LMFDB - Embedded newform 650.4.b.m.599.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,4,Mod(599,650)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$38.3512415037$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{51})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 25x^{2} + 169 $ x^4 - 25*x^2 + 169
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			599.1
Root			$-3.57071 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.4.b.m.599.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} -6.14143i q^{3} -4.00000 q^{4} -12.2829 q^{6} -22.2829i q^{7} +8.00000i q^{8} -10.7171 q^{9} +43.8586 q^{11} +24.5657i q^{12} +13.0000i q^{13} -44.5657 q^{14} +16.0000 q^{16} -99.9800i q^{17} +21.4343i q^{18} -93.2729 q^{19} -136.849 q^{21} -87.7171i q^{22} -154.990i q^{23} +49.1314 q^{24} +26.0000 q^{26} -100.000i q^{27} +89.1314i q^{28} -191.677 q^{29} -213.839 q^{31} -32.0000i q^{32} -269.354i q^{33} -199.960 q^{34} +42.8686 q^{36} +401.697i q^{37} +186.546i q^{38} +79.8386 q^{39} +458.809 q^{41} +273.697i q^{42} -251.799i q^{43} -175.434 q^{44} -309.980 q^{46} +261.071i q^{47} -98.2629i q^{48} -153.526 q^{49} -614.020 q^{51} -52.0000i q^{52} +151.374i q^{53} -200.000 q^{54} +178.263 q^{56} +572.829i q^{57} +383.354i q^{58} -263.899 q^{59} +644.203 q^{61} +427.677i q^{62} +238.809i q^{63} -64.0000 q^{64} -538.709 q^{66} -387.577i q^{67} +399.920i q^{68} -951.860 q^{69} -544.990 q^{71} -85.7371i q^{72} +301.737i q^{73} +803.394 q^{74} +373.091 q^{76} -977.294i q^{77} -159.677i q^{78} +562.991 q^{79} -903.506 q^{81} -917.617i q^{82} -938.689i q^{83} +547.394 q^{84} -503.597 q^{86} +1177.17i q^{87} +350.869i q^{88} -261.514 q^{89} +289.677 q^{91} +619.960i q^{92} +1313.27i q^{93} +522.143 q^{94} -196.526 q^{96} +1371.50i q^{97} +307.051i q^{98} -470.039 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} + 8 q^{6} - 100 q^{9} + 204 q^{11} - 64 q^{14} + 64 q^{16} - 116 q^{19} - 376 q^{21} - 32 q^{24} + 104 q^{26} - 24 q^{29} - 484 q^{31} + 400 q^{36} - 52 q^{39} + 864 q^{41} - 816 q^{44} - 840 q^{46}+ \cdots - 5508 q^{99}+O(q^{100}) $ 4 * q - 16 * q^4 + 8 * q^6 - 100 * q^9 + 204 * q^11 - 64 * q^14 + 64 * q^16 - 116 * q^19 - 376 * q^21 - 32 * q^24 + 104 * q^26 - 24 * q^29 - 484 * q^31 + 400 * q^36 - 52 * q^39 + 864 * q^41 - 816 * q^44 - 840 * q^46 + 300 * q^49 - 2856 * q^51 - 800 * q^54 + 256 * q^56 - 1884 * q^59 + 920 * q^61 - 256 * q^64 + 816 * q^66 - 1008 * q^69 - 1980 * q^71 + 2528 * q^74 + 464 * q^76 - 776 * q^79 - 2300 * q^81 + 1504 * q^84 + 328 * q^86 - 2760 * q^89 + 416 * q^91 - 768 * q^94 + 128 * q^96 - 5508 * 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
$3$	− 6.14143i	− 1.18192i	−0.806701	−	0.590959i	$-0.798749\pi$
$3$	− 6.14143i	− 1.18192i	0.806701	−	0.590959i	$-0.201251\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−12.2829	−0.835743
$7$	− 22.2829i	− 1.20316i	−0.798812	−	0.601581i	$-0.794538\pi$
$7$	− 22.2829i	− 1.20316i	0.798812	−	0.601581i	$-0.205462\pi$
$8$	8.00000i	0.353553i
$9$	−10.7171	−0.396931
$10$	0	0
$11$	43.8586	1.20217	0.601084	−	0.799186i	$-0.294736\pi$
$11$	43.8586	1.20217	0.601084	+	0.799186i	$0.294736\pi$
$12$	24.5657i	0.590959i
$13$	13.0000i	0.277350i
$13$	13.0000i	0.277350i
$14$	−44.5657	−0.850763
$15$	0	0
$16$	16.0000	0.250000
$17$	− 99.9800i	− 1.42639i	−0.700963	−	0.713197i	$-0.747246\pi$
$17$	− 99.9800i	− 1.42639i	0.700963	−	0.713197i	$-0.252754\pi$
$18$	21.4343i	0.280673i
$19$	−93.2729	−1.12622	−0.563112	−	0.826381i	$-0.690396\pi$
$19$	−93.2729	−1.12622	−0.563112	+	0.826381i	$0.690396\pi$
$20$	0	0
$21$	−136.849	−1.42204
$22$	− 87.7171i	− 0.850062i
$23$	− 154.990i	− 1.40512i	−0.711627	−	0.702558i	$-0.752041\pi$
$23$	− 154.990i	− 1.40512i	0.711627	−	0.702558i	$-0.247959\pi$
$24$	49.1314	0.417871
$25$	0	0
$26$	26.0000	0.196116
$27$	− 100.000i	− 0.712778i
$28$	89.1314i	0.601581i
$29$	−191.677	−1.22736	−0.613682	−	0.789553i	$-0.710312\pi$
$29$	−191.677	−1.22736	−0.613682	+	0.789553i	$0.710312\pi$
$30$	0	0
$31$	−213.839	−1.23892	−0.619460	−	0.785028i	$-0.712649\pi$
$31$	−213.839	−1.23892	−0.619460	+	0.785028i	$0.712649\pi$
$32$	− 32.0000i	− 0.176777i
$33$	− 269.354i	− 1.42087i
$34$	−199.960	−1.00861
$35$	0	0
$36$	42.8686	0.198466
$37$	401.697i	1.78483i	0.451218	+	0.892414i	$0.350990\pi$
$37$	401.697i	1.78483i	−0.451218	+	0.892414i	$0.649010\pi$
$38$	186.546i	0.796361i
$39$	79.8386	0.327805
$40$	0	0
$41$	458.809	1.74766	0.873828	−	0.486236i	$-0.161630\pi$
$41$	458.809	1.74766	0.873828	+	0.486236i	$0.161630\pi$
$42$	273.697i	1.00553i
$43$	− 251.799i	− 0.892998i	−0.894784	−	0.446499i	$-0.852671\pi$
$43$	− 251.799i	− 0.892998i	0.894784	−	0.446499i	$-0.147329\pi$
$44$	−175.434	−0.601084
$45$	0	0
$46$	−309.980	−0.993566
$47$	261.071i	0.810238i	0.914264	+	0.405119i	$0.132770\pi$
$47$	261.071i	0.810238i	−0.914264	+	0.405119i	$0.867230\pi$
$48$	− 98.2629i	− 0.295480i
$49$	−153.526	−0.447597
$50$	0	0
$51$	−614.020	−1.68588
$52$	− 52.0000i	− 0.138675i
$53$	151.374i	0.392318i	0.980572	+	0.196159i	$0.0628469\pi$
$53$	151.374i	0.392318i	−0.980572	+	0.196159i	$0.937153\pi$
$54$	−200.000	−0.504010
$55$	0	0
$56$	178.263	0.425382
$57$	572.829i	1.33111i
$58$	383.354i	0.867877i
$59$	−263.899	−0.582316	−0.291158	−	0.956675i	$-0.594041\pi$
$59$	−263.899	−0.582316	−0.291158	+	0.956675i	$0.594041\pi$
$60$	0	0
$61$	644.203	1.35216	0.676080	−	0.736829i	$-0.263677\pi$
$61$	644.203	1.35216	0.676080	+	0.736829i	$0.263677\pi$
$62$	427.677i	0.876049i
$63$	238.809i	0.477572i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−538.709	−1.00470
$67$	− 387.577i	− 0.706718i	−0.935488	−	0.353359i	$-0.885039\pi$
$67$	− 387.577i	− 0.706718i	0.935488	−	0.353359i	$-0.114961\pi$
$68$	399.920i	0.713197i
$69$	−951.860	−1.66073
$70$	0	0
$71$	−544.990	−0.910964	−0.455482	−	0.890245i	$-0.650533\pi$
$71$	−544.990	−0.910964	−0.455482	+	0.890245i	$0.650533\pi$
$72$	− 85.7371i	− 0.140336i
$73$	301.737i	0.483776i	0.970304	+	0.241888i	$0.0777667\pi$
$73$	301.737i	0.483776i	−0.970304	+	0.241888i	$0.922233\pi$
$74$	803.394	1.26206
$75$	0	0
$76$	373.091	0.563112
$77$	− 977.294i	− 1.44640i
$78$	− 159.677i	− 0.231793i
$79$	562.991	0.801791	0.400895	−	0.916124i	$-0.368699\pi$
$79$	562.991	0.801791	0.400895	+	0.916124i	$0.368699\pi$
$80$	0	0
$81$	−903.506	−1.23938
$82$	− 917.617i	− 1.23578i
$83$	− 938.689i	− 1.24138i	−0.784056	−	0.620689i	$-0.786853\pi$
$83$	− 938.689i	− 1.24138i	0.784056	−	0.620689i	$-0.213147\pi$
$84$	547.394	0.711019
$85$	0	0
$86$	−503.597	−0.631445
$87$	1177.17i	1.45064i
$88$	350.869i	0.425031i
$89$	−261.514	−0.311466	−0.155733	−	0.987799i	$-0.549774\pi$
$89$	−261.514	−0.311466	−0.155733	+	0.987799i	$0.549774\pi$
$90$	0	0
$91$	289.677	0.333697
$92$	619.960i	0.702558i
$93$	1313.27i	1.46430i
$94$	522.143	0.572925
$95$	0	0
$96$	−196.526	−0.208936
$97$	1371.50i	1.43561i	0.696243	+	0.717806i	$0.254854\pi$
$97$	1371.50i	1.43561i	−0.696243	+	0.717806i	$0.745146\pi$
$98$	307.051i	0.316499i
$99$	−470.039	−0.477178
$100$	0	0
$101$	259.697	0.255850	0.127925	−	0.991784i	$-0.459168\pi$
$101$	259.697	0.255850	0.127925	+	0.991784i	$0.459168\pi$
$102$	1228.04i	1.19210i
$103$	749.053i	0.716567i	0.933613	+	0.358283i	$0.116638\pi$
$103$	749.053i	0.716567i	−0.933613	+	0.358283i	$0.883362\pi$
$104$	−104.000	−0.0980581
$105$	0	0
$106$	302.749	0.277411
$107$	− 797.133i	− 0.720203i	−0.932913	−	0.360102i	$-0.882742\pi$
$107$	− 797.133i	− 0.720203i	0.932913	−	0.360102i	$-0.117258\pi$
$108$	400.000i	0.356389i
$109$	−648.949	−0.570257	−0.285128	−	0.958489i	$-0.592036\pi$
$109$	−648.949	−0.570257	−0.285128	+	0.958489i	$0.592036\pi$
$110$	0	0
$111$	2466.99	2.10952
$112$	− 356.526i	− 0.300790i
$113$	600.060i	0.499548i	0.968304	+	0.249774i	$0.0803563\pi$
$113$	600.060i	0.499548i	−0.968304	+	0.249774i	$0.919644\pi$
$114$	1145.66	0.941234
$115$	0	0
$116$	766.709	0.613682
$117$	− 139.323i	− 0.110089i
$118$	527.797i	0.411760i
$119$	−2227.84	−1.71618
$120$	0	0
$121$	592.574	0.445210
$122$	− 1288.41i	− 0.956121i
$123$	− 2817.74i	− 2.06559i
$124$	855.354	0.619460
$125$	0	0
$126$	477.617	0.337695
$127$	1918.42i	1.34041i	0.742174	+	0.670207i	$0.233795\pi$
$127$	1918.42i	1.34041i	−0.742174	+	0.670207i	$0.766205\pi$
$128$	128.000i	0.0883883i
$129$	−1546.40	−1.05545
$130$	0	0
$131$	1428.12	0.952484	0.476242	−	0.879314i	$-0.341999\pi$
$131$	1428.12	0.952484	0.476242	+	0.879314i	$0.341999\pi$
$132$	1077.42i	0.710433i
$133$	2078.39i	1.35503i
$134$	−775.154	−0.499725
$135$	0	0
$136$	799.840	0.504307
$137$	1129.19i	0.704187i	0.935965	+	0.352093i	$0.114530\pi$
$137$	1129.19i	0.704187i	−0.935965	+	0.352093i	$0.885470\pi$
$138$	1903.72i	1.17431i
$139$	−180.889	−0.110380	−0.0551898	−	0.998476i	$-0.517576\pi$
$139$	−180.889	−0.110380	−0.0551898	+	0.998476i	$0.517576\pi$
$140$	0	0
$141$	1603.35	0.957635
$142$	1089.98i	0.644149i
$143$	570.161i	0.333422i
$144$	−171.474	−0.0992328
$145$	0	0
$146$	603.474	0.342081
$147$	942.867i	0.529023i
$148$	− 1606.79i	− 0.892414i
$149$	1515.13	0.833050	0.416525	−	0.909124i	$-0.363248\pi$
$149$	1515.13	0.833050	0.416525	+	0.909124i	$0.363248\pi$
$150$	0	0
$151$	−258.970	−0.139567	−0.0697837	−	0.997562i	$-0.522231\pi$
$151$	−258.970	−0.139567	−0.0697837	+	0.997562i	$0.522231\pi$
$152$	− 746.183i	− 0.398180i
$153$	1071.50i	0.566181i
$154$	−1954.59	−1.02276
$155$	0	0
$156$	−319.354	−0.163903
$157$	512.949i	0.260750i	0.991465	+	0.130375i	$0.0416181\pi$
$157$	512.949i	0.260750i	−0.991465	+	0.130375i	$0.958382\pi$
$158$	− 1125.98i	− 0.566952i
$159$	929.654	0.463688
$160$	0	0
$161$	−3453.62	−1.69058
$162$	1807.01i	0.876372i
$163$	− 3299.58i	− 1.58554i	−0.609522	−	0.792769i	$-0.708638\pi$
$163$	− 3299.58i	− 1.58554i	0.609522	−	0.792769i	$-0.291362\pi$
$164$	−1835.23	−0.873828
$165$	0	0
$166$	−1877.38	−0.877787
$167$	− 81.4315i	− 0.0377327i	−0.999822	−	0.0188663i	$-0.993994\pi$
$167$	− 81.4315i	− 0.0377327i	0.999822	−	0.0188663i	$-0.00600570\pi$
$168$	− 1094.79i	− 0.502766i
$169$	−169.000	−0.0769231
$170$	0	0
$171$	999.619	0.447034
$172$	1007.19i	0.446499i
$173$	− 1038.49i	− 0.456386i	−0.973616	−	0.228193i	$-0.926718\pi$
$173$	− 1038.49i	− 0.456386i	0.973616	−	0.228193i	$-0.0732818\pi$
$174$	2354.34	1.02576
$175$	0	0
$176$	701.737	0.300542
$177$	1620.71i	0.688251i
$178$	523.029i	0.220240i
$179$	−2001.70	−0.835834	−0.417917	−	0.908485i	$-0.637240\pi$
$179$	−2001.70	−0.835834	−0.417917	+	0.908485i	$0.637240\pi$
$180$	0	0
$181$	−3417.70	−1.40351	−0.701756	−	0.712417i	$-0.747600\pi$
$181$	−3417.70	−1.40351	−0.701756	+	0.712417i	$0.747600\pi$
$182$	− 579.354i	− 0.235959i
$183$	− 3956.33i	− 1.59814i
$184$	1239.92	0.496783
$185$	0	0
$186$	2626.55	1.03542
$187$	− 4384.98i	− 1.71477i
$188$	− 1044.29i	− 0.405119i
$189$	−2228.29	−0.857587
$190$	0	0
$191$	−2415.03	−0.914899	−0.457450	−	0.889236i	$-0.651237\pi$
$191$	−2415.03	−0.914899	−0.457450	+	0.889236i	$0.651237\pi$
$192$	393.051i	0.147740i
$193$	− 3899.05i	− 1.45420i	−0.686534	−	0.727098i	$-0.740869\pi$
$193$	− 3899.05i	− 1.45420i	0.686534	−	0.727098i	$-0.259131\pi$
$194$	2742.99	1.01513
$195$	0	0
$196$	614.103	0.223798
$197$	− 1664.75i	− 0.602073i	−0.953613	−	0.301037i	$-0.902667\pi$
$197$	− 1664.75i	− 0.602073i	0.953613	−	0.301037i	$-0.0973326\pi$
$198$	940.077i	0.337416i
$199$	4671.64	1.66414	0.832070	−	0.554671i	$-0.187156\pi$
$199$	4671.64	1.66414	0.832070	+	0.554671i	$0.187156\pi$
$200$	0	0
$201$	−2380.28	−0.835283
$202$	− 519.394i	− 0.180913i
$203$	4271.11i	1.47672i
$204$	2456.08	0.842941
$205$	0	0
$206$	1498.11	0.506689
$207$	1661.05i	0.557734i
$208$	208.000i	0.0693375i
$209$	−4090.81	−1.35391
$210$	0	0
$211$	−1405.82	−0.458677	−0.229338	−	0.973347i	$-0.573656\pi$
$211$	−1405.82	−0.458677	−0.229338	+	0.973347i	$0.573656\pi$
$212$	− 605.497i	− 0.196159i
$213$	3347.02i	1.07668i
$214$	−1594.27	−0.509261
$215$	0	0
$216$	800.000	0.252005
$217$	4764.93i	1.49062i
$218$	1297.90i	0.403233i
$219$	1853.10	0.571784
$220$	0	0
$221$	1299.74	0.395611
$222$	− 4933.99i	− 1.49166i
$223$	− 2423.95i	− 0.727890i	−0.931420	−	0.363945i	$-0.881430\pi$
$223$	− 2423.95i	− 0.727890i	0.931420	−	0.363945i	$-0.118570\pi$
$224$	−713.051	−0.212691
$225$	0	0
$226$	1200.12	0.353234
$227$	− 185.926i	− 0.0543626i	−0.999631	−	0.0271813i	$-0.991347\pi$
$227$	− 185.926i	− 0.0543626i	0.999631	−	0.0271813i	$-0.00865315\pi$
$228$	− 2291.31i	− 0.665553i
$229$	−6353.04	−1.83328	−0.916639	−	0.399715i	$-0.869109\pi$
$229$	−6353.04	−1.83328	−0.916639	+	0.399715i	$0.869109\pi$
$230$	0	0
$231$	−6001.98	−1.70953
$232$	− 1533.42i	− 0.433939i
$233$	− 1088.91i	− 0.306168i	−0.988213	−	0.153084i	$-0.951079\pi$
$233$	− 1088.91i	− 0.306168i	0.988213	−	0.153084i	$-0.0489205\pi$
$234$	−278.646	−0.0778446
$235$	0	0
$236$	1055.59	0.291158
$237$	− 3457.57i	− 0.947651i
$238$	4455.68i	1.21352i
$239$	1555.66	0.421036	0.210518	−	0.977590i	$-0.432485\pi$
$239$	1555.66	0.421036	0.210518	+	0.977590i	$0.432485\pi$
$240$	0	0
$241$	520.786	0.139198	0.0695991	−	0.997575i	$-0.477828\pi$
$241$	520.786	0.139198	0.0695991	+	0.997575i	$0.477828\pi$
$242$	− 1185.15i	− 0.314811i
$243$	2848.82i	0.752064i
$244$	−2576.81	−0.676080
$245$	0	0
$246$	−5635.48	−1.46059
$247$	− 1212.55i	− 0.312358i
$248$	− 1710.71i	− 0.438025i
$249$	−5764.89	−1.46721
$250$	0	0
$251$	−7079.53	−1.78030	−0.890151	−	0.455665i	$-0.849401\pi$
$251$	−7079.53	−1.78030	−0.890151	+	0.455665i	$0.849401\pi$
$252$	− 955.234i	− 0.238786i
$253$	− 6797.64i	− 1.68919i
$254$	3836.85	0.947816
$255$	0	0
$256$	256.000	0.0625000
$257$	437.806i	0.106263i	0.998588	+	0.0531315i	$0.0169202\pi$
$257$	437.806i	0.106263i	−0.998588	+	0.0531315i	$0.983080\pi$
$258$	3092.81i	0.746317i
$259$	8950.96	2.14743
$260$	0	0
$261$	2054.23	0.487179
$262$	− 2856.24i	− 0.673508i
$263$	1237.18i	0.290067i	0.989427	+	0.145033i	$0.0463289\pi$
$263$	1237.18i	0.290067i	−0.989427	+	0.145033i	$0.953671\pi$
$264$	2154.83	0.502352
$265$	0	0
$266$	4156.77	0.958150
$267$	1606.07i	0.368127i
$268$	1550.31i	0.353359i
$269$	6507.75	1.47504	0.737518	−	0.675328i	$-0.235998\pi$
$269$	6507.75	1.47504	0.737518	+	0.675328i	$0.235998\pi$
$270$	0	0
$271$	2547.66	0.571069	0.285534	−	0.958368i	$-0.407829\pi$
$271$	2547.66	0.571069	0.285534	+	0.958368i	$0.407829\pi$
$272$	− 1599.68i	− 0.356599i
$273$	− 1779.03i	− 0.394402i
$274$	2258.39	0.497935
$275$	0	0
$276$	3807.44	0.830366
$277$	− 5214.12i	− 1.13100i	−0.824749	−	0.565499i	$-0.808684\pi$
$277$	− 5214.12i	− 1.13100i	0.824749	−	0.565499i	$-0.191316\pi$
$278$	361.777i	0.0780502i
$279$	2291.74	0.491766
$280$	0	0
$281$	7359.26	1.56234	0.781168	−	0.624320i	$-0.214624\pi$
$281$	7359.26	1.56234	0.781168	+	0.624320i	$0.214624\pi$
$282$	− 3206.70i	− 0.677150i
$283$	− 2985.29i	− 0.627057i	−0.949579	−	0.313528i	$-0.898489\pi$
$283$	− 2985.29i	− 0.627057i	0.949579	−	0.313528i	$-0.101511\pi$
$284$	2179.96	0.455482
$285$	0	0
$286$	1140.32	0.235765
$287$	− 10223.6i	− 2.10271i
$288$	342.949i	0.0701682i
$289$	−5083.00	−1.03460
$290$	0	0
$291$	8422.95	1.69678
$292$	− 1206.95i	− 0.241888i
$293$	9607.65i	1.91565i	0.287357	+	0.957824i	$0.407223\pi$
$293$	9607.65i	1.91565i	−0.287357	+	0.957824i	$0.592777\pi$
$294$	1885.73	0.374076
$295$	0	0
$296$	−3213.58	−0.631032
$297$	− 4385.86i	− 0.856880i
$298$	− 3030.26i	− 0.589055i
$299$	2014.87	0.389709
$300$	0	0
$301$	−5610.79	−1.07442
$302$	517.940i	0.0986891i
$303$	− 1594.91i	− 0.302394i
$304$	−1492.37	−0.281556
$305$	0	0
$306$	2143.00	0.400350
$307$	8851.79i	1.64560i	0.568334	+	0.822798i	$0.307588\pi$
$307$	8851.79i	1.64560i	−0.568334	+	0.822798i	$0.692412\pi$
$308$	3909.18i	0.723201i
$309$	4600.25	0.846923
$310$	0	0
$311$	2645.58	0.482370	0.241185	−	0.970479i	$-0.422464\pi$
$311$	2645.58	0.482370	0.241185	+	0.970479i	$0.422464\pi$
$312$	638.709i	0.115897i
$313$	− 6088.30i	− 1.09946i	−0.835342	−	0.549730i	$-0.814731\pi$
$313$	− 6088.30i	− 1.09946i	0.835342	−	0.549730i	$-0.185269\pi$
$314$	1025.90	0.184378
$315$	0	0
$316$	−2251.97	−0.400895
$317$	− 9945.91i	− 1.76220i	−0.472928	−	0.881101i	$-0.656803\pi$
$317$	− 9945.91i	− 1.76220i	0.472928	−	0.881101i	$-0.343197\pi$
$318$	− 1859.31i	− 0.327877i
$319$	−8406.69	−1.47550
$320$	0	0
$321$	−4895.53	−0.851221
$322$	6907.24i	1.19542i
$323$	9325.42i	1.60644i
$324$	3614.02	0.619688
$325$	0	0
$326$	−6599.15	−1.12115
$327$	3985.47i	0.673997i
$328$	3670.47i	0.617889i
$329$	5817.42	0.974847
$330$	0	0
$331$	2735.14	0.454189	0.227095	−	0.973873i	$-0.427077\pi$
$331$	2735.14	0.454189	0.227095	+	0.973873i	$0.427077\pi$
$332$	3754.75i	0.620689i
$333$	− 4305.05i	− 0.708454i
$334$	−162.863	−0.0266810
$335$	0	0
$336$	−2189.58	−0.355510
$337$	− 7468.44i	− 1.20722i	−0.797281	−	0.603608i	$-0.793729\pi$
$337$	− 7468.44i	− 1.20722i	0.797281	−	0.603608i	$-0.206271\pi$
$338$	338.000i	0.0543928i
$339$	3685.23	0.590425
$340$	0	0
$341$	−9378.65	−1.48939
$342$	− 1999.24i	− 0.316100i
$343$	− 4222.03i	− 0.664630i
$344$	2014.39	0.315723
$345$	0	0
$346$	−2076.98	−0.322714
$347$	− 9931.82i	− 1.53651i	−0.640146	−	0.768253i	$-0.721126\pi$
$347$	− 9931.82i	− 1.53651i	0.640146	−	0.768253i	$-0.278874\pi$
$348$	− 4708.69i	− 0.725322i
$349$	−1573.92	−0.241404	−0.120702	−	0.992689i	$-0.538514\pi$
$349$	−1573.92	−0.241404	−0.120702	+	0.992689i	$0.538514\pi$
$350$	0	0
$351$	1300.00	0.197689
$352$	− 1403.47i	− 0.212515i
$353$	10869.2i	1.63884i	0.573194	+	0.819420i	$0.305704\pi$
$353$	10869.2i	1.63884i	−0.573194	+	0.819420i	$0.694296\pi$
$354$	3241.43	0.486667
$355$	0	0
$356$	1046.06	0.155733
$357$	13682.1i	2.02839i
$358$	4003.41i	0.591024i
$359$	7001.85	1.02937	0.514684	−	0.857380i	$-0.327909\pi$
$359$	7001.85	1.02937	0.514684	+	0.857380i	$0.327909\pi$
$360$	0	0
$361$	1840.83	0.268381
$362$	6835.40i	0.992433i
$363$	− 3639.25i	− 0.526202i
$364$	−1158.71	−0.166848
$365$	0	0
$366$	−7912.65	−1.13006
$367$	− 11254.8i	− 1.60081i	−0.599462	−	0.800403i	$-0.704619\pi$
$367$	− 11254.8i	− 1.60081i	0.599462	−	0.800403i	$-0.295381\pi$
$368$	− 2479.84i	− 0.351279i
$369$	−4917.12	−0.693699
$370$	0	0
$371$	3373.05	0.472022
$372$	− 5253.10i	− 0.732152i
$373$	− 9551.67i	− 1.32592i	−0.748657	−	0.662958i	$-0.769301\pi$
$373$	− 9551.67i	− 1.32592i	0.748657	−	0.662958i	$-0.230699\pi$
$374$	−8769.96	−1.21252
$375$	0	0
$376$	−2088.57	−0.286462
$377$	− 2491.80i	− 0.340409i
$378$	4456.57i	0.606406i
$379$	−6124.55	−0.830071	−0.415036	−	0.909805i	$-0.636231\pi$
$379$	−6124.55	−0.830071	−0.415036	+	0.909805i	$0.636231\pi$
$380$	0	0
$381$	11781.9	1.58426
$382$	4830.07i	0.646931i
$383$	2206.62i	0.294394i	0.989107	+	0.147197i	$0.0470252\pi$
$383$	2206.62i	0.294394i	−0.989107	+	0.147197i	$0.952975\pi$
$384$	786.103	0.104468
$385$	0	0
$386$	−7798.10	−1.02827
$387$	2698.56i	0.354459i
$388$	− 5485.99i	− 0.717806i
$389$	8458.16	1.10243	0.551216	−	0.834363i	$-0.314164\pi$
$389$	8458.16	1.10243	0.551216	+	0.834363i	$0.314164\pi$
$390$	0	0
$391$	−15495.9	−2.00425
$392$	− 1228.21i	− 0.158249i
$393$	− 8770.70i	− 1.12576i
$394$	−3329.50	−0.425730
$395$	0	0
$396$	1880.15	0.238589
$397$	− 10377.1i	− 1.31186i	−0.754821	−	0.655931i	$-0.772276\pi$
$397$	− 10377.1i	− 1.31186i	0.754821	−	0.655931i	$-0.227724\pi$
$398$	− 9343.28i	− 1.17672i
$399$	12764.3	1.60153
$400$	0	0
$401$	8962.85	1.11617	0.558084	−	0.829784i	$-0.311537\pi$
$401$	8962.85	1.11617	0.558084	+	0.829784i	$0.311537\pi$
$402$	4760.55i	0.590634i
$403$	− 2779.90i	− 0.343615i
$404$	−1038.79	−0.127925
$405$	0	0
$406$	8542.23	1.04420
$407$	17617.9i	2.14566i
$408$	− 4912.16i	− 0.596049i
$409$	1133.52	0.137039	0.0685193	−	0.997650i	$-0.478173\pi$
$409$	1133.52	0.137039	0.0685193	+	0.997650i	$0.478173\pi$
$410$	0	0
$411$	6934.87	0.832291
$412$	− 2996.21i	− 0.358283i
$413$	5880.41i	0.700620i
$414$	3322.10	0.394378
$415$	0	0
$416$	416.000	0.0490290
$417$	1110.91i	0.130460i
$418$	8181.63i	0.957360i
$419$	−10588.0	−1.23450	−0.617251	−	0.786766i	$-0.711754\pi$
$419$	−10588.0	−1.23450	−0.617251	+	0.786766i	$0.711754\pi$
$420$	0	0
$421$	−8296.85	−0.960484	−0.480242	−	0.877136i	$-0.659451\pi$
$421$	−8296.85	−0.960484	−0.480242	+	0.877136i	$0.659451\pi$
$422$	2811.65i	0.324334i
$423$	− 2797.94i	− 0.321609i
$424$	−1210.99	−0.138705
$425$	0	0
$426$	6694.03	0.761331
$427$	− 14354.7i	− 1.62687i
$428$	3188.53i	0.360102i
$429$	3501.61	0.394077
$430$	0	0
$431$	3010.89	0.336496	0.168248	−	0.985745i	$-0.446189\pi$
$431$	3010.89	0.336496	0.168248	+	0.985745i	$0.446189\pi$
$432$	− 1600.00i	− 0.178195i
$433$	− 3310.31i	− 0.367399i	−0.982982	−	0.183699i	$-0.941193\pi$
$433$	− 3310.31i	− 0.367399i	0.982982	−	0.183699i	$-0.0588073\pi$
$434$	9529.87	1.05403
$435$	0	0
$436$	2595.79	0.285128
$437$	14456.4i	1.58247i
$438$	− 3706.19i	− 0.404312i
$439$	−2597.90	−0.282439	−0.141220	−	0.989978i	$-0.545102\pi$
$439$	−2597.90	−0.282439	−0.141220	+	0.989978i	$0.545102\pi$
$440$	0	0
$441$	1645.36	0.177665
$442$	− 2599.48i	− 0.279739i
$443$	− 16570.8i	− 1.77721i	−0.458671	−	0.888606i	$-0.651675\pi$
$443$	− 16570.8i	− 1.77721i	0.458671	−	0.888606i	$-0.348325\pi$
$444$	−9867.98	−1.05476
$445$	0	0
$446$	−4847.89	−0.514696
$447$	− 9305.07i	− 0.984597i
$448$	1426.10i	0.150395i
$449$	6153.27	0.646750	0.323375	−	0.946271i	$-0.395183\pi$
$449$	6153.27	0.646750	0.323375	+	0.946271i	$0.395183\pi$
$450$	0	0
$451$	20122.7	2.10098
$452$	− 2400.24i	− 0.249774i
$453$	1590.45i	0.164957i
$454$	−371.851	−0.0384402
$455$	0	0
$456$	−4582.63	−0.470617
$457$	12968.9i	1.32748i	0.747961	+	0.663742i	$0.231033\pi$
$457$	12968.9i	1.32748i	−0.747961	+	0.663742i	$0.768967\pi$
$458$	12706.1i	1.29632i
$459$	−9998.00	−1.01670
$460$	0	0
$461$	−12449.4	−1.25776	−0.628878	−	0.777504i	$-0.716485\pi$
$461$	−12449.4	−1.25776	−0.628878	+	0.777504i	$0.716485\pi$
$462$	12004.0i	1.20882i
$463$	2012.71i	0.202028i	0.994885	+	0.101014i	$0.0322086\pi$
$463$	2012.71i	0.202028i	−0.994885	+	0.101014i	$0.967791\pi$
$464$	−3066.83	−0.306841
$465$	0	0
$466$	−2177.83	−0.216493
$467$	4541.22i	0.449984i	0.974361	+	0.224992i	$0.0722356\pi$
$467$	4541.22i	0.449984i	−0.974361	+	0.224992i	$0.927764\pi$
$468$	557.291i	0.0550445i
$469$	−8636.33	−0.850295
$470$	0	0
$471$	3150.24	0.308185
$472$	− 2111.19i	− 0.205880i
$473$	− 11043.5i	− 1.07353i
$474$	−6915.14	−0.670091
$475$	0	0
$476$	8911.36	0.858091
$477$	− 1622.30i	− 0.155723i
$478$	− 3111.33i	− 0.297717i
$479$	−5671.15	−0.540963	−0.270482	−	0.962725i	$-0.587183\pi$
$479$	−5671.15	−0.540963	−0.270482	+	0.962725i	$0.587183\pi$
$480$	0	0
$481$	−5222.06	−0.495022
$482$	− 1041.57i	− 0.0984280i
$483$	21210.2i	1.99813i
$484$	−2370.30	−0.222605
$485$	0	0
$486$	5697.63	0.531790
$487$	− 5379.59i	− 0.500560i	−0.968174	−	0.250280i	$-0.919477\pi$
$487$	− 5379.59i	− 0.500560i	0.968174	−	0.250280i	$-0.0805227\pi$
$488$	5153.62i	0.478061i
$489$	−20264.1	−1.87398
$490$	0	0
$491$	−883.306	−0.0811874	−0.0405937	−	0.999176i	$-0.512925\pi$
$491$	−883.306	−0.0811874	−0.0405937	+	0.999176i	$0.512925\pi$
$492$	11271.0i	1.03279i
$493$	19163.9i	1.75071i
$494$	−2425.09	−0.220871
$495$	0	0
$496$	−3421.42	−0.309730
$497$	12143.9i	1.09604i
$498$	11529.8i	1.03747i
$499$	15251.5	1.36824	0.684119	−	0.729371i	$-0.260187\pi$
$499$	15251.5	1.36824	0.684119	+	0.729371i	$0.260187\pi$
$500$	0	0
$501$	−500.105	−0.0445969
$502$	14159.1i	1.25886i
$503$	− 9087.97i	− 0.805592i	−0.915290	−	0.402796i	$-0.868038\pi$
$503$	− 9087.97i	− 0.805592i	0.915290	−	0.402796i	$-0.131962\pi$
$504$	−1910.47	−0.168847
$505$	0	0
$506$	−13595.3	−1.19443
$507$	1037.90i	0.0909168i
$508$	− 7673.70i	− 0.670207i
$509$	19916.8	1.73438	0.867189	−	0.497979i	$-0.165924\pi$
$509$	19916.8	1.73438	0.867189	+	0.497979i	$0.165924\pi$
$510$	0	0
$511$	6723.57	0.582061
$512$	− 512.000i	− 0.0441942i
$513$	9327.29i	0.802748i
$514$	875.611	0.0751392
$515$	0	0
$516$	6185.61	0.527726
$517$	11450.2i	0.974043i
$518$	− 17901.9i	− 1.51847i
$519$	−6377.80	−0.539411
$520$	0	0
$521$	6154.94	0.517568	0.258784	−	0.965935i	$-0.416678\pi$
$521$	6154.94	0.517568	0.258784	+	0.965935i	$0.416678\pi$
$522$	− 4108.46i	− 0.344488i
$523$	6032.73i	0.504384i	0.967677	+	0.252192i	$0.0811515\pi$
$523$	6032.73i	0.504384i	−0.967677	+	0.252192i	$0.918848\pi$
$524$	−5712.48	−0.476242
$525$	0	0
$526$	2474.35	0.205108
$527$	21379.6i	1.76719i
$528$	− 4309.67i	− 0.355216i
$529$	−11854.9	−0.974349
$530$	0	0
$531$	2828.24	0.231140
$532$	− 8313.54i	− 0.677515i
$533$	5964.51i	0.484712i
$534$	3212.14	0.260305
$535$	0	0
$536$	3100.62	0.249862
$537$	12293.3i	0.987888i
$538$	− 13015.5i	− 1.04301i
$539$	−6733.42	−0.538087
$540$	0	0
$541$	9116.03	0.724452	0.362226	−	0.932090i	$-0.382017\pi$
$541$	9116.03	0.724452	0.362226	+	0.932090i	$0.382017\pi$
$542$	− 5095.33i	− 0.403806i
$543$	20989.6i	1.65884i
$544$	−3199.36	−0.252153
$545$	0	0
$546$	−3558.06	−0.278885
$547$	12356.8i	0.965885i	0.875652	+	0.482942i	$0.160432\pi$
$547$	12356.8i	0.965885i	−0.875652	+	0.482942i	$0.839568\pi$
$548$	− 4516.78i	− 0.352093i
$549$	−6904.01	−0.536714
$550$	0	0
$551$	17878.3	1.38229
$552$	− 7614.88i	− 0.587157i
$553$	− 12545.1i	− 0.964684i
$554$	−10428.2	−0.799736
$555$	0	0
$556$	723.554	0.0551898
$557$	10292.1i	0.782928i	0.920194	+	0.391464i	$0.128031\pi$
$557$	10292.1i	0.782928i	−0.920194	+	0.391464i	$0.871969\pi$
$558$	− 4583.48i	− 0.347731i
$559$	3273.38	0.247673
$560$	0	0
$561$	−26930.0	−2.02672
$562$	− 14718.5i	− 1.10474i
$563$	21015.4i	1.57317i	0.617482	+	0.786585i	$0.288153\pi$
$563$	21015.4i	1.57317i	−0.617482	+	0.786585i	$0.711847\pi$
$564$	−6413.41	−0.478818
$565$	0	0
$566$	−5970.58	−0.443396
$567$	20132.7i	1.49117i
$568$	− 4359.92i	− 0.322074i
$569$	−5207.27	−0.383656	−0.191828	−	0.981429i	$-0.561442\pi$
$569$	−5207.27	−0.383656	−0.191828	+	0.981429i	$0.561442\pi$
$570$	0	0
$571$	16827.8	1.23331	0.616655	−	0.787234i	$-0.288487\pi$
$571$	16827.8	1.23331	0.616655	+	0.787234i	$0.288487\pi$
$572$	− 2280.65i	− 0.166711i
$573$	14831.8i	1.08134i
$574$	−20447.1	−1.48684
$575$	0	0
$576$	685.897	0.0496164
$577$	− 7624.71i	− 0.550123i	−0.961427	−	0.275061i	$-0.911302\pi$
$577$	− 7624.71i	− 0.550123i	0.961427	−	0.275061i	$-0.0886982\pi$
$578$	10166.0i	0.731574i
$579$	−23945.7	−1.71874
$580$	0	0
$581$	−20916.7	−1.49358
$582$	− 16845.9i	− 1.19980i
$583$	6639.06i	0.471633i
$584$	−2413.90	−0.171041
$585$	0	0
$586$	19215.3	1.35457
$587$	− 18164.5i	− 1.27722i	−0.769529	−	0.638612i	$-0.779509\pi$
$587$	− 18164.5i	− 1.27722i	0.769529	−	0.638612i	$-0.220491\pi$
$588$	− 3771.47i	− 0.264511i
$589$	19945.3	1.39530
$590$	0	0
$591$	−10223.9	−0.711601
$592$	6427.15i	0.446207i
$593$	− 13329.0i	− 0.923029i	−0.887133	−	0.461514i	$-0.847306\pi$
$593$	− 13329.0i	− 0.923029i	0.887133	−	0.461514i	$-0.152694\pi$
$594$	−8771.71	−0.605905
$595$	0	0
$596$	−6060.53	−0.416525
$597$	− 28690.5i	− 1.96688i
$598$	− 4029.74i	− 0.275566i
$599$	10495.5	0.715920	0.357960	−	0.933737i	$-0.383472\pi$
$599$	10495.5	0.715920	0.357960	+	0.933737i	$0.383472\pi$
$600$	0	0
$601$	−12575.5	−0.853518	−0.426759	−	0.904365i	$-0.640345\pi$
$601$	−12575.5	−0.853518	−0.426759	+	0.904365i	$0.640345\pi$
$602$	11221.6i	0.759730i
$603$	4153.72i	0.280518i
$604$	1035.88	0.0697837
$605$	0	0
$606$	−3189.82	−0.213825
$607$	− 4144.35i	− 0.277124i	−0.990354	−	0.138562i	$-0.955752\pi$
$607$	− 4144.35i	− 0.277124i	0.990354	−	0.138562i	$-0.0442480\pi$
$608$	2984.73i	0.199090i
$609$	26230.7	1.74536
$610$	0	0
$611$	−3393.93	−0.224720
$612$	− 4286.00i	− 0.283090i
$613$	9975.52i	0.657272i	0.944457	+	0.328636i	$0.106589\pi$
$613$	9975.52i	0.657272i	−0.944457	+	0.328636i	$0.893411\pi$
$614$	17703.6	1.16361
$615$	0	0
$616$	7818.35	0.511381
$617$	− 19244.9i	− 1.25571i	−0.778332	−	0.627853i	$-0.783934\pi$
$617$	− 19244.9i	− 1.25571i	0.778332	−	0.627853i	$-0.216066\pi$
$618$	− 9200.51i	− 0.598865i
$619$	9885.84	0.641915	0.320958	−	0.947094i	$-0.395995\pi$
$619$	9885.84	0.641915	0.320958	+	0.947094i	$0.395995\pi$
$620$	0	0
$621$	−15499.0	−1.00154
$622$	− 5291.16i	− 0.341087i
$623$	5827.29i	0.374744i
$624$	1277.42	0.0819513
$625$	0	0
$626$	−12176.6	−0.777436
$627$	25123.4i	1.60021i
$628$	− 2051.79i	− 0.130375i
$629$	40161.7	2.54587
$630$	0	0
$631$	−5213.18	−0.328896	−0.164448	−	0.986386i	$-0.552584\pi$
$631$	−5213.18	−0.328896	−0.164448	+	0.986386i	$0.552584\pi$
$632$	4503.93i	0.283476i
$633$	8633.76i	0.542119i
$634$	−19891.8	−1.24607
$635$	0	0
$636$	−3718.62	−0.231844
$637$	− 1995.83i	− 0.124141i
$638$	16813.4i	1.04334i
$639$	5840.74	0.361590
$640$	0	0
$641$	−1754.92	−0.108136	−0.0540680	−	0.998537i	$-0.517219\pi$
$641$	−1754.92	−0.108136	−0.0540680	+	0.998537i	$0.517219\pi$
$642$	9791.07i	0.601904i
$643$	− 8477.04i	− 0.519909i	−0.965621	−	0.259955i	$-0.916292\pi$
$643$	− 8477.04i	− 0.519909i	0.965621	−	0.259955i	$-0.0837076\pi$
$644$	13814.5	0.845290
$645$	0	0
$646$	18650.8	1.13592
$647$	23346.4i	1.41861i	0.704901	+	0.709306i	$0.250992\pi$
$647$	23346.4i	1.41861i	−0.704901	+	0.709306i	$0.749008\pi$
$648$	− 7228.05i	− 0.438186i
$649$	−11574.2	−0.700043
$650$	0	0
$651$	29263.5	1.76179
$652$	13198.3i	0.792769i
$653$	− 18810.5i	− 1.12727i	−0.826022	−	0.563637i	$-0.809402\pi$
$653$	− 18810.5i	− 1.12727i	0.826022	−	0.563637i	$-0.190598\pi$
$654$	7970.94	0.476588
$655$	0	0
$656$	7340.94	0.436914
$657$	− 3233.76i	− 0.192026i
$658$	− 11634.8i	− 0.689321i
$659$	−2823.49	−0.166901	−0.0834503	−	0.996512i	$-0.526594\pi$
$659$	−2823.49	−0.166901	−0.0834503	+	0.996512i	$0.526594\pi$
$660$	0	0
$661$	14353.1	0.844586	0.422293	−	0.906459i	$-0.361225\pi$
$661$	14353.1	0.844586	0.422293	+	0.906459i	$0.361225\pi$
$662$	− 5470.27i	− 0.321160i
$663$	− 7982.26i	− 0.467580i
$664$	7509.51	0.438894
$665$	0	0
$666$	−8610.09	−0.500952
$667$	29708.0i	1.72459i
$668$	325.726i	0.0188663i
$669$	−14886.5	−0.860306
$670$	0	0
$671$	28253.8	1.62552
$672$	4379.15i	0.251383i
$673$	2021.69i	0.115795i	0.998323	+	0.0578977i	$0.0184397\pi$
$673$	2021.69i	0.115795i	−0.998323	+	0.0578977i	$0.981560\pi$
$674$	−14936.9	−0.853631
$675$	0	0
$676$	676.000	0.0384615
$677$	− 16670.5i	− 0.946380i	−0.880960	−	0.473190i	$-0.843102\pi$
$677$	− 16670.5i	− 0.946380i	0.880960	−	0.473190i	$-0.156898\pi$
$678$	− 7370.45i	− 0.417493i
$679$	30560.9	1.72727
$680$	0	0
$681$	−1141.85	−0.0642522
$682$	18757.3i	1.05316i
$683$	19539.7i	1.09468i	0.836910	+	0.547341i	$0.184360\pi$
$683$	19539.7i	1.09468i	−0.836910	+	0.547341i	$0.815640\pi$
$684$	−3998.47	−0.223517
$685$	0	0
$686$	−8444.06	−0.469964
$687$	39016.8i	2.16679i
$688$	− 4028.78i	− 0.223250i
$689$	−1967.87	−0.108809
$690$	0	0
$691$	−26847.1	−1.47802	−0.739011	−	0.673694i	$-0.764707\pi$
$691$	−26847.1	−1.47802	−0.739011	+	0.673694i	$0.764707\pi$
$692$	4153.95i	0.228193i
$693$	10473.8i	0.574122i
$694$	−19863.6	−1.08647
$695$	0	0
$696$	−9417.37	−0.512880
$697$	− 45871.7i	− 2.49285i
$698$	3147.83i	0.170698i
$699$	−6687.49	−0.361866
$700$	0	0
$701$	−27878.5	−1.50208	−0.751039	−	0.660258i	$-0.770447\pi$
$701$	−27878.5	−1.50208	−0.751039	+	0.660258i	$0.770447\pi$
$702$	− 2600.00i	− 0.139787i
$703$	− 37467.4i	− 2.01012i
$704$	−2806.95	−0.150271
$705$	0	0
$706$	21738.4	1.15883
$707$	− 5786.79i	− 0.307829i
$708$	− 6482.86i	− 0.344125i
$709$	−27805.6	−1.47286	−0.736431	−	0.676512i	$-0.763491\pi$
$709$	−27805.6	−1.47286	−0.736431	+	0.676512i	$0.763491\pi$
$710$	0	0
$711$	−6033.66	−0.318256
$712$	− 2092.11i	− 0.110120i
$713$	33142.8i	1.74083i
$714$	27364.2	1.43429
$715$	0	0
$716$	8006.81	0.417917
$717$	− 9554.00i	− 0.497630i
$718$	− 14003.7i	− 0.727874i
$719$	33653.6	1.74557	0.872787	−	0.488101i	$-0.162310\pi$
$719$	33653.6	1.74557	0.872787	+	0.488101i	$0.162310\pi$
$720$	0	0
$721$	16691.0	0.862145
$722$	− 3681.65i	− 0.189774i
$723$	− 3198.37i	− 0.164521i
$724$	13670.8	0.701756
$725$	0	0
$726$	−7278.51	−0.372081
$727$	− 1384.32i	− 0.0706210i	−0.999376	−	0.0353105i	$-0.988758\pi$
$727$	− 1384.32i	− 0.0706210i	0.999376	−	0.0353105i	$-0.0112420\pi$
$728$	2317.42i	0.117980i
$729$	−6898.86	−0.350498
$730$	0	0
$731$	−25174.8	−1.27377
$732$	15825.3i	0.799071i
$733$	− 1212.64i	− 0.0611049i	−0.999533	−	0.0305524i	$-0.990273\pi$
$733$	− 1212.64i	− 0.0611049i	0.999533	−	0.0305524i	$-0.00972666\pi$
$734$	−22509.6	−1.13194
$735$	0	0
$736$	−4959.68	−0.248392
$737$	− 16998.6i	− 0.849594i
$738$	9834.23i	0.490519i
$739$	8168.18	0.406592	0.203296	−	0.979117i	$-0.434835\pi$
$739$	8168.18	0.406592	0.203296	+	0.979117i	$0.434835\pi$
$740$	0	0
$741$	−7446.77	−0.369182
$742$	− 6746.10i	− 0.333770i
$743$	− 11712.7i	− 0.578328i	−0.957280	−	0.289164i	$-0.906623\pi$
$743$	− 11712.7i	− 0.578328i	0.957280	−	0.289164i	$-0.0933773\pi$
$744$	−10506.2	−0.517709
$745$	0	0
$746$	−19103.3	−0.937564
$747$	10060.1i	0.492742i
$748$	17539.9i	0.857384i
$749$	−17762.4	−0.866521
$750$	0	0
$751$	29405.0	1.42877	0.714383	−	0.699754i	$-0.246707\pi$
$751$	29405.0	1.42877	0.714383	+	0.699754i	$0.246707\pi$
$752$	4177.14i	0.202559i
$753$	43478.4i	2.10417i
$754$	−4983.61	−0.240706
$755$	0	0
$756$	8913.14	0.428793
$757$	− 13612.9i	− 0.653593i	−0.945095	−	0.326797i	$-0.894031\pi$
$757$	− 13612.9i	− 0.653593i	0.945095	−	0.326797i	$-0.105969\pi$
$758$	12249.1i	0.586949i
$759$	−41747.2	−1.99648
$760$	0	0
$761$	4598.55	0.219050	0.109525	−	0.993984i	$-0.465067\pi$
$761$	4598.55	0.219050	0.109525	+	0.993984i	$0.465067\pi$
$762$	− 23563.7i	− 1.12024i
$763$	14460.4i	0.686111i
$764$	9660.14	0.457450
$765$	0	0
$766$	4413.24	0.208168
$767$	− 3430.68i	− 0.161506i
$768$	− 1572.21i	− 0.0738699i
$769$	9640.19	0.452060	0.226030	−	0.974120i	$-0.427425\pi$
$769$	9640.19	0.452060	0.226030	+	0.974120i	$0.427425\pi$
$770$	0	0
$771$	2688.75	0.125594
$772$	15596.2i	0.727098i
$773$	− 4369.66i	− 0.203319i	−0.994819	−	0.101660i	$-0.967585\pi$
$773$	− 4369.66i	− 0.203319i	0.994819	−	0.101660i	$-0.0324152\pi$
$774$	5397.12	0.250640
$775$	0	0
$776$	−10972.0	−0.507566
$777$	− 54971.7i	− 2.53809i
$778$	− 16916.3i	− 0.779537i
$779$	−42794.4	−1.96825
$780$	0	0
$781$	−23902.5	−1.09513
$782$	30991.8i	1.41722i
$783$	19167.7i	0.874838i
$784$	−2456.41	−0.111899
$785$	0	0
$786$	−17541.4	−0.796032
$787$	− 36670.3i	− 1.66094i	−0.557066	−	0.830468i	$-0.688073\pi$
$787$	− 36670.3i	− 1.66094i	0.557066	−	0.830468i	$-0.311927\pi$
$788$	6658.99i	0.301037i
$789$	7598.03	0.342835
$790$	0	0
$791$	13371.1	0.601036
$792$	− 3760.31i	− 0.168708i
$793$	8374.64i	0.375022i
$794$	−20754.1	−0.927627
$795$	0	0
$796$	−18686.6	−0.832070
$797$	2533.75i	0.112610i	0.998414	+	0.0563050i	$0.0179319\pi$
$797$	2533.75i	0.112610i	−0.998414	+	0.0563050i	$0.982068\pi$
$798$	− 25528.5i	− 1.13246i
$799$	26101.9	1.15572
$800$	0	0
$801$	2802.69	0.123631
$802$	− 17925.7i	− 0.789250i
$803$	13233.8i	0.581581i
$804$	9521.11	0.417641
$805$	0	0
$806$	−5559.80	−0.242972
$807$	− 39966.9i	− 1.74337i
$808$	2077.58i	0.0904566i
$809$	−20002.0	−0.869261	−0.434630	−	0.900609i	$-0.643121\pi$
$809$	−20002.0	−0.869261	−0.434630	+	0.900609i	$0.643121\pi$
$810$	0	0
$811$	45063.8	1.95118	0.975589	−	0.219605i	$-0.0704770\pi$
$811$	45063.8	1.95118	0.975589	+	0.219605i	$0.0704770\pi$
$812$	− 17084.5i	− 0.738358i
$813$	− 15646.3i	− 0.674956i
$814$	35235.7	1.51721
$815$	0	0
$816$	−9824.32	−0.421471
$817$	23486.0i	1.00572i
$818$	− 2267.03i	− 0.0969010i
$819$	−3104.51	−0.132455
$820$	0	0
$821$	−17316.1	−0.736097	−0.368049	−	0.929807i	$-0.619974\pi$
$821$	−17316.1	−0.736097	−0.368049	+	0.929807i	$0.619974\pi$
$822$	− 13869.7i	− 0.588519i
$823$	− 32263.9i	− 1.36653i	−0.730173	−	0.683263i	$-0.760560\pi$
$823$	− 32263.9i	− 1.36653i	0.730173	−	0.683263i	$-0.239440\pi$
$824$	−5992.42	−0.253345
$825$	0	0
$826$	11760.8	0.495414
$827$	33244.3i	1.39784i	0.715198	+	0.698921i	$0.246336\pi$
$827$	33244.3i	1.39784i	−0.715198	+	0.698921i	$0.753664\pi$
$828$	− 6644.20i	− 0.278867i
$829$	−30822.9	−1.29134	−0.645672	−	0.763614i	$-0.723423\pi$
$829$	−30822.9	−1.29134	−0.645672	+	0.763614i	$0.723423\pi$
$830$	0	0
$831$	−32022.2	−1.33675
$832$	− 832.000i	− 0.0346688i
$833$	15349.5i	0.638450i
$834$	2221.83	0.0922490
$835$	0	0
$836$	16363.3	0.676956
$837$	21383.9i	0.883076i
$838$	21175.9i	0.872925i
$839$	−21723.5	−0.893895	−0.446947	−	0.894560i	$-0.647489\pi$
$839$	−21723.5	−0.893895	−0.446947	+	0.894560i	$0.647489\pi$
$840$	0	0
$841$	12351.1	0.506422
$842$	16593.7i	0.679165i
$843$	− 45196.4i	− 1.84655i
$844$	5623.29	0.229338
$845$	0	0
$846$	−5595.88	−0.227412
$847$	− 13204.2i	− 0.535659i
$848$	2421.99i	0.0980795i
$849$	−18333.9	−0.741130
$850$	0	0
$851$	62259.0	2.50789
$852$	− 13388.1i	− 0.538342i
$853$	− 15600.6i	− 0.626206i	−0.949719	−	0.313103i	$-0.898631\pi$
$853$	− 15600.6i	− 0.626206i	0.949719	−	0.313103i	$-0.101369\pi$
$854$	−28709.4	−1.15037
$855$	0	0
$856$	6377.06	0.254630
$857$	34393.9i	1.37091i	0.728113	+	0.685457i	$0.240398\pi$
$857$	34393.9i	1.37091i	−0.728113	+	0.685457i	$0.759602\pi$
$858$	− 7003.21i	− 0.278655i
$859$	−8181.69	−0.324978	−0.162489	−	0.986710i	$-0.551952\pi$
$859$	−8181.69	−0.324978	−0.162489	+	0.986710i	$0.551952\pi$
$860$	0	0
$861$	−62787.3	−2.48523
$862$	− 6021.79i	− 0.237938i
$863$	− 3270.88i	− 0.129017i	−0.997917	−	0.0645087i	$-0.979452\pi$
$863$	− 3270.88i	− 0.129017i	0.997917	−	0.0645087i	$-0.0205480\pi$
$864$	−3200.00	−0.126003
$865$	0	0
$866$	−6620.63	−0.259790
$867$	31216.9i	1.22282i
$868$	− 19059.7i	− 0.745311i
$869$	24692.0	0.963888
$870$	0	0
$871$	5038.50	0.196008
$872$	− 5191.59i	− 0.201616i
$873$	− 14698.5i	− 0.569840i
$874$	28912.7	1.11898
$875$	0	0
$876$	−7412.39	−0.285892
$877$	− 15309.4i	− 0.589466i	−0.955580	−	0.294733i	$-0.904769\pi$
$877$	− 15309.4i	− 0.589466i	0.955580	−	0.294733i	$-0.0952307\pi$
$878$	5195.79i	0.199715i
$879$	59004.7	2.26414
$880$	0	0
$881$	10407.6	0.398003	0.199001	−	0.979999i	$-0.436230\pi$
$881$	10407.6	0.398003	0.199001	+	0.979999i	$0.436230\pi$
$882$	− 3290.71i	− 0.125628i
$883$	− 39646.5i	− 1.51100i	−0.655150	−	0.755499i	$-0.727395\pi$
$883$	− 39646.5i	− 1.51100i	0.655150	−	0.755499i	$-0.272605\pi$
$884$	−5198.96	−0.197805
$885$	0	0
$886$	−33141.7	−1.25668
$887$	33937.4i	1.28467i	0.766422	+	0.642337i	$0.222035\pi$
$887$	33937.4i	1.28467i	−0.766422	+	0.642337i	$0.777965\pi$
$888$	19736.0i	0.745828i
$889$	42748.0	1.61273
$890$	0	0
$891$	−39626.5	−1.48994
$892$	9695.78i	0.363945i
$893$	− 24350.9i	− 0.912510i
$894$	−18610.1	−0.696215
$895$	0	0
$896$	2852.21	0.106345
$897$	− 12374.2i	− 0.460604i
$898$	− 12306.5i	− 0.457321i
$899$	40988.0	1.52061
$900$	0	0
$901$	15134.4	0.559600
$902$	− 40245.4i	− 1.48561i
$903$	34458.3i	1.26988i
$904$	−4800.48	−0.176617
$905$	0	0
$906$	3180.89	0.116642
$907$	− 31969.4i	− 1.17037i	−0.810900	−	0.585185i	$-0.801022\pi$
$907$	− 31969.4i	− 1.17037i	0.810900	−	0.585185i	$-0.198978\pi$
$908$	743.703i	0.0271813i
$909$	−2783.21	−0.101555
$910$	0	0
$911$	160.760	0.00584656	0.00292328	−	0.999996i	$-0.499069\pi$
$911$	160.760	0.00584656	0.00292328	+	0.999996i	$0.499069\pi$
$912$	9165.26i	0.332776i
$913$	− 41169.5i	− 1.49235i
$914$	25937.8	0.938673
$915$	0	0
$916$	25412.2	0.916639
$917$	− 31822.6i	− 1.14599i
$918$	19996.0i	0.718918i
$919$	−7482.72	−0.268588	−0.134294	−	0.990942i	$-0.542877\pi$
$919$	−7482.72	−0.268588	−0.134294	+	0.990942i	$0.542877\pi$
$920$	0	0
$921$	54362.6	1.94496
$922$	24898.8i	0.889368i
$923$	− 7084.87i	− 0.252656i
$924$	24007.9	0.854765
$925$	0	0
$926$	4025.43	0.142855
$927$	− 8027.71i	− 0.284428i
$928$	6133.67i	0.216969i
$929$	34614.1	1.22245	0.611224	−	0.791458i	$-0.290678\pi$
$929$	34614.1	1.22245	0.611224	+	0.791458i	$0.290678\pi$
$930$	0	0
$931$	14319.8	0.504094
$932$	4355.66i	0.153084i
$933$	− 16247.6i	− 0.570122i
$934$	9082.43	0.318187
$935$	0	0
$936$	1114.58	0.0389223
$937$	− 43467.7i	− 1.51551i	−0.652542	−	0.757753i	$-0.726297\pi$
$937$	− 43467.7i	− 1.51551i	0.652542	−	0.757753i	$-0.273703\pi$
$938$	17272.7i	0.601250i
$939$	−37390.9	−1.29947
$940$	0	0
$941$	4890.69	0.169428	0.0847142	−	0.996405i	$-0.473002\pi$
$941$	4890.69	0.169428	0.0847142	+	0.996405i	$0.473002\pi$
$942$	− 6300.47i	− 0.217920i
$943$	− 71110.7i	− 2.45566i
$944$	−4222.38	−0.145579
$945$	0	0
$946$	−22087.1	−0.759104
$947$	20043.4i	0.687776i	0.939011	+	0.343888i	$0.111744\pi$
$947$	20043.4i	0.687776i	−0.939011	+	0.343888i	$0.888256\pi$
$948$	13830.3i	0.473826i
$949$	−3922.58	−0.134175
$950$	0	0
$951$	−61082.1	−2.08278
$952$	− 17822.7i	− 0.606762i
$953$	− 33417.6i	− 1.13589i	−0.823067	−	0.567944i	$-0.807739\pi$
$953$	− 33417.6i	− 1.13589i	0.823067	−	0.567944i	$-0.192261\pi$
$954$	−3244.60	−0.110113
$955$	0	0
$956$	−6222.66	−0.210518
$957$	51629.1i	1.74392i
$958$	11342.3i	0.382519i
$959$	25161.7	0.847250
$960$	0	0
$961$	15935.9	0.534924
$962$	10444.1i	0.350033i
$963$	8542.99i	0.285871i
$964$	−2083.14	−0.0695991
$965$	0	0
$966$	42420.3	1.41289
$967$	48557.9i	1.61481i	0.590000	+	0.807403i	$0.299128\pi$
$967$	48557.9i	1.61481i	−0.590000	+	0.807403i	$0.700872\pi$
$968$	4740.59i	0.157405i
$969$	57271.4	1.89868
$970$	0	0
$971$	2384.30	0.0788009	0.0394005	−	0.999223i	$-0.487455\pi$
$971$	2384.30	0.0788009	0.0394005	+	0.999223i	$0.487455\pi$
$972$	− 11395.3i	− 0.376032i
$973$	4030.71i	0.132804i
$974$	−10759.2	−0.353949
$975$	0	0
$976$	10307.2	0.338040
$977$	− 37231.4i	− 1.21918i	−0.792717	−	0.609590i	$-0.791334\pi$
$977$	− 37231.4i	− 1.21918i	0.792717	−	0.609590i	$-0.208666\pi$
$978$	40528.2i	1.32510i
$979$	−11469.6	−0.374435
$980$	0	0
$981$	6954.87	0.226353
$982$	1766.61i	0.0574082i
$983$	27677.9i	0.898055i	0.893518	+	0.449027i	$0.148229\pi$
$983$	27677.9i	0.898055i	−0.893518	+	0.449027i	$0.851771\pi$
$984$	22541.9	0.730295
$985$	0	0
$986$	38327.8	1.23794
$987$	− 35727.3i	− 1.15219i
$988$	4850.19i	0.156179i
$989$	−39026.3	−1.25477
$990$	0	0
$991$	22073.3	0.707548	0.353774	−	0.935331i	$-0.384898\pi$
$991$	22073.3	0.707548	0.353774	+	0.935331i	$0.384898\pi$
$992$	6842.83i	0.219012i
$993$	− 16797.6i	− 0.536815i
$994$	24287.9	0.775014
$995$	0	0
$996$	23059.6	0.733604
$997$	11349.0i	0.360507i	0.983620	+	0.180253i	$0.0576918\pi$
$997$	11349.0i	0.360507i	−0.983620	+	0.180253i	$0.942308\pi$
$998$	− 30503.0i	− 0.967490i
$999$	40169.7	1.27219

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.4.b.m.599.1		4
5.2	odd	4		130.4.a.f.1.1	✓	2
5.3	odd	4		650.4.a.k.1.2		2
5.4	even	2	inner	650.4.b.m.599.4		4
15.2	even	4		1170.4.a.t.1.2		2
20.7	even	4		1040.4.a.h.1.2		2
65.12	odd	4		1690.4.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.4.a.f.1.1	✓	2	5.2	odd	4
650.4.a.k.1.2		2	5.3	odd	4
650.4.b.m.599.1		4	1.1	even	1	trivial
650.4.b.m.599.4		4	5.4	even	2	inner
1040.4.a.h.1.2		2	20.7	even	4
1170.4.a.t.1.2		2	15.2	even	4
1690.4.a.o.1.1		2	65.12	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -6.14143i q^{3} -4.00000 q^{4} -12.2829 q^{6} -22.2829i q^{7} +8.00000i q^{8} -10.7171 q^{9} +43.8586 q^{11} +24.5657i q^{12} +13.0000i q^{13} -44.5657 q^{14} +16.0000 q^{16} -99.9800i q^{17} +21.4343i q^{18} -93.2729 q^{19} -136.849 q^{21} -87.7171i q^{22} -154.990i q^{23} +49.1314 q^{24} +26.0000 q^{26} -100.000i q^{27} +89.1314i q^{28} -191.677 q^{29} -213.839 q^{31} -32.0000i q^{32} -269.354i q^{33} -199.960 q^{34} +42.8686 q^{36} +401.697i q^{37} +186.546i q^{38} +79.8386 q^{39} +458.809 q^{41} +273.697i q^{42} -251.799i q^{43} -175.434 q^{44} -309.980 q^{46} +261.071i q^{47} -98.2629i q^{48} -153.526 q^{49} -614.020 q^{51} -52.0000i q^{52} +151.374i q^{53} -200.000 q^{54} +178.263 q^{56} +572.829i q^{57} +383.354i q^{58} -263.899 q^{59} +644.203 q^{61} +427.677i q^{62} +238.809i q^{63} -64.0000 q^{64} -538.709 q^{66} -387.577i q^{67} +399.920i q^{68} -951.860 q^{69} -544.990 q^{71} -85.7371i q^{72} +301.737i q^{73} +803.394 q^{74} +373.091 q^{76} -977.294i q^{77} -159.677i q^{78} +562.991 q^{79} -903.506 q^{81} -917.617i q^{82} -938.689i q^{83} +547.394 q^{84} -503.597 q^{86} +1177.17i q^{87} +350.869i q^{88} -261.514 q^{89} +289.677 q^{91} +619.960i q^{92} +1313.27i q^{93} +522.143 q^{94} -196.526 q^{96} +1371.50i q^{97} +307.051i q^{98} -470.039 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} + 8 q^{6} - 100 q^{9} + 204 q^{11} - 64 q^{14} + 64 q^{16} - 116 q^{19} - 376 q^{21} - 32 q^{24} + 104 q^{26} - 24 q^{29} - 484 q^{31} + 400 q^{36} - 52 q^{39} + 864 q^{41} - 816 q^{44} - 840 q^{46}+ \cdots - 5508 q^{99}+O(q^{100}) \) 4 * q - 16 * q^4 + 8 * q^6 - 100 * q^9 + 204 * q^11 - 64 * q^14 + 64 * q^16 - 116 * q^19 - 376 * q^21 - 32 * q^24 + 104 * q^26 - 24 * q^29 - 484 * q^31 + 400 * q^36 - 52 * q^39 + 864 * q^41 - 816 * q^44 - 840 * q^46 + 300 * q^49 - 2856 * q^51 - 800 * q^54 + 256 * q^56 - 1884 * q^59 + 920 * q^61 - 256 * q^64 + 816 * q^66 - 1008 * q^69 - 1980 * q^71 + 2528 * q^74 + 464 * q^76 - 776 * q^79 - 2300 * q^81 + 1504 * q^84 + 328 * q^86 - 2760 * q^89 + 416 * q^91 - 768 * q^94 + 128 * q^96 - 5508 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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