-0.707246\pi$
$809$	−27890.8	−1.21210	−0.606049	+	0.795427i	$0.707246\pi$
$810$	30987.6	1.34419
$811$	−4519.06	−0.195667	−0.0978333	−	0.995203i	$-0.531191\pi$
$811$	−4519.06	−0.195667	−0.0978333	+	0.995203i	$0.531191\pi$
$812$	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	$-0.295429\pi$
$821$	28198.0	1.19868	0.599341	+	0.800494i	$0.295429\pi$
$822$	−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	$-0.533871\pi$
$827$	−5051.86	−0.212419	−0.106210	+	0.994344i	$0.533871\pi$
$828$	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	$-0.223408\pi$
$853$	38049.2	1.52729	0.763645	+	0.645636i	$0.223408\pi$
$854$	−7376.94	−0.295590
$855$	−36016.5	−1.44063
$856$	−10943.1	−0.436948
$857$	−13673.1	−0.544999	−0.272500	−	0.962156i	$-0.587850\pi$
$857$	−13673.1	−0.544999	−0.272500	+	0.962156i	$0.587850\pi$
$858$	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	$-0.249463\pi$
$877$	36791.3	1.41660	0.708298	+	0.705913i	$0.249463\pi$
$878$	−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	$-0.509338\pi$
$887$	−1549.80	−0.0586664	−0.0293332	+	0.999570i	$0.509338\pi$
$888$	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	$-0.455940\pi$
$919$	7688.00	0.275956	0.137978	+	0.990435i	$0.455940\pi$
$920$	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	$-0.676917\pi$
$937$	−30266.7	−1.05525	−0.527625	+	0.849478i	$0.676917\pi$
$938$	11474.6	0.399425
$939$	−40522.4	−1.40830
$940$	133858.	4.64464
$941$	31355.3	1.08624	0.543120	−	0.839655i	$-0.317243\pi$
$941$	31355.3	1.08624	0.543120	+	0.839655i	$0.317243\pi$
$942$	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	$-0.681585\pi$
$953$	−31774.7	−1.08005	−0.540023	+	0.841650i	$0.681585\pi$
$954$	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	$-0.713159\pi$
$967$	−37330.5	−1.24143	−0.620717	+	0.784034i	$0.713159\pi$
$968$	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	$-0.130348\pi$
$997$	57755.5	1.83464	0.917319	+	0.398152i	$0.130348\pi$
$998$	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.m.1.2	✓	14
11.10	odd	2		847.4.a.n.1.13	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.4.a.m.1.2	✓	14	1.1	even	1	trivial
847.4.a.n.1.13	yes	14	11.10	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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
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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