LMFDB - Embedded newform 650.4.b.l.599.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [650,4,Mod(599,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("650.599"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	650.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-16,0,-8,0,0,28,0,-100] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$38.3512415037$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{19})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.2
Root			$2.17945 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.4.b.l.599.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} +3.35890i q^{3} -4.00000 q^{4} +6.71780 q^{6} +26.1534i q^{7} +8.00000i q^{8} +15.7178 q^{9} +5.51229 q^{11} -13.4356i q^{12} -13.0000i q^{13} +52.3068 q^{14} +16.0000 q^{16} -105.025i q^{17} -31.4356i q^{18} +112.383 q^{19} -87.8466 q^{21} -11.0246i q^{22} +74.0767i q^{23} -26.8712 q^{24} -26.0000 q^{26} +143.485i q^{27} -104.614i q^{28} +225.896 q^{29} -205.359 q^{31} -32.0000i q^{32} +18.5152i q^{33} -210.049 q^{34} -62.8712 q^{36} +299.485i q^{37} -224.767i q^{38} +43.6657 q^{39} -416.203 q^{41} +175.693i q^{42} +418.942i q^{43} -22.0492 q^{44} +148.153 q^{46} +419.589i q^{47} +53.7424i q^{48} -341.000 q^{49} +352.767 q^{51} +52.0000i q^{52} -655.786i q^{53} +286.970 q^{54} -209.227 q^{56} +377.485i q^{57} -451.792i q^{58} +221.666 q^{59} +597.687 q^{61} +410.718i q^{62} +411.074i q^{63} -64.0000 q^{64} +37.0305 q^{66} -310.503i q^{67} +420.098i q^{68} -248.816 q^{69} -310.334 q^{71} +125.742i q^{72} +486.859i q^{73} +598.970 q^{74} -449.534 q^{76} +144.165i q^{77} -87.3314i q^{78} -52.3501 q^{79} -57.5703 q^{81} +832.405i q^{82} +1125.85i q^{83} +351.386 q^{84} +837.884 q^{86} +758.761i q^{87} +44.0983i q^{88} +112.466 q^{89} +339.994 q^{91} -296.307i q^{92} -689.780i q^{93} +839.178 q^{94} +107.485 q^{96} +159.963i q^{97} +682.000i q^{98} +86.6411 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} - 8 q^{6} + 28 q^{9} - 100 q^{11} + 64 q^{16} + 188 q^{19} - 456 q^{21} + 32 q^{24} - 104 q^{26} + 520 q^{29} - 804 q^{31} - 352 q^{34} - 112 q^{36} - 52 q^{39} - 1072 q^{41} + 400 q^{44}+ \cdots + 364 q^{99}+O(q^{100}) $ 4 * q - 16 * q^4 - 8 * q^6 + 28 * q^9 - 100 * q^11 + 64 * q^16 + 188 * q^19 - 456 * q^21 + 32 * q^24 - 104 * q^26 + 520 * q^29 - 804 * q^31 - 352 * q^34 - 112 * q^36 - 52 * q^39 - 1072 * q^41 + 400 * q^44 + 488 * q^46 - 1364 * q^49 + 888 * q^51 + 32 * q^54 + 660 * q^59 + 1240 * q^61 - 256 * q^64 + 1264 * q^66 + 16 * q^69 - 1468 * q^71 + 1280 * q^74 - 752 * q^76 + 1848 * q^79 - 1660 * q^81 + 1824 * q^84 + 248 * q^86 + 1496 * q^89 + 3008 * q^94 - 128 * q^96 + 364 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	3.35890i	0.646420i	0.946327	+	0.323210i	$0.104762\pi$
$3$	3.35890i	0.646420i	−0.946327	+	0.323210i	$0.895238\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	6.71780	0.457088
$7$	26.1534i	1.41215i	0.708137	+	0.706075i	$0.249536\pi$
$7$	26.1534i	1.41215i	−0.708137	+	0.706075i	$0.750464\pi$
$8$	8.00000i	0.353553i
$9$	15.7178	0.582141
$10$	0	0
$11$	5.51229	0.151093	0.0755463	−	0.997142i	$-0.475930\pi$
$11$	5.51229	0.151093	0.0755463	+	0.997142i	$0.475930\pi$
$12$	− 13.4356i	− 0.323210i
$13$	− 13.0000i	− 0.277350i
$13$	− 13.0000i	− 0.277350i
$14$	52.3068	0.998541
$15$	0	0
$16$	16.0000	0.250000
$17$	− 105.025i	− 1.49836i	−0.662364	−	0.749182i	$-0.730447\pi$
$17$	− 105.025i	− 1.49836i	0.662364	−	0.749182i	$-0.269553\pi$
$18$	− 31.4356i	− 0.411636i
$19$	112.383	1.35698	0.678488	−	0.734612i	$-0.262636\pi$
$19$	112.383	1.35698	0.678488	+	0.734612i	$0.262636\pi$
$20$	0	0
$21$	−87.8466	−0.912843
$22$	− 11.0246i	− 0.106839i
$23$	74.0767i	0.671568i	0.941939	+	0.335784i	$0.109001\pi$
$23$	74.0767i	0.671568i	−0.941939	+	0.335784i	$0.890999\pi$
$24$	−26.8712	−0.228544
$25$	0	0
$26$	−26.0000	−0.196116
$27$	143.485i	1.02273i
$28$	− 104.614i	− 0.706075i
$29$	225.896	1.44648	0.723238	−	0.690599i	$-0.242653\pi$
$29$	225.896	1.44648	0.723238	+	0.690599i	$0.242653\pi$
$30$	0	0
$31$	−205.359	−1.18979	−0.594896	−	0.803803i	$-0.702807\pi$
$31$	−205.359	−1.18979	−0.594896	+	0.803803i	$0.702807\pi$
$32$	− 32.0000i	− 0.176777i
$33$	18.5152i	0.0976693i
$34$	−210.049	−1.05950
$35$	0	0
$36$	−62.8712	−0.291070
$37$	299.485i	1.33068i	0.746542	+	0.665338i	$0.231712\pi$
$37$	299.485i	1.33068i	−0.746542	+	0.665338i	$0.768288\pi$
$38$	− 224.767i	− 0.959527i
$39$	43.6657	0.179285
$40$	0	0
$41$	−416.203	−1.58536	−0.792682	−	0.609635i	$-0.791316\pi$
$41$	−416.203	−1.58536	−0.792682	+	0.609635i	$0.791316\pi$
$42$	175.693i	0.645477i
$43$	418.942i	1.48577i	0.669420	+	0.742884i	$0.266543\pi$
$43$	418.942i	1.48577i	−0.669420	+	0.742884i	$0.733457\pi$
$44$	−22.0492	−0.0755463
$45$	0	0
$46$	148.153	0.474870
$47$	419.589i	1.30220i	0.758992	+	0.651099i	$0.225692\pi$
$47$	419.589i	1.30220i	−0.758992	+	0.651099i	$0.774308\pi$
$48$	53.7424i	0.161605i
$49$	−341.000	−0.994169
$50$	0	0
$51$	352.767	0.968574
$52$	52.0000i	0.138675i
$53$	− 655.786i	− 1.69961i	−0.527101	−	0.849803i	$-0.676721\pi$
$53$	− 655.786i	− 1.69961i	0.527101	−	0.849803i	$-0.323279\pi$
$54$	286.970	0.723178
$55$	0	0
$56$	−209.227	−0.499271
$57$	377.485i	0.877177i
$58$	− 451.792i	− 1.02281i
$59$	221.666	0.489126	0.244563	−	0.969633i	$-0.421356\pi$
$59$	221.666	0.489126	0.244563	+	0.969633i	$0.421356\pi$
$60$	0	0
$61$	597.687	1.25452	0.627262	−	0.778808i	$-0.284175\pi$
$61$	597.687	1.25452	0.627262	+	0.778808i	$0.284175\pi$
$62$	410.718i	0.841310i
$63$	411.074i	0.822070i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	37.0305	0.0690627
$67$	− 310.503i	− 0.566180i	−0.959093	−	0.283090i	$-0.908641\pi$
$67$	− 310.503i	− 0.566180i	0.959093	−	0.283090i	$-0.0913595\pi$
$68$	420.098i	0.749182i
$69$	−248.816	−0.434115
$70$	0	0
$71$	−310.334	−0.518731	−0.259366	−	0.965779i	$-0.583513\pi$
$71$	−310.334	−0.518731	−0.259366	+	0.965779i	$0.583513\pi$
$72$	125.742i	0.205818i
$73$	486.859i	0.780583i	0.920691	+	0.390292i	$0.127626\pi$
$73$	486.859i	0.780583i	−0.920691	+	0.390292i	$0.872374\pi$
$74$	598.970	0.940930
$75$	0	0
$76$	−449.534	−0.678488
$77$	144.165i	0.213366i
$78$	− 87.3314i	− 0.126773i
$79$	−52.3501	−0.0745550	−0.0372775	−	0.999305i	$-0.511869\pi$
$79$	−52.3501	−0.0745550	−0.0372775	+	0.999305i	$0.511869\pi$
$80$	0	0
$81$	−57.5703	−0.0789716
$82$	832.405i	1.12102i
$83$	1125.85i	1.48889i	0.667685	+	0.744444i	$0.267285\pi$
$83$	1125.85i	1.48889i	−0.667685	+	0.744444i	$0.732715\pi$
$84$	351.386	0.456421
$85$	0	0
$86$	837.884	1.05060
$87$	758.761i	0.935031i
$88$	44.0983i	0.0534193i
$89$	112.466	0.133948	0.0669740	−	0.997755i	$-0.478666\pi$
$89$	112.466	0.133948	0.0669740	+	0.997755i	$0.478666\pi$
$90$	0	0
$91$	339.994	0.391660
$92$	− 296.307i	− 0.335784i
$93$	− 689.780i	− 0.769106i
$94$	839.178	0.920794
$95$	0	0
$96$	107.485	0.114272
$97$	159.963i	0.167441i	0.996489	+	0.0837203i	$0.0266802\pi$
$97$	159.963i	0.167441i	−0.996489	+	0.0837203i	$0.973320\pi$
$98$	682.000i	0.702984i
$99$	86.6411	0.0879572
$100$	0	0
$101$	408.663	0.402609	0.201304	−	0.979529i	$-0.435482\pi$
$101$	408.663	0.402609	0.201304	+	0.979529i	$0.435482\pi$
$102$	− 705.534i	− 0.684885i
$103$	1534.30i	1.46776i	0.679278	+	0.733881i	$0.262293\pi$
$103$	1534.30i	1.46776i	−0.679278	+	0.733881i	$0.737707\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	−1311.57	−1.20180
$107$	− 35.7876i	− 0.0323338i	−0.999869	−	0.0161669i	$-0.994854\pi$
$107$	− 35.7876i	− 0.0323338i	0.999869	−	0.0161669i	$-0.00514630\pi$
$108$	− 573.939i	− 0.511364i
$109$	−717.265	−0.630289	−0.315144	−	0.949044i	$-0.602053\pi$
$109$	−717.265	−0.630289	−0.315144	+	0.949044i	$0.602053\pi$
$110$	0	0
$111$	−1005.94	−0.860176
$112$	418.454i	0.353038i
$113$	901.835i	0.750774i	0.926868	+	0.375387i	$0.122490\pi$
$113$	901.835i	0.750774i	−0.926868	+	0.375387i	$0.877510\pi$
$114$	754.970	0.620258
$115$	0	0
$116$	−903.583	−0.723238
$117$	− 204.331i	− 0.161457i
$118$	− 443.331i	− 0.345864i
$119$	2746.75	2.11592
$120$	0	0
$121$	−1300.61	−0.977171
$122$	− 1195.37i	− 0.887083i
$123$	− 1397.98i	− 1.02481i
$124$	821.436	0.594896
$125$	0	0
$126$	822.148	0.581291
$127$	1763.63i	1.23226i	0.787645	+	0.616129i	$0.211300\pi$
$127$	1763.63i	1.23226i	−0.787645	+	0.616129i	$0.788700\pi$
$128$	128.000i	0.0883883i
$129$	−1407.18	−0.960431
$130$	0	0
$131$	415.497	0.277115	0.138558	−	0.990354i	$-0.455753\pi$
$131$	415.497	0.277115	0.138558	+	0.990354i	$0.455753\pi$
$132$	− 74.0609i	− 0.0488347i
$133$	2939.21i	1.91625i
$134$	−621.007	−0.400349
$135$	0	0
$136$	840.197	0.529752
$137$	574.527i	0.358286i	0.983823	+	0.179143i	$0.0573324\pi$
$137$	574.527i	0.358286i	−0.983823	+	0.179143i	$0.942668\pi$
$138$	497.632i	0.306966i
$139$	−2136.85	−1.30392	−0.651962	−	0.758251i	$-0.726054\pi$
$139$	−2136.85	−1.30392	−0.651962	+	0.758251i	$0.726054\pi$
$140$	0	0
$141$	−1409.36	−0.841768
$142$	620.669i	0.366798i
$143$	− 71.6598i	− 0.0419056i
$144$	251.485	0.145535
$145$	0	0
$146$	973.719	0.551956
$147$	− 1145.38i	− 0.642651i
$148$	− 1197.94i	− 0.665338i
$149$	222.295	0.122222	0.0611111	−	0.998131i	$-0.480536\pi$
$149$	222.295	0.122222	0.0611111	+	0.998131i	$0.480536\pi$
$150$	0	0
$151$	1185.07	0.638674	0.319337	−	0.947641i	$-0.396540\pi$
$151$	1185.07	0.638674	0.319337	+	0.947641i	$0.396540\pi$
$152$	899.068i	0.479763i
$153$	− 1650.76i	− 0.872259i
$154$	288.330	0.150872
$155$	0	0
$156$	−174.663	−0.0896424
$157$	2797.58i	1.42211i	0.703136	+	0.711056i	$0.251783\pi$
$157$	2797.58i	1.42211i	−0.703136	+	0.711056i	$0.748217\pi$
$158$	104.700i	0.0527183i
$159$	2202.72	1.09866
$160$	0	0
$161$	−1937.36	−0.948355
$162$	115.141i	0.0558413i
$163$	− 1556.22i	− 0.747808i	−0.927467	−	0.373904i	$-0.878019\pi$
$163$	− 1556.22i	− 0.747808i	0.927467	−	0.373904i	$-0.121981\pi$
$164$	1664.81	0.792682
$165$	0	0
$166$	2251.69	1.05280
$167$	− 2743.80i	− 1.27139i	−0.771942	−	0.635693i	$-0.780714\pi$
$167$	− 2743.80i	− 1.27139i	0.771942	−	0.635693i	$-0.219286\pi$
$168$	− 702.773i	− 0.322739i
$169$	−169.000	−0.0769231
$170$	0	0
$171$	1766.42	0.789951
$172$	− 1675.77i	− 0.742884i
$173$	− 3563.32i	− 1.56598i	−0.622035	−	0.782989i	$-0.713694\pi$
$173$	− 3563.32i	− 1.56598i	0.622035	−	0.782989i	$-0.286306\pi$
$174$	1517.52	0.661167
$175$	0	0
$176$	88.1967	0.0377732
$177$	744.553i	0.316181i
$178$	− 224.932i	− 0.0947156i
$179$	−481.054	−0.200870	−0.100435	−	0.994944i	$-0.532023\pi$
$179$	−481.054	−0.200870	−0.100435	+	0.994944i	$0.532023\pi$
$180$	0	0
$181$	4736.18	1.94496	0.972479	−	0.232989i	$-0.0748505\pi$
$181$	4736.18	1.94496	0.972479	+	0.232989i	$0.0748505\pi$
$182$	− 679.988i	− 0.276946i
$183$	2007.57i	0.810951i
$184$	−592.614	−0.237435
$185$	0	0
$186$	−1379.56	−0.543840
$187$	− 578.926i	− 0.226392i
$188$	− 1678.36i	− 0.651099i
$189$	−3752.61	−1.44425
$190$	0	0
$191$	1700.58	0.644238	0.322119	−	0.946699i	$-0.395605\pi$
$191$	1700.58	0.644238	0.322119	+	0.946699i	$0.395605\pi$
$192$	− 214.970i	− 0.0808026i
$193$	1455.94i	0.543009i	0.962437	+	0.271505i	$0.0875212\pi$
$193$	1455.94i	0.543009i	−0.962437	+	0.271505i	$0.912479\pi$
$194$	319.925	0.118398
$195$	0	0
$196$	1364.00	0.497085
$197$	− 3617.28i	− 1.30822i	−0.756397	−	0.654112i	$-0.773042\pi$
$197$	− 3617.28i	− 1.30822i	0.756397	−	0.654112i	$-0.226958\pi$
$198$	− 173.282i	− 0.0621951i
$199$	3445.88	1.22750	0.613749	−	0.789501i	$-0.289661\pi$
$199$	3445.88	1.22750	0.613749	+	0.789501i	$0.289661\pi$
$200$	0	0
$201$	1042.95	0.365990
$202$	− 817.325i	− 0.284687i
$203$	5907.94i	2.04264i
$204$	−1411.07	−0.484287
$205$	0	0
$206$	3068.61	1.03786
$207$	1164.32i	0.390947i
$208$	− 208.000i	− 0.0693375i
$209$	619.491	0.205029
$210$	0	0
$211$	−4173.44	−1.36166	−0.680832	−	0.732440i	$-0.738382\pi$
$211$	−4173.44	−1.36166	−0.680832	+	0.732440i	$0.738382\pi$
$212$	2623.14i	0.849803i
$213$	− 1042.38i	− 0.335318i
$214$	−71.5751	−0.0228634
$215$	0	0
$216$	−1147.88	−0.361589
$217$	− 5370.83i	− 1.68017i
$218$	1434.53i	0.445682i
$219$	−1635.31	−0.504585
$220$	0	0
$221$	−1365.32	−0.415572
$222$	2011.88i	0.608236i
$223$	− 1103.24i	− 0.331292i	−0.986185	−	0.165646i	$-0.947029\pi$
$223$	− 1103.24i	− 0.331292i	0.986185	−	0.165646i	$-0.0529709\pi$
$224$	836.909	0.249635
$225$	0	0
$226$	1803.67	0.530878
$227$	2494.39i	0.729333i	0.931138	+	0.364667i	$0.118817\pi$
$227$	2494.39i	0.729333i	−0.931138	+	0.364667i	$0.881183\pi$
$228$	− 1509.94i	− 0.438588i
$229$	2443.33	0.705066	0.352533	−	0.935799i	$-0.385320\pi$
$229$	2443.33	0.705066	0.352533	+	0.935799i	$0.385320\pi$
$230$	0	0
$231$	−484.236	−0.137924
$232$	1807.17i	0.511406i
$233$	− 6343.70i	− 1.78365i	−0.452385	−	0.891823i	$-0.649427\pi$
$233$	− 6343.70i	− 1.78365i	0.452385	−	0.891823i	$-0.350573\pi$
$234$	−408.663	−0.114167
$235$	0	0
$236$	−886.663	−0.244563
$237$	− 175.839i	− 0.0481939i
$238$	− 5493.50i	− 1.49618i
$239$	−2939.07	−0.795449	−0.397725	−	0.917505i	$-0.630200\pi$
$239$	−2939.07	−0.795449	−0.397725	+	0.917505i	$0.630200\pi$
$240$	0	0
$241$	−4466.81	−1.19391	−0.596955	−	0.802275i	$-0.703623\pi$
$241$	−4466.81	−1.19391	−0.596955	+	0.802275i	$0.703623\pi$
$242$	2601.23i	0.690964i
$243$	3680.72i	0.971679i
$244$	−2390.75	−0.627262
$245$	0	0
$246$	−2795.96	−0.724651
$247$	− 1460.99i	− 0.376357i
$248$	− 1642.87i	− 0.420655i
$249$	−3781.60	−0.962448
$250$	0	0
$251$	−5118.30	−1.28711	−0.643554	−	0.765400i	$-0.722541\pi$
$251$	−5118.30	−1.28711	−0.643554	+	0.765400i	$0.722541\pi$
$252$	− 1644.30i	− 0.411035i
$253$	408.332i	0.101469i
$254$	3527.26	0.871339
$255$	0	0
$256$	256.000	0.0625000
$257$	747.695i	0.181478i	0.995875	+	0.0907392i	$0.0289230\pi$
$257$	747.695i	0.181478i	−0.995875	+	0.0907392i	$0.971077\pi$
$258$	2814.37i	0.679127i
$259$	−7832.54	−1.87911
$260$	0	0
$261$	3550.58	0.842052
$262$	− 830.993i	− 0.195950i
$263$	2135.10i	0.500592i	0.968169	+	0.250296i	$0.0805279\pi$
$263$	2135.10i	0.500592i	−0.968169	+	0.250296i	$0.919472\pi$
$264$	−148.122	−0.0345313
$265$	0	0
$266$	5878.42	1.35500
$267$	377.762i	0.0865868i
$268$	1242.01i	0.283090i
$269$	780.553	0.176919	0.0884594	−	0.996080i	$-0.471806\pi$
$269$	780.553	0.176919	0.0884594	+	0.996080i	$0.471806\pi$
$270$	0	0
$271$	−5412.95	−1.21333	−0.606667	−	0.794956i	$-0.707494\pi$
$271$	−5412.95	−1.21333	−0.606667	+	0.794956i	$0.707494\pi$
$272$	− 1680.39i	− 0.374591i
$273$	1142.01i	0.253177i
$274$	1149.05	0.253346
$275$	0	0
$276$	995.265	0.217058
$277$	− 1738.26i	− 0.377047i	−0.982069	−	0.188524i	$-0.939630\pi$
$277$	− 1738.26i	− 0.377047i	0.982069	−	0.188524i	$-0.0603702\pi$
$278$	4273.71i	0.922014i
$279$	−3227.79	−0.692626
$280$	0	0
$281$	3923.08	0.832851	0.416426	−	0.909170i	$-0.363283\pi$
$281$	3923.08	0.832851	0.416426	+	0.909170i	$0.363283\pi$
$282$	2818.71i	0.595220i
$283$	− 356.775i	− 0.0749401i	−0.999298	−	0.0374701i	$-0.988070\pi$
$283$	− 356.775i	− 0.0749401i	0.999298	−	0.0374701i	$-0.0119299\pi$
$284$	1241.34	0.259366
$285$	0	0
$286$	−143.320	−0.0296317
$287$	− 10885.1i	− 2.23877i
$288$	− 502.970i	− 0.102909i
$289$	−6117.16	−1.24510
$290$	0	0
$291$	−537.298	−0.108237
$292$	− 1947.44i	− 0.390292i
$293$	6936.33i	1.38302i	0.722367	+	0.691510i	$0.243054\pi$
$293$	6936.33i	1.38302i	−0.722367	+	0.691510i	$0.756946\pi$
$294$	−2290.77	−0.454423
$295$	0	0
$296$	−2395.88	−0.470465
$297$	790.930i	0.154527i
$298$	− 444.590i	− 0.0864242i
$299$	962.997	0.186259
$300$	0	0
$301$	−10956.8	−2.09813
$302$	− 2370.14i	− 0.451611i
$303$	1372.66i	0.260254i
$304$	1798.14	0.339244
$305$	0	0
$306$	−3301.51	−0.616780
$307$	− 3919.02i	− 0.728568i	−0.931288	−	0.364284i	$-0.881314\pi$
$307$	− 3919.02i	− 0.728568i	0.931288	−	0.364284i	$-0.118686\pi$
$308$	− 576.661i	− 0.106683i
$309$	−5153.58	−0.948792
$310$	0	0
$311$	1817.13	0.331319	0.165659	−	0.986183i	$-0.447025\pi$
$311$	1817.13	0.331319	0.165659	+	0.986183i	$0.447025\pi$
$312$	349.325i	0.0633867i
$313$	− 1464.09i	− 0.264394i	−0.991223	−	0.132197i	$-0.957797\pi$
$313$	− 1464.09i	− 0.264394i	0.991223	−	0.132197i	$-0.0422032\pi$
$314$	5595.17	1.00558
$315$	0	0
$316$	209.400	0.0372775
$317$	4610.16i	0.816821i	0.912798	+	0.408411i	$0.133917\pi$
$317$	4610.16i	0.816821i	−0.912798	+	0.408411i	$0.866083\pi$
$318$	− 4405.44i	− 0.776870i
$319$	1245.20	0.218552
$320$	0	0
$321$	120.207	0.0209012
$322$	3874.71i	0.670588i
$323$	− 11803.0i	− 2.03324i
$324$	230.281	0.0394858
$325$	0	0
$326$	−3112.44	−0.528780
$327$	− 2409.22i	− 0.407432i
$328$	− 3329.62i	− 0.560511i
$329$	−10973.7	−1.83890
$330$	0	0
$331$	2243.30	0.372516	0.186258	−	0.982501i	$-0.440364\pi$
$331$	2243.30	0.372516	0.186258	+	0.982501i	$0.440364\pi$
$332$	− 4503.39i	− 0.744444i
$333$	4707.24i	0.774640i
$334$	−5487.59	−0.899005
$335$	0	0
$336$	−1405.55	−0.228211
$337$	− 6004.40i	− 0.970566i	−0.874357	−	0.485283i	$-0.838717\pi$
$337$	− 6004.40i	− 0.970566i	0.874357	−	0.485283i	$-0.161283\pi$
$338$	338.000i	0.0543928i
$339$	−3029.17	−0.485316
$340$	0	0
$341$	−1132.00	−0.179769
$342$	− 3532.84i	− 0.558580i
$343$	52.3068i	0.00823411i
$344$	−3351.54	−0.525299
$345$	0	0
$346$	−7126.64	−1.10731
$347$	1757.95i	0.271964i	0.990711	+	0.135982i	$0.0434189\pi$
$347$	1757.95i	0.271964i	−0.990711	+	0.135982i	$0.956581\pi$
$348$	− 3035.04i	− 0.467516i
$349$	5774.74	0.885715	0.442857	−	0.896592i	$-0.353965\pi$
$349$	5774.74	0.885715	0.442857	+	0.896592i	$0.353965\pi$
$350$	0	0
$351$	1865.30	0.283654
$352$	− 176.393i	− 0.0267097i
$353$	− 286.265i	− 0.0431625i	−0.999767	−	0.0215812i	$-0.993130\pi$
$353$	− 286.265i	− 0.0431625i	0.999767	−	0.0215812i	$-0.00687006\pi$
$354$	1489.11	0.223574
$355$	0	0
$356$	−449.864	−0.0669740
$357$	9226.05i	1.36777i
$358$	962.108i	0.142036i
$359$	6004.87	0.882799	0.441399	−	0.897311i	$-0.354482\pi$
$359$	6004.87	0.882799	0.441399	+	0.897311i	$0.354482\pi$
$360$	0	0
$361$	5771.05	0.841383
$362$	− 9472.36i	− 1.37529i
$363$	− 4368.63i	− 0.631663i
$364$	−1359.98	−0.195830
$365$	0	0
$366$	4015.14	0.573429
$367$	− 3793.70i	− 0.539590i	−0.962918	−	0.269795i	$-0.913044\pi$
$367$	− 3793.70i	− 0.539590i	0.962918	−	0.269795i	$-0.0869559\pi$
$368$	1185.23i	0.167892i
$369$	−6541.79	−0.922905
$370$	0	0
$371$	17151.0	2.40010
$372$	2759.12i	0.384553i
$373$	− 8923.03i	− 1.23865i	−0.785134	−	0.619326i	$-0.787406\pi$
$373$	− 8923.03i	− 1.23865i	0.785134	−	0.619326i	$-0.212594\pi$
$374$	−1157.85	−0.160083
$375$	0	0
$376$	−3356.71	−0.460397
$377$	− 2936.65i	− 0.401180i
$378$	7505.23i	1.02124i
$379$	10977.5	1.48781	0.743903	−	0.668287i	$-0.232972\pi$
$379$	10977.5	1.48781	0.743903	+	0.668287i	$0.232972\pi$
$380$	0	0
$381$	−5923.86	−0.796557
$382$	− 3401.16i	− 0.455545i
$383$	− 10570.4i	− 1.41024i	−0.709087	−	0.705121i	$-0.750893\pi$
$383$	− 10570.4i	− 1.41024i	0.709087	−	0.705121i	$-0.249107\pi$
$384$	−429.939	−0.0571360
$385$	0	0
$386$	2911.88	0.383966
$387$	6584.85i	0.864926i
$388$	− 639.850i	− 0.0837203i
$389$	−517.213	−0.0674132	−0.0337066	−	0.999432i	$-0.510731\pi$
$389$	−517.213	−0.0674132	−0.0337066	+	0.999432i	$0.510731\pi$
$390$	0	0
$391$	7779.87	1.00625
$392$	− 2728.00i	− 0.351492i
$393$	1395.61i	0.179133i
$394$	−7234.55	−0.925055
$395$	0	0
$396$	−346.564	−0.0439786
$397$	6979.33i	0.882323i	0.897428	+	0.441162i	$0.145433\pi$
$397$	6979.33i	0.882323i	−0.897428	+	0.441162i	$0.854567\pi$
$398$	− 6891.76i	− 0.867972i
$399$	−9872.51	−1.23871
$400$	0	0
$401$	3751.66	0.467204	0.233602	−	0.972332i	$-0.424949\pi$
$401$	3751.66	0.467204	0.233602	+	0.972332i	$0.424949\pi$
$402$	− 2085.90i	− 0.258794i
$403$	2669.67i	0.329989i
$404$	−1634.65	−0.201304
$405$	0	0
$406$	11815.9	1.44437
$407$	1650.85i	0.201055i
$408$	2822.14i	0.342442i
$409$	5059.92	0.611729	0.305864	−	0.952075i	$-0.401055\pi$
$409$	5059.92	0.611729	0.305864	+	0.952075i	$0.401055\pi$
$410$	0	0
$411$	−1929.78	−0.231603
$412$	− 6137.22i	− 0.733881i
$413$	5797.31i	0.690719i
$414$	2328.65	0.276441
$415$	0	0
$416$	−416.000	−0.0490290
$417$	− 7177.48i	− 0.842884i
$418$	− 1238.98i	− 0.144977i
$419$	−1758.72	−0.205057	−0.102529	−	0.994730i	$-0.532693\pi$
$419$	−1758.72	−0.205057	−0.102529	+	0.994730i	$0.532693\pi$
$420$	0	0
$421$	3013.57	0.348866	0.174433	−	0.984669i	$-0.444191\pi$
$421$	3013.57	0.348866	0.174433	+	0.984669i	$0.444191\pi$
$422$	8346.87i	0.962842i
$423$	6595.01i	0.758063i
$424$	5246.29	0.600901
$425$	0	0
$426$	−2084.76	−0.237106
$427$	15631.6i	1.77158i
$428$	143.150i	0.0161669i
$429$	240.698	0.0270886
$430$	0	0
$431$	9218.86	1.03029	0.515147	−	0.857102i	$-0.327737\pi$
$431$	9218.86	1.03029	0.515147	+	0.857102i	$0.327737\pi$
$432$	2295.76i	0.255682i
$433$	14908.9i	1.65468i	0.561701	+	0.827340i	$0.310147\pi$
$433$	14908.9i	1.65468i	−0.561701	+	0.827340i	$0.689853\pi$
$434$	−10741.7	−1.18806
$435$	0	0
$436$	2869.06	0.315144
$437$	8325.00i	0.911301i
$438$	3270.62i	0.356795i
$439$	14586.5	1.58582	0.792910	−	0.609339i	$-0.208565\pi$
$439$	14586.5	1.58582	0.792910	+	0.609339i	$0.208565\pi$
$440$	0	0
$441$	−5359.77	−0.578746
$442$	2730.64i	0.293854i
$443$	− 1461.45i	− 0.156740i	−0.996924	−	0.0783698i	$-0.975029\pi$
$443$	− 1461.45i	− 0.156740i	0.996924	−	0.0783698i	$-0.0249715\pi$
$444$	4023.76	0.430088
$445$	0	0
$446$	−2206.47	−0.234259
$447$	746.667i	0.0790070i
$448$	− 1673.82i	− 0.176519i
$449$	580.289	0.0609923	0.0304961	−	0.999535i	$-0.490291\pi$
$449$	580.289	0.0609923	0.0304961	+	0.999535i	$0.490291\pi$
$450$	0	0
$451$	−2294.23	−0.239537
$452$	− 3607.34i	− 0.375387i
$453$	3980.54i	0.412852i
$454$	4988.79	0.515717
$455$	0	0
$456$	−3019.88	−0.310129
$457$	− 21.7638i	− 0.00222772i	−0.999999	−	0.00111386i	$-0.999645\pi$
$457$	− 21.7638i	− 0.00222772i	0.999999	−	0.00111386i	$-0.000354553\pi$
$458$	− 4886.67i	− 0.498557i
$459$	15069.4	1.53242
$460$	0	0
$461$	−14548.5	−1.46983	−0.734917	−	0.678157i	$-0.762779\pi$
$461$	−14548.5	−1.46983	−0.734917	+	0.678157i	$0.762779\pi$
$462$	968.472i	0.0975269i
$463$	− 5988.97i	− 0.601147i	−0.953759	−	0.300574i	$-0.902822\pi$
$463$	− 5988.97i	− 0.601147i	0.953759	−	0.300574i	$-0.0971781\pi$
$464$	3614.33	0.361619
$465$	0	0
$466$	−12687.4	−1.26123
$467$	− 6294.85i	− 0.623749i	−0.950123	−	0.311875i	$-0.899043\pi$
$467$	− 6294.85i	− 0.623749i	0.950123	−	0.311875i	$-0.100957\pi$
$468$	817.325i	0.0807284i
$469$	8120.72	0.799531
$470$	0	0
$471$	−9396.80	−0.919282
$472$	1773.33i	0.172932i
$473$	2309.33i	0.224489i
$474$	−351.677	−0.0340782
$475$	0	0
$476$	−10987.0	−1.05796
$477$	− 10307.5i	− 0.989409i
$478$	5878.13i	0.562468i
$479$	−5353.36	−0.510649	−0.255325	−	0.966855i	$-0.582182\pi$
$479$	−5353.36	−0.510649	−0.255325	+	0.966855i	$0.582182\pi$
$480$	0	0
$481$	3893.30	0.369063
$482$	8933.61i	0.844222i
$483$	− 6507.39i	− 0.613036i
$484$	5202.46	0.488586
$485$	0	0
$486$	7361.43	0.687081
$487$	− 15339.9i	− 1.42735i	−0.700479	−	0.713673i	$-0.747030\pi$
$487$	− 15339.9i	− 1.42735i	0.700479	−	0.713673i	$-0.252970\pi$
$488$	4781.50i	0.443542i
$489$	5227.19	0.483398
$490$	0	0
$491$	16288.5	1.49713	0.748563	−	0.663063i	$-0.230744\pi$
$491$	16288.5	1.49713	0.748563	+	0.663063i	$0.230744\pi$
$492$	5591.93i	0.512406i
$493$	− 23724.6i	− 2.16735i
$494$	−2921.97	−0.266125
$495$	0	0
$496$	−3285.74	−0.297448
$497$	− 8116.30i	− 0.732526i
$498$	7563.21i	0.680553i
$499$	−5857.06	−0.525447	−0.262723	−	0.964871i	$-0.584621\pi$
$499$	−5857.06	−0.525447	−0.262723	+	0.964871i	$0.584621\pi$
$500$	0	0
$501$	9216.14	0.821850
$502$	10236.6i	0.910123i
$503$	4444.45i	0.393972i	0.980406	+	0.196986i	$0.0631154\pi$
$503$	4444.45i	0.393972i	−0.980406	+	0.196986i	$0.936885\pi$
$504$	−3288.59	−0.290646
$505$	0	0
$506$	816.665	0.0717494
$507$	− 567.654i	− 0.0497246i
$508$	− 7054.52i	− 0.616129i
$509$	19485.0	1.69677	0.848385	−	0.529379i	$-0.177575\pi$
$509$	19485.0	1.69677	0.848385	+	0.529379i	$0.177575\pi$
$510$	0	0
$511$	−12733.0	−1.10230
$512$	− 512.000i	− 0.0441942i
$513$	16125.3i	1.38782i
$514$	1495.39	0.128325
$515$	0	0
$516$	5628.74	0.480216
$517$	2312.90i	0.196753i
$518$	15665.1i	1.32873i
$519$	11968.8	1.01228
$520$	0	0
$521$	12935.7	1.08776	0.543879	−	0.839164i	$-0.316955\pi$
$521$	12935.7	1.08776	0.543879	+	0.839164i	$0.316955\pi$
$522$	− 7101.17i	− 0.595421i
$523$	− 10562.9i	− 0.883142i	−0.897226	−	0.441571i	$-0.854421\pi$
$523$	− 10562.9i	− 0.883142i	0.897226	−	0.441571i	$-0.145579\pi$
$524$	−1661.99	−0.138558
$525$	0	0
$526$	4270.19	0.353972
$527$	21567.7i	1.78274i
$528$	296.244i	0.0244173i
$529$	6679.64	0.548997
$530$	0	0
$531$	3484.10	0.284740
$532$	− 11756.8i	− 0.958127i
$533$	5410.63i	0.439701i
$534$	755.524	0.0612261
$535$	0	0
$536$	2484.03	0.200175
$537$	− 1615.81i	− 0.129846i
$538$	− 1561.11i	− 0.125100i
$539$	−1879.69	−0.150212
$540$	0	0
$541$	−13076.9	−1.03922	−0.519610	−	0.854403i	$-0.673923\pi$
$541$	−13076.9	−1.03922	−0.519610	+	0.854403i	$0.673923\pi$
$542$	10825.9i	0.857957i
$543$	15908.4i	1.25726i
$544$	−3360.79	−0.264876
$545$	0	0
$546$	2284.01	0.179023
$547$	− 12729.9i	− 0.995049i	−0.867450	−	0.497524i	$-0.834243\pi$
$547$	− 12729.9i	− 0.995049i	0.867450	−	0.497524i	$-0.165757\pi$
$548$	− 2298.11i	− 0.179143i
$549$	9394.33	0.730310
$550$	0	0
$551$	25387.0	1.96283
$552$	− 1990.53i	− 0.153483i
$553$	− 1369.13i	− 0.105283i
$554$	−3476.53	−0.266613
$555$	0	0
$556$	8547.41	0.651962
$557$	14621.7i	1.11228i	0.831087	+	0.556142i	$0.187719\pi$
$557$	14621.7i	1.11228i	−0.831087	+	0.556142i	$0.812281\pi$
$558$	6455.58i	0.489761i
$559$	5446.25	0.412078
$560$	0	0
$561$	1944.55	0.146344
$562$	− 7846.16i	− 0.588915i
$563$	− 11104.7i	− 0.831271i	−0.909531	−	0.415636i	$-0.863559\pi$
$563$	− 11104.7i	− 0.831271i	0.909531	−	0.415636i	$-0.136441\pi$
$564$	5637.43	0.420884
$565$	0	0
$566$	−713.549	−0.0529907
$567$	− 1505.66i	− 0.111520i
$568$	− 2482.67i	− 0.183399i
$569$	26150.1	1.92666	0.963331	−	0.268316i	$-0.0864670\pi$
$569$	26150.1	1.92666	0.963331	+	0.268316i	$0.0864670\pi$
$570$	0	0
$571$	−6762.85	−0.495651	−0.247825	−	0.968805i	$-0.579716\pi$
$571$	−6762.85	−0.495651	−0.247825	+	0.968805i	$0.579716\pi$
$572$	286.639i	0.0209528i
$573$	5712.07i	0.416449i
$574$	−21770.2	−1.58305
$575$	0	0
$576$	−1005.94	−0.0727676
$577$	− 58.4468i	− 0.00421693i	−0.999998	−	0.00210847i	$-0.999329\pi$
$577$	− 58.4468i	− 0.00421693i	0.999998	−	0.00210847i	$-0.000671146\pi$
$578$	12234.3i	0.880417i
$579$	−4890.35	−0.351012
$580$	0	0
$581$	−29444.7	−2.10253
$582$	1074.60i	0.0765351i
$583$	− 3614.88i	− 0.256798i
$584$	−3894.88	−0.275978
$585$	0	0
$586$	13872.7	0.977943
$587$	11211.0i	0.788290i	0.919048	+	0.394145i	$0.128959\pi$
$587$	11211.0i	0.788290i	−0.919048	+	0.394145i	$0.871041\pi$
$588$	4581.54i	0.321326i
$589$	−23078.9	−1.61452
$590$	0	0
$591$	12150.1	0.845663
$592$	4791.76i	0.332669i
$593$	− 5124.35i	− 0.354860i	−0.984133	−	0.177430i	$-0.943222\pi$
$593$	− 5124.35i	− 0.354860i	0.984133	−	0.177430i	$-0.0567783\pi$
$594$	1581.86	0.109267
$595$	0	0
$596$	−889.180	−0.0611111
$597$	11574.4i	0.793479i
$598$	− 1925.99i	− 0.131705i
$599$	9235.37	0.629961	0.314981	−	0.949098i	$-0.398002\pi$
$599$	9235.37	0.629961	0.314981	+	0.949098i	$0.398002\pi$
$600$	0	0
$601$	−26519.4	−1.79992	−0.899958	−	0.435977i	$-0.856403\pi$
$601$	−26519.4	−1.79992	−0.899958	+	0.435977i	$0.856403\pi$
$602$	21913.5i	1.48360i
$603$	− 4880.43i	− 0.329596i
$604$	−4740.29	−0.319337
$605$	0	0
$606$	2745.31	0.184028
$607$	− 8337.23i	− 0.557492i	−0.960365	−	0.278746i	$-0.910081\pi$
$607$	− 8337.23i	− 0.557492i	0.960365	−	0.278746i	$-0.0899188\pi$
$608$	− 3596.27i	− 0.239882i
$609$	−19844.2	−1.32041
$610$	0	0
$611$	5454.66	0.361165
$612$	6603.02i	0.436130i
$613$	− 7475.93i	− 0.492578i	−0.969196	−	0.246289i	$-0.920789\pi$
$613$	− 7475.93i	− 0.492578i	0.969196	−	0.246289i	$-0.0792112\pi$
$614$	−7838.04	−0.515175
$615$	0	0
$616$	−1153.32	−0.0754361
$617$	21101.5i	1.37684i	0.725310	+	0.688422i	$0.241696\pi$
$617$	21101.5i	1.37684i	−0.725310	+	0.688422i	$0.758304\pi$
$618$	10307.2i	0.670897i
$619$	−19468.7	−1.26416	−0.632080	−	0.774903i	$-0.717799\pi$
$619$	−19468.7	−1.26416	−0.632080	+	0.774903i	$0.717799\pi$
$620$	0	0
$621$	−10628.9	−0.686831
$622$	− 3634.27i	− 0.234278i
$623$	2941.37i	0.189155i
$624$	698.651	0.0448212
$625$	0	0
$626$	−2928.18	−0.186955
$627$	2080.81i	0.132535i
$628$	− 11190.3i	− 0.711056i
$629$	31453.3	1.99384
$630$	0	0
$631$	−6405.28	−0.404105	−0.202052	−	0.979375i	$-0.564761\pi$
$631$	−6405.28	−0.404105	−0.202052	+	0.979375i	$0.564761\pi$
$632$	− 418.801i	− 0.0263592i
$633$	− 14018.1i	− 0.880208i
$634$	9220.32	0.577580
$635$	0	0
$636$	−8810.87	−0.549330
$637$	4433.00i	0.275733i
$638$	− 2490.41i	− 0.154539i
$639$	−4877.77	−0.301974
$640$	0	0
$641$	−3794.20	−0.233794	−0.116897	−	0.993144i	$-0.537295\pi$
$641$	−3794.20	−0.233794	−0.116897	+	0.993144i	$0.537295\pi$
$642$	− 240.414i	− 0.0147794i
$643$	22637.5i	1.38839i	0.719786	+	0.694196i	$0.244240\pi$
$643$	22637.5i	1.38839i	−0.719786	+	0.694196i	$0.755760\pi$
$644$	7749.43	0.474177
$645$	0	0
$646$	−23606.1	−1.43772
$647$	− 26719.3i	− 1.62356i	−0.583964	−	0.811780i	$-0.698499\pi$
$647$	− 26719.3i	− 1.62356i	0.583964	−	0.811780i	$-0.301501\pi$
$648$	− 460.562i	− 0.0279207i
$649$	1221.89	0.0739033
$650$	0	0
$651$	18040.1	1.08609
$652$	6224.89i	0.373904i
$653$	− 26866.7i	− 1.61007i	−0.593226	−	0.805036i	$-0.702146\pi$
$653$	− 26866.7i	− 1.61007i	0.593226	−	0.805036i	$-0.297854\pi$
$654$	−4818.44	−0.288098
$655$	0	0
$656$	−6659.24	−0.396341
$657$	7652.36i	0.454409i
$658$	21947.4i	1.30030i
$659$	14165.9	0.837368	0.418684	−	0.908132i	$-0.362491\pi$
$659$	14165.9	0.837368	0.418684	+	0.908132i	$0.362491\pi$
$660$	0	0
$661$	−14186.8	−0.834799	−0.417399	−	0.908723i	$-0.637058\pi$
$661$	−14186.8	−0.834799	−0.417399	+	0.908723i	$0.637058\pi$
$662$	− 4486.60i	− 0.263409i
$663$	− 4585.97i	− 0.268634i
$664$	−9006.77	−0.526401
$665$	0	0
$666$	9414.48	0.547753
$667$	16733.6i	0.971406i
$668$	10975.2i	0.635693i
$669$	3705.66	0.214154
$670$	0	0
$671$	3294.63	0.189549
$672$	2811.09i	0.161369i
$673$	− 23199.0i	− 1.32876i	−0.747395	−	0.664380i	$-0.768696\pi$
$673$	− 23199.0i	− 1.32876i	0.747395	−	0.664380i	$-0.231304\pi$
$674$	−12008.8	−0.686294
$675$	0	0
$676$	676.000	0.0384615
$677$	28477.3i	1.61665i	0.588738	+	0.808324i	$0.299625\pi$
$677$	28477.3i	1.61665i	−0.588738	+	0.808324i	$0.700375\pi$
$678$	6058.34i	0.343170i
$679$	−4183.56	−0.236451
$680$	0	0
$681$	−8378.42	−0.471456
$682$	2264.00i	0.127116i
$683$	29441.0i	1.64938i	0.565583	+	0.824691i	$0.308651\pi$
$683$	29441.0i	1.64938i	−0.565583	+	0.824691i	$0.691349\pi$
$684$	−7065.68	−0.394975
$685$	0	0
$686$	104.614	0.00582240
$687$	8206.91i	0.455769i
$688$	6703.07i	0.371442i
$689$	−8525.21	−0.471386
$690$	0	0
$691$	10587.3	0.582863	0.291431	−	0.956592i	$-0.405868\pi$
$691$	10587.3	0.582863	0.291431	+	0.956592i	$0.405868\pi$
$692$	14253.3i	0.782989i
$693$	2265.96i	0.124209i
$694$	3515.89	0.192308
$695$	0	0
$696$	−6070.09	−0.330584
$697$	43711.5i	2.37545i
$698$	− 11549.5i	− 0.626295i
$699$	21307.8	1.15298
$700$	0	0
$701$	−2837.65	−0.152891	−0.0764456	−	0.997074i	$-0.524357\pi$
$701$	−2837.65	−0.152891	−0.0764456	+	0.997074i	$0.524357\pi$
$702$	− 3730.60i	− 0.200573i
$703$	33657.1i	1.80569i
$704$	−352.787	−0.0188866
$705$	0	0
$706$	−572.530	−0.0305205
$707$	10687.9i	0.568544i
$708$	− 2978.21i	− 0.158090i
$709$	−20574.7	−1.08984	−0.544921	−	0.838487i	$-0.683440\pi$
$709$	−20574.7	−1.08984	−0.544921	+	0.838487i	$0.683440\pi$
$710$	0	0
$711$	−822.828	−0.0434015
$712$	899.729i	0.0473578i
$713$	− 15212.3i	− 0.799026i
$714$	18452.1	0.967161
$715$	0	0
$716$	1924.22	0.100435
$717$	− 9872.03i	− 0.514195i
$718$	− 12009.7i	− 0.624233i
$719$	23117.7	1.19909	0.599544	−	0.800342i	$-0.295349\pi$
$719$	23117.7	1.19909	0.599544	+	0.800342i	$0.295349\pi$
$720$	0	0
$721$	−40127.3	−2.07270
$722$	− 11542.1i	− 0.594948i
$723$	− 15003.6i	− 0.771768i
$724$	−18944.7	−0.972479
$725$	0	0
$726$	−8737.27	−0.446653
$727$	− 34214.3i	− 1.74545i	−0.488216	−	0.872723i	$-0.662352\pi$
$727$	− 34214.3i	− 1.74545i	0.488216	−	0.872723i	$-0.337648\pi$
$728$	2719.95i	0.138473i
$729$	−13917.6	−0.707085
$730$	0	0
$731$	43999.2	2.22622
$732$	− 8030.29i	− 0.405475i
$733$	9083.99i	0.457742i	0.973457	+	0.228871i	$0.0735034\pi$
$733$	9083.99i	0.457742i	−0.973457	+	0.228871i	$0.926497\pi$
$734$	−7587.40	−0.381548
$735$	0	0
$736$	2370.45	0.118718
$737$	− 1711.59i	− 0.0855456i
$738$	13083.6i	0.652592i
$739$	13479.4	0.670971	0.335485	−	0.942045i	$-0.391100\pi$
$739$	13479.4	0.670971	0.335485	+	0.942045i	$0.391100\pi$
$740$	0	0
$741$	4907.30	0.243285
$742$	− 34302.0i	− 1.69713i
$743$	− 27595.0i	− 1.36253i	−0.732036	−	0.681266i	$-0.761430\pi$
$743$	− 27595.0i	− 1.36253i	0.732036	−	0.681266i	$-0.238570\pi$
$744$	5518.24	0.271920
$745$	0	0
$746$	−17846.1	−0.875859
$747$	17695.8i	0.866742i
$748$	2315.70i	0.113196i
$749$	935.966	0.0456602
$750$	0	0
$751$	1384.29	0.0672618	0.0336309	−	0.999434i	$-0.489293\pi$
$751$	1384.29	0.0672618	0.0336309	+	0.999434i	$0.489293\pi$
$752$	6713.42i	0.325550i
$753$	− 17191.9i	− 0.832013i
$754$	−5873.29	−0.283677
$755$	0	0
$756$	15010.5	0.722123
$757$	− 16314.4i	− 0.783301i	−0.920114	−	0.391650i	$-0.871904\pi$
$757$	− 16314.4i	− 0.783301i	0.920114	−	0.391650i	$-0.128096\pi$
$758$	− 21955.1i	− 1.05204i
$759$	−1371.55	−0.0655916
$760$	0	0
$761$	1939.46	0.0923856	0.0461928	−	0.998933i	$-0.485291\pi$
$761$	1939.46	0.0923856	0.0461928	+	0.998933i	$0.485291\pi$
$762$	11847.7i	0.563251i
$763$	− 18758.9i	− 0.890063i
$764$	−6802.31	−0.322119
$765$	0	0
$766$	−21140.8	−0.997192
$767$	− 2881.65i	− 0.135659i
$768$	859.878i	0.0404013i
$769$	11834.6	0.554961	0.277480	−	0.960731i	$-0.410501\pi$
$769$	11834.6	0.554961	0.277480	+	0.960731i	$0.410501\pi$
$770$	0	0
$771$	−2511.43	−0.117311
$772$	− 5823.76i	− 0.271505i
$773$	33715.5i	1.56878i	0.620271	+	0.784388i	$0.287023\pi$
$773$	33715.5i	1.56878i	−0.620271	+	0.784388i	$0.712977\pi$
$774$	13169.7	0.611595
$775$	0	0
$776$	−1279.70	−0.0591992
$777$	− 26308.7i	− 1.21470i
$778$	1034.43i	0.0476684i
$779$	−46774.3	−2.15130
$780$	0	0
$781$	−1710.65	−0.0783764
$782$	− 15559.7i	− 0.711529i
$783$	32412.6i	1.47935i
$784$	−5456.00	−0.248542
$785$	0	0
$786$	2791.22	0.126666
$787$	− 25730.6i	− 1.16543i	−0.812676	−	0.582716i	$-0.801990\pi$
$787$	− 25730.6i	− 1.16543i	0.812676	−	0.582716i	$-0.198010\pi$
$788$	14469.1i	0.654112i
$789$	−7171.57	−0.323593
$790$	0	0
$791$	−23586.0	−1.06021
$792$	693.129i	0.0310976i
$793$	− 7769.94i	− 0.347943i
$794$	13958.7	0.623897
$795$	0	0
$796$	−13783.5	−0.613749
$797$	19638.7i	0.872823i	0.899747	+	0.436412i	$0.143751\pi$
$797$	19638.7i	0.872823i	−0.899747	+	0.436412i	$0.856249\pi$
$798$	19745.0i	0.875897i
$799$	44067.2	1.95117
$800$	0	0
$801$	1767.72	0.0779766
$802$	− 7503.32i	− 0.330363i
$803$	2683.71i	0.117940i
$804$	−4171.80	−0.182995
$805$	0	0
$806$	5339.33	0.233337
$807$	2621.80i	0.114364i
$808$	3269.30i	0.142344i
$809$	−6908.35	−0.300228	−0.150114	−	0.988669i	$-0.547964\pi$
$809$	−6908.35	−0.300228	−0.150114	+	0.988669i	$0.547964\pi$
$810$	0	0
$811$	11961.9	0.517929	0.258964	−	0.965887i	$-0.416619\pi$
$811$	11961.9	0.517929	0.258964	+	0.965887i	$0.416619\pi$
$812$	− 23631.8i	− 1.02132i
$813$	− 18181.6i	− 0.784324i
$814$	3301.70	0.142168
$815$	0	0
$816$	5644.27	0.242143
$817$	47082.2i	2.01615i
$818$	− 10119.8i	− 0.432558i
$819$	5343.96	0.228001
$820$	0	0
$821$	44644.7	1.89782	0.948909	−	0.315549i	$-0.102189\pi$
$821$	44644.7	1.89782	0.948909	+	0.315549i	$0.102189\pi$
$822$	3859.56i	0.163768i
$823$	− 46052.2i	− 1.95052i	−0.221063	−	0.975260i	$-0.570953\pi$
$823$	− 46052.2i	− 1.95052i	0.221063	−	0.975260i	$-0.429047\pi$
$824$	−12274.4	−0.518932
$825$	0	0
$826$	11594.6	0.488412
$827$	− 24966.6i	− 1.04979i	−0.851168	−	0.524894i	$-0.824105\pi$
$827$	− 24966.6i	− 1.04979i	0.851168	−	0.524894i	$-0.175895\pi$
$828$	− 4657.29i	− 0.195473i
$829$	−1562.61	−0.0654665	−0.0327333	−	0.999464i	$-0.510421\pi$
$829$	−1562.61	−0.0654665	−0.0327333	+	0.999464i	$0.510421\pi$
$830$	0	0
$831$	5838.65	0.243731
$832$	832.000i	0.0346688i
$833$	35813.4i	1.48963i
$834$	−14355.0	−0.596009
$835$	0	0
$836$	−2477.96	−0.102515
$837$	− 29465.9i	− 1.21683i
$838$	3517.44i	0.144997i
$839$	13249.3	0.545191	0.272596	−	0.962129i	$-0.412118\pi$
$839$	13249.3	0.545191	0.272596	+	0.962129i	$0.412118\pi$
$840$	0	0
$841$	26639.9	1.09229
$842$	− 6027.14i	− 0.246685i
$843$	13177.2i	0.538372i
$844$	16693.7	0.680832
$845$	0	0
$846$	13190.0	0.536031
$847$	− 34015.5i	− 1.37991i
$848$	− 10492.6i	− 0.424901i
$849$	1198.37	0.0484428
$850$	0	0
$851$	−22184.8	−0.893639
$852$	4169.53i	0.167659i
$853$	− 39445.1i	− 1.58332i	−0.610959	−	0.791662i	$-0.709216\pi$
$853$	− 39445.1i	− 1.58332i	0.610959	−	0.791662i	$-0.290784\pi$
$854$	31263.1	1.25269
$855$	0	0
$856$	286.300	0.0114317
$857$	14494.7i	0.577749i	0.957367	+	0.288874i	$0.0932810\pi$
$857$	14494.7i	0.577749i	−0.957367	+	0.288874i	$0.906719\pi$
$858$	− 481.396i	− 0.0191545i
$859$	−22349.5	−0.887724	−0.443862	−	0.896095i	$-0.646392\pi$
$859$	−22349.5	−0.887724	−0.443862	+	0.896095i	$0.646392\pi$
$860$	0	0
$861$	36562.0	1.44719
$862$	− 18437.7i	− 0.728528i
$863$	12406.4i	0.489360i	0.969604	+	0.244680i	$0.0786829\pi$
$863$	12406.4i	0.489360i	−0.969604	+	0.244680i	$0.921317\pi$
$864$	4591.51	0.180794
$865$	0	0
$866$	29817.8	1.17004
$867$	− 20546.9i	− 0.804856i
$868$	21483.3i	0.840083i
$869$	−288.569	−0.0112647
$870$	0	0
$871$	−4036.55	−0.157030
$872$	− 5738.12i	− 0.222841i
$873$	2514.26i	0.0974740i
$874$	16650.0	0.644387
$875$	0	0
$876$	6541.25	0.252293
$877$	− 28964.4i	− 1.11523i	−0.830099	−	0.557616i	$-0.811716\pi$
$877$	− 28964.4i	− 1.11523i	0.830099	−	0.557616i	$-0.188284\pi$
$878$	− 29173.0i	− 1.12134i
$879$	−23298.4	−0.894013
$880$	0	0
$881$	33358.6	1.27569	0.637844	−	0.770166i	$-0.279827\pi$
$881$	33358.6	1.27569	0.637844	+	0.770166i	$0.279827\pi$
$882$	10719.5i	0.409235i
$883$	17482.9i	0.666303i	0.942873	+	0.333151i	$0.108112\pi$
$883$	17482.9i	0.666303i	−0.942873	+	0.333151i	$0.891888\pi$
$884$	5461.28	0.207786
$885$	0	0
$886$	−2922.90	−0.110832
$887$	− 39666.1i	− 1.50153i	−0.660568	−	0.750766i	$-0.729685\pi$
$887$	− 39666.1i	− 1.50153i	0.660568	−	0.750766i	$-0.270315\pi$
$888$	− 8047.51i	− 0.304118i
$889$	−46124.9	−1.74014
$890$	0	0
$891$	−317.344	−0.0119320
$892$	4412.94i	0.165646i
$893$	47154.9i	1.76705i
$894$	1493.33	0.0558664
$895$	0	0
$896$	−3347.63	−0.124818
$897$	3234.61i	0.120402i
$898$	− 1160.58i	− 0.0431281i
$899$	−46389.7	−1.72100
$900$	0	0
$901$	−68873.6	−2.54663
$902$	4588.46i	0.169378i
$903$	− 36802.6i	− 1.35627i
$904$	−7214.68	−0.265439
$905$	0	0
$906$	7961.07	0.291930
$907$	− 16181.4i	− 0.592388i	−0.955128	−	0.296194i	$-0.904283\pi$
$907$	− 16181.4i	− 0.592388i	0.955128	−	0.296194i	$-0.0957174\pi$
$908$	− 9977.57i	− 0.364667i
$909$	6423.28	0.234375
$910$	0	0
$911$	22957.0	0.834906	0.417453	−	0.908699i	$-0.362923\pi$
$911$	22957.0	0.834906	0.417453	+	0.908699i	$0.362923\pi$
$912$	6039.76i	0.219294i
$913$	6206.00i	0.224960i
$914$	−43.5276	−0.00157524
$915$	0	0
$916$	−9773.33	−0.352533
$917$	10866.6i	0.391328i
$918$	− 30138.9i	− 1.08358i
$919$	7028.37	0.252279	0.126140	−	0.992012i	$-0.459741\pi$
$919$	7028.37	0.252279	0.126140	+	0.992012i	$0.459741\pi$
$920$	0	0
$921$	13163.6	0.470961
$922$	29097.1i	1.03933i
$923$	4034.35i	0.143870i
$924$	1936.94	0.0689619
$925$	0	0
$926$	−11977.9	−0.425075
$927$	24115.9i	0.854444i
$928$	− 7228.66i	− 0.255703i
$929$	−6014.14	−0.212398	−0.106199	−	0.994345i	$-0.533868\pi$
$929$	−6014.14	−0.212398	−0.106199	+	0.994345i	$0.533868\pi$
$930$	0	0
$931$	−38322.8	−1.34906
$932$	25374.8i	0.891823i
$933$	6103.56i	0.214171i
$934$	−12589.7	−0.441057
$935$	0	0
$936$	1634.65	0.0570836
$937$	− 35125.0i	− 1.22464i	−0.790612	−	0.612318i	$-0.790237\pi$
$937$	− 35125.0i	− 1.22464i	0.790612	−	0.612318i	$-0.209763\pi$
$938$	− 16241.4i	− 0.565354i
$939$	4917.74	0.170910
$940$	0	0
$941$	−22190.2	−0.768736	−0.384368	−	0.923180i	$-0.625581\pi$
$941$	−22190.2	−0.768736	−0.384368	+	0.923180i	$0.625581\pi$
$942$	18793.6i	0.650030i
$943$	− 30830.9i	− 1.06468i
$944$	3546.65	0.122281
$945$	0	0
$946$	4618.66	0.158737
$947$	− 43661.7i	− 1.49822i	−0.662445	−	0.749111i	$-0.730481\pi$
$947$	− 43661.7i	− 1.49822i	0.662445	−	0.749111i	$-0.269519\pi$
$948$	703.354i	0.0240969i
$949$	6329.17	0.216495
$950$	0	0
$951$	−15485.1	−0.528010
$952$	21974.0i	0.748090i
$953$	11102.5i	0.377381i	0.982037	+	0.188691i	$0.0604243\pi$
$953$	11102.5i	0.377381i	−0.982037	+	0.188691i	$0.939576\pi$
$954$	−20615.0	−0.699618
$955$	0	0
$956$	11756.3	0.397725
$957$	4182.51i	0.141276i
$958$	10706.7i	0.361084i
$959$	−15025.8	−0.505953
$960$	0	0
$961$	12381.3	0.415605
$962$	− 7786.60i	− 0.260967i
$963$	− 562.502i	− 0.0188228i
$964$	17867.2	0.596955
$965$	0	0
$966$	−13014.8	−0.433482
$967$	36593.6i	1.21693i	0.793581	+	0.608465i	$0.208214\pi$
$967$	36593.6i	1.21693i	−0.793581	+	0.608465i	$0.791786\pi$
$968$	− 10404.9i	− 0.345482i
$969$	39645.2	1.31433
$970$	0	0
$971$	34303.3	1.13372	0.566861	−	0.823813i	$-0.308158\pi$
$971$	34303.3	1.13372	0.566861	+	0.823813i	$0.308158\pi$
$972$	− 14722.9i	− 0.485840i
$973$	− 55886.0i	− 1.84134i
$974$	−30679.8	−1.00929
$975$	0	0
$976$	9563.00	0.313631
$977$	14168.1i	0.463948i	0.972722	+	0.231974i	$0.0745184\pi$
$977$	14168.1i	0.463948i	−0.972722	+	0.231974i	$0.925482\pi$
$978$	− 10454.4i	− 0.341814i
$979$	619.946	0.0202386
$980$	0	0
$981$	−11273.8	−0.366917
$982$	− 32577.0i	− 1.05863i
$983$	49509.4i	1.60641i	0.595700	+	0.803207i	$0.296875\pi$
$983$	49509.4i	1.60641i	−0.595700	+	0.803207i	$0.703125\pi$
$984$	11183.9	0.362326
$985$	0	0
$986$	−47449.2	−1.53255
$987$	− 36859.5i	− 1.18870i
$988$	5843.94i	0.188179i
$989$	−31033.8	−0.997794
$990$	0	0
$991$	45392.1	1.45502	0.727511	−	0.686096i	$-0.240677\pi$
$991$	45392.1	1.45502	0.727511	+	0.686096i	$0.240677\pi$
$992$	6571.48i	0.210327i
$993$	7535.01i	0.240802i
$994$	−16232.6	−0.517974
$995$	0	0
$996$	15126.4	0.481224
$997$	40408.9i	1.28361i	0.766866	+	0.641807i	$0.221815\pi$
$997$	40408.9i	1.28361i	−0.766866	+	0.641807i	$0.778185\pi$
$998$	11714.1i	0.371547i
$999$	−42971.5	−1.36092

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.4.b.l.599.2		4
5.2	odd	4		650.4.a.r.1.2		2
5.3	odd	4		130.4.a.d.1.1	✓	2
5.4	even	2	inner	650.4.b.l.599.3		4
15.8	even	4		1170.4.a.ba.1.2		2
20.3	even	4		1040.4.a.i.1.2		2
65.38	odd	4		1690.4.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.4.a.d.1.1	✓	2	5.3	odd	4
650.4.a.r.1.2		2	5.2	odd	4
650.4.b.l.599.2		4	1.1	even	1	trivial
650.4.b.l.599.3		4	5.4	even	2	inner
1040.4.a.i.1.2		2	20.3	even	4
1170.4.a.ba.1.2		2	15.8	even	4
1690.4.a.s.1.1		2	65.38	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	650.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +3.35890i q^{3} -4.00000 q^{4} +6.71780 q^{6} +26.1534i q^{7} +8.00000i q^{8} +15.7178 q^{9} +5.51229 q^{11} -13.4356i q^{12} -13.0000i q^{13} +52.3068 q^{14} +16.0000 q^{16} -105.025i q^{17} -31.4356i q^{18} +112.383 q^{19} -87.8466 q^{21} -11.0246i q^{22} +74.0767i q^{23} -26.8712 q^{24} -26.0000 q^{26} +143.485i q^{27} -104.614i q^{28} +225.896 q^{29} -205.359 q^{31} -32.0000i q^{32} +18.5152i q^{33} -210.049 q^{34} -62.8712 q^{36} +299.485i q^{37} -224.767i q^{38} +43.6657 q^{39} -416.203 q^{41} +175.693i q^{42} +418.942i q^{43} -22.0492 q^{44} +148.153 q^{46} +419.589i q^{47} +53.7424i q^{48} -341.000 q^{49} +352.767 q^{51} +52.0000i q^{52} -655.786i q^{53} +286.970 q^{54} -209.227 q^{56} +377.485i q^{57} -451.792i q^{58} +221.666 q^{59} +597.687 q^{61} +410.718i q^{62} +411.074i q^{63} -64.0000 q^{64} +37.0305 q^{66} -310.503i q^{67} +420.098i q^{68} -248.816 q^{69} -310.334 q^{71} +125.742i q^{72} +486.859i q^{73} +598.970 q^{74} -449.534 q^{76} +144.165i q^{77} -87.3314i q^{78} -52.3501 q^{79} -57.5703 q^{81} +832.405i q^{82} +1125.85i q^{83} +351.386 q^{84} +837.884 q^{86} +758.761i q^{87} +44.0983i q^{88} +112.466 q^{89} +339.994 q^{91} -296.307i q^{92} -689.780i q^{93} +839.178 q^{94} +107.485 q^{96} +159.963i q^{97} +682.000i q^{98} +86.6411 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} - 8 q^{6} + 28 q^{9} - 100 q^{11} + 64 q^{16} + 188 q^{19} - 456 q^{21} + 32 q^{24} - 104 q^{26} + 520 q^{29} - 804 q^{31} - 352 q^{34} - 112 q^{36} - 52 q^{39} - 1072 q^{41} + 400 q^{44}+ \cdots + 364 q^{99}+O(q^{100}) \) 4 * q - 16 * q^4 - 8 * q^6 + 28 * q^9 - 100 * q^11 + 64 * q^16 + 188 * q^19 - 456 * q^21 + 32 * q^24 - 104 * q^26 + 520 * q^29 - 804 * q^31 - 352 * q^34 - 112 * q^36 - 52 * q^39 - 1072 * q^41 + 400 * q^44 + 488 * q^46 - 1364 * q^49 + 888 * q^51 + 32 * q^54 + 660 * q^59 + 1240 * q^61 - 256 * q^64 + 1264 * q^66 + 16 * q^69 - 1468 * q^71 + 1280 * q^74 - 752 * q^76 + 1848 * q^79 - 1660 * q^81 + 1824 * q^84 + 248 * q^86 + 1496 * q^89 + 3008 * q^94 - 128 * q^96 + 364 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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